JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 15, 2013 0651098283

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Volume 15, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

January 2013

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC(eight times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Editor’s Assistant:Dr.Razvan Mezei,Lander University,SC 29649, USA.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications 1) George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory,Real Analysis, Wavelets, Neural Networks,Probability, Inequalities. 2) J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago,IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis

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23) Hrushikesh N.Mhaskar Department Of Mathematics California State University Los Angeles,CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory,Splines, Wavelets, Neural Networks 24) M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations,Optimization,

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314-935-6007 e-mail: [email protected] Systems Theory, Semantic Control, Partial Differential Equations, Calculus of Variations, Optimization and Artificial Intelligence, Operations Research, Math.Programming

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14) Augustine O.Esogbue School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta,GA 30332 404-894-2323 e-mail: [email protected] Control Theory,Fuzzy sets, Mathematical Programming, Dynamic Programming,Optimization

33) Xin-long Zhou Fachbereich Mathematik, Fachgebiet Informatik Gerhard-Mercator-Universitat Duisburg

15) Christodoulos A.Floudas Department of Chemical Engineering Princeton University

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Princeton,NJ 08544-5263 609-258-4595(x4619 assistant) e-mail: [email protected] OptimizationTheory&Applications, Global Optimization

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16) J.A.Goldstein Department of Mathematical Sciences The University of Memphis Memphis,TN 38152 901-678-3130 e-mail:[email protected] Partial Differential Equations, Semigroups of Operators

34) Xiang Ming Yu Department of Mathematical Sciences Southwest Missouri State University Springfield,MO 65804-0094 417-836-5931 e-mail: [email protected] Classical Approximation Theory, Wavelets

17) H.H.Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany 011-49-203-379-3542 e-mail:[email protected] Approximation Theory, Computer Aided Geometric Design 18) John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA [email protected] Ordinary and functional differential equations, difference equations, impulsive systems, differential inclusions, dynamic equations on time scales , control theory and their applications 19) Weimin Han Department of Mathematics University of Iowa Iowa City, IA 52242-1419 319-335-0770 e-mail: [email protected] Numerical analysis, Finite element method, Numerical PDE, Variational inequalities, Computational mechanics

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35) Lotfi A. Zadeh Professor in the Graduate School and Director, Computer Initiative, Soft Computing (BISC) Computer Science Division University of California at Berkeley Berkeley, CA 94720 Office: 510-642-4959 Sec: 510-642-8271 Home: 510-526-2569 FAX: 510-642-1712 e-mail: [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic 36) Ahmed I. Zayed Department Of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected] Shannon sampling theory, Harmonic analysis and wavelets, Special functions and orthogonal polynomials, Integral transforms

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 10-22, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A note on solving the fuzzy Sylvester matrix equation A. Sadeghi

a,1

, Ahmad I. M. Ismail a , A. Ahmad

a School

a

and M. E. Abbasnejad

b

of Mathematical Sciences, Universiti Sains Malaysia 11800 USM, Penang, Malaysia

b Statistical

Machine Learning Group, NICTA and the Australian National University 7 London Circuit, Tower A, Canberra, ACT 2601, Australia

Abstract In this work, we present theoretical analysis of the solution of Fuzzy Sylvester e + XB e = C. e The necessary and sufficient Matrix Equation (FSME) in the form AX conditions for the existence of fuzzy solutions are proposed and some operators to finding the solution of FSME are exploited. Furthermore, an iterative scheme which can solve two 𝑛 × 𝑛 system instead one 2𝑛 × 2𝑛 system is presented to solving extended fuzzy linear system. Several numerical examples are performed to illustrate developed theory.

keywords: Fuzzy Sylvester matrix equation, Fuzzy linear system, Iterative method. 2000 AMS Subject Classification: 15A06

1

Introduction

The matrix equation AX + XB = C,

(1.1)

where A ∈ ℝn×n , B ∈ ℝm×m and C ∈ ℝn×m are known matrices and X ∈ ℝn×m is an unknown matrix is called a Sylvester matrix equation. This equation is of considerable importance in many applications such as system theory, control theory, matrix problems arising in partial differential equations and some techniques in ordinary deferential equations [8], [9], [14]. The solution of the Sylvester matrix equations uses two types of methods. The first type of methods is based on the transforming the coefficient matrices to Schur or Hessenberg form and then solving corresponding linear system directly. The 1

Corresponding author ( E-mail address: [email protected])

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SADEGHI ET AL: SYLVESTER MATRIX EQUATION

second type of method are iterative scheme methods focusing on large sparse systems [8], [9], [14]. Now, consider the matrix equation e + XB e = C, e AX

(1.2)

e ∈ ℝn×m is an unknown fuzzy matrix and C e ∈ ℝn×m is a known fuzzy matrix. where X This paper will investigate the solution of (1.2) which is called a “fuzzy Sylvester matrix equation”. It is known that the systems of linear equations play major role in various areas such as science, engineering and economics. The concept of fuzzy numbers and arithmetic operations associated with these numbers was first presented by Zadeh [20]. Friedman et al. in [12] proposed a model for solving an n × n fuzzy linear system of equation where the coefficient matrix is crisp and the right-hand side column is an arbitrary vector of fuzzy number. They used the embedding method and replaced the original n × n fuzzy linear system by 2n × 2n crisp function linear system. Authors such as Abbasbandy et al. in [1], [2], Allahviranloo et al. in [3], [4] and Dehghan and Hashemi in [10] have extended the work of Friedman. Recently, some authors focus on solving fuzzy matrix equations. Zengtai and Xiaobin e = B. e They used in [13] investigated the fuzzy matrix equation in the form of AX generalized matrix inverse and presented the least square solution to the fuzzy matrix equation. Allahviranloo et al. in [5] presented a two stage method for computing fuzzy e = C. e They applied the Kronecker product to transform this linear matrix equation AXB fuzzy linear matrix equation to a non-square system and used the embedding approach. Salkuyeh [18] investigated the fuzzy Sylvester matrix equation where the crisp matrices A and B are M-matrices. He has solved the associated system by accelerated over relaxation method. In this paper we investigate analytically and practically the fuzzy Sylvester matrix e + XB e =C e where A is an n × n matrix, B is a m × m matrix and C e is equation AX an m × n fuzzy matrix. Some operators such as Kronecker product, Kronecker sum and vec-operator are exploited to transform the fuzzy Sylvester matrix equation to mn × mn fuzzy linear system. An iterative method which has feasible property will be introduced for solving the associated fuzzy linear system. Furthermore, an interesting application of fuzzy Sylvester matrix equation is introduced. Throughout this note, the Kronecker product of two matrices denoted by A ⊗ B, the vector of each matrix A ∈ Cm×n presented by vec(A) ∈ Cmn , and the spectrum of an arbitrary matrix W introduced by σ(W ). The outline of the paper is as follows: In Section 2 we will present basic definitions and concepts. In Section 3 we will discuss about the necessary and sufficient condition for existence of solution of FSME. In section 4, an algorithm will be presented for solving fuzzy linear system. Numerical examples will be given in Section 5 and the conclusions

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SADEGHI ET AL: SYLVESTER MATRIX EQUATION

are in Section 6.

2

Basic concepts

In this section, some brief preliminary concepts of the fuzzy number arithmetic and fuzzy Sylvester matrix equation are given. Definition 2.1: [10] A fuzzy number is defined by an ordered pair of functions (u(r), u(r)), 0 ≤ r ≤ 1, which satisfy the following conditions: 1. u(r) is a bounded left continuous non-decreasing function over [0, 1]; 2. u(r) is a bounded left continuous non-increasing function over [0, 1]; 3. u(r) ≤ u(r), 0 ≤ r ≤ 1. A crisp number α is represented by u(r) = u(r) = α, 0 ≤ r ≤ 1. For an arbitrary fuzzy number u = (u(r), u(r)), v = (v(r), v(r)) and λ ∈ ℝ, arithmetic operators can be defined as [10]: u = v iff u(r) = v(r) and u(r) = v(r), u + v = (u(r) + v(r), u(r) + v(r)) u−v = (u(r) − v(r), u(r) − v(r)) (λu(r), λu(r)), λ ≥ 0, λu = (λu(r), λu(r)), λ < 0.

(2.1)

e + XB e =C e is called a fuzzy Sylvester Definition 2.2: The Sylvester matrix equation AX equation (FSME) if matrices A = (aij ), 1 ≤ i, j ≤ n and B = (bij ), 1 ≤ i, j ≤ m are crisp e = (cij ), 1 ≤ i ≤ n, 1 ≤ j ≤ m is a known fuzzy matrices and right-hand side matrix C e = (xij ), 1 ≤ i ≤ n, 1 ≤ j ≤ m is an unknown fuzzy matrix. Note that matrix, where X the ijth equation of this system is: ∑ k

aik xkj +

∑

xiℓ bℓj = cij ,

1 ≤ i ≤ n, 1 ≤ j ≤ m.

(2.2)

ℓ

By writing the matrices in (1.2) in terms of their columns, it is easily seen by equating ith column we have the relation e i = Ax + Axi + Xb

m ∑ j=1

12

bji xj = ci

(2.3)

SADEGHI ET AL: SYLVESTER MATRIX EQUATION

These equations can be written as the mn × mn fuzzy linear system as follows:      A + b11 I b21 I ··· bm1 I x1 c1       b12 I A + b22 I · · · bm2 I   x2   c2    .  =  . . .. .. .. ...   .   .  . . .   .   .  b1m I

b2m I

···

A + bmm I

xm

(2.4)

cm

Clearly, the system of equation (2.4) can be replaced in the form e = vec(C), e (A ⊕ B t )vec(X)

(2.5)

where (A ⊕ B t ) is Kronecker sum of matrices A and B that can be defined as (A ⊕ B t ) = (Im ⊗ A) + (B t ⊗ In ). In continue of this paper, we use vec(W ) = w (small and bold) for given matrix W ∈ Cm×n . Now, in the following definition we define a solution of the FSME. e = (xij ), 1 ≤ i ≤ n, 1 ≤ j ≤ m given Definition 2.3: A fuzzy number matrix X by xij = (xij (r), xij (r)), 1 ≤ i ≤ n, 1 ≤ j ≤ m is called a solution of the FSEM if: ∑ ∑ ∑ ∑ aik xkj + xiℓ bℓj (r) = aik xkj (r) + xiℓ bℓj (r) = cij (r), k

ℓ

k

ℓ

∑

∑

∑

∑

aik xkj +

k

xiℓ bℓj (r) =

ℓ

aik xkj (r) +

k

(2.6) xiℓ bℓj (r) = cij (r).

ℓ

From (3.1) in particular , if aij ≥ 0, for 1 ≤ i ≤ n, 1 ≤ j ≤ m and bij ≥ 0, for 1 ≤ i ≤ n, 1 ≤ j ≤ m or if aii + bjj ≥ 0, for 1 ≤ i ≤ n, 1 ≤ j ≤ m, we simply get: ∑ ∑ ∑ ∑ aik xkj + xiℓ bℓj (r) = aik xkj (r) + xiℓ (r)bℓj = cij (r), k

ℓ

k

ℓ

∑

∑

∑

∑

k

aik xkj +

ℓ

xiℓ bℓj (r) =

aik xkj (r) +

k

(2.7) xiℓ (r)bℓj = cij (r).

ℓ

In general, an arbitrary equation may include a linear combination of xij ’s and xij ’s for ( c ) either cij or cij . Hence, where the right hand side vector is the function vector C = −c , one can solve a crisp linear system in order to solve the system given by (2.6) and (2.7).

3

Fuzzy Sylvester Matrix Equation

In this section we discuss some aspects related to the FSME. First, in the following theorems we are ready to state the condition which the FSME must satisfy to have a unique

13

SADEGHI ET AL: SYLVESTER MATRIX EQUATION

fuzzy solution. For this purpose, we need to find the eigenvalues and eigenvectors of A⊕B. Theorem 3.1: [16] Let A ∈ ℝn×n have eigenvalues λi , i = 1, . . . , n and B ∈ ℝn×n have eigenvalues µj , j = 1, . . . , m. Then the kronecker sum A ⊕ B have mn eigenvalues λ1 + µ1 , . . . , λ1 + µm , λ2 + µ1 , . . . , λ2 + µm , . . . , λn + µ1 , . . . , λn + µm . Moreover, if x1 , . . . , xp are linearly independent right eigenvectors of A corresponding to λ1 , . . . , λp (p ≤ n), and z1 , . . . , zp are linearly independent right eigenvectors of B corresponding to µ1 , . . . , µq (q ≤ m) , then zj ⊗ xi ∈ ℝmn are linearly independent right eigenvectors of A ⊕ B corresponding to λi + µj , i = 1, . . . , p, j = 1, . . . , q. We can now prove the following theorem. e is m × n Theorem 3.2: Suppose that A ∈ ℝn×n , B ∈ ℝm×m are crisp matrices, and C e + XB e = C e has a unique fuzzy matrix. Then the fuzzy Sylvester matrix equation AX fuzzy solution if and only if A and −B have no eigenvalue in common. In other words, σ(A) ∩ σ(−B) = ∅. proof: According to definition of Kronecker sum, the equation (1.2) can be written as follows: (A ⊕ B t )e x=e c, e + XB e =C e has a Suppose that FSME has a unique fuzzy solution. The equation AX unique fuzzy solution if and only if (A ⊕ B t ) is a nonsingular matrix. But we know that (A ⊕ B t ) is a nonsingular matrix if and only if it has non zero eigenvalues. According to Theorem 3.1 we then have (A ⊕ B t )(λi + µj ) = (λi + µj )(e x⊗e z) where λi ∈ σ(A), i = 1, . . . , n and µj ∈ σ(B), j = 1, . . . , m. Hence we conclude σ(A) ∩ σ(−B) = ∅. Conversely, suppose that σ(A) ∩ σ(−B) = ∅. We would like to show that the equation ( ) ( ) e e =C e has a unique solution. Let us put T = A Ce and Z = Im Xe . It is AX + XB 0 −B 0 In clear that ) ( ) )−1 ( )( ( e e e X A 0 X A C I I m m = Z −1 T Z = −B In 0 −B 0 In 0 −B ( e ) ( 0 ) C Thus the two matrices A0 −B and A0 −B are similar and have the same set of eigenvalues. According to [15], this condition is equivalent to the condition for existence of a

14

SADEGHI ET AL: SYLVESTER MATRIX EQUATION

e + XB e = C. e  unique solution of AX In this part, we are going to rearrange the FSME so that the unknowns are xij , −xij , for 1 ≤ i ≤ n, 1 ≤ j ≤ m. For this purpose, first we compute the coefficient matrix Q = (A ⊕ B t ). Clearly, we can state that Q = (qi,j ) ∈ ℝmn×mn ,

i, j = 1, 2, . . . , mn

Where, the 2mn × 2mn matrix S = (sij ) is determined as follows: If qij ≥ 0 for i, j = 1, . . . , mn, then we can put si,j = qij and si+n,j = qij . If qij < 0 for i, j = 1, . . . , mn, then we can put si,j+n = −qij and si+n,j = −qij . e + XB e =C e is extended to the following crisp linear system Thus, the system AX SX = C (3.1) ( ( )t )t where S = (sij ) for 1 ≤ i, j ≤ 2mn and X = x −x , C = c −c . The structure of S implies that sij ≥ 0 for 1 ≤ i, j ≤ 2mn and ( ) S1 S2 S= (3.2) S2 S1 where S1 contains the positive entries of (A ⊕ B t ), S2 contains the absolute values of the negative entries of (A ⊕ B t ). Thus we have A ⊕ B t = S1 − S2 , which implies that the rest of the entries are zero. As can be seen, the Sylvester matrix equation is a general crisp system of linear equations and we can solve 2mn × 2mn system for X . But, we should answer this important question: “Does a crisp function linear system have a solution when the FSME has a solution?” The next example shows that it is possible for SX = C to have e + XB e =C e has a unique solution. no solution or an infinite number of solutions even if AX e + XB e =C e where Example 3.1: Consider the fuzzy Sylvester matrix equation AX ( ) ( ) ( ) 1 0 0 1 (−1 + r, 1 − r) (−2 + 3r, 4 − 3r) e= A= , B= , C . 1 −1 0 0 (−5 + 6r, 2 − r) (−2 + 2r, 5 − 5r) The coefficient of the left hand side system  1 0 1 −1  Q= 1 0 0 1

 0 0 0 0  , 1 0 1 −1

( ) is a nonsingular matrix and hence (A ⊕ B t )x = c has a unique solution, while S = SS21 SS12 is singular, because S1 and S2 are singular matrices. It can be seen that a FSME may

15

SADEGHI ET AL: SYLVESTER MATRIX EQUATION

have no solution or an infinite number of solution. If A+ and B+ contain the positive entries of A and B respectively and A− and B− contain the negative entries of A and B respectively, it is obvious that A = A+ − A− and B = B+ − B− . So, by simplifying the expression (Im ⊗ A) + (B t ⊗ In ) = S1 − S2 , then S1 and S2 can be obtained as S1 = (A+ ⊕ B+t ),

t S2 = (A− ⊕ B− ),

(3.3)

S1 + S2 = (A+ + A− ) ⊕ (B+ + B− )t .

(3.4)

Thus, it can be concluded

Theorem 3.3: [12] The matrix S is nonsingular if and if only the matrices S1 − S2 and S1 + S2 are both nonsingular. ( ) Corollary 3.1: Suppose that A, B are square matrices. Then S = SS21 SS12 is nonsingular matrix if and only if matrices (A ⊕ B t ), (A+ + A− ) ⊕ (B+ + B− )t are nonsingular. Corollary 3.2: Let matrix S be in the form introduced S = S −1

( S1

S2 S2 S1

)

. Then we have

( ) 1 ((A+ + A− ) ⊕ (B+ + B− )t )−1 + (A ⊕ B t )−1 ((A+ + A− ) ⊕ (B+ + B− )t )−1 − (A ⊕ B t )−1 = 2 ((A+ + A− ) ⊕ (B+ + B− )t )−1 − (A ⊕ B t )−1 ((A+ + A− ) ⊕ (B+ + B− )t )−1 + (A ⊕ B t )−1 (3.5)

Corollary 3.3: If a crisp linear system does not have a solution, the associated fuzzy Sylvester linear system does not have one too. In order to solve the fuzzy linear system SX = C directly, S −1 must be calculated (whenever it exists). The next theorem provides information related to the structure of S −1 . Theorem 3.4: [12] The unique solution X of Equation (3.4) is a fuzzy vector for arbitrary C if and only if S −1 is nonnegative. i.e, (sij )−1 ≥ 0. By using result of Theorem 3.4 and corollary 3.2, we are ready to present necessary and sufficient condition for the existence of the solution to the system SX = C. Theorem 3.5: The necessary and sufficient conditions for SX = C to have a solution is that ((A+ + A− ) ⊕ (B+ + B− )t )−1 x = u and (A ⊕ B t )x = v should have a solution, where u = c − c and v = c + c.

16

SADEGHI ET AL: SYLVESTER MATRIX EQUATION

proof: According to [17] a necessary and sufficient condition for non-square system SX = C to be consistent is SS − C = C where S − is a generalized inverse of matrix S. If we consider square system, then it can be written SS −1 C = C. Now SS −1 C = C if and only if:  1  [((A+ + A− ) ⊕ (B+ + B− )t )((A+ + A− ) ⊕ (B+ + B− )t )−1 (c + c)  2     +(A ⊕ B t )−1 (A ⊕ B t )(c + c)] = c, 1   [((A+ + A− ) ⊕ (B+ + B− )t )((A+ + A− ) ⊕ (B+ + B− )t )−1 (c − c)  2    +(A ⊕ B t )−1 (A ⊕ B t )(c − c)] = −c, i.e., if and only if  ((A + A ) ⊕ (B + B )t )((A + A ) ⊕ (B + B )t )−1 (c − c) = c − c, + − + − + − + − (A ⊕ B t )(A ⊕ At )−1 (c + c) = (c + c), i.e., if and only if

 ((A + A ) ⊕ (B + B )t )x = u + − + − (A ⊕ B t )x = v

where u = c − c and v = c + c are consistent.  Since c and c are linear combinations of cij and cij , then c and c are bounded and left continuous. Hence, we should have the following properties for sufficient condition for one of solution of (3.1) to be fuzzy solution vector of (2.5) [5]: 1. c(r) monotonically increasing. 2. c(r) monotonically decreasing. 3. For 0 ≤ r ≤ 1 we have c(r) ≤ c(r). Now, we are ready to define fuzzy solution to the linear system. Remark 3.6: More recently Allahviranloo et al. in [6] has proposed that the fuzzy solution of the fuzzy linear systems, introduced by Friedman et al. [12] may not be a fuzzy numbers vector. In other words, in that case it may be at least one of vector’s components is not fuzzy number. It should be emphasized that in this work the authors considered the fuzzy solutions which are fuzzy vector numbers.

4

New iteration for solving fuzzy linear systems

System (3.1) can be solved directly by using of the inverse of S. In this section, an iteration to solve fuzzy linear system (3.1) is presented.

17

SADEGHI ET AL: SYLVESTER MATRIX EQUATION

Suppose that A ∈ ℝn×n be a full-rank matrix to be determined, Ax = y is the linear system and γ be the step-size or convergence factor. Ding and Chen in [11] proposed a family of iterative methods for solving linear systems which limk→∞ x(k) = x as follows: 2 (4.1) ∥A∥2 ( x ) ( c ) Now suppose that we take G = S T , A = S, X = −x , and C = −c then the following recursion converges to X : x(k+1) = x(k) + γG[y − Ax(k) ],

X (k+1) = X (k) + γS T [C − SX (k) ],

0 1 and center 0. Theorem 2 Let f ∈ H (DR ) and f : [R, ∞) ∪ DR → C be bounded on [0, ∞). If 1 ≤ r < for all |z| ≤ r and n ∈ N we have

R 2

then

∞ X 3 m Kn,q (f ; z) − f (z) − 1 − 2z f 0 (z) − z (1 − z) f 00 (z) ≤ 10 |am | m (m − 1) (2r) . n2 2 (n + 1) 2 (n + 1) m=2 As an application of Theorem 2 we present the order of approximation for complex q-Kantorovich operators. Theorem 3 Let f ∈ H (DR ) and f : [R, ∞) ∪ DR → C be bounded on [0, ∞). If 1 ≤ r < f is not a constant function then the estimate kKn (f ) − f kr ≥

1 Cr (f ) , n

R 2

and if

n ∈ N,

holds, where the constant Cr (f ) depends on f and r but it is independent of n.

2

Auxiliary results

Lemma 4 For all n ∈ N, m ∈ N∪ {0}, z ∈ C we have Kn (em ; z) =

m   X m nk 1 Sn (ek ; z). m k m−k+1 (n + 1) k=0

Proof. The recurrence formula can be derived by direct computation: m  Z 1 t m−k ∞ j m  X j+t (nz) X m j t dt = e−nz m dt k j! n+1 j! 0 0 (n + 1) j=0 j=0 k=0  ∞ j m  X jk (nz) X m −nz =e dt m k (n + 1) (m − k + 1) j! j=0 k=0  m  ∞ j  k X X (nz) j nk m −nz e = k (n + 1)m (m − k + 1) j! n j=0 k=0   m X m 1 nk = S (e ; z). m k (m − k + 1) n k (n + 1)

Kn (em ; z) = e−nz

∞ j Z X (nz)

1



k=0

Lemma 5 For all z ∈ C we have m

|Kn (em ; z)| ≤ (2r) ,

m ∈ N.

j

Proof. Indeed, using the inequality |Sn (ej ; z)| ≤ (2r) from [3] p. 115, we get |Kn (em ; z)| ≤

 m  X m j=0

j

nj |Sn (ej ; z)| (m − j + 1) m

≤ |Sn (em ; z)| ≤ (2r) .

33

(2)

N. I. Mahmudov, M. Kara: Sz´asz-Kantorovich Operators

Lemma 6 We have 1 n + z, 2 (n + 1) n + 1 1 2n n2 2 Kn (e2 ; z) = + z + 2 2 2z , 3 (n + 1) (n + 1) (n + 1)   n−1 1 1 2 2 Kn (e1 − xe0 ) ; z = 2 + 2z + 2z . 3 (n + 1) (n + 1) (n + 1) Kn (e0 ; z) = 1,

Kn (e1 ; z) =

Lemma 7 For all n, m ∈ N, z ∈ C we have Kn (em+1 ; z) = +

z 0 K (em ; z) + zKn (em ; z) n n m+1 X m + 1 1 (n + 1)

m

nj (m − j + 2)

j

j=0



1 j − n + 1 (m + 1) n

 Sn (ej ; z) .

(3)

Proof. We know that, (see [3] p. 115) 0

Sn (ej ; z) = −nSn (ej ; z) +

n Sn (ej+1 ; z) . z

Taking the derivative of the formula (2) and using the above formula we have  m  X 0 nj 1 n m Kn (em ; z) = (−nSn (ej ; z) + Sn (ej+1 ; z) m z (n + 1) j=0 j (m − j + 1)   m  m  X X nj nj 1 n 1 m m S (e ; z) − n Sn (ej ; z) = n j+1 m m z (n + 1) j=0 j (m − j + 1) (n + 1) j=0 j (m − j + 1) m+1 X m  1 z 0 nj−1 Kn (em ; z) = Sn (ej ; z) − zKn (em ; z). m n (n + 1) j=1 j − 1 (m − j + 2)

It follows that Kn (em+1 ; z) = +

−

z 0 K (em ; z) + zKn (em ; z) n n m+1 X m + 1 1 (n + 1)

m+1

j

j=0

nj Sn (ej ; z) (m − j + 2)

m+1 X m  1 nj−1 Sn (ej ; z) m (n + 1) j=1 j − 1 (m − j + 2)

z 0 1 K (em ; z) + zKn (em ; z) + m+1 n n n (m + 2)   m+1 X m + 1 1 nj 1 j + − Sn (ej ; z) m j (m − j + 2) n + 1 (m + 1) n (n + 1) j=1 =

= +

z 0 K (em ; z) + zKn (em ; z) n n m+1 X m + 1 1 (n + 1)

m

nj (m − j + 2)

j

j=0



1 j − n + 1 (m + 1) n

Here we used the identity 

m j−1



 =

m+1 j

34



j . (m + 1)

 Sn (ej ; z) .

N. I. Mahmudov, M. Kara: Sz´asz-Kantorovich Operators

Define En,m (z) := Kn (em ; z) − em (z) −


m2 − 2mz z m−1 . 2(n + 1)

Lemma 8 Let n, m ∈ N, we have the following recurrence formula z m − 1 m−1 1 1 0 (Kn (em−1 ; z) − em−1 (z)) + zEn,m−1 (z) + z − z m−1 + zm n n (n + 1) 2(n + 1) n+1    m  X 1 nj j j m + 1 − − Sn (ej ; z) . m m mn (n + 1) j=0 j (m − j + 1)

En,m (z) =

Proof. It is immediate that En,m (z) is a polynomial of degree less than or equal to m and that En,0 (z) = 0. Using the formula (3) we get
0 o zn 0 (Kn (em−1 ; z) − em−1 (z)) + z m−1 En,m (z) = n     2  (m − 1) − 2 (m − 1) z z m−2  + z En,m−1 (z) + z m−1 +   2(n + 1)
   m  X m2 − 2mz z m−1 1 nj 1 j m m + − S (e ; z) − z − . n j m−1 j (m − j + 1) n + 1 mn 2(n + 1) (n + 1) j=0   2 (m − 1) − 2 (m − 1) z z m−1 m − 1 m−1 z 0 z + zEn,m−1 + = (Kn (em−1 ; z) − em−1 (z)) + n n 2(n + 1)
   m  2 j X m − 2mz z m−1 1 n 1 j m + − Sn (ej ; z) − m−1 j (m − j + 1) n + 1 mn 2(n + 1) (n + 1) j=0 which is the desired recurrence formula.

3

Proofs of the main results

Proof of Theorem 1. Using the recurrence formula (3) we obtain the following relationship: z 0 K (em−1 ; z) + z (Kn (em−1 ; z) − em−1 (z)) n n    m  X 1 nj j j m + 1 − − Sn (ej ; z) . m m mn (n + 1) j=0 j (m − j + 1)

Kn,q (em ; z) − em (z) =

(4)

We can easily estimate the sum in the above formula as follows:    m  j X 1 n j j m 1− − Sn (ej ; z) (n + 1)m j (m − j + 1) m mn j=0      m−1  j X m 1 n m−1 1 − j − j |Sn (ej ; z)| ≤ m j m−j m−j+1 m mn  (n + 1)  j=0 +

nm−1 m Sn (em ; z) (n + 1)

≤

2m (n + 1) + nm−1 2m + 1 m m (2r) ≤ (2r) . m n+1 (n + 1)

m−1

(5)

It is known that by a linear transformation, the Bernstein inequality in the closed unit disk becomes 0 |Pm (z)| ≤

m 35 kPm kr , for all |z| ≤ r, r ≥ 1, r

N. I. Mahmudov, M. Kara: Sz´asz-Kantorovich Operators

where Pm (z) is a complex polynomial of degree ≤ m. From the above recurrence formula (4) we get (2m + 1) |z| 0 m |Kn,q (em ; z) − em (z)| ≤ (2r) Kn (em−1 ; z) + |z| |Kn (em−1 ; z) − em−1 (z)| + n n+1 (2m + 1) r m−1 m kKn (em−1 )kr + r |Kn (em−1 ; z) − em−1 (z)| + (2r) ≤ n r n+1 3m m ≤ r |Kn,q (em−1 ; z) − em−1 (z)| + (2r) . n Writing the last inequality for k = 1, 2, ..., we easily obtain, m

m−1

m−2

(2r) (2r) (2r) 3m + r 3 (m − 1) + r2 n n n 3m (m + 1) m (m + m − 1 + ... + 1) ≤ (2r) . 2n

|Kn (em ; z) − em (z)| ≤ m

=

3 (2r) n

+ ... + rm−1

(2r) 3 n (6)

Since Kn (f ; z) is analytic in DR , we can write Kn (f ; z) =

∞ X

am Kn (em ; z) ,

z ∈ DR ,

m=0

which, together with estimate (6) immediately implies for all |z| ≤ r |Kn (f ; z) − f (z)| ≤

∞ X

|am | |Kn (em ; z) − em (z)| ≤

m=0

∞ 3 X m |cm | m (m + 1) (2r) . 2n m=1

Proof of Theorem 2. A simple calculation and applying the recurrence formula (3) lead us to the following relationship z m − 1 m−1 0 (Kn (em−1 ; z) − em−1 (z)) + zEn,m−1 (z) + z n n (n + 1) ! 1 1 nm−1 m Sn (em ; z) + (z − Sn (em ; z)) + 1− m−1 n+1 n+1 (n + 1)

En,m (z) =


nm−2 1 Sn (em−1 ; z) − z m−1 m (m − 1) Sn (em−1 ; z) + 2 (n + 1) 2 (n + 1) ! 1 nm−1 − 1− Sn (em−1 ; z) m−1 2 (n + 1) (n + 1)   m−2 X m 1 nj j j + 1− − Sn (ej ; z) m m mn (n + 1) j=0 j (m − j + 1) −

:=

9 X

Ik .

k=1

From the proof of Theorem 1.8.4 of [3], we have |z m − Sn (em ; z)| ≤

6 (m − 1) m−1 (2r) . n

It follows that m−1

1 6 (m − 1) (2r) |z m − Sn (em ; z)| ≤ , n+1 n (n + 1) m−1 3 (m − 2) (2r)m−2 1 z |I7 | ≤ − Sn36(em−1 ; z) ≤ . 2 (n + 1) n (n + 1)

|I4 | ≤

N. I. Mahmudov, M. Kara: Sz´asz-Kantorovich Operators

Applying the inequality 1−

k Y

xj ≤

j=1

k X

(1 − xj ) ,

0 ≤ xj ≤ 1,

j = 1, ..., k

j=1

we have 1 |I5 | ≤ n+1

1−

1 |I8 | ≤ 2 (n + 1)

!

nm−1 (n + 1)

1−

|Sn (em ; z)| ≤

m−1

(n + 1)

2

!

nm−1 (n + 1)

m−1

m−1

|Sn (em−1 ; z)| ≤

(2r)

m

m−1 2 (n + 1)

2

(2r)

m−1

.

For I9 we have m−2 X m − 2 m(m − 1) 1 nj j j |I9 | ≤ 1− − |Sn (ej ; z)| m j (m − j)(m − j − 1) (m − j + 1) m mn (n + 1) j=0 ≤

2m(m − 1) (n + 1) m (n + 1)

m−2 m−2

(2r)

≤

2m(m − 1) (n + 1)

2

(2r)

m−2

.

Thus |En,m (z)| ≤ +

r m − 1 m−1 6 (m − 1) 0 m−1 (Kn (em−1 ; z) − em−1 (z)) + r |En,m−1 (z)| + r + (2r) n n (n + 1) n (n + 1) m−1

m

2

(n + 1) m−1

(2r) +

m−2

m−1 2

(2r)

m−1

+

3 (m − 2) (2r) n (n + 1)

2 (n + 1) 2m(m − 1) m−1 m−2 + + 2 (2r) 2 (2r) 2 (n + 1) (n + 1) r m − 1 3m (m − 1) 8m(m − 1) m m ≤ (2r) + r |En,m−1 (z)| + 2 (2r) n r 2n (n + 1) 2

≤ r |En,m−1 (z)| +

10m (m − 1) m (2r) n2 2

|En,m (z)| ≤ r |En,m−1 (z)| +

10m (m − 1) m (2r) n2

As a consequence, we get 3

|En,m (z)| ≤

10m (m − 1) m (2r) . n2

P∞ Note that since f (4) = m=4 am m (m − 1) (m − 2) (m − 3) z m−4 and the series is absolutely convergent for all |z| < R, it easily follows the finiteness of the involved constants in the statement. Proof of Theorem 3. For all z ∈ DR and n ∈ N we get    1 (1 − 2z) 0 z 00 (1 − 2z) 0 z 00 Kn (f ; z)−f (z) = f (z) + f (z) + n Kn (f ; z) − f (z) − f (z) − f (z) . n 2 2 2n 2n We apply kF + Gkr ≥ |kF kr − kGkr | ≥ kF kr − kGkr to get kKn (f ) − f kr ≥

1 n

   

(1 − 2z) 0

z 00 37

− n Kn (f ; z) − f (z) − (1 − 2z) f 0 (z) − z f 00 (z) . f (z) + f (z)



2 2 2n 2n r r

N. I. Mahmudov, M. Kara: Sz´asz-Kantorovich Operators

Taking into account that by hypothesis f is not a polynomial of degree 0 in DR , we get ||e1 (1 − e1 )f 00 − e1 f 0 ||r > 0. Indeed, supposing the contrary it follows that (1 − 2z)f 0 (z) + zf 00 (z) = 0 for all |z| ≤ r, that is (zf 0 (z))0 − 2zf 0 (z) = 0 for all |z| ≤ r. The last equality is equivalent to zf 0 (z) = Ce2z for all e2z , for all |z| ≤ r with z 6= 0. But since f is |z| ≤ r with z 6= 0. Therefore we get f 0 (z) = C z analytic in Dr , we necessarily have C = 0, which implies f 0 (z) = 0 and f (z) = c for all z ∈ Dr , a contradiction with the hypothesis. Now, by Theorem 2 we have ∞ (1 − 2z) 0 z 00 10 X 3 m |am | m (m − 1) (2r) → 0 as n → ∞. n Kn (f ; z) − f (z) − f (z) − f (z) ≤ 2n 2n n m=2 Consequently, there exists n1 (depending only on f and r) such that for all n ≥ n1 we have





(1 − 2z) 0

(1 − 2z) 0 z 00 z 00

f (z) + f (z) − n Kn (f ; z) − f (z) − f (z) − f (z)

2 2 2n 2n r r

0 00 1 z (1 − 2z) ≥ f (z) + f (z)

, 2 2 2 r which implies

1 z 00 (1 − 2z) 0

kKn (f ) − f kr ≥ f (z) + f (z)

, 2n 2 2 r

for all n ≥ n1 .

For 1 ≤ n ≤ n1 − 1 we have kKn (f ) − f kr ≥

1 1 (n kKn (f ) − f kr ) = Mr,n (f ) > 0, n n

which finally implies that 1 Cr (f ) , n

o n 0

z 00 for all n, with Cr (f ) = min Mr,1 (f ) , ..., Mr,n1 −1 (f ) , 21 (1−2z) f (z) + f (z)

. 2 2 kKn (f ) − f kr ≥

r

References [1] J. Favard, Sur les multiplicateurs d’interpolation, J. Math. Pures Appl. 23 (1944) 219–247. Series 9. [2] V. Totik, Uniform approximation by positive operators on infinite intervals, Anal. Math. 10 (1984) 163–182. [3] S.G. Gal, Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, Vol. 8, World Scientific Publishing Co, 2009. [4] S.G. Gal, Approximation and geometric properties of complex Favard-Szasz-Mirakian operators in compact diks, Comput. Math. Appl., 56 (2008), 1121-1127. [5] S.G. Gal, Approximation of analytic functions without exponential growth conditions by complex Favard-Sz´ asz-Mirakjan operators. Rend. Circ. Mat. Palermo (2) 59 (2010), no. 3, 367–376. [6] N. I. Mahmudov, Convergence properties and iterations for q-Szasz polynomials in compact disks, Computers & Mathematics with Applications, Volume 60, Issue 6, September 2010, Pages 1784-1791.

38

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 39-44, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

IDENTITIES ON THE MODIFIED q-EULER AND q-BERNSTEIN POLYNOMIALS AND NUMBERS WITH WEIGHT SEOG-HOON RIM AND JOOHEE JEONG

Abstract. Recently, the modified q-Euler numbers and polynomials with weight α are introduced in [15]. In this paper, we give some interesting identities on the modified q-Euler numbers and polynomials with weight α and q-Bernstein polynomials. These results are different from those in [11].

1. Introduction Let p be a fixed odd prime number. Throughout, this paper Zp , Qp , and Cp will respectively denote the ring of p−adic integers, the field of p−adic rational numbers and the completion Sof the algebraic closure of Qp . Let N be the set of natural numbers and Z+ = N {0}. Let vp be the normalized exponential valuation of Cp with |p|p = 1/p. Let q ∈ C with |1 − q|p < 1. As a notation of q-numbers, x [x]q is defined by [x]q = 1−q 1−q ,(cf. [1–16]) In this paper, we assume that α ∈ Qp . Note that limq→1 [x]q =x. Let C(Zp ) be the space of continuous function on Zp . For f ∈ C(Zp ), the q-Bernstein operator is introduced in [2, 3, 6, 11, 15] as follows :   n X k n B(α) (f : x) = f ( ) [x]kqα [1 − x]n−k 1 n qα n k k=0 (1) n X k e (α) = f ( )B (x, q). n k,n k=0

(α) Here,Bn (f

: x) is called the weighted q-Bernstain operator of order n for f . For k, n ∈ Z+ , the weighted q-Bernstein polynomial of degree n is defined by   n (α) e (2) Bk,n (x) = [x]kqα [1 − x]n−k , (see [4, 7, 11]). 1 qα k For f ∈ C(Zp ), the p-adic invariant q-integral on Zp is defined by Kim as follows: Z I−q (f ) = f (x)dµ−q (x) Zp N

(3)

pX −1 1 = lim f (x)µ−q (x + pN Zp ) N →∞ [pN ]−q x=0 N

pX −1 1 = lim f (x)(−q)x , N →∞ [pN ]−q x=0

(see [5]).

Let f1 (x) = f (x + 1). Then we see that (4)

I−q (q −x f1 ) + I−q (q −x f ) = [2]q f (0)

39

(see [5, 6])

RIM AND JEONG

IDENTITIES ON THE MODIFIED Q-EULER AND Q-BERNSTEIN POLYNOMIALS

In [15], the modified q-Euler numbers and polynomials with weight α are defined as follows: (α) Een,q =

Z

q −x [x]nqα dµ−q (x),

for n ∈ Z+

Zp

(5)

n   αl X n [2]q l q (−1) = (1 − q α )n l 1 + q αl l=0

= [2]q

∞ X

(−1)m [m]nqα .

m=0

From (5), we can derive the following relations on modified q-Euler polynomials with weight α and modified q-Euler numbers with weight α:

(6)

(α) Een,q (x) =

n   X n l=0

l

(α)

αlx e (−1)l [x]n−l El,q qα q

= ([x]qα + q αx Eeq(α) )n , (α) (α) with the usual convention about replacing (Eeq )n by Een,q . (see [11, 15]) (α) By (5) and (6), we get the following recurrence relation for Een,q :

[2]q (α) Ee0,q = , 2 (7) (q α Eeq(α)

n

(α) + 1) + Een,q =

( [2]q , if n = 0, 0, if n > 0, (α)

(α)

with the usual convention about replacing (Eeq )n by Een,q . In this paper, we give some interesting properties between the weighted qBernstein polynomial and the modified q-Euler numbers with weight by using p-adic invariant q-integral on Zp . From these properties, we derive identities on the modified q-Euler polynomials with weight and the weighted q-Bernstein polynomials. These results are different from those in [11].

2. Identities on the modified q-Euler polynomials and Bernstein polynomials The relation of reflection symmetry of the modified q-Euler polynomials with weight α, we evaluate the following p-adic q-integral on Zp , we have (8)

(α) (α) Een,q−1 (1 − x) = (−1)n q αn Een,q (x). (see [15])

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IDENTITIES ON THE MODIFIED q-EULER AND q-BERNSTEIN POLYNOMIALS

From (8), we have (α) (α) n Een,q (2) = ([2]qα + q 2α Een,q ) = (q α (q α Eeq(α) + 1) + 1)n n   X n αl α e(α) = q (q Eq + 1)l l l=0 n   X n αl α e(α) (α) α α e(α) 1 e = E0,q + nq (q Eq + 1) + q (q Eq + 1)l l l=2   X n   [2]q n αl e(α) [2]q (α) + nq α + Ee1,q + q El,q = 2 2 l l=2 n   X n αl e α [2]q + = nq q El,q , for n > 0. 2 l

(9)

l=0

Let n ∈ N with n ≥ 2. Then, by (7) and (9), we obtain the following: For n ∈ N with n ≥ 2, we have [2]q (α) (α) = (q Eeq(α) + 1)n = Een,q . Een,q (2) − nq α 2 From (2), we have  Z Z n −x e (α) q Bk,n (x, q)dµ−q (x) = q −x [x]kqα [1 − x]n−k q −α dµ−q (x) k Zp Zp     n−k Z n X n−k l (−1) q −x [x]k+l = q α dµ−q (x) (11) l k Z p l=0   n−k   n X n−k (α) = (−1)l Eek+l,q . k l

(10)

l=0

Thus by (11), we have a proposition: Proposition 1. For n, k ∈ Z+ , we have Z q

−x

e (α) (x, q)dµ−q (x) = B k,n

Zp

    n−k n X n−k (α) (−1)l Eek+l,q . k l l=0

By the definition of the weighted q-Bernstein polynomials in (2), we see that e (α) (x, q) = B e (α) (1 − x, 1/q), B k,n n−k,n

(12)

where k, n ∈ Z+ and x ∈ Z+ . From (12), we have Z Z e (α) (x, q)dµ−q (x) = e (α) (1 − x, 1/q)dµ−q (x) q −x B q −x B k,n n−k,n Zp

(13)

Zp

 X Z k   n k k+l = (−1) q −x [1 − x]n−l q −α dµ−q (x) k l Z p l=0  X k   n k (α) (−1)k+l Een−l,q−1 (2). = k l l=0

Therefore, by (13), we obtain the following theorem :

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RIM AND JEONG

IDENTITIES ON THE MODIFIED Q-EULER AND Q-BERNSTEIN POLYNOMIALS

Theorem 2. For n, k ∈ Z+ , we have Z q

−x

e (α) (x, q)dµ−q (x) = B k,n

Zp

 X k   n k (α) (−1)k+l Een−l,q−1 (2). k l l=0

By comparing the coefficients on the both sides of proposition and Theorem 2, we obtain the following corollary : Corollary 3. For n, k ∈ Z+ , we have n−k X

 k   X n−k k (α) (α) (−1)k+l Eek+l,q = (−1)k+l Een−l,q−1 (2). l l

l=0

l=0

From (10) and Corollary 3, we obtain the following corollary : Corollary 4. For n, k ∈ Z+ with n > k, we have n−k X l=0

 k   X n−k k (α) l e(α) (−1) Ek+l,q = (−1)k+l Een−l,q−1 (2). l l l=0

In particular, if k = 0 , we have n   X n [2]qα (α) (α) (α) (−1)l Eel,q = Een,q−1 (2) = Een,q + nq −α . 2 l l=0

For m, n, k ∈ Z+ with m, n > k. Then we can derive the following equation from the p-adic invariant q-integral on Zp : (14) Z

e (α) (x, q)B e (α) (x, q)dµ−q (x) q −x B k,n k,m

Zp

   Z n m n+m−2k q −x [x]2k dµ−q (x) q α [1 − x]q −α k k Zp    X  Z 2k  n m 2k l+2k = (−1) q −x [1 − x]n+m−l dµ−q (x) q −α k k l Z p l=0    X  2k  n m 2k (α) = (−1)l+2k Een+m−l,q−1 (2). k k l

=

l=0

Therefore, by (14), we obtain the following theorem : Theorem 5. For k, n, m ∈ Z+ , with m, n > k, we obtain Z q Zp

−x

e (α) (x, q)B e (α) (x, q)dµ−q (x) = B k,n k,m

   X  2k  n m 2k (α) (−1)l+2k Een+m−l,q−1 (2). l k k l=0

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RIM AND JEONG

IDENTITIES ON THE MODIFIED q-EULER AND q-BERNSTEIN POLYNOMIALS

From (14), we have Z e (α) (x, q)B e (α) (x, q)dµ−q (x) q −x B k,n k,m Zp

   X  2k    n m 2k (α) (−1)l+2k 2 + Een+m−l,q−1 k k l l=0    X   2k n m 2k (α) = (−1)l+2k Een+m−l,q−1 . k k l

=

(15)

l=0

Let k, n, m ∈ Z+ . By (3)and (4), we get Z e (α) (x, q)B e (α) (x, q)dµ−q (x) q −x B k,n k,m Zp

   Z n m n+m−2k = q −x [x]2k dµ−q (x) q α [1 − x]q −α k k Zp    n+m−2k X n + m − 2k  n m (α) (−1)l Eel+2k,q . = l k k

(16)

l=0

Therefore, by (16), we obtain the following theorem : Theorem 6. For k, n, m ∈ Z+ with m, n > k, we obtain Z q

−x

e (α) (x, q)B e (α) (x, q)dµ−q (x) = B k,n k,m

Zp

   n+m−2k X n + m − 2k  n m (α) (−1)l Eel+2k,q . k k l l=0

By comparing the coefficients on the both sides of proposition and Theorem 6 and (16), we obtain the following corollary : Corollary 7. For m, n, k ∈ Z+ with m, n > k, we have n+m−2k X  l=0

  2k  X n + m − 2k 2k (α) (α) e (−1)El+2k,q = (−1)2k+l Een+m−l,q−1 , l l

if

k > 0.

l=0

In particular, if k = 0 , we have n+m X n + m  [2]q (α) (α) + Een+m,q−1 . (−1)l Eel,q = (n + m)q −α 2 l l=0

Continuing this process, we obtain the following theorem : Theorem 8. For k, n1 , n2 , ..., ns ∈ Z+ with n1 + n2 + ... + ns > sk, we obtain n1 +n2 +...+n X s −skn1 + n2 + ... + ns − sk  (α) (−1)l Eel+sk,q l l=0 sk   X sk (α) = (−1)sk+l Een1 +n2 +...+ns −l,q−1 , if k > 0. l l=0

43

RIM AND JEONG

IDENTITIES ON THE MODIFIED Q-EULER AND Q-BERNSTEIN POLYNOMIALS

In particular, if k = 0 , we have  n1 +nX 2 +...+ns  n1 + n2 + ... + ns (α) (−1)l Eel,q = l l=0

(n1 + · · · + ns )q −α

[2]q (α) + Een1 +n2 +...+ns ,q−1 . 2

References [1] M. Acikgoz, Y. Simsek, On multiple interpolation function of the N¨ orlund-type q-Euler polynomials, Abst. Appl. Anal. 2009 (2009), Article ID 382574, 14 pages. [2] S.Araci, D. Erdal and J.J Seo, A study on the fermionic p-adic q-integral representation on Zp associated with weighted q-Bernstein and q-Genocchi polynomials,Abstract and Applied Analysis 2011 (2011), Article in press http://hindawi.com/26592680. [3] S.Araci, D. Erdal and D-J,Kang, Some New Properties on the q-Genocchi numbers and Polynomials associated with q-Bernstein polynomials, Honam Mathematical J. 33 (2011) no. 2, 261-270 [4] A. Bayad, T. Kim, B. Lee and S.-H. Rim, Some Identities on Bernstein Polynomials associated with q-Euler polynomials, Abstract and Applied Analysis, vol 2011, Article ID 294715, 10 pages 2011. [5] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288–299. [6] T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys. 16 (2009), 484-491. [7] T Kim, A note on q-Bernstein polynomials, Russ. J. Math. Phys. 18 (2011), no.2, 41-50. [8] T. Kim, Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Phys. A: Math. Theor. 43 (2010), 255201, 11 pages. [9] T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys. 15 (2007), 51-57. [10] T.Kim, An application of polylogarithms in analogs of Genocchi numbers, Russ. J. Math. Phys. 17 (2010), no. 2, 218-225. [11] T. Kim, S.H. Lee, B. Lee and S.-H. Rim, Identities of the weighted q-Euler and q-Bernstein polynomials and numbers, submitted. [12] T. Kim, B. Lee, J. Choi, Y.H. Kim, A new approach of q-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 14 (2011), no. 1, 7-14. [13] H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. 18 (2009), 41–48. [14] H. Ozden, Y. Simsek, S.-H. Rim, I. N. Cangul, Multivariate interpolation functions of higherorder Euler numbers and their applications, Abstract and Applied Analysis, 2008 (2008), Article ID 390897, 16 pages. [15] S.-H. Rim and J. Jeong, A Note on The Modified q-Euler Numbers and Polynomials with Weight α, International Mathematical Forum 6 no.65 (2011), 3245–3250 [16] C. S. Ryoo, Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstein polynomials, Proceedings of the Jangieon Mathematical Society, 14 (2011), 239– 348. Seog-Hoon Rim. Department of Mathematics Education, Kyungpook National University, Daegu 702-701, Republic of Korea, E-mail address: [email protected] Joohee Jeong. Department of Mathematics Education, Kyungpook National University, Daegu 702-701, Republic of Korea, E-mail address: [email protected]

44

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 45-54, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

GENERALIZED TERNARY BI-DERIVATIONS ON TERNARY BANACH ALGEBRAS: A FIXED POINT APPROACH MADJID ESHAGHI GORDJI, GWANG HUI KIM, JUNG RYE LEE∗ , AND CHOONKIL PARK Abstract. Using fixed point method, we investigate the Hyers-Ulam stability of generalized bi-derivations on ternary Banach algebras.

1. Introduction and preliminaries The stability problem of functional equations had been first raised by Ulam (cf. [35]): Let (G1 , .) be a group and let (G2 , ∗) be a metric group with the metric d(., .). Given ϵ > 0, dose there exist a δ > 0, such that if a mapping h : G1 → G2 satisfies the inequality d(h(x.y), h(x) ∗ h(y)) < δ for all x, y ∈ G1 , then there exists a homomorphism H : G1 −→ G2 with d(h(x), H(x)) < ϵ for all x ∈ G1 ? Ulam [34] discusses: The notion of stability of mathematical theorems considered from a rather general point of view: When is it true that by changing a little the hypothesis of a theorem one can still assert that the thesis of the theorem remains true or approximately true? In the other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [15] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Th.M. Rassias [28] introduced a new functional inequality that we call CauchyRassias inequality and succeeded to extend the result of Hyers by weakening the condition for the Cauchy difference to be unbounded: if there exist an ϵ ≥ 0 and 0 ≤ p < 1 such that ∥f (x + y) − f (x) − f (y)∥ ≤ ϵ(∥x∥p + ∥y∥p ) for all x, y ∈ E1 , then there exists a unique additive mapping T : E1 → E2 such that ∥f (x) − T (x)∥ ≤

2ϵ ∥x∥p |2 − 2p |

2010 Mathematics Subject Classification. Primary 39B52, 47B47, 47H10, 17A40, 39B82, 20N10. Key words and phrases. Hyers-Ulam stability; bi-derivation; ternary Banach algebra; fixed point. ∗ Corresponding author. 1

45

GORDJI ET AL: TERNARY BANACH ALGEBRAS

for all x ∈ E1 (see [16, 17, 29, 30, 31]). In 1991, Gajda [13] solved the problem for 1 < p, which was raised by Th.M. Rassias. In fact, the result of Th.M. Rassias is valid for 1 < p; moreover, Gajda gave an example that a similar stability result does not hold for ˇ p = 1. Another example was given by Th.M. Rassias and P. Semrl [32]. In 1994, a generalization of the Th.M. Rassias’ theorem was obtained by Gˇavruta as follows [14]. Suppose (G,+) is an abelian group, E is a Banach space, and that the so-called admissible control function φ : G × G → ℝ satisfies φ(x, ˜ y) := 2

−1

∞ ∑

2−n φ(2n x, 2n y) < ∞

n=0

for all x, y ∈ G. If f : G → E is a mapping with ∥f (x + y) − f (x) − f (y)∥ ≤ φ(x, y) for all x, y ∈ G, then there exists a unique mapping T : G → E such that T (x + y) = T (x) + T (y) and ∥f (x) − T (x)∥ ≤ φ(x, ˜ x) for all x, y ∈ G. Ternary algebraic operations were considered in the 19-th century by several mathematicians such as A. Cayley [3] who introduced the notion of cubic matrix which in turn was generalized by Kapranov, Gelfand and Zelevinskii in 1990 ([19]). The comments on physical applications of ternary structures can be found in [1, 20, 21, 33, 36]. A C ∗ -ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z)  [xyz] of A3 into A, which is C-linear in the outer variables, conjugate Clinear in the middle variable, and associative in the sense that [xy[zvw]] = [x[wzy]v] = [[xyz]wv], and satisfies ∥[xyz]∥ ≤ ∥x∥.∥y∥.∥z∥ and ∥[xxx]∥ = ∥x∥3 (see [3], [6]–[12] and [24]). A C- bilinear mapping δ : A × A → A is called a ternary bi-derivation if it satisfies δ([abc], d) = [δ(a, d)bc] + [aδ(b, d)c] + [abδ(c, d)], δ(a, [bcd]) = [δ(a, b)cd] + [bδ(a, c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A (see [2]). A C-bilinear mapping D : A × A → A is called a ternary generalized bi-derivation if there exists a bi-derivation δ : A × A → A such that D([abc], d) = [D(a, d)bc] + [aδ(b, d)c] + [abδ(c, d)], D(a, [bcd]) = [D(a, b)cd] + [bδ(a, c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A. Let X and Y be be real or complex linear spaces. For a mapping f : X × X → Y, consider the functional equation f (a + b, c − d) + f (a − b, c + d) = 2f (a, c) − 2f (b, d).

46

(1.1)

GORDJI ET AL: TERNARY BANACH ALGEBRAS

Bae and W. Park [2] investigated the Hyers-Ulam stability of ternary bi-derivations for bi-additive mappings satisfying (1.1). We recall a fundamental result in fixed point theory. Theorem 1.1. ([5]) Suppose that a complete generalized metric space (X , d) and a strictly contractive mapping J : X → X with Lipschits constant 0 < L < 1 are given. Then, for a given element x ∈ X , exactly one of the following assertions is true: either (1) d(J n x, J n+1 x) = ∞ for all n ≥ 0 or (2) there exists n0 such that d(J n x, J n+1 x) < ∞ for all n ≥ n0 . Actually, if (a2 ) holds, then the sequence J n x is a convergent to a fixed point x∗ of J and (3) x∗ is the unique fixed point of J in Λ := {y ∈ X , d(J n0 x, y) < ∞}; (4) d(y, x∗ ) ≤ d(y,Jy) for all y ∈ Λ. 1−L In this paper, we investigate the Hyers-Ulam stability of ternary generalized bi-derivations associated with the functional equation f (x − y, t) + f (x, t − s) = 2f (x, t) − f (y, t) − f (x, s). 2. Main results From now on, we assume that A is a ternary Banach algebra. For a given mapping f : A × A → A, we define the difference operator Eλ,µ f : A4 → A by Eλ,µ f (a, b, c, d) = f (λa − λb, µc) + f (λa, µc − µd) − λµ(2f (a, c) − f (b, c) − f (a, d)) for all λ, µ ∈ T1 := {λ ∈ C; |λ| = 1} and all a, b, c, d ∈ A. We establish the stability of ternary generalized bi-derivations. Theorem 2.1. Let f, g : A × A → A be mappings such that g(0, 0) = f (0, 0) = 0. Let φ : A4 → [0, ∞) be a function such that max{∥Eλ,µ f (a, b, c, d)∥, ∥Eλ,µ g(a, b, c, d)∥} ≤ φ(a, b, c, d),

(2.1)

max{∥f ([abc], d) − [f (a, d)bc] − [af (b, d)c] − [abf (c, d)]∥, ∥f (a, [bcd]) − [f (a, b)cd] − [bf (a, c)d] − [bcf (a, d)]∥}

(2.2)

≤ φ(a, b, c, d), max{∥g([abc], d) − [g(a, d)bc] − [af (b, d)c] − [abf (c, d)]∥, ∥g(a, [bcd]) − [g(a, b)cd] − [bf (a, c)d] − [bcf (a, d)]∥} ≤ φ(a, b, c, d), 1 φ(2n a, 2n b, 2n c, 2n d) = 0 n→∞ 4n lim

47

(2.3)

GORDJI ET AL: TERNARY BANACH ALGEBRAS

for all λ, µ ∈ T1 and all a, b, c, d ∈ A. If there exists an L < 1 such that Ψ(a, b) ≤ 4LΨ( a2 , 2b ) for all a, b ∈ A, where Ψ(a, b) := φ(0, a, 2b, 0) + φ(a, −a, 2b, b) + φ(0, 0, 2b, 0) + 3(φ(a, 0, b, −b) +φ(a, 0, 0, b) + φ(a, 0, 0, 0) + φ(0, 0, b, 0)), then there is a unique ternary bi-derivation δ : A × A → A and a unique ternary generalized bi-derivation D : A × A → A (related to δ) such that max{∥g(a, c) − D(a, c)∥, ∥f (a, c) − δ(a, c)∥} ≤ for all a, c ∈ A.

a c L Ψ( , ) 1−L 2 2

Proof. It follows from (2.1) that ∥Eλ,µ f (a, b, c, d)∥ ≤ φ(a, b, c, d),

(2.4)

Setting λ = µ = 1 in (2.4), we have ||f (a − b, c) + f (a, c − d) − 2f (a, c) + f (b, c) + f (a, d)|| ≤ φ(a, b, c, d), ||g(a − b, c) + g(a, c − d) − 2g(a, c) + g(b, c) + g(a, d)|| ≤ φ(a, b, c, d)

(2.5) (2.6)

for all a, b, c, d ∈ A. Letting a = b = d = 0 in (2.5), shows that ||f (0, c)|| ≤ φ(0, 0, c, 0). Letting b = c = d = 0 in (2.5), we get ∥f (a, 0)∥ ≤ φ(a, 0, 0, 0). Setting b = −a, c = 2d in (2.5), we get ||f (2a, 2d) + 2f (a, d) − 2f (a, 2d) + f (−a, 2d)|| ≤ φ(a, −a, 2d, d).

(2.7)

Putting a = d = 0 in (2.5), we conclude that ||f (−b, c) + f (b, c)|| ≤ φ(0, b, c, 0) + φ(0, 0, c, 0).

(2.8)

Letting b = a, c = 2d in (2.8), we get ||f (−a, 2d) + f (a, 2d)|| ≤ φ(0, a, 2d, 0) + φ(0, 0, 2d, 0).

(2.9)

By (2.7) and (2.9), we obtain ||f (2a, 2d) + 2f (a, d) − 3f (a, 2d)|| ≤ φ(0, a, 2d, 0) + φ(a, −a, 2d, d) + φ(0, 0, 2d, 0). (2.10) Setting b = 0 and d = −c in (2.5), we get ||f (a, 2c) − f (a, c) + f (0, c) + f (a, −c)|| ≤ φ(a, 0, c, −c).

(2.11)

Letting b = c = 0 in (2.5), we have ∥f (a, d) + f (a, −d)∥ ≤ φ(a, 0, 0, d) + φ(a, 0, 0, 0).

48

(2.12)

GORDJI ET AL: TERNARY BANACH ALGEBRAS

Substituting d by c in (2.12) shows that ||f (a, −c) + f (a, c)|| ≤ φ(a, 0, 0, c) + φ(a, 0, 0, 0).

(2.13)

By (2.11) and (2.13), we obtain ||f (a, 2c) − 2f (a, c)|| ≤ φ(a, 0, c, −c) + φ(a, 0, 0, c) + φ(a, 0, 0, 0) + φ(0, 0, c, 0). (2.14) Letting c = d in (2.14) and multiplying both sides by 3, we get ||3f (a, 2d)−6f (a, d)|| ≤ 3(φ(a, 0, d, −d)+φ(a, 0, 0, d)+φ(a, 0, 0, 0)+φ(0, 0, d, 0)). (2.15) By (2.15) and (2.10), we obtain ||f (2a, 2d) − 4f (a, d)|| ≤ Ψ(a, d). Hence

1 1 a d ∥ f (2a, 2d) − f (a, d)∥ ≤ Ψ(a, d) ≤ LΨ( , ) 4 4 2 2

(2.16)

for all a, d ∈ A. Consider the set X := {g| g : A × A → A, g(0) = 0} and introduce the generalized metric on X a d d(g, h) := inf{t ∈ ℝ+ : ∥g(a, d) − h(a, d)∥ ≤ tΨ( , ), ∀a, d ∈ A}. 2 2 It is easy to show that (X, d) is a complete generalized metric space. Now we consider the linear mapping J : X → X such that 1 J(g)(a, d) := g(2a, 2d) 4 for all g ∈ X, a, d ∈ A. Let g, h ∈ X be given such that d(g, h) = ϵ. Then a d ∥g(a, d) − h(a, d)∥ ≤ ϵΨ( , ) 2 2 for all a, d ∈ X. Hence 1 1 ∥Jg(a, d) − Jh(a, d)∥ = ∥g(2a, 2d) − h(2a, 2d)∥ ≤ ϵΨ(a, d) ≤ Lϵ 4 4 for all a, d ∈ A. So d(g, h) = ϵ implies that d(J(g), J(h)) ≤ Lϵ. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ X. It follows from (2.16) that d(f, Jf ) ≤ L. By alternative fixed point, J has a unique fixed point in the set X1 := {h ∈ X : d(f, h) < ∞}. Let δ be the fixed point of J, that is, δ(2a, 2d) = 4δ(a, d) for all a, d ∈ A, and there exists a t ∈ (0, ∞) such that a d ∥δ(a, d) − f (a, d)∥ ≤ tΨ( , ) 2 2

49

GORDJI ET AL: TERNARY BANACH ALGEBRAS

for all a, d ∈ A. On the other hand, we have lim d(J n f, δ) = 0.

n→∞

It follows that

1 f (2n a, 2n d) = δ(a, d) n→∞ 4n 1 for all a, d ∈ A. It follows from d(f, δ) ≤ 1−L d(f, Jf ) that lim

d(f, δ) ≤

(2.17)

L . 1−L

This implies that L a c Ψ( , ) 1−L 2 2 for all a, c ∈ A. It follows from (2.4) and (2.17) that ∥f (a, c) − δ(a, c)∥ ≤

(2.18)

∥δ(x − y, t) + δ(x, t − s) − 2δ(x, t) + δ(y, t) + δ(x, s)∥ 1 = lim n ∥f (2n (x − y), 2n t) n→∞ 4 + f (2n x, 2n (t − s)) − 2f (2n x, 2n t) + f (2n y, 2n t) + f (2n x, 2n s) 1 ≤ lim n φ(2n x, 2n y, 2n t, 2n s) n→∞ 4 =0 for all x, y, t, s ∈ A. So δ(x − y, t) + δ(x, t − s) = 2δ(x, t) − δ(y, t) − δ(x, s) for all x, y, t, s ∈ A. By [12, Lemma 2.2], the mapping δ : A × A → A is C-bilinear. By (2.6), we can use the same method as above to show that the limit 1 g(2n a, 2n d) n→∞ 4n exists for all a, d ∈ A. Moreover, we can show that D : A × A → A is C-bilinear mapping satisfying a c L ∥g(a, c) − D(a, c)∥ ≤ Ψ( , ) 1−L 2 2 for all a, c ∈ A. It follows from (2.2) that D(a, d) = lim

∥δ([abc], d) − [δ(a, d)bc] − [aδ(b, d)c] − [abδ(c, d)]∥ 1 = lim n (∥f ([2n a2n b2n c], 2n d) − [f (2n a, 2n d)2n b2n c] n→∞ 4 − [2n af (2n b, 2n d)2n c] − [2n a2n bf (2n c, 2n d)]∥) 1 ≤ lim n φ(2n a, 2n b, 2n c, 2n d) = 0 n→∞ 4

50

GORDJI ET AL: TERNARY BANACH ALGEBRAS

for all a, b, c, d ∈ A. This means that δ([abc], d) = [δ(a, d)bc] + [aδ(b, d)c] + [abδ(c, d)] for all a, b, c, d ∈ A. Similarly, we can show that δ(a, [bcd]) = [δ(a, b)cd] + [bδ(a, c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A. Hence δ is a bi-derivation. On the other hand, by (2.2), we have ∥D([abc], d) − [D(a, d)bc] − [aδ(b, d)c] − [abδ(c, d)]∥ 1 = lim n (∥g([2n a2n b2n c], 2n d) − [g(2n a, 2n d)2n b2n c] n→∞ 4 − [2n af (2n b, 2n d)2n c] − [2n a2n bf (2n c, 2n d)]∥) 1 ≤ lim n φ(2n a, 2n b, 2n c, 2n d) = 0 n→∞ 4 for all a, b, c, d ∈ A. It follows that D([abc], d) = [D(a, d)bc] + [aδ(b, d)c] + [abδ(c, d)] for all a, b, c, d ∈ A. Similarly, we can show that D(a, [bcd]) = [D(a, b)cd] + [bδ(a, c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A. This means that D is a generalized bi-derivation related to δ.



Corollary 2.2. Let p ∈ (0, 2) and q ∈ (0, ∞) be real numbers. Suppose that f, g : A × A → A with g(0, 0) = f (0, 0) = 0 satisfying max{∥Eλ,µ f (a, b, c, d)∥, ∥Eλ,µ g(a, b, c, d)∥, ∥f ([abc], d) − [f (a, d)bc] − [af (b, d)c] − [abf (c, d)]∥, ∥f (a, [bcd]) − [f (a, b)cd] − [bf (a, c)d] − [bcf (a, d)]∥, ∥g([abc], d) − [g(a, d)bc] − [af (b, d)c] − [abf (c, d)]∥, ∥g(a, [bcd]) − [g(a, b)cd] − [bf (a, c)d] − [bcf (a, d)]} ≤ q(∥a∥p + ∥b∥p + ∥c∥p + ∥d∥p ) for all a, b, c, d ∈ A. Then there exist a unique ternary bi-derivation δ : A × A → A and a unique ternary generalized bi-derivation D : A × A → A (related to δ) such that 5q (∥a∥p + ∥c∥p ) max{∥g(a, c) − D(a, c)∥, ∥f (a, c) − δ(a, c)∥} ≤ 4 − 2p for all a, c ∈ A. Proof. It follows from Theorem 2.1 by putting φ(a, b, c, d) := q(∥a∥p + ∥b∥p + ∥c∥p + ∥c∥p ) for all a, b, c, d ∈ A and L = 2p−2 .



51

GORDJI ET AL: TERNARY BANACH ALGEBRAS

Theorem 2.3. Let f, g : A × A → A be mappings such that g(0, 0) = f (0, 0) = 0. Let φ : A4 → [0, ∞) be a mapping satisfying (2.1), (2.2) and (2.3). Let lim 4n φ(2−n a, 2−n b, 2−n c, 2−n d) = 0

n→∞

for all λ, µ ∈ T1 and all a, b, c, d ∈ A. If there exists an L < 1 such that Ψ(a, b) ≤ L Ψ(2a, 2b) for all a, b ∈ A, where Ψ(a, b) is defined in Theorem 2.1. Then there exist 4 a unique ternary bi-derivation δ : A × A → A and a unique ternary generalized biderivation D : A × A → A (related to δ) such that max{∥g(a, c) − D(a, c)∥, ∥f (a, c) − δ(a, c)∥} ≤

L Ψ(a, c) 4 − 4L

for all a, c ∈ A. 

Proof. The proof is similar to the proof of Theorem 2.1.

Corollary 2.4. Let p ∈ (2, ∞) and q ∈ (0, ∞) be real numbers. Suppose that f, g : A × A → A with g(0, 0) = f (0, 0) = 0 satisfying max{∥Eλ,µ f (a, b, c, d)∥, ∥Eλ,µ g(a, b, c, d)∥, ∥f ([abc], d) − [f (a, d)bc] − [af (b, d)c] − [abf (c, d)]∥, ∥f (a, [bcd]) − [f (a, b)cd] − [bf (a, c)d] − [bcf (a, d)]∥, ∥g([abc], d) − [g(a, d)bc] − [af (b, d)c] − [abf (c, d)]∥, ∥g(a, [bcd]) − [g(a, b)cd] − [bf (a, c)d] − [bcf (a, d)]} ≤ q(∥a∥p + ∥b∥p + ∥c∥p + ∥d∥p ) for all a, b, c, d ∈ A. Then there exist a unique ternary bi-derivation δ : A × A → A and a unique ternary generalized bi-derivation D : A × A → A (related to δ) such that max{∥g(a, c) − D(a, c)∥, ∥f (a, c) − δ(a, c)∥} ≤

5q (∥a∥p + ∥c∥p ) 2p − 4

for all a, c ∈ A. Proof. It follows from Theorem 2.3 by taking φ(a, b, c, d) := q(∥a∥p + ∥b∥p + ∥c∥p + ∥c∥p ) for all a, b, c, d ∈ A and L = 22−p .



Acknowledgement This work was supported by the Daejin University Research Grants in 2012.

52

GORDJI ET AL: TERNARY BANACH ALGEBRAS

References [1] V. Abramov, R. Kerner and B. Le Roy, Hypersymmetry a Z3 graded generalization of supersymmetry, J. Math. Phys. 38 (1997), 1650. [2] J. Bae and W. Park, Approximate bi-homomorphisms and bi-derivations in C ∗ -ternary algebras, Bull. Korean Math. Soc. 47 (2010) 195–209. [3] A. Cayley, On the 34 concomitants of the ternary cubic, Amer. J. Math. 4 (1881) 1–15. [4] S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias type. Hadronic Press, Palm Harbor, Florida, 2003. [5] J.B. Diaz, B. Margolis, A fixed point theorem of the alternative for the contractions on generaliuzed complete metric space, Bull. Amer. Math. Soc. 74 (1968) 305–309. [6] A. Ebadian, N. Ghobadipour, M. Eshaghi Gordji, A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C ∗ -ternary algebras, J. Math. Phys. 51 (2010), 10 pages, doi:10.1063/1.3496391. [7] M. Eshaghi Gordji and N. Ghobadipour, Stability of (𝛼, 𝛽, 𝛾)-derivations on Lie C ∗ -algebras, Int. J. Geom. Methods Mod. Phys. 7 (2010), No. 7, 1–10. [8] M. Eshaghi Gordji and M.S. Moslehian, A trick for investigation of approximate derivations, Mathematical Communications (to appear). [9] M. Eshaghi Gordji, H. Khodaei and R. Khodabakhsh, On approximate 𝑛-ary derivations, Int. J. Geom. Methods Mod. Phys. 8 (2011), No. 3, 1–16. [10] M. Eshaghi Gordji and N. Ghobadipour, Nearly generalized Jordan derivations, Math. Slovaca 61 (2011) 55–62. [11] M. Eshaghi Gordgi and N. Ghobadipour, Approximately quartic homomorphisms on Banach algebras, Word Applied Sciences Journal (to appear). [12] M. Eshaghi Gordji and A. Fazeli, Stability and superstability of ∗-bihomomorphisms on C ∗ -ternary algebras (preprint). [13] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991) 431–434. [14] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431–436. [15] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941) 222–224. [16] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Boston, 1998. [17] D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992) 125– 153. [18] G. Isac and Th. M. Rassias, Stability of 𝜓-additive mappings: applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996) 219–228. [19] M. Kapranov, I. M. Gelfand and A. Zelevinskii, Discrimininants, Resultants and Multidimensional Determinants, Birkh¨ auser, Berlin, 1994. [20] R. Kerner, Ternary algebraic structures and their applications in physics, Univ. P. M. Curie preprint, Paris (2000), http://arxiv.org/list/math-ph/0011. [21] R. Kerner, The cubic chessboard: Geometry and Physics, Class. Quantum Grav. 14 (1997), A203. [22] G.J. Murphy, C ∗ -Algebras and Operator Theory, Acad. Press, 1990. [23] C. Park, Linear ∗-derivations on JB ∗ -algebras, Acta Math. Sci. Ser. B Engl. Ed. 25 (2005) 449–454. [24] C. Park and M. Eshaghi Gordji, Comment on ‘Approximate ternary Jordan derivations on Banach ternary algebras [Bavand Savadkouhi et al., J. Math. Phys. 50 (2009), 042303]’, J. Math. Phys. 51 (2010), 044102, 7 pages. [25] C. Park, Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach,Fixed Point Theory and Applications 2008 (2008) Article ID 493751, 9 pages. [26] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003) 91–96.

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[27] J. R¨ 𝑎𝑡𝑧, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003) 191–200. [28] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [29] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000) 23–130. [30] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000) 264–284. [31] Th.M. Rassias (Ed.), Functional Equations, Inequalities and Applications, Kluwer Academic, Dordrecht, Boston and London, 2003. ˇ [32] Th.M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992) 989–993. [33] G. L. Sewell, Quantum Mechanics and its Emergent Macrophysics, Princeton Univ. Press, Princeton, NJ, 2002. MR1919619 (2004b:82001). [34] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ., New York, 1960. [35] S. M. Ulam, Problems in Modern Mathematics,Chapter VI, Science ed. Wiley, New York, 1940. [36] H. Zettl, A characterization of ternary rings of operators, Adv. Math. 48 (1983) 117-143. MR0700979 (84h:46093). Madjid Eshaghi Gordji Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran E-mail address: [email protected] Gwang Hui Kim Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Korea E-mail address: [email protected] Jung Rye Lee Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea E-mail address: [email protected] Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected]

54

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 55-64, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

DETERMINANT FORM AND A TEST OF CONVERGENCE FOR THE NEWTON-PADE APPROXIMATIONS ALEXEY LUKASHOV AND CEVDET AKAL

Abstract. One of the main goals of the paper is to give a new extension of Hadamard’s formula for numerator and denominator of Newton-Pad´ e approximations in Newton form. As a byproduct, we’ll give a test of convergence for the Newton-Pad´ e approximations. The test is an analogue of M.B. Balk’s test of convergence of Pad´ e approximations and its proof heavily uses those determinant forms.

1. Introduction

The Newton-Pad´e approximants are a particular case of the multipoint Pad´e approximants, corresponding to the situation when the sets of interpolation points are nested. Given a formal power series f (z) and a sequence {βn : n ∈ N } ⊂ C with the corresponding polynomials (1.1)

ωn (z) =

n Y

(z − βj ) , β0 (z) = 1,

j=1

the linear Frobenius-Pad´e approximation of degree [m, n] is built as a ratio Rm,n = Pm /Qn of the polynomials of degree ≤ m and ≤ n, respectively, such that Qn f −Pm is divisible by ωm+n+1 . First formula for Rm,n was given by Cauchy (concerning its version for the general case see [14], [13]). Computational aspects are discussed in many papers, let us indicate [10] and recent paper [16] , [18] and [6] as an example. Cauchy’s formula may be considered as a generalization of the Lagrange interpolation formula. The main goals of the paper are a new extension of the Hadamard’s formula for Pm and Qn , and a convergence criterion based on the identity (3.6). The work contains also a substantial amount of auxiliary algebraic results, such as connection formulas for the expansion coefficients of a polynomial in terms of two different bases ωk and ωk0 . Here we’ll consider generalizations of the Newton interpolation formula for rational case. So f will be considered as a formal Newton series. Definition 1.1. [7] A formal Newton series (FNS) is an ordered triple ∞

∞

∞

[{αn }0 , {βn }0 , {fn }0 ] , Key words and phrases. Newton-Pade approximations, Newton series, Test of convergence.

55

ALEXEY LUKASHOV AND CEVDET AKAL

where α0 , α1 , α2 , · · · and β1 , β2 , β3 , · · · are complex numbers (not necessarily distinct) and for each n = 0, 1, 2, · · · , fn is the polynomial (1.2)

fn (z) =

n X

αk ωk (z) ,

k=0

where (1.3)

ω0 (z) = 1;

ωk (z) =

k Y

(z − βj ) , k = 1, 2, 3, · · · ,

j=1

and where z is a complex variable. The αn , βn , and fn are called, respectively, the nth Newton coefficient, interpolation point, and partial sum of [{αn } , {βn } , {fn }] and a FNS is said to converge at z if the sequence of partial sums {fn (z)} is convergent. Definition 1.2. The Newton-Pad´e approximation M P

(1.4)

RM,N (z) =

PM (z) = i=0 N P QN (z)

ci ωi (z) bi ωi (z)

i=0

with degree of numerator (resp. denominator) less than or equal to M (resp. N ), is the function of the form (1.4) such that (1.5)

QN (z) · f (z) − PM (z) = dM +N +1 ωM +N +1 (z) + · · · .

Different formulas for the Newton-Pad´e approximations were obtained by G. Claessens [4], A. Draux [5], M.A. Gallucci, W.B. Jones [7], D.D. Warner [17]. In [7] the approximations were given as a rational function R in the form M P

RM,N (z) =

ci ωi (z)

i=0 N P

. λi

zi

i=0

In their paper a determinant form for the denominator was given. In [4] the approximations were considered in the form (1.4), but only recurrence formulas for numerator and denominator coefficients were given. Recently a very elegant formula for a special rational interpolation problem which also generalizes the Newton interpolation formula was given in [6, Theorem 2.2]. One of the main goals of this paper is to give an explicit determinant form for the numerator and denominator of Newton-Pad´e approximations in the form (1.4). Convergence of multipoint Pad´e approximations was considered by many authors (see, for instance; [1],[2],[8], [9] and survey [15]). But to the best of our knowledge there are very few papers with investigations of that convergence from algebraic properties of the Newton series coefficients (see [11], [12]). As a byproduct, we’ll give a test of convergence for the Newton-Pad´e approximations. The test is an analogue of M.B. Balk’s test of convergence of Pad´e approximations [3] and its proof heavily uses those determinant forms.

56

DETERMINANT FORM FOR NEWTON-PADE APPROXIMATIONS

´ approximations 2. Explicit formulas for Newton-Pade n

n

Lemma 2.1. Let {zk }k=1 and {zk0 }k=1 be two finite sequences of (not necessarily distinct) complex numbers. If ωn (z) =

n Y

(z − zk ) ,

ω0 (z) = 1

k=1

and ωn0 (z) = (z − z10 ) (z − z20 ) · · · (z − zn0 ) , then ωn0 (z) =

n X

cin ωi (z) ,

i=0

where cin

i+1 X i+1 X

=

i+1 X

···

k1 =1 k2 =k1

n−i Y

 zkj − zk0 j +j−1 ,

kn−i =kn−i−1 j=1

0, 1, · · · , n − 1, cn,n = 1

i =

Proof. By induction, suppose it is valid for n = m − 1. Writing down 0 ωm (z)

0 0 = ωm−1 (z) (z − zm )=

m−1 X

0 ci,m−1 ωi (z) (z − zm )

i=0

=

m−1 X

0 ci,m−1 ωi (z) [(z − zi+1 ) + (zi+1 − zm )]

i=0

one gets (2.1)

0 ci,m = ci−1,m−1 + ci,m−1 (zi+1 − zm ),

i = 1, 2, · · · , m − 1.

So, for i = 0 we have
0 0 0 c0,m = c0,m−1 (z1 − zm ) = (z1 − z10 ) · · · z1 − zm−1 (z1 − zm ). Now, by (2.1) and by the induction hypothesis,

ci,m =

i i X X k1 =1 k2 =k1

+

i+1 X

i+1 X

k1 =1 k2 =k1

···

i X

···

m−i Y

zkj − zk0 j +j−1



km−i =km−i−1 j=1

i+1 X

0 (zi+1 − zm )

m−i−1 Y  j=1

km−i−1 =km−i−2

57

 zkj − zk0 j +j−1 .

ALEXEY LUKASHOV AND CEVDET AKAL

Combining in all sums, we obtain ci,m

i i X X

=

···

i X

m−i Y

k1 =1 k2 =k1

zkj − zk0 j +j−1



(zi+1 − zj0 )

j=i+2

i+1 X

···

m Y

(zi+1 − zj0 ) + · · · +

j=i+2

k1 =1

i =

m Y

(zk1 − zk0 1 )

i+1 X i+1 X

=

i+1 X

km−i−1 =km−i−2 km−i =km−i−1 j=1

k1 =1 k2 =k1

+

i X

m−i Y

 zkj − zk0 j +j−1 ,

km−i =km−i−1 j=1

1, 2, · · · , m − 1;

m = 1, 2, · · · , n.

So, the lemma is proved.



Corollary 2.2. Let k and l be two integers. Then the identity l+k X

ωk (z) · ωl (z) =

(i)

Ωk,l ωi (z)

i=max(l,k)

holds, where (i)

Ωk,l

i+1 X

i+1 X

=

i+1 X

···

j1 =max(l,k)+1 j2 =j1


zj1 − zj1 −max(l,k) · · ·

jk+l−i =jk+l−i−1


× zjk+l−i − zjk+l−i +k+l−i−1−max(l.k) .

Proof. This is an immediate application of Lemma 3.



Lemma 2.3. If (2.2)

f (z) =

∞ X

al ωl (z)

l=0

is a Newton series with finitely many non-zero terms and (2.3)

QN (z) =

N X

bk ωk (z) ,

k=0

then f (z) QN (z) =

∞ X

ei,N ωi (z) ,

i=0

where N −(N −i)+

(2.4)

ei,N =

X

i X

βi,k bk, βi,k =

k=0

(i)

al Ωk,l .

l=i−k

Proof. Applying corollary 4, we get f (z) QN (z) =

∞ X l=0

al

N X

bk ωl (z)ωk (z) =

k=0

∞ X l=0

58

al

N X k=0

bk

l+k X i=max(l.k)

(i)

Ωk,l ωi (z).

DETERMINANT FORM FOR NEWTON-PADE APPROXIMATIONS

Let us split the summation in three parts as follows, f (z) QN (z) =

N X

al

l=0

l X

l+k X

bk

k=0

i=l

∞ X

+

(2.5)

(i)

Ωk,l ωi (z) +

al

l=N +1

N X

N X

al

l=0 N X

l+k X

bk

k=0

l+k X

bk

k=l+1

(i)

Ωk,l ωi (z)

i=k

(i)

Ωk,l ωi (z).

i=l

Then by changing the order of summations in each term of (2.5), we obtain N X

al

l=0

+

N X

l X

bk

l+k X

k=0

i X

(i)

Ωk,l ωi (z) =

N X

X

(i)

al bk Ωk,l ωi (z)

i=N +1 k=i−N l=i−k

2N X

+

bi/2c

2N X

(i)

al bk Ωk,l ωi (z) +

i=0 k=d(i+1)/2e l=k

(2.6)

(i)

al bk Ωk,l ωi (z)

i=0 k=0 l=i−k

i=l i X

N bi/2c i X X X

N X

N X

(i)

al bk Ωk,l ωi (z),

i=N +1 k=d(i+1)/2e l=k

N X

N X

al

l=0

bk

k=l+1

(2.7)

l+k X

(i)

Ωk,l ωi (z) =

i X

k−1 X

(i)

al bk Ωk,l ωi (z)

i=0 k=d(i+1)/2e l=i−k

i=k 2N X

+

N X

N X

k−1 X

(i)

al bk Ωk,l ωi (z),

i=N +1 k=d(i+1)/2e l=i−k

and ∞ X

al

l=N +1

(2.8)

bk

k=0

2N X

+

N X

l+k X i=l

N X

2N X

i−N X−1

i X

i=N +1

k=0

l=i−k

(i)

Ωk,l ωi (z) =

i X

∞ X

(i)

al bk Ωk,l ωi (z) +

(i)

al bk Ωk,l ωi (z)

i N X X

(i)

al bk Ωk,l ωi (z).

i=2N +1 k=0 l=i−k

i=N +1 k=i−N l=N +1

By combining (2.7) and the second and fourth terms of (2.6) we get, taking into account (2.5) and (2.8) the equality f (z) QN (z) =

N X i i X X

(i)

al bk Ωk,l ωi (z) +

i=0 k=0 l=i−k

+

2N X i=N +1

(2.9)

+

i−N X−1

i X

k=0

l=i−k

∞ X

N X

N X

N X

(i)

al bk Ωk,l ωi (z)

i=N +1 k=d(i+1)/2e l=i−k 2N X

(i)

al bk Ωk,l ωi (z) +

i X

2N X

N X

i X

(i)

al bk Ωk,l ωi (z)

i=N +1 k=i−N l=N +1 (i) al bk Ωk,l ωi (z)

i=2N +1 k=0 l=i−k

+

2N X

bi/2c

X

N X

i=N +1 k=i−N l=i−k

59

(i)

al bk Ωk,l ωi (z).

ALEXEY LUKASHOV AND CEVDET AKAL

Combining second, third, fourth and sixth terms of (2.9) we obtain f (z) QN (z) =

N X i i X X i=0 k=0 l=i−k

(i)

al bk Ωk,l ωi (z)

i=N +1 k=0 l=i−k

∞ X

+

(2.10)

2N X N X i X

(i)

al bk Ωk,l ωi (z) + N X

i X

(i)

al bk Ωk,l ωi (z),

i=N +1 k=0 l=i−k

and combining second and third terms of (2.10) we finally get f (z) QN (z) =

∞ N −(N i X X−i)+ X i=0

 where x+ =

k=0

(i)

al bk Ωk,l ωi (z) ,

l=i−k

x, x > 0 . 0, x ≤ 0



Now, we may give the following definition. Definition 2.4. Let f (z) =

∞ P

al ωl (z) and g(z) =

∞ P

cl ωl (z) be formal Newton

l=0

l=0

series with interpolation points {zi } and let r be a complex number. We define: ∞ P (a) (f + g)(z) = (al + cl )ωl (z), l=0

(b) (r · f )(z) =

∞ P

(c) If

l=0 N P

(r · cl )ωl (z),

QN (z) =

f (z) QN (z) =

bk ωk (z) , k=0 i ∞ N −(N P−i)+ P P

i=0

k=0

l=i−k

(i)

al bk Ωk,l ωi (z),

Definition 2.5. The Hankel-Newton determinants for a formal Newton series (2.2) are defined by γM,0 γM,1 ··· γM,N γ γM +1,1 · · · γM +1,N HM,N = M +1,0 . · · · ··· ··· ··· γM +N,0 γM +N,1 · · · γM +N,N Theorem 2.6. Let f (z) =

∞ P

al ωl (z) be a formal Newton series with interpolation

l=0

points {zi } and let M and N be (fixed) nonnegative integers. Then, (A) If PM (z) = c0 ω0 (z) + c1 ω1 (z) + . . . + cM ωM (z) and QN (z) = b0 ω0 (z) + b1 ω1 (z) + . . . + bN ωN (z), then a necessary and sufficient condition that the formal Newton series f (z) QN (z) − PM (z) be of the form (2.11)

f (z) QN (z) − PM (z) = dN +M +1 ωN +M +1 + dN +M +2 ωN +M +2 + . . .

60

DETERMINANT FORM FOR NEWTON-PADE APPROXIMATIONS

is that the coefficients cj and bj satisfy the system of equations (2.12)

b0 γ0,0 = c0 b0 γ1,0 + b1 γ1,1 = c1 .. .

.. .

.. .

.. .

.. .

b0 γM −1,0 + b1 γM −1,1 + . . . + bM −1 γM −1,M −1 = cM −1 (2.13)

b0 γM,0 + b1 γM,1 + . . . + bN γM,N = cM b0 γM +1,0 + b1 γM +1,1 + . . . + bM γM +1,N = 0 .. .

.. .

.. .

.. .

.. .

b0 γM +N,0 + b1 γM +N,1 + . . . + bM γM +N,N = 0 where the γi,j are defined by (2.4) and γi,j = 0 for i < j. (B) A nontrivial solution, b0 , · · · , bN , c0 , · · · , cM , to the system of equations (2.13) is determined uniquely (up to a nonzero multiplicative constant) if and only if HM,N 6= 0. (C) If cM , b0 , · · · , bN satisfy (2.13), then γM,0 ··· γM,j−1 cM γM,j+1 ··· γM,N γ · · · γM +1,j−1 0 γM +1,j+1 · · · γM +1,N , (2.14) bj · HM,N = M +1,0 · · · · · · · · · · · · ··· ··· · · · γM +N,0 · · · γM +N,j−1 0 γM +N,j+1 · · · γM +N,N j = 0, 1, · · · , N. (D) In particular, if HM,N 6= 0 and b0 , · · · , bN satisfy (2.12), then we can choose cM = 1 and obtain γM,0 ··· γM,j−1 1 γM,j+1 ··· γM,N γM +1,0 · · · γM +1,j−1 0 γM +1,j+1 · · · γM +1,N ··· ··· ··· ··· ··· ··· · · · γM +N,0 · · · γM +N,j−1 0 γM +N,j+1 · · · γM +N,N (2.15) bj = , HM,N j = 0, 1, · · · , N. Proof is quite analogous to the proof of Theorem 2 and Lemma 4 from [7] and is omitted here. Corollary 2.7. If HM,N 6= 0, then the following determinant representations for the denominator and numerator hold: ω0 (z) ω1 (z) · · · ωN (z) γ γM +1,1 · · · γM +1,N 1 M +1,0 γM +2,0 γM +2,1 · · · γM +2,N QN (z) = HM,N ··· ··· · · · ··· γM +N,0 γM +N,1 · · · γM +N,N and PM (z) =

M X k=0

where ck ’s are given in (2.13).

61

ck ωk (z)

ALEXEY LUKASHOV AND CEVDET AKAL

´ approximations 3. Test of convergence for Newton-Pade Firstly, recall a definition of convergence of double sequences. ∞

Definition 3.1. The double sequence {αµ,ν }µ,ν=0 of complex numbers converges ∞ if all sequences {αµk ,νk }k=1 with non-repeating points (µk , νk ) converge and have the same limit. Secondly, we will use the following theorem. ∞

Theorem 3.2. [3] If all subsequences {αµk ,νk }k=0 , such that 1)µk ≤ µk+1, 2)νk ≤ νk+1 , 3) (µk+1 − µk ) + (νk+1 − νk ) = 1, ∞

converge and have the same limit then the double sequence {αµ,ν }µ,ν=0 converges. Theorem 3.3. Suppose that for any increasing sequences {µn } , {νn } such that  µn ≤ µn+1 ≤ µn + 1,  νn ≤ νn+1 ≤ νn + 1, (3.1)  (µn+1 − µn ) + (νn+1 − νn ) ≥ 1. and Hµn ,νn 6= 0, there exists a natural number N such that the series (3.2)

∞ X n=N

×

µn

(−1)

Hµn +1,νn Qνn+1 (zµn +νn +2 ) ωµn +νn +1 (z) Hµn ,νn

1 , Qνn (z) Qνn+1 (z)

converges uniformly in a domain G which contains points z1, z2, . . .. Then the double sequence of the Newton-Pad´e approximations {Rµ,ν (z)} with poles z1, z2, . . . , converges in G to f (z) . Proof. Consider the expression (3.3)

Pµn+1 (z) Qνn (z) − Pµn (z) Qνn+1 (z) = [f (z) Qνn (z) − Pµn (z)] Qνn+1 (z)   − f (z) Qνn+1 (z) − Pµn+1 (z) Qνn (z)

The left hand side is a polynomial of ωk (z) with k ≤ µn + νn + 1. The second term in the right hand side by the definition of Newton-Pad´e approximations starts from the term ωk (z) with k ≥ µn+1 + νn+1 + 1 (> µn + νn + 1), and the first term starts with the term ωk (z) with k = µn + νn + 1. Hence, in both sides of (3.3) there is only one non-zero term with ωµn +νn +1 (z) .  Now, since Hµn ,νn 6= 0, the solution of (2.12) and (2.13) with M = µn , N = νn is determined uniquely (up to a nonzero multiplicative constant). The coefficient of ωµn +νn +1 (z) in Pµn+1 (z) Qνn (z) − Pµn (z) Qνn+1 (z) is equal to the coefficient of ωµn +νn +1 (z) in the expression of [f (z) Qνn (z) − Pµn (z)] Qνn+1 (z) .

62

DETERMINANT FORM FOR NEWTON-PADE APPROXIMATIONS

By Lemma 7, the coefficient of ωµn +νn +1 (z) in f (z) Qνn (z) − Pµn (z) is equal to eµn +νn +1,νn and by straightforward calculations it is equal to µn

(3.4)

(−1)

Hµn +1,νn . Hµn ,νn

By Corollary 6, the coefficient of ωµn +νn +1 (z) in [f (z) Qνn (z) − Pµn (z)] Qνn+1 (z) is equal to µn

(3.5)

(−1) µn

= (−1)

νn+1 Hµn +1,νn X +νn +1 ) bk Ω(µµnn+ν n +1,k Hµn ,νn

Hµn +1,νn Hµn ,νn µn

= (−1)

k=0 νn+1 k X Y

bk

k=0

(zµn +νn +2 − zj )

j=1

Hµn +1,νn Qνn+1 (zµn +νn +2 ) . Hµn ,νn

So, (3.6)

Pµn+1 (z) Qνn (z) − Pµn (z) Qνn+1 (z) µ Hµn +1,νn = (−1) n Qνn+1 (zµn +νn +2 ) ωµn +νn +1 (z) . Hµn ,νn

Note that formula (3.6) may be considered as a complete analogue of the well-known formula valid for classical Pad´e approximants, see e.g. [1, chapter 3, (5.16)]. Hence, (3.7) µn

= (−1)

Rµn+1, νn+1 (z) − Rµn ,νn (z) Hµn +1,νn Qνn+1 (zµn +νn +2 ) ωµn +νn +1 (z) Hµn ,νn 1 × Qνn (z) Qνn+1 (z)

From here it is clear if series (3.2) converges at a point z, then at that point the sequence {Rµn ,νn (z)} converges too. Denote the limit function by F (z) . From the supposition of theorem it follows that Qνn (z) 6= 0 in G, starting from some number N. Hence, all functions Rµn ,νn (z) are analytic in G for n ≥ N. Since the sequence {Rµn ,νn } converges uniformly on compacts in G, the function F (z) is also analytic in G. Moreover, for any point z ∈ G dk F (z) dk Rµn ,νn (z) = k n→∞ dz dz k for any natural number k. Taking z1, z2, . . . , for z values, we get that F (zj ) = f (zj ) , j = 1, 2, . . . . Hence F (z) = f (z) identically in G. Application of Theorem 11 finishes the proof. lim

Remark 3.4. M.B. Balk [3] considered a large class of functions satisfying suppositions of his test of convergence for the Pad´e approximations. A continuity argument shows that Theorem 12 may be applied for any function of that class and poles zi , |zi | < ε with sufficiently small ε > 0.

63

ALEXEY LUKASHOV AND CEVDET AKAL

References 1. G.A. Baker, P. Graves-Morris, Pade approximants, vol.1, 1981. 2. G.A. Baker, P. Graves-Morris, Pade approximants, vol.2 1981. 3. M.B.Balk, The Pad´ e interpolation process for certain analytic functions (Russian), in: Markushevich, A.I. (Ed.), Issledovaniya po sovremennym problemam teorii funktzii kompleksnogo peremennogo, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960, pp.234-257. 4. G. Claessens, The rational Hermite interpolation problem and some related recurrence formulas, Comput. Math. Appl. 2 (1976) 117-123. 5. A. Draux, Formal orthogonal polynomials and Newton-Pad´ e approximants, Numer. Alg. 29 (2002) 67-74. 6. A. M. Fu, A. Lascoux, A Newton type rational interpolation formula, Adv. Appl. Math. 41 (2008) 452-458 7. M.A. Gallucci, W.B. Jones, Rational approximations corresponding to Newton series (NewtonPad´ e approximants), J. Approx. Theory 17 (1976) 366-392. 8. A. A. Gonchar, G. L´ opez Lagomasino, On Markov’s theorem for multipoint Pad´ e approximants, Mat. Sb. 105(4) (1978) 512–524. 9. A. A. Gonchar, E. A. Rakhmanov, Equilibrium distributions and degree of rational approximation of analytic functions, Mat. Sb. 134(3) (1987) 306–352. 10. M. H. Gutknecht, The multipoint Pad´ e table and general recurrences for rational interpolation, Acta Appl. Math. 33 (1993) 165-194. 11. A. A. Kandayan, Multipoint Pad´ e approximations of the beta function, Math. Notes 85(2) (2009) 176-189. 12. G. L´ opez Logomasino, Survey on multipoint Pad´ e approximation to Markov type meramorphic functions and asymptotic properties of the orthogonal polynomial generated by them, in: C. Brezinski, A. Draux, A. Magnus, P. Maroni, A. Ronveaux (Eds.), Orthogonal polynomials and applications, Lecture Notes in Mathematics, 1171, Springer, Berlin, 1985, pp. 309-316. 13. J. Meinguet, On the solubility of the Cauchy interpolation problem, Approximation Theory, ed. Talbot, A., Academic Press, London 1970, 535-600. 14. H. E. Salzer, An osculatory extension of Cauchy’s rational interpolation formula, Zamm-Z. Angew. Math. Mech. 64(1) (1984) 45-50. 15. H. Stahl, Convergence of rational interpolants, Bull. Belg. Math. Soc. Simon Stevin (1996) suppl. 11-32. 16. S. Tang, L. Zou, C. Li, Block based Newton-like blending osculatory rational interpolation, Anal. Theory Appl. 26(3) (2010) 201–214. 17. D. D. Warner, Hermite interpolation with rational functions, Ph.D. Thesis, Univ. of California (1974). 18. Q. Zhao, J. Tan, Block-based Thiele-like blending rational interpolation, J. Comput. Appl. Math. 195 (2006) 312–325. Alexey Lukashov, Department of Mathematics, Fatih University, 34500, Istanbul, Turkey and N.G. Chernyshevsky Saratov State University, Saratov, Russia E-mail address: [email protected] Cevdet Akal, Department of Mathematics, Fatih University, 34500, Istanbul, Turkey E-mail address: [email protected]

64

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 65-72, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

On the mean value of character sums in short intervals∗ Jing Gao Department of Mathematical Sciences, Xi’an Jiaotong University Xi’an 710049, Shaanxi, P. R. China E-mail: [email protected]

Huaning Liu Department of Mathematics, Northwest University Xi’an 710069, Shaanxi, P. R. China E-mail: [email protected]

Abstract £ q¤ £ qIn¤ this paper, we study the mean value of character sums in short intervals 1, 4 and 1, 3 , and give a few asymptotic formulae, by using the Fourier expansion for character sums, and the mean value theorems of Dirichlet L- functions. Key words Character sums; mean value; short intervals. AMS subject classfications 11L40; 11L26.

§1. Introduction Let q > 1 be an integer, and χ be a Dirichlet character modulo q. The character sums play an important role in number theory, so many authors studied the properties of the character sums and obtained lots of beautiful results. For example, for χ any non-principal character modulo p (prime) and any positive integer x, G. P´olya [4] and I. M. Vinogradov [6] proved that ¯ x ¯ ¯X ¯ √ ¯ ¯ χ(a) ¯ ¯ ≤ c p log p. ¯ ¯ a=1

The above inequality was established by D. A. Burgess [1] with the constant c = 1 as follows: ¯ N +x ¯ ¯ X ¯ 2 ¯ ¯ χ(a)¯ ¿ x1−1/r p(r+1)/4r log p, ¯ ¯ ¯ a=N +1

where x and r are arbitrary positive integers, and N is any integer. ∗

Supported by the National Natural Science Foundation of China under Grant No.10901128, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No.20090201120061, the Natural Science Foundation of the Education Department of Shaanxi Province of China under Grant No.09JK762, and the Fundamental Research Funds for the Central University.

65

Jing Gao and Huaning Liu Let χ be a primitive character modulo q. For r = 1 or 2, and every ² > 0, D. A. Burgess [1] showed that

¯ ¯ +H ¯ NX ¯ 2 ¯ ¯ χ(n)¯ ¿ H 1−1/r q (r+1)/4r +² . ¯ ¯ ¯ n=N +1

A. V. Sokolovskii [5] proved that there exists a positive integer x, such that ¯ ¯ ¯x+[q/2] ¯ ¯ X ¯ p 1 √ ¯ χ(n)¯¯ > 1 − 8(log q)/q · √ · q. ¯ 2 2 ¯ n=x ¯ For general non-principal character χ modulo q, and any positive integer h, D. A. Burgess [2] gave the estimate

¯2 ¯ h q ¯X ¯ X ¯ ¯ χ(n + m) ¯ < qh. ¯ ¯ ¯

n=1 m=1

Furthermore, he [3] obtained the following estimate for primitive character sums: ¯ ¯4 q ¯X h ¯ X∗ X ¯ ¯ χ(n + m)¯ ≤ 8d7 (q)q 2 h2 , ¯ ¯ ¯ χ mod q n=1 m=1

where

P∗

denotes the summation over all the primitive characters modulo q, and d(q) is the

χ mod q

Dirichlet divisor function.

£ ¤ In this paper, we shall study the mean value of character sums in short intervals 1, 4q and £ q¤ 1, 3 . This problem may be interesting since it relates to the existed works. Now we present the main results. Theorem 1.1. For any positive integer q > 1, we have ´3 ³ ¯ ¯4 ¯ 1 1 − X∗ ¯¯ X Y 2 ¯ ¡ 2+² ¢ p ¯ ¯ = 5 q 2 J(q) + O q χ(a) ¯ ¯ 144 1 + p12 ¯ χ mod q ¯a≤ q p|2q χ(−1)=1

and

X∗ χ mod q χ(−1)=−1

¯ ¯4 ¯ ¯ ¯X ¯ ¯ ¯ χ(a) ¯ ¯ ¯a≤ q ¯ 4

4

 ³ ´3  1  1 − Y 2  ¡ 2+² ¢ 5 2 p   q J(q) + O q ,    24 1 + p12 p|2q ´3 ³ =  1  1 − Y  ¡ ¢ 5 2 p2   + O q 2+² , q J(q)   1  144 1+ p|q

if q is odd;

if q is even,

p2

X ³q´ µ where J(q) = φ(d) denotes the number of all primitive characters modulo q. d d|q

Theorem 1.2. For any positive integer q > 1, we have ´3 ³ ¯4 ¯ ¯ ¯ 1 1 − X∗ ¯ X Y ¯ ¡ ¢ 5 2 p2 ¯ χ(a)¯¯ = q J(q) + O q 2+² ¯ 1 256 1 + p2 ¯ χ mod q ¯a≤ q p|3q χ(−1)=1

3

66

Character sums in short intervals and X∗ χ mod q χ(−1)=−1

³ ´3 ¯ ¯4 ¯X ¯ 1 1 − Y ¯ ¯ ¡ ¢ 45 2 p2 ¯ χ(a)¯¯ = q J(q) + O q 2+² . ¯ 1 256 1 + p2 ¯a≤ q ¯ p|3q 3

§2. Some lemmas To prove the theorems, we need the following lemmas. Lemma 2.1. Let χ be a primitive character modulo q, we have  1   τ (χ)L(1, χχ04 ), if χ(−1) = 1; X π χ(a) = (1 + χ(2))(2 − χ(2))   τ (χ)L(1, χ), if χ(−1) = −1, a≤ 4q 2πi where

χ04

is the non-principal real character modulo 4, and τ (χ) =

q X

χ(a)e2πia/q is the Gauss

a=1

sum.

Proof. Firstly we suppose that χ(−1) = 1. The Fourier expansion for primitive character sums, which was first used by P´olya [4], is the following:  +∞  τ (χ) X χ(n) sin(2πnλ)   , if χ(−1) = 1;   π X n n=1 χ(a) = +∞  τ (χ)L(1, χ) τ (χ) X χ(n) cos(2πnλ)  a≤λq  − , if χ(−1) = −1.   πi πi n n=1

Then we have

X

+∞

a≤ 4

n=1

τ (χ) X χ(n) sin χ(a) = π n q

¡ πn ¢ 2

.

Since  1,    ³ πn ´ 0, sin = −1,  2   0,

if if if if

n ≡ 1 mod 4; n ≡ 2 mod 4; n ≡ 3 mod 4; n ≡ 4 mod 4.

= χ04 (n), we have

X a≤ 4q

χ(a) =

1 τ (χ)L(1, χχ04 ), π

if χ(−1) = 1.

Now we consider the case χ(−1) = −1. From (2.1) we also get X

+∞

a≤ 4

n=1

τ (χ)L(1, χ) τ (χ) X χ(n) cos χ(a) = − πi πi n q

67

¡ πn ¢ 2

.

(2.1)

Jing Gao and Huaning Liu Since

¡ πn ¢

+∞ X χ(n) cos

2

=

n

n=1

n +∞ X χ(n)(−1) 2

 =

n=1







+∞ +∞ +∞ +∞ X  X χ(2)  χ(n) X χ(n)  χ(n) X χ(n)    = χ(2) 2  − − 2  n n  2  n n  n=1 2|n

=

n

n=1 2|n

+∞

χ(2) X χ(n)(−1)n = 2 n

n=1 2-n

n=1 2|n

n=1

χ(4) − χ(2) L(1, χ), 2

we have

X

χ(a) =

a≤ 4q

(1 + χ(2))(2 − χ(2)) τ (χ)L(1, χ). 2πi

This proves Lemma 2.1. Lemma 2.2. Let χ be a primitive character modulo q, we have  √   3 τ (χ)L(1, χχ0 ), if χ(−1) = 1; X 3 2π χ(a) = 3   τ (χ)L(1, χχ03 ), if χ(−1) = −1, a≤ 3q 2πi where χ03 is the non-principal real character modulo 3, and χ03 denotes the principal character modulo 3. Proof. Firstly we consider the case χ(−1) = 1. By (2.1) we have ¡ ¢ +∞ X τ (χ) X χ(n) sin 2πn 3 . χ(a) = π n q n=1

a≤ 3

Since µ sin

we have

X

2πn 3

  

¶ =

√

3 2 √,

3 2 ,

−   0, √ 3 0 = χ (n), 2 3

if n ≡ 1 mod 3; if n ≡ 2 mod 3; if n ≡ 3 mod 3.

√ χ(a) =

a≤ 3q

3 τ (χ)L(1, χχ03 ), 2π

if χ(−1) = 1.

Now we suppose that χ(−1) = −1. From (2.1) we can get ¡ ¢ ¡ ¡ ¢¢ +∞ +∞ X τ (χ)L(1, χ) τ (χ) X χ(n) cos 2πn τ (χ) X χ(n) 1 − cos 2πn 3 3 χ(a) = − = . πi πi n πi n q n=1

a≤ 3

n=1

Since µ 1 − cos

2πn 3

 

¶ =



68

3 2, 3 2,

0,

if n ≡ 1 mod 3; if n ≡ 2 mod 3; if n ≡ 3 mod 3.

Character sums in short intervals 3 0 = χ (n), 2 3 we have

X

3 τ (χ)L(1, χχ03 ). 2πi

χ(a) =

a≤ 3q

This completes the proof of Lemma 2.2. Lemma 2.3. Let q and r be integers with q ≥ 2 and (r, q) = 1. Then we have the identity X∗

X

χ(r) =

χ mod q

³q´

µ

d

d|(q,r−1)

φ(d),

where µ(q) is the M¨ obius function. Proof. This is Lemma 4 of [7]. Lemma 2.4. For any positive integer q > 1, we have ³ X∗ ¯ ¯ ¯L(1, χχ04 )¯4 = χ mod q χ(−1)=1

|L(1, χ)|4 =

χ mod q χ(−1)=−1

X∗

ζ 4 (2) 2ζ(4)

χ(2j ) |L(1, χ)|4 =

χ mod q χ(−1)=−1

χ mod q χ(−1)=1

2ζ(4)

J(q)

1

p|2q

J(q)

´3

1 p2 + p12

Y 1−

³

X∗

X∗

ζ 4 (2)

p|q

3 1 4 (j + 1) + 2 ¡ ¢3 2j+1 1 − 14

(2.2)

´3

1 p2 + p12

Y 1− 1

+ O (q ² ) ;

·

+ O (q ² ) ;

ζ 4 (2) ζ(4)

J(q)

(2.3)

³ Y 1− p|2q

1

´3

1 p2 + p12

for j ≥ 0, and q odd; ³ ´3 1 1 − 4 Y 2 ¯ ¯ p ¯L(1, χχ03 )¯4 = ζ (2) J(q) + O (q ² ) ; 2ζ(4) 1 + p12

X∗ χ mod q χ(−1)=−1

+ O (q ² ) ,

(2.4) (2.5)

p|3q

³ ´3 1 1 − 4 Y 2 ¯ ¯ p ¯L(1, χχ03 )¯4 = ζ (2) J(q) + O (q ² ) . 1 2ζ(4) 1 + p2

(2.6)

p|3q

Proof. We only prove (2.4), since similarly we can deduce others. For any non-principal character χ modulo q, and parameter N ≥ q, by Abel’s identity we get X χ(n) +∞ X X χ(n) Z +∞ N 0 and that our assumption holds for n 1. That is; x12n

16

=e

n Y2 i=0

(1 + (4i + 1) bd) (1 + (4i + 2) ac) (1 + (4i) ec) (1 + (4i + 3) bd) (1 + (4i) ac) (1 + (4i + 2) ec) 3 75

;

ELSAYED: RATIONAL DIFFERENCE EQUATIONS

x12n x12n x12n x12n

15

14

13

12

= d = c = b = a

n Y2

i=0 n Y2

i=0 n Y2

i=0 n Y2 i=0

x12n x12n x12n x12n x12n x12n x12n

11

10

9

8

7

6

5

= =

ec b(1+ec)

bd a(1+bd)

(1 + (4i) bd) (1 + (4i + 1) ac) (1 + (4i + 3) ec) (1 + (4i + 2) bd) (1 + (4i + 3) ac) (1 + (4i + 1) ec)

;

(1 + (4i + 3) bd) (1 + (4i) ac) (1 + (4i + 2) ec) (1 + (4i + 1) bd) (1 + (4i + 2) ac) (1 + (4i + 4) ec)

;

(1 + (4i + 2) bd) (1 + (4i + 3) ac) (1 + (4i + 1) ec) (1 + (4i + 4) bd) (1 + (4i + 1) ac) (1 + (4i + 3) ec)

;

(1 + (4i + 1) bd) (1 + (4i + 2) ac) (1 + (4i + 4) ec) (1 + (4i + 3) bd) (1 + (4i + 4) ac) (1 + (4i + 2) ec)

;

n Y2

i=0 n Y2

i=0 n Y2

=

ab(1+ec) e(1+ac)

=

i=0 n Y2 aec(1+bd) bd(1+2ec) i=0

=

de(1+ac) a(1+ec)(1+2bd)

= =

(1 + (4i + 4) bd) (1 + (4i + 1) ac) (1 + (4i + 3) ec) (1 + (4i + 2) bd) (1 + (4i + 3) ac) (1 + (4i + 5) ec)

;

(1 + (4i + 3) bd) (1 + (4i + 4) ac) (1 + (4i + 2) ec) (1 + (4i + 5) bd) (1 + (4i + 2) ac) (1 + (4i + 4) ec)

;

(1 + (4i + 2) bd) (1 + (4i + 3) ac) (1 + (4i + 5) ec) (1 + (4i + 4) bd) (1 + (4i + 5) ac) (1 + (4i + 3) ec)

;

(1 + (4i + 5) bd) (1 + (4i + 2) ac) (1 + (4i + 4) ec) (1 + (4i + 3) bd) (1 + (4i + 4) ac) (1 + (4i + 6) ec)

bd(1+2ec) e(1+bd)(1+2ac)

n Y2

i=0 n Y2

ac(1+ec)(1+2bd) d(1+ac)(1+3ec)

i=0 n Y2

;

(1 + (4i + 4) bd) (1 + (4i + 5) ac) (1 + (4i + 3) ec) (1 + (4i + 6) bd) (1 + (4i + 3) ac) (1 + (4i + 5) ec)

;

(1 + (4i + 3) bd) (1 + (4i + 4) ac) (1 + (4i + 6) ec) (1 + (4i + 5) bd) (1 + (4i + 6) ac) (1 + (4i + 4) ec)

;

(1 + (4i + 6) bd) (1 + (4i + 3) ac) (1 + (4i + 5) ec) (1 + (4i + 4) bd) (1 + (4i + 5) ac) (1 + (4i + 7) ec)

i=0

:

Now, it follows from Eq.(3) that x12n

4

=

x12n

x12n 7 x12n 9 6 (1 + x12n 7 x12n 9 )

bd (1+2bd)

n Y2

(1+(4i+2)bd) (1+(4i+6)bd)

i=0

= bd(1+2ec) e(1+bd)(1+2ac)

n Y2

(1+(4i+3)bd)(1+(4i+4)ac)(1+(4i+6)ec) (1+(4i+5)bd)(1+(4i+6)ac)(1+(4i+4)ec)

1+

bd (1+2bd)

i=0

=

bd (1+(4n 2)bd) bd(1+2ec) e(1+bd)(1+2ac)

1+

bd (1+(4n 2)bd)

n Y2 i=0

n Y2

(1+(4i+5)bd)(1+(4i+6)ac)(1+(4i+4)ec) (1+(4i+3)bd)(1+(4i+4)ac)(1+(4i+6)ec)

i=0

4 76

:

(1+(4i+2)bd) (1+(4i+6)bd)

!

ELSAYED: RATIONAL DIFFERENCE EQUATIONS

Hence, we have x12n

4

=e

n Y1 i=0

(1 + (4i + 1) bd) (1 + (4i + 2) ac) (1 + (4i) ec) (1 + (4i + 3) bd) (1 + (4i) ac) (1 + (4i + 2) ec)

:

Also, from Eq. (3) we see that x12n 6 x12n 8 x12n 3 = x12n 5 (1 + x12n 6 x12n 8 ) ac (1+2ac)

=

n 2 Y (1+(4i+2)ac) ( (1+(4i+6)ac) ) i=0

0

i=0

=

i=0

d(1 + ac)(1 + 3ec) (1 + ec)(1 + 2bd) (1 + (4n

1) ac)

Hence, we have x12n

1

n 2 n 2 Y Y C (1+(4i+6)bd)(1+(4i+3)ac)(1+(4i+5)ec) B ac(1+ec)(1+2bd) ac 1+ @ ( (1+(4i+4)bd)(1+(4i+5)ac)(1+(4i+7)ec) ) ( (1+(4i+2)ac) d(1+ac)(1+3ec) (1+2ac) (1+(4i+6)ac) )A

3

=d

n Y1 i=0

n Y2

(1+(4i+4)bd)(1+(4i+5)ac)(1+(4i+7)ec) (1+(4i+6)bd)(1+(4i+3)ac)(1+(4i+5)ec)

:

i=0

(1 + (4i) bd) (1 + (4i + 1) ac) (1 + (4i + 3) ec) (1 + (4i + 2) bd) (1 + (4i + 3) ac) (1 + (4i + 1) ec)

:

Similarly, we can easily obtain the other relations. Thus, the proof is completed. Theorem 2 Eq.(3) has a unique equilibrium point which is the number zero and this equilibrium point is not locally asymptotically stable. Proof: For the equilibrium points of Eq.(3), we can write x=

x2 : x (1 + x2 )

Then we have x4 = 0: Thus the equilibrium point of Eq.(3) is x = 0: Let f : (0; 1)3 ! (0; 1) be a function de…ned by vw f (u; v; w) = : u(1 + vw) Therefore it follows that vw w v fu (u; v; w) = ; fv (u; v; w) = ; 2 ; fw (u; v; w) = 2 u (1 + vw) u (1 + vw) u (1 + vw)2 we see that fu (x; x; x) =

1;

fv (x; x; x) = 1;

The proof follows by using Theorem A. 5 77

fw (x; x; x) = 1:

ELSAYED: RATIONAL DIFFERENCE EQUATIONS

3

xn 2 xn 4 xn 1 ( 1+xn 2 xn 4 )

On the Di¤erence Equation xn+1 =

We obtain in this section the solution of the second di¤erence equation in the form xn+1 =

xn 2 xn 4 ; xn 1 ( 1 + xn 2 xn 4 )

(4)

n = 0; 1; :::;

where the initial values are arbitrary non zero real numbers with x0 x 2 ; x 1 x 3 ; x 2 x 1:

4

Theorem 3 Let fxn g1 n= 4 be a solution of Eq.(4). Then every solution of Eq.(4) is periodic solution with period twelve and for n = 0; 1; 2; ::: x12n

4

= e; x12n

3

= d; x12n

2

= c; x12n

bd ab( ; x12n+3 = a( 1 + bd) e( de( 1 + ac) = ; x12n+6 = a( 1 + ec) e(

x12n+2 = x12n+5

1

ec ; b( 1 + ec) aec( 1 + bd) x12n+4 = ; bd ac x12n+7 = : d( 1 + ac)

= b; x12n = a; x12n+1 =

1 + ec) ; 1 + ac) bd ; 1 + bd)

Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n 1. That is; x12n

16

= e; x12n

x12n

10

=

7

=

x12n

15

= d; x12n

14

bd ; x12n 9 = ab( a( 1+bd) e( de( 1+ac) ; x12n 6 = e( a( 1+ec)

= c; x12n

13

= b; x12n

= a; x12n

12

1+ec) ; 1+ac)

x12n

8

=

aec( 1+bd) ; bd

bd ; 1+bd)

x12n

5

=

ac : d( 1+ac)

11

=

ec ; b( 1+ec)

Now, it follows from Eq.(4) that x12n

4

=

x12n

3

=

x12n

x12n 7 x12n 9 6 ( 1+x12n 7 x12n

x12n

x12n 6 x12n 8 5 ( 1+x12n 6 x12n

9)

8)

=

de( 1+ac) ab( 1+ec) a( 1+ec) e( 1+ac) de( 1+ac) ab( 1+ec) bd 1+ a( 1+ec) e( 1+ac) e( 1+bd))

=

aec( 1+bd) bd e( 1+bd) bd aec( 1+bd) ac 1+ e( bd d( 1+ac) 1+bd) bd

(

(

)

)

= =

bd bd ( e( 1+bd))

1+bd)

ac

ac ( d( 1+ac)

1+ac)

= e;

= d:

Similarly, we can easily obtain the other relations. Thus, the proof is completed. p Theorem 4 Eq.(4) has three equilibrium points which are 0; 2.and these equilibrium points are not locally asymptotically stable. Proof: For the equilibrium points of Eq.(4), we can write x2 x= : x ( 1 + x2 ) 6 78

6=

ELSAYED: RATIONAL DIFFERENCE EQUATIONS

Then we have x2 x2

2 = 0: p Thus the equilibrium points of Eq.(4) are 0; 2: 3 Let f : (0; 1) ! (0; 1) be a function de…ned by vw : f (u; v; w) = u ( 1 + vw) Therefore it follows that vw w v fu (u; v; w) = ; fv (u; v; w) = ; 2 ; fw (u; v; w) = 2 u ( 1 + vw) u ( 1 + vw) u ( 1 + vw)2 we see that fu (x; x; x) =

1;

fv (x; x; x) =

1;

fw (x; x; x) =

1:

The proof follows by using Theorem A. Theorem 5 Eq.(4) has a periodic solutions of period four i¤ x0 x 2 = x 1 x 3 = x 2 x 4 = 2; and will be take the form fx 4 ; x 3 ; x 2 ; x 1 ; x 4 ; x 3 ; x 2 ; x 1 ; ::::g. Proof: First suppose that there exists a prime period four solution of Eq.(4) of the form x 4 ; x 3 ; x 2 ; x 1 ; x 4 ; x 3 ; x 2 ; x 1 ; :::: : Then we see from the form of solution of Eq.(4) that x12n

4

= e; x12n

3

x12n+2 =

bd a( 1+bd)

x12n+5 =

de( 1+ac) a( 1+ec)

= d; x12n

2

= c; x12n

= c; x12n+3 =

ab( 1+ec) e( 1+ac)

= d; x12n+6 =

1

= b; x12n = e; x12n+1 =

= b; x12n+4 =

bd e( 1+bd))

aec( 1+bd) bd

= c; x12n+7 =

ec b( 1+ec)

= d;

= e;

ac = b: d( 1 + ac)

Then a = e; bd = ec = 2: Second suppose that x0 x

2

= x 1x

3

= x 2x

4

= 2:

Then we see from the solution of Eq.(4) that x12n 4 = e; x12n 3 = d; x12n 2 = c; x12n 1 = b; x12n = a = e; 1+ec) ec = d; x12n+2 = a( bd = c; x12n+3 = ab( = b; x12n+1 = b( 1+ec) 1+bd) e( 1+ac) x12n+4 = x12n+6 =

aec( 1+bd) 1+ac) = e; x12n+5 = de( = d; bd a( 1+ec) bd = b: = c; x12n+7 = d( ac e( 1+bd) 1+ac)

Thus we have a period four solution and the proof is complete. The proofs of the theorems in the following sections are similar to that are presented in the previous sections and so will be omitted. 7 79

ELSAYED: RATIONAL DIFFERENCE EQUATIONS

On the Di¤erence Equation xn+1 =

4

xn 2 xn 4 xn 1 (1 xn 2 xn 4 )

In this section we get the solution of the following di¤erence equation xn 2 xn 4 xn+1 = ; n = 0; 1; :::; xn 1 (1 xn 2 xn 4 )

(5)

where the initial values are arbitrary non zero real numbers. Theorem 6 Let fxn g1 n= x12n x12n x12n x12n

4

3

2

1

= e = d = c = b

x12n = a

n Y1

(1 (1

i=0 n Y1

i=0 n Y1

i=0 n Y1

i=0 n Y1 i=0

x12n+1 = x12n+2 =

ec b(1 ec)

bd a(1 bd)

4

(4i + 1) bd) (1 (4i + 3) bd) (1

; ;

(1 (4i + 3) bd) (1 (4i) ac) (1 (4i + 2) ec) (1 (4i + 1) bd) (1 (4i + 2) ac) (1 (4i + 4) ec)

;

(1 (1

(4i + 2) bd) (1 (4i + 4) bd) (1

(4i + 3) ac) (1 (4i + 1) ac) (1

(4i + 1) ec) (4i + 3) ec)

;

(1 (1

(4i + 1) bd) (1 (4i + 3) bd) (1

(4i + 2) ac) (1 (4i + 4) ac) (1

(4i + 4) ec) (4i + 2) ec)

;

n Y1

i=0 n Y1

i=0 n Y1

ab(1 ec) e(1 ac)

x12n+4 =

i=0 n Y1 aec(1 bd) bd(1 2ec) i=0

x12n+5 =

de(1 ac) a(1 ec)(1 2bd)

x12n+7 =

(4i + 2) ac) (1 (4i) ec) (4i) ac) (1 (4i + 2) ec)

(1 (4i) bd) (1 (4i + 1) ac) (1 (4i + 3) ec) (1 (4i + 2) bd) (1 (4i + 3) ac) (1 (4i + 1) ec)

x12n+3 =

x12n+6 =

be a solution of Eq.(5). Then for n = 0; 1; :::

bd(1 2ec) e(1 bd)(1 2ac)

(1 (1

(4i + 4) bd) (1 (4i + 2) bd) (1

(4i + 1) ac) (1 (4i + 3) ac) (1

(4i + 3) ec) (4i + 5) ec)

;

(1 (1

(4i + 3) bd) (1 (4i + 5) bd) (1

(4i + 4) ac) (1 (4i + 2) ac) (1

(4i + 2) ec) (4i + 4) ec)

;

(1 (1

(4i + 2) bd) (1 (4i + 4) bd) (1

(4i + 3) ac) (1 (4i + 5) ac) (1

(4i + 5) ec) (4i + 3) ec)

;

(1 (1 n Y1

i=0 n Y1

ac(1 ec)(1 2bd) d(1 ac)(1 3ec)

i=0 n Y1 i=0

(4i + 5) bd) (1 (4i + 3) bd) (1

(4i + 2) ac) (1 (4i + 4) ac) (1

(4i + 4) ec) (4i + 6) ec)

;

(1 (1

(4i + 4) bd) (1 (4i + 6) bd) (1

(4i + 5) ac) (1 (4i + 3) ac) (1

(4i + 3) ec) (4i + 5) ec)

;

(1 (1

(4i + 3) bd) (1 (4i + 5) bd) (1

(4i + 4) ac) (1 (4i + 6) ac) (1

(4i + 6) ec) (4i + 4) ec)

;

(1 (1

(4i + 6) bd) (1 (4i + 4) bd) (1 8 80

(4i + 3) ac) (1 (4i + 5) ac) (1

(4i + 5) ec) (4i + 7) ec)

:

ELSAYED: RATIONAL DIFFERENCE EQUATIONS

Theorem 7 Eq.(5) has a unique equilibrium point which is the number zero and this equilibrium point is not locally asymptotically stable.

5

On the Di¤erence Equation xn+1 =

xn 2 xn 4 xn 1 ( 1 xn 2 xn 4 )

Here we obtain a form of the solutions of the equation xn 2 xn 4 xn+1 = ; n = 0; 1; :::; xn 1 ( 1 xn 2 xn 4 ) where the initial values are non zero real numbers with x0 x 2 ; x 1 x 3 ; x 2 x

(6) 4

6=

1:

Theorem 8 Let fxn g1 n= 4 be a solution of Eq.(6). Then every solution of Eq.(6) is periodic solution with period twelve and for n = 0; 1; 2; ::: x12n

4

= e; x12n

3

= d; x12n

2

= c; x12n

x12n+2 =

bd ; a( 1 bd)

x12n+3 =

ab( 1 ec) ; e( 1 ac)

x12n+5 =

de( 1 ac) ; a( 1 ec)

x12n+6 =

bd ; e( 1 bd)

1

= b; x12n = a; x12n+1 =

x12n+4 = x12n+7 =

ec ; b( 1 ec)

aec( 1 bd) ; bd ac : d( 1 ac)

Theorem 9 Eq.(6) has a unique equilibrium point which is x = 0 and this equilibrium point is not locally asymptotically stable. Theorem 10 Eq.(6) has a periodic solutions of period four i¤ x0 x 2 = x 1 x 3 = x 2 x 4 = 2; and will be take the form fx 4 ; x 3 ; x 2 ; x 1 ; x 4 ; x 3 ; x 2 ; x 1 ; ::::g. Acknowledgements This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support. References [1] R. P. Agarwal and E. M. Elsayed, Periodicity and stability of solutions of higher order rational di¤erence equation, Adv. Stud. Cont. Math., 17(2) (2008), 181-201. [2] C. Cinar, On the positive solutions of the di¤erence equation xn+1 = axn 1 =(1 + bxn xn 1 ); Appl. Math. Comp., 156 (2004), 587-590. [3] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the di¤erence equation xn+1 = axn bxn =(cxn dxn 1 ); Adv. Di¤er. Equ., 2006 (2006), ID 82579,1–10. [4] E. M. Elabbasy, H. QkEl-Metwally and E. M. Elsayed, On the di¤erence equations xn+1 = xn k = + i=0 xn i ; J. Conc. Appl. Math., 5 (2) (2007), 101-113. [5] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Disc. Dyn. Nat. Soc., Volume 2011, Article ID 982309, 17 pages. [6] E. M. Elsayed, Solutions of rational di¤erence system of order two, Math. Comp. Mod., 55 (2012), 378–384. [7] E. A. Grove and G. Ladas, Periodicities in Nonlinear Di¤erence Equations, Chapman & Hall / CRC Press, 2005. 9 81

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 82-98, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A New Perturbed Iterative Algorithm with Mixed Errors for Solving System of Generalized Nonlinear Variational Inclusions Narin Petrot† and Javad Balooee‡ †

Department of Mathematics, Faculty of Science

Naresuan University, Phitsanulok, 65000, Thailand †

Centre of Excellence in Mathematics,

CHE, Si Ayutthaya Road, Bangkok 10400, Thailand E-mail: [email protected] ‡

Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran E-mail: [email protected]

‡

The corresponding author: [email protected] (Javad Balooee)

Abstract. A new system of generalized nonlinear variational inclusions with A-maximal m-relaxed η-accretive (so called (A, η)-accretive [22]) mappings in q-uniformly smooth Banach spaces is introduced and studied. By using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings due to Lan et al., the existence and uniqueness of solution for this system of generalized nonlinear variational inclusions is verified and a new perturbed iterative algorithm with mixed errors for solving the aforementioned system is constructed. Also the convergence of the sequences generated by the our algorithm in q-uniformly smooth Banach spaces is proved. The results presented in this paper extend and improve some known results in the literature. Keywords : A-maximal m-relaxed η-accretive; Perturbed iterative algorithm with mixed errors; A system of generalized nonlinear variational inclusions; Variational convergence; Resolvent operator technique. 2010 MSC : Primary 47H05; Secondary 49J40.

1

Introduction

Variational inequalities theory, which was introduced by Stampacchia [26] can be viewed as a natural generalization of the variational principles. Because of its wide applications, variational inequalities have been generalized and extended in several directions using the novel and new techniques in the past years. Among these generalizations, variational inclusion introduced and studied by Hassouni and Moudafi [12] is of interest and importance. It provides us with a unified, natural, novel innovative and general technique to study a wide class of problems arising in different branches of mathematical and engineering sciences, see for example [2, 4, 9]. It is known that one of the most important and interesting problems in the theory of variational inequality is the development of an efficient and implementable algorithm for solving

82

1

PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

various variational inequalities and variational inclusions. In recent years, many numerical methods have been developed for solving various classes of variational inequalities and variational inclusions in Euclidean spaces or Hilbert spaces, such as the projection methods and its variant forms, linear approximation, descent method, Newton’s method and the method based on auxiliary principle technique. In particular, the method based on the resolvent operator technique is a generalization of projection method and has been widely used to solve variational inclusions. Some new and interesting problems, which are called systems of variational inequality problems have been introduced and studied. Pang [24], showed that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium and the general equilibrium programming problems can be modeled as a variational inequality. He decomposed the original variational inequality into a system of variational inequalities which are easy to solve and studied the convergence of such methods. Kassay and Kolumban [16] introduced a system of variational inequalities and proved an existence theorem by the Ky Fan lemma. Subsequently, for examples, Argawal et. al. [1], Cho et. al [5], [7], [8], Cho and Petrot[6], Kim and Kim [19], Kumam et. al. [20] and Petrot[25] introduced a new system of generalized nonlinear (quasi)-variational inequalities and obtained some existence and uniqueness results of solution for these system of generalized nonlinear (quasi)-variational inequalities in Hilbert spaces. On the other hand, it is known that accretivity of the underlying operator plays indispensable roles in the theory of variational inequality and its generalizations. In 2001, Huang and Fang [14] were the first to introduce generalized m-accretive mapping and give the definition of the resolvent operator for generalized m-accretive mappings in Banach spaces. They also proved some properties of the resolvent operator for generalized m-accretive mappings in Banach spaces. In [22], Lan et al. first introduced a new concept of (A, η)-accretive mappings, which generalizes the existing monotone or accretive operators and studied some properties of (A, η)-accretive mappings and defined resolvent operators associated with (A, η)-accretive mappings. They also investigated a class of variational inclusions using the resolvent operator associated with (A, η)-accretive mappings. Subsequently, Lan [21] by using the concept of (A, η)-accretive mappings and the new resolvent operator technique associated with (A, η)-accretive mappings, introduced and studied a system of general mixed quasivariational inclusions involving (A, η)-accretive mappings in Banach spaces and constructed a perturbed iterative algorithm with mixed errors for this system of nonlinear (A, η)-accretive variational inclusions in q-uniformly smooth Banach spaces. Recently, Jin [15], by using the concept of (A, η)-accretive mappings and the resolvent operator technique associated with (A, η)-accretive mappings due to Lan et al., introduced and studied a new class of nonlinear variational inclusion systems with (A, η)accretive mappings in Banach spaces and constructed some new iterative algorithms to approximate the solutions of the mentioned nonlinear variational inclusion systems. Inspired and motivated by recent research works in these fields, in this paper, we shall introduce and study a new system of generalized nonlinear variational inclusions with A-maximal m-relaxed η-accretive (so-called (A, η)-accretive) mappings in q-uniformly smooth Banach spaces. By using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings due to Lan et al., we construct a new perturbed iterative algorithm with mixed errors for approximating the solutions of this system of generalized nonlinear variational inclusions in Banach spaces. Further, we shall prove the existence and uniqueness of solution and the convergence of the sequence generated by the our algorithm in q-uniformly smooth Banach spaces. Our results improve and extend the corresponding results of [10, 11, 13, 15, 17, 27, 28, 29] and many other recent works.

83

2

PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

2

Preliminaries

Let X be a real Banach space with dual space X ∗ , h·, ·i be the dual pair between X and X ∗ and CB(X) denote the family of all nonempty closed bounded subsets of X. The generalized duality mapping Jq : X ( X ∗ is defined by Jq (x) = {f ∗ ∈ X ∗ : hx, f ∗ i = kxkq , kf ∗ k = kxkq−1 },

∀x ∈ X,

where q > 1 is a constant. In particular, J2 is the usual normalized duality mapping. It is known that, in general, Jq (x) = kxkq−2 J2 (x) for all x 6= 0 and Jq is single-valued if X ∗ is strictly convex. In the sequel, we always assume that X is a real Banach space such that Jq is single-valued. If X is a Hilbert space, then J2 becomes the identity mapping on X. The modulus of smoothness of X is the function ρX : [0, ∞) → [0, ∞) defined by 1 ρX (t) = sup{ (kx + yk + kx − yk) − 1 : kxk ≤ 1, kyk ≤ t}. 2 A Banach space X is called uniformly smooth if lim

t→0

ρX (t) = 0. t

X is called q-uniformly smooth if there exists a constant c > 0 such that ρX (t) ≤ ctq ,

q > 1.

Note that Jq is single-valued if X is uniformly smooth. Concerned with the characteristic inequalities in q-uniformly smooth Banach spaces, Xu [30] proved the following result. Lemma 2.1. The real Banach space X is q-uniformly smooth if and only if there exists a constant cq > 0 such that for all x, y ∈ X, kx + ykq ≤ kxkq + qhy, Jq (x)i + cq kykq . ˆ Definition 2.2. A set-valued mapping T : X ( X is called ξ-H-Lipschitz continuous if there exists a constant ξ > 0 such that ˆ (x), T (y)) ≤ ξkx − yk, ∀x, y ∈ X, H(T ˆ is the Hausdorff pseudo-metric, i.e., for any two nonempty subsets A, B of X, where H ˆ H(A, B) = max{sup d(x, B), sup d(y, A)}, x∈A

y∈B

where d(u, K) = inf ku − vk. v∈K

ˆ is restricted to closed bounded subsets CB(X), then H ˆ is It should be pointed that, if the domain of H the Hausdorff metric. Definition 2.3. Let X be a q-uniformly smooth Banach space, T, A : X → X and η : X × X → X be three single-valued mappings. (a) T is said to be accretive if hT (x) − T (y), Jq (x − y)i ≥ 0,

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PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

(b) T is called strictly accretive if T is accretive and hT (x) − T (y), Jq (x − y)i = 0, if and only if x = y; (c) T is said to be r-strongly accretive if there exists a constant r > 0 such that hT (x) − T (y), Jq (x − y)i ≥ rkx − ykq ,

∀x, y ∈ X;

(d) T is called m-relaxed accretive if there exists a constant m > 0 such that hT (x) − T (y), Jq (x − y)i ≥ −mkx − ykq ,

∀x, y ∈ X;

(e) T is called k-cocoercive if there exists a constant k > 0 such that hT (x) − T (y), Jq (x − y)i ≥ kkT (x) − T (y)kq ,

∀x, y ∈ X;

(f) T is said to be γ-relaxed cocoercive if there exists a constant γ > 0 such that hT (x) − T (y), Jq (x − y)i ≥ −γkT (x) − T (y)kq ,

∀x, y ∈ X;

(g) T is said to be (ζ, ς)-relaxed cocoercive if there exist constants ζ, ς > 0 such that hT (x) − T (y), Jq (x − y)i ≥ −ζkT (x) − T (y)kq + ςkx − ykq ,

∀x, y ∈ X;

(h) T is said to be %-Lipschitz continuous if there exists a constant % > 0 such that kT (x) − T (y)k ≤ %kx − yk,

∀x, y ∈ X;

(i) η is said to be τ -Lipschitz continuous if there exists a constant τ > 0 such that kη(x, y)k ≤ τ kx − yk,

∀x, y ∈ X.

Definition 2.4. Let X be q-uniformly smooth Banach space and T : X × X → X be a single-valued mapping. Then T is said to be (α, β)-Lipschitz continuous if there exist constants α, β > 0 such that kT (x, y) − T (x0 , y 0 )k ≤ αkx − x0 k + βky − y 0 k,

∀x, y, x0 , y 0 ∈ X.

Definition 2.5. Let X be a q-uniformly smooth Banach space, η : X × X → X and H, A : X → X be three single-valued mappings and M : X ( X be a set-valued mapping. Then, M is said to be: (a) accretive if hu − v, Jq (x − y)i ≥ 0,

∀x, y ∈ X, u ∈ M x, v ∈ M y;

hu − v, Jq (η(x, y))i ≥ 0,

∀x, y ∈ X, u ∈ M x, v ∈ M y;

(b) η-accretive if

(c) strictly η-accretive if M is η-accretive and the equality holds if and only if x = y; (d) r-strongly η-accretive if there exists a constant r > 0 such that hu − v, Jq (η(x, y))i ≥ rkx − ykq ,

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PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

(e) α-relaxed η-accretive if there exists a constant α > 0 such that hu − v, Jq (η(x, y))i ≥ −αkx − ykq ,

∀x, y ∈ X, u ∈ M x, v ∈ M y;

(f) m-accretive if M is accretive and (I + λM )(X) = X for all λ > 0, where I denotes the identity mapping on X; (g) generalized m-accretive if M is η-accretive and (I + λM )(X) = X for all λ > 0; (h) H-accretive if M is accretive and (H + λM )(X) = X for all λ > 0; (i) (H, η)-accretive if M is η-accretive and (H + λM )(X) = X for all λ > 0. Remark 2.6. It should be noticed that (1) The class of generalized m-accretive mappings was first introduced by Huang and Fang [14] and includes that of m-accretive mappings as a special case. The class of H-accretive mappings was first introduced and studied by Fang and Huang [11] and also includes that of m-accretive mappings as a special case. (2) When X = H, parts of (a)-(i) of Definition 2.5 reduce to the definitions of monotone operators, η-monotone operators, strictly η-monotone operators, strongly η-monotone operators, relaxed η-monotone operators, maximal monotone operators, maximal η-monotone operators, H-monotone operators, (H, η)monotone operators, respectively. Definition 2.7. Let A : X → X, η : X × X → X be two single-valued mappings and M : X ( X be a set-valued mapping. Then M is called A-maximal m-relaxed η-accretive (so-called (A, η)-accretive [22]) if M is m-relaxed η-accretive and (A + λM )(X) = X for any λ > 0. Remark 2.8. For appropriate and suitable choices of m, A, η and the space X, it is easy to see that Definition 2.7 includes a number of definitions of monotone operators and accretive mappings (see [22]). In [22], Lan et al. showed that (A + ρM )−1 is a single-valued mapping if M : X ( X is a A-maximal m-relaxed η-accretive mapping and A : X → X a r-strongly η-accretive mapping. Based on this fact, we η,M can define the resolvent operator Rρ,A associated with an A-maximal m-relaxed η-accretive mapping M as follows: Definition 2.9. Let A : X → X be a strictly η-accretive mapping and M : X ( X be an A-maximal η,M m-relaxed η-accretive mapping. The resolvent operator Rρ,A : X → X associated with A and M is defined by η,M Rρ,A (x) = (A + ρM )−1 (x), ∀x ∈ X. Proposition 2.10. [22] Let X be a q-uniformly smooth Banach space, η : X × X → X be τ -Lipschitz continuous, A : X → X be an r-strongly η-accretive mapping and M : X ( X be an A-maximal m-relaxed η,M τ q−1 η-accretive mapping. Then the resolvent operator Rρ,A : X → X is r−ρm -Lipschitz continuous, i.e., η,M η,M kRρ,A (x) − Rρ,A (y)k ≤

τ q−1 kx − yk, r − ρm

r where ρ ∈ (0, m ) is a constant.

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3

A new system of generalized nonlinear variational inclusions

In this section, we introduce a new system of generalized nonlinear variational inclusions in q-uniformly smooth Banach spaces and investigate some its special cases. System 3.1. Let X be an q-uniformly smooth Banach space and Ai , fi , gi : X → X, Bi , ηi : X × X → X (i = 1, 2) be nonlinear single-valued mappings. Further, suppose that for each i = 1, 2, Mi : X × X ( X is an Ai -maximal mi -relaxed ηi -accretive mapping such that for all u ∈ X, gi (u) ∈ dom(Mi (·, z)) for each z ∈ X. For any given a, b ∈ X and ρi > 0 (i = 1, 2), our problem is finding x, y ∈ X such that ( a ∈ A1 (g1 (x)) − A1 (g2 (y)) + ρ1 (B1 (f2 (y), f1 (x)) + M1 (g1 (x), x)), b ∈ A2 (g2 (y)) − A2 (g1 (x)) + ρ2 (B2 (f1 (x), f2 (y)) + M2 (g2 (y), y)). This system is called a system of generalized nonlinear variational inclusions with A-maximal m-relaxed η-accretive mappings. Remark 3.2. For appropriate and suitable choices of X, Ai , Bi , ηi , fi , gi , Mi , ρi (i = 1, 2), a and b, one can obtain many known classes of variational inequalities and variational inclusions as special cases of System 3.1. The following are some special cases of problem. Case I: If for each i = 1, 2, Bi : X → X and Mi : X ( X are univariate nonlinear mappings, fi ≡ I, the identity mapping, gi = g and a = b = 0, then System 3.1 collapses to the following system of nonlinear variational inclusions with A-maximal m-relaxed η-accretive mappings: Find x, y ∈ X such that  0 ∈ A (g(x)) − A (g(y)) + ρ (B(y) + M (g(x))), 1 1 1 1 (3.1) 0 ∈ A (g(y)) − A (g(x)) + ρ (B(x) + M (g(y))). 2

2

2

2

The problem (3.1) was introduced and studied by Jin [15]. Case II: If for each i = 1, 2, Ai = A, Mi = M , ηi = η, ρi = ρ and x = y, then the system (3.1) reduces to the following nonlinear variational inclusion problem: Find x ∈ X such that (3.2)

0 ∈ B(x) + M (g(x)).

In view of Remark 2.8, one can easy to see that the problem (3.2) contains the variational inclusions with H-accretive mappings or H-monotone operators in [10, 11] as special cases. Case III: If X = H is a Hilbert space, for each i = 1, 2, Mi : H ( H is an univariate nonlinear operator, fi = gi ≡ I, Ai = A, Bi = B and a = b = 0, then System 3.1 reduces to the following system of nonlinear variational inclusions with A-maximal m-relaxed η-monotone operators: Find (x, y) ∈ H × H such that  0 ∈ A(x) − A(y) + ρ (B(y, x) + M (x)), 1 (3.3) 0 ∈ A(y) − A(x) + ρ (B(x, y) + M (y)), 2

which was introduced and studied by Wang and Wu [29].

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Case IV: Taking x = y in (3.3), the system (3.3) reduces to the following nonlinear variational inclusion problem: Find x ∈ H such that 0 ∈ B(x, x) + M (x).

(3.4)

The problem (3.4) considered and studied by Verma [28]. Case V: If for each i = 1, 2, Mi = M : X ( X be H-accretive mapping and Ai = H, then the system (3.1) changes into the following system of variational inclusions with H-accretive mappings: Find x, y ∈ X such that  0 ∈ H(g(x)) − H(g(y)) + ρ (B(y) + M (g(x))), 1 (3.5) 0 ∈ H(g(y)) − H(g(x)) + ρ (B(x) + M (g(y))). 2

The system (3.5) was introduced and studied by He et al. [13]. Some special cases of the system (3.5) were studied by Verma [27]. Case VI: If H ≡ I, then the system (3.5) reduces to the following system considered by Kazmi and Bhat [17]:  0 ∈ g(x) − g(y) + ρ (B(y) + M (g(x))), 1 (3.6) 0 ∈ g(y) − g(x) + ρ (B(x) + M (g(y))). 2

For applications of such variational inclusions, see [12, 18]. Some special cases of the system (3.6) can be found in [13] and the references therein.

4

Existence of solution and uniqueness

In this section, we prove the existence and uniqueness theorem for solution of System 3.1. For our main results, we have the following lemma which offers a good approach to solve System 3.1. Lemma 4.1. Let X, Ai , Bi , fi , gi , ηi , Mi , ρi (i = 1, 2), a and b be the same as in System 3.1. Further, suppose that Ai is ri -strongly ηi -accretive mapping and Mi is Ai -maximal mi -relaxed ηi -accretive mapping for i = 1, 2. Then an element (x∗ , y ∗ ) ∈ X ×X is a unique solution for System 3.1 if and only if (x∗ , y ∗ ) ∈ X ×X satisfies  g (x∗ ) = Rη1 ,M1 (·,x∗ ) [(1 − λ )A (g (x∗ )) + λ (A (g (y ∗ )) − ρ B (f (y ∗ ), f (x∗ )) + a)], 1 1 1 1 1 1 2 1 1 2 1 λ1 ρ1 ,A1 (4.1) g (y ∗ ) = Rη2 ,M2 (·,y∗ ) [(1 − λ )A (g (y ∗ )) + λ (A (g (x∗ )) − ρ B (f (x∗ ), f (y ∗ )) + b)], 2

λ2 ρ2 ,A2

2

2

2

2

2

1

2

2

1

2

where η ,M (·,x∗ )

1 Rλ11 ρ1 ,A 1

η ,M (·,y ∗ )

= (A1 + λ1 ρ1 M1 (·, x∗ ))−1 ,

2 Rλ22 ρ2 ,A 2

= (A2 + λ2 ρ2 M2 (·, y ∗ ))−1

and λi > 0 (i = 1, 2) are two constants. Theorem 4.2. Let X, Ai , Bi , fi , gi , ηi , Mi , ρi (i = 1, 2), a and b be the same as in System 3.1. Also, suppose that for each i = 1, 2, (a) Ai is ri -strongly ηi -accretive and βi -Lipschitz continuous mapping;

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(b) Bi and fi are (ξi , σi )-Lipschitz continuous and ςi -Lipschitz continuous mappings, respectively; (c) gi is (κi , θi )-relaxed cocoercive and πi -Lipschitz continuous mapping; i (d) there is constants λi ∈ (0, ρirm ) and µi > 0 such that i

η ,M (.,x)

i kRλii ρi ,A i

η ,M (.,y)

i (z) − Rλii ρi ,A i

(z)k ≤ µi kx − yk,

∀x, y, z ∈ X;

p τ q−1 (|1−λ1 |β1 π1 +λ1 ρ1 σ1 ς1 ) λ τ q−1 (β π +ρ ξ ς ) q 1 − qθ1 + (cq + qκ1 )π1q + 1 + 2 2 r2 −λ22 ρ12 m22 2 1 < 1; r1 −λ1 ρ1 m1 p λ τ q−1 (β π +ρ ξ ς ) τ q−1 (|1−λ2 |β2 π2 +λ2 ρ2 σ2 ς2 ) (f) µ2 + q 1 − qθ2 + (cq + qκ2 )π2q + 2 + 1 1 r1 −λ11 ρ21 m11 1 2 < 1, r2 −λ2 ρ2 m2 where cq > 0 is a constant guaranteed by Lemma 2.1. Then System 3.1 admits a unique solution. (e) µ1 +

Proof. For any given λ1 > 0, λ2 > 0, define Ψλ1 , Φλ2 : X × X → X by    Ψλ1 (x, y)      = x − g (x) + Rη1 ,M1 (·,x) [(1 − λ )A (g (x)) + λ (A (g (y)) − ρ B (f (y), f (x)) + a)], 1 1 1 1 1 1 2 1 1 2 1 λ1 ρ1 ,A1 (4.2)   Φλ2 (x, y)      = y − g (y) + Rη2 ,M2 (·,y) [(1 − λ )A (g (y)) + λ (A (g (x)) − ρ B (f (x), f (y)) + b)] 2

λ2 ρ2 ,A2

2

2

2

2

2

1

2

2

1

2

for all (x, y) ∈ X × X. Now, define k · k∗ on X × X by k(x, y)k∗ = kxk + kyk,

∀(x, y) ∈ X × X.

It is easy to see that (X × X, k · k∗ ) is a Banach space. Also, define a mapping Fλ1 ,λ2 : X × X → X × X by (4.3)

Fλ1 ,λ2 (x, y) = (Ψλ1 (x, y), Φλ2 (x, y)),

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PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

It follows from Proposition 2.10 and the condition (d) that, for all (x, y), (x0 , y 0 ) ∈ X × X, kΨλ1 (x, y) − Ψλ1 (x0 , y 0 )k η ,M (·,x)

1 = kx − g1 (x) + Rλ11 ρ1 ,A [(1 − λ1 )A1 (g1 (x)) + λ1 (A1 (g2 (y)) 1

η ,M (·,x0 )

1 − ρ1 B1 (f2 (y), f1 (x)) + a)] − (x0 − g1 (x0 ) + Rλ11 ρ1 ,A 1

[(1 − λ1 )A1 (g1 (x0 ))

+ λ1 (A1 (g2 (y 0 )) − ρ1 B1 (f2 (y 0 ), f1 (x0 )) + a)])k ≤ kx − x0 − (g1 (x) − g1 (x0 ))k η ,M (·,x)

1 + kRλ11 ρ1 ,A [(1 − λ1 )A1 (g1 (x)) + λ1 (A1 (g2 (y)) 1

η ,M (·,x0 )

1 − ρ1 B1 (f2 (y), f1 (x)) + a)] − Rλ11 ρ1 ,A 1

[(1 − λ1 )A1 (g1 (x0 ))

+ λ1 (A1 (g2 (y 0 )) − ρ1 B1 (f2 (y 0 ), f1 (x0 )) + a)]k ≤ kx − x0 − (g1 (x) − g1 (x0 ))k η ,M (·,x)

(4.4)

1 + kRλ11 ρ1 ,A [(1 − λ1 )A1 (g1 (x)) + λ1 (A1 (g2 (y)) 1

η ,M (·,x0 )

1 − ρ1 B1 (f2 (y), f1 (x)) + a)] − Rλ11 ρ1 ,A 1

[(1 − λ1 )A1 (g1 (x))

+ λ1 (A1 (g2 (y)) − ρ1 B1 (f2 (y), f1 (x)) + a)]k η ,M (·,x0 )

1 + kRλ11 ρ1 ,A 1

[(1 − λ1 )A1 (g1 (x)) + λ1 (A1 (g2 (y)) η ,M (·,x0 )

1 − ρ1 B1 (f2 (y), f1 (x)) + a)] − Rλ11 ρ1 ,A 1

[(1 − λ1 )A1 (g1 (x0 ))

+ λ1 (A1 (g2 (y 0 )) − ρ1 B1 (f2 (y 0 ), f1 (x0 )) + a)]k ≤ kx − x0 − (g1 (x) − g1 (x0 ))k + µ1 kx − x0 k n τ1q−1 + |1 − λ1 |kA1 (g1 (x)) − A1 (g1 (x0 ))k r 1 − λ 1 ρ1 m 1 + λ1 (kA1 (g2 (y)) − A1 (g2 (y 0 ))k o + ρ1 kB1 (f2 (y), f1 (x)) − B1 (f2 (y 0 ), f1 (x0 ))k) . By Lemma 2.1, there exists cq > 0 such that kx − x0 − (g1 (x) − g1 (x0 ))kq ≤ kx − x0 kq − qhg1 (x) − g1 (x0 ), Jq (x − x0 )i + cq kg1 (x) − g1 (x0 )kq . It follows from (κ1 , θ1 )-relaxed cocoercivity and π1 -Lipschitz continuity of g1 that (4.5)

kx − x0 − (g1 (x) − g1 (x0 ))kq ≤ kx − x0 kq − qθ1 kx − x0 kq + (cq + qκ1 )π1q kx − x0 kq = (1 − qθ1 + (cq + qκ1 )π1q )kx − x0 kq .

Since A1 is β1 -Lipschitz continuous and for each i = 1, 2, gi is πi -Lipschitz continuous, we have (4.6)

kA1 (g1 (x)) − A1 (g1 (x0 ))k ≤ β1 kg1 (x) − g1 (x0 )k ≤ β1 π1 kx − x0 k

and (4.7)

kA1 (g2 (y)) − A1 (g2 (y 0 ))k ≤ β1 kg2 (y) − g2 (y 0 )k ≤ β1 π2 ky − y 0 k.

By using (ξ1 , σ1 )-Lipschitz continuity of B1 and ςi -Lipschitz continuity of fi for i = 1, 2, we get (4.8)

kB1 (f2 (y), f1 (x)) − B1 (f2 (y 0 ), f1 (x0 ))k ≤ ξ1 kf2 (y) − f2 (y 0 )k + σ1 kf1 (x) − f1 (x0 )k ≤ ξ1 ς2 ky − y 0 k + σ1 ς1 kx − x0 k.

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Combining (4.4)-(4.8), we obtain kΨλ1 (x, y) − Ψλ1 (x0 , y 0 )k ≤ ϕ1 kx − x0 k + φ1 ky − y 0 k,

(4.9) where

ϕ1 = µ1 +

q q

1 − qθ1 + (cq + qκ1 )π1q +

and φ1 =

τ1q−1 (|1 − λ1 |β1 π1 + λ1 ρ1 σ1 ς1 ) r1 − λ1 ρ1 m1

λ1 τ1q−1 (β1 π2 + ρ1 ξ1 ς2 ) . r1 − λ1 ρ1 m1

Similarly, one can easily get kΦλ2 (x, y) − Φλ2 (x0 , y 0 )k ≤ ϕ2 kx − x0 k + φ2 ky − y 0 k,

(4.10) where

φ2 = µ2 +

q τ q−1 (|1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 ) q 1 − qθ2 + (cq + qκ2 )π2q + 2 , r2 − λ2 ρ2 m2 ϕ2 =

λ2 τ2q−1 (β2 π1 + ρ2 ξ2 ς1 ) . r 2 − λ 2 ρ2 m 2

It follows from (4.9) and (4.10) that kΨλ1 (x, y) − Ψλ1 (x0 , y 0 )k + kΦλ2 (x, y) − Φλ2 (x0 , y 0 )k ≤ ϑ(kx − x0 k + ky − y 0 k),

(4.11) where

ϑ = max{ϕ1 + ϕ2 , φ1 + φ2 }. By using (4.3) and (4.11), we obtain kFλ1 ,λ2 (x, y) − Fλ1 ,λ2 (x0 , y 0 )k∗ ≤ ϑk(x, y) − (x0 , y 0 )k∗ .

(4.12)

In view of the conditions (e) and (f), we know that 0 ≤ ϑ < 1 and so it follows from (4.12) that Fλ1 ,λ2 is a contraction mapping. According to Banach fixed point theorem, there exists a unique (x∗ , y ∗ ) ∈ X × X such that Fλ1 ,λ2 (x∗ , y ∗ ) = (x∗ , y ∗ ). The relations (4.2) and (4.3) imply that  g (x∗ ) = Rη1 ,M1 (·,x∗ ) [(1 − λ )A (g (x∗ )) + λ (A (g (y ∗ )) − ρ B (f (y ∗ ), f (x∗ )) + a)], 1 1 1 1 1 1 2 1 1 2 1 λ1 ρ1 ,A1 g (y ∗ ) = Rη2 ,M2 (·,y∗ ) [(1 − λ )A (g (y ∗ )) + λ (A (g (x∗ )) − ρ B (f (x∗ ), f (y ∗ )) + b)]. 2

λ2 ρ2 ,A2

2

2

2

2

2

1

2

2

1

2

Now, Lemma 4.1 guarantees that (x∗ , y ∗ ) is a unique solution of System 3.1 and this is desired result.

5

Variational convergence and algorithm

In this section, by using resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings, we construct a new perturbed iterative algorithm with mixed errors for solving the system of generalized nonlinear variational inclusions in q-uniformly smooth Banach spaces. We also establish the convergence of the iterative sequence generated by our suggested iterative algorithm. Definition 5.1. Let Mn , M : X ( X (n ≥ 0) be set-valued mappings. We say that the sequence {Mn } is G graph-convergent to M (denote by Mn −→ M ) if, for all (x, u) ∈ Gph(M ), there exists (xn , un ) ∈ Gph(Mn ) such that xn → x and un → u as n → ∞, where Gph(M ) is defined as follows: Gph(M ) = {(x, u) ∈ X × X : u ∈ M (x)}. 10

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Theorem 5.2. [3] Suppose that ηn , η : X × X → X (n ≥ 0) are τn -Lipschitz continuous and τ -Lipschitz continuous, respectively, An : X → X is rn -strongly ηn -accretive and βn -Lipschitz continuous and A : X → X is r-strongly η-accretive mapping. For n ≥ 0, let Mn , M : X ( X be An -maximal mn -relaxed ηn accretive and A-maximal m-relaxed η-accretive mappings, respectively. Further, assume that, for any given q−1 q−1
∞
∞ βn τn τn constant ρ > 0, the sequences rn −ρm and are bounded and lim An (x) = A(x) for any r −ρm n n=0 n n n=0 n→∞ x ∈ X. Then, for any given constant ρ > 0, the sequence {Mn } is graph-convergent to M if and only if ηn ,Mn η,M Rρ,A (z) → Rρ,A (z) for all z ∈ X. n Remark 5.3. The equality (4.1) can be written as follows:   p = (1 − λ1 )A1 (g1 (x∗ )) + λ1 (A1 (g2 (y ∗ )) − ρ1 B1 (f2 (y ∗ ), f1 (x∗ )) + a),     qˆ = (1 − λ )A (g (y ∗ )) + λ (A (g (x∗ )) − ρ B (f (x∗ ), f (y ∗ )) + b), 2 2 2 2 2 1 2 2 1 2 η1 ,M1 (·,x∗ ) ∗  g (x ) = R (p),  1 λ1 ρ1 ,A1    g (y ∗ ) = Rη2 ,M2 (·,y∗ ) (ˆ q ). 2

λ2 ρ2 ,A2

The above fixed point formulation enables us to construct the following perturbed iterative algorithm with mixed errors. Algorithm 5.4. Let X, Ai , Bi , fi , gi , ηi , Mi , ρi (i = 1, 2), a and b be the same as in System 3.1 such that for each i = 1, 2, gi be an onto mapping. Further, for each n ≥ 0 and i = 1, 2, let ηn,i : X ×X → X, An,i : X → X be single-valued mappings and Mn,i : X × X ( X be any nonlinear mapping such that, for all s ∈ X, Mn,i (·, s) : X ( X be an An,i -maximal mn,i -relaxed ηn,i -accretive mapping with g(X)∩dom(Mn,i (·, s)) 6= ∅. Step 1. For any given (p0 , qˆ0 ) ∈ X × X, choose (x0 , y0 ) ∈ X × X such that η

,M

(·,x0 )

0,1 g1 (x0 ) = Rλ0,1 1 ρ1 ,A0,1

(p0 ),

η

,M

(·,y0 )

0,2 g2 (y0 ) = Rλ0,2 2 ρ2 ,A0,2

(ˆ q0 ).

Step 2. For all n = 0, 1, 2, · · · , let  ,Mn,1 (·,xn )  g1 (xn ) = Rληn,1 (pn ),  1 ρ1 ,An,1    ηn,2 ,Mn,2 (·,yn )   g2 (yn ) = Rλ2 ρ2 ,An,2 (ˆ qn ),     p n+1 = (1 − αn )pn + αn [(1 − λ1 )A1 (g1 (xn )) + λ1 (A1 (g2 (yn )) (5.2)   − ρ1 B1 (f2 (yn ), f1 (xn )) + a)] + αn en + ln ,      qˆn+1 = (1 − αn )qn + αn [(1 − λ2 )A2 (g2 (yn )) + λ2 (A2 (g1 (xn ))      − ρ2 B2 (f1 (xn ), f2 (yn )) + b)] + αn jn + kn , where {αn } is a sequence in [0, 1] with

∞ P

αn = ∞ and {en }, {ln }, {jn }, {kn } ⊂ X are errors to take into

n=0

account a possible inexact computation of the resolvent operator point satisfying the following conditions:   en = e0n + e00n , jn = jn0 + jn00 , lim k(e0n , jn0 )k∗ = 0; n ∞ ∞ (5.3) P P 00 00  k(ln , kn )k∗ < ∞. k(en , jn )k∗ < ∞,  n=0

n=0

Step 3. If {pn+1 }, {ˆ qn+1 }, {xn }, {yn } satisfy (5.2) to a sufficient degree of accuracy, then we stop here. Otherwise, set n := n + 1 and consider Step 2.

11

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PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

Remark 5.5. Let X, fi , gi , Bi (i = 1, 2), a and b be the same as in System 3.1. (1) If for each i = 1, 2 and n ≥ 0, ηn,i = ηi , An,i = Ai , Mn,i = Mi : X ( X is an univariate nonlinear mapping, αn = 1 and λi = 1, then Algorithm 5.4 reduces to Algorithm 3.1 in [15]. (2) If for each i = 1, 2 and n ≥ 0, ηn,i = η, An,i = A, Mn,i = M : X ( X is an univariate nonlinear mapping, ηi = η, αn = 1 and xn = yn , then Algorithm 5.4 reduces to Algorithm 3.2 in [15]. Lemma 5.6. Let {an }, {bn } and {cn } be three nonnegative real sequences satisfying the following condition: there exists a natural number n0 such that an+1 ≤ (1 − tn )an + bn tn + cn , where tn ∈ [0, 1],

∞ P

∞ P

tn = ∞, lim bn = 0 and n→∞

n=0

∀n ≥ n0 ,

cn < ∞. Then lim an = 0. n→0

n=0

Proof. The proof directly follows from Lemma 2 in Liu [23]. Theorem 5.7. Suppose that X, Ai , Bi , fi , gi , ηi , Mi , ρi (i = 1, 2), a and b are the same as in Theorem 4.2 and all the conditions of Theorem 4.2 hold. Assume that, for each i = 1, 2 and n ≥ 0, gi is an onto mapping and ηn,i , An,i , Mn,i are the same as in Algorithm 5.4. Further, for each i = 1, 2 and n ≥ 0, let (a) ηn,i : X × X → X be τn,i -Lipschitz continuous; (b) An,i : X → X be rn,i -strongly ηn,i -accretive and βn,i -Lipschitz continuous; G

(c) lim An,i (x) = Ai (x) and Mn,i (·, x) −→ Mi (·, x), for any x ∈ X; n→∞

βn,i τ q−1

τ q−1

)∞ and ( rn,i −λi ρn,i )∞ be bounded and (d) the sequences ( rn,i −λn,i i ρi mn,i n=0 i mn,i n=0

τ1q−1 r1 −λ1 ρ1 m1

=

τ2q−1 r2 −λ2 ρ2 m2 ;

i (e) there exists constant λi ∈ (0, ρirm ) such that i

η

,M

n,i kRλn,i i ρi ,An,i

(·,x)

η

,M

n,i (z) − Rλn,i i ρi ,An,i

(·,y)

(z)k ≤ µn,i kx − yk,

∀x, y, z ∈ X;

(f) τn,i → τi , rn,i → ri , mn,i → mi , µn,i → µi , as n → ∞. Then the iterative sequence {(xn , yn )}∞ n=0 generated by Algorithm 5.4 converges strongly to the unique solution ∗ ∗ (x , y ) of System 3.1. Proof. According to Theorem 4.2, System 3.1 admits a unique solution (x∗ , y ∗ ) ∈ X × X. It follows from Remark 5.3 that  η ,M1 (·,x∗ )   g1 (x∗ ) = Rλ11 ρ1 ,A (p),  1    g (y ∗ ) = Rη2 ,M2 (·,y∗ ) (ˆ q ), 2 λ2 ρ2 ,A2 (5.4)   p = (1 − λ1 )A1 (g1 (x∗ )) + λ1 (A1 (g2 (y ∗ )) − ρ1 B1 (f2 (y ∗ ), f1 (x∗ )) + a),     qˆ = (1 − λ )A (g (y ∗ )) + λ (A (g (x∗ )) − ρ B (f (x∗ ), f (y ∗ )) + b). 2

2

2

2

2

1

12

93

2

2

1

2

PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

From (5.2)-(5.4) and the assumptions, we have kpn+1 − pk ≤ (1 − αn )kpn − pk + αn {|1 − λ1 |kA1 (g1 (xn )) − A1 (g1 (x∗ ))k + λ1 (kA1 (g2 (yn )) − A1 (g2 (y ∗ ))k + ρ1 kB1 (f2 (yn ), f1 (xn )) − B1 (f2 (y ∗ ), f1 (x∗ ))k)} + αn (ke0n k + ke00n k) + kln k n ≤ (1 − αn )kpn − pk + αn |1 − λ1 |β1 π1 kxn − x∗ k

(5.5)

o + λ1 (β1 π2 kyn − y ∗ k + ρ1 ξ1 ς2 kyn − y ∗ k + ρ1 σ1 ς1 kxn − x∗ k) + αn ke0n k + ke00n k + kln k n = (1 − αn )kpn − pk + αn (|1 − λ1 |β1 π1 + λ1 ρ1 σ1 ς1 )kxn − x∗ k o + λ1 (β1 π2 + ρ1 ξ1 ς2 )kyn − y ∗ k + αn ke0n k + ke00n k + kln k. By using the same argument, we can prove that n kˆ qn+1 − qˆk ≤ (1 − αn )kˆ qn − qˆk + αn (|1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 )kyn − y ∗ k o + λ2 (β2 π1 + ρ2 ξ2 ς1 )kxn − x∗ k + αn kjn0 k + kjn00 k + kkn k.

(5.6)

On the other hand, we find that kxn − x∗ k ≤ kxn − x∗ − (g1 (xn ) − g1 (x∗ ))k η

,M

n,1 + kRλn,1 1 ρ1 ,An,1

η

,M

n,1 + kRλn,1 1 ρ1 ,An,1

η

(·,xn ) (·,xn )

η

η

,M

n,1 (p) − Rλn,1 1 ρ1 ,An,1

∗

,M

,M

n,1 (pn ) − Rλn,1 1 ρ1 ,An,1

(·,xn )

(p)k

∗

(·,x )

(p)k

∗

(·,x )

η ,M (·,x )

n,1 1 + kRλn,1 (p) − Rλ11 ρ1 ,A (p)k 1 ρ1 ,An,1 1 q   ≤ µn,1 + q 1 − qθ1 + (cq + qκ1 )π1q kxn − x∗ k

q−1 τn,1 + kpn − pk + khn k rn,1 − λ1 ρ1 mn,1

and kyn − y ∗ k ≤ kyn − y ∗ − (g2 (yn ) − g2 (y ∗ ))k η

,M

n,2 + kRλn,2 2 ρ2 ,An,2

+

(·,yn )

η

,M

n,2 (ˆ qn ) − Rλn,2 2 ρ2 ,An,2

η ,Mn,2 (·,yn ) kRλn,2 (ˆ q) 2 ρ2 ,An,2 η ,Mn,2 (·,y ∗ ) (ˆ q) kRλn,2 2 ρ2 ,An,2

−

(·,yn )

(ˆ q )k

η ,Mn,2 (·,y ∗ ) Rλn,2 (ˆ q )k 2 ρ2 ,An,2 η ,M (·,y ∗ )

2 − Rλ22 ρ2 ,A (ˆ q )k + 2 q   ≤ µn,2 + q 1 − qθ2 + (cq + qκ2 )π2q kyn − y ∗ k

+

q−1 τn,2 kˆ qn − qk + kdn k, rn,2 − λ2 ρ2 mn,2

where η

,M

n,1 hn = Rλn,1 1 ρ1 ,An,1

(.,x∗ )

η ,M (·,x∗ )

1 (p) − Rλ11 ρ1 ,A 1

η

,M

n,2 dn = Rλn,2 2 ρ2 ,An,2

(p),

(·,y ∗ )

η ,M (·,y ∗ )

2 (ˆ q ) − Rλ22 ρ2 ,A 2

(ˆ q ).

Hence we have kxn − x∗ k ≤ (5.7)

q−1 τn,1 p
kpn − pk (rn,1 − λ1 ρ1 mn,1 ) 1 − µn,1 − q 1 − qθ1 + (cq + qκ1 )π1q 1 p + khn k q 1 − µn,1 − 1 − qθ1 + (cq + qκ1 )π1q

13

94

PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

and q−1 τn,2 p kyn − y k ≤ qn − qˆk
kˆ rn,2 − λ2 ρ2 mn,2 )(1 − µn,2 − q 1 − qθ2 + (cq + qκ2 )π2q 1 p + kdn k. q 1 − µn,2 − 1 − qθ2 + (cq + qκ2 )π2q ∗

(5.8)

Combining (5.5)-(5.8), we obtain kpn+1 − pk ≤ (1 − αn )kpn − pk + αn

n

q−1 λ1 (β1 π2 + ρ1 ξ1 ς2 )τn,2 p qn − qˆk
kˆ (rn,2 − λ2 ρ2 mn,2 ) 1 − µn,2 − q 1 − qθ2 + (cq + qκ2 )π2q

q−1 (|1 − λ1 |β1 π1 + λ1 ρ1 σ1 ς1 )τn,1 p
kpn − pk (rn,1 − λ1 ρ1 mn,1 ) 1 − µn,1 − q 1 − qθ1 + (cq + qκ1 )π1q |1 − λ1 |β1 π1 + λ1 ρ1 σ1 ς1 p + khn k 1 − µn,1 − q 1 − qθ1 + (cq + qκ1 )π1q o λ1 (β1 π2 + ρ1 ξ1 ς2 ) p + kd k + αn ke0n k + ke00n k + kln k n 1 − µn,2 − q 1 − qθ2 + (cq + qκ2 )π2q

+

and kˆ qn+1 − qˆk ≤ (1 − αn )kˆ qn − qˆk + αn

n

q−1 λ2 (β2 π1 + ρ2 ξ2 ς1 )τn,1 p
kpn − pk (rn,1 − λ1 ρ1 mn,1 ) 1 − µn,1 − q 1 − qθ1 + (cq + qκ1 )π1q

q−1 (|1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 )τn,2 p qn − qˆk
kˆ (rn,2 − λ2 ρ2 mn,2 ) 1 − µn,2 − q 1 − qθ2 + (cq + qκ2 )π2q |1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 p kdn k + 1 − µn,2 − q 1 − qθ2 + (cq + qκ2 )π2q o λ2 (β2 π1 + ρ2 ξ2 ς1 ) p kh k + αn kjn0 k + kjn00 k + kkn k. + n 1 − µn,1 − q 1 − qθ1 + (cq + qκ1 )π1q

+

Therefore, it follows that k(pn+1 , qˆn+1 ) − (p, qˆ)k∗ = kpn+1 − pk + kˆ qn+1 − qˆk ≤ (1 − αn )(kpn − pk + kˆ qn − qˆk)   q−1 n |1 − λ1 |β1 π1 + λ1 ρ1 σ1 ς1 + λ2 (β2 π1 + ρ2 ξ2 ς1 ) τn,1 p + αn
kpn − pk (rn,1 − λ1 ρ1 mn,1 ) 1 − µn,1 − q 1 − qθ1 + (cq + qκ1 )π1q 

(5.9)

 q−1 |1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 + λ1 (β1 π2 + ρ1 ξ1 ς2 ) τn,2 p + qn − qˆk
kˆ (rn,2 − λ2 ρ2 mn,2 ) 1 − µn,2 − q 1 − qθ2 + (cq + qκ2 )π2q |1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 + λ1 (β1 π2 + ρ1 ξ1 ς2 ) p + kdn k 1 − µn,2 − q 1 − qθ2 + (cq + qκ2 )π2q o |1 − λ1 |β1 π1 + λ1 ρ1 σ1 ς1 + λ2 (β2 π1 + ρ2 ξ2 ς1 ) p + kh k n 1 − µn,1 − q 1 − qθ1 + (cq + qκ1 )π1q + αn k(e0n , jn0 )k∗ + k(e00n , jn00 )k∗ + k(ln , kn )k∗ ≤ [1 − (1 − ϑ(n))αn ]k(pn , qˆn ) − (p, qˆ)k∗ + αn Γ(n)k(hn , dn )k∗ + αn k(e0n , jn0 )k∗ + k(e00n , jn00 )k∗ + k(ln , kn )k∗ , 14

95

PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

where ϑ(n) = max

q−1 (|1 − λ1 |β1 π1 + λ1 ρ1 σ1 ς1 + λ2 (β2 π1 + ρ2 ξ2 ς1 ))τn,1 p
, (rn,1 − λ1 ρ1 mn,1 ) 1 − µn,1 − q 1 − qθ1 + (cq + qκ1 )π1q q−1 o (|1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 + λ1 (β1 π2 + ρ1 ξ1 ς2 ))τn,2 p q
q (rn,2 − λ2 ρ2 mn,2 ) 1 − µn,2 − 1 − qθ2 + (cq + qκ2 )π2

n

and Γ(n) = max

n |1 − λ |β π + λ ρ σ ς + λ (β π + ρ ξ ς ) 1 1 1 1 1 1 1 2 2 1 2 2 1 p , 1 − µn,1 − q 1 − qθ1 + (cq + qκ1 )π1q |1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 + λ1 (β1 π2 + ρ1 ξ1 ς2 ) o p . 1 − µn,2 − q 1 − qθ2 + (cq + qκ2 )π2q

In view of the assumptions, ϑ(n) → ϑ and Γ(n) → Γ as n → ∞, where n (|1 − λ |β π + λ ρ σ ς + λ (β π + ρ ξ ς ))τ q−1 1 1 1 1 1 1 1 2 2 1 1 p 2 2 1 ϑ = max
, (r1 − λ1 ρ1 m1 ) 1 − µ1 − q 1 − qθ1 + (cq + qκ1 )π1q (|1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 + λ1 (β1 π2 + ρ1 ξ1 ς2 ))τ2q−1 o p
(r2 − λ2 ρ2 m2 ) 1 − µ2 − q 1 − qθ2 + (cq + qκ2 )π2q and

n |1 − λ |β π + λ ρ σ ς + λ (β π + ρ ξ ς ) 1 1 1 1 1 1 1 2 2 1 2 2 1 p Γ = max , q q 1 − µ1 − 1 − qθ1 + (cq + qκ1 )π1 |1 − λ2 |β2 π2 + λ2 ρ2 σ2 ς2 + λ1 (β1 π2 + ρ1 ξ1 ς2 ) o p . 1 − µ2 − q 1 − qθ2 + (cq + qκ2 )π2q

2 ρ2 m2 1 ρ1 m1 = r2 −λ < ∞. Then, for ϑˆ = 21 (ϑ + 1), there exists n0 ≥ 1 It is clear that ϑ < 1 and Γ < r1 −λ τ1q−1 τ2q−1 ˆ for all n ≥ n0 . Therefore, it follows from (5.9) that, for all n ≥ n0 , such that ϑ(n) < ϑ,

k(pn+1 , qˆn+1 ) − (p, qˆ)k∗ ˆ n )k(pn , qˆn ) − (p, qˆ)k∗ + αn Γk(hn , dn )k∗ ≤ (1 − (1 − ϑ)α (5.10)

+ αn k(e0n , jn0 )k∗ + k(e00n , jn00 )k∗ + k(ln , kn )k∗ 0

0

ˆ n )k(pn , qˆn ) − (p, qˆ)k∗ + αn (1 − ϑ) ˆ Γk(hn , dn )k∗ + ken , jn )k∗ = (1 − (1 − ϑ)α 1 − ϑˆ + k(e00n , jn00 )k∗ + k(ln , kn )k∗ . Clearly, it follows from

∞ P n

αn = ∞ that

∞ P ˆ n = ∞. Since {Mn,i }∞ is graph convergence to Mi (1 − ϑ)α n=0 n

for i = 1, 2, by the assumptions and Theorem 5.2, we know that k(hn , dn )k∗ → 0 as n → 0. Now, (5.3) and Lemma 5.6 imply that k(pn , qˆn ) − (p, qˆ)k∗ → 0 as n → ∞, that is, lim (pn , qˆn ) = (p, qˆ). In view of n→∞ that khn k → 0 and kdn k → 0 as n → ∞, by the inequalities (5.7) and (5.8), it follows that xn → x∗ and yn → y ∗ as n → ∞. Thus the sequence {(xn , yn )} generated by Algorithm 5.4, converges strongly to the unique solution (x∗ , y ∗ ) of System 3.1. This completes the proof. Remark 5.8. The conditions (e) and (f) of Theorem 4.2 hold for some suitable value of constants, for example, µi = 0.07, q = 2, cq = 0.09, τi = λi = ςi = 1, θi = 0.12, κi = 0.06, πi = 0.2, ρi = ξi = 0.06, βi = 0.0025, mi = 0.5, ri = 0.7 and σi = 0.04 (i = 1, 2). Remark 5.9. Theorems 4.2 and 5.7 extend and improve a lot of the corresponding results in [10, 11, 13, 15, 17, 27, 28, 29].

15

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PETROT, BALOOEE: NONLINEAR VARIATIONAL INCLUSIONS

6

Acknowledgments

The first author is supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand.

References [1] R. P. Agarwal, Y. J. Cho and N. Petrot, Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces, Fixed Point Theory Appl. Vol. 2011, 2011:31 doi:10.1186/16871812-2011-31. 1 [2] S. Adly, Perturbed algorithm and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl. 201(1996), 609–630. 1 [3] M. Alimohammady, J. Balooee, Y. J. Cho and M. Roohi, New perturbed finite step iterative algorithms for a system of extended generalized nonlinear mixed quasi-variational inclusions, Comput. Math. Appl. 60 (2010), 2953-2970. 5.2 [4] C. Baiocchi and A. Capelo, Variational and Quasi-variational Inequalities, Applications to Free Boundary Problems, John Wiley and Sons, New York, 1984. 1 [5] Y. J. Cho, Y. P. Fang, N. J. Huang and H. J. Hwang, Algorithms for systems of nonlinear variational inequalities, J. Korean Math. Soc. 41(2004), 489–499. 1 [6] Y. J. Cho and N. Petrot, Approximate solvability of a system of nonlinear relaxed cocoercive variational inequalities and Lipschitz continuous mapping in Hilbert spaces, Advan. Nonlinear Variat. Inequal. 13(2010), 91-101. 1 [7] Y. J. Cho, I. K. Argyros and N. Petrot, Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems, Comput. Math. Appl. 60(2010), 2292–2301. 1 [8] Y. J. Cho, N. Petrot and S. Suantai, Fixed point theorems for nonexpansive mappings with applications to generalized equilibrium and system of nonlinear variational inequalities problems, J. Nonlinear Anal. Optim. 1(2010), 45–53. 1 [9] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. 1 [10] Y. P. Fang and N. J. Huang, H-monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput. 145(2003), 795–803. 1, 3, 5.9 [11] Y. P. Fang and N. J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17(2004), 647–653. 1, 2.6, 3, 5.9 [12] A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185(1994), 706–712. 1, 3 [13] X. F. He, J. Lou and Z. He, Iterative methods for solving variational inclusions in Banach spaces, J. Comput. Appl. Math. 203(2007), 80–86. 1, 3, 5.9

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[14] N. J. Huang and Y. P. Fang, Generalized m-accretive mappings in Banach spaces, J. Sichuan Univ. 38(2001), 591–592. 1, 2.6 [15] M. M. Jin, Iterative algorithms for a new system of nonlinear vriational inclusions with (A, η)-accretive mappings in Banach spaces, Comput. Math. Appl. 54(2007), 579–588. 1, 3, 5.5, 5.9 [16] G. Kassay and J. Kolumban, System of multi-valued variational inequalities, Publ. Math. Debrecen 54(1999), 267–279. 1 [17] K. R. Kazmi and M. I. Bhat, Iterative algorithm for a system of nonlinear variational-like inclusions, Comput. Math. Appl. 48(2004), 1929–1935. 1, 3, 5.9 [18] N. Kikuchi and J. T. Oden, Contact Problem in Elasticity, A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, PA, 1988. 3 [19] J. K. Kim and D. S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Nonlinear and Convex Anal. 11(2004), 235–243. 1 [20] P. Kumam, N. Petrot and R. Wangkeeree, Existence and iterative approximation of solutions of generalized mixed quasi-variational-like inequality problem in Banach spaces, Appl. Math. Comput. 217 (2011), 7496-7503. 1 [21] H. Y. Lan, Stability of iterative processes with errors for a system of nonlinear (A, η)-accretive variational inclusions in Banach spaces, Comput. Math. Appl. 56(2008), 290–303. 1 [22] H. Y. Lan, Y. J. Cho and R. U. Verma, Nonlinear relaxed cocoercive variational inclusions involving (A, η)-accretive mappings in Banach spaces, Comput. Math. Appl. 51(2006), 1529–1538. (document), 1, 2.7, 2.8, 2, 2.10 [23] L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194(1995), 114–125. 5 [24] J. S. Pang, Asymmetric variational inequality problems over product sets: Applications and iterative methods, Math. Prog. 31(1985), 206–219. 1 [25] N. Petrot, A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems, Appl. Math. Lett. 23(2010), 440-445. 1 [26] G. Stampacchia, Formes bilineaires sur les ensemble convexes, C. R. Acad. Sci. Paris 285(1964), 4413– 4416. 1 [27] R. U. Verma, A system of generalized auxiliary problems principle and a system of variational inequalities, Math. Inequal. Appl. 4 (2001), 443–453. 1, 3, 5.9 [28] R. U. Verma, A-monotone nonlinear relaxed cocoercive variational inclusions, Central Europ. J. Math. 5(2007), 386–396. 1, 3, 5.9 [29] Z. Wang and C. Wu, A system of nonlinear variational inclusions with (A, η)-monotone mappings, J. Inequal. Appl. Vol. 2008, Art. ID 681734, 6 pages, doi:10.1155/2008/681734. 1, 3, 5.9 [30] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16(1991), 1127–1138.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 99-110, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Fixed Point Theorems for Set-valued Contractions in Ordered Cone Metric Spaces Narin Petrot† and Javad Balooee‡ †

Department of Mathematics, Faculty of Science

Naresuan University, Phitsanulok, 65000, Thailand †

Centre of Excellence in Mathematics,

CHE, Si Ayutthaya Road, Bangkok 10400, Thailand E-mail: [email protected] ‡

Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran E-mail: [email protected]

‡

The corresponding author: [email protected] (Javad Balooee)

Abstract. At the present paper, the concept of a new contraction for set-valued maps in cone metric spaces with regular cone is introduced and two fixed point theorems for a such contraction are established. Two examples are given to show that our results are genuine generalizations of Wardowski’s theorems. Keywords : Fixed point, Endpoint; Cone metric space; Set-valued contraction; Ordered Banach space. 2010 MSC : Primary 47H10; Secondary 54C60, 54H25

1

Introduction and preliminaries

Fixed point theory has a basic role in applications of some branches of mathematics. The theory of fixed points of set-valued mappings is a generalization of the theory of fixed points of mappings in a sense. Huang and zhang [7] introduced the concept of cone metric spaces as a generalization of metric spaces. Indeed, they defined cone metric spaces by substituting an ordered normed space instead of the real numbers. They and also subsequently several authors proved some fixed point theorems of contractive mappings in such spaces, for instance see [1, 2, 7–10, 15–19]. In 1969, Nadler [13] succeeded to introduce a notion of the fixed point theory of set-valued contractions in a metric space and subsequently the mentioned theory was developed in different directions by many authors, in particular, by Y. J. Cho et. al. [4], Y. Feng-S. Liu [5], S. Hirunworakit-N. Petrot [6], D. Klim-D. Wardowski [11], N. Mizoguchi-W. Takahashi [12], S. Reich [14], Suwannawit-Petrot [17], and many others. Recently, D. Wardowski in [18] inspired by the idea of contraction for set-valued maps in metric spaces succeeded to introduce it in cone metric spaces. In this paper, we shall prove two fixed point and endpoint theorems for set-valued maps in cone metric spaces with regular cone, by introducing a new contraction condition. In section 3, comparisons and examples are given. We initiate our discussion by introducing some preliminaries and notations.

99 1

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Definition 1.1. [3] Let E be a real Banach space and C be a non-empty subset of E. C 6= {θ}, where θ denotes the zero element of E, is called a cone, if (a) C is closed, (b) λx + µy ∈ C for all x, y ∈ C and non-negative real numbers λ and µ, (c) C ∩ (−C) = {θ}. For a given cone C ⊆ E, we can define a partial ordering with respect to C by x  y or y  x if and only if y − x ∈ C. The real Banach space E equipped with the partial order induced by C is denoted by (E, ). The symbol x  y or y  x will stand for y − x ∈ int(C), where int(C) denotes the interior of C. We shall write x ≺ y to indicate x  y but x 6= y. Proposition 1.2. [15] Suppose C is a cone in E. Then (a) If e  f and f  g, then e  g. (b) If e  f and f  g, then e  g. (c) If e  f and f  g, then e  g. (d) If a ∈ C and a  e for each e ∈ int(C), then a = θ. Proposition 1.3. [2] Suppose e ∈ int(C), θ  an and an → θ. Then there exists N ∈ N such that an  e for all n ≥ N . Definition 1.4. The cone C is called normal if there is a number k > 0 such that for all x, y ∈ E, θ  x  y implies kxk ≤ kkyk.

(1.1)

The least positive number satisfying (1.1) is called the normal constant of C. Example 1.5. [3, 16] Let E = CR1 ([0, 1]) with the norm kf k = kf k∞ + kf 0 k∞ . The cone C = {f ∈ E : f ≥ θ} is a non-normal cone. The cone C is called regular if every increasing sequence in E which is bounded from above is convergent in it. That is, if {xn } is a sequence in E such that x1  x2  . . .  xn  . . .  y for some y ∈ E, then there is x ∈ E such that kxn − xk → 0. Equivalently the cone C is regular if and only if every decreasing sequence in E which is bounded from below is convergent in it [7]. Proposition 1.6. [7, 16] Every regular cone is normal. Example 1.7. [16] Let E = CR ([0, 1]) with the supremum norm and C = {f ∈ E : f ≥ θ}. Then, C is a cone with normal constant M = 1. Now, consider the following sequence of elements of E which is decreasing and bounded from below but it is not convergent in E; x  x2  x3  . . .  θ. Therefore, the converse of Proposition 1.6 is not true.

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Definition 1.8. [7] Let X be a non-empty set, E be a real Banach space and C be a cone in E. The mapping d : X × X −→ (E, ) is called a cone metric on X, if (a) d(x, y)  θ for all x, y ∈ X and d(x, y) = θ if and only if x = y, (b) d(x, y) = d(y, x) for all x, y ∈ X, (c) d(x, y)  d(x, z) + d(z, y) for all x, y, z ∈ X. In this case, (X, d) is called a cone metric space. Example 1.9. [2] Let X = R, E = Rn and C = {(x1 , . . . , xn ) ∈ Rn : xi ≥ 0}. It is easy to see that d : X × X −→ E defined by d(x, y) = (|x − y|, k1 |x − y|, . . . , kn−1 |x − y|) is a cone metric on X, where ki ≥ 0 for all i ∈ {1, . . . , n − 1}. Example 1.10. [16] Let E = l1 , C = {{xn }n≥1 ∈ E : xn ≥ 0, for all n ∈ N}, (X, ρ) be a metric space and d : X × X −→ E defined by d(x, y) = { ρ(x,y) 2n }n≥1 . Then (X, d) is a cone metric space. Remark 1.11. It should be noticed that Example 1.9 for n = 2 goes back to Example 1 in [7], and Example 1.10 ensures us that the class of cone metric spaces is bigger than the class of metric spaces. Definition 1.12. [7] Let (X, d) be a cone metric space. The sequence {xn } in X is called (a) Cauchy if for every e in E with e  θ, there is N ∈ N in which for all m, n > N , d(xm , xn )  e, (b) Convergent if for every e in E with e  θ, there is N ∈ N such that for all n ≥ N , d(xn , x)  e for some fixed x in X. Clearly every convergent sequence is a Cauchy sequence [7]. A cone metric space (X, d) is said to be complete if every Cauchy sequence in X is convergent in it. Definition 1.13. Let (X, d) be a cone metric space. A set A ⊆ X is called (a) closed if for any sequence {xn } ⊆ A convergent to x, we have x ∈ A, (b) sequentially compact if for any sequence {xn } ⊆ A, there exists a subsequence {xnk } of {xn } such that {xnk } is convergent to an element of A.

2

Main results

Denote N (X) a collection of all nonempty subsets of X, C(X) a collection of all nonempty closed subsets of X and K(X) a collection of all nonempty sequentially compact subsets of X. Definition 2.1. Let X be a non-empty set. An element x ∈ X is said to be a fixed point of a set-valued mapping T −→ N (X) if x ∈ T x. If T x = {x}, then x is called a endpoint of T . We denote the sets of all fixed points and endpoints of T by Fix(T ) and End(T ), respectively. Definition 2.2. A function f : X −→ R is called lower semi-continuous, if for any {xn } ⊆ X and x ∈ X, xn → x =⇒ f (x) ≤ lim inf f (xn ). n→∞

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If (X, d) be a cone metric space and T : X −→ C(X), then for x ∈ X, we denote D(x, T x) = {d(x, z) : z ∈ T x} and S(x, T x) = {u ∈ D(x, T x) : kuk = inf{kvk : v ∈ D(x, T x)}}. D. Wardowski in [18] proved the following two theorems about fixed points and endpoints of set-valued mappings in complete cone metric spaces which we may read as follows: Theorem 2.3. [18] Let (X, d) be a complete cone metric space, C be a normal cone with normal constant K and let T : X −→ C(X). Assume that a function I : X −→ R defined by I(x) = inf kd(x, y)k, x ∈ X is lower y∈T x

semicontinuous. If there exist λ ∈ [0, 1), b ∈ (λ, 1] such that ∀x ∈ X ∃y ∈ T x ∃v ∈ D(y, T y) ∀u ∈ D(x, T x)

{[bd(x, y)  u] ∧ [v  λd(x, y)]},

(2.1)

then Fix(T ) 6= ∅. Theorem 2.4. [18] Let (X, d) be a complete cone metric space, C be a normal cone with normal constant K and let T : X −→ K(X). Assume that a function I : X −→ R of the form I(x) = inf kd(x, y)k, x ∈ X is lower y∈T x

semicontinuous. The following hold: (a) If there exist λ ∈ [0, 1), b ∈ (λ, 1] such that ∀x ∈ X ∃y ∈ T x ∃v ∈ S(y, T y) ∀u ∈ S(x, T x)

{[bd(x, y)  u] ∧ [v  λd(x, y)]},

then Fix(T ) 6= ∅. (b) If there exist λ ∈ [0, 1), b ∈ (λ, 1] such that ∀x ∈ X ∀y ∈ T x ∃v ∈ S(y, T y) ∀u ∈ S(x, T x)

{[bd(x, y)  u] ∧ [v  λd(x, y)]},

then Fix(T ) = End(T ) 6= ∅. Now, we prove a theorem which can be considered as a generalization of Theorem 2.3 of D. Wardowski [18] for cone metric spaces with regular cone. Theorem 2.5. Let (X, d) be a complete cone metric space, C be a regular cone in E and let T : X −→ C(X). Assume that the following conditions hold: (a) The map f : X −→ R, defined by f (x) = inf kd(x, y)k, x ∈ X, is lower semi-continuous, y∈T x

(b) there exists a constant b ∈ (0, 1] and a function ϕ : C −→ [0, b) satisfying lim sup ϕ(r) < b for each t ∈ C,

(2.2)

r→t

(c) for any x ∈ X there is y ∈ T x and v ∈ D(y, T y) such that for each u ∈ D(x, T x) we have bd(x, y)  u

(2.3)

v  ϕ(d(x, y))d(x, y).

(2.4)

and

Then Fix(T ) 6= ∅.

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Proof. We select a sequence {xn } in X in the following way. Let x1 ∈ X be arbitrary and fixed. Take any u1 ∈ D(x1 , T x1 ). By (c), there exist x2 ∈ T x1 and u2 ∈ D(x2 , T x2 ) such that bd(x1 , x2 )  u1

(2.5)

u2  ϕ(d(x1 , x2 ))d(x1 , x2 ) and ϕ(d(x1 , x2 )) < b.

(2.6)

and

By using (2.5) and (2.6), we get u1 − u2  bd(x1 , x2 ) − ϕ(d(x1 , x2 ))d(x1 , x2 ) = [b − ϕ(d(x1 , x2 ))]d(x1 , x2 ) and u2 

ϕ(d(x1 , x2 )) u1 . b

(2.7)

Again by (c), for x2 ∈ X, there exist x3 ∈ T x2 and u3 ∈ D(x3 , T x3 ) satisfying bd(x2 , x3 )  u2

(2.8)

u3  ϕ(d(x2 , x3 ))d(x2 , x3 ) and ϕ(d(x2 , x3 )) < b.

(2.9)

and

It follows from (2.7)-(2.9) that u2 − u3  bd(x2 , x3 ) − ϕ(d(x2 , x3 ))d(x2 , x3 ) = [b − ϕ(d(x2 , x3 ))]d(x2 , x3 ) and u3 

ϕ(d(x2 , x3 )) ϕ(d(x2 , x3 ))ϕ(d(x1 , x2 )) u2  u1 . b b2

From (2.8) and (2.6), we obtain d(x2 , x3 ) 

ϕ(d(x1 , x2 )) 1 u2  d(x1 , x2 )  d(x1 , x2 ). b b

Inductively, for xn ∈ X, there exist xn+1 ∈ T xn and un+1 ∈ D(xn+1 , T xn+1 ) such that bd(xn , xn+1 )  un

(2.10)

and un+1  ϕ(d(xn , xn+1 ))d(xn , xn+1 )

and ϕ(d(xn , xn+1 )) < b.

(2.11)

Applying (2.10) and (2.11), we get un − un+1  bd(xn , xn+1 ) − ϕ(d(xn , xn+1 ))d(xn , xn+1 ); i.e., un − un+1  [b − ϕ(d(xn , xn+1 ))]d(xn , xn+1 ),

(2.12)

and d(xn , xn+1 )  d(xn−1 , xn ).

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Thus {d(xn , xn+1 )} is a decreasing sequence. Regularity of C guarantees that the mentioned sequence is convergent. Accordance to (2.2), there exists q ∈ [0, b) such that lim sup ϕ(d(xn , xn+1 )) = q. n→∞

Therefore, for any b0 ∈ (q, b), there exists n0 ∈ N such that ϕ(d(xn , xn+1 )) < b0 , for all n > n0 .

(2.13)

Consequently, using (2.12), for all n > n0 , we have un − un+1  αd(xn , xn+1 ),

(2.14)

where α = b − b0 . Further, by (2.10)-(2.12), for all n > n0 , conclude that ϕ(d(xn , xn+1 )) un b ϕ(d(xn , xn+1 )) · · · ϕ(d(x1 , x2 )) u1  ···  bn ϕ(d(xn , xn+1 )) · · · ϕ(d(xn0 +1 , xn0 +2 )) ϕ(d(xn0 , xn0 +1 )) · · · ϕ(d(x1 , x2 )) = u1 · bn−n0 b n0 b0 ϕ(d(xn0 , xn0 +1 )) · · · ϕ(d(x1 , x2 )) ≺ ( )n−n0 · u1 ; b bn 0

un+1  ϕ(d(xn , xn+1 ))d(xn , xn+1 ) 

i.e., b0 n−n0 ϕ(d(xn0 , xn0 +1 )) · · · ϕ(d(x1 , x2 )) ) · u1 , b b n0 for all n ∈ N. Let now m, n ∈ N be such that m > n > n0 . By (2.14) and (2.15) it follows that un+1 ≺ (

d(xm , xn ) 

m−1 X

d(xj , xj+1 ) 

j=n

(2.15)

m−1 1 1 X (uj − uj+1 ) = (un − um ) α j=n α

1 b0 n−n0 −1 b0 ϕ(d(xn0 , xn0 +1 )) · · · ϕ(d(x1 , x2 )) [( ) − ( )m−n0 −1 ] · u1 . α b b bn 0 Suppose that e  θ be given. Since b0 < b, the right side of the above inequality tends to zero as n → ∞. Thus, by Proposition 1.3 there exists N ∈ N such that 

1 b0 n−n0 −1 b0 ϕ(d(xn0 , xn0 +1 )) · · · ϕ(d(x1 , x2 )) [( ) − ( )m−n0 −1 ] · u1  e, α b b b n0 for all n, m ≥ N and then by part (a) of Proposition 1.2 we have d(xm , xn )  e, for all n, m ≥ N , consequently {xn } is a Cauchy sequence in X. From completeness of X conclude that xn → p, as n → ∞, for some p ∈ X. Accordance to the definition of D(x, T x), from un ∈ D(xn , T xn ) it follows that there exists a sequence {zn } such that for any n ∈ N, zn ∈ T xn and un = d(xn , zn ). In view of the convergence of the sequence {un } and lower semicontinuity of the function f deduce that inf kd(p, y)k ≤ lim inf inf kd(xn , y)k ≤ lim inf kd(xn , zn )k = 0,

y∈T p

n→∞ y∈T xn

n→∞

so inf kd(p, y)k = 0.

(2.16)

y∈T p

Now, we claim that p ∈ T p. To prove this, on the contrary, suppose that p ∈ / T p. It follows from (2.16) that there exists a sequence {yn } ⊆ T p such that lim kd(p, yn )k = 0, and hence n→∞

lim d(p, yn ) = θ.

(2.17)

n→∞

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Then for any m, n ≥ 0, d(ym , yn )  d(ym , p) + d(p, yn ). Let again e  θ be given. By (2.17) and Proposition 1.3 there exists N ∈ N such that d(ym , p)  d(yn , p)  2e . Thus d(ym , yn ) 

e 2

and

e e + = e, 2 2

hence {yn } is a Cauchy sequence in X. Completeness of X guarantees that yn → w for some w ∈ X and because of the closedness of T p deduce that w ∈ T p. Then, for any n ∈ N, d(p, w)  d(p, yn ) + d(yn , w). Let e0  θ be given. In view of (2.17) and the fact that yn → w, as n → ∞, conclude that there exists N ∈ N 0 0 such that d(yn , p)  e2 and d(yn , w)  e2 , so d(p, w) 

e0 e0 + = e0 . 2 2

Now, part (d) of Proposition 1.2 implies that d(p, w) = θ, consequently p = w, which is a contradiction. Accordingly p ∈ T p and this is the desired result. Corollary 2.6. Let (X, d) be a complete cone metric space, C be a regular cone in E and let T : X −→ X be a single valued mapping. Assume that the function f : X −→ R defined by f (x) = kd(x, T (x))k for all x ∈ X, is lower semi-continuous. If there exist b ∈ (0, 1] and a function ϕ : C −→ [0, b) satisfying lim sup ϕ(r) < b for each t ∈ C

(2.18)

r→t

such that d(T x, T 2 x)  ϕ(d(x, T x))d(x, T x), for each x ∈ X, then Fix(T ) 6= ∅. Corollary 2.7. [11] Let (X, d) be a complete metric space and let T : X −→ C(X). Assume that the following conditions hold. (a) The map f : X −→ R, defined by f (x) = D(x, T x), x ∈ X, is lower semi-continuous, (b) there exists a constant b ∈ (0, 1) and a function ϕ : [0, +∞) −→ [0, b) satisfying lim sup ϕ(r) < b for each t ∈ [0, +∞),

(2.19)

r→t

and for any x ∈ X there is y ∈ T x satisfying the following conditions: bd(x, y) ≤ D(x, T x) and D(y, T y) ≤ ϕ(d(x, y))d(x, y). Then Fix(T ) 6= ∅.

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Corollary 2.8. Let (X, d) be a complete metric space and let T : X −→ X. Assume that the function f : X −→ R, which is defined by f (x) = d(x, T x), x ∈ X, is lower semi-continuous. If there exists b ∈ (0, 1] and a function ϕ : [0, +∞) −→ [0, b) satisfying lim sup ϕ(r) < b for each t ∈ [0, +∞),

(2.20)

r→t

and d(T x, T 2 x) ≤ ϕ(d(x, T x))d(x, T x) for each x ∈ X, then Fix(T ) 6= ∅. Corollary 2.9. [18] Let (X, d) be a complete metric space and let T : X −→ X. Assume that the function f : X −→ R, which is defined by f (x) = d(x, T x), x ∈ X, is lower semi-continuous. If there exists λ ∈ (0, 1] such that d(T x, T 2 x) ≤ λd(x, T x) for each x ∈ X, then Fix(T ) 6= ∅. Theorem 2.10. Let (X, d) be a complete cone metric space, C be a regular cone in E and let T : X −→ K(X). Assume that the map f : X −→ R, defined by f (x) = inf kd(x, y)k, x ∈ X, is lower semi-continuous and there y∈T x

exists a constant b ∈ (0, 1] and a function ϕ : C −→ [0, b) satisfying lim sup ϕ(r) < b for each t ∈ C. r→t

(a) If for any x ∈ X there exists y ∈ T x and v ∈ S(y, T y) such that for each u ∈ S(x, T x) we have bd(x, y)  u

and

v  ϕ(d(x, y))d(x, y),

then Fix(T ) 6= ∅. (b) If for any x ∈ X and for any y ∈ T x there exists v ∈ S(y, T y) such that for each u ∈ S(x, T x) we have bd(x, y)  u

(2.21)

and v  ϕ(d(x, y))d(x, y), then Fix(T ) = End(T ) 6= ∅. Proof. (a) As it has been established in the proof of part (i) of Theorem 3.2 in [18], S(x, T x) 6= ∅ for all x ∈ X. In view of that S(x, T x) ⊆ D(x, T x) for all x ∈ X, similar to Theorem 2.3 one can deduce that Fix(T ) 6= ∅. (b) By part (a), we get the existence of p ∈ X such that p ∈ T p. Take any w ∈ T p. Then by (2.21) for all u ∈ S(p, T p), we have bd(p, w)  u. It follows from p ∈ T p that θ ∈ S(p, T p) and so bd(p, w)  θ whence deduce that p = w. Therefore we get T p = {p}. Remark 2.11. It should be noticed that (a) Theorems 2.5 and 2.10 are extended versions of Theorems 2.3 and 2.4 respectively, but for regular cones. (b) Similar to Corollaries 2.6-2.9, one can find some corollaries for Theorem 2.4.

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3

Comparisons and examples In this section, we present two examples which support our results.

Example 3.1. Let X = [0, 1], E = R2 with maximal norm and C = {(x, y) ∈ E : x, y ≥ 0}. Consider the function d : X × X −→ E defined by d(x, y) = (|x − y|, β|x − y|), where β ∈ (0, 1). It is easy to see that the pair (X, d) is a complete cone metric space. Let T : X −→ C(X) be such that ( 13 { 12 x2 } x ∈ [0, 28 ) ∪ ( 13 28 , 1], T (x) = 15 1 13 { 84 , 4 } x = 28 . Assume that b =

ϕ(t) =

    

3 4

and let ϕ : C −→ [0, b) be defined by

3 2 ktk 372 567 1 2

ktk ∈ [0, 27 ) ∪ ( 27 , 12 ), ktk = 72 , ktk ∈ [ 21 , ∞).

Clearly, ( f (x) = inf kd(x, y)k = y∈T x

x − 12 x2 3 14

13 x ∈ [0, 13 28 ) ∪ ( 28 , 1], 13 x = 28 ,

13 1 2 13 and so f is lower semi-continuous. Moreover, T x = { 12 x2 }, for any x ∈ [0, 28 ) ∪ ( 13 28 , 1]. If 2 x 6= 28 then taking y = 21 x2 , d(x, y) = (x − 12 x2 , β(x − 21 x2 )), D(x, T x) = {d(x, 12 x2 )} = {(x − 12 x2 , β(x − 12 x2 ))} and D(y, T y) = {d( 12 x2 , 18 x4 )} = {( 12 x2 − 18 x4 , β( 12 x2 − 18 x4 ))}. Then

bd(x, y)  u, for each u ∈ D(x, T x) and v  ϕ(d(x, y))d(x, y),

1 1 for v = d( x2 , x4 ) ∈ D(y, T y). 2 8

Because, 1 1 1 1 1 1 v = d( x2 , x4 ) = ( x2 − x4 , β( x2 − x4 )) 2 8 2 8 2 8 1 2 2 β 2 1 1 2 = ( (x − ( x ) ), (x − ( x2 )2 )) 2 2 2 2 1 1 1 β 1 1 = ( (x + x2 )(x − x2 ), (x + x2 )(x − x2 )) 2 2 2 2 2 2 3 1 2 1 2 3β 1 2 1  ( (x − x )(x − x ), (x − x )(x − x2 )) 2 2 2 2 2 2 3 1 2 1 2 1 2 = (x − x )(x − x , β(x − x )). 2 2 2 2 If x − 21 x2 6= 27 then 32 (x − 21 x2 )(x − 12 x2 , β(x − 21 x2 )) = ϕ(d(x, y))d(x, y) and if x − 12 x2 = 27 then 23 (x − 12 x2 )(x − 1 2 1 2 372 1 2 1 2 2 x , β(x − 2 x ))  567 (x − 2 x , β(x − 2 x )) = ϕ(d(x, y))d(x, y). Therefore in both cases 1 1 v = d( x2 , x4 )  ϕ(d(x, y))d(x, y). 2 8 1

1

13 1 2 13 15 1 26 2 26 2 13 13 If x ∈ [0, 13 28 ) ∪ ( 28 , 1] such that y = 2 x = 28 and hence T y = { 84 , 4 }, d(x, y) = (( 28 ) − 28 , β(( 28 ) − 28 )) and 15 13 1 D(y, T y) = {d( 13 for each v ∈ D(y, T y). Let now x = 13 28 , 84 ), d( 28 , 4 )}. Then v  ϕ(d(x, y))d(x, y) 28 . Then 15 1 13 15 13 1 2 2β T x = { 84 , 4 } and D(x, T x) = {d( 28 , 84 ), d( 28 , 4 )}, thus for y = 15 ∈ T x, d(x, y) = ( , ) and D(y, T y) = 84 7 7

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PETROT, BALOOEE: ORDERED CONE METRIC SPACES

2

1 15 {d( 15 84 , 2 842 )}. One can verify that bd(x, y)  u, we have

v = d(

2

1 15 for each u ∈ D(x, T x) and for v = d( 15 84 , 2 842 ) ∈ D(y, T y)

15 1 152 372 2 2β 15 1 152 15 1 152 )=( − , β( − ) , ( , ) = ϕ(d(x, y))d(x, y). 2 2 84 2 84 84 2 84 84 2 842 567 7 7

Therefore, all hypothesis of Theorem 2.5 are satisfied and Fix(T ) = {0}. Next, let us observe that b ∈ (0, 34 ) and λ ∈ [0, b), then for x = 1 we have T x = { 12 } so taking y = 12 , follows d(x, y) = ( 12 , β2 ) and D(y, T y) = {d( 21 , 18 )}. Consequently, 1 1 3 3β 3 1 β d( , ) = ( , ) = ( , )  λd(x, y). 2 8 8 8 4 2 2 Accordingly, in this case T does not satisfy the contractive condition (2.1) of Theorem 2.3. If b ∈ ( 34 , 1) and λ ∈ 13 1 15 13 15 2 2β (0, b) then for x = 28 , we have T x = { 15 84 , 4 }. Consider the case that y = 84 . Since d(x, y) = d( 28 , 84 ) = ( 7 , 7 ) 15 13 1 and D(x, T x) = {d( 13 28 , 84 ), d( 28 , 4 )}, conclude that bd(x, y) 

3 3 13 15 3 2 2β d(x, y) = d( , ) = ( , ) 4 4 28 84 4 7 7 3 3β 13 1 =( , ) = d( , ). 14 14 28 4

1 Consequently, for u = d( 13 28 , 4 ) ∈ D(x, T x) we have bd(x, y)  u, hence, in this case the inequality (2.1) in 1 3 3β 1 1 Theorem 2.3 does not hold. If choose y = 14 ∈ T x then d(x, y) = d( 13 28 , 4 ) = ( 14 , 14 ) and D(y, T y) = {d( 4 , 32 )} such that

1 1 7 7β 3 3β 13 1 v = d( , ) = ( , )( , ) = d( , ) = d(x, y)  λd(x, y). 4 32 32 32 14 14 28 4 So, in this case the inequality (2.1) of Theorem 2.3 does not hold. Therefore, for x = 13 28 , there is not y ∈ T x which satisfies (2.1). Hence the map T does not satisfy hypothesis of Theorem 2.3 of Wardowski [18]. The following example shows that the lower semicontinuity of the function f in Theorem 2.3 cannot be omitted. Example 3.2. Let X, E, C and d be as in Example 3.1. Consider T : X −→ C(X) defined by  13 13 1 2   { 2 x } x ∈ (0, 28 ) ∪ ( 28 , 1], T (x) = {1} x = 0,   15 1 { 84 , 4 } x = 13 28 . Also, assume that b =

3 4

and the map ϕ be defined as in Example 3.1. Then

 1 2   x − 2x f (x) = inf kd(x, y)k = 1 y∈T x   3 14

13 x ∈ [0, 13 28 ) ∪ ( 28 , 1], x = 0, 13 x = 28 .

Clearly, f is not lower semicontinuous. From Example 3.1 conclude that for any x ∈ (0, 1] there exists y ∈ T x and v ∈ D(y, T y) such that the contractive conditions (2.3) and (2.4) hold. Let now x = 0. Then T x = {1} and so taking y = 1 we have d(x, y) = (1, β), D(x, T x) = {d(0, 1)} = {(1, β)} and D(y, T y) = {d(1, 12 )} = {( 21 , β2 )}. Obviously, bd(x, y)  u,

for each u ∈ D(x, T x)

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PETROT, BALOOEE: ORDERED CONE METRIC SPACES

and v  ϕ(d(x, y))d(x, y),

1 for v = d(1, ) ∈ D(y, T y). 2

Thus for any x ∈ X there exist y ∈ T x and v ∈ D(y, T y) such that the contractive conditions (2.3) and (2.4) hold, but Fix(T ) = ∅.

4

Acknowledgments

The first author is supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand.

References [1] M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341, 416–420 (2008) [2] M. Alimohammady, J. Balooee, S. Radojevi´c, V. Rakoˇcevi´c and M. Roohi, Conditions of regularity in cone metric spaces, Appl. Math. Comput. (2011), doi:10.1016/j.amc.2011.01.010. 1.3, 1.9 [3] C. D. Aliprantice and R. Tourky, Cones and Duality, Graduate Studies in Mathematics. Vol. 84, American Mathematical Society, Providence, Rhode Island (2007) 1.1, 1.5 [4] Y. J. Cho, S. Hirunworakit and N. Petrot, Set-valued fixed-point theorems for generalized contractive mappings without the Hausdorff metric, Applied Mathematics Letters 24 (2011) 1959-1967. 1 [5] Y. Feng and S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 317, 103–112 (2006) 1 [6] S. Hirunworakit and N. Petrot, Some fixed point theorems for contractive multivalued mappings induced by generalized distance in metric spaces, Fixed Point Theory and Applications 2011, 24 2011:78. 1 [7] L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332, 1468–1476 (2008) 1, 1, 1.6, 1.8, 1.11, 1.12 [8] Z. Kadelburg, S. Radenovi´c and V. Rako˘cevi´c, A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett. 24, 370–374 (2011) [9] M. Khani and M. Pourmahdian, On the metrizability of cone metric spaces, Topology and its Applications 158, 190–193 (2011) [10] D. Klim and D. Wardowski, Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces, Nonlinear Anal. 71, 5170–5175 (2009) [11] D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334, 132–139 (2007) 1, 2.7 [12] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141, 177–188 (1989) 1 [13] J. S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30, 475–488 (1969) 1

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[14] S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5, 26–42 (1972) 1 [15] S. H. Rezapour, M. Drafshpour and R. Hamlbarani, A review on topological properties of cone metric spaces. Analysis, Topology and Applications 2008, Vrnjacka Banja, Serbia, from May 30 to June 4, (2008) 1.2 [16] S. H. Rezapour and R. Hamlbarani, Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl. 345(2), 719–724 (2008) 1.5, 1.6, 1.7, 1.10 [17] J. Suwannawit and N. Petrot, Common fixed point theorem for hybrid generalized multivalued, Thai Journal of Mathematics 9(2) (2011), 417-427. 1 [18] D. Wardowski, Endpoints and fixed points of set-valued contractions in cone metric spaces, Nonlinear Anal. 71, 512–516 (2009) 1, 2, 2.3, 2.4, 2, 2.9, 2, 3.1 [19] D. Wardowski, On set-valued contractions of Nadler type in cone metric spaces, Appl. Math. Lett. 24, 275–278 (2011)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 111-117, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

ARITHMETIC PROPERTIES OF q-BARNES POLYNOMIALS A. BAYAD, T. KIM, W. J. KIM, AND S. H. LEE

Abstract In this paper, we introduce and investigate the q-analogues of Barnes numbers and polynomials. The main purpose of this paper is to establish Fourier expansion of these q-Barnes polynomials and from this study we connect q-Barnes numbers to values of Dirichlet-Hurwitz L-function evaluating at non-negative positive integers. 1. Introduction and preliminaries Throughout this paper we use the following notation: N = {0, 1, ...} set of naturals numbers. Let q ∈ R and x a variable, the q-Bernoulli polynomials Bn (x; q) are defined by the generating function ∞ X tn t xt B (x; q) e = , (q = 1, |t| < 2π), (q 6= 1, |t| < |log(−q)|). n qet − 1 n! n=0

(1)

The q-Bernoulli numbers Bn (q) are given by Bn (q) := Bn (0; q). These polynomials were introduced by Apostol, see [1, 13] . These polynomials are a natural extension of the classical Bernoulli polynomials : Bn (x) = Bn (x; 1) , see [12] . They have many applications in mathematics. Recently, first author proves their most important property Fourier expansion which is given by q x Bn (x; q) =

∗ −n! X e2πikx  n , log(q) (2πi)n k∈Z k − 2πi

for q ∈ C\{0} and , for 0 < x < 1 if n = 1, 0 ≤ x ≤ 1 if n ≥ 2. Here

(2) ∗ X k∈Z

if q = 1 and

∗ X k∈Z

=

X

=

X k∈Z\{0}

if q 6= 1. See [2, 3, 13]. This identity is the foundation of

k∈Z

the theory of q-Bernoulli polynomials and their relations to special values of the Riemann zeta function and Dirichlet L-functions. Let us define the Barnes and the q-Barnes polynomials and numbers. Let N → positive integer and a N = (a1 , ..., aN ), where a1 , ..., aN are complex with strictly 2010 Mathematics Subject Classification : 11B68, 11B83, 11B99, 11M32 . Key words and phrases : Bernoulli numbers and polynomials, q-Bernoulli numbers and polynomials, Barnes numbers and polynomials, q-Barnes numbers and polynomials, Dirichlet-Hurwitz L-function. 1

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BAYAD ET AL: Q-BARNES POLYNOMIALS

positive real part. The Barnes polynomials and numbers are given by tN N Y

xt

e

aj t

e

−1

=

∞ X

t → Bn (x| a N )

n!

n=0




n

 , |t| < min

2π 2π ,··· , a1 aN

 , see [5, 6, 8, 9, 14, 15, 17] .

j=1

The main interest of these numbers is that they give the values at non negative ∞ X χ(n) (Re(s) > 1) is the L-series integers of Dirichlet L-series: if L(s, χ) = ns n=1 attached to χ of conductor f , then we have the formula in [4] ∗ X

1 −1 N −1 am ...am L(m1 , χ)...L(mN , χ) = 1 N

m1 ,...,mN

  f (2πi)n (−1)N χ(−1)Gχ X t → χ(t)B ¯ | aN , n n! f f t=1 where

∗ X

X

=

m1 ,...,mN

(3)

.

m1 +..+mN =n,m1 ,...,mN ≥0 (−1)m1 =...=(−1)mN =χ(−1)

Note that in case N = 1, a1 = 1, the numbers in the right side of the equality (13) correspond to the generalized Bernoulli numbers Bm,χ which are defined by the generating function f X

χ(a)

a=1

∞ X t tn 2π at B e = , | t |< . n,χ f t e −1 n! f n=0

(4)

From the equation (3) we have , for n ≥ 0 , the well-known formula L(−n, χ) = −

Bn+1,χ , see [4, 18] . n+1

(5)

→

Let N positive integer , a N = (a1 , ..., aN ), where a1 , ..., aN are complex with strictly positive real part and let q ∈ C, | q |< 1. We introduce and investigate the following → q-Barnes polynomials Bn,q (x| a N ) defined by tN N Y

qeaj t − 1

ext =


∞ X n=0

→

Bn,q (x| a N )

tn , |t + log(q)| < min n!



2π 2π ,··· , a1 aN

 . (6)

j=1 →

→

and Bn,q ( a N ) = Bn,q (0| a N ) are the so called q-Barnes numbers. This paper can now be summarized as a generalization of these facts to the qBarnes polynomials and numbers. More precisely, the main purpose of this paper is to prove the extension of the properties (2) and ( 3) to q-Barnes numbers and polynomials.

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BAYAD ET AL: Q-BARNES POLYNOMIALS

2. Statement and proof of main results For λ ∈ C\{0} and q ∈ C, we can write tN N Y

eλxt = λ−N


qeλaj t − 1

j=1

(λt)N N  Y

ex(λt) .

(7)

 qeaj (λt) − 1

j=1

Their Taylor expansions are given as follows ∞ X

→

Bn,q (λx|λ a N )

n=0

∞ X tn λ n tn → = λ−N Bn,q (x| a N ) . n! n! n=0

(8)

Then, by comparing the coefficients of both sides of the equation (8), we obtain the homogeneity equation Proposition 1 (Homogeneity). For any a1 , ..., aN are complex with strictly positive real part and λ ∈ C\{0}, we have →

→

Bn,q (λx | λ a N ) = λn−N Bn,q (x | a N ), (n ≥ 1).

(9)

Now we state our main results. Theorem 2. Let a1 , ...aN are complex with strictly positive real part. Then →

Bn,q (x | a N ) = n! where X =

x AN

X

mN −1 1 −1 am ...aN 1

m1 +..+mN =n m1 ,··· ,mN ≥0

Bm1 (X; q) BmN (X; q) ... . m1 ! mN !

(10)

and AN = a1 + ... + aN .

Proof of Theorem 2: Writing X =

x AN

where AN = a1 + ... + aN . We have tN

N Y

ext

=


qeaj t − 1

N Y 1 ai teX(ai t) a1 ...aN i=1 qeai t − 1

j=1

Then we get tN N Y

ext =

qeaj t − 1




j=1 ∞ X

X

n=0

m1 +···+mN =n

1 −1 am 1

Bm1 (X; q) N −1 · · · am N m1 !

BmN (X; q) ··· mN !

! tn .

(11)

In other way tN N Y

aj t

qe

ext = −1




∞ X Bn,q → (x| a N )tn . n! n=0

j=1

113

(12)

BAYAD ET AL: Q-BARNES POLYNOMIALS

By comparing the right sides of the equations (11) and (11) we obtain →

Bn,q (x | a N ) = n! m

X

mN −1 1 −1 am ...aN 1

1 +..+mN =n

Bm1 (X; q) BmN (X; q) ... , (n ∈ N). m1 ! mN !

This yields our theorem.  Theorem 3 (Fourier expansion). Let a1 , ..., aN are complex with strictly positive real part and set AN = a1 + ... + aN and X = AxN . Then for any n ≥ 1 and |X| < 1 we have →

q x Bn,q (x | a N ) = ∗ ∗∗ X X e ((k1 + ... + kN )X) (−1)N n! mN −1 m1 −1  m1  mN . a ...a n 1 N (2πi) log(q) ... kN − log(q) m1 +..+mN =n k1 ,...,kN ∈Z k1 − 2πi 2πi Here

∗∗ X

means that k1 , ..., kN ∈ Z\{ log(q) 2πi } and, in the non-absolutely con-

k1 ,...,kN ∈Z

vergent case mi = 1, for any 1 ≤ i ≤ N, the sum

∗∗ X

to be interpreted as a Cauchy

ki ∈Z ∗ X

principal value for each i, and

means that m1 , ..., mN ∈ N with the

m1 +..+mN =n 0

usual convention the sum

X

e (ki X) = −1 if mi = 0.

ki ∈Z\{0}

Proof of Theorem 3: Using the equation (2) and Theorem 2, we can write →

Bn,q (x| a N ) n! ∗∗ ∗ mN −1 m1 −1 N X X a1 · · · aN (−1) m1 ! · · · mN ! m1 ! · · · mN ! (2πi)m1 +···+mN m1 +···+mN =n k1 ,··· ,kN ∈Z

qx

=

=

(−1)N (2πi)n

∗ X

∗∗ X

mN −1 1 −1 am · · · aN 1

m1 +···+mN =n

 k1 ,··· ,kN ∈Z



e ((k1 + · · · + kN )X)  m1 mN , k1 − · · · kN − log(q) 2πi log(q) 2πi

e ((k1 + · · · + kN )X) m1  mN . · · · kN − log(q) k1 − 2πi log(q) 2πi

This gives the theorem.  Let f an integer ≥ 2 and χ a Dirichlet character modulo f . As usual we define the L-series by X χ(k) , 1. L(s, x, χ) = (x + k)s k∈Z,k6=−x

In this L-series we relax the summation over all k ∈ Z, k 6= −x. But it’s easy to see that is related to the classical Dirichlet L-series where the summation is over

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BAYAD ET AL: Q-BARNES POLYNOMIALS

k ∈ N, k 6= −x. We recall the definition of the Gauss sum associated to the character χ is Gχ =

f X

t χ(t)e( ). f t=1

By homogeneity Proposition 1, without loss of generality we can assume for the following theorems that: a1 + · · · + aN = 1. Theorem 4 (Values of L-function at non-negative integers). Let f ≥ 2 be a natural number, a1 , ...aN with real part strictly positive real and a1 + ... + aN = 1 , χ a non trivial Dirichlet character modulo f ≥ 2. Then we have X 1 −1 N −1 am ...am L(m1 , α, χ)...L(mN , α, χ) = 1 N m1 +...+mN =n m1 ,...,mN ≥0

  f t t → (2πi)n (−1)N χ(−1)Gχ X f χ(t)q ¯ Bn,q | aN , n! f f t=1 where α =

log(q) 2πi ,

with the usual convention L(0, χ) =

(13)

−1 2 .

Proof of Theorem 4: Using Theorem 3, we have   f X t t → f B | a χ(t)q ¯ N n f t=1 =

∗ X

(−1)N n! (2πi)n

m1 +..+mN =n f X

∗∗ X

mN −1 1 −1 am ...aN 1

1 

k1 ,...,kN ∈Z



t χ(t)e ¯ (k1 + ... + kN ) f t=1

k1 −

log(q) 2πi

m1

 ... kN −

log(q) 2πi

mN

 .

Since f X

  t = χ(k)Gχ¯ χ(t)e ¯ k f t=1

we have f X

t

f B χ(t)q ¯ n

t=1

=

(−1)N n! ¯ (2πi)n Gχ



t → |aN f



∗ X

1 −1 N −1 am ...am 1 N

m1 +..+mN =n

=

(−1)N n! ¯ (2πi)n Gχ

∗ X m1 +..+mN =n

∗∗ X

 k1 ,...,kN ∈Z

k1 −

χ(k1 + ... + kN ) m1  mN . ... kN − log(q) 2πi

log(q) 2πi

0

χ(k1 )

X

1 −1 N −1 am ...am 1 N

 k1 ,...,kN ∈Z\{0}

115

k1 −

log(q) 2πi

m1 · · · 

χ(kN ) kN −

log(q) 2πi

mN

BAYAD ET AL: Q-BARNES POLYNOMIALS

While

0

χ(ki )

X  k∈Z\{0}

k1 −

log(q) 2πi

mi = L(mi , α, χ),

Therefore, we arrive at   f X t t → (−1)N (n!) f χ(t)q ¯ Gχ¯ Bn |aN = f (2πi)n t=1 =

2N (−1)N (n!) (2πi)n

X

log(q) . 2πi

1 −1 N −1 am ...am 1 N

N Y

L(mi , α, χ)

i=1

m1 +..+mN =n m1 ,...,mN ≥0

X

Gχ¯

where α =

N −1 a1m1 −1 ...am L(m1 , χ)...L(mN , χ). N

m1 +..+mN =n χ(−1)=(−1)mi ,m1 ,...,mN ≥0

Using the relation Gχ¯ = χ(−1)q/Gχ , see [18] chap.4 p.29-37), we obtain the following formula X mN −1 1 −1 ...aN L(m1 , α, χ)...L(mN , α, χ) = am 1 m1 +...+mN =n m1 ,...,mN ≥0

(2πi)n (−1)N χ(−1)Gχ n! f

f X

t

f B χ(t)q ¯ n,q

t=1



t → | aN f

 .

Hence, we obtain our desired theorem.  Remark 1. (1) Taking q = 1 we recover the main results of Bayad and Kim in [4]. (2) If we take N = 1 and q = 1 we obtain results of Bayad [2, 3]. (3) In case N = 1, q = a1 = 1 the Theorem 4 and functional equation of L-series gives us the main property on the values of Dirchlet L-series (5). References [1] T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), p. 161-167. [2] A. Bayad, Special values of Lerch zet function and their Fourier expansions, Adv. Studies in Contemp. Math Volume 21 (2011), No.1, pp. 1–4. [3] A. Bayad, Fourier expansions for Apostol-Bernoulli, Aposol-Euler and ApostolGenocchi polynomials, Mathematics Computation (2011), In press. [4] A. Bayad, T. Kim, Results on values of Barnes polynomials, Preprint. [5] A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russian Journal of Mathematical Physics, Volume 18, no. 2, (2011), 133-143. [6] A. Bayad, Y. Simsek, Values of twisted Barnes Zeta functions at negative integers, Preprint. [7] K.Dilcher, L.Louise, Arithmetic Properties of Bernoulli Numbers and Polynomials, Journal of Number Theory, Volume 92, (2), (2002), pp.330-347. [8] E. Friedman; S. Ruijsenaars, Shintani-Barnes zeta and gamma functions, Advances in Mathematics 187 (2) (2004), pp. 362395.

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[9] K. Katayama, Barne’s Multiple function and Apostol’s Generalized Dedekind Sum, Tokyo J. Math. Vol 27, N◦1,2004, 57–74. [10] T. Kim, Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Phys. A, 43(25), Article ID 255201, 11 pages, 2010 [11] T. Kim, On a p-adic interpolation function for the q-extension of the generalized Bernoulli polynomials and its derivative, Discrete Mathematics, Volume 309( 6), (2009), pp.1593-1602. [12] T. Kim, Non-Archimedean qq-integrals associated with multiple Changhee qBernoulli polynomials, Russian Journal of Mathematical Physics, Volume 10, (2003), 91-98. [13] Q-M. Luo, Fourier expansions and integral representations for the ApostolBernoulli and Apostol-Euler polynomials, Math. Comp, Volume 78 ( 2009), No. 268, p. 2193-2208. [14] K. Ota, On Kummer-Type Congruences for Derivatives of Barnes Multiple Bernoulli Polynomials, Journal of Number Theory, Volume 92(1), (2002) ,pp.136. [15] S. Ruijsenaars, On Barnes’ Multiple Zeta and Gamma Functions, Advances in Mathematics 156,(2000), 107–132. [16] Y. Simsek, Twisted -Bernoulli numbers and polynomials related to twisted qzeta function and L-function, Journal of Mathematical Analysis and Applications, Volume 324, (2), (2006), pp.790-804. 6. [17] M. Spreafico, On the Barnes double zeta and Gamma functions, Journal of Number Theory, Volume 129, (9), (2009),pp.2035-2063. [18] L.C. Washington, Introduction to Cyclotomic Fields,, Springer (1982). ´partement de mathe ´matiques, Universite ´ d’Evry Abdelmejid Bayad. De Val d’Essonne, Bd. F. Mitterrand, 91025 Evry Cedex, France, E-mail address: [email protected] Taekyun Kim. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: [email protected] Won-Joo Kim. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: [email protected] Sang Hun Lee. Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: [email protected]

117

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 118-132, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Error analysis of a C 0 discontinuous Galerkin method for Kirchhoff plates∗ Xuehai Huanga , Jianguo Huangb,c

†

a College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China b Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China c Division of Computational Science, E-Institute of Shanghai Universities, Shanghai 200235, China

Abstract This paper discusses a priori error analysis of a C 0 discontinuous Galerkin method (CDG) for Kirchhoff plates (cf. [17]). With the help of error estimates of a L2 orthogonal projection operator and an interpolation operator due to [1,13], we obtain the error estimate for the CDG method in energy norm, following the technique in [6]. The error estimate in H 1 norm is also offered via the duality argument. Several numerical results are performed to support the theory obtained. Keywords. Kirchhoff plates; C 0 discontinuous Galerkin method (CDG); interpolation operator; error analysis

1

Introduction

Discontinuous Galerkin (DG) methods have been developed thoroughly for solving biharmonic equations and Kirchhoff plate bending problems, with the interior penalty (IP) method as a typical one (cf. [2, 3, 5, 12, 18–20, 22]). One main advantage of DG methods for fourth order problems is that it requires less regularity of finite element spaces by including some edge terms into the discrete variational formulation. In [12], a C 0 IP formulation was devised for Kirchhoff plates and quasi-optimal error estimates were obtained for smooth functions. In [5], a rigorous error analysis for the previous method was derived under the weak regularity assumption for the solution [11, 15], and a post-processing procedure was also produced that can generate C 1 approximate solutions from the obtained C 0 approximate solutions. A drawback of the forgoing method is the presence of a dimensionless penalty parameter which must be chosen suitably large to guarantee stability, but it can not be precisely quantified a priori. Based on this observation, a new C 0 DG (CDG) method was designed in [23] for which the stability condition can be precisely quantified. The fully discontinuous IP method was investigated systematically in [18–20, 22] for biharmonic problems, where the subdivision mesh size and the degree of polynomials at each individual element can vary arbitrarily, very suitable for the design of hp-adaptive algorithms. However, due to the fact that nodal variables on the edges of elements may take different values, the total number of degrees of freedom is much more than that of usual finite element methods. In [17], by taking numerical traces in terms of a discrete stability identity, a class of stable CDG methods were proposed for Kirchhoff plate bending problems, and the resulting local CDG (LCDG) method from the CDG method does not contain any to-be-determined parameters and is more convenient to implement in actual computation than that in [23]. Based on Ciarlet-Raviart method, a mixed discontinuous Galerkin method for biharmonic equation was designed in [16] using the idea of interior penalty method. Since the global mass matrix of this method is block diagonal, the corresponding linear system can be reduced to a smaller linear system consisting only one unknown approximation uh of u, just as the LCDG method. By rewriting biharmonic equation as a first-order system and using single ∗

The work of the first author was partly supported by the NNSFC (Grant nos. 11126226, 11171257) and Zhejiang Provincial Natural Science Foundation of China (Y6110240, LY12A01015). The work of the second author was partly supported by the NNSFC (Grant nos. 11171219, 11161130004) and E–Institutes of Shanghai Municipal Education Commission (E03004). † Corresponding author. E-mail address: [email protected].

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HUANG, HUANG: KIRCHHOFF PLATES

face-hybridizable technique in [9], Cockburn, Dong and Guzm´an derived a hybridizable and superconvergent DG method in [8] which improves the convergence rate of the approximation to ∆u with order O(hk+1/2 ). The main goal of this paper is to establish a priori error analysis of the CDG method in [17]. It is mentioned that the error analysis was just obtained in [17] for the LCDG method. With the help of error estimates of a L2 projection operator and an interpolation operator due to [1, 13], we obtain the error estimate for general CDG method in energy norm, following the technique in [6]. In addition, we also get the error estimate in H 1 norm by using duality argument. Finally, we offer some numerical results to illustrate the accuracy of our CDG method, to demonstrate our theoretical results. The rest of this paper is organized as follows. Some basic definitions and symbols are given in Section 2, where the CDG method for Kirchhoff plates is also introduced. We obtain the a prior error estimates in Section 3. In the final section, we provide several numerical results to demonstrate our theoretical results.

2

The CDG method for Kirchhoff plates

Let Ω ⊂ R2 be a bounded polygonal domain and f ∈ L2 (Ω). The mathematical model of a clamped Kirchhoff plate under a vertical load f ∈ L2 (Ω) reads [14, 21] ½ ∇ · (∇ · M(u)) + f = 0 in Ω, (2.1) u = ∂n u = 0 on ∂Ω, where n is the unit outward normal to ∂Ω, ∇ is the usual gradient operator acting on tensor fields (cf. [21]), and M(u) := (1 − ν)K(u) + νtr(K(u))I, K(u) := (Kij (u))2×2 , Kij (u) := −∂ij u, 1 ≤ i, j ≤ 2, with I a second order identity tensor, tr the trace operator acting on second order tensors, and ν ∈ (0, 0.5). Introduce an auxiliary second order tensor field σ by σ := (1 − ν)K(u) + νtr(K(u))I. Then, Problem (2.1) can be reformulated as the following second-order system:  1 ν    1 − ν σ − 1 − ν 2 (trσ)I = K(u) in Ω, ∇ · (∇ · σ) = −f in Ω,    u = ∂n u = 0 on ∂Ω.

(2.2)

A CDG method based on (2.2) is introduced in [17] for solving problem (2.1). For recalling the method and later requirement, let us first introduce some definitions and symbols. Denote by S the space of all second order symmetric tensors. Given a bounded domain G ⊂ R2 and a non-negative integer m, let H m (G) be the usual Sobolev space of functions on G; let H m (G, X) be a Sobolev space of functions taking values in a finite dimensional vector space X, with X being S or R2 . The corresponding norm and semi-norm are denoted respectively by k · km,G and | · |m,G . If G is Ω, we abbreviate them by k · km and | · |m , respectively. Let H0m (G) be the closure of C0∞ (G) with respect to the norm k · km,G . We also denote by H(div, G, S) the Sobolev space consisting of all L2 (G, S) functions whose divergence are square-integrable. Denote by Pm (G) the set of all polynomials in G with the total degree no more than m; P m (G, X) stands for the tensor field analogue of Pm (G), with X being S or R2 . 2 119

HUANG, HUANG: KIRCHHOFF PLATES

Let {Th }h be a regular family of triangulations of Ω [4, 7]; h := maxK∈Th hK and hK := diam(K). Let Eh be the union of all edges of the triangulation Th and Ehi the union of all interior edges of the triangulation Th . For any e ∈ Eh , its length is denoted by he . Based on the triangulation Th , let © ª Σ := τ ∈ L2 (Ω, S) : τ |K ∈ H(div, K, S) ∀ K ∈ Th , © ª V := v ∈ H01 (Ω) : v|K ∈ H 2 (K) ∀ K ∈ Th . The corresponding finite element spaces are given by © ª Σh := τ ∈ L2 (Ω, S) : τ |K ∈ S 1 (K) ∀ K ∈ Th , ª © Vh := v ∈ H01 (Ω) : v|K ∈ S2 (K) ∀ K ∈ Th , where, for a triangle K ∈ Th , S 1 (K) ⊂ P k−1 (K, S) and S2 (K) are two finite-dimensional spaces of polynomials in K containing P l (K, S) and Pk (K), respectively, with k ≥ 1 and l ≥ 0. To guarantee uniqueness of the solution to the CDG method developed later on, we always assume that 1 ν ∇2h Vh ⊂ Σh , Σh − trΣh I ⊂ Σh , (2.3) 1−ν 1 − ν2 where ∇2h Vh |K := ∇2 (Vh |K ) for any K ∈ Th . For a function v ∈ L2 (Ω) with v|K ∈ H m (K) for all K ∈ Th , define ¶1/2 ¶1/2 µ X µ X 2 2 |v|m,K . kvkm,K , |v|m,h = kvkm,h = K∈Th

K∈Th

The above definition can be extended to a tensor field naturally. Throughout this paper, we use “. · · · ” to indicate that “≤ C · · · ”, where C is a positive generic constant independent of h and other parameters, which may take different values at different appearances. Let ∇v ⊗ n be a matrix whose ij-th component is nj ∂i v for two vectors ∇v and n. The symbol n ⊗ ∇v is defined in the samePmanner. For two second order tensors τ and σ, define their double dot product by σ : τ = 2i,j=1 σij τij , and denote the norm |τ | = (τ : τ )1/2 . For a vector a, its norm is |a| = (a · a)1/2 . Denote the averages and jumps of a scalar or tensor field on an interior edge e shared by triangles K + and K − as follows. 1 {v} = (v + + v − ), [v] = v + n+ + v − n− , 2 1 {∇v} = (∇v + + ∇v − ), [∇v] = ∇v + · n+ + ∇v − · n− , 2 1 {τ } = (τ + + τ − ), [τ ] = τ + · n+ + τ − · n− , 2 where n+ and n− are unit outward normals to the common edge e of the triangles K + and K − , respectively. If an edge e lies on the boundary ∂Ω, the above terms are defined by {v} = v, {∇v} = ∇v,

[v] = vn, [∇v] = ∇v · n,

{τ } = τ , [τ ] = τ · n, where n = N is the unit outward normal vector on ∂Ω. Similarly, define a second order tensor jump J·K for a vector ∇v by 1 J∇vK = (∇v + ⊗ n+ + n+ ⊗ ∇v + + ∇v − ⊗ n− + n− ⊗ ∇v − ), 2 1 J∇vK = (∇v ⊗ n + n ⊗ ∇v), if e ∈ Eh ∩ ∂Ω. 2 3 120

if e ∈ Ehi ,

HUANG, HUANG: KIRCHHOFF PLATES

With the above definitions and symbols in mind, the C 0 discontinuous Galerkin (CDG) method for Problem (2.1) reads (cf. [17]): Find (σ h , uh ) ∈ Σh × Vh such that a(σ h , τ ) + b(uh , τ ) = 0, −b(v, σ h ) + c(uh , v) = F (v),

(2.4) (2.5)

for all (τ , v) ∈ Σh × Vh , where ¶ Z µ Z 1 ν a(σ, τ ) := σ:τ− trσtrτ dx + C22 [σ] · [τ ]ds, 1 − ν2 Ω 1−ν Ehi Z X Z τ : ∇2 vdx − {τ } : J∇vKds, b(v, τ ) := K

K∈Th

Z

Eh

c(u, v) := C11 J∇uK : J∇vKds, Eh Z F (v) := f vdx. Ω

The so called LCDG method is a special case of CDG method (2.4)-(2.5) for C22 = 0. In [17], CDG method (2.4)-(2.5) has been proved to be uniquely solvable.

3

Error estimates

In this section, we are going to derive error estimates for CDG method (2.4)-(2.5). To this end, let P h be the usual L2 orthogonal projection operator from Σ onto the finite element space Σh . Introduce a finite element space Wh ⊂ Vh by © ª Wh := v ∈ H01 (Ω) : v|K ∈ Pk (K), ∀K ∈ Th . Then we define the second interpolation operator Ih : V ∩ H 2 (Ω) → Wh in the following way (cf. [1, 13]). Given w ∈ V ∩ H 2 (Ω), for any element K ∈ Th , any vertex a of K, and any edge e of K, Ih w ∈ Wh satisfies that Ih w(a) = w(a), Z (w − Ih w)vds = 0 ∀ v ∈ Pk−2 (e), Z

e

K

(w − Ih w)vdx = 0

∀ v ∈ Pk−3 (K).

For simplicity, we still write P h and Ih for P h |K and Ih |K . According to Proposition 5.1 in [10], for all w ∈ V ∩ H 2 (Ω) and K ∈ Th , there holds Z v · ∇(w − Ih w)dx = 0 ∀ v ∈ P k−2 (K, R2 ). K

Then noting the fact that Σh |K ⊂ P k−1 (K, S), we have for all w ∈ V and K ∈ Th , Z (∇ · τ ) · ∇(w − Ih w)dx = 0 ∀ τ ∈ Σh .

(3.1)

K

The following lemma collects some error estimates for interpolation operators P h and Ih , given in [1, 4, 7, 10, 13]. 4 121

HUANG, HUANG: KIRCHHOFF PLATES

Lemma 3.1 Let τ ∈ H m+1 (K, S) and v ∈ H m+3 (K), with m ≥ 0. Then 1/2

min{m,l}+1

|τ − P h τ |0,K + hK kτ − P h τ k0,∂K . hK 2 X

1/2

kτ km+1,K ,

min{m+1,k−1}+1

hi−1 K |v − Ih v|i,K + hK k∇ (v − Ih v)k0,∂K . hK

kvkm+3,K .

i=1

To develop error analysis, we first rewrite our method (2.4)-(2.5) in a compact way: Find (σ h , uh ) ∈ Σh × Vh such that A(σ h , uh ; τ , v) = F (v), (3.2) for all (τ , v) ∈ Σh × Vh where A(σ, u; τ , v) := a(σ, τ ) + b(u, τ ) − b(v, σ) + c(u, v). In what follows, we always let (σ, u) to be the solution of the original problem (2.2). Then using (2.2) and integration by parts, we know A(σ, u; τ , v) = F (v) ∀(τ , v) ∈ Σh × Vh .

(3.3)

Subtracting (3.2) from (3.3) implies the following Galerkin orthogonality. A(σ − σ h , u − uh ; τ , v) = 0

∀(τ , v) ∈ Σh × Vh .

For (τ , v) ∈ Σ × V , we define two seminorms as follows. |(τ , v)|2A : = A(τ , v; τ , v) ¶ Z Z µ ¡ ¢ 1 ν 2 2 2 dx + C |[τ ]| + C |J∇vK| ds = |τ |2 − (trτ ) 22 11 1 − ν2 Ehi Ω 1−ν Z + C11 |J∇vK|2 ds, ∂Ω

Z |(τ , v)|2B

:=

µ

¶ ¶ Z µ 1 1 2 2 2 |{τ }| + C11 |J∇vK| ds. C22 |[τ ]| + |{∇v}| ds + χ C11 Eh 2

Ehi

Here for each interior edge e, ½ χ(x) :=

he , if C22 (x) = 0, C22 (x), otherwise.

For (σ, u) ∈ H m+1 (Ω, S) × H m+3 (Ω) and (τ , v) ∈ H t+1 (Ω, S) × H t+3 (Ω), with m, t ≥ 0, ½ P5 if (σ, u) 6= (τ , v), i=1 Si (σ, u; τ , v), KA (σ, u; τ , v) := S1 (σ, u; σ, u) + S2 (σ, u; σ, u) + S5 (σ, u; σ, u), if (σ, u) = (τ , v),

5 122

(3.4)

HUANG, HUANG: KIRCHHOFF PLATES

where 1 

 S1 := 

X

2

2 min{m,l}+2 kσk2m+1,K  hK





∂K 2 min{m,l}+1 C22 hK kσk2m+1,K 

2

X



∂K 2 min{t,l}+1 C22 hK kτ k2t+1,K 

X

2

2 min{m+1,k−1} hK kuk2m+3,K 



K∈Th

1 X

2

2 min{t,l}+2 kτ k2t+1,K  hK

,

K∈Th

1  X

2

2 min{m,l}+2 kσk2m+1,K  hK



1 X

2

2 min{t+1,k−1} hK kvk2t+3,K 

,

K∈Th

K∈Th

1 

 X

,

K∈Th

1 



S5 := 

1

2

K∈Th

S4 := 

,

1  X

 S3 := 

2

2 min{t,l}+2 hK kτ k2t+1,K 

K∈Th

K∈Th

S2 := 

1 X

2

∂K 2 min{m+1,k−1}+1 hK kuk2m+3,K  C11



1 X

2

∂K 2 min{t+1,k−1}+1 C11 hK kvk2t+3,K 

,

K∈Th

K∈Th

Cii∂K := sup{Cii (x) : x ∈ ∂K}, i = 1, 2. And we define another functional by ! Ã Ã ! X 1 2 min{m,l}+1 ∂K 2 + C22 KB (σ, u) = hK kσk2m+1,K e ∂K C 11 K∈Th ¶ ¶ X µ 2 min{m+1,k−1}+1 µ 1 ∂K 2 + hK C11 + ∂K kukm+3,K . χ e K∈Th

e ∂K = inf{C11 (x) : x ∈ ∂K}, χ Here C e∂K = inf{χ(x) : x ∈ ∂K}. 11 Now we are ready to establish error estimates for our CDG method (2.4)-(2.5) (or equivalently, (3.2)). Lemma 3.2 For any (σ, u), (τ , v) ∈ Σ × V , we assume that for each K ∈ Th , (σ, u)|K ∈ H m+1 (K, S) × H m+3 (K) and (τ , v)|K ∈ H t+1 (K, S) × H t+3 (K). Then A(σ − P h σ, u − Ih u; τ − P h τ , v − Ih v) . KA (σ, u; τ , v). Proof. For simplicity, write ξσ := σ − P h σ, ξu := u − Ih u, ξτ := τ − P h τ , ξv := v − Ih v. Then A(σ − P h σ, u − Ih u; τ − P h τ , v − Ih v) = a(ξσ , ξτ ) + b(ξu , ξτ ) − b(ξv , ξσ ) + c(ξu , ξv ). We next bound each quantity on the right side of the above identity separately. By the Cauchy-

6 123

HUANG, HUANG: KIRCHHOFF PLATES

Schwarz inequality and Lemma 3.1, ¯ ¯ ¯ ¯ ¶ Z Z µ X ¯ ¯X ν 1 ¯ ¯ dx + C [ξ ] · [ξ ]ds ξσ : ξτ − trξ trξ |a(ξσ , ξτ )| = ¯ 22 σ τ σ τ ¯ 1 − ν2 e ¯ ¯K∈Th K 1 − ν e∈Ehi X p X p k C22 [ξσ ]k0,e k C22 [ξτ ]k0,e kξσ k0,K kξτ k0,K + . K∈Th

1 

 .

X

2

kξσ k20,K 



e∈Ehi

X

1

2

kξτ k20,K 

K∈Th

K∈Th

1  1 2 2 X p X p 2 2 k C22 [ξτ ]k0,e  + k C22 [ξσ ]k0,e   

e∈Ehi

e∈Ehi

.S1 (σ, u; τ , v) + S2 (σ, u; τ , v). By the Cauchy-Schwarz inequality and Lemma 3.1, ¯ ¯ ¯ ¯ Z Z ¯ ¯X 2 |b(ξu , ξτ )| = ¯¯ ∇ ξu : ξτ dx − J∇ξu K : {ξτ }ds¯¯ Eh ¯K∈Th K ¯  1  1 2 ¶ 2 X µ X ¡ ¢ 1 2 2 2 2 2 . k∇ ξu k0,K + k∇ξu k0,∂K   kξτ k0,K + hK kξτ k0,∂K  hK K∈Th

K∈Th

. S3 (σ, u; τ , v). Similarly, we can get |b(ξv , ξσ )| . S4 (σ, u; τ , v), 1  1 2 2 X X ∂K 2 ∂K 2     |c(ξu , ξv )| . C11 k∇ξu k0,∂K . S5 (σ, u; τ , v). C11 k∇ξv k0,∂K 

K∈Th

K∈Th

This proves the result for (σ, u) 6= (τ , v). If (σ, u) = (τ , v), it is routine that A(σ − P h σ, u − Ih u; τ − P h τ , v − Ih v) = a(ξσ , ξσ ) + c(ξu , ξu ), so we can derive the desired result readily via the estimates for a(ξσ , ξσ ) and c(ξu , ξu ) just obtained. Lemma 3.3 For any (σ, u) ∈ Σ × V, (τ , v) ∈ Σh × Vh , there holds A(τ , v; σ − P h σ, u − Ih u) . |(τ , v)|A |(σ − P h σ, u − Ih u)|B . Proof. Setting ξσ := σ − P h σ, ξu := u − Ih u, we have by a direct manipulation that |A(τ , v; ξσ , ξu )| ≤ |a(τ , ξσ )| + |b(v, ξσ )| + |b(ξu , τ )| + |c(v, ξu )| =: T1 + T2 + T3 + T4 . According to (2.3) and the Cauchy-Schwarz inequality, ÃZ T1 ≤

! 1 ÃZ 2

Ehi

C22 |[τ ]| ds

!1

2

2

2

Ehi

C22 |[ξσ ]| ds 7 124

≤ |(τ , v)|A |(ξσ , ξu )|B .

HUANG, HUANG: KIRCHHOFF PLATES

Since we have expressed as

R K

ξσ : ∇2 vdx = 0 from the inclusion property (2.3), the quantity T2 can be ¯Z ¯ ¯ ¯ T2 = ¯¯ J∇vK : {ξσ }ds¯¯ , Eh

from which and the Cauchy-Schwarz inequality we obtain µZ T2 ≤

2

Eh

C11 |J∇vK| ds

¶ 1 µZ 2 Eh

1 |{ξσ }|2 ds C11

¶1

2

≤ |(τ , v)|A |(ξσ , ξu )|B .

For bounding T3 , we have by integration by parts, (3.1), and the Cauchy-Schwarz inequality that ¯ ¯ ¯ ¯Z X Z ¯ ¯ ¯ T3 = ¯ {∇ξu } · [τ ]ds − ∇ξu · (∇ · τ )dx¯¯ ¯ ¯ Ehi K∈Th K ¯Z ¯ ÃZ ! 1 ÃZ !1 2 2 ¯ ¯ 1 ¯ ¯ χ|[τ ]|2 ds = ¯ {∇ξu } · [τ ]ds¯ ≤ |{∇ξu }|2 ds . ¯ Ei ¯ Ei Ei χ h

h

h

Moreover, it follows from the inverse inequality that Z Z Z 2 2 χ|[τ ]| ds ≤ C22 |[τ ]| ds + he |[τ ]|2 ds ≤ |(τ , v)|2A + kτ k20 . |(τ , v)|2A . Ehi

Ehi

Ehi

Thus, combining the last two estimates implies T3 . |(τ , v)|A |(ξσ , ξu )|B . Finally, we have ¯ ¯Z ¯ ¯ C11 J∇vK : J∇ξu Kds¯¯ T4 = ¯¯ Eh µZ ¶ 1 µZ 2 2 ≤ C11 |J∇vK| ds Eh

Eh

¶1 2

C11 |J∇ξu K| ds

2

≤ |(τ , v)|A |(ξσ , ξu )|B .

The theorem follows directly from the Cauchy-Schwarz inequality and the above estimates for Ti , 1 ≤ i ≤ 4. Lemma 3.4 Assume that for each K ∈ Th , (σ, u)|K ∈ H m+1 (K, S) × H m+3 (K). Then, for any (σ, u) ∈ Σ × V, (τ , v) ∈ Σh × Vh , there holds A(τ , v; σ − P h σ, u − Ih u) . |(τ , v)|A KB (σ, u). Proof. Use the error estimates of P h and Ih to get |(σ − P h σ, u − Ih u)|B . KB (σ, u), which together with Lemma 3.3 implies A(τ , v; σ − P h σ, u − Ih u) . |(τ , v)|A |(σ − P h σ, u − Ih u)|B . |(τ , v)|A KB (σ, u).

8 125

HUANG, HUANG: KIRCHHOFF PLATES

Lemma 3.5 Let (σ, u) ∈ Σ × V be the solution of (2.2) and (σ h , uh ) ∈ Σh × Vh be the solution of (3.2). If for any K ∈ Th , (σ, u)|K ∈ H m+1 (K, S) × H m+3 (K), then 1/2

|(σ − σ h , u − uh )|A . KA (σ, u; σ, u) + KB (σ, u).

(3.5)

Proof. From the triangle inequality and Lemma 3.2, it follows that |(σ − σ h , u − uh )|A ≤ |(σ − P h σ, u − Ih u)|A + |(P h σ − σ h , Ih u − uh )|A 1/2

. KA (σ, u; σ, u) + |(P h σ − σ h , Ih u − uh )|A .

(3.6)

By Galerkin orthogonality (3.4), the definition of A, and Lemma 3.4, we get |(P h σ − σ h , Ih u − uh )|2A = A(P h σ − σ h , Ih u − uh ; P h σ − σ h , Ih u − uh ) = A(P h σ − σ, Ih u − u; P h σ − σ h , Ih u − uh ) = A(σ h − P h σ, Ih u − uh ; σ − P h σ, Ih u − u) . |(P h σ − σ h , Ih u − uh )|A KB (σ, u). Then we obtain |(P h σ − σ h , Ih u − uh )|A . KB (σ, u).

(3.7)

Combining (3.6) and (3.7) implies (3.5). We assume that the stabilization coefficients C11 and C22 are defined as follows: C11 (x) = ζhαe ,

for x ∈ e and e ∈ Eh ,

C22 (x) = ηhβe ,

for x ∈ e and e ∈ Ehi ,

with ζ > 0, η ≥ 0, −1 ≤ α ≤ 0 ≤ β ≤ 1 independent of the mesh size. We introduce two notations as follows: ˆ ˆ µ? = max{−α, β}, µ? = min{−α, β}, where βˆ = 1 if η = 0; otherwise, βˆ = β. Theorem 3.1 Let (σ, u) ∈ Σ × V be the solution of (2.2) and (σ h , uh ) ∈ Σh × Vh be the solution of (3.2). If (σ, u) ∈ H m+1 (Ω, S) × H m+3 (Ω) with m ≥ 0, then |(σ − σ h , u − uh )|A . hmin{m+(1+µ? )/2,l+(1+µ? )/2,k−(1+µ

? )/2}

kukm+3 .

(3.8)

Proof. According to the definition of KA and the regularity assumption for the solution of (2.2), we have KA (σ, u; σ, u) = S1 (σ, u; σ, u) + S2 (σ, u; σ, u) + S5 (σ, u; σ, u) X 2 min{m,l}+2 X ∂K 2 min{m,l}+1 ≤ hK kσk2m+1,K + C22 hK kσk2m+1,K K∈Th

+ ³

X

K∈Th ∂K 2 min{m+1,k−1}+1 C11 hK kuk2m+3,K

K∈Th

´ ˆ . h2 min{m,l}+2 + h2 min{m,l}+1+β + h2 min{m+1,k−1}+1+α kuk2m+3 ˆ

ˆ

. hmin{2m+1+β,2l+1+β,2k−1+α} kuk2m+3 .

9 126

(3.9)

HUANG, HUANG: KIRCHHOFF PLATES

Similarly, we can obtain 2 KB (σ, u)

! ! 1 ∂K 2 = + C22 kσkm+1,K e ∂K C 11 K∈Th ¶ ¶ X µ 2 min{m+1,k−1}+1 µ 1 ∂K 2 + hK C11 + ∂K kukm+3,K χ e X

Ã

Ã

2 min{m,l}+1 hK

K∈Th

.h

min{2m+1+µ? ,2l+1+µ? ,2k−1−µ? }

kuk2m+3 .

(3.10)

Hence, (3.8) follows readily from Lemma 3.5 and (3.9)-(3.10). Theorem 3.2 Let Ω be a convex bounded polygonal domain. Let (σ, u) ∈ Σ × V be the solution of (2.2) and (σ h , uh ) ∈ Σh × Vh be the solution of (3.2). If (σ, u) ∈ H m+1 (Ω, S) × H m+3 (Ω) with m ≥ 0, then |u − uh |1 . hmin{m+(1+µ? )/2,l+(1+µ? )/2,k−(1+µ

? )/2}+min{(1+µ

? )/2,k−(1+µ

? )/2}

kukm+3 .

Proof. We proceed by the duality argument. Let (e σ, u e) be the solution of the following auxiliary problem:  1 ν e − 1−ν e )δ = K(e (trσ u) in Ω,  1−ν σ (3.11) e ) = 4(u − uh ) ∇ · (∇ · σ in Ω,  u e = ∂n u e=0 on ∂Ω. Since Ω is a convex polygonal domain, we have u e ∈ H 3 (Ω) ∩ H01 (Ω) with the bound (cf. [11,15]) ke uk3,Ω . k∆(u − uh )k−1,Ω .

(3.12)

Similar to (3.3), we can get after a direct manipulation that Z ∇(u − uh ) · ∇vdx, ∀(τ , v) ∈ Σh × Vh . A(e σ, u e; τ , v) = Ω

Now taking (τ , v) = (σ h − σ, u − uh ), we have by the definition of A and Galerkin orthogonality (3.4) that |u − uh |21 = A(e σ, u e; σ h − σ, u − uh ) = A(σ − σ h , u − uh ; −e σ, u e) e −σ e, u = A(σ − σ h , u − uh ; P h σ e − Ih u e) e −σ e, u = A(P h σ − σ h , Ih u − uh ; P h σ e − Ih u e) e −σ e, u + A(σ − P h σ, u − Ih u; P h σ e − Ih u e).

(3.13)

Since (P h σ − σ h , Ih u − uh ) ∈ Σh × Vh , it follows from Lemma 3.4 and inequality (3.7) that e −σ e, u A(P h σ − σ h , Ih u − uh ; P h σ e − Ih u e) .|(P h σ − σ h , Ih u − uh )|A KB (e σ, u e) .KB (σ, u)KB (e σ, u e).

(3.14)

And in view of Lemma 3.2, e −σ e, u e, u A(σ − P h σ, u − Ih u; P h σ e − Ih u e) . KA (σ, u; σ e).

(3.15)

The combination of (3.13)-(3.15) together gives e, u |u − uh |21 . KB (σ, u)KB (e σ, u e) + KA (σ, u; σ e). 10 127

(3.16)

HUANG, HUANG: KIRCHHOFF PLATES

Using (3.10) with m = 0 and regularity (3.12) we know KB (e σ, u e) . hmin{(1+µ? )/2,k−(1+µ

? )/2}

ke uk3 . hmin{(1+µ? )/2,k−(1+µ

? )/2}

|u − uh |1 .

(3.17)

And arguing as in the derivation of (3.9) yields ˆ

e, u KA (σ, u; σ e) . hmin{min{m,l}+1+β,k,min{m,l,k−1+α}+min{1,k−1}+1} kukm+3 |u − uh |1 .

(3.18)

Hence, we have by (3.16)-(3.18) and (3.10) that |u − uh |1 . hmin{m+(1+µ? )/2,l+(1+µ? )/2,k−(1+µ

? )/2}+min{(1+µ

? )/2,k−(1+µ

? )/2}

kukm+3 ,

from which the theorem follows. In the end of this section, we will show some theoretical results for the most remarkable cases in Theorems 3.1-3.2. For (σ, u) ∈ H m+1 (Ω, S) × H m+3 (Ω) with m ≥ 0, Table 1 and Table 2 list the convergence orders in h for k = l − 2 and k = l − 1, respectively. Table 1: Convergence orders in h of some remarkable cases for l = k − 2. C11 O(1/h) O(1/h) O(1) O(1)

C22 O(1) O(h), 0 O(1) O(h), 0

|(σ − σ h , u − uh )|A min{m, k − 2} + 1/2 min{m, k − 2} + 1 min{m, k − 2} + 1/2 min{m, k − 2} + 1/2

|u − uh |1 min{m, k − 2} + 1 min{m, k − 2} + 2 min{m, k − 2} + 1 min{m, k − 2} + 1

Table 2: Convergence orders in h of some remarkable cases for l = k − 1. C11 O(1/h) O(1/h) O(1) O(1)

4

C22 O(1) O(h), 0 O(1) O(h), 0

|(σ − σ h , u − uh )|A min{m + 1/2, k − 1} min{m + 1, k − 1} min{m, k − 1} + 1/2 min{m + 1/2, k − 1}

|u − uh |1 min{m + 1/2, k − 1} + min{1/2, k − 1} min{m + 1, k − 1} + min{1, k − 1} min{m, k − 1} + min{1, k} min{m + 1/2, k − 1} + min{1/2, k − 1}

Numerical results

In this section, we perform some numerical results to show the behavior and the accuracy of our method proposed here. Set Ω = (−1, 1) × (−1, 1), ν = 0.3, and f (x1 , x2 ) = 24(1 − x21 )2 + 24(1 − x22 )2 + 32(3x21 − 1)(3x22 − 1). We can check that the exact solution of (2.1) in this case is u(x1 , x2 ) = (1 − x21 )2 (1 − x22 )2 . We triangulate Ω into Th uniformly. For any K ∈ Th , we take S 1 (K) = P l (K, S) and S2 (K) = Pk (K) where l = k − 2, k − 1 with l ≥ 0. Set ζ = η = 1. The numerical results for α = −1 and β = 1 are given in Figure 1, from which we may observe that the convergence rates of |u − uh |1 11 128

HUANG, HUANG: KIRCHHOFF PLATES

and |(σ − σ h , u − uh )|A are O(hk−1+min{1,k−1} ) and O(hk−1 ), respectively, which coincide with the theoretical results in Theorems 3.1-3.2. Some numerical results for α = 0 and β = 0 are listed in Figure 2, which shows that the convergence rates of |u − uh |1 and |(σ − σ h , u − uh )|A are O(h) and O(h1/2 ), respectively, for l = 0 and k = 1 or k = 2, just as Theorems 3.1-3.2 indicate. When k = 2 and l = 1, the convergence rates of |u − uh |1 and |(σ − σ h , u − uh )|A are O(h2 ) and O(h3/2 ), respectively. These results again agree with our theoretical estimates. It is mentioned that the CDG method for k = 1 has positive convergence rate while the LCDG method exhibits no convergence (cf. [17]). The numerical results for α = −1 and β = 0 are shown in Figure 3. We can see that the convergence rates of |u − uh |1 and |(σ − σ h , u − uh )|A for k = 2, l = 0 and k = 1, l = 0 are consistent with the theoretical results. However, the convergence rates of |u − uh |1 and |(σ − σ h , u − uh )|A for k = 2, l = 1 are O(h2 ) and O(h3/2 ), 1/2 order higher than the convergence rates from our theoretical results.

ln |u − uh |1

2 0 −2 −4 −6

ln |(σ − σ h , u − uh )|A

−8 0.5

k = 2, l = 1 k = 2, l = 0 k = 1, l = 0 1

2 1 1.5

2

2.5

3

3.5

3

3.5

4 2 0 −2 −4 0.5

k = 2, l = 1 k = 2, l = 0 k = 1, l = 0 1

1 1 1.5

2

2.5

ln(1/h) Figure 1: The convergence rates of ln |u − uh |1 and ln |(σ − σ h , u − uh )|A in ln scale for different k and l when α = −1 and β = 1.

12 129

HUANG, HUANG: KIRCHHOFF PLATES

ln |u − uh |1

2 0

1

−4 −6 −8 0.5

ln |(σ − σ h , u − uh )|A

1

−2 k = 2, l = 1 k = 2, l = 0 k = 1, l = 0 1

2 1 1.5

2

2.5

3

3.5

4 2

0.5

0 −2 −4 0.5

1 k = 2, l = 1 k = 2, l = 0 k = 1, l = 0 1

1.5 1 1.5

2

2.5

3

3.5

ln(1/h) Figure 2: The convergence rates of ln |u − uh |h and ln |(σ − σ h , u − uh )|A in ln scale for different k and l when α = 0 and β = 0.

ln |u − uh |1

2 0

1

−4 −6 −8 0.5

ln |(σ − σ h , u − uh )|A

1

−2 k = 2, l = 1 k = 2, l = 0 k = 1, l = 0 1

2 1 1.5

2

2.5

3

3.5

3

3.5

4 2

0.5

0 −2 −4 0.5

1 k = 2, l = 1 k = 2, l = 0 k = 1, l = 0 1

1.5 1 1.5

2

2.5

ln(1/h) Figure 3: The convergence rates of ln |u − uh |h and ln |(σ − σ h , u − uh )|A in ln scale for different k and l when α = −1 and β = 0.

13 130

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References [1] I. Babuˇska, J. Osborn, and J. Pitk¨aranta. Analysis of mixed methods using mesh dependent norms. Math. Comp., 35(152):1039–1062, 1980. [2] I. Babuˇska and M. Zl´amal. Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal., 10:863–875, 1973. [3] G. A. Baker. Finite element methods for elliptic equations using nonconforming elements. Math. Comp., 31(137):45–59, 1977. [4] S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods (Third Edition). Springer, New York, 2008. [5] S. C. Brenner and L.-Y. Sung. C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput., 22/23:83–118, 2005. [6] P. Castillo, B. Cockburn, I. Perugia, and D. Sch¨otzau. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal., 38(5):1676– 1706, 2000. [7] P. G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. [8] B. Cockburn, B. Dong, and J. Guzm´an. A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. J. Sci. Comput., 40(1-3):141–187, 2009. [9] B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal., 47(2):1319–1365, 2009. [10] M. I. Comodi. The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing. Math. Comp., 52(185):17–29, 1989. [11] M. Dauge. Elliptic Boundary Value Problems on Corner Domains, volume 1341 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988. [12] G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Engrg., 191(34):3669–3750, 2002. [13] R. S. Falk and J. E. Osborn. Error estimates for mixed methods. RAIRO Anal. Num´er., 14(3):249–277, 1980. [14] K. Feng and Z.-C. Shi. Mathematical Theory of Elastic Structures. Springer-Verlag, Berlin, 1996. [15] P. Grisvard. Singularities in Boundary Value Problems. Masson, Paris, 1992. [16] T. Gudi, N. Nataraj, and A. K. Pani. Mixed discontinuous Galerkin finite element method for the biharmonic equation. J. Sci. Comput., 37(2):139–161, 2008. [17] J. Huang, X. Huang, and W. Han. A new C 0 discontinuous Galerkin method for Kirchhoff plates. Comput. Methods Appl. Mech. Engrg., 199(23-24):1446–1454, 2010.

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[18] I. Mozolevski and P. R. B¨osing. Sharp expressions for the stabilization parameters in symmetric interior-penalty discontinuous Galerkin finite element approximations of fourthorder elliptic problems. Comput. Methods Appl. Math., 7(4):365–375, 2007. [19] I. Mozolevski and E. S¨ uli. A priori error analysis for the hp-version of the discontinuous Galerkin finite element method for the biharmonic equation. Comput. Methods Appl. Math., 3(4):596–607, 2003. [20] I. Mozolevski, E. S¨ uli, and P. R. B¨osing. hp-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation. J. Sci. Comput., 30(3):465–491, 2007. [21] J. N. Reddy. Theory and Analysis of Elastic Plates and Shells (Second Edition). CRC Press, New York, 2006. [22] E. S¨ uli and I. Mozolevski. hp-version interior penalty DGFEMs for the biharmonic equation. Comput. Methods Appl. Mech. Engrg., 196(13-16):1851–1863, 2007. [23] G. N. Wells and N. T. Dung. A C 0 discontinuous Galerkin formulation for Kirchhoff plates. Comput. Methods Appl. Mech. Engrg., 196(35-36):3370–3380, 2007.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 133-141, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

The generalized LR-fuzzy numbers and its application in fully fuzzy linear systems† Zeng-tai Gonga,∗ , Kun Liua,b a College of Mathematics and Information Science, Northwest Normal University, Lanzhou Gansu,730070, P.R. China b College of Mathematics and Statistics, Longdong University, Qingyang Gansu,745000, P.R. China

Abstract: The fully fuzzy linear systems (shown as FFLS) plays an essential role in fuzzy modelling, which can formulate uncertainty in actual environment. Recently, Dehghan et al. [M. Dehghan, B. Hashemi, M. Ghatee, Computational methods for solving fully fuzzy linear systems, Applied Mathematics ˜x = ˜b, and Computation 179 (2006) 328-343.] gave the positive fuzzy solution of nonnegative FFLS A˜ ˜ ˜ where A and b are a fuzzy matrix and a fuzzy vector, respectively. When the positive fuzzy solution does not exist or the fuzzy solution has negative spreads, it has not been investigated any further in the existing literatures. In this paper, we extend the definition of LR-fuzzy numbers, which we call generalized LRfuzzy numbers, and develop its fuzzy arithmetic operations. Moreover, we study the fuzzy approximate solutions of FFLS based on GLR-fuzzy numbers, and analyze its solvability. An numerical example for calculating the fuzzy solution to FFLS is given. Keywords: LR-fuzzy numbers; Fuzzy matrix; Fully fuzzy linear systems (FFLS) AMS subject classifications. 08A72, 26E50, 03E72. 1 Introduction One field of applied mathematics that has many applications in various areas of science is solving a system of linear equations. Systems of simultaneous linear equations are very important topics, such as systems analysis, operations research, engineering and sciences. When the estimation of the system coefficients is imprecise and only some vague knowledge about the actual values of the parameters is available, it is necessary to set up a model whose data is only approximately known. Fuzzy set theory introduced in 1965 by Zadeh [1] is a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. The main directions of development have been diverse and its application to the very varied real problems includes fuzzy linear systems [2, 3, 4, 5, 6, 7, 8], fuzzy differential equation systems [9, 10, 11, 12], and so on. However, this uncertainty and vagueness or subjective information can be represented by means of fuzzy subsets of the real line, known as fuzzy numbers. Many authors have studied fuzzy numbers, their properties, arithmetic operations with fuzzy numbers, and other related fields [13, 14, 15]. A good overview of fuzzy numbers was prepared by Dubois et al. [16]. A specialized book on fuzzy arithmetic with fuzzy numbers was written by Kaufmann et al. [17]. At the same time, in order to use fuzzy numbers in any system and get fast computational formulas for the operations of fuzzy numbers, Jain had already attempted algebraic operations on fuzzy sets having supports in 1976. Jain’s method is inexact and rather impractical, since it often involves too many computations. Dubois and Prade then introduced the concept of LR-fuzzy numbers, and extended the usual algebraic operations on real numbers to fuzzy numbers by the use of a fuzzification principle in 1978 [18]. As can be seen from the approximate arithmetic operations on LR-fuzzy numbers, so-called LR-fuzzy numbers became quite popular, because of their good interpretability and relatively easy handing for fuzzy function. However, it is noteworthy that the left and right spreads of LR-fuzzy numbers defined by Dubois and Prade, are all positive. This restriction is that almost every existing method for solving FFLS † This work is supported by the National Natural Science Foundation of China (No. 71061013) and the Scientific Research Project of Northwest Normal University (No. NWNU-KJCXGC-03-61) of China. ∗ Corresponding author. E-mail: [email protected], [email protected](Z.T.Gong) and [email protected](K.Liu).

133

Zeng-tai Gong and Kun Liu: The generalized LR-fuzzy numbers and its application...

˜ right-hand side vector ˜b and unknown solution vector presumes the non negativity of coefficient matrix A, x ˜. Yet, it can appear in the process of practical calculation that the left or right spreads of unknown solution vector x ˜ may be negative, this is a contradiction with the definition of LR-fuzzy numbers. In this context, Dehghan et al. [5] called it as dummy solution of FFLS, which is not acceptable. Moreover, Dehghan et al. [4] also pointed out that modification or interpretation of the fuzzy vector with negative ˜x = ˜b, is of importance. However, it has not been investigated any further in spreads which satisfies A˜ the existing literatures. Therefore, in the present paper, we will focus on these studies employing the ˜x = ˜b throughout the method of Dehghan et al. [4], finding some fuzzy vector x ˜ which satisfies FFLS A˜ whole article, where A˜ and ˜b are an arbitrary fuzzy matrix and an arbitrary fuzzy vector, respectively. In this paper, firstly, considering that the possibility of the left or right spreads of LR-fuzzy numbers generated in the practical operation is negative, we propose the concept of generalized LR-fuzzy numbers (shown as GLR-fuzzy numbers) combined with the actual graphics to the left shape function L(·) or right shape function R(·) of LR-fuzzy numbers. We also define the approximate arithmetic operations on GLRfuzzy numbers. Secondly, we transform an n × n FFLS into a 3n × 3n crisp linear system of equations and obtain the fuzzy approximate solution by use of Dehghan’ method based on the approximate arithmetic operations on GLR-fuzzy numbers. Then, we analyze the solvability of FFLS in detail. The rest of this paper is organized as follows: In Section 2, we review some basic results on fuzzy numbers and arithmetic operations with LR-fuzzy numbers. In Section 3, we define the GLR-fuzzy numbers and investigate the fuzzy arithmetic operators with such quantities. In Section 4, we study the fuzzy approximate solutions of FFLS based on GLR-fuzzy numbers, and analyze its solvability. In Section 5, we present an example based on GLR-fuzzy numbers. A brief conclusion is outlined in Section 6. 2 Preliminaries Given a set X 6= φ, a fuzzy subset of X is a mapping u : X → [0, 1] and obviously any classical subset A of X can be considered as a fuzzy subset of X defined by µA (X) → [0, 1], µA (X) = 1, if x ∈ A, µA (X) = 0, if x ∈ x \ A (see [1]). Let us denote by RF the class of fuzzy subsets of real axis (i.e. u : R → [0, 1]) satisfying the following properties: (i) ∀u ∈ RF , u is normal, i.e. ∃x0 ∈ R with u(x0 ) = 1; (ii) ∀u ∈ RF , u is convex fuzzy set (i.e. u(tx + (1 − t)y ≥ min{u(x), u(y)}, ∀t ∈ [0, 1], x, y ∈ R); (iii) ∀u ∈ RF , u is upper semi-continuous on R; (iv) {x ∈ R : u(x) > 0} is compact, where A denotes the closure of A. Then RF is called the space of fuzzy real numbers [19]. According to Definition 2.7 of [4], a fuzzy number M is called positive (negative), denoted by M > 0 (M < 0), if its membership function µM (x) satisfies µM (x) = 0, ∀x < 0 (∀x > 0). Notice that a fuzzy number could be neither positive, nor negative. A special type of representation for fuzzy numbers which increases computational efficiency is LRfuzzy numbers [18, 20]. Definition 2.1. fuzzy number M is said to be an LR-fuzzy number if ½ Am−x L( α ), x ≤ m, α > 0, µM (x) = R( x−m β ), x ≥ m, β > 0, where m is the mean value of M , and α and β are left and right spreads, respectively, and a function L(·) the left shape function satisfying: (i) L(x) = L(−x); (ii) L(0) = 1 and L(1) = 0; (iii) L(x) is non-increasing on [0, ∞). Naturally, a right shape function R(·) is similarly defined as L(·). Using its mean value and left and right spreads, and shape functions, such an LR-fuzzy number M is symbolically written M = (m, α, β)LR . Remark 2.1. Every LR-fuzzy number is also a fuzzy number, where the support is included in [m − α, m + β]. Crisp real numbers can formally be represented as M = (m, 0, 0)LR . With this convention, we may allow the spreads to be 0. Remark 2.2. An LR-fuzzy number M = (m, α, β)LR is positive (negative), if and only if m − α > 0 (m + β < 0). 134

Zeng-tai Gong and Kun Liu: The generalized LR-fuzzy numbers and its application...

Remark 2.3. If L(x) and R(x) be linear functions, then the corresponding LR fuzzy number is said to be a triangular fuzzy number. Note that we use a fixed function L(·) and a fixed function R(·) for all fuzzy numbers in each problem. This metric is equivalent to the one by Puri and Ralescu [42] and Kaleva [43]. Definition 2.2. Two LR-fuzzy numbers M = (m, α, β)LR and N = (n, γ, δ)LR are said to be equal, if and only if m = n, α = γ and β = δ. Definition 2.3. An LR-fuzzy number M = (m, α, β)LR is said to be a subset of the LR-fuzzy number N = (n, γ, δ)LR , if and only if m − α ≥ n − γ and m + β ≤ n + δ. The fuzzy arithmetic operations of fuzzy numbers have grown in importance during recent years as a tool of advance in fuzzy optimization and control theory. Based on the extension principle, Dubois and Prade designed exact formulas for ⊕ and ª together with the approximate formulas for ⊗ and scalar multiplication [18, 20] to LR-fuzzy numbers, whose advantage comes from the fact that arithmetic operations with them can be performed in a relatively easy manner. For two LR-fuzzy numbers M = (m, α, β)LR and N = (n, γ, δ)LR the formula for ⊕ becomes: • Addition M ⊕ N = (m, α, β)LR ⊕ (n, γ, δ)LR = (m + n, α + γ, β + δ)LR . For an LR-fuzzy number M = (m, α, β)LR and a RL-fuzzy number N = (n, γ, δ)RL the formulas for opposite and ª become: • Opposite −M = −(m, α, β)LR = (−m, , β, α)RL . • Subtraction Let M = (m, α, β)LR and N = (n, γ, δ)RL be two LR and RL fuzzy numbers, respectively. M ª N = (m, α, β)LR ª (n, γ, δ)RL = (m − n, α + δ, β + γ)LR . When M < 0 and N > 0 or M < 0 and N < 0, it is easy to get similar formulas for ⊗. Thus, the formulas for ⊗ of two symmetric fuzzy numbers can be summarized as follows: • M ultiplication If M > 0 and N > 0 then M ⊗ N = (m, α, β)LR ⊗ (n, γ, δ)LR ∼ = (mn, mγ + nα, mδ + nβ)LR . If M < 0 and N > 0 then M ⊗ N = (m, α, β)RL ⊗ (n, γ, δ)LR ∼ = (mn, nα − mδ, nβ − mγ)RL . If M < 0 and N < 0 then M ⊗ N = (m, α, β)LR ⊗ (n, γ, δ)LR ∼ = (mn, −nβ − mδ, −nα − mγ)RL . • Scalar multiplication ( (λm, λα, λβ)LR , λ > 0, λ ⊗ M = λ ⊗ (m, α, β)LR ∼ = (λm, −λβ, −λα)RL , λ < 0. 3 GLR-fuzzy numbers and its fuzzy arithmetic operations As can be seen from the approximate arithmetic operations on LR-fuzzy numbers, so-called LR-fuzzy numbers became quite popular, because of their good interpretability and relatively easy handing for fuzzy function. However, it is noteworthy that the left and right spreads of LR-fuzzy numbers defined by Dubois and Prade, are all positive. Yet, it can appear that the left or right spreads of an LR-fuzzy number may be negative in the solution of the fuzzy model equations or a fuzzy system of linear equations, which have not be discussed in the existing literatures. And it is beyond the situation defined by Dubois and Prade [18, 20]. See the following examples. Example 3.1. Solve the following FFLS (See Test 3.2 in [4]): ½ (5, 1, 1) ⊗ (x1 , y1 , z1 ) ⊕ (6, 1, 2) ⊗ (x2 , y2 , z2 ) = (50, 10, 17), (7, 1, 0) ⊗ (x1 , y1 , z1 ) ⊕ (4, 0, 1) ⊗ (x2 , y2 , z2 ) = (48, 5, 7). · ¸ 1 , 0) (4, 11 The solution is x ˜= . 1 (5, 11 , 12 ) Clearly, the solution of FFLS consists of LR-fuzzy numbers. We change the value of a few elements of the right-hand side column vector of the above system as follows: Example 3.2. Solve the following FFLS: 135

Zeng-tai Gong and Kun Liu: The generalized LR-fuzzy numbers and its application...

½

(5, 1, 1) ⊗ (x1 , y1 , z1 ) ⊕ (6, 1, 2) ⊗ (x2 , y2 , z2 ) = (50, 8, 13), (7, 1, 0) ⊗ (x1 , y1 , z1 ) ⊕ (4, 0, 1) ⊗ (x2 , y2 , z2 ) = (48, 5, 4). · ¸ 5 (4, 11 , −5 ) 66 The solution is x ˜= . −1 (5, −6 11 , 11 ) One can see that the left and right spreads of the solution are negative. In view of this situation, Dehghan et al. [5] called it as dummy solution, which is not acceptable. They also pointed out that ˜x = ˜b, is of modification or interpretation of the fuzzy vector with negative spreads which satisfies A˜ importance. However, it has not been investigated any further in the existing literatures. We try to extend the definition of LR-fuzzy numbers and its fuzzy arithmetic operations in the following. 3.1 GLR-fuzzy numbers Definition 3.1. (GLR-fuzzy numbers)A fuzzy number M is said to be a GLR-fuzzy number combined with the actual graphics to the left shape function L(·) or right shape function R(·) of LR-fuzzy numbers if its membership satisfies one of the following conditions: ½ function m−x L( α ), x ≤ m, α > 0, (i) µM (x) = R( x−m β ), x ≥ m, β > 0, symbolically, we ½ write M = (m, α, β)GLR ; 0, x < m, α < 0, (ii) µM (x) = R( x−m ), x ≥ m, β > 0, max{−α,β} symbolically, we write M = (m, 0, max{−α, β})GLR ; ½ m−x L( max{α,−β} ), x ≤ m, α > 0, (iii) µM (x) = 0, x > m, β < 0, symbolically, we write M = (m, max{α, −β}, 0)GLR ; ½ L( m−x ), x < m, α < 0, −β (iv) µM (x) = x−m R( −α ), x ≥ m, β < 0, symbolically, we write M = (m, −β, −α)GLR , where m is the mean value of M , and α and β are left and right spreads, respectively, and L(·), R(·) : [0, ∞) −→ [0, 1] are two continuous, decreasing functions fulfilling: (i) L(0) = R(0) = 1; (ii) L(1) = R(1) = 0; (iii) L(x), R(x) are non-increasing on [0, ∞). 3.2 Fuzzy arithmetic operations of GLR-fuzzy numbers For two GLR-fuzzy numbers M = (m, α, β)GLR and N = (n, γ, δ)GLR , the formulas for the extended addition become: • Extended addition (i) If α > 0, β > 0, γ > 0 and δ > 0 then M ⊕ N = (m + n, α + γ, β + δ)GLR . (ii) If α < 0, β > 0, γ > 0 and δ > 0 then M ⊕ N = (m + n, γ, max{−α, β} + δ)GLR . (iii) If α > 0, β < 0, γ > 0 and δ > 0 then M ⊕ N = (m + n, max{α, −β} + γ, δ)GLR . (iv) If α < 0, β < 0, γ > 0 and δ > 0 then M ⊕ N = (m + n, γ − β, δ − α)GLR . For a GLR-fuzzy number M = (m, α, β)GLR and a GRL-fuzzy number N = (n, γ, δ)GRL , the formulas for ª become: • Extended subtraction (i) If α > 0, β > 0, γ > 0 and δ > 0 then M ª N = (m, α, β)GLR ª (n, γ, δ)GRL = (m − n, α + δ, β + γ)GLR . (ii) If α < 0, β > 0, γ > 0 and δ > 0 then M ª N = (m, α, β)GLR ª (n, γ, δ)GRL = (m − n, δ, max{−α, β} + γ)GLR . (iii) If α > 0, β < 0, γ > 0 and δ > 0 then M ª N = (m, α, β)GLR ª (n, γ, δ)GRL = (m − n, max{α, −β} + δ, γ)GLR . 136

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(iv) If α < 0, β < 0, γ > 0 and δ > 0 then M ª N = (m, α, β)GLR ª (n, γ, δ)GRL = (m − n, δ − β, γ − α)GLR . For two GLR-fuzzy numbers M = (m, α, β)GLR and N = (n, γ, δ)GLR , the formulas for the approximate multiplication ⊗ become: • Extended multiplication (i) If M > 0, N > 0, α > 0, β > 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GLR ⊗ (n, γ, δ)GLR ∼ = (mn, mγ + nα, mδ + nβ)GLR . (ii) If M > 0, N > 0, α < 0, β > 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GLR ⊗ (n, γ, δ)GLR ∼ = (mn, mγ, mδ + n · max{−α, β})GLR . (iii) If M > 0, N > 0, α > 0, β < 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GLR ⊗ (n, γ, δ)GLR ∼ = (mn, mγ + n · max{α, −β}, mδ)GLR . (iv) If M > 0, N > 0, α < 0, β < 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GLR ⊗ (n, γ, δ)GLR ∼ = (mn, mγ − nβ, mδ − nα)GLR . (v) If M < 0, N > 0, α > 0, β > 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GRL ⊗ (n, γ, δ)GLR ∼ = (mn, nα − mδ, nβ − mγ)GRL . (vi) If M < 0, N > 0, α < 0, β > 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GRL ⊗ (n, γ, δ)GLR ∼ = (mn, −mδ, n · max{−α, β} − mγ)GRL . (vii) If M < 0, N > 0, α > 0, β < 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GRL ⊗ (n, γ, δ)GLR ∼ = (mn, n · max{α, −β} − mδ, −mγ)GRL . (viii) If M < 0, N > 0, α < 0, β < 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GRL ⊗ (n, γ, δ)GLR ∼ = (mn, −mδ − nβ, −mγ − nα)GRL . (ix) If M < 0, N < 0, α > 0, β > 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GLR ⊗ (n, γ, δ)GLR ∼ = (mn, −nβ − mδ, −nα − mγ)GRL . (x) If M < 0, N < 0, α < 0, β > 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GLR ⊗ (n, γ, δ)GLR ∼ = (mn, −mδ − n · max{−α, β}, −mγ)GRL . (xi) If M < 0, N < 0, α > 0, β < 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GLR ⊗ (n, γ, δ)GLR ∼ = (mn, −mδ, −mγ − n · max{α, −β})GRL . (xii) If M < 0, N < 0, α < 0, β < 0, γ > 0 and δ > 0 then M ⊗ N = (m, α, β)GLR ⊗ (n, γ, δ)GLR ∼ = (mn, nα − mδ, nβ − mγ)GRL . • Extended calar multiplication (i) If α > 0 and ( β > 0 then (λm, λα, λβ)GLR , λ > 0, ∼ λ⊗M = (λm, −λβ, −λα)GRL , λ < 0. (ii) If α < 0 and ( β > 0 then (λm, 0, λ · max{−α, β})GLR , λ > 0, λ⊗M ∼ = (λm, −λ · max{−α, β}, 0)GRL , λ < 0. (iii) If α > 0 and ( β < 0 then (λm, λ · max{α, −β}, 0)GLR , λ > 0, λ⊗M ∼ = (λm, 0, −λ · max{α, −β})GRL , λ < 0. (iv) If α < 0 and ( β < 0 then (λm, −λβ, −λα)GLR , λ > 0, λ⊗M ∼ = (λm, λα, λβ)GRL , λ < 0. 4 The fuzzy approximate solution of FFLS (4.2) In this section, we discuss the fuzzy approximate solution of FFLS (4.2) employing Dehghan’s methed based on the above mentioned GLR-fuzzy numbers and its fuzzy arithmetic operations. Let us first recall some basic concepts to fuzzy matrix and the fully fuzzy linear system as follows: ˜ Definition 4.1.[20] A matrix A˜ = af ij is called a fuzzy matrix if each element of A is a fuzzy number. A˜ will be positive (negative) and denoted by A˜ > 0 (A˜ < 0) if each element of A˜ be positive (negative). Similarly nonnegative and nonpositive fuzzy matrices will be defined. We may represent m × n fuzzy matrix A˜ = (f aij )m×n , that af ij = (aij , αij , βij ), with new notation A˜ = (A, M, N ), where A = (aij ), M = (αij ) and N = (βij ) are three crisp m × n matrices with the same 137

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˜ size of A. ˜ = (bf Definition 4.2.[4] Let A˜ = (f aij ) and B ij ) be two m × n and n × p fuzzy matrices. Then, their ˜ ˜ = C˜ = (f approximate multiplication was defined as A ⊗ B cij ) which is m × p matrix, where PL f cf f ij = ik ⊗ bkj . k=1,··· ,n a Definition 4.3. Consider the m × n linear system of equations:    

(af f1 ) ⊕ (af f2 ) ⊕ · · · ⊕ (af fn ) = be1 , 11 ⊗ x 12 ⊗ x 1n ⊗ x (af f1 ) ⊕ (af f2 ) ⊕ · · · ⊕ (af fn ) = be2 , 21 ⊗ x 22 ⊗ x 2n ⊗ x ··························· (ag f1 ) ⊕ (ag f2 ) ⊕ · · · ⊕ (ag fn ) = bf m1 ⊗ x m2 ⊗ x mn ⊗ x m.

The matrix form of the above equations is ³ ´ ^ ^ . A ⊗ x ] = b , m×n n×1 m×1

  . 

(4.1)

(4.2)

˜x = ˜b, where the coefficient matrix A˜ = (f or simply A˜ aij ) = (aij , αij , βij ) = (A, M, N )(1 ≤ i ≤ m, 1 ≤ j ≤ n) is an m × n fuzzy matrix and x ˜ = (xei ) = (xi , yi , zi ) = (x, y, z)(1 ≤ i ≤ n), ˜b = (bej ) = (bj , gj , hj ) = (b, h, g) (1 ≤ j ≤ n) are two fuzzy vectors. This system is called a fully fuzzy linear system (show as FFLS (4.2)). Also, if each element of A˜ and ˜b is an nonnegative GLR-fuzzy number, then the equation (4.2) is called an nonnegative FFLS. Definition 4.4. The fuzzy vector x ˜ = (x, y, z) = ((x1 , y1 , z1 ), (x2 , y2 , z2 ), · · · , (xn , yn , zn ))T denotes a fuzzy approximate solution of FFLS (4.2), if and only if Am×n xn×1 = bm , Am×n yn×1 + Mm×n xn×1 = gm×1 , Am×n zn×1 + Nm×n xn×1 = hm×1 , where the left and right spreads yi , zi (1 ≤ i ≤ n) are arbitrary real number. Remark 4.1.[5] Let A˜ = (A, M, N ) ≥ 0, ˜b = (b, g, h) ≥ 0, x ˜ = (x, y, z) ≥ 0. Moreover, if x ≥ 0 and x − y ≥ 0, then x ˜ = (x, y, z) is said to be a consistent solution of nonnegative FFLS or forabbreviation, consistent solution. Otherwise, it will be called dummy solution. ˜ ˜b when x ≥ 0 and x − y ≥ 0, x Remark 4.2. Generally, for arbitrary A, ˜ is called an nonnegative fuzzy approximate solution of FFLS (4.2); when x ≤ 0 and x+z ≤ 0, x ˜ is called a nonpositive fuzzy approximate solution of FFLS (4.2); otherwise, x ˜ is called a fuzzy approximate solution of FFLS (4.2). Then, we give the existence condition of the fuzzy solution to n × n FFLS (4.2) by converting an n × n FFLS into a 3n × 3n crisp linear system of equations as follows: Theorem 4.1. Consider n × n FFLS (4.2). Let A˜ = (A, M, N ) and ˜b = (b, g, h) be a fuzzy matrix and a fuzzy vector consisting of GLR-fuzzy numbers, respectively. Then, the following conclusions satisfy: 1) If A is an nonsingular crisp matrix, so x ˜ = (A−1 b, A−1 (g − M A−1 b), A−1 (h − N A−1 b)) is a unique fuzzy approximate solution to FFLS (4.2). 2) If A−1 is an nonnegative inverse matrix, and ˜b is an nonnegative fuzzy vector, moreover, let g ≤ M A−1 b, h ≥ N A−1 b. −1 So, x ˜ = (A b, 0, max{−A−1 (g − M A−1 b), A−1 (h − N A−1 b)}) is an nonnegative fuzzy approximate solution to FFLS (4.2). 3) If A−1 is an nonnegative inverse matrix, and ˜b is an nonnegative fuzzy vector, moreover, let g ≤ M A−1 b, h ≤ N A−1 b, (N A−1 + I)b ≥ h. −1 So, x ˜ = (A b, − A−1 (h − N A−1 b), − A−1 (g − M A−1 b)) is an nonnegative fuzzy approximate solution to FFLS (4.2). 4) If A−1 is an nonpositive inverse matrix, and ˜b is an nonpositive fuzzy vector, moreover, let g ≤ M A−1 b, h ≤ N A−1 b, (M A−1 + I)b ≤ g. −1 So, x ˜ = (A b, A−1 (g − M A−1 b), A−1 (h − N A−1 b)) is an nonnegative fuzzy approximate solution to FFLS (4.2). 5) If A−1 is an nonpositive inverse matrix, and ˜b is an nonpositive fuzzy vector, moreover, let g ≥ M A−1 b, h ≤ N A−1 b. So, x ˜ = (A−1 b, max{−A−1 (g − M A−1 b), A−1 (h − N A−1 b)}, 0) is an nonnegative fuzzy approximate solution to FFLS (4.2). 138

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6) If A−1 is an nonpositive inverse matrix, and ˜b is an nonpositive fuzzy vector, moreover, let g ≥ M A−1 b, h ≥ N A−1 b, (N A−1 + I)b ≤ h. So, x ˜ = (A−1 b, − A−1 (h − N A−1 b), − A−1 (g − M A−1 b)) is an nonnegative fuzzy approximate solution to FFLS (4.2). Proof. 1) Since A is an nonsingular matrix, A−1 exists, and x = A−1 b is a solution of the equation Ax = b. Furthermore, y = A−1 (g − M A−1 b), z = A−1 (h − N A−1 b). That is, x ˜ = (A−1 b, A−1 (g − M A−1 b), A−1 (h − N A−1 b)) is a unique fuzzy approximate solution of FFLS (4.2). 2) Since A−1 is an nonnegative inverse matrix, and ˜b is an nonnegative fuzzy vector, x = A−1 b ≥ 0. On the other hand, since g ≤ M A−1 b, h ≥ N A−1 b, we have y = A−1 (g − M A−1 b) ≤ 0, z = A−1 (h − N A−1 b) ≥ 0. According to Definition 3.1 (ii), take y = 0. We have x − y = A−1 b − 0 = A−1 b ≥ 0. That is, x ˜ = (A−1 b, 0, max{−A−1 (g − M A−1 b), A−1 (h − N A−1 b)}) is an nonnegative fuzzy approximate solution of FFLS (4.2). 3) Since A−1 is an nonnegative inverse matrix, and ˜b is an nonnegative fuzzy vector, we have x = −1 A b ≥ 0. On the other hand, g ≤ M A−1 b, h ≤ N A−1 b, (N A−1 + I)b ≥ h, we have y = A−1 (g − M A−1 b) ≤ 0, z = A−1 (h − N A−1 b) ≤ 0. According to Definition 3.1 (iv), we may exchange the position of y and z, thus, we have x − z = A−1 b − A−1 (h − N A−1 b) = A−1 [(N A−1 + I)b − h] ≥ 0. That is, x ˜ = (A−1 b, − A−1 (h − N A−1 b), − A−1 (g − M A−1 b)) is an nonnegative fuzzy approximate solution of FFLS (4.2). The proof is completed, and the proofs of 4), 5), 6) are similar.¤ Corollary 4.1. Consider n × n FFLS (4.2). Let A˜ = (A, M, N ) and ˜b = (b, g, h) be a fuzzy matrix and a fuzzy vector consisting of GLR-fuzzy numbers, respectively. Then, the following conclusions satisfy: 1) If A−1 is an nonpositive inverse matrix, and ˜b is an nonnegative fuzzy vector, moreover, let g ≥ M A−1 b, h ≤ N A−1 b. So, x ˜ = (A−1 b, 0, max{−A−1 (g − M A−1 b), A−1 (h − N A−1 b)}) is an nonpositive fuzzy approximate solution to FFLS (4.2). 2) If A−1 is an nonpositive inverse matrix, and ˜b is an nonnegative fuzzy vector, moreover, let g ≤ M A−1 b, h ≤ N A−1 b, (I − N A−1 )b + h ≥ 0. −1 So, x ˜ = (A b, A−1 (h − N A−1 b), A−1 (g − M A−1 b)) is an nonpositive fuzzy approximate solution to FFLS (4.2). 3) If A−1 is an nonnegative inverse matrix, and ˜b is an nonpositive fuzzy vector, moreover, let g ≤ M A−1 b, h ≤ N A−1 b, (I − M A−1 )b + g ≤ 0. −1 So, x ˜ = (A b, −A−1 (h − N A−1 b), −A−1 (g − M A−1 b)) is an nonpositive fuzzy approximate solution to FFLS (4.2). 4) If A−1 is an nonnegative inverse matrix, and ˜b is an nonpositive fuzzy vector, moreover, let g ≥ M A−1 b, h ≤ N A−1 b. −1 So, x ˜ = (A b, max{A−1 (g − M A−1 b), −A−1 (h − N A−1 b)}, 0) is an nonpositive fuzzy approximate solution to FFLS (4.2). Proof. The proof is similar to Theorem 4.1, it is omitted here.¤ Theorem 4.2. Consider FFLS (4.2). 1) If A˜ is an nonsingular square fuzzy matrix, so, FFLS(4.2) has a unique fuzzy approximate solution. 2) If A˜ is an nonsquare fuzzy matrix and FFLS (4.2) is consistent, so, FFLS(4.2) has infinite fuzzy approximate solutions. 139

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3) If A˜ is an nonsquare fuzzy matrix and FFLS(4.2) is inconsistent, so, FFLS(4.2) has not fuzzy approximate solution. However, it has the least square fuzzy approximate solution and the minimum least square fuzzy approximate solution. Theorem 4.3. The FFLS (4.2) has a unique fuzzy approximate solution, if and only if the rank of fuzzy ˜ ˜b), i.e., matrix A˜ equals to that of fuzzy matrix (A, Rank(A) = Rank(A, b) = Rank(A, g − M x) = Rank(A, h − N x). ˜x = ˜b into a 3n×3n crisp linear system of equations Proof. The proof is obvious by converting an FFLS A˜ Ax = b, Ay + M x = g, Az + N x = h.¤ ˜x = ˜b exists fuzzy approximate solution if and only if the rank of Corollary 4.2. The FFLS (4.2) A˜ matrix A equals to that of fuzzy matrix (A, b). 5 Numerical example In this section, we solve a fully fuzzy linear system applying the GLR-fuzzy numbers and its fuzzy arithmetic operations. Example 5.1. Solve the following  FFLS:      (4, 3, 2) (5, 2, 1) (3, 0, 3) x ˜1 (−71, 54, 76)  (7, 4, 3) (10, 6, 5) (2, 1, 1)   x ˜2  =  (118, 115, 129)  . (6, 2, 2) (7, 1, 2) (15, 5, 4) x ˜3 (−155, 89, 151) Since ¯ ¯ ¯ ¯ ¯ 4 5 3 ¯ ¯ −71 5 3 ¯ ¯ ¯ ¯ ¯ det(A) = ¯¯ 7 10 2 ¯¯ = 46, det(A(1) ) = ¯¯ 118 10 2 ¯¯ = −12928, ¯ 6 7 15 ¯ ¯ −155 7 15 ¯ ¯ ¯ ¯ ¯ ¯ 4 −71 3 ¯ ¯ 4 5 −71 ¯ ¯ ¯ ¯ ¯ 2 ¯¯ = 9544, det(A(3) ) = ¯¯ 7 10 118 ¯¯ = 242, det(A(2) ) = ¯¯ 7 118 ¯ 6 −155 15 ¯ ¯ 6 7 −155 ¯ we may calculate 242 = −281.04348, x2 = 9544 x1 = −12928 46 46 = 207.47826, x3 = 46 = 5.26087. We have     482.17392 414.82609 h − M x =  −10.95651  , g − N x =  −70.52173  . 417.30435 277.08696 So, we have ¯ ¯ ¯ 482.17392 5 3 ¯ ¯ ¯ det(A0(1) ) = ¯¯ −10.95651 10 2 ¯¯ = 57821.2176, ¯ 417.30435 7 15 ¯ ¯ ¯ ¯ 4 482.17392 3 ¯ ¯ ¯ det(A0(2) ) = ¯¯ 7 −10.95651 2 ¯¯ = −39877.391, ¯ 6 417.30435 15 ¯ ¯ ¯ ¯ 4 5 482.17392 ¯ ¯ ¯ det(A0(3) ) = ¯¯ 7 10 −10.95651 ¯¯ = −3239.3044, ¯ 6 7 417.30435 ¯ we may calculate y1 = 1256.98299, y2 = −866.8998, y3 = −70.419661. Similarly, we obtain det(A00(1) ) = 54682.782, det(A00(2) ) = −37938.609, det(A00(3) ) = −3318.6956. So, we have z1 = 1188.7561, z2 = −824.75237, z3 = −72.145557. Thus,   (−281.04348, 1256.98299, 1188.7561) −866.8998, −824.75237)  . x ˜ =  (207.47826, (5.26087, −70.419661, −72.145557) According  to Definition 3.1, the fuzzy approximatesolution is (−281.04348, 1256.98299, 1188.7561) 824.75237, 866.8998)  . x ˜ =  (207.47826, (5.26087, 72.145557, 70.419661) 140

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6 Conclusion In this work, we proposed the generalized LR-fuzzy numbers and its fuzzy arithmetics operations. The fuzzy approximate solutions of FFLS with arbitrary coefficient matrix and arbitrary right-hand column vector are investigated using Dehghan’s method. We also analyzed the solvability of FFLS. It is obvious that employing GLR-fuzzy numbers can broaden the application domain of fuzzy linear equations in scientific applications.

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SEQUENCE SPACES DEFINED BY A MUSIELAK-ORLICZ FUNCTION IN 2-NORMED SPACES KULDIP RAJ, AJAY K. SHARMA AND SUNIL K. SHARMA

Abstract. In the present paper we introduce some sequence spaces defined by a Musielak-Orlicz function M = (Mk ) in 2-normed spaces. We study some topological properties and prove some inclusion relations between these spaces.

1. Introduction and Preliminaries The concept of 2-normed space was initially introduced by G¨ahler [2] as an interesting linear generalization of a normed linear space which was subsequently studied by many others see ([3], [16]) and references therein. Recently a lot of activities have started to study sumability and related topics in these linear spaces see ([4],[17]) and many others. Let X be a vector space of dimension d, where 2 ≤ d < ∞. A 2-norm on X is a function ||., .|| : X × X → R which satisfies the following : (1) ||x, y|| = 0 if and only if x and y are linearly dependent; (2) ||x, y|| = ||y, x||; (3) ||αx, y|| = |α|||x, y||, α ∈ R; (4) ||x, y + z|| ≤ ||x, y|| + ||x, z||, for all x, y, z ∈ X. The pair (X, ||., .||) is then called a 2-normed space see [3]. For example, we may take X = R2 equipped with Euclidean 2-norm as ||x, y||E = the area of the parallelogram spanned by the vectors x and y which may be given explicitly by the formula   x11 x12 ||x1 , x2 ||E = abs . x21 x22 Then, clearly (X, ||., .||E ) is a 2-normed space. Recall that (X, ||., .||) is a 2-Banach space if every cauchy sequence in X is convergent to some x in X. Let w be the set of all sequences of real or complex numbers and l∞ , c and c0 be the sequence spaces of bounded, convergent and null sequences x = (xk ), respectively. A sequence x ∈ l∞ is said to be almost convergent if all Banach limits of x coincide. Lorentz [7] proved that n n o 1X cˆ = x = (xk ) : lim xk+s exists, uniformly in s . n n k=1

Maddox ([8], [9]) has defined x to be strongly almost convergent to a number L if n

lim n

1X |xk+s − L| = 0, n

uniformly in s.

k=1

2000 Mathematics Subject Classification. 40A05,46A45. Key words and phrases. Paranorm space, Difference sequence space, Orlicz function, Musielak-Orlicz function, 2-normed space. 1

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KULDIP RAJ, AJAY K. SHARMA AND SUNIL K. SHARMA

Let p = (pk ) be a sequence of strictly positive real numbers. Nanda [12] has defined the following sequence spaces : n n o 1X |xk+s − L|pk = 0, uniformly in s , [ˆ c, p] = x = (xk ) : lim n n k=1

n n 1X |xk+s |pk = 0, [ˆ c, p]0 = x = (xk ) : lim n n

o uniformly in s

k=1

and

n n o 1X [ˆ c, p]∞ = x = (xk ) : sup |xk+s |pk < ∞ . s,n n k=1

Kizmaz [5] defined the sequence space n o X(∆) = x = (xk ) : (∆xk ) ∈ X for X = l∞ , c or c0 , where ∆x = (∆xk ) = (xk − xk+1 ) for all k ∈ N. Et and Colak [1] generalized the above sequence spaces to the sequence spaces n o X(∆m ) = x = (xk ) : (∆m xk ) ∈ X for X = l∞ , c or c0 , where m ∈ N, ∆0 x = (xk ), ∆x = (xk − xk+1 ), ∆m x = (∆m xk ) = (∆m−1 xk − ∆m−1 xk+1 ) f or all k ∈ N. The generalized difference has the following binomial representation,   m X m v m (−1) ∆ xk = xk+v v v=0

for all k ∈ N. An orlicz function M : [0, ∞) → [0, ∞) is a continuous, non-decreasing and convex function such that M (0) = 0, M (x) > 0 for x > 0 and M (x) −→ ∞ as x −→ ∞. Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x = (xk ), then ∞ o n  |x |  X k 0 : M ≤1 . ρ k=1

Also, it was shown in [6] that every Orlicz sequence space lM contains a subspace isomorphic to lp (p ≥ 1). The ∆2 − condition is equivalent to M (Lx) ≤ LM (x), for all L with 0 < L < 1. An Orlicz function M can always be represented in the following integral form Z x M (x) = η(t)dt 0

where η is known as the kernel of M , is right differentiable for t ≥ 0, η(0) = 0, η(t) > 0, η is non-decreasing and η(t) → ∞ as t → ∞. A sequence M = (Mk ) of Orlicz functions is called a Musielak-Orlicz function see ([10],[11]). A sequence N = (Nk ) of Orlicz functions defined by Nk (v) = sup{|v|u − Mk : u ≥ 0}, k = 1, 2, ...

143

SEQUENCE SPACES

3

is called the complementary function of the Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM and its subspace hM are defined as follows n o tM = x ∈ w : IM (cx) < ∞, for some c > 0 , n o hM = x ∈ w : IM (cx) < ∞, for all c > 0 , where IM is a convex modular defined by ∞ X IM (x) = Mk (xk ), x = (xk ) ∈ tM . k=1

We consider tM equipped with the Luxemburg norm o n x ≤1 ||x|| = inf k > 0 : IM k or equipped with the Orlicz norm  o n1 1 + IM (kx) : k > 0 . ||x||0 = inf k Let X be a linear metric space. A function p : X → R is called paranorm, if (1) p(x) ≥ 0, for all x ∈ X, (2) p(−x) = p(x), for all x ∈ X, (3) p(x + y) ≤ p(x) + p(y), for all x, y ∈ X, (4) if (λn ) is a sequence of scalars with λn → λ as n → ∞ and (xn ) is a sequence of vectors with p(xn − x) → 0 as n → ∞, then p(λn xn − λx) → 0 as n → ∞. A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and the pair (X, p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [18], Theorem 10.4.2, P-183). For more details about sequence spaces see ([13], [14], [15]) and references therein. Let M = (Mk ) be a Musielak-Orlicz function, (X, ||., .||) be a 2-normed space and p = (pk ) be a bounded sequence of positive real numbers. By S(2 − X) we denote the space of all sequences defined over (X, ||., .||). In the present paper we define the following sequence spaces : n h i n ipk 1 X h  ∆m xk+s − L cˆ, M, p, ||., .|| (∆m ) = x = (xk ) ∈ S(2−X) : lim Mk || , z|| = 0, n n ρ k=1 o uniformly in s for some ρ > 0 and L > 0 , h

n i ipk n 1 X h  ∆m xk+s Mk || , z|| = 0, cˆ, M, p, ||., .|| (∆m ) = x = (xk ) ∈ S(2 − X) : lim n n ρ 0 k=1 o uniformly in s, for some ρ > 0 ,

and n h i n ipk 1 X h  ∆m xk+s cˆ, M, p, ||., .|| (∆m ) = x = (xk ) ∈ S(2 − X) : sup Mk || , z|| < ∞, ρ ∞ s,n n k=1 o for some ρ > 0 . h i h i When M(x) = x for all k, the spaces cˆ, M, p, ||., .|| (∆m ), cˆ, M, p, ||., .|| (∆m ) and 0 h i h i h i h i cˆ, M, p, ||., .|| (∆m ) reduces to cˆ, p, ||., .|| (∆m ), cˆ, p, ||., .|| (∆m ) and cˆ, p, ||., .|| (∆m ), ∞

0

144

∞

4

KULDIP RAJ, AJAY K. SHARMA AND SUNIL K. SHARMA

respectively. h i h i If p = (pk ) = 1 for all k, the spaces cˆ, M, p, ||., .|| (∆m ), cˆ, M, p, ||., .|| (∆m ) and h i h i h i h0 i cˆ, M, p, ||., .|| (∆m ) reduces to cˆ, M, ||., .|| (∆m ), cˆ, M, ||., .|| (∆m ) and cˆ, M, ||., .|| (∆m ), ∞ 0 ∞ respectively. The following inequality will be used throughout the paper. If 0 ≤ pk ≤ sup pk = H, K = max(1, 2H−1 ) then |ak + bk |pk ≤ K{|ak |pk + |bk |pk }

(1.1)

for all k and ak , bk ∈ C. Also |a|pk ≤ max(1, |a|H ) for all a ∈ C. The main aim of this paper is to study some sequence spaces by using difference operator and a Musielak-Orlicz function. We also make an effort to study some topological properties and inclusion relations between these sequence spaces. 2. Main Results Theorem 2.1 If M = (Mk ) be a Musielak-Orlicz function iand p =h(pk ) be a bounded h i sequence of positive real numbers, then the spaces cˆ, M, p, ||., .|| (∆m ), cˆ, M, p, ||., .|| (∆m ) ∞ h i and cˆ, M, p, ||., .|| (∆m ) are linear spaces over the field of complex number C. 0 h i Proof. Let x = (xk ), y = (yk ) ∈ cˆ, M, p, ||., .|| (∆m ) and α, β be any scalars. Then 0 there exist positive numbers ρ1 and ρ2 such that n ipk 1 X h  ∆nm xk+s Mk || , z|| = 0, uniformly in s lim n n ρ1 k=1

and lim n

n ipk 1 X h  ∆nm xk+s Mk || , z|| = 0, uniformly in s. n ρ2 k=1

Let ρ3 = max(2|α|ρ1 , 2|β|ρ2 ). Since M = (Mk ) is non-decreasing convex function, by using inequality (1.1), we have n h  ∆m (αx ipk X k+s + βyk+s ) 1 M || , z|| k n ρ3 k=1

≤ ≤

n ipk 1 X h  ∆m αxk+s ∆m βyk+s Mk || , z|| + || , z|| n ρ3 ρ3

1 n

k=1 n h X

 ∆m x ipk ∆m yk+s k+s Mk || , z|| + || , z|| ρ1 ρ2

k=1 n h X

1 ≤ K n

k=1

n  ∆m x ipk ipk 1 X h  ∆m xk+s k+s Mk || , z|| +K Mk || , z|| ρ1 n ρ2 k=1

→ 0 as n → ∞, uniformly in s. h i So that αx + βy ∈ cˆ, M, p, ||., .|| (∆m ). This completes the proof. Similarly, we can 0 h h i i prove that cˆ, M, p, ||., .|| (∆m ) and cˆ, M, p, ||., .|| (∆m ) are linear spaces. ∞

145

SEQUENCE SPACES

5

Theorem 2.2 Let M = (Mk ) be Musielak-Orlicz h ifunction, p = (pk ) be a bounded sequence of positive real numbers. Then cˆ, M, p, ||., .|| (∆m ) is a paranormed space with 0 respect to the paranorm defined by n h  ∆m x n pn  1 X ipk  H1 o k+s Mk || g(x) = inf ρ H : , z|| ≤1 , n ρ k=1

where H = max(1, supk pk < ∞). h i Proof. Clearly g(x) ≥ 0 for x = (xk ) ∈ cˆ, M, p, ||., .|| (∆m ). Since Mk (0) = 0, we 0

get g(0) = 0. Conversely, suppose that g(x) = 0, then n h ipk  H1 o n pn  1 X  ∆m x k+s , z|| ≤ 1 = 0. inf ρ H : Mk || n ρ k=1

This implies that for a given  > 0, there exists some ρ (0 < ρ < ) such that n h 1 X

n

k=1

 ∆m x ipk  H1 k+s Mk || , z|| ≤ 1. ρ

Thus n h n h  ∆m x ipk  H1  1 X  ∆m x ipk  H1 1 X k+s k+s Mk || , z|| Mk || ≤ ≤ 1, , z|| n  n ρ k=1

k=1

for each n. Suppose that xk 6= 0 for each k ∈ N . This implies that ∆m xk+s 6= 0, for each m k, s ∈ N . Let  → 0, then || ∆ x k+s , z|| → ∞. It follows that n h 1 X

n

k=1

ipk  H1  ∆m x k+s Mk || , z|| →∞ 

which is a contradiction. Therefore, ∆m xk+s = 0 for each k and thus xk = 0 for each k ∈ N . Let ρ1 > 0 and ρ2 > 0 be such that n h 1 X

n

k=1

 ∆m x ipk  H1 k+s Mk || , z|| ≤1 ρ1

and n h 1 X

n

k=1

 ∆m x ipk  H1 k+s Mk || , z|| ≤1 ρ2

146

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KULDIP RAJ, AJAY K. SHARMA AND SUNIL K. SHARMA

for each n. Let ρ = ρ1 + ρ2 . Then by using Minkoski’s inequality, we have n h  X ipk  H1  ∆m (x k+s + yk+s ) 1 M || , z|| k n ρ k=1

≤

n h 1 X  ∆m x ipk  H1 m k+s + ∆ yk+s Mk || , z|| n ρ1 + ρ2 k=1

≤

n h 1 X

n

≤



+



k=1

 ∆m x   ∆m y ipk  H1 ρ1 ρ2 k+s k+s Mk || , z|| + Mk || , z|| ρ1 + ρ2 ρ1 ρ1 + ρ2 ρ2

n ipk  H1 ρ1  1 X h  ∆m xk+s Mk || , z|| ρ1 + ρ2 n ρ1 k=1

n ipk  H1 ρ2  1 X h  ∆m yk+s Mk || , z|| ρ1 + ρ2 n ρ2 k=1

≤ 1. Since ρ’s are non-negative, so we have n h ipk  H1 o n pn  1 X  ∆m x m k+s + ∆ yk+s H , z|| Mk || ≤1 , g(x + y) = inf ρ : n ρ k=1

n h n pn  1 X o  ∆m x ipk  H1 k+s ≤ inf ρ1H : ≤1 Mk || , z|| n ρ1 k=1

n h n pn  1 X o ipk  H1  ∆m y k+s +inf ρ2H : ≤1 . Mk || , z|| n ρ2 k=1

Therefore, g(x + y) ≤ g(x) + g(y). Finally, we prove that the scalar multiplication is continuous. Let λ be any complex number. By definition, n h n pn  1 X ipk  H1 o  ∆m λx k+s g(λx) = inf ρ H : , z|| ≤1 . Mk || n ρ k=1

Then

n h n 1 X ipk  H1 o  ∆m x pn k+s g(λx) = inf (|λ|t) H : , z|| Mk || ≤1 . n t

where t =

ρ |λ| .

Since |λ|

pn

≤ max(1, |λ|

sup pn

g(λx) ≤ max(1, |λ|

k=1 sup pn

), we have

n h ipk  H1 n pn  1 X  ∆m x o k+s H ) inf t : Mk || , z|| ≤1 . n t k=1

So, the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof of the theorem. Theorem 2.3 Let M = (Mk ) be Musielak-Orlicz function. Then the following statements : h h are equivalent i i m (i) cˆ, p, ||., .|| (∆ ) ⊆ cˆ, M, p, ||., .|| (∆m ); h i∞ h i∞ (ii) cˆ, p, ||., .|| (∆m ) ⊆ cˆ, M, p, ||., .|| (∆m ); 0

∞

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SEQUENCE SPACES

7

n

m 1X [Mk (t)]pk < ∞, where t = || ∆ xρ k+s , z|| > 0. n n k=1 h i h i Proof. (i) =⇒ (ii) is obvious. Since cˆ, p, ||., .|| (∆m ) ⊆ cˆ, p, ||., .|| (∆m ). 0 ∞ h i h i (ii) =⇒ (iii). Suppose cˆ, p, ||., .|| (∆m ) ⊆ cˆ, M, p, ||., .|| (∆m ) and let (iii) does not 0 ∞ hold. Then for some t > 0 n 1X sup [Mk (t)]pk = ∞, n n

(iii) sup

k=1

and therefore there is a sequence (ni ) of positive integers such that ni 1 X [Mk (i−1 )]pk > i, ni

(2.1)

i = 1, 2, ...

k=1

Define x = (xk ) by n i−1 , 1 ≤ k ≤ n , i = 1, 2, ... i 0, k ≥ ni . h i h i Then x ∈ cˆ, p, ||., .|| (∆m ) but x 6∈ cˆ, M, p, ||., .|| (∆m ) which contradicts (ii). Hence xk =

∞

0

(iii) must hold. h i h i (iii) =⇒ (i). Suppose x ∈ cˆ, p, ||., .|| (∆m ) and x 6∈ cˆ, M, p, ||., .|| (∆m ). Then ∞

(2.2)

sup n m

Let t = || ∆

xk+s , z|| ρ

∞

n ipk 1 X h  ∆m xk+s Mk || , z|| = ∞. n ρ k=1

for each k and fixed s, then from eqn.(2.2), we have n

1X [Mk (t)]pk = ∞, n

sup n

k=1

which contradicts (iii). Hence (i) must hold. Theorem 2.4 Let 1 ≤ pk ≤ sup pk < ∞. Then the following statements are equivak

lenth: i h i (i) cˆ, M, p, ||., .|| (∆m ) ⊆ cˆ, p, ||., .|| (∆m ); h i0 h i0 (ii) cˆ, M, p, ||., .|| (∆m ) ⊆ cˆ, p, ||., .|| (∆m ); n

∞

0

1X [Mk (t)]pk > 0, (iii) inf n n

t > 0.

k=1

Proof. (i) =⇒ (ii) is obvious. h i h i (ii) =⇒ (iii) Suppose cˆ, M, p, ||., .|| (∆m ) ⊆ cˆ, p, ||., .|| (∆m ) and let (iii) does not 0 ∞ hold. Then n

(2.3)

inf n

1X [Mk (t)]pk = 0, n k=1

148

t > 0.

8

KULDIP RAJ, AJAY K. SHARMA AND SUNIL K. SHARMA

We can choose an index sequence (ni ) such that ni 1 X [Mk (i)]pk < i−1 , ni

i = 1, 2, ...

k=1

Define the sequence x = (xk ) by n i, 1 ≤ k ≤ n , i = 1, 2, ... i xk = 0, k ≥ ni . h i h i Thus by eqn.(2.3), x ∈ cˆ, M, p, ||., .|| (∆m ) but x 6∈ cˆ, p, ||., .|| (∆m ) which contra∞

0

dicts (ii). Hence (iii) must hold. h i (iii) =⇒ (i) Let x ∈ cˆ, M, p, ||., .|| (∆m ). That is, 0 n ipk 1 X h  ∆m xk+s (2.4) lim Mk || , z|| = 0, uniformly in s. n n ρ k=1 h i Suppose (iii) hold and x 6∈ cˆ, p, ||., .|| (∆m ). Then for some number 0 > 0 and index m

n0 , we have || ∆

xk+s , z|| ρ

0

≥ 0 , for some s > s0 and 1 ≤ k ≤ n0 . Therefore h  ∆m x ipk k+s [Mk (0 )]pk ≤ Mk || , z|| ρ

and consequently by eqn.(2.4) n

1X [Mk (0 )]pk = 0, n n k=1 h i h i which contradicts (iii). Hence cˆ, M, p, ||., .|| (∆m ) ⊆ cˆ, p, ||., .|| (∆m ). lim

0

0

Theorem 2.5 Let M = (Mk ) be Musielak-Orlicz function. Let 1 ≤ pk ≤ sup pk < ∞. k

Then h

i h i cˆ, M, p, ||., .|| (∆m ) ⊆ cˆ, p, ||., .|| (∆m ) ∞

0

hold if n

(2.5)

1X [Mk (t)]pk = ∞, t > 0. lim n n k=1

Proof.

h i h i Suppose cˆ, M, p, ||., .|| (∆m ) ⊆ cˆ, p, ||., .|| (∆m ) and let eqn.(2.5) does ∞

0

not hold. Therefore there is a number t0 > 0 and an index sequence (ni ) such that (2.6)

ni 1 X [Mk (t0 )]pk ≤ N < ∞, i = 1, 2, ... ni k=1

Define the sequence x = (xk ) by n t , 1 ≤ k ≤ n , i = 1, 2, ... 0 i xk = 0, k ≥ ni .

149

SEQUENCE SPACES

9

h i h i Clearly, x ∈ cˆ, M, p, ||., .|| (∆m ) but x 6∈ cˆ, p, ||., .|| (∆m ). Hence eqn.(2.5) must hold. ∞ 0 h i Conversely, if x ∈ cˆ, M, p, ||., .|| (∆m ), then for each s and n ∞

1 n

(2.7)

n h X

ipk  ∆m x k+s Mk || , z|| ≤ N < ∞. ρ

k=1

h

i Suppose that x 6∈ cˆ, p, ||., .|| (∆m ). Then for some number 0 > 0 there is a number s0 0

||

∆m xk+s , z|| ≥ 0 , ρ

f or s ≥ so .

Therefore

h  ∆m x ipk k+s [Mk (0 )]pk ≤ Mk || , z|| , ρ and hence for each k and s we get n

1X [Mk (o )]pk ≤ N < ∞, n k=1

for some N > 0, which contradicts eqn. (2.5). Hence h i h i cˆ, M, p, ||., .|| (∆m ) ⊆ cˆ, p, ||., .|| (∆m ). ∞

0

This completes the proof. Theorem 2.6 Suppose M = (Mk ) be Musielak-Orlicz function and let 1 ≤ pk ≤ sup pk < k

∞. Then h

i h i cˆ, p, ||., .|| (∆m ) ⊆ cˆ, M, p, ||., .|| (∆m ) ∞

0

hold if n

(2.8)

lim n

1X [Mk (t)]pk = 0, t > 0. n k=1

h i h i Proof. let cˆ, p, ||., .|| (∆m ) ⊆ cˆ, M, p, ||., .|| (∆m ). Suppose that eqn.(2.8) does ∞ 0 not hold. Then for some t0 > 0, n

(2.9)

1X [Mk (t)]pk = L 6= 0. lim n n k=1

Define x = (xk ) by k−m X



 m+k−v−1 k−v v=0 h i h i for k = 1, 2, .... Then x 6∈ cˆ, M, p, ||., .|| (∆m ) but x ∈ cˆ, p, ||., .|| (∆m ). Hence eqn. xk = t

(−1)m

∞

0

(2.8) must hold. h i Conversely,let x ∈ cˆ, p, ||., .|| (∆m ). Then for every k and s, we have ∞

||

∆m xk+s , z|| ≤ N < ∞. ρ

150

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KULDIP RAJ, AJAY K. SHARMA AND SUNIL K. SHARMA

Therefore

h ipk ∆m xk+s Mk || , z|| ≤ [Mk (N )]pk ρ

and

n n ipk 1 X h  ∆m xk+s 1X Mk || , z|| [Mk (N )]pk = 0. ≤ lim n n n n ρ k=1 k=1 h i m Hence x ∈ cˆ, M, p, ||., .|| (∆ ). This completes the proof.

lim

0

References [1] M. Et and R. Colak, On some generalized difference sequence spaces, Soochow J. Math., 21 (4)(1995), 377-386. [2] S. G¨ ahler, 2- metrische Raume und ihre topologishe Struktur , Math. Nachr., 26 (1963), 115-148. [3] H. Gunawan and Mashadi, On finite dimensional 2-normed spaces, Soochow J. Math., 27(3) (2001), 321-329. [4] M. Gurdal and S. Pehlivan, Statistical convergence in 2-normed spaces, Southeast Asian Bull. Math., 33 (2) (2009), 257-264. [5] H. Kizmaz, On certain sequence spaces, Cand. Math. Bull., 24 (2)(1981), 169-176. [6] J. Lindenstrauss and L. Tzafriri, On Orlicz seequence spaces, Israel J. Math., 10, (1971), 379-390 . [7] G. G. Lorentz, A contribytion to the theory of divergent series, Act. Math., 80 (1948), 167-190. [8] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. 18 (1967), 345-355. [9] I. J. Maddox, A new type of convergence, Math. Proc. Camb. Phil. Soc., 83 (1978), 61-64. [10] L. Maligranda, Orlicz spaces and interpolation, Seminars in Mathematics 5, Polish Academy of Science, 1989. [11] J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 1034,(1983). [12] S. Nanda, Strongly almost convergent sequences, Bull. Call. Math. Soc., 76 (1984), 236-240. [13] K. Raj, A. K. Sharma and S. K. Sharma, A Sequence space defined by Musielak-Orlicz functions, Int. Journal of Pure and Appl. Mathematics, Vol.67 (2011), 475-484 . [14] K. Raj, S. K. Sharma and A. K. Sharma, Difference sequence spaces in n-normed spaces defined by Musielak-Orlicz functions, Armenian journal of Mathematics, 3 (2010), pp. 127-141. [15] K. Raj and S. K. Sharma, Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz functions, Acta Univ. Sapientiae Mathematica 3 (2011), pp. 97-109. [16] W. Raymond, Y. Freese and J. Cho, Geometry of linear 2-normed spaces, N. Y. Nova Science Publishers, Huntington, 2001. [17] A. Sahiner, M. Gurdal, S. Saltan and H. Gunawan, Ideal Convergence in 2-normed spaces, Taiwanese J. Math., 11(5) 2007, 1477-1484. [18] A. Wilansky, summability through Functional Analysis, North- Holland Math. stud. (1984). School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J&K, India E-mail address: [email protected] E-mail address: aksju [email protected] E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 152-162, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

ON THE EXISTENCE AND UNIQUENESS OF SOLUTION OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS R. DARZI1, B. MOHAMMADZADEH2, A. NEAMATY3, D. BALEANU4,5,6 1

Department of Mathematics, Neka Branch, Islamic Azad University, P.O.Box 48411-86114, Neka, Iran, E-mail:[email protected]

2

Department of Mathematics, Neka Branch, Islamic Azad University, P.O.Box 48161-19318, Sari, Iran, E-mail:[email protected]

3

Department of Mathematics, University of Mazandaran, P.O.Box 47416-95447, Babolsar, Iran, E-mail: [email protected]

4

5

Department of Mathematics and Computer Science, Faculty of Art and Sciences, Cankaya University, 06810, Yenimahalle, Ankara, Turkey, E-mail: [email protected]

Department of Chemical and Materials Engineering, Faculty of Engineering King Abdulaziz University, P.O. Box: 80204, Jeddah, 21589, Saudi Arabia 6

Institute of Space Sciences, Magurele-Bucharest, Romania

Abstract. In this paper, we investigate the existence and uniqueness of solution for fractional boundary value problem for nonlinear fractional differential equation ( )

(

( ))

with the integral boundary conditions

{

( )

( )

∫

(

( ))

( )

( )

∫

(

( ))

( )

( )

where denotes Caputo derivative of order by using the fixed point theory. We apply the contraction mapping principle and Krasnoselskii’s fixed point theorem to obtain some new existence and uniqueness results. Two examples are given to illustrate the main results. Key words: Fractional boundary value problem; Integral boundary conditions; Fixed point theory.

152

DARZI ET AL: FRACTIONAL DIFFERENTIAL EQUATIONS

1. INTRODUCTION In recent years, fractional calculus is one of the interest issues that attract many scientists, specially mathematics and engineering sciences. Many natural phenomena can be present by boundary value problems of fractional differential equations. Many authors in different fields such as chemical physics, fluid flows, electrical networks, viscoelasticity, try to modeling of these phenomena by boundary value problems of fractional differential equations [1-11]. For achieve extra information in fractional calculus, specially boundary value problems, reader can refer to more valuable papers or books that are written by authors [5-22]. In boundary value problems, one of the most important factors that cause to write different papers is the variety of boundary condition selection. One of these situations is integral boundary conditions. Integral boundary conditions have various applications in applied fields such as underground water flow, population dynamics, blood flow problems, thermoelasticity, etc. (for more details see Refs. [23-28] and references therein). These conditions have different aspects, for example, two, three, multi-point and nonlocal integral boundary conditions. There are some papers dealing with existence and uniqueness solution of boundary value problems of fractional order with integral boundary conditions see [29-31]. In this paper, we obtain new existence and uniqueness results for boundary value problems of fractional differential equations. Indeed, we study the existence and uniqueness of solution for fractional boundary value problem for nonlinear fractional differential equation ( )

(

( ))

(1.1)

with the integral boundary conditions ( )

( )

∫

(

( ))

{ ( )

( )

( ∫ ( )

( ))

( )

(

)

where denotes Caputo derivative of order is continuously differentiable function, ( ) and with Constructing a special bounded, convex and closed subset on Banach space ( ) and employing principle contraction mapping and Banach fixed point, we prove the uniqueness of the solution. Then, by defining the operators and on , which will be introduced later, and using Krasnoselskii’s fixed point theorem [32], we show that the problem has at least one solution. For convenience of presentation, we now present below hypothesis to be used in the rest of the paper. (

)

(

) There exist ( )

is a continuously differentiable function. (

)) such that 153

DARZI ET AL: FRACTIONAL DIFFERENTIAL EQUATIONS

(

)

∫ (

)

( )

and ( ) (

∫ (

)

( )

) For (

∫ (

)

)

(

)

and ∫ (

( ) (

) For

(

( ))

( ))|

(

) There exists positive constants

) | ( )

( )) | (

(

(

)

)) | (

(

)

( ))|

and

| ( )

( )| ( ) (

( )|

), such that

( )|(

)

)| ))

(

where

{

} and

{

}.

2. BASIC DEFINITIONS AND PRELIMINARIES

We now give definitions, lemmas and theorems that will be used in the remainder of this paper. Definition 2.1.([7,8]) The Riemann-Liouville fractional integral of order , of a ) is defined by function ( ( )

( )

∫(

)

( )

,

(2.1)

provided that right-hand side is point wise defined on ( Definition 2.2.([7,8]) The Caputo function ( ) is defined by ( )

(

)

∫(

)

( )(

)

fractional

, n-1
0 (i.e., x(T ; u) = x(T )). And one says that the system is approximately controllable when x(T ; u) belongs to a ”small” neighborhood of x(T ). Thus, many authors have been interested in the controllability. Recently, Kwun et al. [3] proved ε-approximate controllability for the semilinear fuzzy integrodifferential equations. In [8], Mahmudov proved approximated controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. Goreac [2] studied a kalman-type condition for stochastic approximate controllability. In this paper, we study approximate controllability for fuzzy differential equations driven by Liu process. We consider the following fuzzy differential equation driven by Liu process:  dx(t, θ) = Ax(t, θ)dt + f (t, x(t, θ))dC(t) + Bu(t)dt, t ∈ [0, T ], (1) x(0) = x0 ∈ EN , ∗ This

study was supported by research funds from Dong-A University. author. E-mail: [email protected] (Y.C. Kwun), [email protected] (J.S. Kim), [email protected] (H.E. Youm), [email protected] (J.H. Park) † Corresponding

163

Y.C. Kwun, J.S. Kim, H.E. Youm, J.H. Park

where the state function x(t, θ) takes values in X(⊂ EN ) and another bounded space Y (⊂ EN ). EN is the set of all upper semi-continuously convex fuzzy numbers on R, (Θ, P, Cr) is credibility space, A is fuzzy coefficient, the state function x : [0, T ] × (Θ, P, Cr) → X is a fuzzy process, f : [0, T ] × X → X is regular fuzzy function, u : [0, T ] × (Θ, P, Cr) → Y is control function, B is a linear bounded operator from Y to X. C(t) is a standard Liu process, x0 ∈ EN is initial value.

2

Preliminaries

In this chapter, we give basic definitions, terminologies, notations and Lemmas which are most relevant to our investigated and are needed in later chapters. All undefined concepts and notions used here are standard. A fuzzy set of Rn is a function u : Rn → [0, 1]. For each fuzzy set u, we denote by [u]α = {x ∈ Rn : u(x) ≥ α} for any α ∈ [0, 1], its α-level set. Let u, v be fuzzy sets of Rn . It is well known that [u]α = [v]α for each α ∈ [0, 1] implies u = v. Let E n denote the collection of all fuzzy sets of Rn that satisfies the following conditions: (1) u is normal, i.e., there exists an x0 ∈ Rn such that u(x0 ) = 1; (2) u is fuzzy convex, i.e., u(λx + (1 − λ)y) ≥ min{u(x), u(y)} for any x, y ∈ Rn , 0 ≤ λ ≤ 1; (3) u(x) is upper semi-continuous, i.e., u(x0 ) ≥ limk→∞ u(xk ) for any xk ∈ Rn (k = 0, 1, 2, · · ·), xk → x0 ; (4) [u]0 is compact. n n Definition 2.1 [3] Define D : EN × EN → [0, ∞) by the equation

D(u, v)

=

sup dH ([u]α , [v]α ), 0 0. Lemma 2.15 [1] Let C(t) be a standard Liu process, and let h(t; c) be a continuously differentiable function. Define x(t) = h(t; C(t)). Then we have the following chain rule dx(t) =

∂h(t; C(t)) ∂h(t; C(t)) dt + dC(t). ∂t ∂C

166

Approximate Controllability for Fuzzy Differential Equations

Lemma 2.16 [1] Let f (t) be continuous fuzzy process, the following inequality of fuzzy integral holds Z d Z d |f (t)|dt, f (t)dC(t) ≤ K c

c

where K = K(θ) is defined in Lemma 2.14.

3

Existence of Solutions for Fuzzy Differential Equation

In this section, instead of longer notation x(t, θ), sometimes we use the symbol x(t). we consider the existence and uniquencess of solutions for fuzzy differential equation (1)(u ≡ 0).  dx(t) = Ax(t)dt + f (t, x(t))dC(t), t ∈ [0, T ], (2) x(0) = x0 ∈ EN , where the state function x(t) takes values in X(⊂ EN ). EN is the set of all upper semi-continuously convex fuzzy numbers on R, (Θ, P, Cr) is credibility space, A is fuzzy coefficient, the state function x : [0, T ] × (Θ, P, Cr) → X is a fuzzy process, f : [0, T ] × X → X is regular fuzzy function, C(t) is a standard Liu process, x0 ∈ EN is initial value. Lemma 3.1 [7] Let g be a function of two variables and let at be an integrable uncertain process. Then the uncertain differential equation dxt = at xt dt + g(t, xt )dCt has a solution where

xt = yt−1 zt  Z t  yt = exp − as ds 0

and zt is the solution of uncertain differential equation dzt = yt g(t, y −1 zt )dCt with initial value z0 = x0 . Using Lemma 3.1, we show that for fuzzy coefficient A equation (2) has solution. Lemma 3.2 For x(0) = x0 , if x(t) is solution of the equation (2), then x(t) is given by Z t x(t) = S(t)x0 + S(t − s)f (s, x(s))dC(s), t ∈ [0, T ], 0

where S(t) is continuous with S(0) = I, |S(t)| ≤ c, c > 0, for all t ∈ [0, T ].

167

Y.C. Kwun, J.S. Kim, H.E. Youm, J.H. Park

Proof For fuzzy coefficient A, the following define inverse of S(t) S −1 (t) = e−At . Then we have that dS −1 (t) = −Ae−At dt = −AS −1 (t)dt. It follows from the integration by parts that d(S −1 (t)x(t)) = d(S −1 (t))x(t) + S −1 (t)dx(t) = −AS −1 (t)dtx(t) + S −1 (t)Ax(t)dt + S −1 (t)f (t, x(t))dC(t). That is,

d(S −1 (t)x(t)) = S −1 (t)f (t, x(t))dC(t).

Defining z(t) = S −1 (t)x(t), we obtain x(t) = S(t)z(t) and dz(t) = S −1 (t)f (t, S(t)z(t))dC(t). Furthermore, since S(0) = I, the initial value z0 = x0 . We have Z t z(t) = x0 + S −1 (s)f (s, S(s)z(s))dC(s). 0

Therefore the equation (2) has the following solution Z t x(t) = S(t)x0 + S(t − s)f (s, x(s))dCs , t ∈ [0, T ], 0

where we write S(t − s) instead of S(t)S −1 (s). Assume the following: (H1) For x, y ∈ C([0, T ] × (Θ, P, Cr ), X), t ∈ [0, T ], there exists positive number m such that dH ([f (t, x)]α , [f (t, y)]α ) ≤ mdH ([x]α , [y]α ) and f (0, X{0} (0)) ≡ 0. (H2) 2cmKT ≤ 1. Theorem 3.3 If hypotheses (H1) and (H2) hold, for every x0 ∈ EN , then the equation (2) has unique solution x ∈ C([0, T ] × (Θ, P, Cr ), X). Proof For each ξ(t) ∈ C([0, T ] × (Θ, P, Cr ), X), t ∈ [0, T ] define Z t (Ψξ)(t) = S(t)x0 + S(t − s)f (s, ξ(s))dC(s). 0

Thus, Ψξ : [0, T ] × (Θ, P, Cr ) → C([0, T ] × (Θ, P, Cr ), X) is continuous, and Ψ : C([0, T ] × (Θ, P, Cr ), X) → C([0, T ] × (Θ, P, Cr ), X).

168

Approximate Controllability for Fuzzy Differential Equations

It is obvious that fixed points of Ψ is solution for the equation (2). For ξ(t), η(t) ∈ C([0, T ] × (Θ, P, Cr ), X), by Lemma 2.16 and hypothesis (H1), we have dH ([(Ψξ)(t)]α , [(Ψη)(t)]α ) h Z t iα h Z t iα  = dH S(t − s)f (s, ξ(s))dC(s) , S(t − s)f (s, η(s))dC(s) 0 0 Z t ≤ cmK dH ([ξ(s)]α , [η(s)]α )ds. 0

Therefore, D((Ψξ)(t), (Ψη)(t))

=

sup dH ([(Ψξ)(t)]α , [(Ψη)(t)]α ) α∈(0,1]

Z

t

sup dH ([ξ(s)]α , [η(s)]α )ds

≤ cmK

0 α∈(0,1]

Z

t

D(ξ(s), η(s))ds.

= cmK 0

Hence, for a.s. θ ∈ Θ, by Lemma 2.11,     E H1 (Ψξ, Ψη) = E sup D((Ψξ)(t), (Ψη)(t)) t∈[0,T ]

Z



≤ E cmK sup t∈[0,T ]

t

D(ξ(s), η(s))ds



0



 ≤ cmKT E H1 (ξ, η) . By hypothesis (H2), Ψ is a contraction mapping. By the Banach fixed point theorem, equation (2) has a unique fixed point x ∈ C([0, T ] × (Θ, P, Cr ), X).

4

Approximate Controllability for Fuzzy Differential Equation

In this section, we study approximate controllability for fuzzy differential equation (1). We consider solution for the equation (1), for each u in Y . 

Rt Rt x(t) = S(t)x0 + 0 S(t − s)f (s, x(s))dC(s) + 0 S(t − s)Bu(s)ds, x(0) = x0 ∈ EN ,

(3)

where S(t) is continuous with S(0) = I, |S(t)| ≤ c, c > 0, for all t ∈ [0, T ]. We define the continuous linear operator Se from X to C([0, T ]×(Θ, P, Cr ), X) by Z t

S(t − s)p(s)ds, p ∈ X, 0 ≤ t ≤ T.

e Sp(t) = 0

169

Y.C. Kwun, J.S. Kim, H.E. Youm, J.H. Park

We define a solution mapping W from Y to C([0, T ] × (Θ, P, Cr ), X) by (W u)(t) = x(t; u). The reachable sets of a nonlinear system are used to be compared to the reachable sets of its corresponding linear system(f ≡ 0 in (3)). Put n KT (0) = z(T ) ∈ C([0, T ] × (Θ, P, Cr ), X) : Z T o z(T ) = S(T )x0 + S(T − s)Bu(s)ds, u ∈ Y 0

and the reachable set KT (f ) in X by n KT (f ) = x(T ; u) ∈ X : x(T ; u) = S(T )x0 Z T Z t o + S(T − s)f (s, x(s))dC(s) + S(t − s)Bu(s)ds . 0

0

Lemma 4.1 Let u, x0 ∈ EN . Then under hypothesis (H1), the solution mapping (W u)(t) = x(t; u) of (3) satisfies a.s. θ,   E H1 (x, X{0} )    ≤ E cH1 (x0 , X{0} ) + cT H1 (B, X{0} ) · H1 (u, X{0} ) exp(cmKT ) . Proof From hypothesis (H1), Lemma 2.16 and equation (3), we have Z t Z t



kx(t)k = S(t)x0 + S(t − s)f (s, x(s))dC(s) + S(t − s)Bu(s)ds 0 0 Z t Z t ≤ ckx0 k + cmK kx(s)kds + c kBkku(s)kds 0 0 Z t ≤ ckx0 k + ckBkku(t)kt + cmK kx(s)kds. 0

From Gronwall’s inequality, we have kx(t)k ≤ (ckx0 k + ckBkku(t)kt) exp(cmKT ). Hence   E H1 (x, X{0} )   = E sup kx(t)k t∈[0,T ]

≤E



 sup (ckx0 k + ckBkku(t)kt) exp(cmKT )

≤E



t∈[0,T ]

  cH1 (x0 , X{0} ) + cT H1 (B, X{0} ) · H1 (u, X{0} ) exp(cmKT ) .

170

Approximate Controllability for Fuzzy Differential Equations

Lemma 4.2 Let u1 and u2 be in EN . Then under hypothesis (H1) and Lemma 2.16, the solution mapping (W u)(t) = x(t; u) of (3) satisfies a.s. θ,     E H1 (x(t; u1 ), x(t; u2 )) ≤ cT exp(cmKT )E H1 (Bu1 , Bu2 ) . Proof From hypothesis (H1), Lemma 2.16 and equation (3), we have dH ([x(t; u1 )]α , [x(t; u2 )]α ) Z t iα h Z t S(t − s)Bu1 (s)ds , = dH S(t − s)f (s, x(s; u1 ))dC(s) + 0 0 Z t iα  hZ t S(t − s)Bu2 (s)ds S(t − s)f (s, x(s; u2 ))dC(s) + 0 0 Z t   dH [x(s; u1 ))]α , [x(s; u2 ))]α ds ≤ cmK 0 Z t   +c dH [Bu1 (s)]α , [Bu2 (s)]α ds. 0

Hence, by Gronwall’s inequality, we get dH ([x(t; u1 )]α , [x(t; u2 )]α ) ≤ ctdH ([Bu1 ]α , [Bu2 ]α ) exp(cmKT ), D(x(t; u1 ), x(t; u2 ))

=

sup dH ([x(t; u1 )]α , [x(t; u2 )]α ) α∈(0,1]

≤ ct exp(cmKT )D(Bu1 , Bu2 ). Hence   E H1 (x(t; u1 ), x(t; u2 ))

= E



 sup D(x(t; u1 ), x(t; u2 )) t∈[0,T ]

  ≤ cT exp(cmKT )E H1 (Bu1 , Bu2 ) . Definition 4.3 The equation (3) is called approximately controllable on [0, T ] if KT (f ) = X, that is for any given ε > 0 and ξT ∈ X there exists some control v ∈ Y such that a.s. θ,    e (s, x(s; v)) + SBv e E H1 ξT − S(T )x0 , Sf < ε. The range space of the operator B is denoted by XB and its closure is denoted by X B . We assume the following hypotheses; For every arbitrary given ε > 0 and p(·)  ∈ X, thereexists some u(·) ∈ Y such that e SBu) e (H3) E H1 (Sp, < ε,     (H4) E H1 (Bv, X{0} ) ≤ q1 E H1 (p, X{0} ) where q1 is a positive constant independent of p(·),

171

Y.C. Kwun, J.S. Kim, H.E. Youm, J.H. Park

(H5) the constant q1 satisfies q1 cmT < 1, 1 − cmKT (H6) {S(t) : t ∈ R+ } is compact. First of all, we introduce some necessary lemmas before proving the main theorem. Lemma 4.4 Under hypotheses (H3)-(H6), we have KT (0) = X. Proof It is sufficient to prove X ⊂ KT (0) i.e., for any given ε > 0, ξT ∈ X there exists v ∈ Y such that    e E H1 ξT − S(T )x0 , SBv < ε. As ξT ∈ X, S(T )x0 ∈ X, there exists p ∈ C([0, T ] × (Θ, P, Cr) : X) such that T

Z

S(T − s)p(s)ds,

η1 = 0

where η1 = ξT − S(T )x0 . By hypotheses (H3) and (H4) there exists a function v ∈ Y such that Z

T

Z

T

S(T − s)p(s)ds =

η1 = 0

S(T − s)Bv(s)ds. 0

Since η1 = ξT − S(T )x0 , Z

T

S(T − s)Bu(s)ds.

ξT = S(T )x0 + 0

Therefore ξT ∈ KT (0). Hence X ⊂ KT (0). Lemma 4.5 Let x(t; u0 ) and suppose that (H1)-(H6) hold. Then for T > 0 there exists a constant 0 < m < 1 such that a.s. θ    E H1 f (x(t; v1 )), f (x(t; v2 )) ≤

172

  cmT E H1 (Bv1 , Bv2 ) . 1 − cmKT

Approximate Controllability for Fuzzy Differential Equations   Proof By hypothesis (H5), S(t) is compact, we put H1 S(t), X{0} ≤ c.   H1 f (x(t; v1 )), f (x(t; v2 )) ≤ mH1 (x(t; v1 ), x(t; v2 )) Z t Z t S(t − s)Bv1 (s)ds, ≤ mH1 S(t − s)f (x(t; v1 ))dC(s) + 0 0 Z t Z t  S(t − s)f (x(t; v2 ))dC(s) + S(t − s)Bv2 (s)ds 0 0   ≤ cmKT H1 f (x(t; v1 )), f (x(t; v2 )) + cmT H1 (Bv1 , Bv2 ). So we obtain    E H1 f (x(t; v1 )), f (x(t; v2 )) ≤

  cmT E H1 (Bv1 , Bv2 ) 1 − cmKT

where cmKT < 1. Theorem 4.6 Under hypotheses (H1)-(H6), a.s. θ, KT (0) = KT (f ) proved 1 that T < (q1 +K)cm Proof Since, by Lemma 4.4, KT (0) = X, it is sufficient to show that KT (0) ⊂ KT (f ). Let ξT ∈ KT (0). Then for any given ε > 0 there exists u ∈ Y such that    ε e E H1 ξT − S(T )x0 , SBu < 2. 2 Assume v1 ∈ Y is arbitrary given. By hypotheses (H4)-(H5), there exists some v2 ∈ Y such that    ε e e (x(s; v1 )) + SBv e 2 E H1 SBu, Sf < 2. 2 Since from (H3)-(H4) and Lemma 4.5, we can take w2 ∈ Y such that    ε e (x(s; v2 )), Sf e (x(s; v1 )) + SBw e 2 E H1 Sf < 3 2 and so       e 2 , X{0} E H1 SBw ≤ q1 mE H1 f (x(s; v2 )), f (x(s; v1 ))   q1 cmT E H1 (Bv1 , Bv2 ) . ≤ 1 − cmKT

173

Y.C. Kwun, J.S. Kim, H.E. Youm, J.H. Park

Thus we may define v3 = v2 − w2 in Y , which has the following property;   e e 3 H1 ξT − S(T )x0 , Sg(x(s; v2 )) + SBv  e 2, = H1 ξT − S(T )x0 + sef (x(s; v1 )) + SBw  e (x(s; v1 )) + SBv e 2 + Sf e (x(s; v1 )) Sf   e (x(s; v1 )) + SBv e 2 ≤ H1 ξT − S(T )x0 , Sf   e 2 + Sf e (x(s; v1 )), Sf e (x(s; v2 )) +H1 SBw 1 1 < 2 + 3 ε. 2 2 Define vn = vn−1 − wn−1 . Then, by induction, we have    1 1  e (x(s; vn )) + SBv e n+1 + · · · + ε E H1 ξT − S(T )x0 , Sf < 22 2n+1 ε < . 2 Also, we get that   E H1 (Bvn+1 , Bvn ) ≤

  q1 cmT E H1 (Bvn , Bvn−1 ) . 1 − cmKT

1 Since T < (q1 +K)cm , the sequence {Bvn } is cauchy sequence and so convergent to some point in X. Therefore, for any given ε > 0, there exists some integer N such that for all N ≤ n we have    ε e n+1 , SBv)n e E H1 SBv < 2

and so    e (x(s; vn )) + SBv e n E H1 ξT − S(T )x0 , Sf    e (x(s; vn )) + SBv e n+1 ≤ E H1 ξT − S(T )x0 , Sf    e n+1 , SBv)n e +E H1 SBv 1 1  ε ≤ 2 + · · · + n+1 ε + ≤ ε, 2 2 2 for all N ≤ n. That is, ξT ∈ KT (f ). Therefore, the equation (3) is approximately controllable on [0, T ].

References [1] W. Fei, Uniqueness of solutions to fuzzy differential equations driven by Liu’s process with non-Lipschitz coefficients, International Conference on Fuzzy Systems and Knowledge Discovery (2009), 565-569.

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Approximate Controllability for Fuzzy Differential Equations

[2] D. Goreac, A kalman-type condition for stochastic approximate controoability, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 183-188. [3] Y. C. Kwun, J. S. Kim, M. J. Park and J. H. Park, ε-Approximate controllability for the semilinear fuzzy integrodifferential equations, Journal of Computational Analysis and Applications, 13 (2011), 1171-1179. [4] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, 10 (2002), 445-450. [5] B. Liu, A survey of credibility theory, Fuzzy Optim. Decis. Making, 5 (2006), 387-408. [6] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16. [7] B. Liu, Uncertainty theory, http : //orsc.edu.cn/liu/ut.pdf. [8] N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, Siam J. Control Optim., 42 (2003), 1604-1622.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.1, 176-187, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Approximation properties of the modification of Kantorovich type q-Sz´ asz operators Qing-Bo Caia,b,∗, Xiao-Ming Zengb and Zhenlu Cuic a

School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China b

c

Department of Mathematics, Xiamen University, Xiamen 361005, China

Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville, North Carolina 28301, USA E-mail: [email protected], [email protected], [email protected]

Abstract. In this paper, we introduce a generalization of modification of Kantorovich type q-Sz´ asz operators Kn,q based on the concept of q-integral, these operators are different from those proposed by Mahmudov and Gupta [Mahmudov, N., Gupta, V.: On certain q-analogue of Sz´asz Kantorovich operators, J. Appl. Math. Comput, 10.1007/s12190-010-0441-4 (2010)]. We investigate weighted statistical approximation properties and establish a local approximation theorem, we also give a convergence theorem for the Lipschitz continuous functions. Furthermore, we give the relationship between the derivative of q-Sz´ asz Mirakjan operators and the operators Kn,q . Keywords: Kantorovich type operators, q-integral, q-Sz´asz operators, weighted statistical approximation, rate of convergence. 2000 Mathematics Subject Classification: 41A10, 41A25, 41A36.

1

Introduction

In recent years, the applications of q-calculus in the approximation theory is one of the main research area. After q-Bernstein polynomials were introduced by Phillips [1] in 1997, several researchers have studied in this field and obtained many approximation properties of different operators, we mention some of them as [1]-[8]. Very recently, Mahmudov and Gupta [2] introduced the following operators : ∗ Kn,q (f ; x)

= [n]q

∞ X

Z k

q sn,k (q; qx)

k=0 ∗

Corresponding author.

176

[k+1]q q k [n]q [k]q q k−1 [n]q

f (t)dt,

(1)

Q. B. Cai, X. M. Zeng and Z. L. Cui: Approximation properties of the Kn,q operators where sn,k (q; x) = eq (−[n]q x)q

k(k−1) 2

[n]kq xk , n = 1, 2, ... . [k]q !

(2)

In the present paper, instead of using the Riemann integral in the operators, we use the q-Jackson integral in the operators for the sake of harmony of q-operators’ notation. We define a new kind of Kantorovich type q-Sz´ asz operator Kn,q based on the q-Jackson integral as follows. For f ∈ C[0, ∞), q ∈ (0, 1) and n ∈ N, the Kantorovich type q-Sz´asz operators Kn,q are given by Z [k+1]q ∞ X q k [n]q k Kn,q (f ; x) = [n]q q sn,k (q; qx) f (t)dq t, (3) [k] q q k−1 [n]q

k=0

where sn,k (q; x) is defined in (2). It can be seen that for q → 1− , the operators (3) become the ones studied in [9]. In [2], Mahmudov and Gupta studied the local approximation properties as well as ∗ . Different from their work, in weighted approximation properties of the operators Kn,q the present paper we mainly study the weighted statistical approximation properties of the operators Kn,q and some other approximation properties of the operators Kn,q . The main contents of this paper are organized as follows. In Section 2, we give some lemmas which are need for proving our theorems. In Section 3, we investigate weighted statistical approximation properties of the operators Kn,q . In Section 4, we establish a local approximation theorem of the operators, and we also obtain a convergence theorem of the operators Kn,q for the Lipschitz continuous functions. In Section 5, we establish a relationship between the derivative of q-Sz´asz Mirakjan operators and the operators Kn,q . Now we mention certain definitions based on q-integers, details can be found in [10, 11]. For any fixed real number 0 < q ≤ 1 and each nonnegative integer k, we denote q-integers by [k]q , where ( 1−q k 1−q ,

[k]q :=

k,

q= 6 1, q = 1.

Also q-factorial and q-binomial coefficients are defined as follows: ( " # [n]q ! [k]q [k − 1]q ...[1]q , k = 1, 2, ..., n [k]q ! := , := [k]q ![n − k]q ! 1, k = 0, k

(n ≥ k ≥ 0).

q

The q-Jackson integral is defined as Z 0

a

f (x)dq x := (1 − q)a

∞ X

f (aq n )q n , (a > 0)

n=0

provided the sums converge absolutely. One can define the q-Jackson integral in a generic intercal [a, b] as Z a Z b Z b f (x)dq x, f (x)dq x − f (x)dq x = a

0

0

177

Q. B. Cai, X. M. Zeng and Z. L. Cui: Approximation properties of the Kn,q operators

and it is easy to see that Z b ∞ X £ ¤ f (x)dq x = (1 − q) bf (q j b) − af (q j a) q j . a

j=0

The q-analogs eq (x) and Eq (x) of the exponential function are given as ∞ X xk 1 1 eq (x) := = , |x| < , |q| < 1, [k]q ! (1 − (1 − q)x)∞ 1 − q q k=0

Eq (x) :=

∞ X

q k(k−1)/2

k=0

xk = (1 + (1 − q)x)∞ q , |q| < 1, [k]q !

where (1 −

x)∞ q

:=

∞ Y

(1 − q j x).

j=0

It is easily observed that eq (x)Eq (−x) = eq (−x)Eq (x) = 1. For 0 < q < 1 the q-Sz´asz Mirakjan operators are defined as µ ¶ ∞ X [k]q Sn,q (f ; x) = sn,k (q; x)f , x ∈ [0, ∞), q k−1 [n]q

(4)

k=0

where sn,k (q; x) is defined in (2). It is clear that sn,k (q; x) ≥ 0, for all 0 < q < 1 and x ≥ 0 and moreover ∞ ∞ X X k(k−1) ([n]q x)k 1 sn,k (q; x) = q 2 = 1. (5) Eq ([n]q x) [k]q ! k=0

k=0

2

Some auxiliary results In this section we give the following lemmas, which are need for proving our theorems:

Lemma 2.1. For s=0, 1, 2, ... , and 0 < q < 1, we have à ! Ps Pi i [k]s−i+j q k(i−j)+(s−i) q Z [k+1]q i=0 j=0 j k 1 q [n]q ts d q t = . [k]q [s + 1]q q k(s+1) [n]s+1 q k−1 q

[n]q

Proof. Using the definition of q-Jackson integrals, we have " ¶ µ ¶s+1 # Z [k+1]q ∞ µ X [k]q [k + 1]q s+1 q k [n]q s − q (s+1)j t dq t = (1 − q) k [n] k−1 [n] [k]q q q q q k−1 q

j=0

[n]q

= =

− (q[k]q )s+1 1 − q [k + 1]s+1 q 1 − q s+1 (q k [n]q )s+1 Ps i s−i 1 i=0 [k + 1]q (q[k]q ) , [s + 1]q (q k [n]q )s+1

using [k + 1]q = [k]q + q k and simple computations, we get the desired result.

178

(6)

Q. B. Cai, X. M. Zeng and Z. L. Cui: Approximation properties of the Kn,q operators

Lemma 2.2. The following conclusions hold: Kn,q (1; x) = 1,

(7)

1 , [2]q [n]q x2 [2]q ([2]q + q) 1 Kn,q (t2 ; x) = + x+ , q [3]q q[n]q [3]q [n]2q µ ¶ ¡ ¢ 1 1 + 2q + 3q 2 1 2 Kn,q (t − x) ; x = − 1 x2 + x+ , q [2]q [3]q q[n]q [3]q [n]2q Kn,q (t; x) = x +

Proof. Using (5) and Lemma 2.1, we get the equality Kn,q (1, x) = 1 easily. Indeed, by Lemma 2.1, we have Kn,q (t; x) = [n]q = [n]q

∞ X k=0 ∞ X

Z k

q snk (q; qx) q k snk (q; qx)

k=0

=

∞ X

[k+1]q q k [n]q [k]q q k−1 [n]q

tdq t

[2]q [k]q + q k q 2k [2]q [n]2q ∞

snk (q; qx)

X [k]q 1 + snk (q; qx) k [2]q [n]q q [n]q k=0

k=0

using (5), we obtain Kn,q (t; x) =

∞ X k=1

=

∞ X k=0

= x+

k−1 (qx)k−1 (qx) k(k−1) [n]q 1 1 1 q 2 + k E([n]q qx) [k − 1]q [2]q [n]q q k k k(k−1) [n]q (qx) 1 1 q 2 x+ Eq ([n]q qx) [k]q ! [2]q [n]q

1 . [2]q [n]q

Similarly, using Lemma 2.1, we get Kn,q (t2 ; x) = [n]q

∞ X

Z k

q snk (q; qx)

k=0

= [n]q

∞ X

q k snk (q; qx)

k=0

=

∞ X k=0

snk (q; qx)

[k+1]q q k [n]q [k]q q k−1 [n]q

t2 d q t

[3]q [k]2q + q k (1 + [2]q )[k]q + q 2k [3]q q 3k [n]3q

∞ ∞ X [k]2q (1 + [2]q )[k]q X 1 + + , s (q; qx) snk (q; qx) nk 2 2k k 2 [3]q [n]2q [n]q q [3]q q [n]q k=0

k=0

179

(8) (9) (10)

Q. B. Cai, X. M. Zeng and Z. L. Cui: Approximation properties of the Kn,q operators using [k]q = [k − 1]q + q k−1 and (5), we have Kn,q (t2 ; x) =

∞ X

k−2 (qx)k k(k−1) [n]q [k − 1]q + q k−1 1 q 2 Eq ([n]q qx) [k − 1]q ! q 2k

k=1 ∞ X

+

k=1

=

∞ X

k−2 (qx)2 k(k−1) [n]q 1 1 q 2 Eq ([n]q qx) [k − 2]q ! q 2k

k=2 ∞ X

+

k=1

+

∞ X k=0

=

∞ X

k−1 (qx)k−1 (qx) k(k−1) [n]q 1 + [2]q 1 1 + q 2 Eq ([n]q qx) [k − 1]q ! [3]q [n]2q [3]q q k [n]q

k−1 (qx)k−1 (qx) k(k−1) [n]q q k−1 1 q 2 Eq ([n]q qx) [k − 1]q ! q 2k [n]q k k k(k−1) [n]q (qx) (1 + [2]q )x 1 1 q 2 + Eq ([n]q qx) [k]q ! [3]q [n]q [3]q [n]2q

k k 2 k(k−1) [n]q (qx) (qx) 1 1 q 2 Eq ([n]q qx) [k]q ! q3

k=0 ∞ X

+

k k k(k−1) [n]q (qx) (qx) 1 + [2]q 1 1 1 q 2 + x+ 2 Eq ([n]q qx) [k]q ! q [n]q [3]q [n]q [3]q [n]2q

k=0 2 [2]q ([2]q x

+ q) 1 x+ . q [3]q q[n]q [3]q [n]2q ¡ ¢ Finally, using (7)-(9) and Kn,q (t − x)2 ; x = Kn,q (t2 ; x) − 2xKn,q (t; x) + x2 , we have ¶ µ ¡ ¢ 1 + 2q + 3q 2 1 1 2 − 1 x2 + . x+ Kn,q (t − x) ; x = q [2]q [3]q q[n]q [3]q [n]2q =

+

Lemma 2.2 is proved. Remark 2.3. From Lemma 2.2, it is observed that for q → 1− , we obtain Kn,1− (1; x) = 1, 1 , 2n 2x 1 Kn,1− (t2 ; x) = x2 + + 2, n 3n

Kn,1− (t; x) = x +

which are moments for modified Sz´ asz-Mirakjan-Kantorovich operators in [9].

3

Weighted statistical approximation properties

In this section, we present the statistical approximation properties of the operator Kn,q by using a Korovkin-type theorem proved in [12]. Let K be a subset of N, the set of all natural numbers. The density of K is defined P by δ(K) := limn n1 nk=1 χK (k) provided the limit exists, where χK is the characteristic

180

Q. B. Cai, X. M. Zeng and Z. L. Cui: Approximation properties of the Kn,q operators

function of K. A sequence x := {xn } is called statistically convergent to a number L if, for every ε > 0, δ{n ∈ N : |xn − L| ≥ ε} = 0. Let A := (ajn ), j, n = 1, 2, ... be an infinite summability matrix. For a given sequence x := {xn }, the A−transform of x, denoted by P Ax := ((Ax)j ), is given by (Ax)j = ∞ k=1 ajn xn provided the series converges for each j. We say that A is regular if limn (Ax)j = L whenever lim x = L. Assume that A is a non-negative regular summability matrix. A sequence x = {xn } is called A-statistically P convergent to L provided that for every ε > 0, limj n:|xn −L|≥ε ajn = 0. We denote this limit by stA − limn xn = L (see [13]). For A = C1 , the Ces` aro matrix of order one, A-statistical convergence reduces to statistical convergence. It is easy to see that every convergent sequence is statistically convergent but not conversely. A real function ρ(x) is called a weight function if it is continuous on R and lim ρ(x) = |x|→∞

∞, ρ(x) ≥ 1 for all x ∈ R. Let (see [3]) Bρ (R) := {f : R → R : |f (x)| ≤ Mf ρ(x), Mf is a positive constant depending only on f }, Cρ (R) := {f ∈ Bρ (R) : f is continuous on R}. (x)| Endowed with the norm || · ||ρ , where ||f ||ρ := sup |fρ(x) , Bρ (R) and Cρ (R) are Banach spaces. Using A−statistical convergence, Duman and Orhan proved the following Korovkintype theorem.

Theorem 3.1. (see [12]) Let A = (ajn ) be a nonnegative regular summability matrix and let Ln be a sequence of positive linear operators from Cρ1 (R) into Bρ2 (R), where ρ1 and ρ2 satisfy ρ1 (x) = 0, lim |x|→∞ ρ2 (x) then stA − lim ||Ln f − f ||ρ2 = 0 f or all f ∈ Cρ1 (R) n

if and only if stA − lim ||Ln Fv − Fv ||ρ1 = 0 f or all v = 0, 1, 2, n

where Fv =

xv ρ

1 (x) ,v 1+x2

= 0, 1, 2.

We consider the weight functions ρ1 (x) = 1 + x2 , ρ2 (x) = 1 + x2+α , α > 0. Further on, we consider a sequence q := {qn } for 0 < qn < 1 satisfying st − lim qn = 1, n

(

(11)

1/2; if n = m2 , (m = 1, 2, 3...) ¡ ¢ e1/n 1 − n1 ; if n 6= m2 . (see [4]). We can deduce that it satisfies the conditions (11) for statistically convergence but it does not work for ordinary case. If ei = ti , t ∈ R+ , i = 0, 1, 2, ... stands for the ith monomial, then we have for example, define the sequence q = {qn } by qn =

181

Q. B. Cai, X. M. Zeng and Z. L. Cui: Approximation properties of the Kn,q operators Theorem 3.2. Let q := {qn } be a sequence satisfying (11), then for allf ∈ Cρ1 (R+ ), we have st − lim ||Kn,qn f − f ||ρ2 = 0. (12) n

Proof. Obviously st − lim ||Kn,qn (e0 ) − e0 ||ρ1 = 0. n

By (8) we have

1

|Kn,qn (e1 ; x) − e1 (x)| 1 [2] [n] = qn q2n ≤ . 1 + x2 1+x [2]qn [n]qn Now for a given ε > 0, let us define the following sets: ½ ¾ 1 U := {k : ||Kn,qk (e1 ) − e1 ||ρ1 ≥ ε}, U1 := k : ≥ε . [2]qk [k]qk Then one can see that U ⊆ U1 , so we have

½

δ{k ≤ n : ||Kn,qk (e1 ) − e1 ||ρ1

¾ 1 ≥ε . ≤δ k≤n: [2]qk [k]qk

1 = 0, which implies that the right-hand [2]qn [n]qn side of the above inequality is zero, thus we have Since st − lim qn = 1, we have st − lim n

n

st − lim ||Kn,qn (e1 ) − e1 ||ρ1 = 0. n

Finally, by (9), we get |Kn,qn (e2 ; x) − e2 (x)| 1 + x2

=

¯ 2 ¯x ¯ qn + µ

≤ Now we let αn :=

[2]qn ([2]qn +qn ) [3]qn qn [n]qn x

+

1 [3]qn [n]2qn

−

¯ ¯

¯ x2

1 + x2 [2]qn ([2]qn + qn ) 1 1 −1 + . + qn [3]qn qn [n]qn [3]qn [n]2qn ¶

[2]qn ([2]qn + qn ) 1 1 − 1, βn := , γn := . qn [3]qn qn [n]qn [3]qn [n]2qn

Since st − lim qn = 1, one can see that n

st − lim αn = st − lim βn = st − lim γn = 0. n

n

n

(13)

For ε > 0, we define the following four sets

n n εo εo , V2 := k : βk ≥ , V := {k : ||Kn,qk (e2 ) − e2 ||ρ1 ≥ ε}, V1 := k : αk ≥ 3 3 n εo V3 := k : γk ≥ . 3 Hence, from (13) we obtain the right-hand side of the above inequality is zero, so we have δ{k ≤ n : ||Kn,qk (e2 ) − e2 ||ρ1 ≥ ε} = 0,

thus st − lim ||Kn,qn (e2 ) − e2 ||ρ1 = 0. n Then the proof of Theorem 3.2 is obtained by Theorem 3.1 with A = C1 , where C1 is a Ces` aro matrix of order one.

182

Q. B. Cai, X. M. Zeng and Z. L. Cui: Approximation properties of the Kn,q operators

4

Local approximation properties

Let CB [0, ∞) be the space of all real-valued continuous bounded functions f on [0, ∞), endowed with the norm ||f || = supx∈[0,∞) |f (x)|. The Peetre’s K−functional is defined by K2 (f ; δ) = inf

©

2 g∈CB

ª ||f − g|| + δ||g 00 || ,

2 = {g ∈ C [0, ∞) : g 0 , g 00 ∈ C [0, ∞)} . By [14, p.177, Theorem 2.4], where δ > 0 and CB B B there exits an absolute constant C > 0 such that √ K2 (f ; δ) ≤ Cω2 (f ; δ), (14)

where ω2 (f ; δ) = sup

sup |f (x + 2h) − 2f (x + h) + f (x)|

0 0. Hence ( ) 1 1 N (Jg(x) − Jh(x), Lεt) = N g (lx) − h (lx) , Lεt l l = N (g (lx) − h (lx) , lLεt) lLt lLt ≥ ≥ lLt + ψ (lx) lLt + lLψ(x) t = t + ψ(x) for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.3) that

( ) 1 t t N f (x) − f (lx), 2 ≥ l l t + ψ(x) for all x ∈ X and all t > 0. Thus d(f, Jf ) ≤ l12 . By Theorem 1.4, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e., A (lx) = lA(x)

(2.8)

for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.8) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − A(x), µt) ≥ t + ψ(x) for all x ∈ X; (2) d(J n f, A) → 0 as n → ∞. This implies the equality N - lim

1

n→∞ ln

f (ln x) = A(x)

for all x ∈ X;

457

Fuzzy stability of additive-quadratic functional equation (3) d(f, A) ≤

1 1−L d(f, Jf ),

which implies the inequality d(f, A) ≤

1 . l2 − l2 L

This implies that the inequality (2.7) holds. The rest of the proof is similar to the proof of Theorem 2.1.



Corollary 2.4. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm ∥ · ∥. Let f : X → Y be an odd mapping satisfying (2.6). Then A(x) := N -limn→∞ l1𝑛 f (ln x) exists for each x ∈ X and defines an additive mapping A : X → Y such that (l − lp )t N (f (x) − A(x), t) ≥ (l − lp )t + θ∥x∥p for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.3 by taking φ(x1 , · · · , xl ) := θ

l ∑

∥xj ∥p

j=1

for all x1 , · · · , xl ∈ X. Then we can choose L = l

p−1

and we get the desired result.



3. Hyers-Ulam stability of the functional equation (0.1): an even case In this section, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces for an even case. Using fixed point method, we prove the Hyers-Ulam stability of the functional equation Cf (x1 , · · · , xl ) = 0 in fuzzy Banach spaces: an even case. Theorem 3.1. Let φ : X l → [0, ∞) and ψ(x) := φ(x, · · · , x) be functions such that there exists an | {z } l times

L < 1 with φ(x1 , · · · , xl ) ≤ lL2 φ (lx1 , · · · , lxl ) for all x1 , · · · , xl ∈ X. Let f : X → Y be an even ( ) mapping satisfying f (0) = 0 and (2.1). Then Q(x) := N -limn→∞ l2n f lx𝑛 exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(l3 − l3 L)t (l3 − l3 L)t + Lψ(x)

(3.1)

for all x ∈ X and all t > 0. Proof. Letting x1 = · · · = xl = x in (2.1), we get ( ) t t N lf (lx) − l3 f (x), t ≥ = t + φ(x, · · · , x) t + ψ(x) | {z } l times

for all x ∈ X. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping J : S → S such that (x) Jg(x) := l2 g l for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ψ(x)

458

(3.2)

C. Park, A. Najati, S. Y. Jang for all x ∈ X and all t > 0. Hence

( (x) (x) ) N (Jg(x) − Jh(x), Lεt) = N l2 g − l2 h , Lεt l l ( ( ) ) ( ) x x L = N g −h , 2 εt ≥ l l l ≥

Lt l2 Lt L + 2 l l2 ψ(x)

Lt l2 Lt l2

+ψ

(x) l

t = t + ψ(x)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (3.2) that ( ( x ) Lt ) 2 N f (x) − l f , 3 ≥ l l

L l2 t L l2 t

+ψ

(x) ≥ l

t t + ψ (x)

(3.3)

for all x ∈ X. So d(f, Jf ) ≤ lL3 . By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., (x) 1 Q = 2 Q(x) (3.4) l l for all x ∈ X. Since f : X → Y is even, Q : X → Y is an even mapping. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (3.4) such that there exists a µ ∈ (0, ∞) satisfying N (f (x) − Q(x), µt) ≥

t t + ψ(x)

for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality N - lim l2n f

(x)

n→∞

for all x ∈ X; (3) d(f, Q) ≤

1 1−L d(f, Jf ),

= Q(x)

ln

which implies the inequality d(f, Q) ≤

l3

L . − l3 L

This implies that the inequality (3.1) holds. The rest of the proof is similar to the proof of Theorem 2.1.



Corollary 3.2. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm ∥ · ∥. Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.6). Then Q(x) := N ( ) limn→∞ l2n f lx𝑛 exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(lp − l2 )t (lp − l2 )t + θ∥x∥p

for all x ∈ X and all t > 0.

459

Fuzzy stability of additive-quadratic functional equation Proof. The proof follows from Theorem 3.1 by taking φ(x1 , · · · , xl ) := θ

l ∑

∥xj ∥p

j=1

for all x1 , · · · , xl ∈ X. Then we can choose L = l

2−p

and we get the desired result.



Similarly, we can obtain the following. We will omit the proof. Theorem 3.3. Let φ : X l → [0, ∞) and ψ(x) := φ(x, · · · , x) be functions such that there exists an | {z } l times ( ) x L < 1 with φ(x1 , · · · , xl ) ≤ l2 Lφ xl1 , · · · , ll for all x1 , · · · , xl ∈ X. Let f : X → Y be an even 1 mapping satisfying f (0) = 0 and (2.1). Then Q(x) := N -limn→∞ l2𝑛 f (ln x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − A(x), t) ≥

(l3

(l3 − l3 L)t − l3 L)t + ψ(x)

for all x ∈ X and all t > 0. Corollary 3.4. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm ∥ · ∥. Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.6). Then 1 Q(x) := N -limn→∞ l2𝑛 f (ln x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (l2 − lp )t N (f (x) − Q(x), t) ≥ 2 (l − lp )t + θ∥x∥p for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 3.3 by taking φ(x1 , · · · , xl ) := θ

l ∑

∥xj ∥p

j=1

for all x1 , · · · , xl ∈ X. Then we can choose L = l

p−2

and we get the desired result.



Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), and S. Y. Jang was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2011-004872) and the Ulsan Metropolitan Office of Education. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705. [3] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [4] L. C˘adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003).

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C. Park, A. Najati, S. Y. Jang [5] L. C˘adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [6] L. C˘adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). [7] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [8] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [9] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [10] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [11] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [12] M. Eshaghi Gordji, A characterization of (σ, τ )-derivations on von Neumann algebras, Politehn. Univ. Bucharest Sci. Bull. Ser. A–Appl. Math. Phys. 73 (2011), No. 1, 111–116. [13] M. Eshaghi Gordji, A. Bodaghi and C. Park, A fixed point approach to the stability of double Jordan centralizers and Jordan multipliers on Banach algebras, Politehn. Univ. Bucharest Sci. Bull. Ser. A–Appl. Math. Phys. 73 (2011), No. 2, 65–74. [14] M. Eshaghi Gordji, H. Khodaei and R. Khodabakhsh, General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A–Appl. Math. Phys. 72 (2010), No. 3, 69–84. [15] M. Eshaghi Gordji and M.B. Savadkouhi, Approximation of generalized homomorphisms in quasiBanach algebras, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17 (2009), No. 2, 203–213. [16] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [17] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [18] P. G˘avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [19] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [20] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [21] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [22] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [23] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [24] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [25] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [26] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [27] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [28] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [29] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729.

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Fuzzy stability of additive-quadratic functional equation [30] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791–3798. [31] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). [32] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [33] C. Park, Inner product spaces and quadratic functional equations, (preprint). [34] C. Park, Y. Cho and M. Han, Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [35] C. Park and J. Cui, Generalized stability of C ∗ -ternary quadratic mappings, Abstract Appl. Anal. 2007, Art. ID 23282 (2007). [36] C. Park and A. Najati, Homomorphisms and derivations in C ∗ -algebras, Abstract Appl. Anal. 2007, Art. ID 80630 (2007). [37] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [38] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [39] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [40] Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babes-Bolyai XLIII (1998), 89–124. [41] Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378. [42] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [43] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130. ˇ [44] Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. ˇ [45] Th.M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325–338. [46] Th.M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234–253. [47] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [48] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [49] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 463-470, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Common fixed point results for generalized quasicontractions in tvs-cone metric spaces Hui-Sheng Ding a,∗, Mirko Jovanovi´c b , Zoran Kadelburg c , Stojan Radenovi´c d a

College of Mathematics and Information Science, Jiangxi Normal University Nanchang, Jiangxi 330022, People’s Republic of China

b

University of Belgrade, Faculty of Electrical Engineering, Bul. kralja Aleksandra 73, 11000 Beograd, Serbia c d

University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Beograd, Serbia

University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Beograd, Serbia

Abstract The quasicontraction result of X. Zhang [X. Zhang, Fixed point theorem of generalized quasicontractive mapping in cone metric spaces, Comput. Math. Appl., 62 (2011), 1627–1633] is generalized by omitting the assumption of normality of the underlying cone, and much shorter proof is given. Respective common fixed point results, as well as the result for Fisher’s quasicontraction, are also proved. Examples are provided showing that our results are proper generalizations of the known ones. Keywords: Tvs-cone metric space; generalized quasicontraction; nonnormal cone; common fixed point. Mathematics Subject Classification: 47H10, 54H25.

1

Introduction

Fixed point and common fixed point results and their applications in metric spaces and their generalizations have received much attention in the past 90 years since the basic Banach’s result. ´ c [1], Das and Naik [2] Among the most famous are the following quasicontraction results of Ciri´ and Fisher [3]. Theorem 1.1. Let (X, d) be a complete metric space and let f : X → X. Suppose that there exists k ∈ [0, 1) such that for all x, y ∈ X, d(f x, f y) ≤ k max{d(x, y), d(x, f x), d(y, f y), d(x, f y), d(y, f x)}

(1.1)

holds. Then f has a unique fixed point u and, for each x ∈ X, the iterative sequence {f n x} converges to u. Theorem 1.2. Let (X, d) be a metric space, and let f, g : X → X be two selfmappings such that f X ⊂ gX and gX is a complete subset of X. Suppose that there exists k ∈ [0, 1) such that for all x, y ∈ X, d(f x, f y) ≤ k max{d(gx, gy), d(gx, f x), d(gy, f y), d(gx, f y), d(gy, f x)} ∗ Corresponding author. E-mail addresses: [email protected] (H. S. Ding), [email protected] (M. Jovanovi´ c), [email protected] (Z. Kadelburg), [email protected] (S. Radenovi´ c).

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holds. Then f and g have a unique point of coincidence. If, moreover, f and g are weakly compatible, then they have a unique common fixed point u and for each x ∈ X the corresponding Jungck sequence converges to u. Theorem 1.3. Let (X, d) be a complete metric space, and let f, g : X → X be two selfmappings. Suppose that there exists k ∈ [0, 1) such that for all x, y ∈ X, d(f x, gy) ≤ k max{d(x, y), d(x, f x), d(y, gy), 21 [d(x, gy) + d(y, f x)]} holds. Then f and g have a unique common fixed point. Moreover, each fixed point of f is a fixed point of g and conversely. Theorem 1.4. Let (X, d) be a complete metric space and let f : X → X be a continuous selfmap. Suppose that there exist k ∈ [0, 1) and positive integers p, q such that for all x, y ∈ X there are some 0 ≤ r, r0 ≤ p and 0 ≤ s, s0 ≤ q so that 0

0

d(f p x, f q y) ≤ k max{d(f r x, f s y), d(f r x, f r x), d(f s y, f s y)}. Then f has a unique fixed point u ∈ X and, for each x ∈ X, the iterative sequence {f n x} converges to u. L.G. Huang and X. Zhang introduced cone metric spaces in [5], replacing the set of real numbers by an ordered Banach space as the codomain of the metric. Thus, they reconsidered the notion of K-metric space that was used earlier (see, e.g., [13]). A lot of known metric fixed point and common fixed point results were subsequently proved in this new setting (see, e.g., [5–7, 9, 10]). Recently, X. Zhang [14] considered the following modification of the quasicontractive condition (1.1): There exists k ∈ [0, 1) such that for all x, y ∈ X there is some sx,y ∈ co {θ, d(x, y), d(x, f x), d(y, f y), d(x, f y), d(y, f x)} so that d(f x, f y) ¹ k sx,y .

(1.2)

Note that it is equivalent to use the set co {θ, d(f x, f y), d(x, y), d(x, f x), d(y, f y), d(x, f y), d(y, f x)} instead of co {θ, d(x, y), d(x, f x), d(y, f y), d(x, f y), d(y, f x)}. It is easy to see that in the frame of metric spaces condition (1.2) is equivalent to condition (1.1). However, in arbitrary cone metric spaces, new condition is weaker and Zhang showed by an example that the corresponding fixed point result is stronger. His proof, though, was rather involved and, moreover, he used additional assumption that the underlying cone was normal. We prove in this article that normality condition can be omitted and, in the same time, Zhang’s result can be proved in a much easier way. We use the method of Minkowski functional introduced in [10]. In the same way, generalized common fixed point theorems for two mappings are easily derived, by reducing them to their known metric counterparts. The same is true for Fisher’s quasicontraction. At the end, we provide examples showing that our results are proper generalizations of the known ones.

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2

Preliminaries

We recall some properties of cones and tvs-cone metric spaces. The details and proofs can be found in [5, 7, 9, 10, 12]. Let E be a real Hausdorff topological vector space (tvs for short) with the zero vector θ. A proper nonempty and closed subset K of E is called a (convex) cone if K + K ⊂ K, λK ⊂ K for λ ≥ 0 and K ∩ (−K) = {θ}. We shall always assume that the cone K has a nonempty interior int K (such cones are called solid ). Each cone K induces a partial order ¹ on E by x ¹ y ⇔ y − x ∈ K. x ≺ y will stand for x ¹ y and x 6= y, while x ¿ y will stand for y − x ∈ int K. The pair (E, K) is an ordered topological vector space. For a pair of elements x, y in E such that x ¹ y, put [x, y] = { z ∈ E : x ¹ z ¹ y }. The sets of the form [x, y] are called order-intervals. It is easily verified that order-intervals are convex. A subset A of E is said to be order-convex if [x, y] ⊂ A, whenever x, y ∈ A and x ¹ y. Ordered topological vector space (E, K) is order-convex if it has a base of neighborhoods of θ consisting of order-convex subsets. In this case the cone K is said to be normal. In the case of a normed space this condition means that the unit ball is order-convex, which is equivalent to the condition that there is a number k such that x, y ∈ E and θ ¹ x ¹ y implies that kxk ≤ kkyk. The smallest constant k satisfying the last inequality is called the normal constant of K. If V is an absolutely convex and absorbing subset of a tvs E, its Minkowski functional qV is defined by E 3 x 7→ qV (x) = inf{ λ > 0 : x ∈ λV }. It is a semi-norm on E (i.e., qV (x + y) ≤ qV (x) + qV (y) for x, y ∈ E and qV (λx) = |λ|qV (x) for x ∈ E, λ a scalar). If V is an absolutely convex neighborhood of θ in E, then qV is continuous and {x ∈ E : qV (x) < 1} = int V ⊂ V ⊂ V = {x ∈ E : qV (x) ≤ 1}. Note also that qV (co A) ⊂ co (qV (A)) for arbitrary A ⊂ E. Let now (E, K) be an ordered tvs and let e ∈ int K. Then [−e, e] = (K − e) ∩ (e − K) = {z ∈ E : −e ¹ z ¹ e} is an absolutely convex neighborhood of θ; its Minkowski functional q[−e,e] will be denoted by qe . It is an increasing function on K. Indeed, let θ ¹ x1 ¹ x2 ; then {λ : x1 ∈ λ[−e, e]} ⊃ {λ : x2 ∈ λ[−e, e]} and it follows that qe (x1 ) ≤ qe (x2 ). If the cone K is solid and normal, qe is a norm in E. Let X be a nonempty set and (E, K) an ordered tvs. A function d : X × X → E is called a tvs-cone metric and (X, d) is called a tvs-cone metric space if the following conditions hold: (C1) θ ¹ d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y; (C2) d(x, y) = d(y, x) for all x, y ∈ X; (C3) d(x, z) ¹ d(x, y) + d(y, z) for all x, y, z ∈ X. Let x ∈ X and {xn } be a sequence in X. Then it is said that: (i) {xn } tvs-cone converges to x if for every c ∈ E with θ ¿ c there exists a natural number n0 d

such that d(xn , x) ¿ c for all n > n0 ; we denote it by d-limn→∞ xn = x or xn → x as n → ∞; (ii) {xn } is a tvs-cone Cauchy sequence if for every c ∈ E with θ ¿ c there exists a natural number n0 such that d(xm , xn ) ¿ c for all m, n > n0 ; (iii) (X, d) is tvs-cone complete if every tvs-Cauchy sequence is tvs-convergent in X. The following result was proved in [10].

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Theorem 2.1. Let (X, d) be a tvs-cone metric space over a solid cone K and let e ∈ int K. Let qe be the corresponding Minkowski functional of [−e, e]. Then dq = qe ◦ d is a (real-valued) metric on X. Moreover: dq d 1◦ xn → x ⇔ xn → x. 2◦ {xn } is a d-Cauchy sequence if and only if it is a dq -Cauchy sequence. 3◦ (X, d) is complete if and only if (X, dq ) is complete. Hence, topologies induced on X by d and dq are equivalent, i.e., these spaces have the same collections of closed, resp. open sets, and the same continuous functions (provided the underlying cone is solid).

3

Main results

We will first show that the main theorem from [14] can be proved in a much easier way. Moreover, our result is more general, since normality of the cone is not assumed. Theorem 3.1. Let (X, d) be a complete tvs-cone metric space and let f : X → X. Suppose that there exists k ∈ [0, 1) such that for all x, y ∈ X there is some sx,y ∈ co {θ, d(x, y), d(x, f x), d(y, f y), d(x, f y), d(y, f x)} so that d(f x, f y) ¹ k sx,y .

(3.1)

Then f has a unique fixed point u ∈ X and, for each x ∈ X, the iterative sequence {f n x} converges to u. Proof. Let e ∈ int K and let qe and dq have the meaning as in Theorem 2.1. Applying Minkowski functional qe , we obtain from (3.1) that for fixed x, y ∈ X, dq (f x, f y) = qe (d(f x, f y)) ≤ k qe (sx,y ). Here, qe (sx,y ) ∈ qe (co {θ, d(x, y), d(x, f x), d(y, f y), d(x, f y), d(y, f x)}) ⊂ co (qe ({θ, d(x, y), d(x, f x), d(y, f y), d(x, f y), d(y, f x)})) = co {qe (θ), qe (d(x, y)), qe (d(x, f x)), qe (d(y, f y)), qe (d(x, f y)), qe (d(y, f x))} = co {0, dq (x, y), dq (x, f x), dq (y, f y), dq (x, f y), dq (y, f x)} = [0, M ], where M = max{dq (x, y), dq (x, f x), dq (y, f y), dq (x, f y), dq (y, f x)}. We have proved that dq (f x, f y) ≤ k max{dq (x, y), dq (x, f x), dq (y, f y), dq (x, f y), dq (y, f x)}. Thus, f satisfies quasicontraction condition (1.1) in the metric space (X, dq ) and by Theorem 1.1 there is a unique fixed point u ∈ X of f . Moreover, for each x ∈ X, the Picard sequence {f n x} converges to u in metric dq , and hence (by Theorem 2.1) also in tvs-cone metric d. Now we generalize the previous result to the case of two mappings. Two variants of this generalization are possible (see further Theorems 3.2 and 3.3).

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Theorem 3.2. Let (X, d) be a tvs-cone metric space, and let f, g : X → X be two selfmappings such that f X ⊂ gX and gX is a complete subset of X. Suppose that there exists k ∈ [0, 1) such that for all x, y ∈ X there is some sx,y ∈ co {θ, d(gx, gy), d(gx, f x), d(gy, f y), d(gx, f y), d(gy, f x)} so that d(f x, f y) ¹ k sx,y .

(3.2)

Then f and g have a unique point of coincidence. If, moreover, f and g are weakly compatible, then they have a unique common fixed point u ∈ X and for each x ∈ X the corresponding Jungck sequence converges to u. Proof. In order to apply Theorem 1.2, it is enough to show that for arbitrary x, y ∈ X, dq (f x, f y) ≤ k max{dq (gx, gy), dq (gx, f x), dq (gy, f y), dq (gx, f y), dq (gy, f x)}. Similarly as in the previous proof, using (3.2), this follows from dq (f x, f y) = qe (d(f x, f y)) ≤ k qe (sx,y ), where qe (sx,y ) ∈ qe (co {θ, d(gx, gy), d(gx, f x), d(gy, f y), d(gx, f y), d(gy, f x)}) ⊂ co (qe ({θ, d(gx, gy), d(gx, f x), d(gy, f y), d(gx, f y), d(gy, f x)})) = co {0, dq (gx, gy), dq (gx, f x), dq (gy, f y), dq (gx, f y), dq (gy, f x)} = [0, M ], where M = max{dq (gx, gy), dq (gx, f x), dq (gy, f y), dq (gx, f y), dq (gy, f x)}. The conclusion about the Jungck sequence again follows using Theorem 2.1. Theorem 3.3. Let (X, d) be a complete tvs-cone metric space, and let f, g : X → X be two selfmappings. Suppose that there exists k ∈ [0, 1) such that for all x, y ∈ X there is some tx,y ∈ co {θ, d(x, y), d(x, f x), d(y, gy), 12 [d(x, gy) + d(y, f x)]} so that d(f x, gy) ¹ k tx,y .

(3.3)

Then f and g have a unique common fixed point in X. Moreover, each fixed point of f is a fixed point of g and conversely. Proof. Here, Theorem 1.3 is applied after checking that (3.3) implies that for all x, y ∈ X, dq (f x, gy) = qe (d(f x, gy)) ≤ k qe (tx,y ), where qe (tx,y ) ∈ qe (co {θ, d(x, y), d(x, f x), d(y, gy) 21 [d(x, gy) + d(y, f x)]}) ⊂ co {qe (θ), qe (d(x, y)), qe (d(x, f x)), qe (d(y, gy)), 12 [qe (d(x, gy)) + qe (d(y, f x))]} = co {0, dq (x, y), dq (x, f x), dq (y, gy), 21 [dq (x, gy) + dq (y, f x)]} = [0, M ],

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and M = max{0, dq (x, y), dq (x, f x), dq (y, gy), 21 [dq (x, gy) + dq (y, f x)]}. Remark 3.4. This result also answers in positive and in a more general way, B. Samet’s second question from [11, Remark 2.3]. Finally, fixed point result for Fisher’s quasicontraction (which was derived in [4]) can also be easily proved by our method. Theorem 3.5. Let (X, d) be a complete tvs-cone metric space and let f : X → X be a continuous selfmap. Suppose that there exist k ∈ [0, 1) and positive integers p, q such that for all x, y ∈ X there are some 0 ≤ r, r0 ≤ p and 0 ≤ s, s0 ≤ q and 0

0

ux,y ∈ co {θ, d(f r x, f s y), d(f r x, f r x), d(f s y, f s y)} so that d(f p x, f q y) ¹ k ux,y .

(3.4)

Then f has a unique fixed point u ∈ X and, for each x ∈ X, the iterative sequence {f n x} converges to u. Proof. In order to apply Theorem 1.4 we have to prove that 0

0

dq (f p x, f q y) ≤ k max{dq (f r x, f s y), dq (f r x, f r x), dq (f s y, f s y)} where 0 ≤ r, r0 ≤ p and 0 ≤ s, s0 ≤ q. Applying Minkowski functional qe to inequality (3.4) we get that dq (f p x, f q y) = qe (d(f p x, f q y)) ≤ kqe (ux,y ), where 0

0

0

0

qe (ux,y ) ∈ qe (co {θ, d(f r x, f s y), d(f r x, f r x), d(f s y, f s y)}) ⊂ co (qe {θ, d(f r x, f s y), d(f r x, f r x), d(f s y, f s y)}) 0

0

= co {0, dq (f r x, f s y), dq (f r x, f r x), dq (f s y, f s y)} = [0, M ], 0

0

M = max{dq (f r x, f s y), dq (f r x, f r x), dq (f s y, f s y)}, 0 ≤ r, r0 ≤ p and 0 ≤ s, s0 ≤ q. Using Theorems 1.4 and 2.1 the conclusion follows.

4

Examples

X. Zhang gave an example (see [14, Example 1]) showing that his fixed point result for generalized quasicontractions is more general than the classical one. We provide a similar example, but for the case of a nonormal cone. Hence, our Theorem 3.1 is more general than both Theorem 1.1 and Zhang’s [14, Theorem 3]. Example 4.1. Let E = CR1 [0, 1] with kuk = kuk∞ + ku0 k∞ , u ∈ E and let K = {u ∈ E : u(t) ≥ 0 on [0, 1]}. It is well known that this cone is solid but it is not normal. Now consider the space E = CR1 [0, 1] endowed with the strongest locally convex topology t∗ . Then K is also t∗ -solid (it has the nonempty t∗ -interior), but not t∗ -normal. (For details see [9, Example 2.2]). Let X = {a, b, c} and define a tvs-cone metric d : X × X → K by d(a, b)(t) = α(t) = 2 + 3t,

d(b, c)(t) = β(t) = 5 − 3t,

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d(a, c)(t) = γ(t) = 3,

DING ET AL: TVS-CONE METRIC SPACES

d(x, y) = d(y, x) and d(x, x) = θ for x, y ∈ X. Then (X, d) is a complete tvs-cone metric space over the nonnormal cone K. µ ¶ a b c Consider now the mapping f : X → X given by f = . Denote a c a Mx,y = {d(x, y), d(x, f x), d(y, f y), d(x, f y), d(y, f x)} for x, y ∈ X. Then Ma,b = {α, θ, β, γ, α}, Ma,c = {γ, θ, γ, θ, γ}, Mb,c = {β, β, γ, α, θ}. We will check that f is not a quasicontraction in a tvs-cone metric space (X, d), but it is a generalized quasicontraction in the sense of Zhang [14]. Take x = a, y = b. We have that d(f a, f b) = d(a, c) = γ and there is no sa,b ∈ Ma,b satisfying d(f a, f b) ¹ ksa,b with some k ∈ [0, 1). Take now k ∈ [ 67 , 1) and show that for all x, y ∈ X there exists sx,y ∈ co ({θ} ∪ Mx,y ) such that d(f x, f y) ¹ ksx,y . The only nontrivial cases are x = a, y = b and x = b, y = c. If x = a, y = b, then d(f a, f b) = d(a, c) = γ and d(f a, f b)(t) = γ(t) = 3 ≤ k( 21 α(t) + 12 β(t)) = k · 27 .

(4.1)

If x = b, y = c, then again d(f b, f c) = d(c, a) = γ and (4.1) again holds. Thus, f satisfies all requirements of Theorem 3.1 and it has a unique fixed point u = a. We provide now an example showing how our Theorem 3.2 can be used for proving the existence of a common fixed point of two mappings, while the classical result cannot. Similar examples can be constructed for Theorems 3.3 and 3.5. Example 4.2. Let E = R2 , K = {(x, y) : x, y ≥ 0}, X = {a, b, c} and let d : X × X → K be defined by d(a, b) = (5, 2), d(b, c) = (2, 5), d(a, c) = (3, 3), d(x, y) = d(y, x) and d(x, x) = (0, 0) = θ for x, y ∈ X. Then (X, d) is a complete tvs-cone metric space over the (normal) cone K. Consider the following two mappings f, g : X → X: µ f=

a a

¶ b c , a c

µ g=

a b a c

¶ c . b

Then, e.g., d(f a, f c) = d(a, c) = (3, 3) and one cannot find constant k ∈ [0, 1) such that d(f a, f c) ¹ kua,c with ua,c ∈ Ma,c = {d(ga, gc), d(ga, f a), d(gc, f c), d(ga, f c), d(gc, f a)} = {d(a, b), d(a, a), d(b, c), d(a, c), d(a, b)} = {(5, 2), θ, (2, 5), (3, 3)}. However, taking any k with kua,c holds. Indeed,

6 7

< k < 1, there exists ua,c ∈ co ({θ} ∪ Ma,c ) such that d(f a, f c) ¹

d(f a, f c) = (3, 3) ¹ k[ 12 · (5, 2) +

1 2

· (2, 5)] = k[ 12 · d(ga, gc) +

1 2

· (gc, f c)].

It is easy to check that the same conclusion holds for other pairs of elements of X. Hence, f and

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DING ET AL: TVS-CONE METRIC SPACES

g satisfy all the conditions of Theorem 3.2 and they have a unique common fixed point a.

5

Acknowledgements

Hui-Sheng Ding acknowledges the support from the NSF of China, the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), the Jiangxi Provincial Education Department (GJJ12173), and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University. The second, third and fourth author are thankful to the Ministry of Science and Technological Development of Serbia.

References ´ c, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 [1] Lj.B. Ciri´ (1974), 267–273. [2] K.M. Das, K.V. Naik, Common fixed point theorems for commuting maps on metric spaces, Proc. Amer. Math. Soc. 77 (1979), 369–373. [3] B. Fisher, Quasi-contractions on metric spaces, Proc. Am. Math. Soc. 75 (1979), 321–325. ´ c maps with a general contractive iterate at a point and [4] Lj. Gaji´c, D. Ili´c, V. Rakoˇcevi´c, On Ciri´ Fisher’s quasicontractions in cone metric spaces, Apll. Math. Comput. 216 (2010), 2240–2247. [5] L.G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 323 (2007), 1468–1476. [6] D. Ili´c, V. Rakoˇcevi´c, Quasi-contraction on a cone metric space, Appl. Math. Lett. 22 (2009), 728–731. [7] S. Jankovi´c, Z. Kadelburg, S. Radenovi´c, On cone metric spaces: A survey, Nonlinear Anal. TM&A 74 (2011), 2591–2601. [8] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976), 261–263. [9] Z. Kadelburg, S. Radenovi´c, V. Rakoˇcevi´c, Topological vector space-valued cone metric spaces and fixed point theorems, Fixed Point Theory Appl. (2010), Article ID 170253, 17 pages, doi:10.1155/2010/170253. [10] Z. Kadelburg, S. Radenovi´c, V. Rakoˇcevi´c, A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett. 24 (2011), 370–374. ´ c’s type in cone metric [11] B. Samet, Common fixed point under contractive condition of Ciri´ spaces, Appl. Anal. Discrete Math. 5 (2011), 159–164. [12] H.H. Schaefer, Topological Vector Spaces, 3rd ed., Berlin-Heidelberg-New York: SpringerVerlag, 1971. [13] P.P. Zabre˘ıko, K-metric and K-normed spaces: survey, Collectanea Math. 48, 4–6 (1997), 825–859. [14] X. Zhang, Fixed point theorem of generalized quasicontractive mapping in cone metric spaces, Comput. Math. Appl. 62 (2011), 1627–1633.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 471-480, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

SOME PROPERTIES OF BERNOULLI AND EULER POLYNOMIALS ARISING FROM THE p-ADIC INTEGRAL ON Zp D.S. KIM, T. KIM, S.H. LEE, D.V. DOLGY Abstract. In this paper we give some interesting identities on the Bernoulli and Euler polynomials. To derive our identities, we investigate some properties and equations on the p-adic integral on Zp .

1. Introduction Let p be a fixed odd prime number. Throughout this paper Zp , Qp and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp . Let vp be the normalized exponential valuation of Cp with |p|p = p−vp (p) = p−1 . Let U D(Zp ) be the space of uniformly differentiable function on Zp . For f ∈ U D(Zp ), p-adic integers on Zp for f in the sense of bosonic is defined by Z

(1)

N

p −1 1 X f (x), I1 (f ) = f (x)dµ(x) = lim N N →∞ p Zp

(see [10]).

x=0

The fermionic p-adic integral on Zp is also defined by Kim as follows: Z (2)

I−1 (f ) =

Zp

f (x)dµ−1 (x) = lim

N →∞

N −1 pX

f (x)(−1)x ,

x=0

From (1) and (2), we have the following equations : (3)

I1 (f1 ) − I1 (f ) = f 0 (0),

(see [11]),

and (4)

I−1 (f1 ) + I−1 (f ) = 2f (0),

(see [9]),

where f1 (x) = f (x + 1). The Euler numbers of the first kind are defined by ∞

(5)

X tn t Et , E = e = n et + 1 n! n=0

471

(see [10]).

D.S. KIM, T. KIM, S.H. LEE, D.V. DOLGY

with the usual convention about replacing E n by En , The Euler numbers of the second kind are defined by

(see [1-15]).

∞

n X 1 2 ∗t = t = E . n cosh t e + e−t n!

(6)

n=0

∗ Note that E2k = 0 and E2k+1 = 0, for k ∈ N. As is well known, Bernoulli polynomials are defined by ∞

X t tn xt B(x) e = e t = B , n et − 1 n!

(7)

(see [14]).

n=0

with the usual convention about replacing B n (x) by Bn (x). In the special case, x = 0, Bn (0) = Bn are called the n-th Bernoulli numbers. Now, the Euler polynomials are also defined by n µ ¶ X n n−` En (x) = (8) x E` , (see [11-15]), ` `=0

where E` are the `-th Euler numbers of the first kind. From (7), we can derive the following equation : n µ ¶ X n n−` (9) x B` , (see [11-15]). Bn (x) = ` `=0

The following identities for the Bernoulli polynomials are known in (see [6,7]) : for k ∈ N ¶ k µ X B2j (x) 2k 1 1 22j (10) = 22k (x − )2k − , 2j − 1 2j 2 2k + 1 j=1

and (11)

¶ k µ X 2k + 1 B2j (x) j=1

2j − 1

2j

=

x2k+1 − (x − 1)2k+1 1 − . 2 2k + 1

For k ∈ Z+ = N ∪ {0} , let us know that ¶ k µ X x2k+1 + (x − 1)2k+1 2k + 1 B2j+1 (x) (12) = , 2j 2j + 1 2 j=0

and (13)

¶ k µ X 2k + 1 2j B2j+1 (x) 1 2 = 22k (x − )2k+1 , 2j 2j + 1 2

(see [6,7]).

j=0

In this paper, we derive some properties and new identities for the Bernoulli and Euler polynomials.To derive our identities we use p-adic integral equation on Zp .

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SOME PROPERTIES OF BERNOULLI AND EULER POLYNOMIALS

2. Identities for the Bernoulli and Euler polynomials From (1) and (3), we note that Z Z Z 2tx tx (14) e dµ(x) = e dµ(x) Zp

Zp

Zp

etx dµ−1 (x).

Thus, by (14), we have Z Z Z 1 1 1 2t(x+ 12 ) tx (15) e dµ(x) = e dµ(x) − e2tx dµ(x). 2t Zp t Zp 2t Zp By (15), we get 1 2 2 = t − 2t . sinh t e −1 e −1 From (15), we can also derive the following p-adic integral equation on Zp : Z Z Z 1 n n−1 n n−1 (16) 2 (x + ) dµ(x) = x dµ(x) − 2 xn dµ(x). 2 Zp Zp Zp Therefore (3),(7) and (15), we obtain the following proposition: Proposition 1 . For n ∈ Z+ , we have 1 1 Bn ( ) = ( n−1 − 1)Bn . 2 2 Taking

R Zp

dµ(x) on both sides in (10), we have

I1 =

¶ Z k µ X 2k 1 B2j (x)dµ(x) 22j 2j Zp 2j − 1 j=1

¶ Z 2j µ ¶ k µ X X 2k 2j 2j 1 = 2 B2j−` x` dµ(x) 2j 2j − 1 ` Zp

(17)

j=1

=

2j k X X j=1 `=0

`=0

¶µ ¶ 1 2j 2k 2j 2 B2j−` B` , 2j 2j − 1 `

On the other hand,

µ

Z

1 1 (x − )2k dµ(x) − 2 2k + 1 Zp (18) 1 1 . = 22k B2k (− ) − 2 2k + 1 By (3), we easily see that Z Z m (19) (x + y) dµ(y) − (x + y − 1)m dµ(y) = m(x − 1)m−1 . I1 = 22k

Zp

Zp

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D.S. KIM, T. KIM, S.H. LEE, D.V. DOLGY

Thus, by (19), we have Bm (x) − Bm (x − 1) = m(x − 1)m−1 ,

(m ∈ Z+ ).

Let us take x = 21 . Then we get 1 1 1 Bm (− ) = Bm ( ) + m( )m−1 (−1)m 2 2 2 1 1 = ( m−1 − 1)Bm + m( )m−1 (−1)m . 2 2

(20)

Therefore, by (17),(18) and (20), we obtain the following theorem. Theorem 2 . For k ∈ N , we have µ ¶µ ¶ 2j k X X 22j−1 2k 2j 1 B2j−` B` = (2 − 22k )B2k + 4k − . j 2j − 1 ` 2k + 1 j=1 `=0

By (4),(5) and (6), we easily see that Z ∞ X 1 1 tn (21) e2t(x+ 2 ) dµ−1 (x) = = En∗ . cosh t n! Zp n=0

Thus , by (21), we have Z n

(22)

2

1 (x + )n dµ−1 (x) = En∗ . 2 Zp

By (22), we get 1 2n En ( ) = En∗ . 2

(23)

From (4), we note that Z Z n (24) (y + x) dµ−1 (y) + (y + x − 1)n dµ−1 (y) = 2(x − 1)n . Zp

Zp

By (4),(5) and (24), we get (25)

En (x) + En (x − 1) = 2(x − 1)n ,

f or

n ∈ Z+ .

Therefore, by (25), we obtain the following proposition. Proposition 3 . For n ∈ Z+ , we have 1 1 1 En ( ) + En (− ) = (−1)n ( )n−1 . 2 2 2

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SOME PROPERTIES OF BERNOULLI AND EULER POLYNOMIALS

Let us take the fermionic p-adic integral on Zp in (10) as follows: ¶ Z 2j µ ¶ k µ X X 2k 2j 2j−1 1 I2 = 2 B2j−` x` dµ−1 (x) 2j − 1 j ` Zp j=1 `=0 (26) ¶ 2j µ ¶ k µ X X 2k 2j 2j−1 1 B2j−` E` . = 2 2j − 1 j ` j=1

`=0

On the other hand,

Z

1 1 (x − )2k dµ−1 (x) − 2 2k + 1 Zp (27) 1 1 = 22k E2k (− ) − . 2 2k + 1 By (27) and Proposition 3, we get 1 1 I2 = 22k E2k (− ) − 2 2k + 1 1 1 1 (28) = 22k {−E2k ( ) + 2(− )2k } − 2 2 2k + 1 1 1 . = −22k E2k ( ) + 2 − 2 2k + 1 From (23) and (28), we note that 1 1 I2 = −22k E2k ( ) + 2 − 2 2k + 1 (29) 1 ∗ = −E2k +2− . 2k + 1 Therefore, by (26) and (29), we obtain the following theorem. 2k

I2 = 2

Theorem 4 . For k ∈ N, we have ¶ 2j µ ¶ k µ X X 2k 2j 1 2j−1 1 ∗ 2 B2j−` E` = −E2k +2− . 2j − 1 j ` 2k + 1 j=1

`=0

Let us consider the p-adic integral on Zp in (12) as follows: ¶ Z k µ X 2k + 1 1 I3 = B2j+1 (x)dµ(x) 2j 2j + 1 Zp j=0

(30)

=

¶ k µ X 2k + 1 j=0

2j

Z 2j+1 X µ2j + 1¶ 1 B2j+1−` x` dµ(x), 2j + 1 ` Zp `=0

k 2j+1 X X µ2k + 1¶ 1 µ2j + 1¶ = B2j+1−` B` . 2j + 1 ` 2j j=0 `=0

475

D.S. KIM, T. KIM, S.H. LEE, D.V. DOLGY

On the other hand,

Z 1 I3 = (x2k+1 + (x − 1)2k+1 )dµ(x) 2 Zp Z 1 = (x2k+1 − (x + 2)2k+1 )dµ(x) 2 Z p (31) 1 1 = B2k+1 − B2k+1 (2) 2 2 1 = − (2k + 1 + δ0,k ). 2 Therefore, by (30) and (31), we obtain the following proposition. Proposition 5 . For k ∈ Z+ , we have k 2j+1 X X j=0 `=0

µ ¶µ ¶ 1 2k + 1 2j + 1 1 B2j+1−` B` = − (2k + 1 + δ0,k ). 2j + 1 2j ` 2

Now we consider the fermionic p-adic integral on Zp in (12) as follows: ¶ Z 2j+1 k µ X X µ2j + 1¶ 2k + 1 1 B2j+1−` x` dµ−1 (x), I4 = 2j + 1 ` 2j Zp j=0 `=0 (32) µ ¶ µ ¶ 2j+1 k X X 2j + 1 2k + 1 1 = B2j+1−` E` . 2j 2j + 1 ` j=0

`=0

On the other hand, by (12), we get Z 1 I4 = (x2k+1 + (x − 1)2k+1 )dµ−1 (x) 2 Zp Z 1 = (x2k+1 − (x + 2)2k+1 )dµ−1 (x) 2 Zp (33) 1 = (E2k+1 − E2k+1 (2)) 2 1 = (E2k+1 − (2 + E2k+1 − δ0,2k+1 )) 2 = −1. Therefore, by (32) and (33), we obtain the following corollary. Corollary 6 . For k ∈ Z+ , we have k 2j+1 X X j=0 `=0

µ ¶µ ¶ 2k + 1 2j + 1 1 B2j+1−` E` = −1 2j + 1 2j `

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SOME PROPERTIES OF BERNOULLI AND EULER POLYNOMIALS

Let us take p-adic integral on both sides in equation (11). I5 =

¶ Z k µ X 2k + 1 1 B2j (x)dµ(x) 2j − 1 2j Zp j=1

(34)

¶ Z 2j µ ¶ k µ X 2k + 1 1 X 2j = B2j−` x` dµ(x), 2j− 2j ` Zp j=1

=

`=0

¶ 2j µ ¶ 2k + 1 1 X 2j B2j−` B` , ` 2j − 1 2j

k µ X j=1

f or

k ∈ N.

`=0

On the other hand, by (11), we get Z 1 1 (x2k+1 − (x − 1)2k+1 )dµ(x) − I5 = 2 Zp 2k + 2 Z 1 1 = (x2k+1 + (x + 2)2k+1 )dµ(x) − 2 Zp 2k + 2 1 1 (35) = (B2k+1 + B2k+1 (2)) − 2 2k + 2 1 1 = (B2k+1 + B2k+1 + δ1,2k+1 ) − 2 2k + 2 1 2k + 1 − . = 2 2k + 2 Therefore, by (34) and (35), we obtain the following theorem. Theorem 7 . For k ∈ N , we have µ ¶µ ¶ 2j k X X 1 2k + 1 2j k(2k + 3) B2j−` E` = . 2j 2j − 1 ` 2k + 2 j=1 `=0

Let us consider the p-adic integral on Zp in equation (13) as follows : ¶ Z k µ X 2k + 1 2j 1 I6 = 2 B2j+1 (x)dµ(x) 2j 2j + 1 Zp j=0

(36)

Z k 2j+1 X X µ2k + 1¶µ2j + 1¶ 1 2j = B2j+1−` x` dµ(x), 2 2j + 1 2j ` Zp j=0 `=0

k 2j+1 X X µ2k + 1¶µ2j + 1¶ 1 = B2j+1−` B` . 22j 2j + 1 2j ` j=0 `=0

477

D.S. KIM, T. KIM, S.H. LEE, D.V. DOLGY

On the other hand, by (13) and (20), we get Z 1 2k I6 = 2 (x − )2k+1 dµ(x) 2 Zp 1 = 22k B2k+1 (− ) (37) 2 1 2k 1 2k = 2 {( ) − 1)B2k+1 − (2k + 1)( )2k } 2 2 = (1 − 22k )B2k+1 − (2k + 1). Therefore, by (36) and (37), we obtain the following theorem. Theorem 8 . For k ∈ Z+ , we have k 2j+1 X X j=0 `=1

µ ¶µ ¶ 1 2j + 1 1 2j 2k + 1 2 B2j+1−` B` = (22k−1 − )δ0,k − (2k + 1). 2j + 1 2j ` 2

Now we consider the fermionic p-adic integral on Zp in (13) as follows:

(38)

¶ Z 2j+1 k µ X X µ2j + 1¶ 2k + 1 2j 1 I7 = 2 B2j+1−` x` dµ−1 (x) 2j 2j + 1 ` Zp j=0

=

`=0

k 2j+1 X Xµ j=0 `=0

¶µ ¶ 2j 2k + 1 2j + 1 2 B2j+1−` E` . 2j ` 2j + 1

On the other hand, by (13), we get Z 1 2k I7 = 2 (x − )2k+1 dµ−1 (x) 2 Zp 1 = 22k E2k+1 (− ) 2 1 1 = 22k (−E2k+1 ( ) + 2(− )2k+1 ) 2 2 (39) 1 2k = −2 E2k+1 ( ) − 1 2 1 1 2k+1 E2k+1 ( ) − 1 =− 2 2 2 1 ∗ = − E2k+1 − 1 2 = −1. Therefore, by (38) and (39), we obtain the following corollary.

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SOME PROPERTIES OF BERNOULLI AND EULER POLYNOMIALS

Corollary 9 . For k ∈ Z+ , we have k 2j+1 X X µ2k + 1¶µ2j + 1¶ 22j B2j+1−` E` = −1. 2j ` 2j + 1 j=0 `=0

References [1] S. Araci, D. Erdal, D-J. Kang, Some new properties on the q-Genocchi numbers and polynomials ssociated with q-Bernstein polyomials, Honam. Math. J. 33 (2011), 261270. [2] A. Bayad, Modular properties of elliptic Bernoulli and Euler functions, Adv. Stud. Contemp. Math. 20 (2010), no. 3, 389-401. [3] A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials , Russ. J. Math. Phys. 18 (2011), no. 2, 133-143. [4] L. Carlitz, Note on the integral of the product of several Bernoulli polynomials , J. London Math. Soc. 34 (1959), 361-363. [5] L. Carlitz, Some arithmetic properties of generalized Bernoulli numbers,Bull. Amer. Math. Soc. 65 ,(1959), 68-69. [6] H. Cohen, Number Theory, Volume II. Analytic and Modern Tools ,Graduate Texts in Mathematics, 240, Springer, New York , (2007). [7] H. Cohen, Number Theory, Volume I. Tools and Diophantine Equations,Graduate Texts in Mathematics, 239, Springer, New York , (2007). [8] D. Ding, J. Yang, Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math. 20 (2010), 7-21. [9] T.Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys. 16 (2009), no. 4, 484-491. [10] T.Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on Zp , Russ. J. Math. Phys. 16 (2009), 93-96. [11] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288-299. [12] H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. 18 (2009), 41-48. [13] C. S. Ryoo, Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstein polynomials, Proc. Jangjeon Math. Soc. 14 (2011), no. 2, 239-248. [14] Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math. 6 (2008), no. 2, 251-278. [15] Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers, Adv. Stud. Contemp. Math. 11(2005), 205-218.

D.S. Kim Department of Mathematics, Sogang University, Seoul 121-742, S. Korea

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D.S. KIM, T. KIM, S.H. LEE, D.V. DOLGY

T. KIM Department of Mathematics, Kwangwoon University, Seoul 139-701, S.Korea E-mail: [email protected]

S.H. Lee Division of General Education, Kwangwoon University, Seoul 139-701, S. Korea E-mail: [email protected]

D.V. Dolgy Hanrimwon,Kwangwoon University, Seoul 139-701, S. Korea

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 481-497, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Representations for Drazin inverse of block matrix∗ Jelena Ljubisavljevi´c†, Dragana S. Cvetkovi´c-Ili´c‡

Abstract In this paper we offer new representations for Drazin inverse of block matrix, which recover some representations from current literature on this subject. 2000 Mathematics Subject Classification: 15A09 Key words: Drazin inverse; block matrix; representation.

1

Introduction

Let A be a square complex matrix. By rank(A) we denote the rank of matrix A. The index of matrix A, denoted by ind(A), is the smallest nonnegative integer k such that rank(Ak+1 ) = rank(Ak ). For every matrix A ∈ Cn×n , such that ind(A) = k, there exists the unique matrix Ad ∈ Cn×n , which satisfies following relations: Ak+1 Ad = Ak , Ad AAd = Ad , AAd = Ad A. Matrix Ad is called the Drazin inverse of matrix A (see [1]). In the case ind(A) = 1, the Drazin inverse of A is called the group inverse of A, denoted by A# or Ag . The case ind(A) = 0 is valid if and only if A is nonsingular, so in that case Ad reduces to A−1 . Throughout this paper we suppose that P A0 = I, where I is identity matrix, and k−j ∗ = 0, for k ≤ j. i=1 The theory of Drazin inverse of square matrix has numerous applications, such as in singular differential equations and singular difference equations, ∗

supported by Grant No. 174007 of the Ministry of Science and Technological Development, Republic of Serbia † University of Niˇs, Faculty of Medicine ‡ University of Niˇs, Faculty of Science and Mathematics

481

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

Markov chains and iterative methods (see [2, 4, 5, 6, 8, 9]). An application of the Drazin inverse of a 2 × 2 block matrix can be found in [2, 3, 7]. In 1979 Campbell and Meyer[4] posed the problem of finding an explicit representation for the Drazin inverse of 2 × 2 complex matrix   A B M= , (1.1) C D in terms of its blocks, where A and D are square matrices, not necessarily of the same size. Until now, there has been no formula for M d without any side conditions for blocks of matrix M . However, many papers studied special cases of this open problem and offered a formula for M d under some specific conditions for blocks of M . Here we list some of them: (i) B = 0 (or C = 0) (see [10, 11]); (ii) BC = 0, BD = 0 and DC = 0 (see [6]); (iii) BC = 0, DC = 0 (or BD = 0) and D is nilpotent (see [7]); (iv) BC = 0 and DC = 0 (see [12]); (v) CB = 0 and AB = 0 (or CA = 0) (see [12, 13]); (vi) BCA = 0, BCB = 0, DCA = 0 and DCB = 0 (see [14]); (vii) ABC = 0, CBC = 0, ABD = 0 and CBD = 0 (see [14]); (viii) BCA = 0, BCB = 0, ABD = 0 and CBD = 0 (see [15]); (ix) BCA = 0, DCA = 0, CBC = 0, and CBD = 0 (see [15]); (x) BCA = 0, BD = 0 and DC = 0 (or BC is nilpotent) (see [16]); (xi) BCA = 0, DC = 0 and D is nilpotent (see [16]); (xii) ABC = 0, DC = 0 and BD = 0 (or BC is nilpotent, or D is nilpotent) (see [17]); (xiii) BCA = 0 and BD = 0 (see [18]); (xiv) ABC = 0 and DC = 0 (or BD = 0) (see [18, 19]). In this paper we derive representations for M d which recover representations from previous list.

482

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

2

Key lemmas

In order to prove our main results, we first state some lemmas. Lemma 2.1 [14] Let P, Q ∈ Cn×n be such that ind(P ) = r and ind(Q) = s. If P QP = 0 and P Q2 = 0 then   (P + Q)d = Y1 + Y2 + Y1 (P d )2 + (Qd )2 Y2 − Qd (P d )2 − (Qd )2 P d P Q, where Y1 =

s−1 X

Qπ Qi (P d )i+1 , Y2 =

i=0

r−1 X

(Qd )i+1 P i P π .

(2.1)

i=0

Cn×n

Lemma 2.2 [14] Let P, Q ∈ be such that ind(P ) = r and ind(Q) = s. 2 If QP Q = 0 and P Q = 0 then   d 2 d 2 d d 2 d 2 d d (P + Q) = Y1 + Y2 + P Q Y1 (P ) + (Q ) Y2 − Q (P ) − (Q ) P , where Y1 and Y2 are defined by (2.1). Lemma 2.3 [20] Let M ∈

Cn×n



0 B C 0

be such that M =



, B ∈ Cp×(n−p) ,

C ∈ C(n−p)×p . Then d

M =



0 B(CB)d (CB)d C 0

 .

Deng and Wei [21] gave representations for the Drazin inverse of upper anti-triangular block matrix under some specific conditions. Here we state these results and some additional facts, which we will be useful to prove our results. Consider the block matrix of a form (1.1), where D = 0:   A B M= . (2.2) C 0 Lemma 2.4 [21] Let M ∈ Cn×n be matrix of a form (2.2). If ABC = 0, then   ΦA ΦB Md = , CΦ CΦ2 AB where Φ = (A2 + BC)d =

tX 1 −1

(BC)π (BC)i (Ad )2i+2 +

i=0

and t1 = ind(BC), ν1 =

νX 1 −1

((BC)d )i+1 A2i Aπ (2.3)

i=0

ind(A2 ).

483

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

Remark 1 Let M be matrix of a form (2.2). If conditions of Lemma 2.4 are satisfied, we have that:   (A2 + BC)k A (A2 + BC)k B 2k+1 M = , for k ≥ 1 C(A2 + BC)k C(A2 + BC)k−1 AB and M

2k

 =

(A2 + BC)k (A2 + BC)k−1 AB 2 k−1 C(A + BC) A C(A2 + BC)k−1 B k X

Notice that (A2 + BC)k =

 , for k ≥ 1.

(BC)k−j A2j , for k ≥ 0. Also, (A2 + BC)π =

j=0

Aπ − BCΦ = (BC)π − ΦA2 . We can check that k

Φ =

tX 1 −1

π

i

d 2i+2k

(BC) (BC) (A )

νX 1 −1

+

i=0

d i+k

((BC) )

k−1 X A A − ((BC)d )k−i (Ad )2i , 2i

i=0

π

i=1

for k ≥ 1. Therefore we have   k+1 Φ A Φk+1 B d 2k+1 , for k ≥ 0 (M ) = CΦk+1 CΦk+2 AB and d 2k

(M )

 =

Φk Φk+1 AB k+1 CΦ A C(Φk+1 B

 , for k ≥ 1.

Lemma 2.5 [21] Let M ∈ Cn×n be as in (2.2). If BCA = 0, then   AΩ ΩB d M = , CΩ CAΩ2 B where 2

d

Ω = (A + BC) =

tX 1 −1

d 2i+2

(A )

i

π

(BC) (BC) +

i=0

νX 1 −1 i=0

and t1 = ind(BC), ν1 = ind(A2 ).

484

Aπ A2i ((BC)d )i+1 (2.4)

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

Remark 2 Let M be matrix of a form (2.2). If conditions of Lemma 2.5 hold, we have that:   A(A2 + BC)k (A2 + BC)k B 2k+1 M = , for k ≥ 1 C(A2 + BC)k CA(A2 + BC)k−1 B and M 2k =



(A2 + BC)k A(A2 + BC)k−1 B 2 k−1 CA(A + BC) C(A2 + BC)k−1 B k X

Clearly, (A2 + BC)k =

 , for k ≥ 1.

A2j (BC)k−j , for k ≥ 0. Also (A2 + BC)π =

j=0

Aπ − ΩBC = (BC)π − A2 Ω. Furthermore, we have that k

Ω =

tX 1 −1

d 2i+2k

(A )

i

π

νX 1 −1

(BC) (BC) +

i=0

k−1 X Aπ A2i ((BC)d )i+k − (Ad )2i ((BC)d )k−i ,

i=0

i=1

for k ≥ 1. Hence we get that   AΩk+1 Ωk+1 B , for k ≥ 0 (M d )2k+1 = CΩk+1 CAΩk+2 B and d 2k

(M )

 =

Ωk AΩk+1 B k+1 CAΩ CΩk+1 B

 , for k ≥ 1.

In following two lemmas we present two new representations for Drazin inverse of lower anti-triangular block matrix. Consider the block matrix of a form (1.1) such that A = 0:   0 B . (2.5) M= C D Lemma 2.6 Let M ∈ Cn×n be matrix of a form (2.5). If DCB = 0, then   BΨ2 DC BΨ d M = , ΨC ΨD where Ψ = (D2 +CB)d =

tX 2 −1

νX 2 −1

i=0

i=0

(CB)π (CB)i (Dd )2i+2 +

and t2 = ind(CB), ν2 = ind(D2 ).

485

((CB)d )i+1 D2i Dπ (2.6)

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

Proof. First, notice that from DCB = 0 we have that matrices D2 and CB satisfy the conditions of Lemma 2.1. Hence we get (D2 + CB)d =

tX 2 −1

νX 2 −1

i=0

i=0

(CB)π (CB)i (Dd )2i+2 +

((CB)d )i+1 D2i Dπ .

Consider the splitting of matrix M       0 B 0 0 0 B M= = + := P + Q. C D 0 D C 0 Since DCB = 0 we have that P Q2 = 0. Also, we have P QP = 0. Therefore matrices P and Q satisfy the conditions of Lemma 2.1 and   (P +Q)d = Y1 +Y2 + Y1 (P d )2 + (Qd )2 Y2 − Qd (P d )2 − (Qd )2 P d P Q, (2.7) where Y1 , Y2 are as in (2.1). Clearly,     (BC)k 0 0 B(CB)k 2k+1 , for k ≥ 0. Q2k = , Q = (CB)k C 0 0 (CB)k Furthermore, by Lemma 2.3 we have   B((CB)d )k+1 0 , for k ≥ 1, (Qd )2k = 0 ((CB)d )k   0 B((CB)d )k+1 d 2k+1 , for k ≥ 0. (Q ) = ((CB)d )k+1 C 0 After computing, we get 

tX 2 −1

 π

i

d 2i+2

(CB) (CB) (D )  0 B  i=0  Y1 =  tX 2 −1  0 (CB)π (CB)i (Dd )2i+1

  ,  

(2.8)

i=0

   Y2 =   

0 (CB)d C

B

νX 2 −1

 d i+1

((CB) )

i=0 νX 2 −1

2i

D D

π

((CB)d )i+1 D2i+1 Dπ

  .  

(2.9)

i=0

After substituting (2.8) and (2.9) into (2.7) we get that the statement of the lemma is valid. 2

486

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

Remark 3 Let M be matrix of a form (2.5) such that DCB = 0. Then   B(D2 + CB)k−1 DC B(D2 + CB)k 2k+1 M = , for k ≥ 1 (D2 + CB)k C (D2 + CB)k D and M 2k =

B(D2 + CB)k−1 C B(D2 + CB)k−1 D (D2 + CB)k−1 DC (D2 + CB)k



It can be checked easily that (D2 + CB)k =

k X

 , for k ≥ 1.

(CB)k−j D2j , for k ≥ 0, and

j=0

(D2 + CB)π = Dπ − CBΨ = (CB)π − ΨD2 . Also, we have that Ψk =

tX 2 −1

νX 2 −1

i=0

i=0

(CB)π (CB)i (Dd )2i+2k +

k−1 X ((CB)d )i+k D2i Dπ − ((CB)d )k−i (Dd )2i , i=1

for k ≥ 1. Therefore we get   BΨk+2 DC BΨk+1 , for k ≥ 0 (M d )2k+1 = Ψk+1 C Ψk+1 D and (M d )2k =



BΨk+1 C BΨk+1 D Ψk+1 DC Ψk

 , for k ≥ 1.

Using the similar method as in the proof of Lemma 2.6 we can get the following result. Lemma 2.7 Let M ∈ Cn×n be as in (2.5). If CBD = 0, then   BDΓ2 C BΓ d M = , ΓC DΓ where Γ=

tX 2 −1

d 2i+2

(D )

i

π

(CB) (CB) +

i=0

νX 2 −1 i=0

and t2 = ind(CB), ν2 = ind(D2 ).

487

Dπ D2i ((CB)d )i+1

(2.10)

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

Proof. Since CBD = 0, using Lemma 2.1 we get (2.10). Now, if we split matrix M as       0 B 0 B 0 0 M= = + := P + Q, C D C 0 0 D we have that QP Q = 0 and P 2 Q = 0. Hence, the conditions of Lemma 2.2 are satisfied. After applying Lemma 2.2 and Lemma 2.3 we complete the proof.2 Remark 4 Let M be as in (2.5) and let CBD = 0. Then   BD(D2 + CB)k−1 C B(D2 + CB)k 2k+1 M = , for k ≥ 1 (D2 + CB)k C D(D2 + CB)k and M

Clearly

2k

 =

(D2

+

B(D2 + CB)k−1 C BD(D2 + CB)k−1 (D2 + CB)k−1 C (D2 + CB)k CB)k

=

k X

 , for k ≥ 1.

D2j (CB)k−j , for k ≥ 0, and (D2 + CB)π =

j=0

Dπ − ΓCB = (CB)π − D2 Γ. In addition, we can get that k

Γ =

tX 2 −1

d 2i+2k

(D )

i

π

νX 2 −1

(CB) (CB) +

i=0

π

2i

d i+k

D D ((CB) )

i=0

k−1 X − (Dd )2i ((CB)d )k−i , i=1

for k ≥ 1. Also, we can get that   BDΓk+2 C BΓk+1 d 2k+1 (M ) = , for k ≥ 0 Γk+1 C DΓk+1 and (M d )2k =

3



BΓk+1 C BDΓk+1 DΓk+1 C Γk

 , for k ≥ 1.

Representations

Consider the block matrix M of a form (1.1). Djordjevi´c and Stanimirovi´c [6] gave explicit representation for M d under conditions BC = 0, BD = 0 and DC = 0. This result was extended to a case BC = 0, DC = 0 (see [12]). As another generalization of these results, Yang and Liu [14] gave the

488

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

representation for M d under conditions BCA = 0, BCB = 0, DCA = 0 and DCB = 0. In the next theorem we derive an explicit representation for M d under conditions BCA = 0, DCA = 0 and DCB = 0. Therefore we can see that the condition BCB = 0 from [14] is superfluous. Theorem 3.1 Let M be matrix of a form (1.1) such that BCA = 0, DCA = 0 and DCB = 0. Then   Ad + Σ 0 C BΨ + AΣ0   M d =  ΨC + CAΣ1 C + C(Ad )2 , d D + CΣ0 d 2 2 −CA (BΨ D + ABΨ )C where     Σk = V1 Ψk + (Ad )2k V2 D + A V1 Ψk + (Ad )2k V2 , for k = 0, 1, V1 =

νX 1 −1

Aπ A2i BΨi+2 ,

(3.1)

(3.2)

i=0

V2 =

µX 1 −1

µ1 X

i=0

i=0

(Ad )2i+4 B(D2 + CB)i Dπ −

(Ad )2i+2 B(CB)i Ψ,

(3.3)

ν1 = ind(A2 ), µ1 = ind(D2 + CB) and Ψ is defined by (2.6). Proof. Consider the splitting of matrix M       A 0 0 B A B := P + Q. + = M= 0 0 C D C D Since BCA = 0 and DCA = 0 we get P 2 Q = 0 and QP Q = 0. Hence matrices P and Q satisfy the conditions of Lemma 2.2 and (P + Q)d = Y1 + Y2 + P QY1 (P d )2 + P Qd Y2 − P QQd (P d )2 − P Qd P d , (3.4) where Y1 and Y2 are as in (2.1). By the assumption of the theorem DCB = 0 we have that matrix P satisfy the conditions of Lemma 2.6. After applying Lemma 2.6 and using Remark 3, we get   (V1 D + AV1 )C Aπ BΨ + A(V1 D + AV1 ) Y1 = , (3.5) ΨC ΨD  d  A + (V2 D + AV2 )C BΨ − Aπ BΨ + A(V2 D + AV2 ) Y2 = , (3.6) 0 0

489

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

where V1 and V2 are defined by (3.2) and (3.3), respectively. After substituting (3.5) and (3.6) into (3.4) and computing all elements of (3.4) we obtain the result. 2 As a direct corollary of the previous theorem we get the following result. Corollary 3.1 Let M be as in (1.1). If DCB = 0 and CA = 0 then  d  A + Σ0 C BΨ + AΣ0 d M = , ΨC ΨD where Σ0 is defined by (3.1) and Ψ is given in (2.6). Notice that Corollary 3.1, therefore and Theorem 3.1 is also a generalization of representation for M d under conditions CB = 0 and CA = 0 which is given in [13]. The next result is a corollary of Theorem 3.1. we can get the  Also,   A B 0 0 := P + Q + following result using the splitting M = C 0 0 D and applying Lemma 2.1 and Lemma 2.5. Corollary 3.2 Let M be matrix of a form (1.1). If BCA = 0 and DC = 0 then   AΩ ΩB + RD d M = , CΩ Dd + CR where R = (R1 + R2 )D + A(R1 + R2 ), R1 =

R2 =

µX 2 −1 i=0 νX 2 −1

Aπ (A2 + BC)i B(Dd )2i+4 −

µ2 X

Ω(BC)i B(Dd )2i+2 ,

i=0

Ωi+2 BD2i Dπ ,

i=0

ν2 = ind(D2 ), µ2 = ind(A2 + BC) and Ω is defined by (2.4). We remark that Corollary 3.2, hence and Theorem 3.1 is also extension of results from [16], where beside conditions BCA = 0 and DC = 0 additional condition BD = 0 (or D is nilpotent) is required. Castro–Gonz´alez et al. (see [16]) gave explicit representation for M d under conditions BCA = 0, BD = 0 and BC is nilpotent (or DC = 0).

490

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

This result was extended to a case when BCA = 0 and BD = 0 (see [18]). The following theorem is extension of these results. Theorem 3.2 Let M be matrix of a form (1.1) such that BCA = 0, ABD = 0 and CBD = 0. Then   ΩB + BD(F1 Ω + (Dd )2 F2 )B  AΩ + B(F1 + F2 )  Md =  +B(Dd )2 − BDd (CA + DC)Ω2 B  , (3.7) CΩ + D(F1 + F2 ) Dd + (F1 + F2 )B where F1 =

F2 =

νX 2 −1

Dπ D2i (CA + DC)Ωi+2 ,

i=0 µX 2 −1

µ2 X

i=0

i=0

(Dd )2i+4 (CA + DC)(A2 + BC)i (BC)π −

(Dd )2i+2 (CA + DC)A2i Ω,

ν2 = ind(D2 ), µ2 = ind(A2 + BC) and Ω is defined by (2.4). Proof. If we split matrix M as     0 0 A B := P + Q. + M= 0 D C 0 we have that QP Q = 0 and P 2 Q = 0. Hence, matrices P and Q satisfy the conditions of Lemma 2.2. Since BCA = 0, matrix P satisfies conditions of Lemma 2.5. Using the similar method as in the proof of Theorem 3.1, after applying Lemma 2.2, Lemma 2.5 and using Remark 2, we get that (3.7) holds. 2 Notice that Theorem 3.2 is also generalization of representation from [15] where additional condition BCB = 0 is required. In [15] a formula for M d is given under conditions BCA = 0, DCA = 0, CBD = 0 and CBC = 0. In the next theorem we offer a representation for M d under conditions BCA = 0, DCA = 0 and CBD = 0, without additional condition CBC = 0. Theorem 3.3 Let M be as in (1.1). If BCA = 0, DCA = 0 and CBD = 0 then   Ad + (G1 + G2 )C BΓ + A(G1 + G2 )   Md =  , ΓC + CA(G1 Γ + (Ad )2 G2 )C DΓ + C(G + G ) 1 2 +C(Ad )2 − CAd (AB + BD)Γ2 C

491

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

where G1 =

νX 1 −1

Aπ A2i (AB + BD)Γi+2 ,

(3.8)

i=0

G2 =

µX 1 −1

d 2i+4

(A )

µ1 X (AB+BD)(D +CB) (CB) − (Ad )2i+2 (AB+BD)D2i Γ, 2

i

π

i=0

i=0

(3.9) ν1 = ind(A2 ), µ1 = ind(D2 + CB) and Γ is given in (2.10). Proof. Using the splitting of matrix M     0 B A 0 M= + := P + Q, C D 0 0 we get that conditions of Lemma 2.2 are satisfied. Also, we have that matrix P satisfies the conditions of Lemma 2.7. Using these lemmas and Remark 4, similarly as in the proof of Theorem 3.1, we get that the statement of the theorem is valid. 2 Corollary 3.3 Let M be matrix of a form (1.1). If CBD = 0 and CA = 0, then  d  A + (G1 + G2 )C BΓ + A(G1 + G2 ) d M = , ΓC DΓ where Γ, G1 and G2 are defined by (2.10), (3.8) and (3.9) respectively. We can see that Theorem 3.3 and Corollary 3.3 are also extensions of representation for M d under conditions CB = 0 and CA = 0 (see [13]). In [12] a representation for M d is offered under conditions AB = 0 and CB = 0. This result was extended in [14], where a formula for M d is given under conditions ABC = 0, ABD = 0, CBD = 0 and CBC = 0. In our following result we derive the representation for M d under conditions ABC = 0, ABD = 0 and CBD = 0, without additional condition CBC = 0. Theorem 3.4 Let M be matrix of a form (1.1). If ABC = 0, ABD = 0 and CBD = 0. Then   BΓ + BΘ1 AB + (Ad )2 B d  A + BΘ0 −B(Γ2 CA + DΓ2 C)Ad B  (3.10) Md =  , D d + Θ0 B

ΓC + Θ0 A

492

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

where     Θk = K1 (Ad )2k + Γk K2 A+D K1 (Ad )2k + Γk K2 , for k = 0, 1, (3.11) K1 =

µX 1 −1

Dπ (D2 + CB)i C(Ad )2i+4 −

i=0

µ1 X

Γ(CB)i C(Ad )2i+2 ,

(3.12)

i=0

K2 =

νX 1 −1

Γi+2 CA2i Aπ ,

(3.13)

i=0

ν1 = ind(A2 ), µ1 = ind(D2 + CB) and Γ is defined by (2.10). Proof. We can split matrix M as M = P + Q, where     A 0 0 B P = , Q= . 0 0 C D According to assumptions of the theorem, we have that P QP = 0 and P Q2 = 0. Hence we can apply Lemma 2.1 and we have   (P + Q)d = Y1 + Y2 + Y1 (P d )2 + (Qd )2 Y2 − Qd (P d )2 − (Qd )2 P d P Q, (3.14) where Y1 and Y2 are defined by (2.1). Since CBD = 0, matrix Q satisfies condition of Lemma 2.7. After applying Lemma 2.7 and facts from Remark 4 we get   Ad + B(K1 A + DK1 ) 0 , (3.15) Y1 = ΓC − ΓCAπ + (K1 A + DK1 )A 0   B(K2 A + DK2 ) BΓ , (3.16) Y2 = ΓCAπ + (K2 A + DK2 )A DΓ where K1 and K2 are given in (3.12) and (3.13), respectively. Now, by substituting (3.16) and (3.15) into (3.14) we get that (3.10) holds. 2 Notice that Theorem 3.4 is also an extension of a case when ABC = 0 and BD = 0 (see [19]). The following result is direct corollary of Theorem 3.4. Corollary 3.4 Let M be given by (1.1). If CBD = 0 and AB = 0 then  d  A + BΘ0 BΓ Md = , ΓC + Θ0 A DΓ where Γ and Θ0 are defined by (2.10) and (3.11) respectively.

493

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

As another extension of a result from [12], where formula for M d is given under conditions AB = 0 and CB = 0, we offer the following theorem and its corollary. Theorem 3.5 Let M be matrix of a form (1.1). If ABC = 0, ABD = 0 and DCB = 0 then   d )2 + ΨN )AB BΨ + B(N (A 1 2 d  A + B(N1 + N2 ) +(Ad )2 B − BΨ2 (CA + DC)Ad B  Md =  , ΨC + (N1 + N2 )A

ΨD + (N1 + N2 )B (3.17)

where N1 =

µX 1 −1

µ1 X ΨD2i (CA+DC)(Ad )2i+2 , (CB)π (D2 +CB)i (CA+DC)(Ad )2i+4 − i=0

i=0

(3.18) N2 =

νX 1 −1

Ψi+2 (CA + DC)A2i Aπ ,

(3.19)

i=0

ν1 = ind(A2 ), µ1 = ind(D2 + CB) and Ψ is defined by (2.6). Proof. Using the splitting     0 B A 0 := P + Q, + M= C D 0 0 we get that matrices P and Q satisfy the conditions of Lemma 2.1. Furthermore, matrix Q satisfies the conditions of Lemma 2.6. After applying these lemmas, using Remark 3 and computing, we get that (3.17) holds. 2 Next corollary follows immediately from Theorem 3.5. Corollary 3.5 Let M be given by (1.1). If DCB = 0 and AB = 0 then  d  A + B(N1 + N2 ) BΨ d M = , ΨC + (N1 + N2 )A ΨD where Ψ, N1 and N2 are defined by (2.6), (3.18) and (3.19), respectively. Cvetkovi´c and Milovanovi´c (see [17]) offered a representation for M d under conditions ABC = 0, DC = 0 , with third condition BD = 0 (or BC is nilpotent, or D is nilpotent). Cvetkovi´c - Ili´c (see [18]) extended this

494

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

result and gave a formula for M d under conditions ABC = 0 and DC = 0, without any additional condition. In our next result we replace second condition DC = 0 from [18] with two weaker conditions. Therefore, we can get results from [17, 18] as direct corollaries. Theorem 3.6 Let M be matrix of a form (1.1), such that ABC = 0, DCA = 0 and DCB = 0. Then   ΦA + (U1 + U2 )C ΦB + (U1 + U2 )D   Md =  , CΦ + C(U1 (Dd )2 + ΦU2 )DC d D + C(U1 + U2 ) d 2 2 d +(D ) C − CΦ (AB + BD)D C where U1 =

U2 =

µX 2 −1 i=0 νX 2 −1

π

2

i

d 2i+4

(BC) (A + BC) (AB + BD)(D )

−

µ2 X

ΦA2i (AB + BD)(Dd )2i+2

i=0

Φi+2 (AB + BD)D2i Dπ ,

i=0

ν2 = ind(D2 ), µ2 = ind(A2 + BC) and Φ is defined by (2.3). Proof. If we split matrix M as     A B 0 0 := P + Q, + M= C 0 0 D we have P QP = 0 and P Q2 = 0. Also, matrix P satisfies conditions of Lemma 2.4. After applying Lemma 2.1, Lemma 2.4, Remark 1 and computing we get that the statement of the theorem is valid. 2

References [1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd Edition, Springer Verlag, New York, 2003. [2] S. L. Campbell, Singular Systems of Differential Equations, Pitman, London, 1980. [3] S. L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear and Multilinear Algebra, 14 (1983) 195– 198.

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J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

[4] S. L. Campbell, C. D. Meyer, Generalized Inverse of Linear Transformations, Pitman, London, 1979; Dover, New York, 1991. [5] X. Chen, R.E. Hartwig, The group inverse of a triangular matrix, Linear Algebra Appl., 237/238 (1996) 97–108. [6] D. S. Djordjevi´c, P. S. Stanimirovi´c, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J., 51(126)(2001) 617– 634. [7] R. E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2 × 2 block matrix, SIAM J. Matrix Anal. Appl., 27 (2006) 757–771. [8] R. E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207–217 [9] Y. Wei, X. Li, F. Bu, F. Zhang, Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices-application of perturbation theory for simple invariant subspaces, Linear Algebra Appl., 419 (2006) 765-771. [10] R. E. Hartwig, J. M. Shoaf, Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices, Austral. J. Math., 24(A) (1977) 10–34. [11] C. D. Meyer, N. J. Rose, The index and the Drazin inverse of block triangular matrices, SIAM J. Appl. Math., 33 (1977) 1–7. [12] D. S. Cvetkovi´c–Ili´c, A note on the representation for the Drazin inverse of 2 × 2 block matrices, Linear Algebra Appl., 429 (2008) 242–248 [13] D. S. Cvetkovi´c–Ili´c, J. Chen, Z. Xu, Explicit representation of the Drazin inverse of block matrix and modified matrix, Linear and Multilinear Algebra, 57.4 (2009) 355–364. [14] H. Yang, X. Liu, The Drazin inverse of the sum of two matrices and its applications, J. Comput. Appl. Math., 235 (2011) 1412-1417. [15] J. Ljubisavljevi´c, D. S. Cvetkovi´c–Ili´c, Additive results for the Drazin inverse of block matrices and applications, J. Comput. Appl. Math. 235 (2011) 3683–3690. [16] N. Castro–Gonz´alez, E. Dopazo, M. F. Mart´ınez–Serrano, On the Drazin Inverse of sum of two operators and its application to operator matrices, J. Math. Anal. Appl. 350 (2009) 207–215.

496

J.Ljubisavljevi´c, D.S.Cvetkovi´c–Ili´c: Representations for the Drazin inverse

[17] A. S. Cvetkovi´c, G. V. Milovanovi´c, On Drazin inverse of operator matrices, J. Math. Anal. Appl., 375 (2011) 331-335. [18] D. S. Cvetkovi´c–Ili´c, New additive results on Drazin inverse and its applications, Appl. Math. Comput., 218(7) (2011) 3019–3024. [19] C. Bu, K. Zhang, The Explicit Representations of the Drazin Inverses of a Class of Block Matrices, Electron. J. Linear Algebra, 20 (2010) 406–418. [20] M. Catral, D. D. Olesky, P. Van Den Driessche, Block representations of the Drazin inverse of a bipartite matrix, Electron. J. Linear Algebra, 18 (2009) 98-107. [21] C. Deng, Y. Wei, A note on the Drazin inverse of an anti-triangular matrix, Linear Algebra Appl., 431 (2009) 1910-1922.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 498-506, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

The Solution of Fisher-Kolomogrov-Petrovskii-Piskunov equation by the Homotopy Perturbation Method M. Ghoreishi∗, A.I.B.Md.Ismail∗ and A. Rashid†

Abstract In this paper, the Homotopy Perturbation Method (HPM) has been implemented for solving the extended Fisher-Kolomogrov-Petrovskii-Piskunov (FKPP) equation. The HPM is an approximate method which yields an analytical solution in the form of a convergent power series with components, which can be easily computed. We show that the HPM can generate highly accurate solutions for the FKPP equation. Keywords: Homotopy Perturbation Method, Fisher-Kolomogrov-Petrovskii-Piskunov equation.

1

Introduction

The objective of this paper is to apply the HPM for obtaining the approximate solution of the generalized FKPP equation [2, 10]: Z D t − t−s e τ uxx (x, s)ds, ut (x, t) = q(x, t) + f (u) + λux (x, t) + µuxx (x, t) + (1.1) τ 0 with initial condition u(x, 0) = p(x),

(1.2)

α0 u(a, t) + β0 ux (a, t) = g0 (t),

(1.3)

α1 u(b, t) + β1 ux (b, t) = g1 (t),

(1.4)

and the Robin boundary conditions

where (x, t) ∈ (a, b) × (0, T ). Here p(x), q(x, t), g0 (t), g1 (t) and f (u) are prescribed functions, whilst α0 , α1 , β0 , β1 , λ, τ , D and µ are given constants. Recently, a numerical method based on finite differences and spline collocation was presented by Khuri and Sayfy [10] for the FKPP equation. The solutions obtained indicated that their method was reliable and yielded results compatible with the exact solution. The equation was also studied numerically using the method of lines by Branco et al. [2]. They also studied aspects relating to stability and convergence. In recent years, the application of the HPM in linear and nonlinear problems has been developed by scientists and engineers [3, 12, 15, 16]. This method was first proposed and developed by He in [6, 7, 8]. According to [9], a combination of the perturbation method and the homotopy method is called the HPM, which has eliminated the limitations of traditional perturbation methods. This method offers certain advantages over finite difference methods. Finite difference methods use discretization which increase to rounding off errors and the consequent loss of accuracy. In addition, finite difference methods require large computer memory and time. The HPM method does not involve discretization of the variables and hence is free from rounding off errors and does not require large computer memory or time. The paper is organized as follows: In section 2, we have presented a description of the HPM as has been expounded by other researchers, in particular [1]. In section 3, we have employed the HPM for solving two examples of the equation and compare the approximate solution obtained with the exact solution. Finally, the conclusion is given in section 4. ∗ School

of Mathematical Science, Universiti Sains Malaysia, 11800, Penang, Malaysia. of Mathematics, Gomal University, Dera Ismail Khan, Pakistan, E-Mail:

† Department

1 498

[email protected]

GHOREISHI ET AL: HOMOTOPY PERTURBATION METHOD

2

A Description of the HPM

To illustrate the basic idea and principles of He’s homotopy perturbation method considered the nonlinear differential equation [1]: A(u) − f (r) = 0, with boundary conditions

µ

∂u B u, ∂n

r ∈ Ω,

(2.1)

r ∈ Γ,

(2.2)

¶ = 0,

where A is a general differential operator, B is a boundary operator, f (r) is a known analytic function, ∂ Γ is the boundary of the domain Ω and ∂n denotes differentiation of the normal vector drawn outwards from Ω. According to [1] the operator A can be divided into two parts L and N , where L is linear and N is nonlinear, and thus Eq. (2.1) can be rewritten as L(u) + N (u) − f (r) = 0.

(2.3)

By using homotopy technique, a homotopy υ(r, p) : Ω × [0, 1] −→ R satisfying H(υ, p) = (1 − p)[L(υ) − L(u0 )] + p[A(υ) − f (r)] = 0,

(2.4)

H(υ, p) = L(υ) − L(u0 ) + pL(u0 ) + p[N (υ) − f (r)] = 0,

(2.5)

or can be constructed [1]. Here p ∈ [0, 1] is an embedding parameter, and u0 is the initial value of Eq. (2.1) which satisfies the boundary conditions. It can be seen that H(v, 0) = L(v) − L(u0 ) = 0,

(2.6)

H(v, 1) = L(v) + N (v) − f (r) = 0.

(2.7)

The changing process of p from zero to one is just that of υ(r, p) changing from u0 (r) to u(r) [1]. If the parameter p is considered as a ”small parameter”, applying the standard perturbation technique, it can be assumed that the solution of Eq. (2.5) can be given as a power series in p as v=

∞ X

pi vi ,

(2.8)

i=0

and setting p = 1 results in the approximate solution of Eq. (1.1), viz u = lim v = p→1

∞ X

vi .

(2.9)

i=0

The fundamental idea of the HPM is to introduce an implicit small parameter (p) which takes values from 0 to 1. According to [5, 11, 13, 14], when p = 0, the system of equations usually reduces to a sufficiently simple form, which usually admits a simple solution. As p gradually increases to one, the problem undergoes a sequence of deformations, the solution of each of which is close to that at the previous stage of the deformation. Finally at p = 1, the problem takes the original form and the final stage of deformation gives the desired solution. By applying homotopy technique on the Fisher-KolomogrovPetrovskii-Piskunov, then Eq. (1.1) yields à Ã∞ ! ∞ ∞ X X X pi (vi )t − (u0 )t = p −(u0 )t + f pi vi + λ pi (vi )x i=0

i=0

i=0

! ∞ Z D X t − t−s i e τ p (vi )xx (x, s)ds + q(x, t) . + τ i=0 0

(2.10)

Equating the coefficients of like powers of p in (2.10), we can obtain a set of differential equations with initial conditions. The solutions of the obtained differential equations are the components of (2.9). The HPM has eliminated limitations of traditional perturbation methods whilst retaining the full advantages [1]. The convergence of the series in (2.9) has been studied and discussed in [1, 6]. 2 499

GHOREISHI ET AL: HOMOTOPY PERTURBATION METHOD

3

Illustrative Examples

The HPM will be demonstrated on two examples of FKPP equation. For our computation, let the expression m−1 X ψm (x, t) = uk (x, t), (3.1) k=0

denote the m-term HPM approximation to u(x, t). We compare the approximate analytical solution obtained from HPM for the FKPP equations with the exact solution. We define Em (x, t) to be the absolute error between the exact solution and m-term approximate HAM solution ψm (x, t) as follows Em (x, t) = |u(x, t) − ψm (x, t)|.

(3.2)

Example 1: For the first example, we consider the equation (1.1) with f (u) = u, λ = 1/2, µ = 3/2, and D = τ = 1. Thus we have [10]: Z t 1 3 ut = u − ux + uxx + e−(t−s) uxx (x, s)ds + q(x, t), (3.3) 2 2 0 where (x, t) ∈ (0, 2π)×(0, 1), q(x, t) =

1 2

cos(x+t)− 21 e−t sin x+sin x cos t, subject to the initial condition u(x, 0) = sin x,

(3.4)

u(0, t) = 0, u(2π, t) + ux (2π, t) = cos t.

(3.5) (3.6)

and boundary conditions as follows

It can be verified that the exact solution is u(x, t) = sin x cos t [10]. To solve equation (3.3) with initial condition (3.4), using the homotopy perturbation technique, the following homotopy: ¶ µ µ ¶ Z t ∂v 3 ∂v ∂u0 1 − +p − v + vx − vxx − e−(t−s) vxx (x, s)ds − q(x, t) = 0, (1 − p) (3.7) ∂t ∂t ∂t 2 2 0 or equivalently µ ¶ Z t ∂v ∂u0 ∂u0 1 3 − =p − + v − vx + vxx + e−(t−s) vxx (x, s)ds + q(x, t) = 0 ∂t ∂t ∂t 2 2 0

(3.8)

is constructed. Suppose the solution of Eq. (3.8) has the form of the infinite series in (2.8). By substituting (2.8) into (3.8), we have à ∞ ∞ ∞ X X 1X i i p (vi )x + p (vi )t − (u0 )t =p −(u0 )t + pi vi − 2 i=0 i=0 i=0 ! (3.9) Z t ∞ ∞ X 3X i i −(t−s) p (vi )xx + p e (vi )xx (x, s)ds + q(x, t) = 0. 2 i=0 0 i=0

3 500

GHOREISHI ET AL: HOMOTOPY PERTURBATION METHOD

Comparing coefficients of terms with identical powers of p, leads to the following: (v0 )t − (u0 )t = 0,

Z t 1 3 e−(t−s) (v0 )xx (x, s)ds + q(x, t), (v1 )t = −(u0 )t + v0 − (v0 )x + (v0 )xx + 2 2 0 Z t 3 1 e−(t−s) (v1 )xx (x, s)ds (v2 )t = v1 − (v1 )x + (v1 )xx + 2 2 0 .. . Z t 3 1 e−(t−s) (vj−1 )xx (x, s)ds, (vj )t = vj−1 − (vj−1 )x + (vj−1 )xx + 2 2 0 with initial conditions vi (x, 0) = 0,

i = 1, 2, · · ·

For simplicity we take v0 (x, t) = sin x. So we can obtain recursive formula as ¶ Z t Z tµ 1 3 −(t−s) e (v0 )xx (x, s)ds + q(x, t) dt, v1 (x, t) = −(u0 )t + v0 − (v0 )x + (v0 )xx + 2 2 0 0 ¶ Z tµ Z t 3 1 vj (x, t) = vj−1 − (vj−1 )x + (vj−1 )xx + e−(t−s) (vj−1 )xx (x, s)ds dt, 2 2 0 0 for j = 2, 3, · · · . Thus, we can obtain the components of the infinite series (2.9) as follows: v0 (x, t) = sin x, v1 (x, t) = −0.5t cos t − 1.5t sin t + 0.5 sin(t + x) + sin x(−0.5 cosh t + sin t + 0.5 sinh t), v2 (x, t) = (−0.75(−1.21525 + t)(0.5486 + t) + cos t) cos x + (1 + t(−1.5 + t) + cos t) sin x + cosh t(−0.5 cos t + (−2 − 0.5t) sin x) + (0.5 cos x + (2 + 0.5t) sin x) sinh t. and so on. Table 1 shows the absolute error between the solution obtained using HPM with six terms and the exact solution for various x ∈ (0, 2π) and t ∈ (0, 1). The errors are very small in this table. The results provide very strong evidence that the homotopy perturbation technique is not only easy to use but also enable accurate approximate solution of the FKPP equation to be obtained. We have verified that the overall errors can be made smaller by adding new terms of the HPM series (2.9). Table 1: Absolute error E6 for various values of x and t. x/t 0 2π/5 4π/5 6π/5 8π/5 2π

0

0.4

5.32907×10−15 9.32587×10−15 4.66294×10−15 4.88498×10−15 5.88418×10−15 5.32407×10−15

0.6

2.54337×10−9 1.33096×10−9 1.68988×10−9 2.42537×10−9 1.90928×10−10 2.54337×10−9

7.10227×10−8 4.62650×10−8 4.24293×10−8 7.24878×10−8 2.37061×10−9 7.10227×10−8

1 4.81678×10−6 4.16812×10−6 2.24074×10−6 5.55298×10−6 1.19119×10−6 4.81678×10−6

Note that the HPM provides the solution of equation (3.3) by using the initial condition (3.4) only. It is also to be noted that the boundary conditions (3.5) and (3.6) can be used only to justify the solution obtained [4]. Hence, in Table 2 we have obtained the following absolute error for the boundary conditions Table 2: Absolute error boundary condition (3.5) and (3.6) for various values of t ∈ (0, 1). Error/t 0

E19 0 E20

0

0.4

0.6

4.66294×10−15

2.54329×10−9

7.10227×10−8

3.55271×10−15

3.16895×10−9

9.65919×10−8

4 501

1 4.81678×10−6 7.63434×10−6

GHOREISHI ET AL: HOMOTOPY PERTURBATION METHOD

0

0

where E19 and E20 are obtained as 0

E19 = |ψ6 (0, t)|, 0

E20 = |ψ6 (2π, t) + (ψ6 )x (2π, t) − cos t|, In Figure 1 and Figure 2, we have shown the absolute error E6 for various x ∈ (0, 2π) at t = 0, 0.4, 0.6 and t = 1 while this error has been shown in Figure 3 for various t ∈ (0, 1) at x = 0 and x = 9π/5. Absolute error E6 at t=0

Absolute error E6 at t=0.4

4. ´ 10-15

2. ´ 10-9

3. ´ 10-15

1.5 ´ 10-9

E6

2.5 ´ 10-9

E6

5. ´ 10-15

2. ´ 10-15

1. ´ 10-9

1. ´ 10-15

5. ´ 10-10

0

0 0

1

2

3

4

5

6

0

1

2

3

x

4

5

6

5

6

x

Figure 1: Absolute error E6 for various x ∈ (0, 2π) at t = 0 and t = 0.4.

Absolute error E6 at t=0.6

Absolute error E6 at t=1

-8

7. ´ 10

5. ´ 10-6 -8

6. ´ 10

4. ´ 10-6

5. ´ 10-8

3. ´ 10-6

E6

E6

4. ´ 10-8 -8

3. ´ 10

2. ´ 10-6

2. ´ 10-8 1. ´ 10-6

1. ´ 10-8 0

0 0

1

2

3

4

5

6

0

x

1

2

3

4 x

Figure 2: Absolute error E6 for various x ∈ (0, 2π) at t = 0.6 and t = 1. Example 2: −1 We consider the nonlinear Eq. (1.1) with f (u) = u2 , λ = 0, µ = 2π 2 , and D = τ = 2, then we obtained. Z t t−s 1 e− 2 uxx (x, s)ds + q(x, t), ut = u2 − 2 uxx + (3.10) 2π 0 where (x, t) ∈ (0, 1) × (0, 0.5), q(x, t) = −e−t/2 sin πx + π 2 te−t/2 sin πx − e−t sin2 πx, subject to the initial condition u(x, 0) = sin πx, (3.11) and boundary conditions as follows u(0, t) + ux (0, t) = πe−t/2 , u(1, t) + ux (1, t) = −πe 5 502

−t/2

(3.12) .

(3.13)

GHOREISHI ET AL: HOMOTOPY PERTURBATION METHOD

Absolute error E6 at x=0

Absolute error E6 at x=9А5

1.5 ´ 10-6

6. ´ 10-7

E6

8. ´ 10-7

E6

2. ´ 10-6

1. ´ 10-6

4. ´ 10-7

5. ´ 10-7

2. ´ 10-7

0

0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

t

0.2

0.4

0.6

0.8

1.0

t

Figure 3: Absolute error E6 for various t ∈ (0, 1) at x = 0 and x = 9π/5.

It can be verified that the exact solution is u(x, t) = e−t/2 sin πx [10]. To solve equation (3.10) with initial condition (3.11), according to the homotopy perturbation technique, we get the following homotopy equation: ¶ µ ¶ µ Z t 1 ∂v ∂u0 ∂v − t−s 2 2 e − − v + 2 vxxx − vxx (x, s)ds − q(x, t) = 0, (3.14) (1 − p) +p ∂t ∂t ∂t 2π 0 or equivalently; µ ¶ Z t t−s ∂v ∂u0 ∂u0 1 − =p − + v 2 − 2 vxx + e− 2 vxx (x, s)ds + q(x, t) = 0, ∂t ∂t ∂t 2π 0

(3.15)

is constructed. Suppose the solution of Eq. (3.15) has the form of the infinite series in (2.8). By substituting (2.8) into (3.15), we have  Ã ∞ !2 ∞ ∞ X X 1 X  (vi )xx + (vi )t − (u0 )t =p −(u0 )t + vi − 2 2π i=0 i=0 i=0 (3.16) ! Z t ∞ X t−s + pi e− 2 (vi )xx (x, s)ds + q(x, t) = 0. i=0

0

By comparing coefficients of terms with identical powers of p, we have: (v0 )t − (u0 )t = 0,

Z t t−s 1 (v ) + e− 2 (v0 )xx (x, s)ds + q(x, t), 0 xx 2 2π 0 Z t t−s 1 e− 2 (v1 )xx (x, s)ds, (v2 )t = 2v0 v1 − 2 (v1 )xx + 2π 0 Z t t−s 1 (v3 )t = v12 + 2v0 v2 − 2 (v2 )xx + e− 2 (v2 )xx (x, s)ds, 2π 0 Z t t−s 1 (v4 )t = 2v1 v2 + 2v0 v3 − 2 (v3 )xx + e− 2 (v3 )xx (x, s)ds, 2π 0 .. . (v1 )t = −(u0 )t + v02 −

with initial conditions vi (x, 0) = 0,

i = 1, 2, · · · 6

503

GHOREISHI ET AL: HOMOTOPY PERTURBATION METHOD

For simplicity we take v0 (x, t) = u0 (x, t) = sin πx. So we can obtain recursive formula as ¶ Z t Z tµ 1 − t−s 2 2 e (v0 )xx (x, s)ds + q(x, t) dt, v1 (x, t) = −(u0 )t + (v0 ) − 2 (v0 )xx + 2π 0 0 Ã ! Z t Z t j−i−1 X t−s 1 e− 2 (vj−1 )xx (x, s)ds dt, vj (x, t) = vi vj−i−1 − 2 (vj−1 )xx + 2π 0 0 i=0 for j = 2, 3, · · · . Thus, we can obtain the components of the infinite series (2.9) as follows v0 (x, t) = sin πx, 1 v1 (x, t) = e−t sin πx(e−t (4π 2 − 1)(t − 4) − 4et/2 (π 2 (t + 4) − 1) + 2(1 + et (t − 1)) sin πx, 2 1 v2 (x, t) = e−t (2(4 + 16π 2 − 16et/2 (π 2 (t + 14) − 1)) + et (−20 + 12t − 3t2 + 4π 2 (52 − 20t 8 + 3t2 ))) cos 2πx + (−12 + et (28 − 20t + 7t2 + 16π 4 (48 − 12t + t2 ) − · · · .. . and so on. The absolute error between solution obtained using HPM with five terms and the exact solution have been shown in Table 3 for various x ∈ (0, 1) and t ∈ (0, 0.5). It can be seen that the errors are very small. Only five terms were used in evaluating the approximate solutions. It was verified that the overall errors can be made smaller by adding new terms to HPM series (2.9). Table 3: Absolute error E5 for various values of x and t. x/t

0

0.1

0.3

0.4

0.1 0.3 0.5

3.27298×10−8 1.67329×10−10 7.86859×10−9 3.10176×10−8

2.42881×10−8 5.30104×10−7 7.44754×10−7 2.17058×10−7

2.44835×10−6 1.54711×10−5 5.52748×10−5 1.67071×10−5

3.07153×10−5 9.25156×10−5 6.98455×10−5 7.33535×10−4

0.7 0.9 1

9.21583×10−10 3.31014×10−8 3.27298×10−8

7.68673×10−7 4.97243×10−7 2.42881×10−8

5.52822×10−5 3.14663×10−5 2.44835×10−6

6.98531×10−5 9.25156×10−5 3.07153×10−5

0

Note that the HPM provides the solution of equation (3.10) by using the initial condition (3.11) only. It is also to be noted that the boundary conditions (3.12) and (3.13) can be used only to justify the solution obtained [4]. Hence, in Table 4 we have obtained the following absolute error for boundary conditions Table 4: Absolute error boundary condition (3.12) and (3.13) for various values of t ∈ (0, 0.5). Error/t

0

0

0.1

3.21285×10−8 3.33310×10−8

E26 0 E27 0

0.3

3.11288×10−6 3.06430×10−6

3.78200×10−4 3.73304×10−4

0.4 3.28723×10−4 3.90153×10−4

0

where E26 and E27 are obtained as 0

E26 = |ψ5 (0, t) + (ψ5 )x (0, t) − πe−t/2 |, 0

E27 = |ψ5 (1, t) + (ψ5 )x (1, t) + πe−t/2 |, In Figure 4 and Figure 5, we have displayed the absolute error between solution obtained using HPM with five terms and the exact solution for various x and t.

7 504

GHOREISHI ET AL: HOMOTOPY PERTURBATION METHOD

Absolute error E5 at t=0.1

Absolute error E5 at t=0.4 0.0007

8. ´ 10-7

0.0006 0.0005

6. ´ 10-7

E5

E5

0.0004

-7

4. ´ 10

0.0003 0.0002

2. ´ 10-7 0.0001 0

0.0000 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

x

0.6

0.8

1.0

0.4

0.5

x

Figure 4: Absolute error E5 for various x ∈ (0, 1) at t = 0.1 and t = 0.4.

Absolute error E5 at x=0

Absolute error E5 at x=0.9 0.00020

0.00007 0.00006

0.00015

0.00005

E5

E5

0.00004

0.00010

0.00003 0.00002

0.00005

0.00001 0

0.00000 0.0

0.1

0.2

0.3

0.4

0.5

0.0

t

0.1

0.2

0.3 t

Figure 5: Absolute error E5 for various t ∈ (0, 0.5) at x = 0 and x = 0.9.

8 505

GHOREISHI ET AL: HOMOTOPY PERTURBATION METHOD

4

Conclusion

In this paper, we have introduced that the HPM as a technique that can be used to solve the generalized Fisher-Kolomogrov-Petrovskii-Piskunov (FKPP) equation. The relevant formulas have been derived. The method was tested on two examples of the FKPP equation and it was shown that the HPM is highly accurate and rapidly convergent method.

References [1] J. Biazar, H. Ghazvini, Exact solutions for nonlinear Schr¨oodinger equations by He’s homotopy perturbation method, Physics Letters A, 366 (2007) 79-84. [2] J. R. Branco, J. A. Ferreira, P. de Oliveira, Numerical methods for the generalized FisherKolomogrov-Piskunov-Piskunov equation, Applied Numerical Mathematics, 57 (2007) 89-102. [3] L. Cveticanin, Homotopy perturbation method for pure nonlinear differential equation, Chaos Solitons Fractals, 30 (5) (2006) 1221-1230. [4] M. Dehghan, Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications, Applied Mathematics and Computation, 157 (2004) 549-560. [5] D.D. Ganji, H. B. Rokni, M.G. Sfahani, S.S. Ganji, Approximate traveling wave solutions for coupled WhithamBroerKaup shallow water, Advances in Engineering Software, 41 (78) 956-961. [6] J.H. He, Homotopy perturbation technique, Computational Methods in Applied Mechanics and Engineering, 178 (1999) 257-262. [7] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal Nonlinear Mechanics, 35 (1) (2000) 37-43. [8] J.H. He, Some asymptotic methods for strongly nonlinear equations, International, Journal of Modern Physics B, 20 (2006) 1141-1199. [9] J.H. He, A new perturbation technique which is also valid for large parameters, Journall of Sound Vibration, 229 (2000) 1257-1263. [10] S.A. Khuri, A. Sayfy, A numerical approach for solving an extended Fisher-Kolomogrov-PetrovskiiPiskunov equation, Journal of Computational and Applied Mathematics, 233 (2010) 2081-2089. [11] S. Momani, G. H. Erjaee, M. H. Alnasr, The modified homotopy perturbation method for solving strongly nonlinear oscillators, Computers and Mathematics with Applications, 58 (2009) 2209-2220. [12] F. Shakeri, M. Dehghan, Solution of the delay differential equations via homotopy perturbation method, Mathematics Computer Modelling, Mathematical and Computer Modelling, 48 (5) (2008) 1-8. [13] A. Yildirim, Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method, Computers and Mathematics with Applications, 56 (2008) 3175-3180. [14] A. Yildirim, Application of He’s homotopy perturbation method for solving the Cauchy reactiondiffusion problem, Computers and Mathematics with Applications, 57 (2009) 612-618. [15] M. Ghoreishi, A.I.B.Md. Ismail, A. Rashid, The coupled viscous Burgers equations with fractionaltime derivative by the homotopy perturbation method, Journal of Computational Analysis and Applications, 13(6) (2011) 1054-1066. [16] M. Ghoreishi, A.I.B.Md. Ismail, A. Rashid, Numerical Solution of Klein-Gordon-Zakharov Equations using Chebyshev Cardinal Functions, Journal of Computational Analysis and Applications, 14 (3) (2012) 574-582.

9 506

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 507-525, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A study on the limiting ratio of consecutive terms in a class of bivariate generating functions Shun-Pin Hsua a

Department of Electrical Engineering, National Chung Hsing University Taichung 402, Taiwan [email protected]

Abstract A class of bivariate quadratic functions and its induced generating functions are studied in the paper. The author shows that under some conditions, the sequence of quotients obtained from the ratios of the consecutive terms in the generating function converges. In particular, letting the quotient be 1 yields a tangent line to the original bivariate quadratic curve at a point in the x-axis. The point is the only one at which the line intersects the curve, and is either the only one or the closer one (to the origin) at which the curve intersects the x-axis. Numerical examples are provided to illustrate the work. Keywords: quadratic bivariate function, limiting ratio, generating function

1. Preliminaries Since the introduction by Abraham de Moivre in 1730 [7, p.86], generating functions have been a successful tool in the analysis of sequences. With this tool an exact formula, a recurrent formula or an asymptotic formula might

507

HSU: BIVARIATE GENERATING FUNCTIONS

be obtained for a given sequence. Classical and comprehensive applications of generating functions can be found in [1, 2, 3, 4, 9] and the references therein. In this paper we study a class of generating functions that serve as the normalizers of polynomials in the following sense. Consider a complex polynomial fn (z) : = an z n + an−1 z n−1 + · · · + a1 z + 1

(1)

= (−1) (θ1 z − 1)(θ2 z − 1) · · · (θn z − 1) . n

Let qk (z) := bk z k + bk−1 z k−1 + · · · + b1 z + 1 satisfying

k+n ∑

fn (z)qk (z) = 1 +

ck,i z i

(2)

(3)

i=k+1

for some constant ck,i . As k tends to infinity the generating function qk (z) works as a normalizing factor for fn (z). That is, lim qk (z) =

k→∞

1 . fn (z)

(4)

For the univariate case, Jin [5, Sec.2] developed a low-complexity algorithm to compute the Kalantari family [6] of zero bounds of the polynomial based on the sequence of coefficients of the generating functions. In fact, under some condition the sequence of ratios of the coefficients has a convergence property (cf. [8, p.110]). Specifically, suppose in (1) |θ1 | > |θ2 | ≥ |θ3 | ≥ · · · ≥ |θn |

(5)

and bk ’s in (2) are nonzero. We then have lim

k→∞

bk bk−1

508

= θ1 .

(6)

HSU: BIVARIATE GENERATING FUNCTIONS

To see this, note that ai in (1) can be expressed with ∑

ai = (−1)i

θ1t1 θ2t2 · · · θntn

(7)

t1 +···+tn =i,t1 ,··· ,tn ∈{0,1}

for i ∈ {1, 2, · · · n}. Observe (see e.g. [5, Remark 3.4]) that ∑

min{n,k}

bk = −

aj bk−j

(8)

j=1

where b0 = 1. A combinatorial analysis yields ∑

bk =

θ1t1 θ2t2 · · · θntn .

t1 +···+tn =k,t1 ,··· ,tn ∈{0,1,···k}

Note that by assumption bk−1 is nonzero. As a result, ∑ θ1t1 θ2t2 · · · θntn bk t1 +···+tn =k,t1 ,··· ,tn ∈{0,1,···k} ∑ = bk−1 θ1t1 θ2t2 · · · θtn n

t1 +···+tn =k−1,t1 ,··· ,tn ∈{0,1,···k−1}

∑

θ2t2 · · · θntn

t2 +···+tn =k,t2 ,··· ,tn ∈{0,1,···k}

∑

= θ1 +

θ1t1 θ2t2 · · · θntn

t1 +···+tn =k−1,t1 ,··· ,tn ∈{0,1,···k−1}

∑

= θ1 + θ1

θ˜2t2 · · · θ˜ntn

t2 +···+tn =k,t2 ,··· ,tn ∈{0,1,···k}

∑

t2 +···+tn ≤k−1,t2 ,··· ,tn ∈{0,1,···k−1}

509

θ˜2t2 · · · θ˜ntn

(9)

HSU: BIVARIATE GENERATING FUNCTIONS

where θ˜k = θk /θ1 . Since |θ˜∗ | := max{|θ˜2 |, · · · , |θ˜n |} < 1 by assumption, we have

∑ t t ˜ 2 · · · θ˜ n θ 2 n t2 +···+tn =k,t2 ,··· ,tn ∈{0,1,···k} ∑ ≤ |θ˜2 |t2 · · · |θ˜n |tn t2 +···+tn =k,t2 ,··· ,tn ∈{0,1,···k}

≤

∑

|θ˜∗ |k

t2 +···+tn =k,t2 ,··· ,tn ∈{0,1,···k}

( ) k + n − 2 ˜∗ k = θ −→ 0 n−2 as k tends to infinity. We conclude that lim

k→∞

bk bk−1

= θ1 .

(10)

If we let the limit of the ratios of the consecutive terms in qk (z) be 1, then bk z k = θ1 z . k→∞ bk−1 z k−1

1 = lim

(11)

The solution z of (11) is the polynomial’s zero with the smallest modulus. In the following sections we consider the bivariate quadratic case and prove the existence of similar convergence result. In particular, we show that this convergence leads to an interesting geometric relation between the original bivariate quadratic function and the limiting ratio of consecutive terms of the induced generating functions. 2. A convergence proof Observe that for the special case n = 2 in (3), we can express for (2) ⌊ k2 ⌋ ( ) ∑ k − i k−2i k a1 (−a2 )i bk = (−1) i i=0

510

(12)

HSU: BIVARIATE GENERATING FUNCTIONS

where ⌊x⌋ is the largest integer not exceeding x. This can be seen by the following induction. Namely, by (8)   −a 1 bk =  a2 − a

if k = 1

,

(13)

if k = 2

2

1

which satisfies (12). For k = 3, 4, · · · , ) k + 1 − i k+1−2i a1 (−a2 )i i

⌊ k+1 ⌋( 2

−a1 bk+1 − a2 bk = − a1 (−1)

∑

k+1

i=0

∑ (k − i) k − a2 (−1) (−a2 )i ak−2i 1 i i=0 ⌊ k2 ⌋

(14)

Actually, ⌊ k+1 ⌋( 2 k

(14) =(−1)

) k + 1 − i k+2−2i a1 (−a2 )i i

∑ i=0

⌊ k2 ⌋+1 (

− a2 (−1)

k

) k − (i − 1) k−2(i−1) a1 (−a2 )i−1 i−1

∑ i=1

⌋( ⌊ k+1 2

=(−1)k ak+2 1

+ (−1)

k

∑ i=1

) k + 1 − i k+2−2i a1 (−a2 )i i

(15)

⌊ k2 ⌋+1 (

+ (−1)

k

∑ i=1

) k + 1 − i k+2−2i a1 (−a2 )i i−1

∑ (k + 2 − i) ak+2−2i (−a2 )i 1 i i=0

⌊ k+2 ⌋+1 2 k+2

=(−1)

=bk+2 . Now we define r := −a2 /a21 and ⌊ k2 ⌋ ( ) ∑ k−i i r . Tk (r) := i i=0

511

(16)

HSU: BIVARIATE GENERATING FUNCTIONS

By (12) bk = (−a1 )k Tk (r) .

(17)

Note that bk+2 = −a1 bk+1 − a2 bk . We have Tk+2 (r) − Tk+1 (r) − rTk = 0 . As a result,

  ( 1 )k (˜ c1 + c˜2 k) 2 Tk (r) =  c rk + c rk 1 1

where r1 =

1+

2 2

(18)

if a21 = 4a2 ,

,

(19)

otherwise

√ √ 1 + 4r 1 − 1 + 4r , r2 = 2 2

(20)

and c1 , c2 , c˜1 , c˜2 can be found using the condition (13). Let z = 1. Replace a1 and a2 with dx + ey and ax2 + bxy + cy 2 respectively, where a, b, c, d, e are all real. (3) becomes ( (ax2 + bxy + cy 2 + dx + ey + 1) 1 +

k ∑

) Bi (x, y)

i=1

=1 + Ck,k+1 (x, y) + Ck,k+2 (x, y) where Bi (x, y) =

i ∑

ci,j xj y i−j , i ∈ {1, 2, · · · k} ,

j=0

Ck,i (x, y) =

i ∑

(k)

c˜i,j xj y i−j , i ∈ {k + 1, k + 2} ,

j=0 (k)

and ci,j , c˜i,j are some constants. Write Bk (x, y) Rk (x, y) = γ1,k x + γ2,k y + Bk−1 (x, y) Bk−1 (x, y)

512

(21)

HSU: BIVARIATE GENERATING FUNCTIONS

where γ1,k : = γ2,k

ck,k

, ck−1,k−1 ck,k−1 ck−1,k−1 − ck,k ck−1,k−2 := . c2k−1,k−1

(22)

A convergence property concerning γ1,k and γ2,k follows. Lemma 2.1. Assume d2 > 4a, and γ1,k , γ2,k are nonzero for each natural number k. We then have k→∞

lim γ2,k

k→∞

√

1 − 4a/d2 , 2 √ b/d − e(1 + 1 − 4a/d2 )/2 √ := γ2 = . 1 − 4a/d2

lim γ1,k := γ1 = (−d)

1+

Remark 2.1. There are two division methods to derive γ1,k and γ2,k . With the first method, as shown in (22), γ1,k is the ratio of the coefficient of xk in Bk (x, y) to the coefficient of xk−1 in Bk−1 (x, y). With the second method γ2,k is the ratio of the coefficient of y k in Bk (x, y) to the coefficient of y k−1 in Bk−1 (x, y). We adopt the first division method throughout the paper. Similar result can be derived for the case using the second division method. Proof. Under the assumption, the coefficient of xk in Bk (x, y) is ⌊ k2 ⌋ ( ) ∑ k − i k−2i k d (−a)i = (−d)k Tk (ˆ r) = (−d)k (c1 rˆ1k + c2 rˆ2k ) (−1) i i=0

where rˆ1 =

1+

√ √ 1 + 4ˆ r r 1 − 1 + 4ˆ , rˆ2 = 2 2

and rˆ = −a/d2 . As a result, (−d)k Tk (ˆ r) k−1 k→∞ (−d) Tk−1 (ˆ r) √ 1 + 1 − 4a/d2 . = (−d)ˆ r1 = (−d) 2

lim γ1,k : = lim

k→∞

513

(23)

HSU: BIVARIATE GENERATING FUNCTIONS

To show the second part, note that the coefficient of xk−1 y in Bk (x, y) is  k  ⌊2⌋ ( ⌊ k2 ⌋ ( )( ) ) ( ) ∑ ∑ k − i k − 2i k−2i−1 k − i k−2i i d e(−a)i + d (−a)i−1 (−b) (−1)k  i 1 i 1 i=0 i=1  k  k ⌊2⌋ ( ⌊2⌋ ( )( ) ) e ∑ k − i k − 2i i b ∑ k−i =(−d)k  rˆ − 2 iˆ ri−1  . (24) d i=0 i 1 d i=1 i Since

( ) ( )( ) k−i k − i k − 2i , = (i + 1) i 1 i+1

(25)

( ) k−i i Sk (ˆ r) := (i + 1) rˆ i + 1 i=0

(26)

we can define

⌊2⌋ ∑ k

and write (24) as

( k

(−d)

e b Sk (ˆ r) − 2 Sk−1 (ˆ r) d d

) .

(27)

Observe that γ2,k =

(−d)k−1 Tk−1 (−d)k ( de Sk −

= (−d)

e (Tk−1 Sk d

−

b S ) − (−d)k−1 ( de Sk−1 − db2 Sk−2 )(−d)k Tk d2 k−1 ((−d)k−1 Tk−1 )2 Tk Sk−1 ) − db2 (Tk−1 Sk−1 − Tk Sk−2 ) . 2 Tk−1

Since Sk (ˆ r) is the derivative of Tk+1 (ˆ r) with respect to rˆ, namely, d Tk+1 (ˆ r) dˆ r k+1 (c1 rˆ1k − c2 rˆ2k ) , =√ 1 + 4ˆ r

Sk (ˆ r) =

2 the coefficients of rˆ12k−2 in Tk−1 , Tk−1 Sk − Tk Sk−1 and Tk−1 Sk−1 − Tk Sk−2 are

514

HSU: BIVARIATE GENERATING FUNCTIONS

√ √ c21 , c21 rˆ1 / 1 + 4ˆ r and c21 / 1 + 4ˆ r respectively. We thus conclude that ( ) −d e b lim γ2,k = √ rˆ1 − 2 k→∞ d 1 + 4ˆ r d ) ( 1 b =√ − eˆ r1 r d 1 + 4ˆ √ b/d − e(1 + 1 − 4a/d2 )/2 √ = . 1 − 4a/d2 3. A geometric analysis Now we present the geometric relation between the curve C and the line L where C:

ax2 + bxy + cy 2 + dx + ey + 1 = 0 ,

L:

γ1 x + γ2 y = 1 .

Theorem 3.1. Under the conditions of Lemma 2.1, L intersects C only at the point (1/γ1 , 0). The point is either the only one or the closer one (to the origin) at which C intersects the x-axis. In particular, L is a tangent line to C at the point. Proof. Under the conditions, L intersects the x-axis only at the point (1/γ1 , 0). If a = 0, C intersect the x-axis only at the point (−1/d, 0), which is exactly (1/γ1 , 0). If a ̸= 0, C intersects the x-axis at (p1 , 0) and (p2 , 0) where √ √ −d + |d| 1 − 4a/d2 d2 − 4a = , p1 = 2a 2a √ √ −d − |d| 1 − 4a/d2 −d − d2 − 4a p2 = = . 2a 2a −d +

515

HSU: BIVARIATE GENERATING FUNCTIONS

If |p1 | < |p2 | then we have p1 = (−d)

1−

√

1 − 4a/d2 1 = . 2a γ1

Now we show that (1/γ1 , 0) is the only point at which L intersects C. Note that if the line y = mx + k intersects the curve C at only one point, the determinant of ax2 + bx(mx + k) + c(mx + k)2 + dx + e(mx + k) + 1 = 0

(28)

should be 0. Equivalently, m, k should satisfy k 2 (b2 − 4ac) + km(2be + 4cd − 4be) + m2 (e2 − 4c)

(29)

+ k(2bd − 4ae) + m(2de − 4b) + d2 − 4a = 0 . By the definition of L, we shall prove that as √ r) d(1 + 4ˆ r + 1 + 4ˆ m= , 2(b/d − eˆ r1 ) √ 1 + 4ˆ r k= , b/d − eˆ r1 (29) is satisfied. To merge some parameters, let √ b r b 1 + 1 + 4ˆ D1 : = − eˆ r1 = − e d d 2) ( b e b 1 + D2 e = −e = − − D2 d 2 d 2 2 where

√ D2 := 1 + 4ˆ r=

√

d2 − 4a , d2

(30)

d(1 + D2 )D2 D2 , k= . 2D1 D1

(31)

then m=

516

HSU: BIVARIATE GENERATING FUNCTIONS

We need to check D22 (b2

− 4ac) +

d D22 (1 2

( + D2 )(2be + 4cd − 4be) +

D22

d (1 + D2 ) 2

)2 (e2 − 4c)

d + D1 D2 (2bd − 4ae) + D1 D2 (1 + D2 )(2de − 4b) + (d2 − 4a)D12 = 0 2 or d2 D22 (b2 − 4ac) + D22 (1 + D2 )d(2cd − be) + D22 (D22 + 2D2 + 1)(e2 − 4c) 4 (( ) ) b e e − + D2 (1 + D2 ) − D2 d(de − 2b) d 2 2 ) )2 (( ) ) (( b e e e b e 2 − − + D2 − D2 (2bd − 4ae) + (d − 4a) − D2 = 0 . d 2 2 d 2 2 To see this, note that the coefficient of D23 is d(2cd − be) + 2(e2 − 4c)

d2 e − d(de − 2b) = 0 4 2

(32)

and the coefficient of D2 is ( ( ) ) ( ) b e b e b e − (2bd − 4ae) + − d(de − 2b) − e − (d2 − 4a) d 2 d 2 d 2 ) ( b e − (2bd − 4ae + d2 e − 2bd − d2 e + 4ae) = 0 . = d 2 (33) The coefficient of D24 is (e2 − 4c)d2 /4. The coefficient of D22 , defined as K, is d2 e K := b2 − 4ac + d(2cd − be) + (e2 − 4c) − (2bd − 4ae) 4 2 ( ) b e e2 e + − d(de − 2b) + (d2 − 4a) − d(de − 2b) d 2 4 2

(34)

and the coefficient of D20 is ( (d − 4a) 2

517

b e − d 2

)2 .

(35)

HSU: BIVARIATE GENERATING FUNCTIONS

Since D22 = (d2 − 4a)/d2 , it remains to show that ( )2 de (d2 − 4a)(e2 − 4c) +K+ b− = 0. 4 2

(36)

To confirm (36), observe that (d2 − 4a)(e2 − 4c) d2 e2 d2 2 2 2 +b + − bde + b − 4ac + d(2cd − be) + (e − 4c) 4 4 ( )4 de de e2 2 − e(bd − 2ae) + b − (de − 2b) + (d − 4a) − (de − 2b) = 0 . 2 4 2 Finally, we show that L is actually a tangent line to C at the point (1/γ1 , 0). Define g(x, y) := ax2 + bxy + cy 2 + dx + ey + 1

(37)

and its derivative [

] ∂g ∂g Dg := = [2ax + by + d bx + 2cy + e] . ∂x ∂y The associated tangent line passing (1/γ1 , 0) is ( ) −(2a/γ1 ) + d 1 y= x− b/γ1 + e γ1

(38)

(39)

or equivalently b + γ1 e b + γ1 e x+ y = 1. b/γ1 + e 2a/γ1 + d

(40)

Since 1/γ1 is a zero of ax2 + dx + 1, we have 2a + d = −(2γ1 + d) , γ1

(41)

which implies that the line described by (40) is the same as γ1 x + γ2 = 1, and the proof is concluded.

518

HSU: BIVARIATE GENERATING FUNCTIONS

Corollary 3.2. If the bivariate quadratic function is decomposable, namely ax2 + bxy + cy 2 + dx + ey + 1 = 0 = (α1 x + β1 y + 1)(α2 x + β2 y + 1) , (42) the tangent line L becomes either α1 x + β1 y + 1 = 0 or α2 x + β2 y + 1 = 0, depending on the used division methods mentioned in Remark 2.1. Example 3.1. Consider the bivariate quadratic function −x2 + 4xy + y 2 + x + y + 1 = 0 .

(43)

Note that the curve has two x-intercepts and the one closer to the origin is √ at x = (1 − 5)/2. By (40) the tangent line to the curve at this x-intercept is b + γ1 e b + γ1 e x+ y=1 b/γ1 + e 2a/γ1 + d where 2

4 + 1−√5 b + γ1 e √ ≈ −1.6180 , = b/γ1 + e 4 · 1−2 5 + 1 4 + 1−2√5 b + γ1 e √ ≈ 1.0625 . = 2a/γ1 + d −2 · 1−2 5 + 1 The curve along with the tangent line are shown in the upper right part of Figure 1. Table 3.1 shows the coefficients of x term and y term in the quotients of ratios of the consecutive terms in the associated generating function of (43). These values form the upper left plot of Figure 1. Before 20 iterations the sequence of coefficients for x term converges to -1.6180, and for y term 1.0652, as expected.

519

HSU: BIVARIATE GENERATING FUNCTIONS

Table 1: the coefficients of 𝑥-term and 𝑦-term in the quotients of ratios of the consecutive terms in the associated generating function of the function in Example 3.1.

x term

y term

x term

y term

1

-2.0000

4.0000

11

-1.6181

1.0660

2

-1.5000

0

12

-1.6180

1.0649

3

-1.6667

1.6667

13

-1.6180

1.0654

4

-1.6000

0.8000

14

-1.6180

1.0652

5

-1.6250

1.1875

15

-1.6180

1.0653

6

-1.6154

1.0118

16

-1.6180

1.0652

7

-1.6190

1.0884

17

-1.6180

1.0653

8

-1.6176

1.0554

18

-1.6180

1.0652

9

-1.6182

1.0694

19

-1.6180

1.0652

10

-1.6180

1.0635

20

-1.6180

1.0652

520

HSU: BIVARIATE GENERATING FUNCTIONS

Example 3.2. Consider the bivariate quadratic function −4x2 + 4xy − y 2 + x + y + 1 = 0 .

(44)

Note that the curve also has two x-intercepts and the one closer to the origin √ is at x = (1 − 17)/8. The tangent line to the curve at this x-intercept is b + γ1 e b + γ1 e x+ y=1 b/γ1 + e 2a/γ1 + d where 8

4 + 1−√17 b + γ1 e √ = ≈ −2.5616 , b/γ1 + e 4 · 1−8 17 + 1 8 4 + 1−√ b + γ1 e 17 √ = ≈ 0.3489 . 1− 17 2a/γ1 + d −8 · 8 + 1

The lower right part of Figure 1 shows the curve and the tangent line. Table 3 provides the coefficients of x term and y term in the quotients of ratios of the consecutive terms in the associated generating function of (44). These values are plotted in the lower left of Figure 1. After 30 iterations the sequence of coefficients for x term converges to -2.5616, and for y term 0.3489, as expected.

4. Conclusion In this paper we study a class of generating functions induced by a bivariate quadratic function. Under some condition we show that the sequence of the quotients resulted from the ratios of consecutive terms in the bivariate generating function converges. Assigning the limiting quotient to 1 we obtain a tangent line to the associated bivariate quadratic curve at a point

521

HSU: BIVARIATE GENERATING FUNCTIONS

Table 2: the coefficients of 𝑥-term and 𝑦-term in the quotients of ratios of the consecutive terms in the associated generating function of the function in Example 3.2.

x term

y term

x term

y term

x term

y term

1

-5.0000

7.0000

12

-2.5549

0.2837

23

-2.5616

0.3494

2

-1.8000 -1.3200

13

-2.5656

0.3917

24

-2.5615

0.3486

3

-3.2222

2.8519

14

-2.5591

0.3210

25

-2.5616

0.3491

4

-2.2414 -0.8573

15

-2.5631

0.3670

26

-2.5615

0.3487

5

-2.7846

1.4672

16

-2.5606

0.3372

27

-2.5616

0.3490

6

-2.4365 -0.3204

17

-2.5621

0.3564

28

-2.5616

0.3488

7

-2.6417

0.8576

18

-2.5612

0.3440

29

-2.5616

0.3489

8

-2.5142

0.0226

19

-2.5618

0.3520

30

-2.5616

0.3489

9

-2.5910

0.5767

20

-2.5614

0.3469

31

-2.5616

0.3489

10

-2.5438

0.2002

21

-2.5616

0.3501

32

-2.5616

0.3489

11

-2.5724

0.4487

22

-2.5615

0.3481

33

-2.5616

0.3489

522

HSU: BIVARIATE GENERATING FUNCTIONS

in the x-axis. For the tangent line, the point is the only one at which it intersects the curve. For the curve, the point is the only one or the closer one (measured from the origin) at which it intersects the x-axis. Acknowledgments This work was supported by the National Science Council of Taiwan under Grant NSC 100-2221-E-005-071. References [1] E. A. Bender, Asymptotic methods in enumeration, SIAM Rev., 16, 485–515 (1974). [2] L. Comtet, Advanced Combinatorics: The art of finite and infinite expansions, Boston, D. Reidel Publ. Co., 1974. [3] P. Doubilet, G-C. Rota, and R. Stanley, On the foundations of combinatorial theory. VI. The idea of generating function, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 1972, vol. 2, 267–318. [4] I. P. Goulden and D. M. Jackson, Combinatorial enumeration, John Wiley and Sons, 1983. [5] Y. Jin, On efficient computation and asymptotic sharpness of Kalantari’s bounds for zeros of polynomials, Math. Comp., 75, 1905–1912 (2006). [6] B. Kalantari, An infinite family of bounds on zeros of analytic functions and relationship to Smale’s bound, Math. Comp., 74, 841–852 (2005).

523

HSU: BIVARIATE GENERATING FUNCTIONS

[7] D. E. Knuth, The Art of Computer Programming, Volume 1: Fundamental Algorithms, 3rd ed., Addison-Wesley, 1997. [8] R. Stanley, Generating functions, in: Studies in Combinatorics, Math. Asso. Amer. (Gian-Carlo Rota, ed), 1978, pp. 100–141. [9] H. S. Wilf, Generatingfunctionology, 3rd ed., A K Peters Ltd, 2006.

524

HSU: BIVARIATE GENERATING FUNCTIONS

the quotients of ratios in Example 3.1 4

the curve in Example 3.1 and its tangent line 3 2

2

1 1.0652

1

y

3

0

−1.618x+1.0652y=1

0 −1

−1

−2 −1.618

−2

5

10 15 number of step(s)

−3

20

the quotients of ratios in Example 3.2 8

−2

0 x

2

the curve in Example 3.2 and its tangent line 3

6

2

4

1 y

2

0

0.3489

0

−1

−2

−2.5616

−2

−4 5

10 15 20 25 number of step(s)

−3

30

−2.5616x+0.3489y=1 −2

0 x

2

Figure 1: Illustrations of Example 3.1 and 3.2. The upper right figure shows the curve −𝑥2 +4𝑥𝑦 +𝑦 2 +𝑥+𝑦 +1 = 0, along with its tangent line whose coefficients are the limiting ratios of consecutive terms of the associated generating function, as shown in the upper left figure. Similar figures are in the lower half for the curve: −4𝑥2 + 4𝑥𝑦 − 𝑦 2 + 𝑥 + 𝑦 + 1 = 0.

525

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 526-533, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

ISOMETRIES OF COMPOSITION AND DIFFERENTIATION OPERATORS FROM BLOCH TYPE SPACE TO Hα∞ GENG-LEI LI AND ZE-HUA ZHOU∗

Abstract. In this paper, we characterize the isometries of the products of composition and differentiation operators from the Bloch type space to the space of all weighted bounded analytic functions in the disk.

1. Introduction Let D be the unit disk of the complex plane, and S(D) the set of analytic selfmaps of D. The algebra of all holomorphic functions with domain D will be denoted by H(D). For 0 < β < ∞, by Hβ∞ (D) denote the space of all weighted bounded holomorphic functions on the unit disk with the norm kf kβ = sup(1 − |z|2 )β |f (z)|. z∈D

We recall that the Bloch type space B α (α > 0) consists of all f ∈ H (D) such that 2 kf kBα = sup(1 − |z| )α |f 0 (z)| < ∞, z∈D

then k·kBα is a complete semi-norm on B α , which is M¨obius invariant. It is well known that B α is a Banach space under the norm kf k = |f (0)| + kf kBα . Let ϕ ∈ S(D), the composition operator Cϕ induced by ϕ is defined by (Cϕ f )(z) = f (ϕ(z)) for z ∈ D and f ∈ H(D). Such operators act boundedly on many classical spaces of analytic functions, we recommend the interested readers refer to the books [29] by Shapiro and [10] by Cowen and MacCluer, which are excellent sources for the developments in the theory of composition operators up to the middle of last decade, and the recent papers [12, 13, 30, 32, 34, 35] listed in the bibliography. Let D be the differentiation operator on H(D), that is Df (z) = f 0 (z). For f ∈ H(D), the products of composition and differentiation operators DCϕ and Cϕ D are defined by Cϕ D(f ) = f 0 (ϕ) DCϕ (f ) = (f ◦ ϕ)0 = f 0 (ϕ)ϕ0 2010 Mathematics Subject Classification. Primary: 47B38; Secondary: 30H30, 47B33,47G20. Key words and phrases. Composition operator, Bloch type space, differentiation operator, isometry. ∗ Corresponding author. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 10971153, 10671141). 1

526

LI, ZHOU: ISOMETRIES ON BLOCH SPACE

It is of interest to provide function-theoretic characterizations for when ϕ induces a bounded or compact composition operator on various spaces. The boundedness and compactness of DCϕ on the Hardy space were discussed by Hibschweiler and Portnoy in [16] and by Ohno in [28]. A related but independent question is that of describing all isometries (surjective or not) of a given space among the composition operators. Let X and Y be two Banach spaces, recall that a linear isometry is a linear operator T from X to Y such that kT f kY = kf kX for all f ∈ X. In [3], Banach raised the question about concerning the form of an isometry on a specific Banach space. In most cases the isometries of a space of analytic functions on the disk or the ball have the canonical form of weighted composition operators, which is also true for most symmetric function spaces. For example, the surjective isometries of Hardy and Bergman spaces are certain weighted composition operators, See [14, 17, 18, 19, 24]. The description of all isometric composition operators is known for the Hardy space H 2 (e.g., see [8]), and the BMOA space (e.g., see [20, 21]). An analogous statement for the Bergman space A2α with standard radial weights has recently been obtained in [7], and there is a unified proof for all Hardy spaces and also for arbitrary Bergman spaces with reasonable radial weights [26]. For the Dirichlet space and Bloch space, the reader is referred to [15, 25, 27]. The surjective isometries of the Bloch space are characterized in [11]. Trivially, every rotation ϕ induces an isometry Cϕ of B. It has recently been shown in [31] that for composition operators, which induce isometries of B, the conditions ϕ(0) = 0 and D ⊂ C(ϕ) must hold. Here, C(ϕ) denotes the (global) cluster set of ϕ, that is, the set of all points a ∈ C such that there exists a sequence {zn } in D with the properties |zn | → 1 and ϕ(zn ) → a as n → ∞. Plenty of information on cluster sets is contained in [9]. Continued the work, in 2008, Bonet, Lindstr¨om and Wolf [4] discussed isometric weighted composition operators on weighted Banach spaces of type H ∞ . In 2008, Cohen and Colonna [6] discussed the spectrum of an isometric composition operators on the Bloch space of the polydisk. Li and Ruan [22] gave characterization for an isometric isomorphism on little Bloch space, VMOA and holomorphic Besov space over the unit ball Bn . In 2009, Allen and Colonna [1] investigated the isometric composition operators on the Bloch space in Cn . They [2] also discussed the isometries and spectra of multiplication operators on the Bloch space in the disk. Isometries of weighted spaces of holomorphic functions on unbounded domains were discussed by Boyd and Rueda in [5]. In 2010, Li and Zhou discussed the isometries on products of composition and integral operators on Bloch type space in [23]. Base on those foundations, we investigate the isometries of the products of composition operators and differentiation operators from Bloch type space to the space of all weighted bounded analytic functions in the disk. 2. Notation and Lemmas To begin the discussion, let us introduce some notation and state a couple of lemmas. For a ∈ D, the involution ϕa which interchanges the origin and point a, is defined by a−z . ϕa (z) = 1 − az

527

LI, ZHOU: ISOMETRIES ON BLOCH SPACE

For z, w in D, the pseudo-hyperbolic distance between z and w is given by z−w , ρ(z, w) = |ϕz (w)| = 1 − zw and the hyperbolic metric is given by Z |dξ| 1 1 + ρ(z, w) β (z, w) = inf = log , 2 γ 2 1 − ρ(z, w) 1 − |ξ| γ

where γ is any piecewise smooth curve in D from z to w. The following lemma is well known [33]. Lemma 1. For all z, w ∈ D, we have 1 − ρ2 (z, w) =

(1 − |z|2 )(1 − |w|2 ) |1 − zw|

2

.

For ϕ ∈ S(D), the Schwarz-Pick lemma shows that ρ (ϕ(z), ϕ(w)) ≤ ρ(z, w), and if equality holds for some z 6= w, then ϕ is an automorphism of the disk. It is also well known that for ϕ ∈ S(D), Cϕ is always bounded on B. A little modification of Lemma 1 in [4] shows the following lemma. Lemma 2. There exists a constant C > 0 such that  α  α 2 2 f 0 (z) − 1 − |w| f 0 (w) ≤ C kf kBα · ρ(z, w) 1 − |z| for all z, w ∈ D and f ∈ B α . Throughout the remainder of this paper, C will denote a positive constant, the exact value of which will vary from one appearance to the next. 3. Main theorems Theorem 1. Let ϕ ∈ S(D). Then the operator DCϕ : B α → Hβ∞ is an isometry in the semi-norm if and only if the following conditions hold: (1−|z|2 )β |ϕ0 (z)| (A) sup (1−|ϕ(z)|2 )α ≤ 1; z∈D

For every a ∈ D, there at least exists a sequence {zn } in D, such that (1−|zn |2 )β |ϕ0 (zn )| lim ρ(ϕ(zn ), a) = 0 and lim (1−|ϕ(z )|2 )α = 1. (B)

n→∞

n→∞

n

Proof. We prove the sufficiency first. By condition (A), for every f ∈ B α , we have ||DCϕ f ||β

=

2

sup (1 − |z| )β |ϕ0 (z)| |f 0 (ϕ(z))| z∈D

=

sup

(1 − |z|2 )β |ϕ0 (z)| 2

(1 − |ϕ(z)| )α kf kBα .

z∈D

≤

2

(1 − |ϕ(z)| )α |f 0 (ϕ(z))|

Next we show that the property (B) implies ||DCϕ f ||β ≥ ||f ||Bα . Given any f ∈ B α , then ||f ||Bα = lim (1 − |am |2 )α |f 0 (am )| for some sequence m→∞

{am } ⊂ D. For any fixed m, by property (B), there is a sequence {zkm } ⊂ D such that 2 β 0 m (1 − |z m k | ) |ϕ (zk )| ρ(ϕ(zkm ), am ) → 0 and →1 2 m (1 − |ϕ(zk )| )α

528

LI, ZHOU: ISOMETRIES ON BLOCH SPACE

as k → ∞. By Lemma 2, for all m and k, (1 − |ϕ(zkm )|2 )α f 0 (ϕ(zkm )) − (1 − |am |2 )α f 0 (am ) ≤ C||f ||Bα · ρ(ϕ(zkm ), am ). Hence (1 − |ϕ(zkm )|2 )α |f 0 (ϕ(zkm ))| ≥ (1 − |am |2 )α |f 0 (am )| − C||f ||Bα · ρ(ϕ(zkm ), am ) Therefore, ||DCϕ f ||β

=

sup

(1 − |z|2 )β |ϕ0 (z)| (1 −

z∈D

2 |ϕ(z)| )α

2

(1 − |ϕ(z)| )α |f 0 (ϕ(z))|

2

≥ lim sup

β 0 m (1 − |z m k | ) |ϕ (zk )|

k→∞

=

2

(1 − |ϕ(zkm )| )α

(1 − |ϕ(zkm )|2 )α |f 0 (ϕ(zkm ))|

(1 − |am |2 )α |f 0 (am )|.

The inequality ||DCϕ f ||β ≥ ||f ||Bα follows by letting m → ∞. From the above discussions, we have kDCϕ f kβ = kf kBα , which means that DCϕ is an isometry operator in the semi-norm from B α to Hβ∞ . Now we turn to the necessity. For any a ∈ D, we begin by taking test function Z z (1 − |a|2 )α fa (z) = dt. (1) ¯t)2α 0 (1 − a It is clear that fa0 (z) =

(1−|a|2 )α (1−¯ az)2α .

(1 − |z|2 )α |fa0 (z)| =

Using Lemma 1, we have

(1 − |z|2 )α (1 − |a|2 )α = (1 − ρ2 (a, z))α . |1 − a ¯z|2α

(2)

So kfa kBα = sup(1 − |z|2 )α |fa0 (z)| ≤ 1.

(3)

z∈D

2 2α

(1−|a| ) On the other hand, since (1 − |a|2 )α |fa0 (a)| = (1−|a| 2 )2α = 1, we have kfa kB α = 1. By isometry assumption, for any a ∈ D, we have

1

= ||fϕ(a) ||Bα = kDCϕ fϕ(a) kβ =

sup z∈D

(1 − |z|2 )β |ϕ0 (z)| 2

(1 − |ϕ(z)| )α

0 2 (1 − |ϕ(z)| )α fϕ(a) (ϕ(z))

2

≥

(1 − |a| )β |ϕ0 (a)| 2

(1 − |ϕ(a)| )α

.

So (A) follows by noticing a is arbitrary. Since ||DCϕ fa ||β = ||fa ||Bα = 1, there exists a sequence {zm } ⊂ D such that (1 − |zm |2 )β |(DCϕ fa )(zm )| = (1 − |zm |2 )β |fa0 (ϕ(zm ))||ϕ0 (zm )| → 1

(4)

as m → ∞. It follows from (A) that (1 − |zm |2 )β |fa0 (ϕ(zm ))||ϕ0 (zm )| (1 − |zm |2 )β |ϕ0 (zm )| 2 = (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| 2 (1 − |ϕ(zm )| )α 2

≤ (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| .

529

(5) (6)

LI, ZHOU: ISOMETRIES ON BLOCH SPACE

Combining (4) and (6), it follows that 2

1 ≤ lim inf (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| m→∞

2

≤ lim sup(1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| ≤ 1. m→∞

The last inequality follows by (2) since ϕ(zm ) ∈ D. Consequently, 2

lim (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| = lim (1 − ρ2 (ϕ(zm ), a))α = 1.

m→∞

m→∞

(7)

That is, lim ρ(ϕ(zm ), a) = 0. m→∞

Combining (4), (5) and (7), we know lim

(1 − |zm |2 )β |ϕ0 (zm )|

m→∞

2

(1 − |ϕ(zm )| )α

= 1.

This completes the proof of this theorem.



Theorem 2. ϕ ∈ S(D). Then the operator Cϕ D : B α → Hβ∞ is an isometry in the semi-norm if and only if the following conditions hold: (1−|z|2 )β (C) sup (1−|ϕ(z)| 2 α ≤ 1; ) z∈D

(D) For every a ∈ D, there at least exists a sequence {zn } in D, such that (1−|zn |2 )β lim ρ(ϕ(zn ), a) = 0 and lim (1−|ϕ(z = 1. )|2 )α

n→∞

n→∞

n

Proof. We prove the sufficiency first. By condition (C), for every f ∈ B α we have ||Cϕ Df ||β

=

2

sup (1 − |z| )β |f 0 (ϕ(z))| z∈D

=

sup

(1 − |z|2 )β 2

≤

2

(1 − |ϕ(z)| )α kf kBα .

z∈D

(1 − |ϕ(z)| )α |f 0 (ϕ(z))|

Next we show that the property (D) implies ||Cϕ Df ||β ≥ ||f ||Bα . Given any f ∈ B α , then ||f ||Bα = lim (1−|am |2 )α |f 0 (am )| for some sequence {am } ⊂ D. For any fixed m→∞

m, by property (II), there is a sequence {zkm } ⊂ D such that 2

ρ(ϕ(zkm ), am ) → 0 and

β (1 − |z m k | ) 2

(1 − |ϕ(zkm )| )α

→1

as k → ∞. By Lemma 2, for all m and k, (1 − |ϕ(zkm )|2 )α f 0 (ϕ(zkm )) − (1 − |am |2 )α f 0 (am ) ≤ C||f ||Bα · ρ(ϕ(zkm ), am ). Hence (1 − |ϕ(zkm )|2 )α |f 0 (ϕ(zkm ))| ≥ (1 − |am |2 )α |f 0 (am )| − C||f ||Bα · ρ(ϕ(zkm ), am )

530

LI, ZHOU: ISOMETRIES ON BLOCH SPACE

Therefore, ||Cϕ Df ||β

=

sup z∈D

(1 − |z|2 )β

2

2

(1 − |ϕ(z)| )α

(1 − |ϕ(z)| )α |f 0 (ϕ(z))| 2

≥ lim sup k→∞

=

β (1 − |z m k | ) 2

(1 − |ϕ(zkm )| )α

(1 − |ϕ(zkm )|2 )α |f 0 (ϕ(zkm ))|

(1 − |am |2 )α |f 0 (am )|

The inequality ||Cϕ Df ||β ≥ ||f ||Bα follows by letting m → ∞. From the above discussions, we have kCϕ Df kβ = kf kBα , which means that Cϕ D is an isometry operator in the semi-norm from B α to Hβ∞ . Now we turn to the necessity. For any a ∈ D, using the same test function fa defined by (1) which satisfies kfa k = 1. By isometry assumption, for any a ∈ D, we have 1

= ||fϕ(a) ||Bα = kCϕ Dfϕ(a) kβ =

(1 − |z|2 )β

sup

2

(1 − |ϕ(z)| )α

z∈D

0 2 (ϕ(z)) (1 − |ϕ(z)| )α fϕ(a)

2

≥

(1 − |a| )β 2

(1 − |ϕ(a)| )α

.

So (C) follows by noticing a is arbitrary. Since ||Cϕ Dfa ||β = ||fa ||Bα = 1, there exists a sequence {zm } ⊂ D such that (1 − |zm |2 )β |(Cϕ Dfa )(zm )| = (1 − |zm |2 )β |fa0 (ϕ(zm ))| → 1

(8)

as m → ∞. It follows from (C) that (1 − |zm |2 )β |fa0 (ϕ(zm ))| (1 − |zm |2 )β 2 = (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| 2 (1 − |ϕ(zm )| )α 2

≤ (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| .

(9) (10)

Combining (8) and (10), it follows that 2

1 ≤ lim inf (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| m→∞

2

≤ lim sup(1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| ≤ 1. m→∞

The last inequality follows by (2) since ϕ(zm ) ∈ D. Consequently, 2

lim (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| = lim (1 − ρ2 (ϕ(zm ), a))α = 1.

m→∞

m→∞

(11)

That is, lim ρ(ϕ(zm ), a) = 0. m→∞

Combining (8), (9) and (11), we know lim

m→∞

(1 − |zm |2 )β |ϕ0 (zm )| 2

(1 − |ϕ(zm )| )α

This completes the proof of this theorem.

531

= 1. 

LI, ZHOU: ISOMETRIES ON BLOCH SPACE

Remark If α = 1, β = 0, then B α will be Bloch space B and Hβ∞ will be H ∞ . The similar results from the Bloch space B to H ∞ corresponding to Theorems 1 and 2 also hold. References [1] R. F. Allen and F. Colonna, Isometric composition operators on the Bloch space in C n , J. Math. Anal. Appl., 355 (2009), 675-688. [2] R. F. Allen and F. Colonna, Isometries and spectra of multiplication operators on the Bloch space, Bull. Aust. Math. Soc., 79 (2009),147-160. [3] S. Banach, Theorie des Operations Lineares, Chelsea, Warzaw, 1932. [4] J.Bonet, M. Lindstr¨ om and E. Wolf, Isometric weighted composition operators on weighted Banach spaces of type H ∞ , Proc. Amer. Math. Soc., 136(12)(2008), 4267-4273. [5] C. Boyd and P. Rueda, Isometries of weighted spaces of holomorphic functions on unbounded domains, Proceedings of the Royal Society of Edinburgh Section A-mathematics, 139A (2009), 253-271. [6] J.M.Cohen and F. Colonna, Isometric Composition Operators on the Bloch Space in the Polydisk, Contemporary Mathematics, Banach Spaces of Analytic Functions, 454 (2008), 9-21. [7] B. J. Carswell and C. Hammond, Composition operators with maximal norm on weighted Bergman spaces, Proc. Amer. Math. Soc., 134 (2006), 2599-2605. [8] B. A. Cload, Composition operators: hyperinvariant subspaces, quasi-normals and isometries, Proc. Amer. Math. Soc., 127 (1999),1697-1703. [9] E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge University Press, Cambridge, 1966. [10] C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton , FL, 1995. [11] J. Cima and W. R. Wogen, On isometries of the Bloch spaces, Illinois J. Math., 24 (1980), 313-316. [12] Z.S.Fang and Z.H.Zhou, Differences of composition operators on the space of bounded analytic functions in the polydisc, Abstract and Applied Analysis, 2008 (2008), Article ID 983132, 10 pages. [13] Z. S. Fang and Z. H. Zhou, Differences of composition operators on the space of bounded analytic functions in the polydisc, Bull.Aust.Math.Soc., 79(2009),465-471. [14] W. Hornor and J. E. Jamison, Isometries of some Banach spaces of analytic functions, Integral Equations Operator Theory ,41(2001), 410-425. [15] T. Hosokawa, S. Ohno, Differences of composition operators on the Bloch space, J. Operator Theory, 57(2007), 229-242. [16] R. A. Hibschweiler and N.Portony, Composition followed by differtition between Bergman and Hardy spaces, The Rocky Mountain Journal of Mathematics, 35(2005), 843-855. [17] C.J. Kolaski, Isometries of weighted Bergman spaces, Can.J. Math., 34 (1982), 910-915. [18] C. J. Kolaski, Isometries of some smooth spaces of analytic functions, Complex Variables, 10 (1988), 115-122. [19] A. Koranyi and S. Vagi, On isometries of H p of bounded symmetruc domains, Cann. J. Math., 28(1976), 334-340. [20] S. Y. Li, Composition operators and isometries on holomorphic function spaces over domains in Cn . AMS/IP Stud Adv Math. 39 (2007), 161-174. [21] J. Laitila, Isometric composition operators on BMOA, Math. Nachr., to appear. [22] S. Y. Li and Y. B. Ruan, On characterizations of isometries on function spaces, Science in China Series A: Mathematics, 51(4)(2008),620-631. [23] Geng-Lei Li and Ze-Hua Zhou,Isometries on Products of composition and integral operators on Bloch type space, Journal of Inequalities and Applications, 2010 (2010), 12p., Article ID 18495. [24] A. Matheson, Isometries into function algebras, Houston J. Math., 30 (2004), 219-230 . [25] M. J. Mart´tn and D. Vukoti´ c, Isometries of the Bloch space among the composition operators, Bull. London Math. Soc., 39 (2007), 151-155. [26] M. J. Mart´tn and D. Vukoti´ c, Isometries of some classical function spaces among the composition operators, Recent advances in operator-related function theory, ed. A. L. Matheson,

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LI, ZHOU: ISOMETRIES ON BLOCH SPACE

[27] [28] [29] [30] [31] [32] [33] [34] [35]

M. I. Stessin, and R. M. Timoney, Contemporary Mathematics, 393 American Mathematical Society, Providence, RI, 2006, 133-138. M. J. Mart´tn and D. Vukoti´ c, Isometries of the Dirichlet space among the composition operators, Proc. Amer. Math. Soc., 134 (2006), 1701-1705. Ohno, Product of composition and differtition between Hardy spaces, Bulletin of the Australian Mathematical Society, 73(2006), 235-243. J.H. Shapiro, Composition operators and classical function theory, Spriger-Verlag, 1993. J. Xiao, Composition operators associated with Bloch-type spaces, Complex Variables 46 (2001), 109-121. C. Xiong, Norm of composition operators on the Bloch space, Bull. Austral. Math. Soc., 70 (2004), 293-299. X. J. Zhang, J. C. Liu, Composition operators from weighted Bergman spaces to µ-Bloch spaces. Chinese Ann. Math. Ser A, 28(2)(2007), 255-266. K. H. Zhu, Spaces of holomorphic functions in the unit ball, Springer-Verlag (GTM 226), 2004. Z. H. Zhou and J. H. Shi, Compactness of composition operators on the Bloch space in classical bounded symmetric domains, Michigan Math. J., 50(2002),381-405. H. G. Zeng, Z. H. Zhou, An estimate of the essential norm of a composition operator from F (p, q, s) to Bα in the unit ball, J. Inequal. Appl. 2010 (2010). 18p., Article ID 132970.

Geng-Lei Li, Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, P.R. China. E-mail address: [email protected]

Ze-Hua Zhou, Department of Mathematics, Tianjin University, Tianjin 300072, P.R. China. E-mail address: [email protected]

533

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 534-543, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

An Expanded Mixed Finite Element Method for Sobolev Equation Na Li 1

1

, Fuzheng Gao

2, ∗

, Tiande Zhang

2

Basic Subject Department, Shandong Women’s University Jinan, Shandong 250300, China

2

School of Mathematics, Shandong University, Jinan, 250100, China

September 28, 2012

Abstract In this paper we shall present an expanded mixed finite element method (EMFEM) for a class of Sobolev equation. The optimal L2 error estimates for the semi-discrete and fully discrete schemes are obtained. Also Some numerical results are given to verify our analysis for the schemes.

1

Introduction.

In this paper, we will present a numerical scheme based on the EMFEM for the Sobolev equation ut + f (u)x − µuxxt = 0 (x, t) ∈ I × (0, T ]

(1)

where I = [X1 , X2 ], and the first order derivative of the function f (u) with respect to independent variable u is bounded. Here, µ is a given positive constant. We consider the equation (1) with the homogeneous boundary conditions u(X1 , t) = 0,

t > 0,

u(X2 , t) = 0,

t > 0,

and the initial condition u(x, 0) = u0 (x),

x ∈ I,

where u0 (x) is a prescribed function. Equation (1) has a wide range of applications in many mathematical and physical problems, for example, thermodynamics, shear in second-order fluids, and consolidation of clay. If f (u) = αu + βu2 with given numbers α and ∗ Corresponding

author: [email protected] (F. Z. Gao)

534

N. LI, F.GAO, T.ZHANG

β, (1) is referred as the regularized long-wave equation, or Benjamin-BonaMahony(BBM) equation. Up to now, there are some different schemes studied to solve this kind of equations (see[1, 2, 3, 4, 5]). Gao, Qiu and Zhang (see [5]) proposed a LDG method by defining three auxiliary variables, and transforming spatial derivative into temporal derivative in [5], with that the main computation changed to solve an elliptic equation. In this paper we will adopt those tricks to obtain optimal error for the semi-discrete and fully-discrete schemes by the expanded mixed finite element method. The mixed finite element method (MFE) , which is a finite element method with constrained conditions or Lagrangian multipliers, plays an important role in the research of the numerical solution for the higher order partial differential equation(PDE) or the PDE including two(or more) unknown functions. Its general theory was first proposed by Babuska [6] and Brezzi [7] in the early 1970. In the early 1980s, Falk and Osborn [8] improved their theory and expanded the adaptability of the MFE method. So far, the MFE method(see [9, 10]) are wildly used for the modeling of fluid flow and transport, as they provide accurate and locally mass conservative velocities and handle well discontinuous coefficients. The rest content of this paper is organized as follows. In Sect.2, we will describe the semi-discrete scheme for (1), present the uniqueness theorem, and obtain the optimal error estimates. In Sect.3, we give the fully discrete scheme and the optimal error estimates. In Sect.4, one numerical experiment is presented to verify our error estimates are optimal. Throughout this article we use C(without or with subscript) to denote a generic constant independent of the discretization parameters, which has different values in different appearances. We also adopt the standard definitions and notations of Sobolev spaces and their norms in [11] and [12].

2

2.1

The Semi-discrete Scheme Based on The Expanded Mixed Finite Element Method The Semi-discrete Scheme and Uniqueness of The Solution

By introducing three auxiliary variables w = ut , p = wx , q = ux ,

535

An Expanded Mixed FEM for Sobolev Equation

(1) can be rewritten into the following equivalent first-order differential system with regard to the solution (u, q, w, p). It reads that  u =w (x, t) ∈ I × (0, T ]   t    qt = p (x, t) ∈ I × (0, T ]       w − µpx = −f ′ (u)q (x, t) ∈ I × (0, T ] u (2)  p − w = 0 (x, t) ∈ I × (0, T ]  x      u(x, 0) = u0 (x) x∈I     q(x, 0) = u0,x (x) x∈I Multiplying by v and integrating each equation of (2) in I, respectively, we get that  (ut , v) = (w, v) ∀v ∈ L2      (qt , v) = (p, v) ∀v ∈ H01       (w, v) − µ(px , v) = −(f ′ (u)q, v) ∀v ∈ L2 u (3)  ∀v ∈ H01   (p, v) + (w, vx ) = 0     u(x, 0) = u0 (x) x∈I     q(x, 0) = u0,x (x) x∈I Let Qh ⊂ L2 (I), Rh ⊂ H01 (I) be finite-dimensional spaces. Approximating (u, q, w, p) by (uh , qh , wh , ph ), we get the expanded mixed finite element scheme: find (uh (·, t), qh (·, t), wh (·, t), ph (·, t)) : [0, T ] → Qh × Rh × Qh × Rh , such that  (uh,t , vh ) = (wh , vh ) ∀vh ∈ Qh      ∀vh ∈ Rh  (qh,t , vh ) = (ph , vh ) (4) (wh , vh ) − µ(ph,x , vh ) = −(fu′ (uh )qh , vh ) ∀vh ∈ Qh       (ph , vh ) + (wh , vh,x ) = 0 ∀vh ∈ Rh Theorem 1 The solution of (4) exists uniquely. Proof. Suppose ψ1 (x), ψ2 (x), ..., ψm (x) be the basic functions of Qh , ϕ1 (x), ϕ2 (x), ..., ϕn (x) be the basic functions of Rh , and writing uh (x, t) = m n m n P P P P ai ψi (x), qh (x, t) = bi ϕi (x), wh (x, t) = ci ψi (x), ph (x, t) = di ϕi (x), i=1

i=1

i=1

i=1

with that the computation is changed into solve ordinary differential equations as follows  da A = Ac    dt    D db = Dd dt (5)  Ac − µBd = F      Dd + B T c = 0

536

N. LI, F.GAO, T.ZHANG

where a, b, c, d, F denote vectors and A, B, D are matrixes. Each meaning in the above equations is clear, A, D are positive definite matrixes, we can get that (5) has unique solution in [0, T ], and finally we come to the conclusion that (4) has unique solution.

2.2

The Error Estimates

Introducing projection: find (u, q, w, p) : [0, t] → Qh × Rh × Qh × Rh , such that  ∀vh ∈ Qh (u − u, vh ) = 0       (q − q, vh ) = 0 ∀vh ∈ Rh (6)  (px − ph,x , vh ) = 0 ∀vh ∈ Qh      (p − p, v ) + (w − w , v ) = 0 ∀v ∈ R h h h,x h h we can get k ξ − ξ k0 ≤ Ch k ξ k1 ≤ Ch2 k ξ k2 ,

ξ = u, q, w, p.

Subtracting (4) from (3) gives  (ut − uh,t , vh ) = (w − wh , vh )       (qt − qh,t , vh ) = (p − ph , vh )

∀vh ∈ Qh ∀vh ∈ Rh

 (w − wh , vh ) − µ(px − ph,x , vh ) =      (p − p , v ) + (w − w , v ) = 0 h h h h,x

−(fu′ (u)q

−

fu′ (uh )qh , vh )

∀vh ∈ Qh

(7)

∀vh ∈ Rh

using (6), we have (ut − uh,t , vh ) = (w − wh , vh )

∀vh ∈ Qh

(8)

(q t − qh,t , vh ) = (p − ph , vh )

∀vh ∈ Rh

(9)

(w − wh , vh ) − µ(px − ph,x , vh ) = − (w − w, vh )− (fu′ (u)q − fu′ (uh )qh , vh ) (p − ph , vh ) + (w − wh , vh,x ) = 0

(10) ∀vh ∈ Qh ∀vh ∈ Rh

(11)

Theorem 2 Let (u, q, w, p) and (uh , qh , wh , ph ) is the solution of (3) and (4), respectively. (u, q, w, p) satisfies (6), assume that (u, q, w, p) ∈ L2 ×H01 ×L2 ×H01 , the semi-discrete scheme has prior error estimate as follows k u − uh kL∞ (0,T ;H 0 (I)) + k q − qh kL∞ (0,T ;H 0 (I)) + k w − wh kL∞ (0,T ;H 0 (I)) + k p − ph kL∞ (0,T ;H 0 (I)) ≤

k (u − uh )(0) k0 + k (q − qh )(0) k0 +Ch2 (k u kL∞ (0,T ;H 2 (I)) + k q kL∞ (0,T ;H 2 (I)) + k w kL∞ (0,T ;H 2 (I)) + k p kL∞ (0,T ;H 2 (I)) ).

537

An Expanded Mixed FEM for Sobolev Equation

Proof. Let vh = px − ph,x in (10), we can get that (w − wh , px − ph,x ) − µ(px − ph,x , px − ph,x ) = −(w − w, px − ph,x ) − (fu′ (u)q −fu′ (uh )qh , px − ph,x ), Let vh = p − ph in (11), we can get (p − ph , p − ph ) + (w − wh , px − ph,x ) = 0, using the two equations and Lemmas above, we have, for ε relatively small, that k p − ph k20 +µ k px − ph,x k20 =

(w − w, px − ph,x ) + (fu′ (u)q − fu′ (uh )qh , px − ph,x )

≤

C(k w − w k0 k px − ph,x k0 + k q − qh k0 k px − ph,x k0 )

≤

C(h4 k w k22 +h4 k q k22 + k q − qh k20 ) + ε k px − ph,x k20 ,

k p − ph k20 + k px − ph,x k20 ≤ C(h4 k w k22 +h4 k q k22 + k q − qh k20 ), then k p − ph k0 + k px − ph,x k0 ≤ C(h2 k w k2 +h2 k q k2 + k q − qh k0 ), Let vh = w − wh in (10), we have (w − wh , w − wh ) − µ(px − ph,x , w − wh ) = −(w − w, w − wh ) − (fu′ (u)q −fu′ (uh )qh , w − wh ), k w − wh k20 = µ(px − ph,x , w − wh ) − (w − w, w − wh ) − (fu′ (u)q − fu′ (uh )qh , w − wh ) ≤ C(k px − ph,x k0 +h2 k w k2 +h2 k q k2 + k q − qh k0 ) k w − wh k0 , with that k w − wh k0

≤

C(k px − ph,x k0 +h2 k w k2 +h2 k q k2 + k q − qh k0 )

≤

C(h2 k w k2 +h2 k q k2 + k q − qh k0 ),

Let vh = u − uh in (8), we have (ut − uh,t , u − uh ) = (w − wh , u − uh ), then d k u − uh k20 dt

≤

C k w − wh k0 k u − uh k0

≤

C(k w − wh k20 + k u − uh k20 )

≤

C(h4 k w k22 + k w − wh k20 +h4 k u k22 + k u − uh k20 )

≤

C(h4 k u k22 +h4 k q k22 +h4 k w k22 + k q − qh k20 + k u − uh k20 ).

538

N. LI, F.GAO, T.ZHANG

Similarly, Let vh = q − qh in (9), we get d k q − qh k20 ≤ C(h4 k q k22 +h4 k w k22 +h4 k p k22 + k q − qh k20 ). dt Obviously d (k u − uh k20 + k q − qh k20 ) ≤ dt

Ch4 (k u k22 + k q k22 + k w k22 + k p k22 ) +C(k u − uh k20 + k q − qh k20 ).

Using Gronwall T heorem, we have k u − uh k20 + k q − qh k20

≤

k (u − uh )(0) k20 + k (q − qh )(0) k20 +Ch4 ( k u k22 + k q k22 + k w k22 + k p k22 ),

so k u − uh kL∞ (0,T ;H 0 (I)) + k q − qh kL∞ (0,T ;H 0 (I)) + k w − wh kL∞ (0,T ;H 0 (I)) + k p − ph kL∞ (0,T ;H 0 (I)) ≤

k (u − uh )(0) k0 + k (q − qh )(0) k0 +Ch2 (k u kL∞ (0,T ;H 2 (I)) + k q kL∞ (0,T ;H 2 (I)) + k w kL∞ (0,T ;H 2 (I)) + k p kL∞ (0,T ;H 2 (I)) ).

3

The Fully discrete Scheme Based on The Expanded Mixed Finite Element Method

Let 0 = t0 < t1 < ... < tN = T be a partition of the domain [0, T ], ∆tk = tk − tk−1 , k = 1, 2, ..., N, (ukh , qhk , whk , pkh ) is the approximation of (u, q, w, p) at tk . Then, using forward difference quotient in place of time derivative, we get the fully discrete scheme: find (ukh , qhk , whk , pkh ) such that  k+1 k u −u  ( h ∆t h , vh ) = (whk , vh )       qhk+1 −qhk ( ∆t , vh ) = (pkh , vh )    (whk , vh ) − µ(pkh,x , vh ) = −(fu′ (ukh )qhk , vh )     k (ph , vh ) + (whk , vh,x ) = 0

∀vh ∈ Qh ∀vh ∈ Rh

(12)

∀vh ∈ Qh ∀vh ∈ Rh

Let ρu = u−u, θu = u−uh , ρq = q−q, θq = q−qh , ρw = w−w, θw = w−wh , ρp = p − p, θp = p − ph ,

539

An Expanded Mixed FEM for Sobolev Equation

then (

θuk+1 − θuk uk+1 − uk k , vh ) = (θw , vh ) + (ρkw , vh ) − (uk+1 − , vh )∀vh ∈ Qh (13) t ∆t ∆t

(

θqk+1 − θqk q k+1 − q k , vh ) = (θpk , vh ) + (ρkp , vh ) − (qtk+1 − , vh ) ∀vh ∈ Rh (14) ∆t ∆t

k k (θw , vh ) − µ(θp,x , vh ) = −(ρkw , vh ) − (fu′ (uk )q k − fu′ (ukh )qhk , vh ) ∀vh ∈ Qh (15) k (θpk , vh ) + (θw , vh,x ) = 0

∀vh ∈ Rh (16)

k Let vh = θp,x in (15), we have k k k k k k (θw , θp,x ) − µ(θp,x , θp,x ) = −(ρkw , θp,x ) − (fu′ (uk )q k − fu′ (ukh )qhk , θp,x ),

Let vh = θpk in (16), using the equation above, we can get k k (θpk , θpk ) + (θw , θp,x ) = 0,

for ε relatively small, k k ) ) + (fu′ (uk )q k − fu′ (ukh )qhk , θp,x (ρkw , θp,x

k k ) = , θp,x (θpk , θpk ) + µ(θp,x

then

≤

k C(h2 k w k2 +h2 k q k2 + k θqk k0 ) k θp,x k0

≤

k C(h4 k w k22 +h4 k q k22 + k θqk k20 ) + ε k θp,x k20 ,

k k20 ≤ C(h4 k w k22 +h4 k q k22 + k θqk k20 ), k θpk k20 + k θp,x

k Let vh = θw in (15), we have k 2 k k k θw k0 ≤ C(k θp,x k0 + k ρkw k0 + k q k − qhk k0 ) k θw k0 , k k θw k0 ≤ C(h2 k w k2 +h2 k q k2 + k θqk k0 ),

Let vh = θuk+1 in (13), we get k

θuk+1

k0

≤ k

θuk

Z

k0 +∆t k

k θw

k0 +∆t k

ρkw

k0 +∆t

tn+1

k tn

∂2u k0 dt ∂t2

≤ k θuk k0 +C∆t(h2 k q k2 +h2 k w k2 + k θqk k0 ) + Z

tn+1

∆t

k tn

∂2u k0 dt, ∂t2

similarly, Let vh = θqk+1 in (14), we get k θqk+1 k0

≤ k θqk k0 +C∆t(h2 k q k2 +h2 k w k2 +h2 k p k2 + k θqk k0 ) + Z

tn+1

k

∆t tn

∂2q k0 dt, ∂t2

540

N. LI, F.GAO, T.ZHANG

so k θuk+1 k0 + k θqk+1 k0

≤

C[k θu0 k0 + k θq0 k0 +h2 (k q k2 + k w k2 + k p k2 )] Z +∆t(

tn+1

t0

k

∂2u k0 dt + ∂t2

Z

tn+1

k t0

∂2q k0 dt). ∂t2

Theorem 3 Under the assumptions of Theorem 2, the fully discrete scheme has prior error estimate as following max k uk − ukh k0 + max k q k − qhk k0 + max k wk − whk k0 +

1≤k≤N

1≤k≤N

1≤k≤N

max k pk − pkh k0

1≤k≤N

≤

C[k u0 − u0,h k0 + k q0 − q0,h k0 +h2 (k q kL∞ (0,T ;H 0 (I)) + k w kL∞ (0,T ;H 0 (I)) )+ k p kL∞ (0,T ;H 0 (I)) )] + ∆t(k k

4

∂2u kL1 (0,T ;H 0 (I)) + ∂t2

∂2q kL1 (0,T ;H 0 (I)) ). ∂t2

Numerical Examples

In this section, we consider the following example by using the scheme given in (4).  ut + ux + uux − uxxt = 0    u(±100, t) = 0    u(x, 0) = 3asech2 (bx)

x ∈ (−100, 100), t > 0 t>0

(17)

x ∈ (−100, 100)

where a is an arbitrary constant and b = (1/2)(a/(1+a))(1/2) . The exact solution is u(x, t) = 3asech2 (bx − ct + φ), where c = (1/2)(a(a + 1))(1/2) , and φ is an arbitrary constant. 1 1 , τ = 4096 , uh and qh are very good We choose φ = 0, a = 0.03, h = 64 approximations of u and q, which are showed in the following two figures.

541

An Expanded Mixed FEM for Sobolev Equation

* Approximate solution uh −− Exact solution u 0.09 0.08 0.07 0.06

uh

0.05 0.04 0.03 0.02 0.01 0 −1

−0.5

0 x/100

0.5

1

* Approximate solution qh −− Exact solution q 0.6

0.4

0.2

qh

0

−0.2

−0.4

−0.6

−0.8 −1

−0.5

0 x/100

0.5

1

The discrete L2 norms of error are listed in Tables 1-2. From these results we can see that the error in the discrete norms have the optimal order.

τ /h2 h = 21 h = 41 h = 81 1 h = 16

= 1/4 1 τ = 16 1 τ = 64 1 τ = 256 1 τ = 1024

Table 1: t=0.25 0.01988025 0.00355500 0.00142838 0.00060856

k u − uh k0,h t=0.50 t=0.75 0.01988021 0.01988016 0.00354435 0.00353454 0.00135176 0.00129912 0.00047695 0.00037632

τ /h2 h = 21 h = 41 h = 81 1 h = 16

= 1/4 1 τ = 16 1 τ = 64 1 τ = 256 1 τ = 1024

Table 2: k q − qh k0,h t=0.25 t=0.50 0.04340493 0.04335041 0.00675072 0.00674950 0.00122853 0.00123664 0.00023734 0.00028585

542

t=0.75 0.04326232 0.00674930 0.00125064 0.00035341

t=1.0 0.01988010 0.00352558 0.00127347 0.00033592

t=1.0 0.04314177 0.00675033 0.00127063 0.00043262

N. LI, F.GAO, T.ZHANG

5

Concluding Remarks

Sobolev equations (1) are solved using the EMFEM, by introducing three auxiliary variables. The performance of the method is examined on one numerical experiment. It is seen that the error norms are satisfactorily small and the method successfully models the motion. Furthermore, from the numerical results, it seems certain that some invariants could be obtained, we will discuss the content of this respect in the ongoing work. Acknowldgements The authors’ research is supported by the Scientific Research Award Fund for Excellent Middle-Aged and Yang Scientists of Shandong Province grant BS2009HZ015.

References [1] C. E. Seyler, D. C. Fenstermacler. A Symmetric Regularized Long Wave Equation[J]. Phys. Fluids., 1984, 27(1): 4-7. [2] Y. D. Shang, B. L. Guo. Analysis of Chebyshev Pseudospectral Method for Multidimensional Regularized SRLW Equations[J]. Applied Mathematics and Mechanics, 2003, 24(10): 1035-1048. [3] K. Omarani. The convergence of fully discrete Galerkin approximation for the BenjaminBona-Mahony (BBM) equation[J]. Applied Mathematics and Computation, 2006, 180: 614-621. [4] A. Cesar, S. Gomez, H. Alvaro, Salas, Bernardo Acevedo Frias. New periodic and soliton solutions for the Generalized BBM and Burgers-BBM equations. Applied Mathematics and Computation, 2010, 217: 1430-1434. [5] F. Z. Gao, J. X. Qiu, Q. Zhang. Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation[J]. J Sci Comput, 2009, 41: 436-460. [6] I. Babuska. Error-bounds for finite element method. Numer. Math., 1970, 16: 322-333. [7] F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian mulipliers. Rev. Francaise Automat. Informat. Recherche Operationalle Ser. Rouge, 1974, 8: 129-151. [8] R. S. Falk and J. E. Osborn. Error estimates for mixed methods. RAIRO Anal. Numer., 1980, 14: 249-277. [9] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Method, Springer Ser. Comput. Math. 15, Spinger-Verlag, Berlin, 1991. [10] J. E. Roberts and J. M. Thomas. Mixed and Hybrid Methods. Handb. Numer.Anal. 2, P. Ciarlet and J. Lions, eds., Elsevier/North Holland, Amsterdam, 1991. [11] R. Adams. Sobolev Spaces. Academic Press, 1975. [12] P. G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. [13] J. L. Bona, W. G. Pritchard, L. R. Scott. Numerical schemes for a model of nonlinear dispersive waves. J. Comp. Phys. 1985, 60: 167-196.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 544-551, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

SOME IDENTITIES FOR THE FROBENIUS-EULER NUMBERS AND POLYNOMIALS TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM

Abstract. In this paper we give some new identities on the Frobenius-Euler numbers and polynomials arising from the generating function of FrobeniusEuler polynomials.

1. Introduction The Bernoulli polynomials are defined by means of ∞ X t tn xt e = Bn (x) , t e −1 n! n=0

(1)

(see [1-10]),

and the Euler polynomials are also defined by ∞ X 2 tn xt e = E (x) , n et + 1 n! n=0

(2)

(see [1-10]),

In the special case x = 0, Bn (0) = Bn and En (0) = En are called the n-th Bernoulli numbers and the n-th Euler numbers. By (1) and (2), we get n   n   X X n n En−l xl . Bn−l xl , and En (x) = Bn (x) = (3) l l l=0

l=0

From (1), (2) and (3), we can derive the recursion formulae for Bn and En : (4)

B0 = 1,

Bn (1) − Bn = δ1,n ,

and E0 = 1,

En (1) + En = 2δ0,n ,

where δk,n is Kronecker’s symbol. It may be of interest in the connection to the formula for a product of two Bernoulli polynomials: (5) Bm (x)Bn (x) =

X m  r

   n Bm+n−2r (x)B2r m!n! n+ m +(−1)m+1 Bm+n , 2r 2r m + n − 2r (m + n)! (m + n ≥ 2).

Nielsen also gave similar formulas for Em (x)En (x) and Em (x)Bn (x), (see [3,4]). For u(6= 1) ∈ C, the Frobenius-Euler polynomials are defined by (6)

∞ X 1 − u xt tn H(x|u)t e = e = H (x|u) , n et − u n! n=0

544

TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM

with the usual convention about replacing H n (x|u) by Hn (x|u) (see [5]). In the special case, x = 0, Hn (0|u) = Hn (u) are called the n-th Frobenius-Euler numbers. Thus, by (2) and (6), we get Hn (x| − 1) = En (x). Let α 6= 1, β 6= 1 and αβ 6= 1. Then, from (6), Carlitz derived the following equation: (7) m   (1 − α)(1 − β) α(1 − β) X m Hm (x|α)Hn (x|β) = Hm+n (x|αβ) + Hr (α)Hm+n−r (x|αβ) 1 − αβ 1 − αβ r=1 r n   β(1 − α) X n + Hs (β)Hm+n−s (x|αβ), (see [3]). 1 − αβ s=1 s

It may be of interest in this connection to recall the formula for a product of Bernoulli and Frobenius-Euler polynomials ([3]): m   X m Bm (x)Hn (x|α) = mHm+n−1 (x|α) + Br Hm+n−r (x|α) r r=0 (8) n   mα X n + Hs (α)Hm+n−s−1 (x|α). 1 − α s=1 s In particular, let α 6= 1 with αα−1 = 1. Then, we have (9) m−1 X

 m Bm+n−r (x) Hr+1 (α) m+n−r r+1 r=0   n−1 X n Bm+n−s (x) + Cm,n , − (1 − α−1 ) Hs+1 (α−1 ) m+n−s s+1 s=0

Hm (x|α)Hn (x|α−1 ) = −(1 − α)

m!n! where Cm,n = (−1)n+1 (m+n+1)! (1 − α)Hm+1 (α) (see [3,4]). In this paper, we derive some new identities for the product of Frobenius-Euler polynomials from the generating functions of Frobenius-Euler polynomials and differential operator.

2. On identities of Frobenius-Euler numbers and polynomials Consider x as a fixed parameter and set (10)

Fx = Fx (t|u)

∞ X tn 1 − u xt H(x|u)t e = e = Hn (x|u) , t e −u n! n=0

where u ∈ C with u 6= 1. From (10), we can derive the following equation: (11)

et Fx − uFx = (1 − u)ext .

Let us define the differential operator D by D = operator Dk in (11), then we have (12)

d dt .

If we take the k-th differential

et (D + I)k Fx − uDk Fx = (1 − u)xk ext ,

where k ∈ N and I is identity operator.

545

SOME IDENTITIES FOR THE FROBENIUS-EULER NUMBERS AND POLYNOMIALS

By multiplying e−t on both sides of (12), we get (D + I)k Fx − e−t uDk Fx = (1 − u)xk e(x−1)t .

(13)

Taking Dm on both sides of (13), we have Dm (D + I)Fx − e−t uDk (D − I)m Fx = (1 − u)xk (x − 1)m e(x−1)t .

(14)

By multiplying et on both sides of (14), we get et Dm (D + I)k Fx − uDk (D − I)m Fx = (1 − u)xk (x − 1)m ext .

(15)

Let G[0] (not G(0)) be the constant term in a Laurent series of G(t). Then, from (15), we have (16) k   X k j

j=0

m   X

m et Dk−m−j Fx [0] − u (−1)j Dk+m−j Fx [0] = (1 − u)xk (x − 1)m . j j=0

By (10), we get
DN Fx (0) = HN (x|u),

(17)

and


et DN Fx [0] = HN (x|u)

From (16) and (17), we note that (18) max{k,m}

X j=0

    m k j Hk+m−j (x|u) − u (−1) Hk+m−j (x|u) = (1 − u)xk (x − 1)m . j j

Therefore, by (18), we obtain the following theorem. Theorem 1. For m, k ∈ Z+ = N ∪ {0}, we have    max{k,m}  X m k Hk+m−j (x|u) − u (−1)j Hk+m−j (x|u) = (1 − u)xk (x − 1)m . j j j=0 From (16), we note that Hm (x|u) =

(19)

m   X m l=0

l

Hl (u)xm−l .

Thus, by (19), we get (20)

m−1 X m − 1  dHm (x|u) =m Hl (u)xm−1−l = mHm−1 (x|u). dx l l=0

By (20), we get (21)

1

Z

Hm (x|u)dx = 0


1 Hm+1 (1|u) − Hm+1 (u) . m+1

From the definition of Frobenius-Euler numbers and (8), we can derive the following recurrence relation: (22)

H0 (u) = 1,

Hn (1|u) − uHn (u) = 0

546

if n > 0.

TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM

Thus by (21) and (22), we get Z 1 (23) Hm (x|u)dx = 0

1 (u − 1)Hm+1 (u) m+1

for m ∈ Z+ .

Let us take the integral from 0 to 1 in (18): (24) max{k,m}

(u − 1)

X j=0

Hk+m+1−j (u) k+m+1−j

    Z 1 k j m − u(−1) = (1 − u) xk (x − 1)m dx. j j 0

For α > 0, the gamma function is defined by Z ∞ (25) e−t tα−1 dt, Γ(α) =

(see [7]),

0

and Γ(n + 1) = nΓ(n) = n(n − 1)Γ(n − 1) = · · · = n!Γ(1) = n!. Let α > 0, β > 0. Then the beta function defined by Z 1 Γ(α)Γ(β) (26) . B(α, β) = tα−1 (1 − t)β−1 dt = Γ(α + β) 0 From (24) and (26), we have max{k,m}

(27)

X j=0

Hk+m+1−j (u) k+m+1−j

    k m j − (−1) u = (−1)m+1 B(k + 1, m + 1). j j

Therefore, by (27), we obtain the following theorem. Theorem 2. For k, m ∈ Z+ , we have max{k,m}

X j=0

Hk+m+1−j (u) k+m+1−j

    k m (−1)m+1 j − (−1) u =
. j j (k + m + 1) k+m k

Let m, k ∈ N. In [7], it is known that (28) max{k,m}

X j=1

   k Bk+m+1−j (x) (−1)m+1 j+1 m + (−1) = xk (x − 1)m +
. j j k+m+1−j (k + m + 1) k+m k

Let us take x = 0 in (28). Then we have max{k,m}

(29)

X j=1

   k m Bk+m+1−j (−1)m+1 + (−1)j+1 =
. j j k+m+1−j (k + m + 1) k+m k

Thus, by Theorem 2 and (29), we obtain the following corollary.

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SOME IDENTITIES FOR THE FROBENIUS-EULER NUMBERS AND POLYNOMIALS

Corollary 3. For m, k ∈ N, we have   X k Hk+m+1−j (u) Bk+m+1−j − j k+m+1−j k+m+1−j j=1

max{k,m} 

max{k,m}

X

+

(−1)j+1

j=1

   m uHk+m+1−j (u) Bk+m+1−j (u − 1)Hk+m+1 (u) − = . j k+m+1−j k+m+1−j k+m+1

In [7], we see that (30) max{k,m}

X j=1

   k Ek+m+1−j (−1)m+1 2Ek+m+1 j m + (−1) = ,
− k+m j j k+m+1−j k+m+1 (k + m + 1) k

where m, k ∈ N. From Theorem 2 and (30), we have   max{k,m} X Hk+m+1−j (u) k  m Hk+m+1 (u) − u(−1)j +(1 − u) k + m + 1 − j j j k+m+1 j=1 (31)   max{k,m}  X 2Ek+m+1 k Ek+m+1−j j m + , = + (−1) k + m + 1 − j k +m+1 j j j=1 Therefore, by (31), we obtain the following corollary. Corollary 4. For m, k ∈ N, we have  max{k,m} max{k,m}   X X m k Hk+m+1−j (u) Ek+m+1−j − + (−1)j+1 j k + m + 1 − j k + m + 1 − j j j=1 j=1   Ek+m+1−j 2Ek+m+1 Hk+m+1−j (u) Hk+m+1 (u) + + . × u = (u − 1) k+m+1−j k+m+1−j k+m+1 k+m+1 Let m = k + 1 in Corollary 4. Then we have  k+1   k+1 X k  H2k+2−j (u) X k+1 E2k+2−j − + (−1)j+1 j 2k + 2 − j 2k + 2 − j j j=1 j=1 (32)   H2k+2−j (u) E2k+2−j H2k+2 (u) × u + for k ∈ N, = (u − 1) 2k + 2 − j 2k + 2 − j 2k + 2 and, by Corollary 3, we get  k+1   k+1 X k  H2k+2−j (u) X B2k+2−j k+1 − + (−1)j+1 j 2k + 2 − j 2k + 2 − j j j=1 j=1 (33)   H2k+2−j (u) B2k+2−j H2k+2 (u) × u − = (u − 1) . 2k + 2 − j 2k + 2 − j 2k + 2 Thus, from (32) and (33), we obtain the following corollary.

548

TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM

Corollary 5. For k ∈ N, we have k+1 X j=1

k j



 k+1   X B2k+2−j E2k+2−j j+1 k + 1 − + (−1) 2k + 2 − j 2k + 2 − j j j=1   E2k+2−j B2k+2−j + = 0. × 2k + 2 − j 2k + 2 − j

By Corollary 5, we can derive the following equation: k

[2]  X j=1

  [k 2]  X k B2k−2j+2 k E2k−2j+1 E2k+1 − = . 2j − 1 2k − 2j + 2 j=1 2j 2k − 2j + 1 2k + 1

By Theorem 1 and (20), we get max{k,m}

    k m −u (−1)j (k + m − j)Hk+m−j−1 (x|u) j j
= (1 − u)xk−1 (x − 1)m−1 (m + k)x − k .

X

(34)

j=0

Let m = k. Then, by (34), we get (35) k   X k j=0

j

−u

   k (−1)j (2k − j)H2k−j−1 (x|u) = (1 − u)xk−1 (x − 1)k−1 (2kx − k). j

From (35), we note that (u − 1)

(36)

k   X k j=0

j

   k j −u (−1) H2k−j (u) = 0 j

for k ∈ N.

Therefore, by (36), we obtain the following theorem. Theorem 6. For k ∈ N, we have (u − 1)

k   X k j=0

j

−u

   k (−1)j H2k−j (u) = 0. j

From Theorem 1, we have (37) k

k

(1 − u)

[2]   X k j=0

2j

H2k−2j (x|u) + (1 + u)

[2]  X j=0

where [·] is Gauss’ symbol.

549

 k H2k−2j−1 (x|u) = (1 − u)xk (x − 1)k , 2j + 1

SOME IDENTITIES FOR THE FROBENIUS-EULER NUMBERS AND POLYNOMIALS

Let us take the integral from 0 to 1 on both sides of (37). Then we have k

[2]   X k H2k−2j+1 (u)

(u − 1)

j=0

(38)

2j

2k − 2j + 1

k

− (1 + u)

[2]  X j=0

 k H2k−2j (u) 2j + 1 2k − 2j

(−1)k = (−1)k B(k + 1, k + 1) =
. (2k + 1) 2k k Therefore, by (38), we obtain the following theorem. Theorem 7. For k ∈ Z+ , we have k

(u − 1)

[2]   X k H2k−2j+1 (u) j=0

2k − 2j + 1

2j

k

− (1 + u)

[2]  X j=0

 k H2k−2j (u) (−1)k =
. 2j + 1 2k − 2j (2k + 1) 2k k

In the special case, u = −1, we have the following equation (39). k

[2]   X k E2k−2j+1 (−1)k+1 =
. 2j 2k − 2j + 1 (4k + 2) 2k k j=0

(39)

Let k ∈ N. Then, from Theorem 1, we have    k+1 Xk  k+1 j H2k+1−j (u) − u (−1) H2k+1−j (u) = 0. (40) j j j=0 Thus, by (40), we get (41)

[ k+1 2 ]

(1 − u)

X j=0

  [ k+1 2 ] X k k H2k+1−2j (u) + (1 + u) H2k−2j (u) = 0. 2j 2j + 1 j=0

By (41), we get [ k+1 2 ]

X j=0

  [ k+1 2 ] k u+1 X k H2k+1−2j (u) = H2k−2j (u). 2j u − 1 j=0 2j + 1

Acknowledgements. This paper was supported by the research grant of Kwangwoon University in 2013, and the Kyungpook National University Research Fund 2012.

References

550

TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM

[1] S. Araci, D. Erdal, J. Seo, A study on the fermionic p-adic q-integral on Zp associated with weighted q-Bernstein q-Genocchi polynomials , Abstract and Applied Analysis, 2011(2011), Article ID 649248, 10 pages. [2] A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys, 18 (2011), 133-143. [3] L. Carlitz, The product of two Eulerian polynomials, Mathematics Magazine 36(1963), 37-41. [4] L. Carlitz, Eulerian numbers and polynomials, Mathematics Magazine 36(1963), 37-41. [5] I. N. Cangul, Y. Simsek, A note on interpolation functions of the FrobeniousEuler numbers, Application of Mathematics in Technical and Natural Sciences, 59-67, AIP Conf. Proc., 1301, Amer. Inst. Phys., Melville, N.Y., 2010. [6] K.W. Hwang, D.V. Dolgy, T. Kim, S.H. Lee, On the higher-order q-Euler numbers and polynomials with weight , Discrete Dyn. Nat. Soc. 2011(2011), Art. ID 354329, 12 pages. [7] T. Kim, B. Lee, S. H. Lee, S.-H. Rim, Identities for the Bernoulli and Euler numbers and polynomials (communicated). [8] T. Kim, New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys. 17 (2010), 218-225. 484-491. [9] T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys. 16 (2009), 484-491. [10] Y. Simsek, Special functions related to Dedekind-type DC-sums and their applications, Russ. J. Math. Phys. 17 (2010), 495–504.

Taekyun Kim Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail: [email protected].

Byungje Lee Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139701, Republic of Korea.

Sang-Hun Lee Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea.

Seog-Hoon Rim Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea, E-mail: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 552-556, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

ON THE SECOND KIND (h, q)-EULER POLYNOMIALS OF HIGHER ORDER C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea (h,k)

Abstract : In this paper, we define the second kind Euler numbers En,q

(h,k)

and polynomials En,q (x) of order k. We also give the generating functions of the second kind Euler numbers and polynomials of order k. By applying the Mellin transformation to these generating functions, we construct the multiple (h, q)-Euler zeta function and Barnes’ type multiple (h, q)-Euler zeta function of order k which are interpolated higher order second kind Euler numbers and polynomials at negative integers, respectively. Key words : Euler numbers, Euler polynomials, the second kind Euler numbers and polynomials, the second kind (h, q)-Euler numbers and polynomials, (h, q)-Euler zeta function

1. Introduction Euler numbers and polynomials were studied by many authors (see for details [1-6]). Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Qp denotes the field of rational numbers, N denotes the set of natural numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . In this paper we assume that q ∈ Cp with |1 − q|p < 1 as an indeterminate. For g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, the fermionic p-adic invariant integral on Zp of the function g ∈ U D(Zp ) is defined by  I−1 (g) =

Zp

g(x)dμ−1 (x) = lim

N p −1

N →∞

g(x)(−1)x , see [1, 2].

(1.1)

x=0

From (1.1), we note that 



Zp

g(x + 1)dμ−1 (x) +

Zp

g(x)dμ−1 (x) = 2g(0).

(1.2)

(h)

First, we introduce the second kind (h, q)-Euler numbers En,q . The second kind (h, q)-Euler numbers (h) En,q are defined by the generating function: Fq(h) (t) =

∞ n  2et (h) t = , E n,q q h e2t + 1 n=0 n!

(|h log q + 2t| < π).

(1.3)

(h)

We introduce the second kind (h, q)-Euler polynomials En,q (x) as follows: Fq(h) (x, t) =

∞  2et tn xt (h) e . = E (x) n,q q h e2t + 1 n! n=0

552

(1.4)

RYOO: (h,q)-EULER POLYNOMIALS

2. The second kind (h, q)-Euler polynomials of higher order In this section, we assume that q ∈ Cp and h ∈ Z. We introduce the second kind Euler (h,k) polynomials of higher order, En,q (x). We use the notation m  k1 =0

m 

···

=

kn =0

m 

.

k1 ···kn =0

The binomial formulae are known as   n    n(n − 1) . . . (n − i + 1) n n n i (−a) , where = (1 − a) = , i i i! i=0 and

  n  n    −n n+i−1 i 1 −n i (−a) a = (1 − a) = i i (1 − a)n i=0 i=0

Now, using multiple of p-adic integral, we introduce the second kind (h, q)-Euler polynomials of (h,k) higher order En,q (x): For k ∈ N and h ∈ Z, we define   ∞  tn (h,k) En,q (x) = ··· q hx1 +hx2 +···+hxk e(x+2x1 +2x2 +···+2xk +k)t dμ−1 (x1 ) · · · dμ−1 (xk ). (2.1) n! Z Z p p n=0   k times By using Taylor series of e(2x+1)t in the above equation, we obtain

   ∞  tn hx1 +···+hxk n ··· q (x + 2x1 + · · · + 2xk + k) dμ−1 (x1 ) · · · dμ−1 (xk ) n! Zp Zp n=0 =

∞ 

(h,k) En,q (x)

n=0

tn n!

tn in the above equation, we arrive at the following theorem. n! Theorem 1. For positive integers n, k, and h ∈ Z, we have   (h,k) En,q (x) = ··· q hx1 +···+hxk (x + 2x1 + · · · + 2xk + k)n dμ−1 (x1 ) · · · dμ−1 (xk ).

By comparing coefficients

Zp

(2.2)

Zp

(h,k)

By (1.4), the second kind (h, q)-Euler polynomials of higher order, En,q (x) are defined by means of the following generating function k  ∞  2et tn (h,k) xt (h,k) . (2.3) (x, t) = e = E (x) Fq n,q q h e2t + 1 n! n=0 (h,k)

By using (2,1), the second kind (h, q)-Euler numbers of higher order, En,q following generating function 

2et h q e2t + 1

k =

∞ 

(h,k) En,q

n=0

tn , n!

|t + h log q|
0, we can derive the following Eq. (3.1) form the Mellin transformation (k) of Fq (x, t). 1 Γ(s)

 0

∞

∞ 

ts−1 Fq(k) (x, −t)dt = 2k

a1 ,··· ,ak

(−q h )a1 +···+ak (2a1 + · · · + 2ak + k + x)s =0

(3.1)

For s, x ∈ C with R(x) > 0, we define Barnes’ type multiple (h, q)-Euler zeta function as follows:

555

RYOO: (h,q)-EULER POLYNOMIALS

Definition 1. For s, x ∈ C with R(x) > 0 and h ∈ Z, we define ∞ 

ζq(h,k) (s, x) = 2k

a1 ,··· ,ak

(−q h )a1 +···+ak . (2a1 + · · · + 2ak + k + x)s =0

(3.2)

For s = −l in (3.2) and using (2.6), we arrive at the following theorem. Theorem 7. For positive integer l, we have (h,k)

ζq(h,k) (−l, x) = El,q

(x).

By (2.4), we have ∞ 

(h,k) t En,q

n!

n=0



n

=

2et q h e2t + 1

k

 ∞   m+k−1 (−q h )m e(2m+k)t . =2 m m=0 k

By using Taylor series of e(2m+k)t in the above, we have

  ∞ ∞ ∞  n    m+k−1 tn (h,k) t k h m n (−q ) (2m + k) = . 2 En,q m n! n=0 n! n=0 m=0 By comparing coefficients

tn n!

in the above equation, we have  ∞   m+k−1 (h,k) (−q h )m (2m + k)n . = 2k En,q m m=0

(3.3)

By using (3.3), we define the multiple (h, q)-Euler zeta function as follows: Definition 2. For s ∈ C and h ∈ Z, we define  ∞   m + k − 1 (−q h )m (h,k) k ζq (s) = 2 . m (2m + k)s m=0 (h,k)

(3.4)

(h,k)

(s) interpolates the number En,q at negative integers. Substituting s = −n The function ζq with n ∈ Z+ into (3.4), and using (3.3), we obtain the following theorem: Theorem 8. Let n ∈ Z+ , We have (h,k) ζq(h,k) (−n) = En,q .

REFERENCES 1. Kim, T.(2008). Euler numbers and polynomials associated with zeta function, Abstract and Applied Analysis, Art. ID 581582. 2. Kim, T., Jang, L. C., Pak, H. K.(2001). A note on q-Euler and Genocchi numbers , Proc. Japan Acad., v.77 A, pp. 139-141. 3. Liu, G.(2006). Congruences for higher-order Euler numbers, Proc. Japan Acad., v.82 A, pp. 30-33. 4. Ryoo, C.S.(2010). Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, v.12, pp. 828-833. 5. Ryoo, C.S., Kim, T., Agarwal, R.P.(2006). A numerical investigation of the roots of qpolynomials, Inter. J. Comput. Math., v.83, pp. 223-234. 6. C. S. Ryoo, C.S.(2011). A Note on the second kind q-Euler polynomials of higher order, Applied Mathematics Sciences, v.5, pp. 3421-3427.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 557-571, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Further exact traveling wave solutions to the (2+1)-dimensional Boussinesq and Kadomtsev-Petviashvili equation M. Ali Akbara,b,1 , Norhashidah Hj. Mohd. Alia and Syed Tauseef Mohyud-Dinc a School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia b Department of Applied Mathematics, University of Rajshahi, Bangladesh c Department of Mathematics, HITEC University Taxila Cantt Pakistan Abstract In the real world, the modelling of various intricate physical phenomena, the higher dimensional nonlinear evolution equations come into further attractive in many branches of physical sciences. In this article, by the use of the Riccati equation to the (G′ /G)–expansion method, we construct more new exact traveling wave solutions of the (2+1)-dimensional Boussinesq and Kadomtsev-Petviashvili equation in a unified way involving arbitrary parameters. The obtained solutions are expressed in terms of hyperbolic, trigonometric and rational functions. When the parameters take special values, the solitary wave are derived from the traveling waves. The obtained solutions may be imperative and significant for the explanation of some practical physical problems. PACS numbers: 02.30.Jr, 05.45.Yv, 02.30.Ik. Keywords: The (G′ /G)-expansion method, the Riccati equation, the traveling wave solutions, the Boussinesq and Kadomtsev-Petviashvili equation.

1

Introduction

After the observation of solitonary phenomena by John Scott Russell in 1834 [1] and since the KdV equation was solved by Gardner et al. [2] by the inverse scattering method, finding exact solutions of nonlinear evolution equations (NLEEs) has turned out to be one of the enthusiastic and much lucrative areas of research. The appearance of solitary wave solutions in nature is somewhat frequent. Bell-shaped sech-solutions and kink-shaped tanh-solutions model wave phenomena in fluids, chemical kinematics, solid state physics, condensed matter physics, plasmas, optical fibers, electrical circuits, bio-genetics, elastic media etc. The traveling wave solutions of the KdV equation and the Boussinesq equation which describe water waves are well-known examples. Apart from their physical relevance, the closed-form solutions of NLEEs if 1

Email : ali [email protected]

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Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

available facilitate the numerical solvers in comparison, and aids in the stability analysis. In soliton theory, there are many methods and techniques to deal with the problem of solitary wave solutions for NLEEs, such as, the Backlund transformation method [3], the Hirota’s bilinear transformation method [4], the variational iteration method [5], the homogeneous balance method [6], the tanh-function method [7], the Jacobi elliptic function method [8], the F-expansion method [9], the variable separation method [10], the Lie group symmetry method [11], the homotopy analysis method [12, 13], the homotopy perturbation method [14], the Adomian decomposition method [15], the first integration method [16], the Exp-function method [17-19], the (G′ /G)-expansion method [20-30] and so on. It is significant to observe that there exist some fundamental relationships among numerous complex nonlinear partial differential equations and some basic and soluble nonlinear ordinary differential equations (ODEs), such as the sine-Gordon equation, the sinh-Gordon equation, the Riccati equation, the Weierstrass elliptic equation etc. Therefore, it is natural to use the solutions of these nonlinear ODEs to construct exact solutions of various intricate nonlinear partial differential equations. Based on the relationships of complex nonlinear partial differential equations and ODEs, a number of methods, such as, the sinh-Gordon equation expansion method [31], the generalized F-expansion method [32, 33], the Riccati equation expansion method [34, 35], the projective Riccati equation method [36, 37], the algebraic method [38] etc. have been developed. In the present article, we combine the Riccati equation with the (G′ /G)–expansion method, and construct more new exact traveling wave solutions of the (2+1)-dimensional Boussinesq and Kadomtsev-Petviashvili equation in a uniform way.

2

Description of the (G′ /G)-expansion method combined with the Riccati equation

Suppose the general nonlinear partial differential equation Φ(𝑢, 𝑢t , 𝑢x , 𝑢t t , 𝑢t x , 𝑢x x , · · · ) = 0,

(1)

where 𝑢 = 𝑢(𝑥, 𝑡) is an unknown function, Φ is a polynomial in 𝑢(𝑥, 𝑡) and its partial derivatives in which the highest order partial derivatives and the nonlinear terms are involved. The main steps of the (G′ /G)-expansion method combined with the Riccati equation is as follows: Step 1: The travelling wave variable 𝑢(𝑥, 𝑡) = U (𝜉), 𝜉 = 𝑥 − c 𝑡,

558

(2)

Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

where 𝑉 is the speed of the traveling wave, allows us to convert the Eq. (1) into an ODE: 𝜓(U, U ′ , U ′′ , · · · ) = 0, (3) where the superscripts stands for the ordinary derivatives with respect to 𝜉. Step 2: If Eq. (3) is integrable, integrate term by term one or more times, yields constant(s) of integration. Step 3: Suppose the traveling wave solution of Eq. (3) can be expressed by a polynomial in (G′ /G) as follows: U (𝜉) =

m ∑

( 𝑎n

n=0

G′ G

)n , 𝑎m ̸= 0,

(4)

where G = G(𝜉) satisfies the Riccati equation, G′ = r + q G2 ,

(5)

where 𝑎n (𝑛 = 0, 1, 2, · · · , 𝑚), r and q are arbitrary constants to be determined later. The Riccati Eq. (5) has the following twenty one solutions [40]. Family 1: When r and q have same sign and r q ̸= 0, the solutions of Eq. (5) are: G1 =

1 √ √ [ q r tan ( q r𝜉)] , q

1 √ √ G2 = − [ q r cot ( q r𝜉)] , q G3 =

1 √ √ √ [ q r (tan (2 q r𝜉) ± sec (2 q r𝜉))] , q

1 √ √ √ G4 = − [ q r (cot (2 q r𝜉) ± csc (2 q r𝜉))] , q [ ( ( ) ( ))] 1 √ 1√ 1√ q r tan q r𝜉 − cot q r𝜉 , G5 = 2q 2 2 ( √ )] √ [√ 2 (A − B 2 ) − A cos 2 q r𝜉 qr ( √ ) G6 = , q A sin 2 q r𝜉 + B ( √ )] √ [√ 2 (A − B 2 ) + A sin 2 q r𝜉 qr ( √ ) G7 = , q A cos 2 q r𝜉 + B

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Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

where A and B are two non-zero real constants and satisfies the condition A2 −B 2 > 0. ( √ ) −r cos 2 q r𝜉 ( √ ) √ , G8 = √ q r sin 2 q r𝜉 ± q r ( √ ) r sin 2 q r𝜉 ( √ ) √ , G9 = √ q r cos 2 q r𝜉 ± q r ( √ ) ( √ ) 2 r sin 12 q r𝜉 cos 12 q r𝜉 ( √ ) √ G10 = √ . 2 q r cos2 12 q r𝜉 − q r Family 2: When r and q possess opposite sign and r q ̸= 0, the solutions of Eq. (5) are: (√ )] 1 [√ − q r tanh −q r𝜉 , G11 = − q G12 = − G13 = −

(√ )] 1 [√ −q r coth −q r𝜉 , q

( ( √ ) ( √ ))] 1 [√ −q r tanh 2 −q r𝜉 ± 𝑖 sec ℎ 2 −q r𝜉 , q

( ( √ ) ( √ ))] 1 [√ −q r coth 2 −q r𝜉 ± csc ℎ 2 −q r𝜉 , q [ ( ( ) ( ))] 1 √ 1√ 1√ =− −q r tanh −q r𝜉 + coth −q r𝜉 , 2q 2 2 [√ ] √ √ (A2 + B 2 ) − A cosh (2 −q r𝜉) −q r √ G16 = , q A sinh (2 −q r𝜉) + B [√ ] √ √ −q r B 2 − A2 + A sinh (2 −q r𝜉) √ G17 = − , q A cosh (2 −q r𝜉) + B

G14 = − G15

where A and B are two non-zero real constants and satisfies the condition B 2 −A2 > 0. √ r cosh (2 − q r𝜉) √ √ G18 = √ , − q r sinh (2 − q r𝜉) ± 𝑖 −q r √ r sinh (2 −q r𝜉) √ √ G19 = √ , −q r cosh (2 −q r𝜉) ± −q r ( √ ) ( √ ) 2 r sinh 21 −q r𝜉 cosh 12 −q r𝜉 ) √ ( √ G20 = √ . 2 −q r cosh2 12 −2 q r𝜉 − −q r

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Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

Family 3: When q ̸= 0 but r = 0, the solution of Eq. (5) is: G21 = −

1 , q𝜉 +𝑑

where 𝑑 is an arbitrary constant. Step 4: To determine the positive integer 𝑚, substitute solution Eq. (4) along with Eq. (5) into Eq. (3) and consider the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in Eq. (3). Step 5: Substituting Eq. (4) together with Eq. (5) into Eq. (3) together with the value of 𝑚 obtained in step 4, we obtain polynomials in Gi and G−i (𝑖 = 0, 1, 2, 3 · · · ) and setting each coefficient of the resulted polynomial to zero, yields a set of algebraic equations for 𝑎n q, r and c. Step 6: Suppose the value of the constants 𝑎n q, r and c can be obtained by solving the set of algebraic equations obtained in step 5. Since the general solutions of Eq. (5) are known (arranged in step 3), substituting 𝑎n q, r and c into Eq. (4), we obtain new exact traveling wave solutions of the nonlinear evolution Eq. (1).

3

Application of the method to the Boussinesq and KadomtsevPetviashvili equation

In this section, we apply the proposed approach of the (G′ /G)-expansion method to construct new exact traveling wave solutions of the Boussinesq and KadomtsevPetviashvili equation which is an important nonlinear equation in mathematical physics. Let us consider the (2+1)-dimensional Boussinesq and Kadomtsev-Petviashvili equation, 𝑤t = 𝑤x x x + 𝑤y y y + 6 (𝑢 𝑤)x + 6 (v 𝑤)y (6) 𝑢 y = 𝑤x

(7)

v x = 𝑤y

(8)

We solve the Boussinesq and Kadomtsev-Petviashvili equation by the method described in section 2. In order to obtain traveling wave solutions of Eqs. (6)-(8), similar to step 1, we assume that 𝑢(𝑥 , 𝑦, 𝑡) = U (𝜉), v(𝑥 , 𝑦, 𝑡) = 𝑉 (𝜉), 𝑤(𝑥 , 𝑦, 𝑡) = W (𝜉), 𝜉 = 𝑥 + 𝑦 − c 𝑡.

(9)

Therefore, Eqs. (6)-(8) are converted into the following ODEs: 2 W ′′′ + cW ′ + 6 (U W )′ + 6 (𝑉 W )′ = 0,

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(10)

Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

U′ − W′ = 0

(11)

𝑉 ′ − W′ = 0

(12)

Eqs. (10)-(12) are integrable, therefore, integrating we obtain 2 W ′′ + c W + 6 U W + 6 𝑉 W + 𝑔1 = 0

(13)

U − W − 𝑔2 = 0

(14)

𝑉 − W − 𝑔3 = 0

(15)

According to step 3, the solution of Eqs. (13)-(15) can be expressed by a polynomial in (G′ /G) as follows: U (𝜉) = 𝑎0 + 𝑎1 (G′ /G) + 𝑎2 (G′ /G)2 + · · · + 𝑎m (G′ /G)m , 𝑎m ̸= 0

(16)

𝑉 (𝜉) = 𝑏0 + 𝑏1 (G′ /G) + 𝑏2 (G′ /G)2 + · · · + 𝑏n (G′ /G)n , 𝑏n ̸= 0

(17)

W (𝜉) = c0 + c1 (G′ /G) + c2 (G′ /G)2 + · · · + cl (G′ /G)l , cl ̸= 0

(18)

and

where 𝑎i , (𝑖 = 0, 1, 2, · · · , 𝑚), 𝑏j , (𝑗 = 0, 1, 2, · · · , 𝑛) and ck , (𝑘 = 0, 1, 2, · · · , l) are constants to be determined and G = G(𝜉) satisfies the Riccati Eq. (5). Considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in Eqs. (13)-(15), we obtain 𝑚 = 𝑛 = l = 2 . Therefore, solution Eqs. (16)-(18) respectively become U (𝜉) = 𝑎0 + 𝑎1 (G′ /G) + 𝑎2 (G′ /G)2 , 𝑎2 ̸= 0

(19)

𝑉 (𝜉) = 𝑏0 + 𝑏1 (G′ /G) + 𝑏2 (G′ /G)2 , 𝑏2 ̸= 0

(20)

W (𝜉) = c0 + c1 (G′ /G) + c2 (G′ /G)2 , c2 ̸= 0.

(21)

By means of Eq. (5), Eqs. (19)-(21) can be rewritten respectively as, U (𝜉) = 𝑎0 + 𝑎1 (r G−1 + q G) + 𝑎2 (r G−1 + q G)2 ,

(22)

𝑉 (𝜉) = 𝑏0 + 𝑏1 (r G−1 + q G) + 𝑏2 (r G−1 + q G)2 ,

(23)

W (𝜉) = c0 + c1 (r G−1 + q G) + c2 (r G−1 + q G)2 .

(24)

and

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Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

Substituting Eqs. (22)-(24) into Eqs. (13)-(15), the left hand sides of these equations are converted into polynomials in Gi and G−i , (𝑖 = 0, 1, 2, · · · ). Setting each coefficient of these polynomials to zero, we obtain a set of simultaneous algebraic equations (we will omit to display them for simplicity) for 𝑎0 , 𝑎1 , 𝑎2 , 𝑏0 , 𝑏1 , 𝑏2 , c0 , c1 , c2 and c. Solving the over-determined set of algebraic equations by using the symbolic computation software, such as Maple, we obtain 𝑎2 = 𝑏2 = c2 = −1, 𝑎1 = 𝑏1 = c1 = 0, 𝑎0 = 𝑎0 , 𝑏0 = 𝑏0 , c0 = c0 , 𝑔1 = −32 r q c0 +12 c20 , 𝑔2 = 𝑎0 − c0 , 𝑔3 = 𝑏0 − c0 , 𝑔2 = 𝑎0 − c0 , and c = −6𝑎0 − 6𝑏0 − 12c0 + 32 r q, (25) where 𝑎0 , 𝑏0 , c0 , r and q are arbitrary constants. Now on the basis of the solutions of the Riccati Eq. (5), we obtain the following families of solutions of Eqs. (6)-(8). Family 1: When r and q have same sign and r q ̸= 0, the periodic form solutions of Eqs. (6)-(8) are: √ 𝑢1 = 𝑎0 − 4 q r csc2 (2 q r 𝜉), √ v1 = 𝑏0 − 4 q r csc2 (2 q r 𝜉), √ 𝑤1 = c0 − 4 q r csc2 (2 q r 𝜉), where 𝜉 = 𝑥+𝑦 −(32 r q −6𝑎0 −6𝑏0 −12c0 ) 𝑡 and 𝑎0 , 𝑏0 , c0 , q, r are arbitrary constants. √ 𝑢3 = 𝑎0 − 4 q r sec2 (2 q r 𝜉), √ v3 = 𝑏0 − 4 q r sec2 (2 q r 𝜉), √ 𝑤3 = c0 − 4 q r sec2 (2 q r 𝜉), { } 2  √ √ √ √ 2 q rA A + B sin(2 q r 𝜉) − A2 − B 2 cos(2 q r 𝜉) }  , 𝑢 6 = 𝑎0 −  { √ }{ √ √ 2 2 A sin(2 q r 𝜉) + B A cos(2 q r 𝜉) − A − B { } 2  √ √ √ √ 2 q rA A + B sin(2 q r 𝜉) − A2 − B 2 cos(2 q r 𝜉) }  , v6 = 𝑏0 −  { √ }{ √ √ 2 2 A sin(2 q r 𝜉) + B A cos(2 q r 𝜉) − A − B { } 2  √ √ √ √ 2 q rA A + B sin(2 q r 𝜉) − A2 − B 2 cos(2 q r 𝜉) }  , 𝑤6 = c0 −  { √ }{ √ √ A sin(2 q r 𝜉) + B A cos(2 q r 𝜉) − A2 − B 2 { } 2  √ √ √ √ 2 q r A A + B cos(2 q r 𝜉) + A2 − B 2 sin(2 q r 𝜉) }  , 𝑢7 = 𝑎0 −  { √ }{ √ √ A cos(2 q r 𝜉) + B A sin(2 q r 𝜉) + A2 − B 2

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Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

{ } 2  √ √ √ √ 2 2 2 q r A A + B cos(2 q r 𝜉) + A − B sin(2 q r 𝜉) }  , v7 = 𝑏0 −  { √ }{ √ √ A cos(2 q r 𝜉) + B A sin(2 q r 𝜉) + A2 − B 2 { } 2  √ √ √ √ 2 q r A A + B cos(2 q r 𝜉) + A2 − B 2 sin(2 q r 𝜉) }  , 𝑤7 = c0 −  { √ }{ √ √ 2 2 A cos(2 q r 𝜉) + B A sin(2 q r 𝜉) + A − B where A and B are two non-zero real constants satisfies the condition A2 − B 2 > 0. ( )2 √ qr { } 𝑢10 = 𝑎0 − , √ √ √ 2 sin(( q r𝜉)/2) cos(( q r𝜉)/2) 2 cos2 (( q r𝜉)/2) − 1 (

)2 √ qr { } v10 = 𝑏0 − , √ √ √ 2 sin(( q r𝜉)/2) cos(( q r𝜉)/2) 2 cos2 (( q r𝜉)/2) − 1 ( )2 √ qr { } 𝑤10 = c0 − . √ √ √ 2 sin(( q r𝜉)/2) cos(( q r𝜉)/2) 2 cos2 (( q r𝜉)/2) − 1 The solutions corresponding to G2 , G4 , G5 and G9 are identical to the solution 𝑢1 and the solution corresponding to G8 is identical to the solution 𝑢3 . Family 2: When r and q possess opposite sign and r q ̸= 0, the soliton and soliton-like solutions of Eqs. (6)-(8) are: √ 𝑢11 = 𝑎0 + 4 q r csc ℎ2 (2 −q r 𝜉) , √ v11 = 𝑏0 + 4 q r csc ℎ2 (2 −q r 𝜉) , √ 𝑤11 = c0 + 4 q r csc ℎ2 (2 −q r 𝜉) , where 𝜉 = 𝑥+𝑦 −(32 r q −6𝑎0 −6𝑏0 −12c0 ) 𝑡 and 𝑎0 , 𝑏0 , c0 , q, r are arbitrary constants. √ 𝑢13 = 𝑎0 − 4 q r sec ℎ2 (2 −q r 𝜉) , √ v13 = 𝑏0 − 4 q r sec ℎ2 (2 −q r 𝜉) , √ 𝑤13 = c0 − 4 q r sec ℎ2 (2 −q r 𝜉) . { } 2  √ √ √ √ 2 −q rA A − B sinh(2 −q r 𝜉) − A2 + B 2 cosh(2 −q r 𝜉) { }  , 𝑢16 = 𝑎0 −  √ √ √ {A sinh(2 −q r 𝜉) + B} A cosh(2 −q r 𝜉) − A2 + B 2 { } 2  √ √ √ √ 2 −q rA A − B sinh(2 −q r 𝜉) − A2 + B 2 cosh(2 −q r 𝜉) { }  , v16 = 𝑏0 −  √ √ √ {A sinh(2 −q r 𝜉) + B} A cosh(2 −q r 𝜉) − A2 + B 2

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Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

{ } 2  √ √ √ √ 2 2 2 −q rA A − B sinh(2 −q r 𝜉) − A + B cosh(2 −q r 𝜉) { }  , 𝑤16 = c0 −  √ √ √ {A sinh(2 −q r 𝜉) + B} A cosh(2 −q r 𝜉) − A2 + B 2 { } 2  √ √ √ √ 2 −q r A A + B cosh(2 −q r 𝜉) − B 2 − A2 sinh(2 −q r 𝜉)  , { } 𝑢17 = 𝑎0 −  √ √ √ 2 2 {A cosh(2 −q r 𝜉) + B} A sinh(2 −q r 𝜉) + B − A { } 2  √ √ √ √ 2 −q r A A + B cosh(2 −q r 𝜉) − B 2 − A2 sinh(2 −q r 𝜉)  , { } v17 = 𝑏0 −  √ √ √ 2 2 {A cosh(2 −q r 𝜉) + B} A sinh(2 −q r 𝜉) + B − A { } 2  √ √ √ √ 2 −q r A A + B cosh(2 −q r 𝜉) − B 2 − A2 sinh(2 −q r 𝜉)  . { } 𝑤17 = c0 −  √ √ √ {A cosh(2 −q r 𝜉) + B} A sinh(2 −q r 𝜉) + B 2 − A2 where A and B are two non-zero real constants and satisfies the condition B 2 −A2 > 0. ( )2 √ −q r { } 𝑢20 = 𝑎0 − , √ √ √ 2 sinh(( −q r𝜉)/2) cosh(( −q r𝜉)/2) 2 cosh2 (( −q r𝜉)/2) − 1 )2 √ −q r { } v20 = 𝑏0 − , √ √ √ 2 sinh(( −q r𝜉)/2) cosh(( −q r𝜉)/2) 2 cosh2 (( −q r𝜉)/2) − 1 ( )2 √ −q r { } 𝑤20 = c0 − . √ √ √ 2 sinh(( −q r𝜉)/2) cosh(( −q r𝜉)/2) 2 cosh2 (( −q r𝜉)/2) − 1 (

The solutions corresponding to G12 , G14 , G15 and G19 are identical to the solution 𝑢11 and the solution corresponding to G18 is identical to the solution 𝑢13 . Family 3: When r = 0 and q ̸= 0, the solution of Eqs. (6)-(8) is: )2 ( q 𝑢21 = 𝑎0 − , q𝜉 +𝑑 )2 ( q v21 = 𝑏0 − , q𝜉 +𝑑 ( )2 q 𝑤21 = c0 − , q𝜉 +𝑑 where 𝑑 is an arbitrary constant. Because of the arbitrariness of the parameters 𝑎0 , 𝑏0 , c0 , q, r and 𝑑 in the above families of solution, the physical quantities 𝑢, v and 𝑤 may possess rich structures.

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Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

4

Comparison

Zheng [39] investigated solutions of the Boussinesq and Kadomtsev-Petviashvili equation by using the (G′ /G)-expansion method, where he used the second order linear ordinary differential equation as an supplementary equation. √If we set C1 = 0 (or √ C2 = 0) in Zheng’s solution and 𝑎0 = 𝑏0 = c0 = 0, −q r = 14 𝜆2 − 4 µ in our solutions then Zheng’s solutions 𝑢1 , v1 and q1 are identical to our solutions 𝑢11 , v11 and 𝑤11 (or 𝑢13 , v13 and 𝑤13 ) respectively. Again √if we set C1 = 0 (or C2 = 0) in Zheng’s √ 1 solution and 𝑎0 = 𝑏0 = c0 = 0, q r = 4 4 µ − 𝜆2 in our solutions then Zheng’s solutions 𝑢2 , v2 and q2 are identical to our solutions 𝑢1 , v1 and 𝑤1 (or 𝑢3 , v3 and 𝑤3 ) respectively. Similarly, it can be shown that Zheng’s solutions 𝑢3 , v3 and q3 (𝑢7 , v7 and q7 ) are identical to our solutions 𝑢21 , v21 and 𝑤21 . Further, it can be shown that Zheng’s solutions 𝑢5 , v5 and q5 (𝑢6 , v6 and q6 ) are identical to our solutions 𝑢11 , v11 and 𝑤11 (𝑢13 , v13 and 𝑤13 ). Zheng did not obtain any further solutions, but apart from these solutions in this article we obtain further new solutions.

5

Graphical representations

Graph is an influential tool for communication and it illustrates clearly the solutions of the problems. Therefore, some graphs of the solutions are given below. The graphs readily have shown the periodic and solitary wave forms of the solutions.

Fig. 1: Periodic solution corresponding to 𝑢1 , v1 and 𝑤1 for 𝑎0 = 𝑏0 = c0 = 1, q = 2, r = 1, 𝑡 = 1.

566

Fig. 2: Periodic solution corresponding to 𝑢3 , v3 and 𝑤3 for 𝑎0 = 𝑏0 = c0 = 1, q = 2, r = 1, 𝑡 = 1.

Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

6

Fig. 3: Periodic solution corresponding to 𝑢6 , v6 and 𝑤6 for 𝑎0 = 𝑏0 = c0 = 1, q = 2, r = 1, 𝑡 = 1, A = 2 and B = 1.

Fig. 4: Periodic solution corresponding to 𝑢7 , v7 and 𝑤7 for 𝑎0 = 𝑏0 = c0 = 1, q = 2, r = 1 𝑡 = 1, A = 2 and B = 1.

Fig. 5: Solutions corresponding to 𝑢13 , v13 and 𝑤13 for 𝑎0 = 𝑏0 = c0 = 1, q = −2, r = 1, 𝑡 = 0.

Fig. 6: Solutions corresponding to 𝑢16 , v16 and 𝑤16 for 𝑎0 = 𝑏0 = c0 = 1, q = −2, r = 1, 𝑡 = 0, A = 1 and B = 2.

Conclusion

The (G′ /G)-expansion method is an advance mathematical tool for investigating exact solutions of nonlinear partial differential equations associated with complex physical

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Akbar et. al.: Exact solutions of Boussinesq and Kadomtsev-Petviashvili equation

phenomena wherein, in general the second order linear ordinary differential equation is employed as an auxiliary equation. But, in this article, we utilize the Riccati equation as an auxiliary equation; in consequence we obtain further new exact solutions of the Boussinesq and Kadomtsev-Petviashvili equation in a unified way. The obtained exact solutions may be important and significant to reveal the internal mechanism of some complicated physical phenomena. The algorithm presented in this article is effective and more powerful and it can be used for other kind of nonlinear evolution equations in mathematical physics.

Acknowledgement This work is supported by the research grant under the Government of Malaysia and the authors acknowledge the support.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 572-583, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

ORTHOGONAL STABILITY OF A CUBIC-QUARTIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN SPACES JUNG RYE LEE, CHOONKIL PARK, YEOL JE CHO, AND DONG YUN SHIN∗ Abstract. Using the fixed point method, we prove the Hyers-Ulam stability of the orthogonally cubic-quartic functional equation 𝑓 (2𝑥 + 𝑦) + 𝑓 (2𝑥 − 𝑦)

=

3𝑓 (𝑥 + 𝑦) + 𝑓 (−𝑥 − 𝑦) + 3𝑓 (𝑥 − 𝑦) + 𝑓 (𝑦 − 𝑥) +18𝑓 (𝑥) + 6𝑓 (−𝑥) − 3𝑓 (𝑦) − 3𝑓 (−𝑦) (0.1)

for all 𝑥, 𝑦 with 𝑥 ⊥ 𝑦 in non-Archimedean Banach spaces, where ⊥ is the orthogonality in the sense of R¨atz.

1. Introduction and preliminaries In 1897, Hensel [26] introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [13, 19, 35, 36, 45]). A valuation is a function |·| from a field K into [0, ∞) such that 0 is the unique element having the 0 valuation, |rs| = |r| · |s| and the triangle inequality holds, i.e., |r + s| ≤ |r| + |s|,

∀r, s ∈ K.

A field K is called a valued field if K carries a valuation. Throughout this paper, we assume that the base field is a valued field and hence call it simply a field. The usual absolute values of ℝ and C are examples of valuations. Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|},

∀r, s ∈ K,

then the function | · | is called a non-Archimedean valuation and the field is called a nonArchimedean field. Clearly, |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N. A trivial example of a non-Archimedean valuation is the function | · | taking everything except for 0 into 1 and |0| = 0. Definition 1.1. ([44]) Let X be a vector space over a field K with a non-Archimedean valuation | · |. A function ∥ · ∥ : X → [0, ∞) is called a non-Archimedean norm if it satisfies the following conditions: (a) ∥x∥ = 0 if and only if x = 0; (b) ∥rx∥ = |r|∥x∥ for all r ∈ K, x ∈ X; 2010 Mathematics Subject Classification. Primary 39B55, 46S10, 39B72, 39B52, 54E40, 47H10, 47S10, 26E30, 12J25, 46H25. Key words and phrases. Hyers-Ulam stability, orthogonally cubic-quartic functional equation, fixed point, non-Archimedean normed space, orthogonality space. ∗ Corresponding author.

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(c) the strong triangle inequality ∥x + y∥ ≤ max{∥x∥, ∥y∥},

∀x, y ∈ X,

holds. Then (X, ∥ · ∥) is called a non-Archimedean normed space. Definition 1.2. (1) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called a Cauchy sequence if, for any ε > 0, there is a positive integer N such that ∥xn − xm ∥ ≤ ε for all n, m ≥ N . (2) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is said to be convergent if, for any ε > 0, there are a positive integer N and x ∈ X such that ∥xn − x∥ ≤ ε for all n ≥ N . Then we call x ∈ X the limit of the sequence {xn }, which is denote by limn→∞ xn = x. (3) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. Assume that X is a real inner product space and f : X → ℝ is a solution of the orthogonal Cauchy functional equation f (x + y) = f (x) + f (y), where ⟨x, y⟩ = 0. By the Pythagorean theorem ,f (x) = ∥x∥2 is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus orthogonal Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space. Pinsker [51] characterized the orthogonally additive functionals on an inner product space when the orthogonality is the ordinary one in such spaces. Sundaresan [62] generalized this result to arbitrary Banach spaces equipped with the Birkhoff-James orthogonality. The orthogonal Cauchy functional equation f (x + y) = f (x) + f (y),

x ⊥ y,

where ⊥ is an abstract orthogonality relation, was first investigated by Gudder and Strawther [25]. They defined ⊥ by a system consisting of five axioms and described the general semi-continuous real-valued solution of the conditional Cauchy functional equation. In 1985, R¨atz [58] introduced a new definition of orthogonality by using more restrictive axioms than of Gudder and Strawther. Moreover, he investigated the structure of orthogonally additive mappings. R¨atz and Szab´o [59] investigated the problem in a rather more general framework. Let us recall the orthogonality in the sense of J. R¨atz [58]. Suppose that X is a real vector space with dim X ≥ 2 and ⊥ is a binary relation on X with the following properties: (O1 ) totality of ⊥ for zero: x ⊥ 0 and 0 ⊥ x for all x ∈ X; (O2 ) independence: if x, y ∈ X − {0} and x ⊥ y, then x and y are linearly independent; (O3 ) homogeneity: if x, y ∈ X and x ⊥ y, then αx ⊥ βy for all α, β ∈ ℝ; (O4 ) Thalesian property: if P is a 2-dimensional subspace of X, x ∈ P and λ ∈ ℝ+ , which is the set of nonnegative real numbers, then there exists y0 ∈ P such that x ⊥ y0 and x + y0 ⊥ λx − y0 .

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The pair (X, ⊥) is called an orthogonality space. By an orthogonality normed space we mean an orthogonality space having a normed structure. Some interesting examples are as follows: (1) The trivial orthogonality on a vector space X defined by (O1 ) and, for any non-zero elements x, y ∈ X, x ⊥ y if and only if x and y are linearly independent. (2) The ordinary orthogonality on an inner product space (X, ⟨·, ·⟩) given by x ⊥ y if and only if ⟨x, y⟩ = 0. (3) The Birkhoff-James orthogonality on a normed space (X, ∥ · ∥) defined by x ⊥ y if and only if ∥x + λy∥ ≥ ∥x∥ for all λ ∈ ℝ. The relation ⊥ is called symmetric if x ⊥ y implies that y ⊥ x for all x, y ∈ X. Clearly, Examples (1) and (2) are symmetric, but Example (3) is not. It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [1]–[4], [8, 9, 21, 31]). The stability problem of functional equations originated from the following question of Ulam [64]: Under what condition does there exist an additive mapping near an approximately additive mapping? In 1941, Hyers [27] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1978, Th.M. Rassias [53] extended the theorem of Hyers by considering the unbounded Cauchy difference ∥f (x + y) − f (x) − f (y)∥ ≤ ε(∥x∥p + ∥y∥p ) (ε > 0, p ∈ [0, 1)). The result of Rassias has provided a lot of influence in the development of what we now call generalized Hyers-Ulam stability or Hyers-Ulam stability of functional equations. During the last decades several stability problems of functional equations have been investigated in the spirit of Hyers-Ulam-Rassias. The readers refer to [18, 28, 33, 57] and references therein for detailed information on stability of functional equations. Ger and Sikorska [24] investigated the orthogonal stability of the Cauchy functional equation f (x + y) = f (x) + f (y), namely, they showed that, if f is a mapping from an orthogonality space X into a real Banach space Y and ∥f (x + y) − f (x) − f (y)∥ ≤ ε for all x, y ∈ X with x ⊥ y and for some ε > 0, then there exists exactly one orthogonally additive mapping g : X → Y such that ∥f (x) − g(x)∥ ≤ 16 ε for all x ∈ X. 3 The first author treating the stability of the quadratic equation was Skof [61] by proving that, if f is a mapping from a normed space X into a Banach space Y satisfying ∥f (x + y) + f (x − y) − 2f (x) − 2f (y)∥ ≤ ε for some ε > 0, then there is a unique quadratic mapping g : X → Y such that ∥f (x) − g(x)∥ ≤ 2ε . Cholewa [15] extended the Skof’s theorem by replacing X by an abelian group G. Skof’s result was later generalized by Czerwik [16] in the spirit of Hyers-Ulam-Rassias. The stability problem of functional equations has been extensively investigated by some mathematicians (see [12, 17, 50, 49], [54]–[56]). The orthogonally quadratic equation f (x + y) + f (x − y) = 2f (x) + 2f (y), x ⊥ y

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was first investigated by Vajzovi´c [65] when X is a Hilbert space, Y is the scalar field, f is continuous and ⊥ means the Hilbert space orthogonality. Later, Drljevi´c [22], Fochi [23], Moslehian [41, 42] and Szab´o [63] generalized this result. See also [43, 46]. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies the following conditions: (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. We recall a fundamental result in fixed point theory. Theorem 1.3. [5, 20] Let (X, d) be a complete generalized metric space and J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then, for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−α d(y, Jy) for all y ∈ Y . In 1996, Isac and Th.M. Rassias [29] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [11, 6, 7, 34, 39, 47, 48, 52]). In [32], Jun and Kim considered the following cubic functional equation f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x).

(1.1)

It is easy to show that the function f (x) = x3 satisfies the functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping. In [37], Lee et al. considered the following quartic functional equation f (2x + y) + f (2x − y) = 4f (x + y) + 4f (x − y) + 24f (x) − 6f (y).

(1.2)

It is easy to show that the function f (x) = x4 satisfies the functional equation (1.2), which is called a quartic functional equation and every solution of the quartic functional equation is called a quartic mapping. For more results on the stability of quartic functional equations, see [3], [10], [14], [40] and [60]. This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the orthogonally cubic-quartic functional equation (0.1) in non-Archimedean orthogonality spaces for an odd mapping. In Section 3, we prove the Hyers-Ulam stability of the orthogonally cubic-quartic functional equation (0.1) in non-Archimedean orthogonality spaces for an even mapping. Throughout this paper, assume that (X, ⊥) is a non-Archimedean orthogonality space and that (Y, ∥ · ∥Y ) is a real non-Archimedean Banach space. Assume that |2| ̸= 1.

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ORTHOGONAL STABILITY OF FUNCTIONAL EQUATION

2. Stability of the orthogonally cubic-quartic functional equation: an odd mapping case In this section, applying some ideas from [24, 28], we deal with the stability problem for the orthogonally cubic-quartic functional equation Df (x, y) := f (2x + y) + f (2x − y) − 3f (x + y) − f (−x − y) −3f (x − y) − f (y − x) − 18f (x) − 6f (−x) + 3f (y) + 3f (−y) = 0 for all x, y ∈ X with x ⊥ y in non-Archimedean Banach spaces: an odd mapping case. Definition 2.1. A mapping f : X → Y is called an orthogonally cubic mapping if f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x) for all x, y ∈ X with x ⊥ y. Theorem 2.2. Let φ : X 2 → [0, ∞) be a function such that there exists an α < 1 with ( ) x y φ(x, y) ≤ |8|αφ , (2.1) 2 2 for all x, y ∈ X with x ⊥ y. Let f : X → Y be an odd mapping satisfying ∥Df (x, y)∥Y ≤ φ(x, y)

(2.2)

for all x, y ∈ X with x ⊥ y. Then there exists a unique orthogonally cubic mapping C : X → Y such that 1 ∥f (x) − C(x)∥Y ≤ φ(x, 0) (2.3) |16| − |16|α for all x ∈ X. Proof. Putting y = 0 in (2.2), we get ∥2f (2x) − 16f (x)∥Y ≤ φ(x, 0) for all x ∈ X since x ⊥ 0. So, we have



1 1

f (x) − f (2x) ≤ φ(x, 0)

8 |16| Y for all x ∈ X. Consider the set S := {h : X → Y } and introduce the generalized metric on S:

(2.4)

(2.5)

d(g, h) = inf {µ ∈ ℝ+ : ∥g(x) − h(x)∥Y ≤ µφ (x, 0) , ∀x ∈ X} , where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [38, Lemma 2.1]). Now, we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 8 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then we have ∥g(x) − h(x)∥Y ≤ φ (x, 0)

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J. LEE, C. PARK, Y. CHO, AND D. SHIN

for all x ∈ X and hence ∥Jg(x) − Jh(x)∥Y



1

1

= g (2x) − h (2x)

≤ αφ (x, 0) 8 8 Y for all x ∈ X. So d(g, h) = ε implies that d(Jg, Jh) ≤ αε. This means that d(Jg, Jh) ≤ αd(g, h) for all g, h ∈ S. It follows from (2.5) that d(f, Jf ) ≤ a mapping C : X → Y satisfying the following: (1) C is a fixed point of J, i.e.,

1 . |16|

By Theorem 1.3, there exists

C (2x) = 8C(x)

(2.6)

for all x ∈ X. The mapping C is a unique fixed point of J in the set M = {g ∈ S : d(h, g) < ∞}. This implies that C is a unique mapping satisfying (2.6) such that there exists a µ ∈ (0, ∞) satisfying ∥f (x) − C(x)∥Y

≤ µφ (x, 0)

for all x ∈ X; (2) d(J n f, C) → 0 as n → ∞. This implies the equality 1 lim f (2n x) = C(x) n→∞ 8n for all x ∈ X; 1 d(f, Jf ), which implies the inequality (3) d(f, C) ≤ 1−α d(f, C) ≤

1 . |16| − |16|α

This implies that the inequality (2.3) holds. It follows from (2.1) and (2.2) that 1 ∥DC(x, y)∥Y = lim ∥Df (2n x, 2n y)∥Y n→∞ |8|n |8|n αn 1 n n ≤ lim φ(2 x, 2 y) ≤ lim φ(x, y) = 0 n→∞ |8|n n→∞ |8|n for all x, y ∈ X with x ⊥ y. So DC(x, y) = 0 for all x, y ∈ X with x ⊥ y. Since f is odd, C is odd. Hence C : X → Y is an orthogonally cubic mapping, i.e., C(2x + y) + C(2x − y) = 2C(x + y) + 2C(x − y) + 12C(x) for all x, y ∈ X with x ⊥ y. Thus C : X → Y is a unique orthogonally cubic mapping satisfying (2.3). This completes the proof.  From now on, in Corollaries, assume that (X, ⊥) is a non-Archimedean orthogonality normed space.

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Corollary 2.3. Let θ be a positive real number and p a real number with 0 < p < 3. Let f : X → Y be an odd mapping satisfying ∥Df (x, y)∥Y ≤ θ(∥x∥p + ∥y∥p )

(2.7)

for all x, y ∈ X with x ⊥ y. Then there exists a unique orthogonally cubic mapping C : X → Y such that |2|p θ ∥f (x) − C(x)∥Y ≤ ∥x∥p |16|(|2|p − |2|3 ) for all x ∈ X. Proof. The proof follows from Theorem 2.2 by taking φ(x, y) = θ(∥x∥p + ∥y∥p ) for all x, y ∈ X with x ⊥ y. Then we can choose α = |2|3−p and we get the desired result.  Theorem 2.4. Let f : X → Y be an odd mapping satisfying (2.2) for which there exists a function φ : X 2 → [0, ∞) such that α φ(x, y) ≤ φ (2x, 2y) |8| for all x, y ∈ X with x ⊥ y. Then there exists a unique orthogonally cubic mapping C : X → Y such that α ∥f (x) − C(x)∥Y ≤ φ (x, 0) (2.8) |16| − |16|α for all x ∈ X. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now, we consider the linear mapping J : S → S such that ( ) x Jg(x) := 8g 2 for all x ∈ X. It follows from (2.4) that d(f, Jf ) ≤ d(f, C) ≤

α . |16|

So, we have

α . |16| − |16|α

Thus we obtain the inequality (2.8). The rest of the proof is similar to the proof of Theorem 2.2.  Corollary 2.5. Let θ be a positive real number and p a real number with p > 3. Let f : X → Y be an odd mapping satisfying (2.7). Then there exists a unique orthogonally cubic mapping C : X → Y such that ∥f (x) − C(x)∥Y ≤

|2|p θ ∥x∥p 3 p |16|(|2| − |2| )

for all x ∈ X. Proof. The proof follows from Theorem 2.4 by taking φ(x, y) = θ(∥x∥p + ∥y∥p ) for all x, y ∈ X with x ⊥ y. Then we can choose α = |2|p−3 and we get the desired result. 

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3. Stability of the orthogonally cubic-quartic functional equation: an even mapping case In this section, applying some ideas from [24, 28], we deal with the stability problem for the orthogonally cubic-quartic functional equation given in the previous section: an even mapping case. Definition 3.1. A mapping f : X → Y is called an orthogonally quartic mapping if f (2x + y) + f (2x − y) = 4f (x + y) + 4f (x − y) + 24f (x) − 6f (y) for all x, y ∈ X with x ⊥ y. Theorem 3.2. Let φ : X 2 → [0, ∞) be a function such that there exists an α < 1 with (

x y φ(x, y) ≤ |16|αφ , 2 2

)

for all x, y ∈ X with x ⊥ y. Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.2). Then there exists a unique orthogonally quartic mapping P : X → Y such that ∥f (x) − P (x)∥Y ≤

1 φ(x, 0) |32| − |32|α

for all x ∈ X. Proof. Putting y = 0 in (2.2), we get ∥2f (2x) − 32f (x)∥Y ≤ φ(x, 0)

(3.1)

for all x ∈ X, since x ⊥ 0. So, we have



1

f (x) − f (2x)

16

Y

≤

1 φ(x, 0) |32|

for all x ∈ X. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now, we consider the linear mapping J : S → S such that 1 g (2x) 16 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2. Jg(x) :=



Corollary 3.3. Let θ be a positive real number and p a real number with 0 < p < 4. Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.7). Then there exists a unique orthogonally quartic mapping P : X → Y such that ∥f (x) − P (x)∥Y ≤

|2|p θ ∥x∥p p 4 |32|(|2| − |2| )

for all x ∈ X. Proof. The proof follows from Theorem 3.2 by taking φ(x, y) = θ(∥x∥p + ∥y∥p ) for all x, y ∈ X with x ⊥ y. Then we can choose α = |2|4−p and we get the desired result. 

579

ORTHOGONAL STABILITY OF FUNCTIONAL EQUATION

Theorem 3.4. Let f : X → Y be an even mapping satisfying (2.2) and f (0) = 0 for which there exists a function φ : X 2 → [0, ∞) such that φ(x, y) ≤

α φ (2x, 2y) 16

for all x, y ∈ X with x ⊥ y. There exists a unique orthogonally quartic mapping P : X → Y such that α ∥f (x) − P (x)∥Y ≤ φ (x, 0) (3.2) |32| − |32|α for all x ∈ X. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now, we consider the linear mapping J : S → S such that ( )

Jg(x) := 16g

x 2

α for all x ∈ X. It follows from (3.1) that d(f, Jf ) ≤ |32| . So, we obtain the inequality (3.2). The rest of the proof is similar to the proof of Theorem 2.2. 

Corollary 3.5. Let θ be a positive real number and p a real number with p > 4. Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.7). Then there exists a unique orthogonally quartic mapping P : X → Y such that ∥f (x) − P (x)∥Y ≤

|2|p θ ∥x∥p |32|(|2|4 − |2|p )

for all x ∈ X. Proof. The proof follows from Theorem 3.4 by taking φ(x, y) = θ(∥x∥p + ∥y∥p ) for all x, y ∈ X with x ⊥ y. Then we can choose α = |2|p−4 and we get the desired result.  (−x) (−x) Let fo (x) = f (x)−f and fe (x) = f (x)+f . Then fo is an odd mapping and fe is an 2 2 even mapping such that f = fo + fe . The above corollaries can be summarized as follows:

Theorem 3.6. Assume that (X, ⊥) is a non-Archimedean orthogonality normed space. Let θ be a positive real number and p a real number with 0 < p < 3 (resp., p > 4). Let f : X → Y be a mapping satisfying f (0) = 0 and (2.7). Then there exist an orthogonally cubic mapping C : X → Y and an orthogonally quartic mapping P : X → Y such that (

∥f (x) − C(x) − P (x)∥Y

≤ (

(resp.

∥f (x) − C(x) − P (x)∥Y

≤

)

1 1 + |2|p θ∥x∥p 3 p |16|(|2| − |2| ) |32|(|2|4 − |2|p ) )

1 1 + |2|p θ∥x∥p ) |16|(|2|p − |2|3 ) |32|(|2|p − |2|4 )

for all x ∈ X.

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J. LEE, C. PARK, Y. CHO, AND D. SHIN

Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), and D. Y. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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[22] F. Drljevi´c, On a functional which is quadratic on A-orthogonal vectors, Publ. Inst. Math. (Beograd) 54 (1986), 63–71. [23] M. Fochi, Functional equations in A-orthogonal vectors, Aequat. Math. 38 (1989), 28–40. [24] R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Polish Acad. Sci. Math. 43 (1995), 143–151. [25] S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific J. Math. 58 (1975), 427–436. [26] K. Hensel, Ubereine news Begrundung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein 6 (1897), 83–88. [27] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. [28] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨auser, Basel, 1998. [29] G. Isac and Th.M. Rassias, Stability of 𝜓-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [30] R.C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291–302. [31] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292. [32] K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878. [33] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Florida, 2001. [34] Y. Jung and I. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl. 306 (2005), 752–760. [35] A.K. Katsaras and A. Beoyiannis, Tensor products of non-Archimedean weighted spaces of continuous functions, Georgian Math. J. 6 (1999), 33–44. [36] A. Khrennikov, Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models, Mathematics and its Applications 427, Kluwer Academic Publishers, Dordrecht, 1997. [37] S. Lee, S. Im and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), 387–394. [38] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [39] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [40] M. Mohammadi, Y.J. Cho, C. Park, P. Vetro and R. Saadati, Random stability of an sdditivequadratic-quartic functional equation, J. Inequal. Appl. Vol. 2010, Article ID 754210, pp. 18. [41] M.S. Moslehian, On the orthogonal stability of the Pexiderized quadratic equation, J. Differ. Equat. Appl. 11 (2005), 999–1004. [42] M.S. Moslehian, On the stability of the orthogonal Pexiderized Cauchy equation, J. Math. Anal. Appl. 318 (2006), 211–223. [43] M.S. Moslehian and Th.M. Rassias, Orthogonal stability of additive type equations, Aequat. Math. 73 (2007), 249–259. [44] M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal. 69 (2008), 3405–3408. [45] P.J. Nyikos, On some non-Archimedean spaces of Alexandrof and Urysohn, Topology Appl. 91 (1999), 1–23. [46] L. Paganoni and J. R¨atz, Conditional function equations and orthogonal additivity, Aequat. Math. 50 (1995), 135–142. [47] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007, Art. ID 50175 (2007).

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[48] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. 2008, Art. ID 493751 (2008). [49] C. Park, Y.J. Cho and H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in non-Archimedean spaces, J. Comput. Anal. Appl. 14(2012), 526–535. [50] C. Park and J. Park, Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping, J. Differ. Equat. Appl. 12 (2006), 1277–1288. [51] A.G. Pinsker, Sur une fonctionnelle dans l’espace de Hilbert, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 20 (1938), 411–414. [52] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [53] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [54] Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babe¸s-Bolyai Math. 43 (1998), 89–124. [55] Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378. [56] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [57] Th.M. Rassias (ed.), Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003. [58] J. R¨atz, On orthogonally additive mappings, Aequat. Math. 28 (1985), 35–49. [59] J. R¨atz and Gy. Szab´o, On orthogonally additive mappings 𝐼𝑉 , Aequat. Math. 38 (1989), 73–85. [60] R. Saadati, Y.J. Cho and J. Vahidi, The stability of the quartic functional equation in various spaces, Comput. Math. Appl. 60(2010), 1994–2002. [61] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [62] K. Sundaresan, Orthogonality and nonlinear functionals on Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 187–190. [63] Gy. Szab´o, Sesquilinear-orthogonally quadratic mappings, Aequat. Math. 40 (1990), 190–200. [64] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960. ¨ [65] F. Vajzovi´c, Uber das Funktional H mit der Eigenschaft: (𝑥, 𝑦) = 0 ⇒ H(𝑥 + 𝑦) + H(𝑥 − 𝑦) = 2H(𝑥) + 2H(𝑦), Glasnik Mat. Ser. III 2 (22) (1967), 73–81. Jung Rye Lee, Department of Mathematics, Daejin University, Kyeonggi 487-711, Republic of Korea E-mail address: [email protected] Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea E-mail address: [email protected] Yeol Je Cho Department of Mathematics Education and RINS, Gyeongsang National University, Chinju 660-701, South Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 130-743, South Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.3, 584-592, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A NOTE ON q-BERNSTEIN POLYNOMIALS ASSOCIATED WITH p-ADIC INTEGRAL ON Zp TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM Abstract. In [7], Kim proved some interesting properties on Phillips q-Bernstein polynomials. In this paper, we investigate some identities on Euler and Bernoulli numbers associated with Phillips q-Bernstein polynomials.

1. Introduction Let p be a fixed odd prime number. Throughout this paper Zp , Qp and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp . Let | · |p be p-adic absolute value which is normally defined by |p|p = p1 .when one talks of q-extension, q is variously considered as an inderterminate, a complex numbers q ∈ C, or p-adic numbers q ∈ Cp . If q ∈ C, then we always assume that |q| < 1. If q ∈ Cp , we always assume that |1 − q|p < 1. For each x ∈ Cp (or C), the q-basic numbers are defined by [x]q =

1 − qx , 1−q

[n]q ! = [n]q [n − 1]q ...[2]q [1]q

(n ∈ N).

Now, we use the notation of Gaussian q-binomial coefficient in the form   [n]q ! [n]q [n − 1]q ...[n − k + 1]q n = = , (n, k ∈ N). k q [k]q ![n − k]q ! [k]q ! Note that     n n(n − 1)...(n − k + 1) n lim = = . q−→1 k q k k! The Gaussian binomial coefficient satisfies the following recursion formula :           n+1 n n n k n n−k = +q =q + . k k−1 q k q k−1 q k q q The q-binomial formulae are known as (1)

(1 − x)nq =

n n   Y X ` n (1 − q i−1 x) = q (2) (−1)` x` , ` q i=1

`=0

584

TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM

and (2)

 ∞  X 1 n+`−1 = x` , (1 − x)nq ` q

(see [1-15]).

`=0

Let C[0, 1] be the space of contious functions on [0, 1]. Phillips’ q-Bernstein polynomials are given by Bn,q (f |x) =

(3)

n X

Bk,n (x, q)f (

k=0 n k k q x (1




[k]q ), [n]q

x)qn−k

where Bk,n (x, q) = − and f ∈ C[0, 1]. Some interesting identities and formulae of Phillips’ q-Bernstein polynomials are given in [13]. Let C(Zp ) be the space of continous functions on Zp . In this paper, we assume that q ∈ Cp with |1 − q|p < 1. For f ∈ C(Zp ), the fermionic p-adic integral on Zp is defined by Kim as follows (see [5-12]): N

pX −1 1 Iq (f ) = f (x)dµq (x) = lim N f (x)(−q)x , N →∞ [p ] −q Zp

Z

(4)

x=0

and (5)

qIq (f1 ) + Iq (f ) = [2]q f (0),

where f1 (x) = f (x + 1).

In this paper, we give some identities on the q-Euler numbers and polynomials associated with q-Bernstein polynomials by using the ferminoic p-adic q-integral on Zp . 2. Some identities on q-Euler numbers Let us take f (x) = etx . Then, by (5), we get Z [2]q xt (6) e . et(x+y) dµq (y) = t qe + 1 Zp Now, we define the q-Euler polynomials as follows: ∞

(7)

X [2]q xt tn Eq (x)t e = e = E (x) , n,q qet + 1 n! n=0

with the usual convention about replacing Eqn (x) by En,q (x). In the special case, x = 0, En,q (0) = En,q are called the n-th q-Euler numbers. From (7), we note that (8)

En,q (x) =

n X

En,q xn−` = (x + Eq )n ,

`=0

585

A NOTE ON q-BERNSTEIN POLYNOMIALS ASSOCIATED WITH p-ADIC INTEGRAL ON Zp

with the usual convention about replacing Eqn by En,q . By (7) and (8), we get the recurrence relation for the q-Euler numbers : (9)

qEn,q (1) + En,q = [2]q δ0,n

where n ∈ Z+ = N ∪ {0},

where δn,k is a Kronecker symbol. From (4) and (7), we note that Z Z (10) (y + x)n dµq−1 (y). (y + 1 − x)n dµq (y) = (−1)n Zp

Zp

Therefore, by (10), we obtain the following lemma. Lemma 1. For n ∈ Z , we have En,q (1 − x) = (−1)n En,q−1 (x). By (9), we get (11)

2

q En,q (2) = q

n   X n `=0

`

qEn,q (1)

!  X n   n n = q [2] − q En,q ` ` `=0

= q[2]q − qEn,q (1) = q[2]q + En,q − [2]q δ0,n . Therefore, by (11), we obtain the following corollary. Corollary 2. For n ∈ Z+ , we have q 2 En,q (2) − q[2]q + [2]q δ0,n = En,q . By Lemma 1 and Corollary 2, we get (12)

(−1)n En,q−1 (−1) = En,q (2) = En,g + q[2]q + [2]q δ0,n .

As is well known, the Frobenius-Euler polynomials are defined by ∞

(13)

1 − u xt X tn e = H (x, u) , n et − u n!

(u 6= 1).

n=0

In the spcial case, x = 0, Hn (0, u) = Hn (u) are called the n-th FrobeniusEuler numbers. From (7) and (13), we have ∞

n [2]q xt 1 + q −1 xt X −1 t e = e = H (x, q ) , n qet + 1 et + q −1 n! n=0

586

TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM

and by (7), we get Hn (x, −q −1 ) = En,q (x).

(14)

Therefore, by (14), we obtain the following corollary. Corollary 3. For n ∈ Z+ , we have Hn (x, −q −1 ) = En,q (x).

For f ∈ C(Zp ), the q-Bernstein operator is defined by Bn,q (f |x) =

(15)

n X

Bk,n (x, q)f (

k=0

[k]q ), [n]q


n

where Bk,n (x, q) = k q xk (1 − x)n−k for n, k ∈ Z+ . In the paper Bk,n (x, q) q are called the q-Bernstein polynomials of degree n. Now, we define the q-difference operator as follows : (16)

4nq = (E − I)nq =

n Y (E − Iq i−1 ),

(see [7]).

i=1

where (Eh)(x) = h(x + 1), and I is identity operator. From (15) and (16), we get n   X 0 n (17) ), (see [7]). xk 4kq f ( Bn,q (f |x) = [n]q k q k=0

Let us take the fermionic p-adic q-integral on both sides of (17). Then we have Z n   X 0 n 4kq f ( (18) Bn,q (f |x)dµq (x) = )Ek,q . k [n] q Zp q k=0

From (15), we have (19) Z Bn,q (f |x)dµq (x) = Zp

=

n n−k X X n − k  k=0 `=0 n n−k X X k=0 `=0

` n−k `

(−1)` q



Z

! xk+` dµq (x) f (

Zp

(−1)` Ek+`,q f ( q

[k]q ). [n]q

Therefore, by (18) and (19), we obtain the following proposition :

587

[k]q ) [n]q

A NOTE ON q-BERNSTEIN POLYNOMIALS ASSOCIATED WITH p-ADIC INTEGRAL ON Zp

Proposition 4. For f ∈ C(Zp ), we have n n−k X X n − k 

`

k=0 `=0

=

(−1)` Ek+`,q f ( q

n  X k=0

[k]q ) [n]q

 n 0 4kq f ( )Ek,q . k [n]q

The second kind q-stirling numbers are defined by (20)

k   k ∞ X k−j q −( 2 ) X tn k−j k (−1) q ( 2 ) e[j]q t = S(n, k|q) . [k]q ! j q n!

n=0

j=0

From (20), we have (21)

S(n, k|q) =

k   k j k q −( 2 ) X [k − j]nq (−1)j q (2) [k]q ! j q

j=0

−(k2)

=

  k X k−j q k [j]n (−1)k−j q ( 2 ) [k]q ! j q q j=0

−(k2)

=

q 4k 0n , [k]q ! q

(see [7]).

By (15), (20) and (21), we get m [n]m q Bn,q (x |x)

(22)

=

n   X n k=0

k

k

xk [k]q !q (2) S(m, k|q). q

From (22), we have (23)

[n]m q

Z

Bn,q (xm |x)dµq (x) =

Zp

n   X n

k

k=0

[k]q !S(m, k|q)Ek,q . q

Therefore, by (23), we obtain the following theorem. Theorem 5. For m, n ∈ Z+ , we have n n−k X X n − k  k=0 `=0

`

(−1)` Ek+`,q [k]m q = q

n   X n k=0

588

k

[k]q !S(m, k|q)Ek,q . q

TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM

From (15), we have   n Bk,n (x, q) = xk (1 − x)n−k k q   n−k X n − k  ` n k = x (−1)` q (2) x` k q ` q `=0     n−k X n−k ` n = (−1)` q (2) x`+k ` q k q `=0     n X `−k n−k n = (−1)`−k q ( 2 ) x` . `−k q k q

(24)

`=k

By simple calculation, we get         n n−k n ` = . k q `−k q ` q k q

(25)

From (24) and (25), we have n     X `−k n ` Bk,n (x, q) = (−1)`−k q ( 2 ) x` . ` q k q

(26)

`=k

Let us take the fermionic p-adic q-integral on both sides of (26). Then we have n     X `−k n ` Bk,n (x, q)dµq (x) = (−1)`−k q ( 2 ) E`,q . ` q k q Zp

Z (27)

`=k

By (15), we get

(28)

n X k=1

n k q
n Bk,n (x, q) 1 q




  n X [k]q n xk (1 − x)n−k q [n]q k q k=1  n  X n−1 = xk (1 − x)n−k q k−1 q k=1 n−1 X  n − 1 =x xk (1 − x)qn−k−1 k q =

k=0

= x.

589

A NOTE ON q-BERNSTEIN POLYNOMIALS ASSOCIATED WITH p-ADIC INTEGRAL ON Zp

By the same method of (28), we see that
 k n n  X X n−2 2 q
(29) xk (1 − x)n−k q n Bk,n (x, q) = k − 2 q k=2 2 q k=2 n−2 X  n − 2 2 =x xk (1 − x)qn−k−2 k q k=0

= x2 . Continuing this process, we get
k n X i q i
(30) n Bk,n (x, q) = x , k=i

f or k, i ∈ Z+ .

i q

Let us take the fermionic p-adic q-integral on both sides of (30).
Z k n X i q
(31) Bk,n (x, q)dµq (x). Ei,q = n k=i

i q

Zp

By (27) and (31), we get (32)

Ei,q =

n X k=i

k n     X ` n i q (`−k)
(−1)`−k q 2 q E`,q . n k q k q i q `=k




Therefore, by (32), we obtain the following theorem. Theorem 6. For m, i, k ∈ Z+ , we have

Ei,q =

n X k=i

k n     X ` n i q (`−k)
(−1)`−k q 2 q E`,q . n k q k q i q `=k




Let Bn (f |x) = limq−→1 Bn,q (f |x). From (15), we have Bn (f |x) =

(33)

n X k=0


n

k Bk,n (x)f ( ), n

where Bk,n (x) = k xk (1 − x)n−k . Here, Bn (f |x) is called the Bernstein polynomials of degree n. From the definition of Bn (f |x), we note that Bn (x) = Bn−k,n (1 − x). Let us consider the fermionic p-adic q-integral on Zp in (33).

590

TAEKYUN KIM, BYUNGJE LEE, SANG-HUN LEE, SEOG-HOON RIM

Now, we get Z (34)

Bk,n (x)dµq (x)

IJ = Zp

 Z n = xk (1 − x)n−k dµq (x) k Zp  X Z k   n k ` (1 − x)n−` dµq (x) = (−1) k ` Z p `=0  X k   n k = (−1)n En−`,q (−1). k ` `=0

By (12), we get (35)

 X k   n k IJ = En−`,q−1 (2) k ` `=0  X  k   [2]q n k = En−`,q−1 + 2 + [2]q−1 δ0,n−` . k ` q `=0

From (34), we note that   n−k   Z n X n−k ` (36) IJ = (−1) xk+` dµq (x) k ` Z p `=0     n−k n X n−k (−1)` Ek+`,q . = ` k `=0

Therefore, by (35) and (36), we obtain the following theorem. Theorem 7. For m, k ∈ Z+ with n > k, we have k   X k `=0

`

En−`,q−1 +

[2]q 2k 2 q

=

n−k X `=0

 n−k (−1)` Ek+`,q . `

References [1] S. Bernstein, D´ emonstration du th´ eor´ eme de Weierstrass,fond´ es sur le Calcul des probabilities, Commun.Soc. Math. Kharkow(2). 13 (1912-13), 1-2. [2] H. Exton, q-hypergeometric functions and applications, Ehis Horwood, 1983. [3] N. K. Govil, V. Gupta, Convergence of q-Meger-K¨ onig-Zeher-Purrmeyer operators, Adv. Stud. Contem. Math 19 (2009), 97-108. [4] F. H. Jackson, A basic sine and cosine with symbolical solutions of certain differential equations , Proc. Edinburgh Math. Soc. 22 (1904), 28-39. [5] T. Kim, Some formulae for the q-Berenstein polynomials and q-deformed binomial distributions, J. Comput. Anal. Appl. 14 (2012), 917-933.

591

A NOTE ON q-BERNSTEIN POLYNOMIALS ASSOCIATED WITH p-ADIC INTEGRAL ON Zp

[6] T. Kim, New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys. 17 (2010), no. 2, 218-225. [7] T. Kim, q-extension of the Euler formula and trigonometric functions, Russ. J. Math. Phys. 14 (2007), 275-278. [8] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288-299. [9] T. Kim, q-Benoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys. 15 (2008), 51-57. [10] T. Kim, A note on q-Bernstein polynomials, Russ. J. Math. Phys. 18 (2011), No.1, 73-82. [11] T. Kim, J. Choi, Y. H. Kim, q-Bernstein polynomials associated with q-stirling numbers and Carlitz’s q-Bernoulli numbers, Abstract and Applied Analysis 2010 (2010), Ariticle ID 150975, 412-422. [12] B. A. Kupershmidt, Reflection symmetries of q-Bernoulli polynomials, J. Nonlinear. Math. Phys. 12(2005). 412-422. [13] G. M. Phillips, Bernstein polynomials based on the q-integers, Annals of Numerical Analysis 4 (1997), 511-514. [14] C. S. Ryoo, A note on the weighted q-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 21(2011), No.1, 47-54. [15] C. S. Ryoo, On the generalited Barnes’ type multiple q-Euler polynomials twisted by ramified roots of unity, Proc. Jangjeon. Math. Soc. 13 (2010), 255-263.

B. Lee Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, S.Korea

S. H. Lee Division of General Educations, Kwangwoon University, Seoul 139-701, S.Korea E-mail: [email protected]

T. KIM Department of Mathematics, Kwangwoon University, Seoul 139-701, S.Korea E-mail: [email protected]

S-H. Rim Department of Mathematics Education, Kyungpook National University, Taegu 702-701, S. Korea E-mail: [email protected]

592

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 3, 2013

On the Logarithms of Circulant Matrices, Chengbo Lu,…….………………………………402 On the Identities for the Bernoulli and Euler Polynomials, D. S. Kim, T. Kim, S. H. Lee, and B. Lee,...............................................................................................................................................413 Solutions of the Stiff-String Model with an Iterative Method, Chao Min, Qing-you Liu, Lie-hui Zhang, and Nan-jing Huang,……………………………………………………………… ..424 Product of Extended Cesàro Operator and Composition Operator from the Logarithmic BlochType Space to
(; ; ) Space on the Unit Ball, Yu-Xia Liang, Ze-Hua Zhou, and Ren-Yu Chen,…………………………………………………………………………………………432 Some New Identities on the Twisted Bernoulli and Euler Polynomials, Dmitry V. Dolgy, Taekyun Kim, Byungje Lee, and Sang-Hun Lee,…………………………………………..441 Fixed Points and Fuzzy Stability of an Additive-Quadratic Functional Equation, Choonkil Park, Abbas Najati, and Sun Young Jang,………………………………………………………..452 Common Fixed Point Results for Generalized Quasicontractions in TVS-Cone Metric Spaces, Hui-Sheng Ding, Mirko Jovanović, Zoran Kadelburg, Stojan Radenović,………………..463 Some Properties of Bernoulli and Euler Polynomials Arising from the p-Adic Integral on ℤ , D.S. Kim, T. Kim, S.H. Lee, and D.V. Dolgy,………………………………………………471 Representations for Drazin Inverse of Block Matrix, Jelena Ljubisavljević, and Dragana S. Cvetković-Ilić,………………………………………………………………………………481 The Solution of Fisher-Kolomogrov-Petrovskii-Piskunov Equation by the Homotopy Perturbation Method, M. Ghoreishi, A.I.B.Md.Ismail, and A. Rashid,……………………498 A Study on the Limiting Ratio of Consecutive Terms in a Class of Bivariate Generating Functions, Shun-Pin Hsu,……………………………………………………………………507 Isometries of Composition and Differentiation Operators from Bloch Type Space to  , GengLei Li and Ze-Hua Zhou,……………………………………………………………………526 An Expanded Mixed Finite Element Method for Sobolev Equation, Na Li, Fuzheng Gao, and Tiande Zhang,……………………………………………………………………………….534

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 3, 2013 (continued) Some Identities for The Frobenius-Euler Numbers and Polynomials, Taekyun Kim, Byungje Lee, Sang-Hun Lee, and Seog-Hoon Rim,………………………………………………………544 On the Second Kind (h,q)-Euler Polynomials of Higher Order, C. S. Ryoo,……………….552 Further Exact Traveling Wave Solutions to the (2+1)-Dimensional Boussinesq and KadomtsevPetviashvili Equation, M. Ali Akbar, Norhashidah Hj. Mohd. Ali, and Syed Tauseef MohyudDin,…………………………………………………………………………………………..557 Orthogonal Stability of a Cubic-Quartic Functional Equation in Non-Archimedean Spaces, Jung Rye Lee, Choonkil Park, Yeol Je Cho, and Dong Yun Shin,………………………………572 A Note on q-Bernstein Polynomials Associated With p-Adic Integral on ℤ , Taekyun Kim, Byungje Lee, Sang-Hun Lee, and Seog-Hoon Rim,……………………………………….584

Volume 15, Number 4 ISSN:1521-1398 PRINT,1572-9206 ONLINE

May 2013

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC(eight times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Editor’s Assistant:Dr.Razvan Mezei,Lander University,SC 29649, USA.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 604-611, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Positive Solutions to a Nonlinear 2nth Order Boundary Value Problems on Time Scales Ilkay Yaslan Karaca Department of Mathematics Ege University, 35100 Bornova, Izmir, Turkey [email protected]

Abstract In this paper, we consider a 2nth -order boundary value problem for dynamic equations on time scales. We establish criteria for the existence of one or more positive solutions by using fixed point theorem of cone expansion and compression type. We also give an example to illustrate our results. 2000 Mathematics subject Classification : 39A10 Keywords: positive solutions, fixed-point theorems, time scales, dynamic equations, cone.

1

Introduction We are concerned with the following boundary value problem

(

2n

2n−2

2n−1

(−1)n y ∆ (t) + λQ(t, y, y ∆ , ..., y ∆ ) = λP (t, y, y ∆ , ..., y ∆ ), 2i 2i ∆2i ∆2i+1 ∆2i αi+1 y (η) + βi+1 y (a) = y (a), γi+1 y ∆ (η) = y ∆ (σ(b)),

t ∈ [a, b], 0 ≤ i ≤ n − 1,

(1.1)

i −1)(a−βi ) i where n ≥ 1, a < η < σ(b), βi ≥ 0, 0 ≤ αi < σ(b)−γi η+(γ , 1 < γi < σ(b)−a+β and λ is positive σ(b)−η η−a+βi parameter. Throughout this paper we let T be any time scale (nonempty closed subset of R) and [a, b] be a subset of T such that [a, b] = {t ∈ T, a ≤ t ≤ b}. We assume that σ(b) is right dense so that σ j (b) = σ(b) for j ≥ 1 and there exist continuous functions f : (0, ∞) → [0, ∞), p, q, p1 , q1 : [a, σ(b)] → R such that

(H1) The maps y ∈ R2n → P (t, y) ∈ R and y ∈ R2n−1 → Q(t, y) ∈ R are continuous for all t ∈ [a, σ(b)], (H2) For y ∈ [0, ∞), q(t) ≤

Q(t,y,y1 ,...,y2n−2 ) f (y)

≤ q1 (t),

p(t) ≤

P (t,y,y1 ,...,y2n−1 ) f (y)

≤ p1 (t),

(H3) q(t) − p1 (t) is nonnegative for t ∈ [a, σ(b)]. The existence and multiplicity of positive solutions for three-point boundary value problems have been studied by many authors. For example, see [3, 5, 6, 7, 8] and references therein. Recently, using the fixed point theorem of cone expansion and compression type, Liu, Ume, Anderson and Kang [6] gave several sufficient conditions for the existence of positive solutions for the nonlinear third-order ordinary differential equation x000 (t) + λα(t)f (t, x(t)) = 0, a < t < b under the boundary conditions x(a) = x00 (a) = x0 (b) = 0. Hu, Xiao and Liang [4] studied singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Using similar arguments and techniques, the obtained criteria for the existence of positive solutions. The purpose of this paper motivated by the papers mentioned above is to fill in the gap in this area. In this paper, a few sufficient conditions for the existence of at least one or more positive solutions to the BVP (1.1). Moreover, an example to illustrate our main result is given.

1 604

KARACA: BVP ON TIME SCALES

2

Preliminaries and Lemmas For 1 ≤ i ≤ n, let Gi (t, s) be Green’s function for the boundary value problems (

2

−y ∆ (t) = 0, ∆ αi y(η) + βi y (a) = y(a), γi y(η) = y(σ(b)).

t ∈ [a, b],

(2.1)

Then, for 1 ≤ i ≤ n,  Gi (t, s) =

Gi1 (t, s), Gi2 (t, s),

a ≤ s ≤ η, η < s ≤ σ(b),

(2.2)

where

Gi1 (t, s) =

Gi2 (t, s) =

1 di

1 di

    

[γi (t − η) + σ(b) − t](σ(s) + βi − a),

σ(s) ≤ t,

[γi (σ(s) − η) + σ(b) − σ(s)](t + βi − a) + αi (η − σ(b))(t − σ(s)),    

t ≤ s,

 [σ(s)(1 − αi ) + αi η + βi − a](σ(b) − t) + γi (η − a + βi )(t − σ(s)),    

σ(s) ≤ t,

[t(1 − αi ) + αi η + βi − a](σ(b) − σ(s)),

   

t ≤ s,

and di = (γi − 1)(a − βi ) + (1 − αi )σ(b) + η(αi − γi ). Lemma 2.1 [1] For 1 ≤ i ≤ n, the Green’s function Gi (t, s) in (2.2) possesses the following property; (t, s) ∈ (a, σ(b)) × (a, b).

Gi (t, s) > 0,

Lemma 2.2 [1] For 1 ≤ i ≤ n, the Green’s function Gi (t, s) in (2.2) satisfies Gi (t, s) ≤ max{Gi (σ(b), s), Gi (σ(s), s)},

(t, s) ∈ [a, σ(b)] × [a, b].

Lemma 2.3 [4] For 1 ≤ i ≤ n, the Green’s function Gi (t, s) in (2.2) the following inequality yields; t−a , Gi (t, s) ≥ min{ σ(b)−a

σ(b)−t } max{Gi (σ(b), s), Gi (σ(s), s)} γi (σ(b)−a)

(t, s) ∈ [a, σ(b)] × [a, b].

Corollary 2.1 [4] For 1 ≤ i ≤ n, the Green’s function Gi (t, s) in (2.2) the following inequality yields; Gi (t, s) ≥ hi (t)kGi (., s)k, t−a where hi (t) = min{ σ(b)−a ,

(t, s) ∈ [a, σ(b)] × [a, b],

σ(b)−t }, γi (σ(b)−a)

kGi (., s)k = maxt∈[a,σ(b)] |Gi (t, s)|,

s ∈ [a, b].

Lemma 2.4 [1] For G as in (2.2), take H1 (t, s) := G1 (t, s), and recursively define Z Hj (t, s) =

σ(b)

Hj−1 (t, r)Gj (r, s)∆r a

for 2 ≤ j ≤ n. Then Hn (t, s) is Green’s function for the homogenous problem (

2n

αi+1 y

∆

2i

(η) + βi+1 y

(−1)n y ∆ (t) = 0, 2i 2i 2i (a) = y ∆ (a), γi+1 y ∆ (η) = y ∆ (σ(b)),

∆

2i+1

Lemma 2.5 [4] If we define K = Πn−1 j=1 Kj ,

L = Πn−1 j=1 Lj

then the Green’s function Hn (t, s) in Lemma 2.4 satisfies

2 605

t ∈ [a, b], 0 ≤ i ≤ n − 1.

KARACA: BVP ON TIME SCALES

h1 (t)LkGn (., s)k ≤ Hn (t, s) ≤ KkGn (., s)k,

(t, s) ∈ [a, σ(b)] × [a, b]

where Kj :=

R σ(b)

Lj :=

R σ(b)

a

kGj (., s)k∆s > 0,

1≤j ≤n−1

(2.3)

and

a

kGj (., s)khj+1 (s)∆s > 0,

1 ≤ j ≤ n − 1.

(2.4)

Let B be a real Banach space and P be a cone in B. Set Pr = {y ∈ P : kyk < r},

∂Pr = {y ∈ P : kyk = r},

Pr,s = {y ∈ P : r ≤ kyk ≤ s}, where kyk := maxt∈[a,σ(b)] |y(t)|. Throughout this paper, we assume that {am }m≥1 and {bm }m≥1 are strictly decreasing and strictly increasing sequences, respectively, a1 < b1 , limm→∞ am = a, limm→∞ bm = σ(b), and p, q are constants with a < p < q < σ(b), Dm =

L K

am −a min{ δ1 (σ(b)−a) ,

σ(b)−bm }, γ1 (σ(b)−a)

H = min{h1 (p), h1 (q)}, R = [HL k = (K

R σ(b) a

Rq p

Am = [a, am ] ∪ [bm , σ(b)], kGn (., s)k(q(s) − p1 (s))∆s]−1 ,

kGn (., s)k(q1 (s) − p(s))∆s)−1 ,

M (t) = supy∈Pt λK l(β, r) = max{β,

R σ(b) a

kGn (., s)k(q1 (s) − p(s))f (y(s))∆s,

k M (β)}, k−r

t > 0.

0 < β, r < k.

Assume that (H4)

R σ(b) a

kGn (., s)k∆s < +∞, and

Rq p

kGn (., s)k∆s > 0,

(H5)f : (0, +∞) → R+ is continuous and limm→∞ supy∈Pc,d λK

R Am

kGn (., s)k(q1 (s) − p(s))f (y(s))∆s = 0

for all c, d with 0 < c < d. Lemma 2.6 Let (H1) − (H5) hold and c, d be a fixed constants with 0 < c < d. Define an integral operator Tλ : Pc,d → P by Tλ y(t) = λ

R σ(b) a

2n−2

Hn (t, s)[Q(s, y, y ∆ , ..., y ∆

2n−1

) − P (s, y, y ∆ , ..., y ∆

)]∆s

(2.5)

for t ∈ [a, σ(b)], y ∈ Pc,d . Then Tλ is completely continuous. Proof. We consider the Banach space B = C[a, σ(b)] equipped with a norm k.k defined by kyk = maxt∈[a,σ(b)] |y(t)|. Define a cone by P = {y ∈ B : y(t) ≥

h1 (t)L kyk, t ∈ [a, σ(b)]}. K

First of all we claim that

3 606

KARACA: BVP ON TIME SCALES

supy∈Pc,d λK

R σ(b) a

kGn (., s)k(q1 (s) − p(s))f (y(s))∆s < ∞.

(2.6)

It follows from (H5) that there exists some positive integer m0 satisfying supy∈Pc,d λK

R Am0

kGn (., s)k(q1 (s) − p(s))f (y(s))∆s ≤ 1.

(2.7)

It is easy to see that for each y ∈ Pc,d and t ∈ [am0 , bm0 ], we have h1 (t)L kyk K

d ≥ ky(t)k ≥ y(t) ≥

cL K

≥

−a

a

m0 min{ δ1 (σ(b)−a) ,

σ(b)−bm0 } γ1 (σ(b)−a)

= cDm0 .

(2.8)

Set B = λK max{(q1 (t) − p(t))f (y(t)) : am0 ≤ t ≤ bm0 , cDm0 ≤ y(t) ≤ d}. Using (2.7) and (2.8), we conclude that supy∈Pc,d λK

R σ(b) a

≤ supy∈Pc,d λK

R Am0

≤1+B

[am0 ,bm0 ]

R [am0 ,bm0 ]

R σ(b) a

kGn (., s)k(q1 (s) − p(s))f (y(s))∆s

R

+ supy∈Pc,d λK ≤1+B

kGn (., s)k(q1 (s) − p(s))f (y)∆s

kGn (., s)k(q1 (s) − p(s))f (y(s))∆s

kGn (., s)k∆s

kGn (., s)k∆s < +∞,

that is, (2.6) holds. By Lemma 2.5, (H4) and (H5) ensure that Tλ is well defined and is nonnegative. For each (y, t) ∈ Pc,d × [a, σ(b)], by Lemma 2.5 we know that kTλ yk = supt∈[a,σ(b)] λ ≤ λK Tλ y(t) = λ

R σ(b) a

R σ(b)

a

2n−2

Hn (t, s)[Q(s, y, y ∆ , ..., y ∆ 2n−2

2n−1

) − P (s, y, y ∆ , ..., y ∆ 2n−1

kGn (., s)k[Q(s, y, y ∆ , ..., y ∆

) − P (s, y, y ∆ , ..., y ∆

2n−2

2n−1

Hn (t, s)[Q(s, y, y ∆ , ..., y ∆

a

≥ λLh1 (t) ≥

R σ(b)

R σ(b) a

) − P (s, y, y ∆ , ..., y ∆ 2n−2

kGn (., s)k[Q(s, y, y ∆ , ..., y ∆

)]

)]∆s,

)]∆s 2n−1

) − P (s, y, y ∆ , ..., y ∆

)]∆s

h1 (t)L kTλ yk K

which gives Tλ y ∈ P. Next we claim that Tλ : Pc,d → P is completely continuous. It follows from (H4) and (H5) that for any y ∈ Pc,d kTλ yk = supt∈[a,σ(b)] λ ≤1+B 0, there exists some positive integer n satisfying Z  sup λK kGn (., s)k(q1 (s) − p(s))f (y(s))∆s < . 4 y∈Pc,d Am

4 607

KARACA: BVP ON TIME SCALES

It follows from the uniform continuity of Green’s function Hn on [a, σ(b)] × [a, b] that there exists some constant δ > 0 satisfying |Hn (t, s) − Hn (r, s)| ≤ µ−1 ,

t, r ∈ [a, σ(b)], s ∈ [a, b] with |t − r| < δ,

where µ = 2λ(σ(b) − a)[1 + max(v,ω)∈[an ,bn ]×[cDn ,d] (q1 (v) − p(v))f (w)]. |Tλ y(t) − Tλ y(r)| ≤λ

R σ(b)

=λ

R

+λ

a

Am0

) − P (s, y, y ∆ , ..., y ∆

2n−1

2n−2

) − P (s, y, y ∆ , ..., y ∆

)]∆s

2n−2

2n−1

|Hn (t, s) − Hn (r, s)|[Q(s, y, y ∆ , ..., y ∆

R [am0 ,bm0 ]

R

≤ 2Kλ

2n−2

|Hn (t, s) − Hn (r, s)|[Q(s, y, y ∆ , ..., y ∆

Am0

)]∆s

2n−1

|Hn (t, s) − Hn (r, s)|[Q(s, y, y ∆ , ..., y ∆

) − P (s, y, y ∆ , ..., y ∆

kGn (., s)k(q1 (s) − p(s))f (y(s))∆s + λµ−1 

R [am0 ,bm0 ]

)]∆s

(q1 (s) − p(s))f (y(s))∆s < 

for all (y, t, r) ∈ Pc,d × [a, σ(b)] × [a, σ(b)] with |t − r| < , that is, {Tλ y : y ∈ Pc,d } is equicontinuous on [a, σ(b)]. We will show that Tλ : Pc,d → P is continuous. Let {yn }n≥1 be any sequence in Pc,d with limn→∞ yn = y ∈ Pc,d . Condition (H5) implies that for each  > 0 there exists some positive integer m0 satisfying Z  kGm0 (., s)k(q1 (s) − p(s))f (y(s))∆s < . sup λK 4 y∈Pc,d Am 0

Since f is uniformly continuous on [cDm0 , d], there exists some positive constant δ satisfying |f (t) − f (r)| < ν,

t, r ∈ [cDm0 , d], with |t − r| < δ,

R σ(b) where ν = (2λK a kGn (., s)k(q1 (s) − p(s))∆s)−1 . From limn→∞ yn = y ∈ Pc,d , we can choose a positive integer N > m0 satisfying kyn − yk < δ for n > N. Clearly, cDm0 ≤ min{yn (t), y(t)} ≤ max{yn (t), y(t)} ≤ d, ∀t ∈ [am0 , bm0 ], n ≥ 1. In light of lines (H4) − (H5), we infer that for any n > N, kTλ yn − Tλ yk ≤ λK

R σ(b)

= λK

R

+λK

a

Am0

kGn (., s)k(q1 (s) − p(s))|f (yn (s)) − f (y(s))|∆s kGn (., s)k(q1 (s) − p(s))|f (yn (s)) − f (y(s))|∆s

R [am0 ,bm ] | 0

kGn (., s)k(q1 (s) − p(s))|f (yn (s)) − f (y(s))|∆s

≤ 2λK supu∈Pc,d

R Am0

kGn (., s)k(q1 (s) − p(s))f (u(s))∆s + νλK

R [am0 ,bm0 ]

kGn (., s)k(q1 (s) − p(s))∆s < ,

which yields that limn→∞ Tλ yn = Tλ y. That is, Tλ is continuous on Pc,d . It follows from the Arzela-Ascoli theorem that Tλ : Pc,d → P is completely continuous. This completes the proof. The following fixed-point theorem will be useful in the proof of the existence of positive solutions in the next section.

5 608

KARACA: BVP ON TIME SCALES

Lemma 2.7 (Fixed-Point Theorem of Cone Expansion and Compression Type)([2]) Let (B, k.k) be a real Banach space and let P ⊂ B be a cone in B. Assume that T : Pc,d → P is a completely continuous operator such that either (a) T y 6≤ y for y ∈ ∂Pc , and T y 6≥ y for y ∈ ∂Pd , or (b) T y 6≥ y for y ∈ ∂Pc , and T y 6≤ y for y ∈ ∂Pd . Then T has a fixed point y ∈ P with c < kyk < d.

3

Existence of Positive Solutions

Now we are ready to establish a few sufficient conditions for the existence of one or more positive solutions of the boundary value problem (1.1) − (1.2). Theorem 3.1 Assume there exist positive constants β, θ1 , θ2 with θ1 < θ2 , r < k and θ2 ≥ l(β, r) such that (i) f (y) >

θ1 R , λ

(ii) f (y)
min{h1 (p), h1 (q)}Lθ1 R

Rq p

kGn (., s)k[q(s) − p1 (s)]∆s = θ1 ,

which is a contradiction. Therefore, (3.3) holds. We next show that Tλ y 6≥ y,

y ∈ ∂Pθ2 .

Suppose that there exists some y1 ∈ ∂Pθ2 , with Tλ y1 ≥ y1 , which gives that Tλ y1 (t) ≥ y1 (t) for t ∈ [a, σ(b)]. So Tλ y1 − y1 ∈ P, which implies that Tλ y1 (t) − y1 (t) ≥

h1 (t)L kTλ y1 (t) K

− y1 (t)k ≥ 0,

t ∈ [a, σ(b)].

Put

6 609

(3.4)

KARACA: BVP ON TIME SCALES

B(y1 ) = {t ∈ [a, σ(b)] : y1 (t) > β}

y¯1 (t) = min{y1 (t), β}, t ∈ [a, σ(b)].

and

(t)L Notice that y¯1 ∈ C([a, σ(b)], [0, ∞)), θ2 h1K ≤ y1 (t) ≤ ky1 k = θ2 for t ∈ [a, σ(b)] and there exists some t0 ∈ [a, σ(b)] satisfying y1 (t0 ) = θ2 . Consequently, we deduce that y¯1 (t) = min{y1 (t), β} ≤ min{θ2 , β} = β for t ∈ [a, σ(b)] and y¯1 (t0 ) = min{θ2 , β} = β. Hence ky¯1 k = β. Since

(t)L y¯1 (t) = min{y1 (t), β} ≥ min{θ2 h1K , β} ≥

h1 (t)L β, K

t ∈ [a, σ(b)],

it follows that y¯1 ∈ ∂Pβ . In light of inequality (3.2) and Lemma 2.5, we deduce that Tλ y1 (t) = λ

R σ(b) a

2n−2

≤ λK

R σ(b)

≤ λK

R σ(b)

= λK

R

+λK

a

a

+λK

R σ(b)

a

2n−2

2n−1

) − P (s, y1 , y1∆ , ..., y1∆

)]∆s

kGn (., s)k[q1 (s) − p(s)]f (y1 (s))∆s

[a,σ(b)]−B(y1 )

R σ(b)

)]∆s

kGn (., s)k[q1 (s) − p(s)]f (y1 (s))∆s

R

a

) − P (s, y1 , y1∆ , ..., y1∆

kGn (., s)k[Q(s, y1 , y1∆ , ..., y1∆

B(y1 )

< rθ2 K

2n−1

Hn (t, s)[Q(s, y1 , y1∆ , ..., y1∆

kGn (., s)k[q1 (s) − p(s)]f (y1 (s))∆s

kGn (., s)k[q1 (s) − p(s)]∆s

kGn (., s)k[q1 (s) − p(s)]f (y¯1 (s))∆s

≤ rθ2 k−1 + M (β) ≤ θ2 .

(3.5)

In view of (3.4) and (3.5), we have ky1 k ≥ kTλ y1 k > r = ky1 k, which is a contradiction. Therefore, by Lemma 2.7, we see that Tλ has at least one fixed point in P. Therefore, boundary value problem (1.1) − (1.2) has at least one positive solution. The proof of Theorem 3.1 is complete. Corollary 3.1 Assume there exist positive constants β, θ1 , θ2 , ..., θn+1 with θ1 < θ2 < ... < θn+1 (n = 1, 2, ... ), r < k, θ2 ≥ l(β, r) such that (i) f (y) > (ii) f (y)
r.

It is clear that (H1) − (H5) are satisfied. Thus by Theorem 3.1, the BVP (3.6) has at least one positive solution y satisfying 1 < kyk < 106 .

References [1] D.R. Anderson and I.Y. Karaca, Higher-order three-point boundary value problem on time scales, Comput. Math. Appl. 56 (2008), 2429-2443. [2] D.J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, Mass, USA, 1988. [3] L. Hu, Positive solutions to singular third-order three-point boundary value problems on time scales, Math. Comput. Modelling 51 (2010), 606615. [4] L.G. Hu, T.J. Xiao and J. Liang, Positive solutions to singular and delay higher-order differential equations on time scales, Boundary Value Problems 2009 (2009), Article ID 937064, 19 pages. [5] I.Y. Karaca, Nonlinear triple-point problems with change of sign, Comput. Math. Appl. 55 (2008), 691703. [6] Z. Liu, J.S. Ume, D.R. Anderson and S.M. Kang, Twin monotone positive solutions to a singular nonlinear third-order differential equation, J. Math. Anal. Appl. 334 (2007) 299-313. [7] Y. Sang and Z.L. Wei, Existence of solutions to a semipositone third-order three-point BVP on time scales, Acta Math. Sci. Ser. A Chin. Ed. 31 (2011), 455465. [8] I. Yaslan, Multiple positive solutions for nonlinear three-point boundary value problems on time scales, Comput. Math. Appl. 55 (2008), 18611869.

8 611

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 612-621, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Fixed points and the random stability of a mixed type cubic, quadratic and additive functional equation M. Eshaghi Gordji1 , Masumeh Ghanifard2 , Hamid Khodaei3 and Choonkil Park4∗ 1,2,3

Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran; 4 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea Abstract. Mihet¸ and Radu, investigated the random stability problems for the Cauchy functional equation and the Jensen functional equation via fixed point method. In this paper, we prove the Hyers-Ulam stability of a mixed type cubic, quadratic and additive functional equation f (x + ky) + f (x − ky) = k 2 f (x + y) + k 2 f (x − y) + 2(1 − k 2 )f (x) for a fixed integer k with k ̸= ±1 in random normed spaces via fixed point method.

1. Introduction The purpose of this paper is to give a comprehensive text to the study of nonlinear random analysis such as the study of fixed point theory, approximation theory and stability of functional analysis in random normed spaces. The notion of random normed space goes back to Sherstnev [41] (see also [15, 16, 40]). For useful examples on random normed spaces and random analysis, see [4]. Traditionally when we investigate stability of functional equations the problem posed by Ulam [43] in 1940 and the well-known theorem of Hyers [18] which came within a year are taken as a starting point. Following Ulam and Hyers a great number of papers on the subject have been published, generalizing Ulam, s problem in various directions. One of these possible generalizations is to allow the Cauchy difference to be unbounded, to be controlled by a function, not necessarily by a constant. Perhaps Aoki in 1950 was the first author treating this problem [3]. He proved that if a mapping f : X → Y between two Banach spaces satisfies ∥f (x + y) − f (x) − f (y)∥ ≤ φ(x, y) for all x, y ∈ X, where φ(x, y) = K(∥ x ∥𝑝 + ∥ y ∥𝑝 ) with (K ≥ 0, 0 ≤ p < 1), then there exists a unique additive mapping A : X → Y such that K ∥ f (x) − A(x) ∥≤ ∥ x ∥𝑝 1 − 2𝑝−1 for all x ∈ X. In (D.G. Bourgin [5], 1951) as well as in (Th.M. Rassias [38], 1978), (G.L. Forti [12, 13], 1980) and (Gavruta [14], 1994) the stability problem with unbounded Cauchy differences is considered. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) (1.1) 0

2010 Mathematics Subject Classification: 39B52, 46S50, 47H10. Keywords: Fixed point method; Random normed spaces; Hyers-Ulam stability; Mixed type functional equation. ∗ Corresponding author. 0 E-mail: 1 [email protected], 2 [email protected], 3 [email protected], 4 [email protected] 0

612

M. Eshaghi Gordji, M. Ghanifard, H. Khodaei, C. Park is related to symmetric bi-additive function [1, 19]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic mapping. It is well known that a mapping f between real vector spaces is quadratic if and only if there exits a unique symmetric bi-additive mapping B such that f (x) = B(x, x) for all x (see [1, 19]). The bi-additive mapping B is given by B(x, y) = 14 (f (x + y) − f (x − y)). The Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for mappings f : A → B, where A is a normed space and B is a Banach space [42] (see [2, 8, 9, 10, 36]). Jun and Kim [20] introduced the following cubic functional equation f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x)

(1.2)

and they established the general solution and the Hyers-Ulam stability for the functional equation (1.2). They proved that a mapping f between two real vector spaces X and Y is a solution of (1.2) if and only if there exists a unique mapping C : X × X × X → Y such that f (x) = C(x, x, x) for all x ∈ X, moreover, C is symmetric for each fixed one variable and is additive for fixed two variables. 1 The mapping C is given by C(x, y, z) = 24 (f (x + y + z) + f (x − y − z) − f (x + y − z) − f (x − y + z)) for all x, y, z ∈ X. Eshaghi and Khodaei [11] have established the general solution and investigated the Hyers-Ulam stability for a mixed type of cubic, quadratic and additive functional equation with f (0) = 0, f (x + ky) + f (x − ky) = k 2 f (x + y) + k 2 f (x − y) + 2(1 − k 2 )f (x)

(1.3)

in quasi-Banach spaces, where k is a nonzero integer with k ̸= ±1. Obviously, the function f (x) = ax + bx2 + cx3 is a solution of the functional equation (1.3) on ℝ. Interesting new results concerning mixed functional equations have recently been obtained by Najati et al. [32, 33, 34], Jun and Kim [21, 22] as well as for the fuzzy stability of a mixed type of additive and quadratic functional equation by Park [35]. The first result on the stability of Cauchy functional equation in the setting of fuzzy normed spaces has been given in [31]. In 2008, Mihet¸ and Radu, [26] applied the fixed alternative method to study the stability of the Cauchy and Jensen functional equations on the random normed space, the stability of some functional equations in the framework of fuzzy normed spaces or random normed spaces has been investigated (see e.g., [24]–[31]). In this paper, by using the idea of Mihet¸ and Radu, we will prove the Hyers-Ulam stability problem of the functional equation (1.3) in random normed spaces via fixed point method. 2. Preliminaries We start our work with the following notion of fixed point theory. For the proof, refer to [23], see also [39], chapter 5. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to [38, 44]. In 2003, Radu [37] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative (see also [6, 7]). Let (X, d) be a generalized metric space. An operator T : X → X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that d(T x, T y) ≤ Ld(x, y) for all x, y ∈ X. If the Lipschitz constant L is less than 1, then the operator T is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Margolis and Diaz. Theorem 2.1. (Cf. [23, 37].) Suppose that we are given a complete generalized metric space (Ω, d) and a strictly contractive mapping T : Ω → Ω with Lipschitz constant L. Then for each given x ∈ Ω, either d(T 𝑚 x, T 𝑚+1 x) = ∞ for all m ≥ 0, or there exists a natural number m0 such that • d(T 𝑚 x, T 𝑚+1 x) < ∞ for all m ≥ m0 ; • the sequence {T 𝑚 x} is convergent to a fixed point y ∗ of T ; 613

Fixed points and the random stability of a functional equation • y ∗ is the unique fixed point of T in Λ = {y ∈ Ω : d(T 𝑚0 x, y) < ∞}; • d(y, y ∗ ) ≤

1 1−L d(y, T y)

for all y ∈ Λ.

We recall (see, e.g. Schweizer and Sklar, [40]) that a distribution function Λ is a mapping from [0, ∞) into [0, 1], nondecreasing and left-continuous, with Λ(0) = 0. The class of all distribution functions Λ, with lim𝑥→∞ Λ(x) = 1, is denoted by D+ . An element of D+ is { 0, if t = 0, ε0 (t) = 1, if t > 0. Suppose that X is a real vector space, Λ is a mapping from X into D+ (for any x in X, Λ(x) is denoted by Λ𝑥 ) and T is a t-norm. The triple (X, Λ, T ) is called a random normed space (briefly RN-space) if and only if the following conditions are satisfied: (RN1 ) Λ𝑥 (t) = ε0 (t) for all t > 0 if and only if x = 0; 𝑡 (RN2 ) Λ𝛼𝑥 (t) = Λ𝑥 ( |𝛼| ) for all x ∈ X and all α ∈ ℝ α ̸= 0;

(RN3 ) Λ𝑥+y (t + s) ≥ T (Λ𝑥 (t), Λy (s)) for all x, y ∈ X and all t, s ≥ 0. Recall that a triangular norm (t-norm) is a mapping T : [0, 1] × [0, 1] → [0, 1], which is associative, commutative and increasing in each variable, with T (a, 1) = a for all a ∈ [0, 1]. The most important t-norms are TM (a, b) = min{a, b}, TL (a, b) = max{a + b − 1, 0} and TP (a, b) = ab. 𝑡 Every normed space (X, ∥ . ∥) defines a random normed space (X, Λ, TM ), where Λu (t) = 𝑡+∥u∥ for all t > 0 and TM is the minimum t-norm. If the t-norm T is such that sup0 0. A sequence {x𝑛 } is called a Cauchy sequence if lim𝑛,𝑚→∞ Λ𝑥𝑛 −𝑥m (t) = 1 for all t > 0. The random normed space (X, Λ, T ) is complete if every Cauchy sequence in X is convergent. Theorem 2.2. [40]. If (X, Λ, T ) is an RN-space and {x𝑛 } is a sequence such that x𝑛 → x, then lim𝑛→∞ Λ𝑥𝑛 (t) = Λ𝑥 (t) almost everywhere. From now on, we will assume that X, V1 and V2 are vector spaces and (Y, Λ, TM ) is a complete RN-space. For convenience, we use the following abbreviation for a given mapping f : X → Y and a nonzero integer k with k ̸= ±1, Df (x, y) := f (x + ky) + f (x − ky) − k 2 f (x + y) − k 2 f (x − y) − 2(1 − k 2 )f (x) for all x, y ∈ X. 3. Fixed points and the random stability of the functional equation (1.3) First, we prove the Hyers-Ulam stability of the functional equation (1.3) in RN-spaces via fixed point method for an even case. Lemma 3.1. [11]. If an even mapping f : V1 → V2 satisfies (1.3), then f (x) is quadratic. Theorem 3.2. Let Φ : X × X → D+ be a mapping (Φ(x, y) is denoted by Φ𝑥,y ) such that, for some 0 < α < k2 , Φ𝑘𝑥,𝑘y (αt) ≥ Φ𝑥,y (t) (3.1) for all x, y ∈ X and all t > 0. Suppose that an even mapping f from X into a complete random normed space (Y, Λ, TM ) with f (0) = 0 satisfies the inequality ΛDf (𝑥,y) (t) ≥ Φ𝑥,y (t) 614

(3.2)

M. Eshaghi Gordji, M. Ghanifard, H. Khodaei, C. Park for all x, y ∈ X and all t > 0. Then there exists a unique quadratic mapping Q : X → Y such that ( ) Λf (𝑥)−Q(𝑥) (t) ≥ Φ0,𝑥 2(k 2 − α)t (3.3) for all x ∈ X and all t > 0. Proof. Letting x = 0 in (3.2), we get Λ2f (𝑘y)−2𝑘2 f (y) (t) ≥ Φ0,y (t)

(3.4)

for all y ∈ X and all t > 0. Thus we have Λ f (𝑘x) −f (𝑥) (t) ≥ Φ0,𝑥 (2k 2 t)

(3.5)

𝑘2

for all x ∈ X and all t > 0. Let S be the set of all even mappings h : X → Y with h(0) = 0 and introduce a generalized metric on S as follows: { } d(h, k) = inf u ∈ ℝ+ : Λh(𝑥)−𝑘(𝑥) (ut) ≥ Φ0,𝑥 (t), ∀x ∈ X, ∀t > 0 where, as usual, inf ∅ = +∞. The proof of the fact that (S, d) is a complete generalized metric space, which can be shown in [6, 17, 26, 27]. Now we consider the mapping J : S → S defined by h (kx) k2 for all h ∈ S and x ∈ X. Let f, g ∈ S such that d(f, g) < ε. Then ( αu ) ( αu ) ΛJg(𝑥)−Jf (𝑥) t = Λ t = Λg(𝑘𝑥)−f (𝑘𝑥) (αut) ≥ Φ0,𝑘𝑥 (αt) ≥ Φ0,𝑥 (t), f (𝑘x) g(𝑘x) − 𝑘2 k2 k2 𝑘2 that is, if d(f, g) < ε we have d(Jf, Jg) < 𝑘𝛼2 ε. This means that α d(Jf, Jg) ≤ 2 d(f, g) k for all f, g ∈ S, that is, J is a strictly contractive self-mapping on S with the Lipschitz constant It follows from (3.5) that ( ) t ΛJf (𝑥)−f (𝑥) ≥ Φ0,𝑥 (t) 2k 2 Jh(x) :=

𝛼 𝑘2 .

for all x ∈ X and all t > 0, which implies that d(Jf, f ) ≤ 2𝑘12 . Due to Theorem 2.1, there exists a unique mapping Q : X → Y such that Q is a fixed point of J, i.e., Q (kx) = k 2 Q(x) for all x ∈ X. Also, d(J 𝑚 g, Q) → 0 as m → ∞, which implies the equality f (k 𝑚 x) = Q(x) 𝑚→∞ k 2𝑚 for all x ∈ X. If we replace x, y with k 𝑚 x, k 𝑚 y in (3.2), respectively, and divide by k 2𝑚 , then it follows from (3.1) that ( 2 )𝑚 ) (( 2 )𝑚 ) ( ( ) k k t ≥ Φ𝑥,y t (3.6) Λ Df (𝑘m x,𝑘m y) (t) ≥ Φ𝑘m 𝑥,𝑘m y k 2𝑚 t = Φ𝑘m 𝑥,𝑘m y α𝑚 α α 𝑘2m lim

for all x, y ∈ X and all t > 0. Letting m → ∞ in (3.6), we find that Λ∆Q(𝑥,y) (t) = 1 for all t > 0, which implies DQ(x, y) = 0. So Q satisfies (1.3). By Lemma 3.1, the mapping Q : X → Y is quadratic. According to the fixed point alternative, since Q is the unique fixed point of J in the set Ω = {g ∈ S : d(f, g) < ∞}, Q is the unique mapping such that Λf (𝑥)−Q(𝑥) (ut) ≥ Φ0,𝑥 (t) for all x ∈ X and all t > 0. Using the fixed point alternative, we obtain that 1 1 1 , d(f, Jf ) ≤ 2 = 2 d(f, Q) ≤ 1−L 2k (1 − L) 2k (1 − 𝑘𝛼2 ) 615

Fixed points and the random stability of a functional equation which implies the inequality

( Λf (𝑥)−Q(𝑥)

t 2(k 2 − α)

) ≥ Φ0,𝑥 (t)

for all x ∈ X and all t > 0. So Λf (𝑥)−Q(𝑥) (t) ≥ Φ0,𝑥 (2(k 2 − α)t) for all x ∈ X and all t > 0.



Now, we prove the Hyers-Ulam stability of the functional equation (1.3) in the RN -spaces via fixed point method for an odd case. Lemma 3.3. [11, 34]. If an odd mapping f : V1 → V2 satisfies (1.3), then the mapping f1 : V1 → V2 defined by f1 (x) = f (2x) − 8f (x) is additive. Theorem 3.4. Let Φ : X × X → D+ be a mapping such that, for some 0 < α < 2, Φ2𝑥,2y (αt) ≥ Φ𝑥,y (t)

(3.7)

for all x, y ∈ X and all t > 0. Suppose that an odd mapping f : X → Y satisfies (3.2) for all x, y ∈ X and all t > 0. Then the limit ) 1 ( A(x) = lim 𝑚 f (2𝑚+1 x) − 8f (2𝑚 x) 𝑚→∞ 2 exists for all x ∈ X and A : X → Y is a unique additive mapping satisfying ˜ 𝑥 ((2 − α)t) Λf (2𝑥)−8f (𝑥)−A(𝑥) (t) ≥ Φ (3.8) for all x ∈ X and all t > 0, where ( ( ( ( ( 2 ) ( 2 2 )) k k (k − 1) ˜ Φ𝑥 (t) = T T T T Φ𝑥,2𝑥 t , Φ(2𝑘−1)𝑥,𝑥 t , 25 24 ( ) ( 2 2 ))) ( 2 2 k (k − 1) k (k − 1) T Φ(2𝑘+1)𝑥,𝑥 t , Φ𝑥,𝑥 t , 4 2 24 ( ( 2 ) ( 2 2 ))) ( ( ( ( 2 ) )) ( 2 2 k −1 k (k − 1) k k (k − 1) T Φ2𝑥,2𝑥 t , Φ t , T T T Φ t , Φ t , 𝑥,3𝑥 𝑥,𝑥 (𝑘−1)𝑥,𝑥 22 22 26 25 ) ( 2 2 ))) ( 2 )))) ( ( 2 2 k (k − 1) k −1 k (k − 1) t , Φ t , Φ t T Φ(𝑘+1)𝑥,𝑥 𝑥,2𝑥 2𝑥,𝑥 25 25 23 for all x ∈ X and all t > 0. Proof. It follows from (3.2) and the oddness of f that Λf (𝑘y+𝑥)−f (𝑘y−𝑥)−𝑘2 f (𝑥+y)−𝑘2 f (𝑥−y)+2(𝑘2 −1)f (𝑥) (t) ≥ Φ𝑥,y (t)

(3.9)

for all x, y ∈ X and all t > 0. Putting y = x in (3.9), we have Λf ((𝑘+1)𝑥)−f ((𝑘−1)𝑥)−𝑘2 f (2𝑥)+2(𝑘2 −1)f (𝑥) (t) ≥ Φ𝑥,𝑥 (t)

(3.10)

for all x ∈ X and all t > 0. It follows from (3.10) that Λf (2(𝑘+1)𝑥)−f (2(𝑘−1)𝑥)−𝑘2 f (4𝑥)+2(𝑘2 −1)f (2𝑥) (t) ≥ Φ2𝑥,2𝑥 (t)

(3.11)

for all x ∈ X and all t > 0. Replacing x and y by 2x and x in (3.9), respectively, we get Λf ((𝑘+2)𝑥)−f ((𝑘−2)𝑥)−𝑘2 f (3𝑥)−𝑘2 f (𝑥)+2(𝑘2 −1)f (2𝑥) (t) ≥ Φ2𝑥,𝑥 (t)

(3.12)

for all x ∈ X and all t > 0. Setting y = 2x in (3.9), we get Λf ((2𝑘+1)𝑥)−f ((2𝑘−1)𝑥)−𝑘2 f (3𝑥)−𝑘2 f (−𝑥)+2(𝑘2 −1)f (𝑥) (t) ≥ Φ𝑥,2𝑥 (t)

(3.13)

for all x ∈ X and all t > 0. Putting y = 3x in (3.9), we obtain Λf ((3𝑘+1)𝑥)−f ((3𝑘−1)𝑥)−𝑘2 f (4𝑥)−𝑘2 f (−2𝑥)+2(𝑘2 −1)f (𝑥) (t) ≥ Φ𝑥,3𝑥 (t) 616

(3.14)

M. Eshaghi Gordji, M. Ghanifard, H. Khodaei, C. Park for all x ∈ X and all t > 0. Replacing x and y by (k + 1)x and x in (3.9), respectively, we get Λf ((2𝑘+1)𝑥)−f ((−𝑥)−𝑘2 f ((𝑘+2)𝑥)−𝑘2 f (𝑘𝑥)+2(𝑘2 −1)f ((𝑘+1)𝑥) (t) ≥ Φ(𝑘+1)𝑥,𝑥 (t)

(3.15)

for all x ∈ X and all t > 0. Replacing x and y by (k − 1)x and x in (3.9), respectively, one gets Λf ((2𝑘−1)𝑥)−f (𝑥)−𝑘2 f ((𝑘−2)𝑥)−𝑘2 f (𝑘𝑥)+2(𝑘2 −1)f ((𝑘−1)𝑥) (t) ≥ Φ(𝑘−1)𝑥,𝑥 (t)

(3.16)

for all x ∈ X and all t > 0. Replacing x and y by (2k + 1)x and x in (3.9), respectively, we obtain Λf ((3𝑘+1)𝑥)−f (−(𝑘+1)𝑥)−𝑘2 f (2(𝑘+1)𝑥)+2(𝑘2 −1)f ((2𝑘+1)𝑥)−𝑘2 f (2𝑘𝑥) (t) ≥ Φ(2𝑘+1)𝑥,𝑥 (t)

(3.17)

for all x ∈ X and all t > 0. Replacing x and y by (2k − 1)x and x in (3.9), respectively, we have Λf ((3𝑘−1)𝑥)−f (−(𝑘−1)𝑥)−𝑘2 f (2(𝑘−1)𝑥)+2(𝑘2 −1)f ((2𝑘−1)𝑥)−𝑘2 f (2𝑘𝑥) (t) ≥ Φ(2𝑘−1)𝑥,𝑥 (t) for all x ∈ X and all t > 0. It follows from (3.10), (3.12), (3.13), (3.15) and (3.16) that ( ( ( ( 2 ) ( 2 2 )) k k (k − 1) Λf (3𝑥)−4f (2𝑥)+5f (𝑥) (t) ≥ T T T Φ𝑥,𝑥 t , Φ t , (𝑘−1)𝑥,𝑥 24 23 ( ) ( 2 2 ))) ( 2 )) ( 2 2 k (k − 1) k (k − 1) k −1 T Φ(𝑘+1)𝑥,𝑥 t , Φ t , Φ t 𝑥,2𝑥 2𝑥,𝑥 23 23 2

(3.18)

(3.19)

for all x ∈ X and all t > 0. It follows from (3.10), (3.11), (3.13), (3.14), (3.17) and (3.18) that ( ( ( ( 2 ) ( 2 2 )) k k (k − 1) Λf (4𝑥)−2f (3𝑥)−2f (2𝑥)+6f (𝑥) (t) ≥ T T T Φ𝑥,2𝑥 t , Φ t , (2𝑘−1)𝑥,𝑥 24 23 ( ( 2 2 ) ( 2 2 ))) k (k − 1) k (k − 1) (3.20) T Φ(2𝑘+1)𝑥,𝑥 t , Φ t , 𝑥,𝑥 23 23 ) ( 2 2 ))) ( ( 2 k (k − 1) k −1 t , Φ𝑥,3𝑥 t T Φ2𝑥,2𝑥 2 2 for all x ∈ X and all t > 0. Finally, using (3.19) and (3.20), we obtain that ˜ 𝑥 (t) Λf (4𝑥)−10f (2𝑥)+16f (𝑥) (t) ≥ Φ

(3.21)

for all x ∈ X and all t > 0, where ( ( ( ( ( 2 ) ( 2 2 )) k k (k − 1) ˜ Φ𝑥 (t) = T T T T Φ𝑥,2𝑥 t , Φ(2𝑘−1)𝑥,𝑥 t , 25 24 ( ) ( 2 2 ))) ( 2 2 k (k − 1) k (k − 1) T Φ(2𝑘+1)𝑥,𝑥 t , Φ t , 𝑥,𝑥 24 24 ) ( 2 2 ))) ( ( ( ( 2 ) ( 2 2 )) (3.22) ( ( 2 k (k − 1) k (k − 1) k k −1 t , Φ t , T T T Φ t , Φ t , T Φ2𝑥,2𝑥 𝑥,3𝑥 𝑥,𝑥 (𝑘−1)𝑥,𝑥 22 22 26 25 ( 2 2 ( ) ( 2 2 ))) ( 2 )))) k (k − 1) k (k − 1) k −1 T Φ(𝑘+1)𝑥,𝑥 t , Φ t , Φ t 𝑥,2𝑥 2𝑥,𝑥 25 25 23 for all x ∈ X and all t > 0. Let f1 : X → Y be a mapping defined by f1 (x) := f (2x) − 8f (x) for all x ∈ X. It follows from (3.21) that Λ f1 (2x) −f 2

1 (𝑥)

˜ 𝑥 (2t) (t) ≥ Φ

(3.23)

for all x ∈ X and all t > 0. Similar to the proof of Theorem 3.2, let S be the set of all odd mappings h : X → Y and introduce a generalized metric on S as follows: { } ˜ 𝑥 (t), ∀x ∈ X, ∀t > 0 d(h, k) = inf u ∈ ℝ+ : Λh(𝑥)−𝑘(𝑥) (ut) ≥ Φ 617

Fixed points and the random stability of a functional equation So (S, d) is a generalized complete metric space. We consider the function J : S → S defined by Jh(x) := h(2𝑥) for all h ∈ S and x ∈ X. Let f, g ∈ S such that d(f, g) < ε. Then 2 ( αu ) ˜ 2𝑥 (αt) ≥ Φ ˜ 𝑥 (t). ΛJg(𝑥)−Jf (𝑥) t = Λg(2𝑥)−f (2𝑥) (αut) ≥ Φ 2 So if d(f, g) < ε, then we have d(Jf, Jg) < 𝛼2 ε. This means that d(Jf, Jg) ≤ 𝛼2 d(f, g) for all f, g ∈ S, that is, J is a strictly contractive self-mapping on S with the Lipschitz constant 𝛼2 . It follows from (3.23) that ( ) t ˜ 𝑥 (t) ΛJf1 (𝑥)−f1 (𝑥) ≥Φ 2 for all x ∈ X and all t > 0, which implies that d(Jf1 , f1 ) ≤ 21 . Due to Theorem 2.1, there exists a unique mapping A : X → Y such that A is a fixed point of J, m 𝑥) i.e., A (2x) = 2A(x) for all x ∈ X and the sequence {J 𝑛 } converges to A , i.e., lim𝑚→∞ f1 (2 = A(x) 2m for all x ∈ X. It follows from (3.2) and (3.7) that ΛDA(𝑥,y) (t) = Λ Df1 (2m x,2m y) (t) 2m

= Λ ∆f (2m+1 x,2m+1 y) −8 ∆f (2m x,2m y) (t) m 2m ( )) ( ( 2) t t ≥ T Λ ∆f (2m+1 x,2m+1 y) , Λ ∆f (2m x,2m y) 2 2 2m 2m−3 ( ) ≥ T Φ2m+1 𝑥,2m+1 y (2𝑚−1 t), Φ2m 𝑥,2m y (2𝑚−4 t) ( (( )𝑚 𝑚 ) (( )𝑚 𝑚 )) 2 α t 2 α t = T Φ2m 𝑥,2m y , Φ2m 𝑥,2m y 2 α 2 α 24 (( )𝑚 )) ( (( )𝑚 ) t 2 t 2 , Φ𝑥,y ≥ T Φ𝑥,y 2 α 2 α 24

(3.24)

for all x, y ∈ X and all t > 0. Letting m → ∞ in (3.24), we get that ΛDA(𝑥,y) (t) = 1 for all t > 0, which implies DA(x, y) = 0. Thus A satisfies (1.3). By Lemma 3.3, the mapping A : X → Y is additive. The rest of the proof is similar to the proof of Theorem 3.2 and we omit the details.  Lemma 3.5. [11, 34]. If an odd mapping f : V1 → V2 satisfies (1.3), then the mapping f3 : V1 → V2 defined by f3 (x) = f (2x) − 2f (x) is cubic. Theorem 3.6. Let Φ : X × X → D+ be a mapping which, for some 0 < α < 23 , satisfies (3.7) for all x, y ∈ X and all t > 0. Suppose that an odd mapping f : X → Y satisfies (3.2) for all x, y ∈ X and all t > 0. Then the limit ) 1 ( C(x) = lim 3𝑚 f (2𝑚+1 x) − 2f (2𝑚 x) 𝑚→∞ 2 exists for all x ∈ X and C : X → Y is a unique cubic mapping satisfying ˜ 𝑥 ((23 − α)t) Λf (2𝑥)−2f (𝑥)−C(𝑥) (t) ≥ Φ (3.25) ˜ 𝑥 (t) is defined as in Theorem 3.4. for all x ∈ X and all t > 0, where Φ Proof. Similar to proof of Theorem 3.4, we obtain ˜ 𝑥 (t) Λf (4𝑥)−10f (2𝑥)+16f (𝑥) (t) ≥ Φ for all x ∈ X and t > 0. Let f3 : X → Y be a mapping defined by f3 (x) := f (2x) − 2f (x) for all x ∈ X. Thus (3.21) implies that ( ) ˜ 𝑥 23 t Λ f3 (2x) −f (𝑥) (t) ≥ Φ (3.26) 23

3

for all x ∈ X and all t > 0. The rest of the proof is similar to the proof of Theorem 3.4. Lemma 3.7. [11]. If an odd mapping f : V1 → V2 satisfies (1.3), then f is cubic-additive. 618



M. Eshaghi Gordji, M. Ghanifard, H. Khodaei, C. Park Theorem 3.8. Let Φ : X × X → D+ be a mapping which, for some 0 < α < 2, satisfies (3.7) for all x, y ∈ X and all t > 0. Suppose that an odd mapping f : X → Y satisfies (3.2) for all x, y ∈ X and all t > 0. Then there exists an additive mapping A : X → Y and a cubic mapping C : X → Y such that ˜ 𝑥 (3(2 − α)t), Φ ˜ 𝑥 (3(23 − α)t) Λf (𝑥)−A(𝑥)−C(𝑥) (t) ≥ T (Φ

(3.27)

˜ 𝑥 (t) is defined as in Theorem 3.4. for all x ∈ X and all t > 0, where Φ Proof. By Theorems 3.4 and 3.6, there exist an additive mapping Ao : X → Y and a cubic mapping Co : X → Y such that ˜ 𝑥 ((2 − α)t) Λf (2𝑥)−8f (𝑥)−A0 (𝑥) (t) ≥ Φ (3.28) and ˜ 𝑥 ((23 − α)t) Λf (2𝑥)−2f (𝑥)−C0 (𝑥) (t) ≥ Φ

(3.29)

for all x ∈ X and t > 0. It follows from (3.28) and (3.29) that ( ) Λf (𝑥)+ 61 A0 (𝑥)− 16 C0 (𝑥) (t) ≥ T Λf (2𝑥)−8f (𝑥)−A0 (𝑥) (3t), Λf (2𝑥)−2f (𝑥)−C0 (𝑥) (3t) for all x ∈ X and t > 0. So we get (3.27) by letting A(x) = − 61 A0 (x) and C(x) = 61 C0 (x).



The main result of the paper is the following: Theorem 3.9. Let Φ : X × X −→ D+ be a mapping which, for some 0 < α < 2, satisfies (3.7) for all x, y ∈ X and all t > 0. Suppose that a mapping f : X → Y with f (0) = 0 satisfies (3.2) for all x, y ∈ X and all t > 0. Then there exist a cubic mapping C : X → Y , a quadratic mapping Q : X → Y and an additive mapping A : X → Y such that ( ( ( ) ( )) Λf (𝑥)−C(𝑥)−Q(𝑥)−A(𝑥) (t) ≥T T Φ0,𝑥 (k 2 − α)t , Φ0,𝑥 (k 2 − α)t , ( ( ( ) ( )) 3 3 3 ˜ ˜ Φ T T (2 − α)t , Φ𝑥 (2 − α)t , 𝑥 (3.30) 2 2 ( ( ) ( )))) ˜ 𝑥 3 (2 − α)t , Φ ˜ 𝑥 3 (23 − α)t T Φ 2 2 ˜ 𝑥 (t) is defined as in Theorem 3.4. for all x ∈ X and all t > 0, where Φ Proof. Let fe (x) = 12 (f (x) + f (−x)) for all x ∈ X. Then fe (0) = 0, fe (−x) = fe (x) and ΛDfe (𝑥,y) (t) = Λ Df (x,y)+Df (−x,−y) (t) ≥ T (ΛDf (𝑥,y) (t), ΛDf (−𝑥,−y) (t)) 2

≥ T (Φ𝑥,y (t), Φ−𝑥,−y (t)) = T (Φ𝑥,y (t), Φ𝑥,y (t)) for all x, y ∈ X and t > 0. By Theorem 3.2, there exists a quadratic mapping Q : X → Y such that Λfe (𝑥)−Q(𝑥) (t) ≥ T (Φ0,𝑥 (2(k 2 − α)t), Φ0,𝑥 (2(k 2 − α)t))

(3.31)

for all x ∈ X and all t > 0. On the other hand, let fo (x) = 21 (f (x) − f (−x)) for all x ∈ X. Then fo (0) = 0, fo (−x) = −fo (x). By Theorem 3.8, there exists an additive mapping A : X → Y and a cubic mapping C : X → Y such that ˜ 𝑥 (3(2 − α)t), Φ ˜ 𝑥 (3(23 − α)t)), T (Φ ˜ 𝑥 (3(2 − α)t), Φ ˜ 𝑥 (3(23 − α)t))) (3.32) Λfo (𝑥)−A(𝑥)−C(𝑥) (t) ≥ T (T (Φ for all x ∈ X and t > 0. Hence (3.30) follows from (3.31) and (3.32).



Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299). 619

Fixed points and the random stability of a functional equation

References [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989. [2] M. Adam and S. Czerwik, On the stability of the quadratic functional equation in topological spaces, Banach J. Math. Anal. 1 (2007), 245–251. [3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [4] R.P. Agarwal, Y. Cho, C. Park and R. Saadati, Random topological structure and functional analysis (preprint). [5] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951) 223–237. [6] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Mathematische Berichte 346 (2004), 43–52. [7] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Ineq. Pure Appl. Math. 4 (2003), Article ID 4, 7 pages. [8] Y.J. Cho, J.I. Kang and R. Saadati, Fixed points and stability of additive functional equations on the Banach algebras, J. Comput. Anal. Appl. 14(2012), 1103–1111. [9] Y.J. Cho, C. Park, Th.M. Rassias and R. Saadati, Inner product spaces and functional equations, J. Comput. Anal. Appl. 13(2011), 296–304. [10] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [11] M. Eshaghi Gordji and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Analysis.–TMA 71 (2009), 5629–5643. [12] G.L. Forti, An existence and stability theorem for a class of functional equations, Stochastica 4 (1980) 23–30. [13] G.L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), 143–190. [14] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [15] O. Hadˇzi´c and E. Pap, Fixed Point Theory in PM-Spaces, Kluwer Academic Publishers, Dordrecht, 2001. [16] O. Hadˇzi´c and E. Pap, New classes of probabilistic contractions and applications to random operators, in: Y. Cho, J. Kim, S. Kong (Eds.), Fixed Point Theory and Application, vol. 4, Nova Science Publishers, Hauppauge, New York, 2003, pp. 97–119. [17] O. Hadˇzi´c, E. Pap and V. Radu, Generalized contraction mapping principles in probabilistic metric spaces, Acta Math. Hungar. 101 (2003), 131–148. [18] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [19] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), 368–372. [20] K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 267–278. [21] K. Jun and H. Kim, Ulam stability problem for a mixed type of cubic and additive functional equation, Bull. Belg. Math. Soc.–Simon Stevin 13 (2006), 271–285. [22] H. Kim, On the stability problem for a mixed type of quartic and quadratic functional equation, J. Math. Anal. Appl. 324 (2006), 358–372. 620

M. Eshaghi Gordji, M. Ghanifard, H. Khodaei, C. Park [23] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on the generalized complete metric space, Bull. Amer. Math. Soc. 126 (1968), 305–309. [24] D. Mihet¸, The fixed point method for fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 160 (2009), 1663–1667. [25] D. Mihet¸, The Hyers-Ulam stability for two functional equations in a single variable, Banach J. Math. Anal. 2 (2008), 48–52. [26] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [27] D. Mihet¸ and V. Radu, Generalized pseudo-metrics and fixed points in probabilistic metric spaces, Carpathian J. Math. 23 (2007), 126–132. [28] D. Mihet¸, R. Saadati and S.M. Vaezpour, The stability of the quartic functional equation in random normed spaces, Acta Appl. Math. (in press). [29] D. Mihet¸, R. Saadati and S.M. Vaezpour, The stability of an additive functional equation in Menger probabilistic φ-normed spaces, Math. Slovak (in press). [30] A.K. Mirmostafaee, A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces, Fuzzy Sets and Systems 160 (2009), 1653–1662. [31] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [32] A. Najati and M.B. Moghimi, Stability of a functional equation deriving from quadratic and additive function in quasi-Banach spaces, J. Math. Anal. Appl. 337 (2008), 399–415. [33] A. Najati and Th.M. Rassias, Stability of a mixed functional equation in several variables on Banach modules, Nonlinear Analysis.–TMA (in press). [34] A. Najati and G. Zamani Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl. 342 (2008), 1318–1331. [35] C. Park, Fuzzy stability of a functional equation associated with inner product spaces, Fuzzy Sets and Systems 160 (2009), 1632–1642. [36] C. Park, Y.J. Cho and H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in non-Archimedean spaces, J. Comput. Anal. Appl. 14(2012), 526–535. [37] V. Radu, The fixed point alternative and the stability of functional equations, Sem. Fixed Point Theory 4 (2003), 91–96. [38] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [39] I.A. Rus, Principles and Applications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979. [40] B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, Series in Probability and Applied Mathematics, 1983, Second Ed. Dover Publications, 2005. ˇ [41] A.N. Serstnev, On the motion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280–283 (English translation in Soviet Math. Dokl. 4 (1963), 388–390). [42] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano. 53 (1983), 113–129. [43] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publishers, New York, 1960. [44] E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, SpringerVerlag, New York, 1986.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 622-632, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Weakly Set Valued Generalized Vector Variational Inequalities George A. Anastassiou Department of Mathematical Sciences The University of Memphis,Memphis,TN 38152,USA [email protected] Salahuddin Department of Mathematics Aligarh Muslim University, Aligarh-202002, India [email protected] Abstract In this paper we discuss the weak and strong solutions for setvalued generalized vector variational inequalities in real topological vector spaces. These two sets of solutions coincide whenever the mapping T is single valued but not setvalued. We use the Ferro Minimax Theorem to prove the existence of strong solutions for weakly setvalued generalized vector variational inequality problems. Keywords: Setvalued generalized vector variational inequalities, weak solutions, strong solutions, Ferro minimax theorem. Mathematics Subject Classification: 49J40.

1

Introduction

Vector Variational inequality theory conceived by Giannessi [7] has emerged as a powerful tool for a wide class of vector optimization problems and vector equilibrium problems [1]. In 2000, Chen and Hou [3] summarized representative existence results of solutions for vector variational inequalities and pointed out that the most of the existence results in this area touch upon a weak version of variational inequality and its generalizations. The existence of solutions for strong vector variational inequalities is still open problem, see [5, 8, 15, 16]. In recent years, much attention has been paid to generalized vector quasi equilibrium problems and generalized vector quasi-variational inequality problems see, [2, 9, 10, 13] with moving cone C(·) where C(·) is a set valued map between topological vector spaces X and Y such that for every x ∈ X, C(x) is a closed convex cone of Y. When dealing with such moving cone some authors have used the assumptions of the upper semicontinuity of C(·) and for W (·) = Y \intC(·). In fact, this assumption is a very strong one, at least for the case of Y being a normed space. This observation is clear from the remark [14] that for a setvalued map C : X → 2Y from a topological space X to a normed space Y such that for all x ∈ X, C(x) is a closed ( not necessarily convex) cone, the use of C at x ∈ X is equivalent to the fact that C(x) ⊂ C(x0 ) for x in some neighbourhood of x0 . Hence if we additionally assume that intC(x0 ) 6= ∅ and such that C(x) is convex for all x near x0 for all x0 , then use of both C and W at x0 means that C(x) = C(x0 ) for all x in

622

ANASTASSIOU, SALAHUDDIN: VECTOR VARIATIONAL INEQUALITIES

some neighbourhood of x0 . Let X, Y be arbitrary real Hausdorff topological vector spaces. Let L(X, Y ) denotes the space of all continuous linear mappings from X to Y . Let K be a nonempty set of X, C : K * Y a set valued mapping such that for each x ∈ K, C(x) is a proper closed convex pointed cone with apex at the origin and int C(x) 6= ∅. The mappings g : K → K, A : K × L(X, Y ) → L(X, Y ), T : K * 2L(X,Y ) and h : K × K → Y are given. For each x ∈ K, we define the relations ≤C(x) and 6≤C(x) as follows: (i) z ≤C(x) y ⇔ y − z ∈ C(x), (ii) z 6≤C(x) y ⇔ y − z 6∈ C(x). Similarly we can define the relations ≤intC(x) and 6≤int C(x) if we replace the set C(x) by intC(x). If the mapping C(x) is constant, then we write C(x) as C. We consider the following weakly setvalued generalized vector variational inequalities for finding x0 ∈ K for some s0 ∈ T (x0 ) such that hA(x0 , s0 ), y − g(x0 )i + h(g(x0 ), y) 6≤int

C(x0 )

0, ∀ y ∈ K.

(1.1)

We note that s0 depends on y, that is to find an x0 ∈ K with some s0 ∈ T (x0 ) such that hA(x0 , s0 ), y − g(x0 )i + h(g(x0 ), y) 6≤int

C(x0 )

0, ∀ y ∈ K.

(1.2)

We call this solutions a strong solution of the weakly setvalued generalized vector variational inequality problems. P Definition 1.1 [11] Let Ω be P a vector space, a topological vector space, K a nonempty convex subset of Ω, C : K → a set valued mapping such that for each x ∈ K, C(x) is a proper closed P convex pointed cone with apex at the origin and intC(x) 6= ∅. For any x ∈ K, ψ : K → is said to be (i) C(x)-convex iff ψ(tx1 + (1 − t)x2 ) ≤C(x) tψ(x1 ) + (1 − t)ψ(x2 ) for every x1 , x2 ∈ K and t ∈ [0, 1], (ii) properly quasi C(x)-convex iff we have either ψ(tx1 + (1 − t)x2 ) ≤C(x) ψ(x1 ) or ψ(tx1 + (1 − t)x2 ) ≤C(x) ψ(x2 ) for every x1 , x2 ∈ K and t ∈ [0, 1]. P Definition 1.2 [11] Let Ω be P a vector space, a topological vector space, K a nonempty convex subset of Ω, C : K → a set-valued mapping such that for each x ∈ K, C(x) is a proper closed convex P pointed cone with apex at the origin and intC(x) 6= ∅. Let A be a nonempty subset of , then for any fixed x ∈ K : (i) a point z ∈ A is called a minimal point of A with respect to the cone C(x) iff A ∩ (z − C(x)) = {z}; M inC(x) A is the set of all minimal point of A with respect to the cone C(x); 2 623

ANASTASSIOU, SALAHUDDIN: VECTOR VARIATIONAL INEQUALITIES

(ii) a point z ∈ A is called a maximal point of A with respect to the cone C(x) iff A ∩ (z + C(x)) = {z}; M axC(x) A is the set of all maximal points of A with respect to the cone C(x); (iii) a point z ∈ A is called a weakly minimal point of A with respect to the cone C(x) C(x) iff A ∩ (z − intC(x)) = ∅; M inw A is the set of all weakly minimal point of A with respect to the cone C(x); (iv) a point z ∈ A is called a weakly maximal point of A with respect to the cone C(x) iff C(x) A ∩ (z + intC(x)) = ∅; M axw A is the set of all weakly maximal point of A with respect to the cone C(x). Definition 1.3 Let X, Y be real topological vector spaces. The set valued mapping T : X → Y is a closed mapping iff the following holds: the net (xα ) → x0 , (yα ) → y0 , yα ∈ T (xα ) ⇒ y0 ∈ T (x0 ). Lemma 1.4 [4] Let K be a nonempty subset of a Hausdorff topological vector space X. Let G : K * X be a KKM mapping such that for any y ∈ K, G(y) is closed and G(y ∗ ) is compact for some y ∗ ∈ K. Then there exists x∗ ∈ K such that x∗ ∈ G(y) for all y ∈ K. Lemma 1.5 [12] Let X, Y, Z be real topological vector spaces, let K and C be two nonempty subsets of X and Y respectively. Let F : K × C → Z, S : K → Y be set valued mappings. If both F and S are upper semicontinuous with nonempty compact values, then the multivalued mapping T : K → Z defined by [

T (x) =

F (x, y) = F (x, S(x))

y∈S(x)

is upper semicontinuous with nonempty compact values.

2

Weak solutions for weakly setvalued generalized vector variational inequalities

Theorem 2.1 Let X, Y be the real Hausdorff topological vector spaces, K a nonempty closed convex subset of X, C : K * Y a set-valued mapping such that for each x ∈ K, C(x) is a proper closed convex pointed cone with apex at the origin and int C(x) 6= ∅. Given the mappings A : K × L(X, Y ) → L(X, Y ), h : K × K → Y, g : K → K, T : K * 2L(X,Y ) and υ : K × K → Y , suppose that (i) g is continuous and convex; (ii) h is convex with first variable and continuous in both variables; (iii) 0 ≤C(x) υ(x, x) for all x ∈ K; 3 624

ANASTASSIOU, SALAHUDDIN: VECTOR VARIATIONAL INEQUALITIES

(iv) for each x ∈ K, there is an s ∈ T (x) such that for all y ∈ K υ(x, y) − hA(x, s), y − g(x)i + h(g(x), y) ≤C(x) 0; (v) for each x ∈ K, the set {y ∈ K : 0 6≤C(x) υ(x, y)} is convex; (vi) there is a nonempty compact convex subset D of K such that for every x ∈ K\D, there is a y ∈ D such that for all s ∈ T (x) hA(x, s), y − g(x)i + h(g(x), y) ≤intC(x) 0; (vii) for each y ∈ K, the set {x ∈ K : hA(x, s), y − g(x)i + h(g(x), y) ≤intC(x) 0, for all s ∈ T x} is open in K. Then there exists x0 ∈ K, s0 ∈ T (x0 ) which is a solution of the weakly setvalued generalized vector variational inequalities. That is, there is x0 ∈ K, s0 ∈ T (x0 ) such that hA(x0 , s0 ), y − g(x0 )i + h(g(x0 ), y) 6≤intC(x0 ) 0, for all y ∈ K. Proof. Define the set valued mapping Ω : K * D by Ω(y) = {x ∈ D, s ∈ T (x) : hA(x, s), y − g(x)i + h(g(x), y) 6≤intC(x) 0, ∀y ∈ K}. From condition (vii), we have for each y ∈ K, the set Ω(y) is closed in K and hence it is compact in D because of the compactness of D. Next we claim that the Tthen the T family {Ω(y) : y ∈ K} has the finite intersection property Ω(y) is Ω(y) is nonempty and any element in the intersection whole intersection y∈K

y∈K

a solution of (1.1). For any given nonempty finite subset N of K, let DN = conv{D ∪ N } be the convex hull of D ∪ N then DN is a compact convex subset of K. Define the set valued mappings P, R : DN * DN respectively by P (y) = {x ∈ DN , s ∈ T (x) : hA(x, s), y − g(x)i + h(g(x), y) 6≤intC(x) 0}, R(y) = {x ∈ DN : 0 ≤C(x) υ(x, y)}, for each y ∈ DN . From the conditions (iii) and (iv), we have 0 ≤C(y) υ(y, y) for all y ∈ DN and for each y ∈ K, there is an s ∈ T (y) such that υ(y, y) − hA(y, s), y − g(y)i + h(g(y), y) ≤C(y) 0. 4 625

(3.1)

ANASTASSIOU, SALAHUDDIN: VECTOR VARIATIONAL INEQUALITIES

Hence 0 ≤C(y) hA(y, s), y − g(y)i + h(g(y), y) and then y ∈ P (y) for all y ∈ DN . We can easily see that P has closed values in DN . Since for each y ∈ DN , Ω(y) = P (y)∩D. If we prove that whole intersection of the family {P (y) : y ∈ DN } is nonempty, we can deduce that the family {Ω(y) : y ∈ K} has the finite intersection property because N ⊂ DN and due to the condition (vi). In order to deduce the conclusion of our theorem we can apply Lemma 1.1 if we claim that P is a KKM mapping. Indeed if P is not a KKM mapping neither is R since R(y) ⊂ P (y) for each y ∈ DN , then there is a nonempty finite subset M of DN such that [ conv M 6⊂ R(u). u∈M

Thus there is an element u0 ∈ conv M ⊂ DN such that u0 6∈ R(u) for all u ∈ M , that is 0 6≤C(u0 ) υ(u0 , u) for all u ∈ M. By (v) we have u0 ∈ convM ⊂ {y ∈ K : 0 6≤C(u0 ) υ(u0 , y)} and hence 0 6≤C(u0 ) υ(u0 , u0 ) which contradicts (3.1). Hence R is a KKM mapping and so is P . Therefore there exists an x0 ∈ K, s0 ∈ T (x0 ) which is a solution of problem (1.1). This completes the proof. Let condition (vii) be replaced by a stronger conditions as follows: Let the mappings A : K × L(X, Y ) → L(X, Y ), g : K → K and h : K × K → Y be continuous and T : K * L(X, Y ) be upper semicontinuous with nonempty compact values. Then from Lemma 1.2, we know that the condition (vii) of Theorem 2.1 is always true. Hence we have the following corollary. Corollary 2.2 Let X, Y be the real Hausdorff topological vector spaces, K a nonempty closed convex subset of X, C : K * Y a set valued mapping such that for each x ∈ K, C(x) is a proper closed convex pointed cone with apex at the origin and intC(x) 6= ∅. Let A : K × L(X, Y ) → L(X, Y ), h : K × K → Y and g : K → K be the continuous mappings. Let T : K * 2L(X,Y ) be the upper semicontinuous with nonempty compact values and υ : K × K → Y . Suppose that the condition (i)-(vi) of Theorem 2.1 holds. Then there exists x0 ∈ K, s0 ∈ T (x0 ) which is solutions of problem (1.1). That is, there is x0 ∈ K, s0 ∈ T (x0 ) such that hA(x0 , s0 ), y − g(x0 )) + h(g(x0 ), y) 6≤intC(x0 ) 0, for all y ∈ K. Theorem 2.3 Let X, Y, K, C, A, h, g, T be same as in Theorem 2.1. Assume that for each x ∈ K, such that (i) h is C(x)-convex in K with first variable; 5 626

ANASTASSIOU, SALAHUDDIN: VECTOR VARIATIONAL INEQUALITIES

(ii) for each x ∈ K there is s ∈ T (x) such that hA(x, s), x − g(x)i + h(g(x), x) 6≤intC(x) 0; (iii) there is a nonempty compact convex subset D of K such that for every x ∈ K\D there is y ∈ D such that for all s ∈ T (x), hA(x, s), y − g(x)i + h(g(x), y) ≤intC(x) 0; (iv) for each y ∈ K, the set {x ∈ K, s ∈ T (x) : hA(x, s), y − g(x)i + h(g(x), y) ≤intC(x) 0} is open in K. Then there is x0 ∈ K, s0 ∈ T (x0 ) which is weak solutions of problem (1.1). Proof. For any given nonempty finite subset N of K let DN = conv(D ∪ N ). Then DN is a nonempty compact convex subset of K. Define S : DN * DN as in the proof of the Theorem 2.1 and for each y ∈ K, let Ω(y) = {x ∈ D, s ∈ T (x) : hA(x, s), y − g(x)i + h(g(x), y) 6≤intC(x) 0}. We note that for each x ∈ DN , S(x) is nonempty and closed since x ∈ S(x) by conditions (i) and (iii). For each y ∈ K, Ω(y) is compact in D. Next we claim that S is a KKM mapping. Indeed if not there is a nonempty finite subset M of DN such that [ convM 6⊂ S(x). x∈M

Then there is an

x∗

∈ convM ⊂ DN such that

hA(x∗ , s), x − g(x∗ )i + h(g(x∗ ), x) ≤intC(x∗ ) 0, for all x ∈ M and s ∈ T (x∗ ). Since h is C(x∗ )-convex with first variable, the mapping x → hA(x∗ , s), x − g(x∗ )i + h(g(x∗ ), x), ∀s ∈ T (x∗ ) is C(x∗ )-convex on DN . Hence we can deduce that hA(x∗ , s), x∗ − g(x∗ )i + h(g(x∗ ), x∗ ) ≤intC(x∗ ) 0, for all s ∈ T (x∗ ). This contradicts the condition (iii). Therefore S is a KKM mapping by Lemma 1.1, we have \ S(x) 6= ∅. x∈DN

Note that for any u ∈

T

S(x), we have u ∈ D by condition (iv). Hence we have

x∈DN

\ y∈N

Ω(y) =

\

(S(y) ∩ D) 6= ∅

y∈N

for each nonempty finite subset N of K. Therefore the whole intersection T nonempty. Let x0 ∈ Ω(y). Then (x0 , s0 ) is a solution of problem (1.1). y∈K

6 627

T y∈K

Ω(y) is

ANASTASSIOU, SALAHUDDIN: VECTOR VARIATIONAL INEQUALITIES

3

Existence of Strong Solutions for Problem (1.1)

Theorem 3.1 Let X be a real topological vector space, Y, K, C, D, A, h, g and υ be the same as in Theorem 2.1. Under the assumptions of Theorem 2.1, we have a weak solution x0 of the problem (1.1) with s0 ∈ T (x0 ). In addition, if K is compact, x → Y \{−intC(x)} a closed mapping on K, T (x0 ) is convex, h is C(x0 )-convex and continuous on K, the mappings A : K × L(X, Y ) → L(X, Y ), g : K → K are continuous, T : K * 2L(X,Y ) is upper semicontinuous with nonempty compact values and the mapping s → −hA(x0 , s), x− g(x0 )i is properly quasi C(x0 )-convex on T (x0 ) for each x ∈ K. Assume that [ [ 0) M inC(x (L!! ) M axC(x0 ) {hA(x0 , s), x − g(x0 )i + h(g(x0 ), x)} ⊂ w x∈K

s∈T (x0 )

[

0) M inC(x w

{hA(x0 , s), x − g(x0 )i + h(g(x0 ), x)} + C(x0 ), ∀s ∈ T (x0 ).

x∈K

Assume (i) for any fixed x ∈ K, if δ ∈ M axC(x0 )

[

{hA(x0 , s), x − g(x0 )i + h(g(x0 ), x)}

s∈T (x0 )

and δ cannot be compared with hA(x0 , s0 ), x − g(x0 )i + h(g(x0 ), x) which does not equal to δ then δ 6≤intC(x0 ) 0. (ii) if M axC(x0 )

[

0) M inC(x w

[

{hA(x0 , s), x − g(x0 )ih(g(x0 ), x)} ⊂ Y \(−intC(x0 )),

x∈K

s∈T (x0 )

there exists s ∈ T (x0 ) such that [ 0) M inC(x {hA(x0 , s), x − g(x0 )i + h(g(x0 ), x)} ⊂ Y \(−intC(x0 )). w x∈K

Then x0 is a strong solution of the problem (1.1), that is there exists s0 ∈ T (x0 ) such that hA(x0 , s0 ), x − g(x0 )i + h(g(x0 ), x) 6≤intC(x0 ) 0, for all x ∈ K.

Furthermore the set of all strong solutions of problem (1.1) is compact. 7 628

ANASTASSIOU, SALAHUDDIN: VECTOR VARIATIONAL INEQUALITIES

Proof. Since h is C(x0 )-convex on K, the mapping x → hA(x0 , s), x − g(x0 )i + h(g(x0 ), x) is C(x0 )-convex on K. Since the mapping s → −hA(x0 , s), x − g(x0 )i is properly quasi C(x0 )-convex on T (x0 ) for each x ∈ K so is the mapping s → −hA(x0 , s), x − g(x0 )i + h(g(x0 ), x) for each x ∈ K. From Theorem 2.1, we know that x0 ∈ K such that (1.1) holds for all x ∈ K and for some s0 ∈ T (x0 ). Then [ [ {hA(x0 , s), x − g(x0 )i + h(g(x0 ), x)}, M axC(x0 ) ∀γ ∈ M inC(x0 ) x∈K

s∈T (x0 )

by (ii), we have γ 6≤intC(x0 ) 0. From condition (L!! ) and the convexity of T (x0 ), the Ferro Minimax Theorem [6] tells us for every [ [ 0) α ∈ MaxC(x0 ) MinC(x {hA(x0 , s0 ), x − g(x0 )i + h(g(x0 ), x)}, α 6≤intC(x0 ) 0. w x∈K

s∈T (x0 )

This implies that [ [ 0) MaxC(x0 ) MinC(x {hA(x0 , s0 ), x − g(x0 )i + h(g(x0 ), x)} ⊂ Y \(−intC(x0 )). w x∈K

s∈T (x0 )

From (ii) there is an s0 ∈ T (x0 ) such that [ 0) MinC(x {hA(¯ x, s¯), x − g(¯ x)i + h(g(¯ x), x)} ⊂ Y \(−intC(x0 )). w x∈K

Hence we know that ∀ρ ∈

[

{hA(x0 , s0 ), x − g(x0 )i + h(g(x0 ), x)},

x∈K

therefore ρ 6≤int

C(x0 )

0.

Hence there exists s0 ∈ T (x0 ) such that hA(x0 , s0 ), x − g(x0 )i + h(g(x0 ), x) 6≤intC(x0 ) 0 for all x ∈ K. Hence x0 is a strong solution of the problem (1.1). Finally to see that the solution set of the problem (1.1) is compact, it is sufficient to show 8 629

ANASTASSIOU, SALAHUDDIN: VECTOR VARIATIONAL INEQUALITIES

that the solution set is closed due to the coercivity condition (vi) of the Theorem 2.1. To this end let Γ denote the solution of the problem (1.1). Suppose that {xn } ⊂ Γ which converges to ρ. Fix any y ∈ K, for each n there is sn ∈ T (xn ) such that hA(xn , sn ), y − g(xn )i + h(g(xn ), y) 6≤intC(xn ) 0.

(3.1)

Since T is upper semicontinuous with nonempty compact values and the set {xn } ∪ {p} is compact, it follows that T ({xn } ∪ {p}) is compact. Therefore without loss of generality we may assume that the sequence {sn } converges to some s. Then s ∈ T (p) and h(g(xn ), y) − hA(xn , sn ), y − g(xn )i 6∈ intC(xn ). This implies that h(g(xn ), y) − hA(xn , sn ), y − g(xn )i ∈ Y \intC(xn ). We note that h(g(xn ), y) − hA(xn , sn ), y − g(xn )i = h(g(xn ), y) − hA(xn , sn ) − A(x, s), y − g(xn )i − hA(x, s), y − g(xn )i = h(g(xn ), y) − hA(xn , sn ) − A(x, s), y − g(xn )i − hA(x, s), (y − g(xn )) − (y − g(p))i − hA(x, s), y − g(p)i.

(3.2)

Since {xn } ∪ {p} is compact and g is continuous, g({xn } ∪ {p}) is also compact. Hence it is bounded. Thus hA(xn , sn ) − A(x, s), y − g(xn )i → 0 as n → ∞. hA(x, s), (y − g(xn )) − (y − g(p))i = hA(x, s), g(p) − g(xn )i → 0 as n → ∞ by the continuity of g. Since h is continuous and x → Y \(intC(x)) is a closed mapping on K, from (3.2) we have h(g(p), y) − hA(x, s), y − g(p)i = lim h(g(xn ), y) − hA(xn , sn ), y − g(xn )i ∈ Y \intC(p). x→∞

Then we obtain hA(x, s), y − g(p)i + h(g(p), y) 6≤intC(p) 0. Hence p ∈ Γ and Γ is closed. Corollary 3.2 Let X be a real Banach space, let Y, K, C, D, A, h, g and T be as in Theorem 2.1. Under the Assumption of Theorem 2.2, we have a weak solution x0 of the problem (1.1) with s0 ∈ T (x0 ). In addition, if K is compact, x → Y \(−intC(x)) a closed mapping on K, T (x0 ) is convex, the mapping s → −hA(x0 , s), x − g(x0 )i is properly quasi C(x0 )convex on T (x0 ) for each x ∈ K and h : K ×K → Y, A : K ×L(X, Y ) → L(X, Y ), g : K → K are continuous mappings, T : K * 2L(X,Y ) is upper semi continuous with nonempty compact values. Assume that the condition (L!! ) and the conditions (i)-(iv) in Theorem 3.1 hold, then x0 is a strong solution of the problem (1.1), that is there exists s ∈ T (x0 ) such that hA(x0 , s0 ), x − g(x0 )i + h(g(x0 ), y) 6≤intC(x0 ) 0, for all x ∈ K. Furthermore, the set of all strong solutions of the problem (1.1) is compact. 9 630

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References [1] E. Blum, W. Oettli, Form Optimization and Variational inequalities to Equilibrium problems, Math. Stud., 63 (1994), 123-145. [2] G. Y. Chen, X. Q. Yang and H. Xu, A nonlinear scalarization functions and generalized quasi vector equilibrium problem, J. Global Optim., 32 (2005), 451-466. [3] G. Y. Chen and S. H. Hou, Existence of solutions for vector variational inequalities, In F. Gianness (ed.) Vector variational inequalities and vector equilibria, Math. Theo. 73-86, Nonconvex Optimization and Applications, Vol. 38, Kluwer Academic, Dordrecht, 2000. [4] K. Fan, A generalization of Tychonoff fixed point theorems, Math. Ann., 142 (1961) 305-310. [5] Y. P. Fang, N. J. Huang, Strong vector variational inequalities in Banach spaces, Appl. Math. Lett., 19(2006), 362-368. [6] F. Ferro, A minimax theorem for vector valued functions, J. Optim. Theory Appl., 60 (1989) 19-31. [7] F. Giannessi, Theorem of alternative, quadratic program and complementarity problems: In R.W. Cottle, F. Giannessi, J.L. Lions (eds.) Variational Inequalities and Complementarity Problems, Wiley, New York, 1980, 151-186. [8] N. J. Huang and Y. P. Fang, On vector variational inequalities in reflexive Banach spaces, J. Global Optim., 32 (2005), 495-505. [9] S. L. Li, K. L. Toe and X. Q. Yang, Generalized vector quasi equilibrium problems, Math. Methods Oper. Res. 61, (2005), 385-397. [10] S. L. Li, K. L. Toe and X. Q. Yang, Gap functions and existence of solutions to generalized vector quasi equilibrium problems, J. Global Optim., 34 (2006), 427-440. [11] Y. C. Lin, On F-implicit generalized vector variational inequalities, J. Optim. Theory Appl., 142 (2009), 557-568. [12] L. J. Lin, Z. T. Yu, On some equilibrium problems for multimapping, J. Comput. Appl. Math., 129, (2001), 171-183. [13] K. L. Lin, D. P. Yang and J. C. Yao, Generalized vector variational Inequalities, J. Optim. Theo. Appl., 92 (1997), 117-125. [14] D. T. Luc and J. P. Penot, Convergence of asymptotic directions, Trans. Amer. Math. Soc., 353(2001), 4093-4121. [15] L. C. Zeng, J. C. Yao, Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces, J. Global Optim., 36 (2006), 483-497. 10 631

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[16] L. C. Zeng, Y. C. Lin, J. C. Yao, On weak and strong solutions of F-implicit generalized variational inequalities with applications, Appl. Math. Lett., 19 (2004), 684-689.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 633-646, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A new bivariate interpolation by rational triangular patch Qinghua Sun1 , Fangxun Bao1,∗, Yunfeng Zhang2 , Qi Duan1 1 2

School of Mathematics, Shandong University, Jinan, 250100, China

School Computer Sciences & Technology, Shandong Economic University Jinan, 250014, China

October 7, 2012

Abstract In this paper, a new approach is proposed to construct bivariate rational spline interpolation over triangulation, based on scattered data in parallel lines. The main advantage of this method have two points: (1) the interpolation function has a simple and explicit mathematical representation with some parameters α, β, µ and ν; (2) the shape of the interpolating surface can be modified by using the parameters for the unchanged interpolating data. Moreover, a local shape control method is employed to control the shape of surfaces. Keywords: Rational spline, bivariate interpolation, scattered data, triangular surface, shape constraint

1

Introduction

Spline interpolation is a useful and powerful tool in computer-aided geometric design (CAGD). In order to meet the needs of the ever-increasing model complexity and incorporate manufacturing requirements, many kinds of spline interpolation methods have been proposed, such as polynomial spline, triangular spline, β-spline, Box spline, vertex spline, etc [2, 4, 5, 6, 8, 14, 19, 22]. These methods are effective and applied widely in shape design of industrial products. However, one of the disadvantages of the spline method is that it can not be used to modify the local shape of the interpolating surfaces for unchanged interpolating data. Thus, constructing the interpolating function, which satisfies the following conditions, will be necessary in CAGD: (a) the interpolating functions achieve simple and explicit representations, so that these representations can be conveniently used for both practical application and theoretical analysis; ∗ Corresponding

author: [email protected]

633

Q. SUN, F. BAO, Y. ZHANG AND Q. DUAN

(b) the parameters of constructed curves and surfaces can be modified without changing the given data. In recent years, the study of univariate rational spline interpolation with parameters has received attention in the literature, and many results have been established [1, 7, 9, 10, 16, 17, 20]. Motivated by the univariate rational spline interpolation, the bivariate rational spline, which has simple and explicit mathematical representation with parameters, has been studied. Since the parameters in the interpolation function are selective according to the control need, the constrained control of the shape becomes possible. In [11, 12, 13], several bivariate spline interpolations have been constructed over rectangular mesh, and properties have been derived also. In [18], the preserving positivity of a rational bicubic spline interpolation with parameters over a rectangular grid is discussed. In [23], convexity control of a bivariate rational interpolating spline surfaces is studied. However, in many practical problems, the rectangular mesh is very difficult to be calculated, because only the scattered data can be obtained to achieve an interpolation. Hence, it is necessary to construct the bivariate interpolation function over the triangulation lattice. There are many publications contributing to the bivariate spline over triangulation. For example, in [2, 5, 15], the structure of bivariate spline spaces is investigated, and the important applications of Bernstein-B´ezier techniques in QρCAGD are discussed; In [3], the approximation order from the space S := k,∆ of piecewise polynomial functions is studied, etc. The all above bivariate spline over triangulation are in fact polynomial splines. Here we are concerned with bivariate rational spline interpolations with a simple and explicit mathematical representation, which can be modified by using parameters. In many fields, such as geological exploration, forging technology and medical, etc., it has been detected that the scattered data are usually arranged in parallel lines. For this kind of scattered data, it is easy to obtain their triangulation, as shown in Fig 1. In this paper, we deal with the construction problem of the bivariate spline interpolation over this kind of triangulation net. A new approach to construct bivariate rational spline interpolation over triangulation is proposed, whose constructed interpolation function comprises a simple and explicit mathematical representation with the new parameters α, β, µ and ν; Also, a local shape control method of interpolating surface is developed. This paper is organized as follows. In Section 2, a new bivariate rational spline interpolation with parameters is constructed over triangulation. In Section 3, the bounded property of the interpolation function is discussed. Sections 4 is about the error estimates of the interpolation function. Section 5 deals with the shape control method of the interpolating surface, and numerical examples are presented to show the performance of the method.

2

Interpolation

Let {(xi , yi , fi , di , ei ), i = 1, 2, · · · , n} be the given scattered data arranged in i ,yi ) , and ei = parallel lines: L1 , L2 , · · · , Lm , where fi = f (xi , yi ), di = ∂f (x ∂x

634

A NEW BIVARIATE INTERPOLATION ∂f (xi ,yi ) ∂y

(see Fig. 1).

Figure 1: Triangulation of interpolating region Ω. For a triangle T1 = △V1 V2 V3 ,with vertices {Vi = (xi , yi ), i = 1, 2, 3} and y1 = y2 , let γ11 = ∠V3 V1 V2 , and γ12 is the angle between line V2 V3 and the extension line of V1 V2 (see Fig. 2).

Figure 2: An element Ω∗ of subdivision. Denote h = x2 − x1 , l = y3 − y1 . For any point Q in the line V1 V2 , let τ1 = ∠V3 QV2 , thus, V1 Q = x3 − x1 − l cot τ1 , and for any point V (x, y) in the x3 −x1 −l cot τ1 1 3 , and η = y−y line V3 Q, cot τ1 = x−x y−y3 . Let θ = h l . A rational cubic interpolation function is defined over the interval [x1 , x2 ] as [10]: p(x) =

pT1 (x) , qT1 (x)

where pT1 (x) = (1 − θ)3 αf1 + θ(1 − θ)2 V + θ2 (1 − θ)W + θ3 f2 β, qT1 (x) = (1 − θ)2 α + 2θ(1 − θ) + θ2 β, with V = (α + 2)f1 + αhd1 , W = (β + 2)f2 − βhd2 ,

635

(1)

Q. SUN, F. BAO, Y. ZHANG AND Q. DUAN

and α > 0, β > 0. Obviously, the interpolation function p(x) on [x1 , x2 ] is unique for the given data (xi , fi , di ), i = 1, 2 and the parameter α, and which satisfies p(xi ) = fi , p′ (xi ) = di ; i = 1, 2. Using the x-direction interpolation function p(x), we define the bivariate rational interpolation function P (x, y) on the triangle domain T1 as follows: PT1 (x, y) =

pT1 (x, y) , qT1 (y)

(2)

where pT1 (x, y) = (1 − η)3 µp(x) + η(1 − η)2 VT1 + η 2 (1 − η)WT1 + νη 3 UT1 , qT1 (y) = (1 − η)2 µ + 2η(1 − η) + η 2 ν, with VT1 = (µ + 2)p(x) + µ((1 − θ)(le1 + d1 (x3 − x1 )) + θ(le2 + d2 (x3 − x2 )), WT1 = d3 (x3 − x1 − hθ)(η − 1)ν + (ν + 2)f3 − lνe3 , UT1 = (d3 (x3 − x1 − hθ)(η − 1) + f3 )ν. and µ > 0, ν > 0. Define PT1 (x3 , y3 ) = f3 ,

∂PT1 (x3 , y3 ) ∂PT1 (x3 , y3 ) = d3 , = e3 , ∂x ∂y

then the interpolation function PT1 (x, y) defined by (2) satisfies PT1 (xi , yi ) = fi ,

∂PT1 (xi , yi ) ∂PT1 (xi , yi ) = di , = ei ; i = 1, 2, 3. ∂x ∂y

When α = 0, β = 0, µ = 0 and ν = 0, the interpolation function PT1 defined by (2) becomes PT1 (x, y) = (1 − θ)(1 − η)f1 + θ(1 − η)f2 + ηf3 .

(3)

It is obviously that Eq. (3) is an equation of the plane determined by three points (x1 , y1 , f1 ), (x2 , y2 , f2 ) and (x3 , y3 , f3 ) Let T2 , T3 and T4 be the triangular domains which have S common S edges S V1 V3 , V1 V2 and V2 V3 with T1 respectively. Denote that Ω∗ = T1 T2 T3 T4 , then the subregion Ω∗ of interpolating region Ω is called element of subdivision (see Fig. 2). Similar to (2), we can define the interpolation functions over T2 , T3 and T4 , and it is easy to obtained the following theorem. Theorem 1. For the given set of interpolating data {(xi , yi , fi , di , ei ), i = 1, 2, 3, · · · , n}, let P (x, y) be the interpolation function defined by (2) on the triangulation of interpolating region Ω. Let Lk and Lk+1 are two adjacent parallel

636

A NEW BIVARIATE INTERPOLATION (k,k+1)

(k,k+1)

(k,k+1)

(k,k+1)

lines, denoting αTi , βTi and µTi , νTi are the positive parameters in PTi (x, y), where PTi (x, y) is the interpolation function on the triangular domain Ti which is between Lk and Lk+1 , then a sufficient condition for the function P (x, y) to be continuous in the whole interpolating region Ω is that the (k,k+1) (k,k+1) positive parameters µTi are equal to constant and νTi are equal to con(k,k+1)

(k,k+1)

stant for all i = 1, 2, · · · , no matter what the parameters αTi and βTi might be. 2 ) , (x, y) ∈ R2 . Example 1. Let the interpolated function f (x, y) = cos3 π(x+y 4 Note that the interpolation function defined S by S (2) is local, we only consider an element of subdivision, which is T = T1 T2 T3 , where T1 , T2 and T3 are the triangular domains, T1 with vertexes V1 (0.5, 0.5), V2 (0.65, 0.5) and V3 (0.6, 0.7), T2 with vertexes V1 (0.5, 0.5), V3 (0.6, 0.7) and V4 (0.45, 0.7), T3 with vertexes V1 (0.5, 0.5), V2 (0.65, 0.5) and V5 (0.6, 0.3). Denoting the interpolation function defined by (2) in the domain T by P (x, y). Figure 3 and Figure 4 show the graphs of the surfaces f (x, y) and P (x, y) respectively, Figure 5 shows the surface of the error f (x, y) − P (x, y).

0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2

0.45 0.3

0.4

0.5 0.5

0.55 0.6

0.6 0.7

0.65

Figure 3: Graph of the surface f (x, y).

3

Properties of the interpolation

Note that the interpolation defined by (2) is local, without loss of generality, in the following, we only consider its properties in a triangular domain T1 . Denoting m1 = m3 = x3 − x1 , m2 = x3 − x2 . From (1) and (2), the interpolation function PT1 (x, y) defined by (2) can be written as follows: 3 X PT1 (x, y) = [ai (θ, η)fi + bi (θ, η)hdi + ci (θ, η)(mi di + lei )], (4) i=1

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Q. SUN, F. BAO, Y. ZHANG AND Q. DUAN

0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2

0.45 0.3

0.4

0.5 0.55

0.5

0.6

0.6

0.7

0.65

Figure 4: Graph of the surface P (x, y).

−3

x 10 2 1.5 1 0.5 0 −0.5 −1 −1.5 0.7

0.6

0.65 0.5

0.6 0.4

0.55 0.3

0.5 0.2

0.45

Figure 5: Graph of the surface f (x, y) − P (x, y). where (1 − θ)2 (2θ + α)(1 − η)2 (2η + µ) , qT1 (x) · qT1 (y) θ2 (2(1 − θ) + β)(1 − η)2 (2η + µ) a2 (θ, η) = , qT1 (x) · qT1 (y) η 2 (2(1 − η) + ν) a3 (θ, η) = , qT1 (y) θ(1 − θ)2 (1 − η)2 (2η + µ)α b1 (θ, η) = , qT1 (x) · qT1 (y) θ2 (1 − θ)(1 − η)2 (2η + µ)β b2 (θ, η) = − , qT1 (x) · qT1 (y) θη 2 (1 − η)ν η(1 − η)2 (1 − θ)µ b3 (θ, η) = , c1 (θ, η) = , qT1 (y) qT1 (y) θη(1 − η)2 µ η 2 (1 − η)ν c2 (θ, η) = , c3 (θ, η) = − . qT1 (y) qT1 (y) a1 (θ, η) =

638

A NEW BIVARIATE INTERPOLATION

Terms ai (θ, η), bi (θ, η), ci (θ, η)(i = 1, 2, 3) are called the basis of the bivariate interpolation function defined by (2), and which satisfy the following equation: a1 (θ, η) + a2 (θ, η) + a3 (θ, η) = 1,

(5)

For the interpolation function PT1 (x, y) defined by (2), there is the following unity property. Theorem 2. If f (x, y) = 1, (x, y) ∈ Ω, PT1 (x, y) is its interpolation function on T1 defined by (2), no matter what positive numbers the parameters α, β take, the unity property holds, namely Z Z 1 PT1 (x, y)dxdy = hl. 2 T1 Denoting M1 = max{|fi |, i = 1, 2, 3},

(6)

M2 = max{h|di |, |mi di |, i = 1, 2, 3}.

(7)

M3 = max{l|ei |, i = 1, 2, 3},

(8)

For the given interpolating data, the values of the rational bivariate interpolation function PT1 (x, y) defined by (2) are bounded in the triangular domain T1 as described by the following property. Theorem 3. For the given data {(xi , yi , fi , ei , di ), i = 1, 2, 3}, let PT1 (x, y) be the interpolation function on the triangular domain T1 defined by (2). Whatever the positive values of the parameters α and β might be, the values of PT1 (x, y) in the domain T1 satisfy |PT1 (x, y)| ≤ M1 + 2M2 + M3 , where M1 , M2 and M3 are defined by (6),(8) and (7) respectively. Proof: From (4) and Eq. (5), it is easy to derive that |PT1 (x, y)| ≤ [a1 (θ, η) + a2 (θ, η) + a3 (θ, η)]M1 + [b1 (θ, η) − b2 (θ, η) + b3 (θ, η)]M2 +[c1 (θ, η) + c2 (θ, η) − c3 (θ, η)](M2 + M3 ) η(1 − η)((1 − η)µ + ην)(M2 + M3 ) (1 − η)2 (2η + µ) + η 2 (1 − η)ν M2 + ≤ M1 + (1 − η)2 µ + 2η(1 − η) + η 2 ν (1 − η)2 µ + 2η(1 − η) + η 2 ν ≤ M1 + 2M2 + M3 .

2

The proof is completed .

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Q. SUN, F. BAO, Y. ZHANG AND Q. DUAN

4

Error estimates of the interpolation

For the error estimation of the bivariate interpolation function defined by (2), without loss of generality, it is only necessary to consider the triangular domain T1 . We assume that f (x, y) ∈ C 2 , and PT1 (x, y) is the interpolation function defined by (2) over T1 . Denoting k

∂f (x, y) ∂P ∂PT1 (x, y) ∂f k = max | |, k k = max | |. ∂y (x,y)∈T1 ∂y ∂y (x,y)∈T1 ∂y

Using the Taylor expansion and the Peano-Kernel Theorem [21], it is can be obtained that |f (x, y) − PT1 (x, y)| ≤ |f (x, y) − f (x, y1 )| + |PT1 (x, y1 ) − PT1 (x, y)| + |f (x, y1 ) − PT1 (x, y1 )| Z x2 2 ∂f ∂P ∂ f (τ, y1 ) ≤ l(k k + k k) + | Rx [(x − τ )+ ]dτ | ∂y ∂y ∂x2 x1 Z x2 ∂ 2 f (x, y1 ) ∂f ∂P ≤k k |Rx [(x − τ )+ ]|dτ + l(k k + k k), ∂x2 ∂y ∂y x1 where k ∂

2

f (x,y1 ) k ∂x2

=

max | ∂

2

x∈[x1 ,x2 ]

f (x,y1 ) |, ∂x2



with

r(τ ), t(τ ),

Rx [(x − τ )+ ] =

x1 < τ < x; x < τ < x2 ,

and r(τ ) = x − τ − a2 (θ, 0)(x2 − τ ) − b2 (θ, 0)h, x1 < τ < x; t(τ ) = −a2 (θ, 0)(x2 − τ ) − b2 (θ, 0)h, x < τ < x2 . Thus, by simple integral calculation, it can be derived that Z x2 |Rx [(x − τ )+ ]|dτ = h2 U (θ, α), x1

where U (θ, α) =

u(θ, α) v(θ, α)

(9)

with u(θ, α) = θ2 (1 − θ)2 [(4(1 − θ)2 + β 2 )(2θ + α) + (4θ2 + α2 )(2 − 2θ + β)], v(θ, α) = 2((1 − θ)2 α + 2θ(1 − θ) + θ2 β)(2θ + α)(2 − 2θ + β). For the fixed α, let (x)

BT1 = max U (θ, α). θ∈[0,1]

640

(10)

A NEW BIVARIATE INTERPOLATION

Then the following theorem can be obtained. Theorem 4. Let f (x, y) ∈ C 2 , and PT1 (x, y) be its interpolation function defined by (2) on the triangular domain T1 . Whatever the positive values of the parameters α and β might be, the error of the interpolation function satisfies |f (x, y) − PT1 (x, y)| ≤ l(k

∂f ∂P ∂ 2 f (x, y1 ) (x) k+k k) + h2 k kBT1 , ∂y ∂y ∂x2

(x)

where BT1 and U (θ, α) are defined by (9) and (10) respectively.

5

Shape control of the interpolating surface

The shape of the interpolating surface on the interpolating region depends on the interpolating data. Generally speaking, because of the uniqueness of the interpolation function, the shape of interpolating surface is fixed for the unchanged interpolating data. However, for the bivariate interpolation function defined by (2), since there are some parameters, the shape of the interpolating surface can be modified by selecting suitable parameters. In the follows, we consider a local shape control method of the piecewise bivariate interpolation function defined by (2). Let Ω be the interpolating region, T1 ∈ Ω is a triangular domain, and let PT1 (x, y) be the interpolation function on T1 defined by (2). If the interpolating surface on the part of T1 protrude at some points, for example, the interpolating surface is too high or too low at the point (x∗, y ∗ ), where (x∗ , y ∗ ) ∈ T1 \∂T1 , the value of PT1 (x∗ , y ∗ ) can be constrained to be down or up by selecting suitable parameters as described by the following theorem. Theorem 5. For the given data {(xi , yi , fi , ei , di ), i = 1, 2, 3}, let PT1 (x, y) be the interpolation function on the triangular domain T1 defined by (2), and let fi < M , where M is a real number, then for (x∗ , y ∗ ) ∈ T1 \∂T1, there must exist the positive parameters α, β, µ and ν, such that PT1 (x∗ , y ∗ ) < M . Proof: Let (θ, η) be the local coordinate of the point (x∗ , y ∗ ), then θ=

y ∗ − y1 (x3 − x1 )(y ∗ − y3 ) − l(x∗ − x3 ) , η = . h(y ∗ − y3 ) l

Since (x∗ , y ∗ ) ∈ T1 \∂T1, θ ∈ (0, 1) and η ∈ (0, 1). Denoting Ai = fi − M, Bi = hdi , Ci = mi di + lei . From (4) and Eq. (5), it is easy to show that PT1 (x∗ , y ∗ ) < M is equivalent to 3 X

[ai (θ, η)Ai + bi (θ, η)Bi + ci (θ, η)Ci ] < 0.

i=1

641

(11)

Q. SUN, F. BAO, Y. ZHANG AND Q. DUAN

Thus, if there exist the positive parameters α, β, µ and ν to satisfy the following inequalities (12)-(14), then (11) holds. φ1 (θ, η) = a1 (θ, η)A1 + b1 (θ, η)B1 < 0,

(12)

φ2 (θ, η) = a2 (θ, η)A2 + b2 (θ, η)B2 < 0,

(13)

φ3 (θ, η) = a3 (θ, η)A3 + b3 (θ, η)B3 +

P3

i=1 ci (θ, η)Ci

< 0.

(14)

It is easy to derive that φ1 (θ, η) =

(1 − θ)2 (1 − η)2 (2η + µ) ϕ1 (θ, η), qT1 (x)qT1 (y)

where ϕ1 (θ, η) = 2θA1 + α(A1 + θB1 ). Thus, the inequality (12) is equivalent to ϕ1 (θ, η) < 0. Since A1 < 0, no matter what (A1 + θB1 ) can be, there exists positive parameter α such that (12) holds. Similarly, φ2 (θ, η) =

θ2 (1 − η)2 (2η + µ) ϕ2 (θ, η), qT1 (x)qT1 (y)

where ϕ2 (θ, η) = 2(1 − θ)A2 + β(A2 − (1 − θ)B2 ). Since A2 < 0, no matter what (A2 − (1 − θ)B2 ) can be, there exists positive parameter β such that (13) holds. φ3 (θ, η) =

η ϕ3 (θ, η), qT1 (x)qT1 (y)

where ϕ3 (θ, η)

= 2η(1 − η)A3 + µ(1 − η)2 ((1 − θ)C1 + θC2 ) +νη(A3 + θ(1 − η)B3 − (1 − η)C3 ).

Since A3 < 0, it is easy to show that no matter what ((1 − θ)(1 − η)2 C1 + θ(1 − η)2 C2 ) and η(A3 + θ(1 − η)B3 − (1 − η)C3 ) can be, there exist positive parameters µ and ν such that (14) holds. Which completes the proof. 2 Theorem 5 is called the up-constrained theorem. In the similar way, we have the down-constrained theorem as follows. Theorem 6. For the given data {(xi , yi , fi , ei , di ), i = 1, 2, 3}, let PT1 (x, y) be the interpolation function on the triangular domain T1 defined by (2), and let

642

A NEW BIVARIATE INTERPOLATION

fi > N , where N is a real number, then for (x∗ , y ∗ ) ∈ T1 \∂T1 , there must exist the positive parameters α, β, µ and ν, such that PT1 (x∗ , y ∗ ) > N . Example 2. Note that the interpolation function defined by (2) is local, we consider a triangular domain. Given the interpolating data shown in Table 2, let P (x, y) be the interpolation function defined by (2), it is possible to compute directly its expression in the interpolating region T , where T is a triangular domain with vertices V1 (0.5, 0.5), V2 (1.0, 0.5) and V3 (0.8, 1.0). Table 1: Interpolating data (x, y) (0.5, 0.5) (1.0, 0.5) (0.8, 1.0)

f (x, y) 2 2.5 3.5

∂f (xi ,yi ) ∂x

∂f (xi ,yi ) ∂y

1.5 −1 −0.5

0.5 −0.5 −2

In the general case, one can select any positive real numbers as the values of α, β, µ and ν. Now, we take α = 0.4, β = 0.8, µ = 0.6 and ν = 0.5, and denote the interpolation function by P1 (x, y). Fig. 6 shows the graph of the surface P1 (x, y). It is easy to compute that P1 (0.8, 0.95) = 3.52817, and the local coordinate of the point (0.8, 0.95) is that (θ, η) = (0.6, 0.9).

3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1

1 0.9

0.9 0.8

0.8 0.7

0.7 0.6

0.6 0.5

0.5

Figure 6: Graph of the surface P1 (x, y). Let M = 3.525, for the given interpolating data, inequalities (12)-(14) can be simplified as −1.075α − 1.83 < 0, −0.825β − 0.82 < 0, −0.0045 + 0.0025µ + 0.0675ν < 0. Obviously, α = 0.6, β = 0.5, µ = 0.2 and ν = 0.058 satisfy these inequalities.

643

Q. SUN, F. BAO, Y. ZHANG AND Q. DUAN

We denote the interpolation function by P2 (x, y), Fig. 7 shows the graph of the surface P2 (x, y). From Fig. 6, it can be seen that the values of P1 (x, y) at the point (0.8, 0.95) and in its neighbourhood are constrained to be less than 3.525.

3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1

1 0.9

0.9 0.8

0.8 0.7

0.7 0.6

0.6 0.5

0.5

Figure 7: Graph of the surface P2 (x, y).

6

Conclusions

In this paper, a new approach is proposed to construct bivariate rational spline interpolation over triangulation, based on the scattered data in parallel lines. The interpolation function has a simple and explicit mathematical expression with some free parameters α, β, µ and ν, and the shape of the interpolating surface can be modified by selecting suitable parameter for the unchanged interpolating data according to the control need. This kind of bivariate rational interpolation can not be found in literature. For each pitch of the interpolating surface, the value of the interpolation function depends on the interpolating data. Theorem 3 shows that the interpolation is stable. It is difficult to derive the error estimate formula of the bivariate interpolation function over triangulation. In this paper, it is worked out in Theorem 4. This is because that the interpolation function has the convenient basis functions: ai (θ, η), bi (θ, η), ci (θ, η), i = 1, 2, 3. Theorem 5 and 6 give the sufficient conditions for the shape control of bivariate rational interpolating surfaces, and the method selecting the suitable parameters is given in the proving process of Theorem.

Acknowledgment This research was supported by the National Nature Science Foundation of China (grant no.61070096).

644

A NEW BIVARIATE INTERPOLATION

References [1] F. Bao, Q. Sun and Q. Duan, Point control of the interpolating curve with a rational cubic spline, J. Vis. Commun. Image R. vol.20, no.4, pp.275-280, 2009. [2] C. de Boor, A practical guide to splines, revised ed. New York, Berlin, Heidelberg: Springer, 2001. [3] C. de Boor and R.Q. Jia, A sharp upper bound on the approximation order of smooth bivariate pp function, J. Approx. Theory, vol.72, no.1, pp.24-33, 1993. [4] P.E. B´ezier, The Mathematical Basis of The UNISURF CAD System, Butterworth, London, 1986. [5] C.K. Chui, Multivariate Splines, SIAM, 1988. [6] P. Comninos, An interpolating piecewise bicubic surface with shape parameters, Computers & Graphics, vol.25, no.3, pp.463-481, 2001. [7] R. Delbourgo, Shape preserving interpolation to convex data by rational functions with quadratic numerator and linear denominator, IMA J. Numerical Analysis, vol.9, no.1, pp.123-136, 1989. [8] P. Dierck and B. Tytgat, Generating the B´ezier points of BETA-spline curve, Computer-Aided Geometric Design, vol.6, no.4, pp.279-291, 1989. [9] Q. Duan, F. Bao, S. Du and E.H. Twizell, Local control of interpolating rational cubic spline curves, Computer-Aided Design, vol.41, no.11, pp.825829, 2009. [10] Q. Duan, K. Djidjeli, W.G. Price and E.H. Twizell, The approximation properties of some rational cubic splines, Int. J. Computer Mathematics, vol.72, no.2, 155-166, 1999. [11] Q. Duan, S. Li, F. Bao and E.H. Twizell, Hermite interpolation by piecewise rational surface, Applied Mathematics and Computation, vol.198, no.1, pp.59-72, 2008. [12] Q. Duan, Y. Zhang and E.H. Twizell, A bivariate rational interpoaltion and the properties, Applied Mathematics and Computation, vol.179, no.1, pp.190-199, 2006. [13] Q. Duan, H. Zhang, A. Liu and H. Li, A bivariate rational interpolation with a bi-quadratic denominator, J. Computational and Applied Mathematics, vol.195, no.1-2, pp.24-33, 2006. [14] G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide (fourth ed.), Academic press, New York, 1997.

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[15] G. Farin, Triangular Berstein-B´ezier patches, Computer-Aided Geometric Design, vol.3, no.2, pp.83-127, 1986. [16] J.A. Gregory, M. Sarfraz and P.K. Yuen, Interactive curve design using C 2 rational splines, Computers & Graphics, vol.18, no.2, pp.153-159, 1994. [17] X. Han, Convexity preserving piecewise rational quartic interpolation, SIAM J. Numerical Analysis, vol.46, no.2, pp.920-929, 2008. [18] M.Z. Hussain and M. Sarfraz, Positivity-preserving interpolation of positive data by rational cubics, J. Computational and Applied Mathematics, vol.218, no.2, pp.446-458, 2008. [19] R. M¨ uller, Universal parametrization and interpolation on cubic surfaces, Computer-Aided Geometric Design, vol.19, no.7, pp.479-502, 2002. [20] M. Sarfraz, A C 2 rational cubic spline which has linear denominator and shape control, Ann. Univ. Sci. Budapest, vol.37, no.1, pp.53-62, 1994. [21] M.H. Schultz,Spline Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1973. [22] R. Wang, Multivariate Spline Functions and Their Applications, Kluwer Academic Publishers, Beijing/New York/Dordrecht/Boston/London, 2001. [23] Y. Zhang, Q. Duan and E.H. Twizell, Convexity control of a bivariate rational interpolating spline surfaces, Computers & Graphics, vol.31, no.5, pp.679-687, 2007.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 647-654, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A NOTE ON THE MODIFIED CARLITZ’S q-BERNOULLI NUMBERS AND POLYNOMIALS JIN-WOO PARK1 , DMITRY V. DOLGY2 , TAEKYUN KIM3 , SANG-HUN LEE4 , AND SEOG-HOON RIM5

Abstract. In [1], Carlitz introduced q-extension of Bernoulli numbers and polynomials and Kim gave the Witt’s formula for the Carlitz’s q-Bernoulli numbers and polynomials(see [7, 8]). In this paper we consider the modified q-Bernoulli numbers and polynomials which are slightly different Carlitz’s qBernoulli numbers and polynomials.

1. Introduction Let p be a fixed prime number. Throughout this paper Zp , Qp and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp . Let vp be the normalized exponential valuation with |p|p = p−vp (p) = p1 . When we talk of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp . If q ∈ C, we assume |q| < 1. If 1 q ∈ Cp , we assume |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. We use x the notation [x]q = [x : q] = 1−q 1−q . Thus, we note that limq→1 [x]q = x. In complex case [1], Carlitz defined a set of numbers ηk = ηk (q) induced by η0 = 1, (qη + 1)k − ηk =



1 if k = 1, 0 if k > 1,

with the usual convention of replacing η k by ηk . These numbers are q-analogue of ordinary Bernoulli number Bk , but they do not remain finite for q = 1. So, Carlitz reconstructed the sequences as follows: β˜0,q = 1, and q(q β˜q + 1)k − β˜k,q =



1 if k = 1, 0 if k > 1,

(1.1)

with the usual convention about replacing (β˜q )n by β˜n,q (see [1]). The sequence β˜n,q are aclled the n-th Carlitz’s q-Bernoulli polynomials. He also defined the qBernoulli polynomials as follows: β˜n,q (x) =

n   X n l=0

l

β˜l,q q lx [x]n−l , (n ∈ Z+ = N ∪ {0}). q

647

(1.2)

J. W. PARK, D. V. DOLGY, T. KIM, S. H. LEE, AND S. H. RIM

Let U D(Zp ) be the space of uniformly differentiable functions on Zp . For f ∈ U D(Zp ), the p-adic invariant integral on Zp is defined by N pX −1

Z I0 (f ) =

f (x)dµ0 (x) = lim

N →∞

Zp

1 = lim N N →∞ p

N pX −1

f (x)µ0 (x + pN Zp )

x=0

(1.3)

f (x) (see [7, 8]).

x=0

The q-analogue of p-adic invariant integral on Zp is defined by Kim as follows: N pX −1

Z Iq (f ) =

f (x)dµq (x) = lim

N →∞

Zp

= lim

N →∞

1 [pN ]q

N pX −1

f (x)µq (x + pN Zp )

x=0

(1.4)

x

f (x)q (see [7, 8]).

x=0

In [8], it is known that dµq (x) =

q−1 x q dµ0 (x). logq

(1.5)

By (1.3), (1.4) and (1.5), we get Z Z q−1 q x f (x)dµ0 (x). f (x)dµq (x) = logq Zp Zp

(1.6)

Thus, by (1.6), we get Z Z logq x q f (x)dµ0 (x) = f (x)dµq (x) (see [8]). q − 1 Zp Zp Now, we redefine I(f ) from I0 (f ) as follows: N

p −1 1 X I(f ) = q f (x)dµ0 (x) = lim N f (x)q x . N →∞ p Zq x=0

Z

x

(1.7)

In this paper, we consider the modified Carlitz’s q-Bernoulli numbers and polynomials which are slightly different Carlitz q-Bernoulli numbers and polynomials. These numbers and properties are derived from (1.7). 2. Modified Carlitz’s q-Bernoulli numbers From (1.7), we note that Z qI(f1 ) = q x+1 f (x + 1)dµ0 (x)

(2.1)

Zp 0

= (logq)f (0) + f (0), where f1 (x) = f (x + 1). By (2.1), we get q 2 I(f2 ) = q 2

Z

q x f (x + 2)dµ0 (x) = q q

Zp

Z =

Z

! f1 (x + 1)q x dµ0 (x)

Zp

q x f (x)dµ0 (x) + logq{qf (1) + f (0)} + {f 0 (0) + qf 0 (1)},

Zp

648

A NOTE ON THE MODIFIED CARLITZ’S q-BERNOULLI NUMBERS AND POLYNOMIALS

where f2 (x) = f1 (x + 1) = f (x + 2). Continuing this process, we obtain the following theorem. Theorem 2.1. For n ∈ Z+ , we have Z Z n−1 n−1 X X n x q fn (x)q dµ0 (x) = q x f (x)dµ0 (x) + logq q l f (l) + q l f 0 (l), Zp

Zp

l=0

l=0

where fn (x) = f (x + n). Now, we consider the following sequence which can be derived by the p-adic integral on Zp of (1.7): Z βn,q = q y [x + y]nq dµ0 (y) (n ∈ Z+ ). (2.2) Zp

Here, βn,q (x) are called the n-th q-Bernoulli polynomials which are slightly different Carlitz’s q-Bernoulli polynomials. In the special case, x = 0, βn,q = βn,q (0) are called the n-th q-Bernoulli numbers which are slightly different Carlitz’s q-Bernoulli numbers. From (2.2), we can derive the following equation: Z n   X n 1 l lx (−1) q q (l+1)y dµ0 (y) βn,q (x) = (1 − q)n l Z p l=0 n   X logq n l+1 = (−1)l−1 q lx (2.3) (1 − q)n l 1 − q l+1 l=0 n   X n l+1 logq (−1)l−1 q lx = , n+1 l (1 − q) [l + 1]q l=0

and βn,q (x) =

n   X n

l

l=0

[x]n−l q lx βl,q . q

(2.4)

By Theorem 2.1, we get Z Z n x q [x + 1]q q dµ0 (x) − [x]nq q x dµ0 (x) Zp

Zp n

= (logq)[0] +



n[0]n−1 q

logq q−1

(2.5)

 .

Therefore, by (2.2) and (2.5), we obtain the following recurrence relation: Theorem 2.2. For n ∈ Z+ , we have qβn,q (1) − βn,q =

  logq logq q−1



0

if n = 0 if n = 1 if n > 1.

By (2.4), we obtain the following corollary. Corollary 2.3. For n ∈ Z+ , we have q(qβq + 1)n − βn,q =

  logq logq q−1



649

0

if n = 0 if n = 1 if n > 1,

J. W. PARK, D. V. DOLGY, T. KIM, S. H. LEE, AND S. H. RIM

with the usual convention of replacing βqn by βn,q . Let f (x) = [x]m q . From Theorem 2.1, we have q n βm,q (n) − βm,q = logq

n−1 X

q i [i]m q +m

i=0

n−1 X

q 2i [i]m−1 q



i=0

log q q−1

 .

(2.6)

Therefore, by (2.6), we obtain the sums powers of consecutive q-integers: Theorem 2.4. For m ∈ Z+ , and n ∈ N, we have n

q βm,q (n) − βm,q = logq

n−1 X

q

i

[i]m q

+m

i=0

n−1 X

q

2i

[i]m−1 q



i=0

log q q−1

 .

From (2.2), we note that Z q −y [1 − x + y]nq−1 dµ0 (y) βn,q−1 (1 − x) = Zp

Z n   X 1 n l −l+lx = (−1) q q −y−ly dµ0 (y) (1 − q −1 )n l Z p l=0 n   (−1) log q X n 1+l (−1)l q −l+lx = −1 n l (1 − q ) 1 − q −1−l l=0 n   log q X n 1+l = (−1)n q n+1 (−1)l q lx . (1 − q)n l 1 − q l+1

(2.7)

l=0

Therefore, by (2.3) and (2.7), we obtain the following theorem. Theorem 2.5. For n ∈ Z+ , we have βn,q−1 (1 − x) = (−1)n q n+1 βn,q (x). From (2.4) and Corollary 2.3, we have n   X n l 2 2 2 n 2 q βn,q (2) = q (q βq + q + 1) = q q (qβq + 1)l l l=0     n   X n l n 2 log q q βl,q +q = q(log q + β0,q ) + q β1,q + q−1 l 1 l=0 n   X n l 2 log q = q log q + nq +q q βl,q q−1 l l=0

log q + q(qβq + 1)n = q log q + nq q−1   log q log q = q log q + nq 2 + (log q)δ0,n + δ1,n + βn,q , q−1 q−1 where δn,k is Kronecker symbol. Therefore, by (2.8), we obtain the following theorem. 2

Theorem 2.6. For n ∈ Z+ , we have 2

2

q βn,q (2) = (q + δ0,n ) log q + nq + δ1,n

650






log q q−1

 + βn,q .

(2.8)

A NOTE ON THE MODIFIED CARLITZ’S q-BERNOULLI NUMBERS AND POLYNOMIALS

Now, we set Z I1 = q x βn,q−1 (1 − x)dµ0 (x) Zp n X l X n−l X

    Z n l n−l (q − 1)k q −l (−1)m βl,q−1 [x]k+m q x dµ0 (x) q l k m Zp l=0 k=0 m=0     n l n−l XX X n l n − l −l = q (−1)m βl,q−1 βk+m,q (q − 1)k . l k m l=0 k=0 m=0 (2.9)

=

From Theorem 2.5, we note that Z I1 = (−1)n q n+1 q x βn,q (x)dµ0 (x) Zp

= (−1)n q n+1

n   X n l=0

q x q lx [x]n−l dµ0 (x) q

Zp

Z l   X n l βl,q (q − 1)m q x [x]qm+n−l dµ0 (x) l m Z p m=0 l=0     n l X n X l = (−1)n q n+1 βl,q (q − 1)m βm+n−l,q . l m m=0 = (−1)n q n+1

n  X

l

Z βl,q



(2.10)

l=0

Therefore, by (2.9) and (2.10), we obtain the following theorem. Theorem 2.7. For n ∈ Z+ , we have  n X l X n−l    X n l n−l (q − 1)k q −l (−1)m βl,q−1 βk+m,q l k m l=0 k=0 m=0 n X l    X n l = (−1)n q n+1 (q − 1)m βl,q βm+n−l,q . l m m=0 l=0

Let k, n ∈ Z+ . Then the q-Bernstein polynomials of degree n are defined by Kim:   n Bk,n (n, q) = [x]kq [1 − x]n−k (2.11) q −1 , (see [1–13]). k By (2.6), we get Bn−k,n (1 − x, q −1 ) = Bk,n (x, q). Let us take p-adic integral on both sides of (2.11). Now, we set  Z Z n x x I2 = q Bk,n (x, q)dµ0 (x) = [x]kq [1 − x]n−k q −1 q dµ0 (x) k Zp Zp   k   Z X n k l n−k n−k = (−1) q q [x − 1]qn−k+l q x dµ0 (x) k l Z p l=0   k   X n k l n−k n−k = (−1) q q βn−k+l,q (−1). k l l=0

651

(2.12)

J. W. PARK, D. V. DOLGY, T. KIM, S. H. LEE, AND S. H. RIM

By Theorem 2.5 and (2.12), we get   k   1 n X k I2 = (−1)l βn−k+l,q−1 (2). q k l

(2.13)

l=0

On the other   n−k   Z n X n−k l x I2 = (−1) [x]k+l q q dµ0 (x) l k Z p l=0   n−k   X n n−k = (−1)l βk+l,q . k l

(2.14)

l=0

Therefore, by (2.13) and (2.14), we obtain the following theorem. Theorem 2.8. For n, k ∈ Z+ , we have k   n−k X X n − k  k l (−1) βn−k+l,q−1 (2) = q (−1)l βl+k,q . l l l=0

l=0

Let n, k ∈ Z+ with n ≥ k + 2. Then, (2.8) and Theorem 2.8, we get   k   X kq log q k n−k 2 δ1,k + q δ0,k + (−1)l βn−k+l,q−1 − q log q 1 + 1−q 1−q l l=0 n−k X n − k  (−1)l βl+k,q . =q l

(2.15)

l=0

Therefore, by (2.15), we obtain the following corollary. Corollary 2.9. For n, k ∈ Z+ with n ≥ k + 2, we have   k   X n−k kq log q k 2 δ1,k + q − q log q 1 + δ0,k + (−1)l βn−k+l,q−1 1−q 1−q l l=0 n−k X n − k  =q (−1)l βl+k,q . l l=0

Let d ∈ N. We set
X = Xd = lim ZdpN Z , ← − N

[

∗

X =

a + dpZp ,

0 < a < dp (a, p) = 1 N

a + dp Zp = {x ∈ X | x ≡ a (mod dpN )}, where a ∈ Z lies in 0 ≤ a < dpN (see [1–13]). Let χ be a primitive Dirichlet character with conductor d ∈ N. Then we define the generated q-Bernoulli polynomial attached to χ as follows; Z βn,χ,q (x) = χ(y)q y [x + y]nq dµ0 (y). (2.16) X

652

A NOTE ON THE MODIFIED CARLITZ’S q-BERNOULLI NUMBERS AND POLYNOMIALS

In the special case, x = 0, βn,χ,q (0) = βn,χ,q will be called the n-th generalized q-Bernoulli numbers attached to χ. From (2.16), we note that  P N −1  Z dp y n χ(y)q [x + y] q x=0  βn,χ,q (x) = χ(y)q y [x + y]nq dµ0 (y) = lim  N →∞ dpN X =

d−1 X a=0

N

p −1 1 X dy q [x + a + dy]nq N →∞ dpN y=0

χ(a)q a lim

N

 n p −1 d−1 [d]nq X 1 X dy x + a = χ(a)q a lim N q +y N →∞ p d a=0 d qd y=0

(2.17)

n  Z d−1 [d]nq X x+a χ(a)q a +y dµ0 (y) q dy d a=0 d Zp qd   d−1 [d]nq X x+a a χ(a)q βn,qd = . d a=0 d

=

Therefore, by (2.16) and (2.17), we obtain the following theorem. Theorem 2.10. For d ∈ N, and n ∈ Z+ , we have βn,χ,q (x) =

  d−1 [d]nq X x+a χ(a)q a βn,qd . d a=0 d

References [1] L. Carltz, q-Bernoulli numbers and polynomials, Duke Math. J., 15(1948), 987-1000. 19:121128(2010). [2] L. -C. Jang, C. -S. Ryoo, A note on the multiple twisted Carlitz’s type q-Bernoulli polynomials, Abstr. Appl. Annal., 2008, Art. ID 498173, 7pp. [3] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9(2002), no. 3, 288-299. [4] T. Kim, On p-adic q-L-function and sums of powers, Discerete Math., 252(2002), no. 1-3, 179-187. [5] T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys., 15(2008), no. 1, 51-57. [6] T. Kim, On the weighted q-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math., 21(2011), no. 2, 207-215. [7] T. Kim, A note on q-Bernstein polynomials, Russ. J. Math. Phys., 18(2011), no. 1, 73-82. [8] T. Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory, 76(1999), no. 2, 320-329. [9] H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math., 18(2009), no. 1, 41-48. [10] S. -H. Rim, E. -J. Moon, J. -H. Jin, S. -J. Lee, On the symmetric properties for the generalized Genocchi polynomials, J. Comput. Anal. Appl., 13(2011), no. 7, 1240-1245. [11] C. S. Ryoo, A numerical investigation of the structure of the roots of q-Bernoulli polynomials, J. Appl. Math. Comput., 23(2007), no. 1-2, 205-214. [12] Y. Simsek, Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function, J. Math. Anal. Appl., 324(2006), no. 2, 790-804. [13] Y. Simsek, Generating functions of the twisted Beronulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math., 16(2008), no. 2, 251-278.

653

J. W. PARK, D. V. DOLGY, T. KIM, S. H. LEE, AND S. H. RIM 1 Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea. E-mail address: [email protected] 2

Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected] 3 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected] 4

Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected] 5 Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea. E-mail address: [email protected]

654

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 655-664, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

New q-analogue of modified Bessel function and the quantum algebra Eq (2) M. Mansour1 and M. M. Al-Shomrani2 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 1

[email protected]

2 [email protected]

. Abstract In this paper, a new q-analogue of modified Bessel function and its corresponding generating function are given. Also, we used the generating function method to deduce its q−difference equation which gives us the differential equation of the ordinary modified Bessel function when q → 1 and we deduced its recurrence relation. Finally, we established the relation between the two-dimensional quantum algebra Eq (2) and the new q−analogue of modified Bessel function using the realization of the standard Euclidean algebra in one real variable and q−exponentials. 2010 Mathematics Subject Classification: 33D45, 81R50, 22E70. Key Words: q−exponential functions, q−Bessel functions, generating function, q−difference equation, recurrence relation, quantum algebra, irreducible representation, Lie algebra, Lie group.

1

Introduction.

Qk−1 k The q−shifted factorials are defined by (a; q) = 1, (a; q) = 0 k i=o (1 − aq ) and (a; q)∞ = Q∞ i i=0 (1 − aq ), where a and q be real numbers such that 0 < q < 1 [5]. Exton [3] defines a family of q−exponential functions as ∞ X q λn(n−1) n z , E(λ, z; q) = [n]q ! n=0 (q;q)n z where [n]q ! = (1−q) n and limq→1 E(λ, (1 − q)z; q) = e . If we put z = get the following formula n ∞ X q a(2 ) n Eq (x, a) = x , (q; q)n n=0 1

x 1−q

and replace λ by 2a, we (1)

Permanent address: M. Mansour, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 655

M. Mansour, M. M. Al-Shomrani / New q-analogue of modified Bessel function

which satisfies the functional relation [1] Eq (γx, a) = Eq (qγx, a) + γxEq (q a γx, a);

γ ∈ R.

In terms of the q−Jackson q−difference operator [7] Dq f (x) = it may be represented by Dq Eq (γx, a) =

f (x) − f (qx) (1 − q)x

(2)

γ Eq (q a γx, a). 1−q

(3)

As special cases of the function Eq (x, a), there are two important q-analogues of the exponential function given by: ∞ X xn (4) eq (x) = (q, q) n n=0 and

n ∞ X q ( 2 ) xn Eq (x) = . (q, q) n n=0

(5)

In [6] the q−Bessel functions are defined by: ¶ µ (q n+1 ; q)∞ ³ x ´n x2 0, 0 = | q; − , 2 ϕ1 q n+1 (q; q)∞ 2 4 ¶ µ (q n+1 ; q)∞ ³ x ´n q n+1 x2 − (2) Jn (x; q) = , | q; − 0 ϕ1 q n+1 (q; q)∞ 2 4 Ã ! n+1 (q n+1 ; q)∞ ³ x ´n q 2 x2 0 (3) Jn (x; q) = | q; − , 1 ϕ1 q n+1 (q; q)∞ 2 4 Jn(1) (x; q)

(6) (7) (8)

where r ϕs is the basic hypergeometric function [5] µ r ϕs

a1 , ..., ar | q; z b1 , ..., bs (1)

¶ =

∞ X (a1 , ..., ar ; q)k ³ k=0

(b1 , ..., bs ; q)k

k

(−1) q

k (k−1) 2

´1+s−r

zk . (q; q)k

(9)

(2)

The two functions Jn (x; q) and Jn (x; q) are q−analogues of the ordinary Bessel function (3) and the function Jn (x; q) is a q−analogue of the ordinary modified Bessel function. In the following, we are going to define a new q−analogue of ordinary modified Bessel function. A fundamental relationship between Lie groups and certain special functions is that special functions appear as basis vectors and matrix elements corresponding to local multiplier representations of Lie groups [9]. In analogy with the ordinary Lie theory, the new q−analogue of modified Bessel function is shown to arise as matrix elements of q−exponentials of the generators in a representation of the two-dimensional quantum algebra Eq (2). An expansion formula is algebraically derived using this model.

656

M. Mansour, M. M. Al-Shomrani / New q-analogue of modified Bessel function

2

A q−analogue of modified Bessel function and its generating function. (4)

Definition 2.1. The q−Bessel function Jn (x; q) is defined by: (x/2)n (4) Jn (x; q) = 0 ϕ2 (q; q)n

Ã

3(n+1)

−q 2 x2 − n+1 | q; 0, q 4

! (10)

3k ∞ (x/2)n X q 2 (k+n) (x2 /4)k , (q; q)n k=0 (q n+1 ; q)k (q; q)k

=

(11)

which converges absolutely for all x. (4)

Lemma 2.2. The function Jn (x; q) is a q-analogue of the ordinary modified Bessel function. Proof. (

∞ ³ x ´2k (1 − q) ³ x ´n X 3k(k+n) (1 − q)2k (4) 2 lim Jn ((1 − q)x; q) = lim q q→1 q→1 (q, q)n 2 k=0 (q n+1 ; q)k (q; q)k 2 ∞ ³ x ´2k 1 1 ³ x ´n X = (1)n 2 k=0 (n + 1)k (1)k 2 ∞ ³ x ´n X ³ x ´2k 1 = 2 k=0 Γ(n + k + 1)Γ(k + 1) 2 n

)

= In (x), where In (x) is the ordinary modified Bessel function. Lemma 2.3. If n is an integer then (4)

J−n (x; q) = Jn(4) (x; q). (4)

Proof. From the definition of the function Jn (x; q), we have (4) J−n (x; q)

3k(k−n) ∞ X q 2 (q −n+k+1 ; q)∞ ³ x ´2k−n = , (q; q)∞ (q; q)k 2 k=n

which will take the following formula, for s = k − n (4) J−n (x; q)

3s(s+n) ∞ X q 2 (q s+1 ; q)∞ ³ x ´2s+n . = (q; q) (q; q) 2 ∞ s+n s=0

Then using the q−shifted factorials (q s+1 ; q)∞ = (q n+s+1 ; q)∞ (q s+1 ; q)n , (q; q)s+n = (q; q)s (q s+1 ; q)n , 657

(12)

M. Mansour, M. M. Al-Shomrani / New q-analogue of modified Bessel function

we get (4) J−n (x; q)

3s(s+n) ∞ X q 2 (q n+s+1 ; q)∞ ³ x ´2s+n = (q; q)∞ (q; q)s 2 s=0

= Jn(4) (x; q).

(4)

Lemma 2.4. The function Jn (x; q) is an even (odd) function if the integer n is even (odd), since Jn(4) (−x; q) = (−1)n Jn(4) (x; q). (13) (4)

The following theorem will give us the generating function of the function Jn (x; q). Theorem 1. The function à g(x, t; q) = Eq

3

q 4 xt 3 , 2 2

!

à Eq

3

q4x 3 , 2t 2

! (14)

(4)

is the generating function of the function Jn (x; q) with ∞ X

g(x, t; q) =

q

3n2 4

Jn(4) (x; q)tn .

n=−∞

Proof.

3 3 r s ∞ X ∞ X q 2 [(2 )+(2 )]+ 4 (s+r) ³ x ´s+r r−s g(x, t; q) = t (q; q) (q; q) 2 r s r=0 s=0

Setting s = r − n in the last series, we get r−n 3 3 r ∞ X ∞ X q 2 [(2 )+( 2 )]+ 4 (2r−n) ³ x ´2r−n n g(x, t; q) = t . (q; q)r (q; q)r−n 2 n=−∞ r=0

Then the coefficient of tn for n ≤ 0 is cn

r−n 3 3 r ∞ X q 2 [(2 )+( 2 )]+ 4 (2r−n) ³ x ´2r−n = (q; q)r (q; q)r−n 2 r=0

= q

3n2 4

µ 2 ¶r 3 ∞ x (x/2)−n X q 2 r(r−n) , 1−n (q; q)−n r=0 (q; q)r (q ; q)r 4

where (q; q)r−n = (q; q)−n (q 1−n ; q)r for n ≤ 0. Then cn = q

3n2 4

(4)

J−n (x; q) = q

Similarly for n ≥ 0. 658

3n2 4

Jn(4) (x; q).

(15)

M. Mansour, M. M. Al-Shomrani / New q-analogue of modified Bessel function

(4)

3

q−Difference equation of the function Jn (x; q). (4)

Now we will deduce the q−difference equation of the function Jn (x; q) by using the generating function method [2]. From equations (14) and (15), we get à 3 ! à 3 ! ∞ X 3n2 q 4 xth 3 q 4 xh 3 Eq , Eq , = q 4 Jn(4) (x; q)tn ; h ∈ R − {0}. (16) 2 2 2t 2 n=−∞ Keeping the q−difference operator Dq of the last equation, we find à 7 ! à 9 ! à 9 ! 3 3 q4h q 4 xth 3 q 4 xh 3 q 4 th q 4 xth 3 Eq , Eq , + Eq , 2(1 − q)t 2 2 2t 2 2(1 − q) 2 2 ! à 3 ∞ X 3n2 q 4 xh 3 = q 4 Dq Jn(4) (xh; q)tn . ×Eq , 2t 2 n=−∞ Using equation (16), we get 3

3 n+1

n+1

3n2

3n(n−1)

3

3 q 4 + 2 ( 2 )+ 2 (4) 5 q 4+ 4 q 4 (4) Jn+1 (q 4 xh; q) + Jn−1 (q 4 xh; q) = Dq Jn(4) (xh; q). 2(1 − q) 2(1 − q) h

Then

3 o −3n 3 q 4 h n 5n+2 (4) 5 (4) 4 4 4 4 = q Jn+1 (q xh; q) + q Jn−1 (q xh; q) . 2(1 − q) Similarly, we can prove that 3 o −5n+2 5 q 4 h n 3n (4) 3 (4) (4) 4 4 4 4 q Jn+1 (q xh; q) + q Jn−1 (q xh; q) . Dq Jn (xh; q) = 2(1 − q)

Dq Jn(4) (xh; q)

(17)

(18)

By using equations (17) and (18), we get 4n+1 (4) √ (4) q 2 Jn+1 ( qxh; q) + Jn−1 (xh; q) 3n

(4)

= q 2 Jn+1 (xh; q) + q But (4) √ Jn−1 ( qxh; q)

= =

(

√

qxh n−1 ) 2

(q; q)∞ q

= q

1−n 2

(4) √ Jn−1 ( qxh; q).

µ√ ¶ 3k ∞ X qxh 2k q 2 (k+n−1) (q n+k ; q)∞ (q; q)k 2 k=0

µ ¶2k 3k ∞ ( xh )n−1 X q 2 (k+n−1) (q n+k ; q)∞ k xh 2 (q − 1 + 1) (q; q)∞ k=0 (q; q)k 2

n−1 2

n−1 2

(4) Jn−1 (xh; q)

−

q

3(k+1)(k+n) ∞ 2 ( xh )n−1 X q 2 (q; q)∞ k=0 (q; q)k

n−1 2

×(q ( = q

(19)

n−1 2

) 3n 2 xh 3 q (4) Jn−1 (xh; q) − Jn (q 4 xh; q) . 2 659

µ n+k+1

; q)∞

xh 2

¶2k+2

M. Mansour, M. M. Al-Shomrani / New q-analogue of modified Bessel function

Then q

1−n 2

(4) √ Jn−1 ( qxh; q)

3n

−

(4) Jn−1 (xh; q)

3 q 2 xh =− Jn (q 4 xh; q). 2

(20)

Equations (19) and (20) give the relation −xh (4) 3 (4) (4) √ Jn+1 ( qxh; q) − Jn+1 (xh; q) = J (q 4 xh; q). 2 n From equations (17) and (21), we get ( ) 3(1−n) 5(n+1) £ n ¤ 5 hq 4 q 4 (4) Dq + q 2 δq − 1 Jn (xh; q) = Jn(4) (q 4 xh; q), (1 − q)x 2(1 − q) √ where δq f (x) = f ( qx). If we define the operator Mn,q by ( ) 3(1−n) ¤ £ n q 4 Mn,q = Dq + q 2 δq − 1 , (1 − q)x q

1+n 2

(21)

(22)

(23)

then equation (22) takes the formula 5(n+1)

Mn,q Jn(4) (xh; q)

5 hq 4 = Jn(4) (q 4 xh; q). 2(1 − q)

Also, from equations (18) and (20) we get ) ( −3(1+n) 5(1−n) h −n i 5 q 4 hq 4 (4) (4) Dq + q 2 δq − 1 Jn (xh; q) = Jn−1 (q 4 xh; q). (1 − q)x 2(1 − q) If we define the operator Nn,q by

(

Nn,q =

) −3(1+n) h −n i q 4 q 2 δq − 1 , Dq + (1 − q)x

(24)

(25)

(26)

then equation (25) takes the formula 5(1−n)

Nn,q Jn(4) (xh; q)

5 hq 4 (4) Jn−1 (q 4 xh; q). = 2(1 − q)

(27)

Now, by using the two operators Mn,q and Nn,q we can obtain the q−difference equation of the (4) function Jn (x; q): 5 5 q 4 h2 (4) Mn−1,q Nn,q Jn (xh; q) = Jn(4) (q 2 xh; q) (28) 2 4(1 − q) If we put h = 1 − q and take the limit as q → 1, then we get µ ¶µ ¶ 1 d n−1 n 1 d 1 − + (29) y(x) = y(x), 2 dx 2x 2 dx 2x 4 which takes the formula

¡ ¢ x2 y 00 (x) + xy 0 − n2 + x2 y = 0,

which is the differential equation of the ordinary modified Bessel function. 660

(30)

M. Mansour, M. M. Al-Shomrani / New q-analogue of modified Bessel function

4

(4)

The recurrence relations of the function Jn (x; q).

Lemma 4.1. Jn(4) (x; q) =

¢ (4) 5(n+1) 2¡ (4) 1 − q n+1 Jn+1 (q −3/4 x; q) + q 2 Jn+2 (x; q). x

(31)

Proof. Jn(4) (x; q)

µ ¶ 3k ∞ ¡ ¢ x2 k (x/2)n X q 2 (k+n) n+1 n+1 n+k−1 = 1−q +q −q (q; q)n k=0 (q; q)k (q n+1 ; q)k+1 4 ³  ´2 k − 43 3k ∞ (k+n+1) q x n+1 X ¢ (x/2) q2 2¡   = 1 − q n+1   n+2 x (q; q)n+1 k=0 (q; q)k (q ; q)k 4 + q

n+1 (x/2)

n+2

∞ X

q

µ

3k (k+n) 2

x2 4

¶k−1

(q; q)n+2 k=1 (q; q)k−1 (q n+3 ; q)k−1 ¢ (4) 2¡ = 1 − q n+1 Jn+1 (q −3/4 x; q) x µ 2 ¶k 3k ∞ n+2 X 5(n+1) (x/2) q 2 (k+n+2) x + q 2 (q; q)n+2 k=0 (q; q)k (q n+3 ; q)k 4 ¢ (4) 5(n+1) 2¡ (4) = 1 − q n+1 Jn+1 (q −3/4 x, a; q) + q 2 Jn+2 (x; q). x

(4)

Lemma 4.2. The function Jn (x; q) satisfies the relation ¡

2 1−q

n+1

¢h

(4) Jn+1 (q −3/4 x; q)

−q

n 4

(4) Jn+1 (x; q)

i =q

2n+1 2

³ 1−q

3n+4 2

´

1 3n (4) (4) xJn+2 (x; q)− q 4 x2 Jn+1 (x; q). 2 (32)

By using equations (31) and (32), we get (4)

Lemma 4.3. The function Jn (x; q) has the recurrence relation · ¸ ¢ n n x 2¡ (4) (4) (4) n+1 Jn (x; q) = q 4 1−q −q2 Jn+1 (x; q) + q n+1/2 Jn+2 (x; q), x 2

(33)

(3)

which is the same recurrence relation of Jn (x; q).

5

(4)

The quantum algebra approach to Jn (x; q).

The two-dimensional quantum algebra Eq (2) is determined by its generators which obey the commutation relations [H, E+ ] = E+ ,

[H, E− ] = −E− , 661

[E− , E+ ] = 0.

(34)

M. Mansour, M. M. Al-Shomrani / New q-analogue of modified Bessel function

We consider irreducible representations (ω) of Eq (2) characterized by the positive real number ω. The spectrum of H corresponding to (ω) is the set of integers Z and the complex representation space has basis vectors fm , m ∈ Z such that E± fm = ωfm+1 ,

Hfm = mfm ,

E+ E− fm = ω 2 fm ,

(35)

where C = E+ E− is an invariant operator called Casimir operator. A simple realization of (ω) is given by the operators d ω H = z , E+ = ωz, E− = (36) dz z acting on the space of all linear combinations of the functions z m , z a complex variable, m ∈ Z, with basis vectors fm (z) = z m . In the ordinary Lie theory, matrix elements Tsm of the complex motion group in the representation (ω) are typically defined by the expansions [4], [8], [9] e

αE+ βE− τ H

e

e

fm =

∞ X

Tsm (α, β, τ )fs .

(37)

s=−∞

The group multiplication property of the operators on the left-hand side of equation (37) leads to addition theorems for the matrix elements. We replace the usual exponential function mapping from the Lie algebra to the Lie group by the q−exponential mapping Eq (x, 3/2). For convenience in the computations to follow we shall limit ourselves to the case where τ = 0. We employ the model (36) to define the following q−analog of matrix elements of (ω) Eq (αE+ , 3/2)Eq (βE− , 3/2)fm =

∞ X

Tsm (α, β)fs .

s=−∞

Now

µ Eq

3 αωz, 2

¶

µ Eq

ω 3 β , z 2

¶

3 r t ∞ X q 2 [(2 )+(2 )] r+t r t m−t+r z = ω αβz (q; q)r (q; q)t r,t=0

m

Setting s = m − t + r in the last series, we get m+r−s µ ¶ µ ¶ 3 r ∞ X ∞ X 3 q 2 [(2 )+( 2 )] m+2r−s r m+r−s s ω 3 m Eq αωz, ω αβ z Eq β , z = 2 z 2 (q; q)r (q; q)m+r−s s=−∞ r=0 Then the coefficient of z s for m ≥ s is ∞ X q 2( 2 ) q 2 r(r+m−s−1) ¡ 2 ¢r m−s (ωβ) ω αβ Tsm (α, β) = (q; q)m−s (q; q)r (q m−s+1 ; q)r r=0 £ r ¤3 3 m−s ∞ ³ 3 ´r X (−1)r q (2 ) q 2( 2 ) m−s (m−s) 2 2 = (ωβ) −q ω αβ (q; q)m−s (q; q)r (q m−s+1 ; q)r r=0 µ ¶ 3 m−s 3 q 2( 2 ) − m−s (m−s) 2 = (ωβ) | q; −q 2 ω αβ . 0 ϕ2 0, q m+1 (q; q)m−s 3 m−s

3

662

(38)

M. Mansour, M. M. Al-Shomrani / New q-analogue of modified Bessel function

Then µ ¶ 3 m−s 3 q 2( 2 ) − m−s (m−s) 2 Tsm (α, β) = (ωβ) | q; −q 2 ω αβ , m ≥ s. 0 ϕ2 0, q m+1 (q; q)m−s µ ¶ m−s 2 p −3 3 β (4) (m−s)2 4 = q Jm−s (2q 4 ω αβ; q), αβ > 0; m ≥ s α µ ¶ s−m p −3 3 α 2 (4) 2 (s−m) = q4 Js−m (2q 4 ω αβ; q), αβ > 0; m ≥ s, β (4)

(4)

where Jn (x; q) = J−n (x; q). Also, µ ¶ 3 s−m 3 q 2( 2 ) − s−m (s−m) 2 | q; −q 2 Tsm (α, β) = (ωβ) ω αβ , s ≥ m. 0 ϕ2 0, q m+1 (q; q)s−m µ ¶ s−m p −3 3 α 2 (4) 2 = q 4 (s−m) Js−m (2q 4 ω αβ; q), αβ > 0; s ≥ m. β The two cases can be combined into the single expression Tsm (α, β) = q

3 (s−m)2 4

µ ¶ s−m p −3 α 2 (4) Js−m (2q 4 ω αβ; q), β

αβ > 0

(39)

valid for all integers integral values of m, s. Then µ Eq

3 αωz, 2

¶

¶ µ ¶ s−m µ ∞ X p −3 3 α 2 (4) ω 3 2 m (s−m) Eq β , z = q4 Js−m (2q 4 ω αβ; q) z s , z 2 β s=−∞

αβ > 0; m ∈ Z. (40)

3 4

x , 2

In particular, for m = 0, α = β = q and ω = we get ! Ã 3 ! Ã 3 ∞ X 3s2 q4x 3 q 4 xz 3 , Eq , = q 4 Js(4) (x; q) z s , Eq 2 2 2z 2 s=−∞

(41)

(4)

which is the generating function of Js (x; q). Acknowledgement This Project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 3 − 104/430. The authors, therefore, acknowledge with thanks DSR support for Scientific Research.

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M. Mansour, M. M. Al-Shomrani / New q-analogue of modified Bessel function

[2] G. Dattoli and A. Torre, q−Bessel functions: the point of view of the generating function method, Rendiconti di Mathematica, Serie V II. Volume 17, Roma, 329-345, 1997. [3] H. Exton q−Hypergeometric functions and applications (Chichester:Ellis Horwood) 1983. [4] R. Floreanini and L. Vinet, Addition formulas for q−Bessel functions, J. Math. Phys. 33, 2984-2988, 1992. [5] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, 2nd edition 2004. [6] F.H. Jackson, The application of basic numbers to Bessel’s and Legendre’s functions, Proc. London Math. Soc. 2, 192-220, 1903-1904. [7] F.H. Jackson, On q−functions and certain difference operator, Transactions Royal Society of Edinburgh 46, 253-281, 1908. [8] E. G. Kalnins, S. Mukherjee and W. Miller, Models of q-algebra representations: The group of plane motions, 25, 513-527, 1994. [9] W. Miller, Lie theory and special functions, Academic Press, 1968.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 665-677, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A NEW SYSTEM OF GENERAL VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS FOR A COUNTABLE FAMILY OF STRICT PSEUDO-CONTRACTIONS IN BANACH SPACES∗∗ RATTANAPORN WANGKEEREE* AND RABIAN WANGKEEREE Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand Abstract. The purpose of this paper is to introduce a new system of general variational inequalities in Banach spaces and an iterative method for finding a common element of the set of solutions of the system of general variational inequalities for two relaxed cocoercive mappings and the set of common fixed points for a countable family of strict pseudo-contractions. The strong convergence theorems of the proposed iterative method are obtained without the control condition weakly sequentially continuous on the duality mapping in Banach spaces. Keywords: General variational inequality; Relaxed cocoercive mapping, Strict pseudo-contraction, Weakly sequentially continuous, Fixed point, and Banach space. AMS Subject Classification: 47H05, 47H09, 47J25, 65J15.

1. Introduction Variational inequalities are being used as a mathematical programming tool in modeling a wide class of obstacle, unilateral, free, moving, equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. Using the projection technique one can establish the equivalence between the variational inequalities and the fixed point problem. This equivalence has played an important role in developing several numerical techniques for solving variational inequalities and the related optimization problem. For the physical formulation, applications, numerical methods and other aspects of the variational inequalities, see [11, 21, 22, 23, 24, 25, 26] and the references therein. Related to the variational inequalities is the problem of finding the common fixed points of the strict pseudo-contractions, which is the subject of current interest in functional analysis. It is natural to unify these two problems and find the common elements of the set of the solution of variational inequality and the set of the common fixed points of the strict pseudo-contractions. Let E be a real Banach space and UE = {x ∈ E : kxk = 1}. A Banach space E is said to be uniformly convex if, for any ǫ ∈ (0, 2], there exists δ > 0 such that, for any x, y ∈ UE ,

x + y

kx − yk ≥ ǫ implies

≤ 1 − δ. 2 It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space E is said to be smooth if the limit kx + tyk − kxk lim t→0 t exists for all x, y ∈ UE . It is also said to be uniformly smooth if the limit is attained uniformly for all x, y ∈ UE . The norm of E is said to be Fr´echet differentiable if, for any x ∈ UE , the above limit is attained uniformly for all y ∈ UE . The modulus of smoothness of E is defined by 1 ρ(τ ) = sup{ (kx + yk + kx − yk) − 1 : x, y ∈ E, kxk = 1, kyk = τ }, 2 ∗∗ This research was partially supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand and Naresuan university. ∗ Corresponding author: Email address: [email protected] (Rattanaporn Wangkeeree) and [email protected] (Rabian Wangkeeree).

1

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R. WANGKEEREE AND R. WANGKEEREEE

where ρ : [0, ∞) → [0, ∞) is a function. It is known that E is uniformly smooth if and only if ) limτ →0 ρ(τ = 0. Let q be a fixed real number with 1 < q ≤ 2. A Banach space E is said to be τ q-uniformly smooth if there exists a constant c > 0 such that ρ(τ ) ≤ cτ q for all τ > 0. From [5], we know the following property: Let q be a real number with 1 < q ≤ 2 and let E be a Banach space. Then E is q-uniformly smooth if and only if there exists a constant K ≥ 1 such that kx + ykq + kx − ykq ≤ 2(kxkq + kKykq ),

∀x, y ∈ E.

The best constant K in the above inequality is called the q-uniformly smoothness constant of E (see [5] for more details). Let E be a real Banach space and E ∗ the dual space of E. Let h·, ·i denote the pairing between E ∗ and E ∗ . For q > 1, the generalized duality mapping Jq : E → 2E is defined by Jq (x) = {f ∈ E ∗ : hx, f i = kxkq , kf k = kxkq−1 },

∀x ∈ E.

In particular, J = J2 is called the normalized duality mapping. It is known that Jq (x) = kxkq−2 J(x) for all x ∈ E. If E is a Hilbert space, then J = I (: the identity mapping). Note that (1) E is a uniformly smooth Banach space if and only if J is single-valued and uniformly continuous on any bounded subset of E. p (2) All Hilbert spaces, Lp (or lp ) spaces (p ≥ 2) and the Sobolev spaces Wm (p ≥ 2) are 2-uniformly p p p smooth, while L (or l ) and Wm spaces (1 < p ≤ 2) are p-uniformly smooth. (3) Typical examples of both uniformly convex and uniformly smooth Banach spaces are Lp , where p > 1. More precisely, Lp is min{p, 2}-uniformly smooth for any p > 1. Further, we have the following properties of the generalized duality mapping Jq : (i) Jq (x) = kxkq−2 J2 (x) for all x ∈ E with x 6= 0, (ii) Jq (tx) = tq−1 Jq (x) for all x ∈ E and t ∈ [0, ∞), (iii) Jq (−x) = −Jq (x) for all x ∈ E. It is known that if E is smooth, then J is single-valued, which is denoted by j . Recall that the duality mapping j is said to be weakly sequentially continuous if for each sequence {xn } ⊂ E with xn −→ x weakly, we have j(xn ) −→ j(x) weakly-∗. We know that if E admits a weakly sequentially continuous duality mapping, then E is smooth. For the details, see [13]. Let C be a nonempty closed convex subset of a smooth Banach space E. Recall that the following definitions of a nonlinear mapping A : C −→ E. Definition 1.1. Let A : C −→ E be a mapping. (i) A is said to be accretive if hAx − Ay, j(x − y)i ≥ 0 for all x, y ∈ C. (ii) A is said to be α-strongly accretive if there exists a constant α > 0 such that hAx − Ay, j(x − y)i ≥ αkx − yk2 for all x, y ∈ C. (iii) A is said to be α-inverse-strongly accretive or α-cocoercive if there exists a constant α > 0 such that hAx − Ay, j(x − y)i ≥ αkAx − Ayk2 for all x, y ∈ C. (iv) A is said to be α-relaxed cocoercive if there exists a constant α > 0 such that hAx − Ay, j(x − y)i ≥ −αkAx − Ayk2 for all x, y ∈ C. (v) A is said to be (α, β)-relaxed cocoercive if there exists positive constants α and β such that hAx − Ay, j(x − y)i ≥ (−α)kAx − Ayk2 + βkx − yk2 for all x, y ∈ C.

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A NEW SYSTEM OF VARIATIONAL INEQUALITIES

3

Remark 1.2. (1). Every α-strongly accretive mapping is an accretive mapping. (2). Every α-strongly accretive mapping is an (β, α)-relaxed cocoercive mapping for any positive constant β but the converse is not true in general. Then the class of relaxed cocoercive operators is more general than the class of strongly accretive operators. (3). Evidently, the definition of the inverse-strongly accretive operator is based on that of the inverse-strongly monotone operator in real Hilbert spaces (see, for example, [6]). (4). The notion of the cocoercivity is applied in several directions, especially to solving variational inequality problems using the auxiliary problem principle and projection methods [35]. Several classes of relaxed cocoercive variational inequalities have been studied in [33, 34]. Let C be a nonempty closed and convex subset of a smooth Banach space E. We introduce the following system of general variational inequalities involving two different nonlinear mappings A, B : C −→ E : Find (x∗ , y ∗ ) ∈ C × C such that ( hλAy ∗ + x∗ − y ∗ , j(x − x∗ )i ≥ 0, ∀x ∈ C, (1.1) hµBx∗ + y ∗ − x∗ , j(x − y ∗ )i ≥ 0, ∀x ∈ C, where λ and µ are two positive real numbers. As special cases of the problem (1.1), we have the following: (i) If A = B, then the problem (1.1) is reduced to the following: Find (x∗ , y ∗ ) ∈ C × C such that ( hλAy ∗ + x∗ − y ∗ , j(x − x∗ )i ≥ 0, ∀x ∈ C, hµAx∗ + y ∗ − x∗ , j(x − y ∗ )i ≥ 0, ∀x ∈ C,

(1.2)

where λ and µ are two positive real numbers. This system of variational inequalities was considered and studied by Noor [22, 21] using the auxiliary principle technique. (ii) If λ = µ = 1, then the problem (1.1) is reduced to the following: Find (x∗ , y ∗ ) ∈ C × C such that ( hAy ∗ + x∗ − y ∗ , j(x − x∗ )i ≥ 0, ∀x ∈ C, (1.3) hBx∗ + y ∗ − x∗ , j(x − y ∗ )i ≥ 0, ∀x ∈ C. This system of variational inequalities was considered and studied by Yao, Noor, Noor, Liou, and Yaqoob [39]. (iii) In real Hilbert spaces, the problem (1.1) is reduced to the following: Find (x∗ , y ∗ ) ∈ C × C such that ( hλAy ∗ + x∗ − y ∗ , x − x∗ i ≥ 0, ∀x ∈ C, (1.4) hµBx∗ + y ∗ − x∗ , x − y ∗ i ≥ 0, ∀x ∈ C, where λ and µ are two positive real numbers. The system (1.4) is introduced and studied by Ceng, Wang and Yao [10]. To illustrate the applications of this system, Zhu and Marcotte [41] considered the problem of finding x∗ ∈ C such that hA(x∗ ), x − x∗ i ≥ 0, ∀x ∈ E = C ∩ {x ∈ H : B(x) ≤ 0},

(1.5)

where A is strongly monotone on E and B(x) = {f1 (x), f2 (x), . . . , fm (x)} is a constraint mapping explicitly defined by the convex, Lipschitz continuous and continuously differentiable functions fi , i = 1, 2, . . . , m. Assume that there exists x0 ∈ C such that fi (x0 ) < 0, i = 1, 2, . . . , m (Slaters constraint qualification). Then the variational inequality (1.5) is equivalent to the KuhnTucker-like system ( hA(x∗ ) + (∇B(x∗ ))′ y ∗ , x − x∗ i ≥ 0, ∀x ∈ C, (1.6) hµ − B(x∗ ), y − y ∗ i ≥ 0, ∀y ≥ 0, which is exactly the system of variational inequalities (1.4).

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(iv) If A = B , λ = µ = 1, and x∗ = y ∗ then the problem (1.1) is reduced to the following: Find x∗ ∈ C such that hAx∗ , j(x − x∗ )i ≥ 0, ∀x ∈ C.

(1.7)

The problem (1.7) is very interesting as it is connected with the fixed point problem for nonlinear mappings, the problem of finding a zero point of an accretive operator and so on. For the problem of finding a zero point of an accretive operator by the proximal point algorithm, see Kamimura and Takahashi [16] and the references therein. In [2], Aoyama, Iiduka and Takahashi [2] first considered the such problem in Banach spaces. In order to find a solution of problem (1.7), they proved the following theorem which is generalized simultaneously theorems of [6] and [12]. Theorem AIT. Let E be a uniformly convex and 2-uniformly smooth Banach space and C a nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C, α > 0 and A an α-inverse strongly-accretive operator of C into E with S(C, A) 6= ∅, where S(C, A) = {x∗ ∈ C : hAx∗ , J(x − x∗ )i ≥ 0, x ∈ C}. If {λn } and {αn } are chosen such that λn ∈ [a, Kα2 ] for some a > 0 and αn ∈ [b, c] for some b, c with 0 < b < c < 1, then the sequence {xn } defined by the following manners:    x1 = x ∈ C, (1.8) yn = QC (xn − λn Axn ),   xn+1 = αn xn + (1 − αn )yn , n ≥ 1, converges weakly to some element z of S(C, A), where K is the 2-uniformly smoothness constant of E. On the other hand, in [14], Hao obtained a strong convergence theorem for approximating the solutions of the generalized variational inequality problem (1.7) by using the following iterative algorithm:    x1 = u ∈ C, (1.9) yn = βn xn + (1 − βn )QC (I − λn A)xn ,   xn+1 = αn u + (1 − αn )yn , n ≥ 1. where QC is a sunny nonexpansive retraction from E onto C, {αn }, {λn } and {βn } are appropriate real sequences in [0, 1]. Very recently, motivated by Aoyama, Iiduka and Takahashi [2] and Hao [14], for solving the problem (1.3), Yao, Noor, Noor, Liou, and Yaqoob [39] established the equivalence between the system of variational inequalities (1.3) and a fixed point problem involving the nonexpansive mapping. This alternative equivalent formulation is used to suggest and analyze a modified extragradient method. Using the demi-closedness principle for nonexpansive mappings, they obtained the following strong convergence theorem of the proposed iterative method under some suitable conditions. Theorem YNNLY-A. Let C be a nonempty closed convex subset of a real smooth Banach space E. Let QC be the sunny nonexpansive retraction from E onto C. Let A, B : C −→ E be α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let G : C −→ C be a mapping defined by G(x) = QC [QC (x − Bx) − AQC (x − Bx)], ∀x ∈ C.

(1.10)

Then (i).[39, Lemma 3.2] If E is real 2-uniformly smooth Banach space, α ≥ K 2 and β ≥ K 2 , then G is nonexpansive. (ii). [39, Lemma 3.3] For given x∗ , y ∗ ∈ C, (x∗ , y ∗ ) is a solution of the problem (1.3) if and only if x∗ is a fixed point the mapping G : C −→ C defined by (1.10), where y ∗ = QC (x∗ − Bx∗ ). Theorem YNNLY-B [39, Theorem 3.1]. Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E which admits a weakly sequentially continuous duality mapping and the smooth constant K. Let QC be the sunny nonexpansive retraction from E onto C. Let A, B : C −→ E be α-inverse-strongly accretive with α ≥ K 2 and β-inverse-strongly accretive with β ≥ K 2 , respectively. Suppose the set of fixed point Ω of the mapping G : C −→ C defined by (1.10)

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A NEW SYSTEM OF VARIATIONAL INEQUALITIES

is nonempty. For fixed u ∈ C, let the sequence {xn } be generated iteratively by  x1 ∈ C, chosen arbitrary,     y = Q (x − Bx ), n C n n  z = Q (y − Ay n C n n ),    xn+1 = αn u + βn xn + γn zn , n ≥ 1,

5

(1.11)

where the real sequences {αn }, {βn }, and {γn } ⊂ (0, 1) satisfy the following conditions: (i) αn + βn + γn = 1, P ∀n ≥ 1; ∞ (ii) limn−→∞ αn = 0, n=1 αn = +∞; (iii) 0 < lim inf n−→∞ βn ≤ lim supn−→∞ βn ≤ 1. Then {xn } defined by (1.11) converges strongly to QΩ u, where QΩ is the sunny nonexpansive retraction of C onto Ω. All of the above bring us the following conjectures?. Question (i) Could we weaken the condition uniformly convex on Banach spaces?. (ii) Could we remove the control condition ”weakly sequentially continuous” on the duality mapping in Theorem YNNLY-B?. (iii) Could we construct an iterative algorithm to approximate a common element of the set of solutions of general variational inequalities (1.1) for two relaxed cocoercive mappings and the set of common fixed points of a countable family of strict pseudo-contractions in Banach spaces?. In this paper, motivated by Aoyama, Iiduka and Takahashi [2], Hao [14], and Yao, Noor, Noor, Liou, and Yaqoob [39], we introduce a new system of general variational inequalities in Banach spaces. We establish the equivalence between the system of variational inequalities (1.1) for two relaxed cocoercive mappings and fixed point problems involving a nonexpansive mapping. This alternative equivalent formulation is used to suggest and analyze a new iterative approximation method for solving the system of general variational inequalities for two relaxed cocoercive mappings and fixed point problems for a countable family of strict pseudo-contractions. The strong convergence theorems of the proposed iterative method are obtained without the control condition ”weakly sequentially continuous” of the duality mapping on Banach spaces. The results presented in the paper improve some recent results of Aoyama, Iiduka and Takahashi [2], Hao [14], and Yao, Noor, Noor, Liou, and Yaqoob [39]. 2. Preliminaries Now we collect some useful lemmas for proving the convergence results. Lemma 2.1. [1, Lemma 2.3] Let {an } be a sequence of nonnegative real numbers satisfying the property: an+1 ≤ (1 − αn )an + αn cn + bn , ∀n ≥ 0, where {αn }, {bn}, {cn } satisfy the restrictions: ∞ X

αn = ∞;

n=0

∞ X

bn < ∞; and lim sup cn ≤ 0. n→∞

n=0

Then limn→∞ an = 0. Lemma 2.2. ([30]) Let {xn } and {yn } be bounded sequences in a Banach space E and {βn } a sequence in [0, 1] with 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1. Suppose that xn+1 = (1 − βn )yn + βn xn for all n ≥ 0 and lim sup(kyn+1 − yn k − kxn+1 − xn k) ≤ 0. n→∞

Then limn→∞ kyn − xn k = 0. A mapping T : C −→ C is said to be ε-strictly pseudo-contractive, if there exists a constant ε ∈ [0, 1) such that kT x − T yk2 ≤ kx − yk2 + εk(I − T )x − (I − T )yk2 , ∀x, y ∈ C.

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Note that the class of ε-strictly pseudo-contractive mappings strictly includes the class of nonexpansive mappings which are mappings T on C such that kT x − T yk ≤ kx − yk, for all x, y ∈ C. That is, T is nonexpansive if and only if T is 0-strict pseudo-contractive. We denote by F (T ) := {x ∈ C : T x = x} the set of fixed points of T . Definition 2.3. A countable family of mapping {Tn : C −→ C}∞ i=1 is called a family of uniformly ε-strict pseudo-contractions, if there exists a constant ε ∈ [0, 1) such that kTn x − Tn yk2 ≤ kx − yk2 + εk(I − Tn )x − (I − Tn )yk2 , ∀x, y ∈ C, ∀n ≥ 1. Lemma 2.4. ([9]) Let E be a strictly convex Banach space. Let T1 and T2 be two nonexpansive mappings from E into itself with a common fixed point. Define a mapping S by Sx = λT1 x + (1 − λ)T2 x,

∀x ∈ E,

where λ is a constant in (0, 1). Then S is nonexpansive and F (S) = F (T1 ) ∩ F (T2 ). Lemma 2.5. ([40]) Let E be a real 2-uniformly smooth Banach space and T : E → E a ε-strict pseudo-contraction. Then S := (1 − ε/K 2 )I + ε/K 2 T is nonexpansive and F (T ) = F (S). Lemma 2.6. ([37]) Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds: kx + yk2 ≤ kxk2 + 2hy, jxi + 2kKyk2,

∀x, y ∈ E.

Let D be a nonempty subset of C. A mapping Q : C −→ D is said to be sunny if Q(Qx + t(x − Qx)) = Qx, whenever Qx + t(x − Qx) ∈ C for x ∈ C and t = 0. A mapping Q : C −→ D is called a retraction if Qx = x for all x ∈ D. Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. The following lemma concerns the sunny nonex- pansive retraction. Lemma 2.7. ([28, 8]) Let C be a closed convex subset of a smooth Banach space E. Let D be a nonempty subset of C and Q : C −→ D be a retraction. Then Q is sunny and nonexpansive if and only if hu − Qu, j(y − Qu)i ≤ 0 for all u ∈ C and y ∈ D. Definition 2.8. Let {Sn } be a family of mappings from a subset C of Banach space E into E with ∩∞ n=1 F (Sn ) 6= ∅. We say that {Sn } satisfies the P U -condition if for each bounded subset D of C, there exists a continuous increasing and convex function h : R+ −→ R+ such that h(0) = 0 and

lim

sup h(kSk z − Sl zk) = 0.

k,l−→∞ z∈D

(2.1)

Remark 2.9. The example of a sequence of mappings satisfying P U -condition is supported by Example 3.5. Lemma 2.10. [27, Lemma 3.1] Suppose that there exists a continuous increasing function h : R+ −→ R+ satisfying (2.1). Then (i). For each x ∈ C, {Sn x} is a convergent sequence in C. (ii). Let the mapping S : C −→ C be defined by Sx = lim Sn x, for all x ∈ C. n−→∞

(2.2)

Then limn−→∞ supz∈D h(kSz − Sn zk) = 0 for each bounded subset D of C. Remark 2.11. If {Sn } satisfies the P U -condition, then the facts (i) and (ii) in Lemma 2.10 hold.

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A NEW SYSTEM OF VARIATIONAL INEQUALITIES

7

Lemma 2.12. [15, Lemma 3.2] Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E with the smooth constant K. Let A : C −→ E be a LA -Lipchitzian and relaxed (c, d)-cocoercive mapping. Then k(I − λA)x − (I − λA)yk2 ≤ (1 + 2λcL2A − 2λd + 2λ2 K 2 L2A )kx − yk2 . If λ ≤

d−cL2A K 2 L2A

(2.3)

, then I − λA is nonexpansive. 3. Main Results

In this section, we state and prove our main results. Lemma 3.1. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E with the smooth constant K. Let QC be the sunny nonexpansive retraction from E onto C. Let A : C −→ E be a LA -Lipchitzian and relaxed (c, d)-cocoercive mapping and B : C −→ H a LB d′ −c′ L2 d−cL2 Lipchitzian and relaxed (c′ , d′ )-cocoercive mapping, where λ ≤ K 2 L2A and µ ≤ K 2 L2 B . Define the A B mapping G by G(x) = QC [QC (x − µBx) − λAQC (x − µBx)], ∀x ∈ C. (3.1) Then G is nonexpansive. Proof. From Lemma 2.12, we deduce that I − λA, I − µB and QC are nonexpansive mappings. Then, for any x, y ∈ C, we obtain kG(x) − G(y)k2

= kQC [QC (x − µBx) − λAQC (x − µBx)] − QC [QC (y − µBy) − λAQC (y − µBy)]k2 ≤ k(I − λA)QC (I − µB)x − (I − λA)QC (I − µB)yk2 ≤ kx − yk2 .

Hence G is nonexpansive on C.



Lemma 3.2. Let C be a nonempty closed convex subset of a real smooth Banach space E. Let QC be the sunny nonexpansive retraction from E onto C. For given x∗ , y ∗ ∈ C, (x∗ , y ∗ ) is a solution of the problem (1.1) if and only if x∗ is a fixed point the mapping G : C −→ C defined by (3.1), where y ∗ = QC (x∗ − µBx∗ ), λ, µ are positive constants and A, B : C −→ H are possibly nonlinear mappings. Proof. We can rewrite the problem (1.1) as ( hx∗ − (y ∗ − λAy ∗ ), j(x − x∗ )i ≥ 0, ∀x ∈ C, hy ∗ − (x∗ − µBx∗ ), j(x − y ∗ )i ≥ 0, ∀x ∈ C.

(3.2)

Applying Lemma 2.7, we can deduce that (3.2) is equivalent to x∗ = QC (y ∗ − λAy ∗ ) and y ∗ = QC (x∗ − µBx∗ ), which is equivalent to x∗ = QC (QC (x∗ − µBx∗ ) − λAQC (x∗ − µBx∗ )). Hence x∗ is a fixed point the mapping G defined by (3.1). This completes the proof.



Throughout this paper, the set of fixed points of the mapping G is denoted by GV I(A, B, C). Theorem 3.3. Let C be a nonempty closed convex subset of a strictly convex and 2-uniformly smooth Banach space E with the smooth constant K. Let QC be the sunny nonexpansive retraction from E onto C. Let A : C −→ E be a LA -Lipchitzian and relaxed (c, d)-cocoercive mapping and B : C −→ H a LB -Lipchitzian and relaxed (c′ , d′ )-cocoercive mapping. Let {Tn : C −→ C}∞ i=1 be a countable family of uniformly ε-strict pseudo-contractions such that Ω := ∩∞ F (T ) ∩ GV I(A, B, C) 6= ∅. Define a n n=1 mapping Sn : C −→ C by ε ε Sn x = (1 − 2 )x + 2 Tn x for all x ∈ C and n ≥ 1. K K Let the sequence {xn } be generated iteratively by  x1 = u ∈ C, chosen arbitrary,     y = Q (x − µBx ), n C n n (3.3)  z = Q (y − λAy ), n C n n    xn+1 = αn u + βn xn + γn [νSn xn + (1 − ν)zn ], n ≥ 1,

671

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R. WANGKEEREE AND R. WANGKEEREEE d−cL2

d′ −c′ L2

where ν ∈ (0, 1), λ ∈ (0, K 2 L2A ) and µ ∈ (0, K 2 L2 B ) and the real sequences {αn }, {βn }, and {γn } ⊂ A B (0, 1) satisfy the following conditions: (D1) αn + βn + γn = 1; P ∞ (D2) limn−→∞ αn = 0, n=1 αn = +∞; (D3) 0 < lim inf n−→∞ βn ≤ lim supn−→∞ βn < 1. Suppose that {Sn } satisfies the P U -condition. Let the mapping S : C −→ C be defined by (2.2) and suppose that F (S) = ∩∞ n=1 F (Sn ). Then {xn } defined by (3.3) converges strongly to QΩ u, where QΩ is the sunny nonexpansive retraction of C onto Ω. Proof. Take x∗ ∈ Ω. Then x∗ = QC [QC (x∗ − µBx∗ ) − λAQC (x∗ − µBx∗ )]. Putting y ∗ = PC (x∗ − µBx∗ ), we have x∗ = QC (y ∗ − µAy ∗ ). From nonexpansivity of QC , I − λA and I − µB, we have kzn − x∗ k = ≤

kQC (yn − λAyn ) − QC (y ∗ − λAy ∗ )k k(I − λA)yn − (I − λA)y ∗ k

≤ =

kyn − y ∗ k kQC (xn − µBxn ) − QC (x∗ − µBx∗ )k

≤ ≤

k(I − µB)xn − (I − µB)x∗ k kxn − x∗ k.

(3.4)

For each n ∈ N, set tn = νSn xn + (1 − ν)zn . It follows from Lemma 2.5 that Sn is a nonexpansive mapping such that F (Sn ) = F (Tn ) for all n ≥ 1 ∞ and hence ∩∞ n=1 F (Sn ) = ∩n=1 F (Tn ). Hence ktn − x∗ k = ≤ ≤

kνSn xn + (1 − ν)zn − x∗ k νkSn xn − x∗ k + (1 − ν)kzn − x∗ k kxn − x∗ k.

(3.5)

It follows that kxn+1 − x∗ k

= kαn u + βn xn + γn tn − x∗ k ≤ αn ku − x∗ k + βn kxn − x∗ k + γn ktn − x∗ k ≤ αn ku − x∗ k + (1 − αn )kxn − x∗ k ≤ max{ku − x∗ k, kxn − x∗ k}.

(3.6)

It follows from the simple induction that kxn −x∗ k ≤ max{ku−x∗ k, kx1 −x∗ k} for all n ≥ 1. Therefore, {xn } is bounded. Hence {yn }, {zn } and {tn } are also bounded. We observe that kzn+1 − zn k

= kQC (yn+1 − λAyn+1 ) − QC (yn − λAyn )k ≤ k(yn+1 − λAyn+1 ) − (yn − λAyn )k ≤ kyn+1 − yn k = kQC (xn+1 − µBxn+1 ) − QC (xn − µBxn )k ≤ k(xn+1 − µBxn+1 ) − (xn − µBxn )k ≤ kxn+1 − xn k.

(3.7)

It follows from (3.7) that ktn+1 − tn k = =

kνSn+1 xn+1 + (1 − ν)zn+1 − (νSn xn + (1 − ν)zn )k kνSn+1 xn+1 − νSn+1 xn + (1 − ν)zn+1 + νSn+1 xn − νSn xn − (1 − ν)zn k

≤ ≤

νkSn+1 xn+1 − Sn+1 xn k + (1 − ν)kzn+1 − zn k + νkSn+1 xn − Sn xn k νkxn+1 − xn k + (1 − ν)kxn+1 − xn k + νωn

=

kxn+1 − xn k + νωn ,

(3.8)

672

A NEW SYSTEM OF VARIATIONAL INEQUALITIES

9

where ωn = kSn+1 xn − Sn xn k. Next, we will prove that limn−→∞ ωn = 0. Indeed, Since {xn } is bounded, there exists a bounded subset D of C such that {xn } ⊂ D. We observe that 1 1 1 1 ωn = kSn+1 xn − Sn xn k ≤ kSn+1 xn − Sxn k + kSxn − Sn xn k. 2 2 2 2 Since {Sn } satisfies PU-condition, then there exists an increasing, continuous and convex function h : R+ −→ R+ satisfying (2.1). Then 1 1 1 h( ωn ) ≤ h(kSn+1 xn − Sxn k) + h(kSxn − Sn xn k) 2 2 2 1 1 = sup h(kSn+1 z − Szk) + sup h(kSn z − Szk). (3.9) 2 z∈D 2 z∈D Applying Lemma 2.10 to the above inequality, we obtain that 1 lim h( ωn ) = 0. n−→∞ 2 The properties of the function h implies that lim ωn = 0.

(3.10)

n−→∞

Putting xn+1 = (1 − βn )en + βn xn ,

∀n ≥ 1,

(3.11)

one sees that en+1 − en

= =

αn u + γn tn αn+1 u + γn+1 tn+1 − 1 − βn+1 1 − βn   αn+1 αn γn+1 − u+ (tn+1 − tn ) 1 − βn+1 1 − βn 1 − βn+1   γn+1 γn + − tn . 1 − βn+1 1 − βn

Combining (3.11) and (3.12), we have ken+1 − en k − kxn+1 − xn k

≤

≤

(3.12)

αn+1 αn γn+1 1 − βn+1 − 1 − βn kuk + 1 − βn+1 (kxn+1 − xn k + νωn ) γn+1 γn − ktn k − kxn+1 − xn k + 1 − βn+1 1 − βn αn+1 αn γn+1 (3.13) 1 − βn+1 − 1 − βn (kuk + ktn k) + 1 − βn+1 νωn .

It follows from the conditions (D2), (D3) and (3.10) that lim sup(ken+1 − en k − kxn+1 − xn k) ≤ 0. n→∞

Hence, from Lemma 2.2, it follows that lim ken − xn k = 0.

n→∞

(3.14)

From (3.11), it follows that kxn+1 − xn k = (1 − βn )ken − xn k. Using (3.14) and the condition (D3), one sees that lim kxn+1 − xn k = 0.

n→∞

(3.15)

On the other hand, one has xn+1 − xn = αn (u − tn ) + (1 − γn )(tn − xn ). It follows that (1 − γn )ktn − xn k ≤ kxn+1 − xn k + αn ku − tn k. From the conditions 0 < lim inf n−→∞ γn ≤ lim supn−→∞ γn < 1, limn−→∞ αn = 0 and (3.15), one sees that lim ktn − xn k = 0. (3.16) n→∞

673

10

R. WANGKEEREE AND R. WANGKEEREEE

Next, we prove that lim suphu − QΩ u, j(xn − QΩ u)i ≤ 0, n−→∞

where QΩ is the sunny nonexpansive retraction from E onto Ω. Define a mapping Wn by Wn y = νSn y + (1 − ν)QC [(I − λA)QC (I − µB)y], ∀y ∈ C, ∀n ≥ 1. In view of Lemma 3.1 and Lemma 2.4, we see that Wn is a nonexpansive mapping satisfying F (Wn ) = F (Sn ) ∩ F (QC [(I − λA)QC (I − µB)y]) = F (Sn ) ∩ F (G).

(3.17)

This implies that ∞ ∞ ∩∞ n=1 F (Wn ) = ∩n=1 [F (Sn ) ∩ F (G)] = (∩n=1 F (Sn )) ∩ F (G) = F (S) ∩ F (G).

From (3.16), it follows that lim kWn xn − xn k = 0.

n→∞

(3.18)

On the other hand, since {Sn } satisfies the PU-condition, we have lim

sup h(kWk y − Wl yk) =

k,l−→∞ y∈D

lim

sup h(νkSk y − Sl yk) = 0.

k,l−→∞ y∈D

In virtue of Lemma 2.10, we obtain that {Wn y} is a convergent sequence for all y ∈ C. So, let W be a mapping from C into itself defined by W y = lim Wn y, for all y ∈ C.

(3.19)

lim sup h(kW y − Wn yk) = 0.

(3.20)

n−→∞

Using Lemma 2.10, we have n−→∞ y∈D

Since W y = lim Wn y = lim [νSn y + (1 − ν)QC [(I − λA)QC (I − µB)y]] = νSy + (1 − ν)Gy. n−→∞

n−→∞

The nonexpansivity of S and G and Lemma 2.4 imply that W is nonexpansive such that F (W ) = F (S) ∩ F (G) = ∩∞ n=1 F (Wn ) = Ω. Next, we observe that 1 1 1 h( kW xn −xn k) ≤ h(kW xn −Wn xn k)+ h(kWn xn −xn k) ≤ sup h(kW y −Wn yk)+h(kWn xn −xn k). 2 2 2 y∈D Applying (3.20) and (3.18) to the last inequality, we have 1 lim h( kW xn − xn k) = 0. n−→∞ 2 It follows from the properties of h that lim kW xn − xn k = 0.

n−→∞

(3.21)

(3.22)

Let QΩ be the sunny nonexpansive retraction of C onto Ω. Now we show that lim suphu − QΩ u, j(xn − QΩ u)i ≤ 0.

(3.23)

n−→∞

Let ut be the fixed point of the contraction z 7→ tu + (1 − t)W z, where t ∈ (0, 1). That is, ut = tu + (1 − t)W ut . It follows that kut − xn k = k(1 − t)(W ut − xn ) + t(u − xn )k. On the other hand, we have kut − xn k2 ≤ (1 − t)2 kW ut − xn k2 + 2thu − xn , j(ut − xn )i ≤ (1 − 2t + t2 )kut − xn k2 + fn (t) + 2thu − ut , j(ut − xn )i + 2tkut − xn k2 , where fn (t) = (2kut − xn k + kxn − W xn k)kxn − W xn k → 0 as n → 0.

674

(3.24)

A NEW SYSTEM OF VARIATIONAL INEQUALITIES

It follows that hut − u, j(ut − xn )i ≤

11

t 1 kut − xn k2 + fn (t). 2 2t

In view of (3.24), we arrive at lim suphut − u, j(ut − xn )i ≤ n→∞

t M, 2

(3.25)

where M > 0 is an appropriate constant such that M ≥ kut − xn k2 for all t ∈ (0, 1) and n ≥ 1. Letting t → 0 in (3.25), we have lim sup lim suphut − u, j(ut − xn )i ≤ 0. t→0

n→∞

So, for any ξ > 0, there exists a positive number δ1 with t ∈ (0, δ1 ) such that lim suphut − u, j(ut − xn )i ≤ n→∞

ξ . 2

(3.26)

On the other hand, we see that QF (W ) u = limt→0 ut and F (W ) = Ω. It follows that ut → QΩ u as t → 0. This implies that there exists δ2 > 0, for t ∈ (0, δ2 ), such that |hu − QΩ u, j(xn − QΩ u)i − hut − u, j(ut − xn )i| ≤ |hu − QΩ u, j(xn − QΩ u)i − hu − QΩ u, j(xn − ut )i| + |hu − QΩ u, j(xn − ut )i − hut − u, j(ut − xn )i| ≤ |hu − QΩ u, j(xn − QΩ u) − j(xn − ut )i| + |hut − QΩ u, j(xn − ut )i| ≤ ku − QΩ ukkj(xn − QΩ u) − j(xn − ut )k + kut − QΩ ukkxn − ut k ξ . 2 Choosing δ = min{δ1 , δ2 }, it follows that, for each t ∈ (0, δ),
0k for every (iii) n=1 k=1 |βn+1 − βnk | < ∞. It follows from [1, Theorem 4.1], we have (1) Each Tn is a nonexpansive mapping. (2) For any bounded subset D of E, whereP {βnk } n (i) k=1

∞ X

sup{kTn+1 z − Tn zk : z ∈ B} < ∞.

n=1

Using [27, Remark 3.2], we obtain that lim

sup h(kTk z − Tl zk) = 0

k,l−→∞ z∈D

for any continuous increasing and convex function h : R+ −→ R+ such that h(0) = 0. Hence {Tn } satisfies the PU-condition. Acknowledgements. The first author was partially supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand and Naresuan University. References 1. K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal., 67(2007) 2350-2360. 2. K. Aoyama, H. Iiduka, W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006, 1-13 (2006). 3. M. Aslam Noor, Some algorithms for general monotone mixed variational inequalities. Math. Comput. Model. 29, 1-9, (1999). 4. M. Aslam Noor, Some development in general variational inequalities. Appl. Math. Comput. 152, 199-277 (2004) 5. K. Ball, E.A. Carlen and E.H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math. 115 (1994), 463-482. 6. F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228. 7. F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure. Math. 18 (1976), 78-81. 8. R.E. Bruck, Nonexpansive retracts of Banach spaces. Bull. Am. Math. Soc. 76, 384-386 (1970) 9. R.E. Bruck, Properties of fixed point sets of nonexpansive mappings in Banach spaces, Tras. Amer. Math. Soc. 179 (1973), 251-262. 10. L.C. Ceng, C.Y. Wang and J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008), 375-390. 11. R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984). 12. E.G. Golshtein and N.V. Tretyakov, Modified Lagrangians in convex programming and their generalizations. Math. Program. Study 10, 86-97, (1979). 13. J.P. Gossez, E. Lami Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings. Pac. J. Math. 40, 565-573, (1972).

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14. Y. Hao, Strong convergence of an iterative method for inverse strongly accretive operators, J. Inequl. Appl. 2008 (2008), Article ID 420989. 15. U. Kamraksa and R. Wangkeeree, A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces, Journal of Inequalities and Applications, (2010), Article ID Doi:10.1155/2010/190126. 16. S. Kitahara and W. Takahashi, Image recovery by convex combinations of sunny nonexpansive retractions, Topol. Meth. Nonlinear Anal. 2 (1993), 333-342. 17. F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive type mappings in Banach spaces, SIAM Journal on Optimization, vol. 19, no. 2, pp. 824-835, 2008. 18. A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46-55. 19. A. Moudafi and M. Thera, Proximal and dynamical approaches to equilibrium problems, Lecture note in Economics and Mathematical Systems, Springer-Verlag, New York, 477 (1999), 187-201. 20. M.A. Noor, Generalized set-valued variational inclusions and resulvent equations, J. Math. Anal. Appl. 228 (1998), 206-220. 21. M. Aslam Noor, Some development in general variational inequalities. Appl. Math. Comput. 152, 199- 277 (2004). 22. Y. Yao, M. A. Noor, On viscosity iterative methods for variational inequalities. J. Math. Anal. Appl. 325, 776-787 (2007). 23. M.A. Noor, New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217229 (2000). 24. M. Aslam Noor, K. Inayat Noor, Self-adaptive projection algorithms for general variational inequalities. Appl. Math. Comput. 151, 659-670 (2004). 25. M. Aslam Noor, K. Inayat Noor, Th.M. Rassias, Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285-312 (1993). 26. M. Aslam Noor, General variational inequalities. Appl. Math. Lett. 1, 119121 (1988) 27. S. Plubtieng, K. Ungchittrakool, Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications, Nonlinear Analysis 72 (2010) 2896-2908. 28. S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl. 44 (1973), 57-70. 29. D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). 30. T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochne integrals, J. Math. Anal. Appl. 305 (2005), 227-239. 31. W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Analysis: Theory, Methods and Applications, vol. 70,1, 45-57, 2009. 32. S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces , J. Math. Anal. Appl. impress. 33. R.U. Verma, General over-relaxed proximal point algorithm involving A-maximal relaxed monotone mappings with applications, Nonlinear Anal. TMA (2009), doi:10.1016/j.na.2009.01.184. 34. R.U. Verma, A-monotonicity and applications to nonlinear variational inclusion problems, J. Appl. Math. Stoch. Anal. 17 (2) (2004) 193-195. 35. R.U. Verma, Approximation-solvability of a class of A-monotone variational inclusion problems, J. Korea Soc. Ind. Appl. Math. 8 (1) (2004) 55-66. 36. Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279-291. 37. H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), 1127-1138. 38. H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), 240-256. 39. Y. Yao, M.A. Noor, K.I. Noor, Y.C. Liou and H. Yaqoob, Modified extragradient methods for a system of variational inequalities in Banach spaces, Acta Appl Math (2010) 110: 1211-1224. 40. H. Zhou, Convergence theorems for λ-strict pseudo-contractions in 2-uniformly smooth Banach spaces, Nonlinear Anal. 69 (2008) 3160-3173. 41. D. L. Zhu,P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim. 6, 774-726 (1996). 42. L. Zhao, S.S. Chang and M. Liu, Viscosity approximation algorithms of common solutions for fixed points of infinite nonexpansive mappings and quasi-variational inclusion problems, Commun. Appl. Nonlinear Anal. 15 (2008) 83-98. 43. S.S. Zhang, J.H.W. Lee and C.K. Chan, Algorithms of common solutions for quasi variational inclusion and fixed point problems, Appl. Math. Mech. 29 (2008) 571-581.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 678-685, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Best Proximity Points For Cyclical Contraction Mappings With 0−Boundedly Compact Decompositions T. Abdeljawada,b , J. O. Alzabutb,1 , A. Mukheimerb , Y. Zaidanb,c a Department

of Mathematics, C ¸ ankaya University, 06530, Ankara, Turkey of Mathematics and Physical Sciences, Prince Sultan University P. O. Box 66833, Riyadh 11586, Saudi Arabia c Department of Mathematics, University of Wisconsin–Fox Valley

b Department

Menasha, WI 54952, USA

Abstract. The existence of best proximity points for cyclical type contraction mappings is proved in the category of partial metric spaces. The concept of 0−boundedly compact is introduced and used in the cyclical decomposition. Some possible generalizations to the main results are discussed. Further, illustrative examples are given to demonstrate the effectiveness of our results. Keywords. Partial metric space; Best proximity point; Cyclic mapping; Banach contraction principle; Boundedly compact set; 0−Compact set; φ−Cyclical contraction.

1

Introduction and Preliminaries

In [1], Petru¸sel proved some periodic point results for cyclic contraction maps. Later on in [2], some results were established on best proximity points of cyclic contraction maps. Indeed, the authors raised a question about the existence of a best proximity point for a cyclic contraction map on a reflexive Banach space. In [3], an affirmative answer to this question was provided. Specifically, some results on the existence and convergence of best proximity points of cyclic contraction maps defined on reflexive Banach spaces were presented. Moreover, the authors introduced the notion of cyclic ϕ−contraction maps. It is worth mentioning that all the above results were proved under the category of metric spaces; see [4] for more details. The notion of partial metric space was first introduced in [5]. In particular, the author defined the partial metric space and proved its version of Banach fixed point theorem. After then, some generalizations of this result were proved in [6, 7, 8, 9, 10, 11, 12, 13, 14]. Following this trend, this paper carries out the investigation and prove the existence of best proximity points for cyclical type contraction mappings in partial metric space. Before proceeding to the main results, we recall some definitions and properties in the framework of partial metric spaces. A partial metric space (PMS) is a pair (X, p : X × X → R+ ) (where R+ denotes the set of all non negative real numbers) such that (P1) p(x, y) = p(y, x) (symmetry); (P2) If 0 ≤ p(x, x) = p(x, y) = p(y, y) then x = y (equality); (P3) p(x, x) ≤ p(x, y) (small self–distances); (P4) p(x, z) + p(y, y) ≤ p(x, y) + p(y, z) (triangularity); for all x, y, z ∈ X. For a partial metric p on X, the function ps : X × X → R+ given by ps (x, y) = 2p(x, y) − p(x, x) − p(y, y) 1 Corresponding

Author E-Mail Address: [email protected]

678

(1)

ABDELJAWAD ET AL: BEST PROXIMITY POINTS

is a (usual) metric on X. Each partial metric p on X generates a T0 topology τp on X with a base of the family of open p−balls {Bp (x, ε) : x ∈ X, ε > 0}, where Bp (x, ε) = {y ∈ X : p(x, y) < p(x, x) + ε} for all x ∈ X and ε > 0. Definition 1. [5] (i) A sequence {xn } in a PMS (X, p) converges to x ∈ X if and only if p(x, x) = limn→∞ p(x, xn ). (ii) A sequence {xn } in a PMS (X, p) is called a Cauchy if and only if limn,m→∞ p(xn , xm ) exists (and finite). (iii) A PMS (X, p) is said to be complete if every Cauchy sequence {xn } in X converges, with respect to τp , to a point x ∈ X such that p(x, x) = limn,m→ ∞ p(xn , xm ). (iv) A mapping f : X → X is said to be continuous at x0 ∈ X if for every ε > 0 there exists δ > 0 such that f (Bp (x0 , δ)) ⊂ Bp (f (x0 ), ε). Lemma 1. [5] (a1) A sequence {xn } is Cauchy in a PMS (X, p) if and only if {xn } is Cauchy in a metric space (X, ps ). (a2) A PMS (X, p) is complete if and only if the metric space (X, ps ) is complete. Moreover, lim ps (x, xn ) = 0 ⇔ p(x, x) = lim p(x, xn ) =

n→∞

n→∞

lim p(xn , xm ).

n,m→ ∞

(2)

Lemma 2. Let (X, p) be a partial metric space and T : X → X be a continuous self–mapping. Assume {xn } ∈ X such that xn → z as n → ∞. Then lim p(T xn , T z) = p(T z, T z).

n→∞

Proof. Let  > 0 be given. Since T is continuous at z find δ > 0 such that T (Bp (z, δ)) ⊆ Bp (T z, ). Since xn → z then limn→∞ p(xn , z) = p(z, z) and hence find n0 ∈ N such that p(z, z) ≤ p(xn , z) < p(z, z) + δ for all n ≥ n0 . That is, xn ∈ Bp (z, δ) for all n ≥ n0 . Thus T (xn ) ∈ Bp (T z, ) and so p(T z, T z)) ≤ p(T xn , T z) < p(T z, T z) +  for all n ≥ n0 . This shows our claim. A sequence {xn } is called 0−Cauchy if limm,n→∞ p(xn , xm ) = 0. The partial metric space (X, p) is called 0–complete if every 0–Cauchy sequence in x converges to a point x ∈ X with respect to p and p(x, x) = 0. Clearly, every complete partial metric space is 0–complete. The converse need not be true; see [14] for more details. Example 1. Let X = Q ∩ [0, ∞) with the partial metric p(x, y) = max{x, y} where Q is the set of rational numbers. Then (X, p) is a 0–complete partial metric space which is not complete. The following theorem is the core of the extensions to the partial metric space case. Theorem 1. [5, 14] Let(X, p) be a 0−complete partial metric space and f : X → X be such that p(f (x), f (y)) ≤ αp(x, y) ∀ x, y ∈ Xand α ∈ [0, 1). Then there exists a unique u ∈ X such that u = f (u) and p(u, u) = 0. Let ρp = inf{p(x, y) : x, y ∈ X} and define Xp = {x ∈ X : p(x, x) = ρp }.

679

ABDELJAWAD ET AL: BEST PROXIMITY POINTS

Theorem 2. [15] Let (X, p) be a complete metric space, α ∈ [0, 1) and T : X → X be a given mapping. Suppose that for each x, y ∈ X the following condition holds p(x, y) ≤ max{αp(x, y), p(x, x), p(y, y)}. Then (1) the set Xp is nonempty; (2) there is a unique u ∈ Xp such that T u = u; (3) for each x ∈ Xp the sequence {T n x}n≥1 converges to u with respect to the metric ps . Definition 2. Let A and B be two nonempty closed subsets of a partial metric space (X, p) such that X = A ∪ B. A mapping T : X → X is called cyclical contraction if it satisfies (C1) T (A) ⊆ B and T (B) ⊆ A; (C2) ∃ 0 < α < 1 : p(T x, T y) ≤ αp(x, y) + (1 − α)p(A, B), ∀ x ∈ A and ∀ y ∈ B, where p(A, B) = inf {p(x, y) : x ∈ A, y ∈ B}. A point x ∈ X = A ∪ B is called a best proximity point of T if p(x, T x) = p(A, B). The proof of the following lemma can be easily achieved by using the partial metric topology. Lemma 3. A subset A of a partial metric space is closed if and only if x ∈ A whenever xn ∈ A satisfies xn → x as n → ∞. Definition 3. A set A in a partial metric space (X, p) is called 0−compact if for any sequence {xn } in A there exists a subsequence {xnk } and x ∈ A such that limn→∞ p(xnk , x) = p(x, x) = 0. Clearly a closed subset of a 0–compact set is 0-compact. Lemma 4. [8, 10] Assume xn → z as n → ∞ in a PMS (X, p) such that p(z, z) = 0. Then limn→∞ p(xn , y) = p(z, y) for every y ∈ X. The paper is organized as follows: In Section 2 we present the 0−boundedly compactness concept in partial metric space and use it to prove an existence result of best proximity points for cyclical contraction mappings. In Section 3 we extend our results to φ−cyclical contraction mappings. On the other hand, we show that the extension is not possible if we use a partial type contraction cyclical map generalizing the one used in Theorem 2. Some examples are given through the presentation of our main results.

2

Best Proximity Points for Cyclical Contraction Mappings

Definition 4. A set A in a partial metric space (X, p) is called 0–boundedly compact if for any bounded sequence {xn } in A there exists a subsequence {xnk } and x ∈ A such that limn→∞ p(xnk , x) = p(x, x) = 0. Theorem 3. Let A and B be nonempty subsets of a partial metric space (X, p). Suppose that T : A ∪ B → A ∪ B is a cyclical contraction map. Then starting with any arbitrary x0 ∈ A ∪ B we have p(xn , T xn ) → p(A, B), where xn+1 = T xn , f or n ≥ 0.

680

ABDELJAWAD ET AL: BEST PROXIMITY POINTS

Proof. p(xn , xn+1 ) ≤ αp(xn−1 , xn ) + (1 − α)p(A, B) ≤ α(αp(xn−2 , xn−1 ) + (1 − α)p(A, B)) + (1 − α)p(A, B) . . . ≤ αn p(x1 , x0 ) + (1 − αn )p(A, B) Thus, limn→∞ p(xn , xn+1 ) = p(A, B) Theorem 4. Let A and B be nonempty closed subsets of a partial metric space (X, p). Suppose T : A ∪ B → A ∪ B is a cyclical contraction map. Let x0 ∈ A and define xn+1 = T xn , f or n ≥ 0. Suppose {x2n } has a convergent subsequence {x2nk } that converges to x ∈ A with p(x, x) = 0. Then p(x, T x) = p(A, B). Proof. In view of the the definition of p(A, B), we have 0 ≤ p(A, B) ≤ p(x, x2nk −1 ) ≤ p(x, x2nk ) + p(x2nk , x2nk −1 ). Then, the assumption on {x2nk } and Theorem 3 imply taht p(x, x2nk −1 ) → p(A, B). Further, we have p(A, B) ≤ p(x2nk , T x) ≤ p(x2nk −1 , x). By lemma 4, we conclude that p(x, T x) = p(A, B). Theorem 5. Let A and B be nonempty subsets of a partial metric space (X, p) and suppose that T : A ∪ B → A ∪ B is a cyclical contraction map. Let x0 ∈ A ∪ B and define xn+1 = T xn , f or n ≥ 0. Then the sequences {x2n } and {x2n+1 } are bounded. Proof. Take x0 ∈ A (the proof when x0 ∈ B is similar). Suppose, on the contrary, that {x2n+1 } is unbounded. Then there exists N0 such that and p(T 2 x0 , T 2N0 −1 x0 ) ≤ M, n o 0 ,T x0 ) 2 + p(A, B), p(T x , T x ) . Using the cyclic contraction property and where M > max 2p(x 2 0 0 1/α −1 rearranging the terms, we obtain p(T 2 x0 , T 2N0 +1 x0 ) > M

M − p(A, B) + p(A, B) < p(x0 , T 2N0 −1 x0 ) α2 ≤ p(x0 , T 2 x0 ) + p(T 2 x0 , T 2N0 −1 x0 ) ≤ 2p(x0 , T x0 ) + M. 0 ,T x0 ) Solving for M we get M < 2p(x 1/α2 −1 + p(A, B). However, this contradiction shows that {x2n+1 } is bounded. Now we use this result and Theorem 3 to show that {x2n } is bounded. More precisely,

p(x2n , x2m ) ≤ p(x2n , x2n+1 ) + p(x2n+1 , x2m+1 ) + p(x2m+1 , x2m ) ≤ L for some positive real number L. The proof is complete.

681

ABDELJAWAD ET AL: BEST PROXIMITY POINTS

Theorem 6. Let A and B be nonempty closed subsets of a partial metric space (X, p) and T : A ∪ B → A ∪ B be a cyclical contraction map. If either A or B is 0−boundedly compact, then there exists x ∈ A ∪ B such that p(x, T x) = p(A, B). That is, x is a best proximity point of T . The proof of the above statement is straightforward and follows from Theorem 4 and Theorems 5. The following example shows that the generalization to partial metric space case is meaningful. Example 2. Let A = [0, 1], B = [3, 4] ∪ { 32 } and X = A ∪ B. Define p : X × X → [0, ∞) by p(x, y) =

  |x − y|

,

 max{x, y} ,

x, y ∈ [0, 1] otherwise

.

(3)

Then (X, p) is a partial metric space and A is 0−boundedly compact. Define T : X → X by  3 , 0≤x≤1    2 1 , x = 32 T (x) = . (4) 2    x−2 , 3≤x≤4 2 Then T is a cyclical contraction map. We can easily see that p(1/2, T (1/2)) = p(A, B) = 3/2. Therefore x = 1/2 is a best proximity point. However, if we consider the metric d(x, y) = |x − y| on whole of X, one can deduce that T is not a contraction map. Here is an example of a compact space which is not 0−compact. Example 3. Let X = [0, 1] and A = [1/2, 1]. Define p : X → [0, ∞) by  , x, y ∈ [0, 1/2)  |x − y| p(x, y) = .  max{x, y} , otherwise Then (X, p) is a complete partial metric space and A is compact (boundedly compact) but not 0−compact (0−boundedly compact). Indeed, if {xn } is a sequence in A, then using that A is compact under the usual metric, there exists a subsequence {xnk } and x ∈ A such that |xnk −x| → 0. Since |xnk − x| = ps (xnk , x) = 2p(xnk , x) − p(xnk , xnk ) − p(x, x), then by Lemma 1 we obtain limn→∞ p(xnk , x) = p(x, x) = x 6= 0.

3

More Generalizations

Definition 5. Let A and B be nonempty closed subsets of a partial metric space (X, p) and ϕ : [0, ∞) → [0, ∞) be a strictly increasing map. The map T : A ∪ B → A ∪ B is called a cyclical ϕ−contraction map if (C2) in Definition 2 is replaced by the following contraction principle p(T x, T y) ≤ p(x, y) − ϕ(p(x, y)) + ϕ(p(A, B)), ∀ x ∈ A and ∀ y ∈ B. In what follows, we provide generalization for the results obtained in Section 2 based on the case of cyclical ϕ−contraction maps.

682

ABDELJAWAD ET AL: BEST PROXIMITY POINTS

Theorem 7. Let A and B be nonempty subsets of a partial metric space (X, p) and ϕ : [0, ∞) → [0, ∞) be a strictly increasing map. Suppose that T : A ∪ B → A ∪ B is a cyclical ϕ−contraction map. For x0 ∈ A ∪ B, define xn+1 = T xn for each n ≥ 1. Then p(xn , xn+1 ) → p(A, B) as n → ∞. Theorem 8. Let A and B be nonempty subsets of a partial metric space (X, p) and ϕ : [0, ∞) → [0, ∞) be a strictly increasing map. Suppose that T : A ∪ B → A ∪ B is a cyclical ϕ-contraction map. For x0 ∈ A ∪ B, define xn+1 = T xn for each n ≥ 1. Suppose that {x2n } has a convergent subsequence {x2nk } that converges to x ∈ A with p(x, x) = 0. Then p(x, T x) = p(A, B). Theorem 7 and Theorem 8 can be proved using the same approach followed in [3]. Theorem 9. Let A and B be nonempty subsets of a partial metric space (X, p) and ϕ : [0, ∞) → [0, ∞) be a strictly increasing map. Suppose that T : A ∪ B → A ∪ B is a cyclical φ−contraction map. Let x0 ∈ A ∪ B and define xn+1 = T xn , n ≥ 1. Then the sequences {x2n } and {x2n+1 } are bounded. The proof of Theorem 9 can be carried out as in [4]. By employing Theorem 7, Theorem 8 and Theorem 9, one can immediately state the following general theorem. Theorem 10. Let A and B be nonempty closed subsets of a partial metric space (X, p) and ϕ : [0, ∞) → [0, ∞) be a strictly increasing map. Suppose that T : A ∪ B → A ∪ B is a cyclical ϕ−contraction map. If either A or B is 0−boundedly compact, then there exists x ∈ A ∪ B such that p(x, T x) = p(A, B). In what follows, we present an example that shows that the above theorems can not be extended when the cyclical contractive condition is replaced by p(T x, T y) ≤ max{αp(x, y), p(x, x), p(y, y)} + (1 − α)p(A, B).

(5)

Condition (5) is considered to be the best proximity version of the contraction type mapping introduced in [15]. Mappings satisfying (5) might be called partial cyclical contractions. Example 4. Let X = A ∪ B where A = [0, 1] ∪ {5} and B = [3, 4]. Define p : X → [0, ∞) by  , x, y ∈ A  |x − y| p(x, y) = .  max{x, y} , otherwise Then (X, p) is a partial metric space. Define T  4    x−2 T (x) = 2    5

: X → X by ,

x∈A

, 3 3 + 3(1 − α) = 6 − 3α.

Thus, condition 5 is satisfied. Case 2. x = 5, y = 3 p(T x, T y) = p(4, 5) = 5. Further, max{αp(x, y), p(x, x), p(y, y)} + (1 − α)p(A, B)

= max{5α, 0, 3} + 3(1 − α) = 6 − 3α

and therefore, condition 5 is satisfied. Case 3. 0 ≤ x ≤ 1, y = 3 p(T x, T y) = p(4, 5) = 5. Since max{αp(x, y), p(x, x), p(y, y)} + (1 − α)p(A, B)

= max{3α, 0, 3} + 3(1 − α) = 6 − 3α

then condition 5 is satisfied. Case 4. 0 ≤ x ≤ 1, 3 < y ≤ 4 

y−2 p(T x, T y) = p 4, 2

 = 4.

Further, max{αp(x, y), p(x, x), p(y, y)} + (1 − α)p(A, B)

= max{αy, 0, y} + 3(1 − α) > 6 − 3α.

Hence, condition 5 is satisfied. It remains to show that there are no proximity points for T . To do this, we consider the following cases. Case 1. 0 ≤ x ≤ 1 : p(x, T x) = p(x, 4) = max{x, 4} = 4 6= p(A, B) = 3. Case 2. x = 5 : p(5, T 5) = p(5, 4) = max{5, 4} = 5 6= p(A, B) = 3. Case 3. x = 3 : p(3, T 3) = p(3, 5) = 5 6= p(A, B) = 3.
Case 4. 3 < x ≤ 4 : p(x, T x) = p x, x−2 = x 6= p(A, B) = 3. 2

References [1] G. Petrusel, Cyclic representations and periodic points, Stud. Univ. Babes–Bolyai Math 50 (3) (2005), 107–112. [2] A. A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2) (2006), 1001–1006. [3] M. A. Al–Thaqafi, N. Shahzad, Convergence and existence results f or best proximity points, Nonlinear Anal. 70 (2009), 3665—3671.

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[4] Sh. Rezapour, M. Derafshpour, N. Shahzad, Best proximity points of φ-cyclic contractions on reflexive Banach spaces, Fixed Point Theory Appl., (2010 ) doi: 10.1155/2010/946178. 7 pages, Article ID 946178. [5] S. G. Matthews, Partial metric topology, in Proceedings of the 11–th Summer Conference on General Topology and Applications, vol. 728, pp. 183–197, The New York Academy of Sciences, Gorham, Me, USA, Augusts 1995. [6] S. Oltra, O. Valero, Banach’s fixed point theorem for partial metric spaces, Rend. Istit. Mat. Univ. Trieste 36 (1–2) (2004), 17–26. [7] O. Valero, On Banach fixed point theorems for partial metric spaces, Appl. Gen. Topol. 6 (2) (2005), 229–240. [8] T. Abdeljawad, E. Karapinar, K. Ta¸s, Existence and uniqueness of a common fixed point on partial metric spaces, Appl. Math. Lett. 24 (11) (2011), 1900–1904. [9] T. Abdeljawad, E. Karapinar, K. Ta¸s, A generalized contraction principle with control functions on partial metric spaces, Comput. Math. Appl., In Press, Corrected Proof, doi:10.1016/j.camwa.2011.11.035, 2011. [10] T. Abdeljawad, Fixed points for generalized weakly contractive mappings in partial metric spaces, Math. Comput. Modelling 54 (11-12) (2011), 2923–2927. [11] I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology Appl. 157 (18) (2010), 2778–2785. [12] I. Altun and A. Erduran, Fixed Point Theorems for Monotone Mappings on Partial Metric Spaces, Fixed Point Theory Appl., vol. 2011, Article ID 508730, 10 pages, 2011. doi:10.1155/2011/508730. [13] W. Shatanawi, B. Samet, M. Abbas, Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces, Math. Comput. Modelling 55 (3-4) (2012), 680–687. [14] S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl., (2010 ) doi: 10.1155/2010/493298. 6 pages, Article ID 493298. [15] D. Ili´ c, V. Pavlovi´ c and V. Rako˘ cevi´ c, Some new extensions of Banach’s contraction principle to partial metric spaces, Appl. Math. Lett. 24 (2011), 1326–1330.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 686-691, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

NUMERICAL COMPUTATIONS OF THE DISTRIBUTION OF THE ZEROS OF THE SECOND KIND (h, q)-EULER POLYNOMIALS C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea (h)

Abstract : In [4, 5], we introduced the second kind (h, q)-Euler numbers En,q and polynomials (h) En,q (x). In this paper, we observe the behavior of complex roots of the second kind (h, q)-Euler (h)

polynomials En,q (x), using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the second kind (h, q)-Euler polynomials (h) (h) En,q (x). Finally, we give a table for the solutions of the second kind (h, q)-Euler polynomials En,q (x). Key words : the second kind Euler numbers and polynomials, the second kind (h, q)-Euler numbers and polynomials, complex roots, numerical experiments 1. Introduction Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, R denotes the set of real numbers, and C denotes the set of complex numbers. First, we introduce the second kind Euler numbers En and polynomials En (x). The second kind Euler numbers En are defined by the generating function: F (t) =

∞  2et tn = En . 2t e + 1 n=0 n!

(1.1)

We introduce the second kind Euler polynomials En (x) as follows: F (x, t) =

∞ tn 2et xt  e = En,q (x) . 2t e +1 n! n=0

(1.2)

Recently, many authors have studied the Euler numbers and polynomials, the second kind Euler numbers and polynomials. The Euler numbers and polynomials, the second kind Euler numbers and polynomials posses many interesting properties and arising in many areas of mathematics (h) and physics. In [4, 5], we constructed the second kind (h, q)-Euler numbers En,q and polynomials (h) En,q (x). By using these numbers and polynomials, we obtained the recurrence identities the second kind (h, q)-Euler polynomials and the alternating sums of powers of consecutive (h, q)-odd integers (h) and some interesting properties. In order to study the second kind (h, q)-Euler numbers En,q and (h) polynomials En,q (x), we must understand the structure of the second kind (h, q)-Euler numbers (h) (h) En,q and polynomials En,q (x). Therefore, using computer, a realistic study for the second kind (h)

(h)

(h, q)-Euler numbers En,q and polynomials En,q (x) is very interesting. It is the aim of this paper to observe an interesting phenomenon of ‘scattering’ of the zeros of (h) the second kind (h, q)-Euler polynomials En,q (x) in complex plane. The outline of this paper is as (h) (h) follows. In Section 2, introduce the second kind (h, q)-Euler numbers En,q and polynomials En,q (x). (h) In Section 3, we describe the beautiful zeros of the second kind (h, q)-Euler polynomials En,q (x) using a numerical investigation. Finally, we investigate the roots of the second kind (h, q)-Euler (h)

polynomials En,q (x). Also we carried out computer experiments for doing demonstrate a remarkably (h) regular structure of the complex roots of the second kind (h, q)-Euler polynomials En,q (x). 686

RYOO: ZEROS OF (h, q)-EULER POLYNOMIALS

2. The second kind (h, q)-Euler numbers and polynomials (h)

(h)

In this section, we introduce the second kind (h, q)-Euler numbers En,q and polynomials En,q (x) and investigate their properties. Let q be a complex number with |q| < 1 and h ∈ Z. By the meaning (h) (h) of (1.1) and (1.2), let us define the second kind (h, q)-Euler numbers En,q and polynomials En,q (x) as follows(see [4, 5]): Fq(h) (t) = Fq(h) (x, t) =

∞ n  2et (h) t = , E n,q q h e2t + 1 n=0 n!

(2.1)

∞  2et tn xt (h) e = En,q (x) . h 2t q e +1 n! n=0

(h)

(2.2)

(h)

Observe that if q → 1, then En,q (x) = En (x), En,q = En . (h) By using computer, the second kind (h, q)-Euler numbers En,q can be determined explicitly. A few of them are 2 4q h 2 (h) (h) E0,q = , E = − + , 1,q 1 + qh (1 + q h )2 1 + qh 16q 2h 16q h 2 (h) E2,q = − + , h 3 h 2 (1 + q ) (1 + q ) 1 + qh 96q 3h 144q 2h 52q h 2 (h) + − + . E3,q = − h 4 h 3 h 2 (1 + q ) (1 + q ) (1 + q ) 1 + qh (h)

The following elementary properties of the second kind q-Euler polynomials En,q (x) are readily derived from (2.1) and (2.2). We, therefore, choose to omit the details involved. More studies and results in this subject we may see references [4], [5]. By the above definition, we obtain ∞  l=0

(h)

El,q (x)

∞ ∞ n   2et tl tm (h) t = h 2t ext = En,q xm l! q e +1 n! m=0 m! n=0   ∞ l n   tl−n (h) t xl−n En,q = n! (l − n)! l=0 n=0   ∞ l     l tl (h) l−n En,q x . = n l! n=0 l=0

l

t , we have the following theorem. l! Proposition 1. For any positive integer n, we have n    n (h) (h) Ek,q xn−k . (x) = En,q k

By using comparing coefficients

k=0

(h)

Proposition 2. The second kind (h, q)-Euler numbers En,q are defined by  2, if n = 0, h (h) n (h) n q (Eq + 1) + (Eq − 1) = 0, if n > 0,  n (h) (h) with the usual convention about replacing Eq by En,q in the binomial expansion. (h)

The second kind (h, q)-Euler polynomials En,q (x) can be determined explicitly. A few of them are

2 4q h 2 2x (h) , E1,q (x) = − + + , h 1+q (1 + q h )2 1 + qh 1 + qh 16q 2h 16q h 2 8q h x 4x 2x2 (h) − + − + + . E2,q (x) = h 3 h 2 h h 2 h (1 + q ) (1 + q ) 1+q (1 + q ) 1+q 1 + qh (h)

E0,q (x) =

687

RYOO: ZEROS OF (h, q)-EULER POLYNOMIALS

Theorem 3. For any positive integer n, we have ∞ 

(h) En,q (x)

n=0

∞  tn =2 (−1)n q hn e(2n+x+1)t . n! n=0

Theorem 4(Difference equation). For any positive integer n, we have (h) (h) q h En,q (x + 2) + En,q (x) = 2(1 + x)n .

Theorem 5(Theorem of complement). For any positive integer n, we have (h)

(h) En,q (x) = (−1)n q −h En,q−1 (−x).

(h)

3. Zeros of the second kind (h, q)-Euler polynomials En,q (x) In this section, we assume that q ∈ C, with 0 < q < 1 and h ∈ N. We investigate the zeros (h) (h) of the second (h, q)-Euler polynomials En,q (x). We investigate the beautiful zeros of the En,q (x) (h)

by using a computer. We plot the zeros of the second kind (h, q)-Euler polynomials En,q (x) for n = 30, h = 5, q = 1/5, 2/5, 3/5, 4/5 and x ∈ C(Figure 1). We plot the zeros of the second kind

Imx

15

15

10

10

5

5

Imx

0

0

-5

-5

-10

-10

-10

-5

0

5

10

15

-10

-5

Rex

Imx

15

15

10

10

5

5

Imx

0

-5

-10

-10

-5

0

5

10

15

5

10

15

0

-5

-10

0

Rex

5

10

15

Rex

-10

-5

0

Rex

(5)

Figure 1: Zeros of E30,q (x) for q = 1/5, 2/5, 3/5, 4/5 (h)

(h, q)-Euler polynomials En,q (x) for n = 30, h = 10, 15, 20, 25, q = 1/2 and x ∈ C(Figure 2). Plot (h) of real zeros of En,q (x) for 1 ≤ n ≤ 30, h = 5, q = 1/2, 4/5 structure are presented(Figure 3). Our (h) numerical results for approximate solutions of real zeros of En,q (x) are displayed(Tables 1, 2). 688

RYOO: ZEROS OF (h, q)-EULER POLYNOMIALS

6

6

4

4

2

2

Imx 0

Imx 0

-2

-2

-4

-4

-6

-6 -4

-6

-2

0

2

4

6

-4

-6

-2

Rex

6

6

4

4

2

2

Imx 0

Imx 0

-2

-2

-4

-4

-6

-6 -4

-6

-2

0

0

2

4

6

2

4

6

Rex

2

4

6

-4

-6

Rex

-2

0

Rex

(h)

Figure 2: Zeros of E30,1/2 (x) for h = 10, 15, 20, 25

(h)

Table 1. Numbers of real and complex zeros of En,1/2 (x) degree n

h=2 real zeros complex zeros

h=5 real zeros complex zeros

1

1

0

1

0

2

2

0

2

0

3

3

0

1

2

4

2

2

2

2

5

3

2

3

2

6

4

2

2

4

7

5

2

3

4

8

4

4

4

4

9

3

6

3

6

10

4

6

4

6

11

3

8

3

8

12

4

8

4

8

13

5

8

5

8

We observe a remarkably regular structure of the complex roots of the second kind (h, q)-Euler (h) polynomials En,q (x). We hope to verify a remarkably regular structure of the complex roots of the 689

RYOO: ZEROS OF (h, q)-EULER POLYNOMIALS

30

30

20

20

n

n

10

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

10

0

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Rex

0

Rex (5)

Figure 3: Real zeros of En,q (x) for q = 1/2, 4/5, 1 ≤ n ≤ 30

(h)

second kind (h, q)-Euler polynomials En,q (x)(Table 1). Next, we calculate an approximate solution (h) satisfying En,q (x), q = 1/2, x ∈ R. The results are given in Table 2.

(2)

Table 2. Approximate solutions of En,1/2 (x) = 0, x ∈ R degree n

x

1

−0.600000000 −1.400000000,

2 3

−0.325604236, −1.5750355,

5 6 7

−1.04629976,

1.70749372,

4

−2.3075000, −2.413540,

0.953793481

1.633725201

0.412159430,

−0.85141325,

−2.190476,

0.200000000

2.24041323

1.14858713,

−0.116593525,

2.77313789

1.88561086,

3.22312848 (h)

Finally, we shall consider the more general problems. In general, how many roots does En,q (x) (h) have? This is open problem. Prove or disprove: En,q (x) = 0 has n distinct solutions. Find the (h)

numbers of complex zeros CE (h) (x) of En,q (x), Im(x) = 0. Since n is the degree of the polynomial (h)

n,q

En,q (x), the number of real zeros RE (h) (x) lying on the real plane Im(x) = 0 is then RE (h) (x) = n,q n,q n − CE (h) (x) , where CE (h) (x) denotes complex zeros. See Table 1 for tabulated values of RE (h) (x) and n,q n,q n,q CE (h) (x) . Observe that the structure of the zeros of the second Euler polynomials En (x) resembles n,q

(h)

the structure of the zeros of the second kind (h, q)-Euler polynomials En,q (x) as q → 1(see Figures 1, 2, 3). The author has no doubt that investigation along this line will lead to a new approach employing numerical method in the field of research of the second kind (h, q)-Euler polynomials (h) En,q (x) to appear in mathematics and physics. The reader may refer to [3, 5] for the details.

690

RYOO: ZEROS OF (h, q)-EULER POLYNOMIALS

h En,q x

h En,q x

0.6

0.6 q

0.8

q

0.8

0.4

0.4

0.2

0.2

-2

-1

0 x

1

2

-2

-1

h En,q x

0 x

1

2

1

2

h En,q x

0.6

0.6 q

0.8

q

0.8

0.4

0.4

0.2

0.2

-2

-1

0 x

1

2

-2

-1

0 x

(h)

Figure 4: Zero contour of En,q (x)

(h)

The plot above shows En,q (x) for real 1/10 ≤ q ≤ 9/10 and −2 ≤ x ≤ 2, with the zero contour indicated in black(Figure 4). In Figure 4(top-left), we choose n = 2 and h = 3. In Figure 4(topright), we choose n = 3 and h = 3. In Figure 4(bottom-left), we choose n = 4 and h = 3. In Figure 4(bottom-right), we choose n = 5 and h = 3. REFERENCES 1. Kim, T. (2007). q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., v. 14, pp. 15-27. 2. Kim, T. (2008). Euler numbers and polynomials associated with zeta function, Abstract and Applied Analysis, Art. ID 581582. 3. Ryoo, C. S. (2010). Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, v.12, pp. 828-833. 4. Ryoo, C. S. (2011). A note on the second kind (h, q)-Euler polynomials, Far East Journal of Mathematical Sciences, v.49, pp. 35-41. 5. Ryoo, C. S. (2012). On the symmetric properties for the second kind (h, q)-Euler polynomials, Journal of Computational Analysis and Applications, v.14, pp. 785-791.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 692-698, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Fixed points and stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras Jung Rye Lee Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea e-mail: [email protected]

Sung Jin Lee∗ Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea e-mail: [email protected]

Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea e-mail: [email protected] Abstract. Using fixed point method, we prove the Hyers-Ulam stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras. Keywords: fuzzy Banach algebra, fixed point, Hyers-Ulam stability, Cauchy-Jensen functional equation.

1. Introduction and preliminaries The theory of fuzzy space has much progressed as developing the theory of randomness. Some mathematicians have defined fuzzy norms on a vector space from various points of view [2, 16, 22, 24, 27, 38]. Following Cheng and Mordeson [7], Bag and Samanta [2] gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [23] and investigated some properties of fuzzy normed spaces [3]. We use the definition of fuzzy normed spaces given in [2, 27, 28] to investigate a fuzzy version of the Hyers-Ulam stability for the Cauchy-Jensen functional equation in the fuzzy normed algebra setting. Definition 1.1. [2, 27, 28, 29] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1; (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. Definition 1.2. [2, 27, 28, 29] (1) Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N -limn→∞ xn = x. (2) Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

∗

02010 Mathematics Subject Classification: Primary 47H10, 46S40, 39B52, 26E50. Corresponding author.

692

J. Lee, S. Lee, C. Park We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [3]). Definition 1.3. Let X be an algebra and (X, N ) a fuzzy normed space. (1) The fuzzy normed space (X, N ) is called a fuzzy normed algebra if N (xy, st) ≥ N (x, s) · N (y, t) for all x, y ∈ X and all positive real numbers s and t. (2) A complete fuzzy normed algebra is called a fuzzy Banach algebra. Example 1.4. Let (X, k · k) be a normed algebra. Let

 N (x, t) =

t t+kxk

0

t > 0, x ∈ X t ≤ 0, x ∈ X.

Then N (x, t) is a fuzzy norm on X and (X, N (x, t)) is a fuzzy normed algebra. Definition 1.5. Let (X, N ) and (Y, N ) be fuzzy normed algebras. Then a multiplicative R-linear mapping H : (X, N ) → (Y, N ) is called a fuzzy algebra homomorphism. The stability problem of functional equations was originated from a question of Ulam [37] concerning the stability of group homomorphisms. Hyers [18] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [34] for linear mappings by considering an unbounded Cauchy difference. The paper of Th.M. Rassias [34] has provided a lot of influence in the development of what we call the Hyers-Ulam stability or the HyersUlam-Rassias stability of functional equations. A generalization of the Th.M. Rassias theorem was obtained by G˘ avruta [17] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called a quadratic functional equation. The Hyers-Ulam stability of the quadratic functional equation was proved by Skof [36] for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa [11] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [12] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [9, 10, 13, 14, 19, 21]). Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. We recall a fundamental result in fixed point theory. Theorem 1.6. [4, 15] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [20] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5, 6, 8, 26], [30]–[33]).

693

Cauchy-Jensen functional equation in fuzzy Banach algebras In this paper, we prove the Hyers-Ulam stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras by using fixed point method. Throughout this paper, assume that (X, N ) is a fuzzy normed algebra and that (Y, N ) is a fuzzy Banach algebra. 2. Hyers-Ulam stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras: fixed point method Using fixed point method, we prove the Hyers-Ulam stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras. Theorem 2.1. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < L ϕ(2x, 2y, 2z) 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying     rx + ry N 2f + rz − rf (x) − rf (y) − 2rf (z), t 2

1 2

with

ϕ(x, y, z) ≤

N (f (xy) − f (x)f (y), t)

≥ ≥

for all x, y, z ∈ X, all t > 0 and all r ∈ R. Then H(x) := N -limn→∞ 2n f defines a fuzzy algebra homomorphism H : X → Y such that N (f (x) − H(x), t) ≥

t , t + ϕ (x, y, z) t t + ϕ (x, y, 0) x 2n




(2.1) (2.2)

exists for each x ∈ X and

(1 − L)t (1 − L)t + ϕ (x, 0, 0)

(2.3)

for all x ∈ X and all t > 0. Proof. Letting r = 1 and y = z = 0 in (2.1), we get     x t N 2f − f (x), t ≥ 2 t + ϕ (x, 0, 0)

(2.4)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: d(g, h) = inf{µ ∈ R+ : N (g(x) − h(x), µt) ≥

t , ∀x ∈ X, ∀t > 0}, t + ϕ (x, 0, 0)

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see the proof of [25, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ϕ (x, 0, 0) for all x ∈ X and all t > 0. Hence N (Jg(x) − Jh(x), Lεt)

= ≥



N 2g

Lt 2

+

x 2

 

− 2h

Lt 2 ϕ x2 , 0, 0

x , Lεt = N g 2

 


≥

Lt 2



+

x 2

  

Lt 2 L ϕ (x, 0, 0) 2

=

−h

x L , εt 2 2

 

t t + ϕ (x, 0, 0)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S.

694



J. Lee, S. Lee, C. Park It follows from (2.4) that d(f, Jf ) ≤ 1. By Theorem 1.6, there exists a mapping H : X → Y satisfying the following: (1) H is a fixed point of J, i.e., H

x 2

 

1 H(x) 2

=

(2.5)

for all x ∈ X. The mapping H is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that H is a unique mapping satisfying (2.5) such that there exists a µ ∈ (0, ∞) satisfying t t + ϕ (x, 0, 0)

N (f (x) − H(x), µt) ≥ for all x ∈ X; (2) d(J n f, H) → 0 as n → ∞. This implies the equality N - lim 2n f



n→∞

for all x ∈ X; (3) d(f, H) ≤

1 d(f, Jf ), 1−L

x 2n



= H(x)

which implies the inequality 1 . 1−L

d(f, H) ≤ This implies that the inequality (2.3) holds. By (2.1),



N 2k+1 f



rx + ry rz + k 2k+1 2



− 2k rf



x 2k



− 2k rf



y 2k





− 2k+1 rf

z 2k



t



, 2k t ≥

t+ϕ

x , y, z 2k 2k 2k




for all x, y, z ∈ X, all t > 0 and all r ∈ R. So



N 2k+1 f



rx + ry rz + k 2k+1 2



− 2k rf



x 2k



− 2k rf



y 2k



for all x, y, z ∈ X, all t > 0 and all r ∈ R. Since limk→∞ all r ∈ R,





N 2H







z 2k

t 2k k t + L ϕ(x,y,z) 2k 2k

t 2k

 

,t ≥

t 2k

+

Lk ϕ (x, y, z) 2k

= 1 for all x, y, z ∈ X, all t > 0 and

rx + ry + rz − rH(x) − rH(y) − 2rH(z), t = 1 2





rx+ry 2

for all x, y, z ∈ X, all t > 0 and all r ∈ R. Thus 2H mapping H : X → Y is additive and R-linear. By (2.2), N 4k f

− 2k+1 rf

xy 4k



x 2k



x 2k



− 2k f



− 2k f



y 2k



y 2k

 

· 2k f



· 2k f






+ rz − rH(x) − rH(y) − 2rH(z) = 0. So the

t



, 4k t ≥

t+ϕ

x , y ,0 2k 2k




for all x, y ∈ X and all t > 0. So



N 4k f



xy 4k



for all x, y ∈ X and all t > 0. Since limk→∞

t 4k k t + L ϕ(x,y,0) 4k 2k

,t ≥

t 4k t 4k

+

Lk ϕ (x, y, 0) 2k

= 1 for all x, y ∈ X and all t > 0,

N (H(xy) − H(x)H(y), t) = 1 for all x, y ∈ X and all t > 0. Thus H(xy) − H(x)H(y) = 0. So the mapping H : X → Y is a fuzzy algebra homomorphism, as desired. 

695

Cauchy-Jensen functional equation in fuzzy Banach algebras Theorem 2.2. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 with   x y z ϕ(x, y, z) ≤ 2Lϕ , , 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying (2.1) and (2.2). Then H(x) := N -limk→∞ exists for each x ∈ X and defines a fuzzy algebra homomorphism H : X → Y such that (1 − L)t (1 − L)t + Lϕ(x, 0, 0)

N (f (x) − H(x), t) ≥

1 f 2k




2k x

(2.6)

for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. Consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2 for all x ∈ X. It follows from (2.4) that



N f (x) −

1 1 t t f (2x), t ≥ ≥ 2 2 t + ϕ(2x, 0, 0) t + 2Lϕ(x, 0, 0)



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L. Hence d(f, H) ≤

L , 1−L

which implies that the inequality (2.6) holds. The rest of the proof is similar to the proof of Theorem 2.1.



3

Theorem 2.3. Let ϕ : X → [0, ∞) be a function such that there exists an L
0 and all r ∈ R. Then H(x) := N -limn→∞ 2n f defines a fuzzy algebra homomorphism H : X → Y such that N (f (x) − H(x), t) ≥

t t + ϕ (x, y, z) x 2n




(2.7)

exists for each x ∈ X and

(2 − 2L)t (2 − 2L)t + Lϕ (x, x, x)

(2.8)

for all x ∈ X and all t > 0. Proof. Letting r = 1 and y = z = x in (2.7), we get N (f (2x) − 2f (x), t) ≥

t t + ϕ (x, x, x)

(2.9)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: d(g, h) = inf{µ ∈ R+ : N (g(x) − h(x), µt) ≥

t , ∀x ∈ X, ∀t > 0}, t + ϕ (x, x, x)

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see the proof of [25, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 2g 2 for all x ∈ X.

696

J. Lee, S. Lee, C. Park It follows from (2.9) that x L t , t ≥ 2 2 t + ϕ (x, x, x) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L2 . Hence



N f (x) − 2f

 

d(f, H) ≤



L , 2 − 2L

which implies that the inequality (2.8) holds. The rest of the proof is similar to the proof of Theorem 2.1.



Theorem 2.4. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 with   x y z , , ϕ(x, y, z) ≤ 2Lϕ 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying (2.2) and (2.7). Then H(x) := N -limk→∞ exists for each x ∈ X and defines a fuzzy algebra homomorphism H : X → Y such that N (f (x) − H(x), t) ≥

1 f 2k




2k x

(2 − 2L)t (2 − 2L)t + ϕ(x, x, x)

for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.3. Consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2 for all x ∈ X. It follows from (2.9) that   1 1 t N f (x) − f (2x), t ≥ 2 2 t + ϕ(x, x, x) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 12 . The rest of the proof is similar to the proof of Theorem 2.1.



Conclusions We have introduced the concept of fuzzy algebras and we have proved the Hyers-Ulam stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras. Acknowledgments This work was supported by the Daejin University Research Grant in 2013. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705. [3] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [4] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [5] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [6] L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). [7] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [8] Y. Cho, J. Kang and R. Saadati, Fixed points and stability of additive functional equations on the Banach algebras, J. Comput. Anal. Appl. 14(2012), 1103–1111.

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Cauchy-Jensen functional equation in fuzzy Banach algebras [9] Y. Cho, C. Park, Th.M. Rassias and R. Saadati, Inner product spaces and functional equations, J. Comput. Anal. Appl. 13(2011), 296–304. [10] Y. Cho, C. Park and R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Letters 23 (2010), 1238–1242. [11] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [12] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [13] S. Czerwik, The stability of the quadratic functional equation. in: Stability of mappings of Hyers-Ulam type, (ed. Th.M. Rassias and J.Tabor), Hadronic Press, Palm Harbor, Florida, 1994, 81-91. [14] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [15] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [16] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [17] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [18] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [19] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [20] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [21] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [22] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [23] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [24] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [25] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [26] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [27] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [28] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. [29] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791–3798. [30] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). [31] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [32] C. Park, Y. Cho and H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in non-Archimedean spaces, J. Comput. Anal. Appl. 14(2012), 526–535. [33] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [34] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [35] R. Saadati and C. Park, Non-Archimedean L-fuzzy normed spaces and stability of functional equations, Computers Math. Appl. 60 (2010), 2488–2496. [36] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [37] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [38] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 699-706, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Intra-orbit separation of orbits of tree maps

∗

Zhanhe Chen† Taixiang Sun and Guangwang Su College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, P.R. China

Abstract In this paper, we will generalize the notion of separation index, which was introduced by G. Manjunath et al (2006), to tree maps, and use the notion to characterize the intra-orbit separation for the orbits of continuous transitive maps on a tree. For a transitive map 𝑓 , we will show: (i) if the separation index 𝛾 of 𝑓 is positive, then for every 0 < τ < 𝛾, (𝑥𝑠 , 𝑥𝑡 ) is a Li-Yorke pair of modulus τ , where 𝑥𝑠 and 𝑥𝑡 be any two distinct points on a dense orbit; (ii) there is a natural number 𝑛 such that the separation index of 𝑓 𝑛 is greater than zero. Keywords Tree map, transitive map, separation index, separating orbit Mathematics Subject Classification (2000): 37B05, 37B20, 54H20.

1

Introduction

A dynamical system is a pair (X, 𝑓 ), where X is a topological space called phase space and 𝑓 is a continuous self-map on X. In recent years, some researchers discuss dynamics of various discrete systems with different phase spaces in a variety of contexts and have obtained some remarkable results (see for example [1–4]). As a special topological space, a tree is a connected space that is the union of finitely many intervals (by an interval we mean any space homeomorphic to the closed interval [0, l]), no two of which intersect in more than one point. In this note, we are interested in a special dynamical system (T, 𝑓 ) whose phase space T is a tree. The dynamics of continuous self-maps on trees have been studied by several authors (see for example [5–8]). In paper [9], the authors introduced the notion of separation index to characterize the intraorbit separation for the orbits of continuous transitive maps on a compact interval and discuss some separation properties of orbits of transitive points. In this note, we will generalize the notion of separation index to tree maps and use it to illustrate the intra-orbit separation for the orbits of continuous tree maps. The structure of this paper is as follows. In the next Section, some notions and some simple but helpful lemmas are given. The main results and some corollaries and examples are shown in Section 3. ∗

Project Supported by NNSF of China (11126342, 11261005) and NSF of Guangxi (2011GXNSFA018135,

2010GXNSFA013109) and the Scientific Research Foundation of GuangXi University (Grant No. XBZ110565). † Corresponding author. E-mail: [email protected]

699

CHEN ET AL: INTRA-ORBIT SEPARATION

2

Some notions and lemmas

Let (X, 𝑓 ) be a dynamical system, we denote by C 0 (X) the set of continuous self-maps on X. For any subset A of X, we use 𝑑𝑖𝑎𝑚(A), C𝑎r𝑑(A) and ∂A to denote the diameter of A, the cardinality of A, the boundary of A, respectively. A point 𝑥 of X satisfying 𝑓 n (𝑥) = 𝑥 and 𝑓 k (𝑥) ̸= 𝑥(1 ≤ 𝑘 ≤ 𝑛 − 1) is called a periodic point of 𝑓 with period 𝑛, briefly called a 𝑛periodic point of 𝑓 . A 1-periodic point is generally called a fixed point. The set of fixed points, 𝑛-periodic points and periodic points of 𝑓 is denoted by F (𝑓 ), Pn (𝑓 ) and P (𝑓 ) respectively. Some other related notions appeared in the sequel can be found in numerous references (see for example [1, 9]). We now recall some useful notations concerning tree. A subtree of a tree T is a subset of T , which is a tree itself. A continuous map from a tree into itself is said to be a tree map. In the sequel, we always use the sign T to denote a tree and 𝑑(·, ·) to denote the metric on T . Let 𝑥 ∈ T , the number of connected components of T \ {𝑥} is called the valence of 𝑥 in T and will be denoted by ValT (𝑥). A point of T with valence one is called an end of T , and a point of valence greater than 2 is called a branched point of T . The set of ends of T and the set of branched points of T is denoted by E(T ) and B(T ) respectively. The closure of each connected component of T \ B(T ) is called an edge of T . Each point belongs to 𝑥 ∈ T \ E(T ) is said to be an interior point of T and denote by 𝐼𝑛𝑡(T ) the set of interior points of T . For any subset A of T , we denote by [A] the smallest connected closed subset of T containing A. For any 𝑥, 𝑦 ∈ T , we use [𝑥, 𝑦] to denote [{𝑥, 𝑦}]. Define (𝑥, 𝑦] = [𝑥, 𝑦] − {𝑥} and (𝑥, 𝑦) = (𝑥, 𝑦] − {𝑦}. For any 𝑥 ̸= 𝑦, denote by Tx (𝑦) the component of T − {𝑥} containing 𝑦. The following results which are partially shown in some papers [5–7,11] is very useful for us. Lemma 2.1 (Blokh [5, 11]) Let𝑓 ∈ C 0 (T ), c, 𝑑 ∈ T , c ̸= 𝑑 and Y a subtree of T . (1) If 𝑓 ([c, 𝑑]) ⊃ [c, 𝑑] and (𝑑, 𝑓 (𝑑)) ∩ [c, 𝑑] = ∅, then [c, 𝑑] ∩ F (𝑓 ) ̸= ∅. (2) If (𝑎, 𝑓 (𝑎)] ∩ Y ̸= ∅ for any point 𝑎 ∈ Y − F (𝑓 ), then Y ∩ F (𝑓 ) ̸= ∅. Lemma 2.2 (Ye [6, 7]) Let 𝑓 be a transitive tree map on a tree T , then P (𝑓 ) = T .

3

Intra-orbit separation for tree maps

In this section, we will generalize the notion of intra-orbit separation for the orbits of continuous transitive maps on a compact interval in [9] to tree maps. The purpose of this paper is to generalize the notion of separation index of tree maps and to illustrate the intra-orbit separation for the orbits of continuous tree maps. To characterize the intra-orbit separation for a tree map, we first introduce the following notion for a tree map. Definition 3.1 Let 𝑓 ∈ C 0 (T ), define a separation set D(𝑛, 𝑓 ) of 𝑓 n for each 𝑛 ∈ N as follows: D(𝑛, 𝑓 ) = {𝑥 ∈ T : 𝑓 n (𝑥) ̸= 𝑥 and there is 𝑦 ∈ T − Tx (𝑓 n (𝑥)) such that 𝑓 n (𝑦) ∈ / Ty (𝑥)}. In the sequel, if no confusion arise, we denote the separation set D(𝑛, 𝑓 ) by D(𝑛). 700

CHEN ET AL: INTRA-ORBIT SEPARATION

Remark 3.1 It’s obvious that D(𝑛) ∩ E(T ) = ∅ for each natural number 𝑛. However, the set ∂D(𝑛) ∩ E(T ) may be nonempty. Proposition 3.1 Let 𝑓 ∈ C 0 (T ), then D(1) ̸= ∅ if and only if F (𝑓 ) is not connected. Proof

=⇒ Let D(1) ̸= ∅, then there are two points 𝑥, 𝑦 ∈ T such that 𝑥 ∈ (𝑦, 𝑓 (𝑥)) and

𝑦 ∈ [𝑓 (𝑦), 𝑥). So we have that F (𝑓 ) ∩ Tx (𝑓 (𝑥)) ̸= ∅ and F (𝑓 ) ∩ [𝑦, 𝑥) ̸= ∅, from Lemma 2.1. Therefore, F (𝑓 ) is not connected. ⇐= Let F (𝑓 ) is not connected, then there are two fixed points 𝑝, q of 𝑓 , such that (𝑝, q) ∩ F (𝑓 ) = ∅. Take any 𝑥 ∈ (𝑎, 𝑏), it’s obvious that 𝑥 ∈ D(1).



Corollary 3.1 Let 𝑓 ∈ C 0 (T ) and D(1) ̸= ∅, then C𝑎r𝑑(F (𝑓 )) ≥ 2. Corollary 3.2 Let 𝑓 be a transitive tree map on a tree T . Then D(1) ̸= ∅ if and only if C𝑎r𝑑(F (𝑓 )) ≥ 2. It’s known that 𝐼𝑛𝑡(T ) ∩ F (𝑓 ) ̸= ∅ for a transitive tree map 𝑓 on a tree T (see the proof of Proposition 3.2 in [8]). So one can easily to show the following corollaries. Corollary 3.3 Let 𝑓 be a transitive tree map on a tree T and E(T ) ∩ F (𝑓 ) ̸= ∅, then D(1) ̸= ∅. Corollary 3.4 Let 𝑓 be a transitive tree map on a tree T and D(1) = ∅, then the unique fixed point of 𝑓 is contained in the interior of T . Proposition 3.2 Let 𝑓 ∈ C 0 (T ) and D(1) ̸= ∅. Then for any 𝑥 ∈ D(1), there are two points 𝑎, 𝑏 ∈ T such that 𝑥 ∈ (𝑎, 𝑏) and (𝑎, 𝑏) ⊂ D(1). In particular, 𝐼𝑛𝑡(D(1)) ̸= ∅. Proof

Let 𝑥 ∈ D(1), then there is 𝑦 ∈ T −Tx (𝑓 (𝑥)) such that 𝑓 (𝑦) ∈ / Ty (𝑥). By the continuity

of 𝑓 , there exists a neighborhood 𝑁 (𝑥) of 𝑥 such that 𝑓 ((𝑦, 𝑓 (𝑥)) ∩ 𝑁 (𝑥)) ⊂ Tx (𝑓 (𝑥)) \ 𝑁 (𝑥). It’s easy to see, by the definition of D(1), that (𝑦, 𝑓 (𝑥)) ∩ 𝑁 (𝑥) ⊂ D(1), which implies the 

conclusion.

Remark 3.2 We know that ∂A ∩ A ⊂ B(T ) for any connected subset A of a tree T . So Proposition 3.2 implies that ∂D(1) ⊂ F (𝑓 ) ∪ B(T ) and ∂D(1) ∩ D(1) ⊂ B(T ). Furthermore, the ends of the subtree consisted by the closure of each component of D(1) are all fixed points of 𝑓 . The following corollaries are simple consequences of Proposition 3.2. Corollary 3.5 Let 𝑓 ∈ C 0 (T ) and D(1) ̸= ∅, then each component of D(1) has nonempty interior. Corollary 3.6 Let 𝑓 ∈ C 0 (T ) and D(1) ̸= ∅. If ∂D(1) = E(T ), then 𝑓 is not transitive. Proposition 3.3 Let 𝑓 ∈ C 0 (T ) and D(1) ̸= ∅, then D(1) ⊂ F (𝑓 n )∪D(𝑛) and D(1)−F (𝑓 n ) ̸= ∅ for each 𝑛 ∈ N. 701

CHEN ET AL: INTRA-ORBIT SEPARATION

Proof

Let 𝑥 ∈ D(1) \ F (𝑓 n ), then 𝑓 n (𝑥) ̸= 𝑥 and there exists 𝑦 ∈ T \ Tx (𝑓 (𝑥)) satisfying

𝑦 ∈ [𝑓 (𝑦), 𝑥). Consider two cases 𝑓 n (𝑥) ∈ / Tx (𝑦) or 𝑓 n (𝑥) ∈ Tx (𝑦). It’s easy to see, by Lemma 2.1, that 𝑥 ∈ D(𝑛) for both cases. Let 𝑥 ∈ D(1) and 𝑛 ∈ N, then, by Lemma 2.1, there exist 𝑝 ∈ F (𝑓 ) ∩ (T \ Tx (𝑓 (𝑥))) and a sequence of points 𝑥−j ∈ (𝑥−(j+1) , 𝑥) ⊂ (𝑝, 𝑥) satisfying that 𝑓 j (𝑥−j ) = 𝑥−(j−1) for each 1 ≤ 𝑗 ≤ 𝑛 − 1, where 𝑥0 = 𝑥. It’s obvious that 𝑥−j ∈ D(1) \ Pj (𝑓 ) for each 1 ≤ 𝑗 ≤ 𝑛 − 1, which implies D(1) − F (𝑓 n ) ̸= ∅.



Remark 3.3 Proposition 3.3 shows that D(1) ̸= ∅ implies D(𝑛) ̸= ∅. However, if D(𝑁 ) ̸= ∅ for some 𝑁 ∈ N doesn’t imply that D(𝑛) ̸= ∅ for each 𝑛 ≥ 𝑁 . In fact, even though D(𝑚)∩D(𝑛) ̸= ∅ for some 𝑚, 𝑛 ∈ N, 𝑚 ̸= 𝑛, the set D(𝑚+𝑛) maybe empty set. This can be shown by the following example. Example 3.1 Let 𝑖2 = −1, T1 = {𝑧 ∈ C : (𝑧 + 1)𝑖 ∈ [0, 1]}, T2 = {𝑧 ∈ C : (𝑧 − 1)𝑖 ∈ [0, 1]}, T3 = {𝑧 ∈ C : (𝑧 + 1)𝑖 ∈ [−1, 0]}, T4 = {𝑧 ∈ C : (𝑧 − 1)𝑖 ∈ [−1, 0]}, T5 = {𝑧 ∈ C : 𝑖𝑧 ∈ [−1, 0]}, T6 = {𝑧 ∈ C : 𝑧 ∈ [−1, 0]}, T7 = {𝑧 ∈ C : 𝑧 ∈ [0, 1]} and T = ∪7k=1 Tk . We now define a continuous linear mapping 𝑓 : T −→ T with 𝑓 (±1 − 𝑖) = ∓1 − 𝑖, 𝑓 (−1 + 𝑖) = 1 + 𝑖, 𝑓 (1 + 𝑖) = 𝑖, 𝑓 (𝑖) = −1 + 𝑖, 𝑓 (−1) = 1, 𝑓 (1) = 0 and 𝑓 (0) = 0. One can show that F (𝑓 ) = {0}, P2 (𝑓 ) = {±1 − 𝑖}, P3 (𝑓 ) = {−1 ± 𝑖, 𝑖}, P5 (𝑓 ) = ∅ and −1 ∈ D(2) ∩ D(3) which implies that 𝛾(2) > 0, 𝛾(3) > 0, but 𝛾(5) = 0.



Proposition 3.4 Let 𝑓 ∈ C 0 (T ) and D(1) ̸= ∅, then for any 𝑥 ∈ D(1) and every natural numbers 𝑘, 𝑛 with 𝑘 ≥ 𝑛 − 1 ≥ 0, there exists a point 𝑦 ∈ D(𝑛) \ {𝑥} such that 𝑓 k (𝑦) = 𝑥. In particular, D(1) ⊂ 𝑓 k (D(𝑛)) for any integers 𝑘, 𝑛 with 𝑘 ≥ 𝑛 − 1 ≥ 0. Proof

Let 𝑛 ∈ N, 𝑘 ≥ 𝑛 − 1 and 𝑥 ∈ D(1). Then, by the proof of Proposition 3.3, there exist

a point 𝑧 ∈ T \ Tx (𝑓 (𝑥)) such that 𝑓 k (𝑧) = 𝑥 and 𝑧 ∈ D(𝑛), which implies the conclusion.



Remark 3.4 The converse of Proposition 3.4 is not true. For example, let T = {𝑧 ∈ C : 𝑧 3 ∈ [0, 1], define 𝑓 : T −→ T as following { 1 − |2𝑥 − 1|, if 𝑥 ∈ [0, 1]; 𝑓 (𝑧) = 0, if 𝑥 ∈ T \ [0, 1]. One can show that

1 3·2k−2

∈ D(𝑛) and 𝑓 k ( 3·21k−2 ) =

4 3

∈ / D(1) for any integer 𝑘 ≥ 𝑛 − 1 ≥ 0.

Corollary 3.7 Let 𝑓 ∈ C 0 (T ) and D(1) ̸= ∅, then D(1) − P (𝑓 ) ̸= ∅. Proof

Suppose that D(1) − P (𝑓 ) = ∅, then D(1) ⊂ P (𝑓 ). Take any 𝑥 ∈ D(1), then 𝑥 ∈ Pn (𝑓 )

for some integer 𝑛 ≥ 2. By Proposition 3.4, there exists 𝑦−n ∈ D(1) such that 𝑓 n (𝑦−n ) = 𝑥. Notice that 𝑦−n ∈ P (𝑓 ) and 𝑥 ∈ Pn (𝑓 ). It follows that 𝑦−n ∈ Pn (𝑓 ), a contradiction. Proposition 3.5 Let 𝑓 ∈ C 0 (T ) is turbulent, then D(1) ̸= ∅.

702



CHEN ET AL: INTRA-ORBIT SEPARATION

Proof

If 𝑓 is turbulent, it is to say, there exist [𝑥, 𝑦] ⊂ T and [𝑎, 𝑏], [c, 𝑑] ⊂ [𝑥, ](𝑎 ̸= 𝑏, c ̸= 𝑑)

with (𝑎, 𝑏) ∩ (c, 𝑑) = ∅ such that [𝑎, 𝑏], [c, 𝑑] ⊂ 𝑓 ([𝑎, 𝑏]) ∩ 𝑓 ([c, 𝑑]) (see [12]). Without loss of generality, we may suppose that 𝑏 ∈ (𝑎, c] and c ∈ [𝑏, 𝑑). Then there exist 𝑥 ∈ [𝑎, 𝑏] and 𝑦 ∈ [c, 𝑑] such that 𝑓 (𝑥) = 𝑎 and 𝑓 (𝑦) = 𝑑. So the desired result follows from Lemma 2.1 and Proposition 

3.1.

Remark 3.5 We can easily construct an example to show that the converse of Proposition 3.5 doesn’t hold. To characterize the intra-orbit separation property for a tree map, we give the following definition. Definition 3.2 Let 𝑓 ∈ C 0 (T ), the separation index of 𝑓 n for 𝑛 ∈ N is defined as { 𝑠𝑢𝑝{𝑑(𝑥, 𝑓 n (𝑥)) : 𝑥 ∈ D(𝑛)}, if D(𝑛) ̸= ∅; 𝛾(𝑛) = 0, if D(𝑛) = ∅. The quantity 𝛾(1) will be referred to as the separation index of 𝑓 . Proposition 3.6 Let 𝑓 ∈ C 0 (T ) and 𝛾(1) > 0, then 𝛾(1) = 𝑑(𝑥, 𝑓 (𝑥)) for some 𝑥 ∈ D(1) and 𝛾(𝑛) > 𝛾(1) for every natural number 𝑛 > 1. Proof

If 𝛾(1) > 0, by Definition 3.2, there is a sequence {𝑥n }∞ n=1 ⊂ D(1) such that 𝛾(1) =

lim 𝑑(𝑥n , 𝑓 (𝑥n )). Without loss of generality, we may suppose that lim (𝑥n , 𝑓 (𝑥n )) = (𝑥, 𝑓 (𝑥)).

n→∞

n→∞

Notice that 𝑑(𝑥, 𝑓 (𝑥)) = 𝛾(1) > 0, so there is a neighborhood 𝑁 (𝑥) of 𝑥 such that 𝑓 (𝑦) ∈

Tx (𝑓 (𝑥)) \ 𝑁 (𝑥) for all 𝑦 ∈ 𝑁 (𝑥). Take a point 𝑥M ∈ 𝑁 (𝑥) with (𝑥M , 𝑥) ∩ B(T ) = ∅ for some natural number M . Since 𝑥M ∈ D(1), we have a point 𝑦M ∈ T \ TxM (𝑓 (𝑥m )) to satisfy 𝑦M ∈ [𝑓 (𝑦M ), 𝑥M ). It follows that 𝑥 ∈ D(1) by the choice of 𝑁 (𝑥) and 𝑥m . Let 𝑛 ∈ N and 𝑥 ∈ D(1) with 𝑑(𝑥, 𝑓 (𝑥)) = 𝛾(1). Then, by the proof of Proposition 3.3, there exist a point 𝑧 ∈ T \ Tx (𝑓 (𝑥)) such that 𝑓 n−1 (𝑧) = 𝑥 and 𝑧 ∈ D(𝑛). Hence 𝑑(𝑧, 𝑓 n (𝑧)) > 

𝑑(𝑥, 𝑓 (𝑥)) = 𝛾(1), which implies 𝛾(𝑛) > 𝛾(1).

Definition 3.3 The orbit of 𝑥 under 𝑓 , denoted by Or𝑏f (𝑥) (simply by Or𝑏(𝑥) if no confusion arise), is said to be a separating orbit with an instability constant τ > 0, if for every pair of distinct points 𝑥i , 𝑥j ∈ Or𝑏(𝑥), there is an integer 𝑛 ≥ 0 such that 𝑑(𝑓 n (𝑥i ), 𝑓 n (𝑥j )) > τ . Theorem 3.1 Let 𝑓 be a transitive tree map with a positive separation index 𝛾 and let 𝑥s and 𝑥t be any two distinct points on a dense orbit. Then for every 0 < τ < 𝛾, (𝑥s , 𝑥t ) is a Li-Yorke pair of modulus τ . Proof

Let 𝑥s and 𝑥t be any two distinct points on a dense orbit Or𝑏(𝑥0 ) = {𝑥k := 𝑓 k (𝑥0 )}∞ n=0 .

Without loss of generality, let 𝑠 > 𝑡, then 𝑓 i (𝑥t ) = 𝑥s , where 𝑖 = 𝑠 − 𝑡. By Proposition 3.6, there is a point 𝑝 ∈ T such that 𝑑(𝑝, 𝑓 i (𝑝)) > 𝛾 > τ > 0. Since 𝑥0 is a transitive point, there exists nm (𝑥 ) = 𝑝. So we have lim 𝑓 nm (𝑥 ) = 𝑓 i (𝑝). a subsequence {𝑓 nm (𝑥t )}∞ t s m=1 such that lim 𝑓 m→∞

m→∞

Since 𝑑(𝑝, 𝑓 i (𝑝)) > 𝛾 > τ , we have a positive number 𝜉 > τ and a natural number M such 703

CHEN ET AL: INTRA-ORBIT SEPARATION

that 𝑑(𝑓 nm (𝑥s ), 𝑓 nm (𝑥t )) > 𝜉 for all 𝑚 ≥ M . Therefore, lim supn→∞ 𝑑(𝑓 n (𝑥s ), 𝑓 n (𝑥t )) > τ . Let nk q ∈ F (𝑓 ), then there exists a subsequence {𝑓 nk (𝑥t )}∞ k=1 such that lim 𝑓 (𝑥t ) = q. So we have k→∞

lim 𝑓 nk (𝑥s ) = 𝑓 i (q) = q. Therefore, lim inf n→∞ 𝑑(𝑓 n (𝑥s ), 𝑓 n (𝑥t )) = 0. This complete the proof

k→∞



of the proposition.

Corollary 3.8 Let 𝑓 be a transitive tree map with a positive separation index 𝛾. Then orbits of all transitive points are separating orbits with instability constant τ > 0 for any 0 < τ < 𝛾. We now discuss the intra-orbit separation properties for transitive tree maps. First, we recall a known result concerning a transitive tree map, which can be found in Theorem 3.1 of [7] or Proposition 3.1 of [13]. Proposition 3.7 ( [7, 13]) Let 𝑓 : T −→ T be a transitive tree map. Then exactly one of the following two statements holds: (a) 𝑓 is totally transitive. (b) There exist 𝑘 > 1, an interior fixed point 𝑦 of 𝑓 with ValT (𝑦) ≥ 𝑘 and non-degenerate closed subtrees T1 , · · · , Tk of T such that T = ∪ki=1 Ti , Ti ∩ Tj = {𝑦} for all 𝑖 ̸= 𝑗, 𝑓 (Ti ) = Ti+1(mod k) and 𝑓 k |Ti is transitive for 𝑖 = 1, · · · , 𝑘. Moreover, 𝑦 is the unique fixed point of 𝑓 . Theorem 3.2 Let 𝑓 be a transitive tree map on a tree T , then exactly one of the following two statements holds: (i) The map 𝑓 is totally transitive, and there is a natural number 𝑛 such that the separation index of 𝑓 n is greater than zero. (ii) There exist 𝑘 > 1 and non-degenerate closed subtrees T1 , · · · , Tk of T with T = ∪ki=1 Ti and Ti ∩ Tj = {𝑦} for all 𝑖 ̸= 𝑗, where 𝑦 is the unique fixed point of 𝑓 , such that 𝑔i := 𝑓 k |Ti is transitive with positive separation index for all 𝑖 = 1, · · · , 𝑘. Proof

According to Proposition 3.7, we need only to cosider the following two cases:

(i) The map 𝑓 is totally transitive. since 𝑓 is transitive, P (𝑓 ) = T by Lemma 2.2. Then it’s easy to show that Pn (𝑓 ) ̸= ∅ for some integer 𝑛 > 1. Note that F (𝑓 ) ̸= ∅, so C𝑎r𝑑(F (𝑓 n )) ≥ 𝑛 + 1. By Corollary 3.2, 𝛾(𝑛) > 0. (ii) 𝑓 k |Ti is transitive for all 𝑖 = 1, · · · , 𝑘, where T = ∪ki=1 Ti , Ti ∩ Tj = {𝑦} for all 𝑖 ̸= 𝑗, and 𝑦 is the unique fixed point of 𝑓 . By Corollary 3.3, 𝑔 k := 𝑓 k |Ti has positive separation index for all 𝑖 = 1, · · · , 𝑘.



Corollary 3.9 Let 𝑓 be a transitive tree map on a tree T and the separation index of 𝑓 is greater than zero, then 𝑓 is totally transitive. Remark 3.6 (1) Form Corollary 3.9 or Proposition ??, tree maps 𝑔i (1 ≤ 𝑖 ≤ 𝑘) in Theorem 3.2(ii) are totally transitive. (2) Theorem 3.2 shows that the separation index of a tree map, which is transitive but not totally transitive, is zero. However, for a totally transitive tree map, the separation index of it may be greater than zero (see Example 3.2). (3) In Theorem 3.2, both the number 𝑛 and the number 𝑘 maybe equal to 2 (see Example 3.3 and Example 3.4). Of course, the converse of Corollary 3.9 does not hold. 704

CHEN ET AL: INTRA-ORBIT SEPARATION

(4) In Theorem 3.2, both the number 𝑛 and the number 𝑘 maybe greater than any natural number 𝑁 (see Example 3.5 and Example 3.6). Example 3.2 Let T = {𝑧 ∈ C : 𝑧 3 ∈ [0, 1]} and 𝑔j : [ 3j , j+1 3 ] −→ [0, 1] be a continuous map defined as 𝑔j (r) = 1 − |6r − 2𝑗 − 1| for each 𝑗 = 0, 1, 2. We now define a map 𝑓 : T −→ T by 𝑓 (r𝑒kθ ) = 𝑔j (r)𝑒(k+j)θ for every 𝑗, 𝑘 ∈ {0, 1, 2}, where 𝜃 =

2πi 3

and 𝑖2 = −1.

One can show that 𝑓 is totally transitive and C𝑎r𝑑(F (𝑓 )) = 4 ≥ 2, and so 𝑓 has positive 

separation index.

Example 3.3 Let T = {𝑧 ∈ C : 𝑧 3 ∈ [0, 1]}, 𝑔0 : [0, 12 ] −→ [0, 1] defined as 𝑔0 (r) = 1 − |4r − 1| and 𝑔1 : [ 21 , 1] −→ [0, 1] defined as 𝑔1 (r) = 2r − 1 are continuous maps. We now define a map 𝑓 : T −→ T by 𝑓 (r𝑒kθ ) = 𝑔χ(r) (r)𝑒(k+1+χ(r))θ for every 𝑘 ∈ {0, 1, 2}, where 𝜃 =

2πi 3 ,

{

𝑖2 = −1 and χ(r) := χ[ 1 ,1] (r) = 2

1,

if r ∈ [ 21 , 1]

0 , if r ∈ [0, 12 )

is

the characteristic function of [ 12 , 1]. One can show that 𝑓 is totally transitive, F (𝑓 ) = {0} and C𝑎r𝑑(P2 (𝑓 )) = 12 ≥ 2. So the separation index of 𝑓 is zero and 𝑓 2 has positive separation index.



Example 3.4 Let T1 = {𝑧 ∈ C : (𝑧 − 1)3 ∈ [0, 1]}, T2 = {𝑧 ∈ C : (𝑧 + 1)3 ∈ [0, 1]} and T = T1 ∪ T2 . We now define a map 𝑓 : T −→ T by 𝑓 ((−1)k+1 + r𝑒kθ ) = (−1)k + 𝑔χ(r) (r)𝑒(k+1+2χ(r))θ for every 𝑘 ∈ {0, 1, · · · , 5}, where 𝜃 =

πi 2 N, 𝑖

= −1, and 𝑔0 , 𝑔1 , χ are continuous functions defined

as Example 3.3. One can show that 𝑓 is transitive but not totally transitive for 𝑓 2 (Tj ) = Tj (𝑗 = 1 or 2), F (𝑓 ) = {0}, and ± 35 𝑒kθ ∈ P2 (𝑓 ) which implies C𝑎r𝑑(F (𝑓 2 )) ≥ 2. So the separation index of 𝑓 

is zero and 𝑓 2 has positive separation index.

Example 3.5 Given natural number 𝑁 ≥ 3, let T = {𝑧 ∈ C : 𝑧 N ∈ [0, 1]} and 𝑔1 : [ 21 , 1] −→ [0, 21 ] defined as 𝑔1 (r) = r − 12 . We now define a map 𝑓 : T −→ T by 𝑓 (r𝑒kθ ) = 𝑔χ(r) (r)𝑒(k+1−χ(r))θ for every 𝑘 ∈ {0, 1, · · · , 𝑁 − 1}, where 𝜃 =

2πi N ,

𝑖2 = −1 and 𝑔0 , χ are continuous functions

defined as Example 3.3. One can show that 𝑓 is totally transitive, F (𝑓 ) = {0}, Pn (𝑓 ) = ∅ for each 2 ≤ 𝑛 ≤ 𝑁 − 1 and 25 𝑒kθ ∈ PN (𝑓 ) for each 0 ≤ 𝑘 ≤ 𝑁 − 1 which implies C𝑎r𝑑(F (𝑓 N )) ≥ 2. So the separation index of 𝑓 n is zero for any 1 ≤ 𝑛 ≤ 𝑁 − 1 and 𝑓 N has positive separation index.

705



CHEN ET AL: INTRA-ORBIT SEPARATION

Example 3.6 Fixed natural number 𝑁 ≥ 3, let T = {𝑧 ∈ C : 𝑧 N ∈ [0, 1]} and 𝑔 : [0, 1] −→ [0, 1] defined as 𝑔(r) = 1 − |2r − 1| be a continuous maps. We now define a map 𝑓 : T −→ T by 𝑓 (r𝑒kθ ) = 𝑔(r)𝑒(k+1)θ for every 𝑘 ∈ {0, 1, · · · , 𝑁 − 1}, where 𝜃 =

2πi N

and 𝑖2 = −1.

One can show that 𝑓 is transitive but not totally transitive for 𝑓 N (Tj ) = Tj , where Tj = {r𝑒jθ : 0 ≤ r ≤ 1} and 0 ≤ 𝑗 ≤ 𝑁 − 1. On the other hand, we have that F (𝑓 ) = {0}, Pn (𝑓 ) = ∅ for each 2 ≤ 𝑛 ≤ 𝑁 − 1 and C𝑎r𝑑(F (𝑓 N ))

2 kθ 3𝑒

∈ PN (𝑓 ) for each 0 ≤ 𝑘 ≤ 𝑁 − 1 which implies

≥ 2. Therefore, the separation index of 𝑓 n is zero for any 1 ≤ 𝑛 ≤ 𝑁 − 1 and 𝑓 N 

has positive separation index.

References [1] L. Block, W. Coppel. Dynamics in one dimension. New York: Springer-Verlag (1992). [2] J. Mai, T. Sun. The 𝜔-limit set of a graph map. Topology and its Applications, 154(2007), 2306-2311. [3] J. Mai, T. Sun. Non-wandering points and the depth for graph maps. Science in China Series A: Mathematics, 50(12)(2007), 1808-1814. [4] F. Blanchard, B. Host, A. Maass. Topological complexity. Ergod. Th. & Dynam. Sys., 20(2000), 641-662. [5] A.M. Blokh. Periods implying almost all periods for tree maps. Nonlinearity, 5(1992), 13751382. [6] X. Ye. The center and the depth of the center of a tree map. Bull. Austral. Math. Soc., 48(1993), 347-350. [7] X. Ye. Topological entropy of transitive maps of a tree. Ergod. Th. & Dynam. Sys., 20(2000), 289-314. [8] L. Alseda, S. Baldwin, J. Llibre, M. Misiurewicz. Entropy of transitive tree maps. Topology, 36(2)(1997), 519-532. [9] G. Manjunath, S.S. Ganesh, G.V. Anand. Intra-orbit separation of dense orbits of interval maps. Aequationes Math., 72(2006), 89-99. [10] S. Silverman. On maps with dense orbits and the definition of chaos. Rocky Mountain Jour. Math.,22(1992), 353-375. [11] A.M. Blokh. Trees with snowflakes and zero entropy maps. Topology, 33(1994), 379-396. [12] T. Sun. 𝜔-limit sets and turbulent of tree maps. ACTA Mathematica Sinica(Chinese), 45(2)(2002), 253-260. [13] L. Alseda, S. Kolyada, J. Llibre, L. Snoha. Entropy and periodic points for transitive maps. Trans. Amer. Math. Soc., 351(4)(1999), 1551-1573. 706

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 707-713, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A GENERALIZED SYSTEM MIXED VARIATIONAL INEQUALITY INVOLVING THREE DIFFERENCE RELAXED COCOERCIVE OPERATORS NAWITCHA ONJAI-UEA AND POOM KUMAM‡

Abstract. In this paper, we introduce and consider a new generalized system of nonlinear mixed variational inequality involving three difference relaxed cocoercive operators. This work is to use the resolvent operator technique to find the common solutions for a new generalized system of relaxed cocoercive mixed variational inequality problems and fixed point problems for Lipschitz mappings in Hilbert spaces. Our result in this paper improves and generalizes some know corresponding results in the literature.

1. Introduction The theory of variational inequalities, variational inclusions, and equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. This theory combines theoretical and algorithmic advances with novel domain of applications. Analysis of these problems requires a blend of techniques from convex analysis, functional analysis and numerical analysis. The quasi-variational inequalities is equivalent to the fixed-point problem using the projection technique. This equivalent formulation has been used to develop iterative methods for solving the quasi-variational inequality and its various variant forms. These alternative formulations have played a very significant role in the developments of numerical methods, sensitivity analysis, dynamical systems and other aspects of variational inequalities. The important generalization of variational inequalities, called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in mechanics, optimization, nonlinear programming, economics, finance, engineering and applied sciences. Let H be a real Hilbert space whose inner product and norm are denoted by h·, ·i andk·k, respectively. Let A1 , A2 , A3 : H × H × H → H be nonlinear operators. Let ϕ : H → (−∞, +∞] be a proper convex lower semi-continuous function on H. We consider the following problem: Find x∗ , y ∗ , z ∗ ∈ H such that   hsA1 (y ∗ , x∗ , z ∗ ) + x∗ − y ∗ , x − x∗ i + ϕ(x) − ϕ(x∗ ) ≥ 0, ∀x ∈ C, s > 0, hrA2 (z ∗ , x∗ , y ∗ ) + y ∗ − z ∗ , x − y ∗ i + ϕ(x) − ϕ(y ∗ ) ≥ 0, ∀x ∈ C, r > 0, (F)  htA3 (x∗ , y ∗ , z ∗ ) + z ∗ − x∗ , x − z ∗ i + ϕ(x) − ϕ(z ∗ ) ≥ 0, ∀x ∈ C, t > 0. The inequality (F) is called generalized system for nonlinear mixed variational inequality problem involving three different nonlinear operators, the solution of (F) denoted by (GSN M V IP ). Some special cases of the problem (F): (I) If A3 = 0, A1 = A2 = A are bifunctions from H × H → H and z ∗ = x∗ , then the problem (F) reduces to the following generalized system for a relaxed cocoercive mixed variational inequality problem (SM V IP (A, ϕ)), considered by He and Gu [3] and Petrot [13]: finding x∗ , y ∗ ∈ H such that

(1.1)

hsA(y ∗ , x∗ ) + x∗ − y ∗ , x − x∗ i + ϕ(x) − ϕ(x∗ ) ≥ 0, ∀x ∈ C, s > 0, hrA(x∗ , y ∗ ) + y ∗ − x∗ , x − y ∗ i + ϕ(x) − ϕ(y ∗ ) ≥ 0, ∀x ∈ C, r > 0.

2000 Mathematics Subject Classification. : 47H10, 47H19, 49J40. Key words and phrases. A new generalized system mixed variational inequality problem; relaxed cocoercive mappings ; Fixed point problem; Hilbert spaces. † This research was partially supported by the Centre of Excellence in Mathematics, under the Commission on Higher Education, Ministry of Education, Thailand. ‡ Corresponding author email: [email protected]. 707 1

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N. ONJAI-UEA AND P. KUMAM

(II) If A3 = 0, A1 , A2 are bifunctions from H × H → H, z ∗ = x∗ and C is a closed convex subset of H and ϕ(x) = δC (x), x ∈ C, where δC (x) is the indicator function of C defined by  if x ∈ C,  0, δC =  +∞, if otherwise, then the problem (1.1) reduces to the following system of nonlinear variational inequalities involving two different nonlinear operators (SNVID) considered by Huang and Noor [4]: finding x∗ , y ∗ ∈ C such that hsA1 (y ∗ , x∗ ) + x∗ − y ∗ , x − x∗ i ≥ 0, ∀x ∈ C, s > 0, hrA2 (x∗ , y ∗ ) + y ∗ − x∗ , x − y ∗ i ≥ 0, ∀x ∈ C, r > 0. (III) If A1 = A2 = A, then problem SNVID reduces to the following system of variational inequalities (SVIP(A,C)) considered by Verma [17] and Chan, Lee and Chan [2]: finding x∗ , y ∗ ∈ C such that hsA(y ∗ , x∗ ) + x∗ − y ∗ , x − x∗ i ≥ 0, ∀x ∈ C, s > 0, hrA(x∗ , y ∗ ) + y ∗ − x∗ , x − y ∗ i ≥ 0, ∀x ∈ C, r > 0. If A : C → H is univariate nonlinear operator, the problem SNVID is equivalent to finding u ∈ C such that hA(u), v − ui ≥ 0, ∀v ∈ C, which is known as the classical variational inequality introduced and studied by Stampacchia [14] in 1964. Recently, Kumam et al. [5], studied a relaxed extragradient approximation method for solving a system of variational inequalities over the fixed-point sets of nonexpansive mapping and also obtained a strong convergence theorem. Moreover, the proposed algorithm can be applied for instance to solving the classical variational inequality problems.Various kinds of iterative algorithms to solve the variational inequalities and variational inclusions have been developed by many authors. There exists a vast literature [15, 16, 17, 18, 19] on the approximation solvability of nonlinear variational inequalities as well as nonlinear variational inclusions using projection type methods, resolvent operator type methods or averaging techniques. For the recent applications, numerical methods and formulations, (see for example [7]-[12], [14]-[16], [19]-[20]) and the references therein. Very recently, Wang et al [22], introduced and studied a new system of generalized mixed quasivariational inclusion problem (SGMQVI) in uniformly smooth Banach spaces which includes some previous variational inequalities as special cases. Furthermore, the existence and uniqueness theorems of solutions for the problem (SGMQVI) are established by using resolvent techniques. They also proposed some new iterative algorithms for solving the problem (SGMQVI). Strong convergence of the iterative sequences generated by the corresponding iterative algorithms are proved under suitable conditions. Motivated and inspired by the recent research works in this fascinating area, the purpose of this paper is to introduce and study a new system of generalized nonlinear mixed variational inequality problem with three difference (µ, ν)-cocoercive mappings in Hilbert spaces. Using the resolvent operator technique, we suggest three-step iterative methods for solving this system and fixed point problem of a nonlinear Lipschitz mapping include several relate problems as special cases. Our results improve and extend the corresponding results of recent works. 2. Preliminaries Next, we will use the following definitions and lemmas. Definition 2.1. A nonlinear mapping T : H → H is said to be a κ-Lipschitzian mapping if there exists a positive constant κ such that kT x − T yk ≤ κkx − yk, ∀x, y ∈ H. In the case κ = 1, the mapping T is known as a nonexpansive mapping. If T is a mapping, we will denote by F (T ) the set of fixed points of T , that is, F (T ) = {x ∈ H : T x = x}. 708

A GENERALIZED SYSTEM MIXED VARIATIONAL INEQUALITY

3

Definition 2.2. Let M ⊂ H × H be a set-valued mapping. Then M is called monotone if for any (x1 , y1 ), (x2 , y2 ) ∈ M, hy1 − y2 , x1 − x2 i ≥ 0. A monotone operator M ⊂ H × H is called maximal if M is not properly contained in any other monotone operator. Definition 2.3. ([1]) If M is a maximal monotone operator on H, then, for any λ > 0, the resolvent operator associated with M is defined by JM (u) = (I + λM )−1 (u), for all u ∈ H, where I is the identity mapping on H. It is well known that a monotone operator is maximal if and only if its resolvent operator is defined everywhere. Furthermore, the resolvent operator is single-valued and nonexpansive. In particular, it is well known that the subdifferential ∂ϕ of a proper convex lower semi-continuous function ϕ : H → (−∞, +∞] is a maximal monotone operator; see [6]. Moreover, we have the following interesting characterization. Now, we recall some classes of the nonlinear mappings: Definition 2.4. The mapping A : H → H is said to be: (i) ν-strongly monotone if there exists a constant ν > 0 such that hAx − Ay, x − yi ≥ νkx − yk2 , ∀x, y ∈ H; (ii) µ-cocoercive if there exists a constant µ > 0 such that hAx − Ay, x − yi ≥ µkAx − Ayk2 , ∀x, y ∈ H; (iii) relaxed µ-cocoercive if there exists a constant µ > 0 such that hAx − Ay, x − yi ≥ (−µ)kAx − Ayk2 , ∀x, y ∈ H; (iv) relaxed (µ, ν)-cocoercive if there exists a constant µ, ν > 0 such that hAx − Ay, x − yi ≥ (−µ)kAx − Ayk2 + νkx − yk2 , ∀x, y ∈ H. From definition 2.4, obviously, we see that the class of the relaxed (µ, ν)-cocoercive mappings is the most general class. In this work, we will consider a kind of mapping which can be viewed as a generalization of a relaxed (µ, ν)-cocoercive mapping as follows: Definition 2.5. A mapping A : H × H → H is said to be relaxed (µ, ν)-cocoercive if there exist constant µ, ν > 0 such that, for each x, x0 ∈ H, hA(x, y) − A(x0 , y 0 ), x − x0 i ≥ (−µ)kA(x, y) − A(x0 , y 0 )k2 + νkx − x0 k2 , ∀y, y 0 ∈ H. Definition 2.6. A mapping A : H × H → H is said to be τ -Lipschitz in the first variable if there exist constant τ > 0 such that, for each x, x0 ∈ H, kA(x, y) − A(x0 , y 0 )k ≤ τ kx − x0 k, ∀y, y 0 ∈ H. Lemma 2.7. ([21]) Let {an } and {bn } be two nonnegative real sequences satisfying the following conditions: an+1 ≤ (1 − ln )an + bn , ∀n ≥ n0 , P∞ for some n0 ∈ N, ln ∈⊂ (0, 1) with n=0 ln = ∞, bn = o(ln ), Then limn→∞ an = 0. Lemma 2.8. ([1]) For a given u, z ∈ H satisfies the inequality hu − z, x − ui + λϕ(x) − λϕ(u) ≥ 0, ∀x ∈ H if and only if u = Jϕ (z)

709

4

N. ONJAI-UEA AND P. KUMAM

3. Main Results First, we establish the equivalence between the system of variational inequalities and fixed point problems. For this purpose, we recall the following well known result. Lemma 3.1. Let H be a real Hilbert space, A1 , A2 , A3 : H × H × H → H be mappings and let s, r, t be any positive real numbers. Then (x∗ , y ∗ , z ∗ ) ∈ H is a solution to problem (1.1) if and only if (x∗ , y ∗ , z ∗ ) ∈ H is a solution to the following system of operator equations:  ∗  x = Jϕ [y ∗ − sA1 (y ∗ , z ∗ , x∗ )], y ∗ = Jϕ [z ∗ − rA2 (z ∗ , x∗ , y ∗ )],  ∗ z = Jϕ [x∗ − tA3 (x∗ , y ∗ , z ∗ )], where Jϕ = (I + ∂ϕ)−1 . Proof. In fact, (x∗ , y ∗ , z ∗ ) ∈ H is a solution of problem (F) if and only if for all x ∈ H and s, r, t > 0, hsA1 (y ∗ , z ∗ , x∗ ) + x∗ − y ∗ , x − x∗ i + ϕ(x) − ϕ(x∗ ) ≥ 0, ∀x ∈ C, s > 0, hrA2 (z ∗ , x∗ , y ∗ ) + y ∗ − z ∗ , x − y ∗ i + ϕ(x) − ϕ(y ∗ ) ≥ 0, ∀x ∈ C, r > 0, htA3 (x∗ , y ∗ , z ∗ ) + z ∗ − x∗ , x − z ∗ i + ϕ(x) − ϕ(z ∗ ) ≥ 0, ∀x ∈ C, t > 0. if and only if for all x ∈ H and s, r, t > 0, hx∗ − (y ∗ − sA1 (y ∗ , z ∗ , x∗ )), x − x∗ i + ϕ(x) − ϕ(x∗ ) ≥ 0, ∀x ∈ C, s > 0, hy ∗ − (z ∗ − rA2 (z ∗ , x∗ , y ∗ )), x − y ∗ i + ϕ(x) − ϕ(y ∗ ) ≥ 0, ∀x ∈ C, r > 0, hz ∗ − (x∗ − tA3 (x∗ , y ∗ , z ∗ )), x − z ∗ i + ϕ(x) − ϕ(z ∗ ) ≥ 0, ∀x ∈ C, t > 0. By Lemma 2.8, if and only if  ∗  x = Jϕ [y ∗ − sA1 (y ∗ , z ∗ , x∗ )], y ∗ = Jϕ [z ∗ − rA2 (z ∗ , x∗ , y ∗ )],  ∗ z = Jϕ [x∗ − tA3 (x∗ , y ∗ , z ∗ )]. This complete the proof. ∗

∗

 ∗

∗

∗

∗

Remark 3.2. If (x , y , z ) is a solution of problem (F) and {x , y , z } ⊂ F (T ), then, it follows that  ∗  x = T (x∗ ) = T Jϕ [y ∗ − sA1 (y ∗ , z ∗ , x∗ )], s > 0, y ∗ = T (y ∗ ) = T Jϕ [z ∗ − rA2 (z ∗ , x∗ , y ∗ )], r > 0,  ∗ z = T (z ∗ ) = T Jϕ [x∗ − tA3 (x∗ , y ∗ , z ∗ )], t > 0. Algorithm 3.3. Let r, s, t > 0 that appeared in the problem (F), and let us have x0 , y0 , z0 ∈ H; compute the sequences {xn }, {yn }, {zn } such that   zn = (1 − γn )xn + γn T Jϕ [xn − tA3 (xn , yn , zn )], yn = (1 − βn )xn + βn T Jϕ [zn − rA2 (zn , xn , yn )],  xn+1 = (1 − αn )xn + αn T Jϕ [yn − sA1 (yn , zn , xn )]. Remark 3.4. ([13]) If A(·, ·, ·) : H × H × H → H is three-variable relaxed (µ, ν)-cocoercive and τ -Lipschitz in the first variable, we defined a function p : (0, +∞) → (−∞, +∞) by p(s) = 1 + 2sµτ 2 − 2sν + s2 τ 2 , ∀s ∈ (0, +∞). Consequently, if T : H → H is a κ-Lipschitz mapping, we put  p  p(s), if p(s) > 0, (3.1) θs =  1 if p(s) < 0. 1+κ , If A3 = 0, A1 = A2 = A are bifunctions from H × H → H, γn = 0 and zn = xn , then Algorithm 3.3 is reduced to the following. Algorithm 3.5. For arbitrarily chosen initial points x0 , y0 ∈ H, compute the sequences {xn } and {yn } by  yn = (1 − βn )xn + βn T Jϕ [xn − rA(xn , yn )], xn+1 = (1 − αn )xn + α710 n T Jϕ [yn − sA(yn , xn )].

A GENERALIZED SYSTEM MIXED VARIATIONAL INEQUALITY

5

If C is closed convex subset of H, and the function ϕ(·) is the indicator function of C in H, then it is well known that Jϕ = PC , the projection operator of H, onto the closed convex set C, then Algorithm 3.5 is reduced to the following. Algorithm 3.6. For arbitrarily chosen initial points x0 , y0 ∈ H, compute the sequences {xn } and {yn } by  yn = (1 − βn )xn + βn T PC [xn − rA(xn , yn )], xn+1 = (1 − αn )xn + αn T PC [yn − sA(yn , xn )]. Now, we stat and prove the main result of this work. Theorem 3.7. Let H be a real Hilbert space. Let A1 , A2 , A3 : H ×H ×H → H be three-variable relaxed (µ1 , ν1 ), (µ2 , ν2 ), (µ3 , ν3 )-cocoercive mappings and τ1 , τ2 , τ3 -Lipschitz mappings, in the first argument respectively, and T : H → H be a κ-Lipschitz mapping. Assume that {αn }, {βn }, {γn } ∈ [0, 1] and the following conditions hold: P∞ (C1) n=0 αn = ∞; (C2) limn→∞  limn→∞  γn = 1;2     βn = 1 2and 2(ν2 −µ2 τ2 ) 2(ν3 −µ3 τ32 ) 2(ν1 −µ1 τ1 ) , r ∈ 0, and t ∈ 0, ; (C3) s ∈ 0, 2 2 2 τ1 τ2 τ3 (C4) κθ < 1, where θ = max{θs , θr , θt }. If GSN M V IP ∩ F (T ) 6= ∅, then the sequences {xn }, {yn } and {zn } generated by Algorithm 3.3 converge strongly to x∗ , y ∗ and z ∗ , respectively, such that (x∗ , y ∗ , z ∗ ) is a solution of problem (F) and {x∗ , y ∗ , z ∗ } ⊂ F (T ). Proof. From Remark 3.2 and Algorithm 3.3, we compute kzn − z ∗ k = ≤ = ≤ ≤

k(1 − γn )xn + γn T Jϕ [xn − tA3 (xn , yn , zn )] − z ∗ k (1 − γn )kxn − z ∗ k + γn kT Jϕ [xn − tA3 (xn , yn , zn )] − z ∗ k (1 − γn )kxn − z ∗ k + γn kT Jϕ [xn − tA3 (xn , yn , zn )] − T Jϕ [x∗ − tA3 (x∗ , y ∗ , z ∗ )]k (1 − γn )kxn − z ∗ k + γn κkJϕ [xn − tA3 (xn , yn , zn )] − Jϕ [x∗ − tA3 (x∗ , y ∗ , z ∗ )]k (1 − γn )kxn − x∗ k + (1 − γn )kx∗ − z ∗ k + γn κkxn − x∗ − t[A3 (xn , yn , zn ) − A3 (x∗ , y ∗ , z ∗ )]k.

By assumption that A3 is relaxed (µ3 , ν3 )-cocoercive and τ3 -Lipschitz mapping in the first variable, we can compute the following; kxn − x∗ − t[A3 (xn , yn , zn ) − A3 (x∗ , y ∗ , z ∗ )]k2

= kxn − x∗ k2 − 2thxn − x∗ , A3 (xn , yn , zn ) − A3 (x∗ , y ∗ , z ∗ )i + t2 kA3 (xn , yn , zn ) − A3 (x∗ , y ∗ , z ∗ )k2 ≤

kxn − x∗ k2 − 2t[−µ3 kA3 (xn , yn , zn ) − A3 (x∗ , y ∗ , z ∗ )k2 + ν3 kxn − x∗ k2 ] + t2 τ32 kxn − x∗ k2

≤

kxn − x∗ k2 + 2tµ3 τ32 kxn − x∗ k2 − 2tν3 kxn − x∗ k2 + t2 τ32 kxn − x∗ k2

=

(1 + 2tµ3 τ32 − 2tν3 + t2 τ32 )kxn − x∗ k2

≤ θt2 kxn − x∗ k2 So, we obtain kzn − z ∗ k (3.2)

≤ (1 − γn )kxn − x∗ k + (1 − γn )kx∗ − z ∗ k + γn κθt kxn − x∗ k ≤ (1 − γn + γn κθ)kxn − x∗ k + (1 − γn )kx∗ − z ∗ k.

Now, we estimate kyn − y ∗ k = ≤ = ≤ ≤

k(1 − βn )(xn − y ∗ ) + βn (T Jϕ [zn − rA2 (zn , xn , yn )] − y ∗ )k (1 − βn )kxn − y ∗ k + βn kT Jϕ [zn − rA2 (zn , xn , yn )] − y ∗ k (1 − βn )kxn − y ∗ k + βn kT Jϕ [zn − rA2 (zn , xn , yn )] − T Jϕ [z ∗ − rA2 (z ∗ , x∗ , y ∗ )]k (1 − βn )kxn − y ∗ k + βn κkJϕ [zn − rA2 (zn , xn , yn )] − Jϕ [z ∗ − rA2 (z ∗ , x∗ , y ∗ )]k (1 − βn )kxn − x∗ k + (1 − βn )kx∗ − y ∗ k + βn κkzn − z ∗ − r[A2 (zn , xn ,711 yn ) − A2 (z ∗ , x∗ , y ∗ )]k.

6

N. ONJAI-UEA AND P. KUMAM

Similarly, by assumption that A2 is relaxed (µ2 , ν2 )-cocoercive and τ2 -Lipschitz mapping in the first variable, we obtain that kzn − z ∗ − r[A2 (zn , xn , yn ) − A2 (z ∗ , x∗ , y ∗ )]k2

≤ (1 + 2rµ2 τ22 − 2rν2 + r2 τ22 )kzn − z ∗ k2 ≤ θr2 kzn − z ∗ k.2

(3.3) From (3.2) and (3.3), we have kyn − y ∗ k

(3.4)

≤ (1 − βn )kxn − x∗ k + (1 − βn )kx∗ − y ∗ k + βn κθr kzn − z ∗ k ≤ (1 − βn )kxn − x∗ k + (1 − βn )kx∗ − y ∗ k + βn κθ[(1 − γn + γn κθ)kxn − x∗ k + (1 − γn )kx∗ − z ∗ k] = (1 − βn + βn κθ(1 − γn + γn κθ))kxn − x∗ k + (1 − βn )kx∗ − y ∗ k + βn κθ(1 − γn )kx∗ − z ∗ k.

From Algorithm 3.3, we compute that kxn+1 − x∗ k = ≤ ≤ (3.5) ≤

k(1 − αn )xn + αn T Jϕ [yn − sA1 (yn , zn , xn )] − x∗ k (1 − αn )kxn − x∗ k + αn kT Jϕ [yn − sA1 (yn , zn , xn )] − T Jϕ [y ∗ − sA1 (y ∗ , z ∗ , x∗ )]k (1 − αn )kxn − x∗ k + αn κkJϕ [yn − sA1 (yn , zn , xn )] − Jϕ [y ∗ − sA1 (y ∗ , z ∗ , x∗ )]k (1 − αn )kxn − x∗ k + αn κkyn − y ∗ − s[A1 (yn , zn , xn ) − A1 (y ∗ , z ∗ , x∗ )]k.

Similarly, by assumption that A1 is relaxed (µ1 , ν1 )-cocoercive and τ1 -Lipschitz mapping in the first variable, we can compute the following; kyn − y ∗ − s[A1 (yn , zn , xn ) − A1 (y ∗ , z ∗ , x∗ )]k2

≤ (1 + 2sµ1 τ12 − 2sν1 + s2 τ12 )kyn − y ∗ k2 ≤ θs2 kyn − y ∗ k.2

(3.6) From (3.4)-(3.6), we have kxn+1 − x∗ k

(3.7)

≤ (1 − αn )kxn − x∗ k + αn κθs kyn − y ∗ k ≤ (1 − αn )kxn − x∗ k + αn κθs [(1 − βn + βn κθ(1 − γn + γn κθ))kxn − x∗ k + (1 − βn )kx∗ − y ∗ k + βn κθ(1 − γn )kx∗ − z ∗ k]
≤ 1 − αn (1 − κθ(1 − βn + βn κθ(1 − γn + γn κθ))) kxn − x∗ k   + αn κθ (1 − βn )kx∗ − y ∗ k + (1 − γn )βn κθkx∗ − z ∗ k .

∗ Set an = kx  n − x k, ln = αn (1 − κθ(1 − βn + βn κθ(1 − γn + γn κθ))) and bn = αn κθ (1 − βn )kx∗ − y ∗ k + (1 − γn )βn κθkx∗ − z ∗ k , it follows that

(3.8)

kxn+1 − x∗ k

≤ (1 − ln )kxn − x∗ k + bn .

Since, κθ < 1, we have ln ∈ (0, 1) for all n ∈ N. From the condition (C2) implies that bn = o(ln ); in addition to the P∞condition (C4), it to see that ln > αn (1 − κθ) for all n ∈ N and by condition (C1), we obtain that n=0 ln = ∞. Therefore, from Lemma 2.7 and conditions (C1)-(C4) are satisfied and so, kxn − x∗ k → 0 as n → ∞ i.e., xn → x∗ as n → ∞. Consequently, by condition (C2), (3.2) and (3.4), we obtain zn → z ∗ and yn → y ∗ as n → ∞, respectively. This complete this proof.  If A1 = A2 = A3 = A, γn = 0 and zn = xn , then the following theorem can be obtained from Theorem 3.7 directly. Theorem 3.8. Let H be a real Hilbert space. Let A : H × H → H are two-variable relaxed (µ, ν)cocoercive and τ -Lipschitz mappings, in the first argument, and T : H → H be a κ-Lipschitz mapping. Assume that {αn }, {βn } ∈ [0, 1] and the following conditions hold: P∞ (C1) n=0 αn = ∞; (C2) limn→∞  βn = 1;2  ) (C3) r, s ∈ 0, 2(ν−µτ ; τ2 (C4) κθ < 1, where θ = max{θs , θr }. If SM V IP (A, ϕ) ∩ F (T ) 6= ∅ then the sequences {xn } and {yn } generated by Algorithm 3.5 converge strongly to x∗ and y ∗ , respectively, such that (x∗ , y ∗ ) ∈ SM V IP (A, ϕ) and {x∗ , y ∗ } ⊂ F (T ). 712the result of Petrot in [13]. Remark 3.9. Theorem 3.8 generalize and improve

A GENERALIZED SYSTEM MIXED VARIATIONAL INEQUALITY

7

If C is closed convex subset of H, and the function ϕ(·) is the indicator function of C in H, then it is well known that Jϕ = PC , the projection operator of H, onto the closed convex set C, then the following theorem can be obtained from Theorem 3.8 directly. Theorem 3.10. Let H be a real Hilbert space. Let A : H × H → H are two-variable relaxed (µ, ν)cocoercive and τ -Lipschitz mappings, in the first argument, and T : H → H be a κ-Lipschitz mapping. Assume that {αn }, {βn } ∈ [0, 1] and the following conditions (C1)-(C4) in Theorem 3.8 holds. If SV IP (A, C) ∩ F (T ) 6= ∅ then the sequences {xn } and {yn } generated by Algorithm 3.6 converge strongly to x∗ and y ∗ , respectively, such that (x∗ , y ∗ ) ∈ SV IP (A, C) and {x∗ , y ∗ } ⊂ F (T ). Remark 3.11. Theorem 3.10 generalize and improve the result of Chang, Lee and Chan in [2]. ACKNOWLEDGEMENTS This research is partially supported by the “ Centre of Excellence in Mathematics”, the Commission on High Education, Thailand. Moreover, Mr. Nawitcha Onjai-uea is supported by the “ Centre of Excellence in Mathematics”, the Commission on High Education for the Ph.D. Program at KMUTT. References 1. H. Brezis, Ope’rateurs maximaux monotone et semi-groupes de contractions dans les espaces de Hilbert, in: NorthHolland Mathematics Studies. 5. Notas de matematica, vol. 50, North-Holland, Amsterdam, 1973. 2. S.S. Chang, H.W. Joseph Lee, C.K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. Math. Lett. 20 (2007) 329–334. 3. Z. He, F. Gu, Generalized system for relaxed cocoercive mixed variational inequalities in Hilbert spaces, Appl. Math. Comput. 214 (2009) 26–30. 4. Z. Huang, M.A. Noor, An explicit projection method for a system of nonlinear variational inequalities with different (γ, r)-cocoercive mappings, Appl. Math. Comput. 190 (2007) 356–361. 5. W. Kumam, P.Junlouchai, and P. Kumam , Generalized Systems of Variational Inequalities and Projection Methods for Inverse-Strongly Monotone Mappings, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 976505, 24 pages doi:10.1155/2011/976505 6. G.J. Minty, On the monotonicity of the gradient of a convex function, Pacific J. Math. 14 (1964) 243–247. 7. M. A. Noor, General variational inequalities, Appl. Math. Lett. 1 (1988) 119–121. 8. M.A. Noor, General variational inequalities and nonexpansive mappings, J. Math. Anal. Appl. 331 (2007) 810–822. 9. M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000) 217–229. 10. M. A. Noor, New extragradient-type methods for general variational inequalities, J. Math. Anal. Appl. 277 (2003) 379–395. 11. M. A. Noor, Projection-proximal methods for general variational inequalities, J. Math. Anal. Appl. 316 (2006) 53–62. 12. M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004) 199–277. 13. N. Petrot, A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems, Appl. Math. Lett. 23 (2010) 440–445. 14. G. Stampacchia, Formes bilinearies coercivities sur les ensembles convexes, C.R. Acad. Sci. Paris 258 (1964) 4413– 4416. 15. R.U. Verma, A class of projection-contraction methods applied to monotone variational inequality, Appl. Math. Lett. 13 (2000), 55–62. 16. R.U. Verma, General convergence analysis for two-step projection methods and application to variational problems, Appl. Math. Lett. 18 (2005) 1286–1292. 17. R.U. Verma, Generalized system for relaxed cocoercive variational inequalities and its projection methods, J. Optim. Theory Appl. 121 (2004) 203–210. 18. R.U. Verma, A-monotone nonlinear relaxed cocoercive variational inclusions, Central European J. Math. 5 (2)(2007), 386-396. 19. R.U. Verma, Projection methods, algorithms, and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41 (2001) 1025–1031. 20. N.C. Wong, D.R. Sahu, J.C. Yao, Solving variational inequalities involving nonexpansive type mappings, Nonlinear Anal. 69 (2008) 4732–4753. 21. X.L. Weng, Fixed point iteration for local strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 113 (1991) 727–731. 22. Z. Wan, J.-w. Chen, H. Sun, and L. Yuan, A New System of Generalized Mixed Quasivariational Inclusions with Relaxed Cocoercive Operators and Applications, Journal of Applied Mathematics, Volume 2011 (2011), Article ID 961038, 26 pages doi:10.1155/2011/961038 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand. E-mail address: [email protected](P.Kumam) and [email protected] (N.Onjai-uea) Centre of Excellence in Mathematics, CHE, Sriayudthaya Rd., Bangkok 10140, Thailand.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 714-721, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Block preconditioned AOR methods for H -matrices linear systems Xue-Zhong Wang School of Mathematics and Statistics, Hexi University, Zhangye, Gansu, 734000 P.R. China

Abstract: In this paper, we consider block preconditioned AOR iterative methods for solving linear system Ax = b . When A is an H -matrix, we study the convergence of our methods and give some comparison results of the spectral radius. Numerical example is also given to illustrate our methods. Key words: H -matrix; block AOR iterative methods; preconditioner; convergence; the spectral radius., 2000 MR Subject Classification: 65F10, 65F50

1. Introduction For the linear system Ax = b,

(1.1)

where A is an n × n square matrix, and x and b are n-dimensional vectors. The basic iterative method for solving equation (1.1) is M x k +1 = N x k + b, k = 0, 1, · · · , (1.2) where A = M − N , and M is nonsingular. Thus (1.2) can be written as x k +1 = T x k + c , k = 0, 1, · · · , where T = M −1 N , c = M −1b . Let us consider the following partition of A    A =  

A 11 A 21 .. . Am1

A 12 A 22 .. . Am2

... ... .. . ...

A 1m A 2m .. . Amm

   ,  

(1.3)

where the blocks A i i ∈ C n i ×n i , i = 1, . . . , m , are nonsingular, and n 1 + n 2 + . . . + n m = n. Usually we split A into A = D − L −U, where D = diag(A 11 , . . . , A m m ), −L and −U are strictly block lower and strictly block upper triangular parts of A, respectively. Then, the iteration matrix of the AOR method for A is given by Lr α = (D − r L)−1 ((1 − α)D + (α − r )L + αU ), where α and r are real parameters with α , 0. Notice that if we take some specific values of r and α, then we obtain the successive overrelaxation (SOR), Gauss-Seidel and Jacobi methods. Transforming the original system (1.1) into the preconditioned form PAx = Pb,

(1.4)

M p x k +1 = N p x k + Pb, k = 0, 1, · · · ,

(1.5)

then, we can define the basic iterative scheme:

Email address: [email protected] (Xue-Zhong Wang)

714

Wang:Block Preconditioned AOR Methods where PA = M p − N p , and M p is nonsingular. Thus (1.5) can also be written as x k +1 = T x k + c , k = 0, 1, · · · , where T = M p−1 N p , c = M p−1 Pb . Similar to the original system (1.1), we call the basic iterative methods corresponding to the preconditioned system the preconditioned iterative methods, such as the preconditioned Jacobi method, the preconditioned Gauss-Seidel method, etc. If A is an M -matrix, Alanelli et al. in [1], considered the preconditioner P = Q+S, whereQ =diag(L −1 11 , I 22 , . . . , I m m ) and S is given by   O11 O12 · · · O1m   −1 O22 · · · O2m   −A 21 L 11  , S = .. .. .. ..  . . . .   −1 −A m 1 L 11 Om 2 · · · Om m with L 11 being the lower triangular matrix in the LU triangular decomposition of A 11 . Let Ωi =diag(ωi , ωi , . . . , ωi ) ∈ R n i ×n i , i = 2, 3, . . . , m , we consider the preconditioner P1 = Q 1 + S 1 , where Q 1 =diag(A −1 11 , I 22 , . . . , I m m ) and    S1 =   

O11 −Ω2 A 21 A −1 11 .. . −Ωm A m 1 A −1 11

O12 O22 .. . Om 2

··· ··· .. . ···

O1m O2m .. . Om m

   .  

−1 Let P1 A = (Q 1 + S 1 )(D − L − U ) = D˜ − L˜ − U˜ , where D˜ =diag(I 11 , A 22 − Ω2 A 21 A −1 11 A 12 , . . . , A m m − Ωm A m 1 A 11 A 1m ),

 L˜ =

Oi j , (Ωi − I i i )A i 1 ,  −A i j + Ωi A i 1 A −1 11 A 1j , 

j ≥i 1 < i ≤ m, j = 1 i = 3, 4, . . . , m , j > 2

 U˜ =

Oi j , −A −1 11 A 1j  −A i j + Ωi A i 1 A −1 11 A 1j , 

i ≥j i = 1, j > 1 , i < j, j >2

where Oi j are some n i × n j zero matrices. ˜ −1 exists, and then, it is possible to define the AOR iteration matrix for P1 A. If D˜ is nonsingular, then (D˜ − r L) Namely ˜ −1 ((1 − α)D˜ + (α − r )L˜ + αU˜ ). L˜r ω = (D˜ − r L) Let |A| denote the matrix whose elements are the moduli of the elements of the given matrix. We call 〈A〉 = (a¯ i j ) is comparison matrix if (a¯ i j ) = |a i j | for i = j , if (a¯ i j ) = −|a i j | for i , j . For (1.3), under the above definition, we have   〈A 11 〉 −|A 12 | . . . −|A 1m |   〈A 22 〉 . . . −|A 2m |   −|A 21 |  = 〈D〉 − |L| − |U |, 〈A〉 =  .. .. .. ..   . . . .   −|A m 1 | −|A m 2 | . . . 〈A m m 〉 where 〈D〉 =diag(〈A 11 〉, 〈A 22 〉, . . . , 〈A m m 〉), −|L| and −|U | are strictly block lower and strictly block upper triangular parts of 〈A〉, respectively.

715

Wang:Block Preconditioned AOR Methods Notice that the preconditioner of the matrix 〈A〉 corresponding to P1 is P2 = Q 2 +S 2 , where Q 2 =diag(〈A 11 〉−1 , I 22 , . . . , I m m ), and   O11 O12 · · · O1m   O22 · · · O2m   Ω2 |A 21 |〈A 11 〉−1  , S2 =  .. .. .. ..  . . . .   −1 Ωm |A m 1 |〈A 11 〉 Om 2 · · · Om m −1 −1 ¯ let P2 〈A〉 = (Q 2 +S 2 )〈A〉 = D¯ − L¯ −U¯ , where D=diag(I 11 , 〈A 22 〉−Ω2 |A 21 |〈A 11 〉 |A 12 |, . . . , 〈A m m 〉−Ωm |A m 1 |〈A 11 〉 |A 1m |), ¯ ¯ −L and −U are strictly block lower and strictly block upper triangular parts of P2 〈A〉, respectively. ¯ −1 exists, and then, it is possible to define the AOR iteration matrix for P2 〈A〉. If D¯ is nonsingular, then (D¯ − r L) Namely ¯ −1 ((1 − α)D¯ + (α − r )L¯ + αU¯ ). L¯r α = (D¯ − r L) (1.6)

Alanelli et al. in [1], showed the preconditioned Gauss-Seidel, the preconditioned SOR, the preconditioned Jacobi methods with preconditioner P are better than original methods. Our work in the presentation are to prove convergence of the preconditioned AOR method with preconditioner P1 and give more comparison results of the spectral radius for the case when A is an H -matrix. 2. Preliminaries A matrix A is called nonnegative (positive) if each entry of A is nonnegative (positive), respectively. We denote them by A ≥ 0 (A > 0). Similarly, for n-dimensional vector x , by identifying then with n × 1 matrix, we can also define x ≥ 0 (x > 0). Additionally, we denote the spectral radius of A by ρ(A). A T denotes the transpose of A. A matrix A = (a i j ) is called a Z -matrix if for any i , j , a i j ≤ 0. A Z -matrix is a nonsingular M -matrix if A is nonsingular and A −1 ≥ 0, If 〈A〉 is a nonsingular M -matrix , then A is called an H -matrix. A = M − N is said to be a splitting of A if M is nonsingular, A = M − N is said to be regular if M −1 ≥ 0 and N ≥ 0, M -splitting if M is an M -matrix and N ≥ 0, and weak regular if M −1 ≥ 0 and M −1 N ≥ 0, respectively. Some basic properties are given below, which will be used in the proof of our theorems. Lemma 2.1 [2]. Let A be a Z-matrix. Then the following statements are equivalent: (a) A is an M-matrix. (b) There is a positive vector x such that Ax > 0. (c) A −1 ≥ 0. (d) All principal submatrices of A are M-matrices. (e) All principal minors are positive. Lemma 2.2 [3, 4]. Let A = M − N be an M-splitting of A. Then ρ(M −1 N ) < 1 if and only if A is an M-matrix. Lemma 2.3 [2]. Let A and B be two n × n matrices with 0 ≤ B ≤ A. Then, ρ(B ) ≤ ρ(A). Lemma 2.4 [5]. If A is an H −matrix, then |A −1 | ≤ 〈A〉−1 . Lemma 2.5 [6]. Suppose that A 1 = M 1 − N 1 and A 2 = M 2 − N 2 are weak regular splitting of monotone matrices A 1 and A 2 respectively. Such that M 2−1 ≥ M 1−1 . If there exists a positive vector x such that 0 ≤ A 1 x ≤ A 2 x , then for the monotone norm associated with x , k M 1−1 N 1 kx ≤k M 2−1 N 2 kx . (2.1) In particular, if M 1−1 N 1 has a positive perron vector, then ρ(M 1−1 N 1 ) ≤ ρ(M 2−1 N 2 ).

(2.2)

Moreover if x is a Perron vector of M 1−1 N 1 and strictly inequality holds in (2.1), then strictly inequality holds in (2.2).

716

Wang:Block Preconditioned AOR Methods Lemma 2.6. If A and B are two n × n matrices, then 〈A − B 〉 ≥ 〈A〉 − |B |. Proof. It is easy to see that, |a i j −b i j | ≥ |a i j |−|b i j |, for i = j , and −|a i j −b i j | ≥ −|a i j |−|b i j |, for i , j . Therefore, 〈A − B 〉 ≥ 〈A〉 − |B | is true. Lemma 2.7. If A is an H-matrix with unit diagonal elements, then k〈A〉−1 k∞ > 1 and 1 + 2s i k〈A〉−1 k∞ > 1, i = 2, 3, . . . , m , s i (2k〈A〉−1 k∞ − 1) where s i denotes the maximum row sum of |A i 1 |, i = 2, 3, . . . , m , respectively. Proof. Let 〈A〉 = I − B , From 〈A〉 is an M-matrix, then B ≥ 0 and ρ(B ) < 1, and thus we have ∞ X

〈A〉−1 =

Bk ≥ I

K =0

and then

k〈A〉−1 k∞

> 1, furthermore, we have 1 + 2s i k〈A〉−1 k∞ 2k〈A〉−1 k∞ > > 1, i = 2, 3, . . . , m . s i (2k〈A〉−1 k∞ − 1) (2k〈A〉−1 k∞ − 1)

3. Convergence results Theorem 3.1. Let A be a nonsingular H-matrix with unit diagonal elements, if 0 < ωi
0, it follows from A is an H -matrix. Let  i =1 A −1  11 A 1j , (I i i − Ωi )A i 1 , 1 < i ≤ m, j = 1 . P1 A =  ot he r w i s e A i j − Ωi A i 1 A −1 11 A 1j , For i = 1 (〈P1 A〉r )n 1

= ≥ ≥ = >

For i = 2, . . . , m (〈P1 A〉r )n i

〈A −1 11 A 11 〉r n 1 − rn 1 − rn 1 −

m P j =2 m P

j =2 〉−1 e

〈A 11 On1 .

m P j =2

|A −1 11 A 1j |r n j

|A −1 11 ||A 1j |r n j 〈A 11 〉−1 |A 1j |rn j n1

=

−|(I i i − Ωi )A i 1 |rn 1 + 〈A i i − Ωi A i 1 A −1 11 A 1i 〉r n i m P −1 − |A i j − Ωi A i 1 A 11 A 1j |rn j

≥

−|(I i i − Ωi )A i 1 |rn 1 + 〈A i i 〉rn i − Ωi |A i 1 A −1 11 A 1i |r n i m m P P − |A i j |rn j − Ωi |A i 1 A −1 11 A 1j |r n j

j =2,j ,i

j =2,j ,i

717

j =2,j ,i

Wang:Block Preconditioned AOR Methods

≥

−|(I i i − Ωi )A i 1 |rn 1 + 〈A i i 〉rn i − Ωi |A i 1 ||A −1 11 ||A 1i |r n i m m P P − |A i j |rn j − Ωi |A i 1 ||A −1 11 ||A 1j |r n j

≥

−|(I i i − Ωi )A i 1 |rn 1 + 〈A i i 〉rn i − Ωi |A i 1 |〈A 11 〉−1 |A 1i |rn i m m P P − |A i j |rn j − Ωi |A i 1 |〈A 11 〉−1 |A 1j |rn j

j =2,j ,i

j =2,j ,i

j =2,j ,i

=

−|(I i i − Ωi )A i 1 |rn 1 + 〈A i i 〉rn i − −Ωi |A i 1 |〈A 11 〉−1

m P

j =2,j ,i

m P j =2,j ,i

|A i j |rn j

|A 1j |rn j ,

j =2,

if 0 < ωi ≤ 1, then (〈P1 A〉r )n i

−|A i 1 |rn 1 + Ωi |A i 1 |rn 1 + 〈A i i 〉rn i −

≥

−Ωi |A i 1 |〈A 11 〉−1

m P

m P

e ni + Ωi |A i 1 |rn 1 − Ωi |A i 1 |〈A 11 〉−1 e ni + Ωi |A i 1 |〈A 11 〉−1 (〈A 11 〉|rn1 −

=

j =2,j ,i

|A i j |rn j

|A 1j |rn j

j =2,

=

m P

|A 1j |rn j

j =2, m P

|A 1j |rn j )

j =2,

e ni + Ωi |A i 1 |〈A 11 〉−1 e n 1 On i ,

= > if 1 < ωi
1, we can obtain (〈P1 A〉r )n i

≥ =

e ni + 2(|A i 1 | − Ωi |A i 1 |)k〈A〉−1 k∞ e n1 + Ωi |A i 1 |e n1 e ni + 2k〈A〉−1 k∞ |A i 1 |e n1 − Ωi |A i 1 |(2k〈A〉−1 k∞ − 1)e n1 ,

let

1 + 2s i i k〈A〉−1 k∞ , s i i (2k〈A〉−1 k∞ − 1) where s i i denotes one of the row sun of |A i 1 |, then f (s i i ) is nonincreasing about s i i , and then f (s i i ) =

f (s i i ) = and thus, when ωi


e n i + 2k〈A〉−1 k∞ |A i 1 |e n 1 − Ωi |A i 1 |(2k〈A〉−1 k∞ − 1)e n 1 On i .

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Wang:Block Preconditioned AOR Methods Therefore, 〈P1 A〉 is an M -matrix, and P1 A is an H -matrix. Theorem 3.2. If A is a nonsingular H -matrix with unit diagonal elements, 0 ≤ r ≤ α ≤ 1, with α , 0 and 0 < ωi
0. Therefore, P2 〈A〉 is a nonsingular M -matrix. If P2 〈A〉 is a nonsingular M -matrix, then (P2 〈A〉)T is also a nonsingular M -matrix, By Lemma 2.1, there exists a vector x > 0, such that (P2 〈A〉)T x > 0, i.e, 〈A〉T (Q 2 + S 2 )T x > 0, let y = (Q 2 + S 2 )T x , then y > 0, from 〈A〉T (Q 2 + S 2 )T x = 〈A〉T y > 0, we can get A T is a nonsingular H -matrix, and then, A is a nonsingular H -matrix.

719

Wang:Block Preconditioned AOR Methods Theorem 4.2. Let A be a nonsingular H -matrix with unit diagonal elements, 0 < ωi ≤ 1, i = 2, 3. . . . , m , and 0 ≤ r ≤ α ≤ 1 with α , 0. Let Lˆr α denote the AOR iteration matrix for 〈A〉. Then ρ(L¯r α ) ≤ ρ(Lˆr α ). Proof. Since 〈A〉 is a nonsingular M -matrix. By Theorem 4.1, P2 〈A〉 is a nonsingular M -matrix, and thus 〈A〉 and P2 〈A〉 are two monotone matrices. Let ˆ − Nˆ = 〈A〉 = M

1 1 (〈D〉 − r |L|) − ((1 − α)〈D〉 + (α − r )|L| + α|U |), α α

ˆ = 1 (〈D〉 − r |L|), Nˆ = 1 ((1 − α)〈D〉 + (α − r )|L| + α|U |). Then the AOR iteration matrix for 〈A〉 as follows. where M α α ˆ −1 Nˆ = (〈D〉 − r |L|)−1 ((1 − α)〈D〉 + (α − r )|L| + α|U |), Lˆr α = M and let

1 1 ¯ − ((1 − α)D¯ + (α − r )L¯ + αU¯ ), (D¯ − r L) α α ¯ = 1 (D¯ − r L), ¯ N¯ = 1 ((1 − α)D¯ + (α − r )L¯ + αU¯ ). Then the AOR iteration matrix for P2 〈A〉 is (1.6). where M α α ˆ and M ¯ are lower triangular M -matrices and Since 〈A〉 and P2 〈A〉 are M -matrices, we can get M ¯ − N¯ = P2 〈A〉 = M

(1 − α)I + (α − r )〈D〉−1 |L| + α〈D〉−1 |U | > 0 and (1 − α)I + (α − r )D¯ −1 L¯ + αD¯ −1U¯ > 0. ˆ − Nˆ and P2 〈A〉 = M ¯ − N¯ are two weak regular splittings. By simple calculation, we have Therefore, 〈A〉 = M ¯ = M

1 1 ¯ ≤ (〈D〉 − r |L|) = M ˆ (D¯ − r L) α α

¯ −1 ≥ M ˆ −1 ≥ 0, let x = 〈A〉−1 e > 0, then (P2 〈A〉 − 〈A〉)x = (Q 2 + S 2 )e > 0, since M ¯ −1 ≥ M ˆ −1 ≥ 0, we have and thus M ¯ −1 (P2 〈A〉)x = (I − M ¯ −1 N¯ )x ≥ M ˆ −1 〈A〉x = (I − M ˆ −1 Nˆ x ), M it follows that ¯ −1 N¯ )||x ≤ ||M ˆ −1 Nˆ ||x . ||M ˆ − Nˆ is a weak regular splitting, there exists a positive perron vector y , by Lemma 2.5, the following As 〈A〉 = M inequality hold: ¯ −1 N¯ ) ≤ ρ(M ˆ −1 Nˆ ) ρ(M ie, ρ(L¯r α ) ≤ ρ(Lˆr α ). Theorem 4.3. Let A be a nonsingular H -matrix with unit diagonal elements, 0 < ωi ≤ 1, i = 2, 3. . . . , m , and 0 ≤ r ≤ α ≤ 1 with α , 0. Then, ρ(L¨r α ) ≤ ρ(L¯r α ). Proof. Let

1 1 ˜ − r |L|) − ((1 − α)〈D〉 ˜ + (α − r )|L| + α|U |), (〈D〉 α α then the AOR iteration matrix for 〈P1 A〉 is L¨r α which is defined in the proof of Theorem 3.2. and let 〈P1 A〉 =

P2 〈A〉 =

1 1 ¯ − ((1 − α)D¯ + (α − r )L¯ + αU¯ ), (D¯ − r L) α α

then the AOR iteration matrix for P2 〈A〉 is (1.6). It is easy to know the above two splittings are weak regular splittings. Furthermore, by Lemma 2.6, we have the following result ˜ 〈D〉

= ≥ ≥ =

〈A i i − Ωi A i 1 A −1 11 A 1i 〉 −1 〈A i i 〉 − Ωi |A i 1 A 11 A 1i | 〈A i i 〉 − Ωi |A i 1 |〈A 11 〉−1 |A 1i | ¯ D.

720

Wang:Block Preconditioned AOR Methods From 〈P1 A〉 and P2 〈A〉 are two M -matrices, we have ˜ −1 ≤ D¯ −1 0 ≤ 〈D〉 and then

L¨r α

= = ≤ =

˜ − r |L|) ˜ −1 ((1 − α)〈D〉 ˜ + (α − r )|L| ˜ + α|U˜ |) (〈D〉 −1 −1 ˜ ˜ ˜ −1 |L| ˜ + α〈D〉 ˜ −1 |U˜ |) (I − r 〈D〉 |L|) ((1 − α)I + (α − r )〈D〉 −1 −1 −1 −1 ¯ ¯ ¯ ¯ ¯ (I − r D |L|) ((1 − α)I + (α − r )D |L| + αD |U¯ |) L¯r α .

Therefore, by Lemma 2.3, ρ(L¨r α ) ≤ ρ(L¯r α ). Combining the above Theorems, we can obtain the following conclusion: Theorem 4.4. Let A be a nonsingular H -matrix with unit diagonal elements, 0 < ωi ≤ 1, i = 2, 3. . . . , m , and 0 ≤ r ≤ α ≤ 1 with α , 0. Then, ρ(L˜r α ) ≤ ρ(L¨r α ) ≤ ρ(L¯r α ) ≤ ρ(Lˆr α ) < 1. 5. Numerical example For randomly generated nonsingular H -matrices for n = 100, 200, 300, 400, 500 with n 1 = n 2 = . . . = n m = 5, we have determined the spectral radius of the iteration matrices of AOR method mentioned previously with preconditioner P1 . We report the spectral radius of the corresponding iteration matrix by ρ. In Table 1, the meaning of notations ρ(L˜r α ), ρ(L¯r α ), ρ(L¨r α ), ρ(Lˆr α ) and ρ(Lr α ) denotes the spectral radius of P1 A, P2 〈A〉, 〈P1 A〉, 〈A〉, and A, respectively. The m parameters Ωi , i = 1, 2, ..., m , are taken from the m equal-partitioned points of the interval [0, 1]. Table 1: Comparison of spectral radius with preconditioner P1

α, r α = 0.8 r=0.5 α=1 r=0.8

α=1 r=1

N 100 400 500 200 300 500 100 200 300 400 500

ρ(L˜r α ) 0.4916 0.4819 0.4958 0.3671 0.3219 0.3381 0.1595 0.2132 0.2110 0.2471 0.2490

ρ(L¨r α ) 0.6669 0.6581 0.6692 0.6184 0.5717 0.5754 0.4500 0.5093 0.4560 0.5153 0.5203

ρ(L¯r α ) 0.7595 0.8628 0.8654 0.7782 0.8198 0.8476 0.7506 0.8047 0.7457 0.8085 0.8395

ρ(Lˆr α ) 0.8092 0.8772 0.8791 0.8187 0.8493 0.8672 0.8002 0.8370 0.8011 0.8410 0.8617

ρ(Lr α ) 0.6174 0.5890 0.6007 0.5070 0.4580 0.4615 0.2969 0.3463 0.3640 0.4002 0.3964

From Table 1, we can conclude that the spectral radius of the preconditioned AOR method with preconditioner P1 is the best among others, which further illustrate the Theorem 4.4 is true. References [1] M. Alanelli, A. Hadjidimos, Block Gauss elimination followed by a classical iterative method for the solution of linear systems, J. Comput. Appl. Math., 163: 381-400 (2004). [2] A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994. [3] W. Li, Z. Y. You, The multi-parameters overrelaxation method, J. Comput. Math., 16(4): 231-238 (1998). [4] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1981. [5] L. Yu. Kolotilina, Two-sided bounds for the inverse of an H-matrix, Lin. Alg. Appl., 225: 117-123 (1995). [6] M. Neumann, R. J. Plemmons, Convergence of parallel multisplitting iterative methods for M -matrices, Lin. Alg. Appl., 88/89: 559-573 (1987).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 722-729, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A NOTE ON THE q-BERNOULLI NUMBERS AND q-BERNSTEIN POLYNOMIALS JIN-WOO PARK1 , HONG KYUNG PAK2 , SEOG-HOON RIM3 , TAEKYUN KIM4 , AND SANG-HUN LEE5

Abstract. In this paper we consider the q-extension of Bernstein numbers and polynomials which are slightly different Carlitz’s q-Bernstein numbers and polynomials. From those q-Bernoulli numbers and polynomials, we derive same interesting identities between q-Bernoulli numbers and q-Bernstein polynomials.

1. Introduction Let p be a fixed prime number. Throughout this paper Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p1 and let Z+ = N ∪ {0}. 1

Assume that q is an indeterminate in Cp with |1 − q| < p− p−1 so that q x = exp(x log q). Let U D(Zp ) be the space of uniformly differentiable function on Zp . For f ∈ U D(Zp ), the p-adic integral on Zp is defined as n

p −1 1 X I(f ) = f (x)dµ(x) = lim n f (x), (see [1–21]). n→∞ p Zp x=0

Z

(1.1)

From (1.1), we have I(f1 ) = I(f ) + f 0 (0), where f1 (x) = f (x + 1).

(1.2)

As is well known, Bernoulli polynomials are defined by he generating function as follows: ∞ X tn t xt B(x)t B (x) e = e = , (see [10–18]), (1.3) n et − 1 n! n=0 with the usual convention about replacing B n (x) by Bn (x). In the special case x = 0, Bn (0) = Bn are called the n-th Bernoulli numbers. Let us take f (x) = etx . Then, from (1.2), we have Z ∞ X t tn et(x+y) dµ(y) = t ext = Bn (x) . (1.4) e −1 n! Zp n=0 Thus, by (1.4), we get Witt’s formula for the n-th Bernoulli polynomials: Z (x + y)n dµ(y) = Bn (x), (n ∈ Z+ ). Zp

Now, we assume that q-number is defined as [x]q =

722

1−q x 1−q

and [0]q = 0.

(1.5)

J. W. PARK, H. W. PAK, S. H. RIM, T. KIM, AND S. H. LEE

In [7], L. Carltz defined q-Bernoulli numbers as follows:  1 if n = 1 n β0,q = 1, q(qβq + 1) − βn,q = 0 if n > 1, with the usual convention about replacing βqn by βn,q . By (1.6), we get n   X 1 n l βn,q = (−1)l , (see [7, 8]). (1 − q)n l [l + 1]q

(1.6)

(1.7)

l=0

He also defined q-Bernoulli polynomials by n   n   X X 1 n n l lx l + 1 βn,q (x) = (−1) q = [x]n−l q lx βl,q . q (1 − q)n l [l + 1]q l l=0

(1.8)

l=0

In 1994, the modified q-Bernoulli numbers are defined by  q−1 1 if n = 1 , (qBq + 1)n − Bn,q = B0,q = 0 if n > 1, log q

(1.9)

with the usual convention about replacing Bqn by Bn,q (see [14, 16]). Thus, from (1.9), we have n   X 1 n l (−1)l . Bn,q = l (1 − q)n [l]q l=0

For f ∈ U D(Zp ), the q-Bernstein operator is defined by   n X k Bn,q (f |x) = f [x]kq [1 − x]qn−k n l=0   n X k f Bk,n (x, q), (see [2, 6]). = n

(1.10)

k=0

Here Bn,q (f |x) is called the q-Bernstein operator for f of order n. For n, k ∈ Z+ , the q-Bernstein polynomials of degree n is defined as   n Bk,n (x, q) = [x]kq [1 − x]n−k (see [2]). (1.11) 1 qα k In this paper we consider q-Bernoulli numbers and polynomials which are slightly different Carltz q-Bernoulli numbers and polynomials. From those q-Bernoulli numbers and polynomials, we derive some interesting identities between q-Bernstein polynomials and q-Bernoulli numbers. 2. q-Bernoulli numbers and q-Bernstein polynomials In view point of q-extension of (1.5), we consider the q-Bernoulli polynomials as follows: Z ˜ βn,q (x) = [x + y]nq dµ(y) for n ∈ Z+ . (2.1) Zp

Thus, from (2.1), we have β˜n,q (x) =

n   X log q n l (−1)l−1 q l−1 . (1 − q)n−1 l [l]q l=0

723

(2.2)

A NOTE ON THE q-BERNOULLI NUMBERS AND q-BERNSTEIN POLYNOMIALS

In the special case, x = 0, β˜n,q (0) = β˜n,q are called the n-th q-Bernoulli numbers. By (2.2), we set n    n X n ˜ βn,q (x) = [x]n−l q lx β˜l,q = [x]q + q x β˜q , (2.3) q l l=0

with the usual convention about replacing β˜qn by β˜n,q . From (1.2), we can derive the following equation:   Z Z log q n n n−1 . [x + y]q dµ(x) − [x]q dµ(x) = n[0]q q−1 Zp Zp

(2.4)

By (2.4), we get β˜0,q = 1, β˜n,q (1) − β˜n,q =



log q q−1

if n = 1, if n > 1.

0

(2.5)

From (2.3) and (2.5), we can derive recurrence relation for the q-Bernoulli numbers:   log q δ1,n , (2.6) β˜0,q = 1, q β˜q + 1 − β˜n,q = q−1 where δk,n is a Kronecker symbol. From (1.1) and (1.2), we can derive the following equation: I(fn ) − I(f ) =

n−1 X

f 0 (l), where fn (x) = f (x + n) and n ∈ N.

(2.7)

l=0

Thus, we have   n−1 β˜n,q (n) − β˜m,q q − 1 X m−1 = l , for m, n ∈ N. m log q

(2.8)

l=0

By (2.7), we get β˜n,q (2) = (1 + q + q 2 β˜q )n =

n   X n l=0

=n =n

q log q + q−1

n   X n l=0

l

l

q l β˜l,q (1)

q l βl,q

q log q log q + δ1,n + β˜n,q . q−1 q−1

Therefore, by (2.9), we obtain the following theorem. Theorem 2.1. For n ∈ Z+ , we have log q β˜n,q (2) = (nq + δ1,n ) + β˜n,q . q−1 In particular, n ∈ N, we have log q β˜n,q (2) = nq + β˜n,q . q−1

724

(2.9)

J. W. PARK, H. W. PAK, S. H. RIM, T. KIM, AND S. H. LEE

Now, we consider the following q-Bernoulli polynomials: Z ˜ βn,q−1 (1 − x) = [1 − x + y]nq−1 dµ(y) Zp n

n

(2.10)

Z

= q (−1)

[x +

y]nq dµ(y).

Zp

Therefore, by (1.10) and (2.10), we obtain the following. Theorem 2.2. For n ∈ Z+ , we have β˜n,q−1 (1 − x) = q n (−1)n β˜n,q (x). It is easy to show that Z Z [1 − x]nq−1 dµ(x) = (1 − [x]q )n dµ(x) Zp

Zp n n

(2.11)

Z [x −

= (−1) q

1]nq dµ(x).

Zp

By (2.10) and (2.11), we get Z [1 − x]nq−1 dµ(x) = (−1)n q n βn,q (−1) = βn,q−1 (2).

(2.12)

Zp

Thus, from (2.12), we have Z Z q log q −1 n −1 [1 − x]q−1 dµ(x) = (nq + δ1,n ) + [x]nq−1 dµ(x) 1−q Zp Zp Z = (1 − [x]q )n dµ(x).

(2.13)

Zp

From (1.9), we have  Z Z n Bk,n (x, q)dµ(x) = [x]kq [1 − x]n−k q−1 dµ(x) k Zp Zp     n−k Z n X n−k = (−1)l [x]k+l q dµ(x) k l Zp l=0     n−k n X n−k (−1)l β˜k+l,q . = k l

(2.14)

l=0

Therefore, by (2.14), we obtain the following theorem. Theorem 2.3. For n, k ∈ Z+ , we have   n−k   Z n X n−k Bk,n (x, q)dµ(x) = (−1)l β˜k+l,q . k l Zp l=0

By the definition of q-Bernstein polynomials of degree n, we get   1 Bk,n (x, q) = Bn−k,n 1 − x, , q where k, n ∈ Z+ , and x ∈ Zp .

725

(2.15)

A NOTE ON THE q-BERNOULLI NUMBERS AND q-BERNSTEIN POLYNOMIALS

From (2.12) and (2.15), we note that   Z Z 1 Bk,n (x, q)dµ(x) = Bn−k,n 1 − x, dµ(x) q Zp Zp  X Z k   n k = (−1)k+l [1 − x]n−l q −1 dµ(x) k l Z p l=0  X k   n k = (−1)k+l β˜n−l,q−1 (2). k l

(2.16)

l=0

Therefore, by (2.16), we obtain the following theorem Theorem 2.4. For n, k ∈ Z+ , we have  X Z k   n k Bk,n (x, q)dµ(x) = (−1)k+l β˜n−l,q−1 (2). k l Zp l=0

In particular, if n > k + 1, then we have   X  Z k   n k log q (n − l) . Bk,n (x, q)dµ(x) = (−1)k+l β˜n−l,q−1 − 1−q k l Zp

(2.17)

l=0

By Theorem 2.3 and Theorem 2.4, we obtain the following corollary. Corollary 2.5. For n, k ∈ Z+ with n > k + 1, we have   n−k k   X n − k  X k log q (n − l) . (−1)l β˜k+l,q = (−1)k+l β˜n−l,q−1 − 1−q l l l=0

l=0

In particular, k = 0, we have n   X n log q (−1)l β˜k+l,q = β˜n,q−1 − n . l 1−q l=0

Let m, n, k ∈ Z+ with m, n > k + 1. Then we can derive the following equation: Z Bk,n (x, q)Bk,m (x, q)dµ(x) Zp

   Z n m n+m−2k = [x]2k dµ(x) q [1 − x]q −1 k k Zp    X  Z 2k  n m 2k (−1)l+2k = [1 − x]n+m−l dµ(x) q −1 k k l Z p l=0    X  2k  n m 2k = (−1)l+2k β˜n+m−l,q−1 (2). k k l

(2.18)

l=0

Therefore, by (2.18), we obtain the following theorem. Theorem 2.6. Let n, m, k ∈ Z+ with n, m > k + 1, we have    X  Z 2k  n m 2k Bk,n (x, q)Bk,m (x, q)dµ(x) = (−1)l+2k β˜n+m−l,q−1 (2). k k l Zp l=0

726

J. W. PARK, H. W. PAK, S. H. RIM, T. KIM, AND S. H. LEE

From (2.18), we have Z Bk,n (x, q)Bk,m (x, q)dµ(x) Zp

   X    2k  n m 2k (n + m − l) log q = (−1)l+2k + β˜n+m−l,q−1 . k k l q−1

(2.19)

l=0

Let n, m, k ∈ Z+ . Then (2.13), we get Z Bk,n (x, q)Bk,m (x, q)dµ(x) Zp

   Z n m n+m−2k = [x]2k dµ(x) q [1 − x]q −α k k Zp    n+m−2k X n + m − 2k  n m = (−1)l β˜l+2k,q . k k l

(2.20)

l=0

Therefore, by (2.19) and (2.20), we obtain the following theorem. Theorem 2.7. Let n, m, k ∈ Z+ with n, m > k + 1. Then n+m−2k X n + m − 2k  (−1)l β˜l+2k,q l l=0    2k  X 2k (n + m − l) log q + β˜n+m−l,q−1 . = (−1)l+2k q−1 l l=0

In particular, k = 0, we get n+m X  n + m log q (−1)l β˜l,q = (n + m) + β˜n+m,q−1 . l q−1 l=0

By induction hypothesis, we obtain the following theorem. Theorem 2.8. For n1 , n2 , . . . , ns , k ∈ Z+ with n1 , . . . , ns > k + 1 and s ∈ N, we have n1 +···+n Xs −sk n1 + · · · + ns − sk  (−1)l β˜l+sk,q l l=0   sk   X sk (n1 + · · · + ns − l) log q l+sk ˜ = (−1) + βn1 +···+ns −l,q−1 . l q−1 l=0

In particular, k = 0, we get n1 +···+n X s n 1 + · · · + n s  (n1 + · · · + ns ) log q (−1)l β˜l+sk,q = + β˜n1 +···+ns ,q−1 . l q−1 l=0

Acknowledgements. This paper was supported by the research grant of Kwangwoon University in 2013, and the Kyungpook National University Research Fund 2012.

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A NOTE ON THE q-BERNOULLI NUMBERS AND q-BERNSTEIN POLYNOMIALS

References [1] S. Araci, DS. Erdal and J. J. Seo, A study on the Fermionic p-Adic q-Integral Representation on Zp Associated with weighted q-Bernstein and q-Genocchi Polynomials, Abstract and Applied Analysis 2011(2011), Article ID 649248, 10 pages. [2] S. Araci, J. Seo and D. Erdal, New Construction Weighted (h, q)-Genocchi Numbers and Polynomials Related to Zeta Type Functions, Discrete Dynamics in Nature and Society 2011(2011), Article ID 478490, 7 pages. [3] S. Araci, N. Aslan and J. Seo, A note on the weighted wtisted Dirichelet’s wype q-Euler numbers and polynomials, Honam Math. J., 33(2011), no. 3, 311-320. [4] A. Bayad, Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Math. Comp., 80(2011), no. 276, 2219-2221. [5] A. Bayad, Modular properties of elliptic Bernoulli and Euler functions, Adv. Stud. Contemp. Math., 20(2010), no. 3, 389-401. [6] A. Bayad and T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math., 20(2010), no. 2, 274-253. [7] L. Carlitz, q-Bernoulli numbers and polynomials, Duke. Math. j., 15(1948), 987-1000. [8] L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., 76(1954), 332-350. [9] I. N. Cangul V. Kurt, H. Ozden and Y. Simsek, On the higher-order w −q-Genocchi numbers, Adv. Stud. Contemp. Math., 19(2009), 39-57. [10] K. W. Hwang, D. V. Dolgy, T. Kim and S. H. Lee, On the higher-order q-Euler numbers and polynomials with weight α, Discrete Dyn. Nat. Soc. 2011(2011), Article ID 354329, 12 pages. [11] T. Kim, An identity of wymmetry for the generalized Euler polynomials, J. Comput. Anal. Appl., 13(2011), no. 7, 1292-1296. [12] T. Kim, Some formulae for the q-Bernstein polynomials and q-deformed binomial distributions, J. Comput. Anal. Appl., 14(2012), no. 5, 917-933. [13] T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stiriling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys., 16(2009), no. 4, 484-491. [14] T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys., 15(2008), no. 1, 51-571. [15] T. Kim, q-generalized Euler numbers and polynomials, Russ. J. Math. Phys., 13(2006), no. 3, 293-398. [16] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9(2009), no. 3, 288-299. [17] C. S. Ryoo, Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstein polynomials, Proc. Jangjeon Math. Soc., 14(2011), no. 2, 239-248. [18] C. S. Ryoo, Some relations between twisted q-Euler numbers and Bernstein polynomials, Adv. Stul. Contemp. Math., 21(2011), no. 2, 217-223. [19] Y. Simsek, Complete sum of products of (h, q)-extension of Euler polynomials and numbers, J. Difference Equ. Appl., 16(2010), no. 11, 1331-1348. [20] Y. Simsek, Special functios related to Dedekind-type DC-sums and their applications, Russ. J. Math. Phys., 17(2010), no. 4, 495-508. [21] Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers, Adv. Stul. Contemp. Math., 11(2005), no. 2, 205-218.

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J. W. PARK, H. W. PAK, S. H. RIM, T. KIM, AND S. H. LEE 1 Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea. E-mail address: [email protected] 2 Department of Mathematics, Daegu Haany University, Kyungsan 712-715, Republic of Korea. E-mail address: [email protected] 3 Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea. E-mail address: [email protected] 4 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected] 5 Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 730-737, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

NEW CLASSES OF GENERALIZED SEQUENCE SPACES DEFINED BY AN ORLICZ FUNCTION KULDIP RAJ, SEEMA JAMWAL AND SUNIL K. SHARMA

Abstract. In this paper we introduce new sequence spaces defined by an Orlicz function on a seminormed space. We also make an effort to study some topological properties and inclusion relation between resulting sequence spaces.

1. Introduction and Preliminaries Let w(X) denotes the space of all sequences with elements in (X, q), where (X, q) denote a seminormed space, seminormed by q. The zero sequence is denoted by θ = (0, 0, 0, · · · ), where θ is zero element in (X, q). An Orlicz function M : [0, ∞) → [0, ∞) is a continuous, non-decreasing and convex function such that M (0) = 0, M (x) > 0 for x > 0 and M (x) −→ ∞ as x −→ ∞. Lindenstrauss and Tzafriri [5] used the idea of Orlicz function to define the following sequence space, ∞  |x |  n o X k M `M = x ∈ w : 0 : ≤1 . ρ k=1

Also, it was shown in [5] that every Orlicz sequence space `M contains a subspace isomorphic to `p (p ≥ 1). In the later stage different Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [8], Esi and Et [3], Tripathy and Mahanta [13], Mursaleen [7]and many others. Recently, different Orlicz sequence spaces were introduced and studied by Raj, Sharma and Sharma [9], Raj and Sharma ([10], [11]) and references therein. A sequence space E is said to be solid if (αk xk ) ∈ E, whenever (xk ) ∈ E for all sequences (αk ) of scalars such that |αk | ≤ 1 for all k ∈ N. A sequence space E is said to be monotone if E contains the canonical pre images of all its step spaces. The sequence space m(ϕ) was introduced by Sargent [12]. He studied some of its properties and obtained its relationship with the space lp . Later on, it was investigated from sequence space point of view and related with summability theory by Bilgin [1], Esi [2], Tripathy and Mahanta [13] and many others. Let C denote the space whose elements are the sets  of distinct positive integers. Given any elements σ of C, we denote by c(σ) the sequence cn (σ) which is such that cn (σ) = 2000 Mathematics Subject Classification. 40A05, 46A45, 46E30. Key words and phrases. Orlicz function, sequence space, seminormed space, solid, monotone. 1

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KULDIP RAJ, SEEMA JAMWAL AND SUNIL K. SHARMA

1 if n ∈ σ, cn (σ) = 0 otherwise. Further ∞ n o X Cs = σ ∈ C : cn (σ) ≤ s , n=1

the set of those σ whose support has cardinality at most s, and ϕ  o n  k ≤ 0 (k = 1, 2, · · · ) , Φ = ϕ = ϕk ∈ `0 : ϕ1 > 0, ∆ϕk ≥ 0 and ∆ k where ∆ϕk = ϕk − ϕk−1 , where {ϕk } are real sequences see [6]. For ϕ ∈ Φ, Sargent [12] define the following sequence space n  1 X  o  m(ϕ) = x = xk ∈ `0 : sup sup |xk | < ∞ . s≥1 σ∈Cs ϕs k∈σ

The space m(ϕ) was extended to m(ϕ, p) by Tripathy and Sen [14] as follows: n  1 X  o  |xk |p < ∞ . m(ϕ, p) = x = xk ∈ `0 : sup sup s≥1 σ∈Cs ϕs k∈σ

The space m(ϕ, p) equipped with the norm  1 X  p1 |xk |p s≥1 σ∈Cs ϕs

||x|| = sup sup

k∈σ

is a Banach space see [4]. For a given infinite matrix A = (aik )i,k≥1 the operators Ai are defined for any integer i ≥ 1, by ∞ X aik xk , Ai (x) = k=1

where x = (xk )k≥1 , the series intervening on the right hand being convergent. Let (X, q) be a seminormed space, M be an Orlicz function and p = (pi ) be a bounded sequence of positive real numbers. Then we define the following classes of sequences in this paper: n   A (x) pi o i l∞ (M, A, q, p) = x = (xk ) ∈ w(X) : sup M q < ∞, for some ρ > 0 , ρ i≥1 ∞ n   A (x) pi o X i l1 (M, A, q, p) = x = (xk ) ∈ w(X) : M q < ∞, for some ρ > 0 ρ i=1

and n m(M, A, ϕ, q, p) = x = (xk ) ∈ w(X) :

sup s≥1,σ∈Cs

1 X   Ai (x) pi M q < ∞, ϕs i∈σ ρ

o for some ρ > 0 . If we take, A = I(the identity matrix) and p = (pi ) = 1, then we obtain the spaces l∞ (M, q), l1 (M, q) and m(M, ϕ, q) which were studied by Tripathy and Mahanta [13]. The following inequality will be used throughout the paper. If 0 < h = inf pk ≤ pk ≤ sup pk = H, D = max(1, 2H−1 ) then (1.1)

|ak + bk |pk ≤ D{|ak |pk + |bk |pk }

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NEW CLASSES OF GENERALIZED SEQUENCE SPACES

3

for all k and ak , bk ∈ C. Also |a|pk ≤ max(1, |a|H ) for all a ∈ C. The main aim of the present paper is to study some topological properties and prove some inclusion relations between above defined classes of sequences. 2. Main Results Theorem 2.1. If M is an Orlicz function and p = (pi ) be a bounded sequence of positive real numbers, then the spaces l∞ (M, A, q, p), l1 (M, A, q, p) and m(M, A, ϕ, q, p) are linear spaces over the field of complex number C. Proof. Let x = (xk ), y = (yk ) ∈ m(M, A, ϕ, q, p) and α, β ∈ C. Then there exist positive real numbers ρ1 , ρ2 > 0 such that 1 X   Ai (x) pi M q 0, we have ∞ X

  A (x) pi i < ∞. M q ρ i=1

Since (ϕn ) is a monotonic increasing, so we have ∞ 1 X   Ai (x) pi 1 X   Ai (x) pi 1 X   Ai (x) pi M q ≤ M q ≤ M q < ∞. ϕs i∈σ ρ ϕ1 i∈σ ρ ϕ1 i=1 ρ

Hence, sup s≥1,σ∈Cs

1 X   Ai (x) pi M q < ∞. ϕs i∈σ ρ

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KULDIP RAJ, SEEMA JAMWAL AND SUNIL K. SHARMA

Thus, x = (xk ) ∈ m(M, A, ϕ, q, p). Therefore, l1 (M, A, q, p) ⊂ m(M, A, ϕ, q, p). Next, let x = (xk ) ∈ m(M, A, ϕ, q, p). Then for some ρ > 0, we have sup s≥1,σ∈Cs

1 X   Ai (x) pi M q < ∞. ϕs i∈σ ρ

Hence, sup s≥1,σ∈Cs

1 X   Ai (x) pi M q < ∞ (on taking cardinality of σ to be 1). ϕs i∈σ ρ

Thus, x = (xk ) ∈ l∞ (M, A, q, p). Therefore, m(M, A, ϕ, q, p) ⊂ l∞ (M, A, q, p). This completes the proof of the theorem. Theorem 2.3. If M is an Orlicz function and p = (pi ) be a bounded sequence of positive real numbers, then the space m(M, A, ϕ, q, p) is a seminormed space, seminormed by n g(x) = inf ρ > 0 :

sup s≥1,σ∈Cs

o 1 X   Ai (x) pi ≤1 . M q ϕs i∈σ ρ

Proof. Clearly, g(x) ≥ 0 for all x = (xk ) ∈ m(M, A, ϕ, q, p) and g(θ) = 0. Let x = (xk ), y = (yk ) ∈ m(M, A, ϕ, q, p). Then there exist ρ1 > 0 and ρ2 > 0 such that sup s≥1,σ∈Cs

1 X   Ai (x) pi M q ≤1 ϕs i∈σ ρ1

and sup s≥1,σ∈Cs

1 X   Ai (y) pi M q ≤ 1. ϕs i∈σ ρ2

Let ρ = ρ1 + ρ2 . Then, we have 1 X   Ai (x + y) pi sup M q ρ s≥1,σ∈Cs ϕs i∈σ 1 X   Ai (x + y) pi M q ρ1 + ρ2 s≥1,σ∈Cs ϕs i∈σ n   A (x)    A (y) opi X ρ1 ρ2 1 i i M q + M q ≤ sup ρ1 ρ1 + ρ2 ρ2 s≥1,σ∈Cs ϕs i∈σ ρ1 + ρ2        ρ X p i 1 Ai (x) ρ2 1 X   Ai (y) pi 1 sup M q + sup M q ≤ ρ1 + ρ2 s≥1,σ∈Cs ϕs i∈σ ρ1 ρ1 + ρ2 s≥1,σ∈Cs ϕs i∈σ ρ2 =

sup

≤ 1. Since the ρ’s are non-negative, so we have n g(x + y) = inf ρ > 0 : sup

o 1 X   Ai (x + y) pi M q ≤1 ρ s≥1,σ∈Cs ϕs i∈σ n   o 1 X Ai (x) pi ≤ inf ρ1 > 0 : sup M q ≤1 ρ1 s≥1,σ∈Cs ϕs i∈σ n   o 1 X Ai (y) pi + inf ρ2 > 0 : sup M q ≤1 . ρ2 s≥1,σ∈Cs ϕs i∈σ

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NEW CLASSES OF GENERALIZED SEQUENCE SPACES

5

Thus, g(x + y) ≤ g(x) + g(y). Next, for λ ∈ C, without loss of generality, λ 6= 0, then n o 1 X   Ai (λx) pi g(λx) = inf ρ > 0 : sup M q ≤1 ρ s≥1,σ∈Cs ϕs i∈σ o n   ρ 1 X Ai (x) pi ≤ 1 , where r = = inf ρ > 0 : sup M q . r |λ| s≥1,σ∈Cs ϕs i∈σ This completes the proof of the theorem. Theorem 2.4. If M is an Orlicz function and p = (pi ) be a bounded sequence of positive real numbers, then (a) the space l∞ (M, A, q, p) is a seminormed space, seminormed by o n   A (x) pi i ≤1 , f (x) = inf ρ > 0 : sup M q ρ i≥1 (b) the space l1 (M, A, q, p) is a seminormed space, seminormed by ∞ o n   A (x) pi X i ≤1 . h(x) = inf ρ > 0 : M q ρ i=1 Proof. It is easy to prove inview of Theorem 2.3, so we omit the details. Theorem 2.5. Let M be an Orlicz function and p = (pi ) be a bounded sequence of posiϕs tive real numbers. Then m(M, A, ϕ, q, p) ⊂ m(M, A, ψ, q, p) if and only if sups≥1 ψ < ∞. s ϕs Proof. Suppose sup < ∞ and x = (xk ) ∈ m(M, A, ϕ, q, p). Then, we have for some s≥1 ψs ρ>0 1 X   Ai (x) pi M q < ∞. sup ρ s≥1,σ∈Cs ϕs i∈σ Thus, sup s≥1,σ∈Cs

1 X   Ai (x) pi  1 X   Ai (x) pi  ϕs  ≤ sup sup < ∞. M q M q ψs i∈σ ρ ρ s≥1 ψs s≥1,σ∈Cs ϕs i∈σ

Therefore, x = (xk ) ∈ m(M, A, ψ, q, p). Hence, m(M, A, ϕ, q, p) ⊂ m(M, A, ψ, q, p). ϕs Conversely, let m(M, A, ϕ, q, p) ⊂ m(M, A, ψ, q, p). Suppose that sup = ∞. Then s≥1 ψs ϕsi there exists a sequence of naturals {si } such that lim = ∞. Let x = (xk ) ∈ i→∞ ψsi m(M, A, ϕ, q, p). Then there exists ρ > 0 such that 1 X   Ai (x) pi < ∞. M q sup ρ s≥1,σ∈Cs ϕs i∈σ Now, we have 1 X   Ai (x) pi  ϕs  1 X   Ai (x) pi  sup M q ≥ sup sup M q = ∞. ρ ρ s≥1,σ∈Cs ψs i∈σ s≥1 ψs s≥1,σ∈Cs ϕs i∈σ Therefore, x = (xk ) ∈ / m(M, A, ψ, q, p), which is a contradiction. Hence sup s≥1

ϕs < ∞. ψs

Corollary 2.6. Let M be an Orlicz function and p = (pi ) be a bounded sequence of posiϕs tive real numbers. Then m(M, A, ϕ, q, p) = m(M, A, ψ, q, p) if and only if sups≥1 ψ 0 such that   A (x) pi 1 X i M1 q < ∞. sup ρ s≥1,σ∈Cs ϕs i∈σ Let0 <  δ. Since M satisfies ∆2 -condition, we have X X X (2.1) M (yk ) ≤ M (1) yk ≤ M (2) yk . 1

1

1

For yk > δ

yk yk ≤1+ , δ δ since M is non-decreasing and convex, so yk
1 1 2yk
M (yk ) < M 1 + < M (2) + M . δ 2 2 δ Since M also satisfies ∆2 -condition, so 1 yk yk 1 yk M (yk ) < K M (2) + K M (2) = K M (2). 2 δ 2 δ δ Hence, X X (2.2) M (yk ) ≤ max(1, Kδ −1 M (2)) yk . yk
0 such that   A (x) pi 1 X i sup M1 q 0 and x0 > 0 be fixed. Then for each rx 0 > 0, there exists a positive integer n0 such that  g(xj − xl ) < , for all j, l ≥ n0 . rx0 This implies o n  1 X   Ai (xj − xl ) pi ≤1 < M q (2.4) inf ρ : sup , for all j, l ≥ n0 . ϕ ρ rx 0 s≥1,σ∈Cs s i∈σ We have for all j, l ≥ n0 and by (2.4) 1 X   Ai (xj − xl ) pi M q ≤1 sup h(xj − xl ) s≥1,σ∈Cs ϕs i∈σ   A (xj − xl ) pi 1 i ≤1 M q ϕ1 h(xj − xl )   A (xj − xl ) pi i =⇒ M q ≤ ϕ1 , for all j, l ≥ n0 . h(xj − xl )
We can find r > 0 such that rx20 η x20 > ϕ1 , where η is the kernel associated with orlicz function M , such that   A (xj − xl ) pi rx0 x0
i M q ≤ η j l h(x − x ) 2 2 rx0   j l pi =⇒ q(Ai (x − x )) < . = . 2 rx0 2 Hence Ai (xj )j≥1 is a Cauchy sequence in (X, q), which is complete. Therefore for each k ∈ N, there exists xk ∈ X and x = (xk ) such that q(Ai (xj − x))pi → 0 as j → ∞. Using the continuity of M and q is seminorm, so for some ρ > 0, we have 1 X   limi→∞ Ai (xj − xl ) pi M q ≤1 sup ρ s≥1,σ∈Cs ϕs i∈σ =⇒

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KULDIP RAJ, SEEMA JAMWAL AND SUNIL K. SHARMA

=⇒

sup s≥1,σ∈Cs

1 X   Ai (xj − xl ) pi M q ≤ 1. ϕs i∈σ ρ

Now, taking the infimum of such ρ’s by (2.4), we get o n 1 X   Ai (xj − xl ) pi ≤ 1 < , for all j ≥ n0 . inf ρ > 0 : sup M q ρ s≥1,σ∈Cs ϕs i∈σ Since m(M, A, ϕ, q, p) is a linear space and (x − xj ) are in m(M, A, ϕ, q, p), so it follows that x = xj + (x − xj ) ∈ m(M, A, ϕ, q, p). Hence m(M, A, ϕ, q, p) is complete. This completes the proof of the theorem. 3. Conclusions If one considers a normed linear space (X, ||.||) instead of seminormed space (X, q), then one will get m(M, A, ϕ, p, ||.||), which will be normed linear space, normed by o n 1 X   Ai (x)  pi M || || ≤1 . ||x|| = inf ρ > 0 : sup ρ s≥1,σ∈Cs ϕs i∈σ The space m(M, A, ϕ, p, ||.||) will be a Banach space if (X, ||.||) is a Banach space. The most of the results proved in the previous section will be true for this space too. Also, giving particular values the matrix A, we obtain some sequence spaces which were defined earlier by some authors. For instance, if we take A = I (identity matrix) and p = (pi ) = 1, we obtain the space m(M, ϕ, q) which was defined and studied by Tripathy and Mahanta. References [1] Bilgin, T., The sequence space l(p, f, q, s) on seminormed spaces, Bull. Calcutta Math. Soc., 80(1994), 295-304. [2] Esi, A., On a new class of new type difference sequence spaces related to the space lp , Far East J. Math. Sci., 13 (2004), 167-172. [3] Esi, A. and ET, M., Some new sequence spaces defined by a sequence of Orlicz functions, Indian J. Pure Appl. Math., 31 (2000), 967-972. [4] Khan, V. A., A new type of difference sequence spaces, Applied Sciences, 12 (2010), 102-108. [5] Lindenstrauss, J. and Tzafriri, L., On Orlicz sequence spaces, Israel J. Math., 10 (1971), 345-355. [6] Mursaleen, M., Some geometric properties of a sequence space related to lp , Bull. Aust. Math. Soc. 67 (2003), 343-347. [7] Mursaleen, M., Generalized spaces of difference sequences, J. Math. Anal. Appl., 203 (1996), 738745. [8] Parashar, S. D. and Choudhary, B., Sequence spaces defined by Orlicz function, Indian J. Pure Appl. Math., 25 (1994), 419-428. [9] Raj, K., Sharma, S. K. and Sharma, A. K., Some difference sequence spaces in n-normed spaces defined by Musielak-Orlicz function, Armen. J. Math., 3 (2010), 127-141. [10] Raj, K. and Sharma, S. K., Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz functions, Acta Univ. Sapientiae Math., 3 (2011), 97-109. [11] Raj, K. and Sharma, S. K., Some Multiplier Double Sequence spaces, Acta Mathematica Vietnamica (To Appear). [12] Sargent, W. L. C., Some sequence spaces related to the lp spaces, J. London Math. Soc. 35 (1960), 161-171. [13] Tripathy, B. C. and Mahanta, S., On a class of sequences related to the lp space defined by Orlicz functions, Soochow J. Math. 29 (2003), 379-391. [14] Tripathy, B. C. and Sen, M., On a new class of sequences related to the space lp , Tamkang J. Math. 33 (2002), 167-171. School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J&K, India E-mail address: [email protected] E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 738-745, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Stability and superstability of quadratic ∗-derivations on fuzzy Banach ∗-algebras Sun Young Jang

1

and Young Cho

2∗

1 Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea e-mail: [email protected] 2∗ Faculty of Electrical and Electronics Engineering Ulsan College West Campus, Ulsan 680-749, Korea e-mail: [email protected]

Abstract. In this paper, we establish the functional equations of quadratic ∗-derivations and prove the stability of quadratic ∗-derivations on fuzzy Banach ∗-algebras. We also prove the superstability of quadratic ∗-derivations. 1. Introduction Suppose that A is a Banach ∗-algebra. A linear mapping δ : D(δ) → A is said to be a derivation on A if δ(ab) = δ(a)b + aδ(b) for all a, b ∈ A, where D(δ) is a domain of δ and D(δ) is dense in A. If δ satisfies the additional condition δ(a∗ ) = δ(a)∗ for all a ∈ A, then δ is called a ∗-derivation on A. It is well-known that if A is a C ∗ -algebra and D(δ) is A, then the derivation δ is bounded. A C ∗ -dynamical system is a triple (A, G, α) consisting of a C ∗ -algebra A, a locally compact group G, and a pointwise norm continuous homomorphism α of G into the group Aut(A) of ∗automorphisms of A. Every bounded ∗-derivation δ arises as an infinitesimal generator of a dynamical system for R. In fact, if δ is a bounded ∗-derivation of A on a Hilbert space H , then there exists an element h in the enveloping von Neumann algebra A00 such that δ(x) = adih (x) for all x ∈ A. If for each t ∈ R αt is defined by αt (x) = eith xe−ith for each x ∈ A, then αt is a ∗-automorphism of A induced by unitaries Ut = eith for each t ∈ R. The action α : R → Aut(A), t → αt , is a strongly continuous one-parameter group of ∗-automorphisms of A. For several reasons the theory of bounded derivations of C ∗ -algebras is important in the quantumn mechanics ([3, 4, 10]). A functional equation is called stable if any function satisfying a functional equation “approximately” is near to a true solution of the functional equation. We say that a functional equation is superstable if every approximate solution is an exact solution of it (see [2]). In 1940, Ulam [16] proposed the following question concerning stability of group homomorphisms: under what condition is there an additive mapping near an approximately additive mapping? Hyers [8] answered the problem of Ulam for the case where G1 and G2 are Banach spaces. A generalized version of the theorem of Hyers for an approximately linear mapping was given by Th. M. Rassias [14]. Since then, the stability problems of various functional equations have been extensively investigated by a number of authors (for instances, [1, 6, 7, 9, 12, 14]). The functional equation f (x + y) + f (x − y)

=

2f (x) + 2f (y)

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [15].

∗

0 2010 Mathematics Subject Classification: 39B52, 47B47, 46L05, 39B72 Corresponding author

738

Approximate quadratic ∗-derivation We use the definition of fuzzy normed spaces given in [5, 11] to investigate the stability of quadratic ∗-derivation in the fuzzy Banach ∗-algebra setting. Definition 1.1. [5, 11] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1; (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. Furthermore we can make (X, N ) a fuzzy normed ∗-algebra if we add (N7 ) and (N8 ) as follows: (N7 ) N (xy, st) ≥ min{N (x, s), N (y, t)}; (N8 ) N (x, t) = N (x∗ , t). The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [11]. Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N -limn→∞ xn = x. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X(see [5]). In this paper, we introduce functional equations of quadratic ∗-derivations and prove the stability of quadratic ∗-derivations on fuzzy Banach ∗-algebras. We also prove the superstability of quadratic ∗-derivations. 2. Stability of quadratic ∗-derivations on fuzzy Banach ∗-algebras In this section, we establish the stability of quadratic ∗-derivations on a fuzzy Banach ∗-algebras A. Definition 2.1. Let A be a ∗-normed algebra. A mapping δ : A → A is a quadratic ∗-derivation on A if δ satisfies the following properties: 1) δ is a quadratic mapping, 2) δ is quadratic homogeneous, that is, δ(λa) = λ2 δ(a) for all a ∈ A and all λ ∈ C, 3) δ(ab) = δ(a)b2 + a2 δ(b) for all a, b ∈ A, 4) δ(a∗ ) = δ(a)∗ for all a ∈ A. Theorem 2.2. Suppos that A is a fuzzy Banach ∗-algebra, s ∈ {−1, 1} and that f : A → A is a mapping with f (0) = 0 for which there exist functions ϕ : A4 → [0, ∞) and ψ : A2 → [0, ∞) such that ∞ X ϕ(2sk a, 2sk b, 2sk c, 2sk d) < ∞, (2.1) ϕ(a, ˜ b, c, d) := 22sk k=0

lim 2−2ns ψ(2ns a, 2ns b) = 0

n→∞

739

(2.2)

S. Jang and Y. Cho

lim N (f (λa + λb + cd) + f (λa − λb + cd) − 2λ2 f (a) − 2λ2 f (b) − 2f (c)d2 − 2c2 f (d),

t→∞

tϕ(a, b, c, d)) = 1,

(2.3) lim N (f (a)∗ − f (a∗ ), tψ(a, a∗ )) = 1

t→∞

(2.4)

for all a, b, c, d ∈ A and for all λ ∈ T. Also, if for each fixed a ∈ A the mapping t → f (ta) from R to A is continuous, then there exists a unique quadratic ∗-derivation δ on A satisfying 1 ˜ a, 0, 0)) = 1, lim N (f (a) − δ(a), tϕ(a, 4

t→∞

(2.5)

for all a ∈ A. Proof. Let s = 1. Putting a = b, c = d = 0 and λ = 1 in (2.3), for a given  > 0 we can find some t0 > 0 such that N (f (2a) − 4f (a), tϕ(a, a, 0, 0)) > 1 −  for all a ∈ A and all t ≥ t0 . One can use induction to show that N

 f (2n a) 4n

−

n−1  f (2m a) 1 X −2k k k , 2 tϕ(2 a, 2 a, 0, 0) 4m 4 k=m

 n−1  X f (2k+1 a) f (2k a) 1 n−1 X −2k k k =N 2 tϕ(2 a, 2 a, 0, 0) (2.6) − , 22k 4 22(k+1) k=m k=m   f (2k+1 a) f (2k a) −2(k+1) k k ≥ minm≤k≤n−1 {N , 2 tϕ(2 a, 2 a, 0, 0) }>1− − 22k 22(k+1) for all n > m ≥ 0 and all a ∈ A. n a) It follows from (2.6) that the sequence { f (2 22n } is Cauchy. Since A is complete, this sequence is convergent. Define f (2n a) . n→∞ 22n

δ(a) := N − lim

(2.7)

We have δ(0) = 0 because f (0) = 0. Replacing a and b by 2n a and 2n b, c = d = 0, respectively, in (2.3), we can find some t0 > 0 such that N

 f (2n(λa + λb))

+

n n f (2n(λa − λb)) 2 f (2 a) 2 f (2 b) − 2λ − 2λ , 22n 22n 22n

22n ϕ(2n a, 2n b, 0, 0)  t >1− 22n

for all a, b ∈ A and t ≥ t0 . Fix t temporarily. Since limn→∞ 41n ϕ(2n a, 2n b, 0, 0) = 0, there exists n n b,0,0) n0 > 0 such that t ϕ(2 a,2 < 2t for all n ≥ n0 . To show the quadratic property of δ we consider 22n the following equation N (δ(λa + λb) + δ(λa − λb) − 2λ2 δ(a) − 2λ2 δ(b), t)  f (2n (λa + λb)) t   f (2n (λa − λb)) t  ≥ min{N δ(λa + λb) − , , N δ(λa − λb) − , , 22n 8 22n 8  f (2n a) t   2 f (2n b) t  N 2λ2 δ(a) − 2λ2 2n , , N 2λ δ(b) − 2λ2 2n , , 2 8 2 8  f (2n (λa + λb)) f (2n (λa − λb))  n n 2 f (2 a) 2 f (2 b) t N + − 2λ − 2λ , }. 22n 22n 2n 22n 2

740

Approximate quadratic ∗-derivation The first four terms on the second and third lines of the above inequality tend to 1 as n → ∞. Furthermore from the following computation  f (2n (λa + λb)) f (2n (λa − λb))  n n 2 f (2 a) 2 f (2 b) t N + − 2λ − 2λ , 22n 22n 22n 22n 2   f (2n (λa + λb)) f (2n (λa − λb)) n f (2 a) f (2n b) + − 2λ2 2n − 2λ2 2n , t2−2n ϕ(2n a, 2n b, 0, 0) > 1 −  ≥N 2n 2n 2 2 2 2 the last term on the right hand side of the above inequality is greater than 1 − . Hence   N δ(λa + λb) + δ(λa − λb) − 2λ2 δ(a) − 2λ2 δ(b), t > 1 −  for all t > 0. So by (N2 ) we can obtain δ(λa + λb) + δ(λa − λb) = 2λ2 δ(a) + 2λ2 δ(b)

(2.8)

for all a, b ∈ A and all λ ∈ T. Putting λ = 1 in (2.8), we obtain that δ is a quadratic mapping. Setting b := a in (2.8), we get δ(2λa) = 4λ2 δ(a) for all a ∈ A and all λ ∈ T. We can say that δ(λa) = λ2 δ(a) for all a ∈ A and all λ ∈ T. Under the assumption that f (ta) is continuous in t ∈ R for each fixed a ∈ A , by the same reasoning as in the proof of [7], δ(λa) = λ2 δ(b) for all a ∈ A and all λ ∈ R. Hence  λ λ2 λ2 |λ|a = δ(λa) = δ δ(|λ|a) = |λ|2 δ(a) 2 |λ| |λ| |λ|2 for all a ∈ A and all λ ∈ C(λ 6= 0).This means that δ is quadratic homogeneous. Furthermore it follows from (2.6) with m = 0 and (2.7) that limt→∞ N (f (a) − δ(a), 41 tϕ(a, e a, 0, 0)) = 1. It is known that the quadraic mapping δ satisfying (2.5) is unique. Replacing c and d by 2n c and 2n d and putting a = b = 0 in (2.3), there exists t0 > 0 such that  f (2n c2n d) f (2n c2n d) 22n c2 f (2n d) f (2n c)22n d2 ϕ(0, 0, 2n c, 2n d)  N + − 2 − 2 , 24n 24n 24n 24n 24n  f (22n cd) 22n c2 f (2n d) f (2n c) 22n d2 t ϕ(0, 0, 2n c, 2n d)  =N − 2n − , >1− 24n 2 22n 22n 22n 2 24n n

c,2 for all c, d ∈ A for all t ≥ t0 . Fix t temporarily. Since limn→∞ ϕ(0,0,2 22n n n c,2 d) such that t ϕ(0,0,2 ≤ 4t for all n ≥ n0 . we consider the following 24n

n

d)

= 0, we can find n0 > 0

N (δ(cd) − c2 δ(d) − δ(c)d2 , t)  22n c2 f (2n d) t  f (2n c2n d) t   2 , , N c δ(d) − , , ≥ min{N δ(cd) − 24n 4 24n 4  f (2n c)22n d2 t   f (2n c2n d) 22n c2 f (2n d) f (2n c)22n d2 t  N δ(c)d2 − ,N }. , − − , 24n 4 24n 24n 24n 4 The first three terms on the right hand side of the above inequality tend to 1 as n → ∞. Furthermore the last term is greater than 1 −  because  f (2n c2n d) 22n c2 f (2n d) f (2n c)22n d2 t  N − − , 24n 24n 24n 4  f (22n cd) 22n c2 f (2n d) f (2n c) 22n d2 t ϕ(0, 0, 2n c, 2n d)  ≥N − 2n − , > 1 − . 24n 2 22n 22n 22n 2 24n So we have N (δ(cd) − c2 δ(d) − δ(c)d2 , t) > 1 − 

741

S. Jang and Y. Cho for all t > 0. By the definition of the fuzzy norm we have δ(cd) − c2 δ(d) − δ(c)d2 = 0. Replacing a and a∗ by 2n a and 2n a∗ , respectively, in (2.4) for a given  > 0 we can find t0 > 0 such that N (2−2n f (2n a)∗ − 2−2n f (2n a∗ ), t2−2n ψ(2n a, 2n a∗ )) > 1 − , for all a ∈ A all t ≥ t0 . Since limn→∞ 2−2n ψ(2n a, 2n a∗ ) = 0, there exists some n0 > 0 such that 2n tψ(2n a, 2n a∗ ) < t22 for all n ≥ n0 and t > 0. Hence N (δ(a)∗ − δ(a∗ ), t)  t 22n t  t  ∗ , N δ(a ) − 2−2n f (2n a∗ ), , N (f (2n a)∗ − f (2n a∗ ), }. ≥ min{N δ(a)∗ − 2−2n f (2n a)∗ , 4 4 2 The first two terms on the right hand side of the above inequality tend to 1 as n → ∞. Furthermore the last term is greater than N (f (2n a)∗ − f (2n a∗ ), tψ(2n a, 2n a∗ )), which is greater than 1 − . Since N (δ(a)∗ − δ(a∗ ), t) > 1 −  for all t > 0, we get δ(a)∗ = δ(a)∗ for all a ∈ A. Thus δ is a quadratic ∗-derivation on A. The proof in the case that s = −1 is similar.  Next, we prove the superstability of quadratic ∗-derivations on fuzzy Banach ∗-algebras. Definition 2.3. Suppose that A is a ∗-normed algebra and s ∈ {−1, 1}. Let δ : A → A be a mapping for which there exist functions ψi : A × A → [0, ∞)(1 ≤ i ≤ 3) and φ : A → A satisfying lim n−2s ψi (ns a, b) = lim n−2s ψi (a, ns b) = 0

(2.9)

lim N (a2 δ(b) − φ(a)b2 , tψ1 (a, b)) = 1

(2.10)

lim N (φ(a)(cd)2 − a2 (δ(c)d2 − c2 δ(d)), tψ2 (a, cd)) = 1

(2.11)

lim N (aδ(b∗ ) − φ(a)b∗ , tψ3 (a, b)) = 1

(2.12)

n→∞

n→∞

such that t→∞

t→∞

t→∞

for all a, b, c, d ∈ A. Then δ is called a (ψ, φ)-approximate quadratic ∗-derivation on A. Theorem 2.4. Suppose that A is a fuzzy Banach ∗-algebra with approximate unit and s ∈ {−1, 1}. Let δ : A → A be a (φ, ψ)-approximate quadratic ∗-derivations on A . Then δ ∗-derivations on A . Proof. We asume that (2.9) holds. We first show that δ is quadratic homogeneous. To do this, pick λ ∈ C and a, z ∈ A and  > 0 is given. By (2.10) there exists t0 > 0 such that   2 N n−2s (n2s z 2 δ(λa) − φ(ns z)(λa) ), n−2s tψ1 (ns z, λa) > 1 − ,   N n−2s (λ2 φ(ns z)a2 − λ2 n2s z 2 δ(a)), n−2s t|λ|2 ψ1 (ns z, a) > 1 − 

742

Approximate quadratic ∗-derivation for all t ≥ t0 . Fix t temporarily. By (2.9) there exists n0 > 0 such that n−2s tψ1 (ns z, λa) ≤ n−2s t|λ|2 ψ1 (ns z, a) ≤ 2t for all n ≥ n0 . By (N5 ) we have

t 2

and

  N z 2 (δ(λa) − λ2 δ(a)), t   = N n−2s (n2s z 2 δ(λa) − λ2 n2s z 2 δ(a)), t   = N n−2s (n2s z 2 δ(λa) − φ(ns z)(λa)2 ) + n−2s (λ2 φ(ns z)a2 − λ2 n2s z 2 δ(a)), t  t t   −2s 2 ≥ min{N n−2s (n2s z 2 δ(λa) − φ(ns z)(λa)2 ), , N n (λ φ(ns z)a2 − λ2 n2s z 2 δ(a)), } 2 2   ≥ min{N n−2s (n2s z 2 δ(λa) − φ(ns z)(λa)2 ), n−2s tψ1 (ns z, λa) ,   N n−2s (λ2 φ(ns z)a2 − λ2 n2s z 2 δ(a)), n−2s t|λ|2 ψ1 (ns z, a) }.

   Since N n−2s (n2s z 2 δ(λa) − φ(ns z)(λa)2 ), n−2s tψ1 (ns z, λa) > 1 −  and N n−2s (λ2 φ(ns z)a2 −  λn2s z 2 δ(a)), n−2s t|λ|2 ψ1 (ns z, a) > 1 − , we have N (z 2 (δ(λa) − λ2 δ(a)), t) > 1 −  for all t > 0.And by (N2 ) it leads us to have a conclusion that z 2 (δ(λa) − λ2 δ(a)) = 0 for all z ∈ A. Let {ei }i∈I be an approximate unit of A. Then {f (ei )|i ∈ I} is also an approximate unit of A for every polynomial f . Considering ei instead of z in the above inequality, we conclude that δ(λa) = λ2 δ(a) for all λ ∈ C. Next, we are going to show that δ is quadratic. By (2.10) there exists t0 > 0 such that

  N n−2s (n2s z 2 δ(a + b) − φ(ns z)(a + b)2 ), n−2s tψ1 (ns z, a + b) > 1 − ,

  N n−2s (n2s z 2 δ(a − b) − φ(ns z)(a − b)2 ), n−2s tψ1 (ns z, a − b) > 1 − ,

  N 2n−2s ( φ(ns z)a2 − n2s z 2 δ(a)), n−2s 2tψ1 (ns z, a) > 1 − , and   N 2n−2s ( φ(ns z)b2 − n2s z 2 δ(b)), n−2s 2tψ1 (ns z, b) > 1 −  for all t ≥ t0 . For fixed t temporarily there exists n0 > 0 such that n−2s tψ1 (ns z, a + b) ≤ n−2s tψ1 (ns z, a − b) ≤ 4t , 2n−2s tψ1 (a, ns z) ≤ 4t , and 2n−2s tψ1 (b, ns z) ≤ 4t for all n ≥ n0 .

743

t 4,

S. Jang and Y. Cho To show the quadraticity of δ we consider the following:   N z 2 (δ(a + b) + δ(a − b) − 2δ(a) − 2δ(b)), t  = N n−2s (n2s z 2 δ(a + b) − φ(ns z)(a + b)2 ) + n−2s (n2s z 2 δ(a − b) − φ(ns z)(a − b)2 )  +2n−2s ( φ(ns z)a2 − n2s z 2 δ(a)) + 2n−2s ( φ(ns z)b2 − n2s z 2 δ(b)), t  t ≥ min{N n−2s (n2s z 2 δ(a + b) − φ(ns z)(a + b)2 ), , 4    t t , N 2n−2s ( φ(ns z)a2 − n2s z 2 δ(a)), , N n−2s (n2s z 2 δ(a − b) − φ(ns z)(a − b)2 ), 4 4   t N 2n−2s ( φ(ns z)b2 − n2s z 2 δ(b)), } 4   ≥ min{N n−2s (n2s z 2 δ(a + b) − φ(ns z)(a + b)2 ), n−2s ψ1 (ns z, a + b) ,   N n−2s (n2s z 2 δ(a − b) − φ(ns z)(a − b)2 ), n−2s ψ1 (ns z, a − b) ,     N n−2s ( φ(ns z)a2 − n2s z 2 δ(a)), 2n−2s ψ1 (ns z, a) , N 2n−2s ( φ(ns z)b2 − n2s z 2 δ(b)), 2n−2s ψ1 (ns z, b) } forall a, b, z ∈ A. Since all terms of the finalinequality of the above inequality are larger than 1 − , N z 2 (δ(a + b) + δ(a − b) − 2δ(a) − 2δ(b)), t > 1 −  for all t > 0. So we can have that z 2 (δ(a + b) + δ(a − b) − 2δ(a) − 2δ(b)) = 0 for all a, b, z ∈ A. By using the approximate unit instead of z as the similar discussion in the above, we have δ(a + b) + δ(a − b) − 2δ(a) − 2δ(b) = 0 for all a, b ∈ A. Next we are going to show the derivation property of δ. By (2.10) there exists t0 > 0 such that   N n−2s (n2s z 2 δ(ab) − φ(ns z)(ab)2 ), n−2s tψ1 (ns z, ab) > 1 − , and   N n−2s (φ(ns z)(ab)2 − n2s z 2 δ(a)b2 − n2s z 2 a2 δ(b)), n−2s tψ2 (ns z, ab) > 1 −  for t ≥ t0 . For fixed t temporarily there exists n0 > 0 such that n−2s tψ1 (ns z, ab) ≤ 2t and n−2s tψ2 (ns z, ab) ≤ 2t for n ≥ n0 .   N z 2 (δ(ab) − (δ(a)b2 − a2 δ(b))), t   = N n−2s (n2s z 2 δ(ab) − φ(ns z)(ab)2 ) + n−2s (φ(ns z)(ab)2 − n2s z 2 δ(a)b2 − n2s z 2 a2 δ(b)), t  t t   −2s , N n ( φ(ns z)(ab)2 − n2s z 2 δ(a)b2 − n2s z 2 a2 δ(b)), } ≥ min{N n−2s (n2s z 2 δ(ab) − φ(ns z)(ab)2 ), 2 2   ≥ min{N n−2s (n2s z 2 δ(ab) − φ(ns z)(ab)2 ), n−2s tψ1 (ns z, ab) ,   N n−2s (φ(ns z)(ab)2 − n2s z 2 δ(a)b2 − n2s z 2 a2 δ(b)), n−2s tψ2 (ns z, ab) } forall a, b, z ∈ A. Since all termsof the final inequality of the above inequality are larger than 1 − , N z 2 (δ(ab) − (δ(a)b2 + a2 δ(b)), t > 1 −  for all t > 0. So we have z 2 (δ(ab) − (δ(a)b2 + a2 δ(b)) = 0 for all a, b, z ∈ A. By using approximate unit instead of z we have that δ(ab) = δ(a)b2 + a2 δ(b) for all a, b ∈ A. On the involution we can have the same computation in theorem 2.1. Thus δ is a quadratic ∗-derivation on A. 

744

Approximate quadratic ∗-derivation Acknowledgement S. Y. Jang was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-20110004872) and have written during visiting the Research Institute of Mathematics, Seoul Natinal Univerity.

References [1] J. Aczl and J. Dhombres,Functional Equations in Serveral Variables, Cambrdge Univ. Press, Cambridge, 1989. [2] J. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1979), 242–246. [3] O. Bratteli, Derivation, Dissipation and Gruop Actions on C ∗ -Algebras, Lecture Notes in Math., Vol. 1229, Springer-Verlag, Belin, 1986. [4] O. Bratteli, A. Kishimoto and D. W. Robinson, Approximately inner derivations, Math. Scan. 103 (2008), 141-160. [5] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705. [6] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math.27(1984), [7] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hambrug 62 (1992), 59-64 [8] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. [9] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser,Basel, 1998. [10] S. G. Lee and S. Y. Jang, Unbounded derivations on compact actions of C ∗ -algebras, Commun. Korean Math. Soc. 5 (1990), 79–86. [11] M. Mirzavaziri and M. S. Moslehain, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37(2006), 361-376. [12] M. S. Moslehian, F. Rahbarnia and P. K. Sahoo, Approximate double centralizers are exact double centralizers, Bol. Soc. Mat. Mexicana 13 (2007), 111-122. [13] T. W. Parlem, Banach Algebras and General Theory of ∗-Algebras, Vol.I, Algebras and Banach Algebras, Encyclopedia of Mathematics and Applications 49, Cambridge University Press, Cambridge, 1994. [14] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [15] F. Skof, Proriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis.Milano 53(1983),113-129 [16] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 746-752, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

THE TOPOLOGICAL STRUCTURE ON SOFT SETS ZHAOWEN LI, HAIYAN CHEN, AND NINGHUA GAO Abstract. Soft set theory is a new mathematical tool to deal with uncertain problems. This paper is devoted to investigating relationships between soft sets and topologies. Based on new granulation structures called soft approximation spaces, soft rough approximations are introduced. We give some properties of soft rough approximation. New types of soft sets such as keeping intersection, keeping union and topological soft sets are defined and supported by some illustrative examples. We obtain the topological structure on soft sets and reveal that every topological space on the initial universe is a soft approximating space.

1. Introduction Most of traditional methods for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. However, Many practical problems within fields such as economics, engineering, environmental science, medical science and social science involve data that contain uncertainties. We can not use traditional methods because of various types of uncertainties present in these problems. There are several theories: probability theory, fuzzy set theory [18] and rough set theory [15], which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. For example, probability theory can deal only with stochastically stable phenomena. To overcome these kinds of difficulties, Molodtsov [14] proposed a completely new approach, which is called soft set theory, for modeling uncertainty. Recently works on soft set theory are progressing rapidly. Maji et al. [12, 13] discussed the application of of soft sets in decision making problems and introduced fuzzy soft sets. Roy et al. [16] presented a fuzzy soft set theoretic approach towards decision making problems. Jiang et al. [9] extended soft sets with description logics. Aktas and Ca˘ g man [1] defined soft groups. Feng et al. [5, 6] investigated the relationship among soft sets, rough sets and fuzzy sets. Shabir et al. [17] investigated soft topological spaces. Ge et al. [7] discussed the relationship between topological spaces and soft sets. The purpose of this paper is to investigate the problem on soft sets combined with topologies. We introduce several new types of soft sets, obtain the topological structure on soft sets and reveal that every topological space on the initial universe is a soft approximating space. 2000 Mathematics Subject Classification. 03E47; 54A05; 54C60. Key words and phrases. Soft sets; Soft approximation spaces; Soft rough approximations; Topologies; Soft approximating spaces. This work is supported by the National Natural Science Foundation of China (No. 11061004, 10971186, 71140004), the Natural Science Foundation of Guangxi Province in China (No. 2011GXNSFA018125) and the Science Research Project of Guangxi University for Nationalities (No. 2011QD015). 1

746

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ZHAOWEN LI, HAIYAN CHEN, AND NINGHUA GAO

2. Preliminaries In this section, we briefly recall several basic concepts of soft sets. Definition 2.1 ([14]). Let U be an initial universe and let E be a set of all possible parameters. Let 2U denote the power set of U and let A ⊆ E. A pair (f, A) is called a soft set over U , if f is a mapping given by f : A → 2U . We denote (f, A) by fA . In other words, a soft set over U is a parametrized family of subsets of the universe U . For e ∈ A, f (e) may be considered as the set of e-approximate elements of the soft set fA . In this paper, we only consider the case where both U and E are nonempty finite sets. Definition 2.2 ([11]). let A, B ⊆ E and let fA and gB be two soft sets over U . (1) fA is called a soft subset of gB , if A ⊆ B and f (e) = g(e) for each e ∈ A. e gB . We denote it by fA ⊆ e fA . We denote it by fA ⊃ e gB . (2) fA is called a soft super set of gB , if gB ⊆ Definition 2.3 ([11]). let A, B ⊆ E and let fA and gB be two soft sets over U . fA and gB are called soft equal, if A ⊆ B and f (e) = g(e) for each e ∈ A. We denote it by fA = gB . e gB and fA ⊃ e gB . Obviously, fA = gB if and only if fA ⊆ Definition 2.4 ([6, 7]). A soft set fA over U is called full, if

S

f (a) = U .

a∈A

Definition 2.5. Let fA be a soft set over U . (1) fA is called topological, if {f (a) : a ∈ A} is a topology on U . (2) fA is called keeping union, if for each a, b ∈ A, there exists c ∈ A such that f (a) ∪ f (b) = f (c). (3) fA is called keeping intersection, if for each a, b ∈ A, there exists c ∈ A such that f (a) ∩ f (b) = f (c). Example 2.6. Let U = {h1 , h2 , h3 , h4 , h5 }, A = {a1 , a2 , a3 , a4 } and let fA be a soft set over U , defined as follows f (a1 ) = {h1 , h2 , h5 }, f (a2 ) = ∅, f (a3 ) = {h3 }, f (a4 ) = {h3 , h4 }. We have f (a3 ) ∩ f (a4 ) = {h3 } = f (a3 ). f (a1 )∩f (a2 ) = f (a1 )∩f (a3 ) = f (a1 )∩f (a4 ) = f (a2 )∩f (a3 ) = f (a2 )∩f (a4 ) = ∅ = f (a2 ). Then fA is full and keeping intersection. But f (a1 ) ∪ f (a3 ) = {h1 , h2 , h3 , h5 } 6= f (a) (∀a ∈ A). Thus fA is not keeping union. Example 2.7. Let U = {h1 , h2 , h3 , h4 , h5 }, A = {a1 , a2 , a3 , a4 } and let fA be a soft set over U , defined as follows f (a1 ) = {h1 }, f (a2 ) = {h2 }, f (a3 ) = {h1 , h2 }, f (a4 ) = U. Then fA is full and keeping union. But fA is not keeping intersection.

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THE TOPOLOGICAL STRUCTURE ON SOFT SETS

3

Example 2.8. Let U = {h1 , h2 , h3 , h4 , h5 }, A = {a1 , a2 , a3 , a4 } and let fA be a soft set over U , defined as follows f (a1 ) = {h1 }, f (a2 ) = {h1 , h4 }, f (a3 ) = {h1 , h3 , h4 }, f (a4 ) = X. Obviously, fA is full, keeping intersection and keeping union. But fA is not topological. From Example 2.6, 2.7 and 2.8, we have the following relationships:

fA is topological

fA is full, keeping intersection and keeping union

fA is full and keeping union

fA is full and keeping intersection

3. Soft rough approximations In this section, we consider a pair of soft rough approximations which are presented by Feng et al. in [5, 6] and give their properties. Definition 3.1 ([5, 6]). Let fA be a soft set over U . Then the pair P = (U, fA ) is called a soft approximation space. Based on the soft approximation space P , we define the following two operations: For X ∈ 2U , aprP (X) = {u ∈ U : ∃a ∈ A, s.t. u ∈ f (a) ⊆ X}, aprP (X) = {u ∈ U : ∃a ∈ A, s.t. u ∈ f (a) and f (a) ∩ X 6= ∅}. aprP (X) and aprP (X) are called the soft P -lower approximation and the soft P -upper approximation of X, respectively. In general, we refer to aprP (X) and aprP (X) as soft rough approximations of X with respect to P . Proposition 3.2 ([5, 6]). Let fA be a soft set over U and let P = (U, fA ) be a soft approximation space. Then the following properties hold for X, Y ∈ 2U . S (1) aprP (X) = {f (a) : a ∈ A and f (a) ⊆ X} ⊆ X; S aprP (X) = {f (a) : a ∈ A and f (a) ∩ X 6= ∅}. S (2) aprP (∅) = aprP (∅) = ∅; aprP (U ) = aprP (U ) = f (a). a∈A

(3) X ⊆ Y ⇒ aprP (X) ⊆ aprP (Y );

X ⊆ Y ⇒ aprP (X) ⊆ aprP (Y ).

(4) aprP (X ∪ Y ) = aprP (X) ∪ aprP (Y ).

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ZHAOWEN LI, HAIYAN CHEN, AND NINGHUA GAO

(5) aprP (aprP (X)) = aprP (X);

aprP (aprP (X)) = aprP (X).

Proposition 3.3. Let fA be a soft set over U and let P = (U, fA ) be a soft approximation space. Then the following properties hold. (1) If fA is full, then aprP (X) ⊆ X ⊆ aprP (X) for any X ∈ 2U . (2) If fA is full, then aprP (U ) = aprP (U ) = U . (3) If fA is keeping intersection, then aprP (X ∩ Y ) = aprP (X) ∩ aprP (Y ) for any X, Y ∈ 2U . (4) If fA is full and keeping union, then aprP (X) = U for any X ∈ 2U \ {∅}. Proof. (1) By Proposition 3.2, aprP (X) ⊆ X. Suppose X − aprP (X) 6= ∅. Pick

Since fA is full, U =

S

u ∈ X − aprP (X) 6= ∅. f (a). So u ∈ f (a) for some a ∈ A. u ∈ X implies that

a∈A

f (a) ∩ X 6= ∅. Thus u ∈ aprP (X) 6= ∅, contradiction. Hence X ⊆ aprP (X). (2) This holds by (1). (3) Obviously, aprP (X ∩ Y ) ⊆ aprP (X) ∩ aprP (Y ). Suppose aprP (X) ∩ aprP (Y ) − aprP (X ∩ Y ) 6= ∅. Pick u ∈ aprP (X) ∩ aprP (Y ) − aprP (X ∩ Y ). Then there exist a, b ∈ A such that u ∈ f (a) ⊆ X and u ∈ f (b) ⊆ Y . Since fA is keeping intersection, then f (a) ∩ f (b) = f (c) for some c ∈ A. This implies that u ∈ f (c) ⊆ X ∩ Y . Thus u ∈ aprP (X ∩ Y ), contradiction. Thus aprP (X ∩ Y ) ⊇ aprP (X) ∩ aprP (Y ). Therefore, aprP (X ∩ Y ) = aprP (X) ∩ aprP (Y ). S (4) Since fA is full and keeping union, then U = f (a) = f (a∗ ) for some a∈A

a∗ ∈ A. For each X ∈ 2U \ {∅} and each u ∈ U , u ∈ f (a∗ ) and f (a∗ ) ∩ X = X 6= ∅, then aprP (X) = U . ¤ 4. The topological structure on soft sets In this section, we obtain the topological structure on soft sets. Definition 4.1 ([2, 3]). Let τ ⊆ 2U . τ is called a generalized topology on U , if (i) ∅ ∈ τ , (ii) {Aα : α ∈ Γ} ⊆ τ implies ∪{Aα : α ∈ Γ} ∈ τ . Moreover, the pair (U, τ ) is called a generalized topological space and every member of τ is called a generalized open subset of U . Theorem 4.2. Let fA be a soft set over U and let P = (U, fA ) be a soft approximation space. Then (1) {X ∈ 2U : aprP (X) = X} is a generalized topology on U (2) If fA is full, keeping intersection, then {X ∈ 2U : aprP (X) = X} is a topology on U . (3) If fA is full and keeping union, then {X ∈ 2U : aprP (X) = X} = {∅, U } is a indiscrete topology on U .

749

THE TOPOLOGICAL STRUCTURE ON SOFT SETS

5

Proof. (1) By Proposition 3.2, ∅ ∈ τf . Let Xα ∈ τf for each α ∈ Γ. Denote X = ∪{Xα : α ∈ Γ}. Since Xα ⊆ X for each α ∈ Γ, then Xα = aprP (Xα ) ⊆ aprP (X) by Proposition 3.2. So X = ∪{Xα : α ∈ Γ} ⊆ aprP (X). By Proposition 3.2, aprP (X) ⊆ X. Thus aprP (X) = X. This implies ∪{Aα : α ∈ Γ} ∈ τf . Hence τf is a generalized topology on U . (2) By Proposition 3.2 and 3.3, we have U ∈ τ and X ∩ Y ∈ τf whenever X, Y ∈ τf . By (1), τf is a generalized topology on U . Thus τ is a topology on U . (3) This holds by Proposition 3.2 and 3.3. ¤ Definition 4.3. Let fA be a full and keeping intersection soft set over U and let P = (U, fA ) be a soft approximation space. Then {X ∈ 2U : aprP (X) = X} is called the topology induced by fA on U . We denote it by τf . The following Theorem 4.3 gives the topological structure on soft sets (i.e., the structure of topologies induced by soft sets). Theorem 4.4. Let fA be a full and keeping intersection soft set over U and let τf be the topology induced by fA on U . Then (1) {aprP (X) : X ∈ 2U } ⊆ τf = {aprP (X) : X ∈ 2U }. (2) τf ⊇ {f (a) : a ∈ A}. (3) If fA is topological, then τf = {f (a) : a ∈ A}. (4) aprP is an interior operator of τf . Proof. (1) By Proposition 3.2, we have {aprP (X) : X ∈ 2U } ⊆ τf . Obviously, τf ⊆ {aprP (X) : X ⊆ U }. Let Y ∈ {aprP (X) : X ∈ 2U }. Then Y = aprP (X) for some X ∈ 2U . By Proposition 3.2, aprP (aprP (X)) = aprP (X). This implies that Y ∈ τf . Thus τf ⊇ {aprP (X) : X ∈ 2U }. Hence {aprP (X) : X ∈ 2U } ⊆ τf = {aprP (X) : X ∈ 2U }. (2) Obviously, aprP (f (a)) ⊆ f (a) f or each a ∈ A. Let x ∈ f (a). Since x ∈ f (a) ⊆ f (a), x ∈ aprP (f (a)). Then f (a) ⊆ aprP (f (a)). So f (a) ∈ τf . Thus {f (a) : a ∈ A} ⊆ τf . (3) Let X ∈ τf . For each x ∈ X, X = aprP (X), there exists ax ∈ A such S S S that x ∈ f (ax ) ⊆ X. Then X = {x} ⊆ f (ax ) ⊆ X. So X = f (ax ). x∈X x∈X S x∈X Since fA is topological, then f (ax ) = f (a) f or some a ∈ A. This implies x∈X

X ∈ {f (a) : a ∈ A}. Thus τf ⊆ {f (a) : a ∈ A}. By (1), τf ⊇ {f (a) : a ∈ A}. Hence τf = {f (a) : a ∈ A}. (4) It suffices to show that aprP (X) = int(X) f or each X ∈ 2U . By (1), aprP (X) ∈ τf . By Proposition 3.2, aprP (X) ⊆ X. Thus aprP (X) ⊆ int(X). Conversely. For each Y ∈ τf with Y ⊆ X, we have Y = aprP (Y ) ⊆ aprP (X) by S Proposition 3.2. Thus int(X) = {Y : Y ∈ τf and Y ⊆ X} ⊆ aprP (X). Hence aprP (X) = int(X). ¤ 5. Soft sets induced by topologies In this section, we consider soft sets induced by topologies. Definition 5.1. Let τ be a topology on U . Put τ = {Ua : a ∈ A}, where A is the set of indexes. Define a mapping fτ : A → 2U by fτ (a) = Ua for each a ∈ A. Then (fτ )A is called the soft set induced by τ on U .

750

6

ZHAOWEN LI, HAIYAN CHEN, AND NINGHUA GAO

Definition 5.2. Let (U, µ) be a topological space. If there exists a full and keeping intersection soft set fA over U such that τf = µ, then (U, µ) is called a soft approximating space. The following Proposition 5.3 can easily be proved. Proposition 5.3. (1) Let τ be a topology on U and let (fτ )A be the soft set induced by τ on U . Then (fτ )A is a topological soft set over U . (2) Let τ1 and τ2 be two topologies on U and let (fτ1 )A1 and (fτ1 )A2 be two soft e (fτ1 )A2 . sets induced respectively by τ1 and τ2 on U . If τ1 ⊆ τ2 , then (fτ1 )A1 ⊆ Proposition 5.4. Let τ be a topology on U , let (fτ )A be the soft set induced by τ on U and let τfτ be the topology induced by (fτ )A on U . Then τ = τfτ . Proof. Put τ = {Ua : a ∈ A}, then fτ : A → 2U is a mapping, where fτ (a) = Ua for each a ∈ A. By Proposition 5.3, (fτ )A is topological. By Theorem 4.3, τfτ = {fτ (a) : a ∈ A}. Hence τfτ = τ . ¤ Theorem 5.5. Every topological space on the initial universe is a soft approximating space. Proof. This holds by Proposition 5.4.

¤

Theorem 5.6. Let (U, τ ) be a topological space. Then there exists a topological soft set fA over U such that aprP (X) = int(X) f or each X ∈ 2U , where P = (U, fA ) is a soft approximation space. Proof. Put τ = {Ua : a ∈ A}, where A is the set of indexes. Define a mapping f : A → 2U by f (a) = Ua f or each a ∈ A. By Proposition 5.3, fA is topological. Let X ∈ 2U . For each x ∈ aprP (X), x ∈ f (a) ⊆ X for some a ∈ A. So x ∈ Ua ⊆ X with Ua ∈ τ . This implies that x ∈ int(X). Conversely. For each x ∈ int(X), there exists an open neighborhood W of x in U such that W ⊆ X. So W = Ua for some a ∈ A. This implies that x ∈ f (a) ⊆ X. Thus x ∈ aprP (X). Hence aprP (X) = int(X). ¤ Theorem 5.7. Let fA be a full and keeping intersection soft set over U , let τf be the topology induced by fA on U and let (fτf )B be the soft set induced by τf on U . Then e (fτf )B . (1) fA ⊆ (2) If fA is topological, then fA = (fτf )B . Proof. (1) By Theorem 4.3, τf ⊇ {f (a) : a ∈ A}. Denote τf = {Ua : a ∈ B}, where A ⊆ B and Ua = f (a) f or each a ∈ A. Thus fτf is a mapping given by fτf : A → 2U , where fτf (a) = Ua for each a ∈ B. e (fτf )B . Hence fA ⊆ (2) Since fA is topological, then by Theorem 4.3, we have A = B. Hence fA = (fτf )B . ¤

751

THE TOPOLOGICAL STRUCTURE ON SOFT SETS

7

References [1] H.Aktas, N.Ca˘ g man, Soft sets and soft groups, Inform. Sci., 177(2007), 2726-2735. [2] A.Cs´ asz´ ar, Generalized topology, generalized continuity, Acta Math. Hungar., 96(2002), 351357. [3] A.Cs´ asz´ ar, Generalized open sets, Acta Math. Hungar., 75(1997), 65-87. [4] R.Engelking, General topology, Polish Scientific Publishers, Warszawa, 1977. [5] F.Feng, C.Li, B.Davvaz, M.Irfan Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Comput., 14(2010), 899-911. [6] F.Feng, X.Liu, V.Leoreanu-Fotea, Y.Jun, Soft sets and soft rough sets, Inform. Sci., 181(2011), 1125-1137. [7] X.Ge, Z.Li, Y.Ge, Topological spaces and soft sets, J. Comput. Anal. Appl., 13(2011), 881885. [8] Y.Jiang, Y.Tang, Q.Chen, J.Wang, S.Tang, Extending soft sets with description logics, Comput. Math. Appl., 59(2010), 2087-2096. [9] Z.Li, T.Xie, Q.Li, Topological structure of generalized rough sets, Comput. Math. Appl., 63(2012), 1066-1071. [10] Z.Li, K.Zhang, The topological structure on generalized approximation spaces, Acta Math. Appl. Sinica, 33(2010), 750-757. [11] P.K.Maji, R.Biswas, A.R.Roy, Soft set theory, Comput. Math. Appl., 45(2003), 555-562. [12] P.K.Maji, R.Biswas, A.R.Roy, Fuzzy soft sets, J. Fuzzy Math., 9(2001), 589-602. [13] P.K.Maji, A.R.Roy, R.Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl., 44(2002), 1077-1083. [14] D.Molodtsov, Soft set theory-first result, Comput. Math. Appl., 37 (1999), 19-31. [15] Z.Pawlak, Rough Sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers, Boston, 1991. [16] A.R.Roy, P.K.Maji, A fuzzy soft set theoretic approach to decision making problems, J. Comput. Appl. Math., 203(2007), 412-418. [17] M.Shabir, M.Naz, On soft topological spaces, Comput. Math. Appl., 561(2011), 1786-1799. [18] L.A.Zadeh, Fuzzy sets, Inform. Control, 8(1965), 338-353. College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China E-mail address: [email protected] College of Mathematics and Information Science, Guangxi University, Guangxi 530004 P.R.China College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China

752

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 753-763, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A Caputo Fractional Order Boundary Value Problem with Integral Boundary Conditions 1

Azizollah Babakhani and 2 Thabet Abdeljawad

∗

September 23, 2012

1 Department

of Basic Science Babol University of Technology Bobol 47148-71167, Iran. Email: [email protected] and 2 Department

of Mathematics and Computer Science C ¸ ankaya University, 06530 Ankara, Turkey. [email protected] Abstract

In this paper, we discuss existence and uniqueness of solutions to nonlinear fractional order ordinary differential equations with integral boundary conditions in an ordered Banach space. We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function. The nonlinear alternative of the Leray- Schauder type Theorem is the main tool used here to establish the existence and the Banach contraction principle to show the uniqueness of the solution under certain conditions. The compactness of solutions set is also investigated and an example is included to show the applicability of our results.

Keywords. Boundary value problem, differential equations, integral boundary conditions, fixed point.

1

Introduction

During the last two decades fractional calculus has started to appear with many important applications in biology [1], physics ([2], [6], [16], [17], [23]) and chemistry [3]. As surveys for the theory of fractional integration and differentiation we refer the reader to the books ∗

Corresponding author 753

BABAKHANI, ABDELJAWAD: FRACTIONAL BVP

([4], [5], [7], and [10]). For more recent details about the theory of fractional dynamical systems and interpretations of fractional integration and differentiation see ([8], [9], [11], [13],[19], [22], [24] and [25]). As a part of the general theory of fractional dynamical systems, the existence theory for fractional initial and boundary value problems has also appeared (see [12], [18] and the survey books mentioned above). For the existence theory in the delay case we, for example, refer to ([14], [15], [26], and [27]). For the basic tools in fixed point theory necessary to obtain our result see ([20], [21]). This paper is concerned with the existence of solutions for the fractional order boundary value problem c

Dα y(t) = f (t, y(t)), a.e. t ∈ (0, 1), Z 1 y(0) = 0, y(1) = g(s)y(s) ds,

(1a) (1b)

0

where c Dα is the Caputo fractional derivative of order α ∈ (1, 2], f : [0, 1] × R −→ R is a given function and g : [0, 1] −→ R an integrable function.

2

Basic Tools

We dedicate this section to recall and introduce some notations, definitions and preliminary facts that will be used in the remainder of this paper. We shall denote by R the real line, by R+ the interval [0, ∞). We mean by C([0, 1], R) is the Banach space of all continuous functions from [0, 1] into R with the norm kyk∞ = sup{|y(t)| : 0 ≤ t ≤ 1}. Let also AC 1 ([0, 1]), R) be the space of all functions y : [0, 1] −→ R, whose first derivative y 0 is absolutely continuous on [0, 1]. The set AC 1 ([0, 1]), R) is a Banach space when it is furnished with the norm k . k∞ defind by kyk∞ = sup{|y 0 (t)| : t ∈ I}. Finally, we denote L1 ([0, 1], R) denote the Banach space of functions y : [0, 1] −→ R that R1 are Lebesgue integrable with norm kykL1 = 0 |y(t)| dt. Definition 2.1 A map f : [0, 1] −→ R is said to be L1 −Caratheodory if (i) t −→ f (t, u) is measurable for each u ∈ R; (ii) t −→ f (t, u) is continuous for almost each t ∈ [0, 1]; (iii) for every r > 0 there exists hr ∈ L1 ([0, 1], R) such that |f (t, u)| ≤ hr (t) for a.e. t ∈ [0, 1] and all |u| ≤ r. Definitions of Caputo and Remann−Liouville fractional derivative/integral and their relation are given bellow.

754

BABAKHANI, ABDELJAWAD: FRACTIONAL BVP

Definition 2.2 For a function u defined on an interval [a, b], the Remann−Liouville fractional integral of f of order α > 0 is defined by Z t 1 α (t − s)α−1 y(s) ds, t > a, Ia+ y(t) = Γ(α) a and Remann−Liouville fractional derivative of y(t) of order α > 0 defined by Daα+ y(t) =

dn  n−α I y(t) , + dtn a

where n − 1 < α ≤ n, while Caputo fractional derivative of x of order α > 0 is defined by n o c α y (n) (t) . Da+ y(t) = Ian−α + The relation between Caputo fractional derivative and Riemanna−Lioville fractional derivative is given by Daα+ y(t) = c Dαa+ y(t) +

n−1 X j=0

x(j) (a) (t − a)j−α Γ(j − α + 1)

(2)

We denote c Dαa+ y(t) as c Dαa y(t) and Iaα+ y(t) as Iaα y(t). Further c Dα0+ y(t) and I0α+ y(t) are referred as c Dα y(t) and I α y(t), respectively. Lemma 2.3 (Lemma 2.22 [7]). Let α > 0. Then I α (c Dα y(t)) = y(t) + c0 + c1 t + c2 t2 + · · · + cr−1 tr−1 for arbitrary ci ∈ R, i = 0, 1, · · · , r − 1, r = [α] + 1.

3

Existence and Uniqueness Results

Definition 3.1 A function y ∈ AC 1 ((0, 1), R), is said to be a solution of (1a) − (1b) if y satisfies (1a) − (1b). R1

In what follows we assume that g∗ = result.

0

s g(s)ds 6= 1. We need the following auxiliary

Lemma 3.2 Let σ ∈ L1 ([0, 1], R). Then the function defined by Z

1

H(t, s)σ(s) ds

y(t) =

(3)

0

is the unique solution of the boundary value problem c

Dα y(t) = σ(t),

a.e. t ∈ (0, 1), Z 1 y(0) = 0, y(1) = g(s)y(s) ds, 0

755

(4a) (4b)

BABAKHANI, ABDELJAWAD: FRACTIONAL BVP

where H(t, s) = G(t, s) +

Z

t 1−

R1 0

sg(s)ds

1

G(r, s)g(r)dr

(5)

0

and

G(t, s) =

 (t−s)α−1 −t(1−s)α−1  ,  Γ(α) 

if 0 ≤ s ≤ t ≤ 1;

  − t(1−s)α−1 , Γ(α)

if 0 ≤ t ≤ s ≤ 1.

Proof. Let y be a solution of the problem (4a)-(4b). If we apply I α to (4a) and make use of Lemma 2.3, we obtain Z t 1 y(t) = −c0 − c1 t + (t − s)α−1 σ(s)ds. (6) Γ(α) 0 The boundary condition (4b) yields c0 = 0 and, Z 1 Z 1 1 (1 − s)α−1 σ(s)ds − g(s)y(s) ds. c1 = Γ(α) 0 0 Hence Z t (t − s)α−1 σ(s) t(1 − s)α−1 σ(s) ds + ds y(t) = tg(s)y(s) ds − Γ(α) Γ(α) 0 0 0  Z 1 Z t (t − s)α−1 − t(1 − s)α−1 = tg(s)y(s) ds + σ(s)ds Γ(α) 0 0 Z 1 t(1 − s)α−1 σ(s) − ds Γ(α) t Z

1

Z

1

and Z y(t) =

1

Z

0

1

G(t, s)σ(s) ds,

tg(s)y(s) ds +

(7)

0

where

G(t, s) =

 (t−s)α−1 −t(1−s)α−1  ,  Γ(α) 

if 0 ≤ s ≤ t ≤ 1;

  − t(1−s)α−1 , Γ(α)

if 0 ≤ t ≤ s ≤ 1.

Now, multiply equation (7) by g and integrate over [0, 1], to get  Z 1  Z 1 Z 1 Z 1 g(s)y(s)ds = g(s) s g(r)y(r) dr + G(s, r)σ(r) dr ds 0 0 0 0 Z 1  Z 1 Z 1  Z 1 = sg(s)ds g(s)y(s)ds + g(s) G(s, r)σ(r)dr ds. 0

0

756

0

0

BABAKHANI, ABDELJAWAD: FRACTIONAL BVP

Thus, R1

1

Z

g(s)y(s)ds =

0

hR

g(s)

1

0

1 0 G(s, r)σ(r)dr R1 − 0 sg(s)ds

i

ds .

Substituting in (7) we have Z

t

1

G(t, s)σ(s)ds +

y(t) =

R1 0

g(s)

hR

1

0

1 0 G(s, r)σ(r)dr R1 − 0 sg(s)ds

i

ds .

Therefore Z

1

H(t, s)σ(s)ds.

y(t) = 0

Which completes the proof. ∗ Set g = |1 − g∗ |. Note that the function G(t, s) is continuous on [0, 1] × [0, 1]. Let M be such that M−1 = max {|G(t, s)| : (t, s) ∈ [0, 1] × [0, 1]} .

(8)

Our first result reads. Theorem 3.3 Assume that f is an L1 −Caratheodory function. Also assume there exists γ ∈ L1 ([0, 1], R+ ) such that |f (t, y1 ) − f (t, y2 )| ≤ γ(t)|y1 − y2 | for all y1 , y2 ∈ R and t ∈ [0, 1]. If ( ) kgkL1 kγkL1 1 + ∗ < M. g Then the BVP (1a)-(1b) has unique solution. Proof. Transfer the problem (1a)-(1b) into a fixed point problem. Consider the operator F : C([0, 1], R) −→ C([0, 1], R) define by: Z 1 (F y)(t) = H(t, s)f (s, y(s))ds, t ∈ [0, 1]. 0

We shall show that F is a contraction. Indeed, consider y1 , y2 ∈ C([0, 1], R). Then we have for each t ∈ [0, 1] Z 1 |(F y1 )(t) − (F y1 )(t)| ≤ |H(t, s)|, |f (s, y1 (s)) − f (s, y2 (s))|ds 0 Z 1 ≤ G(t, s)γ(s)|y1 (s) − y2 (s)|ds 0 Z Z 1 1 1 γ(s)|y1 (s) − y2 (s)| |g(r)| |G(r, s)|dsdr. + ∗ 0 g 0 757

BABAKHANI, ABDELJAWAD: FRACTIONAL BVP

Therefore 1 kF (y1 ) − F (y1 )k ≤ M

( kγkL1 +

kgkL1 kγkL1

)

∗

g

ky1 − y2 k∞ ,

showing that, F is a contraction and hence F has a unique fixed point which is a solution to (1a)-(1b). The proof is completed. Example 3.4 The following boundary value problem: My t4 , 1 + y2 Z y(0) = 0, y(1) =

c

3

D 2 y(t) =

a.e. t ∈ (0, 1),

(9a)

Msy(s)ds,

(9b)

1

0

has a unique solution on [0, 1]. Where M−1 = max {|G(t, s)| : (t, s) ∈ [0, 1] × [0, 1]} and  √ √  √2 [ t − s − t 1 − s], if 0 ≤ s ≤ t ≤ 1;   π G(t, s) =   − √2t √1 − s, if 0 ≤ t ≤ s ≤ 1. π

We can easily show that this example is applied in Theorem 3.3. Here, we have f (t, y) =

Mty , 1 + y2

(t, y) ∈ [0, 1] × R,

γ(t) = Mt4 , g(t) = Mt, kγkL1 =

∗ M M M , kgkL1 = and g = . 5 2 3

Then |f (t, y1 ) − f (t, y2 )| ≤ γ(t)|y1 − y2 | and ) ( M kgkL1 kγkL1 1 + ∗ = < M. 2 g We now present an existence result for the problem (1a)-(1b). Theorem 3.5 Suppose the hypotheses (H1 ) The function f : [0, 1] × R −→ R is an L1 −Caratheodory, (H2 ) There exist functions ϕ, ψ ∈ L1 ([0, 1], R+ ) and λ ∈ (0, 1) such that |f (t, u)| ≤ ϕ(t)|u|λ + ψ(t) for each (t, u) ∈ [0, 1] × R, are satisfied. Then the BVP (1a)(1b) has at least one solution. Moreover the solution set Ω = {y ∈ C([0, 1], R) : y is solution of the (1a) − (1b)} is compact. 758

BABAKHANI, ABDELJAWAD: FRACTIONAL BVP

Proof. Transform the BVP (1a)-(1b) into a fixed point problem. Consider the operator F as defined in Theorem 3.3. We shall show that F satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps. Step 1. F is continuous. Let {ym } be a sequence such that ym → y in C([0, 1], R), as m → ∞. Then 1

Z |F (ym )(t) − F (y)(t)| ≤

|H(t, s)| |f (s, ym (s)) − f (s, y(s))|ds. 0

Since f is L1 −Caratheodory and g ∈ L1 ([0, 1], R), then using Eqn. (6) and Eqn. (8) we obtain 1 kf (., ym (.)) − f (., y(.))kL1 M kgkL1 ∗ kf (., ym (.)) − f (., y(.))kL1 . Mg

kF (ym ) − F (y)k∞ ≤ +

Hence kF (ym ) − F (y)k∞ −→ 0 as m −→ ∞. Step 2. F maps bounded sets into bounded sets in C([0, 1], R). Indeed, it is enough to show that there exists a positive constant ρ such that for each y ∈ BR = {y ∈ C([0, 1], R) : kyk∞ ≤ R} one has kF (y)k∞ ≤ ρ. Let y ∈ BR . Then for each t ∈ [0, 1], we have Z

1

H(t, s)f (s, y(s))ds.

F (y)(t) = 0

By (H2 ) we have for each t ∈ [0, 1] Z |F (y)(t)| ≤

1

|H(t, s)| |f (s, y(s))|ds 0

≤

o kgk 1 n o 1 n L kψkL1 + Rλ kϕkL1 + kψkL1 + Rλ kϕkL1 . ∗ M Mg

Then for each y ∈ BR we have kF (y)k∞

kψkL1 + Rλ kϕkL1 ≤ M

( 1+

kgkL1 ∗

) := ρ.

g

Step 3. F maps bounded sets into equicontinuous sets of C([0, 1], R). Let t1 , t2 ∈ [0, 1], t1 < t2 and BR be a bounded set of C([0, 1], R) as in step 2. Let y ∈ BR and t ∈ [0, 1] we have Z

1

|F (y)(t2 ) − F (y)(t1 )| ≤

|H(t2 , s) − H(t1 , s)|ψ(s)ds Z 1 λ + R |H(t2 , s) − H(t1 , s)|ϕ(s)ds. 0

0

759

BABAKHANI, ABDELJAWAD: FRACTIONAL BVP

As t2 −→ t1 the right-hand side of the above inequality tends to zero. Then F (BR ) is equicontinuous. As a consequence of Step 1 to 3 together with Arzela-Ascoli theorem we can conclude that F : C([0, 1], R) −→ C([0, 1], R) is completely continuous. Step 4. A priori bounds on solutions. Let y = µF (y) for some µ ∈ (0, 1). This implies by (H2) that for each t ∈ [0. 1] we have Z 1 1 1 |y(t)| ≤ ϕ(s)|y(s)|λ ds + kψkL1 M 0 M Z kgkL1 kgkL1 1 ϕ(s)|y(s)|λ ds. + kψk + 1 L ∗ ∗ Mg Mg 0 Then ky(t)k ≤ +

1 1 kϕkL1 kykλ∞ + kψkL1 M M kgkL1 kgkL1 λ ∗ kψkL1 + ∗ kϕkL1 kyk∞ . Mg Mg

If kyk∞ > 1, we have 1−λ

kyk

1 M

≤

( kϕkL1 + kψkL1 +

kϕkL1 + kψkL1 M

=

1+

kgkL1 ∗

g

kgkL1

kψkL1 +

kgkL1 ∗

g

) kϕkL1

!

∗

.

g

Thus ( kyk ≤ ∗

kϕkL1 + kψkL1 M

1+

kgkL1

!)

1 1−λ

∗

:=L .

∗

g

∗

Hence kyk∞ ≤ max{1, L} :=M. Set ∗

U = {y ∈ C([0, 1], R) : kyk∞ 0 such that kym k∞ < L, for all m ≥ 1 and the set {ym }∞ m=1 is equicontinuous in C([0, 1], R), hence by Arzela-Ascoli 760

BABAKHANI, ABDELJAWAD: FRACTIONAL BVP

theorem [20], [28] we can conclude that there exists a subsequence of {ym }∞ m=1 converging to y in C([0, 1], R). Using that fast that f is an L1 −Carathedory we can prove that Z

1

H(t, s)f (s, y(s))ds,

y(t) =

t ∈ [0, 1].

0

Therefore Ω is compact. Example 3.6 Consider the following boundary value problem   p t 1 + 3 y 2 sin t e c 32 D y(t) = , a.e. t ∈ (0, 1), 1 + y2 Z 1 y(0) = 0, y(1) = e−s y(s)ds.

(10a) (10b)

0

In this example we have   p et 1 + 3 y 2 sin t f (t, y) =

1 + y2

,

(t, y) ∈ [0, 1] × R.

It is easy to show that the conditions (H1 ), (H2 ) are satisfied with 2 λ = , ψ(t) = et , ϕ(t) = et sin t, t ∈ [0, 1]. 3 Therefore, by Theorem 3.5, the BVP (8a)-(8b) has at least one solution on [0, 1]. Moreover, its solutions set is compact.

References [1] W. G. Glockle, T. F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995) 46-53. [2] R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000. [3] F. Metzler, W. Schick, H. G. Kilian, T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phy. 103 (1995) 7180-7186. [4] K. B. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations Wiely New york, 1993. [5] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999. 761

BABAKHANI, ABDELJAWAD: FRACTIONAL BVP

[6] D. Baleanu, A. K. Golmankhaneh, R. Nigmatullin, R., Fractional Newtonian mechanics, Central European Journal of Physics, Vol. 8 (2010) 120-125. [7] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, Yverdon, 1993. [8] B. N. Lundstrom, M. H. Higgs, W. J. Spain and A. L. Fairhall, Fractional differentiation by neocortical pyramidal neurons, Nature Neuroscience, Vol. 11, No. 11 (2008). [9] I. Poudlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. bf 5 (2002) 367-386. [10] A. A. Kilbas, Hari M. Srivastava, Juan J. Trujillo, Theory and applications of fractional differential equations, in: Nrth-Holland Mathematics Studies, vol. bf 204, Elsevier Science B. V, Amesterdam, 2006. [11] V. Lashmikantham, S. Leela, J. Vasundhara, Theory of fractional dynamic systems, Cambridge Academic Publishers, Cambridge, 2009. [12] R. P Agarwal, M. Benchohra, S. Hamani, Asurvey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. vol. bf 109 (2010) 973-1033. [13] Varsha Daftardar- Gejji, A. Babakhani, Analysis of a System of Fractional Differential Equations, Journal of Mathematical Analysis and Applications, 293 (2004) 511-522. [14] A. Babakhani, E. Enteghami, Existence of positive solutions for multiterm fractional differential equations of finite delay with polynomial coefficients, Abstract and Applied Analysis, Vol. 2009, Article ID 768920, (2009). [15] A. Babakhani, Positive solutions for system of nonlinear fractional differential equations in two dimensions with delay, Vol. 2010 (2010), Article number 536317. [16] D. Baleanu, J. J. Trujillo, A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives, Communications in Nonlinear Science and Numerical Simulation, vol. 15 no. 5 (2010) 1111-1115. [17] D. Baleanu, J. J. Trujillo, New applications of fractional variational principles, Rep. Math. Phys. 61 (2008) 331-335. [18] Jinhua Wang, Honjun Xiang, and Zhigang Liu, Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations, Hindawi Publishing Corporation, Inter. Jour. Diff. Equ., Vol 2010, Article ID 186928, (2010). [19] A. Arara, M. Benchohra, N. Hamidi, J. J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Anal., 72 (2010) 580-586.

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[20] R. R. Goldberg, Methods of Real Analysis, Oxford and IFH Publishing Company, New Delhi, 1970. [21] A. Granas, J. Dugundji, Fixed point theory, Springer Verlag, 2003. [22] C. Li, W. Deng, Remarks on fractional derivatives, Appl. Math. Comput., 187 (2007) 777-t84. [23] P. Butzer and L. Westphal, An introduction to fractional calculus. Hilfer, R. (ed.), Applications of fractional calculus in physics, Singapore: World Scientific. (2000), 1-85. MR1890105(2003g:26007). Zbl 0987.26005. [24] R. Gorenflo and S. Vessella, Abel integral equations. Analysis and applications, Lecture Notes in Mathematics, 1461 Springer-Verlag, Berlin, 1991. MR1095269(92e:45003). Zbl 0717.45002. [25] Rabha W. Ibrahim and Shaher Momani, Upper and lower bounds of solutions for fractionla integral equations, Surveys in Mathematics and its Applications, Vol. 2 (2007), 145-156. [26] Maraaba (Abdeljawad) T, Baleanu D, Jarad F, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, Journal of Mathematical Physics, 49 8 Article Number: 083507, (2008). [27] Maraaba (Abdeljawad) T, Baleanu D, Jarad F, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives,Science in China Series A-Mathematics , 51, 10, 1775-1786, ( 2008). [28] M.C. Joshi, R.K. Bose, Some Topics in Nonlinear Functional Analysis, Wiley Eastern, New Delhi, 1985.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.4, 764-777, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

SOME APPLICATIONS OF THE CHOQUET INTEGRAL WITH RESPECT TO A MONOTONE SET FUNCTION ON THE SET OF INTERVAL-VALUED NECESSITY MEASURES LEE-CHAE JANG

Abstract. Based on the concept of an interval-valued necessity measure which is motivated by the goal to represent an reasonable necessity measure, we consider interval-valued necessity measures which a decision maker ranks them according to their Choquet integrals. We also discuss some axiomatizations of such preferences. In particular, we provide the imprecise preference representation theorems for interval-valued necessity measures.

1. Introduction Many researchers, such as Zadeh([26]), Merigo-Casanovas([15]), Choquet ([5]), MurofushiSugeno ([17, 18]), Narukawa-Murofushi-Sugeno([19]), Narukawa([20]), Bykzkan-Ruan([3]), Feng-Nguyen([6]), Grabisch-Marichal-Mesiar-Pap([7]), Jiang-Tang-Tang([13]), Xu([25]), and Chateauneuf-Jaffray([4]) have been studied distance measures and Choquet integrals as a tools to aggregate interacting criteria, and such as Aubin ([1]), Aumann ([2]) and PucciVitillaro ([21]) have been studied interval-valued functions and Aumann integrals, such as Jang-Kil-Kim-Kwon ([8]), Jang-Kwon ([9]), Jang ([10-12]), Li-Sheng ([16]), Lopez GarciaLopez Diaz ([14]), Schjaer-Jacobsen ([23]), Weichselberger ([24]), and Zhang-Guo-Liu ([27]) have been suggested to use intervals in order to represent uncertainty, for examples, closed set-valued functions, interval-valued fuzzy measures, distance measures, fuzzy random variables, interval-valued quantifications, economic uncertainty, and interval-valued probabilities, etc. Recently, Rebille([22]) has been studied decision making problem over necessity measures according to their Choquet integrals. The main idea of this paper is the concept of interval-valued necessity measures which is associated with the representation of reasonable necessity measures. In section 2, we list definitions and basic properties of a monotone set function, a necessity measure, and the Choquet integral. In section 3, we define the new concept of interval-valued necessity measures and investigate the Choquet integral with respect to a monotone set function of them. In section 4, we consider interval-valued necessity measures which a decision maker ranks them according to their Choquet integrals. We also discuss some axiomatizations of such preferences. In particular, we provide the imprecise representation theorems for interval-valued necessity measures. In section 5, we summarize the main conclusions of this paper and discuss some future researches.

1991 Mathematics Subject Classification. 28E10, 28E20, 03E72, 26E50 11B68. Key words and phrases. capacity, interval-valued necessity measure, Choquet integral, decision maker. 1

764

2

LEE-CHAE JANG

2. Definitions and Preliminaries Let Ω be a non-empty set and B stands for a σ-algebra of subsets of Ω. A real-valued set function 𝛽 : B −→ ℝ is called a monotone set function if for all A, B ∈ B, 𝛽(A) ≤ 𝛽(B) whenever A ⊂ B. A monotone set function 𝛽 is said to be normalized if 𝛽(Ω) = 1. If 𝛽(∅) = 0, then a normalized monotone set function 𝛽 is a capacity or a fuzzy measure. A function 𝑓 : Ω −→ [0, 1] is said to be measurable if for every r ∈ [0, 1], {𝑤 ∈ Ω∣𝑓 (𝑤) > r} ∈ B. B[0,1] (Ω, B) will denote the set of all measurable functions from Ω to [0, 1]. It is easy to see that if A ⊂ Ω and χA is the characteristic function of A, then χA ∈ B[0,1] (Ω, B) if and only if A ∈ B. Definition 2.1. ([8-12, 17-20]) (1) Let 𝛽 be a monotone set function on B and 𝑓 ∈ B[0,1] (Ω, B). The Choquet integral of 𝑓 with respect to 𝛽 is defined by ∫ ∫ 1 (C) 𝑓 𝑑𝛽 = 𝛽f (r)𝑑r (1) 0

where µf (r) = µ({𝑥 ∈ X∣𝑓 (𝑥) > r}) and the integral on the right-hand side is the Lebesgue integral. (2) A measurable function 𝑓 is said to be integrable if the Choquet integral of 𝑓 on Ω exists and its value is finite. Let B∗[0,1] (Ω, B) = {𝑓 ∈ B[0,1] (Ω, B) ∣ 𝑓 is integrable on Ω}. Note that if we take Ω = {𝑤1 , 𝑤2 , ⋅ ⋅ ⋅ , 𝑤𝑛 } and B = (Ω) is the power set of Ω and 𝑓 is measurable function on Ω, then ∫ 𝑛 ∑ (C) 𝑓 𝑑𝛽 = 𝑓 (𝑤(𝑖) )[𝛽(A(𝑖) − 𝛽(A(𝑖+1) )], (2) 𝑖=1

where (⋅) is a permutation on {1, 2, ⋅ ⋅ ⋅ , 𝑛} such that 0 ≤ 𝑓 (𝑤(1) ) ≤ ⋅ ⋅ ⋅ ≤ 𝑓 (𝑤(𝑛) ), A(𝑖) = {𝑤(𝑖) , 𝑤(𝑖+1) , ⋅ ⋅ ⋅ , 𝑤(𝑛) }, and A(𝑛+1) = ∅. From (2), we can directly derive ∫ 𝑛 ∑ (C) 𝑓 𝑑𝛽 = [𝑓 (𝑤(𝑖) ) − 𝑓 (𝑤(𝑖−1) )]𝛽(A(𝑖) ), (3) 𝑖=1

where 𝑓 (𝑤(0) ) = 0. Definition 2.2. ([8-12, 17-20]) Let 𝑓, 𝑔 ∈ B[0,1] (Ω, B). We say that 𝑓 and 𝑔 are comonotonic, in symbol, 𝑓 ∼ 𝑔 if for every pair 𝑢, v ∈ Ω, 𝑓 (𝑢) < 𝑓 (v) =⇒ 𝑔(𝑢) ≤ 𝑔(v).

(4)

Remark that we need the above comonotonicity of two functions in order to prove the linearity of the Choquet intgeral(see[17-20]). We introduce some properties of comonotonicity and the Choquet integral as follows: Theorem 2.1. ([8-12, 17-20]) Let 𝑓, 𝑔, ℎ ∈ B[0,1] (Ω, B). Then we have (1) 𝑓 ∼ 𝑓 , (2) 𝑓 ∼ 𝑔 =⇒ 𝑔 ∼ 𝑓 , (3) 𝑓 ∼ 𝑎 for all 𝑎 ∈ [0, 1], and (4) 𝑓 ∼ 𝑔 and 𝑔 ∼ ℎ =⇒ 𝑓 ∼ 𝑔 + ℎ.

765

SOME APPLICATIONS OF THE CHOQUET INTEGRAL ...

3

Theorem 2.2. ([8-12, 17-20]) Let 𝑓, 𝑔 ∈ B∗[0,1] (Ω, B) and 𝛽 be a monotone set function. Then we have ∫ ∫ (1) if 𝑓 ≤ 𝑔, then (C) 𝑓 𝑑𝛽 ≤ (C) 𝑔𝑑𝛽, ∫ ∫ (2) if E1 , E2 ∈ B and E1 ⊂ E2 , then (C) E1 𝑓 𝑑𝛽 ≤ (C) E2 𝑓 𝑑𝛽, (3) if 𝑓 ∼ 𝑔 and 𝑎, 𝑏 ∈ 𝐼, then ∫ ∫ ∫ (C) (𝑎𝑓 + 𝑏𝑔)𝑑𝛽 = 𝑎(C) 𝑓 𝑑𝛽 + 𝑏(C) 𝑔𝑑𝛽, and (4) if we define (𝑓 ∨ 𝑔)(𝑤) = 𝑓 (𝑤) ∨ 𝑔(𝑤) and (𝑓 ∧ 𝑔)(𝑤) = 𝑓 (𝑤) ∧ 𝑔(𝑤) for all 𝑤 ∈ Ω, then ∫ ∫ ∫ (C) 𝑓 ∨ 𝑔𝑑𝛽 ≥ (C) 𝑓 𝑑𝛽 ∨ 𝑔𝑑𝛽, and

∫ (C)

∫ 𝑓 ∧ 𝑔𝑑𝛽 ≤ (C)

∫ 𝑓 𝑑𝛽 ∧

𝑔𝑑𝛽.

From Theorem 2.2 (3), we see that the Choquet integral satisfies the comonotonic affinity as follows: for all 𝑓, 𝑔 ∈ B∗[0,1] (Ω, B) and 𝑎 ∈ [0, 1], ∫ ∫ ∫ 𝑓 ∼ 𝑔 =⇒ (C) (𝑎𝑓 + (1 − 𝑎)𝑔)𝑑𝛽 = 𝑎(C) 𝑓 𝑑𝛽 + (1 − 𝑎) 𝑔𝑑𝛽. (5)

Assume that Ω is a non-empty finite set and B = ℘(Ω) is the power set of Ω. Note that a familiar object in fuzzy set theory is the one of necessity measure or its dual version a possibility measure. A necessity measure is a set function ν : ℘(Ω) −→ [0, 1] such that ν(∅) = 0, ν(Ω) = 1 and for all family of subsets {A𝑖 }𝑖∈I it holds, ν(∩𝑖∈I A𝑖 ) = inf ν(A𝑖 ), 𝑖∈I

(6)

where the index set 𝐼 is arbitrary. By duality we can define a possibility measure ν d associated to ν through ν d (A) = 1 − ν(Ac ), where Ac id the complement of a set A. Then, it is easy to see that for all family of subsets {A𝑖 }𝑖∈I , ν d (∪𝑖∈I A𝑖 ) = sup ν d (A𝑖 ).

(7)

𝑖∈I

Definition 2.3. ([22]) A family F of subsets of Ω is said to be a filter if (i) ∅ ̸∈ F and Ω ∈ F, (ii) for all A, B ∈ ℘(Ω), A, B ∈ F =⇒ A ∩ B ∈ F, (iii) for all A, B ∈ ℘(Ω), A ∈ F, A ⊂ B =⇒ B ∈ F. Since Ω is a nonempty finite set, it ie easy to easy that any filter is principle, that is, there exists A(̸= ∅) such that F = Au = {B∣A ⊂ B ⊂ Ω}, where Au stands for the upset generated by A. We denote the set of filters by F(Ω). Theorem 2.3. ([22]) Let ν : ℘(Ω) −→ [0, 1] be a function. Then ν is a necessity measure if and only if ν ∈ F[0,1] (Ω, F(Ω)).

766

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LEE-CHAE JANG

We recall that the notation of comonotonicity for necessity measures have been introduced in [22] under the name of agreement for necessity measures as follows: Definition 2.4. ([22]) Two necessity measures ν, 𝜔 are said to be agree, in symbol, ν ∼ 𝜔 if for all A, B ∈ ℘(Ω), ν(A) < ν(B) =⇒ 𝜔(A) ≤ 𝜔(B).

(8)

Theorem 2.4. ([22]) Let ν and 𝜔 be necessity measures and 𝛼 ∈ (0, 1). Then, ν and 𝜔 are agree if and only if 𝛼ν + (1 − 𝛼)𝜔 is a necessity measure. ν : ℘(Ω) −→ [0, 1] be a function. In [22], we see that if ν is a necessity measure, then there is a unique decomposition of ν over unanimity games known as the Mobius transform: 𝑛 ∑ ν= 𝛼𝑖 𝑢A𝑖 , (9) 𝑖=1

∑𝑛 where 𝛼1 , ⋅ ⋅ ⋅ , 𝛼𝑛 > 0, 𝑖=1 𝛼𝑖 = 1, Ω ⊃ A1 ⫌ ⋅ ⋅ ⋅ ⫌ A𝑛 ̸= ∅, and 𝑢A denote an unanimity game, that is, an elementary belief function with support A defined by for all B ⊂ Ω, { 1 if A ⊂ B 𝑢A (B) = (10) 0 otherwise. We note that the above necessity measure ν in (9) can be expressed as follows: 𝑛 ∑ ν= 𝛼𝑖 χA𝑖 u ,

(11)

𝑖=1

∑𝑛 where 𝛼1 , ⋅ ⋅ ⋅ , 𝛼𝑛 > 0, 𝑖=1 𝛼𝑖 = 1, and ∅ ̸= A1 u ⫋ ⋅ ⋅ ⋅ ⫋ A𝑛 u . Remark that for A, B ⊂ Ω, we have Au ∩ B u = (A ∪ B)u , and Au = ∩w∈A {𝑤}u , and that if we take X(𝑤) = 𝛽({𝑤}u ), then 𝛽({𝑤}u ) = min X(𝑤). w∈A

(12)

Then we introduce the following strong integral representation theorem. Note that a monotone set function is said to be minitive if for all nonempty subsets A, B ⊂ Ω, 𝛽(Au ∩ B u ) = min{𝛽(Au ), 𝛽(B u )}. ∑𝑛 Theorem 2.5. ([22]) (1) If ν = 𝑖=1 𝛼𝑖 χA𝑖 u is above necessity measure in (11) and 𝛽 is a monotone set function, then we have ∫ 𝑛 ∑ (13) (C) ν𝑑𝛽 = 𝛼𝑖 𝛽(Au𝑖 ). ∑𝑛

𝑖=1

(2) Let ν = 𝑖=1 𝛼𝑖 χA𝑖 be above necessity measure in (11) and 𝛽 a monotone set function. If 𝛽 is minitive and we take X(𝑤) = 𝛽({𝑤}u ) for all 𝑤 ∈ Ω, then we have ∫ ∫ (C) ν𝑑𝛽 = (C) X𝑑ν. (14) u

Let 𝑁 𝑒c(Ω) be the set of all necessity measures. We discuss a binary relation ⪰ on 𝑁 𝑒c(Ω) and note that ν ≻ 𝜔 for ν ⪰ 𝜔 and not(𝜔 ⪰ ν), ν ∼ = 𝜔 for ν ⪰ 𝜔 and 𝜔 ⪰ ν.

767

SOME APPLICATIONS OF THE CHOQUET INTEGRAL ...

5

Definition 2.5. ([22]) (1) A binary relation ⪰ on 𝑁 𝑒c(Ω) is said to be complete if for all ν, 𝜔 ∈ 𝑁 𝑒c(Ω), we have ν ⪰ 𝜔 or ν ⪯ 𝜔.

(15)

(2) ⪰ is said to be transitive if for all 𝑢, ν, 𝜔 ∈ 𝑁 𝑒c(Ω), 𝑢 ⪰ ν and ν ⪯ 𝜔 =⇒ 𝑢 ⪯ 𝜔.

(16)

(3) ⪰ is said to be a weak order(WO) if it is complete and transitive. (4) ⪰ is said to be monotonicity(MON) if for all ν, 𝜔 ∈ 𝑁 𝑒c(Ω), ν ≤ 𝜔 =⇒ ν ⪯ 𝜔,

(17)

where ν ≤ 𝜔 means for all A ⊂ Ω, ν(A) ≤ 𝜔(A). (5) ⪰ is said to be agreement(AGR) if for all 𝑢, ν, 𝜔 ∈ 𝑁 𝑒c(Ω), 𝑢 ∼ 𝜔, and ν ∼ 𝜔, then we have 𝑢∼ (18) = 𝜔 =⇒ 𝛼𝑢 + (1 − 𝛼)𝜔 ∼ = 𝛼ν + (1 − 𝛼)𝜔. (6) ⪰ is said to be Archimedean(ARCH) if for all ν, 𝜔 ∈ 𝑁 𝑒c(Ω), ν ≺ 𝜔 =⇒ ∃𝛼 ∈ (0, 1) such that ν ≺ 𝛼𝜔 + (1 − 𝛼)𝑢Ω ,

(19)

∃𝛼 ∈ (0, 1) such that 𝛼𝜔 + (1 − 𝛼)𝑢Ω ⪯ ν ⪯ 𝜔 =⇒ ∃𝛼′ ∈ (𝛼, 1) such that 𝛼′ 𝜔 + (1 − 𝛼′ )𝑢Ω ⪯ ν.

(20)

and

(7) ⪰ is said to be not degenerate(NDEG) if there are ν, 𝜔 ∈ 𝑁 𝑒c(Ω) such that ν ≻ 𝜔. (8) ⪰ is said to be inclusion(INCL) if for all A, B ⊂ Ω, 𝑢A ⪰ 𝑢B =⇒ 𝑢A∪B ∼ (21) = 𝑢B . {

Note that we take δA (B) =

1 if 𝑤 ∈ A 0 otherwise,

(22)

{

and

1 if A = Ω 0 otherwise, From Definition 2.5, we introduce the following preference representation theorems. 𝑢Ω (A) =

(23)

Theorem 2.6. ([22]) Let ⪰ be a binary relation on 𝑁 𝑒c(Ω). If ⪰ satisfies (WO), (MON), (AGR), (ARCH), and (NDEG), then there exists a monotone set function 𝛽 : F(Ω) −→ [0, 1] such that for all ν, 𝜔 ∈ 𝑁 𝑒c(Ω), ∫ ∫ ν ⪰ 𝜔 ⇐⇒ (C) ν𝑑𝛽 ≥ (C) 𝜔𝑑𝛽. (24) Moreover, there exists an 𝑤1 ∈ Ω such that for all ν ∈ 𝑁 𝑒c(Ω), ∫ ∫ ν∼ = ν𝑑𝛽δw1 + (1 − ν𝑑𝛽)𝑢Ω ,

(25)

𝛽({𝑤1 }u ) = 1, and 𝛽({Ω}) = 0. Conversely, if the binary relation is represented by a Choquet integral with respect to a monotone set function 𝛽 : F(Ω) −→ [0, 1] such that 𝛽({Ω}) = 0 and 𝛽({𝑤1 }u ) = 1 for some 𝑤1 ∈ Ω, then ⪰ satisfies (WO), (MON), (AGR), (ARCH), and (NDEG).

768

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LEE-CHAE JANG

3. Interval-valued necessity measures and the Choquet integral In this section, we consider the concept of bounded closed interval (intervals, for short) in [0, 1], that is, 𝐼([0, 1]) = {[𝑎l , 𝑎r ]∣𝑎l , 𝑎r ∈ [0, 1] and 𝑎l ≤ 𝑎r }.

(26)

For any 𝑎 ∈ [0, 1], we define 𝑎 = [𝑎, 𝑎]. Obviously, 𝑎 ∈ 𝐼([0, 1]) (see[10-14, 16, 23-26]). Definition 3.1. If 𝑎 ¯ = [𝑎l , 𝑎r ], ¯𝑏 = [𝑏l , 𝑏r ] ∈ 𝐼([0, 1]) and 𝑘 ∈ [0, 1], then we define arithmetic, maximum, minimum, order, inclusion, and strong order operations as follows: (1) 𝑎 ¯ + ¯𝑏 = [𝑎l + 𝑏l , 𝑎r + 𝑏r ], (2) 𝑘¯ 𝑎 = [𝑘𝑎l , 𝑘𝑎r ], ¯ (3) 𝑎 ¯𝑏 = [𝑎l 𝑏l , 𝑎r 𝑏r ], (4) 𝑎 ¯ ∨ ¯𝑏 = [𝑎l ∨ 𝑏l , 𝑎r ∨ 𝑏r ], (5) 𝑎 ¯ ∧ ¯𝑏 = [𝑎l ∧ 𝑏l , 𝑎r ∧ 𝑏r ], (6) 𝑎 ¯ ≤ ¯𝑏 if and only if 𝑎l ≤ 𝑏l and 𝑎r ≤ 𝑏r , (7) 𝑎 ¯ < ¯𝑏 if and only if 𝑎 ¯ ≤ ¯𝑏 and 𝑎 ¯ ̸= ¯𝑏, and l l ¯ (8) 𝑎 ¯ ⊂ 𝑏 if and only if 𝑏 ≤ 𝑎 and 𝑎r ≤ 𝑏r . (9) 𝑎 ¯ 0, 𝑏1 , ⋅ ⋅ ⋅ , 𝑏𝑚 > 0, ∅ ̸= B1u & ⋅ ⋅ ⋅ & B𝑛u .

𝑖=1

∑𝑛 𝑖=1

𝑎𝑖 = 1 =

∑𝑚

𝑗=1 𝑏𝑗

(34)

and ∅ ̸= Au1 & ⋅ ⋅ ⋅ & Au𝑛 , and

Proof. Let ν¯ = [ν l , ν r ] be an interval-valued necessity measure. By Theorem 3.1 (1), ν and ν r are necessity measures. From (11), there exist 𝑎1 ,∑ ⋅ ⋅ ⋅ , 𝑎𝑛 > 0, 𝑏1 , ∑ ⋅ ⋅ ⋅ , 𝑏𝑚 > 0, 𝑛 𝑚 ∅ = ̸ Au1 & ⋅ ⋅ ⋅ & Au𝑛 , and ∅ = ̸ B1u & ⋅ ⋅ ⋅ & B𝑛u such that 𝑎 = 1 = 𝑖 𝑖=1 𝑗=1 𝑏𝑗 and ∑𝑛 ∑𝑚 u and u. 𝑎 χ 𝑏 χ 𝑖 A 𝑗 B 𝑖=1 𝑗=1 𝑖 j l

We define the concept of agreement of interval-valued necessity measures ν¯ and 𝜔 ¯ as follows: Definition 3.4. Let ν¯ = [ν l , ν r ] and 𝜔 ¯ = [𝜔 l , 𝜔 r ] be interval-valued necessity measures. Then, ν¯ and 𝜔 ¯ are said to be agree, in symbol, ν¯ ∼ 𝜔 ¯ if for all A, B ∈ ℘(Ω), ν¯(A) 0, ∅ ̸= B1u & ⋅ ⋅ ⋅ & B𝑛u .

∑𝑛 𝑖=1

𝑗=1

𝑎𝑖 = 1 =

∑𝑚

𝑗=1 𝑏𝑗

and ∅ ̸= Au1 & ⋅ ⋅ ⋅ & Au𝑛 , and

Proof. (1) Note that 𝛼¯ ν +(1−𝛼)¯ 𝜔 = [𝛼ν l +(1−𝛼)𝜔 l , 𝛼ν r +(1−𝛼)𝜔 r ] is an interval-valued necessity measure. By (34) and (5), we have ∫ (C) [𝛼¯ ν + (1 − 𝛼)¯ 𝜔 ]𝑑𝛽 [ ] ∫ ∫ = 𝛼(C) [ν l + (1 − 𝛼)𝜔 l ]𝑑𝛽, 𝛼(C) [ν r + (1 − 𝛼)𝜔 r ]𝑑𝛽 [ ] ∫ ∫ ∫ ∫ l l r r = 𝛼(C) ν 𝑑𝛽 + (1 − 𝛼)(C) 𝜔 𝑑𝛽, 𝛼(C) ν 𝑑𝛽 + (1 − 𝛼)(C) 𝜔 𝑑𝛽 [ ] [ ] ∫ ∫ ∫ ∫ l l r r = 𝛼 (C) ν 𝑑𝛽, 𝜔 𝑑𝛽 + (1 − 𝛼) (C) ν 𝑑𝛽, 𝜔 𝑑𝛽 ∫ ∫ = 𝛼(C) ν¯𝑑𝛽 + (1 − 𝛼)(C) 𝜔 ¯ 𝑑𝛽. (2) By Theorem 3.2 and Theorem 2.5 (1), we can easily obtain the result. Remark that if we define the Choquet integral of a measurable function X : Ω −→ [0, 1] with respect to an interval-valued necessity measure ν¯ = [ν l , ν r ] as follows: [ ] ∫ ∫ ∫ (C) X𝑑¯ ν = (C) X𝑑ν l , (C) X𝑑ν r . (40) By using (37) and Theorem 2.5(2), we obtain the following theorem. Theorem 3.6. Let ν¯ = [ν l , ν r ] be an interval-valued necessity measure and 𝛽 is a monotone set function. If 𝛽 is minitive and we take X(𝑤) = 𝛽({𝑤}u ) for all 𝑤 ∈ Ω, then we have ∫ ∫ (C) ν¯𝑑𝛽 = (C) X𝑑¯ ν. (41)

772

10

LEE-CHAE JANG

Proof. By Theorem 3.1 (1), ν∫l and ν r are ∫necessity measures. So, by Theorem 2.5(2), ∫ l ∫ l (C) ν 𝑑𝛽 = (C) X𝑑ν and (C) ν r 𝑑𝛽 = (C) X𝑑ν r . Thus, by Definition 3.5 and (37), [ ] ∫ ∫ ∫ (C) ν¯𝑑𝛽 = (C) ν l 𝑑𝛽, (C) ν l 𝑑𝛽 [ ] ∫ ∫ = (C) X𝑑ν l , (C) X𝑑ν r ∫ = (C) X𝑑¯ ν.

4. Imprecise preference representation theorems In this section, we will provide a simple axiomatization of imprecise preference that can be represented through the Choquet integral with respect to a fuzzy measure of interval-valued necessity measures. Let 𝐼𝑁 𝑒c(Ω) be the set of all interval-valued necessity measures. By using Definition 2.4, we consider the concept of imprecise binary relation ⊒ on 𝐼𝑁 𝑒c(Ω) as follows: Definition 4.1. (1) An imprecise binary relation ⊒ on 𝐼𝑁 𝑒c(Ω) is defined by for all ν¯ = [ν l , ν r ], 𝜔 ¯ = [𝜔 l , 𝜔 r ] ∈ 𝐼𝑁 𝑒c(Ω), ν¯ ⊒ 𝜔 ¯ if and only if ν l ⪰ 𝜔 l and ν r ⪰ 𝜔 r .

(42)

(2) For ν¯ = [ν l , ν r ], 𝜔 ¯ = [𝜔 l , 𝜔 r ] ∈ 𝐼𝑁 𝑒c(Ω), ν¯ ∼ ¯ if and only if ν l ∼ =𝜔 = 𝜔 l and ν r ∼ = 𝜔r . From Definition 4.1 and Definition 2.4, we obtain the following theorems which are able to state imprecise preference representation theorems. Theorem 4.1. Let ⊒ be an imprecise binary relation on 𝐼𝑁 𝑒c(Ω). If If a binary relation ⪰ on 𝑁 𝑒c(Ω) satisfies (WO), (MON), (AGR), (ARCH), and (NDEG) in Definition 2.5, then there exists a monotone set function 𝛽 : F(Ω) −→ [0, 1] such that for all ν¯ = [ν l , ν r ], 𝜔 ¯ = [𝜔 l , 𝜔 r ] ∈ 𝐼𝑁 𝑒c(Ω), ∫ ∫ ν¯ ⊒ 𝜔 ¯ ⇐⇒ (C) ν¯𝑑𝛽 ≥ (C) 𝜔 ¯ 𝑑𝛽. (43) Moreover, there exists an 𝑤1 ∈ Ω such that for all ν¯ = [ν l , ν r ] ∈ 𝐼𝑁 𝑒c(Ω), [∫ ] ∫ ∫ ∫ ν¯ ∼ ν l 𝑑𝛽δw1 + (1 − ν l 𝑑𝛽)𝑢Ω , ν r 𝑑𝛽δw1 + (1 − ν r 𝑑𝛽)𝑢Ω , = where

{ δA (B) =

and

{ 𝑢Ω (A) =

(44)

if 𝑤 ∈ A otherwise,

(45)

1 if A = Ω 0 otherwise,

(46)

1 0

for all A ∈ ℘(Ω).

773

SOME APPLICATIONS OF THE CHOQUET INTEGRAL ...

11

Proof. Assume that a binary relation ⪰ on 𝑁 𝑒c(Ω) satisfies (WO), (MON), (AGR), (ARCH), and (NDEG) in Definition 2.5. By Theorem 2.6, there exists a monotone set function 𝛽 : F(Ω) −→ [0, 1] such that for all ν, 𝜔 ∈ 𝑁 𝑒c(Ω), ∫ ∫ ν ⪰ 𝜔 ⇐⇒ (C) ν𝑑𝛽 ≥ (C) 𝜔𝑑𝛽. (47) Thus by Definition 4.1 (1), we have ν¯ ⊒ 𝜔 ¯

⇐⇒ ν l ⪰∫𝜔 l and ν r ⪰ 𝜔∫r ∫ ∫ l l r ⇐⇒ (C) ν 𝑑𝛽 ≥ (C) 𝜔 𝑑𝛽 and (C) ν 𝑑𝛽 ≥ (C) 𝜔 r 𝑑𝛽 ∫ ∫ ⇐⇒ (C) ν¯𝑑𝛽 ≥ (C) 𝜔 ¯ 𝑑𝛽.

From (25) in Theorem 2.6, there exists 𝑤1 ∈ Ω such that for all ν ∈ 𝑁 𝑒c(Ω), ∫ ∫ ν∼ (C) ν𝑑𝛽δ + (1 − ν𝑑𝛽)𝑢Ω . = w1 Thus, by Definition 4.1(2), we have [∫ ] ∫ ∫ ∫ l r ∼ l l r r ν¯ = [ν , ν ] = ν 𝑑𝛽δw1 + (1 − ν 𝑑𝛽)𝑢Ω , ν 𝑑𝛽δw1 + (1 − ν 𝑑𝛽)𝑢Ω .

Theorem 4.2. Suppose that the binary relation ⪰ is represented by the Choquet integral with respect to a monotone set function 𝛽 : F(Ω) −→ [0, 1] such that 𝛽({Ω}) = 0 and 𝛽({𝑤1 }u ) = 1 for some 𝑤1 ∈ Ω. If an imprecise binary relation ⊒ is defined by for all ν¯ = [ν l , ν r ], 𝜔 ¯ = [𝜔 l , 𝜔 r ] ∈ 𝐼𝑁 𝑒c(Ω), ∫ ∫ ¯ ⇐⇒ (C) ν¯𝑑𝛽 ≥ (C) 𝜔 ν¯⪰𝜔 ¯ 𝑑𝛽, (48) then we have (1) transtivity: for all 𝑢 ¯, ν¯, 𝜔 ¯ ∈ 𝐼𝑁 𝑒c(Ω), 𝑢 ¯ ⊒ ν¯ and ν¯ ⊒ 𝜔 ¯ ⇐⇒ 𝑢 ¯⊒𝜔 ¯,

(49)

(2) monotonicity(MON): for all 𝑢 ¯, ν¯ ∈ 𝐼𝑁 𝑒c(Ω), 𝑢 ¯ ≤ ν¯ ⇐⇒ 𝑢 ¯ ⊒ ν¯, where 𝑢 ¯ ≤ ν¯ means 𝑢l ≤ ν l and 𝑢r ≤ ν l . (3) agreement(ARG): if 𝑢 ¯, ν¯, 𝜔 ¯ ∈ 𝐼𝑁 𝑒c(Ω) and 𝑢 ¯∼𝜔 ¯ and ν¯ ∼ 𝜔 ¯ , then we have ∼ ∼ 𝑢 + (1 − 𝛼)¯ 𝜔 = 𝛼¯ ν + (1 − 𝛼)¯ 𝜔. 𝑢 ¯ = ν¯ ⇐⇒ 𝛼¯

(50)

(51)

(4) not degenerate(NDEG): there exist 𝑢 ¯, ν¯ ∈ 𝐼𝑁 𝑒c(Ω) such that 𝑢 ¯ ⊒ ν¯. Proof. (1) Let 𝑢 ¯ = [𝑢l , 𝑢r ], ν¯ = [ν l , ν r ], 𝜔 ¯ = [𝜔 l , 𝜔 r ] ∈ 𝐼𝑁 𝑒c(Ω). If 𝑢 ¯ ⊒ ν¯ and ν¯ ⊒ 𝜔 ¯ , by Definition 4.1, then we have (𝑢l ⪰ ν l and 𝑢r ⪰ ν r ) and (ν l ⪰ 𝜔 l and ν r ⪰ 𝜔 r ). Thus, (𝑢l ⪰ ν l and ) and ν l ⪰ 𝜔 l ) and (𝑢r ⪰ ν r and ν r ⪰ 𝜔 r ). By (47) and the converse of Theorem 2.6, 𝑢l ⪰ 𝜔 l and 𝑢r ⪰ 𝜔 r . That is, 𝑢 ¯⊒𝜔 ¯. (2) Let 𝑢 ¯ = [𝑢l , 𝑢r ], ν¯ = [ν l , ν r ] ∈ 𝐼𝑁 𝑒c(Ω) and 𝑢 ¯ ≤ ν¯. Then, 𝑢l ≤ ν l and 𝑢r ≤ ν r . By (47) and the converse of Theorem 2.6, 𝑢l ⪯ ν l and 𝑢r ⪯ ν r . That is, 𝑢 ¯ ⊑ ν¯.

774

12

LEE-CHAE JANG

(3) Let 𝑢 ¯ = [𝑢l , 𝑢r ], ν¯ = [ν l , ν r ], 𝜔 ¯ = [𝜔 l , 𝜔 r ] ∈ 𝐼𝑁 𝑒c(Ω) and 𝑢 ¯ ∼ 𝜔 ¯ and ν¯ ∼ 𝜔 ¯ . Then, l r r l l (𝑢 ∼ 𝜔 and 𝑢 ∼ 𝜔 ) and (ν ∼ 𝜔 and ν r ∼ 𝜔 r ). If 𝑢 ¯∼ = ν¯, then we see that l

𝑢l ∼ 𝜔 l and ν l ∼ 𝜔 l , and 𝑢l ∼ = νl,

(52)

𝑢r ∼ 𝜔 r and ν r ∼ 𝜔 r , and 𝑢r ∼ = νr.

(53)

∼ 𝛼ν +(1−𝛼)𝜔 l Therefore, by (52) and (53) and the converse of Theorem 2.6, 𝛼𝑢 +(1−𝛼)𝜔 = r r ∼ r r and 𝛼𝑢 + (1 − 𝛼)𝜔 = 𝛼ν + (1 − 𝛼)𝜔 . That is, 𝛼¯ 𝑢 + (1 − 𝛼)¯ 𝜔∼ ν + (1 − 𝛼)¯ 𝜔. = 𝛼¯ (4) Since the binary relation ⪰ is represented by the Choquet integral with respect to a monotone set function 𝛽, by the converse of Theorem 2.6, ⪰ satisfies (NDEG). Thus, there exist 𝑢, 𝜔 ∈ 𝑁 𝑒c(Ω) such that 𝑢 ⪰ ν. If we take 𝑢 ¯ = [ 52 𝑢, 𝑢] and ν¯ = [ 35 ν, ν], then 2 3 𝑢 ¯ = [ 5 𝑢, 𝑢] ⊒ [ 5 ν, ν] = ν¯. l

Lemma 4.3. If we define 𝑢 ¯(A,B) (C) =

{

l

[𝑢A (C), 𝑢B (C)] if 𝑢A (C) ≤ 𝑢B (C) 0 otherwise,

l

(54)

then we have

⎧ if A ⊂ C and 𝑢A (C) ≤ 𝑢B (C) ⎨ 1 [0, 1] if B ⊂ C and A ⫅̸ C and 𝑢A (C) ≤ 𝑢B (C) 𝑢 ¯(A,B) (C) = ⎩ 0 otherwise.

(55)

Proof. We will prove the result in the following four cases: If 𝑢A ≤ 𝑢B and A ⊂ C, then we have 𝑢A (C) = 1 ≤ 𝑢B (C). Thus, 𝑢 ¯(A,B) (C) = 1. If 𝑢A ≤ 𝑢B and A ⊈ C and B ⊂ C, then we have 𝑢A (C) = 0 and 𝑢B (C) = 1. Thus, 𝑢 ¯(A,B) (C) = [0, 1]. If 𝑢A ≤ 𝑢B and A ⊈ C and B ⊈ C, then we have 𝑢A (C) = 0 = 𝑢B (C). Thus, 𝑢 ¯(A,B) (C) = 0. If 𝑢A > 𝑢B , by the definition of 𝑢 ¯(A − 2 + 𝑛 − 1, 𝑝 = 1, 2, . . . . 𝜃− 𝑝 D 𝑝

Proof. Here we only prove (3.9) and the proof of (3.10) is similar. Since ( ) ∫ γ ∫ γ (−λ) sin2𝑠 vθ 1 Bγ,v,𝑠 2λ 2λ −λ e −λ 2 𝜃 Dv,𝑠 (𝜃) sin 𝜃 𝑑𝜃 = ′ 𝜃 sin 𝜃 𝑑𝜃 = . A γ,v,𝑠 0 A ′ γ,v,𝑠 sin2𝑠−1 θ2 0 We have ∫ (−λ) Bγ,v,𝑠

=

(

γ

𝜃

−λ

0

∫

vθ 2 sin2𝑠−1 θ2

vγ 2

= π 2𝑠−1

sin2𝑠 (

0

2𝑡 v

)

∫ sin2λ 𝜃 𝑑𝜃 ≤ 0

)− 𝑛−2 2 −2𝑠+1

( sin2𝑠 𝑡

γ

sin2𝑠 vθ 2 𝜃−λ ( )2𝑠−1 𝜃𝑛−2 𝑑𝜃 θ π

2𝑡 v

)𝑛−2

2 𝑑𝑡 v

( ( ) 𝑛2 +1−2𝑠 ∫ vγ )2𝑠 2 𝑛 𝑛 sin 𝑡 2 𝑡2 = π 2𝑠−1 𝑑𝑡 ≤ C9 v 2𝑠−1− 2 , v 𝑡 0 where 𝑛

∫

C9 = π 2𝑠−1 2 2 +1−2𝑠

∞

( 𝑛

𝑡2 0

sin 𝑡 𝑡

781

)2𝑠 𝑑𝑡, 2𝑠 >

𝑛 + 1, 2

(3.10)

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

and ∫ (−λ) Bγ,v,𝑠

(

γ

=

𝜃

sin2𝑠

vθ 2 2𝑠−1 θ sin 2

−λ

0

22𝑠+𝑛−3 π 𝑛−2

=

= C𝑛,𝑠 v

∫

(

vγ 2

0 2𝑠−1− 𝑛 2

(−λ)

2𝑡 v

)

∫

γ

𝜃 𝑑𝜃 ≥

2λ

sin

𝜃

−λ

0

) 𝑛2 −2𝑠

sin2𝑠 vθ 2 ( θ )2𝑠−1

(

2 3𝑛

𝑛 2 22 sin 𝑡 𝑑𝑡 = 𝑛−2 v 2𝑠−1− 2 v π

2𝜃 π ∫

)𝑛−2

2𝑠

𝑑𝜃 (

vγ 2

𝑛

𝑡2 0

sin 𝑡 𝑡

)2𝑠 𝑑𝑡

. ∫γ

𝑛

Then we have Bγ,v,𝑠 ≈ v 2𝑠−1− 2 . Because of ∫

1 ′

A

γ,v,𝑠

e 𝑘,𝑠 (𝜃) sin2λ 𝜃𝑑𝜃 = 1, that is D

0

sin2𝑠

vθ 2 sin2𝑠−1 θ2

γ 0

sin𝑛−2 𝜃 𝑑𝜃 = 1.

So we have A

∫

′

sin2𝑠

vθ 2 2𝑠−1 θ 0 sin 2 ∫ vγ 2𝑠 2

=

γ,v,𝑠

γ

0

vπ

where C10 = 2𝑛−2𝑠 π 2𝑠−1 A

∫ γ,v,𝑠

(

sin 𝑡 ( 2𝑡 )2𝑠−1

=

′

∫

∫∞ 0

vθ 2 2𝑠−1 θ 0 sin 2 ∫ vγ 2𝑠 2

=

sin 𝑡 ( 𝑡 )2𝑠−1

= 0

2𝑡 v

)𝑛−2

𝑡𝑛−1

0

sin2𝑠 vθ ( θ ) 2 𝜃𝑛−2 𝑑𝜃 π

2 𝑑𝑡 = 2𝑛−2𝑠 π 2𝑠−1 v 2𝑠−𝑛 v

( sin 𝑡 )2𝑠 𝑡

𝑛−2

sin (

v

4𝑡 πv

(

vγ 2

𝑡 0

𝑛−1

sin 𝑡 𝑡

)2𝑠 𝑑𝑡 ≤ C10 v 2𝑠−𝑛 ,

)𝑛−2 vθ ( 2𝜃 2 𝜃 𝑑𝜃 π sin2𝑠−1 θ2 ( )2𝑠 ∫ vγ 2 sin 𝑡 22𝑛−3 2𝑠−𝑛 𝑛−1 𝑡 v π 𝑛−2 𝑡 0 sin2𝑠

γ

𝜃 𝑑𝜃 ≥

)𝑛−2

∫

𝑑𝑡, 2𝑠 > 𝑛, and we can also see that ∫

sin2𝑠

γ

γ

sin𝑛−2 𝜃 𝑑𝜃 ≤

0

2 𝑑𝑡 = v

𝑑𝑡 = C𝑠,𝑛 v 2𝑠−𝑛 .

∫γ e v,𝑠 (𝜃) sin2λ 𝜃 𝑑𝜃 ≈ v λ , 2𝑠 > 𝑛, completing the proof. So we have A γ,v,𝑠 ≈ v 2𝑠−𝑛 . Then 0 𝜃−λ D e v,𝑠 (𝜃) defined by (2.8), 2𝑠 > r + 𝑛, r ≥ 0, we have Lemma 3.2. For D ∫ γ e v,𝑠 (𝜃) sin2λ 𝜃 𝑑𝜃 ≈ v −r . 𝜃r D (3.11) ′

0

Proof. It is easy to obtain that ∫

γ

e v,𝑠 (𝜃) sin 𝜃 D r

2λ

𝜃 𝑑𝜃 =

0

We have ∫ (r) Bγ,v,𝑠

(

γ

=

𝜃

r

0

sin2𝑠

vθ 2 2𝑠−1 θ sin 2

∫

1

𝜃

A ′ γ,v,𝑠

)

(

γ r

0

∫ sin2λ 𝜃 𝑑𝜃 ≤ 0

γ

sin2𝑠

vθ 2 2𝑠−1 θ sin 2

)

(r)

sin2λ 𝜃 𝑑𝜃 =

Bγ,v,𝑠 . A ′ γ,v,𝑠

sin2𝑠 vθ 2 𝜃r ( )2𝑠−1 𝜃𝑛−2 𝑑𝜃 θ π

( )r+𝑛−2𝑠−1 ∫ vγ 2 2𝑡 2 2𝑠 v𝜃 r+𝑛−2𝑠−1 2𝑠−1 2𝑠−1 𝜃 sin = π 𝑑𝜃 = π sin2𝑠 𝑡 𝑑𝑡 ≤ C11 v 2𝑠−r−𝑛 , 2 v v 0 0 )2𝑠 ( ∫∞ 𝑑𝑡, 2𝑠 > r + 𝑛, r ≥ 0 and where C11 = π 2𝑠−1 2r+𝑛−2𝑠 0 𝑡r+𝑛−1 sin𝑡 𝑡 ) )2λ ∫ γ ( ∫ γ 2𝑠 vθ ( sin2𝑠 vθ 2𝜃 2λ (r) r r sin 2 2 Bγ,v,𝑠 = 𝜃 sin 𝜃 𝑑𝜃 ≥ 𝜃 ( )2𝑠−1 𝜃 𝑑𝜃 θ π sin2𝑠−1 θ2 0 0 2 ( )r+𝑛−2𝑠−1 ( )2𝑠 ∫ vγ ∫ vγ 2 2 22λ+2𝑠−1 2𝑡 2 22𝑛+r−3 2𝑠−r−𝑛 sin 𝑡 2𝑠 r+𝑛−1 = sin 𝑡 𝑑𝑡 = v 𝑡 𝑑𝑡 π 2λ v v π 2λ 𝑡 0 0 ∫

γ

≥ C𝑛,𝑠 v 2𝑠−r−𝑛 .

782

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

′

(−λ)

Then we have Bγ,v,𝑠 ≈ v 2𝑠−r−𝑛 . We have already obtained that A γ,v,𝑠 ≈ v 2𝑠−𝑛 . Therefore, ∫γ r e v,𝑠 (𝜃) sin2λ 𝜃 𝑑𝜃 ≈ v −r , completing the proof. 𝜃 D 0 Lemma 3.3. Let 𝑓 ∈ 𝐿𝑝 (D(𝑥0 , 𝛾)), 1 ≤ 𝑝 ≤ ∞, 0 < 𝜃 ≤ π. (i) For 1 ≤ 𝑚1 < 𝑚, Sθ𝑚 = Sθ𝑚1 Sθ𝑚−𝑚1 ; (ii) For 𝑚 ≥ 1, ∥Sθ𝑚 (𝑓 )∥D,𝑝 ≤ ∥𝑓 ∥D,𝑝 ; (iii) For 𝑚 ≥ 1, ∥Sθ𝑚 (𝑓 ) − 𝑓 ∥D,𝑝 ≤ 𝑚∥Sθ (𝑓 ) − 𝑓 ∥D,𝑝 ; 𝑚 (iv) ∥Jv,𝑠 (𝑓 )∥D,𝑝 ≤ ∥𝑓 ∥D,𝑝 . Proof. With the help of (2.4), we see that (∞ ) ∞ ∑ ∑( )𝑚1 )𝑚−𝑚1 ( λ 𝑚1 𝑚−𝑚1 λ Sθ (Sθ (𝑓 )) = Q𝑗 (cos 𝜃) Y𝑗 Q𝑖 (cos 𝜃) Y𝑖 (𝑓 ) 𝑗=0

=

𝑖=0

∞ ∑ (

Qλ𝑗 (cos 𝜃)

)𝑚

Y𝑗 (𝑓 ) = Sθ𝑚 (𝑓 ), 1 ≤ 𝑚1 < 𝑚.

𝑗=0

Then (i) is valid. (ii) and (iii) can be deduced from the definition of Sθ𝑚 (𝑓 ). For (iv), we use (ii) to see that

∫ γ

∫ γ

𝑚 𝑚 e v,𝑠 (𝜃) sin2λ 𝜃𝑑𝜃 e v,𝑠 (𝜃) sin2λ 𝜃𝑑𝜃 ≤ ∥𝑓 ∥D,𝑝 . ∥Jv,𝑠 (𝑓 )∥D,𝑝 = S (𝑓 ) D ≤ ∥Sθ𝑚 (𝑓 )∥D,𝑝 D θ

0

0

D,𝑝

The proof of Lemma 3.3 is completed. The following Lemma 3.4 is quoted from [2]. e ∆ e 2 𝑔 ∈ 𝐿𝑝 (D(𝑥0 , 𝛾)), 1 ≤ 𝑝 ≤ ∞, Sθ (𝑔; 𝑥) is defined by (2.5), we Lemma 3.4. For any 𝑔, ∆𝑔, have ∫ θ ∫ 𝑡 e 𝑥) 𝑑𝑢 Sθ (𝑔; 𝑥) − 𝑔(𝑥) = sin−2λ 𝑡 𝑑𝑡 sin2λ 𝑢 Su (∆𝑔; 0

and e 𝑥) − ∆𝑔(𝑥) e Su (∆𝑔; =

∫

0

u

sin−2λ σ 𝑑σ

0

∫

σ

e 2 𝑔; 𝑥) 𝑑τ. sin2λ τ Sτ (∆

0

The following Lemma 3.5 will play an important role in the proof of Bernstein-type inequality 𝑚 for Jv,𝑠 (𝑓 ). e ∆ e 2 𝑔 ∈ 𝐿𝑝 (D(𝑥0 , 𝛾)), 1 ≤ 𝑝 ≤ ∞, and 2𝑠 > r + 𝑛, r ≥ 0, there Lemma 3.5. For any 𝑔, ∆𝑔, exist constants A, B and C12 which are independent of 𝑛 and 𝑔, such that

𝑚 e e 2 𝑔∥D,𝑝 + C12 (𝑚 − 1)v −2 ∥∆𝑔∥ e D,𝑝 , ≤ C12 v −4 ∥∆ (3.12)

Jv,𝑠 (𝑔) − 𝑔 − 𝛼(v)∆𝑔

D,𝑝

where 0 < vA2 ≤ 𝛼(v) ≤ vB2 . Proof. Now we will prove (3.12) by induction on 𝑚 and it is easy to verify that } { ∫ θ ∫ 𝑡 −2λ 2λ −2 sin 𝑡 𝑑𝑡 sin 𝑢 𝑑𝑢 < ∞, sup 𝜃 θ>0

0

(3.13)

0

together with the help of Lemma 3.4, we see ∫ γ 1 e v,𝑠 (𝜃)(Sθ (𝑔; 𝑥) − 𝑔(𝑥)) sin2λ 𝜃 𝑑𝜃 Jv,𝑠 (𝑔; 𝑥) − 𝑔(𝑥) = D 0

∫

γ

=

e v,𝑠 (𝜃) sin2λ 𝜃 𝑑𝜃 D

0

∫

∫

θ 0

sin−2λ 𝑡 𝑑𝑡

𝑡

e 𝑥) 𝑑𝑢 sin2λ 𝑢 Su (∆𝑔;

0

∫ 𝑡 ∫ γ ∫ θ 2λ −2λ 2λ 2λ e e e = ∆𝑔(𝑥) Dv,𝑠 (𝜃) sin 𝜃 𝑑𝜃 sin 𝑡 𝑑𝑡 sin 𝑢 𝑑𝑢 + Dv,𝑠 (𝜃) sin 𝜃 𝑑𝜃 sin−2λ 𝑡 𝑑𝑡 0 0 0 0 0 ∫ 𝑡 ( ) e 𝑥) − ∆𝑔(𝑥) e e × sin2λ 𝑢 Su (∆𝑔; 𝑑𝑢 =: 𝛼(v)∆𝑔(𝑥) + Ψ(𝑔; 𝑥), γ

∫

∫

θ

0

783

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

where 𝛼(v) = vC2 , C > 0, and satisfies 0 < vA2 ≤ 𝛼(v) ≤ vB2 . With the help of H¨older-Minkowski’s inequality, Lemma 3.2 and Lemma 3.4, we have

∫

∫ θ ∫ 𝑡

γ ( )

2λ −2λ 2λ e e e ∥Ψ(𝑔)∥D,𝑝 = Dv,𝑠 (𝜃) sin 𝜃𝑑𝜃 sin 𝑡𝑑𝑡 sin 𝑢 Su (∆𝑔) − ∆𝑔 𝑑𝑢

0

0 0 D,𝑝 ∫ γ ∫ θ ∫ 𝑡 ∫ u ∫ σ

e v,𝑠 (𝜃) sin2λ 𝜃𝑑𝜃 e 2 𝑔) ≤ D sin−2λ 𝑡𝑑𝑡 sin2λ 𝑢 𝑑𝑢 sin−2λ σ𝑑σ sin2λ τ 𝑑τ

Sτ (∆

0

e 2 𝑔∥D,𝑝 ≤ ∥∆

0

∫

γ

e v,𝑠 (𝜃) sin2λ 𝜃𝑑𝜃 D

∫

0

e 2 𝑔∥D,𝑝 ≤ C∥∆

∫

0 θ

sin−2λ 𝑡𝑑𝑡

0

γ

0

∫

0

D,𝑝

sin−2λ σ𝑑σ

0

∫

D,𝑝

σ

sin2λ τ 𝑑τ 0

. D,𝑝

e 2 𝑔∥D,𝑝 . By using the equality ≤ C12 v −4 ∥∆

∫ 1 Jv,𝑠 (𝑔; 𝑥) − 𝑔(𝑥)

u

sin2λ 𝑢 𝑑𝑢 0

e v,𝑠 (𝜃) sin2λ 𝜃 𝑑𝜃 ≤ C12 v −4 e 2 𝑔 𝜃4 D

∆



1 e So, Jv,𝑠 (𝑔) − 𝑔 − 𝛼(v)∆𝑔

0

∫

𝑡

γ

e v,𝑠 (𝜃)(Sθ (𝑔; 𝑥) − 𝑔(𝑥)) sin2λ 𝜃 𝑑𝜃 D

= 0

∫

γ

e v,𝑠 (𝜃) sin2λ 𝜃 𝑑𝜃 D

= 0

∫

θ

−2λ

sin

∫ 𝑡 𝑑𝑡

0

𝑡

e 𝑥) 𝑑𝑢, sin2λ 𝑢 Su (∆𝑔;

0

we see that

∫

∫ θ ∫ 𝑡

γ

2λ −2λ 2λ e e = D (𝜃) sin 𝜃 𝑑𝜃 sin 𝑡 𝑑𝑡 sin 𝑢 S ( ∆𝑔) 𝑑𝑢

v,𝑠 u D,𝑝

0

0 0 D,𝑝 ∫ γ ∫ θ ∫ 𝑡

e v,𝑠 (𝜃) sin2λ 𝜃𝑑𝜃 e e D,𝑝 . (3.14) ≤ D sin−2λ 𝑡𝑑𝑡 sin2λ 𝑢 𝑑𝑢 ≤ C12 v −2 ∥∆𝑔∥

Su (∆𝑔)

1

Jv,𝑠 (𝑔) − 𝑔

0

0

D,𝑝

0

Assuming for 𝑚 = 𝑘, 𝑘 ≥ 1, (3.12) is valid, i.e.

𝑘 e e 2 𝑔∥D,𝑝 + C12 (𝑘 − 1)v −2 ∥∆𝑔∥ e D,𝑝 , ≤ C12 v −4 ∥∆

Jv,𝑠 (𝑔) − 𝑔 − 𝛼(v)∆𝑔

(3.15)

D,𝑝

then for the case 𝑚 = 𝑘 + 1, we see that 𝑘+1 𝑘+1 𝑘 𝑘 e e Jv,𝑠 (𝑔; 𝑥) − 𝑔(𝑥) − 𝛼(v)∆𝑔(𝑥) = Jv,𝑠 (𝑔; 𝑥) − Jv,𝑠 (𝑔; 𝑥) + Jv,𝑠 (𝑔; 𝑥) − 𝑔(𝑥) − 𝛼(v)∆𝑔(𝑥) 𝑘 1 𝑘 e = Jv,𝑠 (Jv,𝑠 (𝑔; 𝑥) − 𝑔(𝑥)) + Jv,𝑠 (𝑔; 𝑥) − 𝑔(𝑥) − 𝛼(v)∆𝑔(𝑥),

which yields



𝑘+1

𝑘 1 𝑘 e e = Jv,𝑠 (Jv,𝑠 (𝑔) − 𝑔) + Jv,𝑠 (𝑔) − 𝑔 − 𝛼(v)∆𝑔

Jv,𝑠 (𝑔) − 𝑔 − 𝛼(v)∆𝑔

D,𝑝 D,𝑝

1

𝑘

−2 e −4 e 2 e

≤ C12 v ∥∆𝑔∥D,𝑝 + C12 v ∥∆ 𝑔∥D,𝑝 ≤ Jv,𝑠 (𝑔) − 𝑔 D,𝑝 + Jv,𝑠 (𝑔) − 𝑔 − 𝛼(v)∆𝑔 D,𝑝

+C12 (𝑘 − 1)v

−2

e D,𝑝 = C12 v −4 ∥∆ e 2 𝑔∥D,𝑝 + C12 𝑘v −2 ∥∆𝑔∥ e D,𝑝 , ∥∆𝑔∥

where we use Lemma 3.3 (iv) in the first inequality, (3.14) and the induction assumption (3.15) in the second inequality. This completes the proof of Lemma 3.5.

4

Main result

{ 𝑚 }∞ Theorem 4.1. Let Jv,𝑠 be 𝑚-th Jackson-type operators on the spherical cap D(𝑥0 , 𝛾). Then v=1 for 𝑓 ∈ 𝐿𝑝 (D(𝑥0 , 𝛾)), 1 ≤ 𝑝 ≤ ∞, and 2𝑠 > 2 + 𝑛, it holds that ) (

𝑚

1 2

Jv,𝑠 (𝑓 ) − 𝑓 ≈ 𝜔 𝑓, D,𝑝 v D,𝑝

784

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

Proof. We have already obtained that ∫ γ 𝑚 e v,𝑠 (𝜃)(Sθ𝑚 (𝑔; 𝑥) − 𝑔(𝑥)) sin2λ 𝜃 𝑑𝜃 Jv,𝑠 (𝑔; 𝑥) − 𝑔(𝑥) = D 0

which implies (explained below)

∫ γ

𝑚

e v,𝑠 (𝜃)(Sθ𝑚 (𝑔) − 𝑔) sin2λ 𝜃 𝑑𝜃

Jv,𝑠 (𝑔) − 𝑔

= D

D,𝑝 0 D,𝑝 ∫ γ e v,𝑠 (𝜃) sin2λ 𝜃 ∥Sθ (𝑔) − 𝑔)∥ ≤ 𝑚 D D,𝑝 𝑑𝜃 0 ( ) ∫ γ ∫ θ ∫ 𝑡



2λ −2λ 2λ e v,𝑠 (𝜃) sin 𝜃 e 𝑑𝑡𝑑𝑢 𝑑𝜃 ≤ 𝑚 D sin 𝑡 sin 𝑢 Su (∆𝑔) 0

0

{

≤ 𝑚 sup 𝜃−2 θ>0

∫

θ

sin−2λ 𝑡 𝑑𝑡

0

∫

0

𝑡

D,𝑝

}



e sin2λ 𝑢 𝑑𝑢 ∆𝑔

0

∫ D,𝑝

γ

e v,𝑠 (𝜃) sin2λ 𝜃 𝑑𝜃 ≤ Cv −2 e 𝜃2 D

∆𝑔

0

, D,𝑝

where (iii) of Lemma 3.3 is used in the first inequality, the second inequality is deduced from Lemma 3.4, (ii) of Lemma 3.3 is used in the third inequality, and the last inequality is deduced from (3.12) and Lemma

𝑚 3.2.

Recalling that Jv,𝑠 (𝑓 ) D,𝑝 ≤ ∥𝑓 ∥D,𝑝 , we can see



𝑚

−2 e

Jv,𝑠 (𝑓 ) − 𝑓 ≤ 2∥𝑓 − 𝑔∥ + Cv

∆𝑔 D,𝑝 D,𝑝

D,𝑝

) ( 1 , ≤ CK 𝑓, 2 v D,𝑝

𝑚 ( ) (𝑓 ) − 𝑓 D,𝑝 ≤ C𝜔 2 𝑓, v1 D,𝑝 . which implies from (1.2) that Jv,𝑠 𝑚,q In order to prove the lower bound, we introduce an operator Jv,𝑠 given by 𝑚,q Jv,𝑠 (𝑓 ; 𝑥)

:=

𝑠(v−1) (∫ γ ∑ 𝑘=0

( ) e v,𝑠 (𝜃) Qλ𝑘 (cos 𝜃) 𝑚 sin2λ 𝜃 𝑑𝜃 D

)q Y𝑘 (𝑓 ; 𝑥).

0

With the help of orthogonality of projection operator Y𝑘 in (2.4), we get 𝑚,q+l Jv,𝑠 𝑓

=

𝑠(v−1) (∫ γ ∑ 𝑘=0



×Y𝑘 

) ( e v,𝑠 (𝜃) Qλ𝑘 (cos 𝜃) 𝑚 sin2λ 𝜃 𝑑𝜃 D

)q

0 𝑠(v−1) (∫ γ ∑ z=0

( ) e v,𝑠 (𝜃) Qλz (cos 𝜃) 𝑚 sin2λ 𝜃 𝑑𝜃 D

)l



( 𝑚,l ) 𝑚,q Yz (𝑓 ) = Jv,𝑠 Jv,𝑠 (𝑓 ) .

0

𝑚,q Here, we let 𝑔 = Jv,𝑠 𝑓 and obtain that

∥𝑓 − 𝑔∥D,𝑝

q ∑



𝑚,τ −1

𝑚 𝑚,q 𝑚,τ

Jv,𝑠 (𝑓 ) D,𝑝 = 𝑓 − Jv,𝑠 (𝑓 ) D,𝑝 ≤ (𝑓 ) − Jv,𝑠 (𝑓 ) D,𝑝 ≤ q 𝑓 − Jv,𝑠 τ =1

𝑚,0 where we take Jv,𝑠 (𝑓 ) = 𝑓 .



e 𝑚,q Next, we prove the estimation ∆J v,𝑠 (𝑓 )

D,𝑝

≤

A 2 2C12 C13 v

∥𝑓 ∥D,𝑝 , where A and C12 are the

same as that in Lemma 3.5. In fact, we have

𝑠(v−1)

(∫ γ )q

∑



𝑚

e 𝑚,q 2λ λ e

≤ 𝑘(𝑘 + 𝑛 − 2) Dv,𝑠 (𝜃) (Q𝑘 (cos 𝜃)) sin 𝜃𝑑𝜃 Y𝑘 (𝑓 )

∆Jv,𝑠 (𝑓 )

D,𝑝 0

𝑘=0

785

. D,𝑝

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap λ 𝑚 { } P (cos θ) 𝑚 Since (see [14]) Qλ𝑘 (cos 𝜃) ≡ 𝑘P λ (1) ≤ C14 min (𝑘𝜃)−λ , 1 , we use Lemma 3.1 (3.9) and 𝑘 obtain that for 𝑘𝜃 ≥ 1 and 𝜃 ≤ π2

𝑠(v−1)

(∫ γ )q

∑

𝑛−2

e 𝑚,q 2λ − 2 q −λ e

≤ C15 𝑘(𝑘 + 𝑛 − 2)𝑘 Dv,𝑠 (𝜃)𝜃 sin 𝜃 𝑑𝜃 Y𝑘 (𝑓 )

∆Jv,𝑠 (𝑓 )

D,𝑝 0

𝑘=0

D,𝑝

≤ C16 v

𝑛−2 2 q

∥𝑓 ∥D,𝑝

∞ ∑

𝑘

2− 𝑛−2 2 q

≤ C17 v

𝑛−2 2 q

∥𝑓 ∥D,𝑝 ,

𝑘=0

provided that q > (3.10) that

6 𝑛−2 ,

so

∑∞ 𝑘=0

𝑘 2−

𝑛−2 2 q

< ∞. For the case 𝑘𝜃 ≤ 1, it follows from Lemma 3.1



𝑠(v−1)

(∫ γ )q 1

∑

) 2 ( −q 2λ 2 λ 𝑚 q e

𝜃 𝑘(𝑘 + 𝑛 − 2) |(Q𝑘 (cos 𝜃))| sin 𝜃 𝑑𝜃 Y𝑘 (𝑓 ) ≤ Dv,𝑠 (𝜃)𝜃

D,𝑝 0

𝑘=0

D,𝑝

𝑠(v−1)

(∫ ) q γ

∑

e v,𝑠 (𝜃)𝜃− q2 (𝑘𝜃) q2 sin2λ 𝜃 𝑑𝜃 Y𝑘 (𝑓 ) ≤ C18 D

0

𝑘=0

D,𝑝

𝑠(v−1)

) (∫ q γ

∑

A e v,𝑠 (𝜃)𝜃− q2 sin2λ 𝜃 𝑑𝜃 Y𝑘 (𝑓 ) ≤ C19 C13 v 2 ∥𝑓 ∥D,𝑝 , D ≤ C20 v 2 ∥𝑓 ∥D,𝑝 =

2C12 0

𝑘=0



e 𝑚,q

∆Jv,𝑠 (𝑓 )

D,𝑝

where A and C12 are the same as that in Lemma 3.5. Therefore, when q >

e 𝑚,q

∆Jv,𝑠 (𝑓 )

D,𝑝

≤

6 𝑛−2 ,

it holds that

A C13 v 2 ∥𝑓 ∥D,𝑝 . 2C12

6 6 In the next, without loss of generality, we assume that l > 𝑛−2 and q > l + 𝑛−2 . With the help of Lemma 3.5, we see that



e 𝑚,q

𝑚,q e 𝛼(v) ∆J (𝑓 ) = ∆J (𝑓 )

𝛼(v)

v,𝑠 v,𝑠 D,𝑝 D,𝑝



𝑚,q

−4 2 𝑚,q −2 e 𝑚,q e J (𝑓 ) ≤ Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 + C12 v ∆ + C (𝑚 − 1)v ∆J (𝑓 )

12 v,𝑠 v,𝑠 D,𝑝 D,𝑝



𝑚

AC



13 𝑚,q−l 𝑚,q e v,𝑠 e v,𝑠 v −2 ∆J (𝑓 ) + C12 (𝑚 − 1)v −2 ∆J (𝑓 ) ≤ q Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 + 2 D,𝑝 D,𝑝

𝑚

AC

13 −2 e 𝑚,q v ∆Jv,𝑠 (𝑓 ) ≤ q Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 + 2 D,𝑝

) AC13 −2

e ( 𝑚,q

e 𝑚,q 𝑚,q−l v ∆ Jv,𝑠 (𝑓 ) − Jv,𝑠 (𝑓 ) + C12 (𝑚 − 1)v −2 ∆J + v,𝑠 (𝑓 ) 2 D,𝑝 D,𝑝

𝑚

AC

13 −2 e 𝑚,q v ∆Jv,𝑠 (𝑓 ) ≤ q Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 + 2 D,𝑝

AC13 C21

𝑚,l −2 e 𝑚,q

Jv,𝑠 (𝑓 ) − 𝑓 + + C (𝑚 − 1)v ∆J (𝑓 )

12 v,𝑠 D,𝑝 2 D,𝑝

𝑚

AC

13 −2 e 𝑚,q ≤ q Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 + v ∆Jv,𝑠 (𝑓 ) 2 D,𝑝

AC13 C21

e 𝑚,q 𝑚 l Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 + C12 (𝑚 − 1)v −2 ∆J + v,𝑠 (𝑓 ) 2 D,𝑝

𝑚

C

23 −2 e 𝑚,q ≤ C22 Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 + v ∆Jv,𝑠 (𝑓 ) . 2 D,𝑝



1 e 𝑚,q 22 𝑚 Taking 𝛼(v) = Cv23 ≤ 2C Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 . By the definition of 2 , one has v 2 ∆Jv,𝑠 (𝑓 ) C23 D,𝑝

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M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

K-functional, we see ( )

𝑚,q

1 1

e 𝑚,q K 𝑓, 2 ≤ Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 + 2 ∆J v,𝑠 (𝑓 ) v D,𝑝 v D,𝑝

𝑚

𝑚

2C 22 𝑚 ≤ q Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 + Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 ≤ C24 Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 , C23

𝑚

( 1) 2 which together with (2.6) implies 𝜔 𝑓, 𝑘 D,𝑝 ≤ C Jv,𝑠 (𝑓 ) − 𝑓 D,𝑝 . The proof of Theorem 4.1 is complete.

References [1] E. Belinsky, F. Dai, Z. Ditzian, Multivariate approximating averages, J. Approx. Theory, 125 (2003), 85-105. [2] H. Berens, P. L. Butzer, S. Pawelke, Limitierungsverfahren von reihen mehrdimensionaler kugelfunktionen und deren saturationsverhalten, Publ. Res. Inst. Math. Sci. Ser. A, 4(2) (1968), 201-268. [3] P. L. Butzer, H. Johnen, Lipschitz spaces on compact manifolds, J. Funct. Anal., 7 (1971), 242-266. [4] H. Berens, L. Q. Li, On the de la Vallee-Poussin means on the sphere, Results in Math., 24(1-2) (1993), 12-26. [5] F. L. Cao, X. F. Guo, Approximation by Jackson-type operator on the sphere, Math. Commum., 15(2) (2010), 331-346. [6] F. Dai, Z. Ditzian, Jackson inequality for Banach spaces on the sphere, Acta Math. Hungar. 118 (2008), 171-195. [7] F. Dai, Z. Ditzian, Jackson theorem in 𝐿𝑝 , 0 < 𝑝 < 1, for functions on the sphere, J. Approx. Theory, 162(3) (2010), 382-391. [8] F. Dai, Jackson inequality for doubling weights on the sphere, Constr. Approx. 24 (2006), 91-112. [9] L. Q. Li, R. Y. Yang, Approximation by spherical Jackson polynomials, Journal of Beijing Normal University (Natural Science), 27 (1991), 1-12 (Chinese). [10] I. P. Lizorkin, S. M. Nikol’ski˘ı, A theorem concerning approximation on the sphere, Anal. Math., 9 (1983), 207-221. [11] P. L. Lizorkin, S. M. Nikol’ski˘ı, Function spaces on the sphere that are connected with approximation theory, Mat. Zametki, 41 (1987), 509-515 (English Transl. Math. Notes, 41 (1987) 286-291). [12] K. V. Rustamov, On equivalence of different moduli of smoothness on the sphere, Proc. Stekelov Inst. Math., 204 (3) (1994), 235-260. [13] E. M. Stein, G. Weiss, An introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, NJ, 1971. [14] K. Y. Wang, L. Q. Li, Harmonic analysis and approximation on the unit sphere, Science Press, Beijing, 2000. [15] Y. G. Wang, F. L. Cao, The direct and converse inequality for Jackson-type operators on spherical cap, J. Inequal. Appl., 2009, Volume 2009, Article ID 205298, 16 pages. [16] R. Y. Yang, F. L. Cao, J. Y. Xiong, The strong converse inequality for de la Vall´ee Poussin means on the sphere, Arxiv preprint arXiv:1105.4062, 2011

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 4, 2013 Positive Solutions to a Nonlinear 2nth Order Boundary Value Problems on Time Scales, Ilkay Yaslan Karaca,…….………………………………………………………………………604 Fixed Points and the Random Stability of a Mixed Type Cubic, Quadratic and Additive Functional Equation, M. Eshaghi Gordji, Masumeh Ghanifard, Hamid Khodaei, and Choonkil Park,………………………………………………………………………………………612 Weakly Set Valued Generalized Vector Variational Inequalities, George A. Anastassiou and Salahuddin,………………………………………………………………………………622 A New Bivariate Interpolation by Rational Triangular Patch, Qinghua Sun, Fangxun Bao, Yunfeng Zhang, and Qi Duan,………………………………………………………….633 A Note On The Modified Carlitz’s q-Bernoulli Numbers and Polynomials, Jin-Woo Park, Dmitry V. Dolgy, Taekyun Kim, Sang-Hun Lee, and Seog-Hoon Rim,………………………647 New q-Analogue of Modified Bessel Function and the Quantum Algebra Eq(2), M. Mansour and M. M. Al-Shomrani,……………………………………………………………….……655 A New System of General Variational Inequality and Fixed Point Problems for a Countable Family of Strict Pseudo-Contractions in Banach Spaces, Rattanaporn Wangkeeree and Rabian Wangkeeree,……………………………………………………………………………665 Best Proximity Points For Cyclical Contraction Mappings With 0−Boundedly Compact Decompositions, T. Abdeljawad, J. O. Alzabut, A. Mukheimer, and Y. Zaidan,……678 Numerical Computations of the Distribution of the Zeros of the Second Kind (h, q)-Euler Polynomials, C. S. Ryoo,………………………………………………………………686 Fixed Points and Stability of the Cauchy-Jensen Functional Equation in Fuzzy Banach Algebras, Jung Rye Lee, Sung Jin Lee, and Choonkil Park,……………………………….…….692 Intra-Orbit Separation of Orbits of Tree Maps, Zhanhe Chen, Taixiang Sun, and Guangwang Su,………………………………………………………………………………………699 A Generalized System Mixed Variational Inequality Involving Three Difference Relaxed Cocoercive Operators, Nawitcha Onjai-Uea and Poom Kumam,……………………707 Block Preconditioned AOR Methods for H-matrices Linear Systems, Xue-ZhongWang,714

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 4, 2013 (continued) A Note on the q-Bernoulli Numbers and q-Bernstein Polynomials, Jin-Woo Park, Hong Kyung Pak, Seog-Hoon Rim, Taekyun Kim, and Sang-Hun Lee,…………………………………722 New Classes of Generalized Sequence Spaces Defined by an Orlicz Function, Kuldip Raj, Seema Jamwal and Sunil K. Sharma,……………………………………………………………….730 Stability and Superstability of Quadratic ∗ −Derivations on Fuzzy Banach ∗ −Algebras, Sun Young Jang, and Young Cho,……………………………………………………………….738 The Topological Structure on Soft Sets, Zhaowen Li, Haiyan Chen, and Ninghua Gao,…746 A Caputo Fractional Order Boundary Value Problem with Integral Boundary Conditions, Azizollah Babakhani and Thabet Abdeljawad,……………………………………………753 Some Applications of the Choquet Integral with Respect to a Monotone Set Function on the Set of Interval-Valued Necessity Measures, Lee-Chae Jang,………………………………….764 The Equivalent Theorem for Jackson-type Operators on Spherical Cap, Ming Li, Feilong Cao, and Zhixiang Chen,………………………………………………………………………….778

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Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC(eight times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Editor’s Assistant:Dr.Razvan Mezei,Lander University,SC 29649, USA.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 800-806, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Some order relations induced by fuzzy subsets Jeong Soon Han1 , Young Hee Kim2 and Keum Sook So3∗ 1

2

Department of Applied Mathematics, Hanyang University, Ahnsan, 426-791, Korea Department of Mathematics, Chungbuk National University, Cheongju, 361-763, Korea 3

Department of Mathematics, Hallym University, Chuncheon, 200-702, Korea

Abstract. In this paper, we introduce the notion of groupoids belongs to fuzzy subalgebras, and show that the induced poset of a groupoid belongs to fuzzy subalgebras is an ordinal sum of anti-chains, and obtain several properties of induced posets by fuzzy sets.

1. Introduction. J. Neggers ([6]) has defined a pogroupoid and he obtained a functorial connection between posets and pogroupoids and associated structure mappings. J. Neggers and H. S. Kim ([7]) demonstrated that a pogroupoid (X, ·) is modular* if and only if its associated poset (X, ≤) is (C2 + 1)-free , a condition which corresponds naturally to the notion of sublattice (in the sense of Kelly-Rival [1, 3]) isomorphic to N5 , and that this is equivalent to the associativity of the pogroupoid. J. Neggers and H. S. Kim ([8]) showed that the Jacobi form is 0 precisely when the pogroupoid (X, ·) is a semigroup, i.e., modular*, precisely when the associated poset (X, ≤) is (C2 + 1)-free . Moreover, they showed that a pg-algebra KS over a field K is a Lie algebra with respect to the commutator product if and only if its associated poset (X, ≤) is (C2 + 1)-free . H. S. Kim and J. Neggers ([4]) showed that, for given a pogroupoid (X, ·), the associated poset (X, ≤) is (C2 + 1)-free if and only if the relation Bµ is transitive for any fuzzy subset µ of X. Also they determined the set C(X, ·) of fuzzy subsets µ such that µ(x · y) = µ(y · x) for all x, y ∈ X. Recently, H. S. Kim and J. Neggers ([7]) introduced the notion of Bin(X) and showed that it forms a semigroup. In this paper, we introduce the notion of groupoids belongs to fuzzy subalgebras, and show that the induced poset of a groupoid belongs to fuzzy subalgebras is an ordinal sum of anti-chains, and obtain several properties of induced posets by fuzzy sets. 2. Preliminaries. The concept of a fuzzy set was introduced by L. A. Zadeh ([13]). A fuzzy subset of a set X is a function µ : X → [0, 1]. The applications of fuzzy concepts to posets and groupoids have been investigated by several authors (including [9, 12]). A groupoid (X, ·) is said to be a pogroupoid if (i) x · y ∈ {x, y}; (ii) x · (y · x) = y · x; (iii) (x · y) · (y · z) = (x · y) · z for all x, y, z ∈ X. For a given pogroupoid (X, ·), its associated poset (X, ≤) is defined by the condition x ≤ y iff y · x = x · y = y. On the other hand, for a given poset (X, ≤) its associated pogroupoid (X, ·) is defined by y · x = y if x ≤ y, y · x = x otherwise. This means that there is a natural isomorphism between 0

2010 Mathematics Subject Classification:03E72, 06A06, 06F35. Keywords: fuzzy subset, (induced) poset, pogroupoid, (C2 + 1)-free , fuzzy subset, dispersed, polynomial. ∗ The corresponding author Tel.: +82 33 248 2011, Fax: +82 33 256 2011 (K. S. So). 0 E-mail: [email protected]; [email protected]; [email protected] 0

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J. S. Han, Y. H. Kim and K. S. So the category of pogroupoids and the category of posets. We call a pogroupoid modular* if it is a semigroup ([7]). Given a poset (X, ≤) it is Q-free if there is no full subposet (P, ≤) of (X, ≤) which is order isomorphic to the poset (Q, ≤). If Cn denotes a chain of length n and if n denotes an antichain of cardinal number n , while + denotes the disjoint union of posets, then the poset (C2 + 1) (or C2 + C1 ) has Hasse-diagram:

and may also be represented as {p ≤ q, p ◦ r, q ◦ r}, where a ◦ b denotes the relation of not being comparable (i.e., a ◦ b iff a ≤ b and b ≤ a are both false) ([10]). A poset Z is called the ordinal sum of two posets X and Y , written Z = X ⊕ Y , if z1 ≤ z2 in Z implies z1 ∈ X and z2 ∈ Y , z1 ≤ z2 in X or z1 ≤ z2 in Y . Theorem 2.1. ([7]) A pogroupoid (X, ·) is a semigroup if and only if its associated poset (X, ≤) is (C2 + 1)-free. H. S. Kim and J. Neggers ([4]) introduced the notion of Bin(X) for a given set X, and defined a product (X, ) = (X, ∗)  (X, ◦) where x  y = (x ∗ y) ◦ (y ∗ x) for any x, y ∈ X. Given a set, we let Bin(X) the collection of all groupoids defined on X. Theorem 2.2. ([4]) (Bin(X), ) is a semigroup. Furthermore, the left-zero semigroup is an identity for this operation. Theorem 2.3. ([11]) Let (X, ≤) be a pogroupoid and suppose its associated poset (X, ≤) is (C2 + 1)-free . Let µ : X → [0, 1] be a fuzzy subset of X. Define x Bµ y ⇐⇒ µ(x · y) < µ(y · x), x, y ∈ X. Then (X, Bµ ) is a poset, called the induced poset of X by µ.

3. Groupoids belongs to fuzzy subalgebras. Given a non-empty set X, let µ : X → [0, 1] be a fuzzy subset of X. We say a groupoid (X, ·) belongs to µ if µ is a fuzzy subalgebra of (X, ·), i.e., µ(x · y) ≥ min{µ(x), µ(y)} for any x, y ∈ X. We denote Θ(µ) := {(X, ·) ∈ Bin(X)|(X, ·) belongs to µ}. Example 3.1. Define a binary operation “∗” on X = [0, 1] by x ∗ y := max{0, x − y} for all x, y ∈ X. Then it is a d/BCK-algebra. If we define a map µ : X → [0, 1] by µ(x) := 1 − x, then (X, ∗) ∈ Θ(µ). In fact, if we let x ≤ y in X, then µ(x) ≥ µ(y). Since x ∗ y = max{0, x − y} ≤ x, we obtain µ(x ∗ y) ≥ µ(x) ≥ min{µ(x), µ(y)} for any x, y ∈ X. Let (X, ·) ∈ Θ(µ) and let x, y ∈ X. Define a relation “≤µ ” on X by x ≤µ y ⇐⇒ x = y or µ(x) < µ(y) Hence, if x 6= y and µ(x) = µ(y), then x ◦ y, i.e., x and y are not comparable. 801

Some order relations induced by fuzzy subsets Proposition 3.2. If (X, ·) ∈ Θ(µ), then (X, ≤µ ) is a poset. Proof. Straightforward.



We call such a poset (X, ≤µ ) an induced poset from a groupoid (X, ·) ∈ Θ(µ). Theorem 3.3. If (X, ·) ∈ Θ(µ), then the poset (X, ≤µ ) is an ordinal sum of anti-chains. Proof. For any x, y ∈ X, we have either µ(x) < µ(y) or µ(x) = µ(y). If µ(x) < µ(y), then x ≤µ y, and if µ(x) = µ(y), then x ◦ y. Hence the poset (X, ≤µ ) is an ordinal sum of anti-chains.



Example 3.4. Let X := {0, 1, 2, 3, 4, } be a set with the following table: ·

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

1 1 2 3 4

2 2 2 2 2

3 3 3 3 3

4 4 4 4 4

(X, ·) If we define a fuzzy subset µ : X → [0, 1] by 0 ≤ µ(0) < µ(1) < µ(2) = µ(3) < µ(4) ≤ 1, then (X, ·) ∈ Θ(µ) and (X, ≤µ ) is a poset with the following Hasse diagram:

(X, ≤µ )

This means that (X, ≤µ ) = {0} ⊕ {1} ⊕ {2, 3} ⊕ {4}, an ordinal sum of anti-chains.

Theorem 3.5. Let (X, ≤µ ) be an induced poset from a groupoid (X, ·) ∈ Θ(µ) and let (X, µ ) be the associated pogroupoid of (X, ≤µ ). Then (X, µ ) is a semigroup and (X, µ ) ∈ Θ(µ). Proof. If (X, µ ) is the associated pogroupoid of the poset (X, ≤µ ), then x µ y ∈ {x, y} for any x, y ∈ X. It follows that µ(x µ y) ≥ min{µ(x), µ(y)}, i.e., (X, µ ) ∈ Θ(µ). By Theorem 3.3, the induced poset (X, ≤µ ) is an ordinal sum of anti-chains. Since every ordinal sum of anti-chains is (C2 + 1)-free, by applying Theorem 2.1, we obtain that (X, µ ) is a semigroup. 802



J. S. Han, Y. H. Kim and K. S. So Given (X, ·1 ), (X, ·2 ) ∈ Θ(µ), we define a relation “≤µ ” as follows: (X, ·1 ) ≤µ (X, ·2 ) ⇐⇒ µ(x ·1 y) ≤ µ(x ·2 y) for any x, y ∈ X. Example 3.6. Let X := {0, 1, 2, 3, 4} be a set with the following tables: ·1

0 1 2 3 4

·2

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 0 2 3 4

0 0 0 3 3

0 0 0 0 2

0 0 0 0 0

(X, ·1 )

0 0 2 3 4

0 0 0 3 4

0 0 0 0 4

0 0 0 0 0

(X, ·2 )

Then (X, ·1 , 0) and (X, ·2 , 0) are BCK-algebras (see [5]). If we define a map µ : X → [0, 1] by 0 ≤ µ(4) < µ(3) < µ(2) < µ(1) < µ(0) ≤ 1, then (X, ·2 ) ≤µ (X, ·1 ), since µ(4 ·2 2) = µ(4) < µ(3) = µ(4 ·1 2) and µ(4 ·2 3) = µ(4) < µ(2) = µ(4 ·1 3). By routine calculations, we can see that µ is a fuzzy subalgebra of both (X, ·1 ) and (X, ·2 ), i.e., (X, ·1 ), (X, ·2 ) ∈ Θ(µ). Given a fuzzy subset µ : X → [0, 1], we say a groupoid (X, ·) strongly belongs to µ if µ is a fuzzy subalgebra of (X, ·), i.e., µ(x · y) ≥ max{µ(x), µ(y)} for any x, y ∈ X. We denote ΘS (µ) := {(X, ·) ∈ Bin(X)|(X, ·) strongly belongs to µ}. Proposition 3.7. Let (X, ≤µ ) be the induced poset from any groupoid (X, ·) ∈ Θ(µ) and let (X, µ ) be the associated pogroupoid of (X, ≤µ ). Then µ(x µ y) = max{µ(x), µ(y)}, ∀x, y ∈ X, and (X, µ ) ∈ ΘS (µ).

Proof. Given x, y ∈ X with x 6= y, if x ≤µ y, then µ(x) < µ(y). Since (X, µ ) is a pogroupoid, we have y µ x = x µ y = y. Hence µ(x µ y) = µ(y µ x) = µ(y) > µ(x), proving µ(x µ y) = max{µ(x), µ(y)}. Assume x ◦ y in (X, ≤µ ). Since (X, µ ) is a pogroupoid, we have µ(x) = µ(y) and x µ y = y, and hence µ(x µ y) = µ(y) = max{µ(x), µ(y)}, proving the proposition.



Corollary 3.8. Let (X, ≤µ ) be the induced poset from any groupoid (X, ·) ∈ Θ(µ) and let (X, µ ) be the associated pogroupoid of (X, ≤µ ). Then (X, µ ) is minimal in the poset (ΘS (µ), ≤µ ).

Proof. For any (X, ∗) ∈ ΘS (µ), we have µ(x ∗ y) ≥ max{µ(x), µ(y)} = µ(x µ y) for any x, y ∈ X, proving that (X, µ ) ≤µ (X, ∗).

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Some order relations induced by fuzzy subsets 4. Induced poset by fuzzy subset. The relation ≤µ discussed in section 3 need not be anti-symmetric, i.e., there exist (X, ·1 ), (X, ·2 ) ∈ Θ(µ) such that (X, ·1 ) 6= (X, ·2 ), but (X, ·1 ) ≤µ (X, ·2 ) and (X, ·2 ) ≤µ (X, ·1 ). In fact, in Example 3.6, if we define a map µ : X → [0, 1] by 0 ≤ µ(2) = µ(3) = µ(4) < µ(1) < µ(0) ≤ 1, then (X, ·1 ), (X, ·2 ) ∈ Θ(µ) such that (X, ·1 ) ≤µ (X, ·2 ) and (X, ·2 ) ≤µ (X, ·1 ), but (X, ·1 ) 6= (X, ·2 ), since 4 ·1 2 = 3 6= 4 = 4 ·2 2. Given (X, ·1 ), (X, ·2 ) ∈ Θ(µ), we define a relation “∼µ ” on Θ(µ) as follows: (X, ·1 ) ∼µ (X, ·2 ) ⇐⇒ (X, ·1 ) ≤µ (X, ·2 ), (X, ·2 ) ≤µ (X, ·1 ) Then it is easy to see that the relation ∼µ is an equivalence relation on Θ(µ). Define an equivalence class containing (X, ·1 ) ∈ Bin(X) by [(X, ·1 )] := {(X, ·2 ) ∈ Θ(µ) | (X, ·1 ) ∼µ (X, ·2 )} and let Θ/µ := {[(X, ·1 )] | (X, ·1 ) ∈ Θ(µ)} and let ΘS /µ := {[(X, ·1 )] | (X, ·1 ) ∈ ΘS (µ)} If we define a relation “ ≤[µ] ” on Θ/µ (or ΘS /µ) by [(X, ·1 )] ≤[µ] [(X, ·2 )] ⇐⇒ (X, ·α ) ≤µ (X, ·β ) for any (X, ·α ) ∈ [(X, ·1 )] and any (X, ·β ) ∈ [(X, ·2 )]. Then it is easy to see that the relation ≤[µ] is a partial order on Θ/µ (or ΘS /µ). Proposition 4.1. Let (X, ≤µ ) be the induced poset from any groupoid (X, ·) ∈ Θ(µ) and let (X, µ ) be the associated pogroupoid of (X, ≤µ ). Then (ΘS /µ, ≤[µ] ) is a subposet of (Θ/µ, ≤[µ] ) containing [(X, µ )] as the unique minimal element.

Proof. Given [(X, ·1 )] ∈ ΘS /µ, if (X, ·2 ) ∈ [(X, ·1 )], then (X, ·2 ) ∼µ (X, ·1 ) and hence µ(x ·1 y) = µ(x ·2 y) for any x, y ∈ X. It follows from Proposition 3.7 that µ(x ·2 y) = µ(x ·1 y) ≥ max{µ(x), µ(y)} = µ(x µ y), proving that (X, µ ) ≤µ (X, ·2 ). This proves that [(X, µ )] ≤[µ] [(X, ·1 )].



Given [(X, ·)] ∈ Θ/µ, we define a relation “Bµ(·) ” on X by x Bµ(·) y

⇐⇒

µ(x · y) < µ(y · x)

Proposition 4.2. Let (X, ≤µ ) be the induced poset from any groupoid (X, ·), where [(X, ·)] ∈ Θ(µ), and let (X, µ ) be the associated pogroupoid of (X, ≤µ ). Then (X, Bµ( µ ) ) is an anti-chain. Proof. Assume that there exist x, y ∈ X such that x Bµ( µ ) y, x 6= y. Then µ(x µ y) < µ(y µ x). By Proposition 3.7, it follows that max{µ(x), µ(y)} < max{µ(y), µ(x)}, a contradiction. Given a fuzzy subset µ : X → [0, 1], we define a set [Θ]/µ as follows: [Θ]/µ := {[(X, ·)] | (X, ·) ∈ Θ(µ), (X, Bµ(·) ) is a poset } By Proposition 4.2, [Θ]/µ is a non-empty set. Proposition 4.3. If (X, ·1 ) ∈ Θ(µ), then Bµ(·1 ) =Bµ(·2 ) for any (X, ·2 ) ∈ [(X, ·1 )]. 804



J. S. Han, Y. H. Kim and K. S. So Proof. Straightforward.



Proposition 4.3 shows that the relation x Bµ(·) y is independent of the representative of the equivalence class [(X, ·)]. Given (X, ·) ∈ Θ(µ), we let R(X) := {(X, Bµ(·) )| Bµ(·) is a relation on X}. By Proposition 4.2, we obtain R(X) is a non-empty set. The map ϕ : Bin(X) → R(X) defined by ϕ((X, ·)) := (X, Bµ(·) ) induces an assignment ϕ˜ : [Θ]/µ → R(X) where ϕ([(X, ˜ ·)]) := (X, Bµ(·) ). Example 4.4. Let (X, ·) be the right-zero semigroup, i.e., x · y = y for any x, y ∈ X. Then [(X, ·)] ∈ [Θ]/µ for any fuzzy subset µ of X. In fact, if µ : X → [0, 1] is a fuzzy subset of X, then µ(x · y) = µ(y) ≥ min{µ(x), µ(y)}, i.e., (X, ·) ∈ Θ(µ). If x Bµ(·) y, then µ(y) < µ(x), since (X, ·) is the right-zero semigroup. The relation Bµ(·) is irreflexive, since there is no x ∈ X such that µ(x) < µ(x). If x Bµ(·) y, y Bµ(·) z, then µ(z) < µ(y) < µ(x), proving that x Bµ(·) z. This proves that (X, Bµ(·) ) is a poset, i.e., [(X, ·)] ∈ [Θ]/µ.

Proposition 4.5. Let [(X, ·)] ∈ [Θ]/µ. If (X, Mµ(·) ) is a pogroupoid of the associated poset (X, Bµ(·) ), then [(X, Mµ(·) )] ∈ [Θ]/µ. Proof. If [(X, ·)] ∈ [Θ]/µ, then (X, Bµ(·) ) is a poset. Define a binary operation “Mµ(·) ” on X by ( y if x Bµ(·) y y Mµ(·) x := x otherwise Then (X, Mµ(·) ) is a pogroupoid, i.e., [(X, Mµ(·) )] ∈ [Θ]/µ.



Proposition 4.6. Every pogroupoid (X, Mµ(·) ) induced from a commutative groupoid (X, ·) is the right-zero semigroup for any fuzzy subset µ : X → [0, 1]. Proof. If (X, ·) is a commutative groupoid, then µ(x · y) = µ(y · x), for any x, y ∈ X and for any fuzzy subset µ : X → [0, 1]. It follows that (X, Bµ(·) ) is an anti-chain. Hence the associated pogroupoid (X, Mµ(·) ) satisfies x Mµ(·) y = y for any x, y ∈ X, i.e., (X, Mµ(·) ) is the right-zero semigroup.



Example 4.7. In Example 3.1, we have seen that (X, ∗) ∈ Θ(µ). We show that (X, Bµ(∗) ) is a chain. In fact, if we let x < y in X, then µ(x∗y) = µ(0) = 1, µ(y ∗x) = 1−(y −x) = 1+x−y. Hence µ(y ∗x) < µ(x∗y), i.e., y Bµ(∗) x. We claim that [(X, Bµ(∗) )] is a minimal element of ([Θ]/µ, ≤∗ (µ)). Since (X, Bµ(∗) ) is a chain, either x Bµ(∗) y or y Bµ(∗) x for any x, y ∈ X. Assume that x Bµ(∗) y. By the definition of Mµ(∗) , we have x Mµ(∗) y = y = y Mµ(∗) x, which implies that µ(x Mµ(∗) y) = µ(y) = µ(y Mµ(∗) x). Now, x Bµ(∗) y implies µ(y ∗ x) < µ(x ∗ y), and hence x ∗ y < y ∗ x. It follows that max{0, x − y} < max{0, y − x} and x < y. Since µ is order reversing, µ(y) < µ(x). This shows that µ(y) = min{µ(x), µ(y)}. Hence µ(x Mµ(∗) y) = min{µ(x), µ(y)} ≤ µ(x ·α y) = µ(x · y) for any (X, ·α ) ∈ [(X, ·)] ∈ [Θ]/µ, proving that [(X, Bµ(∗) )] is a minimal element of ([Θ]/µ, ≤∗ (µ)).

Proposition 4.8. If µ : X → [0, 1] is a constant function, then (X, Bµ(·) ) is an anti-chain for any (X, ·) ∈ [Θ]/µ. 805

Some order relations induced by fuzzy subsets Proof. Assume that there exist x 6= y in X such that x Bµ(·) y. Then µ(x · y) = µ(y · x) < µ(x · y), a contradiction, since µ is a constant function. Hence (X, Bµ(·) ) is an anti-chain.



Proposition 4.9. Let (X, ·) ∈ Θ(µ) such that if x · y = y · x then x = y. If µ : X → [0, 1] is an 1-1 function, then (X, Bµ(·) ) is a chain. Proof. For any x, y ∈ X with x 6= y, we have x · y 6= y · x. Since µ is an 1-1 function, we obtain µ(x · y) 6= µ(y · x), which implies that either y Bµ(·) x or x Bµ(·) y. Hence (X, Bµ(·) ) is a chain.



An algebra (X; ∗, 0) of type (2,0) is said to be a strong d-algebra ([2]) if it satisfies (I) x ∗ x = 0, (II) 0 ∗ x = 0 and (III) x ∗ y = y ∗ x implies x = y, for all x, y ∈ X. Obviously, every strong d-algebra is a d-algebra, but the converse need not be true in general. Example 4.10. Let (X, ∗, 0) be a strong d-algebra and let (X, ∗) ∈ Θ(µ). If µ is an 1-1 function, then (X, Bµ(∗) ) is a chain.

References [1] G. Gr¨atzer, General lattice theory, Academic Press, New York, 1978. [2] J. S. Han, H. S. Kim and J. Neggers, Strong and ordinary d-algebras, J. Multiple-Valued Logic and Soft Computing 16 (2010), 331-339 [3] D. Kelly and I. Rival, Planar lattices, Canad. J. Math. 27 (1975), 636-665. [4] H. S. Kim and J. Neggers, The semigroups of binary systems and some perspectives, Bull. Korean Math. Soc. 45 (2008), 651-661. [5] J. Meng and Y. B. Jun, BCK-algebras Kyungmoonsa Co., Seoul 1994. [6] J. Neggers, Partially ordered sets and groupoids, Kyungpook Math. J. 16 (1976), 7-20. [7] J. Neggers and Hee Sik Kim, Modular semigroups and posets, Semigroup Forum 53 (1996), 57-62. [8] J. Neggers and Hee Sik Kim, Algebras associated with posets, Demonstratio Math. 34 (2001), 13-23. [9] J. Neggers and Hee Sik Kim, Fuzzy posets on sets, Fuzzy Sets and Sys. 117 (2001), 391-402. [10] J. Neggers and Hee Sik Kim, Basic Posets, World Scientific Pub. Co., Singapore, 1998. [11] J. Neggers and Hee Sik Kim, Fuzzy pogroupoids, Informations Sciences 175 (2005), 108-119. [12] P. Venugopalan, Fuzzy ordered sets, Fuzzy Sets and Sys. 46 (1992), 221-226. [13] L. Zadeh, Fuzzy Sets, Inform. and Control, 8 (1965), 338-353.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 807-816, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Multi-attribute group decision-making method based on generalized aggregation operators in trapezoidal fuzzy linguistic variables Peide Liu,⋆

Xingying Wu

School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan Shandong 250014, China [email protected]

Abstract. With respect to multi-attribute group decision-making problems in which attribute values are trapezoidal fuzzy linguistic variables, this paper presents a new multi-attribute group decision-making method based on generalized operators in trapezoidal fuzzy linguistic variables. Firstly, we introduce the concept of the trapezoidal fuzzy linguistic variable, and its properties, operational laws and sorting method based on possibility degree. Then, we propose generalized operators in trapezoidal fuzzy linguistic variables, including generalized trapezoidal fuzzy linguistic weighted averaging operator, generalized trapezoidal fuzzy linguistic ordered weighted averaging operator and generalized trapezoidal fuzzy linguistic hybrid averaging operator. Furthermore, the multi-attribute group decision-making method based on these operators is developed. Finally, an illustrative example is given to illustrate the decision-making steps, and to demonstrate its practicality and effectiveness. Keywords: trapezoidal fuzzy linguistic variables; generalized aggregation operators; multi-attribute group decision-making

1

Introduction

Due to the ambiguity of people’s thinking and the complexity of objective things, the attribute values of multi-attribute group decision-making problems cannot be always expressed as real numbers. Especially, for the qualitative information, they are easily expressed by the linguistic variables. Since Zadeh (1975) proposed the concept of linguistic variables firstly, the significant achievements about the theory research and applications of the linguistic variables have been made. Liang and Chen (2008) proposed trapezoid fuzzy linguistic weighted averaging operator(TFLWA), and gave the ranking method based on the possibility degree of trapezoidal fuzzy linguistic variables. Liu and Su (2010) proposed an extend TOPSIS method to solve the multiple attribute decision making problems with trapezoid fuzzy linguistic information. Han and Liu (2011) proposed an extend ⋆

The corresponding author

807

2

Peide Liu,

Xingying Wu

VIKOR method to solve the multiple attribute decision making problems with trapezoid fuzzy linguistic information. In recent years, information aggregation operators were attracting widespread concern in multi-attribute group decision-making. Xu (2007) defined the concept of triangular fuzzy linguistic variables, and proposed the fuzzy linguistic averaging operator(FLA), fuzzy linguistic weighted averaging operator(FLWA), fuzzy linguistic ordered weighted averaging operator(FLOWA), and induced fuzzy linguistic ordered weighted averaging operator(IFLOWA), the developed group decision making method based on these operators. Meng (2010) proposed the multiattribute decision-making methods based on trapezoidal fuzzy linguistic weighted geometric averaging operator and trapezoidal fuzzy linguistic hybrid harmonic averaging operator. Yager (2004) proposed generalized ordered weighted averaging operator by combining the generalized operators proposed by Dyckhoff and Pedrycz (1984) and OWA operator. The generalized aggregation operators can generalized arithmetic averaging operators, geometric averaging operators and harmonic averaging operators. However, they can only process the crisp numbers, and cannot deal with trapezoid fuzzy linguistic information. In this paper, we apply the generalized aggregation operators to multi-attribute decision-making problems in which attribute values take the form of trapezoidal fuzzy linguistic variables and propose a multi-attribute decision-making method.

2 2.1

Preliminaries Trapezoidal fuzzy linguistic variables

Definition 1 (Liang and Chen 2008). Let Se = (𝑠a , 𝑠b , 𝑠c , 𝑠d ) be trapezoidal fuzzy linguistic variables. Where 𝑠a , 𝑠b , 𝑠c and 𝑠d are elements of the extended linguistic set defined by Xu (2007), and 𝑎, 𝑏, c, 𝑑 ∈ R+ with 𝑎 ≤ 𝑏 ≤ c ≤ 𝑑. 𝑠a and 𝑠d indicate the lower and e respectively. Specially, if any two of 𝑎, 𝑏, c, 𝑑 are equal, then upper values of S, Se is reduced to a triangular fuzzy linguistic variable; if any three of 𝑎, 𝑏, c, 𝑑 are equal, then Se is reduced to an uncertain linguistic variable. f1 = (𝑠a , 𝑠b , 𝑠c , 𝑠d ) and S f2 = (𝑠e , 𝑠f , 𝑠g , 𝑠h ) be two trapezoidal fuzzy Let S e linguistic variables, and S = (𝛼, 𝛽, 𝛾, 𝜂) be a trapezoidal fuzzy number, 𝛼 < 𝛽 < 𝛾 < 𝜂, 𝛼, 𝛽, 𝛾, 𝜂 ∈ R+ . Then the operational laws of trapezoidal fuzzy linguistic variables are shown as follows: f1 ⊕ S f2 = (𝑠a , 𝑠b , 𝑠c , 𝑠d ) ⊕ (𝑠e , 𝑠f , 𝑠g , 𝑠h ) = (𝑠a+e , 𝑠b+f , 𝑠c+g , 𝑠d+h ) S

(1)

f1 ⊗ S f2 = (𝑠a , 𝑠b , 𝑠c , 𝑠d ) ⊗ (𝑠e , 𝑠f , 𝑠g , 𝑠h ) = (𝑠ae , 𝑠bf , 𝑠cg , 𝑠dh ) S

(2)

f1 = 𝜆(𝑠a , 𝑠b , 𝑠c , 𝑠d ) = (𝑠λa , 𝑠λb , 𝑠λc , 𝑠λd ), 𝜆 ≥ 0 𝜆S

(3)

f1 = (𝛼, 𝛽, 𝛾, 𝜂) ⊗ (𝑠a , 𝑠b , 𝑠c , 𝑠d ) = (𝑠αa , 𝑠βb , 𝑠γc , 𝑠ηd ) Se ⊗ S

(4)

808

Multi-attribute group decision-making

3

Definition 2 (Liang and Chen 2008). The distance between the two trapef1 , S f2 is defined as zoidal fuzzy linguistic variables S √ f1 , S f2 ) = [(𝑎 − 𝑒)2 + (𝑏 − 𝑓 )2 + (c − 𝑔)2 + (𝑑 − ℎ)2 ]/4 𝑑(S (5) f1 = (𝑠a , 𝑠b , 𝑠c , 𝑠d ) and S f2 = Definition 3 (Liang and Chen 2008). Let S (𝑠e , 𝑠f , 𝑠g , 𝑠h ) be two trapezoidal fuzzy linguistic variables, the possibility degree f1 > S f2 is defined as of S f1 > S f2 ) = min{max{ (c + 𝑑) − (𝑒 + 𝑓 ) , 0}, 1} 𝑝(S f1 ) + l𝑒𝑛(S f2 ) l𝑒𝑛(S

(6)

f1 ) = (c + 𝑑) − (𝑎 + 𝑏), l𝑒𝑛(S f2 ) = (𝑔 + ℎ) − (𝑒 + 𝑓 ). where, l𝑒𝑛(S So, for the given set of trapezoidal fuzzy linguistic variables Sei = (𝑠ia , 𝑠ib , 𝑠ic , 𝑠id ), 𝑖 = 1, 2, . . . , 𝑛, we can compare them in pairs, and build the matrix of possibility fj ), 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑛. Let degree = (𝑝ij )n∗n , where 𝑝ij = 𝑝(Sei > S ∑P n 𝑦i = j=1 𝑝ij and then we can get the sorting vector 𝑦 = (𝑦1 , 𝑦2 , . . . , 𝑦n ). Finally, we can rank the trapezoidal fuzzy linguistic variables Sei (𝑖 = 1, 2, . . . , 𝑛) according to 𝑦i . The bigger 𝑦i is, the bigger the trapezoidal fuzzy linguistic variable Sei is. 2.2

Generalized weighted averaging operators

Definition 4 (Yager 2004). The generalized weighted averaging operator of dimension 𝑛 is a mapping: Rn → R that ∑nhas an associated weight vector 𝑤 = (𝑤1 , 𝑤2 , . . . , 𝑤n )T such that 𝑤j > 0 and j=1 𝑤j = 1. (𝑎1 , 𝑎2 , . . . , 𝑎n ) is a set of real number. Furthermore, GW A(𝑎1 , 𝑎2 , . . . , 𝑎n ) = (

n ∑

1

𝑤j 𝑎λj ) λ

(7)

j=1

3 3.1

Generalized Operators in trapezoidal fuzzy linguistic variables Generalized trapezoidal fuzzy linguistic weighted averaging operator

Definition 5. The generalized trapezoidal fuzzy linguistic weighted averaging operator of dimension 𝑛 is 𝑎 mapping GTFLWA: Rn → R that has an ∑ weight vecn tor 𝑤 = (𝑤1 , 𝑤2 , . . . , 𝑤n )T of (𝑎e1 , 𝑎e2 , . . . , 𝑎f n ), such that 𝑤j ≥ 0 and j=1 𝑤j = 1. (𝑎e1 , 𝑎e2 , . . . , 𝑎f n ) is a set of trapezoidal fuzzy linguistic variables, and 𝑎ej can be expressed as [𝑎aj , 𝑎bj , 𝑎cj , 𝑎dj ], 𝑎aj , 𝑎bj , 𝑎cj , 𝑎dj are the elements of extended linguistic set. If

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GT F 𝐿W A(𝑎e1 , 𝑎e2 , . . . , 𝑎f n) = (

n ∑

n n ∑ ∑ 1 1 1 𝑤j e 𝑎λj ) λ = [( 𝑤j (𝑎aj )λ ) λ , ( 𝑤j (𝑎bj )λ ) λ ,

j=1

j=1

j=1

n 1 1 ∑ ( 𝑤j (𝑎cj )λ ) , ( 𝑤j (𝑎dj )λ ) λ ] 𝜆 j=1 j=1 n ∑

(8) Then GTFLWA is called the generalized trapezoidal fuzzy linguistic weighted averaging operator. Some special cases of the GTFLWA operator are shown as follows. (1) If 𝜆 = 1, then the GTFLWA operator reduces to the trapezoidal fuzzy linguistic weighted averaging (TFLWA) operator. n T F 𝐿W A(𝑎e1 , 𝑎e2 , . . . , 𝑎f n ) = ⊕j=1 𝑤j 𝑎ej = [

n ∑

𝑤j 𝑎aj ,

j=1

n ∑

𝑤j 𝑎bj ,

j=1

n ∑

𝑤j 𝑎cj ,

j=1

n ∑

𝑤j 𝑎dj ]

j=1

(9) (2) If 𝜆 → 0, then the GTFLWA operator reduces to the trapezoidal fuzzy linguistic weighted geometric averaging (TFLWGA) operator.

n wj T F 𝐿W GA(𝑎e1 , 𝑎e2 , . . . , 𝑎f =[ n ) = ⊗j=1 (𝑎ej )

n ∏

(𝑎aj )wj ,

j=1

n ∏

(𝑎bj )wj ,

j=1

n ∏

(𝑎cj )wj ,

j=1 n ∏

(𝑎dj )wj ]

j=1

(10) (3) If 𝜆 = −1, then the GTFLWA operator reduces to the trapezoidal fuzzy linguistic weighted harmonic averaging (TFLWHA) operator. T F 𝐿W HA(𝑎e1 , 𝑎e2 , . . . , 𝑎f n) =

1 1 w = [ ∑n ⊕nj=1 afjj j=1

wj aa j

, ∑n

1

wj j=1 abj

, ∑n

1

∑n

1

wj j=1 acj wj j=1 ad j

3.2

, (11) ]

Generalized trapezoidal fuzzy linguistic ordered weighted averaging operator

Definition 6. The generalized trapezoidal fuzzy linguistic ordered weighted averaging operator of dimension 𝑛 is a mapping GTFLOWA: Rn → R that ∑nhas an associated weight vector 𝜔 = (𝜔1 , 𝜔2 , . . . , 𝜔n )T such that 𝜔j ≥ 0 and j=1 𝜔j = 1.

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(𝑎e1 , 𝑎e2 , . . . 𝑎f n ) is a set of trapezoidal fuzzy linguistic variables, and 𝑎ej can be expressed as [𝑎aj , 𝑎bj , 𝑎cj , 𝑎dj ], 𝑎aj , 𝑎bj , 𝑎cj , 𝑎dj are the elements of extended linguistic set. If GT F 𝐿OW A(𝑎e1 , 𝑎e2 , . . . , 𝑎f n) = ( (

n ∑

n ∑

j=1 n ∑

1 𝜔j (𝑎ba(j) )λ ) λ , (

j=1

1

𝜔j e 𝑎λa(j) ) λ = [( 1 𝜔j (𝑎ca(j) )λ ) λ , (

j=1

n ∑ j=1 n ∑

1

𝜔j (𝑎aa(j) )λ ) λ , (12) 1 𝜔j (𝑎da(j) )λ ) λ ]

j=1

Then GTFLOWA is called the generalized trapezoidal fuzzy linguistic ordered weighted averaging operator. Where, (σ(1), σ(2), . . . , σ(𝑛)) is a permutation of (1, 2, . . . , 𝑛), such that e 𝑎σ(j−1) ≥ e 𝑎σ(j) , 𝑗 = 2, . . . , 𝑛. (1) If 𝜆 = 1, then the GTFLOWA operator reduces to the trapezoidal fuzzy linguistic ordered weighted averaging (TFLOWA) operator. n T F 𝐿OW A(𝑎e1 , 𝑎e2 , . . . , 𝑎f 𝑎σ(j) = [ n ) = ⊕j=1 𝜔j e

n ∑

𝜔j 𝑎aσ(j) ,

j=1

n ∑

𝜔j 𝑎bσ(j) ,

j=1

n ∑

𝜔j 𝑎cσ(j) ,

j=1 n ∑

𝜔j 𝑎dσ(j) ]

j=1

(13) (2) If 𝜆 → 0, then the GTFLOWA operator reduces to the trapezoidal fuzzy linguistic ordered weighted geometric averaging (TFLOWGA) operator. n T F 𝐿OW GA(𝑎e1 , 𝑎e2 , . . . , 𝑎f 𝑎σ(j) )ωj = [ n ) = ⊗j=1 (e

n ∏

(𝑎aσ(j) )ωj ,

n ∏

(𝑎bσ(j) )ωj ,

j=1 n ∏

j=1 n ∏

j=1

j=1

(𝑎cσ(j) )ωj ,

(𝑎dσ(j) )ωj ] (14)

(3) If 𝜆 = −1, then the GTFLOWA operator reduces to the trapezoidal fuzzy linguistic ordered weighted harmonic averaging (TFLOWHA) operator. T F 𝐿OW HA(𝑎e1 , 𝑎e2 , . . . , 𝑎f n) =

1 1 1 ωj = [ ∑ n ωj , ∑ n ωj , ⊕nj=1 eaσ(j) j=1 aa j=1 ab σ(j)

∑n

1

ωj j=1 acσ(j)

σ(j)

, ∑n

1

ωj j=1 ad σ(j)

(15)

]

GTFLOWA operator has the properties similar to GOWA operator shown as follows.

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′

′

Theorem 1 (Commutativity). If e 𝑎1 , e 𝑎2 , . . . , e 𝑎n is any permutation of (𝑎e1 , 𝑎e2 , . . . , 𝑎f n ), ′ ′ ′ then GT F 𝐿OW A(𝑎e1 , 𝑎e2 , . . . , 𝑎f ) = GT F 𝐿OW A(e 𝑎 , e 𝑎 , . . . , e 𝑎 ). n n 1 2 𝑎(e 𝑎 = [𝑎a , 𝑎b , 𝑎c , 𝑎d ]) Theorem 2 (Idempotency). If 𝑎ej (𝑎ej = [𝑎aj , 𝑎bj , 𝑎cj , 𝑎dj ]) = e for all 𝑗, then GT F 𝐿OW A(𝑎e1 , 𝑎e2 , . . . , 𝑎f 𝑎. n) = e ′ Theorem 3 (Monotonicity). If 𝑎ej ≤ 𝑎ej for all 𝑗, then GT F 𝐿OW A(𝑎e1 , 𝑎e2 , . . . , 𝑎f n) ≤ ′

′

′

GT F 𝐿OW A(𝑎e1 , 𝑎e2 , . . . , 𝑎f n ). Theorem 4 (Boundedness). Let e 𝑎− = minj [𝑎aj , 𝑎bj , 𝑎cj , 𝑎dj ], e 𝑎+ = maxj = a b c d − 𝑎 ≤ GT F 𝐿OW A(𝑎e1 , 𝑎e2 , . . . , 𝑎f 𝑎+ . [𝑎j , 𝑎j , 𝑎j , 𝑎j ] for all 𝑗, then e n) ≤ e 3.3

Generalized trapezoidal fuzzy linguistic hybrid averaging operator

Definition 7. Generalized trapezoidal fuzzy linguistic hybrid averaging operator of dimension 𝑛 is a mapping: Rn → R∑ has an associated weighted vector 𝜔 = n (𝜔1 , 𝜔2 , . . . , 𝜔n )T such that 𝜔j ≥ 0 and j=1 𝜔j = 1. (𝑎e1 , 𝑎e2 , . . . , 𝑎f n ) is a set of trapezoidal fuzzy linguistic variables, and 𝑎ej can be expressed as [𝑎aj , 𝑎bj , 𝑎cj , 𝑎dj ], 𝑎aj , 𝑎bj , 𝑎cj , 𝑎dj are the elements of extended linguistic set. If GT F 𝐿HA(𝑎e1 , 𝑎e2 , . . . , 𝑎f n) = ( (

n ∑ j=1

1 𝜔j (𝑏bσ(j) )λ ) λ , (

n ∑

j=1 n ∑

𝜔je𝑏λσ(j) ) λ = [( 1

1 𝜔j (𝑏cσ(j) )λ ) λ , (

j=1

n ∑ j=1 n ∑

1

𝜔j (𝑏aσ(j) )λ ) λ , (16) 1 𝜔j (𝑏dσ(j) )λ ) λ ]

j=1

Then GTFLHA is called the generalized trapezoidal fuzzy linguistic hybrid averaging operator. Where 𝑏ej = [𝑏aj , 𝑏bj , 𝑏cj , 𝑏dj ] = 𝑛𝑤j 𝑎ej , (δ(1), δ(2), . . . , δ(𝑛)) is a permutation of (1, 2, . . . , 𝑛), such that e𝑏δ(j−1) ≥ e𝑏δ(j) for all 𝑗 = 2, . . . , 𝑛, 𝑤 = (𝑤1 , 𝑤2 , . . . , 𝑤n )T ∑1, n is weight vector of 𝑎i (𝑖 = 1, 2, . . . , 𝑛), 𝑤j ≥ 0, j=1 𝑤j = 1. Some special cases of the GTFLHA operator are shown as follows. (1) If 𝜆 = 1, then the GTFLHA operator reduces to the trapezoidal fuzzy linguistic hybrid averaging (TFLHA) operator. n e T F 𝐿HA(𝑎e1 , 𝑎e2 , . . . , 𝑎f n ) = ⊕j=1 𝜔j 𝑏σ(j) = [

n ∑ j=1

𝜔j 𝑏aσ(j) ,

n ∑ j=1

𝜔j 𝑏bσ(j) ,

n ∑ j=1 n ∑

𝜔j 𝑏cσ(j) , 𝜔j 𝑏dσ(j) ]

j=1

(17)

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(2) If 𝜆 → 0, then the GTFLHA operator reduces to trapezoidal fuzzy linguistic hybrid geometric averaging (TFLHGA) operator. T F 𝐿HGA(𝑎e1 , 𝑎e2 , . . . , 𝑎f n) =

n ∏

e𝑏ωj = [ σ(j)

j=1

n ∏

(𝑏aσ(j) )ωj ,

j=1

n ∏

(𝑏bσ(j) )ωj ,

j=1

n ∏ j=1 n ∏

(𝑏cσ(j) )ωj , (𝑏dσ(j) )ωj ]

j=1

(18) (3) If 𝜆 = −1, then the GTFLHA operator reduces to the trapezoidal fuzzy linguistic hybrid harmonic averaging (TFLHHA) operator. T F 𝐿HHA(𝑎e1 , 𝑎e2 , . . . , 𝑎f n ) = ∑n

1 ωj

j=1 e bσ(j)

= [ ∑n

1 ωj

j=1 ba σ(j)

, ∑n

1 ωj

j=1 bb σ(j)

, ∑n

1

∑n

1

ωj j=1 bcσ(j) ωj j=1 bd σ(j)

, ]

(19)

4

Multi-attribute group decision-making method based on generalized operators in trapezoidal fuzzy linguistic variables

Let C = {C1 , C2 , . . . , Cm } be a discrete set of alternatives, and G = {G1 , G2 , . . . , Gn } be the set of attributes. 𝑤 = (𝑤1 , 𝑤2 , . . . , 𝑤n ) is∑the weighting vector of n the attribute Gj (𝑗 = 1, 2, . . . , 𝑛), where 𝑤j ∈ [0, 1], j=1 𝑤j = 1. There are 𝑝 decision-makers {𝑒1 , 𝑒2 , . . . , 𝑒p } that have weight vector (𝑑1 , 𝑑2 , . . . , 𝑑p ), 𝑑k ∈ ∑p kb kc kd 𝑎kij )m×n = [𝑎ka [0, 1], k=1 𝑑j = 1. Suppose that Ak = (e ij , 𝑎ij , 𝑎ij , 𝑎ij ]m×n is the k decision-making matrix, 𝑘 = 1, 2, . . . , 𝑝, where e 𝑎ij is the preference value, which takes the form of trapezoidal fuzzy linguistic variables given by the decisionmaker 𝑒k for the alternative Ci ∈ C with respect to the attribute Gj ∈ G. Then we use the attribute weights, the decision maker weights, and the attribute values to rank the alternatives. The steps of the decision making are shown as follows. Step 1 Combine the evaluation information of each decision maker to the group evaluation information. According to the different attribute values Ak (e 𝑎kij )m×n which were given by different experts under different alternative, we can integrate personal opinion into the result of group decision-making A = (e 𝑎ij )m×n using GTFLHA operator. Where e 𝑎ij = GT F 𝐿HA(e 𝑎1ij , e 𝑎2ij , . . . , e 𝑎pij )

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Step 2 Calculate the comprehensive evaluation value We can use the GTFLHA operator to calculate the comprehensive evaluation value. e 𝑎i = GT F 𝐿HA(e 𝑎i1 , e 𝑎i2 , . . . , e 𝑎in ) (21) Step 3 Build the matrix of possibility degree P = (𝑝ij )m∗m and get the sorting vector 𝑦 = (𝑦1 , 𝑦2 , . . . , 𝑦m ), where 𝑝ij = 𝑝(𝑎ei > 𝑎ej ). Step 4 Rank all alternatives. According to 𝑦i , we can give the ranking result of all alternatives. The bigger 𝑦i is, the better the alternative Ci is.

5

Illustrative examples

Suppose an organization plans to select ERP system. There are five potential ERP systems Ci (𝑖 = 1, 2, . . . , 5) as candidates. The company employs three external experts to aid this decision-making. The weight vector of the experts is D = (0.3, 0.4, 0.3). There are four attributes which are used to evaluate the alternatives: (1) function and technology G1 , (2) strategic fitness G2 , (3) vendor´ s ability G3 , (4) vendor´ s reputation G4 . The five possible alternatives are evaluated in trapezoidal fuzzy linguistic variables, and the decision making matrix Ak = (e 𝑎kij )5×4 are shown as Tables 1-3. The linguistic set used by decisionmakers is S = (𝑠1 , 𝑠2 , . . . , 𝑠9 ). (For convenience, the linguistic variable used by the number of subscript, for example, 𝑠2 is expressed by 2). Table 1. Decision Information Given by Expert 1

A1 A2 A3 A4 A5

G1 (2,3,4,5) (1,2,3,4) (3,3,4,5) (6,7,8,9) (4,5,6,7)

G2 (3,3,4,5) (4,5,6,7) (2,3,4,5) (4,5,6,7) (1,1,1,1)

G3 (4,5,6,7) (3,4,5,6) (4,5,6,7) (3,3,4,5) (2,3,4,5)

G4 (3,4,5,6) (4,5,6,7) (2,3,4,5) (2,3,4,5) (3,3,4,5)

Table 2. Decision Information Given by Expert 2

A1 A2 A3 A4 A5

G1 (3,3,4,5) (4,5,6,7) (2,3,4,5) (4,5,6,7) (1,1,1,1)

G2 (2,3,4,5) (1,2,3,4) (3,3,4,5) (6,7,8,9) (4,5,6,7)

G3 (3,4,5,6) (4,5,6,7) (2,3,4,5) (2,3,4,5) (3,3,4,5)

G4 (4,5,6,7) (3,4,5,6) (4,5,6,7) (3,3,4,5) (2,3,4,5)

The steps of decision making are shown as follows.

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Table 3. Decision Information Given by Expert 3

A1 A2 A3 A4 A5

G1 (4,5,6,7) (2,3,4,5) (6,7,8,9) (3,3,4,5) (1,2,3,4)

G2 (1,1,1,1) (3,3,4,5) (4,5,6,7) (2,3,4,5) (4,5,6,7)

G3 (2,3,4,5) (4,5,6,7) (3,3,4,5) (4,5,6,7) (3,4,5,6)

G4 (3,3,4,5) (3,4,5,6) (2,3,4,5) (2,3,4,5) (4,5,6,7)

Step 1 Calculate the group decision making matrix by (20) (In order to simplify the calculation, we adopt 𝜆 = 1 and 𝜔 = (0.25, 0.25, 0.25, 0.25). It means this is  a TFLWA operator), then we can get  (3.0, 3.6, 4.6, 5.6) (1.9, 2.3, 3.0, 3.7) (3.0, 4.0, 5.0, 6.0) (3.4, 4.1, 5.1, 6.1)  (2.5, 3.5, 4.5, 5.5) (2.5, 3.2, 4.2, 5.2) (3.7, 4.7, 5.7, 6.7) (3.3, 4.3, 5.3, 6.3)     A=  (3.5, 4.2, 5.2, 6.2) (3.0, 3.6, 4.6, 5.6) (2.9, 3.6, 4.6, 5.6) (2.8, 3.8, 4.8, 5.8) .  (4.3, 5.0, 6.0, 7.0) (4.2, 5.2, 6.2, 7.2) (2.9, 3.6, 4.6, 5.6) (2.4, 3.0, 4.0, 5.0)  (1.7, 2.4, 3.0, 3.6) (3.0, 3.7, 4.4, 5.1) (2.7, 3.3, 4.3, 5.3) (2.9, 3.6, 4.6, 5.6) Step 2 Calculate the comprehensive evaluation value, we adopt TFLWHM operator (i.e., in (21), 𝜆 = −1 and 𝜔 = (0.25, 0.25, 0.25, 0.25), and get 𝑎e1 = (2.5, 3.1, 4.0, 4.9), 𝑎e2 = (2.8, 3.7, 4.7, 5.7), 𝑎e3 = (3.1, 3.8, 4.8, 5.8), 𝑎e4 = (3.5, 4.3, 5.3, 6.3), 𝑎e5 = (2.4, 3.1, 3.8, 4.6). Step 3 Calculate the possibility degree 𝑝ij = 𝑝(𝑎ei > 𝑎ej )(𝑖 = 1, 2, 3, 4, 5) according to the (6), we can get  the matrix of possibility degree P = (𝑝ij )5∗5 . 0.5 0.4 0.3 0.2 0.6  0.6 0.5 0.5 0.3 0.7     P =  0.7 0.5 0.5 0.3 0.7 .  0.8 0.7 0.6 0.5 0.9  0.4 0.3 0.3 0.1 0.5 Then we can calculate the sorting vector, and get 𝑦 = (1.9, 2.7, 2.8, 3.5, 1.6). Step 4 Rank all alternatives. According to 𝑦i , we can give the ranking result of all alternatives shown as follows C4 ≻ C3 ≻ C2 ≻ C1 ≻ C5 . In order to prove the effectiveness of multi-attribute group decision-making method of generalized operators based on trapezoidal fuzzy linguistic variables, we adopt the extended TOPSIS method proposed by Meng (2010), and get the sorting result: C4 ≻ C3 ≻ C2 ≻ C1 ≻ C5 . Obviously, they have the same ranking results.

6

Conclusion

The generalized aggregation operators can generalized arithmetic averaging operators, geometric averaging operators and harmonic averaging operators. However, they can only process the crisp numbers, and cannot deal with trapezoid fuzzy linguistic information This paper proposes the generalized operators for trapezoid fuzzy linguistic variables, and applies them to the multi-attribute

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group decision-making problems in which attribute values are trapezoidal fuzzy linguistic variables and presents a decision making method. But this paper only considers the situation that attributes are independent and attribute weights are known, In the future, we will further study the situation that the attributes are correlated and attribute weights are unknown. Acknowledgments This paper is supported by the National Natural Science Foundation of China (No. 71271124 and 71071003), the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 10YJA630073 and 09YJA630088), the Natural Science Foundation of Shandong Province (No.ZR2011FM036), and Graduate Education Innovation Projects of Shandong Province (SDYY12065).

References 1. Dyckhoff, H.; Pedrycz, W. (1984). Generalized means as model of compensative connectives, Fuzzy Sets and Systems, 14:143-154. 2. Han, Z.S.; Liu, P.D. (2011). Multiple Attribute Decision Making Method Based on trapezoid Fuzzy Linguistic Variables, Fuzzy Systems and Mathematics, 25(3):119127. 3. Liang, X.C.; Chen, S.F. (2008). Multi-attribute decision-making method based on trapezoidal fuzzy linguistic variables, Journal of Southeast University, 24(4):478481 4. Liu P., Su, Y. (2010). The extended TOPSIS based on trapezoid fuzzy linguistic variables, Journal of Convergence Information Technology, 5(4):38-53. 5. Meng F.K. (2010). Research on the methods of multiple attribute decision making based on trapezoid fuzzy linguistic variables. A dissertation submitted for the degree of master. Shandong economic university. (In Chinese) 6. Xu Z.S. (2007). Group decision making with triangular fuzzy linguistic variables, Proceeding of Intelligent Data Engineering and Automated Learning ,LNCS, 4881:17-26. 7. Yager, R.R. (2004). Generalized OWA aggregation operators, Fuzzy Optimization and Decision Making, 3(1): 93-107. 8. Zadeh L.A. (1975). The Concept of a Linguistic Variable and Its application to approximate reasoning, Information Sciences, 8(1):199-249

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 817-825, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A new impulsive integro-differential inequality and its applications Diwang Linb∗

Huali Wanga , a

School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China b

Department of Mathematics and Physics, Xiamen University of Technology, Xiamen, 361024, China

Abstract In this paper, a new impulsive integro-differential inequality is presented, which can be widely applied to many fields in real world. Then, based on the equality, we analyze the exponential stability of some impulsive differential systems with discrete and continuously distributed delays, which illustrate the effectiveness of our results. Keywords: Impulsive; Distributed delays; Discrete delays; Exponential stability; Integrodifferential inequality

1. Introduction As is well known, the theory of differential and integral inequalities plays an important role in the qualitative and quantitative study of functional differential equations [1, 2, 4–6, 8–19]. These inequality methods give directly the estimate of the solutions of the systems, but their proofs are rather technical. In the last 40 years, various inequalities have been fully established and used for investigating dynamical behavior of differential equations. For example, in [7], Halanay inequality was first addressed. Since it is the effective tool for analyzing the stability of time delay system, many generalized Halanay inequalities were presented and attracted considerable attention in theoretical research and engineering applications (see [8,13,14,18] and relevant references therein). At the same time, impulsive differential inequalities also have a rapid development. In [10], Lakshmikantham et al. first described impulsive differential inequality and its application. Recently, it attracts many authors’ interesting and many advanced results have been reported [5, 12, 15, 17, 19]. Since time delays and impulses are ubiquitous in real world, the inequalities with impulses and time delays should be developed. ∗

817

Corresponding author. E-mail address: [email protected], [email protected]

H. Wang, D. Lin It should be notice that, so far, most existing works on delayed inequalities focus on the case of discrete time-delays. However, distributed time delays widely exist in many dynamical systems involving such areas as neural networks, population dynamics and so on. For instance, a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, it is desired to model them by introducing continuously distributed delays over a certain duration of time, such that the distant past has less influence compared to the recent behavior of the state. Consequently, when studying the natural modeling, both the discrete and distributed time delays should be taken into account. To the best of our knowledge, few authors studied the impulsive inequalities with discrete and distributed time delays [12]. Motivated by the above discussions, the main aim of this paper is to establish a new impulsive differential inequality with discrete and distributed delays. Then, we apply it to the exponential stability analysis of some impulsive functional differential systems with discrete and continuously distributed delays. The results extend and improve the earlier publications.

2. Main results In this section, we give a new impulsive integro-differential inequality with discrete and distributed delays. Theorem 2.1.

Let τ denote nonnegative constant and functions p(t), q(t), r(t) ∈

P C(R, R+) satisfy the following equation    D + m(t) ≤ −p(t)m(t) + q(t)

Z t−τ ≤s≤t

  m(t ) ≤ a m(t− ) + b m(t− − τ ), k ∈ Z , k k k + k k

σ

k(s)m(t − s)ds, t 6= tk , t ≥ t0 ,

sup m(s) + r(t) 0

(2.1) where 0 < σ ≤ +∞, ak , bk ∈ R, p(t) ≥ p0 > 0, k(·) ∈ P C([0, σ], R+) satisfies R∞ k(s)eη0 s ds < ∞ for some positive constant η0 > 0 in the case when σ = +∞. 0 Moreover, when σ = +∞, the interval [t−σ, t] is understood to be replaced by (−∞, t]. Assume that

Z

σ

(i) For any t ≥ t0 , 0 ≤ q(t) + r(t) constant.

k(s)ds ≤ qp(t), where 0 ≤ q < 1 is a 0 818

A new impulsive integro-differential inequality (ii) There exist constants M > 0, η > 0 such that n Y

max



1, ak + bk eλτ



≤ Meη(tn −t0 ) , n ∈ Z+ ,

k=1

where λ ∈ (0, η0) satisfies Z λ = inf {λ(t) : λ(t) − p(t) + q(t)e

λ(t)τ

σ

+ r(t)

t≥t0

k(s)eλ(t)s ds = 0}.

(2.2)

0

Then m(t) ≤ M m(t e 0 )e−(λ−η)(t−t0 ) , t ≥ t0 , where m(t e 0) =

sup

(2.3)

m(s).

t0 −max{σ,τ }≤s≤t0

Proof. First of all, we shall show that there exists some λ ∈ (0, η0 ) such that the inequality (2.2) holds. Denote

Z H(λ) = λ − p(t) + q(t)e

λτ

+ r(t)

σ

k(s)eλs ds.

(2.4)

0

By assumption (i), for any given fixed t ≥ t0 , we see that Z σ H(0) = −p(t) + q(t) + r(t) k(s)ds < 0, 0

and

Z ′

H (λ) = 1 + τ q(t)e

λτ

σ

+ r(t)

sk(s)eλs ds > 0. 0

Therefore, λ exists and λ ≥ 0. We have to prove λ > 0. Suppose this is not true. Let 1 ln q}, then there is t˜ > t0 such that λ(t˜) < ǫ and ǫ < min{(1 − 1q )p0 , − max{τ,σ} λ(t˜)τ

λ(t˜) − p(t˜) + q(t˜)e

Z

σ

˜

k(s)eλ(t)s ds = 0.

+ r(t˜) 0

Now we have

Z σ ˜ λ(t˜)τ ˜ ˜ ˜ ˜ 0 = λ(t) − p(t) + q(t)e + r(t) k(s)eλ(t)s ds 0 Z σ < ǫ − p(t˜) + q(t˜)eǫτ + r(t˜) k(s)eǫs ds 0

1 < ǫ − p(t˜) + p(t˜) q 1 = ǫ − (1 − )p(t˜) < 0, q 819

which is a contradiction.

(2.5)

H. Wang, D. Lin In order to prove that (2.3) holds for all t ≥ t0 , we need to show that for t ∈ [tk−1 , tk ), m(t) ≤ m(t e 0)

 k−1 Y

 {1, am + bm e } e−λ(t−t0 ) , λτ

m=0

where a0 = 1, b0 = 0. First, it is obvious that m(t) ≤ m(t e 0 ), for t ∈ [t0 − max{σ, τ }, t0 ]. Next we shall show that for t ∈ [t0 , t1 ) m(t) ≤ m(t e 0 )e−λ(t−t0 ) .

(2.6)

In order to do this, let (

m(t)eλ(t−t0 ) , t ≥ t0 ,

L(t) =

t0 − max{τ, σ} ≤ t ≤ t0 .

m(t),

Now we only need to show that L(t) ≤ m(t e 0 ), t ∈ [t0 , t1 ). Suppose on the contrary, then there exists some t ∈ [t0 , t1 ) such that L(t) > m(t e 0 ). Let t∗ = inf{t ∈ [t0 , t1 ), L(t) > m(t e 0 )}, then L(t∗ ) = m(t e 0 ), L(t) ≤ m(t e 0 ), t ∈ [t0 − max{τ, σ}, t∗], and D + L(t∗ ) ≥ 0. For simplicity, let m(θt∗ ) = supt∗ −τ ≤s≤t∗ m(s), θt∗ ∈ [t∗ − τ, t∗ ]. Calculating the upper right-hand Dini derivative of L(t) along the solution of (2.1), it can be deduced that D + L(t)|t=t∗ = D + m(t∗ )eλ(t −t0 ) + λm(t∗ )eλ(t −t0 )  ≤ − p(t∗ )m(t∗ ) + q(t∗ ) sup m(s) + r(t∗ ) ∗

∗

t∗ −τ ≤s≤t∗

 σ ∗ ∗ ∗ k(s)m(t − s)ds eλ(t −t0 ) + λm(t∗ )eλ(t −t0 )

Z 0



λ(t∗ −t0 )

(λ − p(t∗ ))m(t∗ ) + q(t∗ )

= e

Z



σ

sup

k(s)m(t − s)ds ∗

r(t ) ∗

0

Z

< [−q(t )e ∗

sup

σ

k(s)eλs ds]m(t∗ )eλ(t −t0 ) + q(t∗ ) 0 Z σ λ(t∗ −t0 ) ∗ λ(t∗ −t0 ) m(s)e + r(t )e k(s)m(t∗ − s)ds

λτ

∗

− r(t ) ∗

t∗ −τ ≤s≤t∗

0 λ(t∗ −t0 )

= −q(t∗ )eλτ m(t∗ )e Z λ(t∗ −t0 )

m(t )e ∗

m(s)+

t∗ −τ ≤s≤t∗

+ q(t∗ )

sup t∗ −τ ≤s≤t∗

σ

820 λs

∗

∗ −t ) 0

m(s)eλ(t Z

σ

λ(t∗ −t0 )

k(s)m(t∗ − s)ds

k(s)e ds + r(t )e 0

− r(t∗ )

0

A new impulsive integro-differential inequality

≤ −q(t∗ )eλτ L(t∗ ) + q(t∗ )eλ(t −θt∗ ) m(θt∗ )eλ(θt∗ −t0 ) − r(t∗ )L(t∗ ) Z σ Z σ λs ∗ k(s)e ds + r(t ) k(s)eλs L(t∗ − s)ds ∗

0

0

≤ −q(t∗ )eλτ L(t∗ ) + q(t∗ )eλτ L(θt∗ ) − r(t∗ )L(t∗ ) Z σ Z σ λs ∗ ∗ k(s)e ds + r(t )L(t ) k(s)eλs ds 0

(2.7)

0

= 0, which is a contradiction. So we have proven L(t) ≤ m(t e 0 ) for all t ∈ [t0 , t1 ). By (2.1) and (2.6), we note that − λ(t1 −t0 ) L(t1 ) = m(t1 )eλ(t1 −t0 ) ≤ [a1 m(t− 1 ) + b1 m(t1 − τ )]e

≤ (a1 + b1 eλτ )m(t e 0) ≤ max{1, a1 + b1 eλτ }m(t e 0 ). Next we claim that for t ∈ [t1 , t2 ), m(t) ≤ m(t e 0)

Y 1

max



1, am + bm eλτ



 e−λ(t−t0 )

m=0

(2.8)

= m(t e 0 ) max{1, a1 + b1 eλτ }e−λ(t−t0 ) , i.e. L(t) ≤ m(t e 0 ) max{1, a1 + b1 eλτ }. Suppose not, then there exists some t ∈ [t1 , t2 ) such that L(t) > m(t e 0 ) max{1, a1 + b1 eλτ }. Let tˆ = inf{t ∈ [t1 , t2 ), L(t) > m(t e 0 ) max{1, a1 + b1 eλτ }}, then L(tˆ) = m(t e 0 ) max{1, a1 + b1 eλτ }, L(t) ≤ m(t e 0 ) max{1, a1 + b1 eλτ }, t ∈ [t0 − max{τ, σ}, ˆt], and D + L(tˆ) ≥ 0.

(2.9)

Following proof similar to (2.7), we can calculate D + L(t)|t=tˆ < 0, which is a contradiction with (2.9). Hence, we get that (2.8) holds for all t ∈ [t1 , t2 ). Therefore, we get 821

L(t) ≤ m(t e 0 ) max{1, a1 + b1 eλτ }, t ∈ [t0 − max{τ, σ}, t2 ),

H. Wang, D. Lin and − λ(t2 −t0 ) L(t2 ) = m(t2 )eλ(t2 −t0 ) ≤ [a2 m(t− 2 ) + b2 m(t2 − τ )]e

≤ m(t e 0 ) max{1, a1 + b1 eλτ }(a2 + b2 eλτ ) Y  2  λτ ≤ m(t e 0) max 1, am + bm e .

(2.10)

m=1

Using the same argument as (2.8), together with (2.10), we can get for t ∈ [t2 , t3 ), Y  2  −λ(t−t0 ) λτ max 1, am + bm e e . m(t) ≤ m(t e 0) m=0

By the method of induction, we can conclude that for t ∈ [tk−1 , tk ], k ∈ Z+ ,   k−1 Y λτ max{1, am + bm e } e−λ(t−t0 ) . m(t) ≤ m(t e 0) m=0

Applying condition (ii), m(t) ≤ M m(t e 0 )e−(λ−η)(t−t0 ) , t ≥ t0 . So (2.3) holds. The proof of Theorem 2.1 is complete. Remark 2.1. It should be noted that we only require that the function k(·) satisfies the assumption: k(·) ∈ P C([0, σ], R+ ) is integrable if σ < +∞. Remark 2.2. In the case p(t) = p, q(t) = q, r(t) = r, the above result reduces to the inequality in [12]. Remark 2.3. Assume that k(s) = 0, we can easily obtain the following result. Let τ denote nonnegative constant and functions p(t), q(t), r(t) ∈

Corollary 2.1.

P C(R, R+) satisfy the following equation   D + m(t) ≤ −p(t)m(t) + q(t) 

sup m(s), t 6= tk , t ≥ t0 , t−τ ≤s≤t

− m(tk ) ≤ ak m(t− k ) + bk m(tk − τ ), k ∈ Z+ ,

where ak , bk ∈ R, p(t) ≥ p0 > 0. Assume that (i) For any t ≥ t0 , 0 ≤ q(t) ≤ qp(t), where 0 ≤ q < 1 is a constant. (ii) There exist constants M > 0, η > 0 such that n Y

k=1

max



1, ak + bk eλτ



822≤

Meη(tn −t0 ) , n ∈ Z+ ,

A new impulsive integro-differential inequality where λ > 0 satisfies λ = inf {λ(t) : λ(t) − p(t) + q(t)eλ(t)τ = 0}. t≥t0

Then m(t) ≤ M m(t e 0 )e−(λ−η)(t−t0 ) , t ≥ t0 , where m(t e 0) =

sup

m(s).

t0 −τ ≤s≤t0

3. Applications In this section, we shall apply the above impulsive delayed inequality to study the exponential stability for the following nonlinear impulsive functional differential system with diacrete and continuously distributed delays. Consider the following system    x(t) ˙ = −p(t)x(t) + q(t)  

Z

σ

k(s)x(t − s)ds, t 6= tk , t ≥ t0 ,

sup x(s) + r(t) 0

t−τ ≤s≤t

k ∈ Z+ ,

− x(tk ) = ak x(t−  k ) + bk x(tk − τ ),    xt0 = ϕ,

t0 > 0, (3.1)

where ϕ ∈ Ψ, Ψ is an open set in P C([− max{τ, σ}, 0], Rn). For each t ≥ t0 , xt ∈ Ψ is defined by xt (s) = x(t + s), s ∈ [− max{τ, σ}, 0]. Define P CB(t) = {xt ∈ Ψ | xt is bounded }. For ψ ∈ P CB(t), the norm of ψ is defined by ||ψ|| =

sup

|ψ(θ)|.

− max{τ,σ}≤θ≤0

p(t), q(t), r(t) ∈ P C(R, R+ ), τ > 0, 0 < σ ≤ +∞, ak , bk ∈ R+ , p(t) ≥ p0 > 0, k(·) ∈ R∞ P C([0, σ], R+) satisfies 0 k(s)eη0 s ds < ∞ for some positive constant η0 > 0 in the case when σ = +∞. Moreover, when σ = +∞, the interval [t − σ, t] is understood to be replaced by (−∞, t]. Definition 3.1. ( [5]). Let x(t) = x(t, t0 , ϕ) be a solution of system (3.1) through (t0 , ϕ). Then the trivial solution of (3.1) is said to be globally exponentially stable if for any t0 > 0, there exist contants λ > 0 and M ≥ 1 such that kx(t, t0 , ϕ)k ≤ Mkϕke−λ(t−t0 ) , t ≥ t0 . Theorem 3.1. Suppose (i) For any t ≥ t0 , 0 ≤ q(t) + r(t)

Z

σ 823 0

k(s)ds ≤ qp(t), where 0 ≤ q < 1 is a

H. Wang, D. Lin constant. (ii) There exist constants M ≥ 1, η > 0 such that n Y

max



1, ak + bk eλτ



≤ Meη(tn −t0 ) , n ∈ Z+ ,

k=1

where λ ∈ (0, η0) satisfies Z λ = inf {λ(t) : λ(t) − p(t) + q(t)e

λ(t)τ

σ

+ r(t)

k(s)eλ(t)s ds = 0}.

t≥t0

0

Then the solution of system (3.1) is globally exponentially stable. Proof. Choose a Lyapunov function for system (3.1) as V (t) = |x(t)|. If t 6= tk , calculating D + V (t) along the solution of (3.1), we can easily get D V (t) ≤ −p(t)V (t) + q(t)Ve (t) + r(t)

Z

+

σ

k(s)V (t − s)ds, . 0

where Ve (t) =

sup

. In addition,

t−max{τ,σ}≤s≤t − − − V (tk ) = |x(tk )| ≤ ak |x(t− k )| + bk |x(tk − τ )| = ak V (tk ) + bk V (tk − τ ).

Then, by Theorem 2.1, we have kx(t, t0 , ϕ)k ≤ Mkϕke−(λ−η)(t−t0 ) , t ≥ t0 . Therefore,the trivial solution of system (3.1) is globally exponentially stable. Remark 3.1. It should be noted that when the variable coefficient of (3.1) satisfy the conditions in Theorem 3.1, we can obtain the exponential stability. To some extent, it illustrates the effectiveness of our results.

References [1] A. Anokhin, L. Berezansky, E. Braverman, Exponential stability of linear delay impulsive differential equations, Journal of Mathematical Analysis and Applications 193 (1995) 923–941. [2] D. D. Bainov, P. S. Simenov, Systems with824 Impulse Effect: Stability Theory and Applications, Chichester, Ellis Horwood Limited, 1989.

A new impulsive integro-differential inequality [3] C.T.H. Baker, A. Tang, Generalized Halanay inequalities for Volterra functional differential equations and discretised versions, in: C. Corduneanu, I.W. Sandberg (Eds.), Volterra Equations and Applications, Gordon and Breach, Amsterdam, 1999. [4] L. Berezansky, L. Idels, Exponential stability of some scalar impulsive delay differential equation, Communications Application Mathematics Analysis 2 (1998) 301–309. [5] X. Fu, X. Li, Global exponential stability and global attractivity of impulsive Hopfield neural networks with time delays, Journal of Computational and Applied Mathematics 231 (2009) 187–199. [6] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. [7] A. Halanay, Differential Equations, Stability, Oscillation, Time lags, NewYork, Academic Press, 1996. [8] M. Jiang, Y. Shen, X. Liao, On the global exponential stability for functional differential equations, Communications in Nonlinear Science and Numerical Simulation 10 (2005) 705–713. [9] G. Ladas, P. Stavroulakis, On delay differential inequalities of first order, Funkcialaj Ekvacioj 25 (1982) 105–113. [10] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [11] V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, Academic press, New York, 1969. [12] X. Li, Existence and global exponential stability of periodic solution for impulsive Cohen– Grossberg-type BAM neural networks with continuously distributed delays, Applied Mathematics and Computation 215 (2009) 292–307. [13] J. Nieto, Differential inequalities for functional perturbations of first-order ordinary differential equations, Applied Mathematics Letters 15 (2002) 173–179. [14] H. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, Journal of Mathematical Analysis and Applications 270 (2002) 143–149. [15] H. Wang, C. Ding, A new nonlinear impulsive delay differential inequality and its applications, Journal of Inequalities and Applications 2011, 2011:11. [16] D. Xu, S. Long, Attracting and quasi-invariant sets of non-autonomous neural networks with delays, Neurocomputing 77 (2012) 222C-228. [17] L. Xu, D. Xu, Exponential stability of nonlinear impulsive neutral integro-differential equations, Nonlinear Analysis 69 (2008) 2910–2923. [18] D. Xu, L. Xu, New results for studying a certain class nonlinear delay differential systems, IEEE Transactions on Automatic Control 55 (2010) 1641–1645. [19] D. Xu, Z. Yang, Impulsive delay differential inequality and stability of neural networks, 825 Journal of Mathematical Analysis and Applications 305 (2005) 107–120.

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 826-832, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

On the convergence of a modified q-Gamma operators Qing-Bo Caia,b,∗and Xiao-Ming Zengb a

School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China b

Department of Mathematics, Xiamen University, Xiamen 361005, China E-mail: [email protected]; [email protected]

Abstract. In this paper, we introduce a modified q-Gamma operators based on the concept of q-integral, these operators preserve not only constant functions but also linear functions. We establish direct and local approximation theorems of the operators and obtain the estimates on the rate of convergence and weighted approximation of the operators. 2000 Mathematics Subject Classification: 41A10, 41A25, 41A36. Key words and phrases: q-integral, modified q-Gamma operators, weighted approximation, rate of convergence.

1

Introduction

In recent years, the applications of q-calculus in the approximation theory is one of the main area of research. After q-Bernstein polynomials were introduced by Phillips [9] in 1997, many researchers have studied in this field, we mention some of them as [2], [5], [9, 10]. In 2007, Karsli [7] introduced and estimated the rate of convergence for functions with derivatives of bounded variation on [0, ∞) of a new Gamma type operators as follows: Ln (f ; x) =

(2n + 3)!xn+3 n!(n + 2)!

Z 0

∞

tn f (t)dt, (x + t)2n+4

x > 0.

(1)

¡ ¢ x From Lemma 1 in [7], we have Ln (1; x) = 1, Ln (t; x) = x− n+2 , Ln t2 ; x = x2 . Apparently, these operators reproduce constant functions but not linear functions. It seems there have no papers mentioned about the q analogue of these operators defined in (1). In present paper, we will introduce a modified q-Gamma operators G∗n,q based on the concept of q-integral which will be defined in (4). The advantage of these ∗

Corresponding author.

826

Q. CAI and X. ZENG new operators is that they reproduce not only constant functions but also linear functions. We will establish direct and local approximation theorems of the operators G∗n,q and give the estimates on the rate of convergence and weighted approximation of these operators. Before introducing the operators, we mention certain definitions based on q-integers, detail can be found in [4, 6]. For any fixed real number 0 < q ≤ 1 and each nonnegative integer k, we denote q-integers by [k]q , where ( [k]q =

1−q k 1−q ,

q= 6 1; q = 1.

k,

Also q-factorial and q-binomial coefficients are defined as follows: ( [k]q ! =

"

#

[k]q [k − 1]q ...[1]q , k = 1, 2, ...; , 1, k = 0,

n k

= q

[n]q ! , [k]q ![n − k]q !

(n ≥ k ≥ 0).

The q-improper integrals are defined as (see [8]) Z

∞/A

0

µ n¶ n ∞ X q q , A > 0, f (x)dq x = (1 − q) f A A −∞

provided the sums converge absolutely. The q-Beta integral is defined by Z Bq (t; s) = K(A; t)

∞/A

xt−1 dq x, (1 + x)t+s q

0

(2)

¡ ¢t Q −1 1 where K(x; t)= x+1 xt 1 + x1 q (1 + x)1−t (a + q j b), τ > 0. and (a + b)τq = τj=0 q In particular for any positive integer m, n K(x, n) = q

n(n−1) 2

, K(x, 0) = 1 and Bq (m; n) =

Γq (m)Γq (n) , Γq (m + n)

(3)

where Γq (t) is the q-Gamma function satisfies the following functional equations: Γq (t + 1) = [t]q Γq (t),

Γq (1) = 1.

(see [2]). For f ∈ C[0, ∞), q ∈ (0, 1) and n ∈ N, we introduce a modified q-Gamma operators as follows: G∗n,q (f ; x) =

³ n+1 ´n+3 n(n+1) q [n+2]q q 2 Z [2n + 3]q ! x [n+1]q [n]q ![n + 2]q !

0

827

∞/A

³

tn q n+1 [n+2]q [n+1]q x

+t

´2n+4 f (t)dq t. (4) q

A MODIFIED q-GAMMA OPERATORS

2

Some preliminary results In order to obtain the approximation properties, We need the following lemmas:

Lemma 2.1. For any k ∈ N, k ≤ n + 2 and q ∈ (0, 1), we have G∗n,q (tk ; x)

[n + k]q ![n − k + 2]q ! k−k2 = q 2 [n]q ![n + 2]q !

µ

[n + 2]q x [n + 1]q

¶k .

(5)

Proof. Using the properties of q-Beta integral, we have ³ ´ G∗n,q tk ; x ³ n+1 ´n+3 n(n+1) q [n+2]q [2n + 3]q ! x q 2 Z ∞/A [n+1]q tn+k = ³ n+1 ´2n+4 dq t [n]q ![n + 2]q ! q [n+2]q 0 x + t [n+1]q q

n(n+1) 2

=

[2n + 3]q !q [n]q ![n + 2]q !

=

[2n + 3]q !q 2 [n]q ![n + 2]q !

n(n+1)

= =

n(n+1) 2

Z

∞/A

tn+k ´ ´2n+4 dq t n+1 ³ [n+1]q t q n+1 [n+2]q 0 x 1 + n+1 [n+1]q q [n+2]q x q ³ ´n+k [n+1]q t µ n+1 ¶k Z ∞/A ¶ µ n+1 q [n + 2]q x [n + 1]q t q [n+2]q x ³ ´2n+4 dq [n + 1]q q n+1 [n + 2]q x [n+1]q t 0 1 + qn+1 [n+2] qx ³

µ

q

¶k

q n+1 [n + 2]q x Bq (n + k + 1; n − k + 3) [n + 1]q K(A; n + k + 1) ¶ µ k [n + k]q ![n − k + 2]q ! k−k2 [n + 2]q x q 2 . [n]q ![n + 2]q ! [n + 1]q [2n + 3]q !q [n]q ![n + 2]q !

Lemma 2.1 is proved. Lemma 2.2. The following equalities hold: [n + 2]2q 2 G∗n,q (1; x) = 1, G∗n,q (t; x) = x, G∗n,q (t2 ; x) = x , q[n + 1]2q à ! ¡ ¢ [n + 2]2q . ∗ 2 Gn,q (t − x) ; x = − 1 x2 = αn,q (x). q[n + 1]2q

(6) (7)

¡ ¢ Proof. From Lemma 2.1, take k = 0, 1, 2, we get (6). Since G∗n,q (t − x)2 ; x = G∗n,q (t2 ; x)− 2xG∗n,q (t; x) + x2 , and using (6), we obtain (7) easily. Remark 2.3. Let n ∈ N and x ∈ [0, ∞), then for every q ∈ (0, 1), by Lemma 2.2, we have G∗n,q (t − x; x) = 0.

828

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Q. CAI and X. ZENG

3

Local approximation

In this section we establish direct and local approximation theorems in connection with the operators G∗n,q (f ; x). We denote the space of all real valued continuous bounded functions f defined on the interval [0, ∞) by CB [0, ∞). The norm || · || on the space CB [0, ∞) is given by ||f || = sup {|f (x)| : x ∈ [0, ∞)}. Further let us consider Peetre’s K−functional: K2 (f ; δ) = inf

©

g∈W 2

ª ||f − g|| + δ||g 00 || ,

where δ > 0 and W 2 = {g ∈ CB [0, ∞) : g 0 , g 00 ∈ CB [0, ∞)}. For f ∈ CB [0, ∞), the modulus of continuity of second order is defined by ω2 (f ; δ) = sup

sup |f (x + 2h) − 2f (x + h) + f (x)|,

0 0 such that ³ √ ´ K2 (f ; δ) ≤ Cω2 f ; δ , δ > 0.

(9)

Our first result is a direct local approximation theorem for the operators G∗n,q (f ; x). Theorem 3.1. For q ∈ (0, 1), x ∈ [0, ∞) and f ∈ CB [0, ∞), we have à |G∗n,q (f ; x)

− f (x)| ≤ Cω2

f;

r

αn,q (x) 2

! ,

(10)

where αn,q (x) is defined in (7) and C is a positive constant. Proof. Let g ∈ W 2 , by Taylor’s expansion, we have Z 0

g(t) = g(x) + g (x)(t − x) +

t

(t − u)g 00 (u)du, x, t ∈ [0, ∞).

x

Using (6), we get µZ G∗n,q (g; x)

= g(x) +

G∗n,q

t

¶ (t − u)g (u)du; x . 00

x

Thus, we have |G∗n,q (g; x)

¯ ¯ ¶ µZ t ¶¯ µ¯Z t ¯ ∗ ¯ ¯ ¯ 00 ∗ 00 ¯ ¯ ¯ − g(x)| = ¯Gn,q (t − u)g (u)du; x ¯ ≤ Gn,q ¯ (t − u)|g (u)|du¯¯ ; x x

x

¡ ¢ ≤ G∗n,q (t − x)2 ; x ||g 00 || = αn,q (x)||g 00 ||,

829

(11)

A MODIFIED q-GAMMA OPERATORS

where αn,q (x) is defined in (7). On the other hand, using Lemma 2.2, we have ³ n+1 ´n+3 n(n+1) q [n+2]q [2n + 3]q ! x q 2 Z ∞/A [n+1]q tn |f (t)| ∗ |Gn,q (f ; x)| ≤ ³ n+1 ´2n+4 dq t ≤ ||f ||. [n]q ![n + 2]q ! q [n+2]q 0 x + t [n+1]q q

(12)

Now (11) and (12) imply |G∗n,q (f ; x) − f (x)| ≤ |G∗n,q (f − g; x) − (f − g)(x)| + |G∗n,q (g; x) − g(x)| ≤ 2||f − g|| + αn,q (x)||g 00 ||. Hence, taking infimum on the right hand side over all g ∈ W 2 , we get µ ¶ αn,q (x) ∗ . |Gn,q (f ; x) − f (x)| ≤ 2K2 f ; 2 By (9), for every q ∈ (0, 1), we have à |G∗n,q (f ; x) − f (x)| ≤ Cω2

f;

r

αn,q (x) 2

! .

This completes the proof of Theorem 3.1. Remark 3.2. Let q = {qn } be a sequence satisfying 0 < qn < 1 and limn qn = 1, we have limn αn,qn = 0, this gives us the pointwise rate of convergence of the operators G∗n,qn (f ; x) to f (x).

4

Rate of convergence and Weighted approximation

Let Bx2 [0, ∞) be the set of all functions f defined on [0, ∞) satisfying the condition |f (x)| ≤ Mf (1+x2 ), where Mf is a constant depending only on f . We denote the subspace of all continuous functions belonging to Bx2 [0, ∞) by Cx2 [0, ∞). Also, let Cx∗2 [0, ∞) be f (x) the subspace of all functions f ∈ Cx2 [0, ∞), for which limx→∞ 1+x 2 is finite. The norm on (x)| Cx∗2 [0, ∞) is ||f ||x2 = supx∈[0,∞) |f . We denote the usual modulus of continuity of f on 1+x2 the closed interval [0, a], (a > 0) by

ωa (f ; δ) = sup

sup |f (t) − f (x)|.

|t−x|≤δ x,t∈[0,a]

Obviously, for function f ∈ Cx2 [0, ∞), the modulus of continuity ωa (f ; δ) tends to zero. Theorem 4.1. Let f ∈ Cx2 [0, ∞), q ∈ (0, 1) and ωa+1 (f ; δ) be the modulus of continuity on the finite interval [0, a + 1] ⊂ [0, ∞), where a > 0. Then we have µ q ¶ ∗ 2 ||Gn,q (f ) − f ||C[0,a] ≤ 6Mf (1 + a )αn,q (a) + 2ωa+1 f ; αn,q (a) , (13) where αn,q (a) is defined in (7).

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Q. CAI and X. ZENG

Proof. For x ∈ [0, a] and t > a + 1, we have |f (t) − f (x)| ≤ Mf (2 + x2 + t2 ) ≤ Mf [2 + 3x2 + 2(t − x)2 ], hence, we obtain |f (t) − f (x)| ≤ 6Mf (1 + a2 )(t − x)2 .

(14)

For x ∈ [0, a] and t ≤ a + 1, we have µ |f (t) − f (x)| ≤ ωa+1 (f ; |t − x|) ≤

|t − x| 1+ δ

¶ ωa+1 (f ; δ), δ > 0.

(15)

From (14) and (15), we get µ

|t − x| |f (t) − f (x)| ≤ 6Mf (1 + a )(t − x) + 1 + δ 2

¶

2

ωa+1 (f ; δ).

(16)

For x ∈ [0, a] and t ≥ 0, by Schwarz’s inequality and Lemma 2.2, we have |G∗n,q (f ; x) − f (x)| ≤ G∗n,q (|f (t) − f (x)|; x) µ ¶ ¡ ¢ 1q ∗ 2 ∗ 2 2 ≤ 6Mf (1 + a )Gn,q (t − x) ; x + ωa+1 (f ; δ) 1 + Gn,q ((t − x) ; x) δ µ ¶ q 1 2 = 6Mf (1 + a )αn,q (a) + ωa+1 (f ; δ) 1 + αn,q (a) , δ p where αn,q (a) is defined in (7). By taking δ = αn,q (a), we get the assertion of Theorem 4.1. Finally, we will discuss the weighted approximation theorem. Theorem 4.2. Let the sequence q = {qn } satisfy 0 < qn < 1 and qn → 1 as n → ∞, for f ∈ Cx∗2 [0, ∞), we have lim ||G∗n,qn (f ) − f ||x2 = 0. (17) n→∞

Proof. By using the Korovkin theorem in [3], we see that it is sufficient to verify the following three conditions lim ||G∗n,qn (tv ; x) − xv ||x2 , v = 0, 1, 2.

n→∞

Since G∗n,qn (1; x) = 1 and G∗n,qn (t; x) = x, (17) holds true for v = 0 and v = 1. For v = 2, we have |G∗n,qn (t2 ; x) − x2 | 1 + x2 x∈[0,∞) ! Ã [n + 2]2qn x2 − 1 sup = 2 qn [n + 1]2qn x∈[0,∞) 1 + x

||G∗n,qn (t2 ; x) − x||x =

≤

sup

[n + 2]2qn − 1. qn [n + 1]2qn

831

(18)

A MODIFIED q-GAMMA OPERATORS [n+2]2

Since limn→∞ qn = 1, we get limn→∞ qn [n+1]qn2 − 1 = 0, so the condition of (18) holds for qn v = 2 as n → ∞, then the proof of Theorem 4.2 is completed.

Acknowledgement This work is supported by the Project of the Educational Office of Fujian Province of China (Grant No. JK2011041), the National Natural Science Foundation of China (Grant No. 61170324) and the Natural Science Foundation of Fujian Province of China (Grant No. 2010J01012).

References [1] R. A. DeVore, G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993. [2] A. De Sole, V. G. Kac, On integral representation of q-gamma and q-beta functions, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, (9)Mat.Appl., 16(1)(2005), 11-29. [3] A. D. Gadjiev, Theorems of the type of P. P. Korovkin type theorems, Math. Zametki, 20(5)(1976), 781-786, (English Translation, Math. Notes 20(5-6)(1976), 996-998). [4] G. Gasper, M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its applications, Cambridge University press, Cambridge, UK., 35, 1990. [5] V. Gupta, T. Kim, On the rate of approximation by q modified Beta operators, J. Math. Anal. Appl., 377(2011), 471-480. [6] V. G. Kac, P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. [7] H. Karsli, Rate of convergence of a new Gamma Type Operators for functions with derivatives of bounded variation, Math. Comput. Modelling, 45(5-6)(2007), 617-624. [8] T. H. Koornwinder, q-Special Functions, a Tutorial, in: M. Gerstenhaber, J. Stasheff(Eds.), Deformation Theory and Quantum Gruoups with Applications to Mathematial Physics, Contemp. Math., 134 (1992), Amer. Math. Soc., 1992. [9] G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4(1997), 511-518. [10] H. Wang, Korovkin-type theorem and application, J. Approx. Theory, 132(2)(2005), 258-264.

832

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 833-843, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Approximation by Complex Schurer-Stancu Operators in Compact Disks Mei-Ying Ren1,∗, Xiao-Ming Zeng2,∗ 1

Department of Mathematics and Computer Science, Wuyi University, Wuyishan 354300, China

2

Department of Mathematics, Xiamen University, Xiamen 361005, Chnia E-mail: [email protected],

[email protected]

Abstract. In this paper we introduce a class of complex Schurer-Stancu operators and study the approximation properties of these operators. We obtain the order of simultaneous approximation and a Voronovskaja-type result with quantitative estimate for these complex Schurer-Stancu operators attached to analytic functions on compact disks. More important, our results show the overconvergence phenomenon for these complex operators. Keywords: complex Schurer-Stancu operators; linear positive operators; simultaneous approximation; Voronovskaja-type result Mathematical subject classification: 30E10, 41A25, 41A35

1. Introduction In 1986 some approximation properties of complex Bernstein polynomials in compact disks were initially studied by Lorentz [1]. A Voronovskaja-type result with quantitative estimate for complex Bernstein polynomials in compact disks was obtained by Gal [2]. Also, in [3-10] similar results for complex BernsteinKantorovich polynomials, Bernstein-Stancu polynomials, Kantorovich-Schurer polynomials, Kantorovich-Stancu polynomials, Bernstein-Durrmeyer polynomials and genuine Durrmeyer-Stancu polynomials were obtained. The goal of this paper is to obtain approximation results for the complex Schurer-Stancu operators (introduced and studied in the case of real variable in [11]) defined for any fixed p ∈ {0, 1, 2, ...}, 0 ≤ α ≤ β by (α,β) Sn,p (f ; z) =

n+p X j=0

n+p j

 z j (1 − z)n+p−j f (

j+α ), z ∈ C. n+β

It is clearly, in the case p = α = β = 0, these complex operators become the complex Bernstein operators. ∗ Corresponding

Author Xiao-Ming Zeng and Mei-Ying Ren.

833

REN and ZENG

Note that, all these results put in evidence the overconvergence phenomenon for the complex Schurer-Stancu operators, that is the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.

2. Preliminaries In the sequel, we shall need the following auxiliary results. S S Lemma 1. For fixed p ∈ N {0}, 0 ≤ α ≤ β, let ek (z) = z k , k ∈ N {0}, z ∈ C and 1 ≤ r. Then for all |z| ≤ r and n ∈ N, we have 1 rk 2k(k − 1 + 2β) [(p + 1)r]k + [(p + 1)r]k − . n+β+p n+β n+β

(α,β) |Sn,p (ek ; z) − ek (z)| ≤

(α,β)

Proof. To estimate |Sn,p (ek ; z) − ek (z)|, for fixed n ∈ N, we consider two possible cases: (1) 0 ≤ k ≤ n + p ; (2) k > n + p. Denoting by ∆k the finite difference of order k, as in the case of the classical Berstein polynomials, we can easily deduce the representation formula (α,β) Sn,p (f ; z) =

n+p X j=0

n+p j

 ∆j1/(n+β) f (α/(n + β))ej (z).

(α,β)

Case (1). If k = 0, we have Sn,p (ek ; z)−e  k (z) = 0,  therefore, let us suppose n + p (p,α,β) that 1 ≤ k ≤ n + p. Denote Cn,j,k = ∆j1/(n+β) ek (α/(n + β)), j   n+p (p,α,β) j = 0, 1, ...n + p, then Cn,j,k = [α/(n + β), (α + 1)/(n + β), ..., (α + j n+p P (p,α,β) (α,β) j)/(n + β); ek ](j!)/(n + β)j , Sn,p (ek ; z) = Cn,j,k ej (z). j=0

(p,α,β)

Since ek is convex of any order, it follows that all Cn,j,k property of the usual divided difference and n+p k k P (p,α,β) P (p,α,β) Cn,j,k = Cn,j,k = (n+α+p) (n+β)k . j=0

(α,β) Sn,p (f ; 1)

≥ 0. By using the

= f ( n+α+p n+β ), we get

j=0

For any |z| ≤ r with 1 ≤ r, we can write X k (p,α,β) (α,β) |Sn,p (ek ; z) − ek (z)| = Cn,j,k ej (z) − ek (z) j=0

=

(p,α,β) |(Cn,k,k

− 1)ek (z) +

k−1 X j=0

(p,α,β)

Cn,j,k ej (z)|

(n + p − k + 1)(n + p − k + 2) · · · (n + p) ≤ − 1 rk k (n + β)

834

COMPLEX SCHURER-STANCU OPERATORS 

 (n + α + p)k (n + p − k + 1)(n + p − k + 2) · · · (n + p) k + − r (n + β)k (n + β)k   (n + α + p)k (n + p − k + 1)(n + p − k + 2) · · · (n + p) k ≤2 − r (n + β)k (n + β)k   (n + α + p)k − 1 rk . + (n + β)k In view of 0 ≤ α ≤ β and 1 −

k Q

xj ≤

j=1

k P

(1 − xj ), 0 ≤ xj ≤ 1, j = 1, ..., k,

j=1

we get 

 ≤

(n + p − k + 1)(n + p − k + 2) · · · (n + p) (n + α + p)k − (n + β)k (n + β)k

n+β+p n+β



k X  k  n+p−k+j k(k − 1 + 2β) k 1− = (p + 1) . n + β + p 2(n + β + p) j=1

On the other hand, we get k−1 P

(n + β + p)k (n + α + p)k − 1 ≤ −1= (n + β)k (n + β)k (n + β)k−1 ≤

k−1 P



j=0

(n +

β)k

k j

j=0



k j

 (n + β)j pk−j

(n + β)k

 pk−j =

 1  (p + 1)k − 1 . n+β

As a conclusion , we have (α,β) |Sn,p (ek ; z) − ek (z)| ≤

1 rk k(k − 1 + 2β) [(p + 1)r]k + [(p + 1)r]k − . n+β+p n+β n+β

Case (2). For k > n + p ≥ 1 and |z| ≤ r with 1 ≤ r, we get (α,β) (α,β) |Sn,p (ek ; z) − ek (z)| ≤ |Sn,p (ek )(z)| + rk ≤

≤

(n + α + p)k n+p r + rk (n + β)k

2(n + α + p)k k 2(k − 1 + 2β) r ≤ 2[(p + 1)r]k ≤ [(p + 1)r]k . k (n + α) n+β+p

Combining it with the above Case 1, we get the desired inequality. Lemma 2. For fixed p ∈ N z ∈ C, we have (α,β) Sn,p (ek+1 ; z) =

S S {0}, 0 ≤ α ≤ β, let ek (z) = z k , k ∈ N {0},

z(1 − z) (α,β) α + (n + p)z (α,β) [Sn,p (ek ; z)]′ + Sn,p (ek ; z), z ∈ C. n+β n+β

835

REN and ZENG



 n+p Proof. Differentiating the sum = (j + α) z j (1 − j j=0 z)n+p−j and then dividing the formula by (n + β)k+1 , through simple calculation, we get the recurrence formula in statement. n+p P

(α,β) sk,n,p (z)

k

3. Main results The first main result is expressed by the following upper estimates. S Theorem 1. For fixed p ∈ N {0}, 0 ≤ α ≤ β and R > p + 1, let us denote DR = {z ∈ C; |z| < R} and let us suppose that f : DR → C is analytic in ∞ P DR , i.e., f (z) = ck z k for all z ∈ DR . Assume that 1 ≤ r and r(p + 1) < R. k=0

(i) For all |z| ≤ r and n ∈ N, we have (α,β) (α,β) (f ; z) − f (z)| ≤ Mr,p,n (f ), |Sn,p

where n ∞ P (α,β) k Mr,p,n (f ) = |ck | 2k(k−1+2β) n+β+p [r(p + 1)] +

1 n+β [r(p

k=1

+ 1)]k −

rk n+β

o < ∞.

(ii) if 1 ≤ r < r1 ≤ r1 (p + 1) < R, then for all |z| ≤ r and m, n ∈ N, we have (α,β) |[Sn,p (f ; z)](m) − f (m) (z)| ≤

(α,β)

Mr1 ,p,n (f )m!r1 , (r1 − r)m+1

(α,β)

where Mr1 ,p,n (f ) is defined as in (i) above. Proof. (i) Denote ek (z) = z k . By the hypothesis that f (z) is analytic in ∞ P DR , i.e., f (z) = ck z k for all z ∈ DR , we easily get (as in the case of the k=0 (α,β)

Bernstein polynomials) Sn,p (f ; z) =

∞ P

(α,β)

ck Sn,p (ek ; z), which implies

k=0 (α,β) |Sn,p (f ; z) − f (z)| ≤

∞ X

(α,β) |ck | · |Sn,p (ek ; z) − ek (z)|,

k=1

as

(α,β) Sn,p (e0 ; z)

= e0 (z) = 1. By lemma 1, we can get the desired inequality. ∞ P Since f (z) is analytic in DR , we have f (2) (z) = ck k(k − 1)z k−2 and the k=2

series is absolutely convergent in |z| ≤ r(p + 1), it follows that (α,β)

∞ P

|ck |k(k −

k=2

1)[r(p + 1)]k−2 < ∞, which implies that Mr,p,n (f ) < ∞. (ii) For the simultaneous approximation, denoting by Γ the circle of radius r1 > r and center 0, since for any |z| ≤ r and υ ∈ Γ, we have |υ − z| ≥ r1 − r, by Cauchy’s formulas, it follows that for all |z| ≤ r and m, n ∈ N, we have Z (α,β) m! Sn,p (f ; υ) − f (υ) (α,β) (m) (m) dυ |[Sn,p (f ; z)] − f (z)| = 2π Γ (υ − z)m+1

836

COMPLEX SCHURER-STANCU OPERATORS

≤ Mr(p,α,β) (f ) 1 ,n

(p,α,β)

m! 2πr1 Mr1 ,n (f )m!r1 = , 2π (r1 − r)m+1 (r1 − r)m+1

which proves the theorem. The following Voronovskaja-type result with a quantitative estimate holds. S Theorem 2. For fixed p ∈ N {0}, 0 ≤ α ≤ β and R > p + 1, let us denote DR = {z ∈ C; |z| < R} and let us suppose that f : DR → C is analytic in ∞ P DR , i.e., f (z) = ck z k for all z ∈ DR . For all |z| ≤ 1 and n ∈ N, we have k=0

(α,β) (α,β) (f ) z(1 − z) ′′ Mp (β − p)z − α ′ S f (z) − f (z) ≤ , (f ; z) − f (z) + n,p n+β 2(n + β) (n + β)2 where 0 < Mp(α,β) (f ) =

∞ X

(α,β)

|ck |(k − 1)[Ak,β (p + 1) + Bk,p ](p + 1)k−2 < ∞,

k=2 (α,β)

Ak,β = 2(k − 1)[2(k − 1)(k − 2 + 2β) + 1], Bk,p 4k 2 − 13k + 10 + (5k − 8)α + (|β − p| + α)2 ].

= (k − 1)[(5k − 6)|β − p| +

Proof. Denote ek (z) = z k . Since f is analytic in DR , i.e., f (z) = (α,β)

for all z ∈ DR , we can easily obtain Sn,p (f ; z) =

∞ P k=0

(α,β)

∞ P

ck z k

k=0 (α,β)

ck Sn,p (ek ; z). (α,β)

Taking into account that Sn,p (e0 ; z) = 1 and Sn,p (e1 ; z) = e1 (z) − (β−p)z−α , we obtain n+β (α,β) |Sn,p (f ; z) − f (z) +

≤

∞ X

(β − p)z − α ′ z(1 − z) ′′ f (z) − f (z)| n+β 2(n + β)

(α,β) |ck | · |Sn,p (ek ; z) − ek (z) +

k=2

[(β − p)z − α]kz k−1 k(k − 1)(1 − z)z k−1 − |. n+β 2(n + β)

Denote (α,β)

(α,β) Ek,n,p (z) = Sn,p (ek ; z) − ek (z) +

[(β − p)z − α]kz k−1 k(k − 1)(1 − z)z k−1 − . n+β 2(n + β)

(α,β)

(α,β)

It is obvious that Ek,n,p (z) is a polynomial of degree ≤ k and that E0,n,p (z) =

(α,β) E1,n,p (z)

= 0. Using of the recurrence in Lemma 2, for all k ≥ 2, n ∈ N and z ∈ D1 , by simple calculation, we can get (α,β)

Ek,n,p (z) =

z(1 − z) (α,β) α + (n + p)z (α,β) (α,β) [Ek−1,n,p (z)]′ + Ek−1,n,p (z) + Gk,n,p (z), n+β n+β

837

REN and ZENG

where (α,β)

Gk,n,p (z) = + +

(k − 1)z k−1 (k − 2)(1 − z) {[−α + (β − p)z] − } n+β 2

(k − 1)z k−2 {[−(k − 2)((β − p)z − α) − (β − p)z](1 − z) (n + β)2

(k − 2)(1 − z)[k − 2 − (k − 1)z] − [α + (n + p)z][(β − p)z − α] 2  (k − 2)(1 − z)[α + (n + p)z] + := T1 + T2 . 2

In view of α + (n + p)z = [α − (β − p)z] + (n + β)z, by simple calculation, we obtain (k − 1)z k−2 {[(−3k + 4)(β − p) − (k − 1)(k − 2)]z(1 − z) 2(n + β)2 +(k − 2)(3α + k − 2)(1 − z) + 2[(β − p)z − α]2 .

T2 = −T1 +

So, for all k ≥ 2, n ∈ N and z ∈ D1 , we have k−1 {(3k − 4)|β − p| + (k − 1)(k − 2) (n + β)2 +(k − 2)(3α + k − 2) + (|β − p| + α)2 .

(α,β)

|Gk,n,p (z)| = |T1 + T2 | ≤

Denote kf k1 = max{f (z); |z| ≤ 1}, by the Bernstein inequality and the Lemma 1, we obtain (α,β)

|Ek,n,p (z)| ≤

2 (α,β) (α,β) (α,β) |[E (z)]′ | + (p + 1)|Ek−1,n,p (z)| + |Gk,n,p (z)| n + β k−1,n,p

2(k − 1) (α,β) (α,β) kEk−1,n,p (z)k1 + |Gk,n,p (z)| n+β 2(k − 1) h (α,β) (α,β) ≤ (p + 1)|Ek−1,n,p (z)| + kSn,p (ek−1 ; ·) − ek−1 k1 n+β  [(β − p)e1 − α](k − 1)ek−2 (k − 1)(k − 2)(1 − e1 )ek−2 (α,β) +k − k1 + |Gk,n,p (z)| n+β 2(n + β) (α,β)

≤ (p + 1)|Ek−1,n,p (z)| +

(α,β)

≤ (p + 1)|Ek−1,n,p (z)| + +

(p + 1)k−1 {2(k − 1)[2(k − 1)(k − 2 + 2β) + 1]} (n + β)2

k−1 [(5k − 6)|β − p| + 4k 2 − 13k + 10 + (5k − 8)α + (|β − p| + α)2 ] (n + β)2 (α,β)

(α,β)

:= (p + 1)|Ek−1,n,p (z)| +

Bk,p (p + 1)k−1 Ak,β + , 2 (n + β) (n + β)2 (α,β)

where Ak,β = 2(k − 1)[2(k − 1)(k − 2 + 2β) + 1], Bk,p p| + 4k 2 − 13k + 10 + (5k − 8)α + (|β − p| + α)2 ].

838

= (k − 1)[(5k − 6)|β −

COMPLEX SCHURER-STANCU OPERATORS (α,β)

(α,β)

(α,β)

In view of E0,n,p (z) = E1,n,p (z) = 0 for any z ∈ C, also, Ak,β and Bk,p are increasing as functions of k, by writing the last inequality for k = 2, 3, ..., we easily, step by step, obtain the following: (α,β) |Ek,n,p (z)|

≤

k k (p + 1)k−1 X (p + 1)k−2 X (α,β) ≤ Aj,β + B (n + β)2 j=2 (n + β)2 j=2 j,p

(p + 1)k−2 (p + 1)k−1 (α,β) (k − 1)A + (k − 1)Bk,p . k,β (n + β)2 (n + β)2

This inequality combined with that one at the beginning of the proof immediately implies the required estimate in statement. ∞ P Note that, since f (4) (z) = ck k(k − 1)(k − 2)(k − 3)z k−4 and the series k=4

is absolutely convergent for all |z| ≤ p + 1, it easily follows the finiteness of the involved constants in the statement. Remark 1. If in the hypothesis of Theorem 2 we consider 1 ≤ r with r(p+ 1) < R, then by similar reasoning, we obtain an estimate of the form (α,β) (α,β) Sn,p (f ; z) − f (z) + (β − p)z − α f ′ (z) − z(1 − z) f ′′ (z) ≤ Mr,p (f ) n+β 2(n + β) (n + β)2 valid for all |z| ≤ r, n ∈ N and p ∈ N constant independent of n and z.

S (α,β) {0}, 0 ≤ α ≤ β, where Mr,p (f ) is a

Next the exact order of approximation by complex Schurer-Stancu operators is obtained. S Theorem 3. For fixed p ∈ N {0}, 0 ≤ α ≤ β and β > p, Let R > p + 1, DR = {z ∈ C; |z| < R}. Suppose that f : DR → C is analytic in DR . Also, let 1 ≤ r and r(p + 1) < R. If f is not a polynomial of degree 0, then for any r ∈ [1, R), we have (α,β) kSn,p (f ; ·) − f kr ≥

(α,β)

Cr,p (f ) , n ∈ N, n+β (α,β)

where kf kr = max{|f (z)|; |z| ≤ r} and the constant Cr,p (f ) depends only on f, p, α, β and r. (α,β)

Proof. Denote ek (z) = z k , keeping the notation there for Ek,n,p (z), for all z ∈ DR and n ∈ N, we have  1 z(1 − z) ′′ (α,β) Sn,p (f ; z) − f (z) = f (z) − [(β − p)z − α]f ′ (z) n+β 2  1 2 (α,β) + [(n + β) Ek,n,p (z)] . n+β

839

REN and ZENG

In view of the property kF + Gkr ≥ |kF kr − kGkr | ≥ kF kr − kGkr , it follows (α,β) kSn,p (f ; ·) − f kr ≥

1 n+β

 e1 (1 − e1 ) ′′ k f − [(β − p)e1 − α]f ′ kr 2

 1 (α,β) 2 − [(n + β) kEk,n,p kr ] . n+β Taking into account that by hypothesis f is not a polynomial of degree 0 1 ) ′′ f − [(β − p)e1 − α]f ′ kr > 0. Indeed, supposing the in DR , we get k e1 (1−e 2 contrary, it follows that z(1−z) f ′′ (z) − [(β − p)z − α]f ′ (z) = 0 for all z ∈ Dr . 2 Clearly, the analyticity of f implies that f is a constant (since contrariwise z(1−z) ′′ f (z) − [(β − p)z − α]f ′ (z) is a polynomial of degree at least 1, which 2 cannot be equal to 0), a contradiction to the hypothesis. (α,β) (α,β) By the Remark 1 after the Theorem 2, we get (n+β)2 kEk,n,p kr ≤ Mr,p (f ). 1 → 0 as n → ∞, therefore, there exists an index n0 Taking into account n+β 1 ) ′′ f − depending only on f, p, α, β and r such that for all n ≥ n0 , we have k e1 (1−e 2 (α,β) 1 1 ) ′′ ′ [(β −p)e1 −α]f ′ kr − n+β [(n+β)2 kEk,n,p kr ] ≥ 12 k e1 (1−e f −[(β −p)e −α]f kr , 1 2

(α,β)

which implies kSn,p (f ; ·) − f kr ≥ all n ≥ n0 .

e1 (1−e1 ) ′′ 1 f 2(n+β) k 2

(α,β)

For 1 ≤ n ≤ n0 −1, we have kSn,p (f ; ·)−f kr ≥ (α,β)

− [(β − p)e1 − α]f ′ kr , for

(α,β) Wr,n,p (f ) , n+β

(α,β)

where Wr,n,p (f ) =

(n + β) · kSn,p (f ; ·) − f kr > 0. As a conclusion, we have (α,β) kSn,p (f ; ·) − f kr ≥ (α,β)

(α,β)

(α,β)

Cr,p (f ) , f or all n ∈ N, n+β (α,β)

1 ) ′′ f −[(β −p)e1 − where Cr,p (f ) = min{Wr,1,p (f ), . . . , Wr,n0 −1,p (f ), 21 k e1 (1−e 2 α]f ′ kr }, this completes the proof.

Combining now Theorem 3 with Theorem 1, we get the following result. S Corollary 1. For fixed p ∈ N {0}, 0 ≤ α ≤ β and β > p, Let R > p + 1, DR = {z ∈ C; |z| < R}. Suppose that f : DR → C is analytic in DR . Also, let 1 ≤ r and r(p + 1) < R. If f is not a polynomial of degree 0, then for any r ∈ [1, R), we have (α,β) kSn,p (f ; ·) − f kr ≍

1 , n ∈ N, n+β

where kf kr = max{|f (z)|; |z| ≤ r} and the constants in the equivalence depend only on f , p, r and α, β but are independent on n. S Theorem 4. For fixed p ∈ N {0}, 0 ≤ α ≤ β and β > p, Let R > p + 1, DR = {z ∈ C; |z| < R}. Assume that f : DR → C is analytic in DR . Also, let

840

COMPLEX SCHURER-STANCU OPERATORS

m ∈ N, 1 ≤ r < r1 ≤ r1 (p + 1) < R. If f is not a polynomial of degree ≤ m − 1, then we have (α,β) k[Sn,p (f ; ·)](m) − f (m) kr ≍

1 , n ∈ N, n+β

where kf kr = max{|f (z)|; |z| ≤ r} and the constants in the equivalence depend only on f , p, r, r1 and m, α, β but are independent on n. Proof. Taking into account the upper estimate in Theorem 1, it remains to prove the lower estimate only. Denoting by Γ the circle of radius r1 > r and center 0, by Cauchy’s formula it follows that for all |z| ≤ r and m, n ∈ N, we have Z (α,β) Sn,p (f ; υ) − f (υ) m! (α,β) dυ. [Sn,p (f ; z)](m) − f (m) (z) = 2πi Γ (υ − z)m+1 (α,β)

Keeping the notation there for Ek,n,p (z), for all υ ∈ Γ and n ∈ N, we have  υ(1 − υ) ′′ 1 (α,β) Sn,p (f ; υ) − f (υ) = f (υ) − [(β − p)υ − α]f ′ (υ) n+β 2  1 2 (α,β) [(n + β) Ek,n,p (υ)] , + n+β which replaced in the above Cauchy’s formula implies ( (m) 1 z(1 − z) ′′ (α,β) (m) (m) [Sn,p (f ; z)] − f (z) = f (z) − [(β − p)z − α]f ′ (z) n+β 2 m! 1 · + n + β 2πi

Z Γ

)

(α,β)

(n + β)2 Ek,n,p (υ) (υ − z)m+1

dυ

.

Let e1 (z) = z, passing now to k · kr , we have (  (m)

e (1 − e )

1

1 1 (α,β) k[Sn,p (f ; ·)]m − f (m) kr ≥ f ′′ − [(β − p)e1 − α]f ′

n+β 2

) Z (α,β) (n + β)2 Ek,n,p (υ) 1

m!

− dυ .

n + β 2πi Γ (υ − ·)m+1

r

r

Since for any |z| ≤ r and υ ∈ Γ, we have |υ − z| ≥ r1 − r, so, by the Remark 1 after the proof of Theorem 2, we can get

(α,β)

m! Z (n + β)2 E (α,β) (υ) m! 2πr1 Mr1 ,p (f )m!r1

k,n,p (α,β) dυ ≤ · M (f ) = .

r ,p 1

2πi Γ

(υ − ·)m+1 2π (r1 − r)m+1 (r1 − r)m+1 r

h i(m)

e1 (1−e1 ) ′′

′

> Also by the hypothesis on f , we have f − [(β − p)e1 − α]f 2

0. Indeed , let m = 1, supposing the contrary it follows that

841

r z(1−z) ′′ f (z) − [(β − 2

REN and ZENG

p)z − α]f ′ (z) is a constant. Clearly, this is possible only if f is a constant ( since contrariwise, then z(1−z) f ′′ (z) − [(β − p)z − α]f ′ (z) is a polynomial of degree at 2 least 1, which cannot be equal to a constant ), which implies f is a polynomial of degree ≤ m − 1, a contradiction. Now let m ≥ 2, supposing that

 (m)

e (1 − e )

1 1 f ′′ − [(β − p)e1 − α]f ′

= 0,

2 r

f ′′ (z)−[(β −p)z −α]f ′(z) is a polynomial of degree ≤ m − 1 it follows that z(1−z) 2 (without free term), that is z(1−z) f ′′ (z) − [(β − p)z − α]f ′ (z) = Qm−1 (z) = 2 m−1 P ak z k . This is an inhomogeneous linear differential equation, therefore its k=1

∗ (z), where B(z) is the general general solution is of the form f (z) = B(z)+Pm−1 z(1−z) ′′ solution of the homogeneous equation 2 f (z) − [(β − p)z − α]f ′ (z) = 0 and ∗ Pm−1 (z) is a particular solution of the inhomogeneous differential equation. On the one hand, according to the proof of Theorem 3, we have B(z) = c, where c is a arbitrary complex constant. On the other hand, we seek the particm−1 P ∗ k ∗ ular solution under the form Pm−1 (z) = ak z which replaced in the inhomok=1

geneous equation produces, by simple calculation (identification of coefficients) a compatible linear algebraic system of m − 1 equations with m − 1 unknown m−1 P ∗ k a∗k , this shows that f (z) is of the form f (z) = c + ak z , a contradiction. k=1

h i(m)

e1 (1−e1 ) ′′

′

> f − [(β − p)e − α]f Therefore, for all m ∈ N, we have 1

2

r 0. Continuing by reasoning exactly as in the proof of Theorem 3, we immediately get the desired conclusion.

Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 61170324), the Natural Science Foundation of Fujian Province of China (Grant No. 2010J01012), the National Defense Basic Scientific Research Program of China (Grant No. B1420110155) and the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant No. JA12324).

References [1] G.G. Lorentz, Bernstein Polynomials, 2nd ed., Chelsea Publ., New York, 1986. [2] S.G. Gal, Voronovskajas theorem and iterations for complex Bernstein polynomials in compact disks, Mediterr. J. Math. 5 (3) (2008) 253-272. [3] S.G. Gal, Approximation by complex Bernstein-Kantorovich and StancuKantorovich polynomials and their iterates in compact disks, Rev. Anal. Numer. Theor. Approx. (Cluj) 37 (2) (2008) 159-168.

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COMPLEX SCHURER-STANCU OPERATORS

[4] S.G. Gal, Exact orders in simultaneous approximation by complex BernsteinStancu polynomials, Rev. Anal. Num´er. Th´eor. Approx. (Cluj) 37 (1) (2008) 47-52. [5] G.A. Anastassiou, S.G. Gal, Approximation by complex Bernstein-Schurer and Kantorovich-Schurer polynomials in compact disks, Comput. Math. Appl. 58 (4) (2009) 734-743. [6] S.G. Gal, Exact orders in simultaneous approximation by complex Bernstein polynomials, J. Concr. Appl. Math. 7 (3) (2009) 215-220. [7] S.G. Gal, Approximation by complex Bernstein-Stancu polynomials in compact disks, Results Math. 53 (3-4) (2009) 245-256. [8] G.A.Anastassiou, S.G. Gal, Approximation by complex Bernstein-Durrmeyer polynomials in compact disks, Mediterr. J. Math. 7 (4) (2010) 471-482. [9] S.G. Gal, Approximation by complex genuine Durrmeyer type polynomials in compact disks, Appl. Math. Comput. 217 (2010) 1913-1920. [10] N.I.Mahmudov, V.Gupta, Approximation by genuine Durrmeyer-Stancu polynomials in compact disks, Math. Comput. Model. 55 (2012) 278-285 [11] D.Bˇ arbosu, Schuer-Stancu type operators, Studia Univ. “ Babes-Bolyai” . XLVIII (3) (2003) 31-35.

843

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 844-851, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

TRIPLE FIXED POINT THEOREMS FOR WEAK (ψ − φ)-CONTRACTIONS ERDAL KARAPINAR, KISHIN SADARANGANI

Abstract. The notion of coupled fixed point is introduced in by Bhaskar and Lakshmikantham in [2]. Very recently, the concept of the tripled fixed point is introduced by Berinde and Borcut [1]. They also proved some triple fixed point theorems. In this manuscript, by using the weak (ψ − φ)-contraction, the results of Berinde and Borcut [1] are generalized.

1. Introduction and Preliminaries Very recently, Berinde and Borcut [1] introduced the concept of tripled fixed point and proved some related theorems. The notion of triple fixed point is a generalization of the concept of double fixed point which was introduced by Bhaskar and Lakshmikantham [2] in 2006. In this remarkable work, they ´ c in proved the existence and uniqueness of the double fixed point. After that, Lakshmikantham and Ciri´ [7] generalized these results by using g-monotone mappings. Various results on coupled fixed point have been obtained since then (see e.g. [3, 4, 6, 5, 9, 8]). In this manuscript, by using the (φ − ψ)-contraction, the results of Berinde and Borcut [1] are generalized. The weak φ-contraction was introduced by Alber and Guerre-Delabriere [10] in 1997. The notion of (φ − ψ)-contraction is a generalization of a weak φ-contraction (see e.g. [11, 12, 13]) Let (X, d) be a metric space and X 2 := X × X. Then the mapping ρ := X 2 × X 2 :→ [0, ∞) such that ρ((x1 , y1 ), (x2 , y2 )) := d(x1 , x2 ) + d(y1 , y2 ) forms a metric on X 2 . A sequence ({xn }, {yn }) ∈ X 2 is said to be a double sequence of X. Definition 1. (See [2]) Let (X, ≤) be partially ordered set and F : X × X → X. F is said to have mixed monotone property if F (x, y) is monotone nondecreasing in x and is monotone non-increasing in y, that is, for any x, y ∈ X, x1 ≤ x2 ⇒ F (x1 , y) ≤ F (x2 , y), for x1 , x2 ∈ X, and y1 ≤ y2 ⇒ F (x, y2 ) ≤ F (x, y1 ), for y1 , y2 ∈ X. Definition 2. (see [2]) An element (x, y) ∈ X × X is said to be a couple fixed point of the mapping F : X × X → X if F (x, y) = x and F (y, x) = y. Throughout this paper, let (X, ≤) be partially ordered set and d be a metric on X such that (X, d) is a complete metric space. Further, the product spaces X × X satisfy the following: (u, v) ≤ (x, y) ⇔ u ≤ x, y ≤ v; for all (x, y), (u, v) ∈ X × X.

(1.1)

The following two results of Bhaskar and Lakshmikantham in [2] were extended to class of cone metric spaces in [5]: Theorem 3. Let F : X × X → X be a continuous mapping having the mixed monotone property on X. Assume that there exists a k ∈ [0, 1) with k d(F (x, y), F (u, v)) ≤ [d(x, u) + d(y, v)] , for all u ≤ x, y ≤ v. (1.2) 2 If there exists x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ) and F (y0 , x0 ) ≤ y0 , then, there exists x, y ∈ X such that x = F (x, y) and y = F (y, x). Theorem 4. Let F : X × X → X be a mapping having the mixed monotone property on X. Suppose that X has the following properties: 2000 Mathematics Subject Classification. 47H10,54H25,46J10, 46J15. Key words and phrases. Fixed point theorems, weak (ψ − φ)-contraction, partially ordered, Triple Fixed Point. 844 1

2

E. KARAPINAR, K.SADARANGANI

(i) if a non-decreasing sequence {xn } → x, then xn ≤ x, ∀n; (i) if a non-increasing sequence {yn } → y, then y ≤ yn , ∀n. Assume that there exists a k ∈ [0, 1) with k [d(x, u) + d(y, v)] , for all u ≤ x, y ≤ v. (1.3) 2 If there exists x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ) and F (y0 , x0 ) ≤ y0 , then, there exists x, y ∈ X such that x = F (x, y) and y = F (y, x). d(F (x, y), F (u, v)) ≤

2. Triple Fixed Point Theorems Let (X, ≤) be partially ordered set and (X, d) be a complete metric space. We consider the following partial order on the product space X 3 = X × X × X: (u, v, w) ≤ (x, y, z) if and only if x ≥ u, y ≤ v, z ≥ w,

(2.1)

3

where (u, v, w), (x, y, z) ∈ X . Regarding this partial order, we state the definition of the following mapping. Definition 5. (See [1]) Let (X, ≤) be partially ordered set and F : X 3 → X. We say that F has the mixed monotone property if F (x, y, z) is monotone non-decreasing in x and z, and it is monotone non-increasing in y, that is, for any x, yx, z ∈ X x1 , x2 ∈ X, x1 ≤ x2 ⇒ F (x1 , y, z) ≤ F (x2 , y, z), y1 , y2 ∈ X, y1 ≤ y2 ⇒ F (x, y1 , z) ≥ F (x, y2 , z), z1 , z2 ∈ X, z1 ≤ z2 ⇒ F (x, y, z1 ) ≤ F (x, y, z2 ).

(2.2)

Definition 6. (See [1]) An element (x, y, z) ∈ X 3 is called a triple fixed point of F : X 3 → X if F (x, y, z) = x and F (y, x, y) = y and F (z, y, x) = z

(2.3)

3

For a metric space (X, d), the function ρ : X → [0, ∞), given by, ρ((x, y, z), (u, v, w)) := d(x, u) + d(y, v) + d(z, w) forms a metric space on X 3 , that is, (X 3 , ρ) is a metric induced by (X, d). Let Φ denote the all functions φ : [0, ∞) → [0, ∞) which is non-decreasing and satisfies that limt→r φ(t) > 0 for all r > 0 and limt→0+ φ(t) = 0. Let Ψ denote the all functions ψ : [0, ∞) → [0, ∞) which satisfy (i) ψ(t) = 0 if and only if t = 0, (ii) ψ is continuous and non-decreasing, (iii) ψ(s + t) = ψ(s) + ψ(t), ∀s, t ∈ [0, ∞). The aim of this paper is to prove the following theorem. Theorem 7. Let (X, ≤) be partially ordered set and (X, d) be a complete metric space. Let F : X 3 → X be a continuous mapping having the mixed monotone property on X. Assume that there exist constants a, b, c ∈ [0, 1) such that a + 2b + c < 1 with ψ(d(F (x, y, z), F (u, v, w))) ≤ ψ([ad(x, u)+bd(y, v)+cd(z, w)])−φ ([ad(x, u) + bd(y, v) + cd(z, w)]) (2.4) for all x ≥ u, y ≤ v, z ≥ w, where φ ∈ Φ, ψ ∈ Ψ. Suppose that there exist x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ),

z0 ≤ F (z0 , y0 , x0 ).

Suppose either (a) F is continuous, or (b) X has the following property: (i) if non-decreasing sequence xn → x (respectively, zn → z), then xn ≤ x (respectively, zn ≤ z) for all n, (ii) if non-increasing sequence yn → y, then yn ≥ y for all n, then there exist x, y, z ∈ X such that F (x, y, z) = x, F (y, x, y) = y, F (z, y, x) = z. 845

TRIPLE FIXED POINT THEOREMS FOR WEAK (ψ − φ)-CONTRACTIONS

3

Proof. We construct a sequence {(xn , yn , zn )} in the following way: Set x1 = F (x0 , y0 , z0 ) ≥ x0 , y1 = F (y0 , x0 , y0 ) ≤ y0 , z1 = F (z0 , y0 , x0 ) ≥ z0 , and by the mixed monotone property of F ,for n ≥ 1, inductively we get xn = F (xn−1 , yn−1 , zn−1 ) ≥ xn−1 ≥ · · · ≥ x0 , yn = F (yn−1 , xn−1 , yn−1 ) ≤ yn−1 ≤ · · · ≤ y0 , zn = F (zn−1 , yn−1 , xn−1 ) ≥ zn−1 ≥ · · · ≥ z0 .

(2.5)

Due to (2.4) and (2.5), we have ψ(d(x1 , x2 ))

ψ(d(y1 , y2 ))

ψ(d(z1 , z2 ))

= ψ(d(F (x0 , y0 , z0 ), F (x1 , y1 , z1 ))) ≤ ψ ([ad(x0 , x1 ) + bd(y0 , y1 ) + cd(z0 , z1 )]) −φ ([ad(x0 , x1 ) + bd(y0 , y1 ) + cd(z0 , z1 )]) ≤ ψ ([ad(x0 , x1 ) + bd(y0 , y1 ) + cd(z0 , z1 )])

(2.6)

= ψ(d(F (y0 , x0 , y0 ), F (y1 , x1 , y1 ))) ≤ ψ ([ad(y0 , y1 ) + bd(x0 , x1 ) + cd(y0 , y1 )]) −φ ([ad(y0 , y1 ) + bd(x0 , x1 ) + cd(y0 , y1 )]) ≤ ψ ([bd(x0 , x1 ) + (a + c)d(y0 , y1 )])

(2.7)

= ψ(d(F (z0 , y0 , x0 ), F (z1 , y1 , x1 ))) ≤ ψ ([ad(z0 , z1 ) + bd(y0 , y1 ) + cd(x0 , x1 )]) −φ ([ad(z0 , z1 ) + bd(y0 , y1 ) + cd(x0 , x1 )]) ≤ ψ ([ad(z0 , z1 ) + bd(y0 , y1 ) + cd(x0 , x1 )])

(2.8)

Regarding (2.4) together with (2.6),(2.7),(2.8) we have ψ(d(x2 , x3 ))

ψ(d(y2 , y3 ))

ψ(d(z2 , z3 ))

= ψ(d(F (x1 , y1 , z1 ), F (x2 , y2 , z2 ))) ≤ ψ ([ad(x1 , x2 ) + bd(y1 , y2 ) + cd(z1 , z2 )]) −φ ([ad(x1 , x2 ) + bd(y1 , y2 ) + cd(z1 , z2 )]) ≤ ψ ([ad(x1 , x2 ) + bd(y1 , y2 ) + cd(z1 , z2 )])

(2.9)

= ψ(d(F (y1 , x1 , y1 ), F (y2 , x2 , y2 ))) ≤ ψ ([ad(y1 , y2 ) + bd(x1 , x2 ) + cd(y1 , y2 )]) −φ ([ad(y1 , y2 ) + bd(x1 , x2 ) + cd(y1 , y2 )]) ≤ ψ ([bd(x1 , x2 ) + (a + c)d(y1 , y2 )])

(2.10)

= ψ(d(F (z1 , y1 , x1 ), F (z2 , y2 , x2 ))) ≤ ψ ([ad(z1 , z2 ) + bd(y1 , y2 ) + cd(x1 , x2 )]) −φ ([ad(z1 , z2 ) + bd(y1 , y2 ) + cd(x1 , x2 )]) ≤ ψ ([ad(z1 , z2 ) + bd(y1 , y2 ) + cd(x1 , x2 )])

(2.11)

Recursively we have ψ(d(xn , xn+1 )) ≤ ψ ([ad(xn−1 , xn ) + bd(yn−1 , yn ) + cd(zn−1 , zn )]) −φ ([ad(xn−1 , xn ) + bd(yn−1 , yn ) + cd(zn−1 , zn )]) ≤ ψ ([ad(xn−1 , xn ) + bd(yn−1 , yn ) + cd(zn−1 , zn )])

(2.12)

ψ(d(yn , yn+1 )) ≤ ψ ([ad(yn−1 , yn ) + bd(xn−1 , xn ) + cd(yn−1 , yn )]) −φ ([bd(xn−1 , xn ) + (a + c)d(yn−1 , yn )]) ≤ ψ ([bd(xn−1 , xn ) + (a + c)d(yn−1 , yn )])

(2.13)

ψ(d(zn , zn+1 )) ≤ ψ ([ad(zn−1 , zn ) + bd(yn−1 , yn ) + cd(xn−1 , xn )]) −φ ([ad(zn−1 , zn ) + bd(yn−1 , yn ) + cd(xn−1 , xn )]) ≤ ψ ([ad(zn−1 , zn ) + bd(yn−1 , yn ) + cd(xn−1 , xn )])

(2.14)

Since ψ is a non-decreasing, then (2.12)-(2.14) imply that d(xn , xn+1 ) ≤ ad(xn−1 , xn ) + bd(yn−1 , yn ) + cd(zn−1 , zn )

(2.15)

d(yn , yn+1 ) ≤ bd(xn−1 , xn ) + (a + c)d(yn−1 , yn )

(2.16)

d(zn , zn+1 ) ≤ cd(xn−1 , xn ) + bd(yn−1 , yn ) + ad(zn−1 , zn )

(2.17)

846

4

E. KARAPINAR, K.SADARANGANI

For the simplify the notation, we need to define the following matrices:  (n)    2  (n) m11 m12 a b c a + b2 + c2 2ab + 2bc 2ac (n) (n)  and M n =  (a + c)2 + b2 bc M =  b (a + c) 0 , M 2 =  bc + 2ab  m21 m22 2 2 2 (n) (n) c b a b + 2ac 2ab + 2bc a +c m31 m32       d(xn , xn+1 ) ψ(d(xn , xn+1 )) φ(d(xn , xn+1 )) Dn+1 =  d(yn , yn+1 )  , (Dn+1 )ψ =  ψ(d(yn , yn+1 ))  and (Dn+1 )φ =  φ(d(yn , yn+1 ))  d(zn , zn+1 ) ψ(d(zn , zn+1 )) φ(d(zn , zn+1 )) (n)

Since a + 2b + c < 1, then for M n = (mij )3×3 we have (n)

(n)

(n)

R(n) (i) = mi1 + mi2 + mi3 < 1 for all n ∈ N and i = 1, 2, 3, (n) (n) (n) C (n) (j) = m1j + m2j + m3j < 1 for all n ∈ N and j = 1, 2, 3.

(2.18)

By induction we can easily prove it. Indeed, assume that (2.18) holds. Then, the following product yields the result.   (n+1)     (n) (n) (n) (n+1) (n+1) m11 m12 m13 m11 m12 m13 a b c  (n)  (n+1) (n) (n)  (n+1) (n+1)  (2.19)  m21  =  b (a + c) 0   m21 m22 m23  m22 m23 (n) (n) (n) (n+1) (n+1) (n+1) c b a m31 m32 m33 m31 m32 m33   (n) (n) (n) (n) (n) (n) (n) (n) (n) am11 + bm21 + cm31 am12 + bm22 + cm32 am13 + bm23 + cm33   (n) (n) (n) (n) (n) (2.20) =  m(n) m22 (a + c) + bm12 m23 (a + c) + bm13  21 (a + c) + bm11 (n) (n) (n) (n) (n) (n) (n) (n) (n) am31 + bm21 + cm11 am32 + bm22 + cm12 am33 + bm23 + cm13 By using matrix notation and having in mind (2.5) and (2.18), the inequalities (2.6) - (2.11) imply that (Dn+1 )ψ ≤ (M Dn )ψ − (M Dn )φ ≤ (M Dn )ψ for all n ∈ N. (2.21) Since ψ is nondecreasing, then Dn+1 ≤ M Dn ≤ M n+1 D0 for all n ∈ N.

(2.22)

Thus, from by (2.15)-(2.17), we obtain that d(xn , xn+1 ) + d(yn , yn+1 ) + d(zn , zn+1 ) ≤ [a + b + c]d(xn−1 , xn ) + [a + 2b + c]d(yn−1 , yn ) + [a + c]d(zn−1 , zn )]

(2.23)

Set δn+1 = d(xn , xn+1 ) + d(yn , yn+1 ) + d(zn , zn+1 ). Due to (2.23) we have, δn+1 ≤ δn ,

(2.24)

and hence the sequence {δn } is non-increasing and bounded below by 0. Thus, there exists a δ ≥ 0 such that lim δn = δ. (2.25) n→∞

We shall show that δ = 0. Suppose, to the contrary, that δ > 0. Due to (2.12)-(2.14), we conclude that ψ(d(xn+1 , xn+2 )) + ψ(d(yn+1 , yn+2 )) + ψ(d(zn+1 , zn+2 )) ≤ ψ([ad(zn , zn+1 ) + bd(xn , xn+1 ) + cd(yn , yn+1 )]) +ψ([bd(xn , xn+1 ) + (a + c)d(yn , yn+1 )]) +ψ([ad(zn , zn+1 ) + bd(yn , yn+1 ) + cd(xn , xn+1 )]) −φ (ad(xn , xn+1 ) + bd(yn , yn+1 ) + cd(zn , zn+1 )]) −φ([bd(xn , xn+1 ) + (a + c)d(yn , yn+1 )]) −φ([ad(zn , zn+1 ) + bd(yn , yn+1 ) + cd(xn , xn+1 )])

(2.26)

From the property (iii) of ψ, we have ψ([d(xn+1 , xn+2 ) + d(yn+1 , yn+2 ) + d(zn+1 , zn+2 )]) ≤ ψ([a + b + c]d(xn , xn+1 ) + [a + 2b + c]d(yn , yn+1 ) + [a + c]d(zn , zn+1 )) −φ (ad(xn , xn+1 ) + bd(yn , yn+1 ) + cd(zn , zn+1 )]) −φ([bd(xn , xn+1 ) + (a + c)d(yn , yn+1 )]) −φ([ad(zn , zn+1 ) + bd(yn , yn+1 ) + cd(xn , xn+1 )]) 847

(2.27)

 (n) m13 (n)  m23  (n) m33

TRIPLE FIXED POINT THEOREMS FOR WEAK (ψ − φ)-CONTRACTIONS

5

Since a + 2b + c < 1 and the functions ψ, φ are non-decreasing, then (2.27) yields that ψ([d(xn+1 , xn+2 ) + d(yn+1 , yn+2 ) + d(zn+1 , zn+2 )]) ≤ ψ(d(xn+1 , xn+2 ) + d(yn+1 , yn+2 ) + d(zn+1 , zn+2 )) −φ (ad(xn , xn+1 ) + bd(yn , yn+1 ) + cd(zn , zn+1 )]) −φ([bd(xn , xn+1 ) + (a + c)d(yn , yn+1 )]) −φ([ad(zn , zn+1 ) + bd(yn , yn+1 ) + cd(xn , xn+1 )])

(2.28)

ψ(δn+1 ) ≤ ψ(δn ) − ∆n

(2.29)

∆n = φ (ad(xn , xn+1 ) + bd(yn , yn+1 ) + cd(zn , zn+1 )]) +φ([bd(xn , xn+1 ) + (a + c)d(yn , yn+1 )]) +φ([ad(zn , zn+1 ) + bd(yn , yn+1 ) + cd(xn , xn+1 )])

(2.30)

Again by (2.30), we have where

Letting n → ∞ in (2.29) and having in mind that we suppose ψ is continuous, limt→r φ(t) > 0 for all r > 0 and limt→0+ φ(t) = 0, we have ψ(δ) ≤ ψ(δ) − lim ∆n , (2.31) n→

which is a contradiction. Hence ψ(δ) = 0 and δ = 0, that is, lim δn = lim [d(xn , xn−1 ) + d(yn , yn−1 ) + d(zn , zn−1 ) + d(wn , wn−1 )] = 0.

n→∞

n→∞

(2.32)

Now, we shall prove that {xn },{yn } and {zn } are Cauchy sequences. Suppose, to the contrary, that at least one of {xn },{yn } and {zn } is not Cauchy. So, there exists an ε > 0 for which we can find subsequences {xn(k) }, {xn(k) } of {xn } and {yn(k) }, {yn(k) } of {yn } and {zn(k) }, {zn(k) } of {zn } with n(k) > m(k) ≥ k such that d(xn(k) , xm(k) ) + d(yn(k) , ym(k) ) + d(zn(k) , zm(k) ) ≥ ε.

(2.33)

Additionally, corresponding to m(k), we may choose n(k) such that it is the smallest integer satisfying (2.33) and n(k) > m(k) ≥ k. Thus, d(xn(k)−1 , xm(k) ) + d(yn(k)−1 , ym(k) ) + d(zn(k)−1 , zm(k) ) < ε.

(2.34)

By using triangle inequality and having (2.33) and (2.34) in mind ε

≤ tk = d(xn(k) , xm(k) ) + d(yn(k) , ym(k) ) + d(zn(k) , zm(k) ) ≤ d(xn(k) , xn(k)−1 ) + d(xn(k)−1 , xm(k) ) + d(yn(k) , yn(k)−1 ) + d(yn(k)−1 , ym(k) ) +d(zn(k) , zn(k)−1 ) + d(zn(k)−1 , zm(k) ) < d(xn(k) , xn(k)−1 ) + d(yn(k) , yn(k)−1 ) + d(zn(k) , zn(k)−1 ) + ε.

(2.35)

Letting k → ∞ in (2.35) and using (2.32) lim tk = lim d(xn(k) , xm(k) ) + d(yn(k) , ym(k) ) + d(zn(k) , zm(k) ) = ε

k→∞

k→∞

(2.36)

Again by triangle inequality, tk

= d(xn(k) , xm(k) ) + d(yn(k) , ym(k) ) + d(zn(k) , zm(k) ) ≤ d(xn(k) , xn(k)+1 ) + d(xn(k)+1 , xm(k)+1 ) + d(xm(k)+1 , xm(k) ) +d(yn(k) , yn(k)+1 ) + d(yn(k)+1 , ym(k)+1 ) + d(ym(k)+1 , ym(k) ) +d(zn(k) , zn(k)+1 ) + d(zn(k)+1 , zm(k)+1 ) + d(zm(k)+1 , zm(k) ) ≤ δn(k)+1 + δm(k)+1 + d(xn(k)+1 , xm(k)+1 ) + d(yn(k)+1 , ym(k)+1 ) +d(zn(k)+1 , zm(k)+1 )

(2.37)

Since n(k) > m(k), then xn(k) ≥ xm(k) and yn(k) ≤ ym(k) , zn(k) ≥ zm(k) .

(2.38)

Hence from (2.38), (2.5) and (2.4), we have, ψ(d(xn(k)+1 , xm(k)+1 ))

= ψ(d(F (xn(k) , yn(k) , zn(k) ), F (xm(k) , ym(k) , zm(k) ))
≤ ψ [ad(xn(k) , xm(k) ) + bd(yn(k) , ym(k) ) + cd(zn(k) , zm(k) )]
−φ [ad(xn(k) , xm(k) ) + bd(yn(k) , ym(k) ) + cd(zn(k) , zm(k) )] 848

(2.39)

6

E. KARAPINAR, K.SADARANGANI

ψ(d(yn(k)+1 , ym(k)+1 ))

ψ(d(zn(k)+1 , zm(k)+1 ))

= ψ(d(F (yn(k) , xn(k) , yn(k) ), F (ym(k) , xm(k) , ym(k) )))
≤ ψ [ad(yn(k) , ym(k) ) + bd(xn(k) , xm(k) ) + cd(yn(k) , ym(k) )]
−φ [ad(yn(k) , ym(k) ) + bd(xn(k) , xm(k) ) + cd(yn(k) , ym(k) )]

(2.40)

= ψ(d(F (zn(k) , yn(k) , xn(k) ), F (zm(k) , ym(k) , xm(k) )))
≤ ψ [ad(zn(k) , zm(k) ) + bd(yn(k) , ym(k) ) + cd(xn(k) , xm(k) )]
−φ [ad(zn(k) , zm(k) ) + bd(yn(k) , ym(k) ) + cd(xn(k) , xm(k) )]]

(2.41)

From (2.39)-(2.41) and property (iii) of ψ,we get that ψ(d(xn(k)+1 , xm(k)+1 ) + d(yn(k)+1 , ym(k)+1 ) + d(xn(k)+1 , xm(k)+1 )) ≤ ψ([a + b + c]d(xn(k) , xm(k) ) + [a + 2b + c]d(yn(k) , ym(k) ) + [a + c]d(z
n(k) , zm(k) )) −φ ad(xn(k) , xm(k) ) + bd(yn(k) , ym(k) ) + cd(zn(k) , zm(k) )
−φ [ad(yn(k) , ym(k) ) + bd(xn(k) , xm(k) ) + cd(yn(k) , ym(k) )] −φ([ad(zn(k) , zm(k) ) + bd(yn(k) , ym(k) ) + cd(xn(k) , xm(k) )])

(2.42)

Regarding that φ is non-decreasing and 0 < a + 2b + c < 1, then (2.42) becomes ψ(d(xn(k)+1 , xm(k)+1 ) + d(yn(k)+1 , ym(k)+1 ) + d(xn(k)+1 , xm(k)+1 )) ≤ ψ([a + b + c]d(xn(k) , xm(k) ) + [a + 2b + c]d(yn(k) , ym(k) ) + [a + c]d(z
n(k) , zm(k) )) −φ ad(xn(k) , xm(k) ) + bd(yn(k) , ym(k) ) + cd(zn(k) , zm(k) ) ≤ ψ(tk ) − 3φ(tk )

(2.43)

Letting k → ∞ and having in mind (2.43) we get a contradiction. This shows that {xn },{yn } and {zn } are Cauchy sequences. Since X is complete metric space, there exists x, y, z ∈ X such that limn→∞ xn = x and limn→∞ yn = y, limn→∞ zn = z.

(2.44)

Suppose now the assumption (a) holds. Then by (2.5) and (2.44), we have x

= lim xn = lim F (xn−1 , yn−1 , zn−1 ) n→∞

n→∞

= F ( lim xn−1 , lim yn−1 , lim zn−1 ) n→∞

n→∞

n→∞

(2.45)

= F (x, y, z) Analogously, we also observe that y = lim yn = lim F (yn−1 , xn−1 , yn−1 ) = F (y, x, y) n→∞

n→∞

z = lim zn = lim F (zn−1 , yn−1 , xn−1 ) = F (z, y, x) n→∞

(2.46)

n→∞

Thus, we have F (x, y, z) = x, F (y, x, y) = y, F (z, y, x) = z. Suppose now the assumption (b) holds. Since {xn }, {zn } are non-decreasing and xn → x, zn → z and also {yn } is non-increasing and yn → y, then by assumption (b) we have xn ≥ x, yn ≤ y, zn ≥ z, for all n. Having in mind that φ is non-decreasing and by using triangle inequality, (2.4) yields that ψ(d(x, F (x, y, z))) ≤ ψ(d(x, xn+1 ) + d(xn+1 , F (x, y, z))) = ψ(d(x, xn+1 )) + ψ(d(F (xn , yn , zn ), F (x, y, z))) ≤ ψ([ad(xn , x) + bd(yn , y) + cd(zn , z)]) −φ ([ad(xn , x) + bd(yn , y) + cd(zn , z)])

(2.47)

Taking n → ∞ in (2.47) and using (2.44), we get that d(x, F (x, y, z)) = 0. Thus, x = F (x, y, z). Analogously, we get that F (y, x, y) = y, F (z, y, x) = z. Thus, we proved that F has a triple fixed point.

849



TRIPLE FIXED POINT THEOREMS FOR WEAK (ψ − φ)-CONTRACTIONS

7

Corollary 8. Let (X, ≤) be partially ordered set and (X, d) be a complete metric space. Let F : X 3 → X be a mapping having the mixed monotone property on X. Assume that there exist constants a, b, c ∈ [0, 1) such that a + 2b + c < 1 with d(F (x, y, z), F (u, v, w)) ≤ [ad(x, u) + bd(y, v) + cd(z, w)]

(2.48)

for all x ≥ u, y ≤ v, z ≥ w. Assume also that there exist x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ),

z0 ≤ F (x0 , y0 , z0 )

Suppose either (a) F is continuous, or (b) X has the following property: (i) if non-decreasing sequence xn → x (respectively, zn → z), then xn ≤ x (respectively, zn ≤ z) for all n, (ii) if non-increasing sequence yn → y, then yn ≥ y for all n, then there exist x, y, z ∈ X such that F (x, y, z) = x, F (y, x, y) = y, F (z, y, x) = z. Proof. It is sufficient to take ψ(t) = t and φ(t) = (1 − k)t in previous theorem.



3. Uniqueness of Triple Fixed Point In this section we shall prove the uniqueness of triple fixed point. For a product X 3 of a partial ordered set (X, ≤) we define a partial ordering in the following way: For all (x, y, z), (u, v, r) ∈ X × X × X (x, y, z) ≤ (u, v, r) ⇔ x ≤ u, y ≥ v, z ≤ r.

(3.1)

We say that (x, y, z) is equal (u, v, r) if and only if x = u, y = v, z = r. Theorem 9. In addition to hypothesis of Theorem 7, suppose that for all (x, y, z), (u, v, r) ∈ X × X × X, there exists (a, b, c) ∈ X × X × X that is comparable to (x, y, z) and (u, v, r), then F has a unique triple fixed point. Proof. The set of triple fixed point of F is not empty due to Theorem 7. Assume, now, (x, y, z) and (u, v, r) are the triple fixed point of F , that is, F (x, y, z) = x, F (y, x, y) = y, F (z, y, x) = z,

F (u, v, r) = u, F (v, u, v) = v, F (r, v, u) = r,

We shall show that (x, y, z) and (u, v, r) are equal. By assumption, there exists (α, β, ζ) ∈ X × X × X × X that is comparable to (x, y, z) and (u, v, r). Define sequences {αn }, {βn } and {ζn } such that α = α0 ,

β = β0 ,

ζ = ζ0 ,

and

αn = F (αn−1 , βn−1 , ζn−1 ), βn = F (βn−1 , αn−1 , βn−1 ), (3.2) ζn = F (ζn−1 , βn−1 , αn−1 ), for all n. Since (x, y, z) is comparable with (a, b, c), we may assume that (x, y, z) ≥ (α, β, ζ) = (α0 , β0 , ζ0 ). Recursively, we get that (x, y, z, w) ≥ (αn , βn , ζn ) for all n. (3.3) By (3.3) and (2.4), we have ψ(d(x, αn+1 ))

ψ(d(βn+1 , y))

ψ(d(z, ζn+1 ))

= ψ(d(F (x, y, z), F (αn , βn , ζn ))) ≤ ψ([ad(x, αn ) + bd(y, βn ) + cd(z, ζn )]) −φ ([ad(x, αn ) + bd(y, βn ) + cd(z, ζn )])

(3.4)

= ψ(d(F (αn , dn , ζn , βn ), F (y, x, y))) ≤ ψ([ad(βn , y) + bd(αn , x) + cd(βn , y)]) −φ ([ad(βn , y)) + bd(αn , x) + cd(βn , y)])

(3.5)

= ψ(d(F (z, y, x), F (ζn , βn , αn )) ≤ ψ([ad(z, ζn ) + bd(y, βn ) + cd(x, αn )]) −φ ([ad(z, ζn ) + bd(y, βn ) + cd(x, αn )])

(3.6)

850

8

E. KARAPINAR, K.SADARANGANI

Set γn = d(x, αn )+d(y, βn )+d(z, ζn ). By property (iii) of ψ and recalling that φ, ψ are non-decreasing, then (3.4)-(3.6) imply that ψ(d(x, αn+1 ) + d(y, βn+1 ) + d(z, ζn+1 )) ≤ ψ([d(x, αn ) + d(y, βn ) + d(z, ζn )]) (3.7) −3φ ([d(x, αn ) + d(y, βn ) + d(z, ζn )]) Hence we have ψ(γn+1 ) ≤ ψ(γn ) − 3φ(γn ), for all n. (3.8)  Hence, the sequence {γn } is decreasing and bounded below. Thus, there exists γ ≥ 0 such that lim γn = γ.

n→∞

Now, we shall show that γ = 0. Suppose, to the contrary, that γ > 0. Letting n → ∞ in (3.8), we obtain that γ ≤ γ − 3(φ(γ)) which is a contradiction. Therefore, γ = 0. That is, limn→∞ γn = 0. Consequently, we have limn→∞ d(x, αn ) = 0, limn→∞ d(z, ζn ) = 0,

limn→∞ d(y, βn ) = 0,

(3.9)

Analogously, we show that limn→∞ d(u, αn ) = 0, limn→∞ d(v, βn ) = 0, limn→∞ d(r, ζn ) = 0,

(3.10)

Combining (3.9) and (3.10) yield that (x, y, z) and (u, v, r) are equal. References [1] V. Berinde and M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Analysis, 74(15), 4889–4897 (2011). [2] Bhaskar, T.G., Lakshmikantham, V.: Fixed Point Theory in partially ordered metric spaces and applications Nonlinear Analysis, 65, 1379–1393 (2006). [3] N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Analysis, 74, 983-992(2011). [4] B. Samet, Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces, Nonlinear Analysis, 74(12), 45084517(2010). [5] E. Karapınar, Couple Fixed Point on Cone Metric Spaces, Gazi University Journal of Science, 24(1),51-58(2011). [6] E. Karapınar, Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl., 59 (12), 3656-3668(2010). ´ c, L.: : Couple Fixed Point Theorems for nonlinear contractions in partially ordered metric [7] Lakshmikantham, V., Ciri´ spaces Nonlinear Analysis, 70, 4341-4349 (2009). [8] Binayak S. Choudhury, N. Metiya and A. Kundu, Coupled coincidence point theorems in ordered metric spaces, Ann. Univ. Ferrara, 57, 1-16(2011). [9] B.S. Choudhury, A. Kundu : A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. TMA 73, 25242531 (2010). [10] Alber,Ya. I., Guerre-Delabriere, S.: Principle of weakly contractive maps in Hilbert space In: I. Gohberg and Yu. Lyubich, Editors, New Results in Operator Theory, Advances and Appl. 98, Birkh¨ auser, Basel ,722,(1997). [11] P. N Dhutta, B.S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl, (2008) Article ID 406368. [12] E. Karapınar, Fixed point theory for cyclic weak φ-contraction Appl.Math. Lett., 24(6),(2011) 822-825. [13] M. Pacurar, I.A. Rus Fixed point theory for cyclic ϕ-contractions, Nonlinear Amal. 72 (2010) 1181-1187. erdal karapınar, ˙ Department of Mathematics, Atılım University, 06836, Incek, Ankara, Turkey E-mail address: [email protected] E-mail address: [email protected] Kishin Sadarangani, Department of Mathematics, University of Las Palmas de Gran Canaria, Campus Universitario de Tafira, 35017 Las Palmas de Gran Canaria, Spain E-mail address: [email protected]

851

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 852-857, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Form of solutions and periodicity for systems of di¤erence equations H. El-Metwally1 and E. M. Elsayed2 . 1 Department of Mathematics, Rabigh College of Science and Art, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia. 2 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 1;2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 1 E-mail: [email protected]. 2 E-mail: [email protected]. Abstract This paper is devoted to get the form of the solutions and the periodic nature of the following systems of rational di¤erence equations xn+1 =

yn 2 yn 2 xn

1

with initial conditions.x

2;

x

1;

1 yn

x0 ; :y

;

yn+1 =

2;

y

1;

1

xn 2 xn 2 yn

1 xn

;

y0 are real numbers.

Keywords: di¤erence equations, recursive sequence, stability, periodic solution, system of di¤erence equations. Mathematics Subject Classi…cation: 39A10. ——————————————————

1

Introduction

Di¤erence equations is a hot topic in that they are widely used to investigate equations arising in mathematical models describing real-life situations such as population biology, probability theory, and genetics. Recently, rational di¤erence equations have appealed more and more scholars for their wide applications. For details, see [1-9]. However, there are few literatures on the systems of two or three rational di¤erence equations [10–17]. In this article, we investigate the behavior of the solutions for the following systems of di¤erence equations xn+1 = with initial conditions x

2

yn 2 yn 2 xn

1 2;

x

1;

1 yn

x0 ; :y

2;

;

yn+1 =

y

1;

1 xn

;

y0 are real numbers.

yn 2 1 yn 2 xn

First system: xn+1 =

xn 2 xn 2 yn

1

; 1 yn

xn 2 1+xn 2 yn

yn+1 =

1 xn

In this section, we investigate the solutions for the following system of di¤erence equations xn+1 =

yn 2 yn 2 xn

1

1 yn

;

yn+1 =

xn 1 + xn

2

2 yn 1 xn

;

(1)

where the initial conditions are arbitrary real numbers with x 2 y 1 x0 6= 1; 6= y 2 x 1 y0 6= 1. The following theorem is devoted to the form of the solutions for System (1).

1 2

and

Theorem 1 Every solution fxn ; yn g of System (1) is periodic with period twelve and, for n = 0; 1; 2; :::; has the form x12n

2

= x

2;

x12n

1

=x

1;

x12n = x0 ; x12n+1 =

y (1 + y

x12n+2

=

y0 y 1 (1 + x 2 y 1 x0 ) ; x12n+3 = (1 + 2x 2 y 1 x0 ) ( 1 + y 2x

x12n+4

=

x

x12n+8

=

y

2;

x12n+5 =

x

1;

x12n+6 =

1 y0 )

x0 ; x12n+7 =

1 (1

+ x 2 y 1 x0 ) y0 ; x12n+9 = (1 + 2x 2 y 1 x0 ) ( 1 + y 2x 1

852

1 y0 )

2

2 x 1 y0 )

;

;

; y 2 (1 + y 2 x

1 y0 )

;

EL-METWALLY, ELSAYED: SOLVING DIFFERENCE EQUATIONS

and y12n

x 2 ; (1 + x 2 y 1 x0 ) x0 (1 + 2x 2 y 1 x0 ) x 1 (1 + y 2 x 1 y0 ); y12n+3 = ; (1 + x 2 y 1 x0 ) y 2 ( 1 + y 2 x 1 y0 ) y 1 ; y12n+5 = ; (1 + y 2 x 1 y0 ) (1 + 2x 2 y 1 x0 ) x 2 (1 + 2x 2 y 1 x0 ) y0 (1 + y 2 x 1 y0 ) ; y12n+7 = ; ( 1 + y 2 x 1 y0 ) (1 + x 2 y 1 x0 ) x0 x 1 ( 1 + y 2 x 1 y0 ); y12n+9 = : (1 + x 2 y 1 x0 )

= y

2

y12n+2

=

y12n+4

=

y12n+6

=

y12n+8

=

2;

y12n

1

=y

1;

y12n = y0 ; y12n+1 =

Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n 1, that is x12n

14

= x

2;

x12n

10

=

y 1 (1 + x 2 y 1 x0 ) ; (1 + 2x 2 y 1 x0 )

x12n

8

=

x

x12n

4

=

y

x12n

2;

=x

13

x12n

1;

=

7

x12n

x

1;

= x0 ; x12n

12

11

x12n

9

=

y0 ( 1 + y 2x

x12n

6

=

x0 ; x12n

1 (1

+ x 2 y 1 x0 ) ; x12n (1 + 2x 2 y 1 x0 )

3

y0 ( 1 + y 2x

=

y

=

(1 + y 1 y0 )

5

=

1 y0 )

2

2 x 1 y0 )

;

;

y 2 (1 + y 2 x

1 y0 )

;

;

and y12n

14

= y

2;

y12n

y12n

10

= x

1 (1

+y y

y12n

7

=

y12n

4

= x

(1+2x 1(

13

1;

2 x 1 y0 );

1

2y

=y

1 x0 )

1+y

y12n

y12n

; y12n

6

y12n

3

y12n

11

x0 (1+2x 2 y 1 x0 ) (1+x 2 y 1 x0 ) ;

y0 (1+y ( 1+y

=

2 x 1 y0 );

=

9

= y0 ;

12

2x

1 y0 )

2x

1 y0 )

=

; y12n x0 2y

(1 + x

=

x 2 (1 + x 2 y

y12n

x

=

5

1 x0 )

8

=

;

1 x0 ) y 2 ( 1+y 2 x 1 y0 ) ; (1+y 2 x 1 y0 )

2 (1+2x (1+x 2 y

2y

1 x0 )

1 x0 )

;

:

Now it follows from System (1) that x12n

2

= =

y12n

2

= =

1

x (1+x

2y

2 (1+2x

1 x0 )

1

4 y12n 3

2 x 1 y0 )

= 1

2y

1 x0 ) x 2 y 1 x0 (1+x 2 y 1 x0 )

x12n 5 1 + x12n 5 y12n 4 x12n y 2 (1 + y

2 (1+2x 2 y 1 x0 ) (1+x 2 y 1 x0 ) x 2 (1+2x 2 y 1 x0 ) y 1 (1+x 2 y 1 x0 ) (1+2x 2 y 1 x0 ) (1+x 2 y 1 x0 )

x

y12n 5 y12n 5 x12n

=x

x0 2 y 1 x0 )

2; y

= 3

(1+y

1+ (1+y

y 2 x 1 y0 (1+y 2 x 1 y0 )

1

(1+x

y

2 x 2 x 1 y0 )

=y

1(

2 2 x 1 y0 )

1+y

2x

1 y0 ) (

1+y

y0 2x

1 y0 )

2:

Similarly, we can prove the other relations. Then the proof is so complete. Example 1. Here we consider an interesting numerical example for System (1) with the initial conditions x 2 = 0:9, x 1 = 0:4, x0 = 0:3, y 2 = 5, y 1 = 7 and y0 = 2: (See Fig.1). plot of X(n+1)=Y(n-2)/(-1-Y(n-2)X(n-1)Y(n)),Y(n+1)=X(n-2)/(1+X(n-2)Y(n-1)X(n)) 8 X(n) Y(n) 6

4

x(n),y(n)

2

0

-2

-4

-6

0

5

10

15

20

25 n

30

Figure 1.

2

853

35

40

45

50

EL-METWALLY, ELSAYED: SOLVING DIFFERENCE EQUATIONS

3

yn 2 1 yn 2 xn

Second system: xn+1 =

xn 2 1+xn 2 yn

; yn+1 =

1 yn

1 xn

In this section, we study the solutions for the following system of di¤erence equations xn+1 =

yn 2 yn 2 xn

1

1 yn

;

xn 2 1 + xn 2 yn

yn+1 =

1 xn

;

(2)

where n 2 N0 and the initial conditions are arbitrary real numbers: Theorem 2 Assume that fxn ; yn g be a solution for System (2). Then for n = 0; 1; 2; :::; x6n

2

= x

2

n Y1

( 1+(6i)x 2 y 1 x0 )( 1+(6i+3)x 2 y 1 x0 ) ( 1+(6i+1)x 2 y 1 x0 )( 1+(6i+4)x 2 y 1 x0 ) ;

i=0

x6n

1

= x

1

n Y1

(1+(6i+1)y (1+(6i+2)y

2x

1 y0 )(1+(6i+4)y

2x

1 y0 )

2x

1 y0 )(1+(6i+5)y

2x

1 y0 )

( 1+(6i+2)x ( 1+(6i+3)x

2y

1 x0 )(

2y

1 x0 )(

;

i=0

x6n

= x0

n Y1

1+(6i+5)x 1+(6i+6)x

2y

1 x0 )

2y

1 x0 )

;

i=0

x6n+1

=

y 2 ( 1 y 2x

1 y0 )

n Y1

(1+(6i+3)y (1+(6i+4)y

2x

1 y0 )(1+(6i+6)y

2x

1 y0 )

2x

1 y0 )(1+(6i+7)y

2x

1 y0 )

;

i=0

x6n+2

y 1 ( 1+x 2 y 1 x0 ) ( 1+2x 2 y 1 x0 )

=

n Y1

( 1+(6i+4)x ( 1+(6i+5)x

2y

1 x0 )(

2y

1 x0 )(

1+(6i+7)x 1+(6i+8)x

2y

1 x0 )

2y

1 x0 )

;

i=0

x6n+3

y0 (1+2y 2 x 1 y0 ) (1+3y 2 x 1 y0 )

=

n Y1

(1+(6i+5)y (1+(6i+6)y

2x

1 y0 )(1+(6i+8)y

2x

1 y0 )

2x

1 y0 )(1+(6i+9)y

2x

1 y0 )

;

i=0

y6n

2

= y

2

n Y1

(1+(6i)y 2 x 1 y0 )(1+(6i+3)y 2 x 1 y0 ) (1+(6i+1)y 2 x 1 y0 )(1+(6i+4)y 2 x 1 y0 ) ;

i=0

y6n

1

= y

1

n Y1

( 1+(6i+1)x ( 1+(6i+2)x

2y

1 x0 )(

2y

1 x0 )(

1+(6i+4)x 1+(6i+5)x

2y

1 x0 )

2y

1 x0 )

;

i=0

y6n

= y0

n Y1

(1+(6i+2)y (1+(6i+3)y

2x

1 y0 )(1+(6i+5)y

2x

1 y0 )

2x

1 y0 )(1+(6i+6)y

2x

1 y0 )

;

i=0

y6n+1

=

x 2 ( 1+x 2 y

1 x0 )

n Y1

( 1+(6i+3)x ( 1+(6i+4)x

2y

1 x0 )(

2y

1 x0 )(

1+(6i+6)x 1+(6i+7)x

2y

1 x0 )

2y

1 x0 )

;

i=0

y6n+2

n Y1

x 1 (1+y 2 x 1 y0 ) (1+2y 2 x 1 y0 )

=

(1+(6i+4)y (1+(6i+5)y

2x

1 y0 )(1+(6i+7)y

2x

1 y0 )

2x

1 y0 )(1+(6i+8)y

2x

1 y0 )

;

i=0

y6n+3

x0 ( 1+2x 2 y 1 x0 ) ( 1+3x 2 y 1 x0 )

=

n Y1

( 1+(6i+5)x ( 1+(6i+6)x

2y

1 x0 )(

2y

1 x0 )(

1+(6i+8)x 1+(6i+9)x

2y

1 x0 )

2y

1 x0 )

:

i=0

Proof: For n = 0 the result holds. Now suppose that n > 1 and that our assumption holds for n 1; that is x6n

8

= x

2

n Y2

( 1+(6i)x 2 y 1 x0 )( 1+(6i+3)x 2 y 1 x0 ) ( 1+(6i+1)x 2 y 1 x0 )( 1+(6i+4)x 2 y 1 x0 ) ;

i=0

x6n

7

= x

1

n Y2

(1+(6i+1)y (1+(6i+2)y

2x

1 y0 )(1+(6i+4)y

2x

1 y0 )

2x

1 y0 )(1+(6i+5)y

2x

1 y0 )

( 1+(6i+2)x ( 1+(6i+3)x

2y

1 x0 )(

2y

1 x0 )(

;

i=0

x6n

6

= x0

n Y2

1+(6i+5)x 1+(6i+6)x

2y

1 x0 )

2y

1 x0 )

;

i=0

x6n

5

=

y 2 ( 1 y 2x

1 y0 )

n Y2

(1+(6i+3)y (1+(6i+4)y

2x

1 y0 )(1+(6i+6)y

2x

1 y0 )

2x

1 y0 )(1+(6i+7)y

2x

1 y0 )

;

i=0

x6n

4

=

y 1 ( 1+x 2 y 1 x0 ) ( 1+2x 2 y 1 x0 )

n Y2

( 1+(6i+4)x ( 1+(6i+5)x

2y

1 x0 )(

2y

1 x0 )(

1+(6i+7)x 1+(6i+8)x

2y

1 x0 )

2y

1 x0 )

i=0

x6n

3

=

y0 (1+2y 2 x 1 y0 ) (1+3y 2 x 1 y0 )

n Y2

(1+(6i+5)y (1+(6i+6)y

i=0

3

854

2x

1 y0 )(1+(6i+8)y

2x

1 y0 )

2x

1 y0 )(1+(6i+9)y

2x

1 y0 )

;

;

EL-METWALLY, ELSAYED: SOLVING DIFFERENCE EQUATIONS

y6n

= y

8

2

n Y2

(1+(6i)y 2 x 1 y0 )(1+(6i+3)y 2 x 1 y0 ) (1+(6i+1)y 2 x 1 y0 )(1+(6i+4)y 2 x 1 y0 ) ;

i=0

y6n

= y

7

1

n Y2

( 1+(6i+1)x ( 1+(6i+2)x

2y

1 x0 )(

2y

1 x0 )(

1+(6i+4)x 1+(6i+5)x

2y

1 x0 )

2y

1 x0 )

;

i=0

y6n

= y0

6

n Y2

(1+(6i+2)y (1+(6i+3)y

2x

1 y0 )(1+(6i+5)y

2x

1 y0 )

2x

1 y0 )(1+(6i+6)y

2x

1 y0 )

;

i=0

y6n

x 2 ( 1+x 2 y

=

5

n Y2

1 x0 )

( 1+(6i+3)x ( 1+(6i+4)x

2y

1 x0 )(

2y

1 x0 )(

1+(6i+6)x 1+(6i+7)x

2y

1 x0 )

2y

1 x0 )

;

i=0

y6n

x 1 (1+y 2 x 1 y0 ) (1+2y 2 x 1 y0 )

=

4

n Y2

(1+(6i+4)y (1+(6i+5)y

2x

1 y0 )(1+(6i+7)y

2x

1 y0 )

2x

1 y0 )(1+(6i+8)y

2x

1 y0 )

;

i=0

y6n

x0 ( 1+2x 2 y 1 x0 ) ( 1+3x 2 y 1 x0 )

=

3

n Y2

( 1+(6i+5)x ( 1+(6i+6)x

2y

1 x0 )(

2y

1 x0 )(

1+(6i+8)x 1+(6i+9)x

2y

1 x0 )

2y

1 x0 )

:

i=0

It follows from System (2) that x6n

2

=

1

=

0

y6n 5 y6n 5 x6n

4 y6n 3

x 2 ( 1+x 2 y

1 x0 ) i=0

x

1 x0 )

y 1 ( 1+x 2 y 1 x0 ) ( 1+2x 2 y 1 x0 ) x0 ( 1+2x 2 y 1 x0 ) ( 1+3x 2 y 1 x0 )

x 2 ( 1+x 2 y

=

nQ2

x 2 ( 1+x 2 y

1

B B B B B B B B B B B @

2(

=

nQ2

1 x0 ) i=0

; ( 1+(6i+3)x ( 1+(6i+4)x

nQ2

i=0 nQ2

i=0 nQ2 i=0

2y

1 x0 )(

2y

1 x0 )(

( 1+(6i+3)x ( 1+(6i+4)x

2y

1+(6i+6)x 1+(6i+7)x

1 x0 )

1+(6i+6)x 2 y 1 x0 )( 1+(6i+7)x

( 1+(6i+4)x ( 1+(6i+5)x

2y

1 x0 )(

2y

1 x0 )(

( 1+(6i+5)x ( 1+(6i+6)x

2y

1 x0 )(

2y

1 x0 )(

nQ2

( 1+(6i+3)x ( 1+(6i+4)x (1 (6n 3)x 2 y 1 x0 x i=0

1 x0 ) 1 x0 )

1 x0 )(

2y

1 x0 )(

2y

1 x0 )(

2y

2y

1 x0 )

2y

1 x0 )

1+(6i+7)x 1+(6i+8)x

2y

1 x0 )

2y

1 x0 )

1+(6i+8)x 1+(6i+9)x

2y

1 x0 )

2y

1 x0 )

( 1+(6i+3)x 2 y 1 x0 )( 1+(6i+6)x ( 1+(6i+4)x 2 y 1 x0 )( 1+(6i+7)x x 2 y 1 x0 1 ( 1+(6n 3)x y x ) 2 1 0

1+(6n 3)x 2 y ( 1+x 2 y 1 x0 )

2y 2y

2y

1 x0 )

2y

1 x0 )

1+(6i+6)x 1+(6i+7)x

2y

1 x0 )

2y

1 x0 )

1 x0 )

1 C C C C C C C C C C C A

:

Then we see that x6n

2

=x

2

n Y1

( 1+(6i)x 2 y 1 x0 )( 1+(6i+3)x 2 y 1 x0 ) ( 1+(6i+1)x 2 y 1 x0 )( 1+(6i+4)x 2 y 1 x0 ) :

i=0

Again we see from System (2) that y6n

1

x6n 4 1 + x6n 4 y6n

=

3 x6n 2

y 1 ( 1+x 2 y 1 x0 ) ( 1+2x 2 y 1 x0 )

( 1+(6i+4)x 2 y 1 x0 )( 1+(6i+7)x 2 y ( 1+(6i+5)x 2 y 1 x0 )( 1+(6i+8)x 2 y i=0 n 2 y 1 ( 1+x 2 y 1 x0 ) Q ( 1+(6i+4)x 2 y 1 x0 )( 1+(6i+7)x ( 1+2x 2 y 1 x0 ) ( 1+(6i+5)x 2 y 1 x0 )( 1+(6i+8)x i=0 nQ2 x0 ( 1+2x 2 y 1 x0 ) ( 1+(6i+5)x 2 y 1 x0 )( 1+(6i+8)x ( 1+3x 2 y 1 x0 ) ( 1+(6i+6)x 2 y 1 x0 )( 1+(6i+9)x i=0 nQ1 ( 1+(6i)x 2 y 1 x0 )( 1+(6i+3)x 2 y 1 x0 ) x 2 ( 1+(6i+1)x 2 y 1 x0 )( 1+(6i+4)x 2 y 1 x0 ) i=0

0

=

y

B B B 1+B B B @

1 ( 1+x 2 y 1 x0 ) ( 1+2x 2 y 1 x0 )

=

nQ2 i=0

1+ y

1(

1+x

2y

( 1+(6i+4)x ( 1+(6i+5)x

Then 1

=y

1

n Y1

2y

1 x0 )(

2y

1 x0 )(

x 2 y 1 x0 ( 1+(6n 2)x 2 y

1 x0 )( 1+(6n 2)x ( 1+2x 2 y 1 x0 )

=

y6n

nQ2

2y

1 x0 )

1+(6i+7)x 1+(6i+8)x

2y

1 x0 )

2y

1 x0 )

1 x0 ) 1 x0 ) 2y

1 x0 )

2y

1 x0 )

2y

1 x0 )

2y

1 x0 )

1 C C C C C C A

1 x0 )

nQ2 i=0

( 1+(6i+4)x ( 1+(6i+5)x

1 + (6n

2)x

( 1+(6i+1)x ( 1+(6i+2)x

2y

1 x0 )(

2y

1 x0 )(

2 y 1 x0

+x

2y

1 x0 )(

2y

1 x0 )(

1+(6i+7)x 1+(6i+8)x

2y

1 x0 )

2y

1 x0 )

2 y 1 x0

1+(6i+4)x 1+(6i+5)x

2y

1 x0 )

2y

1 x0 )

:

i=0

Similarly we can prove the other relations. This completes the proof. Lemma 1. Let fxn ; yn g be a solution for System (2) with the initial conditions x x 1 , x0 , y 2 , y 1 , y0 2 R, then the following statements are true: 4

855

2,

EL-METWALLY, ELSAYED: SOLVING DIFFERENCE EQUATIONS

(i) If x 2 = 0; y 1 6= 0; x0 6= 0; then we have x6n 2 = y6n+1 = x0 ; x6n+2 = y 1 ; y6n 1 = y 1 ; y6n+3 = x0 : (ii) If x 1 = 0; y 2 6= 0; y0 6= 0; then we have x6n 1 = y6n+2 = 0 y 2 ; x6n+3 = y0 ; y6n 2 = y 2 ; y6n = y0 : (iii) If x0 = 0; y 1 6= 0; x 2 6= 0; then we have x6n = y6n+3 = 0 x 2 ; x6n+2 = y 1 ; y6n 1 = y 1 ; y6n+1 = x 2 : (iv) If y 2 = 0; x 1 6= 0; y0 6= 0; then we have y6n 2 = x6n+1 = 0 x 1 ; x6n+3 = y0 ; y6n = y0 ; y6n+2 = x 1 : (v) If y 1 = 0; x0 6= 0; x 2 6= 0; then we have y6n 1 = x6n+2 = 0 x 2 ; x6n = x0 ; y6n+1 = x 2 ; y6n+3 = x0 : (vi) If y0 = 0; y 2 6= 0; x 1 6= 0; then we have y6n = x6n+3 = 0 x 1 ; x6n+1 = y 2 ; y6n 2 = y 2 ; y6n+2 = x 1 :

0 and x6n = and x6n+1 = and x6n

2

=

and x6n

1

=

and x6n

2

=

and x6n

1

=

Proof: The proof follows by direct substitutions in the obtained form of the solutions for System (2) in Theorem 2. Example 2. Figure (2) shows the behavior of the solution for System (2) with the initial conditions x 2 = 5, x 1 = 0:4, x0 = 0:13, y 2 = 0:3, y 1 = 0:9 and y0 = 2. plot of X(n+1)=Y(n-2)/(-1-Y(n-2)X(n-1)Y(n)),Y(n+1)=X(n-2)/(-1+X(n-2)Y(n-1)X(n)) 5 x(n) y(n) 4

3

2

x(n),y(n)

1

0

-1

-2

-3

-4

0

10

20

30

40

50

60

70

n

Figure 2. The following theorems can be treated similarly to the previous results.

4

Third system: xn+1 =

yn 2 1 yn 2 xn

1 yn

; yn+1 =

xn 2 1 xn 2 yn

1 xn

In this section, we obtain the form of the solutions for the following system of di¤erence equations xn 2 yn 2 ; yn+1 = ; (3) xn+1 = 1 yn 2 xn 1 yn 1 xn 2 yn 1 xn where the initial conditions are arbitrary real numbers such that x y 2 x 1 y0 6= 1; 6= 12 :

2 y 1 x0

6=

1 and

Theorem 3 Every solution fxn ; yn g for System (3) is periodic with period twelve and has the form 8 9 y 2 y0 (1+2y 2 x 1 y0 ) > < x 2 ; x 1 ; x0 ; (1+y 2 x 1 y0 ) ; y 1 (1 x 2 y 1 x0 ); (1+y 2 x 1 y0 ) ; > = x 1 x 2 (1+x 2 y 1 x0 ) x0 (1 x 2 y 1 x0 ) y 2 (1+2y 2 x 1 y0 ) fxn g = ; ; ; ; ; (1 x 2 y 1 x0 ) (1+2y 2 x 1 y0 ) (1+x 2 y 1 x0 ) (1+y 2 x 1 y0 ) > > : ; y0 y 1 (1 + x 2 y 1 x0 ); (1+y 2 x 1 y0 ) ; x 2 ; x 1 ; x0 ; :::

and

fyn g =

5

(

y

x 2 2 ; y 1 ; y0 ; (1 x 2 y x 2 (1 x 2 y 1 x0 ) ;

1 (1+y 2 x 1 y0 ) ; x (1+2y ; (1+x x20y 1 x0 ) ; y 2 ; y 1 ; 1 x0 ) 2 x 1 y0 ) x 1 (1+y 2 x 1 y0 ) x0 (1+2y 2 x 1 y0 ) ; (1+x 2 y 1 x0 ) ; y 2 ; y 1 ; y0 ; :::

Fourth system: xn+1 =

yn 2 1 yn 2 xn

1 yn

; yn+1 =

y0 ;

xn 2 1 xn 2 yn

)

:

1 xn

We get, in this section, the form of the solutions for the following system of di¤erence equations xn 2 yn 2 ; yn+1 = ; (4) xn+1 = 1 yn 2 xn 1 yn 1 xn 2 yn 1 xn where n 2 N0 and the initial conditions are arbitrary real numbers such that x 1 and y 2 x 1 y0 = 6 1: 1

Theorem 4 Let fxn ; yn gn=

2

is a solution of System (4) then 5

856

2 y 1 x0

6=

EL-METWALLY, ELSAYED: SOLVING DIFFERENCE EQUATIONS

1

2

is a periodic solution with period six i.e. xn+6 = xn ; yn+6 = yn for

1

2

has the following form

1. fxn ; yn gn= all n 2. 2. fxn ; yn gn= x6n

2

= x

x6n+2

=

y6n

= y

2;

y

x6n

1 (1

1

+x

=x

1;

x6n = x0 ;

2 y 1 x0 );

x6n+3 =

x6n+1 = (1 + y

y0 2x

y (1 + y 1 y0 )

2

2 x 1 y0 )

;

;

and 2

y6n+2

=

2;

x

y6n 1 (1

1

+y

=y

1;

y6n = y0 ;

2 x 1 y0 );

y6n+3 =

y6n+1 = (1 + x

x0 2y

x (1 + x 1 x0 )

2

2 y 1 x0 )

;

:

Lemma 2. All solutions for System (4) are periodic of period three i¤ y 2 x 1 y0 = 2, y 2 = x 2 , y 1 = x 1 and y0 = x0 ; and has the form f:::; x 2 ; x 1 ; x0 ; x 2 ; x 1 ; x0 ; :::g.

References [1] R. P. Agarwal and E. M. Elsayed, On the solution of fourth-order rational recursive sequence, Advanced Studies in Contemporary Mathematics, 20 (4), (2010), 525–545. [2] N. Battaloglu, C. Cinar and I. Yalç¬nkaya, The dynamics of the di¤erence equation, ARS Combinatoria, 97 (2010), 281-288. [3] E. M. Elabbasy , H. El-Metwally and E. M. Elsayed, Global behavior of the solutions of di¤erence equation, Advances in Di¤erence Equations 2011, 2011:28 doi:10.1186/1687-1847-2011-28. [4] E. M. Elsayed, Dynamics of recursive sequence of order two, Kyungpook Mathematical Journal, 50 (2010), 483-497. [5] E. M. Elsayed, Behavior of a rational recursive sequences, Studia Univ. " BabesBolyai ", Mathematica, LVI (1) (2011), 27–42. [6] E. M. Elsayed, Solution of a recursive sequence of order ten, General Mathematics, 19 (1) (2011), 145–162. [7] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Disc. Dyn. Nat. Soc., Volume 2011, Article ID 982309, 17 pages. [8] E. M. Elsayed, On the solution of some di¤erence equations, European J. Pure Appl. Math., 4 (3) (2011), 287-303. [9] E. M. Elsayed, On the dynamics of a higher order rational recursive sequence, Commun. Math. Anal., 12 (1) (2012), 117–133. [10] E. M. Elsayed, Solutions of rational di¤erence system of order two, Math. Comput. Mod., 55 (2012), 378–384. [11] E. M. Elsayed, M. M. El-Dessoky and A. Alotaibi, On the solutions of a general system of di¤erence equations, Disc. Dyn. Nat. Soc., Volume 2012, Article ID 892571, 12 pages. [12] M. E. Erdo¼ gan, C. Cinar and I. Yalç¬nkaya, On the dynamics of the recursive sequence, Comp. Math. Appl., 61 (2011), 533-537. [13] A. S. Kurbanli, C. Cinar and I. Yalç¬nkaya, On the behavior of positive solutions of the system of rational di¤erence equations, Math. Comp. Mod., 53 (2011), 1261-1267. [14] N. Touafek and E. M. Elsayed, On the solutions of systems of rational di¤erence equations, Math. Comput. Mod., 55 (2012), 1987–1997. [15] I. Yalcinkaya, On the global asymptotic stability of a second-order system of di¤erence equations, Disc. Dyn. Nat. Soc., Vol. 2008, Article ID 860152 (2008), 12 pages. [16] I. Yalç¬nkaya, On the global asymptotic behavior of a system of two nonlinear di¤erence equations, ARS Combinatoria, 95 (2010), 151-159. [17] I. Yalcinkaya, C. Cinar and M. Atalay, On the solutions of systems of di¤erence equations, Advances in Di¤erence Equations, 2008 (2008) Article ID 143943, 9 pages.

6

857

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 858-867, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Scaling Bini’s Algorithm for Fast Inversion of Triangular Toeplitz Matrices∗ Jie Huang†, Ting-Zhu Huang‡

Skander Belhaj§

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P. R. China

University of Tunis El Manar, ENIT-LAMSIN, BP 37, 1002, Tunis, Tunisia

Abstract In this paper, motivated by Lin, Ching and Ng [Theoretical Computer Science, 315:511523 (2004)], a scaling version of Bini’s algorithm [SIAM J. Comput., 13:268-276 (1984)] for an approximate inversion of a triangular Toeplitz matrix is proposed. The scaling algorithm introduces a new scale parameter and is mathematically equivalent to the original Bini’s. Its computational cost is about two fast Fourier transforms of n-vectors (FFTs(n)), equal to that of Bini’s. We also improve the accuracy of the proposed approach by embedding the n-by-n triangular Toeplitz matrix into an (n + n0 )-by-(n + n0 ) triangular (banded) Toeplitz matrix, where n0 is a positive integer. The complexity of the resulting revised scaling Bini’s algorithm is about two FFTs(2n). Several numerical examples are given to illustrate the effectiveness and stability of the proposed methods. Key words: Bini’s algorithm; Toeplitz matrix; Fast Fourier transform; Inverse; Triangular matrix AMSC (2010): 65F05; 65F30

1

Introduction

An n-by-n matrix Tn = [tj,k ; j, k = 0, 1, · · · , n − 1] is said to be Toeplitz if it is constant along its diagonals, i.e., tj,k = tj−k . Such matrices arise in a variety of applications in mathematics and engineering, such as signal and image processing [6] and minimum realization problems in control theory; see Bunch [5] and the references therein. In this paper, we focus our attention on fast inversion of an n-by-n lower triangular Toeplitz matrix, i.e.,   t0  t1  t0   Tn =  . , .. ..  ..  . . tn−1 · · · t1

t0

where tj , j = 0, 1, · · · , n − 1 are real with t0 6= 0. ∗ This research is supported by NSFC (60973015, 61170311), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020) and Sichuan Province Sci. & Tech. Research Project (12ZC1802). † E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail address: [email protected]

858

Jie Huang, Ting-Zhu Huang, and Skander Belhaj For fast inversion of triangular Toeplitz matrices, Morf in [13] noted that the divide-andconquer strategy yields an algorithm using O(n log n) operations (the same order as a convolution using the fast Fourier transform (FFT)). Then Commmenges and Monsion [7] proposed an algorithm requiring O(n log n) operations for inversion of triangular Toeplitz matrices, more precisely, about 10 fast Fourier transforms (FFTs) of n-vectors (FFT(n)). After several years of intensive study, the approximate approach using the fast Fourier transform for inversion of triangular Toeplitz matrices has been studied by Bini [2] and Georgiev [9]. Considering triangular Toeplitz matrices versus polynomials [14, 3, 4, 15, 16, 17], many available techniques for polynomial division can be used, such as Knuth Sieveking-Kung [11] and Bini & Pan [3]. Recently, Lin, Ching and Ng put forward an approximate inversion method for triangular Toeplitz matrices based on trigonometric polynomial interpolation [12]. Moreover, they proposed a revised Bini’s algorithm for a triangular Toeplitz matrix inversion. In this paper, we propose scaling and revised scaling Bini’s methods for fast inversion of triangular Toeplitz matrices. The basic idea is to introduce a new scale parameter used by Lin, Ching and Ng and choose a distinct numerical value. The proposed algorithms are mathematically equivalent to the original ones without scaling, respectively. Therefore, there is not additional computational cost. Several numerical examples are given to illustrate the effectiveness and stability of the proposed algorithms. The outline of the paper is as follows. In section 2, we provide an improvement on a result of Bini for computing the approximate inversion of a lower triangular Toeplitz matrix and then give a scaling algorithm. We present also the revised version of Bini’s algorithm which brings us back to derive the revised scaling Bini’s Algorithm [12]. Section 3 uses some numerical experiments to demonstrate the efficiency and necessity of this modification. Finally, section 4 draws some conclusions.

2 2.1

The scaling version of Bini’s algorithm Inversion based on Bini’s method

Before we move into fast algorithms for triangular Toeplitz matrix inversion, we first present a common special, but very important case of Toeplitz matrices: circulant matrix. An nby-n Toeplitz matrix Cn ([Cn ]jk = cj−k ) is said to be circulant if it satisfies c−k = cn−k for k = 1, 2, · · · , n − 1. There are many important properties of a circulant matrix. Of particular importance to us is that all circulant matrices can be diagonalized by the Fourier matrix, and the multiplication of a circulant matrix to a vector can be done in O(n log n) operations by using FFT [10, 8]. In this event, Bini [2] proposed an approximate approach for fast inversion of triangular Toeplitz matrix Tn . For all n ≥ 1, let Hn = [hjk ]nj,k=1 be the lower shift matrix with ones on the first subdiagonal and zeros elsewhere. We see that n−1 X Tn = tj Hnj . j=0 (ε)

(ε)

(ε)

The basic idea of Bini’s algorithm is to use Hn = [hjk ]nj,k=1 to approximate Hn , where hjk = hjk (ε)

for (j, k) 6= (1, n) and h1n = εn , here εn is a small positive number. It follows that Tn can be

859

Scaling Bini’s Algorithm for Triangular Toeplitz Inversion

approximated by the circulant matrix Tn(ε) =

n−1 X

tj (Hn(ε) )j .

j=0 (ε)

Let Dn

= diag(1, ε, · · · , εn−1 ), Dn = diag(d), where d =

(ε) −1

(ε) −1

(ε) −1

√

(ε)

nFn Dn [tj ]n−1 j=0 , then we have (ε)

(Tn ) by using the decomposition (Tn ) = (Dn ) Fn∗ Dn−1 Fn Dn , where Fn is the n-by-n Fourier matrix. On the other hand, it is well-known that the inverse of a triangular Toeplitz matrix is also a triangular Toeplitz matrix. Hence, the Bini’s Algorithm for computing the first (ε) −1

column b(ε) of (Tn ) can be concluded as follows. Algorithm 1. Bini’s algorithm Step 0: Choose ε ∈ (0,1). Compute t˜j = tj εj , for j = 0, 1, · · · , n − 1. √ Step 1: Compute d = ( nFn )t˜. Step 2: Compute c = [cj ]n−1 = [1/dj ]n−1 j=0 j=0 . √ ∗ Step 3: Compute f = (Fn / n)c. (ε) j n−1 Step 4: Compute b(ε) = [bj ]n−1 j=0 = [fj /ε ]j=0 . We note that the computational cost of Bini’s algorithm is about two FFT(n). Revised Bini’s algorithm [12, Algorithm 2] was proposed to obtain a faster and more accurate approximate inverse by embedding the n-by-n triangular Toeplitz matrix into an (n + n0 )-by(n + n0 ) triangular (banded) Toeplitz matrix, where n0 is a positive integer. For simplicity, they set n0 = n and stated the revised algorithm as follows. Algorithm 2. Revised Bini’s algorithm Step 0: Choose ε ∈ (0,1). Compute t˜j = tj εj , for j = 0, 1, · · · , n − 1, and set t˜j = 0 for j = n, n + 1, · · · , 2n − 1. √ Step 1: Compute d = ( 2nF2n )[t˜j ]2n−1 j=0 . 2n−1 Step 2: Compute c = [cj ]j=0 = [1/dj ]2n−1 j=0 . √ ∗ Step 3: Compute f = (F2n / 2n)c. (ε) j n−1 Step 4: Compute b(ε) = [bj ]n−1 j=0 = [fj /ε ]j=0 . Clearly the computational cost of Algorithm 2 is about two FFT(2n), about twice that of Algorithm 1. In Algorithms 1 and 2, special attention should be paid to the choice of parameter ε. Recall (ε) (ε) that we use Hn to approximate Hn , so that Tn is well approximated by Tn . Thus, theoretically, n the smaller ε , the more accurate the approximate inverse will be. However, notice that if εn (ε) is close to zero, Dn will be very ill-conditioned for large n, the computed vector c by Step 2 in Algorithms 1 and 2 will be therefore not accurate, consequently, the computed b(ε) will not be a meaningful approximation solution. Hence, it is both necessary and important to choose a suitable value of parameter ε to balance the two facts. In fact, it has been pointed out in [12] that for a more accurate numerical inverse and a decreased rounding error, ε = (0.5 × 10−8 )1/n and ε = 10−5/n are good choices for Bini’s and revised Bini’s algorithm, respectively, which is consistent with our numerical tests.

2.2

Inversion based on scaling Bini’s Algorithm

In this subsection we present our main algorithm. By introducing a simple scale parameter which is crucial in practice based on the Bini’s algorithm for a triangular Toeplitz matrix

860

Jie Huang, Ting-Zhu Huang, and Skander Belhaj

inversion, we obtain a faster and more accurate approximate inversion. Recall that the inverse of triangular Toeplitz matrix is also triangular Toeplitz, i.e.,  Tn−1

  = 

−1

t0 t1 .. .

t0 ..

. . .. · · · t1 t0

tn−1

   

   = 

t0 ρt1 .. .

  (ρ) Moreover, from [12], for any ρ, define Tn :=  

ρn−1 tn−1

−1

(Tn(ρ) )

  = 

b0 ..

bn−1 





b0 b1 .. .

b0 ρb1 .. . ρn−1 bn−1

. . .. · · · b1 b0

  . 

 t0 ..

. . .. · · · ρt1 t0

  , then we have 

 b0 ..

. . .. · · · ρb1 b0

  . 

Using this scaling technique, i.e., ρ = 2−18/n ∈ (0, 1), the authors in [12] improves the accuracy of trigonometric polynomial interpolation based approximate solution remarkably. Thus with a similar approach, we give a scaling version of Bini’s algorithm, called the scaling Bini’s algorithm. Algorithm 3. Scaling Bini’s algorithm Step 0: Choose a suitable value ε0 ∈ (0, 1). Choose a suitable ρ and compute t˜j = εj0 tj ρj , j = 0, 1, · · · , n − 1. √ Step 1: Compute d = ( nFn )t˜. Step 2: Compute c = [cj ]n−1 = [1/dj ]n−1 j=0 j=0 . √ ∗ Step 3: Compute f = (Fn / n)c. j n−1 Step 4: Compute [ˆbj ]n−1 j=0 = [fj /(ε0 ρ) ]j=0 . We note that the scaling Bini’s algorithm is mathematically equivalent to the original Bini’s by assuming ε = ρε0 . Here our contribution is mainly to choose a distinct numerical value of ρ (numerically we assume ε0 = ε), to improve the numerical stability of Bini’s algorithm. Thus, the computational cost of Algorithm 3 is the same to that of Algorithm 1, about two FFT(n), equal to that of Bini’s algorithm (Algorithm 1) and half of the revised one (Algorithm 2). Moreover, we have the following two remarks. Remark 1 Although the two parameters ε0 and ρ appear together, their properties and behavior are quite different form both the computational and the theoretical point of view. The parameter (ρ) ε0 is used to obtain a circulant approximation of Tn , while the aim of ρ is to improve the accuracy of the computed first column of Tn−1 as a scale parameter of Algorithm 1. In addition, we assume ε0 = (0.5 × 10−8 )1/n in Algorithm 3 in the rest of paper since it is always a good choice and is not dependent on the choice of ρ unless it is meaningful. Remark 2 It is important to choose a suitable value of parameter ρ since only a suitable ρ can improve the accuracy of the computed inverse. Recall that the authors in [12] limited ρ ∈ (0, 1) to make the parameter ρ meaningful. Moreover, they noted that ρ = 2−18/n is a good choice. Here, we note that the parameter ρ in Algorithm 3 does not have to be limit in (0,1) strictly

861

Scaling Bini’s Algorithm for Triangular Toeplitz Inversion

and the computed results can also be improved remarkably with ρ: |ρ − 1| < δ, and lim ρ = 1, n→∞

where δ(< 0.1) is a small positive number. Moreover, our numerical tests show that ρ = 223/n is a good choice. To estimate the accuracy of [ˆbj ]n−1 j=0 computed by Algorithm 3, we give a useful lemma and a followed theorem, which can be obtained immediately from [12, Theorem 2]. (ρ,ε0 ) n−1 ]j=0

Lemma 1 Let b(ρ,ε0 ) = [bj (ρ)

inverse of Tn

(ρ)

and b(ρ) = [bj ]n−1 j=0 be the first columns of the approximate

from Bini’s algorithm and the exact inversion, respectively. Then (ρ,ε0 )

bj More precisely, (ρ,ε0 )

bj

(ρ)

− bj

= εn0

∞ X

(ρ)

− bj

= O(εn0 ).

(k−1)n (ρ) bj+kn ,

ε0

j = 0, 1, . . . , n − 1.

k=1 n−1 Theorem 1 Let ˆb = [ˆbj ]n−1 j=0 and b = [bj ]j=0 be the first columns of the approximate inverse of Tn computed by Algorithm 3 and the exact inversion, respectively. Then,

ˆbj − bj = O (εn ρn ) , j = 0, 1, · · · , n − 1. 0 n−1 Proof. From the assumption of ˆb = [ˆbj ]n−1 j=0 and b = [bj ]j=0 , we have

ˆbj = b(ρ,ε0 ) /ρj , j

(ρ)

bj = bj /ρj ,

for j = 0, 1, · · · , n − 1. Then Lemma 1 gives ˆbj − bj = εn 0

∞ X

(k−1)n kn

ε0

ρ bj+kn = (ε0 ρ)n

k=1

∞ X

(ε0 ρ)(k−1)n bj+kn ,

k=1

for j = 0, 1, . . . , n − 1; the desired result. 2 j From the above result, numerically we see that if ρ is too small, computing 1/ρ will bring in very large rounding error for large j. However, recall from [12] that if ρ is too large, then bn−1 ρn−1 may be infinite for large n, which leads to a meaningless approximate inverse. So we have to choose a suitable value of ρ to balance these two facts. It is worth noting that the optimal value of ρ for all situations is unlikely to exist. Nonetheless, in many applications, ρn = 223/n is a good choice for Algorithm 3 to improve the accuracy of the approximate solution to a great extent. Based on Lin, Ching and Ng [12] techniques, we can further improve the accuracy of our scaling Bini’s algorithm by embedding the n-by-n triangular Toeplitz matrix into an (n + n0 )by-(n + n0 ) triangular (banded) Toeplitz matrix, where n0 is a positive integer. For simplicity, we set n0 = n and give the revised scaling Bini’s Algorithm as follows. Algorithm 4. Revised scaling Bini’s algorithm Step 0: Choose a suitable value ε0 ∈ (0, 1). Choose a suitable ρ and compute t˜j = εj0 tj ρj , j = 0, 1, · · · , n − 1. Set t˜j = 0 for √ j = n, n + 1, · · · , 2n − 1. Step 1: Compute d = ( 2nF2n )[t˜j ]2n−1 j=0 .

862

Jie Huang, Ting-Zhu Huang, and Skander Belhaj 2n−1 Step 2: Compute c = [cj ]2n−1 j=0√ = [1/dj ]j=0 . ∗ / 2n)c. Step 3: Compute f = (F2n n−1 Step 4: Compute [ˆbj ]j=0 = [fj /(ε0 ρ)j ]n−1 j=0 . We remark that the revised scaling Bini’s algorithm is also mathematically equivalent to the revised Bini’s algorithm from [12] by replacing ε with two functional different parameters ε0 and ρ. The computational cost of Algorithm 4 is about two FFT(2n). To end this section, we give a simple numerical example to further illustrate the effect of rounding error of the scaling Bini’s algorithm. Let Tn be the lower triangular Toeplitz matrix with the first column given by

tj = 0.5j , j = 0, 1, · · · , n − 1. Let b and ˆb are the first columns of the exact inverse of Tn and the approximate inversion from Algorithm 3, respectively. We assume ε0 = (0.5 × 10−8 )1/n , ρ = 223/n (scaling Bini) and ρ = 1 (Bini), respectively. The errors in the numerical results ˆb of our scaling Bini’s algorithm for ρ = 223/4096 and ρ = 1 are shown in Fig. 1. Having n form 2 to 4096, Fig. 2 depicts the relative errors for ρ = 223/n and ρ = 1. It also shows, for n > 40 and ρ = 223/n , the accuracy of the numerical inversion can be improved to a great degree, which is consistent with the theoretical analysis. From Figs. 1 and 2, it is worthwhile stressing that setting ρ = 223/n instead of 1 only, without any additional computational cost, the approximate inverse is shown to be more stable and efficient.

Figure 1: log10 (|ˆb − b|) for tj = 0.5j , j = 0, 1, · · · , 4095 for ρ = 223/4096 and ρ = 1.

3

Numerical experiments

We shall demonstrate the effectiveness of the scaling and revised scaling Bini’s algorithms. In particular, we will compare its performance with Bini’s algorithm, the revised Bini’s algo-

863

Scaling Bini’s Algorithm for Triangular Toeplitz Inversion

ˆ

1 j Figure 2: log10 ( kb−bk kbk1 ) for tj = 0.5 , j = 0, 1, · · · , n, n = 1, 2, · · · , 4095.

rithm and the algorithm based on trigonometric polynomial interpolation [12]. Seven different sequences of lower triangular Toeplitz matrices are tested. They are (1) tj = 1/(j + 1)3 , j = 0, 1, · · · , n − 1, (2) tj = 0.1j , j = 0, 1, · · · , n − 1, (3) tj = 0.5j , j = 0, 1, · · · , n − 1, (4) tj = 0.9j , j = 0, 1, · · · , n − 1, (5) t0 = 1, t1 = 1/2 and tj = 0, j = 2, 3, · · · , n − 1. (6) t0 = 1, t1 = −1/2, t2 = 1/2 and tj = 0, j = 3, · · · , n − 1. (7) t0 = 1, t1 = −1, t2 = 1/2 and tj = 0, j = 3, · · · , n − 1. We note that the test sequence (1) comes from [12], and sequences (2)-(4) are the lower part of the well-known matrices, a class of Toeplitz test matrices: ai−j = η |i−j| , i = 1, · · · , m; j = 1, · · · , n, the parameter η ∈ (0, 1). Here we choose η = 0.1, 0.5 and 0.9, respectively. Moreover, sequences (5)-(7) which are not diagonally dominant will give some more information. In the following tables, we show the relative accuracy of the approximate inverse

kˆb−bk1 kbk1 ,

where ˆb is the first column of the approximate inverse based on Bini’s algorithm and b is the inverse computed by the divide-and-conquer approach. The second, third and fourth rows display the accuracy of the computed inverses of Bini’s algorithm, the revised Bini’s algorithm and the algorithm based on trigonometric polynomial interpolation [12, Algorithm 1; with ρ = 2−18/n ], respectively. The fifth and sixth rows displays also the accuracy of the computed inverses of our scaling and revised scaling Bini’s Algorithms, respectively. From Tables 1 through 7 we see that all approximate inverses of the five methods are very accurate. When not requiring very high order of accuracy, such as in the Gauss-Seidel iteration for Toeplitz systems, Bini’s algorithm is suitable. In some applications, such as in [1], we need an

864

Jie Huang, Ting-Zhu Huang, and Skander Belhaj

approximation inversion as accurate as possible. In these cases, the revised, scaling and revised scaling Bini’s algorithms are more preferred. In addition, we remark that our scaling Bini’s algorithm requires two FFT(n), a slightly lower computational cost, which is equal to that of Bini’s algorithm and about half cost of the revised or revised scaling Bini’s algorithms. n Bini Revised Interpolation Scaling Revised scaling

128 4.57e-009 1.37e-012 3.00e-011 6.41e-009 8.69e-010

256 3.54e-009 1.75e-012 3.77e-011 7.92e-010 1.34e-011

512 4.73e-009 1.96e-012 4.65e-011 9.85e-011 2.01e-013

1024 5.59e-009 2.15e-012 7.49e-011 1.23e-011 2.94e-015

2048 7.37e-009 3.06e-012 1.16e-010 1.53e-012 1.30e-016

4096 7.71e-009 3.55e-012 1.62e-010 1.92e-013 5.97e-017

8192 8.38e-009 3.44e-009 2.24e-010 2.41e-014 1.25e-016

Table 1: Accuracy for tj = 1/(j + 1)3 , j = 0, 1, · · · , n − 1 n Bini Revised Interpolation Scaling Revised scaling

128 2.71e-009 1.15e-012 1.70e-011 1.06e-015 2.95e-017

256 3.10e-009 1.61e-012 2.68e-011 1.19e-015 3.74e-017

512 3.86e-009 1.51e-012 4.64e-011 9.39e-016 3.43e-017

1024 4.24e-009 1.87e-012 5.44e-011 1.18e-015 2.92e-017

2048 5.19e-009 2.11e-012 8.69e-011 1.15e-015 3.43e-017

4096 5.48e-009 1.82e-012 1.22e-010 1.13e-015 3.19e-017

8192 5.21e-009 1.67e-012 1.73e-010 1.16e-015 3.42e-017

4096 8.14e-009 2.72e-012 1.31e-010 1.89e-015 1.13e-016

8192 8.28e-009 3.11e-012 2.03e-010 2.28e-015 5.86e-017

Table 2: Accuracy for tj = 0.1j , j = 0, 1, · · · , n − 1 n Bini Revised Interpolation Scaling Revised scaling

128 6.45e-009 2.61e-012 1.93e-011 1.27e-015 4.95e-017

256 6.21e-009 2.20e-012 3.39e-011 1.69e-015 5.33e-017

512 6.66e-009 1.97e-012 3.75e-011 2.11e-015 1.11e-016

1024 7.24e-009 2.45e-012 6.82e-011 1.95e-015 1.16e-016

2048 7.14e-009 2.66e-012 8.61e-011 1.90e-015 6.62e-017

Table 3: Accuracy for tj = 0.5j , j = 0, 1, · · · , n − 1

4

Conclusions

In this paper we have presented a scaling Bini’s algorithm for approximate inversion of triangular Toeplitz matrices and performed extensive experiments to verify the performance for some triangular Toeplitz matrices. In particular, comparing with the initial Bini’s algorithm, the scaling one improves the accuracy of the numerical solution remarkably without additional computational cost. We also improve the accuracy of the proposed approach by the revised scaling Bini’s algorithm which requires two FFTs of 2n-vectors.

865

Scaling Bini’s Algorithm for Triangular Toeplitz Inversion

n Bini Revised Interpolation Scaling Revised scaling

128 8.06e-009 4.27e-012 4.50e-011 5.83e-008 1.35e-008

256 6.81e-009 6.74e-012 6.28e-011 8.12e-014 2.95e-016

512 1.41e-008 1.02e-011 1.04e-010 5.17e-015 3.54e-016

1024 1.39e-008 9.87e-012 1.43e-010 5.25e-015 3.42e-016

2048 2.01e-008 8.31e-012 3.07e-010 6.11e-015 3.27e-016

4096 2.31e-008 8.85e-012 3.66e-010 6.08e-015 2.33e-016

8192 2.30e-008 9.43e-012 5.01e-010 6.56e-015 2.64e-016

4096 7.01e-009 2.31e-012 4.78e-010 1.64e-015 1.20e-016

8192 7.11e-009 2.56e-012 1.99e-011 1.74e-015 4.30e-017

Table 4: Accuracy for tj = 0.9j , j = 0, 1, · · · , n − 1

n Bini Revised Interpolation Scaling Revised scaling

128 4.77e-009 1.75e-012 1.28e-006 1.39e-015 6.00e-017

256 6.70e-009 1.32e-012 1.28e-006 1.53e-015 6.62e-017

512 6.08e-009 1.83e-012 1.28e-006 1.66e-015 1.07e-016

1024 5.91e-009 2.53e-012 6.65e-009 1.55e-015 1.16e-016

2048 7.41e-009 2.20e-012 4.78e-010 1.76e-015 1.10e-016

Table 5: Accuracy for t0 = 1, t1 = 1/2, and tj = 0, j = 2, 3, · · · , n − 1

n Bini Revised Interpolation Scaling Revised scaling

128 6.64e-009 2.50e-012 1.25e-006 1.66e-015 6.50e-017

256 4.90e-009 2.00e-012 1.25e-006 1.85e-015 9.85e-017

512 6.40e-009 2.95e-012 1.25e-006 2.05e-015 1.01e-016

1024 8.78e-009 3.23e-012 5.84e-009 1.97e-015 1.31e-016

2048 1.01e-008 2.85e-012 4.42e-010 2.08e-015 1.03e-016

4096 9.10e-009 2.86e-012 4.42e-010 2.05e-015 1.25e-016

8192 1.00e-008 3.03e-012 1.25e-010 2.11e-015 5.78e-017

Table 6: Accuracy for t0 = 1, t1 = −1/2, t2 = 1/2 and tj = 0, j = 3, · · · , n − 1

n Bini Revised Interpolation Scaling Revised scaling

128 1.15e-008 2.60e-012 1.49e-006 2.58e-015 8.12e-017

256 8.64e-009 2.70e-012 1.49e-006 2.94e-015 1.59e-016

512 1.24e-008 3.63e-012 1.49e-006 2.44e-015 7.62e-017

1024 1.16e-008 4.09e-012 8.47e-009 2.73e-015 2.25e-016

2048 1.97e-008 3.93e-012 6.22e-010 3.24e-015 1.68e-016

4096 1.17e-008 3.84e-012 6.22e-010 2.79e-015 1.84e-016

8192 1.82e-008 3.57e-012 1.79e-010 3.20e-015 9.51e-017

Table 7: Accuracy for t0 = 1, t1 = −1, t2 = 1/2 and tj = 0, j = 3, · · · , n − 1.

866

Jie Huang, Ting-Zhu Huang, and Skander Belhaj

References [1] S. Belhaj, A fast method to block-diagonalize a Hankel matrix, Numer. Algorithms, 47:15-34 (2008). [2] D. Bini, Parallel solution of certain Toeplitz linear systems, SIAM J. Comput., 13:268-276 (1984). [3] D. Bini, V. Pan, Improved parallel polynomial division, SIAM J. Comput., 22:617-627 (1993). [4] D. Bini, V. Pan, Polynomial division and its computational complexity, SIAM J. Comput., 2:179-203 (1986).

[5] J. Bunch, Stability of methods for solving Toeplitz systems of equations, SIAM J. Sci. Stat. Comput., 6:349-364 (1985).

[6] R. Chan and M.K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38:427-482 (1996).

[7] D. Commenges, M. Monsion, Fast inversion of triangular Toeplitz matrices, IEEE Trans. Auto. Control, 29:250-251 (1984).

[8] P. Davis, Circulant Matrices, John Wiley & Sons, New York, 1979, p.73. [9] R.E. Georgiev, Inversion of triangular Toeplitz matrices by using the fast fourier transform, J. New Gener. Comput. Sys., 2:247-256 (1989).

[10] R.M. Gray, Toeplitz and Circulant Matrices: A review, Foundations and Trends in Communications and Information Theory, 2:155-239 (2006).

[11] D.E. Knuth, The Art of Computer Programming: Seminumerical Algorithms, Vol. 2. AddisonWesley, 1981, p.420-435.

[12] F.R. Lin, W.K. Ching, M.K. Ng, Fast inversion of triangular Toeplitz matrices, Theor. Comp. Sci., 315:511-523 (2004).

[13] M. Morf, Doubling algorithms for Toeplitz and related equations, Proc. IEEE Int. Conf. on Acoust. Speech and Singual Process, 3:954-959 (1980).

[14] V. Pan, A. Sadikou, E. Landowne, Polynomial division with a remainder by means of evaluation and interpolation, Inform. Process. Lett., 44:149-153 (1992).

[15] V. Pan, E. Landowne, A. Sadikou, Univariate polynomial division with a remainder by means of evaluation and interpolation, Proceedings of the Third IEEE Symposium on Parallel and Distributed Processing, December 1991, Dallas, TX, USA, 1991, pp. 212-217.

[16] V. Pan, Complexity of computations with matrices and polynomials, SIAM Rev., 34:225-262 (1992). [17] A. Sch¨onhage, Asymptotically fast algorithms for the numerical multiplication and division of polynomials with complex coefficients, Proc. EUROCAM, Marseille, 1982.

867

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 868-879, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Approximation with modi…ed Gadjiev-Ibragimov operators in C[0,A] Nazmiye GONUL, Erdal COSKUN [email protected], [email protected] Bülent Ecevit University Faculty of Arts and Sciences Department of Mathematics 67100 ZONGULDAK / TURKEY October 9, 2012

Abstract In this work,we introduce a modi…cation of the Gadjiev-I·bragimov operators as an approximation process in continuous functions on [0,A]. The rates of convergence of this generalization are obtained by means of modulus of continuity. Keywords: Linear positive operators, Gadjiev-Ibragimov operators, Korovkin theorems,Modulus of continuity, Rate of convergence. AMS Class.[2000] 41A36;47A58.

1

Introduction

The problem of the approximation of continuous functions by sequence of linear positive operators have been investigated in many papers [2],[1],[8],[10],[14]. Gadjiev and Ibragimov, de…ned a general sequence of positive operators and studied some approximation properties of this operators [8],[5].Their work was motivated by the development of a general expression that cover other Bernstein type operators [4], [9]. C [0; A] be set of all real valued continuous functions on [0; A] for A < 1 with the …nite norm, kf k := max jf (x)j x2[0;A]

The classical Gadjiev -Ibragimov operators are de…ned by following form: n 2 N and A be positive real numbers, f'n (t)gand f n (t)g be the sequence of functions in C [0; A] such that 'n (0) = 0; n (0) 6= 0,n 2 N for each t 2 [0; A]. Let also f n gbe a sequence of positive numbers such that lim nn = 1 and n !1

868

GONUL, COSKUN: GADJIEV-IBRAGIMOV OPERATORS

lim

n !1 n

1 2

n (0)

= 0. Assume that a sequence of functions of three variables

fKn (x; t; u)g satis…es the following conditions; 1 )Each function of this family is an entire analytic function with respect to u for …xed x; t 2 [0; A]. 2 ) Kn (x; 0; 0) = 1 for any x 2 [0; A] and for all n 2 N, 3 ) Each x 2 [0; A] and #; n 2 N , for a u1 2 R; ) ( # @ 0 ( 1)# # Kn (x; t; u) @u t=0;u=u1 4 ) There exist a number m making (n + m) 2 N0 such that @# Kn (x; t; u) @u#

=

@# @u#

nx

t=0;u=u1

1 1

Kn+m (x; t; u) t=0;u=u1

Consider the sequence of positive linear operators;

Ln (f; x) =

1 X

#=0

f

# n2

n (0)

(

@# Kn (x; t; u) @u#

t=0; u=

n

n (t)

)

(

# n (0))

n

#!

It is shown in [5], [3] that this sequence of operators in special case, consist of the well known Bernstein,Szass, Bernstein –Cholodowsky and Baskakov operators. In [6] for these operators were given a modi…cation and the some results in [3],[5], [12], [13] were also developed. Some approximation properties of modi…ed Gadjiev-Ibragimov operators given [11] and [6] in the setting of polynomial weighted function space.The following theorem is called Korovkin’s Theorem. Theorem 1 fLn g be the sequence of linear positive operators on C[a; b]. If lim kLn (g; x)

n!1

g(x)kC[a;b] = 0

holds for g(x) = 1; x and x2 , then it holds for every f in C[a; b]. For every f 2 C[0; A] modulus of continuity ! introduced in [9]; [10] by the formula !(f; ) = sup jf (t) f (x)j x [0;A];jt xj

Some elementary properties of !(f; ) are collected in the following. Lemma 2 [7] For every C[0; A]; (i) !(f; ) is a monotonically increasing function of ; (ii) lim !(f; ) = 0: !0

(iii) For each positive value of !(f;

)

(1 + )!(f; ):

869

0:

GONUL, COSKUN: GADJIEV-IBRAGIMOV OPERATORS

(iv) For every f 2 C[0; A] and x; t 2 [0; A]: jf (t)

f (x)j

!(f; ) 1 +

jt

xj

:

Now we de…ned a modi…ed Gadjiev- Ibragimov operator and introduce the construction of operators.In section three we present some auxilary result and give some examples. Last section we compute the rate of convergence for the kLn (f; x) f (x)kC[0;A] with the help of modulus of continuity.

2

De…nitions and Construction of Operators

Let f n g; f n g be a sequences of positive numbers such that; limn!1 n = 1 , limn!1 n = 0 and limn!1 n n = 1:Assume that Kn;# (x) be a function n n satisfying the following conditions; 1 ) ( 1)# Kn;# (x) 0 for # 2 N0 ; n 2 N0 and x 2 [0; A]: 2 ) 1 X ( Kn;# (x) #=0

# n)

#!

=1

for n 2 N0 and x 2 [0; A]: 3 ) There exists an integer m such that for all x 2 [0; A] and for some nonnegative integer n + m Kn;# (x) =

nxKn+m;#

1 (x)

Consider the family of linear operators; Ln (f; x) =

1 X

# n

f

#=0

Kn;# (x)

# n)

(

#!

:

(1)

It is clear that the operators Ln (f; x) are positive and linear. Remark 1 By choosing Kn;# (x) = ( 1)# Then we have n = 1 and operators as follows.

n

(nx)# e #!

nx

;m = 0

= n, thus the operators (1) turn out to be Szass

Ln (f; x) =

1 X

f

#=0

870

# n

(nx)# e #!

nx

GONUL, COSKUN: GADJIEV-IBRAGIMOV OPERATORS

3

Main Results

Lemma 3 Ln de…ned by (1) is a linear positive operator from C[0; A] into C[0; A] and having the properties Ln (1; x) = 1 n

Ln (t; x) =

(2)

nx

(3)

n 2 n

Ln t2 ; x =

n 2 nx: n

n (n + m) +

x

n

(4)

Theorem 4 Let fLn g be the sequence of linear positive operators de…ned by (1). Then for each f 2 C[0; A]; lim kLn (f; x)

f (x)kC[0;A] = 0:

n!1

Proof. To prove the theorem it is su¢ cient to show that the conditions of Korovkin Theorem should be satis…ed.Clearly we can write lim kLn (1; x)

1kC[0;A] = 0:

n!1

Using Lemma and condition limn!1 max jLn (t; x)

x(

xj = max

x2[0;A]

n n

x2[0;A]

n

n = 1 ve get n

jAj

1)

n

n

n

1 :

n

From this and using de…nition of norm in C[0; A] we get lim kLn (t; x)

n!1

xkC[0;A] = 0:

Finally , should show the following equality, lim

n!1

Ln (t2 ; x)

x2

C[0;A]

= 0:

From Lemma it follows that Ln (t2 ; x)

x2 =

= x2

2 n n(n

+ m)x2 2 n

2 n 2 n(n n

+ m)

+

1 +n

n nx 2 n

x2

n 2x n

Hence max Ln (t2 ; x)

x2[0;A]

x2 = max x2 x2[0;A]

871

2 n 2 n(n n

+ m)

1+n

n 2x n

GONUL, COSKUN: GADJIEV-IBRAGIMOV OPERATORS

2 n 2 n(n n

A2

n 2 n

1 + n

+ m)

A:

Then using de…nition of norm in C[0; A] we get

lim

n!1

Ln (t2 ; x)

x2

2 n 2 n(n n

lim A2

C[0;A]

n!1

+ m)

1 + lim n n!1

n 2 A: n

So we may write that lim

n!1

= lim

n!1

2 n 2 n(n n nn

+ m) = lim

n!1

lim

nn

2 n 2 2n n n

+ lim

n n!1

n!1

n

lim A2

2 n 2 n(n n

+ lim

n!1

n lim

n!1

n

2 n 2 nm n n

m=1

n

and

n!1

+ m)

1

=0

Consequently we write lim

n!1

Ln (t2 ; x)

x2

C[0;A]

=0

Therefore, the desired result follows from Korovkin theorems, i.e. for all f 2 C[0; A]; lim kLn (f; x)

n!1

f (x)kC[0;A] = 0 2

1+x We give the graphics of approximation of functions sin(2x) by e2x+1 and e2x3 +3 modi…ed Gadjiev-I·bragimov polynomials (1) (see Figs. 2:1 and Figs.2:2 respectively).

872

GONUL, COSKUN: GADJIEV-IBRAGIMOV OPERATORS

sin(2x)

Fig.2.1 Approximation of f (x) = e2x+1 by Ln (f; x). (For interpretation of the references to colour in this …gure legend, the reader is referred to the web version of this article.)

2

Fig.2.2 Approximation of f (x) = 1+x by Ln (f; x). (For interpretation of the e2x3 +3 references to colour in this …gure legend, the reader is referred to the web version of this article.)

873

GONUL, COSKUN: GADJIEV-IBRAGIMOV OPERATORS

4

Rates of convergence

In this section we want to …nd the rate of convergence of the sequence of operators fLn gn N : Theorem 5 If f 2 C[0; A] then the inequality

kLn (f; x)

0

K! @f ;

f (x)kC[0;A]

s

2 n

n

1

n

+

n

+

n

1

1 A A n

(5)

holds for su¢ ciently large n , where K is a constant independent of n. Proof. From 2 ) we get Ln (1; x) =

1 X

Kn;# (x)

# n)

(

=1

#!

#=0

Using linearity and positivity of operators (1) we get jLn (f; x)

f (x)j

1 X

f(

#

f (x) Kn;# (x)

)

# n)

(

#!

n

#=0

#

From the de…nition of modulus of continuity by choosing t = 0 1 # x # n A f ( ) f (x) !(f; ) @1 +

n

we have

n

for every

n

> 0:Thus we obtain

jLn (f; x)

1 X

f (x)j

!(f;

#=0

!(f;

n)

n)

(

0

#

@1 +

n

n

1 1 X # n

x

1

A Kn;# (x) (

x Kn;# (x)

(

n

#=0

# n)

#! # n)

#!

We de…ned

M=

1 X

#=0

# n

2

x

! 12

Kn;# (x)

(

# n)

#!

1 2

Kn;# (x)

Now applying Cauchy Schwartz inequality we obtain

874

(

# n)

#!

1 2

:

)

+1

(6)

GONUL, COSKUN: GADJIEV-IBRAGIMOV OPERATORS

M

"

1 X #

2

x Kn;# (x)

# n)

(

#!

n

#=0

# 12

:

Substituting these inequalities in (Eq:6) we have

jLn (f; x)

f (x)j

2

x Kn;# (x)

#

(

n)

#!

#=0

So we can write

jLn (f; x)

8 " 1

L[2](f, x)",font=[TIMES, ROMAN,11],color=black]): >Y2:=textplot([0.89,0.08,”- - - - - - > L[4] ( f, x)",font=[TIMES,ROMAN,11],color=black]): >Y3:=textplot([0.95,0.099,- - - - - - - - > L[10](f,x)",font=[TIMES,ROMAN,11],color=black]): >Y4:=textplot([0.879,0.104,"- - - - - - - - - -> L[20]( f,x)",font=[TIMES,ROMAN,11],color=black]): >Y5:=textplot([0.866,0.110,"- - - - - - - - - - -> L[29]( f,x)",font=[TIMES,ROMAN,11],color=black]): >Y6:=textplot([0.6101,0.117,"- - - - - - - - - - -> f(x)",font=[TIMES,ROMAN,11]],color=black): >display([p1,p2,p3,p4,p5,p6,Y1,Y2,Y3,Y4,Y5,Y6]); Appendix B >restart; >f:=x->1/exp(2*x+5); >n:=1: >for k from 0 to 9 do >n:=10*n; >alpha(n):=1:

877

GONUL, COSKUN: GADJIEV-IBRAGIMOV OPERATORS

>beta(n):=n: >delta(n):=evalf(simplify(sqrt(((n*alpha(n)/beta(n))-1)^2+alpha(n) /beta(n)+1/(1*beta(n))))); >omega(f,delta(n)):=evalf(simplify(maximize(abs(expand (f(x+h)-f(x))),x=0..1,y=0..10,h=0..delta(n)))): >errorL:= 20*omega(f,delta(n)); >end do;

7

Acknowledgment

The authors would like to thank to the referee for his/ her valuable suggestions which improved the paper considerably.

References [1] A. Aral, Approximation by Ibragimov-Gadjiev operators in polynomial weighted space, Proc. of IMM of NAS of Azerbaijan, 19 35–44 (2003). [2] A. D.Gadjiev, The convergence problem for a sequence of positive linear operators on bounded sets and theorems analogous to that of P. P. Korovkin, Soviet Math. Dokl. 15 45–56 (1974). [3] A. D. Gadjiev, H. Hac¬salihoglu, On convergence of the sequences of linear positive operators, Ankara University, 1995. [4] A. D. Gadjiev, R. O. Efendiev, E. Ibikli, Generalized Bernstein-Chlodowsky polynomials, Rocky Mountain J. Math. 28 1267–1277 (1998). [5] A. D.Gadjiev, On P. P. Korovkin type theorems, Math. Zamet. (in Russian) 20 781–786 (1976). [6] A. D. Gadjiev, E. Ibikli, Weighted Approximation by BernsteinChlodowsky Polynomials, Indian J. Pure Ap. Math. 30 83–87 (1999). [7] A. D. Gadjiev, N. Ispir, On a sequence of linear positive operators in weighted space. Proc. of IMM of NAS of Azerbaijan,11 45–56 (1999). [8] I.I.Ibragimov, A. D.Gadjiev, On a sequence of linear positive operators, Soviet Math. Dokl. 11 1092–1095 (1970). [9] O. Dogru, On a certain family of linear positive operators, Turkish Journal of Mathematics, 21-4 387-399 (1997). [10] O. Dogru, On weighted approximation of continuous functions by linear positive operators on in…nite intervals, Mathematica Cluj, 41 39–46 (1999). [11] O. Dogru, Approximation properties of a generalization of positive linear operators, Journal of Mathematical Analysis and Applications, 342 161–170 (2008).

878

GONUL, COSKUN: GADJIEV-IBRAGIMOV OPERATORS

[12] T. Co¸skun, Weighted approximation of unbounded continuous functions by sequences of linear positive operators, Indian J.Pure Appl.Math. 34 477– 485 (2003). [13] T. Co¸skun, On a construction of positive linear operators for approximation of continuous functions in the weighted spaces, Journal of Computational Analysis and Applications, 13 756–770 (2011). [14] T. Co¸skun, On the order of weighted approximation by positive linear operators, Turkish Journal of Mathematics, 36 113–120 (2012).

879

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 880-885, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Some identities on the twisted q-Euler numbers with weight (α, β) and q-Bernstein polynomials with weight α C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper, by using fermionic p-adic q-integral on Zp , we give some interesting relationship between the twisted q-Euler numbers with weight (α, β) and the q-Bernstein polynomials with weight α. Key words : Euler numbers and polynomials, twisted q-Euler numbers and polynomials with weight (α, β), Bernstein polynomials, q-Bernstein polynomials

1. Introduction In this paper we investigate some relations between the q-Bernstein polynomials and the twisted q-Euler numbers with weight (α, β). From these relations, we derive some interesting identities on the twisted q-Euler numbers with weight (α, β). Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of n Qp , respectively. Let N be the set of natural numbers and Z+ = N ∪ {0}. Let Cpn = {w|wp = 1} be the cyclic group of order pn and let Tp = lim Cpn = Cp∞ = ∪n≥0 Cpn n→∞

be the locally constant space. For w ∈ Tp , we denote by φw : Zp → Cp the locally constant function 1 s x −→ wx . The p-adic absolute value is defined by |x|p = r , where x = pr ( r ∈ Q and s, t ∈ Z p t with (s, t) = (p, s) = (p, t) = 1). In this paper we assume that q ∈ Cp with |q − 1|p < 1 as an indeterminate. The q-number is defined by [x]q =

1 − qx , see [1-10]. 1−q

Note that limq→1 [x]q = x. For f ∈ U D(Zp ) = {f |f : Zp → Cp is uniformly differentiable function}, the fermionic p-adic q-integral on Zp is defined by Kim as follows: 

N

p −1 1+q  f (x)dμ−q (x) = lim f (x)(−q)x , see [1-2] . I−q (f ) = pN N →∞ 1 + q Zp x=0

(1.1) (α,β)

For α ∈ Z, β ∈ Z, w ∈ Tp , and q ∈ Cp with |1 − q|p ≤ 1, twisted q-Euler numbers En,q,w with weight (α, β) are defined by  (α,β) En,q,w = φw (x)[x]nqα dμ−qβ (x). (1.2) Zp

In the special case, x = 0, weight (α, β).

(α,β) En,q,w (0)

=

(α,β) En,q,w

are called the n-th twisted q-Euler numbers with

880

RYOO: TWISTED q-EULER NUMBERS

2. Some identities on the twisted q-Euler numbers with weight (α, β) In this section, we investigate some identities on the twisted q-Euler numbers with weight (α, β). By using p-adic q-integral on Zp and (1.2), we obtain, 

N

Zp

φw (x)[x]nqα dμ−qβ (x)

p −1 1 x = lim [x]nqα wx (−1)x q β N →∞ [pN ]−q β x=0  n  n   1 n 1 (−1)l = [2]qβ . l 1 − qα 1 + wq αl+β

(2.1)

l=0

We set (α,β) (t) Fq,w

∞ 

=

(α,β) En,q,w

n=0

tn . n!

(2.2)

By (2.1) and (2.2), we obtain (α,β) Fq,w (t) =

∞ 

tn n!  ∞ 

(α,β) En,q,w

n=0

= [2]qβ = [2]qβ

n=0 ∞ 

1 1 − qα

 n  n   n 1 tn l (−1) αl+h l 1 + wq n!

(2.3)

l=0

(−1)m wm q βm e[m]qα t .

m=0

Since [x + y]qα = [x]qα + q αx [y]qα , we obtain  (α,β) En,q,w (x) = φw (y)[y + x]qα tdμ−qβ (y) Zp

= =

n    n l=0 n   l=0

q αxl [x]n−l qα

l

 Zp

φw (y)[y]lqα dμ−qβ (y)

(2.4)

 n αxl n−l (α,β) q [x]qα El,q,w . l

Therefore, we obtain the following theorem. Theorem 1. For n ∈ Z+ and w ∈ Tp , we have (α,β) En,q,w (x)

= [2]qβ

∞ 

(−1)m wm q βm [x + m]nqα .

m=0

Furthermore, (α,β) (x) En,q,w

n    n αxl n−l (α,β) (α,β) n q [x]qα El,q,w = ([x]qα + q αx Eq,w = ) , l l=0

(α,β)

(α,β)

with usual convention about replacing (Eq,w )n by En,q,w . (α,β)

Let Fq,w (t, x) =

∞

(α,β)

n=0 En,q,w (x)

tn . Then we see that n!

(α,β) Fq,w (t, x) = [2]qβ

∞ 

(−1)m wm q βm e[x+m]qα t .

m=0 (α,β)

(α,β)

In the special case, x = 0, let Fq,w (t, 0) = Fq,w (t).

881

(2.5)

RYOO: TWISTED q-EULER NUMBERS

By (2.1), we get (α,β)

(α,β) En,q−1 ,w−1 (1 − x) = (−1)n wq αn En,q,w (x).

(2.6)

From (2.3) and (2.5), we note that (α,β) (α,β) (t, 1) + Fq,w (t) = [2]qβ . wq β Fq,w

(2.7)

By (2.7), we get the following recurrence formula: (α,β)

E0,q,w =

[2]qβ (α,β) (α,β) , and q β wEn,q,w (1) + En,q,w = 0 if n > 0. 1 + qβ w

(2.8)

By (2.8) and Theorem 1, we obtain the following theorem. Theorem 2. For n ∈ Z+ and w ∈ Tp , we have β

q w(q

α

(α,β) Eq,w

n

+ 1) +

(α,β) En,q,w



[2]qβ , 0,

=

(α,β)

if n = 0, if n > 0,

(α,β)

with usual convention about replacing (Eq,w )n by En,q,w . By (2.4), Theorem 1, and Theorem 2, we obtain (α,β) (2) − q 2β w2 En,q,w

[2]qβ 2β 2 [2]qβ β q w − q w 1 + qβ w 1 + qβ w

n    [2]qβ β n αl α (α,β) q (q Eq,w + 1)l − =q w q w l 1 + qβ w l=1 n    [2]qβ β n αl (α,β) q El,q,w − q w = −q β w l 1 + qβ w 2β

2

l=1

β

(α,β) (α,β) (1) = En,q,w if n > 0. = −q wEn,q,w

Therefore, we obtain the following theorem. Theorem 3. For n ∈ N, we have     [2]qβ [2]qβ 1 1 (α,β) (α,β) + . En,q,w + En,q,w (2) = q 2β w2 1 + qβ w 1 + qβ w qβ w By (2.6), we see that  w

Zp

[1 −

x]nq−α wx dμ−qβ (x)

n αn

= (−1) q

 w

Zp

[x − 1]nqα wx dμ−qβ (x)

(2.9)

(α,β)

(α,β) (−1) = En,q−1 ,w−1 (2). = (−1)n q αn wEn,q,w

Therefore, we obtain the following theorem. Theorem 4. For n ∈ Z+ , we have  (α,β) [1 − x]nq−α wx dμ−qβ (x) = En,q−1 ,w−1 (2). w Zp

Let n ∈ N. By Theorem 3 and Theorem 4, we have  w [1 − x]nq−α wx dμ−qβ (x) Zp

(α,β)

= q 2β w2 En,q−1 ,w−1 + w



[2]qβ 1 + qβ w

882



+ q β w2



[2]qβ 1 + qβ w



(2.10) .

RYOO: TWISTED q-EULER NUMBERS

From (2.10), we have  Zp

[1 −

x]nq−α wx dμ−qβ (x)

=q

2β

(α,β) wEn,q−1 ,w−1

 +

[2]qβ 1 + qβ w



β

+q w



[2]qβ 1 + qβ w

 .

Therefore, we obtain the following corollary. Corollary 5. For n ∈ N, we have  (α,β) [1 − x]nq−α wx dμ−qβ (x) = q 2β wEn,q−1 ,w−1 + [2]qβ . Zp

For x ∈ Zp , the p-adic q-Bernstein polynomials with weight α of degree n are given by   n (α) [x]kqα [1 − x]n−k Bk,n (x, q) = (2.11) q −α , where n, k ∈ Z+ . k By (2.11), we get the symmetry of q-Bernstein polynomials as follows: (α)

(α)

Bk,n (x, q) = Bn−k,n (1 − x, q −1 ).

(2.12)

Thus, by Corollary 5, (2.11), and (2.12), we see that  Zp

(α) Bk,n (x, q)wx dμ−qβ (x)

 = Zp

(α)

Bn−k,n (1 − x, q −1 )wx dμ−qβ (x)

  k  

 n k (α,β) (−1)k+l q 2β wEn−l,q−1 ,w−1 + [2]qβ . = k l

(2.13)

l=0

Let us take the fermionic q-integral on Zp for the q-Bernstein polynomials with weight α of degree n as follows:  Zp

  n x [x]kqα [1 − x]n−k q −α w dμ−q β (x) k Zp    n−k  n  n−k (α,β) (−1)l El+k,q,w . = k l

(α)

Bk,n (x, q)wx dμ−qβ (x) =

(2.14)

l=0

Therefore, by (2.13) and (2.14), we obtain the following theorem. Theorem 6. Let n, k ∈ Z+ with n > k. Then we have    k  

 k n (α) (α,β) x (−1)k+l q 2β wEn−l,q−1 ,w−1 + [2]qβ . Bk,n (x, q)w dμ−qβ (x) = l k Zp l=0

Moreover,  k  

  n−k k (α,β) (α,β) (−1)l El+k,q,w = (−1)k+l q 2β wEn−l,q−1 ,w−1 + [2]qβ . l l

n−k  l=0

l=0

Let n1 , n2 , k ∈ Z+ with n1 + n2 > 2k. Then we get  (α) (α) Bk,n1 (x, q)Bk,n2 (x, q)wx dμ−qβ (x) Zp

      2k  n2 n1 2k l+2k 1 +n2 −l (−1) = [1 − x]qn−α wx dμ−qβ (x) l k k Z p l=0      2k 

 n2 2k n1 (α,β) (−1)l+2k q 2β wEn1 +n2 −l,q−1 ,w−1 + [2]qβ . = l k k l=0

883

(2.15)

RYOO: TWISTED q-EULER NUMBERS

Therefore, by (2.15), we obtain the following theorem. Theorem 7. For n1 , n2 , k ∈ Z+ with n1 + n2 > 2k, we have  (α) (α) Bk,n1 (x, q)Bk,n2 (x, q)wx dμ−qβ (x) Zp

=

     2k 

 n2 n1 2k (α,β) (−1)l+2k q 2β wEn1 +n2 −l,q−1 ,w−1 + [2]qβ . l k k l=0

From the binomial theorem, we can derive the following equation.  (α) (α) Bk,n1 (x, q)Bk,n2 (x, q)wx dμ−qβ (x) Zp

     n1 +n 2 −2k  n2 n1 l n1 + n2 − 2k x = (−1) [x]2k+l q α w dμ−q β (x) l k k Z p l=0     n1 +n  2 −2k  n2 n1 n1 + n2 − 2k (α,β) E2k+l,q,w . = (−1)l l k k

(2.16)

l=0

Thus, by (2.16) and Theorem 7, we obtain the following corollary. Corollary 8. Let n1 , n2 , k ∈ Z+ with n1 + n2 > 2k. Then we have n1 +n 2 −2k 

l

(−1)

l=0

=

 n1 + n2 − 2k (α,β) E2k+l,q l



 2k   2k l=0

l

 (α,β) (−1)l+2k q 2β wEn1 +n2 −l,q−1 ,w−1 + [2]qβ .

For x ∈ Zp and s ∈ N with s ≥ 2, let n1 , n2 , . . . , ns , k ∈ Z+ with n1 + · · · + ns > sk. Then we take the fermionic p-adic q-integral on Zp for the q-Bernstein polynomials with weight α of degree n as follows:  (α) (α) Bk,n1 (x, q) · · · Bk,ns (x, q)wx dμ−qβ (x)

  Zp s−times     n1 ns n1 +···+ns −sk x (2.17) = ··· [x]sk w dμ−qβ (x) q [1 − x]q −α k k Zp     sk  

 ns n1 sk (α,β) (−1)l+sk q 2β wEn1 +···+ns −l,q−1 ,w−1 + [2]qβ . ··· = l k k l=0

Therefore, by (2.17), we obtain the following theorem. Theorem 9. For s ∈ N with s ≥ 2, let n1 , n2 , . . . , ns , k ∈ Z+ with n1 + · · · + ns > sk. Then we get  (α) (α) Bk,n1 (x, q) · · · Bk,ns (x, q)wx dμ−qβ (x)

  Zp s−times (2.18)     sk  

 ns sk n1 (α,β) l+sk 2β (−1) ··· q wEn1 +···+ns −l,q−1 ,w−1 + [2]qβ . = l k k l=0

884

RYOO: TWISTED q-EULER NUMBERS

By the definition of q-Bernstein polynomials with weight α and the binomial theorem, we easily get



(α)

(α)

Bk,n1 (x, q) · · · Bk,ns (x, q)wx dμ−qβ (x)

  s−times       n1 +···+n s −sk ns n1 + · · · + ns − sk n1 ··· (−1)l [x]sk+l wx dμ−qβ (x) = q l k k Z p l=0      n1 +···+n  −sk s  ns n1 n1 + · · · + ns − sk (α,β) Esk+l,q,w . ··· = (−1)l l k k Zp

l=0

Therefore, we have the following corollary. Corollary 10. For w ∈ Tp , s ∈ N with s ≥ 2, let n1 , n2 , . . . , ns , k ∈ Z+ with n1 + · · · + ns > sk. Then we have   n1 +···+n s −sk n1 + · · · + ns − sk (α,β) Esk+l,q,w (−1)l l l=0 sk

  sk (α,β) (−1)l+sk q 2β wEn1 +···+ns −l,q−1 ,w−1 + [2]qβ . = l l=0

References [1] T. Kim, A note on q-Bernstein polynomials, Russ. J. Math. phys., 18(2011), 41-50. [2] T. Kim, q-Volkenborn integration, Russ. J. Math. phys. 9(2002), 288-299. [3] T. Kim, J. Choi, Y. H. Kim, C. S. Ryoo, On the fermionic p-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials, J. Inequal. Appl., 2010(2010), Article ID 864247, 12 pages. [4] T. Kim, J. Choi, Y.-H. Kim, Some identities on the q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli numbers, Adv. Stud. Contemp. Math., 20(2010), 335-341. [5] T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math., 20(2010), 23-28. [6] H. Y. Lee, N. S. Jung, and C. S. Ryoo, Some Identities of the Twisted q-Genocchi Numbers and Polynomials with Weight α and q-Bernstein Polynomials with Weight α, Abstract and Applied Analysis, 2011(2011), Article ID 123483, 9 pages [7] L. C. Jang, W.-J. Kim, Y. Simsek, A study on the p-adic integral representation on Zp associated with Bernstein and Bernoulli polynomials, Advances in Difference Equations, 2010(2010), Article ID 163217, 6 pages. [8] Y. Simsek, O. Yurekli, V. Kurt, On interpolation functions of the twisted generalized Frobinuous-Euler numbers, Adv. Stud. Contemp. Math., 14(2007), 49-68. [9] C. S. Ryoo, On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity, Proc. Jangjeon Math. Soc., 13(2010), 255-263. [10] C. S. Ryoo, A note on the weighted q-Euler numbers and polynomials, Adv. Stud. Contemp. Math., 21(2011), 47-54.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 886-891, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Doubly-accelerated Steffensen’s methods with memory and their applications on solving nonlinear ODEs

∗

Quan Zheng†, Fengxi Huang, Xiuhui Guo, Xiaoli Feng College of Sciences, North China University of Technology, Beijing 100144, China

Abstract: In this paper, a parametric accelerated Steffensen’s method and three doubly-accelerated Steffensen’s methods with memory are designed for solving nonlinear equations efficiently. Their orders of convergence from 2.414 up to 2.831 and 3 are proved theoretically and demonstrated numerically. Each of the accelerated Steffensen’s methods only uses two new evaluations of the function without derivatives per step and is applicable on solving systems of nonlinear equations and nonlinear ODEs. Keywords: Nonlinear equation; Newton’s method; Steffensen’s method; Derivative free; Super convergence

1

Introduction In scientific computation, Newton’s method (NM, see [1, 2]): xn+1 = xn −

f (xn ) , f 0 (xn )

n = 0, 1, 2, . . . ,

(1)

is widely used for root-finding, where x0 is an initial guess of the root. However, when the derivative f 0 is unavailable or is expensive to be obtained, the derivative-free method is necessary. If the derivative f 0 (xn ) is replaced by the divided difference f [xn , xn +f (xn )] =

f (xn +f (xn ))−f (xn ) f (xn )

in (1), Steffensen’s method (SM, see [1, 2]) is obtained. NM/SM converges quadratically and √ requires two function evaluations per iteration. The efficiency index of them is 2 = 1.414. Moreover, a parametric Steffensen’s method (PSM) was suggested in Section 8.4 in [2]: xn+1 = xn −

f (xn ) , f [xn , xn + βn f (xn )]

n = 0, 1, 2, . . . ,

(2)

where βn are arbitrary parameters. PSM gives SM when βn ≡ 1. By defining βn from the 1 at the previous iteration recursively as the iteration procalculation − f [xn−1 ,xn−1 +β n−1 f (xn−1 )]

ceeds, we have a self-accelerating Steffensen’s method (SASM, see Section 8.6 in [2], [3]). SASM only uses two new evaluations of the function per step to achieve super convergence of order p √ √ 1 + 2 ≈ 2.414, and has efficiency index 1 + 2 ≈ 1.554. ∗ †

Supported by Beijing Natural Science Foundation (No. 1122014). E-mail: [email protected] (Q. Zheng).

886

ZHENG et al: DOUBLY-ACCELERATED STEFFENSEN’S METHODS

Two-step self-accelerating Steffensen’s methods were derived in [3-6]. Optimal Steffensentype families without memory for solving nonlinear equations were introduced in [6-9], Steffensentype methods and their applications in the solution of nonlinear systems and nonlinear differential equations were discussed in the literature (see [1, 2, 5, 10]). This paper is organized as follows. In Section 2, a parametric accelerated Steffensen’s method with memory is proposed and its error equation is obtained.

In Section 3, three doubly-

accelerated Steffensen’s method with memory are derived from it. In Section 4, numerical examples and applications are demonstrated.

2

A parametric accelerated Steffensen’s method By the first-order Newtonian interpolatory polynomial N1 (x) = f (xn ) + f [xn , zn ](x − xn ) at

points xn and zn = xn + βn f (xn ), we have f (x) = N1 (x) + R1 (x), where R1 (x) = f (x) − N1 (x) = f [xn , zn , x](x − xn )(x − zn ). So, with some µn ≈ f [xn , zn , x], e2 (x) = f (xn ) + f [xn , zn ](x − xn ) + µn (x − xn )(x − zn ) N should be better than N1 (x) to approximate f (x). Therefore, we suggest xn+1 = xn − i.e., a two-parameter Steffensen’s method (TPSM): xn+1 = xn −

f (xn ) , n = 0, 1, 2, . . . , f [xn , zn ] + µn (xn − zn )

e2 (xn ) N e 0 (xn ) , N 2

(3)

where zn = xn + βn f (xn ), {βn } and {µn } are bounded constant sequences. This method gives PSM when µn ≡ 0. TPSM can be proved to satisfy the error equation en+1 = [(1 + βn f 0 (a))c2 − µn βn ]e2n + O(e3n ), where ck =

f (k) (a) k!f 0 (a) ,

(4)

en = xn − a, n = 0, 1, 2, . . ..

By defining µ0 = 0 and µn =

1+βn f [xn ,zn ] βn f [xn ,zn ] f [zn−1 , xn , zn ](n

tion proceeds without any new evaluation, the factor [(1 +

> 0) recursively as the itera-

βn f 0 (a))c2

− µn βn ] in (4) tends to

zero. Thus, TPSM achieves super convergence by self-accelerating and gives a new parametric accelerated Steffensen’s method (PASM): xn+1 = xn −

f [xn , zn ] + (1 +

f (xn ) 1 βn f [xn ,zn ] )(f [zn−1 , xn ]

− f [zn−1 , zn ])

,

n = 1, 2, · · · .

(5)

Theorem 2.1 Let f : D → < be a sufficiently differentiable function with a simple root a ∈ D, D ⊂ < be an open set, x0 be close enough to a, then PASM satisfies the following error equation en+1 = −(1 + βn f 0 (a))(1 + βn−1 f 0 (a))c3 en−1 e2n + O(e2n−1 e2n ),

887

(6)

ZHENG et al: DOUBLY-ACCELERATED STEFFENSEN’S METHODS

where ck =

f (k) (a) k!f 0 (a) ,

√ en = xn − a, n = 0, 1, 2, . . ., and achieves convergence of order at least 1 + 2.

Proof. By the definition of divided difference and Taylor formula, we also have f [zn−1 , xn , zn ] =

f 00 (a) f 000 (a) z + e + O((ezn−1 )2 ). 2! 3! n−1

1 + βn f 0 (a) f 00 (a) (1 + βn−1 f 0 (a))f 000 (a) ( + en−1 ) + O(e2n−1 ). βn f 0 (a) 2! 3! √ Plugging it into (4), we have (6). The order 1 + 2 ≈ 2.414 is from solving s2 − 2s − 1 = 0. ¤ e2 (x) If one uses N2 (x) = f (xn ) + f [xn , zn ](x − xn ) + f [xn , zn , yn ](x − xn )(x − zn ) instead of N µn =

where yn comes from PSM, then an optimal fourth-order Steffensen’s method in [8] is derived by Zheng-Li-Huang (ZLHM), and it can also be accelerated by choosing βn .

3

Three doubly-accelerated Steffensen’s methods Further, from PASM without any new evaluation, as SASM deriving from PSM, we propose

three doubly-accelerated Steffensen’s methods, i.e., compute PASM (5) with: 1 , (DASM1) f [xn−1 , zn−1 ] 1 (2) βn := β n = − , (DASM2) f [xn−1 , xn ] 1 (3) βn := β n = − , (DASM3) f [zn−1 , xn ] (1) βn := β n = −

(7) (8) (9)

respectively, for n = 1, 2, · · · . Theorem 3.1. Let f : D → < be a sufficiently differentiable function with a simple root a ∈ D, D ⊂ < be an open set, x0 be close enough to a, then DASM1 and DASM2 achieve the convergence of order 2.831, and DASM3 achieves third-order convergence. Proof. For DASMs, denoting ezn := zn − a, if zn converges to a with order p > 1 as: ezn = Cn epn + o(epn ), and if xn converges to a with order r > 2 as: en+1 = Dn ern + o(ern ), then p rp rp ezn = Cn (Dn−1 ern−1 )p + o(erp n−1 ) = Cn Dn−1 en−1 + o(en−1 ), 2

2

2

r en+1 = Dn (Dn−1 ern−1 )r + o(ern−1 ) = Dn Dn−1 ern−1 + o(ern−1 ).

By Taylor formula, particularly for DASM1, we also have ezn = en − en+1 = en −

f [xn−1 ,zn−1 ,a]en−1 +f [zn−1 ,xn ,a](ezn−1 −en ) en f [xn−1 ,zn−1 ] f [xn ,a]en

f [xn ,a]en f [xn−1 ,zn−1 ]

=

f [xn ,zn ]+(1−

f [xn−1 ,zn−1 ] f [x ,a]e )f [zn−1 ,xn ,zn ] f [x n ,z n ] f [xn ,zn ] n−1 n−1

888

r+1 = c2 Dn−1 er+1 n−1 + o(en−1 ),

ZHENG et al: DOUBLY-ACCELERATED STEFFENSEN’S METHODS

f [xn ,a] f [x ,a] − f [x n,z ] )f [zn−1 ,xn ,zn ]en n n n−1 ,zn−1 ] f [x ,a] f [x ,a] f [xn ,zn ]+( f [x n,z − f [x n,z ] )f [zn−1 ,xn ,zn ]en ] n n n−1 n−1 f [x ,a] f [x ,a] f [x ,a] f [xn ,zn ,a](1− f [x n,z )+( f [x n,z − f [x n,z ] )f [zn−1 ,xn ,zn ] ] n n n−1 n−1 n−1 n−1 ] 2 f [x ,a] f [xn ,a] n f [xn ,zn ]+( f [x n,z − )f [z ,xn ,zn ]en n−1 f [xn ,zn ] n−1 n−1 ]

= en

f [xn ,zn ,a](zn −a)+( f [x

= e

[xn ,zn ](f [xn−1 ,zn−1 ]−f [xn ,a])+f [zn−1 ,xn ,zn ]f [xn ,a](f [xn ,zn ]−f [xn−1 ,zn−1 ]) = e2n f [xn ,zn ,a]f f [xn ,zn ]2 f [xn−1 ,zn−1 ]+(f [xn ,zn ]−f [xn−1 ,zn−1 ])f [xn ,a]f [zn−1 ,xn ,zn ]en ,zn ]−f [zn−1 ,xn ,zn ]f [xn ,a])f [xn−1 ,zn−1 ]+(f [zn−1 ,xn ,zn ]−f [xn ,zn ,a])f [xn ,zn ]f [xn ,a] = e2n (f [xn ,zn ,a]f [xfn[x 2 n ,zn ] f [xn−1 ,zn−1 ]+(f [xn ,zn ]−f [xn−1 ,zn−1 ])f [xn ,a]f [zn−1 ,xn ,zn ]en (f [zn−1 ,xn ,zn ]f [xn ,zn ,a]ezn −f [zn−1 ,xn ,zn ,a]f [xn ,zn ]ezn−1 )f [xn−1 ,zn−1 ]+f [zn−1 ,xn ,zn ,a]f [xn ,zn ]f [xn ,a]ezn−1 f [xn ,zn ]2 f [xn−1 ,zn−1 ]+(f [xn ,zn ]−f [xn−1 ,zn−1 ])f [xn ,a]f [zn−1 ,xn ,zn ]en z z f [z ,x ,z ]f [x ,z n n ,a]f [xn−1 ,zn−1 ]en +(f [xn ,a]−f [xn−1 ,zn−1 ])f [zn−1 ,xn ,zn ,a]f [xn ,zn ]en−1 n−1 n n e2n f [xn ,zn ]2 f [xn−1 ,zn−1 ]+(f [xn ,zn ]−f [xn−1 ,zn−1 ])f [xn ,a]f [zn−1 ,xn ,zn ]en 2 e2r+p+1 + o(e2r+p+1 ). −c2 c3 Cn−1 Dn−1 n−1 n−1

= e2n = =

For DASM2, we also have ezn = en −

f [xn ,a]en f [xn−1 ,xn ]

r+1 = c2 Dn−1 er+1 n−1 + o(en−1 ),

f [xn ,a]en f [xn−1 ,xn ] f [x ,a] f [xn ,zn ]+(1− f [x ,z ] )f [zn−1 ,xn ,zn ] f [x n ,x ] en n n n−1 n z z 2 f [zn−1 ,xn ,zn ]f [xn ,zn ,a]f [xn−1 ,xn ]en +(f [xn ,a]−f [xn−1 ,xn ])f [zn−1 ,xn ,zn ,a]f [xn ,zn ]en−1 2 n f [xn ,zn ] f [xn−1 ,xn ]+(f [xn ,zn ]−f [xn−1 ,xn ])f [xn ,a]f [zn−1 ,xn ,zn ]en

en+1 = en − = e

2 e2r+p+1 + o(e2r+p+1 ). = −c2 c3 Cn−1 Dn−1 n−1 n−1

So, comparing the exponents of en−1 in expressions of ezn and en+1 for DASM1 and DASM2 respectively, we obtain the same system of two equations:   rp = r + 1,  r2 = 2r + p + 1. From its non-trivial solution r =

√ 3

√ √ √ 3 (316+12 249)2 +40+4 316+12 249 √ √ 3 6 316+12 249

≈ 2.831 and p ≈ 1.353, we

prove that the convergence of DASM1 and DASM2 is of order 2.831. For DASM3, we have ezn =

f [xn ,zn−1 ,a] z f [zn−1 ,xn ] en−1 en

p+r = c2 Cn−1 Dn−1 ep+r n−1 + o(en−1 ),

f [xn ,a]en f [zn−1 ,xn ] f [x ,a] )f [zn−1 ,xn ,zn ] f [z n ,x ] en f [xn ,zn ] n−1 n z z 2 f [zn−1 ,xn ,zn ]f [xn ,zn ,a]f [zn−1 ,xn ]en +(f [xn ,a]−f [zn−1 ,xn ])f [zn−1 ,xn ,zn ,a]f [xn ,zn ]en−1 n f [xn ,zn ]2 f [zn−1 ,xn ]+(f [xn ,zn ]−f [zn−1 ,xn ])f [xn ,a]f [zn−1 ,xn ,zn ]en z z 2 2 f [zn−1 ,xn ,zn ]f [xn ,zn ,a]f [zn−1 ,xn ]en −f [zn−1 ,xn ,a]f [zn−1 ,xn ,zn ,a]f [xn ,zn ](en−1 ) 2 n f [xn ,zn ] f [zn−1 ,xn ]+(f [xn ,zn ]−f [zn−1 ,xn ])f [xn ,a]f [zn−1 ,xn ,zn ]en

en+1 = en −

f [xn ,zn ]+(1−

= e = e

2 D 2 e2r+2p + o(e2r+2p ), = −c2 c3 Cn−1 n−1 n−1 n−1

and then

By its non-trivial solution p =

  rp = p + r,  r2 = 2r + 2p. 3 2

and r = 3, DASM3 achieves third-order convergence.

889

¤

ZHENG et al: DOUBLY-ACCELERATED STEFFENSEN’S METHODS

PASM and DASMs each only uses two new evaluations of the function per step. PASM has p √ efficiency index 1 + 2 ≈ 1.554. The efficiency index of DASM1 and DASM2 is 1.683, and that √ of DASM3 is 3 ≈ 1.732. Whereas, two accelerated methods proposed by Petkovi´c-Ili´c-Dˇzuni´c (PIDMs, see [6]) each uses three new evaluations of the function per iteration to achieve the p √ √ 3 super fourth-order convergence of order 2+ 6 and its efficiency index is only 2 + 6 ≈ 1.645. Moreover, each of DASMs uses six or seven multiplications/divisions per iteration while each of PIDMs uses eight or nine. ZLHM can be accelerated to have analogous properties as PIDMs.

4

Numerical examples Example 1. The numerical results of NM, SM, SASM, PASM, DASM1, DASM2 and

DASM3 in Table 1 agree with the theoretical analysis, the computational order of convergence is defined by COC =

log(|en |/|en−1 |) log(|en−1 |/|en−2 |) ,

β0 = β 0 = β 0 = β 0 = 1 in SASM and DASMs, and

1 x−2 2 (e

f1 (x) =

− 1), a = 2, x0 = 2.5,

2

f2 (x) = ex + sin x − 1, a = 0, x0 = 0.25, f3 (x) = e−x

2 +x+2

− 1, a = −1, x0 = −0.85,

f4 (x) = e−x − arctan x − 1, a = 0, x0 = 0.2. Table 1. Numerical results for solving fi (x), i = 1, 2, 3, 4 Results

NM

SM

SASM

PASM

DASM1

DASM2

DASM3

f1 : |e6 | COC f2 : |e6 | COC f3 : |e6 | COC f4 : |e6 | COC

.19785e-40 2.00000 .23328e-44 2.00000 .18813e-51 2.00000 .35988e-79 2.00000

.88156e-29 2.00000 8.8156e-30 2.00000 .15758e-18 2.00000 .96290e-84 2.00000

.50439e-84 2.41412 5.0400e-85 2.41400 .12013e-86 2.41405 .16834e-248 2.41600

.61847e-83 2.41538 .17647e-103 2.41140 .93815e-82 2.41702 .73025e-219 2.41472

.10662e-200 2.82941 .27693e-256 2.82789 .22874e-284 2.82691 .46660e-561 2.82646

.51551e-224 2.82871 .10066e-287 2.82697 .15797e-310 2.82588 .10795e-497 2.82984

.69537e-263 3.00000 .17476e-321 3.00000 .67793e-346 3.00000 .42720e-643 3.00000

Example 2. Consider to solve the following nonlinear ODE by finite difference method: ( x00 (t) + x3/2 (t) = 0, t ∈ (0, 1), x(0) = x(1) = 0. Taking nodes ti = ih, where h = N1 and N = 10, we have a system of nine nonlinear equations:  3/2   2x1 − h2 x1 − x2 = 0,  3/2

−xi−1 + 2xi − h2 xi − xi+1 = 0, i = 2, 3, · · · , 8,    −x + 2x − h2 x3/2 = 0. 8 9 9 SM is carried out as follows:    xn+1 = xn − J(xn , Hn )−1 F (xn ), n = 0, 1, 2, · · · .  J(xn , Hn ) = (F (xn + Hn e1 ) − F (xn ), · · · , F (xn + Hn eN −1 ) − F (xn ))Hn−1 ,    H = diag(f (x ), f (x ), · · · , f (x )). n

1

n

2

n

N −1

890

n

ZHENG et al: DOUBLY-ACCELERATED STEFFENSEN’S METHODS

And other methods are carried out by using similar approximations of the divided differences. The numerical results are in Table 2, where x0 = (40, 80, 100, 120, 140, 130, 100, 80, 40)’, x∗ = (33.5739120483377998..., 65.2024509236543787..., 91.5660200355396017..., 109.1676242966423523..., 115.3630336377466172..., 109.1676242966423523..., 91.5660200355396017..., 65.2024509236543787..., 33.5739120483377998...)’. Table 2 The finite difference method for solving x00 + x3/2 = 0, x(0) = x(1) = 0 Methods

n

1

2

3

4

5

6

NM

kxn − x∗ k2 kF (xn )k2 kxn − x∗ k2 kF (xn )k2 kxn − x∗ k2 kF (xn )k2 kxn − x∗ k2 kF (xn )k2 kxn − x∗ k2 kF (xn )k2 kxn − x∗ k2 kF (xn )k2 kxn − x∗ k2 kF (xn )k2

.40882e-1 .24453 4.8552 .37077 4.8552 .37077 4.8552 .37077 4.8552 .37077 4.8552 .37077 4.8552 .37077

.47895e-1 .23685e-2 .64055e-1 .31892e-2 .11027e-1 .54534e-3 .63584e-2 .54579e-3 .78674e-3 .82005e-4 .29285e-3 .46762e-4 .59470e-3 .49853e-4

.67632e-5 .33390e-6 .11495e-4 .56743e-6 .58355e-8 .28807e-9 .53756e-8 .52474e-9 .48628e-12 .13065e-12 .11595e-12 .26770e-13 .16546e-13 .52434e-14

.13490e-12 .6659e-14 .37036e-12 .18275e-13 .33191e-23 .16384e-24 .42148e-19 .24074e-20 .15823e-32 .83237e-34 .26922e-34 .16356e-35 .15352e-38 .82296e-40

.53672e-28 .26493e-29 .38446e-27 .18970e-28 .55918e-60 .27602e-61 .13880e-42 .37446e-43 .78586e-84 .51206e-85 .82638e-88 .64610e-89 .54974e-107 .21506e-107

.84957e-59 .41936e-60 .41429e-57 .20442e-58 .90270e-149 .44559e-150 .18835e-87 .93331e-89 .38375e-207 .83504e-208 .14241e-216 .21690e-217 .10528e-283 .56953e-285

SM SASM PASM (βn ≡ 1) DASM1 DASM2 DASM3

References [1] J.M. Ortega, W.G. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [2] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. [3] Q. Zheng, J. Wang, P. Zhao, L. Zhang, A Steffensen-like method and its higher-order variants, Appl. Math. Comput. 214 (2009) 10-16. [4] W. Bi, H. Ren, Q. Wu, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput. 209 (2009) 206-210. [5] Q. Zheng, P. Zhao, L. Zhang, W. Ma, Variants of Steffensen-secant method and applications, Appl. Math. Comput. 216 (2010) 3486-3496. [6] M.S. Petkovi´c, S. Ili´c, J. Dˇzuni´c, Derivative free two-point methods with and without memory for solving nonlinear equations, Appl. Math. Comput. 217 (2010) 1887-1895. [7] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651. [8] Q. Zheng, J. Li, F. Huang, An optimal Steffensen-type family for solving nonlinear equations, Appl. Math. Comput. 217 (2011) 9592-9597. [9] F. Soleymani, S.K. Vanani, Optimal Steffensen-type methods with eighth order of convergence, Comput. Math. Appl. 62 (2011) 4619-4626. [10] V. Alarc´on, S. Amat, S. Busquier, D. J. L´opez, A Steffensen’s type method in Banach spaces with applications on boundary-value problems, J. Comput. Appl. Math. 216 (2008) 243-250.

891

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 892-902, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

On modi…ed implicit Mann iteration method involving strictly hemicontractive mappings in smooth Banach spaces Nawab Hussain Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia [email protected]

Arif Ra…q Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan aara…[email protected]

Abstract In this paper we prove that the modi…ed implicit Mann iteration process can be applied to approximate the …xed point of strictly hemicontractive mappings in certain Banach spaces. Key words: Implicit Mann type iteration method, Strictly hemicontractive mappings, Strongly pseudocontractive mappings, Local strongly pseudocontractive mappings, Continuous mappings, Lipschitz mappings, Smooth Banach spaces.

1

INTRODUCTION and PRELIMINARIES

Let K be a nonempty subset of an arbitrary Banach space X and X be its dual space. The symbols D(T ); R(T ) and F (T ) stand for the domain, the range and the set of …xed points of T (for a single-valued map T : X ! X; x 2 X is called a …xed point of T i¤ T (x) = x):We denote by J the normalized duality mapping from E to 2E de…ned by J(x) = ff 2 E : hx; f i = kxk2 = kf k2 g; where h:; :i denotes the duality pairing. In a smooth Banach space J is singlevalued (and denoted by j). 892

HUSSAIN, RAFIQ: IMPLICIT MANN ITERATION METHOD

Remark 1 1. X is called uniformly smooth if X is uniformly convex. 2. In a uniformly smooth Banach space, J is uniformly continuous on bounded subsets of X. Let T be a self-mapping of K. De…nition The mapping T is called Lipshitzian if there exists L > 0 such that kT x

T yk 6 L kx

yk ;

for all x; y 2 K. If L = 1, then T is called non-expansive and if 0 6 L < 1; T is called contraction. De…nition 1 [ 3, 5] 1. The mapping T is said to be pseudocontractive if the inequality kx

yk 6k x

y + t((I

T )x

(I

T )y k;

(1.1)

holds for each x; y 2 K and for all t > 0. 2. T is said to be strongly pseudocontractive if there exists t > 1 such that kx

yk

k(1 + r)(x

y)

rt(T x

T y)k

(1.2)

for all x; y 2 D(T ) and r > 0: 3. T is said to be local strongly pseudocontractive if for each x 2 D(T ) there exists a tx > 1 such that kx

yk

k(1 + r)(x

y)

rtx (T x

T y)k

(1.3)

for all y 2 D(T ) and r > 0: 4. T is said to be strictly hemicontractive if F (T ) 6= ; and if there exists t > 1 such that kx

qk

k(1 + r)(x

q)

rt(T x

q)k

(1.4)

for all x 2 D(T ); q 2 F (T ) and r > 0: Clearly, each strongly pseudocontractive operator is local strongly pseudocontractive. Chidume [3] established that the Mann iteration sequence converges strongly to the unique …xed point of T in case T is a Lipschitz strongly pseudocontractive mapping from a bounded closed convex subset of Lp (or lp ) into 893

HUSSAIN, RAFIQ: IMPLICIT MANN ITERATION METHOD

itself. Schu [17] generalized the results in [3] to both uniformly continuous strongly pseudo-contractive mappings and real smooth Banach spaces. Park [14] extended the results in [3] to both strongly pseudocontractive mappings and certain smooth Banach spaces. Rhoades [15] proved that the Mann and Ishikawa iteration methods may exhibit di¤erent behaviors for di¤erent classes of nonlinear mappings. Afterwards, several generalizations have been made in various directions (see for example [4, 10-11, 13-14, 19]). In 2001, Xu and Ori [19] introduced the following implicit iteration process for a …nite family of nonexpansive mappings fTi : i 2 Ig (here I = f1; 2; : : : ; N g), with f n g a real sequence in (0; 1), and an initial point x0 2 K: x1 = (1 x2 = (1 .. . xN = (1 xN +1 = (1 .. .

1 )x0

+ 2 )x1 +

1 T1 x1 ; 2 T2 x2 ;

N )xN 1

+ N +1 )xN +

N TN xN ; N +1 TN +1 xN +1 ;

which can be written in the following compact form: xn = (1

n )xn 1

+

n Tn xn ;

for all n

1;

(XO)

where Tn = Tn (mod N ) (here the mod N function takes values in I). Xu and Ori [19] proved the weak convergence of this process to a common …xed point of the …nite family de…ned in a Hilbert space. They further remarked that it is yet unclear what assumptions on the mappings and/or the parameters f n g are su¢ cient to guarantee the strong convergence of the sequence fxn g. In [13], Osilike proved the following results. Theorem 1 Let E be a real Banach space and K be a nonempty closed convex subset of E. Let fTi : i 2 Ig be N strictly pseudocontractive self-mappings of K with F =

N \

i=1

conditions:

F (Ti ) 6= ;. Let f (i)

0
0 and any bounded subset K; there exists > 0 such that ksx + (1 s)yk2 (1 2s) kyk2 + 2s Re hx; j(y)i + 2s (2.1) for all x; y 2 K and s 2 [0; ]:

Remark 3 1. If X is uniformly smooth, then (1) in Lemma 1 holds. 2. If X is a Hilbert space, then (2) in Lemma 1 holds. Lemma 2 [4] Let T : D(T ) X ! X be an operator with F (T ) 6= ;: Then T is strictly hemicontractive if and only if there exists t > 1 such that for 895

HUSSAIN, RAFIQ: IMPLICIT MANN ITERATION METHOD

all x 2 D(T ) and q 2 F (T ); there exists j(x Re hx

T x; j(x

q)i

q) 2 J(x

1 kx t

1

q) satisfying

qk2 :

(2.2)

Lemma 3 [11] Let X be an arbitrary normed linear space and T : D(T ) X ! X be an operator. (1) If T is a local strongly pseudocontractive operator and F (T ) 6= ;; then F (T ) is a singleton and T is strictly hemicontractive. (2) If T is strictly hemicontractive, then F (T ) is a singleton. 1 1 Lemma 4 [11] Let { n g1 n=0 ; { n gn=0 and { n gn=0 be nonnegative real sequences and let 0 > 0 be a constant satisfying

n+1

where 0

1 P

n=0

:

n

= 1;

n

(1

n) n

1 for all n

0

+

+

n

0 and

n;

1 P

n=0

n

n

0;

< 1: Then, lim sup n!1

n

We now prove our main results. Theorem 2 Let X be a smooth Banach space satisfying any of the Axioms (1)-(3) of Lemma 1. Let K be a nonempty closed bounded convex subset of X and T : K ! K be a continuous strictly hemicontractive mapping. Suppose that f n g1 lim n = 0 and n=0 be a sequence in [0; 1] satisfying conditions (i) n!1 P1 (ii) n=0 n = 1:

1 For a sequence fvn g1 n=0 in K, suppose that fxn gn=0 is the sequence generated from an arbitrary x0 2 K by

xn = (1 and satisfying lim kvn n!1

n )xn 1

+

n T vn ;

n

1;

(2.3)

xn k = 0:

Then the sequence fxn g1 n=0 converges strongly to a unique …xed point q of T: Proof By [5, Corollary 1], T has a unique …xed point q in K. It follows from Lemma 3 that F (T ) is a singleton. That is, F (T ) = fqg for some q 2 K: Now for k = 1t ; where t satis…es (2:2): Set M = 1 + diamK: For all n M = sup kxn n 1

0; it is easy to verify that qk + sup kT vn n 1

896

qk :

(2.4)

HUSSAIN, RAFIQ: IMPLICIT MANN ITERATION METHOD

Also

kvn

qk2

kvn xn k2 + kxn qk2 +2M kvn xn k kxn qk kvn xn k2 + kxn qk2 +2M kvn xn k :

(2.5)

Consider

kxn

qk2 = k(1 = k(1 (1 kxn

qk2 q) + n (T vn n ) (xn 1 qk2 + n kT vn n ) kxn 1 qk2 + M 2 n ; n )xn 1

1

+

(2.6)

n T vn

2

q)k qk2

where the …rst inequality holds by the convexity of k:k2 : Substituting (2.6) in (2.5) to get kvn

qk2

kxn

qk2 + kvn

1

xn k2 + 2M kvn

xn k + M 2

n:

(2.7)

Using (2:3) and Lemma 6, we infer that

kxn

qk2 = k(1 qk2 n )xn 1 + n T vn = k(1 q) + n (T vn q)k2 n ) (xn 1 (1 2 n ) kxn 1 qk2 + 2 n Re(T vn q; j(xn 1 q)) +2 n = (1 2 n ) kxn 1 qk2 + 2 n Re(T vn q; j(vn q)) +2 n Re(T vn q; j(xn 1 q) j(vn q)) + 2 n (1 2 n ) kxn 1 qk2 + 2k n kvn qk2 +2 n kT vn qk kj(xn 1 q) j(vn q)k + 2 n (1 2 n ) kxn 1 qk2 + 2k n kvn qk2 +2M n n + 2 n ;

(2.8)

where n

= kj(xn

1

q)

j(vn

q)k :

Since J is uniformly continuous on any bounded subsets of X; and 897

(2.9)

HUSSAIN, RAFIQ: IMPLICIT MANN ITERATION METHOD

kxn

vn k

1

kxn 1 xn k + kxn vn k = kxn 1 (1 n )xn 1 n T vn k + kxn = n kxn 1 T vn k + kxn vn k 2M n + kxn vn k ! 0;

as n ! 1; implies

vn k

! 0 as n ! 1: (2.10) For given any > 0 and the bounded subset K, there exists a > 0 satisfying (2:1): Note that the condition lim kvn xn k = 0; (2.10) and (i) ensure that n!1 there exists an N such that n

xn k2 + 2M kvn

kvn

xn k

1

; g; 2 (1 k) 6M 2 k Now substituting (2.7) in (2.8) to obtain n

kxn

qk2

< minf ;

(1

2 (1 2

+2M k (1

for all n

3k

k) 2 n

+ 2k

2 (1

k)

; n

6M

qk2 + 2M

n ) kxn 1 n

n

(2.11)

;

n n

2

kvn

xn k + 2M kvn 2

n ) kxn 1

qk + 3

N :

+2

n

(2.12)

xn k

n;

N:

Put = kxn 1 qk ; k) n ; n = 2(1 3 0 = ; 2(1 k) n = 0; n

we have from (2.12) n+1

Observe that

P1

n=0

n

from Lemma 10 that

(1

= 1;

n) n

n

+

1 and

lim sup kxn

n!1

0

n

+

1 P

n=0

qk2

Letting 0 ! 0+ , we obtain that lim sup kxn n!1 xn ! q as n ! 1: 898

n

n;

n

1:

< 1 for all n 0

1: It follows

:

qk2 = 0; which implies that

HUSSAIN, RAFIQ: IMPLICIT MANN ITERATION METHOD

Corollary 1 Let X be a smooth Banach space satisfying any of the Axioms (1)-(3) of Lemma 1. Let K be a nonempty closed bounded convex subset of X and T : K ! K be a Lipschitz strictly hemicontractive mapping. Suppose that f n g1 lim n = 0 and n=0 be a sequence in [0; 1] satisfying conditions (i) n!1 P1 (ii) n=0 n = 1:

From arbitrary x0 2 K, de…ne the sequence fxn g by the implicit iteration process (2.3). Then the sequence fxn g1 n=0 converges strongly to a unique …xed point q of T: Corollary 2 Let X be a smooth Banach space satisfying any of the Axioms (1)-(3) of Lemma 1. Let K be a nonempty closed bounded convex subset of X and T : K ! K be a continuous strictly hemicontractive mapping. Suppose that f n g1 lim n = 0 and n=0 be a sequence in [0; 1] satisfying conditions (i) n!1 P1 (ii) n=0 n = 1:

From arbitrary x0 2 K, de…ne the sequence fxn g by the implicit iteration process (XO). Then the sequence fxn g1 n=0 converges strongly to a unique …xed point q of T: Corollary 3 Let X be a smooth Banach space satisfying any of the Axioms (1)-(3) of Lemma 1. Let K be a nonempty closed bounded convex subset of X and T : K ! K be a Lipschitz strictly hemicontractive mapping. Suppose that f n g1 lim n = 0 and n=0 be a sequence in [0; 1] satisfying conditions (i) n!1 P1 (ii) n=0 n = 1:

From arbitrary x0 2 K, de…ne the sequence fxn g by the implicit iteration process (XO). Then the sequence fxn g1 n=0 converges strongly to a unique …xed point q of T: Remark 3 Similar results can be found for the iteration processes involved error terms, we omit the details. Remark 4 Theorem 2 and Corollary 2 extend and improve Theorem 1 in the following way: We do not need the assumption lim inf d(xn ; F ) = 0 as in Theorem 1. n!1

3

Applications for multi-Step …xed point iterations

Let K be a nonempty closed convex subset of a real normed space E and T1 ; T2 ; :::; Tp : K ! K (p 2) be a family of selfmappings. 899

HUSSAIN, RAFIQ: IMPLICIT MANN ITERATION METHOD

Algorithm 1 For a given x0 2 K, compute the sequence fxn g by the implicit iteration process of arbitrary …xed order p 2; xn = (1 yni = (1 ynp 1 = (1

1 n )xn 1 + n T1 yn ; i i i+1 n )xn 1 + n Ti+1 yn ; p 1 p 1 n )xn 1 + n Tp xn ;

i = 1; 2; :::; p n 1;

2; (3.1)

which is called the multi-step implicit iteration process, where f [0; 1], i = 1; 2; :::; p 1.

n g,

f

i ng

For p = 3, we obtain the following three-step implicit iteration process: Algorithm 2 For a given x0 2 K, compute the sequence fxn g by the iteration process xn = (1 yn1 = (1 yn2 = (1

n )xn 1 1 n )xn 1 2 n )xn 1

+ + +

where f n g , f 1n g and f certain conditions.

1 n T1 yn ; 1 2 n T2 yn ; 2 n T3 xn ; 2 ng

; n

(3.2)

1;

are three real sequences in [0; 1] satisfying some

For p = 2, we obtain the following two-step implicit iteration process: Algorithm 3 For a given x0 2 K, compute the sequence fxn g by the iteration process xn = (1 yn1 = (1

n )xn 1 1 n )xn 1

where f n g and f conditions.

1 ng

If T1 = T; T2 = I; process:

+ +

1 n T1 yn ; 1 n T2 xn ;

n

(3.3)

1;

are two real sequences in [0; 1] satisfying some certain 1 n

= 0 in (3.3), we obtain the implicit Mann iteration

Algorithm 4 For any given x0 2 K, compute the sequence fxn g by the iteration process xn = (1 where f

ng

n )xn 1

+

n T xn ;

n

1;

(3.4)

is a real sequence in [0; 1] satisfying some certain conditions.

Theorem 3 Let K be a nonempty closed bounded convex subset of a smooth Banach space X and T1 ; T2 ; ; Tp (p 2) be selfmappings of K. Let T1 be a 900

HUSSAIN, RAFIQ: IMPLICIT MANN ITERATION METHOD

[0; 1], continuous strictly hemicontractive mapping. Let f n gn 0 , f in gn 0 P i = 1; 2; :::; p 1 be real sequences in [0; 1] satisfying n 0 n = 1; lim n = 0 and n!1 lim

1 n

Then fxn gn

n!1

= 0: For arbitrary x0 2 K; de…ne the sequence fxn gn 0

converges strongly to the common …xed point in

p T

i=1

0

by (3.1).

F (Ti ) 6= ;.

Proof By applying Theorem 2 under the assumption that T1 is continuous strictly hemicontractive, we obtain Theorem 3 which proves strong convergence of the iteration process de…ned by (3.1). Consider by taking T1 = T and vn = yn1 ;

kvn

xn k = yn1

xn

= (1

1 n )xn 1

= (1

1 n ) (xn 1

(1 (1

1 n ) kxn 1

1 n ) kxn 1

+

1 2 n T2 yn

xn ) +

xn 1 n 1 n

xn k +

T2 yn2

xn

T2 yn2

xn

1 n:

xn k + 2M

Now from (3.5) and the condition lim n!1 lim kvn xn k = 0:

1 n

(3.5)

= 0; it can be easily seen that

n!1

Corollary 4 Let K be a nonempty closed bounded convex subset of a smooth Banach space X and T1 ; T2 ; ; Tp (p 2) be selfmappings of K. Let T1 be a Lipschitz strictly hemicontractive mapping. Let f n gn 0 , f in gn 0 [0; 1], P i = 1; 2; :::; p 1 be real sequences in [0; 1] satisfying n 0 n = 1; n!1 lim n = 0 and n!1 lim

1 n

Then fxn gn

= 0: For arbitrary x0 2 K; de…ne the sequence fxn gn 0

converges strongly to the common …xed point in

p T

i=1

0

by (3.1).

F (Ti ) 6= ;.

Acknowledgements The …rst author gratefully acknowledges the support from the Deanship of Scienti…c Research (DSR) at King Abdulaziz University (KAU) during this research. References (1) S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal. TMA 30 (7) (1997), 4197–4208. (2) S. S. Chang, Y. J. Cho, B. S. Lee, S. M. Kang, Iterative approximations of …xed points and solutions for strongly accretive and strongly pseudocontractive mappings in Banach spaces, J. Math. Anal. Appl. 224 (1998), 149–165. (3) C. E. Chidume, Iterative approximation of …xed points of Lipschitzian strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 99 (2) (1987), 901

HUSSAIN, RAFIQ: IMPLICIT MANN ITERATION METHOD

(4)

(5) (6) (7)

(8)

(9)

(10)

(11)

(12) (13)

(14)

(15) (16) (17) (18) (19)

283–288. C. E. Chidume, M. O. Osilike, Fixed point iterations for strictly hemicontractive maps in uniformly smooth Banach spaces, Numer. Funct. Anal. Optimiz. 15 (1994), 779–790. K. Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 365-374. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. N. Hussain, Asymptotically pseudo-contractions, Banach operator pairs and best simultaneous approximations, Fixed Point Theory and Applications, Volume 2011, Article ID 812813 11 pages N. Hussain, R. Chugh, V. Kumar, A. Ra…q, On the rate of convergence of Kirk-type iterative schemes, Journal of Applied Mathematics, Volume 2012 (2012), Article ID 526503, 22 pages N. Hussain, A. Ra…q, B. Damjanovi´c and R. Lazovic, On rate of convergence of various iterative schemes, Fixed Point Theory and Applications, 2011, Volume 2011, Number 1, 45 Z. Liu and S. M. Kang, Stability of Ishikawa iteration methods with errors for strong pseudocontractions and nonlinear equations involving accretive operators in arbitrary real Banach spaces, Math. and Computer Modell. 34 (2001), 319–330. Z. Liu, S. M. Kang, S. H. Shim, Almost stability of the Mann iteration method with errors for strictly hemicontractive operators in smooth Banach spaces, J. Korean Math. Soc. 40 (1) (2003), 29–40. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510. M. O. Osilike, Implicit iteration process for common …xed points of a …nite family of strictly pseudocontractive maps., J. Math. Anal. Appl. 294 (1) (2004), 73-81. J. A. Park, Mann iteration process for the …xed point of strictly pseudocontractive mapping in some Banach spaces, J. Korean Math. Soc. 31 (1994), 333–337. B. E. Rhoades, Comments on two …xed point iteration methods, J. Math. Anal. Appl. 56 (1976), 741–750. S. Reich, Constructive techniques for accretive and monotone operators, Applied Nonlinear Analysis, Academic Press (1979), 335-345. J. Schu, Iterative construction of …xed points of strictly pseudocontractive mappings, Applicable Anal. 40 (1991), 67–72. X. Weng, Fixed point iteration for local strictly pseudo-contractive mapping, Proc. Amer. Math. Soc. 113 (1991), no. 3, 727–731. H. K. Xu, R. Ori, An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001), 767-773.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 903-910, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Normal fuzzy filters in BE-algebras Sun Shin Ahn1 and Young Hie Kim2∗ 1 2

Department of Mathematics Education, Dongguk University, Seoul 100-715, Korea

Bangmok College of General Education, Myongji University, Youngin-Si, Gyeonggi-Do 449-728, Korea

Abstract. In this paper, we introduce the notion of normal fuzzy filters and maximal fuzzy filters in a BEalgebra. We prove that every maximal fuzzy filter of X is normal and if µ is a maximal fuzzy filter of X, then Xµ = {𝑥 ∈ X∣µ(𝑥) = µ(1)} is a maximal filter of X.

1. Introduction Y. Imai and K. Is´eki introduced two classes of abstract algebras: BCK-algebras and BCIalgebras ([5,6]). It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. In [3,4] Q. P. Hu and X. Li introduced a wide class of abstract algebras: BCH-algebras. They have shown that the class of BCI-algebras is a proper subclass of the class of BCH-algebras. J. Neggers and H. S. Kim ([12]) introduced the notion of d-algebras which is another generalization of BCK-algebras, and also they introduced the notion of Balgebras ([13,14]), i.e., (I) x ∗ x = 0; (II) x ∗ 0 = x; (III) (x ∗ y) ∗ z = x ∗ (z ∗ (0 ∗ y)), for any x, y, z ∈ X, which is equivalent in some sense to the groups. Moreover, Y. B. Jun, E. H. Roh and H. S. Kim ([10]) introduced a new notion, called a BH-algebra, which is a generalization of BCH/BCI/BCK-algebras, i.e., (I); (II) and (IV) x ∗ y = 0 and y ∗ x = 0 imply x = y for any x, y ∈ X. A. Walendziak obtained an another equivalent axioms for B-algebra ([15]). H. S. Kim, Y. H. Kim and J. Neggers ([9]) introduced the notion a (pre-) Coxeter algebra and showed that a Coxeter algebra is equivalent to an abelian group all of whose elements have order 2, i.e., a Boolean group. C. B. Kim and H. S. Kim ([7]) introduced the notion of a BM -algebra which is a specialization of B-algebras. They proved that the class of BM -algebras is a proper subclass of B-algebras and also showed that a BM -algebra is equivalent to a 0-commutative B-algebra. In [8], H.S. Kim and Y. H. Kim introduced the notion of a BE-algebra as a generalization of a BCK-algebra. Using the notion of upper sets they gave an equivalent condition of the filter in BE-algebras. In [1,2], S. S. Ahn and K. S. So introduced the notion of ideals in BE-algebras, and then discussed and proved several characterizations of such ideals. Also they generalized the notion of upper sets in BE-algebras, and discussed several properties of the characterizations of generalized upper sets An (u, v) while relating them to the structure of ideals in transitive and self distributive BE-algebras. 0

2010 Mathematics Subject Classification: 06F35; 03G25; 08A72. Keywords: (fuzzy) filter; BE-algebra; normal; maximal. The corresponding author. 0 E-mail: [email protected]; [email protected] 0

∗

903

Sun Shin Ahn and Young Hie Kim

In this paper, we introduce the notion of normal fuzzy filters and maximal fuzzy filters in a BE-algebra. We prove that every maximal fuzzy filter of X is normal and if µ is a maximal fuzzy filter of X, then Xµ = {x ∈ X∣µ(x) = µ(1)} is a maximal filter of X. 2. Preliminaries We recall some definitions and results (See[8]). Definition 2.1 An algebra (X; ∗, 1) of type (2, 0) is called a BE-algebra if (BE1) (BE2) (BE3) (BE4)

x ∗ x = 1 for all x ∈ X; x ∗ 1 = 1 for all x ∈ X; 1 ∗ x = x for all x ∈ X; x ∗ (y ∗ z) = y ∗ (x ∗ z) for all x, y, z ∈ X (exchange)

We introduce a relation “≤” on X by x ≤ y if and only if x ∗ y = 1. A non-empty subset A of X is said to be a subalgebra of a BE-algebra X if it is closed under the operation “ ∗ ”. Noticing that x ∗ x = 1 for all x ∈ X, it is clear that 1 ∈ A. Proposition 2.2. If (X; ∗, 1) is a BE-algebra, then x ∗ (y ∗ x) = 1 for any x, y ∈ X. Definition 2.3. Let (X; ∗, 1) be a BE-algebra and let F be a non-empty subset of X. Then F is said to be a filter of X if (F1) 1 ∈ F ; (F2) x ∗ y ∈ F and x ∈ F imply y ∈ F . Proposition 2.4. Let (X; ∗, 1) be a BE-algebra and let F be a filter of X. If x ≤ y and x ∈ F for any y ∈ X, then y ∈ F . 3. Fuzzy filters In what follows, let X denote a BE-algebra unless otherwise specified. Definition 3.1. A fuzzy set µ in X is called a fuzzy filter of X if it satisfies: for any x, y ∈ X, (UF1) µ(1) ≥ µ(x); (UF2) µ(y) ≥ min{µ(x ∗ y), µ(x)}. Theorem 3.2. Let J be a filter of a BE-algebra X and let µ be a fuzzy set in X defined by { t if x ∈ J, µ(x) := 0 if x ∈ /J where t is a fixed number in (0, 1). Then µ is a fuzzy filter of X. Proof. Since 1 ∈ J, we have µ(1) = t. Hence µ(1) ≥ µ(x) for all x ∈ X. Let x, y ∈ X. If x, y ∈ / J, then µ(x) = µ(y) = 0. Therefore µ(y) = min{µ(x ∗ y), µ(x)}. If x, y ∈ J, then µ(x) = µ(y) = t. Therefore µ(y) = min{µ(x ∗ y), µ(x)} = t. Suppose that y ∈ J and x ∈ / J. Then µ(y) = t and 904

Normal fuzzy filters in BE-algebras

µ(x) = 0. Therefore µ(y) = t ≥ min{µ(x ∗ y), µ(x)} = 0. The case that x ∈ J, y ∈ / J and x ∗ y ∈ J can not happen, since J is a filter of X. This completes the proof.  Example 3.3. Let X := {1, a, b, c, d, 0} be a BE-algebra with the following table: ∗ 1 a b c d 0

1 1 1 1 1 1 1

a a 1 1 a 1 1

b b a 1 b a 1

c c c c 1 1 1

d d c c a 1 1

0 0 d c b a 1

Then F1 := {1, a, b} and F2 := {1, c} are filters of X, but F3 := {1, a}, F4 := {1, b} are not a filter of X, since a ∗ b = a ∈ F3 , b ∈ / F3 and b ∗ a = 1 ∈ F4 , a ∈ / F4 . Define a map µ : X → [0, 1] by µ(1) > µ(c) > µ(a) = µ(b) = µ(d) = µ(0). Then it is easy to see that µ is a fuzzy filter of X. Also if we define a map ν : X → [0, 1] by ν(1) = ν(a) = ν(b) = 0.7 > 0.1 = µ(c) = µ(d) = µ(0), then ν is a fuzzy filter of X. Proposition 3.4. Let F be a non-empty subset X and let µF be a fuzzy set in X defined by { t if x ∈ F , µF (x) := 0 if x ∈ /F for all x ∈ X and for any t ∈ [0, 1]. Define a subset XµF := {x ∈ X∣µF (x) = µF (1)}. Then (i) µF is a fuzzy filter of X if and only if F is a filter of X. (ii) XµF = F if and only if F is a filter of X. Proof. (i) Suppose that µF is a fuzzy filter of X. Given x ∈ X, by (UF1), we have µF (1) ≥ µF (x). Since µF (x) = t for any x ∈ F , µF (1) should be t, i.e., µF (1) = t. Hence 1 ∈ F . If x ∗ y ∈ F and x ∈ F , then µF (x ∗ y) = t and µF (x) = t. By (UF2), we have µF (y) ≥ min{µF (x ∗ y), µF (x)} = t, i.e., µF (y) ≥ t. Hence µF (y) = t and so y ∈ F . Thus F is a filter of X. Conversely, assume that F is a filter of X. By (F1), µF (1) = t. Hence µF (1) ≥ µF (x), ∀x ∈ X. It remains to prove (UF2). Assume that there exist x, y ∈ X such that µF (y) < min{µF (x ∗ y), µF (x)}, Since µF is a two-valued function, we have µF (y) = 0, µF (x ∗ y) = µF (x) = t. This means x ∗ y, x ∈ F , but y∈ / F , proving that F is not a filter of X, a contradiction. (ii) Straightforward.  4. Normal fuzzy filters Definition 4.1. A fuzzy filter µ of X is said to be normal if there exists x ∈ X such that µ(x) = 1. 905

Sun Shin Ahn and Young Hie Kim

Example 4.2. Let X be a BE-algebra as in Example 3.2. Define a fuzzy set µ in X by { 1 if x ∈ F µ(x) = 0.7 if x ∈ /F for all x ∈ X. Then µ is a normal fuzzy filter of X. Theorem 4.3. A fuzzy filter µ of X is normal if and only if µ(1) = 1. 

Proof. Straightforward.

Theorem 4.4. µ be a fuzzy filter of X. Then the fuzzy set µ ¯ of X defined by µ ¯(x) := µ(x) + 1 − µ(1) for all x ∈ X is a normal fuzzy filter of X containing µ. Proof. Let µ be a fuzzy filter of X and let x, y ∈ X. Then µ ¯(1) = µ(1)+1−µ(1) ≥ µ(x)+1−µ(1) = µ ¯(x) for all x ∈ X, since µ(1) ≥ µ(x). Since µ ¯(y) = µ(y) + 1 − µ(1) ≥ min{µ(x ∗ y), µ(x)} + 1 − µ(1) = min{µ(x ∗ y) + 1 − µ(1), µ(x) + 1 − µ(1)} = min{¯ µ(x ∗ y), µ ¯(x)}, µ ¯ is a fuzzy filter of X. On the other hand, µ ¯(1) = µ(1) + 1 − µ(1) = 1. By Theorem 4.3, µ ¯ is normal. Clearly µ ⊆ µ ¯.  Theorem 4.5. A fuzzy filter µ of X is normal if and only if µ ¯ = µ. Proof. Suppose that µ is a normal fuzzy filter of X. Then µ ¯(x) = µ(x) + 1 − µ(1) = µ(x) for any x ∈ X by Theorem 4.3, and so µ ¯ ⊆ µ. Therefore µ = µ ¯. The sufficiency part is trivial.  ¯=µ Theorem 4.6. If µ is a fuzzy filter of X, then µ ¯. ¯(x) = µ Proof. For any x ∈ X, we get µ ¯(x) + 1 − µ ¯(1) = µ ¯(x).



Theorem 4.7. Let µ be a fuzzy filter of X. If there exists a fuzzy filter η of X satisfying η¯ ⊂ µ, then µ is normal. Proof. Assume that there exists a fuzzy filter η of X such that η¯ ⊂ µ. Then 1 = η¯(1) ≤ µ(1) and hence µ(1) = 1. Thus µ is normal.  Proposition 4.8. Let µ be a fuzzy filter of X. If there exists a fuzzy filter η of X satisfying η¯ ⊂ µ, then µ ¯ = µ. Proof. It follows from Theorem 4.5 and Theorem 4.7. Theorem 4.9. Let µ and η be fuzzy filters of X. Then followings hold. (i) (ii) (iii) (iv)

if µ ⊂ η and µ(1) = η(1), then Xµ ⊆ Xη . if µ and η are normal and µ ⊆ η, then Xµ = Xη . Xµ ⊆ Xµ¯ . if there exists x0 ∈ X such that µ ¯(x0 ) = 0, then µ(x0 ) = 0. 906



Normal fuzzy filters in BE-algebras

Proof. (i) Suppose that µ ⊂ η and µ(1) = η(1). If x ∈ Xµ , then η(1) = µ(1) = µ(x) ≤ η(x). Noting that η(x) ≤ η(1) for all x ∈ X, we get η(x) = η(1). This means that x ∈ Xη . (ii) It follows from Theorem 4.3 and Theorem 4.9-(i). (iii) By Theorem 4.4, µ ¯ is a normal fuzzy filter of X containing µ. Let x ∈ Xµ = {x ∈ X∣µ(x) = µ(1)}. Since µ ⊆ µ ¯, we have µ(x) = µ(1) ≤ µ ¯(x) = µ(1). Hence x ∈ Xµ¯ = {x ∈ X∣¯ µ(x) = µ ¯(1)}, which shows that Xµ ⊆ Xµ¯ . (iv) If there exists x0 ∈ X such that µ ¯(x0 ) = 0, then µ ¯(x0 ) = µ(x0 ) + 1 − µ(1) = 0 and so µ(x0 ) = µ(1) − 1 ≤ 0. Since µ(x0 ) ≥ 0, it follows that µ(x0 ) = 0.  Theorem 4.10. Let µ be a fuzzy filter of X and let f : [0, µ(1)] → [0, 1] be an increasing function. Define a fuzzy set µf : X → [0, 1] by µf (x) := f (µ(x)) for all x ∈ X. Then µf is a fuzzy filter of X. In particular, if f (µ(1)) = 1, then µf is normal, and if f (t) ≥ t for all t ∈ [0, µ(1)], then µ ⊆ µf . Proof. Let x ∈ X. Then µf (1) = f (µ(1)) ≥ f (µ(x)) = µf (x) since f is an increasing function. Since µ is a fuzzy filter of X, we have µ(y) ≥ min{µ(x ∗ y), µ(x)} for all x, y ∈ X. Hence f (µ(y)) ≥ min{f (µ(x ∗ y)), f (µ(x))} by using the fact that f is an increasing function, i.e., µf (y) ≥ min{µf (x ∗ y), µf (x)}. Therefore µf is a fuzzy filter of X. Since f (µ(1)) = 1, we have µf (1) = f (µ(1)) = 1. By applying Theorem 4.3, we show that µf is normal. Suppose that f (t) ≥ t for all t ∈ [0, µ(1)]. Then µf (x) = f (µ(x)) for all x ∈ X. This means that µ ⊆ µf .  Lemma 4.11. If µ and η are two fuzzy filters of X, then so is µ ∩ η. Moreover, if µ and η are normal, then so is µ ∩ η. 

Proof. Straightforward.

Given a proper filter F of X, it is easily verified that the characteristic function χF is a normal fuzzy filter of X. Let E and F be two proper filters of X. Obviously, E ⊆ F if and only if χE ⊆ χF . Let P (X) be the set of all proper fuzzy filters of X. Let N (X) be the set of all normal fuzzy filters µ of X such that N (X) is a poset under the set inclusion. We define two functions f : P (X) → N (X) and g : N (X) → P (X) by f (F ) := χF and g(µ) := Xµ , respectively. Then we have the following. Corollary 4.12. Two functions f and g defined above are injective and surjective respectively. Proof. It is straightforward from gf = 1P (X) and f g(µ) = f (Xµ ) = χXµ = µ for all u ∈ N (X).  Theorem 4.13. Let X be a BE-algebra. (i) If E, F ∈ P (X), then χE∩F = χE ∩ χF . (ii) If µ, η ∈ N (X), then Xµ∩η = Xµ ∩ Xη . (iii) ∀E, F ∈ P (X) and ∀µ, η ∈ N (X), f (E ∩ F ) = f (E) ∩ f (F ) and g(µ ∩ η) = g(µ) ∩ g(η). 907

Sun Shin Ahn and Young Hie Kim

Proof. (i) If x ∈ E ∩ F , then χE∩F (x) = 1. However, we also get x ∈ E and x ∈ F . Hence χE (x) = 1 = χF (x). If x ∈ / E ∩ F , then x ∈ / E or x ∈ / F and hence χE∩F (x) = 0. Therefore min{χE (x), χF (x)} = 0. This means that χE∩F = χE ∩ χF . (ii) Xµ∩η ={x ∈ X∣ min{µ(x), η(x)} = 1} ={x ∈ X∣µ(x) = 1, η(x) = 1} ={x ∈ X∣µ(x) = 1} ∩ {x ∈ X∣η(x) = 1} =Xµ ∩ Xη . (iii) By (i) and (ii), we have f (E ∩ F ) = χE∩F = χE ∩ χF = f (E) ∩ f (F ) and g(µ ∩ η) = χµ∩η = χµ ∩ χη = g(µ) ∩ g(η).  Corollary 4.14. (N (X), ⊆) is a meet-semilattice. Proof. Obviously, (N (X), ⊆) is a poset having the smallest element χ{1} , and the greatest element 1 given by 1(x) := 1 for all x ∈ X.  Theorem 4.15. Let µ be a non-constant normal fuzzy filter of X such that it is a maximal element of (N (X), ⊆). Then Im(µ) = {0, 1}. Proof. Note that µ(1) = 1 since µ is normal. Assume that there exists a ∈ X such that 1 0 < µ(a) < 1. Define a fuzzy set η : X → [0, 1] by η(x) := {µ(x) + µ(a)} for all x ∈ X. Then 2 clearly µ is well-defined. For any x ∈ X, we have 1 1 η(1) = {µ(1) + µ(a)} ≥ {µ(x) + µ(a)} = η(x). 2 2 Let x, y ∈ X. Then 1 η(y) = {µ(y) + µ(a)} 2 1 ≥ {min{µ(x ∗ y), µ(x)} + µ(a)} 2 1 1 = min{ {µ(x ∗ y) + µ(a)}, {µ(x) + µ(a)}} 2 2 = min{η(x ∗ y), η(x)}. Therefore η is a fuzzy filter of X. It follows from Theorem 4.4 that η¯ ∈ N (X) where η¯(x) = 1 1 η(x) + 1 − η(1) for all x ∈ X. Note that η¯(a) = η(a) + 1 − η(1) = {µ(a) + µ(a)} + 1 − {µ(1) + 2 2 1 µ(a)} = {µ(a) + 1} > µ(a) and η¯(a) < 1 = η¯(1). Thus η¯ is a non-constant normal fuzzy filter 2 and µ is not a maximal element of (N (X), ⊆). This is a contradiction. Thus Im(µ) = {0, 1}.  Definition 4.16. Let µ be a fuzzy filter of X. Then µ is said to be maximal if it is non-constant and µ ¯ is a maximal element of the poset (N (X), ⊆). Theorem 4.17. Every maximal fuzzy filter of X is normalized and Im(µ) = {0, 1}. 908

Normal fuzzy filters in BE-algebras

Proof. Assume that µ is a maximal fuzzy filter of X. Then µ ¯ is a non-constant maximal element of the poset (N (X), ⊆). Using Theorem 4.4 and Theorem 4.5, we have Im(µ) = {0, 1}. Note that µ ¯(x) = 1 if and only if µ(x) = µ(1) and µ ¯(x) = 0 if and only if µ(x) = µ(1) − 1. By Theorem 4.9-(iv), we get µ(x) = 0. It follows that µ(1) = 1. This shows that µ is normal and hence µ = µ ¯ follows from Theorem 4.15 that Im(µ) = {0, 1}.  In general, if µ is a fuzzy normal filter, then χXµ ⊆ µ. But if µ is a fuzzy filter, then we have the following. Corollary 4.18. If µ is a maximal fuzzy filter of X, then χXµ = µ. Proof. Obviously, χXµ ⊂ µ and χXµ takes only 0 and 1. If µ(x) = 0, then x ∈ / Xµ and χXµ (x) = 0. Thus µ ⊆ χXµ . If µ(x) = 1, then x ∈ Xµ and hence χXµ (x) = 1. This means that XXµ = µ.  Definition 4.19. A filter F of X is said to be maximal if (i) F is a proper subset of X, (ii) F ⊆ G for some filter G of X, then either F = G or G = X. Example 4.20. Let X := {1, a, b, c} be a BE-algebra with the following Cayley table: ∗ 1 a b c

1 1 1 1 1

a a 1 1 a

b b b 1 c

c c c 1 1

Then {1, a}, {1, a, c} are proper filters of X, and {1, a, b} is a unique maximal filter of X. Theorem 4.21. If µ is a maximal fuzzy filter of X, then Xµ is a maximal filter of X. Proof. Xµ is a proper filter of X, since µ is non-constant. Assume that F is a filter of X containing Xµ . Then χXµ ⊆ χF . Thus we get µ = χXµ ⊆ χF by Corollary 4.18. Because µ and χF are normal and µ = µ ¯ is a maximal element of the poset (N (X), ⊆), it follows that µ = χF or χF = 1, where 1 : X → [0, 1] is a fuzzy set defined by 1(x) := 1 for all x ∈ X. It means that F = X. If µ = χF , then Xµ = XχF = F by Proposition 3.4. This means that Xµ is a maximal filter of X.  References [1] S. S. Ahn and K. K. So, On ideals and upper sets in BE-algebras, Sci. Math. Jpn. 68 (2008), 279-285. [2] S. S. Ahn and K. K. So, On generalized upper sets in BE-algebras, Bull. Korean Math. Soc. 46 (2009), 281-287. [3] Q. P. Hu and X. Li, On BCH-algebras, Math. Seminar Notes 11 (1983), 313-320. [4] Q. P. Hu and X. Li, On proper BCH-algebras, Math. Jpn. 30 (1985), 659-661. [5] K. Is´eki and S. Tanaka, An introduction to theory of BCK-algebras, Math. Japonica 23 (1978), 1-26. [6] K. Is´eki, On BC𝐼-algebras, Math. Seminar Notes 8 (1980), 125-130. [7] C. B. Kim and H. S. Kim, On BM -algebras, Sci. Math. Jpn. 63(3) (2006), 421-427. [8] H. S. Kim and Y. H. Kim, On BE-algebras, Sci. Math. Jpn. 66(2007), 113-116. 909

Sun Shin Ahn and Young Hie Kim [9] H. S. Kim, Y. H. Kim and J. Neggers, Coxeters and pre-Coxeter algebras in Smarandache setting, Honam Math. J. 26(4) (2004) 471-481. [10] Y. B. Jun, E. H. Roh and H. S. Kim, On BH-algebras, Sci. Math. Jpn. 1(1998), 347-354. [11] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa, Seoul, 1994. [12] J. Neggers and H. S. Kim, On 𝑑-algebras, Math. Slovaca 49 (1999), 19-26. [13] J. Neggers and H. S. Kim, On B-algebras, Mate. Vesnik. 54 (2002), 21-29. [14] J. Neggers and H. S. Kim, A fundamental theorem of B-homomorphism for B-algebras, Int. Math. J. 2 (2002), 215-219. [15] A. Walendziak, Some axiomatizations of B-algebras, Math. Slovaca 56 (2006), 301-306.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 911-916, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Comment on “Nearly ternary cubic homomorphism in ternary Fr´ echet algebras” [Shagholi et al., J. Computat. Anal. Appl. 13 (2011), 1097-1105] Choonkil Park1 , Jung Rye Lee2∗ and Dong Yun Shin3 1

Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea 2 Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea 3

Department of Mathematics, University of Seoul, Seoul 130-743, Korea

Abstract. Shagholi et al. [17] proved the Hyers-Ulam stability of ternary cubic homomorphisms in ternary Fr´echet algebras. But there are fatal errors in the proofs of the main results. In this paper, we correct the statements of the main results and prove the corrected results.

1. Introduction The stability problem of functional equations originated from a question of Ulam [18] in 1940, concerning the stability of group homomorphisms. In 1941, Hyers [10] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Th. M. Rassias [15] provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. Jun and Kim [11] introduced the following functional equation f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x)

(1.1)

and established the general solution and the Hyers-Ulam stability for the functional equation (1.1). The function f (x) = x3 satisfies the functional equation (1.1), which is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic mapping. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [1, 2, 5, 6, 7, 8, 12, 13, 14, 16]). Definition 1.1. ([17]) A mapping H : A → B is called a ternary cubic homomorphism in ternary algebras A, B if (1) H is a cubic mapping, (2) H([x, y, z]) = [H(x), H(y), H(z)], for all x, y, z ∈ A. Recently, M. Eshaghi Gordji and M. Bavand Savadkouhi [4] investigated approximate cubic homomorphisms in Banach algebras. Definition 1.2. A topological vector space X is a Fr´echet space if it satisfies the following three properties: (1) it is complete as a uniform space, 0

2010 Mathematics Subject Classification: 46K05; 39B82; 47B47. Keywords: Hyers-Ulam stability; cubic functional equation; Fr´echet algebra; ternary cubic homomorphism. ∗ The corresponding author. 0 E-mail: [email protected]; [email protected]; [email protected] 0

911

C. Park, J. Lee, D. Shin (2) it is locally convex, (3) its topology can be induced by a translation invariant metric, i.e., a metric d : X × X → R such that d(x, y) = d(x + a, y + a) for all a, x, y ∈ X. For more detailed definitions of such terminologies, we can refer to [3]. Note that a ternary algebra is called ternary Fr´echet algebra if it is a Fr´echet space with a metric d. If we multiply both sides of the inequality (2.6) in the proof of [17, Theorem 2.1] by

1 16 ,

then we get

1 φ(x, 0, 0) d2 (2f (2x), 16f (x)) ≤ . 16 16

(1.2)

By the triangle inequality, we get 1 d2 (2f (2x), 16f (x)) ≤ d2 16



 f (2x) , f (x) . 8

(1.3)

The inequalities (1.2) and (1.3) do not guarantee the inequality (2.7) in the proof of [17, Theorem 2.1]. In [17], there are a lot of similar fatal errors in the proofs of the main results. In this paper, we correct the statements of the main results and prove the corrected results. 2. Hyers-Ulam stability of ternary cubic homomorphisms in Fr´ echet algebras In this section, we prove the Hyers-Ulam stability of ternary cubic homomorphisms in ternary Fr´echet algebras. Theorem 2.1. Let A be a ternary Fr´echet algebra with metric d and B a ternary Banach algebra with norm k · k. Let f : A → B be a mapping with f (0) = 0 for which there exists a function φ : A3 → [0, ∞) such that ˜ y, z) := φ(x,

∞ X 1 φ(2j x, 2j y, 2j z) < ∞, 3j 2 j=0

(2.1)

kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ φ(x, y, 0),

(2.2)

kf ([x, y, z]) − [f (x), f (y), f (z)]k ≤ φ(x, y, z)

(2.3)

for all x, y, z ∈ A. Then there exists a unique ternary cubic homomorphism H : A → B such that kf (x) − H(x)k ≤

1 ˜ φ(x, 0, 0) 16

(2.4)

for all x ∈ A. Proof. Putting y = 0 in (2.2), we get k2f (2x) − 16f (x)k ≤ φ(x, 0, 0) and so

f (2x)

φ(x, 0, 0)

23 − f (x) ≤ 24 for all x ∈ A. Using the Rassias’ method on (2.5) ([9]), one can use induction on n to show that

n−1 X φ(2j x, 0, 0)

f (2n x)

≤ 1 − f (x)

23n

16 23j j=0 912

(2.5)

(2.6)

Ternary cubic homomorphism in ternary Fr´echet algebras for all x ∈ A and all nonnegative integers n. Hence

n+m−1

f (2n+m x) f (2m x) 1 X φ(2j x, 0, 0)

23(n+m) − 23m ≤ 16 23j j=m n

x) for all nonnegative integers n and m with n ≥ m and all x ∈ A. It follows from (2.1) that the sequence { f (2 23n }

is Cauchy. Due to the completeness of B, this sequence is convergent. So one can define the mapping H : A → B by f (2n x) n→∞ 23n

H(x) := lim

for all x ∈ A. Replacing x, y by 2n x, 2n y, respectively, in (2.2) and multiplying both sides of (2.2) by

(2.7) 1 23n ,

we get

kH(2x + y) + H(2x − y) − 2H(x + y) − 2H(x − y) − 12H(x)k 1 kf (2n (2x + y)) + f (2n (2x − y)) − 2f (2n (x + y)) − 2f (2n (x − y)) − 12f (2n x)k n→∞ 23n φ(2n x, 2n y, 0) =0 ≤ lim n→∞ 23n for all x, y ∈ A. So = lim

H(2x + y) + H(2x − y) = 2H(x + y) + 2H(x − y) + 12H(x) for all x, y ∈ A. Moreover, it follows from (2.6) and (2.7) that kf (x) − H(x)k ≤

1 ˜ φ(x, 0, 0) 16

for all x ∈ A. It follows from (2.3) that 1 kf ([2n x, 2n y, 2n z]) − [f (2n x), f (2n y), f (2n z)]k 29n φ(2n x, 2n y, 2n z) ≤ lim =0 n→∞ 29n

kH([x, y, z]) − [H(x), H(y), H(z)]k = lim

n→∞

for all x, y, z ∈ A. So H([x, y, z]) = [H(x), H(y), H(z)] for all x, y, z ∈ A. Now, let H 0 : A → B be another ternary cubic homomorphism satisfying (2.4). Then we have 1 kH(2n x) − H 0 (2n x)k 23n 1 ≤ 3n (kH(2n x) − f (2n x)k + kf (2n x) − H 0 (2n x)k) 2 1 ˜ n ≤ φ(2 x, 0, 0) 8 · 23n which tends to zero as n → ∞ for all x ∈ A. So we can conclude that H(x) = H 0 (x) for all x ∈ A. This proves the kH(x) − H 0 (x)k =

uniqueness of H. Thus the mapping H : A → B is a unique ternary cubic homomorphism satisfying (2.4).



Corollary 2.2. Let A be a ternary Fr´echet algebra with metric d and B a ternary Banach algebra with norm k · k. Let 0 < p < 3, and let f : A → B be a mapping such that kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ d(x, 0)p + d(y, 0)p , kf ([x, y, z]) − [f (x), f (y), f (z)]k ≤ d(x, 0)p + d(y, 0)p + d(z, 0)p 913

C. Park, J. Lee, D. Shin for all x, y, z ∈ A. Then there exists a unique ternary cubic homomorphism H : A → B such that kf (x) − H(x)k ≤

d(x, 0)p 2(8 − 2p )

holds for all x ∈ X. Proof. Note that d(2x, 0) ≤ 2d(x, 0). It follows from Theorem 2.1 by putting φ(x, y, z) = d(x, 0)p +d(y, 0)p +d(z, 0)p for all x, y, z ∈ A.



Theorem 2.3. Let A be a ternary Banach algebra with norm k · k and B a ternary Fr´echet algebra with metric d. Let f : A → B be a mapping for which there exists a function φ : A3 → [0, ∞) such that ∞ X

29j φ

x

y z , , < ∞, 2j 2j 2j

j=0

(2.8)

d(f (2x + y) + f (2x − y), 2f (x + y) + 2f (x − y) + 12f (x)) ≤ φ(x, y, 0),

(2.9)

d(2f ([x, y, z]), [2f (x), 2f (y), 2f (z)]) ≤ φ(x, y, z)

(2.10)

for all x, y, z ∈ A. Then there exists a unique ternary cubic homomorphism H : A → B such that  x , 0, 0 d(2f (x), H(x)) ≤ φ˜ 2

(2.11)

for all x ∈ A. Here ˜ y, z) := φ(x,

∞ X

23j φ

x

j=0

2j

,

y z , 2j 2j

for all x, y, z ∈ A. Proof. By (2.8), φ(0, 0, 0) = 0. Putting x = y = 0 in (2.9), we get f (0) = 0. Putting y = 0 in (2.9), we have d(2f (2x), 16f (x)) ≤ φ(x, 0, 0) for all x ∈ A. Replacing x by

x 2

(2.12)

in (2.12), we get  x   x d 23 · 2f , 2f (x) ≤ φ , 0, 0 2 2

for all x ∈ A. By induction on n, we have  n−1  x  x  X d 23n · 2f n , 2f (x) ≤ 23j φ j+1 , 0, 0 2 2 j=0

(2.13)

for all x ∈ A and all nonnegative integers n. Hence   d 23(n+m) · 2f

x 2n+m



, 23m · 2f

 x  n+m−1  x  X ≤ 23j φ j+1 , 0, 0 m 2 2 j=m

for all non-negative integers n and m with n ≥ m and all x ∈ A. It follows from (2.8) that the sequence
{23n · 2f ( 2xn )} is Cauchy. Since B is complete, the sequence {23n · 2f 2xn } is convergent. So one can define the mapping H : A → B by H(x) := lim 23n · 2f n→∞

914

x 2n

(2.14)

Ternary cubic homomorphism in ternary Fr´echet algebras for all x ∈ A. Replacing x, y by

y x 2n , 2n ,

respectively, in (2.9) and multiplying both sides by 23n , we get

d(H(2x + y) + H(2x − y), 2H(x + y) + 2H(x − y) + 12H(x))             x  2x − y x+y x−y 2x + y 3n 3n + 2f ,2 4f + 4f + 24f n = lim d 2 2f n→∞ 2n 2n 2n 2n 2           x  2x + y 2x − y x + y x − y ≤ lim 23n d 2f + 2f , 4f + 4f + 24f n n→∞ 2n 2n 2n 2n 2 x y  3n ≤ lim 2 · 2φ n , n , 0 = 0 n→∞ 2 2 for all x, y ∈ A. So H(2x + y) + H(2x − y) = 2H(x + y) + 2H(x − y) + 12H(x) for all x, y ∈ A. It follows from (2.13) and (2.14) that d(2f (x), H(x)) ≤ φ˜

x 2

 , 0, 0

for all x ∈ A. By (2.10), we have d(H([x, y, z]), [H(x), H(y), H(z)])  h x y z i h x y   z i 9n = lim d 29n · 2f , , , 2 2f , 2f , 2f n n n n n n n→∞  h x2 y2 z2i h  x  2  y  2  z i2 9n , n , n , 2f n , 2f n , 2f n ≤ lim 2 d 2f n n→∞ 2 2 2  x y2 z2  2 9n ≤ lim 2 φ n , n , n = 0 n→∞ 2 2 2 for all x, y, z ∈ A. So H([x, y, z]) = [H(x), H(y), H(z)] for all x, y, z ∈ A. To prove the uniqueness of H, let H 0 : A → B be another ternary cubic homomorphism satisfying (2.11). Then we have  x   x d(H(x), H 0 (x)) = d 23n H n , 23n H 0 n 2    2x    x  x  x 3n 3n ≤ d 2 H n , 2 · 2f n + d 23n H 0 n , 23n · 2f n 2  x 2   x  2  x    2x  3n 3n 0 ≤ 2 d H n , 2f n +2 d H , 2f n 2n 2  2x 2 3n+1 ˜ ≤2 φ n+1 , 0, 0 , 2 which tends to zero as n → ∞ for all x ∈ A. So we can conclude that H(x) = H 0 (x) for all x ∈ A.



Corollary 2.4. Let A be a ternary Banach algebra with norm k · k and B a ternary Fr´echet algebra with metric d. Let p, θ be positive real numbers with p > 3, and let f : A → B be a mapping such that d(f (2x + y) + f (2x − y), 2f (x + y) + 2f (x − y) + 12f (x)) ≤ θ(kxkp + kykp ), d(2f ([x, y, z]), [2f (x), 2f (y), 2f (z)]) ≤ θ(kxkp + kykp + kzkp ) for all x, y, z ∈ A. Then there exists a unique ternary cubic homomorphism H : A → B such that θ d(2f (x), H(x)) ≤ p kxkp 2 −8 holds for all x ∈ A. 915

C. Park, J. Lee, D. Shin Proof. Note that d(2x, 0) ≤ 2d(x, 0). It follows from Theorem 2.3 by putting φ(x, y, z) = kxkp + kykp + kzkp for all x, y, z ∈ A.

 Acknowledgments

This work was supported by the Daejin University Research Grant in 2013.

References [1] I. Chang, Stability of higher ring derivations in fuzzy Banach algebras, J. Computat. Anal. Appl. 14 (2012), 1059–1066. [2] I. Cho, D. Kang and H. Koh, Stability problems of cubic mappings with the fixed point alternative, J. Computat. Anal. Appl. 14 (2012), 132–142. [3] M. Eshaghi Gordji, Stability of an additive-quadratic functional equation of two variables in F-spaces, J. Nonlinear Sci. Appl. 2 (2009), 251–259. [4] M. Eshaghi Gordji and M. Bavand Savadkouhi, On approximate cubic homomorphisms, Adv. Difference Equ. 2009, Article ID 618463, 11 pages (2009). [5] M. Eshaghi Gordji, M. Bavand Savadkouhi and M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 12 (2010), 454–462. [6] M. Eshaghi Gordji and A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Computat. Anal. Appl. 13 (2011), 724–729. [7] M. Eshaghi Gordji, R. Farokhzad Rostami and S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ algebras, J. Computat. Anal. Appl. 13 (2011), 734–742. [8] M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi, E. Rashidi and M. Aghaei, Ternary Jordan derivations on C ∗ -ternary algebras, J. Computat. Anal. Appl. 12 (2010), 463–470. [9] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [10] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [11] K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 267–278. [12] H.A. Kenary, J. Lee and C. Park, Non-Archimedean stability of an AQQ-functional equation, J. Computat. Anal. Appl. 14 (2012), 211–227. [13] C. Park, Y. Cho and H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 14 (2012), 526–535. [14] C. Park, S. Jang and R. Saadati, Fuzzy approximate of homomorphisms, J. Computat. Anal. Appl. 14 (2012), 833–841. [15] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [16] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. [17] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. [18] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940.

916

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 917-927, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

On Joint Distributions of Order Statistics from Nonidentically Distributed Discrete Vectors 1,2

B. Y¨ uzba¸sı1 and M. G¨ ung¨or2 Department of Econometrics, University of Inonu, 44280 Malatya, Turkey E-mail: 1 [email protected] and 2 [email protected]

Abstract. In this study, the joint distributions of order statistics arising from innid discrete random vectors are expressed in the form of an integral. Then, the results related to pf and df are given.

Mathematics Subject Classification: 62G30, 62E15. Key words: Order statistics, discrete random vector, probability function, distribution function.

1. Introduction Several identities and recurrence relations for probability density function(pdf ) and distribution function(df ) of order statistics of independent and identically distributed(iid ) random variables were established by numerous authors including Arnold et al.[1], Balasubramanian and Beg[4], David[14], and Reiss[21]. Furthermore, Arnold et al.[1], David[14], Gan and Bain[15], and Khatri[18] obtained the probability function(pf ) and df of order statistics of iid random variables from a discrete parent. Balakrishnan[2] showed that several relations and identities that have been derived for order statistics from continuous distributions also hold for the discrete case. Nagaraja[19] explored the behavior of higher order conditional probabilities of order statistics in a attempt to understand the structure of discrete order statistics. Nagaraja[20] considered some results on order statistics of a random sample taken from a discrete population. Corley[12] defined a multivariate generalization of classical order statistics for random samples from a continuous multivariate distribution. Expressions for generalized joint densities of order statistics of iid random variables in terms of Radon-Nikodym derivatives with respect to product measures based on df were derived by Goldie and Maller[16]. Guilbaud[17] expressed the probability of the functions of independent but not

917

YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

necessarily identically distributed(innid ) random vectors as a linear combination of probabilities of the functions of iid random vectors and thus also for order statistics of random variables. Recurrence relationships among the distribution functions of order statistics arising from innid random variables were obtained by Cao and West[10]. In addition, Vaughan and Venables[22] derived the joint pdf and marginal pdf of order statistics of innid random variables by means of permanents. Balakrishnan[3], and Bapat and Beg[8] obtained the joint pdf and df of order statistics of innid random variables by means of permanents. Using multinomial arguments, the pdf of Xr:n+1 (1 ≤ r ≤ n) was obtained by Childs and Balakrishnan[11] by adding another independent random variable to the original n variables X1 , X2 , ..., Xn . Also, Balasubramanian et al.[7] established the identities satisfied by distributions of order statistics from non-independent non-identical variables through operator methods based on the difference and differential operators. In a paper published in 1991, Beg[9] obtained several recurrence relations and identities for product moments of order statistics of innid random variables using permanents. Recently, Cramer et al.[13] derived the expressions for the distribution and density functions by Ryser’s method and the distributions of maxima and minima based on permanents. In the first of two papers, Balasubramanian et al.[5] obtained the distribution of single order statistic in terms of distribution functions of the minimum and maximum order statistics of some subsets of {X1 , X2 , ..., Xn } where Xi ’s are innid random variables. Later, Balasubramanian et al.[6] generalized their previous results[5] to the case of the joint distribution function of several order statistics. In this study, the joint distributions of p order statistics of innid discrete random vectors are expressed in form of an integral. As far as we know, these approaches have not been considered in the framework of order statistics from innid discrete random vectors. From now on, the subscripts and superscripts are defined in the first place in which they are used and these definitions will be valid unless they are redefined.

Consider x= x(1) , x(2) , ..., x(b) and y= y (1) , y (2) , ..., y (b) (b = 1, 2, ..., n) are vector, then it can be written as
x≤y if x(s) ≤ y (s) (s = 1, 2, ..., b) and x∓y = x(1) ∓y (1) , x(2) ∓y (2) , ..., x(b) ∓y (b) .   (1) (2) (b) Let ξi = ξi , ξi , ..., ξi (i = 1, 2, ..., n) be n innid discrete random vectors which components of ξi are independent. The expression   (s) (s) (s) Xr:n = Zr:n ξ1 , ξ2 , ..., ξn(s)

(1)

is stated as the rth order statistic of the sth components of ξ1 , ξ2 , ..., ξn . From (1), the ordered values of the sth components of ξ1 , ξ2 , ..., ξn are espressed as (s)

(s)

(s) X1:n ≤ X2:n ≤ ... ≤ Xn:n .

918

(2)

YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

From (2), we can write (1) (2) (b) Xr:n = Xr:n , Xr:n , ..., Xr:n




(1 ≤ r ≤ n) .

  (1) (2) (b) (s) Also, xw = xw , xw , ..., xw , (xw = 0, 1, 2, ...)

(w = 1, 2, ..., p; p = 1, 2, ..., n).

(s)

Let fi and Fi be pf and df of ξi , respectively. The pf and df of Xr1 :n , Xr2 :n , ..., Xrp  (1 ≤ r1 < r2< ... < rp ≤ n) will be :n (s) (s) (s) (s) (s) (s) (s) given. Let X = Xr1 :n , Xr2 :n , ..., Xrp :n and x(s) = x1 , x2 , ..., xp . For R R P P , and instead of , notational convenience we write (s)

mp ,kp ,...,m1 ,k1

(s)

(s)

x1 , x2 ,...,xp (s) x2

(s) x1

P

(s) x3

P

(s) x1 =0

P

(s) (s) x2 =x1 (s)

(s) xp

P

...

(s) (s) xp =xp−1

(s) (s) x3 =x2

(s)

(s)

Fir (x1 )

Fir (x2 )

Firp (xp )

R

R

R

1

(s) Fir (x1 −) 1

,

2

...

(s) Fir (x2 −) 2

n−r Pp

rp −rP p−1 −1

r3 −r P2 −1

r2 −r P1 −1

r2 −r P1 −1

rP 1 −1

mp =0

kp =0

m2 =0

k2 =0

m1 =0

k1 =0

...

Firp (xp )

2

1

R

R

and

0

,

(s)

(s)

(s)

Fir (x1 ) Fir (x2 )

(s) Firp (xp −)

V

R

...

in the expressions below,

(s,p−1) vir p−1

(s,1) vir 1

  x P (s) respectively x0 = 0 . Also, F (x) = f (x) = f (0) +f (1) +... + f (x) . x=0

2. Theorems for Probability and Distribution Functions In this section, the theorems related to pf and df of Xr1 :n , Xr2 :n , ..., Xrp :n will be given. We will now express the following theorem for the joint pf of order statistics of innid discrete random vectors. Theorem 2.1. fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) =

b Y s=1

where x1 < x2 < ... < xp ,

P

 D

XZ

p+1 Y

rY w −1

 P

 p h i Y (s,w) (s,w−1)  (s,w) vi` − vi` dvirw ,

w=1 `=rw−1 +1

w=1

(3) denotes the sum over all n! permutations (i1 , i2 , ..., in )

P

of (1, 2, ..., n), D =

p+1 Q w=1

(s,p+1)

v i`

(s,w)

= 1 and vi`

[(rw − rw−1 − 1)!]−1 , r0 = 0, rp+1 = n + 1, vi`

(s,0)

(s,w)

(s)

= [virw − Firw (xw −)]

Proof. It can be written

919

(s)

fi` (xw ) (s)

firw (xw )

(s)

+ Fi` (xw −).

= 0,

YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

 fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) = P Xr1 :n = x1 , Xr2 :n = x2 , ..., Xrp :n = xp  = P X(1) = x(1) , X(2) = x(2) , ..., X(b) = x(b) b Y

=

s=1 b Y

=

s=1 b Y

=

 P X(s) = x(s) n o (s) (s) (s) (s) (s) P Xr(s) = x , X = x , ..., X = x 1 2 r2 :n rp :n p 1 :n fr1 ,r2 ,...,rp :n



(s) (s) x1 , x2 , ..., x(s) p



.

s=1

Consider the event o n (s) (s) (s) (s) (s) , ..., X = x . , X = x Xr(s) = x 2 1 rp :n p r2 :n 1 :n The above event can be realized mutually exclusive as follows: r1 − 1 − k1 (s) observations are less than x1 , kw +1+mw (w = 1, 2, . . . , p) observations  are equal to  (s) (s) (s) xw , rξ − 1 − kξ − mξ−1 − rξ−1 (ξ = 2, 3, ..., p) observations are in interval xξ−1 , xξ (s)

and n − mp − rp observations exceed xp . The probability function of the above event can be written as   n o (s) (s) (s) (s) (s) (s) (s) (s) fr1 ,r2 ,...,rp :n x1 , x2 , ..., x(s) = P X = x , X = x , ..., X = x . 1 2 p r1 :n r2 :n rp :n p The following expression can be written from the last identity.

fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) =

b Y

X

   rw−1 +mw−1 p+1 X Y Y (s)   C1 fi`1 (xw−1 )

s=1 mp ,kp ,...,m1 ,k1

 ·

rw −1−k Y w

w=1

P

h

`1 =rw−1 +1

 i (s)  Fi`2 (x(s) w −) − Fi`2 (xw−1 )

`2 =rw−1 +mw−1 +1

·

rY w −1

# fi`3 (x(s) w )

`3 =rw −kw

p Y

firw (x(s) w ),

(4)

w=1

p Q [(rw − 1 − kw − mw−1 − rw−1 )!] [(kw + 1 + mw )!]−1 , w=1 w=1     (s) (s) m0 = 0, kp+1 = 0, mw−1 + kw ≤ rw − rw−1 − 1, Fi` x0 = 0, Fi` xp+1 − = 1 and     (s) (s) (s) Fi` xw − = P Xi` < xw .

where

C1

=

 p+1 Q

−1

920



YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

(4) can be written as fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) =

b Y

X

C1

s=1 mp ,kp ,...,m1 ,k1 p+1 imw−1 Y  (s) kw h (s) 1 − yw−1 yw

!

p Y

p Y (kw + 1 + mw )! kw ! mw ! w=1

X P

! dyw(s)



w=1

w=1



Y

0

0

(s)

fi`1 (xw−1 ) #

fi`3 (x(s) w )

`3 =rw −kw

`2 =rw−1 +mw−1 +1

0

`1 =rw−1 +1 rY w −1

i h (s)  −) − F (x ) Fi`2 (x(s) i w−1 w `2

...

rw−1 +mw−1

w=1

Z1







rw −1−k Y w

·

p+1 Y



! Z1 Z1

p Y

firw (x(s) w ).

w=1

The above identity can be expressed as fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) =

b Y

X

C

1 1 XZ Z

s=1 mp ,kp ,...,m1 ,k1

" p+1 Y



rw−1 +mw−1

Y 

w=1

·

P



yw(s) fi`3 (x(s) w )

 i h (s)  Fi`2 (x(s) w −) − Fi`2 (xw−1 )

p

Y

dyw(s) firw (x(s) w ),

(5)

[(rw − 1 − kw − mw−1 − rw−1 )!]

−1



w=1 (s,w)

0

w=1

 p+1 Q

In (5), if vij

0

`2 =rw−1 +mw−1 +1

#

`3 =rw −kw

where C =

...

rw −1−k Y w

  (s) (s) 1 − yw−1 fi`1 (xw−1 ) 

`1 =rw−1 +1

rY w −1

0

Z1

p Q

[mw !kw !]−1 .

w=1 (s)

(s)

(s)

= yw fij (xw ) + Fij (xw −), the following identity is obtained. (s)

fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) =

Fir (x1 ) 1

b Y

X

C

Z

X

s=1 mp ,kp ,...,m1 ,k1

P

(s)

(s)

Firp (xp )

Fir (x2 ) 2

Z

Z ...

(s)

(s)

Fir (x1 −) Fir (x2 −) 1

2

(s)

Firp (xp −)

 " p+1 rw−1 +mw−1 h i Y Y (s) (s,w−1)   Fi`1 (xw−1 ) − vi` 1

w=1

 ·

`1 =rw−1 +1 rw −1−k Y w

 h i (s)  Fi`2 (x(s) w −) − Fi`2 (xw−1 )

`2 =rw−1 +mw−1 +1

·

rY w −1

h

# p i Y (s,w) (s,w) vi` − Fi`3 (x(s) dvirw . w −) 3

`3 =rw −kw

921

w=1

(6)

YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

Considering ξ Q

 n X n X X τ =0 ξ=0

`1 =1

(1) Gi` 1



!

n−τ Q `2 =ξ+1

(2) Gi` 2

n Q

(3)

`3 =n−τ +1

Gi`

3

ξ! (n − τ − ξ)!τ !

P n

i 1 X Y h (1) (3) (2) = Gi` + Gi` + Gi` , n! P `=1 where τ + ξ ≤ n and using the above identity for each kw and mw−1 , in (6), we get

fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) =

b Y

D

s=1



p+1 Y

rY w −1

h





(s)

XZ P



(s,w−1)

Fi` xw−1 − vi`



(s)



(s,w)

+ Fi` x(s) w − − Fi` xw−1 + vi`


− F i`

 i
 x(s) w −

w=1 `=rw−1 +1

·

p Y w=1

Thus, the proof is completed. RR R R (s) (s) (s) Moreover, if x1 ≤ x2 ≤ ... ≤ xp , it should be written ... instead of in RR R (s,p) (s,2) (s,1) (3), where ... is to be carried out over the region: vir1 ≤ vir2 ≤ ... ≤ virp , (s)

(s,1)

Fir1 (x1 −) ≤ vir1 (s,p)

(s)

(s)

(s,2)

≤ Fir1 (x1 ), Fir2 (x2 −) ≤ vir2

(s)

(s)

≤ Fir2 (x2 ), ..., Firp (xp −) ≤

(s)

virp ≤ Firp (xp ). We will now express the following theorem to obtain the joint df of order statistics of innid discrete random vectors. Theorem 2.2. Fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) =

b Y s=1

 D

XZ

p+1 Y

rY w −1

 P

V

w=1 `=rw−1 +1

 p h i Y (s,w) (s,w−1)  (s,w) vi` − vi` dvirw . w=1

(7) Proof. It can be written

922

(s,w)

dvirw .

YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

Fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) = =

b Y

Fr1 ,r2 ,...,rp :n

s=1 b Y



X

(s) (s) x1 , x2 , ..., x(s) p

fr1 ,r2 ,...,rp :n





(s) (s) x1 , x2 , ..., x(s) p



.

s=1 x(s) , x(s) ,...,x(s) p 2 1

The above identity can be expressed as

Fr1 ,r2 ,...,rp :n (x1 , x2 , ..., xp ) =

b Y

 X

D

XZ

p+1 Y

rY w −1



s=1 x(s) , x(s) ,...,x(s) p 1 2

 p h i Y (s,w) (s,w−1)  (s,w) vi` − vi` dvirw .

w=1 `=rw−1 +1

P

w=1

Thus, the proof is completed. 3. Results for Probability and Distribution Functions In this section, the results related to pf and df of Xr1 :n , Xr2 :n , ..., Xrp :n will be given. We will now express the following result for pf of the rth order statistic of innid discrete random vectors. Result 3.1. 

Fir

1



(1)

fr1 :n x1



(1)

x1



Z

X 1 = (r1 − 1)! (n − r1 )! P Fir

1

  (1) x1 −

rY 1 −1 `=1

! (1,1)

v i`

n h Y

(1,1)

1 − v i`

! i

(1,1)

dvir1 .

`=r1 +1

(8) Proof. In (3), if b = 1, p = 1, (8) is obtained. In the following two results, the pf ’s of minimum and maximum order statistics of innid discrete random vectors are given, respectively. Result 3.2.   (1) Fi1 x1



(1)

f1:n x1



=

1 (n − 1)!

Z

X P Fi1



 (1) x1 −

923

! n h i Y (1,1) (1,1) 1 − vi` dvi1 . `=2

(9)

YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

Proof. In (8), if r1 = 1, (9) is obtained. (1,1)

Specially, in (9), by taking n = 2 and vi2

(1)

(1)

(1,1)

− Fi1 (x1 −)]

= [vi1

fi2 (x1 )

+

(1)

fi1 (x1 )

(1)

Fi2 (x1 −), the following identity is obtained.   (1) f1:2 x1   (1) Fi1 x1

=

Z

X P

( [vi1

(1)

− Fi1 (x1 −)]

  (1) Fi1 x1 −

X

(1,1)

vi1

−

P

X

1 2

)#

(1)

(1,1)

1−

" =

"

 2 f (x(1) ) i2 (1,1) 1 vi1 (1) fi1 (x1 )

fi2 (x1 ) (1) fi1 (x1 )

(1)

(1) fi2 (x1 ) (1,1) (1) vi1 Fi1 (x1 −) (1) fi1 (x1 )

+

(1,1)

+ Fi2 (x1 −)

dvi1

#Fi1 (1,1)

  (1) x1

(1)

− vi1 Fi2 (x1 −)   (1) Fi1 x1 −

 h i 1 (1) (1) (1) (1) (1) (1) (1) = − fi2 (x1 ) Fi1 (x1 ) + Fi1 (x1 −) + fi2 (x1 )Fi1 (x1 −) − fi1 (x1 )Fi2 (x1 −) 2 P h i 1 (1) (1) (1) (1) (1) (1) (1) (1) = f1 (x1 ) − f2 (x1 ) F1 (x1 ) + F1 (x1 −) + f2 (x1 )F1 (x1 −) − f1 (x1 )F2 (x1 −) 2 i h 1 (1) (1) (1) (1) (1) (1) (1) (1) +f2 (x1 ) − f1 (x1 ) F2 (x1 ) + F2 (x1 −) + f1 (x1 )F2 (x1 −) − f2 (x1 )F1 (x1 −) 2 i i h h 1 1 (1) (1) (1) (1) (1) (1) (1) (1) = f1 (x1 ) − f2 (x1 ) F1 (x1 ) + F1 (x1 −) + f2 (x1 ) − f1 (x1 ) F2 (x1 ) + F2 (x1 −) . 2 2 (1) fi1 (x1 )

Morever, the above identity in the iid case can be expressed as f1:2



(1) x1



=

(1) 2f (x1 )

(1) f (x1 )

−

h

(1) F (x1 )

+

i

(1) F (x1 −)

h

i (1) (1) (1) (1) = 2f (x1 ) − f (x1 ) 2F (x1 ) − f (x1 ) (1)

(1)

(1)

(1)

= 2f (x1 ) − 2f (x1 )F (x1 ) + f 2 (x1 ). This result is obtained, if i = 1, n = 2 in equation (2) in [18]. Also, the above (1) identity for x1 = 1 can be written as f1:2 (1) = 2f (1) − 2f (0)f (1) − f 2 (1). Result 3.3.   (1) Fin x1



(1)

fn:n x1



=

1 (n − 1)!

Z

X P



(1)

Fin x1 −

924



n−1 Y `=1

! (1,1)

vi`

(1,1)

dvin .

(10)

YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

Proof. In (8), if r1 = n, (10) is obtained. (1)

(1)

(1)

In the following result, we will give the joint pf of X1:n , X2:n , ..., Xp:n . (1)

(1)

(1)

Result 3.4. If x1 ≤ x2 ≤ ... ≤ xp , 

(1)

(1)

f1,2,...,p:n x1 , x2 , ..., x(1) p



Z Z

X 1 = (n − p)! P

n h Y

Z ...

(1,p)

1 − v i`

! p i Y

(1,w)

dviw

,

w=1

`=p+1

(11) where

RR

...

(1,1)

R

(1)

is to be carried out over region: vi1 (1,1)

(1)

(1)

(1,p)

(1,2)

≤ ... ≤ vip ,

≤ vi2

(1)

(1,2)

(1)

≤ Fi2 (x2 ), ..., Fip (xp −) ≤ Fi1 (x1 −) ≤ vi1 ≤ Fi1 (x1 ), Fi2 (x2 −) ≤ vi2 (1,p) (1) vip ≤ Fip (xp ). RR R R Proof. In (3), if b = 1, r1 = 1, r2 = 2, ..., rp = p and ... instead of , (11) is obtained. We will now give three results for the df of single order statistic of innid discrete random vectors. Result 3.5. Fir

  (1) Fr1 :n x1 =

X 1 (r1 − 1)! (n − r1 )! P

1

  (1) x1

Z

rY 1 −1

! (1,1)

vi`

`=1

0

n h Y

(1,1)

1 − vi`

i

! (1,1)

dvir1 .

`=r1 +1

(12) Proof. In (7), if b = 1, p = 1, (12) is obtained. Result 3.6.   (1) Fi1 x1



(1)

F1:n x1



X 1 = (n − 1)! P

Z

! n h i Y (1,1) (1,1) 1 − vi` dvi1 .

(13)

`=2

0

Proof. In (12), if r1 = 1, (13) is obtained. Result 3.7.   (1) Fin x1

Fn:n



(1) x1



X 1 = (n − 1)! P

Z 0

Proof. In (12), if r1 = n, (14) is obtained.

925

n−1 Y `=1

! (1,1) vi`

(1,1)

dvin .

(14)

YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

(1)

(1)

(1)

In the following result, we will give the joint df of X1:n , X2:n , ..., Xp:n . Result 3.8. (1)

(1)

Fi1 (x1 ) Fi2 (x2 )

F1,2,...,p:n



(1) (1) x1 , x2 , ..., x(1) p



1 = (n − p)!

X P

Z

(1)

Fip (xp )

Z

Z

... 0

(1,1) 1

vi

(1,p−1) p−1

vi

n h Y

1−

(1,p) vi`

! p i Y w=1

`=p+1

(15) Proof. In (7), if b = 1, r1 = 1, r2 = 2, ...,rp = p, (15) is obtained.

References [1] B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, A first course in order statistics, John Wiley and Sons Inc., New York (1992).

[2] N. Balakrishnan, Order statistics from discrete distributions, Commun. Statist. Theory Meth., 15 (1986), 657-675.

[3] N. Balakrishnan, Permanents, order statistics, outliers and robustness, Rev. Mat. Complut., 20 (2007), 7-107.

[4] K. Balasubramanian and M. I. Beg, On special linear identities for order statistics, Statistics, 37 (2003), 335-339.

[5] K. Balasubramanian, M. I. Beg and R. B. Bapat, On families of distributions closed under extrema, Sankhy¯ a Ser. A, 53 (1991), 375-388.

[6] K. Balasubramanian, M. I. Beg and R. B. Bapat, An identity for the joint distribution of order statistics and its applications, J. Statist. Plann. Inference, 55 (1996), 13-21.

[7] K. Balasubramanian, N. Balakrishnan and H. J. Malik, Identities for order statistics from non-independent non- identical variables, Sankhy¯a Ser. B, 56 (1994), 67-75.

[8] R. B. Bapat and M. I. Beg, Order statistics for nonidentically distributed variables and permanents, Sankhy¯ a Ser. A, 51 (1989), 79-93.

[9] M. I. Beg, Recurrence relations and identities for product moments of order statistics corresponding to nonidentically distributed variables, Sankhy¯a Ser. A, 53 (1991), 365-374.

[10] G. Cao and M. West, Computing distributions of order statistics, Commun. Statist. Theory Meth., 26 (1997), 755-764.

[11] A. Childs and N. Balakrishnan, Relations for order statistics from non-identical logistic random variables and assessment of the effect of multiple outliers on bias of linear estimators, J. Statist. Plann. Inference, 136 (2006), 2227-2253.

926

(1,w)

dviw

.

YUZBASI, GUNGOR: DISTRIBUTIONS OF ORDER STATISTICS

[12] H. W. Corley, Multivariate order statistics, Commun. Statist. Theory Meth., 13 (1984), 1299-1304.

[13] E. Cramer, K. Herle and N. Balakrishnan, Permanent Expansions and Distributions of Order Statistics in the INID Case, Commun. Statist. Theory Meth., 38 (2009), 2078-2088.

[14] H. A. David, Order statistics, John Wiley and Sons Inc., New York (1970). [15] G. Gan and L. J. Bain, Distribution of order statistics for discrete parents with applications to censored sampling, J. Statist. Plann. Inference, 44 (1995), 37-46.

[16] C. M. Goldie and R. A. Maller, Generalized densities of order statistics, Statist. Neerlandica, 53 (1999), 222-246.

[17] O. Guilbaud, Functions of non-i.i.d. random vectors expressed as functions of i.i.d. random vectors, Scand. J. Statist., 9 (1982), 229-233.

[18] C. G. Khatri, Distributions of order statistics for discrete case, Ann. Inst. Statist. Math., 14 (1962), 167-171.

[19] H. N. Nagaraja, Structure of discrete order statistics, J. Statist. Plann. Inference, 13 (1986), 165-177.

[20] H. N. Nagaraja, Order statistics from discrete distributions, Statistics, 23 (1992), 189-216. [21] R. -D. Reiss, Approximate distributions of order statistics, Springer-Verlag, New York (1989). [22] R. J. Vaughan and W. N. Venables, Permanent expressions for order statistics densities, J. Roy. Statist. Soc. Ser. B, 34 (1972), 308-310.

927

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 928-935, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

1

On differential superordinations of analytical functions with negative coefficients Irina Dorca1 and Daniel Breaz2 Abstract. In this paper we study the differential superordinations of analytical functions with negative coefficients, making use of generalized S˘al˘agean and Ruscheweyh operators. 2010 MSC: 30C45, 30A20, 34A40. Key words and phrases: differential superordination, differential operator, negative coefficients, convex function, best subordinator.

1

Introduction

Let H(U ) be the set of functions which are regular in the unit disc U , A = {f ∈ H(U ) : f (0) = f 0 (0) − 1 = 0} and S = {f ∈ A : f is univalent in U }. In [6], the subfamily T of S is introduced as following: T = {f (z) : f (z) = z −

(1)

∞ X

aj z j , aj ≥ 0, n ∈ N? , z ∈ U }

j=n+1

Let (2) An = {f ∈ H(U ) , f (z) = z −

∞ X

aj z j , aj ≥ 0, n ∈ N, ..., z ∈ U } ,

j=n+1

(3) H[a, n] = {f ∈ H(U ) , f (z) = a −

∞ X

aj z j , aj ≥ 0, n ∈ N, z ∈ U } ,

j=n+1

for a ∈ C and n ∈ N . If f and g are analytic functions in U , we say that f is superordinate to g , g ≺ f , if there is a function w analytic in U , where w(0) = 0 , |w(z)| < 1 , for all z ∈ U such that f (0) = g(0) and g(U ) ⊆ f (U ) . Let ψ : C2 × U → C and h analytic in U . If p and ψ(p(z), zp0 (z); z) are univalent in U and satisfies the first-order differential superordination (4)

h(z) ≺ ψ(p(z), zp0 (z); z) , f or z ∈ U ,

928

DORCA, BREAZ: DIFFERENTIAL SUPERORDINATIONS

2

then p is called a function of differential superordination. The analytic function q is called a subordination of the differential superordination solutions, or more simple a subordination, if q ≺ p for all p which satisfy (4). An univalent subordination q˜ that satisfies q ≺ g˜ for all subordinations (4) is said to be the best subordination (4). The best subordination is unique up to a rotation of U .

2

Preliminary results

Let Dn be the S˘ al˘ agean differential operator (see [5]) Dn : A → A, n ∈ N : D0 f (z) = f (z), D1 f (z) = Df (z) = zf 0 (z), Dn f (z) = D(Dn−1 f (z)). ∞ P

Remark 2.1 If f ∈ T, n ∈ N? , f (z) = z −

aj z j , aj ≥ 0, z ∈ U then

j=n+1

Dn f (z) = z −

∞ P

j n aj z j .

j=n+1

Definition 2.1 [2] Let β, λ ∈ R, β ≥ 0, λ ≥ 0 and f (z) = z +

∞ X

aj z j . We

j=2

denote by

Dλβ

the linear operator defined by

Dλβ : A → A Dλβ f (z) = z +

∞ X

β

(1 + (j − 1)λ) aj z j .

j=2

Remark 2.2 In ([1]) we have considered the following operator concerning the functions of form (1): If f ∈ T, n ∈ N − {0} , β, λ ∈ R , β ≥ 0 , f (z) = ∞ P z− aj z j , aj ≥ 0, z ∈ U then j=n+1

Dλβ : A → A ,

(5)

Dλβ f (z) = z −

∞ X

[1 + (j − 1)λ]β aj z j .

j=n+1

Definition 2.2 [4] For f ∈ An , An = {f ∈ H(U ) , f (z) = z+ 0, n ∈ N, z ∈ U } , m , n ∈ N , the operator Rm is defined by

∞ P

aj z j , aj ≥

j=n+1 Rm : An

→ An ,

R0 f (z) = f (z) , R1 f (z) = zf 0 (z) , (m + 1)Rm+1 f (z) = z(Rm f (z))0 + mRm f (z) , z ∈ U . Remark 2.3 If f ∈ An , f (z) = z +

∞ P j=n+1

z+

∞ P j=n+1

m Cm+j−1 aj z j , z ∈ U .

929

aj z j , n ∈ N , then Rm f (z) =

DORCA, BREAZ: DIFFERENTIAL SUPERORDINATIONS

3

Definition 2.3 [3] We denote by Q the set of functions that are analytic and injective on U − E(f ) , where E(f ) = {ζ ∈ ∂U : lim f (z) = ∞} , and f 0 (ζ) 6= 0 z→ζ

for ζ ∈ ∂U − E(f ) . The subclass of Q for which f (0) = a is denoted by Q(a) . Lemma 2.1 [3] Let h be a convex function with h(0) = a , and let γ ∈ C−{0} be a complex number with Reγ ≥ 0 . If p ∈ H[a, n] ∩ Q , p(z) + γ1 zp0 (z) is univalent in U and h(z) ≺ p(z) + γ1 zp0 (z) , f or z ∈ U , then q(z) ≺ p(z) , f or z ∈ U , where q(z) =

γ nz γ/n

Rz

h(t)tγ/n−1 dt , f or z ∈ U . The function q is convex and is

0

the best subordination. Lemma 2.2 [3] Let q be a convex function and let h(z) = q(z)+ γ1 zq 0 (z) , f or z ∈ U , where Reγ ≥ 0 . If p ∈ H[a, n] ∩ Q , p(z) + γ1 zp0 (z) is univalent in U and q(z) + γ1 zp0 (z) ≺ p(z) + γ1 zp0 (z) , f or z ∈ U , then q(z) ≺ p(z) , f or z ∈ U , where q(z) =

γ nz γ/n

Rz

h(t)tγ/n−1 dt , f or z ∈ U . The function q is the best sub-

0

ordination. The propose of this paper is to define the best subordination for the functions with negative coefficients, making use of generalized S˘al˘agean and Ruscheweyh operators.

3

Main results

Definition 3.1 The operator (5) can be also written as follows: Let f ∈ T , of form (1), then Dλ0 f (z) = f (z), Dλ1 f (z) = (1 − λ)f (z) + λzf 0 (z) = Dλ f (z), . . . Dλβ f (z) = (1 − λ)Dλβ−1 f (z) + λz(Dλβ f (z))0 = Dλ (Dλβ−1 f (z)) , z ∈ U . Remark 3.1 If f ∈ An , f (z) = z−

∞ P

aj z j , then Rβ f (z) = z−

j=n+1

∞ P

j=n+1

β Cβ+j−1 aj z j ,

β ∈ R, z ∈ U . Definition 3.2 Let λ ≥ 0 and n ∈ N , β ∈ R . Denote by DRλβ the operator given by Hadamard product (the convolution product) of the S˘ al˘ agean operator

930

DORCA, BREAZ: DIFFERENTIAL SUPERORDINATIONS

4 defined in (5) and the Ruscheweyh operator Rβ , DRλβ : An → An , DRλβ f (z) = (Dλβ ? Rβ )f (z) . Remark 3.2 If λ = 1 we obtain the Hadamard product DRβ of the S˘ al˘ agean operator Dβ and Ruscheweyh operator Rβ introduced above. Theorem 3.1 Let h be a convex function, h(0) = 1 . Let λ ≥ 0 , β ∈ R , n ∈ h i 1 (β + 1) · DRλβ+1 f (z) − β(1 − λ) · DRλβ f (z) N , f ∈ An and consider that z(βλ+1) is univalent and (DRλβ f (z))0 ∈ H[1, n] ∩ Q . If (6) h(z) ≺

h i 1 (β + 1) · DRλβ+1 f (z) − β(1 − λ) · DRλβ f (z) , z ∈ U, z(βλ + 1)

then q(z) ≺ (DRλβ f (z))0 , z ∈ U , where q(z) =

βλ+1 zλn βλ+1 nλ

Rz

h(t)t

(β−n)λ+1 nλ

dt . The

0

function q is convex and it is the best subordinator. Proof. We take p(z) = (DRλβ f (z))0 = 1 −

β Cβ+j−1 [1 + (j − 1)λ]β ja2j z j−1 , P β p(0) = 0 for f ∈ T , of form (1) and we obtain p(z)−zp0 (z) = 1− Cβ+j−1 [1+

P

j≥n+1

j≥n+1

(j − 1)λ]β j 2 a2j z j−1 , which means that 1 β(1 − λ) β+1 DRλβ+1 f (z) − (β − 1 + )(DRλβ f (z))0 − DRλβ f (z) zλ λ zλ   X β λ λ(j − 1) 2 j−1 0 β and p(z)+ zp (z) = 1− Cβ+j−1 [1+(j−1)λ] j 1 + aj z , βλ + 1 βλ + 1

p(z) − zp0 (z) =

j≥n+1

which means that p(z) +

λ 0 βλ+1 zp (z)

=

β+1 β+1 f (z) z(βλ+1) DRλ

−

β(1−λ) β z(βλ+1) DRλ f (z) .

It is obviously that p ∈ H[1, n] . Further, we obtain from (6) that h(z) ≺ p(z) +

λ 0 βλ+1 zp (z) ,

z∈U.

If we take γ = β + λ1 in Lemma 2.1 we obtain q(z) ≺ p(z) , z ∈ U i.e. q(z) ≺ Rz (β−n)λ+1 βλ+1 (DRλβ f (z))0 , z ∈ U , where q(z) = zλn h(t)t nλ dt . The function q is βλ+1 nλ

0

convex and it is the best subordinator. Theorem 3.2 Let q be a convex function in U , and let h be defined as h(z) = h λ 1 q(z)+ βλ+1 zq 0 (z) , β ∈ R , n ∈ N . If f ∈ An , supposing that z(βλ+1) (β + 1) · DRλβ+1 f (z) − β(1 − λ) · is univalent and (DRλβ f (z))0 ∈ H[1, n]∩ Q and satisfies the following differential superordination h(z) = q(z)+

h i λ 1 zq 0 (z) ≺ (β + 1) · DRλβ+1 f (z) − β(1 − λ) · DRλβ f (z) , z ∈ U , βλ + 1 z(βλ + 1)

(7)

931

DORCA, BREAZ: DIFFERENTIAL SUPERORDINATIONS

5

then q(z) ≺ (DRλβ f (z))0 , z ∈ U , where q(z) =

βλ+1 zλn βλ+1 nλ

Rz

h(t)t

(β−n)λ+1 nλ

dt . The

0

function q is convex and it is the best subordinator. Proof. We consider p(z) = (DRλβ f (z))0 and proceed in the same way as we done before. Therefore, (7) becomes q(z) +

λ 0 βλ+1 zq (z)

≺ p(z) +

λ 0 βλ+1 zp (z) ,

z∈U.

1 λ

If we take γ = β + in Lemma 2.2 we have q(z) ≺ p(z) , z ∈ U , i.e. q(z) = Rz (β−n)λ+1 βλ+1 h(t)t nλ dt ≺ (DRλβ f (z))0 and q is the best subordinator. βλ+1 zλn nλ

0

Theorem 3.3 Let h be a convex function, h(0) = 1 . Let λ ≥ 0 , β ∈ R , n ∈ N , f ∈ An and consider that (DRλβ f (z))0 is univalent and

β DRλ f (z) z

∈ H[1, n] ∩

Q . If h(z) ≺ (DRλβ f (z))0 , z ∈ U ,

(8) then q(z) ≺

β DRλ f (z) z

, z ∈ U , where q(z) =

1 1 nz n

Rz

1

h(t)t n −1 dt . The function q

0

is convex and it is the best subordinator. DRβ f (z)

λ Proof. Let p(z) = , p ∈ H[1, n] . We obtain p(z) + zp0 (z) = 1 − z P β Cβ+j−1 [1 + (j − 1)λ]β ja2j z j−1 = [DRλβ f (z)]0 , z ∈ U . Then (8) becomes

j≥n+1

h(z) ≺ p(z) + zp0 (z) , z ∈ U . For γ = 1 in Lemma 2.1 we obtain q(z) ≺ β Rz 1 DRλ f (z) p(z) , z ∈ U , q(z) ≺ , z ∈ U , where q(z) = 1 1 h(t)t n −1 dt . The z nz n 0

function q is convex and it is the best subordinator. Theorem 3.4 Let q be a convex function in U , and let h be defined as h(z) = q(z)+zq 0 (z) , β ∈ R , n ∈ N . If f ∈ An , supposing that (DRλβ f (z))0 is univalent and

β DRλ f (z) z

(9) then q(z) ≺

∈ H[1, n]∩Q and satisfies the following differential superordination h(z) = q(z) + zq 0 (z) ≺ (DRλβ f (z))0 , z ∈ U , β DRλ f (z) z

, z ∈ U , where q(z) =

1 1 nz n

Rz

1

h(t)t n −1 dt . The function q

0

is convex and it is the best subordinator. Proof. We take the same assumption as before for p(z) and, after differentiating, the relation (9) becomes q(z) + zq 0 (z) ≺ p(z) + zp0 (z) , z ∈ U . Considering that γ = 1 in Lemma 2.2 we obtain q(z) ≺ p(z) , z ∈ U , i.e.q(z) = β Rz 1 f (z) DRλ 1 h(t)t n −1 dt ≺ . z ∈ U . Therefore, q is the best subordinator. 1 z nz n 0

932

DORCA, BREAZ: DIFFERENTIAL SUPERORDINATIONS

6 1−(2σ−1)z 1−z

Theorem 3.5 Let h =

be a convex function in U , σ ∈ [0, 1) . Let

λ ≥ 0 , β ∈ R , n ∈ N , f ∈ An and suppose that (DRλβ f (z))0 is univalent and β DRλ f (z) z

∈ H[1, n] ∩ Q . If h(z) ≺ (DRλβ f (z))0 , z ∈ U ,

(10) then q(z) ≺

β DRλ f (z) z

2(1−σ)

, z ∈ U , where q(z) = (2σ − 1) −

1 nz n

Rz 0

1

t n −1 1−t dt

z ∈ U.

The function q is convex and it is the best subordinator. Proof. After we take the same assumption for p(z) as in Theorem 3.3, the differ1−(2σ−1)z 1−z

≺ p(z) + zp0 (z) , z ∈ U . z 1 R 1 By using Lemma 2.1 for γ = 1 , we have q(z) ≺ p(z) , i.e., q(z) = nz n h(t)t n −1 dt =

ential superordination (10) becomes h(z) =

nz

1 n

Rz 0

0

t

1 n −1

1−(2σ−1)t dt 1−t

q(z) = (2σ − 1) −

β+1 DRλ f (z) β DRλ f (z)

= p(z) , z ∈ U , i.e.q(z) ≺

2(1−σ) 1 nz n

Rz 0

1

t n −1 1−t dt

≺

β DRλ f (z) z

. z ∈ U , where

z ∈ U. The function q is convex

and it is the best subordinator. Theorem 3.6 Let h be a convex function, h(0) = 1 . Let λ ≥ 0 , β ∈ R , n ∈  0 β+1 zDRλ f (z) DRβ+1 f (z) N , f ∈ An and suppose that is univalent and DRλβ f (z) ∈ β DR f (z) λ

λ

H[1, n] ∩ Q . If h(z) ≺

(11) then q(z) ≺

zDRλβ+1 f (z)

β+1 DRλ f (z) β DRλ f (z)

!0 , z∈U,

DRλβ f (z)

, z ∈ U , where q(z) =

1 1 nz n

Rz

1

h(t)t n −1 dt . The function

0

q is convex and it is the best subordinator. Proof. Let p(z) =

β+1 DRλ f (z) β DRλ f (z)

∞ P

1−

j=n+1 ∞ P

=

1−

j=n+1

(DRβ+1 f (z))0

λ Furthermore, p0 (z) = DR β λ f (z   0 β+1 zDRλ f (z) p(z) + zp0 (z) = . DRβ f (z)

β+1 Cβ+j [1+(j−1)λ]β+1 a2j z j β Cβ+j−1 [1+(j−1)λ]β a2j z j

) − p(z)

β (DRλ f (z))0 β DRλ f (z)

∈ H[1, n] .

and

λ

Then the relation (11) becomes h(z) ≺ p(z) + zp0 (z) , z ∈ U . If σ = 1 in zDRβ+1 f (z)

Lemma 2.1, we obtain q(z) ≺ p(z) , z ∈ U , i.e. q(z) ≺ DRλβ f (z) , z ∈ U , λ Rz 1 1 −1 n where q(z) = h(t)t dt . The function q is convex and it is the best 1 nz n 0

subordinator. Theorem 3.7 Let q be a convex function in U , and let h be defined as h(z) =  0 β+1 zDRλ f (z) q(z) + zq 0 (z) , β ∈ R , n ∈ N . If f ∈ An , supposing that is β DR f (z) λ

933

DORCA, BREAZ: DIFFERENTIAL SUPERORDINATIONS

7

univalent and

β+1 DRλ f (z) β DRλ f (z)

∈ H[1, n] ∩ Q and satisfies the following differential

superordination 0

h(z) = q(z) + zq (z) ≺

(12) then q(z) ≺

β+1 DRλ f (z) β DRλ f (z)

zDRλβ+1 f (z)

!0 , z∈U,

DRλβ f (z)

, z ∈ U , where q(z) =

1 1 nz n

Rz

1

h(t)t n −1 dt . The function

0

q is convex and it is the best subordinator. DRβ+1 f (z)

Proof. Let p(z) = DRλβ f (z) ∈ H[1, n] . Differentiating, we obtain λ  0 β+1 zDRλ f (z) 0 p(z) + zp (z) = , z∈U. β DR f (z) λ

Considering γ = 1 in Lemma 2.2, we obtain q(z) ≺ p(z) , z ∈ U , i.e. q(z) = Rz 1 DRβ+1 f (z) 1 h(t)t n −1 dt ≺ DRλβ f (z) , z ∈ U , so q is convex and it is the best subor1

nz n 0

λ

dinator. 1−(2σ−1)z 1−z

be a convex function in U , σ ∈ [0, 1) . Let  0 β+1 zDRλ f (z) λ ≥ 0 , β ∈ R , n ∈ N , f ∈ An and suppose that is univalent DRβ f (z) Theorem 3.8 Let h =

λ

and

β+1 DRλ f (z) β DRλ f (z)

∈ H[1, n] ∩ Q . If h(z) ≺

(13) then q(z) ≺

β+1 DRλ f (z) β DRλ f (z)

zDRλβ+1 f (z)

!0

DRλβ f (z)

, z∈U,

, z ∈ U , where q(z) = (2σ − 1) −

2(1−σ) 1 nz n

Rz 0

1

t n −1 1−t dt

z ∈ U.

The function q is convex and it is the best subordinator. Proof. The proof of Theorem 3.8 is similar with the one of Theorem 3.6 and the differential superordination (13) becomes h(z) =

1−(2σ−1)z 1−z

≺ p(z) +

zp0 (z) , z ∈ U . By using Lemma 2.1 for γ = 1 , we have q(z) ≺ p(z) , i.e., z z 1 R 1 1 R 1 q(z) = nz n h(t)t n −1 dt = nz n t n −1 1−(2σ−1)t dt = p(z) , z ∈ U , i.e.q(z) ≺ 1−t 0

β+1 DRλ f (z) β DRλ f (z)

0

, z ∈ U , where q(z) = (2σ − 1) − 2(1−σ) 1 nz n

Rz 0

1

t n −1 1−t dt

≺

β+1 DRλ f (z) β DRλ f (z)

z ∈ U.

The function q is convex and it is the best subordinator. Remark 3.3 We obtain particular cases if λ = 1 , β = n , n ∈ N , where we use the S˘ al˘ agean operator defined in Remark 2.1. The proof is similar. Acknowledgment. This work was partially supported by the strategic project POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007-2013 co-financed by the European Social Fund-Investing in People.

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DORCA, BREAZ: DIFFERENTIAL SUPERORDINATIONS

8

References [1] M. Acu, I. Dorca, S. Owa, On some starlike functions with negative coefficients, Proceedings of the Interational Coference on Theory and Applications of Mathematics and Informatics, ICTAMI 2011, Alba Iulia (2011), 101-112. [2] M. Acu, S. Owa, Note on a class of starlike functions, Proceeding Of the International Short Joint Work on Study on Calculus Operators in Univalent Function Theory - Kyoto (2006), 1-10. [3] S. S. Miller, P. T. Mocanu, Subordonants of Differential Superordinations, Complex Variables, 48(10), (2003), 815-826. [4] St. Ruscheweyh, New criteria of univalent functions, Proc. Amer. Math. Soc., 49 (1975), 109-115. [5] G. S. S˘ al˘ agean, Geometria Planului Complex, Ed. Promedia Plus, Cluj - Napoca, (1999). [6] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 5(1975), 109-116. 1

University of Pite¸sti, Department of Mathematics, Arge¸s, Romˆ ania.

E-mail address: [email protected] 2

”1 Decembrie 1918” University of Alba Iulia, Department of Mathematics,

Alba Iulia, Romˆ ania. E-mail address: [email protected]

935

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 936-946, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Adaptive regulation with almost disturbance decoupling for nonlinearly parameterized systems with control coefficients Ling-Ling Lv1 *, Lei Zhang,2 Hai-Bin Su1 , An-Fu Zhu1 1. College of Electric Power, North China Institute of Water Conservancy and Hydroelectric Power, Zhengzhou 450011, P. R. China 2. Institute of Data and Knowledge Engineering, Henan University, Kaifeng, 475001, P. R. China.

Abstract The problem of almost disturbance decoupling (ADD) for a class of inherently nonlinear systems with nonlinear parameterization is considered. The controlled systems are beyond triangular form and possess uncontrollable linearization. The performance of ADD is characterized in terms of 𝐿2 − 𝐿2𝑝 . By using the tool of adding a power integrator combined with the parameter separation technique, under a set of growth conditions a smooth one-dimensional adaptive controller is explicitly constructed to attenuate the influence of the disturbance on the output with an arbitrary degree of accuracy.

I. I NTRODUCTION When a system is subjected to an unknown disturbance, a reasonable method is to design certain feedback control law such that the influence of the disturbance on the output is attenuated to an arbitrary degree of accuracy. This is well-known as the problem of almost disturbance decoupling (ADD). The earliest papers on the problem of ADD for nonlinear systems may be [1] and [2] in the late 1980s. In [2] the solution is explicitly constructed by applying singular perturbation methods. A drawback of this result is that internal stability, which is crucial for a meaningful application or a practical implementation, is not taken into account. Such a problem was solved later in [3] by applying a recursive design technique. The result in [3] was later generalized to a larger class of nonlinear minimum systems in [4]. These two results in [3] and [4] were further extended to a class of nonminimum-phase nonlinear systems in [5]. The proposed approach in [5] required that the unstable part of the zero-dynamics was not affected by the disturbance. Such a restriction was relaxed in the results of [6] and [7]. The construction of controll law in [7] is based on a recursive Lyapunov-based design approach. In the above-mentioned literature on the ADD problem, most of the considered nonlinear systems are feedback (partial) linearizable and/or linear in control input. Recently, the ADD problem was addressed in [8] for a class of inherently nonlinear systems. The class of the systems is in the form of a chain of power integrators perturbed by a lower-triangular vector field. The controller was explicitly constructed by applying the so-called technique of adding a power integrator developed in [9]. When the ADD problem for nonlinear systems with unknown parameters is dealt with, a natural idea is to design a adaptive control law to solve this problem. However, only a few results on adaptive regulation with almost adaptive decoupling for nonlinear systems are available in the existing literature. In [10] and [11], adaptive controllers are designed to guarantee arbitrary disturbance attenuation on the output tracking error for smooth reference signals for uncertain systems with output depending nonlinearities. Very recently, in [13] the ADD problem was discussed for power integrator lower triangular nonlinear systems. The adaptive control law was explicitly constructed by employing the adaptive adding a power integrator technique proposed [14]. A common feature of [10], [11] and [13] is that the unknown parameter vector enters linearly in the state space equation. So far the ADD problem for nonlinearly parameterized systems has not been addressed. In this paper we will deal with ADD problem for a class of inherently nonlinear systems with nonlinear parameterization. With the aid of the parameter separation technique proposed in [15], an explicit design approach for a one-dimensional adaptive controller that solves the ADD problem is derived by using the adaptive adding one power integrator technique. For simplicity, throughout this paper we use 𝐼[𝑚, 𝑛] to denote the set {𝑚, 𝑚 + 1, ⋅ ⋅ ⋅ , 𝑛} for two integers 𝑚 < 𝑛. For a [ ]𝑇 . ∥⋅∥ is used to denote the Euclid group of scalars 𝑥𝑖 , 𝑖 ∈ 𝐼[1, 𝑗], we use 𝑥[𝑗] to denote the vector 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑗 norm of a vector. II. P ROBLEM F ORMULATION We consider the following nonlinear system with an unknown parameter vector 𝜃: ⎧ ⎨ 𝑥˙ 𝑖 = 𝑑𝑖 (𝑥, 𝑢, 𝜃)𝑥𝑖+1 + 𝑔𝑖 (𝑥[𝑖] , 𝜃)𝑤 + 𝜙𝑖 (𝑥[𝑖] , 𝜃), 𝑖 ∈ 𝐼[1, 𝑛 − 1] 𝑥˙ 𝑛 = 𝑑𝑛 (𝑥, 𝑢, 𝜃)𝑢 + 𝑔𝑛 (𝑥[𝑛] , 𝜃)𝑤 + 𝑢𝑗 𝜙𝑛 (𝑥[𝑛] , 𝜃) ⎩ 𝑦 = ℎ(𝑥1 )

(1)

*Corresponding author. Email address: lingling [email protected] (No. U1204605), the Scientific Research Key Project Fund of Henan Provincial This work is supported by the National Natural Science Foundation of China936 Education Commission (No. 12B120007) and the High-Level Talents Research Startup Program of North China Institute of Water Conservancy (No. 201013).

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems where 𝑢 ∈ ℝ, 𝑥 = 𝑥[𝑛] ∈ ℝ𝑛 , 𝑦 ∈ ℝ and 𝑤 ∈ ℝ𝑠 are the control input, system state, system output and disturbance signal, respectively; 𝑝𝑖 , 𝑖 ∈ 𝐼[1, 𝑛], are positive integers, and 𝑔𝑖 (⋅), 𝑖 ∈ 𝐼[1, 𝑛] and ℎ(⋅), are smooth functions with ℎ(0) = 0; 𝑑𝑖 (⋅), 𝜙𝑖,𝑗 (⋅), 𝑖 ∈ 𝐼[1, 𝑛], 𝑗 ∈ 𝐼[0, 𝑝𝑖 − 1] are continuous functions. The problem of adaptive regulation with almost disturbance decoupling for the system (1) can be addressed as follows. Adaptive Regulation with Almost Disturbance Decoupling (ARADD): Consider the system (1). Given any real number 𝛾 > 0, find, if possible, a smooth adaptive controller { ⋅ ˆ 𝜓(0, 0) = 0 𝜃ˆ = 𝜓(𝑥[𝑛] , 𝜃), (2) ˆ 𝑢(0, 0) = 0 𝑢 = 𝑢(𝑥[𝑛] , 𝜃), such that the closed-loop system (1) – (2) satisfies the following: 1) when 𝑤 = 0, the closed-loop system is globally stable in the sense of Lyapunov, and globally asymptotical regulation of the state is achieved, i.e., lim𝑡→∞ 𝑥[𝑛] (𝑡) = 0. 2) for any disturbance 𝑤 ∈ 𝐿2 , the response of the closed-loop system starting from the initial state 𝑥(0) = 0 is such that ∫ 𝑡 ∫ 𝑡 2𝑝1 2 2 ∣𝑦(𝑠)∣ 𝑑𝑠 ≤ 𝛾 ∥𝑤(𝑠)∥ 𝑑𝑠, for any 𝑡 ≥ 0. 0

0

Since the system (1) is in a nontriangular form and possesses uncontrollable linearization, the problem of ARADD is difficult to be dealt with. We need the following assumptions imposed on the system (1). Assumption A1: 𝑝1 ≥ 𝑝2 ≥ ⋅ ⋅ ⋅ ≥ 𝑝𝑛 are odd integers. Assumption A2: For any 𝑖 ∈ 𝐼[1, 𝑛],

𝑔𝑖 (𝑥[𝑖] , 𝜃) ≤ 𝜑𝑖 (𝑥[𝑖] , 𝜃) where 𝜑𝑖 (⋅) is nonnegative and smooth. Assumption A3: There exist smooth functions 𝛼𝑖 (𝑥[𝑖] ) > 0 and 𝑑𝑖 (𝑥[𝑖+1] , 𝜃), such that 𝛼𝑖 (𝑥[𝑖] ) ≤ 𝑑𝑖 (𝑥, 𝑢, 𝜃) ≤ 𝑑𝑖 (𝑥[𝑖+1] , 𝜃), 𝑖 ∈ 𝐼[1, 𝑛]. Assumption A4: For any 𝑖 ∈ 𝐼[1, 𝑛], 𝑗 ∈ 𝐼[1, 𝑝𝑖 − 1], 𝑖 ∑ 𝑝 −𝑗 𝜙𝑖,𝑗 (𝑥[𝑖] , 𝜃) ≤ 𝛽𝑖,𝑗 (𝑥[𝑖] , 𝜃) ∣𝑥𝑘 ∣ 𝑖 , 𝑘=1

where 𝛽𝑖,𝑗 (⋅) is a nonnegative continuous function. As the end of this section, we provide some useful lemmas. The first two lemmas are a slight extension of the well-known Young’s inequality, and will be repeatedly used in the design of the adaptive controller. The proof has been given in [9] and [8]. Lemma 1: For any positive integers 𝑚, 𝑛, and any real-valued function 𝛾(𝑥, 𝑦) > 0, the following inequality holds: 𝑚 𝑛 𝑚 𝑛 𝑚+𝑛 𝑚+𝑛 ∣𝑥∣ ∣𝑦∣ ≤ 𝛾(𝑥, 𝑦) ∣𝑥∣ + 𝛾 −𝑚/𝑛 (𝑥, 𝑦) ∣𝑦∣ . 𝑚+𝑛 𝑚+𝑛 Lemma 2: Let 𝑥, 𝑦 and 𝑧, be real variables. Assume that 𝑔1 : ℝ2 → ℝ is a smooth function. Then, for any positive integers 𝑚, 𝑛 and real number 𝑁 > 0, there exist a nonnegative smooth function ℎ1 : ℝ3 → ℝ such that the following relation holds: 𝑚+𝑛

∣𝑥∣ 𝑚+𝑛 + ∣𝑦∣ ℎ1 (𝑥, 𝑦, 𝑧). 𝑁 The following lemma provides the parameter separation principle. It is this principle that enables us to deal with nonlinear parameterization. A constructive proof of the result can be found in [15]. Lemma 3: For any real-valued continuous function 𝑓 (𝑥, 𝑦), where 𝑥 ∈ ℝ𝑚 , 𝑦 ∈ ℝ𝑛 , there are smooth scalar functions 𝑎(𝑥) ≥ 1 and 𝑏(𝑦) ≥ 1, such that ∣𝑓 (𝑥, 𝑦)∣ ≤ 𝑎(𝑥)𝑏(𝑦). ∣𝑥𝑚 [(𝑦 + 𝑥𝑔1 (𝑥, 𝑧))𝑛 − (𝑥𝑔1 (𝑥, 𝑧))𝑛 ]∣ ≤

III. M AIN R ESULTS In this section we solve the problem of adaptive regulation with almost disturbance decoupling for the system (1). Using the adding a power integrator technique as the design tool, we will explicitly construct a control Lyapunov function and minimum-order adaptive controller that solves the problem. Before giving the main result, we gives a useful lemma, which is proved in [15] with the aid of the parameter separation technique provided in Lemma 3. Lemma 4: For nonlinear functions 𝑑𝑖 (⋅) and 𝜙𝑖,𝑗 (⋅) satisfying Assumptions A3 and A4, respectively, there exist a constant Θ1 ≥ 1, and smooth functions 𝛼 ¯ 𝑖 (𝑥[𝑖+1] ) ≥ 1, 𝑒¯𝑖 (𝜃) ≥ 1, 𝛾𝑖 (𝑥[𝑖] ) ≥ 0, such that ¯ 𝑖 (𝑥[𝑖+1] )¯ 𝑒𝑖 (𝜃) ≤ 𝛼 ¯ 𝑖 (𝑥[𝑖+1] )Θ1 𝛼𝑖 (𝑥[𝑖] ) ≤ 𝑑𝑖 (𝑥, 𝑢, 𝜃) ≤ 𝛼 937

(3)

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems 𝑝∑ 𝑖 ∑ 𝑖 −1 𝑗 1 𝑝 ≤ 𝛼𝑖 (𝑥[𝑖] ) ∣𝑥𝑖+1 ∣𝑝𝑖 + 𝛾𝑖 (𝑥[𝑖] )Θ1 𝑥 𝜙 (𝑥 , 𝜃) ∣𝑥𝑗 ∣ 𝑖 . 𝑖,𝑗 [𝑖] 𝑖+1 2 𝑗=0 𝑗=1

(4)

Now we are in position to present the main results. Theorem 1: Under the condition of Assumptions A1 – A4, the ARADD problem for system (1) is solvable by a onedimensional smooth adaptive controller { ⋅ ˆ = 𝜓(𝑥[𝑛] , Θ), ˆ Θ ˆ ∈ ℝ, 𝜓(0, 0) = 0, Θ (5) ˆ 𝑢 = 𝑢(𝑥[𝑛] , Θ), 𝑢(0, 0) = 0. Proof: The proof is constructive by applying the adding a power integrator technique [9], parameter separation technique [15] and Lemma 4. First we define a new unknown parameter Θ=

𝑛 ∑ [ ] Θ1 + 𝜔 ¯ 𝑖2 (𝜃) ≥ 1,

(6)

𝑖=1

where Θ1 is defined in 4 and 𝜔 ¯ 𝑖 (𝜃) will be determined later. ˜ = Θ − Θ, ˆ where Θ(𝑡) ˆ Step 1: Define Θ is the estimate of Θ to be designed later. By Lemma 3, there exist smooth functions 𝜔1 (𝑥1 ) ≥ 1 and 𝜔 ¯ 1 (𝜃) ≥ 1 such that 𝜑1 (𝑥1 , 𝜃) ≤ 𝜔1 (𝑥1 )¯ 𝜔1 (𝜃). (7) Now we Let the following Lyapunov function for the 𝑥1 -subsystem of (1) 𝑝1 +1 1 ˜2 ˆ = 𝑥1 𝑉1 (𝑥1 , Θ) + Θ . 𝑝1 + 1 2

(8)

By Assumptions A2, A3, A4, and relations (7), (4) and (6), we have for some nonnegative smooth function 𝜌0 (𝑥1 ) ˆ + 𝑦 2𝑝1 − 𝛽 ∥𝑤∥2 𝑉˙ 1 (𝑥1 , Θ) ⎡ ⎣𝑑1 (𝑥, 𝑢, 𝜃)𝑥2𝑝1

=

𝑥𝑝11

≤

𝑥𝑝11 𝑥𝑝21 𝑑1 (𝑥, 𝑢, 𝜃)

+ 𝑔1 (𝑥1 , 𝜃)𝑤 +

𝑝∑ 1 −1

⎤ 𝑥𝑗2 𝜙1,𝑗 (𝑥1 , 𝜃)⎦

2

+ 𝑦 2𝑝1 − 𝛽 ∥𝑤∥

𝑗=0

+

∣𝑥𝑝11 ∣ 𝜔1 (𝑥1 )¯ 𝜔1 (𝜃) ∥𝑤∥

1 + 𝛾1 (𝑥1 )Θ1 𝑥2𝑝 1

⋅ 1 𝑝 𝑝 1 ˜Θ ˆ − 𝛽 ∥𝑤∥2 𝜌 (𝑥 ) − Θ + 𝛼1 (𝑥1 ) ∣𝑥2 ∣ 1 ∣𝑥1 ∣ 1 + 𝑥2𝑝 0 1 1 2 𝜔 ¯ 2 (𝜃) 1 ≤ 𝑑1 (𝑥, 𝑢, 𝜃)𝑥1𝑝1 𝑥𝑝21 + 𝑥12𝑝1 𝜔12 (𝑥1 ) 1 + Θ1 𝛾1 (𝑥1 )𝑥2𝑝 1 4𝛽 ⋅ 1 𝑝 𝑝 1 ˜Θ ˆ + 𝛼1 (𝑥1 ) ∣𝑥2 ∣ 1 ∣𝑥1 ∣ 1 + 𝑥2𝑝 𝜌 (𝑥 ) − Θ 0 1 1 2 [ 2 ] 𝜔 1 (𝑥1 ) 1 ≤ 𝑑1 (𝑥, 𝑢, 𝜃)𝑥𝑝11 𝑥𝑝21 + 𝑥2𝑝 + 𝛾 (𝑥 ) Θ 1 1 1 4𝛽 ⋅ 1 𝑝 𝑝 ˜ Θ. ˆ + 𝛼1 (𝑥1 ) ∣𝑥2 ∣ 1 ∣𝑥1 ∣ 1 + 𝑥2𝑝1 𝜌0 (𝑥1 ) − Θ 2 By choosing smooth virtual controller ]1 [ ˆ 𝑝1 2𝑛 + 2𝜌1 (𝑥1 , Θ) ∗ 𝑥2 = −𝑥1 𝛼1 (𝑥1 )

[

with ˆ = 𝜌0 (𝑥1 ) + 𝜌1 (𝑥1 , Θ)

𝜔12 (𝑥1 ) + 𝛾1 (𝑥1 ) 4𝛽

(9)

]√ ˆ 2 + 1, Θ

we have ˆ + 𝑦 2𝑝1 − 𝛽 ∥𝑤∥2 𝑉˙ 1 (𝑥1 , Θ) ≤

1 −𝑛𝑥2𝑝 1

+

𝑑1 (𝑥, 𝑢, 𝜃)𝑥1𝑝1 𝑥𝑝21

[ ⋅] 1 1 𝑝1 𝑝1 𝑝1 ∗𝑝1 ˆ ˆ ˜ + 𝛼1 (𝑥1 ) ∣𝑥2 ∣ ∣𝑥1 ∣ − 𝛼1 (𝑥1 )𝑥1 𝑥2 + Ψ1 (𝑥1 , Θ) − Θ Θ 2 2 [

where ˆ = 𝑥2𝑝1 Ψ1 (𝑥1 , Θ) 1

] 𝜔12 (𝑥1 ) + 𝛾1 (𝑥1 ) ≥ 0. 4𝛽 938

(10)

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems 1 By applying (3) and the fact that −𝑥1𝑝1 𝑥∗𝑝 ≥ 0, we have 2

1 1 1 1 1 − 𝛼1 (𝑥1 )𝑥𝑝11 𝑥∗𝑝 ≤ −𝑑1 (𝑥, 𝑢, 𝜃)𝑥𝑝11 𝑥∗𝑝 − 𝛼1 (𝑥1 ) 𝑥𝑝11 𝑥∗𝑝 . 2 2 2 2 2 Hence, it follows from (3) that ˆ + 𝑦 2𝑝1 − 𝛽 ∥𝑤∥2 𝑉˙ 1 (𝑥1 , Θ) [ ⋅] 1 𝑝1 𝑝1 𝑝1 ( 𝑝1 ∗𝑝1 ) ∗𝑝1 1 ˆ ˆ ˜ + 𝑑 (𝑥, 𝑢, 𝜃)𝑥 𝑥 − 𝑥 + 𝛼 𝑥 − 𝑥 + Ψ (𝑥 , Θ) − Θ Θ ≤ −𝑛𝑥2𝑝 (𝑥 ) ∣𝑥 ∣ 1 1 1 1 1 1 1 2 2 2 2 2 1 [ ] [ ⋅ ]( ) 1 𝑝1 𝑝1 ∗𝑝1 1 ˆ ˆ ˜ + 𝜂1 , ≤ −𝑛𝑥2𝑝 + ∣𝑥 ∣ 𝑥 − 𝑥 𝑒 ¯ (𝜃)¯ 𝛼 (𝑥 ) + 𝛼 (𝑥 ) + Ψ (𝑥 , Θ) − Θ Θ 1 1 1 [2] 1 1 1 1 1 2 2 2 with 𝜂1 = 0. Step 2: Consider the (𝑥1 , 𝑥2 )-subsystem of the system (1). Let 𝜉2 = 𝑥2 − 𝑥∗2 . Then [ ] 𝑝∑ 2 −1 ∂𝑥∗2 𝜉˙2 = 𝑑2 (𝑥, 𝑢, 𝜃)𝑥𝑝32 + 𝑔2 (𝑥[2] , 𝜃) − 𝑔1 (𝑥1 , 𝜃) 𝑤 + 𝑥𝑗3 𝜙2,𝑗 (𝑥[𝑖] , 𝜃) ∂𝑥1 𝑗=0 ⎛ ⎞ 𝑝 −1 1 ∑ ∂𝑥∗ ∂𝑥∗2 ˆ⋅ − 2 ⎝𝑑1 (𝑥, 𝑢, 𝜃)𝑥𝑝21 + 𝑥𝑗2 𝜙1,𝑗 (𝑥1 , 𝜃)⎠ − Θ. ˆ ∂𝑥1 ∂Θ 𝑗=0

According to Lemma 4, in view of the expression (9) and the fact that 1 ≤ 𝑒¯1 (𝜃) ≤ Θ1 and 𝑝1 ≥ 𝑝2 , we have by applying Lemma 1 ⎛ ⎞ 𝑝∑ 𝑝∑ 1 −1 ∗ 2 −1 𝑗 ∂𝑥 𝑝1 𝑗 2 ⎝ ⎠ 𝑑1 (𝑥, 𝑢, 𝜃)𝑥2 + 𝑥2 𝜙1,𝑗 (𝑥1 , 𝜃) 𝑥3 𝜙2,𝑗 (𝑥[𝑖] , 𝜃) − ∂𝑥 1 𝑗=0 𝑗=0 ≤

1 𝑝 𝑝 𝑝 𝛼 (𝑥[2] ) ∣𝑥3 ∣ 2 + 𝛾2 (𝑥[2] )Θ1 (∣𝑥1 ∣ 2 + ∣𝑥2 ∣ 2 ) 2 2 [ ] ∂𝑥∗2 1 𝑝2 𝑝1 𝑝1 + 𝛼 ¯ 1 (𝑥[𝑖+1] )¯ 𝑒1 (𝜃) ∣𝑥2 ∣ + 𝛼1 (𝑥1 ) ∣𝑥2 ∣ + 𝛾1 (𝑥1 )Θ1 ∣𝑥1 ∣ ∂𝑥1 2

(11)

1 𝑝 ˆ 1 (∣𝜉1 ∣𝑝2 + ∣𝜉2 ∣𝑝2 ) . 𝛼 (𝑥[2] ) ∣𝑥3 ∣ 2 + 𝜆2 (𝜉[2] , Θ)Θ 2 2 In addition, due to Assumption A2 we have

∗ ∗



𝑔2 (𝑥[2] , 𝜃) − ∂𝑥2 𝑔1 (𝑥1 , 𝜃) ≤ 𝜑2 (𝑥2 , 𝜃) + 𝜑1 (𝑥1 , 𝜃) ∂𝑥2 .

∂𝑥1 ∂𝑥1 ≤

ˆ ≥ 1 and 𝜔 By Lemma 3, there are two smooth functions 𝜔2 (𝜉[2] , Θ) ¯ 2 (𝜃) ≥ 1 such that

∗

ˆ 𝜔2 (𝜃).

𝑔2 (𝑥[2] , 𝜃) − ∂𝑥2 𝑔1 (𝑥1 , 𝜃) ≤ 𝜔2 (𝜉[2] , Θ)¯

∂𝑥1

(12)

Now we construct the following Lyapunov function 2𝑝1 −𝑝2 +1 ˆ = 𝑉1 (𝜉1 , Θ) ˆ + 𝜉2 . 𝑉2 (𝜉[2] , Θ) 2𝑝1 − 𝑝1 + 1

With the relations (11) and (12), the time derivative of 𝑉2 along the solution of (1) satisfies ˆ + 𝑦 2𝑝1 − 2𝛽 ∥𝑤∥2 𝑉˙ 2 (𝜉[2] , Θ) ] [ [ ⋅] 1 1 𝑝2 2𝑝1 −𝑝2 𝑝1 𝑝1 ∗𝑝1 1 ˆ ˆ ˜ (𝑥 ) + (𝑥 ) ∣𝑥 ∣ ≤ −𝑛𝑥2𝑝 𝛼 𝛼 + Ψ (𝑥 , Θ) − Θ Θ + ∣𝑥 ∣ 𝑥 − 𝑥 𝑒 ¯ (𝜃)¯ 𝛼 (𝑥 ) + 𝜉 1 3 1 1 1 1 1 [2] [2] 2 1 2 2 2 1 2 2 ( ) ˆ 𝜔2 (𝜃) ∥𝑤∥ − 𝛽 ∥𝑤∥2 +𝜉22𝑝1 −𝑝2 𝑥𝑝32 𝑑2 (𝑥, 𝑢, 𝜃) + 𝜉22𝑝1 −𝑝2 𝜔2 (𝑥[2] , Θ)¯ (13) ∗ ⋅

ˆ ˆ 1 (∣𝜉1 ∣𝑝2 + ∣𝜉2 ∣𝑝2 ) − 𝜉 2𝑝1 −𝑝2 ∂𝑥2 Θ. +𝜉22𝑝1 −𝑝2 𝜆2 (𝜉[2] , Θ)Θ 2 ˆ ∂Θ

939

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems ˆ In view of the fact that 𝑒¯1 (𝜃) ≤ Θ and Θ1 ≤ Θ, by applying Lemma 1 there are two nonnegative smooth functions 𝜌ˆ2 (𝜉[2] , Θ) ˆ and 𝜌˜2 (𝜉[2] , Θ) such that [ ] 1 𝑝1 𝑝1 ∗𝑝1 ∣𝑥1 ∣ 𝑥2 − 𝑥2 𝑒¯1 (𝜃)¯ 𝛼1 (𝑥[2] ) + 𝛼1 (𝑥1 ) 2 [ ] 1 𝑝1 𝑝1 ∗𝑝1 ≤ ∣𝑥1 ∣ 𝑥2 − 𝑥2 𝛼 ¯ 1 (𝑥[2] ) + 𝛼1 (𝑥1 ) Θ 2 [ ] 2𝑝1 𝜉1 2𝑝1 ˆ Θ ≤ + 𝜉2 𝜌ˆ2 (𝜉[2] , Θ) (14) ˆ 2 )(1 + 𝜂 2 ) 3(1 + Θ 1 [ ] √ 𝜉12𝑝1 𝜉12𝑝1 2𝑝1 2𝑝1 ˆ ˜ ˆ ˆ 2, ≤ + 𝜉 𝜌 ˆ (𝜉 , Θ) Θ + + 𝜉 𝜌 ˆ (𝜉 , Θ) 1+Θ 2 [2] 2 [2] 2 2 ˆ 2 )(1 + 𝜂 2 ) 6 3(1 + Θ 1 ˆ 1 (∣𝜉1 ∣𝑝2 + ∣𝜉2 ∣𝑝2 ) 𝜉22𝑝1 −𝑝2 𝜆2 (𝜉[2] , Θ)Θ ˆ (∣𝜉1 ∣𝑝2 + ∣𝜉2 ∣𝑝2 ) Θ ≤ 𝜉22𝑝1 −𝑝2 𝜆2 (𝜉[2] , Θ) [ ] √ 2𝑝1 𝜉 𝜉12𝑝1 1 ˆ ˆ2 + ˆ Θ. ˜ + 𝜉22𝑝1 𝜌˜2 (𝜉[2] , Θ) 1+Θ + 𝜉22𝑝1 𝜌˜2 (𝜉[2] , Θ) ≤ ˆ 2 )(1 + 𝜂 2 ) 6 3(1 + Θ 1

(15)

By the fact that 𝜔 ¯ 22 (𝜃) ≤ Θ and the completion of square, it is derived that ) 2𝑝1 −𝑝2 ( 2 𝜔2 (𝜃) ∥𝑤∥ − 𝛽 ∥𝑤∥ 𝜔2 (𝑥[2] )¯ 𝜉2 𝜉22𝑝1 −2𝑝2 𝜔22 (𝑥[2] ) 4𝛽 √ 2𝑝1 −2𝑝2 2 2𝑝1 −2𝑝2 2 𝜔2 (𝑥[2] ) 𝜔2 (𝑥[2] ) 2𝑝1 𝜉2 ˆ 2 + Θ𝜉 ˜ 2𝑝1 𝜉2 ≤ 𝜉2 1+Θ . 2 4𝛽 4𝛽 Substituting (14), (15) and (16) into (13) yields ≤ Θ𝜉22𝑝1

(16)

ˆ + 𝑦 2𝑝1 − 2𝛽 ∥𝑤∥2 𝑉˙ 2 (𝜉[2] , Θ)

[ ⋅] 𝜉12𝑝1 1 𝑝2 2𝑝1 −𝑝2 1 ˜ + 𝜉 2𝑝1 −𝑝2 𝑥𝑝2 𝑑2 (𝑥, 𝑢, 𝜃) ˆ ˆ (𝑥 ) ∣𝑥 ∣ Θ ≤ −𝑛𝑥2𝑝 + + 𝛼 + Ψ (𝑥 , Θ) − Θ 𝜉 3 1 1 [2] 2 3 1 2 3 2 2 ] [ √ 2𝑝1 −2𝑝2 2 ∗ ⋅ ˆ 𝜔2 (𝑥[2] , Θ) ˆ + 𝜌ˆ2 (𝜉[2] , Θ) ˆ + 𝜉2 ˆ 2 − 𝜉 2𝑝1 −𝑝2 ∂𝑥2 Θ ˆ +𝜉22𝑝1 𝜌˜2 (𝜉[2] , Θ) 1+Θ 2 ˆ 4𝛽 ∂Θ ( [ )] ˆ 𝜉22𝑝1 −2𝑝2 𝜔22 (𝑥[2] , Θ) 2𝜉12𝑝1 2𝑝1 ˆ ˜ ˆ + 𝜉2 𝜌˜2 (𝜉[2] , Θ) + Θ + + 𝜌ˆ2 (𝜉[2] , Θ) ˆ 2 )(1 + 𝜂 2 ) 4𝛽 3(1 + Θ 1 Define ˆ = Ψ1 (𝑥1 , Θ) ˆ + Ψ2 (𝑥[2] , Θ)

2𝜉12𝑝1 + 𝜉22𝑝1 ˆ 2 )(1 + 𝜂 2 ) 3(1 + Θ 1

) ˆ 𝜉22𝑝1 −2𝑝2 𝜔22 (𝑥[2] , Θ) ˆ ˆ 𝜌˜2 (𝜉[2] , Θ) + + 𝜌ˆ2 (𝜉[2] , Θ) , 4𝛽

(

∗

ˆ = 𝜂1 + 𝜉 2𝑝1 −𝑝2 ∂𝑥2 , 𝜂2 (𝜉[2] , Θ) 2 ˆ ∂Θ ( ) ˆ 𝜃 Π2 𝜉[2] , Θ,

∗ ˆ 2𝑝1 −𝑝2 ∂𝑥2 = −Ψ2 (𝑥[2] , Θ)𝜉 2 ˆ ∂Θ )] [ ( 2𝑝1 −2𝑝2 2 2𝑝1 ˆ 𝜉 𝜔 (𝑥 , Θ) 2𝜉1 [2] 2 2𝑝1 2 ˆ ˆ + + 𝜌ˆ2 (𝜉[2] , Θ) 𝜂1 − + 𝜉2 𝜌˜2 (𝜉[2] , Θ) ˆ 2 )(1 + 𝜂 2 ) 4𝛽 3(1 + Θ 1

Then the relation (17) can be rewritten as ˆ + 𝑦 2𝑝1 − 2𝛽 ∥𝑤∥2 𝑉˙ 2 (𝜉[2] , Θ)

≤

(17)

𝜉12𝑝1 1 𝑝2 2𝑝1 −𝑝2 2𝑝 −𝑝 𝑝 1 −𝑛𝑥2𝑝 + (𝑥 ) ∣𝑥 ∣ + 𝛼 𝜉 + 𝜉2 1 2 𝑥32 𝑑2 (𝑥, 𝑢, 𝜃) 3 [2] 2 1 2 3 2 ]√ [ 𝜉22𝑝1 −2𝑝2 𝜔22 (𝑥[2] ) 2𝑝1 ˆ2 ˆ ˆ 1+Θ +𝜉2 𝜌˜2 (𝜉[2] , Θ) + 𝜌ˆ2 (𝜉[2] , Θ) + 4𝛽 [ ⋅ ][ ] ( ) 940 , Θ) ˆ ˆ ˜ + 𝜂2 (𝜉 ˆ + Π2 𝜉[2] , Θ ˆ . + Ψ2 (𝑥[2] , Θ) − Θ Θ [2]

(18)

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems ˆ such that It follows from (10) and (18) that there exists a nonnegative smooth function 𝛼 ˜ 2 (𝜉[2] , Θ) ˆ ≤ (𝜉 2𝑝1 + 𝜉 2𝑝1 )˜ ˆ Ψ2 (𝑥[2] , Θ) 𝛼2 (𝜉[2] , Θ). 1 2 ˆ such that With this, by applying 1 it is not difficult to show that there exists a smooth nonnegative function 𝜌ˇ2 (𝜉[2] , Θ) ( ) 2 2𝑝1 ˆ 𝜃 ˆ Π2 𝜉[2] , Θ, ≤ 𝜉 + 𝜉 2𝑝1 𝜌ˇ2 (𝜉[2] , Θ) 3 1 ( 2 )√ 2𝑝1 −2𝑝2 2 ˆ 𝜉 𝜔 (𝑥 , Θ) [2] 2 2𝑝1 2 ˆ + ˆ +𝜉2 𝜌˜2 (𝜉[2] , Θ) + 𝜌ˆ2 (𝜉[2] , Θ) 1 + 𝜂12 . 4𝛽 By letting

[

] ] 2𝑝1 −2𝑝2 2 √ ˆ [√ 𝜉 𝜔 (𝑥 , Θ) [2] 2 2 ˆ = 𝜌˜2 (𝜉[2] , Θ) ˆ + 𝜌ˆ2 (𝜉[2] , Θ) ˆ + ˆ 2 + 1 + 𝜂 2 + 𝜌ˇ2 (𝜉[2] , Θ), ˆ 𝜌2 (𝜉[2] , Θ) 1+Θ 1 4𝛽

we have ˆ + 𝑦 2𝑝1 − 2𝛽 ∥𝑤∥2 𝑉˙ 2 (𝜉[2] , Θ) 1 𝑝 ≤ −(𝑛 − 1)𝑥12𝑝1 + 𝛼2 (𝑥[2] ) ∣𝑥3 ∣ 2 𝜉22𝑝1 −𝑝2 + 𝜉22𝑝1 −𝑝2 𝑥𝑝32 𝑑2 (𝑥, 𝑢, 𝜃) 2 [ ⋅ ]( ) 2𝑝1 ˆ ˆ ˆ ˜ + 𝜂2 (𝜉[2] , Θ) ˆ . +𝜉2 𝜌2 (𝜉[2] , Θ) + Ψ2 (𝑥[2] , Θ) − Θ Θ By choosing virtual controller

[ ˆ 𝑥∗3 (𝜉[2] , Θ)

= −𝜉2

ˆ 2(𝑛 − 1) + 2𝜌2 (𝜉[2] , Θ) 𝛼2 (𝑥[2] )

] 𝑝1

2

,

we have ˆ + 𝑦 2𝑝1 − 2𝛽 ∥𝑤∥2 𝑉˙ 2 (𝜉[2] , Θ) ( ) 1 𝑝 ≤ −(𝑛 − 1) 𝜉12𝑝1 + 𝜉22𝑝1 + 𝛼2 (𝑥[2] ) ∣𝑥3 ∣ 2 𝜉22𝑝1 −𝑝2 + 𝜉22𝑝1 −𝑝2 𝑥𝑝32 𝑑2 (𝑥, 𝑢, 𝜃) 2 [ ⋅ ]( ) 1 ∗𝑝2 2𝑝1 −𝑝2 ˆ ˆ ˜ + 𝜂2 (𝜉[2] , Θ) ˆ . + Ψ2 (𝑥[2] , Θ) − Θ Θ − 𝛼2 (𝑥[2] )𝑥3 𝜉2 2 2 2𝑝1 −𝑝2 ≥ 0, it is derived that Due to the fact that −𝑥∗𝑝 3 𝜉2

≤

ˆ + 𝑦 2𝑝1 − 2𝛽 ∥𝑤∥2 𝑉˙ 2 (𝜉[2] , Θ) ⋅ ]( ( ) [ ) ˆ −Θ ˆ Θ ˜ + 𝜂2 (𝜉[2] , Θ) ˆ −(𝑛 − 1) 𝜉12𝑝1 + 𝜉22𝑝1 + Ψ2 (𝑥[2] , Θ) ( ) 1 2 + 𝛼2 (𝑥[2] ) + 𝑒2 (𝜃)¯ 𝛼2 (𝑥[3] ) 𝜉22𝑝1 −𝑝2 𝑥𝑝32 − 𝑥∗𝑝 . 3 2

Inductive Step: Suppose for system (1) with dimension 𝑘, there are a set of smooth virtual controller 𝑥∗𝑖 , 𝑖 ∈ 𝐼[2, 𝑘 + 1] in the form of ⎧ ]𝑝 1 [ ⎨ ∗ ˆ 𝑖−1 2(𝑛−𝑖+2)+2𝜌𝑖−1 (𝜉[𝑖−1] ,Θ) 𝑥𝑖 = −𝜉𝑖−1 𝛼𝑖−1 (𝑥[𝑖−1] ) , 𝑖 ∈ 𝐼[1, 𝑘 + 1] (19) ⎩ 𝜉 = 𝑥 − 𝑥∗ , 𝑖 𝑖 𝑖 ˆ 𝑖 ∈ 𝐼[2, 𝑘 + 1], being nonnegative smooth functions, such that with 𝑥∗1 = 0 and 𝜌𝑖−1 (𝜉[𝑖−1] , Θ), ˆ + 𝑦 2𝑝1 − 𝑘𝛽 ∥𝑤∥2 𝑉˙ 𝑘 (𝜉[𝑘] , Θ) ] [ 𝑘 ∑ 1 𝑘 𝑘 𝛼𝑘 (𝑥[𝑘+1] ) 𝜉𝑘2𝑝1 −𝑝𝑘 𝑥𝑝𝑘+1 ≤ −(𝑛 − 𝑘 + 1) 𝜉𝑖2𝑝1 + 𝛼𝑘 (𝑥[𝑘] ) + 𝑒¯𝑘 (𝜃)¯ − 𝑥∗𝑝 𝑘+1 2 𝑖=1 [ ⋅ ]( ) ˆ −Θ ˆ Θ ˜ + 𝜂𝑘 (𝜉[𝑘] , Θ) ˆ + Ψ𝑘 (𝑥[𝑘] , Θ) where ˆ = 𝑉𝑘 (𝜉[𝑘] , Θ)

𝑘 ∑ 𝜉𝑖2𝑝1 −𝑝𝑖 +1 1 ˜2 + Θ . 2𝑝 − 𝑝 + 1 2 1 𝑖 𝑖=1

941

(20)

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems ˆ such that Moreover, there exists a smooth nonnegative function 𝛼 ˜ 𝑘 (𝜉[𝑘] , Θ) 𝑘 ∑ ˆ ≤ 𝛼 ˆ ˜ 𝑘 (𝜉[𝑘] , Θ) 𝜉𝑖2𝑝1 . Ψ𝑘 (𝑥[𝑘] , Θ)

(21)

𝑖=1

A direct calculation gives 𝑝𝑘+1 −1

𝜉˙𝑘+1

𝑝

𝑘+1 = 𝑑𝑘+1 (𝑥, 𝑢, 𝜃)𝑥𝑘+2 + 𝑔𝑘+1 (𝑥[𝑘+1] , 𝜃)𝑤 +

∑

𝑥𝑗𝑘+2 𝜙𝑘+2,𝑗 (𝑥[𝑘+1] , 𝜃)

𝑗=0

−

𝑘 ∑

∂𝑥∗𝑘+1

𝑖=1

∂𝑥𝑖

𝑥˙ 𝑖 −

∂𝑥∗𝑘+1 ⋅ ˆ ˆ ∂Θ

Θ.

In view of the expressions (19) and (4), Assumption A1 and the fact that 1 ≤ 𝑒¯𝑖 (𝜃) ≤ Θ1 , we have by applying Lemma 1 ⎛ ⎞ 𝑝𝑘+1 −1 𝑝∑ 𝑘 𝑖 −1 ∗ ∑ ∑ 𝑗 ∂𝑥𝑘+1 𝑝𝑖 𝑗 ⎝ ⎠ (𝑥, 𝑥 𝜙 (𝑥 , 𝜃) 𝑥 𝜙 (𝑥 , 𝜃) − 𝑑 𝑢, 𝜃)𝑥 + 𝑖 [𝑘+1] [𝑖] 𝑖+1 𝑖+1 𝑖,𝑗 𝑘+2 𝑘+2,𝑗 ∂𝑥𝑖 𝑗=0 𝑖=1 𝑗=0 ≤

≤

𝑘+1 ∑ 1 𝑝 𝑝 𝛼𝑘+1 (𝑥[𝑘+1] ) ∣𝑥𝑘+2 ∣ 𝑘+1 + 𝛾𝑘+1 (𝑥[𝑘+1] )Θ1 ∣𝑥𝑖 ∣ 𝑘+1 2 𝑖=1 ⎛ ⎞ 𝑖 𝑘 ∗ ∑ ∑ 𝑝𝑖 1 ∂𝑥𝑘+1 𝑝 𝑝 ⎝ ¯ 𝑖 (𝑥[𝑖+1] )¯ + 𝑒𝑖 (𝜃) 𝑥𝑖+1 + 𝛼𝑖 (𝑥[𝑖] ) ∣𝑥𝑖+1 ∣ 𝑖 + 𝛾𝑖 (𝑥[𝑖] )Θ1 ∣𝑥𝑗 ∣ 𝑖 ⎠ ∂𝑥𝑖 𝛼 2 𝑖=1 𝑗=1 𝑘+1 ∑ 𝑝 1 𝑝 ˆ 1 𝛼𝑘+1 (𝑥[𝑘+1] ) ∣𝑥𝑘+2 ∣ 𝑘+1 + 𝜆𝑘+1 (𝜉[𝑘+1] , Θ)Θ ∣𝜉𝑖 ∣ 𝑘+1 . 2 𝑖=1

In addition, due to Assumption A2 we have

∗ 𝑘 𝑘

∑ ∑ ∂𝑥 ∂𝑥∗𝑘+1

𝑔𝑖 (𝑥[𝑖] , 𝜃) ≤ 𝜑𝑘+1 (𝑥[𝑘+1] , 𝜃) + 𝜑𝑖 (𝑥[𝑖] , 𝜃) 𝑘+1 .

𝑔𝑘+1 (𝑥[𝑘+1] , 𝜃) −

∂𝑥𝑖 ∂𝑥𝑖 𝑖=1

𝑖=1

ˆ and 𝜔 By Lemma 3 there exist smooth nonnegative functions 𝜔𝑘+1 (𝜉[𝑘+1] , Θ) ¯ 𝑘+1 (𝜃) such that ∗ 𝑘 ∑ ∂𝑥 ˆ 𝜔𝑘+1 (𝜃). 𝜑𝑘+1 (𝑥[𝑘+1] , 𝜃) + 𝜑𝑖 (𝑥[𝑖] , 𝜃) 𝑘+1 ≤ 𝜔𝑘+1 (𝜉[𝑘+1] , Θ)¯ ∂𝑥 𝑖 𝑖=1 Now we construct the following Lyapunov function 2𝑝 −𝑝

ˆ = 𝑉𝑘 (𝜉[𝑘] , Θ) ˆ + 𝑉𝑘+1 (𝜉[𝑘+1] , Θ)

+1

1 𝑘+1 𝜉𝑘+1 . 2𝑝1 − 𝑝𝑘+1 + 1

With the above relations, the time derivative of 𝑉𝑘+1 along the solutions of the (𝑘 + 1)-dimensional system (1) satisfies ˆ + 𝑦 2𝑝1 − (𝑘 + 1)𝛽 ∥𝑤∥2 𝑉˙ 𝑘+1 (𝜉[𝑘+1] , Θ) ] [ 𝑘 ∑ 1 𝑘 𝑘 ≤ −(𝑛 − 𝑘 + 1) 𝛼𝑘 (𝑥[𝑘+1] ) 𝜉𝑘2𝑝1 −𝑝𝑘 𝑥𝑝𝑘+1 𝜉𝑖2𝑝1 + 𝛼𝑘 (𝑥[𝑘] ) + 𝑒¯𝑘 (𝜃)¯ − 𝑥∗𝑝 𝑘+1 2 𝑖=1 [ ⋅ ]( ) ˆ −Θ ˆ Θ ˜ + 𝜂𝑘 (𝜉[𝑘] , Θ) ˆ + 𝜉 2𝑝1 −𝑝𝑘+1 𝑑𝑘+1 (𝑥, 𝑢, 𝜃)𝑥𝑝𝑘+1 + Ψ𝑘 (𝑥[𝑘] , Θ) 𝑘+1 𝑘+2 ) ( 𝑘+1 ∑ 𝑝 2𝑝1 −𝑝𝑘+1 1 𝑝𝑘+1 𝑘+1 ˆ 𝛼 (𝑥[𝑘+1] ) ∣𝑥𝑘+2 ∣ + 𝜆𝑘+1 (𝜉[𝑘+1] , Θ)Θ1 ∣𝜉𝑖 ∣ + 𝜉𝑘+1 2 𝑘+1 𝑖=1 ∂𝑥∗ ⋅ 2𝑝1 −𝑝𝑘+1 ˆ 𝜔𝑘+1 (𝜃) ∥𝑤∥ − 𝛽 ∥𝑤∥2 − 𝜉 2𝑝1 −𝑝𝑘+1 𝑘+1 Θ. ˆ + 𝜉𝑘+1 𝜔𝑘+1 (𝜉[𝑘+1] , Θ)¯ 𝑘+1 ˆ ∂Θ

942

(22)

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems In view of the fact that 1 ≤ 𝑒¯𝑘 (𝜃) ≤ Θ1 and Θ1 ≤ Θ, by applying Lemma 1 there are two nonnegative smooth functions ˆ and 𝜌˜𝑘+1 (𝜉[𝑘+1] , Θ) ˆ such that 𝜌ˆ𝑘+1 (𝜉[𝑘+1] , Θ) [ ] 1 𝑘 𝑘 𝛼𝑘 (𝑥[𝑘] ) + 𝑒¯𝑘 (𝜃)¯ 𝛼𝑘 (𝑥[𝑘+1] ) 𝜉𝑘2𝑝1 −𝑝𝑘 𝑥𝑝𝑘+1 − 𝑥∗𝑝 𝑘+1 2 [ ] 1 𝑘 𝑘 ≤ 𝛼𝑘 (𝑥[𝑘] ) + 𝛼 ¯ 𝑘 (𝑥[𝑘+1] ) 𝜉𝑘2𝑝1 −𝑝𝑘 𝑥𝑝𝑘+1 − 𝑥∗𝑝 𝑘+1 Θ 2 [ ] ∑𝑘 2𝑝1 2𝑝1 𝑖=1 𝜉𝑖 ˆ + 𝜉𝑘+1 𝜌ˆ𝑘+1 (𝜉[𝑘+1] , Θ) Θ (23) ≤ ˆ 2 )(1 + 𝜂 2 (𝜉[𝑘] , Θ)) ˆ 3(1 + Θ 𝑘 [ ] ∑𝑘 √ 2𝑝1 1 ∑𝑘 2𝑝1 2𝑝1 2𝑝1 𝑖=1 𝜉𝑖 ˆ ˜ ˆ ˆ 2 + 1, ≤ + 𝜉 𝜌 ˆ (𝜉 , Θ) Θ + 𝜉 + 𝜉 𝜌 ˆ (𝜉 , Θ) Θ 𝑘+1 𝑘+1 [𝑘+1] [𝑘+1] 𝑘+1 𝑘+1 ˆ 2 )(1 + 𝜂 2 (𝜉[𝑘] , Θ)) ˆ 𝑖=1 𝑖 6 3(1 + Θ 𝑘 𝑘+1 ∑ 𝑝 2𝑝1 −𝑝𝑘+1 ˆ ∣𝜉𝑖 ∣ 𝑘+1 𝜉𝑘+1 𝜆𝑘+1 (𝜉[𝑘+1] , Θ)Θ1 𝑖=1 𝑘+1 ∑ 𝑝 2𝑝1 −𝑝𝑘+1 ˆ ∣𝜉𝑖 ∣ 𝑘+1 ≤ Θ 𝜉𝑘+1 𝜆𝑘+1 (𝜉[𝑘+1] , Θ) 𝑖=1

≤

1 ∑𝑘 2𝑝1 ˆ 𝜉 2𝑝1 + 𝜉𝑘+1 𝜌˜𝑘+1 (𝜉[𝑘+1] , Θ) 𝑖=1 𝑖 6

[ √ ˆ2 + 1 + Θ

(24) ]

∑𝑘

3(1 +

2𝑝1 𝑖=1 𝜉𝑖 ˆ 2 )(1 + 𝜂 2 (𝜉[𝑘] , Θ)) ˆ Θ 𝑘

+

2𝑝1 ˆ 𝜉𝑘+1 𝜌˜𝑘+1 (𝜉[𝑘+1] , Θ)

˜ Θ.

2 Due to 𝜔 ¯ 𝑘+1 (𝜃) ≤ Θ, it is easily obtained that 2𝑝 −𝑝𝑘+1

ˆ 𝜔𝑘+1 (𝜃) ∥𝑤∥ − 𝛽 ∥𝑤∥2 𝜔𝑘+1 (𝜉[𝑘+1] , Θ)¯ 2𝑝1 −2𝑝𝑘+1 2 ˆ 𝜔𝑘+1 (𝜉[𝑘+1] , Θ) 2𝑝1 𝜉𝑘+1 Θ ≤ 𝜉𝑘+1 4𝛽 2𝑝 −2𝑝 2𝑝1 −2𝑝𝑘+1 2 2 ˆ √ ˆ 𝜉 1 𝑘+1 𝜔𝑘+1 (𝜉[𝑘+1] , Θ) 𝜔𝑘+1 (𝜉[𝑘+1] , Θ) 2𝑝1 𝜉𝑘+1 ˜ + 𝜉 2𝑝1 𝑘+1 ˆ 2 + 1. Θ Θ ≤ 𝜉𝑘+1 𝑘+1 4𝛽 4𝛽 Substituting (23), (24) and (25) into (22) yields 1 𝜉𝑘+1

(25)

ˆ + 𝑦 2𝑝1 − (𝑘 + 1)𝛽 ∥𝑤∥2 𝑉˙ 𝑘+1 (𝜉[𝑘+1] , Θ) ≤

1 ∑𝑘 1 2𝑝1 −𝑝𝑘+1 𝑝 𝜉𝑖2𝑝1 + 𝛼𝑘+1 (𝑥[𝑘+1] ) ∣𝑥𝑘+2 ∣ 𝑘+1 𝜉𝑘+1 𝑖=1 3 2 𝑖=1 [ ⋅ ]( ) 2𝑝1 −𝑝𝑘+1 𝑝𝑘+1 ˆ ˆ ˜ + 𝜂𝑘 (𝜉[𝑘] , Θ) ˆ +𝜉𝑘+1 𝑑𝑘+1 (𝑥, 𝑢, 𝜃)𝑥𝑘+2 + Ψ𝑘 (𝑥[𝑘] , Θ) − Θ Θ (26) ] [ 2𝑝 −2𝑝 2 ˆ √ 𝜉 1 𝑘+1 𝜔𝑘+1 (𝜉[𝑘+1] , Θ) ∂𝑥∗ ⋅ 2𝑝1 ˆ 2 + 1 − 𝜉 2𝑝1 −𝑝𝑘+1 𝑘+1 Θ ˆ ˆ + 𝜌˜𝑘+1 (𝜉[𝑘+1] , Θ) ˆ + 𝑘+1 Θ +𝜉𝑘+1 𝜌ˆ𝑘+1 (𝜉[𝑘+1] , Θ) 𝑘+1 ˆ 4𝛽 ∂Θ ( )] [ ∑𝑘 2𝑝1 −2𝑝𝑘+1 2 ˆ 𝜉𝑘+1 𝜔𝑘+1 (𝜉[𝑘+1] , Θ) 2 𝑖=1 𝜉𝑖2𝑝1 2𝑝1 ˜ ˆ + 𝜌˜𝑘+1 (𝜉[𝑘+1] , Θ) ˆ + Θ. + 𝜉𝑘+1 𝜌ˆ𝑘+1 (𝜉[𝑘+1] , Θ) + ˆ 2 )(1 + 𝜂 2 (𝜉[𝑘] , Θ)) ˆ 4𝛽 3(1 + Θ 𝑘

−(𝑛 − 𝑘 + 1)

𝑘 ∑

𝜉𝑖2𝑝1 +

Define ˆ Ψ𝑘+1 (𝑥[𝑘+1] , Θ) =

ˆ + Ψ𝑘 (𝑥[𝑘] , Θ) ( 2𝑝1 +𝜉𝑘+1

∑𝑘 2 𝑖=1 𝜉𝑖2𝑝1 ˆ 2 )(1 + 𝜂 2 (𝜉[𝑘] , Θ)) ˆ 3(1 + Θ 𝑘

2𝑝 −2𝑝𝑘+1

ˆ + 𝜌˜𝑘+1 (𝜉[𝑘+1] , Θ) ˆ + 𝜌ˆ𝑘+1 (𝜉[𝑘+1] , Θ)

ˆ = 𝜂𝑘 (𝜉[𝑘] , Θ) ˆ + 𝜉 2𝑝1 −𝑝𝑘+1 𝜂𝑘+1 (𝜉[𝑘+1] , Θ) 𝑘+1 ˆ Π𝑘+1 (𝜉[𝑘+1] , Θ)

1 𝜉𝑘+1

2 ˆ (𝜉[𝑘+1] , Θ) 𝜔𝑘+1 4𝛽

∂𝑥∗𝑘+1 , ˆ ∂Θ

∑𝑘 ∂𝑥∗𝑘+1 2 𝑖=1 𝜉𝑖2𝑝1 ˆ − 𝜂𝑘 (𝜉[𝑘] , Θ) ˆ ˆ 2 )(1 + 𝜂 2 (𝜉[𝑘] , Θ)) ˆ ∂Θ 3(1 + Θ 𝑘 [ ] 2𝑝1 −2𝑝𝑘+1 2 ˆ 𝜔𝑘+1 (𝜉[𝑘+1] , Θ) 𝜉 2𝑝1 𝑘+1 ˆ ˆ ˆ −𝜉𝑘+1 𝜌ˆ𝑘+1 (𝜉[𝑘+1] , Θ) + 𝜌˜𝑘+1 (𝜉[𝑘+1]943 , Θ) + 𝜂𝑘 (𝜉[𝑘] , Θ). 4𝛽

ˆ 2𝑝1 −𝑝𝑘+1 = −Ψ𝑘+1 (𝑥[𝑘+1] , Θ)𝜉 𝑘+1

) ,

(27)

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems Then the expression (26) can be rewritten as ˆ + 𝑦 2𝑝1 − (𝑘 + 1)𝛽 ∥𝑤∥2 𝑉˙ 𝑘+1 (𝜉[𝑘+1] , Θ) 1 ∑𝑘 1 2𝑝1 −𝑝𝑘+1 2𝑝1 −𝑝𝑘+1 𝑝𝑘+1 𝑝 𝜉𝑖2𝑝1 + 𝛼𝑘+1 (𝑥[𝑘+1] ) ∣𝑥𝑘+2 ∣ 𝑘+1 𝜉𝑘+1 𝑑𝑘+1 (𝑥, 𝑢, 𝜃)𝑥𝑘+2 + 𝜉𝑘+1 𝑖=1 3 2 𝑖=1 [ ] 2𝑝1 −2𝑝𝑘+1 2 ˆ √ 𝜉𝑘+1 𝜔𝑘+1 (𝜉[𝑘+1] , Θ) 2𝑝1 ˆ ˆ ˆ2 + 1 +𝜉𝑘+1 𝜌ˆ𝑘+1 (𝜉[𝑘+1] , Θ) + 𝜌˜𝑘+1 (𝜉[𝑘+1] , Θ) + Θ 4𝛽 [ ⋅ ]( ) ˆ ˆ ˜ + 𝜂𝑘+1 (𝜉[𝑘+1] , Θ) ˆ + Π𝑘+1 (𝜉[𝑘+1] , Θ). ˆ + Ψ𝑘+1 (𝑥[𝑘+1] , Θ) − Θ Θ

≤

−(𝑛 − 𝑘 + 1)

𝑘 ∑

𝜉𝑖2𝑝1 +

ˆ such that It follows from (21) and (27) that there exists a nonnegative smooth function 𝛼 ˜ 𝑘+1 (𝜉[𝑘+1] , Θ) ∑𝑘+1 2𝑝 ˆ ≤ 𝛼 ˆ ˜ 𝑘+1 (𝜉[𝑘+1] , Θ) 𝜉𝑖 1 . Ψ𝑘+1 (𝑥[𝑘+1] , Θ) 𝑖=1

ˆ such that With this in mind, it is difficult to show that there exists a smooth nonnegative function 𝜌ˇ𝑘+1 (𝜉[𝑘+1] , Θ)

≤

ˆ Π𝑘+1 (𝜉[𝑘+1] , Θ) ∑ 𝑘 2 2𝑝1 ˆ 𝜉 2𝑝1 + 𝜉𝑘+1 𝜌ˇ𝑘+1 (𝜉[𝑘+1] , Θ) 𝑖=1 𝑖 3 ( 2𝑝1 +𝜉𝑘+1

2𝑝 −2𝑝𝑘+1

ˆ + 𝜌˜𝑘+1 (𝜉[𝑘+1] , Θ) ˆ + 𝜌ˆ𝑘+1 (𝜉[𝑘+1] , Θ)

1 𝜉𝑘+1

2 𝜔𝑘+1 (⋅)

4𝛽

)√

ˆ + 1. 𝜂𝑘2 (𝜉[𝑘] , Θ)

By letting

=

ˆ 𝜌𝑘+1 (𝜉[𝑘+1] , Θ) [ ] ] 2𝑝1 −2𝑝𝑘+1 2 √ ˆ [√ 𝜉𝑘+1 𝜔𝑘+1 (𝜉[𝑘+1] , Θ) ˆ + 𝜌˜𝑘+1 (𝜉[𝑘+1] , Θ) ˆ + ˆ 2 + 1 + 𝜂 2 (𝜉[𝑘] , Θ) ˆ +1 𝜌ˆ𝑘+1 (𝜉[𝑘+1] , Θ) Θ 𝑘 4𝛽 ˆ +ˇ 𝜌𝑘+1 (𝜉[𝑘+1] , Θ),

we have ˆ + 𝑦 2𝑝1 − (𝑘 + 1)𝛽 ∥𝑤∥2 𝑉˙ 𝑘+1 (𝜉[𝑘+1] , Θ) ≤

1 2𝑝1 −𝑝𝑘+1 2𝑝1 −𝑝𝑘+1 𝑝𝑘+1 𝑝 𝜉𝑖2𝑝1 + 𝛼𝑘+1 (𝑥[𝑘+1] ) ∣𝑥𝑘+2 ∣ 𝑘+1 𝜉𝑘+1 𝑑𝑘+1 (𝑥, 𝑢, 𝜃)𝑥𝑘+2 + 𝜉𝑘+1 2 𝑖=1 [ ⋅ ]( ) 2𝑝1 ˆ + Ψ𝑘+1 (𝑥[𝑘+1] , Θ) ˆ −Θ ˆ Θ ˜ + 𝜂𝑘+1 (𝜉[𝑘+1] , Θ) ˆ . 𝜉𝑘+1 𝜌𝑘+1 (𝜉[𝑘+1] , Θ)

−(𝑛 − 𝑘)

𝑘 ∑

It is easy to see that the smooth virtual controller [ 𝑥∗𝑘+2

= −𝜉𝑘+1

ˆ 2 (𝑛 − 𝑘) + 2𝜌𝑘+1 (𝜉[𝑘+1] , Θ) 𝛼𝑘+1 (𝑥[𝑘+1] )

]𝑝 1

𝑘+1

renders ˆ + 𝑦 2𝑝1 − (𝑘 + 1)𝛽 ∥𝑤∥2 𝑉˙ 𝑘+1 (𝜉[𝑘+1] , Θ) ≤

1 − 𝛼𝑘+1 2 ∗𝑝

1 2𝑝1 −𝑝𝑘+1 𝑝𝑘+1 2𝑝1 −𝑝𝑘+1 𝑝 𝑑𝑘+1 (𝑥, 𝑢, 𝜃)𝑥𝑘+2 𝜉𝑖2𝑝1 + 𝛼𝑘+1 (𝑥[𝑘+1] ) ∣𝑥𝑘+2 ∣ 𝑘+1 𝜉𝑘+1 + 𝜉𝑘+1 2 𝑖=1 [ ⋅ ]( ) 2𝑝 −𝑝 ∗𝑝 ˆ −Θ ˆ Θ ˜ + 𝜂𝑘+1 (𝜉[𝑘+1] , Θ) ˆ . (𝑥[𝑘+1] )𝑥 𝑘+1 𝜉 1 𝑘+1 + Ψ𝑘+1 (𝑥[𝑘+1] , Θ)

−(𝑛 − 𝑘)

2𝑝 −𝑝𝑘+1

1 𝑘+1 𝜉𝑘+1 Due to −𝑥𝑘+2

𝑘+1 ∑

𝑘+2

𝑘+1

≥ 0, we have from (4)

1 1 ∗𝑝𝑘+1 2𝑝1 −𝑝𝑘+1 2𝑝1 −𝑝𝑘+1 ∗𝑝𝑘+1 ∗𝑝𝑘+1 2𝑝1 −𝑝𝑘+1 − 𝛼𝑘+1 (𝑥[𝑘+1] )𝑥𝑘+2 𝑥𝑘+2 ≤ − 𝛼𝑘+1 (𝑥[𝑘+1] ) 𝑥𝑘+2 ≤ −𝑑𝑘+1 (𝑥, 𝑢, 𝜃)𝜉𝑘+1 𝜉𝑘+1 𝜉𝑘+1 . 2 2

944

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems Hence, it is derived from ((4) that ˆ + 𝑦 2𝑝1 − (𝑘 + 1)𝛽 ∥𝑤∥2 𝑉˙ 𝑘+1 (𝜉[𝑘+1] , Θ) [ ] 𝑘+1 ∑ 2𝑝 1 2𝑝 −𝑝 𝑝𝑘+1 ∗𝑝𝑘+1 1 ≤ −(𝑛 − 𝑘) 𝜉𝑖 + 𝛼𝑘+1 (𝑥[𝑘] ) + 𝑒¯𝑘+1 (𝜃)¯ 𝛼𝑘+1 (𝑥[𝑘+2] ) 𝜉𝑘 1 𝑘+1 𝑥𝑘+2 − 𝑥𝑘+2 2 𝑖=1 [ ⋅ ]( ) ˆ ˆ ˜ + 𝜂𝑘+1 (𝜉[𝑘+1] , Θ) ˆ . + Ψ𝑘+1 (𝑥[𝑘+1] , Θ) − Θ Θ The aforementioned inductive argument shows that (20) holds for 𝑘 = 𝑛. In fact, in the 𝑛-th step, one can construct explicitly ˜ and a smooth a global change of coordinates (𝜉1 , 𝜉2 , ⋅ ⋅ ⋅ , 𝜉𝑛 ), a positive-definite and proper Lyapunov function 𝑉𝑛 (𝜉[𝑛] , Θ) controller ]1/𝑝𝑛 [ ˆ 2 + 2𝜌 (𝜉 , Θ) 𝑛 [𝑛] ∗ ˆ = −𝜉𝑛 𝑢 (𝜉[𝑛] , Θ) 𝛼𝑛 (𝑥[𝑛] ) ˆ ≥ 0 and Ψ𝑘+1 (𝜉[𝑘+1] , Θ), ˆ such that for some smooth functions 𝜌𝑛 (𝜉[𝑛] , Θ)

≤

˜ + 𝑦 2𝑝1 − 𝑛𝛽 ∥𝑤∥2 𝑉˙ 𝑛 (𝜉[𝑛] , Θ) ( 𝑛 ⋅ )( ) ∑ ˆ −Θ ˆ ˜ + 𝜂𝑛 (𝜉[𝑛] , Θ) ˆ . − 𝜉𝑖2𝑝1 + 𝜉𝑛2𝑝1 −𝑝𝑛 (𝑢𝑝𝑛 − 𝑢∗𝑝𝑛 ) + Ψ𝑛 (𝜉[𝑛] , Θ) Θ 𝑖=1

Therefore, the one-dimensional smooth adaptive controller { ⋅ ˆ = Ψ𝑛 (𝜉[𝑛] , Θ) ˆ Θ ∗ ˆ 𝑢 = 𝑢 (𝜉[𝑛] , Θ) is such that ˆ + 𝑦 2𝑝1 − 𝑛𝛽 ∥𝑤∥2 ≤ − 𝑉˙ 𝑛 (𝜉[𝑛] , Θ)

(28) 𝑛 ∑

𝜉𝑖2𝑝1 .

(29)

𝜉𝑖2𝑝1 .

(30)

𝑖=1

Set 𝛽 = 𝛾 2 /𝑛, we have ˆ + 𝑦 2𝑝1 − 𝛾 2 ∥𝑤∥2 ≤ − 𝑉˙ 𝑛 (𝜉[𝑛] , Θ)

𝑛 ∑ 𝑖=1

When 𝑤 = 0, it is derived that ˆ ≤− 𝑉˙ 𝑛 (𝜉[𝑛] , Θ)

𝑛 ∑

𝜉𝑖2𝑝1 .

(31)

𝑖=1

According to the classical Lyapunov stability theory, it is known that the closed-loop system is global stable at the equilibrium ˆ = (0, 0). Since the Lyapunov function 𝑉𝑛 (𝜉[𝑛] , Θ) ˆ is positive definite and proper, it follows from (31) and La Salle’s (𝜉[𝑛] , Θ) invariance principle that { } all the bounded trajectories of the closed-loop system approach the largest invariant set contained ˆ : 𝑉˙ 𝑛 = 0 . Hence, lim𝑡→∞ 𝜉[𝑛] (𝑡) = 0. This, combined with (19) with 𝑘 = 𝑛, implies lim𝑡→∞ 𝑥[𝑛] (𝑡) = 0. in (𝜉[𝑛] , Θ) Moreover, note that 𝑉𝑛 (⋅) is positive definite with 𝑉𝑛 (0) = 0. It follows from (30) that ∫ 𝑡 ∫ 𝑡 2𝑝1 2 2 ∣𝑦(𝑠)∣ d𝑠 ≤ 𝛾 ∥𝑤∥ d𝑠, ∀𝑡 ≥ 0, when 𝑥(0) = 0. 0

0

This completes the proof of the theorem. The proof of Theorem 1 is constructive, thus the design procedure of the adaptive controller solving the ARADD problem is actually given. When 𝑤 = 0, Theorem 1 recovers the global stabilization results obtained in [15]. In addition, by combining Theorem 1 with Lemma 4, the following result is easily derived. Corollary 1: Under Assumptions A1, A2, A3, and (4), the problem of ARADD for the system (1) is solvable by the one-dimensional smooth adaptive controller (5). IV. C ONCLUSION For a class of inherently nonlinear systems with nonlinear parameterization, we have formulated the problem of adaptive regulation with almost disturbance decoupling. The considered system is not in triangular form, and possesses uncontrollable linearization. Under a set of growth conditions, an explicit design procedure for the adaptive smooth controller solving the ADD problem was provided with the aid of the adding a power integrator technique proposed in [9]. A significant feature of the obtained adaptive dynamical compensator is its minimum-property. The results of this paper exploit a new application of the parameter separation technique proposed recently in [15].945

Ling-Ling Lv etal: ADD for nonlinearly parameterized systems R EFERENCES [1] A. Saberi, P. Sannuti. Global stabilization with almost disturbance decoupling of a class of uncertain nonlinear systems. International Journal of Control, 1988, 47:717-727. [2] R. Marino, W. Respondek, A. J. Van der Schaft. Almost disturbanc decoupling for single-input single-output nonlinear systems. IEEE Transactions on Automatic Control, Sept. 1989, 34(9): 1013-1017. [3] R. Marino, W. Respondek, A. J. Van der Schaft. Nonlinear 𝐻∞ almost disturbanc decoupling. Systems & Control Letters, 1994, 23: 159-168. [4] A. Isidori. A note on almost disturbance decoupling for nonlinear minimum phase systems. Systems & Control Letters, 1996, 27: 191-194. [5] A. Isidori. Global almost disturbance decoupling with stability for non minimum-phase single-input single-output nonlinear systems. Systems & Control Letters, 1996, 28: 115-122. [6] Z. Lin. Almost disturbance decoupling with global asymptotic stability for nonlinear systems with disturbance-affected unstable zero dynamics. Systems & Control Letters, 1998, 33: 163-169. [7] W. Su, L. Xie, C. E. de Souza. Global robust disturbance attenuation and almost disturbance decoupling for uncertain cascaded nonlinear systems. Automatica, 1999, 35: 697-707. [8] C. Qian, W. Lin. Almost disturbance decoupling for a class of high-order nonlinear systems. IEEE Transactions on Automatic Control, June 2000, 45(6): 1208-1214. [9] W. Lin, C. Qian. Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems. Systems & Control Letters, 2002, 39: 339-351. [10] Z. Ding. Almost disturbance decoupling of uncertain nonlinear output feedback systems. IEE Pro.-Control Theory Appl., 1999, 146(2): 220-226. [11] R. Marino, P. Tomei. Adaptive output feedback regulation with almost disturbance decoupling for nonlinearly parameterized systems. International Journal of Robust and Nonlinear Control, 2000, 10: 655-669. [12] R. Marino, P. Tomei. Adaptive output feedback tracking with almost disturbance decoupling for a class of nonlinear systems. Automatica, 2000, 36: 1871-1877. [13] W. Bi. Robust adaptive disturbance attenuation for high-order uncertain nonlinear systems (In Chinese). J. Sys. Sci. & Math. Scis., 2006, 26(2):193-205. [14] W. Lin, C. Qian. Adaptive regulation of high-order lower-triangular systems: an adding a power integrator technique. Systems & Control Letters, 2002, 39: 353-364. [15] W. Lin, C. Qian. Adaptive control of nonlinearly parameterized systems: the smooth feedback case. IEEE Transactions on Automatic Control, Aug. 2002, 47(8): 1249-1266.

946

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 947-966, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A fixed-point approach to locating polynomial zeros Shun-Pin Hsua a

Department of Electrical Engineering, National Chung Hsing University Taichung 402, Taiwan [email protected]

Abstract We propose an iterative algorithm based on the fixed-point identification for polynomial zerofinding. With the analysis involving the generating function, we show under mild conditions that the algorithm yields a sequence converging globally to the reciprocal of the zero with the smallest modulus. Also, we prove that this sequence can be modified to form the sequence of lower bounds that are asymptotically sharp for the zero moduli. In particular, we show that obtaining the sequence of lower bounds with our algorithm effectively avoids the numeric overflow that might happen in other similar approach. Numerical examples are provided to illustrate our work. Keywords: polynomial zeros, fixed-point analysis,, generating function

1. INTRODUCTION Polynomial zerofinding is one of the most classical problems in the history of mathematics. With the significant results by Abel, Ruffini and Galois in the early nineteen century, it is justified that the iterative method is needed for the general zerofinding problem [1]. A comprehensive collection of avail-

947

HSU: POLYNOMIAL ZEROS

able algorithms can be found in McNamee’s bibliography paper [2, 3, 4]. Some important ideas behind these algorithms are explained in the book of Henrici [5] and the survey paper of Pan [6]. Currently no algorithm has the overwhelming superiority over others and the dominating commercial softwares use different algorithms (e.g., Jenkins-Traub algorithm [7, 8] by Mathematica, Laguerre algorithm [9] by Numerical Recipes, and balanced companion matrix algorithm [10] by Matlab). In this paper we pursue the solution with the approach of fixed-point identification. This conceptually intuitive and theoretically interesting approach is not widely used since the associated global convergence property is not clear. For example, consider the equation f (z) = (z − 1)(z − 3) = z 2 − 4z + 3 = 0. If we make use of the relation z = (z 2 + 3)/4, the zero of f becomes the fixed point of g(z) = (z 2 + 3)/4 and a natural recursive sequence to approximate this fixed point is 2 zj−1 +3 . zj = 4

(1)

where j is a nonnegative integer. However, the convergence in (1) is not guaranteed. A simple graphic analysis has that this sequence converges only locally. Specifically, as j → ∞    1 if z0 ∈ (−3, 3)   zj −→ 3 if z0 = −3 or 3 .     ∞ o.w.

(2)

If instead we make use of the relation 1 4−z = , z 3

948

(3)

HSU: POLYNOMIAL ZEROS

the zero of f becomes the fixed point of g(z) = 3/(4 − z). A natural recursive sequence for finding this point is zj =

3 . 4 − zj−1

(4)

Note that 3 3 3(zj+1 − zj ) − = 4 − zj+1 4 − zj (4 − zj+1 )(4 − zj ) 3 (zj+1 − zj ) . = 13 − 4zj

zj+2 − zj+1 =

(5)

To avoid the trivial cases, suppose zj ̸= 4. Also, zj+1 ̸= zj , namely, zj ̸= 1, 3. Define A := [5/2, 4] and Ac := (−∞, ∞)\A. Eg. (5) implies that   < |z − z | if z ∈ Ac j+1 j j |zj+2 − zj+1 | .  > |z − z | if z ∈ A j+1

j

(6)

j

Observe that Ac is an invariant set, meaning that zj ∈ Ac implies zj+1 ∈ Ac for any nonnegative integer j. As a result an initial condition z0 in Ac incurs a sequence {zj }∞ i=1 convergent to 1. If z0 is in A, it is easy to see that there must be a finite integer k such that zk ∈ Ac due to (6), and thus the induced sequence {zj }∞ i=1 is again convergent to 1. Since a detailed comparison with current zerofinding algorithms is worthy of the separate work, this paper concentrates only on the study of the global convergence property of this fixed-point approach. An interesting application of our scheme is to resolve the overflow issue in calculating the sequence of asymptotically sharp bounds of zeros using Jin’s algorithm [14]. The detailed theoretical derivations along with numerical examples are presented in the next section, followed by a conclusion in section 3.

949

HSU: POLYNOMIAL ZEROS

2. MAIN RESULTS 2.1. the algorithm Consider a polynomial f (z) of degree n written as f (z) = an z n + an−1 z n−1 + · · · + a1 z + 1

(7)

where ai ’s are complex numbers. We would like to devise an approach based on fixed-point identification for its zerofinding. For the convenience of further discussion, we target at the reciprocal of zero instead. Write y = 1/z and let fˆ(y) := an y −n + an+1 y −(n−1) + · · · + a1 y −1 + 1 = 0 .

(8)

We then have y = g(y) := −(an y −(n−1) + an−1 y −(n−2) + · · · + a2 y −1 + a1 ) .

(9)

Clearly, the fixed point of g(y) is the reciprocal of the zero of f (z). A natural iterative method to find the fixed point is to use the following assignment: y

−d

=

d ∏

−1 yj−s

(10)

s=1

where d ∈ {1, 2, · · · , (n − 1)}, j is some positive integer and yj−s ̸= 0, and compute ..

.

an−1 + .

a3 + a2 +

yj−1

950

yj−(n−1) ..

yj−3 yj−2

yj = −a1 −

an

.

(11)

HSU: POLYNOMIAL ZEROS

Example 2.1. In the special case that n = 2, yj = −a1 −

a2 . yj−1

(12)

We can thus write y∞ in a continued fraction form y∞ = −a1 −

a2 −a1 −

a2 −a1 −

.

(13)

a2 . −a1 . .

It is well known that in this case the sequence {yj }∞ i=1 converges if the zeros of f (z) are repeated or have different moduli. Let N be the set of natural numbers and N := {1, 2, · · · , n}. Define for k ∈ N that qk (z) := bk z k + bk−1 z k−1 + · · · + b1 z + 1 , which satisfies f (z)qk (z) = 1 +

n ∑

ck,r z k+r .

(14)

(15)

r=1

We then have the following result: Lemma 2.1. For k ∈ N and r ∈ N , ck,r in (15) can be written as the product of ar and yj ’s generated by (11) if the appropriate initial values are given and yj ̸= 0 for each j. Proof. We first show that bk in (14) can be decomposed into the product of yj ’s. For integer k ≥ n, let bk := [b1 b2 · · · bk ]T . Clearly bk should satisfy T(k) bk = ak

951

HSU: POLYNOMIAL ZEROS

where ak := −[a1 a2 · · · an 0 · · · 0]T ∈ ℝk . T(k) is a Toeplitz matrix with its (i, j)th component    0   tij :=

if i < j or i > j + n ,

1 if i = j ,     a i−j otherwise .

(k) Note that |T(k) |=1. By Cramer’s rule, for j ∈ {1, 2, · · · , k}, bj = Tj (k)

where Tj

is the matrix T(k) with the jth column replaced by ak . Let Ij1 ,j2

be the identity matrix with the j1 th column and j2 th column interchanged. Then we can write (k) bk = (−1) Tk Ik,k−1 Ik−1,k−2 · · · I2,1 (k) ˆ = (−1)k T k−1

where



ˆ (k) T

        =        

(16)

 a1

1

a2 a1 1 .. . . . a2 . . . . . .. .. . . an .. . . an . . . . .. .. . .

..

.

..

.

..

. a1

an · · ·

1

a2

               1   a1

.

(17)

k×k

Note that the unmarked components in (17) are 0’s. To find the determinant ˆ (k) , we can apply the column reduction operation to T ˆ (k) to obtain a of T lower triangular matrix, say L, with the jth diagonal term lj . It is easy to

952

HSU: POLYNOMIAL ZEROS

see that l1 = a1 and .. a3 +

as−1 +

.

as −lj−(s−1) . ..

−lj−3 −lj−2 −lj−1

a2 + lj = a1 +

(18)

where j = 2, 3, · · · , and s = min{j, n}. Let yj = −lj for j ∈ {1, 2, · · · n − 1}, then yj in (11) becomes −lj in (18) for j ∈ {n, n + 1, · · · }. As a result, k k (k) ∏ ∏ ˆ bk = (−1) T = (−lj ) = yj . k

j=1

(19)

j=1

To show the case for ck,r , we combine (43), (16) and (17) to obtain (k+1) ˆ ck,r = (−1)k T r ∈ {1, 2, · · · , n} , r

(20)

ˆ (k+1) is T ˆ (k+1) with the first column replaced by where T r [ar ar+1 · · · an 0 · · · 0]T . ˆ (k+1) yields a triangular Applying the column reduction operation again to T r (r)

matrix L(r) with the jth diagonal term lj . We thus have ck,r = ar

k+1 ∏(

(r)

−lj

) .

(21)

j=2

In particular, for j ∈ {n + 1, n + 2, · · · }, an (r) −lj−(n−1)

an−1 + ..

.

..

a3 + a2 + (r) lj

.

(r)

−lj−3 (r)

−lj−2

= a1 +

(r)

−lj−1

953

.

(22)

HSU: POLYNOMIAL ZEROS

Since the recursive form in (22) is the same as that in (18) we thus complete the proof. 2.2. convergence analysis Before giving a formal proof on the global convergence of {yj }∞ j=1 , we analyze its recursive scheme in (11) to gain some insight into such property. Note that f (z) = an z n + an−1 z n−1 + · · · + a1 z + 1 = (−1)n (λ1 z − 1)(λ2 z − 1) · · · (λn z − 1) where |λ1 | ≥ |λ2 | ≥ · · · ≥ |λn |. (11) can then be written as   ∑ . . . ∑ k̸=l̸=m −λk λl λm +   λ λ + k l ∑  k̸=l yj−2   yj = −  −λk +    yj−1 k  

(23)

(24)

where k ̸= l ̸= m means that the summation is over the indices k, l, m that are different from each other. Assume yj is nonzero for each j. We then have for u ∈ N .



∑  ∑ k̸=l λk λl +  yj − λu = −  −λk +   k̸=u [

∑

(

 . . . k̸=l̸=m −λk λl λm +   yj−2    yj−1 

∑

∑

)

λk λl (yj−2 − λu ) yj−1 yj−2 k̸=u ] ∑ ∏ n−1 −λ λ λ (y − λ ) (−1) λ (y − λ ) k l m j−3 u k u j−(n−1) k̸=l̸=m̸=u k̸=u + + ··· + . yj−1 yj−2 yj−3 yj−1 yj−2 · · · yj−(n−1) =−

−λk

yj−1 − λu yj−1

+

954

k̸=l̸=u

HSU: POLYNOMIAL ZEROS

Namely, ∑

∑

−λk

k̸=l̸=u,v λk λl (yj−1 − λu ) + (yj−2 − λu ) yj−1 yj−1 yj−2 ∏ (−1)n−2 k̸=u,v λk + ··· + (yj−(n−2) − λu ) yj−1 yj−2 · · · yj−(n−2) ∑ [ λv k̸=u,v λk −λv (yj−1 − λu ) + (yj−2 − λu ) =− yj−1 yj−1 yj−2 ] ∏ (−1)n−1 λv k̸=u,v λk +··· + (yj−(n−1) − λu ) . yj−1 yj−2 · · · yj−(n−1)

y j − λu +

k̸=u,v

where v ∈ {1, 2, · · · n} and v ̸= u. If we write the vectors yj := [yj yj−1 · · · yj−(n−2) ]T λu := [λu λu · · · λu ]T ]T [ ∑ ∑ ∏ (−1)n−2 k̸=u,v λk k̸=l̸=u,v λk λl k̸=u,v −λk ··· cu,v (yj ) := 1 yj−1 yj−1 yj−2 yj−1 yj−2 · · · yj−(n−2)

and define the distance measure between yj and λu as

then we have

dv (yj , λu ) := cTu,v (yj )(yj − λu ) ,

(25)

λv dv (yj−1 , λu ) dv (yj , λu ) = yj−1

(26)

where u, v ∈ N and u ̸= v. By (10) and (23), if {yj }∞ j=1 converges to some constant λ then λ ∈ {λ1 , · · · λn }. Suppose λ = λu ̸= λ1 . By (25) limj→∞ dv (yj , λu ) = 0 for each v (̸= u) and each initial value y1 making yj ’s nonzero. However, for (26) we choose v = 1 and y1 such that d1 (y1 , λu ) ̸= 0.

955

HSU: POLYNOMIAL ZEROS

Consequently j−1 ∏ λ1 lim dv (yj , λu ) = lim zk d1 (y1 , λu ) j→∞ j→∞ k=1

̸= 0 and the contradiction results. The above analysis shows that if {yj }∞ j=1 converges, it must converge to λ1 . In the following we study the sufficient condition for the global convergence of the sequence, and in particular we confirm that under the condition the sequence converges to the reciprocal of the polynomial’s zero with the smallest modulus. The proof is based on the combinatorial result of zeros of the polynomial (cf. [11, p.110]). Lemma 2.2. Suppose 1. in (23) |λ1 | > |λ2 | ≥ |λ3 | ≥ · · · ≥ |λn | , 2. in (11) yj is initiated with −lj defined in (18) for j ∈ {1, 2, · · · n − 1} , and 3. yj ’s are nonzero for each j. We then have lim yj = λ1 .

j→∞

(27)

Proof. Expressing aj ’s in (44) with λj ’s in (23) yields aj = (−1)j

∑

λt11 λt22 · · · λtnn .

(28)

t1 +···+tn =j,t1 ,··· ,tn ∈{0,1}

Using a combinatorial analysis we have bj =

∑ t1 +···+tn =j,t1 ,··· ,tn ∈{0,1,···j}

956

λt11 λt22 · · · λtnn .

(29)

HSU: POLYNOMIAL ZEROS

As a result,

∑ yj =

bj bj−1

=

λt11 λt22 · · · λtnn

t1 +···+tn =j,t1 ,··· ,tn ∈{0,1,···j}

∑

λt11 λt22 · · · λtnn

t1 +···+tn =j−1,t1 ,··· ,tn ∈{0,1,···j−1}

∑

λt22 · · · λtnn

t2 +···+tn =j,t2 ,··· ,tn ∈{0,1,···j}

∑

= λ1 +

λt11 λt22 · · · λtnn

t1 +···+tn =j−1,t1 ,··· ,tn ∈{0,1,···j−1}

∑

= λ1 + λ1

˜ t2 · · · λ ˜ tn λ 2 n

t2 +···+tn =j,t2 ,··· ,tn ∈{0,1,···j}

∑

˜ t2 · · · λ ˜ tn λ 2 n

t2 +···+tn ≤j−1,t2 ,··· ,tn ∈{0,1,···j−1}

˜ k = λk /λ1 . Since |λ ˜ ∗ | := max{|λ ˜ 2 |, · · · , |λ ˜ n |} < 1 by assumption, we where λ have

∑ t2 tn ˜ ˜ λ · · · λ 2 n t2 +···+tn =j,t2 ,··· ,tn ∈{0,1,···j} ∑ ˜ 2 |t2 · · · |λ ˜ n |tn |λ ≤ t2 +···+tn =j,t2 ,··· ,tn ∈{0,1,···j}

≤

∑

˜ ∗ |j |λ

t2 +···+tn =j,t2 ,··· ,tn ∈{0,1,···j}

( ) j + n − 2 ˜ ∗ j = λ −→ 0 n−2 as j tends to infinity. Note that bj−1 is nonzero, we conclude that lim yj → λ1 .

j→∞

(30)

The global convergence property of {yj }∞ j=1 is shown in the following. Theorem 2.3. Suppose in Lemma 2.2 only condition (1) and (3) are satisfied. The conclusion holds still.

957

HSU: POLYNOMIAL ZEROS

Proof. If {˜ yj }∞ j=1 is a nonzero sequence generated by (11) with initial values y˜j , j ∈ {1, 2, · · · , n − 1}, then there exist a ˜j , j ∈ {1, 2, · · · , n − 1} such that for j > n (−1)j |T˜(j) | = y˜1 y˜2 · · · y˜j

(31)

where T˜(j) is the matrix Tˆ(j) defined in (17), with the first column replaced by [˜ a1 a ˜2 · · · a ˜n−1 an 0 · · · 0]T ∈ Rj . By Laplace expansion, ) ( n−1 (j−n) (j−s) ∑ ˆ ˆ (−1)j |T˜(j) | = (−1)j (−1)s−1 a ˜s T + (−1)n−1 an T =−

( n−1 ∑

s=1

)

a ˜s bj−s + an bj−n

s=1

where bj is defined in (16). As a result, ∑n−1 a ˜s bj−s + an bj−n (−1)j |T˜(j) | y˜j = = ∑n−1s=1 j−1 (j−1) ˜s bj−1−s + an bj−1−n (−1) |T˜ | s=1 a ∑n−1 bj−s ˜s bj−n + an bj−n s=1 a = · ∑n−1 bj−1−s bj−n−1 ˜s bj−1−n + an s=1 a ∑n−1 a ˜s λn−s + an 1 −→ λ1 · ∑s=1 n−1 n−s ˜ s λ1 + a n s=1 a

(32)

= λ1 . as j tends to infinity. Remark 2.1. Similar results can be derived if we write f˜(z) = z n + a ˜n−1 z n−1 + · · · + a ˜1 z + a ˜0 ˜ 1 )d˜1 (z − λ ˜ 2 )d˜2 · · · (z − λ ˜ k )d˜k . = (z − λ

(33) (34)

Specifically, for r = 0, 1, · · · n and some integer j, let zr =

r−1 ∏

zj−(n−r+s) ,

s=0

958

(35)

HSU: POLYNOMIAL ZEROS

then we have zj zj−1 · · · zj−(n−1) + a ˜n−1 zj−1 zj−2 · · · zj−(n−1) + · · · + a ˜1 zj−(n−1) + a ˜0 = 0 , (36) and ..

a ˜1 +

.

a ˜n−3 + a ˜n−2 + zj = −˜ an−1 −

a ˜0 zj−(n−1) . ..

zj−3 zj−2

zj−1

.

(37)

Suppose the sequence {zj }∞ j=1 is initiated with non-zero values z−1 , z−2 , · · · , z−(n−1) ˜ 1 | > |λ ˜ 2 | ≥ |λ ˜ 3 | ≥ · · · ≥ |λ ˜ k | , then the sequence and zj ̸= 0 for each j. If |λ ˜ {zj }∞ j=1 converges to λ1 . 2.3. a remark on the condition for convergence We show in Lemma 2.1 that the global convergence of sequence {yj }∞ j=1 in (11) depends on the uniqueness of some zero modulus of the polynomial. In the following we prove that this uniqueness property can always be achieved. Theorem 2.4. Given a polynomial f (z), there exists a complex number c such that f (z + c) has a unique (repeated zeros are counted only once) zero with the smallest modulus. Proof. Suppose the zeros of f (z) are rm eiθm where m ∈ {1, 2, · · · n}, i = √ −1, θm ∈ [0, 2π) and r1 ≥ r2 ≥ · · · ≥ rn > 0 .

959

(38)

HSU: POLYNOMIAL ZEROS

∗

Let r = mini=1,2···n ri and I := {i|ri = r}. Let c = |c|eiθ . Then the zeros of ∗

f (z + c) are rm eiθm − |c|eiθ . Note that rm eiθm − |c|eiθ∗ ≥ rm eiθm − |c|eiθ∗ ≥ r − |c| .

(39) (40)

The equality holds as rm = r, 0 ≤ |c| ≤ r, and θ∗ = θm where m ∈ I. We thus conclude the proof. Example 2.2. Finding the nth root of a positive number r is fundamental in many theoretical and practical problems. Note that a direct application of our algorithm will not yield a convergent sequence since all zeros of z n −r are equimodular. However, following the proof in Theorem 2.4 we can reassign the locations of zeros such that the sequence defined in (11) or (37) converges globally. For example, the equation (z − c)n − r = 0

(41)

where c is an positive number, must have a unique zero with the maximum modulus. A simple choice is to assign c = 1. The sequence in (37) becomes ( ) ( ) (−1)n nn − r n−1 n (−1) + n−1 zj−(n−1) . . . . ( ) .. − n3 + (n ) zj−3 ( ) + 2 n zj−2 (42) − zj = zj−1 1 √ n and the sequence {zj }∞ r+1 if zj ̸= 0 for each j. We j=1 converges to illustrate the recursive calculation with r = 100 and n = 5 and show in Table 1 some numerical values generated from the process. Note that the

960

HSU: POLYNOMIAL ZEROS

Table 1: Approximating

j zj -1

5

√ 5 100 with the sequence {𝑧𝑗 − 1}∞ 𝑗=1

10

20

30

1.7458 2.3771 2.5340 2.5093

40

50

2.5121 2.5119

used initial values [z0 , z−1 , z−2 , z−3 ] are [4, 3, 2, 1]. The table shows that it √ takes 50 steps to approximate 5 100 (≈ 2.5119) with error less than 10−4 .

2.4. application to the zero bound problem ck,r defined in (15) plays a crucial role in bounding the zero of f (z) in (7). Indeed, let γk,r := |ck,r |1/(k+r) . Kalantari [12, 13] showed that τk /γk is a lower bound for the moduli of zeros of f (z) where τk is the unique positive zero of tk + t − 1 and γk := maxr∈{1,··· ,n} {γk,r }. Jin [14] showed that these bounds are asymptotically sharp, and proposed an algorithm with complexity O(kn) to calculate ck,r . Namely, k ∑

ck,r =

ak+r−j bj

(43)

j=max{0,k+r−n}

for r ∈ N , where b0 = 1 and ∑

min{n,s}

bs = −

aj bs−j

(44)

j=1

for s ∈ {1, 2, · · · , k}. Theoretically, Jin’s elegant results promise a computationally cheap method to obtain the zero bound with arbitrary sharpness. However, the derivation of ck,r in (43) raises an overflow issue. Generally speaking, |ck,r | might become very small or very large as k ≥ 100 (see Example 2.3) and can not be dealt with by ordinary computer softwares in a

961

HSU: POLYNOMIAL ZEROS

straightforward way. Fortunately, Lemma 2.1 allows one to obtain γk,r by first calculating log γk,r with a fashion of Kalman filter [15, p.214]. That is, (r)

let yj

(r)

be −lj in (22). We have 1 log |ck,r | k+r ) 1 ( (r) (r) (r) = log |ar | + log y1 + log y2 + · · · + log yk k+r ) 1 ( (r) = (k + r − 1) log γk−1,r + log yk k + r ( ) 1 1 (r) (45) log yk = 1− log γk−1,r + k+r k+r

log γk,r =

where k ∈ N and r ∈ N . Clearly, calculating log γk,r (and then γk,r ) with the above iteration scheme effectively avoids the problem of numeric overflow if (r) log yk is bounded. Example 2.3. Consider the polynomial f (z) = 3z 5 + (1 + 5i)z 4 + 4z 3 + 2z 2 + (2 − i)z + 1 , where i =

(46)

√ −1. The corresponding zeros are

z1 = −0.1504 − 2.1895i , z2 = −0.3209 + 0.6511i , z3 = 0.5693 + 0.6135i , z4 = −0.0273 − 0.5677i , z5 = −0.4040 − 0.1740i . Note that z5 is the unique zero with the smallest modulus. The growth of |ck,1 | is shown in the left part of Figure 1. Observe that as k ≥ 102 , |ck,1 | is scaled up to the level of 1036 . The right part of Figure 1 illustrates the (r)

sequences of moduli of {lj } in (22) for r = 1, 2, · · · , 5. The sequences are (1)

properly initiated so that (21) holds. Table 2 lists the sequence of yj = −lj . After j = 40, yj reaches the value −2.0878+0.8995i, which approximates the reciprocal of the 1/z5 as expected by Theorem 2.3.

962

HSU: POLYNOMIAL ZEROS

Table 2: The sequence of {𝑦𝑗 } generated by (11) satisfying c𝑘,1 = 𝑎1

j

yj

𝑗=1

𝑦𝑗 .

j

yj

1

-1.2000+1.4000i

15 -2.1071+0.8783i

29 -2.0874+0.9001i

2

-1.5294+0.8824i

16 -2.1091+0.9054i

30 -2.0872+0.8994i

3

-1.9434+0.3019i

17 -2.0906+0.9161i

31 -2.0877+0.8990i

4

-2.5805+0.4244i

18 -2.0773+0.9073i

32 -2.0881+0.8993i

5

-2.2974+1.1534i

19 -2.0789+0.8949i

33 -2.0881+0.8996i

6

-2.0327+1.1969i

20 -2.0880+0.8916i

34 -2.0878+0.8997i

7

-1.9156+0.9666i

21 -2.0934+0.8969i

35 -2.0877+0.8996i

8

-1.9641+0.7842i

22 -2.0915+0.9025i

36 -2.0877+0.8994i

9

-2.1167+0.7657i

23 -2.0870+0.9030i

37 -2.0878+0.8994i

10 -2.1922+0.8918i

24 -2.0851+0.9001i

38 -2.0879+0.8995i

11 -2.1328+0.9673i

25 -2.0865+0.8978i

39 -2.0879+0.8995i

12 -2.0576+0.9491i

26 -2.0885+0.8980i

40 -2.0878+0.8995i

13 -2.0408+0.8964i

27 -2.0891+0.8995i

41 -2.0878+0.8995i

14 -2.0732+0.8664i

28 -2.0883+0.9004i

42 -2.0878+0.8995i

963

j

∏𝑘

yj

HSU: POLYNOMIAL ZEROS

|lj(r) |, r ∈ {1, 2 · · · , 5}

|ck,1 |

40

10

6 5

30

10

4 20

10

3 2.2734

2

10

10

1 0

10

0

1

10

10

0

2

10

0

10

20

30

j

k

Figure 1: Illustrations of Example 2.3. The left part of the figure shows the growth |c𝑘,1 | (r)

with 𝑘. The right part shows the sequences |l𝑗 | satisfying (21) for r = 1, 2, · · · , 5. All the sequences converge to 2.2734, which is 1/|𝑧5 |.

3. Conclusions A novel iterative method is proposed to find the zeros of a complex polynomial. The scheme is based on the fixed-point identification of the related function. Under some mild conditions the proposed algorithm is shown to generate a sequence convergent globally to the reciprocal of the zero with the smallest modulus. In addition, this sequence can be modified to form a sequence of lower bounds of zero moduli. Compared to its counterpart that might lead to numeric overflow, the new method requires only the standard computation resource. Numerically examples are provided to illustrate the results.

964

40

HSU: POLYNOMIAL ZEROS

Acknowledgments This work was supported by the National Science Council of Taiwan under Grant NSC 100-2221-E-005-071. References [1] H. Eves, An Introduction to the History of Mathematics, Brooks Cole, 1990. [2] J. M. McNamee, A bibliography on roots of polynomials, J. Comput. Appl. Math. 47 (1993) 391–394. [3] J. M. McNamee, An updated supplementary bibliography on roots of polynomials, J. Comput. Appl. Math. 110 (1999) 305–306. [4] J. M. McNamee, A 2002 update of the supplementary bibliography on roots of polynomials, J. Comput. Appl. Math. 142 (2002) 433–434. [5] P. Henrici, Applied and computational complex analysis. Vol. 1, John Wiley & Sons Inc., New York, 1988. [6] V. Y. Pan, Solving a polynomial equation: some history and recent progress, SIAM Rev. 39 (1997) 187–220. [7] M. A. Jenkins, J. F. Traub, A three-stage algorithm for real polynomials using quadratic iteration., SIAM J. Numer. Anal. 7 (1970) 545–566. [8] M. A. Jenkins, J. F. Traub, A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration, Numer. Math. 14 (1970) 252–263.

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[9] W. T. V. W. H. Press, S. A. Teukolsky, B. P. Flannery, Numerical recipes in Fortran 90, 2nd Edition, Cambridge University Press, Cambridge, 1996. [10] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985. [11] R. P. Stanley, Generating functions, in: Studies in combinatorics, Vol. 17, Math. Assoc. America, 1978, pp. 100–141. [12] B. Kalantari, An infinite family of bounds on zeros of analytic functions and relationship to Smale’s bound, Math. Comp. 74 (2005) 841–852. [13] B. Kalantari, Polynomial root-finding and polynomiography, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. [14] Y. Jin, On efficient computation and asymptotic sharpness of Kalantari’s bounds for zeros of polynomials, Math. Comp. 75 (2006) 1905–1912. [15] G. Strang, Introduction to Linear Algebra, 4th Edition, Wellesley Cambrisge Press, MA, 2009.

966

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 967-976, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

On the solution of rational systems of difference equations M. Mansour1,2 , M. M. El-Dessoky1,2 and E. M. Elsayed1,2 1 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected], [email protected], [email protected]. Abstract In this paper, we deal with the periodic nature and the form of the solutions of the following systems of rational difference equations xn+1 =

xn−1 yn−1 , yn+1 = , ±xn−1 yn − g ±yn−1 xn − f

with a nonzero real numbers initial conditions and f, g are nonzero real numbers with f, g 6= 1.

Keywords: difference equations, periodic solutions, system of difference equations. Mathematics Subject Classification: 39A10. ––––––––––––––––––––––

1

Introduction

Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on. So, recently there has been an increasing interest in the study of qualitative analysis of rational difference equations and systems of difference equations. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. see [1]—[16] and the references cited therein. 1

967

MANSOUR ET AL: RATIONAL SYSTEMS OF DIFFERENCE EQUATIONS

Periodic solutions of a difference equations have been investigated by many researchers, and various methods have been proposed for the existence and qualitative properties of the solution. The periodicity of the positive solutions of the system of rational difference equations m pyn xn+1 = , yn+1 = , yn xn−1 yn−1 was studied by Cinar in [4]. The behavior of positive solutions of the following system xn−1 yn−1 , yn+1 = . xn+1 = 1 + xn−1 yn 1 + yn−1 xn has been studied by Kurbanli et al. [11]. In [14] Yalçınkaya investigated the sufficient condition for the global asymptotic stability of the following system of difference equations tn zn−1 + a zn tn−1 + a , tn+1 = . zn+1 = tn + zn−1 zn + tn−1 Also, Yalçınkaya [15] has obtained the sufficient conditions for the global asymptotic stability of the system of two nonlinear difference equations xn + yn−1 yn + xn−1 , yn+1 = . xn+1 = xn yn−1 − 1 yn xn−1 − 1

Similar nonlinear systems of rational difference equations were investigated see [7]-[8], [12]-[13], [16]. Our aim in this paper is to investigate the periodic nature and get the form of the solutions of the following systems of rational difference equations xn−1 yn−1 , yn+1 = . xn+1 = ±xn−1 yn − g ±yn−1 xn − f

with a nonzero real numbers initial conditions and f, g are nonzero real numbers with f, g 6= 1. Definition (Periodicity) A sequence {xn }∞ n=−k is said to be periodic with period p if xn+p = xn for all n ≥ −k.

2

On the System: xn+1 =

xn−1 xn−1 yn −g ,

yn+1 =

yn−1 yn−1 xn −f .

In this section, we study the solutions of the system of two difference equations xn−1 yn−1 , yn+1 = , n = 0, 1, ..., (1) xn+1 = xn−1 yn − g yn−1 xn − f

with nonzero real initials conditions x−1 , x0 , y−1 , y0 and f, g are nonzero real numbers with f, g 6= 1. 2

968

MANSOUR ET AL: RATIONAL SYSTEMS OF DIFFERENCE EQUATIONS

Theorem 1 Suppose that {xn , yn } are solutions of system (1). Also, assume that x−1 , x0 , y−1 and y0 are arbitrary nonzero real numbers. Then ¶n ¶n µ µ 1−f 1−f x2n−1 = d , x2n = c , 1−g 1−g and

y2n−1

¶n 1−g , y2n = a , 1−f ´ ³ ´ ³ g 1−f g and ad = . = d , x0 = c, bc = 1−f 1−g 1−f µ

1−g =b 1−f

where y−1 = b, y0 = a, x−1

¶n

µ

Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n − 1. That is; ¶n−1 ¶n−1 µ µ 1−f 1−f x2n−3 = d , x2n−2 = c , 1−g 1−g and

µ

1−g y2n−3 = b 1−f Now, it follows from Eq.(1) that x2n−1 =

=

y2n−1 =

=

, y2n−2

µ

1−g =a 1−f

¶n−1

.

³ ´n−1 ´n−1 ³ 1−f 1−f d d 1−g 1−g x2n−3 = ³ ´n−1 ³ = ´n−1 x2n−3 y2n−2 − g da − g 1−g d 1−f a 1−f −g 1−g ³ ´n−1 ´n−1 ´n−1 ³ ³ 1−f 1−f ¶n µ d d 1−f d 1−g 1−g 1−g 1−f ³ ´ ´= ³ ´ =d =³ , 1−f g 1−f g−g+f g 1−g 1−g − g 1−f 1−f 1−f ³ ´n−1 ³ ´n−1 1−g 1−g b b 1−f 1−f y2n−3 = ³ = ´n−1 ³ ´n−1 y2n−3 x2n−2 − f bc − f 1−g 1−f b 1−f c 1−g −f ³ ´n−1 ´n−1 ´n−1 ³ ³ 1−g 1−g 1−g ¶n µ b 1−f b 1−f b 1−f 1−g ³ ´ ´= ³ ´ =b =³ , 1−f g 1−f g−f +f g 1−f 1−f −f 1−g

and

¶n−1

1−g

1−g

³

1−f 1−g

´n−1

³

1−f 1−g

´n−1

c c x2n−2 ´ = ³ = ³ ´n−1 ³ ´n 1−g x2n−2 y2n−1 − g 1−f 1−g cb 1−f − g c 1−g b 1−f − g ³ ³ ´n−1 ´n−1 ´n−1 ³ 1−f 1−f c c (1 − f ) c 1−f 1−g 1−g 1−g ´³ ´ ´ =³ = = ³ 1−f g 1−g 1−f g 1 − fg − g + fg −g −g 1−g 1−f 1−f

x2n =

3

969

MANSOUR ET AL: RATIONAL SYSTEMS OF DIFFERENCE EQUATIONS

y2n =

=

=

´n−1

¶n 1−f =c , 1−g 1−g ³ ´n−1 ´n−1 ³ 1−g 1−g a 1−f a 1−f y2n−2 ³ ´ = ³ = ´n−1 ³ ´n y2n−2 x2n−1 − f 1−g 1−f −f ad 1−f a 1−f d 1−g − f 1−g ³ ³ ´n−1 ´n−1 ³ ´n−1 1−g 1−g 1−g a 1−f a 1−f a 1−f (1 − g) ³ ´³ ´ ´ =³ = 1−f g 1−f 1−f g 1 − fg − f + gf −f −f 1−f 1−g 1−g ´n−1 ³ 1−g µ ¶n (1 − g) a 1−f 1−g =a . 1−f 1−f c

=

³

1−f 1−g

(1 − f )

µ

The proof is complete. Theorem 2 The following statements are true: (a) If f 6= g, then there exists unboundedness solution of system (1). (b) If f, g > 1 or f, g < 1, then all solution of system (1) are positive, and when f > 1(f < 1) and g < 1(g > 1), then all solution of system (1) are oscillating. (c) Every solutions of system (1) are periodic with prime period two solutions if and only if f = g and will take the form x2n−1 = d, x2n = c,

and

y2n−1 = b, y2n = a.

Proof: (a),(b) The proof in these cases follows directly from the form of the solutions which given in Theorem 1. (c) First suppose that there exists a prime period two solution of system (1) of the form {xn }+∞ {yn }+∞ n=−1 = {d, c, d, c, ...} , n=−1 = {b, a, b, a, ...} . Then we see from the form of solution of System (1) that ¶n ¶n ¶n ¶n µ µ µ µ 1−f 1−f 1−g 1−g d=d , c=c , b=b , a=a . 1−g 1−g 1−f 1−f Then f = g. Second suppose that f = g. Then we see from the solution of system (1) that x2n−1 = d, x2n = c,

and y2n−1 = b, y2n = a,

Thus we have a period two solution which completes the proof. 4

970

MANSOUR ET AL: RATIONAL SYSTEMS OF DIFFERENCE EQUATIONS

Example 1. Consider the difference system (1) with f = 1.3, g = 1.2 and the initial conditions x−1 = 0.3, x0 = 1.1, y−1 = 28/11 and y0 = 56/9. (See Fig. 1). plot of X(n+1)=X(n−1)/(X(n−1)Y(n)−g),Y(n+1)=Y(n−1)/(Y(n−1)X(n)−f) 20 x(n) y(n)

18 16 14

x(n),y(n)

12 10 8 6 4 2 0

0

2

4

6

8

10

12

14

16

18

n

Figure 1. Example 2. For the initial conditions x−1 = 7, x0 = 1.3, y−1 = 5/7, y0 = 50/13 and f = 4, g = 4. when we take the system (1). (See Fig. 2). plot of X(n+1)=X(n−1)/(X(n−1)Y(n)−g),Y(n+1)=Y(n−1)/(Y(n−1)X(n)−f) 7 x(n) y(n) 6

x(n),y(n)

5

4

3

2

1

0

0

5

10

15 n

20

25

30

Figure 2.

3

On the System: xn+1 =

xn−1 xn−1 yn −g ,

yn+1 =

yn−1 −yn−1 xn −f .

In this section, we study the solutions of the system of two difference equations xn+1 =

xn−1 yn−1 , yn+1 = , xn−1 yn − g −yn−1 xn − f

n = 0, 1, ...,

(2)

with a nonzero real numbers initial conditions and f, g are nonzero real numbers with f, g 6= −1.

5

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MANSOUR ET AL: RATIONAL SYSTEMS OF DIFFERENCE EQUATIONS

Theorem 3 Suppose that {xn³ , yn } are ´ solutions ³ of´system (2) with y−1 = b, y0 = 1−f g g−1 . Then a, x−1 = d , x0 = c and bc = 1+g , ad = f1+f x2n−1 =

d , (ad − g)n

and y2n−1 =

x2n = c (−1)n (bc + f)n ,

b (−1)n , y2n = a (ad − g)n . (bc + f )n

Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n − 1. That is; x2n−3 =

d , x2n−2 = c (−1)n−1 (bc + f)n−1 , (ad − g)n−1

and y2n−3 =

b (−1)n−1 n−1 . n−1 , y2n−2 = a (ad − g) (bc + f )

Now, we see from Eq.(2) that x2n−1

y2n−1

x2n−3 = = x2n−3 y2n−2 − g

d (ad−g)n−1 d a (ad (ad−g)n−1

y2n−3 = = −y2n−3 x2n−2 − f

n−1

− g)

−g

=

b(−1)n−1 (bc+f )n−1

−

b(−1)n−1 c(−1)n−1 (bc+f )n−1 −f (bc+f )n−1

d (ad−g)n−1

ad − g

=

d , (ad − g)n

b(−1)n−1 (bc+f )n−1

b (−1)n = = , −bc − f (bc + f )n

and n−1 x2n−2 c(−1)n−1 (bc+f )n−1 )n−1 = = c(−1) −bc(bc+f n−1 n−1 b(−1)n −g c(−1) (bc+f ) −g x2n−2 y2n−1 − g (bc+f ) (bc+f )n n−1 n c (−1) (bc + f ) = c (−1)n (bc + f)n , = −bc − bcg − f g y2n−2 a (ad − g)n−1 a(ad−g)n−1 = a(ad−g)n−1 d −f = = ad −y2n−2 x2n−1 − f (ad−g)n −f (ad−g)

x2n =

y2n

a (ad − g)n = a (ad − g)n . = ad − adf + fg

The proof is complete. The proof of the following Theorem is similar to the proof of Theorem 2. Theorem 4 The following statements are true: 6

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MANSOUR ET AL: RATIONAL SYSTEMS OF DIFFERENCE EQUATIONS

(a) If ad − g 6= 1, bc + f 6= −1, then there exists unboundedness solution of system (1). (b) If ad > g, bc + f < 0, then all solution of system (2) are positive, and when ad < g and bc + f > o, then all solution of system (2) are oscillating. (c) Every solutions of system (2) are periodic with prime period two solutions if and only if ad − g = 1, bc + f = −1 and will take the form {xn }+∞ n=−1 = {d, c, d, c, ...} ,

{yn }+∞ n=−1 = {b, a, b, a, ...} .

Example 3. If we consider the difference system (2) with the initial conditions x−1 = 0.4, x0 = 1.1, y−1 = −16/121 y0 = 8/21 and f = 1.1, g = 1.2. (See Fig. 3). plot of X(n+1)=X(n−1)/(X(n−1)Y(n)−g),Y(n+1)=Y(n−1)/(−Y(n−1)X(n)−f) 2 x(n) y(n) 1.5

1

x(n),y(n)

0.5

0

−0.5

−1

−1.5

−2

0

10

20

30

40

50

60

70

n

Figure 3. The following cases can be proved similarly.

4

On the System: xn+1 =

xn−1 −xn−1 yn −g ,

yn+1 =

yn−1 yn−1 xn −f .

we study in this section, the solutions of the system of two difference equations xn−1 yn−1 , yn+1 = , n = 0, 1, ..., (3) xn+1 = −xn−1 yn − g yn−1 xn − f with a nonzero real numbers initial conditions and f, g 6= −1.

Theorem 5 Suppose that {xn , yn } are solutions of system (3). Then ¶n ¶n µ µ 1+f 1+f n n , x2n = c (−1) , x2n−1 = d (−1) 1+g 1+g and n

y2n−1 = b (−1) where y−1 = b, y0 = a, x−1

µ

1+g 1+f

¶n

n

µ

1+g 1+f

¶n

, y2n = a (−1) , ´ ³ ´ ³ g−1 g , and ad = 1−f . = d , x0 = c, bc = f1+g 1+f 7

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MANSOUR ET AL: RATIONAL SYSTEMS OF DIFFERENCE EQUATIONS

Theorem 6 The following statements are true: (a) If f 6= g, then the system (3) has unboundedness solution. (b) If f, g > −1 or f, g < −1, then all solution of system (3) are oscillating, and when f > −1(f < −1) and g < −1(g > −1), then all solution of system (3) are positive. (c) Every solutions of system (3) are periodic with prime period four solutions if and only if f = g and will take the form x4n−1 = d, x4n = c, x4n+1 = −d, x4n+2 = −c and y4n−1 = b, y4n = a, y4n+1 = −b, y4n+2 = −a. (d) Every solutions of system (3) are periodic with prime period two solutions if and only if f + g = −2 and will take the form x2n−1 = d, x2n = c,

and

y2n−1 = b, y2n = a.

Example 4. Figure 4 shows the solutions of system (3) when we assume that x−1 = −0.3, x0 = −3, y−1 = −1/6, y0 = 5/3 and f = 1.5, g = 1.5. plot of X(n+1)=X(n−1)/(−X(n−1)Y(n)−g),Y(n+1)=Y(n−1)/(Y(n−1)X(n)−f) 3 x(n) y(n) 2

x(n),y(n)

1

0

−1

−2

−3

0

5

10

15 n

20

25

30

Figure 4.

5

On the System: xn+1 =

xn−1 −xn−1 yn −g ,

yn+1 =

yn−1 −yn−1 xn −f .

In this section, we study the solutions of the system of two difference equations xn+1 =

xn−1 yn−1 , yn+1 = , −xn−1 yn − g −yn−1 xn − f

n = 0, 1, ...,

(4)

with nonzero real initials conditions x−1 , x0 , y−1 , y0 and f, g are nonzero real numbers with f, g 6= 1. 8

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MANSOUR ET AL: RATIONAL SYSTEMS OF DIFFERENCE EQUATIONS

Theorem 7 Suppose that {xn , yn } are solutions of system (4). Also, assume that x−1 , x0 , y−1 and y0 are arbitrary nonzero real numbers. Then x2n−1 = and y2n−1 = where y−1 = b, y0 = a, x−1

d , (−ad − g)n

x2n = c (−bc − f )n ,

b n n , y2n = a (−ad − g) . (−bc − f ) ´ ³ ´ ³ −1 g , and ad = 1+f . = d , x0 = c, bc = gf1−g 1−f

Theorem 8 The following statements are true:

(a) If ad + g 6= −1, bc + f 6= −1, then there exists unboundedness solution of system (4). (b) If ad + g > 0, bc + f > 0, then all solution of system (4) are oscillating, and when ad + g < 0 and bc + f < 0, then all solution of system (4) are positive. (c) Every solutions of system (4) are periodic with prime period two solutions if and only if ad + g = −1, bc + f = −1 and will take the form {xn }+∞ n=−1 = {d, c, d, c, ...} ,

{yn }+∞ n=−1 = {b, a, b, a, ...} .

(d) Every solutions of system (4) are periodic with prime period four solutions if and only if ad + g = 1, bc + f = 1 and will take the form {xn }+∞ n=−1 = {d, c, −d, −c, d, c, ...} ,

{yn }+∞ n=−1 = {b, a, −b, −a, b, a, ...} .

Example 5. See Figure 5 to see the behavior of solutions of system (4) when we put x−1 = 0.4, x0 = −7, y−1 = 6/35, y0 = 13/4 and f = 0.2, g = 0.2. plot of X(n+1)=X(n−1)/(−X(n−1)Y(n)−g),Y(n+1)=Y(n−1)/(−Y(n−1)X(n)−f) 6 x(n) y(n) 4

x(n),y(n)

2

0

−2

−4

−6

−8

2

4

6

8

10

12 n

Figure 5. 9

975

14

16

18

20

MANSOUR ET AL: RATIONAL SYSTEMS OF DIFFERENCE EQUATIONS

Acknowledgements This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

References [1] R. P. Agarwal, Difference Equations and Inequalities, 1st edition, Marcel Dekker, New York, 1992, 2nd edition, 2000. [2] R. P. Agarwal and E. M. Elsayed, On the solution of fourth-order rational recursive sequence, Adv. Stud. Contemp. Math., 20 (4), (2010), 525—545. [3] N. Battaloglu, C. Cinar and I. Yalçınkaya, The dynamics of the difference equation, ARS Combinatoria, 97 (2010), 281-288. [4] C. Cinar, I. Yalçinkaya and R. Karatas, On the positive solutions of the difference equation system xn+1 = m/yn , yn+1 = pyn /xn−1 yn−1 , J. Inst. Math. Comp. Sci., 18 (2005), 135-136. [5] E. M. Elabbasy , H. El-Metwally and E. M. Elsayed, Global behavior of the solutions of difference equation, Adv. Differ. Equ., 2011, 2011:28. [6] H. El-Metwally and E. M. Elsayed, Solution and behavior of a third rational difference equation, Utilitas Mathematica, 88 (2012), 27—42. [7] E. M. Elsayed, Solutions of rational difference system of order two, Math. Comp. Mod., 55 (2012), 378—384. [8] E. M. Elsayed, On the solutions of a rational system of difference equations, Fasciculi Mathematici, 45 (2010), 25—36. [9] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Disc. Dyn. Nat. Soc., Volume 2011, Article ID 982309, 17 pages. [10] E. M. Elsayed and M. M. El-Dessoky, Dynamics and behavior of a higher order rational recursive sequence, Adv. Differ. Equ., 2012, 2012:69. [11] A. Kurbanli, C. Cinar and I. Yalçınkaya, On the behavior of positive solutions of the system of rational difference equations, Math. Comp. Mod., 53 (2011), 1261-1267. [12] M. Mansour, M. M. El-Dessoky and E. M. Elsayed, The form of the solutions and periodicity of some systems of difference equations, Disc. Dyn. Nat. Soc., Volume 2012, Article ID 406821, 17 pages. [13] A. Y. Ozban, On the system of rational difference equations xn+1 = a/yn−3 , yn+1 = byn−3 /xn−q yn−q , Appl. Math. Comp., 188(1) (2007), 833-837. [14] I. Yalçınkaya, On the global asymptotic stability of a second-order system of difference equations, Disc. Dyn. Nat. Soc., Vol. 2008, Article ID 860152, 12 pages. [15] I. Yalçınkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria, 95 (2010), 151-159. [16] I. Yalçınkaya, C. Cinar and M. Atalay, On the solutions of systems of difference equations, Adv. Differ. Equ., Vol. 2008, Article ID 143943, 9 pages.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.5, 977-984, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A note on the shared set and normal family Junfeng Xu1

Xiaobin Zhang2

1.Department of Mathematics, Wuyi University, Jiangmen, Guangdong 529020, P.R.China E-mail: [email protected] 2.College of Mathematics, Civil Aviation University of China, Tianjin, 300300, P.R.China E-mail: [email protected]

Abstract Let F be a family of holomorphic functions on D, and let 𝑎1 , 𝑎2 be two distinct finite complex numbers and S = {𝑎1 , 𝑎2 }, and let M be a positive number. If, for each 𝑓 ∈ F , |𝑓 (𝑘) (𝑧)| ≤ M whenever 𝑓 ∈ S for 𝑘 = 1, 2, then F is normal in D.

2010 MSC: Primary 30D45. Secondly 30D35. Keywords and phrases: normal family; share set; meromorphic function;

1

Introduction and main results

Let C denote the complex plane and 𝑓 (𝑧) be a non-constant meromorphic function in C. It is assumed that the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as the characteristic function T (r, 𝑓 ), the proximate function 𝑚(r, 𝑓 ), the counting function 𝑁 (r, 𝑓 ) (see, e.g. [4, 6, 14, 15]), and S(r, 𝑓 ) denotes any quantity that satisfies the condition S(r, 𝑓 ) = o(T (r, 𝑓 )) as r → ∞ outside of a possible exceptional set of finite linear measure. Let D be a domain in C, and F be a family of meromorphic functions defined in D. Then F is said to be normal on D, in the sense of Montel, if for any sequence {𝑓n } ⊂ F there exists a subsequence {𝑓nj } such that {𝑓nj } converges spherically locally uniformly on D to a meromorphic function or ∞. (see. [11]) Let 𝑓 (𝑧) and 𝑔(𝑧) be two nonconstant meromorphic functions in the complex plane C and let 𝑎 be a complex number. If 𝑔(𝑧) = 𝑎 whenever 𝑓 (𝑧) = 𝑎, we write 𝑓 (𝑧) = 𝑎 ⇒ 𝑔(𝑧) = 𝑎. If 𝑓 (𝑧) = 𝑎 ⇒ 𝑔(𝑧) = 𝑎 and 𝑔(𝑧) = 𝑎 ⇒ 𝑓 (𝑧) = 𝑎, we write 𝑓 (𝑧) = 𝑎 ⇔ 𝑔(𝑧) = 𝑎 and say that 𝑓 and 𝑔 share the value 𝑎 IM (ignoring multiplicity). If 𝑓 − 𝑎 and 𝑔 − 𝑎 have the same zeros with the same multiplicities, we write 𝑓 (𝑧) = 𝑎 𝑔(𝑧) = 𝑎 and say that 𝑓 and 𝑔 share the value a CM (counting multiplicity).(see [14]). We denote 1 1 by 𝑁k) (r, f −a ) (or 𝑁 k) (r, f −a ) ) the counting function for zeros of 𝑓 − 𝑎 with multiplic1 1 ity ≤ 𝑘 (ignoring multiplicities), and by 𝑁(k (r, f −a ) (or 𝑁 (k (r, f −a ) ) the counting function for zeros of 𝑓 − 𝑎 with multiplicity ≥ 𝑘 (ignoring multiplicities). Moreover we set 1 1 1 1 1 𝑁k (r, f −a ) = 𝑁 (r, f −a ) + 𝑁 (2 (r, f −a ) + 𝑁 (3 (r, f −a ) + · · · + 𝑁 (k (r, f −a ).

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XU, ZHANG: SHARED SET AND NORMAL FAMILY

Definition. Let 𝑓 and 𝑔 be two meromorphic function in a domain D, 𝑎1 , 𝑎2 be two distinct finite complex numbers. We say that 𝑓 and 𝑔 share the set S = {𝑎1 , 𝑎2 }, if 𝑓 −1 (S) = 𝑔 −1 (S). Schwick[12] was the first to draw a connection between values shared by functions in F (and their derivatives) and the normality of the family F. Specially, he showed that if there exist three distinct complex numbers 𝑎1 , 𝑎2 , 𝑎3 such that 𝑓 and 𝑓 ′ share 𝑎j (𝑗 = 1, 2, 3) in D for each 𝑓 ∈ F , then F is normal in D. Pang and Zalcman [10] extended this result as follows. Theorem A. Let F be a family of meromorphic functions in a domain D, and let 𝑎, 𝑏, c, 𝑑 be complex numbers such that c ̸= 𝑎 and 𝑑 ̸= 𝑏. If for each 𝑓 ∈ F we have 𝑓 (𝑧) = 𝑎 ⇔ 𝑓 ′ (𝑧) = 𝑏 and 𝑓 (𝑧) = c ⇔ 𝑓 ′ (𝑧) = 𝑑, then F is normal in D. Fang[2], Liu and Pang [8] used the idea of sharing set to extend the result of Schwick and obtain: Theorem B. Let F be a family of meromorphic functions in a domain D, and let 𝑎1 , 𝑎2 , 𝑎3 be three distinct finite complex numbers. If 𝑓 and 𝑓 ′ share the set S = {𝑎1 , 𝑎2 , 𝑎3 }, then F is normal in D. Recently, Chen[1] continued to investigate the problem and proved the following. Theorem C. Let F be a family of meromorphic functions on D, all of whose poles are of multiplicity at least 3, let 𝑎1 , 𝑎2 , 𝑎3 be three distinct finite complex numbers and S = {𝑎1 , 𝑎2 , 𝑎3 }, and let M be a positive number. If, for each 𝑓 ∈ F , |𝑓 ′ (𝑧)| ≤ M whenever 𝑓 ∈ S, then F is normal in D. Chen gave an example to show the set S with three elements is the best possible. Example 1.[3, 9] Let S = {−1, 1}. Set F = {𝑓n (𝑧) : 𝑛 = 2, 3, 4, . . .}, where 𝑓n (𝑧) =

𝑛 + 1 nz 𝑛 − 1 −nz 𝑒 + 𝑒 , D = {𝑧 : |𝑧| < 1}. 2𝑛 2𝑛

Then, for any 𝑓n ∈ F, we have 𝑛2 [𝑓n2 (𝑧) − 1] = 𝑓n′2 (𝑧) − 1. Thus 𝑓n and 𝑓n′ share S CM, but F is not normal in D. Naturally, one can asked whether the condition on 𝑓 ′ (𝑧) can be replaced by 𝑓 (k) (𝑧) in Theorem C? In the note, we proved the following two results. Theorem 1. Let F be a family of meromorphic functions on D, and let 𝑚, q be positive integers with 𝑚(3q − 4) − 4 > 0, (1.1)

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XU, ZHANG: SHARED SET AND NORMAL FAMILY

and all of whose poles are of multiplicity at least 3, let 𝑎1 , 𝑎2 , · · · , 𝑎q be three distinct finite complex numbers and S = {𝑎1 , 𝑎2 , · · · , 𝑎3 }, and let M be a positive number. If, for each 𝑓 ∈ F, |𝑓 (k) (𝑧)| ≤ M whenever 𝑓 ∈ S(𝑘 = 1, 2, . . . , 𝑚), then F is normal in D. Let q = 3 and 𝑘 = 1 in Theorem 1, we can get Theorem C. Let q = 2, we can get the following result. Corollary 1. Let F be a family of meromorphic functions on D, and all of whose poles are of multiplicity at least 3, let 𝑎1 , 𝑎2 be two distinct finite complex numbers and S = {𝑎1 , 𝑎2 }, and let M be a positive number and 𝑚(≥ 3) be a positive integer. If, for each 𝑓 ∈ F , |𝑓 (k) (𝑧)| ≤ M whenever 𝑓 ∈ S(𝑘 = 1, 2, . . . , 𝑚), then F is normal in D. Next, one can ask whether the multiplicity of poles of 𝑓 (𝑧) at least 3 can be reduced. We can obtain the following result by a slight different method. Theorem 2. Let F be a family of meromorphic functions on D, and let 𝑚, 𝑛 be positive integers with (𝑘 + 1)(𝑚 + 𝑛) < 2, (1.2) 𝑚𝑛 and all of whose zeros are of multiplicity at least 𝑚 and whose poles are of multiplicity at least 𝑛, let 𝑎1 , 𝑎2 , 𝑎3 be three distinct finite complex numbers and S = {𝑎1 , 𝑎2 , 𝑎3 }, and let M be a positive number. If, for each 𝑓 ∈ F , |𝑓 (k) (𝑧)| ≤ M whenever 𝑓 ∈ S, then F is normal in D. Remark 2. In Theorem 1, if 𝑘 = 1, 𝑚 = 3, then we can take 𝑛 = 2, this is, the condition that the poles are of multiplicity at least 3 is not necessary in Theorem C. In fact, we can prove the following more general result. Theorem 3. Let F be a family of meromorphic functions on D, and let 𝑚, 𝑛 be positive integers with (𝑘 + 1)(𝑚 + 𝑛) < q − 1, (1.3) 𝑚𝑛 and all of whose zeros are of multiplicity at least 𝑚 and whose poles are of multiplicity at least 𝑛, let 𝑎1 , 𝑎2 ,· · · , 𝑎q be q distinct finite complex numbers and S = {𝑎1 , 𝑎2 , · · · , 𝑎q }, and let M be a positive number. If, for each 𝑓 ∈ F, |𝑓 (k) (𝑧)| ≤ M whenever 𝑓 ∈ S, then F is normal in D.

2

lemmas

Lemma 1 ([10]). Let F be a family of meromorphic functions on the unit disc ∆, all of whose zeros have the multiplicity at least 𝑘, and suppose that there exists A ≥ 1 such that |𝑓 (k) (𝑧)| ≤ A wherever 𝑓 (𝑧) = 0, 𝑓 ∈ F. Then if F is not normal, there exist, for each 0≤𝛼≤𝑘: (𝑎) a number r, 0 < r < 1, (𝑏) points 𝑧n , |𝑧n | < r, (c) functions 𝑓n ∈ F , and (𝑑) positive numbers 𝜌n → 0

979

XU, ZHANG: SHARED SET AND NORMAL FAMILY

such that 𝜌−α n 𝑓n (𝑧n + 𝜌n 𝜉) = 𝑔n (𝜉) → 𝑔(𝜉) locally uniformly with respect to the spherical metric, where 𝑔(𝜉) is a non-constant meromorphic function on C, all of whose zeros have multiplicity at least 𝑘, such that 𝑔 ♯ (𝜉) ≤ 𝑔 ♯ (0) = 𝑘A + 1. In particular, if 𝑔 is an entire function, it is of exponential type. Here, as usual, 𝑔 ♯ (𝑧) =

|𝑔 ′ (𝑧)| 1 + |𝑔(𝑧)|2

is the spherical derivative. Lemma 2. Let 𝑓 (𝑧) be a non-constant meromorphic function, 𝑠, 𝑘 be two positive integers. Then 𝑁s (r,

1

1 ) ≤ 𝑘𝑁 (r, 𝑓 ) + 𝑁 (r, ) + S(r, 𝑓 ). s+k 𝑓 𝑓 (k)

This lemma can be obtained immediately from the proof of Lemma 2.3 in [5] which is the special case 𝑝 = 2. 1 1 ) = 𝑁 (r, f (k) ), we can get Note that 𝑁1 (r, f (k) 𝑁 (r,

1 𝑓 (k)

1 ) ≤ 𝑘𝑁 (r, 𝑓 ) + 𝑁1+k (r, ) + S(r, 𝑓 ) 𝑓 1 1 1 ≤ 𝑘𝑁 (r, 𝑓 ) + 𝑁 (r, ) + 𝑁 (2 (r, ) + 𝑁 (3 (r, ) 𝑓 𝑓 𝑓 1 + · · · + 𝑁 (k+1 (r, ) + S(r, 𝑓 ) 𝑓 1 ≤ 𝑘𝑁 (r, 𝑓 ) + (𝑘 + 1)𝑁 (r, ) + S(r, 𝑓 ). 𝑓

(2.1)

The above equation (2.1) play an important role in the proof of Theorem 3. Lemma 3 ([11]). (Marty’s Theorem) Let F be a family of meromorphic functions on a domain Ω is normal if and only if for each compact subset K ⊆ Ω, there exists a constant C = C(𝑘) such that the spherical derivative 𝑓 ♯ (𝑧) =

|𝑓 ′ (𝑧)| ≤ C, 1 + |𝑓 (𝑧)|2

𝑧 ∈ K, 𝑓 ∈ F,

this is, 𝑓 ♯ (𝑧) is locally bounded.

3

Proof of Theorem 1

Without loss of generality, we may assume D = ∆ = {𝑧 : |𝑧| < 1}. Suppose that F is not normal in D. By Lemma 1, there exist: (𝑎) a number r, 0 < r < 1, (𝑏) points 𝑧n , |𝑧n | < r, (c) functions 𝑓n ∈ F, and (𝑑) positive numbers 𝜌n → 0

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XU, ZHANG: SHARED SET AND NORMAL FAMILY

such that 𝑔n (𝜉) = 𝑓n (𝑧n + 𝜌n 𝜉) → 𝑔(𝜉) locally uniformly with respect to the spherical metric, where 𝑔(𝜉) is a non-constant meromorphic function on C, all of whose zeros have multiplicity at least 𝑘 and whose poles have multiplicity at least 3. We claim that every 𝑎i (𝑖 = 1, 2, . . . , q) points of 𝑔 have multiplicity at least 𝑚 + 1. Without loss of generality, we may assume 𝑔(𝜉0 ) = 𝑎1 . Clearly, 𝑔(𝜉) ̸≡ 𝑎1 . Then by Hurwitz’s theorem there exists 𝜉n , 𝜉n → 𝜉0 , such that, for 𝑛 sufficiently large, 𝑎1 = 𝑔(𝜉0 ) = 𝑔n (𝜉n ) = 𝑓n (𝑧n + 𝜌n 𝜉n ). (k)

(k)

Then |𝑓n (𝑧n + 𝜌n 𝜉n )| ≤ M because |𝑓n (𝑧)| ≤ M whenever 𝑓n (𝑧) ∈ S for 𝑘 = 1, 2, . . . , 𝑚. It now follows that |𝑔n(k) (𝜉n )| = |𝜌kn 𝑓n(k) (𝑧n + 𝜌n 𝜉n )| ≤ 𝜌kn M for 𝑘 = 1, 2, . . . , 𝑚. (k) Since 𝑔 (k) (𝜉0 ) = lim 𝑔n (𝜉n ) = 0, (𝑘 = 1, 2, . . . , 𝑚). This implies that 𝑔 − 𝑎1 have n→∞ multiplicity at least 𝑚 + 1. Hence the claim holds. By Nevanlinna’s second fundamental theorem, we have 1 1 1 ) + 𝑁 (r, ) + · · · + 𝑁 (r, ) + 𝑁 (r, 𝑔) + S(r, 𝑔) 𝑔 − 𝑎1 𝑔 − 𝑎2 𝑔 − 𝑎q ] 1 1 1 1 [ 𝑁 (r, ) + 𝑁 (r, ) + · · · + 𝑁 (r, ) + 𝑁 (r, 𝑔) + S(r, 𝑔) 𝑚+1 𝑔 − 𝑎1 𝑔 − 𝑎2 𝑔 − 𝑎q 1 q T (r, 𝑔) + 𝑁 (r, 𝑔) + S(r, 𝑔) 𝑚+1 3 q 1 T (r, 𝑔) + T (r, 𝑔) + S(r, 𝑔) 𝑚+1 3 q 1 ( + )T (r, 𝑔) + S(r, 𝑔)T (r, 𝑔) + S(r, 𝑔). 𝑚+1 3

(q − 1)T (r, 𝑔) ≤ 𝑁 (r, ≤ ≤ ≤ ≤ i.e,

𝑚(3q − 4) − 4 T (r, 𝑔) ≤ S(r, 𝑔), 3(𝑚 + 1) This implies that 𝑔 is a constant, a contradiction. This completes the proof of Theorem 1.

4

Proof of Theorem 3

Without loss of generality, we may assume D = ∆ = {𝑧 : |𝑧| < 1}. Suppose that F is not normal in D. By Lemma 1, there exist: (𝑎) a number r, 0 < r < 1, (𝑏) points 𝑧n , |𝑧n | < r, (c) functions 𝑓n ∈ F, and (𝑑) positive numbers 𝜌n → 0 such that 𝑔n (𝜉) = 𝑓n (𝑧n + 𝜌n 𝜉) → 𝑔(𝜉) locally uniformly with respect to the spherical

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XU, ZHANG: SHARED SET AND NORMAL FAMILY

metric, where 𝑔(𝜉) is a non-constant meromorphic function on C, all of whose zeros have multiplicity at least 𝑘 and whose poles have multiplicity at least 2. Obviously, 𝑔 (k) (𝜉) ̸≡ 0, for otherwise 𝑔 would be a polynomial of degree less than 𝑘, and so could not have the multiplicity at least 𝑘. We claim that 𝑔 (k) (𝜉) = 0 whenever 𝑔(𝜉) = 𝑎i (𝑖 = 1, 2, · · · , q). Without loss of generality, we may assume 𝑔(𝜉0 ) = 𝑎1 . Clearly, 𝑔(𝜉) ̸≡ 𝑎1 . Then by Hurwitz’s theorem there exists 𝜉n , 𝜉n → 𝜉0 , such that, for 𝑛 sufficiently large, 𝑎1 = 𝑔(𝜉0 ) = 𝑔n (𝜉n ) = 𝑓n (𝑧n + 𝜌n 𝜉n ). (k)

(k)

Then |𝑓n (𝑧n + 𝜌n 𝜉n )| ≤ M because |𝑓n (𝑧)| ≤ M whenever 𝑓n (𝑧) ∈ S. It now follows that |𝑔n(k) (𝜉n )| = |𝜌kn 𝑓n(k) (𝑧n + 𝜌n 𝜉n )| ≤ 𝜌kn M. (k)

Since 𝑔 (k) (𝜉0 ) = lim 𝑔n (𝜉n ) = 0. This implies that 𝑔 (k) (𝜉) = 0 whenever 𝑔(𝜉) = 𝑎1 . n→∞ Hence the claim holds. By Nevanlinna’s second fundamental theorem, we have 1 1 1 ) + 𝑁 (r, ) + · · · + 𝑁 (r, ) + 𝑁 (r, 𝑔) + S(r, 𝑔) 𝑔 − 𝑎1 𝑔 − 𝑎2 𝑔 − 𝑎m 1 𝑁 (r, (k) ) + 𝑁 (r, 𝑔) + S(r, 𝑔) 𝑔 1 𝑁1+k (r, ) + (𝑘 + 1)𝑁 (r, 𝑔) + S(r, 𝑔) 𝑔 𝑘+1 1 𝑁 (r, 𝑔) + S(r, 𝑔) (𝑘 + 1)𝑁 (r, ) + 𝑔 𝑛 𝑘+1 1 𝑘+1 𝑁 (r, ) + 𝑁 (r, 𝑔) + S(r, 𝑔) 𝑚 𝑔 𝑛 (𝑘 + 1)(𝑚 + 𝑛) T (r, 𝑔) + S(r, 𝑔). 𝑚𝑛

(q − 1)T (r, 𝑔) ≤ 𝑁 (r, ≤ ≤ ≤ ≤ ≤ i.e,

(q − 1)𝑚𝑛 − (𝑘 + 1)(𝑚 + 𝑛) T (r, 𝑔) ≤ S(r, 𝑔), 𝑚𝑛 This implies that 𝑔 is a constant, a contradiction. This completes the proof of Theorem 3.

5

Final Remark

For the holomorphic function, using the ideas of the shared set to obtain the normal family have been studied by Fang, L¨ u and Xu, Li. Theorem D.([2]) Let F be a family of holomorphic functions in a domain D, and let 𝑎1 , 𝑎2 , 𝑎3 be three distinct complex numbers. If 𝑓 and 𝑓 ′ share the set S = {𝑎1 , 𝑎2 , 𝑎3 }, then F

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XU, ZHANG: SHARED SET AND NORMAL FAMILY

is normal in D. Theorem E.([9]) Let F be a family of functions holomorphic in a domain D, let 𝑎 and 𝑏 be two distinct finite complex numbers with 𝑎 + 𝑏 ̸= 0. If for all 𝑓 ∈ F, 𝑓 and 𝑓 ′ share S = {𝑎, 𝑏} CM, then F is normal in D. Theorem F.([7]) Let F be a family of functions holomorphic in a domain, and let 𝑘(≥ 2) be a positive integer. Let 𝑎 and 𝑏 be two distinct finite complex numbers. If for all 𝑓 ∈ F , all zeros of 𝑓 (𝑧) are of multiplicity at least 𝑘, and 𝑓 and 𝑓 (k) share S = {𝑎, 𝑏} CM in D , then F is normal in D. From the proof of Theorem 1, we can obtain that Corollary 2. Let F be a family of holomorphic functions on D, and let 𝑎1 , 𝑎2 be two distinct finite complex numbers and S = {𝑎1 , 𝑎2 }, and let M be a positive number. If, for each 𝑓 ∈ F , |𝑓 (k) (𝑧)| ≤ M whenever 𝑓 ∈ S for 𝑘 = 1, 2, then F is normal in D. Proof. We can obtain every 𝑎i (𝑖 = 1, 2, . . . , q) points of 𝑔 have multiplicity at least 3 by Theorem 1. By Nevanlinna’s second fundamental theorem, we have 1 1 ) + 𝑁 (r, ) + S(r, 𝑔) 𝑔 − 𝑎1 𝑔 − 𝑎2 ] 1[ 1 1 𝑁 (r, ) + 𝑁 (r, ) + S(r, 𝑔) 3 𝑔 − 𝑎1 𝑔 − 𝑎2 2 T (r, 𝑔) + S(r, 𝑔). 3

T (r, 𝑔) ≤ 𝑁 (r, ≤ ≤ i.e,

1 T (r, 𝑔) ≤ S(r, 𝑔), 3 This implies that 𝑔 is a constant, a contradiction. This completes the proof of Corollary 2. Remark 3. From Example 1, we know 𝑓n ∈ S, then |𝑓n′ | ≤ 1. But from 𝑓n′′ = 𝑛2 𝑓n , we know if 𝑓n ∈ S, then |𝑓n′′ | → ∞ as 𝑛 → ∞. Hence our condition |𝑓 ′′ (𝑧)| ≤ M whenever 𝑓 ∈ S is necessary in Corollary 2. Example 2. Let F = {𝑓n (𝑧) : 𝑓n (𝑧) = 𝑛(𝑒z − 𝑒−z ), 𝑛 = 1, 2, . . . , 𝑧 ∈ ∆}, where ∆ is a unit disk. Thus 𝑓 and 𝑓 ′′ share every complex number in ∆. Therefore, if 𝑓n√ (𝑧) ∈ S = {1, −1}, z ′ ′′ then |𝑓n (𝑧)| ≤ 1. But if 𝑓n (𝑧) = ±1, we have |𝑓n (𝑧)| = |2𝑛𝑒 ∓ 1| = 4𝑛2 + 1 → ∞ as 𝑛 → ∞. While |𝑓 ′ (0)| 𝑓n♯ (0) = = 2𝑛 → ∞. 1 + |𝑓 (0)|2 By Marty’s Theorem, we have that F is not normal at 𝑧 = 0. Hence our condition |𝑓 ′ (𝑧)| ≤ M whenever 𝑓 ∈ S is necessary in Corollary 2.

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Acknowledgements This work was supported by National Natural Science Foundation of China (No. 11126327, 11171184), the Science Research Foundation of CAUC, China (No.2011QD10X), NSF of Guangdong Province(No. S2011010000735) and STP of Jiangmen, China(No.[2011]133).

References [1] J. F. Chen, Shared sets and normal families of meromorphic functions, Rocky Mountain J. Math. vol. 40, no. 5 (2010), 1473-1479. [2] M. L. Fang, A note on sharing values and normality, J. Math. Study, 29(1996), 29-32. [3] M.L. Fang and L. Zalcman, Normal families and uniqueness theorems for entire functions, J. Math. Anal. Appl. 280 (2003), 273-283 [4] W. K. Hayman, Meromorphic Functions. Oxford University Press, London, 1964. [5] I. Lahiri, Uniqueness of a meromorphic function and its derivative,J. Inequal. Pure Appl. Math., 5(1) (2004), Art. 20. [6] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, New York, 1993. Basel, Boston, 1985. [7] Y. T. Li, Sharing set and normal families of entire functions, Results. Math. Doi 10.1007/s00025-011-0216-8. [8] X. J. Liu and X. C. Pang, Shared values and normal function, Acta Mathematica Sinca, Chinese Series, 50(2007), 409-412. [9] F. L¨ u and J. F. Xu, Sharing set and normal families of entire functions and their derivatives, Houston J. Math. vol. 34, no. 4 (2008), 1213-1223. [10] X.C. Pang and L. Zalcman, Normal families and shared values, Bull. London. Math. Soc. 32 (2000) 325–331. [11] J. Schiff, Normal families, Springer, New York, 1993. [12] W. Schwick, Sharing values and normality, Arch math.(Basel) 59(1992), 50-54. [13] Q.F. Wu, B. Xiao and W.J. Yuan, Normality of composite analytic functions and sharing an analytic function, Fixed Point Theory and Applications, Volume 2010 (2010), Article ID 417480, 9 pages. [14] C.C. Yang and H.X. Yi, Uniqueness theory of meromorphic functions. Kluwer Acad. Publ. Dordrecht, 2003. [15] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 5, 2013

Some Order Relations Induced by Fuzzy Subsets, Jeong Soon Han, Young Hee Kim, and Keum Sook So, …….…………………………………………………………………………………800 Multi-Attribute Group Decision-Making Method Based on Generalized Aggregation Operators in Trapezoidal Fuzzy Linguistic Variables, Peide Liu and Xingying Wu,…………………….807 A New Impulsive Integro-Differential Inequality and its Applications, Huali Wang and Diwang Lin,……………………………………………………………………………………………...817 On the Convergence of a Modified q-Gamma Operators, Qing-Bo Cai and Xiao-Ming Zeng,.826 Approximation by Complex Schurer-Stancu Operators in Compact Disks, Mei-Ying Ren and Xiao-Ming Zeng,………………………………………………………………………………833 Triple Fixed Point Theorems for Weak ( , )-Contractions, Erdal Karapinar and Kishin Sadarangani,……………………………………………………………………………………844 Form of Solutions and Periodicity for Systems of Difference Equations, H. El-Metwally and E. M. Elsayed,…………………………………………………………………………………….852 Scaling Bini's Algorithm for Fast Inversion of Triangular Toeplitz Matrices, Jie Huang, TingZhu Huang, and Skander Belhaj,………………………………………………………………858 Approximation with Modified Gadjiev-Ibragimov Operators in C[0,A], Nazmiye Gonul and Erdal Coskun,…………………………………………………………………………………..868 Some Identities on the Twisted q-Euler Numbers with Weight (α, β) and q-Bernstein Polynomials with Weight α, C. S. Ryoo,………………………………………………………880 Doubly-Accelerated Steffensen's Methods with Memory and their Applications on Solving Nonlinear ODEs, Quan Zheng, Fengxi Huang, Xiuhui Guo, and Xiaoli Feng,………………886 On Modified Implicit Mann Iteration Method Involving Strictly Hemicontractive Mappings in Smooth Banach Spaces, Nawab Hussain, Arif Rafiq,……………………………………..…..892 Normal Fuzzy Filters in BE-Algebras, Sun Shin Ahn and Young Hie Kim,…………………903 Comment on "Nearly ternary cubic homomorphism in ternary Fréchet algebras" [Shagholi et al., J. Computat. Anal. Appl. 13 (2011), 1097-1105], Choonkil Park, Jung Rye Lee, and Dong Yun Shin,……………………………………………………………………………………………911

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 5, 2013 (continued) On Joint Distributions of Order Statistics from Nonidentically Distributed Discrete Vectors, B. Yüzbaşı and M. Güngör,…………………………………………………………………..…917 On Differential Superordinations of Analytical Functions with Negative Coefficients, Irina Dorca and Daniel Breaz,……………………………………………………………………..928 Adaptive Regulation with Almost Disturbance Decoupling for Nonlinearly Parameterized Systems with Control Coefficients, Ling-Ling Lv, Lei Zhang, Hai-Bin Su, and An-Fu Zhu,936 A Fixed-Point Approach to Locating Polynomial Zeros, Shun-Pin Hsu,……………………947 On the Solution of Rational Systems of Difference Equations, M. Mansour, M. M. El-Dessoky, and E. M. Elsayed,……………………………………………………………………………967 A Note on the Shared Set and Normal Family, Junfeng Xu and Xiaobin Zhang,……………977

Volume 15, Number 6 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC(eight times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Editor’s Assistant:Dr.Razvan Mezei,Lander University,SC 29649, USA.

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17) H.H.Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany 011-49-203-379-3542 e-mail:[email protected] Approximation Theory, Computer Aided Geometric Design 18) John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA [email protected] Ordinary and functional differential equations, difference equations, impulsive systems, differential inclusions, dynamic equations on time scales , control theory and their applications 19) Weimin Han Department of Mathematics University of Iowa Iowa City, IA 52242-1419 319-335-0770 e-mail: [email protected] Numerical analysis, Finite element method, Numerical PDE, Variational inequalities, Computational mechanics

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35) Lotfi A. Zadeh Professor in the Graduate School and Director, Computer Initiative, Soft Computing (BISC) Computer Science Division University of California at Berkeley Berkeley, CA 94720 Office: 510-642-4959 Sec: 510-642-8271 Home: 510-526-2569 FAX: 510-642-1712 e-mail: [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic 36) Ahmed I. Zayed Department Of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected] Shannon sampling theory, Harmonic analysis and wavelets, Special functions and orthogonal polynomials, Integral transforms

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 996-1005, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Weighted superposition operators in some analytic function spaces A. El-Sayed Ahmed 1,2 and S. Omran1,3 1 Taif University, Faculty of Science, Math. Dept. Taif, Saudi Arabia 2 Sohag University, Faculty of Science, Math. Dept. Egypt e-mail: [email protected] 3 South Valley University, Faculty of Science, Math. Dept. Egypt

Abstract In this paper, we study boundedness and the compactness of weighted superposition operators between weighted logarithmic Bloch and Zygmund spaces. Moreover, we characterize all entire functions that transform a weighted logarithmic Bloch-type space into another by weighted superposition operators.

1

Introduction

Let D = {z ∈ C : |z| < 1} be the open unit disk in the complex plane C and H(D) denote the class of all analytic functions on D. Let X and Y be two metric spaces of analytic functions on the unit disk D and suppose that φ denotes a complex-valued function of the plan C. The superposition operator Sφ on X is defined by Sφ (f ) = φ ◦ f, f ∈ X. If Sφ f ∈ Y for f ∈ X, we say that φ acts by superposition from X into Y. We see that if X contains linear functions, φ must be an entire function. For a fixed u ∈ H(D), we define the operator Su,φ = uSϕ on H(D) as follows: Su,φ (f ) = uSϕ f = u(φ ◦ f ), f ∈ H(D). The operator Su,φ will be called the weighted superposition operator. This operator generalizes the superposition operator Sφ (f ) and the multiplication operator Mu f = uf. To the best of our knowledge, the operator Su,φ is introduced in the present paper for the first time. The graph of Su,φ is usually closed but, since the operator is nonlinear, this is not enough to assure its boundedness. Nonetheless, for a number of important spaces X, Y, such as Hardy, Bergman, Dirichlet, Bloch, etc., the mere action Su,φ : X → Y implies that φ must belong to a very special class of entire functions, which in turn implies boundedness. Our goal is to study the following questions: (a) Which entire functions can transform one space into another? (b) Are there spaces (of the type considered) which are transformed one into another by specified classes of entire functions.? (c) When does ϕ induces a superposition operator form one space into another? When it is bounded.? Such questions have been extensively studied for real valued functions (cf. [2], for example). In the context of analytic functions, the question was investigated for the Hardy and Bergman spaces and ´ the Nevanlinna class by Alvarez, M´arquez and Vukoti´c [1] as well as by C´amera and Gim´enez [7, 8]. p The Bergman space A is the space of all Lp functions (with respect to Lebesgue area measure) which are analytic in the unit disk. C´amera and Gim´enez prove that Sφ (Ap ) ⊂ Aq if and only if φ is a AMS: Primary 47 B 33 , Secondary 46 E 15. Key words and phrases: Weighted Bloch-type space, superposition operator, entire function.

996

1

AHMED, OMRAN: WEIGHTED SUPERPOSITION OPERATORS

2 polynomial of degree at most p/q; note that our notation is different from theirs. Next, they show that such operators are necessarily continuous, bounded and locally Lipschitz. They also consider similar problems for superposition operators acting from Bergman spaces into the Nevanlinna area class, etc. Their method is based on choosing certain Ap “test functions” with the largest possible range and applying suitable Cauchy estimates. Later, Buckley and Vukotic considered superposition operators from Besov spaces into Bergman spaces in [5], univalent interpolation in Besov spaces and superposition into Bergman spaces in [6] and those between the conformally invariant Qp spaces and Bloch-type spaces in [15]. Wen Xu studied superposition operators on Bloch-type spaces in [16]. Very recently in [12], for any pair of numbers (s, p) with 0 6 s < ∞ and 0 < p 6 ∞, the authors characterized superposition operators which map the conformally invariant Qs space into the Hardy space H p , and also those which map H p into Qs . In this paper we study boundedness and compactness of weighted superpositions on weighted logarithmic Bloch-type spaces and on Zygmund space too. Recall that the well known Bloch space (cf. [4]) is defined as follows: 2

B = {f : f is analytic in D and sup(1 − |z| )|f 0 (z)| < ∞}; z∈D

the little Bloch space B0 (cf. [4]) is a subspace of B consisting of all f ∈ B such that lim (1 − |z|2 )|f 0 (z)| = 0.

|z|→1−

For 0 < α < ∞, the space of all analytic functions f ∈ D such that µ ¶ 2 2 α α kf kBlog = sup (1 − |z| ) log |f 0 (z)| < ∞, 1 − |z|2 z∈D α α is called weighted logarithmic α-Bloch space Blog (see [3]). If α = 1 the space Blog is just the weighted α α α Bloch space Blog . The little weighted Bloch space Blog,0 is a subspace of Blog consisting of all f ∈ Blog such that µ ¶ 2 2 α lim (1 − |z| ) log |f 0 (z)| = 0. 1 − |z|2 |z|→1

From a theorem of Zygmund [11] and the closed graph theorem, we have an analytic function f belongs to the Zygmund space Z if and only if sup(1 − |z|2 )|f 00 (z)| < ∞. z∈D

It is easy to see that the Zygmund space Z is a Banach space under the norm k.kZ , where kf kZ = |f (0)| + |f 0 (0)| + sup(1 − |z|2 )|f 00 (z)|.

(1)

z∈D

We call the Zygmund space of D, denoted by Z0 , is the closed subspace of Z consisting of functions f with lim (1 − |z|2 )|f 00 (z)| = 0. |z|→1

From (1) it is easy to obtain |f 0 (z) − f 0 (0)| ≤ Ckf kZ log

1 1 − |z|2

(2)

Conformally invariant spaces of the disk. It is a standard fact that the set of all disk automorphisms (i.e., of all one-to-one analytic maps ϕ of D onto itself), denoted Aut(D), coincides with the set of all M¨obius transformations of D onto itself: Aut(D) = {λϕa : |λ| = 1; a ∈ D}, 997

AHMED, OMRAN: WEIGHTED SUPERPOSITION OPERATORS

3 a−z where ϕa (z) = 1−¯ az are the automorphisms: ϕa (ϕa (z)) ≡ z. The space X of analytic functions in D, equipped with a semi-norm ρ, is said to be conformally invariant or M¨obius invariant if whenever f ∈ X, then also f ◦ ϕ ∈ X for any ϕ ∈ Aut(D) and, moreover, ρ(f ◦ ϕ) ≤ Cρ(f ) for some positive constant C and all f ∈ X.

Definition 1.1 In topology, a geometrical object or space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed into every other while preserving the two endpoints in question. Definition 1.2 A path from a point x to a point y in a topological space X is a continuous function f from the unit interval [0, 1] to X with f (0) = x and f (1) = y. A path-component of X is an equivalence class of X under the equivalence relation defined by x is equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwise connected or 0-connected) if there is only one path-component, i.e. if there is a path joining any two points in X. Remark 1.1 Every path-connected space is connected, but the reverse is not always true. Recall that a linear operator T : X → Y is said to be compact if it takes bounded sets in X to sets in Y which have compact closure. For Banach spaces X and Y of the space of all analytic functions H(D), we call that T is compact from X to Y if and only if for each bounded sequence (xn ) in X, the sequence (T xn ) ∈ Y contains a subsequence converging to some limit in Y.

2

Weighted logarithmic Bloch space

Let the letter Ω denote a planar domain and ∂Ω its boundary.A univalent function in D is an analytic function which is one-to-one in the disk. By the Riemann mapping theorem [13], for any given simply connected domain Ω (other than the plane itself) there is such a function f (called a Riemann map) that takes D onto Ω and the origin to a prescribed point. Denoting by dist(w, ∂Ω) the Euclidean distance of the point w to the boundary of the domain Ω, the Riemann map f has the following property: µ ¶ µ ¶ 1 2 2 2 α 0 2 α 0 (1 − |z| ) |f (z)| log ≤ dist(f (z), ∂Ω) ≤ (1 − |z| ) |f (z)| log , (3) 4 1 − |z|2 1 − |z|2 for all z ∈ D. This estimate plays an important role in the geometric theory of functions. In particular, α (3) tells us that a function f univalent in D belongs to Blog if and only if the image domain f (D) does not contain arbitrarily large disks. The auxiliary construction of a conformal map onto a specific weighted Bloch domain with the maximal (logarithmic) growth along a certain polygonal line displayed below might be of some independent interest. Thus, we state it separately as a lemma. Loosely speaking, such a domain can be imagined as a “highway from the origin to infinity” of width 2δ. Somewhat similar constructions of simply connected domains as the images of functions in various function spaces can be found in the recent papers [5] and [10]. Now, we give some auxiliary results which are incorporated in the following lemmas. Lemma 2.1 For each positive number δ and for every sequence wn of complex number such that w0 = 0, |w1 | ≥ 5δ, | arg w1 − θ0 | < π4 , arg wn & θ0 , or arg wn % θ0 and ½ |wn | ≥ max 3|wn−1 | ,

n−1 X

¾ |wk − wk−1 |

for all n ≥ 2,

(4)

k=1

there exists a domain Ω with the following properties: (i) Ω is simply connected; ∞ S (ii) Ω contains the infinite polygonal line L = [wn−1 , wn ], where [wn−1 , wn ] denotes the line segment from wn−1 to wn ;

n=1

998

AHMED, OMRAN: WEIGHTED SUPERPOSITION OPERATORS

4 (iii) there exists a conformal mapping f of D onto Ω which takes the origin to a prescribed point belongs α to Blog ; (iv) dist(w, ∂D) = δ for each point w on L, where f % denotes the increasing functions and f & denotes the decreasing functions. Proof: It is clear from (4) that |wn | % ∞, as n → ∞. We construct the domain Ω as follows. First connect the points wn by a polygonal line L as indicated in the statement. Let D(z, δ) = {w : |z −w| < δ} and define [ Ω = {D(z, δ) : z ∈ L}, i.e. let Ω be a δ-thickening of L. In other words, Ω is the union of simply connected cigar-shaped domains [ Cn = {D(z, δ) : z ∈ [wn−1 , wn ]}. By our choice of wn , it is easy to check inductively that |wn − wk | ≥ 5δ whenever n > k. Since our construction implies that Cn ⊂ {w : |wn−1 | − δ < |w| < |wn | + δ}, we see immediately that (a) for all m and n, Cm ∩ Cn 6= ∅, if and only if |m − n| ≤ l; (b) for all n, Cn ∩ Cn+1 is either D(wn , δ) or the interior of the convex hull of D(wn , δ) ∪ {an } for some N S point an outside of D(wn , δ), where D(wn , δ) is the closure of D(wn , δ). Thus, each ΩN = Cn is also n=1

simply connected. Since

∞ [

Ω=

ΩN

and

ΩN ⊂ ΩN +1 for all N,

N =1

we conclude that Ω is also simply connected (see [10]). By construction, dist(w, ∂Ω) ≤ δ for all w in Ω, hence any Riemann map onto Ω will belong to Bω . It is also clear that (iv) holds. The following lemma was proved by Tjani in [14]: Lemma 2.2 [14] Let X, Y be two Banach spaces of analytic functions on D. Suppose that (i) the point evaluation functionals on X are continuous. (ii) the closed unit ball of X is a compact subset of X in the topology of uniform convergence on compact sets. (iii) T : X → Y is continuous when X and Y are given the topology of uniform convergence on compact sets. Then T is a compact operator if and only if given a bounded sequence (fn ) in X such that fn → 0 uniformly on compact sets, then the sequence (T fn ) converges to zero in the norm of Y. Now, we prove the following results. α Lemma 2.3 Let X = Blog . Then (i) Every bounded sequence (fn ) ∈ X is uniformly bounded on compact sets. (ii) For any sequence (fn ) on X such that kfn kX → 0, fn − fn (0) → 0 uniformly on compact sets.

Proof: If z ∈ D(0, r), 0 < r < 1, then we have ¯Z1 ¯ ¯ ¯ 0 ¯ |fn (z) − fn (0)| = ¯ fn (zt)zdt¯¯

Z1 ≤ kfn k

α Blog

0

0

|z|dt ¶ µ 2 2 2 α (1 − |z| t ) log 1−|z|2

α ≤ C kfn kBlog ≤ C kfn kX .

Hence the result follows.

999

AHMED, OMRAN: WEIGHTED SUPERPOSITION OPERATORS

5 α . Lemma 2.4 Let 0 < α < ∞, u ∈ H(D) and φ be an analytic self-map of D. Let X, Y = Blog or Blog Then Su,φ : X → Y is a compact operator if and only if Su,φ : X → Y is bounded and any bounded sequence (fn )n∈IN ∈ X with fn → 0 uniformly on compact sets as n → ∞, we have kSu,φ fn kY → 0 as n → ∞.

Proof: We will show that (i), (ii), and (iii) of Lemma 2.2 hold for our spaces. By Lemma 2.3 it is easy to see that (i) and (iii) hold. To show that (ii) holds, let (fn ) be a sequence in the closed unit ball of X. Then by Lemma 2.3, (fn ) is a uniformly bounded on compact sets. Therefore, by Montel’s theorem (see [9]), there is a subsequence (fnk ), where (n1 < n2 < n3 < . . .) such that fnk → h uniformly bounded on compact sets, for some h ∈ H(D). Thus we only need to show that h ∈ X. α If X = Blog , we have that µ ¶ µ ¶ 2 2 0 2 α 0 2 α |h (z)|(1 − |z| ) log = lim |fnk (z)|(1 − |z| ) log k→∞ 1 − |z|2 1 − |z|2 α ≤ lim kfnk kBlog < ∞, k→∞

where we used Fatou’s theorem [13] and our hypothesis. Therefore, Lemma 2.2 yields that Su,φ : X → Y is a compact operator if and only if for any bounded sequence (fn ) ∈ X with fn → 0 uniformly on compact sets as n → ∞, |fn (f (0))| + kSu,φ fn kY → 0 as n → ∞, which is clearly equivalent to the statement of this lemma. This completes the proof of the lemma. α Theorem 2.1 Assume that α > 0. Then, the closed set η in Blog,0 is compact if and only if it is bounded and satisfies µ ¶ 2 lim sup (1 − |z|2 )α |f 0 (z)| log = 0. (5) 1 − |z|2 |z|→1 f ∈η

Proof. Suppose η is compact. If ε > 0, then the balls centered at the elements of η with radii 2ε cover η, α so by compactness there exist f1 , ..., fn ∈ η such that for every f ∈ η, we have kf − fj kBlog < 2ε for some 1 ≤ j ≤ n, and consequently µ ¶ µ ¶ 2 2 ε 2 α 0 (1 − |z|2 )α |f 0 (z)| log ≤ (1 − |z| ) |f (z)| log + , j 2 2 1 − |z| 1 − |z| 2 for all z ∈ D. For each j, there exists an rj ∈ (0, 1) such that µ ¶ 2 ε 2 α 0 (1 − |z| ) |fj (z)| log < 1 − |z|2 2 whenever rj < |z| < 1. Setting r = max{r1 , ..., rn }, we have µ ¶ 2 ε (1 − |z|2 )α |f 0 (z)| log < 1 − |z|2 2 whenever r < |z| < 1 and f ∈ η. This proves that (5) holds. α Now suppose that η ⊂ Blog,0 is closed, bounded and satisfies (5). Then η is a normal family. If (fn ) is a sequence in η, by passing to a subsequence (which we do not relabel) we may assume that fn → f α . Let ε > 0 be given. uniformly on compact subsets of D. We are done once we show that fn → f in Blog,0 ¶ µ 2 ≤ 2ε , for all r < |z| < 1 and all By (5) there exists an r ∈ (0, 1) such that (1 − |z|2 )α |g 0 (z)| log 1−|z| 2 0 0 g ∈ η. Since fn0 → f 0 uniformly on ¶ compact subsets of D, it follows that fn → ¶ on D, and thus µ f pointwise µ 2 2 ≤ 2ε , for all r < |z| < 1. Hence (1 − |z|2 )α log 1−|z| |fn0 (z) − f 0 (z)| ≤ also (1 − |z|2 )α |f 0 (z)| log 1−|z| 2 2

¯ there exists an IN such that |f 0 (z) − f 0 (z)| ≤ ε for ε, for all r < |z| < 1. Since fn0 → f 0 uniformly on rD, n all |z| ≤ r and n ≥ N. It follows that ¶ µ 2 2 α |fn0 (z) − f 0 (z)| ≤ ε (1 − |z| ) log 1 − |z|2 1000

AHMED, OMRAN: WEIGHTED SUPERPOSITION OPERATORS

6 α . Since η is closed, it follows that f ∈ η. This proves for all z ∈ D and all n ≥ N. Thus fn → f in Blog that the set η is compact.

3

Superposition operators on Zygmund space

Now we are ready to state and prove the main results in this section. α Theorem 3.1 Let 0 < α < ∞, u, f ∈ H(D) and φ be an analytic self-map of D. Then Su,φ : Z → Blog is bounded if and only if µ ¶µ ¶ 2 1 L := sup (1 − |z|2 )α |u(z)| log log < ∞. (6) 1 − |z|2 1 − |f (z)|2 z∈D

Proof: Suppose that (6) holds. Then for arbitrary z ∈ D and f ∈ Z, we have µ ¶ µ ¶ ¯¡ ¢0 ¯ 2 2 2 α 0 (1 − |z|2 )α ¯ Su,φ f (z)¯ log = (1 − |z| ) |u(z)||φ (f (z))| log 1 − |z|2 1 − |z|2 µ ¶µ ¶ 2 1 log . ≤ CkφkZ (1 − |z|2 )α |u(z)| log 1 − |z|2 1 − |f (z)|2 α From this, (6) and since Su,φ f (0) = 0, it follows that Su,φ : Z → Blog is bounded. α Conversely, assume that Su,φ : Z → Blog is bounded. Let

·µ h(z) = (z − 1) 1 + log

1 1−z

¶2

¸ +1

and put φa (z) = for any a ∈ D such that

√1 2

µ ¶−1 1 h(¯ az) log a ¯ 1 − |a|2

(7)

< |a| < 1. Then we have φ0a (z) =

and φ00a (z) =

µ log

1 1−a ¯z

¶2 µ log

1 1 − |a|2

¶−1

µ ¶2 µ ¶−1 1 1 2¯ a log log 1−a ¯z 1−a ¯z 1 − |a|2

which implies that φ00a (z) = for

√1 2

< |a| < 1 and

sup 1 √ 1 − |f (zk )|, k ∈ N. 2 1003

AHMED, OMRAN: WEIGHTED SUPERPOSITION OPERATORS

9 1 . Hence, Then, for every nonnegative integer s there is at most one zk such that 1 − 21s ≤ f (zk ) < 1 − 2s+1 iθ there is m0 ∈ N such that for any Carleson window S = {re : 0 < 1 − r < l(S), |θ − θ0 | < l(S)} and s ∈ N, there are at most m0 elements in {f (zk ) ∈ S : 2−(s+1) l(S) < 1 − |f (zk )| < 2−s l(S)}. Therefore, (f (zk ))k∈N is an interpolating sequence for Blog . Now, suppose that φ ∈ Blog such that Z f (z0 ) 1 φ(z) = log dξ, k ∈ N. 1 − |ξ|2 0

Then from the definition of weighted logarithmic Bloch functions and Zygmund functions, we see that φ ∈ Z. Then, we obtain µ ¶ µ ¶ 2 2 2 α 0 2 α 0 (1 − |zk | ) |(Su,φ ) (zk )| log = (1 − |zk | ) |u(zk )||φ (f (zk ))| log 1 − |zk |2 1 − |zk |2 ¶µ ¶ µ 1 2 = (1 − |zk |2 )α |u(zk )| log log 1 − |zk |2 1 − |f (zk )|2 ≥ ε0 > 0. α Since lim |f (zk )| = 1 implies that lim |zk | = 1, from the above inequality we obtain that Su,φ 6∈ Blog,0 , k→∞

which is a contradiction.

k→∞

In the next result, we consider the following operator: 0 Su,φ (f ) = u0 Sϕ f = u0 (φ ◦ f ), f ∈ H(D) 0 α α Theorem 3.4 Let 0 < α < ∞, u, f ∈ H(D) and φ be an analytic self-map of D. Then Su,φ : Blog → Blog is a compact operator if and only if 0 α kSu,φ ϕa kBlog →0

as

|a| → 1− .

0 α α Proof: First, we suppose that Su,φ : Blog → Blog is a compact operator. Then, we have {ϕa (z) : a ∈ D} α is a bounded set in Blog and ϕa − a → 0 uniformly on compact sets as |a| → 1− . Thus by Lemma 2.4, α lim kSu,φ ϕa kBlog = 0.

(15)

|a|→1−

α Conversely, suppose that (15) holds and let (φn ) be a bounded sequence in Blog such that φn → 0 α uniformly on compact sets, as n → ∞. We will show that limn→0 kSu,φ fn kBlog = 0. Let λ > 0 be given 0 0 α α < λ. < λ. Then, we have kSu,φ ϕf (z0 ) kBlog and fix 0 < δ < 1 such that if |a| > δ, then kSu,φ ϕa kBlog Hence, for any n ∈ N and z0 ∈ D such that |f (z0 )| > δ, we have µ ¶ 2 0 α kSu,φ fn kBlog = sup |φ0 (fn (z0 ))| |u0 (z0 )|(1 − |z0 |2 )α log 1 − |z0 |2 z∈D µ ¶ 2 < ε |u0 (z0 )|(1 − |z0 |2 )α log 1 − |z0 |2 α ≤ ε ||u||Blog ≤ εconst. (16)

Since the set A = {w : |w| ≤ δ} is a compact subset of D and φ0n → 0 uniformly on compact sets and supw∈A |φ0n (w)| → 0 as n → ∞. Therefore we may choose n0 large enough so that |(φ0 (fn ))| < ε, for any n > n0 and any z ∈ D such that |f (z)| ≤ δ. Then, for n ≥ n0 , we have 0 α kSu,φ fn kBlog < ε const.

Thus (16) and (17) yield Thus (18) yield that compact operator.

0 kSu,φ

(17)

0 α kSu,φ fn kBlog < ε const. ∀ n ≥ n0 . α fn kBlog → 0 as n → ∞. Hence by Lemma 2.2,

(18) 0 kSu,φ

α fn kBlog →

α Blog

is a

Acknowledgements. The authors are grateful to Taif University Saudi Arabia for its financial support of this research under number 1184/432/1. 1004

AHMED, OMRAN: WEIGHTED SUPERPOSITION OPERATORS

10

References ´ M. A. M´arquez and D. Vukoti´c, Superposition operators between the Bloch space and [1] V. Alvarez, Bergman spaces, Ark. Mat. 42(2004), 205-216. [2] J. Appell and P.P. Zabrejko, Nonlinear Superposition Operators, Cambridge University Press, Cambridge, 1990. [3] K. R. M. Attele, Toeplitz and Hankel operators on Bergman one space, Hokkaido Math. J. 21(2)(1992), 279-293. [4] R. Aulaskari and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, Complex Analysis and its Applications (Eds Y. Chung-Chun et al.), Pitman Research Notes in Mathematics 305, Longman (1994), 136-146. [5] S. M. Buckley, J. L. Fern´andez and D. Vukoti´c, Superposition operators on Dirichlet type spaces, in: Papers on Analysis: A Volume Dedicated to Olli Martio on the Occasion of his 60th Birthday, Rep. Univ. Jyvaskyl¨a Dept. Math. Stat. 83, pp. 4161, University of Jyv¨askyl¨a, Jyv¨askyl¨a, 2001. [6] S. M. Buckley and D. Vukoti´c, Univalent interpolation in Besov spaces and superposition into Bergman spaces, Potential Anal 29(2008) 1-16. [7] G. A. C´amera, Nonlinear superposition on spaces of analytic functions, in: Harmonic Analysis and Operator Theory (Caracas, 1994), Contemp. Math. 189, pp. 103-116, Amer. Math. Soc., Providence, RI, 1995. [8] G. A. C´amera and J. Gimenez, The nonlinear superposition operator acting on Bergman spaces, Compos. Math. 93 (1994), 23-35. [9] J. B. Conway, Functions of one complex variable, Second Edition, Springer-Verlag, New York, 1978. [10] J. J. Donaire, D. Girela and D. Vukoti´c, On univalent functions in some M¨obius invariant spaces, J. Reine Angew. Math. 553 (2002), 43-72. [11] P. Duren, Theory of H P spaces, Academic Press, New York, 1973. [12] D. Girela and M. A. M´ arquez, Superposition operators between Qp spaces and Hardy spaces, J. Math. Anal. Appl. 364(2)(2010), 463-472. [13] W. Rudin, Real and Complex Analysis, New York, 1987. [14] M. Tjani, Compact composition operators on Besov spaces, Trans. Amer. Math. Soc. 355(11)(2003), 4683-4698. [15] C. Xiong, Superposition operators between Qp spaces and Bloch-type spaces, Complex Var. Theory Appl. 50(12)(2005), 935-938. [16] W. Xu, Superposition operators on Bloch-type spaces, Comput. Methods Funct. Theory 7(2)(2007), 501-507.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1006-1014, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

FUZZY FIXED POINTS OF CONTRACTIVE FUZZY MAPPINGS AKBAR AZAM1 AND MUHAMMAD ARSHAD2 Abstract. We prove the existence of fuzzy …xed points of a general class of fuzzy mappings satisfying a contractive condition on a metric space with the Hausdor¤ metric on the family of fuzzy sets and apply it to obtain fuzzy …xed points of fuzzy locally contractive mappings.

1. Introduction and Preliminaries Heilpern [16] …rst introduced the concept of fuzzy mappings and established a …xed point theorem for fuzzy contraction mappings. Afterwards many researcher (e.g.,see [1, 2, 3, 4, 9, 10, 21, 22, 23, 24] and reference therein) extended the result of Heilpern and studied …xed point theorems for fuzzy generalized contractive mappings. Recently in ([1, 2]), the authors obtained Heilpern …xed points of fuzzy contractive and fuzzy locally contractive mappings on a compact metric space with the d1 -metric for fuzzy sets. In [4] the authors studied …xed point theorems of a wider class of fuzzy mappings and obtained some d1 -metric …xed point results of the literature as corollaries. In the present paper we prove theorems concerning common …xed points of the same wider class [4] of fuzzy contractive and fuzzy locally contractive mappings and obtain some d1 -metric …xed point results of [2] as corollaries. Our results also generalize/fuzzify several other known results (e.g., see [7, 13, 16, 18, 25]). Let (X; d) be a metric space and CB(X) = fA : A is nonempty closed and bounded subset of Xg, C(X) = fA : A is nonempty compact subset of Xg: For A; B 2 CB(X) and " > 0 the sets N d ("; A) and d EA;B are de…ned as follows: N d ("; A) = fx 2 X : d(x; a) < " for d some a 2 Ag; EA;B = f" 2 A : N d ("; B); B : N d ("; A)g; where d(x; A) = inffd(x; y) : y 2 Ag. The Hausdor¤ metric dH on CB(X) 2000 Mathematics Subject Classi…cation. 46S40; 47H10; 54H25. Key words and phrases. Fuzzy …xed point; contractive type mappings; fuzzy set; fuzzy mapping. 1

1006

2

A. AZAM AND M. ARSHAD

d induced by d is de…ned as dH (A; B) = inf EA;B : For x; y 2 X, an "chain from x to y is a …nite set of points x0 ; x1 ; x2 ; ; xn such that x = x0 ; xn = y and d(xj ; xj+1 ) " for all j = 0; 1; 2; ; n 1: A fuzzy set in X is a function with domain X and values in [0; 1]. If A is a fuzzy set and x 2 X; then the function values A(x) is called the membership grade of x in A. The -level set of A, denoted by A, and is de…ned by A = fx : A(x) g if 2 (0; 1]; 0

A = fx : A(x) 0g: Here B denotes the closure of the set B: A fuzzy set A in a metric linear space X is said to be an approximate quantity if and only if A is compact and convex in X for each 2 [0; 1] and supA(x) = 1: x2X

The family of all approximate quantities in a metric linear space X is denoted by W (X). We denote the fuzzy set fxg by fxg unless and until it is stated, where A is the characteristic function of the crisp set A. Let F (X) be the collection of all fuzzy sets in a metric space X and E(X) = fA 2 F (X)g : A 2 CB(X); 8 2 [0; 1]g: EC (X) = fA 2 F (X)g : A 2 C(X); 8 2 [0; 1]g: For A; B 2 F (X) , A B means A(x) B(x) for each x 2 X: If there exists an 2 [0; 1] such that A; B 2 CB(X) then de…ne P (A; B) =

inf

x2 A; y2 B

d(x; y);

D (A; B) = dH ( A; B): If A; B 2 CB(X) for each 2 [0; 1] then de…ne P (A; B) = supP (A; B); D(A; B) = supD (A; B): If d is another metric on X then P (A; B) =

Now de…ne d1 dH ) as

inf

x2 A; y2 B

d (x; y);

D (A; B) = dH ( A; B): : E(X) E(X) ! R (induced by the Hausdor¤ metric

d1 (A; B) = dH ( A; B): We note that d1 is a metric on E(X ) and the completeness of (X; d) implies that (CB(X); dH ) and (E(X); d1 ) are complete. Moreover (X; d) 7! (CB(X); dH ) 7! (E(X); d1 );

are isometrics embeddings by means x ! fxg (crisp set) and A ! A respectively. Let X be an arbitrary set, Y be a metric space. A mapping T is called fuzzy mapping if T is a mapping from X into

1007

FUZZY FIXED POINTS

3

F (Y ). A fuzzy mapping T is a fuzzy subset on X Y with membership function T (x)(y). The function T (x)(y) is the grade of membership of y in T (x). A point x 2 X is said to be fuzzy …xed point of a fuzzy mapping T if fxg T (x): Lemma 1. [25] Let (X; d) be a metric space and A; B 2 CB(X) with dH (A; B) < "; then for each a 2 A there exists an element b 2 B such that d(a; b) < ": Lemma 2. [25] Let (X; d) be a metric space and A; B 2 CB(X);then for each a 2 A; d(a; B) d(A; B): In section 2 we extend Edelstein …xed point theorem to fuzzy mappings. Section 3 deals with the study of fuzzy …xed point theorems for locally contractive mappings. We extend the concept of locally contractive mappings of Edelstein [12, 13] (see also [1, 3, 6, 7, 18, 20, 25]) to locally contractive fuzzy mappings and obtained a fuzzy …xed points for such mappings. 2. FIXED POINTS OF FUZZY CONTRACTIVE MAPS One very pretty and signi…cant …xed point theorem, originally due to Edelstein [13] is that if (X; d) is a compact metric space and T : X ! X is a contractive mapping ( i.e d(T x; T y) < d(x; y) for each x; y 2 X): Then there exists a unique …xed point of T. Edelstein …xed point theorem was further studied/extended by Da¤er and Kaneko[11], Hu and Rosen [18]. Beg [5] proved random analogue of this result and obtained random …xed points of contractive random mappings. Recently Grabiec [15] and Mihet [24] extended this result to fuzzy metric spaces. In the following theorem, we extend the above result to a general class of fuzzy mappings. Theorem 1. Let (X; d) is a compact metric space and T : X ! X is a fuzzy mapping such that for each x 2 X there exists (x) 2 (0; 1] such that (x) T (x) is nonempty, compact and x,y2 X; x 6= y; dH (

(x)

T (x);

(y)

T (y) < d(x; y):

Then there exists x 2 X such that x 2

(z )

T (x ):

Proof. For each x 2 X; pick (x) 2 (0; 1] such that (x) T (x) is nonempty, compact and de…ne a real valued function g : X ! R by g(x) = d(x; (x) T (x)): It follows that, g(x) = d(x;

(x)

T (x))

d(x; y) + d(y;

d(x; y) + d(y; (y)

T (y)) + dH (

1008

(x) (x)

T (x))

T (x);

(y)

T (y)):

4

A. AZAM AND M. ARSHAD

That is, g(x)

g(y)

d(x; y) + dH (

(x)

T (x);

(y)

T (y)):

By symmetry, we obtained, jg(x)

g(y)j

d(x; y) + dH (

(x)

T (x);

(y)

T (y)):

It follows that g(x) = d(x; (x) T (x)) is continuous. By compactness, this function attains a minimum, say at x : Now, by compactness of (x ) T (x ); we can choose x1 2 (x ) T (x ) such that, (x )

d(x ; x1 ) = d(x ; Then x 2

(z )

T (x )) = g(x ):

T (x ); otherwise,

g(x1 ) = d(x1 ;

(x1 )

T (x1 ))

< d(x ; x1 ) = d(x ;

dH ( (x )

(x )

T (x );

(x1 )

T (x1 ))

T (x )) = g(x ):

Which is a contradiction to the minimality of g(x) at x : It completes the proof. Example 1. Let X = [0; 1); d(x; y) = jx A : (0; 1) ! F (X) be de…ned as follows:

yj; whenever x; y 2 X and

8 1 if 0 t < x8 > > < 1 x if 8 t x4 2 A(x)(t) = 1 if x < t x > > : 03 if x4 < t < 1

Now, de…ne T : X ! F (X) as follows: T (x) =

f0g if x = 0 A(x) if x 6= 0 1

Then, if x 6= 0; 1 T (x) = [0; x8 ]; which is not compact and 2 T (x) = ft 2 X : T (x)(t) = 21 g = [0; x3 ]: Thus all conditions of Theorem 1 are 1 satis…ed to obtain 0 2 2 T (0) while previously known result [4, Theorem 2.1] is not applicable to obtain it. Corollary 1. Let (X; d) is a compact metric space and T : X ! EC (X ) is a fuzzy mapping such that for each x; y 2 X; x 6= y d(T (x); T (y)) < d(x; y): Then there exists x 2 X such that x 2 T (x ):

1009

FUZZY FIXED POINTS

5

Proof. Let x2 X, by hypothesis 1 T (x) is nonempty compact subset of X for each x. Thus dH (1 T (x);1 T (y))

D(T (x)T (y)) d1 (T (x)T (y)) < d(x; y):

Apply theorem 1 to obtain x 2 X such that x 2 1 T (x ); hencefxg T (x ): 3. FUZZY LOCALLY CONTRACTIVE MAPS In this section we established fuzzy …xed point theorem for locally contractive fuzzy mappings. The following lemma is recorded from [27]. Lemma 3. [27] Let (X; d) is a compact conected metric space. Then for each " > 0 and x; y 2 X there exists an -chain from x to y and n 1 X the mapping d : X X ! R de…ned by d (x; y) = inff d(xj ; xj+1 ) : j=1

x0 ; x1 ; x2 ; ; xn is an - chain from x to yg is a metric on X equivalent to d. Furthermore, for x; y 2 X and " > 0 there exists an "-chain n 1 X x = x0 ; x1 ; x2 ; ; xn = y such that d (x; y) = d(xj ; xj+1 ): j=1

Theorem 2. Let (X; d) is a compact conected metric space and T : X ! F (X) is a fuzzy mapping such that the following conditions are satis…ed: (i) For each x 2 X there exists

(x) 2 (0; 1] such that

(x)

T (x) is

nonempty, compact and (ii) each x of X belongs to an open set U such that for each y; z 2 U; y 6= z dH ( (y) T (y); (z) T (z)) < d(y; z): Then there is a new metric d for X equivalent to d such that for each x; y 2 X dH ( (x) T (x); (y) T (y)) < d(x; y) and there exists x 2 X such that x 2

(z )

T (x ):

Proof. First, by Lemma 3 for each " > 0 and each pair of points p; q 2 X there exists an "-chain p = x0 ; x1 ; x2 ; ; xn = q from p to q. Next use compactness of X to …nd > 0 such that if x 6= y and d(x; y) < , then dH ( (x) T (x); (y) T (y)) < d(x; y):

1010

6

A. AZAM AND M. ARSHAD

Now let d = d 2 that is for p; q 2 X d(p; q) = inff

n 1 X

d(xj ; xj+1 ) : x0 ; x1 ; x2 ;

; xn is an

j=0

2

chain from p to qg:

By lemma 3, d is a metric on X equivalent to d and there exists an chain p = x0 ; x1 ; x2 ; ; xn = q from p to q such that 2 d (p; q) =

n 1 X

d(xj ; xj+1 ):

j=0

Now, d(xj ; xj+1 ) dH (

2

(xj )

implies that


0:

Assume that Mj = d(xj ; xj+1 ) dH ( (xj ) T (xj ); (xj+1 ) T (xj+1 )) for j = 0; 1; 2; ; n 1: It further implies that Mj > 0 and (2) Mj dH ( (xj ) T (xj ); (xj+1 ) T (xj+1 )) < d(xj ; xj+1 ) for j = 0; 1; 2; ; n 1: 2 Consider an arbitrary element y0 2 (x0 ) T (x0 ):In the view of inequality (2) along with Lemma 2 we may choose y1 2 (x1 ) T (x1 ) such that d(y0 ; y1 ) < d(x0 ; x1 ) M20 : Similarly, we may choose y2 2 (x2 ) T (x2 ) M1 such that d(y1 ; y2 ) < d(x1 ; x2 ) : Continuing in this fashion we 2 produce a set of points y0 ; y1 ; y2 ; ; yn where yj 2 (xj ) T (xj ) such Mj 1 that d(yj 1 ; yj ) < d(xj 1 ; xj ) for j = 0; 1; 2; ; n 1: Obviously 2 y0 ; y1 ; y2 ; ; yn is an 2 chain formy0 to yn : Thus n 1 X d (y0 ; yn ) = inff d(xj ; xj+1 ) : x0 ; x1 ; x2 ; j=0

n 1 X

d(yj ; yj+1 )

j=0

n 1 X (d(xj ; xj+1 ) j=0

Mj ): 2

1011

; xn is an

2

chain from y0 to yn g:

FUZZY FIXED POINTS

Since d (p; q) =

n 1 X

7

d(xj ; xj+1 ):

j=0

Therefore, d (y0 ; yn )

d (p; q)

n 1 X M ( 2j ): Assume that k = d (p; q) j=0

n 1 X M ( 2j ); then k > 0 and y0 2 N d (k;

(xn )

T (xn )): Hence

j=0

(x0 )

(3)

N d (k;

T (x0 )

(xn )

T (xn )):

Now consider an arbitrary element zn 2 (xn ) T (xn ): Again in the view of inequality (2) along with Lemma 2, we may choose zn 1 2 (xn 1 ) T (xn 1 ) such that d(zn 1 ; zn ) d(x0 ; x1 ) ( Mn2 1 ): Then by the same procedure we obtain an 2 chain z0 ; z1 ; z2 ; ; zn from z0 to zn where, d (z0 ; zn ) Thus zn 2 N d (k; (4)

(x0 )

d (p; q)

n 1 X Mj ( ) = k: 2 j=0

T (x0 )): Hence

(xn )

T (xn )

N d (k;

(x0 )

T (x0 )):

In the view of inequalities (3) and (4), it follows that k 2 E d(x0 ) T (x0 ); Thus dH ( (x0 ) T (x0 ); (xn ) T (xn )) < k: It further implies that dH (

(p)

T (p);

(q)

T (q)) < d (p; q)

Hence for all x; y; dH (

(p)

T (x);

(q)

(xn ) T (x ) n

n 1 X Mj ) < d (p; q): ( 2 j=0

T (y)) < d (x; y):

Now by lemma 3 there exists x 2 X such that x 2

(z )

T (x ):

Corollary 2. Let (X; d) is a compact conected metric space and T : X ! EC (X) is a fuzzy mapping such that the following condition is satis…ed: each x 2 X belongs to an open set U such that for each y; z 2 U; y 6= z d1 (T (y); T (z)) < d(y; z): Then there exists x 2 X such that fx g T x : Here by providing following theorem, we achieve set-valued version of Edelstein Theorems.

1012

:

8

A. AZAM AND M. ARSHAD

Theorem 3. Let (X; d) is a compact metric space and S : X ! C(X) be a set valued mapping such that either for each x; y 2 X; x 6= y dH (S(x); S(y)) < d(x; y):

Then there exists x 2 X such that x 2 S(x ): Proof. Consider a fuzzy mapping T : X ! F (X) de…ned by as follows: T (x)(t) =

9 10 1 10

9

t 2 S(x) t 62 S(x):

Then 10 T (x) = S(x) hence by Theorem 1 and Theorem 2 there exists 9 x 2 X such that x 2 10 T (x ) = S(x ): References [1] A. Azam, I. Beg, Common …xed points of fuzzy maps, Math. Comp. Modelling 49 (2009) 1331-1336. [2] A. Azam, M. Arshad and I. Beg, Fixed points of fuzzy contractive and fuzzy locally contractive maps, Chaos, Solitons & Fractals 42 (2009), 2836-2841. [3] A. Azam, M. Arshad, A note on "Fixed point theorems for fuzzy mappings" by P. Vijayaraju and M. Marudai, Fuzzy Sets and Systems 161 (2010), 1145-1149. [4] A. Azam, M. Arshad and P. Vetro, On a pair of fuzzy -c contractive mappings, Math. Comp. Modelling 52 (2010), 207-214. [5] I. Beg, Random Edelstein theorem, Bull. Greek Math. Soc. 45 (2001), 31-41. [6] I. Beg. and A. Azam, Fixed points of multivalued locally contractive mappings, Boll. Un. Mat. Ital. (4A) 7 (1990), 227-233. [7] I. Beg and A. Azam, Fixed points of asymptotically regular multivalued mappings, J. Austral. Math. Soc. (Series A) 53 (1992), 313-226. [8] I. Beg and N. Shahzad, Common random …xed points of random multivalued operators on metric spaces, Boll. U. M. I. 7 (9A) (1995), 493-503. [9] R. K. Bose and D. Sahani, Fuzzy mappings and …xed point theorems, Fuzzy Sets and Systems 21 (1987), 53-58. [10] A. Chitra, A note on the …xed points of fuzzy maps on partially ordered topological spaces, Fuzzy Sets and Systems 19 (1986), 305-308. [11] P. Z. Da¤er and H . Kaneko, Multivalued f - contractive mappings, Boll. U. M. I. 8-A, (1994), 233-241. [12] M. Edelstein, An extension of Banach’s contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7-10. [13] M. Edelstein, On …xed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79. [14] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and System 64 (1994) 395-399. [15] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988), 385-389. [16] S. Heilpern, Fuzzy mappings and …xed point theorems, J. Math. Anal. Appl. 83 (1981), 566-569. [17] R.D. Holmes, On …xed and periodic points under certain set of mappings, Canad. Math. Bull. 12 (1969), 813-822.

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FUZZY FIXED POINTS

9

[18] T. Hu and H. Rosen, Locally contractive and expansive mappings, Proc. Amer. Math. Soc. 86 (1982), 656-662. [19] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975), 336-344. [20] S. Leader, A …xed point principle for locally expansive multifunctions, Fund. Math.106 (1980), 99-104. [21] B. S. Lee, Fixed points for nonexpansive fuzzy mappings in locally convex spaces, Fuzzy Sets and System 76 (1995), 247-251. [22] B. S. Lee and S. J. Cho, A …xed point theorem for contractive type fuzzy mappings, Fuzzy Sets and Systems 61 (1994), 309-312. [23] B. S. Lee, G. M. Lee, S. J. Cho and D. S. Kim, Generalized common …xed point theorems for a sequence of fuzzy mappings, Internat. J. Math. & Math. Sci. 17 (3) (1994) 437-440. [24] D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems 158 (2007), 915-921. [25] S. B. Nadler, Multivalued contraction mappings, Paci…c J. Math. 30 (1969), 475- 488. [26] D. Qiu, L. Shu and J. Guan, Common …xed point theorems for fuzzy mappings under contraction condition, Chaos, Solitons & Fractals 41 (2009), 360-367. [27] C. Waters, A …xed point theorem for locally nonexpansive mappings in normed space, Proc. Amer. Math. Soc. 97 (1986), 695-699. 1

Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad, Pakistan, 2 Department of Mathematics, Faculty of Basic and Applied Sciences, International Islamic University, H-10, Islamabad, 44000, Pakistan. E-mail address: [email protected], [email protected]

1014

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1015-1025, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

On explicit solutions to a polynomial equation and its applications to constructing wavelets∗

D. H. Yuan

1,2

, Y. Feng3 , Y. F. Shen2 , S. Z. Yang2,† Abstract

In this paper, we address the problem of finding appropriate polynomial solution for a polynomial equation, which is corresponding to construct an orthonormalscaling filter mM (ξ) with the dilation factor 4 and proposed in [J.Math.Anal.Appl. 317(1):364-379]. By constructing methods, we present explicitly solutions for this system. As application, we obtain orthonormal scaling function with dilation factor 4. In particular, we give some examples of constructing real and complex scaling function. Keywords: polynomial equation, orthonormal wavelet bases, scaling function, binomial theorem

1

Introduction

The usual method of constructing a compactly supported orthonormal wavelet bases of L2 (R), with dilation factor 4, is the construction of a mother scaling function Φ(·). This scaling function is an L2 -solution of the following refinement equation: Φ(X) =

N X

αn Φ(4X − n), X ∈ R, {αn }n ⊂ R.

(1)

n=0

Note that the orthonormality of the translates of Φ(·) implies that the trigonometric polynomial N 1X m0 (ξ) = αn einξ 4 n=0 ∗ This work was supported by the National Natural Science Foundation of China (Grant No.11071152), the Natural Science Foundation of Guangdong Province (Grant Nos.10151503101000025 and S2011010004511) 1 Dept. of Math.,Hanshan Norm. Univ., Chaozhou, Guangdong, 521041, China. 2 Dept. of Math.,Shantou Univ., Shantou, Guangdong, 521041, China. 3 School of Computer and Inform. Tech., Xinyang Norm. Univ., Henan, 464000, China. † Corresponding Author. Email: [email protected]

1

1015

YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

satisfies the orthogonality condition |m0 (ξ)|2 + |m0 (ξ + π/2)|2 + |m0 (ξ + π)|2 + |m0 (ξ + 3π/2)|2 = 1, ∀ξ ∈ [0, 2π]. (2) It is well known that in order Φ(·) ∈ L2 (R), it is necessary that m0 (ξ) =



1 + eiξ 2

M 

1 + e2iξ 2

M

L(eiξ )

(3)

for some positive integers M , where L(eiξ ) is some the trigonometric polynomial. By using (3), one concludes that |mM (ξ)|2 : = |m0 (ξ)|2 = cos2M (ξ) (1 + cos(ξ))M QM (cos(ξ)), QM (·) = |L(ei· )|2 /2M

(4)

Let X = cos(ξ), then by substituting (4) into (2), one concludes that the polynomial QM has to satisfy the following equation:

[

]

X 2M (1 + X)M QM (X) + (1 − X)M QM (−X) + (1 − X 2 )M i h p p p p × (1 + 1 − X 2 )M QM ( 1 − X 2 ) + (1 − 1 − X 2 )M QM (− 1 − X 2 ) = 1, ∀X ∈ [−1, 1].

(5)

Note that function (1 + X)M QM (X) + (1 − X)M QM (−X) is even with respect to X, we can denote it by the symbol HM (X 2 ). Therefore, the equation (5) can be rewrite as X 2M HM (X 2 ) + (1 − X 2 )M HM (1 − X 2 ) = 1, ∀X ∈ [−1, 1].

(6)

X M HM (X) + (1 − X)M HM (1 − X) = 1, ∀X ∈ [0, 1].

(7)

or

By Bezout lemma, Equation (7) has a unique solution of degree M − 1 HM (X), HM (X) := PM (X) :=

M−1 X k=0

2M − 1 k



X k (1 − X)M−1−k .

(8)

The problem of solving (2) is converted to the problem of finding an appropriate polynomial QM satisfying (1 + X)M QM (X) + (1 − X)M QM (−X) = PM (X 2 ), X ∈ [−1, 1], QM (X) > 0, X ∈ [−1, 1] where PM (X) is the solution of degree M − 1 of Equation X M PM (X) + (1 − X)M PM (1 − X) = 1, ∀X ∈ [0, 1]. 2

1016

(9) (10)

YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

Karoui [1] proposed that the above system Eqs (9) and (10) can be solved numerically by converting it to a system of quadratic equations. However, he pointed out that this methods are feasible for small enough M but large M . Moreover, the numerical method can not provide us with an explicit solution that depends on M . In this paper, we address the problem of finding appropriate polynomial solution for the system Eqs (9)and (10). We present some explicit solutions for this system by constructing methods in section 2. As application, we obtain orthonormal scaling function with dilation factor 4 in section 3. In particular, some real or complex scaling functions are given in section 4.

2

Main results

In this section, we solve the system Eqs (9), (10) and provide some explicitly solutions for this system. To this purpose, we need the following lemma, which is Theorem 2.4 in [2]. Lemma 1 For given  nonnegative integers N and l with l < N , let PN,l (X) : Pl N +l = k=0 X k (1 − X)l−k . Then k   P N −1+k (I) PN,l (X) = lk=0 X k, k (II) PN,l (X) > 0 for all x ∈ R if and only if l is an even number. Remark 1 Note that PM (X) defined in (8) is PM,M−1 (X) defined in Lemma 1. Therefore, PM (X) > 0 if and only if M is an even number. Firstly, we give a solution of the system Eqs (9), (10) of degree M −1. Define f (X) =



1 (1 − X)(1 − 2X)2

M

(11)

and let TM (X) be the (M − 1)th-degree Taylor polynomial of the function f at X = 0. We have the following theorem. Theorem 1 For any integer M > 1, let f (X) be the function defined in (11) and TM (X) the (M − 1)th-degree Taylor polynomial of f at X = 0. Then
be M TM 1−X /2 is the unique solution of the system Eqs (9), (10) of degree M −1. 2 Proof. By the definition of TM (X) in (11), it is evident that TM (0) = 1 and all the coefficients of TM (X) are nonnegative. Therefore, it is straightforward to see that TM (X) > 1 for all X > 0. Note that     1−X 1+X (1 + X)M TM /2M + (1 − X)M TM /2M 2 2

3

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YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

is an even function with respect to X on [−1, 1] and denote it by
P M (X 2 ). It is easy to deduce that deg(P M (·)) 6 M − 1. To prove TM 1−X /2M is the 2 unique solution of the system Eqs (9), (10) of degree M − 1, we have to show P M (X 2 ) > 0 for all X ∈ [−1, 1] and P M (X 2 ) = PM (X 2 ). In fact, since 1+X 1−X > 0, > 0, ∀X ∈ [−1, 1] 2 2 and TM (X) > 1 for all X > 0, one obtains P M (X 2 ) > 0 for all X ∈ [−1, 1]. Now we are ready to prove P M (X 2 ) = PM (X 2 ). Denote A(ξ) := cos2M (ξ/2) cos2M (ξ)TM (sin2 (ξ/2)). Han and Ji [3] proved that A(ξ) + A(ξ + π/2) + A(ξ + π) + A(ξ + 3π/2) = 1.

(12)

By the definition A and P M , we have B(ξ) : = A(ξ) + A(ξ + π) = cos2M (ξ) cos2M (ξ/2)TM (sin2 (ξ/2))
+ sin2M (ξ/2)TM (cos2 (ξ/2)) = cos2M (ξ)P M (1 − cos2M (ξ)) = X 2M P M (X 2 )

A(ξ + π/2) + A(ξ + 3π/2) = B(ξ + π/2) = cos2M (ξ + π/2)P M (1 − cos2M (ξ + π/2)) = (1 − X 2 )M P M (1 − X 2 ) with X = cos(ξ). Now by Eq. (12) and the above two identities, we conclude that X 2M P M (1 − X 2 ) + (X 2 )M P M (1 − X 2 ) = 1, ∀X ∈ [0, 1].

(13)

Taking Y = X 2 in (13), we get Y M P M (Y ) + (1 − Y )M P M (1 − Y ) = 1, ∀Y ∈ [0, 1]. Recall deg(P M (·)) 6 M − 1, the above relation implies that P M (Y ) must be the polynomial PM (Y ). Thus, we prove P M (X 2 ) = PM (X 2 ). Since
there is a unique solution of (9) with degree M − 1, we claim that /2M is the unique solution of the system Eqs (9) and (10) of degree TM 1−X 2 M − 1.  The following Theorem 2 and 3 provide solutions QM of the system Eqs (9), (10) with deg(QM ) > M for fixed M . 4

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YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

Theorem 2 For any odd integer M > 1, denote Q0M (X) =

 l−M  2M−1−l 2M−1  1 X 1+X 1−X 2M − 1 . l 2M 2 2 l=M

Then, QM (X) = Q0M (X)PM (X 2 ).

(14)

is a solution of the system Eqs (9), (10) and the degree of QM (X) is 3M − 3. Proof. Note that 2M−1 X 

l  2M−1−l 1+X 1−X 1= = 2 2 l=0       2M−1 l−M 2M−1−l X 1+X 1−X 2M − 1 M 1 = (1 + X) l 2M 2 2 l=M      M−1−l M−1 l X 1+X 1−X 2M − 1 M 1 + (1 − X) (15) l 2M 2 2 

1+X 1−X + 2 2

2M−1

2M − 1 l



l=0

and



2M − 1 l



=



2M − 1 2M − 1 − l



,

we have Q0M (−X)

 l  M−1−l M−1  1 X 1+X 1−X 2M − 1 = M . l 2 2 2 l=0

Thus, we conclude from (15) that M

(1 + X)

Q0M (X) + (1 − X)

M

Q0M (−X) = 1.

Multiplying the above equation by PM (X 2 ), we obtain M

(1 + X)

M

Q0M (X)PM (X 2 ) + (1 − X)

Q0M (−X)PM (X 2 ) = PM (X 2 ).

Therefore QM (−X) = Q0M (−X)PM ((−X)2 ) = Q0M (−X)PM (X 2 ). Thus QM (X) is a solution of Eq (9). Let N = M and l = M − 1. Since M is an odd integer, then from Lemma 1 PN,l (X) = PM (X) > 0 for all X ∈ R. Hence   1+X 2M Q0M (X) = PM > 0, PM (X 2 ) > 0 2 5

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YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

or QM (X) = Q0M (X)PM (X 2 ) > 0 Therefore, QM (X) is a explicit solution for this system Eqs (9), (10). Note that the degree of Q0M (X) is M − 1 and the degree of PM (X 2 ) is 2M − 2, we obtain the desired result.  Theorem 3 For any even integer M > 1, denote Q0M (X)

=

2M+1 X

1 2M+1

l=M+1





2M + 1 l

1+X 2

l−(M+1) 

1−X 2

2M+1−l

.

Then, QM (X) = (1 + X)Q0M (X)PM (X 2 ).

(16)

is a solution of the system Eqs (9), (10) and the degree of QM (X) is equal to 3M − 1. Proof. Note that 1=



M

= (1 + X)

2M+1

2M+1 X 

l 

2M+1−l 1−X 2 l=0  l−(M+1)  2M+1−l 2M+1  1+X X 1+X 1−X 2M + 1 l 2M+1 2 2

1+X 1−X + 2 2

=

2M + 1 l



1+X 2

l=M+1

+ (1 − X)

M

 l  M−l M  1 − X X 2M + 1 1+X 1−X . l 2M+1 2 2

(17)

l=0

Denote Q0M (X)

By



2M+1 X

1+X := M+1 2

2M + 1 l



=

l=M+1



Q0M (−X) =



2M + 1 l

2M + 1 2M + 1 − l





1+X 2

l−(M+1) 

1−X 2

2M+1−l

for l = 0, 1, · · · , M , we obtain

 l  M−l M  1+X 1−X 1 − X X 2M + 1 . l 2M+1 2 2 l=0

Thus, we conclude from (17) that (1 + X)M Q0M (X) + (1 − X)M Q0M (−X) = 1. Multiplying the above equation by PM (X 2 ), we obtain M

(1 + X)

M

Q0M (X)PM (X 2 ) + (1 − X) 6

1020

Q0M (−X)PM (X 2 ) = PM (X 2 ),

.

YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

and QM (−X) = Q0M (−X)PM ((−X)2 ) = Q0M (−X)PM (X 2 ). Thus QM (X) is a solution of Eq (9). Let N = M + 1 and l = M . Since M is a even integer, then from Lemma 1 PN,l (X) = PM+1 (X) > 0 for all X ∈ R. Hence   1 1+X Q0M (X)PM (X 2 ) = M+1 PM+1 PM (X 2 ) > 0, ∀X ∈ R. (18) 2 2 Note that 1 + X > 0 for X ∈ [−1, 1], we obtain QM (X) = (1 + X)Q0M (X)PM (X 2 ) > 0, X ∈ [−1, 1] Therefore, QM (X) is a explicit solution for this system Eqs (9), (10). Note that the degree of Q0M (X) is M and the degree of PM (X 2 ) is 2M − 2, we obtain the desired result. 

3

Applications

To proceed further, we need the following version of Cohen’s condition for wavelet filters with dilation factor 4, see [4]. Note that this condition ensures the orthogonality of the translates for the scaling function and consequently the stability of the associated wavelet basis of L2 (R). Cohen’s condition. Let m0 (·) be a scaling filter with dilation factor 4. Assume that there exists a compact set κ such that (I) κ contains a neighborhood of the orign; (II) |κ| = 2π, and ∀ξ ∈[−π,  π], ∃k ∈ Z, satisfies ξ + 2πk ∈ κ; (III) inf k≥1 inf ξ∈κ |m0

ξ 4k

| > 0.

Theorem 4 For any odd integers M > 1, let QM be the polynomial given by (14). Then any scaling filter mM (ξ) given by |mM (ξ)|2 = cos2M (ξ) (1 + cos(ξ))

M

QM (cos(ξ))

generates orthonormal scaling function with dilation factor 4. Proof. Let M > 1 be an odd integer, from Theorem 2, then QM (X) is continuous and positive for all X ∈ R. Therefore there exists X0 ∈ [−1, 1] such that QM (X0 ) > 0 and QM (X) > QM (X0 ), ∀ X ∈ [−1, 1]. Consequently, the only roots of SM (X) := X 2M (1 + X)M QM (X) inside [−1, 1] are −1, 0 or equivalently, mM (ξ) vanishes only at π, π/2. Let κ = [−π, π], then
2 1 ξ/4k ∈ [−π/4, π/4] and mM ξ/4k > M QM (X0 ) > 0. 2

So that mM (ξ) satisfies Cohen’s condition. Thus we can obtain the desired result.  7

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YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

Theorem 5 For any even integers M > 1, let QM be the polynomial given by (16). Then any scaling filter mM (ξ) given by |mM (ξ)|2 = cos2M (ξ) (1 + cos(ξ))

M

QM (cos(ξ))

generates orthonormal scaling function with dilation factor 4. Proof. Since Q0M (X)PM (X 2 ) is continuous and positive for all X ∈ R from (18), then there exists X0 ∈ [−1, 1] such that QM (X0 ) > 0 and QM (X) > QM (X0 ), ∀ X ∈ [−1, 1]. Consequently, the only roots of SM (X) := X 2M (1 + X)M QM (X) = X 2M (1 + X)M+1 Q0M (X)PM (X 2 ) inside [−1, 1] are −1, 0 or equivalently, mM (ξ) vanishes only at π, π/2. Let κ = [−π, π], then for any even integers M > 1,
2 ξ/4k ∈ [−π/4, π/4] and mM ξ/4k >

1 QM (X0 ) > 0. 2M+1

Therefore, mM (ξ) satisfies Cohen’s condition.

4



Examples

4.1

Construction of real scaling functions

When the scaling filter mM (ξ) is obtained, one can obtain high-pass filters by the algorithm given in [5]. Therefore, we present here only the expression of scaling filter mM (ξ). With the notation Z = eiξ , we give the following explicit expressions of mM (ξ) for M = 3, 4, 5 by using the Riesz Lemma [6]. m3 (ξ) =0.055356 + 0.13423Z + 0.210734Z 2 + 0.303099Z 3 + 0.247787Z 4 + 0.136913Z 5 + 0.0487845Z 6 − 0.0713197Z 7 − 0.064884Z 8 − 0.0172573Z 9 − 0.00801157Z 10 + 0.017604Z 11 + 0.0117441Z 12 − 0.0038828Z 13 − 0.00150388Z 14 + 0.000620209Z 15 m4 (ξ) =0.026537 + 0.0823424Z + 0.155013Z 2 + 0.252279Z 3 + 0.276618Z 4 + 0.219042Z 5 + 0.132719Z 6 − 0.00902654Z 7 − 0.0776167Z 8 − 0.0605182Z 9 − 0.0424561Z 10 + 0.00986602Z 11 + 0.0305473Z 12 + 0.0085857Z 13 + 0.00319259Z 14 − 0.00278122Z 15 − 0.0059109Z 16 + 0.000491937Z 17 + 0.00153169Z 18 − 0.000337657Z 19 − 0.000174237Z 20 + 0.0000561525Z 21 8

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YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

m5 (ξ) =0.0128164 + 0.0483373Z + 0.106319Z 2 + 0.193386Z 3 + 0.257265Z 4 + 0.257891Z 5 + 0.20645Z 6 + 0.081691Z 7 − 0.0345989Z 8 − 0.0768662Z 9 − 0.082805Z 10 − 0.0299905Z 11 + 0.0246983Z 12 + 0.0258706Z 13 + 0.0212312Z 14 + 0.00499153Z 15 − 0.0123916Z 16 − 0.00578444Z 17 − 0.000321503Z 18 − 0.0000588937Z 19 + 0.00200214Z 20 + 0.000582568Z 21 − 0.000852807Z 22 − 0.0000249286Z 23 + 0.000208516Z 24 − 0.0000313929Z 25 − 0.000020983Z 26 + 0.0000055653Z 27

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0

0

0

−0.5

−0.5 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

−0.5 0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

8

9

Figure 1: (Left 1)Graph of real scaling function Φ3 (ξ), (Left 2)Graph of real scaling function Φ4 (ξ), (Left 3)Graph of real scaling function Φ5 (ξ).

4.2

Construction of complex scaling functions

Han and Ji [3] pointed out that constructing compactly supported symmetric orthonormal real-valued wavelets with a dilation factor greater than two such that these wavelets have high vanishing moments is a challenging task. However, by considering complex wavelets, one can construct compactly supported symmetric orthonormal complex wavelets with dilation 4 with arbitrarily high vanishing moments. For any odd positive integer M , Basing on a positive polynomial of degree M − 1, Han and Ji provided a family of compactly supported symmetric orthonormal complex wavelets with dilation 4 with M vanishing moments. For any odd positive integer M , note that our polynomial QM (X) defined by (14) is positive for all X ∈ R, we can also construct symmetric orthonormal complex wavelets with dilation 4. Once the scaling filter mcM (ξ) is obtained, one can get the symmetric orthonormal complex wavelets with dilation 4 using Theorem 1 and Algorithm 1 in [3]. Hence, we present the scaling filters mcM (ξ) for M = 3, 5.

9

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YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

mc3 (ξ) = (−0.000691392 + 0.00581844i)Z (−7) − (0.0108436 + 0.00128853i)Z (−6) − (0.0184532 + 0.0370257i)Z (−5) − (0.0129805 + 0.0441327i)Z (−4) + (0.0196523 − 0.0557696i)Z (−3) + (0.0969841 − 0.0344487i)Z (−2) + (0.182313 + 0.0727629i)Z (−1) + (0.24402 + 0.0940838i) + (0.24402 + 0.0940838i)Z 1 + (0.182313 + 0.0727629i)Z 2 + (0.0969841 − 0.0344487i)Z 3 + (0.0196523 − 0.0557696i)Z 4 − (0.0129805 + 0.0441327i)Z 5 − (0.0184532 + 0.0370257i)Z 6 − (0.0108436 + 0.00128853i)Z 7 − (0.000691392 − 0.00581844i)Z 8 mc5 (ξ) = (−0.000153418 − 0.0002185575)Z (−13) + (0.000694473 − 0.000487492i)Z (−12) + (0.00272565 + 0.00162133i)Z (−11) − (0.000244253 − 0.00521591i)Z (−10) + (−0.00780018 + 0.00739194i)Z (−9) − (0.0180613 − 0.00705253i)Z (−8) + (−0.0313129 − 0.0119401i)Z (−7) − (0.0208646 + 0.046018i)Z (−6) + (0.0166215 − 0.0749529i)Z (−5) + (0.0594812 − 0.0956152i)Z (−4) + (0.111141 − 0.0639564i)Z (−3) + (0.136773 + 0.0155634i)Z (−2) + (0.125668 + 0.0930182i)Z (−1) + (0.125332 + 0.163325i) + (0.125332 + 0.163325i)Z 1 + (0.125668 + 0.0930182i)Z 2 + (0.136773 + 0.0155634i)Z 3 + (0.111141 − 0.0639564i)Z 4 + (0.0594812 − 0.0956152i)Z 5 + (0.0166215 − 0.0749529i)Z 6 − (0.0208646 + 0.046018i)Z 7 − (0.0313129 + 0.0119401i)Z 8 − (0.0180613 − 0.00705253i)Z 9 − (0.00780018 − 0.00739194i)Z 10 − (0.000244253 − 0.00521591)Z 11 + (0.00272565 + 0.00162133i)Z 12 + (0.000694473 − 0.000487492i)Z 13 − (0.000153418 + 0.000218557i)Z 14

10

1024

YUAN ET AL: POLYNOMIAL EQUATION AND WAVELETS

1 Real Image

Real Image

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−4

−3

−2

−1

0

1

2

3

4

Figure 2: (Left)Graph of complex scaling function Φc3 (ξ), (Right)Graph of complex scaling function Φc5 (ξ).

References [1] A. Karoui, A family of orthonormal wavelet bases with dilation factor 4, J. Math. Anal. Appl. 317(1) (2006) 364-379. [2] Y. Shen, S. Li, Q. Mo, Complex wavelets and framelets from pseudo splines, J. Fourier Anal. Appl. 16(6) (2010) 885-900. [3] B. Han, H. Ji, Compactly supported orthonormal complex wavelets with dilation 4 and symmetry, Appl. Comput. Harmon. Anal. 26(3)(2009) 422431. [4] A. Cohen, I. Daubechies, J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992) 485-560. [5] W. Lawton, S. L. Lee, Z.W. Shen, An algorithm for matrix extension and wavelet construction, Math. Comp. 214 (1996) 723-737. [6] C. K. Chui, An introduction to wavelets, Bosten: Academic Press, 1992.

11

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1026-1035, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Numerical solution of fully fuzzy linear matrix equations† Kun Liua,b , Zeng-Tai Gonga,∗ a College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China b College of Mathematics and Statistics, Longdong University, Qingyang 745000, China

Abstract In this paper, we investigate the numerical solution of fully fuzzy linear matrix equations AX = B. The fuzzy solution to the fully fuzzy matrix equations are expressed by the location index solution and the left and right fuzziness index function of this location index solution. The necessary and sufficient conditions for the existence of fuzzy solution and the solvability to fully fuzzy matrix equations are also discussed. Some numerical examples are given to illustrate the efficiency of the proposed method. Keywords: Fuzzy numbers; Fuzzy arithmetic; Parametric form; Fully fuzzy linear matrix equations; Fuzzy solution. 1. Introduction The concept of fuzzy numbers and fuzzy arithmetic operations with these fuzzy numbers were first introduced and investigated by Zadeh [17, 42], Dubois and Prade [21]. One major application of the fuzzy number is treating linear systems whose parameters are all or partially represented by fuzzy numbers. Fuzzy systems are used to study a variety of problems ranging from fuzzy topological spaces [15] to control chaotic systems (for example, in [22, 38]), fuzzy metric spaces [32], fuzzy linear systems (see [2, 4 − 7, 11, 12, 18, 36, 37]), fuzzy differential equations (see [1, 8, 9, 14, 16, 27, 28, 34]), particle physics [35] and so on. One field of applied mathematics that has many applications in various areas of science is solving a system of linear equations. In many problems in various areas of science, which can be solved by solving a system of linear equations. Some of the system parameters are vague or imprecise, and fuzzy mathematics is better than crisp mathematics for mathematical modeling of these problems. It is immensely important to develop numerical procedures that would appropriately treat a system of linear equations where some elements of the system are fuzzy, is called fuzzy system. A general model for solving a fuzzy linear system whose coefficient matrix is crisp and its right column is an arbitrary fuzzy vector was first proposed by Friedman et al. [23] and his colleagues used the embedding method and replaced the original fuzzy linear system by a crisp linear system and then they solved it. And studied duality in fuzzy linear systems Ax = Bx + y, where A, B are real n × n matrices, both the unknown vector x and the constant y are vectors consisting of n fuzzy numbers in [29]. A large number of researches have been produced about how to solve numerically fuzzy linear systems (see [5 − 7, 18]) and so on. Asady et al. [13], who merely considered the full row rank system, used the same method to solve the m×n fuzzy linear system for m ≤ n. Later, Zheng and Wang in [39, 43] discussed the m × n general fuzzy linear system and the inconsistent fuzzy linear system by using generalized inverses of the coefficient matrix. Also, Wang et al. [40] presented an iterative algorithm for solving dual linear system of the form x = Ax + u, where A is real n × n matrix, the unknown vector x and the constant u are all vectors consisting of fuzzy numbers and Abbasbandy [3] investigated the existence of a minimal solution of general dual fuzzy linear equation system by means of matrix generalized inverse theory. At the same time, Muzziloi et al. [31] considered fully fuzzy linear systems of the form A1 x + b1 = A2 x + b2 where A1 , A2 are coefficient matrices consisting of fuzzy numbers and b1 , b2 are vectors consisting of fuzzy numbers, respectively. And Dehghan et al. [19, 20] considered fully fuzzy linear systems Ax = b where A †

Supported by the Natural Scientific Fund of China (No.71061013) and the Scientific Research Project of Northwest Normal University (No. NWNU-KJCXGC-03-61) of China. ∗ Corresponding Author: Zeng-Tai Gong. Tel.:+86 09317971430. E-mail addresses: [email protected], [email protected](Zeng-Tai Gong) and [email protected](Kun Liu). 1026

Kun Liu and Zeng-tai Gong : Numerical solution of fully fuzzy linear matrix equations

and b are a square matrix of fuzzy coefficient and a fuzzy vector, respectively. They also discussed the iterative solution of fully fuzzy linear systems. It is well known that the fuzzy linear matrix equations (shown as FLME) has widely used in the control theory and control engineering. However, few works have been done over the past decades. Allahviranloo et al. [10] studied fuzzy linear matrix equations of the form AXB = C where A ∈ Rm×n , B ∈ Rr×e and A, B and C are given matrices where C is a fuzzy matrix, and X is the unknown matrix by applying the Kronecker product to transform this system to nonsquare system. Gong et al. [25] investigated the ˜ =B ˜ where A is a crisp real matrix and B, ˜ X ˜ m × n inconsistent fuzzy matrix equation of the form AX are matrices consisting of fuzzy numbers by using generalized inverse of the matrix. In this paper, we attempt to find a fuzzy solution of fully fuzzy linear matrix equations of the form AX = B based on a new arithmetic calculation in [30], where A and B are matrices consisting of fuzzy numbers. To this end, we split the general fuzzy linear matrix equations into a location index linear matrix equations and two fuzziness matrix functions. First, we obtain a location index solution by solving a location index linear matrix equations A0 X = B0 . Second, we select two fuzziness matrix functions max{A∗ , B∗ }, max{A∗ , B ∗ } as the left fuzziness index function and the right fuzziness index function of this location index solution. Thus, a fuzzy solution is expressed by means of a location index number and two fuzziness index function. Using this new fuzzy number arithmetic the solution of fully fuzzy linear matrix equations is not only obtained easily, but also some restrictions, which assumed the solution is positive fuzzy matrix in the existing literature, are overcame. The outline of the paper is as follows: In Section 2, we recall some important fundamental results. In Section 3, the numerical solution of fully fuzzy linear matrix equations AX = B are discussed. The necessary and sufficient conditions for the existence of fuzzy solution and the solvability to fully fuzzy matrix equations are also discussed. In Section 4, some numerical examples are given to illustrate our proposed method. The conclusion is drawn in Section 5. 2. Preliminaries In this section, we give some definitions and introduce the notation which will be used throughout the paper. Let us denote by RF the class of fuzzy subsets of the real axis (i.e.,u : R → [0, 1]) (see [21]) satisfying the following properties: (1) u is normal, i.e., there exists s0 ∈ R such that u(s0 ) = 1, (2) u is a convex fuzzy set (i.e., u(ts + (1 − t)r) ≥ min{u(s), u(r)}, ∀t ∈ [0, 1], r, s ∈ R), (3) u is upper semicontinuous on R, (4) cl{s ∈ R | u(s) > 0} is compact where cl denotes the closure of a subset. Then RF is called the space of fuzzy numbers. Obviously R ∈ RF . For 0 < α ≤ 1, set [u]α = {s ∈ R | u(s) ≥ α} and [u]0 = cl{s ∈ R | u(s) > 0}. Then from (1)-(4) it follows that if u belongs to RF then the α−level set [u]α is a non-empty compact interval for all 0 ≤ α ≤ 1. An equivalent parametric form of an arbitrary fuzzy number is also given in Goetschel and Voxman [24] as follows: Definition 2.1. A fuzzy number u in parametric form is a pair (u, u) of functions u(r), u(r), 0 ≤ r ≤ 1, which satisfies the requirements: (1) u(r) is a bounded monotonic increasing left continuous function, (2) u(r) is a bounded monotonic decreasing left continuous function, (3) u(r) ≤ u(r), 0 ≤ r ≤ 1. A crisp number x is simply represented by (u(r), u(r)) = x, 0 ≤ r ≤ 1 and called singleton. For arbitrary two fuzzy numbers x = (x(r), x(r)), y = (y(r), y(r)) and k, the addition, subtraction and scalar multiplication are defined by the extension principle [41] and can be equivalently represented as follows: Definition 2.2. Let x = (x(r), x(r)), y = (y(r), y(r)) ∈ RF , 0 ≤ r ≤ 1 and real number k. (1) x = y iff x(r) = y(r) and x(r) = y(r), (2) x + y = (x(r) + y(r), x(r) + y(r)), (3) x − y = (x(r) − y(r), x(r) − y(r)),  (kx(r), kx(r)), k ≥ 0, (4) kx = (kx(r), kx(r)), k < 0. 1027

Kun Liu and Zeng-tai Gong : Numerical solution of fully fuzzy linear matrix equations

The collection of all the fuzzy numbers with addition and scalar multiplication as defined above is a convex cone. Definition 2.3. For arbitrary fuzzy numbers u = (u(r), u(r)), v = (v(r), v(r)) ∈ RF , the quantity D(u, v) = sup {max[| u(r) − v(r) |, | u(r) − v(r) |]} 0≤r≤1

is called the distance between u and v. This metric is equivalent to the one by Puri and Ralescu [33] and Kaleva [26]. For later use, we introduce a lattice L and fuzzy number (u0 , u∗ , u∗ ), with a parametric form and a new fuzzy arithmetic (see [30]). Definition 2.4. A lattice L as L = {h | h : [0, 1] → [0, ∞) is nondecreasing and left continuous}. The order in L is the natural order defined by h ≤ g if and only if h(r) ≤ g(r) for all r ∈ [0, 1]. It is easy to show that [h ∨ g] = max{h(r), g(r)}, [h ∧ g] = min{h(r), g(r)}, where h ∨ g and h ∧ g are supremum and infimum of h and g. Definition 2.5. For arbitrary fuzzy number u = (u(r), u(r)), the number u0 = 21 (u(1) + u(1)) is said to be a location index number of u, and two nondecreasing left continuous functions u∗ = u0 − u, u∗ = u − u0 are called the left fuzziness index function and the right fuzziness index function, respectively. According to Def. 2.5, every fuzzy number can be represented by (u0 , u∗ , u∗ ). Definition 2.6. For arbitrary fuzzy numbers u = (u0 , u∗ , u∗ ) and v = (v0 , v∗ , v ∗ ), the four arithmetical operations are defined by uv = (u0 v0 , u∗ ∨ v∗ , u∗ ∨ v ∗ ), where uv is either of u + v, u − v, u · v, uv . The arithmetic is determined by the operations on both location index and fuzziness index functions. The location index number is taken in the ordinary arithmetic, whereas the fuzziness index functions are considered to follow the lattice rule which is least upper bound in the lattice L. Here and after this, we operate all fuzzy arithmetic calculation using the above definition. Definition 2.7. A matrix A = (aij ), i, j = 1, 2, · · · n, is called a fuzzy matrix, if each element of A is a fuzzy number with a parametric form as Def. 2.5, we represent A = (aij ) that aij = ((aij )0 , (aij )∗ , (aij )∗ ) where (aij )0 is location index matrix of aij and (aij )∗ , (aij )∗ are left fuzziness index matrix function and right fuzziness index matrix function, respectively. Definition 2.8. A vector b = (bi ) is called a fuzzy vector, if each element of b is a fuzzy number, with new notation b = ((bi )0 , (bi )∗ , (bi )∗ ) i = 1, 2, · · · n, where (bi )0 is location index vector of bi and (bi )∗ , (bi )∗ are left fuzziness index vector function and right fuzziness index vector function, respectively. 3. Fully Fuzzy linear matrix equations In this section, we will investigate the fuzzy solution of fully fuzzy linear matrix equations of the form AX = B based on a new arithmetic calculation in [30]. We give the necessary and sufficient conditions for the existence of a fuzzy solution of fully fuzzy linear matrix equations and analyze the solvability of fully fuzzy linear matrix equations AX = B. Definition 3.1. The model equations      a11 a12 · · · a1n x11 x12 · · · x1n b11 b12 · · · b1n  a21 a22 · · · a2n   x21 x22 · · · x2n   b21 b22 · · · b2n       (3.1)  ...................  ...................  =  .................. , an1 an2 · · · ann xn1 xn2 · · · xnn bn1 bn2 · · · bnn where the left coefficient matrix A = (aij ) (1 ≤ i, j ≤ n), and the right-hand matrix B = (bij ) (1 ≤ 1028

Kun Liu and Zeng-tai Gong : Numerical solution of fully fuzzy linear matrix equations

i, j ≤ n) are all fuzzy number matrices which be defined as Def 2.7, is called a fully fuzzy linear matrix equations (FFLME). Using matrix notation, we have AX = B. (3.2) Let A be a nonsingular fuzzy matrix. Then a fuzzy number matrix X = (X0 , X∗ , X ∗ ) is called a unique fuzzy solution of FFLME (3.1) if A0 X0 = B0 , A∗ ∨ X∗ = B∗ , A∗ ∨ X ∗ = B ∗ . Actually, X = (X0 , X∗ , X ∗ ) is also equivalent to X = (x1 , x2 · · · , xn ) if Axl = bl , l = 1, 2, · · · , n,

(3.3)

where xl = (((x1l )0 , (x1l )∗ , (x1l )∗ ), ((x2l )0 , (x2l )∗ , (x2l )∗ ), · · · , ((xnl )0 , (xnl )∗ , (xnl )∗ ))T , l = 1, 2, · · · , n, is the lth column of unknown matrix X, and B = (b1 , b2 · · · , bn ), given by bl = (((b1l )0 , (b1l )∗ , (b1l )∗ ), ((b2l )0 , (b2l )∗ , (b2l )∗ ), · · · , ((bnl )0 , (bnl )∗ , (bnl )∗ ))T , l = 1, 2, · · · , n, is the lth column of the right-hand fuzzy matrix B. For the Eq. (3.3), we also can write as the following form: n X

aij xl = bl , i, l = 1, 2, · · · , n.

(3.4)

j=1

Theorem 3.1. If X = (X0 , X∗ , X ∗ ) is a fuzzy solution of FFLME (3.1), then X0 = (A0 )−1 B0 , X∗ = max{A∗ , B∗ }, X ∗ = max{A∗ , B ∗ }. Proof. By virtue of Def 2.7, FFLME AX = B can write (A0 , A∗ , A∗ )(X0 , X∗ , X ∗ ) = (B0 , B∗ , B ∗ ). By Def 2.6, we have (A0 X0 , A∗ ∨ X∗ , A∗ ∨ X ∗ ) = (B0 , B∗ , B ∗ ). It follows A0 X0 = B0 , A∗ ∨ X∗ = B∗ , A∗ ∨ X ∗ = B ∗ . This implies X0 = (A0 )−1 B0 , X∗ = max{A∗ , B∗ }, X ∗ = max{A∗ , B ∗ }. The proof is completed.  Theorem 3.2. If the fuzzy linear matrix equations (3.1) has a fuzzy solution, then the following conditions hold: n P (i). (aij )0 xl = (bil )0 , i, l = 1, 2, · · · , n, has a solution as a crisp linear equation system; j=1

(ii). max {(aij )∗ } ≤ (bil )∗ , i, l = 1, 2, · · · , n, where bil ∈ B; 1≤j≤n

(iii). max {(aij )∗ } ≤ (bil )∗ , i, l = 1, 2, · · · , n, where bil ∈ B. 1≤j≤n

Proof. Let xl (l = 1, 2, · · · , n) be a fuzzy solution of Eq. (3.4). That is n P aij xl = bl . j=1

Therefore, n P ((aij )0 , (aij )∗ , (aij )∗ ) · ((xjl )0 , (xjl )∗ , (xjl )∗ ) j=1

= ((bil )0 , (bil )∗ , (bil )∗ ). This implies n P ( (aij )0 (xjl )0 , max {(aij )∗ , (xjl )∗ }, max {(aij )∗ , (xjl )∗ }) 1≤j≤n ((bil )0 , (bil )∗ , (bil )∗ ).

j=1

1≤j≤n

= It follows n P (aij )0 (xjl )0 = (bil )0 , j=1

1029

Kun Liu and Zeng-tai Gong : Numerical solution of fully fuzzy linear matrix equations

max {(aij )∗ , (xjl )∗ } = (bil )∗ ,

1≤j≤n

max {(aij )∗ , (xjl )∗ } = (bil )∗ .

1≤j≤n

By the Def. 2.6, we easily get (i)-(iii). The proof is completed.  Corollary 3.1. If the fully fuzzy linear matrix equations (3.1) has a fuzzy solution, then the following conditions hold: (i). A0 X0 = B0 has a solution as a crisp linear equation system; (ii). (xil )∗ ≤ max{ max {(aij )∗ }, (bil )∗ }, i, l = 1, 2, · · · , n; 1≤j≤n

(iii). (xil )∗ ≤ max{ max {(aij )∗ }, (bil )∗ }, 1≤j≤n

i, l = 1, 2, · · · , n.

Proof. By virtue of the proof of Th. 3.2, it is obviously.  n P (aij )0 xl = (bl )0 , l = 1, 2, · · · , n has a crisp solution (xl )0 , l = 1, 2, · · · , n, and Theorem 3.3. If j=1

max {(aij )∗ } = (bil )∗ , max {(aij )∗ } = (bil )∗ .

1≤j≤n

1≤j≤n

Then  ((x11 )0 , (x11 )∗ , (x11 )∗ ) ((x12 )0 , (x12 )∗ , (x12 )∗ ) · · · ((x1n )0 , (x1n )∗ , (x1n )∗ )  ((x21 )0 , (x21 )∗ , (x21 )∗ ) ((x22 )0 , (x22 )∗ , (x22 )∗ ) · · · ((x2n )0 , (x2n )∗ , (x2n )∗ )   X=  ...........................................................................  ((xn1 )0 , (xn1 )∗ , (xn1 )∗ ) ((xn2 )0 , (xn2 )∗ , (xn2 )∗ ) · · · ((xnn )0 , (xnn )∗ , (xnn )∗ ) 

with (xl )∗ ≤ (bl )∗ , (xl )∗ ≤ (bl )∗ , l = 1, 2, · · · , n, is a solution of Eq. (3.3). Proof. Let (xl )0 , l = 1, 2, · · · , n be a crisp solution of the equation n P (aij )0 xl = (bl )0 . j=1

Then, for any xjl = ((xjl )0 , (xjl )∗ , (xjl )∗ ) with (xl )∗ ≤ (bl )∗ , (xl )∗ ≤ (bl )∗ , l = 1, 2, · · · , n, we easily get n P aij xjl j=1

=

n P

((aij )0 , (aij )∗ , (aij )∗ ) · ((xjl )0 , (xjl )∗ , (xjl )∗ )

j=1 n P

=(

(aij )0 (xjl )0 , max {(aij )∗ , (xjl )∗ }, max {(aij )∗ ), (xjl )∗ }) 1≤j≤n

1≤j≤n ((bl )0 , max {(aij )∗ , (xjl )∗ }, max {(aij )∗ ), (xjl )∗ }). 1≤j≤n 1≤j≤n j=1

=

By virtue of the requirements max {(aij )∗ , (xjl )∗ } 1≤j≤n

= max{ max (aij )∗ , max (xjl )∗ } 1≤j≤n

1≤j≤n

= max{(bil )∗ , max (xjl )∗ } 1≤j≤n

= (bil )∗ . Similarly, max {(aij )∗ , (xjl )∗ } = (bil )∗ . 1≤j≤n

Hence, we complete the proof.  Theorem 3.4. The fully fuzzy linear matrix equations AX = B has a fuzzy solution if and only if Rank(A0 ) = Rank(A0 , B0 ). Proof. Let X = (x1 , x2 · · · , xn ), B = (b1 , b2 · · · , bn ), where xl = (((x1l )0 , (x1l )∗ , (x1l )∗ ), ((x2l )0 , (x2l )∗ , (x2l )∗ ), · · · , ((xnl )0 , (xnl )∗ , (xnl )∗ ))T , l = 1, 2, · · · , n, 1030

Kun Liu and Zeng-tai Gong : Numerical solution of fully fuzzy linear matrix equations

bl = (((b1l )0 , (b1l )∗ , (b1l )∗ ), ((b2l )0 , (b2l )∗ , (b2l )∗ ), · · · , ((bnl )0 , (bnl )∗ , (bnl )∗ ))T , l = 1, 2, · · · , n. The fully fuzzy linear matrix equations AX = B is equivalent to the following fully fuzzy linear equations Axl = bl , l = 1, 2, · · · , n. Take A = (A0 , A∗ , A∗ ), bl = ((bl )0 , (bl )∗ , (bl )∗ ). If Rank(A0 ) = Rank(A0 , B0 ), we have Rank(A0 ) = Rank(A0 , (bl )). Since Rank(A0 ) ≤ Rank(A0 , (bl )0 ) ≤ Rank(A0 , B0 ), we know all fully fuzzy linear equations Axl = bl have fuzzy solution, that is, the fully fuzzy linear matrix equations AX = B has a fuzzy solution. Conversely, suppose that the fully fuzzy linear matrix equations AX = B is solvable, this mean each fully fuzzy linear equations Axl = bl , l = 1, 2, · · · , n has a fuzzy solution. Take xl = (((x1l )0 , (x1l )∗ , (x1l )∗ ), ((x2l )0 , (x2l )∗ , (x2l )∗ ), · · · , ((xnl )0 , (xnl )∗ , (xnl )∗ ))T , and A = (a1 , a2 , · · · , an )T , where aj = (((a1j )0 , (a1j )∗ , (a1j )∗ ), ((a2j )0 , (a2j )∗ , (a2j )∗ ), · · · , ((anj )0 , (anj )∗ , (anj )∗ )). Using the fully fuzzy linear equation Axl = bl . we get (a1j )0 (x1l )0 + (a2j )0 (x2l )0 + · · · + (anj )0 (xnl )0 = (bjl )0 , j, l = 1, 2, · · · , n. It shows that (bjl )0 can be expressed by the linear combination of (a1 )0 , (a2 )0 , · · · , (an )0 , i.e., Rank(A0 ) = Rank(A0 , (bl )0 ), l = 1, 2, · · · , n. Thus Rank(A0 ) = Rank(A0 , B0 ). The proof is completed.  Theorem 3.5. If X = (X0 , X∗ , X ∗ ) is a fuzzy solution of the fully fuzzy linear matrix equations AX = B, then the fuzzy matrix X = (X(r), X(r)) obtained by X(r) = X0 − (X)∗ , X(r) = X0 + (X)∗ , (0 ≤ r ≤ 1) is a fuzzy solution of the fully fuzzy linear matrix equations AX = B. Proof. By Def. 2.5, it is clear.  Theorem 3.6. That the fully fuzzy linear matrix equations AX = B has fuzzy solution is equivalent to the following condition, i.e., A0 xl = (bl )0 for all l = 1, 2, · · · , n has solution. Theorem 3.7. That the fully fuzzy linear matrix equations AX = B has a fuzzy solution is equivalent to that rows (columns) (bl )0 of matrix B0 has the same linear relation as rows (columns) (al )0 of the matrix A0 . Theorem 3.8. If the matrix equation A0 xl = (bl )0 for all l = 1, 2, · · · , n does not have the solution, then the fully fuzzy linear matrix equations AX = B must do not have any one either. Corollary 3.2. Under the condition of Rank(A0 ) = Rank(A0 , B0 ), if Rank(A0 ) = n, then the fully fuzzy linear matrix equations AX = B has a unique fuzzy solution, else the fully fuzzy linear matrix equations AX = B has an infinite fuzzy solutions. Corollary 3.3. The fully fuzzy linear matrix equations AX = B has a unique fuzzy solution is equivalent to that the matrix equation A0 xl = (bl )0 for all l = 1, 2, · · · , n has a unique solution. Moreover, we could express a fuzzy solution for the fully fuzzy linear matrix equations AX = B by algorithm as follows: Algorithm: • By means of Def. 2.5, we will transform AX = B into (A0 , A∗ , A∗ )X = (B0 , B∗ , B ∗ ), where A = (A(r), A(r)), B = (B(r), B(r), 0 ≤ r ≤ 1. • By applying the block forms of matrix, we rewrite (A0 , A∗ , A∗ )X = (B0 , B∗ , B ∗ ) as (A0 , A∗ , A∗ )((xl )0 , (xl )∗ , (xl )∗ ) = ((bl )0 , (bl )∗ , (bl )∗ ), l = 1, 2, · · · , n, where xl , bl denote the lth column of unknown X and right-hand fuzzy number matrix B, respectively. • By simple calculations of A0 (xl )0 = (bl )0 , l = 1, 2, · · · , n, we first find location index vector (xl )0 of xl , l = 1, 2, · · · , n. • Using max{ max {(aij )∗ }, (bil )∗ } = (xl )∗ , max{ max {(aij )∗ }, (bil )∗ } = (xl )∗ , l = 1, 2, · · · , n, then we 1≤j≤n

1≤j≤n 1031

Kun Liu and Zeng-tai Gong : Numerical solution of fully fuzzy linear matrix equations

obtain left fuzziness index and right fuzziness index vector function (xl )∗ , (xl )∗ of xl , l = 1, 2, · · · , n. • The fuzzy solution X = (((xl )0 , (xl )∗ , (xl )∗ ), ((x2 )0 , (x2 )∗ , (x2 )∗ ), · · · , ((xn )0 , (xn )∗ , (xn )∗ ))T are derived. 4. Numerical examples In this section, we employ some examples to illustrate the utility of algorithm. Example 4.1. Consider the following fully fuzzy linear matrix equations      (1 + r, 3 − r) (−1 + r, 1 − r) (3 + r, 5 − r) (2 + r, 4 − r) x11 x12 = . (r, 2 − r) (2 + r, 4 − r) (1 + r, 3 − r) (r, 2 − r) x21 x22 The coefficient matrix and right-hand matrix respectively are     (3 + r, 5 − r) (2 + r, 4 − r) (1 + r, 3 − r) (−1 + r, 1 − r) A= , B= . (1 + r, 3 − r) (r, 2 − r) (r, 2 − r) (2 + r, 4 − r) So we easily get    4 3 2 0 (A)0 = , (B)0 = . 2 1 1 3 Furthermore, we maycalculate          4 3 x11 2 4 3 x12 0 = , = , x21 x22 2 1 1 2 1 3 thus we obtain location index number of x11 , x21 , x12 , x22 as follows: (x11 )0 = 21 , (x21 )0 = 0, (x12 )0 = 92 , (x22 )0 = −6. At the same time, we can calculate    1−r 1−r 1−r 1−r (A)∗ = , (B)∗ = , 1−r 1−r 1−r 1−r     1−r 1−r 1−r 1−r (A)∗ = , (B)∗ = , 1−r 1−r 1−r 1−r thus we obtain left fuzziness index and right fuzziness index of x11 , x21 , x12 , x22 as follows: (x11 )∗ = 1 − r, (x21 )∗ = 1 − r, (x12 )∗ = 1 − r, (x22 )∗ = 1 − r, (x11 )∗ = 1 − r, (x21 )∗ = 1 − r, (x12 )∗ = 1 − r, (x22 )∗ = 1 − r. Thus thefuzzy solution  is 1  x11 x12 ( 2 , 1 − r, 1 − r) ( 92 , 1 − r, 1 − r) X= = . x21 x22 (0, 1 − r, 1 − r) (−6, 1 − r, 1 − r) The parametric forms of x11 , x21 , x12 , x22 are the following form: x11 (r) = − 12 + r, x21 (r) = −1 + r, x12 (r) = 72 + r, x22 (r) = −7 + r, x11 (r) = 32 − r, x21 (r) = 1 − r, x12 = 11 2 − r, x11 (r) = −5 − r. So the fuzzy can also write  solution  (− 21 + r, 32 − r) ( 72 + r, 11 2 − r) X= . (−1 + r, 1 − r) (−7 + r, −5 − r) Example 4.2. Consider the following fully fuzzy linear matrix equations      (1, 2, 3) (4, 6, 9) (1, 3, 4) x11 x12 x13 (3, 5, 8) (2, 4, 5) (6, 7, 9)  (0, 1, 3) (5, 6, 8) (3, 5, 6)   x21 x22 x23  =  (0, 7, 8) (2, 5, 6) (1, 3, 7)  . x31 x32 x33 (4, 6, 7) (1, 3, 5) (2, 4, 7) (4, 5, 6) (1, 7, 9) (2, 3, 4) The coefficient matrix and right-hand  matrix respectively are    (1, 2, 3) (4, 6, 9) (1, 3, 4) (3, 5, 8) (2, 4, 5) (6, 7, 9) A =  (0, 1, 3) (5, 6, 8) (3, 5, 6)  , B =  (0, 7, 8) (2, 5, 6) (1, 3, 7)  . (4, 5, 6) (1, 7, 9) (2, 3, 4) (4, 6, 7) (1, 3, 5) (2, 4, 7) By simple calculations of the new arithmetic, we have following linear matrix equations for finding location index number of x11 , x12 , x13 , x21 , x22 , x23 , x31 , x32 , x33 :      2 6 3 (x11 )0 (x12 )0 (x13 )0 5 4 7  1 6 5   (x21 )0 (x22 )0 (x23 )0  =  7 5 3  , 5 7 3 (x31 )0 (x32 )0 (x33 )0 6 3 4 1032

Kun Liu and Zeng-tai Gong : Numerical solution of fully fuzzy linear matrix equations

therefore, we have 8 17 5 22 (x11 )0 = 29 , (x12 )0 = − 29 , (x13 )0 = − 62 29 , (x21 )0 = 29 , (x22 )0 = 29 , 33 6 89 (x23 )0 = 99 29 , (x31 )0 = 29 , (x32 )0 = 29 , (x33 )0 = − 29 . At the sametime, we can calculate    2 − 2r 2 − 2r 1 − r 1 − r 2 − 2r 2 − 2r (B)∗ =  7 − 7r 3 − 3r 2 − 2r  , (A)∗ =  1 − r 1 − r 2 − 2r  , 2 − 2r 2 − 2r 2 − 2r 1 − r 6 − 6r 1 − r     1 − r 3 − 3r 1 − r 3 − 3r 1 − r 2 − 2r (A)∗ =  2 − 2r 2 − 2r 1 − r  , (B)∗ =  1 − r 1 − r 4 − 4r  , 1 − r 2 − 2r 1 − r 1 − r 2 − 2r 3 − 3r thus we obtain location left fuzziness index and right fuzziness index of x11 , x21 , x31 , x12 , x22 , x32 , x13 , x23 , x33 as follows: (x11 )∗ = 7 − 7r, (x21 )∗ = 3 − 3r, (x31 )∗ = 2 − 2r, (x11 )∗ = 3 − 3r, (x21 )∗ = 2 − 2r, (x31 )∗ = 4 − 4r, (x12 )∗ = 7 − 7r, (x22 )∗ = 6 − 6r, (x32 )∗ = 6 − 6r, (x12 )∗ = 3 − 3r, (x22 )∗ = 3 − 3r, (x32 )∗ = 4 − 4r, (x13 )∗ = 7 − 7r, (x23 )∗ = 3 − 3r, (x33 )∗ = 2 − 2r, (x13 )∗ = 3 − 3r, (x23 )∗ = 2 − 2r, (x33 )∗ = 4 − 4r. Thus thefuzzy solution is  x11 x12 x13 X =  x21 x22 x23  x x32 x33  8 31  17 ( 29 , 7 − 7r, 3 − 3r) (− 29 , 7 − 7r, 3 − 3r) (− 62 29 , 7 − 7r, 3 − 3r) 22 5 . =  ( 29 , 3 − 3r, 2 − 2r) ( 29 , 6 − 6r, 3 − 3r) ( 99 29 , 3 − 3r, 2 − 2r) 3 6 89 ( 29 , 2 − 2r, 4 − 4r) ( 29 , 6 − 6r, 4 − 4r) (− 29 , 2 − 2r, 4 − 4r) The parametric forms of x11 , x21 , x31 , x12 , x22 , x32 , x13 , x23 , x33 are the following form: x11 (r) = −6.7241 + 7r, x21 (r) = −2.8279 + 3r, x31 (r) = −1.8966 + 2r, x11 (r) = 3.2759 − 3r, x21 (r) = 2.1724 − 2r, x31 (r) = 4.1035 − 4r, x12 (r) = −7.5862 + 7r, x22 (r) = −5.2414 + 6r, x32 (r) = −5.7931 + 6r, x12 (r) = 2.4138 − 3r, x22 (r) = 3.7586 − 3r, x32 (r) = 4.2069 − 4r, x13 (r) = −9.1380 + 7r, x23 (r) = 0.4138 + 3r, x33 (r) = −5.0690 + 2r, x13 (r) = 0.8621 − 3r, x23 (r) = 5.4138 − 2r, x33 (r) = 0.9310 − 4r. So the fuzzy solution can also write  (−6.7241 + 7r, 3.2759 − 3r) (−7.5862 + 7r, 2.4138 − 3r) (−9.1380 + 7r, 0.8621 − 3r) X =  (−2.8279 + 3r, 2.1724 − 2r) (−5.2414 + 6r, 3.7586 − 3r) (0.4138 + 3r, 5.4138 − 2r)  . (−1.8966 + 2r, 4.1035 − 4r) (−5.7931 + 6r, 4.2069 − 4r) (−5.0690 + 2r, 0.9310 − 4r) 5. Conclusion In this paper, the solution of fully fuzzy linear matrix equations of the form AX = B based on a new arithmetic calculation in [30] is given, where A, B are all matrices consisting of fuzzy numbers. By splitting the general fully fuzzy linear matrix equations into a location index linear matrix equations and two fuzziness matrix functions, a fuzzy solution is expressed by means of a location index number and two fuzziness index function. At the same time, necessary and sufficient conditions for the existence of a fuzzy solution are derived. Also, we analyzed the solvability of FFLME AX = B. Using this new fuzzy number arithmetic the solution of fully fuzzy linear matrix equations is not only obtained easily, but also some restrictions, which assumed the solution is positive fuzzy matrix in the existing literature, are overcame.

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Kun Liu and Zeng-tai Gong : Numerical solution of fully fuzzy linear matrix equations

[2] S. Abbasbandy, M. Otadi, M. Mosleh, Minimal solution of general dual fuzzy linear systems, Chaos, Solitions and Fractals 29 (2008) 638-652. [3] S. Abbasbandy, M. Otadi, M. Mosleh, Minimal solution of general dual fuzzy linear systems, Chaos, Solitons and Fractals, 37 (2008) 1113-1124. [4] T. Allahviranloo, A comment on fuzzy linear systems, Fuzzy Sets and Systems 140 (2003) 559. [5] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation 153 (2004) 493-502. [6] T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equations, Applied Mathematics and Computation 162 (2005) 189-196. [7] T. Allahviranloo, The Adomian decomposition method for fuzzy system of linear equations, Applied Mathematics and Computation 163 (2005) 553-563. [8] T. Allahviranloo, E. Ahmady, N. Ahmady, N th-order fuzzy linear differential equations, Information Sciences 178 (2008) 1309-1324. [9] T. Allahviranloo, N. A. Kiani, N. Motamedi, Solving fuzzy differential equations by differential transformation method, Information Sciences 179 (2009) 956-966. [10] T. Allahviranloo, N. Mikaeilvand, M. Barkhordary, Fuzzy linear matrix equation, Fuzzy Optimization and Decision Making 8 (2009) 165-177. [11] T. Allahviranloo, S. Salahshour, Fuzzy symmetric solutions of fuzzy linear systems, Journal of Computational and Applied Mathematics 235 (2011) 4545-4553. [12] T. Allahviranloo, S. Salahshour, M. Khezerloo, Maximal and minimal symmetric solutions of fully fuzzy linear systems, Journal of Computational and Applied Mathematics 235 (2011) 4652-4662. [13] B. Asady, S. Abbasbandy, M. Alavi, Fuzzy general linear systems, Applied Mathematics and Computation 169 (2005) 34-40. [14] B. Bede, I. J. Rudas, A. L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Information Sciences 177 (2007) 1648-1662. [15] M. Caldas, S. Jafari, θ-Compact fuzzy topological spaces, Chaos, Solitons and Fractals 25 (2005) 229-232. [16] Y. Chalco-Cano, H. Rom´ an-Flores, Comparation between some approaches to solve fuzzy differential equations, Fuzzy Sets and Systems 160 (2009) 1517-1527. [17] S. S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, Systems, Man and Cybernetics, IEEE Transactions 2 (1972) 30-33. [18] M. Dehghan, B. Hashemi, Iterative solution of fuzzy linear systems, Applied Mathematics and Computation 175 (2006) 645-674. [19] M. Dehghan, B. Hashemi, M. Ghatee, Computational methods for solving fully fuzzy linear systems, Applied Mathematics and Computation 179 (2006) 328-343. [20] M. Dehghan, B. Hashemi, M. Ghatee, Solution of the full fuzzy linear systems using iterative techniques, Chaos, Solitons and Fractals 34 (2007) 316-336. [21] D. Dubois, H. Prade, Operations on fuzzy numbers, International Journal of Systems Science 9 (1978) 613-626. [22] G. Feng, G. Chen, Adaptive control of discrete-time chaotic systems: a fuzzy control approach, Chaos, Solitons and Fractals 23 (2005) 459-467. [23] M. Friedman, M. Ma, A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems 96 (1998) 201-209. [24] R. Goetschel, W. Voxman, Elementary calculus, Fuzzy Sets and Systems 18 (1986) 31-43. [25] Z. T. Gong, X. B. Guo, Inconsistent fuzzy matrix equations and its fuzzy least squares solutions, Applied Mathematical Modelling 35 (2011) 1456-1469. [26] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317. 1034

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[27] V. Lupulescu, Initial value problem for fuzzy differential equations under dissipative conditions, Information Sciences 178 (2008) 4523-4533. [28] M. Ma, M. Friedman, A. Kandel, Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems 105 (1999) 133-138. [29] M. Ma, M. Friedman, A. Kandel, Duality in fuzzy linear systems, Fuzzy Sets and Systems 109 (2000) 55-58. [30] M. Ma, M. Friedman, A. Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems 108 (1999) 83-90. [31] S. Muzzioli, H. Reynaerts, Fuzzy linear systems of the form A1 x + b1 = A2 x + b2 , Fuzzy Sets and Systems 157 (2006) 939-951. [32] J. H. Park, Intuitionistic fuzzy metric space, Chaos, Solitons and Fractals 22 (2004) 1039-1046. [33] M. L. Puri, D. A. Ralescu, Differentials for fuzzy functions, Journal of Mathematical Analysis and Applications 91 (1983) 552-558. [34] S. J. Song, C. X. Wu, Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations, Fuzzy Sets and Systems 110 (2000) 55-67. [35] Y. Tanaka, Y. Mizuno, T. Kado, Chaotic dynamics in the Friedman equation, Chaos, Solitons and Fractals 24 (2005) 407-422. [36] Z. F. Tian, L. J. Hu, D. Greenhalgh, Perturbation analysis of fuzzy linear systems, Information Sciences 180 (2010) 4706-4713. [37] K. Wang, G. L. Chen, Y. M. Wei, Perturbation analysis for a class of fuzzy linear systems, Journal of Computational and Applied Mathematics 224 (2009) 54-65. [38] J. Wang, G. D. Qiao, B. Deng, H∞ Variable universe adaptive fuzzy control for chaotic system, Chaos, Solitons and Fractals 24 (2005) 1075-1086. [39] K. Wang, B. Zheng, Inconsistent fuzzy linear systems, Applied Mathematics and Computation 181 (2006) 973-981. [40] X. Wang, Z. Zhong, M. Ma, Iteration algorithms for solving a system of fuzzy linear equations, Fuzzy Sets and Systems 119 (2001) 121-128. [41] R. R. Yager, D. P. Filev, SLIDE: a simple adaptive defuzzification method, IEEE Transactions on Fuzzy Systems 1 (1993)69-78. [42] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Information Science 8 (1975) 199-249. [43] B. Zheng, K. Wang, General fuzzy linear systems, Applied Mathematics and Computation 181 (2006) 1276-1286.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1036-1045, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Korovkin type approximation theorem for statistical A-summability of double sequences a)

M. Mursaleena) and Abdullah Alotaibib) Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India b)

Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia [email protected]; [email protected]

Abstract. In this paper, we prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical A–summability. We also construct an example by MeyerK¨onig and Zeller operators to show that our result is stronger than those of previously proved by other authors. Keywords and phrases: Double sequence; density; statistical convergence; A–statistical convergence; statistical A–summability; positive linear operator; Korovkin type approximation theorem. AMS subject classification (2000): 41A10, 41A25, 41A36, 40A30, 40G15.

1. Introduction and preliminaries The concept of statistical convergence for sequences of real numbers was introduced by Fast [8] and further studied many others. Let K ⊆ N and Kn = {k ≤ n : k ∈ K} .Then the natural density of K is defined by δ(K) = limn n−1 |Kn | if the limit exists, where |Kn | denotes the cardinality of Kn . A sequence x = (xk ) of real numbers is said to be statistically convergent to L provided that for every  > 0 the set K := {k ∈ N : |xk − L| ≥ } has natural density zero, i.e. for each  > 0, 1 lim |{j ≤ n : |xj − L| ≥ }| = 0. n n By the convergence of a double sequence we mean the convergence in the Pringsheim’s sense [18]. A double sequence x = (xjk ) is said to be Pringsheim’s convergent (or P -convergent) if for given  > 0 there exists an integer N such that |xjk − `| <  whenever j, k > N . In this case, ` is called the Pringsheim limit of x = (xjk ) and it is written as P − lim x = `. A double sequence x = (xjk ) is said to be bounded if there exists a positive number M such that |xjk | < M for all j, k. Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. The idea of statistical convergence for double sequences was introduced and studied by Moricz [13] and Mursaleen and Edely [17], independently in the same year. Let K ⊆ N × N be a two-dimensional set of positive integers and let Km,n = {(j, k) : j ≤ m, k ≤ n}. Then the two-dimensional analogue of natural density can be defined as follows.

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MURSALEEN, ALOTAIBI: KOROVKIN THEOREM

In case the sequence (K(m, n)/mn) has a limit in Pringsheim’s sense, then we say that K has a double natural density and is defined as P − lim m,n

K(m, n) = δ (2) {K}. mn

For example, let K = {(i2 , j 2 ) : i, j ∈ N}. Then K(m, n) δ {K} = P − lim ≤ P − lim m,n m,n mn (2)

√ √ m n = 0, mn

i.e. the set K has double natural density zero, while the set {(i, 2j) : i, j ∈ N} has double natural density 21 . A real double sequence x = (xjk ) is said to be statistically convergent to the number L if for each  > 0, the set {(j, k), j ≤ m and k ≤ n :| xjk − L |≥ } has double natural density zero. In this case we write st(2) - lim xjk = L. j,k

Remark 1.1. Note that if x = (xjk ) is P -convergent then it is statistically convergent but not conversely. See the following example. Example 1.1. The double sequence x = (xjk ) defined by  1 , if j and k are squares; xjk = 0 , otherwise .

(1.1.1)

Then x is statistically convergent to zero but not P -convergent. Let C[a, b] be the space of all functions f continuous on [a, b]. We know that C[a, b] is a Banach space with norm kf kC[a,b] := sup |f (x)|, f ∈ C[a, b]. x∈[a,b]

The classical Korovkin approximation theorem states as follows (see [10]): Let (Tn ) be a sequence of positive linear operators from C[a, b] into C[a, b]. Then limn kTn (f, x) − f (x)kC[a,b] = 0, for all f ∈ C[a, b] if and only if limn kTn (fi , x) − fi (x)kC[a,b] = 0, for i = 0, 1, 2, where f0 (x) = 1, f1 (x) = x and f2 (x) = x2 . Recently, such type of approximation theorems have been proved by many authors by using the concept of statistical convergence and its variants, e.g. [2]–[6], [12], [14]– [16] and [20]. In [1] and [11] authors have used the concept of almost convergence. In y x x 2 [21] the Korovkin theorem was proved by using the test function 1, 1−x , 1−y , ( 1−x ) + y 2 ( 1−y ) . In this paper, we extend the result of [21] by using the notion of statistical A–summability of double sequences and show that our result is stronger.

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MURSALEEN, ALOTAIBI: KOROVKIN THEOREM

2. Statistical A–summability
, m, n, i, j ∈ N, be a four dimensional matrix and x = (xij ) be a Let A = amn ij double sequence. Then the double (transformed) sequence, Ax := (ymn ), is denoted by ∞,∞

X

ymn :=

amn ij xij ,

i=1,,j=1

where it is assumed that the summation exists as a Pringsheim limit (of the partial sums) for each (m, n) ∈ N2 = N × N. Also the sums ymn are called A−means of the double sequence x. We say that a sequence x is A−summable to the limit ` if the A−means exist for all m, n ∈ N in the sense of Pringsheim convergence, lim

p,q X

p,q→∞

amn ij xij = ymn

i,,j

and lim ymn = `.

m,n→∞

A two dimensional matrix transformation is said to be regular if it maps every convergent sequence into a convergent sequence with the same limit. The well-known conditions for two dimensional matrix to be regular are known as Silverman-Toeplitz conditions. In 1926, Robinson [19] presented a four dimensional analogue of the regularity by considering an additional assumption of boundedness. This assumption was made because a double P −convergent sequence is not necessarily bounded. The definition and the characterization of regularity for four dimensional matrices is known as RobinsonHamilton regularity, or briefly, RH−regularity (see Robinson [19], Hamilton [9]). Recall that a four dimensional matrix A is said to be RH−regular or boundedregular (see Robinson [19], Hamilton [9]) if it maps every bounded P −convergent sequence into a P − convergent sequence with the same P −limit. The Robinson-Hamilton
conditions state that a four dimensional matrix A = amn is RH− regular if and only if ij 2 (RH1 ) P − limm,n amn ij = 0 for each (i, j) ∈ N ,

amn ij = 1

(RH2 ) P − limm,n

P

(RH3 ) P − limm,n

P

mn aij = 0 for each i ∈ N,

(RH4 ) P − limm,n

P

mn aij = 0 for each j ∈ N,

(RH5 )

P

(i,j)∈N2

(i,j)∈N2

j∈N

i∈N

mn aij is P −convergent,

P < A (RH6 ) there exist finite positive integers A and B such that i,j>B amn ij holds for every (m, n) ∈
N2 . Now, let A = amn be a nonnegative RH−regular summability matrix, and let ij 2 K ⊆ N . Then the A−density of K is given by X (2) δA {K} := P − lim amn ij m,n

(i,j)∈K

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MURSALEEN, ALOTAIBI: KOROVKIN THEOREM

provided that the limit on the right-hand side exists in Pringsheim’s sense. A real double sequence x = (xij ) is said to be A− statistically convergent to a number L if for every  > 0, (2)  δA (i, j) ∈ N2 : |xij − L| ≥  = 0 (2)

In this case, we write stA − lim xm,n = L. Clearly, a P − convergent double sequence m,n

is A−statistically convergent to the same limit but the converse need not be true. For example, take A = C (1, 1), which is the double Ces`aro matrix, and the double sequence w = (wij ) be defined as in Example 1.1. Then this sequence is statistically convergent (that is, C(1, 1)-statistically convergent) to 0 but not P -convergent, since A-density coincides with double natural density and C(1, 1)-statistical convergence coincides with the notion of statistical convergence for double sequences (see Mursaleen and Edely [17]), i.e., the double natural density of K is given by (2)

δC(1,1) {K} := δ2 {K} := P − lim m,n

1  (m, n) ∈ N2 : (m, n) ∈ K , mn

and x is statistically convergent to L if for each  > 0 P − lim m,n

1  (i, j) ∈ N2 , i ≤ m, j ≤ n : |xij − L| ≥  = 0. mn (2)

In this case we will write stC(1,1) − lim x = L or briefly st(2) − lim x = L. If A = I, the four dimensional identity matrix, then A− statistical convergence coincides with Pringsheim’s convergence. Statistical A−summability of a double sequence for a nonnegative RH−regular summability matrix has recently been defined in [2] and proved that it is stronger than A–statistical convergence for bounded double sequences. In [7], Edely and Mursaleen have given the notion
of statistical A− summability for single sequences. Let A = amn be a nonnegative RH−regular summability matrix and x = (xij ) ij be a double sequence. We say that x is statistically A−summable to L if for every  > 0, 
δ (2) (m, n) ∈ N2 : |ymn − L| ≥  = 0. So, if x is statistically A–summable to L then for every  > 0, P − lim m,n

1 |{(i, j) , i ≤ m, j ≤ n : |yij − L| ≥ }| = 0. mn

Note that if a double sequence is bounded and A−statistically convergent to L, then it is A−summable to L; hence it is statistically A−summable to L but not conversely (see [2]). 3. Main result Let I = [0, A], J = [0, B], A, B ∈ (0, 1) and K = I × J. We denote by C(K) the space of all continuous real valued functions on K. This space is a equipped with norm kf kC(K) := sup |f (x, y)|, f ∈ C(K). (x,y)∈K

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MURSALEEN, ALOTAIBI: KOROVKIN THEOREM

Let Hω (K) denote the space of all real valued functions f on K such that r s x 2 t y 2 | f (s, t) − f (x, y) |≤ ω(f ; ( − ) +( − ) ), 1−s 1−x 1−t 1−y where ω is the modulus of continuity, i.e. ω(f ; δ) =

{|f (s, t) − f (x, y)| :

sup

p (s − x)2 + (t − y)2 ≤ δ}.

(s,t),(x,y)∈K

It is to be noted that any function f ∈ Hω (K) is continuous and bounded on K. The following result was given by Ta¸sdelen and Eren¸cin [21]. Theorem A. Let (Tj,k ) be a double sequence of positive linear operators from Hω (K) into C(K). Then for all f ∈ Hω (K)



P - lim Tj,k (f ; x, y) − f (x, y) = 0. (1)

j,k→∞

C(K)

if and only if



P - lim

Tj,k (fi ; x, y) − fi j,k→∞

= 0 (i = 0, 1, 2, 3),

(2)

C(K)

where f0 (x, y) = 1, f1 (x, y) = and f3 (x, y) = (

y x , f2 (x, y) = , 1−x 1−y

x 2 y 2 ) +( ). 1−x 1−y

We prove the following result:
Theorem 3.1. Let A = amn jk be nonnegative RH–regular summability matrix method. Let (Tj,k ) be a double sequence of positive linear operators from Hω (K) into C(K). Then for all f ∈ Hω (K)

∞,∞

X mn

(2)

st - lim ajk Tj,k (f ; x, y) − f (x, y) = 0. (3.1.0)

m,n→∞

if and only if

C(K)

j,k=1,1

∞,∞

X mn

st - lim ajk Tj,k (1; x, y) − 1

m,n→∞ (2)

j,k=1,1

= 0,

(3.1.1)

C(K)

∞,∞

X mn s x

; x, y) − = 0, st - lim a T ( j,k jk

m,n→∞ 1 − s 1 − x C(K) j,k=1,1

∞,∞

X mn

t y (2)

st - lim a T ( ; x, y) − = 0, j,k jk

m,n→∞ 1 − t 1 − y C(K) j,k=1,1 (2)

(3.1.2)

(3.1.3)

∞,∞

X mn

s t x y 2 2 2 2 ) +( ) ; x, y) − (( ) +( ) ) st - lim ajk Tj,k ((

m,n→∞ 1−s 1−t 1−x 1−y (2)

= 0.

C(K)

j,k=1,1

(3.1.4)

1040

MURSALEEN, ALOTAIBI: KOROVKIN THEOREM y y 2 x 2 x , 1−y , ( 1−x ) + ( 1−y ) belongs to Hω (K), conditions (3.1.1)– Proof. Since each 1, 1−x (3.1.4) follow immediately from (3.1.0). Let f ∈ Hω (K) and (x, y) ∈ K be fixed. Then after using the properties of f, a simple calculation gives that

| Tj,k (f ; x, y)−f (x, y) |≤ Tj,k (| f (s, t)−f (x, y) |; x, y)+ | f (x, y) || Tj,k (f0 ; x, y)−f0 (x, y) | 2N 4N ) | Tj,k (f0 ; x, y) − f0 (x, y) | + 2 | Tj,k (f1 ; x, y) − f1 (x, y) | 2 δ δ 4N 2N + 2 | Tj,k (f2 ; x, y) − f2 (x, y) | + 2 | Tj,k (f3 ; x, y) − f3 (x, y) | δ δ ≤ ε + M { | Tj,k (f0 ; x, y) − f0 (x, y) | + | Tj,k (f1 ; x, y) − f1 (x, y) | ≤ ε + (ε + N +

+ | Tj,k (f2 ; x, y) − f2 (x, y) | + | Tj,k (f3 ; x, y) − f3 (x, y) | }, where N =k f kC(K) and B 2 4N 4N 2N A 2 A B 2N ) +( ) ), 2 ( ), 2 ( ), 2 }. (( 2 δ 1−A 1−B δ 1−A δ 1−B δ P∞,∞ mn Now replacing Tj,k (f ; x, y) by j,k=1,1 ajk Tj,k (f ; x, y) and taking sup(x,y)∈K , we get M = max{ε + N +

∞,∞

X mn

a T (f ; x, y)−f (x, y) j,k jk

C(K)

j,k=1,1

 ∞,∞

X mn

≤ ε+M a T (f ; x, y)−f (x, y) j,k 0 0 jk



∞,∞

X mn

+ a T (f ; x, y)−f (x, y) j,k 1 1 jk

C(K)

j,k=1,1

C(K)

j,k=1,1

∞,∞

X mn

+ a T (f ; x, y)−f (x, y) j,k 2 2 jk

C(K)

j,k=1,1

∞,∞

X mn

+ a T (f ; x, y) − f (x, y) j,k 3 3 jk

 .

(3.1.5)

C(K)

j,k=1,1

For a given r > 0 choose ε > 0 such that ε < r . Define the following sets

∞,∞

X mn

D := {(j, k), j ≤ m and k ≤ n : a T (f ; x, y) − f (x, y) ≥ r}, j,k jk

C(K)

j,k=1,1

∞,∞

X mn

D1 := {(j, k), j ≤ m and k ≤ n : a T (f ; x, y) − f (x, y) j,k 0 0 jk

≥

r−ε }, 4K

∞,∞

X mn

D2 := {(j, k), j ≤ m and k ≤ n : a T (f ; x, y) − f (x, y) j,k 1 1 jk

≥

r−ε }, 4K

∞,∞

X mn

D3 := {(j, k), j ≤ m and k ≤ n : a T (f ; x, y) − f (x, y) j,k 2 2 jk

≥

r−ε }, 4K

∞,∞

X mn

D4 := {(j, k), j ≤ m and k ≤ n : ajk Tj,k (f3 ; x, y) − f3 (x, y)

≥

r−ε }. 4K

j,k=1,1

j,k=1,1

j,k=1,1

j,k=1,1

C(K)

C(K)

C(K)

C(K)

Then from (2.1.5), we see that D ⊂ D1 ∪ D2 ∪ D3 ∪ D4 and therefore δ (2) {D} ≤ δ (2) {D1 } + δ (2) {D2 } + δ (2) {D3 } + δ (2) {D4 }. Hence conditions (3.1.1)–(3.1.4) imply the condition (3.1.0).

1041

MURSALEEN, ALOTAIBI: KOROVKIN THEOREM

This completes the proof of the theorem. If we replace the matrix A by identity double matrix in Theorem 3.1, then we immediately get the following result: Corollary 3.2 [6]. Let (Tj,k ) be a double sequence of positive linear operators from Hω (K) into C(K). Then for all f ∈ Hω (K)



(2)

st - lim T (f ; x, y) − f (x, y) = 0. (3.2.0) j,k

j,k→∞

C(K)

if and only if



st - lim T (1; x, y) − 1 j,k

(2)

j,k→∞

= 0,

(3.2.1)

C(K)



x s

; x, y) − = 0, st - lim Tj,k ( j,k→∞ 1−s 1 − x C(K)



y t

st(2) - lim ; x, y) − T ( = 0, j,k j,k→∞ 1−t 1 − y C(K) (2)

(3.2.2) (3.2.3)



t x y s 2 2 2 2 ) +( ) ; x, y) − (( ) +( ) ) = 0. (3.2.4) st - lim Tj,k ((

j,k→∞ 1−s 1−t 1−x 1 − y C(K) (2)

4. Example and the concluding remark We show that the following double sequence of positive linear operators satisfies the conditions of Theorem 3.1 but does not satisfy the conditions of Corollary 3.2 and Theorem A. Example 4.1. Consider the following Meyer-K¨onig and Zeller operators: Bm,n (f ; x, y) := (1−x)

m+1

(1−y)

n+1

 ∞ X ∞ X f j=0 k=0

j k , j+m+1 k+n+1



m+j j



 n+k j k xy , k

(4.1.1) where f ∈ Hω (K), and K = [0, A] × [0, B], A, B ∈ (0, 1). Since, for x ∈ [0, A], A ∈ (0, 1),  ∞  X m+j j 1 x, = (1 − x)m+1 j j=0 it is easy to see that Bm,n (f0 ; x, y) = f0 (x, y). Also, we obtain m+1

Bm,n (f1 ; x, y) = (1 − x)

n+1

(1 − y)

∞ X ∞ X j=0 k=0

1042

   j m+j n+k j k xy m+1 j k

MURSALEEN, ALOTAIBI: KOROVKIN THEOREM

= (1 − x)

m+1

n+1

(1 − y)

x

∞ X ∞ X j=0 k=0

= (1 − x)m+1 (1 − y)n+1 x

  1 (m + j)! n + k j−1 k x y m + 1 m!(j − 1)! k

1 1 x , = (1 − x)m+2 (1 − y)n+1 (1 − x)

and similarly Bm,n (f2 ; x, y) =

y . (1 − y)

Finally, we get Bm,n (f3 ; x, y) = (1−x)

m+1

n+1

(1−y)

∞ X ∞ X j=0

m+1

= (1 − x)

(1 − y)

m+1

+(1 − x)

m+1

= (1 − x)

n+1

(1 − y)

n+1

n+1

(1 − y)

   j k 2 m+j n+k j k 2 {( ) +( )} xy m+1 n+1 j k k=0

  ∞ ∞ x XX j (m + j)! n + k j−1 k x y m + 1 j=0 k=0 m + 1 m!(j − 1)! k   ∞ ∞ y XX k m + j (n + k)! j k−1 xy n + 1 j=0 k=0 n + 1 j n!(k − 1)!

  ∞ X ∞ X x (m + j + 1)! n + k j−1 k {x x y m + 1 j=0 k=0 (m + 1)!(j − 1)! k

  ∞ X ∞  X m+j+1 n+k j k + xy } j k j=0 k=0 m+1

+(1 − x)

(1 − y)

n+1

  ∞ ∞ X X (n + k + 1)! m + j j k−1 y {y xy n + 1 j=0 k=0 (n + 1)!(k − 1)! j

  ∞ X ∞  X n+k+1 m+j j k + xy } k j j=0 k=0 m+2 x 2 1 x n+2 y 2 1 y ( ) + + ( ) + m+1 1−x m+11−x n+1 1−y n+11−y x 2 y 2 →( ) +( ). 1−x 1−y

=

Therefore the conditions of Theorem A are satisfied and we get for all f ∈ Hω (K) that



P - lim C(K)

Tj,k (f ; x, y) − f (x, y) = 0. j,k→∞

Now, take A = C(1, 1) and define w = (wmn ) by wmn = (−1)m for all n. Then this sequence is neither P –convergent nor A–statistically convergent but st(2) – lim Aw = 0 (since P –lim Aw = 0). Let Lm,n : Hω (K) → C(K) be defined by Lm,n (f ; x, y) = (1 + wmn )Bm,n (f ; x, y).

1043

MURSALEEN, ALOTAIBI: KOROVKIN THEOREM

It is easy to see that the sequence (Lm,n ) satisfies the conditions (3.1.1), (3.2.2), (3.1.3) and (3.1.4). Hence by Theorem 3.1, we have (2)

stC(1,1) - lim kLm,n (f ; x, y) − f (x, y)k = 0. m,n→∞

On the other hand, the sequence (Lm,n ) does not satisfy the conditions of Theorem A and Corollary 3.2, since (Lm,n ) is neither P –convergent nor A–statistically convergent. That is, Theorem A and Corollary 3.2 do not work for our operators Lm,n . Hence our Theorem 3.1 is stronger than Theorem A and Corollary 3.2.

References [1] G. A. Anastassiou, M. Mursaleen and S. A. Mohiuddine, Some approximation theorems for functions of two variables through almost convergence of double sequences, Jour. Comput. Analy. Appl., 13 (2011) 37–40. [2] C. Belen, M. Mursaleen and M. Yildirim, Statistical A–summability of double sequences and a Korovkin type approximation theorem, Bull. Korean Math. Soc., 49(4) (2012) 851–861. [3] K. Demirci and S. Karaku¸s, Statistical A–summability of positive linear operators, Math. Comput. Model., 53 (2011) 189–195. [4] K. Demirci and S. Karaku¸s, Korovkin-type approximation theorem for double sequences of positive linear operators via statistical A–summability, Results. Math., doi: 10.1007/s00025-011-0140-y. [5] F. Dirik and K. Demirci, Korovkin type approximation theorem for functions of two variables in statistical sense, Turk. J. Math., 34 (2010) 73–83. [6] F. Dirik and K. Demirci, A Korovkin type approximation theorem for double sequences of positive linesr operators of two variables in A–statistical sense, Bull. Korean Math. Soc., 47 (2010) 825–837. [7] O. H. H. Edely and M. Mursaleen, On statistical A–summability, Math. Comput. Model., 49 (2009) 672–680. [8] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951) 241–244. [9] H. J. Hamilton, Transformations of multiple sequences, Duke Math. J., 2 (1936) 29–60. [10] P. P. Korovkin, Linear Operators And Approximation Theory, Hindustan Publ. Co., Delhi, 1960. [11] S. A. Mohiuddine, An application of almost convergence in approximation theorems, Appl. Math. Lett., 24 (2011) 1856–1860. [12] S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical summability (C, 1) and a Korovkin type approximation theorem, Jour. Ineq. Appl. 2012, 2012:172 doi:10.1186/1029-242X-2012-172.

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MURSALEEN, ALOTAIBI: KOROVKIN THEOREM

[13] F. Moricz, Statistical convergence of multiple sequences, Arch. Math., 81 (2003) 82–89. [14] M. Mursaleen and A. Alotaibi, Statistical summability and approximation by de la Vall´ee-Poussin mean, Appl. Math. Lett., 24 (2011) 320–324 [Erratum: Appl. Math. Letters, 25 (2012) 665]. [15] M. Mursaleen and A. Alotaibi, Statistical lacunary summability and a Korovkin type approximation theorem, Ann. Univ. Ferrara, 57(2) (2011) 373–381. [16] M. Mursaleen and A. Alotaibi, Korovkin type approximation theorem for functions of two variables through statistical A-summability, Adv. Difference Equ., 2012, 2012:65, doi:10.1186/1687-1847-2012-65. [17] M. Mursaleen and Osama H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003) 223–231. [18] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Z., 53 (1900) 289–321. [19] G. M. Robison, Divergent double sequences and series, Trans. Amer. Math. Soc., 28 (1926) 50–73. [20] H. M. Srivastava, M. Mursaleen and Asif Khan, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. Comput. Model., 55 (2012) 2040–2051. [21] F. Ta¸sdelen and A. Eren¸cin, The generalization of bivariate MKZ operators by multiple generating functions, J. Math. Anal. Appl., 331 (2007) 727–735.

1045

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1046-1056, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

The properties of logistic function and applications to neural network approximation∗ Zhixiang Chen1

Feilong Cao2†

1. Department of Mathematics, Shaoxing University, Shaoxing 312000, Zhejiang Province, P R China 2. Department of Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang Province, P R China

Abstract This paper discusses some analytic properties of logistic function and estimates some approximation errors by operators with logistic function. Firstly, an equation of partitions of unity for the logistic function is given. Then, two kinds of quasi-interpolation type neural network operators are constructed to approximate univariate and bivariate functions, respectively. Also, the errors of the approximation are estimated by means of the modulus of continuity of function. Moreover, for approximated functions with high order derivatives, the approximation errors by the constructed operator are estimated. Keywords: logistic function, approximation, modulus of continuity, neural network operators MSC: 41A25, 41A63

1

Introduction

A function σ defined on ℝ is called a sigmoid function if the following conditions are satisfied: lim σ(x) = a,

𝑥→+∞

lim σ(x) = b, a ̸= b.

𝑥→−∞

Sigmoid function is a kind of important function, which usually is taken as activation function of neural networks. A familiar example of sigmoid function is the logistic function defined by σ(x) =

1 . 1 + e−𝑥

(1.1)

Feed-forward neural networks (FNNs) with one hidden layer are mathematically expressed as N𝑛 (x) =

𝑛 ∑

c𝑗 σ(⟨a𝑗 · x⟩ + b𝑗 ), x ∈ ℝ𝑠 ,

(1.2)

𝑗=1

where σ is the activation function of the networks, for 1 ≤ j ≤ n, b𝑗 ∈ ℝ are the thresholds, a𝑗 ∈ ℝ𝑠 , c𝑗 ∈ ℝ are input and output weights, respectively, and ⟨a𝑗 · x⟩ is the inner product of a𝑗 and x. It is well-known that FNNs are universal approximators. Theoretically, any continuous function defined on a compact set can be approximated to any desired degree of accuracy by increasing the number of hidden neurons (see [1]-[7]), which is called density problem. Yet, a related and important problem is that of complexity: determining the number of neurons required to guarantee that all functions (belonging to a certain class) can be approximated to the prescribed degree of accuracy. We refer the reader to the related literature, for example, [8]-[12]. ∗ This

research was supported by the National Natural Science Foundation of China(Nos. 61179041, 61272023) author. Email: [email protected]

† Corresponding

1046

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

Recently, in [13] we discussed some properties of logistic function (1.1), and obtained an equation of partition of unity for the function. Also, we constructed quasi-interpolation type neural network operators, and gave the estimates error of approximation. The further properties on the logistic function were studied in [14] where a class of rational quasi-interpolation type neural network operators was constructed by using the logistic function. To approximate multivariate functions, a class of rational function type quasi-interpolation neural network operators with hyperbolic tangent activation function was constructed in [15] and [16]. In this paper, motivated by inspiring articles [14]-[16], we further investigate the properties of logistic function, and give some applications to neural network approximation.

2

Analytic properties of logistic function and applications to neural networks

It is easy to see that 1 e𝑥 1 1 1 2 = 𝑥 =1− 𝑥 , = 𝑥 − , +1 e +1 e + 1 e𝑥 + 1 e − 1 e2𝑥 − 1

e−𝑥

) 2 1 1 1 lim − 2𝑥 = = 𝑥 . 𝑥→0 e𝑥 − 1 e −1 2 e + 1 𝑥=0 ∑∞ Applying expansion (see P. 97 of [17]): ex𝑥−1 = 𝑛=0 B𝑛!𝑛 x𝑛 , where B𝑛 is Bernoulli number, we have ∞ ∑ x 2x (1 − 2𝑛 )B𝑛 𝑛 − = x , 𝑥 2𝑥 e − 1 e − 1 𝑛=1 n! (

and

which leads to

∞ ∑ 1 2 (1 − 2𝑛 )B𝑛 𝑛−1 − = x . 𝑥 2𝑥 e − 1 e − 1 𝑛=1 n!

From the above arguments, we get σ(x) = (1 + B1 ) +

(22 − 1)B2 (23 − 1)B3 2 (2𝑛 − 1)B𝑛 𝑛−1 x+ x + ··· + x + ···. 2! 3! n!

Considering B3 = B5 = · · · = B2𝑘+1 = 0, k = 1, 2, . . . , and (see Section 23.1.15 of [18]) 1 2(2k)! B1 = − , |B2𝑘 | > ̸= 0, k = 1, 2, . . . , 2 (2π)2𝑘 we therefore have ∞

σ(x) =

1 ∑ + b2𝑘−1 x2𝑘−1 , b2𝑘−1 ̸= 0, k = 1, 2, . . . . 2

(2.3)

𝑘=1

Thus, we have proved that Theorem 1. Logistic function σ defined by (1.1) has property: σ(0) ̸= 0, σ (2𝑘−1) (0) ̸= 0 for k = 1, 2, . . .. We denote the space of continuous functions defined on [a, b] by C([a, b]), which is endowed with the uniform norm. By M¨ untz Theorem (see Example 13 in P. 192 of [19]), we know that span{1, x, x3 , . . . , x2𝑘−1 , . . .} is dense in C([0, 1]). Let Λ𝑛 := span{1, x, x3 , . . . , x2𝑛−1 }, n ∈ N, and E𝑛 (f, Λ𝑛 ) := inf g∈Λ𝑛 ∥f − g∥ be the best approximation to f ∈ C([0, 1]) from Λ𝑛 . Now we give a approximation theorem by FNNs:

1047

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

Theorem 2. Let f ∈ C([0, 1]), n ∈ N. Then for any ε > 0, there exist FNNs: N𝑛 (x) =

𝑛 ∑

a𝑘 σ(b𝑘 x)

𝑘=0

where a𝑘 , b𝑘 ∈ ℝ, such that (

3

e2 |f (x) − N𝑛 (x)| ≤ 80ω f, n

) + ε, x ∈ [0, 1],

where ω(f, ·) is modulus of continuity defined by ω(f, t) = sup𝑥,y∈[0,1],|𝑥−y|≤𝑡 |f (x) − f (y)|. Proof. For f ∈ C([0, 1]), there exists p ∈ Λ𝑛 , such that ∥f − p∥ ≤ E𝑛 (f, Λ𝑛 ). From [20], it follows that ( ) 3 e2 E𝑛 (f, Λ𝑛 ) ≤ 80ω f, , n which implies

(

3

e2 ∥f − p∥ ≤ 80ω f, n

) .

From Proposition 1 of [21], it follows that for any ε > 0, there exist real numbers (a𝑖 )0≤𝑖≤𝑛 and (b𝑖 )0≤𝑖≤𝑛 such that 𝑛 ∑ a𝑘 σ(b𝑘 x) ≤ ε, x ∈ [0, 1]. p(x) − 𝑘=0 ∑𝑛 Therefore, for any f ∈ C([0, 1]), there exist FNNs form as N𝑛 (x) = 𝑘=0 a𝑘 σ(b𝑘 x), such that ) ( 3 𝑛 ∑ e2 a𝑘 σ(b𝑘 x) ≤ 80ω f, + ε, x ∈ [0, 1]. f (x) − n 𝑘=0

This completes the proof of Theorem 2.  The following theorem called M¨ untz denseness Theorem on [−1, 1] can be found in [19](see Example 21 in P. 205 of [19]). Theorem M. Suppose Λ := (λ𝑖 )∞ 𝑖=1 is a sequence of distinct nonnegative integers. Then span{1, xλ1 , xλ2 , . . .} is dense in C([−1, 1]) if and only if ∑ λ𝑖 is even

1 = ∞ and λ𝑖

∑ λ𝑖 is odd

1 = ∞. λ𝑖

From Theorem M, we know that the condition C([0, 1]) in Theorem 2 can not be altered to C([−1, 1]). x ˜ (x) satisfies Now we consider the derivatives of σ: σ ˜ (x) := (σ(x))′ = (exe+1)2 . Obviously, σ lim σ ˜ (x) =

𝑥→ −∞

lim σ ˜ (x) = 0,

𝑥→ +∞

and the function σ ¯ (x) := σ(x) + σ ˜ (x) has expansion: σ ¯ (x) = and satisfies: (i) lim σ ¯ (x) = 1, 𝑥→ +∞ (𝑘)

3 (22 − 1)B2 3(24 − 1)B4 2 + x+ x + ···, 4 2! 4!

lim σ ¯ (x) = 0;

𝑥→ −∞

(ii) σ ¯ (0) ̸= 0, k = 0, 1, 2, . . .. Hence, combining Proposition 1 of [21] and the above deduction, we get

1048

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

Proposition 1. Let f ∈ C([−1, 1]), n ∈ N. Then for any ε > 0, there exist FNNs: N𝑛 (x) =

𝑛 ∑

a𝑘 σ ¯ (b𝑘 x),

𝑘=0

such that |f (x) − N𝑛 (x)| ≤ E𝑛 (f ) + ε, where E𝑛 (f ) denotes the best approximation to f from the space of polynomials with degree at most n. Remark 1. Proposition 1 can be generalized to the case of f ∈ C([−1, 1]d ), d ∈ N.

3

Partitions of unity of logistic function and applications to neural networks

Set

1 (σ(x + 1) − σ(x − 1)), x ∈ ℝ. 2 Then, ψ(x) has properties (see [13], [14]): (i) ψ(x) ∑∞ > 0; (ii) ∑𝑘=−∞ ψ(x − k) = 1, ∀x ∈ ℝ; ∞ (iii) 𝑘=−∞ ψ(nx − k) = 1, ∀x ∈ ℝ, n ∈ N; (iv) ψ(x) is a density function; (v) ψ(x) is even: Φ(−x) = Φ(x); (vi) ψ(x) is decreasing on ℝ+ . For n ∈ N, we set e𝑛𝑥 φ(x) ˜ := σ(nx) = 𝑛𝑥 , e +1 and 1 1 ˜ − ), Φ(x) := φ(x ˜ + ) − φ(x 2 2 then, Φ(x) has properties as follows. Theorem 3. Φ(x) is even function. Moreover, for x > 0, Φ is positive and strictly decreasing. Proof. Since 1 1 1 e𝑛(𝑥+ 2 ) 1 e𝑛(𝑥− 2 ) φ(x ˜ + ) = 𝑛(𝑥+ 1 )+1 , φ(x ˜ − ) = 𝑛(𝑥− 1 )+1 , 2 2 2 2 e e ψ(x) :=

by straightforward calculations, we have 1 1 φ(−x ˜ + ) − φ(−x ˜ − ) = 2 2 =

1

1

e𝑛(−𝑥+ 2 ) 𝑛(−𝑥+ 12 )+1

e

1 1 e𝑛(𝑥− 2 )+1

−

−

e𝑛(𝑥+ 2 ) 1

𝑛(−𝑥− 12 )+1

e

1 1 e𝑛(𝑥+ 2 )+1

1

=

e𝑛(−𝑥− 2 )

1

−

e𝑛(𝑥− 2 ) 1

e𝑛(𝑥+ 2 ) + 1 e𝑛(𝑥− 2 ) + 1 1 1 ˜ − ). = φ(x ˜ + ) − φ(x 2 2 So Φ(−x) = Φ(x). That is to say that Φ(x) is even function. From 1 1 e𝑛(𝑥+ 2 ) e𝑛𝑥 e𝑛𝑥 (e 2 − e− 2 ) φ(x ˜ + ) − φ(x ˜ − ) = 𝑛(𝑥+ 1 ) − 𝑛(𝑥− 1 ) = 𝑛(𝑥+ 1 ) , 1 2 2 2 + 1 2 + 1 2 + 1)(e𝑛(𝑥− 2 ) + 1) e e (e 1

𝑛

we get Φ(x) > 0.

1049

𝑛

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

Finally, we will prove that when x > 0, Φ(x) is strictly decreasing. In fact, from standard calculations, it follows that ( )′ 1 1 1 e𝑛(𝑥+ 2 ) = n(e𝑛(𝑥+ 2 ) + 1)−2 e𝑛(𝑥+ 2 ) =: Λ1 , 1 𝑛(𝑥+ ) 2 + 1 e ( )′ 1 1 1 e𝑛(𝑥− 2 ) = n(e𝑛(𝑥− 2 ) + 1)−2 e𝑛(𝑥− 2 ) =: Λ2 . 1 𝑛(𝑥− ) 2 + 1 e Then

( Λ1 − Λ2

= ne𝑛𝑥

𝑛

e2 1

)

e− 2

𝑛

−

1

(e𝑛(𝑥+ 2 ) + 1)2 (e𝑛(𝑥− 2 ) + 1)2 𝑛 2𝑛𝑥 −𝑛 (e − 1)(e 2 − e 2 ) = ne𝑛𝑥 𝑛(𝑥+ 1 ) < 0, 1 2 + 1)2 (e𝑛(𝑥− 2 ) + 1)2 (e 

which shows that Φ(x) is strictly decreasing for x > 0. This finishes the proof of Theorem 3. In addition, from 1 1 e𝑛𝑥 (e 2 − e− 2 ) φ(x ˜ + ) − φ(x ˜ − ) = 𝑛(𝑥+ 1 ) , 1 2 2 2 + 1)(e𝑛(𝑥− 2 ) + 1) (e 𝑛

𝑛

we can obtain 𝑛

e2 Φ(x) < 𝑛𝑥 . e ∑∞ Theorem 4. For x ∈ ℝ, there holds 𝑖=−∞ Φ(x − i) = 1. Proof. Obviously, for x ∈ ℝ, ∞ ∞ ( ∑ ∑ 1 Φ(x − i) = φ(x ˜ + − i) − φ(x ˜ − 2 𝑖=−∞ 𝑖=−∞ ∞ ( ∑ 1 1 = φ(x ˜ + − i) − φ(x ˜ − 2 2 𝑖=0 −1 ( ∑ 1 + φ(x ˜ + − i) − φ(x ˜ − 2 𝑖=−∞

(3.4)

) 1 − i) 2 ) − i) ) 1 − i) . 2

Since 𝑗 ( ∑ 𝑖=0

) 1 1 φ(x ˜ + − i) − φ(x ˜ − − i) 2 2

1 1 1 3 = φ(x ˜ + ) − φ(x ˜ − ) + φ(x ˜ − ) − φ(x ˜ − ) + ··· 2 2 2 2 1 1 − j) − φ(x ˜ − − j) 2 2 1 1 = φ(x ˜ + ) − φ(x ˜ − − j), 2 2 + φ(x ˜ +

we have ) ) 𝑗 ( −1 ( ∑ ∑ 1 1 1 1 φ(x ˜ + − i) − φ(x ˜ − − i) = ˜ − + i) φ(x ˜ + + i) − φ(x 2 2 2 2 𝑖=−𝑗 𝑖=1 3 1 5 3 = φ(x ˜ + ) − φ(x ˜ + ) + φ(x ˜ + ) − φ(x ˜ + ) + ··· 2 2 2 2 1 1 ˜ − + j) + φ(x ˜ + + j) − φ(x 2 2 1 1 = φ(x ˜ + + j) − φ(x ˜ + ). 2 2

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M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

From lim φ(x ˜ −

𝑗→∞

1 − j) = 0, 2

lim φ(x ˜ +

𝑗→∞

1 + j) = 1, 2

∑∞ we get 𝑖=−∞ Φ(x − i) = 1. Thus, the ∑ proof of Theorem 4 is completed.  ∞ Particularly, for any n ∈ N, we have 𝑖=−∞ Φ(nx−i) = 1, x ∈ ℝ. Therefore, we can construct operator for f ∈ C([−1, 1]): ( ) 𝑛 ∑ k F𝑛 (f, x) := f Φ(nx − k). n 𝑘=−𝑛

Now we prove the the following estimates of approximation error by the operator. Theorem 5. Let 0 < α < 1, n ∈ N, and 2n1−𝛼 − 3 > 0. Then for any f ∈ C([−1, 1]), there holds ( ) ( ) 1−α 3 1 e−𝑛(𝑛 − 2 ) −𝑛 2 |f (x) − F𝑛 (f, x)| ≤ ω f, 𝛼 + 4 e ∥f ∥. + n n Proof. Obviously, we have |f (x) − F𝑛 (f, x)|

= ≤ ≤

( ) ∞ 𝑛 ∑ ∑ k Φ(nx − k) − f Φ(nx − k) f (x) n 𝑘=−∞ 𝑘=−𝑛 𝑛 ( ( )) ∑ ∑ k Φ(nx − k) + |f (x)|Φ(nx − k) f (x) − f −𝑛 n |𝑘|≥𝑛+1 ( ) 𝑛 ∑ ∑ f (x) − f k Φ(nx − k) + ∥f ∥ Φ(nx − k) n |𝑘|≥𝑛+1

𝑘=−𝑛

=: ∆1 + ∥f ∥∆2 , where the fact Φ(x) > 0 proved in Theorem 3 is used. Next we estimate △1 and △2 , respectively. For 0 < α < 1, we have ( ) ( ) ∑ ∑ k k ∆1 = f (x) − f n Φ(nx − k) + f (x) − f n Φ(nx − k) 𝑘 1 𝑘 1 𝑘:|𝑥− 𝑛 |≤ 𝑛α

𝑘:|𝑥− 𝑛 |> 𝑛α

) ∑ ( ∞ ∑ 1 Φ(nx − k) + 2∥f ∥ Φ(nx − k) ≤ ω f, 𝛼 n 𝑘 𝑘=−∞ 𝑘:|𝑥− 𝑛 |> 𝑛1α ( ) ∑ 1 ≤ ω f, 𝛼 + 2∥f ∥ Φ(nx − k). n 1−α 𝑘:|𝑛𝑥−𝑘|>𝑛

From (4) and the fact that Φ(x) is strictly decreasing proved by Theorem 3, we obtain ∫ +∞ ∫ +∞ 𝑛 ∑ 1−α 3 e2 2 Φ(nx − k) ≤ 2 Φ(x)dx ≤ 2 dx = e−𝑛(𝑛 − 2 ) . 𝑛𝑥 e n 1−α 1−α 𝑛 −1 𝑛 −1 1 𝑘 𝑘:|𝑥− 𝑛 |> 𝑛α

Since −n ≤ nx ≤ n, |k| ≥ n + 1, then |nx − k| ≥ 1, and ( ) ( ∫ ∞ ∫ 𝑛 ∆2 ≤ 2 Φ(1) + Φ(x)dx ≤ 2 e− 2 + 1

Above arguments lead to

1

∞

) 𝑛 𝑛 e2 dx ≤ 4e− 2 . e𝑛𝑥

) ( 1−α 3 𝑛 4 1 ω f, 𝛼 + e−𝑛(𝑛 − 2 ) ∥f ∥ + 4e− 2 ∥f ∥ n n ( ) ) ( 1−α 3 1 e−𝑛(𝑛 − 2 ) −𝑛 2 ∥f ∥. ≤ ω f, 𝛼 + 4 e + n n

|f (x) − F𝑛 (f, x)| ≤

1051

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

The proof of Theorem 5 is completed.  Let f ∈ CB (ℝ) be the set of continuous and bounded functions on ℝ. We construct operator for f ∈ CB (ℝ): ( ) ∞ ∑ k ¯ F𝑛 (f, x) := f Φ(nx − k). n 𝑘=−∞

Then

∞ ∞ ∑ (k) ∑ |F¯𝑛 (f, x) − f (x)| = f Φ(nx − k) − f (x)Φ(nx − k) n 𝑘=−∞ 𝑘=−∞ ( ) ∞ ( ) ∑ ∑ f k − f (x) Φ(nx − k) = f (x) − f k Φ(nx − k) ≤ n n 𝑘 𝑘=−∞ 𝑘:|𝑥− 𝑛 |≤ 𝑛1α ( ) ( ) ∑ ∑ k 1 + Φ(nx − k) f (x) − f n Φ(nx − k) ≤ ω f, n𝛼 + 2∥f ∥ 𝑘 |𝑛𝑥−𝑘|>𝑛1−α 𝑘:|𝑥− 𝑛 |> 𝑛1α ( ) 1 4∥f ∥ −𝑛(𝑛1−α − 3 ) 2 . ≤ ω f, 𝛼 + e n n

Hence, we get Theorem 6. Let 0 < α < 1, n ∈ N, and 2n1−𝛼 − 3 > 0. Then for any f ∈ CB (ℝ), we have ( ) 1−α 3 1 4 |F¯𝑛 (f, x) − f (x)| ≤ ω f, 𝛼 + e−𝑛(𝑛 − 2 ) ∥f ∥. n n Let Ψ(x) := Ψ(x1 , x2 ) := Φ(x1 )Φ(x2 ), x = (x1 , x2 ) ∈ ℝ2 . Then

∞ ∑ 𝑘=−∞

and

∞ ∑

∞ ∑

Ψ(x − k) :=

∞ ∑

(3.5)

Ψ(x1 − k1 , x2 − k2 ) = 1,

𝑘1 =−∞ 𝑘2 =−∞ ∞ ∑

Ψ(nx − k) :=

𝑘=−∞

∞ ∑

Ψ(nx1 − k1 , nx2 − k2 ) = 1.

𝑘1 =−∞ 𝑘2 =−∞

For f (x1 , x2 ) ∈ C([−1, 1]2 ), we introduce operator: G𝑛 (f ; x1 , x2 ) :=

𝑛 ∑

𝑛 ∑

( f

𝑘1 =−𝑛 𝑘2 =−𝑛

k1 k2 , n n

) Ψ(nx1 − k1 , nx2 − k2 ) =:

𝑛 ∑ 𝑘=−𝑛

( ) k f Ψ(nx − k). n

We are interesting in the error f (x1 , x2 ) − G𝑛 (f ; x1 , x2 ) and will prove the following estimates. Theorem 7. Let f ∈ C([−1, 1]2 ), 0 < α < 1, n ∈ N, and 2n1−𝛼 − 3 > 0. Then ) ( ) ( 1 −𝑛(𝑛1−α − 3 ) 1 1 −𝑛 2 2 + e ∥f ∥, |G𝑛 (f ; x1 , x2 ) − f (x1 , x2 )| ≤ ω f ; 𝛼 , 𝛼 + 24 e n n n where ω (f ; δ1 , δ2 ) is the modulus of continuity of f defined by ω (f ; δ1 , δ2 ) =

sup 𝑥,y∈[−1,1]2 ,|𝑥𝑖 −y𝑖 |≤δ𝑖

|f (x) − f (y)|.

Proof. It is easy to see that G𝑛 (f ; x1 , x2 ) − f (x1 , x2 ) =

𝑛 ∑ 𝑘=−𝑛

( ) ∞ ∑ k Ψ(nx − k) − f (x) Ψ(nx − k) f n 𝑘=−∞

1052

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap ( ( ) ) 𝑛 ∑ k1 k2 f , − f (x1 , x2 ) Ψ(nx1 − k1 , nx2 − k2 ) n n

𝑛 ∑

=

𝑘1 =−𝑛 𝑘2 =−𝑛



∑

−f (x) 

∑

∑

Ψ(nx − k) +

|𝑘1 |≤𝑛 |𝑘2 |>𝑛

∑

∑

Ψ(nx − k) +

|𝑘1 |>𝑛 |𝑘2 |≤𝑛

∑

 Ψ(nx − k)

|𝑘1 |>𝑛 |𝑘2 |>𝑛

=: ∆3 − f (x)(∆4 + ∆5 + ∆6 ). From the deductive process of Theorem 5, we can obtain that ∑ ∑ ∑ ∑ ∆4 = Ψ(nx1 − k1 , nx2 − k2 ) = Φ(nx1 − k1 )Φ(nx2 − k2 ) |𝑘1 |≤𝑛 |𝑘2 |>𝑛

≤

∑

|𝑘1 |≤𝑛 |𝑘2 |>𝑛 −𝑛 2

Φ(nx2 − k2 ) ≤ 4e

,

|𝑘2 |>𝑛

∑

∆5 =

∑

Ψ(nx1 − k1 , nx2 − k2 ) ≤ 4e− 2 , 𝑛

|𝑘1 |>𝑛 |𝑘2 |≤𝑛

and ∆6 =

∑

∑

Ψ(nx1 − k1 , nx2 − k2 ) =

∑

Φ(nx1 − k1 )

So, we have

Φ(nx2 − k2 ) ≤ 16e−𝑛 .

|𝑘2 |>𝑛

|𝑘1 |>𝑛

|𝑘1 |>𝑛 |𝑘2 |>𝑛

∑

∆4 + ∆5 + ∆6 ≤ 24e− 2 . 𝑛

On the other hand, 𝑛 ∑ |∆3 | =

𝑘1 =−𝑛 𝑘2 =−𝑛

≤

∑

𝑘1 :|𝑥1 −

+

+

|≤ 𝑛1α

𝑘1 𝑛

|> 𝑛1α

𝑘1 𝑛

|≤ 𝑛1α

𝑘1 𝑛

|> 𝑛1α 𝑘2 :|𝑥2 −

∑ 𝑘1 :|𝑥1 −

+

𝑘1 𝑛

∑ 𝑘1 :|𝑥1 −

∑ 𝑘1 :|𝑥1 −

Ψ(nx1 − k1 , nx2 − k2 ) f (x1 , x2 ) − f ( ) ∑ f (x1 , x2 ) − f k1 , k1 Ψ(nx − k) n n 𝑘 𝑘2 :|𝑥2 − 𝑛2 |≤ 𝑛1α ( ) ∑ f (x1 , x2 ) − f k1 , k1 Ψ(nx − k) n n 𝑘 𝑘2 :|𝑥2 − 𝑛2 |≤ 𝑛1α ( ) ∑ f (x1 , x2 ) − f k1 , k1 Ψ(nx − k) n n 𝑘 𝑘2 :|𝑥2 − 𝑛2 |> 𝑛1α ) ( ∑ f (x1 , x2 ) − f k1 , k1 Ψ(nx − k) n n 𝑘 1

( 𝑛 ∑

(

2 𝑛

k1 k1 , n n

))

|> 𝑛α

( ) 1 1 8∥f ∥ −𝑛(𝑛1−α − 3 ) 8∥f ∥ −𝑛(2𝑛1−α −3) 2 + ≤ ω f; 𝛼 , 𝛼 + e e n n n n2 ( ) 1 1 16∥f ∥ −𝑛(𝑛1−α − 3 ) 2 . ≤ ω f; 𝛼 , 𝛼 + e n n n Collecting the above estimates, we have ) ( ( ) 1 1 1 −𝑛(𝑛1−α − 3 ) −𝑛 2 |G𝑛 (f ; x1 , x2 ) − f (x1 , x2 )| ≤ ω f ; 𝛼 , 𝛼 + 24 e 2 + e ∥f ∥. n n n This completes the proof of Theorem 7.  Finally, we will discuss the high order of approximation by means of the smoothness of f . That is, we will prove the following Theorem 8.

1053

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

Theorem 8. Let 0 < α < 1 and 2n1−𝛼 − 3 > 0. Then for any f ∈ C N ([−1, 1]), N ∈ N, we have ( ) 𝑛 3 16 −𝑛(𝑛1−α − 3 ) 2 |F𝑛 (f, x) − f (x)| ≤ 4e− 2 ∥f ∥ + + e ∥f ∥N n𝛼 n ( ) 1 2(N +2) ∥f (N ) ∥ −𝑛(𝑛1−α − 3 ) (N ) 1 2 , + ω f , 𝛼 + e n n𝛼N N ! nN ! where ∥f ∥N = max{∥f ′ ∥, ∥f ′′ ∥, . . . , ∥f (N ) ∥}. Proof. Applying Taylor’s formula with integral remainder: f

( ) ∑ ( )𝑗 ∫ 𝑘 ( N ) ( 𝑘 − t)N −1 𝑛 k f (𝑗) (x) k = −x + f (N ) (t) − f (N ) (x) 𝑛 dt, n j! n (N − 1)! 𝑥 𝑗=0

we have 𝑛 ∑

f

𝑘=−𝑛

( ) k Φ(nx − k) = n

(

N 𝑛 ∑ f (𝑗) (x) ∑

j!

𝑗=0

𝑘=−𝑛

∫

𝑛 ∑

+

Φ(nx − k)

Φ(nx − k) 𝑛 ∑

= f (x)

f (N ) (t) − f (N ) (x)

Φ(nx − k) +

𝑘=−𝑛 𝑛 ∑

(

𝑥

𝑘=−𝑛

+

𝑘 𝑛

)𝑗 k −x n

𝑘 𝑛

Φ(nx − k)

(

j!

Φ(nx − k)

𝑘=−𝑛

f (N ) (t) − f (N ) (x)

𝑥

𝑘=−𝑛

(

𝑛 N ∑ f (𝑗) (x) ∑ 𝑗=1

∫

) ( 𝑘 − t)N −1 𝑛 dt (N − 1)! )𝑗 k −x n

) ( 𝑘 − t)N −1 𝑛 dt. (N − 1)!

Therefore, 𝑛 ∑ 𝑘=−𝑛

( ( ) )𝑗 N 𝑛 ∑ ∑ f (𝑗) (x) ∑ k k Φ(nx − k) + Φ(nx − k) f Φ(nx − k) − f (x) = −f (x) −x n j! n 𝑗=1

+

|𝑘|>𝑛

∫

𝑛 ∑

𝑘 𝑛

Φ(nx − k)

𝑘=−𝑛

(

𝑥

𝑘=−𝑛

) ( 𝑘 − t)N −1 f (N ) (t) − f (N ) (x) 𝑛 dt =: Ξ1 + Ξ2 + Ξ3 . (N − 1)!

The estimate of ∆2 implies immediately |Ξ1 | ≤ 4e− 2 ∥f ∥. Moreover, 𝑛

|Ξ2 | ≤

N 𝑛 ∑ |f (𝑗) (x)| ∑ 𝑗=1

j!

𝑘=−𝑛

𝑗 k Φ(nx − k) − x . n

Noting the fact 𝑛𝑘 − x ≤ 2 and using the estimate of ∆1 proved in Theorem 5, we obtain that 𝑛 ∑ 𝑘=−𝑛

≤

𝑗 k Φ(nx − k) − x ≤ n

∑ 𝑘 𝑘:| 𝑛 −𝑥|≤ 𝑛1α

𝑗 k Φ(nx − k) − x + n

∑ 𝑘 𝑘:| 𝑛 −𝑥|> 𝑛1α

1 2(𝑗+1) −𝑛(𝑛1−α − 3 ) 2 . + e n𝛼𝑗 n

Thus, |Ξ2 | ≤

) ( N ∑ |f (𝑗) (x)| 1 2(𝑗+1) −𝑛(𝑛1−α − 3 ) 2 . + e j! n𝛼𝑗 n 𝑗=1

From the expansion of e𝑥 : e𝑥 = 1 + x +

x2 x𝑘 + ··· + + ···, 2! k!

1054

𝑗 k Φ(nx − k) − x n

M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

and the inequality (see Section 3.6.6 of [22]): e𝑥 ≤ 1 + x + it follows that

N ∑ 𝑗=1

x2 x3 + , 2 2(3 − x)

(0 ≤ x < 3),

N ∑ 1 3 2(𝑗+1) < , ≤ 16. 𝛼𝑗 𝛼 j!n n j! 𝑗=1

Therefore,

( |Ξ2 | ≤

) 3 16 −𝑛(𝑛1−α − 3 ) 2 + e ∥f ∥N . n𝛼 n

To estimate Ξ3 , we use the result (see P. 72-73 of [23]): ∫ 𝑘 { ( ) 𝑛( ) ( 𝑘 − t)N −1 ω f (N ) , 𝑛1α 𝑛αN1 N ! , (N ) (N ) 𝑛 f (t) − f (x) dt ≤ (N +1) 𝑥 (N − 1)! ∥f (N ) ∥ 2 N ! , and deduce that ( ) 1 (N ) 1 |Ξ3 | ≤ ω f , 𝛼 n n𝛼N N !

∑

Φ(nx − k) + ∥f (N ) ∥

𝑘 𝑘:| 𝑛 −𝑥|≤ 𝑛1α

) ( 1 2(N +2) ∥f (N ) ∥ −𝑛(𝑛1−α − 3 ) (N ) 1 2 . + e ≤ ω f , 𝛼 n n𝛼N N ! nN !

2(N +1) N!

| 𝑛𝑘 − x| ≤ | 𝑛𝑘 − x| >

∑

1 𝑛α , 1 𝑛α

Φ(nx − k)

𝑘 𝑘:| 𝑛 −𝑥|> 𝑛1α

Combining the estimates of Ξ1 , Ξ2 and Ξ3 leads to ( ) 𝑛 3 16 −𝑛(𝑛1−α − 3 ) 2 |F𝑛 (f, x) − f (x)| ≤ 4e− 2 ∥f ∥ + + e ∥f ∥N n𝛼 n ) ( 1 2(N +2) ∥f (N ) ∥ −𝑛(𝑛1−α − 3 ) (N ) 1 2 . + e + ω f , 𝛼 n n𝛼N N ! nN ! This finishes Remark Remark Remark 2, d ∈ N).

the proof of Theorem 8.  2. For f ∈ C([−1, 1]2 ), we can establish the same result as Theorem 6. 3. For f ∈ C N ([−1, 1]2 ), a similar result to Theorem 8 can be established. 4. In fact, we can establish corresponding results in C([−1, 1]d ) and C N ([−1, 1]d )(d >

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M. Li, F. L. Cao, & Z. X. Chen: Jackson-type Operators on Spherical Cap

[6] T. P. Chen, H. Chen, R. W. Liu, Approximation capability in C(R𝑛 ) by multilayer feedforward networks and related problems, IEEE Trans. Neural Networks, 6 (1995) 25-30. [7] T. P. Chen, H. Chen, Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks, IEEE Trans. Neural Networks, 6 (1995) 904-910. [8] A. R. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inform. Theory, 39 (1993) 930-945. [9] D. B. Chen, Degree of approximation by superpositions of a sigmoidal function, Approx. Theory & Appl., 9 (1993) 17-28. [10] S. Suzuki, Constructive function approximation by three-layer neural networks, Neural Networks, 11 (1998) 1049-1058. [11] Y. Makovoz, Uniform approximation by neural networks, J. Approx. Theory, 95 (1998) 215228. [12] F. L. Cao, T. F. Xie, Z. B. Xu, The estimate for approximation error of neural networks: A constructive approach, Neurocomputing, 71 (2008) 626-630. [13] Z. X. Chen, F. L. Cao, The approximation operators with sigmoidal functions, Computers and Mathematics with Applications, 58 (2009) 758-765. [14] G. A. Anastassiou, Multivariate sigmoidal neural network approximation, Neural Networks, 24 (2011) 378-386. [15] G. A. Anastassiou, Univariate hyperbolic tangent neural network approximation, Mathematical and Computer Modelling, 53 (2011) 1111-1132. [16] G. A. Anastassiou, Multivariate hyperbolic tangent neural network approximation, Computers and Mathematics with Applications, 61 (2011) 809-821. [17] E. M. Stein, R. Shakarchi, Fourier Analysis An Introduction, Princetion University Press, Princetion and Oxford, 2003. [18] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover Publ., New York, 1968. [19] P. Borwein, T. Erd´elyi, Polynomials and Polynomial Inequalities, Springer-Verlag, 1995. [20] D. Leviatan, Improved estimates in M¨ untz-Jackson theorems, in: Progress in Approximation Theory, Academic Press, New York, 1991. [21] J. G. Attali, G. Pag`es, Approximations of functions by a multilayer perceptron: a new approach, Neural Networks, 10 (1997) 1069-1081. [22] D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970. [23] G. A. Anastassiou, Quantitative Approximations, Chapman & Hall/CRC, Boca Raton, New York, 2001.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1057-1068, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

ORTHOGONAL STABILITY OF AN ADDITIVE FUNCTIONAL EQUATION IN BANACH MODULES OVER A C ∗ -ALGEBRA HASSAN AZADI KENARY, CHOONKIL PARK, AND DONG YUN SHIN∗ Abstract. Using fixed point method, we prove the Hyers-Ulam stability of the following additive functional equation m ∑ i=1

(

f

mai +

m ∑

)

aj

+f

(m ) ∑

j=1,j̸=i

ai

i=1

= 2f

(m ) ∑ ai

i=1

in Banach modules over a unital C ∗ -algebra and in non-Archimedean Banach modules over a unital C ∗ -algebra.

1. Introduction and preliminaries Assume that X is a real inner product space and f : X → ℝ is a solution of the orthogonal Cauchy functional equation f (x + y) = f (x) + f (y), ⟨x, y⟩ = 0. By the Pythagorean theorem f (x) = ∥x∥2 is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus orthogonal Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space. G. Pinsker [53] characterized orthogonally additive functionals on an inner product space when the orthogonality is the ordinary one in such spaces. K. Sundaresan [65] generalized this result to arbitrary Banach spaces equipped with the Birkhoff-James orthogonality. The orthogonal Cauchy functional equation f (x+y) = f (x)+f (y), x ⊥ y, in which ⊥ is an abstract orthogonality relation, was first investigated by S. Gudder and D. Strawther [30]. They defined ⊥ by a system consisting of five axioms and described the general semi-continuous realvalued solution of conditional Cauchy functional equation. In 1985, J. R¨atz [60] introduced a new definition of orthogonality by using more restrictive axioms than of S. Gudder and D. Strawther. Moreover, he investigated the structure of orthogonally additive mappings. J. R¨atz and Gy. Szab´o [61] investigated the problem in a rather more general framework. Let us recall the orthogonality in the sense of J. R¨atz; cf. [60]. Suppose X is a real vector space (algebraic module) with dim X ≥ 2 and ⊥ is a binary relation on X with the following properties: (O1 ) totality of ⊥ for zero: x ⊥ 0, 0 ⊥ x for all x ∈ X; (O2 ) independence: if x, y ∈ X − {0}, x ⊥ y, then x, y are linearly independent; (O3 ) homogeneity: if x, y ∈ X, x ⊥ y, then αx ⊥ βy for all α, β ∈ ℝ; (O4 ) the Thalesian property: if P is a 2-dimensional subspace of X, x ∈ P and λ ∈ ℝ+ , which is the set of nonnegative real numbers, then there exists y0 ∈ P such that x ⊥ y0 and x + y0 ⊥ λx − y0 . The pair (X, ⊥) is called an orthogonality space (module). By an orthogonality normed space (normed module) we mean an orthogonality space (module) having a normed (normed 2010 Mathematics Subject Classification. Primary 39B55, 46S10, 47H10, 39B52, 47S10, 30G06, 46H25, 46L05, 12J25. Key words and phrases. Hyers-Ulam stability, orthogonally Cauchy-Jensen additive functional equation, fixed point, non-Archimedean Banach module over C ∗ -algebra, orthogonality space. ∗ Corresponding author.

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H. AZADI KENARY, C. PARK, AND D.Y. SHIN

module) structure. Assume that if A is a C ∗ -algebra and X is a module over A and if x, y ∈ X, x ⊥ y, then ax ⊥ by for all a, b ∈ A. Some interesting examples are (i) The trivial orthogonality on a vector space X defined by (O1 ), and for non-zero elements x, y ∈ X, x ⊥ y if and only if x, y are linearly independent. (ii) The ordinary orthogonality on an inner product space (X, ⟨., .⟩) given by x ⊥ y if and only if ⟨x, y⟩ = 0. (iii) The Birkhoff-James orthogonality on a normed space (X, ∥.∥) defined by x ⊥ y if and only if ∥x + λy∥ ≥ ∥x∥ for all λ ∈ ℝ. The relation ⊥ is called symmetric if x ⊥ y implies that y ⊥ x for all x, y ∈ X. Clearly examples (i) and (ii) are symmetric but example (iii) is not. It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [1]–[3], [5, 14, 35, 36, 44]). The stability problem of functional equations originated from the following question of Ulam [67]: Under what condition does there is an additive mapping near an approximately additive mapping? In 1941, Hyers [32] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1978, Th.M. Rassias [55] extended the theorem of Hyers by considering the unbounded Cauchy difference ∥f (x + y) − f (x) − f (y)∥ ≤ ε(∥x∥p + ∥y∥p ), (ε > 0, p ∈ [0, 1)). During the last decades several stability problems of functional equations have been investigated in the spirit of Hyers-Ulam-Rassias. The reader is referred to [11, 33, 37, 59] and references therein for detailed information on stability of functional equations. R. Ger and J. Sikorska [29] investigated the orthogonal stability of the Cauchy functional equation f (x + y) = f (x) + f (y), namely, they showed that if f is a mapping from an orthogonality space X into a real Banach space Y and ∥f (x + y) − f (x) − f (y)∥ ≤ ε for all x, y ∈ X with x ⊥ y and some ε > 0, then there exists exactly one orthogonally additive mapping g : X → Y such that ∥f (x) − g(x)∥ ≤ 16 3 ε for all x ∈ X. The first author treating the stability of the quadratic equation was F. Skof [64] by proving that if f is a mapping from a normed space X into a Banach space Y satisfying ∥f (x + y) + f (x − y) − 2f (x) − 2f (y)∥ ≤ ε for some ε > 0, then there is a unique quadratic mapping g : X → Y such that ∥f (x) − g(x)∥ ≤ 2ε . P.W. Cholewa [8] extended the Skof’s theorem by replacing X by an abelian group G. The Skof’s result was later generalized by S. Czerwik [9] in the spirit of Hyers-Ulam-Rassias. The stability problem of functional equations has been extensively investigated by some mathematicians (see [6, 7, 10, 51], [16]–[18], [40], [56]–[58], [63]). The orthogonally quadratic equation f (x + y) + f (x − y) = 2f (x) + 2f (y), x ⊥ y was first investigated by F. Vajzovi´c [68] when X is a Hilbert space, Y is the scalar field, f is continuous and ⊥ means the Hilbert space orthogonality. Later, H. Drljevi´c [15], M. Fochi [28], and Gy. Szab´o [66] generalized this result. In 1897, Hensel [31] introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [12, 39, 41, 43]). Definition 1.1. By a non-Archimedean field we mean a field K equipped with a function (valuation) | · | : K → [0, ∞) such that for all r, s ∈ K, the following conditions hold: (1) |r| = 0 if and only if r = 0; (2) |rs| = |r||s|; (3) |r + s| ≤ max{|r|, |s|}. Definition 1.2. Let X be a vector space over a scalar field K with a non-Archimedean nontrivial valuation | · | . A function || · || : X → R is a non-Archimedean norm (valuation) if it satisfies the following conditions:

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ORTHOGONAL STABILITY OF ADDITIVE FUNCTIONAL EQUATION

(1) ||x|| = 0 if and only if x = 0; (2) ||rx|| = |r|||x|| (r ∈ K, x ∈ X); (3) The strong triangle inequality (ultrametric); namely, ||x + y|| ≤ max{||x||, ||y||}, x, y ∈ X. Then (X, ||.||) is called a non-Archimedean space. Assume that if A is a C ∗ -algebra and X is a module over A, which is a non-Archimedean space, and if x, y ∈ X, x ⊥ y, then ax ⊥ by for all a, b ∈ A. Then (X, ||.||) is called an orthogonality non-Archimedean module. Due to the fact that ||xn − xm || ≤ max{||xj+1 − xj || : m ≤ j ≤ n − 1}

(n > m).

Definition 1.3. A sequence {xn } is Cauchy if and only if {xn+1 − xn } converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. We recall a fundamental result in fixed point theory. Theorem 1.4. [13] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point ∗ ∗ y of J; (3) y is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; (4) 1 d(y, y ∗ ) ≤ 1−α d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [34] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [4],[19]–[27],[45]–[52], [54]). This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the orthogonally additive functional equation in Banach modules over a unital C ∗ -algebra. In Section 3, we prove the Hyers-Ulam stability of the orthogonally additive functional equation in non-Archimedean Banach modules over a unital C ∗ -algebra. 2. Stability of the orthogonally additive functional equation in Banach modules over a C ∗ -algebra Throughout this section, assume that A is a unital C ∗ -algebra with unit e and unitary group U (A) := {u ∈ A | u∗ u = uu∗ = e}, (X, ⊥) is an orthogonality normed module over A and (Y, ∥.∥Y ) is a Banach module over A. In this section, applying some ideas from [29, 33], we deal with the stability problem for the orthogonally additive functional equation m ∑ i=1



f mxi +

m ∑



xj  + f

(m ∑ i=1

j=1,j̸=i

)

xi

= 2f

(m ∑

)

xi

i=1

for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. Theorem 2.1. Let φ : X m → [0, ∞) be a function such that there exists an α < 1 with (

φ(x1 , x2 , · · · , xm ) ≤ mαφ

1059

x x2 xm , ,··· , m m m

)

(2.1)

H. AZADI KENARY, C. PARK, AND D.Y. SHIN

for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. Let f : X → Y be a mapping satisfying

  (m ) ( m )

∑

m ∑ ∑ ∑

m

  f muxi + uxj + f uxi − 2uf xi

i=1

i=1 i=1 j=1,j̸=i

≤ φ(x1 , · · · , xn )

(2.2)

Y

for all u ∈ U (A) and all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. If for each x ∈ X the mapping f (tx) is continuous in t ∈ ℝ, then there exists a unique orthogonally additive and A-linear mapping L : X → Y such that ∥f (x) − L(x)∥Y ≤

1 ψ (x) m − mα

(2.3)

for all x ∈ X, where ψ(x) = φ(x, 0, · · · , 0). Proof. Putting x1 = x and x2 = · · · = xm = 0 and u = e in (2.2), since x ⊥ 0, we get



f (x) − f (mx) ≤ ψ(x)

m Y m

(2.4)

for all x ∈ X. Consider the set S := {h : X → Y } and introduce the generalized metric on S: d(g, h) = inf {µ ∈ ℝ+ : ∥g(x) − h(x)∥Y ≤ µψ (x) , ∀x ∈ X} , where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [42]). Now we consider the linear mapping J : S → S such that 1 g (mx) m for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then ∥g(x) − h(x)∥Y ≤ εψ (x) for all x ∈ X. Hence Jg(x) :=

g (mx)

∥Jg(x) − Jh(x)∥Y =

m

−

h (mx)

≤ ψ (mx) ≤ mαψ (x) ≤ αψ (x) m Y m m

for all x ∈ X. So d(g, h) = ε implies that d(Jg, Jh) ≤ αε. This means that d(Jg, Jh) ≤ αd(g, h) for all g, h ∈ S. It follows from (2.4) that 1 . m By Theorem 1.4, there exists a mapping L : X → Y satisfying the following: (1) L is a fixed point of J, i.e., d(f, Jf ) ≤

L (mx) = mL(x)

(2.5)

for all x ∈ X. The mapping L is a unique fixed point of J in the set M = {g ∈ S : d(h, g) < ∞}. This implies that L is a unique mapping satisfying (2.5) such that there exists a µ ∈ (0, ∞) satisfying ∥f (x) − L(x)∥Y ≤ µψ (x) for all x ∈ X; (2) d(J k f, L) → 0 as k → ∞. This implies the equality 1 ( k ) f m x = L(x) k→∞ mk lim

for all x ∈ X; (3) d(f, L) ≤

1 1−α d(f, Jf ),

which implies the inequality d(f, L) ≤

1 . m − mα

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ORTHOGONAL STABILITY OF ADDITIVE FUNCTIONAL EQUATION

This implies that (2.3) holds true. Let u = e in (2.2). It follows from (2.1) and (2.2) that

  (m ) ( m )

∑

m ∑ ∑ ∑

m

mxi +  L x + L x − 2L x j i i

i=1

i=1 i=1 j=1,j̸=i Y

   ( ) (m )

∑

m m m ∑ ∑ ∑

1 k k k    = lim k f m mxi + xj +f m xi − 2f m xi

k→∞ m

i=1 i=1 i=1 j=1,j̸=i

Y

φ(mk x1 , mk x2 , · · · , mk xm ) ≤ lim k→∞ mk k n m α φ(x1 , · · · , xm ) ≤ lim =0 k→∞ mk for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. So m ∑



L mxi +

i=1

m ∑



xj  + L

(m ∑

)

xi − 2L

i=1

j=1,j̸=i

(m ∑

)

xi

=0

i=1

for all x1 , · · · , xn ∈ X with x1 ⊥ xj for all i ̸= j. Hence L : X → Y is an orthogonally additive mapping. Let x2 = · · · = xn = 0 in (2.2). It follows from (2.1) and (2.2) that ∥L(mux) − muL(x)∥Y

∥f (mk+1 ux) − mf (mk ux)∥Y k→∞ mk

f (mk+1 ux) f (mk ux)

= m lim −

k+1 k

k→∞ m m Y =

lim

mk αn ψ(x) ψ(mk x) ≤ lim k→∞ k→∞ mk mk n = lim α ψ(x) = 0 ≤

lim

k→∞

for all x ∈ X and all u ∈ U (A). So muL

(

x m

for all x ∈ X and all u ∈ U (A). Hence

)

− L(ux) = 0 (

x L(ux) = muL m

)

= uL(x)

(2.6)

for all u ∈ U (A) and all x ∈ X. By the same reasoning as in the proof of [55, Theorem], we can show that L : X → Y is ℝ-linear, since the mapping f (tx) is continuous in t ∈ ℝ for each x ∈ X and L : X → Y is additive. Since L is ℝ-linear and each∑a ∈ A is a finite linear combination of unitary elements (see [38, Theorem 4.1.7]), i.e., a = m j=1 λj uj (λj ∈ C, uj ∈ U (A)), it follows from (2.6) that 







m ∑ j=1

|λj | ·

m ∑

(

)

m ∑ λj λj L(ax) = L  λj uj x = L  |λj | · uj x = |λj |L uj x |λj | |λj | j=1 j=1 j=1

=

m ∑

m ∑ λj uj L(x) = λj uj L(x) = aL(x) |λj | j=1

1061

H. AZADI KENARY, C. PARK, AND D.Y. SHIN λ

for all x ∈ X. It is obvious that |λjj | uj ∈ U (A). Thus L : X → Y is a unique orthogonally additive and A-linear mapping satisfying (2.3).  Corollary 2.2. Let θ be a positive real number and p a real number with 0 < p < 1. Let f : X → Y be a mapping satisfying

  (m ) ( m )

∑

m ∑ ∑ ∑

m

  f muxi + uxj + f uxi − 2uf xi

i=1

i=1 i=1 j=1,j̸=i

Y

≤θ

(m ∑

)

∥xi ∥

p

(2.7)

i=1

for all u ∈ U (A) and all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. If for each x ∈ X the mapping f (tx) is continuous in t ∈ ℝ, then there exists a unique orthogonally additive and A-linear mapping L : X → Y such that θ∥x∥p ∥f (x) − L(x)∥Y ≤ m − mp for all x ∈ X. Proof. The proof follows from Theorem 2.1 by taking φ(x1 , x2 , · · · , xn ) = θ

( n ∑

)

∥xi ∥

p

i=1

for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. Then we can choose α = mp−1 and we get the desired result.  Theorem 2.3. Let f : X → Y be a mapping satisfying (2.2) for which there exists a function φ : X m → [0, ∞) such that (

)

xm αφ (x1 , x2 , · · · , xm ) x1 x2 , ,··· , ≤ φ m m m m for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. If for each x ∈ X the mapping f (tx) is continuous in t ∈ ℝ, then there exists a unique orthogonally additive and A-linear mapping L : X → Y such that αψ(x) ∥f (x) − L(x)∥Y ≤ (2.8) m − mα for all x ∈ X. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping J : S → S such that ( ) x Jg(x) := mg m for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then ∥g(x) − h(x)∥Y ≤ εψ (x) for all x ∈ X. Hence

( ) ( ) ( )

x x

≤ mψ x ≤ mαψ (x) ≤ αψ (x) ∥Jg(x) − Jh(x)∥Y = − mh mg

m m Y m m for all x ∈ X. So d(g, h) = ε implies that d(Jg, Jh) ≤ αε. This means that d(Jg, Jh) ≤ αd(g, h) for all g, h ∈ S. It follows from (2.4) that

( ) ( )

mf x − f (x) ≤ ψ x ≤ α ψ(x).

m m m Y Therefore α d(f, Jf ) ≤ . m

1062

ORTHOGONAL STABILITY OF ADDITIVE FUNCTIONAL EQUATION

By Theorem 1.4, there exists a mapping L : X → Y satisfying the following: (1) L is a fixed point of J, i.e., ( ) x 1 L = L(x) (2.9) m m for all x ∈ X. The mapping L is a unique fixed point of J in the set M = {g ∈ S : d(h, g) < ∞}. This implies that L is a unique mapping satisfying (2.9) such that there exists a µ ∈ (0, ∞) satisfying ∥f (x) − L(x)∥Y ≤ µψ (x) for all x ∈ X; (2) d(J k f, L) → 0 as k → ∞. This implies the equality ( ) x lim mk f = L(x) k→∞ mk for all x ∈ X; 1 (3) d(f, L) ≤ 1−α d(f, Jf ), which implies the inequality α d(f, L) ≤ . m − mα This implies that (2.8) holds true. The rest of the proof is similar to the proof of Theorem 2.1.  Corollary 2.4. Let θ be a positive real number and p a real number with p > 1. Let f : X → Y be a mapping satisfying (2.7). If for each x ∈ X the mapping f (tx) is continuous in t ∈ ℝ, then there exists a unique orthogonally additive and A-linear mapping L : X → Y such that θ∥x∥p ∥f (x) − L(x)∥Y ≤ p m −m for all x ∈ X. Proof. The proof follows from Theorem 2.3 by taking φ(x1 , x2 , · · · , xn ) = θ

(m ∑

)

∥xi ∥p

i=1

for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. Then we can choose α = m1−p and we get the desired result.  3. Stability of the orthogonally additive functional equation in non-Archimedean Banach modules over a C ∗ -algebra Throughout this section, assume that A is a unital C ∗ -algebra with unit e and unitary group U (A) := {u ∈ A | u∗ u = uu∗ = e}, (X, ⊥) is an orthogonality non-Archimedean normed module over A and (Y, ∥.∥Y ) is a non-Archimedean Banach module over A. Assume that |m| ̸= 1. In this section, applying some ideas from [29, 33], we deal with the stability problem for the orthogonally Jensen functional equation. Theorem 3.1. Let φ : X m → [0, ∞) be a function such that there exists an α < 1 with ( ) x x2 xm φ(x1 , x2 , · · · , xm ) ≤ |m|αφ , ,··· , (3.1) m m m for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. Let f : X → Y be a mapping satisfying (2.2). If for each x ∈ X the mapping f (tx) is continuous in t ∈ ℝ, then there exists a unique orthogonally additive and A-linear mapping L : X → Y such that ψ (x) ∥f (x) − L(x)∥Y ≤ (3.2) |m| − |m|α

1063

H. AZADI KENARY, C. PARK, AND D.Y. SHIN

for all x ∈ X. Proof. It follows from (2.4) that



f (x) − f (mx) ≤ ψ(x)

m Y |m|

(3.3)

for all x ∈ X. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping J : S → S such that g(mx) Jg(x) := m for all x ∈ X. It follows from (3.3) that d(f, Jf ) ≤ |m|. By Theorem 1.4, there exists a mapping L : X → Y satisfying the following: (1) d(J k f, L) → 0 as k → ∞. This implies the equality ) 1 ( lim k f mk x = L(x) k→∞ m for all x ∈ X; 1 (2) d(f, L) ≤ 1−α d(f, Jf ), which implies the inequality d(f, L) ≤

1 . |m| − |m|α

This implies that (3.2) holds true. It follows from (3.1) and (2.2) that

  (m ) ( m )

∑

m ∑ ∑ ∑

m

  L muxi + uxj + L uxi − 2uL xi

i=1

i=1 i=1 j=1,j̸=i Y

  

m m ∑ ∑ 1

f mk muxi + uxj  = lim

k k→∞ |m| i=1 j=1,j̸=i ) ) (m (m

∑ ∑

k k m xi +f m uxi − 2uf

i=1

≤

φ(mk x1 , mk x2 , · · · lim k→∞ |m|k

i=1 , mk xm )

Y

|m|k αn φ(x1 , · · · , xm ) =0 k→∞ |m|k

≤ lim

for all u ∈ U (A) and all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. So m ∑ i=1



L muxi +

m ∑



uxj  + L

(m ∑ i=1

j=1,j̸=i

)

uxi

= 2uL

(m ∑

)

xi

i=1

for all u ∈ U (A) and all x1 , · · · , xn ∈ X with xi ⊥ xj for all i ̸= j. Hence L : X → Y is an orthogonally additive mapping. The rest of the proof is similar to the proof of Theorem 2.1.  Corollary 3.2. Let θ be a positive real number and p a real number with p > 1. Let f : X → Y be a mapping satisfying (2.7). If for each x ∈ X the mapping f (tx) is continuous in t ∈ ℝ, then there exists a unique orthogonally additive and A-linear mapping L : X → Y such that θ∥x∥p ∥f (x) − L(x)∥Y ≤ |m| − |m|p+1

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ORTHOGONAL STABILITY OF ADDITIVE FUNCTIONAL EQUATION

for all x ∈ X. Proof. The proof follows from Theorem 3.1 by taking φ(x1 , x2 , · · · , xn ) = θ

( n ∑

)

∥xi ∥

p

i=1

for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. Then we can choose α = |m|p−1 and we get the desired result.  Theorem 3.3. Let f : X → Y be a mapping satisfying (2.2) and for which there exists a function φ : X m → [0, ∞) such that (

φ

x1 x2 xm , ,··· , m m m

)

≤

αφ (x1 , x2 , · · · , xm ) |m|

for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. If for each x ∈ X the mapping f (tx) is continuous in t ∈ ℝ, then there exists a unique orthogonally additive and A-linear mapping L : X → Y such that αψ(x) ∥f (x) − L(x)∥Y ≤ (3.4) |m| − |m|α for all x ∈ X. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping J : S → S such that ( ) x Jg(x) := mg m for all x ∈ X. It follows from (2.4) that d(f, Jf ) ≤ proofs of Theorems 2.1 and 3.1.

α |m| .

The rest of the proof is similar to the 

Corollary 3.4. Let θ be a positive real number and p a real number with 0 < p < 1. Let f : X → Y be a mapping satisfying (2.7). If for each x ∈ X the mapping f (tx) is continuous in t ∈ ℝ, then there exists a unique orthogonally Jensen and A-linear mapping L : X → Y such that |m|θ∥x∥p ∥f (x) − L(x)∥Y ≤ |m|p+1 − |m| for all x ∈ X. Proof. The proof follows from Theorem 3.3 by taking φ(x1 , x2 , · · · , xn ) = θ

( n ∑

)

∥xi ∥

p

i=1

for all x1 , · · · , xm ∈ X with xi ⊥ xj for all i ̸= j. Then we can choose α = |m|1−p and we get the desired result.  Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF2012R1A1A2004299). D. Y. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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[58] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [59] Th.M. Rassias (ed.), Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003. [60] J. R¨ atz, On orthogonally additive mappings, Aequationes Math. 28 (1985), 35–49. [61] J. R¨ atz and Gy. Szab´ o, On orthogonally additive mappings IV , Aequationes Math. 38 (1989), 73–85. [62] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. [63] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Computat. Anal. Appl. 13 (2011), 1106–1114. [64] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [65] K. Sundaresan, Orthogonality and nonlinear functionals on Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 187–190. [66] Gy. Szab´ o, Sesquilinear-orthogonally quadratic mappings, Aequationes Math. 40 (1990), 190–200. [67] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960. ¨ [68] F. Vajzovi´c, Uber das Funktional H mit der Eigenschaft: (x, y) = 0 ⇒ H(x+y)+H(x−y) = 2H(x)+2H(y), Glasnik Mat. Ser. III 2 (22) (1967), 73–81. Hassan Azadi Kenary Department of Mathematics, Yasouj University, Yasouj 75914-353, Iran E-mail address: [email protected], [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 130-743, Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1069-1084, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

SOME CHARACTERIZATIONS AND CONVERGENCE PROPERTIES OF THE CHOQUET INTEGRAL WITH RESPECT TO A FUZZY MEASURE OF FUZZY COMPLEX VALUED FUNCTIONS

Lee-Chae Jang Department of Computer Engineering, Konkuk University, Chungju 380-701, Korea e-mail: [email protected] Abstract. In this paper, we consider Choquet integrals with respect to a fuzzy measure and fuzzy complex valued functions. We define the Choquet integral with respect to a fuzzy measure of a fuzzy complex valued functions and investigate their characterizations. Furthermore, we discuss some convergence properties of the Choquet integral with respect to a fuzzy measure of an integrably bounded fuzzy complex valued measurable function.

§1. Introduction Choquet integrals, introduced in [8,9,10], has emerged as an interesting extension of the Lebesgue integral. Puri and Ralescu [11] have been studied Lebesgue integral with respect to a classical measure of closed set-valued measurable functions. In the papers [4-7], we defined interval-valued Choquet integrals and have studied some convergence theorems for Choquet integrals with respect to a fuzzy measure of interval-valued measurable functions under some sufficient conditions. Zhang, Guo and Liu [14] restudied Choquet integrals with respect to a fuzzy measure of closed set-valued measurable functions. Burkley [1-3] introduced the concept of fuzzy complex numbers, the differentiability and integrability of fuzzy complex valued functions on a complex plane C. Wang and Li [11] have researched generalized Lebesgue integrals with respect to a complex valued fuzzy measure of fuzzy complex valued functions. 2000 AMS Subject Classification: 28E10, 03E72, 26E50 keywords and phrases : Choquet integrals, fuzzy measures, fuzzy complex numbers, fuzzy complex valued functions Typeset by AMS-TEX

1

1069

JANG: CHOQUET INTEGRAL

In this paper, we define the Choquet integral with respect to a fuzzy measure of a fuzzy complex valued function and discuss their properties. In particular, we prove some convergence theorems for the Choquet integrals of a fuzzy complex valued function. In section 2, we list the definitions and various properties of fuzzy measures and Choquet integrals. In section 3, we introduce fuzzy complex numbers and fuzzy complex valued functions. We define Choquet integrals with respect to a fuzzy measure of a fuzzy complex valued functions and discuss some of their some characterizations. In section 4, we discuss some convergence properties of the Choquet integrals of integrably bounded fuzzy complex valued functions. In section 5, we give a brief summery results and some conclusions. §2. Definitions and Preliminaries Throughout this paper, we assume that (X, ℑ(X)) is a measurable space and denote ¯ + = [0, ∞]. We list the definitions of fuzzy measures and Choquet ℝ+ = [0, ∞) and ℝ integrals(see [4-12]). ¯ + is called a fuzzy measure if (i) Definition 2.1. (1) A set function µ : ℑ(X) −→ ℝ µ(∅) = 0 and (ii) µ(A) ≤ µ(B) whenever A, B ∈ ℑ(X) and A ⊂ B. (2) If µ(X) < ∞, µ is said to be finite. (3) A set function µ is said to be lower semi-continuous if for each increasing sequence {A𝑛 } in ℑ(X), µ(∪∞ 𝑛=1 A𝑛 ) = lim µ(A𝑛 ). 𝑛→∞

(4) A set function µ is said to be lower semi-continuous if for each decreasing sequence{A𝑛 } in ℑ(X) with µ(A1 ) < ∞, µ(∩∞ 𝑛=1 A𝑛 ) = lim µ(A𝑛 ). 𝑛→∞

(5) If µ is both lower semi-continuous and upper semi-continuous, it is said to be semi-continuous. We remark that fuzzy measures are known to be the generalization of classical measures where additivity is replaced by the weaker condition of monotonicity and that fuzzy measures are not assumed to be semi-continuous. We introduce the Choquet integral proposed by M. Sugeno(see [8]) as follows. Definition 2.2. (1) The Choquet integral with respect to a fuzzy measure µ of a measurable function 𝑓 : X −→ ℝ+ on A ∈ ℑ(X) is defined by ∫ ∫ ∞ (C) 𝑓 𝑑µ = µ({𝑥|𝑓 (𝑥) > r} ∩ A)𝑑r A

0

where the integral on the right-hand side is the Lebesgue integral. 2

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JANG: CHOQUET INTEGRAL

(2) A measurable function 𝑓 is said to be C-integrable if the Choquet integral of 𝑓 on X can be defined and its value is finite. ∫ ∫ Instead of (C) X 𝑓 𝑑µ, we will write (C) 𝑓 𝑑µ. We consider the (decreasing) distribution function Gf (r) = µ({𝑥|𝑓 (𝑥) > r}) of a measurable function 𝑓 for any r ∈ ℝ+ = [0, ∞). Definition 2.3. Let µ be a fuzzy measure on ℑ(X) and 𝑓 a measurable function. We say that 𝑓 and 𝑔 are comonotonic, in symbol, 𝑓 ∼ 𝑔 if 𝑓 (𝑥) < 𝑓 (𝑥′ ) =⇒ 𝑔(𝑥) ≤ 𝑔(𝑥′ ) for all 𝑥, 𝑥′ ∈ X. Now we introduce the following basic properties of the comonotonicity and the Choquet integral. Theorem 2.4. [8-10, 12]) Let 𝑓, 𝑔, and ℎ be measurable functions. Then we have (1) 𝑓 ∼ 𝑓 , (2) 𝑓 ∼ 𝑔 =⇒ 𝑔 ∼ 𝑓 , (3) 𝑓 ∼ 𝑎 for all 𝑎 ∈ ℝ+ , (4) 𝑓 ∼ 𝑔 and 𝑔 ∼ ℎ =⇒ 𝑓 ∼ 𝑔 + ℎ. Theorem 2.5. [8-10, 12]) ∫ Let 𝑓 and ∫𝑔 be C-integrable functions. Then we have (1) if 𝑓 ≤ 𝑔, then (C) 𝑓 𝑑µ ≤ (C) 𝑔𝑑µ, ∫ ∫ (2) if E1 ⊂ E2 and E1 , E2 ∈ ℑ(X), then (C) E1 𝑓 𝑑µ ≤ (C) E2 𝑓 𝑑µ, (3) if 𝑓 ∼ 𝑔 and 𝑎, 𝑏 ∈ ℝ+ , then ∫ (C)

∫ (𝑎𝑓 + 𝑏𝑔)𝑑µ = 𝑎(C)

∫ 𝑓 𝑑µ + 𝑏(C)

𝑔𝑑µ,

(4) if we define (𝑓 ∨ 𝑔)(𝑥) = 𝑓 (𝑥) ∨ 𝑔(𝑥) and (𝑓 ∧ 𝑔)(𝑥) = 𝑓 (𝑥) ∧ 𝑔(𝑥) for all 𝑥 ∈ X, then ∫ ∫ ∫ (C) 𝑓 ∨ 𝑔𝑑µ ≥ (C) 𝑓 𝑑µ ∨ (C) 𝑔𝑑µ ∫

and (C)

∫ 𝑓 ∧ 𝑔𝑑µ ≤ (C)

∫ 𝑓 𝑑µ ∧ (C)

𝑔𝑑µ

Throughout this paper, 𝐼(ℝ+ ) is the class of all closed intervals in ℝ+ , that is, 𝐼(ℝ+ ) = {[𝑎− , 𝑎+ ]|𝑎− , 𝑎+ ∈ ℝ+ and 𝑎− ≤ 𝑎+ }. For any 𝑎 ∈ ℝ+ , we define 𝑎 = [𝑎, 𝑎]. Obviously, 𝑎 ∈ 𝐼(ℝ+ )(see[4-7]). 3

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Definition 2.6. If 𝑎 ¯ = [𝑎− , 𝑎+ ], ¯𝑏 = [𝑏− , 𝑏+ ] ∈ 𝐼(ℝ+ ) and c ∈ ℝ+ , then we define the following operations: (1) 𝑎 ¯ + ¯𝑏 = [𝑎− + 𝑏− , 𝑎+ + 𝑏+ ]. (2) 𝑘¯ 𝑎 = [c𝑎− , c𝑎+ ]. (3) 𝑎 ¯¯𝑏 = [𝑎− 𝑏− , 𝑎+ 𝑏+ ]. (4) 𝑎 ¯ ∧ ¯𝑏 = [𝑎− ∧ 𝑏− , 𝑎+ ∧ 𝑏+ ]. (5) 𝑎 ¯ ∨ ¯𝑏 = [𝑎− ∨ 𝑏− , 𝑎+ ∨ 𝑏+ ]. (6) 𝑎 ¯ ≤ ¯𝑏 if and only if 𝑎− ≤ 𝑏− and 𝑎+ ≤ 𝑏+ . (7) 𝑎 ¯ < ¯𝑏 if and only if 𝑎 ¯ ≤ ¯𝑏 and 𝑎 ¯ ̸= ¯𝑏. (8) 𝑎 ¯ ⊂ ¯𝑏 if and only if 𝑏− ≤ 𝑎− and 𝑎+ ≤ 𝑏+ . + + Definition 2.7. If 𝑎 ¯ = [𝑎− 𝑘 , 𝑎𝑘 ] ∈ 𝐼(ℝ ) for 𝑘 = 1, 2, · · · , then we define the following operations: − + ∞ (1) ∧∞ ¯𝑘 = [∧∞ 𝑘=1 𝑎 𝑘=1 𝑎𝑘 , ∧𝑘=1 𝑎𝑘 ]. − + ∞ (2) ∨∞ ¯𝑘 = [∨∞ 𝑘=1 𝑎 𝑘=1 𝑎𝑘 , ∨𝑘=1 𝑎𝑘 ].

Theorem 2.8. For 𝑎 ¯, ¯𝑏, c¯ ∈ 𝐼(ℝ+ ), we have (1) idempotent law: 𝑎 ¯∧𝑎 ¯=𝑎 ¯, 𝑎 ¯∨𝑎 ¯=𝑎 ¯, ¯ ¯ (2) commutative law: 𝑎 ¯∧𝑏=𝑏∧𝑎 ¯, 𝑎 ¯ ∨ ¯𝑏 = ¯𝑏 ∨ 𝑎 ¯, (3) associative law: (¯ 𝑎 ∧ ¯𝑏) ∧ c¯ = 𝑎 ¯ ∧ (¯𝑏 ∧ c¯), (4) absorption law: 𝑎 ¯ ∧ (¯ 𝑎 ∨ ¯𝑏) = 𝑎 ¯ ∨ (¯ 𝑎 ∧ ¯𝑏) = 𝑎 ¯, ¯ ¯ (5) distributive law: 𝑎 ¯ ∧ (𝑏 ∨ c¯) = (¯ 𝑎 ∧ 𝑏) ∨ (¯ 𝑎 ∧ c¯), 𝑎 ¯ ∨ (¯𝑏 ∧ c¯) = (¯ 𝑎 ∨ ¯𝑏) ∧ (¯ 𝑎 ∨ c¯). W note that (𝐼(ℝ+ ), 𝑑H ) is a metric space, where a mapping 𝑑H : 𝐼(ℝ+ ) × ¯ + is the Hausdorff metric defined by 𝐼(ℝ+ ) −→ ℝ 𝑑H (A, B) = max{sup inf |𝑥 − 𝑦|, sup inf |𝑥 − 𝑦|} 𝑥∈A y∈B

y∈B 𝑥∈A

for all A, B ∈ 𝐼(ℝ+ ). By the definition of the Hausdorff metric, it is easy to see that for any 𝑎 ¯ = [𝑎− , 𝑎+ ], ¯𝑏 = [𝑏− , 𝑏+ ] ∈ 𝐼(ℝ+ ), we have 𝑑H (¯ 𝑎, ¯𝑏) = max{|𝑎− − 𝑏− |, |𝑎+ − 𝑏+ |}. Note that for a sequence of closed intervals {¯ 𝑎𝑛 } converges to 𝑎 ¯, in symbols 𝑑H − lim𝑛→∞ 𝑎 ¯𝑛 = 𝑎 ¯ if lim𝑛→∞ 𝑑H (¯ 𝑎𝑛 , 𝑎 ¯) = 0 and that 𝑑H − lim𝑛→∞ 𝑎 ¯𝑛 = 𝑎 ¯ if and only − − + + if lim𝑛→∞ 𝑎𝑛 = 𝑎 and lim𝑛→∞ 𝑎𝑛 = 𝑎 . In the following definition, we introduce fuzzy numbers and some operations on them which are used in the next sections. Definition 2.9. A fuzzy set 𝑢 e on ℝ+ is called a fuzzy number if it satisfies the following conditions; (i) (normality) 𝑢 e(𝑥) = 1 for some 𝑥 ∈ ℝ+ , 4

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(ii) (fuzzy convexity) for every 𝜆 ∈ (0, 1], 𝑢 eλ = {𝑥 ∈ ℝ+ | 𝑢 e(𝑥) ≥ 𝜆} ∈ 𝐼(ℝ+ ), where 𝑢 eλ is the level set of 𝑢 e. Let F 𝑁 (ℝ+ ) denote the class of all fuzzy numbers. We define the following basic operations on F 𝑁 (ℝ+ )(see[8,9,12]); for every 𝑢 e, ve ∈ F 𝑁 (ℝ+ ) and 𝑘 ∈ ℝ+ , (e 𝑢 + ve)λ = 𝑢 eλ + veλ , (𝑘e 𝑢)λ = 𝑘e 𝑢λ , (e 𝑢ve)λ = 𝑢 eλ veλ , 𝑢 e ≤ ve if and only if 𝑢 eλ ≤ veλ , for all 𝜆 ∈ (0, 1], 𝑢 e < ve if and only if 𝑢 e ≤ ve and 𝑢 e ̸= ve, 𝑢 e ⊂ ve if and only if 𝑢 eλ ⊂ veλ , for all 𝜆 ∈ (0, 1]. §3. Choquet integrals of fuzzy complex fuzzy functions In this section, we consider a fuzzy number and fuzzy complex numbers(see[1-3,13]). Definition 3.1. Let e 𝑎, e𝑏 ∈ F 𝑁 (ℝ+ ). We define a double ordered fuzzy numbers (e 𝑎, e𝑏) as follows: (e 𝑎, e𝑏) : C+ −→ [0, 1] 𝑧 = 𝑥 + 𝑦𝑖 7−→ (e 𝑎, e𝑏)(𝑧) = e 𝑎(𝑥) ∧ 𝑦e(𝑦), where C+ = {𝑥 + 𝑦𝑖|𝑥, 𝑦 ∈ ℝ+ }. Then the mapping (e 𝑎, e𝑏) determines a fuzzy complex number, where e 𝑎 and e𝑏 is called a real part and an imaginary part of (e 𝑎, e𝑏), respectively. We note that if we put C = (e 𝑎, e𝑏), then e 𝑎 = R𝑒C and e𝑏 = 𝐼𝑚C. Let F C𝑁 (C+ ) be the class of all fuzzy complex numbers on C+ , writing C ≡e 𝑎 + e𝑏𝑖. Note that if c = 𝑎+𝑏𝑖 is a nonnegative complex number, then its membership function is { 1 if 𝑥 = 𝑎, 𝑦 = 𝑏 c(𝑧) = 0 otherwise where 𝑧 = 𝑥+𝑦𝑖 ∈ C+ . Clearly, c ∈ F C𝑁 (C+ ), that is, a fuzzy complex number is also a generalization of an ordinary complex number. We recall that if C1 , C2 ∈ F C𝑁 (C+ ) and we define C1 ∗ C2 = (R𝑒C1 ∗ R𝑒C2 , 𝐼𝑚C1 ∗ 𝐼𝑚C2 ) for an operation ∗ ∈ {+, −, ×, ∧, ∨}, then clearly we have C1 ∗ C2 ∈ F C𝑁 (C+ ). 5

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Definition 3.2. Let C1 , C2 ∈ F C𝑁 (C+ ). Then we define the following order and equality operations: (1) C1 ≤ C2 if and only if R𝑒C1 ≤ R𝑒C2 and 𝐼𝑚C1 ≤ 𝐼𝑚C2 . (2) C1 < C2 if and only if C1 ≤ C2 and C1 ̸= C2 . (3) C1 = C2 if and only if C1 ≤ C2 and C2 ≤ C1 . (4) C1 ⊂ C2 if and only if R𝑒C1 ⊂ R𝑒C2 and 𝐼𝑚C1 ⊂ 𝐼𝑚C2 . From Definition 3.2, it is easy to see that if we define 𝜆-cut set Cλ = {𝑧 = 𝑥 + 𝑦𝑖 ∈ C |(R𝑒C)(𝑥) ≥ 𝜆 and (𝐼𝑚C)(𝑦) ≥ 𝜆}, then it is a closed rectangle region in C+ . Now, we consider fuzzy complex valued functions as follows(see [13]). +

Definition 3.3. If a mapping 𝑓e : C+ −→ F C𝑁 (C+ ) is defined by 𝑧 = 𝑥 + 𝑦𝑖 7−→ 𝑓e(𝑧) = (R𝑒𝑓e, 𝐼𝑚𝑓e)(𝑧) = R𝑒𝑓e(𝑥) ∧ 𝐼𝑚𝑓e(𝑦), then 𝑓e is called a fuzzy complex valued function on C+ . We note that for any 𝜆 ∈ (0, 1], let 𝑓eλ (𝑧) ≡ (𝑓e(𝑧))λ = ((R𝑒𝑓e(𝑥))λ , (𝐼𝑚𝑓e(𝑦))λ ), for all 𝑧 = 𝑥 + 𝑦𝑖 ∈ C+ , e+ e e− e+ where (R𝑒𝑓e)λ ≡ [(R𝑒𝑓e)− λ , (R𝑒𝑓 )λ ] and (𝐼𝑚𝑓 )λ ≡ [(𝐼𝑚𝑓 )λ , (𝐼𝑚𝑓 )λ ] for all 𝜆 ∈ (0, 1] and that 𝑓e is said to be measurable if for any 𝜆 ∈ (0, 1], (R𝑒𝑓e)λ and (𝐼𝑚𝑓e)λ are measurable. We introduce Choqeut integral of interval-valued measurable functions as follows(see [4-7,14]). Definition 3.4. ([4-7, 14]) Let (ℝ+ , ℑ(ℝ+ )) be a measurable space. A closed setvalued function F : X −→ 𝐼(ℝ+ ) is said to be measurable if for any open set O ⊂ ℝ+ , F −1 (O) = {𝑥 ∈ ℝ+ |F (𝑥) ∩ O ̸= ∅} ∈ ℑ(ℝ+ ). Definition 3.5. ([4-7, 14]) (1) Let F be a closed set-valued function and A ∈ ℑ(ℝ+ ). The Choquet integral of F on A is defined by { } ∫ ∫ (C) F 𝑑µ = (C) 𝑓 𝑑µ | 𝑓 ∈ Sc (F ) , A

A

where Sc (F ) is the family of measurable selections of F . ∫ (2) A closed set-valued functions F is said to be integrable if (C) F 𝑑µ ̸= ∅. (3) A closed set-valued function F is said to be integrably bounded if there exists a integrable function 𝑔 such that ∥ F (𝑥) ∥= 𝑠𝑢𝑝r∈F (𝑥) |r| ≤ 𝑔(𝑥) for all 𝑥 ∈ ℝ+ .

6

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Theorem 3.6. ([14 Theorem 3.16(iii)]) Let µ be a semi-continuous fuzzy measure. If F = [𝑓 − , 𝑓 + ] : ℝ+ −→ 𝐼(ℝ+ ) is an integrably bounded interval-valued measurable function, then [ ] ∫ ∫ ∫ − + (C) F 𝑑µ = (C) 𝑓 𝑑µ, (C) 𝑓 𝑑µ .

Theorem 3.7. ([13])If 𝑓e1 and 𝑓e2 are fuzzy complex valued measurable functions, then 𝑓e1 ± 𝑓e2 and 𝑓e1 · 𝑓e2 are fuzzy complex valued measurable functions, where 𝑓e1 ± 𝑓e2 = (R𝑒𝑓e1 ± R𝑒𝑓e2 , 𝐼𝑚𝑓e1 ± 𝐼𝑚𝑓e2 ) and 𝑓e1 · 𝑓e2 = (R𝑒𝑓e1 · R𝑒𝑓e2 , 𝐼𝑚𝑓e1 · 𝐼𝑚𝑓e2 ). Now, we define the Choquet integral with respect to a fuzzy measure of a fuzzy complex valued function as follows. Definition 3.8. Let µ be a semi-continuous fuzzy measure on (ℝ+ , ℑ(ℝ+ )) and 𝑓e = (R𝑒𝑓e, 𝐼𝑚𝑓e) a fuzzy complex valued measurable function. (1) For every A, B ∈ ℑ(ℝ+ ), the Choquet integral with respect to µ to 𝑓e on A × B is defined by ( ) ( ) ∫ ∫ ∫ (C) 𝑓e𝑑µ = (C) (R𝑒𝑓e)λ 𝑑µ, (C) (𝐼𝑚𝑓e)λ 𝑑µ A×B

λ

A

B

for all 𝜆 ∈ (0, 1].

( ) ∫ (2) If there exists 𝑢 e ∈ F C𝑁 (C+ ) such that (e 𝑢)λ = (C) A×B 𝑓e𝑑µ for all 𝜆 ∈ λ

(0, 1], then 𝑓e is said to be integrable on A × B. (3) 𝑓e is said to be integrably bounded if for any 𝜆 ∈ (0, 1], (R𝑒𝑓e)λ and (𝐼𝑚𝑓e)λ are integrably bounded. ∫ ∫ Instead of (C) R+ ×R+ 𝑓e𝑑µ, we will write (C) 𝑓e𝑑µ. If we set A × B = ℝ+ × ℝ+ , then we denote ( ) ( ) ∫ ∫ ∫ = (C) (R𝑒𝑓e)λ 𝑑µ, (C) (𝐼𝑚𝑓e)λ 𝑑µ . (C) 𝑓e𝑑µ λ

In order to prove the existence of the Choquet integral of 𝑓e, we need the Choquet integral of a fuzzy complex valued measurable function to satisfy the following lemma. Lemma 3.9 ([7,10]). Let {[𝑎λ , 𝑏λ ]|𝜆 ∈ (0, 1]} be a family of nonempty closed intervals in 𝐼(ℝ+ ). If (i) for all 0 < 𝜆1 ≤ 𝜆2 , [𝑎λ1 , 𝑏λ1 ] ⊃ [𝑎λ2 , 𝑏λ2 ] and (ii) for any increasing sequence {𝜆𝑘 } in (0, 1] converging to 𝜆, [𝑎λ , 𝑏λ ] = ∩∞ 𝑘=1 [𝑎λ𝑘 , 𝑏λ𝑘 ]. Then + there exists a unique fuzzy number 𝑢 e ∈ F 𝑁 (ℝ ) such that the family [𝑎λ , 𝑏λ ] represents the 𝜆-level sets of a fuzzy number 𝑢 e. 7

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Conversely, if [𝑎λ , 𝑏λ ] are the 𝜆-level sets of a fuzzy number 𝑢 e ∈ F 𝑁 (ℝ+ ), then the conditions (i) and (ii) are satisfied. From Theorem 3.6 and Definition 3.8, we obtain the following theorem. Theorem 3.10. Let µ be a semi-continuous fuzzy measure on ℑ(ℝ+ ). If an integrably bounded fuzzy complex valued measurable function 𝑓e = (R𝑒𝑓e, 𝐼𝑚𝑓e) is measurable, then for any 𝜆 ∈ (0, 1], ∫ (C) ∫

and (C)

[ ] ∫ ∫ − + (R𝑒𝑓e)λ 𝑑µ = (C) (R𝑒𝑓e)λ 𝑑µ, (C) (R𝑒𝑓e)λ 𝑑µ

[ ] ∫ ∫ − + e e e (𝐼𝑚𝑓 )λ 𝑑µ = (C) (𝐼𝑚𝑓 )λ 𝑑µ, (C) (𝐼𝑚𝑓 )λ 𝑑µ .

Lemma 3.11. If {𝜆𝑘 } is an increasing sequence in (0, 1] converging to 𝜆 and µ is lower semi-continuous, then we have e− lim µ({𝑥|(R𝑒𝑓e)− λ𝑛 (𝑥) > 𝛼}) = µ({𝑥|(R𝑒𝑓 )λ (𝑥) > 𝛼}),

𝑛→∞

e+ lim µ({𝑥|(R𝑒𝑓e)+ λ𝑛 (𝑥) > 𝛼}) = µ({𝑥|(R𝑒𝑓 )λ (𝑥) > 𝛼}),

𝑛→∞

e− lim µ({𝑥|(𝐼𝑚𝑓e)− λ𝑛 (𝑥) > 𝛼}) = µ({𝑥|(𝐼𝑚𝑓 )λ (𝑥) > 𝛼}),

𝑛→∞

and

e+ lim µ({𝑥|(𝐼𝑚𝑓e)+ λ𝑛 (𝑥) > 𝛼}) = µ({𝑥|(𝐼𝑚𝑓 )λ (𝑥) > 𝛼}.

𝑛→∞

Under same condition for {𝜆𝑘 } in Lemma 3.11, we have ∞ e− lim µ({𝑥|(R𝑒𝑓e)− λ𝑛 (𝑥) > 𝛼}) = µ(∩𝑛=1 {𝑥|(R𝑒𝑓 )λ𝑛 (𝑥) > 𝛼}),

𝑛→∞

∞ e+ lim µ({𝑥|(R𝑒𝑓e)+ λ𝑛 (𝑥) > 𝛼}) = µ(∩𝑛=1 {𝑥|(R𝑒𝑓 )λ𝑛 (𝑥) > 𝛼}),

𝑛→∞

∞ e− lim µ({𝑥|(𝐼𝑚𝑓e)− λ𝑛 (𝑥) > 𝛼}) = µ(∩𝑛=1 {𝑥|(𝐼𝑚𝑓 )λ𝑛 (𝑥) > 𝛼}),

𝑛→∞

and

∞ e+ lim µ({𝑥|(𝐼𝑚𝑓e)+ λ𝑛 (𝑥) > 𝛼}) = µ(∩𝑛=1 {𝑥|(𝐼𝑚𝑓 )λ𝑛 (𝑥) > 𝛼}.

𝑛→∞

Thus, by Lemma 3.11, we can obtain the following theorem. 8

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Theorem 3.12. Let µ be a semi-continuous fuzzy measure. If a fuzzy complex valued function 𝑓e is integrably bounded and {𝜆𝑘 } is an increasing sequence in (0, 1] converging to 𝜆, then we have (i) for any 0 < 𝜆1 ≤ 𝜆2 ≤ 1, (

∫ (C)

𝑓e𝑑µ

)

( ) ∫ e ⊃ (C) 𝑓 𝑑µ

λ1

,

λ2

and (ii) for any increasing sequence {𝜆𝑘 } in (0, 1] converging to 𝜆, (

∫ (C)

R𝑒𝑓e𝑑µ

) =

∩∞ 𝑘=1

(

∫ (C)

R𝑒𝑓e𝑑µ

)

λ

and

λ𝑘

( ) ( ) ∫ ∫ ∞ e e (C) 𝐼𝑚𝑓 𝑑µ = ∩𝑘=1 (C) 𝐼𝑚𝑓 𝑑µ λ

. λ𝑘

e+ e e− e+ Proof. (i) Note that (R𝑒𝑓e)λ1 = [(R𝑒𝑓e)− λ1 , (R𝑒𝑓 )λ1 ] ⊂ (R𝑒𝑓 )λ2 = [(R𝑒𝑓 )λ2 , (R𝑒𝑓 )λ2 ] implies e− e+ e+ (R𝑒𝑓e)− λ1 ≤ (R𝑒𝑓 )λ2 and (R𝑒𝑓 )λ1 ≤ (R𝑒𝑓 )λ2 e+ e e− e+ and that (𝐼𝑚𝑓e)λ1 = [(𝐼𝑚𝑓e)− λ1 , (𝐼𝑚𝑓 )λ1 ] ⊂ (𝐼𝑚𝑓 )λ2 = [(𝐼𝑚𝑓 )λ2 , (𝐼𝑚𝑓 )λ2 ] implies e− e+ e+ (𝐼𝑚𝑓e)− λ1 ≤ (𝐼𝑚𝑓 )λ2 and (𝐼𝑚𝑓 )λ1 ≤ (𝐼𝑚𝑓 )λ2 . Thus, by Theorem 2.4(1) and Definition 2.5 (8) and Theorem 3.10, we obtain the followings: (

∫ (C)

R𝑒𝑓e𝑑µ

)

∫ = (C)

(R𝑒𝑓e)λ1 𝑑µ

λ1

[ ] ∫ ∫ − + e e = (C) (R𝑒𝑓 )λ1 𝑑µ, (C) (R𝑒𝑓 )λ1 𝑑µ [ ] ∫ ∫ − + ⊃ (C) (R𝑒𝑓e)λ2 𝑑µ, (C) (R𝑒𝑓e)λ2 𝑑µ ( ) ∫ ∫ e e = (C) (R𝑒𝑓 )λ2 𝑑µ = (C) R𝑒𝑓 𝑑µ . λ2

Similarly, we obtain the followings. ( ) ∫ (C) 𝐼𝑚𝑓e𝑑µ

( ) ∫ ⊃ (C) 𝐼𝑚𝑓e𝑑µ

λ1

λ2

9

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JANG: CHOQUET INTEGRAL

(ii) Let {𝜆𝑘 } be an increasing sequence in (0, 1] converging to 𝜆. Then, by Definition 2.5 (4) and the monotone convergence theorem for Lebesgue integral, we can obtain the followings. ∫ ∫ ∞ − (C) (R𝑒𝑓e)λ 𝑑µ = µ({𝑥|(R𝑒𝑓e)− λ (𝑥) > 𝛼})𝑑𝛼 0 ∫ ∞ = lim µ({𝑥|(R𝑒𝑓e)− λ𝑛 (𝑥) > 𝛼})𝑑𝛼 𝑛→∞ 0 ∫ ∞ = lim µ({𝑥|(R𝑒𝑓e)− λ𝑛 (𝑥) > 𝛼})𝑑𝛼 𝑛→∞ 0 ∫ ∫ ∞ e = lim (C) (R𝑒𝑓 )λ𝑛 𝑑µ = ∩𝑛=1 (C) (R𝑒𝑓e)− λ𝑛 𝑑µ. 𝑛→∞

Similarly, we obtain the following three equalities. ∫ ∫ + ∞ e (C) (R𝑒𝑓 )λ 𝑑µ = ∩𝑛=1 (C) (R𝑒𝑓e)+ λ𝑛 𝑑µ, ∫ (C) ∫

and (C)

(𝐼𝑚𝑓e)− λ 𝑑µ (𝐼𝑚𝑓e)+ λ 𝑑µ

=

∩∞ 𝑛=1 (C)

=

∩∞ 𝑛=1 (C)

∫ ∫

(𝐼𝑚𝑓e)− λ𝑛 𝑑µ, (𝐼𝑚𝑓e)+ λ𝑛 𝑑µ.

Thus we have ( ) [ ] ∫ ∫ ∫ − + e e e (C) R𝑒𝑓 𝑑µ = (C) (R𝑒𝑓 )λ 𝑑µ, (C) (R𝑒𝑓 )λ 𝑑µ λ [ ] ∫ ∫ − + ∞ ∞ e e = ∩𝑛=1 (C) (R𝑒𝑓 )λ𝑛 𝑑µ, ∩𝑛=1 (C) (R𝑒𝑓 )λ𝑛 𝑑µ [ ] ∫ ∫ − + ∞ e e = ∩𝑛=1 (C) (R𝑒𝑓 )λ𝑛 𝑑µ, (R𝑒𝑓 )λ𝑛 𝑑µ ( ) ∫ ∫ ∞ ∞ = ∩𝑛=1 (C) (R𝑒𝑓e)λ𝑛 𝑑µ = ∩𝑛=1 (C) R𝑒𝑓e𝑑µ

.

λ𝑛

By the same method of the above equality’s proof for R𝑒𝑓e, we can obtain ( ) ( ) ∫ ∫ ∞ (C) 𝐼𝑚𝑓e𝑑µ = ∩𝑛=1 (C) R𝑒𝑓e𝑑µ . λ

λ𝑛

From Theorem 3.12, we can obtain the following Remark which is the existence of the Choquet integral with respect to a fuzzy measure of an integrably bounded fuzzy complex valued measurable function. 10

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Remark 3.13. By Theorem 3.12 and Lemma 3.11, there exists a fuzzy number 𝑢 e, ve ∈ + F 𝑁 (C ) such that ( ) ( ) ∫ ∫ e e (e 𝑢)λ = (C) R𝑒𝑓 𝑑e µ and (e v )λ = (C) 𝐼𝑚𝑓 𝑑e µ . λ

λ

for all 𝜆 ∈ (0, 1]. If we put C = (e 𝑢, ve), then C ∈ F C𝑁 (C+ ) and (( ) ( ) ) ( ) ∫ ∫ ∫ Cλ = (e 𝑢λ , veλ ) = (C) R𝑒𝑓e𝑑e µ , (C) 𝐼𝑚𝑓e𝑑e µ = (C) 𝑓e𝑑e µ . λ

λ

λ

That is, if a fuzzy complex valued function 𝑓e is integrably bounded, then 𝑓e is integrable. Thus, we have the following basic properties of Choquet integrals of fuzzy complex valued measurable functions. Theorem 3.14. Let µ be a semi-continuous fuzzy measure. The Choquet of integrably bounded fuzzy complex valued measurable functions has the following properties: for any two fuzzy complex valued measurable ∫ ∫ functions 𝑤𝑖𝑑𝑒𝑡𝑖l𝑑𝑒𝑓 and 𝑤𝑖𝑑𝑒𝑡𝑖l𝑑𝑒𝑔, then e e (1) if 𝑓 ≤ 𝑔e, then (C) 𝑓 𝑑µ ≤ (C) 𝑔e𝑑µ, (2) if we define (𝑓e∨ 𝑔e)(𝑧) = 𝑓e(𝑧) ∨ 𝑔e(𝑧) and (𝑓e∧ 𝑔e)(𝑧) = 𝑓e(𝑧) ∧ 𝑔e(𝑧) for all 𝑧 ∈ C+ , then ∫ ∫ ∫ e e (C) 𝑓 ∨ 𝑔e𝑑µ ≥ (C) 𝑓 𝑑µ ∨ (C) 𝑔e𝑑µ ∫

and (C)

𝑓e ∧ 𝑔e𝑑µ ≤ (C)

∫

𝑓e𝑑µ ∧ (C)

∫ 𝑔e𝑑µ

§4. Some convergence properties of the fuzzy complex valued Choquet integral In this section, we introduce some convergence properties of the Choquet integral, for examples, Denneberg’s convergence theorem and monotone convergence theorem for Choquet integrals with respect to a fuzzy measure of real-valued measurable functions(see [11,12]). Definition 4.1 ([10]). A sequence {𝑓𝑛 } of measurable functions is said to converge to 𝑓 in distribution, in symbols G − lim𝑛→∞ 𝑓𝑛 = 𝑓 , if lim Gf𝑛 (r) = Gf (r), e.c.,

𝑛→∞

where ”e.c.” stands for ”except at most countably many values of r”. 11

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Theorem 4.2 ([10]). If {𝑓𝑛 } is a sequence of measurable functions that converges to 𝑓 in distribution and if 𝑔 and ℎ are integrable functions such that Gh ≤ Gf𝑛 ≤ Gg e.c., 𝑛 = 1, 2, · · · , then 𝑓 is integrable and

∫ lim (C)

𝑛→∞

∫ 𝑓𝑛 𝑑µ = (C)

𝑓 𝑑µ.

Theorem 4.3 ([9]). (1) If a fuzzy measure µ is semi-continuous and {𝑓𝑛 } is an increasing sequence of measurable functions which converges to 𝑓 , µ − 𝑎.𝑒., then we have ∫ ∫ lim (C) 𝑓𝑛 𝑑µ = (C) 𝑓 𝑑µ, 𝑛→∞

where ”P is µ − 𝑎.𝑒.” means µ({𝑥 ∈ ℝ+ |P (𝑥) is not true }) = 0. (2) If a fuzzy measure µ is upper semi-continuous and {𝑓𝑛 } is an decreasing sequence of measurable functions which converges to 𝑓 , µ − 𝑎.𝑒., and if there exists an integrable function 𝑔 such that 𝑓1 ≤ 𝑔, then we have ∫ ∫ lim (C) 𝑓𝑛 𝑑µ = (C) 𝑓 𝑑µ. 𝑛→∞

We discuss some convergence theorems for Choquet integrals with respect to a fuzzy measure of fuzzy complex valued measurable functions and define the new metric on F C𝑁 (C+ ). ¯ + is defined by Definition 4.4. A mapping D : F C𝑁 (C+ ) × F C𝑁 (C+ ) −→ ℝ D(C1 , C2 ) = max{△(R𝑒C1 , R𝑒C2 ), △(𝐼𝑚C1 , 𝐼𝑚C2 )}, where △(e 𝑢, ve) = supλ∈(0,1] 𝑑H (e 𝑢λ , veλ ) for all 𝑢 e, ve ∈ F 𝑁 (ℝ+ ). Note that (F C𝑁 (C+ , D) is a metric space. By using this metric D, we define the concept of convergence of a sequence in (F C𝑁 (C+ , D). Definition 4.5. A sequence {C𝑛 } of fuzzy complex numbers in F C𝑁 (C+ ) is said to converge to a fuzzy complex number C in the metric D, in symbols D − lim𝑛→∞ C𝑛 = C, if l𝑖𝑚𝑛→∞ D(C𝑛 , C) = 0. From the definition of metric D on F C𝑁 (C+ ), we can define the following definitions. 12

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Definition 4.6. A sequence {𝑓e𝑛 } of integrably bounded fuzzy complex valued measurable functions on F C𝑁 (C+ ) is said to converges to 𝑓e in distribution, in syme + e − bols G − lim𝑛→∞ 𝑓e𝑛 = 𝑓e if four sequences {(R𝑒𝑓e𝑛 )− λ }, {(R𝑒𝑓𝑛 )λ }, {(𝐼𝑚𝑓𝑛 )λ }, and e− e+ e− e+ {(𝐼𝑚𝑓e𝑛 )+ λ } converge to {(R𝑒𝑓 )λ }, {(R𝑒𝑓 )λ }, {(𝐼𝑚𝑓 )λ }, and {(𝐼𝑚𝑓 )λ } in distribution, respectively. By using Definition 4.6 and Theorem 2.5 and the definition of the metric D, we can obtain the following theorem under some sufficient conditions which is Dennebergtype convergence theorem for Choquet integral with respect to a fuzzy measure of integrably bounded fuzzy complex valued functions. Theorem 4.7. Assume that a fuzzy complex valued function 𝑓e is integrably bounded and µ is a semi-continuous fuzzy measure. If {𝑓e𝑛 } is a sequence of fuzzy complex valued measurable functions that converges to 𝑓e in distribution, and if 𝑔 and ℎ are integrable functions such that e + e − e + ℎ ≤ (R𝑒𝑓e𝑛 )− λ ≤ (R𝑒𝑓𝑛 )λ ≤ 𝑔 and ℎ ≤ (𝐼𝑚𝑓𝑛 )λ ≤ (𝐼𝑚𝑓𝑛 )λ ≤ 𝑔 for all 𝜆 ∈ (0, 1] and a.c. for 𝑛 = 1, 2, · · · , then 𝑓e is integrably bounded and ∫ D − lim (C) 𝑛→∞

𝑓e𝑛 𝑑µ = (C)

∫

𝑓e𝑑µ.

Proof. Clearly, if we take 𝑧 = 𝑥 + 𝑖𝑦 ∈ C+ , then we have e e+ ∥(R𝑒𝑓e)λ (𝑥)∥ ≤ (R𝑒𝑓e)+ λ ≤ 𝑔(𝑥) and ∥(𝐼𝑚𝑓 )λ (𝑥)∥ ≤ (𝐼𝑚𝑓 )λ ≤ 𝑔(𝑥), e + for all 𝜆 ∈ (0, 1]. Thus, 𝑓e is integrably bounded. Since ℎ ≤ (R𝑒𝑓e𝑛 )− λ ≤ (R𝑒𝑓𝑛 )λ ≤ 𝑔 e + ≤ G(Refe𝑛 )+ ≤ Gg and Gh ≤ and ℎ ≤ (𝐼𝑚𝑓e𝑛 )− λ ≤ (𝐼𝑚𝑓𝑛 )λ ≤ 𝑔, Gh ≤ G(Refe𝑛 )− λ λ G(I𝑚fe𝑛 )− ≤ G(I𝑚fe𝑛 )+ ≤ Gg . Then, by Definition 4.6 and Theorem 4.2, we obtain λ

λ

∫ lim (C)

𝑛→∞

∫ lim (C)

𝑛→∞

∫

lim (C)

𝑛→∞

∫

and lim (C)

𝑛→∞

(R𝑒𝑓e𝑛 )− λ 𝑑µ = (C) (R𝑒𝑓e𝑛 )+ λ 𝑑µ = (C) (𝐼𝑚𝑓e𝑛 )− λ 𝑑µ = (C) (𝐼𝑚𝑓e𝑛 )+ λ 𝑑µ = (C) 13

1081

∫ ∫ ∫

∫

(R𝑒𝑓e)− λ 𝑑µ, (R𝑒𝑓e)+ λ 𝑑µ, (𝐼𝑚𝑓e)− λ 𝑑µ, (𝐼𝑚𝑓e)+ λ 𝑑µ,

JANG: CHOQUET INTEGRAL

for all 𝜆 ∈ (0, 1]. Thus, by the definition of the metric ∆, we have ( ) ∫ ∫ e e ∆ (C) R𝑒𝑓𝑛 𝑑µ, (C) R𝑒𝑓 𝑑µ ( ) ∫ ∫ = sup 𝑑H (C) (R𝑒𝑓e𝑛 )λ 𝑑µ, (C) (R𝑒𝑓e)λ 𝑑µ λ∈(0,1]

{

∫

= sup max |(C) λ∈(0,1]

∫ |(C)

(R𝑒𝑓e𝑛 )− λ 𝑑µ − (C)

(R𝑒𝑓e𝑛 )+ λ 𝑑µ

∫ − (C)

∫

(R𝑒𝑓e)− λ 𝑑µ|, }

(R𝑒𝑓e)+ λ 𝑑µ|

−→ 0, for all 𝜆 ∈ (0, 1] as 𝑛 → ∞ and (

) ∫ e e ∆ (C) 𝐼𝑚𝑓𝑛 𝑑µ, (C) 𝐼𝑚𝑓 𝑑µ ( ) ∫ ∫ = sup 𝑑H (C) (𝐼𝑚𝑓e𝑛 )λ 𝑑µ, (C) (𝐼𝑚𝑓e)λ 𝑑µ ∫

λ∈(0,1]

{ ∫ ∫ − e = sup max |(C) (𝐼𝑚𝑓𝑛 )λ 𝑑µ − (C) (𝐼𝑚𝑓e)− λ 𝑑µ|, λ∈(0,1]

∫ |(C)

(𝐼𝑚𝑓e𝑛 )+ λ 𝑑µ

∫ − (C)

(𝐼𝑚𝑓e)+ λ 𝑑µ|

}

−→ 0. Therefore, by Definition 4.4, we obtain ∫

∫ e D − lim (C) 𝑓𝑛 𝑑µ = (C) 𝑓e𝑑µ 𝑛→∞ { ( ) ∫ ∫ e e = lim max ∆ (C) R𝑒𝑓𝑛 𝑑µ, (C) R𝑒𝑓 𝑑µ , 𝑛→∞ ( )} ∫ ∫ e e ∆ (C) R𝑒𝑓𝑛 𝑑µ, (C) R𝑒𝑓 𝑑µ = 0.

Finally, we can obtain monotone convergence theorems for Choquet integrals with respect to a fuzzy measure of integrably bounded fuzzy complex valued functions as follows. 14

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JANG: CHOQUET INTEGRAL

Theorem 4.8. Assume that 𝑓e is integrably bounded and that a fuzzy measure µ is semi-continuous. (1) If {𝑓e𝑛 } is an increasing sequence of integrably bounded fuzzy complex valued measurable functions that converges to 𝑓e in the metric D,then we have ∫ ∫ D − lim (C) 𝑓e𝑛 𝑑µ = (C) 𝑓e𝑑µ. 𝑛→∞

(2) If {𝑓e𝑛 } is a decreasing sequence of integrably bounded fuzzy complex valued measurable functions that converges to 𝑓e in the metric D and if there exists an integrabe function 𝑔 such that e + e − e + (R𝑒𝑓e𝑛 )− λ ≤ (R𝑒𝑓𝑛 )λ ≤ 𝑔 and (𝐼𝑚𝑓𝑛 )λ ≤ (𝐼𝑚𝑓𝑛 )λ ≤ 𝑔,

µ − 𝑎.𝑒.,

for all 𝜆 ∈ (0, 1] and for all 𝑛 = 1, 2, · · · ,, then we have ∫ ∫ e D − lim (C) 𝑓𝑛 𝑑µ = (C) 𝑓e𝑑µ. 𝑛→∞

Proof. Note that if {𝑓e𝑛 } is an increasing sequence of fuzzy complex valued measurable functions that converges to 𝑓e in the metric D, then four increasing see + e − e + e− quences {(R𝑒𝑓e𝑛 )− λ }, {(R𝑒𝑓𝑛 )λ }, {(𝐼𝑚𝑓𝑛 )λ }, and {(𝐼𝑚𝑓𝑛 )λ } converge to {(R𝑒𝑓 )λ }, e− e+ {(R𝑒𝑓e)+ µ − 𝑎.𝑒., respectively for all 𝜆 ∈ (0, 1]. By λ }, {(𝐼𝑚𝑓 )λ }, and {(𝐼𝑚𝑓 )λ }, Theorem 4.3 (1), we have ∫ ∫ − lim (C) (R𝑒𝑓e𝑛 )λ 𝑑µ = (C) (R𝑒𝑓e)− λ 𝑑µ, 𝑛→∞

∫ lim (C)

𝑛→∞

∫

lim (C)

𝑛→∞

∫

and lim (C)

𝑛→∞

(R𝑒𝑓e𝑛 )+ λ 𝑑µ = (C) (𝐼𝑚𝑓e𝑛 )− λ 𝑑µ = (C) (𝐼𝑚𝑓e𝑛 )+ λ 𝑑µ = (C)

∫ ∫ ∫

(R𝑒𝑓e)+ λ 𝑑µ, (𝐼𝑚𝑓e)− λ 𝑑µ, (𝐼𝑚𝑓e)+ λ 𝑑µ,

for all 𝜆 ∈ (0, 1]. Thus, by Definition 4.4 and the same method of the proof of Theorem 4.7, we have ( ) ∫ ∫ e e lim D (C) 𝑓𝑛 𝑑µ, (C) 𝑓 𝑑µ = 0. 𝑛→∞

(2) The proof is similar to the proof of (1). §5. Conclusions 15

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In this paper, by using, we use the Choquet integral with respect to a fuzzy measure instead of the Lebesgue integral with respect to a classical measure, we define the new concept of the Choquet integral with respect to a fuzzy measure of fuzzy complex valued functions in Definition 3.8 and Theorems 3.10, 3.12. In Definitions 4.4, 4.5, 4.6, and Theorems 4.7, 4.8, we investigate the existence of the fuzzy complex valued Choquet integral and some convergence properties of the Choquet integrals of integrably bounded fuzzy complex valued functions. In the future, we will study a probability measure approach to rank fuzzy complex numbers and the theoretical fundamentals of leaning theory based on fuzzy complex random samples, etc. Acknowledgement: This paper was supported by Konkuk University in 2013. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

J.J. Buckley, Fuzzy complex numbers, Fuzzy Sets and Systems 33 (1989), 333-345. J.J. Buckley, Fuzzy complex analysis I, Fuzzy Sets and Systems 41 (1991), 269-284. J.J. Buckley, Fuzzy complex analysis II, Fuzzy Sets and Systems 49 (1992), 171-179. L.C. Jang, B.M. Kil, Y.K. Kim, J.S. Kwon, Some properties of Choquet integrals of setvalued functions, Fuzzy Sets and Systems 91 (1997), 61-67. L.C. Jang, J.S. Kwon, On the representation of Choquet integrals of set-valued functions and null sets, Fuzzy Sets and Systems 112 (2000), 1 233-239. L.C. Jang, A note on the monotone interval-valued set function defined by the intervalvalued Choquet integral, Comm. Korean Math. Soc. 22(2) (2007), 227-234. L.C. Jang, T. Kim, J.D. Jeon, and W.J. Kim, On Choquet integrals of measurable fuzzy number-valued functions, Bull. Koran Math. Soc. 41(1) (2004), 95-107. T. Murofushi and M. Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems 29 (1989), 201-227. T. Murofushi and M. Sugeno, A theory of fuzzy measure representations, the Choquet integral, and null sets, J. math. Anal. Appl. 159 (1991), 532-549. T. Murofushi, M.Sugeno, and M. Suzaki, Autocontinuity, convergence in measure, and convergence in distribution, Fuzzy Sets and Systems 92 (1997), 197-203. M.L. Puri and D.A. Ralescu, Fuzzy random variable, J. Math. Anal. Appl. 114 (1986), 409-422. M. Sugeno, Y. Narukawa and T. Murofushi, Choquet integral and fuzzy measures on locally compact space, Fuzzy Sets and Systems 99 (1998), 205-211. G. Wang and X. Li, Generalized Lebesgue integrals of fuzzy complex valued functions, Fuzzy Sets and Systems 127 (2002), 363-370. D. Zhang, C. Guo and D. Liu, Set-valued Choquet integrals revisited, Fuzzy Sets and Systems 147 (2004), 475-485.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1085-1095, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

INTUITIONISTIC FUZZY STABILITY OF EULER-LAGRANGE TYPE QUARTIC MAPPINGS HEEJEONG KOH1 , DONGSEUNG KANG1 AND IN GOO CHO2∗

Abstract. We investigate some stability results and intuitionistic fuzzy continuities concerning the following Euler-Lagrange type quartic functional equation 1 f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) 2 1 = a(a + 1)2 f (x + y) + (a2 − 1)2 (f (x) + f (y)) 2 in intuitionistic fuzzy normed spaces.

1. Introduction In 1965, Zadeh [19] introduced the theory of fuzzy sets. After the pioneering work of Zadeh, there has been a great effort to obtain fuzzy analogues of classical theories. It has useful applications in various fields such as population dynamics, chaos control, computer programming, nonlinear dynamical systems, nonlinear operators, etc. Also, many mathematicians considered the fuzzy metric spaces in different view. In particular, In 1984, Katsaras [8] defined a fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. Stability problem of a functional equation was first originated by S.M. Ulam [18] concerning the stability of group homomorphisms. It was answered by Hyers [5] on the assumption that the spaces are Banach spaces and generalized by T. Aoki [1] for the stability of the additive mapping involving a sum of powers of p-norms and Th.M. Rassias [16] for the stability of the linear mapping by considering the Cauchy difference to be unbounded. During the last three decades, several stability problems of a large variety of functional equations have been extensively studied and generalized by a number of authors [3], [4], [6], [16], and [2] and various fuzzy stability results have been studied in [9], [10], [11], and [12]. In particular, J. M. Rassias [15] introduced the Euler-Lagrange type quadratic functional equation (1.1)

f (rx + sy) + f (sx − ry) = (r2 + s2 )[f (x) + f (y)] ,

for fixed reals r, s with r 6= 0 , s 6= 0 . Also, K-W. Jun and H-M. Kim [7] proved the Hyers-Ulam-Rassias stability of a Euler-Lagrange type cubic mapping as follows: (1.2)

f (ax + y) + f (x + ay) = (a + 1)(a − 1)2 [f (x) + f (y)] + a(a + 1)f (x + y) ,

2000 Mathematics Subject Classification. 39B52. Key words and phrases. stability problem, Euler-Lagrange functional equation, quartic functional equation, intuitionistic fuzzy stability, intuitionistic fuzzy continuity. * Corresponding author. 1

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HEEJEONG KOH, DONGSEUNG KANG, IN GOO CHO

where a 6= 0 , ±1 , for all x, y ∈ X . In this paper, we investigate the stability problem for the Euler-Lagrange type quartic functional equation as follows: 1 (1.3) f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) 2 1 2 = a(a + 1) f (x + y) + (a2 − 1)2 (f (x) + f (y)) , 2 for fixed integer a with a 6= 0, ±1 . In fact, f (x) = x4 is a solution of (1.3) by virtue of the identity 1 (ax + y)4 + (x + ay)4 + a(a − 1)2 (x − y)4 2 1 2 4 = a(a + 1) (x + y) + (a2 − 1)2 (x4 + y 4 ) . 2 In this paper, we investigate some stability results and intuitionistic fuzzy continuities concerning the equation (1.3) in intuitionistic fuzzy normed spaces. Definition 1.1. A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-norm if it satisfies the following conditions: (1) * is associative and commutative, (2) * is continuous, (3) a ∗ 1 = a for all a ∈ [0, 1] , (4) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d , for each a, b, c, d ∈ [0, 1] . Definition 1.2. A binary operation ♦ : [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-conorm if it satisfies the following conditions: (1) ♦ is associative and commutative, (2) ♦ is continuous, (3) a♦0 = a for all a ∈ [0, 1] , (4) a♦b ≤ c♦d whenever a ≤ c and b ≤ d , for each a, b, c, d ∈ [0, 1] . Saadati and Park introduced the concept of intuitionistic fuzzy normed space; [17]. Definition 1.3. The five-tuple (X, µ, ν, ∗, ♦) is called an intuitionistic fuzzy normed space(for short, IFNS) if X is a vector space, ∗ is a continuous t-norm, ♦ is continuous t-conorm, and µ and ν are fuzzy sets on X × (0, 1) satisfying the following conditions. For all x, y ∈ X and s, t > 0 , (1) µ(x, t) + ν(x, y) ≤ 1 , (2) µ(x, t) > 0 , (3) µ(x, t) = 1 if and only if x = 0 , t (4) µ(αx, t) = µ(x, |α| ) for each α 6= 0 , (5) µ(x, t) ∗ µ(y, s) ≤ µ(x + y, t + s) , (6) µ(x, ·) : (0, ∞) → [0.1] is continuous, (7) limt→∞ µ(x, t) = 1 and limt→0 µ(x, t) = 0 , (8) ν(x, t) < 1 , (9) ν(x, t) = 0 if and only if x = 0 , t ) for each α 6= 0 , (10) ν(αx, t) = ν(x, |α| (11) ν(x, t)♦ν(y, s) ≥ ν(x + y, t + s) , (12) ν(x, ·) : (0, ∞) → [0.1] is continuous, (13) limt→∞ ν(x, t) = 0 and limt→0 ν(x, t) = 1 . In this case (µ, ν) is said to be an intuitionistic fuzzy norm. Also, they investigated the concepts of convergence and Cauchy sequences in an intuittionistic fuzzy normed space as follows:

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3

Let (X, µ, ν, ∗, ♦) be an IFNS. A sequence (xk ) is said to be intuittionistic fuzzy convergent to L ∈ X if limk→∞ µ(xk − L, t) = 1 and limk→∞ ν(xk − L, t) = 0 , for all t > 0 . A sequence (xk ) is said to be intuittionistic fuzzy Cauchy sequence if limk→∞ µ(xk+p − xk , t) = 1 and limk→∞ ν(xk+p − xk , t) = 0 , for all t > 0 and p = 1, 2, · · · . Also, (X, µ, ν, ∗, ♦) is said to be complete if every intuitionistic fuzzy Cauchy sequence in (X, µ, ν, ∗, ♦) is intuitionistic fuzzy convergent in (X, µ, ν, ∗, ♦) . 2. Intuitionistic Fuzzy Stability Throughout this section, let X be a linear space and let Y be a intuitionistic fuzzy Banach space. Let a be a fixed integer with a 6= 0, ±1 , For convenience, we use the following abbreviation: 1 (2.1) Da f (x, y) := f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) 2 1 − a(a + 1)2 f (x + y) − (a2 − 1)2 (f (x) + f (y)) , 2 for all x, y ∈ X . Theorem 2.1. Let a be an integer with a 6= 0, ±1 , and let X be a linear space and let (Z, µ0 , ν 0 ) be an intuitionistic fuzzy normed space(IFNS). Let φ : X × X → Z be a function such that for some 0 < α < a4 (2.2)

µ0 (φ(ax, 0), t) ≥ µ0 (αφ(x, 0), t) and ν 0 (φ(ax, 0), t) ≤ ν 0 (αφ(x, 0), t) ,

and limn→∞ µ0 (φ(an x, an y), a4n t) = 1 and limn→∞ ν 0 (φ(an x, an y), a4n t) = 0 , for all x, y ∈ X and t > 0 . Suppose (Y, µ, ν) is an intuitionistic fuzzy Banach space and f : X → Y is a φ-approximately mapping such that f (0) = 0 and   (2.3) µ Da f (x, y), t ≥ µ0 (φ(x, y), t) and   ν Da f (x, y), t ≤ ν 0 (φ(x, y), t)

(2.4)

for all t > 0 and all x, y ∈ X . Then there exists a unique Euler-Lagrange type quartic mapping Q : X → Y such that 1 (2.5) µ(Q(x) − f (x), t) ≥ µ0 (φ(x, 0), (a4 − α)t) , 2 and 1 (2.6) ν(Q(x) − f (x), t) ≤ ν 0 (φ(x, 0), (a4 − α)t) , 2 for all x ∈ X and all t > 0 . Proof. By letting y = 0 in inequalities (2.3) and (2.4), we have (2.7) µ(f (ax)−a4 f (x), t) ≥ µ0 (φ(x, 0), t) and ν(f (ax)−a4 f (x), t) ≤ ν 0 (φ(x, 0), t) , that is, (2.8)

µ(

f (ax) t − f (x), 4 ) ≥ µ0 (φ(x, 0), t) , a4 a

ν(

f (ax) t − f (x), 4 ) ≤ ν 0 (φ(x, 0), t) , a4 a

and (2.9)

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HEEJEONG KOH, DONGSEUNG KANG, IN GOO CHO

for all x ∈ X and t > 0 . For each n ∈ N , letting x = an x in inequalities (2.8) and (2.9), we get  f (an+1 x) f (an x) t  µ a4n ( 4(n+1) − ), 4 ≥ µ0 (φ(an x, 0), t) a4n a a  f (an+1 x) f (an x) t  ν a4n ( 4(n+1) − ), 4 ≤ ν 0 (φ(an x, 0), t) . a4n a a By using the inequality (2.2), these previous inequalities imply that  f (an+1 x) f (an x) t  t − , ≥ µ0 (φ(an x, 0), t) = µ0 (φ(x, 0), n ) µ a4n α a4(n+1) a4(n+1)  f (an+1 x) f (an x) t  t ν − , ≤ ν 0 (φ(x, 0), n ) , 4n 4(n+1) 4(n+1) a α a a for all x ∈ X , t > 0 , and n ≥ 0 . Now, switching t by αn t in the previous inequalities, we have   f (an+1 x) f (an x) 1 α n − , ( ) t ≥ µ0 (φ(x, 0), t) , µ a4n a4 a4 a4(n+1)  f (an+1 x) f (an x) 1 α  ν , 4 ( 4 )n t ≤ ν 0 (φ(x, 0), t) , − 4n 4(n+1) a a a a for all x ∈ X , t > 0 , and n ≥ 0 . Then n−1  f (an x)   n−1  X f (ak+1 x) f (ak x) n−1 X 1 α X 1 α k k µ ( − f (x), ( ) t = µ − ), ( ) t a4n a4 a4 a4k a4 a4 a4(k+1) k=0

k=0

k=0

n−1 Y

 f (ak+1 x) f (ak x) 1 α  k ≥ µ − , ( ) t ≥ µ0 (φ(x, 0), t) , 4(k+1) a4k a4 a4 a k=0 and n−1  f (an x)   n−1  X 1 α X f (ak+1 x) f (ak x) n−1 X 1 α k k ν ( − − f (x), ( ) t = ν ), ( ) t a4n a4 a4 a4k a4 a4 a4(k+1) k=0

k=0

n−1 a

 f (ak+1 x)

k=0

k

f (a x) 1 α k , 4 ( 4 ) t ≤ ν 0 (φ(x, 0), t) , a4k a a k=0 Qn `n for all x ∈ X , t > 0 , and n ≥ 1 , where j=1 aj = a1 ∗ · · · ∗ an and j=1 aj = a1 ♦ · · · ♦an . For any integer s ≥ 0 , replacing x with as x in the previous inequalities, we have ≤

ν

a4(k+1)

−



n−1  f (an+s x) f (as x) X 1 α k  µ a4s [ 4(n+s) − ], ( ) t ≥ µ0 (φ(as x, 0), t) , a4s a4 a4 a k=0

and n−1 f (an+s x) f (as x) X 1 α k  ν a [ 4(n+s) − ], ( ) t ≤ ν 0 (φ(as x, 0), t) , a4s a4 a4 a k=0



4s

that is,  f (an+s x) f (as x) 1 n−1  X 1 α t k µ − , ( ) t ≥ µ0 (φ(x, 0), s ) , a4s a4s a4 a4 α a4(n+s) k=0

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EULER-LAGRANGE TYPE QUARTIC MAPPINGS

5

and  f (an+s x) f (as x) 1 n−1  X 1 α t k µ − , ( ) t ≥ µ0 (φ(x, 0), s ) , 4s 4s 4 4 4(n+s) a a a a α a k=0 for all x ∈ X , t > 0 , n ≥ 0 , and s ≥ 0 . Now, switching t by αs t , we get   f (an+s x) f (as x) 1 n−1 X αs α , 4s ( 4 )k t − µ 4s 4 4(n+s) a a a a a k=0  f (an+s x) f (as x) n+s−1  X 1 α k = µ , ( ) t ≥ µ0 (φ(x, 0), t) , − a4s a4 a4 a4(n+s) k=s and ν

= ν

 f (an+s x) a4(n+s)

−

n−1 f (as x) 1 X αs α k  , ( ) t a4s a4s a4 a4 k=0

 f (an+s x) a4(n+s)

s

−

f (a x) , a4s

n+s−1 X k=s

1 α k  ( ) t ≤ ν 0 (φ(x, 0), t) , a4 a4

for all x ∈ X , t > 0 , n ≥ 0 , and s ≥ 0 . By putting t with

Pn+s−1t k=s

1 a4

( aα4 )k

, we have

 f (an+s x) f (as x)  t µ − , t ≥ µ0 (φ(x, 0), Pn+s−1 1 α ) , 4s 4(n+s) k a a k=s a4 ( a4 )

(2.10) and (2.11)

ν

 f (an+s x) a4(n+s)

−

t f (as x)  , t ≤ ν 0 (φ(x, 0), Pn+s−1 a4s k=s

for all x ∈ X , t > 0 , n ≥ 0 , and s ≥ 0 . Since 0 < α < a4 , 0

k=s

),

P∞  α k k=0

a4

< ∞ . Hence

0

t

limt→∞ µ (φ(x, 0), Pn+s−1

1 α k a4 ( a4 )

1 a4

( aα4 )k

) = 1 , and limt→∞ ν (φ(x, 0), Pn+s−1t 1 k=s

a4

( aα4 )k

)=

0 . Let ε > 0 and δ > 0 . Then there exists a t0 > 0 such that µ0 (φ(x, 0), Pn+s−1t0 1 ( α )k ) ≥ 1 − ε , and ν 0 (φ(x, 0), Pn+s−1t0 1 ( α )k ) ≤ ε . Since k=s k=s a4 a4 a4 a4 P∞ t0  α k Pn+s−1 t0  α k < ∞ , there exists a n0 ∈ N such that k=s a4 a4 < δ , for k=0 a4 a4  n  f (a x) all n + s > s ≥ n0 . Hence the sequence is a Cauchy sequence in (Y, µ, ν) . a4n  n  x) Since (Y, µ, ν) is a Banach space, the sequence f (a converges. Hence we can a4n define a function Q : X → Y by f (an x) , n→∞ a4n for all x ∈ X . Letting s = 0 in the inequalities (2.10) and (2.11), we have  f (an x)  t µ − f (x), t ≥ µ0 (φ(x, 0), Pn−1 1 α ) , 4n k a k=0 4 ( 4 ) Q(x) = lim

a

and ν

 f (an x) a4n

 t − f (x), t ≤ ν 0 (φ(x, 0), Pn−1 1

a

α k k=0 a4 ( a4 )

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for all t > 0 and n > 0 . Hence we have µ(Q(x) − f (x), t)

f (an x) f (an x) t t = µ(Q(x) − + − f (x), + ) a4n a4n 2 2   f (an x) f (an x) t  t ≥ µ Q(x) − , ∗ µ − f (x), a4n 2 a4n 2   1 t ≥ µ0 φ(x, 0),   , 2 Pn−1 1 α k k=0 a4

a4

and ν(Q(x) − f (x), t)

f (an x) f (an x) t t = ν(Q(x) − + − f (x), + ) a4n a4n 2 2  f (an x)  t f (an x) t  , ∗ ν − f (x), ≤ ν Q(x) − a4n 2 a4n 2   1 t 0 ≤ ν φ(x, 0),   , 2 Pn−1 1 α k k=0 a4

a4

that is, 1 µ(Q(x) − f (x), t) ≥ µ0 (φ(x, 0), (a4 − α)t) , 2 and 1 ν(Q(x) − f (x), t) ≤ ν 0 (φ(x, 0), (a4 − α)t) , 2 as n → ∞ . Respectively, replacing x , y , and t by an x , an y , and a4n t in inequalities (2.3) and (2.4), we have  D f (an x, an y)  a µ , t ≥ µ0 (φ(an x, an y), a4n t) , a4n and  D f (an x, an y)  a ν , t ≤ ν 0 (φ(an x, an y), a4n t) , a4n for all x ∈ X , t > 0 , and n ∈ N . Since limn→∞ µ0 (φ(an x, an y), a4n t) = 1 and limn→∞ ν 0 (φ(an x, an y), a4n t) = 0 , the mapping Q : X → Y satisfies the equation (1.3), that is, it is the Euler-Lagrange type quartic mapping. It only remains to show that the mapping Q : X → Y is unique. Assume Q0 : X → Y is another Euler-Lagrange type quartic mapping satisfying the inequalities (2.5) and (2.6). It is easy to show that Q(an x) = a4n Q(x) and Q0 (an x) = a4n Q0 (x) , for all n ∈ N .    Q(an x) Q0 (an x)  µ Q(x) − Q0 (x), t = µ − ,t a4n a4n  Q(an x) f (an x) t   f (an x) Q0 (an x) t  ≥ µ − , ∗µ − , a4n a4n 2 a4n a4n 2   a4n (a4 − α)  a4 − α  a4 n  0 n 0 ≥ µ φ(a x, 0), t ≥ µ φ(x, 0), t , 4 4 α and    a4 − α  a4 n  ν Q(x) − Q0 (x), t ≤ ν 0 φ(x, 0), t , 4 α

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for all x ∈ X and all t > 0 . Since limn→∞



a4 α

n

7

= ∞,

 a4 − α  a4 n  a4 − α  a4 n  lim µ0 φ(x, 0), t = 1 and lim ν 0 φ(x, 0), t = 0. n→∞ n→∞ 4 α 4 α Hence     µ Q(x) − Q0 (x), t = 1 and ν Q(x) − Q0 (x), t = 0 , 

for all x ∈ X and all t > 0 . We may conclude that Q(x) = Q0 (x) , for all x ∈ X , that is, the mapping Q : X → Y is unique, as desired.  Theorem 2.2. Let a be an integer with a 6= 0, ±1 , and let X be a linear space and let (Z, µ0 , ν 0 ) be an intuitionistic fuzzy normed space(IFNS). Let φ : X × X → Z be a function such that for some α > a4 x x (2.12) µ0 (φ( , 0), t) ≥ µ0 (φ(x, 0), αt) and ν 0 (φ( , 0), t) ≤ ν 0 (φ(x, 0), αt) , a a and limn→∞ µ0 (φ(a−n x, a−n y), a−4n t) = 1 and limn→∞ ν 0 (φ(a−n x, a−n y), a−4n t) = 0 , for all x, y ∈ X and t > 0 . Suppose (Y, µ, ν) is an intuitionistic fuzzy Banach space and f : X → Y is a φ-approximately mapping with f (0) = 0 satisfying the inequalities (2.3) and (2.4). Then there exists a unique Euler-Lagrange type quartic mapping Q : X → Y such that (2.13)

µ(Q(x) − f (x), t) ≥ µ0 (φ(x, 0),

(α − a4 ) t) , 2

ν(Q(x) − f (x), t) ≤ ν 0 (φ(x, 0),

(α − a4 ) t) , 2

and (2.14)

for all x ∈ X and all t > 0 . Proof. Letting x =

x a

in inequalities (2.7) of proof of Theorem 2.1, we have

(2.15) x x µ(f (x) − a4 f ( ), t) ≥ µ0 (φ(x, 0), αt) and ν(f (x) − a4 f ( ), t) ≤ ν 0 (φ(x, 0), αt) , a a for all x ∈ X and t > 0 . Similar to the proof of Theorem 2.1, we can deduce   t (2.16) µ a4(n+s) f (a−(n+s) x) − a4s f (a−s x), t ≥ µ0 (φ(x, 0), Pn+s−1 a4k ) , k=s

αk+1

and (2.17)

  t ν a4(n+s) f (a−(n+s) x) − a4s f (a−s x), t ≤ ν 0 (φ(x, 0), Pn+s−1 k=s

a4k αk+1

),

P∞  4 k for all x ∈ X , t > 0 , and s ≥ 0 and n ≥ 0 . Since α > a4 and k=0 aα < ∞,   the Cauchy criterion for convergence in IFNS implies that a4n f ( axn ) is a Cauchy sequence in the Banach space (Y, µ, ν) . A function Q : X → Y by x Q(x) = lim a4n f ( n ) , n→∞ a for all x ∈ X . Also, letting s = 0 and taking n → ∞ in the inequalities (2.16) and (2.17), we have the inequalities (2.13) and (2.14). The remains follows from the proof of Theorem 2.1. 

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3. Intutionistic fuzzy continuity Throughout this section, let (X, || · ||) be a normed space. In [13], they defined and studied the intuitionistic fuzzy continuity. In this section, we will investigate interesting results of continuous approximately Euler-Lagrange type quartic mappings. Before proceeding the proof, we will state the definition of intuitionistic fuzzy continuity as follows. Definition 3.1. [ [14, Definition 3.1]] Let f : R → X be a function, where R is endowed with the Euclidean topology and X is an intuitionistic fuzzy normed space equipped with intuitionistic fuzzy norm (µ , ν) . Them f is called intuitionistic fuzzy continuous at a point s0 ∈ R if for all ε > 0 and all 0 < α < 1 there exists δ > 0 such that for each s with 0 < |s − s0 | < δ µ(f (sx) − f (s0 x), ε) ≥ α and ν(f (sx) − f (s0 x), ε) ≤ 1 − α . Theorem 3.2. Let a be an integer with a 6= 0, ±1 ,and let X be a normed space and (Z, µ0 , ν 0 ) be an IFNS. Let (Y, µ, ν) be an intuitionistic fuzzy Banach space and f : X → Y be a (p, q)-approximately mapping with f (0) = 0 in the sense that for some p, q and some z0 ∈ Z   (3.1) µ Da f (x, y), t ≥ µ0 ((||x||p + ||y||q )z0 , t) and (3.2)

  ν Da f (x, y), t ≤ ν 0 ((||x||p + ||y||q )z0 , t)

for all t > 0 and all x, y ∈ X . If p, q < 4 , then there exists a unique Euler-Lagrange type quartic mapping Q : X → Y such that 1 (3.3) µ(C(x) − f (x), t) ≥ µ0 (||x||p z0 , (a4 − |a|p )t) , 2 and m2 4 (3.4) ν(C(x) − f (x), t) ≤ ν 0 (||x||p z0 , (a − |a|p )t) , 2 for all x ∈ X and all t > 0 . Furthermore, if for some x ∈ X and all n ∈ N , the mapping g : R → Y defined by g(s) = f (an sx) is intuitionistic fuzzy continuous, then the mappings s 7→ Q(sx) from R to Y is intuitionistic fuzzy continuous. Proof. For x, y ∈ X and for some z0 ∈ Z , we define the function φ : X × X → Z by φ(x, y) = (||x||p + ||y||q )z0 in Theorem 2.1. Since p < 4 , we have α = |a|p < a4 . Hence Theorem 2.1 implies the existence and uniqueness of the Euler-Lagrange type quartic mapping Q : X → Y satisfying inequalities (3.3) and (3.4). Now, we will show the intuitionistic fuzzy continuity. For each x ∈ X , t ∈ R and n ∈ N , we have f (an x) Q(an x) f (an x) µ(Q(x) − , t) = µ( − , t) = µ(Q(an x) − f (an x), a4n t) 4n a a4n a4n a4n (a4 − |a|p ) a4n 4 (a − |a|p )t) = µ0 (||x||p z0 , t) , ≥ µ0 (|a|np ||x||p z0 , 2 2 · |a|np and ν(Q(x) −

f (an x) a4n (a4 − |a|p ) 0 p , t) ≤ ν (||x|| z , t) . 0 a4n 2 · |a|np

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EULER-LAGRANGE TYPE QUARTIC MAPPINGS

9

Let x ∈ X and s0 ∈ R be fixed and ε > 0 and 0 < β < 1 be given. For all s ∈ R with |s − s0 | < 1 , by replacing x with sx in the previous inequalities, µ(Q(sx) −

f (an sx) a4n (a4 − |a|p ) 0 p , t) ≥ µ (||sx|| z , t) 0 a4n 2 · |a|np a4n (a4 − |a|p ) ≥ µ0 (||x||p z0 , t) , 2 · |a|np (1 + |s0 |)p

and ν(Q(sx) −

f (an sx) a4n (a4 − |a|p ) 0 p , t) ≤ ν (||x|| z , t) . 0 a4n 2 · |a|np (1 + |s0 |)p

Since ap < a4 , we have a4n (a4 − |a|p ) = ∞. n→∞ 2 · |a|np (1 + |s0 |)p lim

Hence there exists n0 ∈ N such that   f (an0 sx) ε  f (an0 sx) ε  , ≥ β and ν Q(sx) − , ≤1−β, µ Q(sx) − a4n0 3 a4n0 3 for all |s − s0 | < 1 and s ∈ R . The intuitionistic fuzzy continuity of the mapping t 7→ f (an0 tx) implies that there exists δ < 1 such that for each s with 0 < |s − s0 | < δ , we get µ(

f (an0 sx) f (an0 s0 x) ε f (an0 sx) f (an0 s0 x) ε − , ) ≥ β and ν( − , )≤1−β. a4n0 a4n0 3 a4n0 a4n0 3

Thus f (an0 sx) ε , )∗ a4n0 3 f (an0 sx) f (an0 s0 x) ε f (an0 s0 x) ε µ( − , ) ∗ µ(Q(s0 x) − , )≥β a4n0 a4n0 3 a4n0 3 µ(Q(sx) − Q(s0 x), ε) ≥ µ(Q(sx) −

and ν(Q(sx) − Q(s0 x), ε) ≤ 1 − β , for all s ∈ R with 0 < |s − s0 | < δ , that is, the mapping s 7→ Q(sx) is intuitionistic fuzzy continuous.  Theorem 3.3. Let a be an integer with a 6= 0, ±1 , and let X be a normed space and (Z, µ0 , ν 0 ) be an IFNS. Let (Y, µ, ν) be an intuitionistic fuzzy Banach space and f : X → Y be a (p, q)-approximately mapping with f (0) = 0 satisfying (3.1) and (3.2) for some p, q and some z0 ∈ Z . If p, q > 4 , then there exists a unique Euler-Lagrange type quartic mapping Q : X → Y such that 1 (3.5) µ(Q(x) − f (x), t) ≥ µ0 (||x||p z0 , (|a|p − a4 )t) , 2 and 1 (3.6) ν(Q(x) − f (x), t) ≤ ν 0 (||x||p z0 , (|a|p − a4 )t) , 2 for all x ∈ X and all t > 0 . Furthermore, if for some x ∈ X and all n ∈ N , the mapping g : R → Y defined by g(s) = f (an sx) is intuitionistic fuzzy continuous, then the mappings s 7→ Q(sx) from R to Y is intuitionistic fuzzy continuous.

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Proof. Similar to the proof of Theorem 3.2, we may define the function φ : X ×X → Z by φ(x, y) = (||x||p + ||y||q )z0 . Then we have x x µ0 (φ( , 0), t) = µ0 (||x||p z0 , |a|p t) and ν 0 (φ( , 0), t) = ν 0 (||x||p z0 , |a|p t) , 2 2 for all x ∈ X and all t > 0 . Since p > 4 , we have α = |a|p > a4 . Hence Theorem 2.2 implies the existence and uniqueness of the Euler-Lagrange type quartic mapping Q : X → Y satisfying inequalities (3.5) and (3.6). The remains follow from the proof of Theorem 3.2.  Acknowledgement This work was supported by the University of Incheon Research Grant in 2011. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950) 64–66. [2] J.-H. Bae and W.-G. Park, On the generalized Hyers-Ulam-Rassias stability in Banach modules over a C ∗ −algebra, J. Math. Anal. Appl. 294(2004), 196–205. [3] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [4] Z. Gajda, On the stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431–434. [5] D. H. Hyers, On the stability of the linear equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [6] D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Mathematicae, 44 (1992),125–153. [7] K.-W. Jun and H.-M. Kim, On the stability of Euler-Lagrange type cubic functional equations in quasi-Banach spaces, J. Math. Anal. Appl. 332 (2007), 1335–1350. [8] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12 (1984), 143–154. [9] A.K. Mirmostafaee, M. Mirzavaziri, M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst, 159 (2008), 730–738. [10] A.K. Mirmostafaee, M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst, 159 (2008), 720–729. [11] A.K. Mirmostafaee, M.S. Moslehian, Fuzzy approximately cubic mappings, Inf Sci, 178 (2008), 3791-3798. [12] A.K. Mirmostafaee, M.S. Moslehian, Fuzzy almost quadratic functions, Results Math. doi:10.1007/s00025-007-0278-9. [13] A.K. Mirmostafaee, M.S. Moslehian, Nonlinear operators between intuitionistic fuzzy normed spaces and Frechet derivative , Chaos, Solitons and Fractals, 42 (2009), 10101015. [14] M. Mursaleen, S.A. Mohiuddine, On Stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals, 42 (2009), 2997–3005. [15] J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math., 20 (1992) 185–190. [16] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [17] R. Saadati, J.H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons and Fractals 27 (2006), 331–344. [18] S. M. Ulam, Problems in Morden Mathematics, Wiley, New York (1960). [19] L.A. Zadeh, Fuzzy sets, Inform Control, 8 (1965), 338-353. 1 Department of Mathematical Education, Dankook University, 126, Jukjeon, Suji, Yongin, Gyeonggi, South Korea 448-701 E-mail address: [email protected] (H. Koh) E-mail address: [email protected] (D. Kang)

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2 Graduate School of Education, University of Incheon, 12-1, Songdo, Yeonsu, Incheon, South Korea 406-772 E-mail address: [email protected] (I. G. Cho)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1096-1103, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

STABILITY FOR AN n-DIMENSIONAL FUNCTIONAL EQUATION OF QUADRATIC-ADDITIVE TYPE WITH THE FIXED POINT APPROACH

ICK-SOON CHANG AND YANG-HI LEE

Abstract. In this paper, we investigate the stability of a functional equation 𝑛 ∑ ∑ [f (𝑥𝑖 + 𝑥j ) + f (𝑥𝑖 − 𝑥j )] − (𝑛 − 1) f (2𝑥j ) = 0 j=1

1≤𝑖,j≤𝑛,𝑖̸=j

by using the fixed point methd in the sense of C˘ adariu and Radu.

1. Introduction and peliminaries It is of interest to consider the concept of stability for a functional equation arising when we replace the functional equation by an inequality which acts as a perturbation of the equation. The study of stability problems had been formulated by Ulam [17] during a talk : under what condition does there exists a homomorphism near an approximate homomorphism ? In the following year, Hyers [6] was answered affirmatively the question of Ulam for Banach spaces, which states that if ε ≥ 0 and f : X → Y is a mapping with X a normed space, Y a Banach space such that ||f (x + y) − f (x) − f (y)|| ≤ ε

(1.1)

for all x, y ∈ X , then there exists a unique additive mapping T : X → Y such that ||f (x) − T (x)|| ≤ ε for all x ∈ X . A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki [1] and for the theorem of Hyers for approximately linear mappings it was presented by Rassias [15] by considering the case when the inequality (1.1) is unbounded. Since then, more generalizations and applications of the stability to a number of functional equations and mappings have been investigated (for example, [5], [7], [8]-[14]). In this very active area, almost all subsequent proofs have used the method of Hyers [6]. On the other hand, C˘ adariu and Radu [2] observed that the existence of the solution for a functional equation and the estimation of the difference with the given mapping can be obtained from the fixed point alternative. This method is called a fixed point method. In particular, they [3, 4] applied this method to prove the stability theorems of the additive functional equation f (x + y) − f (x) − f (y) = 0.

(1.2)

f (x + y) + f (x − y) − 2f (x) − 2f (y) = 0.

(1.3)

and the quadratic functional equation

Note that the additive mapping f1 (x) = ax and quadratic mapping f2 (x) = ax2 are solution of the functional equations (1.2) and (1.3). We now take account of the functional equation : ∑

[f (xi + xj ) + f (xi − xj )] − (n − 1)

n ∑

f (2xj ) = 0.

(1.4)

j=1

1≤i,j≤n,i̸=j

2000 Mathematics Subject Classification : 39B52. Keywords and phrases : stability : fixed point method : 𝑛-dimensional quadratic-additive type functional equation. The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002410). 1

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Hence, throughout this paper, we promise that the equation (1.4) is said to be an quadratic-additive type functional equation and every solution of the equation (1.4) is called a quadratic-additive mapping. In this paper, we will deal with the stability of the functional equation (1.4) by using the fixed point method : The stability of (1.4) can be obtained by handling the odd part and the even part of the given mapping. But, in violation of this processing, we can take the desired solution at once instead of splitting the given mapping into two parts. Here and now, we recall the following result of the fixed point theory by Margolis and Diaz : Theorem 1.1. (The alternative of fixed point) ([14] or [16]) Suppose that a complete generalized metric space (X, d), which means that the metric d may assume infinite values, and a strictly contractive mapping J : X → X with the Lipschitz constant 0 < L < 1 are given. Then, for each given element x ∈ X, either d(J n x, J n+1 x) = +∞, ∀n ∈ N ∪ {0}, or there exists a nonnegative integer k such that : (1) d(J n x, J n+1 x) < +∞ for all n ≥ k ; (2) the sequence {J n x} is convergent to a fixed point y ∗ of J ; (3) y ∗ is the unique fixed point of J in Y := {y ∈ X, d(J k x, y) < +∞} ; (4) d(y, y ∗ ) ≤ (1/(1 − L))d(y, Jy) for all y ∈ Y. 2. A general fixed point method Throughout this paper, let V be a real or complex linear space and Y a Banach space. For a given mapping f : V → Y, we use the following abbreviation n ∑ ∑ f (x1 , x2 , · · · , xn ) := [f (xi + xj ) + f (xi − xj )] − (n − 1) f (2xj ) j=1

1≤i,j≤n,i̸=j

for all x1 , x2 , · · · , xn ∈ V. Now we can prove some stability results of the functional equation (1.4). Theorem 2.1. Let φ : V n → [0, ∞) be a given function with φ(x, 0, · · · , 0) = φ(−x, 0, · · · , 0) for all x ∈ V. Suppose that the mapping f : V → Y satisfies ∥Df (x1 , x2 , · · · , xn )∥ ≤ φ(x1 , x2 , · · · , xn )

(2.1)

for all x1 , x2 , · · · , xn ∈ V with f (0) = 0. If there exists a constant 0 < L < 1 such that φ has the property φ(2x1 , 2x2 , · · · , 2xn ) ≤ 2Lφ(x1 , x2 , · · · , xn )

(2.2)

for all x1 , x2 , · · · , xn ∈ V, then there exists a unique quadratic-additive mapping F : V → Y such that φ(x, 0, · · · , 0) (2.3) ∥f (x) − F (x)∥ ≤ 2(n − 1)(1 − L) for all x ∈ V. In particular, F is given by ( ) f (2m x) + f (−2m x) f (2m x) − f (−2m x) F (x) = lim + (2.4) m→∞ 2 · 22m 2 · 2m for all x ∈ V. Proof. Consider the set S := {g : g : V → Y, g(0) = 0} and introduce a generalized metric on S by { } d(g, h) = inf K ∈ ℝ ∥g(x) − h(x)∥ ≤ Kφ(x, 0, · · · , 0) for all x ∈ V . It is easy to see that (S, d) is a generalized complete metric space. Now we define a mapping J : S → S by g(2x) − g(−2x) g(2x) + g(−2x) Jg(x) := + 4 8 for all x ∈ V. Note that g(2m x) − g(−2n x) g(2m x) + g(−2m x) J m g(x) = + m+1 2 2 · 4m

1097

STABILITY FOR AN n-DIMENSIONAL FUNCTIONAL EQUATION OF QUADRATIC-ADDITIVE TYPE

3

for all m ∈ N and x ∈ V. Let g, h ∈ S and let K ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ K. From the definition of d, we have



3(g(2x) − h(2x)) g(−2x) − h(−2x)



∥Jg(x) − Jh(x)∥ ≤

+

8 8 Kφ(2x, 0, · · · , 0) 2 ≤KLφ(x, 0, · · · , 0) ≤

for all x ∈ V, which implies that d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S. That is, J is a strictly contractive self-mapping of S with the Lipschitz constant L. Moreover, by (2.1), we see that

φ(x, 0, · · · , 0) 1 1 3

∥f (x) − Jf (x)∥ =

Df (x, 0, · · · , 0) − Df (−x, 0, · · · , 0) ≤ n−1 8 8 2(n − 1) 1 for all x ∈ V. It means that d(f, Jf ) ≤ 2(n−1) < ∞ by the definition of d. Therefore, according to m Theorem 1.1, the sequence {J f } converges to the unique fixed point F : V → Y of J in the set T = {g ∈ S : d(f, g) < ∞}, which is given by (2.4) for all x ∈ V. Observe that 1 1 d(f, F ) ≤ d(f, Jf ) ≤ , 1−L 2(n − 1)(1 − L) which implies (2.3). By the definition of F, together with (2.1) and (2.4) that

∥DF (x1 , x2 , · · · , xn )∥

m m m m m m

Df (2 x1 , 2 x2 , · · · , 2 xn ) − Df (−2 x1 , −2 x2 , · · · , −2 xn ) = lim m+1 m→∞ 2

Df (2m x1 , 2m x2 , · · · , 2m xn ) + Df (−2m x1 , −2m x2 , · · · , −2m xn ) +

2 · 4m m 2 +1 ≤ lim (φ(2m x1 , · · · , 2m xn ) + φ(−2m x1 , · · · , −2m xn )) m→∞ 2 · 4m =0 for all x1 , x2 , · · · , xn ∈ V, which completes the proof.



We continue our investigation with the following result. Theorem 2.2. Let φ : V n → [0, ∞) with φ(x, 0, · · · , 0) = φ(−x, 0, · · · , 0) for all x, y ∈ V. Suppose that f : V → Y satisfies the inequality (2.1) for all x1 , x2 , · · · , xn ∈ V with f (0) = 0. If there exists 0 < L < 1 such that the mapping φ has the property φ(2x1 , 2x2 , · · · , 2xn ) ≥ 4φ(x1 , x2 , · · · , xn )

(2.5)

for all x1 , x2 , · · · , xn ∈ V, then there exists a unique quadratic-additive mapping F : V → Y such that ∥f (x) − F (x)∥ ≤

Lφ (x, 0, · · · , 0) 4(n − 1)(1 − L)

for all x ∈ V. In particular, F is represented by ( ( ( x ) ( x )) 4m ( ( x ) ( x ))) F (x) = lim 2m−1 f − f − + f + f − m m→∞ 2m 2m 2 2m 2

(2.6)

(2.7)

for all x ∈ V. Proof. Let the set (S, d) be as in the proof of Theorem 2.1. Now we consider the mapping J : S → S defined by ( x) ( (x) ( x )) (x) −g − +2 g +g − Jg(x) := g 2 2 2 2 for all g ∈ S and x ∈ V. We remark that ( x )) 4m ( ( x ) ( x )) ( ( x ) + g m +g − m J m g(x) = 2m−1 g m − g − m 2 2 2 2 2

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I.-S. CHANG AND Y.-H. LEE

and J 0 g(x) = g(x) for all x ∈ V. Let g, h ∈ S and let K ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ K. From the definition of d, we have

( ) (x) ( x) ( x)

x



∥Jg(x) − Jh(x)∥ ≤ 3 g −h −h −

+ g −

2 2 2 2 (x ) ≤ 4Kφ , 0, · · · , 0 ≤ LKφ (x, 0, · · · , 0) 2 for all x ∈ V. So we find that J is a strictly contractive self-mapping of S with the Lipschitz constant L. Also, we see that

(x ) 1

∥f (x) − Jf (x)∥ = , 0, · · · , 0

−Df n−1 2 (x ) 1 L ≤ φ , 0, · · · , 0 ≤ φ (x, 0, · · · , 0) n−1 2 4(n − 1) L for all x ∈ V, which implies that d(f, Jf ) ≤ 4(n−1) < ∞. Therefore, according to Theorem 1.1, the sequence {J m f } converges to the unique fixed point F of J in the set T := {g ∈ S : d(f, g) < ∞}, which is represented by (2.7). Since 1 L d(f, F ) ≤ d(f, Jf ) ≤ 1−L 4(n − 1)(1 − L) the inequality (2.6) holds. From the definition of F, (2.1), and (2.5), we have

∥DF (x1 , x2 , · · · , xn )∥

( (x x ( x xn ) x2 xn ))

1 2 1 = lim 2m−1 Df , m , · · · , m − Df − m , − m , · · · , − m m m→∞ 2 2 2 2 2 2

(x x ( x 4m ( xn ) x2 xn )) 1 2 1 + Df , , · · · , + Df − , − , · · · , −

m 2 ( 2m 2m 2m 2m 2m )( ( 2 ) ( m x1 x2 xn x1 x2 xn )) 4 φ m, m,··· , m + φ − m,− m,··· ,− m ≤ lim 2m−1 + m→∞ 2 2 2 2 2 2 2 =0 for all x1 , x2 , · · · , xn ∈ V. This completes the proof.



3. Applications For the sake of convenience, given a mapping f : V → Y, we set Af (x, y) := f (x + y) − f (x) − f (y) for all x, y ∈ V. Corollary 3.1. Let fk : V → Y, k = 1, 2, be mappings for which there exist functions ϕk : V 2 → [0, ∞), k = 1, 2, such that ∥Afk (x, y)∥ ≤ ϕk (x, y)

(3.1)

for all x, y ∈ V. If fk (0) = 0, ϕk (0) = 0, ϕk (x, y) = ϕk (−x, −y), k = 1, 2, for all x, y ∈ V and there exists 0 < L < 1 such that ϕ1 (2x, 2y) ≤ 2Lϕ1 (x, y),

(3.2)

4ϕ2 (x, y) ≤ Lϕ2 (2x, 2y)

(3.3)

for all x, y ∈ V, then there exist unique additive mappings Fk : V → Y, k = 1, 2, such that ϕ1 (x, x) + ϕ1 (x, −x) , 2(1 − L)

(3.4)

L(ϕk (x, x) + ϕk (x, −x)) 4(1 − L)

(3.5)

∥f1 (x) − F1 (x)∥ ≤ ∥f2 (x) − F2 (x)∥ ≤

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STABILITY FOR AN n-DIMENSIONAL FUNCTIONAL EQUATION OF QUADRATIC-ADDITIVE TYPE

for all x ∈ V. In particular, the mappings F1 , F2 are represented by f1 (2m x) F1 (x) = lim , m→∞ 2m( ) x F2 (x) = lim 2m f2 m m→∞ 2 for all x ∈ V. Proof. Now we note that Dfk (x1 , x2 , · · · , xn ) =

∑

5

(3.6) (3.7)

Ak (xi + xj , xi − xj )

1≤i,j≤n,i̸=j

for all x1 , x2 , · · · , xn ∈ V and k = 1, 2. Put φk (x1 , x2 , · · · , xn ) :=

∑

ϕk (xi + xj , xi − xj )

1≤i,j≤n,i̸=j

for all x1 , x2 , · · · , xn ∈ V and k = 1, 2, then ∥Dfk (x1 , x2 , · · · , xn )∥ ≤ φk (x1 , x2 , · · · , xn ) and φ1 and φ2 satisfies (2.2) and (2.5), respectively. According to Theorem 2.1, there exists a unique mapping F1 : V → Y satisfying (3.4), which is represented by (2.4). Observe that, by (3.1) and (3.2),

f1 (2m x) + f1 (−2m x) 1 m m

= lim lim

m→∞ 2m+1 ∥Af1 (2 x, −2 x)∥ m→∞ 2m+1 1 ϕ1 (2m x, −2m x) 2m+1 Lm ϕ1 (x, −x) = 0 ≤ lim m→∞ 2 ≤ lim

m→∞

as well as

m m

f1 (2m x) + f1 (−2m x)

≤ lim 2 L ϕ1 (x, −x) = 0 lim

m m→∞ m→∞ 2 · 4m 2·4

for all x ∈ V. From these and (2.4), we get (3.6). Moreover, we have

m m

Af1 (2m x, 2m y)

≤ ϕ1 (2 x, 2 y) ≤ Lm ϕ1 (x, y)

m m 2 2 for all x, y ∈ V. Taking the limit as m → ∞ in the above inequality, we get AF1 (x, y) = 0 for all x, y ∈ V. On the other hand, according to Theorem 2.4, there exists a unique mapping F2 : V → Y satisfying (3.5), which is represented by (2.7). Observe that, by (3.1) and (3.3),

(

( −x ) ( x x ) x )

2m−1 lim 22m−1 f2 m + f2 = lim 2 , −

Af

2 m m m→∞ m→∞ 2 2m ( x 2 x 2) ≤ lim 22m−1 ϕ2 m , − m m→∞ 2 2 Lm ≤ lim ϕ2 (x, −x) = 0 m→∞ 2 as well as

( ( −x ) Lm x )

lim 2m−1 f2 m + f2

≤ lim m+1 ϕ2 (x, −x) = 0 m m→∞ m→∞ 2 2 2 for all x ∈ V. From these and (2.5), we get (3.10). Moreover, we have

( x y ) ( x y ) Lm

m

m

2 Af2 m , m ≤ 2 ϕ2 m , m ≤ m ϕ2 (x, y) 2 2 2 2 2 for all x, y ∈ V. Taking the limit as m → ∞ in the above inequality, we get AF2 (x, y) = 0 for all x, y ∈ V. This completes the proof.



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6

I.-S. CHANG AND Y.-H. LEE

Corollary 3.2. Let fk : V → Y, k = 1, 2, be mappings for which there exist functions ϕk : V 2 → [0, ∞), k = 1, 2, such that ∥Qfk (x, y)∥ ≤ ϕk (x, y) for all x, y ∈ V. If fk (0) = 0, ϕk (0) = 0, ϕk (x, y) = ϕi (−x, −y), k = 1, 2, for all x, y ∈ V, and there exists 0 < L < 1 such that the mapping ϕ1 satisfies (3.2) and ϕ2 satisfies (3.3) for all x, y ∈ V, then there exist unique quadratic mappings Fk : V → Y, k = 1, 2, such that ϕ1 (x, x) + ϕ1 (x, −x) + 3ϕ1 (x, 0) + ϕ1 (0, −x) , 4(1 − L)

(3.8)

L(ϕ2 (x, x) + ϕ2 (x, −x) + 3ϕ2 (x, 0) + ϕ2 (0, −x)) 8(1 − L)

(3.9)

∥f1 (x) − F1 (x)∥ ≤ ∥f2 (x) − F2 (x)∥ ≤

for all x ∈ V. In particular, the mappings Fk , k = 1, 2, are represented by f1 (2m x) , 4m( x ) F2 (x) = lim 4m f2 m m→∞ 2

F1 (x) = lim

m→∞

(3.10) (3.11)

for all x ∈ V. Proof. Notice that Dfk (x1 , · · · , xn ) =

∑

1 2

− for all x1 , x2 , · · · , xn ∈ V and k = 1, 2. Put 1 φk (x1 , · · · , xn ) = 2

(Qk (xi , xj ) + Qk (xi , −xj ))

1≤i,j≤n,i̸=j n n−1 ∑ (Qk (xi , xi ) + Qk (xi , −xi )) 2 i=1

∑

(ϕk (xi , xj ) + ϕk (xi , −xj ))

1≤i,j≤n,i̸=j

+

n n−1 ∑ (ϕk (xi , xi ) + ϕk (xi , −xi )) 2 i=1

for all x1 , x2 , · · · , xn ∈ V and k = 1, 2, then φ1 satisfies (2.2) and φ2 satisfies (2.5). Moreover, ∥Dfk (x1 , x2 , · · · , xn )∥ ≤ φk (x1 , x2 , · · · , xn ) for all x1 , x2 , · · · , xn ∈ V and k = 1, 2. According to Theorem 2.1, there exists a unique mapping F1 : V → Y satisfying (3.8) which is represented by (2.4). Observe that

m m

1

f1 (2 x) − f1 (−2 x) m−1 lim x)

= lim m+1 Qf1 (0, −2 m→∞ m→∞ 2 2m+1 1 ≤ lim m+1 ϕ1 (0, −2m−1 x) m→∞ 2 Lm ( x) ≤ lim ϕ1 0, − m→∞ 2 2 =0 as well as

( ) m

f1 (2m x) − f1 (−2m x)

≤ lim L ϕ1 0, − x = 0 lim

m m+1 m→∞ 2 m→∞ 2·4 2

for all x ∈ V. From these and (2.4), we get (3.10) for all x ∈ V. Moreover, we have

m m m

Qf1 (2m x, 2m y)

≤ ϕ1 (2 x, 2 y) ≤ L ϕ1 (x, y)

m m m 4 4 2 for all x, y ∈ V. Taking the limit as m → ∞ in the above inequality, we get QF1 (x, y) = 0 for all x, y ∈ V.

1101

STABILITY FOR AN n-DIMENSIONAL FUNCTIONAL EQUATION OF QUADRATIC-ADDITIVE TYPE

7

On the other hand, according to Theorem 2.2, there exists a unique mapping F2 : V → Y satisfying (3.9) which is represented by (2.7). Observe that

(

( x ) ( x ) x )

4m f2 m − f2 − m =4m Qf2 0, − m+1 2 2 2 ( x ) ≤4m ϕ2 0, − m+1 2) ( x ≤Lm ϕ2 0, − 2 for all x ∈ V. It leads us to get ( ( x ) ( x )) ( ( x ) ( x )) lim 4m f2 m − f2 − m = 0, lim 2m f2 m − f2 − m =0 m→∞ m→∞ 2 2 2 2 for all x ∈ V. From these and (2.7), we obtain (3.11). Moreover, we have

( x y ) ( x y )

m

m m

4 Qf2 m , m ≤ 4 ϕ2 m , m ≤ L ϕ2 (x, y) 2 2 2 2 for all x, y ∈ V. Taking the limit as m → ∞ in the above inequality, we get QF2 (x, y) = 0 for all x, y ∈ V, which completes the proof.  Corollary 3.3. Let X be a normed space and Y a Banach space. Suppose that the mapping f : X → Y satisfies the inequality ∥Df (x1 , x2 , · · · , xn )∥ ≤ ∥x1 ∥𝑝 + ∥x2 ∥𝑝 + · · · + ∥xn ∥𝑝 for all x1 , x2 , · · · , xn ∈ X, where p ∈ (0, 1) ∪ (2, ∞). Then there exists a unique quadratic-additive mapping F : X → Y such that { ∥x∥𝑝 if p > 2, (n−1)(2𝑝 −4) ∥f (x) − F (x)∥ ≤ ∥x∥𝑝 if p < 1 (n−1)(2−2𝑝 ) for all x ∈ X. Proof. This follows from Theorem 2.1 and Theorem 2.2, by putting φ(x1 , x2 , · · · , xn ) := ∥x1 ∥𝑝 + ∥x2 ∥𝑝 + · · · + ∥xn ∥𝑝 for all x1 , x2 , · · · , xn ∈ X with L = 2𝑝−1 < 1 if 0 < p < 1 and L = 22−𝑝 < 1 if p > 2.



Corollary 3.4. Let X be a normd space and Y a Banach space. Suppose that the mapping f : X → Y satisfies the inequality ∥Df (x1 , x2 , · · · , xn )∥ ≤ θ∥x1 ∥𝑝1 ∥x2 ∥𝑝2 · · · ∥xn ∥𝑝𝑛 for all x1 , x2 , · · · , xn ∈ X, where θ ≥ 0 and p1 , p2 , · · · , pn , p1 + p2 + · · · + pn ∈ (0, 1) ∪ (2, ∞). Then f is itself a quadratic additive mapping. Proof. This follows from Theorem 2.1 and Theorem 2.2, by letting φ(x1 , x2 , · · · , xn ) := ∥x1 ∥𝑝1 ∥x2 ∥𝑝2 · · · ∥xn ∥𝑝𝑛 for all x1 , x2 , · · · , xn ∈ X with L = 2𝑝−1 < 1 if 0 < p < 1 and L = 22−𝑝 < 1 if p > 2.



References [1] T. Aoki, On the stability of the linear mapping in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), Art. 4. [3] L. C˘ adariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat.-Inform. 41 (2003), 25–48. [4] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation : a fixed point approach in iteration theory, Grazer Mathematische Berichte, Karl-Franzens-Universit¨ aet, Graz, Graz, Austria 346 (2004), 43–52. [5] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [6] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [7] G.-H. Kim, On the stability of functional equations with square-symmetric operation, Math. Inequal. Appl. 4 (2001), 257–266. [8] H.-M. Kim, On the stability problem for a mixed type of quartic and quadratic functional equation, J. Math. Anal. Appl. 324 (2006), 358–372.

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[9] Y.-H. Lee, On the stability of the monomial functional equation, Bull. Korean Math. Soc. 45 (2008), 397–403. [10] Y.H. Lee and K.W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensen’s equation, J. Math. Anal. Appl. 238 (1999), 305–315. [11] Y.H. Lee and K.W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Pexider equation, J. Math. Anal. Appl. 246 (2000), 627–638. [12] Y.-H. Lee and K.W. Jun, A note on the Hyers-Ulam-Rassias stability of Pexider equation, Korean Math. Soc. 37 (2000), 111–124 [13] Y.-H. Lee and K.W. Jun, On the stability of approximately additive mappings, Proc. Amer. Math. Soc. 128 (2000), 1361–1369. [14] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [15] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [16] I.A. Rus, Principles and applications of fixed point theory, Ed. Dacia, Cluj-Napoca, (1979) (in Romanian). [17] S.M. Ulam, A collection of mathematical problems, Interscience, New York, (1968). Ick-Soon Chang, Department of Mathematics, Mokwon University, Daejeon 302-729, Republic of Korea. E-mail address: [email protected] Yang-Hi Lee, Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea. E-mail address: [email protected]

1103

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1104-1109, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

An identity of the q-Euler polynomials associated with the p-adic q-integrals on Zp C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : We introduce the q-Euler numbers and polynomials. By using these numbers and polynomials, we investigate the alternating sums of powers of consecutive integers. By applying the symmetry of the fermionic p-adic q-integral on Zp , we give recurrence identities the q-Euler polynomials and q-analogue of alternating sums of powers of consecutive integers. 2000 Mathematics Subject Classification - 11B68, 11S40, 11S80. Key words : Euler numbers and polynomials, q-Euler numbers and polynomials, alternating sums. 1. Introduction Throughout this paper, we always make use of the following notations: C denotes the set of complex numbers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assume that |q| < 1. If q ∈ Cp , we normally 1 assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. For g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, the fermionic p-adic q-integral on Zp is defined by Kim as follows: N

p −1 [2]q g(x)dμ−q (x) = lim g(x)(−q)x , see [1-10] . I−q (g) = N N →∞ 1 + q p Zp x=0



(1.1)

If we take g1 (x) = g(x + 1) in (1.1), then we easily see that qI−q (g1 ) + I−q (g) = [2]q g(0).

(1.2)

n,q (x) are defined by For q ∈ Cp with |1 − q|p ≤ 1, the q-Euler polynomials E Fq (x, t) =

∞ 

n n,q (x) t = [2]q ext . E n! qet + 1 n=0

(1.3)

n,q are defined by the generating function: The q-Euler numbers E Fq (t) =

∞ 

n n,q t = [2]q . E n! qet + 1 n=0

(1.4)

n,q and polynomials E n,q (x) are readily The following elementary properties of the q-Euler numbers E derived form (1.1), (1.2), (1.3) and (1.4). We, therefore, choose to omit details involved.

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RYOO: q-EULER POLYNOMIALS

Theorem 1(Witt formula). For q ∈ Cp with |1 − q|p < 1, we have  n,q = E xn dμ−q (x), Zp



n,q (x) = E

n

(x + y) dμ−q (y).

Zp

Theorem 2. For any positive integer n, we have n    n   Ek,q xn−k . En,q (x) = k k=0

2. The alternating sums of powers of consecutive q-integers Let q be a complex number with |q| < 1. By using (1.3), we give the alternating sums of powers of consecutive q-integers as follows: ∞ 

∞ n  n,q t = [2]q = [2]q (−1)n q n ent . E t+1 n! qe n=0 n=0

From the above, we obtain −

∞ 

n n (n+k)t

(−1) q e

+

n=0

∞ 

n−k n−k nt

(−1)

q

n=0

Thus, we have − [2]q

∞ 

e

=

k−1 

(−1)n−k q n−k ent .

n=0

(−1)n q n e(n+k)t + [2]q (−1)−k q −k

n=0

∞ 

(−1)n q n ent

n=0 k−1 

= [2]q (−1)−k q −k

(−1)n q n ent .

n=0

By using (1.3)and (1.4), and (2.1), we obtain ∞ 

  ∞ k−1 ∞    tj tj tj −k −k −k −k n n j   − . (−1) q (−1) q n Ej,q (k) + (−1) q Ej,q = [2]q j! j! j! n=0 j=0 j=0 j=0 By comparing coefficients of

tj in the above equation, we obtain j! k−1 

(−1)n q n nj =

n=0

j,q (k) + E j,q (−1)k+1 q k E . [2]q

By using the above equation we arrive at the following theorem: Theorem 3. Let k be a positive integer and q ∈ C with |q| < 1. Then we obtain Tj,q (k − 1) =

k−1 

(−1)n q n nj =

n=0

j,q (k) + E j,q (−1)k+1 q k E . [2]q

Remark 4. Let k be a positive integer and q ∈ C with |q| < 1. Then we have lim Tj,q (k − 1) =

q→1

k−1 

(−1)n nj =

n=0

1105

(−1)k+1 Ej (k) + Ej , 2

(2.1)

RYOO: q-EULER POLYNOMIALS

where Ej (x) and Ej denote the Euler polynomials and Euler numbers, respectively. Next, we assume that q ∈ Cp . We obtain recurrence identities the q-Euler polynomials and the q-analogue of alternating sums of powers of consecutive integers. By using (1.1), we have n−1 

q n I−q (gn ) + (−1)n−1 I−q (g) = [2]q

(−1)n−1−l q l g(l),

l=0

where gn (x) = g(x + n). If n is odd from the above, we obtain q n I−q (gn ) + I−q (g) = [2]q

n−1 

(−1)n−1−l q l g(l) (cf. [1-5]).

(2.2)

l=0

It will be more convenient to write (2.2) as the equivalent integral form qn



g(x + n)dμ−q (x) +

Zp



g(x)dμ−q (x) = [2]q

Zp

n−1 

(−1)k q k g(k).

(2.3)

(−1)j q j ejt .

(2.4)

k=0

Substituting g(x) = ext into the above, we obtain qn



Zp

e(x+n)t dμ−q (x) +



Zp

ext dμ−q (x) = [2]q

j=0

After some elementary calculations, we have  ext dμ−q (x) = Zp



(x+n)t

Zp

e

(2.5)

[2]q . dμ−q (x) = e qet + 1

Zp

From the above, we get

[2]q , qet + 1 nt

By using (2.4) and (2.5), we have   qn e(x+n)t dμ−q (x) + Zp

n−1 

ext dμ−q (x) =

[2]q (1 + q n ent ) . qet + 1

[2]q Zp ext dμ−q (x) [2]q (1 + q n ent ) = . qet + 1 q (n−1)x entx dμ−q (x) Zp

By substituting Taylor series of ext into (2.4), we obtain    ∞  qn (x + n)m dμ−q (x) + Zp

m=0

=

∞ 

m=0

⎛

⎝[2]q

Zp

n−1  j=0

⎞

(−1)j q j j m ⎠

m

(2.6)



x dμ−q (x)

tm m!

tm . m!

tm in the above equation, we obtain m!   m   n−1   m m−k n k n q x dμ−q (x) + xm dμ−q (x) = [2]q (−1)j q j j m . k Zp Zp j=0

By comparing coefficients

k=0

By using Theorem 3, we have   m    m m−k n k n x dμ−q (x) + xm dμ−q (x) = [2]q Tm,q (n − 1). q k Zp Zp k=0

1106

(2.7)

RYOO: q-EULER POLYNOMIALS

By using (2.6) and (2.7), we arrive at the following theorem: Theorem 5. Let n be odd positive integer. Then we have ∞   m  ext dμ−q (x) Zp m,q (n − 1) t . = T m! q (n−1)x entx dμ−q (x) m=0 Zp Let w1 and w2 be odd positive integers. By (2.5), Theorem 5, and after some elementary calculations, we obtain the following theorem. Theorem 6. Let w1 and w2 be odd positive integers. Then we have ∞  m ew2 xt dμ−qw2 (x) [2]qw2    Zp m t w . = T (w − 1)w 2 m,q 2 [2]q m=0 m! q (w1 w2 −1)x ew1 w2 tx dμ−q (x) Zp

(2.8)

By (1.1), we obtain

e(w1 x1 +w2 x2 +w1 w2 x)t dμ−qw1 (x1 )dμ−qw2 (x2 ) q (w1 w2 −1)x ew1 w2 xt dμ−q (x) Zp ew1 w2 xt Zp ew1 x1 t dμ−qw1 (x1 ) Zp ew2 x2 t dμ−qw2 (x2 ) . = q (w1 w2 −1)x ew1 w2 xt dμ−q (x) Zp Zp

Zp

(2.9)

By using (2.8) and (2.9), after elementary calculations, we obtain    ex2 w2 t dμ−qw2 (x2 ) Zp (w1 x1 +w1 w2 x)t e dμ−qw1 (x1 ) a= q (w1 w2 −1)x ew1 w2 xt dμ−q (x) Zp Zp    ∞ ∞  tm tm [2]qw2   m m  . Em,qw1 (w2 x)w1 Tm,qw2 (w1 − 1)w2 = m! [2]q m=0 m! m=0 By using Cauchy product in the above, we have ⎛ ⎞ m   ∞ m   w m [2] 2 j,qw1 (w2 x)wj Tm−j,qw2 (w1 − 1)wm−j ⎠ t . ⎝ q a= E 1 2 [2]q j=0 j m! m=0 By using the symmetry in (2.10), we obtain  (w2 x2 +w1 w2 x)t

a=

=

Zp



e

∞ 



dμ−qw2 (x2 )

m m,qw1 (w1 x)w2m t E m! m=0



ex1 w1 t dμ−qw1 (x1 ) Zp q (w1 w2 −1)x ew1 w2 xt dμ−q (x) Zp

∞ tm [2]qw1   Tm,qw1 (w2 − 1)w1m [2]q m=0 m!



(2.10)

(2.11)



.

Thus we obtain ⎞ m   m  w m [2] j,qw2 , (w1 x)wj Tm−j,qw1 (w2 − 1)wm−j ⎠ t . ⎝ q 1 E a= 2 1 [2]q j=0 j m! m=0 ∞ 

⎛

By comparing coefficients theorem.

(2.12)

tm in the both sides of (2.11) and (2.12), we arrive at the following m!

1107

RYOO: q-EULER POLYNOMIALS

Theorem 7. Let w1 and w2 be odd positive integers. Then we obtain m    m

j,qw1 (w2 x)wj Tm−j,qw2 (w1 − 1)wm−j E 1 2 j j=0 m    m  Ej,qw2 (w1 x)w2j Tm−j,qw1 (w2 − 1)w1m−j , = [2]qw1 j j=0

[2]

q w2

k,q (x) and Tm,q (k) denote the q-Euler polynomials and the q-analogue of alternating sums where E of powers of consecutive integers, respectively. By using Theorem 2, we have the following corollary. Corollary 8. Let w1 and w2 be odd positive integers. Then we obtain [2]qw1

j    m   j m j=0 k=0

= [2]qw2

k

j

k,qw2 Tm−j,qw1 (w2 − 1) w1m−k w2j xj−k E

j    m   j m j=0 k=0

By using (2.9), we have   a = ew1 w2 xt

j

k,qw1 Tm−j,qw2 (w1 − 1). w1j w2m−k xj−k E

k

ex2 w2 t dμ−qw2 (x2 ) Zp e dμ−qw1 (x1 ) q (w1 w2 −1)x ew1 w2 xt dμ−q (x) Zp Zp   w2  w 1 −1 (w1 t) x1 +w2 x+j w1 (−1)j q w2 j e dμ−qw1 (x1 )



x1 w1 t



[2]qw2 [2]q j=0 Zp ⎛ ⎞   w1 −1 ∞  w2  [2] tn w q 2 n,qw1 w2 x + j ⎝ w1n ⎠ . (−1)j q w2 j E = [2]q j=0 w1 n! n=0

=

(2.13)

By using the symmetry property in (2.13), we also have     ex1 w1 t dμ−qw1 (x1 ) Zp w1 w2 xt x2 w2 t a= e e dμ−qw2 (x2 ) q (w1 w2 −1)x ew1 w2 xt dμ−q (x) Zp Zp w 2 −1



w1 w2



(w2 t) [2]qw1 (−1)j q w1 j e dμ−qw2 (x2 ) [2]q j=0 Zp ⎛ ⎞   w ∞ 2 −1 n  w w [2] n,qw2 w1 x + j 1 wn ⎠ t . ⎝ q 1 = (−1)j q w1 j E 2 [2]q j=0 w2 n! n=0

=



x2 +w1 x+j

(2.14)

tn in the both sides of (2.13) and (2.14), we have the following theorem. n! Theorem 9. Let w1 and w2 be odd positive integers. Then we have

By comparing coefficients

[2]qw2

w 1 −1

=[2]qw1

w 2 −1

j w2 j

n,qw1 E

j w1 j

n,qw2 E

(−1) q

j=0

j=0

(−1) q

1108

w2 w2 x + j w1



w1n

w1 w1 x + j w2



w2n .

 

(2.15)

RYOO: q-EULER POLYNOMIALS

Remark 10. Let w1 and w2 be odd positive integers. If q → 1, we have w 1 −1 j=0

    w 2 −1 w2 w1 n j w1 = w2n . (−1) En w2 x + j (−1) En w1 x + j w1 w 2 j=0 j

Substituting w1 = 1 into (2.15), we arrive at the following corollary. Corollary 11. Let w2 be odd positive integer. Then we obtain   w2 −1 x+j [2]q  j j   w w2n . (−1) q En,q 2 En,q (x) = [2]qw2 j=0 w2

ACKNOWLEDGEMENT This paper has been supported by the 2013 Hannam University Research Fund.

REFERENCES 1. T. Kim, q-Volkenborn integration, Russ. J. Math. phys., 9(2002), 288-299. 2. T. Kim, Note on the Euler numbers and polynomials, Adv. Stud. Contemp. Math., 17(2008), 131-136. 3. T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math., 20(2010), 23-28. 4. T. Kim, Euler numbers and polynomials associated with zeta function, Abstr. Appl. Anal., Art. ID 581582, (2008), pp. 1-11 5. S-H. Rim, T. Kim and C.S. Ryoo, On the alternating sums of powers of consecutive q-integers, Bull. Korean Math. Soc., 43(2006), 611-617. 6. C.S. Ryoo and Y.S. Yoo, A note on Euler numbers and polynomials, Journal of Concrete and Applicable Mathematics, 7(2009), 341-348. 7. C. S. Ryoo, Calculating zeros of the twisted Genocchi polynomials, Adv. Stud. Contemp. Math., 17(2008), 147-159. 8. C. S. Ryoo, Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstein polynomials, Proc. Jangjeon Math. Soc., 14(2011), 239-248. 9. C. S. Ryoo, Some relations between twisted q-Euler numbers and Bernstein polynomials, Adv. Stud. Contemp. Math, 21(2011), 217-223. 10. C.S. Ryoo, Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, 12(4)(1010), 828-833.

1109

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1110-1119, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Approximate septic and octic mappings in quasi-β-normed spaces Tian Zhou Xu* School of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R.China E-mail addresses: [email protected], [email protected]

John Michael Rassias Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens,4, Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece E-mail addresses: [email protected], [email protected], [email protected] Abstract In this paper, we achieve the general solution of the septic and octic functional equations. Moreover, we prove the stability of the septic and octic functional equations in quasi-β-normed spaces. Keywords Quasi-β-normed spaces; Septic mapping; Octic mapping; (β, p)-Banach spaces; Hyers–Ulam stability. MR(2000) Subject Classification: 39B52, 39B82.

1. Introduction and preliminaries The concept of stability for a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Rassias [3] for approximate linear mappings by allowing the Cauchy difference operator CDf (x, y) = f (x + y) − [f (x) + f (y)] to be controlled by ϵ(∥x∥𝑝 + ∥y∥𝑝 ). In 1994, a generalization of Rassias’ theorem was obtained by G˘avrut¸a [4], who replaced ϵ(∥x∥𝑝 + ∥y∥𝑝 ) by a general control function φ(x, y) in the spirit of Rassias’ approach. The reader is referred to [5–20] and references therein for more information on stability of functional equations. In this paper, we achieve the general solutions of the septic functional equation f (x + 4y) − 7f (x + 3y) + 21f (x + 2y) − 35f (x + y) + 35f (x) − 21f (x − y) + 7f (x − 2y) − f (x − 3y) = 5040f (y) (1.1) and the octic functional equation f (x + 4y) − 8f (x + 3y) + 28f (x + 2y) − 56f (x + y) + 70f (x) − 56f (x − y) + 28f (x − 2y) −8f (x − 3y) + f (x − 4y) = 40320f (y).

(1.2)

Moreover, we prove the stability of the septic and octic functional equations in quasi-β-normed spaces. Since f (x) = x7 is a solutions of (1.1), we say it quintic functional equation. Similarly, f (x) = x8 is a solutions of (1.2), we say it septic functional equation. Every solution of the septic or octic functional equation is said to be a septic or an octic mapping, respectively. Let us recall some basic concepts concerning quasi-β-normed spaces (see [9, 16]). Let β be a fix real number with 0 < β ≤ 1 and let K denote either ℝ or C. Let X be a linear space over K. A quasi-β-norm ∥ · ∥ is a real-valued function on X satisfying the following: (1) ∥x∥ ≥ 0 for all x ∈ X and ∥x∥ = 0 if and only if x = 0. (2) ∥λx∥ = |λ|β ∥x∥ for all λ ∈ K and all x ∈ X. (3) There is a constant K ≥ 1 such that ∥x + y∥ ≤ K(∥x∥ + ∥y∥) for all x, y ∈ X. A quasi-β-normed space is a pair (X, ∥ · ∥), where ∥ · ∥ is a quasi-β-norm on X. The smallest possible K is called the modulus of concavity of ∥ · ∥. A quasi-β-Banach space is a complete quasi-β-normed space. A quasi-β-norm ∥ · ∥ is called a (β, p)-norm (0 < p ≤ 1) if ∥x + y∥𝑝 ≤ ∥x∥𝑝 + ∥y∥𝑝 for all x, y ∈ X. In this case, a quasi-β-Banach space is called a (β, p)-Banach space. We can refer to [13] for the concept of quasi-normed spaces and p-Banach spaces. Given a p-norm, the formula d(x, y) := ∥x − y∥𝑝 gives us a translation invariant metric on X. By the AokiRolewicz theorem, each quasi-norm is equivalent to some p-norm. Since it is much easier to work with p-norms than quasi-norms, henceforth we restrict our attention mainly to p-norms. *Corresponding author. The first author was supported by the National Natural Science Foundation of China (NNSFC)(Grant No. 11171022).

1110

T.Z. Xu and J.M. Rassias

2. General solutions to the septic and octic functional equations In this section, let X and Y be vector spaces. Some basic facts on n-additive symmetric mappings can be found in [11, 17, 20]. Theorem 2.1. A function f : X → Y is a solution of the functional equation (1.1) if and only if f is of the form f (x) = A7 (x) for all x ∈ X, where A7 (x) is the diagonal of the 7-additive symmetric map A7 : X 7 → Y . Proof. Assume that f satisfies the functional equation (1.1). Replacing x = y = 0 in equation (1.1), one finds f (0) = 0. Replacing (x, y) with (0, x) and (x, −x) in (1.1), respectively, and adding the two resulting equations, we obtain f (−x) = −f (x). Replacing (x, y) with (4x, x) and (0, 2x) in (1.1), respectively, and subtracting the two resulting equations, we get 7f (7x) − 27f (6x) + 35f (5x) − 21f (4x) + 21f (3x) − 5061f (2x) + 5041f (x) = 0

(2.1)

Replacing (x, y) with (3x, x) in (1.1), and multiplying the resulting equation by 7, one obtains 7f (7x) − 49f (6x) + 147f (5x) − 245f (4x) + 245f (3x) − 147f (2x) − 35231f (x) = 0

(2.2)

for all x ∈ X. Subtracting equations (2.1) and (2.1), we get 22f (6x) − 112f (5x) + 224f (4x) − 224f (3x) − 4914f (2x) + 40272f (x) = 0

(2.3)

Replacing (x, y) with (2x, x) in (1.1), and multiplying the resulting equation by 22, one finds 22f (6x) − 154f (5x) + 462f (4x) − 770f (3x) + 770f (2x) − 111320f (x) = 0

(2.4)

for all x ∈ X. Subtracting equations (2.3) and (2.4), we arrive at 42f (5x) − 238f (4x) + 546f (3x) − 5684f (2x) + 151592f (x) = 0

(2.5)

for all x ∈ X. Replacing (x, y) with (x, x) in (1.1), and multiplying the resulting equation by 42, one finds 42f (5x) − 294f (4x) + 882f (3x) − 1428f (2x) − 210504f (x) = 0

(2.6)

for all x ∈ X. Subtracting equations (2.5) and (2.6), one gets 56f (4x) − 336f (3x) − 4256f (2x) + 362096f (x) = 0

(2.7)

for all x ∈ X. Replacing (x, y) with (0, x) in (1.1), and multiplying the resulting equation by 56, one finds 56f (4x) − 336f (3x) + 784f (2x) − 283024f (x) = 0

(2.8)

for all x ∈ X. Subtracting equations (2.7) and (2.8), we arrive at f (2x) = 27 f (x)

(2.9)

for all x ∈ X. On the other hand, one can rewrite the functional equation (1.1) in the form f (x) + =

1 35 f (x

1 35 f (x

+ 4y) − 15 f (x + 3y) + 35 f (x + 2y) − f (x + y) − 35 f (x − y) + 15 f (x − 2y)

− 3y) + 144f (y)

(2.10)

for all x ∈ X. By Theorems 3.5 and 3.6 in [11], f is a generalized polynomial function of degree at most 6, that is, f is of the form f (x) = A7 (x) + A6 (x) + A5 (x) + A4 (x) + A3 (x) + A2 (x) + A1 (x) + A0 (x),

∀ x ∈ X,

(2.11)

where A0 (x) = A0 is an arbitrary element of Y , and A𝑖 (x) is the diagonal of the i-additive symmetric map A𝑖 : X 𝑖 → Y for i = 1, 2, 3, 4, 5. By f (0) = 0 and f (−x) = −f (x) for all x ∈ X, we get A0 (x) = A0 = 0 and the function f is odd. Thus we have A6 (x) = A4 (x) = A2 (x) = 0. It follows that f (x) = A7 (x)+A5 (x)+A3 (x)+A1 (x). By (2.9) and A𝑛 (rx) = r𝑛 A𝑛 (x) whenever x ∈ X and r ∈ Q, we obtain 27 (A7 (x) + A5 (x) + A3 (x) + A1 (x)) = 27 A7 (x) + 25 A5 (x) + 23 A3 (x) + 2A1 (x). It follows that A5 (x) = A3 (x) = A1 (x) = 0 for all x ∈ X. Hence f (x) = A7 (x).

1111

Approximate septic and octic mappings in quasi-β-normed spaces

Conversely, assume that f (x) = A7 (x) for all x ∈ X, where A7 (x) is the diagonal of the 7-additive symmetric map A7 : X 7 → Y . From A7 (x + y) = A7 (x) + A7 (y) + 7A6,1 (x, y) + 21A5,2 (x, y) + 35A4,3 (x, y) + 35A3,4 (x, y) + 21A2,5 (x, y) + 7A1,6 (x, y), A7 (rx) = r7 A5 (x), A6,1 (x, ry) = rA6,1 (x, y), A5,2 (x, ry) = r2 A5,2 (x, y), A4,3 (x, ry) = r3 A4,3 (x, y), A3,4 (x, ry) = r4 A3,4 (x, y), A2,5 (x, ry) = r5 A2,5 (x, y), and A1,6 (x, ry) = r6 A1,6 (x, y) (x, y ∈ X, r ∈ Q), we see that f satisfies (1.1), which completes the proof of Theorem 2.1.  Theorem 2.2. A function f : X → Y is a solution of the functional equation (1.2) if and only if f is of the form f (x) = A8 (x) for all x ∈ X, where A8 (x) is the diagonal of the 8-additive symmetric map A8 : X 8 → Y . Proof. Assume that f satisfies the functional equation (1.2). Replacing x = y = 0 in equation (1.2), one gets f (0) = 0. Substituting y by −y in (1.2) and subtracting the resulting equation from equation (1.2) and then y by x, we obtain f (−x) = f (x). Replacing (x, y) with (0, 2x) and (4x, x) in (1.2), respectively, we get f (8x) − 8f (6x) + 28f (4x) − 20216f (x) = 0

(2.12)

f (8x) − 8f (7x) + 28f (6x) − 56f (5x) + 70f (4x) − 56f (3x) + 28f (2x) − 40328f (x) = 0

(2.13)

and for all x ∈ X. Subtracting equations (2.12) and (2.13), we find 8f (7x) − 36f (6x) + 56f (5x) − 42f (4x) + 56f (3x) − 20244f (2x) + 40328f (x) = 0

(2.14)

for all x ∈ X. Replacing (x, y) with (3x, x) in (1.2), and multiplying the resulting equation by 8, one obtains 8f (7x) − 64f (6x) + 224f (5x) − 448f (4x) + 560f (3x) − 448f (2x) − 322328f (x) = 0

(2.15)

for all x ∈ X. Subtracting equations (2.14) and (2.15), one gets 28f (6x) − 168f (5x) + 406f (4x) − 504f (3x) − 19796f (2x) + 362656f (x) = 0

(2.16)

for all x ∈ X. Replacing (x, y) with (2x, x) in (1.2), and multiplying the resulting equation by 28, one finds 28f (6x) − 224f (5x) + 784f (4x) − 1568f (3x) + 1988f (2x) − 1130752f (x) = 0

(2.17)

for all x ∈ X. Subtracting equations (2.16) and (2.17), one gets 56f (5x) − 378f (4x) + 1064f (3x) − 21784f (2x) + 1493408f (x) = 0

(2.18)

for all x ∈ X. Replacing (x, y) with (x, x), and multiplying the resulting equation by 56, one finds 56f (5x) − 448f (4x) + 1624f (3x) − 3584f (2x) − 2252432f (x) = 0

(2.19)

for all x ∈ X. Subtracting equations (2.18) and (2.19), we arrive at 70f (4x) − 560f (3x) − 18200f (2x) + 3745840f (x) = 0

(2.20)

for all x ∈ X. Replacing (x, y) with (0, x), and multiplying the resulting equation by 70, one finds 70f (4x) − 560f (3x) + 1960f (2x) − 1415120f (x) = 0

(2.21)

for all x ∈ X. Subtracting equations (2.20) and (2.21), we arrive at f (2x) = 28 f (x)

(2.22)

for all x ∈ X. On the other hand, one can rewrite the functional equation (1.2) in the form f (x) + =

1 70 f (x

4 35 f (x

+ 4y) −

− 3y) −

4 35 f (x

1 70 f (x

+ 3y) + 25 f (x + 2y) − 45 f (x + y) − 54 f (x − y) + 25 f (x − 2y)

− 4y) +

1 576 f (y)

(2.23)

for all x ∈ X. By Theorems 3.5 and 3.6 in [11], f is a generalized polynomial function of degree at most 6, that is f is of the form f (x) = A8 (x) + A7 (x) + · · · + A1 (x) + A0 (x), ∀ x ∈ X, (2.24) where A0 (x) = A0 is an arbitrary element of Y , and A𝑖 (x) is the diagonal of the i-additive symmetric map A𝑖 : X 𝑖 → Y for i = 1, 2, . . . , 8. By f (0) = 0 and f (−x) = f (x) for all x ∈ X, we get A0 (x) = A0 = 0

1112

T.Z. Xu and J.M. Rassias

and the function f is even. Thus we have A7 (x) = A5 (x) = A3 (x) = A1 (x) = 0. It follows that f (x) = A8 (x) + A6 (x) + A4 (x) + A2 (x). By (2.22) and A𝑛 (rx) = r𝑛 A𝑛 (x) whenever x ∈ X and r ∈ Q, we obtain 28 (A8 (x) + A6 (x) + A4 (x) + A2 (x)) = 28 A8 (x) + 26 A6 (x) + 24 A4 (x) + 22 A2 (x). It follows that A6 (x) = A4 (x) = A2 (x) = 0, x ∈ X. Therefore, f (x) = A8 (x). The rest of the proof is similar to the proof of Theorem 2.1.  3. Stability of the septic and octic functional equations Throughout this section, we assume that X is a linear space and Y is a (β, p)-Banach space with (β, p)-norm ∥ · ∥Y . For a given mapping f : X → Y , we define the difference operators D𝑠 f (x, y) := f (x+4y)−7f (x+3y)+21f (x+2y)−35f (x+y)+35f (x)−21f (x−y)+7f (x−2y)−f (x−3y)−5040f (y) and Do f (x, y)

:=

f (x + 4y) − 8f (x + 3y) + 28f (x + 2y) − 56f (x + y) + 70f (x) − 56f (x − y) +28f (x − 2y) − 8f (x − 3y) + f (x − 4y) − 40320f (y)

for all x, y ∈ X. Lemma 3.1(see [16]). Let j ∈ {−1, 1} be fixed, s, a ∈ N with a ≥ 2, and ψ : X → [0, ∞) be a function such that there exists an L < 1 with ψ(a𝑗 x) ≤ a𝑗𝑠β Lψ(x) for all x ∈ X. Let f : X → Y be a mapping satisfying ∥f (ax) − a𝑠 f (x)∥Y ≤ ψ(x)

(3.1)

for all x ∈ X, then there exists a uniquely determined mapping F : X → Y such that F (ax) = a𝑠 F (x) and ∥f (x) − F (x)∥Y ≤

1 ψ(x) a𝑠β |1 − L𝑗 |

(3.2)

for all x ∈ X. Theorem 3.2. Let j ∈ {−1, 1} be fixed, φ : X × X → [0, ∞) be a function such that there exists an L < 1 with φ(2𝑗 x, 2𝑗 y) ≤ 128𝑗β Lφ(x, y) for all x, y ∈ X. Let f : X → Y be a mapping satisfying ∥D𝑠 f (x, y)∥Y ≤ φ(x, y)

(3.3)

for all x, y ∈ X. Then there exists a unique septic mapping S : X → Y such that ∥f (x) − S(x)∥Y ≤

1 φ𝑠 (x) 128β |1 − L𝑗 |

(3.4)

for all x ∈ X, where φ𝑠 (x) =

1 [K 5 φ(4x, x) + K 6 φ(0, 2x) + 7β K 5 φ(3x, x) + 22β K 4 φ(2x, x) + 42β K 3 φ(x, x) 5040β 7 β 3 K 11β K 5 K5 7β K 4 K 10 +( 144 + 720 + 736Kβ )φ(0, 0) + 5040 β + 360β β + 40β β (φ(0, 6x) + φ(6x, −6x)) K 10 K9 K6 7β K 6 + 720β (φ(0, 4x) + φ(4x, −4x)) + ( 240β + 120β + 90β )(φ(0, 2x) + φ(2x, −2x)) β β 6 β 5 K6 K6 +56β K 2 φ(0, x) + ( 11 + 7120Kβ + 730Kβ )(φ(0, x) + φ(x, −x)) + 5040 β (φ(0, 3x) + φ(3x, −3x))]. 2520β

Proof. Replacing x = y = 0 in (3.3), we get 1 φ(0, 0). 5040β Replacing x and y by 0 and x in (3.3), respectively, we get ∥f (0)∥Y ≤

∥f (4x) − 7f (3x) + 21f (2x) − 5075f (x) + 35f (0) − 21f (−x) + 7f (−2x) − f (−3x)∥Y ≤ φ(0, x)

(3.5)

(3.6)

for all x ∈ X. Replacing x and y by x and −x in (3.3), respectively, we have ∥f (−3x) − 7f (−2x) − 35f (0) + 35f (x) − 21f (2x) + 7f (3x) − f (4x) − 5019f (−x)∥Y ≤ φ(x, −x)

(3.7)

for all x ∈ X. By (3.6) and (3.7), we obtain ∥f (x) + f (−x)∥Y ≤

K (φ(0, x) + φ(x, −x)) 5040β

1113

(3.8)

Approximate septic and octic mappings in quasi-β-normed spaces

for all x ∈ X. Replacing x and y by 0 and 2x in (3.3), respectively, we find ∥f (8x) − 7f (6x) + 21f (4x) − 5075f (2x) + 35f (0) − 21f (−2x) + 7f (−4x) − f (−6x)∥Y ≤ φ(0, 2x)

(3.9)

for all x ∈ X. By (3.5), (3.8) and (3.9), one obtains ∥f (8x) − 6f (6x) + 14f (4x) − 5054f (2x)∥Y ≤ Kφ(0, 2x) +

K2 φ(0, 0) 144β

K5 + 720 β (φ(0, 4x)

+

K4 (φ(0, 2x) + φ(2x, −2x)) 240β K5 φ(4x, −4x)) + 5040 β (φ(0, 6x) + φ(6x, −6x))

(3.10)

+

for all x ∈ X. Replacing x and y by 4x and x in (3.3), respectively, we get ∥f (8x) − 7f (7x) + 21f (6x) − 35f (5x) + 35f (4x) − 21f (3x) + 7f (2x) − 5041f (x)∥Y ≤ φ(4x, x)

(3.11)

for all x ∈ X. By (3.10) and (3.11), we obtain ∥7f (7x) − 27f (6x) + 35f (5x) − 21f (4x) + 21f (3x) − 5061f (2x) + 5041f (x)∥Y K3 K5 φ(0, 0) + 240 β (φ(0, 2x) + φ(2x, −2x)) 144β K6 φ(4x, −4x)) + 5040 (φ(0, 6x) + φ(6x, −6x)) β

≤ Kφ(4x, x) + K 2 φ(0, 2x) + 6

K + 720 β (φ(0, 4x) +

(3.12)

for all x ∈ X. Replacing x and y by 3x and x in (3.3), respectively, we get ∥f (7x) − 7f (6x) + 21f (5x) − 35f (4x) + 35f (3x) − 21f (2x) − f (0) − 5033f (x)∥Y ≤ φ(3x, x)

(3.13)

for all x ∈ X. Using (3.5), we have ∥7f (7x) − 49f (6x) + 147f (5x) − 245f (4x) + 245f (3x) + 147f (2x) − 35231f (x)∥Y ≤ 7β Kφ(3x, x) +

K φ(0, 0) 720β

(3.14)

for all x ∈ X. By (3.12) and (3.14), one obtains ∥22f (6x) − 112f (5x) + 224f (4x) − 224f (3x) − 4914f (2x) + 40272f (x)∥Y K6 K4 φ(0, 0) + 240 β (φ(0, 2x) + φ(2x, −2x)) 144β K7 φ(4x, −4x)) + 5040 (φ(0, 6x) + φ(6x, −6x)) + 7β K 2 φ(3x, x) β

≤ K 2 φ(4x, x) + K 3 φ(0, 2x) + 7

K + 720 β (φ(0, 4x) +

(3.15) +

K2 φ(0, 0) 720β

for all x ∈ X. Replacing x and y by 2x and x in (3.3), respectively, we get ∥f (6x) − 7f (5x) + 21f (4x) − 35f (3x) + 35f (2x) − 5061f (x) + 7f (0) − f (−x)∥Y ≤ φ(2x, x)

(3.16)

for all x ∈ X. Using (3.5), (3.8) and (3.16), we have ∥f (6x) − 7f (5x) + 21f (4x) − 35f (3x) + 35f (2x) − 5060f (x)∥Y ≤ Kφ(2x, x) +

K2 φ(0, 0) 720β

+

K3 (φ(0, x) 5040β

(3.17)

+ φ(x, −x))

for all x ∈ X. Hence ∥22f (6x) − 154f (5x) + 462f (4x) − 770f (3x) + 770f (2x) − 111320f (x)∥Y ≤ 22β Kφ(2x, x) +

11β K 2 φ(0, 0) 360β

+

11β K 3 (φ(0, x) 2520β

+ φ(x, −x))

(3.18)

for all x ∈ X. By (3.15) and (3.18), one obtains ∥42f (5x) − 238f (4x) + 546f (3x) − 5684f (2x) + 151592f (x)∥Y K5 K7 φ(0, 0) + 240 β (φ(0, 2x) + φ(2x, −2x)) 144β K8 K8 + 720β (φ(0, 4x) + φ(4x, −4x)) + 5040β (φ(0, 6x) + φ(6x, −6x)) + 7β K 3 φ(3x, x) β 3 β K4 +22β K 2 φ(2x, x) + 11360Kβ φ(0, 0) + 11 (φ(0, x) + φ(x, −x)) 2520β

≤ K 3 φ(4x, x) + K 4 φ(0, 2x) +

+

K3 φ(0, 0) 720β

(3.19)

for all x ∈ X. Replacing x and y by x and x in (3.3), respectively, we have ∥f (5x) − 7f (4x) + 21f (3x) − 35f (2x) − 5005f (x) − 21f (0) + 7f (−x) − f (−2x)∥Y ≤ φ(x, x)

1114

(3.20)

T.Z. Xu and J.M. Rassias

for all x ∈ X. By (3.5), (3.8), and (3.20), we have ∥f (5x) − 7f (4x) + 21f (3x) − 34f (2x) − 5012f (x)∥Y ≤ Kφ(x, x) +

K2 φ(0, 0) 240β

+

K4 (φ(0, x) 720β

+ φ(x, −x)) +

K4 (φ(0, 2x) 5040β

+ φ(2x, −2x))

(3.21)

for all x ∈ X. Hence ∥42f (5x) − 294f (4x) + 882f (3x) − 1428f (2x) − 210504f (x)∥Y 7β K 2 φ(0, 0) 40β

≤ 42β Kφ(x, x) +

+

7β K 4 (φ(0, x) 120β

+ φ(x, −x)) +

K4 (φ(0, 2x) 120β

+ φ(2x, −2x))

(3.22)

for all x ∈ X. By (3.19) and (3.22), we obtain ∥56f (4x) − 336f (3x) − 4256f (2x) + 362096f (x)∥Y K6 K8 φ(0, 0) + 240 β (φ(0, 2x) + φ(2x, −2x)) 144β K9 K9 + 720β (φ(0, 4x) + φ(4x, −4x)) + 5040β (φ(0, 6x) + φ(6x, −6x)) + 7β K 4 φ(3x, x) K4 11β K 4 11β K 5 β 3 + 720 β φ(0, 0) + 22 K φ(2x, x) + 360β φ(0, 0) + 2520β (φ(0, x) + φ(x, −x)) β 3 β 5 K5 +42β K 2 φ(x, x) + 740Kβ φ(0, 0) + 7120Kβ (φ(0, x) + φ(x, −x)) + 120 β (φ(0, 2x) + φ(2x, −2x))

≤ K 4 φ(4x, x) + K 5 φ(0, 2x) +

(3.23)

for all x ∈ X. Replacing x and y by 0 and x in (3.3), respectively, one gets ∥f (4x) − 7f (3x) + 21f (2x) − 5075f (x) + 35f (0) − 21f (−x) + 7f (−2x) − f (−3x)∥Y ≤ φ(0, x)

(3.24)

for all x ∈ X. By (3.5), (3.8) and (3.24), we obtain ∥f (4x) − 6f (3x) + 14f (2x) − 5054f (x)∥Y ≤ Kφ(0, x) +

K2 φ(0, 0) 144β

K5 + 5040 β (φ(0, 3x)

+

K4 (φ(0, x) 240β

+ φ(x, −x)) +

K5 (φ(0, 2x) 720β

+ φ(2x, −2x))

(3.25)

+ φ(3x, −3x))

for all x ∈ X. Thus ∥56f (4x) − 336f (3x) + 784f (2x) − 283024f (x)∥Y ≤ 56β Kφ(0, x) + 5

7β K 2 φ(0, 0) 36β

K + 5040 β (φ(0, 3x)

+

7β K 4 (φ(0, x) 30β

+ φ(x, −x)) +

7β K 5 (φ(0, 2x) 90β

+ φ(2x, −2x))

(3.26)

+ φ(3x, −3x))

for all x ∈ X. By (3.23) and (3.26), we obtain ∥f (2x) − 27 f (x)∥Y ≤ φ𝑠 (x) for all x ∈ X. By Lemma 3.1, there exists a unique mapping S : X → Y such that S(2x) = 27 S(x) and ∥f (x) − S(x)∥Y ≤

1 128β |1

− L𝑗 |

φ𝑠 (x)

for all x ∈ X. It remains to show that S is a septic map. By (3.3), we have ∥D𝑠 f (2𝑗𝑛 x, 2𝑗𝑛 y)/128𝑗𝑛 ∥Y ≤ 128−𝑗𝑛β φ(2𝑗𝑛 x, 2𝑗𝑛 y) ≤ 128−𝑗𝑛β (128𝑗β L)𝑛 φ(x, y) = L𝑛 φ(x, y) for all x, y ∈ X and n ∈ N. So ∥D𝑠 S(x, y)∥Y = 0 for all x, y ∈ X. Thus the mapping S : X → Y is septic.  Corollary 3.3. Let X be a quasi-α-normed space with quasi-α-norm ∥ · ∥X , Y be a (β, p)-Banach space with (β, p)-norm ∥ · ∥Y . Let δ, λ be positive numbers with λ ̸= 7β 𝛼 , and f : X → Y be a mapping satisfying ∥D𝑠 f (x, y)∥Y ≤ δ(∥x∥λX + ∥y∥λX ) for all x, y ∈ X. Then there exists a unique septic mapping S : X → Y such that { δελ ∥x∥λX , λ ∈ (0, 7β 𝛼 ); 128β −2αλ ∥f (x) − S(x)∥Y ≤ 7β 2λα δελ λ ∥x∥ , λ ∈ ( , X 𝛼 ∞); 128β (2λα −128β ) for all x ∈ X, where ελ

=

1 [K 5 (4𝛼λ + 1) + K 6 2𝛼λ + 7β K 5 (3𝛼λ 5040β 10 αλ K9 K6 7β K 6 + 3⋅K720β4 + 3 · 2𝛼λ ( 240 β + 120β + 90β )

1115

3⋅K 10 6αλ 5040β 3K 6 3αλ + 5040β ].

+ 1) + 22β K 4 (2𝛼λ + 1) + 2 · 42β K 3 + β

6

K + 56β K 2 + 3( 11 + 2520β

7β K 6 120β

+

7β K 5 ) 30β

Approximate septic and octic mappings in quasi-β-normed spaces

The following example shows that the assumption λ ̸= 7β 𝛼 cannot be omitted in Corollary 3.3. This example is a modification of the example of Gajda [21] for the additive functional inequality (see also [12] and [16]). Example 3.4. Let ϕ : ℝ → ℝ be defined by

{

ϕ(x) =

x7 , for |x| < 1, 1,

for |x| ≥ 1.

Consider the function f : ℝ → ℝ be defined by f (x) =

∞ ∑

4−7𝑛 ϕ(4𝑛 x)

𝑛=0

for all x ∈ ℝ. Then f satisfies the functional inequality 5168 · 163843 (|x|7 + |y|7 ) 16383

|D𝑠 f (x, y)| ≤

(3.27)

for all x, y ∈ ℝ, but there do not exist a septic mapping S : ℝ → ℝ and a constant d > 0 such that |f (x)−S(x)| ≤ d |x|7 for all x ∈ ℝ. Proof. It is clear that f is bounded by 16384/16383 on ℝ. If |x|7 + |y|7 = 0 or |x|7 + |y|7 ≥ 1/16384, then |D𝑠 f (x, y)| ≤

5168 · 16384 5168 · 163842 ≤ (|x|7 + |y|7 ). 16383 16383

Now suppose that 0 < |x|5 + |y|5 < 1/1024. Then there exists a non-negative integer k such that 1 1 ≤ |x|7 + |y|7 < . 𝑘+2 16384 16384𝑘+1

(3.28)

Hence 16384𝑘 |x|7 < 1/16384, 16384𝑘 |y|7 < 1/16384, and 4𝑛 (x + 3y), 4𝑛 (x + 2y), 4𝑛 (x − 2y), 4𝑛 (x + y), 4𝑛 (x − y), 4𝑛 x, 4𝑛 y ∈ (−1, 1) for all n = 0, 1, . . . , k − 1. Hence, for n = 0, 1, . . . , k − 1, D𝑠 ϕ(4𝑛 x, 4𝑛 y) = 0. From the definition of f and the inequality (3.28), we obtain that |D𝑠 f (x, y)| ≤

∞ ∑ 𝑛=𝑘

4−7𝑛 · 5168 =

5168 · 47(1−𝑘) 5168 · 163843 ≤ (|x|7 + |y|7 ). 16383 16383

Therefore, f satisfies (3.27) for all x, y ∈ ℝ. Now, we claim that the functional equation (1.1) is not stable for λ = 7 in Corollary 3.3 (α = β = p = 1). Suppose on the contrary that there exists a septic mapping S : ℝ → ℝ and constant d > 0 such that |f (x) − S(x)| ≤ d |x|7 for all x ∈ ℝ. Then there exists a constant c ∈ ℝ such that S(x) = cx7 for all rational numbers x. So we obtain that |f (x)| ≤ (d + |c|)|x|5

(3.29)

for all x ∈ Q. Let m ∈ N with m + 1 > d + |c|. If x is a rational number in (0, 4−𝑚 ), then 4𝑛 x ∈ (0, 1) for all n = 0, 1, . . . , m, and for this x we get f (x) =

𝑚 ∞ ∑ ϕ(4𝑛 x) ∑ (4𝑛 x)7 ≥ = (m + 1)x7 > (d + |c|)x7 , 7𝑛 7𝑛 4 4 𝑛=0 𝑛=0

which contradicts (3.29). Theorem 3.5. Let j ∈ {−1, 1} be fixed, φ : X × X → [0, ∞) be a function such that there exists an L < 1 with φ(2𝑗 x, 2𝑗 y) ≤ 256𝑗β Lφ(x, y) for all x, y ∈ X. Let f : X → Y be a mapping satisfying ∥Do f (x, y)∥Y ≤ φ(x, y)

(3.30)

for all x, y ∈ X. Then there exists a unique octic mapping O : X → Y such that ∥f (x) − O(x)∥Y ≤

1116

1 φo (x) 256β |1 − L𝑗 |

(3.31)

T.Z. Xu and J.M. Rassias

for all x ∈ X, where φo (x)

=

6 1 [ K φ(0, 2x) 20160β 2β β 2

7

K + ( 1152 β +

K6 40320β

+

7β K 5 360β

+

7β K 4 90β β

+

35β K 3 576β

+

K6 )φ(0, 0) 630β β 4

+35 K φ(0, x) + 56β K 3 φ(x, x) + K 6 φ(4x, x) + 8 K 4 φ(3x, x) + 28 K φ(2x, x) 9

K7 K 11 )(φ(2x, 2x) + φ(2x, −2x)) + 10080 β (φ(6x, 6x) + φ(6x, −6x)) 1440β K 11 K7 K7 7β K 6 7β K 5 + 80640β (φ(8x, 8x) + φ(8x, −8x)) + ( 180β + 5040β + 180β + 144β )(φ(x, x) + φ(x, −x)) K7 K7 K7 K 10 +( 720 β + 144β )(φ(3x, 3x) + φ(3x, −3x)) + ( 1152β + 2880β )(φ(4x, 4x) + φ(4x, −4x))]. K +( 1440 β +

K7 90β

+

7β K 6 288β

+

Proof. Replacing x = y = 0 in (3.30), we have ∥f (0)∥Y ≤

1 φ(0, 0). 40320β

(3.32)

Replacing y by −y in (3.30), we get ∥f (x − 4y) − 8f (x − 3y) + 28f (x − 2y) − 56f (x − y) + 70f (x) − 56f (x + y) +28f (x + 2y) − 8f (x + 3y) + f (x + 4y) − 40320f (−y)∥Y ≤ φ(x, −y)

(3.33)

for all x, y ∈ X. By (3.30) and (3.33), one gets K (φ(x, x) + φ(x, −x)) 40320β for all x ∈ X. Replacing x and y by 0 and 2x in (3.30), respectively, one obtains ∥f (x) − f (−x)∥Y ≤

(3.34)

∥f (8x) − 8f (6x) + 28f (4x) − 56f (2x) + 70f (0) − 56f (−2x) + 28f (−4x) − 8f (−6x) +f (−8x) − 40320f (2x)∥Y ≤ φ(0, 2x)

(3.35)

for all x ∈ X. By (3.32), (3.34), and (3.35), we have ∥f (8x) − 8f (6x) + 28f (4x) − 20216f (2x)∥Y ≤

K2 K4 K5 K φ(0, 2x) + 1152 β φ(0, 0) + 1440β (φ(2x, 2x) + φ(2x, −2x)) + 2880β (φ(4x, 4x) 2β K6 K6 + 10080 β (φ(6x, 6x) + φ(6x, −6x)) + 80640β (φ(8x, 8x) + φ(8x, −8x))

+ φ(4x, −4x))

(3.36)

for all x ∈ X. Replacing x and y by 4x and x in (3.30), respectively, we get ∥f (8x) − 8f (7x) + 28f (6x) − 56f (5x) + 70f (4x) − 56f (3x) + 28f (2x) + f (0) − 40328f (x)∥Y ≤ φ(4x, x) (3.37) for all x ∈ X. Using (3.32), one gets ∥f (8x) − 8f (7x) + 28f (6x) − 56f (5x) + 70f (4x) − 56f (3x) + 28f (2x) − 40328f (x)∥Y ≤ Kφ(4x, x) +

K φ(0, 0) 40320β

(3.38)

for all x ∈ X. By (3.36) and (3.38), we have ∥8f (7x) − 36f (6x) + 56f (5x) − 70f (4x) + 56f (3x) − 28f (2x) + 40328f (x)∥Y K2 K3 K5 K6 φ(0, 2x) + 1152 β φ(0, 0) + 1440β (φ(2x, 2x) + φ(2x, −2x)) + 2880β (φ(4x, 4x) + φ(4x, −4x)) 2β K7 K2 K7 2 + 10080 β (φ(6x, 6x) + φ(6x, −6x)) + 80640β (φ(8x, 8x) + φ(8x, −8x)) + K φ(4x, x) + 40320β φ(0, 0)

≤

(3.39)

for all x ∈ X. Replacing x and y by 3x and x in (3.30), respectively, and then using (3.32) and (3.34), one obtains ∥8f (7x) − 64f (6x) + 224f (5x) − 448f (4x) + 560f (3x) − 448f (2x) − 322328f (x)∥Y (3.40) K2 K3 ≤ 8β φ(3x, x) + 630 β φ(0, 0) + 5040β (φ(x, x) + φ(x, −x)) for all x ∈ X. Subtracting (3.39) − (3.40), we obtain ∥28f (6x) − 168f (5x) + 406f (4x) − 504f (3x) − 19796f (2x) + 362656f (x)∥Y K3 K4 K6 K7 φ(0, 2x) + 1152 β φ(0, 0) + 1440β (φ(2x, 2x) + φ(2x, −2x)) + 2880β (φ(4x, 4x) + φ(4x, −4x)) 2β 8 8 K K3 K 3 + 10080 β (φ(6x, 6x) + φ(6x, −6x)) + 80640β (φ(8x, 8x) + φ(8x, −8x)) + K φ(4x, x) + 40320β φ(0, 0) K4 K3 +8β Kφ(3x, x) + 630 β φ(0, 0) + 5040β (φ(x, x) + φ(x, −x))

≤

1117

(3.41)

Approximate septic and octic mappings in quasi-β-normed spaces

for all x ∈ X. Replacing x and y by 2x and x in (3.30), respectively, and then using (3.32) and (3.34), we have ∥28f (6x) − 224f (5x) + 784f (4x) − 1568f (3x) + 1988f (2x) − 1130752f (x)∥Y ≤ 28β Kφ(2x, x) +

7β K 2 φ(0, 0) 360β

+

K4 (φ(x, x) 180β

+ φ(x, −x)) +

K4 (φ(2x, 2x) 1440β

+ φ(2x, −2x))

(3.42)

for all x ∈ X. Subtracting (3.41) − (3.42), one gets ∥56f (5x) − 378f (4x) + 1064f (3x) − 21784f (2x) + 1493408f (x)∥Y K4 K5 K7 K8 φ(0, 2x) + 1152 β φ(0, 0) + 1440β (φ(2x, 2x) + φ(2x, −2x)) + 2880β (φ(4x, 4x) + φ(4x, −4x)) 2β 9 9 K K K4 4 + 10080 β (φ(6x, 6x) + φ(6x, −6x)) + 80640β (φ(8x, 8x) + φ(8x, −8x)) + K φ(4x, x) + 40320β φ(0, 0) K4 K5 7β K 3 β 2 +8β K 2 φ(3x, x) + 630 β φ(0, 0) + 5040β (φ(x, x) + φ(x, −x)) + 28 K φ(2x, x) + 360β φ(0, 0) K5 K5 + 180 β (φ(x, x) + φ(x, −x)) + 1440β (φ(2x, 2x) + φ(2x, −2x))

≤

(3.43)

for all x ∈ X. Replacing x and y by x and x in (3.30), respectively, and then using (3.32) and (3.34), we have ∥f (5x) − 8f (4x) + 29f (3x) − 64f (2x) − 40222f (x)∥Y ≤ Kφ(x, x) + 5

K2 φ(0, 0) 720β

K + 5040 β (φ(2x, 2x)

+

K4 (φ(x, x) + φ(x, −x)) 1440β K5 φ(2x, −2x)) + 40320 β (φ(3x, 3x) +

(3.44)

+

φ(3x, −3x))

for all x ∈ X. Multiply each side of (3.44) by 56β , one gets ∥56f (5x) − 448f (4x) + 1624f (3x) − 3584f (2x) − 2252432f (x)∥Y ≤ 56β Kφ(x, x) + K5 + 90 β (φ(2x, 2x)

7β K 4 (φ(x, x) + φ(x, −x)) 180β K5 φ(2x, −2x)) + 720β (φ(3x, 3x) + φ(3x, −3x))

7β K 2 φ(0, 0) 90β

+

+

(3.45)

for all x ∈ X. By (3.43) and (3.45), we have ∥70f (4x) − 560f (3x) − 18200f (2x) + 3745840f (x)∥Y K6 K8 K9 K5 φ(0, 2x) + 1152 β φ(0, 0) + 1440β (φ(2x, 2x) + φ(2x, −2x)) + 2880β (φ(4x, 4x) + φ(4x, −4x)) 2β 10 10 K K K5 5 + 10080 β (φ(6x, 6x) + φ(6x, −6x)) + 80640β (φ(8x, 8x) + φ(8x, −8x)) + K φ(4x, x) + 40320β φ(0, 0) K5 K6 7β K 4 β 3 +8β K 3 φ(3x, x) + 630 β φ(0, 0) + 5040β (φ(x, x) + φ(x, −x)) + 28 K φ(2x, x) + 360β φ(0, 0) K6 K6 7β K 3 β 2 + 180 β (φ(x, x) + φ(x, −x)) + 1440β (φ(2x, 2x) + φ(2x, −2x)) + 56 K φ(x, x) + 90β φ(0, 0) β 5 K6 K6 + 7180Kβ (φ(x, x) + φ(x, −x)) + 90 β (φ(2x, 2x) + φ(2x, −2x)) + 720β (φ(3x, 3x) + φ(3x, −3x))

≤

(3.46)

for all x ∈ X. Replacing x and y by 0 and x in (3.30), respectively, and then using (3.32) and (3.34), we have ∥2f (4x) − 16f (3x) + 56f (2x) − 40432f (x)∥Y ≤ Kφ(0, x) +

K2 φ(0, 0) 576β

K6 + 5040 β (φ(3x, 3x)

+

K4 K5 (φ(x, x) + φ(x, −x)) + 1440 β (φ(2x, 2x) 720β K6 φ(3x, −3x)) + 40320β (φ(4x, 4x) + φ(4x, −4x))

+

+ φ(2x, −2x))

(3.47)

for all x ∈ X. Multiply each side of (3.47) by 35β , one gets ∥70f (4x) − 560f (3x) + 1960f (2x) − 1415120f (x)∥Y ≤ 35β Kφ(0, x) +

β 5 7β K 4 (φ(x, x) + φ(x, −x)) + 7288Kβ (φ(2x, 2x) 144β K6 φ(3x, −3x)) + 1152 β (φ(4x, 4x) + φ(4x, −4x))

35β K 2 φ(0, 0) 576β

K6 + 144 β (φ(3x, 3x)

+

+

+ φ(2x, −2x))

(3.48)

for all x ∈ X. By (3.46) and (3.48), we obtain ∥f (2x) − 28 f (x)∥Y ≤ φo (x) for all x ∈ X. By Lemma 3.1, there exists a unique mapping O : X → Y such that O(2x) = 28 O(x) and ∥f (x) − O(x)∥Y ≤

1 φ(x) ˜ 256β |1 − L𝑗 |

for all x ∈ X. It remains to show that O is an octic mapping. By (3.30), we have ∥Do f (2𝑗𝑛 x, 2𝑗𝑛 y)/256𝑗𝑛 ∥Y ≤ 256−𝑗𝑛β φ(2𝑗𝑛 x, 2𝑗𝑛 y) ≤ 256−𝑗𝑛β (256𝑗β L)𝑛 φ(x, y) = L𝑛 φ(x, y) for all x, y ∈ X and n ∈ N. So ∥Do O(x, y)∥Y = 0 for all x, y ∈ X. Thus the mapping O : X → Y is octic. 

1118

T.Z. Xu and J.M. Rassias

Corollary 3.6. Let X be a quasi-α-normed space with quasi-α-norm ∥ · ∥X , Y be a (β, p)-Banach space with (β, p)-norm ∥ · ∥Y . Let δ, λ be positive numbers with λ ̸= 8β 𝛼 , and f : X → Y be a mapping satisfying ∥Do f (x, y)∥Y ≤ δ(∥x∥λX + ∥y∥λX ) for all x, y ∈ X. Then there exists a unique octic mapping O : X → Y such that { δελ ∥x∥λX , λ ∈ (0, 8β 𝛼 ); 256β −2αλ ∥f (x) − O(x)∥Y ≤ λα 8β 2 δελ λ ∥x∥X , λ ∈ ( 𝛼 , ∞); 256β (2λα −256β ) for all x ∈ X, where ελ

=

6 1 [ K 2𝛼λ + 35β K 2 + 2 · 56β K 3 + K 6 (4𝛼λ + 1) + 8β K 4 (3𝛼λ + 1) + 28β K 4 (2𝛼λ 20160β 2β K9 K7 7β K 6 K7 K7 K7 7β K 6 7β K 5 +4 · 2𝛼λ ( 1440 β + 90β + 288β + 1440β ) + 4( 180β + 5040β + 180β + 144β ) 11 αλ 11 αλ 6 8 K7 K7 K 10 𝛼λ K 7 + 4⋅K + 4⋅K + 4 · 3𝛼λ ( 720 ( 1152β + 2880 β + 144β ) + 4 · 4 β )]. 10080β 80640β

Remark 3.7. The Hyers–Ulam stability for the case of λ =

8β 𝛼

+ 1)

was excluded in Corollary 3.6 (see Example 3.4).

References [1] S.M. Ulam, A Collection of Mathematical Problems, Interscience Publ., New York, 1960. [2] D.H. Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27(1941) 222-224. [3] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72(2)(1978) 297–300. [4] P. G˘ avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications, 184(3)(1994) 431–436. [5] T.Z. Xu, J.M. Rassias, and W.X. Xu, On the stability of a general mixed additive-cubic functional equation in random normed spaces, Journal of Inequalities and Applications, 2010(2010), Article ID 328473, 1–16. [6] M. Mohamadi, Y.J. Cho, C. Park, P. Vetro, and R. Saadati, Random stability of an additive-quadratic quartic functional equation, Journal of Inequalities and Applications, 2010(2010), Article ID 754210, 1–18. [7] L. C˘ adariu and V. Radu, Fixed points and stability for functional equations in probabilistic metric and random normed spaces, Fixed Point Theory and Applications, 2009(2009), Article ID 589143, 1–18. [8] M. Eshaghi Gordji and M.B. Savadkouhi, Stability of mixed type cubic and quartic functional equations in random normed spaces, Journal of Inequalities and Applications, 2009(2009), Article ID 527462, 1–9. [9] J.M. Rassias and H.-M. Kim, Generalized Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces, Journal of Mathematical Analysis and Applications, 356(2009) 302–309. [10] C. Park, Fixed points and the stability of an AQCQ-functional equation in non-Archimedean normed spaces, Abstract and Applied Analysis, 2010(2010), Article ID 849543, 1–15. [11] T.Z. Xu, J.M. Rassias, and W.X. Xu, A generalized mixed quadratic-quartic functional equation, Bull. Malays. Math. Sci. Soc., 35(3)(2012) 633–649. [12] T.Z. Xu and J.M. Rassias, A fixed point approach to the stability of an AQ-functional equation on β-Banach modules. Fixed Point Theory and Applications, 2012(2012), Article ID 32, 1–21. [13] T.Z. Xu, J.M. Rassias, and W.X. Xu, Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces, Acta Mathematica Sinica, English Series, 28(3)(2012) 529–560. [14] T.Z. Xu, Stability of multi-Jensen mappings in non-Archimedean normed spaces, Journal of Mathematical Physics, 53(2012), Article ID 023507, 1–9. [15] T.Z. Xu, On the stability of multi-Jensen mappings in β-normed spaces, Applied Mathematics Letters, 25(2012) 1866–1870. [16] T.Z. Xu, J.M. Rassias, M.J. Rassias, and W.X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spaces, Journal of Inequalities and Applications, 2010(2010), Article ID 423231, 1–23. [17] T.Z. Xu, J.M. Rassias, and W.X. Xu, A generalized mixed additive-cubic functional equation, Journal of Computational Analysis and Applications, 13(7)(2011), 1273–1282. [18] T.Z. Xu, J.M. Rassias, and W.X. Xu, Stability of ageneral mixed additive-cubic equation in F -spaces, Journal of Computational Analysis and Applications, 14(6)(2012), 1026–1037. [19] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011. [20] T.Z. Xu, J.M. Rassias, and W.X. Xu, A generalized mixed type of quartic-cubic-quadratic-additive functional equations, Ukrainian Mathematical Journal, 63(3)(2011) 461–479. [21] Z. Gajda, On stability of additive mappings, International Journal of Mathematics and Mathematical Sciences, 14(1991) 431–434.

1119

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1120-1137, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Power harmonic operators and their applications in group decision making Jin Han Park, Jung Mi Park, Jong Jin Seo Department of Applied Mathematics, Pukyong National University, Busan 608-737, South Korea [email protected](J.H. Park), [email protected](J.J. Seo) Young Chel Kwun∗ Department of Mathematics, Dong-A University, Busan 608-714, South Korea [email protected]

Abstract The power average (PA) operator, power geometric (PG) operator, power ordered weighted average (POWA) operator and power ordered weighted geometric (POWG) operator are the nonlinear weighted aggregation tools whose weighting vectors depend on input arguments. In this paper, we develop a power harmonic (PH) operator and a power ordered weighted harmonic (POWH) operator, and study some properties of these operators. Then we extends the PH and POWH operators to uncertain environments, i.e, develop an uncertain PH (UPH) operator and its weighted form, and uncertain POWH (UPOWH) operator to aggregate the input arguments taking the form of interval numbers. Moreover, we utilize the weighted PH and POWH operators, respectively, to develop an approach to group decision making based on preference relations and utilize the weighted UPH and UPOWH operators, respectively, to develop an approach to group decision making based on uncertain preference relations. Finally, an example is used to illustrate the applicability of both the developed approaches. Keywords: Group decision making, power harmonic (PH) operator, power ordered weighted harmonic (POWH) operator, uncertain PH (UPH) operator, uncertain POWH (UPOWH) operator. 2000 AMS Subject Classifications: 90B50, 91B06, 90C29

1

Introduction

Information aggregation is an essential process of gathering relevant information from multiple sources by using a proper aggregation technique. Many techniques, such as the weighted average operator [1], the weighted geometric mean operator [2], harmonic mean operator [3], weighted harmonic mean ∗ Corresponding author. This study was supported by research funds from Dong-A University.

1120

J.H. Park, J.M. Park, J.J. Seo, Y.C. Kwun

(WHM) operator [3], ordered weighted average (OWA) operator [4], ordered weighted geometric operator [5, 6], weighted OWA operator [7], induced OWA operator [8], induced ordered weighted geometric operator [9], uncertain OWA operator [10], hybrid aggregation operator [11], linguistic aggregation operators [12, 14, 15, 16, 17, 18] and so on have been developed to aggregate data information. However, yet most of existing aggregation operators do not take into account the information about the relationship between the values being fused. Yager [19] introduced a tool to provide more versatility in the information aggregation process, i.e., developed a power average (PA) operator and a power OWA (POWA) operator. In some situations, however, these two operators are unsuitable to deal with the arguments taking the forms of multiplicative variables because of lack of knowledge, or data, and decision makers’ limited expertise related to the problem domain. So, based on this tool, Xu and Yager [20] developed additional new geometric aggregation operators, including the power geometric (PG) operator, weighted PG operator and power ordered weighted geometric (POWG) operator, whose weighting vectors depend upon the input arguments and allow values being aggregated to support and reinforce each other. In this paper, we will develop some new harmonic aggregation operators, including the power-harmonic (PH) operator, weighted PH operator, and power-ordered weighted harmonic (POWH) operator, and apply them to group decision making. In order to do this, the remainder of this paper is arranged in following sections. In Section 2, we first review some aggregation operators, including the PA, PG, POWA and POWG operators. Then, we develop a PH operator and its weighted form based on the PA (or PG) operator and the harmonic mean, and a POWH operator based on the POWA (POWG) operator and the harmonic mean, and investigate some of their properties, such as commutativity, idempotency and boundedness. The relationship among the PA, PG and PH operators and the relationship the POWA, POWG and POWH operators are also discussed. In Section 3, we utilize the weighted PH and POWH operators, respectively, to develop an approach to group decision making. In Section 4, we develop an uncertain PH (UPH) operator and its weighted form and uncertain POWH (UPOWH) operator to aggregate the input arguments, which are expressed in interval numbers, and also study the properties of these operators. In Section 5, we utilize the weighted UPH and UPOWH operators, respectively, to develop an approach to group decision making based on uncertain preference relations. Section 6 illustrates the presented approach with a practical example. Section 7 ends the paper with some concluding remarks.

2

Power harmonic operators

Yager [19] introduced a nonlinear weighted average aggregation operation tool, which is called PA operator, and can be defined as follows: Pn (1 + T (ai ))ai (1) PA(a1 , a2 , . . . , an ) = Pi=1 n i=1 (1 + T (ai )) where n X

T (ai ) =

Sup(ai , aj )

(2)

j=1,j6=i

and Sup(a, b) is the support for a from b, which satisfies the following three properties: 1) Sup(a, b) ∈ [0, 1], 2) Sup(a, b) = Sup(b, a), 3) Sup(a, b) ≥ Sup(x, y) if

1121

Power harmonic operators and their applications

|a − b| < |x − y|. Yager [19], based on the OWA operator [4] and PA operator, also defined a POWA operator as follows: POWA(a1 , a2 , . . . , an ) =

n X

ui aindex(i)

(3)

i=1

where index is an indexing function such that index(i) is the index of the ith largest of the arguments aj (j = 1, 2, . . . , n), and thus aindex(i) is the ith largest argument of aj (j = 1, 2, . . . , n), and ui (i = 1, 2, . . . , n) are a collection of weights such that  ui = g

Ri TV



 −g

Ri−1 TV

 , Ri =

i X

Vindex(j) , T V =

j=1

n X

Vindex(i) ,

i=1

Vindex(i) = 1 + T (aindex(i) )

(4)

where g : [0, 1] → [0, 1] is a basic unit-interval monotone (BUM) function having the following properties: 1) g(0) = 0, 2) g(1) = 1, 3) g(x) ≥ g(y) if x > y, and T (aindex(i) ) denotes the support of the ith largest argument by all the other arguments, i.e., T (aindex(i) ) =

n X

Sup(aindex(i) , aindex(j) )

(5)

j=1,j6=i

where Sup(aindex(i) , aindex(j) ) indicates the support of the jth largest argument for the ith largest argument. Based on the PA operator and the geometric mean, in the following, Xu and Yager [20] defined the PG operator: Pn1+T (ai ) n Y (1+T (ai )) PG(a1 , a2 , . . . , an ) = ai i=1

(6)

i=1

where aj (j = 1, 2, . . . , n) are a collection of arguments, and T (ai ) satisfies the condition (2). Based on the POWA operator and the geometric mean, Xu and Yager [20] also defined the power ordered weighted geometric (POWG) operator as follows: POWG(a1 , a2 , . . . , an ) =

n Y

i auindex(i)

(7)

i=1

which satisfies the conditions (4) and (5), and aindex(i) is the ith largest argument of aj (j = 1, 2, . . . , n). Based on PA operator and the harmonic mean, in the following, we define a PH operator: PH(a1 , a2 , . . . , an ) = Pn

1

i=1

Pn

1+T (ai ) (1+T (ai ))ai

i=1

1122

(8)

J.H. Park, J.M. Park, J.J. Seo, Y.C. Kwun

where aj (j = 1, 2, . . . , n) are a collection of arguments, and T (ai ) satisfies the condition (2). Clearly, the PH operator is a nonlinear weighted harmonic (ai ) aggregation operator, and the weight Pn1+T of the argument ai depends (1+T (a )) i

i=1

on all the input arguments aj (j = 1, 2, . . . , n) and allows the argument values to support each other in the harmonic aggregation process. Pn Lemma 2.1 [22, 23, 24] Letting xi > 0, αi > 0, i = 1, 2, . . . , n, and i=1 αi = 1, then 1 Pn

αi i=1 xi

≤

n Y i=1

(xi )αi ≤

n X

αi xi

(9)

i=1

with equality if and only if x1 = x2 = · · · = xn . By Lemma 2.1, we have the following theorem. Theorem 2.2 Assuming that aj (j = 1, 2, . . . , n) are a collection of arguments, then we have PH(a1 , a2 , . . . , an ) ≤ PG(a1 , a2 , . . . , an ) ≤ PA(a1 , a2 , . . . , an ).

(10)

Now, we discuss some properties of the PH operator. Theorem 2.3 Letting Sup(ai , aj ) = k, for all i 6= j, then n PH(a1 , a2 , . . . , an ) = Pn

1 i=1 ai

(11)

which indicates that when all supports are the same, the PG operator is simply the harmonic mean. Especially, if Sup(ai , aj ) = 0 for all i 6= j, i.e., all the supports are zero, then there is no support in the harmonic aggregation process, and in this case, by the condition (2), we have T (ai ) = 0, i = 1, 2, . . . , n, then 1 + T (ai ) 1 = , i = 1, 2, . . . , n n i=1 (1 + T (ai ))

Pn

(12)

and thus, by (8) and (12), it is clear that the PH operator reduces to the harmonic mean. Theorem 2.4 Let aj (j = 1, 2, . . . , n) be a collection of arguments, then we have the following properties. 1) (Commutativity): If (a01 , a02 , . . . , a0n ) is any permutation of (a1 , a2 , . . . , an ), then PH(a1 , a2 , . . . , an ) = PH(a01 , a02 , . . . , a0n ).

(13)

2) (Idempotency): If aj = a for all j, then PH(a1 , a2 , . . . , an ) = a.

(14)

3) (Boundedness): min ai ≤ PH(a1 , a2 , . . . , an ) ≤ max ai . i

i

1123

(15)

Power harmonic operators and their applications

In (8), all the objects that are being aggregated are of equal importance. In many situations, the weights of the objects should be taken into account, for example, in group decision making, the decision makers usually have different importance and thus, need to be assigned different weights. Suppose that each object that is being aggregated has a weight indicating its importance, then we define the weighted form of (8) as follows: 1

PHw (a1 , a2 , . . . , an ) = Pn

(16)

Pn wi (1+T 0 (a0i )) i=1 wi (1+T (ai ))ai i=1

where 0

T (ai ) =

n X

wj Sup(ai , aj )

(17)

j=1,j6=i

with the condition n X

wi ∈ [0, 1], i = 1, 2, . . . , n,

wi = 1.

(18)

i=1

Obviously, the weighted PH operator has the properties, as described in Theorem 2.2, as well as 2) and 3) of Theorem 2.4. However, Theorem 2.3 and 1) of Theorem 2.4 do not hold for the weighted PH operator. Based on the POWA operator and the harmonic mean, we define a power ordered weighted harmonic (POWH) operator as follows: POWH(a1 , a2 , . . . , an ) = Pn

1

(19)

ui i=1 aindex(i)

which satisfies the conditions (4) and (5), and aindex(i) is the ith largest argument of aj (j = 1, 2, . . . , n). Especially, if g(x) = x, then the POWH operator reduces to the PH operator, In fact, by (4), we have POWH(a1 , a2 , . . . , an ) = Pn

1

=

ui

i=1 aindex(i)

1

= Pn

i=1

= Pn

i=1

Ri TV

−

Ri−1 TV

g

Pn

1


Ri TV

i=1

Ri−1 TV




aindex(i)

1

=

aindex(i)

−g

Pn

Vindex(i) TV

i=1 aindex(i)

1 Pn

1+T (ai ) (1+T (ai ))ai

i=1

= PH(a1 , a2 , . . . , an ).

(20)

By Lemma 2.1, we the following theorem. Theorem 2.5 Assuming that aj (j = 1, 2, . . . , n) are a collection of arguments, then we have POWH(a1 , a2 , . . . , an ) ≤ POWG(a1 , a2 , . . . , an ) ≤ POWA(a1 , a2 , . . . , an ). (21)

1124

J.H. Park, J.M. Park, J.J. Seo, Y.C. Kwun

From Theorem 2.3 and (20), we have the following corollary. Corollary 2.6 Letting Sup(ai , aj ) = k for all i 6= j, and g(x) = x, then we have n POWH(a1 , a2 , . . . , an ) = Pn

(22)

1 i=1 ai

which indicates that when all supports are the same, the POWH operator is simply the harmonic mean. Similar to Theorem 2.4, we have the following theorem. Theorem 2.7 Let aj (j = 1, 2, . . . , n) be a collection of arguments, then we have the following properties. 1) (Commutativity): If (a01 , a02 , . . . , a0n ) is any permutation of (a1 , a2 , . . . , an ), then POWH(a1 , a2 , . . . , an ) = POWH(a01 , a02 , . . . , a0n ).

(23)

2) (Idempotency): If aj = a for all j, then POWH(a1 , a2 , . . . , an ) = a.

(24)

3) (Boundedness): min ai ≤ POWH(a1 , a2 , . . . , an ) ≤ max ai . i

(25)

i

From the above-mentioned theoretical analysis, the difference between the weighted PH and POWH operators is that the weighted PH operator emphasizes the importance of each argument, while the POWH operator weights the importance of the ordered position of each argument.

3

Approach to group decision making

Let us consider a group decision making problem. Let X = {x1 , x2 , . . . , xn } be a finite set of alternatives and let D = {d1 , d2 , . . . , dm } be a set of decision makers, whose vector is w = (w1 , w2 , . . . , wm )T , with wk ≥ 0, k = 1, 2, . . . , m, Pweight m and k=1 wk = 1. The decision maker dk compare each pair of alternatives (k) (xi , xj ) and provides his/her preference value aij over them and constructs the (k)

preference relation Ak on the set X, which is defined as a matrix Ak = (aij )n×n under the following condition: (k)

(k)

(k)

(k)

aij ≥ 0, aij + aji = 1, aii =

1 , for all i, j = 1, 2, . . . , n. 2

(26)

Then, we utilize the weighted PH operator to develop an approach to group decision making based on preference relations, which involves the following steps.

1125

Power harmonic operators and their applications

Approach I. Step 1: Calculate the supports (k)

(k)

(l)

Sup(aij , aij ) = 1 − Pm

(l)

|aij − aij | (k)

(l)

l=1,l6=k |aij − aij |

, l = 1, 2, . . . , m

(27)

which satisfy the support condition 1)-3) in Section 2. Pm (k) (l) (k) (l) Especially, if l=1,l6=k |aij − aij | = 0, then we stipulate Sup(aij , aij ) = 1. Step 2: Utilize the weights wk (k = 1, 2, . . . , m) of the decision makers dk (k) (k = 1, 2, . . . , m) to calculate the weighted support T 0 (aij ) of the preference (k)

(l)

value aij by the other preference values aij (l = 1, 2, . . . , m, and l 6= k) (k)

T 0 (aij ) =

m X

(k)

(l)

wl Sup(aij , aij )

(28)

l=1,l6=k (k)

and calculate the weights vij (k = 1, 2, . . . , m) associated with the preference (k)

values aij (k = 1, 2, . . . , m) (k)

vij

  (k) wk 1 + T 0 (aij )   , k = 1, 2, . . . , m =P m 0 (k) k=1 wk 1 + T (aij )

(29)

Pm (k) (k) where vij ≥ 0, k = 1, 2, . . . , m, and k=1 vij = 1. Step 3: Utilize the weighted PH operator to aggregate all the individual pref(k) erence relations Ak = (aij )n×n (k = 1, 2, . . . , m) into the collective preference relation A = (aij )n×n , where (1)

(2)

1

(m)

aij = PHw (aij , aij , . . . , aij ) =

(k)

vij k=1 a(k) ij

Pm

, i, j = 1, 2, . . . , n.

(30)

Step 4: Utilize the normalizing rank aggregation method (NRAM) [25] given by Pn

j=1 aij Pn i=1 j=1 aij

vi = Pn

, i = 1, 2, . . . , n

(31)

to derive the priority v = (v1 , v2 , . . . , vn )T of A = (aij )n×n , where vi > 0, Pvector n i = 1, 2, . . . , n, and i=1 vi = 1. Step 5: Rank all alternatives xi (i = 1, 2, . . . , n) in accordance with the priority weights vi (i = 1, 2, . . . , n). The more the wight vi , the better the alternative xi will be. In the case where the information about the weights of decision makers is unknown, then we utilize the POWH operator to develop an approach to group decision making based on preference relations, which can be described as follows.

1126

J.H. Park, J.M. Park, J.J. Seo, Y.C. Kwun

Approach II. Step 1: Calculate the supports Sup

index(k) index(l)
aij , aij

index(k) index(l) a − aij ij = 1 − Pm , l = 1, 2, . . . , m (32) index(k) index(l) − aij l=1,l6=k aij index(l)

which indicates the support of the lth largest preference value aij index(k) aij

for the

(s) aij

kth largest preference value of (s = 1, 2, . . . , m). Especially, if Pm index(k) index(l) index(k) index(l) − aij | = 0, then we stipulate Sup(aij , aij ) = 1. l=1,l6=k |aij It is necessary to point out that the support measure is a similarity measure, which can be used to measure the degree that a preference value provided by a decision maker is close to another one provided by other decision maker in index(k) index(l)
a group decision making problem. Thus, Sup aij , aij denotes the index(k)

similarity degree between the kth largest preference value aij largest preference value

index(k)

Step 2: Calculate the support T (aij index(k) aij

and the lth

index(l) aij .

by the other preference values index(k)

T (aij

)=

m X

(l) aij

) of the kth largest preference value

(l = 1, 2, . . . , m, and l 6= k)

index(k)

Sup(aij

index(l)

, aij

)

(33)

l=1,l6=k (k)

and by (4), calculate the weight uij associated with the kth largest preference index(k)

value aij

, where (k)

(k) uij

=g

T Vij =

Rij T Vij

m X

(k−1)

!

Rij T Vij

−g

index(l)

Vij

index(l)

, Vij

! (k)

, Rij =

k X

index(l)

Vij

,

l=1 index(l)

= 1 + T (aij

)

(34)

l=1

Pm (k) (k) where uij ≥ 0, k = 1, 2, . . . , m, and k=1 uij = 1, and g is the BUM function described in Section 2. Step 3: Utilize the POWH operator to aggregate all the individual preference (k) relations Ak = (aij )n×n (k = 1, 2, . . . , m) into the collective preference relation A = (aij )n×n , where (1)

(2)

1

(m)

aij = POWH(aij , aij , . . . , aij ) =

Pm

(k)

uij

, i, j = 1, 2, . . . , n.

(35)

k=1 aindex(k) ij

Step 4: For this step, see Approach I. Step 5: For this step, see Approach I. In the above-mentioned two approaches, Approach I considers the situations where the weighted PH operator to aggregate all the individual preference relations into the collective preference relation and then the NRAM method to

1127

Power harmonic operators and their applications

derive its priority vector, and using this, we can rank and select the given alternatives. While Approach II considers the situations where the information about the weights of decision makers is unknown and utilizes the POWH operator to aggregate all the individual preference relations into collective preference relation, then it also uses the NRAM method to find the final decision result.

4

Uncertain power harmonic operators

In this section, we consider the situations where the input arguments cannot be expressed in exact numerical values, but value range (i.e., interval numbers) can be obtained. We first review some operational laws, which are related to interval numbers [26, 27]. Let a ˜ = [aL , aU ] and ˜b = [bL , bU ] be two interval numbers, where aU ≥ aL > U 0 and b ≥ bL > 0, then we have the following operational laws. 1) a ˜ + ˜b = [aL , aU ] + [bL , bU ] = [aL + bL , aU + bU ]. 2) a ˜˜b = [aL , aU ] · [bL , bU ] = [al bL , aU , bU ]. 3) λ˜ a = λ[aL , aU ] = [λaL , λaU ], where λ > 0. 1 4) a˜ = [aL1,aU ] = [ a1U , a1L ]. L

U

L

U

,a ] 5) a˜˜b = [a = [ abU , abL ]. [bL ,bU ] In order to rank interval numbers, we now introduce a possibility degree formula [28] for the comparison between the interval numbers a ˜ = [aL , aU ] and ˜b = [bL , bU ]     aU − bL ˜ p(˜ a ≥ b) = min max ,0 ,1 (36) aU − aL + bU − bL

where p(˜ a ≥ ˜b) is called the possibility degree of a ˜ ≥ ˜b, which satisfies 0 ≤ p(˜ a ≥ ˜b) ≤ 1, p(˜ a ≥ ˜b) + p(˜b ≥ a ˜) = 1, p(˜ a≥a ˜) = 0.5.

(37)

U Let a ˜j = [aL j , aj ] (j = 1, 2, . . . , n) be a collection of interval numbers, then based on the previous operational laws of interval numbers, we extend the PH operator to uncertain environments and define an UPH operator as follows:

UPH(˜ a1 , a ˜2 , . . . , a ˜n ) = Pn

1

i=1

Pn

1+T (˜ ai ) (1+T (˜ ai ))˜ ai

(38)

i=1

where T (˜ ai ) =

n X

Sup(˜ ai , a ˜j )

(39)

j=1,j6=i

and Sup(˜ a, ˜b) is the support for a ˜ from ˜b, which satisfies the following three properties: 1) Sup(˜ a, ˜b) ∈ [0, 1], 2) Sup(˜ a, ˜b) = Sup(˜b, a ˜), 3) Sup(˜ a, ˜b) ≥ Sup(˜ x, y˜) if ˜ d(˜ a, b) < d(˜ x, y˜), where d is a distance measure. Similar to the PH operator, the UPH operator has the following properties.

1128

J.H. Park, J.M. Park, J.J. Seo, Y.C. Kwun

Theorem 4.1 Letting Sup(˜ ai , a ˜j ) = k for all i 6= j, then n UPH(˜ a1 , a ˜2 , . . . , a ˜n ) = Pn 1

(40)

i=1 a ˜i

which indicates that when all the supports are the same, the UPH operator is simply the uncertain harmonic mean. Theorem 4.2 Let a ˜j (j = 1, 2, . . . , n) be a collection of interval numbers, then we have the following properties. 1) (Commutativity): If (˜ a01 , a ˜02 , . . . , a ˜0n ) is any permutation of (˜ a1 , a ˜2 , . . . , a ˜n ), then UPH(˜ a1 , a ˜2 , . . . , a ˜n ) = UPH(˜ a01 , a ˜02 , . . . , a ˜0n ).

(41)

2) (Idempotency): If a ˜j = a ˜ for all j, then UPH(˜ a1 , a ˜2 , . . . , a ˜n ) = a ˜.

(42)

3) (Boundedness): min a ˜i ≤ UPH(˜ a1 , a ˜2 , . . . , a ˜n ) ≤ max a ˜i . i

i

(43)

If the weights of the objects are taken into account, then we define the weighted form of (38) as follows: UPHw (˜ a1 , a ˜2 , . . . , a ˜n ) = Pn

i=1

1 Pn wi (1+T 0 (˜a0i )) i=1

(44)

wi (1+T (˜ ai ))˜ ai

where T 0 (˜ ai ) =

n X

wj Sup(˜ ai , a ˜j )

(45)

j=1,j6=i

with the condition wi ∈ [0, 1], i = 1, 2, . . . , n,

n X

wi = 1.

(46)

i=1

Obviously, the weighted UPH operator has the properties of 2) and 3) in Theorem 4.2. However, Theorem 4.1 and 1) of Theorem 4.2 do not hold for the weighted UPH operator. Based on the POWH operator and the possibility degree formula, we define a UPOWH operator as follows: UPOWH(˜ a1 , a ˜2 , . . . , a ˜n ) = Pn

1

ui i=1 a ˜index(i)

(47)

where a ˜index(i) is the ith largest interval number of a ˜j (j = 1, 2, . . . , n), and     i X Ri−1 Ri −g , Ri = Vindex(j) , ui = g TV TV j=1 TV =

n X

Vindex(i) , Vindex(j) = 1 + T (˜ aindex(i) )

i=1

1129

(48)

Power harmonic operators and their applications

and T (˜ aindex(i) ) denotes the support of the ith largest interval number by all the other interval numbers, i.e., T (˜ aindex(i) ) =

n X

Sup(˜ aindex(i) , a ˜index(j) )

(49)

j=1

where Sup(˜ aindex(i) , a ˜index(j) ) indicates the support of the jth largest interval number for the ith largest interval number (here, we can use the possibility degree formula (36) to rank interval numbers). Especially, if g(x) = x, then the UPOWH operator reduces to the UPH operator. From Theorem 4.1, we have the following corollary. Corollary 4.3 Letting Sup(˜ aindex(i) , a ˜index(j) ) = k for all i 6= j, and g(x) = x, then n UPOWH(˜ a1 , a ˜2 , . . . , a ˜n ) = Pn

1 i=1 a ˜i

(50)

which indicates that when the supports are the same, the UPOWH operator is simply the uncertain harmonic mean. Similar to Theorem 4.2, we have the following theorem. Theorem 4.4 Let a ˜j (j = 1, 2, . . . , n) be a collection of interval numbers, then we have the following properties. a1 , a ˜2 , . . . , a ˜n ), 1) (Commutativity): If (˜ a01 , a ˜02 , . . . , a ˜0n ) is any permutation of (˜ then UPOWH(˜ a1 , a ˜2 , . . . , a ˜n ) = UPOWH(˜ a01 , a ˜02 , . . . , a ˜0n ).

(51)

2) (Idempotency): If a ˜j = a ˜ for all j, then UPOWH(˜ a1 , a ˜2 , . . . , a ˜n ) = a ˜.

(52)

3) (Boundedness): min a ˜i ≤ UPOWH(˜ a1 , a ˜2 , . . . , a ˜n ) ≤ max a ˜i . i

5

i

(53)

Approach to group decision making based on uncertain preference relations

As mentioned in Section 3, in this section, we will apply the weighted UPH and UPOWH operators to group decision making based on uncertain preference relations. Let X = {x1 , x2 , . . . , xn } be a finite set of alternatives and let D = {d1 , d2 , . . . , dm } be a set of decision makers, whose weight Pm vector is w = (w1 , w2 , . . . , wm )T , with wk ≥ 0, k = 1, 2, . . . , m, and k=1 wk = 1. The decision maker dk compare each pair of alternatives (xi , xj ) and provides (k) L(k) U (k) his/her preference value range a ˜ij = [aij , aij ] over them and constructs

1130

J.H. Park, J.M. Park, J.J. Seo, Y.C. Kwun the uncertain preference relation A˜k on the set X, which is defined as a matrix (k) A˜k = (˜ aij )n×n under the following condition: U (k)

aij

L(k)

L(k)

≥ aij

U (k)

L(k)

U (k)

> 0, aij + aji = 1, aji + aij = 1, 1 L(k) U (k) (54) aii = aii = , i, j = 1, 2, . . . , n. 2 Then, we utilize the weighted UPH operator to develop an approach to group decision making based on uncertain preference relations, which involves the following steps. Approach III. Step 1: Calculate the supports (k) (l)
d a ˜ij , a ˜ij (k) (l) , l = 1, 2, . . . , m (55) Sup(˜ aij , a ˜ij ) = 1 − Pm (k) (l)
˜ij , a ˜ij l=1,l6=k d a which satisfy the support condition 1)-3) in Section 4. Here, without loss of generality, we let U (l) 1 L(l) U (k)
(k) (l)
L(k) (56) d a ˜ij , a ˜ij = aij − aij + aij − aij . 2 Pm (k) (l) (k) (l) Especially, if l=1,l6=k d(˜ aij , a ˜ij ) = 0, then we stipulate Sup(˜ aij , a ˜ij ) = 1. Step 2: Utilize the weights wk (k = 1, 2, . . . , m) of the decision makers (k) dk (k = 1, 2, . . . , m) to calculate the weighted support T 0 (˜ aij ) of the uncer(k)

(l)

tain preference value a ˜ij by the other uncertain preference values a ˜ij (l = 1, 2, . . . , m, and l 6= k) T

0

(k) (˜ aij )

=

m X

(k)

(l)

wl Sup(˜ aij , a ˜ij )

(57)

l=1,l6=k (k)

and calculate the weights v˙ ij (k = 1, 2, . . . , m) associated with the uncertain (k)

preference values a ˜ij (k = 1, 2, . . . , m)   (k) wk 1 + T 0 (˜ aij ) (k)   , k = 1, 2, . . . , m v˙ ij = P m 0 a(k) ) ij k=1 wk 1 + T (˜

(58)

Pm (k) (k) where v˙ ij ≥ 0, k = 1, 2, . . . , m, and k=1 v˙ ij = 1. Step 3: Utilize the weighted UPH operator to aggregate all the individual (k) uncertain preference relations A˜k = (˜ aij )n×n (k = 1, 2, . . . , m) into the collective uncertain preference relation A˜ = (˜ aij )n×n , where (1)

(2)

(m)

a ˜ij = [alij , aU aij , a ˜ij , . . . , a ˜ij ) ij ] = UPHw (˜ 1 = , i, j = 1, 2, . . . , n. (k) Pm v˙ ij (k) k=1 a ˜ij

1131

(59)

Power harmonic operators and their applications

Step 4: Utilize the uncertain NRAM (UNRAM) given by Pn ˜ij j=1 a v˜i = Pn Pn , i = 1, 2, . . . , n ˜ij i=1 j=1 a

(60)

to derive the uncertain priority vector v˜ = (˜ v1 , v˜2 , . . . , v˜n )T of A˜ = (˜ aij )n×n . Step 5: Compare each pair of the uncertain priority weights v˜i (i = 1, 2, . . . , n) by using the possibility degree formula (36) and construct a possibility degree matrix P = (pij )n×n , where pij = p(˜ vi ≥ v˜j ), i, j = 1, 2, . . . , n, which satisfy pij ≥ 0 pij + pji = 1, pii = 0.5, i, j = 1, 2, . . . , n. Summing all the elements in each line of the matrix P , we get pi =

n X

pij , i = 1, 2, . . . , n.

(61)

j=1

Then we rank the uncertain priority weights v˜i (i = 1, 2, . . . , n) in descending order in accordance with pi (i = 1, 2, . . . , n). Step 6: Rank all alternatives xi (i = 1, 2, . . . , n) in accordance with the descending order of the uncertain priority weights v˜i (i = 1, 2, . . . , n). In the case where the information about the weights of decision makers is unknown, then we utilize the UPOWH operator to develop an approach to group decision making based on uncertain preference relations, which can be described as follows. Approach IV. Step 1: Calculate the supports index(k) index(l)
d a ˜ij ,a ˜ij index(k) index(l)
, l = 1, 2, . . . , m (62) Sup a ˜ij ,a ˜ij = 1 − Pm index(k) index(l)
˜ij ,a ˜ij l=1,l6=k d a index(l)

which indicates the support of lth largest uncertain preference value a ˜ij index(k)

(s)

for the kth largest uncertain preference value a ˜ij of a ˜ij (s = 1, 2, . . . , m) (here, we can use Step 5 of Approach III to rank uncertain preference values). Pm index(k) index(l) index(k) Especially, if l=1,l6=k d(˜ aij ,a ˜ij ) = 0, then we stipulate Sup(˜ aij , index(l)

a ˜ij

) = 1. index(k)

Step 2: Calculate the support T (˜ aij ence value and l 6= k)

index(k) a ˜ij

) of the kth largest uncertain prefer(l)

by the other uncertain preference values a ˜ij (l = 1, 2, . . . , m,

index(k)

T (˜ aij

)=

m X

index(k)

Sup(˜ aij

index(l)

,a ˜ij

)

(63)

l=1,l6=k (k)

and by (48), calculate the weight u˙ ij associated with the kth largest uncertain index(k)

preference value a ˜ij (k) u˙ ij

=g

, where ! (k) R˙ ij −g T Vij0

(k−1) R˙ ij T Vij0

!

1132

(k) , R˙ ij =

k X l=1

index(l)

Vij

,

J.H. Park, J.M. Park, J.J. Seo, Y.C. Kwun

T Vij0 =

m X

index(l)

Vij

index(l)

, Vij

index(l)

= 1 + T (˜ aij

)

(64)

l=1

Pm (k) (k) where u˙ ij ≥ 0, k = 1, 2, . . . , m, and k=1 u˙ ij = 1, and g is the BUM function described in Section 2. Step 3: Utilize the UPOWH operator to aggregate all the individual uncer(k) tain preference relations A˜k = (˜ aij )n×n (k = 1, 2, . . . , m) into the collective uncertain preference relation A˜ = (˜ aij )n×n , where (1)

(1)

(m)

U a ˜ij = [aL aij , a ˜ij , . . . , a ˜ij ) ij , aij ] = UPOWH(˜ 1 = , i, j = 1, 2, . . . , n. (k) Pm u˙ ij

(65)

index(k) k=1 a ˜ij

Step 4: For this step, see Approach III. Step 5: For this step, see Approach III. Step 6: For this step, see Approach III.

6

Illustrative example

Four university students share a house, where they intend to have broadband Internet connection installed (adapted from [20, 29]). There are four options available to choose from, which are provided by three Internet service providers. 1) Option 1 (x1 ): 1 Mbps broadband; 2) Option 2 (x2 ): 2 Mbps broadband; 3) Option 3 (x3 ): 3 Mbps broadband; 4) Option 4 (x4 ): 8 Mbps broadband. Since the Internet service and its monthly bill will be shared among the four students dk (k = 1, 2, 3, 4) (whose weight vector w = (0.3, 0.3, 0.2, 0.2)T ), they decide to perform a group decision analysis. Suppose that the students reveal their preference relations for the options independently and anonymously and construct the following preference relations, respectively:     0.5 0.8 0.7 0.4 0.5 0.4 0.5 0.8  0.2 0.5 0.6 0.6   0.6 0.5 0.8 0.9  , A2 =  A1 =  0.3 0.4 0.5 0.8  0.5 0.2 0.5 0.6  0.6 0.4 0.2 0.5 0.2 0.1 0.4 0.5     0.5 0.4 0.7 0.6 0.5 0.7 0.7 0.5  0.6 0.5 0.3 0.7   0.3 0.5 0.4 0.4  A3 =  , A4 =  . 0.3 0.7 0.5 0.6  0.3 0.6 0.5 0.9  0.4 0.3 0.4 0.5 0.5 0.6 0.1 0.5 Since the weights of students are given, we then utilize Approach I to find the decision result. (k) (l) We first utilize (27) to calculate the supports Sup(aij , aij ) (i, j, k, l = (k)

(l)

1, 2, 3, 4, k 6= l), which are contained in the matrices S kl = (S kl (aij , aij ))4×4

1133

Power harmonic operators and their applications

(k = 1, 2, 3, 4), respectively  1 0.429 0.667  0.429 1 0.818 12 S = 0.667 0.818 1 0.556 0.700 0.600  1 0.571 0.667  0.571 1 0.636 14 S = 0.667 0.636 1 0.667 0.500 0.400  1 0.556 1  0.556 1 0.571 S 23 =  1 0.571 1 0.714 0.833 0.600  1 1 0  1 1 0.444 31 S = 0 0.444 1 0.600 0.667 1  1 0.571 1  0.571 1 0.889 S 34 =  1 0.889 1 0.800 0.500 0.400  1 0.857 1  0.857 1 0.714 42 S = 1 0.714 1 0.800 0.800 0.857

 0.556 0.700  , 0.600  1  0.667 0.500  , 0.400  1  0.714 0.833  , 0.600  1  0.600 0.667  , 1  1  0.800 0.500  , 0.400  1  0.800 0.800  , 0.857  1



1  1 13 S = 0.667 0.778  1  0.556 21 S = 0 0.429  1  0.889 S 24 =  1 0.857  1  0.429 32 S = 1 0.600  1  0.571 S 41 =  0 0.400  1  0.571 43 S = 1 0.800

 1 0.667 0.778 1 0.545 0.800  0.545 1 1  0.800 1 1  0.556 0 0.429 1 0.714 0.500  0.714 1 0.600  0.500 0.600 1  0.889 1 0.857 1 0.714 0.667  0.714 1 0.800  0.667 0.800 1  0.429 1 0.600 1 0.667 0.833  0.667 1 0.600  0.833 0.600 1  0.571 0 0.400 1 0.429 0.500  0.429 1 0.571  0.500 0.571 1  0.571 1 0.800 1 0.857 0.700  . 0.857 1 0.571  0.70 0.571 1

Then, we utilize the weight vector w = (0.3, 0.3, 0.2, 0.2)T of the students (k) dk (k = 1, 2, 3, 4) and (28) to calculate the weighted supports T 0 (aij ) (i, j, k = (k)

1, 2, 3, 4) of the preference values aij (i, j, k = 1, 2, 3, 4), which are contained in (k)

the matrices Tk0 = (T 0 (aij ))4×4 (k = 1, 2, 3, 4), respectively    0.700 0.456 0.700 0.443 0.467 0.456  0.456 0.700  0.443 0.700 0.482 0.470  0 , T = T10 =   0.400 0.471 2 0.467 0.482 0.700 0.460  0.456 0.470 0.460 0.700 0.443 0.450    0.800 0.543 0.800 0.543 0.500 0.520  0.543 0.800  0.543 0.800 0.511 0.550  0 0 , T4 =  T3 =  0.500 0.514 0.500 0.511 0.800 0.560  0.520 0.530 0.520 0.550 0.560 0.800

 0.400 0.443 0.471 0.450  0.700 0.460  0.460 0.700  0.500 0.520 0.514 0.530  0.800 0.543  0.543 0.800

(k)

and then utilize (29) to calculate the weights vij (i, j, k = 1, 2, 3, 4) associated with the preference values matrices Vk =  0.293  0.291 V1 =  0.301 0.295

(k) aij

(i, j, k = 1, 2, 3, 4), which are contained in the

(k) (vij )4×4

0.291 0.293 0.298 0.295

(k = 1, 2, 3, 4), respectively    0.301 0.296 0.293 0.294 0.288 0.293  0.293 0.293 0.296 0.292  0.298 0.295  , V2 =  0.293 0.293  0.287 0.296 0.293 0.293  0.293 0.293 0.293 0.292 0.293 0.293

1134

J.H. Park, J.M. Park, J.J. Seo, Y.C. Kwun 

   0.207 0.207 0.205 0.206 0.207 0.208 0.206 0.206  0.208 0.207 0.203 0.208   0.208 0.207 0.203 0.205  V3 =  , V4 =  . 0.206 0.203 0.207 0.208  0.206 0.203 0.207 0.206  0.206 0.208 0.208 0.207 0.206 0.205 0.206 0.207 Based on this, we utilize the weighted PH operator (30) to aggregate all the (k) individual preference relations Ak = (aij )4×4 (k = 1, 2, 3, 4) into the collective preference relation   0.5000 0.5237 0.6248 0.5383  0.3344 0.5000 0.4878 0.6157  A= . 0.3411 0.3499 0.5000 0.6992  0.3460 0.2121 0.2093 0.5000 After this, we utilize the NRAM (31) to derive the priority vector of A v = (0.3003, 0.2661, 0.2596, 0.1740)T . Using this, we get the ranking of the options as follows: x1  x2  x3  x4 .

7

Conclusions

In this paper, based on the PA operator, we have developed several new nonlinear weighted harmonic aggregation operators including the PH operator, weighted PH operator, POWH operator, UPH operator, weighted UPH operator and UPOWH operator. We have studied some desired properties of the developed operators, such as commutativity, idempotency and boundedness. The fundamental idea of these operators is that the weight of each input argument depends on the other input arguments and allows argument values to support each other in the harmonic aggregation process. Moreover, we have applied the developed operators to aggregate all individual preference (or uncertain preference) relations into collective preference (or uncertain preference) under various group decision making environment and then developed some group decision making approaches. The merit of the developed approaches is that they can take all the decision arguments and their relationships into account. In the future, we will develop several applications of the developed aggregation operators in other fields, such as pattern recognition, supply chain management and image processing.

References [1] J.C. Harsanyi, Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility, J. Polit. Econ. 63 (1955) 309-321. [2] J. Acz´el and T.L. Saaty, Procedures for synthesizing ratio judgements, J. Math. Psychol. 27 (1983) 93-102. [3] P.S. Bullen, D.S. Mitrinovi and P.M. Vasi, Means and Their Inequalities, Dordrecht, The Netherlands: Reidel, 1988.

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Power harmonic operators and their applications

[4] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybern. 18 (1988) 183190. [5] F. Chiclana, F. Herrera and E. Herrera-Viedma, Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations, Fuzzy Sets Syst. 122 (2001) 277-291. [6] Z.S. Xu and Q.L. Da, The ordered weighted geometric averaging operators, Int. J. Intell. Syst. 17 (2002) 709-716. [7] V. Torra, The weighted OWA operators, Int. J. Intell. Syst. 12 (1997) 153-166. [8] R.R. Yager and D.P. Filev, Induced ordered weighted averaging operators, IEEE Trans. Syst. Man Cybern. 29 (1999) 141-150. [9] Z.S. Xu and Q.L. Da, An overview of operators aggregating information, Int. J. Intell. Syst. 18 (2003) 953-969. [10] Z.S. Xu and Q.L. Da, The uncertain OWA operators, Int. J. Intell. Syst. 17 (2002) 569-575. [11] Z.S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications, Beijing, China: Tsinghua Univ. Press, 2004. [12] R.R. Yager, Generalized OWA aggregation operator, Fuzzy Optim. Decision Making 3 (2004) 93-107. [13] R.R. Yager, An approach to ordinal decision making, Int. J. Approx. Reasoning 12 (1995) 237-261. [14] F. Herrera, E. Herrera-Viedma and J.L. Verdegay, A sequential selection process in group decision making with a linguistic assessment approach, Inf. Sci. 85 (1995) 223-239. [15] F. Herrera and L. Mart´ınez, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Trans. Fuzzy Syst. 8 (2000) 746-752. [16] Z.S. Xu, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Inf. Sci. 166 (2004) 19-30. [17] J.H. Park, M.G. Gwak and Y.C. Kwun, Linguistic harmonic mean operators and their applications to group decision making, Int. J. Adv. Manuf. Technol. 57 (2011) 411-419. [18] J.H. Park, M.G. Gwak and Y.C. Kwun, Uncertain linguistic harmonic mean operators and their applications to multiple attribute group decision making, Computing 93 (2011) 47-64. [19] R.R. Yager, The power average operator, IEEE Trans. Syst. Man Cybern. A. Syst. Humans 31 (2001) 724-731. [20] Z.S. Xu and R.R. Yager, Power-geometric operators and their use in group decision making, IEEE Trans. Fuzzy Syst. 18 (2010) 94-105.

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[21] Y. Xu, J.M. Merig´o and H. Wang, Linguistic power aggregation operators and their application to multiple attribute group decision making, Appl. Math. Modelling (2011), doi: 10.1016/j.amp.2011.12.002. ¨ [22] O. Holder, Uber einen Mittelwertsatz, G¨ottingen Nachrichten (1889) 3847. [23] J.L. Jensen, Sur les fonctions convexes et les in´egualit´es entre les valeurs moyennes, Acta Math. 30 (1906) 175-193. [24] Wikipedia, http://en.wikipedia.org/wiki/Generalizedmean. [25] Z.S. Xu, Q.L. Da and L.H. Liu, Normalizing rank aggregation method for priority of a fuzzy preference relation and its effectiveness, Int. J. Approx. Reasoning 50 (2009) 1287-1297. [26] R.N. Xu and X.Y. Zhai, Extensions of the analytic hierarchy process in fuzzy environment, Fuzzy Sets Syst. 52 (1992) 251-257. [27] G. Bojadziev and M. Bojadziev, Fuzzy Sets, Fuzzy Logic, Applications, World Scientific, Singapore, 1995. [28] G. Faccinetti, R.G. Ricci and S. Muzzioli, Note on ranking fuzzy triangular numbers, Int. J. Intell. Syst. 13 (1998) 613-622. [29] Y.M. Wang and C. Parkan, Optimal aggregation of fuzzy preference relations with an application to broadband internet service selection, Eur. J. Oper. Res. 187 (2008) 1476-1486.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1138-1149, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

MULTIPLICATIONAL COMBINATIONS AND A GENERAL SCHEME OF SINGLE-STEP ITERATIVE METHODS FOR MULTIPLE ROOTS SIYUL LEE1,∗ AND HYEONGMIN CHOE1 1

Seoul Science High School, Seoul 110-530, Republic of Korea

A BSTRACT. In this paper, a general form of single-step iterative methods for multiple roots of nonlinear equations is derived under a number of assumptions of optimization. Definition of multiplicational combinations and their properties are used upon the optimization procedure. Among all, we construct a family of iterative methods with nine parameters and simplest terms, and we obtain 23 simplest iterative methods within the family, those including all existing methods of single-step scheme. Numerical comparisons between the methods also present interesting and noteworthy results.

1. I NTRODUCTION Solving nonlinear equations is one of the most basic problems of mathematics, yet it is often greatly complicated. Therefore, to develop methods to obtain roots of a nonlinear equation f (x) = 0 has become crucial, especially with advance of computational technology. Newton’s method, defined by f (xn ) xn+1 = xn − 0 (1) f (xn ) makes use of an approximated root to obtain a new approximation with less error. This classical method, however, ceases to be efficient when a multiple root of f is to be obtained. In such cases, one may solve a nonlinear equation u(x) = 0 where u(x) = f (x)/f 0 (x) instead of f (x) = 0, since u(x) has multiple roots of f (x) as its simple roots, see [1, p.126]. When multiplicity m of the desired root of f is known, one may use the modified Newton’s method, xn+1 = xn − m

fn fn0

(2)

(i)

where fn denotes f (i) (xn ) instead of the original Newton’s method (1). The modified Newton’s method for multiple roots is quadratically convergent. More advanced iterative algorithms with cubic or higher order of convergence are actively being developed, in Keywords: Newton’s method, Iterative methods, Single-step methods, Nonlinear equations, Cubic order, Multiple roots. 2010 MSC: 65H05. E-mail addresses: [email protected](S. Lee), [email protected](H. Choe).

1138

S. LEE AND H. CHOE order to improve the computational efficiency. One widely known cubically convergent example is Halley’s method(HM), namely, xn+1 = xn −

fn m+1 0 2m fn

−

fn fn00 2fn0

,

(3)

see [2]. The Euler-Chebyshev method(ECM), xn+1 = xn −

m(3 − m) fn m2 fn 2 fn00 − 2 fn0 2 fn0 3

(4)

is also of cubic convergence, see [1]. Osada in [3] and Chun and Neta in [4], developed other cubically convergent iterative methods, 1 fn 1 f0 xn+1 = xn − m(m + 1) 0 + (m − 1)2 n00 , 2 fn 2 fn

(5)

and xn+1 = xn −

2m2 fn 2 fn00 , m(3 − m)fn fn0 fn00 + (m − 1)2 fn0 3

(6)

OM and CNM in short, respectively. Biazar and Ghanbari in [5] assumed a form of Newton-like methods with four parameters as follows: Afn fn0 2 fn00 + Bfn0 4 + Cfn 2 fn00 2 . (7) xn+1 = xn − fn0 3 fn00 + Dfn fn0 fn00 2 From the error equation of the assumed method, parameters are controlled to make the method cubically convergent. A new method thereby introduced is xn+1 = xn −

fn0 m+3 00 2(m−1) fn

−

m(m+1) fn fn00 2 2(m−1)2 fn0 2

,

(8)

which is to be referred to as Biazar and Ghanbari’s method(BGM). In Section 2.1, we start with basic but essential definitions. We also define multiplicational combinations with restricted derivatives of f , and write a general expression for them. Then, a Newton-like method with nine parameters is constructed under a number of assumptions. In Section 2.2, we derive the error equation of the method and solve for parameters to obtain a cubic convergence. In such a way, we derive a number of Newton-like methods, some of which are introduced previously. Section 3 contains numerical comparisons between the methods introduced or derived. 2. D EVELOPMENT OF METHODS 2.1. Construction of the scheme. Before we begin, the order of convergence and multiple roots must be defined clearly.

1139

SINGLE-STEP ITERATIVE METHODS FOR MULTIPLE ROOTS Definition 1. (See [6]) With α a real number, and n a non-negative integer, if a real sequence {xn } converges to α and for n large enough there exist constants c ≥ 0 and p ≥ 0 that satisfy | xn+1 − α |≤ c | xn − α |p ,

(9)

then the maximum of p is said to be an order of convergence of {xn } to α. Definition 2. (See [7, p.79]) A root α of an equation f (x) = 0 is said to have the multiplicity m if and only if f (α) = 0, f 0 (α) = 0, f 00 (α) = 0, . . . , f (m−1) (α) = 0 and f (m) (α) 6= 0. In this case, f can be written as f (x) = (x − α)m g(x),

(10)

with g(α) 6= 0. Now, as a preparation for the rest of the section, we define a new concept of multiplicational combinations. Definition 3. Let f be a two times differentiable function. With any integers a,b, and c such that a + b + c = 0, Fk,−c = f a f 0b f 00c

(11)

is a multiplicational combination of f , f 0 , and f 00 , with differential order k=b+2c. Multiplicational combinations acquire an important property that will be used importantly for the discussion followed. Theorem 1. If Fk,−c is a multiplicational combination of f , f 0 , and f 00 , with differential order k, Fk,−c = Fk,s = (

f 0 k f 02 s ) ( 00 ) , f ff

(12)

for some integer s = −c. The converse is also true. Proof. Let Fk,−c = f a f 0b f 00c for integers a, b, and c. By Definition 3, a+b+c = 0 and b+2c = k. Solving the system gives a = −k + c and b = k − 2c. Thus Fk,−c = f −k+c f 0k−2c f 00c = (

f 0 k f 02 −c ) ( 00 ) . f ff

(13)

Letting s = −c, we have Fk,s = ( 0

f 0 k f 02 s ) ( 00 ) . f ff

(14)

02

If u = ( ff )k ( fff 00 )s , u = f −k−s f 0k+2s f 00−s = Fk,s ,

(15)

and thus is a multiplicational combination of f , f 0 , and f 00 , with differential order k. This completes the proof. 

1140

S. LEE AND H. CHOE Single-step iterative methods are generally expressed as xn+1 = xn − g(xn ), where g(xn ) denotes an iteration function of xn . For computational efficiency, we only consider g(xn )’s that consist of fn , fn0 , fn00 and a finite number of fundamental arithmetic operations between them. With the assumption, g(xn ) can be written as follows: P a 0 b 00 c a,b,c fn fn fn θ(a, b, c) , (16) g(xn ) = P a 0 b 00 c a,b,c fn fn fn φ(a, b, c) where θ and φ symbolize the linear combination of fn a fn0 b fn00 c ’s in the numerator and the denominator, respectively. It is reasonable to assume that all terms included in the sum are required to have the same arithmetic order, namely, a + b + c. Thus, by an appropriate division, both the numerator and the denominator each reduces to a linear combination of multiplicational combinations. Then by Theorem 1, P f 0 k f 02 s k,s ( f ) ( f f 00 ) θ(k, s) . (17) g(xn ) = P f 0 k f 02 s k,s ( f ) ( f f 00 ) φ(k, s) Here, for optimization(see Remark 1), we assume that the numerator and the denominator each consists of multiplicational combinations of uniform differential order. That is, for integers k1 and k2 , P 0 02 ( ff )k1 s ( fff 00 )s θ(s) g(xn ) = f 0 . (18) P 02 ( f )k2 s ( fff 00 )s φ(s) Theorem 2. For an iteration function defined by (18), if xn+1 = xn −g(xn ) is cubically convergent to α, the root of f (x) = 0 with multiplicity m, it is required that k1 − k2 = −1. Proof. Taylor’s expansion for f about a multiple root α of f (x) = 0 with multiplicity m gives m+1 f (xn ) = f (m) (α)(c0 em + c2 em+2 + · · · ), n + c1 en n 0

f (xn ) = f

(m)

(α){mc0 em−1 n

+ (m +

1)c1 em n

+ (m +

(19) 2)c2 em+1 n

+ · · · },

(20)

and f 00 (xn ) = f (m) (α){m(m − 1)c0 em−2 + (m + 1)mc1 em−1 + (m + 2)(m + 1)c2 em n n n,

(21)

where cn ’s and en are defined as follows: cn =

1 f (m+n) (α) , en = xn − α. (m + n)! f (m) (α)

(22)

Then, fn 1 1 c1 2  m + 1 c21 2 c2  3 = e − e + − e + ··· , n fn0 m m2 c0 n m3 c20 m2 c0 n  3m2 + 1 c2 fn02 m 2 c1 6 c2  2 1 = − e + − e + ··· , n fn fn00 m − 1 (m − 1)2 c0 m(m − 1)3 c20 (m − 1)2 c0 n and thus g(xn ) = en + O(e2n )


k2 −k1


1 + O(en ) = ekn2 −k1 + O(ekn2 −k1 +1 ).

1141

(23) (24) (25)

SINGLE-STEP ITERATIVE METHODS FOR MULTIPLE ROOTS For cubic convergence, we require en+1 = O(e3n ) and thus, g(xn ) = en + O(e3n ).

(26)

From (25) and (26), k2 − k1 = 1, which completes the proof.



Thereby the iteration function g(xn ) reduces to its final form, P F0,s θ(s) fn . g(xn ) = ( 0 ) P s fn s F0,s φ(s)

(27)

There are infinitely many F0,s ’s, however, writing from the simplest terms, five examples of multiplicational combinations of zeroth differential order can be written as 1,

f 02 f 02 f 02 f 02 , ( 00 )−1 , ( 00 )2 , ( 00 )−2 , · · · . 00 ff ff ff ff

(28)

Therefore, we construct a Newton-like method with nine parameters as follows:

xn+1

fn0 2 −1 fn0 2 2 fn0 2 −2  fn0 2  fn A + B( fn fn00 ) + C( fn fn00 ) + D( fn fn00 ) + E( fn fn00 ) = xn − ( 0 ) 02 02 02 02 fn 1 + F ( fn 00 ) + G( fn 00 )−1 + H( fn 00 )2 + I( fn 00 )−2 fn fn

fn fn

fn fn

(29)

fn fn

2.2. Solving for parameters. During the last section, (29) was derived to be the simplest possible form for the cubic order methods. Now we will find which among the form actually acquire the desired order. Theorem 3. Let α be an exact root of f and its multiplicity be m. Let n be an integer with n ≥ 0, xn an approximation after n iterations. Then the Newton-like method defined by (29) is cubically convergent if and only if X A B C D E F

G H I


T

  1 = 0

(30)

with X=

1 m 1 m2

1 m−1 m+1 m(m−1)2

m−1 m2 m−3 m3

m (m−1)2 m+3 (m−1)3

(m−1)2 m3 (m−1)(m−5) m4

2

m − m−1

− m−1 m

m − (m−1) 2

− (m−1) m2

2 − (m−1) 2

2 m2

4m − (m−1) 3

4(m−1) m3

2

!

(31) is satisfied.

1142

S. LEE AND H. CHOE Proof. We use the Taylor’s expansion (19) through (21) of f about α and definition (22) to obtain expressions for the nine terms included in (29). 2 c2  3 fn 1 1 c1 2  m + 1 c21 e + − e + ··· (32) = e − n fn0 m m2 c0 n m3 c20 m2 c0 n fn0 2(m + 2) c2  3 1 m + 1 c21 2  (m + 1)2 c21 e − − e + ··· (33) = e − n fn00 m − 1 m(m − 1)2 c20 n m(m − 1)3 c20 m(m − 1)2 c0 n fn2 fn00 m − 1 m − 3 c1 2  m2 − 3m − 6 c21 2(m − 4) c2  3 − e − e + = e + ··· (34) n fn03 m2 m3 c0 n m4 m3 c0 n c20 m m + 3 c1 2  (m + 2)(m + 3) c21 2(m + 5) c2  3 fn03 − e + e + · · · (35) = e − n fn2 fn00 (m − 1)2 (m − 1)3 c0 n (m − 1)4 (m − 1)3 c0 n c20 fn3 fn002 (m − 1)2 (m − 1)(m − 5) c1 2 = en − e 05 3 fn m m4 c0 n  m3 − 7m2 − 5m + 15 c2 2(m − 1)(m − 7) c  2 1 − e3 + · · · + m5 m4 c0 n c20  3m2 + 1 c2 fn02 m 2 c1 6 c2  2 1 = − e + − e + ··· n fn fn00 m − 1 (m − 1)2 c0 m(m − 1)3 c20 (m − 1)2 c0 n  3m + 1 c2 fn fn00 m − 1 2 c1 6 c2  2 1 = + e + − + e + ··· n fn02 m m2 c0 m3 c20 m2 c0 n  6(m2 + 1) c2 m2 4m c1 fn04 12m c2  2 1 = − e + e + ··· − n fn2 fn002 (m − 1)2 (m − 1)3 c0 (m − 1)4 c20 (m − 1)3 c0 n  2(3m2 − 5) c2 12(m − 1) c  fn2 fn002 (m − 1)2 4(m − 1) c1 2 1 = + − e + − e2 + · · · n 2 04 2 3 4 3 fn m m c0 m m c0 n c0

(36)

(37) (38) (39) (40)

From these equations, an error equation of (29) is easily derived: en+1 = en − K1 en − K2 en 2 + O(e3n ) where K1 =

1 m−1 m (m − 1)2 1 A+ B+ C + D + E m m−1 m2 (m − 1)2 m3 m m−1 m2 (m − 1)2
− F− G− H − I m−1 m (m − 1)2 m2

(41)

(42)

and 1 m+1 m−3 m+3 (m − 1)(m − 5) A+ B+ C+ D+ E m2 m(m − 1)2 m3 (m − 1)3 m4 (43) 2 2 4m 4(m − 1)
− F + 2G − H+ I . (m − 1)2 m (m − 1)3 m3 The condition for (29) to be cubically convergent is K1 = 1 and K2 = 0, which is equivalent to (30). This completes the proof. K2 =

1143

SINGLE-STEP ITERATIVE METHODS FOR MULTIPLE ROOTS 

Any combinations of parameters satisfying (30) would yield a cubic order Newton-like iterative method. However, a combination with all parameters activated will lead to a very complicated method, resulting in a relatively high computational cost. For this reason, it would be the best to let as many parameters as possible be zero, leaving only two of them non-zero. Noting that A, B, C, D, E cannot be all zero at the same time, there are 30 combinations in each of which all parameters except for two of them are zero. Nevertheless, it can be observed that 7 pairs are equivalent, by 02 multiplying an appropriate power of fff 00 to both the numerator and the denominator. Thereby we obtain 23 unique cubic order methods among the family of (29). Letting all parameters but A and B be zero, and solving (30) gives

A=

m(m + 1) (m − 1)2 ,B = − , 2 2

(44)

yielding a method xn+1 = xn −

m(m + 1) fn (m − 1) 2 fn0 + . 2 fn0 2 fn00

(45)

Similarly, 22 other methods obtained are displayed in Table 1. In the left column are combinations of non-zero parameters, and by solving (30) for the parameters, we obtain iterative methods displayed in the right column. parameters iterative method obtained m(3 − m) fn m2 fn2 f 00 − (46) A,C xn+1 = xn − 2 fn0 2 fn03 m(m + 3) fn (m − 1)3 fn03 A,D xn+1 = xn − + (47) 4 fn0 4m fn fn002 002 3 3 m(m − 5) fn m fn fn A,E xn+1 = xn + − (48) 0 4 fn 4(m − 1) fn05 2m2 fn2 fn00 xn+1 = xn + (49) A,F m(m − 3)fn fn0 fn00 − (m − 1)2 fn03 2mfn fn0 A,G xn+1 = xn − (50) (m + 1)fn02 − mfn fn00 4m3 fn3 fn002 A,H xn+1 = xn + 2 (51) m (m − 5)fn2 fn0 fn002 − (m − 1)3 fn05 4m(m − 1)fn fn03 xn+1 = xn − A,I (52) (m − 1)(m + 3)fn04 − m2 fn2 fn002 Table 1. Non-zero parameters and corresponding iterative methods.

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S. LEE AND H. CHOE parameters iterative method obtained (m − 1)(m − 3) fn0 m2 (m + 1) fn2 fn00 B,C xn+1 = xn + − 4 fn00 4(m − 1) fn03 0 (m − 1)2 (m + 1) fn03 (m − 1)(m + 3) fn B,D + xn+1 = xn − 2 fn00 2m fn fn002 0 3 002 3 (m − 1)(m − 5) fn m (m + 1) fn fn B,E xn+1 = xn + − 6 fn00 6(m − 1)2 fn05 2 2(m − 1) fn03 xn+1 = xn − B,G (m − 1)(m + 3)fn02 fn00 − m(m + 1)fn fn002 4m2 (m − 1)fn2 fn0 fn00 B,H xn+1 = xn + 2 m (m − 3)fn2 fn002 − (m − 1)2 (m + 1)fn04 4(m − 1)3 fn05 B,I xn+1 = xn − 2 (m − 1) (m + 5)fn04 fn00 − m2 (m + 1)fn2 fn003 m2 (m + 3) fn2 fn00 (m − 1)2 (m − 3) fn03 C,D xn+1 = xn − + 6(m − 1) fn03 6m fn fn002 2 00 2 3 002 2 m (m − 3) fn fn m (m − 5) fn fn C,E − xn+1 = xn + 03 2(m − 1) fn 2(m − 1)2 fn05 2m3 fn3 fn002 C,F xn+1 = xn + m(m − 1)(m − 5)fn fn03 fn00 − (m − 1)2 (m − 3)fn05 4m4 fn4 fn003 C,H xn+1 = xn + 2 m (m − 1)(m − 7)fn2 fn03 fn002 − (m − 1)3 (m − 3)fn07 (m − 1)2 (m − 5) fn03 m3 (m + 3) fn3 fn002 D,E xn+1 = xn + − 8m fn fn002 8(m − 1)2 fn05 3 2(m − 1) fn05 D,G xn+1 = xn − m(m − 1)(m + 5)fn fn02 fn002 − m2 (m + 3)fn2 fn003 4(m − 1)4 fn07 D,I xn+1 = xn − m(m − 1)2 (m + 7)fn fn04 fn002 − m3 (m + 3)fn3 fn004 2m4 fn4 fn003 E,F xn+1 = xn + m(m − 1)2 (m − 7)fn fn05 fn00 − (m − 1)3 (m − 5)fn07 4m5 fn5 fn004 E,H xn+1 = xn + 2 m (m − 1)2 (m − 9)fn2 fn05 fn002 − (m − 1)4 (m − 5)fn09 Table 1. (continued)

(53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67)

Method (45) is Osada’s method(OM) introduced in (5), (46) is Euler-Chebyshev method(ECM) introduced in (4), (48) is Chun and Neta’s method(CNM) introduced in (6), (49) is Halley’s method(HM) introduced in (3), and (56) is Biazar and Ghanbari’s method(BGM) introduced in (8). Moreover, since these methods are constructed by allowing only two of nine parameters to be non-zero, more can be constructed from (29) by setting various combinations of non-zero parameters, though an excess of non-zero terms would corrupt the computational efficiency. An efficiency index of an iterative method is defined by p1/d where p denotes the order of convergence of an iterative method, and d denotes the number of function evaluations required per each iteration, which is very reasonable considering the definition of the order of convergence. The

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SINGLE-STEP ITERATIVE METHODS FOR MULTIPLE ROOTS efficiency index of methods (45) through (67) is 31/3 = 1.442, which is higher than the Newton’s method (2) or optimal fourth-order iterative methods, with efficiency index 21/2 = 41/4 = 1.414. Note that the third-ordered methods (45) through (67) require one functional and two derivative evaluations per iteration. Remark 1. Summing multiplicational combinations of uniform differential order k preserves the c2 expansion form of ekn (p1 + p2 cc10 en + (p3 c12 + p4 cc20 )e2n + O(e3n )), where pi ’s are constants. While 0 the error equation must be an identity of ci ’s and en , it is optimal to reduce as many terms of ci ’s and en as possible in order to keep the method simple. In fact, all existing single-step methods of cubic convergence are included within (27), or in fact, within (29). Remark 2. The condition for (29) to converge with fourth order, simultaneously derived, is equivalent to an impossible system of equations. Therefore we consider it to be impossible to construct a fourth-order iterative method of single-step scheme, with three or less function evaluations. This limits the efficiency of single-step iterative methods for multiple roots.

3. N UMERICAL C OMPARISONS In this Section, numerical comparisons between cubically convergent methods of family (29) are presented. Test functions used for root-finding are displayed in Table 2, along with each of their approximate root and their multiplicity, and values used as initial points for each test function. test function approximate root multiplicity initial value f1 (x) = (x3 + 4x2 − 10)3 1.36523 m=3 2 1 2 2 2 1.40449 m=2 2.3 2 f2 (x) = (sin x − x + 1) f3 (x) = (x2 − ex − 3x + 2)5 0.25753 m=5 -1 1 f4 (x) = (cos x − x)3 0.73909 m=3 1.7 1 2 m=6 3 2.3 f5 (x) = ((x − 1)3 − 1)6 2 f6 (x) = (xex − sin2 x + 3 cos x + 5)4 -1.20765 m=4 -2 -1 1.89549 m=2 1.7 2 f7 (x) = (sin x − x/2)2 Table 2. Test functions, approximate roots, their multiplicity, and initial values used. Displayed in Table 3 are the number of iterations required to reach | f (xn ) |≤ 10−128 for each method and for each test function and an initial value. In the parentheses are the absolute value of f (xn ) after such iterations. Average numbers of iterations required for these cases are also displayed for each method. All computations were done using Mathematica, inserting inputs with significant figures large enough. Here * denotes where the approximation does not converge into the exact root. From the result, we consider (52) to be the most powerful iterative method among the family, and (50), (56), or (63) are also of considerable quality. It is interesting that though (64)and (65) often fail to converge into the root either temporarily or permanently, other methods have similar speed of convergence, differing by no more than 1 in average number of iterations. In fact, all methods in the comparison required the same number of iterations in two cases, namely, f3 (x), x0 = 1 and f4 (x), x0 = 1.

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S. LEE AND H. CHOE 4. C ONCLUSION Reduced from the most primitive form of iteration functions, a general single-step iterative scheme is constructed under a number of assumptions while maintaining simplicity. Considering only a finite number of multiplicational combinations, 23 cubically convergent iterative methods, those we consider to be the simplest among the scheme, are derived by the method of undetermined coefficients in the error equation. They include all existing single-step iterative methods. The multiplicational combination-based approach allows construction of more methods with consistency, within the same scheme. The numerical comparisons show the quality of the derived methods, and it can be observed from the comparisons that few of these methods have higher quality than the others, though not of significant difference.

f1 (x) f2 (x) f3 (x) f4 (x) methods x0 = 2 x0 = 1 x0 = 2.3 x0 = 2 x0 = −1 x0 = 1 x0 = 1.7 (45)(OM) 5(8e-322) 5(2e-258) 6(3e-343) 5(7e-153) 4(1e-267) 4(7e-286) 5(2e-364) (46)(ECM) 5(5e-371) 5(7e-374) 5(2e-142) 5(1e-190) 4(2e-278) 4(2e-279) 5(5e-378) 5(3e-304) 5(7e-210) 6(9e-317) 5(3e-141) 4(2e-264) 4(3e-289) 5(9e-359) (47) (48)(CNM) 4(2e-133) 4(1e-149) 5(2e-169) 5(1e-227) 4(8e-287) 4(2e-276) 5(4e-386) 5(5e-371) 5(7e-374) 6(4e-377) 5(6e-168) 4(5e-300) 4(2e-273) 5(5e-378) (49)(HM) (50) 4(1e-154) 4(1e-179) 5(8e-172) 5(3e-231) 4(1e-358) 4(1e-267) 4(5e-131) (51) 5(6e-342) 5(1e-321) 6(3e-341) 5(5e-152) 4(8e-287) 4(2e-276) 5(7e-371) (52) 4(7e-195) 4(7e-294) 5(2e-266) 5(1e-335) 3(8e-180) 4(8e-265) 4(1e-134) 5(5e-371) 5(7e-374) 5(7e-166) 5(1e-222) 4(3e-275) 4(5e-281) 5(5e-378) (53) (54) 5(7e-262) 6(3e-324) 6(5e-225) 6(3e-302) 4(6e-261) 4(1e-296) 5(5e-344) (55) 4(4e-149) 4(3e-268) 5(7e-173) 5(3e-253) 4(8e-287) 4(2e-276) 4(6e-131) (56)(BGM) 4(3e-146) 4(1e-131) 5(3e-240) 5(2e-144) 4(3e-309) 4(2e-259) 4(5e-156) (57) 5(5e-371) 5(7e-374) 6(3e-336) 5(2e-149) 4(2e-309) 4(6e-272) 5(5e-378) 5(7e-323) 5(3e-325) 5(5e-131) 6(2e-143) 4(2e-272) 4(1e-255) 4(3e-169) (58) (59) 5(5e-371) 5(7e-374) 5(3e-208) 5(1e-289) 4(1e-272) 4(1e-282) 5(5e-378) (60) 5(5e-371) 5(7e-374) 6(8e-333) 5(1e-147) 4(8e-287) 4(2e-276) 5(5e-378) (61) 5(5e-371) 5(7e-374) 5(1e-164) 5(9e-221) 4(8e-287) 4(2e-276) 5(5e-378) (62) 5(5e-371) 5(7e-374) 5(4e-205) 5(6e-283) 4(6e-282) 4(6e-278) 5(5e-378) (63) 4(1e-174) 4(5e-164) 5(1e-157) 4(5e-147) 4(8e-287) 4(2e-276) 4(2e-134) (64) 5(1e-194) 5(3e-228) 17(4e-357) 74(3e-164) 4(3e-236) 4(1e-249) 4(6e-135) (65) 6(2e-370) 5(3e-185) * 6(2e-187) 4(4e-215) 4(6e-245) 5(6e-383) (66) 4(2e-147) 4(2e-160) 5(1e-152) 5(1e-268) 4(8e-287) 4(2e-276) 4(7e-131) (67) 4(1e-171) 5(8e-390) 5(9e-153) 5(4e-299) 4(8e-287) 4(2e-276) 4(2e-134) Table 3. Numbers of iterations for test functions and initial points given in Table 1, with | f (xn ) | after such iterations.

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SINGLE-STEP ITERATIVE METHODS FOR MULTIPLE ROOTS

(45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67)

f4 (x) x0 = 1 4(1e-237) 4(6e-247) 4(1e-233) 4(2e-252) 4(6e-247) 4(4e-259) 4(6e-242) 4(8e-267) 4(6e-247) 4(2e-223) 4(9e-259) 4(2e-308) 4(6e-247) 4(2e-315) 4(6e-247) 4(6e-247) 4(6e-247) 4(6e-247) 4(5e-266) 4(1e-372) 4(7e-285) 4(1e-258) 4(8e-266)

f5 (x) x0 = 3 5(4e-258) 5(3e-286) 5(3e-246) 5(2e-303) 5(5e-351) 4(7e-140) 5(2e-324) 4(6e-158) 5(2e-277) 5(6e-227) 5(5e-300) 4(4e-215) 5(2e-370) 4(1e-252) 5(3e-269) 5(5e-307) 5(1e-319) 5(1e-306) 5(9e-297) 4(1e-310) 4(2e-201) 5(8e-311) 5(7e-307)

f6 (x) x0 = 2.3 x0 = −2 x0 = −1 4(9e-238) 6(1e-141) 5(3e-317) 4(1e-253) 6(5e-200) 4(4e-150) 4(5e-231) 7(4e-375) 5(1e-261) 4(3e-263) 6(5e-248) 4(1e-184) 4(2e-288) 6(8e-255) 4(3e-181) 4(4e-326) 5(6e-196) 4(5e-361) 4(1e-274) 6(7e-196) 4(1e-152) 4(6e-354) 5(2e-289) 4(5e-259) 4(1e-248) 6(8e-189) 4(7e-140) 4(6e-220) 7(2e-268) 5(5e-167) 4(3e-261) 6(7e-265) 4(8e-205) 3(6e-146) 7(6e-354) 4(4e-160) 4(3e-298) 6(3e-286) 4(3e-192) 3(7e-175) 7(4e-180) 4(6e-132) 4(6e-244) 6(2e-179) 4(1e-129) 4(3e-265) 6(1e-230) 4(4e-171) 4(5e-272) 6(3e-213) 4(8e-161) 4(4e-265) 6(6e-199) 4(4e-151) 4(2e-259) 6(1e-281) 4(4e-242) 4(9e-389) * 5(9e-297) 4(1e-319) * 5(1e-242) 4(2e-267) 6(1e-246) 4(1e-186) 4(3e-265) 6(6e-263) 4(2e-211) Table 3. (continued)

f7 (x) x0 = 1.7 5(3e-227) 5(2e-333) 5(2e-187) 4(2e-146) 5(3e-276) 4(1e-139) 5(8e-236) 4(4e-192) 4(1e-178) 6(2e-221) 5(3e-280) 5(1e-228) 5(3e-245) 5(1e-185) 5(3e-378) 5(5e-248) 4(1e-135) 5(2e-331) 5(7e-186) 6(4e-354) 6(3e-270) 6(1e-381) 5(3e-292)

x0 = 2 4(2e-157) 4(6e-177) 4(4e-151) 4(7e-195) 4(1e-165) 4(1e-195) 4(3e-157) 4(2e-243) 4(1e-193) 4(5e-130) 4(1e-198) 4(1e-152) 4(8e-157) 4(3e-134) 4(9e-230) 4(2e-156) 4(4e-193) 4(3e-228) 4(1e-159) 5(3e-328) 5(7e-294) 4(3e-163) 4(5e-145)

average 4.79 4.64 4.86 4.43 4.71 4.21 4.71 4.14 4.57 5.07 4.43 4.36 4.71 4.57 4.64 4.71 4.57 4.64 4.36 10.85 4.83 4.5 4.5

ACKNOWLEDGEMENT This work was supported by the Individual Research Program of Seoul Science High School in 2011. The authors wish to thank Mr. Googhin Kim and Mr. Changbum Chun for their helpful comments. R EFERENCES [1] J. F. Traub, Iterative methods for the solution of equations, Prentice Hall, New Jersey, 1964. [2] E. Halley, A new, exact and easy method of finding the roots of equations generally and that without any previous reduction, Phil. Trans. Roy. Soc. London 18 (1694), 136-148 [3] N. Osada, An optimal multiple root-finding method of order three, J. Comput. Appl. Math. 51 (1994), 131-133 [4] C. Chun, B. Neta, A third-order modification of Newton’s method for multiple roots, Appl. Math. and Comput. 211 (2009), 474-479 [5] J.Biazar, B.Ghanbari, A new third-order family of nonlinear solvers for multiple roots, Computers and Mathematics with Applications 59 (2010), 3315-3319

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S. LEE AND H. CHOE [6] S. Weerakoon, T. G. I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000), 87-93. [7] R. L. Burden, J. D. Faires, Numerical Analysis 8/e IE, Brooks/Cole Cengage Learning, 2005.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1150-1157, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

COMPACT DIFFERENCES OF VOLTERRA COMPOSITION OPERATORS FROM BERGMAN-TYPE SPACES TO BLOCH-TYPE SPACES ZHI JIE JIANG

Abstract. This paper characterizes the metrically compactness of differences of Volterra composition operators from the weighted Bergman-type space Apu , 0 < p < ∞, to the Bloch-type space Bv∞ of analytic functions on the unit disk D in terms of inducing symbols ϕ1 , ϕ2 : D → D and ψ1 , ψ2 : D → C.

1. Introduction Let D be the open unit disk in the complex plane, H(D) the space of all analytic functions on D, and H ∞ (D) = H ∞ the space of all bounded analytic functions on D with the supremum norm kf k∞ = supz∈D |f (z)|. Let dA(z) = π1 dxdy be the normalized Lebesgue measure on D. A positive continuous function u on [0, 1) is normal, if there exist positive numbers s and t, 0 < s < t, such that u(r)/(1−r)s is decreasing on [0, 1) and limr→1 µ(r)/(1−r)s = 0; u(r)/(1 − r)t is increasing on [0, 1) and limr→1 u(r)/(1 − r)t = ∞. For 0 < p < ∞ and the normal function u, the Bergman-type space Apu (D) = Apu consists of all f ∈ H(D) such that Z up (|z|) p kf kp,u = |f (z)|p dA(z) < ∞. 1 − |z| D When p ≥ 1, the Bergman-type space with the norm k · kp,u becomes a Banach space. If p ∈ (0, 1), it is a Fr´echet space with the translation invariant metric d(f, g) = kf − gkpp,u . Let v be a positive continuous function on D (weight). The weighted-type space Hv∞ (D) = Hv∞ consists of all f ∈ H(D) such that kf kHv∞ = sup v(z)|f (z)| < ∞. z∈D

Hv∞

It is known that is a Banach space. The Bloch-type space Bv∞ (D) = Bv∞ consists of all f ∈ H(D) such that kf kv = sup v(z)|f 0 (z)| < ∞. z∈D

Various kinds of weights and related weighted-type spaces and Bloch-type spaces have been studied, e.g., in [1, 2, 4, 10, 11, 12]. 2000 Mathematics Subject Classification. Primary 47B38; Secondary 47B33, 47B37. Key words and phrases. Volterra composition operator, Bergman-type space, weighted-type space, Bloch-type space, metrically bounded operator, metrically compact operator. 1

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2

ZHI JIE JIANG

Let ϕ be an analytic self-map of D and ψ be an analytic function on D. For f ∈ H(D) the Volterra composition operator Vϕ,ψ is defined by Z z Vϕ,ψ f (z) = (f ◦ ϕ)(ξ)(ψ ◦ ϕ)0 (ξ)dξ, z ∈ D. 0

As a kind of integral-type operator, the Volterra composition operators have been studied in [7, 14, 17]. Let X and Y be topological vector spaces whose topologies are given by translationinvariant metrics dX and dY , respectively, and L : X → Y be a linear operator. It is said that L is metrically bounded if there exists a positive constant K such that dY (Lf, 0) ≤ KdX (f, 0) for all f ∈ X. When X and Y are Banach spaces, the metrically boundedness coincides with the usual definition of bounded operators between Banach spaces. Recall that L : X → Y is metrically compact if it maps bounded sets into relatively compact sets. If X and Y are Banach spaces then metrically compactness becomes usual compactness. For some results in this topic see [3, 5, 9, 16, 18, 19]. Let ϕ1 , ϕ2 be nonconstant analytic self-maps of D and ψ1 , ψ2 ∈ H(D). Differences of Voterra composition operators on H(D) are defined as follows Z z
(Vϕ1 ,ψ1 −Vϕ2 ,ψ2 )(f )(z) = (f ◦ϕ1 )(ξ)(ψ1 ◦ϕ1 )0 (ξ)−(f ◦ϕ2 )(ξ)(ψ2 ◦ϕ1 )0 (ξ) dξ, z ∈ D. 0

Differences of composition operators was studied first on the Hardy space H 2 (D) in [3]. Recently Nieminen [13] has characterized the compactness of difference of weighted composition operators Wϕ1 ,ψ1 − Wϕ2 ,ψ2 on weighted-type space given by standard weights. Lindstr¨om and wolf [9] have generalized Nieminen’s result to more general weights v and u and found an expression for the essential norm kWϕ1 ,ψ1 − Wϕ2 ,ψ2 ke,Hv∞ →Hu∞ , where max{kϕ1 k∞ , kϕ2 k∞ } = 1. Here we continue this line of research and investigate the metrically compactness of differences of Volterra composition operators acting from the weighted Bergmantype space Apu to the Bloch-type space Bv∞ on the open unit disk. These results extend the corresponding results on the single Volterra composition operators (see, for example, [7, 14, 17]). For w ∈ D, let σw be the M¨obius transformation of D defined by σw (z) = (w − z)/(1 − wz). Note that the pseudo-hyperbolic metric ρ(z, w) = |σw (z)|. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation a  b means that there is a positive constant C such that a/C ≤ b ≤ Ca.

2. Auxiliary results The proof of the following lemma is standard, so it will be omitted (see, e.g., Lemma 3 in [15]). Lemma 1. Assume that p > 0, u is a normal function on [0, 1), v is a weight on D, ϕ1 , ϕ2 are analytic self-maps of D, ψ1 , ψ2 are analytic functions on D and the operator Vϕ1 ,ψ1 − Vϕ2 ,ψ2 : Apu → Bv∞ is metrically bounded. Then the operator Vϕ1 ,ψ1 − Vϕ2 ,u2 : Apu → Bv∞ is metrically compact if and only if for every bounded

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COMPACT DIFFERENCES OF VOLTERRA COMPOSITION OPERATORS

3

sequence (fn )n∈N in Apu such that fn → 0 uniformly on every compact subset of D as n → ∞ it follows that lim k(Vϕ1 ,ψ1 − Vϕ2 ,ψ2 )fn kv = 0.

n→∞

The following lemma was proved in [8]. Lemma 2. There exists a constant C > 0 independent of f ∈ Apu such that |f (z)| ≤

Ckf kp,u . u(|z|)(1 − |z|2 )1/p

(1)

Lemma 3. Let p > 0, u is a normal function on [0, 1), v is a weight on D, ϕ is an analytic self-map of D and ψ is an analytic function on D. Then the operator Vϕ,ψ : Apu → Bv∞ is metrically bounded if and only if sup z∈D

v(z)|ϕ0 (z)||ψ 0 (z)| < ∞. u(|ϕ(z)|)(1 − |ϕ(z)|2 )1/p

(2)

Proof. Suppose that Vϕ,ψ : Apu → Bv∞ is metrically bounded. For a fixed w ∈ D, setting (1 − |ϕ(w)|2 )t+1 fw (z) = , u(|ϕ(w)|)(1 − ϕ(w)z)1/p+t+1 then it is easy to show fw ∈ Apu and kfw kp,u ≤ C. Thus CkVϕ,ψ k

≥

kVϕ,ψ fw kv = sup v(z)|ϕ0 (z)||ψ 0 (z)||fw (ϕ(z))| z∈D

≥ v(w)|ϕ0 (w)||ψ 0 (w)||fw (ϕ(w))| v(w)|ϕ0 (w)||ψ 0 (w)| = . u(|ϕ(w)|)(1 − |ϕ(w)|2 )1/p So, we prove that (2) holds. If (2) holds, by Lemma 2, then we have kVϕ,ψ f kv

=

sup v(z)|ϕ0 (z)||ψ 0 (z)||f (ϕ(z))| z∈D

≤ C sup z∈D

It follows that Vϕ,ψ :

Apu

→

Bv∞

v(z)|ϕ0 (z)||ψ 0 (z)| kf kp,u . u(|ϕ(z)|)(1 − |ϕ(z)|2 )1/p

is metrically bounded. 

The next lemma shows that H ∞ ⊆ Apu . Lemma 4. Assume that p > 0 and u is a normal function on [0, 1). Then H ∞ ⊆ Apu . Proof. For f ∈ H ∞ , we assume that |f (z)| ≤ M for all z ∈ D. Then by the definition of the normal function and the Beta function, Z Z p p u (|z|) p p u (|z|) kf kp,u = |f (z)| dA(z) ≤ M dA(z) 1 − |z| D D 1 − |z| Z up (|z|) = M (1 − |z|)ps−1 dA(z) ps (1 − |z|) D Z Z M 2π 1 up (r) = (1 − r)ps−1 rdrdθ ps π 0 0 (1 − r)

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ZHI JIE JIANG

≤ 2M up (0)B(2, ps), where B(2, ps) is the Beta function. Thus we prove that f ∈ Apu .



The following lemma is very useful in the proof of the main result. Lemma 5. Assume that u is a normal function on [0, 1) such that u is continuously differentiable. Then there exists a constant C > 0 such that (3) u(|z|)(1 − |z|2 )1/p f (z) − u(|w|)(1 − |w|2 )1/p f (w) ≤ Ckf kp,u ρ(z, w) for all f ∈ Apu and for all z, w in D. ∞ Proof. By Lemma 3 we have that if f ∈ Apu , then f ∈ Hu(|z|)(1−|z| 2 )1/p and moreover kf ku(|z|)(1−|z|2 )1/p ≤ Ckf kp,u . By the definition of normal function, it follows that u(|z|)(1 − |z|2 )1/p (1 − |z|)1/p+t is increasing on [0, 1), where t is the positive number in the definition of normal function. Then by the proof in [9], we obtain that u(|z|)(1 − |z|2 )1/p satisfies the following so-called Lusky condition (which is due to Lusky [11])

u(1 − 2−n−1 )(1 − (1 − 2−n−1 )2 )1/p > 0. n∈N u(1 − 2−n )(1 − (1 − 2−n )2 )1/p inf

∞ Therefore, by the Lemma 1 in [9], for each f ∈ Hu(|z|)(1−|z| 2 )1/p and z, V ∈ D there exists a C > 0 such that u(|z|)(1 − |z|2 )1/p f (z) − u(|w|)(1 − |w|2 )1/p f (w) ≤ Ckf ku(|z|)(1−|z|2 )1/p ρ(z, w)

≤ Ckf kp,u ρ(z, w). From this inequality estimate (3) follows.



3. Main results In this section we formulate and prove the main result of this paper. Theorem 1. Assume that p > 0, u is a normal function on [0, 1) such that u is continuously differentiable, v is a weight on D, ϕ1 , ϕ2 are nonconstant analytic self-maps of D, ψ1 , ψ2 are analytic functions on D and Vϕ1 ,ψ1 , Vϕ2 ,ψ2 : Apu → Bv∞ are metrically bounded operators. Then the operator Vϕ1 ,ψ1 − Vϕ2 ,ψ2 : Apu → Bv∞ is metrically compact if and only if the following conditions hold: (a) v(z)|ϕ01 (z)||ψ10 (z)| lim 1 ρ(ϕ1 (z), ϕ2 (z)) = 0; |ϕ1 (z)|→1 u(|ϕ (z)|)(1 − |ϕ (z)|2 ) p 1 1 (b) v(z)|ϕ02 (z)||ψ20 (z)| lim 1 ρ(ϕ1 (z), ϕ2 (z)) = 0; |ϕ2 (z)|→1 u(|ϕ (z)|)(1 − |ϕ (z)|2 ) p 2 2 (c) ϕ01 (z)ψ10 (z) ϕ02 (z)ψ20 (z) lim v(z) − 1 1 = 0. min{|ϕ1 (z)|,|ϕ2 (z)|}→1 u(|ϕ1 (z)|)(1 − |ϕ1 (z)|2 ) p u(|ϕ2 (z)|)(1 − |ϕ2 (z)|2 ) p

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5

Proof. Suppose that the operator Vϕ1 ,ψ1 − Vϕ2 ,ψ2 : Apu → Bv∞ is metrically compact. If kϕ1 k∞ < 1, then (a) vacuously holds. Hence assume that kϕ1 k∞ = 1. Suppose to the contrary that (a) is not true. Then there exists a sequence (zn )n∈N such that |ϕ1 (zn )| → 1 as n → ∞ and v(zn )|ϕ01 (zn )||ψ10 (zn )| ρ(ϕ1 (zn ), ϕ2 (zn )) > 0. n→∞ u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 )1/p

δ := lim

(4)

Since |ϕ1 (zn )| → 1 as n → ∞, we can use the proof of Theorem 3.1 in [6] to find functions fn ∈ H ∞ , n ∈ N, such that ∞ X |fn (z)| ≤ 1, for all z ∈ D, (5) n=1

and 1 , n ∈ N. (6) 2n Since fn ∈ H ∞ , by Lemma 4 we have that fn ∈ Apu and kfn kp,u ≤ C for all n ∈ N. Note that form (6) it follows that lim |fn (ϕ1 (zn ))| = 1. Now, we define fn (ϕ1 (zn )) > 1 −

n→∞

kn (z) =

(1 − |ϕ(zn )|2 )t+1 u(|ϕ(zn )|)(1 − ϕ(zn )z)1/p+t+1

,

n ∈ N.

By the proof of Theorem 3.1 in [8], we obtain that that supn∈N kkn kp,u ≤ C. Put gn (z) = fn (z)σϕ2 (zn ) (z)kn (z), n ∈ N. Then clearly gn ∈ Apu with supn∈N kgn kp,u ≤ C and gn → 0 uniformly on compact subsets of D as n → ∞. Since Vϕ1 ,ψ1 −Vϕ2 ,ψ2 : Apu → Bv∞ is metrically compact, by Lemma 1 we get lim k(Vϕ1 ,ψ1 − Vϕ2 ,ψ2 )gn kv = 0.

(7)

n→∞

On the other hand, from the definition of the space Bv∞ , the definition of functions gn and by using (6), we have that k(Vϕ1 ,ψ1 − Vϕ2 ,ψ2 )gn kv ≥v(zn ) ϕ01 (zn )ψ10 (zn )gn (ϕ1 (zn )) − ϕ02 (zn )ψ20 (zn )gn (ϕ2 (zn )) =v(zn ) ϕ01 (zn )ψ10 (zn )fn (ϕ1 (zn ))σϕ2 (zn ) (ϕ1 (zn ))kn (ϕ1 (zn )) 1
v(zn )|ϕ01 (zn )||ψ10 (zn )|ρ(ϕ1 (zn ), ϕ2 (zn )) 1− n . ≥ (8) 1 2 u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 ) p Letting n → ∞ in (8) and using (4), we obtain lim k(Vϕ1 ,ψ1 − Vϕ2 ,ψ2 )gn kv ≥ lim

n→∞

v(zn )|ϕ01 (zn )||ψ10 (zn )|ρ(ϕ1 (zn ), ϕ2 (zn )) 1

n→∞

u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 ) p

= δ > 0,

which contradicts (7). This shows that lim

n→∞

v(zn )|ϕ01 (zn )||ψ10 (zn )| 1

u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 ) p

ρ(ϕ1 (zn ), ϕ2 (zn )) = 0,

for every sequence (zn )n∈N such that |ϕ1 (zn )| → 1 as n → ∞, which implies (a). Condition (b) is proved similarly. Hence we omit it. Now, we prove (c). Suppose to the contrary that (c) does not hold. Then there is a sequence (zn )n∈N such that min{|ϕ1 (zn )|, |ϕ2 (zn )|} → 1 as n → ∞ and ϕ01 (zn )ψ10 (zn ) ϕ02 (zn )ψ20 (zn ) β := lim v(zn ) − (9) 1 1 . n→∞ u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 ) p u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )|2 ) p

1154

6

ZHI JIE JIANG

We may also assume that there is the following limit l := lim ρ(ϕ1 (zn ), ϕ2 (zn )) ≥ 0. n→∞

(10)

Assume that l > 0. Then we have that for sufficiently large n, say n ≥ n0 β ϕ01 (zn )ψ10 (zn ) ϕ02 (zn )ψ20 (zn ) 0 < ≤ v(zn ) 1 − 1 2 2 2 u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )| ) p u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )| ) p  0 0 v(zn )|ϕ1 (zn )||ψ1 (zn )| v(zn )|ϕ02 (zn )||ψ20 (zn )|  2 + ρ(ϕ1 (zn ), ϕ2 (zn )). ≤ 1 l u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 ) p1 u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )|2 ) p (11) Letting n → ∞ in (11) and using (a) and (b), we arrive at a contradiction. Thus, we can assume that l = 0. Let the sequences of functions (fn )n∈N and (kn )n∈N be defined as above. Set hn (z) = fn (z)kn (z), n ∈ N. Then supn∈N khn kp,u ≤ C and hn → 0 uniformly on compact subsets of D as n → ∞. Hence by Lemma 1

Since Vϕ2 ,ψ2 : Apu → M

lim k(Vϕ1 ,ψ1 − Vϕ2 ,ψ2 )hn kv = 0. n→∞ Bv∞ is metrically bounded, then by Lemma v(z)|ϕ02 (z)||ψ20 (z)| := sup < ∞. 2 1/p z∈D u(|ϕ2 (z)|)(1 − |ϕ2 (z)| )

(12) 3 we have that (13)

We have k(V ϕ1 ,ψ1 − Vϕ2 ,ψ2 )hn kv ≥ v(zn ) ϕ01 (zn )ψ10 (zn )hn (ϕ1 (zn )) − ϕ02 (zn )ψ20 (zn )hn (ϕ2 (zn )) =v(zn ) ϕ01 (zn )ψ10 (zn )fn (ϕ1 (zn ))kn (ϕ1 (zn )) − ϕ02 (zn )ψ20 (zn )fn (ϕ2 (zn ))kn (ϕ2 (zn )) ϕ0 (z )ψ 0 (z )f (ϕ (z )) ϕ02 (zn )ψ20 (zn )fn (ϕ1 (zn )) 1 n 1 n 1 n n ≥v(zn ) − u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 )1/p u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )|2 )1/p ϕ0 (z )ψ 0 (z )f (ϕ (z )) 1 n 2 n 2 n n 0 0 − v(zn ) − ϕ (z )ψ (z )f (ϕ (z ))k (ϕ (z )) n n n 2 n n 2 n 2 2 u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )|2 )1/p ϕ01 (zn )ψ10 (zn ) ϕ02 (zn )ψ20 (zn ) 1
≥v(zn ) − 1− n 2 1/p 2 1/p 2 u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )| ) u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )| ) v(zn )|ϕ02 (zn )||ψ20 (zn )| − u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 )1/p hn (ϕ1 (zn )) u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )|2 )1/p − u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )|2 )1/p hn (ϕ2 (zn )) . (14) From (13), applying Lemma 5 to the functions hn with the points z = ϕ1 (zn ) and w = ϕ2 (zn ), and by using the fact supn∈N khn kp,u ≤ C, we get v(zn )|ϕ02 (zn )||ψ20 (zn )| u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 )1/p hn (ϕ1 (zn )) − u(|ϕ2 (zn )|) 2 1/p u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )| ) (1 − |ϕ2 (zn )|2 )1/p hn (ϕ2 (zn )) ≤ CM ρ(ϕ1 (zn ), ϕ2 (zn )). (15) Using (15) in (14), then letting n → ∞ is such obtained inequality and using (12) we obtain that β = 0, which is a contradiction. This proves (c). Now we assume that conditions (a)-(c) hold. Assume (fn )n∈N is a bounded sequence in Apu such that fn → 0 uniformly on compact subsets of D. To prove

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COMPACT DIFFERENCES OF VOLTERRA COMPOSITION OPERATORS

7

that Vϕ1 ,ψ1 −Vϕ2 ,ψ2 : Apu → Bv∞ is a metrically compact operator, in view of Lemma 1, it is enough to show that k(Vϕ1 ,ψ1 − Vϕ2 ,ψ2 )fn kv → 0 as n → ∞. Suppose to the contrary that this is not true. Then for some ε > 0 there is a subsequence (fnk )k∈N of (fn )n∈N such that k(Vϕ1 ,ψ1 − Vϕ2 ,ψ2 )fnk kv ≥ 2ε > 0 for every k ∈ N. We may assume that (fnk )k∈N is (fn )n∈N . Then there is a sequence (zn )n∈N in D such that v(zn ) ϕ01 (zn )ψ10 (zn )fn (ϕ1 (zn )) − ϕ02 (zn )ψ20 (zn )fn (ϕ2 (zn )) ≥ ε > 0, n ∈ N. (16) We may also assume that the sequences (ϕ1 (zn ))n∈N and (ϕ2 (zn ))n∈N converge. If it were max{|ϕ1 (zn )|, |ϕ2 (zn )|} → q < 1, then from (16), since for the test function f (z) ≡ 1 ∈ Apu (by Lemma 4), from the boundedness of the operators Vϕi ,ψi : Apu → Bv∞ , i = 1, 2, we have that ψ1 ◦ ϕ1 , ψ2 ◦ ϕ2 ∈ Bv∞ and since fn (ϕi (zn )) → 0 as n → ∞, i = 1, 2, we would obtain a contradiction. Hence max{|ϕ1 (zn )|, |ϕ2 (zn )|} → 1 as n → ∞. We can suppose that |ϕ1 (zn )| → 1 and ϕ2 (zn ) → z0 as n → ∞. Also, we can suppose that limit in (10) exists. Assume that l > 0. Then by (a) and (b), we get v(zn )|ϕ01 (zn )||ψ10 (zn )| ρ(ϕ1 (zn ), ϕ2 (zn )) = 0 |ϕ1 (zn )|→1 u(|ϕ1 (zn )|)(1 − |ϕ1 (zn )|2 )1/p

(17)

v(zn )|ϕ02 (zn )||ψ20 (zn )| ρ(ϕ1 (zn ), ϕ2 (zn )) = 0. |ϕ2 (zn )|→1 u(|ϕ2 (zn )|)(1 − |ϕ2 (zn )|2 )1/p

(18)

lim

and lim

From (16) and Lemma 2, it follows that 1 v(zn )|ϕ01 (zn )||ψ10 (zn )| 2 p 0 0 as √ 1 |𝑓 (𝑡) − Fn (𝑡, 𝑎)| ≤ C σ(𝑓 (n+1) )(𝑡 − 𝑎)n+ 2 . (21)

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SOME NEW ERROR INEQUALITIES FOR A TAYLOR-LIKE FORMULA

We may find a function 𝑓 : [𝑎, 𝑡] → ℝ such that 𝑓 (n) is absolutely continuous on [𝑎, 𝑡] as  ( ) [ ] [ ] 1 3𝑎 + 𝑡 n (𝑛 − 2)𝑎 − (𝑛 + 2)𝑡 𝑎+𝑡    𝑥+ , 𝑥 ∈ 𝑎, ,  (𝑛 + 1)! 𝑥 − 4 4 2 (n) 𝑓 (𝑥) = ( ) [ ] ( ]  1 𝑎 + 3𝑡 n (𝑛 − 2)𝑡 − (𝑛 + 2)𝑎 𝑎+𝑡   𝑥− 𝑥+ , 𝑥∈ ,𝑡  (𝑛 + 1)! 4 4 2 It follows that 𝑓 (n+1) (𝑥) = Gn (𝑥). It’s easy to find that the left-hand side of the inequality (21) becomes 𝐿.H.S.(21) =

2𝑛3 + 𝑛2 + 2𝑛 − 1 (𝑡 − 𝑎)2n+1 , (2𝑛 + 1)(2𝑛 − 1)(𝑛!)2 42n

and the right-hand side of the inequality (21) is √ 2𝑛3 + 𝑛2 + 2𝑛 − 1 R.H.S.(21) = √ C(𝑡 − 𝑎)2n+1 . (2𝑛 + 1)(2𝑛 − 1) 𝑛! 4n

(22)

(23)

(24)

It follows from (21), (23) and (24) that √ 2𝑛3 + 𝑛2 + 2𝑛 − 1 , C≥√ (2𝑛 + 1)(2𝑛 − 1) 𝑛! 4n √ 2n3 +n2 +2n−1 (2n+1)(2n−1) n! 4n

which prove that the constant √

is the best possible in (20).



Theorem 6. Let 𝑓 : [𝑎, 𝑡] → ℝ be a function such that 𝑓 (n) is absolutely continuous on [𝑎, 𝑡] and 𝑓 (n+1) ∈ 𝐿2 [𝑎, 𝑡], where 𝑛 is an even integer (𝑛 = 2𝑚). Then we have 2(𝑡 − 𝑎)2m+1 𝑓 (2m) (𝑡) − 𝑓 (2m) (𝑎) (25) 𝑓 (𝑡) − F2m (𝑡, 𝑎) − (2𝑚 + 1)! 42m 𝑡−𝑎 √ √ 1 1 4𝑚2 4 1 + − σ(𝑓 (2m+1) )(𝑡 − 𝑎)2m+ 2 . (26) ≤ 2m 2 (2𝑚)! 4 4𝑚 + 1 4𝑚 − 1 (2𝑚 + 1) √ 1 4m2 4 Inequality (25) is sharp in the sense that the constant (2m)!1 42m 4m+1 + 4m−1 − (2m+1) 2 cannot be replaced by a smaller one. Proof. From (2), (7) and (10), we can easily obtain 2(𝑡 − 𝑎)2m+1 𝑓 (2m) (𝑡) − 𝑓 (2m) (𝑎) 𝑓 (𝑡) − F2m (𝑡, 𝑎) − (2𝑚 + 1)! 42m 𝑡−𝑎 ∫ t ∫ t ∫ t 1 (2m+1) (2m+1) = G2m (𝑥)𝑓 (𝑥)𝑑𝑥 − G2m (𝑥)𝑑𝑥 𝑓 (𝑥)𝑑𝑥 𝑡−𝑎 a a a ∫ t ∫ t 1 (2m+1) (2m+1) = [G2m (𝑥) − G2m (𝑦)][𝑓 (𝑥) − 𝑓 (𝑦)]𝑑𝑥𝑑𝑦 2(𝑡 − 𝑎) a a (∫ t ∫ t ) 21 (∫ t ∫ t ) 12 1 2 (2m+1) (2m+1) 2 ≤ [G2m (𝑥) − G2m (𝑦)] 𝑑𝑥𝑑𝑦 [𝑓 (𝑥) − 𝑓 (𝑦)] 𝑑𝑥𝑑𝑦 2(𝑡 − 𝑎) a a a a (∫ [∫ t ]2 ) 12 (∫ t [∫ t ]2 ) 12 t 1 1 = G2m (𝑦)𝑑𝑦 𝑓 (2m) (𝑦)𝑑𝑦 G22m (𝑥)𝑑𝑥 − [𝑓 (2m) (𝑥)]2 𝑑𝑥 − 𝑡 − 𝑎 𝑡 − 𝑎 a a a a

1162

W. J. LIU AND Q. L. ZHANG

√ 1 = (2𝑚)! 42m

1 4𝑚2 4 + − 4𝑚 + 1 4𝑚 − 1 (2𝑚 + 1)2

√

1

σ(𝑓 (2m+1) )(𝑡 − 𝑎)2m+ 2 .

To prove the sharpness of (25), we suppose that (25) holds with a constant C > 0 as 2(𝑡 − 𝑎)2m+1 𝑓 (2m) (𝑡) − 𝑓 (2m) (𝑎) 𝑓 (𝑡) − F2m (𝑡, 𝑎) − (2𝑚 + 1)! 42m 𝑡−𝑎 √ 1 ≤C σ(𝑓 (2m+1) )(𝑡 − 𝑎)2m+ 2 . (27) We may find a function 𝑓 : [𝑎, 𝑏] → R such that 𝑓 (2m) is absolutely continuous on [𝑎, 𝑡] as 𝑓 (n) (𝑥)  ( ) [ ] 1 3𝑎 + 𝑡 2m (2𝑚 − 2)𝑎 − (2𝑚 + 2)𝑡    𝑥+ −  (2𝑚 + 1)! 𝑥 − 4 4 = ( ) [ ]  1 𝑎 + 3𝑡 2m (2𝑚 − 2)𝑡 − (2𝑚 + 2)𝑎    𝑥− 𝑥+ + (2𝑚 + 1)! 4 4

[ ] 2(𝑡 − 𝑎)2m+1 𝑎+𝑡 , 𝑥 ∈ 𝑎, , 2(2𝑚 + 1)! 42m 2 ( ] 2(𝑡 − 𝑎)2m+1 𝑎+𝑡 , 𝑥∈ ,𝑡 . 2(2𝑚 + 1)! 42m 2

It follows that 𝑓 (2m+1) (𝑥) = G2m (𝑥). It’s easy to find that the left-hand side of the inequality (27) becomes [ ] 1 4𝑚2 4 1 𝐿.H.S.(27) = + − (𝑡 − 𝑎)4m+1 , ((2𝑚)!)2 44m 4𝑚 + 1 4𝑚 − 1 (2𝑚 + 1)2 and the right-hand side of the inequality (27) is √ 1 4𝑚2 4 1 + − R.H.S.(27) = C(𝑡 − 𝑎)4m+1 . 2m (2𝑚)! 4 4𝑚 + 1 4𝑚 − 1 (2𝑚 + 1)2

(28)

(29)

(30)

It follows from (27), (29) and (30) that √ 1 4𝑚2 4 1 C≥ + − , 2m (2𝑚)! 4 4𝑚 + 1 4𝑚 − 1 (2𝑚 + 1)2 √ 1 4m2 4 + 4m−1 − (2m+1) which prove that the constant (2m)!1 42m 4m+1 2 is the best possible in (25).  Remark 1. We note that some applications of the classical or perturbed Taylor’s formula with the integral remainder in numerical analysis, for special means and some usual mappings have been given in [7]. The interested reader can also apply the results we obtained here in these mentioned fields. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 40975002, 11126289) and the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 09KJB110005).

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References [1] M. Akkouchi, Improvements of some integral inequalities of H. Gauchman involving Taylor’s remainder, Divulg. Mat. 11 (2) (2003), 115–120. [2] G. A. Anastassiou and S. S. Dragomir, On some estimates of the remainder in Taylor’s formula, J. Math. Anal. Appl., 263 (2001), 246–263. [3] L. Bougoffa, Some estimations for the integral Taylor’s remainder, JIPAM. J. Inequal. Pure Appl. Math. 4 (5) (2003), Article 86, 4 pp. [4] P. Cerone, Generalized Taylor’s formula with estimates of the remainder, in Inequality Theory and Applications, Vol 2, 33–52. Nova Sicence Publ., New York, 2003. [5] S. S. Dragomir, New estimation of the remainder in Taylor’s formula using Gr¨ uss type inequalities and applications, Math. Inequal. Appl., 2 (2) (1999), 183–193. [6] S. S. Dragomir and A. Sofo, A perturbed version of the generalised Taylor’s formula and applications, in Inequality theory and applications. Vol. 4, 71–84, Nova Sci. Publ., New York, 2007. [7] S. S. Dragomir, A. Sofo and P. Cerone, A perturbation of Taylor’s formula with integral remainder, Tamsui Oxf. J. Math. Sci., 17 (1) (2001), 1–21. [8] H. Gauchman, Some integral inequalities involving Taylor’s remainder I, JIPAM. J. Inequal. Pure Appl. Math., 3 (2) (2002), Article 26, 9 pp. [9] H. Gauchman, Some integral inequalities involving Taylor’s remainder. II, JIPAM. J. Inequal. Pure Appl. Math. 4 (1) (2003), Article 1, 5 pp. [10] D.-Y. Hwang, Improvements of some integral inequalities involving Taylor’s remainder, J. Appl. Math. Comput. 16 (1-2) (2004), 151–163. [11] Huy V. N., Ngo Q. A., New inequalities of Ostrowski-like type involving n knots and the L𝑝 -norm of the m-th derivative, Appl. Math. Lett., 22 (2009), 1345–1350. [12] W. J. Liu, Several error inequalities for a quadrature formula with a parameter and applications, Comput. Math. Appl., 56 (2008) 1766–1772. [13] Z. Liu, Note on inequalities involving integral Taylor’s remainder, JIPAM. J. Inequal. Pure Appl. Math. 6 (3) (2005), Article 72, 6 pp. [14] M. Mati´c, J. Pe˘ caric and N. Ujevi´c, On new estimation of the remainder in generalized Taylor’s formula, Math. Inequal. Appl., 2 (3) (1999), 343–361. [15] Y. X. Shi and Z. Liu, Some sharp Simpson type inequalities and applications, Applied Mathematics E-Notes, 9 (2009), 205–215. [16] E. Talvila, Estimates of the remainder in Taylor’s theorem using the Hentstock-Kurzweil integral, Czechoslovak Math. J., 55 (4) (2005), 933–940. [17] N. Ujevi´c, A new generalized perturbed Taylor’s formula, Nonlin. Funct. Anal. Appl., 7 (2) (2002), 255-267. [18] N. Ujevi´c, On generalized Taylor’s formula and some related results, Tamsui Oxford J. Math., 19 (1) (2003), 27-39. [19] N. Ujevi´c, Error Inequalities for a Taylor-like Formula, CUBO A Mathematical Journal, 10 (1) (2008), 11–18. (W. J. Liu) College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: [email protected] (Q. L. Zhang) College of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.6, 1165-1175, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

ADDITIVE FUNCTIONAL INEQUALITIES IN GENERALIZED QUASI-BANACH SPACES LEXIN LI, GANG LU, CHOONKIL PARK, AND DONG YUN SHIN∗ Abstract. In this paper, we investigate the Hyers-Ulam stability of the following function inequalities

( )

ax + by + cz

(0 < |K| < |a + b + c|), ∥af (x) + bf (y) + cf (z)∥ ≤ Kf

K

) (

ax + by +z ∥af (x) + bf (y) + Kf (z)∥ ≤

(0 < K < |a + b + K|)

Kf K in generalized quasi-Banach spaces.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let (G1 , .) be a group and let (G2 , ∗) be a metric group with the metric d(., .). Given ϵ > 0, does there exist a δ0, such that if a mapping h : G1 → G2 satisfies the inequality d(h(x.y), h(x) ∗ h(y)) < δ for all x, y ∈ G1 , then there exists a homomorphism H : G1 → G2 with d(h(x), H(x)) < ϵ for all x ∈ G1 ? In the other words, Under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces. Let f : E → E ′ be a mapping between Banach spaces such that ∥f (x + y) − f (x) − f (y)∥ ≤ δ for all x, y ∈ E, and for some δ > 0. Then there exists a unique additive mapping T : E → E ′ such that ∥f (x) − T (x)∥ ≤ δ for all x ∈ E. Moreover, if f (tx) is continuous in t ∈ ℝ for each fixed x ∈ E, then T is ℝ-linear. In 1978, Th.M. Rassias [3] proved the following theorem. Theorem 1.1. Let f : E → E ′ be a mapping from a normed vector space E into a Banach space E ′ subject to the inequality ∥f (x + y) − f (x) − f (y)∥ ≤ ϵ(∥x∥p + ∥y∥p )

(1.1)

2010 Mathematics Subject Classification. Primary 39B62, 39B52, 46B25. Key words and phrases. Hyers-Ulam stability; additive functional inequality; generalized quasi-Banach space; additive mapping. ∗ Corresponding author.

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L. LI, G. LU, C. PARK, AND D.Y. SHIN

for all x, y ∈ E, where ϵ and p are constants with ϵ > 0 and p < 1. Then there exists a unique additive mapping T : E → E ′ such that 2ϵ ∥f (x) − T (x)∥ ≤ ∥x∥p (1.2) 2 − 2p for all x ∈ E. If p < 0 then inequality (1.1) holds for all x, y ̸= 0, and (1.2) for x ̸= 0. Also, if the function t 7→ f (tx) from ℝ into E ′ is continuous in t ∈ ℝ for each fixed x ∈ E, then T is ℝ-linear. In 1991, Gajda [4] answered the question for the case p > 1, which was raised by Th.M. Rassias. On the other hand, J.M. Rassias [5] generalized the Hyers-Ulam stability result by presenting a weaker condition controlled by a product of different powers of norms. Theorem 1.2. ([6, 7]) If it is assumed that there exist constants Θ ≥ 0 and p1 , p2 ∈ ℝ such that p = p1 + p2 ̸= 1, and f : E → E ′ is a mapping from a norm space E into a Banach space E ′ such that the inequality ∥f (x + y) − f (x) − f (y)∥ ≤ Θ∥x∥p1 ∥y∥p2 for all x, y ∈ E, then there exists a unique additive mapping T : E → E ′ such that Θ ∥f (x) − T (x)∥ ≤ ∥x∥p , 2 − 2p for all x ∈ E. If, in addition, f (tx) is continuous in t ∈ ℝ for each fixed x ∈ E, then T is ℝ-linear More generalizations and applications of the Hyers-Ulam stability to a number of functional equations and mappings can be found in [8]–[22]. In [23], Park et al. investigated the following inequalities

( )

x + y + z

, ∥f (x) + f (y) + f (z)∥ ≤

2f 2 ∥f (x) + f (y) + f (z)∥ ≤ ∥f (x + y + z)∥,

( )

x + y +z ∥f (x) + f (y) + 2f (z)∥ ≤

2f 2 in Banach spaces. Recently, Cho et al. [24] investigated the following functional inequality

( )

x+y+z

∥f (x) + f (y) + f (z) ≤ Kf (0 < |K| < |3|)

K in non-Archimedean Banach spaces. Lu and Park [25] investigated the following functional inequality



(∑ ) N



∑ N



i=1 (xi ) (0 < |K| ≤ N ) f (xi ) ≤ Kf



K i=1

in Fr´ echet spaces. In [26], we investigated the following functional inequalities

( )

x + y + z

∥f (x) + f (y) + f (z)∥ ≤ (0 < |K| < 3),

Kf

K

1166

(1.3)

FUNCTIONAL INEQUALITIES IN GENERALIZED QUASI-BANACH SPACES

( )

x + y ∥f (x) + f (y) + Kf (z)∥ ≤ +z

Kf

K

(0 < K ̸= 2)

(1.4)

and proved the Hyers-Ulam stability of the functional inequalities (1.3) and (1.4) in Banach spaces. We consider the following functional inequalities

( )

ax + by + cz

(0 < |K| < |a + b + c|), (1.5) ∥af (x) + bf (y) + cf (z)∥ ≤ Kf

K

( )

ax + by

∥af (x) + bf (y) + Kf (z)∥ ≤ Kf +z

(0 < K < |a + b + K|), (1.6) K where a, b, c are nonzero real numbers. Now, we recall some basic facts concerning quasi-Banach spaces and some preliminary results. Definition 1.3. ([27, 28]) Let X be a linear space. A quasi-norm is a real-valued function on X satisfying the following: (1) ∥x∥ ≥ 0 for all x ∈ X and ∥x∥ = 0 if and only if x = 0. (2) ∥λx∥ = |λ|∥x∥ for all λ ∈ ℝ and all x ∈ X. (3) There is a constant C ≥ 1 such that ∥x + y∥ ≤ C(∥x∥ + ∥y∥) for all x, y ∈ X. The pair (X, ∥ · ∥) is called a quasi-normed space if ∥ · ∥ is a quasi-norm on X. A quasi-Banach space is a complete quasi-normed space. Baak [29] generalized the concept of quasi-normed spaces. Definition 1.4. ([29]) Let X be a linear space. A generalized quasi-norm is a real-valued function on X satisfying the following: (1) ∥x∥ ≥ 0 for all x ∈ X and ∥x∥ = 0 if and only if x = 0. (2) ∥λx∥ = |λ| · ∥x∥ for all λ ∈ ℝ and all x ∈ X. ∑ ∑ xj ∥ ≤ ∞ (3) There is a constant C ≥ 1 such that ∥ ∞ j=1 C∥xj ∥ for all x1 , x2 , · · · ∈ X j=1 ∑∞ with j=1 xj ∈ X. The pair (X, ∥·∥) is called a generalized quasi-normed space if ∥·∥ is a generalized quasi-norm on X. The smallest possible C is called the modulus of concavity of ∥ · ∥. A generalized quasi-Banach space is a complete generalized quasi-normed space. In this paper, we show that the Hyers-Ulam stability of the functional inequalities (1.5) and (1.6) in generalized quasi-Banach spaces. Throughout this paper, assume that X is a generalized quasi-normed vector space with generalized quasi-norm ∥ · ∥ and that (Y, ∥ · ∥) is a generalized quasi-Banach space. Let C be the modulus of concavity of ∥ · ∥. 2. Hyers-Ulam stability of the functional inequality (1.5) Throughout this section, assume that K is a real number with 0 < |K| < |a + b + c|. Proposition 2.1. Let f : X → Y be a mapping such that

( )

ax + by + cz

Kf ∥af (x) + bf (y) + cf (z)∥ ≤

K for all x, y, z ∈ X. Then the mapping f : X → Y is additive.

1167

(2.1)

L. LI, G. LU, C. PARK, AND D.Y. SHIN

Proof. Letting x = y = z = 0 in (2.1), we get ∥(a + b + c)f (0)∥ ≤ ∥Kf (0)∥. So f (0) = 0. Letting z = 0 and y = − ab x in (2.1), we get

( a )

af (x) + bf − x ≤ ∥Kf (0)∥ = 0

b for all x ∈ X. So f (x) = − ab f (− ab x) for all x ∈ X. Replacing x by −x and letting y = 0 and z = ac x in (2.1), we get

( a )

x ≤ ∥Kf (0)∥ = 0

af (−x) + cf c for all x ∈ X. So f (−x) = − ac f ( ac x) for all x ∈ X. Then we get

b ( a ) c ( a )

∥f (x) + f (−x)∥ =

− a f − b x − a f c x ( a ) ( a ) 1

= af (0) + bf − x + cf x

|a| b c

( ) a · 0 − b ab x + c ac x 1

=0 Kf ≤

|a| K Thus f (x) = −f (−x). ∥f (x) + f (y) − f (x + y)∥ = ∥f (x) + f (y) + f (−x − y)∥

a a b a c ax + ay

= − f (− x) − f (− y) − f ( )

a a a b a c

1

af (− a x) + bf (− a y) + cf ( ax + ay ) =

|a| a b c

) (

a · (− aa x) + b · (− ab x) + c · a(x+y) 1

c =

Kf

= 0.

|a| K Thus f (x + y) = f (x) + f (y) for all x, y ∈ X, as desired.



Theorem 2.2. Assume that a mapping f : X → Y satisfies the inequality

) (

ax + by + cz

+ ϕ(x, y, z),

∥af (x) + bf (y) + cf (z)∥ ≤ Kf

K

(2.2)

where ϕ : X 3 → [0, ∞) satisfies ϕ(0, 0, 0) = 0 and (( ) ) ∞ ( ) ∑ c j a j ( a )j ( a )j e ϕ(x, y, z) := ϕ y, z, x l and all x ∈ X. It means that the sequence {( ac )n f (( ac )n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {( ac )n f (( ac )n x)} converges. We define the mapping A : X → Y by A(x) = limn→∞ {( ac )n f (( ac )n x)} for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞, we get (2.3).

1169

L. LI, G. LU, C. PARK, AND D.Y. SHIN

Next, we show that A : X → Y is an additive mapping.

( n ) ( n ) c n a x −a x

∥A(x) + A(−x)∥ = lim ( ) f +f

n n→∞ a c cn [ ( n ) ( ) c n a x b a an x

≤ C lim ( ) f + f − · n

n→∞ a cn a b c

( n ) ( ) n

a x c a a x

+

f − cn + a f c · cn

( ) ( ) ]

b a an x c a an x

+ f − · n + f · a b c a c cn [ ( n ) ( n ) 1 c n a x a an x a x an+1 x ≤ C lim ( ) ϕ ,− , 0 + ϕ − n , 0, n+1 |a| n→∞ a cn b cn c c )] ( a an x an+1 x + ϕ 0, − , b cn cn+1 = 0 and so A(−x) = −A(x) for all x ∈ X.

( n ) ) ( n ) ( n c n a x a y a (x + y)

∥A(x) + A(y) − A(x + y)∥| = lim ( ) f +f −f

n→∞ a cn cn cn [ ( n ) ( )

c a an x a x b

= C lim ( )n f − f +

n→∞ a cn a b cn

( n ) ( n+1 )

a y c a y

f − + + f

cn a cn+1

( n ) ( ) ( n+1 ) ] nx

b a a c a y a (x + y) + f − + f − n+1 +

f n n c a b c a c [ ( n ) ( ) c n a x a an x an y a an x 1 lim ( ) ϕ , − ( n ), 0 + ϕ , 0, − ( n ) ≤C |a| n→∞ a cn b c cn c c ( n )] n n a (x + y) a a x a a x +ϕ , − ( n ), − ( n ) cn b c c c =0

for all x, y ∈ X. Thus the mapping A : X → Y is additive. Now, we prove the uniqueness of A. Assume that T : X → Y is another additive mapping satisfying (2.3). Then we obtain ( a ) ( a ) c

∥A(x) − T (x)∥ = ( )n A ( )n x − T ( )n x a c c ( a ) ( a ) c [

≤ C · ( )n A ( )n x − f ( )n x a c c

( a ) ( a ) ]

+ T ( )n x − f ( )n x c c ) ] C2 [ e ( a e − a x, a x) ≤ 2C ϕ x, − x, 0 + ϕ(0, |a| b b c which tends to zero as n → ∞ for all x ∈ X. Then we can conclude that A(x) = T (x) for all x ∈ X. This complete the proof. 

1170

FUNCTIONAL INEQUALITIES IN GENERALIZED QUASI-BANACH SPACES

Corollary 2.3. Let p and θ be positive real numbers with p > 1. Let f : X → Y be a mapping satisfying

( )

ax + by + cz

+ θ(∥x∥p + ∥y∥p + ∥z∥p ) ∥af (x) + bf (y) + cf (z)∥ ≤ Kf

K for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤

C cp + ap · p θ∥x∥p |a| c − c(a + b)p−1

for all x ∈ X. 3. Hyers-Ulam stability of the functional inequality (1.6) Throughout this section, assume that K, a, b are nonzero real numbers with 0 < K ̸= 2 and |a + b + K| ≥ K. Proposition 3.1. Let f : X → Y be a mapping such that

( )

ax + by

∥af (x) + bf (y) + Kf (z)∥ ≤ Kf +z

K

(3.1)

for all x, y, z ∈ X. Then the mapping f : X → Y is additive. Proof. Letting x = y = z = 0 in (3.1), we get ∥(K + a + b)f (0)∥ ≤ ∥Kf (0)∥. So f (0) = 0. Letting y = − ab x and z = 0 in (3.1), we get

( a )

af (x) + bf − x ≤ ∥Kf (0)∥ = 0 b for all x ∈ X. So f (x) = − ab f (− ab x) for all x ∈ X. a Replacing x by −x and letting y = 0 and z = K x in (3.1), we get

( a )

x ≤ ∥Kf (0)∥ = 0

af (−x) + Kf K a for all x ∈ X. So f (−x) = − K a f ( K x) for all x ∈ X. Thus we get ( a ) ( a ) 1 1

∥f (x) + f (−x)∥ = x ≤ ∥f (0)∥ = 0

bf − x + Kf |a| b K |a|

for all x ∈ X. So f (−x) = −f (x) for all x ∈ X. Letting z = −x−y in (3.1), we get K

( )

ax + by

af (x) + bf (y) − Kf

=

K ≤ for all x, y ∈ X. Thus

( Kf

ax + by K

( )

−ax − by

af (x) + bf (y) + Kf

K ∥Kf (0)∥ = 0

) = af (x) + bf (y)

1171

(3.2)

L. LI, G. LU, C. PARK, AND D.Y. SHIN

( ) a for all x, y ∈ X. Letting ( y = 0) in (3.2), we get f (x) = K f Kx for all x ∈ X. Letting x = 0 in a Ky b (3.2), we get f (y) = K f b . So

( ) ( )

a

Kx b Ky

∥f (x) + f (y) − f (x + y)∥ = f + f + f (−x − y)

K a K b

(

) ( )

1

af Kx + bf Ky + Kf (−x − y) = 0 =

|K| a b for all x, y ∈ X, as desired.



Theorem 3.2. Assume that a mapping f : X → Y satisfies the inequality

( )

ax + by

∥af (x) + bf (y) + Kf (z)∥ ≤ Kf +z

+ ϕ(x, y, z), K

(3.3)

where ϕ : X 3 → [0, ∞) satisfies ϕ(0, 0, 0) = 0 and (( ) j ( ) j ( ) j ) ∞ ( ∑ a )j K K K e y, z) := x, y, z l and all x ∈ X. It means that the sea n n quence {( K ) f (( K a ) x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sea n K n quence {( K ) f (( a ) x)} converges. So we may define the mapping A : X → Y by A(x) = a n n limn→∞ (( K ) f (( K a ) x)) for all x ∈ X. Moreover, by letting l = 0 and passing the limit m → ∞, we get (3.4). Now, we show that A is additive. ∥A(x) + A(y) − A(x + y)∥

a n

K n K n K n

= lim f (( ) x) + f (( ) y) − f (( ) (x + y))

n→∞ K a a a ) ( ( ) [ ) ) (( n n a n

K K b K

≤ C lim f − x + x f

n→∞ K a K b a

(( )n ) ( ( )n )

a K K K + y + f − y

f

a K a a

( ( )n ) ( ( )n ) (( )n ) ]

a K K b K K K

y + f − x +f (x + y) + f −

K a a K b a a ( ) ( ) ) [ ( a n K n K K n ≤ C lim ϕ 0, − x, x n→∞ K b a a ( ) ( )n ) ( K K K n y, 0, y +ϕ − a a a ( ( ) ( ) ( )n )] K K n K K n K +ϕ − y, − x, (x + y) a a b a a =0 for all x, y ∈ X. So the mapping A : X → Y is an additive mapping.

1173

L. LI, G. LU, C. PARK, AND D.Y. SHIN

Now, we show that the uniqueness of A. Assume that T : X → Y is another additive mapping satisfying (3.4). Then we get ( n ) (( )n ) a n K

K

∥A(x) − T (x)∥ = lim A x − T x

n→∞ K a a (( )n ) (( )n ) (( )n ) (( )n ) ] a n [



K K K K

+ T ≤ C lim A x − f x x − f x



n→∞ K a a a a ( ) ( ) ) ( ( ) ( ) )] [ ( C2 K K n K n K K n K K n ≤ 2C lim ϕe 0, − x, x) + ϕe x, − x, 0 |K| n→∞ a a a a a b a =0 for all x ∈ X. Thus we may conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of A. So the mapping A : X → Y is a unique additive mapping satisfying (3.4).  Corollary 3.3. Let p, θ and K be positive real numbers with p > 1 and |a + b + K| > K. Let f : X → Y be a mapping satisfying

( )

ax + by p p p

∥af (x) + bf (y) + Kf (z)∥ ≤ Kf +z

+ θ(∥x∥ + ∥y∥ + ∥z∥ ) K for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that ( a )p 3a 1 +K K( K) θ∥x∥p ∥f (x) − A(x)∥ ≤ a p a − K K for all x ∈ X. Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF2012R1A1A2004299). D. Y. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

References [1] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ., New York, 1960. [2] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [3] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [4] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [5] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445–446. [6] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130. [7] J.M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss. Math. 7 (1985), 193–196. [8] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press Inc., Palm Harbor, Florida, 2001. [9] G. Lu, C. Park, Hyers-Ulam stability of additive set-valued functional equations, Appl. Math. Lett. 24 (2011), 1312–1316.

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[10] I. Chang, Stability of higher ring derivations in fuzzy Banach algebras, J. Computat. Anal. Appl. 14 (2012), 1059–1066. [11] I. Cho, D. Kang, H. Koh, Stability problems of cubic mappings with the fixed point alternative, J. Computat. Anal. Appl. 14 (2012), 132–142. [12] M. Eshaghi Gordji, M. Bavand Savadkouhi, M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 12 (2010), 454–462. [13] M. Eshaghi Gordji, A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Computat. Anal. Appl. 13 (2011), 724–729. [14] M. Eshaghi Gordji, R. Farokhzad Rostami, S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ algebras, J. Computat. Anal. Appl. 13 (2011), 734–742. [15] M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi, E. Rashidi, M. Aghaei, Ternary Jordan derivations on C ∗ -ternary algebras, J. Computat. Anal. Appl. 12 (2010), 463–470. [16] H.A. Kenary, J. Lee, C. Park, Non-Archimedean stability of an AQQ-functional equation, J. Computat. Anal. Appl. 14 (2012), 211–227. [17] C. Park, Y. Cho, H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in nonArchimedean spaces, J. Computat. Anal. Appl. 14 (2012), 526–535. [18] C. Park, S. Jang, R. Saadati, Fuzzy approximate of homomorphisms, J. Computat. Anal. Appl. 14 (2012), 833–841. [19] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. [20] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Computat. Anal. Appl. 13 (2011), 1106–1114. [21] C. Park, Homomorphisms between Poisson JC ∗ -algebra, Bull. Braz. Math. Soc. 36 (2005), 79–97. [22] C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math. 132 (2008), 87–96. [23] C. Park, Y. Cho, M. Han, Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [24] Y. Cho, C. Park, R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Lett. 23 (2010), 1238–1242. [25] G. Lu, Y. Jiang, C. Park, Functional inequality in Fr´ echet spaces, J. Computat. Anal. Appl. (to appear) [26] G. Lu, C. Park, Additive functional inequalities in Banach spaces (preprint). [27] Y. Benyamini, J. Lindenstrauss, Geometric Nolinear Functional Analysis, Vol. 1, Colloq. Publ. 48, Amer. Math. Soc., Providence, 2000. [28] S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ., Reidel and Dordrecht, 1984. [29] C. Baak, Generalized quasi-Banach spaces, J. Chungcheong Math. Soc. 18 (2005), 215–222. Lexin Li School of Equipment Engineering, Shenyang Ligong University, Shenyang, 110159P.R. China E-mail address: [email protected] Gang Lu Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P.R. China E-mail address: [email protected] Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 130-743, Korea E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 6, 2013

Weighted Superposition Operators in Some Analytic Function Spaces, A. El-Sayed Ahmed and S. Omran,…….……………………………………………………………………………996 Fuzzy Fixed Points of Contractive Fuzzy Mappings, Akbar Azam and Muhammad Arshad,1006 On Explicit Solutions to a Polynomial Equation and its Applications to Constructing Wavelets, D. H. Yuan, Y. Feng, Y. F. Shen, and S. Z. Yang,…………………………………………1015 Numerical Solution of Fully Fuzzy Linear Matrix Equations, Kun Liu and Zeng-Tai Gong,1026 Korovkin Type Approximation Theorem for Statistical A-Summability of Double Sequences, M. Mursaleen and Abdullah Alotaibi,…………………………………………………………1036 The Properties of Logistic Function and Applications to Neural Network Approximation, Zhixiang Chen and Feilong Cao,……………………………………………………………1046 Orthogonal Stability of an Additive Functional Equation in Banach Modules Over a ‫ ∗ ܥ‬−Algebra, Hassan Azadi Kenary, Choonkil Park, and Dong Yun Shin,…………………1057 Some Characterizations and Convergence Properties of the Choquet Integral with Respect to a Fuzzy Measure of Fuzzy Complex Valued Functions, Lee-Chae Jang,…………………1069 Intuitionistic Fuzzy Stability of Euler-Lagrange Type Quartic Mappings, Heejeong Koh, Dongseung Kang, and In Goo Cho,………………………………………………………1085 Stability for an n-Dimensional Functional Equation of Quadratic-Additive Type with the Fixed Point Approach, Ick-Soon Chang and Yang-Hi Lee,…………………………………… 1096 An Identity of the q-Euler Polynomials Associated with the p-Adic q-Integrals on ℤ௣ , C. S. Ryoo,………………………………………………………………………………………1104 Approximate Septic and Octic Mappings in Quasi-ߚ-Normed Spaces,Tian Zhou Xu, J.Rassias,1110 Power Harmonic Operators and Their Applications in Group Decision Making, Jin Han Park, Jung Mi Park, and Jong Jin Seo, Y.C.Kwun,..…………………………………………….1120 Multiplicational Combinations and a General Scheme of Single-Step Iterative Methods for Multiple Roots, Siyul Lee and Hyeongmin Choe,……………………………………….1138

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 6, 2013 (continued) Compact Differences of Volterra Composition Operators from Bergman-Type Spaces to BlochType Spaces, Zhi Jie Jiang,………………………………………………………………1150 Some New Error Inequalities for a Taylor-Like Formula, Wenjun Liu and Qilin Zhang,1158 Additive Functional Inequalities in Generalized Quasi-Banach Spaces, Lexin Li, Gang Lu, Choonkil Park, and Dong Yun Shin,…………………………………………………….1165

Volume 15, Number 7 ISSN:1521-1398 PRINT,1572-9206 ONLINE

November 2013

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

1179

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC(eight times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Editor’s Assistant:Dr.Razvan Mezei,Lander University,SC 29649, USA.

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected]

http//:www.eudoxuspress.com.Annual Subscription Prices:For USA and Canada,Institutional:Print $550,Electronic $350,Print and Electronic $700.Individual:Print $300,Electronic $100,Print &Electronic $350.For any other part of the world add $80 more(postages) to the above prices for Print.No credit card payments. Copyright©2013 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

1180

Editorial Board Associate Editors of Journal of Computational Analysis and Applications 1) George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory,Real Analysis, Wavelets, Neural Networks,Probability, Inequalities. 2) J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago,IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis

20)Margareta Heilmann Faculty of Mathematics and Natural Sciences University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected] Approximation Theory (Positive Linear Operators) 21) Christian Houdre School of Mathematics Georgia Institute of Technology Atlanta,Georgia 30332 404-894-4398 e-mail: [email protected] Probability, Mathematical Statistics, Wavelets

3) Mark J.Balas Department Head and Professor Electrical and Computer Engineering Dept. College of Engineering University of Wyoming 1000 E. University Ave. Laramie, WY 82071 307-766-5599 e-mail: [email protected] Control Theory,Nonlinear Systems, Neural Networks,Ordinary and Partial Differential Equations, Functional Analysis and Operator Theory

22) Burkhard Lenze Fachbereich Informatik Fachhochschule Dortmund University of Applied Sciences Postfach 105018 D-44047 Dortmund, Germany e-mail: [email protected] Real Networks, Fourier Analysis,Approximation Theory

4) Dumitru Baleanu Cankaya University, Faculty of Art and Sciences, Department of Mathematics and Computer Sciences, 06530 Balgat, Ankara, Turkey, [email protected] Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics

5) Carlo Bardaro Dipartimento di Matematica e Informatica

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23) Hrushikesh N.Mhaskar Department Of Mathematics California State University Los Angeles,CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory,Splines, Wavelets, Neural Networks 24) M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations,Optimization,

Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis. 6) Martin Bohner Department of Mathematics and Statistics Missouri S&T Rolla, MO 65409-0020, USA [email protected] web.mst.edu/~bohner Difference equations, differential equations, dynamic equations on time scale, applications in economics, finance, biology. 7) Jerry L.Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics 8) Luis A.Caffarelli Department of Mathematics The University of Texas at Austin Austin,Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1188-1193, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

COMPOSITION OPERATORS FROM HARDY SPACE TO n-TH WEIGHTED-TYPE SPACE OF ANALYTIC FUNCTIONS ON THE UPPER HALF-PLANE ZHI-JIE JIANG AND ZUO-AN LI

Abstract. Motivated by some recent results on composition operators, the boundedness of composition operator from the Hardy space to the n-th weightedtype space on the half plane H = {z ∈ C : Imz > 0} is characterized.

1. Introduction Let H = {z ∈ C : Imz > 0} be the upper half plane in the complex plane C and H(H) the space of all analytic functions in H. For p > 0, the Hardy space H p (H) consists of all f ∈ H(H) such that Z +∞ kf kpH p (H) = sup |f (x + iy)|p dx < ∞. y>0

−∞

When p ≥ 1, the Hardy space with the norm k·kH p (H) becomes a Banach space(even a Hilbert space when p = 2), and when 0 < p < 1, d(f, g) = kf − gkpH p (H) defines a Fr´echet space distance on H p (H). For some details of this space and some operators on it see, e.g. [2], [3], [10] and [12]. Let µ(z) be a positive continuous function on a domain X ⊆ C, and n ∈ N0 (n) be fixed. The n-th weighted-type space on X, denoted by Wµ (X) consists of all f ∈ H(X) such that bW (n) (X) (f ) := sup µ(z)|f (n) (z)| < ∞. µ

z∈X

For n = 0 the space is called the weighted-type space Aµ (X), for n = 1 the Blochtype Bµ (X), and for n = 2 the Zygmund-type space Zµ (X). Some information of these spaces on the unit disc and some operators on them can be found, e.g., in [5], [8], [9], [11], [14] and [16]. This considerable interest in Zygmund-type spaces, as well as a necessity for unification of weighted-type, Bloch-type and Zygmund-type spaces, motivated us to define the n-th weighted-type space. (n) The quantity bW (n) (X) (f ) is a seminorm on the n-th weighted-type space Wµ (X) µ

(n)

and a norm on Wµ (X)/Pn−1 , where Pn−1 is the set of all polynomials whose degrees are less than or equal to n − 1. A natural norm on the n-th weighted-type 2000 Mathematics Subject Classification. Primary 47B38; Secondary 47B33, 47B37. Key words and phrases. Hardy space, upper half plane, n-th weighted-type space, composition operator, boundedness. 1

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2

ZHI-JIE JIANG AND ZUO-AN LI (n)

space Wµ (X) is defined as follows kf kW (n) (X) =

n−1 X

µ

|f (j) (a)| + bW (n) (X) (f ), µ

j=0

where a is an element in X. Under this norm this space becomes a Banach space. (n) For X = H, we obtain the space Wµ (H) on which the following norm can be introduced by kf kW (n) (H) :=

n−1 X

µ

|f (j) (i)| + sup µ(z)|f (n) (z)|, z∈H

j=0

(n)

and for X = D, the unit disc we get the space Wµ (D), and a norm on it is introduced by kf kW (n) (D) :=

n−1 X

µ

|f (n) (0)| + sup µ(z)|f (n) (z)|. z∈D

j=0

Let ϕ be an analytic self-map of X. The composition operator induced by ϕ is defined on H(X) by Cϕ f (z) = f (ϕ(z)), z ∈ X. A natural problem is to characterize the bounded or compact composition operator between two given spaces of analytic functions in terms of function theoretic properties of the induced symbol ϕ. During the past few decades, composition operators have been studied extensively on spaces of analytic functions on the unit disc or the unit ball. One can consult [1] and [13] for the general theory of these operators. As a consequence of the Littlewood’s subordination theorem, it is well known that every composition operator is bounded on Hardy spaces and weighted Bergman spaces of the unit disc. However, when people consider the Hardy space or the Bergman space on the upper half plane, they find that the situation is entirely different. There do exist unbounded composition operators on these spaces. Matache [10] proved that there didn’t exist compact composition operators on Hardy spaces of the upper half plane. Shapiro and Smith [12] also showed that there were no compact composition operators on Bergman spaces of the upper half plane. Because of these facts of composition operators, many authors recently have begun to investigate them on spaces of analytic functions on the upper half plane. The present author in [5] characterized the boundedness of composition operators from the weighted Bergman spaces to the weighted-type, Bloch-type and Zymund-type spaces with the weight µ(z) = Imz on the upper half plane. In [16], Stevi´c generalized the result of [14]. In [6], the present author characterized the boundedness of composition operator from the weighted Bergman space to n-th weighted-type space with µ(z) = Imz and n = 4. Motivated by [5], [6], [14] and [16], here we characterize the boundedness of composition operator from the Hardy space to the n-th weighted-type space on the upper half plane. On the one hand, this paper can be regarded as a generalization of results in [14] and [16]; on the other hand, it also can be regarded as a continuation of investigations of composition operators see, e.g. [4]-[12],[14]-[16].

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COMPOSITION OPERATORS

3

Let Y be a Banach space. Recall that the norm of the composition operator is defined by kCϕ kH p (H)→Y := sup kCϕ f kY . kf kH p (H) ≤1

It is easy to see that this quantity is finite if and only if the operator Cϕ : H p (H) → Y is bounded. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation a  b means that there is a positive constant C such that a/C ≤ b ≤ Ca. 2. Main results In this section, we first quote and prove several auxiliary lemmas. The first lemma was proved in [16]. Lemma 2.1. Suppose that p ≥ 1, n ∈ N and w ∈ H, then the function 1

fw,n (z) =

(Imw)n− p (z − w)n 1

belongs to H p (H) and supw∈H kfw,n kH p (H) ≤ π p . Lemma 2.2. Suppose that p ≥ 1, then there exists a positive constant C independent of f such that kf kH p (H) |f (n) (z)| ≤ C 1 (Imz)n+ p . Proof. For each f ∈ H p (H), it follows from Cauchy’s integral formula that Z +∞ 1 f (t) dt. (1) f (z) = 2πi −∞ t − z Differentiating in (1) under the integral sign n times , we have Z n! +∞ f (t) (n) f (z) = dt. 2π −∞ (t − z)n+1 Then |f (n) (z)| ≤

n! 2π

Z

+∞

−∞

f (t) dt. [(t − x)2 + y 2 ](n+1)/2

By using the change t − x = sy, we have Z +∞ Z +∞ yn ds dt = =: cn < ∞. 2 + y 2 ](n+1)/2 2 )(n+1)/2 [(t − x) (1 + s −∞ −∞ From (3) and applying Jensen’s inequality in (2), we get Z +∞ |f (t)|p yn (n) p |f (z)| ≤ dn dt y np [(t − x)2 + y 2 ](n+1)/2 −∞ Z +∞ |f (t)|p ≤ dn dt np+1 −∞ y kf kpH p (H) = dn np+1 , y

1190

(2)

(3)

4

ZHI-JIE JIANG AND ZUO-AN LI

where dn = (cn n!/2π)p , from which the desired result is obtained. The following lemma was proved in [15]. Lemma 2.3. Suppose that a > 0 and 1 1 a a + 1 Dn (a) = · · · · · · Qn−2 Qn−2 j=0 (a + j) j=0 (a + j + 1) Qn−1 then Dn (a) = j=1 j!

··· 1 ··· a+n+1 ··· ··· Qn−2 ··· j=0 (a + j + n − 1)

,

Before we formulate and prove the main result of this paper, we will need the following classical Fa` adi Bruno’s formula n  (j) Y X n! ϕ (z) kj (f ◦ ϕ)(n) (z) = f (k) (ϕ(z)) , k1 ! · · · kn ! j! j=1 where k = k1 + k2 + · · · + kn , and the sum is over all non-negative integers k1 , k2 , ..., kn satisfying k1 + 2k2 + · · · + nkn = n. For the information related to this formula see [7]. Theorem 2.4. Suppose that p ≥ 1 and ϕ is an analytic self-map of H, then the (n) operator Cϕ : H p (H) → Wµ (H) is bounded if and only if for each k ∈ {1, 2, ..., n} it follows that P Qn  ϕ(j) (z) kj n! µ(z) j=1 k1 !···kn ! j! Ik := sup < ∞, (4) 1 z∈H (Imϕ(z))k+ p where the sum is over all non-negative integers k1 , k2 , ..., kn satisfying k1 + 2k2 + · · · + nkn = n. (n) Moreover, if the operator Cϕ : H p (H) → Wµ (H)/Pn−1 is bounded, then kCϕ kH p (H)→W (n) (H)/Pn−1 

n X

µ

Ik .

(5)

k=1 (n)

Proof. First assume that the operator Cϕ : H p (H) → Wµ (H) is bounded. For a fixed w ∈ H and constants c1 , c2 , ..., cn , set the function fw (z) =

n X j=1

cj n−2+j+

1

(2iImw)n−2+j+ p 2 p

2

(z − w)n−2+j+ p

.

Then by Lemma 2.1 we know that fw ∈ H p (H) for every w ∈ H, and sup kfw kH p (H) ≤ C.

(6)

w∈H

Now we prove that for each k ∈ {1, ..., n}, there are constants c1 , c2 , ..., cn such that 1 fw(k) (w) = fw(l) (w) = 0, l ∈ {1, ..., n} \ {k}. (7) 1 , (2iImw)k+ p

1191

COMPOSITION OPERATORS

5

In fact, by differentiating function fw for each k ∈ {1, ..., n}, the system in (7) becomes c1 + c2 + · · · + cn = 0 

k−2 Y

n+

n+j+

j=0

  2 2 2 c1 + n + 1 + c2 + · · · + 2n − 1 + cn = 0 p p p ···

k−2 k−2 Y Y 2 2 2 c1 + c2 + · · · + cn = 1 n+j+1+ 2n − 1 + p p p j=0 j=0

··· n−2 Y

n+j+

j=0

n−2 n−2 Y Y 2 2 2 c1 + c2 + · · · + cn = 0. (8) n+j+1+ 2n − 1 + p p p j=0 j=0

Applying Lemma 2.3 with a = n + 2/p > 0, we see that the determinant of system (8) is different from zero, from which the claim holds. For each k ∈ {1, ..., n}, we choose the corresponding function which satisfy (7), and write it by fw,k . For each k ∈ {1, ..., n}, the boundedness of the operator (n) adi Bruno’s formula and (6) imply that Cϕ : H p (H) → Wµ (H), Fa` P Qn  ϕ(j) (z) kj n! µ(z) j=1 k1 !···kn ! j! ≤ sup kCϕ fϕ(w),k kW (n) (H) 1 µ w∈H (Imϕ(z))k+ p ≤ CkCϕ kH p (H)→W (n) (H) , (9) µ

where the sum is over all non-negative integers k1 , k2 , ..., kn satisfying k1 + 2k2 + · · · + nkn = n. Now assume that the condition in (4) holds. By Fa`adi Bruno’s formula and Lemma 2.2, we have kCϕ f kW (n) (H) =

n−1 X

µ

|f ◦ ϕ(0)| + sup µ(z)|(Cϕ f )(n) (z)| z∈H

j=0

=

n−1 X X



j=0

j  (s) Y j! ϕ (0) ls f (l) (ϕ(0)) l1 ! · · · lj ! s! s=1

X + sup µ(z) z∈H

≤

j n−1 XX

n  (j) Y n! ϕ (z) kj f (k) (ϕ(z)) k1 ! · · · kn ! j! j=1 j  (s) Y j! ϕ (0) ls l1 ! · · · lj ! s=1 s! P Qn  ϕ(j) (z) kj n! µ(z) j=1 k1 !···kn ! j!

X |f (l) (ϕ(0))|

j=0 l=0

+ Ckf kH p (H)

n X k=1

sup z∈H

1

(Imϕ(z))k+ p

.

(10)

From this, Lemma 2.2 with z = ϕ(0) and the condition in (4), we prove that (n) Cϕ : H p (H) → Wµ (H) is bounded. Moreover, if we consider the bounded operator

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6

ZHI-JIE JIANG AND ZUO-AN LI (n)

Cϕ : H p (H) → Wµ (H)/Pn−1 , then kCϕ kH p (H)→W (n) (H)/Pn−1 ≤ C

n X

µ

k=1

sup

P Qn  ϕ(j) (z) kj n! µ(z) j=1 k1 !···kn ! j!

z∈H

1

(Imϕ(z))k+ p

.

(11)

Combining (9) and (11), we obtain the desired asymptotic relation in (5). Acknowledgement. This work is supported by the Science Foundation of Sichuan Province (Grant No.11ZA120). References [1] C. C. Cowen, B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, 1995. [2] P. Duren, Theory of H p spaces, Pure and Applied Mathematics, Vol.38 Academic Press, New York, 1970. [3] K. Hoffman, Banach spces of analytic functions, Prentice-Hall Series in Morden Analysis Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. [4] Z. J. Jiang, G. F. Cao, Composition operator on Bergman-Orlicz space, Journal of Inequalities and Applications, vol. 2009, Article ID 32124, 14 pages, (2009). [5] Z. J. Jiang, Composition operators from weighted Bergman spaces to some analytic spaces on the half plane, Ars. Combin., 94 (2010), 10-16. [6] Z. J. Jiang, Y. Yang, Products of differentiation and composition from weighted Bergman spaces to some spaces of analytic functions on the upper half-plane, Int. Journal of Math. Analysis., 4 (22) (2010), 1085-1094. [7] W. Johnson, The curious history of Fa` adi Bruno’s formula, Amer. Math. Monthly., 109 (3) (2002), 217-234. [8] S. Li, S. Stevi´ c, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl., 338 (2008), 1282-1295. [9] S. Li, S. Stevi´ c, Volterra-type operators on Zygmund spaces, Journal of Inequalities and Applications, vol. 2007, Article ID 32124, 10 pages, (2007). [10] V. Matache, Composition operators on Hardy spaces of a half-plane, Proc. Amer. Math. Soc., 127 (5) (1999), 1483-1491. [11] S. Ohno, Products of composition and differentiation on Bloch spaces, Bull. Korean Math. Soc., 46 (6) (2009), 1135-1140. [12] J. H. Shapiro, W. Smith, Hardy spaces that support no compact composition operators, J. Functional Analysis., 205 (2003), 62-89. [13] J. H. Shapiro, Composition operators and classical function theory, Springer-Verlag, New York, Heidelberg, Berlin, 1993. [14] S. D. Sharma, A. K. Sharma, S. Ahmed, Composition operators between Hardy and Blochtype spaces of the upper half-plane, Bull. Korean Math. Soc., 43 (3) (2007), 475-482. [15] S. Stevi´ c, Composition operators from weighted Bergman spaces to the n-th weighted spaces on the unit disc, Discrete Dyn. Nat. Soc., Vol. 2009, Artical ID 742019, (2009), 11 page. [16] S. Stevi´ c, Composition operators from the Hardy Space to the Zygmund-type space on the upper half-plane, Abstract and Applied Analysis, vol 2009, Article ID 161528, 8 pages, (2009). Zhi-jie Jiang, School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China E-mail address: [email protected] Zuo-an Li, School of Computer Science, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1194-1201, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Notes on Generalized Gamma, Beta and Hypergeometric Functions Mehmet Bozer and Mehmet Ali Özarslan Eastern Mediterranean University Gazimagusa, TRNC, Mersin 10, Turkey Email: [email protected] and [email protected]

Abstract Recently, some generalizations of the generalized Gamma, Beta, Gauss hypergeometric and Con‡uent hypergeometric functions has been introduced in [11]. In this paper we obtain some integral representations of the above mentioned functions and Mellin transform representation of the generalized Gamma function. Furthermore, some recurrence relations of these functions are given.

Key words : Gamma Function, Beta Function, Hypergeometric Function, Con‡uent Hypergeometric Function, Mellin transform. 2000 Mathematics Subject Classi…cation. 33C45, 33C50.

1

Introduction

In recent years, some extensions of the well known special functions have been considered by several authors [1], [2], [4], [5], [6], [9]. In 1994, Chaudhry and Zubair [1] have introduced the following extension of gamma function Z1 tx p (x) :=

1

exp

t

pt

1

dt;

(1)

0

Re (p) > 0: In 1997, Chaudhry et al. [2] has presented the following extension of Euler’s beta function Bp (x; y) :=

Z

1

tx

1

(1

y 1

t)

0

exp

p t(1

t)

dt;

(2)

(Re(p) > 0; Re(x) > 0; Re(y) > 0) and they proved that this extension has connection with the Macdonald, error and Whittakers function. It is clearly seen that 0 (x) = (x) and B0 (x; y) = B (x; y). Afterwards, Chaudhry et al. [3] used Bp (x; y) to extend the hypergeometric functions (and con‡uent hypergeometric functions) as follows: Fp (a; b; c; z) = p

1 X Bp (b + n; c b) zn (a)n B (b; c b) n! n=0

0 ; Re (c) > Re (b) > 0;

1194

1

BOZER, OZARSLAN: HYPERGEOMETRIC FUNCTIONS

p

(b; c; z) = p

1 X Bp (b + n; c b) z n B (b; c b) n! n=0

0 ; Re (c) > Re (b) > 0;

where ( ) denotes the Pochhammer symbol de…ned by ( )0

( + ) ( )

1 and ( ) :=

and gave the Euler type integral representation Z 1 1 Fp (a; b; c; z) = tb 1 (1 B (b; c b) 0 p > 0 ; p = 0 and jarg (1

c b 1

t)

z)j
Re (b) > 0:

They called these functions as extended Gauss hypergeometric function (EGHF) and extended con‡uent hypergeometric function (ECHF), respectively. They have discussed the di¤erentiation properties and Mellin transforms of Fp (a; b; c; z) and obtained transformation formulas, recurrence relations, summation and asymptotic formulas for this function. Note that F0 (a; b; c; z) = 2 F1 (a; b; c; z) : Note that, very recently, the second au¬thor obtained some representations of these extended functions in terms of a …nite number of well known higher transcendental functions, specially, as an in…nite series containing hypergeometric, con‡uent hypergeometric, Whittaker’s, Lagrange functions, Laguerre polynomials, and products of them [10]. We consider the following generalizations of gamma and Euler’s beta functions Z 1 p ( ; ) (x) := t x 1 1 F1 ; ; t dt (3) p t 0 Re( ) > 0; Re( ) > 0; Re(p) > 0; Re(x) > 0;

Bp(

; )

(x; y) :=

Z

1

tx

1

y 1

(1

t)

1 F1

; ;

0

p t(1

t)

dt;

(4)

(Re(p) > 0; Re(x) > 0; Re(y) > 0; Re( ) > 0; Re( ) > 0): ( ; )

( ; )

( ; )

(x) = p (x), 0 (x) = (x) ; Bp (x; y) = respectively. It is obvious by (1), (3) and (2), (4) that, p ( ; ) ( ; ) ( ; ) Bp (x; y) and B0 (x; y) = B (x; y) : We call the functions p (x) and Bp (x; y) as generalized Euler’s gamma function (GEGF) and generalized Euler’s beta function (GEBF), respectively. On the other hand using the new generalization (4) of beta function the generalized Gauss hypergeometric (GGHF) and generalized con‡uent hypergeometric functions (GCHF) is de…ned by Fp( ; )

(a; b; c; z) :=

1 X

( ; )

(a)n

Bp

n=0

and ( ; ;p) 1 F1

(b; c; z) :=

1 ( X Bp

n=0

; )

(b + n; c B (b; c b)

(b + n; c B (b; c b)

b) z n n! b) z n ; n!

respectively (see [11]). The following integral representations were obtained in [11]: Z 1 p 1 c b 1 Fp( ; ) (a; b; c; z) := tb 1 (1 t) ; ; (1 1 F1 B(b; c b) 0 t (1 t) Re (p) > 0; p = 0 and jarg (1 1195

2

z)j < ; Re (c) > Re (b) > 0;

zt)

a

dt;

BOZER, OZARSLAN: HYPERGEOMETRIC FUNCTIONS

and 1 (b; c; z) := B(b; c

( ; ;p) 1 F1

b)

p

Z

1

tb

1

c b 1 zt

(1

t)

e

1 F1

; ;

0

p t (1

t)

dt;

(6)

0; and Re(c) > Re(b) > 0:

Observe that [3], Fp( and

; )

( ; ;p) 1 F1

( ; )

(a; b; c; z) = Fp (a; b; c; z) ; F0 (p)

(b; c; z) = 1 F1 (b; c; z) =

(a; b; c; z) = 2 F1 (a; b; c; z) ; ( ; ;0)

p

(b; c; z) ; 1 F1

(b; c; z) = 1 F1 (b; c; z) :

In section 2, we obtain some integral representations of generalized beta, Gauss hypergeometric and Con‡uent hypergeometric functions. Mellin transform representation of the generalized Gamma function is also be given. Furthermore, some recurrence relations of the above mentioned functions are presented.

2

New integral representations of GEBF, GGHF and GCHF

It is important and useful to obtain di¤erent integral representations of the new generalized beta function, for later use. Also it is useful to discuss the relationships between classical gamma functions and new ( ; ) (x) by means of Chaudhry’s generalizations. We start with the following integral representation for Bp extended beta function: Theorem 1 For the new generalized beta function, we have Bp( ; )

(x; y) =

( ) ( ) (

)

Z1

Bpt (x; y) t

1

(1

1

t)

dt:

0

Proof. Using the integral representation of con‡uent hypergeometric function, we have Bp( ; )

(x; y) =

( ) ( ) (

)

Z1 Z1

ux

1

u)y

(1

1

exp

pt t u(1 u)

1

(1

t)

1

dtdu:

0 0

From the uniform convergence of the integrals, the order of integration can be interchanged to yield that 8 9 Z1 0 0 if t < 0 5

; ;

p t(1

t)

dt;

(x + 1; y

1) :

BOZER, OZARSLAN: HYPERGEOMETRIC FUNCTIONS

( ; )

is the Heaviside unit function. Hence Bp Bp(

; )

(x; y) has the Mellin transform representation

(x; y) = M ff (t : y; ; ; p) : xg :

Taking derivative of f (t : y; ; ; p); we get @ (f (t : y; ; ; p)) = @t p y H(1 t) (1 t)

1

h

(1

y 1

t) (1

1

1

+

t2

t)

1 F1

t)2

(1

+ (y

1)H(1 + 1;

y 2

t) (1

+ 1;

t) p

t(1

i

1 F1

; ;

p t(1

t)

;

t)

where (t

1 if t = t0 0 if t = 6 t0

t0 ) =

is the Dirac delta function. Since M ff 0 (t) : xg = (1

x)M ff (t) : x

1g

we have (x

1)Bp(

; )

+M = (y

h

1; y) = M

(x p

H(1

1)Bp(

; )

(1 t)

(x; y

1)

t)

1

y 1

t) (1

y 1

t) (1 t2 p

1

+

Bp(

+ (y

+1; +1)

t)

1 F1

t)2

(1

y 2

1) (1

(x

+ 1; p

2; y) +

Bp(

i

1 F1

+ 1;

; ; p

t(1

t)

+1; +1)

p t(1 :x

(x; y

This completes the proof. Remark 8 For

= ; we get the recurrence obtained in [[12], pp.222, Eq(5.65)] xBp (x; y + 1)

yBp (x + 1; y) = p [Bp (x + 1; y (Re(p) > 0)

Theorem 9 We have the following di¤ erence formula for (s Proof. By (3),

( ; ) (s) p

1)

( ; ) (s p

1) =

1)

( ; ) (s) p

p

( +1; +1) (s) p

Bp (x

1; y + 1)] :

:

( +1; +1) (s p

2):

is the Mellin transform of the function f (t : ; ; p) =

1 F1 (

; ; t

1

pt

):

Hence ( ; ) (s) p

= M ff (t : ; ; p) : sg :

Taking derivative of f (t : ; ; p); we get @ (f (t : ; ; p)) = @t

1 + pt

2

1 F1 (

+ 1;

+ 1; t

p ) t

Since M ff 0 (t) : sg = (1

s)M ff (t) : s

we get (1

s)

( ; ) (s p

1) =

( +1; +1) (s) p

This completes the proof. 1199

6

+

p

1g ( +1; +1) (s p

2):

2) :

t)

:x

dt

BOZER, OZARSLAN: HYPERGEOMETRIC FUNCTIONS

Remark 10 When p = 0 and

= , we have the well known identity (s) = (s

1) (s

1): ( ; )

Theorem 11 We have the following di¤ erence formula for Fp (b = (c p

b

B(b

1)B(b; c

1)B(b 1)Fp(

b

b)Fp( +1; +1)

2; c

1; c ; )

(a; b

( ; )

Hence B(b; c

b)Fp

(a; b

1; c; z) b)Fp(

1; x) + azB(b; c p 2; z) + B(b; c b

2; c

b)Fp

fa;b;c (t : z; ; ; p) = H(1

; )

(a; b; c

( ; )

Proof. Observe from (5) that B(b; c

1)Fp(

b

(a; b; c; z) :

; )

(a + 1; b; c; x)

2)Fp( +1; +1)

(a; b; c

2; z) :

(a; b; c; z) is the Mellin transform of the function

t) (1

c b 1

t)

1 F1

; ;

p t (1

t)

(1

zt)

a

:

(a; b; c; z) has the Mellin transform representation b)Fp(

B(b; c

; )

(a; b; c; z) = M ffa;b;c (t : z; ; ; p) : bg :

Taking derivative of fa;b (t : y; ; ; p); we get @ (fa;b (t : z; ; ; p)) = @t +azH(1 p

H(1

c b 1

t) (1

t)

t) (1

t)

h

(1

(1

c b 1

t)

a 1

zt)

(1

c b 1

t) (1

zt)

a

i

(1

zt)

1 F1

; ;

1

1

t2

+

+ (c

p t(1

b

1)H(1

c b 2

t) (1

t)

(1

dt

t) 1 F1

t)2

(1

a

+ 1;

+ 1;

p t(1

t)

:

Since M ff 0 (t) : bg = (1

b)M ff (t) : b

1g

we get (b

1)B(b

+ azB(b; c +

p

B(b; c

1; c

b

b)Fp(

; )

b

1)Fp(

; )

1; c; z) = (c b 1)B(b; c b 1)Fp( ; ) (a; b; c p B(b 2; c b)Fp( +1; +1) (a; b 2; c 2; z)

(a; b

(a + 1; b; c; x)

2)Fp(

+1; +1)

(a; b; c

1; x)

2; z) :

This completes the proof. Similarly, using (6), we get: ( ; ;p)

Theorem 12 We have the following di¤ erence formula for 1 F1 (b

1)B(b

1; c

b

( ; ;p)

1)1 F1

(b

(b; c; z) :

1; c; z) ( ; ;p)

( ; ;p)

= (c b 1)B(b; c b 1)1 F1 (b; c 1; z) + zB(b; c b)1 F1 (b; c; z) p p ( +1; +1;p) ( +1; +1;p) B(b 2; c b)1 F1 (b 2; c 2; z) + B(b; c b 2)1 F1 (b; c Proof. Observe from (6) that B(b; c

( ; ;p)

b)1 F1

fa;b;c (t : z; ; ; p) = H(1

(b; c; z) is the Mellin transform of the function c b 1 zt

t) (1 1200

2; z) :

t)

7

e

1 F1

; ;

p t (1

t)

:

zt)

a

BOZER, OZARSLAN: HYPERGEOMETRIC FUNCTIONS

Hence B(b; c

( ; ;p)

b)1 F1

(b; c; z) has the Mellin transform representation

B(b; c

( ; ;p)

b)1 F1

(b; c; z) = M ffa;b;c (t : z; ; ; p) : bg :

Taking derivative of fa;b (t : y; ; ; p); we get h @ c b 1 zt c (fa;b (t : z; ; ; p)) = (1 t) (1 t) e + (c b 1)H(1 t) (1 t) @t i p c b 1 zt dt +zH(1 t) (1 t) e ; ; 1 F1 t(1 t) 1 p p 1 c b 1 zt + + 1; + 1; H(1 t) (1 t) e : 1 F1 2 2 t (1 t) t(1 t)

b 2 zt

e

Since M ff 0 (t) : bg = (1

we get (b

1)B(b

+ zB(b; c +

p

B(b; c

1; c

b ( ; ;p)

b)1 F1 b

( ; ;p)

1)1 F1

(b; c; z) ( +1; +1;p)

2)1 F1

(b p

b)M ff (t) : b

1; c; z) = (c B(b

(b; c

2; c

b

1g

1)B(b; c

( +1; +1;p)

b)1 F1

b (b

( ; ;p)

1)1 F1 2; c

(b; c

1; z)

2; z)

2; z) :

This completes the proof.

References [1] M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma functions with applications, Journal of Computational and Applied Mathematics 55 (1994) 99-124. [2] M.A. Chaudhry, A. Qadir, M. Ra…que, S.M. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math. 78 (1997) 19–32. [3] M.A. Chaudhry, A. Qadir, H.M. Srivastava, R.B. Paris , Extended hypergeometric and con‡uent hypergeometric functions, Appl. Math. and Comput. 159 (2004) 589–602. [4] M. A. Chaudhry and S. M. Zubair, On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms, Journal of Computational and Applied Mathematics 59 (1995) 253-284. [5] M. A. Chaudhry, N.M. Temme, E.J.M. Veling, Asymptotic and closed form of a generalized incomplete gamma function, Journal of Computational and Applied Mathematics 67 (1996) 371-379. [6] A.R. Miller, Reduction of a generalized incomplete gamma function, related Kampe de Feriet functions, and incomplete Weber integrals, Rocky Mountain J. Math. 30 (2000) 703-714. [7] G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999. [8] M. A. Chaudhry and S. M. Zubair, Extended incomplete gamma functions with applications, J. Math. Anal. Appl. 274 (2002) 725-745. [9] F. AL-Musallam and S.L. Kalla, Futher results on a generalized gamma function occurring in diffraction theory, Integral Transforms and Special Functions, 7 (3-4) (1998) 175-190. [10] Mehmet Ali Özarslan, Some Remarks on Extended Hypergeometric, Extended Con‡uent Hypergeometric and Extended Appell’s Functions, Journal of Comut. Anal. and Appl., 14 (6) (2012), 1148-1153. [11] E. Özergin, M.A. Özarslan and A. Alt¬n, Extension of gamma, beta and hypergeometric functions, Journal of Computational and Applied Mathematics, 235(16), 4601-4610. [12] M. Aslam Chaudry, Syed M Zubair, On a Class of Incomplete Gamma Functions with Applications 2002 by Chapman & Hall/CRC.

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8

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1202-1210, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A modified AOR iterative method for new preconditioned linear systems for L−matrices∗ Guang Zeng†, Li Lei‡ School of Science, East China Institute of Technology, Fuzhou, Jiangxi, 344000, PR China

Abstract In this paper, a preconditioned AOR iterative method is presented with a new preconditioner, and the corresponding convergence and comparison results are given. The optimum parameters and spectral radius for strictly diagonally dominant L-matrices are found. Numerical examples are given to illustrate the efficiency of our method. Keywords: AOR− iteration method; L− matrix; Spectral radius; Optimum parameters; Preconditioner. AMS subject classification: 65F10

1

Introduction

Consider the following linear system Ax = b,

(1.1)

where A ∈ Rn×n , b, x ∈ Rn . For any splitting, A = M − N with det(M ) 6= 0, the basic iterative scheme for solving (1.1) is as follows xk+1 = M −1 N xk + M −1 b, k = 0, 1, · · ·. ∗

The work is supported by the National Natural Science Foundation of China (11226331). G. Zeng was supported by the Doctor Fund of East China Institute of Technology (No. DHBK201201) and L. Lei was suppoted the Principal fund of East China Institute of Technology (No. DHXK1217). Partial results of this paper were represented at the 4th International Conference on Bioinformatics and Biomedical Engineering,Chengdu, China. June 18-20, 2010. † Corresponding author: [email protected] (G. Zeng) ‡ [email protected] (L. Lei)

1202

GUANG ZENG, LI LEI: A MODIFIED AOR ITERATIVE METHOD For simplicity, without loss of generality, we assume throughout this paper that A = I − L − U, where I is the identity matrix, L and U are strictly lower and upper triangular matrices obtained from A, respectively. Thereby the iterative matrix of the classical AOR iterative method in [1] is defined as Lrω = (I − rL)−1 [(1 − ω)I + (ω − r)L + ωU ],

(1.2)

where ω and r are real parameters with ω 6= 0. The spectral radius of the iterative matrix determines the convergence and stability of the method, and the smaller it is, the faster the method converges when the spectral radius is smaller than 1. In order to accelerate the convergence of the iterative method solving (1.1), preconditioned methods are often utilized, which is, which is, P Ax = P b, (1.3) where P is the nonsingular preconditioner. ˆ with Construct P = (I + S)   0 0 ··· 0 0  −α2 a2,1 0 · · · 0 0     −α3 a3,1 0 · · · 0 0  ˆ S=  , and αi ∈ R, i = 2, · · · , n.  .. .. .. .. ..   . . . . .  −αn an,1 0 · · · 0 0 The Equation (1.3) transform to

ˆ = ˆb, Ax

(1.4)

ˆ and ˆb = (I + S)b. ˆ where Aˆ = (I + S)A The coefficient matrix of (1.4) is splited as ˆ −L ˆ − Uˆ , Aˆ = D

(1.5)

ˆ = diag(A), ˆ L ˆ and Uˆ are strictly lower and upper triangular matrices where D ˆ respectively. Through some trivial calculation, we obtain that obtained from A, ˆ = diag(1, 1 − α2 a2,1 a1,2 , · · · , 1 − αn an,1 a1,n ), D and

   ˆ = − L   

0 a2,1 − α2 a2,1 a3,1 − α3 a3,1 .. .

0 0 a3,2 − α3 a3,1 a1,2 .. .

··· ··· ··· .. .

0 0 0 .. .

0 0 0 .. .

an,1 − αn an,1 an,2 − αn an,1 a1,2 an,3 − αn an,1 a1,3 . . . 0

1203

    ,  

GUANG ZENG, LI LEI: A MODIFIED AOR ITERATIVE METHOD and

    Uˆ = −   

0 a1,2 a1,3 0 0 a2,3 − α2 a2,1 a1,3 0 0 0 .. .. .. . . . 0 0 0

··· a1,n · · · a2,n − α2 a2,1 a1,n · · · a3,n − α3 a3,1 a1,n .. ... . ···

    .  

0

Applying the AOR method to the preconditioned linear system (1.4), the corresponding preconditioned AOR iterative method is obtained with iterative matrix ˆ rω = (D ˆ − rL) ˆ −1 [(1 − ω)D ˆ + (ω − r)L ˆ + ω Uˆ ], L

(1.6)

where ω and r are real parameters with ω 6= 0. The rest of the article is organized as follows. In Section 2, we briefly explain some notation and some Lemma which are used to state and to prove our results. In Section 3, we sate our result with its proof. Examples are given to illustrate our main theorem in Section 4.

2

Preliminaries

Some notation and Lemmas as follows are needed in this article. A matrix A is nonnegative(positive) if each entry of A is nonnegative(positive), respectively, which is denoted by A ≥ 0, (A > 0). Let ρ(A) be the spectral radius of A. In addition, A matrix A is irreducible if the directed graph associated to A is strongly connected. Lastly, A matrix A is an L−matrix if ai,i > 0, i = 1, 2, · · · , n and ai,j ≤ 0, for all i, j = 1, 2, · · · , n such that i 6= j. The following Lemma will be useful to prove the main results. Lemma 2.1a([5]). Let A ∈ Rm×n , A = M − N is a splitting of A. Then (a). If M −1 ≥ 0 and N ≥ 0, then A = M − N is a regular splitting; (b). If M −1 ≥ 0 and M −1 N ≥ 0, then A = M − N is a weak regular splitting. Lemma 2.1b([5]). Let A = M1 − N1 = M2 − N2 are two regular splitting for matrix A and suppose that A−1 and N2 ≥ N1 ≥ 0. Then 0 ≤ ρ(M1−1 N1 ) ≤ ρ(M2−1 N2 ) < 1. Lemma 2.2a([6]). Let A ∈ C n×n be a non-negative and irreducible n × n matrix. Then (a). A has a positive real eigenvalue equal to its spectral radius ρ(A); (b). There exists an eigenvector x > 0 corresponding to ρ(A), (c). ρ(A) is a simple eigenvalue of A;

1204

GUANG ZENG, LI LEI: A MODIFIED AOR ITERATIVE METHOD (d). ρ(A) increases when any entry of A increases. Lemma 2.2b([6]). Let A be a non-negative matrix. Then (a). If αx ≤ Ax for some non-negative vector x, x 6= 0, then α ≤ ρ(A). (b). If Ax ≤ βx for some positive vector x, then ρ(A) ≤ β. Moreover, if A is irreducible and if 0 6= αx ≤ Ax ≤ βx, αx 6= Ax, Ax 6= βx for non-negative vector x, then α < ρ(A) < β and x is a positive vector.

3

Main results

Our main goal in this section is to establish the following results with proof. Lemma 3.1. Let A and Aˆ be the coefficient matrices of the linear systems (1.1) and (1.4), where A is an L−matrix for which there exists i such that ai,1 6= 0, i = 2, · · · , n, with ai+1,i ai,i+1 6= 0, i = 1, · · · , n − 1. If 0 ≤ r ≤ ω ≤ 1 (ω 6= 0, r 6= 1) and one of the following conditions is also satisfied simultaneously (a). 0 < αi ≤ 1 if 0 < a1,i ai,1 < 1, or ai,1 6= 0 and a1,i = 0; (b). 0 < αi ≤ 1 if a1,i ai,1 = 1; (c). 0 < αi < a1,i1ai,1 if a1,i ai,1 > 1; (d). αi > 0 if ai,1 = 0, i = 1, 2, · · · , n. ˆ r,ω are nonnegative and irreducible. Then the iterative matrices Lr,ω and L Proof. From (1.2) we have Lr,ω = (1 − ω)I + ω(1 − r)L + ωU + T,

(3.1)

where T = rL[ω −r)L+ωU ]+(r2 L2 +· · ·+rn−1 Ln−1 )[(1−ω)I +(ω −r)L+ωU ] ≥ 0. (3.2) ˆ r,ω are nonnegative and irreducible according to lemma 1 of [4]. 2 Then Lr,ω and L ˆ r,ω be Theorem 3.2. Under the assumptions of Lemma 3.1, and let Lr,ω and L the iterative matrices of the AOR method obtained from (1.1) and (1.4), respectively. Then we have ˆ r,ω ) < ρ(Lr,ω ) < 1, if ρ(Lr,ω ) < 1; (a) ρ(L ˆ r,ω ) = ρ(Lr,ω ) = 1, if ρ(Lr,ω ) = 1; (b) ρ(L ˆ r,ω ) > ρ(Lr,ω ) > 1, if ρ(Lr,ω ) > 1. (c) ρ(L ˆ r,ω are nonnegative and Proof. From Lemma 3.1 it is obvious that Lr,ω and L irreducible. Therefore, according to Lemma 2.2a there is a positive vector x, such that Lr,ω x = λx, (3.3)

1205

GUANG ZENG, LI LEI: A MODIFIED AOR ITERATIVE METHOD where λ = ρ(Lr,ω ), (3.3) can equivalently to [(1 − ω)I + (ω − r)L + ωU ]x = λ(I − rL)x.

(3.4)

Now consider ˆ rω x − λx = (D ˆ − rL) ˆ −1 [(1 − ω)D ˆ + (ω − r)L ˆ + ω Uˆ ]x − λx L ˆ − rL) ˆ −1 [(1 − ω)D ˆ + (ω − r)L ˆ + ω Uˆ − λ(D ˆ − rL)]x ˆ = (D ˆ − rL) ˆ −1 [(1 − ω − λ)D1 + ωU1 − (ω − r + λr)L1 ]x = (D ˆ − rL) ˆ −1 (0, η2 , η3 , · · · , , ηn )T , = (λ − 1)(D

(3.5)

P where ηi = αi ai,1 [r 1 λ = ρ(Lrω ) by Lemma 2.2b again. fore, Lrω x ≥ λx. Furthermore, we get ρ(L The proof of Theorem 3.2 is completed. 2 According to our main result, we have the following corollary. ¯ r,ω be the iterative matrices of the AOR method Corollary 3.3. Let Lr,ω and L obtained from (1.1) and (1.4), respectively. Under the same conditions in Theorem 3.2 except for the ones for αi , i = 2, · · · , n, we have ¯ r,ω ) < ρ(Lr,ω ) < 1, if ρ(Lr,ω ) < 1; (a) ρ(L ¯ r,ω ) = ρ(Lr,ω ) = 1, if ρ(Lr,ω ) = 1; (b) ρ(L ¯ r,ω ) > ρ(Lr,ω ) > 1, if ρ(Lr,ω ) > 1. (c) ρ(L Now we show how the Modified AOR optimum parameters and spectral radius ˆ = I ˆ − L ˆ − U ˆ . We redefine (1.5), are found. For convenience, let A˜ = SA S S S ˆ ˆ ˆ D = I + ISˆ , L = L + LSˆ , U = U + USˆ . Lemma 3.4 Under the assumptions of Lemma 3.1, and A is a strictly diagonally dominant L−matrix. Then Aˆ is a strictly diagonally dominant L−matrix. Proof. We first prove Aˆ is an L−matrix. ˆ =D ˆ −L ˆ − Uˆ Aˆ = (I + S)A

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GUANG ZENG, LI LEI: A MODIFIED AOR ITERATIVE METHOD     =  

1 a2,1 − α2 a2,1 a3,1 − α3 a3,1 .. .

a1,2 1 − α2 a1,2 a2,1 a3,2 − α3 a3,1 a1,2 .. .

an,1 − αn an,1 an,2 − αn an,1 a1,2

a1,3 ··· a1,n a2,3 − α2 a2,1 a1,3 · · · a2,n − α2 a2,1 a1,n 1 − α3 a1,3 a3,1 · · · a3,n − α3 a3,1 a1,n .. .. ... . . .. an,3 − αn an,1 a1,3 . 1 − αn a1,n an,1

    .  

Since A is a strictly diagonally dominant L−matrix, the non-diagonal elements of ˆ we the first line of Aˆ are non-positive. For all the lines from the second line for A, have ( ai,j − αi ai,1 a1,j ≤ 0, if i 6= j a ˆi,j = 1 − αi ai,1 a1,j > 0, if i = j Thus Aˆ is an L-matrix. Below we prove Aˆ is a strictly diagonally dominant matrix. ˆ |a1,2 + a1,3 + · · · + a1,n | < 1| holds, and for the i−th line For the first line of A, |(ai,1 − αi ai,1 ) + (ai,i−1 − αi ai,1 a1,i−1 ) + (ai,i+1 − αi ai,1 a1,i+1 ) + · · · + (ai,n − αi ai,1 a1,n )| = −(ai,1 + · · · + ai,i−1 + ai,i+1 + · · · + ai,n ) + αi ai,1 (1 + · · · + a1,i−1 + a1,i+1 + · · · + a1,n ) ≤ 1 − αi ai,1 a1,i holds too. Then, Aˆ is a strictly diagonally dominant L−matrix. 2 Theorem 3.5. Under the assumptions of Lemma 3.4, ρ(Lr,ω ) < 1, 0 ≤ r ≤ ω ≤ 1 and ω > 0. Then ˆ 1,1 ) ≤ ρ(L ˆ r,ω ) < 1. ρ(L

(3.6)

If r = 1, ω = 1 and α = [1, 1, · · · , 1], then equality holds in (3.6). Proof. From Equation (1.6), we get ˆ r,ω =[(I + I ˆ ) − r(L + L ˆ )]−1 L S S × [(1 − ω)(I + ISˆ ) + (ω − r)(L + LSˆ ) + ω(U + USˆ )], 0

let s denote M2 = [(I + ISˆ ) − r(L + LSˆ )], N2 = [(1 − ω)(I + ISˆ ) + (ω − r)(L + LSˆ ) + ω(U + USˆ )], ˆ according to Lemma 3.4, we known that Aˆ = (I + S)A is a strictly diagonally dominant L−matrix. Hence, (I + ISˆ )−1 ≥ 0 and ρ[(I + ISˆ )−1 (L + LSˆ )] < 1.

1207

GUANG ZENG, LI LEI: A MODIFIED AOR ITERATIVE METHOD Moreover, M2−1 = {I + [r(I + ISˆ )−1 (L + LSˆ )] + [r(I + ISˆ )−1 (L + LSˆ )]2 + · · · } × (I + ISˆ )−1 ≥ 0, (3.7) N2 = [(1 − ω)(I + ISˆ ) + (ω − r)(L + LSˆ ) + ω(U + USˆ )] ≥ 0,

(3.8)

M2 − N2 =[(I + ISˆ ) − r(L + LSˆ )]− [(1 − ω)(I + ISˆ ) + (ω − r)(L + LSˆ ) + ω(U + USˆ )] =ω[(I + ISˆ ) − (L + LSˆ ) − (U + USˆ )] ˆ =ω A.

(3.9)

and

Therefore, ω Aˆ = M2 − N2 is a regular splitting. On the other hand, ˆ 1,1 =[(I + I ˆ ) − (L + L ˆ )]−1 (U + U ˆ ) L S S S =[ω(I + ISˆ ) − ω(L + LSˆ )]−1 ω(U + USˆ ). Let M1 = ω(I + ISˆ ) − ω(L + LSˆ ), N1 = ω(U + USˆ ), −1

since (I + ISˆ )

≥ 0 and ρ[(I + ISˆ )−1 (L + LSˆ )] < 1, we have

1 M1−1 = {I + [(I + ISˆ )−1 (L + LSˆ )] ω + [(I + ISˆ )−1 (L + LSˆ )]2 + · · · } × (I + ISˆ )−1 ≥ 0,

(3.10)

N1 = ω(U + USˆ ) ≥ 0,

(3.11)

M1 − N1 =ω(I + ISˆ ) − ω(L + LSˆ ) − ω(U + USˆ ) =ω[(I + ISˆ ) − (L + LSˆ ) − (U + USˆ )] ˆ =ω A.

(3.12)

and

According to (3.7-12), ω Aˆ = M2 − N2 = M1 − N1 are two different regular splitting ˆ and N2 = (1−ω)(I +I ˆ )+(ω−r)(L+L ˆ )+ω(U +U ˆ ) ≥ ω(U +U ˆ ) = N1 ≥ 0, of ω A, S S S S we can obtain ρ(M1−1 N1 ) ≤ ρ(M2−1 N2 ) < 1 by Lemma 2.1b. Hence, ˆ 1,1 ) ≤ ρ(L ˆ r,ω ) < 1. ρ(L ˆ r,ω ) = ρ(L ˆ 1,1 ) hold. 2 In particular, if r = 1, ω = 1 and α = [1, 1, · · · , 1], then ρ(L

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GUANG ZENG, LI LEI: A MODIFIED AOR ITERATIVE METHOD

4

Numerical experiments

In this section we give some numerical examples to illustrate the results obtained in Section 3. Example 4.1. Consider the matrix A of (1.1),  1 −0.01 −0.5 −0.3  −0.2 1 −0.1 −0.15   −0.1 −0.14 1 −0.05 A=  −0.2 −0.05 −0.11 1   −0.4 −0.03 −0.05 −0.15 −0.08 −0.3 −0.1 −0.1

and given by  −0.05 −0.3 −0.12 −0.14   −0.4 −0.2  . −0.2 −0.1   1 −0.2  −0.3 1

For the modified AOR iterative method, we have the following results. The digital of following table is formed by Matlab R2010a program. Table 1. Numerical illustration of our main results (α2 , α3 , · · · , α6 ) (0.5, 0.5, 0.5, 0.5, 0.5) (0.6, 0.6, 0.6, 0.6, 0.6) (0.8, 0.8, 0.8, 0.8, 0.8) (0.9, 0.9, 0.9, 0.9, 0.9) (1, 1, 1, 1, 1)

ω 0.8 0.75 0.8 0.95 1

r 0.2 0.65 0.6 0.9 1

ρ(Lr,ω ) 0.8890 0.8674 0.8629 0.7999 0.7710

ˆ r,ω ) ρ(L 0.8745 0.8462 0.8317 0.7471 0.7009

ˆ r,ω ) < ρ(Lr,ω ) when ρ(Lr,ω ) < 1. Remark 4.1. From the above table, we know ρ(L ˆ r,ω ) = ρ(L ˆ 1,1 ) hold. So In particular, if r = 1, ω = 1 and α = [1, 1, · · · , 1], then ρ(L the results are in concord with our main results. Example 4.2. Consider the matrix A of (1.1),  1 −0.01 −0.5 −0.3  −0.4 1 −0.2 −0.15   −0.5 −0.14 1 −0.5 A=  −0.2 −0.05 −0.11 1   −0.6 −0.05 −0.06 −0.15 −0.7 −0.3 −0.1 −0.1

and given by −0.05 −0.12 −0.6 −0.2 1 −0.3

−0.3 −0.3 −0.2 −0.1 −0.2 1

    .   

For the modified AOR iterative method, we have the following results.

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GUANG ZENG, LI LEI: A MODIFIED AOR ITERATIVE METHOD Table 2. Numerical illustration of Theorem 3.2 (α2 , α3 , · · · , α6 ) (0.9, 0.1, 0.5, 0.2, 0.8) (0.4, 0.7, 0.8, 0.3, 0.6) (0.7, 0.2, 0.8, 0.3, 0.3) (0.2, 0.4, 0.6, 0.7, 0.3) (0.2, 0.4, 0.6, 0.7, 0.3)

ω 0.05 0.7 0.75 0.8 0.95

r 0.05 0.3 0.65 0.6 0.9

ρ(Lr,ω ) 1.0121 1.1970 1.2706 1.2778 1.4209

ˆ r,ω ) ρ(L 1.0151 1.2686 1.3252 1.3568 1.5515

ˆ r,ω ) > ρ(Lr,ω ) when Remark 4.2. From the above table, it is easy to know that ρ(L ρ(Lr,ω ) > 1. The results are also in concord with Theorem 3.2 and Corollary 3.3.

References [1] A. Hadjidimos, D. Noutsos, M. Tzoumas, More on modifications and improvements of classical iterative schemes for M −matrices, Linear Algebra Appl. 364(2003) 253-279. [2] D.J. Evans, M.M. Martins, M.E. Trigo, The AOR iterative method for new preconditioned linear systems, J Comput Math. 132 (2001) 461-466. [3] A.D. Gunawardena, S.K. Jain, L.Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl. 154-156 (1991) 99-110. [4] C. Li., D.J. Evans, Improving the SOR Method, Technical Rrport 901, Department of computer studies, University of Loughborough, 1994. [5] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1985. [6] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1981. [7] D.M.Young, Iterative Solution of Large Linear Systems, Academic Press, NewYork, London, 1971. [8] B. Robert. Iterative Solution Methods, Appl Numer Math. 2004, 51:437-450.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1211-1222, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

JOCAAA manuscript No. (will be inserted by the editor)

Modern Algorithms of Simulation for Getting Some Random Numbers

G. A. Anastassiou, I. F. Iatan 1

2

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA, e-mail: [email protected] Department of Mathematics and Computer Science, Technical University of Civil Engineering, Bucharest, Romania, e-mail: [email protected]

The date of receipt and acceptance will be inserted by the editor

Abstract In order to carry out the simulation, we need a source of random numbers distributed according to the desired probability distribution. In this paper we have constructed algorithms for generating both continuous and discrete random variables. One simulates a discrete random variable having a geometric distribution, which is used in reliability.We also create some algorithms for generating: a normal continuous variable, other continuous variables having exponential distribution, Weibull distribution, gamma distribution. The aim of this paper is to see that if we have random numbers generated according to some distribution, we may perform a transformation to generate the desired distribution. Key words generalized test likelihood ratio – parametric classi…cation criterion –maximum likelihood estimates –likelihood function 1 Inverse Transform Method We shall describe a method of simulating a discrete random variable that take a …nite number of values, called inverse transform method. According to this method, we can simulate any random variable X if we know its distribution function F and we can calculate the inverse function F 1 : Using this method, we build a Matlab program to simulate the discrete variable X, whose distribution is Send o¤ print requests to: G. A. Anastassiou Correspondence to: University of Memphis, Memphis, TN 38152, USA

1211

2

G. A. Anastassiou, I. F. Iatan

a1 p1

X:

ak pk

am pm

;

where m X

pk = 1:

k=1

Its distribution function is: 8 0; x a1 > > > > > > > > p1 ; x 2 (a1 ; a2 ] > > > > > > < p1 + p2 ; x 2 (a2 ; a3 ] FX (x) = .. > > . > > > > p + p + + pk ; x 2 (ak ; ak+1 ] > 1 2 > > > . > . > > . > : 1 x > am

(1)

and the inverse function will be: FX 1 (u) = ak ;

x 2 (FX (ak

1) ;

FX (ak )] ;

(8) k = 1; m;

where a0 =

1; FX (a0 ) = 0:

The algorithm for simulating the random variable X consists of: – generating a value u uniformly distributed in [0; 1]; – …nding the index k for which FX (ak

1)

> > > 1=6; x 2 (1; 2] > > > > < 2=6; x 2 (2; 3] FX (x) = 3=6; x 2 (3; 4] > > 4=6; x 2 (4; 5] > > > > 5=6; x 2 (5; 6] > > : 1; x > 6:

In the command line of Matlab we shall write: >> a = 1 : 7; >> F = 0 : 1=6 : 1; >> x = simdiscrv(F; a; 7) It will display: u= 0.6557 x= 5

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G. A. Anastassiou, I. F. Iatan

2 Simulation of a random variable having a geometric distribution Let X be a random variable signifying the number of failures until a certain success in a number of independent Bernoulli samples. So, X has the distribution:

X:

0 1 2 p pq pq 2

k pq k

n pq n

and with the mean and respectively the variance: M (X) = V ar (X) =

q p q p2 ;

where p is the probability the probability of having a success, i.e the probability that a random event observable A to occur in a random experience and q = 1 p is the probability to achieve a failure, i.e the probability that the event contrary A to occur. The distribution function of X is:

F (x) = P (X < x) =

x X

pq k = 1

q x+1 ; x = 0; 1; 2;

; n;

k=0

namely it is a discrete distribution function. The name of geometric distribution comes from the fact that P (X = x) = pq x is thew term of a geometric progression. The simulation the random variable X, which has a geometric distribution can be also achieved by means of the inverse transform method, using the formula: X=

log (U ) ; log (q)

(3)

where: – [a] is the integer part of a, – U is a random variable, uniformly distributed in [0; 1].

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Modern Algorithms of Simulation for Getting Some Random Numbers

5

3 Simulation of a random variable with a exponential and a Weibull distribution

A exponential variable X

Exp( ) has the probability density function:

(8)

x

e

f (x) =

0;

;x>0 x 0

2 R; the distribution function: Z x FX (x) = f (t) dt = 1

e

x

; x>0

1

and M (X) = V ar (X) =

1 1 2

:

To simulate a random variable X, which has an exponential distribution we shall use the inverse transform method, hence the algorithm for simulating the random variable X consists in: – generating a value u uniformly distributed in [0; 1], – …nding of

X=F

1

1

(u) =

ln (1

u) :

A Weibull variable (denoted W ( ; ; )) is a random variable, closely related to the exponential random variable and which has the probability density function: f (x) = (8)

(x

)

1

e

(x

)

0;

;x> x

2 R; ; > 0: If X Exp(1) then the Weibull variable is generated using the formula W =

+

X

1

:

(4)

Indeed, we have: P (W < w) = P (X
0;

2

where ( )=

Z

1

x

1

e

x

dx

(6)

0

signi…es the Euler Gamma function, ties: 8 > > < and

> > :

: (0; 1) ! R; which has the proper-

p = (1) = 1 (a + 1) = a (a) ; (8) a > 0 (n + 1) = n!; (8) n 2 N 1 2

M V ar

2 2

= =2 :

For the simulation in Matlab of a random variable formula (5): function x=hip(n) z=randn(n,1); x=sum(z.^2); end

1216

2

we shall use the

Modern Algorithms of Simulation for Getting Some Random Numbers

7

5 Simulation of a random variable with a Gamma distribution A random variable X has has the distribution G( ; ; ) if it has the probability density function:

f (x) =

(

( )

(x

1

) 0;

(x

e

)

x> x

where (8) 2 R; ; > 0 are respectively the parameters of location, scale and form of the variable. We can notice that an exponential variable is a gamma variable G(0; ; 1) and 2 is a gamma variable G(0; 21 ; 2 ): If Y G( ; ; 2 ) and Z G(0; 12 ; 2 ) then we have: Y =

+

Z : 2

(7)

The relation (7) can be justi…ed as follows:

FZ (z) = P (Z < z) = P (2 (Y Z + 2z z = FY + = 2 1

) < z) = P Y < 2

(t

1

)2

e

+

z 2

(t

)

dt

2

and further, using the change of variable w = 2 (t

)

we shall achieve:

FZ (z) =

Z

z 1

2

2

w 2

2

1

e

w 2

dw = 2

Z

z

2

w2 1

1

e

w 2

dw:

2

For the simulation in Matlab of a random variable Y , whose distribution is G ; ; 2 we proceed as follows: – one generates Z = 2 ; – one determines Y using (7). Hence, we have: function y =gam(al,la,n) z=hip(n); y=al+z/(2*la); end

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G. A. Anastassiou, I. F. Iatan

6 Validation of the Generators The validation of the generators one refers both to the formal correctness of the programs and to the checking of the statistical hypothesis of concordance H: X

F (x)

(8)

with regard to distribution function F (x) of the random variable X, over which the simulated selection X1 ; X2 ; ; Xn , of volume n big enough has been made. The validation of the generators involves the following two steps: A) Building the graphical histogram and comparing it with the probability density of X. B) Application of the concordance test 2 to verify the hypothesis (8). The histogram construction is done using the following algorithm: Step 1.We simulate a number n1 ak then then we set: ak+1 = max fak+1 ; Xg and fk = fk + 1; c) if a2 < X ak then we set: p = X ha2 + 2 and fp+1 = fp+1 + 1. Step 6. We represent graphically the selection histogram X1 ; X2 ; ; Xn , as follows: we take on the abscissa the intervals Ii , then we build some rectangles having these intervals as their bases and the relative frequencies fi as their heights. Remark 1 For a discrete random variable X, which takes the values a1 ; a2 ; ; am with the probabilities p1 ; p2 ; ; pm , the probability density function f (x) is de…ned by: pi ; if x = xi ; i = 1; m 0; otherwise

f (x) =

(9)

and the and distribution function is given in (1). With the built histogram, we can apply the test 2 to verify the hypothesis (8). Therefore, we have to compute the statistics 2

=

k X (ni i=1

which has a distribution theorem), where:

2

2

npi ) ; npi

, with k 1 degrees of freedom (see Karl Pearson’s

– k is the number of intervals in the histogram, – ni (8) i = 1; k represent the absolute frequencies, – pi (8) i = 1; k are the probabilities that an observation to belong to the interval Ii and they are expressed by: 8
𝑦 if and only if 𝑥i > 𝑦i , 𝑖 = 1, 2, · · · , 𝑛. In this paper, we consider the following multiobjective programming problem: (P )

M 𝑖𝑛𝑖𝑚𝑖𝑧𝑒 s.t.

𝑓 (𝑥) 𝑔(𝑥) 5 0, 𝑥 ∈ X,

where 𝑓 = (𝑓1 , 𝑓2 , · · · , 𝑓k ) : X → Rk , 𝑔 = (𝑔1 , 𝑔2 , · · · , 𝑔m ) : X → Rm are assumed to be twice differentiable functions over X, an open subset of Rn . Definition 2.1 A feasible point 𝑥 is said to be an efficient solution of the vector minimum problem (P) if there exists no other feasible point 𝑥 such that 𝑓 (𝑥) ≤ 𝑓 (𝑥). Assume that 𝛼 : X × X → R+ \ {0}, 𝜌 ∈ R and 𝑑 : X × X → R+ satisfies 𝑑(𝑥, 𝑥0 ) = 0 ⇔ 𝑥 = 𝑥0 . Let C : X × X × Rn → R be a function which satisfies C(x,x0 ) (0) = 0 for any (𝑥, 𝑥0 ) ∈ X × X. Definition 2.2 [19]A function C : X × X × Rn → R is said to be convex on Rn iff for any fixed (𝑥, 𝑥0 ) ∈ X × X and for any 𝑦1 , 𝑦2 ∈ Rn , one has C(x,x0 ) (𝜆𝑦1 + (1 − 𝜆)𝑦2 ) ≤ 𝜆C(x,x0 ) (𝑦1 ) + (1 − 𝜆)C(x,x0 ) (𝑦2 ), ∀𝜆 ∈ (0, 1).

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S.C.Wang, Second order duality for multiobjective programming

Definition 2.3 [19]A differentiable function ℎ : X → R is said to be (C, 𝛼, 𝜌, 𝑑)-convex at 𝑥0 iff for any 𝑥 ∈ X ℎ(𝑥) − ℎ(𝑥0 ) 𝑑(𝑥, 𝑥0 ) ≥ C(x,x0 ) (∇ℎ(𝑥0 )) + 𝜌 . 𝛼(𝑥, 𝑥0 ) 𝛼(𝑥, 𝑥0 ) The function ℎ is said to be (C, 𝛼, 𝜌, 𝑑)-convex on X iff ℎ is (C, 𝛼, 𝜌, 𝑑)-convex at every point in X. In the sequel, we introduce a class of second order (C, 𝛼, 𝜌, 𝑑)-convexity. Definition 2.4 A twice differentiable function 𝑓i over X is said to be (strict) second order (C, 𝛼, 𝜌, 𝑑)convex at 𝑥0 if for all 𝑥 ∈ X and for all 𝑝 ∈ Rn , 𝑓i (𝑥) − 𝑓i (𝑥0 ) + 12 𝑝T ∇2 𝑓i (𝑥0 )𝑝 𝑑(𝑥, 𝑥0 ) (>) ≥ C(x,x0 ) (∇𝑓i (𝑥0 ) + ∇2 𝑓i (𝑥0 )𝑝) + 𝜌 . 𝛼(𝑥, 𝑥0 ) 𝛼(𝑥, 𝑥0 ) A twice differentiable vector function 𝑓 : X → Rk is said to be second order (C, 𝛼, 𝜌, 𝑑)-convex at 𝑥0 if each of its components 𝑓i is second order (C, 𝛼, 𝜌, 𝑑)-convex at 𝑥0 . Remark 2.1 From the above definition, second order (F, 𝛼, 𝜌, 𝑑)-convexity[5] is a special case of (C, 𝛼, 𝜌, 𝑑)-convexity, since any linear function is also a convex function. The following convention will be followed. If 𝑓 is an k-dimensional vector function, then 𝑓 (𝑢) − ∇𝑓 (𝑢)r− 12 𝑝T ∇2 𝑓 (𝑢)𝑝 denotes the vector of components 𝑓1 (𝑢)−∇𝑓1 (𝑢)r− 21 𝑝T ∇2 𝑓1 (𝑢)𝑝, · · · , 𝑓k (𝑢)− ∇𝑓k (𝑢)r − 21 𝑝T ∇2 𝑓k (𝑢)𝑝. In order to prove the strong duality theorem, we need the following Kuhn-Tucker type necessary conditions [9]. Theorem 2.1 (Kuhn-Tucker type necessary conditions)Assume that 𝑥∗ is an efficient solution for (P) at which Kuhn-Tucker constraint qualification is satisfied. Then there exist 𝜆∗ ∈ Rk and 𝑦 ∗ ∈ Rm such that 𝜆∗T ∇𝑓 (𝑥∗ ) + 𝑦 ∗T ∇𝑔(𝑥∗ ) = 0, 𝑦 ∗T 𝑔(𝑥∗ ) = 0, 𝑦 ∗ = 0, 𝜆∗ ≥ 0.

3.

Second order Mond-Weir type duality

In this section, we consider the following Mond-Weir type second order dual associated with multiobjective problem (P) and establish weak, strong and strict converse duality theorems under second

1225

S.C.Wang, Second order duality for multiobjective programming

order (C, 𝛼, 𝜌, 𝑑)-convexity. (M D)

M 𝑎𝑥𝑖𝑚𝑖𝑧𝑒

s.t.

𝑓 (𝑢) − ∇𝑓 (𝑢)T r − 21 𝑝T ∇2 𝑓 (𝑢)𝑝, k k m m ∑ ∑ ∑ ∑ 𝜆i ∇𝑓i (𝑢) + 𝜆i ∇2 𝑓i (𝑢)𝑝 + 𝑦i ∇𝑔i (𝑢) + 𝑦i ∇2 𝑔i (𝑢)𝑝 = 0, i=1 m ∑ i=1 k ∑ i=1 m ∑

i=1

𝑦i 𝑔i (𝑢) −

m ∑

i=1

𝑦i ∇𝑔i

(𝑢)T r

−

i=1

m ∑

i=1

i=1 1 T 2 𝑦i 2 𝑝 ∇ 𝑔i (𝑢)𝑝 ≥

0,

𝜆i ∇𝑓i (𝑢)T r ≥ 0, 𝑦i ∇𝑔i (𝑢)T r ≥ 0,

i=1

𝑦 = 0, 𝜆 = 0, r ∈ Rn , 𝑦 ∈ Rm , 𝜆 ∈ Rk . Remark 3.1 If r = 0, then (MD) becomes the dual considered in [5]. Theorem 3.1 (Weak duality)Suppose that for all feasible 𝑥 in (P) and all feasible (𝑢, 𝑦, 𝜆, r, 𝑝) in (MD). If 𝑔i (·)(𝑖 = 1, 2, · · · , 𝑚) is second order (C, 𝛼1 , 𝜌1 , 𝑑1 )-convex and 𝑓i (·)(𝑖 = 1, 2, · · · , 𝑘) is second order (C, 𝛼2 , 𝜌2 , 𝑑2 )-convex, and 𝜌1 αd11

m ∑ i=1

𝑦i + 𝜌2 αd22

k ∑

𝜆i ≥ 0,

(3.1)

i=1

then the following cannot hold: 1 𝑓 (𝑥) ≤ 𝑓 (𝑢) − ∇𝑓 (𝑢)T r − 𝑝T ∇2 𝑓 (𝑢)𝑝. 2 Proof. Suppose the conclusion is not true, i.e., 1 𝑓 (𝑥) ≤ 𝑓 (𝑢) − ∇𝑓 (𝑢)T r − 𝑝T ∇2 𝑓 (𝑢)𝑝. 2 In view of (C, 𝛼2 , 𝜌2 , 𝑑2 )-convexity of 𝑓i (·) at 𝑢, we obtain −

k ∑ i=1

λi T α2 ∇𝑓i (𝑢) r

> ≥

k ∑ i=1 k ∑ i=1

𝜆i

fi (x)−fi (u)+ 12 pT ∇2 fi (u)p α2

𝜆i C(x,u) (∇𝑓i (𝑢) + ∇2 𝑓i (𝑢)𝑝) + 𝜌2 αd22

k ∑

(3.2) 𝜆i .

i=1

Let 𝑥 be any feasible solution in (P) and (𝑢, 𝑦, 𝜆, r, 𝑝) be any feasible solution in (MD). Then we have m m m m ∑ ∑ ∑ 1∑ T 2 𝑦i 𝑔i (𝑥) ≤ 0 ≤ 𝑦i 𝑔i (𝑢) − 𝑦i 𝑝 ∇ 𝑔i (𝑢)𝑝 − 𝑦i ∇𝑔i (𝑢)T r. 2 i=1

i=1

i=1

i=1

Using second order (C, 𝛼1 , 𝜌1 , 𝑑1 )-convexity of 𝑔i (·) at 𝑢 and the above inequality, we get −

m ∑ i=1

yi T α1 ∇𝑔i (𝑢) r

≥ ≥

m ∑ i=1 m ∑ i=1

𝑦i

gi (x)−gi (u)+ 21 pT ∇2 gi (u)p α1

𝑦i C(x,u) (∇𝑔i (𝑢) + ∇2 𝑔i (𝑢)𝑝) + 𝜌1 αd11

1226

m ∑ i=1

(3.3) 𝑦i .

S.C.Wang, Second order duality for multiobjective programming

Taking into account convexity of C(x,u) (·), (3.2) and (3.3), one gets −

k ∑ i=1

λi T α2 ∇𝑓i (𝑢) r

−

m ∑ i=1

> (

yi T α1 ∇𝑔i (𝑢) r

k ∑

𝜆i +

i=1

i=1

m ∑

+

yi

i=1 𝑦i )C(x,u) { ∑ k

λi +

i=1

m ∑

(∇𝑓i (𝑢) + ∇2 𝑓i (𝑢)𝑝) yi

i=1

(3.4)

yi

i=1

k ∑

λi

(∇𝑔i (𝑢) + ∇2 𝑔i (𝑢)𝑝)}

m ∑

λi +

𝜌2 αd22

k ∑

i=1

i=1 k ∑ i=1

+

m ∑

𝜆i + 𝜌1 αd11

m ∑

𝑦i .

i=1

From the first, third, fourth dual constraint in (MD) and C(x,u) (0) = 0, we obtain 0 > 𝜌2

k m 𝑑2 ∑ 𝑑1 ∑ 𝜆 i + 𝜌1 𝑦i , 𝛼2 𝛼1 i=1

i=1

which contradicts the condition (3.1). Hence the following cannot hold: 1 𝑓 (𝑥) ≤ 𝑓 (𝑢) − ∇𝑓 (𝑢)T r − 𝑝T ∇2 𝑓 (𝑢)𝑝. 2 Theorem 3.2 (Strong duality) Let 𝑥 be an efficient solution of (P) at which the Kuhn-Tucker constraint qualification is satisfied. Then there exist 𝑦 ∈ Rm and 𝜆 ∈ Rk , such that (𝑥, 𝑦, 𝜆, r = 0, 𝑝 = 0) is feasible for (MD) and the objective values of (P) and (D) are equal. Furthermore, if the assumptions of Weak duality hold for all feasible solutions of (P) and (MD), then (𝑥, 𝑦, 𝜆, r = 0, 𝑝 = 0) is an efficient solution of (MD). Proof. Since 𝑥 is an efficient solution of (P) at which the Kuhn-Tucker constraint qualification is satisfied, then by Theorem 2.1, there exist 𝑦 ∈ Rm and 𝜆 ∈ Rk such that 𝜆T ∇𝑓 (𝑥) + 𝑦 T ∇𝑔(𝑥) = 0, 𝑦 T 𝑔(𝑥) = 0, 𝑦 = 0, 𝜆 ≥ 0. Therefore (𝑥, 𝑦, 𝜆, r = 0, 𝑝 = 0) is feasible for (MD) and the objective values of (P) and (MD) are equal. The efficiency of this feasible solution for (MD) follows from the weak duality theorem. Theorem 3.3 (Strict Converse duality) Let 𝑥 and (𝑢, 𝑦, 𝜆, r, 𝑝) be the efficient solution of (P) and (MD), respectively, such that 𝑓 (𝑥) = 𝑓 (𝑢) − ∇𝑓 (𝑥)T r − 12 𝑝T ∇2 𝑓 (𝑢)𝑝.

(3.5)

If 𝑔i (·)(𝑖 = 1, 2, · · · , 𝑚) is strict second order (C, 𝛼1 , 𝜌1 , 𝑑1 )-convex and 𝑓i (·)(𝑖 = 1, 2, · · · , 𝑘) is second order (C, 𝛼2 , 𝜌2 , 𝑑2 )-convex, and 𝜌1 αd11

m ∑ i=1

𝑦 i + 𝜌2 αd22

k ∑ i=1

Then 𝑥 = 𝑢; that is, 𝑢 is an efficient solution of (P). 1227

𝜆i ≥ 0.

(3.6)

S.C.Wang, Second order duality for multiobjective programming

Proof. Suppose the conclusion is not true, i.e.,𝑥 ̸= 𝑢. In view of (C, 𝛼2 , 𝜌2 , 𝑑2 )-convexity of 𝑓i (·) at 𝑢 and (3.5), we obtain −

k ∑ i=1

λi T α2 ∇𝑓i (𝑢) r

k ∑

=

i=1 k ∑

≥

i=1

𝜆i

fi (x)−fi (u)+ 12 pT ∇2 fi (u)p α2

𝜆i C(x,u) (∇𝑓i (𝑢) + ∇2 𝑓i (𝑢)𝑝) + 𝜌2 αd22

k ∑

(3.7) 𝜆i .

i=1

Let 𝑥 be any feasible solution in (P) and (𝑢, 𝑦, 𝜆, r, 𝑝) be any feasible solution in (MD). Then we have m m m m ∑ ∑ ∑ 1∑ T 2 𝑦 i 𝑔i (𝑥) ≤ 0 ≤ 𝑦 i 𝑔i (𝑢) − 𝑦 i 𝑝 ∇ 𝑔i (𝑢)𝑝 − 𝑦 i ∇𝑔i (𝑢)T r. 2 i=1

i=1

i=1

i=1

Using strict second order (C, 𝛼1 , 𝜌1 , 𝑑1 )-convexity of 𝑔i (·) at 𝑢 and the above inequality, we get −

m ∑ i=1

yi T α1 ∇𝑔i (𝑢) r

m ∑

≥

i=1 m ∑

>

i=1

𝑦i

gi (x)−gi (u)+ 12 pT ∇2 gi (u)p α1

𝑦 i C(x,u) (∇𝑔i (𝑢) + ∇2 𝑔i (𝑢)𝑝) + 𝜌1 αd11

m ∑

(3.8) 𝑦i.

i=1

Taking into account convexity of C(x,u)(·) , (3.7) and (3.8), one gets −

k ∑ i=1

λi T α2 ∇𝑓i (𝑢) r

−

m ∑ i=1

> (

yi T α1 ∇𝑔i (𝑢) r

k ∑

𝜆i +

i=1

i=1 m ∑

+

i=1 k ∑

+ 𝜌2 αd22

k ∑

𝑦 i )C(x,u) { ∑ k

yi

λi +

i=1

i=1

m ∑

(∇𝑓i (𝑢) + ∇2 𝑓i (𝑢)𝑝)

i=1

yi

(3.9)

(∇𝑔i (𝑢) + ∇2 𝑔i (𝑢)𝑝)}

m ∑

k ∑

λi

i=1

i=1

λi +

i=1

m ∑

yi

𝜆i + 𝜌1 αd11

m ∑

𝑦i.

i=1

From the first, third, fourth dual constraint in (MD) and C(x,u) (0) = 0, we obtain 0 > 𝜌2

k m 𝑑2 ∑ 𝑑1 ∑ 𝜆i + 𝜌1 𝑦i, 𝛼2 𝛼1 i=1

i=1

which contradicts the condition (3.6). Hence 𝑥 = 𝑢.

4.

Conclusions

In this paper, we introduce a class of second order (C, 𝛼, 𝜌, 𝑑)-convexity, which includes many other generalized convexity concepts in mathematical programming as special cases. Using the (C, 𝛼, 𝜌, 𝑑)-convexity assumptions on the functions involved, weak, strong and strict converse duality theorems are established for a second order Mond-Weir type multiobjective dual. Our results generalize these existing dual results which were discussed by Ahmad et al. in [5], These results can be further generalized to a class of nondifferentiable multiobjective programming. 1228

S.C.Wang, Second order duality for multiobjective programming

References [1] B.Aghezzaf, Second order mixed type duality in multiobjective programming problems, Journal of Mathematical Analysis and Applications 285(2003) 97-106. [2] I.Ahmad, Sufficiency and duality in multiobjective programming with generalized (F, 𝜌)-convexity, Journal of Applied Analysis 11(2005) 19-33. [3] I.Ahmad, Second order symmetric duality in nondifferentiable mnltiobjective programming, Information Science 173(2005) 23-34. [4] I.Ahmad, Symmetric duality for multiobjective fractional variational problems with generalized invexity, Information Science 176(2006) 2192-2207. [5] I.Ahmad, Z.Husian, Second order (F, 𝛼, 𝜌, 𝑑)-convexity and duality in multiobjective programming, Information Science 176(2006) 3094-3103. [6]

A.Chinchuluun, D.H.Yuan, P.M.Pardalos, Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity, Annals of Operations Research 154(2007) 133-147.

[7] T.R.Gulati, M.A.Islam, Sufficiency and duality in multiobjective programming involving generalized F-convex functions, Journal of Mathematical Analysis and Applications 183(1994) 181-195. [8] M.A.Hanson, B.Mond, Further generalizations of convexity in mathematical programming, Journal of Information and Optimization Sciences 3(1982) 25-32. [9] R.N.Kaul, S.K.Suneja, M.K.Srivastava, Optimality criteria and duality in multiobjective optimization involving generalized invexity, Journal of Optimization Theory and Applications 80(1994) 465-482. [10] Z.A.Liang, H.X.Huang, P.M.Pardalos, Optimality conditions and duality for a class of nonlinear fractional programming problems, Journal of Optimization Theory and Applications 110(2001) 611-619. [11] Z.A.Liang, H.X.Huang, P.M.Pardalos, Efficiency conditions and duality for a class of multiobjective programming problems, Journal of Global Optimization 27(2003) 1-25. [12] X.J.Long, Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with (C, 𝛼, 𝜌, 𝑑)-convexity, Journal of Optimization Theory and Applications 148(2011) 197-208. [13] O.L.Mangassrian, Second and higher order duality in nonliear programming, Journal of Mathematical Analysis and Applications 51(1975) 607-620. [14] B.Mond, Second order duality for nonlinear programs, Opsearch 11(1974) 90-99. [15] B.Mond, J.Zhang, Duality for multiobjective programming involving second order V-invex functions, In: B.M.Glower, V.Jeyakumar(Eds.), Proceedings of the Optimization Miniconference, University of New South Wales, Sydney, Australia, 1995, pp.89-100. [16] V.Preda, On efficiency and duality for multiobjective programs, Journal of Mathematical Analysis and Applications 166(1992) 365-377. [17] J.P.Vial, Strong and weak convexity of sets and functions, Mathematics of Operations Research 8(1983) 231-259.

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[18] X.M.Yang, X.Q.Yang, K.L.Teo, S.H.Hou, Second order duality for nonlinear programming, Indian Journal of Pure and Applied Mathematics 35(2004) 699-708. [19] D.H.Yuan, X.L.Liu, A.Chinchuluun, P.M.Pardalos, Nondifferentiable minimax fractional programming problems with (C, 𝛼, 𝜌, 𝑑)-convexity, Journal of Optimization Theory and Applications 129(2006) 185199. [20] J.Zhang, B.Mond, Second order duality for multiobjective nonlinear programming involving generalized convexity, in: B.M.Glower, B.D.Craven, D.Ralph(Eds.), Proceedings of the Optimization Miniconference III, University of Ballarat, 1997, pp.79-95.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1231-1239, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Fractional Voronovskaya type asymptotic expansions for bell and squashing type neural network operators George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we introduce the normalized bell and squashing type neural network operators of one hidden layer. Based on fractional calculus theory we derive fractional Voronovskaya type asymptotic expansions for the error of approximation of these operators to the unit operator.

2010 AMS Mathematics Subject Classi…cation: 26A33, 41A25, 41A36, 41A60. Keywords and Phrases: Neural Network Fractional Approximation, Voronovskaya Asymptotic Expansion, fractional derivative.

1

Background

We need De…nition 1 Let f : R ! R; > 0, n = d e (d e is the ceiling of the number), such that f 2 AC n ([a; b]) (space of functions f with f (n 1) 2 AC ([a; b]), absolutely continuous functions), 8 [a; b] R. We call left Caputo fractional derivative (see [8], pp. 49-52) the function Z x 1 n 1 (n) D a f (x) = (x t) (1) f (t) dt; (n ) a R1 8 x a, where is the gamma function ( ) = 0 e t t 1 dt, > 0. Notice D a f 2 L1 ([a; b]) and D a f exists a.e.on [a; b], 8 b > a. We set D0a f (x) = f (x), 8 x 2 [a; +1): 1

1231

ANASTASSIOU: FRACTIONAL VORONOVSKAYA ASYMPTOTIC EXPANSION

We also need De…nition 2 (see also [2], [9], [10]). Let f : R ! R; such that f 2 AC m ([a; b]), 8 [a; b] R; m = d e, > 0. The right Caputo fractional derivative of order > 0 is given by Z b m ( 1) m 1 (m) Db f (x) = (J x) f (J) dJ; (2) (m ) x 8 x b. We set Db0 f (x) = f (x), 8 x 2 ( 1; b]: Notice that Db f 2 L1 ([a; b]) and Db f exists a.e.on [a; b], 8 a < b: We mention the left Caputo fractional Taylor formula with integral remainder. Theorem 3 ([8], p. 54) Let f 2 AC m ([a; b]), 8 [a; b] Then Z x m X1 f (k) (x0 ) 1 k (x x0 ) + (x J) f (x) = k! ( ) x0

R, m = d e, 1

D

x0 f

> 0.

(J) dJ;

(3)

k=0

8x

x0 : Also we mention the right Caputo fractional Taylor formula.

Theorem 4 ([2]) Let f 2 AC m ([a; b]), 8 [a; b] f (x) =

m X1 k=0

8x

f (k) (x0 ) (x k!

k

x0 ) +

1 ( )

Z

R, m = d e,

x0

(J

x)

x

1

> 0. Then

Dx0 f (J) dJ;

(4)

x0 :

Convention 5 We assume that D

x0 f

(x) = 0, for x < x0 ;

and Dx0 f (x) = 0, for x > x0 ; for all x; x0 2 R: We mention Proposition 6 (by [3]) i) Let f 2 C n (R), where n = d e, > 0. Then D a f (x) is continuous in x 2 [a; 1): ii) Let f 2 C m (R), m = d e, > 0. Then Db f (x) is continuous in x 2 ( 1; b]: 2

1232

ANASTASSIOU: FRACTIONAL VORONOVSKAYA ASYMPTOTIC EXPANSION

We also mention Theorem 7 ([5]) Let f 2 C m (R), f (m) 2 L1 (R), m = d e, > 0, 2 = N, x; x0 2 R. Then D x0 f (x), Dx0 f (x) are jointly continuous in (x; x0 ) from R2 into R. For more see [4], [6]. We need the following (see [7]). De…nition 8 A function b : R ! R is said to be bell-shaped if b belongs to L1 and its integral is nonzero, if it is nondecreasing on ( 1; a) and nonincreasing on [a; +1), where a belongs to R. In particular b (x) is a nonnegative number and at a b takes a global maximum; it is the center of the bell-shaped function. A bell-shaped function is said to be centered if its center is zero. The function b (x) may have jump discontinuities. In this work we consider only centered bell-shaped functions of compact support [ T; T ], T > 0. Example 9 (1) b (x) can be the characteristic function over [ 1; 1] : (2) b (x) can be the hat function over [ 1; 1], i.e., 8 1 x 0; < 1 + x, b (x) = 1 x; 0 < x 1 : 0, elsewhere.

Here we consider functions f 2 C (R) : We study the following ”normalized bell type neural network operators”(see also related [1], [7]) Pn2 k 1 x nk n2 f n b n ; (5) (Hn (f )) (x) := k= Pn2 1 x nk k= n2 b n

where 0 < < 1 and x 2 R, n 2 N. We …nd a fractional Voronovskaya type asymptotic expansion for Hn (f ) (x) : The terms in Hn (f ) (x) are nonzero i¤ n1

x

k n

T , i.e. x

k n

T n1

i¤ nx

Tn

k

nx + T n :

(6)

In order to have the desired order of numbers n2

nx

Tn

nx + T n

n2 ;

(7)

it is su¢ cient enough to assume that n

T + jxj :

When x 2 [ T; T ] it is enough to assume n 3

1233

(8) 2T which implies (7).

ANASTASSIOU: FRACTIONAL VORONOVSKAYA ASYMPTOTIC EXPANSION

Proposition 10 (see [1]) Let a b, a; b 2 R. Let card (k) ( maximum number of integers contained in [a; b]. Then max (0; (b

a)

1)

card (k)

(b

a) + 1:

0) be the

(9)

Remark 11 We would like to establish a lower bound on card (k) over the interval [nx T n ; nx + T n ]. From Proposition 10 we get that card (k) We obtain card (k)

max (2T n

1; 0) :

1, if 2T n

1

1 i¤ n

So to have the desired order (7) and card (k) we need to consider n max T + jxj ; T

T

1

:

1 over [nx 1

:

T n ; nx + T n ], (10)

Also notice that card (k) ! +1, as n ! +1. Denote by [ ] the integral part of a number. Remark 12 Clearly we have that nx

Tn

nx

nx + T n :

(11)

[nx]

nx

dnxe

(12)

We prove in general that nx

Tn

nx + T n :

Indeed we have that, if [nx] < nx T n , then [nx] + T n < nx; and [nx] + [nx], resulting into [T n ] = 0, which for large enough n is not true. [T n ] Therefore nx T n [nx]. Similarly, if dnxe > nx + T n , then nx + T n nx + [T n ], and dnxe [T n ] > nx, thus dnxe [T n ] dnxe, resulting into [T n ] = 0, which again for large enough n is not true. Therefore without loss of generality we may assume that nx

Tn

[nx]

nx

dnxe

nx + T n :

(13)

Hence dnx T n e [nx] and dnxe [nx + T n ] : Also if [nx] 6= dnxe, then 1 dnxe = [nx] + 1. If [nx] = dnxe, then nx 2 Z; and by assuming n T , we get T n 1 and nx + T n nx + 1, so that [nx + T n ] nx + 1 = [nx] + 1: We need also De…nition 13 Let the nonnegative function S : R ! R, S has compact support [ T; T ], T > 0, and is nondecreasing there and it can be continuous only on either ( 1; T ] or [ T; T ], S can have jump discontinuites. We call S the ”squashing function”, see [1], [7]. 4

1234

ANASTASSIOU: FRACTIONAL VORONOVSKAYA ASYMPTOTIC EXPANSION

Let f 2 C (R). For x 2 R we de…ne the following ”normalized squashing type neural network operators” (see also related [1]) Pn2 k 1 x nk n2 f n S n ; (14) (Kn (f )) (x) := k= Pn2 1 x nk k= n2 S n

0 < < 1 and n 2 N : n It is clear that

max T + jxj ; T

(Kn (f )) (x) :=

1

P[nx+T n

] k k=dnx T n e f n P[nx+T n ] k=dnx T n e S

:

S n1 n1

k n

x x

:

k n

(15)

We …nd a fractional Voronovskaya type asymptotic expansion for (Kn (f )) (x) :

2

Main Results

We present our …rst main result. Theorem 14 Let > 0, N 2 N, N = d e ; f 2 AC N ([a; b]), 8 [a:b] R, with Dx0 f , D x0 f M , M > 0, x0 2 R: Let T > 0, n 2 N : n 1

max T + jx0 j ; T (Hn (f )) (x0 )

1

1

Then

f (x0 ) =

N X1 j=1

f (j) (x0 ) Hn ( j!

j

x0 )

(x0 ) + o

as n ! 1, 0 < " . When N = 1, or f (j) (x0 ) = 0, j = 1; :::; N n as n ! 1, 0 < "

)(

)(

")

;

(16)

where 0 < " : If N = 1, the sum in (16) disappears. The last (16) implies that 2 N X1 f (j) (x0 ) Hn ( n(1 )( ") 4(Hn (f )) (x0 ) f (x0 ) j! j=1

(1

1 n(1

")

[(Hn (f )) (x0 )

j

x0 )

3

(x0 )5 ! 0;

(17)

1; then we derive f (x0 )] ! 0

. Of great interest is the case of

= 12 :

Proof. From [8], p. 54; (3), we get by the left Caputo fractional Taylor formula that Z nk N j 1 X1 f (j) (x0 ) k k 1 k f = x0 + J D x0 f (J) dJ; n j! n ( ) x0 n j=0 (18)

5

1235

ANASTASSIOU: FRACTIONAL VORONOVSKAYA ASYMPTOTIC EXPANSION

k for all x0 x0 + T n 1 , i¤ dnx0 e k [nx0 + T n ], where k 2 Z: n Also from [2]; (4), using the right Caputo fractional Taylor formula we get

k n

f

=

N X1 j=0

f (j) (x0 ) j!

k for all x0 T n 1 n that dnx0 e [nx0 ] + 1: Call

j

k n

1 ( )

+

x0

x0 , i¤ dnx0

Z

x0 k n

Tn e

X

Dx0 f (J) dJ;

(19) [nx0 ], where k 2 Z: Notice

k

[nx0 +T n ]

V (x0 ) :=

1

k n

J

b n1

k n

x0

k=dnx0 T n e

:

Hence we have f

k n

b n1 x0 V (x0 )

k n

=

N X1

f (j) (x0 ) j!

j=0

b n x0 V (x0 ) ( )

x0

b n1

x0 V (x0 )

k n

+ (20)

Z

k n

1

j

k n

k n

1

k n

x0

J

D

x0 f

(J) dJ;

and f

k n

b n1 x0 V (x0 )

k n

=

N X1 j=0

k n

1

b n x0 V (x0 ) ( )

Z

f (j) (x0 ) j!

j

k n

x0

b n1

x0 V (x0 )

k n

+ (21)

x0

1

k n

J k n

Dx0 f (J) dJ;

Therefore we obtain P[nx0 +T n

] k=[nx0 ]+1

N X1 j=0

X

k=[nx0 ]+1

b n1 x0 nk V (x0 ) ( )

P[nx0 ]

k=dnx0 T n e

b n1

x0

k n

=

V (x0 )

0P [nx0 +T n ] f (j) (x0 ) @ k=[nx0 ]+1 j!

[nx0 +T n ]

and

k n

f

k n

x0

b n1

k n

x0

V (x0 ) Z

k n

x0

f

j

k n

k n

b n1

V (x0 )

6

1236

1

J

x0

D

k n

x0 f

=

1

A+

(J) dJ;

(22)

ANASTASSIOU: FRACTIONAL VORONOVSKAYA ASYMPTOTIC EXPANSION

N X1 j=0

f (j) (x0 ) j!

P[nx0 ]

k=dnx0 T n e

P[nx0 ]

k n

k=dnx0 T n e

b n1

b n1

k n

x0

V (x0 ) Z

k n

x0

j

x0

V (x0 ) ( )

x0 k n

(23)

1

k n

J

+

Dx0 f (J) dJ:

We notice here that Pn2

k k= n2 f n Pn2 k= n2 b

(Hn (f )) (x) :=

=

P[nx+T n

] k k=dnx T n e f n P[nx+T n ] k=dnx T n e b

b n1 n1

b n1

n1

k n

; 8 x 2 R:

k n

x

(24)

k n

x

x

k n

x

Adding the two equalities (22), (23) and rewriting it, we obtain T (x0 ) := (Hn (f )) (x0 )

N X1

f (x0 )

j=1

f (j) (x0 ) Hn ( j!

j

x0 )

(x0 ) =

n

(x0 ) ; (25)

where n (x0 ) :=

P[nx0 ]

k=dnx0 T n e

+

k=[nx0 ]+1

Z

b n1 x0 nk V (x0 ) ( )

:

j

n

X

b n

1

x0

k=dnx0 T n e

+

[nx0 +T n ]

X

b n1

x0

k=[nx0 ]+1

8 < M V (x0 ) ( ) :

k n k n

x0 k n

J

Z

x0 k n

Z

k n

k n

x0

b n1

[nx0 +T n ]

b n1

D

x0 f

(J) dJ:

(26)

x0

k=[nx0 ]+1

7

1237

Dx0 f (J) dJ 1

J

D

k n

x0

k n

1

k n

J

k=dnx0 T n e

X

Dx0 f (J) dJ

1

k n

[nx0 ]

X

1

k n

J

1 V (x0 ) ( )

(x0 )j

[nx0 ]

k n

x0

We observe that 8
0 such that hgt (x) − gt (y), x(t) − y(t)i ≥ s(t)kgt (x) − gt (y)k2 ,

∀x(t), y(t) ∈ X , t ∈ Ω;

(ii) γ-relaxed cocoercive in the second argument, if there exists a positive real-valued random variable γ(t) such that hgt (x) − gt (y), x(t) − y(t)i ≥ −γ(t)kgt (x) − gt (y)k2 ,

∀x(t), y(t) ∈ X , t ∈ Ω;

(iii) (β, ²)-relaxed cocoercive in the second argument, if there exist positive real-valued random variables α(t) and ²(t) such that hgt (x) − gt (y), x(t) − y(t)i ≥ −β(t)kgt (x) − gt (y)k2 + ²(t)kx(t) − y(t)k2 , for all x(t), y(t) ∈ X , t ∈ Ω;

1249

L.C. Cai and H.Y. Lan

(iv) µ-Lipschitz continuous in the second argument if there exists a real-valued random variable µ(t) > 0 such that kgt (x) − gt (y)k ≤ µ(t)kx(t) − y(t)k, ∀x(t), y(t) ∈ X , t ∈ Ω. Definition 2.6. Let H : Ω × X → X be a nonlinear (in general) operators. A multi-valued operator M : Ω × X → 2X is said to be (i) monotone in the second argument if hu(t) − v(t), x(t) − y(t)i ≥ 0, ∀(x(t), u(t)), (y(t), v(t)) ∈ Graph(Mt ), where Graph(Mt ) = {(z(t), w(t)) ∈ X × X : w(t) ∈ M (t, x(t)), t ∈ Ω}; (ii) r-strongly monotone in the second argument if there exists a measurable function r : Ω → (0, +∞) such that for any t ∈ Ω, hu(t) − v(t), x(t) − y(t)i ≥ r(t)kx(t) − y(t)k2 , ∀(x(t), u(t)), (y(t), v(t)) ∈ Graph(Mt ); (iii) m-relaxed monotone in the second argument if, there exists a real-valued random variable m(t) > 0 such that for any t ∈ Ω, hu(t) − v(t), x(t) − y(t)i ≥ −m(t)kx(t) − y(t)k2 , ∀(x(t), u(t)), (y(t), v(t)) ∈ Graph(Mt ); (iv) H-maximal monotone if M is monotone in the second argument and R(Ht + ρ(t)Mt ) = X for every t ∈ Ω and ρ(t) > 0. Lemma 2.1. ([1]) Let X be a separable real Hilbert space, H : Ω × X → X be r-strongly monotone in the second argument, and M : Ω × X → 2X be H-maximal monotone. Then the generalized resolvent operator associated with M is defined by Mt Jρ(t),H (x) = (Ht + ρ(t)Mt )−1 (x), ∀x ∈ X , t ∈ Ω t

and is

1 r(t) -Lipschitz

continuous for any t ∈ Ω. Moreover,

Mt Mt kJρ(t),H (Ht (x)) − Jρ(t),H (Ht (y))k ≤ t t

1 kHt (x) − Ht (y)k, ∀x, y ∈ X , t ∈ Ω, r(t) − ρ(t)

where r(t) − ρ(t) > 1 for all t ∈ Ω. Lemma 2.2. Let H, f, M and X be the same as in the problem (1.2). If It (x) = Mt (Ht (x))) for x ∈ X , and for all x1 (t), x2 (t) ∈ X , ρ(t) > 0 and Ht (ft (x)) − Ht (Jρ(t),H t γ(t) > 21 , t ∈ Ω,

Mt Mt (Ht (x2 ))), Ht (ft (x1 )) − Ht (ft (x2 ))i (Ht (x1 ))) − Ht (Jρ(t),H hHt (Jρ(t),H t t Mt Mt (Ht (x2 )))k2 , (Ht (x1 ))) − Ht (Jρ(t),H ≥ γ(t)kHt (Jρ(t),H t t

then Mt Mt (Ht (x2 )))k2 (Ht (x1 ))) − Ht (Jρ(t),H (2γ(t) − 1)kHt (Jρ(t),H t t

+kIt (x1 ) − It (x2 )k2 ≤ kHt (ft (x1 )) − Ht (ft (x2 ))k2 .

1250

Convergence Analysis of the Over-relaxed Proximal Point Algorithms with Errors

Proof. By the assumption, now we know kIt (x1 ) − It (x2 )k2 Mt Mt ≤ kHt (Jρ(t),H (Ht (x1 ))) − Ht (Jρ(t),H (Ht (x2 )))k2 + kHt (ft (x1 )) − Ht (ft (x2 ))k2 t t Mt Mt (Ht (x2 ))), Ht (ft (x1 )) − Ht (ft (x2 ))i −2hHt (Jρ(t),H (Ht (x1 ))) − Ht (Jρ(t),H t t Mt Mt ≤ −(2γ(t) − 1)kHt (Jρ(t),H (Ht (x1 ))) − Ht (Jρ(t),H (Ht (x2 )))k2 t t

+kHt (ft (x1 )) − Ht (ft (x2 ))k2 . This completes the proof.

3

¤

Main Results

In this section, we shall introduce a new class of the over-relaxed proximal point algorithms with errors to approximate solvability of the generalized nonlinear random operator equation (1.2) with H-maximal monotonicity framework. Definition 3.1. An operator M −1 , the inverse of M : X → 2X , is (s, c)-Lipschitz continuous at 0 if for any c ≥ 0, there exist a constant s ≥ 0 and a solution x∗ of 0 ∈ M (x) (equivalently x∗ ∈ M −1 (0)) such that kx − x∗ k ≤ skw − 0k,

∀x ∈ M −1 (w),

where w ∈ Bt = {w : kwk ≤ c, w ∈ X , c > 0}. Algorithm 3.1. Step 1. For all t ∈ Ω, choose an arbitrary initial point x0 (t) ∈ X . Step 2. Choose sequences {αn }, {δn (t)} and {ρn (t)} such that for n ≥ 0 and t ∈ Ω, sequence real-value {αn } ⊂ [0, ∞) and real-value random sequences {δn (t)} and {ρn (t)} are in [0, ∞) satisfying ∞ X

δn (t) < ∞,

ρn (t) ↑ ρ(t),

∀t ∈ Ω.

n=0

Step 3. Let {xn (t)} ⊂ X be generated by the following iterative procedure Ht (ft (xn+1 )) = (1 − αn )Ht (ft (xn )) + αn yn (t) + en (t),

∀n ≥ 0,

(3.1)

where {en (t)} is a random error sequence in X toPtake into account a possible inexact computation of the operator point, which satisfies ∞ n=0 ken (t)k < ∞, and yn (t) satisfies kyn (t) − Ht (JρMnt(t),Ht (Ht (xn )))k ≤ δn (t)kyn (t) − Ht (ft (xn ))k, ∀t ∈ Ω. Step 4. If xn (t) and yn (t) satisfy (3.1) to sufficient accuracy, stop; otherwise, set n := n + 1 and return to Step 2. Algorithm 3.2. For any t ∈ Ω and an arbitrary initial point x0 (t) ∈ X , sequence {xn (t)} ⊂ X is generated by the following iterative procedure Ht (xn+1 ) = (1 − αn )Ht (xn ) + αn yn (t) + en (t),

1251

∀n ≥ 0,

L.C. Cai and H.Y. Lan

where {en (t)} is a random error sequence in X toPtake into account a possible inexact computation of the operator point, which satisfies ∞ n=0 ken (t)k < ∞, and yn (t) satisfies kyn (t) − Ht (JρMnt(t),Ht (Ht (xn )))k ≤ δn (t)kyn (t) − Ht (xn )k, Mt and Jρ(t),H = (Ht + ρn (t)Mt )−1 , {αn }, {δn (t)} and {ρn (t)} are three sequences in [0, ∞) t satisfying ∞ X δn < ∞, ρn (t) ↑ ρ(t), ∀t ∈ Ω. n=0

Remark 3.1 If en (t) ≡ 0 for all t ∈ Ω, then the determinate form of Algorithm 3.2 is reduced to the generalized proximal point algorithm in Theorem 3.2 of [1]. Next, we apply the over-relaxed proximal point algorithm 3.1 to approximate the solution of the problems (1.1) and (1.2), and as a result, we end up showing linear convergence. Theorem 3.1. Let X be a separable real Hilbert space, H : Ω × X → X be rstrongly monotone and κ-Lipschitz continuous in the second argument, f : Ω × X → X is σ-Lipschitz continuous and (β, ²)-relaxed cocoercive in the second argument with the inverse f −1 is µ-expanding and M : Ω × X → 2X be H-maximal monotone. If, in addition, (i) (Ht ◦ ft − Ht + ρ(t)Mt )−1 is (s, c)-Lipschitz continuous in the second argument at 0, where Ht ◦ ft is defined by Ht ◦ ft (x) = H(t, f (t, x(t))) for (t, x) ∈ Ω × X ; (ii) for any t ∈ Ω and x1 (t), x2 (t) ∈ X , there exists a real-value random variable γ(t) > 21 such that Mt Mt hHt (Jρ(t),H (Ht (x1 ))) − Ht (Jρ(t),H (Ht (x2 ))), Ht (ft (x1 )) − Ht (ft (x2 ))i t t Mt Mt ≥ γ(t)kHt (Jρ(t),H (Ht (x2 )))k2 ; (Ht (x1 ))) − Ht (Jρ(t),H t t

(iii) there exists a real-value random variable ρ(t) > 0 such that p  2 r(t) 1 − 2²(t) + β(t)σ 2 (t)  p+ σ (t) + κ(t) < r(t),    2β(t)κ(t)σ(t)ϑ(t) < r(t)( 1 + 4β(t)²(t) − 1), p ϑ(t) = (1 − α)2 + κ2 (t)ε2 (t)[α2 − 2γ(t)α(α − 1)] < 1,    s(t)  ε(t) = √ 2 2 < 1, 2 2

(3.2)

µ (t)ρ (t)+s (t)r (t)(2γ(t)−1)

then (1) the generalized nonlinear random operator equation (1.2) has a unique solution x∗ (t) in X . (2) the sequence {xn (t)} generated by Algorithm 3.1 converges linearly to the solution x∗ (t) with convergence rate 2β(t)κ(t)σ(t)ϑ(t) p < 1, r(t)( 1 + 4β(t)²(t) − 1) where ϑ(t) = ε(t) = √ 2 2

p

(1 − α)2 + κ2 (t)ε2 (t)[α2 − 2γ(t)α(α − 1)], α = lim supn→∞ αn > 1, s(t) , ρn (t) ↑ ρ(t) for all t ∈ Ω. 2 2

µ (t)ρ (t)+s (t)r (t)(2γ(t)−1)

1252

Convergence Analysis of the Over-relaxed Proximal Point Algorithms with Errors

Proof. Firstly, for any given positive real-valued random variable ρ(t), define F : Ω × X → X by Mt Ft (x) = x(t) − ft (x) + Jρ(t),H (Ht (x)), ∀x ∈ H. t

By the assumptions of the theorem and Lemma 2.1, for all x(t), y(t) ∈ X we have kFt (x) − Ft (y)k Mt Mt ≤ kx(t) − y(t) − [ft (x) − ft (y)]k + kJρ(t),H (Ht (x)) − Jρ(t),H (Ht (y))k t t

≤ θ(t)kx(t) − y(t)k, p where θ(t) = 1 − 2²(t) + β(t)σ 2 (t) + σ 2 (t) + κ(t) r(t) . It follows from condition (3.2) that 0 < θ(t) < 1 and so F (t, ·) is a contractive mapping for any t ∈ Ω, which shows that F (t, ·) has a unique fixed point in X . Now, we prove the conclusion (2). Let x∗ (t) be a solution of Eqn. (1.2). Then for any given positive real-valued random variable ρn (t) and n ≥ 0, we have Ht (ft (x∗ )) = (1 − αn )Ht (ft (x∗ )) + αn Ht (JρMnt(t),Ht (Ht (x∗ ))).

(3.3)

Mt ) and under the assumptions, it follows that It (xn ) → For It = Ht ◦ ft − Ht (Jρ(t),H t

−1 Mt 0(n → ∞). Since ρ−1 n (t)It (xn ) ∈ (Ht ◦ ft − Ht + ρn (t)Mt )(ft (Jρ(t),Ht (Ht (xn )))), this

Mt implies ft−1 (Jρ(t),H (Ht (xn ))) ∈ (Ht ◦ ft − Ht + ρn (t)Mt )−1 (ρ−1 n (t)It (xn )). Then, applying t Lemma 2.2, the strong monotonicity of H, and the Lipschitz continuity of H (and hence, H being expanding), and the Lipschitz continuity at 0 of (Ht ◦ ft − Ht + ρn (t)Mt )−1 by −1 Mt setting w = ρ−1 n (t)It (xn ) and x(t) = Ht (Jρ(t),Ht (Ht (xn ))), we know

µ2 kJρMnt(t),Ht (Ht (xn )) − JρMnt(t),Ht (Ht (x∗ ))k2 ≤ kHt−1 (JρMnt(t),Ht (Ht (xn ))) − Ht−1 (JρMnt(t),Ht (Ht (x∗ )))k2 −1 ∗ 2 ≤ s2 (t)kρ−1 n (t)It (xn ) − ρn (t)It (x )k ∗ 2 ≤ s2 (t)ρ−2 n (t){kHt (ft (xn )) − Ht (ft (x ))k Mt Mt (Ht (x∗ ))k2 }, (Ht (xn )) − Jρ(t),H −r2 (t)(2γ(t) − 1)kJρ(t),H t t

which implies Mt Mt (Ht (x∗ ))k ≤ εn (t)kHt (ft (xn )) − Ht (ft (x∗ ))k, (Ht (xn )) − Jρ(t),H kJρ(t),H t t

where εn (t) = √

s(t) µ2 (t)ρ2n (t)+s2 (t)r 2 (t)(2γ(t)−1)

< 1.

For n ≥ 0, let Ht (ft (zn+1 )) = (1 − αn )Ht (ft (xn )) + αn Ht (JρMnt(t),Ht (Ht (xn ))).

1253

(3.4)

L.C. Cai and H.Y. Lan

Thus, by the assumptions of the theorem, (3.3) and and (3.4), now we find the estimate kHt (ft (zn+1 )) − Ht (ft (x∗ ))k2 = k(1 − αn )(Ht (ft (xn )) − Ht (ft (x∗ )))k2 +αn2 kHt (JρMnt(t),Ht (Ht (xn ))) − Ht (JρMnt(t),Ht (Ht (x∗ )))k2 +2hαn [Ht (JρMnt(t),Ht (Ht (xn ))) − Ht (JρMnt(t),Ht (Ht (x∗ )))], (1 − αn )(Ht (ft (xn )) − Ht (ft (x∗ )))i ≤ (1 − αn )2 kHt (ft (xn )) − Ht (ft (x∗ ))k2 +[αn2 + 2γ(t)αn (1 − αn )]κ2 (t)kJρMnt(t),Ht (Ht (xn )) − JρMnt(t),Ht (Ht (x∗ ))k2 ≤ ϑ2n (t)kHt (ft (xn )) − Ht (ft (x∗ ))k2 , p where ϑn (t) = (1 − αn )2 + κ2 (t)ε2n (t)[αn2 − 2γ(t)αn (αn − 1)]. Since Ht (ft (xn+1 )) = (1 − αn )Ht (ft (xn )) + αn yn + en (t),

(3.5)

we have Ht (ft (xn+1 )) − Ht (ft (xn )) = αn [yn − Ht (ft (xn ))] + en (t) and kHt (ft (xn+1 )) − Ht (ft (zn+1 ))k = αn kyn − Ht (JρMnt(t),Ht (Ht (xn )))k + ken (t)k ≤ αn δn (t)kyn − Ht (ft (xn ))k + ken (t)k ≤ δn (t)kHt (ft (xn+1 )) − Ht (ft (x∗ ))k +δn (t)kHt (ft (xn )) − Ht (ft (x∗ ))k + ken (t)k.

(3.6)

In the sequel, we estimate using (3.5) and (3.6) that kHt (ft (xn+1 )) − Ht (ft (x∗ ))k ≤ kHt (ft (xn+1 )) − Ht (ft (zn+1 ))k + kHt (ft (zn+1 )) − Ht (ft (x∗ ))k ≤ δn (t)kHt (ft (xn+1 )) − Ht (ft (x∗ ))k + ken (t)k +(δn (t) + ϑn (t))kHt (ft (xn )) − Ht (ft (x∗ ))k, which implies kHt (ft (xn+1 )) − Ht (ft (x∗ ))k 1 ϑn (t) + δn (t) ≤ kHt (ft (xn )) − Ht (ft (x∗ ))k + ken (t)k. 1 − δn (t) 1 − δn (t)

(3.7)

It follows from (3.7), the strong monotonicity and the Lipschitz continuity of H and f that for any t ∈ Ω and all x(t), y(t) ∈ X , p r(t)( 1 + 4β(t)²(t) − 1) kx(t) − y(t)k ≤ kHt (ft (x)) − Ht (ft (y))k 2β(t) ≤ κ(t)σ(t)kx(t) − y(t)k, and kxn+1 − x∗ k

≤

2β(t)κ(t)σ(t) ϑn (t) + δn (t) p · kxn − x∗ k 1 − δ (t) r(t)( 1 + 4β(t)²(t) − 1) n 1 2β(t) p · + ken (t)k. r(t)( 1 + 4β(t)²(t) − 1) 1 − δn (t)

1254

(3.8)

Convergence Analysis of the Over-relaxed Proximal Point Algorithms with Errors

By (3.8), we know that the {xn } converges linearly to a solution x∗ for 2β(t)κ(t)σ(t)ϑn p . r(t)( 1 + 4β(t)²(t) − 1) Hence, we have lim sup n→∞

2β(t)κ(t)σ(t) ϑn + δn 2β(t)κ(t)σ(t)ϑ(t) p p · = , r(t)( 1 + 4β(t)²(t) − 1) 1 − δn r(t)( 1 + 4β(t)²(t) − 1)

where t ∈ Ω, ϑ(t) = lim sup ϑn (t) = n→∞

ε(t) = lim supn→∞ εn (t) = √

p

(1 − α)2 + κ2 (t)ε2 (t)[α2 − 2γ(t)α(α − 1)],

s(t) , µ2 (t)ρ2 (t)+s2 (t)r 2 (t)(2γ(t)−1)

ρn (t) ↑ ρ(t), α = lim supn→∞ αn .

This completes the proof. ¤ Remark 3.2. The conditions (3.2) in Theorem 3.1 hold for some suitable value of constant or real-valued random variable, for example, α = 1.35, and r(t) = 1.25, ²(t) = 0.4, β(t) = 0.15, σ(t) = 0.025, s(t) = 0.25, κ(t) = 0.98, γ(t) = 1.5262, µ(t) = 0.6, ρ(t) = 0.7348 and the convergence rate θ(t) = 0.0220 < 1 for all t ∈ Ω. From Theorem 3.1, we have the following results as an application of Theorem 3.1. Theorem 3.2. Let H, M and X be the same as in Theorem 3.1. If, in addition, (i) Mt−1 is (s, c)-Lipschitz continuous in the second argument at 0; (ii) for any t ∈ Ω and x1 (t), x2 (t) ∈ X , there exists a real-value random variable γ(t) > 12 such that Mt Mt hHt (Jρ(t),H (Ht (x1 ))) − Ht (Jρ(t),H (Ht (x2 ))), Ht (x1 ) − Ht (x2 )i t t Mt Mt ≥ γ(t)kHt (Jρ(t),H (Ht (x1 ))) − Ht (Jρ(t),H (Ht (x2 )))k2 ; t t

(iii) there exists a real-value random variable ρ(t) > 0 such that    κ(t)ϑ(t)p< r(t), ϑ(t) = (1 − α)2 + κ2 (t)ε2 (t)[α2 − 2γ(t)α(α − 1)] < 1, s(t)   εn (t) = √ < 1, 2 2 2 ρn (t)+s (t)r (t)(2γ(t)−1)

then the sequence {xn (t)} generated by Algorithm 3.2 converges linearly to the solution x∗ (t) of the problem (1.1) with convergence rate κ(t) p 1 − α{2(1 − γ(t)κ2 (t)ε2 (t)) − α[1 − (2γ(t) − 1)κ2 (t)ε2 (t)]} < 1, r(t) where α = lim supn→∞ αn > 1, ε(t) = √

s(t) , ρ2 (t)+s2 (t)r 2 (t)(2γ(t)−1)

ρn (t) ↑ ρ(t) for all t ∈ Ω.

Theorem 3.3. Let H, M and X be the same as in Theorem 3.1. If, in addition, condition (ii) of Theorem 3.2 holds and there exists a real-value random variable ρ(t) ∈ (0, r(t) − 1) such that s κ2 (t)αn [αn − 2γ(t)(αn − 1)] κ(t) (1 − αn )2 + < r(t), (r(t) − ρ(t))2

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L.C. Cai and H.Y. Lan

then the sequence {xn (t)} generated by Algorithm 3.2 converges linearly to the solution x∗ (t) of the problem (1.1) with convergence rate κ(t) p 1 − α{2(1 − γ(t)κ2 (t)ε2 (t)) − α[1 − (2γ(t) − 1)κ2 (t)ε2 (t)]} < 1, r(t) 1 where α = lim supn→∞ αn > 1, ε(t) = r(t)−ρ(t) with r(t) − ρ(t) > 1, ρn (t) ↑ ρ(t) for all t ∈ Ω. Remark 3.3. In Theorem 3.3, we apply Lemma 2.1, the Lipschitz continuity of the generalized resolvent operator associated with M instead, it seems that the conditions in Theorem 3.3 is less than that in Theorem 3.2. Further, if real-valued random variables γ(t) = 1 or en (t) ≡ 1 or κ(t) = 1 (that is, H is nonexpansive) for all t ∈ Ω, then we can obtain corresponding results of Theorems 3.1-3.3. Therefore, the results presented in this paper improve, generalize and unify the corresponding results of recent works.

References [1] R.U. Verma, The over-relaxed proximal point algorithm based on H-maximal monotonicity design and applications, Comput. Math. Appl. 55(11) (2008), 2673-2679. [2] Z.Y. Huang, A remark on the strong convergence of the over-relaxed proximal point algorithm, Comput. Math. Appl. 60(6) (2010), 1616-1619. [3] R.U. Verma, A general framework for the over-relaxed A-proximal point algorithm and applications to inclusion problems, Appl. Math. Lett. 22 (2009), 698-703. [4] R.P. Agarwal and R.U. Verma, Relatively maximal monotone mappings and applications to general inclusions, Appl. Anal. 91(1) (2012), 105-120. [5] R. Ahmad and A.P. Farajzadeh, On random variational inclusions with random fuzzy mappings and random relaxed cocoercive mappings, Fuzzy Sets and Systems 160(21) (2009), 3166-3174. [6] Y.J. Cho, N.J. Huang and S.M. Kang, Random generalized set-valued strongly nonlinear implicit quasi-variational inequalities, J. Inequal. Appl. 5 (2000), 515-531. [7] Y.J. Cho and H.Y. Lan, Generalized nonlinear random (A, η)-accretive equations with random relaxed cocoercive mappings in Banach spaces, Comput. Math. Appl. 55(9) (2008), 2173-2182. [8] M.F. Khan, Salahuddin and R.U. Verma, Generalized random variational-like inequalities with randomly pseudo-monotone multivalued mappings, Panamer. Math. J. 16(3) (2006), 33-46. [9] H.Y. Lan, Approximation solvability of nonlinear random (A, η)-resolvent operator equations with random relaxed cocoercive operators, Comput. Math. Appl. 57(4) (2009), 624-632. [10] H.Y. Lan, Nonlinear random multi-valued variational inclusion systems involving (A, η)accretive mappings in Banach spaces, J. Comput. Anal. Appl. 10(4) (2008), 415-430. [11] H.G. Li and X.B. Pan, Approximation solution for nonlinear set-valued mixed random variational inclusions involving random nonlinear (Aω , ηω )-monotone mappings, Nonlinear Funct. Anal. Appl. 15(3) (2010), 395-410. [12] Z.Q. Liu, P.P. Zheng, T. Cai and S.M. Kang, General nonlinear variational inclusions with H-monotone operator in Hilbert spaces, Bull. Korean Math. Soc. 47(2) (2010), 263-274. [13] F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, NY, 1983.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1257-1265, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

FIXED POINT THEOREM FOR CIRIC’S TYPE CONTRACTIONS IN GENERALIZED QUASI-METRIC SPACES

Luljeta Kikina 1 , Kristaq Kikina 1 and Kristaq Gjino 2 1

Department of Mathematics and Computer Science, University of Gjirokastra, Albania [email protected], [email protected] 2 Department of Mathematics, University of Tirana, Albania [email protected]

Abstract. A fixed point theorem in generalized quasi-metric spaces is proved. The obtained result extends in generalized quasi-metric spaces the Ciric’s fixed point theorem on quasi-contraction mapping. An example shows that the main theorem of this paper provides a larger class of mappings than the Ciric’s fixed point theorem.

Keywords: Cauchy sequence, fixed point, generalized quasi-metric space, quasi-contraction. Mathematics Subject Classification: 47H10, 54H25

1. Introduction and Preliminaries The concept of metric space, as an ambient space in fixed point theory, has been generalized in several directions. Some of such generalizations are: the quasi-metric spaces, the generalized metric spaces and the generalized quasi-metric spaces. The concept of quasi-metric space is treated differently by many authors. In [2], [8], [14], [15], [18], [19], etc the quasi-metric space is in line of metric space in which the triangular inequality d ( x, y ) ≤ d ( x, z ) + d ( z , y ) is replaced by quasi- triangular inequality d ( x, y ) ≤ k[d ( x, z ) + d ( z , y )], k ≥ 1 . In 2000 Branciari [3] introduced the concept of generalized metric spaces (gms) (The d ( x, y ) ≤ d ( x , z ) + d ( z , y ) triangular inequality is replaced by tetrahedral inequality d ( x, y ) ≤ d ( x, z ) + d ( z , w) + d ( w, y ) ). Starting with the paper of Branciari, some classical metric fixed point theorems have been transferred to gms (see [1], [4], [5], [6], [7], [10], [11], [12], [16], [17]) Recently L. Kikina and K. Kikina [9] introduced the concept of generalized quasimetric space (gqms) replacing the tetrahedral inequality d ( x, y ) ≤ d ( x, z ) + d ( z , w) + d ( w, y ) with the quasi-tetrahedral inequality d ( x, y ) ≤ k[d ( x, z ) + d ( z , w) + d ( w, y )] . The metric spaces are a special case of generalized metric spaces and generalized metric spaces are a special case of generalized 1257

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quasi-metric spaces (for k = 1 ). Also, every qms is a gqms, while the converse is not true [9]. Firstly, we will give some known definitions and notations. Let ( X , d ) be a metric space. A mapping T : X → X is said to be a quasi-contraction if there exists 0 ≤ h < 1 such that d (Tx, Ty ) ≤ h max{d ( x, y ), d ( x, Tx), d ( y, Ty ), d ( x, Ty ), d ( y, Tx)} for all x, y ∈ X . In 1974, Ciric [4] introduced these mappings and proved the following fixed point result: Theorem 1.1 (Ciric [4]) Let T be a quasi-contraction on a metric space ( X , d ) and let X be T-orbitally complete metric space. Then (a) T has a unique fixed point α in X , (b) lim T n x = α , and n →∞

(c) d (T n x, α ) ≤ (h n /(1 − h))d ( x, Tx) for every x ∈ X In this paper we extend in generalized quasi-metric spaces the above theorem.

Definition 1.1 [3] Let X be a set and d : X 2 → R + a mapping such that for all x, y ∈ X and for all distinct points z , w ∈ X , each of them different from x and y, one has (a ) d ( x, y ) = 0 if and only if x = y , (b) d ( x, y ) = d ( y, x) , (c) d ( x, y ) ≤ d ( x, z ) + d ( z , w) + d ( w, y ) (Tetrahedral inequality) Then d is called a generalized metric and ( X , d ) is a generalized metric space (or shortly gms). Definition 1.2 [9] Let X be a set. A nonnegative symmetric function d defined on X × X is called a generalized quasi-distance on X if and only if there exists a constant k ≥ 1 such that for all x, y ∈ X and for all distinct points z , w ∈ X , each of them different from x and y the following conditions hold: (i ) d ( x, y ) = 0 ⇔ x = y; (ii ) d ( x, y ) = d ( y, x); (iii ) d ( x, y ) ≤ k[d ( x, z ) + d ( z , w) + d ( w, y )] . Inequality (3) is often called quasi-tetrahedral inequality and k is often called the coefficient of d . A pair ( X , d ) is called a generalized quasi-metric space (or shortly gqms) if X is a set and d is a generalized quasi-distance on X. The set B(a, r ) = {x ∈ X : d ( x, a ) < r} is called “open” ball with center a ∈ X and radius r > 0 . The family τ = {Q ⊂ X : ∀a ∈ Q, ∃r > 0, B(a, r ) ⊂ Q} is a topology on X and it is called induced topology by the generalized quasi-distance d . The following example illustrates the existence of the generalized quasi-metric

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space for an arbitrary constant k ≥ 1 :  1  Example 1.3 [9] Let X = 1 − : n = 1, 2,... ∪ {1, 2} , Define d : X × X → R as follow:  n  0 for x = y 1 1 1  for x ∈ {1, 2} and y = 1 − or y ∈ {1, 2} and x = 1 − , x ≠ y d ( x, y ) =  n n n 3k for x, y ∈ {1, 2}, x ≠ y  1 otherwise Then it is easy to see that ( X , d ) is a generalized quasi-metric space and is not a generalized metric space (for k > 1 ) . 1 Note that the sequence {xn } = {1 − } converges to both 1 and 2 and it is not a Cauchy n 1 1 sequence: d ( xn , xm ) = d (1 − ,1 − ) = 1, ∀n, m ∈ N n m Since B (1, r ) ∩ B (2, r ) ≠ φ for all r > 0 , the ( X , d ) is non a Hausdorff generalized quasi-metric space. 1 1 1 1 The function d is not continuous: 1 = lim d (1 − , ) ≠ d (1, ) = . n →∞ 2 2 n 2 The above example shows that: in a gqms (and for k = 1 in a gms) , contrary to the case of a metric space, the “open” balls B (a, r ) = {x ∈ X : d ( x, a) < r} are not always open sets and, moreover, the generalized quasi-metric d is not always necessarily continuous with respect to its variables. Also, the generalized quasi-metric space is not always a Hausdorff space and a convergent sequence {xn } in gqms is not always a Cauchy sequence. Under these circumstances, not every theorem of fixed points for metric spaces, can be extended in gqms as well. Even in the cases it may be done, the proof of theorem is more complicated and it may requires additional conditions. In [9] is proved: Proposition 1.4 If ( X , d ) is a quasi-metric space, then ( X , d ) is a generalized quasimetric space. The converse proposition doesn’t hold true. Definition 1.5 A sequence {xn } in a generalized quasi-metric space ( X , d ) is called Cauchy sequence if lim d ( xn , xm ) = 0 . n , m →∞

Definition 1.6 Let ( X , d ) be a generalized quasi-metric space. Then: (1) A sequence {xn } in X is said to be convergent to a point x ∈ X (denoted by lim xn = x ) if lim d ( xn , x ) = 0 . n→∞

n →∞

(2) It is called compact if every sequence contains a convergent subsequence.

Definition 1.7 A generalized quasi-metric space ( X , d ) is called complete, if every Cauchy sequence is convergent. Definition 1.8 Let ( X , d ) be a gqms and the coefficient of d is k.

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1 such that k d (Tx, Ty ) ≤ cd ( x, y ) for all x, y ∈ X . Definition 1.9 Let T : X → X be a mapping where X is a gqms. For each x ∈ X , let O(x)={x,Tx,T 2 x,...} which will be called the orbit of T at x. The space X is said to be T-orbitally complete if and only if every Cauchy sequence which is contained in O(x) converges to a point in X. A map T : X → X is called contraction if there exists 0 ≤ c
0 ,

δ (O( x, n)) ≤ Moreover, since

k (1+ h ) 1− kh 2

max{d ( x, Tx), d ( x, T 2 x)}

δ (O( x,1)) ≤ δ (O( x, 2)) ≤ ... ≤ δ (O( x, n)) ≤ ...

we can write

δ (O( x)) ≤

k (1+ h ) 1− kh 2

max{d ( x, Tx), d ( x, T 2 x)}

This completes the proof of the Lemma.

Remark 2.4 If T is a quasi-contraction, note that, in view of Lemma 2.3, O( x) is bounded set: δ (O( x)) < ∞, ∀x ∈ X Lemma 2.5 Let T be a quasi-contraction on generalized quasi-metric space ( X , d ) . Then, for any n ≥ 1, one has δ (O(T n x)) ≤ h nδ (O( x)) where h is the constant associated with the quasi-contraction definition of T. Moreover, we have d (T n x, T n + m x)) ≤ h n k1(1− kh+ h2) max{d ( x, Tx), d ( x, T 2 x)}

for any n ≥ 1 and m ∈ N . Proof. Let n and m (n < m) be any positive integers. Since T is a quasi-contraction, by condition (1), we have

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d (T n x, T m y ) ≤ ≤ h max{d (T n −1 x, T m −1 y ), d (T n −1 x, T n x), d (T m −1 y, T m y ), d (T n −1 x, T m y ), d (T m −1 y , T n x )}

(*)

From the remark to previous lemma we have δ (O( x)) < ∞, ∀x ∈ X . Then it follows from (*) and (2) that δ (O(T n x)) ≤ hδ (O(T n −1 x)), n ∈ N Inductively we get δ (O(T n x)) ≤ h nδ (O( x)) Moreover, for any n ≥ 1 and m ∈ N , we have d (T n x, T n + m x)) ≤ δ (O (T n x)) ≤ h nδ (O( x)) And so, by (6), we get d (T n x, T n + m x)) ≤ h n k1(1− kh+ h2) max{d ( x, Tx), d ( x, T 2 x)} This completes the proof of the Lemma. Now we can state our main theorem. Theorem 2.6 Let ( X , d ) be an T-orbitally complete gqms with the coefficient k ≥ 1 and T : X → X a quasi-contraction with constant h. on a generalized quasi-metric space ( X , d ) with the coefficient k and ( X , d ) be T-orbitally complete. Then (a) T has a unique fixed point α in X, (b) lim T n x = α , for every x ∈ X and n →∞

(c) d (T n x, α )) ≤ h n

k 2 (1+ h ) 1− kh 2

max{d ( x, Tx), d ( x, T 2 x)} , for all n ∈ N

Proof. Define the sequence {xn } as follows: xn = T n x, n ∈ N . We divide the proof into two cases: Case I: Suppose x p = xq for some p, q ∈ N , p ≠ q. Let p > q. Then

T p x = T p − qT q x = T q x i.e. T nα = α where n = p − q and T q x = α . Now, if n > 1 , then we have α = T nα = T rnα , r ∈ N and by Lemma 2.5, we get d (α , T α ) = d (T nα , T n +1α ) = d (T rnα , T rn +1α ) = d (T rn + q x, T rn + q +1 x) ≤ ≤ δ (O (T rn + q x)) ≤ h rn + qδ (O( x)), ∀r ∈ N Since lim h rn + q = 0 , d (α , T α ) = 0 . So T α = α and hence α is a fixed point of T. r →∞

Case II: Assume that xn ≠ xm for all n ≠ m . Then {xn } = {T n x} is a sequence of distinct point. By lemma 2.5, we have d ( xn , xn + m ) = d (T n x, T n + m x) ≤ h n k1(1− kh+ h2) max{d ( x, Tx), d ( x, T 2 x)} Therefore,

lim d ( xn , xn + m ) = 0 n →∞

(7)

It implies that {xn } is a Cauchy sequence in X. Since ( X , d ) is T-orbitally complete, there exists a α ∈ X such that lim xn = α (8) n →∞

We now prove that the limit α is unique. Suppose to the contrary, that is α ′ ≠ α is

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also lim xn . n →∞

Since xn ≠ xm for all n ≠ m , there exists a subsequence { x n p } of {xn } such that

x n p ≠ α and xn p ≠ α ' for all p ∈ N . Without loss of generality, assume that {xn } is this subsequence. Then, by quasi-tetrahedral inequality, we obtain

d (α , α ') ≤ k [ d (α , xn ) + d ( xn , xn +1 ) + d ( xn +1 , α ')] Letting n tend to infinity, by (7) and (8), we get d (α , α ') = 0 and so α = α ' . Let we prove now that α is a fixed point of T . In contrary, if α ≠ T α , then there exists a subsequence { x n p } such that x n ≠ T α and x n ≠ α for all p ∈ N . p

p

By quasi-tetrahedral inequality, we obtain

d (α , T α ) ≤ k [d (α , xn p−1 ) + d ( xn p−1 , xn p ) + d ( xn p , T α )]

Then, if p → ∞ , we get ___

d (α , T α ) ≤ k lim d ( xn p , T α ) p →∞

(9)

From (1), d ( xn , T α ) = d (Txn −1 , T α ) ≤

≤ h max{(d ( xn −1 , α ), d ( xn −1 , Txn −1 ), d (α , T α ), d ( xn −1 , T α ), d (α , Txn −1 )} = = h max{(d ( xn −1 , α ), d ( xn −1 , xn ), d (α , T α ), d ( xn −1 , T α ), d (α , xn )} ___

___

Letting n tend to infinity, by lim d ( xn , T α ) = lim d ( xn −1 , T α ) , we get n →∞

n →∞

___

___

n →∞

n →∞

lim d ( xn , T α ) ≤ h max{(0, 0, d (α , T α ), lim d ( xn −1 , T α ), 0} ≤ hd (α , T α )

(10)

From (9) and (10), ___

___

p →∞

n →∞

d (α , T α ) ≤ k lim d ( xn p , T α ) ≤ k lim d ( xn , T α ) ≤ khd (α , T α ) Since 0 ≤ kh < 1 , we have d (α , T α ) = 0 . So α is a fixed point of T. Let we prove now the uniqueness (for case I and II in the same time). Assume that α ′ ≠ α is also a fixed point of T . From (1) we get d (α , α ') = d (T α , T α ') ≤ h max{(d (α , α '), 0, 0, d (α , α '), d (α ', α )} ≤ hd (α , α ') Since 0 ≤ h < 1 , we have α = α ' . So we have proved (a) and (b). By quasi-tetrahedral inequality and by Lemma 2.5 we obtain

d ( xn , α ) ≤ k [d ( xn , xn + m ) + d ( xn + m , xn + m +1 ) + d ( xn + m +1 , α )] ≤

≤ hn

k 2 (1+ h ) 1− kh 2

max{d ( x, Tx), d ( x, T 2 x)} + kd ( xn + m , xn + m +1 ) + kd ( xn + m +1 , α )

Letting m tend to infinity, by (7) and (8), we obtain the inequality (c).This completes the proof of the theorem. Corollary 2.7 By the theorem 2.6, in special case k = 1 , we obtain an extension of the Cirich's quasi-contraction principle in a generalized metric space presented by B. K. Lahiri and P. Das [12]. We note that in [12] the proof of the main theorem is not correct since it relies in the continuity of the generalized distance d, that it is not true always. We end this paper with an example: 1263

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Example 2.8 Let X = {0, 12 , 34 ,1} and T ( 12 ) = 1 and T ( x) = 0 for x ∈ X − { 12 } .

T:X →X

be

a

mapping

such

that

In the ordinary metric space, the inequality (1) is not satisfied for x = 12 and y = 0 : 1 = d (T 12 , T 0) ≤ h max{d ( 12 , 0), d ( 12 , T 12 ), d (0, T 0), d ( 12 , T 0), d (0, T 12 )} = = h max{ 12 , 12 , 0, 12 ,1} = h While for the mapping T, it can not be applied the Theorem Ciric [5], although there is unique fixed point, the Theorem 2.7 can be applied in gqms ( X , d ) with generalized quasi-distance as follows: for x = y 0  d ( x, y ) = 6 for x, y ∈ { 12 ,1}, x ≠ y 1 otherwise  Then it is easy to see that ( X , d ) is a generalized quasi-metric space and is not a metric space because it lacks the triangular inequality: 6 = d ( 12 ,1) > d ( 12 , 0) + d (0,1) = 1 + 1 = 2 . In this generalized quasi-metric with the coefficient k = 2 , the inequality (1) is satisfied for all x, y ∈ X : If x = y or x, y ∈ X − { 12 } , the left side of the inequality (1’) is zero and consequently

it is true for any h ∈ [0, 12 ) . If x =

1 2

and y ≠ 12 , inequality (1’) takes the form 1 = d (T 12 , Ty ) ≤ h max{d ( 12 , y ), d ( 12 , T 12 ), d ( y, Ty ), d ( 12 , Ty ), d ( y, T 12 )} =

= h max{d ( 12 , y ), 6, d ( y, Ty ), d ( 12 , Ty ), d ( y, T 12 )} = h6 which is true for h ∈ [ 16 , 12 = 1k ) . If x ≠

1 2

and y = 12 , inequality (1’) takes the form of above case.

All the conditions of Theorem 2.7 are satisfied with h = [ 16 , 1k = 12 ) . The mapping T has unique fixed point: Fix(T ) = {0} and, for any x ∈ X , the Picard iteration {xn } defined by

xn = T n x, n = 1, 2,... , converges to 0. The example given above, show that the Theorem 2.7 provides a larger class of mappings than the Theorem 1.1 (Ciric’s Theorem [4]).

References [1] A. Azam and M. Arshad, Kannan fixed point theorem on generalized metric spaces, J. Nonlinear Sci. Appl., 1 (2008), no. 1, 45–48. [2] M. Bramanti, L. Brandolini, Schauder estimatet for parabolic nondivergence operators of Hormander type, J. Differential Equations 234 (2007) 177-245. [3] A. Branciari, A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31–37.

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[4] Lj. B. Ciric, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, no. 2, pp. 267–273, 1974. [5] P. Das, A fixed point theorem on a class of generalized metric spaces, Korean J. Math. Sciences, 1 (2002), 29-33. [6] P. Das and L. K. Dey, A fixed point theorem in a generalized metric space, Soochow Journal of Mathematics, 33 (2007), 33–39. [7] P. Das and L. K. Dey, Fixed point of contractive mappings in Generalized metric spaces, Math. Slovaca 59 (2009), No. 4, 499-504. [8] L. Kikina and K. Kikina, Generalized fixed point theorem for three mappings on three quasi-metric spaces, Journal of Computational Analysis and Applications, Vol.14, no.2, 2012, 228-238. [9] L. Kikina and K. Kikina “Two fixed point theorems on a class of generalized quasimetric spaces”, "Journal of Computational Analysis and Applications”, Vol.14, no.5, 2012, 950-957 [10] L. Kikina and K. Kikina, Fixed points on two generalized metric spaces, Int. Journal of Math. Analysis, Vol. 5, 2011, no. 30, 1459 – 1467. [11] L. Kikina and K. Kikina, A fixed point theorem in generalized metric spaces, Demonstratio Mathematica, accepted to appear. [12] B. K. Lahiri and P. Das, Fixed point of a Ljubomir Ciric quasi-contraction mapping in a generalized metric space, Publ. Math. Debrecen 61 (3-4), 589–594, 2002. [13] D. Mihet, On Kananan fixed point principle in Generalized metric space, J. Nonlinear Sci. Appl., 2 (2009), no. 2, 92-96. [14] B. Pepo, Fixed point for contractive mapping of third order in pseudo –quasi-metric spaces, Indag. Math. (NS) 1 (1990) 473-482 [15] C. Peppo, Fixed point theorems for (ϕ , k , i, j ) − mappings, Nonlinear Anal. 72 (2010) 562-570. [16] B. Samet, Discussion on A fixed point theorem of Banach-Caccioppoli tipe on a class of generalized metric spaces by A. Branciari, Publ. Math. Debrecen 76 (2010), no. 3-4, 493-494. [17] I. R. Sarma, J. M. Rao, S. S. Rao, Contractions over generalized metric spaces, J. Nonlinear Sci. Appl., 2 (2009), no. 3, 108-182. [18] M. I. Vulpe, D. Ostraih, F. Hoiman, The topological structure of a quasimetric space. (Russian) Investigations in functional analysis and differential equations, pp. 14-19, 137, "Shtiintsa", Kishinev, 1981. [19] Q. Xia, The Geodesic Problem in Quasimetric Spaces, J Geom Anal (2009) 19: 452–479

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Explicit formulas on the second kind q-Euler numbers and polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea Abstract : In [3], we introduced the second kind q-Euler numbers En,q and polynomials En,q (x). From these numbers and polynomials, we establish some interesting identities and explicit formulas. Key words : the second kind Euler numbers and polynomials, the second kind q-Euler numbers and polynomials. 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80 1. Introduction Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, R denotes the set of real numbers, C denotes the set of complex numbers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assume that |q| < 1. If q ∈ Cp , we normally 1 assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1.Throughout this paper we use the notation: 1 − qx , cf. [1,2,3,4,5] . [x]q = 1−q Hence, limq→1 [x] = x for any x with |x|p ≤ 1 in the present p-adic case. For g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, Kim[1] defined the p-adic integral on Zp as follows:  I1 (g) = g(x)dμ−1 (x) = lim

N →∞

Zp



g(x)(−1)x .

(1.1)

0≤x k. Then, by (3.2) and Corollary 8, we have  [2x + 1]kq [1 − 2x]n−k q −1 dμ−1 (x) Zp

  n−k n−k−l l k+l−n (−1) = [2]q−1 q [2x + 1]n−l dμ−1 (x) q l Z p l=0 n−k  n − k  (−1)n−k−l [2]lq−1 q k+l−n En−l,q . = l n−k 

l=0

1268

(3.3)

RYOO: 2ND KIND q-EULER NUMBERS

Therefore, by comparing the coefficients on the both sides of (3.2) and (3.3), we obtain the following theorem. Theorem 9. For n, k ∈ Z+ with n > k, we have k      k (−1)k−l [2]lq q k−l 2 − En−l,q−1 l l=0 n−k  n − k  (−1)n−k−l [2]lq−1 q k+l−n En−l,q . = l l=0

By Theorem 9, we have the following corollary. Corollary 10. For n, k ∈ Z+ with n > k, we have k     n − k    n−k k l l (−1) [2]q−1 2 − En−l,q−1 = (−1)n+l [2]lq En−l,q . q l l n

l=0

l=0

By Corollary 8, we have  1 −k 2 −k [2x + 1]kq [1 − 2x]qn−1 [2x + 1]kq [1 − 2x]qn−1 dμ−1 (x) Zp



= Zp

n1 +n2 −2k [2x + 1]2k dμ−1 (x) q [1 − 2x]q −1

  2k   2k 2k−l l 2k−l 1 +n2 −l (−1) = [2]q q [1 − 2x]qn−1 dμ−1 (x) l Z p l=0  2k     2k (−1)l [2]lq q 2k−l 2 − En1 +n2 −l,q−1 . = l

(3.4)

l=0

Let n1 , n2 , k ∈ Z+ with n1 + n2 > 2k. Then we see that  1 −k 2 −k [2x + 1]kq [1 − 2x]qn−1 [2x + 1]kq [1 − 2x]qn−1 dμ−1 (x) Zp



= Zp

n1 +n2 −2k [2x + 1]2k dμ−1 (x) q [1 − 2x]q −1

  n1 + n2 − 2k n1 +n2 −l l 2k+l−n1 −n2 (−1) = [2]q−1 q [2x + 1]qn1 +n2 −l dμ−1 (x) l Z p l=0  n1 +n 2 −2k   n1 + n2 − 2k (−1)n1 +n2 −l [2]lq−1 q 2k+l−n1 −n2 En1 +n2 −l,q . = l n1 +n 2 −2k  

(3.5)

l=0

By comparing the coefficients on the both sides of (3.4) and (3.5), we obtain the following theorem. Theorem 11. Let n1 , n2 , k ∈ Z+ with n1 + n2 > 2k. Then we have  2k   2k

  (−1)l [2]lq−1 2 − En1 +n2 −l,q−1 l l=0  n1 +n 2 −2k   n1 + n2 − 2k (−1)n1 +n2 +l [2]lq En1 +n2 −l,q . = l

q

n1 +n2

l=0

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Let s ∈ N with s ≥ 2. For n1 , n2 , . . . , ns , k ∈ Z+ with n1 + · · · + ns > sk, we have  1 −k s −k dμ−1 (x) [2x + 1]kq [1 − 2x]qn−1 · · · [2x + 1]kq [1 − 2x]qn−1

 Zp  s−times   n1 +···+n −sk s  n1 + · · · + ns − sk (−1)n1 +···+ns −sk−l [2]lq−1 q sk+l−n1 −···−ns = l l=0  × [2x + 1]qn1 +···+ns −l dμ−1 (x) =

(3.6)

Zp

 n1 + · · · + ns − sk (−1)n1 +···+ns −sk−l [2]lq−1 q sk+l−n1 −···−ns l

n1 +···+n s −sk  l=0

× En1 +···+ns −l,q . From the binomial theorem, we note that 

1 −k s −k dμ−1 (x) [2x + 1]kq [1 − 2x]qn−1 · · · [2x + 1]kq [1 − 2x]qn−1 

 s−times  n1 +···+ns −sk [2x + 1]sk dμ−1 (x) = q [1 − 2x]q −1

Zp

Zp

  sk 1 +···+ns −l (−1)sk−l [2]lq q sk−l = [1 − 2x]qn−1 dμ−1 (x) l Z p l=0 sk      sk (−1)sk−l [2]lq q sk−l 2 − En1 +···+ns −l,q−1 . = l sk  

(3.7)

l=0

Therefore, by (3.6) and (3.7), we obtain the following theorem. Theorem 12. Let s ∈ N with s ≥ 2. For n1 , n2 , . . . , ns , k ∈ Z+ with n1 + · · · + ns > sk, we have

 n1 + · · · + ns − sk (−1)n1 +···+ns +l [2]lq En1 +···+ns −l,q l l=0 sk      sk n1 +···+ns (−1)l [2]lq−1 2 − En1 +···+ns −l,q−1 . =q l

n1 +···+n s −sk 

l=0

ACKNOWLEDGEMENT This paper has been supported by the 2013 Hannam University Research Fund.

REFERENCES 1. Kim, T.(2007). q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., v.14, pp. 15-27. 2. Kim, T.(2002). q-Volkenborn integration, Russ. J. Math. Phys., v.9, pp. 288-299. 3. Ryoo, C.S. (2012). On the q-extension of second kind Euler polynomials, Far East Journal Mathematical Sciences, v.61, pp. 17-25. 1270

RYOO: 2ND KIND q-EULER NUMBERS

4. Ryoo, C.S. (2012). A numerical investigation of the structure of the roots of the second kind q-Euler polynomials, Journal of Computational Analysis and Applications, v.14, pp. 321-327. 5. Ryoo, C.S. (2011). A note on the q-Hurwitz Euler zeta functions, Journal of Computational Analysis and Applications, v.13, pp. 1012-1018.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1272-1279, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Second order α-univexity and duality for nondifferentiable minimax fractional programming ∗ Gang Yang†‡ Fu-qiu Zeng

Qing-jie Hu

School of Mathematics and Statistics, Hunan University of Commerce, 410205, Changsha, P.R. China

Abstract In this paper, we introduce the concept of second order α-univexity by generalizing αunivexity and present a second-order dual for a nondifferentiable minimax fractional programming. Under the assumptions on the functions involving second order α-univexity, weak, strong and strict converse duality theorems are obtained in order to establish a connection between the primal problems and dual problems. Our results extend some existing dual results which were discussed previously in the literature [11, 12, 14, 15, 16]. Keywords. Nondifferentiable minimax fractional programming; Second order duality; second order α-univexity MR(2000)Subject Classification: 49N15,90C30

1.

Introduction

In this paper, we consider the following nondifferentiable minimax fractional programming problem: 1

(P )

M inimize

sup

f (x, y) + (xT Bx) 2

1

h(x, y) − (xT Dx) 2 g(x) ≤ 0, x ∈ Rn , y∈Y

s.t.

where Y is a compact subset of Rm , f, h : Rn × Rm → R, g : Rn → Rp are twice continuously differentiable. B and D are n × n symmetric positive semidefinite matrices. It is assumed that for 1 1 each (x, y) in Rn × Rm , f (x, y) + (xT Bx) 2 ≥ 0 and h(x, y) − (xT Dx) 2 > 0. ∗

Project supported by Soft Science Foundation of Hunan Provincial Scientific Department of China(No.2009CK3081)and Humanities and Social Sciences Foundation of Ministry of Education of China(No.12YJAZH173) † Corresponding author. Permanent Address: School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205, China. Tel: +86 731 88688325, Fax: +86 731 88689249. ‡ E-mail address: yanggang [email protected]

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Since Schmitendorf [1] introduced necessary and sufficient optimality conditions for generalized minimax programming, much attention has been paid to optimality conditions and duality theorems for the minimax fractional programming problems in recent years. Yadav and Mukherjee [2] formulated two dual models for (P) and derived duality theorems for the case of convex differentiable minimax fractional programming. Chandra and Kumar [3] pointed out some omissions in the dual formulation of Yadav and Mukherjee and constructed two modified dual problems for minimax fractional programming problem and proved duality results. Liu and Wu [12, 4], and Ahmad [5] obtained sufficient optimality conditions and duality theorems for (P) assuming the functions involved to be generalized convex.Yang and Hou [6] discussed optimality conditions and duality results for (P) involving generalized convexity assumptions. Bector et al [7] discussed second order duality results for minimax programming problems under generalized binvexity. Later on, Liu [8] extended these results involving second order generalized B-invexity. Husain et al [9] formulated two types of second order dual models for minimax fractional programming problems, and derived weak, strong and strict converse duality theorems under η-bonvexity assumptions. Lai and Lee [10] obtained duality theorems for two parameter-free dual models of nondifferentiable minimax fractional programming problem which involve pseudo-quasi convex functions by using optimality conditions given in [11]. Noor,M.A.[17], Noor,M.A. and Noor,K.I. [18], Mishra and Noor,M.A.[13] introduced some classes of α-invex function by relaxing the definition of an invex function. Mishra, Pant and Rautela [14] introduced the concept of strict pseudo α-invex and quasi α-invex functions. Pant and Rautela [19], and Rautela and Pant [20] introduced various generalizations of α-invex and α-univex functions. Recently, Mishra, Pant and Rautela [16] introduced the concepts of α-univex, pseudo α-univex, strict pseudo α-univex and quasi α-univex functions respectively by unifying the notions of α-invex and univex functions, and derived the sufficient optimality conditions and established duality theorems for three different dual models of problem (P). In this paper, a new concept of second order α-univexity is introduced by generalizing αunivexity. Under the assumptions on the functions involving second order α-univexity, weak, strong and strict converse duality theorems about a second-order dual for a nondifferentiable minimax fractional programming are established. Our results extend some existing dual results which were discussed previously in the literature [11, 12, 14, 15, 16].

2.

Preliminaries

Let S = {x ∈ Rn : g(x) ≤ 0} denote the set of all feasible solutions of (P). For each (x, y) ∈ Rn ×Rm , we define J(x) = {j ∈ M = {1, 2, · · · , m} : gj (x) = 0}, 1

Y (x) = {y ∈ Y :

f (x, y) + (xT Bx) 2 1

h(x, y) − (xT Dx) 2

1

= sup z∈Y

f (x, z) + (xT Bx) 2 1

h(x, z) − (xT Dx) 2

},

and s s K(x) = {(s, t, ye) ∈ N × R+ × Rms : 1 ≤ s ≤ n + 1, t = (t1 , t2 , · · · , ts ) ∈ R+ ,

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s ∑

ti = 1, ye = (y 1 , y 2 , · · · , y s ), y i ∈ Y (x), i = 1, 2, · · · , s}.

t=1

In our discussion we shall need the following generalized Schwartz inequality 1

1

⟨x, Av⟩ ≤ ⟨x, Ax⟩ 2 ⟨v, Av⟩ 2 , f or x, v ∈ Rn ,

(2.1)

the equality holds when Ax = λAv, for some λ ≥ 0. Let X(α-invex set) be a subset of Rn , η : X × X → Rn be an n-dimensional vector-valued function and α(x, a) : X × X → R+ \ {0} be a bifunction. Assume that ϕ0 , ϕ1 : R → R, b0 , b1 : X × X × [0, 1] → R+ \ {0}, b(x, a) = lim b(x, a, λ) ≥ 0, and b does not depend on λ if the function λ→0 is differentiable. In the sequel, we introduce a class of second order α-univexity. Definition 2.1 A twice differentiable function f : X → R is said to be second order α-univex at a with respect to b0 , ϕ0 , α and η if there exist functions b0 , ϕ0 , α and η such that, for every x ∈ X, p ∈ Rn , we have 1 b0 (x, a)ϕ0 [f (x) − f (a) + pT ∇2 f (a)p] ≥ ⟨α(x, a)(∇f (a) + ∇2 f (a)p), η(x, a)⟩. 2 Definition 2.2 A twice differentiable function f : Rn → R over X is said to be second order (strictly) pseudo α-univex at a with respect to b0 , ϕ0 , α and η if there exist functions b0 , ϕ0 , α and η such that, for all x ∈ X, p ∈ Rn , 1 ⟨α(x, a)(∇f (a) + ∇2 f (a)p), η(x, a)⟩ ≥ 0 ⇒ b0 (x, a)ϕ0 [f (x) − f (a) + pT ∇2 f (a)p] ≥ (>)0. 2 Definition 2.3 A twice differentiable function f : Rn → R over X is said to be second order quasi α-univex at x0 with respect to b0 , ϕ0 , α and η if there exist functions b0 , ϕ0 , α and η such that, for all x ∈ X, p ∈ Rn , 1 b0 (x, a)ϕ0 [f (x) − f (a) + pT ∇2 f (a)p] > 0 ⇒ ⟨α(x, a)(∇f (a) + ∇2 f (a)p), η(x, a)⟩ > 0. 2 Remark 2.1 It is obvious that the second order α-univexity generalizes the α-univexity in [16]. The following theorem will be needed in the proofs of strong duality theorems: Theorem 2.1 (Necessary conditions)[11]Let x∗ be a solution of (P) satisfying x∗T Bx∗ > 0, x∗T Dx∗ > 0, and let ∇gj (x∗ ), j ∈ J(x∗ ) be linearly independent. There exist (s∗ , t∗ , y ∗ ) ∈ K(x∗ ), λ0 ∈ p R+ , w, v ∈ Rn and µ∗ ∈ R+ such that ∗

s ∑

t∗i {∇f (x∗ , y ∗i ) + Bw − λ0 (∇h(x∗ , y ∗i ) − Dv)} + ∇

m ∑

µ∗j gj (x∗ ) = 0,

j=1

i=1

f (x∗ , y ∗i ) + (x∗T Bx∗ ) 2 − λ0 (h(x∗ , y ∗i ) − (x∗T Dx∗ ) 2 ) = 0, i = 1, 2, · · · , s∗ , 1

1

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m ∑

µ∗j gj (x∗ ) = 0,

j=1

t∗i ≥ 0,

s∗ ∑

t∗i = 1, i = 1, 2, · · · , s∗ .

i=1

w Bw ≤ 1, v T Dv ≤ 1, T

(x∗T Bx∗ ) 2 = x∗T Bw, (x∗T Dx∗ ) 2 = x∗T Dv. 1

3.

1

Second order duality

By utilizing the optimality conditions of the previous section, we formulate the second order dual to (P)as follows: sup λ, (D) max (s,t,y)∈K(z) (z,µ,λ,w,v,p)∈H1 (s,t,y) m × R × Rn × Rn × Rn satisfying where H1 (s, t, y) denotes the set of all (z, µ, λ, w, v, p) ∈ Rn × R+ + s ∑

ti [∇f (z, y i ) + Bw − λ(∇h(z, y i ) − Dv)] + ∇2

i=1

+ ∇

m ∑ j=1

s ∑ i=1

µj gj (z) + ∇2

m ∑

j=1

ti (f (z, y i ) − λh(z, y i ))p

i=1

(3.1)

µj gj (z)p = 0,

j=1

ti [f (z, y i ) + z T Bw − λ(h(z, y i ) − z T Dv)] − 12 pT ∇2 m ∑

s ∑

µj gj (z) − 12 pT ∇2 1

m ∑

s ∑

ti (f (z, y i ) − λh(z, y i ))p ≥ 0,

(3.2)

i=1

µj gj (z)p ≥ 0,

(3.3)

j=1 1

wT Bw ≤ 1, v T Dv ≤ 1, (z T Bz) 2 = z T Bw, (z T Dz) 2 = z T Dv.

(3.4)

If, for a triplet (s, t, y) ∈ K(z), the set H1 (s, t, y) = ∅, then we define the supremum over it to be −∞. Let Z denote the set of all feasible solutions of (D). In this section, we denote s ∑ ψ(.) = ti [f (., y i ) + (.)T Bw − λ(h(., y i ) − (.)T Dv)]. i=1

Theorem 3.1 (Weak Duality)Let x and (z, µ, λ, s, t, w, v, p) be feasible solutions of (P) and (D), respectively. If, for each (z, µ, λ, s, t, w, v, p) ∈ Z, one of the following conditions holds: (i)µT g(.) is second order α-univex at z with respect to b1 , ϕ1 , α, η and ψ(.) is second order α-univex at z with respect to b0 , ϕ0 , α, η with ϕ0 (V ) ≥ 0 ⇒ V ≥ 0 and ϕ1 (V ) ≤ V , (ii)µT g(.) is second order quasi α-univex at z with respect to b1 , ϕ1 , α, η and ψ(.) is second order pseudo α-univex at z with respect to b0 , ϕ0 , α, η with V < 0 ⇒ ϕ0 (V ) < 0 and V ≤ 0 ⇒ ϕ1 (V ) ≤ 0, (iii)µT g(.) is second order strictly pseudo α-univex at z with respect to b1 , ϕ1 , α, η and ψ(.) is second order quasi α-univex at z with respect to b0 , ϕ0 , α, η with V < 0 ⇒ ϕ0 (V ) < 0 and V ≤ 0 ⇒ ϕ1 (V ) ≤ 0, then 1 f (x, y) + (xT Bx) 2 sup 1 ≥ λ. y∈Y h(x, y) − (xT Dx) 2 1275

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Proof. Suppose the conclusion is not true, i.e., 1

sup y∈Y

f (x, y) + (xT Bx) 2 1

h(x, y) − (xT Dx) 2

< λ.

Then, we have 1

1

f (x, y) + (xT Bx) 2 − λ{h(x, y) − (xT Dx) 2 } < 0, ∀y ∈ Y. That is 1

1

ti [f (x, y i ) + (xT Bx) 2 − λ{h(x, y i ) − (xT Dx) 2 }] ≤ 0, i = 1, 2, · · · , s. From (2.1),(3.4) and the above inequality, we obtain s ∑

ti [f (x, y i ) + xT Bw − λ{h(x, y i ) − xT Dv}] ≤

s ∑

1

1

ti [f (x, y i ) + (xT Bx) 2 − λ{h(x, y i ) − (xT Dx) 2 }]

i=1

i=1

< 0 s ∑ ≤ ti [f (z, y i ) + z T Bw − λ{h(z, y i ) − z T Dv}] −

i=1 s 1 T 2 ∑ p ∇ ti (f (z, y i ) 2 i=1

− λh(z, y i ))p.

That is ψ(x) < ψ(z) − 12 pT ∇2 ψ(z)p.

(3.5)

If condition (i) holds, then b0 (x, z)ϕ0 [ψ(x) − ψ(z) + 12 pT ∇2 ψ(z)p] ≥ = ≥ ≥

⟨α(x, z)(∇ψ(z) + ∇2 ψ(z)p), η(x, z)⟩ ⟨α(x, z)(−∇µT g(z) − ∇2 µT g(z)p), η(x, z)⟩ −b1 (x, z)ϕ1 [µT g(x) − µT g(z) + 21 pT ∇2 µT g(z)p] µT g(z) − µT g(x) − 21 pT ∇2 µT g(z)p ≥ 0 (3.6)

Since ϕ0 (V ) ≥ 0 ⇒ V ≥ 0 and b0 > 0, we have 1 ψ(x) ≥ ψ(z) − pT ∇2 ψ(z)p, 2 which contradicts with (3.5). Hence, the assertion is true. If condition (ii) holds, by the positivity of b0 and V < 0 ⇒ ϕ0 (V ) < 0, then from (3.5), we get 1 b0 (x, z)ϕ0 [ψ(x) − ψ(z) + pT ∇2 ψ(z)p] < 0. 2 Using the second order pseudo α-univexity, we can deduce the following inequality ⟨α(x, z)(∇ψ(z) + ∇2 ψ(z)p), η(x, z)⟩ < 0.

(3.7)

Taking into account (3.1), (3.7) and the positivity of α(x, z), we have ⟨(∇µT g(z) + ∇2 µT g(z)p), η(x, z)⟩ > 0.

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(3.8)

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According to µT g(x) ≤ 0, (3.3), the positivity of b1 (x, z) and V ≤ 0 ⇒ ϕ1 (V ) ≤ 0, we have b1 (x, z)ϕ1 [µT g(x) − µT g(z) + 12 pT ∇2 µT g(z)p] ≤ 0.

(3.9)

By the second order quasi α-univexity of µT g(.) and the above inequality, we get ⟨α(x, z)(∇µT g(z) + ∇2 µT g(z)p), η(x, z)⟩ ≤ 0. That is, ⟨(∇µT g(z) + ∇2 µT g(z)p), η(x, z)⟩ ≤ 0, which contradicts with(3.8). For condition (iii), the proof is similar to that of condition (ii). Remark 3.1 If we take ϕ0 , ϕ1 as identity maps, b0 = b1 = 1, α0 = α1 = 1, p = 0 and η(x1 , x0 ) = x1 − x0 in the above theorem, we get Theorem 4.1 in [11]. If we take ϕ0 , ϕ1 as identity maps, b0 = b1 = 1, α0 = α1 = 1, p = 0 and remove the quadratic terms from the numerator and denominator of objective function and from the constraints in the above theorem, we get Theorem 3.1 in [12]. If we take ϕ0 , ϕ1 as identity maps, b0 = b1 = 1, p = 0, we get Theorem 4.1 in [14]. If we take α0 = α1 = 1, p = 0 in the above theorem, we get Theorem 2 in [15]. If we take p = 0 in the above theorem, we get Theorem 4.1 in [16]. Theorem 3.2 (Strong Duality)Let x∗ be an optimal solution of (P) and ∇gj (x∗ ), j ∈ J(x∗ ) be linearly independent, then there exist (s∗ , t∗ , y ∗ ) ∈ K(x∗ ) and (x∗ , u∗ , λ∗ , w∗ , v ∗ , p∗ = 0) ∈ H1 (s∗ , t∗ , y ∗ ) such that (x∗ , u∗ , λ∗ , s∗ , t∗ , w∗ , v ∗ , p∗ = 0) is feasible for (D), and the corresponding objective values of (P) and (D) are equal. If, in addition, the assumptions of Weak Duality hold for all feasible solutions of (P) and (D), then (x∗ , u∗ , λ∗ , s∗ , t∗ , w∗ , v ∗ , p∗ = 0) is an optimal solution of (D). Proof. By Theorem 2.1, there exist (s∗ , t∗ , y ∗ ) ∈ K(x∗ ) and (x∗ , u∗ , λ∗ , w∗ , v ∗ , p∗ = 0) ∈ H1 (s∗ , t∗ , y ∗ ) such that (x∗ , u∗ , λ∗ , s∗ , t∗ , w∗ , v ∗ , p∗ = 0) is feasible for (D) and 1

∗

λ =

f (x∗ , y ∗i ) + (x∗T Bx∗ ) 2 1

g(x∗ , y ∗i ) − (x∗T Dx∗ ) 2

.

The optimality of the feasible solution for (D)can be derived from Theorem 3.1. Remark 3.2 If we take ϕ0 , ϕ1 as identity maps, b0 = b1 = 1, α0 = α1 = 1, p∗ = 0 and η(x1 , x0 ) = x1 − x0 in the above theorem, we get Theorem 4.2 in [11]. If we take ϕ0 , ϕ1 as identity maps, b0 = b1 = 1, α0 = α1 = 1, p∗ = 0 and remove the quadratic terms from the numerator and denominator of objective function and from the constraints in the above theorem, we get Theorem 3.2 in [12]. If we take ϕ0 , ϕ1 as identity maps, b0 = b1 = 1, p∗ = 0, we get Theorem 4.2 in [14]. If we take α0 = α1 = 1, p∗ = 0 in the above theorem, we get Theorem 3 in [15]. If we take p∗ = 0 in the above theorem, we get Theorem 4.2 in [16].

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Theorem 3.3 (Strict Converse Duality) Let x∗ and (z, µ, λ, s, t, w, v, p) be optimal solutions of (P) and (D), respectively. Assume that the hypothesis of Theorem 3.2 is fulfilled, if one of the following conditions holds: s ∑ (i))µT g(.) is second order strictly α-univex at z with respect to b1 , ϕ1 , α, η and ti [f (., y i ) + i=1

⟨., Bw⟩ − λ(h(., y i ) + ⟨., Dv⟩)] is second order strictly α-univex at z with respect to b0 , ϕ0 , α, η with ϕ0 (V ) ≥ 0 ⇒ V ≥ 0 and ϕ1 (V ) ≤ V ; s ∑ (ii)µT g(.) is second order quasi α-univex at z with respect to b1 , ϕ1 , α, η and ti [f (., y i )+⟨., Bw⟩− i=1

λ(h(., y i ) + ⟨., Dv⟩)] is second order strictly pseudo α-univex at z with respect to b0 , ϕ0 , α, η with V < 0 ⇒ ϕ0 (V ) < 0 and V ≤ 0 ⇒ ϕ1 (V ) ≤ 0. Then x∗ = z, that is, z is an optimal solution for (P) and 1

sup y∈Y

f (z, y) + (z T Bz) 2 1

h(z, y) − (z T Dz) 2

= λ.

Proof. Suppose that x∗ ̸= z. From Theorem 3.2, we know that there exist (s∗ , t∗ , y ∗ ) ∈ K(x∗ ) and (x∗ , u∗ , λ∗ , w∗ , v ∗ , p∗ = 0) ∈ H1 (s∗ , t∗ , y ∗ ) such that (x∗ , u∗ , λ∗ , s∗ , t∗ , w∗ , v ∗ , p∗ = 0) is optimal for (D) and 1 f (x∗ , y) + (x∗T Bx∗ ) 2 ∗ (3.10) λ = sup 1 = λ. y∈Y g(x∗ , y) − (x∗T Dx∗ ) 2 The remaining part of the proof is similar to that of Theorem 3.1 in which x is replaced by x∗ and (z, µ, λ, s, t, w, v, p) by (z, µ, λ, s, t, w, v, p), and we get 1

sup y∈Y

f (x∗ , y) + (x∗T Bx∗ ) 2 1

g(x∗ , y) − (x∗T Dx∗ ) 2

> λ,

which contradicts with (3.10). Therefore, we conclude that x∗ = z. Remark 3.3 If we take ϕ0 , ϕ1 as identity maps, b0 = b1 = 1, α0 = α1 = 1, p = 0 and η(x1 , x0 ) = x1 − x0 in the above theorem, we get Theorem 4.3 in [11]. If we take ϕ0 , ϕ1 as identity maps, b0 = b1 = 1, α0 = α1 = 1, p = 0 and remove the quadratic terms from the numerator and denominator of objective function and from the constraints in the above theorem, we get Theorem 3.3 in [12]. If we take ϕ0 , ϕ1 as identity maps, b0 = b1 = 1, p = 0, we get Theorem 4.3 in [14]. If we take α0 = α1 = 1, p = 0 in the above theorem, we get Theorem 4 in [15]. If we take p = 0 in the above theorem, we get Theorem 4.3 in [16].

References [1] Schmitendorf, W.E., Necessary and sufficient conditions for static minimax problem, Journal of Mathematical Analysis and Applications, 57, 683-693(1977). [2]

Yadav, S.R., Mukherjee, R.N., Duality in fractional minimax programming problems, Journal of Austrain Mathematical Society Series A, 31, 484-492(1990). 1278

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[3] Chandra, S., Kumar, V., Duality in fractional minimax programming problem, Journal of Austrain Mathematical Society Series A, 58, 376-386(1995). [4] Liu, J.C., Wu, C.S., On minimax fractional optimality conditions with (F,ρ)-convexity, Journal of Mathematical Analysis and Applications, 219, 36-51(1998). [5] Ahmad, I., Optimality conditions and duality in fractional minimax programming involving generalized ρ-invexity, International Journal of Management System, 19, 165-180(2003). [6] Yang, X.M., Hou, S.H., On minimax fractional optimality conditions and duality with generalized convexity, Journal of Global Optimization, 31, 235-252(2005). [7] Bector, C.R., Chandra, S., Husian, I., Second order duality for a minimax programming problem, Opsearch, 28, 249-263(1991). [8] Liu, J.C., Lee, J.C., Second order duality for minimax programming, Utilitas Mathematics, 56, 5363(1999). [9] Husian, Z., Ahmad, I., Sharma, S., Second order duality for minimax fractional programming, Optimization Letter, 3, 277-286(2009). [10] Lai, H.C., Lee, J.C., On duality theorems for a nondifferentiable minimax fractional programming, Journal of Computational and Applied Mathematics, 146, 115-126(2002). [11] Lai, H.C., Liu, J.C., Tanaka, K., Necessary and sufficient conditions for minimax fractional programming,Journal of Mathematical Analysis and Applications, 230, 311-328(1999). [12] Liu, J.C., Wu, C.S., On minimax fractional optimality conditions with invexity, Journal of Mathematical Analysis and Applications, 219, 21-35(1998). [13] Mishra, S.K., Noor, M.A., On vector variational inequality problems, Journal of Mathematical Analysis and Applications, 311, 69-75(2005). [14] Mishra, S.K., Pant, R.P., Rautela, J.S., Generalized α-invexity and nondifferentiable minimax fractional programming, Journal of Computational and Applied Mathematics, 206, 122-135(2006). [15] Mishra, S.K., Wang, S.Y., Lai, K.K., Shi, J.M., Nondifferentiable minimax fractional programming under univexity, Journal of Computational and Applied Mathematics, 158, 379-395(2003). [16]

Mishra, S.K., Pant, R.P., Rautela, J.S., Generalized α-univexity and duality for nondifferentiable minimax fractional programming, Nonlinear Analysis, 70, 144-158(2009).

[17] Noor, M.A., On generalized preinvex functions and monotonicities, Journal of Inequality in Pure Applied Mathematics, 5, 1-9(2004). [18] Noor, M.A., Noor, K.I., Some characterizations of strongly preinvex functions, Journal of Mathematical Analysis and Applications, 316, 676-706(2006). [19] Pant, R.P., Rautela, J.S., α-invexity and duality in mathematical programming, Journal of Mathematical Analysis and Approximation Theory, 1, 104-114(2006). [20] Rautela, J.S., Pant, R.P., Duality in mathematical programming under α-invexity, Journal of Mathematical Analysis and Approximation Theory, 2, 72-83(2007).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1280-1290, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

SOME PROPERTIES OF THE INTERVAL-VALUED GENERALIZED FUZZY INTEGRAL WITH RESPECT TO A FUZZY MEASURE BY MEANS OF AN INTERVAL-REPRESENTABLE GENERALIZED TRIANGULAR NORM LEE-CHAE JANG

Abstract. We consider the generalized fuzzy integral introduced by Fang[4] and use the concept of interval-valued functions which is used for representing uncertain functions. In this paper, we define the interval-valued generalized fuzzy integral with respect to a fuzzy measure by means of an interval-representable generalized triangular norm of measurable interval-valued functions and investigate some characterizations and convergence properties of them.

1. Introduction Fang[4], Wu-Wang-Ma[22], and Wu-Ma-Song-Zhang[23] introduced the theory of the generalized fuzzy integral(for short, the (G) fuzzy integral) by means of a generalized triangular norm. Many researchers[5,16,17,20,22-26] have been studying fuzzy measure and fuzzy integral theory used in the decision making and information theory. The main idea of this study is the concept of interval-valued functions which is used for representing uncertain functions. Aubin[1], Aumann[2], Beliakov et al.[3], Jang et al.[6-12], Schjear-Jacoben[18], Weichselberger[21], and Zhang et al.[24-26] have been researching various integrals of uncertain functions, for examples, the Lebesgue integral, the fuzzy integral, and the Choquet integral of interval-valued functions, the calculation of economic uncertain, and the theory of interval-probability, etc. In this paper, we define the interval-valued generalized fuzzy integral (for short, (IG) fuzzy integral) with respect to a fuzzy measure by means of an interval-representable generalized triangular norm of measurable interval-valued functions and investigate some characterizations and convergence properties of them. In section 2, we list definitions and basic properties of a fuzzy measure, a generalized triangular norm, and the (G) fuzzy integral with respect to a fuzzy measure by means of a generalized triangular norm of measurable functions. In section 3, we define the (IG) fuzzy integral of interval-valued functions by means of an interval-representable generalized triangular norm of measurable interval-valued functions and investigate some characterizations of them. In section 4, we investigate some convergence properties of the (IG) fuzzy integral with respect to a fuzzy measure by means of an interval-representable generalized triangular norm of measurable interval-valued functions. In section 5, we give a brief summary results and some conclusions. 1991 Mathematics Subject Classification. 28E10, 28E20, 03E72, 26E50 11B68. Key words and phrases. fuzzy measure, interval-representable generalized triangular norm, generalized fuzzy integral, interval-valued function, convergence theorem. 1

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LEE-CHAE JANG

2. Definitions and Preliminaries In this section, we first introduce some definitions and basic properties of a fuzzy measure, the (G) fuzzy integral with respect to a fuzzy measure by means of a generalized triangular norm of measurable functions. Let X be a set, B be a σ−algebra of subsets of X, and (X, B) be a measurable space. Denote F(X) by the set of all nonnegative measurable functions on (X, B) and N = {1, 2, 3, ⋅ ⋅ ⋅}. Definition 2.1. ([3-26]) (1) A set function µ : B → [0, ∞] is called a fuzzy measure if (i) µ(∅) = 0 and (ii) A, B ∈ B and A ⊂ B implies µ(A) ≤ µ(B). It is easy to see that if 𝑚 is the Lebesgue measure on X and we define µ = 𝑚2 , then µ is a fuzzy measure which satisfies the two conditions of Definition 2.1. Since µ is not additive, we can see that this fuzzy measure is not a classical measure. Definition 2.2. ([22,23]) Let D = [0, ∞]2 \ {(0, ∞), (∞, 0)}. The mapping T : D → [0, ∞] is said to be a generalized triangular norm if it satisfies the following conditions (i) T [0, 𝑥] = 0 for all 𝑥 ∈ [0, ∞) and exists an 𝑒 ∈ (0, ∞] such that T [𝑥, 𝑒] = 𝑥 for each 𝑥[0, ∞]. In this case, 𝑒 is said to be the unit element of T , (ii) T [𝑥, 𝑦] = T [𝑦, 𝑥] for all (𝑥, 𝑦) ∈ D, (iii) T [𝑎, 𝑏] ≤ T [c, 𝑑] whenever 𝑎 ≤ c, 𝑏 ≤ 𝑑, and (iv) if {(𝑥𝑛 , 𝑦𝑛 )} ∈ D, (𝑥, 𝑦) ∈ D, 𝑥𝑛 ↘ 𝑥, and 𝑦𝑛 ↗ 𝑦, then T [𝑥𝑛 , 𝑦𝑛 ] −→ T [𝑥, 𝑦].

Remark 2.1. T1 [𝑥, 𝑦] = min {𝑥, 𝑦} and T2 [𝑥, 𝑦] = 𝑘𝑥𝑦(𝑘 > 0) are generalized triangular norms and the identities of T1 and T2 are ∞ and 𝑘1 , respectively (see [4]). Definition 2.3. ([22,23]) Let (X, B, µ) be a fuzzy measure space and T be a generalized triangular norm. If A ∈ B and 𝑓 ∈ F(X), then the (G) fuzzy integral with respect to µ by means of T of 𝑓 on A is defined by ∫ (G) 𝑓 𝑑µ = sup T [𝛼, µA,f (𝛼)], (1) A

𝛼>0

where µA,f (𝛼) = µ(A ∩ {𝑥 ∈ X ∣ 𝑓 (𝑥) ≥ 𝛼}) for all 𝛼 ∈ [0, ∞). We remark that the Sugeno integral defined by M. Sugeno[20] and the (N) fuzzy integrals defined by N. Shilkret[19] are the special kinds of (G) fuzzy integrals and the corresponding generalized triangular norms are T [𝑥, 𝑦] = min{𝑥, 𝑦} and T [𝑥, 𝑦] = 𝑥𝑦, respectively. Recall that lim𝑛→∞ 𝑓𝑛 = inf sup{𝑓𝑛 }, 𝑘≥1 𝑛≥𝑘

(2)

for all {𝑓𝑛 } ⊂ F(X). In [4], the authors have shown the following theorems which are convergence properties of the (G) fuzzy integral.

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Theorem 2.1. ([4]) Let {𝑓𝑛 } ⊂ F (X), 𝑓 ∈ F (X), A ∈ B, and 𝑓𝑛 ↘ 𝑓 on A. Then we have ∫ ∫ lim (G) 𝑓𝑛 𝑑µ = (G) 𝑓 𝑑µ (3) 𝑛→∞

A

A

if and only if the following conditions are satisfied (i) for any given ε > 0 there exists 𝑛0 ∈ N such that (4) µA,f𝑛0 (c0 + ε) < ∞, ∫ where c0 = 𝑠𝑢𝑝{𝑎 > 0 : T [𝑎, ∞] ≤ (G) A 𝑓 𝑑µ} and µA,f𝑛0 (c0 + ε) = µ(A ∩ {𝑥 ∈ X ∣ 𝑓𝑛0 (𝑥) ≥ c0 + ε}) and (ii) for any {𝛼𝑛 } with 𝛼𝑛 ↗ ∞ or 𝛼𝑛 ↘ 0, ∫ lim𝑛→∞ T [𝛼𝑛 , µA,f𝑛 (𝛼𝑛 )] ≤ (G) 𝑓 𝑑µ (5) A

where µA,f𝑛 (𝛼) = µ(A ∩ {𝑥 ∈ X ∣ 𝑓𝑛 (𝑥) ≥ 𝛼}) for all 𝑛 ∈ N and 𝛼 ∈ ℝ+ .

Theorem 2.2. ([4]) Let {𝑓𝑛 } ⊂ F (X), 𝑓 ∈ F (X), µ(A) < ∞, and 𝑓𝑛 ↘ 𝑓 . Then we have ∫ ∫ lim (G) 𝑓𝑛 𝑑µ = (G) 𝑓 𝑑µ (6) 𝑛→∞

A

A

if and only if for any {𝛼𝑛 } with 𝛼𝑛 ↗ ∞,

∫

lim𝑛→∞ T [𝛼𝑛 , µA,f𝑛 (𝛼𝑛 )] ≤ (G)

𝑓 𝑑µ,

(7)

A

where µA,f𝑛 (𝛼) = µ(A ∩ {𝑥 ∈ X ∣ 𝑓𝑛 (𝑥) ≥ 𝛼}) for all 𝑛 ∈ N and 𝛼 ∈ ℝ+ .

3. The (IG) fuzzy integral of measurable interval-valued functions In this section, we consider the intervals and define an interval-valued generalized triangular norm. Let 𝐼(Y ) be the set of all bounded closed intervals (intervals, for short) in Y as follows: 𝐼(Y ) = {𝑎 = [𝑎l , 𝑎r ] ∣ 𝑎l , 𝑎r ∈ Y and 𝑎l ≤ 𝑎r },

(8)

where Y is [0, ∞) or [0, ∞]. For any 𝑎 ∈ ℝ , we define 𝑎 = [𝑎, 𝑎]. Obviously, 𝑎 ∈ 𝐼(ℝ+ ) (see[3, 9-12, 18, 21, 24-26]). +

Definition 3.1. If 𝑎 = [𝑎l , 𝑎r ], 𝑏 = [𝑏l , 𝑏r ], 𝑎𝑛 = [𝑎𝑛,l , 𝑎𝑛,r ], 𝑎𝛼 = [𝑎𝛼,l , 𝑎𝛼,r ] ∈ 𝐼(Y ) for all 𝑛 ∈ N and 𝛼 ∈ [0, ∞), and 𝑘 ∈ [0, ∞), then we define arithmetic, maximum, minimum, order, inclusion, supremum, and infinimum operations as follows: (1) 𝑎 + 𝑏 = [𝑎l + 𝑏l , 𝑎r + 𝑏r ], (2) 𝑘𝑎 = [𝑘𝑎l , 𝑘𝑎r ], (3) 𝑎𝑏 = [𝑎l 𝑏l , 𝑎r 𝑏r ], (4) 𝑎 ∨ 𝑏 = [𝑎l ∨ 𝑏l , 𝑎r ∨ 𝑏r ], (5) 𝑎 ∧ 𝑏 = [𝑎l ∧ 𝑏l , 𝑎r ∧ 𝑏r ], (6) 𝑎 ≤ 𝑏 if and only if 𝑎l ≤ 𝑏l and 𝑎r ≤ 𝑏r , (7) 𝑎 < 𝑏 if and only if 𝑎 ≤ 𝑏 and 𝑎 ̸= 𝑏, (8) 𝑎 ⊂ 𝑏 if and only if 𝑏l ≤ 𝑎l and 𝑎r ≤ 𝑏r ,

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(9) sup𝑛 𝑎𝑛 = [sup𝑛 𝑎𝑛,l , sup𝑛 𝑎𝑛,r ], (10) inf 𝑛 𝑎𝑛 = [inf 𝑛 𝑎𝑛,l , inf 𝑛 𝑎𝑛,r ], (11) sup𝛼 𝑎𝛼 = [sup𝛼 𝑎𝛼,l , sup𝛼 𝑎𝛼,r ], and (12) inf 𝛼 𝑎𝛼 = [inf 𝛼 𝑎𝛼,l , inf 𝛼 𝑎𝛼,r ]. Note that if a mapping 𝑑H : 𝐼(Y ) × 𝐼(Y ) → [0, ∞] is defined by { } 𝑑H (A, B) = 𝑚𝑎𝑥 sup inf ∣𝑥 − 𝑦∣, sup inf ∣𝑥 − 𝑦∣ , 𝑥∈A y∈B

y∈B 𝑥∈A

(9)

for all A, B ∈ 𝐼(Y ), then 𝑑H is called a Hausdorff metric and (𝐼(Y ), 𝑑H ) is a metric space. It is well-known that for every 𝑎 = [𝑎l , 𝑎r ], 𝑏 = [𝑏l , 𝑏r ] ∈ 𝐼(Y ), 𝑑H (𝑎, 𝑏) = 𝑚𝑎𝑥 {∣𝑎l − 𝑏l ∣, ∣𝑎r − 𝑏r ∣} .

(10)

For a sequence of intervals {𝑎𝑛 }, we say that {𝑎𝑛 } converges in the Hausdorff metric to 𝑎, in ¯) = 0. Then, it is easy to see that symbols, 𝑑H − lim𝑛→∞ 𝑎𝑛 = 𝑎 if lim𝑛→∞ 𝑑H (𝑎𝑛 , 𝑎 𝑑H − lim 𝑎𝑛 = 𝑎 if and only if 𝑛→∞

lim 𝑎𝑛,l = 𝑎l and

𝑛→∞

lim 𝑎𝑛,r = 𝑎r .

𝑛→∞

(11)

Now, we consider an interval-representable generalized triangular norm as follows(see [3]): Definition 3.2. Let D = 𝐼([0, ∞])2 \ {(0, ∞), (∞, 0)}. The mapping T : D → 𝐼([0, ∞]) is called an interval-representable generalized triangular norm if there are two generalized triangular norm Tl and Tr such that Tl ≤ Tl and T = [Tl , Tr ].

Theorem 3.1. If we take T1 [𝑥, 𝑦] = min{𝑥, 𝑦} and T2 [𝑥, 𝑦] = 𝑘𝑥 𝑦(𝑘 > 0), then T1 and T2 are interval-representable generalized triangular norms. Proof. If we define T1,l [𝑥, 𝑦] = min{𝑥, 𝑦} and T1,r [𝑥, 𝑦] = min{𝑥, 𝑦}, then, by Remark 2.1, T1,l and T1,r are generalized triangular norms. Thus, by Definition 3.2, we see that T1 = [T1,l , T1,r is an interval-representable generalized triangular norm. Similarly, if we define T2,l [𝑥, 𝑦] = T2,r [𝑥, 𝑦] = 𝑘𝑥𝑦(𝑘 > 0), then, by Remark 2.1, T2,l and T2,r are generalized triangular norms. By Definition 3.2, we see that T2 = [T2,l , T2,r is an interval-representable generalized triangular norm. Let IF(X) the set of all measurable interval-valued functions 𝑓 : X → 𝐼([0, ∞)) \ {∅}. Then we define the (IG) fuzzy integral with respect to a fuzzy measure by means of an interval-representable generalized triangular norm of interval-valued functions as follows. Definition 3.3. Let (X, B, µ) be a fuzzy measure space, T = [Tl , Tr ] be an interval-representable generalized triangular norm, A ∈ B, and 𝑓 = [𝑓l , 𝑓r ] ∈ IF(X). (1) An interval-valued function 𝑓 is said to be measurable if for any open set O ⊂ [0, ∞), 𝑓

−1

(O) = {𝑥 ∈ X ∣ 𝑓 (𝑥) ∩ O ̸= ∅} ∈ B.

(12)

(2) The (IG) fuzzy integral with respect to µ by means of T of 𝑓 on A is defined by ∫ (𝐼G) 𝑓 𝑑µ = sup T[𝛼, µA,f (𝛼)], (13) A

𝛼>0

where µA,f (𝛼) = [µA,fl (𝛼), µA,fr (𝛼)] for all 𝛼 ∈ [0, ∞).

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(3) 𝑓 is said to be integrable on A if ∫ (𝐼G) 𝑓 𝑑µ ∈ P([0, ∞)) \ {∅},

(14)

A

where P([0, ∞)) is the set of all subsets of [0, ∞). Let IF ∗ (X) be the set of all integrable interval-valued functions. We can obtain the following basic characterizations of the (IG) fuzzy integral with respect to a fuzzy measure by means of an interval-representable generalized triangular norm of interval-valued functions. Theorem 3.2. Let (X, B, µ) be a fuzzy measure space and T = [Tl , Tr ] be an intervalrepresentable generalized triangular norm. (1) If 𝑓 , 𝑔 ∈ IF ∗ (X) and 𝑓 ≤ 𝑔, then we have ∫ ∫ (𝐼G) 𝑓 𝑑µ ≤ (𝐼G) 𝑔𝑑µ. (15) A

A

(2) If A ∈ B and 𝑎 ∈ 𝐼([0, ∞)), then we have ∫ (𝐼G) 𝑎𝑑µ = T[𝑎l , µ(A)] ∨ T[𝑎r , [0, µ(A)]].

(16)

A

Proof. (1) Since 𝑓 ≤ 𝑔, 𝑓l ≤ 𝑔l and 𝑓r ≤ 𝑔r . Thus, we have µA,fl (𝛼) ≤ µA,gl (𝛼) and µA,fr (𝛼) ≤ µA,gr (𝛼) for all 𝛼 ∈ [0, ∞). By Definition 3.2, T[𝛼, µA,f (𝛼)]

= [Tl [𝛼, µA,fl (𝛼)], Tr [𝛼, µA,fl (𝛼)] ≤ [Tl [𝛼, µA,gl (𝛼)], Tr [𝛼, µA,gl (𝛼)] = T[𝛼, µA,g (𝛼)].

for all 𝛼 ∈ [0, ∞). Therefore we obtain ∫ (𝐼G) 𝑓 𝑑µ = sup T[𝛼, µA,f (𝛼)]

∫

𝛼>0

A

≤

sup T[𝛼, µA,g (𝛼)] = (𝐼G) 𝛼>0

𝑔𝑑µ. A

(2) Note that if µ is a fuzzy measure and 𝑎 = [𝑎l , 𝑎r ] ∈ [0, ∞), then we have µA,a (𝛼)

= ⎧ [µA,al (𝛼), µA,ar (𝛼)] ⎨ [µ(A), µ(A)] if 𝛼 ∈ (0, 𝑎l ] [0, µ(A)] if 𝛼 ∈ (𝑎l , 𝑎r ] = ⎩ 0 if 𝛼 ∈ (𝑎r , ∞).

Thus, by Definition 3.1 (11) and Definition 3.3(2), we have ∫ (𝐼G) 𝑎𝑑µ A

=

sup T[𝛼, µA,a (𝛼)] 𝛼>0

= =

sup [Tl [𝛼, µA,al (𝛼), Tr [𝛼, µA,ar (𝛼)]

𝛼>0 [

] sup Tl [𝛼, µA,al (𝛼), sup Tr [𝛼, µA,ar (𝛼)

𝛼>0 [𝛼>0 = sup Tl [𝛼, µ(A)], max{ sup Tr [𝛼, µ(A)], 00

= [sup Tr [𝛼, µA,fl (𝛼)], sup Tr [𝛼, µA,fr (𝛼)]] [𝛼>0 ∫ ∫ 𝛼>0 ] = (G) 𝑓l 𝑑µ, (G) 𝑓r 𝑑µ , A

A

∫ where (G) A 𝑓u 𝑑µ is the (G) fuzzy integral with respect to a fuzzy measure by means of a generalized triangular norm Tu of a measurable function 𝑓u for 𝑢 = l, r. Example 3.1. Let Tl [𝑥l , 𝑦l ] = min{min{𝑥l , 𝑦l }, 𝑥l ⋅ 𝑦l } and Tr [𝑥r , 𝑦r ] = max{min{𝑥r , 𝑦r }, 𝑥r ⋅ 𝑦r }, and T[𝑥, 𝑦] = [Tl [𝑥l , 𝑦l ], Tr [𝑥l , 𝑦r ]] be an interval-valued generalized triangular norm for all 𝑥 = [𝑥l , 𝑥r ], 𝑦 = [𝑦l , 𝑦r ] ∈ 𝐼([0, ∞)), and 𝑚 be the Lebesgue measure on [0, ∞). Note that if 𝑥, 𝑦 ⊂ [0, 1], then we have Tl [𝑥l , 𝑦l ] = 𝑥l ⋅ 𝑦l and Tr [𝑥r , 𝑦r ] = min{𝑥r , 𝑦r } [ ] If we take X = [0, 1] and 𝑓 : X −→ 𝐼([0, ∞))\∅ by 𝑓 = 14 𝑥, 2𝑥 for all 𝑥 ∈ X is an interval-valued function,and µ = 𝑚2 , then we have ∫ (𝐼G) 𝑓¯𝑑µ = sup [Tl [𝛼, µfl (𝛼)], Tr [𝛼, µfr (𝛼)]] 𝛼>0 [ ] 1 2 2 = sup {𝛼 ⋅ (1 − 4𝛼) }, sup min{𝛼, (1 − 𝛼) } 2 0 0 : Tl [𝑎, ∞] ≤ (G) A 𝑓l 𝑑µ}, sup{𝑎 > 0 : Tr [𝑎, ∞] ≤ (G) A 𝑓r 𝑑µ} and (ii) for any 𝛼𝑛 with 𝛼𝑛 ↗ ∞ or 𝛼𝑛 ↘ 0, ∫ lim𝑛→∞ T[𝛼𝑛 , µA,f 𝑛 (𝛼𝑛 )] ≤ (𝐼G) 𝑓 𝑑µ. (23) A

Proof. By Theorem 3.3, we have [ ] ∫ ∫ ∫ 𝑓 𝑑µ = (G) 𝑓𝑛,l 𝑑µ, (G) 𝑓𝑛,r 𝑑µ (𝐼G) A

A

for all 𝑛 ∈ N and

[ ] ∫ ∫ 𝑓 𝑑µ = (G) 𝑓l 𝑑µ, (G) 𝑓r 𝑑µ ,

∫ (𝐼G) A

(24)

A

A

A

(25)

∫ ∫ where where (G) A 𝑓𝑛,u 𝑑µ and (G) A 𝑓u 𝑑µ are the (G) fuzzy integrals with respect to a fuzzy measure by means of a generalized triangular norm Tu for 𝑢 = l, r. By (11),(18),(24) and (25), (21) implies that ∫ ∫ lim (G) 𝑓𝑛,l 𝑑µ = (G) 𝑓l 𝑑µ, (26) 𝑛→∞

A

A

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LEE-CHAE JANG

and

∫ lim (G)

𝑛→∞

∫ 𝑓𝑛,r 𝑑µ = (G)

A

𝑓r 𝑑µ.

(27)

A

By Theorem 2.1, (26) holds if and only if the following conditions are satisfied (i) for any given ε > 0 there exists a 𝑛1 ∈ N such that µA,f𝑛1 ,l (c1 + ε) < ∞, ∫ where c1 = sup{𝑎 > 0 : Tl [𝑎, ∞] ≤ (G) A 𝑓l 𝑑µ}. (ii) For any {𝛼𝑛 } with 𝛼𝑛 ↗ ∞ or 𝛼𝑛 ↘ 0,

∫

lim𝑛→∞ Tl [𝛼𝑛 , µA,f𝑛,l (𝛼𝑛 )] ≤ (G)

𝑓l 𝑑µ.

(28)

A

and (27) holds if and only if the following conditions are satisfied (i) for any given ε > 0 there exists a 𝑛2 ∈ N such that µA,f𝑛2 ,r (c2 + ε) < ∞, ∫ where c2 = sup{𝑎 > 0 : Tl [𝑎, ∞] ≤ (G) A 𝑓r 𝑑µ} and (ii) for any {𝛼𝑛 } with 𝛼𝑛 ↗ 𝛼 or 𝛼𝑛 ↘ 0,

∫

lim𝑛→∞ T [𝛼𝑛 , µA,fr𝑛 (𝛼𝑛 )] ≤ (GF )

𝑓r 𝑑µ.

(29)

A

Without loss of the generality, we assume that 𝑛1 ≥ 𝑛2 and c1 ≤ c2 . Thus, 𝑓𝑛1 ,l ≤ 𝑓𝑛2 ,l and 𝑓𝑛1 ,r ≤ 𝑓𝑛2 ,r and hence µA,f𝑛1 ,l (c2 + ε) ≤ µA,f𝑛1 ,l (c1 + ε),

(30)

µA,f𝑛1 ,r (c2 + ε) ≤ µA,f𝑛2 ,r (c2 + ε).

(31)

and If we take c0 = max{c1 , c2 }, then (30) and (31) implies that for any given ε > 0, there exists a 𝑛0 = 𝑛1 ∈ N such that µA,f 𝑛 (c0 + ε) 0 ≤ µA,f 𝑛 (c2 + ε) 1 = [µA,f𝑛1 ,l (c2 + ε), µA,f𝑛1 ,r (c2 + ε)] ≤ [µA,f𝑛1 ,l (c1 + ε), µA,f𝑛2 ,r (c2 + ε)] < [∞, ∞] = ∞. Thus, the condition (22) holds. For any {𝛼𝑛 } with 𝛼𝑛 ↗ ∞ or 𝛼𝑛 ↘ 0, by Theorem 2.1, we have ∫ lim𝑛→∞ Tl [𝛼𝑛 , µA,f𝑛,l (𝛼𝑛 )] ≤ (G) 𝑓l 𝑑µ, (32) A

and

∫ lim𝑛→∞ Tr [𝛼𝑛 , µA,f𝑛,r (𝛼𝑛 )] ≤ (G)

𝑓r 𝑑µ. A

By (32) and (33) and (20) and Theorem 3.3, lim𝑛→∞ T [𝛼𝑛 , [µA,fl (𝛼), µA,fr (𝛼)]] lim ] [ 𝑛→∞ [Tl [𝛼𝑛 , [µA,fl (𝛼)], Tr [𝛼𝑛 , µA,fr (𝛼)]] lim T [𝛼 , [µ (𝛼)], lim [ 𝑛→∞ ]𝑛→∞ Tr [𝛼𝑛 , µA,fr (𝛼)] ∫ l 𝑛 A,f∫l ≤ (G) 𝑓l 𝑑µ, (G) 𝑓r 𝑑µ A

A

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(33)

INTERVAL-VALUED FUZZY INTEGRAL

9

∫ = (𝐼G)

𝑓 𝑑µ. A

Thus, the condition (23) holds. Similarly, we can derive the converse that (22) and (23) implies (21).

Theorem 4.2. Let Tl , Tr be generalized triangular norms and T[𝑥, 𝑦] = [Tl [𝑥l , 𝑦l ], Tr [𝑥r , 𝑦r ]] be an interval-representable generalized triangular norm for all 𝑥 = [𝑥l , 𝑥r ], 𝑦 = [𝑥l , 𝑥r ] ∈ 𝐼([0 < ∞)). If {𝑓 𝑛 } ⊂ IF ∗ (X) and 𝑓 ∈ IF ∗ (X), and A ∈ B, and 𝑓 𝑛 ↘ 𝑓 on A in the Hausdorff metric, then we have ∫ 𝑑H − lim𝑛→∞ 𝑓 𝑛 (𝑥) = (𝐼G) 𝑓 𝑑µ, (34) A

if and only if for any {𝛼𝑛 } with 𝛼𝑛 ↗ ∞,

∫

lim𝑛→∞ T[𝛼𝑛 , µA,f 𝑛 (𝛼𝑛 )] ≤ (𝐼G)

𝑓 𝑑µ.

Proof. By (11),(18),(24) and (25), (34) implies the following two equations: ∫ ∫ lim (G) 𝑓𝑛,l 𝑑µ = (G) 𝑓l 𝑑µ, 𝑛→∞

A

(35)

A

(36)

A

and ∫

∫

lim (GF )

𝑛→∞

𝑓r𝑛 𝑑µ = (GF ) A

𝑓r 𝑑µ.

(37)

A

By Theorem 2.2, (36) and (37) hold if and only if for any {𝛼𝑛 } with 𝛼𝑛 ↗ ∞, ∫ 𝑓l 𝑑µ, lim Tl [𝛼𝑛 , µA,f𝑛,l (𝛼𝑛 )] ≤ (G) 𝑛→∞

(38)

A

and

∫ lim Tr [𝛼𝑛 , µA,f𝑛,r (𝛼𝑛 )] ≤ (G)

𝑛→∞

𝑓r 𝑑µ.

(39)

A

By (38),(39) and Definition 3.1 (9) and (10), we have lim T[𝛼𝑛 , µA,f 𝑛 (𝛼𝑛 )]

𝑛→∞

=

lim [Tl [𝛼𝑛 , µA,f𝑛,l (𝛼𝑛 )], Tr [𝛼𝑛 , µA,f𝑛,r (𝛼𝑛 )]]

𝑛→∞

= [ lim Tl [𝛼𝑛 , µA,f𝑛,l (𝛼𝑛 )], lim Tr [𝛼𝑛 , µA,f𝑛,r (𝛼𝑛 )]] 𝑛→∞ 𝑛→∞ ∫ ∫ ≤ [(G) 𝑓l 𝑑µ, (G) 𝑓r 𝑑µ] A ∫A = (𝐼G) 𝑓 𝑑µ. A

Thus, the condition (35) holds. Similarly, we can derive the converse that (35) implies (34).

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5. Conclusions In this paper, we considered the concept of an interval-representable generalized triangular norm (see [3] and Definition 3.2)and studied some characterizations and convergence properties of the (IG) fuzzy integral with respect to a fuzzy measure by means of an intervalrepresentable generalized triangular norms of measurable interval-valued functions (see Definition 3.3) which is an extension of the (G)fuzzy integral with respect to a fuzzy measure by means of a generalized triangular norm of measurable functions by Fang[4]. From Theorems 3.1 and 3.2, we investigated some characterizations of the (IVG) fuzzy integral with respect to a fuzzy measure on the space of measurable interval-valued functions. Theorem 3.3 are used in the proof of Theorems 4.1 and 4.2. From Theorems 4.1 and 4.2, we discussed some convergence properties of the (IG) fuzzy integral with respect to a fuzzy measure of measurable interval-valued functions. Acknowledgement This paper was supported by Konkuk University in 2013.

References [1] J.P. Aubin, Set-valued Analysis, Birkhauser Boston, (1990). [2] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl., 12 (1965), 1-12. [3] G. Deschrijver, Generalized arithemetic operators and their relatioship to t-norms in interval-valued fuzzy set theory, Fuzzy Sets and Systems 160 (2009), 3080-3102. [4] J. Fang, A note on the convergence theorem of generalized fuzzy integrals, Fuzzy Sets and Systems, 127 (2002), 377-381. [5] M. Ha, C. Wu, Fuzzy measurs and integral theory, Science Press, Beijing, 1998. [6] L.C. Jang, B.M. Kil, Y.K. Kim, J.S. Kwon, Some properties of Choquet integrals of set-valued functions, Fuzzy Sets and Systems, 91 (1997), 61-67. [7] L.C. Jang, J.S. Kwon, On the representation of Choquet integrals of set-valued functions and null sets, Fuzzy Sets and Systems, 112 (2000), 233-239. [8] L.C. Jang, T. Kim, J.D. Jeon,On the set-valued Choquet integrals and convergence theorems(II), Bull. Korean Math. Soc. 40(1) (2003), 139-147. [9] L.C. Jang, Interval-valued Choquet integrals and their apllications, J. Appl. Math. and Computing, 16(12) (2004), 429-445. [10] L.C. Jang, A note on the monotone interval-valued set function defined by the interval-valued Choquet integral, Commun. Korean Math. Soc., 22 (2007), 227-234. [11] L.C. Jang, On properties of the Choquet integral of interval-valued functions, Journal of Applied Mathematics, 2011 (2011), Article ID 492149, 10pages. [12] L.C. Jang, A note on convergence properties of interval-valued capacity functionals and Choquet integrals, Information Sciences, 183 (2012), 151-158. [13] G.Michel, Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems, 69(1995), 279-298. [14] T. Murofushi, M. Sugeno, A theory of fuzzy measures: representations, the Choquet integral, and null sets, J. Math. Anal. Appl., 159 (1991), 532-549. [15] T. Murofushi, M. Sugeno, M. Suzaki, Autocontinuity, convergence in measure, and convergence indistribution, Fuzzy Sets and Systems, 92(1997) 197-203. [16] D.A. Ralescu, M. Sugeno, Fuzzy integral representation , Fuzzy Sets and Systems, 84(1996),127-133. [17] D.A. Ralescu, G. Adams, The fuzzy integral, J. Math. Anal. Appl., 75(2) (1980), 562-570. [18] H. Schjear-Jacobsen, Representation and calculation of economic uncertains: intervals, fuzzy numbers and probabilities, Int. J. of Production Economics, 78(2002), 91-98. [19] N. Shilkret, Maxitive measures and integration, Indag Math, 33(1971), 109-116. [20] M. Sugeno, Theory of fuzzy integrals and its applications, Doctorial Thesis, Tokyo Institute of Techonology, Tokyo,(1974). [21] K. Wechselberger, The theory of interval-probability as a unifying concept for uncertainty, Int. J. Approximate Reasoning, 24(2000), 149-170. [22] C. Wu, S. Wang, M. Ma, Generalized fuzzy integrals: Part1. Fundamental concept, Fuzzy Sets and Systems, 57(1993), 219-226.

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[23] C. Wu, M. Ma, S. Song, S. Zhang Generalized fuzzy integrals: Part3. convergence theorems, Fuzzy Sets and Systems,, Fuzzy Sets and Systems, 70 (1995), 75-87. [24] D. Zhang, C. Guo, D. Lin, Set-valued Choquet integrals revisited, Fuzzy Sets and Systems, 147(2004), 475-485. [25] D. Zhang, C. Guo, On the convergence of sequences of fuzzy measures and generalized convergences theorems of fuzzy integral, Fuzzy Sets and Systems, 72(1995), 349-356. [26] D. Zhang, Z. Wang, Fuzzy integrals of fuzzy-valued functions, Fuzzy Sets and Systems, 54 (1993), 63-67. Department of Computer Engineering, Konkuk University, Chungju 138-701, Korea, e-mail: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1291-1299, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Soft rough sets and their properties Cheng-Fu Yang∗ School of Mathematics and Statistics of Hexi University, Zhangye Gansu,734000, P. R. China

Abstract Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. However, it has been pointed out that classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In this paper, the notion of the soft rough set theory is proposed. Soft rough set theory is a combination of a rough theory and a soft set theory. The complement, relative complement, union, restricted union, intersection, restricted intersection, ”and” and ”or” operations are defined on the soft rough sets. The basic properties of the soft rough sets are also presented and discussed. Keywords: Rough sets; Soft sets; Soft rough sets; Properties MR2000: 08A99.

1

Introduction

Soft set theory was firstly proposed by Molodtsov in 1999 [7]. It is different from traditional tools for dealing with uncertainties, such as the theory of probability [13], the theory of fuzzy sets [16], the theory of rough sets [12]. It has been demonstrated that soft set theory brings about a rich potential for applications in many fields such as function smoothness, Riemann integration, decision making, measurement theory, game theory, etc. Soft set theory has received much attention since its introduction by Molodtsov. The concept and basic properties of soft set theory are presented in [9,7]. Chen et al. [2] presented a new definition of soft set parameterization reduction and compared this definition with the related concept of knowledge reduction in the rough set theory. In fact, the soft set model can also be combined with other mathematical models [15]. For example, by amalgamating the soft sets and algebra, Aktas and Cagman [1] introduce the basic properties of soft sets, compare soft sets to the related concepts of fuzzy sets [16] and rough sets [12], point out that every fuzzy set and every rough set may be considered a soft set, and give a definition of soft groups. Feng et al. [4] defined soft semirings and several related notions to establish a connection between soft sets and semirings. Maji et al. [11] presented the concept of the fuzzy soft set which is based on a combination of the fuzzy set and soft set models. Xu et al. [14] introduce the notion of vague soft sets which is an extension to the soft sets and is based on a combination of the vague set [5] and soft set models. Majumdar and Samanta [8] further generalized the concept of fuzzy soft sets as introduced by Maji et al. [10], in other words, a degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. Jiang et al. [6] presented the concept of the interval-valued intuitionistic fuzzy soft sets by combining the interval-valued intuitionistic fuzzy set and soft set models. The purpose of this paper is to combine the rough sets and soft sets, from which we can obtain a new soft set model: soft rough set theory. ∗ E-mail:

[email protected] (C.F.Yang).Tel.:+86 0936 8280868; fax:+86 0936 8282000.

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The rest of this paper is organized as follows. The following section briefly reviews some background on soft sets, rough sets. At the same time some operations of rough sets are defined. In Section 3, we propose the concepts and operations of soft rough sets and discuss their properties in detail. Finally, in Section 4, we draw the conclusion and present some topics for future research.

2

Preliminaries

Given a non-empty universe U, by P(U) we will denote the power-set on U. If ρ is an equivalence relation on U then foe every x ∈ U, [x]ρ denotes the equivalence class of ρ determined by x. For any X ⊆ U, we write X c to denote the complementation of X in U, that is the set U − X. Definition 2.1 [3]. A pair (U, ρ) where U , ∅ and ρ is an equivalence relation on U, is called an approximation space. Definition 2.2 [3]. For an approximation space (U, ρ), by a rough approximation in (U, ρ) we mean a mapping ρ : P(U) → P(U) × P(U) defined by for every X ∈ P(U), ρ(X) = (ρ(X), ρ(X)), where ρ(X) = {x ∈ X|[x]ρ ⊆ X},

ρ(X) = {x ∈ X|[x]ρ ∩ X , ∅}.

ρ(X) is called a lower rough approximation of X in (U, ρ), where as ρ(X) is called a upper rough approximation of X in (U, ρ). Definition 2.3 [3]. Given an approximation space (U, ρ), a pair (A, B) ∈ P(U) × P(U) is called a rough set in (U, ρ) iff (A, B) = ρ(X) for some X ∈ P(U). Definition 2.4. Let ρ(X) be is a rough set over U with respect to an equivalence nrelation ρ, then the o complement of ρ(X) is denoted by ρc (X) = (ρc (X), ρc (X)), is a rough set, where ρc (X) = x ∈ X c |[x]ρ ⊆ X c , ρc (X) = n o x ∈ X c |[x]ρ ∩ X c , φ . By the definition of rough set, obviously, ρc (X) = ρ(X c ). Definition 2.5. Let ρ(X) and ρ(Y) be two rough sets over U with respect to an equivalence relation ρ, then union of ρ(X) and ρ(Y) denoted by ρ(X) ∪ ρ(Y), is a rough set ρ(Z), where ρ(Z) = {x ∈ X ∪ Y|[x]ρ ⊆ (X ∪ Y)}, ρ(Z) = {x ∈ X ∪ Y|[x]ρ ∩ (X ∪ Y) , ∅}. By the definition of rough set, obviously, ρ(Z) = ρ(X ∪ Y). Definition 2.6. Let ρ(X) and ρ(Y) be two rough sets over U with respect to an equivalence relation ρ, then intersection of ρ(X) and ρ(Y) denoted by ρ(X) ∩ ρ(Y), is a rough set ρ(Z), where ρ(Z) = {x ∈ X ∩ Y|[x]ρ ⊆ (X ∩ Y)}, ρ(Z) = {x ∈ X ∩ Y|[x]ρ ∩ (X ∩ Y) , ∅}. By the definition of rough set, obviously, ρ(Z) = ρ(X ∩ Y). Molodtsov [7] defined the soft set in the following way. Let U be an initial universe of objects and E the set of parameters in relation to objects in U. Parameters are often attributes, characteristics, or properties of objects. Let P(U) denote the power set of U and A ⊆ E. Definition 2.7. A pair hF, Ai is called a soft set over U, where F is a mapping given by F : A → P(U). In other words, the soft set is not a kind of set, but a parameterized family of subsets of the set U. For any parameter ε ∈ A, F(ε) may be considered as the set of ε-approximate elements of the soft set hF, Ai.

3

Soft rough sets and their properties

Definition 3.1. Let U be an initial universe and E be a set of parameters. RS (U) denotes the set of all rough sets of U with respect to an equivalence relation ρ. Let A ⊆ E. A pair hF, Ai is a soft rough set over U, where F is a mapping given by F : A → RS (U).

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In other words, a soft rough set is a parameterized family of rough subsets of U, thus, its universe is the set of all rough sets of U, i.e., RS (U). A soft rough set is also a special case of a soft set because it is still a mapping from parameters to RS (U). Definition 3.2. The union of two soft rough sets hF, Ai and hG, Bi over a common universe U with respect to an equivalence relation ρ is a soft rough set hH, Ci, where C = A ∪ B and ∀ε ∈ C,   F(ε), i f ε ∈ A − B,    G(ε), i f ε ∈ B − A, H(ε) =     F(ε) ∪ G(ε), i f ε ∈ A ∩ B. We write hF, Aie ∪hG, Bi = hH, Ci. Definition 3.3. The intersection of two soft rough sets hF, Ai and hG, Bi over a common universe U with respect to an equivalence relation ρ is a soft rough set hH, Ci, where C = A ∪ B and ∀ε ∈ C,   F(ε), i f ε ∈ A − B,    G(ε), i f ε ∈ B − A, H(ε) =     F(ε) ∩ G(ε), i f ε ∈ A ∩ B. We write hF, Aie ∩hG, Bi = hH, Ci. Definition 3.4. Let E = {e1 , e2 , . . . , en } be a parameter set. The not set of E denoted by eE is defined by eE = {ee1 , ee2 , . . . , een } where eei = not ei . Definition 3.5. Let hF, Ai be a soft rough set over a common universe U with respect to an equivalence relation ρ, then complement of hF, Ai denoted by hF, Aic = hF c , eAi is a soft rough set, and ∀eε ∈eA, F c (eε) = ρc (X) = ρ(X c ), where F(ε) = ρ(X). Definition 3.6. Let hF, Ai and hG, Bi be two soft rough sets over a common universe U with respect to an equivalence relation ρ such that A ∩ B , ∅. The restricted union of hF, Ai and hG, Bi is denoted by hF, AidhG, Bi, and is defined as hF, AidhG, Bi = hH, Ci, where C = A∩ B and ∀ε ∈ C, H(ε) = F(ε)∪G(ε). Definition 3.7. Let hF, Ai and hG, Bi be two soft rough sets over a common universe U with respect to an equivalence relation ρ such that A ∩ B , ∅. The restricted intersection of hF, Ai and hG, Bi is denoted by hF, AiehG, Bi, and is defined as hF, AiehG, Bi = hH, Ci, where C = A∩ B and ∀ε ∈ C, H(ε) = F(ε)∩G(ε). Definition 3.8. Let hF, Ai be a soft rough set over a common universe U with respect to an equivalence relation ρ, then restricted complement of hF, Ai denoted by hF, Air = hF r , Ai is a soft rough set, and ∀ε ∈ A, F r (ε) = ρc (X) = ρ(X c ), where F(ε) = ρ(X). Definition 3.9. A soft rough set hF, Ai over U with respect to an equivalence relation ρ is said to be a null soft rough set denoted by ∅A , if ε ∈ A, F(ε) = ρ(∅). Definition 3.10. A soft rough set hF, Ai over U with respect to an equivalence relation ρ is said to be a absolute soft rough set denoted by ΣA , if ε ∈ A, F(ε) = ρ(U). Theorem 3.11. Let E be a set of parameters, A ⊆ E. If ∅A is a null soft rough set, ΣA a absolute soft rough set, and hFAi and hF, Ei two soft rough sets over a common universe U with respect to an equivalence relation ρ, then (1) hF, Aie ∪hF, Ai = hF, Ai; (2) hF, Aie ∩hF, Ai = hF, Ai; (3)hF, Eie ∪∅A = hF, Ei; (4)hF, Eie ∩∅E = ∅E ; (5)hF, Eie ∪ΣE = ΣE ; (6)hF, Eie ∩ΣA = hF, Ei. Proof. It is easily obtained from Definitions above. Theorem 3.12. Let hF, Ai and hG, Bi be two soft rough sets over a common universe U with respect to an equivalence relation ρ such that A ∩ B , ∅. Then

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(1) (hF, Ai d hG, Bi)r = hF, Air e hG, Bir ; (2) (hF, Ai e hG, Bi)r = hF, Air d hG, Bir . Proof. For ∀ε ∈ A ∩ B, let F(ε) = ρ(X), G(ε) = ρ(Y), and hF, Ai d hG, Bi = hH, Ci. According to definition, H(ε) = F(ε) ∪ G(ε) = ρ(X) ∪ ρ(Y) = ρ(X ∪ Y), and then H r (ε) = ρc (X ∪ Y) = ρ(X c ∩ Y c ). Now hF, Air e hG, Bir = hF r , Ai e hGr , Bi = hK, Ci, where C = A ∩ B. So by definition, we have K(ε) = F r (ε) ∩ Gr (ε) = ρc (X) ∩ ρc (Y) = ρ(X c ∩ Y c ) = H r (ε) ∀ε ∈ C. Hence (hF, Ai d hG, Bi)r = hF, Air e hG, Bir . (2) Let hF, Ai e hG, Bi = hH, Ci where C = A ∩ B , ∅, thus H(ε) = F(ε) ∩ G(ε) = ρ(X) ∩ ρ(Y) = ρ(X ∩ Y) for all ε ∈ C. Since (hF, Ai e hG, Bi)r = hH, Cir = hH r , Ci, by definition, H r (ε) = ρ((X ∩ Y)c ) = ρ(X c ∪ Y c ). Now hF, Air d hG, Bir = hF r , Ai d hGr , Bi = hK, Ci, where C = A ∩ B. So by definition, we have K(ε) = F r (ε) ∪ Gr (ε) = ρc (X) ∪ ρc (Y) = ρ(X c ∪ Y c ) = H r (ε) ∀ε ∈ C. Hence (hF, Ai e hG, Bi)r = hF, Air d hG, Bir . Theorem 3.13. Let hF, Ai and hG, Bi be two soft rough sets over a common universe U with respect to an equivalence relation ρ. Then we have the following: (1) (hF, Aie ∪hG, Bi)c = hF, Aic e ∩hG, Bic ; c c (2) (hF, Aie ∩hG, Bi) = hF, Ai e ∪hG, Bic . Proof. (1) For the convenience, we do following assumptions, ∀ε ∈ A ∪ B : if ε ∈ A − B, then F(ε) = ρ(X); if ε ∈ B − A, then G(ε) = ρ(Y); if ε ∈ A ∩ B, then F(ε) = ρ(Z), G(ε) = ρ(W). Suppose that hF, Aie ∪hG, Bi = hH, A ∪ Bi. Then (hF, Aie ∪hG, Bi)c = hH, A ∪ Bic = hH c , e(A ∪ Bi) = c hH , eA∪eBi). For ∀ε ∈ A ∪ B, we have   F(ε) = ρ(X), i f ε ∈ A − B,    G(ε) = ρ(Y), i f ε ∈ B − A, H(ε) =     F(ε) ∪ G(ε) = ρ(Z ∪ W), i f ε ∈ A ∩ B. Thus   ρ(X c ), i f eε ∈eA−eB,    c ρ(Y c ), i f eε ∈eB−eA, H (eε) =     ρ(Z c ∩ W c ), i f eε ∈eA∩eB. Moreover, let hF, Aic e ∩hG, Bic = hF c , eAie ∩hGce , eBi = hK, eA∪eBi. Then  c  F (eε) = ρ(X c ), i f eε ∈eA−eB,    c c G (eε) = ρ(Y ), i f eε ∈eB−eA, K(eε) =     F c (eε) ∩ Gc (eε) = ρ(Z c ∩ W c ), i f ε ∈eA∩eB. Since H c and K are indeed the same rough-set-valued mapping, we conclude that (hF, Aie ∪hG, Bi)c = ce c hF, Ai ∩hG, Bi as required. (2) The proof is similar to that of (1). Definition 3.14. Let hF, Ai and hG, Bi be two soft rough sets over a common universe U with respect to an equivalence relation ρ. Then ”hF, Ai and hG, Bi” is a soft rough set denoted by hF, Ai ∧ hG, Bi, is defined as hF, Ai ∧ hG, Bi = hH, A × Bi, where H(α, β) = F(α) ∩ G(β), ∀(α, β) ∈ A × B. Definition 3.15. Let hF, Ai and hG, Bi be two soft rough sets over a common universe U with respect to an equivalence relation ρ. Then ”hF, Ai or hG, Bi” is a soft rough set denoted by hF, Ai ∨ hG, Bi, is defined as hF, Ai ∨ hG, Bi = hO, A × Bi, where O(α, β) = F(α) ∪ G(β), ∀(α, β) ∈ A × B. Theorem 3.16. Let hF, Ai and hG, Bi be two soft rough sets over a common universe U with respect to an

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equivalence relation ρ. Then we have the following: (1) (hF, Ai ∧ hG, Bi)c = hF, Aic ∨ hG, Bic ; (2) (hF, Ai ∨ hG, Bi)c = hF, Aic ∧ hG, Bic . Proof. (1) Suppose that hF, Ai ∧ hG, Bi = hH, A × Bi. Then (hF, Ai ∧ hG, Bi)c = hH, A × Bic = hH c , e(A × B)i. For ∀e(α, β) ∈e(A × B), let F(α) = ρ(X), G(β) = ρ(Y). By definition, H(α, β) = F(α) ∩ G(β) = ρ(X ∩ Y). Thus H c (e(α, β)) = ρc (X ∩ Y) = ρ((X ∩ Y)c ) = ρ(X c ∪ Y c ). Since hF, Aic = hF c , eAi and hG, Bic = hGc , eBi, then hF, Aic ∨ hG, Bic = hF c , eAi ∨ hGc , eBi. Assume that hF c , eAi ∨ hGc , eBi = hO, eA×eBi = hO, e(A × B)i, where ∀(eα, eβ) ∈eA×eB, by definition, O(eα, eβ) = F c (eα) ∪ Gc (eβ) = ρc (X) ∪ ρc (Y) = ρ(X c ∪ ρ(Y c ) = ρ(X c ∪ Y c ). Consequently, H c and O are the same operators. Thus, (hF, Ai ∧ hG, Bi)c = hF, Aic ∨ hG, Bic . (2) The proof is similar to that of (1). Theorem 3.17. Let hF, Ai, hG, Bi and hH, Ci be three soft rough sets over a common universe U with respect to an equivalence relation ρ. Then we have the following: (1) hF, Ai ∧ (hG, Bi ∧ hH, Ci) = (hF, Ai ∧ hG, Bi) ∧ hH, Ci; (2) hF, Ai ∨ (hG, Bi ∨ hH, Ci) = (hF, Ai ∨ hG, Bi) ∨ hH, Ci. Proof. (1) Assume that hG, Bi ∧ hH, Ci = hI, B × Ci. For ∀(α, β) ∈ B × C, let G(α) = ρ(Y), H(β) = ρ(Z). By definition, I(α, β) = G(α) ∩ H(β) = ρ(Y ∩ Z). Since hF, Ai∧(hG, Bi∧hH, Ci) = hF, Ai∧hI, B×Ci, we suppose that hF, Ai∧hI, B×Ci = hK, A×(B×C)i. For ∀(δ, α, β) ∈ A × (B × C), let F(δ) = ρ(X), by definition, K(δ, α, β) = F(δ) ∩ I(α, β) = ρ(X) ∩ ρ(Y ∩ Z) = ρ(X ∩ Y ∩ Z). On the other hand, we take (δ, α) ∈ A × B. Suppose that hF, Ai ∧ hG, Bi = hJ, A × Bi, by definition, J(δ, α) = F(δ) ∩ G(α) = ρ(X ∩ Y). Since (hF, Ai∧hG, Bi)∧hH, Ci = hJ, A×Bi∧hH, Ci, we suppose that hJ, A×Bi∧hH, Ci = hO, (A×B)×C)i, where O(δ, α, β) = J(δ, α) ∩ H(β) = ρ(X ∩ Y ∩ Z), (δ, α, β) ∈ (A × B) × C = A × B × C. Consequently, K and O are the same operators. Thus, hF, Ai ∧ (hG, Bi ∧ hH, Ci) = (hF, Ai ∧ hG, Bi) ∧ hH, Ci. Theorem 3.18. Let hF, Ai, hG, Bi and hH, Ci be three soft rough sets over a common universe U with respect to an equivalence relation ρ such that A ∩ B ∩ C , ∅. Then we have the following: (1) hF, Ai e (hG, Bi e hH, Ci) = (hF, Ai e hG, Bi) e hH, Ci; (2) hF, Ai d (hG, Bi d hH, Ci) = (hF, Ai d hG, Bi) d hH, Ci; (3) hF, Ai e (hG, Bi d hH, Ci) = (hF, Ai e hG, Bi) d (hF, Ai e hH, Ci); (4) hF, Ai d (hG, Bi e hH, Ci) = (hF, Ai d hG, Bi) e (hF, Ai d hH, Ci). Proof. In the following, we shall prove (1) and (3); (2) and (4) are proved analogously. For the convenience, we do following assumptions, ∀ε ∈ A ∪ B ∪ C, if ε ∈ A − B − C, then F(ε) = ρ(X1 ); if ε ∈ B − A − C, then G(ε) = ρ(X2 ); if ε ∈ C − A − B, then H(ε) = ρ(X3 ); if ε ∈ A ∩ B − C, then F(ε) = ρ(X4 ), G(ε) = ρ(X5 ); if ε ∈ A ∩ C − B, then F(ε) = ρ(X6 ), H(ε) = ρ(X7 ); if ε ∈ B ∩ C − A, then G(ε) = ρ(X8 ), H(ε) = ρ(X9 ); if ε ∈ A ∩ B ∩ C, then F(ε) = ρ(X10 ), G(ε) = ρ(X11 ), H(ε) = ρ(X12 ). (1) Suppose that hG, BiehH, Ci = hI, Di, where D = B∩C. For ∀ε ∈ D, by definition, I(ε) = F(ε)∩ H(ε) = ρ(X8 ∩ X9 ) or ρ(X11 ∩ X12 ). Since hF, Ai e (hG, Bi e hH, Ci) = hF, Ai e hI, Di, we assume that hF, Ai e hI, Di = hJ, S i, where S = A ∩ D. By definition, for ∀ε ∈ S , J(ε) = F(ε) ∩ I(ε) = ρ(X10 ) ∩ ρ(X11 ∩ X12 ) = ρ(X10 ∩ X11 ∩ X12 ). On the other hand, assume that hF, Ai e hG, Bi = hK, Vi, where V = A ∩ B. For ∀ε ∈ V, K(ε) = F(ε) ∩ G(ε) = ρ(X4 ∩ X5 ) or ρ(X10 ∩ X11 ).

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Since (hF, Ai e (hG, Bi) e hH, Ci = hK, Vi e hH, Ci, we assume that hK, Vi e hH, Ci = hL, Wi, where W = V ∩ C = A ∩ B ∩ C. By definition, for ∀ε ∈ W, L(ε) = K(ε) ∩ H(ε) = ρ(X10 ∩ X11 ) ∩ ρ(X12 ) = ρ(X10 ∩ X11 ∩ X12 ). Therefore, L(ε) = J(ε) for all ∀ε ∈ A ∩ B ∩ C. That is, J and L are the same operators. Thus, hF, Ai e (hG, Bi e hH, Ci) = (hF, Ai e hG, Bi) e hH, Ci. (3) Let hG, Bi d hH, Ci = hI, Di, where D = B ∩ C. For ∀ε ∈ D, by definition, I(ε) = G(ε) ∪ H(ε) = ρ(X8 ∪ X9 ) or ρ(X11 ∪ X12 ). Since hF, Ai e (hG, Bi d hH, Ci) = hF, Ai e hI, Di, we assume that hF, Ai e hI, Di = hK, Vi, where V = A ∩ D. For ∀ε ∈ V = A ∩ B ∩ C, K(ε) = F(ε) ∩ I(ε) = ρ(X10 ) ∩ ρ(X11 ∪ X12 ) = ρ(X10 ∩ (X11 ∪ X12 )). On the other hand, suppose hF, Ai e hG, Bi = hJ, Mi and hF, Ai e hH, Ci = hL, Wi, where M = A ∩ B, W = A∩C. Since (hF, AiehG, Bi)d(hF, AiehH, Ci) = hJ, MidhL, Wi, assume that hJ, MidhL, Wi = hO, Ni, where N = M∩W. For ∀ ∈ N = A∩B∩C, by definition, O(ε) = J(ε)∪L(ε) = (F(ε)∩G(ε))∪(F(ε)∩H(ε)) = (ρ(X10 ) ∩ ρ(X11 )) ∪ (ρ(X10 ) ∩ ρ(X12 )) = ρ(X10 ∩ X11 ) ∪ ρ(X10 ∩ X12 ) = ρ((X10 ∩ X11 ) ∪ (X10 ∩ X12 )) = ρ(X10 ∩ (X11 ∪ X12 )). Therefore, K(ε) = O(ε) for all ∀ε ∈ A ∩ B ∩ C. That is, K and O are the same operators. Thus, hF, Ai e (hG, Bi d hH, Ci) = (hF, Ai e hG, Bi) d (hF, Ai e hH, Ci). Theorem 3.19. Let hF, Ai, hG, Bi and hH, Ci be three soft rough sets over a common universe U with respect to an equivalence relation ρ. Then we have the following: (1) hF, Aie ∩(hG, Bie ∩hH, Ci) = (hF, Aie ∩hG, Bi)e ∩hH, Ci; (2) hF, Aie ∪(hG, Bie ∪hH, Ci) = (hF, Aie ∪hG, Bi)e ∪hH, Ci; Proof. In the following, we shall prove (1), (2) is proved analogously. For the convenience, ∀ε ∈ A ∪ B ∪ C, we do following assumptions: if ε ∈ A − B − C, then F(ε) = ρ(X1 ); if ε ∈ B − A − C, then G(ε) = ρ(X2 ); if ε ∈ C − A − B, then H(ε) = ρ(X3 ); if ε ∈ A ∩ B − C, then F(ε) = ρ(X4 ), G(ε) = ρ(X5 ); if ε ∈ A ∩ C − B, then F(ε) = ρ(X6 ), H(ε) = ρ(X7 ); if ε ∈ B ∩ C − A, then G(ε) = ρ(X8 ), H(ε) = ρ(X9 ); if ε ∈ A ∩ B ∩ C, then F(ε) = ρ(X10 ), G(ε) = ρ(X11 ), H(ε) = ρ(X12 ). (1) Suppose that hG, Bie ∩hH, Ci = hI, Di, where D = B ∪ C. For ∀ε ∈ D, by definition,   G(ε) = ρ(X2 ) or ρ(X5 ), ε ∈ B − C,    H(ε) = ρ(X3 ) or ρ(X7 ), ε ∈ C − B, I(ε) =     G(ε) ∩ H(ε) = ρ(X ∩ X ) or ρ(X ∩ X ), ε ∈ B ∩ C. 8

9

11

12

Since hF, Aie ∩(hG, Bie ∩hH, Ci) = hF, Aie ∩hI, Di, we assume that hF, Aie ∩hI, Di = hJ, S i, where S = A∪D. By definition, for ∀ε ∈ S ,   F(ε) = ρ(X1 ), ε ∈ A − D,    I(ε) = ρ(X ) or ρ(X ) or ρ(X ∩ X ), ε ∈ D − A, J(ε) =  2 3 8 9    F(ε) ∩ I(ε) = ρ(X ∩ X ) or ρ(X ∩ X ∩ X ) or ρ(X ∩ X ), ε ∈ A ∩ D. 4

               =             

ρ(X1 ), ρ(X2 ), ρ(X3 ), ρ(X8 ∩ X9 ), ρ(X4 ∩ X5 ), ρ(X6 ∩ X7 ), ρ(X10 ∩ X11 ∩ X12 ),

5

10

11

ε ∈ A − B − C, ε ∈ B − A − C, ε ∈ C − B − A, ε ∈ B ∩ C − A, ε ∈ A ∩ B − C, ε ∈ A ∩ C − B, ε ∈ A ∩ B ∩ C.

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6

7

YANG:SOFT ROUGH SETS

On the other hand, assume that hF, Aie ∪hG, Bi = hK, Vi, where V = A ∪ B. For ∀ε ∈ V,   F(ε) = ρ(X1 ) or ρ(X6 ), ε ∈ A − B,    G(ε) = ρ(X ) or ρ(X ), ε ∈ B − A, K(ε) =  2 8    F(ε) ∩ G(ε) = ρ(X ∩ X ) or ρ(X ∩ X ), ε ∈ A ∩ B. 4

5

10

11

Since (hF, Aie ∩(hG, Bi)e ∩hH, Ci = hK, Vie ∩hH, Ci, we assume that hK, Vie ∩hH, Ci = hL, Wi, where W = V ∪ C = A ∪ B ∪ C. By definition, for ∀ε ∈ W,   K(ε) = ρ(X1 ) or ρ(X2 ) or ρ(X4 ∩ X5 ), ε ∈ V − C,    H(ε) = ρ(X ), ε ∈ C − V, L(ε) =  3    K(ε) ∩ H(ε) = ρ(X ∩ X ) or ρ(X ∩ X ∩ X ) or ρ(X ∩ X ), ε ∈ C ∩ V. 8

               =             

ρ(X1 ), ρ(X2 ), ρ(X3 ), ρ(X8 ∩ X9 ), ρ(X4 ∩ X5 ), ρ(X6 ∩ X7 ), ρ(X10 ∩ X11 ∩ X12 ),

9

10

11

12

6

7

ε ∈ A − B − C, ε ∈ B − A − C, ε ∈ C − B − A, ε ∈ B ∩ C − A, ε ∈ A ∩ B − C, ε ∈ A ∩ C − B, ε ∈ A ∩ B ∩ C.

Therefore, L(ε) = J(ε) for all ∀ε ∈ A ∪ B ∪ C. That is, J and L are the same operators. Thus, hF, Aie ∩(hG, Bie ∩hH, Ci) = (hF, Aie ∩hG, Bi)e ∩hH, Ci. The following example shows that if e and d of assertions (3) and (4) of theorem 3.18 are replaced by e ∩ and e ∪ respectively, then assertions (3) and (4) of theorem 3.18 do not hold, i.e., hF, Aie ∩(hG, Bie ∪hH, Ci) = (hF, Aie ∩hG, Bi)e ∪(hF, Aie ∩hH, Ci) and hF, Aie ∪(hG, Bie ∩hH, Ci) = (hF, Aie ∪hG, Bi)e ∩(hF, Aie ∪hH, Ci) are both incorrect. Example. Let U = {x1 , x2 , x3 , x4 , x5 , x6 } be an initial universe and E = {e1 , e2 , e3 , e4 } be a set of parameters. Let ρ be an equivalence relation on U such that ρ−equivalence classes are the subsets {x1 , x3 }, {x2 , x4 , x5 } and {x6 }. hF, Ai, hG, Bi and hH, Ci are three soft rough sets over U with respect to an equivalence relation ρ. Here A = {e1 , e2 , e3 }, B = {e1 , e2 , e4 }, C = {e1 , e3 , e4 }. We take X1 = {x1 , x3 }, X2 = {x1 , x6 }, X3 = {x2 , x4 , x5 }, X4 = {x2 , x5 }, X5 = {x1 , x4 }, X6 = {x1 , x2 , x3 }, X7 = {x3 , x6 }, X8 = {x4 , x6 } and X9 = {x1 , x3 , x6 }. Let F(e1 ) = ρ(X1 ) = ({x1 , x3 }, {x1 , x3 }); F(e2 ) = ρ(X2 ) = ({x6 }, {x1 , x3 , x6 }); F(e3 ) = ρ(X3 ) = ({x2 , x4 , x5 }, {x2 , x4 , x5 }); G(e1 ) = ρ(X4 ) = (∅, {x2 , x4 , x5 }); G(e2 ) = ρ(X5 ) = (∅, {x1 , x2 , x3 , x4 , x5 }); G(e4 ) = ρ(X6 ) = ({x1 , x3 }, {x1 , x2 , x3 , x4 , x5 }); H(e1 ) = ρ(X7 ) = ({x6 }, {x1 , x3 , x6 }); H(e3 ) = ρ(X8 ) = ({x6 }, {x2 , x4 , x5 , x6 }); H(e4 ) = ρ(X9 ) = ({x1 , x3 , x6 }, {x1 , x3 , x6 }). Suppose that hG, Bie ∪hH, Ci = hI, Di, where D = B ∪ C. By definition, I(e1 ) = G(e1 ) ∪ H(e1 ) = ρ(X4 ) ∪ ρ(X7 ) = ρ(X4 ∪ X7 ) = ρ({x2 , x3 , x5 , x6 }) = ({x6 }, {x1 , x2 , x3 , x4 , x5 , x6 }); I(e2 ) = G(e2 ) = ρ(X5 ) = (∅, {x1 , x2 , x3 , x4 , x5 }); I(e3 ) = H(e3 ) = ρ(X8 ) = ({x6 }, {x2 , x4 , x5 , x6 }); I(e4 ) = G(e4 ) ∪ H(e4 ) = ρ(X6 ) ∪ ρ(X9 ) = ρ(X6 ∪ X9 ) = ρ({x1 , x2 , x3 , x6 }) = ({x1 , x3 , x6 }, {x1 , x2 , x3 , x4 , x5 , x6 }).

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Since hF, Aie ∩(hG, Bie ∪hH, Ci) = hF, Aie ∩hI, Di, we assume that hF, Aie ∩hI, Di = hJ, S i, where S = A∪D. By definition, J(e1 ) = F(e1 ) ∩ I(e1 ) = ρ(X1 ) ∩ ρ(X4 ∪ X7 ) = ρ(X1 ∩ (X4 ∪ X7 )) = ρ({x3 }) = (∅, {x1 , x3 }); J(e2 ) = F(e2 ) ∩ I(e2 ) = ρ(X2 ) ∩ ρ(X5 ) = ρ(X2 ∩ X5 ) = ρ({x1 }) = (∅, {x1 , x3 }); J(e3 ) = F(e3 ) ∩ I(e3 ) = ρ(X3 ) ∩ ρ(X8 ) = ρ(X3 ∩ X8 ) = ρ({x4 }) = (∅, {x2 , x4 , x5 }); J(e4 ) = I(e4 ) = ρ(X6 ∪ X9 ) = ρ({x1 , x2 , x3 , x6 }) = ({x1 , x3 , x6 }, {x1 , x2 , x3 , x4 , x5 , x6 }). On the other hand, suppose that hF, Aie ∩hG, Bi = hK, Vi and hF, Aie ∩hH, Ci = hL, Wi, where V = A ∪ B, W = A ∪ C. Since (hF, Aie ∩hG, Bi)e ∪(hF, Aie ∩hH, Ci) = hK, Vie ∪hL, Wi, assume that hK, Vie ∪hL, Wi = hO, Ni, where N = V ∪ W. By definition, O(e1 ) = K(e1 ) ∪ L(e1 ) = (F(e1 ) ∩ G(e1 )) ∪ (F(e1 ) ∩ H(e1 )) = ρ(X1 ∩ X4 ) ∪ ρ(X1 ∩ X7 ) = ρ(∅) ∪ ρ({X3 }) = ρ({x3 }) = (∅, {x1 , x3 }); O(e2 ) = K(e2 ) ∪ L(e2 ) = (F(e2 ) ∩ G(e2 )) ∪ F(e2 ) = (ρ(X2 ) ∩ ρ(X5 )) ∪ ρ(X2 ) = ρ((X2 ∩ X5 ) ∪ X2 ) = ρ(X2 ) = ρ({x1 , x6 }) = ({x6 }, {x1 , x3 , x6 }); O(e3 ) = K(e3 ) ∪ L(e3 ) = F(e3 ) ∪ (F(e3 ) ∩ H(e3 )) = ρ(X3 ) ∪ ρ(X3 ∩ X8 ) = ρ(X3 ∪ (X3 ∩ X8 )) = ρ(X3 ) = ρ({x2 , x4 , x5 }) = ({x2 , x4 , x5 }, {x2 , x4 , x5 }); O(e4 ) = K(e4 ) ∪ L(e4 ) = G(e4 ) ∪ H(e4 ) = ρ(X6 ) ∪ ρ(X9 ) = ρ(X6 ∪ X9 ) = ρ({x1 , x2 , x3 , x6 }) = ({x1 , x3 , x6 }, {x1 , x2 , x3 , x4 , x5 , x6 }). Since J(e2 ) , O(e2 ) and J(e3 ) , O(e3 ). That is, J and O are not the same operators. Thus, hF, Aie ∩(hG, Bie ∪hH, Ci) , (hF, Aie ∩hG, Bi)e ∪(hF, Aie ∩hH, Ci). Likewise, we may show that hF, Aie ∪(hG, Bie ∩hH, Ci) = (hF, Aie ∪hG, Bi)e ∩(hF, Aie ∪hH, Ci) is incorrect.

4

Conclusion

In this paper, the notion of the soft rough set theory is proposed. soft rough set theory is a combination of a rough set theory and a soft set theory. The complement, restricted complement, union, restricted union, intersection, restricted intersection, ”and” and ”or” operations are defined on the soft rough sets. The basic properties of the soft rough sets are also presented and discussed. This new extension not only provides a significant addition to existing theories for handling uncertainties, but also leads to potential areas of further field research and pertinent applications. Our work in this paper is completely theoretical. As far as future directions are concerned, these will include the parameterization reduction of the soft rough sets. It is also desirable to further explore the applications of using the soft rough set approach to solve real world problems such as decision making, forecasting, and data analysis.

References [1] H. Aktas, N. Cagman, Soft sets and soft groups, Information Sciences 177(13)(2007)2726-2735. [2] D. Chen, E.C.C. Tsang, D.S. Yeung, X. Wang, The parameterization reduction of soft sets and its applications, Computers & Mathematics with Applications 49(5-6)(2005)757-763. [3] B. Davvaz, Roughness in rings, Information Sciences 164(2004)147-163. [4] F. Feng, Y.B. Jun, X. Zhao, Soft semirings, Computers & Mathematics with Applications 56(10)(2008)2621-2628. [5] W. L. Gau, D. J. Buehrer, Vague sets, IEEE Transactions on Systems, Man and Cybernetics 23(2)(1993)610-614. [6] Y.C. Jiang,Y. Tang, Q.M. Chen, H. Liu, J.c. Tang, Interval-valued intuitionistic fuzzy soft sets and their properties, Computers & Mathematics with Applications 60(3)(2010)906-918. [7] D. Molodtsov, Soft set theory-first results, Computers & Mathematics with Applications 37(4-5)(1999)19-31.

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[8] P. Majumdar, S.K. Samanta, Generalised fuzzy soft sets, Computers & Mathematics with Applications 59(4)(2010)1425-1432. [9] P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Computers & Mathematics with Applications 45(4-5)(2003)555562. [10] P.K. Maji, R. Biswas, A.R. Roy, Fuzzy soft sets, Journal of Fuzzy Mathematics 9(3)(2001)589-602. [11] P.K. Maji, R. Biswas, A.R.Roy, Fuzzy soft sets, Journal of Fuzzy Mathematics 9(3)(2001)589-602. [12] Z. Pawlak, Rough sets, Int. J. Inf. Comp. Sci. 11(1982)341-356. [13] S.R.S. Varadhan, Probability Theory, American Mathematical Society, 2001. [14] W. Xu, J. Ma, S. Wang, G. Hao, Vague soft sets and their properties, Computers & Mathematics with Applications 59(2)(2010)787-794. [15] X.B. Yang, T.Y. Lin, J.Y. Yang, Y. Li, D. Yu, Combination of interval-valued fuzzy set and soft set, Computers & Mathematics with Applications 58(3)(2009)521-527. [16] L.A. Zadeh, Fuzzy sets, Inform. Cont. 8(1965)338-353.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1300-1309, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Rate of convergence of some multivariate neural network operators to the unit, revisited George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract This paper deals with the determination of the rate of convergence to the unit of some multivariate neural network operators, namely the the normalized ”bell” and ”squashing” type operators. This is given through the multidimensional modulus of continuity of the involved multivariate function or its partial derivatives of speci…c order that appear in the righthand side of the associated multivariate Jackson type inequalitiy.

2010 AMS Mathematics Subject Classi…cation: 41A17, 41A25, 41A30, 41A36. Keywords and Phrases: Neural Networks, Positive operators.

1

Introduction

The multivariate Cardaliaguet-Euvrard operators were …rst introduced and studied thoroughly in [3], where the authors among many other interesting things proved that these multivariate operators converge uniformly on compacta, to the unit over continuous and bounded multivariate functions. Our multivariate normalized ”bell” and ”squashing” type operators (1) and (16) were motivated and inspired by the ”bell” and ”squashing” functions of [3]. The work in [3] is qualitative where the used multivariate bell-shaped function is general. However, though our work is greatly motivated by [3], it is quantitative and the used multivariate ”bell-shaped”and ”squashing”functions are of compact support. This paper is the continuation and simpli…cation of [1] and [2], in the multidimensional case.We produce a set of multivariate inequalities giving close upper bounds to the errors in approximating the unit operator by the above 1

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ANASTASSIOU: MULTIVARIATE NEURAL NETWORKS

multidimensional neural network induced operators. All appearing constants there are well determined. These are mainly pointwise estimates involving the …rst multivariate modulus of continuity of the engaged multivariate continuous function or its partial derivatives of some …xed order.

2

Convergence with rates of multivariate neural network operators

We need the following (see [3]) de…nitions. De…nition 1 A function b : R ! R is said to be bell-shaped if b belongs to L1 and its integral is nonzero, if it is nondecreasing on ( 1; a) and nonincreasing on [a; +1), where a belongs to R. In particular b (x) is a nonnegative number and at a, b takes a global maximum; it is the center of the bell-shaped function. A bell-shaped function is said to be centered if its center is zero. De…nition 2 (see [3]) A function b : Rd ! R (d 1) is said to be a ddimensional bell-shaped function if it is integrable and its integral is not zero, and for all i = 1; :::; d; t ! b (x1 ; :::; t; :::; xd ) is a centered bell-shaped function, where ! x := (x ; :::; x ) 2 Rd arbitrary. 1

d

Example 3 (from [3]) Let b be a centered bell-shaped function over R, then (x1 ; :::; xd ) ! b (x1 ) :::b (xd ) is a d-dimensional bell-shaped function. Qd Assumption 4 Here b (! x ) is of compact support B := i=1 [ Ti ; Ti ], Ti > 0 and it may have jump discontinuities there. Let f : Rd ! R be a continuous and bounded function or a uniformly continuous function. In this paper, we study the pointwise convergence with rates over Rd , to the unit operator, of the ”normalized bell” multivariate neural network operators Mn (f ) (! x ) := Pn2

Pn2 kd k1 k1 = n2 ::: kd = n2 f n ; :::; n Pn2 Pn2 1 k1 = n2 ::: k d = n2 b n

b n1 x1

x1 k1 n

k1 n

; :::; n1

; :::; n1 xd

xd kd n

kd n

;

(1) x := (x1 ; :::; xd ) 2 Rd , n 2 N. Clearly Mn is a positive where 0 < < 1 and ! linear operator. The terms in the ratio of multiple sums (1) can be nonzero i¤ simultaneously n1

xi

ki n

Ti , all i = 1; :::; d; 2

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ANASTASSIOU: MULTIVARIATE NEURAL NETWORKS

ki n

i.e., xi

Ti n1

, all i = 1; :::; d, i¤

nxi

Ti n

ki

nxi + Ti n , all i = 1; :::; d:

(2)

To have the order n2 we need n

nxi

Ti n

ki

n2 ;

nxi + Ti n

(3)

Ti + jxi j, all i = 1; :::; d. So (3) is true when we take n

max

i2f1;:::;dg

(Ti + jxi j) :

(4)

When ! x 2 B in order to have (3) it is enough to assume that n T := maxfT1 ; :::; Td g > 0. Consider Iei := [nxi

2T , where

Ti n ; nxi + Ti n ] , i = 1; :::; d; n 2 N.

The length of Iei is 2Ti n . By Proposition 1 of [1], we get that the cardinality of ki 2 Z that belong to Iei := card (ki ) max (2Ti n 1; 0), any i 2 f1; :::; dg: 1

In order to have card (ki ) 1; we need 2Ti n 1 1 i¤ n Ti , any i 2 f1; :::; dg: Therefore, a su¢ cient condition in order to obtain the order (3) along with the interval Iei to contain at least one integer for all i = 1; :::; d is that n o 1 n max Ti + jxi j ; Ti : (5) i2f1;:::;dg

Clearly as n ! +1 we get that card (ki ) ! +1, all i = 1; :::; d. Also notice that card (ki ) equals to the cardinality of integers in [dnxi Ti n e ; [nxi + Ti n ]] for all i = 1; :::; d: Here, [ ] denotes the integral part of the number while. d e denotes its ceiling. From now on, in this article we will assume (5). Furthermore it holds (Mn (f )) (! x) =

P[nx1 +T1 n

] k1 =dnx1 T1 n e

b n1

x1

k1 n

:::

P[nxd +Td n

] kd =dnxd Td n e

f

V (! x)

; :::; n1

xd

kd k1 n ; :::; n

(6)

kd n

all ! x := (x1 ; :::; xd ) 2 Rd ; where V (! x ) := [nx1 +T1 n ]

X

k1 =dnx1 T1 n e

[nxd +Td n ]

:::

X

b n1

x1

kd =dnxd Td n e

3

1302

k1 n

; :::; n1

xd

kd n

:

ANASTASSIOU: MULTIVARIATE NEURAL NETWORKS

Denote by k k1 the maximum norm on Rd , d Ti , all i = 1; :::; d, we get that ! k n

! x

1

T n1

1. So if n1

xi

ki n

;

! where k := (k1 ; :::; kd ) : De…nition 5 Let f : Rd ! R. We call ! 1 (f; h) :=

sup

all ! x ;! y: x ! yk h k!

jf (! x)

f (! y )j ;

(7)

T n1

(8)

1

where h > 0, the …rst modulus of continuity of f: Here is our …rst main result. Theorem 6 Let ! x 2 Rd ; then j(Mn (f )) (! x)

f (! x )j

! 1 f;

:

Inequality (8) is attained by constant functions. Inequality (8) gives Mn (f ) (! x ) ! f (! x ), pointwise with rates, as n ! +1, ! d where x 2 R , d 1: Proof. Next, we estimate j(Mn (f )) (! x) [nx1 +T1 n ]

[nxd +Td n ]

X

:::

k1 =dnx1 T1 n e

b n1

h ! i n! x +T n ! l ! ! m k= nx Tn

X

! l ! ! m k= nx Tn

f

k1 kd ; :::; n n

f

kd =dnxd Td n e

; :::; n1 V (! x)

P

h ! i n! x +T n

X

k1 n

x1

(6) f (! x )j =

! k n

f

xd

kd n

f (! x ) b n1

f (! x) = ! x

! k n

V (! x) ! k n

f (! x) b n1 ! V (x)

4

1303

! x

! !! k n

ANASTASSIOU: MULTIVARIATE NEURAL NETWORKS

h ! i n! x +T n

X

! l ! ! k= nx Tn

That is

! k ! 1 f; ! x n V (! x) m

j(Mn (f )) (! x) [nx1 +T1 n ]

X

1

b n

! 1 f; nT1 V (! x)

f (! x )j

[nxd +Td n ]

:::

k1 =dnx1 T1 n e

X

b n1

k1 n

x1

kd =dnxd Td n e

= ! 1 f;

! !! k : n

! x

1

T n1

; :::; n1

xd

kd n

;

(9)

proving the claim. Our second main result follows. Theorem 7 Let ! x 2 Rd , f 2 C N Rd , N 2 N, such that all of its partial derivatives f e of order N , e : je j = N , are uniformly continuous or continuous are bounded. Then, j(Mn (f )) (! x ) f (! x )j (10) 8 0 19 ! j N d j 0, which holds for all commonly models. A rule finds a µ > 0 by minimizing Φγ (µ) =

(F (µ) + µβ0 + β1 )1+γ , µ

(3.4)

for proper γ > 0, i.e. γ = αα10 . The rule Φ(µ) follows from the equation (3.3) and the derivation method of Φ(µ) is similar to the rule in [7]. If µ∗ > 0 is a local minimizer of Φ(µ), then µ∗ = λ∗ /τ ∗ holds for all minimizers xµ of (1.5), when F is differentiable at µ. Next we use a-Tikh functional based on the iterative decompose method as described in the previous section. Respectively, yµδ , λ∗ , τ ∗ , were expressed as follows, ⎧ δ 2 2  y {∥Bk y − ∥b∥e1 ∥ + µ∥Rk y∥ }, ⎨ yµ = arg min λ∗ = ∥Rk yαδ ∥02 +β0 , (3.5) µ  α1 ⎩ τ∗ = . ∥Bk y δ −∥b∥e1 ∥2 +β1 µ

Equation (3.5) and the numerical experiments in [4] indicate that the quantity δ 2 = τ ∗−1 = (∥Bk y (k) − ∥b∥e1 ∥2 + β1 )α1−1 , which estimates the accurate noise level δ02 . However, for α0 ∼ δ0−d with 0 < d < 2, that is to say α0 ∼ τ d , 0 < d < 1, α0 is positive and it would been required by the convergence. In this where α0 is replaced by α0 τ d , we rewrite the estimate of λ∗ : λ∗ =

α0 τ ∗d . ∥Rp yµδ ∥2 + β0

(3.6)

which help the algorithm as follows faster convergence to the optimal solution. Now, we consider the following alternating iterative algorithm, through combining the equation (3.5) with Tikhonov’s quasi-optimality principle to solve the the projected problem (2.4). The algorithm constructs a finite parameter sequence of {µi }, which convergence to the minimizer of criterion Φγ . Algorithm 1. Alternating iterative algorithm

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Generalized Tikhonov regularization method

DI ZHANG, TING-ZHU HUANG

1. Choose µ0 , kmax , the parameter pairs (α0 , α1 ) and (β0 , β1 ). 2. Apply k LBD steps to A with starting vector b and form the matrix Bp . 3. Apply QR decomposition to LVk and form the matrix Rp . 4. For i = 0, 1, · · · , Imax . 5. Solve for yi+1 by the Tikhonov regularization method yi+1 ∈ arg min{∥Bk y − ∥b∥e1 ∥2 + µk ∥Rk y∥2 }. y

Set xi+1 = Vk yi+1 . 6. Update the parameter λi+1 and τi+1 by τi+1 =

α1 , ∥Bp yi+1 − ∥b∥e1 ∥2 + β1

λi+1 =

d α0 τi+1 , ∥Rp yi+1 ∥2 + β0

−1 set µi+1 = λi+1 τi+1 ,

7. Check the stopping criterion, until i = arg min ∥xi+1 − xi ∥, i

do µ∗ = µi . 8. Compute the regularized solution xkµ∗ . For large-scale problems, we use a projection method to change it into a smallor medium-scale problem. We would point out that we do not specify the solver for the regularization problem in step 5 deliberately. Therefore, the linear system may be solved directly, or solved by other methods, i.e. the conjugate gradient method. Our numerical experiments indicate that an accurate approximate solution suffices.

3.2

Stopping criteria analysis

The stopping rules are easy to find. We could choose the criteria base on the changes or convergence of either the regularization parameter µ or the solution x. We can stop the iteration when |µi − µ0 | < ϵ1 |µ0 |, where ϵ1 is a small tolerance parameter. We note that because the µ0 is often random. A disadvantage of the stopping criterion is that the approximate solution have the greater error relative to the true solution. To circumvent this trouble, we use another stopping criterion. The following lemma which provides a surprising and important observation on the −1 monotonicity of the sequence µi+1 = λi+1 τi+1 which are generated by alternating iterative algorithm. The monotonicity is the key about the demonstration of the convergence of the total algorithm.

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Generalized Tikhonov regularization method

DI ZHANG, TING-ZHU HUANG

Lemma 3.1. For any initial guess µ0 , the sequence xi is generated by the iterative algorithm and converges to a critical point x∗ of (3.1). Moreover, the sequence µk is monotonically convergent, showing that there exists some µ∗ , such that lim µi = µ∗ ,

(3.7)

lim xδi = x∗ .

(3.8)

i→∞

from which it follows that

i→∞

Proof. See [15 Lemma 4.1]. For each µi the corresponding regularized solution is now denoted by yiδ , then xδi = Vk yiδ . For a parameter choice algorithm, we have to choose a certain i as the stopping criteria. This is done by the quasi-optimality principle, the discredited version we use in this paper could be found in [6]: Definition 3.1 (quasi-optimality). For xδi and µi as in (3.5) the regularization parameter µi∗ defined by the quasi-optimality principle is obtained as i∗ = arg min ∥xδi − xδi+1 ∥. i≥0

(3.9)

Notice that because the sequence xi is convergent, then ∥xi − xi+1 ∥ is monotone decreasing, especially in section 4.1 some examples illustrate the convergence property of ∥xi − xi+1 ∥ and show that the sequence µi increase very quickly. So the solution xi∗ approximate equals the solution x∗ . The solution xi∗ is the iterative optimal solutions for any given a max iterations. In other words, it is stopped if the relative change of the iterates solution (x) at the low point. However this method needs calculate the all iteration solutions, so in order to reduce iterative time, we could set a critical value as the minimum jif maximum iterations is very large, or we set a small maximum iterations.

4

Numerical results

In this section, we illustrate the efficiency of the Algorithms 1 when applied parameter selection method to typical large-scale linear ill-posed problems. For this purpose the numerical results can be divided into two parts, Section 4.1 we choose three benchmark linear inverse problems, e.g. baart, shaw, gravity, which are considered as the test problems, from Hansen’s package Regularization Tools [14]. Section 4.2 we consider the restoration problem of a grayscale image as the test problem. In each case we generate triples A, x b, bb, so that Ab x = bb. The size of A is taken to be 256 × 256 and then simulated distinct noisy vector b, b = bb + e, where e was generated by the Matlab randn function with the seed value set to zero. The vector e is scaled to yield a specified noise level ξ = ∥e∥/∥bb∥. The noise level ξ, i.e., ξ = 5 × 10−3 , is considered in section 4.1. In algorithm 1, the initial guess µ0

1324

Generalized Tikhonov regularization method

DI ZHANG, TING-ZHU HUANG is taken to be 1 × 10−6 , and we get access to the i∗ when ∥xδi − xδi+1 ∥ falls below 10−4 ∥xδi+1 ∥. The parameter d is set to d = 14 . The choice of the parameter pairs (α0 , α1 ), (β0 , β1 ) are based on the value of k. In the numerical examples, for the regularization of small dimension inverse problem, we choice α0 = k2 and α1 = k2 . The tridiagonal regularization operator L is a scaled approximation of the second derivative operator. The relative error (ReErr) is used to measure the quality of the regularized solutions of different algorithms. It is defined as follows: ReErr =

∥x − x b∥ . ∥b x∥

The accuracy of the solution xδµ is measured by ReERR. In follows, δ and δat stand for the norms of true noise level and estimated noise level by Algorithm 1.

4.1

Test problems from Hansen’s package

Comparisons are made for the regularized solutions of the Algorithm 1 chosen by different parameter selection method. In this example, numerical results are given to compare the quasi-optimal (q-o) method, L-curve (L-c) method against the optimal (opt) choice of the regularization parameter on several test problems. To illustrate the performance of algorithm on the above test problems, we run 10 realizations and then compute average values of regularization parameters, average relative errors. The optimal regularized solution produces the minimum relative error, the parameter values are are summarized in parentheses, and comparison of ReErr for three parameter selection methods on the projection problem in Table 1. First we observe that the estimated residual noise δat agree very well with the exact Table 1: Numerical results for three problems from Hansen’s MATLAB package.

baart shaw gravity

(δ) δa𝑡 (1.45e-2) 1.56e-2 (1.86e-1) 1.84e-1 (3.74e-1) 3.71e-1

(µa𝑡 ) ReErr (6.98e-3) 1.65e-1 (6.59e-4) 1.47e-1 (3.10e-3) 1.49e-1

(L-c) ReErr (2.64) 4.49e-1 (4.28e-6) 3.37e-1 (3.36e-3) 1.50e-1

(q-o) ReErr (6.16e-5) 1.79e-1 (3.36e-3) 1.68e-1 (1.00e-10) 7.1534

(opt) ReErr (1.13e-6) 1.05e-1 (2.34e-4) 1.49e-1 (3.36e-3) 1.50e-1

one δ. Second observation is that the balancing principle gives an error fairly close to the optimal one. This illustrates clearly the benefit of using iterative method for large-scale inverse problem. The results of the comparison for three problems are displayed in Fig.1- Fig.3, where the figures display the reconstructed solutions and exact solution. In each of figures the third line show the sequence {µi } is monotonic

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Generalized Tikhonov regularization method

DI ZHANG, TING-ZHU HUANG

convergence history of Tikhonov paramer

−2

convergence history of norm(xk+1−xk)

1

10

10

−3

10

0

10

−4

10

−1

10

−5

10

−2

10 −6

10

−3

10 −7

10

−4

−8

10

−9

10

10

−5

10

−10

10

−6

1

2

3

4

5

6

7

8

9

10

11

10

1

2

3

4

5

6

7

8

9

10

11

Figure 1: General Tikhonov for Baart problem. The first four graphs show the approximate solution with three parameter selected methods and the true solution(solid line). Bottom: the convergence analysis of the parameter and the norm of difference of neighbouring approximate solution.

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Generalized Tikhonov regularization method

DI ZHANG, TING-ZHU HUANG

convergence history of Tikhonov paramer

−3

convergence history of norm(xk+1−xk)

1

10

10

−4

10

0

10

−5

10

−1

10 −6

10

−2

10 −7

10

−3

10

−8

10

−4

10

−9

10

−10

10

−5

1

1.5

2

2.5

3

3.5

4

4.5

5

10

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 2: General Tikhonov for Shaw problem. The first four graphs show the approximate solution (red dashed line) with three parameter selected methods and the true solution (solid line). Bottom: the convergence analysis of the parameter and the norm of difference of adjacent to approximate solution.

convergence history of Tikhonov paramer

−2

convergence history of norm(xk+1−xk)

1

10

10

−3

10

0

10

−4

10

−1

10

−5

10

−2

10 −6

10

−3

10 −7

10

−4

−8

10

−9

10

10

−5

10

−10

10

−6

1

1.5

2

2.5

3

3.5

4

4.5

5

10

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 3: General Tikhonov for Gravity problem. The first four graphs show the approximate solution (red dashed line) with three parameter selected methods and the true figurename solution (solid line). Bottom: the convergence analysis of the parameter and the norm of difference of adjacent to approximate solution.

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Generalized Tikhonov regularization method

DI ZHANG, TING-ZHU HUANG

increasing and the relative change of the regularization solutions which are solved by the Algorithm 1 is monotone decreasing. Other quantities are shown in the third line, the sequences {µi } are convergent and the convergent rates are very quickly, such as in Fig.1, at about k = 3 the µi begin to flat. The stopping criterion for Algorithm 1 may be based on this quantities, however we choose some combination of the quantities as the stopping criteria. Combined the Fig.1 and Fig.2 with Table 1, the ReErr of the computed approximate solutions with L-curve parameter choice method, are larger than the other two methods. In Fig.3 the computed approximate solutions with quasi-optimal method is deviating from the optimal solution, therefore the ReErr is the largest of three princples. So we summarize that in three problems the solutions for our method is more close to the true solution.

4.2

Example for grayscale

To test our algorithm on a large-scale problem we consider a denoising problem of a greyscale image cameraman that is represented by an array of 256 × 256 pixels. The pixels are stored columnwise in a vector in R65536 . A block Toeplitz with Toephlitz blocks blurring matrix A ∈ R65536×65536 is determined with Gaussian point spread function and the width sigma= 4.0. Three different relative noise values are generated with ξ = 5 × 10−3 , 5 × 10−3 , 5 × 10−3 . As we can see from the figures, the computed solutions yield images that resemble the true image relatively well. The stopping criterion is important which determined the time cost. The conclusion in this case is that the alternating iterations i of the Algorithm 1 is very small. By comparing with L-curve, quasi-optimal criterion when they achieved the optimal solutions when the ReErr are identical, respectively the iterations are i = 3, i = 2, i = 2 for different perturbation levels. However the quasi-optimal and L-curve have to calculate all approximate solutions, and then choice the best one.

1328

Generalized Tikhonov regularization method

DI ZHANG, TING-ZHU HUANG

ture image

blurring and noisy image

l−curve

quasi−optimal

optimal

a−Tikh image restoration

Figure 4: General Tikhonov for greyscale image. Image restoration with relative noise level 5 × 10−2 .

ture image

blurring and noisy image

l−curve

quasi−optimal

optimal

a−Tikh image restoration

Figure 5: General Tikhonov for greyscale image. Image restoration with relative noise level 5 × 10−3 .

1329

Generalized Tikhonov regularization method

DI ZHANG, TING-ZHU HUANG

ture image

blurring and noisy image

l−curve

quasi−optimal

optimal

a−Tikh image restoration

Figure 6: General Tikhonov for greyscale image. Image restoration with relative noise level 5 × 10−4 .

5

Conclusion

In this work we have presented a method for solving the general Tikhonov regularization on large-scale ill-posed problems. We have shown that determining regularizing parameters based on the k-dimensional subspace, our selection method is convenient. The examples indicate that the combination of a-Tikh parameter choice method and the iterative projection method is perfected. And our computing method involves less computational expense for solving large-scale ill-posed problems.

References [1] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998. [2] H. Zha, Computing the genralized sigular values/vectors of large sparse or structured matrix pairs, Numer. Math., 72 (1996), pp. 391-417. [3] J. Lampe and L. Reichel, Large-scale Tikhovov regularization via reduction by orthogonal projection, Lin. Alg. Appl., 436 (2012), pp. 2845-2865. [4] B.Jin and J.Zou, Augmented Tikhonov regularization, Inverse Problems, 25 (2009), 025001.

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DI ZHANG, TING-ZHU HUANG

[5] D. Calvetti, P. C. Hansen, and L. Reichel, 𝐿-curve curvature bounds via Lanczos bidiagonalization, Electron. Trans. Numer. Anal., 14 (2002), pp. 134-149. [6] F. Bauer and S. kindermann, The quasi-optimality criterion for classical inverse problem, Inverse Problems, 24 (2008),035002. [7] K. Ito and B. Jin, A regularization parameter for nonsmooth Tikhonov regularization, SIAM J. Sci. Comput., Vol. 33, No. 3 (2011), pp. 1415-1438. [8] F. S. V. Baz´ 𝑎n and L. S. Borges, GKB-FP: an algorithm for large-scale discrete ill-posed problems, BIT, 50 (2010), pp. 481-507. [9] M. E. Kilmer and D. P. OLeary, Choosing regularization parameters in iterative methods for ill-posed problems, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 12041221. [10] F. Bauer and M. Reiß, Regularization independent of the noise level: an analysis of quasi-optimality, Inverse Problems, 24 (2008), 055009. [11] L. Wu, A parameter choice method for Tikhonov regularization, Electron. Trans. Numer. Anal., 16 (2003), pp. 107-128. [12] H. D. Simon and H. Zha, Low-rank matrix approximation using the lanczos bidiagonalization process with applications, SIAM J. Sci. Comput., Vol. 21, No. 6 (2000), pp. 2257-2274. [13] J. Wang and N. Zabaras, Hierarchical Bayesian models for inverse problems in heat conduction, Inverse Problems, 21 (2005), pp.183-206. [14] P. C. Hansen, Regularization tools verson 4.1 for Matlab 7.3, Numer. Algorithms, 46 (2007), pp.189-194. [15] K. Ito, B. Jin, and J. Zou, A new choice rule for regularization parameters in Tikhonov regularization, Applicable Analysis, 90 (2011), pp. 1521-1544. [16] D. Calvetti, G. H. Golub, and L. Reichel, Estimation of the L-curve via Lanczos bidiagonalization, BIT, 39 (1999), pp. 603-619. [17] J. Chung, J. G. Nagy, and D. P. OLeary, A weighted-GCV method for Lanczos-hybrid regularization, Electron. Trans. Numer. Anal., 28 (2008), pp. 149-167.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.7, 1332-1343, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Local and global well-posedness of stochastic Zakharov-Kuznetsov equation

3

Hossam A. Ghany1,2 and Abd-Allah Hyder3 1 Department of Mathematics, Helwan University, Cairo, Egypt. 2 Department of Mathematics, Taif University, Hawea,Taif, KSA. Department of Mathematical Engineering, Azhar University, Cairo, Egypt. [email protected]

Abstract. We consider the Cauchy problem for stochastic Zakharov-Kuznetsov equation forced by a random term of additive white noise type. We obtain a local existence and uniqueness result for the solution of this problem. Our proposed technique is based on employing Banach contraction principle method, fixed point theory, Fourier analysis and some basic inequalities. We also get global existence of solution in the function space Zs (T ) . Detailed computations and implemented examples are explicitly provided. Keywords: Stochastic; Well-Posedness; Zakharov-Kuznetsov.

1

Introduction

This paper is devoted to establish local and global well-posedness to stochastic Zakharov-Kuznetsov equation (SZK) forced by a random term of additive white noise type i.e., ( ∂3u ∂3u du + (u ∂u (x, y, t) ∈ R2 × R+ ∂x + ∂x3 + ∂x∂y 2 )dt = ΦdW, (1.1) u(x, y, 0) = u0 (x, y) for all (x, y) ∈ R2 . Where u is a stochastic process on R2 × R+ , W (t) is a cylindrical Wiener process on L2 (R2 ) and Φ is a linear bounded operator not depend on u i.e., the noise ΦdW is additive. The notion of well-posedness will be the usual one in the context of nonlinear dispersive equations, that is, it includes existence, uniqueness, persistence property, and continuous dependence upon the data. Equation (1.1) can be considered as a 2-dimensional generalization of the stochastic KdV equation and arises when modelling the propagation of weakly nonlinear ion-acoustic waves in noisy plasma[1,2,3]. Recently, many researchers pay more attention to study of random waves, which are important subjects of stochastic partial differential equation (SPDE). Wadati [4] first answered the interesting question, How does external noise affect the motion of solitons? and studied the diffusion of soliton of the KdV equation under Gaussian noise, which satisfies a diffusion

1332

GHANY, HYDER: Zakharov-Kuznetsov equation

equation in transformed coordinates. Wadati and Akutsu also studied the behaviors of solitons under the Gaussian white noise of the stochastic KdV equations with and without damping [5]. In addition, a nonlinear partial differential equation which describes wave propagations in random media was presented by Wadati [4]. Debussche and Printems [6,7], de Bouard and Debussche [8,9], Konotop and Vazquez [10], Printems [11], Ghany [12] and others also researched stochastic KdV-type equations. By local well-posedness (LWP) of a stochastic PDE we mean pathwise LWP almost surely. That is, for almost every fixed ω ∈ Ω , the corresponding PDE is LWP. Similarly, global well-posedness (GWP) of a stochastic PDE will be defined as pathwise GWP almost surely. Linares and Pastor [13] studied the initial value problems (IVPs) associated with both the ZK and modified ZK equations. They improved the results in [14,15] by showing that both IVPs are locally well-posed for initial data in H s (R2 ) , s > 0.75 . Moreover, by using the techniques introduced in Birnir at al. [16,17], they proved that the IVP associated with the modified ZK equation is ill-posed, in the sense that the flow-map data-solution is not uniformly continuous, for data in H s (R2 ) , s 6 0 . It should be noted that the method employed in [13,14] to show local well-posedness, was the one developed by Kenig, Ponce, and Vega [18] (when dealing with the generalized KdV equation), which combines smoothing effects, Strichartz-type estimates, and a maximal function estimate together with the Banach contraction principle. This paper is organized as follows: In Section 2, we introduce some notations and some function spaces along with their embeddings and state deterministic linear estimates from [19,20]. In Section 3, we state two Theorems, as main result of our paper, that guarantees and establishes local and global well-posedness for stochastic Zakharov-Kuznetsov equation forced by a random term of additive white noise type. In Section 4, we prove our main results by establishing the type nonlinear estimate on the second iteration for the integral formulation of the mild solution of equation (1.1).

2

Notations and Preliminaries 0

Suppose that S(Rd ) and S (Rd ) denote the Schwartz space and its completion with respect to the family of seminorms kf kk,α := sup {(1 + kxkkRd )|∂ α f (x)|},

α ∈ Nd0 ,

x∈Rd

k ∈ N0 .

For a Banach space X and s ∈ R we denote by H s (Rd ; X) the space of all functions f ∈ 0 S (Rd ; X) such that kf kH s (Rd ;X) :=

Z

Rd

(1 +

kζk2Rd )s/2 kfˆ(ζ)k2X dζ

1/2

0 ; {W (t)}t>0 ) , where (Ω, F, P) a probability space, {Ft }t>0 a filtration on Ω and {W (t)}t>0 a cylindrical Wiener process adapted to {Ft }t>0 . The mild solution of equation (1.1) is given in the form u(t) = U (t)u0 +

Z

t 0

U (t − s)uux ds +

1333

Z

t 0

U (t − s)ΦdW (s)

(2.1)

GHANY, HYDER: Zakharov-Kuznetsov equation

where {U (t)}t>0 is the unitary group of operators generated by the deterministic ZakharovKuznetsov equation, more precisely the solution of the linear equation vt + vxxx + vxyy = 0,

(x, y, t) ∈ R2 × R+

(2.2)

with v(x, y, 0) = v0 (x, y) for all (x, y) ∈ R2 is given by v(x, y, t) = U\ (t)v0 (ζ, η) = eitΦ vˆ0 (ζ, η)

(2.3)

where the phase function Φ is given by Φ(ζ, η) = ζ(ζ 2 + η) . The solution of the linear equation duL + (

∂ 3 uL ∂ 3 uL + )dt = ΦdW ∂x3 ∂x∂y 2

(2.4)

with uL (x, y, 0) = 0 for all (x, y) ∈ R2 is given by uL =

Z

0

t

U (t − s)ΦdW (s)

(2.5)

0 2 d s d Suppose that L0,s 2 := L2 (L (R ); H (R )) denote the space of Hilbert-Schmidt operators from L2 (Rd ) into H s (Rd ) . Its norm is given by X kΦei k2H s (Rd ) kΦkL0,s := 2

i>1

where {ei }i>1 is any orthonormal basis of L2 (Rd ) . For simplicity we will use the following shorter notations: Lp ([0, T ]; Lq (Rd )) := Lpt (Lqx ) and Lq (Rd ; Lp ([0, T ])) := Lqx (Lpt ) . For a fixed ω ∈ Ω we define s s 2 ∞ 2 2 4 ∞ )) ∩ L2ω (L2x,y (L∞ Zs (T ) = {u ∈ L2ω (Ct (Hx,y t )), D ∂x u ∈ Lω (Lx,y (Lt )), ∂x u ∈ Lω (Lt (Lx,y ))} (2.6)

where the Riesz’s operator Ds [21] is defined by s u(ζ, η) = (ζ 2 + η 2 )s u d ˆ(ζ, η), D

3

s∈R

(2.7)

Main Results

In this section we give the precise statement of our results, more precisely, we give two theorems below. Theorem 1 gives the sufficient conditions for obtaining local will posedness of equation (1.1). Theorem 2 concerning the linearized stochastic Zakharov-Kuznetsov equation (2.4). As usual in the context of nonlinear estimation, Theorem 2 is essential for proving Theorem 1. Eventually, one can find that the results of Theorem 1 are true for arbitrary large T , this gives the global well-posedness of equation (1.1). Theorem 1.

1334

GHANY, HYDER: Zakharov-Kuznetsov equation

1 ) ∩ L4 (L2 ) is F − measurable and Φ ∈ L0,1 , then there exAssume that u0 ∈ L2ω (Hx,y 0 ω x,y 2 ists a unique solution of equation (1.1) in Zs (T0 ) almost surely for any T0 and any s with 0.75 < s < 1.

By virtue the arguments of fixed point theory and the following theorem we can easily prove the above theorem. Theorem 2. s Assume that Φ ∈ L0,ˉ for some sˉ > 0.75 then uL is almost surely in Zs (T ) for any T > 0 and 2 any s such that 0.75 < s < sˉ . Moreover there exists a constant C(s, sˉ, T ) such that E[kuL k2Zs (T ) ] 6 C(s, sˉ, T )kΦk2L0,ˉs

(3.1)

2

4

Computations and Proofs

The proof of Theorem 1 will require four key Propositions concerning the above mentioned spaces. In this section we present these Propositions. Proposition 3. s For any s 6 sˉ we have uL ∈ L2ω (L∞ t (Hx,y ) and 2 E[ sup kuL k2Hx,y s ] 6 C(T )kΦk 0,s L

(4.1)

2

06t6T

Proof. We use Itˆ o formula on the functional k.k2Hx,y [16] and deduce s kuL k2Hx,y s

=2

Z

t 0

(Js uL , Js ΦdW (s))L2x,y +

Z

0

t

Tr(Js2 ΦΦ∗ )ds

where the Bessel’s operator Js is defined by

and has the property [21]

2 2 s/2 d J u b(ζ, η) s u(ζ, η) = (1 + ζ + η )

(4.2)

kJs .kL2x,y = k.k2Hx,y s

(4.3)

Now, we write T r(Js2 ΦΦ∗ ) = kΦkL0,s and hence applying a martingale inequality[20] 2

sup t

Z

t 0

(Js uL , Js ΦdW (s))L2x,y 6 3E[(

Z

0

t

0.5 kΦ∗ uL k2Hx,y ] s ds)

1 6 E[sup kuL k2Hx,y s ] + C(T )kΦk 0,s L2 4 t

implies the required result.

1335

(4.4)

GHANY, HYDER: Zakharov-Kuznetsov equation

The proof of the above proposition implies directly the following Corollary. s )) uL ∈ L2ω (Ct (Hx,y

(4.5)

The above Proposition and its corollary give a draws attention regularity property of the solution uL of the linear problem, that is, they decide that uL is a square integrable random variable with s s values in L∞ ˉ. t (Hx,y ) especially in Ct (Hx,y ) for any s 6 s Now, we will give a simple priori estimate of uL by giving the following result: Proposition 4. uL ∈ L2ω (L2x,y (L∞ t ) and E[

Z

R2

sup |uL |2 dxdy] 6 C(ˉ s, T )kΦk2L0,ˉs

(4.6)

2

06t6T

Proof. Let {ei }i>1 be an orthonormal basis for L2 (R2 ) and {hk }k>1 a partition of unity on R2+ such that: ζ a) hk (ζ, η) = h1 ( 2k−1 , 2kη−1 ), (ζ, η) ∈ R2+ , k > 1; b) supphk ⊆ [2k−1 , 2k+1 ]2 , k > 1; c) supph0 ⊆ [−1, 1]2 . We also consider e hk ∈ C ∞ (R2 ) with supphk ⊆ [2k−2 , 2k+2 ] such that e hk > 0 and e hk = 1 on supphk . For k ∈ N , we define the group {Uk (t)}t∈R by itφ \ b U\ k (t)f (ζ, η) = hk (|ζ|, |η|)U (t)f (ζ, η) = e hk (|ζ|, |η|)f (ζ, η)

(4.7)

and the operator Φk by

e d Φd (4.8) k ei (ζ, η) = hk (|ζ|, |η|)Φei (ζ, η) P implies U (t)Φ = k>1 Uk (t)Φk . Then, by using Minkowski’s integral

Since, Uk (t)Φ = Uk (t)Φk inequality we will get Z Z t X Z E[ sup | U (t − s)ΦdW (s)|2 dxdy]0.5 6 E[ R2

t

6 C(T, sˉ)

0

X k>1

2sk kΦk kL0,0 6 C(T, sˉ)( 2

X

k>1

22(s−ˉs)k )0.5 (

k>1

R2

sup |

X k>1

t

Z

0

t

Uk (t − s)Φk dW (s)|2 dxdy]0.5

22ˉsk kΦk k2L0,0 )0.5 6 C(T, s, sˉ)kΦkL0,ˉs 2

2

where, 0.75 < s < sˉ . Remark. From [16] We have used Z t Z sup | Uk (t − s)Φk dW (s)|2 dxdy] 6 C(T, sˉ)22sk kΦk k2L0,0 E[ R2

t

2

0

1336

GHANY, HYDER: Zakharov-Kuznetsov equation

and for 0.75 < s < sˉ we were used X X 22ˉsk kΦk k2L0,0 = 22ˉsk kΦk ei k2Lx,y 2

k

k,i

and X k

s)kΦuL k2Hx,y 22ˉsk kΦk uL k2Lx,y 6 C(ˉ s ˉ

Its well known that, the Riesz’s operator Ds is a powerful tool for checking the regularity of the solutions of nonlinear partial differential equations. Proposition 5 will clarify the success of the solution uL under this checking. Proposition 5. Suppose 0 < δ < inf{ˉ s, 2} , then Dsˉ−δ ∂x uL ∈ L2ω ((L2x,y (L2t )) and E[ sup

Z

T

x,y∈R 0

Proof. Let q =

4 δ

E|

|Dsˉ−δ ∂x uL |2 dt] 6 C(δ, T )kΦk2L0,ˉs

(4.9)

2

. By virtue of the stochastic integral properties[20]:

Z

t

D

U (t − τ )ΦdW (τ )| dt 6

1+ˉ s

0

2

Z tX 0 i>1

|D1+ˉs U (t − τ )Φei |2 dτ

So, we can easily find that: kD1+ˉs uL kqL∞

q 2 x,y (Lω (Lt ))

= sup E[( x,y∈R

Z

6 C sup

Z

T

|

0 T

E[|

x,y∈R 0

Z

6 C sup

T

(

x,y∈R 0

6C

Z

T

(

X

Z

t 0

Z

0

D1+ˉs U (t − τ )ΦdW (τ )|2 dt)q/2 ] t

D1+ˉs U (t − τ )ΦdW (τ )|2 ]q/2 dt

Z tX 0 i>1 Z t

sup

i>1 x,y∈R 0

0

|D1+ˉs U (t − τ )Φei |2 dτ )q/2 dt |D1+ˉs U (t − τ )Φei |2 dτ )q/2 dt

As pointed in [21, Lemma 2.1], we have Z

sup

x,y∈R 0

t

|D1+ˉs U (t − τ )Φei |2 dτ 6 CkDsˉΦei k2L2x,y 6 CkΦei k2Hx,y s ˉ

(4.10)

hence kD

uL kqL∞ (Lq (L2 )) ω x,y t

1+ˉ s

6C

Z

0

T

(

X i>1

q/2 s ˉ ) kΦei kHx,y dt 6 C(T )kΦkL0,ˉs

1337

2

(4.11)

GHANY, HYDER: Zakharov-Kuznetsov equation

Similarly we can derive kDsˉuL kqL2

6 CkΦk2L0,ˉs

q 2 x,y (Lω (Lt ))

(4.12)

2

Inequality (4.11) and [9, proposition A.1] implies δ

D1+ˉs− 2 uL ∈ Lqx,y (Lqω (L2t ))

(4.13)

and δ

δ

kD1+ˉs− 2 uL kLqω (Lqx,y (L2t )) = kD1+ˉs− 2 uL kLqx,y (Lqω (L2t )) 6 CkΦk2L0,ˉs

(4.14)

2

Also we have kuL kqLq (Lq (L2 )) ω x,y t

Z

Z

Z

T

t

E[( | U (t − τ )ΦdW (τ )|2 dt)q/2 ]dxdy 2 0 0 R Z Z T Z t 6C E(| U (t − τ )ΦdW (τ )|2 )q/2 dtdxdy 0 R2 0 Z Z T Z tX ( |U (t − τ )Φei |2 dτ )q/2 dtdxdy 6C =

R2

0

0 i>1

So, kuL kLqω (Lqx,y (L2t )) 6 CkΦkL0,ˉs

(4.15)

2

Hence, kuL kqLq (Lq (L2 )) ω x,y t

6C

Z

R2

(

Z

T

0

Applying Minkowski’s intgral inequality gives, XZ Z 2 kuL kLqω (Lqx,y (L2 )) 6 C ( ( t

i>1

R2

X i>1

T 0

|U (t)Φei |2 dτ )q/2 dxdy

|U (t)Φei |2 dτ )q/2 dxdy)q/2

So, kuL kLqω (Lqx,y (L2t )) 6 C

X i>1

kU (t)Φei k2L∞ (Lqx,y ) 6 CkΦk2L0,ˉs t

(4.16)

2

2 Obviously, equations (4.14) and (4.15) implies that D1+ˉs−δ uL ∈ Lqω (L∞ x,y (Lt )) and

kD1+ˉs−δ uL kLqω (L∞ 6 CkΦk2L0,ˉs 2 x,y (Lt ))

(4.17)

2

Recalling the definition of the Hilbert transform [21] c (ζ, η) = ( ζ + η )fˆ(ζ, η) Hf |ζ| |η|

1338

(4.18)

GHANY, HYDER: Zakharov-Kuznetsov equation

implies, D

sˉ−δ

Z

t

∂x uL = Dsˉ−δ ∂x U (t − τ )ΦdW (τ ) 0 Z t = D1+ˉs−δ ∂x U (t − τ )HΦdW (τ ) 0

Then, 6 CkHΦk2L0 ,ˉs 6 C kΦk2L0 ,ˉs kDsˉ−δ ∂x uL kLqω (L∞ 2 x,y (Lt )) 2

(4.19)

2

Now we can present the last regularity property of the solution uL by giving the following result: Proposition 6. ∂x uL ∈ L2ω (L4t (L∞ x,y ) and E[(

Z

T

sup |∂x uL |4 dt)0.5 ] 6 kΦk2L0,ˉs

(4.20)

2

x,y∈R

0

q Proof. Let  = sˉ − 0.75 and q = 4(1 + 1/) . Noting that D1+ uL ∈ L4t (L∞ x,y (Lω )) we have,

kD

1+

q uL kL4t (L∞ = x,y (Lω ))

6C

Z

Z

T

sup E[|

x,y∈R

0 T

sup [ 0

(

i>1

t

0 Z X t

x,y∈R i>1 XZ T

6 C(T )[

Z

0

0

Dsˉ+1/4 U (t − τ )ΦdW (τ )|q ]4/q |Dsˉ+1/4 U (t − τ )Φei dτ |2 ]4/2 dt 1

sup |Dsˉ+1/4 U (t − τ )Φei |4 dτ ) 2 ]2

x,y∈R

Applying [21, Theorem 2.4] with α = 2, θ = 1, β = 1/2 we get Z

0

T

sup |Dsˉ+1/4 U (t − τ )Φei |4 dτ 6 CkDsˉΦei k4L2x,y

x,y∈R

So, q 6 CkΦkL0,ˉs kD1+ uL kL4t (L∞ x,y (Lω )) 2

Therefore, kuL kL4t (L2x,y (Lqω )) 6 CkΦkL0,0 6 CkΦkL0,ˉs 2

2

By virtue of the above inequalities and [9, proposition A.1] we obtain for all t ∈ [0, T ] that 2/q

kD1+/2 uL kLqx,y (Lqω )) 6 CkuL kL2

q x,y (Lω ))

1339

1−2/q q x,y (Lω )

kD1+ uL kL∞

GHANY, HYDER: Zakharov-Kuznetsov equation

Since q = 4(1 + 1/) > 4 , so kD1+/2 uL kL4ω (L4t (Lqx,y )) 6 CkD1+/2 uL kL4t (Lqx,y (Lqω )) 6 1−2/q

6 CkuL kL4t (L2x,y (Lqω )) kD1+ uL kL4 (L∞ t

q x,y (Lω ))

6 CkΦkL0,ˉs 2

Using Fuibini’s theorem, we have kuL kL4ω (L4t (Lqx,y )) 6 CkuL kL4t (L4ω (Hx,y s )) Z T Z t 1/4 6 C( E[k U (t − τ )ΦdW (τ )k4Hx,y s ˉ ]dt) 0 0 Z T Z tX 1/4 6 C( [ kU (t − τ )Φei k2Hx,y s ˉ dτ ]dt) 0

0 i>1

So, kuL kL4ω (L4t (Lqx,y )) 6 CkΦkL0,ˉs 2

Since q/2 > 1 , Then 6 C(T )kΦkL0,ˉs k∂x uL kL4ω (L4t (L∞ x,y )) 2

(4.21)

Now, Theorem 2 is a direct result from the global results of the above propositions. To prove Theorem 1 i.e., to solve the stochastic Zakharov-Kuznetsov equation forced by a random term of additive white noise (1.1). We will use a fixed point argument in Zs (T ) for some T > 0 and 1 . From Theorem 2, we s ∈ (0.75, 1) , then a priori estimate will give us the global solution in Hx,y have uL ∈ Zs (T0 ), T0 > 0 for almost all ω ∈ Ω . Proposition 7.[21] For any s > 0.75 and any T > 0 there exists C(T, s) nondecreasing with respect to T such that: k

Z

T 0

U (t − τ )(u∂x v)dτ kZs (T ) 6 C(T, s)kukZs (T ) kvkZs (T )

(4.22)

for any u, v ∈ Zs (T ) and s kU (t)u0 kZs (T ) 6 C(T, s)kukHx,y

for all

s u0 ∈ Hx,y

Proof of Theorem 1. Firstly, we introduce the mapping J defined by Z t U (t − τ )(u∂x u)dτ + uL (t) J u(t) = U (t)u0 +

(4.23)

(4.24)

0

Let 0.75 < s < 1 , since Φ ∈ L0,1 so by Theorem 2 and Proposition 5 we have u0 ∈ H s (R2 ) , J 2 maps Zs (T ) into itself. Moreover, let R0 satisfies: R0 > C(T0 , s)ku0 kH s (R2 ) + kuL kZs (T )

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GHANY, HYDER: Zakharov-Kuznetsov equation

and choose T such that: 1

C(T0 , s)T 2 R0 6 1 then, J maps the ball of center 0 and radius 2R0 in Zs (T ) into itself and 1 kJ u − J vkZs (T ) 6 ku − vkZs (T ) 2

(4.25)

for any u, v ∈ Zs (T ) with norm less than 2 R0 . By virtue of fixed point theorem, J has a unique fixed point, denote by u , in this ball. It is obvious that this solution u for Equation (1.1) belongs to the function space Zs (T ) .

5

Concluding Remarks

This paper is devoted to establish some methods like Banach contraction principle and successive approximations method for handling stochastic nonlinear partial differential equations and for proving local and global well-posedness results for their solutions in selected function spaces. In fact, we restricted our efforts in stochastic Zakharov-Kuznetsov equation, but we believe that, similar ideas can be applied to other stochastic nonlinear partial differential equations in mathematical physics, such as the generalized KdV, KdV-Burgers, Modified KdV-Burgers and Swada-Kotera equations. sˉ ) ∩ L4 (L2 ) with 0.75 6 s Also we remark that, if we assume that u0 ∈ L2ω (Hx,y ˉ < 1 and u0 is ω x,y F0 − measurable , then we cannot construct a solution on a fixed interval, even a finite one of the form [0, T0 ] . Moreover, by using a standard truncation argument we can extend our results under the assumption that u0 ∈ H 1 (R2 ) almost surely. Acknowledge. Authors Thanks the reviewers’ for their notes which improved the quality of the paper.

References [1] S. Monro and E. J. Parkes, Stability of solitary wave solutions to a modified ZakharovKuznetsov equation, J. Plasma Phys. 64(2000)411-437. [2] S. Monro and E. J. Parkes, The derivation of a modified Zakharov-Kuznetsov equation and stability of its solutions, J. Plasma Phys. 62(1999)305-322. [3] V. E. Zakharov and E. A. Kuznetsov, On three-dimensional solutions, Sov. Phys. 39(1974)285-293. [4] M. Wadati, Deformation of Solitons in Random Media, J. Phys. Soc. Jpn. ,59(1990)42014203 . [5] M. Wadati and Y. Akutsu, Stochastic Korteweg-de Vries Equation with and without Damping, J. Phys. Soc. Jpn.53(1984)3342-3350.

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[6] A. Debussche and J. Printems, Effect of a Localized Random Forcing Term on the Korteweg-de Vries Equation, Comput.Anal. Appl., 3(2001)183-206. [7] A. Debussche and J. Printems, Numerical simulation of the stochastic Korteweg-de Vries equation , Physica D : Nonlinear Phenomena, 134(1999)200-226. [8] A. de Bouard and A. Debussche, White Noise Driven Korteweg-de Vries Equation , J. Funct. Anal., 169(1999)532-558. [9] A. de Bouard and A.Debussche, On the Stochastic Korteweg-de Vries Equation, J. Funct. Anal., 154(1998)215-251. [10] V. V. Konotop and L. Vzquez, Nonlinear random waves, World Scientific, 1994. [11] J. Printems, The Stochastic Korteweg-de Vries Equation in L2 (R) , J. Differen. Equat. 153(1999)338-373. [12] H. A. Ghany, Exact Solutions for Stochastic Generalized Hirota-Satsuma Coupled KdV Equations, Chin. J. Phys., 49(2011)926-940. [13] F. Linares and A. Pastor, Well-posedness for the two-dimensional modified ZakharovKuznetsov equation, SIAM J. Math Anal. 41(2009)1323-1339. [14] H. A. Biagioni and F. Linares, Well-posedness results for the modied ZakharovKuznetsov equation, in Nonlinear Equations: Methods, Models and Applications, Progr. Nonlinear Differential Equations Appl., 54(2003)181-189. [15] A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differ. Equ. 31 (1995)1002- 1012. [16] B. Birnir, C. E. Kenig, G. Ponce, N. Svanstedt and L. Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Varies and nonlinear Schr¨oodinger equations, J. London. Math. Soc., 53(1996)551-559. [17] B. Birnir, G. Ponce and N. Svanstedt, The ill-posedness of the modified KdV equation, Ann. Inst. H. Poincar´e Anal. Nonlin´eeaire. 13(1996)529-535. [18] C. E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the Contraction principle, Commun. Pure Appl. Math. 46 (1993)527-620. [19] V. G. Mazjia, Sobolev spaces, Springer-Verlag, Berlin,1985. [20] G. D. Prato and J. Zabczyk, Stochastic equations in infinite dimensions, ”Encyclo-

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pedia of math. and Appl.”, Cambridge University Press, Cambridge UK,1992. [21] C. E. Kenig, G. Ponce, and L. Vega, Well-posedness of the IVP for the KdV equation via the Contraction principle, J. Amer. Math. Soc. 4(1991)323-347.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 7, 2013

Composition Operators From Hardy Space to n-Th Weighted-Type Space of Analytic Functions on the Upper Half-Plane, Zhi-Jie Jiang and Zuo-An Li,…………………………………1188 Notes on Generalized Gamma, Beta and Hypergeometric Functions, Mehmet Bozer and Mehmet Ali Özarslan,………………………………………………………………………………1194 A Modified AOR Iterative Method for New Preconditioned Linear Systems for L-Matrices, Guang Zeng and Li Lei,…………………………………………………………………...1202 Modern Algorithms of Simulation for Getting Some Random Numbers, G. A. Anastassiou and I. F. Iatan,………………………………………………………………………………..….1211 Second Order Mond-Weir Type Duality for Multiobjective Programming Involving Second Order (‫ܥ‬, ߙ, ߩ, ݀)-Convexity, Sichun Wang,…………………………………………….....1223 Fractional Voronovskaya Type Asymptotic Expansions for Bell and Squashing Type Neural Network Operators, George A. Anastassiou,……………………………………………..1231 Iterates of Multivariate Cheney-Sharma Operators, Teodora Cătinaș and Diana Otrocol,1240 Convergence Analysis of the Over-relaxed Proximal Point Algorithms with Errors for Generalized Nonlinear Random Operator Equations, Lecai Cai and Heng-you Lan,……1247 Fixed Point Theorem for Ciric’s Type Contractions in Generalized Quasi-Metric Spaces, Luljeta Kikina, Kristaq Kikina and Kristaq Gjino,………………………………………………….1257 Explicit Formulas on the Second Kind q-Euler Numbers and Polynomials, C. S. Ryoo,…1266 Second Order ߙ-Univexity and Duality for Nondifferentiable Minimax Fractional Programming, Gang Yang, Fu-qiu Zeng, and Qing-jie Hu,…………………………………………………1272 Some Properties of the Interval-Valued Generalized Fuzzy Integral With Respect to a Fuzzy Measure by Means of an Interval-Representable Generalized Triangular Norm, Lee-Chae Jang,…………………………………………………………………………………………1280 Soft Rough Sets and Their Properties, Cheng-Fu Yang,……………………………………1291 Rate Of Convergence of Some Multivariate Neural Network Operators to the Unit, Revisited, George A. Anastassiou,……………………………………………………………………..1300

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 7, 2013 (continued) New Approach to the Analogue of Lebesgue-Radon-Nikodym Theorem With Respect to Weighted p-Adic q-Measure On ℤ௣ , Joo-Hee Jeong, Jin-Woo Park, and Seog-Hoon Rim,1310 Generalized Tikhonov Regularization Method for Large-Scale Linear Inverse Problems, Di Zhang and Ting-Zhu Huang,…………………………………………………………..…1317 Local and Global Well-Posedness of Stochastic Zakharov-Kuznetsov Equation, Hossam A. Ghany and Abd-Allah Hyder,……………………………………………………..………1332

Volume 15, Number 8 ISSN:1521-1398 PRINT,1572-9206 ONLINE

December 2013

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1356-1371, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A modified nonlinear Uzawa algorithm for solving symmetric saddle point problems ∗ Jian-Lei Lia†, Zhi-Jiang Zhangb , Ying-Wang a , Li-Tao Zhangc a

College of Mathematics and Information Science, North China University of,

Water Resources and Electric Power, Zhengzhou, Henan, 450011, PR China. b c

Minsheng College of Henan University, Kaifeng, Henan, 475001, PR China.

Department of Mathematics and Physics, Zhengzhou Institute of Aeronautical Industry Management, Zhengzhou, Henan, 450015, PR China .

Abstract In this paper, a modified nonlinear Uzawa algorithm for solving symmetric saddle point problems is proposed, and also the convergence rate is analyzed. The results of numerical experiments are presented when we apply the algorithm to Stokes equations discretized by mixed finite elements. Keywords: Convergence rate; Modified nonlinear Uzawa algorithm; Saddle point problems; Schur complement AMS classification: 65F10

1

Introduction

Let H1 and H2 be finite-dimensional Hilbert spaces with inner product denoted by (·, ·). In this paper, we propose a modified nonlinear Uzawa algorithm for solving systems of linear equations with the following two-by-two block structure: µ ¶ µ ¶µ ¶ µ ¶ x x f A BT A = = , (1) y y g B −C wheref ∈ H1 , g ∈ H2 are given, and x ∈ H1 , y ∈ H2 are unknown. Here A : H1 → H1 is assumed to be linear, symmetric and positive definite operator, B : H1 → H2 is a linear map and B T : H2 → H1 is its adjoint. In addition, C : H2 → H2 is linear symmetric and positive semidefinite. Such system is usually referred to as saddle point problem, which is typically resulted from mixed or hybrid finite element approximations of second-order elliptic problems, or the Stokes equation, including computational fluid dynamics as well as constrained optimization problems [1, 2, 6-11,14]. On the solution methods for saddle point systems there is a very good reference [2]. In [1], Bramble et al, considered the linear system (1) with C = 0 and assumed that the following LBB condition [13] holds, i.e., (BA−1 B T v, v) ≡ sup u∈H1

(v, Bu)2 ≥ c0 kvk2 , ∀v ∈ H2 , (Au, u)

(2)

∗ This research was supported by Doctoral Research Project of NCWU (2001119) and by NSFC of Tianyuan Mathematics Youth Fund (11226337, 11126323). † Corresponding author. E-mail: [email protected]

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Jian-Lei Li etal: Modified nonlinear Uzawa algorithm

for some positive number c0 . A nonlinear Uzawa algorithm is first proposed by defining the nonlinear approximate inverse of A as a map φ : H1 → H1 , i.e., for any ϕ ∈ H1 , φ(ϕ) is an approximation to the solution ξ of Aξ = ϕ. In [3], Cao considered the linear system (1), and assumed that the following stabilized condition [7, 8] holds, i.e., ((BA−1 B T + C)v, v) ≥ c0 kvk2 , ∀v ∈ H2 ,

(3)

for some positive number c0 . Cao proposed another nonlinear Uzawa algorithm by defining T the nonlinear approximate inverse of approximate Schur complement (BQ−1 A B + C) as a map ψ : H2 → H2 , i.e., for any ϕ ∈ H2 , ψ(ϕ) is an approximation to the solution ξ of T (BQ−1 A B + C)ξ = ϕ, where QA is a symmetric positive definite operator. In [4], Lin and Cao proposed another nonlinear Uzawa algorithm by defining the nonlinear approximate inverse of A and the Schur complement (BA−1 B T + C). In [5], Lin and Wei proposed a modified nonlinear Uzawa algorithm and modified the Cao’s results. In this paper, we present another modified nonlinear Uzawa algorithm for solving the system (1). At the same time, its convergence is analyzed. The inexact Uzawa algorithms [1,3,4,6,14] are of interest because they are simple, efficient and have minimal numerical computer memory requirements. this could be important in largescale scientific applications implemented for today’s computing architectures. Therefore, the inexact Uzawa methods are widely used in the engineer community. The paper is organized as follows. In section 2, we review the Uzawa type algorithms mentioned in section 1 and their convergence results. In section 3, we give our modified nonlinear Uzawa algorithm (MNUAS) and analyze convergence results. In section 4, the MNUAS algorithm is applied to solve system (1), which is resulted from the discretization of Stokes equations by mixed finite element method and the results of the numerical experiments are presented. Finally, the conclusions are drawn.

2

The Uzawa algorithms and convergence

First, some notions are given. Let Q be a symmetric and positive definite matrix, we define a inner product ´ ³ 1 1 hv, uiQ = (Qv, u) = Q 2 v, Q 2 u , ∀v, u ∈ H2 , and denote the Euclidean norm by k · k. So ° 1 ° ³ 1 ´ 12 1 1 2 ° ° kvkQ = hv, viQ ≡ Q 2 v, Q 2 v ≡ °Q 2 v ° . 2

Denote residue of x and y as

exi = x − xi , eyi = y − yi .

The Nonlinear Uzawa algorithm (which is related to the approximate inverse of the matrix A, and is called as NUA algorithm) for solving system (1) is as follows ([1,3,4]). Algorithm 1 (NUA algorithm) ([1, 3]) For x0 ∈ H1 and y0 ∈ H2 given, the iterative sequence {(xi , yi )} is defined, QB is a symmetric positive definite operator, for i = 0, 1, ..., by xi+1 = xi + φ(f − Axi − B T yi ), yi+1 = yi +

Q−1 B (Bxi+1

1357

− Cyi − g).

(4) (5)

Jian-Lei Li etal: Modified nonlinear Uzawa algorithm

It is assumed that

kφ(v) − A−1 vkA ≤ δkvkA−1 , ∀v ∈ H1 ,

(6)

for some positive δ < 1. In [1], the authors also pointed out that (6) is a reasonable assumption which is satisfied by the approximate inverse associated with the Preconditioned Conjugate Gradient algorithm (PCG algorithm) [12]. It is assumed that the following inequality (1 − γ)(QB w, w) ≤ ((BA−1 B T + C)w, w) ≤ (QB w, w), ∀w ∈ H2

(7)

holds for some γ in the interval [0, 1). In practice, preconditioners satisfy (7) with γ bounded away from one. The result on the convergence of the NUA algorithm is given as follows [1,3]. Theorem 1 Assume that (6) and (7) hold. Let {(x, y)} be the solution pair of (1), and {(xi , yi )} be defined by the Algorithm 1. Then, xi and yi converge to x and y, respectively, if 1−γ . (8) δ< 3−γ In this case, the following two inequalities hold: µ ¶ δ δ (Aexi , exi ) + (QB eyi , eyi ) ≤ ρ2i (Aex0 , ex0 ) + (QB ey0 , ey0 ) , (9) 1+δ 1+δ and

µ (Aexi , exi )

2i−2

≤ (1 + δ)(1 + 2δ)ρ

where γ + 2δ +

¶ δ y y x x (Ae0 , e0 ) + (QB e0 , e0 ) , 1+δ

(10)

q 2 (γ + 2δ) + 4δ (1 − γ)

. (11) 2 The following Algorithm 2 is the Nonlinear Uzawa method, which is relate to the approxiT mate inverse of the approximate Schur complement matrix BQ−1 A B + C. We call it as NUS algorithm. ρ=

Algorithm2 (NUS algorithm) ([3]) For x0 ∈ H1 and y0 ∈ H2 given, QA is a symmetric positive definite, the iterative sequence {(xi , yi )} is defined, for i = 0, 1, ..., by T xi+1 = xi + Q−1 A (f − Axi − B yi ), yi+1 = yi + ψ(Bxi+1 − Cyi − g),

(12) (13)

where ψ(w) is an approximation to the solution ξ of the system T (BQ−1 A B + C)ξ = w.

It is assumed that (1 − ω)(QA v, v) ≤ (Av, v) ≤ (QA v, v), ∀v ∈ H1 , v 6= 0.

(14)

holds for some ω in the interval [0, 1), and the approximate Schur complement matrix satisfies T −1 kψ(w) − (BQ−1 wk(BQ−1 B T +C) ≤ εkwk(BQ−1 BT +C)−1 , ∀w ∈ H2 . A B + C) A

(15)

A

for some positive ε < 1. Analogous to (6) in [1], (15) is a reasonable assumption [3], which is satisfied by the approximate inverse associated with the Conjugate Gradient algorithm (CG algorithm). In [3], Cao gave the following convergence result.

1358

Jian-Lei Li etal: Modified nonlinear Uzawa algorithm

Theorem 2 Assume that (14) and (15) hold. Let {(x, y)} be the solution pair of (1), and {(xi , yi )} be defined by the Algorithm 2. Then, xi and yi converge to x and y, respectively, if ω
0, I g(t) = Γ(q) 0 (t − s)1−q provided the integral exists. In the sequel, the following lemma plays a pivotal role. Lemma 2.3 For a given y ∈ C([0, T ], R) and 2 < q ≤ 3, the unique solution of the equation c Dq x(t) = y(t), t ∈ [0, T ] subject to the boundary conditions of (1) is given by Z t Z T (t − s)q−1 (T − s)q−1 x(t) = y(s)ds − λ0 ξ1 y(s)ds Γ(q) Γ(q) 0 0 Z T Z T (T − s)q−2 (T − s)q−3 +λ1 η2 y(s)ds + λ2 η1 y(s)ds Γ(q − 1) Γ(q − 2) 0 0 (2) Z T Z T −µ0 ξ1 g0 (s, x(s))ds + µ1 η2 g1 (s, x(s))ds 0

Z +µ2 η1

0 T

g2 (s, x(s))ds, 0

where

i h η1 = ξ3 −λ0 (λ1 + 1)T 2 + 2λ1 (λ0 − 1)tT − (λ0 − 1)(λ1 − 1)t2 , 1 ξ1 = , λ0 − 1

η2 = ξ2 [λ0 T − (λ0 − 1)t], 1 1 ξ2 = , ξ3 = . (λ0 − 1)(λ1 − 1) 2(λ0 − 1)(λ1 − 1)(λ2 − 1)

Proof. For 2 < q ≤ 3, it is well known [6] that the solution of fractional differential equation c Dq x(t) = y(t) can be written as Z t (t − s)q−1 x(t) = y(s)ds − c0 − c1 t − c2 t2 , t ∈ [0, T ], (3) Γ(q) 0 where c0 , c1 , c2 ∈ R are arbitrary constants. Applying the boundary conditions of (1), we get  Z T Z T (T − s)q−1  2  (λ − 1)c + λ T c + λ T c = µ g (t, x(s))ds + λ y(s)ds,  0 0 0 1 0 2 0 0 0   Γ(q) 0 0    Z T Z T  (T − s)q−2 (λ1 − 1)c1 + 2λ1 T c2 = µ1 g1 (s, x(s))ds + λ1 y(s)ds  Γ(q − 1)  0 0   Z T Z T    (T − s)q−3   2(λ2 − 1)c3 = µ2 g2 (s, x(s))ds + λ2 y(s)ds. Γ(q − 2) 0 0

(4)

Solving the system (4), we find the values of c0 , c1 and c2 . Substituting these values in (3), we obtain (2). 1373

BVP FOR FRACTIONAL DIFFERENTIAL EQUATIONS

3

Main results

Let C = C([0, T ], R) denotes the Banach space of all continuous functions from [0, T ] → R endowed with the usual sup-norm ( kxk = supt∈[0,T ] |x(t)|). By Lemma 2.3, the problem (1) can be transformed to a fixed point problem as x = F (x), where F : C → C is given by t

Z T (t − s)q−1 (T − s)q−1 (F x)(t) = f (s, x(s))ds − λ0 ξ1 f (s, x(s))ds Γ(q) Γ(q) 0 0 Z Z T T (T − s)q−3 (T − s)q−2 f (s, x(s))ds + λ2 η1 f (s, x(s))ds +λ1 η2 Γ(q − 1) Γ(q − 2) 0 0 Z T Z T −µ0 ξ1 g0 (s, x(s))ds + µ1 η2 g1 (s, x(s))ds 0 0 Z T +µ2 η1 g2 (s, x(s))ds, t ∈ [0, T ]. Z

(5)

0

For the sake of computational convenience, we introduce Λ1 =

 Tq 1 + |λ0 ξ1 | + |λ1 η2 |qT −1 + |λ2 η1 |q(q − 1)T −2 . Γ(q + 1)

(6)

Our first existence result is based on Leray-Schauder nonlinear alternative.

Theorem 3.1 (Nonlinear alternative for single valued maps)[24]. Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and 0 ∈ U. Suppose that F : U → C is a continuous, compact (that is, F (U ) is a relatively compact subset of C) map. Then either (i) F has a fixed point in U , or (ii) there is a u ∈ ∂U (the boundary of U in C) and λ ∈ (0, 1) with u = λF (u). Theorem 3.2 Assume that f, gj : [0, 1] × R → R are continuous functions and the following conditions hold: (A1 ) there exist a function p ∈ L1 ([0, 1], R+ ), and ψ : R+ → R+ nondecreasing such that |f (t, x)| ≤ p(t)ψ(kxk) for each (t, x) ∈ [0, T ] × R; (A2 ) there exist continuous nondecreasing functions ψj : [0, ∞) → (0, ∞) and functions pj ∈ L1 ([0, T ], R+ ) such that |gj (t, x)| ≤ pj (t)ψj (kxk), j = 0, 1, 2, for each (t, x) ∈ [0, T ] × R; (A3 ) there exists a number M > 0 such that

ψ(M )Ω1 kpkL1 + ψ0 (M )|µ0 ξ1 |kp0 kL1

M > 1, + ψ1 (M )|µ1 η2 |kp1 kL1 + ψ2 (M )|µ2 η1 |kp2 kL1

where Ω1 =

T q−1  1 + |λ0 ξ1 | + |λ1 η2 |(q − 1)T −1 + |λ2 η1 |q(q − 1)(q − 2)T −2 . Γ(q)

Then the boundary value problem (1) has at least one solution on [0, 1]. 1374

B. AHMAD AND S. K. NTOUYAS

Proof. Consider the operator F : C → C defined by (5). It is easy to prove that F is continuous. Next, we show that F maps bounded sets into bounded sets in C([0, T ], R). For a positive number ρ, let Bρ = {x ∈ C([0, T ], R) : kxk ≤ ρ} be a bounded set in C([0, T ], R). Then, for each x ∈ Bρ ,we have Z T (t − s)q−1 (T − s)q−1 ≤ |f (s, x(s))|ds + |λ0 ξ1 | |f (s, x(s))|ds Γ(q) Γ(q) 0 0 Z T Z T (T − s)q−2 (T − s)q−3 +|λ1 η2 | |f (s, x(s))|ds + |λ2 η1 |ψ(kxk) p(s)|ds Γ(q − 1) Γ(q − 2) 0 0 Z T Z T +|µ0 ξ1 | |g0 (s, x(s))|ds + |µ1 η2 | |g1 (s, x(s))|ds Z

|(F x)(t)|

t

0

Z

0 T

+|µ2 η1 | |g2 (s, x(s))|ds 0  q−1 Z T T T q−1 T q−2 T q−3 ≤ ψ(kxk) p(s)ds + |λ0 ξ1 | + |λ1 η2 | + |λ2 η1 | Γ(q) Γ(q) Γ(q − 1) Γ(q − 2) 0 Z T Z T p0 (s)ds + ψ1 (kxk)|µ1 η2 | +ψ0 (kxk)|µ0 ξ1 | p1 (s)ds 0

Z +ψ2 (kxk)|µ2 η1 |

0 T

p2 (s)ds 0

≤ ψ(kxk)Ω1 kpkL1 + ψ0 (kxk)|µ0 ξ1 |kp0 kL1 + ψ1 (kxk)|µ1 η2 |kp1 kL1 +ψ2 (kxk)|µ2 η1 |kp2 kL1 . Thus, kF xk ≤ ψ(ρ)Ω1 kpkL1 + ψ0 (ρ)|µ0 ξ1 |kp0 kL1 + ψ1 (ρ)|µ1 η2 |kp1 kL1 + ψ2 (ρ)|µ2 η1 |kp2 kL1 . Now we show that F maps bounded sets into equicontinuous sets of C([0, T ], R). Let t0 , t00 ∈ [0, T ] with t0 < t00 and x ∈ Bρ , where Bρ is a bounded set of C([0, T ], R). Then we have |(F x)(t00 ) − (F x)(t0 )|  Z t00 00 Z t0  00 (t − s)q−1 − (t0 − s)q−1 (t − s)q−1 ≤ ψ(kxk) p(s)ds + f (s, x(s))ds Γ(q) Γ(q) 0 t0 Z T h (T − s)q−2 +|(1 − λ0 )λ1 ξ2 ||t00 − t0 |ψ(kxk) p(s)ds + |λ2 ξ3 | 2|(1 − λ0 )λ1 |T |t00 − t0 | Γ(q − 1) 0 Z T i (T − s)q−3 2 2 +|(1 − λ0 )(1 − λ1 )||t00 − t0 | ψ(kxk) p(s)ds Γ(q − 2) 0 Z T 00 0 +|(1 − λ0 )|µ1 λ1 ξ2 ||t − t |ψ1 (kxk) p1 (s)|ds 0

Z h i 2 2 +|λ2 ξ3 µ2 | 2|(1 − λ0 )λ1 T |t00 − t0 | + |(1 − λ0 )(1 − λ1 )||t00 − t0 | ψ2 (kxk)

T

p2 (s)ds.

0

Obviously the right hand side of the above inequality tends to zero independently of x ∈ Bρ as t00 −t0 → 0. Therefore it follows by the Ascoli-Arzel´a theorem that F : C([0, T ], R) → C([0, T ], R) is completely continuous. The result will follow from the Leray-Schauder nonlinear alternative (Theorem 3.1) once we have proved the boundedness of the set of all solutions to equations x = λF x for λ ∈ [0, 1]. Let x be a solution. Then, for t ∈ [0, T ], and using the computations in proving that F is bounded, we have  q−1 Z T T T q−1 T q−2 T q−3 |x(t)| ≤ ψ(kxk) + |λ1 ξ1 | + |λ2 η2 | + |λ3 η1 | p(s)ds Γ(q) Γ(q) Γ(q − 1) Γ(q − 2) 0 1375

BVP FOR FRACTIONAL DIFFERENTIAL EQUATIONS T

Z +ψ0 (kxk)|µ1 ξ1 |

Z

T

p0 (s)ds + ψ1 (kxk)|µ2 η2 | 0

Z +ψ2 (kxk)|µ3 η1 |

p1 (s)ds 0

T

p2 (s)ds 0

≤ ψ(kxk)Ω1 kpkL1 + ψ0 (kxk)|µ1 ξ1 |kp0 kL1 + ψ1 (kxk)|µ2 η2 |kp1 kL1 +ψ2 (kxk)|µ3 η1 |kp2 kL1 . Consequently, we have ψ(kxk)Ω1 kpkL1 + ψ0 (kxk)|µ0 ξ1 |kp0 kL1

kxk ≤ 1. + ψ1 (kxk)|µ1 η2 |kp1 kL1 + ψ2 (kxk)|µ2 η1 |kp2 kL1

In view of (A3 ), there exists M such that kxk 6= M . Let us set U = {x ∈ C([0, T ], R) : kxk < M + 1}. Note that the operator F : U → C([0, T ], R) is continuous and completely continuous. From the choice of U , there is no x ∈ ∂U such that x = λF x for some λ ∈ (0, 1). Consequently, by the Leray-Schauder alternative (Theorem 3.1), we deduce that F has a fixed point x ∈ U which is a solution of the problem (1).  Our next result is based on the celebrated fixed point theorem due to Banach. Theorem 3.3 Assume that f, gj : [0, T ] × R → R are continuous functions satisfying the conditions: (A4 ) |f (t, x) − f (t, y)| ≤ L|x − y|, ∀t ∈ [0, T ], L > 0, x, y ∈ R; (A5 ) |gj (t, x) − gj (t, y)| ≤ Lj |x − y|, ∀t ∈ [0, T ], Lj > 0, j = 0, 1, 2, x, y ∈ R. Then the boundary value problem (1) has a unique solution if n o LΛ1 + L0 |µ0 ξ1 | + L1 |µ1 η2 | + L2 |µ2 η1 | T < 1, where Λ1 is given by (6). Proof. Let us fix supt∈[0,T ] |f (t, 0)| = M, supt∈[0,T ] |gj (t, 0)| = Mj , j = 0, 1, 2 and choose n o M Λ1 + M0 |µ0 ξ1 | + M1 |µ1 η2 | + M2 |µ2 η1 | T  n o . r≥ 1 − LΛ1 + L0 |µ0 ξ1 | + L1 |µ1 η2 | + L2 |µ2 η1 | T Then we show that F Br ⊂ Br , where Br = {x ∈ C : kxk ≤ r}. For x ∈ Br , we have (Z Z T t (T − s)q−1 (t − s)q−1 |f (s, x(s))|ds + |λ0 ξ1 | |f (s, x(s))|ds |(F x)(t)| ≤ sup Γ(q) Γ(q) t∈[0,T ] 0 0 Z T Z T (T − s)q−2 (T − s)q−3 +|λ1 η2 | |f (s, x(s))|ds + |λ2 η1 | |f (s, x(s))|ds Γ(q − 1) Γ(q − 2) 0 0 Z T Z T +|µ0 ξ1 | |g0 (s, x(s))|ds + |µ1 η2 | |g1 (s, x(s))|ds 0 0 ) Z T

+|µ2 η1 |

|g2 (s, x(s))|ds 0

Z ≤

sup t∈[0,T ]

0

t

i (t − s)q−1 h |f (s, x(s)) − f (s, 0)| + |f (s, 0)| ds Γ(q) 1376

B. AHMAD AND S. K. NTOUYAS T

i (T − s)q−1 h |f (s, x(s)) − f (s, 0)| + |f (s, 0)ds| ds Γ(q) 0 Z T i (T − s)q−2 h +|λ1 η2 | |f (s, x(s)) − f (s, 0)| + |f (s, 0)ds| ds Γ(q − 1) 0 Z T i (T − s)q−3 h +|λ2 η1 | |f (s, x(s)) − f (s, 0)| + |f (s, 0)| ds Γ(q − 2) 0 Z Th i +|µ0 ξ1 | |g0 (s, x(s)) − g0 (s, 0)| + |g0 (s, 0)| ds Z

+|λ0 ξ1 |

0 Th

Z +|µ1 η2 |

i |g1 (s, x(s)) − g1 (s, 0)| + |g1 (s, 0)| ds

0 Th

Z + |µ2 η1 |

i |g2 (s, x(s)) − g2 (s, 0)| + |g2 (s, 0)| ds

)

0

 Tq 1 + |λ1 ξ1 | + |λ2 η2 |qT −1 + |λ3 η1 |q(q − 1)T −2 Γ(q + 1) +(L0 r + M0 )|µ0 ξ1 |T + (L1 r + M1 )|µ1 η2 |T + (L2 r + M2 )|µ2 η1 |T = (LΛ1 + L0 |µ0 ξ1 |T + L1 |µ1 η2 |T + L2 |µ2 η1 |T )r +(M Λ1 + M0 |µ0 ξ1 |T + M1 |µ1 η2 |T + M2 |µ2 η1 |T ) ≤ r. ≤ (Lr + M )

Now, for x, y ∈ C and for each t ∈ [0, T ], we obtain Z t (t − s)2 |f (s, x(s)) − f (s, y(s))|ds k(F x)(t) − (F y)(t)k ≤ sup 2 t∈[0,T ] 0 Z T (T − s)q−1 +|λ0 ξ1 | |f (s, x(s)) − f (s, y(s))|ds Γ(q) 0 Z T (T − s)q−2 |f (s, x(s)) − f (s, y(s))|ds |λ1 η2 | Γ(q − 1) 0 Z T (T − s)q−3 +|λ2 η1 | |f (s, x(s)) − f (s, y(s))|ds Γ(q − 2) 0 Z T +|µ0 ξ1 | |g0 (s, x(s)) − g0 (s, y(s))|ds 0 T

Z +|µ1 η2 |

|g1 (s, x(s)) − g1 (s, y(s))|ds 0

Z +|µ2 η1 |

)

T

|g2 (s, x(s)) − g2 (s, y(s))|ds 0

LT q  1 + |λ1 ξ1 | + |λ2 η2 |qT −1 + |λ3 η1 |q(q − 1)T −2 Γ(q + 1) +L0 kx − yk|µ0 ξ1 | + L1 kx − yk|µ1 η2 |T + L2 |µ2 η1 |T kx − yk = (LΛ1 + L0 |µ0 ξ1 |T + L1 |µ1 η2 |T + L2 |µ2 η1 |T )kx − yk. ≤

kx − yk

As LΛ1 + (L0 |µ0 ξ1 | + L1 |µ1 η2 | + L2 |µ2 η1 |)T < 1, therefore F is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem).  Our final existence result is based on Krasnoselskii’s fixed point theorem [25]. Lemma 3.4 (Krasnoselskii’s fixed point theorem) [25]. Let M be a closed bounded, convex and nonempty subset of a Banach space X. Let A, B be the operators such that (i) Ax + By ∈ M whenever x, y ∈ M ; (ii) A is compact and continuous and (iii) B is a contraction mapping. Then there exists z ∈ M such that z = Az + Bz. 1377

BVP FOR FRACTIONAL DIFFERENTIAL EQUATIONS

Theorem 3.5 Let f, gj : [0, T ]×R → R be continuous functions satisfying the assumptions (A4 )−(A5 ). In addition we suppose that (A6 ) |f (t, x)| ≤ ν(t), ∀(t, x) ∈ [0, T ] × R, and ν ∈ C([0, T ], R+ ); (A7 ) |gj (t, x)| ≤ νj (t), j = 0, 1, 2, ∀(t, x) ∈ [0, T ] × R, and νj ∈ C([0, T ], R+ ). If  L(Λ1 Γ(q + 1) − T q )  + L0 |µ0 ξ1 | + L1 |µ1 η2 | + L2 |µ2 η1 | T < 1, Γ(q + 1)

(7)

then problem (1) has at least one solution on [0, T ].

Proof. Letting supt∈[0,T ] |ν(t)| = kνk, supt∈[0,T ] |νj (t)| = kνj k, j = 0, 1, 2, we fix r ≥ Λ1 kνk + (|µ0 ξ1 |kν0 k + |µ1 η2 |kν1 k + |µ2 η1 |kν2 k)T and consider Br = {x ∈ C : kxk ≤ r}. We define the operators P and Q on Br as Z

t

(Px)(t) = 0

(t − s)q−1 f (s, x(s))ds, Γ(q) T

Z (Qx)(t)

= −λ0 ξ1 0 T

Z +λ2 η1 0

Z +µ1 η2

T

(T − s)q−1 f (s, x(s))ds + λ1 η2 Γ(q)

Z

(T − s)q−3 f (s, x(s))ds − µ0 ξ1 Γ(q − 2)

Z

T

Z

T

g0 (s, x(s))ds 0

T

g2 (s, x(s))ds, t ∈ [0, T ].

g1 (s, x(s))ds + µ2 η1 0

0

(T − s)q−2 f (s, x(s))ds Γ(q − 1)

0

For x, y ∈ Br , we find that kPx + Qyk

  ≤ Λ1 kνk + |µ0 ξ1 |kν0 k + |µ1 η2 |kν1 k + |µ2 η1 |kν2 k T ≤ r.

Thus, Px + Qy ∈ Br . It follows from the assumption (A4 ) together with (7) that Q is a contraction mapping. Continuity of f implies that the operator P is continuous. Also, P is uniformly bounded on Br as Tq kPxk ≤ kµk. Γ(q + 1) Now we prove the compactness of the operator P. We define sup(t,x)∈[0,T ]×Br |f (t, x)| = fs < ∞, and consequently, for t1 , t2 ∈ [0, T ] with t2 < t1 , we have Z Z t2 fs t1 q−1 q−1 q−1 |(Px)(t2 ) − (Px)(t1 )| ≤ [(t2 − s) − (t1 − s) ]ds + (t2 − s) ds , Γ(q) 0 t1 which is independent of x. Thus, P is equicontinuous. So P is relatively compact on Br . Hence, by the Arzel´a-Ascoli Theorem, P is compact on Br . Thus all the assumptions of Lemma 3.4 are satisfied. So the conclusion of Lemma 3.4 implies that the boundary value problem (1) has at least one solution on [0, T ].  1378

B. AHMAD AND S. K. NTOUYAS

Example 3.6 Consider the following boundary value problem  c 5/2 D x(t) = L(cos t + tan−1 x(t)), t ∈ [0, 1],     Z 1   x(s)   ds,  x(0) + x(1) =   (1 + s)2 0 Z 1 s e x(s) 1  x0 (0) + x0 (1) = 1  + ds,   2 0 1 + 2es 2   Z    00 1 1  x(s) 3   x (0) + x00 (1) = + ds, 3 0 1 + es 4

(8)

where f (t, x) = L(cos t + tan−1 x(t)), g0 (t, x) =

1 3 x(t) et x(t) x(t) , + , g2 (t, x) = + g (t, x) = 1 2 t t (1 + t) 1 + 2e 2 1+e 4

(L to be fixed later), and λ1 = λ2 = λ3 = −1, µ1 = 1, µ2 = 21 , µ3 = 13 . Clearly, ξ1 = −1/2, ξ2 = 1/4, ξ3 = −1/16, η1 = 1/16, η2 = 1/4, |f (t, x) − f (t, y)| ≤ L|x − y|, |g0 (t, x) − g0 (t, y)| ≤ |x − y|, |g1 (t, x) − g1 (t, y)| ≤ |g2 (t, x) − g2 (t, y)| ≤ Λ1 =

1 |x − y|, 3

1 1 1 |x − y|, L0 = 1, L1 = , L2 = . 2 3 2

 Tq 151 √ , 1 + |λ1 ξ1 | + |λ2 η2 |qT −1 + |λ3 η1 |q(q − 1)T −2 = Γ(q + 1) 120 π

and

n o LΛ1 + L0 |µ1 ξ1 | + L1 |µ2 η2 | + L2 |µ3 η1 | < 1 √

π implies that L < 215 604 . Thus, all the conditions of Theorem 3.3 are satisfied. So there exists at least one solution of the problem (8) on [0, 1].

Remark 3.7 The existence results for a third-order nonlinear boundary vale problem of ordinary differential equations with anti-periodic type integral boundary conditions follow as a special case if we take q = 3 in the results of this paper. We emphasize that these results are new. Acknowledgment. The research of B. Ahmad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

References [1] J.R. Graef, L. Kong, Positive solutions for third order semipositone boundary value problems, Appl. Math. Lett. 22 (2009) 1154-1160. [2] J.R. Graef, J.R.L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal. 71 (2009) 1542-1551. [3] Q. Yao, Y. Feng, The existence of solution for a third-order two-point boundary value problem, Appl. Math. Lett. 15 (2002) 227-232. [4] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [5] G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, 2005. [6] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. 1379

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[7] R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Inc., Connecticut, 2006. [8] G.A. Anastassiou, Advances on fractional inequalities, Springer Briefs in Mathematics, Springer, New York, 2011. [9] D. Baleanu, K. Diethelm, E. Scalas, J. J.Trujillo, Fractional calculus models and numerical methods. Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, 2012. [10] R. P. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Difference Equ. 2009, Art. ID 981728, 47 pp. [11] B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. Math. Lett. 23 (2010) 390-394. [12] B. Ahmad, Existence of solutions for fractional differential equations of order with anti-periodic boundary conditions, J. Appl. Math. Comput. 34 (2010), 385-391. [13] B. Ahmad, J.J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory, Topol. Methods Nonlinear Anal. 35 (2010), 295-304. [14] D. Baleanu, O.G. Mustafa, R.P. Agarwal, An existence result for a superlinear fractional differential equation, Appl. Math. Lett. 23 (2010) 1129-1132. [15] E. Hernandez, D. O’Regan, K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal. 73 (2010), no. 10, 3462-3471. [16] G. Wang, S.K. Ntouyas, L. Zhang, Positive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument, Adv. Difference Equ. 2011, 2011:2, 11 pp. [17] B. Ahmad, S.K. Ntouyas, A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order, Electron. J. Qual. Theory Differ. Equ. 2011, No. 22, 1-15. [18] B. Ahmad, J.J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Boundary Value Problems 2011, 2011:36. [19] B. Ahmad, J.J. Nieto, A. Alsaedi, Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions, Acta Math. Sci. 31 (2011), 2122-2130. [20] S. Liang, J. Zhang, Existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval, Math. Comput. Modelling 54 (2011) 1334-1346. [21] N.J. Ford, M.L. Morgado, Fractional boundary value problems: Analysis and numerical methods, Fract. Calc. Appl. Anal. 14, No 4 (2011), 554-567. [22] B. Ahmad, S.K. Ntouyas, A note on fractional differential equations with fractional separated boundary conditions, Abstr. Appl. Anal. 2012, Article ID 818703, 11 pages. [23] B. Ahmad, J.J. Nieto, Anti-periodic fractional boundary value problem with nonlinear term depending on lower order derivative, Fract. Calc. Appl. Anal. 15 (2012), 451-462. [24] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2005. [25] M.A. Krasnoselskii, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk 10 (1955), 123-127.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1381-1390, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Coupled common fixed point theorems for weakly increasing mappings with two variables∗ Hui-Sheng Ding†, Lu Li, Wei Long a

College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, People’s Republic of China

Abstract In this paper, we introduce a notion of weakly increasing mappings with two variables. Several coupled common fixed point theorems for weakly increasing mappings in ordered metric spaces are established. Then, by using a scalarization method, we obtain two coupled common fixed point theorems in ordered cone metric spaces, which extend some earlier results about weakly increasing mappings with one variable. Keywords: Weakly increasing mapping; fixed point; coupled common fixed point; ordered metric space; ordered cone metric space. Mathematics Subject Classification: 47H10, 54H25.

1

Introduction

Recently, there is a large literature about fixed point theorems in cone metric spaces and ordered cone metric spaces, and such problems have attracted more and more authors. We refer the reader to [1–12] and references therein for some of recent developments on such topics. Especially, Altun et al. [3] introduced the notion of weakly increasing mapping, and obtained the following result: Theorem 1.1. Let (X, v, d) be an ordered complete cone metric space, and (f, g) be a weakly increasing pair of self-maps on X w.r.t. v. Suppose that the following conditions hold: ∗

The work was supported by the NSF of China (11101192), the Key Project of Chinese Ministry of

Education (211090), the NSF of Jiangxi Province of China (20114BAB211002), and the Young Talents Programme of Jiangxi Normal University. † Corresponding author. E-mail address: [email protected].

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(i) there exist α, β, γ ≥ 0 such that α + 2β + 2γ < 1 and d(f x, gy) ¹ αd(x, y) + β[d(x, f x) + d(y, gy)] + γ[d(x, gy) + d(y, f x)] for all comparable x, y ∈ X; (ii) f or g is continuous, or (ii0 ) if an nondecreasing sequence {xn } converges to x in X, then xn v x for all n ∈ N. Then f and g have a common fixed point x∗ ∈ X. In fact, Theorem 1.1 can be seen as an “ordered” variant of a result of Abbas and Rhoades [2]. Very recently, Kadelburg et al. [8] generalized Theorem 1.1 and obtained the following theorem: Theorem 1.2. Let (X, v, d) be an ordered complete cone metric space, and (f, g) be a weakly increasing pair of self-maps on X w.r.t. v. Suppose that the following conditions hold: (i) there exist p, q, r, s, t ≥ 0 such that p + q + r + s + t < 1 and q = r or s = t, such that d(f x, gy) ¹ pd(x, y) + qd(x, f x) + rd(y, gy) + sd(x, gy) + td(y, f x) for all comparable x, y ∈ X; (ii) f or g is continuous, or (ii0 ) if an nondecreasing sequence {xn } converges to x in X, then xn v x for all n ∈ N. Then f and g have a common fixed point x∗ ∈ X. The aim of this paper is to make further studies on such problems, and to extend the results in [3, 8]. Inspired by [3, Definition 14], we introduce the following concept of weakly increasing mappings with two variables: Definition 1.3. Let (X, v) be a partially ordered set. Two mappings F, G : X × X → X are said to be weakly increasing if F (x, y) v G(F (x, y), F (y, x)), hold for all (x, y) ∈ X × X.

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G(x, y) v F (G(x, y), G(y, x))

DING ET AL: COMMON FIXED POINT THEOREMS

Example 1.4. Let X = [1, ∞), endowed with the usual ordering ≤. Let F, G : X ×X → X be defined by F (x, y) = x + 2y, G(x, y) = xy 2 . Then, for all (x, y) ∈ X × X, F (x, y) = x + 2y ≤ G(F (x, y), F (y, x)) = G(x + 2y, y + 2x) = (x + 2y)(y + 2x)2 and G(x, y) = xy 2 ≤ F (G(x, y), G(y, x)) = F (xy 2 , x2 y) = xy 2 + 2x2 y. Thus, F and G are two weakly increasing mappings.

2

Main results in ordered metric spaces

Throughout this section, we denote by (X, v, d) an ordered metric space, i.e., v is a partial order on the set X, and d is a metric on X. In addition, we call that (x, y), (u, v) ∈ X × X are comparable if x v u and y v v or u v x and v v y. We will prove several coupled common fixed point theorems for two weakly increasing mappings. Theorem 2.1. Let (X, v, d) be a complete ordered metric space, and F, G : X × X → X be two weakly increasing mappings w.r.t. v. Suppose that the following assumptions hold: (i) there exists λ ∈ [0, 12 ) such that d(F (x, y), G(u, v)) ≤ λ · z for all comparable (x, y), (u, v) ∈ X × X, where z = max{d(x, u), d(y, v), d(x, F (x, y)), d(y, F (y, x)), d(u, G(u, v)), d(v, G(v, u)), d(x, G(u, v)), d(y, G(v, u)), d(u, F (x, y)), d(v, F (y, x))}; (ii) F or G is continuous, or X has the following property: (P) if an nondecreasing sequence {xn } converges to x in X, then xn v x for all n ∈ N. Then F and G has a coupled common fixed point, i.e., there exists (x∗ , y ∗ ) ∈ X × X such that F (x∗ , y ∗ ) = G(x∗ , y ∗ ) = x∗ and F (y ∗ , x∗ ) = G(y ∗ , x∗ ) = y ∗ .

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Proof. Take x0 , y0 ∈ X. Define two sequences {xn }, {yn } in X as follows: x2n+1 = F (x2n , y2n ), x2n+2 = G(x2n+1 , y2n+1 ), y2n+1 = F (y2n , x2n ) and y2n+2 = G(y2n+1 , x2n+1 ) for all n ≥ 0. Since F and G are weakly increasing, we have x1 = F (x0 , y0 ) v G(F (x0 , y0 ), F (y0 , x0 )) = G(x1 , y1 ) = x2 v F (G(x1 , y1 ), G(y1 , x1 )) = F (x2 , y2 ) = x3 v · · · , y1 = F (y0 , x0 ) v G(F (y0 , x0 ), F (x0 , y0 )) = G(y1 , x1 ) = y2 v F (G(y1 , x1 ), G(x1 , y1 )) = F (y2 , x2 ) = y3 v · · · . So the sequences {xn }, {yn } are nondecreasing. Since (x2n+1 , x2n+2 ) = (F (x2n , y2n ), G(x2n+1 , y2n+1 )),

(y2n+1 , y2n+2 ) = (F (y2n , x2n ), G(y2n+1 , x2n+1 )),

by the condition (i), we have max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )} ≤ λz,

(2.1)

where z = max{d(x2n , x2n+1 ), d(y2n , y2n+1 ), d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 ), d(x2n , x2n+2 ), d(y2n , y2n+2 )}. Now, we consider the following three cases: 1◦ if z = max{d(x2n , x2n+1 ), d(y2n , y2n+1 )}, then max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )} ≤ λz ≤

λ max{d(x2n , x2n+1 ), d(y2n , y2n+1 )}; 1−λ

2◦ if z = max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )}, then by (2.1), we have max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )} = 0 ≤

λ max{d(x2n , x2n+1 ), d(y2n , y2n+1 )}; 1−λ

3◦ if z = max{d(x2n , x2n+2 ), d(y2n , y2n+2 )}, it follows from (2.1) that max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )} ≤ λz ≤ λ[max{d(x2n , x2n+1 ), d(y2n , y2n+1 )} + max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )}], which means that max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )} ≤

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λ max{d(x2n , x2n+1 ), d(y2n , y2n+1 )}. 1−λ

DING ET AL: COMMON FIXED POINT THEOREMS

Thus, in all cases, we have max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )} ≤

λ max{d(x2n , x2n+1 ), d(y2n , y2n+1 )}. 1−λ

By a similar proof, one can also show that max{d(x2n , x2n+1 ), d(y2n , y2n+1 )} ≤

λ max{d(x2n−1 , x2n ), d(y2n−1 , y2n )}. 1−λ

So we get max{d(xn , xn+1 ), d(yn , yn+1 )} ≤ Since λ ∈ [0, 12 ), 0 ≤

λ 1−λ

λ max{d(xn−1 , xn ), d(yn−1 , yn )}. 1−λ

< 1. Then, by a standard proof, one can conclude that

{xn }, {yn } are Cauchy sequences. Thus, there exist x∗ , y ∗ ∈ X such that xn → x∗ and yn → y ∗ as n → ∞. In order to show that x∗ , y ∗ is a coupled common fixed point of F and G, we consider the following three cases: Case I. F is continuous. Obviously, x∗ = F (x∗ , y ∗ ) and y ∗ = F (y ∗ , x∗ ). Noticing that d(x∗ , G(x∗ , y ∗ )) = d(F (x∗ , y ∗ ), G(x∗ , y ∗ )),

d(y ∗ , G(y ∗ , x∗ )) = d(F (y ∗ , x∗ ), G(y ∗ , x∗ )),

by (i’), we obtain max{d(x∗ , G(x∗ , y ∗ )), d(y ∗ , G(y ∗ , x∗ ))} ≤ λ max{d(x∗ , G(x∗ , y ∗ )), d(y ∗ , G(y ∗ , x∗ ))}, which yields that x∗ = G(x∗ , y ∗ ), y ∗ = G(y ∗ , x∗ ). Case II. G is continuous. The proof is similar to that of Case I. Case III. X has the property (P). In view of xn v x∗ and yn v y ∗ for all n ∈ N, one can use (i’) to obtain the following: max{d(F (x∗ , y ∗ ), x2n+2 ), d(F (y ∗ , x∗ ), y2n+2 )} ≤ λz ∗ , where z ∗ = max{d(x∗ , x2n+1 ), d(y ∗ , y2n+1 ), d(x∗ , F (x∗ , y ∗ )), d(y ∗ , F (y ∗ , x∗ )), d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 ), d(x∗ , x2n+2 ), d(y ∗ , y2n+2 ), d(x2n+1 , F (x∗ , y ∗ )), d(y2n+1 , F (y ∗ , x∗ ))}.

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Letting n → ∞, we get max{d(x∗ , F (x∗ , y ∗ )), d(y ∗ , F (y ∗ , x∗ ))} ≤ λ max{d(x∗ , F (x∗ , y ∗ )), d(y ∗ , F (y ∗ , x∗ ))}. Thus x∗ = F (x∗ , y ∗ ) and y ∗ = F (y ∗ , x∗ ). Analogously to the above proof, one can also obtain that x∗ = G(x∗ , y ∗ ) and y ∗ = G(y ∗ , x∗ ). Theorem 2.2. Suppose that all the assumptions of Theorem 2.1 except for (i) are satisfied, and the following assumption holds: (i’) there exists λ ∈ [0, 1) such that d(F (x, y), G(u, v)) ≤ λ · w for all comparable (x, y), (u, v) ∈ X × X, where w = max{d(x, u), d(y, v), d(x, F (x, y)), d(y, F (y, x)), d(u, G(u, v)), d(v, G(v, u)), d(x, G(u, v)) + d(u, F (x, y)) d(y, G(v, u)) + d(v, F (y, x)) , }. 2 2 Then, the conclusion of Theorem 2.1 also holds. Proof. Let {xn }, {yn } be as in the proof of Theorem 2.1. By using (i’) and the construction of {xn }, {yn }, one can conclude max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )} ≤ λw,

(2.2)

where w = max{d(x2n , x2n+1 ), d(y2n , y2n+1 ), d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 ), Noting that

and

d(x2n , x2n+2 ) d(y2n , y2n+2 ) , }. 2 2

d(x2n , x2n+2 ) ≤ max{d(x2n , x2n+1 ), d(x2n+1 , x2n+2 )} 2 d(y2n , y2n+2 ) ≤ max{d(y2n , y2n+1 ), d(y2n+1 , y2n+2 )}, 2

it follows that w = max{d(x2n , x2n+1 ), d(y2n , y2n+1 ), d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )}. We also note that if w = max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )}, then (2.2) yields w = 0, and thus w = max{d(x2n , x2n+1 ), d(y2n , y2n+1 )}. So we conclude w = max{d(x2n , x2n+1 ), d(y2n , y2n+1 )}. 1386

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Then, (2.2) equals to max{d(x2n+1 , x2n+2 ), d(y2n+1 , y2n+2 )} ≤ λ max{d(x2n , x2n+1 ), d(y2n , y2n+1 )}. Similarly, one can also obtain max{d(x2n , x2n+1 ), d(y2n , y2n+1 )} ≤ λ max{d(x2n−1 , x2n ), d(y2n−1 , y2n )}. So we get max{d(xn , xn+1 ), d(yn , yn+1 )} ≤ λ max{d(xn−1 , xn ), d(yn−1 , yn )}. Then, by a standard proof, one can conclude that {xn }, {yn } are Cauchy sequences. Thus, there exist x∗ , y ∗ ∈ X such that xn → x∗ and yn → y ∗ as n → ∞. The remaining proof is similar to that of Theorem 2.1. So we omit the details.

Example 2.3. Let X = {1, 2}, v= {(1, 1), (2, 2)}, d(x, y) = |x−y|, and F = G : X ×X → X defined by F (1, 2) = F (1, 1) = 1, F (2, 1) = F (2, 2) = 2. It is easy to verify that all the assumptions of Theorem 2.1-2.2 are satisfied. So F has a coupled fixed point. In fact, (1, 2) is obviously a coupled fixed point of F .

3

Ordered cone metric space cases

In this section, we suppose that E is a Banach space, P is a convex cone in E with intP 6= ∅, ¹ is the partial ordering induced by P , e ∈ intP , and ξe : E → R is defined by ξe (y) = inf{r ∈ R : y ∈ re − P },

y ∈ E.

First, let us recall some definitions about cone metric space. For more details, we refer the reader to [1–12]. and references therein. Definition 3.1. Let X be a nonempty set and P be a cone in a Banach space E. Suppose that a mapping d : X × X → E satisfies: (i) θ ¹ ρ(x, y) for all x, y ∈ X and ρ(x, y) = θ if and only if x = y ,where θ is the zero element of P ; 1387

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(ii) ρ(x, y) = ρ(y, x) for all x, y ∈ X; (iii) ρ(x, y) ¹ ρ(x, z) + ρ(z, y) for all x, y, z ∈ X. Then ρ is called a cone metric on X and (X, ρ) is called a cone metric space. Definition 3.2. Let (X, ρ) be a cone metric space, and {xn }, {yn } be sequences in X. (i) Let x ∈ X. If ∀c À θ, there exists N ∈ N such that for all n > N , ρ(xn , x) ¿ c, then we call that {xn } converges to x, and we denote it by lim xn = x or xn → x, n→∞

n → ∞. (ii) If ∀c À θ, there exists N ∈ N such that for all n, m > N , ρ(xn , xm ) ¿ c, then {xn } is called a Cauchy sequence in X. (iii) (X, ρ) is called complete if every Cauchy sequence in (X, ρ) is convergent. (iv) A mapping F : X × X → X is called continuous if xn → x and yn → y imply that F (xn , yn ) → F (x, y) as n → ∞. Next, let us recall some properties about the scalarization function ξe . Theorem 3.3. The following statements are true: (a) ξe (·) is positively homogeneous and continuous on E; (b) y, z ∈ E with y ¹ z implies ξe (y) ≤ ξe (z); (c) ξe (y + z) ≤ ξe (y) + ξe (z) for all y, z ∈ E; (d) if (X, ρ) is a complete cone metric space, then (X, ξe ◦ ρ) is a complete metric space; (e) xn → x in (X, ρ) ⇐⇒ xn → x in (X, ξe ◦ ρ), as n → ∞. Proof. (a)-(b) has been prove in [7]. (e) can be seen from the proof of [7, Theorem 2.2]. Now, by using the scalarization function ξe , one can deduce many results on cone metric spaces from our theorems in Section 2. For example, we have the following theorem: Theorem 3.4. Let (X, v, ρ) be an ordered complete cone metric space, i.e., v is a partial order on the set X, and ρ is a complete cone metric on X with the underlying cone P . Suppose that F, G : X × X → X are two weakly increasing mappings w.r.t. v satisfying the following assumptions:

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(H1) there exists λ ∈ [0, 21 ) and z ∈ {ρ(x, u), ρ(y, v), ρ(x, F (x, y)), ρ(y, F (y, x)), ρ(u, G(u, v)), ρ(v, G(v, u)), ρ(x, G(u, v)), ρ(y, G(v, u)), ρ(u, F (x, y)), ρ(v, F (y, x))} such that d(F (x, y), G(u, v)) ¹ λ · z for all comparable (x, y), (u, v) ∈ X × X; (H2) F or G is continuous, or X has the following property: (P) if an nondecreasing sequence {xn } converges to x in X, then xn v x for all n ∈ N. Then F and G has a coupled common fixed point. Proof. Let d = ξe ◦ ρ. By (d) of Theorem 3.3, (X, d) is a complete metric space. Moreover, by (H1) and (a)-(c) of Theorem 3.3, one can show that (i) of Theorem 2.1 holds. In addition, by (H2) and (e) of Theorem 3.3, we know that (ii) of Theorem 2.1 holds. So Theorem 2.1 yields the conclusion. Theorem 3.5. Suppose that all the assumptions of Theorem 3.4 except for (H1) are satisfied, and the following assumption holds: (H1’) there exists λ ∈ [0, 1) and z ∈ {ρ(x, u), ρ(y, v), ρ(x, F (x, y)), ρ(y, F (y, x)), ρ(u, G(u, v)), ρ(v, G(v, u)), ρ(x, G(u, v)) + ρ(u, F (x, y)) ρ(y, G(v, u)) + ρ(v, F (y, x)) , } 2 2 such that d(F (x, y), G(u, v)) ¹ λ · z for all comparable (x, y), (u, v) ∈ X × X. Then F and G has a coupled common fixed point. Proof. By using Theorem 2.2, one can get the conclusion by a similar proof to that of Theorem 3.4.

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References [1] M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008), 416–420. [2] M. Abbas, B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 21 (2008), 511–515. [3] I. Altun, B. Damjanovi´c, D. Djori´c, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. 23 (2010), 310–316. [4] I. Altun, V. Rako˘cevi´c, Ordered cone metric spaces and fixed point results, Comput. Math. Appl. 60 (2010), 1145-1151. [5] H. S. Ding, L. Li, Coupled fixed point theorems in partially ordered cone metric spaces, Filomat 25:2 (2011), 137–149. [6] H. S. Ding, L. Li, S. Radenovi´c, Coupled coincidence point theorems for generalized nonlinear contraction in ordered metric spaces, preprint. [7] W.-S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259–2261. [8] Z. Kadelburg, M. Pavlovi´c, S. Radenovi´c, Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comput. Math. Appl. 59 (2010), 3148–3159. [9] Erdal Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. 59 (2010) 3656–3668. [10] W. Long, B. E. Rhoades, Coupled coincidence points for two mappings in metric spaces and cone metric spaces, preprint. [11] Zoran D. Mitrovi´c, A coupled best approximations theorem in normed spaces, Nonlinear Anal. 72 (2010), 4049–4052. [12] W. Shatanawi, Partially ordered cone metric spaces and coupled fixed point results, Comput. Math. Appl. 60 (2010), 2508–2515.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1391-1402, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

On strictly and semistrictly quasi α−preinvex functions∗ Wan Mei Tang College of Computer and Information Science, Chongqing Normal University, Chongqing 401331, PR China

Abstract In this paper, two new classes of generalized convex functions are introduced, which are called strictly quasi α−preinvex functions and semistrictly quasi α−preinvex functions, respectively. The characterization of quasi α−preinvex functions is established under the condition of lower semicontinuity, or upper semicontinuity or semistrict quasi α−preinvexity. Furthermore, the characterization of semistrictly quasi α−preinvex functions is also obtained under the condition of quasi α−preinvexity or lower semiconitinuity. A similar result can also be obtained for strictly quasi α−preinvex functions. Finally, an important result stating that ‘a local minimum of either a strictly quasi α−preinvex functions or a semistrictly quasi α−preinvex functions over α−invex set is also a global minimum’ is established. Keywords: Convex programming; Quasi α−preinvex functions; Semistrictly quasi α−preinvex functions; Strictly quasi α−preinvex functions; Semicontinuity.

1

Introduction

Convexity and generalized convexity play a central role in mathematical economics, engineering and optimization theory. Therefore, the research on convexity and generalized convexity is one of most important aspects in mathematical programming. In recent years, the concept of convexity has been generalized and extended in several directions using novel and innovative techniques. An important and significant generalization of convexity is the introduction of invexity, preinvexity, semistrictly preinvexity and (semistrictly, strictly) prequasi-invexity, see [1–10] and references therein. Recently, Jeyakumar and Mond in [11, 12] introduced and studied another class of generalized convex functions, which is known as strongly α−invex function. Noor and Noor in [13] introduced a new class of generalized convex functions, which is called the strongly α−preinvex functions, and established the equivalence among the strongly α−preinvex functions, strongly α−invex functions and strongly αη−monotonicity of their differential under some suitable conditions. Fan and Guo in [14] have studied the relationships among (pseudo, quasi) α−preinvexity, (strict, strong, pseudo, quasi) α−invexity and (strict, strong, pseudo, quasi) αη−monotonicity in a systematic way. In this paper, we introduce two new classes of generalized convex functions, which are called strictly quasi α−preinvex functions and semistrictly quasi α−preinvex functions. We establish the relationships between the quasi α−preinvex functions, strictly quasi α−preinvex functions and semistrictly quasi α−preinvex functions under some suitable and appropriate conditions. Finally, we prove that for general mathematical programming problem, when object function are strictly quasi α−preinvex and semistrictly quasi α−preinvex , a local minimum of a strictly quasi α−preinvex and semistrictly quasi α−preinvex functions over an invex set are also a global minimum. The paper is organized as follows. in Section 2, two new concepts concerning strictly quasi α−preinvex functions and semistrictly quasi α−preinvex functions are introduced. In Section 3, the characterization of quasi α−preinvex functions are introduced under the condition of lower semicontinuity or upper semicontinuity or semistrict quasi α−preinvexity. The characterization of strictly quasi α−preinvex functions are introduced in Section 4. Applications of two new types of generalized convex functions are given in Section 5. 0 This work is partially supported by Nanyang Technological University (NTU), Singapore, Temasek Laboratories @ NTU, the National Natural Science Foundation of China (Grant No. 10771228), the Natural Science Foundation Project of CQ CSTC (Grant No. CSTC, 2010BB2090), Education Commission project Research Foundation of Chongqing (Grant No. KJ110617), PR China, and the Program for Innovative Research Team in Higher Educational Institutions of Chongqing, PR China. Corresponding author: Wan Mei Tang E-mail:[email protected](W.M.Tang)

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2

Preliminaries

Let H be a real Hilbert space with inner product h., .i and norm k.k and K be a nonempty subset of H. Let f : K −→ H and α : K × K −→ R \ {0} be two real-valued functions and η(., .) : K × K −→ R be a vector-valued mapping. Firstly, we recall the following well-known results and concepts. Definition 2.1[13] . Let y ∈ K. Then the set K is said to be α-invex at y with respect to η(., .) and α(., .), if, for all x ∈ K, t ∈ [ 0, 1], y + tα(x, y)η(x, y) ∈ K. K is said to be an α-invex set with respect to η and α if K is α-invex at each y ∈ K. The α-invex set K is also called αη-connected set. Note that the convex set with α(x, y) = 1 and η(x, y) = x − y is an invex set, but the converse is not true. From now on, unless otherwise specified, we assume that K is a nonempty α−invex set with respect to η and α. Definition 2.2[13] . The function f on the α-invex set K is said to be α-preinvex with respect to α and η, if f (y + tα(x, y)η(x, y)) ≤ tf (x) + (1 − t)f (y),

∀x, y ∈ K, t ∈ [0, 1].

Remark 2.1[13] . Every convex function is a preinvex function, but the converse is not true. For example, the function f (x) = −|x| is not a convex function, but it is a preinvex function with respect to η and α(x, y) = 1, where ( x − y, if x ≤ 0, y ≤ 0 and x ≥ 0, y ≥ 0, η(x, y) = y − x, otherwise. Definition 2.3[13] . The function f on the α-invex set K is said to be quasi α-preinvex with respect to α and η, if f (y + tα(x, y)η(x, y)) ≤ max{f (x), f (y)},

∀x, y ∈ K, ∀t ∈ [0, 1].

Definition 2.4[15] . The function f on the α-invex set K is said to be strongly quasi α-preinvex with respect to α and η, if there exists a constant β > 0 such that f (y + λα(x, y)η(x, y)) ≤ max{f (x), f (y)} − βλ(1 − λ)kη(x, y)k2 ,

∀x, y ∈ K, ∀λ ∈ [0, 1].

We now introduce two new kinds of generalized convex function termed strictly quasi α−preinvex functions and semistrictly quasi α−preinvex functions as follows. Definition 2.5. The function f on the α-invex set K is said to be strictly quasi α-preinvex with respect to α and η, if for any x, y ∈ K, x 6= y, such that f (y + λα(x, y)η(x, y)) < max{f (x), f (y)},

∀λ ∈ (0, 1).

Definition 2.6. The function f on the α-invex set K is said to be semistrictly quasi α-preinvex with respect to α and η, if for any x, y ∈ K, f (x) 6= f (y), such that f (y + λα(x, y)η(x, y)) < max{f (x), f (y)},

∀λ ∈ (0, 1).

Remark 2.2. It is obvious that strict quasi α−preinvexity implies semistrict quasi α−preinvexity as well as quasi α−preinvexity. However, quasi α−preinvexity does not imply semistrict quasi α−preinvexity, and semistrict quasi α−preinvexity does not imply quasi α−preinvexity. Example 2.1. This example illustrates that a quasi α−preinvex function is not a semistrictly quasi α−preinvex ( −x, if x > 0, function. Let f (x) = and η(x, y) = x − y, and 0, if x ≤ 0,  1,     1, α(x, y) =  −1,    −1,

if x ≥ 0, y ≥ 0, if x ≤ 0, y ≤ 0, if x ≤ 0, y ≥ 0, if x ≥ 0, y ≤ 0.

Then, it is easy to verify that f is a quasi α−preinvex function with respect to α and η. However, let y = −1, x = 1, λ = 21 , we have f (y) = f (−1) = 0 > −1 = f (1) = f (x). That is f (y) 6= f (x). And f (y + λα(x, y)η(x, y))

= f ((−1) + (1/2)α(1, −1)η(1, −1)) = f (−2) = 0 = max{f (1), f (−1)} = 0. 1392

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This shows that f is not a semistrictly quasi α−preinvex function for the same α and η. Example 2.2. This example illustrates that a semistrictly quasi α−preinvex function is not a quasi α−preinvex ( −|x|, if |x| ≤ 1, and function. Let f (x) = −1, if |x| ≥ 1,

η(x, y) =

                            

x − y, x − y, x − y, x − y, −1, y − x, y − x, y − x,

if if if if if if if if

x ≥ 0, y ≥ 0, x ≤ 0, y ≤ 0, x > 1, y < −1, x < −1, y > 1, − 1 ≤ x ≤ 0, y ≥ 0, x ≥ 0, −1 ≤ y ≤ 0, 0 ≤ x ≤ 1, y ≤ 0, x ≤ 0, 0 ≤ y ≤ 1.

α(x, y) =

                            

1, 1, 1, 1, x − y, 1, 1, 1,

if x ≥ 0, y ≥ 0, if x ≤ 0, y ≤ 0, if x > 1, y < −1, if x < −1, y > 1, if − 1 ≤ x ≤ 0, y ≥ 0, if x ≥ 0, −1 ≤ y ≤ 0, if 0 ≤ x ≤ 1, y ≤ 0, if x ≤ 0, 0 ≤ y ≤ 1.

Then, it is easy to verify that f is a semistrictly quasi α−preinvex function with respect to α and η. However, let x = 2, y = −2, λ = 21 . Since f (y + λα(x, y)η(x, y))

= f (−2 + 12 α(2, −2)η(2, −2)) = f (0) = 0 > −1 = f (2) = f (−2) = max{f (x), f (y)},

f is not a quasi α−preinvex function for the same α and η. Remark 2.3. Example 2.2 also shows that a semistrictly quasi α−preinvex function is not necessarily a semistrictly prequasi-invex function. Definitions 2.3 to 2.6, with α(x, y) ≡ 1, reduce to those of perquasi-invex, strongly perquasi-invex, strictly prequasi-invex, semistrictly prequasi-invex functions. See references [6, 7, 9] for details. Example 2.3. This ( example illustrates that a quasi α−preinvex function is not a strongly quasi α−preinvex −|x|, if |x| ≤ 1, function. Let f (x) = and −1, if |x| ≥ 1,  x − y,     x − y, η(x, y) =  y − 1,    1 + y,

if if if if

x ≥ 0, y ≥ 0, x ≤ 0, y ≤ 0, x ≤ 0, y ≥ 0, x ≥ 0, y ≤ 0.

 1,     1, α(x, y) =  −1,    −1,

if x ≥ 0, y ≥ 0, if x ≤ 0, y ≤ 0, if x ≤ 0, y ≥ 0, if x ≥ 0, y ≤ 0.

Then, it is easy to verify that f is a quasi α−preinvex function with respect to α and η. However, for any β > 0, if we let x = 1, y = 2, λ = 21 , we get f (y + λα(x, y)η(x, y))

= f (2 + 12 α(1, 2)η(1, 2) = −1 > max{f (1), f (2)} − 21 (1 − 12 )βk(1 − 2)k2 = −1 − 14 β.

Thus, f is not a strongly quasi α−preinvex function for the same α and η. Example 2.4. This example illustrates that a strictly quasi α−preinvex function is not a strongly quasi α−preinvex function. Let f (x) = −|x|, and η(x, y) = x − y, and  1,     1, α(x, y) =  −1,    −1,

if x ≥ 0, y ≥ 0, if x ≤ 0, y ≤ 0, if x ≤ 0, y ≥ 0, if x ≥ 0, y ≤ 0.

Then, it is easy to verify that f is a strictly quasi α−preinvex function with respect to α and η. However, for any β > 0, if we let x = β5 , y = β1 , λ = 21 , we get f (y + λα(x, y)η(x, y))

= f ( β1 + 12 · 1 · ( β5 − β1 )) = − β3 > max{f ( β5 ), f ( β1 )} − 21 (1 − 12 )β( β5 − β1 )2 = − β5 .

Thus, f is not a strongly quasi α−preinvex function for the same α and η. Example 2.5. This example illustrates that a semistrictly quasi α−preinvex function is not a strongly quasi 1393

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( α−preinvex function. Let f (x) =

α(x, y) =

                            

1, 1, 1, 1, −1, −1, −1, −1,

if if if if if if if if

−|x|, −1,

if |x| ≤ 1, and η(x, y) = x − y, and if |x| ≥ 1,

x ≥ 0, y ≥ 0, x ≤ 0, y ≤ 0, x > 1, y < −1, x < −1, y > 1, − 1 ≤ x ≤ 0, y ≥ 0, x ≥ 0, −1 ≤ y ≤ 0, 0 ≤ x ≤ 1, y ≤ 0, x ≤ 0, 0 ≤ y ≤ 1.

Then, it is easy to verify that f is a semistrictly quasi α−preinvex function with respect to α and η. However, for any λ > 0, if we let x = 2, y = −2, λ = 12 , we get f (y + λα(x, y)η(x, y))

= f ((−2) + 12 α(2, −2)η(2, −2)) = f (0) = 0 > max{f (2), f (−2)} − 14 β(2 + 2)2 = −1 − 4β.

Thus, f is not a strongly quasi α−preinvex function for the same α and η. Remark 2.4. From Example 2.4 and 2.5, we know that strongly quasi α−preinvex functions are different from strictly quasi α−preinvex functions and semistrictly quasi α−preinvex functions and quasi α−preinvex functions. We also need the following assumptions introduced in [13]. Condition A f (y + α(x, y)η(x, y)) ≤ f (x),

∀x, y ∈ K.

which plays an important part in studying the properties of the α−preinvex (α−invex) functions. For α(x, y) = 1, Condition A reduces to the following for preinvex functions. Condition B f (y + η(x, y)) ≤ f (x),

∀x, y ∈ K.

For the applications of Condition B see references [9, 16]. Condition C Let η(., .) : K × K −→ R and α(., .) : K × K −→ R\0 satisfy the assumptions η(y, y + λα(x, y)η(x, y)) η(x, y + λα(x, y)η(x, y))

3

= −λη(x, y), = (1 − λ)η(x, y),

∀x, y ∈ K, λ ∈ [ 0, 1].

Characterizations of quasi α−preinvx functions

First of all, we give two important lemmas. Lemma 3.1[15] . Let K be an α-invex set with respect to α and η, for any x, y ∈ K, λ ∈ [0, 1], if α and η satisfy the assumptions η(y, y + λα(x, y)η(x, y)) = −λη(x, y), α(x, y) = α(y, y + λα(x, y)η(x, y)), then ∀λ1 , λ2 ∈ [0, 1] and λ2 < λ1 , the following equalities hold (i) η(y + λ1 α(x, y)η(x, y), y + λ2 α(x, y)η(x, y)) = (λ1 − λ2 η(x, y), (ii) α(x, y) = α(y + λ1 α(x, y)η(x, y), y + λ2 α(x, y)η(x, y)). Lemma 3.2. Let K be an α-invex set with respect to α and η, and Condition A and C hold. Assume that the following conditions are satisfied: (i) there exists a θ ∈ (0, 1) such that, for all x, y ∈ K, f (y + θα(x, y)η(x, y)) ≤ max{f (x), f (y)}

(3.1)

(ii) for any x, y ∈ K, λ ∈ K[0, 1], η(y, y + λα(x, y)η(x, y)) = −λη(x, y), α(x, y) = α(y, y + λα(x, y)η(x, y)). 1394

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Then the set defined by A = {λ ∈ [0, 1]|f (y + λα(x, y)η(x, y)) ≤ max{f (x), f (y)}, ∀x, y ∈ K} is dense in the interval [ 0, 1]. Proof. By contradiction. Suppose that A is not dense in [ 0, 1]. Then, there exists a λ0 ∈ (0, 1) and a neighborhood N (λ0 ) of λ0 such that N (λ0 ) ∩ A = ∅.

(3.2)

From Condition A and (3.1), we have {λ ∈ A|λ ≥ λ0 } 6= ∅ {λ ∈ A|λ ≤ λ0 } 6= ∅. Define λ1 = inf{λ ∈ A|λ ≥ λ0 } λ2 = sup{λ ∈ A|λ ≤ λ0 }

(3.3) (3.4)

Then, by (3.2), we have 0 ≤ λ2 < λ1 ≤ 1. Since {θ, (1 − θ)} ∈ (0, 1), we can choose u1 , u2 ∈ A satisfying u1 ≥ λ1 , u2 ≤ λ2 such that max{θ, (1 − θ)}(u1 − u2 ) < λ1 − λ2 .

(3.5)

Next, let us consider λ = θu1 + (1 − θ)u2 . From λ2 < λ1 and Lemma 3.1, for any x, y ∈ K, we have y + λα(x, y)η(x, y) = y + (θu1 + (1 − θ)u2 )α(x, y)η(x, y) = y + u2 α(x, y)η(x, y) + θα(x, y) · (u1 − u2 )η(x, y) = y + u2 α(x, y)η(x, y) +θα(y + u1 α(x, y)η(x, y), y + u2 α(x, y)η(x, y))η(y + u1 α(x, y)η(x, y), y + u2 α(x, y)η(x, y)). Hence, from (3.1) and the fact that u1 , u2 ∈ A, we get

= ≤ ≤ =

f (y + λα(x, y)η(x, y)) f (y + u2 α(x, y)η(x, y) +θα(y + u1 α(x, y)η(x, y), y + u2 α(x, y)η(x, y))η(y + u1 α(x, y)η(x, y), y + u2 α(x, y)η(x, y))) max{f (y + u1 α(x, y)η(x, y)), f (y + u2 α(x, y)η(x, y))} max{max{f (x), f (y)}, max{f (x), f (y)}} max{f (x), f (y)}.

That is, λ ∈ A. If λ ≥ λ0 , then it follows from (3.5) that λ − u2 = θ(u1 − u2 ) < λ1 − λ2 , and therefore λ < λ1 . Because of λ ≥ λ0 and λ ∈ A this is a contradiction to (3.3). Similarly, λ ≤ λ0 provides a contradiction to (3.4). Hence, A is dense in [ 0, 1]. Theorem 3.1. Let K be an α-invex set with respect to α and η. If the following assumptions hold: (i) Condition A and C are satisfied; (ii) for any x, y ∈ K, θ ∈ [0, 1], α(x, y) = α(x, y + θα(x, y)η(x, y)) = α(y, y + θα(x, y)η(x, y)); (iii) f is an upper semicontinuous function; then f is quasi α−preinvex function on K if and only if exists a θ ∈ (0, 1), such that, for all x, y ∈ K f (y + θα(x, y)η(x, y)) ≤ max{f (x), f (y)}. Proof. The necessity is obvious from Definition of quasi α−preinvex functions. We only prove the sufficiency. Suppose that f is not quasi α−preinvex functions on K. Then, there exist x, y ∈ K and λ ∈ (0, 1) such that f (y + λα(x, y)η(x, y)) > max{f (x), f (y)}.

(3.6) 1395

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Let z = y + λα(x, y)η(x, y), A = {λ ∈ [0, 1]|f (y + λα(x, y)η(x, y)) ≤ max{f (x), f (y)}, ∀x, y ∈ K}. From Lemma 3.2, there exists a {λn } ⊂ A, λn < λ such that λn → λ, n → ∞. λn Define yn = z − 1−λ α(x, z)η(x, z). From Condition C and (ii), we have yn = y + n yn → y,

λ−λn 1−λn α(x, y)η(x, y).

Then,

n → ∞.

Since K is an α−invex set, it follows that, for sufficiently large n, yn ∈ K. Again from Condition C and (ii), we get yn + λn α(x, yn )η(x, yn ) λ − λn α(x, y)η(x, y) 1 − λn λ − λn λ − λn + λn α(x, y + α(x, y)η(x, y))η(x, y + α(x, y)η(x, y)) 1 − λn 1 − λn λ − λn 1−λ =y + α(x, y)η(x, y) + λn · α(x, y)η(x, y) 1 − λn 1 − λn =y +

(3.7)

=y + λα(x, y)η(x, y) =z. By the upper semicontinuity of f on K, for any ε > 0, there exists an N > 0 such that f (yn ) ≤ f (y) + ε,

f or

n > N.

Therefore, from (3.7) and λn ∈ A, we have f (z)

= f (yn + λn α(x, yn )η(x, yn )) ≤ max{f (x), f (yn )} ≤ max{f (x), f (y) + ε}, f or

n > N.

Since ε > 0 is arbitrarily small, we have f (z) ≤ max{f (x), f (y)}, which contradicts the inequality (3.6). Thus, f is a quasi α−preinvex function for same α and η on K. Remark 3.1. By [15, example 3.1], there exist α and η that satisfy both Condition C and the equality α(x, y) = α(x, y + θα(x, y)η(x, y)) = α(y, y + θα(x, y)η(x, y)). For example, when α(x, y) ≡ 1, the Condition C above is exectly the same as Condition C in [5]. Theorem 3.2. Let K be an α-invex set with respect to η and α. If the following assumptions hold: (i) Condition A and C are satisfied; (ii) for any x, y ∈ K, θ ∈ [0, 1], α(x, y) = α(x, y + θα(x, y)η(x, y)) = α(y, y + θα(x, y)η(x, y)); (iii) f is lower semicontinuous functions; then f is quasi α−preinvex functions on K if and only if for any x, y ∈ K, there exists a θ ∈ (0, 1) such that f (y + θα(x, y)η(x, y)) ≤ max{f (x), f (y)}. Proof. The necessity is obvious from Definition of quasi α−preinvex functions. We only prove the sufficiency. By contradiction, we assume that there exist distinct x, y ∈ K and θ ∈ (0, 1) such that f (y + θα(x, y)η(x, y)) > max{f (x), f (y)}. Let z xt

= y + θα(x, y)η(x, y), = z + tα(x, z)η(x, z).

From Condition C and (ii), we have xt

= y + θα(x, y)η(x, y) + tα(x, y + θα(x, y)η(x, y))η(x, y + θα(x, y)η(x, y)) = y + θα(x, y)η(x, y) + tα(x, y) · (1 − θ)η(x, y) = y + [θ + t(1 − θ)]α(x, y)η(x, y). 1396

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Let B = {xt ∈ K|t ∈ (0, 1], f (xt ) ≤ max{f (x), f (y)}}, u = inf {t ∈ (0, 1]|xt ∈ B}. It is obvious that x1 ∈ B from Condition A, but x0 ∈ / B. Thus, xt ∈ / B, 0 ≤ t < u, and there exist tn ≥ u, xtn ∈ B (from Lemma 3.2), such that tn → u,

u → ∞.

Since f is a lower semicontinuous function, we have f (xu ) ≤ lim inf f (xtn ) ≤ max{f (x), f (y)}. n→∞

Hence, xu ∈ B. Similarly, let yt = z + (1 − t)α(y, z)η(y, z). From Condition C and (ii), we have yt = y + tθα(x, y)η(x, y). Let D v

= {yt ∈ K|t ∈ [0, 1), f (yt ) = f (y + tθα(x, y)η(x, y)) ≤ max{f (x), f (y)}}, = sup{t ∈ [0, 1)|yt ∈ D}.

It is obvious that y0 y1 yt

= y ∈ D, = y + θα(y, z)η(y, z) = z ∈ / D, ∈ / D, v < t ≤ 1,

and there exist tn ≤ v, ytn ∈ D (from Lemma 3.2), such that tn → v,

n → ∞.

Since f is a lower semicontinuous function, we have f (yn ) ≤ lim inf f (ytn ) ≤ max{f (x), f (y)}. n→∞

Hence, yv ∈ D. Let θ1 θ2

= vθ, = θ + u − uθ.

Then, 0 ≤ θ1 < θ < θ2 ≤ 1. Now, from Condition C and (ii), we have xu + λα(yv , xu )η(yv , xu ) =y + θ2 α(x, y)η(x, y) + λα(y + θ1 α(x, y)η(x, y), y + θ2 α(x, y)η(x, y)) · η(y + θ1 α(x, y)η(x, y), y + θ2 α(x, y)η(x, y)) =y + θ2 α(x, y)η(x, y) + λα(x, y) · (θ1 − θ2 )η(x, y) =y + [λθ1 + (1 − λ)θ2 ]α(x, y)η(x, y),

∀λ ∈ [0, 1].

Hence, from the definitions of θ1 and θ2 , we have f (xu + λα(yv , xu )η(yv , xu ))

= f {y + [λθ1 + (1 − λ)θ2 ]α(x, y)η(x, y)} > max{f (x), f (y)} ≥ max{f (yv ), f (xu )}, ∀λ ∈ (0, 1),

contradicting the assumptions of the theorem. Theorem 3.3. Let K be an α−invex set with respect to α and η. If the following assumptions hold: (i) Condition C is satisfied; (ii) for any x, y ∈ K, θ ∈ [0, 1], α(x, y) = α(x, y + θα(x, y)η(x, y)) = α(y, y + θα(x, y)η(x, y)); 1397

Tang: strictly and semistrictly quasi α−preinvex functions

(iii) f is a semistrictly α−preinvex functions; Then, f is a quasi α−preinvex function on K if and only if the following condition is satified: there exists a θ ∈ (0, 1) such that, for all x, y ∈ K, f (y + θα(x, y)η(x, y)) ≤ max{f (x), f (y)}.

(3.8)

Proof. The necessity is obvious from Definition of quasi α−preinvex functions. We prove the sufficiency. Suppose that there exist x, y ∈ K and λ ∈ (0, 1) such that f (y + λα(x, y)η(x, y)) > max{f (x), f (y)}. Without loss of generality, assume that f (x) ≥ f (y) and let z = y + λα(x, y)η(x, y). Then, f (z) > f (x).

(3.9)

If f (x) > f (y), it follows from the semistrict quasi α− preinvexity of f that f (z) < f (x), contradicting (3.9). If f (x) = f (y), then (3.9) implies that f (z) > f (x) = f (y).

(3.10)

There are two cases to be considered. Case 1 0 < λ < θ < 1. Let z1 = y + λθ α(x, y)η(x, y). Thus, from Condition C and (ii), we have y + θα(z1 , y)η(z1 , y) = y + θα(y + λθ α(x, y)η(x, y), y)η(y + λθ α(x, y)η(x, y), y) = y + θα(x, y)η(y + λθ α(x, y)η(x, y), y + λθ α(x, y)η(x, y) − λθ α(x, y)η(x, y)) = y + θα(x, y)η(y + λθ α(x, y)η(x, y), y + λθ α(x, y)η(x, y) +α(y, y + λθ α(x, y)η(x, y))η(y, y + λθ α(x, y)η(x, y))) = y − θη(y, y + λθ α(x, y)η(x, y)) = y + θα(x, y)η(x, y) = z. According to (3.8), we have f (z) ≤ max{f (z1 ), f (y)}. From (3.10) and the above inequality, it follows that f (z) ≤ f (z1 ). Let b = have

λ(1−θ) θ(1−λ) .

= = = = =

(3.11)

Since 0 < λ < θ < 1, it is easy to show that 0 < b < 1. Thus, from Condition C and (ii), we

z + bα(x, z)η(x, z) y + λα(x, y)η(x, y) + bα(x, y + λα(x, y)η(x, y))η(x, y + λα(x, y)η(x, y)) y + [λ + b(1 − λ)]α(x, y)η(x, y) y + [λ + λ · (1−θ) θ ]α(x, y)η(x, y) y + λθ α(x, y)η(x, y) z1 .

Since f is a semistrictly quasi α−preinvex function, it follows from inequality (3.10) and the above equality that f (z1 ) < max{f (x), f (z)} = f (z), contradicting (3.11). Case 2 0 < θ < λ < 1. In this case, we still get a contradiction by just exchanging the roles of θ and 1 − θ and the roles of λ and λ − θ in Case 1. Theorem 3.4. Let K be an α-invex set with respect to α and η. If the following assumptions hold: (i) Condition A and C are statisfied; (ii) for any x, y ∈ K, θ ∈ [0, 1], α(x, y) = α(x, y + θα(x, y)η(x, y)) = α(y, y + θα(x, y)η(x, y)); (iii) f is lower semicontinuous functions and if there exists a θ ∈ (0, 1) such that, for every x, y ∈ K, f (x) 6= f (y) implies f (y + (1 − θ)α(x, y)η(x, y)) < max{f (x), f (y)}, ) 1398

(3.12)

Tang: strictly and semistrictly quasi α−preinvex functions

then f is a quasi α−preinvex function for same η and α on K. Proof. By Theorem 3.2, we need only to show that, for each x, y ∈ K, there exists a λ ∈ (0, 1) such that f (y + λα(x, y)η(x, y)) ≤ max{f (x), f (y)},

(3.13)

By contradiction, we assume that there exist x, y ∈ K such that f (y + λα(x, y)η(x, y)) > max{f (x), f (y)},

∀λ ∈ (0, 1).

(3.14)

If f (x) 6= f (y), it follows from (3.12) that f (y + (1 − θ)α(x, y)η(x, y)) < max{f (x), f (y)}, which contradicts (3.14). If f (x) = f (y), then (3.14) implies f (y + λα(x, y)η(x, y)) > f (x) = f (y),

∀λ ∈ (0, 1).

(3.15)

By (3.15), we obtain f (y + λα(x, y)η(x, y) + (1 − θ)α(x, y + λα(x, y)η(x, y))η(x, y + λα(x, y)η(x, y))) = f (y + λα(x, y)η(x, y) + (1 − θ)α(x, y) · (1 − λ)η(x, y)) = f (y + [λ + (1 − θ)(1 − λ)]α(x, y)η(x, y) > f (y), ∀λ ∈ (0, 1).

(3.16)

And, from (3.12) and (3.15), we have f [y + λα(x, y)η(x, y) + (1 − θ)α(x, y + λα(x, y)η(x, y))η(x, y + λα(x, y)η(x, y))] < max{f (x), f (y + λα(x, y)η(x, y))} = f (y + λα(x, y)η(x, y)), ∀λ ∈ (0, 1).

(3.17)

Again by (3.12),(3.16), (3.17), we have

= = < =
θ, then ν = (λ − θ)/(1 − θ) ∈ (0, 1). From Condition C and (ii), we have x + να(x, x)η(x, x) = y + λα(x, y)η(x, y). Since f is semistrictly quasi α−preinvex function on K and (4.7) holds, we have f (y + λα(x, y)η(x, y))

= f (x + να(x, x)η(x, x)) < max{f (x), f (x)} = f (x).

This completes the proof.

5

Applications of Strictly and Semistrictly quasi α−preinvex Functions

Let the problem of minimizing f (x) subject to x ∈ K be denoted by (P ). The following two theorems show that a local minimum of a strictly quasi α−preinvex functions and semistrictly quasi α−preinvex functions over an α−invex set are also a global minimum. Theorem 5.1. Let K be a nonempty α−invex set with respect to α and η, and f be a strictly quasi α−preinvex for the same α and η on K. If x ∈ K is a local minimum to the problem (P ), then x is a global minimum. Proof. Assume that x ∈ K is a local minimum to the problem (P ). Then there exists an ε-neighborhood Nε (x) ⊂ K around x such that f (x) ≤ f (x),

∀x ∈ K ∩ Nε (x).

(5.1)

Suppose that x is not a global minimum of (P ), then there exists a x∗ ∈ K such that f (x∗ ) < f (x). Since K is a nonempty α−invex set with respect to α and η, and f is strictly quasi α−preinvex function, for any λ ∈ (0, 1), x + λα(x∗ , x)η(x∗ , x) ∈ K, we have f (x + λα(x∗ , x)η(x∗ , x)) < max{f (x∗ ), f (x)} < f (x) i.e., for any λ ∈ (0, 1), we have f (x + λα(x∗ , x)η(x∗ , x)) < f (x). Thus, for a sufficiently small λ > 0, we have x + λα(x∗ , x)η(x∗ , x) ∈ K ∩ Nε (x), which is a contradiction to (5.1). This completes the proof. Theorem 5.2. Let K be a nonempty α−invex set with respect to α and η, and f be a semistrictly quasi α−preinvex for the same α and η on K. If x ∈ K is a local minimum to the problem (P ), then x is a global minimum. Proof. Assume that x ∈ K is a local minimum to the problem (P ). Then there exists an ε-neighborhood Nε (x) ⊂ K around x such that f (x) ≤ f (x),

∀x ∈ K ∩ Nε (x).

(5.2)

Suppose that x is not a global minimum of (P ), then there exists an x∗ ∈ K such that f (x∗ ) < f (x). 1401

Tang: strictly and semistrictly quasi α−preinvex functions

Since K is a nonempty α−invex set with respect to η and α, and f is semistrictly quasi α−preinvex function, for any λ ∈ (0, 1), x + λα(x∗ , x)η(x∗ , x) ∈ K, we have f (x + λα(x∗ , x)η(x∗ , x)) < max{f (x∗ ), f (x)} < f (x) i.e., for any λ ∈ (0, 1), we have f (x + λα(x∗ , x)η(x∗ , x)) < f (x). Thus, for a sufficiently small λ > 0, we have x + λα(x∗ , x)η(x∗ , x) ∈ K ∩ Nε (x), which is a contradiction to (5.2). This completes the proof. Remark 5.1. Theorem 5.1 and 5.2 illustrat that strictly quasi α−preinvex functions and semistrictly quasi α−preinvex functions are very important in mathematical programming.

References [1] M. A.Hanson, On sufficiency of the Kuhn–Tucker conditions. Journal of Mathematical Analysis and Applications 1981; 80; 545–550. [2] A. Ben-Israel, B. Mond, What is invexity?. Journal of the Australian Mathematical Society, Series B 1986; 28; 1–9. [3] T. Weir, B. Mond, Preinvex functions in multiple-objective optimization. Journal of Mathematical Analysis and Applications 1988; 136; 29–38. [4] T. Weir, V. Jeyakumar, A class of nonconvex functions and mathematical programming. Bulletin of the Australian Mathematical Society 1988; 38; 177–189. [5] S. R. Mohan, S. K. Neogy, On Invex Sets and Preinvex Functions. Journal of Mathematical Analysis and Applications 1995; 189; 901–908. [6] X. M. Yang, D.Li, Semistrictly preinvex functions. Journal of Mathematical Analysis and Applications 2001; 258(1); 287–308. [7] X. M. Yang, D. Li, On properties of preinvex functions. Journal of Mathematical Analysis and Applications 2001; 256(1); 229–241. [8] X M. Yang, Semistrictly convex functions. Opsearch 1994; 31(1); 15–27. [9] X. M. Yang, X. Q. Yang, K. L. Teo, Characterizations and Applications of Prequasi-Invex Functions. Journal of Optimization Theory and Applications 2001; 110(3); 647–668. [10] W.M. Tang, On Properties of Strongly Prequasi-invex Functions, Journal of Computational Analysis and Applications, vol. 13, N0. 7, 2011; 1297–1308. [11] V. Jeyakumar, B. Mond, On generalized convex mathematical programming. Journal of the Australian Mathematical Society, Series B 1992; 34; 43–53. [12] V. Jeyakumar, Strong and weak invexity in mathematical programming. Methods Operational Research 1985; 55; 109–125. [13] M. A. Noor, K. I. Noor, Some characterizations of strongly preinvex functions. Journal of Mathematical Analysis and Applications 2006; 316; 697–706. [14] L. Y. Fan, Y. L. Guo, On strongly α−preinvex functions. Journal of Mathematical Analysis and Applications 2007; 330; 1412-1425. [15] C. P. Liu, Some characterizations and applications on strongly α−preinvex and strongly α−invex functions. Journal of industrial and management optimization 2008; 4; 727–738. [16] X. M. Yang, X. Q. Yang, K. L. Teo, Criteria for generalized invex monotonicities. European Journal of Operational Research. 2005; 164; 115C119.

1402

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1403-1412, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

ON STABILITY OF FUNCTIONAL INEQUALITIES AT RANDOM LATTICE ϕ-NORMED SPACES SUNG JIN LEE AND REZA SAADATI∗

Abstract. We establish some stability results concerning the following functional inequalities k f (x) + f (y) + f (z) k≤ kf (x + y + z)k and

 

x+y

2f k f (x) + f (y) + 2f (z) k≤ + z

2 in the setting of latticetic random ϕ-normed spaces.

1. Introduction and preliminaries Let L = (L, ≥L ) be a complete lattice, i.e., a partially ordered set in which every nonempty subset admits supremum and infimum, and 0L = inf L, 1L = sup L. The space of latticetic random distribution functions, denoted by ∆+ L , is defined as the set of all left continuous non-decreasing mappings F : R ∪ {−∞, +∞} → L with F (0) = 0L , F (+∞) = 1L . + + + − − DL ⊆ ∆+ L is defined as DL = {F ∈ ∆L : l F (+∞) = 1L }, where l f (x) denotes the left limit of

the function f at the point x. The space ∆+ L is partially ordered by the usual point-wise ordering of functions, i.e., F ≥ G if and only if F (t) ≥L G(t) for all t in R. The maximal element for ∆+ L in this order is the distribution function given by   0L , if t ≤ 0, ε0 (t) =  1 , if t > 0. L The concept of Menger probabilistic ϕ-normed space was introduced by Golet¸ in [1]. Let ϕ be a function defined on the real field R into itself, with the following properties: (a) ϕ(−t) = ϕ(t) for every t ∈ R; (b) ϕ(1) = 1; (c) ϕ is strictly increasing and continuous on [0, ∞), ϕ(0) = 0 and limα→∞ ϕ(α) = ∞; (d) ϕ(st) = ϕ(s)ϕ(t) for every t, s > 0. An example of such functions is: ϕ(t) = |t|p , p ∈ (0, ∞) (see [2, Theorem 1.49]). 2010 Mathematics Subject Classification. Primary 39B82. Key words and phrases. Stability, random Banach space, random ϕ-normed space, lattice, additive functional inequality, additive functional equation, generalized Hyers–Ulam–Rassias stability. ♣ This work was supported by Daejin University Rearch Grants in 2012. ∗ The

corresponding author: [email protected] (Reza Saadati).

Tel :+98 121 2263650, Fax: +98 121 2263650. 1

1403

2

LEE AND SAADATI

Definition 1.1. A latticetic random ϕ-normed space is a triple (X, µ, ∧), where X is a vector space and + µ is a mapping from X into DL (for x ∈ X, the function µ(x) is denoted by µx , and µx (t) is the value

µx at t ∈ R) such that the following conditions hold: (LRN1) µx (t) = ε0 (t) for all t > 0 if and only if x = 0;   t for all x in X, α 6= 0 and t ≥ 0; (LRN2) µαx (t) = µx ϕ(α) (LRN3) µx+y (t + s) ≥L ∧(µx (t), µy (s)) for all x, y ∈ X and t, s ≥ 0. We note that from (LPN2) it follows µ−x (t) = µx (t)

(x ∈ X, t ≥ 0).

It is also worth noting that latticetic random ϕ-normed spaces include, in a natural way, p-normed spaces ([1, 3]). Example 1.2. Let L = [0, 1] × [0, 1] and operation ≥L be defined by: L = {(a1 , a2 ) : (a1 , a2 ) ∈ [0, 1] × [0, 1] and a1 + a2 ≤ 1}, (b1 , b2 ) ≥L (a1 , a2 ) ⇐⇒ a1 ≤ b1 , a2 ≥ b2 ,

∀a = (a1 , a2 ), b = (b1 , b2 ) ∈ L.

Then (L, ≥L ) is a complete lattice (see [4]). In this complete lattice, we denote its units by 0L = (0, 1) and 1L = (1, 0). Let (X, k · k) be a normed space. Let µ be a mapping defined by  kxkp  t , µx (t) = , ∀t ∈ R+ , 0 < p ≤ 1. p t + kxk t + kxkp Then (X, µ, ∧) is a latticetic random ϕ-normed spaces. Note that, here, ϕ(α) = αp . Definition 1.3. Let (X, µ, ∧) be a latticetic random ϕ-normed spaces. (1) A sequence (xn ) in X is said to be convergent to x in X if, for every 0 < t ∈ R the sequence (µxn −x (t)) is order convergent to 1L . (2) A sequence (xn ) in X is called Cauchy sequence if, for every 0 < t ∈ R the sequence (µxn −xm (t)) is order convergent to 1L whenever n, m tend to ∞. (3) A latticetic random ϕ-normed spaces (X, µ, ∧) is said to be complete if and only if every Cauchy sequence in X is order convergent to a point in X. Theorem 1.4. If (X, µ, ∧) is a latticetic random ϕ-normed space and {xn } is a sequence such that xn → x, then limn→∞ µxn (t) = µx (t). Proof. The proof is the same as classical random normed spaces, see [5].



Lemma 1.5. Let (X, µ, ∧) be a latticetic random ϕ-normed space and x ∈ X. If µx (t) = C, for all t > 0, then C = 1L and x = 0. + Proof. Let µx (t) = C for all t > 0. Since Ran(µ) ⊆ DL , we have C = 1L and by (LRN1) we conclude

that x = 0.

 1404

ON STABILITY OF FUNCTIONAL INEQUALITIES

3

The generalized Hyers-Ulam–Rassias stability of the functional inequality (1.1) has been proved by Fechner [6] and Gil´ anyi [7]. Gil´ anyi [8] showed that if f satisfies the functional inequality k 2f (x) + 2f (y) − f (x − y) k≤k f (x + y) k

(1.1)

then f also satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (x − y) + f (x + y), see also [9]. Park, Cho and Han [10] investigated the Cauchy additive functional inequality k f (x) + f (y) + f (z) k≤k f (x + y + z) k

(1.2)

and the Cauchy-Jensen additive functional inequality  k f (x) + f (y) + 2f (z) k≤k 2f

(1.3)

x+y +z 2

 k

and proved the generalized Hyers-Ulam–Rassias stability of the functional inequalities (1.2) and (1.3) in Banach spaces. We also mention here the paper [11]. The stability of the Cauchy additive functional equation in the settings of fuzzy, probabilistic and random normed spaces and random ϕ-normed spaces has been recently investigated by Mirmostafaee, Mirzavaziri and Moslehian [12, 13], Alsina [14], Mihet¸ [15], Mihet¸ and Radu [16] and Mihet¸, Saadati and Vaezpour [3, 17, 18]. The aims of this paper are a synthesis of these two theories, probabilistic normed space [5] and vectorlattice-normed space [19, 20] respectively, named by latticetic random ϕ-normed spaces and to prove the generalized Hyers-Ulam–Rassias stability of the functional inequalities (1.2) and (1.3) in these spaces. For more details on this preliminary part, the reader is referred to [21], [22], [23], [24], [25], [26], [27]. 2. Main results We start our work with the main result in a latticetic random ϕ-normed space. Lemma 2.1. Let X be a linear space, (Z, µ, ∧) be a latticetic random ϕ-normed space and f : X −→ Z be a function such that  (2.1)

µf (x)+f (y)+f (z) (t) ≥L µf (x+y+z)

t ϕ(2)

 (x, y, z ∈ X, t > 0).

Then f is Cauchy additive, i.e., f (x + y) = f (x) + f (y) for all x, y ∈ X. Proof. Putting x = y = z = 0 in (2.1), we obtain     t t ≥L µf (0) µ3f (0) (t) ≥L µf (0) ϕ(3) ϕ(2)

(t > 0).

By Lemma 1.5, it follows that f (0) = 0. Putting y = −x and z = 0 in (2.1), one obtains     t t µf (x)+f (−x) (t) ≥L µf (0) = µ0 = 1L (t > 0), ϕ(2) ϕ(2) hence f (x) = −f (−x)

(x ∈ X). 1405

4

LEE AND SAADATI

Putting z = −x − y in (2.1) we deduce that µf (x)+f (y)−f (x+y) (t)

=

µf (x)+f (y)+f (−x−y) (t)     t t µf (0) = µ0 = 1L . ϕ(2) ϕ(2)

≥L and thus, from (LRN1),

f (x) + f (y) = f (x + y), ∀x, y ∈ X.  Similarly one can prove the following Lemma 2.2. Let X be a linear space, (Z, µ, ∧) be a latticetic random ϕ-normed space and f : X −→ Z be a function such that  µf (x)+f (y)+2f (z) (t) ≥L µ2f ( x+y +z)

(2.2)

2

ϕ(2)t ϕ(3)

 (x, y, z ∈ X, t > 0).

Then f is Cauchy additive. + Theorem 2.3. Let X be a linear space, Φ be a mapping from X 3 to DL ( Φ(x, y, z)(t) is denoted by

Φx,y,z (t)), such that for some 0 < α < ϕ(2), Φ2x,2y,2z (αt) ≥L Φx,y,z (t)

(2.3)

(x, y, z ∈ X, t > 0)

and (Y, µ, ∧) be a complete a latticetic random ϕ-normed space. If f : X → Y is an odd mapping satisfying the inequality (2.4)

∧(µf (x)+f (y)+f (z) (t), µf (x+y+z) (t)) ≥L Φx,y,z (t)

(x, y, z ∈ X, t > 0),

then there exists a unique Cauchy additive mapping A : X → Y such that µf (x)−A(x) (t) ≥L Φx,x,−2x ((ϕ(2) − α)t)

(2.5)

(x ∈ X, t > 0).

Proof. Putting x = y and z = −2x in (2.4) we get (2.6)

µ2f (x)−f (2x) (t)

=

∧(µ2f (x)−f (2x) (t), 1L )

≥L

∧(µ2f (x)−f (2x) (t), µf (0) (t))

≥L

Φx,x,−2x (t)

(x ∈ X, t > 0).

From (2.6) we have  (2.7)

µ f (2x) −f (x) 2

t ϕ(2)

 = µ2f (x)−f (2x) (t) ≥L Φx,x,−2x (t)

Replacing x by 2n x in (2.7), and using (2.3) we obtain   t (2.8) µ f (2n+1 x) − f (2n x) ≥L Φ2n x,2n x,−2n+1 x (t) ϕ(2n+1 ) 2n 2n+1 1406

(x ∈ X, t > 0).

(x ∈ X, t > 0, n ∈ N),

ON STABILITY OF FUNCTIONAL INEQUALITIES

5

that is, µ f (2n+1 x) − f (2n x) (t) ≥L

(2.9)

2n

2n+1

≥L Since

f (2n x) 2n

− f (x) =

Pn−1

k=0 (

f (2k+1 x) 2k+1

µ f (2nn x) −f (x)

t

2

n X k=0

−

Φ2n x,2n x,−2n+1 x (ϕ(2n+1 )t)   ϕ(2n+1 )t Φx,x,−2x (x ∈ X, t > 0, n ∈ N) αn

f (2k x) ), 2k

αk ϕ(2k+1 )

by (2.9) we have

! n−1

≥L (∧)k=0 Φx,x,−2x (t) = Φx,x,−2x (t)

that is, µ f (2nn x) −f (x) (t) ≥L Φx,x,−2x

(2.10)

!

t Pn

αk k=0 ϕ(2k+1 )

2

By replacing x with 2m x in (2.10) we obtain: µ f (2n+m x) − f (2m x) (t) ≥L 2m

2n+m

≥L

(2.11)

≥L  As Φx,x,−2x

t αk k=m ϕ(2)k+1

Pn+m



Φ2m x,2m x,−2m+1 x

Φx,x,−2x

Φx,x,−2x

!

t Pn

αk k=0 ϕ(2)m+k+1

!

t Pn

αm+k k=0 ϕ(2)m+k+1

!

t Pn+m

αk k=m ϕ(2)k+1

.

n

tends to 1L as m, n tend to ∞, we conclude that ( f (22n x) ) is a Cauchy

sequence in (Y, µ, ∧). Since (Y, µ, ∧) is a complete latticetic random ϕ-normed space, this sequence converges to some point A(x) ∈ Y . Fix x ∈ X and put m = 0 in (2.11) to obtain ! t (2.12) µ f (2nn x) −f (x) (t) ≥L Φx,x,−2x Pn , αk 2

k=0 ϕ(2)k+1

from which we obtain for every t, δ > 0 (2.13)

µA(x)−f (x) (t + δ) ≥L ≥L

  ∧ µA(x)− f (2nn x) (δ) , µ f (2nn x) −f (x) (t) 2

2

∧ µA(x)− f (2nn x) (δ) , Φx,x,−2x 2

t Pn

Taking the limit as n −→ ∞ and using (2.13) we get (2.14)

µA(x)−f (x) (t + δ) ≥L Φx,x,−2x (t(ϕ(2) − α)).

Since δ was arbitrary, by taking δ −→ 0 one obtains µA(x)−f (x) (t) ≥L Φx,x,−2x (t(ϕ(2) − α)). 1407

αk k=0 ϕ(2)k+1

!! .

6

LEE AND SAADATI

Now, we show that the mapping A is Cauchy additive:       1 1  t 1 − ϕ(2) t 1 − ϕ(2) ,µ  µA(x)+A(y)+A(z) (t) ≥L ∧ µA(x)− f (2nn x)  (2.15) f (2n y)  A(y)− 2n 2 8 8       1 1 t 1 − ϕ(2) t 1 − ϕ(2) ,µ  , µA(z)− f (2nn z)  f (2n (x+y+z))  A(x+y+z)− 2 2n 8 8    1 t 1 − ϕ(2)  , µ f (2n (x+y+z)) f (2n x) f (2n y) f (2n z)  − 2n − 2n − 2n 2n 2   t , µA(x+y+z) ϕ(2) for all x, y, z ∈ X and for all t > 0. The first four terms on the right-hand side of the above inequality tend to 1L as n −→ ∞. Also, from (LRN3), µ f (2n (x+y+z)) f (2n x) f (2n y) f (2n z) − − − n n n n 2

≥L ≥L ≥L

2

2

2

   1 t 1 − ϕ(2)   2

 ϕ(2)n   ϕ(2)n   1  1  ∧ µf (2n x)+f (2n y)+f (2n z) 1− t , µf (2n (x+y+z)) 1− t 4 ϕ(2) 4 ϕ(2)  ϕ(2)n  1  Φ2n x,2n y,2n z 1− t 4 ϕ(2)  ϕ(2)n  1  1 − t , Φx,y,z 4αn ϕ(2)

that is, the fifth term also tends to 1L when n tends to ∞. Therefore, we have   t µA(x)+A(y)+A(z) (t) ≥L µA(x+y+z) , ϕ(2) hence by Lemma 2.1 we conclude that the mapping A is Cauchy additive. To prove the uniqueness of the Cauchy additive function A, assume that there exists a Cauchy additive function B : X −→ Y which satisfies (2.5). Fix x ∈ X. Clearly A(2n x) = 2n A(x) and B(2n x) = 2n B(x) for all n ∈ N. It follows from (2.5) that µA(x)−B(x) (t)

= ≥L ≥L ≥L

µ A(2nn x) − B(2nn x) (t) 2 2      t t ∧ µ A(2nn x) − f (2nn x) , µ B(2nn x) − f (2nn x) 2 2 2 2 2 2  ϕ(2n )(ϕ(2) − α)t  Φ2n x,2n x,−2n+1 x 2  n  ϕ(2) (ϕ(2) − α)t Φx,x,−2x . α 2

Since α < ϕ(2), we get  lim Φx,x,−2x

n→∞

ϕ(2) α

n

(ϕ(2) − α)t 2



Therefore µA(x)−B(x) (t) = 1L for all t > 0, whence A(x) = B(x). 1408

= 1L . 

ON STABILITY OF FUNCTIONAL INEQUALITIES

7

Corollary 2.4. Consider Example 1.2. If f : X → Y is a mapping such that, for some p < 1,

≥L

∧(µf (x)+f (y)+f (z) (t), µf (x+y+z) (t))  kxkp + kykp + kzkp  t , p p p t + (kxk + kyk + kzk ) t + (kxkp + kykp + kzkp )

(x, y, z ∈ X, t > 0),

then there exists a unique Cauchy additive mapping A : X → Y such that   (2 + 2p )kxkp (2 − 2p )t , . µf (x)−A(x) (t) ≥L (2 − 2p )t + (2 + 2p )kxkp (2 − 2p )t + (2 + 2p )kxkp for all x ∈ X and t > 0. + Proof. Let Φ : X 3 −→ DL be defined by  t kxkp + kykp + kzkp  Φx,y,z (t) = , . p p p t + (kxk + kyk + kzk ) t + (kxkp + kykp + kzkp )

Then the corollary is followed from Theorem 2.3 with α = 2p .



Corollary 2.5. Consider Example 1.2. If f : X → Y is a mapping such that  t ε  , (x, y, z ∈ X, t > 0) ∧(µf (x)+f (y)+f (z) (t), µf (x+y+z) (t)) ≥L t+ε t+ε and f (0) = 0, then there exists a unique Cauchy additive mapping A : X → Y such that  t ε  µf (x)−A(x) (t) ≥L , . t+ε t+ε for all x ∈ X and t > 0. + Proof. Let Φ : X 3 −→ DL be defined by

Φx,y,z (t) =

 t ε  , . t+ε t+ε

Then the corollary is followed from Theorem 2.3 with α = 1.



+ Theorem 2.6. Let X be a linear space, Φ be a mapping from X 3 × [0, ∞) to DL such that for some

0 < α < ϕ(3), (2.16)

Φ3x,3y,3z (αt) ≥L Φx,y,z (t)

(x, y, z ∈ X, t > 0).

Let (Y, µ, ∧) be a complete latticetic random ϕ-normed space. If f : X → Y is an odd mapping such that (2.17)

∧(µf (x)+f (y)+2f (z) (t), µf ( x+y +z) (t)) ≥L Φx,y,z (t)

(x, y, z ∈ X, t > 0),

2

then there exists a unique Cauchy additive mapping A : X → Y such that (2.18)

µf (x)−A(x) (t) ≥L Φx,−3x,x ((ϕ(3) − α)t)

(x ∈ X, t > 0).

Proof. As the proof is similar to that of the preceding theorem, we only sketch it. Putting y = −3x and z = x in (2.17) we get (2.19)

µ3f (x)−f (3x) (t) ≥L Φx,−3x,x (t)

(x ∈ X, t > 0).

1409

8

LEE AND SAADATI

From this relation it follows

µ

(2.20)

f (3n x) −f (x) 3n

(t) ≥L Φx,−3x,x

!

t Pn

αk k=0 ϕ(3)k+1

and then, as in the proof of Theorem 2.3,

µ f (3n+m x) − f (3m x) (t) ≥L Φx,−3x,x

Pn+m

αk k=m ϕ(3)k+1

3m

3n+m

!

t

,

n

proving that, for every x, ( f (33n x) ) is a Cauchy sequence in (Y, µ, ∧). Denote A(x) ∈ Y its limit. From

µ f (3nn x) −f (x) (t) ≥L Φx,−3x,x

(2.21)

3

t

!

Pn

αk k=0 ϕ(3)k+1

and

(2.22)

µA(x)−f (x) (t + δ) ≥L ≥L

  ∧ µA(x)− f (3nn x) (δ) , µ f (3nn x) −f (x) (t) 3

3

∧ µA(x)− f (3nn x) (δ) , Φx,−3x,x 3

t

!!

Pn

αk k=0 ϕ(3)k+1

we obtain

µA(x)−f (x) (t) ≥L Φx,−3x,x (t(ϕ(3) − α)).

The additivity of A follows from

(2.23) µA(x)+A(y)+2A(z) (t) ≥L

,

,

,

      ϕ(2)  1 − ϕ(2) t 1 − ϕ(3) ϕ(3) t ,µ  ∧ µA(x)− f (3nn x)  f (3n y)  A(y)− 3 3n 12 12       ϕ(2) 1 − ϕ(3) t 1 − ϕ(2) ϕ(3) t ,µ   x+y µ2A(z)−2 f (3nn z)  2f (3n ( +z)) 2 3 12 12 2A( x+y 2 +z)− 3n    1 − ϕ(2) ϕ(3) 2t  µ 2f (3n ( x+y +z)) f (3n x) f (3n y) 2f (3n z)  2 3 − 3n − 3n − 3n 3n   ϕ(2)t  µ2A( x+y +z) (x, y, z ∈ X, t > 0) 2 ϕ(3) 1410

ON STABILITY OF FUNCTIONAL INEQUALITIES

9

and  µ 2f (3n ( x+y +z))

f (3n x) f (3n y) 2f (3n z) − 3n − 3n 3n

1−

ϕ(2) ϕ(3)



  2t 

3    n  ϕ(3) t 1 − ϕ(2) 1− ϕ(3) ,µ  ∧ µf (3n x)+f (3n y)+2f (3n z)  2f (3n ( x+y +z)) 2 3    n 1 − ϕ(2) ϕ(3) t ϕ(3)  Φ3n x,3n y,3n z  3    n 1 − ϕ(2) ϕ(3) ϕ(3) t , Φx,y,z  3αn 2 3n

−



≥L

≥L

≥L

ϕ(2) ϕ(3)



3

ϕ(3)n t

  

by using Lemma 2.2. Finally, the uniqueness of the Cauchy additive mapping A subject (2.18) follows from µA(x)−B(x) (t)

= ≥L ≥L ≥L

µ A(3nn x) − B(3nn x) (t) 3 3      t t n n n n ∧ µ A(3n x) − f (3n x) , µ B(3n x) − f (3n x) 3 3 3 3 2 2  ϕ(3)n (ϕ(3) − α)  Φ3n x,−3n+1 x,2n x t 2  n  ϕ(3) (ϕ(3) − α)t Φx,−3x,x . α 2  Acknowledgements

The authors would like to thank the two referees and the area editor for giving useful comments and suggestions for the improvement of this paper. This work was supported by Daejin University Rearch Grants in 2012. References [1] I. Golet¸, Some remarks on functions with values in probabilistic normed spaces, Math. Slovaca 57 (2007), No. 3, 259-270. [2] Pl. Kannappan, Functional Equations and Inequalities with Applications Springer, Dordrecht-Heidelberg-London-New York, 2009. [3] D. Mihet¸, R. Saadati, S.M. Vaezpour, The stability of an additive functional equation in Menger probabilistic ϕ-normed spaces, Math. Slovaca 61, No. 5 (2011), 817–826. [4] G. Deschrijver and E. E. Kerre. On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems 23 (2003), 227–235. [5] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, North Holand, New York, 1983. [6] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [7] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [8] A. Gil´ anyi, Eine zur Parallelogrammgleichung aquivalente Ungleichung, Aequationes Math. 62 (2001), 303–309.

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[9] J. R¨ otz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. [10] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [11] C. Park, Fixed points in functional inequalities. J. Inequal. Appl. 2008, Art. ID 298050, 8 pp. [12] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers–Ulam–Rassias theorem, Fuzzy Sets and Syst, 159 (2008), 720–729. [13] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Syst., 159 (2008), 730–738. [14] C. Alsina, On the stability of a functional equation arising in probabilistic normed spaces, in: General Inequalities, vol. 5, Oberwolfach, 1986, Birkhuser, Basel, 1987, 263-271. [15] D. Mihet¸, The fixed point method for fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, Fuzzy Sets and Systems 160 (2009), no. 11, 1663–1667. [16] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567-572. [17] D. Mihet¸, R. Saadati, S.M. Vaezpour, The stability of the quartic functional equation in random normed spaces, Acta Appl. Math., 110 (2010), 797–803. [18] R. Saadati, S.M. Vaezpour, Y.J. Cho, A note on the ”On the stability of cubic mappings and quadratic mappings in random normed spaces”, J. Inequal. Appl., Volume 2009, Article ID 214530, 6 pages. [19] A. G. Kusraev, Dominated Operators, Kluwer, Dordrecht, 2000. [20] A. G. Kusraev, Jensen type inequalities for positive bilinear operators, Positivity, in press. [21] S.S. Chang, Y.J. Cho and S.M. Kang, Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publishers, Inc., New York, 2001. [22] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002. [23] O. Hadˇ zi´ c and E. Pap, Fixed Point Theory in PM-Spaces, Kluwer Academic, 2001. [24] D. H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [25] S.-M. Jung, Hyers-Ulam- Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011. [26] Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003. [27] A.N. Sherstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280-283 (in Russian). Sung Jin Lee Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea Reza Saadati Department of Mathematics and Computer Science, Iran University of Science and Technology, Tehran, Iran E-mail address: [email protected] E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1413-1423, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

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1423

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1424-1429, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A note on the second kind generalized q-Euler polynomials C. S. RYOO Department of Mathematics, Hannam University, Daejeon 306-791, Korea Abstract. In this paper we introduce the second kind generalized q-Euler numbers En,χ,q and polynomials En,χ,q (x). We obtain the Witt-type formulae of the second kind generalized q-Euler numbers En,χ,q and polynomials En,χ,q (x) attached to χ. Key words: The second kind Euler numbers and polynomials, the second kind q-Euler numbers and q-Euler polynomials, the second kind generalized q-Euler numbers and polynomials 1. INTRODUCTION Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assume that |q| < 1. If q ∈ Cp , we normally assume 1 that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. Let U D(Zp ) be the space of uniformly differentiable function on Zp . For g ∈ U D(Zp ) the fermionic p-adic invariant q-integral on Zp is defined by Kim as follows: 

N

p −1 1 g(x)dμ−q (x) = lim g(x)(−q)x , see [3, 4] . I−q (g) = N →∞ [pN ]−q Zp x=0

(1.1)

If we take gn (x) = g(x + n) in (1.1), then we see that q n Iq (gn ) + (−1)n−1 Iq (g) = [2]q

n−1 

(−1)n−1−l q l g(l).

(1.2)

l=0

Let a fixed positive integer d with (p, d) = 1, set X = Xd = lim(Z/dpN Z), X1 = Zp , ←− N  ∗ a + dpZp , X = 0 0, 0 < 𝛼 < 2.

(27)

The problem (27) is a linear ordinary differential equation of first order. So, from the initial condition,the solution of it will be 𝑢(𝑥, 𝑡) ≃ exp(𝑡)δ(𝑥). If 𝑘 = 1, the equivalent differential equation of (26) will be ∂ ∂ 𝑢(𝑥, 𝑡) ≃ (𝛼0 + 𝛼1 )𝑢(𝑥, 𝑡) − 𝛼1 𝑢(𝑥, 𝑡), ∂𝑡 ∂𝑥

𝑥 ∈ R, 𝑡 > 0, 0 < 𝛼 < 2.

(28)

Now, we will use the analytic methods for getting the analytic solution of problem (28). To solve equation (28) with initial condition 𝑢(𝑥, 0) = δ(𝑥), according to the homotopy perturbation technique, we construct the following homotopy: ( ) ( ( )) ∂v ∂𝑢0 ∂v ∂ (1 − 𝑝) − +𝑝 − C (𝛼0 + 𝛼1 )v(𝑥, 𝑡) − 𝛼1 v(𝑥, 𝑡) = 0, (29) ∂𝑡 ∂𝑡 ∂𝑡 ∂𝑥 Substituting equation (2) into equation (29), and comparing coefficients of terms with identical powers of 𝑝, leads to: 𝑝0 :

∂v0 ∂𝑢0 − = 0, ∂𝑡 ∂𝑡

( ) ∂v1 ∂v0 ∂ 𝑝 : =− + C (𝛼0 + 𝛼1 )v0 (𝑥, 𝑡) − 𝛼1 v0 (𝑥, 𝑡) , ∂𝑡 ∂𝑡 ∂𝑥 1

𝑝2 :

) ( ∂v2 ∂ = C (𝛼0 + 𝛼1 )v1 (𝑥, 𝑡) − 𝛼1 v1 (𝑥, 𝑡) , ∂𝑡 ∂𝑥

v1 (𝑥, 0) = 0,

v2 (𝑥, 0) = 0,

.. . ( ) ∂v𝑛 ∂ 𝑝 : = C (𝛼0 + 𝛼1 )v𝑛−1 (𝑥, 𝑡) − 𝛼1 v𝑛−1 (𝑥, 𝑡) , ∂𝑡 ∂𝑥 𝑛

v𝑛 (𝑥, 0) = 0,

For simplicity we take v0 (𝑥, 𝑡) = 𝑢0 (𝑥, 𝑡) = δ(𝑥). So we derive the following recurrent relation )) ∫ 𝑡( ( ∂ v1 (𝑥, 𝑡) = C (𝛼0 + 𝛼1 )v0 (𝑥, 𝑡) − 𝛼1 v0 (𝑥, 𝑡) 𝑑𝑡 ∂𝑥 0 ( ) ∂ = C (𝛼0 + 𝛼1 )δ(𝑥) − 𝛼1 δ(𝑥) × 𝑡, ∂𝑥 .. .

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Analytic Approximation of Time-Fractional Diffusion-Wave Equation

H. Fallahgoul, S. M. Hashemiparast

and so on. Therefore )) ∞ ∫ 𝑡( ( ∑ ∂ 𝑢(𝑥, 𝑡) = lim 𝑢𝑛 (𝑥, 𝑡) = δ(𝑥) + C (𝛼0 + 𝛼1 )v𝑘−1 (𝑥, 𝑡) − 𝛼1 v𝑘−1 (𝑥, 𝑡) 𝑑𝑡. 𝑛→∞ ∂𝑥 0 𝑘=1

If 𝑘 = 2, the equivalent differential equation of (6) will be ( ) ∂ ∂2 ≃ C (𝛼0 + 𝛼1 + 𝛼2 )𝑢(𝑥, 𝑡) − (𝛼1 + 2𝛼2 ) 𝑢(𝑥, 𝑡) + 𝛼2 2 𝑢(𝑥, 𝑡) , ∂𝑥 ∂𝑥 𝑥 ∈ R, 𝑡 > 0, 0 < 𝛼 < 2,

∂ 𝑢(𝑥, 𝑡) ∂𝑡

using HPM we get the analytic solution for 𝑘 = 2. So, Algorithm 2 provide a procedure for getting the analytic solution of equation (26).

5

Conclusion

In this paper, we have shown that the HPM can be used successfully for finding the solutions of spacefractional partial differential equation based on connection of FC and OD. It may be concluded that this technique is very powerful and efficient in finding the analytical solutions SFDE and TFDWE. Some experiments supported the theoritical results.

A

Fractional Calculus

Fractional calculus goes back to the beginning of the theory of differential calculus and deals with the generalization of standard integrals and derivatives to a non-integer, or even complex order [14, 16, 15]. In this section we give the basic definitions and some properties of the fractional calculus. More detailed information may be found in the book by Samko et al. [16] and [11]. Let Ω = [𝑎, 𝑏](∞ < 𝑎 < 𝑏 < ∞) be a finite interval on the real axis R. The Riemann-Liouville fractional integrals 𝐼a𝛼+ and 𝐼b𝛼− of order 𝛼 ∈ C (ℜ(𝛼) > 0) are defined by ∫ 𝑥 1 𝑓 (𝑡)𝑑𝑡 𝛼 (𝐼a+ 𝑓 )(𝑥) = (𝑥 > 𝑎, ℜ(𝛼) > 0), Γ(𝛼) a (𝑥 − 𝑡)1−𝛼 and (𝐼b𝛼− 𝑓 )(𝑥)

1 = Γ(𝛼)

∫

b 𝑥

𝑓 (𝑡)𝑑𝑡 (𝑡 − 𝑥)1−𝛼

(𝑥 < 𝑏, ℜ(𝛼) > 0),

respectively. Here Γ(𝛼) is the Gamma function.These integrals are called the left-sided and the right-sided fractional integrals. The Riemann-Liouville fractional derivatives Da𝛼+ 𝑦 and Db𝛼− 𝑦 of order 𝛼 ∈ C (ℜ(𝛼) ≥ 0) are defined by (Da𝛼+ 𝑦)(𝑥) = ( =

1 𝑑 ( )𝑛 Γ(𝑛 − 𝛼) 𝑑𝑥

∫ a

𝑥

𝑑 𝑛 𝑛−𝛼 ) (𝐼b− 𝑦)(𝑥) 𝑑𝑥

𝑦(𝑡)𝑑𝑡 , (𝑥 − 𝑡)𝛼−𝑛+1

1441

(𝑥 > 𝑎, 𝑛 = [ℜ(𝛼)] + 1),

Analytic Approximation of Time-Fractional Diffusion-Wave Equation

H. Fallahgoul, S. M. Hashemiparast

and (Db𝛼− 𝑦)(𝑥) = (− 𝑑 1 (− )𝑛 = Γ(𝑛 − 𝛼) 𝑑𝑥

∫

b 𝑥

𝑑 𝑛 𝑛−𝛼 ) (𝐼b− 𝑦)(𝑥) 𝑑𝑥

𝑦(𝑡)𝑑𝑡 , (𝑡 − 𝑥)𝛼−𝑛+1

(𝑥 < 𝑏, 𝑛 = [ℜ(𝛼)] + 1),

respectively, where [ℜ(𝛼)] means the integral part of ℜ(𝛼). If 0 < ℜ(𝛼) < 1, then ∫ 𝑥 𝑑 𝑦(𝑡)𝑑𝑡 1 (Da𝛼+ 𝑦)(𝑥) = (𝑥 > 𝑎, 0 < ℜ(𝛼) < 1), Γ(1 − 𝛼) 𝑑𝑥 a (𝑥 − 𝑡)𝛼−[ℜ(𝛼)]

(Db𝛼− 𝑦)(𝑥) = −

1 𝑑 Γ(1 − 𝛼) 𝑑𝑥

∫

b

𝑥

𝑦(𝑡)𝑑𝑡 (𝑡 − 𝑥)𝛼−[ℜ(𝛼)]

(𝑥 < 𝑏, 0 < ℜ(𝛼) < 1).

For 𝑓 ∈ cµ , µ ≥ −1, 𝛼, 𝛽 ≥ 0 and 𝛾 > −1 the following properties will be easily obtained: • 𝐼 𝛼 𝐼 β 𝑓 (𝑥) = 𝐼 𝛼+β 𝑓 (𝑥), • 𝐼 𝛼 𝐼 β 𝑓 (𝑥) = 𝐼 β 𝐼 𝛼 𝑓 (𝑥).

References [1] J. M. Blackledge, Application of the fractional diffusion equationfor predicting market behaviour, IAENG International Journal of Applied Mathematics,40 3 (2010) 130–158. [2] D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advectiondispersion equation, Water Resources Res. 36 (2000) 1403–1412. [3] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl.Geophysics 91 (1971) 134–147. [4] F. Demontis and C. van der Mee, Closed Form Solutions to the Integrable Discrete Nonlinear Schrdinger Equation, Journal Of Nonlinear Mathematical Physics (2012). [5] H. Fallahgoul, S. M. Hashemiparast, Y. S. Kim, Svetlozar T. Rachev and Frank J. Fabozzi, Analytic approximation of the pdf of stable and geometric stable distribution. Working paper, Department of Applied Mathematics and Statistics, Stony Brook University, SUNY, (2012). [6] R. Gorenflo and F. Mainardi, Some recent advances in theory and simulation of fractional diffusion process, Journal of Computational and Applied Mathematics 229 (2009) 400–415. [7] S. M. Hashemiparast and H. Fallahgoul, Approximation of Laplace Transform of Fractional Derivatives Via Clenshaw-Curtis Integration, Journal of Computational and Applied Mathematics 229 (2009) 400–415. [8] S. M. Hashemiparast and H. Fallahgoul, Approximation of Fractional Deriva-tives Via Gauss Integration, Annali dell’Universit di Ferrara 57 1 (2011) 67–87. [9] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178 (1999) 257– 262.

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H. Fallahgoul, S. M. Hashemiparast

[10] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int J. Non-Linear Mech., 35 1 (2000) 37–43. [11] A. Kilbas, M. Sirvastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, (Elsevier) (2006). [12] R. Metzler, W. G. Glockle and T. F. Nonnenmacher, Fractional model equation for anomalous diffusion, PhysicaA 15 1 (1994) 13-24. [13] S. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, (PhD thesis, Shanghai Jiao Tong University) (1992). [14] K. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, (London: Academic Press) (1974). [15] I. Podlubny, Fractional Differential Equations,(Academic Press) (1999). [16] S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, (New York: Gordon & Breach) (1993).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1444-1455, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Higher order duality in nondifferentiable fractional programming involving generalized convexity I. Ahmada , Ravi P. Agarwalb,∗ , Anurag Jayswalc a Department

of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.

b Department

of Mathematics, Texas A & M University - Kingsville 700 University Blvd., Kingsville, TX 78363-8202, USA. Email: [email protected]

c Department

of Applied Mathematics, Indian School of Mines, Dhanbad-826004, Jharkhand, India.

Abstract The purpose of this paper is to consider a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective and constraints functions contains a term involving the support function of a compact convex set. Usual duality theorems are established for two types of higher order dual models under the assumptions of higher order (F, α, ρ, d) − V −type I functions. Keywords: Fractional programming; Nondifferentiable programming; Support function; Generalized convexity; Higher order duality

1. Introduction In recent years, optimality conditions and duality theory for nondifferentiable multiobjective fractional programming problems involving different kinds of generalized convexity assumptions have received much attention by many authors [6, 7, 8, 9] and the references therein. Under the assumption of (C, α, ρ, d) convexity, Long [9] established sufficient optimality conditions and duality results for a nondifferentiable multiobjective fractional programming problem in which every component of the objective function contains a term involving the support function of a compact convex set. ∗

Corresponding author.

a

Permanent address: Department of Mathematics, Aligarh Muslim University, Aligarh-202 002, India.

E-mail addresses:

[email protected] (I. Ahmad), [email protected] (Ravi P. Agarwal),

anurag [email protected] (A. Jayswal). The research of the first author partially supported by the Deanship of Scientific Research , King Fahd University of Petroleum and Minerals under internal project IN111015.

1 1444

Ahmad et al: Higher order duality in nondifferentiable fractional programming

Second and higher-order duality in nonlinear programming has been studied in the last few years by many researchers. One practical advantage of second and higher-order duality is that it provides tighter bounds for the value of objective function of the primal problem when approximations are used because there are more parameters involved. Mangasarian [10] first formulated a class of higher-order dual problems for a nonlinear programming problem. Mond and Zhang [11] obtained duality results for various higherorder dual problems under higher-order invexity assumptions. Motivated by the various kinds of generalized convexity Liang et al. [7], introduced a unified form of generalized convexity called (F, α, ρ, d)-convex function. Gulati and Agarwal [2] introduced second order (F, α, ρ, d)-V-type I functions for a multiobjective programming problem and proved duality results involving aforesaid functions. Recently, Suneja et al. [12] formulated higher order Mond-Weir and Schaible type dual programs for a nondifferentiable multiobjective fractional programming problem where the objective functions and the constraints contain support function of compact convex sets in Rn and established weak and strong duality results involving higher order (F, ρ, σ)type I functions. Gulati and Geeta [5] introduced a new class of higher-order (V, α, ρ, d)invex function and established duality results for Schaible type dual of a nondifferentiable multiobjective fractional programming problem. Gulati and Agarwal [4] focus his study on a nondifferentiable multiobjective programming problem where every component of objective and constraint functions contain a term involving the support function of a compact convex set and established duality theorems for Wolfe and unified higher order dual problems involving higher order (F, α, ρ, d)-type I function. Motivated by earlier work of Ahmad [1], Gulati and Agarwal [2] and Suneja et al. [12], we establish higher order duality results for two types of dual models related to nondifferentiable multiobjective fractional programming problem where the objective and the constraints functions contain support functions of compact convex sets in Rn . This paper is organized as follows: In Section 2, we have introduced the concept of higher-order (F, α, ρ, d)-V-type I functions. In Sections 3 and 4, the duality results have been established for higher order Mond-Weir and Schaible type duals of a multiobjective nondifferentiable fractional problem. Finally, conclusion and further development are given in Section 5.

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Ahmad et al: Higher order duality in nondifferentiable fractional programming

2. Preliminaries n its non-negative orthant. If x, y ∈ Let Rn be n-dimensional Euclidean space and R+

Rn then x < y ⇔ xi < yi , i = 1, 2, . . . , n; x 5 y ⇔ xi 5 yi , i = 1, 2, . . . , n and x ≤ y ⇔ xi 5 yi , i = 1, 2, . . . , n and x ̸= y. Definition 2.1. A functional F : X × X × Rn → R (X ⊆ Rn ), is said to be sublinear in its third argument, if ∀ x, x ¯ ∈ X, (i) F (x, x ¯; a1 + a2 ) 5 F (x, x ¯; a1 ) + F (x, x ¯ ; a 2 ) ∀ a 1 , a2 ∈ R n , (ii) F (x, x ¯; αa) = αF (x, x ¯; a) ∀ α ∈ R+ , a ∈ Rn . By (ii), it is clear that F (x, x ¯; 0) = 0. Definition 2.2. Let C be a compact convex set in Rn . The support function of C is defined by s(x|C) = max{xT y : y ∈ C}. A support function, being convex and everywhere finite, has a subdifferential, that is, there exists z ∈ Rn such that s(y|C) = s(x|C) + z T (y − x) for all y ∈ C. The subdifferential of s(x|C) is given by ∂s(x|C) = {z ∈ C : z T = s(x|C)}. For any set D ⊂ Rn , the normal cone to D at a point x ∈ D is defined by ND (x) = {y ∈ Rn | y T (z − x) 5 0 for all z ∈ D}. It is obvious that for a compact convex set C, y ∈ NC (x) if and only if s(y|C) = xT y, or equivalently, x ∈ ∂s(y|C). Consider the following multiobjective programming problem: (P)

Minimize f (x) subject to

x ∈ X 0 = {x ∈ X : h(x) ≤ 0},

where X ⊆ Rn be open, f : X → Rk , h : X → Rm are continuously differentiable functions. Definition 2.3. A point x ¯ ∈ X 0 is an efficient solution of (P) if there exists no x ∈ X 0 such that f (x) ≤ f (¯ x). Lemma 2.1. x0 ∈ X0 is an efficient solution of (P) if and only if x0 is an optimal solution 3 1446

Ahmad et al: Higher order duality in nondifferentiable fractional programming

of Pr (x0 ) for each r = 1, 2, ..., k, Pr (x0 )

minimize fr (x) subject to fi (x) ≤ fi (x0 ), for all i = 1, 2, ..., k, i ̸= r, h(x) ≤ 0, x ∈ X.

Let fi : X → R, hj : X → R, Ki : X × Rn → R and Hj : X × Rn → R be differentiable functions where i = 1, 2, ..., k and j = 1, 2, ..., m. Let d : X × X → R is a pseudo matric. Definition 2.4. The pair of functions (f, h) is said to be higher-order (F, α, ρ, d) − 2 ) and ρ = (ρ1 , ..., ρ1 , ρ2 , ..., ρ2 ), V −type I at u ∈ X, if there exist vectors α = (α11 , ..., αk1 , α12 , ..., αm m 1 k 1

where αi1 , αj2 : X × X → R+ \ {0} and ρ1i , ρ2j ∈ R such that for each x ∈ X0 and for all p, q ∈ Rn , i = 1, 2, ..., k and j = 1, 2, ..., m, fi (x) − fi (u) = F (x, u; αi1 (x, u)(∇fi (u) + ∇p Ki (u, p))) + Ki (u, p) − pT ∇p Ki (u, p) + ρ1i d2 (x, u), − hj (u) = F (x, u; αj2 (x, u)(∇hj (u) + ∇q Hj (u, q))) + Hj (u, q) − q T ∇q Hj (u, q) + ρ2j d2 (x, u). Remark 2.1. (i) If Ki (u, p) = 12 pt ∇2 fi (u)p and Hj (u, q) =

1 t 2 2 q ∇ hj (u)q

for i = 1, 2, ..., k and j =

1, 2, ..., m, then we obtain the second order (F, α, ρ, d)-V-type I introduced by Gulati and Agarwal [2]. (ii) Let Ki (u, p) = 0 and Hj (u, q) = 0 for i = 1, 2, ..., k and j = 1, 2, ..., m. Then above definition becomes that of (F, α, ρ, d)-V-type I [3]. (iii) If αi1 = αi2 = 1 for i = 1, 2, ..., k and j = 1, 2, ..., m, then the higher-order (F, α, ρ, d)V-type I reduces to the higher order (F, ρ, σ)-type I given by Suneja et al. [12]. We now consider the following the multiobjective nondifferentiable fractional program: [ (FP)

minimize subject to

f1 (x) + S(x|C1 ) f2 (x) + S(x|C2 ) fk (x) + S(x|Ck ) , , ..., g1 (x) − S(x|D1 ) g2 (x) − S(x|D2 ) gk (x) − S(x|Dk )

]

hj (x) + S(x|Ej ) 5 0, j = 1, 2, ..., m,

where x ∈ X ⊆ Rn , fi , gi : X → R (i = 1, 2, ..., k) and hj : X → R (j = 1, 2, ..., m) are 4 1447

Ahmad et al: Higher order duality in nondifferentiable fractional programming

continuously differentiable functions. fi (.) + S(.|Ci ) = 0 and gi (.) − S(.|Di ) > 0; Ci , Di and Ej are compact convex sets in Rn and S(x|Ci ), S(x|Di ) and S(x|Ej ) denote the support functions of compact convex sets, Ci , Di and Ej , respectively. Lemma 2.2. If u is an efficient solution of (FP), we have the following results. (FP¯ ϵ)

minimize

fr (x)+S(x|Cr ) gr (x)−S(x|Dr )

subject to fi (x)+S(x|Ci ) gi (x)−S(x|Di )

≤ ϵ¯i , i = 12, ..., k, i ̸= r,

hj (x) + S(x|Ej ) 5 0, j = 1, 2, ..., m, where

ϵ¯i =

fi (u)+S(u|Ci ) gi (u)−S(u|Di ) .

Since gi (x) − S(x|Di ) > 0, for each i = 1, 2, ..., k, therefore (F P ϵ¯) can be rewritten as (FP1 ϵ¯)

minimize

fr (x)+S(x|Cr ) gr (x)−S(x|Dr )

subject to fi (x) + S(x|Ci ) − ϵ¯i (gi (x) − S(x|Di )) ≤ 0, i = 12, ..., k, i ̸= r, hj (x) + S(x|Ej ) 5 0, j = 1, 2, ..., m. Lemma 2.3. u is an efficient solution of (FP) if and only if u solves (FP1 ϵ¯) for each r = 1, 2, ..., k, where ϵ¯i =

fi (u)+S(u|Ci ) gi (u)−S(u|Di ) .

3. Higher order Mond-Weir type dual In connection to (FP) we now consider the following higher order Mond-Weir type dual problem [12]: (MFD)

[

maximize

f1 (u)+uT z1 f2 (u)+uT z2 (u)+uT zk , , ..., gfk (u)−u Tv g1 (u)−uT v1 g2 (u)−uT v2 k k

]

subject to   k m ( f (u) + uT z ) ∑ ∑ i i ∇ λi + µj (hj (u) + uT wj ) gi (u) − uT vi i=1

j=1

+

k ∑

λi ∇p Gi (u, p) +

i=1 m ∑

m ∑

µj ∇q Hj (u, q) = 0,

(1)

j=1

µj {(hj (u) + uT wj ) + Hj (u, q) − q T ∇q Hj (u, q)} = 0,

(2)

j=1 k ∑

( ) λi Gi (u, p) − pT ∇q Gi (u, p) = 0,

i=1

zi ∈ Ci , vi ∈ Di , i = 1, 2, ..., k, wj ∈ Ej , j = 1, 2, ..., m, 5 1448

(3)

Ahmad et al: Higher order duality in nondifferentiable fractional programming

µj = 0, j = 1, 2, ..., m, λi = 0, i = 1, 2, ..., k,

k ∑

λi = 1.

i=1

Theorem 3.1 (Weak duality). Let x and (u, z, v, µ, λ, w, p, q) be feasible solutions to (FP) and (MFD), respectively such that [ (i)

fi (.)+(.)T zi , hj (.) gi (.)−(.)T vi

] + (.)T wj is higher-order (F, α, ρ, d) − V −type I with respect to

Gi and Hj , at u for i = 1, 2, ..., k and j = 1, 2, ..., m, (ii)

k ∑ i=1

λi α1i (x,u)

(iii) λi > 0,

= 1, αj2 (x, u) = 1, j = 1, 2, ..., m,

k ∑ i=1

λi ρ1i α1i (x,u)

+

∑m

2 j=1 µj ρj

= 0.

Then fi (x) + S(x|Ci ) fi (u) + uT zi 5 , i = 1, 2, ..., k, gi (x) − S(x|Di ) gi (u) − uT vi

(4)

fr (u) + uT zr fr (x) + S(x|Cr ) < , for some r = 1, 2, ..., k, gr (x) − S(x|Dr ) gr (u) − uT vr

(5)

and

cannot hold. Proof. Suppose on the contrary that inequalities (4) and (5) hold. Then as λi > 0, xT zi 5 S(x|Ci ), xT vi 5 S(x|Di ) using hypothesis (ii), we get k ∑ i=1

λi ( fi (x) + xT zi fi (u) + uT zi ) − < 0. αi1 (x, u) gi (x) − xT vi gi (u) − uT vi

(6)

Because αj2 (x, u) = 1 for j ∈ M , hypothesis (i) gives ( )) ( ( fi (u) + uT zi ) fi (x) + xT zi fi (u) + uT zi 1 − = F x, u; α (x, + ∇ G (u, p) u) ∇ p i i gi (x) − xT vi gi (u) − uT vi gi (u) − uT vi + Gi (u, p) − pT ∇p Gi (u, p) + ρ1i d2 (x, u).

(7)

( ) ( − (hj (u) + uT wj ) = F x, u; ∇(hj (u) + uT wj ) + ∇q Hj (u, q) + Hj (u, q) − q T ∇q Hj (u, q) + ρ2j d2 (x, u). On multiplying the above inequalities (7) and (8) by

λi α1i (x,u)

and µj , respectively, then

summing the two resultant inequalities, we obtain k ∑ i=1

λi ( fi (x) + xT zi fi (u) + uT zi ) − αi1 (x, u) gi (x) − xT vi gi (u) − uT vi 6 1449

(8)

Ahmad et al: Higher order duality in nondifferentiable fractional programming

k ( ∑ ( ( fi (u) + uT zi ) )) = F x, u; λi ∇ + ∇ G (u, p) p i gi (u) − uT vi i=1

+

k ∑ i=1

−

m ∑

k ∑ λi ( λi ρ1i d2 (x, u) T , (9) G (u, p)−p ∇ G (u, p))+ i p i αi1 (x, u) αi1 (x, u) i=1

(

µj (hj (u) + u wj ) = F x, u; T

j=1

m ∑

) ( µj ∇(hj (u) + uT wj ) + ∇q Hj (u, q))

j=1

+

m ∑

m ( ) ∑ µj Hj (u, q) − q T ∇q Hj (u, q) + µj ρ2j d2 (x, u). (10)

j=1

j=1

Using equation (1) and sublinearity of F , we have k m ( f (u) + uT z ) ∑ ) [ (∑ i i T 0 = F x, u; ∇ λi + µ (h (u) + u w ) j j j gi (u) − uT vi i=1

+

k ∑

j=1

λi ∇p Gi (u, p) +

i=1

m ∑

µj ∇q Hj (u, q)

]

j=1

k ( ∑ ( ( fi (u) + uT zi ) )) + ∇ G (u, p) 5 F x, u; λi ∇ p i gi (u) − uT vi i=1

m ) ( ∑ ( + F x, u; µj ∇(hj (u) + uT wj ) + ∇q Hj (u, q)) .

(11)

j=1

The inequalities (9), (10), (11) and hypothesis (iii) give

05

k ∑ i=1

−

k ∑ i=1

m λi ( fi (x) + xT zi fi (u) + uT zi ) ∑ − − µj (hj (u) + uT wj ) αi1 (x, u) gi (x) − xT vi gi (u) − uT vi j=1

m ∑ ( ) λi ( T Gi (u, p) − p ∇p Gi (u, p)) − µj Hj (u, q) − q T ∇q Hj (u, q) . 1 αi (x, u) j=1

That is, k ∑ i=1

λi ( fi (x) + xT zi fi (u) + uT zi ) − αi1 (x, u) gi (x) − xT vi gi (u) − uT vi =

m ∑

µj (hj (u) + uT wj + Hj (u, q) − q T ∇q Hj (u, q))

j=1

+

k ∑

λi ( Gi (u, p) 1 α (x, u) i=1 i

− pT ∇p Gi (u, p))

From the inequalities (2), (3) and the positivity of αi1 (x, u), i = 1, 2, ..., k, we have k ∑ i=1

λi ( fi (x) + xT zi fi (u) + uT zi ) − = 0, αi1 (x, u) gi (x) − xT vi gi (u) − uT vi

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Ahmad et al: Higher order duality in nondifferentiable fractional programming

which contradicts (6). This completes the proof.

Theorem 3.2 (Strong duality). If u is an efficient solution of (FP), Gi (u, 0) = 0, i = 1, 2, ..., k, Hj (u, 0) = 0, j = 1, 2, ..., m, and a constraint qualification is satisfied for (F P ϵ¯) ¯ ∈ Rk , µ for at least one r = 1, 2, ..., k, then there exist λ ¯ ∈ Rm , z¯i ∈ Rn , v¯i ∈ Rn and ¯ w, w ¯j ∈ Rn , i = 1, 2, ..., k, j = 1, 2, ..., m, such that (u, z¯, v¯, µ ¯, λ, ¯ p = 0, q = 0) is a feasible solution of (MFD) and the corresponding values of the objective functions are equal. Further if the conditions of Weak duality theorem 3.1 are satisfied for each feasible solution x of (FP) and each feasible solution (´ u, z´, v´, µ ´, w, ´ p = 0, q = 0) of (MFD) then ¯ w, (u, z¯, v¯, µ ¯, λ, ¯ p = 0, q = 0) is an efficient solution of (MFD).

Proof. The proof follows along the lines of Theorem 3.2 [12] in light of the discussions given above and hence being omitted.

4. Higher order Schaible type dual Now we consider the following Schaible type higher order dual problem for (FP): (SFD) maximize (γ1 , γ2 , ..., γk ) subject to   k m [ ] ∑ ∑ λi (fi (u) + uT zi ) − γi (gi (u) − uT vi ) + µj (hj (u) + uT wj ) ∇ i=1

j=1

+

k ∑

m ( ) ∑ µj ∇q Hj (u, q) = 0, λi ∇p Ki (u, p) − γi Gi (u, p) +

i=1 k ∑

(12)

j=1

[ ] ( ) λi { (fi (u) + uT zi ) − γi (gi (u) − uT vi ) + Ki (u, p) − γi Gi (u, p)

i=1

( ) −pT ∇p Ki (u, p) − γi Gi (u, p) } = 0, m ∑

µj {(hj (u) + uT wj ) + Hj (u, q) − q T ∇q Hj (u, q)} = 0,

j=1

zi ∈ Ci , vi ∈ Di , i = 1, 2, ..., k, wj ∈ Ej , j = 1, 2, ..., m, µj = 0, j = 1, 2, ..., m, λi ≥ 0, i = 1, 2, ..., k,

k ∑ i=1

γi = 0, i = 1, 2, ..., k. 8 1451

λi = 1,

(13) (14)

Ahmad et al: Higher order duality in nondifferentiable fractional programming

Theorem 4.1 (Weak duality). Let x and (u, γ, z, v, w, µ, λ, p, q) be feasible solutions of (FP) and (SFD), respectively such that (i) (fi (.) + (.)T zi , hj (.) + (.)T wj ) is higher-order (F, α, ρ, d) − V −type I with respect to Ki and Hj and [−(gi (.) − (.)T vi , hj (.) + (.)T wj ] is higher-order (F, α, ρ, d) − V −type I with respect to −Gi and Hj , at u for i = 1, 2, ..., k and j = 1, 2, ..., m, (ii) αi1 (x, u) = αj2 (x, u) = α ˆ (x, u), i = 1, 2, ..., k, j = 1, 2, ..., m, (iii)

k ∑ i=1

λi ρ3i +

∑m

2 j=1 µj ρi

= 0, where ρ3i = ρ1i (1 + γi ).

Then fi (x) + S(x|Ci ) 5 γi , gi (x) − S(x|Di )

i = 1, 2, ..., k,

(15)

for some r = 1, 2, ..., k,

(16)

and fr (x) + S(x|Cr ) < γr , gr (x) − S(x|Dr ) cannot hold. Proof. Suppose on the contrary that inequalities (15) and (16) hold. Then as λi ≥ 0, i = 1, 2, ..., k, using hypothesis (ii), we get k ∑ i=1

) λi ( fi (x) + xT zi − γi (gi (x) − xT vi ) < 0. α ˆ (x, u)

(17)

Since (fi (.) + (.)T zi , hj (.) + (.)T wj ) is higher-order (F, α, ρ, d) − V −type I with respect to Ki and Hj and [−(gi (.) − (.)T vi ), hj (.) + (.)T wj ] is higher-order (F, α, ρ, d) − V −type I with respect to −Gi and Hj , at u for i = 1, 2, ..., k and j = 1, 2, ..., m, we have ( ( ) ((fi (x) + xT zi ) − (fi (u) + uT zi )) = F x, u; αi1 (x, u) ∇(fi (u) + uT zi ) + ∇p Ki (u, p)) + Ki (u, p) − pT ∇p Ki (u, p) + ρ1i d2 (x, u)

(18)

( ( ) (−(gi (x) − xT vi ) + (gi (u) − uT vi )) = F x, u; −αi1 (x, u) ∇(gi (u) − uT vi ) − ∇p Gi (u, p)) − Gi (u, p) + pT ∇p Gi (u, p) + ρ1i d2 (x, u) and

(19)

( ) ( −(hj (u) + uT wj ) = F x, u; αi2 (x, u) ∇(hj (u) + uT wj ) + ∇q Hj (u, q)) + Hj (u, q) − q T ∇q Hj (u, q) + ρ2i d2 (x, u)

9 1452

(20)

Ahmad et al: Higher order duality in nondifferentiable fractional programming

On multiplying (19) by γi and adding with (18),we get [(fi (x) + xT zi ) − γi (gi (x) − xT vi )] − [(fi (u) + uT zi ) − γi (gi (u) − uT vi )] [ ( = F x, u; αi1 (x, u) ∇(fi (u) + uT zi − γi (gi (u) − uT vi )) ))] + ∇p (Ki (u, p) − γi Gi (u, p) + Ki (u, p) − γi Gi (u, p) − pT ∇p (Ki (u, p) − γi Gi (u, p)) + ρ3i d2 (x, u),

(21)

where ρ3i = ρ1i (1 + γi ). Multiplying (21) by λi > 0 and (20) by µj = 0, i = 1, 2, ..., k, j = i, 2, ..., m, and adding, we obtain k ∑

λi {[(fi (x)+x zi )−γi (gi (x)−x vi )]−[(fi (u)+u zi )−γi (gi (u)−u vi )]}− T

T

T

T

i=1

m ∑

µj (hj (u)+uT wj )

j=1

k m [ ( ) ∑ ∑ ( = F x, u; λi αi1 (x, u) ∇(fi (u)+uT zi −γi (gi (u)−uT vi )) + µj αi2 (x, u) ∇(hj (u)+uT wj ) i=1

+

j=1

k ∑

λi αi1 (x, u)

(

m ] )) ∑ ∇p (Ki (u, p) − γi Gi (u, p) + µj αi2 (x, u)∇q Hj (u, q)

i=1

+

k ∑

j=1 m ( ) ∑ ( ) λi Ki (u, p) − γi Gi (u, p) + µj Hj (u, q) − q T ∇q Hj (u, q)

i=1

−

k ∑

j=1

λi p ∇p (Ki (u, p) − γi Gi (u, p)) + T

i=1

k (∑

λi ρ3i

+

i=1

m ∑

) µj ρ2i d2 (x, u).

(22)

j=1

∑ ∑ 2 Using (13), (14) and hypothesis ki=1 λi ρ3i + m j=1 µj ρi = 0, (22) reduces to ∑k T T i=1 λi [(fi (x) + x zi ) − γi (gi (x) − x vi )] k m [ ( ) ∑ ∑ ( = F x, u; λi αi1 (x, u) ∇(fi (u)+uT zi −γi (gi (u)+uT vi )) + µj αi2 (x, u) ∇(hj (u)+uT wj ) i=1

+

k ∑

j=1 m ( ]) )) ∑ λi αi1 (x, u) ∇p (Ki (u, p) − γi Gi (u, p) + µj αi2 (x, u)∇q Hj (u, q)

i=1

As

αi1 (x, u)

=

(23)

j=1

αi2 (x, u)

=α ˆ (x, u), using the sublinearity of F , we have

k ∑ i=1

λi [(fi (x) + xT zi ) − γi (gi (x) − xT vi )] α ˆ (x, u)

k m [ ( ) ∑ ∑ ( = F x, u; λi ∇(fi (u) + uT zi − γi (gi (u) − uT vi )) + µj ∇(hj (u) + uT wj ) i=1

j=1

+

k ∑

(

λi ∇p (Ki (u, p) − γi Gi (u, p)

i=1

))

+

m ∑ j=1

10 1453

]) µj ∇q Hj (u, q)

(24)

Ahmad et al: Higher order duality in nondifferentiable fractional programming

Now by the feasibility condition (12) and the result F (x, u; 0) = 0, we get k ∑ λi [(fi (x) + xT zi ) − γi (gi (x) − xT vi )] = 0, α(x,u) ˆ i=1

which contradicts (17). This completes the proof.

Theorem 4.2 (Strong duality). If u is an efficient solution of (FP) and Ki (u, 0) = 0, Gi (u, 0) = 0, i = 1, 2, ..., k, Hj (u, 0) = 0, j = 1, 2, ..., m, and a constraint qualifica¯ ∈ Rk , µ tion is satisfied for (F P 1 ϵ¯) for at least one r = 1, 2, ..., k, then there exist λ ¯ ∈ Rm , γ¯ ∈ Rk , z¯i ∈ Rn , v¯i ∈ Rn and w ¯j ∈ Rn , i = 1, 2, ..., k, j = 1, 2, ..., m, such that ¯ p = 0, q = 0) is a feasible solution of (SFD). Further if the conditions (u, γ¯ , z¯, v¯, w, ¯ µ ¯, λ, ¯ p = 0, q = 0) is an efof Weak duality theorem 4.1 are satisfied then (u, α ¯ , z¯, v¯, w, ¯ µ ¯, λ, ficient solution of (SFD) and the corresponding values of the objective functions are equal.

Proof. The proof follows along the lines of Theorem 4.2 [12] in light of the discussions given above and hence being omitted.

5. Conclusion In the present analysis, we focus on a Mond-Weir type and Schaible type dual programs of a nondifferentiable multiobjective fractional programming problem in which every component of the objective and constraints functions contains a term involving the support function of a compact convex set and established weak and strong duality theorems under the assumptions of higher order (F, α, ρ, d)-V-type I functions. The question arise whether the duality results developed in this paper still holds for the nondifferentiable minimax fractional programming problem involving the support function of a compact convex set. This will orient the future research of the authors.

References [1] I. Ahmad: Unified higher order duality in nondifferentiable multiobjective programming, Math. Comput. Modelling, 55, 419-425 (2012). [2] Gulati, T.R., Agarwal, D.: Second-order duality in multiobjective programming involving (F, α, ρ, d)-V-type I functions. Num. Funct. Anal. Optim. 28, 1263-1277

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(2007). [3] Gulati, T.R., Ahmad, I., Agarwal, D.: Sufficiency and duality in multiobjective programming under generalized type I functions. J. Optim. Theory Appl. 135, 411427 (2007). [4] Gulati T.R., Agarwal, D.: Optimality and duality in nondifferentiable multiobjective mathematical programming involving higher order (F, α, ρ, d)-type I functions. J. Appl. Math. Comp. 27, 345-364 (2008). [5] Gulati, T.R., Geeta: Duality in nondifferentiable multiobjective fractional programming problem with generalized invexity. J. Appl. Math. Comp. 35, 103-118 (2010). [6] Kim, D.S., Kim, S.J., Kim, M.H.: Optimality and duality for a class of nondifferentiable multiobjective fractional programming problems. J. Optim. Theory Appl. 129, 131-146 (2006). [7] Liang, Z.A., Huang, H.X., Pardalos, P.M.: Optimality conditions and duality for a class of nonlinear fractional programming problems. J. Optim. Theory Appl. 110, 611-619 (2001). [8] Liang, Z.A., Huang, H.X., Pardalos, P.M.: Efficiency conditions and duality for a class of multiobjective fractional programming problems. J. Global Optim. 27, 447471 (2003). [9] Long, X.J.: Optimality Conditions and Duality for Nondifferentiable Multiobjective Fractional Programming Problems with (C, α, ρ, d)-convexity, J. Optim. Theory Appl. 148, 197-208 (2011). [10] Mangasarian, O.L.: Second and higher-order duality in nonlinear programming. J. Math. Anal. Appl. 51, 607-620 (1975). [11] Mond, B., Zhang, J.: Higher-order invexity and duality in mathematical programming, in: J.P. Crouzeix, et al. (Eds.), Generalized Invexity, Generalized Monotonicity: Recent Results, Kluwer Academic, Dordrecht, pp. 357-372 (1998). [12] Suneja, S.K., Srivastava, M. K., Bhatia, M.: Higher order duality in multiobjective fractional programming with support functions. J. Math. Anal. Appl. 347, 8-17 (2008). 12 1455

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1456-1466, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Fuzzy implicative filters of BE-algebras with degrees in the interval (0, 1] Young Bae Jun1 and Sun Shin Ahn2∗ 1

Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju, 660-701, Korea 2

Department of Mathematics Education, Dongguk University, Seoul 100-715, Korea

Abstract. In defining a fuzzy filter and a fuzzy implicative filter in BE-algebras, several degrees are provided, and then related properties are investigated.

1. Introduction In [5], H. S. Kim and Y. H. Kim introduced the notion of a BE-algebra. S. S. Ahn and K. S. So [3,4] introduced the notion of ideals in BE-algebras. S. S. Ahn et al. [1] fuzzified the concept of BE-algebras, investigated some of their properties. In this paper, we provide several degrees in defining a fuzzy filter and a fuzzy implicative filter. It is a generalization of a fuzzy filter. 2. Preliminaries We recall some definitions and results discussed in [3,4,5]. An algebra (X; ∗, 1) of type (2, 0) is called a BE-algebra if (BE1) (BE2) (BE3) (BE4)

x ∗ x = 1 for all x ∈ X; x ∗ 1 = 1 for all x ∈ X; 1 ∗ x = x for all x ∈ X; x ∗ (y ∗ z) = y ∗ (x ∗ z) for all x, y, z ∈ X (exchange)

We introduce a relation “≤” on a BE-algebra X by x ≤ y if and only if x ∗ y = 1. A non-empty subset S of a BE-algebra X is said to be a subalgebra of X if it is closed under the operation “ ∗ ”. Noticing that x ∗ x = 1 for all x ∈ X, it is clear that 1 ∈ S. A BE-algebra (X; ∗, 1) is said to be self distributive if x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z) for all x, y, z ∈ X. Definition 2.1.([5]) Let (X; ∗, 1) be a BE-algebra and let F be a non-empty subset of X. Then F is called a filter of X if (F1) 1 ∈ F ; (F2) x ∗ y ∈ F and x ∈ F imply y ∈ F . 0

2010 Mathematics Subject Classification: 03G25; 06F35; 08A72. Keywords: BE-algebra; enlarged filter; fuzzy enlarged (implicative) filter with degree. The corresponding author. 0 E-mail: [email protected]; [email protected] 0

∗

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Young Bae Jun and Sun Shin Ahn

Example 2.2.([5]) Let X := {1, a, b, c, d, 0} be a BE-algebra with the following table: ∗ 1 a b c d 0

1 1 1 1 1 1 1

b b a 1 b a 1

a a 1 1 a 1 1

c c c c 1 1 1

d d c c a 1 1

0 0 d c b a 1

Then F1 := {1, a, b} is a filter of X, but F2 := {1, a} is not a filter of X, since a ∗ b ∈ F2 and a ∈ F2 , but b ̸∈ F2 . Proposition 2.3. Let (X; ∗, 1) be a BE-algebra and let F be a filter of X. If x ≤ y and x ∈ F for any y ∈ X, then y ∈ F . Proposition 2.4. Let (X; ∗, 1) be a self distributive BE-algebra. Then the following hold: for any x, y, z ∈ X, (i) if x ≤ y, then z ∗ x ≤ z ∗ y and y ∗ z ≤ x ∗ z. (ii) y ∗ z ≤ (z ∗ x) ∗ (y ∗ z). (iii) y ∗ z ≤ (x ∗ y) ∗ (x ∗ z). A BE-algebra (X; ∗, 1) is said to be transitive if it satisfies Proposition 2.4(iii). 3. Fuzzy filters of BE-algebras with degrees in (0, 1] In what follows let X denote a BE-algebra unless specified otherwise. Definition 3.1. A fuzzy subset µ of a BE-algebra X is called a fuzzy filter of X if it satisfies for all x, y ∈ X (d1) µ(1) ≥ µ(x), (d2) µ(x) ≥ min{µ(y ∗ x), µ(y)}. Proposition 3.2. Let µ be a fuzzy filter of a BE-algebra X. Then for any x, y ∈ X, if x ≤ y, then µ(x) ≤ µ(y). 

Proof. Straightforward.

Definition 3.3. Let F be a non-empty subset of a BE-algebra X which is not necessary a filter of X. We say that a subset G of X is an enlarged filter of X related to F if it satisfies: (1) F is a subset of G, (2) 1 ∈ G, (3) (∀y ∈ X)(∀x ∈ F )(x ∗ y ∈ F ⇒ y ∈ G).

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Fuzzy implicative filters of BE-algebras with degrees in the interval (0, 1]

Obviously, every filter is an enlarged filter of X related to itself. Note that there exists an enlarged filter of X related to any non-empty subset F of X. Example 3.4. Let X = {1, a, b, c, d, 0} be a BE-algebra which is given in Example 2.2. Note that F := {1, a} is not a filter since a ∗ b = a ∈ F , a ∈ F and b ∈ / F . Then G := {1, a, b, c} is an enlarged filter of X related to F and G is not a filter of X since b ∗ d = c, b ∈ G and d ∈ / G. In what follows let λ and κ be members of (0, 1], and let n and k denote a natural number and a real number, respectively, such that k < n unless otherwise specified. Definition 3.5. A fuzzy subset µ of a BE-algebra X is called a fuzzy filter of X with degree (λ, κ) if it satisfies: (1) (∀x ∈ X)(µ(1) ≥ λµ(x)), (2) (∀x, y ∈ X)(µ(x) ≥ κmin{µ(y ∗ x), µ(y)}). Note that if λ ̸= κ, then a fuzzy filter with degree (λ, κ) may not be a fuzzy filter with degree (κ, λ), and vice versa. Obviously, every fuzzy filter is a fuzzy filter with degree (λ, κ), but the converse may not be true. Example 3.6. Let X := {1, a, b, c} be a BE-algebra in which the ∗-operation is given by the following table: ∗ 1 a b c 1 1 a b c a 1 1 b b b 1 a 1 a c 1 1 1 1 Define a fuzzy subset µ : X → [0, 1] by ( ) 1 a b c µ= 0.4 0.3 0.7 0.7 Then µ is a fuzzy filter of X with degree ( 47 , 74 ), but it is neither a fuzzy filter of X nor a fuzzy filter of X with degree ( 54 , 45 ) since µ(1) = 0.4  µ(b) = 0.7 and

4 4 4 × 0.4 = × µ(1) = × min{µ(c ∗ a) = µ(1), µ(c)}. 5 5 5 Define a fuzzy subset ν : X → [0, 1] by ( ) 1 a b c ν= 0.6 0.4 0.7 0.7 µ(a) = 0.3 

Then ν is a fuzzy filter of X with degree ( 54 , 35 ), but it is neither a fuzzy filter of X nor a fuzzy filter of X with degree ( 53 , 45 ) since ν(1) = 0.6  ν(c) = 0.7 1458

Young Bae Jun and Sun Shin Ahn

and ν(a) = 0.4  0.48 =

4 4 4 × 0.6 = × ν(1) = × min{ν(c ∗ a) = ν(1), ν(c)}. 5 5 5

Note that a fuzzy filter with degree (λ, κ) is a fuzzy filter if and only if (λ, κ) = (1, 1). Let λ1 and λ2 be members of (0, 1]. If λ1 > λ2 , then every fuzzy filter with degree λ2 , but the converse is not true(See Example 3.6). Proposition 3.7. Every fuzzy filter of a BE-algebra X with degree (λ, κ) satisfies the following assertions. (i) (∀x, y ∈ X)(µ(x ∗ y) ≥ λκµ(y)). (ii) (∀x, y ∈ X)(y ≤ x ⇒ µ(x) ≥ λκµ(y)). Proof. (i) For any x, y ∈ X, we have µ(x ∗ y) ≥κmin{µ(y ∗ (x ∗ y)), µ(y)} =κmin{µ(x ∗ (y ∗ y)), µ(y)} =κmin{µ(x ∗ 1), µ(y)} =κmin{µ(1), µ(y)} ≥κmin{λµ(y), µ(y)} =κλµ(y). (ii) Let x, y ∈ X be such that y ≤ x. Then y ∗ x = 1. Hence we have µ(x) ≥κmin{µ(y ∗ x), µ(y)} =κmin{µ(1), µ(y)} ≥κmin{λµ(y), µ(y)} =λκµ(y) for any x, y ∈ X.



Corollary 3.8. Let µ be a fuzzy filter of a BE-algebra X with degree (λ, κ). If λ = κ, then (i) (∀x, y ∈ X)(µ(x ∗ y) ≥ λ2 µ(y)). (ii) (∀x, y ∈ X)(y ≤ x ⇒ µ(x) ≥ λ2 µ(y)). Denote by F(X) the set of all filters of a BE-algebra X. Note that a fuzzy subset µ of a BE-algebra X is a fuzzy filter of X if and only if (∀t ∈ [0, 1])(U (µ; t) ∈ F (X) ∪ {∅}). But we know that for any fuzzy subset µ of a BE-algebra X there exist λ, κ ∈ (0, 1) and t ∈ [0, 1] such that (1) µ is a fuzzy filter of X with degree (λ, κ), (2) U (µ; t) ∈ / F (X) ∪ {∅}.

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Fuzzy implicative filters of BE-algebras with degrees in the interval (0, 1]

Example 3.9. Consider the fuzzy subset µ of X = {1, a, b, c} which is given Example 3.6. If t ∈ (0.4, 0.6], then U (µ; t) = {1, b, c} is not a filter of X. But µ is a fuzzy filter of X with degree (0.4, 0.6). Theorem 3.10. Let µ be a fuzzy subset of a BE-algebra X. For any t ∈ [0, 1] with t ≤ t max{λ, κ}, if U (µ; t) is an enlarged filter of X related to U (µ; max{λ,κ} ), then µ is a fuzzy filter of X with degree (λ, κ). t Proof. Assume that µ(1) < t ≤ λµ(x) for some x ∈ X and t ∈ (0, λ]. Then µ(x) ≥ λt ≥ max{λ,κ} t t and so x ∈ U (µ; max{λ,κ} ), i.e., U (µ; max{λ,κ} ) ̸= ∅. Since U (µ; t) is an enlarged filter of X related to t U (µ; max{λ,κ} ), we have 1 ∈ U (µ; t), i.e., µ(1) ≥ t. This is a contradiction, and thus µ(1) ≥ λµ(x) for all x ∈ X. Now suppose that there exist a, b, c ∈ X such that µ(a) < κmin{µ(b ∗ a), µ(b)}. If we take t t := κmin{µ(b ∗ a), µ(b)}, then t ∈ (0, κ] ⊆ (0, max{λ, κ}]. Hence b ∗ a ∈ U (µ; κt ) ⊆ U (µ; max{λ,κ} ) t t and b ∈ U (µ; κ ) ⊆ U (µ; max{λ,κ} ). It follows from Definition 3.3(3) that a ∈ U (µ; t) so that µ(a) ≥ t, which is impossible. Therefore

µ(x) ≥ κmin{µ(y ∗ x), µ(y)} for all x, y ∈ X. Thus µ is a fuzzy filter of X with degree (λ, κ).



Corollary 3.11. Let µ be a fuzzy subset of a BE-algebra X. For any t ∈ [0, 1] with t ≤ nk , if U (µ; t) is an enlarged filter of X related to U (µ; nk t), then µ is a fuzzy filter of X with degree ( nk , nk ). Theorem 3.12. Let t ∈ [0, 1] be such that U (µ; t)(̸= ∅) is not necessary a filter of a BE-algebra X. If µ is a fuzzy filter of X with degree (λ, κ), then U (µ; tmin{λ, κ}) is an enlarged filter of X related to U (µ; t). Proof. Since tmin{λ, κ} ≤ t, we have U (µ; t) ⊆ U (µ; tmin{λ, κ}). Since U (µ; t) ̸= ∅, there exists x ∈ U (µ; t) and so µ(x) ≥ t. By Definition 3.5(1), we obtain µ(1) ≥ λµ(x) ≥ λt ≥ tmin{λ, κ}. Therefore 1 ∈ U (µ; tmin{λ, κ}). Let x, y, z ∈ X be such that y ∗ x ∈ U (µ; t) and y ∈ U (µ; t). Then µ(y ∗ x) ≥ t and µ(y) ≥ t. It follows from Definition 3.5(2) that µ(x) ≥κmin{µ(y ∗ x), µ(y)} ≥κt ≥ tmin{λ, κ}. so that x ∈ U (µ; tmin{λ, κ}). Thus U (µ; tmin{λ, κ}) is an enlarged filter of X related to U (µ; t).  Proposition 3.13. Let µ be a fuzzy filter of a BE-algebra X with degree (λ, κ). If the inequality x ≤ y ∗ z holds for any x, y, z ∈ X, then µ(z) ≥ min{κµ(y), λκ2 µ(x)}. 1460

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Proof. Suppose that x ≤ y ∗ z for all x, y, z ∈ X. Then x ∗ (y ∗ z) = 1 and hence we have µ(y ∗ z) ≥κmin{µ(x ∗ (y ∗ z)), µ(x)} =κmin{µ(1), µ(x)} ≥κmin{λµ(x), µ(x)} =κλµ(x). It follows that µ(z) ≥κmin{µ(y ∗ z), µ(y)} ≥κmin{κλµ(x), µ(y)} =min{κµ(y), κ2 λµ(x)} for all x, y, z ∈ X.



Corollary 3.14. Let µ be a fuzzy filter of a BE-algebra X with degree (λ, κ). If λ = κ and the inequality x ≤ y ∗ z holds for any x, y, z ∈ X, then µ(z) ≥ min{λµ(y), λ3 µ(x)} for all x, y, z ∈ X. Corollary 3.15. Let µ be a fuzzy filter of a BE-algebra X. If the inequality x ≤ y ∗ z holds for any x, y, z ∈ X, then µ(z) ≥ min{µ(y), µ(x)} for all x, y, z ∈ X. 4. Fuzzy implicative filters of BE-algebras with degrees in (0, 1] Definition 4.1. A non-empty subset F of a BE-algebra X is called an implicative filter of X if it satisfies (F1) and (F3) x ∗ (y ∗ z) ∈ F and x ∗ y ∈ F imply x ∗ z ∈ F for all x, y, z ∈ X. Example 4.2. Consider a BE-algebra X = {1, a, b, c, d, 0} which is given Example 2.2. It is easy to see that the set F := {1, a, b} is an implicative filter of X. Note that every implicative filter of a BE-algebra X is a filter of X. Definition 4.3. A fuzzy subset µ of a BE-algebra X is called a fuzzy implicative filter of X if it satisfies (d1) and (d3) µ(x ∗ z) ≥ min{µ(x ∗ (y ∗ z)), µ(x ∗ y)} for all x, y, z ∈ X. Definition 4.4. Let F be a non-empty subset of a BE-algebra X which is not necessary an implicative filter of X. We say that a subset G of X is an enlarged implicative filter of X related to F if it satisfies: 1461

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(1) F is a subset of G, (2) 1 ∈ G, (3) (∀x, y, z ∈ X)(x ∗ (y ∗ z) ∈ F and x ∗ y ∈ F ⇒ x ∗ z ∈ G). Obviously, every implicative filter is an enlarged implicative filter of a BE-algebra X related to itself. Note that there exists an enlarged implicative filter of X related to any non-empty subset F of X. Example 4.5. Consider a BE-algebra X = {1, a, b, c, d, 0} which is given in Example 2.2. Note that F := {1, a} is not both a filter and an implicative filter of X. Then G := {1, a, b, c} is an enlarged implicative filter of X related to F . Proposition 4.6. Let F be a non-empty subset of a BE-algebra X. Every enlarged implicative filter of X related to F is an enlarged filter of X related to F. Proof. Let G be an enlarged implicative filter of X related to F . Putting x = 1 in Definition 4.4(3) and use (BE3), we have (∀y, z ∈ X)(1 ∗ (y ∗ z) = y ∗ z ∈ F and 1 ∗ y = y ∈ F ⇒ 1 ∗ z = z ∈ G). Hence G is an enlarged filter of X related to F .



The converse of Proposition 4.6 is not true in general as seen in the following example. Example 4.7. Let X := {1, a, b, c} be a BE-algebra([3]) in which the ∗-operation is given by the following table: ∗ 1 a b c 1 1 a b c a 1 1 a a b 1 1 1 a c 1 1 a 1 Let F := {1} and G := {1, c}. Then G is an enlarged filter of F but it is not an enlarged implicative filter of F since b ∗ (a ∗ c) = 1 ∈ F, b ∗ a = 1 ∈ F and b ∗ c = a ∈ / G. Definition 4.8. A fuzzy subset µ of a BE-algebra X is called a fuzzy implicative filter of X with degree (λ, κ) if it satisfies Definition 3.5(1) (2) (∀x, y, z ∈ X)(µ(x ∗ z) ≥ κmin{µ(x ∗ (y ∗ z)), µ(x ∗ y)}). Note that if λ ̸= κ, then a fuzzy implicative filter with degree (λ, κ) may not be a fuzzy implicative filter with degree (κ, λ), and vice versa. Obviously, every fuzzy implicative filter is a fuzzy implicative filter with degree (λ, κ), but the converse may not be true. Example 4.9. Consider a BE-algebra X = {1, a, b, c, d, 0} which is given in Example 2.2. Define a fuzzy subset µ : X → [0, 1] by ( ) 1 a b c d 0 µ= 0.7 0.8 0.8 0.4 0.5 0.4 1462

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Then µ is a fuzzy implicative filter of X with degree ( 56 , 36 ), but it is neither a fuzzy filter of X nor a fuzzy implicative filter of X with degree ( 36 , 56 ) since µ(1) = 0.7  µ(a) = 0.8 and µ(1 ∗ 0) = µ(0) = 0.4 0.42 =

5 5 × 0.5 = × µ(d) 6 6

5 = × min{µ(1 ∗ (a ∗ 0) = µ(d), µ(1 ∗ a) = µ(a)}. 6 Obviously, every fuzzy implicative filter of a BE-algebra X is a fuzzy implicative filter of X with degree (λ, κ), but the converse may not be true. In fact, the fuzzy implicative filter µ of X with degree ( 36 , 56 ) in Example 4.9 is not a fuzzy implicative filter of X. Note that a fuzzy implicative filter with degree (λ, κ) is a fuzzy implicative filter if and only if (λ, κ) = (1, 1). Proposition 4.10. If µ is a fuzzy implicative filter of a BE-algebra X degree (λ, κ), then µ is a fuzzy filter of X with degree (λ, κ). Proof. Putting x := 1 in Definition 4.8(2), we have µ(z) = µ(1 ∗ z) ≥κmin{µ(1 ∗ (y ∗ z)), µ(1 ∗ y)} =κmin{µ(y ∗ z), µ(y)} for any y, z ∈ X. Thus µ is a fuzzy filter of X with degree (λ, κ).



The converse of Proposition 4.10 is not true in general as seen in the following example. Example 4.11. Consider a BE-algebra X = {1, a, b, c} which is given in Example 4.7. Define a fuzzy subset µ : X → [0, 1] by ( ) 1 a b c µ= 0.6 0.3 0.3 0.7 Then µ is a fuzzy filter of X with degree ( 63 , 74 ), but it is neither a fuzzy filter of X nor a fuzzy implicative filter of X with degree ( 63 , 47 ) since µ(1) = 0.6  µ(c) = 0.7 and µ(b ∗ c) = µ(a) = 0.3 0.34 =

4 4 × 0.6 = × µ(1) 7 7

4 = × min{µ(b ∗ (a ∗ c)) = µ(1), µ(b ∗ a) = µ(1)}. 7 Proposition 4.12. Every fuzzy implicative filter of a BE-algebra X with degree (λ, κ) satisfies the following assertions. (i) (∀x, y ∈ X)(µ(x ∗ y) ≥ λκµ(y)). (ii) (∀x, y ∈ X)(x ≤ y ⇒ µ(y) ≥ λκµ(x)).

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Proof. It follows from Proposition 3.7 and Proposition 4.10.



Corollary 4.13. Let µ be a fuzzy implicative filter of a BE-algebra X with degree (λ, κ). If λ = κ, then (i) (∀x, y ∈ X)(µ(x ∗ y) ≥ λ2 µ(y)). (ii) (∀x, y ∈ X)(x ≤ y ⇒ µ(y) ≥ λ2 µ(x)). Proposition 4.14. Let µ be a fuzzy implicative filter of a BE-algebra X with degree (λ, κ). Then the following are hold: (i) ∀x, y ∈ X)(µ(x ∗ y) ≥ λκµ(x ∗ (x ∗ y))). (ii) (∀x, y, z ∈ X)(µ(y ∗ z) ≥ λκ2 min{µ(x ∗ (y ∗ (y ∗ z))), µ(x)}). Proof. (i) Assume that µ is a fuzzy implicative filter of a BE-algebra X with degree (λ, κ). Putting z := y, y := x in Definition 4.8(2), we have µ(x ∗ y) ≥κmin{µ(x ∗ (x ∗ y)), µ(x ∗ x)} =κmin{µ(x ∗ (x ∗ y)), µ(1)} ≥κmin{µ(x ∗ (x ∗ y)), λµ(x ∗ (x ∗ y))} =κλµ(x ∗ (x ∗ y)) for all x, y ∈ X. Thus (i) holds. (ii) Since µ is a fuzzy filter of X with degree (λ, κ) and using (i), we have µ(y ∗ z) ≥λκµ(y ∗ (y ∗ z)) ≥λκ2 min{µ(x ∗ (x ∗ (y ∗ z))), µ(x)} for any x, y, z ∈ X. Hence (ii) holds.  Corollary 4.15. Let µ be a fuzzy implicative filter of a BE-algebra X with degree (λ, κ). If λ = κ, then (i) (∀x, y ∈ X)(µ(x ∗ y) ≥ λ2 µ(x ∗ (x ∗ y))). (ii) (∀x, y, z ∈ X)(µ(y ∗ z) ≥ κ3 min{µ(x ∗ (y ∗ (y ∗ z))), µ(x)}). Proposition 4.16. Let X be a self distributive BE-algebra X. Then µ is a fuzzy filter of X with degree (λ, κ) if and only if it is a fuzzy implicative filter of X with degree (λ, κ). Proof. Proposition 4.10, a fuzzy implicative filter of a BE-algebra X with degree (λ, κ) is a fuzzy filter of X with degree (λ, κ). Conversely, assume that µ is a fuzzy filter of a BE-algebra X with degree (λ, κ). Since X is a self distributive BE-algebra, we have µ(x ∗ z) ≥κmin{µ((x ∗ y) ∗ (x ∗ z)), µ(x ∗ y)} =κmin{µ(x ∗ (y ∗ z)), µ(x ∗ y)} for any x, y, z ∈ X. Hence X is a fuzzy implicative filter of X with degree (λ, κ). 1464



Young Bae Jun and Sun Shin Ahn

Denote by FI (X) the set of all implicative filters of a BE-algebra X. Note that a fuzzy subset µ of a BE-algebra X is a fuzzy implicative filter of X if and only if (∀t ∈ [0, 1])(U (µ; t) ∈ FI (X) ∪ {∅}). But we know that for any fuzzy subset µ of a BE-algebra X there exist λ, κ ∈ (0, 1) and t ∈ [0, 1] such that (1) µ is a fuzzy implicative filter of X with degree (λ, κ), (2) U (µ; t) ∈ / FI (X) ∪ {∅}. Example 4.17. Let X := {1, a, b, c} be a set table: ∗ 1 1 1 a 1 b 1 c 1

in which the ∗-operation is given by the following a a 1 a 1

b b b 1 b

c c c c 1

Then X is self distributive BE-algebra. Define a fuzzy subset µ : X → [0, 1] by ) ( 1 a b c µ= 0.4 0.3 0.2 0.6 If t ∈ (0.4, 0.6], then U (µ; t) = {1, c} is not an implicative filter of X since 1 ∗ (c ∗ a) = 1 ∈ {1, c}, and 1 ∗ c ∈ {1, c} but 1 ∗ a = a ∈ / {1, c} . But µ is a fuzzy implicative filter of X with degree (0.4, 0.6). Theorem 4.18. Let µ be a fuzzy subset of a BE-algebra X. For any t ∈ [0, 1] with t ≤ t max{λ, κ}, if U (µ; t) is an enlarged implicative filter of X related to U (µ; max{λ,κ} ), then µ is a fuzzy implicative filter of X with degree (λ, κ). t Proof. Assume that µ(1) < t ≤ λµ(x) for some x ∈ X and t ∈ (0, λ]. Then µ(x) ≥ λt ≥ max{λ,κ} t t and so x ∈ U (µ; max{λ,κ} ), i.e., U (µ; max{λ,κ} ) ̸= ∅. Since U (µ; t) is an enlarged filter of X related to t U (µ; max{λ,κ} ), we have 1 ∈ U (µ; t), i.e., µ(1) ≥ t. This is a contradiction, and thus µ(1) ≥ λµ(x) for all x ∈ X. Now suppose that there exist a, b, c ∈ X such that µ(a ∗ c) < κmin{µ(a ∗ (b ∗ c)), µ(a ∗ b)}. If we take t := κmin{µ(a ∗ (b ∗ c)), µ(a ∗ b)}, then t ∈ (0, κ] ⊆ (0, max{λ, κ}]. Hence a ∗ (b ∗ c) ∈ t t ) and a ∗ b ∈ U (µ; κt ) ⊆ U (µ; max{λ,κ} ). It follows from Definition 4.8(2) U (µ; κt ) ⊆ U (µ; max{λ,κ} that a ∗ c ∈ U (µ; t) so that µ(a ∗ c) ≥ t, which is impossible. Therefore

µ(x ∗ z) ≥ κmin{µ(x ∗ (y ∗ z)), µ(x ∗ y)} for all x, y, z ∈ X. Thus µ is a fuzzy implicative filter of X with degree (λ, κ).



Corollary 4.19. Let µ be a fuzzy subset of a BE-algebra X. For any t ∈ [0, 1] with t ≤ nk , if U (µ; t) is an enlarged implicative filter of X related to U (µ; nk t), then µ is a fuzzy implicative filter of X with degree ( nk , nk ). 1465

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Theorem 4.20. Let t ∈ [0, 1] be such that U (µ; t)(̸= ∅) is not necessary an implicative filter of a BE-algebra X. If µ is a fuzzy implicative filter of X with degree (λ, κ), then U (µ; tmin{λ, κ}) is an enlarged implicative filter of X related to U (µ; t). Proof. Since tmin{λ, κ} ≤ t, we have U (µ; t) ⊆ U (µ; tmin{λ, κ}). Since U (µ; t) ̸= ∅, there exists x ∈ U (µ; t) and so µ(x) ≥ t. By Definition 4.8(1), we obtain µ(1) ≥ λµ(x) ≥ λt ≥ tmin{λ, κ}. Therefore 1 ∈ U (µ; tmin{λ, κ}). Let x, y, z ∈ X be such that x ∗ (y ∗ z) ∈ U (µ; t) and x ∗ y ∈ U (µ; t). Then µ(x ∗ (y ∗ z)) ≥ t and µ(x ∗ y) ≥ t. It follows from Definition 4.8(2) that µ(x ∗ z) ≥κmin{µ(x ∗ (y ∗ z)), µ(x ∗ y)} ≥κt ≥ tmin{λ, κ}. so that x ∗ z ∈ U (µ; tmin{λ, κ}). Thus U (µ; tmin{λ, κ}) is an enlarged implicative filter of X related to U (µ; t).  References [1] [2] [3] [4] [5] [6] [7] [8]

S. S. Ahn, Y. H. Kim and K. S. So, Fuzzy BE-algebras, J. Appl. Math. and Informatics 29 (2011), 1049-1057. S. S. Ahn and J. M. Ko, On vague filters in BE-algebras, Commun. Korean Math. Soc. 26 (2011), 417-425. S. S. Ahn and K. K. So, On ideals and upper sets in BE-algebras, Sci. Math. Japon. 68 (2008), 279-285. S. S. Ahn and K. K. So, On generalized upper sets in BE-algebras, Bull. Korean Math. Soc. 46 (2009), 281-287. H. S. Kim and Y. H. Kim, On BE-algebras, Sci. Math. Japon. 66 (2007), 113-116. Y. B. Jun, E. H. Roh and K. J. Lee, Fuzzy subalgebras and ideals of BCK/BC𝐼-algebras with degrees in the interval (0, 1], Fuzzy Sets and Systems, submitted. J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa Co. Seoul (1994). L. A. Zadeh, Fuzzy sets, Inform. Control 56 (2008), 338-353.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1467-1475, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

An AQ-functional equation in paranormed spaces Taek Min Kim Mathematics Branch, Seoul Science High School, Seoul 110-530, Korea e-mail: [email protected]

Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea e-mail: [email protected]

Seo Hong Park∗ Mathematics Branch, Seoul Science High School, Seoul 110-530, Korea e-mail: [email protected] Abstract. In this paper, we prove the Hyers-Ulam stability of an additive-quadratic functional equation in paranormed spaces. Keywords: Hyers-Ulam stability, paranormed space, additive-quadratic functional equation.

1. Introduction and preliminaries The concept of statistical convergence for sequences of real numbers was introduced by Fast [9] and Steinhaus [35] independently and since then several generalizations and applications of this notion have been investigated by various authors (see [10, 19, 22, 23, 33]). This notion was defined in normed spaces by Kolk [20]. We recall some basic facts concerning Fr´ echet spaces. Definition 1.1. [37] Let X be a vector space. A paranorm P : X → [0, ∞) is a function on X such that (1) P (0) = 0; (2) P (−x) = P (x) ; (3) P (x + y) ≤ P (x) + P (y) (triangle inequality) (4) If {t𝑛 } is a sequence of scalars with t𝑛 → t and {x𝑛 } ⊂ X with P (x𝑛 − x) → 0, then P (t𝑛 x𝑛 − tx) → 0 (continuity of multiplication). The pair (X, P ) is called a paranormed space if P is a paranorm on X. The paranorm is called total if, in addition, we have (5) P (x) = 0 implies x = 0. A Fr´ echet space is a total and complete paranormed space. The stability problem of functional equations originated from a question of Ulam [36] concerning the stability of group homomorphisms. Hyers [13] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings ∗

02010

Mathematics Subject Classification: Primary 35A17; 39B52; 39B72. Corresponding author.

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T.M. Kim, C. Park, S.H. Park and by Th.M. Rassias [27] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. In 1990, Th.M. Rassias [28] during the 27𝑡h International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. In 1991, Gajda [11] following the same approach as in Th.M. Rassias [27], gave an affirmative solution to this question ˇ for p > 1. It was shown by Gajda [11], as well as by Th.M. Rassias and Semrl [32] that one cannot prove a Th.M. Rassias’ type theorem when p = 1 (cf. the books of P. Czerwik [5], D.H. Hyers, G. Isac and Th.M. Rassias [14]). In 1982, J.M. Rassias [25] followed the innovative approach of the Th.M. Rassias’ theorem [27] in which he replaced the factor ∥x∥𝑝 + ∥y∥𝑝 by ∥x∥𝑝 · ∥y∥q for p, q ∈ ℝ with p + q ̸= 1. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [34] for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa [3] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [4] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [1, 8, 15, 17, 18, 24, 26], [29]–[31]). Throughout this paper, assume that (X, P ) is a Fr´ echet space and that (Y, ∥ · ∥) is a Banach space. In this paper, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation ( ) ( ) ( ) x+y x−y y−x 2f +f +f = f (x) + f (y) (1.1) 2 2 2 in paranormed spaces. One can easily show that an odd mapping f : X → Y satisfies (1.1) if and only if the odd mapping mapping f : X → Y is an additive mapping, i.e., ( ) x+y 2f = f (x) + f (y). 2 One can easily show that an even mapping f : X → Y satisfies (1.1) if and only if the even mapping f : X → Y is a quadratic mapping, i.e., ( ) ( ) x+y x−y 2f + 2f = f (x) + f (y). 2 2 2. Hyers-Ulam stability of the functional equation (1.1): an odd mapping case For a given mapping f , we define ( ) ( ) ( ) x+y x−y y−x Df (x, y) : = 2f +f +f − f (x) − f (y). 2 2 2 In this section, we prove the Hyers-Ulam stability of the functional equation Df (x, y) = 0 in paranormed spaces: an odd mapping case. Note that P (2x) ≤ 2P (x) for all x ∈ Y .

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AQ-functional equation in paranormed spaces Theorem 2.1. Let ϕ : Y → [0, ∞) be a function such that ∞ (x y) ∑ π(x, y) := 2𝑗 ϕ 𝑗 , 𝑗 < +∞ 2 2 𝑗=0 for all x, y ∈ Y . Let f : Y → X be an odd mapping such that P (Df (x, y)) ≤ ϕ(x, y)

(2.1)

for all x, y ∈ Y . Then there exists a unique additive mapping A : Y → X such that P (f (x) − A(x)) ≤ π(x, 0)

(2.2)

for all x ∈ Y . Proof. Considering f as an odd mapping, we have ( ( ) ) x+y P 2f − f (x) − f (y) ≤ ϕ(x, y) 2 for all x, y ∈ Y . Letting y = 0 in (2.3), we get

(2.3)

( ( x )) P f (x) − 2f ≤ ϕ(x, 0) 2

for all x, y ∈ Y . Hence ( (x) ( x )) 𝑚−1 (x ) ∑ P 2𝑛 f 𝑛 − 2𝑚 f 𝑚 ≤ 2𝑗 ϕ 𝑗 , 0 2 2 2 𝑗=𝑛

(2.4)

holds for all non-negative integers n and m with m > n and all x ∈ Y . It follows from (2.4) that the sequence {2𝑘 f ( 2𝑥𝑘 )} is a Cauchy sequence for all x ∈ Y . Since X is complete, the sequence {2𝑘 f ( 2𝑥𝑘 )} converges. So the mapping A : Y → X can be defined as (x) A(x) := lim 2𝑘 f 𝑘 𝑘→∞ 2 for all x ∈ Y . By (2.1), ( ( x y )) (x y) P (DA(x, y)) = lim P 2𝑘 Df 𝑘 , 𝑘 ≤ lim 2𝑘 ϕ 𝑘 , 𝑘 = 0 𝑘→∞ 𝑘→∞ 2 2 2 2 for all x, y ∈ Y . So DA(x, y) = 0. Since f : Y → X is odd, A : Y → X is odd. So the mapping A : Y → X is additive. Moreover, letting n = 0 and passing the limit m → ∞ in (2.4), we get (2.2). So there exists an additive mapping A : Y → X satisfying (2.2). Now, let T : Y → X be another additive mapping satisfying (2.2). Then we have ( x )) ( (x) P (A(x) − T (x)) = P 2q A q − 2q T q ( ( 2( x ) ( x2 ))) ( ( (x) ( x ))) q ≤ P 2 A q −f q + P 2q T − f 2 2 2q 2q x ≤ 2 × 2q π( q , 0), 2 which tends to zero as q → ∞ for all x ∈ Y . So we can conclude that A(x) = T (x) for all x ∈ Y . This proves the uniqueness of A. Thus the mapping A : Y → X is the unique additive mapping satisfying (2.2). 

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T.M. Kim, C. Park, S.H. Park Corollary 2.2. Let r, θ be positive real numbers with r > 1, and let f : Y → X be an odd mapping such that P (Df (x, y)) ≤ θ(∥x∥r + ∥y∥r )

(2.5)

for all x, y ∈ Y . Then there exists a unique additive mapping A : Y → X such that 2r P (f (x) − A(x)) ≤ r θ∥x∥r 2 −2 for all x ∈ Y . Proof. Letting ϕ(x, y) := θ(∥x∥r + ∥y∥r ) in Theorem 2.1, we obtain the result.



Theorem 2.3. Let ϕ : X → [0, ∞) be a function such that π(x, y) :=

∞ ∑ ) 1 ( 𝑗 ϕ 2 x, 2𝑗 y < ∞ 𝑗 2 𝑗=1

for all x, y ∈ X. Let f : X → Y be an odd mapping such that ∥Df (x, y)∥ ≤ ϕ(x, y)

(2.6)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤ π(x, 0)

(2.7)

for all x ∈ X. Proof. Considering f as an odd mapping, we have

(

)

2f x + y − f (x) − f (y) ≤ ϕ(x, y)

2

(2.8)

for all x, y ∈ X. Letting y = 0 and replacing x by 2x in (2.8), we get ∥2f (x) − f (2x)∥ ≤ ϕ(2x, 0) for all x, y ∈ X. Hence

𝑚−1

1

∑ 1

f (2𝑛 x) − 1 f (2𝑚 x) ≤ ϕ(2𝑗+1 x, 0)

2𝑛

𝑚 𝑗+1 2 2 𝑗=𝑛

(2.9)

holds for all non-negative integers n and m with m > n and all x ∈ X. It follows from (2.9) that the sequence { 21𝑘 f (2𝑘 x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21𝑘 f (2𝑘 x)} converges. So the mapping A : X → Y can be defined as ) 1 ( A(x) := lim 𝑘 f 2𝑘 x 𝑘→∞ 2 for all x ∈ X. By (2.6),

1 ( 𝑘 ) ) 1 ( 𝑘

lim 𝑘 ϕ 2𝑘 x, 2𝑘 y = 0 ∥DA(x, y)∥ = lim 𝑘 Df 2 x, 2 y

≤ 𝑘→∞ 𝑘→∞ 2 2 for all x, y ∈ X, and DA(x, y) = 0 follows. Also, since f : X → Y is odd, A : X → Y is odd. So the mapping A : X → Y is additive. Moreover, letting n = 0 and passing the limit m → ∞ in (2.9), we get (2.7). So there exists an additive mapping A : X → Y satisfying (2.7).

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AQ-functional equation in paranormed spaces Now, let T : X → Y be another additive mapping satisfying (2.7). Then we have



1 1 q q A (2 x) − T (2 x) ∥A(x) − T (x)∥ =

2q q 2



1

1

q q

+ (T (2q x) − f (2q x)) ≤ (A (2 x) − f (2 x))

2q

2q

1 ≤ 2 × q π(2q x, 0), 2 which tends to zero as q → ∞ for all x ∈ X. So we have A(x) = T (x) for all x ∈ X. This proves the uniqueness of A. Thus the mapping A : X → Y is the unique additive mapping satisfying (2.7).  Corollary 2.4. Let r, θ be positive real numbers with r < 1, and let f : X → Y be an odd mapping such that ∥Df (x, y)∥ ≤ θ(P (x)r + P (y)r )

(2.10)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that 2r θP (x)r ∥f (x) − A(x)∥ ≤ 2 − 2r for all x ∈ X. 

Proof. Letting ϕ(x, y) := θ(P (x)r + P (y)r ) in Theorem 2.3, we obtain the result. 3. Hyers-Ulam stability of the functional equation (1.1): an even mapping case

In this section, we prove the Hyers-Ulam stability of the functional equation Df (x, y) = 0 in paranormed spaces: an even mapping case. Note that P (2x) ≤ 2P (x) for all x ∈ Y . Theorem 3.1. Let ϕ : X → [0, ∞) be a function such that ∞ ∑ ) 1 ( 𝑗 π(x, y) := ϕ 2 x, 2𝑗 y < +∞ 𝑗 4 𝑗=1 for all x, y ∈ X. Let f : X → Y be an even mapping such that f (0) = 0 and ∥Df (x, y)∥ ≤ ϕ(x, y)

(3.1)

for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that ∥f (x) − Q(x)∥ ≤ π(x, 0)

(3.2)

for all x ∈ X. Proof. Letting y = 0 in (3.1), we get ∥4f

(x) 2

− f (x)∥ ≤ ϕ(x, 0)

(3.3)

for all x ∈ X. Replacing x by 2𝑗+1 x in (3.3), we get ( ) ( ) ( ) ∥4f 2𝑗 x − f 2𝑗+1 x ∥ ≤ ϕ 2𝑗+1 x, 0 for all x ∈ X. Hence ∥

𝑚 ∑ ) 1 ( 𝑗 1 1 𝑚 𝑛 ≤ f (2 x) − f (2 y) ∥ ϕ 2 x, 0 𝑚 𝑛 𝑗 4 4 4 𝑗=𝑛+1

1471

(3.4)

T.M. Kim, C. Park, S.H. Park for all non-negative integers n and m with m > n and all x ∈ X. ( ) It follows from (3.4) that that the sequence { 41𝑘 f 2𝑘 x } is a Cauchy sequence for all x ∈ X. Since ( ) Y is complete, the sequence { 41𝑘 f 2𝑘 x } converges. So one can define the mapping Q : X → Y by Q(x) := lim

𝑘→∞

1 ( 𝑘 ) f 2 x 4𝑘

for all x ∈ X. By (3.1), ∥DQ(x, y))∥ = lim ∥ 𝑘→∞

) ( ) 1 1 ( Df 2𝑘 x, 2𝑘 y ∥ ≤ lim 𝑘 ϕ 2𝑘 x, 2𝑘 y = 0 𝑘→∞ 4 4𝑘

for all x, y ∈ X. So DQ(x, y) = 0. Since f : X → Y is even, Q : X → Y is even. So the mapping Q : X → Y is quadratic. Moreover, letting n = 0 and passing the limit m → ∞ in (3.4), we get (3.2). So there exists a quadratic mapping Q : X → Y satisfying (3.2). ( ) Let T : X → Y be a quadratic mapping satisfying (3.2). Since T satisfies 4T 𝑥2 = T (x), we have T (x) = 41q T (2q x) for all integer q. Hence 1 1 Q (2q x) − q T (2q x) ∥ q 4 4 1 1 q ≤ ∥ q (Q (2 x) − f (2q x)) ∥ + ∥ q (T (2q x) − f (2q x)) ∥ 4 4 1 q ≤ 2 × q π (2 x, 0) , 4 which tends to zero as q → ∞ for all x ∈ X. So Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. Thus the mapping Q : X → Y is the unique quadrative mapping satisfying (3.2).  ∥Q(x) − T (x)∥ = ∥

Corollary 3.2. Let r, θ be positive real numbers with r < 2, and let f : X → Y be an even mapping such that f (0) = 0 and r

r

∥Df (x, y)∥ ≤ θ(P (x) + P (y) ) for all x, y ∈ Y . Then there exists a unique quadrative mapping Q : X → Y such that ∥f (x) − Q(x)∥ ≤

2r r θP (x) 4 − 2r

for all x ∈ Y . r

r

Proof. Letting ϕ(x, y) := θ(P (x) + P (y) ) in Theorem 3.1, we obtain the result.



Theorem 3.3. Let ϕ : Y → [0, ∞) be a function such that π(x, y) :=

∞ ∑ 𝑗=0

4𝑗 ϕ

(x

y) , < +∞ 2𝑗 2𝑗

for all x, y ∈ Y . Let f : Y → X be an even mapping such that f (0) = 0 and P (Df (x, y)) ≤ ϕ(x, y)

(3.5)

for all x, y ∈ Y . Then there exists a unique quadratic mapping Q : Y → X such that P (f (x) − Q(x)) ≤ π(x, 0) for all x ∈ Y .

1472

(3.6)

AQ-functional equation in paranormed spaces Proof. Letting y = 0 in (3.5), we get ) ( (x) − f (x) ≤ ϕ(x, 0) P 4f 2 for all x ∈ Y . Replacing x by 2𝑥j in (3.7), we get ( ( x ) ( x )) (x ) P 4f 𝑗+1 − f 𝑗 ≤ ϕ 𝑗,0 2 2 2 for all x ∈ Y . Hence ( ( x ) ( x )) 𝑚−1 (x ) ∑ P 4𝑚 f 𝑚 − 4𝑛 f 𝑛 ≤ 4𝑗 ϕ 𝑗 , 0 2 2 2 𝑗=𝑛

(3.7)

(3.8)

for all non-negative integers n and m with m > n and all x ∈ Y . ( ) It follows from (3.8) that that the sequence {4𝑘 f 2𝑥𝑘 } is a Cauchy sequence for all x ∈ Y . Since X ( ) is complete, the sequence {4𝑘 f 2𝑥𝑘 } converges. So one can define the mapping Q : Y → X by (x) Q(x) := lim 4𝑘 f 𝑘 𝑘→∞ 2 for all x ∈ Y . By (3.5), ( ( x y )) (x y) P (DQ(x, y))) = lim P 4𝑘 Df 𝑘 , 𝑘 ≤ lim 4𝑘 ϕ 𝑘 , 𝑘 = 0 𝑘→∞ 𝑘→∞ 2 2 2 2 for all x, y ∈ Y . So DQ(x, y) = 0. Since f : Y → X is even, Q : Y → X is even. So the mapping Q : Y → X is quadratic. Moreover, letting n = 0 and passing the limit m → ∞ in (3.8), we get (3.6). So there exists a quadratic mapping Q : Y → X satisfying (3.6). ( ) Let T : Y → X be a quadratic mapping satisfying (3.6). Since T satisfies 4T 𝑥2 = T (x), we have ( ) T (x) = 4q T 2𝑥q for all integer q. Hence ( (x) ( x )) P (Q(x) − T (x)) = P 4q Q q − 4q T q ( ( 2( x ) ( x2 ))) ( ( (x) ( x ))) ≤ P 4q Q q − f q + P 4q T − f 2 2q 2q ( x2 ) q ≤ 2 × 4 π q,0 , 2 which tends to zero as q → ∞ for all x ∈ X. So Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. Thus the mapping Q : Y → X is the unique quadrative mapping satisfying (3.6).  Corollary 3.4. Let r, θ be positive real numbers with r > 2, and let f : X → Y be an even mapping such that f (0) = 0 and P (Df (x, y)) ≤ θ(∥x∥r + ∥y∥r ) for all x, y ∈ Y . Then there exists a unique quadrative mapping Q : Y → X such that 2r θ∥x∥r P (f (x) − Q(x)) ≤ r 2 −4 for all x ∈ Y . Proof. Letting ϕ(x, y) := θ(∥x∥r + ∥y∥r ) in Theorem 3.3, we obtain the result.



(−𝑥) (−𝑥) Let fo (x) := f (𝑥)−f and fe (x) := f (𝑥)+f . Then fo is odd and fe is even. fo , fe satisfy the 2 2 functional equation (1.1) if and only if f does.

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T.M. Kim, C. Park, S.H. Park Theorem 3.5. Let r, θ be positive real numbers with r > 2. Let f : Y → X be a mapping satisfying f (0) = 0 and (2.5). Then there exist an additive mapping A : Y → X and a quadratic mapping Q : Y → X such that ( r+1 ) 2r+1 2 + θ∥x∥r P (2f (x) − A(x) − Q(x)) ≤ 2r − 2 2r − 4 for all x ∈ Y . Theorem 3.6. Let r, θ be positive real numbers with r < 1. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.10). Then there exist an additive mapping A : X → Y and a quadratic mapping Q : X → Y such that ( r+1 ) 2r+1 2 + θP (x)r ∥2f (x) − A(x) − Q(x)∥ ≤ 2 − 2r 4 − 2r for all x ∈ X. Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), and T.M. Kim and S.H. Park were supported by R & E Program in 2012. References [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [4] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [5] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [6] M. Eshaghi-Gordji, S. Abbaszadeh and C. Park, On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces, J. Inequal. Appl. 2009, Article ID 153084, 26 pages (2009). [7] M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, C. Park and S. Zolfaghari, Stability of an additivecubic-quartic functional equation, Adv. Difference Equat. 2009, Article ID 395693, 20 pages (2009). [8] M. Eshaghi Gordji and M.B. Savadkouhi, Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces, Appl. Math. Letters 23 (2010), 1198–1202. [9] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. [10] J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313. [11] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [12] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [13] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [14] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998.

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AQ-functional equation in paranormed spaces [15] G. Isac and Th.M. Rassias, On the Hyers-Ulam stability of ψ-additive mappings, J. Approx. Theory 72 (1993), 131–137. [16] K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878. [17] K. Jun and Y. Lee, A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations, J. Math. Anal. Appl. 297 (2004), 70–86. [18] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [19] S. Karakus, Statistical convergence on probabilistic normed spaces, Math. Commun. 12 (2007), 11–23. [20] E. Kolk, The statistical convergence in Banach spaces, Tartu Ul. Toime. 928 (1991), 41–52. [21] S. Lee, S. Im and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), 387–394. [22] M. Mursaleen, λ-statistical convergence, Math. Slovaca 50 (2000), 111–115. [23] M. Mursaleen and S.A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Computat. Anal. Math. 233 (2009), 142–149. [24] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [25] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982) 126–130. [26] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268–273. [27] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [28] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [29] Th.M. Rassias (ed.), Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. [30] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [31] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl. 62 (2000), 23–130. ˇ [32] Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. ˇ at, On the statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), [33] T. Sal´ 139–150. [34] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [35] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–34. [36] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [37] A. Wilansky, Modern Methods in Topological Vector Space, McGraw-Hill International Book Co., New York, 1978.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1476-1483, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

A note on special fuzzy differential subordinations using generalized S˘al˘agean operator and Ruscheweyh derivative Alina Alb Lupa¸s Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected] Abstract m In the present paper we establish several fuzzy differential subordinations regardind the operator RDλ,α , m m m m m given by RDλ,α : An → An , RDλ,α f (z) = (1−α)R f (z)+αDλ f (z), where R f (z) denote the Ruscheweyh derivative, Dλm f (z) is the generalized S˘ al˘ agean operator and An = {f ∈ H(U ), f (z) = z+an+1 z n+1 +. . . , z ∈ U } is the class of normalized analytic functions with A1 = A. A certain fuzzy class, denoted by RDF m (δ, λ, α) , of analytic functions in the open unit disc is introduced by means of this operator. By making use of the concept of fuzzy differential subordination we will derive various properties and characteristics of the class m RDF m (δ, λ, α) . Also, several fuzzy differential subordinations are established regarding the operator RDλ,α .

Keywords: fuzzy differential subordination, convex function, fuzzy best dominant, differential operator, generalized S˘al˘agean operator, Ruscheweyh derivative. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction

One of the most recently study methods in the one complex variable functions theory is the admissible functions method known as ”the differential subordination method” introduced by S.S. Miller and P.T. Mocanu in [11], [12] and developed in [13]. The application of this method allows to one obtain some special results and to prove easily some classical results from this domain. More results obtained by the differential subordinations method are differential inequalities. From the development of this method has been written a large number of papers and monographs in the one complex variable functions theory domain. A justification of the introduction of the differential subordinations theory was presented in [14], ”knowing the properties of differential expression for a function we can determine the properties of that function on a given interval.” By publication of the papers [14] and [15] the authors wanted to launch a new research direction in mathematics that combines the notions from the complex functions domain with the fuzzy sets theory. In the same way as mentioned, the author can justify that by knowing the properties of a differential expression on a fuzzy set for a function one can be determined the properties of that function on a given fuzzy set. The author has analyzed the case of one complex functions, leaving as ”open problem” the case of real functions. The author is aware that this new research alternative can be realized only through the joint effort of researchers from both domains. The ”open problem” statement leaves open the interpretation of some notions from the fuzzy sets theory such that each one interpret them personally according to their scientific concerns, making this theory more attractive. The notion of fuzzy subordination was introduced in [14]. In [15] the authors have defined the notion of fuzzy differential subordination. In this paper we will study fuzzy differential subordinations obtained with the differential operator defined in [4]. Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and H(U ) the space of holomorphic functions in U . Let An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U } with A1 = A and H[a, n] = {f ∈ H(U ) : f (z) = a + an z n + an+1 z n+1 n + . . . , z ∈ U 00} for a ∈ C and n ∈oN. (z) Denote by K = f ∈ A : Re zff 0 (z) + 1 > 0, z ∈ U , the class of normalized convex functions in U . 1 1476

Alina Lupas: special fuzzy differential subordinations

In order to use the concept of fuzzy differential subordination, we remember the following definitions: Definition 1.1 [10] A pair (A, FA ), where FA : X → [0, 1] and A = {x ∈ X : 0 < FA (x) ≤ 1} is called fuzzy subset of X. The set A is called the support of the fuzzy set (A, FA ) and FA is called the membership function of the fuzzy set (A, FA ). One can also denote A = supp(A, FA ). Remark 1.1 [8] In the development
work we use the following notations for fuzzy sets: Ff (D) (f (z)) =supp f (D) , Ff (D)
· = {z ∈ D : 0 < Ff (D) f (z) ≤ 1}, Fp(U ) p (z) =supp p (U ) , Fp(U ) · = {z ∈ U : 0 < Fp(U ) (p (z)) ≤ 1}. We give a new definition of membershippfunction on complex numbers set using the module notion of a complex number z = x + iy, x, y ∈ R, |z| = x2 + y 2 ≥ 0. Example 1.1 Let F : C → R+ a function such that FC (z) = |F (z)|, ∀ z ∈ C. Denote by FC (C) = {z ∈ C : 0 < F (z) ≤ 1} = {z ∈ C : 0 < |F (z)| ≤ 1} =supp(C, FC ) the fuzzy subset of the complex numbers set. We call the subset FC (C) = UF (0, 1) the fuzzy unit disk. Definition 1.2 ([14]) Let D ⊂ C, z0 ∈ D be a fixed point and let the functions f, g ∈ H (D). The function f is said to be fuzzy subordinate to g and write f ≺F g or f (z) ≺F g (z), if are satisfied the conditions: 1) f (z0 ) = g (z0 ) , 2) Ff (D) f (z) ≤ Fg(D) g (z), z ∈ D. Definition 1.3 ([15, Definition 2.2]) Let ψ : C3 × U → C and h univalent in U , with ψ (a, 0; 0) = h (0) = a. If p is analytic in U , with p (0) = a and satisfies the (second-order) fuzzy differential subordination Fψ(C3 ×U ) ψ(p(z), zp0 (z) , z 2 p00 (z); z) ≤ Fh(U ) h(z),

z ∈ U,

(1.1)

then p is called a fuzzy solution of the fuzzy differential subordination. The univalent function q is called a fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination, or more simple a fuzzy dominant, if Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U , for all p satisfying (1.1). A fuzzy dominant qe that satisfies Fqe(U ) q˜(z) ≤ Fq(U ) q(z), z ∈ U , for all fuzzy dominants q of (1.1) is said to be the fuzzy best dominant of (1.1). Rz Lemma Corollary 2.6g.2, p. 66]) Let h ∈ A and L [f ] (z) = G (z) = z1 0 h (t) dt, z ∈ U. If  00 1.1 ([13,  (z) 1 Re zh h0 (z) + 1 > − 2 , z ∈ U, then L (f ) = G ∈ K. Lemma 1.2 ([16]) Let h be a convex function with h(0) = a, and let γ ∈ C∗ be a complex number with Re γ ≥ 0. If p ∈ H[a, n] with p (0) = a, ψ : C2 × U → C, ψ (p (z) , zp0 (z) ; z) = p (z) + γ1 zp0 (z) an analytic function in U and   1 0 1 Fψ(C2 ×U ) p(z) + zp (z) ≤ Fh(U ) h(z), i.e. p(z) + zp0 (z) ≺F h(z), z ∈ U, (1.2) γ γ Rz then Fp(U ) p(z) ≤ Fg(U ) g(z) ≤ Fh(U ) h(z), i.e. p(z) ≺F g(z) ≺F h(z), z ∈ U, where g(z) = nzγγ/n 0 h(t)tγ/n−1 dt, z ∈ U. The function q is convex and is the fuzzy best dominant. Lemma 1.3 ([16]) Let g be a convex function in U and let h(z) = g(z) + nαzg 0 (z), z ∈ U, where α > 0 and n is a positive integer. If p(z) = g(0) + pn z n + pn+1 z n+1 + . . . , z ∈ U, is holomorphic in U and Fp(U ) (p(z) + αzp0 (z)) ≤ Fh(U ) h(z), i.e. p(z) + αzp0 (z) ≺F h(z), z ∈ U, then Fp(U ) p(z) ≤ Fg(U ) g(z), i.e. p(z) ≺F g(z), z ∈ U, and this result is sharp. We use the following differential operators. Definition 1.4 (Al Oboudi [9]) For f ∈ An , λ ≥ 0 and n, m ∈ N, the operator Dλm is defined by Dλm : An → An , Dλ0 f (z) = f (z) Dλ1 f (z) = (1 − λ) f (z) + λzf 0 (z) = Dλ f (z) , ... 0 Dλm+1 f (z) = (1 − λ) Dλm f (z) + λz (Dλm f (z)) = Dλ (Dλm f (z)) , 2 1477

z ∈ U.

Alina Lupas: special fuzzy differential subordinations

Remark 1.2 If f ∈ An and f (z) = z +

P∞

j=n+1

aj z j , then Dλm f (z) = z +

P∞

j=n+1

m

[1 + (j − 1) λ] aj z j , z ∈ U .

Remark 1.3 For λ = 1 in the above definition we obtain the S˘ al˘ agean differential operator [18]. Definition 1.5 (Ruscheweyh [17]) For f ∈ An , n, m ∈ N, the operator Rm is defined by Rm : An → An , R0 f (z) = f (z) R1 f (z) = zf 0 (z) , ... 0 (m + 1) Rm+1 f (z) = z (Rm f (z)) + mRm f (z) , z ∈ U. P∞ P ∞ m Remark 1.4 If f ∈ An , f (z) = z + j=n+1 aj z j , then Rm f (z) = z + j=n+1 Cm+j−1 aj z j , z ∈ U . m m Definition 1.6 ([4]) Let α, λ ≥ 0, n, m ∈ N. Denote by RDλ,α the operator given by RDλ,α : An → An , m m m RDλ,α f (z) = (1 − α)R f (z) + αDλ f (z), z ∈ U.

P∞ Remark 1.5 If f ∈ An , f (z) = z + j=n+1 aj z j , then  P∞ m m m RDλ,α f (z) = z + j=n+1 α [1 + (j − 1) λ] + (1 − α) Cm+j−1 aj z j , z ∈ U. m m f (z) = Dλm f (z), z ∈ U. Remark 1.6 For α = 0, RDλ,0 f (z) = Rm f (z), z ∈ U, and for α = 1, RDλ,1 m m 0 For λ = 1, we obtain RD1,α f (z) = Lα f (z) which was studied in [1], [2], [5]. For m = 0, RDλ,α f (z) = m (1 − α) R0 f (z) + αDλ0 f (z) = f (z) = R0 f (z) = Dλ0 f (z), z ∈ U. The operator RDλ,α was studied in [3], [4], [6], [7].

2

Main results

m defined in Definition 1.6 we define the class RDF Using the operator RDλ,α m (δ, λ, α) and we study fuzzy subordinations.
Definition 2.1 [8] Let f (D) =supp f (D) , Ff (D) = {z ∈ D : 0 < Ff (D) f (z) ≤ 1}, where Ff (D) · is the membership function of the fuzzy set f (D) asociated to the function f . The membership function of the fuzzy set (µf ) (D) asociated to the function µf coincide with the membership function of the fuzzy set f (D) asociated to the function f , i.e. F(µf )(D) ((µf ) (z)) = Ff (D) f (z), z ∈ D. The membership function of the fuzzy set (f + g) (D) asociated to the function f + g coincide with the half of the sum of the membership functions of the fuzzy sets f (D), respectively g (D), asociated to the function f , respectively g, i.e. F(f +g)(D) ((f + g) (z)) = Ff (D) f (z)+Fg(D) g(z) , 2

z ∈ D.

Remark 2.1 [8] F(f +g)(D) ((f + g) (z)) can be defined in other ways. Since 0 < Ff (D) f (z) ≤ 1 and 0 < Fg(D) g (z) ≤ 1, it is evidently that 0 < F(f +g)(D) ((f + g) (z)) ≤ 1, z ∈ D. Definition 2.2 Let δ ∈ [0, 1), α, λ ≥ 0 and n, m ∈ N. A function f ∈ An is said to be in the class RDF m (δ, λ, α) if it satisfies the inequality
0 m F(RDm f )0 (U ) RDλ,α f (z) > δ, z ∈ U. (2.1) λ,α

Theorem 2.1 The set RDF m (δ, λ, α) is convex. P∞ Proof. Let the functions fj (z) = z + j=n+1 ajk z j , k = 1, 2, z ∈ U, be in the class RDF m (δ, λ, α). It F is sufficient to show that the function h (z) = η1 f1 (z) + η2 f2 (z) is in the class RDm (δ, λ, α) , with η1 and η2 nonnegative such that η1 + η2 = 1. 0 We have h0 (z) = (µ1 f1 + µ2 f2 ) (z) = µ1 f10 (z) + µ2 f20 (z), z ∈ U , and  0  0  0  0 m m m m RDλ,α h (z) = RDλ,α (µ1 f1 + µ2 f2 ) (z) = µ1 RDλ,α f1 (z) + µ2 RDλ,α f2 (z) . From Definition 2.1 weobtain that   0 0 m m F(RDm h)0 (U ) RDλ,α h (z) = F(RDm (µ1 f1 +µ2 f2 ))0 (U ) RDλ,α (µ1 f1 + µ2 f2 ) (z) = λ,α λ,α   0  0  m m 0 = F(RDm (µ1 f1 +µ2 f2 )) (U ) µ1 RDλ,α f1 (z) + µ2 RDλ,α f2 (z) λ,α

0

0

m m f2 (z)) ) f1 (z)) )+F 0 0 (µ1 (RDλ,α (µ2 (RDλ,α m f m f (µ2 RDλ,α (µ1 RDλ,α 1 ) (U ) 2 ) (U )

F

2

3 1478

=

Alina Lupas: special fuzzy differential subordinations

0

m RDλ,α f1 (z)) +F

( m f (RDλ,α 1 ) (U )

F

0

0

m RDλ,α f2 (z))

( m f (RDλ,α 2 ) (U ) 0

2

.

 0  0 m m f1 (z) ≤ 1 and δ < F(RDm f2 )0 (U ) RDλ,α f2 (z) (δ, λ, α) we have δ < F(RDm f1 )0 (U ) RDλ,α λ,α λ,α 0 0 m m F RDλ,α f1 (z)) +F RDλ,α f2 (z)) 0 0 ( ( m f m f RD (U ) RD (U ) ( λ,α 1 ) ( λ,α 2 ) ≤ 1, z ∈ U . Therefore δ < ≤ 1 and we obtain that 2  0 F F m δ < F(RDm h)0 (U ) RDλ,α h (z) ≤ 1, which means that h ∈ RDm (δ, λ, α) and RDm (δ, λ, α) is convex. λ,α Since f1 , f2 ∈

RDF m

1+z We highlight a fuzzy subset obtained using a convex function. Let the function h (z) = 1−z , z ∈ U . After   00 zh (z) 1+z > 0, so h ∈ K and h (U ) = {z ∈ C : Rez > a short calculation we obtain that Re h0 (z) + 1 = Re 1−z 0}. We define the membership function for the set h (U ) as Fh(U ) (h (z)) = Reh (z), z ∈ U and we have
Fh(U ) h (z) =supp h (U ) , Fh(u) = {z ∈ C : 0 < Fh(U ) (h (z)) ≤ 1} = {z ∈ U : 0 < Rez ≤ 1}. 1 Theorem 2.2 Let g be a convex function in U and let h (z) = g (z) + c+2 zg 0 (z) , where z ∈ U, c > 0. R z c+2 c If f ∈ RDF m (δ, λ, α) and G (z) = Ic (f ) (z) = z c+1 0 t f (t) dt, z ∈ U, then


0 m f (z) ≤ Fh(U ) h (z) , i.e. F(RDm f )0 (U ) RDλ,α λ,α


0 m RDλ,α f (z) ≺F h (z) , z ∈ U,

(2.2)

 0  0 m m G (z) ≤ Fg(U ) g (z), i.e. RDλ,α implies F(RDm G)0 (U ) RDλ,α G (z) ≺F g (z), z ∈ U, and this result is λ,α sharp. Proof. We obtain that z

c+1

Z G (z) = (c + 2)

z

tc f (t) dt.

(2.3)

0

Differentiating (2.3), with respect to z, we have (c + 1) G (z) + zG0 (z) = (c + 2) f (z) and
0 m m m (c + 1) RDλ,α G (z) + z RDλ,α G (z) = (c + 2) RDλ,α f (z) , z ∈ U.

(2.4)

Differentiating (2.4) we have
0 m RDλ,α G (z) +


00
0 1 m m z RDλ,α G (z) = RDλ,α f (z) , z ∈ U. c+2

(2.5)

Using (2.5), the fuzzy differential subordination (2.2) becomes    
0
00 1 1 m m 0 m G(U ) FRDλ,α RDλ,α G (z) + z RDλ,α G (z) zg (z) . ≤ Fg(U ) g (z) + c+2 c+2 If we denote

(2.6)


0 m p (z) = RDλ,α G (z) , z ∈ U,

then p ∈ H [1, n] .  Replacing (2.7) in (2.6) we obtain Fp(U ) p (z) +

1 0 c+2 zp

(z)

(2.7)



  1 ≤ Fg(U ) g (z) + c+2 zg 0 (z) , z ∈ U. Using  0 m 0 RD G (z) ≤ Fg(U ) g (z), z ∈ U, and λ,α G) (U )

Lemma 1.3 we have Fp(U ) p (z) ≤ Fg(U ) g (z) , z ∈ U, i.e. F(RDm  λ,α 0 m g is the best dominant. We have obtained that RDλ,α G (z) ≺F g (z), z ∈ U. Theorem 2.3 Let h (z) = z ∈ U, then

1+(2δ−1)z , 1+z

δ ∈ [0, 1) and c > 0. If α, λ ≥ 0, m ∈ N and Ic (f ) (z) =

  F ∗ Ic RDF m (δ, λ, α) ⊂ RD m (δ , λ, α) ,
R 1 tx+1 where δ ∗ = 2δ − 1 + (c+2)(2−2δ) β c+2 n n − 2 and β (x) = 0 t+1 dt.

c+2 z c+1

Rz 0

tc f (t) dt, (2.8)

Proof. The function h is convex and  using the samesteps as in the proof of Theorem 2.2 we get from the 1 hypothesis of Theorem 2.3 that Fp(U ) p (z) + c+2 zp0 (z) ≤ Fh(U ) h (z) , where p (z) is defined in (2.7).

4 1479

Alina Lupas: special fuzzy differential subordinations

 0 m G (z) ≤ Using Lemma 1.2 we deduce that Fp(U ) p (z) ≤ Fg(U ) g (z) ≤ Fh(U ) h (z) , i.e. F(RDm G)0 (U ) RDλ,α λ,α c+2 R z c+2 −1 1+(2δ−1)t (c+2)(2−2δ) R z t n −1 c+2 dt = (2δ − 1) + Fg(U ) g (z) ≤ Fh(U ) h (z) , where g (z) = c+2 0 t n c+2 1+t 1+t dt. Since g 0 nz

nz

n

n

is convex and g (U ) is symmetric with respect to the real axis, we deduce
0 m G (z) ≥ min Fg(U ) g (z) = Fg(U ) g (1) F(RDm G)(U ) RDλ,α λ,α |z|=1 and δ ∗ = g (1) = 2δ − 1 +

(c+2)(2−2δ) β n

c+2 n

(2.9)


− 2 . From (2.9) we deduce inclusion (2.8).

Theorem 2.4 Let g be a convex function, g(0) = 1 and let h be the function h(z) = g(z) + zg 0 (z), z ∈ U. If α, λ ≥ 0, n, m ∈ N, f ∈ An and satisfies the fuzzy differential subordination
0
0 m m F(RDm f )0 (U ) RDλ,α f (z) ≤ Fh(U ) h(z), i.e. RDλ,α f (z) ≺F h(z), z ∈ U, (2.10) λ,α m f (U ) then FRDλ,α

m RDλ,α f (z) z

≤ Fg(U ) g (z), i.e.

m RDλ,α f (z) z

≺F g(z), z ∈ U, and this result is sharp.

m Proof. By using the properties of operator RDλ,α , we have  P∞ m m m RDλ,α f (z) = z + j=n+1 α [1 + (j − 1) λ] + (1 − α) Cm+j−1 aj z j , z ∈ U. P∞ m m m z+ j=n+1 {α[1+(j−1)λ] +(1−α)Cm+j−1 }aj z j RDλ,α f (z) = = 1 + pn z n + pn+1 z n+1 + ..., z ∈ U. Consider p(z) = z z We deduce that p ∈ H[1, n].  0 m m Let RDλ,α f (z) = zp(z), z ∈ U. Differentiating we obtain RDλ,α f (z) = p(z) + zp0 (z), z ∈ U. Then (2.10)

becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z) = Fg(U ) (g(z) + zg 0 (z)) , z ∈ U. m f (U ) By using Lemma 1.3, we have Fp(U ) p(z) ≤ Fg(U ) g(z), z ∈ U, i.e. FRDλ,α  0 m We obtained that RDλ,α f (z) ≺F h(z), z ∈ U, and this results is sharp.

m RDλ,α f (z) z

≤ Fg(U ) g(z), z ∈ U.

  00 (z) Theorem 2.5 Let h be an holomorphic function which satisfies the inequality Re 1 + zh > − 12 , z ∈ U, h0 (z) and h(0) = 1. If α, λ ≥ 0, n, m ∈ N, f ∈ An and satisfies the fuzzy differential subordination m F(RDm f )0 (U ) RDλ,α f (z) λ,α


0

≤ Fh(U ) h (z) , i.e.

m RDλ,α f (z)

RD m f (z)

RD m f (z)


0

≺F h(z), z ∈ U,

(2.11)

Rz

h(t)t n −1 dt. The

λ,α λ,α m f (U ) ≤ Fq(U ) q (z), i.e. ≺F q(z), z ∈ U, where q(z) = then FRDλ,α z z function q is convex and it is the fuzzy best dominant.

1 1 nz n

0

1

m m j m z+ ∞ RDλ,α f (z) j=n+1 {α[1+(j−1)λ] +(1−α)Cm+j−1 }aj z Proof. = = z z  p(z) = P∞ Let P m ∞ m 1 + j=n+1 α [1 + (j − 1) λ] + (1 − α) Cm+j−1 aj z j−1 = 1 + j=n+1 pj z j−1 , z ∈ U, p ∈ H[1, n].   Rz 00 1 (z) > − 12 , z ∈ U, from Lemma 1.1, we obtain that q(z) = 1 1 0 h(t)t n −1 dt is a Since Re 1 + zh h0 (z)

P

nz n

convex function and verifies the differential equation asscociated to the fuzzy differential subordination (2.11) q (z) + zq 0 (z) = h (z), therefore it is the fuzzy best dominant.  0 m Differentiating, we obtain RDλ,α f (z) = p(z)+zp0 (z), for z ∈ U and (2.11) becomes Fp(U ) (p(z) + zp0 (z)) ≤ m f (U ) Fh(U ) h(z), z ∈ U. Using Lemma 1.2, we have Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U, i.e. FRDλ,α

Fq(U ) q(z), z ∈ U. We have obtained that

m RDλ,α f (z) z

m RDλ,α f (z) z

≤

≺F q(z), z ∈ U.

Corollary 2.6 Let h(z) = 1+(2β−1)z a convex function in U , 0 ≤ β < 1. If α, λ ≥ 0, n, m ∈ N, f ∈ An and 1+z satisfies the fuzzy differential subordination
0
0 m m F(RDm f )0 (U ) RDλ,α f (z) ≤ Fh(U ) h (z) , i.e. RDλ,α f (z) ≺F h(z), z ∈ U, (2.12) λ,α RD m f (z)

RD m f (z)

λ,α λ,α m f (U ) then FRDλ,α ≤ Fq(U ) q (z), i.e. ≺F q(z), z ∈ U, where q is given by q(z) = 2β − 1 + z z 1 −1 R 2(1−β) z t n dt, z ∈ U. The function q is convex and it is the fuzzy best dominant. 1 0 1+t

nz n

5 1480

Alina Lupas: special fuzzy differential subordinations

Proof. We have h (z) = 1+(2β−1)z and h00 (z) = with h (0) = 1, h0 (z) = −2(1−β) 1+z (1+z)2  00      (z) 1−ρ cos θ−iρ sin θ 1−ρ2 1−z 1 Re zh h0 (z) + 1 = Re 1+z = Re 1+ρ cos θ+iρ sin θ = 1+2ρ cos θ+ρ2 > 0 > − 2 .

4(1−β) , (1+z)3

therefore

RD m f (z)

λ,α Following the same steps as in the proof of Theorem 2.5 and considering p(z) = , the differential z 0 m f (U ) (p(z) + zp (z)) ≤ Fh(U ) h(z), z ∈ U. By using Lemma 1.2 for γ = subordination (2.12) becomes FRDλ,α m Rz 1 RDλ,α f (z) m f (U ) 1, we have Fp(U ) p(z) ≤ Fq(U ) q(z), i.e. FRDλ,α ≤ Fq(U ) q(z) and q (z) = 1 1 0 h(t)t n −1 dt = z nz n 1 R z 1 −1 1+(2β−1)t 2(1−β) R z t n −1 1 n t dt = 2β − 1 + dt, z ∈ U. 1 1 1+t 0 1+t nz n 0 nz n  00  zh (z) a convex function in U with h (0) = 1 and Re + 1 > − 12 . Example 2.1 Let h (z) = 1−z 0 1+z h (z)

Let f (z) = z + z 2 , z ∈ U . For n = 1, m = 1, λ = 12 , α = 2, we obtain RD11 ,2 f (z) = −R1 f (z) + 2  0
1 1 1 0 0 2 1 2D 1 f (z) = −zf (z) + 2 2 f (z) + 2 zf (z) = f (z) = z + z , z ∈ U . Then RD 1 ,2 f (z) = f 0 (z) = 1 + 2z 2 2 RD 11 f (z) R ,2 z 2 ln(1+z) 2 and = 1 + z. We have q (z) = z1 0 1−t . z 1+t dt = −1 + z Using Theorem 2.5 we obtain 1 + 2z ≺F

1−z 1+z ,

z ∈ U, induce 1 + z ≺F −1 +

2 ln(1+z) , z

z ∈ U.

Theorem 2.7 Let g be a convex function such that g(0) = 1 and let h be the function h(z) = g(z)+zg 0 (z), z ∈ U. (m+1)(m+2) m+2 m f (U ) ( If α, λ ≥ 0, n, m ∈ N, f ∈ An and the fuzzy differential subordination FRDλ,α RDλ,α f (z) − z 2(1−λ) 1 2 α (m+1)(m+2)− α (m+1)(2m+1)− ] m+2 ] m+1 [ [ (m+1)(2m+1) m+1 m λ2 λ2 RDλ,α f (z) − Dλ f (z) + Dλ f (z) − f (z) + mz RDλ,α z z z   (1−λ)2 α m2 − λ2 z

Dλm f (z)) ≤ Fh(U ) h (z), i.e.

(m + 1) (2m + 1) m2 (m + 1) (m + 2) m+2 m+1 RDλ,α f (z) − RDλ,α f (z) + RDm f (z) − z zh z i λ,α   α (m + 1) (2m + 1) − 2(1−λ) α (m + 1) (m + 2) − λ12 λ2 m+2 Dλ f (z) + Dλm+1 f (z) − z z h i 2 α m2 − (1−λ) 2 λ Dλm f (z) ≺F h(z), z ∈ U, (2.13) z m m holds, then F(RDm f )0 (U ) [RDλ,α f (z)]0 ≤ Fg(U ) g (z), i.e. [RDλ,α f (z)]0 ≺F g(z), z ∈ U. This result is sharp. λ,α Proof. Let


0 0 0 m p(z) = RDλ,α f (z) = (1 − α) (Rm f (z)) + α (Dλm f (z)) (2.14)  P∞ m m j−1 n n+1 = 1 + j=n+1 α [1 + (j − 1) λ] + (1 − α) Cm+j−1 jaj z = 1 + pn z + pn+1 z + .... We deduce that p ∈ H[1, n]. m By using the properties of operators RDλ,α , Rm and Dλm , after a short calculation, we obtain p (z) + zp0 (z) =

(m+1)(m+2) m+2 RDλ,α f z

(z) −

2 (m+1)(2m+1) m+1 m RDλ,α f (z) + mz RDλ,α f z   2

(z) −

(1−λ)2 − λ2

α m α[ ] m+1 (z) + Dλ f (z) − Dλm f (z) . z z Using the notation in (2.14), the fuzzy differential subordination becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z) =  0 m m f (U ) Fg(U ) (g(z) + zg 0 (z)) . By using Lemma 1.3, we have Fp(U ) p(z) ≤ Fg(U ) g(z), z ∈ U, i.e. FRDλ,α RDλ,α f (z) ≤ Fg(U ) g(z), z ∈ U, and this result is sharp. h i 00 (z) Theorem 2.8 Let h be an holomorphic function which satisfies the inequality Re 1 + zh > − 12 , z ∈ U, h0 (z) and h (0) = 1. If α, λ ≥ 0, n, m ∈ N, f ∈ An and satisfies the fuzzy differential subordination 2 α[(m+1)(m+2)− λ12 ] m+2 (m+1)(m+2) m+2 m+1 m m f (U ) ( FRDλ,α RDλ,α f (z) − (m+1)(2m+1) RDλ,α f (z) + mz RDλ,α Dλ f (z) f (z) − z z z α[(m+1)(m+2)− λ12 ] m+2 Dλ f z

α[(m+1)(2m+1)− + z

2(1−λ) λ2

]

2(1−λ) (m+1)(2m+1)− λ2

Dλm+1 f (z) −

  (1−λ)2 α m2 − λ2 z

Dλm f (z)) ≤ Fh(U ) h (z), i.e.

(m + 1) (m + 2) (m + 1) (2m + 1) m2 m+2 m+1 RDλ,α f (z) − RDλ,α f (z) + RDm f (z) − z zh z i λ,α   α (m + 1) (2m + 1) − 2(1−λ) α (m + 1) (m + 2) − λ12 λ2 m+2 Dλ f (z) + Dλm+1 f (z) − z z 6 1481

Alina Lupas: special fuzzy differential subordinations

h α m2 −

(1−λ)2 λ2

z

i Dλm f (z) ≺F h(z),

z ∈ U,

(2.15)

 0  0 m m m f (U ) RDλ,α f (z) ≤ Fq(U ) q (z). i.e. RDλ,α f (z) ≺F q(z), z ∈ U, where q is given by q(z) = then FRDλ,α Rz 1 1 h(t)t n −1 dt. The function q is convex and it is the fuzzy best dominant. 1 0 nz n



Proof. Since Re

1+

zh00 (z) h0 (z)



> − 12 , z ∈ U, from Lemma 1.1, we obtain that q(z) =

1 1 nz n

Rz 0

1

h(t)t n −1 dt is

a convex function and verifies the differential equation asscociated to the fuzzy differential subordination (2.11) q (z) + zq 0 (z) = h (z), therefore it is the fuzzy best dominant.  0 m m Using the properties of operator RDλ,α and considering p (z) = RDλ,α f (z) , we obtain p(z) + zp0 (z) = 2 α[(m+1)(m+2)− λ12 ] m+2 (m+1)(m+2) m+2 m+1 n RDλ,α RDλ,α f (z) − Dλ f (z) + f (z) − (m+1)(2m+1) f (z) + mz RDλ,α z z z 2(1−λ) (m+1)(2m+1)− λ2

α[

  (1−λ)2 α m2 − λ2

]

Dλm+1 f (z) − Dλm f (z) , z ∈ U. z 0 Then (2.15) becomes Fp(U ) (p(z) + zp (z)) ≤ Fh(U ) h(z), z ∈ U. Since p ∈ H[1, n], using Lemma 1.2, we  0 m m f (U ) RDλ,α f (z) ≤ Fq(U ) q(z), z ∈ U. deduce Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U, i.e. FRDλ,α z

Corollary 2.9 Let h(z) =

1+(2β−1)z 1+z

be a convex function in U , where 0 ≤ β < 1. If α, λ ≥ 0, n, m ∈ N, f ∈ An

(m+1)(m+2) m+2 m f (U ) ( RDλ,α f (z) − and satisfies the fuzzy differential subordination FRDλ,α z 2

m + mz RDλ,α f (z) − ≤ Fh(U ) h (z), i.e.

α[(m+1)(m+2)− λ12 ] m+2 Dλ f z

(z) +

α[(m+1)(2m+1)− z

2(1−λ) λ2

]

Dλm+1 f (z) −

(m+1)(2m+1) m+1 RDλ,α f z  

α m2 −

(z)

(1−λ)2 λ2

z

Dλm f (z))

(m + 1) (m + 2) (m + 1) (2m + 1) m2 m+2 m+1 RDλ,α f (z) − RDλ,α f (z) + RDm f (z) − z zh z i λ,α   α (m + 1) (2m + 1) − 2(1−λ) α (m + 1) (m + 2) − λ12 λ2 Dλm+2 f (z) + Dλm+1 f (z) − z z h i 2 α m2 − (1−λ) 2 λ Dλm f (z) ≺F h(z), z ∈ U, (2.16) z  0  0 m m m f (U ) then FRDλ,α RDλ,α f (z) ≤ Fq(U ) q (z) , i.e. RDλ,α f (z) ≺F q(z), z ∈ U, where q is given by q(z) = R z t n1 −1 2β − 1 + 2(1−β) dt, z ∈ U. The function q is convex and it is the fuzzy best dominant. 1 0 1+t nz n

 0 m Proof. Following the same steps as in the proof of Theorem 2.7 and considering p(z) = RDλ,α f (z) , the differential subordination (2.16) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z), z ∈ U. By using Lemma 1.2 for γ =  0  0 m m 1, we have Fp(U ) p(z) ≤ Fq(U ) q(z), i.e. F(RDm f )0 (U ) RDλ,α f (z) ≤ Fq(U ) q (z), i.e. RDλ,α f (z) ≺F q(z), λ,α 1 Rz R z 1 −1 1+(2β−1)t 1 2(1−β) R z t n −1 1 1 −1 n n dt = t dt = 2β − 1 + dt, z ∈ U. and q (z) = 1 0 h(t)t 1 1 1+t 0 0 1+t nz n

Example 2.2 Let h (z) =

nz n

1−z 1+z

nz n

a convex function in U with h (0) = 1 and Re



zh00 (z) h0 (z)

 + 1 > − 12 .

Let f (z) = z + z 2 , z ∈ U . For n = 1, m = 1, λ = 12 , α = 2, we obtain RD11 ,2 f (z) = −R1 f (z) + 2
n+1 n 2D11 f (z) = −zf 0 (z) + 2 21 f (z) + 12 zf 0 (z) = f (z) = z + z 2 and (n + 1) RDλ,α f (z) − (n − 1) RDλ,α f (z) − 2
  n+1 1 n 2 2 2 2 α n + 1 − λ Dλ f (z) − Dλ f (z) = 2RD 1 ,2 f (z) = −2 + 2z, where RD 1 ,2 f (z) = −R f (z) + 2D 1 f (z) = 2 2 2

Rz 2 ln(1+z) − 1 + 3z 2 + 2 12 z + 32 z 2 = −1 + z. We have q (z) = z1 0 1−t . 1+t dt = −1 + z Using Theorem 2.8 we obtain −2 + 2z ≺F

1−z 1+z ,

z ∈ U, induce z + z 2 ≺F −1 +

7 1482

2 ln(1+z) , z

z ∈ U.

Alina Lupas: special fuzzy differential subordinations

References [1] A. Alb Lupa¸s, On special differential subordinations using S˘ al˘ agean and Ruscheweyh operators, Mathematical Inequalities and Applications, Volume 12, Issue 4, 2009, 781-790. [2] Alina Alb Lupa¸s, On a certain subclass of analytic functions defined by Salagean and Ruscheweyh operators, Journal of Mathematics and Applications, No. 31, 2009, 67-76. [3] Alina Alb Lupa¸s, On special differential subordinations using a generalized S˘ al˘ agean operator and Ruscheweyh derivative, Journal of Computational Analysis and Applications, Vol. 13, No.1, 2011, 98-107. [4] A. Alb Lupa¸s, On a certain subclass of analytic functions defined by a generalized S˘ al˘ agean operator and Ruscheweyh derivative, Carpathian Journal of Mathematics, 28 (2012), No. 2, 183-190. [5] A. Alb Lupa¸s, Daniel Breaz, On special differential superordinations using S˘ al˘ agean and Ruscheweyh operators, Geometric Function Theory and Applications’ 2010 (Proc. of International Symposium, Sofia, 27-31 August 2010), 98-103. [6] A. Alb Lupa¸s, On special differential superordinations using a generalized S˘ al˘ agean operator and Ruscheweyh derivative, Computers and Mathematics with Applications 61(2011), 1048-1058, doi:10.1016/j.camwa.2010.12.055. [7] Alina Alb Lupa¸s, Certain special differential superordinations using a generalized S˘ al˘ agean operator and Ruscheweyh derivative, Analele Universitatii Oradea, Fasc. Matematica, Tom XVIII (2011), 167-178. [8] A. Alb Lupa¸s, Gh. Oros, On special fuzzy differential subordinations using S˘ al˘ agean and Ruscheweyh operators, Fuzzy Sets and Systems (to appear). [9] F.M. Al-Oboudi, On univalent functions defined by a generalized S˘ al˘ agean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [10] S.Gh. Gal, A. I. Ban, Elemente de matematic˘ a fuzzy, Oradea, 1996. [11] S.S. Miller, P.T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65(1978), 298-305. [12] S.S. Miller, P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 32(1985), 157-171. [13] S.S. Miller, P.T. Mocanu, Differential Subordinations. Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 225, Marcel Dekker Inc., New York, Basel, 2000. [14] G.I. Oros, Gh. Oros, The notion of subordination in fuzzy sets theory, General Mathematics, vol. 19, No. 4 (2011), 97-103. [15] G.I. Oros, Gh. Oros, Fuzzy differential subordinations, Acta Universitatis Apulensis, No. 30, 2012, 55-64. [16] G.I. Oros, Gh. Oros, Dominant and best dominant for fuzzy differential subordinations, Stud. Univ. BabesBolyai Math. 57(2012), No. 2, 239-248. [17] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [18] G. St. S˘al˘ agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372.

8 1483

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1484-1489, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

On special fuzzy differential subordinations using convolution product of S˘al˘agean operator and Ruscheweyh derivative Alina Alb Lupa¸s Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected] Abstract In this paper we establish several fuzzy differential subordinations regardind the operator defined as Hadamard product of S˘ al˘ agean operator S m and Ruscheweyh derivative Rm , denoted SRm , given by SRm : m m A → A, SR f (z) = (S ∗ Rm ) f (z) and An = {f ∈ H(U ), f (z) = z + an+1 z n+1 + . . . , z ∈ U } is the class of normalized analytic functions with A1 = A. A certain fuzzy class, denoted by SRF m (δ) , of analytic functions in the open unit disc is introduced by means of this operator. By making use of the concept of fuzzy differential subordination we will derive various properties and characteristics of the class SRF m (δ) . Also, several fuzzy differential subordinations are established regarding the operator SRm .

Keywords: fuzzy differential subordination, convex function, fuzzy best dominant, differential operator, convolution product, S˘ al˘ agean operator, Ruscheweyh derivative. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction

In [10] and [11] the authors wanted to launch a new research direction in mathematics that combines the notions from the complex functions domain with the fuzzy sets theory. Also the author can justify that by knowing the properties of a differential expression on a fuzzy set for a function one can be determined the properties of that function on a given fuzzy set. The author has analyzed the case of one complex functions, leaving as ”open problem” the case of real functions. The ”open problem” statement leaves open the interpretation of some notions from the fuzzy sets theory such that each one interpret them personally according to their scientific concerns, making this theory more attractive. The notion of fuzzy subordination was introduced in [10]. In [11] the authors have defined the notion of fuzzy differential subordination. In this paper we will study fuzzy differential subordinations obtained with the differential operator defined in [1]. Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and H(U ) the space of holomorphic functions in U . Let An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U } with A1 = A and H[a, n] = {f ∈ H(U ) : f (z) = a + an z n + an+1 z n+1 n + . . . , z ∈ U 00} for a ∈ C and n ∈oN. (z) Denote by K = f ∈ A : Re zff 0 (z) + 1 > 0, z ∈ U , the class of normalized convex functions in U . In order to use the concept of fuzzy differential subordination, we remember the following definitions: Definition 1.1 [6] A pair (A, FA ), where FA : X → [0, 1] and A = {x ∈ X : 0 < FA (x) ≤ 1} is called fuzzy subset of X. The set A is called the support of the fuzzy set (A, FA ) and FA is called the membership function of the fuzzy set (A, FA ). One can also denote A = supp(A, FA ). Remark 1.1 [5] In the development
work we use the following notations for fuzzy sets: Ff (D) (f (z)) =supp f (D)
, Ff (D) · = {z ∈ D : 0 < Ff (D) f (z) ≤ 1}, p (U ) =supp p (U ) , Fp(U ) · = {z ∈ U : 0 < Fp(U ) (p (z)) ≤ 1}. We give a new definition of membershippfunction on complex numbers set using the module notion of a complex number z = x + iy, x, y ∈ R, |z| = x2 + y 2 ≥ 0. 1 1484

Lupas: Salagean operator and Ruscheweyh derivative

Example 1.1 Let F : C → R+ a function such that FC (z) = |F (z)|, ∀ z ∈ C. Denote by FC (C) = {z ∈ C : 0 < F (z) ≤ 1} = {z ∈ C : 0 < |F (z)| ≤ 1} =supp(C, FC ) the fuzzy subset of the complex numbers set. We call the subset FC (C) = UF (0, 1) the fuzzy unit disk. Definition 1.2 ([10]) Let D ⊂ C, z0 ∈ D be a fixed point and let the functions f, g ∈ H (D). The function f is said to be fuzzy subordinate to g and write f ≺F g or f (z) ≺F g (z), if are satisfied the conditions: 1) f (z0 ) = g (z0 ) , 2) Ff (D) f (z) ≤ Fg(D) g (z), z ∈ D. Definition 1.3 ([11, Definition 2.2]) Let ψ : C3 × U → C and h univalent in U , with ψ (a, 0; 0) = h (0) = a. If p is analytic in U , with p (0) = a and satisfies the (second-order) fuzzy differential subordination Fψ(C3 ×U ) ψ(p(z), zp0 (z) , z 2 p00 (z); z) ≤ Fh(U ) h(z),

z ∈ U,

(1.1)

then p is called a fuzzy solution of the fuzzy differential subordination. The univalent function q is called a fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination, or more simple a fuzzy dominant, if Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U , for all p satisfying (1.1). A fuzzy dominant qe that satisfies Fqe(U ) q˜(z) ≤ Fq(U ) q(z), z ∈ U , for all fuzzy dominants q of (1.1) is said to be the fuzzy best dominant of (1.1). R 1 z Lemma  00 1.1 ([9,  Corollary 2.6g.2, p. 66]) Let h ∈ A and L [f ] (z) = G (z) = z 0 h (t) dt, z ∈ U. If (z) 1 Re zh h0 (z) + 1 > − 2 , z ∈ U, then L (f ) = G ∈ K. Lemma 1.2 ([12]) Let h be a convex function with h(0) = a, and let γ ∈ C∗ be a complex number with Re γ ≥ 0. If p ∈ H[a, n] with p (0) = a, ψ : C2 × U → C, ψ (p (z) , zp0 (z) ; z) = p (z) + γ1 zp0 (z) an analytic function in U and   1 1 Fψ(C2 ×U ) p(z) + zp0 (z) ≤ Fh(U ) h(z), i.e. p(z) + zp0 (z) ≺F h(z), z ∈ U, (1.2) γ γ Rz then Fp(U ) p(z) ≤ Fg(U ) g(z) ≤ Fh(U ) h(z), i.e. p(z) ≺F g(z) ≺F h(z), z ∈ U, where g(z) = nzγγ/n 0 h(t)tγ/n−1 dt, z ∈ U. The function q is convex and is the fuzzy best dominant. Lemma 1.3 ([12]) Let g be a convex function in U and let h(z) = g(z) + nαzg 0 (z), z ∈ U, where α > 0 and n is a positive integer. If p(z) = g(0) + pn z n + pn+1 z n+1 + . . . , z ∈ U, is holomorphic in U and Fp(U ) (p(z) + αzp0 (z)) ≤ Fh(U ) h(z), i.e. p(z) + αzp0 (z) ≺F h(z), z ∈ U, then Fp(U ) p(z) ≤ Fg(U ) g(z), i.e. p(z) ≺F g(z), z ∈ U, and this result is sharp. We use the following differential operators. Definition 1.4 (S˘ al˘ agean [14]) For f ∈ A, m ∈ N, the operator S m is defined by S m : A → A, S 0 f (z) S 1 f (z) m+1 S f (z) P∞ Remark 1.2 If f ∈ A, f (z) = z + j=2 aj z j ,

= f (z) = zf 0 (z), ... 0 = z (S m f (z)) , z ∈ U. P∞ then S m f (z) = z + j=2 j m aj z j , z ∈ U .

Definition 1.5 (Ruscheweyh [13]) For f ∈ A, m ∈ N, the operator Rm is defined by Rm : A → A, R0 f (z) = f (z) R1 f (z) = zf 0 (z) , ... 0 (m + 1) Rm+1 f (z) = z (Rm f (z)) + mRm f (z) , z ∈ U. P∞ P ∞ m aj z j , z ∈ U . Remark 1.3 If f ∈ A, f (z) = z + j=2 aj z j , then Rm f (z) = z + j=n+1 Cm+j−1 Definition 1.6 [1] Let m ∈ N ∪ {0}. Denote by SRm the operator given by the Hadamard product (the convolution product) of the S˘ al˘ agean operator S m and the Ruscheweyh operator Rm , SRm : A → A, SRm f (z) = m m (S ∗ R ) f (z) . P∞ P∞ m Remark 1.4 [1] If f ∈ A, f (z) = z + j=2 aj z j , then SRm f (z) = z + j=2 Cm+j−1 j m a2j z j , z ∈ U . Remark 1.5 The operator SRm was studied in [1], [2], [3], [4]. 2 1485

Lupas: Salagean operator and Ruscheweyh derivative

2

Main results

m Using the operator RDλ,α defined in Definition 1.6 we define the class SRF m (δ) and we study fuzzy subordinations.
Definition 2.1 [5] Let f (D) =supp f (D) , Ff (D) = {z ∈ D : 0 < Ff (D) f (z) ≤ 1}, where Ff (D) · is the membership function of the fuzzy set f (D) asociated to the function f . The membership function of the fuzzy set (µf ) (D) asociated to the function µf coincide with the membership function of the fuzzy set f (D) asociated to the function f , i.e. F(µf )(D) ((µf ) (z)) = Ff (D) f (z), z ∈ D. The membership function of the fuzzy set (f + g) (D) asociated to the function f + g coincide with the half of the sum of the membership functions of the fuzzy sets f (D), respectively g (D), asociated to the function f , respectively g, i.e. F(f +g)(D) ((f + g) (z)) = Ff (D) f (z)+Fg(D) g(z) , 2

z ∈ D.

Remark 2.1 [5] F(f +g)(D) ((f + g) (z)) can be defined in other ways. Since 0 < Ff (D) f (z) ≤ 1 and 0 < Fg(D) g (z) ≤ 1, it is evidently that 0 < F(f +g)(D) ((f + g) (z)) ≤ 1, z ∈ D. Definition 2.2 Let δ ∈ [0, 1) and m ∈ N. A function f ∈ A is said to be in the class SRF m (δ) if it satisfies the inequality 0 F(SRm f )0 (U ) (SRm f (z)) > δ, z ∈ U. (2.1) Theorem 2.1 Let g be a convex function in U and let h (z) = g (z) + Rz c c+2 f ∈ SRF m (δ) and G (z) = Ic (f ) (z) = z c+1 0 t f (t) dt, z ∈ U, then 0

1 0 c+2 zg

(z) , z ∈ U, where c > 0. If

0

F(SRm f )0 (U ) (SRm f (z)) ≤ Fh(U ) h (z) , i.e. (SRm f (z)) ≺F h (z) , 0

z ∈ U,

(2.2)

0

implies F(SRm G)0 (U ) (SRm G (z)) ≤ Fg(U ) g (z), i.e. (SRm G (z)) ≺F g (z), z ∈ U, and this result is sharp. Proof. We have z c+1 G (z) = (c + 2) 0 zG (z) = (c + 2) f (z) and

Rz 0

tc f (t) dt. Differentiating, with respect to z, we obtain (c + 1) G (z)+ 0

(c + 1) SRm G (z) + z (SRm G (z)) = (c + 2) SRm f (z) , z ∈ U.

(2.3)

Differentiating (2.3) we have 0

(SRm G (z)) +

1 00 0 z (SRm G (z)) = (SRm f (z)) , z ∈ U. c+2

(2.4)

Using (2.4), the fuzzy differential subordination (2.2) becomes     1 1 00 0 F(SRm G)0 (U ) (SRm G (z)) + z (SRm G (z)) ≤ Fg(U ) g (z) + zg 0 (z) . c+2 c+2 If we denote

0

p (z) = (SRm G (z))

(2.5)

(2.6)

then p ∈ H [1, n] . Replacing (2.6) in (2.5) we obtain     1 1 0 0 Fp(U ) p (z) + zp (z) ≤ Fg(U ) g (z) + zg (z) , z ∈ U. c+2 c+2 0

Using Lemma 1.3 we have Fp(U ) p (z) ≤ Fg(U ) g (z), z ∈ U , i.e. F(SRm G)0 (U ) (SRm G (z)) ≤ Fg(U ) g (z), z ∈ U, 0 and g is the fuzzy best dominant. We have obtained that (SRm G (z)) ≺F g (z), z ∈ U. Theorem 2.2 Let h (z) =

1+(2β−1)z , 1+z

β ∈ [0, 1) and c > 0. If m ∈ N and Ic is given by Theorem 2.1, then   F ∗ Ic SRF m (δ) ⊂ SRm (δ ) ,

where β ∗ = 2β − 1 + (c + 2) (2 − 2β)

R1

tc+1 dt. 0 t+1

3 1486

(2.7)

Lupas: Salagean operator and Ruscheweyh derivative

Proof. The function h is convex and  using the samesteps as in the proof of Theorem 2.1 we get from the 1 zp0 (z) ≤ Fh(U ) h (z) , where p (z) is defined in (2.6). hypothesis of Theorem 2.2 that Fp(U ) p (z) + c+2 0

Using Lemma 1.2 we deduce that Fp(U ) p (z) ≤ Fg(U ) g (z) ≤ Fh(U ) h (z) , that is F(SRm G)0 (U ) (SRm G (z)) ≤ R z c+1 1+(2β−1)t R z tc+1 Fg(U ) g (z) ≤ Fh(U ) h (z) , where g (z) = zc+2 t dt = 2β −1+ (c+2)(2−2β) dt. Since g is convex c+2 1+t z c+2 0 0 t+1 and g (U ) is symmetric with respect to the real axis, we deduce 0

F(SRm G)0 (U ) (SRm G (z)) ≥ min Fg(U ) g (z) = Fg(U ) g (1)

(2.8)

|z|=1

and β ∗ = g (1) = 2β − 1 + (c + 2) (2 − 2β)

R1

tc+1 dt. 0 t+1

From (2.8) we deduce inclusion (2.7).

Theorem 2.3 Let g be a convex function, g (0) = 1, and let h be the function h (z) = g (z) + zg 0 (z), z ∈ U . If m ∈ N ∪ {0}, f ∈ A and verifies the fuzzy differential subordination 0

0

F(SRm f )0 (U ) (SRm f (z)) ≤ Fh(U ) h (z) , i.e. (SRm f (z)) ≺F h (z) , then FSRm f (U ) SR

m

f (z) z

≤ Fg(U ) g (z), i.e.

SRm f (z) z

z ∈ U,

(2.9)

≺F g (z), z ∈ U, and this result is sharp.

P m m 2 j m P∞ z+ ∞ j=2 Cm+j−1 j aj z m Proof. Consider p (z) = SR zf (z) = = 1 + j=2 Cm+j−1 j m a2j z j−1 . We have p (z) + z 0 0 0 m m zp (z) = (SR f (z)) , z ∈ U . Then F(SRm f )0 (U ) (SR f (z)) ≤ Fh(U ) h (z), z ∈ U, becomes Fp(U ) (p (z) + zp0 (z)) ≤ Fh(U ) h (z) = Fg(U ) (g (z) + zg 0 (z)), z ∈ U . By using Lemma 1.3, we obtain Fp(U ) p (z) ≤ Fg(U ) g (z), z ∈ U , m m i.e. FSRm f (U ) SR zf (z) ≤ Fg(U ) g (z), z ∈ U. We obtain that SR zf (z) ≺F g (z), z ∈ U, and this result is sharp.   00 (z) Theorem 2.4 Let h ∈ H(U ), with h(0) = 1, which verifies the inequality Re 1 + zh > − 12 , z ∈ U. If h0 (z) m ∈ N, f ∈ A and verifies the fuzzy differential subordination 0

0

F(SRm f )0 (U ) (SRm f (z)) ≤ Fh(U ) h(z), i.e. (SRm f (z)) ≺F h(z), m

then FSRm f (U ) SR zf (z) ≤ Fq(U ) q(z), i.e. convex and it is the fuzzy best dominant.

SRm f (z) z

≺F q(z), z ∈ U, where q(z) =

1 z

z ∈ U,

Rz 0

(2.10)

h(t)dt. The function q is

m P∞ P∞ m Proof. Let p(z) = SR zf (z) = 1 + j=2 Cm+j−1 j m a2j z j−1 = 1 + j=2 pj z j−1 , z ∈ U, p ∈ H[1, 1].   Rz 00 (z) > − 21 , z ∈ U, from Lemma 1.1, we obtain that q (z) = z1 0 h(t)dt is a convex function Since Re 1 + zh h0 (z)

and verifies the differential equation asscociated to the fuzzy differential subordination (2.10) q (z) + zq 0 (z) = h (z), therefore it is the fuzzy best dominant. 0 Differentiating, we obtain (SRm f (z)) = p(z) + zp0 (z), z ∈ U, and (2.10) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z), z ∈ U. m Using Lemma 1.3, we have Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U, i.e. FSRm f (U ) SR zf (z) ≤ Fq(U ) q(z), z ∈ U. We m have obtained that SR zf (z) ≺F q(z), z ∈ U. a convex function in U , 0 ≤ β < 1. If m ∈ N, f ∈ A and verifies the Corollary 2.5 Let h(z) = 1+(2β−1)z 1+z fuzzy differential subordination 0

0

F(SRm f )0 (U ) (SRm f (z)) ≤ Fh(U ) h(z), i.e. (SRm f (z)) ≺F h(z), m

z ∈ U,

(2.11)

m

then FSRm f (U ) SR zf (z) ≤ Fq(U ) q(z), i.e. SR zf (z) ≺F q(z), z ∈ U, where q is given by q(z) = 2β − 1 + 2(1−β) ln (1 + z) , z ∈ U. The function q is convex and it is the fuzzy best dominant. z Proof. We have h (z) = 1+(2β−1)z with h (0) = 1, h0 (z) = −2(1−β) and h00 (z) = 1+z (1+z)2  00      2 (z) 1−ρ cos θ−iρ sin θ 1−ρ 1−z 1 Re zh h0 (z) + 1 = Re 1+z = Re 1+ρ cos θ+iρ sin θ = 1+2ρ cos θ+ρ2 > 0 > − 2 . m

4(1−β) , (1+z)3

therefore

Following the same steps as in the proof of Theorem 2.4 and considering p(z) = SR zf (z) , the fuzzy differential subordination (2.11) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z), z ∈ U. m By using Lemma 1.2 for γ = 1 and n = 1, we have Fp(U ) p(z) ≤ Fq(U ) q(z), i.e. FSRm f (U ) SR zf (z) ≤ Fq(U ) q (z) Rz Rz and q (z) = z1 0 h (t) dt = z1 0 1+(2β−1)t dt = 2β − 1 + 2(1−β) ln (1 + z) , z ∈ U. 1+t z 4 1487

Lupas: Salagean operator and Ruscheweyh derivative

zh00 (z) 1 h0 (z) + 1 > − 2 . Let f (z) = z + z 2 , z ∈ U . For n = 1, m = 1, we obtain SR1 f (z) = z + C21 · 2 · 11 · z 2 = z
0 Rz 1 2 ln(1+z) . SR1 f (z) = 1 + 8z and SR zf (z) = 1 + 4z. We have q (z) = z1 0 1−t 1+t dt = −1 + z 2 ln(1+z) 1−z Using Theorem 2.4 we obtain 1 + 8z ≺F 1+z , z ∈ U, induce 1 + 4z ≺F −1 + , z ∈ U. z

Example 2.1 Let h (z) =

1−z 1+z

a convex function in U with h (0) = 1 and Re





+ 4z 2 . Then

Theorem 2.6 Let g be a convex function such that g (0) = 1 and let h be the function h (z) = g (z) + zg 0 (z), z ∈ U . If m ∈ N ∪ {0}, f ∈ A and verifies the fuzzy differential subordination  0 0  zSRm+1 f (z) zSRm+1 f (z) ≤ Fh(U ) h (z) , i.e. ≺F h (z) , z ∈ U, (2.12) FSRm f (U ) SRm f (z) SRm f (z) m+1

SRm+1 f (z) SRm f (z) ≺F g (z), z ∈ U, and this result is sharp. P P∞ m+ m+1 2 j−1 m+1 m+1 2 j m+1 1+ ∞ aj z z+ j=n+1 Cm+j j aj z f (z) j=n+1 Cm+j j P∞ P∞ = . We have p0 (z) = = Proof. Consider p (z) = SR 2 m m m j m SR f (z) z+ j=n+1 Cm+j−1 j aj z 1+ j=n+1 Cm+j−1 j m a2j z j−1   0 0 m (SRm+1 f (z)) f (z))0 zSRm+1 f (z) 0 −p (z)· (SR . Relation (2.12) becomes Fp(U ) (p (z) + zp0 (z)) SRm f (z) SRm f (z) . Then p (z)+zp (z) = SRm f (z) Fh(U ) h (z) = Fg(U ) (g (z) + zg 0 (z)), z ∈ U, and by using Lemma 1.3, we obtain Fp(U ) p (z) ≤ Fg(U ) g (z), z ∈ U , m+1 f (z) SRm+1 f (z) i.e. FSRm f (U ) SR SRm f (z) ≤ Fg(U ) g (z), z ∈ U. We obtained that SRm f (z) ≺F g (z), z ∈ U. f (z) then FSRm f (U ) SR SRm f (z) ≤ Fg(U ) g (z), i.e.

1 Theorem 2.7 Let g be a convex function such that g (0) = 1 and let h be the function h (z) = g (z)+ m+1 zg 0 (z), z ∈ U, m ∈ N. If f ∈ A and the fuzzy differential subordination   1 1 m+1 SR f (z) ≤ Fh(U ) h (z) , i.e. SRm+1 f (z) ≺F h (z) , z ∈ U, (2.13) FSRm f (U ) z z 0

0

holds, then F(SRm f )0 (U ) (SRm f (z)) ≤ Fg(U ) g (z), i.e. (SRm f (z)) ≺F g (z), z ∈ U, and this result is sharp. P∞ 0 m Proof. With notation p (z) = (SRm f (z)) = 1 + j=2 Cm+j−1 j m+1 a2j z j−1 and p (0) = 1, we obtain for P∞ 00 m (SRm f (z)) . f (z) = z + j=2 aj z j , p (z) + zp0 (z) = z1 SRm+1 f (z) + z m+1     1 1 We have Fp(U ) p (z) + m+1 zp0 (z) ≤ Fh(U ) h (z) = Fg(U ) g (z) + m+1 zg 0 (z) , z ∈ U . By using Lemma 0

1.3, we obtain Fp(U ) p (z) ≤ Fg(U ) g (z), z ∈ U , i.e. F(SRm f )0 (U ) (SRm f (z)) ≤ Fg(U ) g (z), z ∈ U, and this result 0 is sharp. We obtained that (SRm f (z)) ≺F g (z), z ∈ U. h i 00 (z) Theorem 2.8 Let h ∈ H(U ) with h(0) = 1, which verifies the inequality Re 1 + zh > − 12 , z ∈ U. If h0 (z) m ∈ N, f ∈ A and satisfies the fuzzy differential subordination   1 1 m+1 SR f (z) ≤ Fh(U ) h(z), i.e. SRm+1 f (z) ≺F h(z), z ∈ U, (2.14) FSRm f (U ) z z 0

0

then F(SRm f )0 (U ) (SRm f (z)) ≤ Fq(U ) q(z), i.e. (SRm f (z)) ≺F q(z), z ∈ U, where q is given by q(z) = Rz m+1 m z m+1 0 h(t)t dt. The function q is convex and it is the fuzzy best dominant.   Rz 00 (z) > − 12 , z ∈ U, from Lemma 1.1, we obtain that q (z) = zm+1 Proof. Since Re 1 + zh h(t)tm dt is 0 m+1 h (z) 0 a convex function and verifies the differential equation associated to the fuzzy differential subordination (2.14) 1 q (z) + m+1 zq 0 (z) = h (z), therefore it is the fuzzy best dominant. 0 m m m m Using  the properties of  operator SR and considering p (z) = (SR f (z)) , we  obtain FSR f (U ) SR f (U ) = 1 0 m+1 zp (z)

, z ∈ U. Then (2.14) becomes Fp(U ) p(z) +

1 0 m+1 zp (z)

≤ Fh(U ) h(z), z ∈ U. Since p ∈ Rz H[1, 1], using Lemma 1.3 for γ = m+1, we deduce Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U, where q(z) = zm+1 h(t)tm dt, m+1 0 0 0 z ∈ U, i.e. F(SRm f )0 (U ) (SRm f (z)) ≤ Fq(U ) q(z), z ∈ U. We have obtained that (SRm f (z)) ≺F q(z), z ∈ U. Fp(U ) p(z) +

Corollary 2.9 Let h(z) = 1+(2β−1)z a convex function in U , 0 ≤ β < 1. If m ∈ N, f ∈ A and verifies the 1+z fuzzy differential subordination   1 1 m+1 FSRm f (U ) SR f (z) ≤ Fh(U ) h(z), i.e. SRm+1 f (z) ≺F h(z), z ∈ U, (2.15) z z 0

0

then F(SRm f )0 (U ) (SRm f (z)) ≤ Fq(U ) q(z), i.e. (SRm f (z)) ≺F q(z), z ∈ U , where q is given by q(z) = R z tm 2β − 1 + 2(1−β)(m+1) dt, z ∈ U. The function q is convex and it is the fuzzy best dominant. z m+1 0 1+t 5 1488

≤

Lupas: Salagean operator and Ruscheweyh derivative

0

m Proof. Following the same steps as in the proof  of Theorem 2.7 and  considering p(z) = (SR f (z)) , the 1 fuzzy differential subordination (2.15) becomes Fp(U ) p(z) + m+1 zp0 (z) ≤ Fh(U ) h(z), z ∈ U. 0

By using Lemma 1.2 for γ = m+1 and n = 1, we have Fp(U ) p(z) ≤ Fq(U ) q(z), i.e. F(SRm f )0 (U ) (SRm f (z)) ≤ Rz R z m 1+(2β−1)t R z tm h(t)tm dt = zm+1 t Fq(U ) q (z) and q (z) = zm+1 dt = 2β − 1 + 2(1−β)(m+1) dt, z ∈ U. m+1 m+1 1+t z m+1 0 0 0 1+t 1−z 1+z

a convex function in U with h (0) = 1 and Re

Using Theorem 2.8 we obtain 1 + 12z ≺F

1−z 1+z ,



zh00 (z) h0 (z)

 + 1 > − 12 .
0 Let f (z) = z + z 2 , z ∈ U . For n = 1, m = 1,we obtain SR1 f (z) = z + 4z 2 . Then SR1 f (z) = 1 + 8z. We obtain also z1 SRm+1 f (z) = z1 SR2 f (z) = 1 + 12z, where SR2 f (z) = z + C32 · 22 · 12 · z 2 + C42 · 32 · 0 · z 3 = z + 12z 2 . Rz 4 ln(1+z) 4 We have q (z) = z22 0 1−t . 1+t tdt = −1 + z − z2

Example 2.2 Let h (z) =

z ∈ U, induce 1 + 8z ≺F −1 +

4 z

−

4 ln(1+z) , z2

z ∈ U.

References [1] Alina Alb Lupa¸s, Certain differential subordinations using S˘ al˘ agean and Ruscheweyh operators, Acta Universitatis Apulensis, No. 29/2012, 125-129. [2] Alina Alb Lupa¸s, A note on differential subordinations using S˘ al˘ agean and Ruscheweyh operators, ROMAI Journal, vol. 6, nr. 1(2010), 1–4. [3] Alina Alb Lupa¸s, Certain differential superordinations using S˘ al˘ agean and Ruscheweyh operators, Analele Universit˘ a¸tii din Oradea, Fascicola Matematica, Tom XVII, Issue no. 2, 2010, 209-216. [4] Alina Alb Lupa¸s, A note on differential superordinations using S˘ al˘ agean and Ruscheweyh operators, Acta Universitatis Apulensis, nr. 24/2010, pp. 201-209. [5] A. Alb Lupa¸s, Gh. Oros, On special fuzzy differential subordinations using S˘ al˘ agean and Ruscheweyh operators, Fuzzy Sets and Systems (to appear). [6] S.Gh. Gal, A. I. Ban, Elemente de matematic˘ a fuzzy, Oradea, 1996. [7] S.S. Miller, P.T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65(1978), 298-305. [8] S.S. Miller, P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 32(1985), 157-171. [9] S.S. Miller, P.T. Mocanu, Differential Subordinations. Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 225, Marcel Dekker Inc., New York, Basel, 2000. [10] G.I. Oros, Gh. Oros, The notion of subordination in fuzzy sets theory, General Mathematics, vol. 19, No. 4 (2011), 97-103. [11] G.I. Oros, Gh. Oros, Fuzzy differential subordinations, Acta Universitatis Apulensis, No. 30/2012, pp. 55-64. [12] G.I. Oros, Gh. Oros, Dominant and best dominant for fuzzy differential subordinations, Stud. Univ. BabesBolyai Math. 57(2012), No. 2, 239-248. [13] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [14] G. St. S˘al˘ agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1490-1495, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Strong differential superordination and sandwich theorem Gheorghe Oros, Roxana S¸endrut¸iu, Adela Venter, Loriana Andrei Department of Mathematics, University of Oradea, Str. Universit˘a¸tii, No.1, 410087 Oradea, Romania E-mail: georgia oros [email protected] Abstract In this paper we study certain strong differential superordinations and give a sandwich theorem, obtained by using a new integral operator introduced in [21].

Keywords. Analytic function, univalent function, starlike function, convex function, strong differential superordination, best dominant, best subordinant. 2000 Mathematical Subject Classification: 30C80, 30C20, 30C40, 34C40.

1

Introduction and preliminaries

The concept of differential subordination was introduced in [11], [12] and developed in [13], by S.S. Miller and P.T. Mocanu. The concept of differential superordination was introduced in [14], like a dual problem of the differential superordination by S.S. Miller and P.T. Mocanu. The concept of strong differential subordination was introduced in [10] by J.A. Antonino and S. Romaguera and developed in [1], [2], [3], [4], [5], [16], [18], [19], [20], [22], [24]. The concept of strong differential superordination was introduced in [17], like a dual concept of the strong differential subordination and developed in [6], [7], [8], [9], [21], [23]. Let H(U × U ) denote the class of analytic function in U × U , U = {z ∈ C : |z| < 1}, U = {z ∈ C : |z| ≤ 1}, ∂U = {z ∈ C : |z| = 1}. For a ∈ C and n ∈ N∗ , let Hζ[a, n] = {f (z, ζ) ∈ H(U × U ) : f (z, ζ) = a + an (ζ) z n + . . . + an+1 (ζ) z n+1 + . . .} with z ∈ U , ζ ∈ U , ak (ζ) holomorphic functions in U , k ≥ n, Aζn = {f (z, ζ) ∈ H(U × U ) : f (z, ζ) = z + an+1 (ζ) z n+1 +an+2 (ζ) z n+2 +. . .} with z ∈ U , ζ ∈ U , ak (ζ) holomorphic functions in U , k ≥ n+1, so Aζ1 = Aζ, Hζu (U ) = {f (z, ζ) ∈ Hζ[a, n] : f (z, ζ) univalent in U, for all ζ ∈ U }, Sζ = {f (z, ζ) ∈ Aζ, f (z, ζ) univalent in U, 0 (z,ζ) for all ζ ∈ U }, denote the class of univalent functions in U × U , S ∗ ζ = {f (z, ζ) ∈ Aζ : Re zff (z,ζ) > 0, z ∈ U, for h 00 i (z,ζ) all ζ ∈ U }, denote the class of normalized starlike functions in U × U , Kζ = {f (z, ζ) ∈ Aζ : Re zff 0 (z,ζ) +1 > 0, z ∈ U, for all ζ ∈ U }, denote the class of normalized convex functions in U × U . r P∞For r ∈ N,k A(r)ζ denote the subclass of the functions f (z, ζ) ∈ (U × U ) of the form f (z, ζ) = z + k=r+1 ak (ζ)z , r ∈ N, z ∈ U, ζ ∈ U and set A(1)ζ = Aζ. To prove our main results, we need the following definitions and lemmas: Definition 1.1 [16], [18] Let f (z, ζ) and F (z, ζ) analytic functions from H(U × U ). The function f (z, ζ) is said to be strongly subordinated to F (z, ζ), or F (z, ζ) is said to be strongly superordinated to f (z, ζ), if there exists a function w analytic in U with w(0) = 0 and |w(z)| < 1, such that f (z, ζ) = F (w(z), ζ). In such a case we write f (z, ζ) ≺≺ F (z, ζ). If F (z, ζ) is univalent then f (z, ζ) ≺≺ F (z, ζ) if and only if f (0, ζ) = F (0, ζ) and f (U × U ) ⊂ F (U × U ). Remark 1.1 If f (z, ζ) ≡ f (z) and F (z, ζ) ≡ F (z), then the strong differential subordination or strong differential superordination becomes the usual notion of differential subordination or differential superordination. Definition 1.2 [14], [16] We denote by Q that are analytic and injective with respect ζ the set of functions q(z, ζ) to z on U \ E(q(z, ζ)), where E(q(z, ζ)) =

ξ ∈ ∂U : lim q(z, ζ) = ∞ z→ξ

The class of Qζ for which q(0, ζ) = a, is denoted by Qζ (a).

1 1490

and q 0 (ξ, ζ) 6= 0, for ξ ∈ ∂U \ E(q(z, ζ)).

Oros et al: Strong differential superordination

We mention that all the derivatives which appear in this paper are considered with respect to variable z. Let ψ : C3 × U × U → C and let h(z, ζ) be univalent in U , for all ζ ∈ U . If p(z, ζ) is analytic in U × U and satisfies the (second-order) strong differential subordination ψ(p(z, ζ), z 0 (z, ζ), z 2 p00 (z, ζ); z, ζ) ≺≺ h(z, ζ), z ∈ U, ζ ∈ U

(1.1)

then p(z, ζ) is called a solution of the strong differential subordination. The univalent function q(z, ζ) is called a dominant of the solutions of the strong differential subordination or simply a dominant, if p(z, ζ) ≺≺ q(z, ζ) for all p(z, ζ) satisfying (1.1). A dominant qe(z, ζ) that satisfies qe(z, ζ) ≺≺ q(z, ζ) for all dominants q(z, ζ) of (1.1) is said to be the best dominant of (1.1). (Note that the best dominant is unique up to a rotation of U ). Let ϕ : C3 × U × U → C and let h(z, ζ) be analytic in U × U . If p(z, ζ) and ϕ(p(z, ζ), zp0 (z, ζ), z 2 p00 (z, ζ); z, ζ) are univalent in U , for all ζ ∈ U and satisfy the (secondorder) strong differential superordination h(z, ζ) ≺≺ ϕ(p(z, ζ), zp0 (z, ζ), z 2 p00 (z, ζ); z, ζ)

(1.2)

then p(z, ζ) is called a solution of the strong differential superordination. An analytic function q(z, ζ) is called a subordinant of the solutions of the differential superordination, or more simply a subordinant, if q(z, ζ) ≺≺ p(z, ζ) for all p(z, ζ) satisfying (1.2). A univalent subordinant qe(z, ζ) that satisfies q(z, ζ) ≺≺ qe(z, ζ) for all subordinants of (1.2) is said to be the best subordinant. (Note that the best subordinant is unique up to a rotation of U ). In order to prove the original results of this paper, we need the following definitions and lemmas. Definition 1.3 [11] For f (z, ζ) ∈ Aζn , n ∈ N∗ , m ∈ N, γ ∈ C, let Lγ be the integral operator given by Lγ : Aζn → Aζn L0γ f (z, ζ) = f (z, ζ) Z γ+1 z 0 Lγ f (z, ζ)tγ−1 dt L1γ f (z, ζ) = zγ 0 Z γ+1 z 1 2 Lγ f (z, ζ)tγ−1 dt, ... Lγ f (z, ζ) = zγ 0 Z γ + 1 z m−1 m Lγ f (z, ζ)tγ−1 dt. Lγ f (z, ζ) = zγ 0 By using Definition 1.3, we can prove the following properties for this integral operator: For f (z, ζ) ∈ Aζn , n ∈ N∗ , m ∈ N, γ ∈ C, we have Lm γ f (z, ζ) = z +

∞ X (γ + 1)m ak (ζ)z k , z ∈ U, ζ ∈ U (γ + k)m

(1.3)

k=n+1

and 0 m−1 z[Lm f (z, ζ) − γLm γ f (z, ζ)]z = (γ + 1)Lγ λ f (z, ζ), z ∈ U, ζ ∈ U .

(1.4)

Definition 1.4 [20] For r ∈ N, f (z, ζ) ∈ A(r)ζ, let H be the integral operator given by H : A(r)ζ → A(r)ζ H 0 f (z, ζ) = f (z, ζ) Z r+1 z 0 1 H f (z, ζ) = H f (t, ζ)dt z Z0 z r+1 H 1 f (t, ζ)dt, ... H 2 f (z, ζ) = z Z0 r + 1 z m−1 m H f (z, ζ) = H f (t, ζ)dt, z ∈ U, ζ ∈ U . z 0 From Definition 1.4 we have H m f (z, ζ) = z t +

∞ X (r + 1)m ak (ζ)z k (r + k)m

(1.5)

k=r+1

and z[H m f (z, ζ)]0z = (r + 1)H m−1 f (z, ζ) − H m f (z, ζ), z ∈ U, ζ ∈ U . 2 1491

(1.6)

Oros et al: Strong differential superordination

Lemma 1.1 [14, Corollary 9.1] Let h1 (z, ζ) R zand h2 (z, ζ) be starlike in U × U , with h1 (0, ζ) = h2 (0, ζ) = 0 and the functions qi (z, ζ) defined by qi (z, ζ) = 0 hi (t, ζ)t−1 dt, for i = 1, 2. If p(z, ζ) ∈ [0, 1] ∩ Qζ and zp0 (z, ζ) is univalent in U × U , then h1 (z, ζ) ≺≺ zp0 (z, ζ) ≺≺ h2 (z, ζ) implies q1 (z, ζ) ≺≺ p(z, ζ) ≺≺ q2 (z, ζ). The functions q1 (z, ζ) and q2 (z, ζ) are convex and they are respectively the best subordinant and best dominant. Lemma 1.2 [15, Theorem 3] Let θ and φ be analytic in a domain D, and let q(z, ζ) be univalent in U , for all ζ ∈ U , with q(0, ζ) = a and q(U × U ) ⊂ D. Let Q(z, ζ) = zq 0 (z, ζ) · φ(q(z)), h(z, ζ) = θ(q(z, ζ)) + Q(z, ζ) and suppose that h 0 i (q(z,ζ)) (i) Re θφ(q(z,ζ)) > 0, and (ii) Q(z, ζ) is starlike in U , for all ζ ∈ U . If p(z, ζ) ∈ [a, 1] ∩ Qζ , p(U × U ) ⊂ D and θ(p(z, ζ)) + zp0 (z, ζ) · φ(z, ζ) is univalent in U , for all ζ ∈ U , then h(z, ζ) ≺≺ θ(p(z, ζ)) + zp0 (z, ζ) · φ(p(z, ζ)) implies q(z, ζ) ≺≺ p(z, ζ), z ∈ U, ζ ∈ U and q(z, ζ) is the best subordinant.

2

Main results

ζz z Theorem 2.1 Let h1 (z, ζ) = ζ−z and h2 (z, ζ) = ζ+z , be starlike in U , for all ζ ∈ U , with h1 (0, ζ) = h2 (0, ζ) = Rz ζ Rz 1 ζ 0, and q1 (z, ζ) = 0 ζ−t dt = ζ ln ζ−z and q2 (z, ζ) = 0 ζ+t dt = ln ζ+z ζ . For m ∈ N, γ ∈ C, f (z, ζ) ∈ Aζ, if 0 z 2 [Lm γ f (z,ζ)] Lm f (z,ζ) γ

∈ [0, 1] ∩ Qζ and

0 m 3 m 00 m 3 m 0 2 2z 2 [Lm γ f (z,ζ)] Lγ f (z,ζ)+z [Lγ f (z,ζ)] Lγ f (z,ζ)−z [(Lγ f (z,ζ)) ] 2 [Lm f (z,ζ)] γ

is univalent in U , for all

ζ ∈ U , then 0 m 3 m 00 m 3 m 0 2 2z 2 [Lm ζz z γ f (z, ζ)] Lγ f (z, ζ) + z [Lγ f (z, ζ)] Lγ f (z, ζ) − z [(Lγ f (z, ζ)) ] ≺≺ ≺≺ 2 ζ −z [Lm f (z, ζ)] ζ + z γ ζ implies ζ ln ζ−z ≺≺

0 z 2 [Lm γ f (z,ζ)] Lm f (z,ζ) γ

(2.1)

≺≺ ln ζ+z ζ , z ∈ U, ζ ∈ U .

ζ The functions q1 (z, ζ) = ζ ln ζ−n and q2 (z, ζ) = ln ζ+z z are convex and they are respectively the best subordinant and best dominant.

Proof. In order to prove the theorem, we shall use Lemma 1.1. zh0 (z,ζ)z zh0 (z,ζ)z ζ ζ We have Re h11 (z,ζ) = Re ζ−z = 12 > 0, z ∈ U, ζ ∈ U and Re h22 (z,ζ) = Re ζ+z =

1 2

> 0, z ∈ U, ζ ∈ U

hence h1 (z, ζ) and h2 (z, ζ) are starlike in U , for all ζ ∈ U . We consider 0 z 2 [Lm γ f (z, ζ)] , z ∈ U, ζ ∈ U . p(z, ζ) = m Lγ f (z, ζ) Using (1.3) in (2.2), we have p(z, ζ) =

 P  (γ+1)m k z 2 z+ ∞ k=2 (γ+k)m ak (ζ)z P (γ+1)m k z+ ∞ k=2 (γ+k)m ak (ζ)z

=

 P  (γ+1)m k−1 z 1+ ∞ k=2 (γ+k)m ak ·ζ·k·z . P (γ+1)m k 1+ ∞ k=2 (γ+k)m ak (ζ)z

(2.2) Since p(0, ζ) = 0,

we have p(z, ζ) ∈ [0, 1]ζ ∩ Qζ . Differentiating (2.2), and after a short calculus we obtain zp0 (z, ζ) =

02 0 m 3 m 00 m z 3 [Lm 2z 2 [Lm γ f (z, ζ)] Lγ f (z, ζ) + z [Lγ f (z, ζ)] Lγ f (z, ζ) γ f (z, ζ)] − . 2 2 [Lm [Lm γ f (z, ζ)] γ f (z, ζ)]

(2.3)

Using (2.3) in (2.1), we have z ζz ≺≺ zp0 (z, ζ) ≺≺ , z ∈ U, ζ ∈ U . ζ −z ζ +z ζ Using Lemma 1.1, we obtain ζ ln ζ−z ≺≺

0 z 2 [Lm γ f (z,ζ)] Lm f (z,ζ) γ

(2.4)

≺≺ ln ζ+z ζ , z ∈ U, ζ ∈ U .

Theorem 2.2 Let m ∈ N, γ ∈ C, λ ∈ C, q(z, ζ) = eλzζ starlike (univalent) function in U , for all ζ ∈ U , with q(0, ζ) = 1, and suppose that (j) Re λzζ > − 12 , (jj) Re λζ > 0. 3 1492

Oros et al: Strong differential superordination

0 Let Q(z, ζ) = λzζe2λzζ and h(z, ζ) = λzζe2λzζ +eλzζ , z ∈ U , ζ ∈ U . If f (z, ζ) ∈ Aζ, [Lm γ f (z, ζ)] ∈ [1, 1]∩Qζ 0 m 0 m 00 and [Lm γ f (z, ζ)] + z[Lγ f (z, ζ)] [Lγ f (z, ζ)] is univalent in U , for all ζ ∈ U , then 0 m 0 m 00 λzζe2λzζ + eλzζ ≺≺ [Lm γ f (z, ζ)] + z[Lγ f (z, ζ)] [Lγ f (z, ζ)]

(2.5)

λzζ 0 is the best subordinant. implies eλzζ ≺≺ [Lm γ f (z, ζ)] , z ∈ U, ζ ∈ U and q(z, ζ) = e

Proof. In order to prove the theorem, we shall use Lemma 1.2. For that, we show that the necessary conditions are satisfied. Let the functions θ : C → C, ϕ : C → C, with Q(w) = w

(2.6)

ϕ(w) = w.

(2.7)

and We check the conditions from the hypothesis of Lemma 1.2. Using (2.6), (2.7), (i) and (ii) we have Re

θ0 (q(z, ζ)) λζeλzζ = Re λzζ = Re λζ > 0, ϕ(q(z, ζ)) e

(2.8)

zQ0 (z, ζ) = Re (1 + 2λzζ) > 0. Q(z, ζ)

(2.9)

and Re We consider

0 (2.10) p(z, ζ) = [Lm γ f (z, ζ)] , z ∈ U, ζ ∈ U . h i 0 P∞ (γ+1)m P∞ (γ+1)m k k−1 Using (1.3) in (2.10), we have p(z, ζ) = z + k=2 (γ+k) . Since = 1 + k=2 (γ+k) m ak (ζ)z m ak (ζ)kz p(0, ζ) = 1, we have p(z, ζ) ∈ [1, 1] ∩ Qζ . Differentiating (2.10) and after a short calculus we obtained 0 m 00 m 0 p(z, ζ) + zp0 (z, ζ)p(z, ζ) = [Lm γ f (z, ζ)] + z[Lγ f (z, ζ)] [Lγ f (z, ζ)] .

(2.11)

Using (2.6) and (2.7), we have θ(p(z, ζ)) = p(z, ζ) and ϕ(p(z, ζ)) = p(z, ζ)

(2.12)

0 m 00 m 0 θ(p(z, ζ)) + zp0 (z, ζ)ϕ(p(z, ζ)) = [Lm γ f (z, ζ)] + z[Lγ f (z, ζ)] [Lγ f (z, ζ)] .

(2.13)

and (2.11) becomes

Using (2.6) and (2.7), we have θ(q(z, ζ)) = q(z, ζ) and ϕ(q(z, ζ)) = q(z, ζ), h(z, ζ) = q(z, ζ) + zq 0 (z, ζ)q(z, ζ) = eλzζ + λzζe2λzζ .

(2.14)

Using (2.13) and (2.14), the strong superordination (2.5) becomes h(z, ζ) ≺≺ θ(p(z, ζ)) + zp0 (z, ζ)ϕ(p(z, ζ)), z ∈ U, ζ ∈ U .

(2.15)

Since (2.8) and (2.9) give the conditions from the hypothesis of Lemma 1.2 and using (2.15) by applying Lemma λzζ 0 1.2 we obtain q(z, ζ) = eλzζ ≺≺ [Lm is the best dominant. γ f (z, ζ)] , z ∈ U, ζ ∈ U and q(z, ζ) = e zζ z Theorem 2.3 Let p ∈ N, m ∈ N, h1 (z, ζ) = 1+zζ , h2 (z, ζ) = 1−zζ be starlike in U , for all ζ ∈ U , with R z h1 (t,ζ) Rz ζ Rz h1 (0, ζ) = h2 (0, ζ) = 0, and q1 (z, ζ) = 0 dt = 0 1+tζ dt = ln(1 + ζz), q2 (z, ζ) = 0 h2 (t,ζ) dt = t t Rz 1 ln(1−zζ) H m f (z,ζ) z[H m f (z,ζ)]0 −(r−1)H m f (z,ζ) dt = − ζ . If zr−1 ∈ [0, 1] ∩ Qζ and is univalent in U , for all ζ ∈ U , z r−1 0 1−tζ then zζ z[H m f (z, ζ)]0 − (r − 1)H m f (z, ζ) z ≺≺ ≺≺ (2.16) 1 + zζ z r−1 1 − zζ H m f (z,ζ) ≺≺ − ln(1−zζ) , z∈ z r−1 ζ ln(1+zζ) q1 (z, ζ) = and q2 (z, ζ) ζ

implies ln(1 + zζ) ≺≺

U, ζ ∈ U .

The functions subordinant and best dominant.

= − ln(1−zζ) are convex and they are respectively the best ζ

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Proof. In order to prove the theorem, we shall use Lemma 1.1. 0 z[h2 (z,ζ)]0 1 1 1 1 (z,ζ)] We have Re z[h h1 (z,ζ) = Re 1+zζ = 2 > 0, z ∈ U, ζ ∈ U and Re h2 (z,ζ) = Re 1−zζ =

1 2

> 0, z ∈ U, ζ ∈ U .

Hence h1 (z, ζ) and h2 (z, ζ) are starlike in U , for ζ ∈ U . We consider H m f (z, ζ) , z ∈ U, ζ ∈ U . (2.17) p(z, ζ) = z r−1 P (r+1)m k P∞ zr + ∞ (r+1)m k=r+1 (r+γ)m ak (ζ)z k−r+1 Using (1.3), we have p(z, ζ) = = z + k=r+1 (r+k) . Since p(0, ζ) = 0, we r−1 m ak (ζ)z z have p(z, ζ) ∈ Aζ. Differentiating (2.17) and after a short calculus, we obtain zp0 (z, ζ) =

z[H m f (z, ζ)]0 − (r − 1)H m f (z, ζ) , z ∈ U, ζ ∈ U . z r−1

(2.18)

Using (2.18) in (2.16), we have zζ z ≺≺ zp0 (z, ζ) ≺≺ , z ∈ U, ζ ∈ U . 1 + zζ 1 − zζ From Lemma 1.1, we obtain ln(1 + zζ) ≺≺

H m f (z,ζ) z r−1

(2.19)

≺≺ − ln(1−zζ) , z ∈ U, ζ ∈ U . The functions q1 (z, ζ) = ζ

ln(1+zζ) and q2 (z, ζ) = − ln(1−zζ) are convex and they are respectively the best subordinant and best dominant. ζ Example 2.1 Let γ = 2, m = 1, f (z, ζ) = z + 5ζz 3 , L12 f (z, ζ) = z+18ζz 3 −36ζ 2 z 4 . From (1+3ζz 2 )2 ζ+z , z ∈ U, ζ ∈ U. ζ

Theorem 2.1, we have

ζz ζ−z

≺≺

z 3 +18ζz 5 −36ζ 2 z 7 (z+3ζz 3 )2

2 3z

+ 2ζz 3 , p(z, ζ) =

≺≺

z ζ+z

implies ζ ln

z+9ζz 2 0 1+6ζz 2 , zp (z, ζ) = ζ z+9ζz 2 ζ−z ≺≺ 1+6ζz 2 ≺≺

References [1] A. Alb Lupa¸s, On special strong differential subordinations using multiplier transformation, Applied Mathematics Letters 25(2012), 624-630, doi:10.1016/j.aml.2011.09.074. [2] A. Alb Lupas, G.I. Oros, Gh. Oros, On special strong differential subordinations using S˘ al˘ agean and Ruscheweyh operators, Journal of Computational Analysis and Applications, Vol. 14, No. 2, 2012, 266270. [3] A. Alb Lupas, G.I. Oros, Gh. Oros, A note on special strong differential subordinations using multiplier transformation, Journal of Computational Analysis and Applications, Vol. 14, No. 2, 2012, 261-265. [4] A. Alb Lupa¸s, On special strong differential subordinations using a generalized S˘ al˘ agean operator and Ruscheweyh derivative, Journal of Concrete and Applicable Mathematics, Vol. 10, No.’s 1-2, 2012, 17-23. [5] A. Alb Lupa¸s, A note on special strong differential subordinations using a multiplier transformation and Ruscheweyh derivative, Journal of Concrete and Applicable Mathematics, Vol. 10, No.’s 1-2, 2012, 24-31. [6] A. Alb Lupa¸s, Certain strong differential superordinations using S˘ al˘ agean and Ruscheweyh operators, Acta Universitatis Apulensis No. 30/2012, 325-336. [7] A. Alb Lupa¸s, A note on strong differential superordinations using S˘ al˘ agean and Ruscheweyh operators, Journal of Applied Functional Analysis, Vol. 7, No.’s 1-2, 2012, 54-61. [8] A. Alb Lupa¸s, Certain strong differential superordinations using a generalized S˘ al˘ agean operator and Ruscheweyh operator, Journal of Applied Functional Analysis, Vol. 7, No.’s 1-2, 2012, 62-68. [9] A. Alb Lupa¸s, Certain strong differential superordinations using a multiplier transformation and Ruscheweyh operator, Analele Universit˘a¸tii din Oradea, Fascicola Matematica, Tom XIX (2012), Issue No. 1, 125-136. [10] J.A. Antonino, S. Romaguera, Strong differential subordination to Briot-Bouquet differential equations, Journal of Differential Equations, 114(1994), 101-105. 5 1494

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[11] S. S. Miller, P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65(1978), 298-305. [12] S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Michig. Math. J., 28(1981), 157-171. [13] S. S. Miller, P. T. Mocanu, Differential subordinations. Theory and applications, Marcel Dekker, Inc., New York, Basel, 2000. [14] S. S. Miller, P. T. Mocanu, Subordinants of differential superordinations, Complex Variables, 48(10)(2003), 815-826. [15] S. S. Miller, P. T. Mocanu, Briot-Bouquet differential superordinations and sandwich theorems, J. Math. Anal. Appl., 329(2007), no. 1, 327-335. [16] G.I. Oros, On a new strong differential subordination, Acta Universitatis Apulensis, 32(2012), 6-17. [17] G.I. Oros, Strong differential superordination, Acta Universitatis Apulensis, 19(2009), 110-116. [18] G.I. Oros, An application of the subordination chains, Fractional Calculus and Applied Analysis, 13(2010), no. 5, 521-530. [19] G.I. Oros, Briot-Bouquet, strong differential subordination, Journal of Computational Analysis and Applications, 14(2012), no. 4, 733-737. [20] G.I. Oros, Gh. Oros, Strong differential subordination, Turkish Journal of Mathematics, 33(2009), 249-257. [21] G.I. Oros, Gh. Oros, Strong differential superordination and sandwich theorem obtained by new integral operators, submitted Mathematical Inequalities and Applications, 2012. [22] G.I. Oros, Gh. Oros, Second order nonlinear strong differential subordinations, Bull. Belg. Math. Soc. Simion Stevin, 16(2009), 171-178. [23] G.I. Oros, First order strong differential superordination, General Mathematics, vol. 15, No. 2-3 (2007), 77-87. [24] Gh. Oros, Briot-Bouquet strong differential superordinations and sandwich theorems, Math. Reports, 12(62)(2010), no. 3, 277-283.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO.8, 1496-1501, 2013, COPYRIGHT 2013 EUDOXUS PRESS, LLC

Strong differential subordinations and superordinations and sandwich theorem Gheorghe Oros, Adela Venter, Roxana S¸endrut¸iu, Loriana Andrei Department of Mathematics, University of Oradea, Str. Universit˘a¸tii, No.1, 410087 Oradea, Romania E-mail: georgia oros [email protected] Abstract In this paper we study certain strong differential subordinations and strong differential superordinations, obtained by using a new integral operator introduced in [21]. We also give some results as a sandwich theorem.

Keywords. Analytic function, univalent function, starlike function, convex function, strong differential subordination, strong differential superordination, best dominant, best subordinant. 2000 Mathematical Subject Classification: 30C80, 30C20, 30C40, 34C40.

1

Introduction and preliminaries

The concept of differential subordination was introduced in [11], [12] and developed in [13], by S.S. Miller and P.T. Mocanu. The concept of differential superordination was introduced in [14], [15] like a dual problem of the differential superordination by S.S. Miller and P.T. Mocanu. The concept of strong differential subordination was introduced in [10] by J.A. Antonino and S. Romaguera and developed in [1], [2], [3], [4], [5], [16], [18], [19], [20], [22], [24]. The concept of strong differential superordination was introduced in [17], like a dual concept of the strong differential subordination and developed in [6], [7], [8], [9], [21], [23]. In [16] the author defines the following classes: Let H(U × U ) denote the class of analytic function in U × U , U = {z ∈ C : |z| < 1}, U = {z ∈ C : |z| ≤ 1}, ∂U = {z ∈ C : |z| = 1}. For a ∈ C and n ∈ N∗ , let Hζ[a, n] = {f (z, ζ) ∈ H(U × U ) : f (z, ζ) = a + an (ζ) z n + . . . + an+1 (ζ) z n+1 + . . .} with z ∈ U , ζ ∈ U , ak (ζ) holomorphic functions in U , k ≥ n, Aζn = {f (z, ζ) ∈ H(U × U ) : f (z, ζ) = z + an+1 (ζ) z n+1 +an+2 (ζ) z n+2 +. . .} with z ∈ U , ζ ∈ U , ak (ζ) holomorphic functions in U , k ≥ n+1, so Aζ1 = Aζ, Hζu (U ) = {f (z, ζ) ∈ Hζ[a, n] : f (z, ζ) univalent in U, for all ζ ∈ U }, Sζ = {f (z, ζ) ∈ Aζ, f (z, ζ) univalent in U, 0 (z,ζ) for all ζ ∈ U }, denote the class of univalent functions in U × U , S ∗ ζ = {f (z, ζ) ∈ Aζ : Re zff (z,ζ) > 0, z ∈ U, for h 00 i (z,ζ) all ζ ∈ U }, denote the class of normalized starlike functions in U × U , Kζ = {f (z, ζ) ∈ Aζ : Re zff 0 (z,ζ) +1 > 0, z ∈ U, for all ζ ∈ U }, denote the class of normalized convex functions in U × U . r P∞For r ∈ N,k A(r)ζ denote the subclass of the functions f (z, ζ) ∈ (U × U ) of the form f (z, ζ) = z + k=r+1 ak (ζ)z , r ∈ N, z ∈ U, ζ ∈ U and set A(1)ζ = Aζ. To prove our main results, we need the following definitions and lemmas: Definition 1.1 [16], [18] Let f (z, ζ) and F (z, ζ) analytic functions from H(U × U ). The function f (z, ζ) is said to be strongly subordinated to F (z, ζ), or F (z, ζ) is said to be strongly superordinated to f (z, ζ), if there exists a function w analytic in U with w(0) = 0 and |w(z)| < 1, such that f (z, ζ) = F (w(z), ζ). In such a case we write f (z, ζ) ≺≺ F (z, ζ). If F (z, ζ) is univalent then f (z, ζ) ≺≺ F (z, ζ) if and only if f (0, ζ) = F (0, ζ) and f (U × U ) ⊂ F (U × U ). If f (z, ζ) ≡ f (z) and F (z, ζ) ≡ F (z), then the strong differential subordination, or strong differential superordination, becomes the usual notion of differential subordination or differential superordination.

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Definition 1.2 [14], [16] We denote by Qζ the set  of functions q(z, ζ) thatare analytic and injective, with respect to z on U \ E(q(z, ζ)), where E(q(z, ζ)) = ξ ∈ ∂U : lim q(z, ζ) = ∞ and are such that q 0 (ξ, ζ) 6= 0, z→ξ

for ξ ∈ ∂U \ E(q(z, ζ)). The class of Qζ for which q(0, ζ) = a, is denoted by Qζ (a). We mention that all the derivatives which appear in this paper are considered with respect to variable z. We shall not indicate that in the paper due to the complexity of the writing. Let ψ : C3 × U × U → C and let h(z, ζ) be univalent in U , for all ζ ∈ U . If p(z, ζ) is analytic in U × U and satisfies the (second-order) strong differential subordination ψ(p(z, ζ), z 0 (z, ζ), z 2 p00 (z, ζ); z, ζ) ≺≺ h(z, ζ), z ∈ U, ζ ∈ U

(1.1)

then p(z, ζ) is called a solution of the strong differential subordination. The univalent function q(z, ζ) is called a dominant of the solutions of the strong differential subordination or simply a dominant, if p(z, ζ) ≺≺ q(z, ζ) for all p(z, ζ) satisfying (1.1). A dominant qe(z, ζ) that satisfies qe(z, ζ) ≺≺ q(z, ζ) for all dominants q(z, ζ) of (1.1) is said to be the best dominant of (1.1). (Note that the best dominant is unique up to a rotation of U ). Let ϕ : C3 × U × U → C and let h(z, ζ) be analytic in U × U . If p(z, ζ) and ϕ(p(z, ζ), zp0 (z, ζ), z 2 p00 (z, ζ); z, ζ) are univalent in U , for all ζ ∈ U and satisfy the (second-order) strong differential superordination h(z, ζ) ≺≺ ϕ(p(z, ζ), zp0 (z, ζ), z 2 p00 (z, ζ); z, ζ)

(1.2)

then p(z, ζ) is called a solution of the strong differential superordination. An analytic function q(z, ζ) is called a subordinant of the solutions of the differential superordination, or more simply a subordinant, if q(z, ζ) ≺≺ p(z, ζ) for all p(z, ζ) satisfying (1.2). A univalent subordinant qe(z, ζ) that satisfies q(z, ζ) ≺≺ qe(z, ζ) for all subordinants of (1.2) is said to be the best subordinant. (Note that the best subordinant is unique up to a rotation of U ). Definition 1.3 [20] For f (z, ζ) ∈ Aζn , n ∈ N∗ , m ∈ N, γ ∈ C, let Lγ be the integral operator given by Lγ : Aζn → Aζn L0γ f (z, ζ) = f (z, ζ) Z γ+1 z 0 Lγ f (z, ζ)tγ−1 dt L1γ f (z, ζ) = zγ Z0 γ+1 z 1 2 Lγ f (z, ζ) = Lγ f (z, ζ)tγ−1 dt, ... zγ 0 Z γ + 1 z m−1 m Lγ f (z, ζ) = Lγ f (z, ζ)tγ−1 dt. zγ 0 By using Definition 1.3, we can prove the following properties for this integral operator: For f (z, ζ) ∈ Aζn , n ∈ N∗ , m ∈ N, γ ∈ C, we have Lm γ f (z, ζ) = z +

∞ X (γ + 1)m ak (ζ)z k , z ∈ U, ζ ∈ U , (γ + k)m

(1.3)

k=n+1

and 0 m−1 z[Lm f (z, ζ) − γLm γ f (z, ζ)]z = (γ + 1)Lγ λ f (z, ζ), z ∈ U, ζ ∈ U .

(1.4)

Definition 1.4 [20] For r ∈ N, f (z, ζ) ∈ A(r)ζ, let H be the integral operator given by H : A(r)ζ → A(r)ζ H 0 f (z, ζ) = f (z, ζ) Z r+1 z 0 1 H f (z, ζ) = H f (t, ζ)dt z Z0 z r+1 H 2 f (z, ζ) = H 1 f (t, ζ)dt, ... z 0 Z r + 1 z m−1 H m f (z, ζ) = H f (t, ζ)dt, z ∈ U, ζ ∈ U . z 0 2 1497

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From Definition 1.4 we have H m f (z, ζ) = z r +

∞ X (r + 1)m ak (ζ)z k (r + k)m

(1.5)

k=p+1

and z[H m f (z, ζ)]0z = (r + 1)H m−1 f (z, ζ) − H m f (z, ζ), z ∈ U, ζ ∈ U .

(1.6)

Lemma 1.1 [15, Corollary 3.1] Let β, γ ∈ C, and q(z, ζ) univalent in U , for all ζ ∈ U , with q(0, ζ) = a. Let zq 0 (z,ζ) h(z, ζ) = q(z, ζ) + βq(z,ζ)+γ and suppose that (i) Re [βq(z, ζ) + γ] > 0, and zq 0 (z,ζ) (ii) βq(z,ζ)+γ is starlike. 0

0

zp (z,ζ) zp (z,ζ) If p(z, ζ) ∈ ζ[a, 1]∩Qζ and p(z, ζ)+ p(z,ζ)+1 is univalent in U , for all ζ ∈ U , then h(z, ζ) ≺≺ p(z, ζ)+ βp(z,ζ)+γ implies q(z, ζ) ≺≺ p(z, ζ) and q(z, ζ) is the best subordinant.

Lemma 1.2 [13, Theorem 3.2b, p.83] Let h(z, ζ) be convex in U , for all ζ ∈ U , and n a positive integer. nzq 0 (z,ζ) Suppose that the differential equation q(z, ζ) + βq(z,ζ)+γ = h(z, ζ) has an univalent solution q(z, ζ) that satisfies q(z, ζ) ≺≺ h(z, ζ). zp0 (z,ζ) If p(z, ζ) ∈ ζ[a, n] satisfies p(z, ζ) + βp(z,ζ)+γ ≺≺ h(z, ζ), then p(z, ζ) ≺≺ q(z, ζ) and q(z, ζ) is the best dominant. Lemma 1.3 [14, Corollary 6.1] Let h1 (z, ζ) and h2 (z, ζ) be convex in U , for all ζ ∈ U , with hR1 (0, ζ) = h2 (0, ζ) = z a. Let γ ∈ C, γ 6= 0, with Re γ ≥ 0, and the functions qi (z, ζ) be defined by qi (z, ζ) = zγγ 0 hi (t, ζ)tγ−1 dt for i = 1, 2. 0 0 If p(z, ζ) ∈ [a, 1]∩Qζ and p(z, ζ)+ zp (z,ζ) is univalent, then h1 (z, ζ) ≺≺ p(z, ζ)+ zp (z,ζ) ≺≺ h2 (z, ζ) implies γ γ q1 (z, ζ) ≺≺ p(z, ζ) ≺≺ q2 (z, ζ), z ∈ U, ζ ∈ U . The functions q1 (z, ζ) and q2 (z, ζ) are convex and they are respectively the best subordinant and best dominant.

2

Main results

1+zζ Theorem 2.1 Let γ ∈ C, with Re γ ≥ 0, and q(z, ζ) = 1−zζ be univalent in U , for all ζ ∈ U , with q(0, ζ) = 1. Let zζ 1 + zζ zζ 2 + 3zζ 1 + zζ (1−zζ)2 + 1+zζ = + = (2.1) h(z, ζ) = 1 − zζ 1 − zζ 2(1 − zζ) 2(1 − zζ) + 1 1−zζ

with

  1 + zζ 2 Re 1 + = Re >0 1 − zζ 1 − zζ

and r(z, t) = starlike in U , for all ζ ∈ U . Lm f (z,ζ) If z[Lγm f (z,ζ)]0 ∈ [1, 1] ∩ Qζ and γ

Lm γ f (z,ζ) 0 z[Lm γ f (z,ζ)]

+

zq 0 (z, ζ) zζ = q(z, ζ) + 1 1 − zζ

0 [Lm γ f (z,ζ)] Lm γ f (z,ζ)

−

00 [Lm γ f (z,ζ)] 0 [Lm γ f (z,ζ)]

(2.3)

− 1 is univalent in U , for all ζ ∈ U , then

00 0 Lm [Lm [Lm 2 + 3zζ γ f (z, ζ) γ f (z, ζ)] γ f (z, ζ)] ≺≺ + − −1 0 0 2(1 − zζ) z[Lm Lm [Lm γ f (z, ζ)] γ f (z, ζ) γ f (z, ζ)]

implies

1+zζ 1−zζ

≺≺

Lm γ f (z,ζ) 0, z[Lm γ f (z,ζ)]

z ∈ U, ζ ∈ U and q(z, ζ) =

1+zζ 1−zζ

(2.2)

(2.4)

is the best dominant.

Proof. In order to prove the theorem, we shall use Lemma 1.1. For that, we show that the necessary conditions are satisfied. Let the functions θ : C → C and φ : C → C with θ(w) = w 3 1498

(2.5)

Oros et al: Strong differential subordinations and superordinations

and

1 , φ(w) 6= 0. (2.6) w+1 We check theconditions from the hypothesis of Lemma 1.1. For β = 1, γ = 1, we have Re [1 · q(z, ζ) + 1] = 1+zζ 2 + 1 = Re 1−zζ > 0, hence condition (i) is satisfied. Re 1−zζ φ(w) =

Let r(z, ζ) =

zq 0 (z,ζ) 1·q(z,ζ)+1

2zζ (1−zζ)2 2 1−zζ

=

=

0

(z,ζ) We have Re zrr(z,ζ) = Re

zζ 1−zζ .

(ii) is satisfied. We consider

zζ (1−zζ)2 zζ 1−zζ

1 > 0, hence condition = Re 1−zζ

Lm γ f (z, ζ) , z ∈ U, ζ ∈ U . 0 z[Lm γ f (z, ζ)]

p(z, ζ) =

(2.7)

Using (1.3) in (2.7), we obtain P∞ P∞ (γ+1)m (γ+1)m k k−1 z + k=n+1 (γ+k) 1 + k=n+1 (γ+k) m ak (ζ)z m ak (ζ)z  p(z, ζ) =  . = P m P∞ ∞ (γ+1) (γ+1)m k−1 k−1 1 + k=n+1 (γ+k) z 1 + k=n+1 (γ+k) m ak (ζ)kz m ak (ζ)kz

(2.8)

Since p(0, ζ) = 1, we have p(z, ζ) ∈ [1, 1] ∩ Qζ . Differentiating (2.7) and after a short calculus we obtain p(z, ζ) +

0 00 [Lm Lm [Lm zp0 (z, ζ) γ f (z, ζ)] γ f (z, ζ) γ f (z, ζ)] + − 1. = − 0 0 p(z, ζ) + 1 z[Lm Lm [Lm γ f (z, ζ)] γ f (z, ζ) γ f (z, ζ)] zp0 (z,ζ) 2+3zζ 2(1−zζ) ≺≺ p(z, ζ) + p(z,ζ)+1 . Lm γ f (z,ζ) 0 , z ∈ U, ζ ∈ U and q(z, ζ) z[Lm γ f (z,ζ)]

(2.9)

Using (2.9) in (2.4), the strong differential superordination becomes From Lemma 1.1, we have q(z, ζ) ≺≺ p(z, ζ), i.e., the best subordinant.

1+zζ 1−zζ

≺≺

=

1+zζ 1−zζ

is

Theorem 2.2 Let h(z, ζ) = ζ−3z ζ+z , be a convex function in U , for all ζ ∈ U , with h(0) = 1. Suppose that the Briot-Bouquet differential equation zq 0 (z, ζ) ζ − 3z q(z, ζ) + = (2.10) q(z, ζ) + 1 ζ +z has an univalent solution q(z, ζ) = If p(z, ζ) =

H

m

f (z,ζ) zr

ζ−z ζ+z ,

that satisfies

ζ−z ζ+z

≺≺

ζ−3z ζ+z .

∈ [1, 1] ∩ Qζ satisfies H m f (z, ζ) z r+1 [H m f (z, ζ)]0 rz r ζ − 3z + − m ≺≺ r m 2 z [H f (z, ζ)] H f (z, ζ) ζ +z

then

H m f (z,ζ) zr

≺≺

ζ−z ζ+z ,

z ∈ U, ζ ∈ U and q(z, ζ) =

ζ−z ζ+z

(2.11)

is the best dominant.

Proof. In order to prove the theorem, we shall use Lemma 1.2. For that, we show that the necessary conditions are satisfied. After a short calculus we obtain     zh00 (z, ζ) ζ −z Re 1 + 0 = Re ≥ 0, z ∈ U, ζ ∈ U . (2.12) h (z, ζ) ζ +z The function q(z, ζ) =

ζ−z ζ+z

is the univalent solution of equation (2.10), hence     2z zq 00 (z, ζ) = Re 1 − ≥ 0. Re 1 + 0 q (z, ζ) ζ +z

We consider p(z, ζ) =

H m f (z, ζ) . zr

(2.13)

(2.14)

P (r+1)m k P∞ zr + ∞ (r+1)m k=r+1 (r+k)m ak (ζ)z k−r = 1+ k=r+1 (r+k) . Since p(0, ζ) = Using (1.5) ˆın (2.14), we obtain p(z, ζ) = m ak (ζ)z zr 1, we have p(z, ζ) ∈ ζ[1, 1] ∩ Qζ . Differentiating (2.14) and after a short calculus we obtain

p(z, ζ) +

zp0 (z, ζ) H m f (z, ζ) z r+1 [H m f (z, ζ)]0 rz r = + − . p(z, ζ) + 1 zr [H m f (z, ζ)]2 H m f (z, ζ) 4 1499

(2.15)

Oros et al: Strong differential subordinations and superordinations

Using (2.15) in (2.11), the strong differential superordination becomes p(z, ζ) + From Lemma 1.1, we have

H m f (z,ζ) zr

Theorem 2.3 Let h1 (z, ζ) =

ζ−z ζ+z ,

≺≺

1−zζ 1+zζ

zp0 (z, ζ) zq 0 (z, ζ) ≺≺ q(z, ζ) + . p(z, ζ) + 1 q(z, ζ) + 1 z ∈ U, ζ ∈ U and q(z, ζ) =

and h2 (z, ζ) = 1 +

z2 ζ

ζ−z ζ+z

(2.16) is the best dominant.

be convex in U , for all ζ ∈ U , with h1 (0, ζ) =

h2 (0, ζ) = 1. Let γ ∈ C, λ 6= 0, with Re γ ≥ 0, and the functions defined by q1 (z, ζ) = −1 + σ1 (z, ζ) is given by Z z γ−1 t dt σ1 (z, ζ) = 1 + tζ 0 γ z2 γ+2 · ζ , z ∈ U, ζ m m f (z,ζ)]0 If f (z, ζ) ∈ Aζ(r), H f (z,ζ)[H rz 2r−1 (2r−1)H m f (z,ζ)(H m f (z,ζ))0 is univalent γrz 2r−1

and q2 (z, ζ) = 1 +

2γζ zγ

· σ1 (z, ζ), where (2.17)

∈ U. ∈ [1, 1]∩Qζ , and

m m m H m f (z,ζ)(H m f (z,ζ))0 f (z,ζ))0 ]2 f (z,ζ))00 + [(Hγrz + H f (z,ζ)(H − 2r−2 γrz 2r−1 γrz 2r−2

in U , for all ζ ∈ U , then

[(H m f (z, ζ))0 ]2 + H m f (z, ζ)(H m f (z, ζ))00 z2 1 − zζ (2 − 2r)H m f (z, ζ)(H m f (z, ζ))0 + ≺≺ 1 + ≺≺ , 1 + zζ γrz 2r−1 γrz 2r−2 ζ implies −1 + 2γζ z γ σ1 (z, ζ) ≺≺

(2.18)

2 H m f (z,ζ)(H m f (z,ζ))0 γ ≺≺ 1 + γ+2 · zζ , where σ1 (z, ζ), given by (2.17), z ∈ U , ζ ∈ U . rz 2r−1 γ z2 −1 + 2γζ z γ σ1 (z, ζ) and q2 (z, ζ) = 1 + γ+2 · ζ are convex and they are respectively

The functions q1 (z, ζ) = the best subordinant and best dominant.

Proof. In order to prove we shall use Lemma 1.3. For that,h we show that h the theorem, i i the necessary zh00 zh00 1−zζ 2 (z,ζ) 1 (z,ζ) conditions are satisfied. Re 1 + h0 (z,ζ) = Re 1+zζ ≥ 0, z ∈ U, ζ ∈ U and Re 1 + h0 (z,ζ) = Re 2 ≥ 0, z ∈ 1

1

U, ζ ∈ U we put p(z, ζ) =

H m f (z, ζ)(H m f (z, ζ))0 , z ∈ U, ζ ∈ U . rz 2r−1 h

P zr + ∞ k=r+1

(2.19)

ih P (γ+1)m a (ζ)z k rz r−1 + ∞ k=r+1 (γ+k)m k

(γ+1)m (γ+k)m

ak (ζ)kz k−1

i

we obtaini hp(z, ζ) = = hUsingP(1.5) in (2.14), i rz2r−1 P∞ ∞ (γ+1)m (γ+1)m k−r k−r 1 + k=r+1 (γ+k)m ak (ζ)z r + k=r+1 (γ+k)m ak (ζ)kz . Since p(0, ζ) = 1, we have p(z, ζ) ∈ ζ[1, 1] ∩ Qζ . Differentiating (2.14), and after a short calculus we obtain p(z, ζ) +

zp0 (z, ζ) (2 − 2r)H m f (z, ζ)(H m f (z, ζ))0 [(H m f (z, ζ))0 ]2 + H m f (z, ζ)(H m f (z, ζ))00 = + . 2r−1 γ γrz γrz 2r−2

(2.20)

Using (2.20) in (2.18), we have 1 − zζ zp0 (z, ζ) z2 ≺≺ p(z, ζ) + ≺≺ 1 + , z ∈ U, ζ ∈ U . 1 + zζ γ ζ Using Lemma 1.3, we have −1 +

2γζ z γ σ1 (z, ζ)

≺≺

H m f (z,ζ)(H m f (z,ζ))0 rz 2r−1

≺≺ 1 +

γ γ+2

Example 2.1 Let γ = 1, m = 1, r = 3, f (z, ζ) = x3 + x4 ζ, H 1 (z, ζ) = p(z, ζ) =

1 16

+

7ζ 30 z

+

16ζ 75

2

z 2 , p(z, ζ) + zp0 (z, ζ) =

From Theorem 2.3, we have 1+

z2 3ζ ,

1−zζ 1+zζ

≺≺

1 6

+

1 6

2

7ζ 16ζ 2 15 z + 25 z , q1 (z, ζ) = 2 16ζ z2 2 25 z ≺≺ 1 + 3ζ implies

+

7ζ 15 z +

2 z

·

z2 ζ .

Rz 0

(2.21)

2ζ 4 1 3 4z + 5 z , ln(1+zζ) z2 , q2 (z, ζ) = 1 + 3ζ . z ln(1+zζ) 2ζ 4 1 3 ≺≺ 4 z + 5 z ≺≺ z

(t3 + t4 ζ)dt =

−1 + −1 +

z ∈ U, ζ ∈ U .

References [1] A. Alb Lupa¸s, On special strong differential subordinations using multiplier transformation, Applied Mathematics Letters 25(2012), 624-630, doi:10.1016/j.aml.2011.09.074.

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Oros et al: Strong differential subordinations and superordinations

[2] A. Alb Lupa¸s, G.I. Oros, Gh. Oros, On special strong differential subordinations using S˘ al˘ agean and Ruscheweyh operators, Journal of Computational Analysis and Applications, Vol. 14, No. 2, 2012, 266270. [3] A. Alb Lupa¸s, G.I. Oros, Gh. Oros, A note on special strong differential subordinations using multiplier transformation, Journal of Computational Analysis and Applications, Vol. 14, No. 2, 2012, 261-265. [4] A. Alb Lupa¸s, A note on special strong differential subordinations using a multiplier transformation and Ruscheweyh derivative, Journal of Concrete and Applicable Mathematics, Vol. 10, No.’s 1-2, 2012, 24-31. [5] A. Alb Lupa¸s, On special strong differential subordinations using a generalized S˘ al˘ agean operator and Ruscheweyh derivative, Journal of Concrete and Applicable Mathematics, Vol. 10, No.’s 1-2, 2012, 17-23. [6] A. Alb Lupa¸s, G.I. Oros, A note on strong differential superordinations using a multiplier transformation and Ruscheweyh operator, Acta Universitatis Apulensis Special Issue ICTAMI 2011, 407-422. [7] A. Alb Lupa¸s, Certain strong differential superordinations using a multiplier transformation and Ruscheweyh operator, Analele Universit˘a¸tii din Oradea, Fascicola Matematica, Tom XIX (2012), Issue No. 1, 125-136. [8] A. Alb Lupa¸s, A note on strong differential superordinations using S˘ al˘ agean and Ruscheweyh operators, Journal of Applied Functional Analysis, Vol. 7, No.’s 1-2, 2012, 54-61. [9] A. Alb Lupa¸s, Certain strong differential superordinations using a generalized S˘ al˘ agean operator and Ruscheweyh operator, Journal of Applied Functional Analysis, Vol. 7, No.’s 1-2, 2012, 62-68. [10] J.A. Antonino, S. Romaguera, Strong differential subordination to Briot-Bouquet differential equations, Journal of Differential Equations, 114(1994), 101-105. [11] S. S. Miller, P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65(1978), 298-305. [12] S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Michig. Math. J., 28(1981), 157-171. [13] S. S. Miller, P. T. Mocanu, Differential subordinations. Theory and applications, Marcel Dekker, Inc., New York, Basel, 2000. [14] S. S. Miller, P. T. Mocanu, Subordinants of differential superordinations, Complex Variables, 48(10)(2003), 815-826. [15] S. S. Miller, P. T. Mocanu, Briot-Bouquet differential superordinations and sandwich theorems, J. Math. Anal. Appl., 329(2007), no. 1, 327-335. [16] G.I. Oros, On a new strong differential subordination, Acta Universitatis Apulensis, 32(2012), 6-17. [17] G.I. Oros, Strong differential superordination, Acta Universitatis Apulensis, 19(2009), 110-116. [18] G.I. Oros, An application of the subordination chains, Fractional Calculus and Applied Analysis, 13(2010), no. 5, 521-530. [19] G.I. Oros, Briot-Bouquet, strong differential subordination, Journal of Computational Analysis and Applications, 14(2012), no. 4, 733-737. [20] G.I. Oros, Gh. Oros, Strong differential subordination, Turkish Journal of Mathematics, 33(2009), 249-257. [21] G.I. Oros, Gh. Oros, Strong differential superordination and sandwich theorem obtained by new integral operators, submitted Mathematical Inequalities and Applications, 2012. [22] G.I. Oros, Gh. Oros, Second order nonlinear strong differential subordinations, Bull. Belg. Math. Soc. Simion Stevin, 16(2009), 171-178. [23] G.I. Oros, First order strong differential superordination, General Mathematics, vol. 15, No. 2-3 (2007), 77-87. [24] Gh. Oros, Briot-Bouquet strong differential superordinations and sandwich theorems, Math. Reports, 12(62)(2010), no. 3, 277-283.

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1502

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 8, 2013

A Modified Nonlinear Uzawa Algorithm for Solving Symmetric Saddle Point Problems, Jian-Lei Li, Zhi-Jiang Zhang, Ying-Wang, and Li-Tao Zhang,………………………………………1356 A Boundary Value Problem of Fractional Differential Equations with Anti-Periodic Type Integral Boundary Conditions, Bashir Ahmad, and S.K. Ntouyas,………………………1372 Coupled Common Fixed Point Theorems for Weakly Increasing Mappings with Two Variables, Hui-Sheng Ding, Lu Li, and Wei Long,…………………………………………………….1381 On Strictly and Semistrictly Quasi ߙ − Preinvex Functions, Wan Mei Tang,………………1391 On Stability of Functional Inequalities at Random Lattice ߮ −Normed Spaces, Sung Jin Lee and Reza Saadati,………………………………………………………………………………1403 On bi-Cubic Functional Equations, A. Fazeli and E. Amini Sarteshnizi,…………………1413 A Note on the Second Kind Generalized q-Euler Polynomials, C. S. RYOO,………….…1424 Analytic Approximation of Time-Fractional Diffusion-Wave Equation Based on Connection of Fractional and Ordinary Calculus, H. Fallahgoul and S. M. Hashemiparast,……………1430 Higher Order Duality in Nondifferentiable Fractional Programming Involving Generalized Convexity, I. Ahmad, Ravi P. Agarwal, and Anurag Jayswal,……………………………1444 Fuzzy Implicative Filters of BE-Algebras with Degrees in the Interval ሺ0, 1ሿ, Young Bae Jun and Sun Shin Ahn,……………………………………………………………………………….1456 An AQ-Functional Equation in Paranormed Spaces, Taek Min Kim, Choonkil Park, and Seo Hong Park,…………………………………………………………………………………..1467 A Note on Special Fuzzy Differential Subordinations Using Generalized Sălăgean Operator and Ruscheweyh Derivative, Alina Alb Lupaș,…………………………………………………1476 On Special Fuzzy Differential Subordinations Using Convolution Product of Sălăgean Operator and Ruscheweyh Derivative, Alina Alb Lupaș,…………………………………………….1484 Strong Differential Superordination and Sandwich Theorem, Gheorghe Oros, Roxana Șendruțiu, Adela Venter, and Loriana Andrei,…………………………………………………………1490

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 15, NO. 8, 2013 (continued)

Strong Differential Subordinations and Superordinations and Sandwich Theorem, Gheorghe Oros, Adela Venter, Roxana Șendruțiu, and Loriana Andrei,……………………………1496