JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 19, 2015 0651098283

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Volume 19, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

July 2015

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 (twelve 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. Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC

28601, 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

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) Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 P.D.E, Control Theory, Functional Analysis, [email protected]

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23) 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 24) 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

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.

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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

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10) Ding-Xuan Zhou Department Of Mathematics City University of Hong Kong

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83 Tat Chee Avenue Kowloon,Hong Kong 852-2788 9708,Fax:852-2788 8561 e-mail: [email protected] Approximation Theory, Spline functions,Wavelets

and Artificial Intelligence, Operations Research, Math.Programming 30) T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece Postal Address: 26 Menelaou St. Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis

11) Sever S.Dragomir School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, MC 8001,AUSTRALIA Tel. +61 3 9688 4437 Fax +61 3 9688 4050 [email protected] Inequalities,Functional Analysis, Numerical Analysis, Approximations, Information Theory, Stochastics.

31) I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3 0651098283

12) Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR-06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

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13) Saber N.Elaydi Department Of Mathematics Trinity University 715 Stadium Dr. San Antonio,TX 78212-7200 210-736-8246 e-mail: [email protected] Ordinary Differential Equations, Difference Equations

33) Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 P.D.E, Control Theory, Functional Analysis, [email protected]

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

34) Gilbert G.Walter Department Of Mathematical Sciences University of Wisconsin-Milwaukee,Box 413, Milwaukee,WI 53201-0413 414-229-5077 e-mail: [email protected] Distribution Functions, Generalised Functions, Wavelets 35) 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

Lotharstr.65,D-47048 Duisburg,Germany e-mail:[email protected] Fourier Analysis,Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory

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

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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

37) 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 38) 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

NEW MEMBERS 39)Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] Computational Mathematics

40) Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S.Korea, [email protected] Functional Equations

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

A new consensus protocol of the multi-agent systems Xin-Lei Fenga,b∗, Ting-Zhu Huangb† a

College of Mathematics and information Science, Leshan Normal University Leshan, Sichuan, 614000, P. R. China b

School of Mathematical Sciences,

University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, PR China

Abstract In consensus protocols of multi-agent system, we can often find state differences between some agents and a reference agent. However, we find that taking the relative states value may not be best. Conversely, using the differences between some agents states and several times state of the reference agent can achieve quickly convergence by choosing a appropriate parameter r. Therefore, in this paper a new consensus protocol taking r times state of reference agent is proposed. Based on this idea, two convergence protocols of single-integrator kinematics are considered, and some sufficient conditions on consensus protocols of double-integrator dynamics are derived. Finally, numerical examples illustrate these theoretical results. Keywords: Consensus protocol; Multi-agent system; Graph; Continuoustime system; Time-delay

1

Introduction

Recently, multi-agent systems have received significant attention due to their potential impacts in numerous civilian, homeland security and military applications, etc. Consensus plays an important role in achieving distributed coordination. The basic idea of consensus is that a team of vehicles reaches an agreement on a common ∗ †

E-mail: [email protected] E-mail: [email protected]

1

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value by negotiating with their neighbors. Consensus algorithms are studied for both single-integrator kinematics [1–5] and high-order-integrator dynamics [6–10]. Formal study of consensus problems in groups of experts originated in management science and statistics in 1960s. Distributed computation over networks has a tradition in systems and control theory starting with the pioneering work of Borkar et. al. Vicsek et. al [11] provided a formal analysis of emergence of alignment in the simplified model of flocking, which has an important influence on the developing of consensus theories of multi-agent systems. On the study of consensus of continnuous-time systems, classical model of consensus is provided by Olfati-Saber and Murray [12], which can be described by the following linear dynamic system: x˙ i (t) = ui (t),

(1)

the consensus protocol is given as X ui (t) = aij (t)(xj (t) − xi (t)),

(2)

j∈Ni (t)

and the model with time-delays is given as X ui (t) = aij (t)(xj (t − τij ) − xi (t − τij )).

(3)

j∈Ni (t)

However, in the movement of agents for protocol (2) and (3), in order to achieve quick convergence, taking the relative state may not be the best, conversely, taking several times state values of reference agents may quickly achieve convergence (in Section 4, this phenomena can be verified by simulation experiments). In this paper, on basis of (2) and (3), we present the following protocol: X ui (t) = aij (t)(xj (t) − rxi (t)), r > 1, (4) j∈Ni (t)

and the corresponding time-delayed protocol is given as X ui (t) = aij (t)(xj (t − τ ) − rxi (t − τ )), r > 1,

(5)

j∈Ni (t)

respectively, where xi (t) = xi (0), for t ∈ [−τ, 0), i = 1, · · · , n. Next we consider double-integrator dynamics consensus protocols based on (4) and (5), respectively. The paper is organized as follows. In Section 2, we introduce basic concepts and preliminary results. In Section 3, we firstly derive sufficient conditions on protocols for two single-integrator kinematics, then we obtain some sufficient conditions on consensus protocols for double-integrator dynamics. In Section 4, numerical examples are presented to illustrate our theoretical results. Conclusions are drawn in Section 5. 2

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2

Preliminaries

Suppose that the multi-agent system under consideration consists of n agents. If there exists an available information channel from the agent j to the agent i, then agent j is a neighbor of agent i. The set of all neighbors of agent i is denoted by Ni (t). A directed graph (digraph) G = (V, E) of order n consists of a set of nodes V = {v1 , . . . , vn } and a set of edges E = V × V . (i, j) is an edge of G if and only if (i, j) ∈ E. Each agent is regarded as a node in a directed graph G. Each available information channel from the agent i to the agent j corresponds to an edge (i, j) ∈ E. The weighted adjacency matrix A ∈ Rn×n is defined as aii = 0 and aij ≥ 0. aij > 0 if only and if (j, i) ∈ E. In this paper, we only consider that the weight aij , i, j = 1, · · · , n, of digraph of multi-agent systems are unchanged. The Laplacian matrix L ∈ Rn×n is defined as X lii = aij , lij = −aij , for i 6= j. j6=i

A graph with the property that (i, j) ∈ E implies (j, i) ∈ E is said to be undirected. Moreover, matrix L is symmetric if an undirected graph has symmetric weights, i.e., aij = aji . A directed path is a sequence of edges in a directed graph with the form (v1 , v2 ), (v2 , v3 ), · · · , where vi ∈ V . A directed graph has a directed spanning tree if there exists at least one node that has a directed path to all other nodes. Lemma 1 [12, 13]. (i) All the eigenvalues of Laplacian matrix L have nonnegative real parts; (ii) Zero is an eigenvalue of L with 1n (where 1n is the n × 1 column vector of all ones) as the corresponding right eigenvector. Furthermore, zero is a simple eigenvalue of L if and only if graph G has a directed spanning tree. Lemma 2. If L ∈ Rn×n is a Laplacian matrix with the directed graph G that has a directed spanning tree, and D ∈ Rn×n is a diagonal matrix with di < 0 (i = 1, · · · , n), then all the eigenvalues of −L + D have negative real parts. Proof. Notice that the directed graph has a directed spanning tree, it follows from Lemma 1 that L has a simple zero eigenvalue and all other P eigenvalues have positive real parts. With the definition of L, we know that lii = j6=i aij . Let λi , i = 1, · · · , n, be eigenvalues of −L + D. By Gerschgorin disk theorem, |λi + lii − di | < lii , i = 1, · · · , n.

(6)

It follows from (6) that all the eigenvalues of −L + D lie on a circle with center at (di − lii , 0) and radius lii . Therefore, λi , i = 1, · · · , n, have negative real parts. 2

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Lemma 3. Let A ∈ Rn×n with eigenvalueµ γi and associated¶ right and left eigen0n×n In vectors si and qi , respectively. Let B = (α > 0). Then the A A − αIn eigenvalues of B are given by p −α − γi + (γi − α)2 + 4γi ξ2i−1 = 2 γi with associated right and left eigenvectors (sTi , ξ2i−1 sTi )T and ( ξ2i−1 qiT , qiT )T , respectively, and p −α − γi − (γi − α)2 + 4γi ξ2i = 2

with associated right and left eigenvectors (sTi , ξ2i sTi )T and ( ξγ2ii qiT , qiT )T , respectively, i = 1, · · · , n; (αqiT , qiT )T is a left eigenvector of zero eigenvalue of B. Proof. Suppose that ξ is an eigenvalue of B with an associated right eigenvector (f T , g T )T , where f, g ∈ Cn . It follows that µ ¶µ ¶ µ ¶ 0n×n In f f =ξ , A A − αIn g g which implies g = ξf and Af + Ag − αg = ξg. Thus (1 + ξ)Af = (ξ 2 − αξ)f. Let f = si . By Asi = γi si , we get γi (1 + ξ) = ξ 2 − αξ. That is, each eigenvalue of A, γi , corresponds to two eigenvalues of B, denoted by p −α − γi ± (γi − α)2 + 4γi ξ2i−1,2i = . 2 Because g = ξf , it follows that the right eigenvectors associated with ξ2i−1 and ξ2i are, respectively, (sTi , ξ2i−1 sTi )T and (sTi , ξ2i sTi )T . A similar analysis can be used to find the left eigenvectors of B associated with ξ2i−1 and ξ2i . 2 √ −α+γu± (γu−α)2 +4u , where ρ, u ∈ C. If α > 0, Re(u) < 0 Lemma 4 [14]. Let ρ± = 2 r 2 and γ > −Re(u) , then Re(ρ± ) < 0, where Re(·) represents the real π |u|| cos( 2 −arctan

Im(u)

)|

part of a number. 4

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3

Consensus of continuous-time systems

In this section, we give our main results. In Subsection 3.1, two convergence protocols of multi-agent systems with single-integrator kinematics are considered. In Subsection 3.2, sufficient conditions on consensus protocols of multi-agent systems with double-integrator dynamics are derived.

3.1

Convergence protocols of multi-agent systems with singleintegrator kinematics

Firstly, we consider the protocol (4). The result is given as follows: Theorem 1. Assume that there are n agents with n ≥ 2. For the protocol (4), the state x(t) of agents converge exponentially to zero if the directed graph G has a directed spanning tree. Proof. From (1) and (4), we can obtain X x˙ i (t) = aij (t)(xj (t) − rxi (t)), r > 1.

(7)

j∈Ni (t)

We take the vector form of the above equation: x(t) ˙ = −Lx(t) − (r − 1)diag(l11 , · · · , lnn )x(t),

(8)

(8) can be rewritten as x(t) ˙ = (−L + D)x(t),

(9)

where D = −(r − 1)diag(l11 , · · · , lnn ). By Lemma 2, all the eigenvalues of −L + D have negative real parts. Therefore, the system is stable. Then we get x(t) = e(−L+D)t x(0). So x(t) converges exponentially to zero, and the convergence is asymptotically achieved. 2 Below we consider the following equation with time-delay: X x˙ i (t) = aij (t)(xj (t − τ ) − rxi (t − τ )), r > 1.

(10)

j∈Ni (t)

Furthermore, (10) can be rewritten as x(t) ˙ = (−L + D)x(t − τ ),

(11) 5

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where D = −(r − 1)diag(l11 , · · · , lnn ). We obtain the following result: Theorem 2. Assume that there are n agents with n ≥ 2 and the directed graph G has a directed spanning tree. For the system (11), the state x(t) of agents asymptotically converge to zero if τ < min i

θi − π2 , i = 1, · · · , n, |λi |

(12)

) denotes the phase of λi . where λi is the eigenvalue of −L + D and θi ∈ ( π2 , 3π 2 e = L − D. By Lemma 2, the eigenvalues of −L e have negative real Proof. Let L π 3π parts, so we can assume that θi ∈ ( 2 , 2 ). Moreover, (11) can be rewritten as e x(t) ˙ = −Lx(t − τ ).

(13)

Taking the Laplace transform of the system (13), and let x(0) = 0, we get e sX(s) = −e−sτ LX(s),

(14)

where X(s) is the Laplace transform of x(t). Thus, we get the characteristic equation about X(s) as follows: e = 0. det(sI + e−sτ L)

(15)

Moreover, (15) is equivalent to n Y (s − e−sτ λi ) = 0,

(16)

i=1

e where λi is the eigenvalue of −L + D, i = 1, · · · , n. Since all the eigenvalues of −L have negative real parts, obviously, s = 0 is not a root of the above equation. Thus, we consider the roots of the following equation 1−

e−sτ λi = 0. s

(17)

Based on the Nyquist stability criterion, the roots of (17) lie on the open left half −iωτ complex plane if and only if the Nyquist curve − e jω λi does not enclose the point (−1, j0) for ω ∈ R. −iωτ −iωτ When ω ∈ (0, ∞), | − e jω λi | = |λωi | and arg(− e jω λi ) = θi − ωτ + π2 are all monotonously decreasing, where arg(·) denotes the phase, and θi ∈ ( π2 , 3π ) denotes 2 e−iωτ λi the phase of λi . Since − jω crosses the negative real axis for the first time at 6

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−iωτ λ i jω0

arg(− e

) = θi − ω0 τ +

complex plane if and only if

π = π, the 2 |λi | < 1. So ω0

roots of (17) all lie on the open left half we get

θi − π2 τ< . |λi | Similarly, when ω ∈ (−∞, 0), we get τ
1, (20) j∈Ni (t)

ui (t) =

X

j∈Ni (t)

aij (t)(xj (t−τ )−xi (t−τ ))+

j∈Ni (t)

X

aij (t)(vj (t−τ )−rvi (t−τ )), r > 1,

j∈Ni (t)

(21) where xi (t) = xi (0), vi (t) = 0, t ∈ [−τ, 0), i = 1, · · · , n. Furthermore, (18), (19) and (20) can be rewritten as x(t) ˙ = v(t),

(22) 7

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v(t) ˙ = −Lx(t) + (−L + D)v(t),

(23)

where D = −(r − 1)diag(l11 , · · · , lnn ). For the consensus protocol (20), we obtain the following theorem: Theorem 3. Assume that there are n agents with n ≥ 2 and the directed graph G has a directed spanning tree. For the consensus protocol (20), consensus is achieved if l11 = l22 = · · · = lnn and s 2 < 1 (when λi 6= 0), i = 1, · · · , n, max π i) i )| |λi || cos( 2 − arctan −Re(λ Im(λi ) where lii is the diagonal element of Laplacian matrix L and λi is eigenvalue of −L. The state convergence values of agents are xi (t) = (r − 1)l11 w1T x(0) + w1T v(0), vi (t) = 0, t → ∞, where w1 is a left eigenvector of zero eigenvalue of L, and (r − 1)l11 w1T 1n = 1. Proof. Equations (22) and (23) can be rewritten in matrix form as à ! µ ¶µ ¶ ˙ x(t) 0n×n In x(t) = . (24) ˙ −L −L − (r − 1)l11 In v(t) v(t) µ

¶ 0n×n In Let Γ = , α = (r − 1)l11 and λi be eigenvalue of −L −L −L − (r − 1)l11 In with associated right and left eigenvectors qi and wi , respectively, i = 1, · · · , n. We can assume without loss of generality that λ1 = 0 and λi , i = 2, · · · , n, have negative real parts. Note from Lemma 3 that each √ eigenvalue λi of L corresponds to −α−λi ±

two eigenvalues of Γ given by ξ2i−1,2i = and ξ2 = (1 − r)l11 . By Lemma 4, if s 2 < 1, π i) )| |λi || cos( 2 − arctan −Re(λ Im(λi )

(λi −α)2 +4λi . 2

Moreover, we get ξ1 = 0

then ξ2i−1,2i have negative real parts with associated right and left eigenvectors given by (qiT , ξ2i−1 qiT )T , (qiT , ξ2i qiT )T , (

λi ξ2i−1

wiT , wiT )T and (

λi T T T w ,w ) , ξ2i i i

respectively, i = 2, · · · , n. Moreover, Γ can be written in Jordan canonical form as SJS −1 , where the columns of S, denoted by sk , k = 1, · · · , 2n, can be chosen to be the right eigenvectors or generalized right eigenvectors of Γ associated with eigenvalue ξk , hTk , k = 8

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1, · · · , 2n, which are the rows of S −1 can be chosen to be the left eigenvectors or generalized left eigenvectors of ξk satisfing hTk sk = 1, hTk sl = 0 (k 6= l), and J is the Jordan block diagonal matrix with ξk being the diagonal entries. Thus, we can choose s1 = (1Tn , 0Tn )T and h1 = ((r − 1)l11 ω1T , ω1T )T . Moreover, we can get µ ¶ µ ¶ x(t) x(0) Γt lim =e v(t) v(0) t→∞   hT1 µ ¶ µ ¶ ¡ ¢ 1 01×n  ..  x(0) = s1 · · · s2n  .  0n×1 0n×n v(0) hT2n µ ¶ µ ¶ ¢ x(0) 1n ¡ T T (r − 1)l11 w1 w1 = , 0 v(0) which implies that xi (t) → (r − 1)l11 w1T x(0) + w1T v(0) and vi (t) → 0 as t → ∞. So the theorem is proved. 2 Remark 1. Our consensus protocol (20) has the following advantages compared with the classic one which is consensus protocol (20) at r = 1: 1) In (20) , if r ≥ 1, then our consensus protocol can include the classic one as a special case; 2) Our consensus protocol has quicker convergence speed; 3) Our consensus protocol has wider convergence range. The advantages of 2) and 3) can be seen in Example 1 and 3 of Section 4, respectively. Recently many new protocols [4,5,7] are proposed on basis of the protocols in [12]. Though our protocol is got by modifying the protocols in [12], these new protocols can also be modified according to our method to improve convergence speed. This will be further considered in our future work. Theorem 4. Assume that there are n agents with n ≥ 2, and the undirected graph G is connected and symmetric. For the protocol (21), consensus is achieved if l11 = l22 = · · · = lnn , p λ2i + ω02 (λi + (r − 1)l11 )2 π and max < 1 (λi 6= 0, i =, · · · , n), τ< i 2(r − 1)l11 ω02 arctan(

ω0 (λi +(r−1)l11 ) ) λi

, lii and λi are the diagonal element and eigenvalue of where ω0 = τ Laplacian matrix L, respectively. The state convergence values of agents are xi (t) = (r − 1)l11 w1T x(0) + w1T v(0), vi (t) = 0, t → ∞, 9

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where w1 is a left eigenvector of zero eigenvalue of L, and (r − 1)l11 w1T 1n = 1. Proof. (18), (19) and (21) can be rewritten as x(t) ˙ = v(t),

(25)

v(t) ˙ = −Lx(t − τ ) + (−L + (1 − r)l11 In )v(t − τ ), r > 1.

(26)

Taking the Laplace transform of the equation (25) and (26), we get sX(s) = V (s),

(27)

sV (s) = −LX(s)e−sτ + (−L + (1 − r)l11 In )V (s)e−sτ ,

(28)

where X(s) and V (s) are the Laplace transforms of x(t) and v(t), respectively, and let x(0) = v(0) = 0. The above equation can be rewritten as s2 X(s) = −LX(s)e−sτ + (−L + (1 − r)l11 In )sX(s)e−sτ .

(29)

Thus, we get the characteristic equation about X(s) as follows: det(s2 In + Le−sτ − (−L + (1 − r)l11 In )se−sτ ) = 0.

(30)

By the undirected graph G is connected and symmetric, we get L is positive semidefinite. Without loss of generality, we can assume that the eigenvalues of L, λ1 = 0 and λi > 0, i = 2, · · · , n. Thus, (30) equals n Y (s2 + e−sτ λi − (−λi + (1 − r)l11 )se−sτ ) = 0.

(31)

i=1

Furthermore, we consider the roots of the following equation s2 + e−sτ λi − (−λi + (1 − r)l11 )se−sτ = 0, i = 1, · · · , n.

(32)

When λ1 = 0, (32) becomes s2 − (1 − r)l11 se−sτ = 0.

(33)

Obviously, s = 0 is a root of the above equation. When s 6= 0, we consider the roots of the following equation 1−

(1 − r)l11 e−sτ = 0. s

(34)

By Nyquist stability criterion, the roots of (34) lie on the open left half complex plane −jωτ 11 e does not enclose the point (−1, j0) for if and only if the Nyquist curve − (1−r)ljω ω ∈ R. Similar to the proof of Theorem 2, we get π τ< . 2(r − 1)l11 10

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When λi 6= 0, we consider the roots of the following equation 1+

λi + s(λi + (r − 1)l11 ) −sτ e = 0. s2

(35)

Similarly, the roots of (35) lie on the open left half complex plane if and only if i +(r−1)l11 ) −jωτ the Nyquist curve z = λi +jω(λ−ω e does not enclose the point (−1, j0) for 2 ω ∈ R. We can get p λ2i + ω 2 (λi + (r − 1)l11 )2 |z| = , ω2 arg(z) = −ωτ + π + arctan

ω(λi + (r − 1)l11 ) . λi

Obviously, |z| is monotonously decreasing for ω ∈ (ω0 , ∞). Since curve z crosses the real axis for the first time at ω0 =

11 ) ) arctan( ω0 (λi +(r−1)l λi

τ

and we can find arg(z) is monotonous function, the roots of (35) lie on the open left half complex plane if we let p λ2i + ω02 (λi + (r − 1)l11 )2 < 1. max i ω02 According to the above analysis, the roots of (30) all lie on the open left half complex plane except for a root at s = 0. Therefore, x(t) and v(t) of the system converge to a steady state, and consensus is achieved. Equations (25) and (26) can be rewritten as à ! µ ¶µ ¶ ˙ x(t) 0n×n In x(t − τ ) = . (36) ˙ −L −L − (r − 1)l11 In v(t − τ ) v(t) Let

µ Γ=

0n×n In −L −L − (r − 1)l11 In

¶

µ , z(t) =

x(t) v(t)

¶ .

Taking the Laplace transform of the equation (36) sZ(s) − z(0) = e−sτ ΓZ(s),

(37)

where Z(s) is the Laplace transform of z(t). Moreover, by the µ Proof of Theorem ¶ 0 01×n −1 3, Γ can be written in Jordan canonical form as SJS = S S −1 , 0n×1 Γ0 11

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where S is the same as one in the Proof of Theorem 3, and the eigenvalues of Γ0 only have negative real parts. Thus lim z(t) = lim sZ(s) = lim s(sIn − e−sτ Γ)−1 z(0) s−→0+ s−→0+ ¶ µ 0 01×n )−1 S −1 z(0) = lim S(In − −sτ s−→0+ 0n×1 e s Γ0 µ ¶ 1 01×n = lim S S −1 z(0) −sτ s−→0+ 0n×1 (In−1 − e s Γ0 )−1   hT1 µ ¶ ¡ ¢ 1 01×n  ..  = s1 · · · s2n  .  z(0) 0n×1 0n×n hT2n µ ¶ µ ¶ ¢ x(0) 1n ¡ T T (r − 1)l11 w1 w1 = , 0 v(0)

t−→∞

which implies that xi (t) → (r − 1)l11 w1T x(0) + w1T v(0) and vi (t) → 0 as t → ∞. So the theorem is proved.2

4

Simulation

In this section, several simulation results are presented to illustrate the proposed protocols introduced in Section 3. Example 1. We consider a system with three agents under the protocol (4) and (20). The corresponding Laplacian matrix and initial value are chosen as       1 0 −1 3 2 0  , x(0) =  2  , and v(0) =  1  . L =  −1 1 0 −1 1 −5 −3 Moreover, the values of parameter r are shown in Figure 1. Note that the digraph G has a directed spanning tree. Figure 1 is the state trajectories of multi-agent system under (4) with different parameters r. From Figure 1, we can see the convergence speed is quicker when parameter r increases. In addition, the protocol (4) has an exponential convergence. Figure 2 is the state trajectories of multi-agent system under consensus protocol (20) with different parameters r. From Figure 2, we can see that the convergence speed of our consensus protocol with parameter r = 1.5 is quicker than r = 1. That is to say, the convergence speed of our consensus protocol is quicker than the classic one.

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3 r=1 r=3 r=5

2 1

position x

0 −1 −2 −3 −4 −5

0

2

4

6

8

10

time t

Figure 1: State trajectories of multi-agent system under protocol (4) with different parameter r 4

4

3 r=1 r=1.5

r=1 r=1.5

3

2 2

1

1 velocity v

position x

0

−1

0

−2

−3

−1

−4 −2 −5

−6

0

1

2

3

4

5

6

7

8

9

−3

10

0

1

2

3

time t

4

5

6

7

8

9

10

time t

(a) Position trajectories

(b) Velocity trajectories

Figure 2: State trajectories of multi-agent system under consensus protocol (20)

4

8

3

6 4

2

position x

position x

2 1

0

0 −2

−1

−4

−2

−3

τ=0.15 τ=0.17 τ=0.182

−6

0

2

4

6

8

−8

10

time t

(a) Under protocol (4)

0

5

10 time t

15

20

(b) Under protocol (5)

Figure 3: State trajectories of multi-agent system 13

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6

5

5 r=1.2 r=1.55 r=3

4 4 3 velocity v

position x

3 2 1 0

1

−1

r=1.2 r=1.55 r=3

−2 −3

2

0

5

10 time t

15

0

−1

20

(a) Position trajectories

0

5

10 time t

15

20

(b) Velocity trajectories

Figure 4: State trajectories of multi-agent system under consensus protocol (20) Example 2. We consider a system with four agents. In Figure 3, we consider the protocol (4) and (5). The corresponding Laplacian matrix and initial values are chosen as     2 0 −2 0 −3  −1 1   0 0   , x(0) =  2  . L=  0 −2 3 −1   4  0 0 −4 4 −1 Then D = diag(−2, −1, −3, −4). Note that the digraph G has a directed spanning tree, then the conditions of Theorem 1 are satisfied. Figure 3 (a) is state trajectories of multi-agent system under protocol (4). By (12), if τ < 0.1706, then global convergence is achieved. Figure 3 (b) is state trajectories of multi-agent system under protocol (5) with different parameters τ . In Figure 3 (b), it can be seen that multi-agent system is divergent with τ = 0.182. It is clear from Figure 3 that the convergence can be asymptotically achieved under the conditions given by Theorem 1 and 2. Example 3. We consider a system with four agents. In Figure 4, we consider the consensus protocol (20). The corresponding Laplacian matrix and initial values are chosen as       2 0 −2 0 −3 2  −2 2     0 0   , x(0) =  2  and v(0) =  1  . L=  0 −1 2 −1   4   3  0 0 −2 2 −1 −1 The values of parameter r are shown in Figure 4. Easy to verify, Γi (i = 1, · · · , 3, namely, when r1 = 1.2, r2 = 1.55, and r3 = 3, respectively.) has a zero eigenvalue 14

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6

5

5

4

4

3

3 velocity v

position x

2 2 1

1 0

0 −1

−1

−2

−2 −3

0

2

4

6

8

−3

10

time t

0

2

4

6

8

10

time t

(a) Position trajectories

(b) Velocity trajectories

Figure 5: State trajectories of agents under consensus protocol(21) and three nonzero eigenvalues which have negative real parts. Accordingly, the left ¡ ¢T eigenvectors of zero eigenvalue of L are taken w11 = 0.5 0.5 1 0.5 , w12 = ¡ ¢T ¡ ¢T 0.1818 0.1818 0.3636 0.1818 and w13 = 0.05 0.05 0.1 0.05 , respectively. The position consensus convergence values xi = (r − 1)l11 w1i , i = 1, · · · , 3, of agents are 5.2, 2.6545, and 1.6. It is clear from Figure 4 that consensus can be asymptotically achieved under the conditions given by Theorem 3. In Figure 4, note that the consensus protocol (20) is not convergent when r = 1. According to Figure 4, we also obtain that the convergence speed is not quicker when parameter r increases. In this example, by plentiful simulations, we find that the convergence speed is quicker when parameter r is equal to 1.5 or so. Therefore, we can choose appropriate parameter r in order to accelerate convergence of consensus protocol. As to how to choose r to obtain the maximal consensus speed, it will be investigated in our future work. Example 4. We consider a system with four agents. In Figure 5, we consider the consensus protocol (21). The corresponding Laplacian matrix and initial values are chosen as       3 −2 0 −1 −1 3  −2 3 −1 0   5   1       L=  0 −1 3 −2  , x(0) =  2  and v(0) =  2  . −1 0 −2 3 −3 3 √ 2 2 λi +ω0 (λi +(r−1)l11 )2 = The values of parameter are r = 4/3 and τ = 0.2. We get maxi ω02 0.9643 < 1 when λi = 6. Easy to verify, the conditions of Theorem 4 can be satisfied. It is clear from Figure 5 that consensus can be asymptotically achieved under the conditions given by Theorem 4. 15

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5

Conclusion

In this paper, a kind of new consensus protocol of the multi-agent systems is presented. We study two protocols of multi-agent systems with single-integrator kinematics, and consider the consensus protocols of multi-agent systems with doubleintegrator dynamics. We derive some sufficient conditions on consensus. Also we find parameter r play an important role in convergence speed. Especially, for consensus protocol (20), we can choose appropriate parameter r in order to accelerate its convergence. How to choose r to obtain the maximal consensus speed is interesting and challenging, which will be further studied in our future work. Finally, numerical examples illustrate our theoretical results. Acknowledgement. This research is supported by NSFC (60973015), A Project supported by Scientific Research Fund of Sichuan Provincial Education Department (13ZB0107), Sichuan Province Sci. & Tech. Research Project (2011JY0002), Research Project of Leshan Normal University(Z1255).

References [1] Lynch N. Distributed Algorithms. San Francisco, CA: Morgan Kaufmann; 1997. [2] Tian Y, Liu C. Consensus of multi-agent systems with diverse input and communication delays. IEEE Trans Auto Control 2008;53(9):2122-2128. [3] Ren W. Collective Motion from Consensus with Cartesian Coordinate Coupling-Part I: Single-integrator Kinematics. In: Proceedings of the 47th IEEE Conference on Decision and Control; 2008. p. 1006-1011. [4] Qiu J, Lu J, Cao J, He H. Tracking analysis for general linearly coupled dynamical systems. Commun Nonlinear Sci Numer Simulat 2011;16:2072-2085. [5] Lu X, Austin F, Chen S. Flocking in multi-agent systems with active virtual leader and time-varying delays coupling. Commun Nonlinear Sci Numer Simulat 2011;16:10141026. [6] Weller S, Mann N. Assessing rater performance without a F gold standard using consensus theory. Med Decision Making 1997;17(1):71-79. [7] Zhu J, Tian Y, Kuang J. On the general consensus protocol of multi-agent systems with double integrator dynamics. Linear Algebra Appl 2009;431:701-715. [8] Tsitsiklis J. Problems in decentralized decision making and computation. Ph D Dissertation, Dept Electr Eng Comput Sci, Lab Inf Decision Syst, Massachusetts Inst Technol, Cambridge, MA; 1984.

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[9] Ren W. Collective Motion from Consensus with Cartesian Coordinate Coupling-Part II: Double-integrator Dynamics. In: Proceedings of the 47th IEEE Conference on Decision and Control; 2008.p. 1012-1017. [10] Su H, Wang X. Second-order consensus of multiple agents with coupling delay. In: Proceedings of the 7th world Congress on Intelligent Control and Automation; 2008.p. 7181-7186. [11] Vicsek T, Cziroo A, Ben-Jacob E, Cohen I, Shochet O. Novel type of phase transition in a system of self-deriven particles. Phys Rev Lett 1995;75(6):1226-1229. [12] Olfati-Saber R, Murray R. Consensus problems in networks of agents with switching topology and time delays. IEEE Trans Auto Control 2004;49(9):1520-1533. [13] Royle G, Godsil C. Algebraic Graph Theory. Springer Graduate Texts in Mathematics 207, Springer; 2001. [14] Ren W, Atkins E. Distributed multi-vehicle coordinated control via local information exchange. Int J Robust Nonlinear Control 2007;17:1002-1033.

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ALMOST DIFFERENCE SEQUENCE SPACES DERIVED BY USING A GENERALIZED WEIGHTED MEAN ˙ ¨ ALI˙ KARAISA AND FARUK OZGER

Abstract. In this study, we define the spaces f s(u, v, 4), f (u, v, 4) and f0 (u, v, 4) that consist of all sequence whose G(u, v, 4)− transforms are in the set of almost convergent sequence and series spaces. Some topological properties of the new almost convergent sets have been investigated as well as γ- and β-duals of the spaces f (u, v, 4) and f s(u, v, 4). Also, the characterization of certain matrix classes on/into the mentioned spaces has exhaustively been examined. Finally, some identities and inclusion relations related to core theorems are established.

1. Introduction To investigate the topological properties of the almost convergent sequence and series spaces and the studies connected with their duals are very recent (see [1, 16, 25, 29]) since the background information about these spaces have not known properly; nevertheless, the role played by the algebraical, geometrical and topological properties of the new Banach spaces which are the matrix domains of triangle matrices in sequence spaces is very well-known (see [7, 8, 9, 10, 11, 12, 13, 14]). The sets f (G) and f0 (G) which are derived by the generalized weighted mean have recently been introduced and studied in [29]. Matrix domains of the double band B(r, s) and triple band B(r, s, t) matrices in the sets of almost null f0 and almost convergent f sequence spaces have been investigated by Ba¸sar and Kiri¸s¸ci [16] and S¨onmez [2], respectively. They have examined some algebraical and topological properties of certain almost null and almost convergent spaces. Following these authors, Kayaduman and S¸eng¨on¨ ul have subsequently introduced some almost convergent spaces which are the matrix domains of the Riesz matrix [26] and Ces`aro matrix of order 1 [25] in the sets of almost null f0 and almost convergent f sequences. Recently, Karaisa and Karabıyık studied some new almost convergent spaces which are the matrix domains of Ar matrix [1]. The general aim of this study is to fill a gap in literature by extending certain topological spaces and to investigate some topological properties in addition to some core theorems. The paper is divided into six sections. The next section contains a foreknowledge on the main argument of this study. Section 3 introduces the almost convergent sequence and series spaces which are the matrix domains of the G(u, v, 4) matrix in the almost convergent sequence and series spaces f s and f , respectively, in addition to give certain theorems related to behavior of some sequences. Section 4 determines the betaand gamma-duals of the spaces f s(u, v, 4) and f (u, v, 4) and contains some stated and proved theorems on the characterization of the matrix mappings on/into the sequence spaces f s(u, v, 4), f (u, v, 4) and f0 (u, v, 4). The final section examines and compares the present results in the context of Banach sequence spaces as studied by other authors. 2. Preliminaries and notations In this section, we start with recalling the most important notations, definitions and make a few remarks on notions which are needed in this study. By w, we shall denote the space of all real or complex valued sequences. We shall write `∞ , c and c0 for the spaces of all bounded, convergent, and null sequences respectively. Also by bs and cs, we denote the spaces of all bounded and convergent series. Let µ and γ be two sequence spaces and A = (ank ) be an infinite matrix of real or complex numbers ank , where n, k ∈ N. Then, we say that A defines a matrix mapping from µ into γ, and we denote it by writing A : µ → γ, if for every sequence x = (xk ) ∈ µ the sequence Ax = (Ax)n , the A−transform of x, is in γ; where (2.1)

(Ax)n =

X

ank xk

(n ∈ N).

k

The notation (µ : λ) denotes the class of all matrices A such that A : µ → λ. Thus, A ∈ (µ : λ) if and only if the series on the right hand side of (2.1) converges for each n ∈ N and every x ∈ µ, and we have 2010 Mathematics Subject Classification. 46A45,40J05,40C05. Key words and phrases. almost convergence; Schauder basis; β- and γ -duals; matrix mappings; core theorems. 1

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Ax = {(Ax)n }n∈N ∈ λ for all x ∈ µ. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. We write e = (1, 1, · · · ) and U for the set of all sequences u = (uk ) such that uk 6= 0 for all k ∈ N. For u ∈ U , let u1 = ( u1k ). Let u, v ∈ U and define the matrix G(u, v) = (gnk ) by   un vk , (k < n), un vn , (k = n), gnk =  0, (k > n) for all k ∈ N, where un depends only on n and vk only on k. The matrix G(u, v), defined above, is called as generalized weighted mean or factorable matrix. The matrix domain µA of an infinite matrix A in a sequence space µ is defined by (2.2)

µA = {x = (xk ) ∈ ω : Ax ∈ µ}

which is a sequence space. Although in the most cases the new sequence space µA generated by the limitation matrix A from a sequence space µ is the expansion or the contraction of the original space µ. A sequence (bk ) in a normed space X is called a Schauder basis for X if and only if for each x ∈ X, there P∞ exists a unique sequence (αk ) of scalars such that x = k=0 αk bk . Let K be a subset of N. The natural density δ(K) of K ⊆ N is limn n−1 |{{k 6 n : k ∈ K}}| provided it exists, where |E| denotes the cardinality of a set E. A sequence x = (xk ) is called statistically convergent (st−convergent) to the number l, denoted st − lim x, if every  > 0, δ({k : |xk − l| > }) = 0, [32]. We write S and S0 to denote the sets of all statistically convergent sequences and statistically null sequences. The statistically convergent sequences were studied by several authors (see [32, 6, 5]). The concepts of the statistical boundedness, statistical limit superior (or briefly st−lim sup) and statistical limit inferior (or briefly st−lim inf) have been introduced by Fridy and Orhan in [32]. They have also studied on the notions of the statistical core (or briefly st−core) of a statistically bounded sequence as the closed interval [st − lim inf, st − lim sup]. We list the following functionals on `∞ : l(x) qσ (x) L∗ (x)

= = =

lim inf xk , L(x) = lim sup xk , k→∞

k→∞

lim sup sup

m X

m→∞ n∈N i=0 m X

lim sup sup

xσi (n) , xn+i .

m→∞ n∈N i=0

The σ−core of a real bounded sequence x is defined as the closed interval [−qσ (−x), qσ (x)] and also the inequality qσ (Ax) 6 qσ (x) holds for all bounded sequences x. The Knopp-core (shortly K−core) of x is the interval [l(x), L(x)] while the Banach core (in short B−core) of x defined by the interval [−L∗ (−x), L∗ (x)]. In particular, when σ(n) = n + 1 because of the equality qσ (x) = L∗ (x), σ−core of x is reduced to the B−core of x (see [21, 27]). The necessary and sufficient conditions for an infinite matrix matrix A to satisfy the inclusion K − core(Ax) ⊆ B − core(x) for each bounded sequences x obtained in [4]. We now focus on the sets of almost convergent sequences. A continuous linear functional φ on `∞ is called a Banach limit if (i) φ(x) > 0 for x = (xk ), xk > 0 for every k, (ii) φ(xσ(k) ) = φ(xk ) where σ is shift operator which is defined on ω by σ(k) = k + 1 and (iii) φ(e) = 1 where e = (1, 1, 1, . . .). A sequence x = (xk ) ∈ `∞ is said to be almost convergent to the generalized limit α if all Banach limits of x are α [22] and denoted by f − lim x = α. In other words, f − lim xk = α uniformly in n if and only if m

1 X xk+n uniformly in n. m→∞ m + 1 lim

k=0

The characterization given above was proved by Lorentz in [22]. We denote the sets of all almost convergent sequences f and series f s by n o f = x = (xk ) ∈ ω : lim tmn (x) = α uniformly in n , m→∞

where tmn (x) =

m X k=0

1 xk+n , t−1,n = 0 m+1

and   f s = x = (xk ) ∈ ω : ∃l ∈ C 3 

m n+k X X

lim

m→∞

k=0 j=0

  xj = l uniformly in n .  m+1

2

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We know that the inclusions c ⊂ f ⊂ `∞ strictly hold. Because of these inclusions, norms k.kf and k.k∞ of the spaces f and `∞ are equivalent. So the sets f and f0 are BK-spaces with the norm kxkf = supm,n |tmn (x)|. Remark that the γ- and β-duals of the set f s were found by Ba¸sar and Kiri¸sc¸i. Now, we give a theorem about non-existence of Schauder basis of the space f s. Theorem 2.1. The set f s has no Schauder basis. Proof. Let us define the matrix S = (snk ) by snk = 1 (0 6 k 6 n), snk = 0 (n > k). Then x ∈ f s if and only if (Sx)n ∈ f for all n. As a result, since the set f has no basis the set f s has no basis too.  3. The sequence space f s(u, v, 4), f (u, v, 4) and their topological properties For a sequence x = (xk ), we denote the difference sequence space by 4x = (xk − xk−1 ). Let u = (uk )∞ k=0 and v = (vk )∞ k=0 be sequence of real numbers such that uk 6= 0 and vk 6= 0 for all k ∈ N. Recently, the difference sequence space of weighted mean µ(u, v, 4) have been introduced in [23], where µ = c0 , c, and `∞ . These sequence space defined as the matrix domains of the triangle W in the space c0 , c, and `∞ , respectively. The matrix G(u, v, 4) = W = (wnk ) is defined by

wnk

  un (vk − vk+1 ), (k < n), un vn , (k = n), =  0, (k > n).

In this section, we define the new sequence space f s(u, v, 4) and f (u, v, 4) derived by using the generalized weighted mean and topological properties of the sets. Much of the study given in this section is, by now, classical and not very difficult, but it is necessary to give. Now, we introduce the new spaces f (u, v, 4), f0 (u, v, 4) and f s(u, v, 4) as the sets of all sequences such that their G(u, v, 4)− transforms are in the spaces f , f0 and f s, respectively, that is ) m k+n 1 XX uk+n vi (xi − xi−1 ) = α uniformly in n , x ∈ ω : ∃α ∈ C 3 lim m→∞ m + 1 i=0

( f (u, v, 4) =

k=0

( f0 (u, v, 4) =

m k+n

1 XX uk+n vi (xi − xi−1 ) = 0 uniformly in n x ∈ ω : lim m→∞ m + 1 i=0

)

k=0

and   f s(u, v, 4) = x = (xk ) ∈ ω : ∃l ∈ C 3 

lim

m→∞

1 m+1

j m n+k X XX

uj vi (xi − xi−1 ) = l uniformly in n

 

.



k=0 j=0 i=0

We can redefine the spaces f s(u, v, 4), f (u, v, 4) and f0 (u, v, 4) by the notation of (2.2) f0 (u, v, 4) = (f0 )G(u,v,4) , f (u, v, 4) = fG(u,v,4) and f s(u, v, 4) = (f s)G(u,v,4) . Define the sequence y = (yk ), which will be frequently used, as the G(u, v, 4)-transform of a sequence x = (xk ), i.e. (3.1)

yk =

k X i=0

uk vi 4xi =

k X

uk ∇vi xi , (∇v = vi − vi+1 ) (k ∈ N).

i=0

Theorem 3.1. The spaces f (u, v, 4) and f s(u, v, 4) has no Schauder basis. Proof. Since, it is known that the matrix domain µA of a normed sequence space µ has a basis if and only if µ has a basis whenever A = (ank ) is a triangle [7, Remark 2.4] and the space f has no Schauder basis by [28, Corollary 3.3] we have f (u, v, 4) has no Schauder basis. Since the set f s has no basis in theorem 2.1, f s(u, v, 4) has no Schauder basis.  Theorem 3.2. The following statements hold. (i): The sets f (u, v, 4) and f0 (u, v, 4) are linear spaces with the co-ordinatewise addition and scalar multiplication which are BK-spaces with the norm m X k 1 X (3.2) kxkf (u,v,4) = sup uk ∇vi xn+i . m + 1 m i=0 k=0

3

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(ii): The set f s(u, v, 4) is a linear space with the co-ordinatewise addition and scalar multiplication which is a BK-space with the norm m j m n+k XX X 1 X uj ∇vi xi . kxkf s(u,v,4) = sup m+1 m k=0 k=0 j=0 i=0 Proof. Since the second part can be similarly proved, we only focus on the first part. Since the sets f and f0 endowed with the norm k.k∞ are BK-spaces (see[30, Example 7.3.2(b)]) and the matrix W = (wnk ) is normal, Theorem 4.3.2 of Wilansky [3, p.61] gives the fact that the spaces f (u, v, 4) and f0 (u, v, 4) are BK-spaces with the norm in (3.2).  Now, we may give the following theorem concerning the isomorphism between our spaces and the sets f , f0 and f s. Theorem 3.3. The sets f (u, v, 4), f0 (u, v, 4) and f s(u, v, 4) are linearly isomorphic to the sets f, f0 and f s respectively; i.e. f (u, v, 4) ∼ = f , f0 (u, v, 4) ∼ = f0 and f s(u, v, 4) ∼ = f s. ∼ f we should show the existence of a linear bijection between the Proof. To prove the fact that f (u, v, 4) = spaces f (u, v, 4) and f . Consider the transformation T defined with the notation of (2.2) from f (u, v, 4) to f by x 7→ y = T x = G(u, v, 4)x. The linearity of T is clear. Further, it is clear that x = θ whenever T x = θ and hence, T is injective. Let y = (yk ) ∈ f (u, v, 4) and define the sequence x = (xk ) by   k−1  k X 1 1 X yi−1 1 1 1 yi − = − yi + yk for each k ∈ N. (3.3) xk = v ui ui−1 u vi vi+1 uk v k i=0 i i=0 i Then, since (3.4)

  yn+i−1 1 yn+i − = for each i ∈ N vi ui ui−1

(4x)i+n

we have m X

k

1 X uk vi (4x)n+i uniformly in n m→∞ m + 1 i=0 k=0   m k X 1 yn+i yn+i−1 1 X = lim uk v i − uniformly in n m→∞ m + 1 i=0 vi ui ui−1

fG(u,v,4) − lim x =

lim

k=0

m

1 X yk+n uniformly in n m→∞ m + 1

=

lim

k=0

= f − lim y which implies that x ∈ f (u, v, 4). As a result, T is surjective. Hence, T is a linear bijection which implies that the spaces f (u, v, 4) and f are linearly isomorphic, as desired. In the same way, it can be shown that f0 (u, v, 4) and f s(u, v, 4) are linearly isomorphic to f0 and f s, respectively and so we omit the details.  Theorem 3.4. Let the spaces f0 (u, v, 4) and f (u, v, 4) be given. Then, (i): f0 (u, v, 4) ⊂ f (u, v, 4) strictly hold. (ii): If (un ) ∈ c and (vk − vk+1 ) ∈ `1 , then `∞ ⊂ f (u, v, 4) holds. Proof. (i) Let x = (xk ) ∈ f0 (u, v, 4) which means that G(u, v, 4) ∈ f0 . Since f0 ⊂ f , G(u, v, 4) ∈ f . This implies that x ∈ f (u, v, 4). Thus, we have f0 (u, v, 4) ⊂ f (u, v, 4). Now, we show that this inclusion is strict. Let us consider the sequence z = {zk } define by, zk =

k−1 X i=0

1 ui



1 1 − vi vi+1

 +

1 for each k ∈ N uk vk

we have that fG(u,v,4) − lim z = lim

m→∞

m X k=0

k

m

X 1 1 X uk vi (4z)n+i = lim = e = (1, 1, · · · ) m→∞ m + 1 i=0 m+1 k=0

which means that z ∈ f (u, v, 4) \ f0 (u, v, 4) that is to say that the inclusion is strict. (ii) Let (un ) ∈ c and (vk − vk+1 ) ∈ `1 , then we have G(u, v, 4) ∈ c for all x ∈ `∞ . Hence, since c ⊂ f , G(u, v, 4) ∈ f which implies that x = (xk ) ∈ f (u, v, 4). This completes the proof.  4

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4. Certain matrix mappings on the sets f s(u, v, 4) and f (u, v, 4) and some duals In this section, we shall characterize some matrix transformations between the spaces of generalized difference almost convergent sequence and almost convergent series in addition to paranormed and classical sequence spaces after giving β− and γ duals of the spaces f s(u, v, 4) and f (u, v, 4). We start with the definition of the beta and gamma duals. −1 If x and y are sequences and X and Y are subsets of ω, then we write x · y = (xk yk )∞ ∗ Y = {a ∈ k=0 , x ω : a · x ∈ Y } and \ M (X, Y ) = x−1 ∗ Y = {a : a · x ∈ Y for all x ∈ X} x∈X

for the multiplier space of X and Y ; in particular, we use the notations X β = M (X, cs) and X γ = M (X, bs) for the β−and γ−duals of X. Lemma 4.1. [31] A = (ank ) ∈ (f : `∞ ) if and only if X (4.1) sup |ank | < ∞. n

k

Lemma 4.2. [31] A = (ank ) ∈ (f : c) if and only if (4.1) holds and there are α, αk ∈ C such that lim ank = αk for all k ∈ N,

n→∞

lim

X

n→∞

lim

X

n→∞

ank = α,

k

∆(ank − αk ) = 0.

k

Theorem 4.3. Let u, v ∈ U , a = (ak ) ∈ w. The γ− dual of the space f (u, v, 4) is the intersection of the sets " ( # )  X n X 1 ak  1 1 ai < ∞ , + − b1 = a = (ak ) ∈ ω : sup uk vk vi vi+1 n n i=k+1   an n)

for all k, n ∈ N. Thus, we deduce from (4.2) that ax = (ak xk ) ∈ bs whenever x = (xk ) ∈ f (u, v, 4) if and only if Ey ∈ `∞ whenever y = (yk ) ∈ f where E = (enk ) is defined in (4.3). Therefore with the help of Lemma 4.1 {f (u, v, 4)}γ = b2 ∩ b1 .  Theorem 4.4. The β− dual of the space f (u, v, 4) is the intersection of the sets n o b3 = a = (ak ) ∈ ω : lim enk exists , n→∞ ( ) n X b4 = a = (ak ) ∈ ω : lim enk exists , n→∞

k=0

( b5

=

)

a = (ak ) ∈ ω : lim

n→∞

X

4 [enk − ak ] < ∞ ,

k

where ak = limn→∞ enk . Then, {f (u, v, 4)}β = ∩5k=1 bk . 5

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Proof. Let us take any sequence a ∈ ω. By (4.2) ax = (ak xk ) ∈ cs whenever x = (xk ) ∈ f (u, v, 4)) if and only if Ey ∈ c whenever y = (yk ) ∈ f where E = (enk ) defined in (4.3), we derive the consequence by Lemma 4.2 that {f (u, v, 4)}β = ∩5k=1 bk .  Theorem 4.5. The γ− dual of the space f s(u, v, 4) is the intersection of the sets ( ) X b6 = a = (ak ) ∈ ω : sup |4enk | < ∞ , n

b7

k

  = a = (ak ) ∈ ω : lim enk = 0 . k→∞

in other words we have {f s(u, v, 4)}γ = b6 ∩ b7 . Proof. This may be obtained in the similar way as mentioned in the proof of Theorem 4.3 with Lemma 4.1 instead of Lemma 4.12(viii). So we omit details.  Theorem 4.6. Defined the set b8 by ( b8 =

) X 2 4 enk < ∞ . a = (ak ) ∈ ω : lim n→∞

k

β

Then, {f s(u, v, 4)} = b3 ∩ b6 ∩ b7 ∩ b8 . Proof. This may be obtained in the similar way as mentioned in the proof of Theorem 4.4 with Lemma 4.2 instead of Lemma 4.12(vii). So, we omit details.  For the sake of brevity the following notations will be used: m 1 X e an+i,k , e a(n, k, m) = m + 1 i=0

a ˜(n, k) =

m

c(n, k, m)

=

1 X cn+i,k , m + 1 i=0

c(n, k) =

n X i=0 n X

a ˜ik ,

cik ,

i=0

P∞ a 1 ( v1k − vk−1 where cnk is defined in (4.9) and e ank = j=k un,j+1 ) for all k, m, n ∈ N. j+1 Assume that the infinite matrices E = (enk ) and A = (ank ) map the sequences x = (xk ) and y = (yk ) which are connected with the relation (3.1) to the sequences ξ = (ξn ) and σ = (σn ), respectively, i.e., X
(4.4) ξn = Ex n = enk xk for all n ∈ N, k

(4.5)

σn = Ay




=

n

X

ank yk for all n ∈ N.

k

One can easily conclude here that the method E is directly applied to the terms of the sequence x = (xk ) while the method A is applied to the G(u, v, ∆)-transform of the sequence x = (xk ). So, the methods E and A are essentially different. Now, suppose that the matrix product AG(u, v, ∆) exists which is a much weaker assumption than the conditions on the matrix A belonging to any matrix class, in general. It is not difficult to see that the sequence in (4.5) reduces to the sequence in (4.4) as follows: σn

=

k XX k

=

XX k

uk ∇vi xi ank

i=0

ui ani ∇vk xk = ξn ,

i=k

where ∇vk = vk − vk−1 . Hence, the matrices E = (enk ) and A = (ank ) are connected with the relation X 1 1 en,i+1 (4.6) ank = ( − ) or vk vk−1 ui+1 i=k X (4.7) enk = ui ani ∇vk xk for all k, n ∈ N. i=k

Note that the methods A and E are not necessarily equivalent since the order of summation may not be reversed. 6

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We now give the following fundamental theorems connected with the matrix mappings on/into the almost convergent spaces f (u, v, ∆) and f s(u, v, ∆): Theorem 4.7. Let µ be any given sequence space and the matrices E = (enk ) and A = (ank ) are connected with the relation (4.6). Then, E ∈ (f (u, v, ∆) : µ) if and only if A ∈ (f : µ) and (enk )k∈N ∈ [f (u, v, ∆)]β for all n ∈ N.

(4.8)

Proof. Suppose that E = (enk ) and A = (ank ) are connected with the relation (4.6) and let µ be any given sequence space and keep in mind that the spaces f (u, v, ∆) and f are norm isomorphic. Let E ∈ (f (u, v, ∆) : µ) and take any sequence x ∈ f (u, v, ∆) and keep in mind that y = G(u, v, ∆)x. Then (enk )k∈N ∈ [f (u, v, ∆)]β that is, (4.8) holds for all n ∈ N and AG(u, v, ∆) exists which implies that (ank )k∈N ∈ `1 = f β for each n ∈ N. Thus, Ay exists for all y ∈ f . Hence, by the equality (4.6) we have A ∈ (f : µ). On the other hand, assume that (4.8) holds and A ∈ (f : µ). Then, we have (ank )k∈N ∈ `1 for all n ∈ N which gives together with (enk )k∈N ∈ [f (u, v, ∆)]β for each n ∈ N that Ex exists. Then, the equality Ex = Ay in (4.6) again holds. Hence Ex ∈ µ for all x ∈ f (u, v, ∆), that is E ∈ (f (u, v, ∆) : µ).  Theorem 4.8. Let µ be any given sequence space and the elements of the infinite matrices F = (fnk ) and C = (cnk ) are connected with the relation n X (4.9) cnk = un vi−k (fi−k,k − fi−k−1,k ) for all k, n ∈ N. i=k

Then, F = (fnk ) ∈ (µ : f (u, v, ∆)) if and only if C ∈ (µ : f ). Proof. Let t = (tk ) ∈ µ and consider the following equality n X un vi ∆(F t)i {G(u, v, ∆)(F t)}n = i=0

=

n X

un vi

i=0

=

k

n XX k

X (fi,k− fi−1,k )tk

un vi−k (fi−k,k − fi−k−1,k )tk

i=k

for all n ∈ N. Whence, it can be seen from here that F t ∈ f (u, v, ∆) whenever t ∈ µ if and only if Ct ∈ f whenever t ∈ µ. This completes the proof.  Theorem 4.9. Let µ be any given sequence space and the matrices E = (enk ) and A = (ank ) are connected with the relation (4.6). Then, E ∈ (f s(u, v, ∆) : µ) if and only if A ∈ (f s : µ) and (enk )k∈N ∈ [f s(u, v, ∆)]β for all n ∈ N. Proof. The proof is based on the proof the Theorem 4.7.



Theorem 4.10. Let µ be any given sequence space and the elements of the infinite matrices F = (fnk ) and C = (cnk ) are connected with the relation (4.9). Then, F = (fnk ) ∈ (µ : f s(u, v, ∆)) if and only if C ∈ (µ : f s). Proof. The proof is based on the proof the Theorem 4.8.



By Theorem 4.7, Theorem 4.8, Theorem 4.9 and Theorem 4.10 we have quite a few outcomes depending on the choice of the space µ to characterize certain matrix mappings. Hence, by the help of these theorems the necessary and sufficient conditions for the classes (f (u, v, ∆) : µ), (µ : f (u, v, ∆)), (f s(u, v, ∆) : µ) and (µ : f s(u, v, ∆)) may be derived by replacing the entries of E and A by those of the entries of A = EG−1 (u, v, ∆) and C = G(u, v, ∆)F , respectively; where the necessary and sufficient conditions on the matrices F and C are read from the concerning results in the existing literature. Lemma 4.11. Let A = (ank ) be an infinite matrix. Then, the following statements hold: i: A ∈ (c0 (p) : f ) if and only if (4.10)

∃N > 1 3 sup m∈N

(4.11)

X

|a(n, k, m)| N −1/pk < ∞ for all n ∈ N,

k

∃αk ∈ C for all k ∈ N 3

lim a(n, k, m) = αk uniformly in n.

m→∞

7

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ii: A ∈ (c(p) : f ) if and only if (4.10) and (4.11) hold and X (4.12) ∃α ∈ C 3 lim a(n, k, m) = α uniformly in n. m→∞

k

iii: A ∈ (`∞ (p) : f ) if and only if (4.10) and (4.11) hold and X (4.13) ∃N > 1 3 lim |a(n, k, m) − αk | N 1/pk = 0 uniformly in n. m→∞

k

Lemma 4.12. Let A = (ank ) be an infinite matrix. Then, the following statements hold. i: [Duran, [33]] A ∈ (`∞ : f ) if and only if (4.1) holds and f − lim ank = αk exists for each fixed k, X lim |a(n, k, m) − αk | = 0 uniformly in n,

(4.14) (4.15)

m→∞

k

ii: [King, [34]] A ∈ (c : f ) if and only if (4.1), (4.14) hold and X (4.16) f − lim ank = α k

iii: [Ba¸sar and C ¸ olak, [19]] A ∈ (cs : f ) if and only if (4.14) holds and X (4.17) sup |∆ank | < ∞ , n∈N

k

iv: [Ba¸sar and C ¸ olak, [19]] A ∈ (bs : f ) if and only if (4.14), (4.17) hold and (4.18)

lim ank = 0 exists for each fixed n, k

q

(4.19)

lim

X

q→∞

k

1 X |∆ [a(n + i, k) − αk ]| = 0 uniformly in n, q + 1 i=0

v: [Duran, [33]] A ∈ (f : f ) if and only if (4.1), (4.14), (4.16) hold and X (4.20) lim |∆ [a(n, k, m) − αk ]| = 0 uniformly in n, m→∞

k

vi: [Ba¸sar, [18]] A ∈ (f s : f ) if and only if (4.14), (4.18), (4.20) and (4.19) hold. vii: [Duran, [15]] A ∈ (f s : c) if and only if (4.2), (4.17), (4.18) hold and X ∆2 ank = α, (4.21) lim n→∞

k

viii: A ∈ (f s : `∞ ) if and only if (4.17) and (4.18) hold. ix: [Ba¸sar and Solak, [17]] A ∈ (bs : f s) if and only if (4.18), (4.19) hold and X (4.22) sup |∆a(n, k)| < ∞ , n∈N

k

f − lim a(n, k) = αk exists for each fixed k,

(4.23)

x: [Ba¸sar, [18]] A ∈ (f s : f s) if and only if (4.19), (4.22), (4.23) hold and (4.24)

lim

q→∞

X k

q 1 X 2 ∆ [a(n + i, k) − αk ] = 0 uniformly in n, q + 1 i=0

xi: [Ba¸sar and C ¸ olak, [19]] A ∈ (cs : f s) if and only if (4.22) and (4.23) hold. xii: [Ba¸sar, [20]] A ∈ (f : cs) if and only if X (4.25) sup |a(n, k)| < ∞ , n∈N

(4.26)

X

k

ank = αk exists for each fixed k,

n

XX

(4.27)

n

(4.28)

lim

m→∞

X

ank = α,

k

|∆ [a(n, k) − αk ]| = 0.

k 8

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Now, we give our main results related to matrix mappings on/into the spaces of almost convergent series f s(∆(m) ) and sequences f (∆(m) ). Corollary 4.13. Let A = (ank ) be an infinite matrix. Then, the following statements hold: i: A ∈ (f s(u, v, ∆) : f ) if and only if {ank }k∈N ∈ [f (u, v, ∆)]β for all n ∈ N and (4.14), (4.18) hold with e ank instead of ank , (4.20) holds with e a(n, k, m) instead of a(n, k, m) and (4.19) holds with a ˜(n, k) instead of a(n, k). ii: A ∈ (f s(u, v, ∆) : c) if and only if {ank }k∈N ∈ [f (u, v, ∆)]β for all n ∈ N and (4.2), (4.17), (4.18) and ( 4.21) hold with e ank instead of ank . iii: A ∈ (f s(∆(m) ) : `∞ ) if and only if {ank }k∈N ∈ [f (u, v, ∆)]β for all n ∈ N and (4.17) and (4.18) hold with e ank instead of ank . iv: A ∈ (f s(u, v, ∆) : f s) if and only if {ank }k∈N ∈ [f (u, v, ∆)]β for all n ∈ N and (4.19), (4.22), (4.23) and (4.24) hold with a ˜(n, k) instead of a(n, k). v: A ∈ (cs : f s(u, v, ∆)) if and only if (4.22) and (4.23) hold with c(n, k) instead of a(n, k). vi: A ∈ (bs : f s(u, v, ∆)) if and only if (4.18) holds with cnk instead of ank , (4.19), (4.22) and (4.23) hold with c(n, k) instead of a(n, k). vii: A ∈ (f s : f s(u, v, ∆)) if and only if (4.19), (4.22), (4.23) and (4.24) hold with c(n, k) instead of a(n, k). Corollary 4.14. Let A = (ank ) be an infinite matrix. Then, the following statements hold: i: A ∈ (c(p) : f (u, v, ∆)) if and only if (4.10), (4.11) and (4.12) hold with c(n, k, m) instead of a(n, k, m). ii: A ∈ (c0 (p) : f (u, v, ∆)) if and only if (4.10) and ( 4.11) hold with c(n, k, m) instead of a(n, k, m). iii: A ∈ (`∞ (p) : f (u, v, ∆)) if and only if (4.10), (4.11) and (4.13) hold with c(n, k, m) instead of a(n, k, m). Corollary 4.15. Let A = (ank ) be an infinite matrix. Then, the following statements hold: i: A ∈ (f (u, v, ∆) : `∞ ) if and only if {ank }k∈N ∈ [f (u, v, ∆)]β for all n ∈ N and (4.1) holds with e ank instead of ank . ii: A ∈ (f (u, v, ∆) : c) if and only if {ank }k∈N ∈ [f (u, v, ∆)]β for all n ∈ N and (4.1), (4.2), (4.2) and (4.2) hold with e ank instead of ank . iii: A ∈ (f (u, v, ∆) : bs) if and only if {ank }k∈N ∈ [f (u, v, ∆)]β for all n ∈ N and (4.22) holds with a ˜(n, k) instead of a(n, k). iv: A ∈ (f (u, v, ∆) : cs) if and only if {ank }k∈N ∈ [f (u, v, ∆)]β for all n ∈ N and (4.2)-(4.25) hold with a ˜(n, k) instead of a(n, k). Corollary 4.16. Let A = (ank ) be an infinite matrix. Then, the following statements hold: i: A ∈ (`∞ : f (u, v, ∆)) if and only if (4.1), (4.14) hold with cnk instead of ank and (4.15) holds with c(n, k, m) instead of a(n, k, m). ii: A ∈ (f : f (u, v, ∆)) if and only if (4.1), (4.14), (4.20) hold with c(n, k, m) instead of a(n, k, m) and (4.16) hold with cnk instead of ank . iii: A ∈ (c : f (u, v, ∆)) if and only if (4.1), (4.14) and (4.16) hold with cnk instead of ank . iv: A ∈ (bs : f (u, v, ∆)) if and only if (4.14), (4.17), (4.18) hold with cnk instead of ank and (4.19) holds with c(n, k) instead of a(n, k). v: A ∈ (f s : f (u, v, ∆)) if and only if (4.14), (4.18) hold with cnk instead of ank , (4.20) holds with c(n, k, m) instead of a(n, k, m) and (4.19) holds with c(n, k) instead of a(n, k). vi: A ∈ (cs : f (u, v, ∆)) if and only if (4.17) and (4.18) hold with cnk instead of ank . Characterization of the classes (f (u, v, ∆) : f∞ ), (f∞ : f (u, v, ∆)), (f s(u, v, ∆) : f∞ ) and (f∞ : f s(u, v, ∆)) is redundant since the spaces of almost bounded sequences f∞ and `∞ are equal. 5. Some core theorems Let us start with the definition of G(u, v, 4)−core of x. Let x ∈ `∞ then G(u, v, 4)−core of x defined by the closed interval [G(x), g(x)], where G(x)

=

lim sup sup m→∞ n∈N

g(x)

=

lim inf sup

n→∞ n∈N

k

m X k=0

m X k=0

1 X uk ∇vi xn+i , m + 1 i=0 k

1 X uk ∇vi xn+i . m + 1 i=0

Hence, it is easy to see that G(u, v, 4)−core of x is α if and only if f (u, v, 4) − lim x = α. 9

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Theorem 5.1. G(u, v, 4) − core(Ax) ⊆ K − core(x), (G(Ax) 6 L(x)) for all x ∈ `∞ if and only if A ∈ (c : f (u, v, 4))reg and m k X X 1 X (5.1) uk ∇vi an+i,j = 1. lim sup m→∞ n∈N m+1 j

i=0

k=0

Proof. Assume that G(u, v, 4) − core(Ax). Let x = (xk ) be a convergent sequences, so we have L(x) = l(x). By this assumption, we have l(x) 6 g(Ax) 6 G(Ax) 6 L(x) Hence, we obtain G(Ax) = g(Ax) = lim x which implies that A ∈ (c : f (4(m) ))reg . Now, let us consider the sequences B = bij (n) of infinite matrices defined by m

bij (n) =

k

1 X X uk ∇vi an+i,j . m + 1 j=0 i=0

Since A ∈ (c : f (u, v, 4))reg , it is easy to see that the conditions of Lemma 2 (see Das, [21]) are satisfied for the matrix sequence A. Hence, there exists y ∈ `∞ such that kyk 6 1 and X (5.2) G(Ay) = lim sup sup |bij (n)|. m→∞

n

j

Hence, x = e = (1, 1, 1 . . .), by using the hypothesis, we can write X X 1 = g(Ae) 6 lim inf sup |bij (n)| 6 lim sup sup |bij (n)| m→∞ n∈N

m→∞ n∈N

j

j

= G(Ay) 6 L(y) 6 kyk 6 1. Which proves necessity of (5.1). On the other hand, let A ∈ (c : f (u, v, 4))reg and (5.1) hold for all x ∈ `∞ . We define any real number µ we write µ+ = max{0, µ} and µ− = max{−µ, 0} then, |µ| = µ+ + µ− and µ = µ+ − µ− . Hence, for any given  > 0, there exists a j0 ∈ N such that xj < L(x) +  for all j > j0 . Then, we can write X X X X bij (n) = (bij (n)xj + (bij (n)+ xj − (bij (n))− xj j

jj0

X

j>j0

|bij (n)| + [L(x) + ]

jj0

|bij (n| + kxk∞

X

[|bij (n)| − bij (n)].

j>j0

Thus, by applying the lim supm supn∈N above equation and using hypothesis, we have G(x) 6 L(x) +  This completes the proof, since  is arbitrary and x ∈ `∞ .  Theorem 5.2. A ∈ (S ∩ `∞ : f (u, v, 4))reg if and only if A ∈ (c : f (u, v, 4))reg and m k X X 1 X (5.3) uk ∇vi an+i,j = 0. lim m→∞ m+1 j∈E

k=0

i=0

for every E ⊆ N with δ(E) = 0. Proof. Firstly, suppose that A ∈ (S ∩ `∞ : f (u, v, 4))reg . Then A ∈ (c : f (u, v, 4))reg immediately follows from the fact that c ⊆ S ∩ `∞ . Now, define sequence s = (sk ) for all x ∈ `∞ as  xk , k ∈ E, sk = 0, k ∈ / E, where E any subset of N Pwith δ(E) = 0. By our assumption, since s ∈ S0 , we have As ∈ f (u, v, 4). On the other hand, since As = k∈E ank sk , the matrix C = (cnk ) defined by  ank , k ∈ E, cnk = 0, k∈ /E for all n, must belong to the class (`∞ : f (u, v, 4)). Hence, the necessity (5.3) follows from Corollary (4.16)(i). Conversely, let A ∈ (c : f (u, v, 4))reg and (5.3) hold for all x ∈ `∞ . Let x any sequence in S ∩ `∞ with 10

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st − lim x = s and write P E = {i : |xi − p| > } for an given  > 0, so that δ(E) = 0. Since A ∈ (c : f (u, v, 4))reg and f (u, v, 4) − lim k ank = 1, we have ! X X f (u, v, 4) − lim(Ax) = f (u, v, 4) − lim ank (xk − p) + p ank k

= f (u, v, 4) − lim

X

k

ank (xk − p) + p

k m

=

lim sup

m→∞ n∈N

X j

k

1 X X uk ∇vi an+i,j (xj − p) + p m+1 i=0 k=0

On the other hand, since we have m X m k k X X 1 X X X 1 uk uk ∇vi an+i,j (xj − p) + kAk ∇vi an+i,j (xj − p) 6 kxk∞ m + 1 m + 1 j i=0 i=0 j∈E k=0 k=0 the condition (5.3) implies that m

lim

m→∞

X j

k

1 X X uk ∇vi an+i,j (xj − p) = 0 uniformly n. m+1 i=0 k=0

Therefore, f (u, v, 4) − lim(Ax) = st − lim x that is A ∈ (S ∩ `∞ : f (u, v, 4))reg which completes the proof.  Theorem 5.3. G(u, v, 4) − core(Ax) ⊆ st − core(x) for all x ∈ `∞ if and only if A ∈ (S ∩ `∞ : f (u, v, 4))reg and (5.1) holds. Proof. Assume that the inclusion G(u, v, 4) − core(Ax) ⊆ st − core(x) holds for each bounded sequence x. Then, G(Ax) 6 st − sup(x) for all x ∈ `∞ . Hence one may easily see that the following inequalities hold: st − inf(x) 6 g(Ax) 6 G(Ax) 6 st − sup(x). If x ∈ (S ∩ `∞ ), then we have st − inf(x) = st − sup(x) = st − lim x. Which implies that st − sup(x) = g(x) = G(x) = f (u, v, 4) − lim(Ax) that is A ∈ (c : f (u, v, 4))reg . On the other hand, let A ∈ (S ∩ `∞ : f (u, v, 4))reg and (5.1) hold for all x ∈ `∞ . Then st − sup(x) is finite. Let E be a subset of N defined by E = {k : xk > st − sup(x) + } for a given  > 0. Then obvious that δ(E) = 0 and xk 6 st − sup(x) + , if k ∈ / E. We define any real number µ we write µ+ = max{0, µ} and µ− = max{−µ, 0} then |µ| = µ+ + µ− and µ = µ+ − µ− , then for fixed positive integer m we can write X X X bij (n) = bij (n)xj + (bij (n))+ xj j

jj0 j∈E

(bij (n))− xj −

j>j0 k∈E /

6 kxk∞

X

(bij (n))+ xj

j>j0

X

|bij (n)| + [st − sup(x) + ]

jj0 j ∈E /

|bij (n)| + kxk∞

j>j0 j∈E

X

[|bij (n)| − bij (n)].

j>j0

Whence, applying lim supm supn∈N to the above equation and using the hypothesis, we obtain G(x) 6 st − sup(x) + . This completes the proof since  is arbitrary and x ∈ `∞ .  6. Conclusion The general aim of this study is to fill a gap in literature by extending certain topological spaces and to investigate some topological properties in addition to some core theorems. Domain of the generalized difference matrix B(r, s) in the almost convergent spaces f and f0 was introduced by Ba¸sar and Kiri¸s¸ci in [16]. One of the nice part of their paper was to find beta- and gamma duals of the set of almost convergent series f s. As a generalization of the spaces Ba¸sar and Kiri¸s¸ci, the sequence space f (B) which is matrix domain of the triple band matrix B(r, s, t) in the almost convergent space f has been examined by S¨onmez. 11

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In the present paper, we study the domains of the triangle matrix G(u, v, 4) in the almost convergent sequence spaces f and f0 and series space f s. Nevertheless, the present results does not compare with the results obtained by S¨onmez [2] and Ba¸sar and Kiri¸s¸ci [16]. But our results are more general and more comprehensive than the corresponding results of Kayaduman and S¸eng¨on¨ ul [25, 26], since the space f (u, v, 4) and f0 (u, v, 4) 1 reduce in the cases vk − vk−1 = rk , un = Rn and vk − vk−1 = 1, un = n1 to the Riesz sequence space fb, fb0 and to the Ces`aro sequence space fe, fe0 , respectively. Acknowledgement We thank the referees for their careful reading of the original manuscript and for the valuable comments. References ¨ Karabıyık, Almost Sequence Spaces Derived by the Domain of the Matrix, Abstr. Appl. Anal., vol. 2013, ID [1] A. Karaisa, U. 783731, 8 pages, 2013. doi:10.1155/2013/783731. [2] A. S¨ onmez, Almost convergence and triple band matrix, Math. Comput. Modelling doi:10.1016/j.mcm.2011.11.079. [3] A. Wilansky, Summability through Functional Analysis, North-Holland Mathematics Studies 85, Amsterdam-New YorkOxford, 1984. [4] C. Orhan, Sublinear Functionals and Knopp’s Core Theorem, internat.J.Math.Sci. 13(1990), 461–468. [5] C. C ¸ akan, H. C ¸ o¸skun, A class of matrices mapping σ−core into A- statistical core, Tamsui Oxford Journal of Math.Sci. 20(2004), 17–25. [6] C. Orhan, S ¸ . Yardımcı Banach and statistical cores of bounded sequence, Czehholslovak Math.Journal 129(2004), 65–72. [7] A.M. Jarrah, E. Malkowsky, BK spaces, bases and linear operators, Ren. Circ. Mat. Palermo II. 52(1990), 177—191. [8] A. Karaisa, On generalized weighted means and compactness of matrix operators, Ann. Funct. Anal., doi: 10.15352/afa/06-3-2, 6(3) 2015, 8–28. ¨ [9] E. Malkowsky, F. Ozger, Compact Operators on Spaces of Sequences of Weighted Means, AIP Conference Proceedings, 1470, (2012) 179–182. ¨ [10] E. Malkowsky, F. Ozger and V. Veliˇ ckovi´ c, Some Spaces Related to Cesaro Sequence Spaces and an Application to Crystallography, MATCH Commun. Math. Comput. Chem., 70(3) (2013) 867-884. ¨ [11] E. Malkowsky, F. Ozger, A note on some sequence spaces of weighted means, Filomat 26:3 (2012), 511—518. [12] F. Ba¸sar, A. Karaisa, Some new paranormed sequence spaces and core theorems, AIP Conf. Proc. 1611, (2014), 380–391. [13] F. Ba¸sar, A. Karaisa, Some New Generalized Difference Spaces of Nonabsolute Type Derived From the Spaces `p and `∞ , The Scientific World Journal, vol. 2013, Article ID 349346, 15 pages, 2013. doi:10.1155/2013/349346. ¨ [14] F. Ozger, F. Ba¸sar, Domain of the double sequential band matrix B(r, s) on some Maddox’s spaces, Acta Mathematica Scientia, 34(2) (2014) 394–408. ¨ urk, On strongly regular dual summability methods, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 32 (1983) 1–5. [15] E. Ozt¨ [16] F. Ba¸sar, M. Kiri¸s¸ci, Almost convergence and generalized difference matrix, Comput. Math. Appl. 61 (2011) 602–611. ˙ Solak, Almost-coercive matrix transformations, Rend. Mat. Appl. (7) 11 (2) (1991) 249–256. [17] F. Ba¸sar, I. [18] F. Ba¸sar, f –conservative matrix sequences, Tamkang J. Math. 22 (2) (1991) 205–212. [19] F. Ba¸sar, R. C ¸ olak, Almost-conservative matrix transformations, Turkish J. Math. 13 (3) (1989) 91–100. ¨ [20] F. Ba¸sar, Strongly-conservative sequence-to-series matrix transformations, Erc. Uni. Fen Bil. Derg. 5 (12) (1989) 888–893. [21] G. Das, Sublinear functionals and a class of conservative matrices, Bulletin of the Institute of Mathematics, Academia Sinica 15(1987), 89–106. [22] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Mathematica. 80(1948), 167—190. [23] H. Polat, V. Karakaya and N. S ¸ im¸sek, Difference sequence spaces derived by using a generalized weighted mean, Applied Mathematics Letters 24, 2011, 608-614. [24] H.I. Miller, C. Orhan, On almost convergent and staistically convergent subsequences, Acta Math. Hungar. 93 (2001) 135–151. [25] K. Kayaduman, M. S ¸ eng¨ on¨ ul, The space of Cesaro almost convergent sequence and core theorems, Acta Mathematica Scientia. 6(2012), 2265–2278. [26] K. Kayaduman, M. S ¸ eng¨ on¨ ul, On the Riesz Almost Convergent Sequences Space, Abstr. Appl. Anal. Volume 2012, Article ID 691694, 18 pages. [27] L. Loone, Knopp’s core and almost-convergence core in the space m, Tartu Riikl. All. Toimetised 335(1975), 148–156. [28] M. Kiri¸sc¸i, F. Ba¸sar, Some new sequence spaces derived by the domain of generalized difference matrix Comput. Math. Appl. 60(5)(2010), 1299–1309. [29] M. Kiri¸s¸ci, Almost convergence and generalized weighted mean, AIP Conference Proceedings 1470(2012), 191–194. [30] J. Boos, Oxford University Press Inc, New York, Classical and Modern Methods in Summability, Oxford University Press Inc, New York 2000 [31] J.A. Sıddıqı, Infinite matrices summing every almost periodic sequence, Pacific Journal of Mathematics. 39 (1971), 235–251. [32] J.A. Fridy, C. Orhan, Statistical limit superior and limit inferior, Proceedings of the American Mathematical Society. 125(1997), 3625–3631. [33] J.P. Duran, Infinite matrices and almost convergence, Math.Z. 128 (1972) 75–83. [34] J.P. King, Almost summable sequences, Proc. Amer. Math. Soc. 17 (1966) 1219–1225. ˙ (A. Karaisa) DEPARTMENT OF MATHEMATICS-COMPUTER SCIENCE, FACULTY OF SCIENCES, NECMETTIN ERBAKAN UNIVERSITY, MERAM YERLES ¸ KESI˙ , 42090 MERAM, KONYA, TURKEY E-mail address, A. Karaisa: [email protected], [email protected] ¨ (F.Ozger) DEPARTMENT OF ENGINEERING SCIENCES, FACULTY OF ENGINEERING AND ARCHITECTURE, ˙ ˙ KATIP ˙ C ˙ ˙ IZM IR ¸ ELEBI˙ UNIVERSITY, 35620, IZM IR, TURKEY ¨ E-mail address, F. Ozger: [email protected] 12

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Ground state solutions for discrete nonlinear Schr¨odinger equations with potentials Guowei Sun

∗

Ali Mai

Department of Applied Mathematics, Yuncheng University Shanxi, Yuncheng 044000, China

Abstract In this paper, we consider discrete nonlinear Schr¨odinger equations with periodic potentials and bounded potentials. Under a more general super-quadratic condition instead of the classical Ambrosetti-Rabinowitz condition, we prove the existence of ground state solutions of the equations by using the critical point theory in combination with the Nehari manifold approach. Key words Solitons; Discrete nonlinear Schr¨odinger equations; Nehari manifold; Ground state solutions; Critical point theory. 2000 Mathematics Subject Classification 39A10, 39A12.

1

Introduction

In this paper, we study discrete solitons for the discrete nonlinear Schr¨odinger (DNLS) equation iψ˙ n = −∆ψn + vn ψn − f (n, ψn ), n ∈ Z,

(1.1)

where ∆ψn = ψn+1 + ψn−1 − 2ψn is the discrete Laplacian in one spatial dimension, potential V = {vn } is a sequence of real number. We assume that f (n, 0) = 0 and the nonlinearity f (n, u) is gauge invariant, i.e., f (n, eiθ u) = eiθ f (n, u), θ ∈ R. ∗

Corresponding author. E-mail address: [email protected]

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Since solitons are spatially localized time-periodic solutions and decay to zero at infinity. Thus ψn has the form ψn = un e−iωt , and lim ψn = 0,

|n|→∞

where {un } is a real valued sequence and ω ∈ R is the temporal frequency. Then (1.1) becomes −∆un + vn un − ωun = f (n, un ), n ∈ Z,

(1.2)

lim un = 0

(1.3)

and |n|→∞

holds. Naturally, if we look for discrete solitons of equation (1.1), we just need to get the solutions of equation (1.2) satisfying (1.3). DNLS equation is one of the most important inherently discrete models, having a crucial role in the modeling of a great variety of phenomena, ranging from solid state and condensed matter physics to biology (see [1–4, 11] and reference therein). For example, Davydov ( [1]) studied the discrete nonlinear Schr¨odinger in molecular biology and Su et al. ( [11]) considered the equation in condensed matter physics. In the past decade, the existence of discrete solitons in DNLS equations has been considered in a number of studies. If the potential is unbounded, the existence of nontrivial solitons of DNLS equations was obtained in [12,16,17] by Nehari manifold method and in [5] by the mountain pass theorem and the fountain theorem, respectively. If the potential is periodic, the existence of discrete solitons for the periodic DNLS equations has been studied, see [6–10,15] and the reference therein. In our paper, we consider two cases of the potentials, one is periodic; the other is nonperiodic corresponding to a potential well in the sense that lim vn exists and is equal to supZ V .

|n|→∞

In some paper (see, e.g. [14]), the following classical Ambrosetti-Rabinowitz superlinear condition is assumed: 0 < µF (n, u) ≤ f (n, u)u, for some µ > 2 and u 6= 0,

(1.4)

where F (n, u) is the primitive function of f (n, u), i.e., Z u F (n, u) = f (n, s)ds. 0

It is easy to see that (1.4) implies that F (n, u) ≥ C|u|µ , for some constant C > 0 and |u| ≥ 1. In this paper, instead of (1.4) we assume the following super-quadratic condition lim F (n, u)/u2 = +∞ uniformly for n ∈ Z.

|u|→∞

(1.5)

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It is well known that (1.4) implies (1.5). It is also well known that many nonlinearities such as f (n, u) = u ln(1 + |u|), do not satisfy (1.4). A crucial role that (1.4) plays is to ensure the boundedness of PalaisSmale sequences. We obtain the existence of ground state solutions without Palais-Smale condition. This paper is organized as follows: In Section 2, we establish the variational framework associated with (1.2). We consider two cases of the potentials in Section 3 and in Section 4, respectively.

2

Preliminaries

In this section, we establish the variational framework associated with (1.2). Let A = −4 + V − ω and E = l2 (Z), and lp ≡ lp (Z) =

 

! p1 u = {un }n∈Z : ∀ n ∈ Z, un ∈ R, kuklp =



X n∈Z

|un |p

0 for all n. (f1 ) f ∈ C(Z × R, R) and f (n, u) = f (n + T, u), and there exist a > 0, p ∈ (2, ∞) such that |f (n, u)| ≤ a(1 + |u|p−1 ), for all n ∈ Z and u ∈ R. (f2 ) lim f (n, u)/u = 0 uniformly for n ∈ Z. |u|→0

(f3 ) lim F (n, u)/u2 = +∞ uniformly for n ∈ Z, where F (n, u) is the primitive function of |u|→∞

f (n, u), i.e.,

u

Z F (n, u) =

f (n, s)ds. 0

(f4 ) u 7→ f (n, u)/|u| is strictly increasing on (−∞, 0) and (0, ∞) for each n ∈ Z. Under the above hypotheses, our results can be stated as follows. Theorem 3.1. Assume that conditions (V1 ), (f1 ) − (f4 ) hold. Then equation (1.2) has a ground state solution. By the condition (V1 ), we may introduce an equivalent norm in E by setting kuk2 := (Au, u), then the functional J can be rewritten as J(u) = where I(u) =

P

1 kuk2 − I(u), 2

F (n, un ).

n∈Z

We assume that (V1 ) and (f1 ) − (f4 ) are satisfied from now on. Lemma 3.1. (a) F (n, u) > 0 and 21 f (n, u)u > F (n, u) for all u 6= 0. (b) J(u) > 0, for all u ∈ N . Proof. (a) From (f2 ) and (f4 ), it is easy to get that F (n, u) > 0, for all u 6= 0. Set H(n, u) = 12 f (n, u)u − F (n, u), by (f4 ), we Z u u f (n, s)ds > H(n, u) = f (n, u) − 2 0

(3.1)

have u f (n, u) f (n, u) − 2 u

Z

u

sds = 0. 0

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So 12 f (n, u)u > F (n, u) for all u 6= 0. (b) For all u ∈ N , by (a), we have X 1 J(u) = J(u) − J 0 (u)u = 2 n∈Z



 1 f (n, u)u − F (n, u) > 0. 2



Lemma 3.2. (a) I 0 (u) = o(kuk) as u → 0. (b) s 7→ I 0 (su)u/s is strictly increasing for all u 6= 0 and s > 0. (c) I(su)/s2 → ∞ uniformly for u on the weakly compact subsets of E \ {0}, as s → ∞. Proof. (a) and (b) are easy to be shown from (f2 ) and (f4 ), respectively. Next we verify (c). Let W ⊂ E \ {0} be weakly compact and let {u(k) } ⊂ W . It suffices to show that if s(k) → ∞ as k → ∞, then so does a subsequence of I(s(k) u(k) )/(s(k) )2 . Passing to a (k) subsequence if necessary, u(k) * u ∈ E \ {0} and un → un for every n, as k → ∞. (k)

Since |s(k) un | → ∞ and u(k) 6= 0, by (f3 ) and (3.1), we have I(s(k) u(k) ) X F (n, ukn ) (k) 2 = (un ) → ∞ as k → ∞. (k) (k) 2 (s(k) )2 n∈Z (s un )



Lemma 3.3. For each w ∈ E \ {0}, there exists a unique sw > 0 such that sw w ∈ N . Proof. Let g(s) := J(sw), s > 0. Since g 0 (s) = J 0 (sw)w = s(kwk2 − s−1 I 0 (sw)w), from (b) of Lemma 3.2, then there exists a unique sw , such that g 0 (s) > 0 whenever 0 < s < sw , g 0 (s) < 0 whenever s > sw and g 0 (sw ) = J 0 (sw w)w = 0. So sw w ∈ N .  Remark 3.1. By (a) and (c) of Lemma 3.2, g(s) > 0 for s > 0 small and g(s) < 0 for s > 0 large. Together with Lemma 3.3, we have sw is a unique maximum of g(s) and sw w is the unique point on the ray s 7→ sw (s > 0) which intersects N . That is, u ∈ N is the unique maximum of J on the ray. Therefore, we may define the mapping m ˆ : E \ {0} → N and m : S → N by setting m(w) ˆ := sw w

and m := m| ˆ S,

where S = {u ∈ E : kuk = 1}. Lemma 3.4. For each compact subset V ⊂ S, there exists a constant CV such that sw ≤ CV for all w ∈ V. (k)

Proof. Suppose by contradiction, sw → ∞ as k → ∞. By Lemma 3.1 and (f3 ), we have (k)

0
0, we may assume wn ∈ S for all n. Write m(w ˆ n ) = swn wn . By Lemma 3.3 and Lemma 3.4, {swn } is bounded, hence swn → s > 0 after passing to a subsequence if needed. Since N is closed and m(w ˆ n) = swn wn → sw, sw ∈ N . Hence sw = sw w = m(w) ˆ by the uniqueness of sw of Lemma 3.3. (b) This is an immediate consequence of (a).  Lemma 3.6. J is coercive on N , i.e., J(u) → ∞ as kuk → ∞, u ∈ N . Proof. Suppose by contradiction, there exists a sequence {u(k) } ⊂ N such that ku(k) k → ∞ (k) and J(u(k) ) ≤ d, for some d ∈ [c, ∞). Let v (k) = kuu(k) k , then there exists a subsequence, still (k)

denoted by the same notation, such that v (k) * v and vn → vn for every n, as k → ∞. First we know that there exist δ > 0 and nk ∈ Z such that |vn(k) | ≥ δ. k

(3.2)

Indeed, if not, then v (k) → 0 in l∞ as k → ∞. For q > 2, (k) 2 kv (k) kqlq ≤ kv (k) kq−2 kl2 l∞ kv

we have v (k) → 0 in all lq , q > 2. Note that by (f1 ) and (f2 ), for any ε > 0, there exists cε > 0 such that |f (n, u)| ≤ ε|u| + cε |u|p−1

and |F (n, u)| ≤ ε|u|2 + cε |u|p .

(3.3)

Then for each s > 0, we have X F (n, svn(k) ) ≤ εs2 kv (k) k2l2 + cε sp kv (k) kplp n∈Z

which implies that

P

(k)

F (n, svn ) → 0 as k → ∞. Therefore

n∈Z

d ≥ J(u(k) ) ≥ J(sv (k) ) =

s2 s2 (k) 2 X kv k − F (n, svn(k) ) → , 2 2 n∈Z

as k → ∞. This is a contradiction if s >

√

(3.4)

2d.

Due to periodicity of coefficients, we know J and N are both invariant under T-translation. Making such shifts, we can assume that 1 ≤ nk ≤ T − 1 in (3.2). Moreover, passing to a subsequence, we can assume that nk = n0 is independent of k. 6 44

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(k)

Next we may extract a subsequence, still denoted by {v (k) }, such that vn → vn for all n ∈ Z. Specially, for n = n0 , inequality (3.2) shows that |vn0 | ≥ δ, so v = 6 0. Since (k) |un | → ∞ as k → ∞, it follows again from (f3 ) and Lemma 3.1 that (k)

J(u(k) ) 1 X F (n, un ) (k) 2 0 ≤ (k) 2 = − (vn ) → −∞ 2 ku k 2 n∈Z (u(k) n )

as k → ∞,

a contradiction again.  ˆ : E \ {0} → R and Ψ : S → R defined by We shall consider the functional Ψ ˆ := J(m(w)) ˆ S. Ψ ˆ and Ψ := Ψ|

(3.5)

ˆ ∈ C 1 (E \ {0}, R) and Lemma 3.7. (a) Ψ ˆ ˆ 0 (w)z = km(w)k Ψ J 0 (m(w))z ˆ kwk

for all w, z ∈ E, w 6= 0.

(b) Ψ ∈ C 1 (S, R) and Ψ0 (w)z = km(w)kJ 0 (m(w))z

for all z ∈ Tw (S) = {v ∈ E : (w, v) = 0}.

(c) If {wn } is a Palais-Smale sequence for Ψ if and only if {m(wn )} is a Palais-Smale sequence for J. (d) w is a critical point of Ψ if and only if m(w) is a nontrivial critical point of J. Moreover, the corresponding values of Ψ and J coincide and inf S Ψ = inf N J. Proof.(a) Let w ∈ E \ {0} and z ∈ E. By Remark 3.1 and the mean value theorem, we obtain ˆ + tz) − Ψ(w) ˆ Ψ(w = J(sw+tz (w + tz)) − J(sw w) ≤ J(sw+tz (w + tz)) − J(sw+tz (w)) = J 0 (sw+tz (w + τt tz))sw+tz tz, where |t| is small enough and τt ∈ (0, 1). Similarly, ˆ + tz) − Ψ(w) ˆ Ψ(w = J(sw+tz (w + tz)) − J(sw w) ≥ J(sw (w + tz)) − J(sw (w)) = J 0 (sw (w + ηt tz))sw tz, where ηt ∈ (0, 1). From the proof of Lemma 3.5, the function w 7→ sw is continuous, combining these two inequalities that ˆ + tz) − Ψ(w) ˆ Ψ(w km(w)k ˆ = sw J 0 (sw w)z = J 0 (m(w))z. ˆ t→0 t kwk

lim

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ˆ is bounded linear in z and continuous in w. It follows Hence the Gˆ ateaux derivative of Ψ 1 ˆ that Ψ is a class of C (see [14], Proposition1.3). (b) follows from (a). Note only that since w ∈ S, m(w) = m(w). ˆ (c) Let {wn } be a Palais-Smale sequence for Ψ, and let un = m(wn ) ∈ N . Since for every wn ∈ S we have an orthogonal splitting E = Twn S ⊕ Rwn . Using (b) we have kΨ0 (wn )k = sup Ψ0 (wn )z = km(wn )k sup J 0 (m(wn ))z = kun k sup J 0 (un )z, z∈Twn S kzk=1

z∈Twn S kzk=1

z∈Twn S kzk=1

Using (b) again, then kΨ0 (wn )k ≤ kun kkJ 0 (un )k = kun k

sup

J 0 (un )(z + tw) kz + twk t∈R

z∈Twn S, z+tw6=0

≤ kun k

J 0 (un )(z) = kΨ0 (wn )k, kzk z∈Twn S\{0} sup

Therefore kΨ0 (wn )k = kun kkJ 0 (un )k.

(3.6)

√

Noticing that c ≤ J(un ) = 12 kun k2 − I(un ) ≤ 12 kun k2 , then kun k ≥ 2c. Together with √ Lemma 3.6, 2c ≤ kun k ≤ supn kun k < ∞. Hence {wn } is a Palais-Smale sequence for Ψ if and only if {un } is a Palais-Smale sequence for J. (d) By (3.6), Ψ0 (w) = 0 if and only if J 0 (m(w)) = 0. The other part is clear.  Proof of Theorem 3.1. If u0 ∈ N satisfies J(u0 ) = c, then m−1 (u0 ) ∈ S is a minimizer of Ψ and therefore a critical point of Ψ, so u0 is a critical point of J by Lemma 3.7. It remains to show that there exists a minimizer u ∈ N of J|N . Let {w(k) } ⊂ S be a minimizing sequence for Ψ. By Ekeland’s variational principle we may assume Ψ(w(k) ) → c, Ψ0 (w(k) ) → 0 as k → ∞, hence J(u(k) ) → c, J 0 (u(k) ) → 0 as k → ∞, where u(k) := m(w(k) ) ∈ N . First, we know that {u(k) } is bounded in N by Lemma 3.6, then there exists a subsequence, still denoted by the same notation, such that u(k) weakly converges to some u ∈ E. We claim that there exist δ > 0 and nk ∈ Z such that |u(k) nk | ≥ δ.

(3.7)

Indeed, if not, then u(k) → 0 in l∞ as k → ∞. From the simple fact that, for q > 2, (k) 2 ku(k) kqlq ≤ ku(k) kq−2 l∞ ku kl2

we have u(k) → 0 in all lq , q > 2. By (4.6), we know X X X (k) (k) p−1 f (n, u(k) ≤ ε |u(k) |u(k) · |u(k) n )un n | · |un | + cε n | n | n∈Z

n∈Z (k)

≤ εku kl2 · ≤ εku(k) kl2 ·

n∈Z (k) ku kl2 + cε ku(k) kp−1 lp · ku klp (k) ku(k) k + cε ku(k) kp−1 lp · ku k, (k)

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which implies that

P

(k)

(k)

f (n, un )un = o(ku(k) k) as k → ∞. Therefore, we have

n∈Z

X
(k) (k) 2 (k) o(ku(k) k) = J 0 (u(k) ), u(k) = ku(k) k2 − f (n, u(k) n )un = ku k − o(ku k). n∈Z

So ku(k) k2 → 0, as k → ∞, contrary to u(k) ∈ N . Since J and J 0 are both invariant under T -translation. Making such shifts, we can assume that 1 ≤ nk ≤ T −1 in (3.7). Moreover passing to a subsequence, we can assume that nk = n0 is independent of k. Extract a subsequence, still denoted by {u(k) }, we have u(k) * u and (k) un → un for all n ∈ Z. Specially, for n = n0 , inequality (3.7) shows that |un0 | ≥ δ, so u 6= 0. Hence u ∈ N . Finally we need to show that J(u) = c. By Lemma 3.1 and Fatou’s lemma, we have    X 1 1 0 (k) (k) (k) (k) (k) (k) c = lim J(u ) − J (u )u = lim f (n, un )un − F (n, un ) k→∞ k→∞ 2 2 n∈Z  X 1 1 ≥ f (n, un )un − F (n, un ) = J(u) − J 0 (u)u = J(u) ≥ c. 2 2 n∈Z Hence J(u) = c. Theorem 3.1 is complete. 

4

The potential well cases

Now we consider −∆un + vn un − ωun = f (un ), n ∈ Z,

(4.1)

for the case where V = {vn } is nonperiodic corresponding to a potential well. Since the nonlinearity is autonomous, the conditions on f needs modified slightly. More precisely, we make the following assumptions. (V2 ) 0 < inf Z V ≤ supZ V = V∞ < ∞ with V∞ := lim vn and vn − ω > 0, for all n. |n|→∞

(f10 )

f ∈ C(R, R) and there exist a > 0, p ∈ (2, ∞) such that |f (u)| ≤ a(1 + |u|p−1 ), for all u ∈ R.

(f20 ) lim f (u)/u = 0. |u|→0

(f30 ) lim F (u)/u2 = +∞, where F (u) is the primitive function of f (u), i.e., |u|→∞

Z F (u) =

u

f (s)ds. 0

(f40 ) u 7→ f (u)/|u| is strictly increasing on (−∞, 0) and (0, ∞). 9 47

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Theorem 4.1. Assume that conditions (V2 ), (f10 ) − (f40 ) hold. Then equation (4.1) has a ground state solution. Consider the functional J˜ defined on E by X 1 ˜ F (un ). J(u) = (Au, u) − 2 n∈Z

(4.2)

By the condition (V2 ), we may introduce an equivalent norm in E by setting kuk2 := (Au, u), then the functional J˜ can be rewritten as 1 ˜ ˜ J(u) = kuk2 − I(u), 2 ˜ = P F (un ). where I(u) n∈Z

We define the Nehari manifold ˜ = {u ∈ E \ {0} : J˜0 (u)u = 0}, N

(4.3)

˜ c˜ = inf J(u).

(4.4)

and ˜ u∈N

We assume that (V2 ) and (f10 ) − (f40 ) are satisfied from now on. The argument is the same as in Lemma 3.1-3.7 with f (u) which does not depend on n, except that the proof of Lemma 3.6 needs to be slightly modified. ˜ , i.e., J(u) ˜ ˜. Lemma 4.1. J˜ is coercive on N → ∞ as kuk → ∞, u ∈ N ˜ such that ku(k) k → ∞ Proof. Suppose by contradiction, there exists a sequence {u(k) } ⊂ N (k) ˜ (k) ) ≤ d, for some d ∈ [c, ∞). Let v (k) = u(k) , then there exists a subsequence, still and J(u ku k (k)

denoted by the same notation, such that v (k) * v and vn → vn for every n, as k → ∞. First we claim that there exist δ > 0 and nk ∈ Z such that |vn(k) | ≥ δ. k

(4.5)

Indeed, if not, then v (k) → 0 in l∞ as k → ∞. For q > 2, (k) 2 kv (k) kqlq ≤ kv (k) kq−2 kl2 l∞ kv

we have v (k) → 0 in all lq , q > 2. Note that by (f10 ) and (f20 ), for any ε > 0, there exists cε > 0 such that |f (u)| ≤ ε|u| + cε |u|p−1

and |F (u)| ≤ ε|u|2 + cε |u|p .

(4.6)

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Then for each s > 0, we have X F (svn(k) ) ≤ εs2 kv (k) k2l2 + cε sp kv (k) kplp n∈Z

which implies that

P

(k)

F (svn ) → 0 as k → ∞. Therefore

n∈Z 2 X s2 ˜ (k) ) ≥ J(sv ˜ (k) ) = s kv (k) k2 − d ≥ J(u F (svn(k) ) → , 2 2 n∈Z

as k → ∞. This is a contradiction if s > (k)

√

(4.7)

2d.

(k)

(k)

Making shifts v¯(k) = {¯ vn } = {vn+zk } such that v¯(k) *¯v = 6 0 and v¯n → v¯n for all n ∈ Z. (k) Since |un | → ∞, as k → ∞, it follows again from (f30 ) and Lemma 3.1 (b) that (k) (k) ˜ (k) ) J(u 1 X F (un ) (k) 2 1 X F (¯ un ) (k) 2 0 ≤ (k) 2 = − (vn ) = − (¯ vn ) → −∞, (k) 2 ku k 2 n∈Z (u(k) 2 n∈Z (¯ un )2 n ) (k)

(k)

as k → ∞ with u¯(k) := {¯ un } = {un+zk }. This is a contradiction again.  Proof of Theorem 4.1. We consider an associated equation −∆un + V∞ un = f (un ), n ∈ Z,

(4.8)

˜∞ = we define the energy functional J˜∞ by replacing V with V∞ , c∞ = inf N∞ J˜∞ (u), here N 0 ˜ {u ∈ E \ {0} : J∞ (u)u = 0}. Since V∞ is constant, by Theorem 3.1, c∞ > 0 is attained at ˜∞ . some u∞ ∈ N If V < V∞ , for all t > 0, we have ˜ ∞ ) < J˜∞ (tu∞ ) ≤ J˜∞ (u∞ ) = c∞ . c˜ ≤ J(tu So we may assume that c˜ < c∞ , since otherwise c˜ = c∞ and the assertion follows from Theorem 3.1. Let {w(k) } ⊂ S be a minimizing sequence for Ψ. By Ekeland’s variational principle we ˜ (k) ) → c˜, J˜0 (u(k) ) → 0 as may assume Ψ(w(k) ) → c˜, Ψ0 (w(k) ) → 0 as k → ∞, hence J(u (k) (k) k → ∞, where u := m(w ). (k)

(k)

First, we know {u(k) } is bounded by Lemma 4.1, making shifts u¯(k) = {¯ un } = {un+zk } such that u¯(k) * u 6= 0, then {zk } is bounded. If not, up to a subsequence if needed, zk → ∞, then u is a critical point of J˜∞ and    X 1 1 ˜0 (k) (k) (k) (k) (k) (k) ˜ c˜ = lim J(u ) − J (u )u = lim f (un )un − F (un ) k→∞ k→∞ 2 2 n∈Z  X 1 1 0 ≥ f (un )un − F (un ) = J˜∞ (u) − J˜∞ (u)u = J˜∞ (u) ≥ c∞ , 2 2 n∈Z 11 49

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this contradicts c˜ < c∞ . Hence the sequence {zk } is bounded, without loss of generality, we ˜. may assume that zk = 0 and therefore u¯(k) = u(k) for all k. Then u 6= 0 and u ∈ N ˜ Finally we need to show that J(u) = c˜. By Lemma 3.1 and Fatou’s lemma, we have    X 1 1 (k) (k) (k) (k) 0 (k) (k) ˜ c˜ = lim J(u ) − J˜ (u )u f (un )un − F (un ) = lim k→∞ k→∞ 2 2 n∈Z  X 1 ˜ − 1 J˜0 (u)u = J(u) ˜ ≥ f (un )un − F (un ) = J(u) ≥ c˜. 2 2 n∈Z ˜ Hence J(u) = c˜. Theorem 4.1 is complete.  Acknowledgments This work is supported by Program for the National Natural Science Foundation of China (No. 11371313) and the Foundation of Yuncheng University(YQ-2014011,SWSX201402).

References [1] A. Davydov, The theory of contraction of proteins under their excitation, J. Theor. Biol. 38 (1973) 559-569. [2] S. Flach, A. Gorbach, Discrete breakers-Advances in theory and applications, Phys. Rep. 467 (2008) 1-116. [3] S. Flash, C. Willis, Discrete breathers, Phys. Rep. 295 (1998) 181-264. [4] D. Henning, G. P. Tsironis, Wave transmission in nonlinear lattics, Phys. Rep. 307 (1999) 333-432. [5] D. Ma, Z. Zhou, Existence and multiplicity results of homoclinic solutions for the DNLS equations with unbounded potentials, Abstra. Appl. Anal. 2012 (2012) 1-15. [6] A. Mai, Z. Zhou, Ground state solutions for the periodic discrete nonlinear Schr¨odinger equations with superlinear nonlinearities, Abstr. Appl. Anal. 2013 (2013) 1-11. [7] A. Mai, Z. Zhou, Discrete solitons for periodic discrete nonlinear Schr¨odinger equations, Appl. Math. Comput. 222 (2013) 34-41. [8] A. Pankov, Gap solitons in periodic discrete nonlinear Schr¨odinger equations, Nonlinearity 19 (2006) 27-40. [9] H. Shi, Gap solitons in periodic discrete Schr¨odinger equations with nonlinearity, Acta Appl. Math. 109 (2010) 1065-1075. [10] H. Shi, H. Zhang, Existence of gap solitons in periodic discrete nonlinear Schr¨odinger equations, J. Math. Anal. Appl. 361 (2010) 411-419. 12 50

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[11] W. Su, J. Schieffer, A. Heeger, Solions in polyacetylene, Phys. Rev. Lett. 42 (1979) 1698-1701. [12] G. Sun, On Standing Wave solutions for discrete nonlinear Schr¨odinger equations, Abstr. Appl. Anal. 2013(2013) 1-6. [13] A. Szulkin, T. Weth, The method of Nehari manifold, in: Handbook of nonconvex Analysis and Applications, D.Y. Gao and D. Motreanu eds, International Press, Boston (2010) 597-632. [14] M. Willem, Minimax Theorems, Birkh¨auser, Boston, 1996. [15] M. Yang, Z. Han, Existence and multiplicity results for the nonlinear Schr¨odingerPoisson systems, Nonlinear Anal. Real World Appl. 13 (2012) 1093-1101. [16] G. Zhang, Breather solutions of the discrete nonlinear Schr¨odinger equations with unbounded potentials, J. Math. Phys. 50 (2009) 1-12. [17] G. Zhang, A. Pankov, Standing waves of the discrete nonlinear Schr¨odinger equations with growing potentials, Commun. Math. Anal. 5 (2008) 38-49.

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On fuzzy approximating spaces

∗

Zhaowen Li† Yu Han‡ Rongchen Cui§ February 14, 2014 Abstract: In this paper, we introduce the concept of fuzzy approximating spaces and obtain a decision condition that fuzzy topological spaces are fuzzy approximating spaces. Keywords: Fuzzy set; Fuzzy approximation space; Fuzzy topology; Fuzzy approximating space; (CC) axiom.

1

Introduction

Rough set theory, proposed by Pawlak [5, 6], is a new mathematical tool for data reasoning. It may be seen as an extension of classical set theory and has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields. The basic structure of rough set theory is an approximation space. Based on it, lower and upper approximations can be induced. Using these approximations, knowledge hidden in information systems may be revealed and expressed in the form of decision rules (see [6]). Topological structure is an important base for knowledge extraction and processing. Therefore, an interesting and natural research topic in rough set theory is to study the relationship between rough sets (resp. fuzzy rough sets) and topologies (resp. topologies). The purpose of this paper is to investigate fuzzy approximating space, i.e., a particular type of fuzzy topological spaces where the given fuzzy topology coincides with the fuzzy topology induced by some reflexive fuzzy relation. ∗ This work is supported by the National Natural Science Foundation of China (No. 11061004) and Guangxi University Science and Technology Research Project (No. 2013ZD020). † Corresponding Author, College of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China. [email protected] ‡ College of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China. [email protected] § College of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China. [email protected]

1

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2

Preliminaries

Throughout this paper, U denotes a finite and nonempty set called the universe. I denotes [0, 1]. F (U ) denotes the set of all fuzzy sets in U . For a ∈ I, a ¯ denotes the constant fuzzy set in U . We recall some basic operations on F (U ) as follows [10]: for any A, B ∈ F (U ), {Aj : j ∈ J} ⊆ F (U ) and λ ∈ I, (1) A = B ⇐⇒ A(x) = B(x) for each x ∈ U . (2) A ⊆ B ⇐⇒ A(x) ≤ B(x) for each x ∈ U . (3) AT= B c ⇐⇒ VA(x) = 1 − B(x) for each x ∈ U . (4) ( A)(x) = A(x) for each x ∈ U . j∈J j∈J S W (5) ( A)(x) = A(x) for each x ∈ U . j∈J

j∈J

(6) (λA)(x) = λ ∧ A(x) for each x ∈ U . A fuzzy set is called a fuzzy point in U , if it takes the value 0 for each y ∈ U except one, say, x ∈ U . If its value at x is λ (0 < λ ≤ 1), we denote this fuzzy point by xλ , where the point x is called its support and λ is called its height (see [3]). S Remark 2.1. A = (A(x)x1 ) (A ∈ F (U )). x∈U

Definition 2.2 ([2]). τ ⊆ F (U ) is called a fuzzy topology on U , if (i) For each a ∈ I, a ¯ ∈ τ. (ii) A ∈ τ, B ∈ τ imply A ∩ B S ∈ τ. (iii) {Aj : j ∈ J} ⊆ τ implies Aj ∈ τ . j∈J

In this case the pair (U, τ ) is called a fuzzy topological space. Every member of τ is called a fuzzy open set in U . Its complement is called a fuzzy closed set in U . We denote τ c = {A ∈ F (U ) : Ac ∈ τ }. It should be pointed out that if (i) in Definition 2.2 is replaced by (i)0 ¯ 0, ¯ 1 ∈ τ, then τ is a fuzzy topology in the sense of Chang [1]. We can see that a fuzzy topology in the sense of Lowen must be a fuzzy topology in the sense of Chang. In this paper, we always consider the fuzzy topology in the sense of Lowen. The interior and closure of A denoted respectively by int(A) and cl(A) for each A ∈ F (U ), are defined as follows: [ \ intτ (A) = {B ∈ τ : B ⊆ A}, clτ (A) = {B ∈ τ c : B ⊇ A}.

3

Fuzzy approximation spaces and fuzzy rough sets Recall that R is a fuzzy relation on U if R ∈ F (U × U ).

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Let R be a fuzzy relation on U . R is called reflexive if R(x, x) = 1 for each x ∈ U . R is called symmetric if R(x, y) = R(y, x) for any x, y ∈ U . R is called transitive if R(x, z) ≥ R(x, y) ∧ R(y, z) for any x, y, z ∈ U . R is called preorder if R is reflexive and transitive. Definition 3.1 ([7]). Let R be a fuzzy relation on U . The pair (U, R) is called a fuzzy approximation space. For each A ∈ F (U ), the fuzzy lower and the fuzzy upper approximation of A with respect to (U, R), denoted by R(A) and R(A) are respectively, defined as follows: ^ _ R(A)(x) = (A(y)∨(1−R(x, y))) (x ∈ U ) and R(A)(x) = (A(y)∧R(x, y)) (x ∈ U ). y∈U

y∈U

The pair (R(A), R(A)) is called the fuzzy rough set of A with respect to (U, R). Remark 3.2. (1) R(x1 )(y) = R(y, x), R((x1 )c )(y) = 1 − R(y, x) (x, y ∈ U ). (2) For each a ∈ I, R(a) ⊆ a ¯ ⊆ R(¯ a); a) = a ¯ = R(¯ a). (3) If R is reflexive, then for each a ∈ I, R(¯ Let (U, R) be a fuzzy approximation space. (U, R) is call reflexive (resp. preorder) if R is reflexive (resp. preorder). Theorem 3.3 ([7, 8]). Let (U, R) be a fuzzy approximation space. Then for any A, B ∈ F (U ), {Aj : j ∈ J} ⊆ F (U ) and λ ∈ I, (1) A ⊆ B =⇒ R(A) ⊆ R(B), R(A) ⊆ R(B). (2) R(Ac ) = (R(A))c , R(Ac ) = (R(A))c . (3) R(AT∩ B) = R(A) T ∩ R(B), R(A S ∪ B) =SR(A) ∪ R(B). (4) R( Aj ) = (R(Aj )), R( Aj ) = (R(Aj )). j∈J

j∈J

j∈J

j∈J

¯ ∪ A) = λ ¯ ∪ R(A), R(λA) = λR(A). (5) R(λ

Theorem 3.4 ([4, 7, 9]). Let (U, R) be a fuzzy approximation space. Then (1) R is ref lexive (2) R is preorder

4

⇐⇒ ⇐⇒ =⇒

∀A ∈ F (U ), R(A) ⊆ A. ∀A ∈ F (U ), A ⊆ R(A). ∀A ∈ F (U ), R(R(A)) = R(A), R(R(A)) = R(A).

Fuzzy approximation spaces and fuzzy topologies In this paper, we denote τR = F ix(R) (i.e., {A ∈ F (U ) : A = R(A)}),

θR = {R(A) : A ∈ F (U )}.

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4.1

Fuzzy topologies induced by approximation spaces

Proposition 4.1. Let (U, R) be a fuzzy approximation space. If R is preorder, then τR = θR . Proof. Obviously, τR ⊆ θR . By Theorem 3.4(2), τR ⊇ θR . So τR = θR . Theorem 4.2 ([7]). Let (U, R) be a preorder fuzzy approximation space. Then (1) θR is a fuzzy topology on U . (2) R is the interior operator of θR . (3) R is the closure operator of θR . Theorem 4.3. Let (U, R) be a reflexive fuzzy approximation space. Then the following properties hold. (1) τR is a fuzzy topology on U . (2) intτR (A) ⊆ R(A) ⊆ A ⊆ R(A) ⊆ clτR (A) (A ∈ F (U )). (3) A ∈ (τR )c ⇐⇒ A = R(A). (4) For each a ∈ I, a ¯ ∈ (τR )c . Proof. (1) This holds by Remark 3.2, Theorem 3.3 and Theorem 3.4(1). (2) For each A ∈ F (U ), by Theorem 3.3(1), [ [ intτR (A) = {B ∈ τR : B ⊆ A} ⊆ {B ∈ τR : R(B) ⊆ R(A)} [ = {B ∈ F (U ) : B = R(B) ⊆ R(A)} ⊆ R(A). By Theorem 3.3(2), clτR (A) = (intτR (Ac ))c ⊇ (R(Ac ))c = R(A). By Theorem 3.4(1), R(A) ⊆ A ⊆ R(A). Thus intτR (A) ⊆ R(A) ⊆ A ⊆ R(A) ⊆ clτR (A). (3) This holds by Theorem 3.3(2). (4) This holds by (3) and Remark 3.2. Definition 4.4. Let (U, R) be a reflexive fuzzy approximation space. τR is called the fuzzy topology induced by (U, R) or R on U .

4.2

Fuzzy approximation spaces induced by fuzzy topologies

Definition 4.5. Let τ be a fuzzy topology on U . Define a fuzzy relation Rτ on U by Rτ (x, y) = clτ (y1 )(x) ((x, y) ∈ U × U ). Then Rτ is called the fuzzy relation induced by τ on U and (U, Rτ ) is called the fuzzy approximation space induced by τ on U .

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Theorem 4.6. Let τ be a fuzzy topology on U and let Rτ be the fuzzy relation induced by τ on U . Then the following properties hold. (1) Rτ is reflexive. (2) Rτ (A) ⊆ intτ (A) ⊆ A ⊆ clτ (A) ⊆ Rτ (A) (A ∈ F (U )). Proof. (1) For each x ∈ U , Rτ (x, x) = clτ (x1 )(x) ≥ (x1 )(x) = 1. Then Rτ is reflexive. (2) For each A ∈ F (U ), by Remark 2.1, [ [ [ clτ (A) = clτ ( (A(y)y1 )) = clτ (A(y)y1 ) = clτ (A(y) ∩ y1 ) ⊆

[

y∈U

y∈U

[

(clτ (A(y)) ∩ clτ (y1 )) =

y∈U

y∈U

(A(y) ∩ clτ (y1 )) =

y∈U

[

(A(y)clτ (y1 )).

y∈U

Then for each x ∈ U , _ _ _ clτ (A)(x) ≤ (A(y)clτ (y1 ))(x) = (A(y)∧clτ (y1 )(x)) = (A(y)∧Rτ (x, y)) = Rτ (A)(x). y∈U

y∈U

y∈U

Thus clτ (A) ⊆ Rτ (A). By Theorem 3.3(2), intτ (A) = (clτ (Ac ))c ⊇ (Rτ (Ac ))c = Rτ (A). Hence Rτ (A) ⊆ intτ (A) ⊆ A ⊆ clτ (A) ⊆ Rτ (A).

5

Fuzzy approximating spaces

As can be seen from Section 4, a reflexive fuzzy relation yields a fuzzy topology. In this section, we consider a interested problem, that is, under which conditions can the given fuzzy topology coincides with the fuzzy topology induced by some reflexive fuzzy relation? Definition 5.1. Let (U, σ) be a fuzzy topological space. If there exists a reflexive fuzzy relation R on U such that τR = σ, then (U, σ) is called a fuzzy approximating space. The following condition for a fuzzy topology τ on U is called (CC) axiom, (CC) axiom: for any λ ∈ I and A ∈ F (U ), clτ (λA) = λclτ (A). Proposition 5.2. Let τ be a fuzzy topology on U . If τ satisfies (CC) axiom, then Rτ is the closure operator of τ . 5

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Proof. For each A ∈ F (U ), by Remark 2.1 and (CC) axiom, [ [ [ clτ (A) = clτ ( (A(y)y1 )) = clτ (A(y)y1 ) = (A(y)clτ (y1 )). y∈U

y∈U

y∈U

Then for each x ∈ U , _ _ _ clτ (A)(x) = (A(y)clτ (y1 ))(x) = (A(y)∧clτ (y1 )(x)) = (A(y)∧Rτ (x, y)) = Rτ (A)(x). y∈U

y∈U

y∈U

Thus Rτ (A) = clτ (A). Hence Rτ is the closure operator of τ . Proposition 5.3. Let R be a preorder fuzzy relation on U . Then τR satisfies (CC) axiom. Proof. For any λ ∈ I and A ∈ F (U ), by Theorem 3.3(5), Proposition 4.1 and Theorem 4.2(3), clτR (λA) = clθR (λA) = R(λA) = λR(A) = λclθR (A) = λclτR (A). Thus τR satisfies (CC) axiom. Proposition 5.4. Let τ be a fuzzy topology on U , let Rτ be the fuzzy relation induced by τ on U and let τRτ be the fuzzy topology induced by Rτ on U . Then τRτ = τ if and only if τ satisfies (CC) axiom. Proof. Necessity. For each A ∈ F (U ), by Theorem 4.6(2), clτ (A) ⊆ Rτ (A). By Theorems 4.6(1) and 4.3(2), clτ (A) = clτRτ (A) ⊇ Rτ (A). Then clτ (A) = Rτ (A). So R is the closure operator of τ . For any a ∈ I and A ∈ F (U ), by Theorem 3.3(5), clτ (λA) = Rτ (λA) = λRτ (A) = λclτ (A). Thus τ satisfies (CC) axiom. Sufficiency. By Theorem 4.6(1), Rτ is reflexive. For any x, y, z ∈ U , put clτ (z1 )(y) = λ. By Remark 2.1, λclτ (y1 ) =

clτ (λy1 ) = clτ (clτ (z1 )(y)y1 ) [ ⊆ clτ ( (clτ (z1 )(t)t1 )) = clτ (clτ (z1 )) = clτ (z1 ). t∈U

Then Rτ (x, y) ∧ Rτ (y, z)

= clτ (y1 )(x) ∧ clτ (z1 )(y) = clτ (y1 )(x) ∧ λ = λ ∧ clτ (y1 )(x) = (λclτ (y1 ))(x) ≤ clτ (z1 )(x) = Rτ (x, z). 6

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Then Rτ is transitive. So Rτ is preorder. For each A ∈ F (U ), by Propositions 4.1 and 4.2(3), clτRτ (A) = clθRτ (A) = Rτ (A). By (CC) axiom and Proposition 5.2, Rτ (A) = clτ (A). So clτRτ (A) = clτ (A). Thus τRτ = τ . Theorem 5.5. Let (U, τ ) be a fuzzy topological space. If τ satisfies (CC) axiom, then (U, τ ) is a fuzzy approximating space. Proof. This holds by Proposition 5.4. Example 5.6. {¯ a : a ∈ I} is a fuzzy approximating space.

References [1] C.Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 24(1968), 182-190. [2] R.Lowen, Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical Analysis and Applications, 56(1976), 621-633. [3] Y.Liu, M.Luo, Fuzzy Topology, World Scientific Publishing, Singapore, 1998. [4] J.Mi, W.Wu, W.Zhang, Constructive and axiomatic approaches for the study of the theory of rough sets, Pattern Recognition Artificial Intelligence, 15(2002), 280-284. [5] Z.Pawlak, Rough sets, International Journal of Computer and Information Science, 11(1982), 341-356. [6] Z.Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991. [7] K.Qin, Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems, 151(2005), 601-613. [8] H.Thiele, Fuzzy rough sets versus rough fuzzy sets–An interpretation and a comparative study using concepts of modal logics, in: Proc. 5th Europ. Congr. on Intelligent Technigues and Soft Computing, Aachen, Germany, September 1997, pp. 159-167. [9] W.Wu, J.Mi, W.Zhang, Generalized fuzzy rough sets, Information Sciences, 151(2003), 263-282. [10] L.A.Zadeh, Fuzzy sets, Information and Control, 8(1965), 338-353.

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On some self-adjoint fractional nite dierence equations Dumitru Baleanu

1

1,2,3 , Shahram Rezapour 4 , Saeid Salehi 4

Department of Chemical and Materials Engineering, Faculty of Engineering King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia 2

Department of Mathematics, Cankaya University,

Ogretmenler Cad. 14 06530, Balgat, Ankara, Turkey 3 4

Institute of Space Sciences, Magurele, Bucharest, Romania

Department of Mathematics, Azarbaijan Shahid Madani University, Azarshahr, Tabriz, Iran

Recently, the existence of solution for the fractional self-adjoint equation ∆νν−1 (p∆y)(t) = h(t) for order 0 < ν ≤ 1 was reported in [9]. In this paper, we investigated the self-adjoint fractional nite dierence equation ∆νν−2 (p∆y)(t) = h(t, p(t + ν − 2)∆y(t + ν − 2)) via the boundary conditions y(ν − 2) = 0, such that ∆y(ν − 2) = 0 and ∆y(ν + b) = 0. Also, we analyzed the self-adjoint fractional nite dierence equation ∆νν−2 (p∆2 y)(t) = h(t, p(t + ν − 3)∆2 y(t + ν − 3)) via the boundary conditions y(ν − 2) = 0, ∆y(ν − 2) = 0, ∆2 y(ν − 2) = 0 and ∆2 y(ν + b) = 0. Finally, we conclude a result about the existence of solution for the general equation ∆νν−2 (p∆m y)(t) = h(t, p(t + ν − m − 1)∆m y(t + ν − m − 1)) via the boundary conditions y(ν − 2) = ∆y(ν − 2) = ∆2 y(ν − 2) = · · · = ∆m y(ν − 2) = 0 and ∆m y(ν + b) = 0 for order 1 < ν ≤ 2. Abstract.

1

Introduction and Preliminaries

One of the trends in fractional calculus [1, 2, 3, 4] is the discrete version of it ( see for example Refs.[5, 6, 7, 8, 7, 19, 9, 11, 12, 13, 14, 15, 16, 17]and the references therein). Some recent applications of the discrete fractional calculus revealed its huge potential applications to solve real world problems (see for example Refs.[19, 20, 21, 22] and the references therein). Very recently, Awasthi provided some results about the fractional self-adjoint equation ∆νν−1 (p∆y)(t) = h(t) for order 0 < ν ≤ 1 ([9]). In this paper we investigate the fractional nite dierence equation ∆νν−2 (p∆y)(t) = h(t, p(t + ν − 2)∆y(t + ν − 2)) in the presence of the boundary conditions y(ν − 2) = 0, ∆y(ν − 2) = 0 b+ν and ∆y(ν + b) = 0, such that b ∈ N0 , t ∈ Nb+2 0 , 1 < ν ≤ 2 and h : Nν−2 × R → R is a map. In addition, we discuss the existence of the solution corresponding to the fractional nite dierence equation ∆νν−2 (p∆2 y)(t) = h(t, p(t + ν − 3)∆2 y(t + ν − 3)) with the boundary conditions y(ν − 2) = 0, ∆y(ν − 2) = 0, ∆2 y(ν − 2) = 0 and ∆2 y(ν + b) = 0, such that 1 < ν ≤ 2, b ∈ N0 , t ∈ Nb+3 and h : Nν+b 0 ν−2 × R → R represents a map. For the second equation we provided an example. Γ(t+1) We recall that tν := Γ(t+1−ν) for all t, ν ∈ R provided that the right-hand side is dened (see for example Ref.[15] and the references therein). In the following we use the notations Na = {a, a + 1, a + 2, . . .} for all a ∈ R and Nba = {a, a + 1, a + 2, . . . , b} for all real numbers a and b such that b − a denotes a natural number. Let ν > 0 with m − 1 < ν < m for some natural number m. Then the ν -th fractional sum of f based at a is

1

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dened by ∆−ν a f (t) =

Pt−ν

− σ(k))ν−1 f (k) for all t ∈ Na+ν , where σ(k) = k + 1 is the forward jump Pt+ν 1 −ν−1 f (k) for all t ∈ Na+N −ν . operator (see [15]). Similarly, we dene ∆νa f (t) = Γ(−ν) k=a (t − σ(k)) 1 Γ(ν)

k=a (t

[9] Let h : Na → R be a mapping and number. Then, the general solution of P m a natural ν−i + ∆−ν the equation ∆νa+ν−m y(t) = h(t) is given by y(t) = m C (t − a) i a h(t) for all t ∈ Na+ν−m , where i=1 C1 , · · · , Cm are arbitrary constants.

Lemma 1.1.

Let h : Nν−m × R → R be a mapping and m a natural number. By using a similar proof, one can check that the general solution of the equation ∆νν−m y(t) = h(t, y(t + ν − m + 1)) is given by y(t) =

m X

Ci tν−i + ∆−ν h(t, y(t + ν − m + 1))

i=1

for all t ∈ Nν−m . In particular, the general solution has the following representation y(t) =

m X

t−ν

Ci t

ν−i

i=1

1 X + (t − σ(s))ν−1 h(s, y(s + ν − m + 1)) Γ(ν) s=0

(1. 1)

Pb ν−1 f (k) = 0 whenever b < a. for all t ∈ Nν−m . By considering the details, note that k=a (t − σ(k)) Also for ν, µ > 0 with m − 1 < ν ≤ m and n − 1 < µ ≤ n, the domain of the operator ∆ is given −ν ν µ ν −µ by D{∆−ν a f } = Na+ν , D{∆a f } = Na+m−ν , D{∆a+n−µ ∆a f } = Na+n+ν−µ , D{∆a+µ ∆a f } = Na+µ+m−ν , −ν ν µ −µ D{∆a+n−µ ∆a f } = Na+n−µ+m−ν and D{∆a+µ ∆a f } = Na+ν+µ (for more details see [9], [14] and [15]). One can nd next result about composing a dierence with a sum in [19]. k−µ Let f : Na → R be a map, k ∈ N0 and µ > k with n − 1 < µ ≤ n. Then ∆k ∆−µ f (t) a f (t) = ∆a k µ k+µ for all t ∈ Na+µ and ∆ ∆a f (t) = ∆a f (t) for all t ∈ Na+n−µ .

Lemma 1.2.

By making use of (1. 1) and the last Lemma for k = 1, we nally report that ∆y(t) =

m X

Ci (ν − i)tν−i−1 +

i=1

t−ν+1 X 1 (t − σ(s))ν−2 h(t, y(t + ν − m + 1)), Γ(ν − 1) s=0

which will be used in proving the main results. Let (X, d) be a metric space, P∞ T : X → X be a self-map and Ψ be the family of nondecreasing functions ψ : [0, ∞) → [0, ∞) such that n=1 ψ n (t) < ∞ for all t > 0. We recall that T is an α-ψ -contractive mapping whenever there exists ψ ∈ Ψ and α : X × X → [0, ∞) such that α(x, y)d(T x, T y) ≤ ψ(d(x, y)) for all x, y ∈ X . Also, T is called α-admissible whenever α(x, y) ≥ 1 implies that α(T x, T y) ≥ 1. In our proofs we used the next result which have been proved in [18]. Theorem 1.3. [18] Let (X, d) be a complete metric space, T : X → X an α-admissible α-ψ -contractive map such that α(x0 , T x0) ≥ 1 for some x0 ∈ X . If xn is a sequence in X such that α(xn , xn+1 ) ≥ 1, for all n and xn → x for some x ∈ X , then α(xn , x) ≥ 1 for all n. Then T has a xed point.

2

Main results

In this section we present the main results of this manuscript. b+ν Let b ∈ N0 , t ∈ Nb+2 0 , 1 < ν ≤ 2 and h : Nν−2 × R → R be a map. Then y0 is a solution of the problem = h(t, p(t+ ν − 2)∆y(t+ ν − 2)) via the boundary conditions y(ν − 2) = 0, ∆y(ν − 2) = 0 and ∆y(ν + b) = 0 if and only if y0 is a solution of the fractional sum equation Lemma 2.1.

∆νν−2 (p∆y)(t)

y(t) =

b X

G(t, s)h(t, p(t + ν − 2)∆y(t + ν − 2)),

s=0

2

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where        

1 G(t, s) = Γ(ν)       

t−1 t−1 X X (k − σ(s))ν−1 −k ν−1 (ν + b − σ(s))ν−1 + ν−1 p(k) p(k)(ν + b)

s ≤ t − ν − 1,

k=s+ν

k=ν−2

t−1 X −k ν−1 (ν + b − σ(s))ν−1 p(k)(ν + b)ν−1 k=ν−2

t − ν ≤ s.

Proof. Let y0 be a solution of the problem ∆νν−2 (p∆y)(t) = h(t, p(t + ν − 2)∆y(t + ν − 2)) via the boundary

conditions y(ν − 2) = 0, ∆y(ν − 2) = 0 and ∆y(ν + b) = 0. If x(t) = (p∆y)(t), then x0 = p∆y0 is a solution of the equation ∆νν−2 x(t) = h(t, x(t + ν − 2)). By using Lemma 1.1, we get t−ν

1 X (t − σ(s))ν−1 h(s, x0 (s + ν − 2)) Γ(ν) s=0

x0 (t) = C1 tν−1 + C2 tν−2 +  1 C1 tν−1 + C2 tν−2 + and so ∆y0 (t) = p(t) ∆y0 (ν − 2) = 0, we have

1 Γ(ν)

Pt−ν

s=0 (t

 1 1 C1 (ν − 2)ν−1 + C2 (ν − 2)ν−2 + 0= p(ν − 2) Γ(ν)

Since (ν − 2)ν−1 = 0 and

P−2

s=0 ((ν

 − σ(s))ν−1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) . Since

(ν−2)−ν

X s=0

 ((ν − 2) − σ(s))ν−1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) .

− 2) − σ(s))ν−1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) = 0, C2 = 0. Hence,

t−ν  1 X 1  C1 tν−1 + (t − σ(s))ν−1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) . p(t) Γ(ν) s=0

∆y0 (t) =

Since ∆y0 (ν + b) = 0, C1 =

−1 (ν+b)ν−1 Γ(ν)

Pb

s=0 (ν

+ b − σ(s))ν−1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) and so

b

∆y0 (t) =

X −tν−1 1  (ν + b − σ(s))ν−1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) ν−1 p(t) (ν + b) Γ(ν) s=0 +

t−ν  1 X (t − σ(s))ν−1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) . Γ(ν) s=0

Thus, we obtain y0 (t) − y0 (ν − 2) =

t−1 X

∆y0 (k) =

k=ν−2

k=ν−2

+

t−1 X b X −k ν−1 (ν + b − σ(s))ν−1 1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) Γ(ν) p(k)(ν + b)ν−1 s=0

t−1 k−ν X X (k − σ(s))ν−1 1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)). Γ(ν) p(k) s=0 k=ν−2

But, we have y0 (ν − 2) = 0. Thus by interchanging the order of summations, we get y0 (t) =

b t−1 1 X X −k ν−1 (ν + b − σ(s))ν−1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) Γ(ν) s=0 p(k)(ν + b)ν−1 k=ν−2

+

t−1−ν t−1 1 X X (k − σ(s))ν−1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) Γ(ν) s=0 p(k) k=s+ν

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=

b X

G(t, s)h(s, p(s + ν − 2)∆y0 (s + ν − 2)).

s=0

Now, let y0 be a solution of the fractional sum equation y(t) = Then, we have

Pb

s=0

G(t, s)h(s, p(s + ν − 2)∆y(s + ν − 2)).

t−1 X b X −k ν−1 (ν + b − σ(s))ν−1 1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) y0 (t) = Γ(ν) p(k)(ν + b)ν−1 k=ν−2 s=0

+

t−1 k−ν X X (k − σ(s))ν−1 1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)). Γ(ν) p(k) s=0 k=ν−2

By using the mentioned details, we get y0 (ν − 2) = 0. Since t−1 k−ν X X (k − σ(s))ν−1  1  h(s, p(s + ν − 2)∆y0 (s + ν − 2)) ∆ Γ(ν) p(k) s=0 k=ν−2

=

t−2 k−ν X X (k − σ(s))ν−2 1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)), Γ(ν − 1) p(k) s=0 k=ν−2

we obtain ∆y0 (ν − 2) = 0. By using a similar calculation, one can get that ∆y0 (ν + b) = 0. Moreover,   t−1 X b X −k ν−1 (ν + b − σ(s))ν−1 1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) ∆ν (p∆y0 )(t) = ∆ν p(t)∆ Γ(ν) p(k)(ν + b)ν−1 s=0 k=ν−2

 t−1 k−ν X X (k − σ(s))ν−1 1 h(s, p(s + ν − 2)∆y0 (s + ν − 2)) + Γ(ν) p(k) s=0 k=ν−2

ν

=∆



p(t) Γ(ν − 1)

t−2 X

b X −(ν − 1)k ν−2 (ν + b − σ(s))ν−1

p(k)(ν + b)ν−1

k=ν−2 s=0

 t−2 X +∆ν p(t)

k=ν−2

Since ∆ν tν−2 = ν

∆



p(t)

Γ(ν−1)t Γ(ν−ν−1)

ν−2−ν

t−2 X

k=ν−2

 h(s, p(s + ν − 2)∆y0 (s + ν − 2))

 1 ∆−ν h(k, p(k + ν − 2)∆y0 (k + ν − 2)) . p(k)

= 0 and

 1 −ν ∆ h(k, p(k + ν − 2)∆y0 (k + ν − 2)) = h(t, p(t + ν − 2)∆y0 (t + ν − 2)), p(k)

a simple calculation shows us that ∆ν (p∆y0 )(t) = h(t, p(t+ν −2)∆y0 (t+ν −2)). This completes the proof. It is important to note that the sum

k X

G(t, s) is bounded, where G is the Green function in last result.

s=0

Denote the upper bound of

Pb

s=0

|G(t, s)| by MG .

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC + Suppose that ψ ∈ Ψ has this property that ψ(2t) ≤ 2ψ(t) for all t ≥ 0 and p : Nb+ν ν−2 → R is b+ν b+ν a bounded function such that p(t) ≤ C for all t ∈ Nν−2 . Let h : Nν−2 × R → R be a map such that Theorem 2.2.

|h(t, a) − h(t, b)| ≤

1 ψ(|a − b|) 2CMG

ν for all a, b ∈ R and t ∈ Nb+ν ν−2 . Then the problem ∆ν−2 (p∆y)(t) = h(t, p(t + ν − 2)∆y(t + ν − 2)) via the boundary conditions y(ν − 2) = 0, ∆y(ν − 2) = 0 and ∆y(ν + b) = 0 has at least one solution.

Proof. Suppose that X is the space of real valued functions on N0 via the norm kyk = max{|y(t)| : t ∈ N0 }. It P

b is known that X is a Banach space. Dene T : X → X by T x(t) = s=0 G(t, s)h(s, p(s + ν − 2)∆x(s + ν − 2)). By using Lemma 2.1, we know that y0 ∈ X is a solution of the problem if and only if y0 is a xed point of T . Let x, y ∈ X and t ∈ N0 . Hence,

|T x(t) − T y(t)| = |

b X

G(t, s)h(s, p(s + ν − 2)∆x(s + ν − 2)) −

s=0

≤

b X

b X

G(t, s)h(s, p(s + ν − 2)∆y(s + ν − 2))|

s=0

|G(t, s)||h(t, p(s + ν − 2)∆x(s + ν − 2)) − h(t, p(s + ν − 2)∆y(s + ν − 2))|

s=0

≤ MG ×

1 ψ(|p(s + ν − 2)∆x(s + ν − 2) − p(s + ν − 2)∆y(s + ν − 2)|) 2CMG

p(s + ν − 2) ψ(|x(s + ν − 1) − x(s + ν − 2) − y(s + ν − 1) + y(s + ν − 2)|) 2C 1 ≤ ψ(kx − yk + kx − yk) ≤ ψ(kx − yk) 2 for all x, y ∈ X . If we dene the function α : X × X → [0, ∞) by α(x, y) = 1 for all x, y ∈ X , then we have α(x, y)d(T x, T y) ≤ ψd(x, y) for all x, y ∈ X . Thus, T is an α-ψ -contractive mapping. It is clear that α(x, y) ≥ 1 implies that α(T x, T y) ≥ 1 for all x, y ∈ X and so T is α-admissible. Also, α(x0 , T x0 ) ≥ 1 for all x0 ∈ X . Also, for each sequence {xn } in X with α(xn , xn+1 ) ≥ 1 and xn → x, we have α(xn , x) ≥ 1 for all n. By using Theorem 1.3, T has a xed point y0 ∈ X and so y0 is a solution of the problem. =

Now, we consider the fractional nite dierence equation ∆νν−2 (p∆2 y)(t) = h(t, p(t + ν − 3)∆2 y(t + ν − 3)) via the boundary conditions y(ν − 2) = 0, ∆y(ν − 2) = 0, ∆2 y(ν − 2) = 0 and ∆2 y(ν + b) = 0, where b ∈ N0 , b+ν t ∈ Nb+3 0 , 1 < ν ≤ 2 and h : Nν−2 × R → R is a map. First, we give next key result. b+3 Let h : Nb+ν and 1 < ν ≤ 2. Then y0 is a solution ν−2 × R → R be a mapping, b ∈ N0 , t ∈ N0 ν 2 2 of the problem ∆ν−2 (p∆ y)(t) = h(t, p(t + ν − 3)∆ y(t + ν − 3)) via the boundary conditions y(ν − 2) = 0, ∆y(ν − 2) = 0, ∆2 y(ν − 2) = 0 and ∆2 y(ν + b) = 0 if and only if y0 is a solution of the fractional sum equation Pb y(t) = s=0 G(t, s)h(t, p(t + ν − 3)∆2 y(t + ν − 3)), where Lemma 2.3.

G(t, s) =

       

1 Γ(ν)       

t−1 X

τ −1 X

t−1 τ −1 X X −k ν−1 (ν + b − σ(s))ν−1 (k − σ(s))ν−1 + ν−1 p(k) p(k)(ν + b) τ =s+1+ν

s ≤ t − ν − 2,

t−1 X

τ −1 X

−k ν−1 (ν + b − σ(s))ν−1 p(k)(ν + b)ν−1

t − ν − 1 ≤ s.

τ =ν−2 k=ν−2

τ =ν−2 k=ν−2

k=s+ν

5

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Proof. Let y0 be a solution of the problem ∆νν−2 (p∆2 y)(t) = h(t, p(t + ν − 3)∆2 y(t + ν − 3)) via the boundary

conditions y(ν − 2) = 0, ∆y(ν − 2) = 0, ∆2 y(ν − 2) = 0 and ∆2 y(ν + b) = 0. If x(t) = (p∆2 y)(t), then x0 = p∆2 y0 is a solution of the equation ∆νν−2 x(t) = h(t, x(t + ν − 3)). By using Lemma 1.1, we get t−ν

x0 (t) = C1 tν−1 + C2 tν−2 +

1 X (t − σ(s))ν−1 h(s, x0 (s, ν − 3)). Γ(ν) s=0

By using the conditions ∆2 y0 (ν − 2) = 0 and ∆2 y0 (ν + b) = 0, we conclude that C2 = 0 and b

X −1 (ν + b − σ(s))ν−1 h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)). C1 = ν−1 (ν + b) Γ(ν) s=0

Thus, we obtain b

∆2 y0 (t) =

X −tν−1 1  (ν + b − σ(s))ν−1 h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)) ν−1 p(t) (ν + b) Γ(ν) s=0 +

t−ν  1 X (t − σ(s))ν−1 h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)) Γ(ν) s=0

and so by applying the condition ∆y0 (ν − 2) = 0 and interchanging the order of summations, we get ∆y0 (t) =

b t−1 1 X X −k ν−1 (ν + b − σ(s))ν−1 h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)) Γ(ν) s=0 p(k)(ν + b)ν−1 k=ν−2

+

t−1−ν t−1 1 X X (k − σ(s))ν−1 h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)). Γ(ν) s=0 p(k) k=s+ν

Again, by applying the condition y0 (ν − 2) = 0 and interchanging the order of summations, we get b t−1 τ −1 1 X X X −k ν−1 (ν + b − σ(s))ν−1 h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)) ν−1 Γ(ν) s=0 τ =ν−2 p(k)(ν + b) k=ν−2

y0 (t) =

+

t−2−ν t−1 τX −1 X (k − σ(s))ν−1 1 X h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)) Γ(ν) s=0 τ =s+1+ν p(k) k=s+ν

=

b X

G(t, s)h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)).

s=0

Now, let y0 be a solution of the fractional sum equation y(t) = Then, we have y0 (t) =

+

Pb

s=0

G(t, s)h(s, p(s + ν − 3)∆2 y(s + ν − 3)).

b t−1 τ −1 1 X X X −k ν−1 (ν + b − σ(s))ν−1 h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)) ν−1 Γ(ν) s=0 τ =ν−2 p(k)(ν + b) k=ν−2 t−2−ν t−1 τ −1 X X 1 X (k − σ(s))ν−1 h(s, p(s + ν − 3)∆2 y0 (s + ν − 3)). Γ(ν) s=0 τ =s+1+ν p(k) k=s+ν

Similar to proof of Lemma 2.1, one can conclude that y(ν − 2) = 0, ∆y(ν − 2) = 0, ∆2 y(ν − 2) = 0 and ∆2 y(ν + b) = 0. Also ∆νν−2 (p∆2 y0 )(t) = h(t, p(t + ν − 3)∆2 y0 (t + ν − 3)). This completes the proof.

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Similar to above results one can get that

b X

G(t, s) is bounded, where G is the Green function in last

s=0

0 result. Denote the bound of the Green function by MG . Also, we can provide next result. + Suppose that ψ ∈ Ψ has this property that ψ(4t) ≤ 4ψ(t) for all t ≥ 0 and p : Nb+ν ν−2 → R is b+ν b+ν a bounded function such that p(t) ≤ C for all t ∈ Nν−2 . Let h : Nν−2 × R → R be a map such that Theorem 2.4.

|h(t, a) − h(t, b)| ≤

1 0 ψ(|a − b|) 4CMG

ν 2 2 for all a, b ∈ R and t ∈ Nb+ν ν−2 . Then the problem ∆ν−2 (p∆ y)(t) = h(t, p(t + ν − 3)∆ y(t + ν − 3)) via the 2 2 boundary conditions y(ν − 2) = 0, ∆y(ν − 2) = 0, ∆ y(ν − 2) = 0 and ∆ y(ν + b) = 0 has at least one solution.

Here, we present an example for the second problem. Example 2.1.

Consider the problem −t 2 ∆1.5 −0.5 (e ∆ y(t)) = tan t +

4 sin(e1.5−t ∆2 y(t − 1.5)) − cosh(t + 11) 4 − t2

via the boundary conditions y(−0.5) = 0, ∆y(−0.5) = 0, ∆2 y(−0.5) = 0 and ∆2 y(4.5) = 0. Let ν = 1.5, 4 sin a 0.5 − 4−t p(z) = e−z , h(t, a) = tan t + cosh(t+11) for all z ∈ N4.5 2 and b = 3. Note that, 0 < p(z) ≤ e −0.5 . Put 0.5 0 C = e . Now, we show that MG = 527.9447. In fact, the Green function is given by

G(t, s) =

       

1 Γ(1.5)       

t−1 X

τ −1 X

t−1 τX −1 X −k 0.5 (4.5 − σ(s))0.5 + p(k)(4.5)0.5 τ =s+2.5

t−1 X

τ −1 X

−k 0.5 (4.5 − σ(s))0.5 p(k)(4.5)0.5

τ =−0.5 k=−0.5

τ =−0.5 k=−0.5

k=s+1.5

(k − σ(s))0.5 p(k)

s ≤ t − 3.5,

t − 2.5 ≤ s.

τ −1 −(−0.5) (3.5) −(0.5) (3.5) −k 3.5 Thus, G(2.5, 0) = 1.5 τ =−0.5 k=−0.5 Γ(1.5)p(k)(4.5)0.5 = 2 × Γ(1.5)p(−0.5)(4.5)0.5 + Γ(1.5)p(0.5)(4.5)0.5 = −1.4655. By some calculations, one can get all values of the Green function which we have arranged those in next table.

P

0.5

P

t G(t, 0) G(t, 1) G(t, 2) G(t, 3)

2.5 -1.4655 -1.2560 -1.0048 -0.6699

0.5

0.5

3.5 4.5 -4.4239 -9.4107 -7.6338 -19.2329 -6.1071 -25.1322 -4.0714 -16.7548

0.5

0.5

0.5

5.5 6.5 -16.6908 -23.9512 -36.3505 -53.467 -55.1956 -85.2591 -58.8741 -100.9934

Table 3.1: Values of the Green function Thus, MG0 = 3s=0 |G(t, s)| = 527.9447. Now, dene the non-decreasing function ψ : [0, ∞) → [0, ∞) by ψ(t) = e−1 t. Then ψ ∈ Ψ and P

|h(t, a) − h(t, b)| =

2 2 1 | sin a − sin b| ≤ t+11 |a − b| ≤ t+11 |a − b|. −t−11 cosh(t + 11) e +e e

Since −0.5 ≤ t ≤ 4.5, e10.5 ≤ et+11 ≤ e15.5 and so |h(t, a) − h(t, b)| ≤

2 e10.5

|a − b| ≤

4e0.5

e−1 1 |a − b| = 0 ψ(|a − b|). × 527.9447 4CMG

Hence by using Theorem 2.4, the problem has at least one solution. 7

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By using the provided technique, one can get the next general result. + Suppose that ψ ∈ Ψ has this property that ψ(2m t) ≤ 2m ψ(t) for all t ≥ 0 and p : Nb+ν ν−2 → R is a bounded b+ν b+ν function such that p(t) ≤ C for all t ∈ Nν−2 . Let h : Nν−2 × R → R be a map such that |h(t, a) − h(t, b)| ≤

1 (m−1)

2m CMG

ψ(|a − b|)

ν m m for all a, b ∈ R and t ∈ Nb+ν ν−2 . Then the problem ∆ν−2 (p∆ y)(t) = h(t, p(t + ν − m − 1)∆ y(t + ν − m − 1)) 2 m via the boundary conditions y(ν − 2) = ∆y(ν − 2) = ∆ y(ν − 2) = · · · = ∆ y(ν − 2) = 0 and ∆m y(ν + b) = 0 (m−1) has at least one solution. Here, MG is the bound of the related Green function.

Acknowledgments Research of the second and third authors was supported by Azarbaijan Shahid Madani University.

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calculus with the nabla operator, Elec. J. Qualitative Theory

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discrete fractional equations, Ph.D. Thesis, Ann Arbor, MI,

An introduction to dierence equations, Springer-Verlag (1996). [11] D. J. H. Garling, D. Gorenstein, T. T. Diesk, P. Walters, Topics in metric xed point theory, Cambridge,

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New York (1990).

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existence results for fractional dierence equations, Int. J. Dynamical Syst.

Sum and dierences compositions in discrete fractional calculus, CUBO. A Math. J. 13 (2011) 8

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[15] M. Holm, The theory of discrete fractional calculus: Arbor, MI, University of Nebraska-Lincoln (2011).

development and applications, Ph.D. Thesis, Ann

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dierence equations, Int. J. Di. Eq. (2012) Article

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[20] G.C. Wu, D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dyn. 75, 283287 (2014). [21] G.C. Wu, D. Baleanu, Discrete chaos of fractional sine and standard maps, Phys. Lett. A (2013) DOI: 10.1016/j.physleta.2013.12.010. [22] G.C. Wu, D. Baleanu, Discrete chaos in fractional delayed logistic map,Nonlinear Dyn. DOI: 10.1007/s11071-014-1250-3

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Local shape control of a weighted interpolation surface Jianxun Pan1 Fangxun Bao2,

∗

1 Woman’s Academy at Shandong, Jinan, 250300, China 2 School of Mathematics, Shandong University, Jinan, 250100, China

Abstract This paper proposes a new weighted bivariate blending rational interpolator only based on function values. The interpolation function with some free parameters has simple and explicit mathematical representation, and it is C 1 -continuous for any positive parameters and weighted coefficient. More important, the shape of the interpolating surfaces can be modified by selecting suitable parameters and weighted coefficient. In order to meet the needs of practical design, a local shape control method is employed to control the shape of interpolating surfaces. In the special case, the method of ”Central Point Value Control” is discussed. Keywords: rational spline, bivariate blending interpolation, weighted interpolation, shape control, computer-aided geometric design

1

Introduction

In various applications such as industrial design and manufacture, atmospheric analysis, geology and medical imaging, etc., it is often necessary to generate a smooth surface that interpolates a prescribed set of data. For most applications, C 1 smoothness is generally sufficient. There are many ways to tackle this problem [2, 3, 4, 9, 12, 13, 18], such as the polynomial spline method, the NURBS method and the B`ezier method. These methods are effective and applied widely in shape design of industrial products. However, one of the disadvantages of the polynomial spline method is that curve and surface can not be adjusted locally for unchanged given data. The NURBS and B´ezier methods are the so-called ”no-interpolating type” methods; this means that the constructed curve and surface do not fit with the given data, so the given points play the role of the control points. Here, we are concerned with the C 1 bivariate rational spline interpolations with a simple and explicit mathematical representation, which can be modified by using new parameters. In fact, in recent years, motivated by the univariate rational spline interpolation [1, 5, 10, 11, 14, 15, 17], the C 1 bivariate rational spline, which has a simple and explicit mathematical representation with parameters, has been studied [6, 7, 8, 19]. Since the parameters in the interpolation function are selective according to the control constrains, the constrained control of the shape becomes possible. However, in order to maintain smoothness of these interpolating surfaces, the parameters of ydirection must be restricted, these parameters can not be selected freely for different interpolating ∗

Corresponding author: [email protected]

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PAN AND BAO: SHAPE CONTROL OF SURFACE

subregion. To overcome this problem, in this paper, a new weighted bivariate blending rational spline is constructed, and a local shape control technique of the interpolating surfaces is also developed. This paper is arranged as follows. In Section 2, a weighted bivariate blending rational interpolator only based on the function values is constructed. Section 3 gives the basis and matric representation of this interpolation. In Section 4, the error estimate formula of the interpolation is derived. Section 5 deal with the local shape control of the interpolating surface, in the special case, the ”Central Point Value Control” technique is developed, and numerical examples are presented to show the performance of the method.

2

Interpolation

Let Ω : [a, b; c, d] be the plane region, and {(xi , yi , fi,j , i = 1, 2, · · · , n + 1; j = 1, 2, · · · , m + 1} be a given set of data points, where a = x1 < x2 < · · · < xn+1 = b and c = y1 < y2 < · · · < ym+1 = d are the knot spacings, fi,j represent f (xi , yj ) at the point (xi , yj ). Let hi = xi+1 −xi , and lj = yj+1 −yj , and for any point (x, y) ∈ [xi , xi+1 ; yj , yj+1 ] in the xy-plane, let θ = (x − xi )/hi and η = (y − yj )/lj . (y) (x) Denote ∆i,j = (fi+1,j − fi,j )/hi , ∆i,j = (fi,j+1 − fi,j )/lj . First, for each y = yj , j = 1, 2, · · · , m + 1, construct the x-direction interpolant curve in [xi , xi+1 ]; this is given by pi,j (x) =

(1 − θ)3 αi,j fi,j + θ(1 − θ)2 Vi,j + θ 2 (1 − θ)Wi,j + θ 3 fi+1,j βi,j , i = 1, 2, · · · , n, (1 − θ)2 αi,j + 2θ(1 − θ) + θ 2 βi,j

(1)

where Vi,j = 2fi,j + αi,j fi+1,j , (x)

Wi,j = (2 + βi,j )fi+1,j − βi,j hi ∆i+1,j , with αi,j > 0, βi,j > 0. This interpolation is called the rational cubic interpolation based only on function values which satisfies (x)

(x)

pi,j (xi ) = fi,j , pi,j (xi+1 ) = fi+1,j , p′i,j (xi ) = ∆i,j , p′i,j (xi+1 ) = ∆i+1,j . For each pair of (i, j), i = 1, 2, · · · , n − 1 and j = 1, 2, · · · , m − 1, using the x-direction interpolation function pi,j (x), define the interpolation function P 1 (x, y) on each rectangular region [xi , xi+1 ; yj , yj+1 ] as follows: 1 (x, y) = (1 − η)3 p (x) + η(1 − η)2 V 1 + η 2 (1 − η)W 1 + η 3 p Pi,j i,j i,j+1 (x), i,j i,j

i = 1, 2, · · · , n − 1; j = 1, 2, · · · , m − 1,

(2)

1 = 2p (x) + p 1 where Vi,j i,j i,j+1 (x), Wi,j = 3pi,j+1 (x) − lj (pi,j+2 (x) − pi,j+1 (x))/lj+1 . It is easy to 1 1 prove that Pi,j (x, y) is C -continuous in the interpolating region [x1 , xn ; y1 , ym ], no matter what the parameters αi,s , βi,s (s = j, j + 1, j + 2) might be, and which satisfies 1 Pi,j (xr , ys ) = fr,s ,

1 (x , y ) 1 (x , y ) ∂Pi,j ∂Pi,j r s r s = ∆(x) , = ∆(y) r,s r,s , r = i, i + 1, s = j, j + 1. ∂x ∂y

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PAN AND BAO: SHAPE CONTROL OF SURFACE

The interpolating scheme above begins in x-direction first. Now, let the interpolation function begins with y-direction first. For each x = xi , i = 1, 2, · · · , n+1, denote the y-direction interpolation in [yj , yj+1 ] by p∗i,j (y) =

∗ + η 2 (1 − η)W ∗ + η 3 f (1 − η)3 γi,j fi,j + η(1 − η)2 Vi,j i,j+1 τi,j i,j , j = 1, 2, · · · , m, 2 2 (1 − η) γi,j + 2η(1 − η) + η τi,j

(3)

where ∗ = 2f Vi,j i,j + γi,j fi,j+1 , (y)

∗ = (2 + τ )f Wi,j i,j i,j+1 − lj τi,j ∆i,j+1 ,

with γi,j > 0, τi,j > 0. For each pair (i, j), i = 1, 2, · · · , n − 1 and j = 1, 2, · · · , m − 1, define the bivariate blending 2 (x, y) on [x , x rational interpolation function Pi,j i i+1 ; yj , yj+1 ] as follows: 2 (x, y) = (1 − θ)3 p∗ (y) + θ(1 − θ)2 V 2,∗ + θ 2 (1 − θ)W 2,∗ + θ 3 p∗ Pi,j i,j i+1,j (y), i,j i,j

i = 1, 2, · · · , n − 1; j = 1, 2, · · · , m − 1,

(4)

2,∗ 2,∗ where Vi,j = 2p∗i,j (y) + p∗i+1,j (y), Wi,j = 3p∗i+1,j (y) − hi (p∗i+2,j (y) − p∗i+1,j (y))/hi+1 . 2 (x, y) is C 1 -continuous in the interpolating region Similarly, the interpolation function Pi,j [x1 , xn ; y1 , ym ], no matter what the parameters γr,j , τr,j (r = i, i + 1, i + 2) might be, and which satisfies 2 Pi,j (xr , ys ) = fr,s ,

2 (x , y ) 2 (x , y ) ∂Pi,j ∂Pi,j r s r s = ∆(x) , = ∆(y) r,s r,s , r = i, i + 1, s = j, j + 1. ∂x ∂y

Where, we let 2 1 (x, y), Pi,j (x, y) = λi,j Pi,j (x, y) + (1 − λi,j )Pi,j

(5)

with the weighted coefficient λi,j ∈ [0, 1], then Pi,j (x, y) is called the weighted bivariate blending rational interpolation on [xi , xi+1 ; yj , yj+1 ]. It is obvious that Pi,j (x, y) defined by (5) is C 1 -continuous in the interpolating region [x1 , xn ; y1 , ym ], no matter what the parameters αi,s , βi,s (s = j, j+1, j+2) and γr,j , τr,j (r = i, i + 1, i + 2) might be, and which satisfies Pi,j (xr , ys ) = fr,s ,

∂Pi,j (xr , ys ) ∂Pi,j (xr , ys ) = ∆(x) = ∆(y) r,s , r,s , r = i, i + 1, s = j, j + 1. ∂x ∂y

Example 1. Let the interpolated function be f (x, y) = sin(x2 − xy + y), (x, y) ∈ [0.2, 1.2; 0.2, 1.2]. Thus, the interpolation function of f (x, y) can be defined in [0.2, 1; 0.2, 1] by (5). Let hi = lj = 0.2, αi,j = αi,j+1 = αi,j+2 = 0.4, βi,j = βi,j+1 = βi,j+2 = 0.8, γi,j = γi+1,j = γi+2,j = 0.2, τi,j = τi+1,j = τi+2,j = 0.6, λi,j = 0.5, and P (x, y) is the interpolation function defined in (5). Figure 1 and Figure 2 show the graphs of the surfaces f (x, y) and P (x, y), respectively. Figure 3 is the graph of error function f (x, y) − P (x, y) with αi,s = 0.4, βi,s = 0.8, γr,j = 0.2, τr,j = 0.6, λi,j = 0.5. Figure 4 shows the graph of error function f (x, y) − P (x, y) with αi,s = 1250, βi,s = 325, γr,j = 202, τr,j = 186, λi,j = 0.5. Figure 3 shows that the interpolator defined by (5) has a good approximation to the interpolated function f (x, y). From Figure 3 and Figure 4, it is easy to see that the errors of the interpolation have only a small floating for a big change of the parameters, it means that the weighted bivariate blending rational interpolation defined by (5) is stable for parameters. Thus, we can finely adjust the surface shape by selecting suitable parameters.

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PAN AND BAO: SHAPE CONTROL OF SURFACE

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1 1

0.1 1 0.8

0.8 0.6 0.4 0.2

0.2

0.3

0.5

0.4

0.7

0.6

0.9

0.8

1

0.6 0.4 0.2

Figure 1: Graph of f (x, y).

0.2

0.3

0.4

0.5

0.8

0.7

0.6

0.9

1

Figure 2: Graph of P (x, y).

−3

−3

x 10

x 10

4

5

2

3

0

1 0 −1

−2 −3 −4 −5 −6

−7

−8 0.2

−9 0.2 0.4

0.4 0.6

0.6 0.8 1

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.8 1

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

Figure 3: Graph of f (x, y) − P (x, y) with Figure 4: Graph of f (x, y) − P (x, y) with αi,s = 0.4, βi,s = 0.8, γr,j = 0.2, τr,j = αi,s = 1250, βi,s = 325, γr,j = 202, τr,j = 0.6, λi,j = 0.5. 186, λi,j = 0.5.

3

Basis and matrix representation of the interpolation

In the following of this paper, for the interpolation defined by (5), consider the equally spaced knots case, namely, for all i = 1, 2, · · · , n and j = 1, 2, · · · , m, hi = hj and li = lj . From (1)-(5), the interpolation function defined in (5) can be written as follows: Pi,j (x, y) =

2 X 2 X

ωr,s (θ, η)fi+r,j+s,

(6)

r=0 s=0

where (x,1) (x,1)

+ (1 − λi,j )ai,j bi,j ,

(x,2) (x,1)

+ (1 − λi,j )ai+1,j bi,j ,

(x,3) (x,1)

+ (1 − λi,j )ai+2,j bi,j ,

(x,1) (x,2)

+ (1 − λi,j )ai,j bi,j ,

(x,2) (x,2)

+ (1 − λi,j )ai+1,j bi,j ,

(x,3) (x,2)

+ (1 − λi,j )ai+2,j bi,j ,

ω0,0 (θ, η) = λi,j ai,j bi,j ω1,0 (θ, η) = λi,j ai,j bi,j ω2,0 (θ, η) = λi,j ai,j bi,j

(y,1) (y,1)

(y,1) (y,2)

ω0,1 (θ, η) = λi,j ai,j+1 bi,j ω1,1 (θ, η) = λi,j ai,j+1 bi,j ω2,1 (θ, η) = λi,j ai,j+1 bi,j

71

(y,1) (y,3) (y,2) (y,1)

(y,2) (y,2) (y,2) (y,3)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

PAN AND BAO: SHAPE CONTROL OF SURFACE (x,1) (x,3)

+ (1 − λi,j )ai,j bi,j ,

(x,2) (x,3)

+ (1 − λi,j )ai+1,j bi,j ,

(x,3) (x,3)

+ (1 − λi,j )ai+2,j bi,j ,

(y,3) (y,1)

ω0,2 (θ, η) = λi,j ai,j+2 bi,j

(y,3) (y,2)

ω1,2 (θ, η) = λi,j ai,j+2 bi,j

(y,3) (y,3)

ω2,2 (θ, η) = λi,j ai,j+2 bi,j with

(1 − θ)2 (2θ + (1 − θ)αi,j ) , (1 − θ)2 αi,j + 2θ(1 − θ) + θ 2 βi,j θ(1 − θ)2 αi,j + θ 2 (2 − θ)βi,j + 2θ 2 (1 − θ) (x,2) ai,j = , (1 − θ)2 αi,j + 2θ(1 − θ) + θ 2 βi,j θ 2 (1 − θ)βi,j (x,3) ai,j = − , (1 − θ)2 αi,j + 2θ(1 − θ) + θ 2 βi,j (1 − η)2 (2η + (1 − η)γi,j ) (y,1) ai,j = , (1 − η)2 γi,j + 2η(1 − η) + η 2 τi,j η(1 − η)2 γi,j + η 2 (2 − η)τi,j + 2η 2 (1 − η) (y,2) ai,j = , (1 − η)2 γi,j + 2η(1 − η) + η 2 τi,j η 2 (1 − η)τi,j (y,3) , ai,j = − (1 − η)2 γi,j + 2η(1 − η) + η 2 τi,j (x,1)

=

(x,1)

= bi,j (η) = (1 − η)2 (1 + η),

(x,2)

= bi,j (η) = η(1 + 2η − 2η 2 ),

(x,3)

= bi,j (η) = −η 2 (1 − η),

(y,1)

= bi,j (θ) = (1 − θ)2 (1 + θ),

(y,2)

= bi,j (θ) = θ(1 + 2θ − 2θ 2 ),

(y,3)

= bi,j (θ) = −θ 2 (1 − θ).

ai,j

bi,j bi,j bi,j bi,j

bi,j bi,j

(x,1)

(x,2) (x,3)

(y,1) (y,2) (y,3)

The terms ωr,s (θ, η), (r = 0, 1, 2; s = 0, 1, 2) are called the basis of the bivariate interpolation defined by (5), which satisfy 2 X 2 X

ωr,s(θ, η) = 1.

(7)

r=0 s=0

Denote (x)

ai,j =



(x)

ai,j+2 = (x)

F2

(y) ai,j

= =

(x,1)

ai,j 





(y)

ai+2,j =

Bi,j =



(x)

(x,1)

(x,2)



(x,3)

(y,1)

(y,2)

ai,j

(y,1)

(y,3)

ai,j

(y,2)

ai+2,j ai+2,j





(x)

, F1

,

(y,3)

ai+2,j

(x,1)

(x,2)

bi,j

(x,3)

bi,j

T

T

ai,j+1 ai,j+1 ai,j+1

=





(x)

(x,2)

=



=



(y,1)

(y)

(y)

, Bi,j =

72



(y,2)

(y,3)

ai+1,j

fi,j fi,j+1 fi,j+2 =



(y,1)

bi,j



,

T

,

fi,j+2 fi+1,j+2 fi+2,j+2

ai+1,j ai+1,j

, F3

(x,3)

fi,j fi+1,j fi+2,j



=

(y)

, F1 T

(x,1)

, F3

(y) ai+1,j

fi+1,j fi+1,j+1 fi+1,j+2 bi,j



(x)

, ai,j+1 =

fi,j+1 fi+1,j+1 fi+2,j+1



=

(y)

F2

(x,3)

ai,j

ai,j+2 ai,j+2 ai,j+2

ai,j



(x,2)

ai,j



(y,2)

(y,3)

bi,j )

T

,

,

fi+2,j fi+2,j+1 fi+2,j+2 bi,j

,

,

T T

T

,

Jianxun Pan et al 68-79

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

PAN AND BAO: SHAPE CONTROL OF SURFACE

and let (x)

Ai,j = (x) Fi,j



(x)

ai,j



(x)

F1 0 0

 = 

(x)

(x)

ai,j+1 ai,j+2 0 (x) F2

0

0 0 (x)

F3



(y)

, Ai,j =







   , F (y) =  i,j  

(y)

ai,j

(y)

ai+1,j

(y)

F1 0 0

0 (y) F2

0

(y)

ai+2,j 0 0 (y)

F3





,

 , 

then the interpolation function Pi,j (x, y) defined by (5) can be represented by matrices as follows: (x)

(x)

(y)

(x)

(y)

(y)

Pi,j (x, y) = λi,j Ai,j Fi,j Bi,j + (1 − λi,j )Ai,j Fi,j Bi,j . Furthermore, Denoting M = max{|fi+r,j+s |, r = 0, 1, 2; j = 0, 1, 2}. From (6), it follows that |Pi,j (x, y)| ≤

2 X 2 X

|ωr,s (θ, η)fi+r,j+s | ≤ M

|ωr,s (θ, η)| r=0 s=0 2 2 2 (x,1) X (x,k) (x,2) X (x,k) (x,3) X (x,k) M |λi,j |(|bi,j | |ai,j | + |bi,j | |ai,j+1 | + |bi,j | |ai,j+2 |) k=0 k=0 k=0 2 2 2 (y,1) X (y,k) (y,2) X (y,k) (y,3) X (x,k) +M |(1 − λi,j )|(|bi,j | |ai,j | + |bi,j | |ai+1,j | + |bi,j | |ai+2,j |). k=0 k=0 k=0 r=0 s=0

≤

2 X 2 X

(8)

It is easy to obtain that 2 X

k=0 2 X

(x,k)

|ai,j+s | = 1 + (y,k)

|ai+r,j | = 1 +

k=0 (ξ,1) |bi,j | (ξ,2) |bi,j |

2θ 2 (1 − θ)βi,j ≤ 3, s = 0, 1, 2, (1 − θ)2 αi,j+s + 2θ(1 − θ) + θ 2 βi,j+s 2η 2 (1 − η)τi,j ≤ 3, r = 0, 1, 2, (1 − η)2 γi+r,j + 2η(1 − η) + η 2 τi+r,j

= (1 − ξ)2 (1 + ξ) ≤ 1, ξ = x, y,

= ξ(1 + 2ξ − 2ξ 2 ) ≤ 1, ξ = x, y, 4 (ξ,3) |bi,j | = ξ 2 (1 − ξ) ≤ , ξ = x, y. 27 Thus, the following bounded theorem can be obtained. Theorem 1. For the bivariate blending interpolation Pi,j (x, y) defined in (6), no matter what positive values the parameters αi,s , βi,s and γr,j , τr,j take, the values of Pi,j (x, y) satisfy |Pi,j (x, y)| ≤

73

58 M. 9

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PAN AND BAO: SHAPE CONTROL OF SURFACE

4

Error estimates of the interpolation

For the error estimation of the interpolator defined in (6), note that the interpolator is local, without loss of generality, it is only necessary to consider the interpolating region [xi , xi+1 ; yj , yj+1 ] in order to process its error estimates. Let f (x, y) ∈ C 2 be the interpolated function, and Pi,j (x, y) is the interpolation function defined by (6) over [xi , xi+1 ; yj , yj+1 ]. Denoting k

∂P ∂f (x, y) ∂Pi,j (x, y) ∂f k = max | |, k k = max | |, ∂y ∂y ∂y ∂y (x,y)∈D (x,y)∈D

where D = [xi , xi+1 ; yj , yj+1 ]. By the Taylor expansion and the Peano-Kernel Theorem [16] gives the following: |f (x, y) − Pi,j (x, y)| ≤ |f (x, y) − f (x, yj )| + |Pi,j (x, yj ) − Pi,j (x, y)| + |f (x, yj ) − Pi,j (x, yj )| Z xi+2 2 ∂P ∂ f (τ, yj ) ∂f k) + | Rx [(x − τ )+ ]dτ | ≤ lj (k k + k ∂y ∂y ∂x2 xi Z xi+2 ∂f ∂P ∂ 2 f (x, yj ) ≤ lj (k k + k k) + k k |Rx [(x − τ )+ ]|dτ, ∂y ∂y ∂x2 xi where k and

∂ 2 f (x,yj ) k ∂x2

Rx1 [(x

= maxx∈[xi ,xi+1 ] |

− τ )+ ] =

=

=

Rx [(x−τ )+ ] = λi,j Rx1 [(x−τ )+ ]+(1−λi,j )Rx2 [(x−τ )+ ],

 (x,3) (x,2)  (x − τ ) − ai,j (xi+1 − τ ) − ai,j (xi+2 − τ ),            

Rx2 [(x − τ )+ ] =

∂ 2 f (x,yj ) |, ∂x2

(x,2) (x,3) −ai,j (xi+1 − τ ) − ai,j (xi+2 − τ ), (x,3) −ai,j (xi+2 − τ ), xi+1 < τ < xi+2 , rx1 (τ ), xi < τ < x, x < τ < xi+1 , s1x (τ ), 1 tx (τ ), xi+1 < τ < xi+2 ,

(y,2)

(x

− τ) − b

(y,3)

(x

− τ ),

i+1 i+2 i,j i,j     −b(y,3) (x xi+1 < τ < xi+2 , i+2 − τ ), i,j  2 xi < τ < x,   rx (τ ),

 

s2x (τ ),

x < τ < xi+1 ,

t2x (τ ),

xi+1 < τ < xi+2 ,

xi < τ < x,

x < τ < xi+1 ,

 (y,2) (y,3)  (x − τ ) − bi,j (xi+1 − τ ) − bi,j (xi+2 − τ ),   

−b

(9)

xi < τ < x,

x < τ < xi+1 ,

Thus, by simple integral calculation, it can be derived that Z

xi+2 xi

|Rx1 [(x − τ )+ ]|dτ = h2i B1 (αi,j , βi,j , θ),

where 2 θ 3 (1 − θ)2 βi,j 1 , B1 (αi,j , βi,j , θ) = θ(1 − θ) + 2 U (αi,j , βi,j , θ)

74

(10)

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PAN AND BAO: SHAPE CONTROL OF SURFACE

with U (αi,j , βi,j , θ) = ((1 − θ)2 αi,j + 2θ(1 − θ) + θ 2 βi,j )((1 − θ)2 αi,j + 2θ(1 − θ) + θβi,j ); Z

xi+2

xi

|Rx2 [(x − τ )+ ]|dτ = h2i B2 (θ),

where B2 (θ) =

θ(1 − 3θ 3 + 2θ 4 ) . 2(1 + θ − θ 2 )

(11)

Let (x)

Bi,j = max {λi,j B1 (αi,j , βi,j , θ) + (1 − λi,j )B2 (θ)}.

(12)

θ∈[0,1]

This leads to the following theorem. Theorem 2. Let f (x, y) ∈ C 2 be the interpolated function, and Pi,j (x, y) be its interpolator defined by (6) in [xi , xi+1 ; yj , yj+1 ]. Whatever the positive values of the parameters αi,s , βi,s and γr,j , τr,j might be, the error of the interpolation satisfies |f (x, y) − Pi,j (x, y)| ≤ lj (k

∂P ∂ 2 f (x, yj ) (x) ∂f k+k k) + h2i k kBi,j , ∂y ∂y ∂x2

(x)

where Bi,j defined by (12). It is easy to derive that B1 (αi,j , βi,j , θ) ≤ 1; B2 (θ) ≤ 0.150902. Thus, we have the following theorem. (x)

Theorem 3. For any positive parameters αi,s , βi,s , γr,j , τr,j and weight coefficient λi,j , Bi,s are bounded, and (x)

0 ≤ Bi,j ≤ 1. ∂ 2 f (x,y

∂ 2 f (x,y

)

)

j+1 j+1 k = maxx∈[xi ,xi+1 ] | |, then the following theorem holds. Similarly, denoting k ∂x2 ∂x2 2 Theorem 4. Let f (x, y) ∈ C be the interpolated function, and Pi,j (x, y) be its interpolator defined by (6) in [xi , xi+1 ; yj , yj+1 ]. Whatever the positive values of the parameters αi,s , βi,s and γr,j , τr,j might be, the error of the interpolation satisfies

|f (x, y) − Pi,j (x, y)| ≤ lj (k

∂P ∂ 2 f (x, yj+1 ) (x) ∂f k+k k) + h2i k kBi,j+1 , ∂y ∂y ∂x2

(x)

where Bi,j+1 = maxθ∈[0,1] {λi,j B1 (αi,j+1 , βi,j+1 , θ)+(1−λi,j )B2 (θ)}, B1 (αi,j , βi,j , θ) defined by (10). Note that the interpolator defined by (6) is symmetric about two variables x and y, we denote 2 (x ,y) 2 (x ,y) ∂Pi,j (x,y) (x,y) ∂f r r |, k ∂P |, and k ∂ f∂y k = maxy∈[yj ,yj+1 ] | ∂ f∂y |, k ∂x k = max(x,y)∈D | ∂f∂x 2 2 ∂x k = max(x,y)∈D | ∂x r = 1, 2, then the following theorem can be obtained.

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PAN AND BAO: SHAPE CONTROL OF SURFACE

Theorem 5. Let f (x, y) ∈ C 2 be the interpolated function, and Pi,j (x, y) be its interpolator defined by (6) in [xi , xi+1 ; yj , yj+1 ]. Whatever the positive values of the parameters αi,s , βi,s and γr,j , τr,j might be, the error of the interpolation satisfies |f (x, y) − Pi,j (x, y)| ≤ hi (k

∂P ∂ 2 f (xr , y) (y) ∂f k+k k) + lj2 k kBr,j , r = 1, 2, ∂x ∂x ∂y 2

(y)

where Br,j = maxη∈[0,1] {λi,j B1 (γr,j , τr,j , η) + (1 − λi,j )B2 (η)}, B1 (αi,j , βi,j , θ) is defined by (10), B2 (θ) is defined by (11).

5

Local shape control of the interpolation

The shape of the interpolating surface on the interpolating region depends on the interpolating data. Generally speaking, when the interpolating data are given, the shape of interpolating surface is fixed. However, for the interpolation defined by (6), since there are parameters and weighted coefficient, the shape of the interpolating surface can be modified. For any point (x, y) in the interpolating region [xi , xi+1 ; yj , yj+1 ], let (θ, η) be its local coordinate. We consider the following local point control issue: the practical design requires the value of the interpolation function at the point (x, y) be equal to a real number M , where M ∈ (min{fi+r,j+s , r = 0, 1; j = 0, 1}, max{fi+r,j+s, r = 0, 1; j = 0, 1}). Thus 2 X 2 X

ωr,s (θ, η)fi+r,j+s = M,

(13)

r=0 s=0

where the local coordinate (θ, η) is given, for the given weighted coefficient λi,j , only the parameters αi,s , βi,s (s = j, j + 1, j + 2) and γr,j , τr,j (r = i, i + 1, i + 2) are unknown. If there exist parameters αi,s , βi,s and γr,j , τr,j which satisfy Eq. (13), then the problem is solved. For example, the central point is more important than other points for shape control of the interpolating surface, we consider the ”Central Point Value Control” problem. Assume αi,j = αi,j+1 = αi,j+2 , βi,j = βi,j+1 = βi,j+2 , and denote by αi and βi respectively; assume γi,j = γi+1,j = γi+2,j , τi,j = τi+1,j = τi+2,j , and denote by γj and τj respectively. Let the weighted coefficient λi,j = 12 . Since (x, y) is the central point in [xi , xi+1 ; yj , yj+1 ], θ = 12 , η = 12 . In this case, Eq. (13) becomes 2(k1 − 32M )αi + 2(k2 − 32M )βi + 2(k1 − 32M )γj + 2(k3 − 32M )τj +k1 αi γj + k3 αi τj + k2 βi γj + k4 βi τj + k0 = 0,

(14)

where k1 = 6fi,j + 9fi,j+1 − fi,j+2 + 9fi+1,j + 12fi+1,j+1 − fi+1,j+2 − fi+2,j − fi+2,j+1 , k2 = 3fi,j + 3fi,j+1 + 15fi+1,j + 24fi+1,j+1 − 3fi+1,j+2 − 4fi+2,j − 7fi+2,j+1 + fi+2,j+2 , k3 = (3fi,j + 15fi,j+1 − 4fi,j+2 + 3fi+1,j + 24fi+1,j+1 − 7fi+1,j+2 − 3fi+2,j+1 + fi+2,j+2 ), k4 = (9fi,j+1 − 3fi,j+2 + 9fi+1,j + 36fi+1,j+1 − 9fi+1,j+2 − 3fi+2,j − 9fi+2,j+1 + 2fi+2,j+2 ), k0 = 24fi,j + 36fi,j+1 − 4fi,j+2 + 36fi+1,j + 48fi+1,j+1 − 4fi+1,j+2 − 4fi+2,j − 4fi+2,j+1 − 128M. If there exist positive solutions αi , βi and γj , τj which satisfy Eq. (14), then the interpolation Pi,j (x, y) defined in (6) with these parameters is the solution of ”Central Point Value Control” problem. This can be stated in the following theorem.

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Theorem 6. The sufficient condition for ”Central Point Value Control” problem having solution is that the number of the variation sign of the sequence {ki , i = 0, 1, · · · , 4; kj − 32M, j = 1, 2, 3} are not equal to zero. Example 2. Let Ω : [0, 2; 0, 2] be plane region, and the interpolating data are given in Table 1. Table 1. The interpolating data (xi , yj ) (0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0) (2, 1) (2, 2) fi,j 2.5 1.5 1 2 1 0.5 1 1.5 0.5 Then the interpolation function Pi,j (x, y) defined by (5) can be constructed in [0, 1; 0, 1] for the given positive parameters αi,s , βi,s , γr,j , τr,j and the weighted coefficient λi,j . In general case, one can select any positive real numbers as the values of αi,s , βi,s and γr,j , τr,j . Let λi,j = 12 and αi = 0.5, βi = 0.8, γj = 0.4, τj = 0.6, for the given interpolation data, denote the interpolation function by P1 (x, y) which is defined over [0, 1; 0, 1]. Figure 5 shows the graph of the weighted bivariate blending rational interpolation surface P1 (x, y). In this case, the function value of central point is that P1 ( 12 , 12 ) = 17269 10560 = 1.63532 · · ·.

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8 0

0.8 0

1 0.8

0.2

1

1 0.4

0.6

0.2

0.8

0.6

0.4

0.4

0.6

0.8

0.2

0.6

0.4

0.2

0.8

0

1

Figure 5: Graph of P1 (x, y).

0

Figure 6: Graph of P2 (x, y).

From the ”Central Point Value Control” equation (13), the value of central point can be controlled through modifying the parameters. For example, if the practical design requires P ( 12 , 12 ) = 1.65. It is easy to test that αi = 0.5, βi = 55 58 , γj = 0.25 and τj = 0.25 satisfy Eq. (14). Denote the interpolation by P2 (x, y). Figure 6 shows the graph of the surface P2 (x, y). Furthermore, if the practical design requires P ( 12 , 12 ) = 1.6. It is easy to examine that αi = 0.5, βi = 45 62 , γj = 0.25 and τj = 1.5 satisfy Eq. (14). Denote the interpolation by P3 (x, y). Figure 7 shows the graph of the surface P3 (x, y).

6

Concluding remarks

An explicit representation of a weighted bivariate blending rational interpolator is presented. The interpolation function is C 1 -continuous in the interpolating region [x1 , xn ; y1 , ym ]. In this interpolation, there are some parameters αi,j , βi,j , γi,j , τi,j and weighted coefficient λi,j . These positive parameters and weighted coefficient can be freely selected according to the needs of practical design.

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2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0

0.2

0.4

0.6

0.4 0.8

1

0

0.6

0.8

1

0.2

Figure 7: Graph of P3 (x, y). The local shape control method of the interpolating surfaces is also developed in this paper. And from (13), there are varied expressions, so manifold control scheme can be derived similar to the process for ”Central Point Value Control” problem. Acknowledgment This research were supported by the National Nature Science Foundation of China (No.61070096) and the Natural Science Foundation of Shandong Province (No.ZR2012FL05).

References [1] F.X. Bao, Q.H. Sun, J.X. Pan, Q. Duan, A blending interpolator with value control and minimal strain energy, Comput & Graph. 34(2010):119-124. [2] P.E. B´ezier, The Mathematical Basis of The UNISURF CAD System, Butterworth, London; 1986. [3] C.K. Chui, Multivariate Spline, SIAM; 1988. [4] P. Dierck, B. Tytgat, Generating the B´ezier points of b-spline curve, Comput Aided Geom Des. 6(1989):279-291. [5] Q. Duan, F. Bao, S. Du, E.H. Twizell, Local control of interpolating rational cubic spline curves, Comput. Aided Des. 41(2009)825-829. [6] Q. Duan, Y.F. Zhang, E.H. Twizell, A bivariate rational interpoaltion and the properties, Apll Math Comput. 179(2006):190-199. [7] Q. Duan, H.L. Zhang, A.K. Liu, H. Li, A bivariate rational interpolation with a bi-quadratic denominator, J Comp Appl Math. 195(2006):24-33. [8] Q. Duan, S. Li, F. Bao, E.H. Twizell, Hermite interpolation by piecewise rational surface, Appl. Math. Comput. 198(2008)59-72. [9] G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, Academic press; 1988.

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[10] J.A. Gregory, M. Sarfraz, P.K. Yuen, Interactive curve design using C 2 rational spline, Comput. & Graph. 18(1994):153-159. [11] J.L. Merrien, P. Sablonni´ere, Rational splines for Hermite interpolation with shape constraints, Comput. Aided Geom. Des. 30(2013):296-309. [12] R. M¨ uller, Universal parametrization and interpolation on cubic surfaces, Comput Aided Geom Des. 19(2002):479-502. [13] L. Piegl, On NURBS: A survey, IEEE Comput Graph Appl. 11(1991):55-71. [14] M. Sarfraz, M.Z. Hussain, M. Hussain, Shape-preserving curve interpolation , Int J Comput Math. 89(2012):35-53. [15] J.W. Schmidt, W. Hess, Positive interpolation with rational quadratic spline, Computing. 38(1987):261-267. [16] M.H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliffs, New Jersey; 1973. [17] L.L. Schumaker, On shape preserving quadratic spline interpolation, SIAM J. Num. Anal. 20(1983)854-864. [18] R.H. Wang, Multivariate Spline Functions and Their Applications, Kluwer Academic Publishers, Beijing/New York/Dordrecht/ Boston/London; 2001. [19] Y.F. Zhang, Q. Duan, E.H. Twizell, Convexity control of a bivariate rational interpolating spline surfaces, Comput & Graph. 31(2007):679-687.

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GENERALIZED ADDITIVE FUNCTIONAL INEQUALITIES IN C ∗ -ALGEBRAS JUNG RYE LEE AND DONG YUN SHIN∗ Abstract. In this paper, we investigate isomorphisms and derivations in real C ∗ algebras associated with the following generalized additive functional inequality ∥af (x) + bf (y) + cf (z)∥ ≤ ∥f (αx + βy + γz)∥.

(0.1)

We moreover prove the Hyers-Ulam stability of homomorphisms and derivations in real C ∗ -algebras associated with the generalized additive functional inequality (0.1).

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [50] concerning the stability of group homomorphisms. Hyers [12] gave a first affirmative answer to the question of Ulam for Banach spaces. Th.M. Rassias [38] provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. Theorem 1.1. [38] 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)∥ ≤ θ(∥x∥p + ∥y∥p )

(1.1)

for all x, y ∈ E, where θ and p are positive real numbers with p < 1. Then there exists an unique additive mapping L : E → E ′ satisfying ∥f (x) − L(x)∥ ≤

2θ ∥x∥p 2 − 2p

for all x ∈ E. Also, if for each x ∈ E the mapping f (tx) is continuous in t ∈ R, then L is R-linear. Th.M. Rassias [39] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. Gajda [8] following the same approach as in Th.M. Rassias [38], gave an affirmative solution to this question for p > 1. It was shown by Gajda [8], as well as by Th.M. Rassias ˇ and Semrl [45] that one cannot prove a Th.M. Rassias’ type theorem when p = 1. ˇ The counterexamples of Gajda [8], as well as of Th.M. Rassias and Semrl [45] have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings, cf. G˘avruta [9], who among others studied the 2010 Mathematics Subject Classification. Primary 39B72; Secondary 46L05. Key words and phrases. Hyers-Ulam stability, generalized additive functional inequality, C ∗ algebra homomorphism, derivation. ∗ Corresponding author.

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Hyers-Ulam stability of functional equations (cf. the books of Czerwik [3, 4], Hyers, Isac and Th.M. Rassias [13]). G˘avruta [9] provided a further generalization of Th.M. Rassias’ Theorem. Isac and Th.M. Rassias [16] applied the Hyers-Ulam stability theory to prove fixed point theorems and study some new applications in Nonlinear Analysis. In [14], Hyers, Isac and Th.M. Rassias studied the asymptoticity aspect of Hyers-Ulam stability of mappings. Several papers have been published on various generalizations and applications of Hyers-Ulam stability to a number of functional equations and mappings (see [1, 5, 6, 18, 19, 20, 21, 23, 24, 26, 28, 29, 30, 32, 34, 40, 41, 42, 43, 47, 48, 49]). Beginning around the year 1980, the topic of approximate homomorphisms and their stability theory in the field of functional equations and inequalities were taken up by several mathematicians (cf. Hyers and Th.M. Rassias [15], Th.M. Rassias [42] and the references therein). Definition 1.2. Let A and B be real C ∗ -algebras. (i) An R-linear mapping H : A → B is called a C ∗ -algebra homomorphism if H(xy) = H(x)H(y) and H(x∗ ) = H(x)∗ for all x, y ∈ A. If a C ∗ -algebra homomorphism H : A → B is bijective, then the C ∗ -algebra homomorphism H : A → B is called a C ∗ -algebra isomorphism. (ii) An R-linear mapping D : A → A is called a derivation if D(xy) = D(x)y+xD(y) for all x, y ∈ A. A real C ∗ -algebra A, endowed with the Lie product [x, y] := Lie C ∗ -algebra (see [25, 27, 33]).

xy−yx 2

on A, is called a

Definition 1.3. Let A and B be Lie C ∗ -algebras. (i) An R-linear mapping H : A → B is called a Lie ∗-homomorphism if H([x, y]) = [H(x), H(y)] and H(x∗ ) = H(x)∗ for all x, y ∈ A. If a Lie ∗-homomorphism H : A → B is bijective, then the Lie ∗-homomorphism H : A → B is called a Lie ∗-isomorphism. (ii) An R-linear mapping D : A → A is called a Lie derivation if D([x, y]) = [x, D(y)] + [D(x), y] for all x, y ∈ A. Gil´anyi [10] showed that if f satisfies the functional inequality ∥2f (x) + 2f (y) − f (x − y)∥ ≤ ∥f (x + y)∥

(1.2)

then f satisfies the quadratic functional equation 2f (x) + 2f (y) = f (x + y) + f (x − y). See also [46]. Fechner [7] and Gil´anyi [11] proved the Hyers-Ulam stability of the functional inequality (1.2). Park, Cho and Han [31] investigated the Jordan-von Neumann type Cauchy-Jensen additive mappings and prove their stability, and Cho and Kim [2] proved the Hyers-Ulam stability of the Jordan-von Neumann type Cauchy-Jensen additive mappings. The purpose of this paper is to prove the Hyers-Ulam stability of generalized additive functional inequalities associated with homomorphisms and derivations in C ∗ -algebras. In Section 2, we investigate isomorphisms in C ∗ -algebras associated with the functional inequality (0.1). In Section 3, we investigate derivations on C ∗ -algebras associated with the functional inequality (0.1).

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In Section 4, we prove the Hyers-Ulam stability of homomorphisms in C ∗ -algebras associated with the functional inequality (0.1). In Section 5, we prove the Hyers-Ulam stability of derivations on C ∗ -algebras associated with the functional inequality (0.1). Throughout this paper, we assume that a, b, c, α, β, γ are nonzero real numbers. Let A be a real C ∗ -algebra with C ∗ -norm ∥ · ∥A , and let B be a real C ∗ -algebra with C ∗ -norm ∥ · ∥B . 2. Isomorphisms in C ∗ -algebras Consider a mapping f : A → B satisfying the following functional inequality ∥af (x) + bf (y) + cf (z)∥B ≤ ∥f (αx + βy + γz)∥B

(2.1)

for all x, y, z ∈ A. In this section, we investigate isomorphisms in C ∗ -algebras associated with the functional inequality (2.1). Theorem 2.1. Let p ̸= 1 and θ be nonnegative real numbers, and let f : A → B be a nonzero bijective mapping satisfying (2.1) and limt∈R,t→0 f (tx) = 0 for all x ∈ A such that 2p ∥f (xy) − f (x)f (y)∥B ≤ θ(∥x∥2p A + ∥y∥A ), ∥f (x∗ ) − f (x)∗ ∥B ≤ 2θ∥x∥pA

(2.2) (2.3)

for all x, y ∈ A. Then the bijective mapping f : A → B is a C ∗ -algebra isomorphism. Proof. By Theorem 2.7 of [22], the mapping f : A → B is R-linear. (i) Assume that p < 1. By (2.2), 1 ∥f (xy) − f (x)f (y)∥B = lim n ∥f (4n xy) − f (2n x)f (2n y)∥B n→∞ 4 4np 2p ≤ lim n θ(∥x∥2p A + ∥y∥A ) = 0 n→∞ 4 for all x, y ∈ A. So f (xy) = f (x)f (y) for all x, y ∈ A. By (2.3), 1 2 · 2np ∗ ∗ n ∗ n ∗ ∥f (x ) − f (x) ∥B = lim n ∥f (2 x ) − f (2 x) ∥B ≤ lim θ∥x∥2p A = 0 n n→∞ 2 n→∞ 2 for all x ∈ A. So f (x∗ ) = f (x)∗ for all x ∈ A. (ii) Assume that p > 1. By a similar method to the proof of the case (i), one can prove that the mapping f : A → B satisfies f (xy) = f (x)f (y), f (x∗ ) = f (x)∗ for all x, y ∈ A. Therefore, the bijective mapping f : A → B is a C ∗ -algebra isomorphism.

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Theorem 2.2. Let p ̸= 1 and θ be nonnegative real numbers, and let f : A → B be a nonzero bijective mapping satisfying (2.1), (2.3) and limt∈R,t→0 f (tx) = 0 for all x ∈ A such that ∥f (xy) − f (x)f (y)∥B ≤ θ · ∥x∥pA · ∥y∥pA

(2.4)

for all x, y ∈ A. Then the bijective mapping f : A → B is a C ∗ -algebra isomorphism. Proof. By Theorem 2.7 of [22], the mapping f : A → B is R-linear. (i) Assume that p < 1. By (2.4), 1 ∥f (xy) − f (x)f (y)∥B = lim n ∥f (4n xy) − f (2n x)f (2n y)∥B n→∞ 4 4np ≤ lim n θ · ∥x∥pA · ∥y∥pA = 0 n→∞ 4 for all x, y ∈ A. So f (xy) = f (x)f (y) for all x, y ∈ A. (ii) Assume that p > 1. By a similar method to the proof of the case (i), one can prove that the mapping f : A → B satisfies f (xy) = f (x)f (y) for all x, y ∈ A. The rest of the proof is similar to the proof of Theorem 2.1.



Remark 2.3. If we replace ∥f (xy) − f (x)f (y)∥B by ∥f ([x, y]) − [f (x), f (y)]∥B in the statements of Theorem 2.1 and Theorem 2.2, then we can show that f : A → B is a Lie ∗-isomorphism instead of a C ∗ -algebra isomorphism, respectively. 3. Derivations on C ∗ -algebras Consider a mapping f : A → A satisfying the following functional inequality ∥af (x) + bf (y) + cf (z)∥A ≤ ∥f (αx + βy + γz)∥A

(3.1)

for all x, y, z ∈ A. In this section, we investigate derivations on C ∗ -algebras associated with the functional inequality (3.1). Theorem 3.1. Let p ̸= 1 and θ be nonnegative real numbers, and let f : A → A be a nonzero mapping satisfying (3.1) and limt∈R,t→0 f (tx) = 0 for all x ∈ A such that 2p ∥f (xy) − f (x)y − xf (y)∥A ≤ θ(∥x∥2p A + ∥y∥A )

(3.2)

for all x, y ∈ A. Then the mapping f : A → A is a derivation on a C ∗ -algebra. Proof. By Theorem 2.7 of [22], the mapping f : A → A is R-linear. (i) Assume that p < 1. By (3.2), 1 ∥f (xy) − f (x)y − xf (y)∥A = lim n ∥f (4n xy) − f (2n x) · 2n y − 2n x · f (2n y)∥A n→∞ 4 4np 2p ≤ n→∞ lim n θ(∥x∥2p A + ∥y∥A ) = 0 4

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for all x, y ∈ A. So f (xy) = f (x)y + xf (y) for all x, y ∈ A. (ii) Assume that p > 1. By a similar method to the proof of the case (i), one can prove that the mapping f : A → A satisfies f (xy) = f (x)y + xf (y) for all x, y ∈ A. Therefore, the mapping f : A → A is a derivation on a C ∗ -algebra, as desired.



Theorem 3.2. Let p ̸= 1 and θ be nonnegative real numbers, and let f : A → A be a nonzero mapping satisfying (3.1) and limt∈R,t→0 f (tx) = 0 for all x ∈ A such that ∥f (xy) − f (x)y − xf (y)∥A ≤ θ · ∥x∥pA · ∥y∥pA

(3.3)

for all x, y ∈ A. Then the mapping f : A → A is a derivation on a C ∗ -algebra. Proof. By Theorem 2.7 of [22], the mapping f : A → A is R-linear. (i) Assume that p < 1. By (3.3), 1 ∥f (4n xy) − f (2n x) · 2n y − 2n x · f (2n y)∥A n→∞ 4n 4np ≤ lim n θ · ∥x∥pA · ∥y∥pA = 0 n→∞ 4

∥f (xy) − f (x)y − xf (y)∥A =

lim

for all x, y ∈ A. So f (xy) = f (x)y + xf (y) for all x, y ∈ A. (ii) Assume that p > 1. By a similar method to the proof of the case (i), one can prove that the mapping f : A → A satisfies f (xy) = f (x)y + xf (y) for all x, y ∈ A. Therefore, the mapping f : A → A is a derivation on a C ∗ -algebra, as desired.



Remark 3.3. If we replace ∥f (xy) − f (x)y − xf (y)∥A by ∥f ([x, y]) − [f (x), y] − [x, f (y)]∥A in the statements of Theorem 3.1 and Theorem 3.2, then we can show that f : A → B is a Lie derivation instead of a derivation on a C ∗ -algebra, respectively. 4. Stability of homomorphisms in C ∗ -algebras In this section, we prove the Hyers-Ulam stability of homomorphisms in C ∗ -algebras. Theorem 4.1. Let f : A → B be a mapping satisfying limt∈R,t→0 f (tx) = 0 for all x ∈ A. When |α| > |β| and 0 < p < 1, or |α| < |β| and p > 1, if there exists a θ ≥ 0 satisfying (2.2) and (2.3) such that ∥αf (x) + βf (y) + γf (z)∥B ≤ ∥f (αx + βy + γz)∥B + θ(∥x∥pA + ∥y∥pA + ∥z∥pA )

84

(4.1)

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for all x, y, z ∈ A, then there exists a unique C ∗ -algebra homomorphism H : A → B satisfying θ(|α|p + |β|p ) ∥x∥pA ∥f (x) − H(x)∥B ≤ p−1 p−1 |α||β|(|β| − |α| )

(4.2)

for all x ∈ A. Proof. Let ξ = − αβ . By Corollary 3.7 of [22], there exists a unique R-linear mapping H : A → B satisfying (4.2). The mapping H : A → B is defined by H(x) := n limn→∞ f (ξξn x) for all x ∈ A. By (2.2), 1 ∥f (ξ 2n xy) − f (ξ n x)f (ξ n y)∥B 2n ξ ξ 2np 2p ≤ n→∞ lim 2n θ(∥x∥2p A + ∥y∥A ) = 0 ξ

∥H(xy) − H(x)H(y)∥B =

lim

n→∞

for all x, y ∈ A. So H(xy) = H(x)H(y) for all x, y ∈ A. By (2.3), 1 ξ np n ∗ n ∗ ∥f (ξ x ) − f (ξ x) ∥ ≤ lim 2θ · ∥x∥pA = 0 B n→∞ ξ n n→∞ ξ n

∥H(x∗ ) − H(x)∗ ∥B = lim for all x ∈ A. So

H(x∗ ) = H(x)∗ for all x ∈ A. Therefore, the mapping H : A → B is a unique C ∗ -algebra homomorphism, as desired.  We prove another stability of generalized additive functional inequalities. Theorem 4.2. Let f : A → B be a mapping satisfying limt∈R,t→0 f (tx) = 0 for all x ∈ A. When |α| > |β| and p > 1, or |α| < |β| and 0 < p < 1, if there exists a θ ≥ 0 satisfying (2.2), (2.3) and (4.1), then there exists a unique C ∗ -algebra homomorphism H : A → B satisfying ∥f (x) − H(x)∥B ≤

θ(|α|p + |β|p ) ∥x∥pA |α||β|(|α|p−1 − |β|p−1 )

(4.3)

for all x ∈ A. Proof. Let ξ = − αβ . By Corollary 3.10 of [22], there exists a unique R-linear mapping H : A → B satisfying (4.3). The mapping H : A → B is defined by H(x) := limn→∞ ξ n f ( ξxn ) for all x ∈ A.

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By (2.2), ∥H(xy) − H(x)H(y)∥B =

lim ξ

2n

n→∞

( ) ( ) ( )

xy x y

− f f

f

ξ 2n ξn ξn

B

ξ 2n 2p ≤ lim 2np θ(∥x∥2p A + ∥y∥A ) = 0 n→∞ ξ for all x, y ∈ A. So H(xy) = H(x)H(y) for all x, y ∈ A. By (2.3), ∥H(x∗ ) − H(x)∗ ∥B

( ) ( )∗ ∗

x x

− f = lim ξ n f

n→∞

ξn ξn

B

for all x ∈ A. So

ξn 2θ · ∥x∥pA = 0 n→∞ ξ np

≤ lim

H(x∗ ) = H(x)∗

for all x ∈ A. Therefore, the mapping H : A → B is a unique C ∗ -algebra homomorphism, as desired.  Remark 4.3. If we replace ∥f (xy) − f (x)f (y)∥B by ∥f ([x, y]) − [f (x), f (y)]∥B in the statements of Theorem 4.1 and Theorem 4.2, then we can show that there is a unique Lie ∗-homomorphism instead of a unique C ∗ -algebra homomorphism, respectively. 5. Stability of derivations on C ∗ -algebras In this section, we prove the Hyers-Ulam stability of derivations on C ∗ -algebras. Theorem 5.1. Let f : A → A be a mapping satisfying limt∈R,t→0 f (tx) = 0 for all x ∈ A. When |α| > |β| and 0 < p < 1, or |α| < |β| and p > 1, if there exists a θ ≥ 0 satisfying (3.2) such that ∥αf (x) + βf (y) + γf (z)∥A ≤ ∥f (αx + βy + γz)∥A + θ(∥x∥pA + ∥y∥pA + ∥z∥pA ) for all x, y, z ∈ A, then there exists a unique derivation D : A → A satisfying θ(|α|p + |β|p ) ∥f (x) − D(x)∥A ≤ ∥x∥pA |α||β|(|β|p−1 − |α|p−1 ) for all x ∈ A.

(5.1)

(5.2)

Proof. Let ξ = − αβ . By Corollary 3.7 of [22], there exists a unique R-linear mapping D : n

A → A satisfying (5.2). The mapping D : A → A is defined by D(x) := limn→∞ f (ξξn x) for all x ∈ A. By (3.2), 1 ∥D(xy) − D(x)y − xD(y)∥A = lim 2n ∥f (ξ 2n xy) − f (ξ n x) · ξ n y − ξ n x · f (ξ n y)∥A n→∞ ξ ξ 2np 2p ≤ lim 2n θ(∥x∥2p A + ∥y∥A ) = 0 n→∞ ξ

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for all x, y ∈ A. So D(xy) = D(x)y + xD(y) for all x, y ∈ A. Therefore, the mapping D : A → B is a unique derivation, as desired.



We prove another stability of generalized additive functional inequalities. Theorem 5.2. Let f : A → B be a mapping satisfying limt∈R,t→0 f (tx) = 0 for all x ∈ A. When |α| > |β| and p > 1, or |α| < |β| and 0 < p < 1, if there exists a θ ≥ 0 satisfying (3.2) and (5.1), then there exists a unique derivation D : A → A satisfying θ(|α|p + |β|p ) ∥f (x) − D(x)∥A ≤ ∥x∥pA (5.3) |α||β|(|α|p−1 − |β|p−1 ) for all x ∈ A. Proof. Let ξ = − αβ . By Corollary 3.10 of [22], there exists a unique R-linear mapping D : A → A satisfying (5.3). The mapping D : A → A is defined by D(x) := limn→∞ ξ n f ( ξxn ) for all x ∈ A. By (3.2), ∥D(xy) − D(x)y − xD(y)∥A =

n→∞

≤

lim n→∞

lim ξ

2n

( ) ( ) ( )

x x y

y xy

− · f − f ·

f

ξ 2n ξn ξn ξn ξ n A

ξ 2n 2p θ(∥x∥2p A + ∥y∥A ) = 0 ξ 2np

for all x, y ∈ A. So D(xy) = D(x)y + xD(y) for all x, y ∈ A. Therefore, the mapping D : A → A is a unique derivation, as desired.



Remark 5.3. If we replace ∥f (xy) − f (x)y − xf (y)∥A by ∥f ([x, y]) − [f (x), y] − [x, f (y)]∥A in the statements of Theorem 5.1 and Theorem 5.2, then we can show that there is a unique Lie derivation instead of a unique derivation on a C ∗ -algebra, respectively. Acknowledgments 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] C. Baak, Cauchy-Rassias stability of Cauchy–Jensen additive mappings in Banach spaces, Acta Math. Sinica 22 (2006), 1789–1796. [2] Y. Cho and H. Kim, Stability of functional inequalities with Cauchy-Jensen additive mappings, Abstr. Appl. Math. 2007 (2007), Art. ID 89180, 13 pages. [3] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong, 2002. [4] S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Florida, 2003.

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[32] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [33] C. Park, J. Hou and S. Oh, Homomorphisms between JC ∗ -algebras and between Lie C ∗ -algebras, Acta Math. Sinica 21 (2005), 1391–1398. [34] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [35] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130. [36] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445–446. [37] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268–273. [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, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378. [41] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [42] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130. [43] Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003. 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] J. R¨atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. [47] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [48] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [49] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [50] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Jung Rye Lee, Department of Mathematics, Daejin University, Kyeonggi 487-711, Republic of Korea E-mail address: [email protected] Dong Yun Shin, Department of Mathematics, University of Seoul, Seoul 130–743, Republic of Korea E-mail address: [email protected]

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Approximation reductions in IVF decision information systems ∗ Hongxiang Tang† February 14, 2014 Abstract: Rough set theory is a mathematical tool to deal with imprecision and uncertainty in data analysis. The lower and upper approximations of a set respectively characterize the non-numeric and numeric uncertain aspects of the available information. In this paper, we study upper and lower approximation reductions in IVF decision information systems by using rough set theory. Keywords: IVF; Decision information system; Similarity relation; Lower approximation reduction; Upper approximation reduction.

1

Introduction

The concept of rough set was originally proposed by Pawlak [1] as a mathematical tool to handle imprecision, vagueness and uncertainty in data analysis. This theory has been amply demonstrated to have usefulness and versatility by successful application in machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields [2, 3, 4, 5]. Rough set theory of deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. By using the concept of lower and upper approximation in rough sets theory, knowledge hidden in information systems may be unravelled and expressed in the form of decision rules. In traditional rough set approach, the values of attributes are assumed to be nominal data, i.e. symbols. In many applications, however, the decision attribute-values can be linguistic terms (i.e. interval value fuzzy sets). The traditional rough set approach would treat these values as symbols, thereby some important information included in these values such as the partial ordering and membership degrees is ignored, which means that the traditional rough set ∗ This work is supported by the National Social Science Foundation of China (No. 12BJL087). † Corresponding Author, School of Management Science and Engineering, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, P.R.China. [email protected]

1

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approach cannot effectively deal with interval value fuzzy initial data. Thus a new rough set model is needed to deal with such data. Interval-valued fuzzy information systems are the combination of traditional rough sets theory and interval-valued fuzzy sets theory. The purpose of this paper is to investigate attribute reductions in interval-valued fuzzy decision information systems.

2

Preliminaries

In this section, we briefly recall some basic concepts about IVF sets and IVF decision information systems. Throughout this paper, “ interval-valued fuzzy ” denote briefly by “ IVF ”. U denotes a finite and nonempty set called the universe. 2U denotes the family of all subsets of U . F (U ) denotes the set of all fuzzy sets in U . I denotes [0, 1] and [I] denotes {[a, b] : a, b ∈ I and a ≤ b}. For a ∈ I, denote a = [a, a].

2.1

IVF sets

Definition 2.1 ([7]). ∀ a, b ∈ [I], define (1) (2) (3)

a = b ⇐⇒ a− = b− , a+ = b+ . a ≤ b ⇐⇒ a− ≤ b− , a+ ≤ b+ ; ac = 1 − a = [1 − a+ , 1 − a− ].

a < b ⇐⇒ a ≤ b, a 6= b.

Definition W 2.2 ([7]). W ∀ {aWi : i+∈ J} ⊆ [I], define (1) ai = [ a− ai ]. i , i∈J i∈J V V − i∈J V + (2) ai = [ ai , ai ]. i∈J

i∈J

i∈J

Definition 2.3 ([7]). A mapping A : U → [I] is called an IVF set on U . Denote A(x) = [A− (x), A+ (x)]

(x ∈ U ).

Then A− (x) (resp. A+ (x)) is called the lower (resp. upper) degree to which x belongs to A. A− (resp. A+ ) is called the lower (resp. upper) IVF set of A. The set of all IVF sets in U is denoted by F (i) (U ). e represents the IVF set which satisfies U e (x) = 1 for each x ∈ U . U Similar to fuzzy sets, some basic operations on F (i) (U ) as follows: ∀ A, B ∈ F (i) (U ), (1) A = B ⇐⇒ A(x) = B(x) for each x ∈ U . (2) A ⊆ B ⇐⇒ A(x) ≤ B(x) for each x ∈ U . e − A ⇐⇒ B(x) = A(x)c or U e (x) − A(x) for each x ∈ U . (3) (3) B = Ac or U (4) (A ∩ B)(x)=A(x) ∧ B(x) for each x ∈ U . (5) (A ∪ B)(x)=A(x) ∨ B(x) for each x ∈ U . Obviously, A = B ⇐⇒ A− = B − and A+ = B + . 2

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2.2

IVF decision information systems

Definition 2.4 ([8]). (U, A ∪ D) is called an IVF decision information system, where U = {x0 , x1 , · · · , xn−1 } is the universe, A is a condition attribute set and D = {dk ∈ F (i) (U ) : k = 1, 2, · · · , r} is a decision attribute set. Denote dk (xi ) = Dik (i = 0, 1, · · · , n − 1, k = 1, 2, · · · , r), D(xi ) =

Di1 Di2 Dir + + ··· + . d1 d2 dr

Example 2.5 ([8]). Table 1 gives an IVF decision information system where U = {x0 , x1 , · · · , x9 }, A = {a1 , a2 , a3 }, D = {d1 , d2 , d3 }. Table 1: An IVF decision information system x0 x1 x2 x3 x4 x5 x6 x7 x8 x9

a1 2 3 2 2 1 1 3 1 2 3

a2 1 2 1 2 1 1 2 1 1 2

a3 3 1 3 3 4 2 1 4 3 1

d1 [0.7,0.9] [0.3,0.5] [0.7,0.8] [0.15,0.2] [0.05,0.1] [0.1,0.2] [0.25,0.4] [0.1,0.2] [0.45,0.6] [0.05,0.1]

d2 [0.15,0.2] [0.5,0.7] [0.3,0.4] [0.5,0.8] [0.2,0.3] [0.35,0.5] [1.0,1.0] [0.25,0.4] [0.25,0.3] [0.8,0.9]

d3 [0.4,0.5] [0.35,0.4] [0.1,0.2] [0.2,0.3] [0.65,0.9] [1.0,1.0] [0.3,0.4] [0.5,0.6] [0.2,0.3] [0.05,0.2]

Obviously, D(x0 ) =

D02 D03 [0.7, 0.9] [0.15, 0.2] [0.4, 0.5] D01 + + = + + . d1 d2 d3 d1 d2 d3

Definition 2.6. Let (U, A ∪ D) be an IVF decision information system. Then B ⊆ A determines an equivalence relation as follows: RB = {(x, y) ∈ U × U : a(x) = a(y) (a ∈ B)}. RB forms a partition U/RB = {[x]B : x ∈ U } of U where [x]B = {y ∈ U : (x, y) ∈ RB }. The lower and upper approximations of X ∈ F (i) (U ) with regard to (U, RB ) as follows: ^ _ X(y), RB (X)(x) = X(y) (x ∈ U ). RB (X)(x) = y∈[x]B

y∈[x]B

3

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Remark 2.7. If y ∈ [x]B , then [y]B = [x]B . So RB (X)(y) = RB (X)(x) and RB (X)(y) = RB (X)(x). Denote RB (X)([x]B ) = RB (X)(x), RB (X)([x]B ) = RB (X)(x)

(∗)

Proposition 2.8. Let (U, A ∪ D) be an IVF decision information system and let C ⊆ B ⊆ A. Then ∀ X ∈ F (i) (U ), e − X) = U e − RB (X). (1) RB (U (2) RC (X) ⊆ RB (X) ⊆ X ⊆ RB (X) ⊆ RC (X). Proof. (1) ∀ x ∈ U , e − X)(x) = RB (U

^

e − X)(y) = (U

y∈[x]B

=

^

^

e (y) − X(y)) (U

y∈[x]B

_

e (y) − U

y∈[x]B

e (x) − =U

X(y)

y∈[x]B

W

e − RB (X))(x). X(y) = (U

y∈[x]B

e − X) = U e − RB (X). Then RB (U (2) Since C ⊆ B, ∀ x ∈ U , [x]C ⊇ [x]B . Then ^ ^ X(y) ≤ X(y) = RB (X)(x). RC (X)(x) = y∈[x]C

y∈[x]B

So RC (X) ⊆ RB (X). Similarly, RB (X) ⊆ RC (X). ∀ x ∈ U , x ∈ [x]B . Then RB (X)(x) =

V

X(y) ≤ X(x). So RB (X) ⊆ X.

y∈[x]B

Similarly, X ⊆ RB (X). Hence RC (X) ⊆ RB (X) ⊆ X ⊆ RB (X) ⊆ RC (X).

3

Approximation reductions in IVF decision information systems

3.1

The similarity relation RD

Definition 3.1. Let S = (U, A ∪ D) be a IVF decision information system where U = {x0 , x1 , · · · , xn }, , A is a condition attribute set D = {d1 , d2 , · · · , dr }. Denote dk (xi ) = Dik (i = 0, 1, · · · , n − 1, k = 1, 2, · · · , r), For i, j ∈ {0, 1, · · · , n − 1}, define ^ RD (xi , xj ) = {1 − Dik ∧ Djk | : k = 1, 2, · · · , r}. 4

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Obviously, RD (xi , xi ) = 1, RD (xi , xj ) = RD (xj , xi ). Then RD is a similarity relation on U . We can obtain the following similar decision class SD (x): SD (x)(y) = RD (x, y) (y ∈ U ). Denote SD (xi ) =

x0 x1 x2 x3 x4 x5 x6 + + + + + + SD (xi )(x0 ) SD (xi )(x1 ) SD (xi )(x2 ) SD (xi )(x3 ) SD (xi )(x4 ) SD (xi )(x5 ) SD (xi )(x6 ) +

x8 x9 x7 + + (i = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), SD (xi )(x7 ) SD (xi )(x8 ) SD (xi )(x9 ) U/RD = {SD (x) : x ∈ U }.

Example 3.2.

In Example 2.5,

SD (x0 )(x1 ) = RD (x0 , x1 )

V V = (1 − D01 ∧ D11 ) (1 − D02 ∧ D12 ) (1 − D03 ∧ D13 ) V = (1 − [0.7, 0.9] ∧ [0.3, 0.5]) (1 − [0.15, 0.2] ∧ [0.5, 0.7]) V (1 − [0.4, 0.5] ∧ [0.35, 0.4]) V V = (1 − [0.3, 0.5]) (1 − [0.15, 0.2]) (1 − [0.35, 0.4]) V V = [0.5, 0.7] [0.8, 0.85]) [0.6, 0.65]

= [0.5, 0.65]. Similarly, SD (x0 )(x0 ) = 1, SD (x0 )(x2 ) = [0.2, 0.3], SD (x0 )(x3 ) = [0.7, 0.8], SD (x0 )(x4 ) = [0.5, 0.6], SD (x0 )(x5 ) = [0.5, 0.6], SD (x0 )(x6 ) = [0.6, 0.7], SD (x0 )(x7 ) = [0.5, 0.6], SD (x0 )(x8 ) = [0.4, 0.55], SD (x0 )(x9 ) = [0.8, 0.85]. Thus SD (x0 ) =

x0 1

+

x1 x2 x3 x4 x5 x6 x7 x8 x9 + + + + + + + + . [0.5, 0.65] [0.2, 0.3] [0.7, 0.8] [0.5, 0.6] [0.5, 0.6] [0.6, 0.7] [0.5, 0.6] [0.4, 0.55] [0.8, 0.85]

We can also calculate SD (xi ) (i = 1, 2, 3, 4, 5, 6, 7, 8, 9). They record in Table 2.

3.2

Upper and lower approximation reductions

Definition 3.3. Let S = (U, A ∪ D) be a IVF decision information system. Denote U/RD = {SD (x) : x ∈ U }. ∀ B ⊆ A, (1) If ∀ x ∈ U , RB (SD (x)) = RA (SD (x)), then B is called a lower approximation consistent set of S; If B is a lower approximation consistent set of S and ∀ C ( B, x ∈ U , RC (SD (x)) 6= RA (SD (x)), then B is called a lower approximation reduction of S. 5

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Table 2: SD (xi ) SD (x0 ) SD (x1 ) SD (x2 ) SD (x3 ) SD (x4 ) SD (x5 ) SD (x6 ) SD (x7 ) SD (x8 ) SD (x9 ) −− SD (x0 ) SD (x1 ) SD (x2 ) SD (x3 ) SD (x4 ) SD (x5 ) SD (x6 ) SD (x7 ) SD (x8 ) SD (x9 )

x0 1 [0.5,0.65] [0.2,0.3] [0.7,0.8] [0.5,0.6] [0.5,0.6] [0.6,0.7] [0.5,0.6] [0.4,0.55] [0.8,0.85] −− x5 [0.5,0.6] [0.5,0.65] [0.6,0.7] [0.5,0.65] [0.1,0.35] 1 [0.5,0.65] [0.4,0.5] [0.7,0.75] [0.5,0.65]

x1 [0.5,0.65] 1 [0.5,0.7] [0.3,0.5] [0.6,0.65] [0.5,0.65]] [0.3,0.5] [0.6,0.65] [0.5,0.7] [0.3,0.5] −− x6 [0.6,0.7] [0.3,0.5] [0.6,0.7] [0.2,0.5] [0.6,0.7] [0.5,0.65] 1 [0.6,0.7] [0.6,0.75] [0.6,0.75]

x2 [0.2,0.3] [0.5,0.7] 1 [0.6,0.7] [0.7,0.8] [0.6,0.7] [0.6,0.7] [0.4,0.75] [0.4,0.55] [0.6,0.7] −− x7 [0.5,0.6] [0.6,0.65] [0.4,0.75] [0.6,0.75] [0.4,0.5] [0.4,0.5] [0.6,0.7] 1 [0.7,0.75] [0.6,0.75]

x3 [0.7,0.8] [0.3,0.5] [0.6,0.7] 1 [0.7,0.8] [0.5,0.65] [0.2,0.5] [0.6,0.75] [0.7,0.75] [0.2,0.5] −− x8 [0.4,0.55] [0.5,0.7] [0.4,0.55] [0.7,0.75] [0.7,0.8] [0.7,0.75] [0.6,0.75] [0.7,0.75] 1 [0.7,0.75]

x4 [0.5,0.6] [0.6,0.65] [0.7,0.8] [0.7,0.8] 1 [0.1,0.35] [0.6,0.7] [0.4,0.5] [0.7,0.8] [0.7,0.8] −− x9 [0.8,0.85] [0.3,0.5] [0.6,0.7] [0.2,0.5] [0.7,0.8] [0.5,0.65] [0.1,0.2] [0.6,0.75] [0.7,0.75] 1

(2) If ∀ x ∈ U , RB (SD (x)) = RA (SD (x)), then B is called a upper approximation consistent set of S; If B is a upper approximation consistent set of S and ∀ C ( B, x ∈ U , RC (SD (x)) 6= RA (SD (x)), then B is called a upper approximation reduction of S. Theorem 3.4. Let S = (U, A ∪ D) be a IVF decision information system. Denote U/RD = {SD (x) : x ∈ U }. (1) B ⊆ A is a lower approximation consistent set of S ⇐⇒ If ∀ x, y, z ∈ U , RA (SD (x))(y) 6= RA (SD (x))(z), then [y]B ∩ [z]B = ∅. (2) B ⊆ A is a upper approximation consistent set of S ⇐⇒ If ∀ x, y, z ∈ U , RA (SD (x))(y) 6= RA (SD (x))(z), then [y]B ∩ [z]B = ∅. Proof. (1) “ =⇒ ” Let B be a lower approximation consistent set of S. Then ∀ x ∈ U , RB (SD (x)) = RA (SD (x)). Thus ∀ x, y, z ∈ U , RB (SD (x))(y) = RA (SD (x))(y), RB (SD (x))(z) = RA (SD (x))(z). Since RA (SD (x))(y) 6= RA (SD (x))(z), we have RB (SD (x))(y) 6= RB (SD (x))(z),

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Then

^

SD (x)(u) 6=

u∈[y]B

^

SD (x)(v).

v∈[z]B

Thus [y]B ∩ [z]B = ∅ “ ⇐= ” Since [y]B = ∪{[z]A : z ∈ [y]B }, we have V V S RB (SD (x))(y) = SD (x)(u) = {SD (x)(u) : u ∈ [z]A } u∈[y]B V V z∈[y] VB = {SD (x)(u) : u ∈ [z]A } = SD (x)(u) z∈[y]B z∈[y]B u∈[z]A V = RA (SD (x))(z). z∈[y]B

If z ∈ [y]B , then [y]B ∩ [z]B 6= ∅. By the Hypothesis, RA (SD (x))(y) = RA (SD (x))(z). So RB (SD (x))(y) = RA (SD (x))(y).

Thus ∀ x ∈ U , RB (SD (x)) = RA (SD (x)). Hence B is a lower approximation consistent set of S. (2) The proof is similar to (1). Definition 3.5. Let S = (U, A ∪ D) be a IVF decision information system. Denote U/RD = {SD (x) : x ∈ U }. ∀ x, y, z ∈ U , denote ½ D(y, z) = ½ D(y, z) =

{ai ∈ A : ai (y) 6= ai (z)}, A,

RA (SD (x))(y) 6= RA (SD (x))(z), RA (SD (x))(y) = RA (SD (x))(z).

{ai ∈ A : ai (y) 6= ai (z)}, A,

RA (SD (x))(y) 6= RA (SD (x))(z), RA (SD (x))(y) = RA (SD (x))(z).

D(y, z) and D(y, z) are called the lower and upper approximation attribute set of y and z in S, respectively. Theorem 3.6. Let S = (U, A ∪ D) be a IVF decision information system. Denote U/RD = {SD (x) : x ∈ U }. Then ∀ B ⊆ A, (1) B is a lower approximation consistent set of S ⇐⇒ ∀ y, z ∈ U , B ∩ D(y, z) 6= ∅. (2) B is a upper approximation consistent set of S ⇐⇒ ∀ y, z ∈ U , B ∩ D(y, z) 6= ∅. Proof. (1) “ =⇒ ” Suppose B is a lower approximation consistent set of S. If RA (SD (x))(y) 6= RA (SD (x))(z), then by Theorem 3.4(1), [y]B ∩ [z]B = ∅. So there exists ai0 ∈ B such that ai0 (y) 6= ai0 (z). Then ai0 ∈ B ∩ {ai ∈ A : ai (y) 6= ai (z)} = B ∩ D(y, z). Thus B ∩ D(y, z) 6= ∅. If RA (SD (x))(y) = RA (SD (x))(z), then B ∩ D(y, z) = B ∩ A = B 6= ∅. 7

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“ ⇐= ” Suppose B ∩ D(y, z) 6= ∅. If RA (SD (x))(y) 6= RA (SD (x))(z), then B ∩ D(y, z) = B ∩ {ai ∈ A : ai (y) 6= ai (z)} 6= ∅. So [y]B ∩ [z]B = ∅. By Theorem 3.4(1), B is a lower approximation consistent set of S. (2) The proof is similar to (1). Example 3.7. Consider Example 3.2. (1) U/RA = {X1 , X2 , X3 , X4 , X5 } = {{x0 , x2 , x8 }, {x1 , x6 , x9 }, {x3 }, {x4 , x7 }, {x5 }}. We need to calculate RA (SD (x0 ))(Xi ). RA (SD (x0 ))(X1 )

=

^

SD (x0 )(y) = SD (x0 )(x0 ) ∧ SD (x0 )(x2 ) ∧ SD (x0 )(x8 )

y∈X1

=

1 ∧ [0.2, 0.3] ∧ [0.4, 0.55] = [0.2, 0.3].

Similarly, RA (SD (x0 ))(X2 ) = [0.5, 0.65], RA (SD (x0 ))(X3 ) = [0.7, 0.8], RA (SD (x0 ))(X4 ) = [0.5, 0.5], RA (SD (x0 ))(X5 ) = [0.5, 0.6]. Denote RA (SD (x0 )) =

X2 X3 X4 X5 X1 + + + + . [0.2, 0.3] [0.5, 0.65] [0.7, 0.8] [0.5, 0.5] [0.5, 0.6]

We can also calculate RA (SD (xi )) (i = 1, 2, 3, 4, 5, 6, 7, 8, 9). They record in Table 3. (2) Put B = {a1 , a2 }. Obviously, U/RB = {X1 , X2 , X3 , X4 ∪ X5 } = {{x0 , x2 , x8 }, {x1 , x6 , x9 }, {x3 }, {x4 , x5 , x7 }}. We can calculate RB (SD (xi )) (i = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9). They record in Table 4. By table 3 and table 4, B = {a1 , a2 } is not a lower approximation consistent set of S. Thus B is not a lower approximation reduction of S.

4

Conclusions

In this paper, we have researched lower approximation reducts and upper approximation reducts in IVF decision information systems. In future work, we will investigate knowledge acquisition in IVF decision information systems..

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Table 3: RA (SD (xi )) RA (SD (x0 )) RA (SD (x1 )) RA (SD (x2 )) RA (SD (x3 )) RA (SD (x4 )) RA (SD (x5 )) RA (SD (x6 )) RA (SD (x7 )) RA (SD (x8 )) RA (SD (x9 ))

X1 [0.2,0.3] [0.5,0.65] [0.2,0.3] [0.6,0.7] [0.5,0.6] [0.5,0.6]] [0.6,0.7] [0.4,0.6] [0.4,0.55] [0.6,0.7]

X2 [0.5,0.65] [0.3,0.5] [0.5,0.7] [0.2,0.5] [0.6,0.65] [0.5,0.65] [0.1,0.2] [0.6,0.65] [0.5,0.7] [0.3,0.5]

X3 [0.7,0.8] [0.3,0.5] [0.6,0.7] 1 [0.7,0.8] [0.5,0.65] [0.2,0.5] [0.6,0.75] [0.7,0.75] [0.2,0.5]

X4 [0.5,0.5] [0.6,0.65] [0.4,0.75] [0.6,0.75] [0.4,0.5] [0.1,0.35] [0.6,0.7] [0.4,0.5] [0.7,0.75] [0.6,0.75]

X5 [0.5,0.6] [0.5,0.65] [0.6.0.7] [0.5,0.65] [0.1,0.35] 1 [0.5,0.65] [0.4,0.5] [0.7,0.75] [0.5,0.65]

Table 4: RB (SD (x1 )) RB (SD (x0 )) RB (SD (x1 )) RB (SD (x2 )) RB (SD (x3 )) RB (SD (x4 )) RB (SD (x5 )) RB (SD (x6 )) RB (SD (x7 )) RB (SD (x8 )) RB (SD (x9 ))

X1 [0.2,0.3] [0.5,0.65] [0.2,0.3] [0.6,0.7] [0.5,0.6] [0.5,0.6]] [0.6,0.7] [0.4,0.6] [0.4,0.55] [0.6,0.7]

X2 [0.5,0.65] [0.3,0.5] [0.5,0.7] [0.2,0.5] [0.6,0.65] [0.5,0.65] [0.1,0.2] [0.6,0.65] [0.5,0.7] [0.3,0.5]

X3 [0.7,0.8] [0.3,0.5] [0.6,0.7] 1 [0.7,0.8] [0.5,0.65] [0.2,0.5] [0.6,0.75] [0.7,0.75] [0.2,0.5]

X4 ∪ X5 [0.5,0.5] [0.5,0.65] [0.4,0.7] [0.5,0.65] [0.1,0.35] [0.1,0.35]] [0.5,0.65] [0.4,0.5]] [0.7,0.75]] [0.5,0.65]

References [1] Z.Pawlak, Rough sets, International Journal of Computer and Information Science, 11(1982), 341-356. [2] Z.Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991. [3] Z.Pawlak, A.Skowron, Rudiments of rough sets, Information Sciences, 177(2007), 3-27. [4] Z.Pawlak, A.Skowron, Rough sets: some extensions, Information Sciences, 177(2007), 28-40. [5] Z.Pawlak, A.Skowron, Rough sets and Boolean reasoning, Information Sciences, 177(2007), 41-73.

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[6] W.Zhang, W.Wu, J.Liang, D.Li, Rough sets theorey and methods, Chinese Scientific Publishers, Beijing, 2001. [7] Y.Cheng, D.Miao, Rule extraction based on granulation order in information system, Expert Systems with Applications, 38(2011), 12249-12261. [8] B.Sun, Z.Gong, D.Chen, Fuzzy rough set theory for the interval-valued fuzzy information systems, Information Sciences, 178(2008), 2794-2815. [9] W.Wu. Attribute reduction based on evidence theory in incomplete decision systems, Information Sciences, 178(2008), 1355-1371. [10] W.Wu, M.Zhang, H.Li, J.Mi. Knowledge reduction in random information systems via Dempster-Shafer theory of evidence, Information Sciences, 174(2005), 143-164. [11] W.Wu, Y.Leung, J.Mi. On generalized fuzzy belief functions in infinite spaces, IEEE transactions on fuzzy systems, 17(2009), 385-397. [12] Z.Li, T.Xie, Q.Li, Topological structure of generalized rough sets, Computers and Mathematics with Applications, 63(2012), 1066-1071. [13] W.Zheng, Y.Liang, W.Wu, Information systems and knowledge discovery, Chinese Scientific Publishers, Beijing, 2003. [14] M.Zhang, L.Xu, W.Zhang, H.Li, A rough set approach to knowledge reduction based on inclusion degree and evidence reasoning theory, Expert Systems with Applications, 20(2003), 298-304.

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The covering method for attribute reductions of concept lattices ∗ †

Neiping Chen

Xun Ge‡

February 14, 2014 Abstract: This paper investigates methods for attribute reductions of concept lattices. We establish the relationship between attribute reductions of the object oriented concept lattice generated by a formal context and covering reductions of the covering approximation space induced by the same formal context, and then present the covering method to compute all attribute reductions of this object oriented concept lattices. Moreover, we give an example to compute all attribute or object reductions of the object oriented concept lattice and all reductions of a formal context by using this method, which illustrates that this method is effective and simple. Keywords: Formal context; Reduction; Concept lattice; Covering method; Covering approximation space.

1

Introduction

Formal concept analysis (FCA), proposed by Wille [7, 12], is an important theory for data analysis and knowledge discovery. Two basic notions in FCA are formal contexts and formal concepts. The set of all formal concepts of a formal context forms a complete lattice, called the concept lattice of the formal context, to reflect the relationship of generalization and specialization among the concepts. Up to now, FCA has been applied to information retrieval, database management systems, software engineering and other aspects (see [2, 4, 6, 9, 13]). Attribute reductions of concept lattices are one of the key issues in FCA or concept lattice theory and there have been many studies on this topic. For example, Ganter et al. [7] discussed the knowledge reduction issue by removing the reducible objects and attributes of a formal context. Zhang et al. [18] proposed a knowledge reduction method in formal contexts from the viewpoint of lattice isomorphism. Liu et al. [10] put forward reduction method for concept ∗ This work is supported by the National Natural Science Foundation of China (No. 11226085, 11061004). † School of Mathematics and Statistics, Hunan University of Commerce, Changsha 410205, China, [email protected] ‡ Corresponding Author, School of Mathematical Sciences, Soochow University, Suzhou 215006, China, [email protected]

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lattices based on rough set theory. Wei et al. [15] investigated the problem of knowledge reduction in decision formal contexts by defining two partial orders between the conditional and the decision concept lattices. Based on granular computing, Wu et al. [14] presented a novel reduction method for decision formal contexts. The purpose of this paper is to give the covering method for attribute reductions of concept lattices.

2

Preliminaries

In this section, we recall some basic concepts of formal contexts, concept lattices and covering approximation spaces. Definition 2.1 ([7]). Let U be a finite set of objects, A be a finite set of features or attributes, and R be a binary relation on U × A (i.e., R ⊆ U × A). The triple (U, A, R) is called a formal context. Remark 2.2. Let (U, A, R) be a formal context. In this paper, we use the following notations. (1) (u, α) ∈ R is also written as uRα, which means that the object u possesses the feature or attribute α. (2) Let u ∈ U and α ∈ A, uR = {α ∈ A : uRα} ⊆ A and Rα = {u ∈ U : uRα} ⊆ U . Thus, uRα ⇐⇒ α ∈ uR ⇐⇒ S u ∈ Rα. S (3) Let X ∈ 2U and B ⊆ A. XR = {uR : u ∈ X} and RB = {Rα : α ∈ B}. 0 (4) Let T B ⊆ A. (U, B, R ) denotes a formal subcontext of (U, A, R), where 0 R = R (U × B). In this paper, all formal context are assumed to be regular, where a formal context (U, A, R) is regular if for each u ∈ U and each α ∈ A, uR 6= ∅, uR 6= A, Rα 6= ∅ and Rα 6= U . Definition 2.3. Let (U, A, R) be a formal context. Then a pair of approximation operators ¤ : 2U −→ 2A and ♦ : 2A −→ 2U are respectively defined as follows, where X ∈ 2U and B ⊂ A. X ¤ = {α ∈ A : ∀u ∈ U (uRα V =⇒ u ∈ X)} = {α ∈ A : Rα T ⊂ X}. S B ♦ = {u ∈ U : ∃α ∈ A(uRα α ∈ B)} = {u ∈ U : uR B 6= ∅} = {Rα : α ∈ B} = RB. Definition 2.4 ([16, 17]). Let (U, A, R) be a formal context, X ∈ 2U and B ⊂ A. The ordered pair (X, B) is called an object oriented concept , if X = B ♦ and B = X ¤ . X and B are called the extension and the intension of the object oriented concept (X, B), respectively. In other words, an object oriented concept is an ordered pair (X, B), where X is the set of objects, each of which possesses at least one attribute in B, and B is the set of attributes, each of which is possessed by one or some objects only in X. 2

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Remark 2.5. Let (U, A, R) be a formal context and X ∈ 2U . It is clear that X is the extension of an object oriented concept of (U, A, R) if and only if X ¤♦ = X. Definition 2.6 ([16]). Let (U, A, R) be a formal context. Put D(U, A, R) = {(X, B) : (X, B) is an object oriented concept of (U, A, R)}. Then D(U, A, R) forms a complete lattice and is called concept W an object oriented V lattice generated by (U, A, R), in which the meet and the join of any two object oriented concepts are respectively defined as follows: _ [ [ [ [ (X1 , B1 ) (X2 , B2 ) = (X1 X2 , (X1 X2 )¤ ) = (X1 X2 , (B1 B2 )♦¤ ), (X1 , B1 )

^

(X2 , B2 ) = ((B1

\

B2 )♦ , B1

\

B2 ) = ((X1

\

X2 )¤♦ , B1

\

B2 ).

Definition 2.7 ([10]). Let (X1 , B1 ) and (X2 , B2 ) be two object oriented concepts of a formal context (U, A, R). If X1 ⊆ X2 , which is equivalent to B1 ⊆ B2 , (X1 , B1 ) is called a sub-concept of (X2 , B2 ), and (X2 , B2 ) is called a superconcept of (X1 , B1 ), i.e., (X1 , B1 ) ≤ (X2 , B2 ). The relation ≤ is the hierarchical order of the concepts. Furthermore, an object oriented concept (X, B) ∈ D(U, A, R) is said to be minimal , if X = X 0 and B = B 0 whenever (X 0 , B 0 ) ∈ D(U, A, R) and (X 0 , B 0 ) ≤ (X, B). Definition 2.8 ([10]). Let D(U, A1 , R1 ) and D(U, A2 , R2 ) be two object oriented concept lattices. If there exists W a bijective function h :WD(U, A1 , R1 ) −→ D(U, A2 , R2V ) such that h((X1 , B1 ) (X V2 , B2 )) = h((X1 , B1 )) h((X2 , B2 )) and h((X1 , B1 ) (X2 , B2 )) = h((X1 , B1 )) h((X2 , B2 )) for whenever (X1 , B1 ), (X2 , B2 ) ∈ D(U, A1 , R1 ), then D(U, A1 , R1 ) and D(U, A2 , R2 ) are said to be isomorphic, which is denoted by D(U, A1 , R1 ) ∼ = D(U, A2 , R2 ). Remark 2.9 ([10]). According to Definition 2.8, we have that the structures and the hierarchical orders of two isomorphic concept lattices are identical. Definition 2.10 ([10]). Let D(U, A, R) be an object oriented concept lattice. (1) Let β ∈ A. Then attribute T β is said to be dispensable in D(U, A, R) , if D(U, A, R) ∼ = D(U, A − {β}, R (U × (A − {β}))); otherwise, attribute α is indispensable in D(U, A, R). (2) Let attributes set B ⊆TA. Then B is called a consistent set of D(U, A, R) , if D(U, A, R) ∼ = D(U, B, R (U × B)). (3) Let attributes set B ⊆ A be a consistent set of D(U, A, R). Then B is called an attributeTreduction of D(U, A, R) , if for whenever β ∈ B, D(U, A, R) ∼ 6 = D(U, B − {β}, R (U × (B − {β}))). Definition 2.11 ([10]). Let D(U, A, R) be an object oriented concept lattice. (1) Let u ∈ U . Then object T v is said to be dispensable in D(U, A, R) , if D(U, A, R) ∼ = D(U − {v}, A, R ((U − {v}) × A)); otherwise, object v is indispensable in D(U, A, R). (2) Let objects set V ⊆TU . Then V is called a consistent set of D(U, A, R) , if D(U, A, R) ∼ = D(V, A, R (V × A)). 3

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(3) Let objects set V ⊆ U be a consistent set of D(U, A, R). Then V is called an object reduction of D(U, A, R) , if for whenever v ∈ V , D(U, A, R) 6∼ = T D(V − {v}, A, R ((V − {v}) × A)). Definition 2.12 ([10]). Let T (U, A, R) be a formal context, V ⊆ U and B ⊆ A. then formal context (V, B, R (V × B)) is called a reduction of (U, A, R) , if V and B are an object reduction and an attribute reduction of the object oriented concept lattice D(U, A, R), respectively. Remark 2.13. Note that there is a dual relation between attribute reductions and object reductions in the object oriented concept lattice D(U, A, R). So this paper only gives some investigations for attribute reductions in D(U, A, R). All results for object reductions in D(U, A, R) can be obtained by the same methods. Definition 2.14 ([1]). Let U , the universe of discourse, be a finite set and C be a family of subsets of U . (1) C is called a covering of U , if U is the union of elements of C. (2) The pair (U, C) is called a covering approximation space, if C is a covering of U . Definition 2.15 ([8]). Let (U, C) be a covering approximation space. Put T = S { F : F ⊆ C}. Then T is called the generalized topology on U induced by C and C is called a base for T . Definition 2.16 ([19, 20]). Let (U, C) be a covering approximation S space. (1) Let K ∈ C. K is called a reducible element of C , if K = F for some F ⊆ C − {K}; otherwise, K is called an irreducible element of C. (2) C is called irreducible , if K is an irreducible element of C for each K ∈ C; otherwise C is called reducible. Let (U, C) be a covering approximation space. It is known that C − {K} is still a covering of U if K is a reducible element of C [20]. It follows that a reduction of a covering can be computed by deleting one reducible element in a step [19]. Definition 2.17 ([19, 20]). Let (U, C) be a covering approximation space. If K1 is a reducible element of C, K2 is a reducible element of C − {K1 }, · · ·, Kn is a reducible element of C − {K1 , K2 , · · · , Kn−1 } and C 0 = C − {K1 , K2 , · · · , Kn } is irreducible, then C 0 is called a reduction (more precisely, covering reduction) of C. In particularly, C is a reduction of C if C is irreducible. Definition 2.18. (1) Let (U, A, R) be a formal context. Put C = {Rα : α ∈ A}. Then (U, C) is a covering approximation space and it is called to be induced by (U, A, R). (2) Let (U, C) be a covering approximation space, where C = {Kα : α ∈ A}. Give a binary relation R on U × A: uRα if and only if u ∈ Kα . Then (U, A, R) is a formal context and it is called to be induced by (U, C).

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Remark 2.19. There is such a case for a formal context (U, A, R): “Rα = Rβ for some α, β ∈ A”. So, for the sake of conveniences, we allow that K = K 0 for some K 0 , K 0 ∈ C for a covering approximation space (U, C) in the discussion of this paper.

3

The main results

In this section, based on the relationship between attribute reductions of D(U, A, R) and covering reductions of (U, C) where (U, A, R) is a formal context, D(U, A, R) is the object oriented concept lattice generated by (U, A, R), (U, C) is the covering approximation spaces induced by (U, A, R), we present the covering method to compute all attribute reductions of D(U, A, R). Let (U, C) be a covering approximation space, where C = {Kα : α ∈ A}. For B ⊆ A, we denote [ [ CB = {Kα : α ∈ B} and CB = {Kα : α ∈ B}. Obviously, CA = C and

S

CA = U .

Proposition 3.1. Let (U, A, R) be a formal context and let (U, C) be a covering approximation space. (1) If (U, A, R) is induced by (U, C), then (U, C) is induced by (U, A, R). (2) If (U, C) is induced by (U, A, R), then (U, A, R) is induced by (U, C). Proof. (1) Let (U, A, R) be induced by (U, C), where C = {Kα : α ∈ A}. Then, for each u ∈ U and each α ∈ A, uRα if and only if u ∈ Kα . Let (U, C 0 ) be a covering approximation space induced by (U, A, R). Then C 0 = {Rα : α ∈ A}. We only need to prove that C = C 0 . Let α ∈ A, it suffices to show that Rα = Kα . In fact, u ∈ Rα if and only if uRα if and only if u ∈ Kα , so Rα = Kα . (2) Let (U, C) be induced by (U, A, R). Then C = {Rα : α ∈ A}. Let (U, A, R0 ) be a formal context induced by (U, C). It suffices to show that R = R0 . In fact, for each u ∈ U and each α ∈ A, (u, α) ∈ R if and only if u ∈ Rα if and only if (u, α) ∈ R0 , so R = R0 . Remark 3.2. Let (U, A, R) be a formal context, and let (U, C) be a covering approximation space induced by S (U, A, R), where C = {Kα : α ∈ A}. Then Kα = Rα for each α ∈ A and {Kα : α ∈ B} = RB for each B ⊆ A. Proposition 3.3. Let (U, C) be a covering approximation space and (U, A, R) be a formal context induced by (U, C). Then the following are equivalent for X ∈ 2U . (1) X is the extension of an object oriented concept of (U, A, R). (2) X ∈ T , where T is the generalized topology on U induced by C. Proof. Let C = {Kα : α ∈ A}. Since (U, A, R) is induced by (U, C), (U, C) is induced by (U, A, R) by Proposition 3.1. So Kα = Rα for each α ∈ A by Remark 3.2. 5

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(1) =⇒ (2): Assume that X is the extension of an object oriented concept ¤♦ of (U, A, R). Then X by Remark 2.8. Note that X ¤ ⊆ A. So X = S S= X ¤ {Rα : α ∈ X } = {Kα : α ∈ X ¤ }. This proves that X is the union of some elements of C, and hence X ∈ T . (2) =⇒ that X ∈ T . Since X ¤ = {β ∈ A : Rβ S⊆ X}, S (1): Assume ¤♦ ¤ X = {Rβ : β ∈ X }. For each β ∈ X ¤ , Rβ ⊆ X, hence X ¤♦ = {Rβ : ¤ β ∈ X that S } ⊆ X. Sine X ∈ T , then there exists a subset B of A such ¤ X = {Kα S : α ∈ B}. For each α ∈ B, Rα = Kα ⊆ X, so α ∈ X , hence S Kα = Rα ⊆ {Rβ : β ∈ X ¤ } = X ¤♦ . It follows that X = {Kα : α ∈ B} ⊆ X ¤♦ . Consequently, X ¤♦ = X. By Remark 2.8, X is the extension of an object oriented concept of (U, A, R). By the proof of Proposition 3.3, we have the following remark. Remark 3.4. Let (U, A, R) be a formal context, and let (U, C) be a covering approximation space induced byS(U, A, R). If X ∈ 2U and B ⊆ A, then (X, B) ∈ D(U, A, R) if and only if X = CB Lemma 3.5. Let (U, A, R) be a formal context and (U, B, R0 ) be a formal subcontext of (U, A, R). For X ∈ 2U and D ⊂ B, put ¤ XB = {α ∈ B : ∀x ∈ U (xR0 α =⇒Vx ∈ X)} = {α ∈ B : R0 α ⊂ X}; T ♦ 0 0 S D0B = {x ∈ U : 0∃α ∈ B(xR α α ∈ D)} = {x ∈ U : xR D 6= ∅} = {R α : α ∈ D} = R D. Then the following T hold. ¤ (1) XB = X ¤ B. ♦ (2) DB = D♦ . ¤ . Then α ∈ B. For each x ∈ U , if (x, α) ∈ R, then Proof. (1) T Let α ∈ XB (x, α) ∈ T R (U × B) = R0 , i.e., xR0 α, T hence x ∈ X. It follows that α ∈ X ¤ . So ¤ ¤ B, then α ∈ B. Since α ∈ X ¤ , for each B. Conversely, let α ∈ X α∈X 0 ¤ x ∈ U , if (x, T α) ∈ R ⊂ R, then x ∈ X. It follows that α ∈ XB . Consequently, ¤ ¤ XB = X B. (2) It is suffices to prove that R0 α = Rα T for each α ∈ D. For each α ∈ DT ⊂ B, (x, α) ∈ R if and only if (x, α) ∈ R (U × B), so R0 α = {x : (x, α) ∈ R (U × B)} = {x : (x, α) ∈ R} = Rα.

Theorem 3.6. Let (U, A1 , R1 ) and (U, A2 , R2 ) be two formal contexts, and let (U, C1 ) and (U, C2 ) be two covering approximation spaces induced by (U, A1 , R1 ) and (U, A2 , R2 ) respectively. Then the following are equivalent. (1) D(U, A1 , R1 ) ∼ = D(U, A2 , R2 ). (2) T1 = T2 , where T1 and T2 are the generalized topologies on U induced by C1 and C2 respectively. Proof. By Remark 2.9, we can use the following equivalent definition of isomorphism between two object oriented concept lattices D(U, A1 , R1 ) and D(U, A2 , R2 ) in the following proof, which was given by Zhang et al. in [18]. D(U, A1 , R1 ) and D(U, A2 , R2 ) are said to be isomorphic if whenever X ∈ 2U , X is the extension

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of an object oriented concept of D(U, A1 , R1 ) if and only if X is the extension of an object oriented concept of D(U, A2 , R2 ). (1) =⇒ (2): Assume that D(U, A1 , R1 ) ∼ = D(U, A2 , R2 ). Let X1 ∈ T1 , then X1 is the extension of an object oriented concept of (U, A1 , R1 ) by Proposition 3.3. Since D(U, A1 , R1 ) ∼ = D(U, A2 , R2 ), there exists the extension X2 of an object oriented concept of (U, A2 , R2 ) such that X1 = X2 . By Proposition 3.3, X2 ∈ T2 , and hence X1 ∈ T2 . This proves that T1 ⊆ T2 . Similarly, T2 ⊆ T1 . It follows that T1 = T2 . (2) =⇒ (1): Assume that T1 = T2 . By Proposition 3.3, X is the extension of an object oriented concept of (U, A1 , R1 ) if and only if X ∈ T1 and X is the extension of an object oriented concept of (U, A2 , R2 ) if and only if X ∈ T2 . It follows that X is the extension of an object oriented concept of (U, A1 , R1 ) if and only if X is the extension of an object oriented concept of (U, A2 , R2 ). So D(U, A1 , R1 ) ∼ = D(U, A2 , R2 ). Corollary 3.7. Let (U, A, R) be a formal context and let (U, C) be a covering approximation space induced by the (U, A, R). Then the following are equivalent for B ⊆ A. (1) B is a consistent set of the object oriented concept lattice D(U, A, R). (2) CB is a base for T , where T is the generalized topologies on U induced by C. Lemma 3.8. Let (U, C) be a covering approximation space, where C = {Kα : α ∈ A}, and let (U, A, R) be a formal context induced by the (U, C). Then the following are equivalent for β ∈ A. (1) β is a dispensable attribute in the object oriented concept lattice D(U, A, R). (2) Kβ is a reducible element of C. Proof. (1) =⇒ (2): Assume that β is a dispensable attribute in D(U, A, R). By [10, Property 3.1], there exists B ⊆ A − {β} such that Rβ = RB It follows that S Kβ = {Kα : α ∈ B} from Remark 3.2. That is Kβ is a reducible element of C. (2) =⇒ (1): It is obtained by reverting the proof of the above (1) =⇒ (2). Lemma 3.9. Let (U, C) be a covering approximation space where C = {Kα : α ∈ A} and let B ⊆ A. Then the following are equivalent. (1) CB is a reduction of C. (2) CB is an irreducible base for T , where T is the generalized topologies on U induced by C. Proof. Let T be the generalized topologies on U induced by C. (1) =⇒ (2): Assume that CB is a reduction of C. Let CB = C−{K1 , K2 , · · · , Kn } described as in Definition 2.14. Then CB is irreducible. Clearly, we only need to prove that Ki is the union of some elements of CB for each i ∈ {1, 2, · · · , n}. In fact, since Kn is a reducible element of C − {K1 , K2 , · · · , Kn−1 }, Kn is the union of some elements of (C − {K1 , K2 , · · · , Kn−1 }) − {Kn } = CB . Similarly, since Kn−1 is a reducible element of C − {K1 , K2 , · · · , Kn−2 }, Kn−1 is the union of

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S some elements of (C − {K1 , K2 , · · · , Kn−2 }) − {Kn−1 } = CB {Kn }. Note that Kn is the union of some elements of CB . So Kn−1 is the union of some elements of CB . And so on, Ki is the union of some elements of CB for i = n−2, n−3, · · · , 2, 1. Thus, the proof is completed. (2) =⇒ (1): Assume that CB is an irreducible base for T . Let C − CB = {K1 , K2 , · · · , Kn }. For each i ∈ {1, 2, · · · , n}, since C ⊆ T and CB is a base for T , Ki is the union of some elements of CB . It follows that Ki is a reducible element of C −{K1 , K2 , · · · , Ki−1 }. On the other hand, CB = C −{K1 , K2 , · · · , Kn } is irreducible. So CB is a reduction of C. The following theorem gives the relationship between attribute reductions and covering reductions. Theorem 3.10. Let (U, A, R) be a formal context and let (U, C) be a covering approximation space induced by the (U, A, R), where C = {Kα : α ∈ A}. Then the following are equivalent for B ⊆ A. (1) B is an attribute reduction of the object oriented concept lattice D(U, A, R). (2) CB is a reduction of C. Proof. Let T be the generalized topologies on U induced by C. (1) =⇒ (2): Assume that B is an attribute reduction of the object oriented concept lattice D(U, A, R). By Lemma 3.9, we only need to prove that CB is an irreducible base for T . Note that B is a consistent set of D(U, A, R). By Corollary 3.7, CB is a base for T . It suffices to prove that CB is irreducible. Whenever β ∈ B. Since T B is an attribute reduction of D(U, A, R), D(U, A, R) 6∼ = D(U, B − {β}, R (U × (B − {β}))), i.e., β is a indispensable attribute in D(U, A, R). By Lemma 3.8, Kβ is a irreducible element of C. Furthermore, Kβ is a irreducible element of CB . So CB is irreducible. (2) =⇒ (1): Assume that CB is a reduction of C. By Lemma 3.9, CB is an irreducible base for T . By Corollary 3.7, B is a consistent set of the object oriented concept lattice D(U, A, R) It suffices to prove that D(U, A, R) 6∼ = T D(U, B−{β}, R (U ×(B−{β}))) for whenever β ∈ B.TIn fact, if not, then there ∼ exists β ∈ B such that D(U, A, (U × (B − {β}))). Note T R) = D(U, B − {β}, R T ∼ that D(U, A, R) D(U, B, R (U × B)). So D(U, B, R (U × B)) ∼ = = D(U, T TB− {β}, R (U × (B − {β}))). i.e., β is a dispensable attribute in D(U, B, R (U × B)). By Lemma 3.8, Kβ is a reducible element of CB . This contradicts that CB is irreducible. Lemma 3.11. Let (U, C) be a covering approximation space. If C1 and C2 are two reductions of C, then C1 = C2 . Proof. Let C1 and C2 be two reductions of C and T be the generalized topologies on U induced by C. Then C1 and C2 are two irreducible base for S T from Lemma 3.9. Let K ∈ C1 , then there is F ⊆ C2 } such that K = F. Since C1 is irreducible, K is not a reducible element of C1 , i.e., K is not a union of some elements of C1 − {K}. It follows that there is K 0 ∈ F such that K 0 is not a union of some elementsSof C1 − {K}. Note that C1 is a base. So there is F 0 ⊆ C1 } such that K 0 = F 0 and K ∈ F 0 . Thus, we have K ⊇ K 0 ⊇ K, hence 8

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K = K 0 ⊆ F ⊆ C2 . This proves that C1 ⊆ C2 . By the same way, we can prove that C2 ⊆ C1 . Consequently, C1 = C2 . Corollary 3.12. Let (U, A, R) be a formal context. If B1 and B2 are two attribute reductions of the object oriented concept lattice D(U, A, R), then RB1 = RB2 , where RBi = {Rβ : β ∈ Bi } for i = 1, 2. In particularly, if B1 and B2 are two attribute reductions of D(U, A, R), then |B1 | = |B2 |, where |A| denotes the cardinal number of A for a set A. By Corollary 3.12, we have the following theorem, which give the covering method to compute all attribute reductions of the object oriented concept lattice D(U, A, R). Theorem 3.13. Let (U, A, R) be a formal context and let E be the family of all attribute reductions of the object oriented concept lattice D(U, A, R). If B = {β1 , β2 , · · · , βn } is an attribute reduction of D(U, A, R), then E = {{α1 , α2 , · · · , αn } : Rαi = Rβi , i = 1, 2, · · · , n}. Just as stated in Remark 2.13, all results for object reductions of the object oriented concept lattice D(U, A, R) can be obtained by the same method, and we omit these results. The following is an example of a formal context (U, A, R), which comes from [10, Table 1]. As an application of results in this paper, we use the covering method to compute easily attribute or object reductions of D(U, A, R) and reductions of (U, A, R), where D(U, A, R) is the object oriented concept lattice generated by (U, A, R). Example 3.14. A formal context (U, A, R) is described as the following Table 1, where U = {1, 2, 3, 4, 5, 6}, A = {a, b, c, d, e, f }, and the number, which lies in the cross of the row labeled by i (i = 1, 2, 3, 4, 5, 6) and the column labeled by t (t = a, b, c, d, e, f ), is 1 or 0 by (i, t) ∈ R or (i, t) 6∈ R. Table 1: A formal context (U, A, R) 1 2 3 4 5 6

a 1 1 0 0 1 0

b 0 1 0 0 1 0

c 0 0 0 1 0 1

d 1 1 1 0 1 0

e 0 1 0 1 1 1

f 1 1 0 0 1 0

(1) (U, A, R) induces a covering approximation space (U, C), where Ka = {1, 2, 5}, Kb = {2, 5}, Kc = {4, 6}, Kd = {1, 2, 3, 5}, Ke = {2, 4,S 5, 6}, Kf = {1, 2, 5}, and C = {Kα : α ∈ A}. It is easy to see that Ke = Kb Kc , Kf = Ka , and {Ka , Kb , Kc , Kd } is irreducible. So {Ka , Kb , Kc , Kd } is a base for 9

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the generalized topology induced by C. By Theorem 3.9, {Ka , Kb , Kc , Kd } is a reduction of C. It follows that {a, b, c, d} is an attribute reduction of the object oriented concept lattice D(U, A, R). Put E is the family of all attribute reductions of D(U, A, R). By Theorem 3.13, E = {{a, b, c, d}, {b, c, d, f }}. (2) (U, A, R) induces a covering approximation space (A, J ), where L1 = {a, d, f }, L2 = {a, b, d, e, f }, L3 = {d}, L4 = {c, e}, L5 = {a, b, d, e, f }, L6 = {c, e}, and J = {Lu : u ∈ U }. It is easy to see that L2 = L5 , L4 = L6 , and {L1 , L2 , L3 , L4 } is irreducible. So {L1 , L2 , L3 , L4 } is a base for the generalized topology induced by J . By Theorem 3.9, {L1 , L2 , L3 , L4 } is a reductions of J . It follows that {1, 2, 3, 4} is an object reduction of the object oriented concept lattice D(U, A, R). Put G is the family of all object reductions of D(U, A, R). Then G = {{1, 2, 3, 4}, {1, 2, 3, 6}, {1, 3, 4, 5}, {1, 3, 5, 6}}. (3) By the above (1) and (2), we can T obtain eight reductions of (U, A, R), i.e., we can obtain a reduction (V, B, R (V × B)) of (U, A, R) for each V ∈ G and each B ∈ E. For example, if we pick {1, 2, 3, 6} ∈ G and {b, c, d, f } ∈ E, then the following reductions of (U, A, R) is obtained. Table 2: A reduction of the formal context (U, A, R) b 0 1 0 0

1 2 3 6

c 0 0 0 1

d 1 1 1 0

f 1 1 0 0

References [1] Z.Bonikowski, E.Bryniarski, U.Wybraniec, Extensions and intentions in the rough set theory, Information Sciences, 107(1998), 149-167. [2] C.Carpineto, G.Romano, A lattice conceptual clustering system and its application to browsing retrieval, Machine Learning, 10(1996), 95-122. [3] J.S.Deogun, J.Saquer, Monotone concepts for formal concept analysis, Discrete Applied Mathematics, 144(2004), 70-78. [4] M.Faid, R.Missaoi, R.Godin, Mining complex structures using context concatenation in formal concept analysis, in: International KRUSE Symposium, Vancouver, BC, 11-13 August, 1997. [5] X.Ge, An application of covering approximation spaces on network security, Computers and Mathematics with Applications, 60(2010), 1191-1199. [6] R.Godin, R.Missaoi, An incremental concept formation approach for learning from databases, in: Formal methods in databases and software engineering, Theoretical Computer Science, 133(1994), 387-419. 10

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[7] B.Ganter, R.Wille, Formal concept analysis: Mathematical Foundations, Springer, Berlin, 1999. [8] X.Ge, S.Yang, Remarks on Si -separation axioms, Filomat, 26(2012), 839844. [9] S.K.Harms, J.S.Deogum, Sequential association rule mining with time lags, Journal of Intelligent Information Systems, 22(1)(2004), 7-22. [10] M.Liu, M.Shao, W.Zhang, C.Wu, Reduction method for concept lattices based on rough set theory and its application, Computers and Mathematics with Applications, 53(2007), 1390-1410. [11] Z.Li, T.Xie, Q.Li, Topological structure for generalized rough sets, Computers and Mathematics with Applications, 63(2012), 1066-1071. [12] R.Wille, Restructuring lattice theory: An approach based on hierarchies of concepts, in: I. Rival (Ed.), Ordered sets, Reidel, Dordrecht, Boston, 1982, pp. 445-470. [13] R.Wille, Knowledge acquisition by methods of formal concept analysis, in: E. Diday (Ed.), Data Analysis, Learning Symbolic and Numeric Knowledge, Nova Science, New York, 1989, pp. 365-380. [14] W.Wu, Y.Leung, J.Mi, Granular computing and knowledge reduction in formal contexts, IEEE Transactions on Knowledge and Data Engineering, 21(10)(2009), 1461-1474. [15] L.Wei, J.Qi, W.Zhang, Attribute reduction theory of concept lattice based on decision formal contexts, Science in China: Series F - Information Sciences, 51(7)(2008), 910-923. [16] Y.Y.Yao, Concept lattices in rough set theory, in: Proceedings of 23rd International Meeting of the North American Fuzzy Information Processing Society, 2004. [17] Y.Y.Yao, A comparative study of formal concept analysis and rough set theory in data analysis, rough sets and current trends in computing, in: Proceedings of 3rd International Conference, 2004. [18] X.Zhang, L.Wei, J.Qi, Attribute reduction theory and approach to concept lattice, Science in China: Series F - Information Sciences, 48(6)(2005), 713-726. [19] W.Zhu, Topological approaches to covering rough sets, Information Sciences, 177(2007), 1499-1508. [20] W.Zhu, F.Wang, Reduction and axiomization of covering generalized rongh sets, Information Sciences, 152(2003), 217-230.

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A new two-step iterative method for solving nonlinear equations Shin Min Kang Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Korea [email protected]

Arif Rafiq Department of Mathematics, Lahore Leads University, Lahore, Pakistan [email protected]

Faisal Ali Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 54000, Pakistan [email protected]

Young Chel Kwun∗ Department of Mathematics, Dong-A University, Pusan 614-714, Korea [email protected]

Abstract In this paper, a new iterative method for solving nonlinear equations is developed by using modified homotopy perturbation method. The convergence analysis of the proposed method is also given. The validity and efficiency of our method is illustrated by applying this new method along with some other existing methods on various test problems. Key words: Homotopy perturbation method; nonlinear equations; iterative methods

∗

Corresponding author.

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1

Introduction

The construction of numerical methods for solving nonlinear equations has been a sweltring subject for many researchers. Consequently, many techniques like modified Adomian decomposition, fixed point, homotopy perturbation etc. have been developed for the purpose. [1–14]. Homotopy perturbation method(HPM) is to continuously transform a simple problem(easy to solve) into the under study problem(difficult to solve). Because of this fact, scientists and engineers have applied HPM extensively to linear and nonlinear problems. HPM was proposed first by He [15] in 1999 and systematical description which is, in fact, a coupling of traditional pertubation method and homotopy in topology [16]. This method then further improved by He and applied to nonlinear oscillator with discontinuities [17], nonlinear wave equation [18], asymptotology[19], boundary value problem [20], limit cycle and bifurcation of nonlinear problems [21] and many other fields. Thus He’s method can be applied to solve various types of nonlinear equations. Subsequently, many kinds of linear and nonlinear problems have been dealt with HPM [22–28]. This paper presents a new numerical technique based on modified homotopy perturbation method to solve linear and nonlinear algebraic equations. The proposed method and a few existing methods are applied for comparison purpose to several test problems in order to reveal the accuracy and convergence of our method. We need the following results in the sequal. Theorem 1 [29]. Suppose f is continuous on [a, b] and differentiable in (a, b).Then there exists a point δ ∈ (a, b) such that f(y) = f(x) + (y − x)f 0 (x) + 21 (y − x)2 f 00(δ). For the solution of nonlinear equation by fixed point iterative method, the equation f (x) = 0 is usually rewritten as

x = g (x) ,

(1)

where (i) there exists [a, b] such that g (x) ∈ [a, b] for all x ∈ [a, b], (ii) there exists L > 0 such that |g 0 (x) | ≤ L < 1 for all x ∈ (a, b). 2

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Considering the following iterative scheme:

xn+1 = g (xn ) ,

n = 0, 1, 2, ...,

(2)

and starting with a suitable initial approximation x0 , we build up a sequence of approximations, say {xn } , for the solution of the nonlinear equation, say α. The scheme will converge to the root α, provided that (i) the initial approximation x0 is chosen in the interval [a, b] , (ii) g has a continuous derivative on (a, b) , (iii) |g 0 (x) | < 1 for all x ∈ [a, b] , (iv) a ≤ g (x) ≤ b for all x ∈ [a, b]. (see [30]) The order of the convergence for the sequence of approximations derived from an iterative method is defined in the literature as: Definition 1. Let {xn } converges to α. If there exists an integer constant p and real positive constant C such that x n+1 − α lim = C, n→∞ (xn − α)p

then p is called the order and C the constant of convergence. To determine the order of convergence of sequence {xn } , let us consider the Taylor expansion of g (xn ) :

g (xn ) = g (x) +

g 0 (x) g 00 (x) g (k) (x) (xn − x) + (xn − x)2 + ... + (xn − x)k + ... 1! 2! k! (3)

using Eq.(1) and Eq.(2) in Eq.(3), we have xn+1 − x = g 0 (x) (xn − x) +

g 00 (x) 2!

(xn − x)2 + ... +

g (k) (x) 3!

(xn − x)k + ...,

and state the following theorem: Theorem 2 [4]. Suppose g ∈ C p [a, b]. If g (k) (x) = 0, for k = 1, 2, ..., p − 1 and g (p) (x) 6= 0, then the sequence {xn } is of order p. 3

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Definition 2. If an iterative method has order of convergence k with number 1 of evaluations in a single iteration e, then (k) e is called the efficiency index of the method. For example, the efficiency index of Newton Raphson methed, Javidi method [31], Noor et al. method [32], Rafiq and Rafiullah method [11] and Algorithm 2 of the present paper are 1.41421, 1.31607, 1.44225, 1.44225 and 1.44225 respectively.

2

Modified homotopy perturbation method

Consider the nonlinear algebraic equation f (x) = 0, x ∈ R.

(4)

We assume that r is a simple zero of Eq.(4) and α is an initial guess sufficiently close to r. Using Theorem 1 around α for Eq.(4), we have

f (α) + (x − α) f 0 (α) +

1 (x − α)2 f 00 (δ) = 0. 2

(5)

where δ lies between x and α. We can rewrite Eq.(5) in the following form

x = c + N (x) ,

(6)

f (α) f 0 (α)

(7)

where

c=α− and

1 f 00 (δ) N (x) = − (x − α) 0 . 2 f (α)

(8)

To illustrate the basic idea of modified homotopy perturbation method, we construct a homotopy Θ : (R × [0, 1]) × R → R for Eq.(6), which satisfies 4

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Θ ($, β, θ) = $−c−βN ($)+β 2 (1 − β) θ = 0,

θ, $ ∈ R

and β ∈ [0, 1] , (9)

where θ is unknown real number and β is embedding parameter. It is obvious that

Θ ($, 0, θ) = $ − c = 0,

(10)

Θ ($, 1, θ) = $ − c − N ($) = 0,

(11)

The embedding parameter β monotonically increases from zero to unity as the trivial problem Θ ($, 0, θ) = $ − c = 0 is continuously deformed to original problem Θ ($, 1, θ) = $ − c − N ($) = 0. The modified HPM uses the homotopy parameter β as an expanding parameter to obtain (see [15] and the references there in) $ = x0 + βx1 + β 2x2 + ...

(12)

The approximate solution of Eq.(4), therefore, can be readily obtained:

r = lim $ = x0 + x1 + x2 + ...

(13)

β→1

The convergence of the series (13) has been proved by He [33]. For the application of modified HPM to Eq.(4), we can write Eq.(9) as follows by expanding N ($) into a Taylor series around x0 :

$−c−β{N (xo )+($ − x0)

00 N 0 (x0) 2 N (x0 ) +($ − x0 ) +...}+β 2 (1 − β) θ = 0. 1! 2! (14)

Substituting Eq.(12) into Eq.(14) yields 

x0 + βx1 + β 2x2 + ... − c − β{N (x0) + x0 + βx1 + β 2x2 + ... − x0 

x0 + βx1 + β 2x2 + ... − xo

2 N 00 (x ) 0

2!

5

115

 N 0 (x ) 0

+ ...} + β 2 (1 − β) θ = 0.

1!

+

(15)

Shin Min Kang et al 111-132

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

By equating the terms with identical powers of β, we have β 0 = x0 − c = 0,

(16)

β 1 = x1 − N (x0) = 0,

(17)

β 2 = x2 − x1N 0 (x0 ) + θ = 0,

(18)

1 β 3 = x3 − x2 N 0 (x0) − x21 N 00 (x0) − θ = 0. 2

(19)

We try to find parameter θ, such that x2 = 0.

(20)

Hence by substituting x1 = N (x0 ) from Eq.(17) into Eq.(18), we have x2 − N (x0) N 0 (x0) + θ = 0.

(21)

Setting x2 = 0 into Eq.(21), we have θ = N (x0 ) N 0 (x0 ) .

(22)

Using Eq.(22), x1 = N (x0) , and x2 = 0 into Eq.(19), we obtain 1 x3 = N 2 (x0 ) N 00 (x0) + N (x0) N 0 (x0) . 2

(23)

1 [f (α)]2 f 00 (δ) N (x0 ) = − , 2 [f 0 (α)]3

(24)

. . . where

6

116

Shin Min Kang et al 111-132

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

f (α) f 00 (δ) N (x0) = 2 , [f 0 (α)] 0

N 00 (x0) = −

(25)

f 00 (δ) . f 0 (α)

(26)

f (α) . f 0 (α)

(27)

Combining Eq.(7) and Eq.(16), we get

x0 = c = α −

Now using equations(24)-(26) in Eq.(17) and Eq.(23), we have

x1 = −

1 [f (α)]2 f 00 (δ) 2 [f 0 (α)]3

(28)

and 1 x3 = − 8

[f (α)]4 [f 00 (δ)]3 + 4 [f (α)]3 [f 0 (α)]2 [f 00 (δ)]2 . [f 0 (α)]7 !

(29)

Using equations(20), (27), (28) and (29) into Eq.(13), we obtain the solution of Eq.(4) as follows: f (α) 1 [f (α)]2 f 00 (δ) r = x0 + x1 + x2 + x3 + ... = α − 0 − − f (α) 2 [f 0 (α)]3 1 8

[f (α)]4 [f 00 (δ)]3 + 4 [f (α)]3 [f 0 (α)]2 [f 00 (δ)]2 [f 0 (α)]7

!

+ ...

(30)

This formulations allows us to suggest the following iterative method for solving nonlinear Eq.(4). Algorithm 1 [32] For a given w0 , calculate the approximate solution wn+1 by the iterative scheme wn+1 = wn − y n = wn −

f (wn ) f 0 (wn )

f (wn ) ; f 0 (wn )

−

2 1 [f (wn )] f 00 (yn ) , 0 2 [f (wn )]3

f 0 (wn ) 6= 0.

Algorith 1 has sbeen suggested by Norr et al. [32] which we derive in this paper by using modified HPM. 7

117

Shin Min Kang et al 111-132

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Algorithm 2. For a given w0, calculate the approximate solution wn+1 by the iterative scheme wn+1 = wn − y n = wn −

3

f (wn ) f 0 (wn )

f (wn ) ; f 0 (wn )

−

2 1 [f (wn )] f 00 (yn ) 0 2 [f (wn )]3

−

1 8



[f (wn )]4 [f 00 (yn )]3 +4[f (wn )]3 [f 0(wn )]2 [f 00 (yn )]2 [f 0(wn )]7



,

f 0 (wn ) 6= 0.

Convergence analysis

We discuss the convergence analysis of algorithm 2. Theorem 3 Consider the nonlinear equation f (x) = 0. Suppose f is sufficently differentiable. Then for the iterative method defined by Algorithm 2, the convergence is at least of order 3.

.

Proof: For the iteration function defined as in Algorithm 2, which has a fixed point r, let us define

y =x−

f(x) , f 0 (x)

(31)

and

h

i2

h

i3

3 2 4 0 f(x) 1 [f(x)]2 f 00(y) 4 [f(x)] [f (x)] f (y) + [f(x)] f (y) g(x) = x− 0 − − . 7 f (x) 2 [f 0 (x)]3 8 [f 0 (x)] (32) 00

00

We obtain the derivatives and the values of derivatives at r by using the software Mathematica. 2

g 0 (x) =

f(x)f 00(x) f(x)f 00(y) 3 [f(x)] f 00(x)f 00 (y) − + + [f 0 (x)]2 [f 0 (x)]2 2 [f 0 (x)]4 h

7f 00 (x)(4 [f(x)]3 [f 0 (x)]2 f (y) 00

8 [f 0 (x)]8 h

i2

h

i3

+ [f(x)]4 f (y) )

i2

00

−

h

i2

2 3 3 0 0 00 [f(x)] f (x)f (y) (12 [f(x)] [f (x)] f (y) + 8 [f(x)] f (x)f (x) f (y) − + 2 [f 0 (x)]5 8 [f 0 (x)]7 3

00

000

8

00

118

00

Shin Min Kang et al 111-132

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

h

4 [f(x)]3 f 0 (x) f (y) 00

i3

3[f (x)]5 f 00(x)[f 00(y)]2 f (3) (y) ) [f 0 (x)]2

+ 8 [f(x)]4 f 00(x)f 00(y)f (3)(y) + 8 [f 0 (x)]7

.

(33)

g 00(x) =

f 00(x) (−2 [f 0 (x)]2 − 2f(x)f 00 (x))f 00 (y) − + f 0(x) 2f 03 (x) h

(4 [f(x)]3 [f 0 (x)]2 f (y)

f(x)(

00

i2

00

7f (3) (x) ) [f 0 (x)]8

8

−

−3f 00(x)f 00(y) f(x)f 00(x)f (3)(y) f (3) (x) 2 [f 00 (x)]2 0 (x)( − ) − 2f(x)f + )+ 3 2 4 5 [f 0 (x)] [f 0 (x)] [f 0 (x)] [f 0 (x)]

h

(7f 00 (x)(12 [f(x)]2 [f 0 (x)]3 f (y)

h

4 [f(x)]3 f 0 (x) f (y) 00

i3

00

i2

h

+ 8 [f(x)]3 f 0 (x)f 00(x) f (y)

4 [f 0 (x)]8

+ 8 [f(x)]4 f 00(x)f 00 (y)f (3)(y) +

2

00

00

(x)

i2

[f 0 (x)]6

f (3) (y)

f 00 (x)f (3) (y) + − f 0 (x)

i2

+

3[f (x)]5 f 00 (x)[f 00 (y)]2 f (3) (y) )) [f 0 (x)]2

3f (3) (x) ) [f 0 (x)]4

2

6f (x) f

00

4 [f 0 (x)]8

(x)] ([f(x)]2 (f 00(y)( 12[f − [f 0(x)]5

h

2

(x)] + [f(x)]4 [f 00(y)]3 )( −56[f + [f 0 (x)]9

h

2f (x) f

00

(x)

i2

f (3) (y)

[f 0 (x)] 3 [f 0 (x)]3

h

(4 [f 0 (x)]2 (6f(x) [f 0 (x)]2 + 3 [f(x)]2 f 00 (x)) f (y)

9

−

f (x)f (3) (x)f (3) (y) + + [f 0 (x)]2

2

119

00

i2

−

h

[f (x)]2 f

00

(x)

i2

[f 0 (x)]4

f (4) (y)

))

−

+ (12 [f(x)]2 [f 0 (x)]2 + 4 [f(x)]3 f 00(x)) [f 00(y)]3

8f 07 (x)

Shin Min Kang et al 111-132

+

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

24[f (x)]4 f 00 (x)[f 00 (y)]2 f (3) (y) f 0(x)

+ 24 [f(x)]2 f 0 (x)(2f 0 (x)f 00 (x) [f 00 (y)]2 + 2f(x)f 00 (x)f 00(y)f (3)(y)) 8f 07 (x)

[f(x)]4 (

h

00

3f 00 (x) f (y)

i2

f (3) (y)

f 0 (x)

−

h

00

6f (x) f (x)

i2 h

00

f (y)

[f 0 (x)]3 7 0

i2

f (3) (y)

+

h

00

3f (x) f (y)

i2

f (3) (x)f (3) (y)

[f 0 (x)]2

+

8 [f (x)]

h

00

6[f (x)]2 f (x)

i2

h

2

f 00 (y)[f (3) (y)]

[f 0 (x)]4

00

3[f (x)]2 [f 00 (x)]2 f (y)

+

[f 0 (x)]4

i2

f (4) (y)

+

7

8 [f 0 (x)] h

i2

h

i2

4 [f(x)]3 ( f (y) (2 f (x) 2f 0 (x)f (3)(x)) + 00

2

[f (x)] ( 0

00

h

2f 00(x)f 00 (y)f (3) (y) f 0 (x)

−

h

00

4f (x) f (x)

00

8f (x) f (x)

8 [f 0 (x)]7 i2

i2

[f 0(x)]

2

+

h

00

i2

2

[f (3) (x)]

+

4

[f 0 (x)]

+

f 00 (y)f (3) (y)

[f 0 (x)]3

+

8 [f 0 (x)]

2[f (x)]2 f (x)

f 00 (y)f (3) (y)

f 0 (x)

7

2f (x)f 00(y)f (3) (x)f (3) (y)

+

h

00

2[f (x)]2 f (x)

i2

f 00 (y)f (4) (y)

[f 0 (x)]4

))). (34)

8 [f 0 (x)]7

2

2

3 [f 00 (x)] f (3)(x) f 00(y)(−6f 0 (x)f 00(x) − 2f(x)f (3) (x)) 2 [f 00 (x)] f (3) (x) 0 g (x) = + − −3f (x)( − )+ f 0 (x) [f 0 (x)]2 2 [f 0 (x)]3 [f 0 (x)]3 [f 0 (x)]2 000

00

00

(y) 3(−2 [f 0 (x)]2 − 2f(x)f 00 (x))( −3f[f 0(x)f + (x)]4

2

00

2

(x)] (3( −56[f + [f 0 (x)]9

7f (3) (x) )(12 [f(x)]2 [f 0 (x)]8

h

[f 0 (x)]3 f (y) 8 10

00

120

i2

f (x)f 00(x)f (3) (y) ) [f 0 (x)]5

+

h

+ 8 [f(x)]3 f 0 (x)f 00(x) f (y) 00

i2

+

Shin Min Kang et al 111-132

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

h

4 [f(x)]3 f 0 (x) f (y) 00

h

i3

+ 8 [f(x)]4 f 00(x)f (y)f (3) (y) +

(4 [f(x)]3 [f 0 (x)]2 f (y)

3

00

[f 0 (x)]2

i2

f (3) (y)

))

+

8

i2

h

i3

3

00

(x)] + [f(x)]4 f (y) )( 504[f − [f 0 (x)]10

(3)

00

8

168f 00(x)f (3) (x) [f 0 (x)]9

h

00

i2

6f(x) f (x) f

00

h

00

12 f (x)

(3)

(y)

6

[f 0 (x)]

+

(4)

7f (4) (x) ) 8

+

−

i2

−6 [f (x)] 6f (x)f (x) f (x) 3f (3)(x) 0 00 (x)(f (y)( + − )−3f(x)f − )− [f 0 (x)]4 [f 0 (x)]3 [f 0 (x)]2 [f 0 (x)]5 [f 0 (x)]4 00

f(x)(

00

00

h

3[f (x)]5 f 00 (x) f (y)

f 00 (x)f (3) (y) f 0 (x)

−

h

00

2f (x) f (x)

i2

f (3) (y)

[f 0 (x)]3

+

f (x)f (3) (x)f (3) (y) [f 0 (x)]2 3 0

h

00

8

8 [f 0 (x)]

h

(12 [f(x)]2 [f 0 (x)]2 4 [f(x)]3 f 00(x)) f (y) 8

8 [f 0 (x)] h

00

24[f (x)]4 f 00 (x) f (y) f 0 (x)

8 [f 0 (x)]8 h

i2

00

i3

00

i2

00

8 [f 0 (x)]8

[f(x)]4 (

00

i2

f 0 (x)

f (3) (y)

i2

f (4) (y)

[f 0 (x)]4

−

h

00

6f (x) f (x)

i2 h

f (3) (y)

+

i2

00

f (y) 3

[f 0 (x)] 8 0

8 [f (x)] 11

121

+

+

24 [f(x)]2 f 0 (x)(2f 0 (x)f 00(x) f (y) + 2f(x)f (y)f (3) (y))

h

00

[f (x)]

(21f (x)(4 [f 0 (x)]2 (6f(x) [f 0 (x)]2 + 3 [f(x)]2 f 00 (x)) f (y)

3f 00 (x) f (y)

+

h

[f (x)]2 f (x)

i2

00

f (3) (y)

+

h

00

3f (x) f (y)

i2

+

f (3) (x)f (3) (y)

[f 0 (x)]2

+

Shin Min Kang et al 111-132

)

)+

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

h

00

6[f (x)]2 f (x)

i2

h

00

00

f (y) f (y) 4

[f 0 (x)]

i2

h

00

3[f (x)]2 f (x)

+

i2 h

00

f (y) 4

[f 0 (x)]

i2

f (4) (y)

8

8 [f 0 (x)] h

3

i2

00

h

00

4 [f(x)] ( f (y) (2 f (x)

h

00

8f (x) f (x)

i2

f 00 (y)f (3) (y)

f 0(x)

+ [f (x)] 0

8 [f 0 (x)]8

2

i2

+ 2f 0 (x)f (3) (x))

00 (y)f (3) (y) 00 ( 2f (x)ff 0 (x) 8 0

h

+

+

00

4f (x) f (x)

−

)

i2

f 00(y)f (3) (y)

[f 0(x)]3

+

8 [f (x)]

2f (x)f 00(y)f (3) (x)f (3) (y) [f 0 (x)]2

2[f (x)]2 [f 00(x)]2 [f (3) (y)]

+

2

4

[f 0 (x)]

+

2[f (x)]2 [f 00 (x)]2 f 00 (y)f (4) (y) )))) [f 0 (x)]4

8 [f 0 (x)]8

h

12 f

2

3f (x)f 00(x)(

([f(x)] (

00

(x)

i2

[f 0 (x)]5

−

3f (3) (x) )f (3) (x) f 0 (x) 4

[

]

[f 0 (x)]2

+ f 00 (y)(

h

00

−60 f (x) [f 0(x)]6

i3

+

−

36f 00(x)f (3) (x) [f 0 (x)]5

−

3f (4) (x) ) [f 0 (x)]4

2

f 00 (x)f (3) (y) 9f 00 (x)( − f 0 (x)

h

2f (x) f

00

(x)

i2

[f 0 (x)] 3

f (3) (y)

+

f (x)f (3) (x)f (3) (y)

[f 0 (x)]2

h

[f (x)]2 f

+

00

(x)

i2

f (4) (y)

(

[f 0 (x)]4

[f 0 (x)]4

h

−3 f

)+

00

(x)

i2

f (3) (y)

−

+

[f 0 (x)]2

h

6f (x) f

[f 0 (x)]3

00

(x)

2f (3) (x)f (3) (y) 6f (x)f − f 0 (x)

(x)f (3) (x)f (3) (y) f (x)f (3) (y)f (4) (x) + + [f 0 (x)]3 [f 0 (x)]2

[f 0(x)]3

h

3f (x) f

00

(x)

i2

f (4) (y)

−

[f 0 (x)] 3

h

6[f (x)]2 f

00

[f 0 (x)]3

2

3 f 00 (x) 3 f (5) (y) [ ] ) [f 0 (x)]6

))

i3

f (4) (y)

[f 0 (x)] 5

+

2

3[f (x)]2 f 00 (x)f (3) (x)f (4) (y) [f (x)] + [f 0 (x)]4

(x)

2

2

3

−(4 [f 0 (x)] ([f 00(y)] (6 [f 0 (x)] +18f(x)f 0 (x)f 00(x)+

12

122

f (3) (y)

[f 0 (x)]4

2

00

i3

Shin Min Kang et al 111-132

+

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

h

i3

3

3 [f(x)]2 f (3)(x))+ f (y) (24f(x) [f 0 (x)] +36 [f(x)]2 f 0 (x)f 00(x)+4 [f(x)]3 f (3)(x))+ 00

h

i2

9f(x)f 00 (x)(12 [f(x)]2 [f 0 (x)]2 + 4 [f(x)]3 f 00(x) f (y) f (3)(y) [f 0 (x)]2 2

h

00

+

i2

12(6f(x) [f 0 (x)] +3 [f(x)]2 f 00(x))(2f 0 (x)f 00(x) f (y) +2f(x)f 00 (x)f 00(y)f (3) (y))+

12 [f(x)]3 f 0 (x)(

2

00

(3)

3f(x) [f (y)] f (x)f 2 [f 0 (x)] 00

h

2

3f (x) [f (y)] f f 0 (x) 00

i2

(3)

(y)

3 [f(x)]2 f (x) [f 00 (y)]2 f (4)(y) 00

[f 0 (x)]4

(3)

(y)

00

2

−

i2

00

3

[f 0 (x)] 2

+

h

6f(x) [f 0 (x)] f (y) f (3)(y)

2

h

6 [f(x)] [f 00 (x)] f 00 (y) f (3)(y) [f 0 (x)]

4

2

i2

+

+

2

+36 [f(x)]2 f 0 (x)([f 00 (y)] (2 [f 00(x)] +2f 0 (x)f (3) (x))+

00 00 (3) 8f(x) [f 00 (x)]2 f 00 (y)f (3) (y) (y) 4f(x) [f 00 (x)]2 f 00 (y)f (3) (y) 2 2f (x)f (y)f 0 + +[f (x)] ( − f 0 (x) f 0 (x) [f 0 (x)]3

h

i2

2 2 00 (3) 2f(x)f 00 (y)f (3)(x)f (3)(y) 2[f(x)] [f (x)] f (y) 2[f(x)]2 [f 00 (x)]2 f 00 (y)f (4) (y) + + ))+ [f 0 (x)]2 [f 0 (x)]4 [f 0 (x)]4

[f(x)]4(

−9 [f 00(x)]2 [f 00 (y)]2 f (3)(y) 18f(x) [f 00 (x)]3 [f 00(y)]2 f (3) (y) 6 [f 00(y)]2 f (3) (x)f (3) (y) + + − 2 4 f 0 (x) [f 0 (x)] [f 0 (x)]

2

18f(x)f (x) [f (y)] f [f 0 (x)]3 00

00

(3)

(x)f

(3)

(y)

+

13

h

18f(x) [f 00(x)]2 f 00 (y) f (3)(y) f 0 (x)

123

i2

−

Shin Min Kang et al 111-132

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.1, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

h

36[f(x)]2 [f 00(x)]3 f 00 (y) f (3) (y) [f 0 (x)]5

i2

h

+

6[f(x)]3 [f 00(x)]3 f 00(y) f (3)(y) [f 0 (x)]6

h

18[f(x)]2f 00 (x)f 00(y)f (3)(y) f (3)(y) [f 0 (x)]4

i3

i2

+

3f(x) [f 00 (y)]2 f (3)(y)f (4)(x) + [f 0 (x)]5

+

9f(x) [f 00(x)]2 [f 00 (y)]2 f (4)(y) 18 [f(x)]2 [f 00 (x)]3 [f 00(y)]2 f (4) (y) − + [f 0 (x)]3 [f 0 (x)]5 2

2

9 [f(x)] f 00 (x) [f 00 (y)] f (3) (x)f (4) (y) + [f 0 (x)]4 3

3

3

3

2

18 [f(x)] [f 00 (x)] f 00(y)f (3) (x)f (4) (y) 3 [f(x)] [f 00(x)] [f 00 (y)] f (5)(y) + )+ [f 0 (x)]6 [f 0 (x)]6

4 [f(x)]3 ( 6f (x)f

00(x)f 00 (y)(2[f 00 (x)]2 +2f 0 (x)f (3) (y))f (3) (y)

[f 0 (x)]6

+ [f 00 (y)]2 (6f 00 (x)f (3)(x) + 2f 0 (x)f (4) (x))

8 [f 0 (x)]7 6f 0 (x)f 00(x)( 2f

00 (x)f 00 (y)f (3) (y)

f 0 (x)

−

4f (x)[f 00 (x)]2 f 00 (y)f (3) (y) [f 0 (x)]3

8 [f 0 (x)]7

2f (x)f 00(y)f (3) (x)f (3) (y) [f 0 (x)]2

+

2[f (x)]2 [f 00 (x)]2 [f (3) (y)] [f 0 (x)]4

2

+

+

2[f (x)]2 [f 00 (x)]2 f 00 (y)f (4) (y) ) [f 0 (x)]4

8 [f 0 (x)]7

[f 0 (x)]2( −6[f

00 (x)]2 f 00 (y)f (3) (y)

[f 0 (x)]2

+

12f (x)[f 00 (x)]3 f 00 (y)f (3) (y) [f 0 (x)]4 7 0

+

4f 00 (y)f (3) (x)f (3) (y) f 0 (x)

8 [f (x)]

12f (x)f 00(x)f 00(y)f (3) (x)f (3) (y) [f 0(x)]3

2

+

6f (x)[f 00 (x)]2 f 00 (y)[f (3) (y)] 3

[f 0 (x)] 7 0

8 [f (x)] 14

124

−

+

−

12[f (x)]2 [f 00 (x)]3 [f (3) (y)] 5

[f 0 (x)]

2

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6[f (x)]2 f 00 (x)f (3) (x)[f (3) (y)]

2

2f (x)f 00(y)f (3) (y)f (4) (x) [f 0 (x)]2

+

[f 0 (x)]4

+

6f (x)[f 00 (x)]2 f 00 (y)f (4) (y) [f 0(x)]3 7

−

12[f (x)]2 [f 00 (x)]3 f 00 (y)f (4) (y) [f 0 (x)]5

8 [f 0 (x)]

6[f (x)]2 f 00 (x)f 00 (y)f (3) (x)f (4) (y) [f 0 (x)]4

6[f (x)]3 [f 00 (x)]3 f (3) (y)f (4) (y) [f 0 (x)]6 7 0

+

+

2[f (x)]3 [f 00 (x)]3 f 00 (y)f (5) (y) ))) [f 0 (x)]6

8 [f (x)]

.

(35) Now, one can easily see that

0

g(r) = r,

00

g (r) = 0 = g (r),

6 [f 00 (r)]2 f (3) (r) 2 [f 00 (r)]2 [f(r)]3 0 g (r) = − 0 −3f (r)( 0 − ). [f 0 (r)2 ] f (r) [f (r)]3 [f 0 (r)]2 (36) 000

But g 000(x) 6= 0 provides that 6[f 00 (r)]2 [f 0 (r)2 ]

−

f (3) (r) f 0(r)

2

00

(r)] − − 3f 0 (r)( 2[f [f 0(r)]3

[f (r)]3 ) [f 0 (r)]2

6= 0

and so, in general, the Algorithm 2 is of order 3. Based on result Eq.(36), by using Taylor’s series expansion of g(xn ) about r and with the help of Eq.(2), we obtain the error equation as follows: xn+1 = g(xn ) = g(r) + g 0 (r)(xx − r) +

g 00(r) g 000(r) (xx − r)2 + (xx − r)3 + O(xx −(37) r)4 . 2! 3!

from Eq.(36) and Eq.(37),we get xn+1 = r +

g 000 (r) 3 en 3!

+ O(e4n ) 2

00

(r)] = r + 16 ( 6[f − [f 0 (r)2] 00

2

(r)] = r + ( [f[f 0 (r) 2] −

f (3) (r) f 0 (r)

f (3) (r) 6f 0 (r)

2

00

(r)] − 3f 0 (r)( 2[f − [f 0 (r)]3 2

00

(r)] − 12 f 0 (r)( 2[f − [f 0 (r)]3

[f (r)]3 ))e3n [f 0 (r)]2

[f (r)]3 ))e3n [f 0 (r)]2

+ O(e4n )

+ O(e4n )

= r + c1e3n + O(e4n ) where en = xn − r and c1 =

[f 00 (r)]2 [f 0 (r)2]

−

f (3) (r) 6f 0(r)

00

2

(r)] − 12 f 0 (r)( 2[f − [f 0(r)]3

[f (r)]3 ) [f 0 (r)]2

Thus,

en+1 = c1 e3n + O(e4n ) 15

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Which shows that Algorithm 2 has convergence order at least 3.

4

Test problems

In this section, we present some examples to illustrate the efficiency of the Algorithm 2 derived in section 2. For each problem, the comparison of our algorithm with Newton-Raphson method (NR), the method of Javidi (JM) [31], the Noor et al. method(NM) [32] and the method of Rafiq and Rafiullah (RR) [11] is given in the form of the tables. Example 1 x5 + x4 + 4x2 − 15 = 0 with x0 = 1.5. The numerical results obtained are given in Table 1. The exact solution prospected is x = 1.34742809896830498171. Table 1

.

Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

5

1.3474280989683050

6.851044e-16

6.616654e-14

10

JM

3

1.3474280989683050

6.851044e-16

1.317722e-10

12

NM

3

1.3474280989683050

6.851044e-16

1.606799e-06

9

RR

3

1.3474280989683050

6.851044e-16

8.567259e-08

9

ALGO2

3

1.3474280989683050

6.851044e-16

1.510379e-06

9

Example 2. sin( x) − 13 x = 0 with x0 = 1.9. In Table 2, we present the numerical results. Table 2 Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

5

2.2788626600758283

1.249294e-17

6.302925e-12

10

JM

4

2.2788626600758283

1.249294e-17

2.108617e-11

16

NM

3

2.2788626600758283

1.249294e-17

9.493745e-07

9

RR

3

2.2788626600758283

1.249294e-17

1.286399e-06

9

ALGO2

3

2.2788626600758283

1.249294e-17

2.454444e-06

9

−x2

Example 3. 10xe − 1 = 0 with x0 = 0.The exact solution prospected is x = 0.10102584831568519737 and the numerical solutions obtained are given in Table 3. Table 3 16

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Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

4

0.1010258483156852

2.552517e-17

3.167545e-14

8

JM

3

0.1010258483156852

2.552517e-17

1.387112e-14

12

NM

3

0.1010258483156852

2.552517e-17

1.261299e-08

9

RR

3

0.1010258483156852

2.552517e-17

1.838993e-08

9

ALGO2

3

0.1010258483156852

2.552517e-17

1.266315e-08

9

2

Example 4. e−x +x+2 − 1 = 0 with x0 = 2.1. The numerical results are given in Table 4 and the exact solution prospected is x = 2.00000000000000000000. Table 4 Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

4

1.9999999999999969

9.300000e-15

5.172728e-08

8

JM

3

2.0000000000000001

3.000000e-16

2.520513e-06

12

NM

3

2.0000000000000000

0.000000e+00

8.242000e-07

9

RR

3

2.0000000000000000

0.000000e+00

9.466958e-10

9

ALGO2

3

2.0000000000000000

0.000000e+00

1.093701e-06

9

Example 5. ln (x2 + x + 2) − x + 1 = 0 with x0 = −1. The exact solution prospected is x = 4.15259073675715827500 and the comparison of numerical results is given in Table 5. Table 5 Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

5

4.1525907367571608

1.520771e-15

2.047030e-07

8

JM

-

-

-

-

Diverges

NM

3

4.1525907367571584

7.528743e-17

2.424885e-05

18

RR

4

4.1525907367571583

1.505894e-17

5.115773e-08

9

ALGO2

3

4.1525907367571576

4.065405e-16

4.809359e-05

9

Example 6. cos (x)−x = 0 with x0 = 1. The numerical solutions obtained are given in Table 6 and the exact solution prospected is x = 0.73908513321516064166. Table 6 17

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Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

4

0.7390851332151606

2.770350e-18

1.701233e-10

8

JM

3

0.7390851332151606

2.770350e-18

1.634240e-08

12

NM

3

0.7390851332151606

2.770350e-18

4.833743e-11

9

RR

3

0.7390851332151606

2.770350e-18

1.997531e-09

9

ALGO2

3

0.7390851332151606

2.770350e-18

5.244611e-11

9

Example 7. e−x + cos(x) = 0 with x0 = 1.75. In Table 7, we present the numerical results. The exact prospected solution is x = 1.74613953040801241765. Table 7 Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

3

1.7461395304080124

2.045916e-17

7.679900e-13

6

JM

2

1.7461395304080124

2.045916e-17

2.016449e-08

8

NM

2

1.7461395304080124

2.045916e-17

1.080398e-08

8

RR

2

1.7461395304080124

2.045916e-17

1.341787e-08

8

ALGO2

2

1.7461395304080124

2.045916e-17

1.080322e-08

8

Example 8. sin−1 (x2 − 1) − 21 x + 1 = 0 with x0 = 0.3. The exact solution prospected is x = 0.59481096839836917752. and the numerical results obtained are given in Table 8. Table 8 Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

4

0.5948109683983692

2.622997e-18

9.482505e-10

8

JM

3

0.5948109683983692

2.622997e-18

8.222903e-08

12

NM

3

0.5948109683983692

7.964943e-18

2.068484e-06

9

RR

3

0.5948109683983692

2.622997e-18

4.993354e-07

9

ALGO2

3

0.5948109683983692

7.964943e-18

2.185594e-06

9

Example 9. x − 2 − exp(−x) = 0 with x0 = 2.0. The numerical solutions obtained are given in Table 9 and the exact solution prospected is x = 2.12002823898764122948. Table 9 18

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Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

3

2.1200282389876412

3.302368e-17

3.651460e-08

6

JM

3

2.1200282389876412

3.302368e-17

1.106344e-12

12

NM

3

2.1200282389876412

3.302368e-17

1.652310e-14

9

RR

3

2.1200282389876412

3.302368e-17

9.357932e-15

9

ALGO2

3

2.1200282389876412

3.302368e-17

1.650296e-14

9

Example 10. x2 − (1− x)5 = 0 with x0 = 0.2. In Table 10, we present the numerical results. The exact solution prospected is x = 0.34595481584824201796. Table 10 Method

No.of Ite.

x[k]

|f(x[k])|

Diff. of two iterations Evaluations

NR

5

0.3459548158482420

3.280895e-18

1.213571e-12

10

JM

4

0.3459548158482420

3.280895e-18

2.047864e-14

16

NM

4

0.3459548158482420

3.280895e-18

2.528477e-13

12

RR

4

0.3459548158482420

3.280895e-18

7.789585e-15

12

ALGO2

4

0.3459548158482420

3.280895e-18

2.329867e-13

12

Example 11. ln(x) = 0 with x0 = 0.5.The exact solution prospected is x = 1.00000000000000000000 and the numerical results obtained are given in Table 11. Table 11 Method

No.of Ite.

x[k]

|f(x[k])|

NR

5

1.0000000000000000

0.000000e+00

3.001887e-09

10

JM

4

1.0000000000000000

0.000000e+00

6.191414e-07

16

NM

4

1.0000000000000000

0.000000e+00

3.906397e-09

12

RR

4

1.0000000000000000

0.000000e+00

1.942042e-10

12

ALGO2

4

1.0000000000000000

0.000000e+00

3.639530e-09

12

5

Diff. of two iterations Evaluations

Conclusion

In order to find the solution of linear and nonlinear algebraic equation, we have introduce a new two-step iterative method based on homotopy perturba19

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tion method. The comparison of the proposed method with Newton-Raphson method (NR), the method of Javidi (JM) [31], Noor et al. method(NM) [32] and the method of Rafiq and Rafiullah(RR) [11] in the previous section reveals the efficiency of our method.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgements This study was supported by research funds from Dong-A University.

References [1] A. Golbabai, M. Javidi, A third-order Newton type method for nonlinear equations based on modified homotopy perturbation method, Appl. Math. Comput. 191 (2007) 199-205. [2] S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 145 (2003) 887-893. [3] E. Babolian, J. Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 132 (2002) 167-172. [4] J. Biazar, A. Amirteimoori, An improvement to the fixed point iterative method, Appl. Math. Comput. 182 (2006) 567-571. [5] M. Basto, V. Semiao, F. L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006) 468-483. [6] C. Chun, Iterative methods improving Newton’s method by the decomposition method, Comput. Math. Appl. 50 (2005) 1559-1568. [7] H. H. H Homeier, On Newton-type methods with cubic convergence, J. Comput. Appl. Math. 176 (2005) 425-432. ¨ [8] A.Y. Ozban, Some new variants of Newton’s method, Appl. Math. Lett. 17 (2004) 677-682.

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[9] A. Golbabai, M. Javidi, A new family of iterative methods for solving system of nonlinear algebric equations, Appl. Math. Comput. 190 (2) (2007) 1717-1722. [10] M. A. Noor, K. I. Noor, Modified iterative methods with cubic convergence for solving nonlinear equations, Appl. Math. Comput. 184(2) (2007) 322-325. [11] A. Rafiq, M. Rafiullah, Some multi-step iterative methods for solving nonlinear equations, Comput. Math. Appl. 58 (2009) 1589-1597. [12] J. H. He, A new iterative method for solving algebric equations, Appl. Math. Comput. 135 (2003) 81-84. [13] E. Babolian, J. Biazar, On the order of convergence of Adomian method, Appl. Math. Comput. 130 (2002) 383-387. [14] E. Babolian, J. Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 132(1) (2002) 167-172 [15] J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178(3-4) (1999) 257-262. [16] J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech. 35(1) (2000) 37-43. [17] J. H. He, The homotopy perturbation method for non-linear oscillators with discontinuities, Appl. Math. Comput. 151(1) (2004) 287-292. [18] J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals 26(3) (2005) 695-700. [19] J. H. He, Asymptotology by homotopy perturbation method, Appl. Math. Comput. 156(3) (2004) 591-596. [20] J. H. He, Homotopy perturbation method for solving boundary problems, Phys. Lett. A. 350(1-2) (2006) 87-88. [21] J. H. He, Limite cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26(3) (2005) 827-833. [22] S. Abbasbandy, Iterated He’s homotopy perturbation method for quardratic Riccati differential equation, Appl. Math. Comput. 175(1) (2006) 581-589. [23] P. D. Ariel, T. Hayat, S. Asghar, Homotopy perturbation method and axisymmetric flow over a stretching sheet, Int. J. Nonlinear Sci. Number. Simul. 7(4) (2006) 399-406. [24] D. D. Gangi, A. Sadighi, Application of He’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, Int. J. Nonlinear Sci. Number. Simul. 7(4) (2006) 411-418. [25] M. Rafei, D. D. Gangi, Explicite solution of Helmholtz equation and fifthorder KdV equation using homotopy perturbation method, Int. J. Nonlinear Sci. Number. Simul. 7(3) (2006) 321-328.

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[26] A. M. Siddiqui, R. Mahmood, Q. K. Ghori, Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. Lett. A 352(4-5) (2006) 404-410. [27] M. Ghasemi, M. Tavassoli, E. Babolian., Numerical solution of the nonlinear Voltra-Fredholn integral equations by using homotopy perturbation metdod, Appl. Math. Comput. 188(1) (2007) 446-449. [28] M. Javidi, A. Golbabai, A numarical solution for solving system of Fredholn integral equations by using homotopy perturbation method, Appl. Math. Comput. 189(2) (2007) 1921-1928. [29] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Book Company, Inc., 1953. [30] E. Isaacson, H. B. Keller, Analysis of Numerical Methods, John Wiley & Son, Inc, New York, USA, 1966. [31] M. Javidi, Iterative methods to nonlinear equations, Appl. Math. Comput. 193 (2007) 360-365. [32] M. A. Noor, K. I. Noor, M. Hassan, Third-order iterative methods free from second derivatives for nonlinear equation, Appl. Math. Comput. 190 (2007) 1551-1556. [33] J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178(3-4) (1999) 257-262.

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Unisoft Filters in R0 -algebras G. Muhiuddina,∗ and Abdullah M. Al-roqib a b

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Abstract. The notion of uni-soft filters in R0 -algebras is introduced, and some related properties are investigated. Characterizations of a uni-soft filter are established, and a new uni-soft filter from old one is constructed.

1. Introduction To solve complicated problem in economics, engineering, and environment, we can not successfully use classical methods because of various uncertainties typical for those problems. Uncertainties can not be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [9]. Maji et al. [8] and Molodtsov [9] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [9] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [8] described the application of soft set theory to a decision making problem. Maji et al. [7] also studied several operations on the theory of soft sets. Chen et al. [2] presented a new definition of soft set parametrization reduction, and compared this definition to the related concept of attributes reduction in rough set theory. R0 -algebras, which are different from BL-algebras, have been introduced by Wang [11] in order to an algebraic proof of the completeness theorem of a formal deductive system [12]. The filter theory in R0 -algebras is discussed in [10].

2010 Mathematics Subject Classification: 06F35, 03G25, 06D72. Keywords: soft set, uni-soft filter, δ-exclusive set, exclusive filter. *Corresponding author. E-mail: [email protected] (G. Muhiuddin), [email protected] (Abdullah M. Al-roqi)

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G. Muhiuddin and Abdullah M. Al-roqi

2

In this paper, we apply the notion of uni-soft property to the filter theory in R0 -algebras. We introduced the concept of (normal) uni-soft filters in R0 -algebras, and investigate related properties. We establish characterizations of a (normal) uni-soft filter, and make a new uni-soft filter from old one. We provide a condition for an uni-soft filter to be normal, and construct a condensational property of a normal uni-soft filter.

2. Preliminaries 2.1. Basic results on R0 -algebras.

Definition 2.1 ([11]). Let L be a bounded distributive lattice with order-reversing involution ¬ and a binary operation → . Then (L, ∧, ∨, ¬, →) is called an R0 -algebra if it satisfies the following axioms: (R1) x → y = ¬y → ¬x, (R2) 1 → x = x, (R3) (y → z) ∧ ((x → y) → (x → z)) = y → z, (R4) x → (y → z) = y → (x → z), (R5) x → (y ∨ z) = (x → y) ∨ (x → z), (R6) (x → y) ∨ ((x → y) → (¬x ∨ y)) = 1.

Let L be an R0 -algebra. For any x, y ∈ L, we define x y = ¬(x → ¬y) and x ⊕ y = ¬x → y. It is proven that and ⊕ are commutative, associative and x ⊕ y = ¬(¬x ¬y), and (L, ∧, ∨, , →, 0, 1) is a residuated lattice. In the following, let xn denote x x · · · x where x appears n times for n ∈ N. We refer the reader to the book [3] for further information regarding R0 -algebras.

Definition 2.2 ([10]). A nonempty subset F of L is called a filter of L if it satisfies (F1) 1 ∈ F, (F2) (∀x ∈ F ) (∀y ∈ L) (x → y ∈ F ⇒ y ∈ F ).

Lemma 2.3 ([10]). Let F be a nonempty subset of L. Then F is a filter of L if and only if it satisfies (1) (∀x ∈ F ) (∀y ∈ L) (x ≤ y ⇒ y ∈ F ). (2) (∀x, y ∈ F ) (x y ∈ F ).

Lemma 2.4 ([10]). Let L be an R0 -algebra. Then the following properties hold:

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Unisoft Filters in R0 -algebras

3

(2.1)

(∀x, y ∈ L) (x ≤ y ⇔ x → y = 1) ,

(2.2)

(∀x, y ∈ L) (x ≤ y → x) ,

(2.3)

(∀x ∈ L) (¬x = x → 0) ,

(2.4)

(∀x, y ∈ L) ((x → y) ∨ (y → x) = 1) ,

(2.5)

(∀x, y ∈ L) (x ≤ y ⇒ y → z ≤ x → z, z → x ≤ z → y) ,

(2.6)

(∀x, y ∈ L) (((x → y) → y) → y = x → y) ,

(2.7)

(∀x, y ∈ L) (x ∨ y = ((x → y) → y) ∧ ((y → x) → x)) ,

(2.8)

(∀x ∈ L) (x ¬x = 0, x ⊕ ¬x = 1) ,

(2.9)

(∀x, y ∈ L) (x y ≤ x ∧ y, x (x → y) ≤ x ∧ y) ,

(2.10)

(∀x, y, z ∈ L) ((x y) → z = x → (y → z)) ,

(2.11)

(∀x, y ∈ L) (x ≤ y → (x y)) ,

(2.12)

(∀x, y, z ∈ L) (x y ≤ z ⇔ x ≤ y → z) ,

(2.13)

(∀x, y, z ∈ L) (x ≤ y ⇒ x z ≤ y z) ,

(2.14)

(∀x, y, z ∈ L) (x → y ≤ (y → z) → (x → z)) ,

(2.15)

(∀x, y, z ∈ L) ((x → y) (y → z) ≤ x → z) .

2.2. Basic results on soft set theory. Soft set theory was introduced by Molodtsov [9] and C ¸ aˇgman et al. [1]. In what follows, let U be an initial universe set and E be a set of parameters. We say that the pair (U, E) is a soft universe. Let P(U ) (resp. P(E)) denotes the power set of U (resp. E). By analogy with fuzzy set theory, the notion of soft set is defined as follows: Definition 2.5 ([1, 9]). A soft set of E over U (a soft set of E for short) is: any function fA : E → P(U ), such that fA (x) = ∅ if x ∈ / A, for A ∈ P(E), or, equivalently, any set FA := {(x, fA (x)) | x ∈ E, fA (x) ∈ P(U ), fA (x) = ∅ if x ∈ / A} , for A ∈ P(E). Definition 2.6 ([1]). Let FA and FB be soft sets of E. We say that FA is a soft subset of FB , denoted e B , if fA (x) ⊆ fB (x) for all x ∈ E. by FA ⊆F

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Definition 2.7 ([5]). For any non-empty subset A of E, a soft set FA of E is said to satisfy the uni-soft property (it is also called a uni-soft set in [5]) if it satisfies: (∀x, y ∈ A) (x ,→ y ∈ A ⇒ fA (x ,→ y) ⊆ fA (x) ∪ fA (y)) where ,→ is a binary operation in E. 3. Uni-soft filters In what follows, denote by S(U, L) the set of all soft sets of L over U where L is an R0 -algebra unless otherwise specified. Definition 3.1. A soft set FL ∈ S(U, L) is called a filter of L based on the uni-soft property (briefly, uni-soft filter of L) if it satisfies: (3.1)

(∀x ∈ L) (fL (1) ⊆ fL (x)) ,

(3.2)

(∀x, y ∈ L) (fL (y) ⊆ fL (x → y) ∪ fL (x)) .

Example 3.2. Let L = {0, a, b, c, 1} be a set with the order 0 < a < b < c < 1, and the following Cayley tables: x ¬x

→

0

a

b

c

1

0

1

0

1

1

1

1

1

a

c

a

c

1

1

1

1

b

b

b

b

b

1

1

1

c

a

c

a

a

b

1

1

1

0

1

0

a

b

c

1

Then (L, ∧, ∨, ¬, →) is an R0 -algebra (see [4]) where x ∧ y = min{x, y} and x ∨ y = max{x, y}. Let FL ∈ S(U, L) be given as follows: FL = {(0, δ1 ), (a, δ1 ), (b, δ1 ), (c, δ2 ), (1, δ2 )} where δ1 and δ2 are subsets of U with δ2 ( δ1 . Then FL is an uni-soft filter of L. Proposition 3.3. Let FL ∈ S(U, L) be a uni-soft filter of L. Then the following properties are valid: (1) FL is order-reversing, that is, (∀x, y ∈ L) (x ≤ y ⇒ fL (y) ⊆ fL (x)). (2) (∀x, y ∈ L) (fL (x → y) = fL (1) ⇒ fL (y) ⊆ fL (x)) . (3) (∀x, y ∈ L) (fL (x y) = fL (x) ∪ fL (y) = fL (x ∧ y)) . (4) (∀x ∈ L)(∀n ∈ N) (fL (xn ) = fL (x)) . (5) (∀x ∈ L) (fL (0) = fL (x) ∪ fL (¬x)) . (6) (∀x, y ∈ L) (fL (x → z) ⊆ fL (x → y) ∪ fL (y → z)) . (7) (∀x, y, z ∈ L) (x y ≤ z ⇒ fL (z) ⊆ fL (x) ∪ fL (y)) . (8) (∀x, y ∈ L) (fL (x) ∪ fL (x → y) = fL (y) ∪ fL (y → x) = fL (x) ∪ fL (y)) .

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(9) (∀x, y ∈ L) (fL (x (x → y)) = fL (y (y → x)) = fL (x) ∪ fL (y)) . (10) (∀x, y, z ∈ L) (fL (x → (¬z → z)) ⊆ fL (x → (¬z → y)) ∪ fL (y → z)) . (11) (∀x, y, z ∈ L) (fL (x → (x → z)) ⊆ fL (x → (y → z)) ∪ fL (x → y)) . Proof. (1) Let x, y ∈ L be such that x ≤ y. Then x → y = 1, and so fL (y) ⊆ fL (x) ∪ fL (x → y) = fL (x) ∪ fL (1) = fL (x) by (3.1) and (3.2). (2) Let x, y ∈ L be such that fL (x → y) = fL (1). Then fL (y) ⊆ fL (x) ∪ fL (x → y) = fL (x) ∪ fL (1) = fL (x) by (3.1) and (3.2). (3) Since x y ≤ x ∧ y for all x, y ∈ L, it follows from (1) that fL (x) ∪ fL (y) ⊆ fL (x y). Using (2.11) and (1), we have fL (y → (x y)) ⊆ fL (x). It follows from (3.2) that fL (x y) ⊆ fL (y → (x y)) ∪ fL (y) ⊆ fL (x) ∪ fL (y). Therefore fL (x y) = fL (x) ∪ fL (y). Since y ≤ x → y and x (x → y) ≤ x ∧ y for all x, y ∈ L, we have fL (x → y) ⊆ fL (y) and fL (x) ∪ fL (y) ⊆ fL (x ∧ y) ⊆ fL (x (x → y)) = fL (x) ∪ fL (x → y) ⊆ fL (x) ∪ fL (y) by (1). Hence fL (x ∧ y) = fL (x) ∪ fL (y) for all x, y ∈ L. (4) It follows from (3). (5) Note that x ¬x = 0 for all x ∈ L. Using (3), we have fL (0) = fL (x ¬x) = fL (x) ∪ fL (¬x) for all x, y ∈ L. (6) Combining (2.15), (1) and (3), we have the desired result. (7) It follows from (1) and (3). (8) Since y ≤ x → y for all x, y ∈ L, it follows from (1) that (3.3)

fL (x) ∪ fL (x → y) ⊆ fL (x) ∪ fL (y).

Since x (x → y) ≤ x ∧ y for all x, y ∈ L, we have (3.4)

fL (x) ∪ fL (y) = fL (x ∧ y) ⊆ fL (x (x → y)) = fL (x) ∪ fL (x → y)

by (3) and (1). Hence fL (x) ∪ fL (y) = fL (x) ∪ fL (x → y). Similarly, fL (y) ∪ fL (y → x) = fL (x) ∪ fL (y) for all x, y ∈ L.

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(9) Using (3), we have fL (x (x → y)) = fL (x) ∪ fL (x → y) and fL (y (y → x)) = fL (y) ∪ fL (y → x) for all x, y ∈ L. It follows from (8) that fL (x (x → y)) = fL (y (y → x)) = fL (x) ∪ fL (y). (10) Note that (x → (¬z → y)) (y → z) = ((x ¬z) → y) (y → z) ≤ (x ¬z) → z = x → (¬z → z) for all x, y, z ∈ L. Using (1) and (3), we have fL (x → (¬z → z)) ⊆ fL ((x → (¬z → y)) (y → z)) = fL (x → (¬z → y)) ∪ fL (y → z) for all x, y, z ∈ L. (11) Note that (x → (y → z)) (x → y) = (y → (x → z)) (x → y) ≤ x → (x → z) for all x, y, z ∈ L. It follows from (1) and (3) that fL (x → (x → z)) ⊆ fL ((x → (y → z)) (x → y)) = fL (x → (y → z)) ∪ fL (x → y) for all x, y, z ∈ L.



Definition 3.4. Let FL ∈ S(U, L) be a unisoft filter of L. Then for any δ ∈ P(U ), the δ-exclusive set of FL is defined by e(FL ; δ) = {x ∈ L | fL (x) ⊆ δ} . If FL is a uni-soft filter of L, every δ-exclusive set e(FL ; δ) is called an exclusive filter of L. We provide characterizations of a uni-soft filter. Theorem 3.5. Let FL ∈ S(U, L). Then FL is a uni-soft filter of L if and only if the following assertion is valid: (3.5)

(∀δ ∈ P(U )) (e(FL ; δ) 6= ∅ ⇒ e(FL ; δ) is a filter of L) .

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Proof. Assume that FL ∈ S(U, L) satisfies (3.5). For any x ∈ L, let fL (x) = δ. Then x ∈ e(FL ; δ). Since e(FL ; δ) is a filter of L, we have 1 ∈ e(FL ; δ) and so fL (1) ⊆ δ = fL (x). For any x, y ∈ L, let fL (x → y) ∪ fL (x) = δ. Then x → y ∈ e(FL ; δ) and x ∈ e(FL ; δ). Since e(FL ; δ) is a filter of L, it follows that y ∈ e(FL ; δ). Hence fL (y) ⊆ δ = fL (x → y) ∪ fL (x). Conversely, suppose that FL is a uni-soft filter of L. Let δ ∈ P(U ) be such that e(FL ; δ) 6= ∅. Then there exists a ∈ e(FL ; δ), and so fL (a) ⊆ δ. It follows from (3.1) that fL (1) ⊆ fL (a) ⊆ δ. Thus 1 ∈ e(FL ; δ). Let x, y ∈ L be such that x → y ∈ e(FL ; δ) and x ∈ e(FL ; δ). Then fL (x → y) ⊆ δ and fL (x) ⊆ δ. It follows from (3.2) that fL (y) ⊆ fL (x → y) ∪ fL (x) ⊆ δ, that is, y ∈ e(FL ; δ). Thus FL is a uni-soft filter of L.



Theorem 3.6. Let FL ∈ S(U, L). Then FL is a uni-soft filter of L if and only if the following assertions are valid: (1) FL is order-reversing, (2) (∀x, y ∈ L) (fL (x y) = fL (x) ∪ fL (y)) . Proof. The necessity follows from (1) and (3) of Proposition 3.3. Conversely, suppose that FL satisfies two conditions (1) and (2). Let x, y ∈ L. Since x ≤ 1, we have fL (1) ⊆ fL (x) by (1). Note that x (x → y) ≤ y for all x, y ∈ L. It follows from (2) and (1) that fL (y) ⊆ fL (x (x → y)) = fL (x) ∪ fL (x → y) for all x, y ∈ L. Therefore FL is a uni-soft filter of L.



Theorem 3.7. Let FL ∈ S(U, L). Then FL is a uni-soft filter of L if and only if the following assertion is valid: (3.6)

(∀x, y, z ∈ L) (fL (x → z) ⊆ fL (y) ∪ fL ((x → y) → z)) .

Proof. Assume that FL is a uni-soft filter of L and let x, y, z ∈ L. Since y ≤ x → y, it follows from Proposition 3.3(1) and (3.2) that fL (x → z) ⊆ fL (z) ⊆ fL (x → y) ∪ fL ((x → y) → z) ⊆ fL (y) ∪ fL ((x → y) → z) for all x, y, z ∈ L. Conversely, suppose that FL satisfies the inclusion (3.6). If we take x = 0 and y = z in (3.6), then fL (1) = fL (0 → z) ⊆ fL (z) ∪ fL ((0 → z) → z) = fL (z) ∪ fL (1 → z) = fL (z) ∪ fL (z) = fL (z)

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for all z ∈ L, and if we put x = 1, y = x and z = y in (3.6), then fL (y) = fL (1 → y) ⊆ fL (x) ∪ fL ((1 → x) → y) = fL (x) ∪ fL (x → y) for all x, y ∈ L. Therefore FL is a uni-soft filter of L.



Theorem 3.8. Let FL ∈ S(U, L). Then FL is a uni-soft filter of L if and only if the following assertion is valid: (3.7)

(∀x, y, z ∈ L) (x ≤ y → z ⇒ fL (z) ⊆ fL (x) ∪ fL (y)) .

Proof. Suppose that FL is a uni-soft filter of L. Let x, y, z ∈ L be such that x ≤ y → z. Then fL (y → z) ⊆ fL (x) by Proposition 3.3(1), and so fL (z) ⊆ fL (y → z) ∪ fL (y) ⊆ fL (x) ∪ fL (y) by (3.2). Conversely, assume that FL satisfies the condition (3.7). Let x, y ∈ L. Since x ≤ 1 = x → 1, we have fL (1) ⊆ fL (x) by (3.7). Note that x → y ≤ x → y. It follows from (3.7) that fL (y) ⊆ fL (x → y) ∪ fL (x). Therefore FL is a uni-soft filter of L.



Proposition 3.9. Every uni-soft filter FL of L satisfies: (3.8)

(∀x, y, z ∈ L) (fL (x → (y → z)) ⊆ fL ((x → y) → z)) .

Proof. Let x, y, z ∈ L. Since 1 = y → (x → y) ≤ ((x → y) → z) → (y → z), we have fL (((x → y) → z) → (y → z)) ⊆ fL (1) by Proposition 3.3(1). It follows from (2.2), Proposition 3.3(1), (3.2) and (3.1) that fL (x → (y → z)) ⊆ fL (y → z) ⊆ fL ((x → y) → z) ∪ fL (((x → y) → z) → (y → z)) ⊆ fL ((x → y) → z) ∪ fL (1) = fL ((x → y) → z) for all x, y, z ∈ L. This completes the proof.



The following example shows that the converse of Proposition 3.9 may not be true in general.

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Example 3.10. Let L = {0, a, b, c, d, 1} be a set with the order 0 < a < b < c < d < 1, and the following Cayley tables: x ¬x

→

0

a

b

c

d 1

0

1

0

1

1

1

1

1

1

a

d

a

d

1

1

1

1

1

b

c

b

c

c

1

1

1

1

c

b

c

b

b

b

1

1

1

d

a

d

a

a

b

c

1

1

1

0

1

0

a

b

c

d 1

Then (L, ∧, ∨, ¬, →) is an R0 -algebra (see [4]) where x ∧ y = min{x, y} and x ∨ y = max{x, y}. Let FL ∈ S(U, L) be given as follows: FL = {(0, δ1 ), (a, δ2 ), (b, δ2 ), (c, δ2 ), (d, δ1 ), (1, δ1 )} where δ1 and δ2 are subsets of U with δ2 ( δ1 . Then FL satisfies the condition (3.8), but FL is not a uni-soft filter of L since fL (1) = δ1 * δ2 = fL (a). Proposition 3.11. For a uni-soft filter FL of L, the following are equivalent: (3.9)

fL (y → x) ⊆ fL (y → (y → x)) for all x, y ∈ L,

(3.10)

fL ((z → y) → (z → x)) ⊆ fL (z → (y → x)) for all x, y, z ∈ L.

Proof. Assume that (3.9) is valid and let x, y, z ∈ L. Using (R4), (2.5) and (2.14), we have z → (y → x) ≤ z → (z → ((z → y) → x)). It follows from (R4), (3.9) and Proposition 3.3(1) that fL ((z → y) → (z → x)) = fL (z → ((z → y) → x)) ⊆ fL (z → (z → ((z → y) → x))) ⊆ fL (z → (y → x)). Conversely, suppose that (3.10) holds. If we use z instead of y in (3.10), then fL (z → x) = fL (1 → (z → x)) = fL ((z → z) → (z → x)) ⊆ fL (z → (z → x)), which proves (3.9).



We make a new uni-soft filter from old one. Theorem 3.12. Let FL ∈ S(U, L). For a subset δ of U, define a soft set FL∗ of L by ( fL (x) if x ∈ e(FL ; δ), ∗ fL : L → P(U ), x 7→ τ otherwise

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where τ is a subset of U such that τ )

∪ x∈e(FL ;δ)

fL (x). If FL is a uni-soft filter of L, then so is FL∗ .

Proof. Assume that FL is a uni-soft filter of L. Then e(FL ; δ) (6= ∅) is a filter of L for all δ ∈ P(U ). Hence 1 ∈ e(FL ; δ), and so fL∗ (1) = fL (1) ⊆ fL (x) ⊆ fL∗ (x) for all x ∈ L. Let x, y ∈ L. If x ∈ e(FL ; δ) and x → y ∈ e(FL ; δ), then y ∈ e(FL ; δ). Hence fL∗ (y) = fL (y) ⊆ fL (x) ∪ fL (x → y) = fL∗ (x) ∪ fL∗ (x → y). If x ∈ / e(FL ; δ) or x → y ∈ / e(FL ; δ), then fL∗ (x) = τ or fL∗ (x → y) = τ. Thus fL∗ (y) ⊆ τ = fL∗ (x) ∪ fL∗ (x → y). Therefore FL∗ is a uni-soft filter of L.



Corollary 3.13. Let FL ∈ S(U, L). For a subset δ of U, define a soft set FL∗ of L by ( fL (x) if x ∈ e(FL ; δ), ∗ fL : L → P(U ), x 7→ U otherwise. If FL is a uni-soft filter of L, then so is FL∗ . Proof. Straightforward.



Theorem 3.14. Any filter of L can be realized as an exclusive filter of some uni-soft filter of L. Proof. Let F be a filter of L. For a nonempty subset δ of U, let FL be a soft set of L defined by ( δ if x ∈ F, fL : L → P(U ), x 7→ U otherwise. Obviously fL (1) ⊆ fL (x) for all x ∈ L. For any x, y ∈ L, if x ∈ F and x → y ∈ F, then y ∈ F. Hence fL (x) ∪ fL (x → y) = δ = fL (y). If x ∈ / F or x → y ∈ / F, then fL (x) = U or fL (x → y) = U. Thus fL (y) ⊆ U = fL (x) ∪ fL (x → y). Therefore FL is a uni-soft filter of L and clearly e(FL ; δ) = F. This completes the proof.

 Acknowledgements

The authors wish to thank the anonymous reviewers for their valuable suggestions. References [1] N. C ¸ aˇgman, F. C ¸ itak and S. Engino˘ glu, Soft set theory and uni-int decision making, Eur. J. Oper. Res. 207 (2010) 848–855. [2] D. Chen, E. C. C. Tsang, D. S. Yeung and X. Wang, The parametrization reduction of soft sets and its applications, Comput. Math. Appl. 49 (2005) 757–763. [3] A. Iorgulescu, Algebras of logic as BCK algebras, Ed.of Academy of Economic Studies, Bucharest 2008. [4] L. Lianzhen and L. Kaitai, Fuzzy implicative and Boolean filters of R0 algebras, Inform. Sci. 171 (2005), 61–71.

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Unisoft Filters in R0 -algebras [5] Y. B. Jun, Union soft sets with applications in BCK/BCI-algebras,

11 Bull. Korean Math. Soc.

(submitted). [6] Y. L. Liu and M. Y. Ren, Normal MP-filters of R0 -algebras, Fuzzy Information and Engineering, Advances in Soft Computing 54 (2009) 113–118. [7] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555–562. [8] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077–1083. [9] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31. [10] D. W. Pei and G. J. Wang, The completeness and application of formal systems L, Sci. China (Ser. E) 32 (2002), no. 1, 56–64. [11] G. J. Wang, Non-Classical Mathematical Logic and Approximate Reasoning, Science Press, Beijing, 2000. [12] G. J. Wang, On the logic foundation of fuzzy reasoning, Inform. Sci. 117 (1999), 47–88.

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The admissible function and non-admissible function in the unit disc ∗ Hong Yan Xua†, Lian Zhong Yangb , and Ting Bin Caoc a

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

b

Department of Mathematics, Shandong University, Jinan, Shandong, 250100, P.R. China

c

Department of Mathematics, Nanchang University Nanchang, Jiangxi, 330031, China

Abstract In this paper, we study the multiple values and uniqueness problem of two nonadmissible functions in the unit disc sharing some values, and also investigate the problem of admissible function and non-admissible function sharing some values, and obtain the following conclusions: 1) Two non-constant non-admissible functions can not be identical if they share four distinct values IM ; 2) An admissible function can share at most three distinct values IM with a non-admissible function. Moreover, we also obtain some theorems which are analogous version of the uniqueness theorems of meromorphic functions sharing some values on the whole complex plane given by Yi and Cao. Key words: uniqueness; meromorphic function; admissible; non-admissible. Mathematical Subject Classification (2010): 30D 35. ∗ The author was supported by the NSF of China(11301233, 61202313), the Natural Science Foundation of Jiang-Xi Province in China (20132BAB211001), and the Foundation of Education Department of Jiangxi (GJJ14644) of China. The second author was supported by the NSF of China (11371225, 11171013). The third author was supported by the NSF of China (11461042). † Corresponding author

1

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1

Introduction and basic notions in the Nevanlinna theory in the unit disc D

Considering meromorphic function f , we shall assume that the readers are familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions such as the proximity function m(r, f ), counting function N (r, f ), characteristic function T (r, f ), the first and second main theorems, lemma on the logarithmic derivatives etc. of Nevalinna theory, (see Hayman [6], Yang [14] and Yi and Yang [16]). For a meromorphic function f , S(r, f ) denotes any quantity satisfying S(r, f ) = o(T (r, f )) for all r outside a possible exceptional set of finite logarithmic measure. In 1926, R. Nevanlinna [9] proved his famous five-value theorem: For two nonconstant meromorphic functions f and g on C, if they have the same inverse images (ignoring multiplicities) for five distinct values, then f (z) ≡ g(z). After his very work, the uniqueness of meromorphic functions with shared values on C attracted many investigations (for references, see the book [16] or some recent papers [1, 3, 5, 8, 13, 12]). It is well-known (r,fi ) that two nonconstant rational functions f1 , f2 satisfy limr→+∞ Tlog r < +∞(i = 1, 2) that share four distinct values IM must be identical. However, for two nonconstant T (r,fi ) meromorphic functions f1 , f2 in the unit disc satisfy limr→1− log < +∞(i = 1, 2), 1 1−r

they may not be identical if f1 , f2 share four distinct values IM (see Corollary 2.3). Hence, we will mainly study the uniqueness of meromorphic functions in the unit disc D in this paper. To state some uniqueness theorems of meromorphic functions in the unit disc D, we need the following basic notations and definitions of meromorphic functions in D(see [2], [4], [7], [11]). Definition 1.1 Let f be a meromorphic function in D and limr→1− T (r, f ) = ∞. Then D(f ) := lim sup r→1−

T (r, f ) − log(1 − r)

is called the (upper) index of inadmissibility of f . If D(f ) = ∞, f is called admissible. Definition 1.2 Let f be a meromorphic function in D and limr→1− T (r, f ) = ∞. Then ρ(f ) := lim sup r→1−

log+ T (r, f ) − log(1 − r)

is called the order (of growth) of f . In 2005, Titzhoff [11] investigated the uniqueness of two admissible functions in the unit disc D by using the Second Main Theorem for admissible functions (see [11, Theorem 3]) and obtained the five values theorem for admissible functions in the unit disc D as follows. Theorem 1.1 (see [11, Theorem 3]). Let f be an admissible meromorphic function in D, q be a positive integer and a1 , a2 , . . . , aq be pairwise distinct complex numbers. Then, 2

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for r → 1− , r 6∈ E, (q − 2)T (r, f ) ≤

q X

 N

j=1

r,

1 f − aj

 + S(r, f ),

R dr < ∞. If the order of where E ⊂ (0, 1) is a possibly occurring exceptional set with E 1−r   1 f is finite, the remainder S(r, f ) is a O log 1−r without any exceptional set. Theorem 1.2 (see [5, 11]). If two admissible function f, g share five distinct values, then f ≡ g. For admissible functions, Theorem 1.1 plays a very important role in studies of the uniqueness problems of meromorphic functions in the unit disc. From Theorem 1.1 (the Second Main Theorem for admissible functions (see [11, Theorem 3]), we can get a lot of results similar to meromorphic functions in the complex plane. However, Theorem 1.1 does not hold for non-admissible functions in the unit disc. Thus it is interesting to consider the uniqueness theory of non-admissible functions in the unit disc. For non-admissible functions, the following theorem also plays a very important role in studying their uniqueness problems. Theorem 1.3 (see [11, Theorem 2]). Let f be a meromorphic function in D and limr→1− T (r, f ) = ∞, q be a positive integer and a1 , a2 , . . . , aq be pairwise distinct complex numbers. Then, for r → 1− , r 6∈ E,   q X 1 1 N r, + S(r, f ). (q − 2)T (r, f ) ≤ + log f − aj 1−r j=1 1 in Theorem 1.3 does Remark 1.1 In contrast to admissible functions, the term log 1−r not necessarily enter the remainder S(r, f ) because the non-admissible function f may   1 have T (r, f ) = O log 1−r .

 Remark 1.2 We can see that S(r, f ) = o log possible exception set when 0 < D(f ) < ∞.

1 1−r



holds in Theorem 1.3 without a

From Theorem 1.3 and Lemma 3.4 in [16], we can get the following result for nonadmissible functions in the unit disc which is used in this paper. Lemma 1.1 Let f (z) be a meromorphic function in D and limr→1− T (r, f ) = ∞, aj (j = 1, 2, . . . , q) be q distinct complex numbers, and kj (j = 1, 2, . . . , q) be positive integers or ∞. If f is a non-admissible function, then   X   q q X kj 1 1 1 (q − 2)T (r, f ) < N k ) r, + N r, k +1 j f − aj k +1 f − aj j=1 j j=1 j + log

1 + S(r, f ), 1−r 3

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and  q − 2 −

q X j=1

   q X 1  kj 1 1 T (r, f ) ≤ N kj ) r, + log + S(r, f ), kj + 1 k + 1 f − a 1 − r j=1 j

where S(r, f ) is stated as in Theorem 2.1. The main purpose of this paper is to investigate the uniqueness problem of nonadmissible functions in the unit disc. In section 2, the uniqueness of two non-admissible functions in D are investigated and some theorems show that the number and weight of sharing values is related to the index of inadmissibility of functions in D. In section 3, the problem of admissible function and non-admissible function sharing some values is studied, and one of those results show that admissible function and non-admissible function can share at most three distinct values IM .

2

Multiple values and uniqueness of non-admissible functions in the unit disc D

We use C to denote the open complex plane, complex plane, and D = {z : |z| < 1} to denote b and ⊆ C. Define elements in C [ E(S, D, f ) = {z ∈ D|fa (z) = 0,

b := C S{∞} to denote the extended C the unit disc. Let S be a set of distinct

counting

multiplicities},

ignoring

multiplicities},

a∈S

E(S, D, f ) =

[

{z ∈ D|fa (z) = 0,

a∈S

where fa (z) = f (z) − a if a ∈ C and f∞ (z) = 1/f (z). Let f, g be two non-constant meromorphic functions in D. If E(S, D, f ) = E(S, D, g), we say f and g share the set S CM (counting multiplicities) in D. If E(S, D, f ) = E(S, D, g), we say f and g share the set S IM (ignoring multiplicities) in D. In particular, b we say f and g share the value a CM in D if E(S, D, f ) = when S = {a}, where a ∈ C, E(S, D, g), and we say f and g share the value a IM in D if E(S, D, f ) = E(S, D, g). If D is replaced by C, we give the simple notation as before, E(S, f ), E(S, f ) and so on(see [14]). Let f (z) be a non-constant meromorphic function in the unit disc, an arbitrary comb and k be a positive integer. We use E k) (a, D, f ) to denote the set of plex number a ∈ C, zeros of f − a in D, with multiplicities no greater than k, in which each zero is counted only once. We say that f (z) and g(z) share the value a in D with reduced weight k, if E k) (a, D, f ) = E k) (a, D, g). b with respect to a meromorphic function f on the We denote the deficiency of a ∈ C unit disc D by δ(a, f ) = δ(0, f − a) = lim inf r→1−

1 m(r, f −a )

T (r, f )

= 1 − lim sup r→1−

1 N (r, f −a )

T (r, f )

,

4

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and denote the reduced deficiency by Θ(a, f ) = Θ(0, f − a) = 1 − lim sup r→1−

1 N (r, f −a )

T (r, f )

.

We now show our main result below which is an analog of a result on the plane C obtained by H.-X. Yi [17] (see also Theorem 3.34 in [16]). Theorem 2.1 Let f1 (z), f2 (z) be non-constant non-admissible functions satisfying 0 < b and let kj (j = D(f1 ), D(f2 ) < ∞, aj (j = 1, 2, . . . , q) be q distinct complex numbers in C, 1, 2, . . . , q) be positive integers or ∞ satisfying k1 ≥ k2 ≥ · · · ≥ kq

(1) and

E kj ) (aj , D, f1 ) = E kj ) (aj , D, f2 ),

(2)

(j = 1, 2, . . . , q).

Furthermore, let Θ(fi ) =

X

Θ(0, fi − a) −

a

q X

Θ(0, fi − aj ), (i = 1, 2),

j=1

and Pm−1 A1 =

j=1

km + 1 Pn−1

A2 =

δ(0, f1 − aj )

j=1

δ(0, f2 − aj ) kn + 1

+

q X kj + δ(0, f1 − aj ) (m − 2)km kn + − + Θ(f1 ) − 2, kj + 1 km + 1 kn + 1 j=m

+

q X kj + δ(0, f2 − aj ) (n − 2)kn km + − + Θ(f2 ) − 2, k + 1 k + 1 k +1 j n m j=n

where m and n are positive integers in {1, 2, . . . , q} and a is an arbitrary complex number. If (3)

min{A1 , A2 } ≥

2 , D(f1 ) + D(f2 )

and

max{A1 , A2 } >

2 . D(f1 ) + D(f2 )

Then f1 (z) ≡ f2 (z). Proof: Suppose that f1 (z) 6≡ f2 (z). We assume that aj (j = 1, 2, . . . , q) are finite complex numbers, otherwise, we will consider a suitable M¨obious transformation. Without loss of generality, we may assume that infinite b satisfy Θ(0, f1 − b) > 0 and b 6= aj (j = P∞ 1, 2, . . . , q). We denote them by bk (k P = 1, 2, . . . , ∞). Obviously, Θ(f1 ) = k=1 Θ(0, f1 − p bk ). Thus there exists a p such that k=1 Θ(0, f1 − bk ) > Θ(f1 ) − ε holds for ε (> 0). From Theorem 1.3 we have (4) (p + q − 2)T (r, f1 ) ≤

p X k=1

N (r,

q X 1 1 1 )+ N (r, ) + log + S(r, f1 ). f1 − bk f − a 1 − r 1 j j=1

5

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By the definition of reduced deficiency, we have (5)

N (r,

1 ) < (1 − Θ(0, f1 − bk )) T (r, f1 ) + S(r, f1 ). f1 − bk

From Lemma 3.4 in [16] and the definition of deficiency, we get N (r,

1 ) ≤ f1 − aj

1 kj 1 1 N k ) (r, )+ N (r, ) kj + 1 j f1 − aj kj + 1 f1 − aj 1 kj 1 )+ (1 − δ(0, f1 − aj )) T (r, f1 ) < N kj ) (r, kj + 1 f1 − aj kj + 1 +S(r, f1 ).

From the above inequalities and (5)-(6), we get ( p ) q X X (1 − Θ(0, f1 − bk )) T (r, f1 ) + (p + q − 2)T (r, f1 ) ≤ j=1

k=1

+

  q X 1 − δ(0, f1 − aj )  

j=1

kj + 1

T (r, f1 ) + log



kj 1 N k ) (r, ) kj + 1 j f1 − aj 1 + S(r, f1 ). 1−r

From (2) we have 1≥

k1 k2 kq 1 ≥ ≥ ··· ≥ ≥ . k1 + 1 k2 + 1 kq + 1 2

Hence we can deduce that q X

km 1 ) N kj ) (R, k +1 f1 − aj j=1 m     m−1 X  kj km − (1 − δ(0, f1 − aj )) T (r, f1 ) +   kj + 1 km + 1 j=1   q X 1 − δ(0, f1 − aj )  1 + T (r, f1 ) + log + S(r, f1 ),   k + 1 1 − r j j=1

(p + q − 2)T (r, f1 ) ≤ (p − Θ(f1 ) + ε) T (r, f1 ) +

and thus, 

 q X (m − 1)km km 1 1 + B1 − ε T (r, f1 ) ≤ N kj ) (r, ) + log + S(r, f1 ), km + 1 k + 1 f − a 1 − r m 1 j j=1

where

Pm−1 B1 =

j=1

δ(0, f1 − aj )

km + 1

+

q X kj + δ(0, f1 − aj ) + Θ(f1 ) − 2. kj + 1 j=m

6

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By similar discussion, we have   q X (n − 1)kn kn 1 1 + B2 − ε T (r, f2 ) ≤ N kj ) (r, ) + log + S(r, f2 ), kn + 1 k +1 f2 − aj 1−r j=1 n where

Pn−1 B2 =

j=1

δ(0, f2 − a1 ) kn + 1

+

q X kj + δ(0, f2 − aj ) + Θ(f2 ) − 2. kj + 1 j=n

Hence    (n − 1)kn (m − 1)km + B1 − ε T (r, f1 ) + + B2 − 2ε T (r, f2 ) km + 1 kn + 1 q q X X km kn 1 1 ≤ )+ ) N kj ) (r, N kj ) (r, k +1 f1 − aj k +1 f2 − aj j=1 m j=1 n 

+2 log

1 + S(r, f1 ) + S(r, f2 ). 1−r

From the condition (2) and the first fundamental theorem, we have   q q X  X 1 1 1 N kj ) (r, N kj ) (r, max ), ) ≤N (r, )  f1 − aj f2 − aj  f1 − f2 j=1

j=1

1 ) + O(1) f1 − f2 ≤T (r, f1 ) + T (r, f2 ) + O(1). ≤T (r,

Therefore, from the above discussion we obtain     (n − 1)kn (m − 1)km + B1 − ε T (r, f1 ) + + B2 − ε T (r, f2 ) km + 1 kn + 1   kn 1 km ≤ + + S(r, f1 ) + S(r, f2 ). (T (r, f1 ) + T (r, f2 )) + 2 log km + 1 kn + 1 1−r hence, 1 + S(r, f1 ) + S(r, f2 ). 1−r    1 Since 0 < D(f1 ), D(f2 ) < ∞, we have S(r, f1 ) = o log 1−r , S(r, f2 ) = o log And from the definition of index, for any ε satisfying   2 (7) 0 < 2ε < min D(f1 ), D(f2 ), max{A1 , A2 } − , D(f1 ) + D(f2 )

(6)

(A1 − ε) T (r, f1 ) + (A2 − ε) T (r, f2 ) ≤ 2 log

1 1−r



.

there exists a sequence {rt } → 1− such that (8)

T (rt , f1 ) > (D(f1 ) − ε) log

1 , 1 − rt

T (rt , f2 ) > (D(f2 ) − ε) log

1 , 1 − rt

7

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for all t → ∞. From (6), (7) and (8), we have (9)

  1 1 [(D(f1 ) − ε)(A1 − ε) + (D(f2 ) − ε)(A2 − ε) − 2] log < o log . 1 − rt 1 − rt

From (9) and ε being arbitrary, the above inequality contradicts to (3). Therefore, we complete the proof of Theorem 2.1. 2 From Theorem 2.1, we obtain the following corollaries. b and kj (j = Corollary 2.1 Let aj (j = 1, 2, . . . , q) be q distinct complex numbers in C, 1, 2, . . . , q) be positive integers or ∞ satisfying (2), and let f1 (z), f2 (z) be non-constant non-admissible functions satisfying 0 < D(f1 ), D(f2 ) < ∞ and (2). Set A1 =

A2 =

q X

(m − 2)km kn kj + − − 2, k + 1 k + 1 k m n+1 j=m j q X

(n − 2)kn km kj + − − 2, k + 1 k + 1 k n m+1 j=n j

where m and n are positive integers in {1, 2, . . . , q}. If min{A1 , A2 } ≥

2 D(f1 ) + D(f2 )

and

max{A1 , A2 } >

2 , D(f1 ) + D(f2 )

then f1 (z) ≡ f2 (z). Proof: Since Θ(fi ) ≥ 0 (i = 1, 2) and δ(0, f1 − aj ) ≥ 0 (j = 1, 2, . . . , q), the assertion follows from Theorem 2.1. 2 b and kj (j = Corollary 2.2 Let aj (j = 1, 2, . . . , q) be q distinct complex numbers in C, 1, 2, . . . , q) be positive integers or ∞ satisfying (2), and let f1 (z), f2 (z) be non-constant non-admissible functions satisfying 0 < D(f1 ), D(f2 ) < ∞ and (2). If A=

q X

kj (m − 3)km + − 2 > 0, k +1 km + 1 j=m j

where m is a positive integers in {1, 2, . . . , q}, then f1 (z) ≡ f2 (z). Proof: Let n = m, Corollary 2.2 follows immediately from Corollary 2.1.

2

b and kj (j = Corollary 2.3 Let aj (j = 1, 2, . . . , q) be q distinct complex numbers in C, 1, 2, . . . , q) be positive integers or ∞ satisfying (2), and let f1 (z), f2 (z) be non-constant non-admissible functions satisfying 0 < D(f1 ), D(f2 ) < ∞ and (2). If q X j=3

kj 2 >2+ , kj + 1 D(f1 ) + D(f2 )

then f1 (z) ≡ f2 (z). 8

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Remark 2.1 From Corollary 2.3, we can get that f1 ≡ f2 can not hold, if two nonadmissible functions f1 , f2 share four distinct values IM and 0 < D(f1 ), D(f2 ) < ∞. 2 1 ≥ k2k+1 ≥ ··· ≥ Note that 1 ≥ k1k+1 following corollary immediately.

kq kq +1

≥

1 2,

from Corollary 2.3, we can get the

b and kj (j = Corollary 2.4 Let aj (j = 1, 2, . . . , q) be q distinct complex numbers in C, 1, 2, . . . , q) be positive integers or ∞ satisfying (2), and let f1 (z), f2 (z) be non-constant non-admissible functions satisfying 0 < D(f1 ), D(f2 ) < ∞ and (2). Set D := min{D(f1 ), D(f2 )}, (i) if D > 1, q = 7 and k7 ≥ 2, then f1 (z) ≡ f2 (z); (ii) if D > 1, q = 6 and k6 ≥ 4, then f1 (z) ≡ f2 (z); (iii) if D > 2 and q = 7, then f1 (z) ≡ f2 (z); (iv) if D > 3, q = 6 and k3 ≥ 2, then f1 (z) ≡ f2 (z); (v) if D > 6, q = 5, k3 ≥ 3 and k5 ≥ 2, then f1 (z) ≡ f2 (z); (vi) if D > 10, q = 5 and k4 ≥ 4, then f1 (z) ≡ f2 (z); (vii) if D > 12, q = 5, k3 ≥ 5 and k4 ≥ 3, then f1 (z) ≡ f2 (z); (viii) if D > 42, q = 5, k3 ≥ 6 and k4 ≥ 2, then f1 (z) ≡ f2 (z).

3

The problem of non-admissible function and admissible function in the unit disc D sharing some values

We now show that an admissible function can share sufficiently many values concerning multiple values with another non-admissible function as follows. Theorem 3.1 If f1 is admissible and f2 is a non-admissible satisfying limr→1− T (r, f2 ) = ∞, aj (j = 1, 2, . . . , q) be q distinct complex numbers, and let kj (j = 1, 2, . . . , q) be posim tive integers or ∞ satisfying (1). Set A3 = B1 + (m−2)k km +1 . Then (2) and A3 > 0 do not hold at same time, where B1 is stated as in Section 2. To prove the above theorem, we require the following lemma. Lemma 3.1 (see [11, Lemma 1]). Let f (z), g(z) satisfy limr→1− T (r, f ) = ∞ and limr→1− T (r, g) = ∞. If there is a K ∈ (0, ∞) with T (r, f ) ≤ KT (r, g) + S(r, f ) + S(r, g), then each S(r, f ) is also an S(r, g). The proof of Theorem 3.1: Suppose that (2) and A3 > 0 can hold at the same time, since f1 (z) is an admissible function, using the same argument as in Theorem 2.1 and from Theorem 1.1 and Lemma 3.4 in [16], for any ε(0 < 2ε < A3 ), we have  (10)

 q X (m − 1)km km 1 + B1 − ε T (r, f1 ) ≤ N kj ) (r, ) + S(r, f1 ), km + 1 k +1 f1 − aj j=1 m

where B1 is stated as in section 2. 9

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From (2), we have (11)

q X

N kj ) (r,

j=1

1 1 1 ) ≤N (r, ) ≤ T (r, ) + O(1) f1 − aj f1 − f2 f1 − f2 ≤T (r, f1 ) + T (r, f2 ) + O(1).

From (10) and (11), we get   km (m − 2)km (12) + B1 − ε T (r, f1 ) ≤ T (r, f2 ) + S(r, f1 ). km + 1 km + 1 Since 0 < ε < A3 , we have (13)

(m−2)km km +1

T (r, f1 ) ≤

+ B1 − ε > 0. From (12), we have

1 km T (r, f2 ) + S(r, f1 ). A3 − ε km + 1

km From A31−ε km +1 > 0 and Lemma 3.1, we can get that each S(r, f1 ) is also an S(r, f2 ). Since f1 (z) is admissible and f2 (z) is non-admissible, we can get T (r, f2 ) = S(r, f1 ). Thus, we have

(14)

T (r, f2 ) = S(r, f1 ) = S(r, f2 ) = o(T (r, f2 )).

This is a contradiction. Hence, we can get that (2) and A3 > 0 do not hold at the same time. Thus, this completes the proof of Theorem 3.1. From Theorem 3.1, we obtain the following corollaries. Corollary 3.1 If f1 is admissible and f2 is a non-admissible satisfying limr→1− T (r, f2 ) = ∞, aj (j = 1, 2, . . . , q) be q distinct complex numbers, and let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1). Then (2) and q X

kj (m − 1)km + −2>0 k + 1 km + 1 j=m+1 j do not hold at same time. Proof: Since Θ(fi ) ≥ 0 (i = 1, 2) and δ(0, f1 − aj ) ≥ 0 (j = 1, 2, . . . , q), the assertion follows from Theorem 3.1. 2 Corollary 3.2 If f1 is admissible and f2 is inadmissible satisfying limr→1− T (r, f2 ) = ∞, aj (j = 1, 2, . . . , q) be q distinct complex numbers. Then (i) f1 (z), f2 (z) can share at most three values a1 , a2 , a3 IM in D; (ii) f1 (z), f2 (z) can not share the following situations in D when a1 with reduced weight k(k ≥ 6), a2 with reduced weight 6, a3 with reduced weight 2 and the fourth value a4 with reduced weight 1; (iii) f1 (z), f2 (z) can not share the following situations in D when a1 with reduced weight k(k ≥ 3), a2 with reduced weight 3 and a3 , a4 with reduced weight 2; (iv) f1 (z), f2 (z) can not share two values a1 , a2 in D with reduced weight k(k ≥ 2), and the other three values a3 , a4 , a5 in D with reduced weight 1; (v) f1 (z), f2 (z) can share at most five values aj (j = 1, 2, . . . , 5) in D with reduced weight 1. 10

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Proof: (i) Suppose that f1 (z), f2 (z) share four values aj (j = 1, 2, 3, 4) IM , that is, kj = ∞(j = 1, 2, 3, 4). Since f1 (z) is admissible, from Theorem 1.1, we have 2T (r, f1 ) ≤

(15)

4 X

 N

j=1

r,

1 f1 − aj

 + S(r, f1 ).

From the assumptions of Corollary 3.2(i), we have (16)

4 X j=1

N

 r,

1 f1 − aj



 ≤N

r,

1 f1 − f2

 ≤ T (r, f1 ) + T (r, f2 ) + O(1).

From (15) and (16), we have T (r, f1 ) ≤ T (r, f2 ) + S(r, f1 ).

(17)

By Lemma 3.1 and similar proof of Theorem 3.1, we have T (r, f2 ) = S(r, f1 ) = S(r, f2 ) = o(T (r, f2 )). From this and limr→1− T (r, f2 ) = ∞, we can get a contradiction. Thus, this completes the proof of Corollary 3.2(i). By similar proof of Corollary 3.2 (i), we can prove (ii), (iii) and (iv) of Corollary 3.2 easily. Here we omit the detail. (v) Suppose that f1 , f2 share six values aj (j = 1, 2, . . . , 6) with reduced weight 1, that is, E 1) (aj , D, f1 ) = E 1) (aj , D, f2 ), (j = 1, 2, . . . , 6), and k1 = k2 = · · · = k6 = 1. Then, we can deduce that 6 X j=2

kj 1 1 − 2 = 5 × − 2 = > 0. kj + 1 2 2

From this and the conclusion of Theorem 3.1, we get a contradiction. Thus, this completes the proof of Corollary 3.2.

4

2

Some Remarks

It is very interesting to consider distinct small functions instead of distinct complex numbers. Li and Qiao[8] proved that Nevanlinna’s five values theorem is also true for five meromorphic functions aj (j = 1, 2, 3, 4, 5) on C which satisfy T (R, aj ) = o(T (R, f1 ) + T (R, f2 )) as R → ∞. For some further results related to small functions, we refer to Yao[15], Yi[18], Thai and Tan[10], Cao and Yi[1, 3]. Thus it may be interesting to consider the substitute of distinct complex numbers by distinct small functions within the results of this paper.

11

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References [1] T. B. Cao, H. X. Yi, On the multiple values and uniqueness of meromorphic functions sharing small functions as targets, Bull. Korean Math. Soc. 44 (4) (2007), 631-640. [2] T. B. Cao, H. X. Yi, The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc, J. Math. Anal. Appl. 319 (2006), 278-294. [3] T. B. Cao, H. X. Yi, Uniquenesstheorems for meromorphic mappings sharing hyperplanes in generalposition, Sci. Sin. Math. 41 (2) (2011), 135-144. (in Chinese) [4] Z. X. Chen, K. H. Shon, The growth of solutions of differential equations with coefficients of small growth in the disc, J. Math. Anal. Appl. 297 (2004), 285-304. [5] M. L. Fang, On the uniqueness of admissible meromorphic functions in the unit disc, Sci. China A 42(1999), 367-381. [6] W. K. Hayman, Meromorphic Functions, Oxford Univ. Press, London, 1964. [7] J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss. 122(2000), 1-54. [8] Y. H. Li, J. Y. Qiao, On the uniqueness of meromorphic functions concerning small functions, Sci. China Ser. A 29 (1999), 891-900. [9] R. Nevanlinna, Le th´eor`eme de Picard-Borel et la th´eorie des fonctions m´eromorphes, Reprinting of the 1929 original, Chelsea Publishing Co. New York, 1974(in Frech). [10] D. D. Thai, T. V. Tan, Meromorphic functions sharing small functions as targets, Internat. J. Math. 16 (4) (2005), 437-451. [11] F. Titzhoff, Slowly growing functions sharing values, Fiz. Mat. Fak. Moksl. Semin. Darb. 8(2005), 143-164. [12] H. Y. Xu, T. B. Cao, Analytic functions in the unit disc sharing values in a sector, Ann. Polon. Math. 103 (2012), 263-275. [13] H. Y. Xu, C. F. Yi, T. B. Cao, The uniqueness problem for meromorphic functions in the unit disc sharing values and set in an angular domain, Math. Scand. 109 (2011), 240-252. [14] L. Yang, Value Distribution Theory, Springer/Science Press, Berlin/Beijing, 1993/1982. [15] W. H. Yao, Two meromorphic functions sharing five small functions in the sense E k) (β, f ) = E k) (β, g), Nagoya Math. J. 167 (2002), 35-54. [16] H. X. Yi, C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press/ Kluwer., Beijing, 2003. [17] H. X. Yi, The multiple values of meromorphic functions and uniqueness, Chinese Ann. Math. Ser. A 10 (4) (1989), 421-427. [18] H. X. Yi, On one problem of uniqueness of meromorphic functions concerning small functions, Pro. Amer. Math. Soc. 130 (2001), 1689-1697.

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February 18, 2014

Global Dynamics of a Certain Two-dimensional Competitive System of Rational Difference Equations with Quadratic Terms M. R. S. Kulenovi´c1 and M. Pilling ‡ Department of Mathematics University of Rhode Island, Kingston, Rhode Island 02881-0816, USA

Abstract. We investigate global dynamics of the following system of difference equations  x2  n   xn+1 = a+y 2 n , n = 0, 1, 2, . . . 2  yn   yn+1 = b+x2 n

where the parameters a, b are positive numbers and the initial conditions x0 , y0 are arbitrary nonnegative numbers. We show that this system has an interesting dynamics which depends on the part of parametric space. The obtained ynamics is very different than the dynamics of the corresponding linear fractional system. In particular, we show that the system always exhibit Allee’s effect. Keywords. Allee effect, Basin of attraction, competitive map, global stable manifold, monotonicity.

AMS 2000 Mathematics Subject Classification: Primary: 39A10, 39A11 Secondary: 37E99, 37D10.

1

Introduction

In this paper we study global dynamics of the following rational system of difference equations  x2n   xn+1 = a+y 2 n , n = 0, 1, 2, . . . (1) 2  yn  yn+1 = b+x2 n

where the parameters a, b are positive numbers and initial conditions x0 , y0 are arbitrary nonnegative numbers. System (1) is a competitive system, and our results are based on recent results about competitive systems in the plane, see [17, 18]. System (1) can be used as a mathematical model for competition in population dynamics. The related competitive systems of the form  xn xn+1 = a+y n , n = 0, 1, 2, . . . (2) yn yn+1 = b+x n and

(

xn+1 yn+1

= =

xn 2 a+yn yn b+x2n

,

n = 0, 1, 2, . . .

(3)

were considered in [4, 6, 7] and global dynamics was derived. Systems (2) and (3) have similar dynamics which depends on the part of parametric space. In general, there are nine dynamic scenarios for systems (2) and (3) depending on the region of parametric space. Dynamics of System (1) is quite different and there are only three dynamic scenarios which describe the global behavior of 1 Corresponding

author, e-mail: [email protected]

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this system given by our main result. The common feature of all three scenarios is the appearance of the so called Allee’s effect, which means that in each case the zero equilibrium has a substantial basin of attraction which contains an open set, see [22] for related competitive system. This feature was not present in any of nine dynamic scenarios for systems (2) and (3). In addition, the non-hyperbolic case for System (1) is characterized by the existence of the unique interior non-hyperbolic equilibrium, while in the case of systems (2) and (3), non-hyperbolic equilibrium points were always all points on one or both coordinate axes. Our main result for System (1) is the following theorem. Theorem 1 All solutions of System (1) are eventually componentwise monotone. (a) Assume that System (1) has no interior equilibrium point, that is assume that D(a, b) > 0, where √ √ √ √
32
34 3 1 2 1 3 3 + 16 D(a, b) = 16b + 128 27 b 9 3 64b + 27 + 1 3 b 9 3 64b + 27 + 1 −8

1 9

+16a

1 9

√ √
5 3 64b3 + 27 + 1 3 +

√ √
4 3 64b3 + 27 + 1 3 +

1 9

√ √
8 3 64b3 + 27 + 1 3

256 4 27 b

+

16 9

√ √ 3b 64b3 + 27.

Then there exist the sets W s (Ex ) and W s (Ey ) which are the graphs of a strictly non-decreasing continuous functions of the first variable on [0, ∞) (and so are manifolds) with the following properties: B(E0 ) = {(x0 , y0 ) : W s (Ey ) se (x0 , y0 ) se W s (Ex )} , B((∞, 0)) = {(x0 , y0 ) : W s (Ex ) se (x0 , y0 )} , B((0, ∞)) = {(x0 , y0 ) : (x0 , y0 ) se W s (Ey )} , B(Ex ) = W s (Ex ),

B(Ey ) = W s (Ey ),

where se denotes south-east ordering defined by (5) below. (b) Assume that System (1) has one interior equilibrium point E, that is assume that the condition 4¯ xy¯ = 1 or D(a, b) = 0 holds. Then E is the non-hyperbolic equilibrium and there are two continuous non-decreasing curves W s (Ex ) and W s (Ey ) emanating from Ex and Ey respectively, with end points at E. Furthermore, there are continuous non-decreasing curves Cu and Cl (which may coincide) emanating from E such that the boundaries of the basins of attraction of (∞, 0) and (0, ∞) are given as: ∂B((∞, 0)) = Cl ∪ W s (Ex ),

∂B((0, ∞)) = Cu ∪ W s (Ey ).

The basins of attraction of different equilibrium points and the boundary points are given as: B(E0 ) = {(x0 , y0 ) : W s (Ey ) se (x0 , y0 ) se W s (Ex )} , B((∞, 0)) = {(x0 , y0 ) : ∂B((∞, 0)) se (x0 , y0 )} , B((0, ∞)) = {(x0 , y0 ) : (x0 , y0 ) se ∂B((0, ∞))} , B(E) = {(x0 , y0 ) : Cu se (x0 , y0 ) se Cl } . (c) Assume that System (1) has two interior equilibrium points E+ and E− , that is assume that D(a, b) < 0. Then E− is a repeller and E+ is a saddle point. There exist three continuous nondecreasing curves W s (Ex ), W s (Ey ) and W s (E+ ) emanating from Ex , Ey , and E+ respectively, with end points at E− . The basins of attraction B(E0 ), B(Ex ) and B(Ey ) are as in (a) and B((∞, 0)) and B((0, ∞)) are as in (b). Furthermore, ∂B((∞, 0)) = W s (Ex ) ∪ W s (E+ ),

157

∂B((0, ∞)) = W s (Ey ) ∪ W s (E+ ).

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Figure 1: Visual illustration of Theorem 1.

2

Preliminaries

A first order system of difference equations  xn+1 = f (xn , yn ) , yn+1 = g(xn , yn )

n = 0, 1, 2, . . .

(4)

where S ⊂ R2 , (f, g) : S → S, f , g are continuous functions is competitive if f (x, y) is non-decreasing in x and non-increasing in y, and g(x, y) is non-increasing in x and non-decreasing in y. If both f and g are nondecreasing in x and y, the system (4) is cooperative. Competitive and cooperative maps are defined similarly. Strongly competitive systems of difference equations or strongly competitive maps are those for which the functions f and g are coordinate-wise strictly monotone. Competitive and cooperative systems have been investigated by many authors, see [7, 8, 13, 15, 16, 17, 18, 20, 21, 23, 24]. Special attention to discrete competitive and cooperative systems in the plane was given in [1, 2, 7, 8, 9, 16, 17, 18, 20, 24]. One of the reasons for paying special attention to two-dimensional discrete competitive and cooperative systems is their applicability and the fact that many examples of mathematical models in biology and economy which involve competition or cooperation are models which involve two species. Another reason is that the theory of twodimensional discrete competitive and cooperative systems is very well developed, unlike such theory for three and higher dimensional systems. Part of the reason for this situation is de MottoniSchiaffino theorem given below, which provides relatively simple scenarios for possible behavior of many two-dimensional discrete competitive and cooperative systems. However, this does not mean that one can not encounter chaos in such systems as has been shown by H. Smith, see [24]. Here we give some basic notions about monotonic maps in plane. We define a partial order se on R2 (so-called south-east ordering) so that the positive cone is the fourth quadrant, i.e. this partial order is defined by:  1   2   1 x x x 6 x2  ⇔ (5) se 2 1 y y1 > y2 . y Similarly, we define the north-east ordering as:  1   2   1 x x x 6 x2 ne ⇔ y2 y1 6 y2 . y1

(6)

A map F is called competitive if it is non-decreasing with respect to se , that is, if the following holds:  1   2   1   2  x x x x F . (7)  ⇒F y1 y2 y1 y2

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For each v = (v 1 , v 2 ) ∈ R2+ , define Qi (v) for i = 1, .., 4 to be the usual four quadrants based at v and numbered in a counterclockwise direction, e.g., Q1 (v) = {(x, y) ∈ R2+ : v 1 ≤ x, v 2 ≤ y}. 2 For S ⊂ R+ let S ◦ denote the interior of S. The following definition is from [24]. Definition 1 Let R be a nonempty subset of R2 . A competitive map T : R → R is said to satisfy condition (O+) if for every x, y in R, T (x) ne T (y) implies x ne y, and T is said to satisfy condition (O−) if for every x, y in R, T (x) ne T (y) implies y ne x. The following theorem was proved by DeMottoni-Schiaffino [9] for the Poincar´e map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps [23]. Theorem 2 Let R be a nonempty subset of R2 . If T is a competitive map for which (O+) holds then for all x ∈ R, {T n (x)} is eventually componentwise monotone. If the orbit of x has compact closure, then it converges to a fixed point of T . If instead (O−) holds, then for all x ∈ R, {T 2n } is eventually componentwise monotone. If the orbit of x has compact closure in R, then its omega limit set is either a period-two orbit or a fixed point. It is well known that a stable period-two orbit and a stable fixed point may coexist, see Hess [12]. The following result is from [24], with the domain of the map specialized to be the cartesian product of intervals of real numbers. It gives a sufficient condition for conditions (O+) and (O−). Theorem 3 Let R ⊂ R2 be the cartesian product of two intervals in R. Let T : R → R be a C 0 competitive map. If T is injective and detJT (x) > 0 for all x ∈ R then T satisfies (O+). If T is injective and detJT (x) < 0 for all x ∈ R then T satisfies (O−). Theorems 2 and 3 are quite applicable as we have shown in [3, 4, 5, 17, 18, 20], in the case of competitive systems in the plane consisting of linear fractional equations. The following result is from [18], which generalizes the corresponding result for hyperbolic case from [17]. Related results have been obtained by H. L. Smith in [23]. Theorem 4 Let R be a rectangular subset of R2 and let T be a competitive map on R. Let x ∈ R be a fixed point of T such that (Q1 (x) ∪ Q3 (x)) ∩ R has nonempty interior (i.e., x is not the NW or SE vertex of R). Suppose that the following statements are true. a. The map T is strongly competitive on int((Q1 (x) ∪ Q3 (x)) ∩ R). b. T is C 2 on a relative neighborhood of x. c. The jacobian of T at x has real eigenvalues λ, µ such that |λ| < µ, where λ is stable and the eigenspace E λ associated with λ is not a coordinate axis. d. Either λ ≥ 0 and T (x) 6= x

and

T (x) 6= x

for all x ∈ int((Q1 (x) ∪ Q3 (x)) ∩ R) ,

or λ < 0 and T 2 (x) 6= x

for all x ∈ int((Q1 (x) ∪ Q3 (x)) ∩ R)

Then there exists a curve C in R such that (i) C is invariant and a subset of W s (x). (ii) the endpoints of C lie on ∂R.

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(iii) x ∈ C. (iv) C the graph of a strictly increasing continuous function of the first variable, (v) C is differentiable at x if x ∈ int(R) or one sided differentiable if x ∈ ∂R, and in all cases C is tangential to E λ at x, (vi) C separates R into two connected components, namely W− := {x ∈ R : ∃y ∈ C with x  y} and W+ := {x ∈ R : ∃y ∈ C with y  x} (vii) W− is invariant, and dist(T n (x), Q2 (x)) → 0 as n → ∞ for every x ∈ W− . (viii) W+ is invariant, and dist(T n (x), Q4 (x)) → 0 as n → ∞ for every x ∈ W+ . The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess, see [17] and [12], and is helpful for determining the basins of attraction of the equilibrium points. Corollary 1 If the nonnegative cone of  is a generalized quadrant in Rn , and if T has no fixed points in the ordered interval I(u1 , u2 ) = {u : u1 se u se u2 } other than u1 and u2 , then the interior of I(u1 , u2 ) is either a subset of the basin of attraction of u1 or a subset of the basin of attraction of u2 . The next results gives the existence and uniqueness of invariant curves emanating from a nonhyperbolic point of unstable type, that is a non-hyperbolic point where second eigenvalue is outside interval [−1, 1]. See [19]. Theorem 5 Let R = (a1 , a2 ) × (b1 , b2 ), and let T : R → R be a strongly competitive map with ¯ ∈ R, and such that T is continuously differentiable in a neighborhood of x ¯. a unique fixed point x ¯ the map T has associated characteristic values µ and ν satisfying Assume further that at the point x 1 < µ and −µ < ν < µ. ¯ mE2 (E− ) = 2¯ x, 2¯ y which is equivalent to 4¯ xy¯ < 1. Similarly, the slope of (E1) at E+ is smaller than the slope of (E2) at E+ , which is quivalent to 4¯ xy¯ > 1. If two equlibrium points E+ and E− coincide at E then 1 mE1 (E− ) = 2¯ = m (E ) = 2¯ x, which is equivalent to 4¯ xy¯ = 1. E2 − y The map associated with the System (1) has the form: !   x2 x a+y 2 T = . y2 y 2 b+y

The Jacobian matrix of T is  JT (x, y) = 

2x a+y 2 2

2xy − (b+x 2 )2

2

2x y − (a+y 2 )2 2y b+x2

 .

The Jacobian matrices of T evaluated in E0 , Ex , Ey are given as ! ! 0 0 2 0 JT (E0 ) = , JT (Ex ) = , JT (Ex ) = 0 0 0 0

161

(11)

0

0

0

2

! .

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By using the equilibrium equation, we obtain that the Jacobian matrix of T evaluated in an equilibrium E(¯ x, y¯) with positive coordinates has the form: ! 2 −2¯ y JT (E) = . (12) −2¯ x 2 The characteristic equation of (12) is given as λ2 − 4λ + 4 − 4¯ xy¯ = 0 and the eigenvalues of (12) are

√ ¯y¯. λ± = 2 ± 2 x

(13)

Theorem 6 System (1) has always three equilibrium points E0 (0, 0), Ex (a, 0), Ey (0, b), where E0 is locally asymptotically stable and Ex (a, 0) and Ey (0, b) are saddle points. In addition, System (1) has exactly one interior equilibrium point E(¯ x, y¯), when 4¯ xy¯ = 1, which is the non-hyperbolic equilibrium point of unstable type. If 4¯ xy¯ > 1, then System (1) has exactly two interior equilibrium points E− (¯ x− , y¯− ), which is repeller and E+ (¯ x+ , y¯+ ), which is a saddle point. Proof. The local character of the equilibrium points E0 (0, 0), Ex (a, 0), Ey (0, b) is obvious. In view of yn2 x2n ≤ 1, n = 0, 1, . . . xn+1 yn+1 = a + yn2 b + x2n we conclude that xn yn ≤ 1 for n = 1, 2, . . . and so x ¯y¯ ≤ 1. In view of Eq.(13) we conclude that λ+ > 2 and λ− ∈ [0, 1) if and only if 4¯ xy¯ > 1, which shows that E+ (¯ x+ , y¯+ ) is a saddle point. Furthermore, λ− > 1 if and only if 4¯ xy¯ < 1, which shows that E− (¯ x− , y¯− ) is a repeller. Finally, λ− = 1 if and only if 4¯ xy¯ = 1, in which case E is a non-hyperbolic point. 2

Figure 2: Visual illustration of Theorem 6.

4

Global Dynamics

In this section we prove that the map which corresponds to System (1) is injective and it satisfies (O+ ) condition. We will also give the global dynamics of the map on the coordinate axes. Theorem 7 The map T which corresponds to System (1) is injective.

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Proof. Indeed,  T

x1 y1



 =T

x2 y2





x21 a+y12 y12 b+x21

⇔





=

x22 a+y22 y22 b+x22

 

which is equivalent to a(x21 − x22 ) = −b(y12 − y22 ) = x22 y12 − x21 y22 , 2

which immediately implies that x1 = x2 and y1 = y2 .

Theorem 8 The map T which corresponds to System (1) satisfies (O+ ) condition. All solutions of System (1) are eventually componentwise monotonic. All bounded solutions converge to an equilibrium point. Proof. Assume that  T

x1 y1



 ≤ne T

x2 y2



 ⇒

x21 a+y12 y12 b+x21





 ≤ne 

x22 a+y22 y22 b+x22

 .

The last inequality is equivalent to a(x21 − x22 ) ≤ x22 y12 − x21 y22

(14)

b(y12 − y22 ) ≤ x21 y22 − x22 y12 .

Suppose x2 < x1 . Then y12 x22 > y22 x21 , which implies y1 > y2 . On the other hand, in view of (14) we obtain b(y12 − y22 ) < 0 and so y1 < y2 , which is a contradiction. Suppose x2 = x1 . Then y1 ≥ y2 , which in view of (14) implies y1 = y2 . Suppose x2 > x1 . If y2 < y1 then x22 > x21 , y22 < y12 and so y12 x22 > y22 x21 , which in view of (14) implies y1 < y2 , which is a contradiction. Consequently, y1 ≤ y2 and so in all cases     x1 x2 ≤ne . y1 y2 Thus we conclude that all solutions of (1) are eventually componentwise monotonic for all values of parameters. 2 Theorem 9 The positive parts of x-axis and y-axis are unstable manifolds of the equilibrium points Ex and Ey respectively. Proof. Since x0 = 0 (resp. y0 = 0) implies xn = 0 (resp. yn = 0) for every n ≥ 1 we conclude that the positive parts of x-axis and y-axis are invariant sets. Furthermore x0 = 0 (resp. y0 = 0) y2

x2

y2

n

n

x2

implies yn+1 = bn (resp. xn+1 = an ) for every n ≥ 1 and so yn = b2n0 −1 (resp. xn = a2n0 −1 ). Thus we have that x0 < a (x0 > a or x0 = a) gives xn → 0 (xn → ∞ or xn = a) as n → ∞. Similarly y0 < b (y0 > b or y0 = b) gives yn → 0 (yn → ∞ or yn = b) as n → ∞. 2 Next we present the proof of Theorem 1. Proof. of Theorem 1 The eventual componentwise monotonicity of solutions of System (1) follows from Theorem 8.

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(a) Local stability of all equilibrium points follows from Theorem 6. The existence and the properties of the global stable manifolds W s (Ex ), W s (Ey ) are guaranteed by Theorems 4 and 7. Thus, the regions B((∞, 0)) and B((0, ∞)) are invariant and in view of Theorem 8 every solution is eventually monotonic. Consequently, in view of Theorem 4 and uniqueness of stable manifold every solution which starts in B((∞, 0)) (resp. B((0, ∞))) is asymptotic to (∞, 0) (resp. (0, ∞)). Let (x0 , y0 ) be an arbitrary initial point between W s (Ex ) and W s (Ey ). First, assume that x0 > x ¯. Then (x0 , yWy ) se (x0 , y0 ) se (x0 , yWx ), where (x0 , yWy ) ∈ W s (Ey ), (x0 , yWx ) ∈ W s (Ex ), and so T n (x0 , yWy ) se T n (x0 , y0 ) se T n (x0 , yWx ). Since T n (x0 , yWy ) → Ey and T n (x0 , yWx ) → Ex as n → ∞, we conclude that T n (x0 , y0 ) eventually enters the ordered interval I(Ey , Ex ). Thus there exists N > 0 such that T N (x0 , y0 ) = (xN , yN ) ∈ intI(Ey , Ex ). Then (0, yN ) se (xN , yN ) se (xN , 0) and so T k (0, yN ) se T k (xN , yN ) se T k (xN , 0) for k ≥ N , which implies that E0 = limk→∞ T k (0, yN ) se limk→∞ T k (xN , yN ) se limk→∞ T k (xN , 0) = E0 and so limk→∞ T k (x0 , y0 ) = E0 . Second, assume that x0 = x ¯ and y0 > 0. Then, by strong monotonicity of T, x1 < x0 . Now, (x1 , yWy ) se (x1 , y1 ) se (x1 , 0), where (x1 , yWy ) ∈ W s (Ey ), which by monotonicity of T , implies that T n (x1 , yWy ) se T n (x1 , y1 ) se T n (x1 , 0). In a similar way as in the proof of the first case we show that T n (x1 , y1 ) eventually enters the ordered interval I(Ey , Ex ), in which case it converges to E0 . Third, assume that x0 < x ¯ and y0 > 0. Then, (x0 , yWy ) se (x0 , y0 ) se (x0 , 0), where (x0 , yWy ) ∈ W s (Ey ), which by monotonicity of T , implies that T n (x0 , yWy ) se T n (x0 , y0 ) se T n (x0 , 0). In a similar way as in the proof of the first case we show that T n (x0 , y0 ) eventually enters the ordered interval I(Ey , Ex ), in which case, it converges to E0 . (b) Local stability of all equilibrium points follows from Theorem 6. The existence and the properties of the global stable manifolds W s (Ex ), W s (Ey ) are guaranteed by Theorems 4 and 7. The existence and the properties of the curves Cl and Cu are guaranteed by Corollary 2. The regions B((∞, 0)) and B((0, ∞)) are invariant and, by Theorem 8 all solutions are eventually componentwise monotonic. Since the basins of attraction of all equilibrium points in those regions are uniquely determined, then every solution which starts in B((∞, 0)) (resp. B((0, ∞))) is asymptotic to (∞, 0) (resp. (0, ∞)). Let (x0 , y0 ) be an arbitrary initial point between W s (Ex ) and W s (Ey ). First, assume that x0 > x ¯. Then (x0 , yWy ) se (x0 , y0 ) se (x0 , yWx ), where (x0 , yWy ) ∈ W s (Ey ), (x0 , yWx ) ∈ W s (Ex ), and so T n (x0 , yWy ) se T n (x0 , y0 ) se T n (x0 , yWx ) for n ≥ 0. Since T n (x0 , yWy ) → Ex and T n (x0 , yWx ) → Ex as n → ∞, we conclude that T n (x0 , y0 ) eventually enters the interior of the ordered interval I(Ey , Ex ), in which case, in a similar way as in the case (a), it converges to E0 . The case x0 ≤ x ¯, is treated in exactly the same as the analogue case of (a). Finally, let (x0 , y0 ) be an arbitrary initial point between the curves Cu and Cl . Then (x0 , yl ) se (x0 , y0 ) se (x0 , yu ), where (x0 , yl ) ∈ C((0, ∞)), (x0 , yu ) ∈ C((∞, 0)), and so T n (x0 , yl ) se T n (x0 , y0 ) se T n (x0 , yu ). Since T n (x0 , yl ) → E and T n (x0 , yu ) → E as n → ∞, we conclude that T n (x0 , y0 ) → E. (c) Local stability of all equilibrium points follows from Theorem 6. The existence and the properties of the global stable manifolds W s (Ex ), W s (Ey ) and W s (E+ ) are guaranteed by Theorems 4 and 7. The proof for the basin of attraction B(E0 ) is same as in the case (a). Assume that an arbitrary initial point (x0 , y0 ) is above ∂B((∞, 0)) in south-east

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ordering. In view of Theorem 8 every solution is eventually componentwise monotonic and so it must be asymptotic to (∞, 0) as the other three equilibrium points have uniquely determined basins of attractions. More precisely, in view of Theorem 4 there exists the unstable manifold W u (E+ ), which is passing through E+ and is continuous, non-increasing curve contained in Q2 (E+ ) ∪ Q4 (E+ ) with the property that all solutions which start below ∂B((∞, 0)) or above ∂B((0, ∞)) are asymptotic to W u (E+ ) as n → ∞. Similar reasoning applies if an arbitrary initial point (x0 , y0 ) is below ∂B((0, ∞)) in south-east ordering. 2 Remark 1 In the special case a = b an immediate checking shows that the line y = x is an invariant set and that all interior equilibrium points (if any) belongs to that line, which in view of Theorem 4 shows that y = x is the stable manifold in that case. In fact, in this case the coordinates of the equilibrium points are computable and we have that for a < 1/4 the interior equilibrium points are √ √ √ √     1 + 1 − 4a 1 + 1 − 4a 1 − 1 − 4a 1 − 1 − 4a , , E+ , , E− 2 2 2 2 and for a = 1/4 the interior equilibrium point is E(1/2, 1/2), while in the case a > 1/4 there is no an interior equilibrium point. In any case the line y = x is an invariant set which in the case a < 1/4 becomes the global stable manifold of E+ . Based on our simulations we pose the following conjecture. Conjecture 1 We conjecture that Cl = Cu in Part b) of Theorem 1.

References [1] S. Basu and O. Merino, On the Behavior of Solutions of a System of Difference Equations, Comm. Appl. Nonlinear Anal. 16 (2009), no. 1, 89–101. [2] A. Brett and M. R. S. Kulenovi´c, Basins of Attraction of Equilibrium Points of Monotone Difference Equations, Sarajevo J. Math., 5(2009), 211–233. [3] Dˇz. Burgi c, S. Kalabuˇsi´c and M. R. S. Kulenovi´c, Nonhyperbolic Dynamics for Competitive Systems in the Plane and Global Period-doubling Bifurcations, Adv. Dyn. Syst. Appl. 3(2008), 229-249. [4] Dˇz. Burgi´c, M. R. S. Kulenovi´c and M. Nurkanovi´c, Global Dynamics of a Rational System of Difference Equations in the plane, Comm. Appl. Nonlinear Anal., 15(2008), 71-84. [5] E. Camouzis, M. R. S. Kulenovi´c, G. Ladas, and O. Merino, Rational Systems in the Plane, J. Difference Equ. Appl. 15 (2009), 303-323. [6] D. Clark and M. R. S. Kulenovi´c, On a Coupled System of Rational Difference Equations, Comput. Math. Appl. 43 (2002), 849-867. [7] D. Clark, M. R. S. Kulenovi´c, and J.F. Selgrade, Global Asymptotic Behavior of a Two Dimensional Difference Equation Modelling Competition, Nonlinear Anal., TMA 52(2003), 1765-1776. [8] J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle , J. Difference Equ. Appl. 10(2004), 1139-1152. [9] P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol. 11 (1981), 319–335.

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[10] J. E. Franke and A.-A. Yakubu, Mutual exclusion verses coexistence for discrete competitive systems, J. Math. Biol. 30 (1991), 161-168. [11] J. E. Franke and A.-A. Yakubu, Geometry of exclusion principles in discrete systems, J. Math. Anal. and Appl. 168 (1992), 385-400. [12] P. Hess, Periodic-parabolic boundary value problems and positivity. Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow; 1991. viii+139 pp. [13] M. Hirsch and H. Smith, Monotone dynamical systems. Handbook of differential equations: ordinary differential equations. Vol. II, 239-357, Elsevier B. V., Amsterdam, 2005. [14] S. Kalabuˇsi´c and M. R. S. Kulenovi´c, Dynamics of Certain Anti-competitive Systems of Rational Difference Equations in the Plane, J. Difference Equ. Appl., 17(2011), 1599–1615. [15] M. R. S. Kulenovi´c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman& Hall/CRC Press, Boca Raton, 2002. [16] M. R. S. Kulenovi´c and O. Merino, Competitive-Exclusion versus Competitive-Coexistence for Systems in the Plane, Discrete Contin. Dyn. Syst. Ser. B 6(2006), 1141-1156. [17] M. R. S. Kulenovi´c and O. Merino, Global Bifurcation for Competitive Systems in the Plane, Discrete Contin. Dyn. Syst. B 12(2009), 133-149. [18] M. R. S. Kulenovi´c and O. Merino, Invariant Manifolds for Competitive Discrete Systems in the Plane, Int. J. of Bifurcations and Chaos, 20(2010), 2471-2486. [19] M. R. S. Kulenovi´c and O. Merino, Invariant Curves for Competitive Discrete Systems in the Plane, (to appear). [20] M. R. S. Kulenovi´c and M. Nurkanovi´c, Asymptotic Behavior of a Linear Fractional System of Difference Equations, J. Ineq. Appl., (2005), 127-143. [21] W. J. Leonard and R. May, Nonlinear aspects of competition between species, SIAM J. Appl. Math. 29 (1975), 243-275. [22] G. Livadiotis and S. Elaydi, General Allee effect in two-species population biology, J. Biol. Dyn., 6(2012), 959–973. [23] H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations 64 (1986), 165-194. [24] H. L. Smith, Planar Competitive and Cooperative Difference Equations, J. Difference Equ. Appl. 3(1998), 335-357.

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HIGHER-ORDER q-DAEHEE POLYNOMIALS YOUNG-KI CHO, TAEKYUN KIM, TOUFIK MANSOUR, AND SEOG-HOON RIM Abstract. Recently, q-Daehee polynomials and numbers are introduced (see [8]). In this paper, we consider the higher-order q-Daehee numbers and polynomials and give some new relations and identities between higher-order q-Daehee polynomials and higher-order q-Bernoulli 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 , respectively. The p-adic norm | · |p is normalized as |p|p = 1/p. Let q be an indeterminate in Cp with |1 − q| < p−1/(p−1) , and let U D(Zp ) be the space of uniformly differentiable functions on Zp . For f ∈ U D(Zp ), the p-adic q-integral on Zp (q-Volkenborn integral on Zp ) is defined by Kim to be (1.1)

Iq (f ) =

Z

f (x)dµq (x) = lim

N →∞

Zp

where [x]q =

1−qx 1−q

(1.3)

[pN ]q

N pX −1

f (x)q x ,

x=0

(see [3]). Let f1 (x) = f (x + 1). Then, by (1.1), we get

(1.2) where f ′ (0) =

1

qIq (f1 ) − Iq (f ) = (q − 1)f (0) + d dx f (x) |x=0

q−1 ′ f (0), log q

(see [3]). From (1.2), we note that Z q−1 q − 1 + log qt . ext dµq (x) = t qe − 1 Zp

Now, we define the q-Bernoulli numbers which are given by (1.4)

q−1+ qet

q−1 log q t

−1

=

X

n≥0

Bn,q

tn , n!

(see [8]). Note that limq→1 Bn,q = Bn the n-th Bernoulli number. From (1.3) and (1.4), we have Z (1.5) xn dµq (x) = Bn,q , n ≥ 0. Zp

We can also define the q-Bernoulli polynomials which are given by the generating function to be (1.6)

q−1+ qet

q−1 log q t xt

−1

e

=

X

n≥0

Bn,q (x)

tn . n!

2000 Mathematics Subject Classification. 05A40. Key words and phrases. Bernoulli numbers, Bernoulli polynomials, Daehee numbers, Daehee polynomials. 1

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Thus, by (1.4) and (1.5), we see that (1.7)

Bn,q (x) =

n   X n Bℓ,q xn−ℓ . ℓ ℓ=0

From (1.2), we can derive the following equation: Z

(1.8)

e(x+y)t dµq (y) =

q−1+

q−1 log q t xt

qet − 1

Zp

e

=

X

Bn,q (x)

n≥0

tn . n!

By (1.8), we get Z

(1.9)

(x + y)n dµq (y) = Bn,q (x),

n ≥ 0.

Zp

For α ∈ N, the higher-order q-Bernoulli polynomials are defined by the generating function to be !α q−1 X q − 1 + log tn qt xt (α) (1.10) e = B (x) . n,q qet − 1 n! n≥0

(α)

(α)

Note that limq→1 Bn,q (x) = Bn (x)the n-th higher-order Bernoulli polynomial which are defined by α P n (α) the generating function to be et t−1 ext = n≥0 Bn (x) tn! (see [1-13]).

(1 + t)x = As is known, the Daehee polynomials are defined by the generating function to be log(1+t) t n t 6=0 Dn (x) n! (see [5]). In [5], the Daehee polynomials of order α are also defined by the generating function to be α  X tn log(1 + t) (1 + t)x = Dn(α) (x) , t n! P

6=0

(see [4, 5, 11, 12]). Recently, q-Daehee polynomials are defined by the generating function to be q−1+

(1.11)

q−1 log q

log(1 + t)

qt + q − 1

(1 + t)x =

X

Dn,q (x)

n≥0

tn n!

(see [8]). From (1.2) and (1.11), we have (1.12)

Z

Zp

x+y

(1 + t)

dµq (y) =

q−1+

q−1 log q

log(1 + t)

qt + q − 1

(1 + t)x =

X

n≥0

Dn,q (x)

tn . n!

In viewpoint of (1.12), we consider the higher-order q-Daehee polynomials and give new relations and identities between higher-order q-Daehee polynomials and higher-order q-Bernoulli polynomials. 2. Higher-order q-Daehee polynomials We assume that t ∈ Cp with |t|p < p−1/(p−1) . For k ∈ N, let us define the higher-order q-Daehee polynomials as follows: !k q−1 X q − 1 + log tn q log(1 + t) (k) (2.1) (1 + t)x = Dn,q (x) . qt + q − 1 n! n≥0

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From (1.2), we can derive the following equation: Z

···

Zp

Z

x1 +···+xk +x

(1 + t)

dµq (x1 ) · · · dµq (xk ) =

q−1+

=

log(1 + t)

qt + q − 1

Zp

(2.2)

q−1 log q

X

(k) Dn,q (x)

n≥0

(1 + t)x

tn . n!

Thus, by (2.2), we get Z Z (k) (2.3) ··· (x1 + · · · + xk + x)n dµq (x1 ) · · · dµq (xk ) = Dn,q (x), Zp

!k

n ≥ 0,

Zp

which implies (k) Dn,q (x) =

(2.4)

n X

S1 (n, ℓ)

Z

···

Zp

ℓ=0

Z

(x1 + · · · + xk + x)n dµq (x1 ) · · · dµq (xk ),

Zp

where S1 (n, ℓ) is the Stirling number of the first kind. We observe that (2.5)

Z

Zp

···

Z

e

(x1 +···+xk +x)t

dµq (x1 ) · · · dµq (xk ) =

q−1+

q−1 log q

log(1 + t)

qt + q − 1

Zp

!k

(1 + t)x .

By (1.10), (2.4) and (2.5), we deduce the following result. Theorem 1. For n ≥ 0 and k ∈ N, (k) Dn,q (x) =

n X

(k)

S1 (n, ℓ)Bℓ,q (x).

ℓ=0

(k)

(k)

When x = 0, Dn,q = Dn,q (0) are called the higher-order q-Daehee numbers. Corollary 1. For n ≥ 0 and k ∈ N, (k) Dn,q

=

X

ℓ1 +···+ℓk =n



 n X n (k) Dℓ1 ,q · · · Dℓk ,q = S1 (n, j)Bj,q , ℓ 1 , . . . , ℓk j=0

(k)

where Bn,q is the n-th q-Bernoulli number of order k. In (2.1), by replacing t by et − 1, we have !k q−1 X X q − 1 + log (et − 1)n qt tx (k) (2.6) e = D (x) = n,q t qe − 1 n! n≥0

m≥0

m X

n=0

!

(k) Dn,q (x)S2 (n, m)

tm , m!

where S2 (n, m) is the Stirling number of the second kind. Therefore, by (1.10) and (2.6), we obtain the following result. Theorem 2. For m ≥ 0 and k ∈ N, (k) Bm,q (x) =

m X

(k)

S2 (m, ℓ)Dℓ,q (x).

ℓ=0

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YOUNG-KI CHO, TAEKYUN KIM, TOUFIK MANSOUR, AND SEOG-HOON RIM

Now, we consider the higher-order q-Daehee polynomials of the second kind as follows: for |t − 2|p < p and t 6= 1, Z Z b n,q (x) = (2.7) D ··· (−x1 − · · · − xk + x)n dµq (x1 ) · · · dµq (xk ) −1/(p−1)

Zp

Zp

=

n X

(−1)n−ℓ S1 (n, ℓ)

=

n X

(−1)n−ℓ S1 (n, ℓ)Bℓ,q (−x),

Z

···

Zp

ℓ=0

Z

(x1 + · · · + xk − x)ℓ dµq (x1 ) · · · dµq (xk )

Zp

(k)

ℓ=0

which gives the following result. Theorem 3. For n ≥ 0 and k ∈ N, b (k) (x) = D n,q

n X

(k)

(−1)n−ℓ S1 (n, ℓ)Bℓ,q (−x).

ℓ=0

From (2.7), we have (2.8)

X

n≥0

n

b n,q (x) t = D n!

Z

···

Z

X x1 + · · · + xk + n − 1

···

Z

(1 − t)−x1 ···−xk +x dµq (x1 ) · · · dµq (xk )

Zp

=

Z

=



n

Zp n≥0

Zp

tn dµq (x1 ) · · · dµq (xk )

Zp

k 1−q 1−t (q − 1 + log(1 − t)) (1 − t)x . q−1+t log q

By (2.8), we get (2.9)

X

n≥0

−t n b n,q (x) (1 − e ) = D n!



=

k e−t q−1 (q − 1 + t) e−xt = q − e−t log q

X

(k) Bm,q (−x)

m≥0

q−1+

q−1 log q t

qet − 1

!k

e−xt

tm . m!

Also, we have (2.10)

X

n≥0

−t n X b n,q (x) (1 − e ) = D n!

m≥0

m X

! tm (k) m−n b Dn,q (x)(−1) . S2 (m, n) m! n=0

Therefore, by (2.9) and (2.10), we obtain the following result. Theorem 4. For m ≥ 0 and k ∈ N, (k) Bm,q (−x)

=

m X ℓ=0

b (k) (x). (−1)m−ℓ S2 (m, ℓ)D ℓ,q

By (2.7), we get (2.11)

(k) b n,q D (x) =

Z

Zp

···

Z

(x1 + · · · + xk + x)(n) dµq (x1 ) · · · dµq (xk ),

Zp

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where x(n) = x(x + 1) · · · (x + n − 1). Thus, from (2.11), we have (k) b n,q (2.12) D (x) = (−1)n

= (−1)n

Z

···

Zp n X

Z

(−x1 − · · · − xk + x)n dµq (x1 ) · · · dµq (xk )

Zp

S1 (n, ℓ)

···

Zp

ℓ=0

n X

Z

Z

(−x1 − · · · − xk + x)ℓ dµq (x1 ) · · · dµq (xk ) Zp

ℓ X

  Z Z ℓ = (−1)n S1 (n, ℓ) xℓ−m ··· (−x1 − · · · − xk )m dµq (x1 ) · · · dµq (xk ) m Zp Zp ℓ=0 m=0   n ℓ XX ℓ (k) = (−1)n S1 (n, ℓ) xℓ−m Bm,q m m=0 ℓ=0

=

n X

(k)

(−1)n−ℓ S1 (n, ℓ)Bℓ,q (−x).

ℓ=0

Therefore, by (2.12), we obtain the following theorem. Theorem 5. For n ≥ 0 and k ∈ N, b (k) (x) = D n,q

n X

(k)

(−1)n−ℓ S1 (n, ℓ)Bℓ,q (−x).

ℓ=0

Now, we observe that (2.13)

n Dn,q (x)

(−1)

b

n!

X x1 + · · · + xk + x = (−1) ··· dµq (x1 ) · · · dµq (xk ) n Zp Zp n≥0  Z Z  −x1 − · · · − xk − x + n − 1 = ··· dµq (x1 ) · · · dµq (xk ) n Zp Zp Z  Z  n  X n−1 −x1 − · · · − xk − x dµq (x1 ) · · · dµq (xk ) = ··· m − 1 Zp m Zp m=0   Z  Z  n X −x1 − · · · − xk − x 1 n−1 m! ··· dµq (x1 ) · · · dµq (xk ) = m! n − m m Zp Zp m=1   n X 1 n − 1 b (k) Dm,q (−x). = m! n − m m=1 n

Z

Z

Therefore, by (2.13), we obtain the following result. Theorem 6. For n ≥ 0 and k ∈ N, (1−)n

  n (k) b n,q D (x) X (−1)ℓ n − 1 b (k) = Dℓ,q (−x). n! ℓ! n−ℓ ℓ=1

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YOUNG-KI CHO, TAEKYUN KIM, TOUFIK MANSOUR, AND SEOG-HOON RIM

Note that, by (2.7), we have

(2.14)

 Z X b n,q (x) Z x1 + · · · + xk + n − 1 − x D = ··· dµq (x1 ) · · · dµq (xk ) n! n Zp Zp n≥0 Z  Z  n  X n−1 x1 + · · · + xk − x = ··· dµq (x1 ) · · · dµq (xk ) m − 1 Zp m Zp m=0   Z  Z  n X x1 + · · · + xk − x 1 n−1 m! ··· dµq (x1 ) · · · dµq (xk ) = m! n − m m Zp Zp m=1   n X 1 n − 1 b (k) Dm,q (−x). = m! n − m m=1

Therefore, by (2.13), we obtain the following theorem. Theorem 7. For n ≥ 0 and k ∈ N,   n (k) b n,q D (x) X 1 n − 1 b (k) Dℓ,q (−x). = n! ℓ! n − ℓ ℓ=1

Acknowledgment This research was supported by Kyungpook National University BK 21 plus fund 2014.

References [1] S. Araci and M. Acikgoz, A note of the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22:3 (2012) 399–406. [2] A. Bayad, Sepcial values of Lerch zeta function and their Fourier expansions, Adv. Stud. Contemp. Math. 21:1 (2011) 1–4. [3] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 19 (2002) 288–299. [4] D.S. Kim and T. Kim, Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials, Adv. Stud. Contemp. Math. 23:4 (2013) 621–636. [5] D.S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. 7:120 (2013) 5969–5976. [6] D.S. Kim, T. Kim, and S.-H. Rim, Some identities arising from Sheffer sequences of special polynomials, Adv. Stud. Contemp. Math. 23:4 (2013) 681–683. [7] D.S. Kim, T. Kim, and J. Seo, A note on Changhee polynomails and numbers, Adv. Stud. Theor. Phys. 7:20 (2013) 993–1003. [8] T. Kim, S.-H. Lee, T. Mansour, and J.-J. Seo, A note on q-Daehee polynomials and numbers, Adv. Stud. Contemp. Math. 24:2 (2014) 131–139. [9] T. Kim, T. Mansour, S.-H. Rim, J.-J. Seo, A note on q-Changhee polynomials and numbers, Adv. Stud. Theor. Phys. 8:1 (2014) 35–41. [10] H. Ozden, I.N. Cangul, and Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. 18 (2009) 41–48. [11] J.-W. Park, S.-H. Rim, and J. Kwon, The twisted Daehee numbers and polynomials, Adv. Diff. Equ. 2014:1 (2014) P. 9. [12] J.-W. Park, S.-H. Rim, and J. Kwon, The hyper-geometric Daehee numbers and polynomials, Turkish J. Analysis and Number Theory 2:1 (2013) 59–62. [13] Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math. 16:2 (2008) 271–278.

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Microwave & Antenna Lab., Kyungpook National University, Taegu 702-701, S. Korea E-mail address: [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, S. Korea E-mail address: [email protected] Department of Mathematics, University of Haifa, 3498838 Haifa, Israel E-mail address: [email protected] Department of Mathematics Education, Kyungpook National University, Taegu 702-701, S. Korea E-mail address: [email protected]

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Approximate fuzzy ternary homomorphisms and fuzzy ternary derivations on fuzzy ternary Banach algebras Ali Ebadian1 , Mohammad Ali Abolfathi2 , Rasoul Aghalary3 , Choonkil Park4 and Dong Yun Shin5∗ 1,2,3

4

Department of Mathematics, Urmia University, P.O. Box 165, Urmia, Iran; Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea; 5 Department of Mathematics, University of Seoul, Seoul 130-743, Korea

Abstract. Using the fixed point method, we prove the Hyers-Ulam stability of ternary homomorphisms and ternary derivations in fuzzy ternary Banach algebras associated to the following additive functional equation n X k X (

k+1 X

k=2 i1 =2 i2 =i1 +1

···

n X

)f (

in−k+1 =in−k +1

n X

i6=1,i6=i1 ,i2 .··· ,in−k+1

xi −

n−k+1 X

xir ) + f (

r=1

n X

xi ) = 2n−1 f (x1 )

i=1

for a fixed integer n with n ≥ 2.

1. Introduction and preliminaries The theory of fuzzy sets was introduced by Zadeh in 1965 [53]. Fuzzy set theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. The fuzzy topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. In 1984, Katsaras [18] introduced an idea of a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. In the same year, Wu and Fang [50] introduced a notion fuzzy normed apace to give a generalization of the Kolmogoroff normalized theorem for fuzzy topological vector spaces. In 1992, Felbin [10] introduced an alternative definition of a fuzzy norm on a vector space with an associated metric of Kaleva and Seikkala type [16]. Some mathematics have define fuzzy normed on a vector form various point of view [25, 41, 51]. In particular, Bag and Samanta [2] following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric of Kramosil and Michalek type [24]. They established a decomposition theorem of fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. A classical equation in the theory of functional equations is the following: “when is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?”. If the problem accepts a solution, we 0

2010 Mathematics Subject Classification: 39B52, 39B82, 45L05, 46S40, 47S40, 47H10. Keywords: Hyers-Ulam stability; fuzzy ternary Banach algebra; additive functional equation; fuzzy ternary homomorphism; fuzzy ternary derivation; fixed point method. ∗ Corresponding author: Dong Yun Shin (email: [email protected]). 0 E-mail: [email protected]; [email protected]; [email protected]; [email protected] 0

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say that the equation is stable. The first problem concerning group homomorphisms was raised by Ulam [49] in 1940. In the next year, Hyers [12] gave a first affirmative answer to the question of Ulam in context of Banach spaces. In 1978, Rassias [45] proved a generalization of the Hyers’ theorem for additive mappings. Furthermore, in 1994, Gˇ avrut¸a [11] provided a further generalization of Rassias’ theorem in which he replaced the bound ε(kxkp + kykp ) in by a general control function ϕ(x, y). Recently, several stability results have been obtained for various equations and mappings with more general domains and ranges have been investigated by a number of authors and there are many interesting results concerning this problem [1, 13, 14, 15, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 42, 43, 46, 47, 48, 52]. In the following, we will give some notations that are needed in this paper. Definition 1.1. Let X be a real vector space. A function N : X × R → [0, 1] is said to be a fuzzy norm on X if for all x, y ∈ X and all t, s ∈ R, (N 1) N (x, t) = 0 for t ≤ 0; (N 2) N (x, t) = 1 for all t > 0 if and only if x = 0; (N 3) N (cx, t) = P (x, |c|t ) for each c 6= 0; (N 4) N (x + y, s + t) ≥ min{N (x, t), N (y, s)}; (N 5) N (x, ·) is a non-decreasing function on R and limt→∞ N (x, t) = 1; (N6) N (x, ·) is continuous on R for x 6= 0. The pair (X, N ) is called a fuzzy normed linear space. One may regard N (x, t) as the truth value of the statement “the norm of x is less than or equal to the real number t”. Example 1.2. Let (X, k · k) be a normed linear space and α, β > 0. Then ( αt , t > 0, x ∈ X, βt+kxk N (x, t) = 0, t ≤ 0, x ∈ X is a fuzzy norm on X. Example 1.3. Let (X, k · k) be a normed linear space. Then  t < 0,  0, t , 0 < t ≤ kxk, x ∈ X, N (x, t) =  kxk 1, t > kxk, x∈X is a fuzzy norm on X. Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent if there exists x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t > 0. In that 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 fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy normed 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 point x0 ∈ X if, for each sequence {xn } converging to x0 in X, the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X , then f is said to be continuous on X [3]. 175

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Ternary algebraic operations have propounded originally in nineteenth century by several mathematicians such as Cayley [5] who introduced the notion of cubic matrix which in turn was generalized by Kapranov, Gelfand and Zelevinskii in 1990 [17]. The application of ternary algebra in supersymmetry is presented in [20] and in Yang-Baxter equation in [37]. Let X be a complex linear space equipped a mapping [·, ·, ·] : X × X × X → X with (x, y, z) → [x, y, z] that is linear in variables x, y, z and satisfies the associative identity, i.e., [x, y, [z, u, v]] = [x, [y, z, u], v] = [[x, y, z], u, v] for all x, y, z, u, v ∈ X. The pair (X, [·, ·, ·]) is called a ternary algebra. The ternary algebra (X, [·, ·, ·]) is called unital if it has an identity element, i.e., an element e ∈ X such that [x, e, e] = [e, e, x] = x for all x ∈ X. The linear space X is called a ternary normed algebra if X is a ternary algebra and there exists a norm k · k on X which satisfies k[x, y, z]k ≤ kxkkykkzk for all x, y, z ∈ X. Whenever the ternary algebra X is unital with unit element e, we repute kek = 1. A normed ternary algebra X is called a ternary Banach algebra if (X, k · k) is a Banach space. Definition 1.4. Let X be a ternary algebra and (X, N ) be a fuzzy normed space. (1) The fuzzy normed space (X, N ) is called a fuzzy ternary normed algebra if N ([x, y, z], tsr) ≥ N (x, t)N (y, s)N (z, r) for all x, y ∈ X and all positive real numbers t, s, r. (2) A complete fuzzy ternary normed algebra is called a fuzzy ternary Banach algebra. Example 1.5. Let (X, k · k) be a ternary normed (Banach) algebra. Let ( t , t > 0, x ∈ X, t+kxk N (x, t) = 0, t ≤ 0, x ∈ X. Then N (x, t) is a fuzzy norm on X and (X, N ) is a fuzzy ternary normed (Banach) algebra. Definition 1.6. Let (X, N ) and (Y, N 0 ) be two fuzzy ternary Banach algebras. (1) The C-linear mapping f : (X, N ) → (Y, N 0 ) is called a fuzzy ternary homomorphism if f ([x, y, z]) = [f (x), f (y), f (z)] for all x, y, z ∈ X. (2) The C-linear mapping f : (X, N ) → (X, N ) is called a fuzzy ternary derivation if f ([x, y, z]) = [f (x), y, z] + [x, f (y), z] + [x, y, f (z)] for all x, y, z ∈ X. Let X be a nonempty 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 for x, y ∈ X ; (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. 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 176

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operator T is called a strictly contractive operator. We recall the following theorem by Diaz and Margolis. Theorem 1.7. ([30, 44]) Let (X, d) be a complete generalized metric space and J : X → X be a strictly contractive mapping with Lipshitz constant L < 1. Then, for each given x ∈ X, either d(J n x, J n+1 x) = ∞ for all n ≥ 0, or there exists a natural number 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 , y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y. In 1996, Isac and Rassias [14] were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [4, 8, 9, 19, 39, 40]). Recently, Ebadian and Ghobadipour [7] considered the Hyers-Ulam stability of double derivations on Banach algebras and Lie *-double derivations on Lie C*-algebras associated with the following additive functional equation k k+1 n X X X ( ··· k=2 i1 =2 i2 =i1 +1

n X in−k+1 =in−k +1

n−1

=2

n X

)f (

xi −

i6=1,i6=i1 ,i2 ,··· ,in−k+1

n−k+1 X

n X xi ) xir ) + f (

r=1

i=1

f (x1 )

(1.1)

for a fixed integer n with n ≥ 2. In this paper, we investigate the Hyers-Ulam stability of fuzzy ternary homomorphisms and fuzzy ternary derivations on fuzzy ternary Banach algebras associated with the additive functional equation (1.1). Throughout this article, assume that (X, N ) and (Y, N 0 ) be fuzzy ternary Banach algebras. 2. Approximate fuzzy ternary homomorphisms in fuzzy ternary Banach algebras In this section, we prove the Hyers-Ulam stability of fuzzy ternary homomorphisms in fuzzy ternary Banach algebras related to the additive functional equation (1.1). Theorem 2.1. Let ϕ : X n → [0, ∞) be a function such that there exists an L < 1 with ϕ(2x1 , 2x2 , · · · , 2xn ) ≤ 2Lϕ(x1 , x2 , · · · , xn )

(2.1)

for all x1 , x2 , · · · , xn ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and n X k k+1 X X N( ( ···

n X

k=2 i1 =2 i2 =i1 +1

in−k+1 =in−k +1

n X

)f (

i6=1,i6=i1 ,i2 ,··· ,in−k+1

µxi −

n−k+1 X

n X +f ( µxi ) − 2n−1 µf (x1 ), t) ≥ i=1

177

µxir )

(2.2)

r=1

t t + ϕ(x1 , x2 , · · · , xn ) Ali Ebadian et al 174-185

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Approximate fuzzy ternary homomorphisms in fuzzy ternary Banach algebras

t (2.3) t + ϕ(x, y, z, 0, · · · , 0) for all x, y, z, x1 , x2 , · · · , xn ∈ X, all µ ∈ T1 := {u ∈ C : |u| = 1} and all t > 0. Then H(x) = N − limm→∞ 21m f (2m x) exists for each x ∈ X, and defines a unique fuzzy ternary homomorphism H : X → Y such that N (f ([x, y, z]) − [f (x), f (y), f (z)], t) ≥

N (f (x) − H(x), t) ≥

2n−1 (1 − L)t 2n−1 (1 − L)t + ϕ(x, x, 0, · · · , 0)

(2.4)

for all x ∈ X and all t > 0. Proof. Consider the set Ω := {g : X → Y, g(0) = 0} and introduce the generalized metric t d(g, h) = inf{η ∈ R+ : N (g(x) − h(x), ηt) ≥ } t + ϕ(x, x, 0, · · · , 0) where inf ∅ = +∞. The proof of the fact that (Ω, d) is a complete generalized metric space can be found in [4]. Now we consider the mapping J : Ω → Ω defined by 1 Jg(x) := g(2x) 2 for all g ∈ Ω and x ∈ X. Let ε > 0 and f, g ∈ Ω be given such that d(g, h) < ε. Then t N (g(x) − h(x), εt) ≥ , t + ϕ(x, x, 0, · · · , 0) for all x ∈ X and all t > 0. Hence 1 1 N (Jg(x) − Jh(x), Lεt) = N ( g(2x) − h(2x), Lεt) 2 2 = N (g(2x) − h(2x), 2Lεt) 2Lt 2Lt + ϕ(2x, 2x, 0, 0, · · · , 0) 2Lt ≥ 2Lt + 2Lϕ(x, x, 0, · · · , 0) 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 for all g, h ∈ Ω, that is, J is a self-mapping of Ω with the Lipschitz constant L. We use the following relation  X  n−k  n−k  X n−k n−k 1+ = = 2n−k k k k=1

k=0

for all n > k. Letting µ = 1 and putting x1 = x2 = x and x3 = x4 = · · · = xn = 0 in (2.2), we have t N (2n−2 f (2x) − 2n−1 f (x), t) ≥ (2.5) t + ϕ(x, x, 0, · · · , 0) for all x ∈ X and all t > 0. Then   1 t t N f (2x) − f (x), n−1 ≥ (2.6) 2 2 t + ϕ(x, x, 0, · · · , 0) 178

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A. Ebadian, M. A. Abolfathi, R. Aghalary, C. Park, D. Shin 1 for all x ∈ X and all t > 0. It follows from (2.6) that d(f, Jf ) = 2n−1 . By Theorem 1.7, there exists a mapping H : X → Y such that the following holds: (1) H is a fixed point of J, that is,

H(2x) = 2H(x)

(2.7)

for all x ∈ X. The mapping H is a unique fixed point of J in the set ∆ = {h ∈ Ω : d(g, h) < ∞}. This implies that H is a unique mapping satisfying (2.7) such that there exists η ∈ (0, ∞) satisfying N (f (x) − H(x), ηt) ≥

t t + ϕ(x, x, 0, · · · , 0)

for all x ∈ X and all t > 0. (2) d(J m f, H) → 0 as m → ∞. This implies the equality 1 f (2m x) = H(x) m→∞ 2m

N − lim

exists for each x ∈ X, 1 d(f, Jf ), which implies inequality (3) d(f, H) ≤ 1−L d(f, H) ≤

1 2n−1 (1

− L)

and so 2n−1 (1 − L)t N (f (x) − H(x), t) ≥ n−1 . 2 (1 − L)t + ϕ(x, x, 0, · · · , 0) Thus (2.4) holds. It follows from (2.1) and (2.2) that k k+1 n X X X ··· N( (

n X

k=2 i1 =2 i2 =i1 +1

in−k+1 =in−k +1

n X

)H(

µxi −

n−k+1 X

µxir )

r=1

i6=1,i6=i1 ,i2 ,··· ,in−k+1

n X + H( µxi ) − 2n−1 H(µx1 ), t) i=1

1 = lim N ( m m→∞ 2 n X

+ f(

n X

(

k+1 k X X

k=2 i1 =2 i2 =i1 +1

n X

···

)f (

in−k+1 =in−k +1

n X i6=1,i6=i1 ,i2 ,··· ,in−k+1

m

µ2 xi −

n−k+1 X

µ2m xir )

r=1

µ2m xi ) − 2n−1 f (µ2m x1 ), t)

i=1

2m t m→∞ 2m t + ϕ(2m x1 , 2m x2 , · · · , 2m xn ) t ≥ lim 1 m m→∞ t + m ϕ(2 x1 , 2m x2 , · · · , 2m xn ) 2 t ≥ lim =1 m m→∞ t + L ϕ(x1 , x2 , · · · , xn ) ≥ lim

179

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Approximate fuzzy ternary homomorphisms in fuzzy ternary Banach algebras

for all x1 , x2 , · · · , xn ∈ X, all t > 0 and all µ ∈ T1 . Thus n X k k+1 X X ( ··· k=2 i1 =2 i2 =i1 +1

n X

)H(

in−k+1 =in−k +1 n−1

=2

n X

µxi −

n−k+1 X

n X µxir ) + H( µxi )

r=1

i6=1,i6=i1 ,i2 ,··· ,in−k+1

i=1

µH(x1 )

for all x1 , x2 , · · · , xn ∈ X. By [21], H : X → Y is Cauchy additive, that is, H(x + y) = H(x) + H(y) for all x, y ∈ X. By a similar method to the proof of [30], one can show that the mapping H : X → Y is C-linear. By (2.3), we have 1 1 N ( 3m f ([2m x, 2p y, 2m z]) − 3m [f (2m x), f (2m y), f (2m y)], t) 2 2 23m t ≥ 3m 2 t + ϕ(2m x, 2m y, 2m z, · · · , 0) 23m t ≥ 3m 2 t + 2m Lm ϕ(x, y, z, · · · , 0) for all x, y, z ∈ X and all t > 0. Since 23m t =1 m→∞ 23m t + 2m Lm ϕ(x, y, z, · · · , 0) for all x, y, z ∈ X and all t > 0, lim

H([x, y, z]) = [H(x), H(y), H(z)] for all x, y, z ∈ X. This means that H is a fuzzy ternary homomorphism. This complete the proof.  Corollary 2.2. Let X be a ternary Banach algebra with norm k · k, δ ≥ 0 and p be a real number with 0 < p < 1. Let f : X → Y be a mapping satisfying n X k k+1 X X N( ( ···

n X

k=2 i1 =2 i2 =i1 +1

in−k+1 =in−k +1

n X

)f (

+f (

µxi −

i6=1,i6=i1 ,i2 ,··· ,in−k+1 n X

n−k+1 X

µxi ) − 2n−1 f (µx1 ), t) ≥

i=1

N (f ([x, y, z]) − [f (x), f (y), f (z)], t) ≥

t+

δ(kxkp

µxir )

(2.8)

r=1

t+δ

t Pn

i=1

t + kykp + kzkp )

kxi kp (2.9)

for all x, y, z, x1 , x2 , · · · , xn ∈ X, all µ ∈ T1 and all t > 0. Then there exists a unique fuzzy ternary homomorphism H : X → Y such that N (f (x) − H(x), t) ≥

2n−1 (1 − 2p−1 )t 2n−1 (1 − 2p−1 )t + 2δkxkp

for all x ∈ X and all t > 0. P Proof. The proof follows from Theorem 2.1 by taking ϕ(x1 , x2 , · · · , xn ) := δ ni=1 kxi kp for all x1 , x2 , · · · , xn ∈ X. It follows from (2.8) that f (0) = 0. Choosing L = 2p−1 , we get the desired result.  180

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Theorem 2.3. Let ϕ : X n → [0, ∞) be a function such that there exists an L < 1 with x x xn  L 1 2 ϕ , ,··· , ≤ ϕ(x1 , x2 , · · · , xn ) 2 2 2 2 for all x1 , x2 , · · · , xn ∈ X. Let f : X → Y
be a mapping satisfying f (0) = 0, (2.2) and (2.3). Then H(x) = N − limm→∞ 2m f 21m x exists for each x ∈ X, and defines a unique fuzzy ternary homomorphism H : X → Y such that N (f (x) − H(x), t) ≥

2n−1 (1 − L)t 2n−1 (1 − L)t + Lϕ(x, x, 0, · · · , 0)

for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space in the proof of Theorem 2.1. Consider the linear mapping J : Ω → Ω defined by x Jg(x) := 2g 2 for all g ∈ Ω and x ∈ X. We can conclude that J is a strictly contractive self-mapping of S with the Lipschitz constant L. It follows from (2.5) that   x Lt t N f (x) − 2f ( ), n−1 ≥ 2 2 t + ϕ(x, x, 0, · · · , 0) L It follows that d(f, Jf ) = 2n−1 . By Theorem 1.7, there exists a mapping H : X → Y such that the following holds: (1) H is a fixed point of J, that is, x 1 = H(x) (2.10) H 2 2 for all x ∈ X. The mapping H is a unique fixed point of J in the set ∆ = {h ∈ Ω : d(g, h) < ∞}. This implies that H is a unique mapping satisfying (2.10) such that there exists η ∈ (0, ∞) satisfying t N (f (x) − H(x), ηt) ≥ t + ϕ(x, x, 0, · · · , 0) for all x ∈ X and all t > 0. (2) d(J m f, H) → 0 as m → ∞. This implies the equality   1 m N − lim 2 f x = H(x) m→∞ 2m

exists for each x ∈ X, 1 (3) d(f, H) ≤ 1−L d(f, Jf ), which implies inequality d(f, H) ≤

L 2n−1 (1 − L)

and so

2n−1 (1 − L)t N (f (x) − H(x), t) ≥ n−1 . 2 (1 − L)t + Lϕ(x, x, 0, · · · , 0) The rest the proof is similar to the proof of Theorem 2.1. 181

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Corollary 2.4. Let X be a ternary Banach algebra with norm k · k, δ ≥ 0 and p be a real number with p > 1. Let f : X → Y be a mapping satisfying (2.8) and (2.9). Then there exists a unique fuzzy ternary homomorphism H : X → Y such that N (f (x) − H(x), t) ≥

2n−1 (1 − 21−p )t 2n−1 (1 − 21−p )t + 22−p δkxkp

for all x ∈ X and all t > 0. P Proof. The proof follows from Theorem 2.3 by taking ϕ(x1 , x2 , · · · , xn ) := δ ni=1 kxi kp for all x1 , x2 , · · · , xn ∈ X. It follows from (2.8) that f (0) = 0. Choose L = 21−p , we get the desired result.  3. Approximate fuzzy ternary derivations on fuzzy ternary Banach algebras In this section, we prove the Hyers-Ulam stability of fuzzy ternary derivations on fuzzy ternary Banach algebras related to the additive bfunctional equation (1.1). Theorem 3.1. Let ϕ : X n → [0, ∞) be a function such that there exists an L < 1 with ϕ(2x1 , 2x2 , · · · , 2xn ) ≤ 2Lϕ(x1 , x2 , · · · , xn ) for all x1 , x2 , · · · , xn ∈ X. Let f : X → X be a mapping satisfying f (0) = 0, (2.2) and t N (f ([x, y, z]) − [f (x), y, z] − [x, f (y), z] − [x, y, f (z)]), t) ≥ (3.1) t + ϕ(x, y, z, · · · , 0) for all x, y, z, x1 , x2 , · · · , xn ∈ X and all t > 0. Then D(x) = N − limm→∞ 21m f (2m x) exists for each x ∈ X, and defines a unique fuzzy ternary derivation D : X → X such that 2n−1 (1 − L)t N (f (x) − D(x), t) ≥ n−1 (3.2) 2 (1 − L)t + ϕ(x, x, 0, · · · , 0) for all x ∈ X and all t > 0. Proof. By the same reasoning as in the proof of Theorem 2.1, the mapping D : X → X is a unique C-linear mapping satisfying (3.2). Now, we show that D : X → X is a fuzzy ternary derivation. By (3.1), we have 1 1 N ( 3m f ([2m x, 2m y, 2m z]) − 3m ([f (2m x), y, z] − [x, f (2m y), z] − [x, y, f (2m z)]), t) 2 2 3m 2 t ≥ 3m m 2 t + ϕ(2 x, 2m y, 2m z, · · · , 0) 23m t ≥ 3m 2 t + 2m Lm ϕ(x, y, z, · · · , 0) for all x, y, z ∈ X and all t > 0. Since 23m t =1 m→∞ 23m t + 2m Lm ϕ(x, y, z, · · · , 0) lim

for all x, y, z ∈ X and all t > 0, D([x, y, z]) = [D(x), y, z] + [x, D(y), z] + [x, y, D(z)] for all x, y, z ∈ X. This means that D is a fuzzy ternary derivation. 182

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Theorem 3.2. Let ϕ : X n → [0, ∞) be a function such that there exists an L < 1 with x1 x2 xn 1 , , · · · , ) ≤ Lϕ(x1 , x2 , · · · , xn ) 2 2 2 2 for all x1 , x2 , · · · , xn ∈ X. Let f : X → X
be a mapping satisfying f (0) = 0, (2.2) and (3.1). Then D(x) = N − limm→∞ 2m f 21m x exists for each x ∈ X, and defines a unique fuzzy ternary derivation D : X → X such that ϕ(

N (f (x) − D(x), t) ≥

2n−1 (1 − L)t 2n−1 (1 − L)t + Lϕ(x, x, 0, · · · , 0)

for all x ∈ X and all t > 0. Acknowledgments C. Park and D. Y. Shin were 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 (NRF-2010-0021792), respectively. References [1] M. A. Abolfathi, A. Ebadian and R. Aghalary, Stability of mixed additive-quadratic Jensen type functional equation in non-Archimedean `-fuzzy normed spaces, Annali Dell ’Universita ’Di Ferrara. 2013 doi: 10.1007/s11565-013-0182-z, 16 pages, (2013). [2] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear space, J. Fuzzy Math. 11 (2003) 687-705. [3] T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Set Syst. 151 (2005) 513-547. [4] L. C˘adriu and V.Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004) 43-52. [5] A. Cayley, On the 34 concomitants of the ternary cubic, Amer. J. Math. 4 (1981) 1-15. [6] S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994) 429-436. [7] A. Ebadian and N. Ghobadipour, A fixed point approach to almost double derivations and Lie *-double derivations, Results. Math. 63 (2013) 409-423. [8] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 45–54. [9] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Nearly generalized derivations on non-Archimdean Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 308–315. [10] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Set Syst. 48 (1992) 239-248. [11] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of the approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436. [12] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941) 222-224. [13] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equation in Several Variables, Birkh¨ause, Basel, 1998. [14] G. Isac and Th. M. Rassias, Stability of ψ-additive mappings: applications to nonlinear analisis, Internat. J. Math. Sci. 19 (1996) 219-228. [15] K. Jun and H. Kim , The generalized Hyers-Ulam-Rassias stability of cubic functional equation, J. Math. Anal. Appl. 274 (2002) 867-878. [16] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Set Syst. 12 (3) (1984) 1-7. 183

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A. Ebadian, M. A. Abolfathi, R. Aghalary, C. Park, D. Shin [42] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [43] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [44] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003) 91-96. [45] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300. [46] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000) 264-284. [47] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [48] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [49] S. M. Ulam, Problem in Modern Mathematics, Chapter V I, Science Editions, Wiley, New York, 1964. [50] C. Wu and J. Fang, Fuzzy generalization of Kolmogoroof ’s theorem, J. Harbin Inst. Technol. 1 (1984) 1-7. [51] J. Z. Xiao and X. H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Set Syst. 133 (2003) 389-399. [52] T. Xu and Z. Yang, Direct and fixed point approaches to the stability of an AQ-functional equation in non-Archimedean normed spaces, J. Comput. Anal. Appl. 17 (2014), 697–706. [53] L.A. Zadeh, Fuzzy sets, Inform. Contr. 8 (1965) 338-353.

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Fuzzy stability of functional equations in n-variable fuzzy Banach spaces Dong Yun Shin1 , Choonkil Park2 , Roghayeh Asadi Aghjeubeh3 and Shahrokh Farhadabadi4∗ 1

Department of Mathematics, University of Seoul, Seoul 130-743, Korea Research Institute for Natural Sciences Hanyang University, Seoul 133-791, Korea 3,4 Department of Mathematics, Urmia University, Urmia, Iran e-mail: [email protected]; [email protected]; [email protected]; shahrokh [email protected] 2

Abstract. In this paper, we prove a generalized fuzzy version of the Hyers-Ulam stability for the functional equations f (ϕ(X)) = φ(X)f (X) + ψ(X)e and f (ϕ(X)) = φ(X)f (X) by using the direct method and the fixed point method, where X denotes an n-variable. Furthermore, we can apply the obtained results to some well-known functional equations such as γ-function, β-function, and Gfunction type’s equations. Keywords: Fuzzy Banach space; Hyers-Ulam stability; Fixed point; Functional equation.

1. Introduction and preliminaries In 1940, Ulam [60] gave a talk and proposed a number of unsolved problems. In particular, he asked the following question: “when and under what condition does a solution of a functional equation near an approximately solution of that exist?” Today, this question is considered as the source of the stability of functional equations. In 1941, Hyers [18] formulated and proved the Ulam’s problem for the Cauchy’s functional equation on Banach spaces. The result of Hyers was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [53] for linear mappings by considering an unbounded Cauchy difference. In 1994, G˘avrut¸a [16] provided a further generalization of Rassias’ theorem in which he replaced the unbounded Cauchy difference by a general control function for the existence of a unique linear mapping. Since then, many interesting results concerning the stability of different functional equations have been obtained by numerous authors (cf. [4, 17, 19, 20, 22, 23, 24, 29, 30, 34, 36, 37, 47, 49, 54, 55, 56, 57]). In this paper, we investigate the following functional equations f (ϕ(X))

= φ(X)f (X) + ψ(X)e,

(1.1)

f (ϕ(X))

= φ(X)f (X)

(1.2)

where ϕ, φ, ψ are given functions, while f is a unknown function and X = (x1 , · · · , xn ), in the fuzzy normed spaces setting. We first state how to construct these equations. The functional equation f (x + 1) = xf (x), is called the gamma type functional equation, which is considered by Jung (cf. [23, 24]). This equation could be generalized the gamma type functional equation f (x + p) = ϕ(x)f (x) and the beta type functional equation f (x + p, y + p) = ϕ(x, y)f (x, y) [27], and also this last case could be extended again to the functional equation (1.1). For more details see [59]. 0

∗

2010 Mathematics Subject Classification: Primary 39B52, 47H10, 46S40, 47S40, 34K36, 26E50. Corresponding author: Sh. Farhadabadi (email: shahrokh [email protected])

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D. Y. Shin, C. Park, R. Asadi Aghjeubeh and Sh. Farhadabadi In 2004, Kim [28] studied the stability of the functional equation (1.1) in n-variables in Banach spaces. Recently, the first author and others have worked on some different functional equations in fuzzy normed spaces by using the direct method and the fixed point method (cf. [21, 35, 46, 50, 58]). In this paper, using the ideas from the papers of Kim (cf. [27, 28]), we adopt the direct and fixed point methods to prove a generalized fuzzy version of the Hyers-Ulam stability for the functional equations (1.1) and (1.2) in n-variables in fuzzy Banach spaces. To more clarify the reader, now we give briefly some useful information, definitions and fundamental results of fixed point theory and then fuzzy normed spaces, respectively. Fixed point theory has a basic role in applications of some branches of mathematics. In 1996, for the first time, Isac and Rassias [20] provided applications of stability theory of functional equations for the proof of new fixed point theorems with applications. For more study about the stability problems of several functional equations, by using fixed point methods, one can refer to (cf. [6, 7, 8, 10, 11, 12, 13, 14, 25, 33, 41, 43, 44, 45, 48, 51, 52]). Definition 1.1. 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 . Theorem 1.2. ([6, 9]) 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 d(y, J y) for all y ∈ Y. (4) d(y, y ∗ ) ≤ 1−L In 1984, Katsaras [26] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy normed on a vector space from various points of view (cf. [2, 15, 26, 32, 38, 61]). In particular, Bag and Samanta [2] following Cheng and Mordeson [5], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [31]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. Definition 1.3. ([2, 38, 39, 40]) Let X be a complex 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 ) if c 6= 0; (N3 ) N (cx, t) = N (x, |c| (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. Example 1.4. Let (X , k · k) be a normed linear space and α, β > 0. Then  αt t > 0, x ∈ X αt+βkxk NX (x, t) = 0 t ≤ 0, x ∈ X

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Fuzzy stability of functional equations in fuzzy Banach spaces is a fuzzy norm on X , which is called the induced fuzzy norm of (X , k · k). In this case (X , NX ) is called an induced fuzzy normed space. Definition 1.5. Let A be a complex algebra and (A, N ) be a fuzzy normed space. The fuzzy norm N is called an algebraic fuzzy norm on A if, (N 7) N (xy, st) ≥ N (x, s) · N (y, t) for all x, y ∈ A and all s, t ∈ R. The pair (A, N ) is called a fuzzy normed algebra. Example 1.6. Let (A, k · k) be a normed algebra. Let  t t > 0, a ∈ A t+kak N (a, t) = 0 t ≤ 0, a ∈ A. Then N (a, t) is a fuzzy norm on A and (A, N (a, t)) is a fuzzy normed algebra. Definition 1.7. ([2, 38, 39, 40]) 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. Definition 1.8. ([2, 38, 39, 40]) Let (X , N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each  > 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 − . 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. A complete fuzzy normed algebra is called a fuzzy Banach algebra. 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]). Throughout this paper, suppose that (B, N ) is a unital fuzzy Banach algebra (with unit e) over a field K, where K is either the field R of real numbers or the field C of complex numbers. β > 0 and  > 0 will be fixed positive real numbers and R+ denotes the set of all nonnegative real numbers. Assume that (C, N ) is a fuzzy Banach space over K. Given a nonempty set S and a mapping ϕ : S n → S n , we define ϕ0 (X) := X and ϕn (X) := ϕ(ϕn−1 (X)) for all positive integers n and all X ∈ S n . In addition, functions φ : S n → K \ {0}, and ε : S n → R+ and a mapping ψ : S n → B are defined. 2. Generalization of Hyers-Ulam stability to a fuzzy version for (1.1) and (1.2): fixed point method In this section, by using the fixed point method, we prove a generalized fuzzy version of Hyers-Ulam stability for the functional equation (1.1) in n-variables and fuzzy Banach spaces. Theorem 2.1. Let ε, ψ, ϕ and φ be given as were explained in the introduction such that there exists an L < 1 with ε(ϕ(X)) ≤ L |φ(X)| ε(X) for all X ∈ S n . Let f : S n → B be a mapping satisfying N (f (ϕ(X)) − φ(X)f (X) − ψ(X)e, t) ≥

188

t t + ε (X)

(2.1)

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D. Y. Shin, C. Park, R. Asadi Aghjeubeh and Sh. Farhadabadi for all X ∈ S n and all t > 0. Then there exists a unique solution H : S n → B of (1.1) such that f (ϕl (X))

H(X) = N - lim Ql−1 l→∞

j=0

N (f (X) − H(X), t) ≥

φ(ϕj (X))

−

l−1 X

ψ(ϕk (X))e , Qk j=0 φ(ϕj (X)) k=0

(1 − L) |φ(X)| t (1 − L) |φ(X)| t + ε(X)

(2.2) (2.3)

for all X ∈ S n and all t > 0. Proof. We consider the set Q := {g : S n → B} and define the following generalized metric d:   t n d(g, h) = inf µ ∈ R+ : N (g(X) − h(X), µt) ≥ , ∀X ∈ S , ∀t > 0 t + ε(X) where, as usual, inf ∅ = +∞. It is easy to show that (Q, d) is complete (see the proof of [42, Lemma 2.1]). Consider the linear mapping Jϕ,ψ,φ : Q → Q, as follows: Jϕ,ψ,φ (g) :=

g(ϕ) − ψe φ

for all g ∈ Q, where the functions ϕ, ψ, φ are defined. For convenience, we write J (g), instead of Jϕ,ψ,φ (g). By the definition of J (g), we can get l−1

X ψ(ϕk (X))e g(ϕl (X)) J l (g(X)) = Ql−1 − Qk j=0 φ(ϕj (X)) j=0 φ(ϕj (X)) k=0

(2.4)

for all g ∈ Q, all X ∈ S n and all positive integers l. From (2.1) and (N3 ), it follows that     t f (ϕ(X)) − ψ(X)e t t N J (f (X)) − f (X), =N − f (X), ≥ |φ(X)| φ(X) |φ(X)| t + ε(X) for all X ∈ S n , which means d (J (f ), f ) ≤

1 |φ(X)|

(2.5)

for all X ∈ S n . Assume that g, h ∈ Q are given with d(g, h) = . Then we have t N (g(X) − h(X), t) ≥ t + ε(X) for all X ∈ S n and all t > 0. By the definition of J (g) and then substituting X and t by ϕ(X) and L |φ(X)| t respectively, we obtain N (J (g(X)) − J (h(X)), Lt)   g(ϕ(X)) − h(ϕ(X)) =N , Lt = N (g(ϕ(X)) − h(ϕ(X)), L |φ(X)| t) φ(X) L |φ(X)| t L |φ(X)| t ≥ ≥ L |φ(X)| t + ε(ϕ(X)) L |φ(X)| t + L |φ(X)| ε(X) t = t + ε(X) for all X ∈ S n and all t > 0, which leads us to the inequality d(J (g), J (h)) ≤ L = Ld(g, h) for all g, h ∈ Q. Therefore, J is a strictly contractive mapping with Lipschitz constant L < 1. According to Theorem 1.2:

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Fuzzy stability of functional equations in fuzzy Banach spaces (1) J has a fixed point, i.e., there exists a mapping H : S n → B such that J (H) = H, and so H(ϕ(X)) − ψ(X)e φ(X) n for all X ∈ S . The mapping H is also the unique fixed point of J in the set H(X) =

(2.6)

M = {g ∈ Q : d(f, g) < ∞}. This implies that H is a unique mapping satisfying (2.6), moreover there exists a µ ∈ (0, ∞) such that t N (f (X) − H(X), µt) ≥ t + ε(X) for all X ∈ S n ; (2) The sequence {J n (g)} converges to H, for each given g ∈ Q. Thus d (J n (f ), H) → 0 as n → ∞. This signifies the equality H(X) = N - lim J n (f (X)) n→∞

for all X ∈ S n . From this and (2.4), we deduce that (2.2) holds; 1 (3) d(g, H) ≤ 1−L d (g, J (g)) for all g ∈ M. So it follows from (2.5) that 1 d (f, J (f )) ≤ 1−L   t ≥ N f (X) − H(X), (1 − L)|φ(X)|

1 , (1 − L)|φ(X)| t t + ε(X)

d(f, H) ≤

for all X ∈ S n and all t > 0. Putting (1 − L) |φ(X)| t instead of t in the above inequality, we get the inequality (2.3). Now, we show that H : S n → B satisfies (1.1) as follows: By (2.4), one can show easily that J n (ϕ(X)) = φ(X)J n+1 (X) + ψ(X)e. By (N4 ), for any fixed t > 0, we have N (H(ϕ(X)) − φ(X)H(X) − ψ(X)e, t)      t t n n+1 ≥ min N H(ϕ(X)) − J (ϕ(X)), , N J (X)φ(X) − H(X)φ(X), , 3 3   t N J n (ϕ(X)) − φ(X)J n+1 (X) − ψ(X)e, 3 in which, every three terms on the right-hand side tend to one as n → ∞. Hence N (H(ϕ(X)) − φ(X)H(X) − ψ(X)e, t) = 1 for all t > 0 and all X ∈ S n , which means by (N2 ) that H(ϕ(X)) − φ(X)H(X) − ψ(X)e = 0 for all X ∈ S n . So the mapping H : S n → B satisfies (1.1), and the proof is complete.  Corollary 2.2. Let ε, ϕ and φ be given in the introduction as were explained such that there exists an L < 1 with ε(ϕ(X)) ≤ L |φ(X)| ε(X) for all X ∈ S n . Let f : S n → C be a mapping satisfying N (f (ϕ(X)) − φ(X)f (X), t) ≥

t t + ε (X)

for all X ∈ S n and all t > 0. Then there exists a unique solution H : S n → C of (1.2) such that f (ϕl (X)) H(X) = N - lim Ql−1 , l→∞ j=0 φ(ϕj (X)) N (f (X) − H(X), t) ≥

(1 − L) |φ(X)| t (1 − L) |φ(X)| t + ε(X)

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D. Y. Shin, C. Park, R. Asadi Aghjeubeh and Sh. Farhadabadi for all X ∈ S n and all t > 0. Proof. Putting ψ(X) = 0, and then applying Theorem 2.1, we get easily the desired result.



The above results include more cases, for example, to obtain simpler results, we can put ϕ(X) = X + P , S = (0, ∞) and B = R. In this case, X = (x1 , x2 , · · · , xn ) and P = (p1 , p2 , · · · , pn ) are supposed. For more details, see [28]. 3. Generalization of Hyers-Ulam stability to a fuzzy version for (1.1) and (1.2): direct method In this section, by using the direct method, we prove a generalized fuzzy version of Hyers-Ulam stability for the functional equations (1.1) and (1.2) in n-variables and fuzzy Banach spaces. Consider the following condition: If ϕ, φ and ε be given as were explained, then ∞ X ε (ϕk (X)) ω(X) := 0, one can assert that there is some t0 > 0 such that N (f (ϕ(X)) − φ(X)f (X) − ψ(X)e, t ε(X)) ≥ 1 − ,   ψ(X)e ε(X) f (ϕ(X)) − f (X) − , t ≥ 1− N φ(X) φ(X) |φ(X)|

(3.4)

for all t ≥ t0 and all X ∈ S n . From the definition of gl and then replacing X by ϕl (X) in (3.4), we obtain ! ε (ϕl (X)) N gl+1 (X) − gl (X), t Ql j=0 |φ (ϕj (X)) |   l−1 Y ε (ϕ (X)) l  =N φ (ϕj (X)) [gl+1 (X) − gl (X)], t |φ (ϕ (X)) | l j=0   f (ϕl+1 (X)) ψ(ϕl (X))e ε(ϕl (X)) =N − f (ϕl (X)) − , t ≥1− (3.5) φ(ϕl (X)) φ(ϕl (X)) |φ(ϕl (X))|

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Fuzzy stability of functional equations in fuzzy Banach spaces for all t ≥ t0 and all X ∈ S n . Using the induction method, we prove that N (gl (X) − f (X), t ωl (X)) ≥ 1 − 

(3.6)

for all t ≥ t0 , all X ∈ S n and all positive integers l. The case l = 1 is the inequality (3.4), and so we assume that the inequality (3.6) holds true for some l. The case l + 1 is an immediate consequence of (3.5) and (3.6). Indeed, we have N (gl+1 (X) − f (X), t ωl+1 (X)) ( ≥ min N

gl+1 (X) − gl (X), t Ql

ε (ϕl (X))

j=0

N

gl (X) − f (X), t

l−1 X

k=0

!

|φ (ϕj (X)) |

ε(ϕk (X)) Qk

j=0

, !)

|φ(ϕj (X))|

≥ min{1 − , 1 − } = 1 − , which ends the induction method. Now we claim that {gl (X)} is a Cauchy sequence. In order to verify that, by (3.5), for l + p > l > 0 and p > 0, we have ! l+p−1 X ε (ϕk (X)) N gl+p (X) − gl (X), t Qk j=0 |φ(ϕj (X))| k=l ( !) min ε (ϕi (X)) ≥ N gi+1 (X) − gi (X), t Qi l ≤i≤l+p−1 j=0 |φ(ϕj (X))| ≥ min {1 − , · · · , 1 − } = 1 − 

(3.7)

for all t ≥ t0 and all X ∈ S n (especially for t = t0 ). The condition (3.1) implies that for a given δ > 0, there is an l0 ∈ N such that l+p−1 X ε(ϕk (X)) t0 < δ Qk j=0 |φ(ϕj (X))| k=l for all l ≥ l0 and all p > 0. From this, (3.7), (N5 ) and (N6 ) for the function N (gl+p (X) − gl (X), ·), we deduce that ! l+p−1 X ε (ϕk (X)) N (gl+p (X) − gl (X), δ) ≥ N gl+p (X) − gl (X), t0 ≥1− Qk j=0 |φ(ϕj (X))| k=l for all l ≥ l0 and all p > 0. Therefore the sequence {gl (X)} is Cauchy in B. Since B is a fuzzy Banach space, the sequence {gl (X)} converges to some g(X) ∈ B, and so we can define for all X ∈ S n , a function g : S n → B by g(X) := N − lim gl (X). l→∞

In other words, liml→∞ N (gl (X) − g(X) , t) = 1 for all t > 0. By the same argument which was done for J : S n → B in the proof of Theorem 2.1, we can easily show that the mapping g : S n → B satisfies (1.1). In continue, we will also show that g : S n → B satisfies the equality (3.3). Let t ≥ t0 , t0 > 0 and 0 <  < 1. Comparing ω with ωl and using (N5 ) and (N6 ) for the function N (f (X) − g(X), ·), we obtain that N (f (X) − g(X), t ω(X) + t0 ) ≥ N (f (X) − g(X), t ωl (X) + t0 ) ≥ min {N (f (X) − gl (X), t ωl (X)) , N (gl (X) − g(X), t0 )}

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D. Y. Shin, C. Park, R. Asadi Aghjeubeh and Sh. Farhadabadi for all X ∈ S n , and so by (3.6) and the fact that liml→∞ N (gl (X) − g(X), t0 ) = 1, we obtain N (f (X) − g(X), t ω(X) + t0 ) ≥ 1 −  if l ∈ N is large enough. Let t0 → 0 and t → ∞. Then the above inequality clearly signifies that (3.3) holds. To finish the proof, it is just necessary to show that the mapping g : S n → B is unique. So let h : S n → B be another mapping satisfying (1.1) and (3.3). It follows from (1.1) for g and h that g(X) − h(X) =

g(ϕ(X)) − h(ϕ(X)) φ(X)

for all X ∈ S n . Substituting X by ϕ(X) continually, we lead to g(X) − h(X) =

g(ϕl (X)) − h(ϕl (X)) Ql−1 j=0 φ (ϕj (X))

(3.8)

for all X ∈ S n . Fix c > 0. Given  > 0, the inequality (3.3) decides some t0 > 0 for h and g such that N (f (X) − g(X), t ω(X)) ≥ 1 − , N (f (X) − h(X), t ω(X)) ≥ 1 −  n

for all t ≥ t0 and all X ∈ S (specially for t = t0 ). From (3.1), we know that there is a large enough integer l0 > 0 such that ∞ X ε (ϕk (X)) c (3.9) t0 < Qk 2 j=0 |φ (ϕj (X)) | k=l for all l ≥ l0 . By (3.8), (3.9) and the fact that ∞ X k=l

ε (ϕk (X)) Qk

j=0

|φ (ϕj (X)) |

∞ X

=

ε (ϕl+k (X)) Qk

! · Ql−1

|φ (ϕl+j (X)) 1 , = ω(ϕl (X)) · Ql−1 |φ (ϕj (X)) | j=0 k=0

j=0

j=0

1 |φ (ϕj (X)) |

we obtain ! g(ϕl (X)) − h(ϕl (X)) N (g(X) − h(X), c) = N , c Ql−1 j=0 φ (ϕj (X)) ! ( g(ϕl (X)) − f (ϕl (X)) c , ≥ min N , Ql−1 2 j=0 φ (ϕj (X)) N

f (ϕl (X)) − h(ϕl (X)) c , Ql−1 2 j=0 φ (ϕj (X))

!)

≥ min {N (g (ϕl (X)) − f (ϕl (X)) , t0 ω (ϕl (X))) , N (f (ϕl (X)) − h (ϕl (X)) , t0 ω (ϕl (X)))} ≥ 1− n

for all X ∈ S and all positive integers l, which shows the uniqueness of g.



Now let ϕ and φ satisfy µ(X) :=

∞ Y k X k=0

1 0 in Theorem 3.1, we have the following corollary:

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Fuzzy stability of functional equations in fuzzy Banach spaces Corollary 3.2. Let ϕ, φ satisfy (3.10) and f : S n → B a mapping satisfying lim N (f (ϕ(X)) − φ(X)f (X) − ψ(X)e, t β) = 1

t→∞

for all X ∈ S n . Then there exists a unique solution g : S n → B of (1.1) such that lim N (f (X) − g(X), t µ(X)) = 1

t→∞

for all X ∈ S n . If we put ψ(X) = 0 in Theorem 3.1 and Corollary 3.2, then we get the results for (1.2). Remark 3.3. All the obtained results could be applied to the gamma type, the G-function, Schr¨oder, and the beta type functional equations. In other words, we can set many different type and suitable functions instead of ϕ(X), ε(X) and φ(X) to obtain more interesting and useful results. For more applications, examples and details, we refer the reader to (cf. [22, 23, 24, 27, 28]). Acknowledgments D. Y. Shin and C. Park were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF2010-0021792) and (NRF-2012R1A1A2004299), respectively. 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] P. Czerwik, Functional Equations and Inequalities in Several Variables, Word Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [5] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [6] 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). [7] 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. [8] 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). [9] 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. [10] M. Eshaghi Gordji, H. Khodaei and J.M. Rassias, Fixed point methods for the stability of general quadratic functional equation, Fixed Point Theory 12 (2011), 71–82. [11] M. Eshaghi Gordji, H. Khodaei, Th.M. Rassias and R. Khodabakhsh, J ∗ -homomorphisms and J ∗ derivations on J ∗ -algebras for a generalized Jensen type functional equation, Fixed Point Theory 13 (2012), 481–494. [12] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 45–54. [13] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Nearly generalized derivations on non-Archimdean Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 308–315. [14] M. Eshaghi Gordji, C. Park and M.B. Savadkouhi, The stability of a quartic type functional equation with the fixed point alternative, Fixed Point Theory 11 (2010), 265–272.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO. 1, 2015 A New Consensus Protocol of the Multi-Agent Systems, Xin-Lei Feng, and Ting-Zhu Huang,10 Almost Difference Sequence Spaces Derived by Using a Generalized Weighted Mean, Ali Karaisa, and Faruk Özger,……………………………………………………………………….27 Ground State Solutions for Discrete Nonlinear Schrödinger Equations with Potentials, Guowei Sun, and Ali Mai,……………………………………………………………………………..….39 On Fuzzy Approximating Spaces, Zhaowen Li, Yu Han, and Rongchen Cui,………………….52 On Some Self-Adjoint Fractional Finite Difference Equations, Dumitru Baleanu, Shahram Rezapour, and Saeid Salehi,……………………………………………………………………..59 Local Shape Control of a Weighted Interpolation Surface, Jianxun Pan, and Fangxun Bao,….68 Generalized Additive Functional Inequalities in C∗-Algebras, Jung Rye Lee, Dong Yun Shin,.80 Approximation Reductions in IVF Decision Information Systems, Hongxiang Tang,…………90 The Covering Method for Attribute Reductions of Concept Lattices, Neiping Chen, Xun Ge,100 A New Two-Step Iterative Method for Solving Nonlinear Equations, Shin Min Kang, Arif Rafiq, Faisal Ali, and Young Chel Kwun,……………………………………………………………111 Unisoft Filters in R0-algebras, G. Muhiuddin, and Abdullah M. Al-roqi,…………………….133 The Admissible Function and Non-Admissible Function in the Unit Disc, Hong Yan Xu, Lian Zhong Yang, and Ting Bin Cao,………………………………………………………………144 Global Dynamics of a Certain Two-dimensional Competitive System of Rational Difference Equations with Quadratic Terms, M. R. S. Kulenović, and M. Pilling,……………………….156 Higher-Order q-Daehee Polynomials, Young-Ki Cho, Taekyun Kim, Toufik Mansour, and Seog-Hoon Rim,……………………………………………………………………………….167 Approximate Fuzzy Ternary Homomorphisms and Fuzzy Ternary Derivations on Fuzzy Ternary Banach Algebras, Ali Ebadian, Mohammad Ali Abolfathi, Rasoul Aghalary, Choonkil Park, and Dong Yun Shin,………………………………………………………………………………..174 Fuzzy Stability of Functional Equations in n-Variable Fuzzy Banach Spaces, Dong Yun Shin, Choonkil Park, Roghayeh Asadi Aghjeubeh, and Shahrokh, Farhadabadi,………………..….186

Volume 19, Number 2 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

August 2015

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 (twelve 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. Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC

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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

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

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and Artificial Intelligence, Operations Research, Math.Programming 30) T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece Postal Address: 26 Menelaou St. Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis

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12) Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR-06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

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33) Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 P.D.E, Control Theory, Functional Analysis, [email protected]

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

34) Gilbert G.Walter Department Of Mathematical Sciences University of Wisconsin-Milwaukee,Box 413, Milwaukee,WI 53201-0413 414-229-5077 e-mail: [email protected] Distribution Functions, Generalised Functions, Wavelets 35) 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

Lotharstr.65,D-47048 Duisburg,Germany e-mail:[email protected] Fourier Analysis,Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory

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

36) 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

37) 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

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

38) 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

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 NEW MEMBERS 39)Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] Computational Mathematics

40) Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S.Korea, [email protected] Functional Equations

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The similarity and application for generalized interval-valued fuzzy soft sets Yan-ping Hea∗, Hai-dong Zhangb a.School of Electrical Engineering, Northwest University for Nationalities, Lanzhou, Gansu, 730030,P. R. China b. School of Mathematics and Computer Science Northwest University for Nationalities, Lanzhou, Gansu, 730030,P. R. China

Abstract In this paper, by combining the interval-valued fuzzy set and fuzzy soft set, we define generalized interval-valued fuzzy soft sets which is an extension to the fuzzy soft set. The complement, union, intersection and sum operations are also investigated. We have further studied the similarity between two generalized interval-valued fuzzy soft sets. Finally, application of generalized interval-valued fuzzy soft sets in decision making problem has been shown. Key words: Interval-valued fuzzy set; Fuzzy soft set; Generalized interval-valued fuzzy soft sets; Similarity measure; Decision making

1

Introduction

Molodtsov [1] initiated a novel concept called soft sets as a new mathematical tool for dealing with uncertainties. The soft set theory is free from many difficulties that have troubled the usual theoretical approaches. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts [2]. Soft set theory has potential applications in many different fields including the smoothness of functions, game theory, operational research, Perron integration, probability theory, and measurement theory [1, 3]. Research works on soft sets are very active and progressing rapidly in these years. Maji et al. [4] defined several operations on soft sets and made a theoretical study on the theory of soft sets. Feng et al. [5] applied soft set theory to the study of semirings and initiated the notion of soft semirings. Furthermore, based on [4], Ali et al. [6] introduced some new ∗

Corresponding author. Address: School of Electrical Engineering Northwest University for Nationalities, Lanzhou, Gansu, 730030, P.R.China. E-mail:he [email protected]

1

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operations on soft sets and improved the notion of complement of soft set. They proved that certain De Morgans laws hold in soft set theory. Qin and Hong [7] introduced the notion of soft equality and established lattice structures and soft quotient algebras of soft sets. Maji et al [8] presented the notion of generalized fuzzy soft sets theory which is based on a combination of the fuzzy set and soft set models. Yang et al. [9] presented the concept of the interval-valued fuzzy soft sets by combining interval-valued fuzzy set [10–12] and soft set models. Feng et al. [13]provided a framework to combine fuzzy sets, rough sets and soft sets all together, which gives rise to several interesting new concepts such as rough soft sets, soft rough sets and soft rough fuzzy sets. Park et al [14] discussed some properties of equivalence soft set relations. By combining the multi-fuzzy set and soft set models, Yang et al. [15] presented the concept of the multi-fuzzy soft set, and provided its application in decision making under an imprecise environment. Shabir [16] presented a new approach to soft rough sets by combining the rough set and soft set. The purpose of this paper is to combine the interval-valued fuzzy sets and generalized fuzzy soft soft, from which we can obtain a new soft set model: generalized interval-valued fuzzy soft set theory. Intuitively, generalized interval-valued fuzzy soft set theory presented in this paper is an extension of interval-valued fuzzy soft set and generalized fuzzy soft set. We have further studied the similarity between two generalized interval-valued fuzzy soft sets. We finally present examples which show that the decision making method of generalized interval-valued fuzzy soft set can be successfully applied to many problems that contain uncertainties. The rest of this paper is organized as follows. The following section briefly reviews some background on soft set, fuzzy soft set and interval-valued fuzzy set. In section 3, the concept of generalized interval-valued fuzzy soft set is presented. The complement, union, intersection and sum operations on the generalized interval-valued fuzzy soft set are then defined. Also their some interesting properties have been investigated. In section 4, similarity between two generalized interval-valued fuzzy soft sets has been discussed. An application of generalized interval-valued fuzzy soft set in decision making problem has been shown in section 5. Section 6 concludes the paper.

2

Preliminaries In this section we give few definitions regarding soft sets.

Definition 2.1 ( [1]) Let U be an initial universe set and E be a universe set of parameters. A pair (F, A) is called a soft set over U if A ⊂ E and F : A → P (U ), where P (U ) is the set of all subsets of U. Definition 2.2 ( [17]) Let U be an initial universe set and E be a universe set of parameters. A pair (F, A) is called a fuzzy soft set over U if A ⊂ E and F : A → F (U ), where F (U ) is the set of all fuzzy subsets of U. 2

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ˆ on an universe U is a mapping Definition 2.3 ( [10]) An interval-valued fuzzy set X such that ˆ : U → Int([0, 1]), X where Int([0,1]) stands for the set of all closed subintervals of [0,1]. For the sake of convenience, the set of all interval-valued fuzzy sets on U is denoted ˆ ∈ IV F (U ), ∀x ∈ U, µ ˆ (x) = [µ− (x), µ+ (x)] is called the by IV F (U ). Suppose that X ˆ ˆ X X X − + ˆ degree of membership an element x to X. µ (x) and µ (x) are referred to as the lower ˆ X

ˆ X

ˆ where 0 ≤ µ− (x) ≤ µ+ (x) ≤ 1. and upper degrees of membership an element x to X ˆ ˆ X X ˆ Yˆ ∈ IV F (U ), then The basic operations on IV F (U ) are defined as follows : for all X, ˆ is denoted by X ˆ C where (1) the complement of X + − µC ˆ (x) = [1 − µX ˆ (x) = 1 − µX ˆ (x), 1 − µX ˆ (x)]; X ˆ ˆ ˆ ∩ Yˆ where (2) the intersection of X and Y is denoted by X − − + + µX∩ ˆ Yˆ (x) = inf [µX ˆ (x), µYˆ (x)] = [inf (µX ˆ (x), µYˆ (x)), inf (µX ˆ (x), µYˆ (x))]; ˆ and Yˆ is denoted by X ˆ ∪ Yˆ where (3) the union of X − − + + µX∪ ˆ Yˆ (x) = sup[µX ˆ (x), µYˆ (x)] = [sup(µX ˆ (x), µYˆ (x)), sup(µX ˆ (x), µYˆ (x))]; ˆ and Yˆ is denoted by X ˆ ⊕ Yˆ where (4) the sum of X − − + + µX⊕ ˆ Yˆ (x) = [inf {1, (µ ˆ (x) + µ ˆ (x))}, inf {1, (µ ˆ (x) + µ ˆ (x))}]; X

3

Y

X

Y

Generalized interval-valued fuzzy soft set

3.1

Concept of generalized interval-valued fuzzy soft set

In this subsection, we give a modified definition of interval-valued fuzzy soft set. Definition 3.1 Let U be an initial universe and E be a set of parameters. The pair (U, E) is called a soft universe. Suppose that F : E → IV F (U ), and f is an interval-valued fuzzy subset of E, i.e.f : E → Int([0, 1]), we say that Ff is a generalized interval-valued fuzzy soft set(GIVFSS, in short) over the soft universe (U, E) if and only if Ff is a mapping given by Ff : E → IV F (U ) × Int([0, 1]), where Ff (e) = (F (e), f (e)), F (e) ∈ IV F (U ), and f (e) ∈ Int([0, 1]) Here for each parameter ei , Ff (ei ) = (F (ei ), f (ei )) indicates not only the range of belongingness of the elements of U in F (ei ) but also the range of possibility of such be belongingness which is represented by f (ei )) . For all e ∈ E, Ff (e) is actually a generalized interval-valued fuzzy set of (U, E), where x ∈ U and e ∈ E, it can be written as: 3

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Ff (e) = (F (e), f (e)), + − + where F (e) = {< x, [µ− F (e) (x), µF (e) (x)] >: x ∈ U }, f (e) = [µf (e) (x), µf (e) (x)].

Remark 3.2 A generalized interval-valued fuzzy soft set is also a special case of a soft set because it is still a mapping from parameters to IV F (U ) × Int([0, 1]). If ∀e ∈ E, ∀x ∈ + − + U, µ− F (e) (x) = µF (e) (x), and µf (e) (x) = µf (e) (x), then Ff will be degenerated to be a generalized fuzzy soft sets [8]. Example 3.3 Let U be a set of three houses under consideration of a decision maker to purchase, which is denoted by U = {h1 , h2 , h3 }. Let E be a parameter set, where E = {e1 , e2 , e3 } = {expensive; beautif ul; in the green surroundings}. Suppose that f (e1 ) = [0.6, 0.9], f (e2 ) = [0.3, 0.5], f (e3 ) = [0.1, 0.2]. We define a function Ff : E → IV F (U ) × Int([0, 1]), given by as follows: Ff (e1 ) = ({

[0.6, 0.7] [0.3, 0.5] [0.6, 0.8] , , , }, [0.6, 0.9]) h1 h2 h3

Ff (e2 ) = ({

Ff (e3 ) = ({

[0.1, 0.3] [0.2, 0.4] [0.8, 1] , , , }, [0.3, 0.5]) h1 h2 h3

[0.4, 0.7] [0.5, 0.8] [0.2, 0.4] , , , }, [0.1, 0.2]) h1 h2 h3

Then Ff is a GIVFSS over (U, E). By the above example, we can see that the precise evaluation for each parameter is unknown while the lower and upper limits of such an evaluation is given. Meanwhile, the precise evaluation for each object on each parameter is unknown while the lower and upper limits of such an evaluation is given. For example, we can describe the degree of three“expensive” houses by the interval [0.6,0.9]. In other words, the concept of “expensive” is vague, but it can be characterized by the interval [0.6,0.9]. In matrix from this can be expressed as   [0.6,0.7] [0.3,0.5] [0.6,0.8] [0.6,0.9]   Ff =  [0.1,0.3] [0.2,0.4] [0.8,1] [0.3,0.5]  [0.4,0.7] [0.5,0.8] [0.2,0.4] [0.1,0.2] where the ith row vector represent Ff (ei ) , the ith column vector represent xi , the last column represent the range of f and it will be called membership interval-valued matrix of Ff . Definition 3.4 Let Ff and Gg be two GIVFSSs over (U, E). Now Ff is said to be a generalized interval-valued fuzzy soft subset of Gg if and only if 4

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(1) f is an interval-valued fuzzy subset of g; (2) F (e) is also an interval-valued fuzzy subset of G(e), ∀e ∈ E. In this case, we write Ff ⊆ Gg . Example 3.5 Consider the GIVFSS Ff over (U, E) given in Example 3.3. Let Gg be another GIVFSS over (U, E) defined as follows: Gg (e1 ) = ({

[0.4, 0.6] [0, 0.4] [0.5, 0.7] , , , }, [0.4, 0.8]) h1 h2 h3

Gg (e2 ) = ({

[0, 0.2] [0.1, 0.3] [0.6, 0.9] , , , }, [0.2, 0.4]) h1 h2 h3

Gg (e3 ) = ({

[0.2, 0.6] [0.3, 0.6] [0, 0.3] , , , }, [0, 0.2]) h1 h2 h3

Clearly, we have Gg ⊆ Ff . Definition 3.6 Let Ff and Gg be two GIVFSSs over (U, E) . Now Ff and Gg are said to be a generalized interval-valued fuzzy soft equal if and only if (1) Ff is a generalized interval-valued fuzzy soft subset of Gg ; (2) Gg is a generalized interval-valued fuzzy soft subset of Ff , which can be denoted by Ff = Gg .

3.2

Operations on generalized interval-valued fuzzy soft set

Definition 3.7 The complement of Ff denoted by FfC and is defined by FfC = Gg , where G(e) = F C (e), g(e) = f C (e). From the above definition, we can see that (FfC )C = Ff . Example 3.8 Consider the GIVFSS Definition 3.7, we have  [0.4,0.6]  C Gg =  [0.8,1] [0.4,0.8]

Gg over (U, E) defined in Example 3.5. Thus, by  [0.6,1] [0.3,0.5] [0.2,0.6]  [0.7,0.9] [0.1,0.4] [0.6,0.8]  [0.4,0.7] [0.7,1] [0.8,1]

Definition 3.9 The union operation on the two GIVFSSs Ff and Gg , denoted by Ff ∪Gg , is defined by a mapping given by Hh : E → IV F (U ) × Int([0, 1]), such that Hh (e) = (H(e), h(e)), where H(e) = F (e) ∪ G(e), h(e) = f (e) ∪ g(e). Definition 3.10 The intersection operation on the two GIVFSSs Ff and Gg , denoted by Ff ∩ Gg , is defined by a mapping given by Hh : E → IV F (U ) × Int([0, 1]), such that Hh (e) = (H(e), h(e)), where H(e) = F (e) ∩ G(e), h(e) = f (e) ∩ g(e). 5

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Definition 3.11 The sum operation on the two GIVFSSs Ff and Gg , denoted by Ff ⊕Gg , is defined by a mapping given by Hh : E → IV F (U ) × Int([0, 1]), such that Hh (e) = (H(e), h(e)), where H(e) = F (e) ⊕ G(e), h(e) = f (e) ⊕ g(e). Example 3.12 Let us consider the GIVFSS Ff in Example 3.3. Let Gg be another GIVFSS over (U, E) defined as follows: Gg (e1 ) = ({

Gg (e2 ) = ({

Gg (e3 ) = ({ Then

[0.5, 0.8] [0, 0.2] [0.2, 0.9] , , , }, [0.5, 1]) h1 h2 h3

[0, 0.4] [0.3, 0.5] [0.9, 1.0] , , , }, [0.4, 0.6]) h1 h2 h3

[0.5, 0.6] [0.6, 0.7] [0.1, 0.3] , , , }, [0.2, 0.4]) h1 h2 h3



[0.6,0.8] [0.3,0.5] [0.6,0.9] [0.6,1.0]  Ff ∪ Gg =  [0.1,0.4] [0.3,0.5] [0.9,1.0] [0.4,0.6] [0.5,0.7] [0.6,0.8] [0.2,0.4] [0.2,0.4]  [0.5,0.7] [0,0.2] [0.2,0.8] [0.5,0.9]  Ff ∩ Gg =  [0,0.3] [0.2,0.4] [0.8,1.0] [0.3,0.5] [0.4,0.6] [0.5,0.7] [0.1,0.3] [0.1,0.2]  [1.0,1.0] [0.3,0.7] [0.8,1] [1.0,1.0]  Ff ⊕ Gg =  [0.1,0.7] [0.5,0.9] [1.0,1.0] [0.7,1.0] [0.9,1.0] [1.0,1.0] [0.3,0.7] [0.3,0.6]

        

Definition 3.13 A GIVFSS is said to a generalized F −empty interval-valued fuzzy soft set, denoted by Fˆ0 , if Fˆ0 : E → IV F (U ) × Int([0, 1]), such that Fˆ0 (e) = (F (e), ˆ0(e)), where ˆ0(e) = [0, 0], ∀e ∈ E. If F (e) = ∅, then the generalized F −empty interval-valued fuzzy soft set is called a generalized empty interval-valued fuzzy soft set, denoted by ∅ˆ0 . Definition 3.14 A GIVFSS is said to a generalized F −universal interval-valued fuzzy soft set, denoted by Fˆ1 , if Fˆ1 : E → IV F (U ) × Int([0, 1]), such that Fˆ1 (e) = (F (e), ˆ1(e)), where ˆ1(e) = [1, 1], ∀e ∈ E. If F (e) = U, then the generalized F −universal interval-valued fuzzy soft set is called a generalized universal interval-valued fuzzy soft set, denoted by Uˆ1 . From the Definition 3.13 and 3.14, obviously we have (1) ∅ˆ0 ⊆ Fˆ0 ⊆ Ff ⊆ Fˆ1 ⊆ Uˆ1 (2) ∅C = Uˆ1 . ˆ 0 6

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Proposition 3.15 Let Ff be a GIVFSS over (U, E), then the following holds: (1) Ff ∪ ∅ˆ0 = Ff , Ff ∩ ∅ˆ0 = ∅ˆ0 , (2) Ff ∪ Uˆ1 = Uˆ1 , Ff ∩ Uˆ1 = Ff , (3) Ff ∪ Fˆ0 = Ff , Ff ∩ Fˆ0 = Fˆ0 , (4) Ff ∪ Fˆ1 = Fˆ1 , Ff ∩ Fˆ1 = Ff , (5) Ff ⊕ ∅ˆ0 = Ff , Ff ⊕ Uˆ1 = Uˆ1 . 2

Proof. Straightforward.

Remark 3.16 Let Ff be a GIVFSS over (U, E), if Ff 6= Uˆ1 or Ff 6= ∅ˆ0 , then Ff ∪ FfC 6= Uˆ1 , and Ff ∩ FfC 6= ∅ˆ0 . Proposition 3.17 Let Ff Gg and Hh be any three GIVFSSs over (U, E) , then the following holds: (1) Ff ∪ Gg = Gg ∪ Ff , (2) Ff ∩ Gg = Gg ∩ Ff , (3) Ff ∪ (Gg ∪ Hh ) = (Ff ∪ Gg ) ∪ Hh , (4) Ff ∩ (Gg ∩ Hh ) = (Ff ∩ Gg ) ∩ Hh , (5) Ff ⊕ Gg = Gg ⊕ Ff . 2

Proof. The properties follow from definition.

Proposition 3.18 Let Ff and Gg be two GIVFSSs over (U, E) . Then De-Morgan’s laws are valid: (1) (Ff ∪ Gg )C = FfC ∩ GC g, C C (2) (Ff ∩ Gg ) = Ff ∪ GC g. Proof. For all e ∈ E, (Ff ∪ Gg )C = ((F (e) ∪ G(e), f (e) ∪ g(e)))C = (F C (e) ∩ GC (e)), (f C (e) ∩ g C (e)) = (F C (e), f C (e)) ∩ (GC (e) ∩ g C (e)) = FfC ∩ GC g 2

Likewise, the proof of (2) can be made similarly. Proposition 3.19 Let Ff , Gg and Hh be any three GIVFSSs over (U, E) . Then, (1) Ff ∪ (Gg ∩ Hh ) = (Ff ∪ Gg ) ∩ (Ff ∪ Hh ), (2) Ff ∩ (Gg ∪ Hh ) = (Ff ∩ Gg ) ∪ (Ff ∩ Hh ),

Proof. The proof follows from definition and distributive property of interval-valued fuzzy set. 2 7

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4

Similarity between two generalized interval-valued fuzzy soft sets

In this section, a measure of similarity between two GIVFSSs has been given. Let U = {x1 , x2 , · · · , xn } be the universal set of elements and E = {e1 , e2 , . . . , em } be the universal set of parameters. Let Ff and Gg ∈ GIV F SS(U, E), where Ff = {(F (ei ), f (ei )), i = 1, 2 · · · , m} and Gg = {(G(ei ), g(ei )), i = 1, 2 · · · , m}. ˆ = {G(ei ), i = 1, 2 · · · , m} are two families Thus Fˆ = {F (ei ), i = 1, 2 · · · , m}, and G ˆ is found first and it is of interval-valued fuzzy sets. Now the similarity between Fˆ and G ˆ Next the similarity between interval-valued fuzzy sets f and g is denoted by M (Fˆ , G). found and is denoted by m(f, g). Then the similarity between the two GIVFSSs Ff and ˆ · m(f, g). Gg is denoted by S(Ff , Gg ) = M (Fˆ , G) ˆ = max Mi (Fˆ , G), ˆ where Here, M (Fˆ , G) i n ˆ = 1 − 1 P max{|µ− (xj ) − µ− (xj )|, |µ+ (xj ) − µ+ (xj )|}. Mi (Fˆ , G) n F (ei ) G(ei ) F (ei ) G(ei ) j=1

Also m(f, g) = 1 −

1 m

m P i=1

+ + − max{|µ− f (ei ) − µg (ei )|, |µf (ei ) − µg (ei )|}.

Example 4.1 Consider the following two {e1 , e2 , e3 }.  [0.6,0.7] [0.3,0.5]  Ff =  [0.1,0.3] [0.2,0.4] [0.4,0.7] [0.5,0.8]

GIVFSSs where U = {x1 , x2 , x3 , x4 } , and E =

 [0.6,0.8] [0.6,0.9] [0.6,0.7]  [0.8,1.0] [0.3,0.5] [0.8,0.9]  [0.2,0.4] [0.1,0.2] [0.4,0.8]



 [0.5,0.6] [0.4,0.8] [0.2,0.3] [0.7,1.0] [0.5,0.8]   Gg =  [0.0,0.2] [0.1,0.5] [0.7,0.8] [0.2,0.6] [0.6,0.8]  [0.5,0.6] [0.6,0.7] [0.3,0.6] [0.2,0.6] [0.3,0.5] Here m(f, g) = 0.8. ˆ = 0.75, M2 (Fˆ , G) ˆ = 0.875, M3 (Fˆ , G) ˆ = 0.8, And M1 (Fˆ , G) ˆ = 0.875. ∴ M (Fˆ , G) Hence the similarity between the two GIVFSSs Ff and Gg will be S(Ff , Gg ) = 0.7. From the above definition, we can easily obtain the following conclusions. Proposition 4.2 Let Ff , Gg and Hh be any three GIVFSSs over (U, E). Then the following holds: (1) S(Ff , Gg ) = S(Gg , Ff ), (2) 0 ≤ S(Ff , Gg ) ≤ 1, (3) Ff = Gg ⇒ S(Ff , Gg ) = 1, (4) Ff ⊆ Gg ⊆ Hh ⇒ S(Ff , Hh ) ≤ S(Gg , Hh ). 8

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5

Application of generalized interval-valued fuzzy soft set

In this section, we define an aggregate interval-valued fuzzy set of a GIVFSS. We also define GIVFSS aggregation operator that produce an aggregate interval-valued fuzzy set from a GIVFSS over (U, E). Definition 5.1 Let Ff be a GIVFSS over (U, E), i.e.Ff ∈ GIVFSS(U, E). Then GIVFSSaggregation operator, denoted by GIV F SSagg , is defined by GIV F SSagg : GIV F SS(U, E) → IV F (U ), GIV F SSagg Ff = H, where µH (u) H={ , u ∈ U} u which is an interval-valued fuzzy set over U . The value H is called aggregate intervalvalued fuzzy set of Ff . Here, the membership degree µH (ui ) of ui is defined as follows + µH (ui ) = [a− i , ai ] = [

M

− µ− f (x)µF (x) (ui ),

M

+ µ+ f (x)µF (x) (ui )]

x∈E

x∈E

From the above definition, it is noted that the GIV F SSagg on the interval-valued fuzzy set is an operation by which several approximate functions of a GIVFSS are combined to produce a single interval-valued fuzzy set that is the aggregate interval-valued fuzzy set of the GIVFSS. Once an aggregate interval-valued fuzzy set has been arrived at, it may be necessary to choose the best single alternative from this set. Therefore, we can construct a GIVFSS-decision making method by the following algorithm. Step 1 Construct a GIVFSS over (U, E), Step 2 Find the aggregate interval-valued fuzzy set H, Step 3 Compute the score ri of ui such that ri =

X

− + + ((a− i − aj ) + (ai − aj )),

ui ∈U

Step 4 Find the largest value in S where S = max{ri }. ui ∈U

Example 5.2 Assume that a company want to fill a position. There are six candidates who form the set of alternatives, U = {u1 , u2 , · · · , u6 }. The hiring committee consider a set of parameters, E = {x1 , x2 , x3 }. The parameters xi (i = 1, 2, 3) stand for “experience”, “computer knowledge” and “young age”, respectively. After a serious discussion each candidate is evaluated from point of view of the goals [0.8,1] [0.6,0.8] and the constraint according to a chose subset f = { [0.5,0.7] x1 , x2 , x3 } of E. Finally, the committee constructs the following GIVFSS over (U, E). 9

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Step 1 Let the constructed GIVFSS, Ff , be as follows, 

 [0.0,0.0] [0.3,0.4] [0.4,0.6] [0.9,1.0] [0.1,0.2] [0.0,0.0] [0.5,0,7]   Ff =  [0.4,0.6] [0.2,0.4] [0.7,0.8] [0.3,0.6] [0.0,0.0] [0.0,0.0] [0.8,1.0]  [0.0,0.2] [0.4,0.6] [0.0,0.0] [0.0,0.0] [0.6,0.8] [0.2,0.5] [0.6,0.8] Step 2 The aggregate interval-valued fuzzy set can be found as H={

[0.32, 0.76] [0.55, 1] [0.76, 1] [0.69, 1] [0.41, 0.78] [0.12, 0.4] , , , , , } u1 u2 u3 u4 u5 u6

Step 3 For all ui ∈ U , compute the score ri of ui such that ri =

P ui ∈U

− + + ((a− i −aj )+(ai −aj )).

Thus, we have r1 = −1.31, r2 = 1.51, r3 = 2.77, r4 = 2.35, r5 = −0.65, r6 = −4.67. Step 4 The decision is anyone of the elements in S where S = max{ri }. ui ∈U

In our example, the candidate u3 is the best choice because max{ri } = {u3 }. This ui ∈U

result is reasonable because we can see that µH (u3 ) ≥ µH (ui ), where i = 1, 2, 4, 5, 6.

6

Conclusion

Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainty. However, it is difficult to be used to represent the fuzziness of problem. In order to handle these types of problem parameters, some fuzzy extensions of soft set theory are presented, yielding fuzzy soft set theory. In this paper, the notion of generalized interval-valued fuzzy soft set theory is proposed. Our generalized intervalvalued fuzzy soft set theory is a combination of a generalized fuzzy soft set theory and an interval-valued fuzzy set theory. In other words, our generalized interval-valued fuzzy soft set theory is an extension of generalized interval-valued fuzzy soft set theory. The complement, union, intersection and sum operations are defined on generalized intervalvalued fuzzy soft sets. The basic properties of the generalized interval-valued fuzzy soft sets are also presented and discussed. Similarity measure of two generalized interval-valued fuzzy soft sets is discussed. Finally, an application of this theory has been applied to solve a decision making problem. In further research, the parameterization reduction of generalized interval-valued fuzzy soft sets is an important and interesting issue to be addressed.

Acknowledgements The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work is supported by the Fundamental Research Funds for the 10

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Central Universities of Northwest University for Nationalities (No. 31920130012) and the Young Research Foundation of Northwest University for Nationalities (No.12XB29).

References [1] D.Molodtsov, Soft set theory-First results,Computers and Mathematics with Applications37(1999): 19-31. [2] H.Aktas, N.Cagman, Soft sets and soft groups, Information Sciences 177(2007):2726-2735. [3] D.Molodtsov, The theory of soft sets. URSS Publishers, Moscow ,2004.(in Russian) [4] P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Computers and Mathematics with Applications 45 (2003) 555-562. [5] F. Feng, Y.B. Jun, X.Z. Zhao, Soft semirings, Computers and Mathematics with Applications 56 (2008) 2621-2628. [6] M.I. Ali, F. Feng, X. Liu, W.K. Min, M. Shabir, On some new operations in soft set theory, Computers and Mathematics with Applications 57 (2009) 1547-1553. [7] K.Y. Qin, Z.Y. Hong, On soft equality, Journal of Computational and Applied Mathematics 234 (2010) 1347-1355. [8] P.K. Maji, S.K. Samanta, Generalized fuzzy soft sets, Computers and Mathematics with Applications 59 (2010) 1425-1432. [9] X.B. Yang, T.Y. Lin, J.Y. Yang, Y.Li, D.Yu, Combination of interval-valued fuzzy set and soft set, Computers and Mathematics with Applications 58(3) (2009) 521-527. [10] M.B. Gorzalczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems 21 (1987) 1-17. [11] G. Deschrijver, E.E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems 133 (2003) 227-235. [12] G. Deschrijver, E.E. Kerre, Implicators based on binary aggregation operators in intervalvalued fuzzy set theory, Fuzzy Sets and Systems 153 (2005) 229-248. [13] F. Feng, C.X. Li, B. Davvaz, M.I. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Computing 14 (2010) 899-911. [14] J. H. Park, O. H. Kima, Y. C. Kwun, Some properties of equivalence soft set relations, Computers and Mathematics with Applications 63 (2012) 1079-1088. [15] Y. Yang , X.Tan, C.C. Meng, The multi-fuzzy soft set and its application in decision making, Applied Mathematical Modelling 37 (2013) 4915-4923. [16] M. Shabir, M.I. Ali, T. Shaheen, Another approach to soft rough sets, Knowledge-Based Systems 40 (2013) 72-80. [17] P.K. Maji, R. Biswas, A.R. Roy, Fuzzy soft set, Journal of Fuzzy Mathematics, 9(3) (2001) 589-602.

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Distribution reductions in α-maximal tolerance level SF decision information systems ∗ Sheng Luo† February 23, 2014 Abstract: In this paper, we introduce α-maximal tolerance levels in SF decision information systems, establish the rough set models and obtain distribution reductions of α-maximal tolerance level SF decision information systems. Keywords: SF decision information system; α-maximal tolerance level; εinconsistent; Distribution reduction.

1

Introduction

Rough set theory, proposed by Pawlak [1], is a new mathematical tool for data reasoning. It may be seen as an extension of classical set theory and has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields [2, 3, 4, 5]. Rough sets are based on an assumption that every object in the universe is associated with some information. Objects characterized by the same information are indiscernible with the available information about them. The indiscernibility relation generated in this way is the mathematical basis for rough set theory. Rough set theory has used successfully in the analysis of data in information systems. Set-valued information systems as important information systems have been gained further research in recent years. Wang et al. [7] and Song et al. [8] studied set-valued information system based on tolerance relations. But compatible class has some shortcomings. Guan et al. [9, 10] introduced the concept of maximum compatible classes and made up for the deficiency of the compatible class. Fuzzy decision information systems are decision information systems under the fuzzy environment. There are a lot of research achievements on fuzzy decision information systems (see [11, 12, 13]). ∗ This work is supported by the National Social Science Foundation of China (No. 12BJL087). † Corresponding Author, School of Business Administration, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, P.R.China. [email protected]

1

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Set-valued fuzzy decision information systems are set-valued decision information systems under the fuzzy environment. α-maximal tolerance level setvalued fuzzy decision information systems are their generalization. In this paper, we investigate deeply attribute reductions in α-maximal tolerance level set-valued fuzzy decision information systems.

2

Preliminaries

Throughout this paper, “set-valued fuzzy” denote briefly by “SF”, U denotes a finite and nonempty set called the universe, 2U denotes the power set of U , F (U ) denotes the set of all fuzzy sets in U , I denotes [0, 1] and B A denotes {f |f : A → B}. For convenience, stipulate U = {x1 , x2 , · · · , xn }, A = {a1 , a2 , · · · , am }, D = {d1 , d2 , · · · , dp }, ε, α ∈ [0, 1]. Definition 2.1 ([14]). (U, A, F, D, G) is called an SF decision information system, where (U, A, F ) is a set-valued information system, i.e., U is the universe, A is the condition attribute set, F = {fa ∈ (2Va − {∅})U : a ∈ A} is the relationship set between U and A, fa is the information function of a, Va is the range of a; D is the decision attribute set; G = {gd ∈ I U : d ∈ D} is the relationship set between U and D, gd ∈ F (U ) is the information function of d. Example 2.2. Table 1 gives an IF decision information system where U = {x1 , x2 , · · · , x8 }, A = {a1 , a2 , a3 }, D = {d1 , d2 , d3 }. Table 1: The SF decision information systems (U, A, F, D, G) U x1 x2 x3 x4 x5 x6 x7 x8

a1 {0, 1} {1, 2} {0} {0} {1} {1} {0, 1} {0, 1}

a2 {2} {1} {0, 1} {2} {1, 2} {1} {0, 2} {0, 1}

a3 {0, 1} {0} {1} {0, 1} {1, 2} {0, 2} {1, 2} {2}

d1 0 0.4 0.2 0.6 0.6 1 0.7 0.5

d2 0.3 0.8 1 0.1 0.8 0.5 0.2 0.3

d3 0.4 0.7 0.1 0.2 0.9 0.5 0.6 0.3

Definition 2.3 ([10]). Let R be a tolerance relation on U . (1) X ∈ 2U is called a compatible class of R, if for any x, y ∈ X, xRy. (2) X ∈ 2U is called a maximum compatible class of R, if X is a compatible class of R and for any x ∈ U − X, there exists y ∈ X such that (x, y) 6∈ R The set of all maximum compatible classes of R is denoted by CCR(U ) For x ∈ U , denote CCR(x) = {K ∈ CCR(U ) : x ∈ K}. Definition 2.4 ([7, 8]). Let (U, A, F, D, G) be a SF decision information system. For B ⊆ A, define a tolerance relation TB of B on U as follows: TB = {(x, y) ∈ U × U : fb (x) ∩ fb (y) 6= ∅ (∀ b ∈ B)}. If xTB y for any x, y ∈ U , then x, y are called compatible on B. 2

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Denote TB (x) = {y ∈ U : xTB y}, CCTB (x) = {K ∈ CCTB (U ) : x ∈ K}. For b ∈ B, x ∈ U , denote Tb = T{b} , Tb (x) = T{b} (x); CCTb (U ) = CCT{b} (U ), CCTb (x) = CCT{b} (x). It is easy to verify that TB =

\

Tb , TB (x) =

b∈B

\

Tb (x).

b∈B

Example 2.5. Consider the SF decision information system (U, A, F, D, G) in Example 2.3. Put C = {a1 , a2 , a3 }. Then TC (x1 ) = {x1 }, TC (x2 ) = {x2 , x3 }, TC (x3 ) = {x2 , x3 , x4 , x6 }, TC (x4 ) = {x3 , x4 , x6 }, TC (x5 ) = {x5 , x6 }, TC (x6 ) = {x3 , x4 , x5 , x6 }; CCTC (x1 ) = {{x1 }}, CCTC (x2 ) = {{x2 , x3 }}, CCTC (x3 ) = {{x2 , x3 }, {x3 , x4 , x6 }}, CCTC (x4 ) = {{x3 , x4 , x6 }}, CCTC (x5 ) = {{x5 , x6 }}, CCTC (x6 ) = {{x3 , x4 , x6 }, {x5 , x6 }}. Definition 2.6 ([15]). Let (U, A, F, D, G) be a SF decision information system. ∀ xi , xj ∈ U , define ^ SD (xi , xj ) = {1 − | fd (xi ) − fd (xj ) | : d ∈ D}, ε SD = {(xi , xj ) ∈ U × U : SD (xi , xj ) > ε},

where ε is called the decision tolerance. ε 1 Remark 2.7. SD is a tolerance relation on U . Specially, SD = {(xi , xj ) ∈ U × U : fd (xi ) = fd (xj ) (∀ d ∈ D)} is an equivalence relation on U .

Denote

ε [xi ]εD = {xj ∈ U : (xi , xj ) ∈ SD }, ε Dε = U/SD = {[xi ]εD : xi ∈ U }.

ε They mean the compatible class of xi on SD and the fuzzy decision classification on U , respectively.

Example 2.8. Consider the SF decision information system (U, A, F, D, G) in Example 2.3. Put D = {d1 , d2 , d3 } and ε = 0.7. Then D0.7 = {[xi ]0.7 : i = 1, 2, · · · , 8}, D where

0.7 0.7 [x1 ]0.7 D = {x1 }, [x2 ]D = {x2 , x5 }, [x3 ]D = {x3 }, 0.7 0.7 [x4 ]0.7 D = {x4 , x8 }, [x5 ]D = {x2 , x5 }, [x6 ]D = {x6 , x7 },

[x7 ]0.7 D = {x6 , x7 , x8 },

[x8 ]0.7 D = {x4 , x7 , x8 }. 3

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3

α-maximal tolerance level SF decision information systems

Definition 3.1. Let (U, A, F, D, G) be a SF decision information system. For xi , xj ∈ U , b ∈ B ⊆ A, α ∈ [0, 1], denote Γb (xi , xj ) =

|CCTb (xi ) ∩ CCTb (xj )| . |CCTb (xi ) ∪ CCTb (xj )|

where | · | means the cardinality. Denote α Tf B = {(xi , xj ) ∈ U × U : Γb (xi , xj ) > α (∀ b ∈ B)}.

Then α is called the maximal tolerance level. α It is easy to verify that Tf B is a tolerance relation on U . For b ∈ B, denote α . Tfα = Tg b

Denote

{b}

α fα Tf B (x) = {y ∈ U : (x, y) ∈ TB }, α fα Tf B (U ) = {TB (x) : x ∈ U }.

1 Obviously Tf B = {(xi , xj ) ∈ U × U : CCTb (xi ) = CCTb (xj ) (∀ b ∈ B)} is an equivalence relation on U .

Example 3.2. Consider the SF decision information system (U, A, F, D, G) in Example 2.3. Put C = {a1 , a2 , a3 }. Then g g 0.4 0.4 0.4 Tg C (x1 ) = {x1 , x4 }, TC (x2 ) = {x2 , x6 }, TC (x3 ) = {x3 }, g g 0.4 0.4 0.4 Tg C (x4 ) = {x1 , x4 }, TC (x5 ) = {x5 }, TC (x6 ) = {x2 , x5 , x6 , x8 }, 0.4 Tg C (x7 ) = {x7 },

0.4 Tg C (x8 ) = {x6 , x8 }.

Proposition 3.3. Let x ∈ U , C ⊆ B ⊆ A. Then α fα Tfα (x) ⊆ Tfα (x) ⊆ Tfα (x). fα (1) Tf B C A A ⊆ TB ⊆ TC , T fα α f (2) TB = Tb . b∈B

Proof. The proof is not difficult, so we omit it. Definition 3.4. Let (U, A, F, D, G) be a α-maximal tolerance level SF decision ε α information system where ε is the decision tolerance. If Tf A ⊆ SD , then it is called a α-maximal tolerance level ε-consistent SF decision information system; Otherwise, it is called a α-maximal tolerance level ε-inconsistent SF decision information system.

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Obviously, inconsistent (resp. consistent) SF decision information systems are 1-maximal tolerance level 1-inconsistent (resp. 1-consistent) SF decision information systems. Example 3.5. Consider the SF decision information system (U, A, F, D, G) in Example 2.3. Note that 0.4 0.7 0.4 0.4 Tg Tg A ⊆ SD , A 6⊆ SD . Then (U, A, F, D, G) are both 0.4-maximal tolerance level 0.4-consistent SF decision information systems and 0.4-maximal tolerance level 0.7-inconsistent SF decision information systems.

4

Distribution reductions

Definition 4.1. Let (U, A, F, D, G) be a α-maximal tolerance level SF decision information system. For B ⊆ A, X ∈ 2U , define α CCTBα (X) = {x ∈ U : Tf B (x) ⊆ X}, α CCTBα (X) = {x ∈ U : Tf B (x) ∩ X 6= ∅}.

Then CCTBα (X) (resp. CCTBα (X)) is called the α-level lower (resp. upper) approximation of X. Proposition 4.2. Let (U, A, F, D, G) be a α-maximal tolerance level SF decision information system. If C ⊆ B ⊆ A, then ∀ X ∈ 2U , CCTCα (X) ⊆ CCTBα (X) ⊆ X ⊆ CCTBα (X) ⊆ CCTCα (X). Proof. This holds by Proposition 3.3. Definition 4.3. Let (U, A, F, D, G) be a α-maximal tolerance level ε-inconsistent SF decision information system and let Dε = {Diε : i 6 r} be the classification of fuzzy decision on U . For B ⊆ A, define LB (α, ε) = (CCTBα (D1ε ), CCTBα (D2ε ), · · · , CCTBα (Drε )), HB (α, ε) = (CCTBα (D1ε ), CCTBα (D2ε ), · · · , CCTBα (Drε )), they are called α-level lower distribution function and α-level upper distribution function, respectively. (1) If LB (α, ε) = LA (α, ε), then B is called a α-level lower ε-distribution coordinate set. (2) If B is a α-level lower ε-distribution coordinate set and ∀ B 0 B, LB 0 (α, ε) 6= LA (α, ε), then B is called a α-level lower distribution ε-reduction. (3) If HB (α, ε) = HA (α, ε), then B is called a α-level upper ε-distribution coordinate set. (4) If B is a α-level upper ε-distribution coordinate set and ∀ B 0 B, HB 0 (α, ε) 6= HA (α, ε), then B is called a α-level upper distribution ε-reduction. 5

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Denote

ε ε ε ε fα (Gα B ) (x) = { D ∈ D : TB (x) ⊆ D }, ε α (MBα )ε (x) = { Dε ∈ Dε : Tf B (x) ∩ D 6= ∅}.

Proposition 4.4. Let (U, A, F, D, G) be a α-maximal tolerance level ε-inconsistent SF decision information system and let Dε be the classification of fuzzy decision on U . (1) ∀ Dε ∈ Dε , ε CCTBα (Dε ) = {x ∈ U : Dε ∈ (Gα B ) (x)}.

(2) ∀ Dε ∈ Dε , CCTBα (Dε ) = {x ∈ U : Dε ∈ (MBα )ε (x)}. (3) If C ⊆ B ⊆ A, then ∀ x ∈ U , ε α ε α ε α ε (Gα C ) (x) ⊆ (GB ) (x), (MC ) (x) ⊇ (MB ) (x). ε α Proof. (1) ∀ x ∈ CCTBα (Dε ), we have Tf B (x) ⊆ D . Then ε Dε ∈ (Gα B ) (x). ε Thus x ∈ {x ∈ U : Dε ∈ (Gα B ) (x)}. ε On the other hand, ∀ x ∈ {x ∈ U : Dε ∈ (Gα B ) (x)}), we have ε Dε ∈ (Gα B ) (x). ε α Then Tf B (x) ⊆ D . Thus

Hence

x ∈ CCTBα (Dε ).

ε CCTBα (Dε ) = {x ∈ U : Dε ∈ (Gα B ) (x)}.

(2) The proof is similar to (1). (3) The proof is not difficult, so we omit it. Theorem 4.5. Let (U, A, F, D, G) be a α-maximal tolerance level ε-inconsistent SF decision information system. Then (1) B ⊆ A is a α-level lower distribution ε-coordinate set ⇐⇒ ε α ε ∀ x ∈ U, (Gα B ) (x) = (GA ) (x).

(2) B ⊆ A is a α-level upper distribution ε-coordinate set ⇐⇒ ∀ x ∈ U, (MBα )ε (x) = (MAα )ε (x).

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Proof. (1) Since Dε is the classification of fuzzy decision on U , we have ε ε α ε ε α ε (Gα B ) (x) ⊆ D ⇐⇒ x ∈ CCTB (D ) ⇐⇒ D ∈ (GB ) (x).

Thus B is a α-level lower distribution ε-coordinate set ⇐⇒ ε ε α ε ε (Gα B ) (x) ⊆ D ⇐⇒ (GA ) (x) ⊆ D

(∀ x ∈ U, Dε ∈ Dε )

ε α ε ⇐⇒ (Gα B ) (x) = (GA ) (x) (∀ x ∈ U ). (2) The proof is similar to (1).

Definition 4.6. Let (U, A, F, D, G) be a α-maximal tolerance level ε-inconsistent SF decision information system and let Dε be the classification of fuzzy decision on U . Define ε ε α ε ε Pα ε = {(xi , xj ) ∈ U × U : ∃ D ∈ D , xi ∈ CCTA (D ), xj 6∈ D }, α

P ε = {(xi , xj ) ∈ U × U : ∃ Dε ∈ Dε , xi 6∈ CCTAα (Dε ), xj ∈ Dε }; ( α {a ∈ A : (xi , xj ) 6∈ Tf (xi , xj ) ∈ P α α a }, ε, W ε (xi , xj ) = ∅, (xi , xj ) 6∈ P α ε. ( α α {a ∈ A : (xi , xj ) 6∈ Tf (xi , xj ) ∈ P ε , α a }, W ε (xi , xj ) = α ∅, (xi , xj ) 6∈ P ε ; α

α Wα ε = (W ε (xi , xj ))n×n ,

α

W ε = (W ε (xi , xj ))n×n .

α

Then W α ε (xi , xj ) (resp. W ε (xi , xj )) is called the α-level lower (resp. upper) α distribution ε-discernibility attribute set of xi and xj on Dε , W α ε (resp. W ε ) is called the α-level lower (resp. upper) distribution ε-discernibility matrix on Dε . Theorem 4.7. Let (U, A, F, D, G) be a α-maximal tolerance level ε-inconsistent SF decision information system. Then (1) B ⊆ A is a α-level lower ε-distribution coordinate set ⇐⇒ α B ∩ Wα ε (xi , xj ) 6= ∅ whenever W ε (xi , xj ) 6= ∅.

(2) B ⊆ A is a α-level upper distribution ε-coordinate set ⇐⇒ α

α

B ∩ W ε (xi , xj ) 6= ∅ whenever W ε (xi , xj ) 6= ∅. Proof. (1) Necessity. Suppose that there exists (xi , xj ) ∈ P α ε such that α B ∩ Wα ε (xi , xj ) = ∅ and W ε (xi , xj ) 6= ∅.

Since B is a α-level lower distribution ε-coordinate set and Dε ∈ Dε , we have B ⊆ A − Wα ε (xi , xj ).

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α This implies that (xi , xj ) ∈ Tf B . Then α xj ∈ Tf B (xi ). ε ε α Note that (xi , xj ) ∈ P α ε . Then xi ∈ CCTA (D ), xj 6∈ D .

Since B is a α-level lower distribution ε-coordinate set, we have CCTBα (Dε ) = CCTAα (Dε ) (∀ Dε ∈ Dε ). ε α Then xi ∈ CCTBα (Dε ). So Tf B (xi ) ⊆ D . ε α Note that xj ∈ Tf B (xi ). Then xj ∈ D . This is a contradiction.

Sufficiency. Suppose that B is not a α-level lower ε-distribution coordinate set. Then LB (α, ε) 6= LA (α, ε). Thus, there exists D0ε ∈ Dε such that CCTBα (D0ε ) 6= CCTAα (D0ε ). B ⊆ A implies that CCTBα (D0ε ) CCTAα (D0ε ). Pick x ∈ CCTAα (D0ε ) − CCTBα (D0ε ). ε α Since Tf B (xi ) * D0 , we have ε α Tf B (xi ) − D0 6= ∅. α This illustrates that there exists xj ∈ U − D0ε such that xj ∈ Tf B (xi ). Then \ α α (xi , xj ) ∈ Tf Tf B = B. b∈B

This implies that

α (xi , xj ) ∈ Tf B (∀ b ∈ B).

α But (xi , xj ) ∈ P α ε . So B ∩ W ε (xi , xj ) = ∅. This is a contradiction.

Hence B is a α-level lower distribution ε-coordinate set. (2) The proof is similar to (1). Definition 4.8. Let (U, A, F, D, G) be a α-maximal tolerance level ε-inconsistent SF decision information system, B ⊆ A, Dε be the fuzzy classification of decision α on U . Suppose that W α ε (xi , xj ) and W ε (xi , xj ) are the α-level lower distribution ε-discernibility attribute set and α-level upper distribution ε-discernibility attribute set of xi and xj on Dε , respectively. Denote ^ _ α Mα { Wα ε = ε (xi , xj ) : (xi , xj ) ∈ P ε }, 8

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α

Mε =

^ _ α α { W ε (xi , xj ) : (xi , xj ) ∈ P ε }.

α

Then Mα ε (resp. Mε ) is called the α-level lower (resp. upper) distribution εdiscernibility function. Theorem 4.9. Let (U, A, F, D, G) be a α-maximal tolerance level ε-inconsistent SF decision information system. The minimal disjunctive normal form of its α-level lower, upper distributionmε-discernibility function are respectively (2)

(1)

Mα ε

qm t _ ^ (1) = ( aln ),

α Mε

m=1 n=1

m=1 n=1

Denote

(i)

qm t _ ^ (2) = ( aln ).

(i)

(i) } (i = 1, 2). Bkm = {aln : n = 1, 2, · · · , qm (1)

(2)

Then ∀ m ≤ t, Bkm (resp. Bkm ) is a α-level lower (resp. upper) distribution ε-reduction. Proof. This hods by using Boolean reasoning theorem. Example 4.10. Consider 0.4-maximal tolerance level 0.7-inconsistent SF decision information systems (U, A, F, D, G) in Example 3.5. The 0.4-level lower distribution 0.7-discernibility attribute matrix is  ∅ ∅ {a2 } ∅ {a3 } ∅ {a3 } {a2 , a3 }  ∅ ∅ {a , a } ∅ ∅ ∅ {a , a } ∅ 1 3 2 3   {a2 } {a , a } ∅ {a } {a , a } {a , a } {a } {a 1 3 2 1 2 1 3 2 3}   ∅ ∅ {a } ∅ C ∅ {a } ∅ 2 3  (W 0.4 0.7 ) =  ∅ {a1 , a2 } C ∅ {a2 , a3 } {a2 } {a2 }  {a3 }  ∅ ∅ {a , a } ∅ {a , a } ∅ {a , a } ∅ 1 3 2 3 2 3   {a3 } {a2 , a3 } {a2 } {a3 } {a2 } {a2 , a3 } ∅ ∅ {a2 , a3 } ∅ {a3 } ∅ {a2 } ∅ ∅ ∅

      .     

The 0.4-level lower distribution 0.7-discernibility function is M0.4 0.7 = a2 ∧ a3 ∧ (a2 ∨ a3 ) ∧ (a1 ∨ a3 ) ∧ (a1 ∨ a2 ) ∧ (a1 ∨ a2 ∨ a3 ) = a2 ∧ a3 . Hence B = {a2 , a3 } is the 0.4-level lower distribution 0.7-reduction.

5

Conclusions

In this paper, we have researched distribution reductions in α-maximal tolerance level ε-inconsistent SF decision information systems systems. In future work, we will investigate knowledge acquisition in α-maximal tolerance level SF decision information systems.

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References [1] Z.Pawlak, Rough sets, International Journal of Computer and Information Science, 11(1982), 341-356. [2] Z.Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991. [3] Z.Pawlak, A.Skowron, Rudiments of rough sets, Information Sciences, 177(2007), 3-27. [4] Z.Pawlak, A.Skowron, Rough sets: some extensions, Information Sciences, 177(2007), 28-40. [5] Z.Pawlak, A.Skowron, Rough sets and Boolean reasoning, Information Sciences, 177(2007), 41-73. [6] W.Zhang, Y.Liang, W.Wu, Information systems and knowledge discovery, Science Press, Beijing, 2003. [7] H.Wang, W.Zhang, H.Li, Knowledge reduction in set-valued decision information System, Computer Engineering and Applications, 2005(6)(2005), 37-38. [8] X.Song, Z.Xie, W.Zhang, Knowledge reduction and rule extraction in setvalued decision information System, Computer Science, 34(4)(2007), 182191. [9] Y.Guan, H.Wang, Set-valued information systems, Information Sciences, 176(2006), 2507-2525. [10] Y.Guan, H.Wang, K.Shi, Set-valued information systems and their attribute reductions, Mathematics in Practice and Theory, 28(2)(2008), 101107. [11] T.Guan, B.Feng, Knowledge reduction methods in fuzzy objective information systems, Journal of Software, 15(10)(2004), 1470-1478. [12] B.Huang, X.Zhou, Z.Hu, VPRS model and its knowledge reduction in fuzzy objective information systems, Systems Engineer ing and Electronics, 29(11)(2007), 1859-1862. [13] X.Yang, J.Yang, Wu C.Wu, Y.Zheng, Research on acquisition of decision rules in fuzzy set-valued information systems, Journal of System Simulation, 20(5)(2008), 1199-1202. [14] Z.Chen, K.Qin, Approximate assignment reduction in set-valued fuzzy decision information system, Journal of Sichuan Normal University, 34(1)(2011), 23-27. [15] A.P.Dempster, Upper and lower probabilities induced by a multivalued mapping, The Annals of Mathematical Statistics, 38(2)(1967), 325-339.

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Separation Problem for Second Order Elliptic Di¤erential Operators on Riemannian Manifolds H. A. Atia. Current Address: Mathematics Department, Rabigh College of Science and Art, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi Arabia. Permennent Address: Zagazig University, Faculty of Science, Mathematics Department, Zagazig, Egypt. E-mail: [email protected] Abstract In this paper we aim to study the separation problem for the second 2 order elliptic di¤erential expression of the form E = 4M +bi @ i2 +q; with @x the real-valued positive continuous functions bi ; on a complete Riemannian manifold (M; g) with metric g; where 4M is the scalar Laplacian on M and q 0 is a locally square integrable function on M . In the terminology of Everitt and Giertz, E is said to be separated 2 in L2 (M ) if for u 2 L2 (M ) such that Eu 2 L2 (M ), we have bi @ iu2 @x and qu 2 L2 (M ) : We give su¢ cient conditions for E to be separated in L2 (M ) : Keywords: Separation, Second order elliptic di¤erential operator, Riemannian manifold AMS Subject Classi…cation: 47F05, 58J99.

1

Introduction

The separation property of di¤erential operators was …rst introduced in 1971 by Everitt and Giertz [9]. Many recent results of separation of di¤erential operators may be found in Brown [6], Mohamed and Atia [15,16], Atia and Mahmoud [3] and Atia [1]. Zayed et al [18] have studied the separation of the Laplace-Beltrami di¤erential operator of the form h i p @ @u 1 det g(x)g (x) Au = p 1 @xi @xj + V (x)u(x); for every x 2 det g(x)

Rn ; in the Hilbert space H = L2 ( ; H1 ) with the operator potential V (x) 2 C 1 ( ; L (H1 )), where L (H1 ) is the space of all bounded linear operators on the Hilbert space H1 and g (x) = gij (x) is the Riemannian matrix and g 1 (x) is the inverse of the matrix g (x) : Atia [2] has proved the separation of a bi-harmonic di¤erential expression of the form A = 44 + q on a complete Riemannian manifold (M; g) with metric g; where 4 is the laplacian on M and 1

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q 0 is a locally square integrable function on M . For more studies about the separation property see for example [4,12,13,14]. Let (M; g) be a Riemannian manifold without boundary (i.e. M is a C 1 manif old without boundary and (gjk ) is a Riemannian metric on M ) and dimM = n. We will assume that M is connected. We will also assume that we are given a positive smooth measure d ; i.e. in any local coordinates x1 ; x2 ; : : : ; xn there exists a strictly positive C 1 density (x) such that d = (x) dx1 dx2 : : : dxn : In the sequel, L2 (M ) is the space of complex-valued square integrable functions on M with the inner product: Z (u; v) = uv d ; (1) M

and k:k is the norm in L2 (M ) corresponding to the inner product (1). We use the notation L2 1 T M for the space of complex-valued square integrable 1-forms on M with the inner product: Z d ; (2) (W; )L2 ( 1 T M ) = W; M

j

where for 1-forms W = Wj dx and = k dxk , we de…ne hW; i = g jk Wj k ; where (g jk ) is the inverse matrix to (gjk ); and = k dxk : (Above we use the standard Einstein summation convention). The notation k:kL2 ( 1 T M ) stands for the norm in L2 1 T M corresponding to the inner product (2). In what follows, by C 1 (M ) we denote the space of smooth functions on M; by Cc1 (M ) the space of smooth compactly supported functions on M , by 1 (M ) the space of smooth 1-forms on M and by 1c (M ) the space of smooth compactly supported 1-forms on M . In the sequel, the operator d : C 1 (M ) ! 1 (M ) is the standard di¤erential and d : 1 (M ) ! C 1 (M ) is the formal adjoint of d de…ned by the identity: (du; w)L2 ( 1 T M ) = (u; d w) ; u 2 Cc1 (M ) and w 2 1 (M ) : By M = d d we will denote the scalar laplacian on M: We consider the second order elliptic di¤erential expression: @2 + q; (3) @xi2 with the real-valued positive continuous functions bi ; where 4M is the scalar Laplacian on M; 0 q 2 L2loc (M ) is a real-valued function and the ex@2 pression bi @xi2 makes sense in a coordinate neighborhood U with coordinates x1 ; x2 ; : : : ; xn . E = 4M + bi

De…nition 1 The set D1 : Let E be as in (3), we will use the notation D1 = fu 2 L2 (M ) : Eu 2 L2 (M )g:

(4)

2

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Remark 2 In general, it is not true that for all u 2 D1 we have M u 2 L2 (M ) ; @2u bi @x and qu 2 L2 (M ) separately. Using the terminology of Everitt and Giertz i2 2

@ [9], we will say that the di¤ erential expression E = 4M + bi @x + q is separated i2 2 in L (M ) when the following statement holds true: for all u 2 D1 we have @2u bi @x and qu 2 L2 (M ) : i2

2

The Main Result We now state the main result.

Theorem 3 Assume that there exists a function 0 v(x) and jdv(x)j

q(x)

v 2 C 1 (M ) such that

cv(x)

3

v 2 (x); 8x 2 M ;

(5) (6)

where c>0 and 0 0 be arbitrary, By the de…nition of Ev ; for

@2u @2u + vu; 4 u + b + vu M i @xi2 @xi2 @2u @2u ; Mu = ( M u; M u) + + ( u; vu) + b M u; bi M i @xi2 @xi2 @2u @2u @2u @2u + bi i2 ; bi i2 + bi i2 ; vu + (vu; M u) + vu; bi i2 @x @x @x @x

2

kEv uk

=

4M u + bi

M u; vu) + (vu; 2

@ u 2 Re bi @x ; vu i2

kEv uk

+

@2u ; vu ; @xi2

for every u 2 Cc1 (M ) : Let every u 2 Cc1 (M ) ; we have

since (

>0:

2

2

2

M u)

2

= kvuk + k

2

2

2

2

@ u ; + bi @x i2

M uk + bi

@2u @xi2

M u; vu)

= kvuk + bi +2 Re (

2

M u; vu)

= kvuk + bi +2 Re (

M u; vu) ;

@ u M u; bi @xi2

and

+2 Re (

= 2 Re (

@2u @xi2

M u; vu)

@2u @xi2

+ 2 Re bi

2

2

@ u @ u ; vu + vu; bi @x bi @x i2 i2 Mu

= 2 Re

2

@ u M u; bi @xi2

+ (vu; vu) ;

= : Then

2

@2u ; vu + 2 Re @xi2

M u; bi

@2u @xi2

2

+

k

M uk

+ 2 Re bi

2

+ (1

)k

M uk

@2u ; vu + 2 Re @xi2

2

M u; bi

@2u @xi2

2

+

k

M uk

+ 2 Re bi

2

+ (1

) Re (

@2u ; vu + 2 Re @xi2

M u;

M u)

M u; bi

@2u @xi2

4

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2

= kvuk + bi +2 Re (

2

@2u @xi2

+

M u; vu)

+ 2 Re bi

+ (1

) Re

M u; Ev u

@2u ; vu + 2 Re @xi2

M u; bi

bi

@2u @xi2

vu

@2u @xi2

2

@2u = kvuk + bi i2 @x 2

+(1 + ) Re

2 M uk

k

+

2 M uk

k

M u; bi

@2u @xi2

+ (1

) Re (

+ (1 + ) Re (

M u; Ev u)

M u; vu)

+ 2 Re bi

@2u ; vu ; @xi2

where(:; :) is as in (1) and k:k is the corresponding norm in L2 (M ) : Since u 2 Cc1 (M ) ; using integration by parts and the product rule, we have Re (

M u; vu)

=

Re (d du; vu) = Re (du; d (vu))L2 (

=

Re(du; vdu)L2 (

=

Re z + W

where z=

Z

1T

M)

1T

M)

+ Re (du; udv)L2 (

1T

M)

(11)

hdu; u (dv)i d ;

(12)

M

and

1

1

W = v 2 du; v 2 du

L2 (

1T

M)

;

(13)

from (11) we get (1 + ) Re (

M u; vu)

= (1 + ) Re z + (1 + )W

(1 + ) jzj + (1 + )W: (14)

We will now estimate jzj ; where z is as in (12). Using Cauchy Schwarz inequality and the inequality 2ab

ka2 + k

1 2

b ;

(15)

where a; b and k are positive real numbers, we get for anyZ > 0 : jzj Z Z 3 1 jduj jdvj juj d ; from (6), we get jzj v 2 jduj juj d = v 2 du jvuj d M 2

M

1 2

2

v du

(1+ ) 2

2

L2 (

1T

M)

+2

2

2

kvuk ; then

M

1

(1+ ) 2

(1+ ) jzj

v 2 du

2 L2 (

1T

M)

2

kvuk ; so from (14), we obtain

(1 + ) Re (

M u; vu)

(1 + ) 2

2

1

v 2 du 1

1T

M)

2

kvuk

2

2

+(1 + ) v 2 du +(1 + ) Re (

L2 (

(1 + ) 2

L2 (

1T

M u; vu)

M)

+ (1 + ) Re

+ 2 Re bi

M u; bi

@2u ; vu ; @xi2

@2u @xi2 (16)

5

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Using (15), for any 2 1 2 kEv uk ; so (1

) jRe (

> 0 we get jRe (

k

2 M uk

+

1 2

(1 + ) Re

)

k

2

j(

M uk

2

M u; Ev u)j

(1

+

)

2

k

M uk

2

kEv uk :

2

@2u M u; bi @xi2

> 0; we get Re

2

+

(17)

@2u M u; bi @xi2

2

2

@ u bi @x i2 M u; bi

and for any

(1

M u; Ev u)j

Similarly, for any 2

M u; Ev u)j

; so @2u @xi2

(1 + ) k 2

M uk

2

+

2

@2u (1 + ) bi i2 2 @x

; (18)

> 0 we get Re bi

@2u ; vu @xi2

2

2

@2u @xi2

bi

1 2 kvuk : 2

(19)

Combining (16), (17), (18) and (19), we obtain kEv uk

2

2

kvuk + bi

@2u @xi2

(1 + ) k 2

bi

@ u @xi2

+

M uk

2

k

M uk

j1

j

2 2

2

j1

2

) v 2 du (1 + ) 2

L2 (

2

j

k

M uk

2

2

kEv uk

@2u (1 + ) b i i2 2 @x

2

1

+(1 + )(1 2

2

1T

2

M)

1

2

kvuk

2

kvuk :

2

(20)

From (20), we get 1+

j1

2

j

kEv uk

2

1

1 +

(1 + ) 2 j1 j 2

+ (1 + )(1 + j1

j

2

2

kvuk

2

(1 + ) 2 )

(1 + ) 2

2

1

v 2 du bi

2 M uk

k L2 (

@2u @xi2

1T

M)

2

:

(21)

The inequalities (8) and (9) will immediately follow from (21) if (1+ )

2

0; > 0; > 0; > 0 and > 1 such that the inequalities (22) hold. This concludes the proof of the lemma. 6

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In the sequel, we will also use the following terms and notations. De…nition 7 Minimal operators S and T : Let E be as in (3) and Ev be as in def 5. We de…ne the minimal operators S and T in L2 (M ) associated to E and Ev by the formulas Su = Eu and T u = Ev u with domains Dom(S)=Dom(T ) =Cc1 (M ). Since S and T are symmetric operators, it follows that S and T is closable, see for example, section fand T f the closures in L2 (M ) V.3.3 in [9] : In what follows, we will denote by S of the operators S and T respectively.

De…nition 8 Maximal operators H and K: Let E be as in (3). We de…ne the maximal operator H in L2 (M ) associated to E by the formula Hu = S u, where S is the adjoint of the operator S in L2 (M ) : In the case when q 2 L2loc (M ) is real-valued, it is well known that Dom(H) = D1 , where D1 is as in (4): Let Ev be as in def. 5, we de…ne the maximal operator K in L2 (M ) associated to Ev by the formula Ku = T u, and we have Dom(K) = fu 2 L2 (M ) : Ev u 2 L2 (M )g: Remark 9 By lemma 5.1 from [17] it follows that Dom(K)

2;2 Wloc (M ):

De…nition 10 Essential self-adjointness of S and T : If (M,g) is a complete Riemannian manifold with metric g and positive smooth measure d and if 0 q 2 L2loc (M ) ; then the operator S is essentially self-adjoint in L2 (M ) (see [5], [10] and [11]). In this case we have Se = S . In particular, since 0 v 2 C 1 (M ) L2loc (M ), it follows that the operator T is 2 essentially self-adjoint in L (M ) : Lemma 11 Assume that (M,g) is connected C 1 Riemannian manifold without boundary, with metric g and positive smooth measure d : Additionally, assume that (M,g) is complete. Assume that 0 v 2 C 1 (M ) satis…es (5). Then the inequalities (8) and (9) hold for all u 2 Dom(K), where K is as in de…nition 8. Proof. Under the hypotheses of this lemma, by de…nition 10 it follows that T is essentially self-adjoint and K = T = Te. In particular, Dom(K) = Dom Te : Let u 2 Dom(K): T hen 9 seq fuk g 2 Cc1 (M ) such that uk ! u and Ev uk ! Teu: Since by lemma (6) the seq fuk g satis…es (8) and (9), it follows that the sequences f 1

M uk g;

2

@ u gand fvuk g are Cauchy sequences fbi @x i2

in L2 (M ) and fv 2 duk g is a Cauchy seq. in L2

1

T M : We will show that

vuk ! vu in L2 (M ) as k ! 1:

(23)

Since fvuk g is a Cauchy seq. in L2 (M ) ; it follows that fvuk g converges to s 2 L2 (M ) : Let be an arbitrary element of Cc1 (M ), then 0 = (vuk ; )

(uk ; v ) ! (s; )

(u; v ) = (s

vu; ) ;

(24)

7

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where (:; :) is as in (1), since Cc1 (M ) is dense in L2 (M ), we get s = vu and (23) is proven. If (M; g) is complete, it is well known that M is essentially self adjoint on Cc1 (M ) and we have the following equality ( M jCc1 (M ) ) = M;max where M;max u = M u with the domain Dom( M;max ) = fu 2 L2 (M ) : M u 2 L2 (M )g; see for example, theorem 3.5 in [8] : Since uk ! u in L2 (M ) and since f M uk g is a Cauchy seq. in L2 (M ) ; by the de…nition of (

M

(

=

M jCc1 (M ) )

jCc1 (M ) )

M;max u; M uk

it follows that u 2 Dom((

M

jCc1 (M ) ) ): Since

we have

!

Mu

in L2 (M ) as k ! 1:

(25) 2 kduk kL2 (

Since f M uk g and fuk g are Cauchy seqs in L2 (M ) ; and since 1T M ) = (duk ; duk )L2 ( 1 T M ) = ( M uk ; uk ) k M uk k kuk k ;it follows that fduk g is a Cauchy seq. in L2 1 T M ; and hence, duk converges to some element w 2 L2 1 T M : Let 2 1c (M ) be arbitrary, then by integration by parts (see, for example, lemma 8.8 in [5]) and remark 9, we get 0

=

(duk ; )L2 (

! (w; )L2 ( =

(w; )L2 (

1T

M)

(uk ; d

1T

M)

(u; d

1T

M)

(du; )L2 (

)

) 1T

M );

(26)

where (:; :) is the inner product in L2 (M ) : From (26) we get du = w L2 1 T M ; and hence duk ! du = w in L2

1

1

Since fv 2 duk g is a Cauchy seq. in L2 1 2

1

1 2

v duk ! v du in L2

T M ; as k ! 1:

2

(27)

T M ; using (27) and (24), we obtain 1

T M

as k ! 1:

(28)

2

Since uk ! u in L2 (M ) and since fbi @@xui2k g is a Cauchy seq. in L2 (M ) ; then we get @ 2 uk @2u bi i 2 ! bi i 2 : (29) @x @x Using (23) ,(25), (28) and (29) and taking limits as k ! 1 in all terms in (8) and (9) (with u replaced by uk ), shows that (8) and (9) hold for all u 2 Dom(K) = Dom(Te): This concludes the proof of the lemma. De…nition 12 Operators R1 and R2 Let S and T be as in def. 7 and let H and K as in def. 8. By def. 10 the operators H = Se and K = Te are non-negative self adjoint in L2 (M ) : Thus, R1 = (H + 1)

and R2 = (K + 1) are bounded linear operators.

1

1

: L2 (M ) ! L2 (M )

: L2 (M ) ! L2 (M ) ;

(30)

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De…nition 13 Positivity preserving property Let (X; ) be a measure space. A bounded linear operator A : L2 (X; ) ! 2 L (X; ) is said to be positivity preserving if every u 2 L2 (X; ) such that u 0 on X, we have Au 0 on X. Remark 14 Let A : L2 (X; ) ! L2 (X; ) be positivity preserving bounded linear operator. Then the following inequality holds for all u 2 L2 (X; ) j(Au)(x)j

A ju(x)j

on X;

(31)

where j:j denotes the absolute value of a complex number. For the proof of (31), see the proof of the inequality (X:103) in [13]. In the sequel, we will use the following proposition. Proposition 15 Assume that (M; g) is (not necessarily complete ) C 1 Riemannian manifold without boundary. Assume that M is connected and oriented. Assume that Q0 2 L2loc (M ) is real valued. Additionally assume that 2

M

sion of

@ + Q0 u; u + bi @x i2

0 8u 2 Cc1 (M ) : Let S0 be the friedrichs exten-

2

M

@ + bi @x + Q0 jCc1 (M ): Assume that i2

Then the operator (S0 + )

1

is a positive real number.

is positivity preserving. (See [12]).

From proposition 15 we obtain the following corollary. Corollary 16 Assume that (M; g) is a complete C 1 Riemannian manifold without boundary. Assume that M is connected and oriented. Assume that 0 q 2 L2loc (M ) and 0 v 2 C 1 (M ). Let R1 and R2 be as in (30). Then the operators R1 and R2 are positivity preserving. Proof. Since (M; g) is complete, by def. 10; it follows that S and T are non-negative essentially self-adjoint operators in L2 (M ). Thus the friedrichs @2 @2 extensions of M + bi @xi2 + v jCc1 (M ) are H M + bi @xi2 + q jCc1 (M ) and and K, respectively. By proposition (15) it follows that R1 and R2 are positivity preserving operators in L2 (M ) : Now we introduce the proof of our main theorem. Proof. of Theorem (3) : By lemma 11 it follows that kvuk

c kKuk

8u 2 Dom (K) :

(32)

Let R2 be as in (30), and let v : L2 (M ) ! L2 (M ) ;denote the maximal multiplication operator corresponding to the function v: From (32) it follows that vR2 : L2 (M ) ! L2 (M ) is a bounded linear operator. Let q : L2 (M ) ! L2 (M ) ;denote the maximal multiplication operator corresponding to the function q. Since vR2 : L2 (M ) ! L2 (M ) ;is a bounded linear operator, by (5) it follows that qR2 : L2 (M ) ! L2 (M ) ; is a bounded linear operator. We will 9

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now show that qR1 is a bounded linear operator: L2 (M ) ! L2 (M ) : First we will show that qR1 f

qR2 f

f or every 0

f 2 L2 (M ) :

(33)

Since R1 f 2 D1 L2 (M ) and since q 2 L2loc (M ) ; it follows that qR1 f 2 1 Lloc (M ) is a function. Let 0 f 2 L2 (M ) be arbitrary. Since R2 f 2 2 2 Dom (K) = fz 2 L (M ) : Ev z 2 L (M )g; by lemma (11) we have ( M (R2 f )) ; 2 R2 f and (v (R2 f )) 2 L2 (M ) : Since qR2 : L2 (M ) ! L2 (M ) is a bounded bi @ @x i2 linear operator, we get (qR2 f ) 2 L2 (M ) : Thus R2 f 2 D1 ; and hence qR2 f = qR1 (H + 1) R2 f :

(34)

By the de…nition of R2 we have qR1 f = qR1 (K + 1) R2 f :

(35)

From (34)and (35) we have qR2 f

qR1 f

= qR1 ((H + 1) R2 f (K + 1) R2 f ) = qR1 ((H K) R2 f ) = qR1 (q v) R2 f :

(36)

Since R2 is positivity preserving in L2 (M ) ; we have R2 f 0: Since vR2 and qR2 are bounded linear operators L2 (M ) ! L2 (M ) and since q v 0; then 0

(q

v) R2 f 2 L2 (M ) :

(37)

Since R1 is positivity preserving in L2 (M ) ; from (37) we get R1 ((q

v) R2 f )

0:

(38)

Since q 0, from (36) and (38) we get qR2 f qR1 f = qR1 (q v) R2 f 0; and (33) is proven. Let w 2 L2 (M ) be arbitrary. Since R1 : L2 (M ) ! L2 (M ) ; is positivity preserving bounded linear operator, by (31) we have the following inequality: jR1 wj R1 jwj : (39)

Since q

0; from (39) and (33) we obtain the inequality, jqR1 wj = q jR1 wj

qR1 jwj

qR2 jwj :

(40)

Since qR2 : L2 (M ) ! L2 (M ) ;is a bounded linear operator, from (40) it follows that, kqR1 wk kqR2 jwjk C3 kwk ; 8w 2 L2 (M ) ;where C3 is a constant. 2 Hence qR1 is a bounded linear operator L2 (M ) ! L2 (M ) : Consequently bi @@xRi21 is a bounded linear operator L2 (M ) ! L2 (M ) : Let u 2 D1 be arbitrary. Then @2u @2 qu = qR1 (u + Eu) ; and bi @x = bi @x R1 (u + Eu) where Eu = M u + i2 i2 2

2

@ u bi @x + qu: Since qR1 : L2 (M ) ! L2 (M ) and bi @@xRi21 : L2 (M ) ! L2 (M ) ;are i2 bounded linear operators, we have

kquk = kqR1 (u + Eu)k C1 ku + Euk C1 (kuk + kEuk) ;

8u 2 D1

(41)

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and bi

@2u @xi2

where C1 ; C2

=

@2 R1 (u + Eu) @xi2 C2 ku + Euk C2 (kuk + kEuk) ; bi

8u 2 D1

(42)

0 are constants. Using (41) and (42) we get k

M uk

=

@2u qu @xi2 @2u kEuk + bi i2 + kquk @x C3 (kuk + kEuk) ; 8u 2 D1 Eu

bi

(43)

where C3 0 is a constant. From (41), (42) and (43), we obtain (7). This concludes the proof of the Theorem. Acknowledgement 17 This paper was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah, under grant No. (16-662D1432). The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

References [1] H. A. Atia, "Separation problem for the Grushin Di¤erential Operator in Banach Spaces", Carpathian Journal of Mathematics 30 (1) (2014) 31-37. [2] H. A. Atia, R. S. Alsaedi and A. Ramady, "Separation of Bi-Harmonic Differential Operators on Riemannian Manifolds", Forum Math. 26 (3) (2014), 953–966. [3] H. A. Atia and R. A. Mahmoud, "Separation of the two dimensional Grushin operator by the disconjugacy property", Applicable Analysis: An In. J. 91 (12) (2012), 2133–2143. [4] K. Kh. Biomatov, Coercive estimates and separation for second order elliptic di¤erential equations, Sov. Math. Dokl. 38 (1) (1989). [5] M. Braverman, O. Milatovic, M. Shubin, Essential self-adjointness of Schrodinger type operators on manifolds, Russ. Math. Surv. 57 (4) (2002) 641-692. [6] R. C. Brown, ”Separation and disconjugacy” J. Inequal. Pure and Appl. Math., 4 (3) (2003) Art. 56. [7] W. A. Coppel, Disconjugacy, Lecture Notes in mathematics, Vol. 220 (A. D. old and B. Echman, Eds), Springer -Veriag, Berlin, Heidelbarg, and New York, 1971. 11

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[8] J. Eichhorn, Elliptic di¤erential operators on non compact manifolds, in: Proceedings of the seminar Analysis of the Karl-Weierstrass-Institute of Mathematics. Teubnet, Leipzig, Berlin, 1986/87, Teubner-Texte Math. 106. 1988, 4-169. [9] W. N. Everitt, M. Giertz, ”Some properties of the domains of certain differential operators”, Proc. London Math. Soc. 23 (1971), 301-324. [10] T. Kato, Perturbation Theory for linear operators, Springer Verlag, New York, 1980. [11] J. Masamune, "Essential self adjointness of Laplacians on Riemannian manifolds with fractal boundary", Communications in Partial Di¤erential Equations 24 (3-4) (1999) 749-757. [12] O. Milatovic, "The form sum and the Friedrichs extension of Schrodingertype operators on Riemannian manifolds", Pro. Am. Math. Soc. 132 (1) (2004) 147-156. [13] O. Milatovic, "Separation property for Schrodinger operators on Riemannian manifolds", Journal of Geometry and Physics 56 (2006) 12831293. [14] O. Milatovic, "A separation property for magnetic Schrodinger operators on Riemannian manifolds", Journal of Geometry and Physics 61 (2011) 1-7. [15] A. S. Mohamed, H. A. Atia, ”Separation of the Sturm-Liouville di¤erential operator with an operator potential”, Applied Mathematics and Computation. 156 (2) (2004), 387-394. [16] A. S. Mohamed, H. A. Atia, ”Separation of the Schrodinger operator with an operator potential in the Hilbert spaces”, Applicable Analysis. 84 (1) (2005) 103-110. [17] M. A. Shubin, "Essential self-adjointness for semi bounded magnetic Schrodinger operators on non-compact manifolds", J. Funct. Anal.186 (2001) 92-116. [18] E. M. E. Zayed, A. S. Mohamed and H. A. Atia, ”Inequalities and separation for the Laplace Beltrami di¤erential operator in Hilbert spaces”, J. Math. Anal. Appl. 336 (2007) 81-92.

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AN INTEGRAL-TYPE OPERATOR FROM WEIGHTED BERGMAN-PRIVALOV SPACES TO BLOCH-TYPE SPACES YONG REN

Abstract Let H(B) denote the space of all holomorphic functions on the unit ball B of Cn . Let ϕ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we study the following integral-type operator, recently introduced by Stevi´c in [16] Z 1 dt Pϕg f (z) = f (ϕ(tz))g(tz) , f ∈ H(B). t 0 The boundedness and compactness of the operator Pϕg from weighted Bergman-Privalov spaces to Bloch-type spaces in the unit ball are studied.

1

Introduction

Let B be the unit ball of Cn , ∂B its boundary, dσ the normalized surface measure on ∂B and dv be the normalized Lebesgue measure on B. We denote by H(B) the class of all holomorphic functions on B. H(B) is a Fr´echet space (locally convex, metrizable and complete) with respect to the compact-open topology. By Montel’s theorem, bounded sets in H(B) are relatively compact and hence bounded sequences in H(B) admit convergent subsequences. Convergence in this space will be referred to as local uniform (l.u.) convergence. A positive continuous function µ on the interval [0, 1) is called normal if there is a δ ∈ [0, 1) and s and t, 0 < s < t such that (see, e.g. [13]) µ(r) µ(r) is decreasing on [δ, 1) and lim = 0; s r→1 (1 − r)s (1 − r) µ(r) µ(r) is increasing on [δ, 1) and lim = ∞. t r→1 (1 − r)t (1 − r) Let p ∈ [1, ∞), α > −1. An f ∈ H(B) is said to belong to the weighted Bergman-Privalov space, denoted by AN pα = AN pα (B), if Z £ ¤p kf kpAN pα = log(1 + |f (z)|) dvα (z) < ∞, B

2000 Mathematics Subject Classification. Primary 47B38, Secondary 30D45. Key words and phrases. Integral-type operator, weighted Bergman-Privalov space, Blochtype space, boundedness, compactness.

1

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where dvα (z) = cα (1 − |z|2 )α dv(z) and cα is chosen such that vα (B) = 1. For each f ∈ AN pα the following inequality holds ([12]) log(1 + |f (z)|) ≤

(1 + |z|)(n+1+α)/p kf kAN pα . (1 − |z|)(n+1+α)/p

(1)

Hence the topology of AN pα is stronger than that of locally uniform convergence. Let µ be a normal function on [0, 1). An f ∈ H(B) is said to belong to the Bloch-type space, denoted by Bµ = Bµ (B), if bµ (f ) := sup µ(|z|) | =

sup kPϕg fw kBµ ≥ kPϕg fw kBµ w∈B |(Pϕg fw )(0)| + sup µ(|z|)| −1, g ∈ H(B), g(0) = 0, µ be a normal function on [0, 1) and ϕ be a holomorphic self-map of B. Then Pϕg : AN pα → Bµ is compact if and only if kgkµ := sup µ(|z|)|g(z)| < ∞

(7)

z∈B

and for every c > 0, h lim

|ϕ(z)|→1

µ(|z|)|g(z)| exp

i

c (1 − |ϕ(z)|2 )

n+1+α p

= 0.

(8)

Proof. Assume that (7) and (8) hold. From these conditions, we easily get (3). Hence Pϕg : AN pα → Bµ is bounded from Theorem 1. Let (fk )k∈N be a sequence in AN pα with supk∈N kfk kAN pα ≤ Q and that fk converges to 0 l.u.. By (8) we have that for every ε > 0 and c > 0, there is a δ ∈ (0, 1) such that i h c 0)

(5.2)

6. FFPDEs under Caputo-type H-differentiability In this section we show the notations and the basic fuzzy fractional partial differential equation associated with the wave reaction-diffusion systems. Let 𝑢(𝑥, 𝑡) be the concentration of a substance distributed in space one dimensional space and 𝜑(𝑥, 𝑡, 𝑢) be a non-linear function, where 𝑡 is the time-variable and 𝑥 is the space-variable. It is important to note that the function 𝜑(𝑥, 𝑡, 𝑢) can represent a term of the so-called Fisher–Kolmogorov equation or a term of the Ginzburg–Landau equation, which appear in field theory and superconductivity. In the first case, the non-linear term is 𝜑(𝑥, 𝑡, 𝑢) = 𝐴𝑢(𝑥, 𝑡)[1 − 𝑢(𝑥, 𝑡)] with 𝐴 a 2 constant, whereas in the second case we have 𝜑(𝑥, 𝑡, 𝑢) = 𝐴𝑢(𝑥, 𝑡) [1 − (𝑢(𝑥, 𝑡)) ] with 𝐵 another constant. Here we discuss only the case 𝜑(𝑥, 𝑡, 𝑢) ≡ 𝜑(𝑥, 𝑡), i.e. linear case, as in Eq. (6.1), [8]. The following fuzzy fractional partial differential equation, the so-called generalized wave reactiondiffusion equation, is considered: 𝛼

𝛼

𝛽

𝑎 ( 𝐶𝐷0+1,𝑡 𝑢) (𝑥, 𝑡) ⊕ 𝑏 ( 𝐶𝐷0+2,𝑡 𝑢) (𝑥, 𝑡) = 𝑐 ( 𝐶𝐷𝑥,−∞ 𝑢) (𝑥, 𝑡) ⊕ 𝜑(𝑥, 𝑡, 𝑢(𝑥, 𝑡)),

𝜕𝑘 𝑢(𝑥,0+ ) 𝜕𝑥 𝑘

= 𝑓𝑘 (𝑥), 𝑘 = 0,1, … , 𝑚 − 1,

𝜕𝑙 𝑢(𝑥,0+ ) 𝜕𝑥 𝑙

= 𝑔𝑙 (𝑥), 𝑙 = 0,1, … , 𝑛 − 1, 𝑙𝑖𝑚 𝑢(𝑥, 𝑡) = 0, 𝑥→±∞

(6.1) (6.2)

where 𝑡 > 0, 𝑥 ∈ ℝ, 𝑚 − 1 < 𝛼1 ≤ 𝑚, 𝑛 − 1 < 𝛼2 ≤ 𝑛, 0 ≤ 𝛽 ≤ 1 and 𝑎, 𝑏, 𝑐 are real constants.

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6.1. Determining algebraic solutions In this subsection, we provide the fuzzy Laplace transform and its inverse to derive solutions of FFPDE (6.1). By taking Laplace transform on the both sides of Eq. (6.1), we get the following: 𝛼

𝛼

𝛽

ℒ𝑡 {𝑎 ( 𝐶𝐷0+1,𝑡 𝑢) (𝑥, 𝑡) ⊕ 𝑏 ( 𝐶𝐷0+2,𝑡 𝑢) (𝑥, 𝑡)} = ℒ𝑡 {𝑐 ( 𝐶𝐷𝑥,−∞ 𝑢) (𝑥, 𝑡) ⊕ 𝜑(𝑥, 𝑡, 𝑢(𝑥, 𝑡))},

(6.3)

Then, based on the types of Caputo-type H-differentiability we have the following cases: Case I. Let us consider 𝐷𝑢(𝑥, 𝑡) is a 𝐶[(𝑖) − 𝛼]-differentiable function, then Eq. (6.3) is extended based on its lower and upper functions as following: 𝛼1 −𝑘−1 (𝑥; 𝛼2 −𝑙−1 (𝑥; 𝑎 (𝑠 𝛼1 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − ∑𝑚−1 𝑓𝑘 𝑟)) + 𝑏 (𝑠 𝛼2 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − ∑𝑛−1 𝑔𝑙 𝑟)) = 𝑘=0 𝑠 𝑙=0 𝑠 𝛽

𝑐 ( 𝐶𝐷𝑥,−∞ ℒ𝑡 {𝑢}) (𝑥, 𝑠; 𝑟) + ℒ𝑡 {𝜑} (𝑥, 𝑠, 𝑢(𝑥, 𝑡); 𝑟), (6.4) 𝛼1 −𝑘−1 𝛼2 −𝑙−1 (𝑥; 𝑎(𝑠 𝛼1 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − ∑𝑚−1 𝑓𝑘 (𝑥; 𝑟)) + 𝑏(𝑠 𝛼2 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − ∑𝑛−1 𝑔𝑙 𝑟)) = 𝑘=0 𝑠 𝑙=0 𝑠 𝛽

𝑐 ( 𝐶𝐷𝑥,−∞ ℒ𝑡 {𝑢}) (𝑥, 𝑠; 𝑟) + ℒ𝑡 {𝜑}(𝑥, 𝑠, 𝑢(𝑥, 𝑡); 𝑟), where 𝜑(𝑥, 𝑡, 𝑢(𝑥, 𝑡); 𝑟) = min{𝜑(𝑥, 𝑡, 𝑢)|𝑢 ∈ [𝑢(𝑥, 𝑡; 𝑟), 𝑢(𝑥, 𝑡; 𝑟)]} , 0 ≤ 𝑟 ≤ 1, 𝜑(𝑥, 𝑡, 𝑢(𝑥, 𝑡); 𝑟) = 𝑚𝑎𝑥{𝜑(𝑥, 𝑡, 𝑢)|𝑢 ∈ [𝑢(𝑥, 𝑡; 𝑟), 𝑢(𝑥, 𝑡; 𝑟)]} , 0 ≤ 𝑟 ≤ 1. Taking the Fourier transform in Eq. (6.4), and in order to solve the linear system (6.3), for simplify we assume that: ℱ𝑥 {ℒ𝑡 {𝑢(𝑥, 𝑡; 𝑟)}} = 𝐻1 (𝜔, 𝑠; 𝑟), 0 ≤ 𝑟 ≤ 1 , { ℱ𝑥 {ℒ𝑡 {𝑢(𝑥, 𝑡; 𝑟)}} = 𝐾1 (𝜔, 𝑠; 𝑟), 0 ≤ 𝑟 ≤ 1 where 𝐻1 (𝜔, 𝑠; 𝑟) and 𝐾1 (𝜔, 𝑠; 𝑟) are solutions of system (6.4). Then, by using the inverse Laplace transform and the inverse Fourier transform, 𝑢(𝑥, 𝑡; 𝑟) and 𝑢(𝑥, 𝑡; 𝑟) are computed as: 𝑢(𝑥, 𝑡; 𝑟) = ℱ𝑥 {ℒ𝑡−1 {𝐻1 (𝜔, 𝑠; 𝑟)}} =

∞ 1 𝑢(𝑥, 0; 𝑟) ∫−∞ 𝐺1 (𝜔, 𝑡)𝑑𝜔 2𝜋

𝜏; 𝑟)𝑑𝜏 𝑑𝜔,

1

∞

𝑡

+ 2𝜋 ∫−∞ ∫0 𝐺1 (𝜔, 𝑡; 𝜏)𝜑(𝜔, 𝑡 − (6.5)

∞

1

1

∞

𝑡

𝑢(𝑥, 𝑡; 𝑟) = ℱ𝑥 {ℒ𝑡−1 {𝐾1 (𝜔, 𝑠; 𝑟)}} = 2𝜋 𝑢(𝑥, 0; 𝑟) ∫−∞ 𝐺1 (𝜔, 𝑡)𝑑𝜔 + 2𝜋 ∫−∞ ∫0 𝐺1 (𝜔, 𝑡; 𝜏)𝜑(𝜔, 𝑡 − 𝜏; 𝑟)𝑑𝜏 𝑑𝜔. These two cases can be considered as particular cases of Eqs. (6.5) A. Classical Green’s function In this case the homogeneous initial condition is 𝑓(𝑥) = 0̃. Substituting this into Eqs. (6.5) 𝑟

1

∞

𝑟

𝑡

[𝑢(𝑥, 𝑡)]𝑟 = (𝐺1 (𝑥, 𝑡|0,0)) = ∫−∞ ∫0 𝐺1 (𝜔, 𝑡; 𝜏) [𝜑(𝜔, 𝑡 − 𝜏), 𝜑(𝜔, 𝑡 − 𝜏)] 𝑑𝜏 𝑑𝜔, 2𝜋 265

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B. The propagator Substituting 𝜑(𝑥, 𝑡; 𝑟) = 0̃ and 𝑓(𝑥; 𝑟) = 𝛿(𝑥; 𝑟) into Eqs. (6.5), we can write 𝑢(𝑥, 𝑡; 𝑟) =

∞ 1 𝑢(𝑥, 0; 𝑟) ∫−∞ 𝐺1 (𝜔, 𝑡)𝑑𝜔, 2𝜋 ∞

1

𝑢(𝑥, 𝑡; 𝑟) = 2𝜋 𝑢(𝑥, 0; 𝑟) ∫−∞ 𝐺1 (𝜔, 𝑡)𝑑𝜔. Case II. Let us consider 𝐷𝑢(𝑥, 𝑡) is a 𝐶[(𝑖) − 𝛼] -differentiable function, then Eq. (6.1) is extended based on its lower and upper functions as following: 𝛼1 −𝑘−1 (𝑥; 𝛼2 −𝑙−1 (𝑥; 𝑓𝑘 𝑟)) + 𝑏 (𝑠 𝛼2 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − ∑𝑛−1 𝑔𝑙 𝑟)) = 𝑎 (𝑠 𝛼1 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − ∑𝑚−1 𝑘=0 𝑠 𝑙=0 𝑠 𝛽

𝑐 ( 𝐶𝐷𝑥,−∞ ℒ𝑡 {𝑢}) (𝑥, 𝑠; 𝑟) + ℒ𝑡 {𝜑} (𝑥, 𝑠, 𝑢(𝑥, 𝑡); 𝑟), (6.6) 𝛼1 −𝑘−1 𝛼2 −𝑙−1 (𝑥; 𝑎(𝑠 𝛼1 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − ∑𝑚−1 𝑓𝑘 (𝑥; 𝑟)) + 𝑏(𝑠 𝛼2 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − ∑𝑛−1 𝑔𝑙 𝑟)) = 𝑘=0 𝑠 𝑙=0 𝑠 𝛽

𝑐 ( 𝐶𝐷𝑥,−∞ ℒ𝑡 {𝑢}) (𝑥, 𝑠; 𝑟) + ℒ𝑡 {𝜑}(𝑥, 𝑠, 𝑢(𝑥, 𝑡); 𝑟), where 𝜑(𝑥, 𝑡, 𝑢(𝑥, 𝑡); 𝑟) = min{𝜑(𝑥, 𝑡, 𝑢)|𝑢 ∈ [𝑢(𝑥, 𝑡; 𝑟), 𝑢(𝑥, 𝑡; 𝑟)]} , 0 ≤ 𝑟 ≤ 1, 𝜑(𝑥, 𝑡, 𝑢(𝑥, 𝑡); 𝑟) = 𝑚𝑎𝑥{𝜑(𝑥, 𝑡, 𝑢)|𝑢 ∈ [𝑢(𝑥, 𝑡; 𝑟), 𝑢(𝑥, 𝑡; 𝑟)]} , 0 ≤ 𝑟 ≤ 1. Taking the Fourier transform in Eq. (6.6), and in order to solve the linear system (6.6), for simplify we assume that: ℱ𝑥 {ℒ𝑡 {𝑢(𝑥, 𝑡; 𝑟)}} = 𝐻2 (𝜔, 𝑠; 𝑟), 0 ≤ 𝑟 ≤ 1 , { ℱ𝑥 {ℒ𝑡 {𝑢(𝑥, 𝑡; 𝑟)}} = 𝐾2 (𝜔, 𝑠; 𝑟), 0 ≤ 𝑟 ≤ 1 where 𝐻2 (𝜔, 𝑠; 𝑟) and 𝐾2 (𝜔, 𝑠; 𝑟) are solutions of system (6.6). Then, by using the inverse Laplace transform and the inverse Fourier transform, 𝑢(𝑥, 𝑡; 𝑟) and 𝑢(𝑥, 𝑡; 𝑟) are computed as: 𝑢(𝑥, 𝑡; 𝑟) = ℱ𝑥 {ℒ𝑡−1 {𝐻2 (𝜔, 𝑠; 𝑟)}} =

∞ 1 𝑢(𝑥, 0; 𝑟) ∫−∞ 𝐺1 (𝜔, 𝑡)𝑑𝜔 2𝜋

𝜏; 𝑟)𝑑𝜏 𝑑𝜔,

1

∞

𝑡

+ 2𝜋 ∫−∞ ∫0 𝐺1 (𝜔, 𝑡; 𝜏)𝜑(𝜔, 𝑡 − (6.7)

𝑢(𝑥, 𝑡; 𝑟) = ℱ𝑥 {ℒ𝑡−1 {𝐾2 (𝜔, 𝑠; 𝑟)}} =

∞ 1 𝑢(𝑥, 0; 𝑟) ∫−∞ 𝐺1 (𝜔, 𝑡)𝑑𝜔 2𝜋

𝜏; 𝑟)𝑑𝜏 𝑑𝜔.

+

1 ∞ 𝑡 ∫ ∫ 𝐺 (𝜔, 𝑡; 𝜏)𝜑(𝜔, 𝑡 2𝜋 −∞ 0 1

−

These two cases can be considered as particular cases of Eqs. (6.7) A. Classical Green’s function In this case the homogeneous initial condition is 𝑓(𝑥) = 0̃. Substituting this into Eqs. (6.7) 𝑟

1

∞

𝑟

𝑡

[𝑢(𝑥, 𝑡)]𝑟 = (𝐺2 (𝑥, 𝑡|0,0)) = ∫−∞ ∫0 𝐺2 (𝜔, 𝑡; 𝜏) [𝜑(𝜔, 𝑡 − 𝜏), 𝜑(𝜔, 𝑡 − 𝜏)] 𝑑𝜏 𝑑𝜔, 2𝜋

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B. The propagator Substituting 𝜑(𝑥, 𝑡; 𝑟) = 0̃ and 𝑓(𝑥; 𝑟) = 𝛿(𝑥; 𝑟) into Eqs. (6.7), we can write 𝑢(𝑥, 𝑡; 𝑟) =

∞ 1 𝑢(𝑥, 0; 𝑟) ∫−∞ 𝐺2 (𝜔, 𝑡)𝑑𝜔, 2𝜋 ∞

1

𝑢(𝑥, 𝑡; 𝑟) = 2𝜋 𝑢(𝑥, 0; 𝑟) ∫−∞ 𝐺2 (𝜔, 𝑡)𝑑𝜔.

7. Examples In this section, we handle two examples with details to solve FFPDEs under Caputo-type Hdifferentiability. Example 7.1 (Fuzzy fractional reaction-diffusion equation). Let us consider the following FFPDE: {

2𝛽

𝐶 𝛼 2𝐶 𝐷02𝛼 𝐷𝑥,−∞ 𝑢(𝑥, 𝑡) + 𝜉 2 𝑢(𝑥, 𝑡) + 𝜑(𝑥, 𝑡) + ,𝑡 𝑢(𝑥, 𝑡) + 2𝜆 𝐷0+ ,𝑡 𝑢(𝑥, 𝑡) = 𝜈 , 𝑢(𝑥, 0) = 0 𝑎𝑛𝑑 𝑢𝑡 (𝑥, 0) = 0 ∈ 𝔼

𝐶

(7.1)

where 𝑡 > 0 and −∞ < 𝑥 < +∞, 1/2 < 𝛼 ≤ 1, 0 < 𝛽 ≤ 1. Applying Laplace transform on the both sides of above equation, we obtain: 2𝛽

𝐶 𝛼 2𝐶 ℒ𝑡 { 𝐶𝐷02𝛼 𝐷𝑥,−∞ 𝑢(𝑥, 𝑡) + 𝜉 2 𝑢(𝑥, 𝑡) + 𝜑(𝑥, 𝑡)}, + ,𝑡 𝑢(𝑥, 𝑡) + 2𝜆 𝐷0+ ,𝑡 𝑢(𝑥, 𝑡)} = ℒ 𝑡 {𝜈

Using 𝐶[(𝑖) − 𝛼]−differentiability, we get 2𝛽

𝜈 2 𝐶𝐷𝑥,−∞ ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) + 𝜉 2 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) + ℒ𝑡 {𝜑}(𝑥, 𝑠; 𝑟) = 𝑠 2𝛼 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − 𝑠 2𝛼−1 𝑓0 (𝑥; 𝑟) + 2𝜆𝑠 𝛼 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − 2𝜆𝑠 𝛼−1 𝑓0 (𝑥; 𝑟), (7.2) 2𝛽

𝜈 2 𝐶𝐷𝑥,−∞ ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) + 𝜉 2 ℒ𝑡 {𝑢}(𝑥𝑠; 𝑟) + ℒ𝑡 {𝜑}(𝑥, 𝑠; 𝑟) = 𝑠 2𝛼 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − 𝑠 2𝛼−1 𝑓0 (𝑥; 𝑟) + 2𝜆𝑠 𝛼 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − 2𝜆𝑠 𝛼−1 𝑓0 (𝑥; 𝑟), Taking the Fourier transform in Eq. (7.2), and the Fourier transform of the fractional derivative 𝜇

we get

ℱ𝑥 { 𝐶𝐷𝑥,−∞ 𝑢(𝑥, 𝑡)} = −|𝜔|𝜇 ℱ𝑥 {𝑢(𝜔, 𝑡)}

(𝑠 2𝛼 + 2𝜆𝑠 𝛼 − 𝜈 2 |𝜔|2𝛽 − 𝜉 2 )ℱ𝑥 {ℒ𝑡 {𝑢}} (𝜔, 𝑠; 𝑟) = 𝑠 2𝛼−1 ℱ𝑥 {𝑓0 } (𝜔; 𝑟) + 2𝜆𝑠 𝛼−1 ℱ𝑥 {𝑓0 } (𝜔; 𝑟)+ℱ𝑥 {ℒ𝑡 {𝜑}} (𝜔, 𝑠; 𝑟), (7.3) (𝑠 2𝛼 + 2𝜆𝑠 𝛼 − 𝜈 2 |𝜔|2𝛽 − 𝜉 2 )ℱ𝑥 {ℒ𝑡 {𝑢}}(𝜔, 𝑠; 𝑟) = 𝑠 2𝛼−1 ℱ𝑥 {𝑓0 }(𝜔; 𝑟) + 2𝜆𝑠 𝛼−1 ℱ𝑥 {𝑓0 }(𝜔; 𝑟)+ℱ𝑥 {ℒ𝑡 {𝜑}}(𝜔, 𝑠; 𝑟), Applying inverse of Laplace transform on the both sides of Eq. (7.3), we get the following: 267

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ℱ𝑥 {ℒ𝑡 {𝑢}} (𝜔, 𝑠; 𝑟) = { ℱ𝑥 {ℒ𝑡 {𝑢}}(𝜔, 𝑠; 𝑟) =

𝑠2𝛼−1 ℱ𝑥 {𝑓0 }(𝜔;𝑟)

+

𝑠2𝛼 +2𝜆𝑠𝛼 +𝛬(𝜔) 𝑠2𝛼−1 ℱ𝑥 {𝑓0 }(𝜔;𝑟) 𝑠2𝛼 +2𝜆𝑠𝛼 +𝛬(𝜔)

+

2𝜆𝑠𝛼−1 ℱ𝑥 {𝑓0 }(𝜔;𝑟) 𝑠2𝛼 +2𝜆𝑠𝛼 +𝛬(𝜔) 2𝜆𝑠𝛼−1 ℱ𝑥 {𝑓0 }(𝜔;𝑟) 𝑠2𝛼 +2𝜆𝑠𝛼 +𝛬(𝜔)

ℱ𝑥 {ℒ 𝑡 {𝜑}}(𝜔,𝑠;𝑟)

+ 𝑠2𝛼 +2𝜆𝑠𝛼 +𝛬(𝜔) +

ℱ𝑥 {ℒ 𝑡 {𝜑}}(𝜔,𝑠;𝑟)

,

𝑠2𝛼 +2𝜆𝑠𝛼 +𝛬(𝜔)

where we have put Λ(𝜔) = −𝜈 2 |𝜔|2𝛽 − 𝜉 2 . Taking the inverse fuzzy Laplace transform and inverse fuzzy Fourier transform, we get ∞

1

(𝑖+1)

−𝑖𝜔𝑥 𝑖 𝛼𝑖 (𝐸2𝛼,𝛼𝑖+1 (−𝛬(𝜔)𝑡 2𝛼 ) + 𝑢(𝑥, 𝑡; 𝑟) = 2𝜋 𝑓0 (𝑥; 𝑟) ∑∞ 𝑖=0(−2𝜆) 𝑡 ∫−∞ 𝑒 (𝑖+1)

2𝜆𝑡 𝛼 𝐸2𝛼,𝛼+𝛼𝑖+1 (−𝛬(𝜔)𝑡 2𝛼 )) 𝑑𝜔 + ∞ 𝑡 1 (𝑖+1) ∑∞ (−2𝜆)𝑖 ∫0 𝜉 𝛼𝑖+(2𝛼−1) ∫−∞ 𝑒 −𝑖𝜔𝑥 𝐸2𝛼,2𝛼+𝛼𝑖 (−𝛬(𝜔)𝜉 2𝛼 )𝑑𝜔 𝜑(𝑥, 𝑡 2𝜋 𝑖=0

− 𝜉; 𝑟)𝑑𝜉 , (7.4)

∞

1

(𝑖+1)

𝑖 𝛼𝑖 −𝑖𝜔𝑥 𝑢(𝑥, 𝑡; 𝑟) = 2𝜋 𝑓0 (𝑥; 𝑟) ∑∞ (𝐸2𝛼,𝛼𝑖+1 (−𝛬(𝜔)𝑡 2𝛼 ) + 𝑖=0(−2𝜆) 𝑡 ∫−∞ 𝑒 2𝜆 𝛼 (𝑖+1) 𝑡 𝐸2𝛼,𝛼+𝛼𝑖+1 (−𝛬(𝜔)𝑡 2𝛼 )) 𝑑𝜔 + 𝑎 𝑡 ∞ 1 (𝑖+1) ∑∞ (−2𝜆)𝑖 ∫0 𝜉 𝛼𝑖+(2𝛼−1) ∫−∞ 𝑒 −𝑖𝜔𝑥 𝐸2𝛼,2𝛼+𝛼𝑖 (−𝛬(𝜔)𝜉 2𝛼 )𝑑𝜔 𝜑(𝑥, 𝑡 2𝜋 𝑖=0

− 𝜉; 𝑟)𝑑𝜉 ,

These two cases can be considered as particular cases of Eqs. (7.4) A. Classical Green’s function In this case the homogeneous initial condition is 𝑓(𝑥) = 0̃. Substituting this into Eqs. (7.4) 𝑟

[𝑢(𝑥, 𝑡)]𝑟 = (𝐺(𝑥, 𝑡|0,0)) = 1 𝑖 𝑡 𝛼𝑖+(2𝛼−1) ∞ −𝑖𝜔𝑥 (𝑖+1) ∑∞ 𝐸2𝛼,2𝛼+𝛼𝑖 (−𝛬(𝜔)𝜉 2𝛼 )𝑑𝜔 [𝜑(𝑥, 𝑡 − 𝜉)]𝑟 𝑑𝜉 , ∫−∞ 𝑒 𝑖=0(−2𝜆) ∫0 𝜉

2𝜋

B. The propagator Substituting 𝜑(𝑥, 𝑡; 𝑟) = 0̃ and 𝑓(𝑥; 𝑟) = 𝛿(𝑥; 𝑟) into Eqs. (7.4), we can write 𝑢(𝑥, 𝑡; 𝑟) =

1 (𝑖+1) 𝑖 𝛼𝑖 ∞ −𝑖𝜔𝑥 𝑓 (𝑥; 𝑟) ∑∞ (𝐸2𝛼,𝛼𝑖+1 (−𝛬(𝜔)𝑡 2𝛼 ) + 𝑖=0(−2𝜆) 𝑡 ∫−∞ 𝑒 2𝜋 0

(𝑖+1)

2𝜆𝑡 𝛼 𝐸2𝛼,𝛼+𝛼𝑖+1 (−𝛬(𝜔)𝑡 2𝛼 )) 𝑑𝜔, 𝑢(𝑥, 𝑡; 𝑟) =

1 𝑖 𝛼𝑖 ∞ −𝑖𝜔𝑥 𝑓 (𝑥; 𝑟) ∑∞ 𝑖=0(−2𝜆) 𝑡 ∫−∞ 𝑒 2𝜋 0

(𝑖+1)

(𝐸2𝛼,𝛼𝑖+1 (−𝛬(𝜔)𝑡 2𝛼 ) +

(𝑖+1)

2𝜆𝑡 𝛼 𝐸2𝛼,𝛼+𝛼𝑖+1 (−𝛬(𝜔)𝑡 2𝛼 )) 𝑑𝜔. For special case 𝐶[(𝑖) − 𝛼], let us consider, 𝛼 = 0.75, 𝛽 = 0.5, 𝜆 = 1/2, 𝜉 = 1,𝜈 = 1/2, [𝑢(𝑥, 0)]𝑟 = [𝑓0 (𝑥)]𝑟 = [−1 + 𝑟, 1 − 𝑟], 𝑡 = 0.001 , then we get the solution for case 𝐶[(𝑖) − 𝛼] as following:

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1

∞

𝑖 0.75𝑖 [𝑢(𝑥, 0.001)]𝑟 = [−1 + 𝑟, 1 − 𝑟] ∑∞ ∫−∞ 𝑒 −𝑖𝜔𝑥 (∑∞ 𝑘=0 𝑖=0(−1) (0.001) 2𝜋

(0.001)0.75 ∑∞ 𝑘=0

1 4

𝛤(𝑖+1+𝑘)(−(− 𝜔−1)(0.001)1.5 )

1 4

𝑘

𝛤(𝑖+1+𝑘)(−(− 𝜔−1)(0.001)1.5 )

+

𝑘!𝛤(𝑖+1)Γ(1.5𝑘+0.75𝑖+1)

𝑘

𝑘!𝛤(𝑖+1)Γ(1.5𝑘+1.75+0.75𝑖)

) 𝑑𝜔, 0 ≤ 𝑟 ≤ 1,

Example 7.2 (Fuzzy fractional reaction-diffusion equation). Let us consider the following FFPDE: 2𝛽

{

𝑏 𝐶𝐷0𝛼+,𝑡 𝑢(𝑥, 𝑡) = 𝑐 𝐶𝐷𝑥,−∞ 𝑢(𝑥, 𝑡) − 𝜈 2 𝑢(𝑥, 𝑡) + 𝜑(𝑥, 𝑡) , 𝑢(𝑥, 0) = 0 𝑎𝑛𝑑 𝑢𝑡 (𝑥, 0) = 0 ∈ 𝔼

(7.5)

where 𝑡 > 0 ve −∞ < 𝑥 < +∞, 0< 𝛼 ≤ 1, 0 < 𝛽 ≤ 1. Applying Laplace transform on the both sides of above equation, we obtain: 2𝛽

ℒ𝑡 {𝑏 𝐶𝐷0𝛼+,𝑡 𝑢(𝑥, 𝑡)} = ℒ𝑡 {𝑐 𝐶𝐷𝑥,−∞ 𝑢(𝑥, 𝑡) − 𝜈 2 𝑢(𝑥, 𝑡) + 𝜑(𝑥, 𝑡)}, Using 𝐶[(𝑖𝑖) − 𝛼]−differentiability, we get 2𝛽 𝑐 𝐶𝐷𝑥,−∞ ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − 𝜈 2 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) + ℒ𝑡 {𝜑} (𝑥, 𝑠; 𝑟) = 𝑏𝑠 𝛼 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − 𝑏𝑠 𝛼−1 𝑓0 (𝑥; 𝑟) , (7.6) { 2𝛽 𝑐 𝐶𝐷𝑥,−∞ ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − 𝜈 2 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) + ℒ𝑡 {𝜑}(𝑥, 𝑠; 𝑟) = 𝑏𝑠 𝛼 ℒ𝑡 {𝑢}(𝑥, 𝑠; 𝑟) − 𝑏𝑠 𝛼−1 𝑓0 (𝑥; 𝑟)

Taking the Fourier transform in Eqs. (7.6), and the Fourier transform of the fractional derivative 𝜇

ℱ𝑥 { 𝐶𝐷𝑥,−∞ 𝑢(𝑥, 𝑡)} = −|𝜔|𝜇 ℱ𝑥 {𝑢(𝜔, 𝑡)}

we get, {

(𝑏𝑠 𝛼 − 𝑐|𝜔|2𝛽 + 𝜈 2 )ℱ𝑥 {ℒ𝑡 {𝑢}} (𝜔, 𝑠; 𝑟) = 𝑏𝑠 𝛼−1 ℱ𝑥 {𝑓0 } (𝜔; 𝑟) + ℱ𝑥 {ℒ𝑡 {𝜑}} (𝜔, 𝑠; 𝑟) , (7.7) (𝑏𝑠 𝛼 − 𝑐|𝜔|2𝛽 + 𝜈 2 )ℱ𝑥 {ℒ𝑡 {𝑢}}(𝜔, 𝑠; 𝑟) = 𝑏𝑠 𝛼−1 ℱ𝑥 {𝑓0 }(𝜔; 𝑟) + ℱ𝑥 {ℒ𝑡 {𝜑}}(𝜔, 𝑠; 𝑟)

Applying inverse of Laplace transform on the both sides of Eq. (7.7), we get the following: ℱ𝑥 {ℒ𝑡 {𝑢}} (𝜔, 𝑠; 𝑟) = { ℱ𝑥 {ℒ𝑡 {𝑢}}(𝜔, 𝑠; 𝑟) =

𝑏𝑠𝛼−1 ℱ𝑥 {𝑓0 }(𝜔;𝑟) 𝑏𝑠𝛼 +𝛬(𝜔) 𝑏𝑠𝛼−1 ℱ𝑥 {𝑓0 }(𝜔;𝑟) 𝑏𝑠𝛼 +𝛬(𝜔)

+ +

ℱ𝑥 {ℒ 𝑡 {𝜑}}(𝜔,𝑠;𝑟) 𝑏𝑠𝛼 +𝛬(𝜔) ℱ𝑥 {ℒ 𝑡 {𝜑}}(𝜔,𝑠;𝑟)

,

𝑏𝑠𝛼 +𝛬(𝜔)

where we have put Λ(𝜔) = −𝑐|𝜔|2𝛽 + 𝜈 2. Taking the inverse fuzzy Laplace transform and inverse fuzzy Fourier transform, we get ∞ 1 𝛬(𝜔) 𝛼 𝑓 (𝑥; 𝑟) ∫−∞ 𝑒 −𝑖𝜔𝑥 𝐸𝛼,1 (− 𝑡 ) 𝑑𝜔 + 2𝜋 0 𝑏 1 1 𝑡 𝛼−1 ∞ −𝑖𝜔𝑥 𝛬(𝜔) 𝛼 𝐸𝛼,𝛼 (− 𝑏 𝜉 ) 𝑑𝜔 𝜑(𝑥, 𝑡 − 𝜉; 𝑟)𝑑𝜉, ∫ 𝜉 ∫−∞ 𝑒 2𝜋 𝑏 0

𝑢(𝑥, 𝑡; 𝑟) =

(7.8) 1

∞

𝑢(𝑥, 𝑡; 𝑟) = 2𝜋 𝑓0 (𝑥; 𝑟) ∫−∞ 𝑒 −𝑖𝜔𝑥 𝐸𝛼,1 (−

1 1 𝑡 𝛼−1 ∞ −𝑖𝜔𝑥 𝐸𝛼,𝛼 ∫ 𝜉 ∫−∞ 𝑒 2𝜋 𝑏 0

𝛬(𝜔) 𝛼 𝑡 ) 𝑑𝜔 𝑏

𝛬(𝜔) (− 𝑏 𝜉 𝛼 ) 𝑑𝜔 𝜑(𝑥, 𝑡

+

− 𝜉; 𝑟)𝑑𝜉.

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These two cases can be considered as particular cases of Eqs. (7.8). A. Classical Green’s function In this case the homogeneous initial condition is 𝑓(𝑥) = 0̃. Substituting this into Eqs. (7.8) 𝑟

1 1

𝑡

∞

[𝑢(𝑥, 𝑡)]𝑟 = (𝐺2 (𝑥, 𝑡|0,0)) = ∫ 𝜉 𝛼−1 ∫−∞ 𝑒 −𝑖𝜔𝑥 𝐸𝛼,𝛼 (− 2𝜋 𝑏 0

𝛬(𝜔) 𝛼 𝜉 ) 𝑑𝜔 [𝜑(𝑥, 𝑡 𝑏

− 𝜉)]𝑟 𝑑𝜉 ,

B. The propagator Substituting 𝜑(𝑥, 𝑡; 𝑟) = 0̃ and 𝑓(𝑥; 𝑟) = 𝛿(𝑥; 𝑟) into Eqs. (7.8), we can write 𝑢(𝑥, 𝑡; 𝑟) =

∞ 1 𝛬(𝜔) 𝛼 𝑓 (𝑥; 𝑟) ∫−∞ 𝑒 −𝑖𝜔𝑥 𝐸𝛼,1 (− 𝑡 ) 𝑑𝜔, 2𝜋 0 𝑏

𝑢(𝑥, 𝑡; 𝑟) =

∞ 𝛬(𝜔) 1 𝑓 (𝑥; 𝑟) ∫−∞ 𝑒 −𝑖𝜔𝑥 𝐸𝛼,1 (− 𝑏 𝑡 𝛼 ) 𝑑𝜔. 2𝜋 0

For special case 𝐶[(𝑖𝑖) − 𝛼], let us consider, 𝛼 = 0.5, 𝛽 = 0.5, 𝑏 = 𝑐 = 1, 𝜈 = 1, [𝑢(𝑥, 0)]𝑟 = [𝑓0 (𝑥)]𝑟 = [−1 + 𝑟, 1 − 𝑟], 𝑡 = 0.001, then we get the solution for case 𝐶[(𝑖𝑖) − 𝛼] as following: 𝑟

[𝑢(𝑥, 0.001)] =

1 [−1 + 2𝜋

𝑟, 1 −

∞ 𝑟] ∫−∞ 𝑒 −𝑖𝜔𝑥

∑∞ 𝑘=0

((𝜔−1)(0.001)0.5 ) 𝛤(0.5𝑘+1)

𝑘

𝑑𝜔 , 0 ≤ 𝑟 ≤ 1,

8. Conclusions In this paper, the fuzzy Laplace transforms and fuzzy Fourier transforms have been studied in order to solve fuzzy fractional partial differential equations (FFPDEs) of order 0 < 𝛼 ≤ 1 under Caputo-type Hdifferentiability. To this end, Caputo-type differentiability based on the Hukuhara difference was introduced and then Laplace transform of fractional derivative was discussed by Derivative theorem (Theorem 4.1). Consequently, we solved some well-known examples in fractional manner to obtain the solutions of FFPDEs under Caputo-type H-differentiability.

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REFERENCES

[1] N. Wiener, Hermitian polynomials and Fourier analysis, J. Math. Phys. 8(1929) 70–73. [2] Allahviranloo T, Abbasbandy S, Salahshour S, Hakimzadeh A. A new method for solving fuzzy linear differential equations. Computing 2011;92:181–97. [3] M. Mazandarani, A. V. Kamyad, Modified fractional Euler method for solving Fuzzy Fractional Initial Value Problem, Commun Nonlinear Sci Numer Simulat 18 (2013) 12–21. [4] A. Ahmadian1, M. Suleiman1, S. Salahshour and D. Baleanu, A Jacobi operational matrix for solving a fuzzy linear fractional differential equation, Ahmadian et al. Advances in Difference Equations 2013, 2013:104. [5] N. Gasilov, S. E. Amrahov, A. G. Fatullayev, A Geometric Approach to Solve Fuzzy Linear Systems of Differential Equations, Applied Mathematics & Information Sciences 5(3) (2011), 484-499. [6] Hsien-Chung Wu, The improper fuzzy Riemann integral and its numerical integration, Information Sciences 111(1998) 109-137. [7] S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun Nonlinear Sci Numer Simulat 17 (2012) 1372–1381. [8] R. Figueiredo Camargo, R. Charnet, and E. Capelas de Oliveira, On some fractional Green’s functions, Journal of Mathematical Physics 50, 043514 (2009). [9] G.A. Anastassiou, Fuzzy Mathematics: Approximation Theory, Springer-Verlag Berlin Heidelberg 251 (2010) 65-73. [10] G.A. Anastassiou, Intelligent Mathematics: Computational Analysis, Springer-Verlag Berlin Heidelberg 5 (2011) 553-574. [11] Zimmermann HJ. Fuzzy set theory and its applications. Dordrecht: Kluwer Academic Publishers; 1991. [12] B. Bede, S.G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems 151 (2005) 581-599. [13] Y. Chalco-Cano, H. Roman-Flores, On new solutions of fuzzy differential equations, Chaos, Solitons and Fractals 38 (2006) 112-119.

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Aspects of univalent holomorphic functions involving Ruscheweyh derivative and generalized Sa˘la˘gean operator Alb Lupa¸s Alina1 and Andrei Loriana2 Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania 1 [email protected], 2 [email protected] Abstract Making use Ruscheweyh derivative and generalized S˘ al˘ agean operator, we introduce a new class of analytic functions RD(γ, λ, α, β) defined on the open unit disc, and investigate its various characteristics. Further we obtain distortion bounds, extreme points and radii of close-to-convexity, starlikeness and convexity and neighborhood property for functions belonging to the class RD(γ, λ, α, β).

Keywords: Analytic functions, univalent functions, radii of starlikeness and convexity, neighborhood property, Salagean operator, Ruscheweyh operator. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction

P∞ Let A denote the class of functions of the form f (z) = z+ j=2 aj z j , which are analytic and univalent in the open P∞ unit disc U = {z : z ∈ C : |z| < 1}. T is a subclass of A consisting the functions of the form f (z) = z − j=2 |aj | z j . P∞ P j For functions f, g ∈ A given by f (z) = z + j=2 aj z j , g(z) = z + ∞ j=2 bj z , we define the Hadamard product (or P∞ convolution) of f and g by (f ∗ g)(z) = z + j=2 aj bj z j , z ∈ U. Definition 1.1 (Al Oboudi [5]) For f ∈ A, λ ≥ 0 and n ∈ N, the operator Dλn is defined by Dλn : A → A,

Dλ0 f (z) = f (z) Dλ1 f (z) = (1 − λ) f (z) + λzf 0 (z) = Dλ f (z) , ... 0 Dλn+1 f (z) = (1 − λ) Dλn f (z) + λz (Dλn f (z)) = Dλ (Dλn f (z)) , z ∈ U. P P n j n j (z) = z + ∞ Remark 1.1 If f ∈ A and f (z) = z + ∞ j=2 aj z , then Dλ fP j=2 [1 + (j − 1) λ] aj z , z ∈ U . P∞ ∞ n j n j If f ∈ T and f (z) = z − j=2 aj z , then Dλ f (z) = z − j=t+1 [1 + (j − 1) λ] aj z , z ∈ U . Remark 1.2 For λ = 1 in the above definition we obtain the Sa˘la˘gean differential operator [8].

Definition 1.2 (Ruscheweyh [7]) For f ∈ A, n ∈ N, the operator Rn is defined by Rn : A → A, R0 f (z) = f (z) R1 f (z) = zf 0 (z) , ... (n + 1) Rn+1 f (z) = z (Rn f (z))0 + nRn f (z) ,

z ∈ U.

P P∞ (n+j−1)! j n j Remark 1.3 If f ∈ A, f (z) = z + ∞ j=2 aj z , then R f (z) = z + j=2 n!(j−1)! aj z , z ∈ U . P P∞ ∞ j If f ∈ T , f (z) = z − j=2 aj z j , then Rn f (z) = z − j=t+1 (n+j−1)! n!(j−1)! aj z , z ∈ U .

n n Definition 1.3 [1], [2] Let γ, λ ≥ 0, n ∈ N. Denote by RDλ,γ the operator given by RDλ,γ : A → A, n f (z) = (1 − γ)Rn f (z) + γDλn f (z), RDλ,γ

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o P∞ P∞ n n n Remark 1.4 If f ∈ A, f (z) = z+ j=2 aj z j , then RDλ,γ f (z) = z+ j=2 γ [1 + (j − 1) λ] + (1 − γ) (n+j−1)! aj z j , n!(j−1)! z ∈ U. n o P∞ P n (n+j−1)! j n a z , then RD f (z) = z − + (1 − γ) γ [1 + (j − 1) λ] aj z j , If f ∈ T , f (z) = z − ∞ j λ,γ j=2 j=t+1 n!(j−1)! z ∈ U. n f (z) = Lnγ f (z) which was introduced in [3]. The operator RD1,γ Following the work of Sh.Najafzadeh and E.Pezeshki [6] we can define the class RD(γ, α, β) as follows. Definition 1.4 For γ, λ ≥ 0, 0 ≤ α < 1 and 0 < β ≤ 1, let RD(γ, λ, α, β) be the subclass of T consisting of functions that satisfying the inequality ¯ ¯ µ,n ¯ ¯ f (z) − 1 RDλ,γ ¯ ¯ (1.1) ¯ −β|γ|, where Lnαf (z) = ¾ α ½ P ∞α P (n+j−1)! (n+j−1)! n j z− ∞ − j=2 (j−1){αj n +(1−α) n!(j−1)! }aj z j j=2 j {αj +(1−α) n!(j−1)! }aj z > −β |γ| . By chossing the and we obtain Re P P (n+j−1)! (n+j−1)! n j n j z− ∞ z− ∞ j=2 {αj +(1−α) n!(j−1)! }aj z j=2 {αj +(1−α) n!(j−1)! }aj z − valkues of z on the real ais and letting z → 1 through real values, the above inequality immediately yields the required condition (2.1). ¯ ¯ n ¯ ¯ z(L f (z))0 Conversely, by applying the hypothesis (2.1) and letting |z| = 1, we obtain ¯ Lnαf (z) − 1¯ = α ¯ ¯P∞ P∞ (n+j−1)! n ¯ j=2 (j−1){αj n +(1−α) (n+j−1)! a zj ¯ j=2 {αj +(1−α) n!(j−1)! }aj ) n!(j−1)! } j ¯ ≤ β|γ|(1− ¯ ≤ β|γ|. Hence, by maximum modulus theoP∞ (n+j−1)! (n+j−1)! n j n ¯ ¯ z−P ∞ 1− j=2 {αj +(1−α) n!(j−1)! }aj j=2 {αj +(1−α) n!(j−1)! }aj z n rem, we have f ∈ T S α (β, γ), which evidently completes the proof of theorem. Finally, the result is sharp with the extremal functions fj be in the class T S nα (β, γ) given by fj (z) = z −

β |γ| n o z j , for j ≥ 2. n (n+j−1)! (j − 1 + β|γ|) α [1 + (j − 1) λ] + (1 − α) n!(j−1)!

(2.2)

Corollary 2.2 Let the function f ∈ T be in the class T S nλ,α (β, γ). Then, we have aj ≤

β|γ| n o , for j ≥ 2. (j − 1 + β|γ|) αj n + (1 − α) (n+j−1)! n!(j−1)!

(2.3)

The equality is attained for the functions f given by (2.2).

3

Growth and distortion theorems

Theorem 3.1 Let the function f ∈ T be in the class T S nα (β, γ). Then for |z| = r, we have r−

β|γ| β|γ| r2 ≤ |f (z)| ≤ r + r2 . (1 + β|γ|) [2n α + (1 − α) (n + 1)] (1 + β|γ|) [2n α + (1 − α) (n + 1)]

with equality for f (z) = z −

β|γ| 2 (1+β|γ|)[2n α+(1−α)(n+1)] z .

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Proof. In view of Theorem 2.1, P we have P∞ n (1 + β|γ|) [2n α + (1 − α) (n + 1)] ∞ j=2 aj ≤ j=2 (j − 1 + β|γ|) [2 α + (1 − α) (n + 1)] aj ≤ β|γ|. P∞ P∞ P ∞ β|γ| r2 and |f (z)| ≥ r − j=2 aj rj ≥ Hence |f (z)| ≤ r + j=2 aj rj ≤ r + r2 j=2 aj ≤ r + (1+β|γ|)[2n α+(1−α)(n+1)] P β|γ| 2 r − r2 ∞ j=2 aj ≥ r − (1+β|γ|)[2n α+(1−α)(n+1)] r . This completes the proof. Theorem 3.2 Let the function f ∈ T be in the class T S nα (β, γ). Then, for |z| = r we have r−

(1 +

2β|γ| 2β|γ| r2 ≤ |f 0 (z)| ≤ r + r2 n + (1 − α) (n + 1)] (1 + β|γ|) [2 α + (1 − α) (n + 1)]

β|γ|) [2n α

with equality for f (z) = z −

β|γ| 2 (1+β|γ|)[2n α+(1−α)(n+1)] z .

P∞ P 2β|γ| j 2 2 0 Proof. We have |f 0 (z)| ≤ r + ∞ j=2 jaj r ≤ r + 2r j=2 aj ≤ r + (1+β|γ|)[2n α+(1−α)(n+1)] r and |f (z)| ≥ P P∞ 2β|γ| 2 r − j=2 jaj rj ≥ r − 2r2 ∞ j=2 aj ≥ r − (1+β|γ|)[2n α+(1−α)(n+1)] r , which completes the proof.

4

Extreme points The extreme points of the class T S nα (β, γ) will be now determined.

Theorem 4.1 Let f1 (z) = z and fj (z) = z − in U . Then f ∈ T P∞ j=1 µj = 1.

S nα (β, γ)

β|γ| j (n+j−1)! z , (j−1+β|γ|){αj n +(1−α) n!(j−1)! }

j ≥ 2. Assume that f is analytic P if and only if it can be expressed in the form f (z) = ∞ j=1 µj fj (z), where µj ≥ 0 and

P P∞ Proof. Suppose that f (z) = ∞ j=1 µj fj (z) with µj ≥ 0 and j=1µµj = 1. Then P∞ P∞ P∞ f (z) = j=1 µj fj (z) = µ1 f1 (z) + j=2 µj fj (z) = µ1 z + j=2 µj z −

P∞

j=2 P∞ j=2

µj

β|γ|

z j . Then

αj n +(1−α) (n+j−1)! n!(j−1)!

(j−1+β|γ|){ P µj = ∞ j=1 µj

P∞

j=2

β|γ|

µj

β|γ| zj (j−1+β|γ|){αj n +(1−α) (n+j−1)! n!(j−1)! }

αj n +(1−α) (n+j−1)! n!(j−1)!

(j−1+β|γ|){ } − µ1 = 1 − µ1 ≤ 1. Thus f ∈ T S nα (β, γ) by Theorem 2.1.

}

¶

= z−

(j−1+β|γ|){αj n +(1−α) (n+j−1)! n!(j−1)! } β|γ|

=

(j−1+β|γ|){αj n +(1−α) (n+j−1)! n!(j−1)! } Conversely, suppose that f ∈ T S nα (β, γ). By using (2.3) we may set and µj = aj β|γ| P∞ for j ≥ 2 and µ1 = 1 − j=2 µj . P∞ P∞ P β|γ| j j Then f (z) = z − ∞ j=1 aj z = z − j=2 µj (j−1+β|γ|){αj n +(1−α) (n+j−1)! } z = µ1 f1 (z) + j=2 µj fj (z) = n!(j−1)! P∞ P∞ j=1 µj fj (z) , with µj ≥ 0 and j=1 µj = 1, which completes the proof.

Corollary 4.2 The extreme points of T S nα (β, γ) are the functions f1 (z) = z and β|γ| j fj (z) = (n+j−1)! z , for j ≥ 2. (j−1+β|γ|){αj n +(1−α) n!(j−1)! }

5

Partial sums

We investigate in this section the partial sums of functions in the class T S nα (β, γ). We shall obtain sharp lower bounds for the real part of its ratios. We shall follow similar works done by Silverman [9] and Khairnar and Moreon [6] about the partial sums of³ analytic ´ functions. In what follows, we will use the well known result that for an 1+ω(z) > 0, z ∈ U, if and only if the inequality |ω (z)| < 1 is satisfied. analytic function ω in U , Re 1−ω(z)

Theorem 5.1 Let f (z) = z − P j z− m j=2 aj z , m ≥ 2. Then

P∞

j=2

aj z j ∈ T S nα (β, γ) and define its partial sums by f1 (z) = z and fm (z) =

Re

and Re

µ

µ

¶

≥1−

¶

≥

f (z) fm (z)

fm (z) f (z)

1 , z ∈ U , m ∈ N, cm+1

cm+1 , 1 + cm+1 280

3

(5.1)

z ∈ U , m ∈ N,

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where cj =

n o (j + β|γ|) αj n + (1 − α) (n+j−1)! n!(j−1)!

(5.3)

β|γ|

This estimates in (5.1) and (5.2) are sharp. Proof. To prove (5.1), it suffices to show that cm+1

³

P P∞ j−1 j−1 1− m −c j=2 aj z j=m+1 aj z P mm+1 j−1 property, we can write 1− j=2 aj z P cm+1 ∞ aj P∞ that ω (0) = 0 and |ω (z)| ≤ 2−2 P m aj −cj=m+1 . m+1 j=2 j=m+1 aj P∞ j=2 cj aj ≤ 1. The above inequality holds because cj

f (z) fm (z)

=

³ − 1−

1+ω(z) 1−ω(z)

1 cm+1

´´

≺

1+z 1−z ,

z ∈ U. By the subordination

for certain ω analytic in U with |ω (z)| ≤ 1. Notice Pm P∞ Now |ω (z)| ≤ 1 if and only if j=2 aj + cm+1 j=m+1 aj ≤

is a non-decreasing sequence. This completes the proof of πi (5.1). Finally, it is observed that equality in (5.1) is attained for the function given by (2.3) when z = re2 m as r → 1− . P P∞ ³ ´ j−1 j−1 1− m +c cm+1 (z) 1+ω(z) j=2 aj z j=m+1 aj z P mm+1 j−1 Similarly, we take (1 + cm+1 ) ffm(z) − 1+c = 1−ω(z) , where |ω (z)| ≤ = 1− j=2 aj z m+1 P∞ Pm P∞ P∞ (1+cm+1 ) j=m+1 aj P P j=2 aj + cm+1 j=m+1 aj ≤ j=2 cj aj ≤ 1. This 2−2 m aj +(1−cm+1 ) ∞ aj . Now |ω (z)| ≤ 1 if and only if j=2

j=m+1

immediately leads to assertion (5.2) of Theorem 5.1. This completes the proof. Using a similar method, we can prove the following theorem.

P j Theorem 5.2 If f ∈ T S nα (β, γ) and define the partial sums by f1 (z) = z and fm (z) = z − m j=2 aj z . Then ³ 0 ´ ³ 0 ´ f (z) cm+1 , z ∈ U , m ∈ N, and Re fm(z) ≥ m+1+c , where cj is given by (5.3). The result is ≥ 1 − cm+1 Re ff0 (z) m+1 m+1 m (z) sharp for every m, with extremal functions given by (2.2).

References [1] A. Abubaker, M. Darus, On a certain subclass of analytic functions involving differential operators, TJMM 3 (2011), No. 1, 1-8. [2] A. Alb Lupa¸s, On special differential subordinations using Sa˘la˘gean and Ruscheweyh operators, Mathematical Inequalities and Applications, Volume 12, Issue 4, 2009, 781-790. [3] A. 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. [4] A. Alb Lupa¸s, D. Breaz, On special differential superordinations using Sa˘la˘gean and Ruscheweyh operators, Geometric Function Theory and Applications’ 2010 (Proc. of International Symposium, Sofia, 27-31 August 2010), 98-103. [5] A. Alb Lupa¸s, Some differential subordinations using Ruscheweyh derivative and Sa˘la˘gean operator, Advances in Difference Equations.2013, 2013:150., DOI: 10.1186/1687-1847-2013-150. [6] S.M. Khairnar, M. Moore, On a certain subclass of analytic functions involving the Al-Oboudi differential operator, J. Ineq. Pure Appl. Math., 10 (2009), 1-22. [7] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [8] G. St. Sa˘la˘gean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372. [9] H. Silverman, Partial sums of starlike and convex functions, J. Math. Anal. Appl., 209 (1997), 221-227.

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New classes containing multiplier transformation 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 α α : A → A, RIn,λ,l f (z) = (1 − In this paper we introduce new classes containig the linear operator RIn,λ,l n n α)R f (z) + αI (n, λ, l) f (z), z ∈ U, where R f (z) is the Ruscheweyh derivative, I (n, λ, l) f (z) the multiplier transformation 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. Characterization and other properties of these classes are studied.

Keywords: differential operator, distortion theorem. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction

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. Definition 1.1 (Ruscheweyh [20]) For f ∈ A, n ∈ N, the operator Rn is defined by Rn : A → A, R0 f (z) R1 f (z) (n + 1) Rn+1 f (z) P j Remark 1.1 If f ∈ A, f (z) = z + ∞ j=2 aj z ,

= f (z) = zf 0 (z) , ... = z (Rn f (z))0 + nRn f (z) , z ∈ U. P (n+j−1)! j then Rn f (z) = z + ∞ j=2 n!(j−1)! aj z , z ∈ U .

Definition 1.2 For f ∈ A, n ∈ N, λ, l ≥ 0, the operator I (n, λ, l) f (z) is defined by the following infinite series ¶n µ ∞ P λ (j − 1) + l + 1 aj z j . I (n, λ, l) f (z) = z + l + 1 j=2 Remark 1.2 It follows from the above definition that

I (0, λ, l) f (z) = f (z), (l + 1) I (n + 1, λ, l) f (z) = (l + 1 − λ) I (n, λ, l) f (z) + λz (I (n, λ, l) f (z))0 ,

z ∈ U.

Remark 1.3 For l = 0, λ ≥ 0, the operator Dλn = I (n, λ, 0) was introduced and studied by Al-Oboudi [16], which is reduced to the Sa˘la˘gean differential operator [21] for λ = 1. α α the operator given by RIn,λ,l : A → A, Definition 1.3 [7] Let α, λ, l ≥ 0, n ∈ N. Denote by RIn,λ,l α f (z) = (1 − α)Rn f (z) + αI (n, λ, l) f (z), z ∈ U. RIn,λ,l P a z j , then Remark 1.4 If f ∈ A, f (z) = z + ∞ j=2 n ³ ´n j o P 1+λ(j−1)+l (n+j−1)! α RIn,λ,l f (z) = z + ∞ + (1 − α) α aj z j , z ∈ U. j=2 l+1 n!(j−1)! This operator was studied also in [13], [14].

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0 1 Remark 1.5 For α = 0, RIm,λ,l f (z) = Rm f (z), where z ∈ U and for α = 1, RIm,λ,l f (z) = I (m, λ, l) f (z), where α m f (z) which was studied in z ∈ U , which was studied in [3], [4], [10], [9]. For l = 0, we obtain RIm,λ,0 f (z) = RDλ,α α m [5], [6], [11], [12], [17], [18] and for l = 0 and λ = 1, we obtain RIm,1,0 f (z) = Lα f (z) which was studied in [1], [2], [8], [15]. n (µ) if and only if Definition 1.4 Let f ∈ A. Then f (z) is in the class Sλ,l,α

⎛ ³ ´0 ⎞ α ⎜ z RIn,λ,l f (z) ⎟ Re ⎝ ⎠ > µ, 0 ≤ µ < 1, z ∈ U. α RIn,λ,l f (z)

n (µ) if and only if Definition 1.5 Let f ∈ A. Then f (z) is in the class Cλ,l,α

⎛∙ ³ ´0 ¸0 ⎞ α ⎟ ⎜ z RIn,λ,l f (z) ⎟ ⎜ Re ⎜ ³ ´0 ⎟ > µ, 0 ≤ µ < 1, z ∈ U. ⎠ ⎝ α f (z) RIn,λ,l

We study the charaterization and distortion theorems, and other properties of these classes, following the paper of M. Darus and R. Ibrahim [19].

2

n General properties of RDλ,α

In this section we study the characterization properties and distortion theorems for the function f (z) ∈ A to n n (µ) and Cλ,l,α (µ) by obtaining the coefficient bounds. belong to the classes Sλ,l,α Theorem 2.1 Let f ∈ A. If

¶n ¾ ½ µ ∞ X 1 + λ (j − 1) + l (n + j − 1)! |aj | ≤ 1 − µ, (j − µ) α + (1 − α) l+1 n! (j − 1)! j=2

0 ≤ µ < 1,

(2.1)

n (µ). The result (2.1) is sharp. then f (z) ∈ Sλ,l,α

n ³ ´n o P∞ 1+λ(j−1)+l (n+j−1)! (j − µ) α + (1 − α) |aj | Proof. Suppose that (2.1) holds. Since 1 − µ ≥ j=2 n ³ o ´n l+1 o n!(j−1)! P∞ n ³ 1+λ(j−1)+l ´n P 1+λ(j−1)+l ≥ µ j=2 α + (1 − α) (n+j−1)! + (1 − α) (n+j−1)! |aj | − ∞ |aj |, then this j=2 j α l+1 n!(j−1)! l+1 n!(j−1)! ¶ µ P∞ 0 1+λ(j−1)+l n (n+j−1)! α 1+ j=2 j {α( z (RIn,λ,l f (z)) ) +(1−α) n!(j−1)! }|aj | l+1 > µ, 0 ≤ µ < 1, > µ. So, we deduce that Re implies that P α 1+λ(j−1)+l n RIn,λ,l f (z) 1+ ∞ ) +(1−α) (n+j−1)! j=2 {α( l+1 n!(j−1)! }|aj | n z ∈ U. We have f (z) ∈ Sλ,l,α (µ), which evidently completes the proof. The assertion (2.1) is sharp and the extremal function is given by P (1−µ) j f (z) = z + ∞ j=2 (j−µ){α( 1+λ(j−1)+l )n +(1−α) (n+j−1)! } z . l+1 n!(j−1)! Corollary 2.2 Let the hypotheses of Theorem 2.1 satisfy. Then |aj | ≤

1−µ

(j−µ){α(

1+λ(j−1)+l n (n+j−1)! +(1−α) n!(j−1)! l+1

)

}

, ∀ j ≥ 2.

Theorem 2.3 Let f ∈ A. If

¶n ¾ ½ µ 1 + λ (j − 1) + l (n + j − 1)! |aj | ≤ 1 − µ, j(j − µ) α + (1 − α) l+1 n! (j − 1)! j=2

∞ X

0 ≤ µ < 1,

n then f (z) ∈ Cλ,l,α (µ). The result (2.2) is sharp.

(2.2)

n ³ ´n o P∞ + (1 − α) (n+j−1)! |aj | ≥ Proof. Suppose that (2.2) holds. Since 1 − µ ≥ j=2 j(j − µ) α 1+λ(j−1)+l l+1 n!(j−1)! o o P∞ n ³ 1+λ(j−1)+l ´n P∞ 2 n ³ 1+λ(j−1)+l ´n (n+j−1)! (n+j−1)! µ j=2 j α + (1 − α) n!(j−1)! |aj | − j=2 j α + (1 − α) n!(j−1)! |aj | then this l+1 l+1 283

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P 1+λ(j−1)+l n 2 1+ ∞ ) +(1−α) (n+j−1)! j=2 j {α( l+1 n!(j−1)! }|aj | > µ. So, we deduce P∞ 1+λ(j−1)+l n 1+ j=2 j {α( ) +(1−α) (n+j−1)! l+1 n!(j−1)! }|aj | n We have f (z) ∈ Cλ,l,α (µ), which evidently completes the proof.

implies that

1, z ∈ U. The assertion (2.2) is sharp and the extremal function is given by P∞ (1−µ) zj . f (z) = z + j=2 1+λ(j−1)+l n j(j−µ){α( ) +(1−α) (n+j−1)! l+1 n!(j−1)! }

that Re

Corollary 2.4 Let the hypotheses of Theorem 2.3 be satisfied. Then |aj | ≤ j ≥ 2.

µ

0 0

α f (z)) ] [z(RIn,λ,l 0 α (RIn,λ,l f (z))

¶

> µ, 0 ≤ µ
0 such that δ() ≥ c2 . Definition 1.7. (1) Let A be a nonempty subset of a metric space X. A mapping T : A → A is said to be asymptotically nonexpansive if there exists a P sequence {kn } ⊂ [1, ∞) with ∞ n=1 (kn − 1) < +∞ such that d(T n x, T ny) ≤ kn d(x, y)

for all x, y ∈ A and n ≥ 1. (2) T is said to be uniformly L-Lipschitzian with a Lipschitzian constant L ≥ 1, i.e., there exists a constant L ≥ 1 such that d(T n x, T ny) ≤ Ld(x, y) for all x, y ∈ A and n ≥ 1. This is a class of mapping introduced by Goebel and Kirk [10], where it is shown that if A is a nonempty bounded closed convex subset of a uniformly convex Banach space and T : A → A is asymptotically nonexpensive, then T has a fixed point and, moreover, the set F ix(T ) of fixed points of T is closed and convex. Remark 1.8. As an application of the Lagrange mean value theorem, we can see that tq − 1 ≤ qtq (t − 1) P for t ≥ 1 and q > 1. This together with ∞ n=1 (kn − 1) < +∞ implies that P∞ q n=1 (kn − 1) < +∞.

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Theorem 1.9. ([23]) If (X, d) is uniformly convex complete metric space, then every decreasing sequence of nonempty closed bounded convex subsets of X has nonempty intersection. Definition 1.10. Let (X, d) be a metric space and Y a topological space. A mapping T : X → X is said to be completely continuous if the image of each bounded set in X is contained in a compact subset of Y. In [3], Beg proved the following results. Theorem 1.11. Let A be a nonempty closed bounded convex subset of a uniformly convex complete metric space (X, d) and T : A → A be an asymptotically nonexpansive mapping. Then T has a fixed point. Theorem 1.12. Let (X, d) be a convex metric space and A be a nonempty convex subset of X. Let L > 0 and T : A
→ A be uniformly L-Lipschitzian.
Let x1 ∈ A. Define yn = W T n xn , xn , 21 , xn+1 = W T n yn , xn , 12 and set cn = d(T n xn , xn ) for all n ∈ N. Then d(xn , T xn ) ≤ cn + cn−1 (L + 3L2 + 2L3 ) for all n ∈ N. Theorem 1.13. Let (X, d) be a 2-uniformly convex metric space having property (B), A be a nonempty closed bounded convex subset of X and T : N A → A be asymptotically nonexpensive with sequence {kn } ∈ [1, +∞) and
P∞ 1 2 n for all n=1 (kn − 1) < +∞. Let x1 ∈ A. Define xn+1 = W T xn , xn , 2 n ∈ N. Then limn→∞ d(xn , T xn ) = 0. Theorem 1.14. Let (X, d) be 2-uniformly convex metric space having property (B), A be a nonempty closed bounded convex subset of X and T : A → A completely continuous asymptotically nonexpensive mapping with sequence P∞ N 2 {kn } ∈ [1, +∞) and n=1 (kn − 1) < +∞. Let x1 ∈ A. Define xn+1 =
1 W T n xn , xn , 2 for all n ∈ N. Then {xn } converges to some fixed point of T. Recently, Agrawal et al. [1] introduced a new iteration process namely Siteration process and studied the iterative approximation problems for fixed points of nearly asymptotically nonexpansive mappings. The purpose of this paper is to construct a two-step type iteration scheme, convergent to the fixed point, for asymptotically nonexpensive mappings defined on a 2-uniformly convex metric space. The results established in this

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5

paper improve the results due to Agrawal et al. [1], Chang [7], Khan and Takahashi [15], Liu and Kang [19], Osilike and Aniagbosor [20], Rhoades [21], Schu [24], [25], Tan and Xu [27], [28] and others. 2. Main Results In the sequel, we will need the following results. The following lemma is now well known. Lemma 2.1. Let {an } and {bn } be sequences of non-negative real numbers P such that an+1 ≤ (1 + bn ) an for all n ≥ 1 and ∞ n=1 bn < ∞. Then limn→∞ an exists. Theorem 2.2. ([2]) Let (X, d) be a uniformly convex metric space having property (B). Then X is 2-unformly convex if and only if there exists a number 

2 c > 0 such that 2 d a, W x, y, 21 + c [d(x, y)]2 ≤ [d(a, x)]2 + [d(a, y)]2 for all a, x, y ∈ X. Now we prove our main results. Lemma 2.3. Let (X, d) be a convex metric space and A be a nonempty convex subset of X. Let L > 0 and T :
A → A be uniformly L-Lipschitzian.
Let x1 ∈ A. Define yn = W T n xn , xn , 21 , xn+1 = W T n yn , T n xn , 12 and set δn = d(T n xn , xn ) for all n ∈ N. Then d(xn , T xn) ≤ δn +

L2 (L + 5)δn−1 4

for all n ∈ N. Proof. For all n ∈ N, set δn = d(xn , T n xn ). Consider d(T xn , xn ) ≤ d(T xn , T n xn ) + d(T n xn , xn ) ≤ Ld(T n−1 xn , xn ) + δn ,    1 n−1 n−1 n−1 n−1 d(T xn , xn ) = d T xn , W T yn−1 , T xn−1, 2 1 1 ≤ d(T n−1 xn , T n−1 yn−1 ) + d(T n−1 xn , T n−1 xn−1 ) 2 2 1 1 ≤ Ld(xn , yn−1 ) + Ld(xn−1 , xn ), 2 2

291

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(2.2)

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d(xn−1 , xn ) = d xn−1 , W



T

n−1

yn−1 , T

1 ≤ d(xn−1 , T n−1 yn−1 ) + 2 1 ≤ d(xn−1 , T n−1 yn−1 ) + 2

n−1

1 xn−1, 2



1 d(xn−1 , T n−1 xn−1 ) 2 1 δn−1 , 2

d(xn−1 , T n−1 yn−1 ) ≤ d(xn−1 , T n−1xn−1 ) + d(T n−1 xn−1 , T n−1 yn−1 ) ≤ δn−1 + Ld(xn−1 , yn−1 )

(2.3)

(2.4)

and 



d(xn−1 , yn−1 ) = d xn−1 , W

T

n−1

1 xn−1 , xn−1 , 2

1 ≤ d(xn−1 , T n−1 xn ) 2 1 = δn−1 . 2

 (2.5)

(2.4) implies d(xn−1 , T

n−1

yn−1 ) ≤



 1 L + 1 δn−1 . 2

(2.6)

Substituting (2.6) in (2.3), we get d(xn−1 , xn ) ≤



 1 L + 1 δn−1 . 4

(2.7)

Also d(xn , yn−1 ) ≤ d(xn , xn−1 ) + d(xn−1 , yn−1 )   1 1 ≤ L + 3 δn−1 . 2 2

(2.8)

Substitution of (2.7) and (2.8) in (2.2) yields 1 d(T n−1 xn , xn ) ≤ L(L + 5)δn−1 . 4 Finally from (2.1) we obtain d(T xn , xn ) ≤ δn +

L2 (L + 5)δn−1 . 4

This completes the proof.



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Theorem 2.4. Let (X, d) be 2-uniformly convex metric space having property (B), A be a nonempty closed convex subset of X and T : A → A be asympP∞ totically nonexpensive with sequence {kn } ∈ [1, +∞)N and n=1 (kn − 1) < +∞ satisfying d(x, T ny) ≤ d(T n x, T ny)

(C)


1 n n for all x, y ∈ A. Assume that, for x 1 ∈ A, we define xn+1 = W T yn , T xn , 2 ,
yn = W T n xn , xn , 12 for all n ∈ N. Then we have limn→∞ d(T n yn , xn ) = 0 = limn→∞ d(T n xn , xn ). Proof. Since T is asymptotically nonexpensive, it possesses a fixed point p ∈ A by Theorem 1.11. Claim. {xn } is bounded. For this claim, we compute as follows    1 1 1 n n d (p, xn+1 ) = d p, W T yn , T xn , ≤ d(p, T n yn ) + d(p, T n xn ) 2 2 2 1 1 = d(T n p, T n yn ) + d(T n p, T n xn ) 2 2 kn ≤ (d(p, yn ) + d(p, xn )) 2 and 



 1 1 1 d (p, yn ) = d p, W T xn , xn , ≤ d(p, T n xn ) + d(p, xn ) 2 2 2 1 1 kn 1 = d(T n p, T n xn ) + d(p, xn ) ≤ d(p, xn ) + d(p, xn ) 2 2 2 2 1 = (kn + 1) d(p, xn ). 2 n

Hence   1 kn d (p, xn+1 ) ≤ (kn + 1) + kn d(p, xn ) 2 2 ≤ (kn (kn + 1) + kn )d(p, xn ) 
 ≤ 1 + (kn − 1) + kn3 − 1 d(p, xn ).

Using Remark 1.8 and Lemma 2.1, it follows that limn→∞ d(p, xn ) exists and the sequence {xn } is bounded. Let M = supn≥1 d(p, xn ).

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With the help of Theorem 2.2, we have 2 [d (p, xn+1 )]2 + c [d(T n yn , T nxn )]2    2 1 n n = 2 d p, W T yn , T xn , + c [d(T n yn , T nxn )]2 2 n

2

n

2

n

n

2

(2.9) n

n

≤ [d(p, T yn )] + [d(p, T xn )] = [d(T p, T yn )] + [d(T p, T xn )]
≤ kn2 [d(p, yn )]2 + [d(p, xn )]2 ,

2

where

2 [d (p, yn )]2 + c [d(T n xn , xn )]2    2 1 n = 2 d p, W T xn , xn ; + c [d(T n xn , xn )]2 2 ≤ [d(p, T n xn )]2 + [d(p, xn )]2 = [d(T n p, T n xn )]2 + [d(p, xn )]2 ≤ kn2 [d(p, xn )]2 + [d(p, xn )]2 implies

1 c 2 2 2 [d (p, yn )] ≤ (1 + kn2 ) [d(p, xn )] − [d(T n xn , xn )] 2 2 (2.10) 1 2 2 ≤ (1 + kn ) [d(p, xn )] . 2 Substituting (2.10) in (2.9), we get   1 2 2 n n 2 2 2 [d (p, xn+1 )] + c [d(T yn , T xn )] ≤ kn (1 + kn ) + 1 [d(p, xn )]2 2

implies c 2 [d(T n yn , T nxn )] 2
≤ kn2 (1 + kn2 ) + kn2 [d(p, xn )]2 − [d(p, xn+1 )]2


≤ 1 + kn2 − 1 + kn5 − 1 [d(p, xn )]2 − [d(p, xn+1 )]2


≤ kn2 − 1 + kn5 − 1 M 2 + [d(p, xn )]2 − [d(p, xn+1 )]2 .

Thus m

2 c X d(T j yj , T j xj ) 2 j=1 ≤M

2

m X j=1

kj2

m


X
5 − 1 + kj − 1 + [d(p, xj )]2 − [d(p, xj+1 )]2 . j=1

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Hence

∞ X 

d(T j yj , T j xj )

j=1

2

9

< +∞.

It implies that lim d(T n yn , T nxn ) = 0.

n→∞

By using condition (C), we have d(T n yn , xn ) ≤ d(T n yn , T nxn ) → 0 as n → ∞. Now

d(T n xn , xn ) ≤ d(T n xn , T nyn ) + d(T n yn , xn ) → 0 as n → ∞.

This completes the proof.

(2.11) 

Theorem 2.5. Let (X, d) be a 2-uniformly convex metric space having property (B), A be a nonempty closed bounded convex subset of X and T : A → A be completely continuous asymptotically nonexpensive mapping with sequence P {kn } ∈ [1, +∞)N and ∞ (C). Let n=1 (kn − 1) < +∞
satisfyingn the condition
1 1 n n x1 ∈ A. Define xn+1 = W T yn , T xn ; 2 , yn = W T xn , xn ; 2 for all n ∈ N. Then {xn } converges to the fixed point of T. Proof. With the help of Lemma 2.3 and (2.11), we have lim d(xn , T xn) = 0.

n→∞

Since T is completely continuous and {xn } is bounded, there exists a subsequence {xnk } of {xn } such that {T xnk } converges. Therefore {xnk } converges from limn→∞ d(xn , T xn) = 0. Let limk→∞ xnk = p. It follows from the continuity of T and limn→∞ d(xn , T xn ) = 0 that p = T p. We know that limn→∞ d (p, xn ) exists. But limk→∞ d (p, xnk ) = 0. This implies limn→∞ d (p, xn ) = 0, i.e., limn→∞ xn = p. This compleets the proof.  Remark 2.6. 1. Theorem 1.12 is proved for 2-step iteration process, while Theorems 1.13 and 1.14 are proved for 1-step iteration process. 2. In Theorem 2.4, we do not need the boundedness assumption, so the boundedness assumption in Theorem 1.13 is superfluous. 3. In Theorems 2.4 and 2.5, the condition (C) is not new; it is due to Liu et al. [18].

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Acknowledgment This study was supported by research funds from Dong-A University. References [1] R. P. Agarwal, D. O’Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex. Anal., 8 (2007), 61–79. [2] I. Beg, Inequalities in metric spaces with applications, Topol. Methods Nonlinear Anal., 17 (2001), 183–190. [3] I. Beg, An iteration scheme for asymptotically nonexpansive mappings on uniformly convex metric space, Nonlinear Anal. Forum, 6 (2001), 27–34. [4] I. Beg, N. Shahzad and M. Iqbal, Fixed point theorems and best approximation in convex metric space, Approx. Theory Appl., 8 (1992), 97–105. [5] S. C. Bose, Weak convergence to the fixed point of an asymptotically nonexpansive map, Proc. Amer. Math. Soc., 68 (1978), 305–308. [6] F. E. Browder, Nonexpansive nonlinear operators in Banach space, Proc. Nat. Acad. Sci. USA, 54 (1965), 1041–1044. [7] S. S. Chang, On the approximation problem of fixed points for asymptotically nonexpansive mappings, Indian J. Pure Appl. Math., 32 (2001), 1297–1307. ´ c, On some discontinuous fixed point theorems in convex metric spaces, [8] Lj. B. Ciri´ Czechoslovak Math. J., 43(118) (1993), 319–326. [9] L. Gaji´c and M. Stojakovic, A remark on Kaneko report on general contractive type conditions for multivalued mappings in Takahashi convex metric spaces, Zb. Rad. Prirod.Mat. Fak. Ser. Mat., 23 (1993), 61–66. [10] K. Goebel and W.A. Krik, Fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171–174. [11] D. G¨ ohde, Zum prinzip der kontraktiven abbildung, Math. Nachr., 30 (1965), 251–258. ´ [12] J. Gornicki, Weak convergence theorem for asymptotically nonexpanssive mappings, Comment. Math. Univ. Carolin., 30 (1989), 249–252. [13] M. D. Guay, K. L. Singh and J. H. M. Whitfield, Fixed point theorems for nonexpansive mappings in convex metric spaces, Lecture Notes in Pure and Appl. Math. 80, Dekker, New York, 1982, 179–189. [14] J. S. Jung, Y. J. Cho and D. R. Sahu, Existence and convergence for fixed points of nonlipschitzian mappings in Banach spaces without uniform convexity, Commun. Korean Math. Soc., 15 (2000), 275–284. [15] S. H. Khan and W. Takahashi, Iterative approximation of fixed points of asymptotically nonexpansive mappings with compact domains, Panamer. Math. J., 11 (2001), 19–24. [16] W. A. Kirk, Fixed point theorems for nonlipschitzian mappings of asymptotically nonexpansive type, Israel J. Math., 11 (1974), 339–346. [17] T. C. Lim and H. K. Xu, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal., 22 (1994), 1345–1355.

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NEW ITERATIVE SCHEME FOR THE APPROXIMATION OF FIXED POINTS

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[18] Z. Liu, C. Feng, J. S. Ume and S. M. Kang, Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings, Taiwanese J. Math., 11 (2007), 27–42. [19] Z. Liu and S. M. Kang, Weak and strong convergence for fixed points of asymptotically nonexpansive mappings, Acta Math. Sin., 20 2004, 1009–1018. [20] M. O. Osilike and S. C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling, 32 (2000), 1181–1191. [21] B. E. Rhoades, Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl., 183 (1994), 118–120. [22] T. Shimizu and W. Takahashi, Fixed point theorems in certain convex metric spaces, Math. Japon., 37 (1992), 855–859. [23] T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal., 8 (1996), 197–203. [24] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mapping, J. Math. Anal. Appl., 159 (1991), 407–413. [25] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153–159. [26] W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai Math. Sem. Rep., 22 (1970), 142–149. [27] K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301–308. [28] K. K. Tan and H. K. Xu, Fixed point iteration process for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 122 (1994), 733–734. [29] H. K. Xu, Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Anal., 16 (1991), 1139–1146.

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Global Behavior and Periodicity of Some Difference Equations E. M. Elsayed1,3 and H. El-Metwally2,3 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Rabigh College of Science and Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia. 3

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: 1 [email protected], 2 [email protected]. Abstract In this paper we study the boundedness, investigate the global convergence and the periodicity of the solutions to the following recursive sequence xn+1 = axn +

bx2n−1 + cxn−2 xn−3 , n = 0, 1, ... , dx2n−1 + exn−2 xn−3

where the parameters a, b, c, d and e are positive real numbers and the initial conditions x−3 , x−2 , x−1 and x0 ∈ (0, ∞). Also we give some numerical examples of some special cases of considered equation and presented some rleated graphs and figures using Matlab.

Keywords: recursive sequence, boundedness, global stability, periodic solutions. Mathematics Subject Classification: 39A10 –––––––––––––––––

1

Introduction

Our goal in this paper is to investigate the global stability character and the periodicity of the solutions of the recursive sequence xn+1 = axn +

bx2n−1 +cxn−2 xn−3 , dx2n−1 +exn−2 xn−3

(1)

1

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where the parameters a, b, c, d and e are positive real numbers and the initial conditions x−3 , x−2 , x−1 , and x0 ∈ (0, ∞). Also we give some numerical examples of some special cases of Eq. (1) and presented some related graphs and figures using Matlab. The study of the nonlinear rational difference equations of a higher order is quite challenging and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order. Recently there has been a lot of interest in studying the global attractivity, the boundedness character and the periodicity nature of nonlinear difference equations where for example: DeVault et al. [3] have studied the global stability and the periodic character of solutions of the equation yn+1 =

p+yn−k qyn +yn−k .

Elabbasy et al. [4] investigated the global stability, periodicity character and gave the solution of special case of the following recursive sequence xn+1 = axn −

bxn cxn −dxn−1 .

Elabbasy et al. [5] investigated the global stability, periodicity character and gave the solution of some special cases of the difference equation xn+1 =

dxn−l xn−k cxn−s −b

+ a.

Yang et al. [15] investigated the invariant intervals, the global attractivity of equilibrium points, and the asymptotic behavior of the solutions of the recursive sequence n−1 +bxn−2 xn+1 = ax c+dxn−1 xn−2 , For some related work see [1—25]. Let I be an interval of real numbers and let F : I k+1 → I be a continuously differentiable function. Consider the difference equation yn+1 = F (yn , yn−1 , ..., yn−k ) , n = 0, 1, ...,

(2)

with y−k , ..., y0 ∈ I. Recall that the point y ∈ I is called an equilibrium point of Eq.(2) if F (y, y, ..., y) = y. That is, yn = y for n ≥ 0, is a solution of Eq.(2), or equivalently, y is a fixed point of F . Let y be an equilibrium point of Eq.(2). Then the linearized equation of Eq.(2) about y is given by wn+1 =

k P

pi wn−i , n = 0, 1, ...,

(3)

i=0

2

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where pi = Eq.(3) is

∂f ∂yn−i (y, ..., y), i

= 0, 1, 2..., k and the characteristic equation of

λ(k+1) − p1 λk − p2 λ(k−1) − ... − pk λ − p(k+1) = 0. Theorem A [9]: Assume that p, q ∈ R and k ∈ {0, 1, 2, ...}. Then |p| + |q| < 1 is a sufficient condition for the asymptotic stability of the difference equation un+1 + pun + qun−k = 0, n = 0, 1, ... . Remark: Theorem A can be easily extended to a general linear equations of the form (4) un+k + p1 un+k−1 + ... + pk un = 0, n = 0, 1, ... where p1 , p2 , ..., pk and k ∈ {1, 2, ...}. Then Eq.(4) is asymptotically stable proPk vided that i=1 |pi | < 1. Theorem B [6]: Let {yn }∞ n=−k be a solution of Eq.(2), and suppose that there exist constants A ∈ I and B ∈ I such that A ≤ yn ≤ B for all n ≥ −k. Let ∞ 0 be a limit point of the sequence {yn }n=−k . Then the following statements are true: (i) There exists a solution {Ln }∞ n=−∞ of Eq.(2), called a full limiting sequence of {yn }∞ n=−k , such that L0 = 0 , and such that for every N ∈ {...,−1,0,1,...} LN is a limit point of {yn }∞ n=−k . ∞ (ii) For every i0 ≤ −k, there exists a subsequence {yri }∞ i=0 of {yn }n=−k such that LN = lim yri +N for every N ≥ i0 . i→∞

Theorem C [10]: Assume for the difference equation (2) that F (Y ) > 0 ¯ ¯ k P ¯ ¯ ∂F for all 0 6= Y ∈ I k+1 where Y = (y0 , y1 , ..., yk ).If yj ¯ ∂y (Y ) ¯ ≤ F (Y ) for j j=0

all Y ∈ I k+1 , then Eq.(2) has stability trichotomy, that is exactly one of the following three cases holds for all solutions of Eq.(2): (i) lim yn = ∞ for all (y−k , y−k+1 , ..., y−1 , y0 ) 6= (0, 0, ...0). n→∞

(ii) lim yn = 0 for all initial points and y = 0 is the only equilibrium point of n→∞

Eq.(2). (iii) lim yn = y for all (y−k , y−k+1 , ..., y−1 , y0 ) 6= (0, 0, ...0) and y is the only n→∞

positive equilibrium of Eq.(2).

2

Local Stability of the Equilibrium Point of Eq.(1)

This section deals with study the local stability character of the equilibrium point of Eq.(1) b+c . Then if a < 1, Eq.(1) The equilibrium points of Eq.(1) given by x = ax + d+e b+c has the unique positive equilibrium point x = (1−a)(d+e) . Let f : (0, ∞)4 −→

(0, ∞) be a continuous function defined by f (u, v, w, t) = au +

bv2 +cwt dv 2 +ewt .

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Therefore it follows that ∂f (u,v,w,t) ∂u

= a,

∂f (u,v,w,t) ∂w

=

∂f (u,v,w,t) ∂v

=

2(be−dc)vwt (dv 2 +ewt)2 ,

(cd−be)v 2 t ∂f (u,v,w,t) (dv 2 +ewt)2 , ∂t

=

(cd−be)v2 w (dv2 +ewt)2 .

Then we see that ∂f (x,x,x,x) ∂u ∂f (x,x,x,x) ∂w ∂f (x,x,x,x) ∂t

= a = −a3 , =

(cd−be) (d+e)2 x

=

(cd−be) (d+e)2 x

∂f (x,x,x,x) ∂v

=

=

(cd−be)(1−a) (d+e)(b+c)

=

(cd−be)(1−a) (d+e)(b+c)

2(be−cd) (d+e)2 x

=

2(be−cd)(1−a) (d+e)(b+c)

= −a2 ,

= −a1 , = −a0 .

Then the linearized equation of Eq.(1) about x is yn+1 + a3 yn + a2 yn−1 + a1 yn−2 + a0 yn−3 = 0,

(5)

whose characteristic equation is λ4 + a3 λ3 + a2 λ2 + a1 λ + a0 = 0.

(6)

Theorem 1 Assume that 4 |(be − dc)| < (d + e)(b + c). Then the positive equilibrium point of Eq.(1) is locally asymptotically stable. Proof: It is follows by Theorem A that, Eq.(5) is asymptotically stable if all roots of Eq.(6) lie in the open disc |λ| < 1 that is if

and so

|a3 | + |a2 | + |a1 | + |a0 | < 1. ¯ ¯ ¯ ¯ ¯ 2(be−cd)(1−a) ¯ ¯ 2(cd−be)(1−a) ¯ |a| + ¯ (d+e)(b+c) ¯ + ¯ (d+e)(b+c) ¯ < 1, ¯ ¯ ¯ ¯ 4 ¯ (be−dc)(1−a) (d+e)(b+c) ¯ < (1 − a),

a < 1,

or 4 |be − dc| < (d + e)(b + c). The proof is complete.

3

Boundedness and Existence of Unbounded Solutions of Eq.(1)

Here we study the boundedness nature and persistence of solutions of Eq.(1). Theorem 2 Assume a < 1. Then every solution of Eq.(1) is bounded and persists. Proof: Let {xn }∞ n=−3 be a solution of Eq.(1). It follows from Eq.(1) that xn+1 = axn +

bx2n−1 +cxn−2 xn−3 dx2n−1 +exn−2 xn−3

= axn +

bx2n−1 dx2n−1 +exn−2 xn−3

+

cxn−2 xn−3 . dx2n−1 +exn−2 xn−3

4

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Then xn+1 ≤ axn +

bx2n−1 dx2n−1

+

cxn−2 xn−3 exn−2 xn−3

= axn +

b d

+

c e

for all

n ≥ 1.

By using a comparison, we see that be+cd ed(1−a)

lim supxn ≤ n→∞

= M.

(7)

Thus the solution is bounded from above. Now we wish to show that there exists m > 0 such that xn ≥ m for all n ≥ 1. The transformation xn = y1n , will reduce Eq.(1) to the equivalent form yn+1 =

2 yn (dyn−2 yn−3 +eyn−1 ) 2 2 a(dyn−2 yn−3 +eyn−1 )+yn (byn−2 yn−3 +cyn−1 )

.

It follows that yn+1

≤ ≤

2 yn (dyn−2 yn−3 +eyn−1 )

(

2 yn byn−2 yn−3 +cyn−1

dyn−2 yn−3 byn−2 yn−3

+

1 yn

≥

2 eyn−1 2 cyn−1

dyn−2 yn−3 2 byn−2 yn−3 +cyn−1

=

) ≤

d b

+

e c

=

be+cd bc

+

=H

2 eyn−1 2 byn−2 yn−3 +cyn−1

for all n ≥ 1.

Thus we obtain xn =

1 H

=

bc be+cd

= m for all n ≥ 1.

(8)

From (7) and (8) we see that m ≤ xn ≤ M

for all n ≥ 1.

Therefore every solution of Eq.(1) is bounded and persists. Theorem 3 Assume that a ≥ 1. Then Eq.(1) possesses unbounded solutions. Proof: Let {xn }∞ n=−3 be a solution of Eq.(1) and set A =

bx2n−1 +cxn−2 xn−3 . dx2n−1 +exn−2 xn−3

min{b,c} Then it is easy to show that max{d,e} ≤ A < db + ec . Therefore xn+1 = axn + A. Then ½ n n a x0 + A(aa−1−1) , if a 6= 1, for all n ≥ 1. xn = if a = 1. x0 + A n,

Then, for a ≥ 1, we obtain lim xn = ∞, and this completes the proof. n→∞

4

Existence of Periodic Solutions

In this section we study the existence of periodic solutions of Eq.(1). The following theorem states the necessary and sufficient conditions that this equation has periodic solutions of prime period two. 5

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Theorem 4 Eq.(1) has positive prime period two solutions if and only if (i) (b − c)(d − e)(1 + a) + 4(bae + cd) > 0, d > e,

b > c.

Proof: First suppose that there exists a prime period two solution ..., p, q, p, q, ... of Eq.(1). We will prove that Condition (i) holds. We see from Eq.(1) that p = aq +

bp2 +cpq dp2 +epq

= aq +

bp+cq dp+eq ,

q

bq 2 +cqp dq 2 +eqp

= ap +

bq+cp dq+ep .

= ap +

Then dp2 + epq dq 2 + epq

= adpq + aeq 2 + bp + cq, = adpq + aep2 + bq + cp.

(9) (10)

Subtracting (9) from (10) gives d(p2 − q 2 ) = −ae(p2 − q 2 ) + (b − c)(p − q). Since p 6= q, it follows that p+q =

(b−c) (d+ae) .

(11)

Again, adding (9) and (10) yields d(p2 + q 2 ) + 2epq (d − ae)(p + q 2 ) + 2(e − ad)pq 2

= 2adpq + ae(p2 + q 2 ) + (b + c)(p + q), = (b + c)(p + q). (12)

It follows by (11), (12) and the relation p2 + q 2 = (p + q)2 − 2pq for all p, q ∈ R, that 2(e − d)(1 + a)pq

=

2(bae+cd)(b−c) . (d+ae)2

pq

=

(bae+cd)(b−c) (d+ae)2 (e−d)(1+a) .

(13)

Now it is clear from Eq.(11) and Eq.(13) that p and q are the two distinct roots of the quadratic equation ³ ´ ³ ´ (b−c) (bae+cd)(b−c) t2 − (d+ae) t + (d+ae) = 0, 2 (e−d)(1+a) ³ ´ (bae+cd)(b−c) = 0, (14) (d + ae)t2 − (b − c)t + (d+ae)(e−d)(1+a) 6

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and so

[b − c]2 +

4(bae+cd)(b−c) (d−e)(1+a)

> 0.

or (b − c)(d − e)(1 + a) + 4(bae + cd) > 0. Therefore Inequality (i) holds. Second suppose that Inequality (i) is true. We will show that Eq.(1) has a prime period two solution. Assume that q b−c+ζ b−c−ζ , q = 2(d+ae) , where ζ = [b − c]2 − 4(bae+cd)(b−c) p = 2(d+ae) (e−d)(1+a) . We see from Inequality (i) that

(b − c)(d − e)(1 + a) + 4(bae + cd) > 0, b > c, d > e, which equivalents to (b − c)2 > 4(bae+cd)(b−c) (e−d)(1+a) . Therefore p and q are distinct real numbers. Set x−3 = p, x−2 = q, x−1 = p and x0 = q. We wish to show that x1 = x−1 = p and x2 = x0 = q. It follows from Eq.(1) that x1 = aq +

2

bp +cqp dp2 +eqp

= aq +

bp+cq dp+eq

=a

³

b−c−ζ 2(d+ae)

´

+

    b−c−ζ b−c+ζ b 2(d+ae) +c 2(d+ae)    . b−c+ζ b−c−ζ d 2(d+ae) +e 2(d+ae)

Dividing the denominator and numerator by 2(d + ae) gives x1 =

ab−ac−aζ 2(d+ae)

+

b(b−c+ζ)+c(b−c−ζ) d(b−c+ζ)+e(b−c−ζ)

=

ab−ac−aζ 2(d+ae)

+

(b−c)[(b+c)+ζ] (d+e)(b−c)+(d−e)ζ .

Multiplying the denominator and numerator of the right side by (d + e)(b − c) − (d − e)ζ gives x1

=

ab−ac−aζ 2(d+ae)

+

(b−c)[(b+c)+ζ][(d+e)(b−c)−(d−e)ζ] [(d+e)(b−c)+(d−e)ζ][(d+e)(b−c)−(d−e)ζ]

=

ab−ac−aζ 2(d+ae)

(b−c){(d+e)(b2 −c2 )+ζ[(d+e)(b−c)−(d−e)(b+c)]−(d−e)ζ 2 } (d+e)2 (b−c)2 −(d−e)2 ζ 2   4(bae+cd)(b−c) (b−c) (d+e)(b2 −c2 )+2ζ(eb−cd)−(d−e)(b−c)2 − (1+a) ab−ac−aζ   4(bae+cd)(b−c) 2(d+ae) + (d+e)2 (b−c)2 −(d−e)2 [b−c]2 − (e−d)(1+a)

+ =

=

ab−ac−aζ 2(d+ae)

+

    2(bae+cd) (b−c) 2(b−c) dc+eb− (1+a) +2ζ(eb−cd)   (e−d)(bae+cd) 4(b−c) ed(b−c)+ (1+a)

Multiplying the denominator and numerator of the right side by (1 + a) we obtain x1

=

ab−ac−aζ 2(d+ae)

+

(b−c)[(dc+eb)(1+a)−2(bae+cd)]+ζ(1+a)(eb−cd) 2[ed(b−c)(1+a)+(e−d)(bae+cd)]

=

ab−ac−aζ 2(d+ae)

+

(eb−dc){(b−c)(1−a)+ζ(1+a)} 2(eb−cd)(d+ae)

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=

ab−ac−aζ+(b−c)(1−a)+ζ(1+a) 2(d+ae)

b−c+ζ 2(d+ae)

=

= p.

Similarly as before one can easily show that x2 = q. Then it follows by induction that x2n = q

and

x2n+1 = p

for all

n ≥ −1.

Thus Eq.(1) has the prime period two solution ...,p,q,p,q,... where p and q are the distinct roots of the quadratic equation (14) and the proof is complete.

5

Global Attractor of the Equilibrium Point of Eq.(1)

In this section we investigate the global asymptotic stability of Eq.(1). c b and , the function f (u, v, w, t) defined d e by Eq.(6) has the monotonicity behavior in its arguments. Lemma 5 For any values of the quotient

Proof: The proof follows by some simple computations and it will be omitted. Theorem 6 The equilibrium point x is a global attractor of Eq.(1) if one of the following statements holds (1) be ≥ dc and 2(2f c − be) ≥ e(b − c) (2) be ≤ dc and 2(2gb − cd) ≥ e(c − b)

h

h

be cf cd gb

i2

i2

, where f = d(1 − a).

(15)

, where g = e(1 − a).

(16)

Proof: Let {xn }∞ n=−3 be a solution of Eq.(1) and again let f be a function defined by Eq.(4) We will prove the theorem when Case (1) is true and the proof of Case (2) is similar and will be omitted. Assume that (15) is true, then it results from the calculations after formula (4) that the function f (u, v, w, t) is non-decreasing in u, v and non-increasing in w, t. Thus from Eq.(1), we see that xn+1 = axn +

bx2n−1 +cxn−2 xn−3 dx2n−1 +exn−2 xn−3

≤ axn +

bx2n−1 +c(0) dx2n−1 +e(0)

= axn + db .

Then xn xn+1

≤

b d(1−a)

= axn + ≥

=

b f

=H

for all n ≥ 1.

bx2n−1 +cxn−2 xn−3 dx2n−1 +exn−2 xn−3

cxn−2 xn−3 exn−2 xn−3

=

c e

≥ a(0) +

= h f or all

(17) b(0)+cxn−2 xn−3 d(0)+exn−2 xn−3

n ≥ 1.

(18)

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Then from Eqs.(17) and (18), we see that 0 3/(3d + 1). Then for any nonzero value c, the function f n (f ′ )n1 · · · (f (k) )nk −c has infinitely many zeros. Moreover, if n ≥ 2, the deficient condition can be omitted. It’s natural to ask whether there exists normality criteria corresponding to Theorem H. We consider this problem and obtain Theorem 1.1. Let a ̸= 0 be a constant, n ≥ 2, k ≥ 1, nk ≥ 1, nj (j = 1, 2, · · · , k − 1) be nonnegative integers. Let F be a family of meromorphic functions in the plane domain D such that each f ∈ F has only zeros of multiplicity at least k. For each pair (f, g) ∈ F, if f n (f ′ )n1 · · · (f (k) )nk and g n (g ′ )n1 · · · (g (k) )nk share a, then F is normal in D. Example 1.1. Let D = {z : |z| < 1} and F = {fm } where fm := emz , and for every pair of functions f, g ∈ F, (f n )(k) and (g n )(k) share 0 in D, it is easy to verify that F is not normal at the point z = 0. Example 1.2. Let D = {z : |z| < 1} and F = {fm } where fm := mz +

1 mk(k+1)! . Then F, (f k+1 )(k) and

k+1 )(k) = mk+1 (k + 1)!z + 1, and for every pair of functions f, g ∈ (fm k+1 (g )(k) share 1 in D, it is easy to verify that F is not normal at the point z = 0.

Remark 1.2. In Theorem 1.1, let k = nk = 1, then f n (f ′ )n1 · · · (f (k) )nk = g n (g ′ )n1

· · · (g (k) )nk

=

(g n+1 )′ n+1 ,

(f n+1 )′ n+1

and

this case is a corollary of Theorem E. Examples 1.1 and 1.2 2

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given by Li and Gu show that the condition a ̸= 0 in Theorem E is inevitable and n ≥ k + 2 in Theorem E is sharp. The examples also show that the conditions in Theorem 1.1 are sharp, at least for the case k = nk = 1. If n = 1, we have Theorem 1.3. Let a ̸= 0 be a constant, k > 2, nk ≥ 1, nj (j = 1, 2, · · · , k − 1) be nonnegative integers such that n1 + · · · + nk−1 ≥ 1. Let F be a family of meromorphic functions in the plane domain D such that each f ∈ F has only zeros of multiplicity at least k. For each pair (f, g) ∈ F , if f (f ′ )n1 · · · (f (k) )nk and g(g ′ )n1 · · · (g (k) )nk share a, then F is normal in D. Remark 1.4. In Theorem 1.3, if n1 ≥ 2, the theorem still holds for k = 2. Corollary 1.5. Let a ̸= 0 be a constant, n, k, nk be positive integers such that nk ≥ 2 and nj (j = 1, 2, · · · , k − 1) be nonnegative integers. Let F be a family of holomorphic functions in the plane domain D such that each f ∈ F has only zeros of multiplicity at least k. For each pair (f, g) ∈ F , if f n (f ′ )n1 · · · (f (k) )nk and g n (g ′ )n1 · · · (g (k) )nk share a, then F is normal in D. Remark 1.6. Examples 1.2 shows that Corollary 1.5 fails if n = k = 1 and thus the condition nk ≥ 2 in Corollary 1.5 is inevitable.

2

Preliminary lemmas

Lemma 2.1 ([12]). 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 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 2.2 ([4]). Let f be an entire function and M a positive integer. If f ♯ (z) ≤ M for all z ∈ C, then f has the order at most one. Lemma 2.3 ([19]). Take nonnegative integers n, n1 , · · · , nk with n ≥ 1, nk ≥ 1 and define d = n+n1 +n2 +· · ·+nk . Let f be a transcendental meromorphic function whose zeros have multiplicity at least k. Then for any nonzero value c, the function f n (f ′ )n1 · · · (f (k) )nk − c has infinitely many zeros, provided that n1 + n2 + · · · + nk−1 ≥ 1 and k > 2 if n = 1. Specially, if f is transcendental entire, the function f n (f ′ )n1 · · · (f (k) )nk − c has infinitely many zeros. 3

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Lemma 2.4. Take nonnegative integers n1 , · · · , nk , k with nk ≥ 1, k ≥ 2, n1 + n2 + · · · + nk−1 ≥ 1 and define d = 1 + n1 + n2 + · · · + nk . Let f be a non-constant rational function whose zeros have multiplicity at least k. Then for any nonzero value c, the function f (f ′ )n1 · · · (f (k) )nk − c has at least two distinct zeros. Proof. We shall divide our argument into two cases. Case 1. Suppose that f (f ′ )n1 · · · (f (k) )nk − c has exactly one zero. Case 1.1. If f is a non-constant polynomial, since the zeros of f have multiplicity at least k, we know that f n (f ′ )n1 · · · (f (k) )nk is also a non-constant polynomial, so f (f ′ )n1 · · · (f (k) )nk − c has at least one zero, suppose that f (f ′ )n1 · · · (f (k) )nk = c + B(z − z0 )l ,

(2.1)

where B is a nonzero constant and l > 1 is an integer. Then (2.5) implies that f (f ′ )n1 · · · (f (k) )nk has only simple zeros, which contradicts the assumption that the zeros of f have multiplicity at least k ≥ 2. Case 1.2. If f is a non-constant rational function but not a polynomial. Set f (z) = A

(z − a1 )m1 (z − a2 )m2 · · · (z − as )ms , (z − b1 )l1 (z − b2 )l2 · · · (z − bt )lt

(2.2)

where A is a nonzero constant and mi ≥ k (i = 1, 2, · · · , s), lj ≥ 1 (j = 1, 2, · · · , t). Then f (k) (z) = A

(z − a1 )m1 −k (z − a2 )m2 −k · · · (z − as )ms −k gk (z) . (z − b1 )l1 +k (z − b2 )l2 +k · · · (z − bt )lt +k

(2.3)

For simplicity, we denote m1 + m2 + · · · + ms = M ≥ ks,

(2.4)

l1 + l2 + · · · + lt = N ≥ t.

(2.5)

deg(gk ) ≤ k(s + t − 1),

(2.6)

It is easily obtained that

moreover, gk (z) = (M −N )(M −N −1) · · · (M −N −k+1)z k(s+t−1) +cm z k(s+t−1)−1 +· · ·+c0 . Combining (2.6) and (2.7) yields ′ n1

f (f )

· · · (f

(k) nk

)

d (z

=A

− a1 )dm1 −

∑k j=1

jnj

· · · (z − as )dms −

∑ dl1 + kj=1 jnj

(z − b1 )

∑k j=1

jnj

g(z)

∑ dlt + kj=1 jnj

· · · (z − bt )

where P (z), Q(z), g(z) are polynomials and g(z) =

∏k

nj j=1 gj (z)

=

with deg(g) ≤

t−1). Moreover g(z) = (M −N )d−1 (M −N −1)d−1−n1 · · · (M −N −k+1)nk z

P (z) , (2.7) Q(z) ∑k

j=1 jnj (s+ j=1 jnj (s+t−1)

∑k

+

4

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∑k

dm z j=1 jnj (s+t−1)−1 + · · · + d0 . Then from (2.10) we obtain ′ n1

(f (f )

· · · (f

(k) nk ′

d (z

) ) =A

− a1 )dm1 −

∑k j=1

(z − b1 )

dl1 +

jnj −1

∑k

· · · (z − as )dms −

j=1 jnj +1

∑k j=1

jnj −1

h(z)

∑ dlt + kj=1 jnj +1

· · · (z − bt )

,

(2.8)

∑ where h(z) is a polynomial with deg(h) ≤ ( kj=1 jnj + 1)(s + t − 1). Since f (f ′ )n1 · · · (f (k) )nk − c has exactly one zero, we obtain from (2.11) that f (f ′ )n1 · · · (f (k) )nk = c +

B(z − z0 )l

∑ dl1 + kj=1 jnj

(z − b1 )

∑k

· · · (z − bt )dlt +

j=1

jnj

,

(2.9)

where B is a nonzero constant. Then (f (f ′ )n1 · · · (f (k) )nk )′ =

B(z − z0 )l−1 H(z)

∑ dl1 + kj=1 jnj +1

(z − b1 )

· · · (z − bt )dlt +

∑k j=1

jnj +1

,

(2.10)

where H(z) is a polynomial of the form H(z) = B(l − dN −

k ∑

jnj t)z t + Bt−1 z t−1 + · · · + b0 .

(2.11)

j=1

∑ Case 1.2.1. If l ̸= dN + kj=1 jnj t. From (2.11) we get deg(p) ≥ deg(Q), namely dM −

k ∑

jnj s + deg(g) ≥ dN +

j=1

In view of deg(g) ≤

∑k

j=1 jnj (s

k ∑

jnj t.

(2.12)

j=1

+ t − 1), we deduce from (2.16) that d(M − N ) ≥

k ∑

jnj ,

(2.13)

j=1

which implies M > N . Combining (2.12), (2.14) and (2.15) yields dM − (

k ∑

jnj + 1)s ≤ deg(H) = t ≤ N < M,

(2.14)

j=1

which implies that k ∑ j=1

nj M −

k ∑

jnj s < s,

(2.15)

j=1

5

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this together with (2.8) implies that k−1 ∑

(k − j)nj < 1.

(2.16)

j=1

which is a contradiction since n1 + n2 + · · · + nk−1 ≥ 1. ∑ Case 1.2.2. If l = dN + kj=1 jnj t. If M > N , with similar discussion as above, we get the same contradiction. If M ≤ N , combining (2.12) and (2.14) yields l − 1 ≤ deg(h) ≤ (

k ∑

jnj + 1)(s + t − 1),

(2.17)

j=1

we deduce from (2.21) that dN ≤

k ∑

jnj s + s + t −

j=1

k ∑

jnj
0 such +1) 2(T +1)−αη(η+1) that 0 < α < 2(T . η(η+1) and 0 < β < 2(αη−1) (H2) f ∈ C([0, ∞), [0, ∞)), f is either superlinear or sublinear. Set f0 = lim

u→0+

f (u) , u

f (u) . u→∞ u

f∞ = lim

Then f0 = 0 and f∞ = ∞ correspond to the superlinear case, and f0 = ∞ and f∞ = 0 correspond to the sublinear case. (H3) a ∈ C(NT +1 , [0, ∞)) and there exists t0 ∈ Nη,T +1 such that a(t0 ) > 0. The proof of the main theorem is based upon an application of the following Krasnoselskii’s fixed point theorem in a cone. Theorem 1.1. ([5]). Let E be a Banach space, and let K ⊂ E be a cone. Assume Ω1 , Ω2 are open subsets of E with 0 ∈ Ω1 , Ω1 ⊂ Ω2 , and let A : K ∩ (Ω2 \ Ω1 ) −→ K be a completely continuous operator such that (i) ∥Au∥ 6 ∥u∥, u ∈ K ∩ ∂Ω1 , and ∥Au∥ > ∥u∥, u ∈ K ∩ ∂Ω2 ; or (ii) ∥Au∥ > ∥u∥, u ∈ K ∩ ∂Ω1 , and ∥Au∥ 6 ∥u∥, u ∈ K ∩ ∂Ω2 . Then A has a fixed point in K ∩ (Ω2 \ Ω1 ).

2

Preliminaries We now state and prove several lemmas before stating our main results.

Lemma 2.1. The problem ∆2 u(t − 1) + y(t) = 0, u(0) = β∆u(0),

t ∈ N1,T ,

u(T + 1) = α

η ∑

(2.1)

u(s),

(2.2)

s=1

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has a unique solution ∑ 2(t + β) u(t) = (T − s + 1)y(s) 2(T + 1) − αη(η + 1) − 2β(αη − 1) T

−

s=1 η−1 ∑

α(t + β) 2(T + 1) − αη(η + 1) − 2β(αη − 1)

(η − s)(η − s + 1)y(s)

s=1

t−1 ∑ − (t − s)y(s), t ∈ NT +1 . s=1

Proof. From ∆2 u(t − 1) = ∆u(t) − ∆u(t − 1) and the first equation of (2.1), we get ∆u(t) − ∆u(t − 1) = −y(t), ∆u(t − 1) − ∆u(t − 2) = −y(t − 1), .. . ∆u(1) − ∆u(0) = −y(1). We sum the above equations to obtain ∆u(t) = ∆u(0) −

t ∑

y(s), t ∈ NT .

(2.3)

s=1

We define

q ∑

y(s) = 0; if p > q. Similarly, we sum (2.3) from t = 0 to t = h, and

s=p

by using the boundary condition u(0) = β∆u(0) in (2.2), we obtain h ∑ u(h + 1) = (h + 1 + β)∆u(0) − (h + 1 − s)y(s), h ∈ NT , s=1

by changing the variable from h + 1 to t, we have u(t) = (t + β)∆u(0) −

t−1 ∑

(t − s)y(s), t ∈ NT +1 .

(2.4)

s=1

From (2.4), ) η−1 ∑ η−s ∑ 1 η(η + 1) + βη ∆u(0) − ly(s) u(s) = 2 s=1 l=1 s=1 ( ) η−1 1 1∑ = η(η + 1) + βη ∆u(0) − (η − s)(η − s + 1)y(s) 2 2

η ∑

(

s=1

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Positive solutions of difference-summation boundary value problem ... Again using the boundary condition u(T + 1) = α

η ∑

5

u(s) in (2.2), we obtain

s=1

) ( η−1 T ∑ α∑ 1 (T +1+β)∆u(0)− (T −s+1)y(s) = α η(η+1)+βη ∆u(0)− (η−s)(η−s+1)y(s) 2 2 s=1

s=1

Thus, ∑ 2 (T − s + 1)y(s) 2(T + 1) − αη(η + 1) − 2β(αη − 1) T

∆u(0) =

−

s=1 η−1 ∑

α 2(T + 1) − αη(η + 1) − 2β(αη − 1)

(η − s)(η − s + 1)y(s).

s=1

Therefore, (2.1)-(2.2) has a unique solution ∑ 2(t + β) u(t) = (T − s + 1)y(s) 2(T + 1) − αη(η + 1) − 2β(αη − 1) T

−

s=1 η−1 ∑

α(t + β) 2(T + 1) − αη(η + 1) − 2β(αη − 1)

(η − s)(η − s + 1)y(s)

s=1

t−1 ∑ − (t − s)y(s), t ∈ NT +1 . s=1

 Lemma 2.2. The function    (s + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] + αs(t + β)(1 − s),    1 2(s + β)(T + 1 − t) + αη(t − s)(η + 1 + 2β), G(t, s) = Λ  (t + β)[2(T + 1) − αη(η + 1) + 2s(αη − 1) + αs(1 − s)],     2(t + β)(T + 1 − s),

s ∈ N1,t−1 ∩ N1,η−1 s ∈ Nη,t−1 s ∈ Nt,η−1 s ∈ Nt,T ∩ Nη,T (2.5)

where Λ = 2(T + 1) − αη(η + 1) − 2β(αη − 1) > 0, is the Green’s function of the problem − ∆2 u(t − 1) = 0, u(0) = β∆u(0),

t ∈ N1,T , u(T + 1) = α

η ∑

u(s).

(2.6)

s=1

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Jiraporn Reunsumrit and Thanin Sitthiwirattham

Proof. Suppose t < η. The unique solution of problem (2.1)-(2.2) can be written u(t) = −

t−1 ∑ (t − s)y(s) s=1

[ t−1 ] η−1 T ∑ ∑ 2(t + β) ∑ + (T − s + 1)y(s) + (T − s + 1)y(s) + (T − s + 1)y(s) Λ s=η s=t s=1

[∑ t−1

] η−1 ∑ α(t + β) (η − s)(η − s + 1)y(s) + − (η − s)(η − s + 1)y(s) Λ s=t s=1 ] t−1 [ 1∑ = (s + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] + αs(t + β)(1 − s) y(s) Λ s=1 ] η−1 [ 1∑ 2 + (t + β)[2(T + 1) − αη(η + 1) + 2s(αη − 1) + αs − αs ] y(s) Λ s=t 1∑ 2(t + β)(T + 1 − s)y(s) + Λ s=η T

=

T ∑

G(t, s)y(s).

s=1

Suppose t ≥ η. The unique solution of problem (2.1)-(2.2) can be written u(t) = −

η−1 t−1 ∑ ∑ (t − s)y(s) − (t − s)y(s) s=1

+

2(t + β) Λ

[∑ η−1

s=η

(T − s + 1)y(s) +

t−1 ∑ s=η

s=1

(T − s + 1)y(s) +

T ∑

] (T − s + 1)y(s)

s=t

η−1 ∑

α(t + β) (η − s)(η − s + 1)y(s) Λ s=1 ] η−1 [ ∑ 1 = (s + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] + αs(t + β)(1 − s) y(s) Λ s=1 ] t−1 [ 1∑ + 2(s + β)(T + 1 − t) + αη(t − s)(η + 1 + 2β) y(s) Λ s=η −

1∑ 2(t + β)(T + 1 − s)y(s) Λ s=t T

+

=

T ∑

G(t, s)y(s).

s=1

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Positive solutions of difference-summation boundary value problem ...

7

Then the unique solution of problem (2.1)-(2.2) can be written as u(t) = T ∑



G(t, s)y(s). The proof is complete.

s=1 +1) 2(T +1)−αη(η+1) We observe that the condition 0 < α < 2(T . η(η+1) and 0 < β < 2(αη−1) implies G(t, s) is positive on N1,T × N1,T , which mean that the finite set { } G(t, s) : t ∈ NT +1 , s ∈ N1,T , G(t, t)

take positive values. Then we let } { G(t, s) M1 = min : t ∈ N1,T , s ∈ N1,T G(t, t) { } G(t, s) M2 = max : t ∈ NT +1 , s ∈ N1,T G(t, t)

(2.7)

(2.8)

Lemma 2.3. Let (t, s) ∈ N1,T × N1,T . Then we have G(t, s) ≥ M1 G(t, t.)

(2.9)

where 0 < M1 < 1 is a constant given by  {  (1+β)[2T −αη(η−1)]−α(η−1+β)(η−2)2 2[T −α−αη(η−2)]  min , 2(T   (η+β−1)[2(T +2−η)+αη(η−3)] +2−η)+αη(η−3) ,     2(η+β)+αη(η+1+2β) 2 1    2(T +2−η)+αη(η−3) , 2(T +β)(T +1−η) , T +1−η ,  }    (1+β)[2(T +1−η)+αη(η−1)]−α(T +β)(η−1)2 +α(η−1)(η+β)  ; if α > η1  2(T +β)(T +1−η) { M1 =  (1+β)[2(T +2−η)+αη(η−3)]−α(η−1+β)(η−2)2 2(T +2−η−α)  min , 2T −αη(η−1) ,   (η+β−1)[2T −αη(η−1)]     2(η+β)+αη(η+1+2β) 1 2    2T −αη(η−1) , 2(T +β)(T +1−η) , T +1−η ,  }    (1+β)[2(T +2−η)+αη(η−3)]−α(η−1+β)(η−2)2 2(T +2−η−α)  , ; if 0 < α <  (η+β−1)[2T −αη(η−1)]

2T −αη(η−1)

1 η

(2.10) Proof. In order that (2.9) holds, it is sufficient that M1 satisfies M1 ≤

min

(t,s)∈N1,T ×N1,T

G(t, s) . G(t, t)

(2.11)

Then we may choose { M1 ≤ min

min

(t,s)∈N1,η−1 ×N1,T

G(t, s) G(t, s) , min G(t, t) (t,s)∈Nη,T ×N1,T G(t, t)

325

} .

(2.12)

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Jiraporn Reunsumrit and Thanin Sitthiwirattham

since G(t, s) min (t,s)∈N1,η−1 ×N1,T G(t, t) { (s + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] + αs(t + β)(1 − s) = min min , t∈N1,η−1 s∈N1,t−1 (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] (t + β)[2(T + 1) − αη(η + 1) + 2s(αη − 1) + αs(1 − s)] min , s∈Nt,η−1 (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] } 2(t + β)(T + 1 − s) min s∈Nη,T (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] { ≥ min

t∈N1,η−1

≥

   min    t∈N1,η−1                  min   t∈N1,η−1              

(1 + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] + α(t − 1)(t + β)(2 − t) , (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] 2(T + 1) − αη(η + 1) + 2s(αη − 1) + αs(1 − s) , min s∈Nt,η−1 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t) } 2 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t) {

(1 + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] − α(t + β)(t − 1)2 + α(t + β)(t − 1) , (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] 2(T + 1) − αη(η + 1) + 2t(αη − 1) + α(η − 1)(2 − η) , 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 −}t) 2 1 ; if α > 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t) η { (1 + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] − α(t + β)(t − 1)2 + α(t + β)(t − 1) , (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] 2(T + 1) − αη(η + 1) + 2(η − 1)(αη − 1) + α(2 − η)(η − 1) , 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 } − t) 2 1 ; if 0 < α < 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t) η

 { 2[T − α − αη(η − 2)] (1 + β)[2T − αη(η − 1)] − α(η − 1 + β)(η − 2)2   , , min    (η + β − 1)[2(T + 2}− η) + αη(η − 3)] 2(T + 2 − η) + αη(η − 3)    2 1   ; if α >  2(T + 2 − η) + αη(η − 3) η { ≥ (1 + β)[2(T + 2 − η) + αη(η − 3)] − α(η − 1 + β)(η − 2)2 2(T + 2 − η − α)    , , min   2T − αη(η − 1)  }(η + β − 1)[2T − αη(η − 1)]   2 1    ; if 0 < α < 2T − αη(η − 1) η

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9

Similarly, we get min

(t,s)∈Nη,T ×N1,T

G(t, s) G(t, t)

 { (1 + β)[2(T + 1 − η) + αη(η − 1)] − α(T + β)(η − 1)2 + α(η − 1)(η + β)   min ,    2(T + β)(T}+ 1 − η)    2(η + β) + αη(η + 1 + 2β) 1 1   , ; if α >  T +1−η η { 2(T + β)(T + 1 − η) ≥ 2 + α(η − 1)(η + β) (1 + β)[2(1 + αηT ) − αη(η + 1)] − α(T + β)(η − 1)    , min   2(T + β)(T } + 1 − η)    2(η + β) + αη(η + 1 + 2β) 1 1    , ; if 0 < α < 2(T + β)(T + 1 − η) T +1−η η (2.13) 

The (2.10) is immediate from (2.13)-(2.14)

Lemma 2.4. Let (t, s) ∈ NT +1 × N1,T . Then we have G(t, s) ≤ M2 G(t, t.)

(2.14)

where M2 ≥ 1 is a constant given by  {  (η−2+β)[2(T +2−η)+αη(η−3) 2(T +2−η)+αη(η−3) T +1−η  max , 2[T −α(η−1)2 ] , T −α(η−1)  2  2(1+β)[T −α(η−1)2 ]   }    (η−1+β)[αη(2T −η−1)+2] 2(T −1+β)(T +1−η)+αη(T −η)(η+1+2β)  , ; if α >  2(η+β) 2(η+β) { M2 =  (η−2+β)[2T −αη(η−1) (η−1+β)[2(T +1−η)+αη(η−1) T +1−η  max ,   2(1+β)(T +2−η−α) , T +2−η−α , 2(η+β)   }    2(T −1+β)(T +1−η)+αη(T −η)(η+1+2β)  ; if 0 < α < η1  2(η+β)

1 η

Proof. For k = 0, from (2.5) we get G(0, s) = 2β(T + 1 − s) < 2β(T + 1) = G(0, 0). Then we may choose M2 = 1. For k ∈ N1,T , if (2.14) holds, it is sufficient that M2 satisfies G(t, s) M2 ≥ max . (2.15) (t,s)∈N1,T ×N1,T G(t, t) Then we may choose { M2 ≥ max

max

(t,s)∈N1,η−1 ×N1,T

G(t, s) G(t, s) , max G(t, t) (t,s)∈Nη,T ×N1,T G(t, t)

327

} .

(2.16)

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since G(t, s) max (t,s)∈N1,η−1 ×N1,T G(t, t) { (s + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] + αs(t + β)(1 − s) = max max , t∈N1,η−1 s∈N1,t−1 (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] (t + β)[2(T + 1) − αη(η + 1) + 2s(αη − 1) + αs(1 − s)] max , s∈Nt,η−1 (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] } 2(t + β)(T + 1 − s) max s∈Nη,T (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)]

{ ≤ max

t∈N1,η−1

≤

   max    t∈N1,η−1              

{

(t − 1 + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] , (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] 2(T + 1) − αη(η + 1) + 2(η − 1)(αη − 1) + αt(1 − t) , 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 −}t) 2(T + 1 − η) 1 ; if α > 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t) η { (t − 1 + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] , (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t) , 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t) } 2(T + 1 − η) 1 ; if 0 < α < 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t) η

  max    t∈N1,η−1              

≤

   max            max

(t − 1 + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1)] , (t + β)[2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)] 2(T + 1) − αη(η + 1) + 2s(αη − 1) + αs(1 − s) max , s∈Nt,η−1 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t) } 2(T + 1 − η) 2(T + 1) − αη(η + 1) + 2t(αη − 1) + αt(1 − t)

{

(η − 2 + β)[2(T + 2 − η) + αη(η − 3) 2(T + 2 − η) + αη(η − 3) , , 2(1 + } β)[T − α(η − 1)2 ] 2[T − α(η − 1)2 ] T +1−η 1 ; if α > 2 T − α(η − 1) η { } (η − 2 + β)[2T − αη(η − 1) T +1−η 1 , 1, ; if 0 < α < 2(1 + β)(T + 2 − η − α) T +2−η−α η

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11

Similarly, we get G(t, s) (t,s)∈Nη,T ×N1,T G(t, t)  { (η − 1 + β)[αη(2T − η − 1) + 2]   max ,    2(η + β)  }   2(T − 1 + β)(T + 1 − η) + αη(T − η)(η + 1 + 2β)   , 1 ;  2(η + β) { ≤ (η − 1 + β)[2(T + 1 − η) + αη(η − 1)   , max    2(η + β)  }   2(T − 1 + β)(T + 1 − η) + αη(T − η)(η + 1 + 2β)   , 1 ;  2(η + β) max

if α >

if 0 < α
0, t ∈ NT +1 and min u(t) ≥ σ ∥ u ∥}. t∈N1,T

where σ = in E.

M1 m M2 M

∈ (0, 1), ∥ u ∥= maxt∈NT +1 | u(t) |. It is obvious that K is a cone

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Jiraporn Reunsumrit and Thanin Sitthiwirattham We define the operator F : K → E by (F u)(t) =

T ∑

G(t, s)a(s)f (u(s)), t ∈ NT +1 .

s=1

It is clear that problem (1.1)-(1.2) has a solution u if and only if u ∈ K is a fixed point of operator F . We shall now show that the operator F maps K to itself. For this, let u ∈ K, from (H2 ) − (H3 ), we get (F u)(t) =

T ∑

G(t, s)a(s)f (u(s)) ≥ 0, t ∈ NT +1 .

(3.1)

s=1

from (2.8), we obtain (F u)(t) =

T ∑

G(t, s)a(s)f (u(s)) ≤ M2

s=1

T ∑

G(t, t)a(s)f (u(s))

s=1

≤M2 M

T ∑

a(s)f (u(s)), t ∈ NT +1 .

s=1

Therefore ∥ F u ∥≤ M2 M

T ∑

a(s)f (u(s)).

(3.2)

s=1

Now from (H2 ), (H3 ), (2.7) and (3.2), for t ∈ Nη,T , we have (F u)(t) ≥M1

T ∑

G(t, t)a(s)f (u(s)) ≥ M1 m

s=1

≥

T ∑

a(s)f (u(s))

s=1

M1 m ∥ F u ∥= σ ∥ u ∥ . M2 M

Then min (F u)(t) ≥ σ ∥ u ∥ .

t∈Nη,T

(3.3)

From (3.1)-(3.2), we obtain F u ∈ K, Hence F (K) ⊆ K. So F : k → K is completely continuous. Superlinear case. f0 = 0 and f∞ = ∞. Since f0 = 0, we may choose H1 > 0 so that f (u) 6 ϵ1 u, for 0 < u 6 H1 , where ϵ1 > 0 satisfies ϵ1 M2 M

T ∑

a(s) ≤ 1.

(3.4)

s=1

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13

Thus, if we let Ω1 = {u ∈ E : ∥u∥ < H1 }, then for u ∈ K ∩ ∂Ω1 , we get (F u)(t) ≤M2

T ∑

G(t, t)a(s)f (u(s)) ≤ ϵ1 M2 M

s=1

≤ϵ1 M2 M

T ∑

a(s)u(s)

s=1 T ∑

a(s)∥u∥ ≤ ∥u∥.

s=1

Thus ∥F u∥ ≤ ∥u∥, u ∈ K ∩ ∂Ω1 . b 2 > 0 such that f (u) ≥ ϵ2 u, for u ≥ H b2, Further, since f∞ = ∞, there exists H where ϵ2 > 0 satisfies T ∑ ϵ2 M 1 σ G(η, η)a(s) ≥ 1. (3.5) s=η b

Let H2 = max{2H1 , Hσ2 } and Ω2 = {u ∈ E : ∥u∥ < H2 }. Then u ∈ K ∩ ∂Ω2 implies b2. min u(t) ≥ σ∥u∥ ≥ H t∈Nη,T

Applying (2.7) and (3.5), we get (F u)(η) =M1

T ∑

G(η, s)a(s)f (u(s)) ≥ M1

G(η, η)a(s)f (u(s))

s=η

s=1

≥ε2 M1

T ∑

T ∑

G(η, η)a(s)y(s) ≥ ε2 M1 σ

s=η

T ∑

G(η, η)a(s)∥u∥

s=η

≥∥u∥. Hence, ∥F u∥ ≥ ∥u∥, u ∈ K ∩ ∂Ω2 . By the first part of Theorem 1.1, F has a fixed point in K ∩ (Ω2 \ Ω1 ) such that H1 6 ∥u∥ 6 H2 . Sublinear case. f0 = ∞ and f∞ = 0. Since f0 = ∞, choose H3 > 0 such that f (u) > ϵ3 u for 0 < u 6 H3 , where ε3 > 0 satisfies ϵ3 M 1 σ

T ∑

G(η, η)a(s) > 1.

(3.6)

s=η

Let Ω3 = {u ∈ E : ∥u∥ < H3 },

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then for u ∈ K ∩ ∂Ω3 , we get (F u)(η) ≥M1

T ∑

G(η, η)a(s)f (u(s)) ≥ ϵ3 M1

T ∑

s=η

≥ε3 M1 σ

G(η, η)a(s)y(s)

s=η T ∑

G(η, η)a(s)∥u∥ ≥ ∥u∥.

s=η

Thus, ∥F u∥ > ∥u∥, u ∈ K ∩ ∂Ω3 . b 4 > 0 so that f (u) 6 ϵ4 u for u > H b 4 , where Now, since f∞ = 0, there exists H ϵ4 > 0 satisfies T ∑ ϵ4 M2 M a(s) > 1. (3.7) s=η

Subcase 1. Suppose f is bounded, f (u) ≤ L for all u ∈ [0, ∞) for some L > 0. T ∑ Let H4 = max{2H3 , LM2 M a(s)}. s=1

Then for u ∈ K and ∥u∥ = H4 , we get (F u)(η) ≤M2

T ∑

G(t, t)a(s)f (u(s)) ≤ LM2 M

s=1

T ∑

a(s)

s=1

≤H4 = ∥u∥ Thus (F u)(t) ≤ ∥u∥. b

Subcase 2. Suppose f is unbounded, there exist H4 > max{2H3 , Hσ4 } such that f (u) ≤ f (H4 ) for all 0 < u ≤ H4 . Then for u ∈ K with ∥u∥ = H4 from (2.8) and (3.7), we have (F u)(t) ≤M2

T ∑

G(t, t)a(s)f (u(s)) ≤ M2 M

s=1

≤ϵ4 M2 M

T ∑

a(s)f (H4 )

s=1 T ∑

a(s)H4 ≤ H4 = ∥u∥.

s=1

Thus in both cases, we may put Omega4 = {u ∈ E : ∥u∥ < H4 }. Then ∥F u∥ 6 ∥u∥, u ∈ K ∩ ∂Ω4 . By the second part of Theorem 1.1, A has a fixed point u in K ∩ (Ω4 \ Ω3 ), such that H3 6 ∥u∥ 6 H4 . This completes the sublinear part of the theorem. Therefore, the problem (1.1)-(1.2) has at least one positive solution. 

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4

15

Some examples In this section, in order to illustrate our result, we consider some examples.

Example 4.1

Consider the BVP ∆2 u(t − 1) + t2 uk = 0, 1 u(0) = ∆u(0), 4

t ∈ N1,4 ,

(4.1)

2∑ u(5) = u(s). 3

(4.2)

2

s=1

Set α = 32 , β = 14 , η = 2, T = 4, a(t) = t2 , f (u) = uk . We can show that 2(T + 1) − αη(η + 1) − 2β(αη − 1) =

40 > 0. 6

Case I : k ∈ (1, ∞). In this case, f0 = 0, f∞ = ∞ and (i) of theorem 3.1 holds. Then BVP (4.1)-(4.2) has at least one positive solution. Case II : k ∈ (0, 1). In this case, f0 = ∞, f∞ = 0 and (ii) of theorem 3.1 holds. Then BVP (4.1)-(4.2) has at least one positive solution.

Example 4.2

Consider the BVP

∆2 u(t − 1) + et te (

π sin u + 2 cos u ) = 0, u2

2 u(0) = ∆u(0), 5

t ∈ N1,4 ,

1∑ u(5) = u(s), 3

(4.3)

3

(4.4)

s=1

Set α = 31 , β = 25 , η = 3, T = 4, a(t) = et te , f (u) = We can show that

π sin u+2 cos u . u2

Λ = 2(T + 1) − αη(η + 1) − 2β(αη − 1) = 6 > 0, Through a simple calculation we can get f0 = ∞, f∞ = 0. Thus, by (ii) of theorem 3.1, we can get BVP (4.3)-(4.4) has at least one positive solution.

Acknowledgement(s) : This research (KMUTNB-GEN-56-12) is supported by King Mongkut’s University of Technology North Bangkok, Thailand.

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References [1] V. A. Ilin and E. I. Moiseev, Nonlocal boundary-value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, J. Differential Equations 23(1987), 803-810. [2] C. P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equations, J. Math. Anal. Appl. 168(1992) no.2, 540-551. [3] R.P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, Kluwer Academic Publishers, Dordrecht, 1998. [4] R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999. [5] M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoof, Gronignen, 1964. [6] R.W.Leggett, L.R.Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28(1979), 673-688. [7] Z.Bai,X. Liang,Z. Du, Triple positive solutions for some second-order boundary value problem on a measure chain. Comput. Math. Appl. 53(2007), 1832-1839. [8] X.He, W. Ge, Existence of three solutions for a quasilinear two-point boundary value problem. Comput. Math. Appl. 45(2003), 765769. [9] X. Lin, W. Lin, Three positive solutions of a secound order difference Equations with Three-Point Boundary Value Problem, J.Appl. Math. Comut. 31(2009), 279-288. [10] G. Zhang, R. Medina, Three-point boundary value problems for difference equations, Comp. Math. Appl. 48(2004), 1791-1799. [11] T. Sitthiwirattham, J. Tariboon, Positive Solutions to a Generalized Second Order Difference Equation with Summation Boundary Value Problem. Journal of Applied Mathematics. Vol.2012, Article ID 569313, 15 pages. [12] J. Henderson, H.B. Thompson, Existence of multiple solutions for second order discrete boundary value problems, Comput. Math. Appl. 43 (2002), 1239-1248.

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Essential norm of extended Ces`aro operators from F (p, q, s) space to Bloch-type space Xiaomin Tang Department of Mathematics, Huzhou University, Huzhou, Zhejiang 313000, P. R. China E-mail: [email protected]

Abstract: Let Tg be the extended Ces` aro operator with holomorphic symbol g on the n unit ball of C . In this paper, we characterize the essential norm of Tg as an operator from the F (p, q, s) space to the Bloch-type space Bµ , where µ is a given normal weight. Keywords: F (p, q, s) space, Bloch-type space, Extended Ces` aro operator, Essential norm. 2010 MR Subject Classification: Primary 32AB37, Secondary 47B38.

1

Introduction

Let Cn be the Euclidean space of complex dimension n. For z = (z1 , · · · , zn ) and w = (w1 , · · · , wn ) in Cn , we define hz, wi =

n X

zj wj , |z| =

p

hz, zi.

j=1

Let B = {z ∈ Cn : |z| < 1} be the open unit ball in Cn . When n = 1, the open unit ball is just the open unit disc D. Let H(B) be the family of all holomorphic functions on B. For a ∈ B, let h(z, a) = log |ϕa1(z)| be the Green function with logarithmic singularity at a, where ϕa is the holomorphic n P ∂f (z) the radial automorphism of B which interchanges 0 and a. Denote by 0 such that µ(r) µ(r) ↓ 0, ↑∞ a (1 − r) (1 − r)b β  β eα − 2 p 2 α with as r → 1 . For example, ω(r) = (1 − r ) with 0 < p < ∞, ω(r) = (1 − r ) ln 1−r2 0 < α < ∞ and 0 ≤ β < ∞, and ω(r) = 1/{log log e2 (1 − r2 )−1 } are all normal weights. From now on if we say that a function µ : B −→ (0, ∞) is normal we also assume that it is radial on B, that is µ(z) = µ(|z|) for z ∈ B. A function f ∈ H(B) is said to belong to the Bloch-type space Bµ if kf kB,µ = sup µ(z)|∇f (z)| < ∞, z∈B

and it is said to belong to the little Bloch-type space Bµ,0 if lim µ(z)|∇f (z)| = 0.

|z|→1



∂f (z), · · · ∂z1



∂f (z) ∂zn

Here ∇f (z) = is the complex gradient of f . It is easy to check that , both of Bµ and Bµ,0 are Banach spaces under the norm kf kµ = |f (0)| + kf kB,µ and that Bµ,0 is a closed subspace of Bµ . When µ(r) = 1 − r2 and µ(r) = (1 − r2 )1−α with α ∈ (0, 1), two typical normal weights, the induced spaces Bµ are the Bloch space and the Lipschitz space, respectively. And also, the space B(1−r2 ) log 1/(1−r2 ) is the logarithmic Bloch space which was studied in [3]. It follows from [4] that f ∈ Bµ if and only if sup µ(z)| 0

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and t1 , t2 ∈ [0, T ] such that |t1 − t2 | ≤ ε. With loss of generality we assume that t1 < t2 . By (H1 )-(H4 ) and inequality (3.13), for a fixed function x ∈ X we have.

|(Gx)(t2 ) − (Gx)(t1 )|

≤ |p(t2 ) − p(t1 )| + |(F x)(t2 )(Φx)(t2 ) − (F x)(t1 )(Φx)(t2 )| + |(F x)(t1 )(Φx)(t2 ) − (F x)(t1 )(Φx)(t1 )| Z |f (t2 , x(t2 )) − f (t1 , x(t1 ))| t2 φ(t2 , s, x(s), vx (s)) T ≤ ω (p, ε) + ds Γ(α) (t2 − s)1−α 0 |f (t1 , x(t1 )| α T + [T W (φ, ε, x) + 2εα (m1 (T )χ1 (r0 ) + m2 (T )χ2 (R) + φ1 (T ))] Γ(α + 1) |f (t2 , x(t2 )) − f (t2 , x(t1 ))| − |f (t2 , x(t1 )) − f (t1 , x(t1 ))| ≤ ω T (p, ε) + Γ(α) Z t2 |φ(t2 , s, x(s), vx (s)) − φ(t2 , s, 0, 0)| + |φ(t2 , s, 0, 0)| ds × (t2 − s)1−α 0 |f (t1 , x(t1 )) − f (t1 , 0)| + |f (t1 , 0)| + Γ(α + 1) × [T α W T (φ, ε, x) + 2εα (m1 (T )χ1 (r0 ) + m2 (T )χ2 (R) + φ1 (T ))] n(t2 )|x(t2 ) − x(t1 ))| + ω1T (f, ε) ≤ ω T (p, ε) + Γ(α) Z t2 m1 (t2 )χ1 (|x(s)|) + m2 (t2 )χ2 (|vx (s)|) + φ1 (t2 ) × ds (t2 − s)1−α 0 n(t1 )|x(t1 )| + |f (t1 , 0)| + Γ(α + 1) × [T α W T (φ, ε, x) + 2εα (m1 (T )χ1 (r0 ) + m2 (T )χ2 (R) + φ1 (T ))] n(t2 )ω T (x, ε) + ω1T (f, ε) ≤ ω T (p, ε) + Γ(α + 1) × [m1 (t2 )χ1 (r0 ) + m2 (t2 )χ2 (R) + φ1 (t2 )]tα 2 n(t1 )r0 + |f (t1 , 0)| + Γ(α + 1) × [T α W T (φ, ε, x) + 2εα (m1 (T )χ1 (r0 ) + m2 (T )χ2 (R) + φ1 (T ))]

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STABILITY IN VOLTERRA INTEGRAL EQUATIONS

≤ ω T (p, ε) + + +

11

a1 (t)χ1 (r0 ) + a2 (t)χ2 (L) + c(t) T ω (x, ε) Γ(α + 1)

ω1T (f, ε) [m1 (t2 )χ1 (r0 ) + m2 (t2 )χ2 (R) + φ1 (t2 )]T α Γ(α + 1) n(t1 )r0 + |f (t1 , 0)| Γ(α + 1)

× [T α W T (φ, ε, x) + 2εα (m1 (T )χ1 (r0 ) + m2 (T )χ2 (R) + φ1 (T ))] A1 χ1 (r0 ) + A2 χ2 (R) + C T ≤ ω T (p, ε) + ω (x, ε) Γ(α + 1) ω1T (f, ε) + [m1 (T )χ1 (r0 ) + m2 (T )χ2 (R) + φ1 (T )]T α Γ(α + 1) +

n(T )r0 + f (T ) Γ(α + 1)

× [T α W T (φ, ε, x) + 2εα (m1 (T )χ1 (r0 ) + m2 (T )χ2 (R) + φ1 (T ))], wherein ω1T (f, ε) = sup{|f (t2 , x) − f (t1 , x)| : t1 , t2 ∈ [0, T ], |t1 − t2 | ≤ ε, x ∈ [−r0 , r0 ]}, and φ1 (T )), mi (T ), i = 1, 2, are defined according to (3.14). From the continuity of f and φ on the sets [0, T ] × [−r0 , r0 ] and [0, T ] × [0, T ] × [−r0 , r0 ] × [Q, R], respectively, we have, ω0T (GX) ≤ λω0T (X). Consequently, by taking T → ∞ we have ω T (GX) ≤ λω T (X). Thus based on (3.16) we obtain µ(GX) ≤ λµ(X). Thus, based on Theorem 2.2 operator G has at least one fixed point x ∈ Br . Finally, we show that solutions of system (1.2) are uniformly locally attractive. Let x and y be the fixed points of the operator G. In view of inequality (3.15) we have c(t) |x(t) − y(t)| + p(t) ≤ λ|x(t) − y(t)| + p(t), Γ(α + 1) where λ < 1 is defined in Remark 3.2 and 2a1 (t)r0 χ1 (r0 ) 2a2 (t)r0 χ2 (R) a1 (t)r0 χ1 (2r0 ) p(t) = + + Γ(α + 1) Γ(α + 1) Γ(α + 1) a2 (t)r0 χ2 (2R) b1 (t)χ1 (2r0 ) b2 (t)χ2 (2R) + + . + Γ(α + 1) Γ(α + 1) Γ(α + 1) Consequently |x(t) − y(t)| = |(Gx)(t) − (Gy)(t)| ≤

1 p(t) = 0 1−λ In view of assumption (H4 ) there exists T ≥ 0 such that for t ≥ T we have ηL ε |x(t) − y(t)| ≤ ≤ε (3.17) k + ηL lim |x(t) − y(t)| ≤ lim

t→∞

t→∞

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According to relation above and appealing to Lemma 3.1 there exists T 0 > T such that |vx (t) − vy (t)| ≤ ε, t ≥ T 0. Where vx (t) and vy (t) belong to ball BR .. This compleat the proof of theorem.  4. An example Consider the following quadratic Volterra integral equations of fractional order with control variable Z exp(s−t) p 1 ( 3 x2 (s) + arctan(v(s))) + 1+3t t + tx(t)) t t2 2 10 √ + ds x(t) = 2 1 + 4t Γ(3/2) 0 t−s 2 + cos t + x2 (t) dv 1 + t2 v + = − , t ≥ 0. (4.1) dt 4 + cos t + t2 5 + cos t + 2x2 (t) Note that above equation is a special case of Eq. (1.2) where, t2 , f (t, x) = t + tx(t), 1 + 4t2 exp(s − t) p 1 φ(t, s, x, v) = { 3 x2 (s) + arctan(v(s))} + , 10 1 + 3t2 1 + t2 2 + cos t + x2 η(t) = , g(t, x) = . 2 4 + cos t + t 5 + cos t + 2x2 The function p is continuous and bounded on R+ and kpk = 0.25. Moreover, the function f satisfies assumption (H2 ) with n(t) = t and |f (t, 0)| = f (t, 0) = t. Further, observe that the function φ(t, s, x, v) satisfies assumption (H3 ), where √ 3 m1 (t) = m2 (t) = e−t , χ1 (r) = r2 , χ2 (r) = arctan(r) and φ(t, s, 0, 0) = 1/(1 + 3t2 ) = φ1 (t). Also, we have 3 a1 (t) = a2 (t) = b1 (t) = b2 (t) = t 2 e−t , p(t) =

3

t2 . 1 + 3t2 Thus, it easily seen that functions ai , bi , c and d are bounded as well as limt→∞ ai = 0 and limt→∞ bi = 0, i = 1, 2. In addition, A1 = a1 ( 32 ) = A2 = B1 = B2 = 0.04101... and C = c(1) = D = 0.25. Since |∂g/∂x| < 1, function g satisfies the Lipschitz condition with respect to the second variable. Moreover, simple calculation shows that αL ≥ 51 , αM ≤ 1, gL ≥ 14 gM ≤ 1. Finally, to check that assumption (H6 ) is satisfied let us note that inequality (3.12) has the form c(t) = d(t) =

0.25

+ +

√ 1 3 (0.04101 r2 + 0.04101 arctan(L) + 0.25)r Γ(3/2) √ 3 0.04101 r2 + 0.04101 arctan(L) + 0.25} ≤ r

Let us rewrite the above inequality in the following form I(r) ≤ rΓ(3/2), wherein

√ √ 3 3 I(r) = Γ(3/2)×0.25+0.04101 r2 +0.04101 arctan(L)+0.25)r+0.04101 r2 +0.04101 arctan(L)+0.25.

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STABILITY IN VOLTERRA INTEGRAL EQUATIONS

13

Since arctan(L) < π/2 = 1.5707... and Γ(3/2) ≤ 0.880, for r = 1 we have I(r) ≤ 0.886 × 0.25 + 0.04101 + 0.04101 × 1.5707 + 0.25 + 0.04101 + 0.04101 × 1.5707 + 0.25 = 0.545 < 0.880 ≤ rΓ(3/2). Thus, Eq. (4.1) satisfies assumptions (H1 )-(H6 ). Now, based on Theorem 3.4 we infer that this equation has a solution in the spaceBC(R+ ) belonging to the ball B1 and solutions of this system are uniformly locally attractive. References [1] M. Garh, A. Rao, S.L. Kalla, Fractional generalization of temperature fields problems in oil strata, Mat. Bilten 30 (2006) 71-84. [2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [3] A.A. Kilbas, J.J. Trujillo, Differential equations of fractional order: Methods, results, problems, I, Appl. Anal. 78 (2001) 153-192. [4] R.K. Saxena, S.L. Kalla, On a fractional generalization of free electron laser equation, Appl. Math. Comput. 143 (2003) 89-97. [5] R.K. Saxena, A.M. Mathai, H.L. Haubold, On generalized fractional kinetic equations, Phys. A 344 (2004) 657-664. [6] H.M. Srivastava, R.K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput. 118 (2001) 1-52. [7] J. Bana´s, D. O’Regan, On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order, J. Math. Anal. Appl. 34 (2008) 573-582. [8] S. Lefschetz, Stability of Nonlinear Control System, Academic Press, New York, 1965. [9] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, NewYork, 1993. [10] F. Chen, The permanence and global attractivity of Lotka-Volterra competition system with feedback control, Nonlinear Anal. 7 (2006) 133-143 [11] J. Bana´s, K. Goebel, Measure of noncompactness in the Banach space. Lecture Notes in Pure and Applied Mathematics, vol. 60. New York:Dekker, 1980 [12] J. Bana´s, Measure of noncompactness in the space of continuous temperet functions, Demonstratio Math. 14 (1981) 127-133. Payam Nasertayoob, S. Mansour Vaezpour Department of Mathematics and Computer Sciences, Amirkabir University of Technology, Hafez Ave., P. O. Box 15914, Tehran, Iran Reza Saadati Department of Mathematics and Computer Science, Iran University of Science and Technology, Tehran, Iran 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]

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Some properties of the Hausdorff fuzzy metric on finite sets Chang-qing Lia, ∗ , Liu Yangb, c a School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, Fujian 363000, China b Department of Mathematics, Shaanxi Xueqian Normal University , Xi’an, Shaanxi 710100, China c Department of Mathematics, Shantou University , Shantou, Guangdong 515063, China Email: helen [email protected], [email protected] ¯ September 30, 2014 In the paper, some properties of the Hausdorff fuzzy metric on the family of nonempty finite sets, as precompactness, completeness, compactness and Fboundedness are explored. Also, an illustrative example of a complete fuzzy metric space that does not induce the complete Hausdorff fuzzy metric on the family of nonempty finite sets is given. Keywords: Fuzzy metric, The Hausdorff fuzzy metric, Precompact, Complete, F-bounded. AMS Subject Classifications: 54A40, 54B20, 54E35

1

Introduction

Fuzzy metric, which is an important notion in Fuzzy Topology, have been introduced by many authors from different points of view [2, 3, 9, 11]. In particular, George and Veeramani [3] gave a definition of fuzzy metric with the help of continuous t-norms and proved that the topology induced by the fuzzy metric is first countable and Hausdorff. Later, Gregori and Romaguera [7] proved that the topological space induced by the fuzzy metric is metrizable. This version of fuzzy metric is more restrictive, but it determines the class of spaces that are tightly connected with the class of metrizable topological spaces. Hence it is interesting to study the version of fuzzy metric. Gregori and ∗ Corresponding author. This work was supported by Nation Natural Science Foundation of China (No. 11471153, No. 11471202), the foundation of Fujian Province (No. JA14200, No. 2013J01029, No. JA13198), the foundation of Minnan Normal University (No. SJ1118).

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Romaguera [8] presented a characterization of the given fuzzy metric spaces that are completable. A fuzzy analogue of the Prokhorov metric defined on the set of all probability measures of a compact fuzzy metric space was considered by Repovˇs and Savchenko [15]. Koˇcinac [10] gave some Selection properties of fuzzy metric spaces. Other more contributions to the study of fuzzy metric spaces can be found in [4, 5, 12, 13, 14, 17, 18]. In order to study the hyperspaces in a fuzzy metric space, Rodr´ıguez-L´opez and Romaguera [16] gave a definition of Hausdorff fuzzy metric on the family of nonempty compact sets. In the paper, we study the Hausdorff fuzzy metric on the family of nonempty finite subsets of a given fuzzy metric space and explore some properties of the Hausdorff fuzzy metric, as precompactness, completeness, compactness and F-boundedness. Also, we give an example of a complete fuzzy metric space that does not induce the complete Hausdorff fuzzy metric on the family of nonempty finite subsets of a given fuzzy metric space.

2

Preliminaries

Throughout the paper the letter N shall denote the set of all nature numbers. Our basic reference for general topology is [1]. Definition 2.1 [3] A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if it satisfies the following conditions: (i) ∗ is associative and commutative; (ii) ∗ is continuous; (iii) a ∗ 1 = a for all a ∈ [0, 1]; (iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, and a, b, c, d ∈ [0, 1]. The following are examples of t-norms: a ∗ b = min{a, b}; a ∗ b = a · b; a ∗ b = max{a + b − 1, 0}. Definition 2.2 [3] A 3-tuple (X, M, ∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X ×X ×(0, ∞) satisfying the following conditions for all x, y, z ∈ X and s, t ∈ (0, ∞): (i) M (x, y, t) > 0; (ii) M (x, y, t) = 1 if and only if x = y; (iii) M (x, y, t) = M (y, x, t); (iv) M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s); (v) the function M (x, y, ·) : (0, ∞) → [0, 1] is continuous. If (X, M, ∗) is a fuzzy metric space, we will say that (M, ∗) is a fuzzy metric on X. It was proved in [3] that every fuzzy metric (M, ∗) on X generates a topology τM on X which has a base the family of open sets of the form {BM (x, r, t)|x ∈ X, t > 0, r ∈ (0, 1)}, where BM (x, r, t) = {y ∈ X|M (x, y, t) > 1 − r} for all r ∈ (0, 1) and t > 0. Definition 2.3 [7] A fuzzy metric space (X, M, ∗) is called precompact if for each ∪ r ∈ (0, 1) and t > 0, there is a finite subset A of X such that X = BM (a, r, t). a∈A

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Obviously, each subset of a precompact fuzzy metric space is precompact. Definition 2.4 [5] Let (X, M, ∗) be a fuzzy metric space. (a) A sequence {xn }n∈N in X is called Cauchy if for each r ∈ (0, 1) and t > 0, there exists an N ∈ N such that M (xn , xm , t) > 1 − r for all n, m ≥ N . (b) (X, M, ∗) is called complete if every Cauchy sequence in X is convergent with respect to τM . Definition 2.5 [7] A fuzzy metric space (X, M, ∗) is called compact if (X, τM ) is a compact topological space. Definition 2.6 [3] A fuzzy metric space (X, M, ∗) is said to be F-bounded if there exist t > 0 and 0 < r < 1 such that M (x, y, t) > 1 − r for all x, y ∈ X. Clearly, each subset of an F-bounded fuzzy metric space is F-bounded.

3

Main results

Given a fuzzy metric space (X, M, ∗), we will denote by P(X), Comp(X) and Fin(X), the set of nonempty subsets, the set of nonempty compact subsets and the set of nonempty finite subsets of (X, τM ), respectively. For every B ∈ P(X), a ∈ X and t > 0, let M (a, B, t) := sup M (a, b, t), M (B, a, t) := sup M (b, a, t) b∈B

b∈B

(see Definition 2.4 of [18]). By condition (iii) in Definition 2.2, we observe that M (a, B, t) = M (B, a, t). Definition 3.1 [16] Let (X, M, ∗) be a fuzzy metric space. For every A, B ∈ Comp(X) and t > 0, define HM : Comp(X)× Comp(X) × (0, ∞) → [0, 1] by HM (A, B, t) = min{ inf M (a, B, t), inf M (A, b, t)}. a∈A

b∈B

It was proved in [16] that (Comp(X),HM , ∗) is a fuzzy metric space. (HM , ∗) is called the Hausdorff fuzzy metric on Comp(X). Lemma 3.2 [16] Let (X, M, ∗) be a fuzzy metric space. Then (Comp(X),HM , ∗) is precompact if and only if (X, M, ∗) is precompact. Lemma 3.3 [16] Let (X, M, ∗) be a fuzzy metric space. Then (Comp(X),HM , ∗) is complete if and only if (X, M, ∗) is complete. Lemma 3.4 [7] A fuzzy metric space is compact if and only if it is precompact and complete. From the three preceding lemmas we immediately conclude the following. Corollary 3.5 Let (X, M, ∗) be a fuzzy metric space. Then (Comp(X),HM , ∗) is compact if and only if (X, M, ∗) is compact. In a fuzzy metric space (X, M, ∗), it it clear that HM ({x}, {y}, t) = M (x, y, t) for all x, y ∈ X and t > 0. So we can regard X as a subset of Fin(X).

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Theorem 3.6 Let (X, M, ∗) be a fuzzy metric space. Then (Fin(X), HM , ∗) is precompact if and only if (X, M, ∗) is precompact. Proof Suppose that (Fin(X), HM , ∗) is precompact. Since X ⊆ Fin(X), we conclude that (X, M, ∗) is precompact. Conversely, suppose that (X, M, ∗) is precompact. Then, by Lemma 3.2, we obtain that Comp(X) is precompact. Since Fin(X) ⊆ Comp(X), we deduce that (Fin(X), HM , ∗) is precompact. Lemma 3.7 [3] A sequence {xn }n∈N in a fuzzy metric space (X, M, ∗) converges to x with respect to τM if and only if M (xn , x, t) → 1 as n → ∞. Lemma 3.8 Let (X, M, ∗) be a complete fuzzy metric space and A be a closed subset of (X, τM ). Then (A, M, ∗) is complete. Proof Let {xn }n∈N be a Cauchy sequence in (A, M, ∗). Then {xn }n∈N is also a Cauchy sequence in (X, M, ∗). Since (X, M, ∗) is complete, there exists an x ∈ X such that M (xn , x, t) → 1 as n → ∞. Moreover, notice that A is a closed subset of (X, τM ). It follows that x ∈ A. Thus, (A, M, ∗) is complete. Theorem 3.9 In a fuzzy metric space (X, M, ∗), if (Fin(X), HM , ∗) is complete, then (X, M, ∗) is complete. Proof Suppose that (Fin(X), HM , ∗) is complete. Now we Take F ∈ Fin(X), where F contains at least two points. Let a, b ∈ F and t > 0, with a ̸= b. Put M (a, b, 2t) = ε. Then there exists a ε1 ∈ (ε, 1) such that ε1 ∗ ε1 > ε. Take ε2 ∈ (ε1 , 1). We show that BM (a, 1 − ε1 , t) ∩ BM (b, 1 − ε1 , t) = Ø. In fact, otherwise, we can choose a c ∈ BM (a, 1 − ε1 , t) ∩ BM (b, 1 − ε1 , t). Hence M (a, b, 2t) ≥ M (a, c, t) ∗ M (c, b, t) ≥ ε1 ∗ ε1 > ε = M (a, b, 2t), which is a contradict. Let x ∈ X. If x ∈ BM (a, 1 − ε1 , t), where BM (a, 1 − ε1 , t) is the closure of BM (a, 1 − ε1 , t), then x ∈ / BM (b, 1 − ε1 , t). So M (b, x, t) ≤ ε1 . Hence HM (F, {x}, t) ≤ inf M (y, {x}, t) = inf M (y, x, t) ≤ M (b, x, t) ≤ ε1 . y∈F

y∈F

If x ∈ / BM (a, 1 − ε1 , t), then x ∈ / BM (a, 1 − ε1 , t). So M (a, x, t) ≤ ε1 . Hence HM (F, {x}, t) ≤ ε1 . Whence {{x}|x ∈ X} ∩ HM (F, 1 − ε2 , t) = Ø, which means that X is a closed subset of (Fin(X),τHM ). Consequently, by Lemma 3.8, (X, M, ∗) is complete. The converse of the preceding theorem is false. We illustrate this fact with an example.

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Example 3.10 Let X = {1, 12 , · · · , n1 , · · · , 0} and d be Euclidian metric of X. Denote a ∗ b = a · b for all a, b ∈ [0, 1]. Define the function M by M (x, y, t) =

1 1 + d(x, y)

for all x, y ∈ X and t > 0. Then (X, M, ∗) is a compact fuzzy metric space (see [6]). Therefore, according to Corollary 3.5, (Comp(X),HM , ∗) is compact. Also, since (X, M, ∗) is complete, we have that (Comp(X),HM , ∗) is complete 1 by Lemma 3.3. Put A = {1, 13 , · · · , 2n−1 , . . . , 0}. Then A ∈ Comp(X)\Fin(X). Let r ∈ (0, 1) and t > 0. Then there exists a k ∈ N such that if n ≥ k, 1 1 we get 2n−1 ∈ BM (0, r, t). Since B = {1, 13 , · · · , 2n−3 , 0} ∈ Fin(X), we have HM (A, B, t) > 1 − r. Hence B ∈ BHM (A, r, t). So Fin(X) is not a closed subset of (Comp(X), τHM ), which implies that (Fin(X), HM , ∗) fails to be complete. By Lemma 3.4, Theorem 3.6 and Theorem 3.9, we immediately deduce the following. Corollary 3.11 In a fuzzy metric space (X, M, ∗), if (Fin(X), HM , ∗) is compact, then (X, M, ∗) is compact. Due to Example 3.10, the converse of the above corollary is false obviously. Now, a question arise naturally. Question 3.12 In a complete fuzzy metric space (X, M, ∗), under what condition is (Fin(X), HM , ∗) complete? Theorem 3.13 Let (X, M, ∗) be a fuzzy metric space. Then (Fin(X), HM , ∗) is F-bounded if and only if (X, M, ∗) is F-bounded. Proof Assume that (Fin(X), HM , ∗) is F-bounded. Since X ⊆ Fin(X), we get that (X, M, ∗) is F-bounded. Conversely, assume that (X, M, ∗) is F-bounded. Then there exist r ∈ (0, 1) and t > 0 such that M (x, y, t) > 1 − r whenever x, y ∈ X. Let A, B ∈ Fin(X). Then, for each a ∈ A, there exists a ba ∈ B such that M (a, B, t) = sup{M (a, b, t)|b ∈ B} = max{M (a, b, t)|b ∈ B} = M (a, ba , t). So there exists an a0 ∈ A such that inf M (a, B, t) = inf M (a, ba , t) = min{M (a, ba , t)|a ∈ A} = M (a0 , ba0 , t) > 1−r.

a∈A

a∈A

Similarly, we have that inf M (A, b, t) > 1 − r. Hence HM (A, B, t) > 1 − r. We b∈B

finish the proof.

4

Conclusion

We have studied some properties of the Hausdorff fuzzy metric on the family of nonempty finite subsets of a given fuzzy metric space, as precompactness, 5

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completeness, compactness and F-boundedness. Also, we have given an example to illustrate that a complete fuzzy metric space does not induce the complete Hausdorff fuzzy metric on the family of nonempty finite subsets of a given fuzzy metric space.

References [1] R. Engelking, General Topology, PWN-Polish Science Publishers, warsaw, 1977. [2] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979) 205–230. [3] A. George, P. Veeramani, On some resules in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994) 395–399. [4] A. George, P. Veeramani, Some theorems in fuzzy metric spaces, J. Fuzzy Math. 3 (1995) 933–940. [5] A. George, P. Veeramani, On some resules of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90 (1997) 365–368. [6] V. Gregori, S. Morillas, A. Sapena, Example of fuzzy metrics and applications, Fuzzy Sets and Systems 170 (2011) 95–111. [7] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems 115 (2000) 485–489. [8] V. Gregori, S. Romaguera, Characterizing completable fuzzy metric spaces, Fuzzy Sets and Systems 144 (2004) 411–420. [9] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984) 215–229. [10] L. Koˇcinac, Selection properties in fuzzy metric spaces, Filomat, 26 (2) (2012) 305–312. [11] I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975) 326–334. [12] C. Q. Li, On some results of metrics induced by a fuzzy ultrametric, Filomat, 27 (6) (2013) 1133–1140. [13] C. Q. Li, Some properties of intuitionstic fuzzy metric spaces, Journal of Computational Analysis and Applications, 16 (4) (2014) 670–677. [14] J. H. Park ,Y. B. Park and R. Saadati, Some results in intuitionistic fuzzy metric spaces, Journal of Computational Analysis and Applications 10 (4) (2008) 441-451. [15] D. Repovˇs, A. Savchenko, M. Zarichnyi, Fuzzy Prokhorov metric on the of probability measures, Fuzzy Sets and Systems 175 (2011) 96–104. [16] J. Rodr´ıguez-L´ opez, S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets and Systems 147 (2004) 273–283. [17] A.Savchenko, M.Zarichnyi, Fuzzy ultrametrics on the set of probability measures, Topology 48 (2009) 130–136. [18] P.Veeramani, Best approximation in fuzzy metric spaces, J. Fuzzy Math. 9 (2001) 75–80.

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LOCALLY BOUNDEDNESS AND CONTINUITY OF SUPERPOSITION OPERATORS ON DOUBLE SEQUENCE SPACES Cr0 ¼ B I·RSEN SA GIR AND N I·HAN GÜNGÖR Abstract. Let R be set of all real numbers, N be the set of all natural numbers and N2 = N N. In this paper, we de…ne the superposition operator Pg where g : N2 R ! R by Pg ((xks )) = g (k; s; xks ) for all real double sequence (xks ). Chew & Lee [4] and Petranuarat & Kemprasit [11] have characterized Pg : c0 ! l1 and Pg : c0 ! lq where 1 q < 1, respectively. The main aim of this paper is to construct the necessary and su¢ cient conditions for the boundedness and continuity of Pg : Cr0 ! L1 and Pg : Cr0 ! Lp where 1 p < 1.

1. INTRODUCTION Let be a double sequence spaces which are the vector spaces with coordinatewise addition and scalar multiplication. Let any sequence x = (xks ) 2 . If for any " > 0 there exist N 2 N and l 2 R such that jxks lj < " for all k; s N, then we call that the double sequence x = (xks ) is convergent in Pringsheim’s sense and denoted by p lim xks = l. If the double sequence x = (xks ) converges in Pringsheim’s sense and, in addition, the limits that limxks and limxks exist, then it s

k

is called regularly convergent and denoted by r lim xks . The space Cr0 is de…ned by Cr0 = fx = (xks ) 2 : r lim xks = 0g and it is a Banach space with the norm kxkCr0 = sup jxks j. The space of all k;s2N

bounded double sequences is denoted by Mu and de…ned by ( Mu :=

x = (xks ) 2

)

: kxk1 = sup jxks j < 1 , k;s2N

and it is a Banach space with the norm k:k1 . The spaces Lp , 1 de…ned by 8 9 1 < = X p Lp := x = (xks ) 2 : jxks j < 1 : ;

p < 1 are

k;s=1

where

1 P

k;s=1

=

1 P 1 P

k=1s=1

. Also it is a Banach space with the norm kxkp =

1 P

k;s=1

p

jxks j

! p1

.

p Also, the spaces Cr0 and Lp are shown di¤erent symbols as 0 cR 2 and l2 , recpectively.

2000 Mathematics Subject Classi…cation. 47H30; 46A45. Key words and phrases. Superposition Operators, Continuity, Locally Bounded, Double Sequence Spaces, Regularly Convergent. 1

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¼ B I·RSEN SA GIR AND N I·HAN GÜNGÖR

2

It is known that Lp Lq Cr0 Mu where 1 p < q < 1 . The sequence eks 1 ,k=i and s=j ks is de…ned as eij = 0 ,otherwise . If we consider the sequence snm de…ned by n P m P snm = xks (n, m 2 N), then the pair of ((xks ) ; (snm )) is called a double k=1s=1

series. Also (xks ) is called general term of series and (snm ) is called partial sums sequence. Let v be convergence notions, i.e., in Pringsheim’s sense or regularly convergent. If the partial sums sequence (snm ) is convergent to a real number s in v-sense, i.e. n X m X v lim xks = s, n;m k=1 s=1

then the series ((xks ) ; (snm )) is called convergent to s in the v sense ,or, v convergent and the sum of series equals to s. It’s denoted by 1 X

xks = s.

k;s=1

It is known that if the series is v convergent, then the v limit of the general term 1 P 1 P of the series equals to zero. The remaining term of the series xks is de…ned k=1s=1

by

(1.1)

Rnm =

n 1 X1 X

xks +

k=1 s=m

1 X 1 1 m X X X1 xks . xks +

k=n s=1

k=ns=m

We will demonstrate the formula (1.1) brie‡y with X xks maxfk;sg N

for n = m = N . It is known that if the series is v convergent, then the v limit of the remaining term of series is zero. For more details on double sequence and series, one can referee [1],[2],[3],[7],[9],[10],[13],[16] and the references therein. Locally boundedness and continuity of superposition operators on sequence spaces are discussed by some authors [4],[5],[6],[8],[11],[12],[15]. In [4], Chew Tuan Seng and Lee Peng Yee have given necessary and su¢ cient conditions for the continuity of the superposition operator acting from sequence space c0 into l1 . In [11], Somkit Petranuarat and Yupaporn Kemprasit have given necessary and su¢ cient conditions for continuity of the superposition operator acting from sequence space c0 into lq with 1 q < 1. We extend the de…nition of superposition operators for double sequence spaces as follows. Let X, Y be two double sequence spaces. A superposition operator Pg 1 on X is a mapping from X into de…ned by Pg (x) = (g (k; s; xks ))k;s=1 where the function g : N2 R ! R satis…es (1) g (k; s; 0) = 0 for all k; s 2 N. If Pg (x) 2 Y for all x 2 X, we say that Pg acts from X into Y and write Pg : X ! Y . Moreover, we shall assume the additionally some of the following conditions: (2) g (k; s; :) is continuous for all k; s 2 N (20 ) g (k; s; :) is bounded on every bounded subset of R for all k; s 2 N. It is obvious that if the function g (k; s; :) satis…es the propety (2), then g satis…es

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3

(20 ) from [15]. Also, it is not hard to see that if the function g (k; s; :) is locally bounded on R, then g satis…es (20 ) from [15]. In this paper, we characterize the superposition operator acting from the double sequence space Cr0 into L1 under the hypothesis that the function g (k; s; :) satis…es (20 ). We discuss the continuity and locally boundedness of the superposition operator Pg by using the methods in [4],[15]. Then we generalize our works as the superposition operator acting from the space Cr0 into Lp where 1 p < 1 without assuming that the function g (k; s; :) satis…es (20 ) by using the methods in [11],[15]. 2. SUPERPOSITION OPERATORS OF Cr0 INTO L1

Theorem 1. Let g : N2 R ! R satis…es (20 ). Then Pg : Cr0 ! L1 if and only if 1 there exist > 0 and (cks )k;s=1 2 L1 such that jg (k; s; t)j

cks whenever jtj

for all k; s 2 N.

1

Proof. Assume that there exist > 0 and (cks )k;s=1 2 L1 such that jg (k; s; t)j cks whenever jtj for all k; s 2 N. Let x = (xks ) 2 Cr0 . Hence p lim xks = 0 and the limits that lim xks and lim xks exist. Therefore there exists N 2 N such that s!1

k!1

jxks j

for all k; s 2 N with max fk; sg X

jg (k; s; xks )j

maxfk;sg N

N . Then, we …nd

X

1 X

cks

k;s=1

maxfk;sg N

cks < 1.

So, we get Pg (x) = g (k; s; xks ) 2 L1 . Conversely, suppose that Pg : Cr0 ! L1 . The sets A ( ) and B (k; s; ) is de…ned as A ( ) = ft 2 R : jtj g and

B (k; s; ) = sup fjg (k; s; t)j : t 2 A ( )g

for all k; s 2 N and > 0. So, we see that jg (k; s; t)j B (k; s; ) when1 ever jtj . We will show that there is 1 > 0 such that (B (k; s; 1 ))k;s=1 2 1 P L1 . Assume the contrary,that is, B (k; s; ) = 1 for all > 0. Therefore k;s=1

1 P

k;s=1

B k; s;

1 i

+

1 j

= 1 for each i; j 2 N. Then there exist sequence of positive

integers n0 = 0 < n1 < n2 < such that ni X k=ni

< ni < mj X

1 +1s=mj

and m0 = 0 < m1 < m2 < B k; s;

1 +1

1 1 + i j

< mj
1

for each i; j 2 N and also there exists "ij > 0 such that (2.1)

ni X

k=ni

mj X

1 +1s=mj

1 +1

B k; s;

1 1 + i j

"ij (ni

ni

1 ) (mj

mj

1)

> 1.

Let i; j 2 N be …xed. Since g satis…es (20 ), we see that B k; s; 1i + 1j < 1 for all i; j 2 N with ni 1 + 1 k ni and mj 1 + 1 s mj . Then, there exists

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¼ B I·RSEN SA GIR AND N I·HAN GÜNGÖR

4

xks 2 A

1 i

+

1 j

such that

(2.2)

jg (k; s; xks )j > B k; s;

for each k; s 2 N satisfying ni ni X

k=ni

1

+1

k

mj

X

1 +1s=mj

1 +1

jg (k; s; xks )j >

1 1 + i j

ni and mj mj

ni X

X

1 +1s=mj

k=ni

1 +1s=mj

k=ni

"ij (ni

ni

1

+1

s

mj . So, we …nd

1 1 B k; s; + i j +1

mj X

ni X

>

1

"ij

k=ni

1 +1s=mj

"ij

1 +1

1 1 + i j

B k; s;

1 +1

1 ) (mj

mj X

ni X

mj

1)

> 1 by using (2.1) and (2.2). Therefore we obtain that 0 mj ni 1 1 X 1 1 X X X X X @ jg (k; s; xks )j = k=1 s=1

i=1 j=1

k=ni

1 +1s=mj

Hence we get g (k; s; xks ) 2 = L1 . Since xks 2 A 1 i

1 i

+

1 +1

1 j

1

jg (k; s; xks )jA = 1.

whenever ni

1 +1

k

ni

1 j.

+ Hence, we obtain x = (xks ) 2 Cr0 . and mj 1 + 1 s mj , we …nd jxks j This contradicts the assumption that Pg : Cr0 ! L1 . Then there exists 1 > 0 1 such that (B (k; s; 1 ))k;s=1 2 L1 . If we put cks = B (k; s; 1 ) for all k; s 2 N, the proof is completed. Theorem 2. If Pg : Cr0 ! L1 , then Pg is continuous on Cr0 if and only if g (k; s; :) is continuous on R for all k; s 2 N. Proof. Suppose that Pg is continuous on Cr0 . Let k; s 2 N, t0 2 R and " > 0. Since Pg is continuous at t0 e(ks) 2 Cr0 , there exists > 0 such that (2.3)

z

t0 e(ks)

Cr0


0 such that jt

xks j
0. From (3.4), we see that NX 1 1 1 X

p

sup jg (k; s; t)j < 1;

k=1 s=N1 jtj

1 NX 1 1 X

k=N1

s=1 jtj

Therefore, there exists N 2 N with N NX 1 1 1X

p

sup jg (k; s; t)j < 1;

N2 such that p

sup jg (k; s; t)j
0g : An operator T 2 L (X; Y ) is called a positive (p; q)-absolutely summing operator, 1 p q 1; if there is a constant C > 0 such that for every nite set of elements fxi g1 i n in X + the inequality p

(T (xi ) : (1

i

n))

C"q (xi : (1

i

n)) ;

(1.1)

holds. + The positive (p; q)-absolutely summing norm of T is kT k(p;q) = inf C. The space of positive (p; q)-absolutely summing operators will be denoted by + (p;q) (X; Y ), +

1 p; q 1. This space becomes a Banach space with the norm k k(p;q) given by the in mum of the constants verifying (1:1). Observe that +

kT k(p;q) = sup

p

(T (xi ) : (1 i n)) : fxi g1 i with "q (xi : (1 i n))

n

in X + ; 1

(1.2)

+ kT k(p;q) "q

and p (T (xi ) : (1 i n)) (xi : (1 i n)). As usual, if p = 1 (resp. q = 1), the left (resp. right) hand side of (1:1) is replaced by supi kT (xi )k (resp. supky k 1 supi jy (yi )j). The theory of (p; q) absolutely summing operators is a uni ed theory of various important classes of operators in connection with the classes of nuclear and Hilbert-Schmidt operators. If p = q = 1, then the class is called positive absolutely summing operator and denoted by + (X; Y ) and the + norm de ned on it by k k : 2000 Mathematics Subject Classi cation. 47B10, 46E40. Key words and phrases. Absolutely Summing operators;K• othe Bochner function spaces. 1

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MONA KHANDAQJI, AHMED AL-RAWASHDEH

In the sequel we use the following useful relation which is a simple implication of duality (`p ) = `q : "q (xi : (1

i

n)) =

sup p ( i :(1

i n)) 1

n X i=1

;

i xi

(1.3)

X

with i > 0, (1 p; q < 1) and p1 + 1q = 1 [5]: P Let (T; ; ) be a nite complete measure space and let L0 = L0 (T ) denote the space of all (equivalence classes) of -measurable real-valued functions. For f; g 2 L0 ; f g means that f (t) g (t) almost every where, t 2 T: A Banach space (E; k kE ) is said to be a K•othe space if : (1) For f; g 2 L0 ; jf j jgj and g 2 E imply f 2 E and kf kE kgkE . (2) For each A 2 , if (A) is nite, then A 2 E. See [4]: So K• othe spaces are Banach lattices in the obvious order ( f > 0 if f (t) > 0 almost every where t 2 T ). A K•othe space E has absolutely continuous norm if for each f 2 E and each decreasing sequence (An ) converges P to 0, then k An f kE ! 0. Let E be a K•othe space on the measure space (T; ; ) and (X; k kX ) be a real Banach space. Then E (X) is the space of all equivalence classes of strongly measurable functions f : T ! X, such that kf ( )kX 2 E equipped with the norm jkf kjE(X) = kkf ( )kX kE : The space (E (X) ; jk kjE(X) ) is a Banach space called the K•othe Bochner function space [3]. The most important class of K•othe Bochner function spaces are the LebesgueBochner spaces Lp (X) ; (1 p < 1) and their generalization the Orlicz-Bochner spaces L (X). If X is a Banach lattice, the K•othe Bochner function spaces E (X) are Banach lattices, we put f > 0; when f (t) > 0 almost every where t 2 T: The characterization of the positive absolutely summing operator was introduced on Lebesgue-Bochner spaces LP (X) ; (1 p < 1) and K•othe Bochner function spaces in [1], [2], [6], [7]. In this paper, we shall investigate the space of positive (p; q)-absolutely summing operators for the K•othe Bochner function spaces with a real Banach lattice X, which is analogous with that given by D. Popa in [7], where he showed that + (E (X) ; Y ) = + (E; + (X; Y )). For T 2 L (E (X) ; Y ) ; de ne a function Te : E ! L (X; Y ) by Tef (x) = T (f x) ; f 2 E; x 2 X:

(1.4)

Then Te is clearly linear and bounded operator. 2. Main results

We now introduce the following lemmas which play important role in the proof of our main theorem Lemma 2.1. Let X be a Banach lattice, Y a Banach space and let E be a K• othe 1 1 ef 2 + = 1, then T space. If T 2 + (E (X) ; Y ), for 1 p; q 1 with (p;q) p q + (X; Y ), for all f 2 E: (p;q)

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ABSOLUTELY SUMMING OPERATORS

3

e Proof. Let T 2 + (p;q) (E (X) ; Y ) and T be de ned as in (1:4). First we show + that Tef 2 + (p;q) (X; Y ), for all f 2 E; f > 0. If f 2 E , then for every nite set of elements fxi g1 i n in X + , we have Tef (xi ) : (1

p

k

n)

=

p

(T (f xi ) : (1

i

n))

+

kT k(p;q) "q (f xi : (1

i

n))

Now, using (1:3) "q (f xi : (1

i

n))

=

sup p ( i :(1

=

i n))

1

sup p ( i :(1

=

i n))

1

sup p ( i :(1

= kf kE

n X

n X

1

i n)) 1

= kf kE "q (xi : (1

if

i

>0

( ) xi X

n X

n X

E

i xi

i=1

sup p ( i :(1

i

E(X)

i=1

f()

i n))

; with

i f xi

i=1

X E i xi

i=1

X

n)) :

Hence, p

Te f (xi ) : (1

this implies that Tef 2 f

+

+

f , where f , f

i + (p;q) +

2E

+

n)

kT k(p;q) kf kE "q (xi : (1

i

n)) ;

(X; Y ), for allf 2 E; f > 0: In general, since f = then Tef 2 + (p;q) (X; Y ), for all f 2 E :

Lemma 2.2. Let X be a Banach lattice,Y a Banach space and let E be a K• othe space. Consider a set of simple functions ffi g1 i n in E (X), which is de ned by k X fi = xij Aj , where Aj are the characteristic functions of Aj 's, xij 2 X + , for j=1

i = 1; 2; : : : ; n, j = 1; 2; :::k. Then for each we have "q (fi : (1

i

n)) > "1

i

n X

Aj

i=1

Proof.

For t 2 T; fi (t) =

k X

xij

Aj

> 0 with

i xij

: (1 X

p

(

i

j

: (1

i

n))

1;

!

k) :

(t) ; i = 1; 2; : : : ; n. Without loss of

j=1

generality we may assume that the Aj 's are pairwise disjoint measurable sets of T with [kj=1 Aj = T . Now for each i > 0 with p ( i : (1 i n)) 1; we have

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MONA KHANDAQJI, AHMED AL-RAWASHDEH

form (1:3) we have the following

"q (fi : (1

i

n X

n)) >

i fi

i=1

E(X)

n X

=

i fi

i=1

n X

=

i

i=1

k X

=

= "1

0 @

Aj (

j=1

() X E k X

xij

Aj (

j=1

)

n X

Aj

)A

X E

i xij

i=1

n X

1

X E

i xij

i=1

: (1 X

j

!

k) :

Lemma 2.3. Let X be a Banach lattice,Y a Banach space and let E be a K• othe space with absolutely continuous norm. For 1 p; q 1 with p1 + 1q = 1, if + T 2 L (E (X) ; Y ) and f T 2 + E; + (p;q) (X; Y ) , then T 2 (p;q) (E (X) ; Y ) with +

kT k(p;q)

Proof.

Te

+

.

Lemma 1. Proof. Let T 2 L (E (X) ; Y ) and Te 2 + E; + (p;q) (X; Y ) : It is known that if the K• othe space E has absolutely continuous norm, then the subspace of all simple functions is dense in K•othe Bochner function spaces E (X) ; [3]. So it is su cient to show that T 2 + (p;q) (E (X) ; Y ) on the restriction of T to the subspace of simple functions from E (X) : Choose a set of simple functions ffi : fi > 0g1 i n in E (X), then as in Lemma (2:2) assume that for each k X i = 1; 2; : : : ; n; fi (t) = xij Aj (t); t 2 T with Aj 's are pairwise disjoint meaj=1

surable sets of T with [kj=1 Aj = T , and Aj 's are characteristic functions of Aj 's, xij 2 X + , j = 1; 2; :::k. Since fi 's (i = 1; : : : ; n) represent classes of functions, we may assume that (Aj ) > 0, for each j = 1; 2; :::; k . Now given " > 0 there exist i n)) 1 that satisfy i > 0 with p ( i : (1

"q (xij : (1

i

n))
0 with p ( i : (1 i n)) Using Lemma 2.2, then for each i with "q (fi : (1

(xij )

Aj

Aj

Te

j=1

Aj

! p1

(X; Y ) since Te 2

+ (p;q)

Te

i=1

p

Te

i=1

2

Aj

5 A

Aj xij

j=1

i=1

j=1

3p 1 p1

: (1 X

j

j

k)

n)

(2.3)

Then from inqualities (2:2), (2:3) ;we have p

As k

T kE

((T (fi )) : (1

i

n)) < Te

+

"q (fi : (1

i

n)) + " k

T kE

:

is nite and " is an arbitrary, then p

((T (fi )) : (1

i

Te

n))

+

"q (fi : (1

i

n)) : +

+ This gives us the result that T 2 + Te : (p;q) (E (X) ; Y ) with kT k(p;q) The following theorem explains the relation between the spaces of positive (p; q)absolutely summing operators over the K•othe spaces

Theorem 2.1. Let X be a Banach lattice,Y a Banach space and E be a K• othe space. Then for 1 p; q 1 with p1 + 1q = 1, + (p;q)

(E (X) ; Y )

+ (p;q)

459

E;

+

(X; Y ) :

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MONA KHANDAQJI, AHMED AL-RAWASHDEH

+ e Proof. Let T 2 + (p;q) (E (X) ; Y ). Using Lemma 2.1, T f 2 (p;q) (X; Y ), for all f 2 E . Hence, from equation(1:2) given " > 0, there exist a nite sequence ffi : fi 0g1 i n in E + and (xij ; j 2 i ) in X + ;where i is a nite set with i N such that for each i = 1; 2; : : : ; n, and "q (xij : (j 2 i )) 1 p h i + " Te (fi ) < pp Te (fi ) (xij ) ; (j 2 i ) ; n (p;q)

Therefore, n X

Te (fi )

i=1

n X

p

+

"

(p;q)


0 or ≤ β < 0 on [0, 1]. We set ¯ ¯ ¯ (20) lτ :≡ sup ¯α−1 0 (x) ατ (x) , τ = 0, 1, ..., k. x∈[0,1]

Here TNβ,α (f ), N ∈ N, as in (15), (N → ∞). Consider the mixed fractional linear differential operator L∗λ :=

k X j=0

£ α α ¤ αj (x) λD∗0j + (1 − λ) D1−j .

(21)

Then, for any N ∈ N, there exists a real polynomial QN+1 of degree (N + 1) such that 1) °¡ αj ¢ ° λD + (1 − λ) Dαj (f ) − ∗0

1−

° ¡ αj T β,α (f ) α ¢ λD∗0 + (1 − λ) D1−j (QN +1 (f ))°∞,[0,1] ≤ N , j = 1, ..., ρ, Γ (2 − αj )

and 2)

kf − QN+1 (f )k∞,[0,1]

# l τ +2 . ≤ TNβ,α (f ) Γ (2 − ατ ) τ =1 "

k X

(22)

(23)

Assuming L∗λ f (x) ≥ 0, for all x ∈ [0, 1] we get (L∗λ (QN +1 (f ))) (x) ≥ 0 for all x ∈ [0, 1] . 5

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Proof. Here let k, ρ ∈ N : 0 < k ≤ ρ, and α0 = 0 < α1 < α2 < ... < αk < ... < αρ ≤ 1, that is dα0 e = 0; dαj e = 1, j = 1, ..., ρ. We set ¯ ¯ ¯ lτ := sup ¯α−1 (24) 0 (x) ατ (x) < ∞, 0 ≤ τ ≤ k, x∈[0,1]

with l0 = 1, and

ρN :=

TNβ,α (f )

! lτ +1 . Γ (2 − ατ ) τ =1

Ã

k X

(25)

I. Suppose, throughout [0, 1], α0 (x) ≥ α > 0. Call QN +1 (f ) (x) := PN +1 (f ) (x) + ρN ,

(26)

where PN +1 (f ) (x) as in (14). Then by (19) we obtain °¡ αj ¢ ° λD + (1 − λ) Dαj (f (x) + ρN ) − ∗0 1−

° ¡ αj T β,α (f ) α ¢ , j = 1, ..., ρ. (27) λD∗0 + (1 − λ) D1−j (QN +1 (f ) (x))°∞,[0,1] ≤ N Γ (2 − αj )

And of course it holds

(16)

k(f (x) + ρN ) − QN +1 (f ) (x)k∞,[0,1] ≤ TNβ,α (f ) .

(28)

So that we find (28)

TNβ,α (f )

kf − QN +1 (f )k∞,[0,1] ≤ ρN + TNβ,α (f ) =

(29)

! # " k X lτ lτ β,α β,α + 1 + TN (f ) = TN (f ) +2 , Γ (2 − αj ) Γ (2 − αj ) τ =1 τ =1

Ã

k X

proving (23). From (27) and (9), (11), we get °¡ αj ¢ ° λD + (1 − λ) Dαj (f (x)) − ∗0 1−

° ¡ αj T β,α (f ) α ¢ λD∗0 + (1 − λ) D1−j (QN +1 (f )) (x)°∞,[0,1] ≤ N , j = 1, ..., ρ, (30) Γ (2 − αj )

proving (22). Next we use the assumption L∗λ f (x) ≥ 0, all x ∈ [0, 1] , to get

−1 ∗ ∗ α−1 0 (x) Lλ (QN +1 (f )) (x) = α0 (x) Lλ f (x) + ρN +

6

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k X

α−1 0 (x) αj (x)

j=0

©¡ αj ª ((27),(28)) α ¢ λD∗0 + (1 − λ) D1−j [QN+1 (f ) (x) − f (x) − ρN ] ≥

ρN −

! lτ + 1 TNβ,α (f ) = ρN − ρN = 0. Γ (2 − α ) j τ =1

Ã

k X

(31)

Hence L∗λ (QN +1 (f )) (x) ≥ 0, all x ∈ [0, 1] . II. Suppose, throughout [0, 1], α0 (x) ≤ β < 0. Call QN +1 (f ) (x) := PN +1 (f ) (x) − ρN ,

(32)

where PN +1 (f ) (x) as in (14). Then by (19) we obtain °¡ αj ¢ ° λD + (1 − λ) Dαj (f (x) − ρN ) − ∗0 1−

° ¡ αj T β,α (f ) α ¢ , j = 1, ..., ρ. (33) λD∗0 + (1 − λ) D1−j (QN +1 (f ) (x))°∞,[0,1] ≤ N Γ (2 − αj )

And of course it holds

(16)

k(f (x) − ρN ) − QN +1 (f ) (x)k∞,[0,1] ≤ TNβ,α (f ) .

(34)

Similarly we obtain again (22) and (23). Next we use the assumption L∗λ f (x) ≥ 0, all x ∈ [0, 1] , to get −1 ∗ ∗ α−1 0 (x) Lλ (QN +1 (f )) (x) = α0 (x) Lλ f (x) − ρN + k X j=0

α−1 0 (x) αj (x)

©¡ αj ª ((33),(34)) α ¢ λD∗0 + (1 − λ) D1−j [QN+1 (f ) (x) − f (x) + ρN ] ≤

−ρN +

! lτ + 1 TNβ,α (f ) = −ρN + ρN = 0. Γ (2 − α ) j τ =1

Ã

k X

(35)

Hence L∗λ (QN +1 (f )) (x) ≥ 0, for any x ∈ [0, 1] . Corollary 12 (to Theorem 11, λ = 1 case) Let L∗1 :=

k X j=0

α

αj (x) D∗0j .

Then, for any N ∈ N, there exists a real polynomial QN +1 of degree (N + 1) such that 1) ° αj ° TNβ,α (f ) °D (f ) − Dαj (QN+1 (f ))° , j = 1, ..., ρ, ≤ ∗0 ∗0 ∞,[0,1] Γ (2 − αj )

(36)

2) inequality (23) is again valid. Assuming L∗1 f (x) ≥ 0, for all x ∈ [0, 1] , we get (L∗1 (QN+1 (f ))) (x) ≥ 0 for all x ∈ [0, 1] . 7

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Corollary 13 (to Theorem 11, λ = 0 case) Let L∗0 :=

k X

α

αj (x) D1−j .

j=0

Then, for any N ∈ N, there exists a real polynomial QN +1 of degree (N + 1) such that 1) ° αj ° TNβ,α (f ) °D f − Dαj (QN +1 (f ))° , j = 1, ..., ρ, ≤ 1− 1− ∞,[0,1] Γ (2 − αj )

(37)

2) inequality (23) is again valid. Assuming L∗0 f (x) ≥ 0, we get L∗0 (QN+1 (f )) (x) ≥ 0 for any x ∈ [0, 1] . Finally we give Corollary 14 (to Theorem 11, λ =

1 2

case) Let L∗1 := 2

k X j=0

αj (x)

∙

α

α

j D∗0j +D1− 2

¸

.

Then, for any N ∈ N, there exists a real polynomial QN +1 of degree (N + 1) such that 1) °¡ αj ° ¢ ¡ ¢ 2TNβ,α (f ) ° D + Dαj (f ) − Dαj + Dαj (QN +1 (f ))° , j = 1, ..., ρ, ≤ ∗0 1− ∗0 1− ∞,[0,1] Γ (2 − αj ) (38) 2) inequality (23) is again valid. Assuming L∗1 f (x) ≥ 0, we get L∗1 (QN +1 (f )) (x) ≥ 0 for any x ∈ [0, 1] . 2

2

References [1] G.A. Anastassiou, Fractional Korovkin Theory, Chaos, Solitons and Fractals, 42 (2009), 2080-2094. [2] G.A. Anastassiou, Complete Fractional Monotone Approximation, submitted, 2014. [3] G.A. Anastassiou, O. Shisha, Monotone approximation with linear differential operators, J. Approx. Theory 44 (1985), 391-393. [4] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Vol. 2004, 1st edition, Springer, New York, Heidelberg, 2010. [5] A.M.A. El-Sayed, M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol. 3, No. 12 (2006), 81-95. [6] O. Shisha, Monotone approximation, Pacific J. Math. 15 (1965), 667-671.

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Modified quadrature method with Sidi transformation for three dimensional axisymmetric potential problems with Dirichlet conditions I Xin Luoa,∗, Jin Huangb , Yan-Ying Mab a College

b School

of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, P.R. China of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P.R. China

Abstract This paper is the application of the modified quadrature method for the numerical evaluation of boundary integral equations of three dimensional (3D) axisymmetric Laplace equations with Dirichlet conditions on curved polygonal boundaries. The Sidi transformation is used to remove the logarithmic singularities in the integral kernels and then the modified quadrature method is presented to approximate the weakly integrals. Numerical examples show that the error of numerical solution for the boundary integral equation of the 3D Laplace equation can converge with the order O (h3 ) by use of a Sidi transformation, and the optimal condition number for the according discrete system is O (h−1 ), where h is the uniform mesh step size. Keywords: Laplace equation, axisymmetric, modified quadrature method, convergence, stability. 2010 Mathematics Subject Classification: 65R20, 65N38.

1. INTRODUCTION Consider the Laplace problem with a Dirichlet condition       ∆φ(x) = 0, in V      φ(x) = g(x), on ∂V

(1.1)

where V is an axisymmetric domain, which is formed by rotating a two dimensional bounded simply connected region Ω around z−axis, ∆ is a three dimensional Laplace operator, and g(x) is a known function on ∂V. For practical problems, high precise and validated solutions of (1.1) are required. Some well known methods, such as finite element methods (FEMs) , collocation methods (CMs) and finite difference methods (FDMs) can be used to solve the mentioned equation above [5, 6, 9–11, 13, 15–19]. By ring potential theory, Eq. (1.1) is converted into the first kind boundary integral equation (BIE) and its solution can be represented via the following single layer potential Z 1 φ(y) = σ(x)dS , x = (x1 , x2 , x3 ) ∈ S , y = (y1 , y2 , y3 ) ∈ V, S 4π|x − y|

(1.2)

I The

work is supported by the National Natural Science Foundation of China (11371079). author Email addresses: [email protected] (Xin Luo), [email protected] (Jin Huang), [email protected] (Yan-Ying Ma)

∗ Corresponding

Preprint submitted to Elsevier

September 6, 2014

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where the density function σ(x) is sought on S , and S = ∂V is the surface of the body V. Here |x − y| denotes the Euclidean distance between the point x and y. Using the cylindrical coordinate (ρ, z, θ), Eq. (1.2) can be written as Z 0 φ(y) = ρφ? (x, y)σ(x)dΓ, x = (ρ, z) ∈ Γ, y = (ρ0 , z0 ) ∈ Ω , (1.3) Γ

0

where Γ is the polygonal boundary of the transformed cylindrical coordinate region Ω of region Ω, and Z 2π 1 dθ. φ? (x, y) = φ? (ρ, z; ρ0 , z0 ) = 4π|x − y| 0

(1.4)

By use of the Hankel transformation [21], the fundamental solution of three dimensional axisymmetric Laplace problem can also be represented by φ? (ρ, z; ρ0 , z0 ) =

Q−1/2 (q) , √ 2π ρρ0

(1.5)

where q=

2ρρ0 + (ρ − ρ0 )2 + (z − z0 )2 2ρρ0

and Q−1/2 (q) is the second kind of Legendre function, which has the following asymptotic expression q−1 1 ) + O((q − 1) ln(q − 1)), q → 1. Q−1/2 (q) = − ln( 2 32 Since the single potential function is continuous on the boundary Γ, we have Z g(y) = ρφ? (x, y)σ(x)dΓ, y ∈ Γ.

(1.6)

Γ

Eq. (1.6) is the weakly singular BIE system of the first kind , whose solution exists and is unique as long as CΓ , 1 [7], where CΓ is the logarithmic capacity (i.e., the transfinite diameter). Once the solution σ(x) is solved from (1.6), 0

we can obtain the solution in Ω by (1.3). In this paper, before discretization the boundary integral equation is converted to an equivalent equation over [0, 1] using Sidi transformation[22] at the corners which vary more slowly than arc-length near each corner. This has the effect of producing a transformed equation with a solution which is smooth on [0, 1]. Then the modified quadrature method is applied to deal with the weakly logarithmically singular integrals in (1.6). The modified quadrature method consists of regularization of the kernel together with trapezoidal approximation of the integral, and this method is superior in efficiency to both Galerkin method and the collocation method since each component in the discrete matrix is directly evaluated by the quadrature [3, 4]. The modified quadrature method was first prosed to solve the first kind BIEs of two dimensional Laplace problems in [23] and the experimental O (h3 ) order of convergence was reported in smooth domains. Later, in the case of a circle, Abou El-Seoud was able to prove the existence of solution and an O (h2 ) order of convergence in [24]. Saranen proved the order of convergence O (h3 ) if the solution is smooth enough in [4]. In this paper, our numerical results show the high convergent rate of modified quadrature method for (1.1) by using Sidi transformation is O (h3 ). In addition, the experimental O (h−1 ) order of the optimal condition number for 2

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the according discrete system of (1.6) is reported, which shows the modified quadrature method for weakly singular integral problems possesses an excellent stability. The remainder of this paper is organized as follows: in Section 2, the singularity analysis for the integral kernel in (1.6) is given in detail. In Section 3, modified quadrature method is applied for the BIEs of axisymmetric Laplace equation. In Section 4, some numerical examples are provided to show the efficiency of the proposed methods. 2. SINGULARITY ANALYSIS OF THE INTEGRAL KERNEL Assume that the non-smooth points on the boundary can be seen as the corners on the boundary Γ = (∂Ω/{(0, z) }|z ∈ R) ∪ (S 0 ∩ {(0, z)|z ∈ R}), where S = ∂V is the surface of the body V, and the corners divide the boundary Γ into d segments, i.e., Γ = ∪dj=1 Γ j (d > 1) with CΓ , 1, and Γ j ∈ C 2l+1 ( j = 1, ..., d, l ∈ N). Also assume that Γ j ( j = 1, · · · , d) have the following parametric representations x j (t) = (x1 j , x2 j ) = (ρ j (t), z j (t)) : [0, 1] → Γ j , j = 1, · · · , d,

(2.1)

where |x0j (t)| = [|ρ0j (t)|2 + |z0j (t)|2 ]1/2 ≥ c > 0, x j (0) and x j (1) denote two endpoints of Γ j . Under the parameter mapping (2.1), Eq. (1.6) can be converted into the first kind integral equations including definite integrals over [0, 1] d Z 1 X g(xi (t)) = ρ j (τ)φ∗ (x j (τ), xi (t))|x0j (τ)|σ(x j (τ))dτ, i = 1, · · · , d. j=1

(2.2)

0

In order to degrade the singularities at corners [8, 14], we apply the Sidi transformation [22] to the variable mapping, which is defined by ϕµ (t) =

Θµ (t) ; Θµ (1)

Z Θµ (t) =

t

(sinπu)µ du,

µ ∈ N.

(2.3)

0

From the equality Θµ (t) =

µ−1 1 Θµ−2 (t) − (sinπt)µ−1 cosπt, µ πµ

which can be obtained by integration by parts, we have the recursion relation ϕµ (t) = ϕµ−2 (t) −

Γ(µ)/2 (sinπt)µ−1 cosπt. √ 2 πΓ((µ + 1)/2)

where Γ(z) is the Gamma function. It is easy to see that Θ0µ (t) has µ order derivatives at the point t = 0 and t = 1. Hence, ϕ0µ (t) has also the same order derivatives. For convenience of discussion, we rewrite Eq. (2.2) as the following operator equation KW = F,

(2.4)

where K = [Kij ]di, j=1 , F = ( f1 , f2 , · · · , fd )T , f j = g j (t) = g j (ϕµ (t)), W = (w1 (τ), w2 (τ), · · · , wd (τ))T , w j (τ) = |x0j (τ)|ϕ0µ (τ)σ j (ϕµ (τ)). 3

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In (2.4), the kernel ki j (t, τ) of Kij can be written as ki j (t, τ) = ρ j (τ)T ∗ (x j (τ), xi (t)) " # αi j (t, τ) 1 qi j (t, τ) − 1 = − ln( ) + O((qi j (t, τ) − 1) ln(qi j (t, τ) − 1)) 2π 2 32  αi j (t, τ)  =− ln rρz (t, τ) − ln(8βi j (t, τ)) + E˜ i j (t, τ), ρi (τ) , 0, 2π

(2.5)

where ρ j (τ) = ρ j (ϕµ (τ)), z j (τ) = z j (ϕµ (τ)), ρ (τ)

αi j (t, τ) = [ ρji (t) ]1/2 , βi j (t, τ) = [ρ j (τ)ρi (t)]1/2 , rρ,z (t, τ) = [(ρ j (τ) − ρi (t))2 + (z j (τ) − zi (t))2 ]1/2 , E˜ i j (t, τ) = O(ρ j (τ)(qi j (t, τ) − 1) ln(qi j (t, τ) − 1), qi j (t, τ) = 1 + [(ρ j (τ) − ρi (t))2 + (z j (τ) − zi (t))2 ](2ρ j (τ)ρi (t))−1 . Now we split Kii into a singularity part and a compact perturbation part as follows Kii = Aii + Bii . Let aii be the kernel of singularity operators Aii (i = 1, 2, · · · , d), aii (t, τ) = −

αii (t, τ) −1/2 ln 2e sin(π(t − τ)) , 2π

˜ ii and let a˜ ii be the kernel of A a˜ ii (t, τ) = −

1 −1/2 ln 2e sin(π(t − τ)) . 2π

Thus, we have aii (t, τ) = αii (t, τ)˜aii (t, τ). Denoted by A = diag(A11 , A22 , · · · , Add ) and B = [Bij ]di, j=1 = K − A. Then the integral operator equation (2.4) can be splitted into (A + B)W = F,

(2.6)

where the kernel of Bij is      kii (t, τ) − aii (t, τ), i = j, bi j (t, τ) =     ki j (t, τ), i , j.

(2.7)

˜ ii is defined on the circular contour with radius e−1/2 . From [2, 7, 8], we can see that A ˜ ii has Clearly, the operator A the special property √ X imt ˜ ii σ(t) = 2 ( A |m|−1 σ(m)e ˆ + σ(0)), ˆ t ∈ [0, 2π], π m∈Z ∗ 4

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where Z ∗ = Z\{0}, Z is the set of all integers, and σ(t) ˆ is the Fourier transform of σ(t) ∈ C ∞ (i.e., the p order derivative ˜ ii σ| p+1 = |σ| p , for all σ ∈ H p , where |σ| p of σ is continuous and 2π-periodic for all p ≥ 0). As is easily shown, |A ˜ ii is an isometry from H p to H p+1 [12]. As above, the operators on C ∞ is the norm of the usual space H p . So A Aii (i = 1, · · · , d) are isometry operators from H p [0, 1] to H p+1 [0, 1] for any real number m. Hence, A is also an isometry operator from (H p [0, 1])d to (H p+1 [0, 1])d . Furthermore, Eq. (2.6) becomes (I + A−1 B)W = A−1 F = e F,

(2.8)

Let P j ( j = 1, 2, · · · , d) be the corner points of the boundary Γ. Define a function χ j ∈ [−1, 1]     − 1 as j = 1 or j = d,    χj =   π − θj    as 1 < j < d,  π where θ j is the interior angle of the middle corner P j . Since the boundary Γ is a closed polygon, the solution singularity for (1.6) occurs at the corner points P1 , ..., Pd of the boundary Γ [7, 11, 18]. The solution w j (x) =

∂φ j (x) ∂ν−

−

∂φ j (x) ∂ν+

has a singularity O(|s − s j |β j ) when s = s j , where

|χ |

β j = − (1+|χj j |) , β j ∈ [− 12 , 0), and ν and s j denote the unit normal and the arc parameter at P j respectively. Lemma 2.1. [20] (1) Let the function σ j (s) = sβ j u j (s) (0 > β j ≥ −1/2), where u j (s) is differentiable enough on [0, 1] with u j (0) , 0. Then the function w j (t) can be expressed by w j (t) = c1 u j (0)t(µ+1)β j +µ (1 + O(t2 )) as t → 0+ ,

(2.9)

where c1 is a constant independent of t. (2) Let the function σ j (s) = (1 − s)β j u˜ j (s) (0 > β j ≥ −1/2), where u˜ j (s) is differentiable enough on [0, 1] with u˜ j (1) , 0. Then the function w j (t) can be expressed by w j (t) = c2 u˜ j (1)(1 − t)(µ+1)β j +µ (1 + O((1 − t)2 ))) as t → 1− ,

(2.10)

where c2 is a constant independent of t. By Lemma 2.1 and β j ≥ − 21 , we can obtain (µ + 1)β j + µ ≥ 0 for µ ≥ 1. Furthermore, we have the following remark. Remark 1 Although σ j (s) has singularities at endpoints s = 0 and s = 1, w j (t) has no singularity by Sidi transformation at t = 0 and t = 1. Bellow we study singularity for the kernels bi j (t, τ) (i, j = 1, · · · , d). Obviously, if Γi ∩ Γ j = ∅, bi j (t, τ) are continuous on [0, 1]2 , and if Γi ∩ Γ j , ∅, bi j (t, τ) have singularities at the points (t, τ) = (0, 1) and (t, τ) = (1, 0), where ∅ denotes the empty set. Here, we only discuss the case in that (t, τ) = (1, 0), the singularity analysis for the case in that (t, τ) = (0, 1) can be similarly obtained. Defining the following function b˜ i j (t, τ) = bi j (t, τ) sinµ (πt), µ ≥ 1, Γi ∩ Γ j , ∅.

(2.11) 5

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Theorem 2.2. Let b˜ i j (t, τ) be defined by (2.11), then b˜ i j (t, τ) and

∂k b˜ i j (t,τ) ∂τk µ

(k = 1, 2) are smooth on [0, 1]2 .

Proof. By using the continuity of b˜ ii (t, τ) and the boundness of sin (πt), we can immediately complete the proof for the case i = j. Let Γi ∩ Γ j = Pi = (0, 0) (|i − j| = 1 or d − 1) and θi ∈ (0, 2π) is the corresponding interior angle. Then we have rρ,z (t, τ) = 0, which shows the kernel bi, j (t, τ) has a logarithmically singularity at (t, τ) = (1, 0). Suppose that ai (t) = |xi (t)| and a j (τ) = |x j (τ)|. Then we can obtain b˜ i, j (t, τ) = I1 (t, τ) + I2 (t, τ).

(2.12)

where I1 (t, τ) = −

αi j (t, τ) sinµ (πτ) αi j (t, τ) sinµ (πτ) 2 ln ai (t) + a2j (τ) − 2ai (t)a j (τ) cos θi − ln 8βi j (t, τ) , 2π Θµ (1) 2π Θµ (1)

and I2 (t, τ) = O((qi j (t, τ) − 1) ln(qi j (t, τ) − 1)) Since

sinµ (πτ) Θµ (1)

sinµ (πτ) . Θµ (1)

has order µ zero at τ = 0 and τ = 1, the function I2 (t, τ) is continuous on [0, 1]2 . Let

I1 (t, τ) = I11 (t, τ) + I12 (t, τ),

(2.13)

where I11 (t, τ) = −

αi j (t, τ) sinµ (πτ) 2 ln ai (t) + a2j (τ) − 2ai (t)a j (τ) cos θi 2π Θµ (1)

I12 (t, τ) = −

αi j (t, τ) sinµ (πτ) ln 8βi j (t, τ) . 2π Θµ (1)

and

Obviously, I12 (t, τ) is a smooth function on [0, 1]2 . Now we will prove I11 (t, τ) is also smooth on the same interval. We write I11 (t, τ) = I111 (t, τ) + I112 (t, τ),

(2.14)

where I111 (t, τ) = −

αi j (t, τ) sinµ (πτ) ln 1 − 2ai (t)a j (τ)/(a2i (t) + a2j (τ)) , 2π Θµ (1)

I112 (t, τ) = −

αi j (t, τ) sinµ (πτ) ln[a2i (t) + a2j (τ)]. 2π Θµ (1)

and

By 2ai (t)a j (τ)/(a2i (t) + a2j (τ)) ≤ cosθi < 1, 6

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we know that I111 (t, τ) is smooth on [0, 1]2 . Let 0 < t, τ < ε, then we have ai (t) = O(ε), a j (τ) = O(ε), sinµ (πτ) = O(εµ ) and I112 (t, τ) = O(εµ ln ε) → 0, ε → 0. Hence, I112 (t, τ) is continuous on [0, 1]2 . Furthermore, ∂I111 (t, τ) αi j (t, τ) sinµ−1 (πτ) cos(πτ) 2 ≤ ln ai (t) + a2j (τ) ∂τ 2π Θµ (1) αi j (t, τ) sinµ (πτ) (x1 j (ϕµ (t))x10 j (ϕµ (t)) + (x2 j (ϕµ (t))x20 j (ϕµ (t)))ϕ0µ (t) + 2π Θµ (1) a2i (t) + a2j (τ) 1 ∂αi j (t, τ) sinµ (πτ) + 2π ∂τ Θµ (1) = O(εµ−1 ln ε) + O(εµ )O(ε2µ )/O(ε2µ+1 ) + O(εµ ln ε) = O(εµ−1 ln ε) + O(εµ−1 ) → 0, ε → 0.

(2.15)

Similarly, we have ∂2 I111 (t, τ) ≤ O(εµ−2 ln ε) + O(εµ−2 ) → 0, as ε → 0, ∂τ2 ∂2 I111 (t,τ) and ∂I111∂τ(t,τ) are continuous ∂τ2 ∂I111 (t,τ) are continuous on [0, 1]2 . So, we ∂t

which shows that 2

∂ I111 (t,τ) ∂t2

and

on [0, 1]2 . Similar to the proof mentioned above, we can prove can obtain I111 (t, τ) ∈ C 2 [0, 1]2 for µ ≥ 3 and I(t, τ) ∈ C 2 [0, 1]2

for µ ≥ 3 . For θi = π, we have αi j (t, τ) sinµ (πτ) b˜ i j (t, τ) ≤ (ln ai (t) + a j (τ) π Θµ (1) sinµ (πτ) + ln 8βi j (t, τ)) + O((qi j (t, τ) − 1) ln(qi j (t, τ) − 1) Θµ (1) = O(εµ ln ε) + O(εµ+2 ln ε).

(2.16)

Similar to the case θi ∈ (0, 2π), we also can prove b˜ i j (t, τ) ∈ C 2 [0, 1]2 . 3. MODIFIED QUADRATURE FORMULA AND DISCRETIZATION FOR INTEGRAL OPERATORS A reformulated rule based on the modified quadrature formula [4] to achieve the quadrature values for integrals with the logarithmically singular kernels can be described as follows Z

1

ln |t − τ|σ(τ)dτ = h 0

N X j=1,t,τ j

X |x (t)|
σ(t) − h ln(2|sinπ jh|)σ(t), 2π j=1 0

ln |t − τ j |σ(τ j ) + h ln

N−1

(3.1)

7

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where τ j = jh and the step size is h = 1/N. Now we apply (3.1) to solve the boundary integral equation (1.6). Assume that h j = 1/n j (n j ∈ N, j = 1, · · · , d) is the mesh size of Γ j , t jm = τ jm = (m − 1/2)h j (m = 1, · · · , n j ) are the nodes on Γ j . (1) When i , j, we can use trapezoidal rule [1] to construct the Nyst¨om approxiamation Khij of Kij (Khij w j )(t) = h j

nj X

ki j (t, τ js )w j (τ js ), i, j = 1, · · · , d,

(3.2)

s=1 h

which has the error order at least O (h3 ) by using the Sidi transformation[22]. The trapezoidal rule approximation Bijj for the boundary integral operator Bij is h

(Bijj w j )(t) = h j

nj X

bi j (t, τ jm )w j (τ jm ), t ∈ [0, 1], i, j = 1, ..., d,

(3.3)

m=1

which has the error bounds [10, 22]   2`+1   O (h j ), for Γ j = Γ j or Γ j ∩ Γ j = ∅, ` ∈ N, (B w )(t) − (Bhj w )(t) ≤   ij j j ∞  ij   O (hω j (γ+1)+1 ), for Γi ∩ Γ j ∈ {Pm }, j where     min{β j + 1, 2}, γ even,      ωj =         min{β j + 1, 1}, γ odd. (2) When i = j, integral operator Kii = Aii + Bii has a logarithmically singularity, we use the reformulated formula (3.1) to construct the approximation Ahii and Bhii of Aii and Bii , respectively, (Ahii wi )(t) = −

ni hi αii (t, τis )  X ln 2e−1/2 sin(π(t − τis )) wi (τis ) − γi wi (t) 2π t=1,t,τ is

− ln |

e1/2 xi0 (t)ϕ0µ (t) 2π

 |wi (t) , i = 1, · · · , d.

(3.4)

P i −1 where γi = ni=1 ln(2|sin(πihi )|) and hi = 1/ni (i = 1, · · · , d). The error of (3.4) is (Ah w )(t) − (A w )(t) ≤ c h3 , i i ii i ii i

(3.5)

∞

where ci is a constant number. Accordingly, we have (Bhii wi )(t) = hi

X

bi j (t, τis )wi (τis )

s=1,t,τis

  [(ρ0i (t))2 + (z0i (t))2 ]1/2 xi0 (t)ϕ0µ (t)  hi αii (t, τis )  + | wi (t), ln |8ρi (t)| − ln | 2π 2πe−1/2 i = 1, · · · , d.

(3.6) 8

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Hence, the approxiamation Khii of Kii can be constructed by ni X

(Khii wi )(t) = hi

kii (t, τis )wi (τis ) +

s=1,s,τis

hi αii (t, τis ) ci wi (t) 2π

e1/2 xi0 (t)ϕ0µ (t) hi αii (t, τis ) ln | |wi (t) 2π 2π [(ρ0 (t))2 + (z0 (t))2 ]1/2 x0 (t)ϕ0 (t) αii (t, τis )hi i µ i i w (t), i = 1, · · · , d. ln − i 2π 16πe−1/2 ri (t) +

(3.7)

Thus the approximate equation of (2.4) is Kh Wh = Fh ,

(3.8)

where Kh = [Khij ]di, j=1 , Fh = ( f1h , f2h , · · · , fdh )t , fjh = (g(x j (t j1 )), g(x j (t j2 )), · · · , g(x j (t jd )))t , j = 1, · · · , d, Wh = (wh1 , wh2 , · · · , whd )t , whi = (wh1 (τi1 ), wh2 (τi2 ), · · · , whd ((τini )))t , i = 1, · · · , d. In fact, E.q. (3.8) can be rewritten as (Ih + (Ah )−1 Bh )Wh = F˜ h ,

(3.9)

where Ah = diag(Ah11 , Ah22 , · · · , Ahdd ), Bh = [Bhij ]di, j=1 , and F˜ h = (Ah )−1 Fh . P Obviously, (3.9) has n (= dj=1 n j ) unknowns. Once Wh is solved from (3.6), φ(y) (y ∈ Ω0 ) can be calculated by h

φ (y) =

nj d X X

ρ j (ϕµ (ti ))φ∗ (x j (ϕµ (ti ), y)wi j .

(3.10)

j=1 i=1

4. NUMERICAL EXAMPLES In this section, two numerical examples about the axisymmetric Laplace equation are computed by modified quadrature method in this paper. Let e (x) = φ (x) − φ (x) be the errors at the polar coordinate point x = (ρ, z) by n

modified quadrature method using n (=

Pd

j=1

exact

n

n j ) boundary nodes, and let rn = en /en/2 be the error ratio by modified

quadrature rule, where n = 2π/h.

Example 4.1 Consider a finite solid cylinder with bottom radius R = 2, and the height L = 3. Temperature on the upper base is g = 2, and the temperature on the other sides is 0. The analytical solution is φexact =

∞ X 2g sinhxk0 z/R J0 (xk0 r/R) k=1

xk0 sinhxk0 L/R

J1 (xk0 )

.

where J0 and J1 are 0 order and 1 order Bessel functions respectively. Assume that each boundary Γ j ( j = 1, · · · , 3) be divided into 2k (k = 3, · · · , 6) segments. We use ϕ3 transformation 9

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to compute the points in the subregion Ω1 = {(r, z)|r = 0.1R : By (2.3), we obtain ϕ3 (t) =

1 3 4 cos πt

−

3 4 cosπt

+

1 2,

1 Nr R

: 0.9R, z = 0.1L :

1 Nz L

: 0.9L, where Nr = Nz = 50}.

where t ∈ [0, 1].

The total node number of the whole boundary are n (= 3 × 2k , k = 3, · · · , 6). The errors and error ratio of the polar coordinate points P1 = (0.5, 1), P2 = (0.5, 1.5) and P3 = (0.5, 2) by modified quadrature method are listed in Table 2. From the numerical results, we can see that log2 rn ≈ 3, which shows the experimental convergence order of φ is O (h3 ). The condition numbers for the matrix Kh = Ah + Bh are listed in Table 1, where |λmin | and |λmax | are minimum and maximum eigenvalues respectively. Let Cond. be the traditional 2-norm condition number, from Table 1, we can see Ξk = Cond.|(2k ,2k ,2k ) /Cond.|(2k−1 ,2k−1 ,2k−1 ) ≈ 2 (k = 4, 5, 6), which show the modified quadrature method has excellent stability. Fig. 1 shows that isolines of temperature distribution in the subregion Ω1 and the errors for points along the line 0.04 ≤ r ≤ 1.96; z = 0.06 by modified quadrature rule. Fig. 2 and Fig. 3 show the isolines of errors in the subregion Ω1 using modified quadrature rule. Table 1: The condition number for Example 1

n

(23 ,23 ,23 )

(24 ,24 ,24 )

(25 ,25 ,25 )

(26 ,26 ,26 )

|λmin |

2.912E-3

1.409E-3

6.982E-4

3.164E-4

|λmax |

1.016

1.017

1.017

1.017

Cond

3.490E+2

7.219E+2

1.457E+3

3.215E+3

Ξk

−

2.068

2.018

2.207

Table 2: The errors for Example 1

n

(23 ,23 ,23 )

(24 ,24 ,24 )

(25 ,25 ,25 )

(26 ,26 ,26 )

en (P1 )

5.840E-4

8.299E-5

1.051E-5

1.302E-6

rn (P1 )

−

22.815

22.981

23.012

en (P2 )

6.961E-4

1.016E-4

1.296E-5

1.629E-6

rn (P2 )

−

22.777

22.970

22.992

en (P3 )

3.800E-4

6.858E-5

9.002E-6

1.184E-6

rn (P3 )

2.470

−

2

2.929

2

22.927

Example 4.2 Consider a finite solid cylinder with bottom radius R = 5, and the height L = 7. Temperature on the upper base is g = r2 , and the temperature on the other sides is 0. The analytical solution is φexact = 2R2

∞ X (xk0 )2 − 4 sinhxk0 z/R J0 (xk0 r/R) k=1

(xk0 )3

sinhxk0 L/R

J1 (xk0 )

. 10

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−1

1.4

1.4

1.2

1.2

3

n=3× 2 10

0.6

n=3× 26 −3

10

0.4

0.6

log||EM||

z

0.6

0.4

1.5

n=3× 25

0.

0.8

n=3× 24

−2

0.8

1

1 2 0.8

10

0.6 4 0.

1.2 1

2

2.5

0.2

0.4

−4

10

−5

10

0.2

1

0.2 −6

10 0.5

−7

0.2

0.4

0.6

0.8

1 r

1.2

1.4

1.6

10

1.8

0

0.5

1 0.04≤ r ≤ 1.96; z=0.06

1.5

2

Figure 1: The distributions of temperature when n = 3 × 26 (left) and errors for points along the line 0.04 ≤ r ≤ 1.96; z = 0.06 by ϕ3 transformation

0.005

0.005

0.0 05

2.5

1

0.015

5 00

0. 0

0.

0.005 01 0.

0. 00 0 05 .001

2

0.001

0.02 0.015

2

0.0005

0.005 2.5

0. 01

(right)

1

1

0.5 0.2

05

1.5

0.0005

z

05

0.0 05

0.0

0.00

0.01

z

0.005 1.5

0.5 0.4

0.6

0.8

1 r

1.2

1.4

1.6

1.8

0.2

0.4

0.6

0.8

1 r

1.2

1.4

1.6

1.8

Figure 2: Isolines of errors of temperature by 3 × 23 boundary nodes (left) and Isolines of errors of temperature by 3 × 24 boundary nodes (right)

−00

1.5e 1e−006

5 1e−00 5

0.5 0.2

0.4

0.6

0.8

1 r

1.2

1.4

1.6

0.5 1.8

0.2

0.4

0.6

06 6

5e 3.

−0 e−0

2e−0

−006

06

06

1.5e− 006 1e−00 6

5e−007 0.8

00

6

2.5

1e−006

5e−007

e−

−0 0

−0

6

5

3.5 3e

06

1

1e−00

06

6

2e 1.5e

05

06

−0

00

1.5

2e−0

−0

2e e−

1e−00

006

e−

1.5

6 1e−00

1.5

z

005

2e−

z

006

06

3e−005

3e−005 3e −0 05

6 006 6 6 0 1e−00 − 1.5e 2e−0e−00 2.5

007

4e−0

005

1e−

1

1e−

7

0 5e−0 2

1.5

5e−

3e

−0 3e

2e

1e− 2

006−006 2e− 1.5ee−006 7 1 0 5e−0

06

005

2.5

4.5e−006

05 −0

05

1e−0

0 1e−0

2.5 e−0

05

5

2.5

1 r

1.2

1.4

1.6

1.8

Figure 3: Isolines of errors of temperature by 3 × 25 boundary nodes (left) and Isolines of errors of temperature by 3 × 26 boundary nodes (right)

Let each boundary Γ j ( j = 1, · · · , 3) be divided into 2k (k = 3, · · · , 6) segments. We also use ϕ3 transformation to compute the polar coordinate points in the subregion Ω2 = {(r, z)|r = 0.1R :

1 Nr R

: 0.9R, z = 0.1L :

1 Nz L

11

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0.9L, where Nr = 50, Nz = 50}. The total node number of all piece-wised boundaries are n (= 3 × 2k , k = 3, · · · , 6). The errors and error ratio of P1 = (0.5, 1), P2 = (0.5, 3) and P3 = (0.5, 5) by modified quadrature method are listed in Table 4. From the numerical results, we can see that log2 rn ≈ 3, which shows the same experimental convergence order of φ is O (h3 ). The condition numbers for the matrix Kh = Ah + Bh are listed in Table 3, where |λmin | and |λmax | are minimum and maximum eigenvalues respectively. From Table 3, we can see Ξk = Cond.|(2k ,2k ,2k ) /Cond.|(2k−1 ,2k−1 ,2k−1 ) ≈ 2 (k = 4, 5, 6), which show the modified quadrature method has excellent stability. Fig. 4 shows that isolines of temperature distribution in the subregion Ω2 by using n = 3 × 26 boundary nodes and the errors for points along the line 0.10 ≤ r ≤ 4.00; z = 0.14 by modified quadrature method. Fig. 5 and Fig. 6 show the isolines of errors in the subregion Ω2 using modified quadrature method. Table 3: The condition number for Example 2

Λk

Λ3

Λ4

Λ5

Λ6

|λmin |

2.669E-3

1.302E-3

6.461E-4

2.453E-4

|λmax |

1.034

1.034

1.034

1.034

Cond

3.829E+2

7.942E+2

1.601E+3

4.216E+3

Ξk

−

2.074

2.016

2.633

Table 4: The errors for Example 2

Λk

(23 ,23 ,23 )

(24 ,24 ,24 )

(25 ,25 ,25 )

(26 ,26 ,26 )

en (P1 )

1.249E-3

5.422E-5

6.052E-6

9.154E-7

4.525

2

3.163

2

22.775

rn (P1 )

−

en (P2 )

6.616E-3

6.621E-4

8.005E-5

1.029E-5

rn (P2 )

−

23.321

23.048

22.959

en (P3 )

2.378E-2

3.164E-3

3.916E-4

4.948E-5

rn (P3 )

−

23.113

23.014

22.985

5. CONCLUSIONS In this paper, the modified quadrature method with Sidi transformation are applied for the first kind integral equations of 3D axisymmetric Laplace problems on polygons. The numerical results show that the rate of error convergence for the modified quadrature can achieve O (h3 ). Furthermore, the excellent stability of the modified 12

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−1

8 7

6

4

3

6

5

5.5

10

6 5

4 3

4

5

−2

10

2

3

3

−3

4.5

10

2

1

log||E ||

z

2

2

M

4 3.5

1

3

−4

10

−5

2.5

10

1

1

n=3× 23

2

n=3× 24

−6

10

1.5

n=3× 25 n=3× 26

1 −7

0.5

1

1.5

2

2.5 r

3

3.5

4

10

4.5

0

1

2 3 0.10≤ r ≤ 4.90; z=0.14.

4

5

Figure 4: The distributions of isolines temperature by ϕ3 transformation when n = 3 × 26 and errors for points along the line 0.10 ≤ r ≤ 4.00; z =

z

0.04

z

2 00 0.

02

0.004

0.00 0.002

0.002 0.0 04

4

2 0.0

3

3.5 3

0.00

2

0.04

2 0.0

2.5

2.5

2

2

1.5

1.5

1 0.5

0.004

2

4.5

0.02

3.5

4 0.00 02 0 . 0

06 0.0

4

0.008 06 0.0

0.008

5

0.04 0.06 0.08 0.1 0.12

0.02

06

5.5 0.004

0.02

0.02

0.0

6

4 0.0 2 0 0.

5 4.5

0.02 0.04

0.00.00.06 24

0.06 0.08 0.1

5.5

04 0.

0.0

0.1 4 0.12 0.1 0.08 0.06

0.06 0.08 0.1

0.02

0.04

04 0.

4 600 0..0000186 0 .0 0.000. 0

0.08 0.06

0.12

6

0.02

0.14

1 1

1.5

2

2.5 r

3

3.5

4

4.5

0.5

1

1.5

2

2.5 r

3

3.5

4

4.5

0.0002

4.5 4

0

0.0

00

2

3

3 2.5

0.000

1

2

1.5

1.5

1 0.5

2e−005

3.5

2.5 2

005

2e−

05

0.0001

4e−005

2e−

0 0.0 01 00 0.0003 2

0.0

01 .00

6e−005 05 4e−0

5

−0 2e

3.5

0.0003

0.0001

z

4

5.5

z

4.5

0.0004

0.000120.0001 05 −0 05 6e e−0 05 05 8e−0 4 −0 2e

0.00 01 8e−0 05

6

6e−005

5

08 0.007 005004 3 0.00.0.000002 0.0060 0 .0 0.000 0 1 0.0005 4 00 0 0.000 3 . 0 0 0.00 2 0 0.00

4e−005

5.5

0.001 0.0009

0.0006 0.0005 0.0004

0.00 0.00 08 0 0.0006 7

6

005

Figure 5: Isolines of errors of temperature by 3 × 23 boundary nodes (left) and Isolines of errors of temperature by 3 × 24 boundary nodes (right)

1 1

1.5

2

2.5 r

3

3.5

4

4.5

0.5

1

1.5

2

2.5 r

3

3.5

4

4.5

Figure 6: Isolines of errors of temperature by 3 × 25 boundary nodes (left) and Isolines of errors of temperature by 3 × 26 boundary nodes (right)

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quadrature method has been presented, which shows the condition number of the discrete system of the weakly singularity problems is of the order O (h−1 ).

References [1] A. Sidi and M. Israeli, Quadrature methods for periodic singular and weakly singular Fredholm integral equation, J. Sci. Comput. 3, pp. 201–231 (1988). [2] K. Atkinson and I. H. Sloan, The numerical solution of first-kind Logarithmic-kernel integral equation on smooth open arcs, Math., Comp. 56, pp. 119–139 (1991). [3] J. Saranen and I. Sloan, Quadrature methods for logarithmic-kernel integral equations on closed curves, IMA J. Numer. Anal. 12, pp. 167–187 (1992). [4] J. Saranen, The modified quadrature method for logarithmic-kernel integral equations on closed curves, J. Integral Equations Appl. 3, pp. 575–600 (1991). [5] Y. C. Hon and R. Schaback, Solving the 3D Laplace equation by meshless collocation via harmonic kernels, Adv. Comput Math. 38, pp. 1-C19 (2013). [6] E. P. Stephan and W. L. Wendland, An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems, Applicable Anal. 18, pp. 183–219 (1984). [7] Y. Yan and I. Sloan, On integral equations of the first kind with logarithmic kernels, J. Integral Equations Appl. 1, pp. 517–548 (1988). [8] Y. Yan, The collocation method for first-kind boundary integral equations on polygonal regions, Math. Comp. 54, pp. 139–154 (1990). [9] M. Costabel, V. J. Ervin and E. P. Stephan, On the convergence of collocation methods for Symm’s integral equation on open curves, Math. Comp. 51, pp. 167–179 (1988). [10] P. Davis, Methods of Numerical Integration, Second edition, Academic Press, New York, 1984. [11] K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, UK, 1997. [12] J. Huang and T. L¨u, The mechanical quadrature methods and their extrapolation for solving BIE of Steklov eigenvalue problems, J. Comput. Math. 22, pp. 719-726 (2004). [13] J. L. Hess and A. M. O. Smith, Calculation of potential flow about arbitrary bodies, in Progress in Aeronautical Science, volume 8. Pergamon Press, London, 1967. [14] K. Atkinson, The numerical solution of Laplace’s equation on a wedge, IMA J Numer. Anal. 4, pp. 19–41 (1984). [15] G. A. Chandler, Galerkin’s method for boundary integral equations on polygonal domains, J. Austral. Math Soc. Ser. B. 26, pp. 1–13 (1984). [16] R. Kress, Linear Integral Equations, Springer-Verlag, Berlin, 1989. [17] A. A. Baker and R. T. Fenner, Bounary integral equation analysis of axisymmetric thermoelastic problem, J. Strain Anal. 18, pp. 329–351 (1981). [18] I. H. Sloan and A. Spence, The Galerkin method for integral equations of first-kind with logarithmic kernel: theory, IMA J. Numer. Anal. 8, pp. 105–122 (1988). [19] I. H. Sloan and A. Spence, The Galerkin method for integral equations of first-kind with logarithmic kernel: applications, IMA J. Numer. Anal. 8, pp. 123–140 (1988). [20] J. Huang, G. Zeng, X.-M. He, et al. Splitting extrapolation algorithm for first kind boundary integral equations with singularities by mechanical quadrature methods, Adv. Comput. Math. 36, pp. 79–97 (2012). [21] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 1980. [22] A. Sidi, A new variable transformation for numerical integration, Int Ser Numer Math. 112, pp. 359–373 (1993). [23] S. Christiansen, Numerical solution of an integral equation with a logarithmic kernel, BIT. 11, 276-287 (1971). [24] M. S. Abou El-Seoud, Numerische Behandlung von schwach singularen Integralgleichungen 1. Art. Dissertation Technische Hochschule Darmstadt, Federal Republic of Germany, Darmstadt, 1979.

14

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STABILITY OF THE GENERALIAZED VERSION OF CAUCHY TYPE EQUATION FROM FUNCTIONAL OPERATOR VIEWPOINT CHANG IL KIM1 , GILJUN HAN2 , AND SEONG-A SHIM3*

1

Department of Mathematics Education, Dankook University, 126 Jukjeon, Yongin, Gyeonggi 448-701, Korea [email protected] 2 Department of Mathematics Education, Dankook University, 126 Jukjeon, Yongin, Gyeonggi 448-701, Korea [email protected] 3 Department of Mathematics, Sungshin Women’s University, SeongBuk-Gu, Seoul 136-742, Korea [email protected] Abstract. We introduce a viewpoint to analyze functional inequalities for the generalized version of Hyers-Ulam stability problems. Especially, this article focuses the following generalized version of Cauchy functional inequality containing an general extra term Gf . kf (x + y) − f (x) − f (y) − Gf (x, y)k ≤ φ(x, y) We investigate the functional operator properties of Gf . And by virtue of this observation, we provide a method to produce lots of interesting additive functional equations as applications. Key Words : Generalized Hyres-Ulam stability, Cauchy type additive functional equation, functional operator, Banach space, iterative technique 2010 Mathematics Subject Classification : Primary 39B, Secondary 26D

1. Introduction In 1940, S. M. Ulam proposed the following stability problem (See [9]) : “Let G1 be a group and G2 a metric group with the metric d. Given a constant δ > 0, does there exist a constant c > 0 such that if a mapping f : G1 −→ G2 satisfies d(f (xy), f (x)f (y)) < c for all x, y ∈ G1 , then a unique homomorphism h : G1 −→ G2 exists with d(f (x), h(x)) < δ for all x ∈ G1 ?” In 1941, D. H. Hyers [2] answered this problem under the assumption that the groups G1 and G2 are Banach spaces. Later T. Aoki [1] and Th. M. Rassias [7] generalized the result of Hyers. Following these results, many interesting stability problems have been studied by various mathematicians. *

Corresponding author. 1

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Among these, for additive equations, Th. M. Rassias [7] solved the generalized Hyers-Ulam stability of the functional inequality kf (x + y) − f (x) − f (y)k ≤ (kxkp + kykp ) for some  ≥ 0 and p with p < 1 and all x, y ∈ X, where f : X −→ Y is a function between Banach spaces. In 1982-1989, J. M. Rassias [3, 4, 5] solved the generalized Hyers-Ulam stability of the functional inequality kf (x + y) − f (x) − f (y)k ≤ kxkp kykq for some  ≥ 0 and p, q ∈ R with p + q 6= 1 and for all x, y ∈ X, where f : X −→ Y is a function from a real normed space X to a real Banach space Y . Also K. Ravi, M. Arunkumar and J. M. Rassias introduced the productsum function in [8] and M. J. Rassias proved in [6] the generalized HyersUlam product-sum stability of the Cauchy type additive functional inequality (1.1) kf (x+y)+f (x−y)+f (y −x)−f (x)−f (y)k ≤ (kxkα/2 kykα/2 +kxkα +kykα ) for every x, y ∈ X with  ≥ 0 and α 6= 1, where f : X −→ Y is a function from a real normed space X to a real Banach space Y . Here, we look at the functional inequality (1.1) as the following. kf (x + y) − f (x) − f (y) + Gf (x, y)k ≤ (kxkα/2 kykα/2 + kxkα + kykα ), where Gf (x, y) = f (x − y) + f (y − x). In this inequality, Gf (x, y) can be considered as a functional operator depending not only on x, y ∈ X but also on f : X −→ Y . We can easily know that Gf (x, y) = f (x − y) + f (y − x) = 0 for all additive function f . With this point of view, in general, for a given functional inequality (1.2)

kF (f, x, y)k ≤ φ(x, y)

we split the functional operator F into two parts as F (f, x, y) = S(f, x, y) + Gf (x, y). Here S(f, x, y) is a standard form which characterize the function f satisfying (1.2), and Gf is the extra term. Especially in this paper, we deal with the case of S(f, x, y) = f (x + y) − f (x) − f (y), and are interested in the conditions on Gf as functional operator in this case. In other words, we are interested in what kind of terms can be added to the standard additive functional equation f (x + y) = f (x) + f (y) while the generalized Hyers-Ulam stability still holds for the new functional equation (1.3)

f (x + y) = f (x) + f (y) + Gf (x, y).

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The precise definition of Gf is given in section 2. Finally we remark that in order to the stability holds for the generalized version of the Cauchy type functional equation (1.3), a function f satisfying (1.3) should be additive. This would be an indispensable necessary condition on Gf in the general context. So we have to impose this fact as a condition in Theorem 2.1. However we eliminate this condition in some practical problems in section 3. Furthermore in section 3, by our new observations, we illustrate methods which are easier than tedious calculations to solve the stability problems. And some of the interesting examples made from our observations are listed. 2. Cauchy type additive functional inequalities with general terms Let X be a real normed linear space and Y a real Banach space. For given l ∈ N and any i ∈ {1, 2, · · ·, l}, let σi : X × X −→ X be a binary operation such that σi (rx, ry) = rσi (x, y) for all x, y ∈ X and all r ∈ R. It is clear that σi (0, 0) = 0. Also let F : Y l −→ Y be a linear, continuous function. For a map f : X −→ Y , define Gf (x, y) = F (f (σ1 (x, y)), f (σ2 (x, y)), · · ·, f (σl (x, y))). As mentioned in section 1, we consider the following condition on Gf which should be proved when specific expression for Gf is given. Condition 1. If f satisfies the equation (1.3), then Gf (x, y) = 0 for all x, y ∈ X. We remark that Condition 1 implies Gf (x, y) = 0 for all x, y ∈ X when f is additive and satisfies (1.3). Now we prove the following main stability theorem. Theorem 2.1. Let Gf be a functional operator satisfying Condition 1, and assume that (2.1)

kGf (x, x)k ≤

s X

kai Gf (λi x, 0)k +

i=1

t X

kbj Gf (0, δj x)k

j=1

for some s, t ∈ N, some real numbers ai , bj , λi , δj and for all x ∈ X and all f : X −→ Y . And let f : X −→ Y be a mapping such that the inequality (2.2) kf (x + y) − f (x) − f (y) − Gf (x, y)k ≤ (kxkp kykp + kxk2p + kyk2p ) holds for all x, y ∈ X,  ≥ 0 and p 6= 12 .

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Then there is a unique additive mapping A : X −→ Y such that P P 3 + si=1 |ai ||λi |2p + tj=1 |bi ||δi |2p (2.3) kf (x) − A(x)k ≤ kxk2p |2 − 4p | for all x ∈ X. Proof. First we show that f (0) = 0. Since
Gf (x, y) = F f (σ1 (x, y)), f (σ2 (x, y)), · · ·, f (σl (x, y)) for a continuous linear map F , we can write X Gf (x, y) = ci f (σi (x, y)) 1≤i≤l

P for some ci ∈ R. Since σi (0, 0) = 0, Gf (0, 0) = cf (0), where c = 1≤i≤l ci . We claim that c 6= −1. If c = −1, then it is easily checked that a non-zero constant function f satisfies (1.3). But this contradicts to Condition 1. So, c 6= −1. Setting x = 0, y = 0 in (2.2), we have 0 = f (0) + Gf (0, 0) = (1 + c)f (0). Since c 6= −1, we have f (0) = 0. Now we analyze the condition on Gf . Setting y = 0 (x = 0, y = x, resp.) in (2.2), we get (2.4)

kGf (x, 0)k ≤ kxk2p (kGf (0, x)k ≤ kxk2p , resp.)

for all x ∈ X. Replacing y by x in (2.2), we get kf (2x) − 2f (x) − Gf (x, x)k ≤ 3kxk2p for all x ∈ X. By (2.1) and (2.4), we have kf (2x) − 2f (x)k ≤ 3kxk2p + kGf (x, x)k (2.5)

≤ 3kxk

2p

+

s X

kai Gf (λi x, 0)k +

i=1

 ≤ 3 +

s X

t X

kbj Gf (0, δj x)k

j=1

|ai ||λi |2p +

i=1

t X

 |bj ||δj |2p  kxk2p

j=1

for all x ∈ X. Let M =3+

s X

|ai ||λi |2p +

i=1

t X

|bj ||δj |2p

j=1

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Suppose that p < 21 . Then by (2.5), for any n ∈ N ∪ {0}, we have kf (x) − 2−n f (2n x)k = kf (x) − 2−1 f (2x)k + k2−1 f (2x) − 2−2 f (22 x)k+ (2.6)

· · · +k2−n+1 f (2n−1 x) − 2−n f (2n x)k ≤ M kxk2p

1 − 2(2p−1)n 2 − 4p

for all x ∈ X. For any n ∈ N ∪ {0}, let fn (x) = 2−n f (2n x). Then for n > m in N ∪ {0}, by (2.6), we have kfn (x) − fm (x)k = k2−n f (2n x) − 2−m f (2m x)k = 2−m kf (2m x) − 2−(n−m) f (2n−m (2m x))k ≤ 2−m M k2m xk2p

1 − 2(2p−1)(n−m) 2 − 4p

1 − 2(2p−1)(n−m) m(2p−1) 2 2 − 4p for all x ∈ X. Since 2p − 1 < 0, {fn (x)} is a Cauchy sequence in Y. Since Y is a Banach space, there exists a mapping A : X −→ Y such that ≤ M kxk2p

A(x) = lim fn (x) n→∞

for all x ∈ X. By (2.6), we have M kxk2p 2 − 4p for all x ∈ X. Replacing x by 2n x and y by 2n y in (2.2), we have kf (x) − A(x)k ≤

(2.7)

k2−n f (2n (x + y)) − 2−n f (2n x) − 2−n f (2n y) − 2−n Gf (2n x, 2n y)k ≤ (kxkp kykp + kxk2p + kyk2p )2n(2p−1)

for all x, y ∈ X. Since 2p − 1 < 0, limn→∞ 2n(2p−1) = 0 and by (2.7), we have A(x + y) = A(x) + A(y) + lim 2−n Gf (2n x, 2n y) n→∞

for all x, y ∈ X. Since F is linear and continuous, by the conditions of σi , we have lim 2−n Gf (2n x, 2n y) n→∞

= lim 2−n F (f (σ1 (2n x, 2n y)), f (σ2 (2n x, 2n y)), · · ·, f (σl (2n x, 2n y))) n→∞

= lim F (2−n f (2n σ1 (x, y)), 2−n f (2n σ2 (x, y)), · · ·, 2−n f (2n σl (x, y))) n→∞

= F (A(σ1 (x, y)), A(σ2 (x, y)), · · ·, A(σl (x, y))) = GA (x, y),

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and we have A(x + y) = A(x) + A(y) + GA (x, y) for all x, y ∈ X. Hence A satisfies (1.3) and so GA (x, y) = 0 for all x, y ∈ X by Condition 1. Thus A is additive. Now, we will show that A is unique. Suppose that C : X −→ Y is an additive mapping satisfying (2.3). Since both A and C are additive, A(2n x) = 2n A(x),

C(2n x) = 2n C(x)

for all x ∈ X and all n ∈ N ∪ {0}. By (2.3), we have kA(x) − C(x)k ≤ 2−n kA(2n x) − f (2n x)k + 2−n kf (2n x) − C(2n x)k M ≤ kxk2p 2(2p−1)n+1 −→ 0 2 − 4p as n −→ ∞, because 2p < 1. Hence we can conclude A(x) = C(x) for all x ∈ X. Thus we prove Theorem 2.1 for the case of p < 12 . Suppose that p > 21 . Replacing x by 2−1 x in (2.5), we get kf (x) − 2f (2−1 x)k ≤ M kxk2p 2−2p for all x ∈ X and by similar calculation as (2.6), we get kf (x) − 2n f (2−n x)k M ≤ p kxk2p (1 − 2(1−2p)n ) 4 −2 for all x ∈ X and all n ∈ N. Then {2n f (2−n x)} is a Cauchy sequence in Y . Since Y is a Banach space, there exists a mapping A : X −→ Y such that A(x) = lim 2n f (2−n x) n→∞

for all x ∈ X. Moreover, we have kf (x) − A(x)k ≤

M kxk2p −2

4p

for all x ∈ X. Replacing x by 2−n x and y by 2−n y in (2.2), we have k2n f (2−n (x + y)) − 2n f (2−n x) − 2n f (2−n y) − 2n Gf (2−n x, 2−n y)k ≤ (kxkp kykp + kxk2p + kyk2p )2n(1−2p) for all x, y ∈ X. Since 1 − 2p < 0, limn→∞ 2n(1−2p) = 0 and hence A(x + y) = A(x) + A(y) + GA (x, y) for all x, y ∈ X. The rest of the proof for p >

1 2

is similar to the above proof for p < 21 .

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Surprisingly, if we confine Gf to somewhat special cases, the Condition 1 in Theorem 2.1 is automatically satisfied just after checking some additional conditions on Gf , without considering full equation (1.3). For example, among Gf satisfying X (2.8) Gf (x, y) = ci f (mi x + ni y) 1≤i≤l

for mi , ni ∈ Q and ci ∈ R with X X X X (2.9) ci mi = ci ni = 0, ci |mi | 6= 0 or ci |ni | 6= 0, we can find a class of examples of Cauchy type functional equations satisfying the generalized Hyers-Ulam stability. The following corollary gives an answer to this problem, which contains many interesting examples. Corollary 2.2. Assume that Gf satisfies (2.8), (2.9) and all of the conditions in Theorem 2.1 hold except Condition 1. Further, suppose that for all k ∈ N, there exists µk ∈ R such that (2.10)

kGf (x, (k + 1)x)k ≤ kGf (µk x, kµk x)k

and (2.11)

Gf (−x, −y) = Gf (x, y),

for all x, y ∈ X and all f : X −→ Y . Then there is a unique additive mapping A : X −→ Y such that P P 3 + si=1 |ai ||λi |2p + tj=1 |bi ||δi |2p kxk2p kf (x) − f (0) − A(x)k ≤ |2 − 4p | for all x ∈ X. Proof. P When Gf has the form of (2.8), P if f satisfies (1.3) then f (0) = 0 or i ci = −1. But in the case of i ci = −1, it is easily verified that f (x) − f (0) also satisfies (1.3) and (2.2). Hence it is enough to prove with the assumption f (0) = 0. First, we claim that if f satisfies (1.3) then f is additive. Set y = 0 (x = 0, resp.) in (1.3), then we have (2.12)

Gf (x, 0) = 0

(Gf (0, y) = 0, resp.).

Setting y = x in (1.3), we have Gf (x, x) = f (2x) − 2f (x). So by (2.1) and (2.12), we have (2.13)

f (2x) − 2f (x) = 0.

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From (2.10), we have

(2.14)


kGf (x, kx)k ≤kGf µk−1 x, (k − 1)µk−1 x k .. .
≤kGf µk−1 µk−2 · · · µ1 x, µk−1 µk−2 · · · µ1 x k

for k ≥ 2. By (2.1) and (2.12) we have Gf (x, x) = 0 for all x ∈ X. So from (2.14) we have (2.15)

Gf (x, kx) = 0

for all k ∈ N and all x ∈ X. Setting y = kx in (1.3), we have
Gf (x, kx) = f (k + 1)x − f (x) − f (kx), hence by (2.15),
f (k + 1)x − f (x) − f (kx) = 0.

(2.16)

So, from (2.13) and (2.16) the standard induction argument gives that f (nx) = nf (x) for all positive integer n, and by the same argument in the proof of Theorem 2.3 below, we also have (2.17)

f (rx) = rf (x)

for all positive rational number r. Now, without loss of generality, we may assume that not all mi ’s are P zeroes and ci |mi | 6= 0. Denote by I + , I − the set of all i’s satisfying mi to be positive, negative respectively. Then by (2.17), we have 0 = Gf (x, 0) =

X

ci f (mi x) =

X

cj f (mj x) +

=

X

X

cj mj f (x) +

ck f (mk x)

k∈I −

j∈I +

(2.18)

X

ck (−mk )f (−x).

k∈I −

j∈I +

By (2.9), we have (2.19)

X

cj mj = −

X

ck mk 6= 0,

k∈I −

j∈I +

hence by (2.18) and (2.19), we have f (−x) = −f (x). It means that Gf (−x, −y) = −Gf (x, y), so from (2.11) we have Gf (x, y) = 0 for all x, y ∈ X, that is, f is additive. In fact, we showed that f (rx) = rf (x) for all rational number r. Now the rest part of the proof is the same as that of Theorem 2.1.

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Note that we don’t need continuity condition on f to prove Theorem 2.1. As in [7], if we impose the continuity condition we can also get the similar conclusion in our cases. So we can get the following theorem. Theorem 2.3. Let f : X −→ Y be a mapping satisfying the conditions of Theorem 2.1. Suppose that f (tx) is continuous on R for each x ∈ X. Then there is a unique linear mapping A : X −→ Y satisfying (2.3). Proof. By Theorem 2.1, there is a unique additive mapping A : X −→ Y satisfying (2.3). Let r ∈ R. If r ∈ N, then A(rx) = rA(x) for all x ∈ X, because A is additive. Since A is additive, A(−x) = −A(x) for all x ∈ X and hence if r = 0 or r is a negative integer, then A(rx) = rA(x) for all x ∈ X. For any integer m(6= 0),     1 1 A(x) = A m x = mA x m m and so   1 1 A(x) = A x m m n If r = m for integers n, m(m 6= 0), then n  n (2.20) A x = A(x) m m for all x ∈ X. Suppose that r ∈ R. Then there is a sequence {rn } in Q such that r = limn→∞ rn and since f (tx) is continuous on R for a fixed x ∈ X, by (2.20), we have A(rx) = lim 2−n f (2n rx) n→∞   = lim 2−n lim f (2n rk x) n→∞

k→∞

= lim A(rk x) k→∞

= lim rk A(x) k→∞

= rA(x) for all x ∈ X. Hence A is a linear mapping.



Remark 2.4. In Corollary 2.2, if we impose the continuity condition on f (tx) as in Theorem 2.3, the functional Gf can be extended to X Gf (x, y) = ci f (mi x + ni y) 1≤i≤l

for ci , mi , ni ∈ R,

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3. Applications In this section, by virtue of the theorems in section 2, we prove the generalized Hyers-Ulam stability theorem for some selected functional equations. The first example can be proved by slightly modified application of Corollary 2.2. Consider the following functional equation :   3 f (x + y) − f (2x − 3y) − f (3y − 2x) − f (6y − 4x) + 4f y−x 2 (3.1) =f (x) + f (y) Theorem 3.1. Let f : X −→ Y be a mapping such that  kf (x + y) − f (2x − 3y) − f (3y − 2x) − f (6y − 4x) + 4f

 3 y−x 2

− f (x) − f (y)k ≤ (kxkp kykp + kxk2p + kyk2p ) for all x, y ∈ X, some  ≥ 0, and p 6= 12 . Then there is a unique additive mapping A : X −→ Y such that 3 · 4p + 1 kf (x) − f (0) − A(x)k ≤ p kxk2p 4 |2 − 4p | for all x ∈ X. Proof. In this case,  Gf (x, y) = f (2x − 3y) + f (3y − 2x) + f (6y − 4x) − 4f

 3 y−x . 2

Let fe(x) = f (x) − f (0). Then Gfe(x, 0) =fe(2x) + fe(−2x) + fe(−4x) − 4fe(−x)   1 e e e e x Gfe(x, x) =f (−x) + f (x) + f (2x) − 4f 2 Gfe(x, kx) =fe((2 − 3k)x) + fe((3k − 2)x)    3k − 2 + fe(2(3k − 2)x) − 4fe x 2 Gfe(x, (k + 1)x) =fe((−1 − 3k)x) + fe((3k + 1)x)    3k + 1 e e x + f (2(3k + 1)x) − 4f 2 Now simple calculations give the following.   1 Gfe(x, x) = Gfe − x, 0 2 Gfe x, (k + 1)x) = Gfe (µk x, kµk x) , where µk =

3k+1 3k−2 .

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By the same argument in the proof of Corollary 2.2, we have (3.2)

fe(rx) = rfe(x)

for positive rational number r. So, if fe satisfies (3.1), we have (3.3)

0 =Gfe(x, 0) = fe(2x) + fe(−2x) + fe(−4x) − 4fe(−x)   =2 fe(x) + fe(−x) .

By (3.2) and (3.3), if fe satisfies (3.1) then Gfe(x, y) = 0 for all x, y ∈ X, hence fe is additive. So the proof is completed by Corollary 2.2.  Second example does not satisfy the conditions in Corollary 2.2, but Theorem 2.1 works well for it. Consider the following functional equation : (3.4)

f (x + y) = f (2x − 4y) − f (x) + 5f (y)

Theorem 3.2. Let f : X −→ Y be a mapping such that kf (x + y) − f (2x − 4y) + f (x) − 5f (y)k ≤ (kxkp kykp + kxk2p + kyk2p ) for all x, y ∈ X, some  ≥ 0, and p 6= 12 . Then there is a unique additive mapping A : X −→ Y such that kf (x) − A(x)k ≤

3(4p + 1) kxk2p , 4p |2 − 4p |

for all x ∈ X. Proof. Setting y = 0 in (3.4), we have (3.5)

f (2x) = 2f (x).

Setting y = x in (3.4) and using (3.5), we have (3.6)

f (−x) = −f (x).

Setting x = −2y, y = x in (3.4), and using (3.5), (3.6), we have f (x − 2y) = −4f (x + y) − 2f (y) + 5f (x).  Using (3.5) and calculating (3.4) + 2 × (3.7) , we have (3.7)

f (x + y) = f (x) + f (y), hence f is additive. Now Gf (x, y) = f (2x − 4y) − 2f (x) + 4f (y)

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in this case, so we have the followings. Gf (x, x) = f (−2x) + 2f (x) Gf (x, 0) = f (2x) − 2f (x) Gf (0, x) = f (−4x) + 4f (x) So, we have

x

  x , 0 + Gf 0, , 2 2 hence

 x   x 



kGf (x, x)k ≤ 2 Gf , 0 + Gf 0,

. 2 2 Thus the proof is completed by Theorem 2.1. Gf (x, x) = 2Gf



Next two examples have some nondetermined constants in the functional equations. Consider the following equation : (3.8)

f (x + y) + f (mx − ny) + f (ny − mx) = f (x) + f (y)

for real numbers m, n. If m = n = 0, then we have the result trivially, so without loss of generality we assume that m 6= 0. Theorem 3.3. Let f : X −→ Y be a mapping such that kf (x + y) + f (mx − ny) + f (ny − mx) − f (x) − f (y)k ≤ (kxkp kykp + kxk2p + kyk2p ) for all x, y ∈ X, some  ≥ 0, and p 6= 21 . Then there is a unique additive mapping A : X −→ Y such that kf (x) − A(x)k ≤

3m2p + |m − n|2p kxk2p , m2p |2 − 4p |

for all x ∈ X. Proof. Suppose that f satisfies (3.8). Replacing y by 0 in (3.8), we have f (mx) = −f (−mx)

(3.9)

for all x ∈ X. Replacing x by

x m

(3.10)

f (x) = f (−x)

in (3.9), we have

for all x ∈ X. Hence by (3.10), f is additive. For any x, y ∈ X, let σ1 (x, y) = mx − ny,

σ2 (x, y) = ny − mx

and for any s, t ∈ Y , let F (s, t) = s + t. Then σi (rx, ry) = rσi (x, y) for all x, y ∈ X and all r ∈ R and F : Y 2 −→ Y is a linear, continuous function. Since Gf (x, y) = −f (mx − ny) − f (ny − mx),

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we have Gf (x, x) = −f ((m − n)x) − f ((n − m)x) Gf (x, 0) = −f (mx) − f (−mx) for all x ∈ X. Hence we have Gf (x, x) = Gf (λx, 0) for all x ∈ X, where λ = m−n m . So the proof is completed by Theorem 2.1.



Remark 3.4. When m and n are rational numbers in Theorem 3.3, in virtue of Corollary 2.2, we can get the results by checking the followings. Gf (−x, −y) = −f (−mx + ny) − f (−ny + mx) Gf (x, 0) = −f (mx) − f (−mx) Gf (x, x) = −f ((m − n)x) − f ((n − m)x)  


 Gf x, (k + 1)x = −f m − n(k + 1) x − f n(k + 1) − m x  

 Gf (x, kx) = −f m − nk x − f nk − m x Now simple calculations give the followings. X ci |mi | = −|m| − |m| 6= 0 Gf (−x, −y) = Gf (x, y) Gf (x, x) = Gf (λx, 0)

Gf x, (k + 1)x = Gf µk x, kµk x m−n(k+1) if m 6= nk for any k ∈ N. with λ = m−n m , µk = m−nk If m = nk0 for some k0 ∈ N, then Gf (x, k0 x) = 0. So, we can still have Gf (x, kx) = 0 for all k ∈ N by restarting induction argument in Corollary 2.2 from k0 . Now Corollary 2.2 can be applied. In this case, using Corollary 2.2 looks like more complicated than checking Condition 1. But Corollary 2.2 provides a systematic way to prove the stability theorem. Also, it is more effective to use Corollary 2.2 when Gf has many terms. Furthermore Corollary 2.2 gives a class of functional equations which satisfy generalized Hyers-Ulam stability theorem just by calculating the properties of Gf .

Theorem 3.3 can be regarded as a general version of the following stability problem appeared in [6]. Theorem 3.5. ([6]) Let f : X −→ Y be a mapping such that kf (x + y) + f (x − y) + f (y − x) − f (x) − f (y)k ≤ (kxkp kykp + kxk2p + kyk2p )

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for all x, y ∈ X, some  ≥ 0, and p 6= 12 . Then there is a unique additive mapping A : X −→ Y such that kf (x) − A(x)k ≤

3 kxk2p |2 − 4p |

for all x ∈ X. The last example in this paper looks like a little bit complicated, but it can be proved by Theorem 2.1. Consider the following equation : 2f (x + y) + f (mx) + mf (−x) + f (−x − y) = f (x) + f (y)

(3.11)

for real number m 6= −2. Theorem 3.6. Let f : X −→ Y be a mapping such that k2f (x + y) + f (mx) + mf (−x) + f (−x − y) − f (x) − f (y)k ≤ (kxkp kykp + kxk2p + kyk2p ) for all x, y ∈ X, some  ≥ 0, and p 6= 21 . Then there is a unique additive mapping A : X −→ Y such that kf (x) − A(x)k ≤

(3.12)

5 + 4p kxk2p |2 − 4p |

for all x ∈ X. Proof. Suppose that f satisfies (3.11). Putting x = 0, y = 0 in (3.11), we have f (0) = 0. Putting y = 0 in (3.11) we have (3.13)

f (mx) + mf (−x) + f (x) + f (−x) = 0

for all x ∈ X. Putting x = 0, y = x in (3.11), we have (3.14)

f (x) + f (−x) = 0

for all x ∈ X. Plugging (3.14) into (3.13), we have (3.15)

f (mx) + mf (−x) = 0

for all x ∈ X. Applying (3.14) and (3.15) to (3.11), we have f (x + y) = f (x) + f (y) for all x, y ∈ X. Thus f is additive. Since Gf (x, y) = −f (mx) − mf (−x) − f (x + y) − f (−x − y), we have Gf (x, x) = −f (mx) − mf (−x) − f (2x) − f (−2x) Gf (x, 0) = −f (mx) − mf (−x) − f (x) − f (−x) Gf (0, x) = −f (x) − f (−x)

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for all x ∈ X. Hence kGf (x, x)k =kf (mx) + mf (−x) + f (2x) + f (−2x)k ≤kf (mx) + mf (−x) + f (x) + f (−x)k + kf (x) + f (−x)k + kf (2x) + f (−2x)k =kGf (x, 0)k + kGf (0, x)k + kGf (0, 2x)k for all x ∈ X. Now the proof is completed by Theorem 2.1.



As Remark 3.4, we can show that if m is a rational number in Theorem 3.6, then the result can be obtained after straightforward checking the properties of Gf as in Corollary 2.2, instead of after proving Condition 1. Furthermore, even though the condition m 6= −2 is needed for using Theorem 2.1 directly, if m = −2 then we can actually conclude that there exists a unique additive mapping A : X −→ Y satisfying 5 + 4p kf (x) − f (0) − A(x)k ≤ kxk2p |2 − 4p | for all x ∈ X, by the argument in the proof of Corollary 2.2. And, in all of the theorems in this section, if f (tx) is continuous on R for each x ∈ X then A is linear by Theorem 2.3. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [2] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. [3] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46(1982), 126-130. [4] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108(1984), 445-446. [5] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57(1989), 268-273. [6] M. J. Rassias, Generalized Hyers-Ulam product-sum stability of a Cauchy Type additive functional Equation, Eur. J. Pure Appl. Math. 4(2011), 50-58. [7] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [8] K. Ravi, M. Arunkumar and J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Inter. J. Math. Stat. 3(2008), 36-46. [9] S. M. Ulam, A collection of mathematical problems, Interscience Publisher, Inc., No. 8, New York; Problems in morfern Mathematics, Wiley and Sons, New York, Chapter VI, 1964.

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Inequalities of Simpson Type for Functions Whose Third Derivatives Are Extended s-Convex Functions and Applications to Means Ling Chun1 1

Feng Qi1,2,3

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China

2

Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China 3

Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

E-mail: [email protected], [email protected], [email protected] URL: http://qifeng618.wordpress.com

Received on April 4, 2014; Accepted on August 26, 2014

Abstract In the paper, the authors establish some new inequalities of Simpson type for functions whose third derivatives are extended s-convex functions, and apply these inequalities to derive some inequalities of special means. 2010 Mathematics Subject Classification: Primary 26D15, 26D20; Secondary 41A55, 41A99. Key words and phrases: Simpson type inequality; extended s-convex functions; H¨ older inequality.

1

Introduction

Throughout this paper, we use the following notation R = (−∞, ∞),

R0 = [0, ∞),

and R+ = (0, ∞).

The following definitions are well known in the literature. Definition 1.1. A function f : I ⊆ R → R is said to be convex if f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) holds for all x, y ∈ I and λ ∈ [0, 1]. In [9], H. Hudzik and L. Maligranda considered, among others, the class of functions which are called s-convex in the second sense. 1

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Definition 1.2 ([9]). Let s be a real number s ∈ (0, 1]. A function f : R0 → R0 is said to be s-convex (in the second sense), or say, f belongs to the class Ks2 , if f (λx + (1 − λ)y) ≤ λs f (x) + (1 − λ)s f (y) for all x, y ∈ I and λ ∈ [0, 1]. In [18], the concept of extended s-convex functions was introduced as follows. Definition 1.3 ([18]). For some s ∈ [−1, 1], a function f : I ⊆ R0 → R is said to be extended s-convex if f (λx + (1 − λ)y) ≤ λs f (x) + (1 − λ)s f (y) is valid for all x, y ∈ I and λ ∈ (0, 1). If f : I ⊆ R → R is a convex function on [a, b] ⊆ I with a < b. Then   Z b 1 f (a) + f (b) a+b ≤ f (x) d x ≤ . f 2 b−a a 2

(1.1)

In the literature, Simpson’s inequality may be recited as Theorem 1.1 below. Theorem 1.1 ([6]). If f : [a, b] → R is a four times function on (a, b)

continuously differentiable and its fourth derivative on (a, b) is bounded by f (4) = supx∈(a,b) f (4) (x) < ∞, then

Z b    (b − a)5 f (4) b − a f (a) + f (b) a + b ≤ f (x) d x − + 2f . (1.2) 3 2 2 2880 a In recent decades, many inequalities of Hermite-Hadamard type and Simpson type for various kind of convex functions have been established. Some of them may be recited as follows. Theorem 1.2 ([5, Theorems 2.2 and 2.3]). Let f : I ◦ ⊆ R → R be a differentiable mapping on I ◦ and a, b ∈ I ◦ with a < b. 1. If |f 0 (x)| is convex on [a, b], then Z b f (a) + f (b) (b − a)(|f 0 (a)| + |f 0 (b)|) 1 ≤ − f (x) d x . 2 b−a a 8

(1.3)

2. If the mapping |f 0 (x)|p/(p−1) is convex on [a, b] for p > 1, then  0 1−1/p Z b f (a) + f (b) 1 b−a |f (a)|p/(p−1) + |f 0 (b)|p/(p−1) − f (x) d x ≤ . (1.4) 2 b−a a 2 2(p + 1)1/p Theorem 1.3 ([12, Theorems 1 and 2]). Let f : I ⊆ R → R be differentiable on I ◦ and a, b ∈ I with a < b. If |f 0 |q is convex on [a, b] for q ≥ 1, then   Z b f (a) + f (b) b − a |f 0 (a)|q + |f 0 (b)|q 1/q 1 − f (x) d x ≤ (1.5) 2 b−a a 4 2 and

    Z b b − a |f 0 (a)|q + |f 0 (b)|q 1/q a+b 1 ≤ f − f (x) d x . 2 b−a a 4 2

(1.6)

2

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Theorem 1.4 ([1, Theorems 2]). Let f 00 : I ⊆ R → R be an absolutely continuous function on I ◦ such that f 000 ∈ L([a, b]) and a, b ∈ I ◦ with a < b. If |f 000 | is quasi-convex on [a, b], then Z

b

    b−a a+b f (a) + 4f + f (b) 6 2          000 a + b 000 (b − a)4 a + b f , |f (b)| . (1.7) ≤ + max max |f 000 (a)|, f 000 1152 2 2

f (x) d x −

a

Theorem 1.5 ([3, Theorem 3.1]). Let f : I ⊆ R0 → R0 be differentiable on I ◦ , a, b ∈ I ◦ with a < b, and f 000 ∈ L([a, b]). If |f 000 |q is s-convex on[a, b] for s ∈ (0, 1] and q ≥ 1, then Z b f (a) + f (b)  b − a 0 1 0 f (x) d x − − f (b) − f (a) 2 b−a a 12  1/q  000 1/q (b − a)3 22−s (s + 6 + 2s+2 s) ≤ |f (a)|q + |f 000 (b)|q . (1.8) 192 (s + 2)(s + 3)(s + 4) Theorem 1.6 ([11, Theorems 3]). Let f : I ⊆ R0 → R0 be a differentiable functions on I ◦ such that f 000 ∈ L([a, b]) and a, b ∈ I ◦ with a < b. If |f 000 |q is s-convex in the second sense on [a, b] for some fixed s ∈ (0, 1], then Z

a

b

f (x) d x −

    (b − a)4  000  a+b b−a f (a) + 4f + f (b) ≤ |f (a)| + |f 000 (b)| 6 2 6   −4−s 2 [(1 + s)(2 + s) + 34 + 2s+4 (−2 + s) + 11s + s2 ] . × (s + 1)(s + 2)(s + 3)(s + 4)

Theorem 1.7 ([18, Theorem 3.1]). Let f : I ⊆ R0 → R be a differentiable function on I ◦ , a, b ∈ I with a < b, f 0 ∈ L([a, b]), and 0 ≤ λ, µ ≤ 1, such that |f 0 |q for q ≥ 1 is extended s-convex on [a, b] for some fixed s ∈ [−1, 1]. 1. If −1 < s ≤ 1, then    1/q Z b λf (a) + µf (b) 2 − λ − µ 1 a+b 1 ≤ b−a + f − f (x) d x 2s/q+2 (s + 1)(s + 2) 2 2 2 b−a a 1−1/q h   1 − λ + λ2 × 2(2 − λ)s+2 + [(s + 2)λ − 2]2s+1 + (s + 2)λ − s − 3 |f 0 (a)|q 2 1−1/q h i1/q  1  s+2  0 q 2 + 2λ − (s + 2)λ + s + 1 |f (b)| + −µ+µ 2µs+2 − (s + 2)µ + s 2 i1/q 
0   0 q s+2 s+1 q + 1 |f (a)| + 2(2 − µ) + ((s + 2)µ − 2)2 + (s + 2)µ − s − 3 |f (b)| . (1.9) 2. If s = −1, we have   Z b a+b 1 b−a f − f (x) d x ≤ 3−2/q 2 b−a a 2  1/q  0 1/q 0 q × (2 ln 2 − 1)|f (a)| + |f 0 (b)|q + |f (a)|q + (2 ln 2 − 1)|f 0 (b)|q . (1.10) 3

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For more information on Simpson type inequalities in recent years, please refer to [2, 4, 7, 8, 10, 13, 14, 15, 16, 17, 19] and closely related references therein. In this paper, we will establish some new integral inequalities of Simpson type for extended s-convex functions.

2

An integral identity

In order to prove our main results, we need the following integral identity. Lemma 2.1. Let f : I ⊆ R → R be a third differentiable mappings on I ◦ , a, b ∈ I with a < b, and f 000 ∈ L([a, b]). Then     Z b 1 a+b 1 f (x) d x f (a) + 4f + f (b) − 6 2 b−a a      Z a+b (b − a)3 1 a+b 2 000 000 t(1 − t) f tb + (1 − t) = −f ta + (1 − t) d t. (2.1) 96 2 2 0 Proof. Integrating by parts and changing variable of definite integral yield      Z 1 a+b a+b − f 000 ta + (1 − t) dt t(1 − t)2 f 000 tb + (1 − t) 2 2 0     1 Z 1   a+b 2 a + b 2 00 2 00 (3t − 4t + 1)f tb + (1 − t) = t(1 − t) f tb + (1 − t) dt − b−a 2 2 0 0     1 Z 1   a+b a + b 2 00 + t(1 − t)2 f 00 ta + (1 − t) − (3t − 4t + 1)f ta + (1 − t) dt 2 2 0 0 Z 1   4 a+b 2 0 =− (3t − 4t + 1) d f tb + (1 − t) (b − a)2 2 0 Z 1   a + b − (3t2 − 4t + 1) d f 0 ta + (1 − t) 2 0      Z 1 4 2 a+b 0 a+b =− −f − (6t − 4) d f tb + (1 − t) (b − a)2 2 b−a 0 2     Z 1 2 a+b a+b + f0 − (6t − 4) d f ta + (1 − t) 2 b−a 0 2     Z b 8 a+b 96 = 2f (a) + 8f + 2f (b) − f (x) d x. 3 (b − a) 2 (b − a)4 a Multiplying by is proved.

3

(b−a)3 96

on both sides of the above equations leads to the identity (2.1). Lemma 2.1

Simpson type inequalities for extended s-convex functions

Now we are in a position to establish some new integral inequalities of Simpson type for functions whose third derivatives are extended s-convex functions. 4

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Theorem 3.1. Let f : I ⊆ R0 → R be a third differentiable mappings on I, a, b ∈ I ◦ with a < b, and f 000 ∈ L([a, b]) such that |f 000 |q is extended s-convex on [a, b] for q ≥ 1 and some fixed s ∈ [−1, 1]. Then  1−1/q      1/q Z b 1 1 1 f (a) + 4f a + b + f (b) − 1 f (x) d x ≤ 6 2 b−a a 12 (s + 2)(s + 3)(s + 4)   1/q       q 1/q 000 a + b q (b − a)3 a + b f + 2|f 000 (b)|q + (s + 2) . × 2|f 000 (a)|q + (s + 2) f 000 96 2 2 Proof. By virtue of Lemma 2.1, H¨ older’s integral inequality, and the extended s-convexity of |f 000 |q , we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a       Z (b − a)3 1 a + b 000 a+b dt ≤ f (1 − t) t(1 − t)2 f 000 ta + (1 − t) + + tb 96 2 2 0  Z  Z   q  1/q 1−1/q 1 1 (b − a)3 2 2 s 000 q s 000 a + b ≤ t(1 − t) d t t(1 − t) t |f (a)| + (1 − t) f dt 96 2 0 0  Z 1   1/q   q a + b s 000 q + t(1 − t)2 (1 − t)s f 000 +t |f (b)| dt 2 0   1−1/q  1/q   q 1/q (b − a)3 1 a + b 1 = 2|f 000 (a)|q + (s + 2) f 000 96 12 (s + 2)(s + 3)(s + 4) 2   1/q   q a + b 000 q + (s + 2) f 000 . + 2|f (b)| 2 Theorem 3.1 is proved. Corollary 3.1.1. Under conditions of Theorem 3.1, 1. if q = 1, we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a     000 a + b (b − a)3 000 000 ≤ |f (a)| + (s + 2) f + |f (b)| ; 48(s + 2)(s + 3)(s + 4) 2 2. if q = 1 and s = 1, then     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a     (b − a)3 a + b 000 ≤ |f 000 (a)| + 3 f 000 + |f (b)| ; 2880 2

5

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3. if q = 1 and s = 0, we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a     000 a + b (b − a)3 000 000 ≤ |f (a)| + 2 f + |f (b)| ; 1152 2 4. if q = 1 and s = −1, we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a     000 a + b (b − a)3 000 000 ≤ |f (a)| + f + |f (b)| . 288 2 Corollary 3.1.2. Under conditions of Theorem 3.1, 1. if s = 1, then       Z b 1 (b − a)3 1 1/q f (a) + 4f a + b + f (b) − 1 f (x) d x ≤ 6 2 b−a a 1152 5    q 1/q    1/q  000 a + b q a + b f + 2|f 000 (b)|q × 2|f 000 (a)|q + 3 f 000 + 3 ; 2 2 2. if s = 0, we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a    1/q      3 000 a + b q 000 a + b q 1/q (b − a) 000 q 000 q ≤ + f ; |f (a)| + f + |f (b)| 1152 2 2 3. if s = −1, we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a    q 1/q    1/q  000 a + b q (b − a)3 1/q a + b f + 2|f 000 (b)|q ≤ 2 2|f 000 (a)|q + f 000 + . 1152 2 2 Theorem 3.2. Let f : I ⊆ R0 → R be a third differentiable mappings on I, a, b ∈ I ◦ with a < b, and f 000 ∈ L([a, b]) such that |f 000 |q is extended s-convex on [a, b] for q ≥ 1 and some fixed s ∈ [−1, 1]. Then       Z b 1 (b − a)3 1 1−1/q f (a) + 4f a + b + f (b) − 1 f (x) d x ≤ 6 2 b−a a 96 2 6

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 ×

   000 a + b q 1/q 1 B(s + 2, 2q + 1)|f (a)| + f (2q + s + 1)(2q + s + 2) 2   1/q   000 a + b q 1 f + B(s + 2, 2q + 1)|f 000 (b)|q + , (2q + s + 1)(2q + s + 2) 2 000

q

where B(α, β) is the classical beta function defined by Z B(α, β) =

1

tα−1 (1 − t)β−1 d t,

α, β > 0.

0

Proof. Since |f 000 |q is extended s-convex on [a, b], by Lemma 2.1 and H¨older’s inequality, we obtain     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a       Z (b − a)3 1 a + b 000 a+b 2 000 ≤ ta + (1 − t) (1 − t) t(1 − t) f + f + tb d t 96 2 2 0  Z 1 1−1/q Z 1   q  1/q 3 (b − a) 2q s 000 q s 000 a + b ≤ tdt t(1 − t) t |f (a)| + (1 − t) f dt 96 2 0 0  Z 1   1/q   q a + b s 000 q + t(1 − t)2q (1 − t)s f 000 +t |f (b)| dt 2 0   1−1/q    000 a + b q 1/q (b − a)3 1 1 000 q = f B(s + 2, 2q + 1)|f (a)| + 96 2 (2q + s + 1)(2q + s + 2) 2   1/q   q 000 a + b 1 f + B(s + 2, 2q + 1)|f 000 (b)|q + . (2q + s + 1)(2q + s + 2) 2 Theorem 3.2 is thus proved. Corollary 3.2.1. Under conditions of Theorem 3.2, 1. if s = 1, then      1/q Z b 1 (b − a)3 1 f (a) + 4f a + b + f (b) − 1 f (x) d x ≤ 6 2 b−a a 192 (2q + 1)(q + 1)(2q + 3)     q 1/q   1/q  000 a + b q a + b f + 2|f 000 (b)|q × 2|f 000 (a)|q + (2q + 1) f 000 + (2q + 1) ; 2 2 2. if s = 0, we have      1/q Z b 1 (b − a)3 1 f (a) + 4f a + b + f (b) − 1 ≤ f (x) d x 6 2 b−a a 192 (2q + 1)(q + 1)    q 1/q    1/q  000 a + b q a + b + |f 000 (b)|q f × |f 000 (a)|q + f 000 + ; 2 2

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3. if s = −1, we have      1/q Z b (b − a)3 1 1 ≤ f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a 192 q(2q + 1)  1/q    q 1/q    000 a + b q 000 a + b 000 q 000 q + f . × 2q|f (a)| + f + 2q|f (b)| 2 2 Theorem 3.3. Let q ≥ r, `, m, n > 0, f : I ⊆ R0 → R be a third differentiable mappings on I, a, b ∈ I ◦ with a < b, and f 000 ∈ L([a, b]) such that |f 000 |q is extended s-convex on [a, b] for q > 1 and some fixed s ∈ [−1, 1]. Then       1−1/q Z b 1 (b − a)3 2q − r − 1 3q − ` − 1 f (a)+4f a + b +f (b) − 1 f (x) d x ≤ B , 6 2 b−a a 96 q−1 q−1   q 1/q    a+b 2q − m − 1 + B , × B(r + s + 1, ` + 1)|f 000 (a)|q + B(r + 1, ` + s + 1) f 000 2 q−1  1/q  1−1/q   q 3q − n − 1 a + b 000 q + B(m + s + 1, n + 1)|f (b)| . B(m + 1, n + s + 1) f 000 q−1 2 Proof. Since |f 000 |q is extended s-convex on [a, b], by Lemma 2.1 and H¨older’s inequality, we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a       3 Z 1 a+b (b − a) a + b 000 2 000 (1 − t) + f ≤ ta + (1 − t) + tb d t t(1 − t) f 96 2 2 0 Z 1 1−1/q Z 1  3 (b − a) ≤ t(q−r)/(q−1) (1 − t)(2q−`)/(q−1) d t tr (1 − t)` ts |f 000 (a)|q 96 0 0   q  1/q Z 1 1−1/q a + b (q−m)/(q−1) (2q−n)/(q−1) dt + t (1 − t) d t + (1 − t)s f 000 2 0   1/q   q Z 1  s 000 q m n s 000 a + b × t (1 − t) (1 − t) f +t |f (b)| d t 2 0   1−1/q  (b − a)3 2q − r − 1 3q − ` − 1 = B , B(r + s + 1, ` + 1)|f 000 (a)|q 96 q−1 q−1   q 1/q   1−1/q a + b 2q − m − 1 3q − n − 1 + B(r + 1, ` + s + 1) f 000 + B , 2 q−1 q−1    q 1/q  000 a + b 000 q × B(m + 1, n + s + 1) f . + B(m + s + 1, n + 1)|f (b)| 2 The proof of Theorem 3.3 is thus completed. Corollary 3.3.1. Under conditions of Theorem 3.3, if r = m and ` = n, we have

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  1−1/q Z b f (a) + 4f a+b (b − a)3 + f (b) 1 2q − r − 1 3q − ` − 1 2 f (x) d x ≤ − B , 6 b−a a 96 q−1 q−1    q 1/q a + b × B(r + s + 1, ` + 1)|f 000 (a)|q + B(r + 1, ` + s + 1) f 000 2   1/q   q 000 a + b 000 q + B(r + 1, ` + s + 1) f . + B(r + s + 1, ` + 1)|f (b)| 2 Corollary 3.3.2. Under conditions of Theorem 3.3, 1. if r = ` = m = n, we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a   1−1/q (b − a)3 2q − r − 1 3q − r − 1 ≤ [B(r + 1, r + s + 1)]1/q B , 96 q−1 q−1   1/q   q 1/q    000 a + b q a + b f + |f 000 (b)|q × |f 000 (a)|q + f 000 + ; 2 2 2. if r = ` = m = n and s = 1, then     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a   1−1/q (b − a)3 2q − r − 1 3q − r − 1 1/q ≤ [B(r + 2, r + 1)] B , 96 q−1 q−1  1/q    q 1/q    q 000 a + b 000 a + b 000 q 000 q + f ; × |f (a)| + f + |f (b)| 2 2 3. if r = ` = m = n and s = 0, we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a   1−1/q (b − a)3 2q − r − 1 3q − r − 1 ≤ , B [B(r + 1, r + 1)]1/q 96 q−1 q−1        1/q  000 a + b q 000 a + b q 1/q 000 q 000 q + f ; × |f (a)| + f + |f (b)| 2 2 4. if r = ` = m = n and s = −1, we have     Z b 1 f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a   1−1/q 3 (b − a) 2q − r − 1 3q − r − 1 1/q ≤ [B(r, r + 1)] B , 96 q−1 q−1 9

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   1/q      000 a + b q 000 a + b q 1/q 000 q 000 q + f . × |f (a)| + f + |f (b)| 2 2 Corollary 3.3.3. Under conditions of Theorem 3.3, 1. if r = ` = m = n = 1, then      1−1/q Z b 1 (b − a)3 (q − 1)2 f (a) + 4f a + b + f (b) − 1 f (x) d x ≤ 6 2 b−a a 96 (3q − 2)(4q − 3)  1/q  1/q   q 1/q     000 a + b q a + b 1 f + |f 000 (b)|q + ; |f 000 (a)|q + f 000 × (s + 2)(s + 3) 2 2 2. if r = ` = m = n = 1 and s = 1, then      1−1/q Z b 1 (b − a)3 (q − 1)2 f (a) + 4f a + b + f (b) − 1 ≤ f (x) d x 6 2 b−a a 96 (3q − 2)(4q − 3)  1/q   q 1/q     1/q  000 a + b q a + b 1 f + |f 000 (b)|q + ; |f 000 (a)|q + f 000 × 12 2 2 3. if r = ` = m = n = 1 and s = 0, we have      1−1/q Z b 1 (b − a)3 (q − 1)2 f (a) + 4f a + b + f (b) − 1 ≤ f (x) d x 6 2 b−a a 96 (3q − 2)(4q − 3)   1/q   q 1/q    q 1/q  000 a + b a + b 1 f + |f 000 (b)|q + × |f 000 (a)|q + f 000 ; 6 2 2 4. if r = ` = m = n = 1 and s = −1, we have      1−1/q Z b 1 (b − a)3 (q − 1)2 f (a) + 4f a + b + f (b) − 1 ≤ f (x) d x 6 2 b−a a 96 (3q − 2)(4q − 3)  1/q   q 1/q     1/q  000 a + b q 000 a + b 1 000 q 000 q + f . |f (a)| + f × + |f (b)| 2 2 2 Corollary 3.3.4. Under conditions of Theorem 3.3, 1. if r = ` = m = n = q, we have       Z b 1 (b − a)3 q − 1 1−1/q f (a) + 4f a + b + f (b) − 1 f (x) d x ≤ 6 2 b−a a 96 2q − 1   q 1/q     1/q  000 a + b 000 a + b q 1/q 000 q 000 q × [B(q + 1, q + s + 1)] |f (a)| + f + f ; + |f (b)| 2 2

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2. if r = ` = m = n = q and s = 1, then       Z b (b − a)3 q − 1 1−1/q 1 ≤ f (a) + 4f a + b + f (b) − 1 f (x) d x 6 2 b−a a 96 2q − 1  1/q    q 1/q    000 a + b q  1/q a + b f + |f 000 (b)|q + ; |f 000 (a)|q + f 000 × B(q + 1, q + 2) 2 2 3. if r = ` = m = n = q and s = 0, we have       Z b 1 (b − a)3 q − 1 1−1/q f (a) + 4f a + b + f (b) − 1 f (x) d x ≤ 6 2 b−a a 96 2q − 1  1/q   q 1/q     000 a + b q a + b f + |f 000 (b)|q + ; × [B(q + 1, q + 1)]1/q |f 000 (a)|q + f 000 2 2 4. if r = ` = m = n = q and s = −1, we have       Z b 1 (b − a)3 q − 1 1−1/q f (a) + 4f a + b + f (b) − 1 f (x) d x ≤ 6 2 b−a a 96 2q − 1  1/q    q 1/q    000 a + b q 000 a + b 000 q 1/q 000 q + f . × [B(q, q + 1)] |f (a)| + f + |f (b)| 2 2 Similar to the proof of Theorem 3.3, we may also establish the following theorem. Theorem 3.4. Let q ≥ r, `, m, n ≥ 0, f : I ⊆ R0 → R be a third differentiable mappings on I, a, b ∈ I ◦ with a < b, and f 000 ∈ L([a, b]) such that |f 000 |q is extended s-convex on [a, b] for q > 1 and some fixed s ∈ (−1, 1]. Then       1−1/q Z b 1 (b − a)3 2q − r − 1 3q − ` − 1 f (a)+4f a + b +f (b) − 1 f (x) d x ≤ B , 6 2 b−a a 96 q−1 q−1    q 1/q   a+b 2q − m − 1 + B × B(r + s + 1, ` + 1)|f 000 (a)|q + B(r + 1, ` + s + 1) f 000 , 2 q−1  1/q  1−1/q   q a + b 3q − n − 1 000 q + B(m + s + 1, n + 1)|f (b)| . B(m + 1, n + s + 1) f 000 q−1 2 Corollary 3.4.1. Under conditions of Theorem 3.4, 1. if r = m = 0, then     Z b 1 (b − a)3 f (a) + 4f a + b + f (b) − 1 ≤ f (x) d x 6 2 b−a a 96  1−1/q      000 a + b q 1/q 1 2q − 1 3q − ` − 1 000 q , B(s + 1, ` + 1)|f (a)| + × B f q−1 q−1 ` + s + 1 2  1−1/q   q   1/q  000 a + b 2q − 1 3q − n − 1 1 f + B(s + 1, n + 1)|f 000 (b)|q + B , ; q−1 q−1 n + s + 1 2 11

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2. if ` = n = 0, we have     Z b 1 (b − a)3 f (a) + 4f a + b + f (b) − 1 ≤ f (x) d x 6 2 b−a a 96  1−1/q   q 1/q   a + b 1 2q − r − 1 3q − 1 , |f 000 (a)|q + B(r + 1, s + 1) f 000 × B q−1 q−1 r+s+1 2  1−1/q   q 1/q    2q − m − 1 3q − 1 1 a + b 000 q + B + , B(m+1, s+1) f 000 |f (b)| ; m+s+1 q−1 q−1 2 3. if r = ` = m = n = 0, we have       1−1/q Z b (b − a)3 1 2q − 1 3q − 1 ≤ f (a) + 4f a + b + f (b) − 1 f (x) d x B , 6 2 b−a a 96 q−1 q−1   1/q   q 1/q    q 1/q  000 a + b 000 a + b 1 000 q 000 q × |f (a)| + f + f . + |f (b)| s+1 2 2

4

Applicaions to means

We now consider for positive numbers b > a > 0 the arithmetic mean A(a, b) = generalized logarithmic mean br+1 − ar+1 Lr (a, b) = (r + 1)(b − a) 

a+b 2

and the

1/r ,

r 6= 0, −1.

(4.1)

s+2

x Let s > 0, q ≥ 1 and define f (x) = (s+1)(s+2) for x ∈ R+ . If 1 ≤ s ≤ 2 and (s − 1)q ≤ 1, we have 000   f (tx + (1 − t)y) q ≤ sq t(s−1)q x(s−1)q + (1 − t)(s−1)q y (s−1)q ≤ ts−1 |f 000 (x) q +(1 − t)s−1 f 000 (y)|q .

If 0 < s ≤ 1 and −1 < (s − 1)q, we have 000 f (tx + (1 − t)y) q ≤ ts−1 |f 000 (x)|q + (1 − t)s−1 f 000 (y) q for x, y ∈ R+ and t ∈ (0, 1). If 0 < s ≤ 2 and −1 < (s − 1)q ≤ 1, then the function |f 000 (x)|q = |s|q x(s−1)q is extended (s − 1)-convex on R+ . Applying Theorem 3.1 to the function |f 000 (x)|q = |s|q x(s−1)q yields the following conclusions. Theorem 4.1. Let b > a > 0, q ≥ 1, 0 < s ≤ 2, and −1 < (s − 1)q ≤ 1. Then  1/q 3 1
 1 A as+2 , bs+2 +2As+2 (a, b) −Ls+2 (a, b) ≤ (b − a) s(s + 1)(s + 2) s+2 3 96 (s + 2)(s + 3)(s + 4)  1−1/q  (s−1)q 1/q  1/q 1 × 2a + (s + 2)A(s−1)q (a, b) + (s + 2)A(s−1)q (a, b) + 2b(s−1)q . (4.2) 12

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Specially, if q = 1, then 3 1
 A as+2 , bs+2 +2As+2 (a, b) −Ls+2 (a, b) ≤ (b − a) s(s + 1) s+2 3 48(s + 3)(s + 4) 
 × 2A as−1 , bs−1 +(s + 2)As−1 (a, b) . (4.3) Using Corollary 3.3.3, we have Theorem 4.2. Let b > a > 0, q > 1, 0 < s ≤ 2, and −1 < (s − 1)q ≤ 1. Then
 1−1/q A as+2 , bs+2 +2As+2 (a, b) (b − a)3 s(s + 1)(s + 2) (q − 1)2 s+2 − Ls+2 (a, b) ≤ 3 96 (3q − 2)(4q − 3)  1/q    1/q 1 1/q × a(s−1)q + A(s−1)q (a, b) + A(s−1)q (a, b) + b(s−1)q . (4.4) (s + 2)(s + 3) Letting q = 2 and r = ` = m = n = 0 in Corollary 3.4.1 leads to the following theorem. Theorem 4.3. Let b > a > 0 and

1 2

≤ s ≤ 32 . Then

 1/2 3 1
 A as+2 , bs+2 +2As+2 (a, b) −Ls+2 (a, b) ≤ (b − a) s(s + 2) s + 1 s+2 3 96 105  2(s−1) 1/2  2(s−1) 1/2 2(s−1) × a +A (a, b) + A (a, b) + b2(s−1) . (4.5)

Acknowledgements This work was partially supported by the NNSF under Grant No. 11361038 of China and the Foundation of Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159, China.

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REPRESENTATION OF HIGHER-ORDER EULER NUMBERS USING THE SOLUTION OF BERNOULLI EQUATION CHEON SEOUNG RYOO, HYUCK IN KWON, JIHEE YOON, AND YU SEON JANG*

Abstract. In this paper, we derive the several representations of the solution of the special Bernoulli differential equation. And then we have the relations between Euler numbers and higher-order Euler numbers.

1. Introduction A differential equation is a relationship between a function of time and its derivatives. Differential equations appear naturally in many areas of science and the humanities. In mathematics, an ordinary differential equation of the form dy (1) + P (t)y = Q(t)y α (α ̸= 0, 1) dt is called a Bernoulli equation which is named after Jacob Bernoulli, who discussed it in 1695. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. When α = 2, the Bernoulli equation has the solution which is the functions of the exponential generating function of the Euler numbers. It is well known that the Euler polynomials, En (x), n = 0, 1, · · · are defined by means of the following generating function; ∞ ∑ 2 tn xt (2) e = En (x) . t e +1 n! n=0 [1, 3, 7, 17, 23, 24] We note that, by substituting x = 0 into (2), En (0) = En , n = 0, 1, · · · are the familiar Euler numbers which is defined by ∞ ∑ 2 tn (3) = E (x) . n et + 1 n=0 n! From the definition of En (x) we know that n ( ) ∑ n (4) Ek (x) + En (x) = 2xn , n = 0, 1, · · · , k k=0

Key words and phrases. Bernoulli equation, Euler numbers, higher-order Euler numbers. *Corresponding Author.

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C. S. RYOO, H. I. KWON, JIHEE YOON, AND Y. S. JANG*

where Ek = Ek (0) is the kth Euler number. These can be rewritten as follows: (E(x) + 1)n + En (x) = 2xn , n = 0, 1, · · ·

(5)

by using the symbolic convention about replacing En (x) by (E(x))n . When x = 0, these relations are given by (E + 1)n + En = 2δ0,δ , n = 0, 1, · · · ,

(6)

∑n ( ) where δ0,n is Kronecker symbol and (E +1)n is interpreted as k=0 nk Ek . [2-6, 9-15] This notion can be expanded to the multiple case. The Euler polynomials of order r ∈ N are defined by the generating function to be ( )r ∞ ∑ 2 tn xt (r) (7) e = E (x) . n et + 1 n! n=0 (r)

(r)

When x = 0, En = En (0), n = 0, 1, · · · are also called the Euler numbers of order r. [8, 10, 18, 19-22] Recently, several authors have studied the identities of Euler numbers and polynomials.[312, 18,28] In this paper, we derive the several representations of the solution of the special Bernoulli differential equation. And then we have the relations between Euler numbers and higher-order Euler numbers. We indebted this idea to the paper [19].

2. Representation of Euler numbers As is well known that when α = 2 a special Bernoulli equation dy (8) + y = y2 dt has the solution 1 (9) y(t) = t . e +1 Differentiating the both side of (8) is d2 y dy dy + = 2y . dt2 dt dt

(10) From (8) and (10), we see that

( ) 1 d2 y dy + 3 + 2y . 2! dt2 dt Continuing this process, we have the following lemma.

(11)

y3 =

Lemma 2.1. The solution y(t) of the Bernoulli equation for α = 2 in (8) has the representation as the following; for some real coefficients am , m = 0, 1, · · · r 1 ∑ dm am,r m y(t), r = 0, 1, · · · . (12) y r+1 (t) = r! m=0 dt

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Proof. Suppose that it holds for any k ∈ N. Then for some real constants am,k , m = 0, n1, · · · y k+1 =

(13)

k 1 ∑ dm y am,k m . k! m=0 dt

Differentiating the both side of (12) is (14)

(k + 1)y k

k dy 1 ∑ dm+1 y = am,k m+1 . dt k! m=0 dt

Since dy/dt = y 2 − y, this implies that (15)

y k+2 =

Take (16)

am,k+1

k k ∑ ∑ dm y 1 dm+1 y . am,k m+1 + (k + 1)! m=0 dt dtm m=0

  m=0 a0,k , = am−1,k + (k + 1)am,k , 1 ≤ m ≤ k   ak,k , m = k + 1.

Therefore, we obtain that (17)

y k+2 =

k+1 ∑ 1 dm y am,k+1 m . (k + 1)! m=0 dt



By the mathematical induction, we have the desired result. In the proof of Lemma 2.1 we know that for any given r ∈ N (18)

am+1,r+1 = am,r + ram+1,r , 0 ≤ m ≤ r − 1.

In order to determine the coefficients am,r , m = 0, 1, · · · , we consider the function of t and µ as follows; for |t| < 1 ( r−1 ) ∞ ∑ ∑ tr (19) f (t, µ) = am,r µm . r! r=1 m=0 From (17), we know that ( r−1 ) ) ( r−1 ∞ ∞ r ∑ ∑ ∑ ∑ tr t (20) am+1,r+1 µm = f (t, µ) + ram+1,r µm . r! r! r=1 m=0 r=1 m=0 Then we have the following theorem. Lemma 2.2. Let f (t, µ) be defined in (18). Then we have 1 1 f (t, µ) = (1 − t)µ − . µ µ

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Proof. Observe that ( r−1 ) ∞ ∞ r ∑ ∑ 1∑∑ tr tr µm = ram,r µm ram+1,r r! µ r=1 m=1 r! r=1 m=0 } { r ∞ 1∑ ∑ tr tr m = µ − a0,r am,r µ r=1 m=0 (r − 1)! (r − 1)! (21) } { ∞ r−1 1 t ∑ ∑ tr−1 m µ − = am,r µ r=1 m=0 (r − 1)! 1−t ( ) t 1 = f ′ (t, µ) − . µ 1−t From (19) we have ( r−1 ) ( ) ∞ ∑ ∑ tr t 1 m ′ (22) am+1,r+1 µ = f (t, µ) + f (t, µ) − . r! µ 1−t r=1 m=0 In (22), the left hand side is ) { r−1 } ( r−1 ∞ ∞ ∑ ∑ 1∑ ∑ tr−1 m tr−1 tr m µ = am,r µ − a0,r am+1,r+1 r! µ r=2 m=1 (r − 1)! (r − 1)! r=1 m=0 (23) { ∞ r−1 } 1 ∑∑ t tr−1 m = µ − . am,r µ r=2 m=0 (r − 1)! 1−t This implies that (24)

∞ ∑ r=1

( r−1 ∑

tr am+1,r+1 r! m=0

) µm =

1 µ

{

∂ 1 f (t, µ) − ∂t 1−t

} .

By (22) and (24), we obtain a first order linear differential equation about t d µ 1 (25) f (t, µ) − f (t, µ) = . dt 1−t 1−t Multiplying the integration factor e obtain the solution of (25)

´

µ/(1−t)dt

= 1/(1 − t)µ in the both side of (25) we

1 1 (1 − t)µ − µ µ because f (0, µ) = 0. This is completion of the proof. (26)

f (t, µ) =



As is well known that the Taylor expansion of ln x at x = 1 is ∞ ∑ (−1)n+1 (27) ln x = (x − 1)n , 0 < x < 2. n n=1 From (27), Lemma 2.1 and Lemma 2.2, the coefficients, am,r , m = 0, 1, · · · , in Lemma 2.1 are determined in the following theorem.

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Theorem 2.3. The solution y(t) of the Bernoulli equation for α = 2 in (8) is satisfied the following equations; for r = 1, 2, · · · (28)

y r (t) = (−1)r r

r−1 ∑

1 (m + 1)! m=0

∑ l1 +···+lm+1 =r l1 ,··· ,lm+1 ≥1

dm 1 y(t). l1 · · · lm+1 dtm

Proof. From the definition of f (t, µ) and Lemma 2.2, we know that ( r−1 ) ∞ ∞ ∑ ∑ ∑ (ln(1 − t))n tr = µn−1 . (29) am,r µm r! n=1 n! r=1 m=0 Since ln(1 − t) =

(30)

∞ ∑ (−1)2k+1

k

k=1

tk , −1 < t < 1,

this implies that for −1 < t < 1 (31)

 ∞  r ∑ t µn−1 am,r µm =  r! n=1 r=1 m=0

∞ ∑ r−1 ∑

∑ k1 ,··· ,kn ≥1

 (−1)k1 +···+kn k1 +k2 +···+kn  1 t .  n! k1 · · · kn

In the right side of (30), taking s = k1 + · · · + kn and changing the order of sums, thus we have ∞ ∑ r−1 ∞ ∑ ∞ r ∑ ∑ ∑ s! (−1)s n−1 ts mt = µ am,r µ r! n=1 s=n n! l1 · · · ln s! r=1 m=0 l1 ,··· ,lm+1 ≥1 l1 +···+lm+1 =s

=

(32)

∞ ∑ s ∑

∑

s=1 n=1 l1 ,··· ,lm+1 ≥1 l1 +···+lm+1 =s

=

∞ ∑ s−1 ∑

∑

s=1 n=0 l1 ,··· ,lm+1 ≥1 l1 +···+lm+1 =s

s! (−1)s n−1 ts µ n! l1 · · · ln s! s! (−1)s ts µn . (n + 1)! l1 · · · ln+1 s!

Consequently, we obtain (33)

∞ ∑ r−1 ∑ r=1 m=0

am,r µm

∞ ∑ r−1 ∑ tr = r! r=1 m=0

∑ l1 ,··· ,lm+1 ≥1 l1 +···+lm+1 =r

r! (−1)r tr µm . (m + 1)! l1 · · · lm+1 r!

comparing the both sides in (32), the coefficients am,r , m = 0, 1, · · · , r − 1(r ∈ N) are determined as following ∑ 1 r! . (34) am,r = (−1)r (m + 1)! l1 · · · lm+1 l1 ,··· ,lm+1 ≥1 l1 +···+lm+1 =r

From Lemma 2.1, we have the desired result in the theorem.

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By definition of the Euler numbers, En , n = 0, 1, · · · , we know that ∞ 1∑ tl dm y(t) = E . (35) l+m dtm 2 l! l=0

Therefore, from Theorem 2.3 we have the following corollary. Corollary 2.4. For the Euler numbers, En , n = 0, 1, · · · , the solution y(t) of the Bernoulli equation for α = 2 in (8) is represented as following; for r = 1, 2, · · ·       r−1 ∞  tn  ∑ ∑ ∑ 1 1 r rr En+m , y (t) = (−1)   2 m=0 (m + 1)! l1 · · · lm+1 n!  n=0  l1 ,··· ,lm+1 ≥1   l1 +···+lm+1 =r

where Ek is k-th Euler number. (r)

In the case of higher-order Euler numbers, En , n = 0, 1, · · · (r ∈ N), we know that y r (t) = 2r

(36)

∞ ∑

En(r)

n=0

tn . n!

From Corollary 2.4 we have the following corollary. (r)

Corollary 2.5. The Euler numbers with order r, En , n ∈ N, are represented as following formula; for r = 1, 2, · · · En(r) = (−1)r r2r−1

r−1 ∑

1 (m + 1)! m=0

∑ l1 ,··· ,lm+1 ≥1 l1 +···+lm+1 =r

1 En+m , l1 · · · lm+1

where Ek is k-th Euler number. References [1] M. Abramowitz and I. A. Stegun, Bernoulli and Euler polynomials and the Euler-Maclaurin formula, New York: Dover, 1972. [2] A. A. Aljinovic and J. Pecaric, Generalizations of weighted Euler identity and Ostrowski type inequalities, Adv. Stud. Contemp. Math. 14 (2007), no. 1, 141–151. [3] 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, 247–253. [4] C. P. Chen and L. Lin, An inequality for the generalized-Euler-constant function, Adv. Stud. Contemp. Math. 17 (2008), no. 1, 105–107. [5] D. Ding and J. Yang, Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math. 20 (2010), no. 1, 7–21. [6] G. B. Folland, Fourier Analysis and Its Applications, Pacific Grove, 1992. [7] K. W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, and S. H. Lee, Some theorems on Bernoulli and Euler numbers, Ars Combin. 109 (2013), 285–297. [8] J. H. Jeong, J. H. Jin, J. W. Park, and S. H. Rim, On the twisted weak q-Euler numbers and polynomials with weight 0, Proc. Jangjeon Math. Soc. 16 (2013), no. 2, 157–163.

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[9] D. S. Kim, N. Lee, J. Na, and K. H. Park, Identities of symmetry for higher-order Euler polynomials in three variables (I), Adv. Stud. Contemp. Math. 22 (2012), no. 1, 51–74. [10] D. S. Kim, T. Kim, Some identities of higher order Euler polynomials arising from Euler basis, Integral Transforms Spec. Funct. 24 (2013), no. 9, 734–738. [11] D. S. Kim, T. Kim, S. H. Lee, and S. H. Rim, Umbral calculus and Euler polynomials. Ars Combin. 112 (2013), 293–306. [12] D. S. Kim, T. Kim, Y. H. Kim, and S. H. Lee, Some arithmetic properties of Bernoulli and Euler numbers, Adv. Stud. Contemp. Math. 22 (2012), no. 4, 467–480. [13] H. M. Kim, G. S. Kim, T. Kim, S. H. Lee, D. V. Dolgy, and B. Lee, Identities for the Bernoulli and Euler numbers arising from the p-adic integral on Zp, Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 155–161. [14] T. Kim, q -Volkenborn integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288–299. [15] T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), no. 3, 261– 267. [16] T. Kim, Note on the Euler numbers and polynomials, Adv. Stud. Contemp. Math. 17 (2008), no. 2, 131–136. [17] T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal. 2008, Art. ID 581582, 11 pp. [18] T. Kim, New approach to q-Euler polynomials of higher order. Russ. J. Math. Phys. 17 (2010), no. 2, 218–225. [19] T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 132 (2012), no. 12, 2854–2865. [20] T. Kim, Corrigendum to Identities involving Frobenius-Euler polynomials arising from nonlinear differential equations, J. Number Theory 133 (2013), no. 2, 822–824. [21] T. Kim, A study on the q-Euler numbers and the fermionic q-integral of the product of several type q-Bernstein polynomials on Zp, Adv. Stud. Contemp. Math. 23 (2013), no. 1, 5–11. [22] T. Kim, J. Choi, and Y. H. Kim, A note on the values of Euler zeta functions at positive integers, Adv. Stud. Contemp. Math. 22 (2012), no. 1, 27–34. [23] T. Kim, D. S. Kim, D. V. Dolgy, and S. H. Rim, Some identities on the Euler numbers arising from Euler basis polynomials, Ars Combin. 109 (2013), 433–446. [24] T. Kim, D. S. Kim, A. Bayad, and S. H. Rim, Identities on the Bernoulli and the Euler numbers and polynomials, Ars Combin. 107 (2012), 455–463. [25] T. Kim, B. Lee, S. H. Lee, and S. H. Rim, Identities for the Bernoulli and Euler numbers and polynomials, Ars Combin. 107 (2012), 325–337. [26] Q. M. Luo, q-analogues of some results for the Apostol-Euler polynomials, Adv. Stud. Contemp. Math. 20 (2010), no. 1, 103–113. [27] E. Sen, Theorems on Apostol-Euler polynomials of higher order arising from Euler basis, Adv. Stud. Contemp. Math. 23 (2013), no. 2, 337–345. [28] Y. Simsek, O. Yurekli, and V. Kurt, On interpolation functions of the twisted generalized Frobenius-Euler numbers, Adv. Stud. Contemp. Math. 15 (2007), no. 2, 187–194.

C. S. Ryoo Department of Mathematics Hannam University Daejeon 306-791, Republic of Korea e-mail: [email protected] H. I. Kwon Department of Mathematics Kwangwoon University

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C. S. RYOO, H. I. KWON, JIHEE YOON, AND Y. S. JANG*

Seoul 139-701, Republic of Korea e-mail: [email protected] J. Yoon Department of Mathematics Kwangwoon University Seoul 139-701, Republic of Korea e-mail: affi[email protected] Y. S. Jang* Department of Applied Mathematics Kangnam University Yongin 446-702, Republic of Korea e-mail: [email protected]

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Intuitionistic fuzzy stability of an Euler-Lagrange type quartic functional equation Ali Ebadian1 , Choonkil Park2 and Dong Yun Shin3∗ 1

2

Department of Mathematics, Urmia University, P. O. Box 165, Urmia, Iran Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea 3

Department of Mathematics, University of Seoul, Seoul 130-743, Korea

Abstract. In this paper, we investigate the Hyers-Ulam stability of the following Euler-Lagrange type quartic functional equation f (ax + y) + f (x + ay) +

1 1 a(a − 1)2 f (x − y) = (a2 − 1)2 (f (x) + f (y)) + a(a + 1)2 f (x + y) 2 2

in intuitionistic fuzzy normed spaces, where a 6= 0, a 6= ±1. Furthermore, we investigate intuitionistic fuzzy continuity through the existence of a certain solution of a fuzzy stability problem for approximately EulerLagrange quartic functional equation.

1. Introduction Fuzzy set theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. The fuzzy topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. In 1984, Katsaras [17] introduced an idea of a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. In the same year Wu and Fang [41] introduced a notion fuzzy normed apace to give a generalization of the Kolmogoroff normalized theorem for fuzzy topological vector spaces. In 1992, Felbin [8] introduced an alternative definition of a fuzzy norm on a vector space with an associated metric of Kaleva and Seikkala type [15]. Some mathematics have define fuzzy normed on a vector form various point of view [19, 29, 42]. In particular, Bang and Samanta [4] following Cheng and Mordeson [6] gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric of Kramosil and Michalek type [18]. They established a decomposition theorem of fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [5]. The notion of intuitionistic fuzzy set introduced by Atanassov [3] has been used extensively in many areas of mathematics and sciences. The notion of intuitionistic fuzzy norm [24, 28, 35] is also useful one to deal with the inexactness and vagueness arising in modeling. There are many situations where the norm of a vector is not possible to find and the concept of intuitionistic fuzzy norm seems to be more suitable in such cases, that is, we can deal with such situations by modeling the inexactness through the intuitionistic fuzzy norm. The stability problem for Pexiderized quadratic functional equation, Jensen functional equation, cubic functional equation, functional equations associated with inner product spaces, and additive functional equation was considered in [21, 22, 23, 25, 26, 40], respectively, in the intuitionistic fuzzy normed spaces. 0

Mathematics Subject Classification 2010: Primary 39B82; 39B52; 46S40; 47S40; 54A40. Keywords: Hyers-Ulam stability; Euler-Lagrange type quartic functional equation; intuitionistic fuzzy normed space; intuitionistic fuzzy continuity. ∗ Corresponding author: Dong Yun Shin (email: [email protected]). 0 E-mail: [email protected]; [email protected]; [email protected] 0

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A. Ebadian, C. Park, D. Y. Shin

One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows. If the problem accepts a solution, we say that the equation is stable. The first problem concerning group homomorphisms was raised by Ulam [39] in 1940. In the next year Hyers [10] gave a first affirmative answer to the question of Ulam in context of Banach spaces. Subsequently, the result of Hyers was generalized by Aoki [2] for additive mapping and by Rassias [33] for linear mapping by considering an unbounded Cauchy difference. The result of Rassias has provided a lot of influence during the last three decades in the development of generalization of Hyers-Ulam stability concept. Furthermore, in 1994, Gˇ avrut¸a [9] provided a further generalization of Rassias’ theorem in which he replaced the bound ε(kxkp + kykp ) by a general control function ϕ(x, y). Recently several stability results have been obtained for various equations and mappings with more general domains and ranges have been investigated by a number of authors and there are many interesting results concerning this problem [1, 7, 11, 12, 13, 30, 31, 34, 36, 37]. In particular, Rassias [32] introduced the following Euler-Lagrange type quadratic functional equation f (ax + by) + f (ax − by) = (a2 + b2 )(f (x) + f (y)), for fixed reals a, b with a 6= 0, b 6= 0. Jun and Kim [14] proved the Hyers-Ulam stability of the following Euler-Lagrange type cubic functional equation f (ax + by) + f (bx + ay) = (a + b)(a − b)2 (f (x) + f (y)) + ab(a + b)f (x + y), where a 6= 0, b 6= 0, a ± b 6= 0, for all x, y ∈ X. Also Lee et al. [20] considered the following functional equation f (2x + y) + f (2x − y) = 4(f (x + y) + f (x − y)) + 24f (x) − 6f (y) and established the general solution and the Hyers-Ulam stability for this functional equation. This functional equation is called quartic functional equation and every solution of this quartic equation is said to be a quartic function. Obviously, the function f (x) = x4 satisfies the quartic functional equation. Kang [16] investigated the solution and the Hyers-Ulam stability for the quartic functional equation f (ax + by) + f (ax − by) = a2 b2 (f (x + y) + f (x − y)) + 2a2 (a2 − b2 )f (x) − 2b2 (a2 − b2 )f (y), where a, b ∈ Z \ {0} with a 6= ±1, ±b. Several Euler-Lagrange type functional equations have been investigated by numerous mathematicians [43, 44, 45]. In this paper, we consider the following Euler-Lagrange type quartic functional equation 1 1 f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) = (a2 − 1)2 (f (x) + f (y)) + a(a + 1)2 f (x + y) (1.1) 2 2 for a fixed integer a 6= 0, ±1 and all x, y ∈ X. We determine some stability results concerning the above Euler-Lagrange quartic functional equation in the setting of intuitionistic fuzzy normed spaces (IFNS). Further, we study intuitionistic fuzzy continuity through the existence of a certain solution of a fuzzy stability problem for approximately Euler-Lagrange type quartic functional equation. Throughout this paper, assume that the symbol N denotes the set of all natural numbers. 2. Preliminaries In this section we recall some notations and basic definitions used in this paper. A binary operation ∗ : [0, 1] × [0, 1] −→ [0, 1] is said to be a continuous t-norm if it satisfies the following conditions: 579

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Intuitionistic fuzzy stability of Euler-Lagrange quartic functional equation

(i) ∗ is associative and commutative, (ii) ∗ is continuous, (iii) a ∗ 1 = a for all a ∈ [0, 1], (iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1]. A binary operation  : [0, 1] × [0, 1] −→ [0, 1] is said to be a continuous t-conorm if it satisfies the following conditions: (i)  is associative and commutative, (ii) ∗ is continuous, (iii) a  0 = a for all a ∈ [0, 1], (iv) a  b ≤ c  d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1]. The five-tuple (X , µ, ν, ∗, ) is said to be intuitionistic fuzzy normed spaces (for short, INFS) if X is a vector space, ∗ is a continuous t-norm,  is a continuous t-conorm, and µ, ν are fuzzy sets on X × (0, ∞) satisfying the following conditions. For every x, y ∈ X and s, t > 0, (i) t µ(x, t) + ν(x, t) ≤ 1, (ii) µ(x, t) > 0, (iii) µ(x, t) = 1 if and only if x = 0, (iv) µ(αx, t) = µ(x, |α| ) for each α 6= 0, (v) µ(x, t) ∗ µ(y, s) ≤ µ(x + y, t + s), (vi) µ(x, .) : (0, ∞) → [0, 1] is continuous, (vii) limt→∞ µ(x, t) = 1 and limt→0 µ(x, t) = 0, (viii) ν(x, t) < 1, (ix) ν(x, t) = 0 if and only t ) for each α 6= 0, (v) ν(x, t)  ν(y, s) ≥ ν(x + y, t + s), (vi) if x = 0, (x) ν(αx, t) = ν(x, |α| ν(x, .) : (0, ∞) → [0, 1] is continuous, (vii) limt→∞ ν(x, t) = 0 and limt→0 ν(x, t) = 1. In this case (µ, ν) is called an intuitionistic fuzzy norm. For simplicity in notation, we denote the intuitionistic fuzzy normed space by (X , µ, ν) instead of (X , µ, ν, ∗, ). For example, let (X , k.k) be a normed space, and let a ∗ b = ab and a  b = min{a + b, 1} for all a, b ∈ [0, 1]. For all x ∈ X and every t ∈ R, consider ( t t>0 t+kxk , µ(x, t) = 0, t ≤ 0 and ( ν(x, t) =

kxk t+kxk ,

0,

t>0 t ≤ 0,

then (X , µ, ν) is an IFNS. The concept of convergence and Cauchy sequences in the setting of IFNS were introduced by Saadati and Park [35] and further studied by Mursaleen and Mohiuddine [27]. Let (X , µ, ν) be an IFNS. Then a sequence {xn } is said to be: (i) convergent to L ∈ X with respect to the intuitionistic fuzzy norm (µ, ν) if, for every ε > 0 and t > 0, there exists n0 ∈ N such that µ(xn − L, t) > 1 − ε and ν(xn − L, t) < ε for all n ≥ n0 . In this case, we write (µ, ν) − lim xn = L or xn → L as n → ∞. (ii) a Cauchy sequence with respect to the intuitionistic fuzzy norm (µ, ν) if, for every ε > 0 and t > 0, there exists n0 ∈ N such that µ(xn − xm , t) > 1 − ε and ν(xn − xm , t) < ε for all n, m ≥ n0 . An IFNS (X , µ, ν) is said to be complete if each intuitionistic fuzzy Cauchy sequence in (X , µ, ν) is intuitionistic fuzzy convergent in (X , µ, ν). In this case, (X , µ, ν) is called an intuitionistic fuzzy Banach space. 3. Approximation on intuitionistic fuzzy normed spaces Theorem 3.1. Let X be a linear space and let (Z, µ0 , ν 0 ) be an IFNS. Let ϕ : X × X → Z be a mapping such that, for some |α| > |a|4 , x µ0 (ϕ( , 0), t) ≥ µ0 (ϕ(x, 0), |α|t), a x 0 ν (ϕ( , 0), t) ≤ ν 0 (ϕ(x, 0), |α|t) a 580

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A. Ebadian, C. Park, D. Y. Shin

and let (Y, µ, ν) be an intuitionistic fuzzy Banach space and let f : X → Y be a mapping satisfying f (0) = 0 and a ϕ-approximately quartic mapping in the sense that 1 µ(f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) − (a2 − 1)2 (f (x) + f (y)) 2 1 a(a + 1)2 f (x + y), t) ≥ µ0 (ϕ(x, y), t), − 2 1 ν(f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) − (a2 − 1)2 (f (x) + f (y)) 2 1 a(a + 1)2 f (x + y), t) ≤ ν 0 (ϕ(x, y), t) − 2

(3.1)

for all x, y ∈ X and all t > 0. Then there exists a unique quartic mapping Q : X → Y such that (|α| − |a|4 )t ), 2 (|α| − |a|4 )t ν(Q(x) − f (x), t) ≤ ν 0 (ϕ(x, 0), ) 2

µ(Q(x) − f (x), t) ≥ µ0 (ϕ(x, 0),

(3.2)

for all x ∈ X and all t > 0. Proof. Letting y = 0 in (3.1), we get µ(f (ax) − a4 f (x), t) ≥ µ0 (ϕ(x, 0), t), ν(f (ax) − a4 f (x), t) ≤ ν 0 (ϕ(x, 0), t)

(3.3)

for all x ∈ X and all t > 0, which implies that x µ(a4 f ( ) − f (x), t) ≥ µ0 (ϕ(x, 0), |α|t), a x 4 ν(a f ( ) − f (x), t) ≤ ν 0 (ϕ(x, 0), |α|t). a Replacing x by

x an

(3.4)

in (3.4), we obtain

x ), |a|4n t) ≥ µ0 (ϕ(x, 0), |α|n+1 t), an x x ν(a4(n+1) f ( n+1 ) − a3n f ( n ), |a|4n t) ≤ ν 0 (ϕ(x, 0), |α|n+1 t). a a

µ(a4(n+1) f (

Replacing t by

t |α|n+1

x

an+1

) − a3n f (

(3.5)

in (3.5), we get

µ(a4(n+1) f (

x |a|4n 3n ) − a f ( ), t) ≥ µ0 (ϕ(x, 0), t), an+1 an |α|n+1

ν(a4(n+1) f (

x |a|4n 3n ) − a f ( ), t) ≤ ν 0 (ϕ(x, 0), t). an+1 an |α|n+1

x

x

(3.6)

It follows from n−1

a4n f (

X x x x ) − f (x) = (a4(j+1) f ( j+1 ) − a3j f ( j )) n a a a j=0

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Intuitionistic fuzzy stability of Euler-Lagrange quartic functional equation

and (3.6) that µ(a4n f (

n−1 n−1 X |a|4j Y x x |a|4j x 3j 4(j+1) ) − f (x), t) ≥ ) − a f ( ), t) µ(a f ( an |α|j+1 aj+1 aj |α|j+1 j=0

j=0 0

≥ µ (ϕ(x, 0), t),

(3.7)

n−1 n−1 X |a|4j a x x |a|4j x 3j 4(j+1) ν(a f ( n ) − f (x), t) ≤ ) − a f ( ), t) ν(a f ( a |α|j+1 aj+1 aj |α|j+1 4n

j=0

j=0 0

≤ ν (ϕ(x, 0), t), for all x ∈ X , all t > 0 and n > 0, where n−1 Y

ωj = ω0 ∗ ω2 ∗ ... ∗ ωn−1 ,

j=0

Replacing x by

x am

n−1 a

ωj = ω0  ω2  ...  ωn−1 .

j=0

in (3.7), we obtain

µ(a

4(n+m)

f(

ν(a4(n+m) f (

x an+m

n−1

)−a

4m

x X |a|4(j+m) f ( m ), t) ≥ µ0 (ϕ(x, 0), t), a |α|j+m+1 j=0

x

x ) − a4m f ( m ), n+m a a

n−1 X j=0

|a|4(j+m) t) ≤ ν 0 (ϕ(x, 0), t). |α|j+m+1

Hence µ(a4(n+m) f (

ν(a4(n+m) f (

x

a

) − a4m f ( n+m

n+m−1 X |a|4j x ), t) ≥ µ0 (ϕ(x, 0), t), am |α|j+1 j=m

x

x ) − a4m f ( m ), n+m a a

n+m−1 X j=m

|a|4j t) ≤ ν 0 (ϕ(x, 0), t) |α|j+1

for all x ∈ X , all t > 0, n ≥ 0 and m ≥ 0. Thus µ(a4(n+m) f ( ν(a4(n+m) f (

x an+m x an+m

) − a4m f (

x t ), t) ≥ µ0 (ϕ(x, 0), Pn+m−1 m a

|a|4j |α|j+1

j=m

) − a4m f (

x t ), t) ≤ ν 0 (ϕ(x, 0), Pn+m−1 m a j=m

|a|4j |α|j+1

), )

(3.8)

4j

|a| for all x ∈ X , all t > 0, n ≥ 0 and m ≥ 0. Since |α| > |a|4 and Σ∞ j=0 |α|j < ∞, the sequence a4n f ( axn ) is a Cauchy sequence in (Y, µ, ν). Since (Y, µ, ν) is complete, this sequence converges to some point Q(x) ∈ Y. Fix x ∈ X and put m = 0 in (3.8). Then we obtain

µ(a4n f (

x t ) − f (x), t) ≥ µ0 (ϕ(x, 0), Pn−1 n a

),

ν(a4n f (

x t ) − f (x), t) ≤ ν 0 (ϕ(x, 0), Pn−1 n a

)

|a|4j j=0 |α|j+1

|a|4j j=0 |α|j+1

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for all x ∈ X , all t > 0 and n ≥ 0. Thus we get x t x t ), ) ∗ µ(a4n f ( n ) − f (x), ) an 2 a 2 t ≥ µ0 (ϕ(x, 0), Pn−1 |a|4j ), 2 j=0 |α|j+1

µ(Q(x) − f (x), t) ≥ µ(Q(x) − a4n f (

x t t x ), )  ν(a4n f ( n ) − f (x), ) n a 2 a 2 t 0 ≤ ν (ϕ(x, 0), Pn−1 |a|4j ) 2 j=0 |α|j+1

ν(Q(x) − f (x), t) ≤ ν(Q(x) − a4n f (

for large n. By taking the limit as n → ∞, we obtain (|α| − |a|4 )t ), 2 (|α| − |a|4 )t ν(Q(x) − f (x), t) ≤ ν 0 (ϕ(x, 0), ) 2 µ(Q(x) − f (x), t) ≥ µ0 (ϕ(x, 0),

for all x ∈ X and all t > 0. Replacing x and y by

x an

and

y an

in (3.1), respectively, we obtain

ax + y 1 x−y x y ) + f (x + ay) + a(a − 1)2 a4n f ( n ) − (a2 − 1)2 a4n (f ( n ) + f ( n )) n a 2 a a a x+y x y t 1 2 4n 0 − a(a + 1) a f ( n ), t) ≥ µ (ϕ( n , n ), 4n ), 2 a a a |a| ax + y 1 x−y x y ν(a4n f ( ) + f (x + ay) + a(a − 1)2 a4n f ( n ) − (a2 − 1)2 a4n (f ( n ) + f ( n )) n a 2 a a a 1 x+y x y t 2 4n 0 − a(a + 1) a f ( n ), t) ≤ ν (ϕ( n , n ), 4n ) 2 a a a |a|

µ(a4n f (

for all x ∈ X and all t > 0. Note that x y |α|n t 0 , ), t) = lim µ (ϕ(x, y), ) = 1, n→∞ n→∞ an an |a|4n x y |α|n t lim ν 0 (a4n ϕ( n , n ), t) = lim ν 0 (ϕ(x, y), 4n ) = 0 n→∞ n→∞ a a |a| lim µ0 (a4n ϕ(

for all x ∈ X and all t > 0. We observe that Q fulfills (1.1). Therefore, Q is an Euler-Lagrange type quartic mapping. It is left to show that the quartic mapping Q is unique. Assume that there is another Euler-Lagrange type quartic mapping Q0 : X → Y satisfying (3.2). For each x ∈ X , clearly 583

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Intuitionistic fuzzy stability of Euler-Lagrange quartic functional equation

a4n Q( axn ) = Q(x) and a4n Q0 ( axn ) = Q0 (x) for all n ∈ N. It follows from (3.3) that x x µ(Q(x) − Q0 (x), t) = µ(a4n Q( n ) − a4n Q0 ( n ), t) a a x x t = µ(Q( n ) − Q0 ( n ), 4n ) a a |a| x x t = µ(Q( n ) − f ( n ), ) a a 2|a|4n x x t ) ∗ µ(f ( n ) − Q0 ( n ) − a a 2|a|4n ≥ µ0 (ϕ(x, 0),

|α|n (|α| − |a|4 )t ), 2|a|4n

and similarly ν(Q(x) − Q0 (x), t) ≤ ν 0 (ϕ(x, 0),

αn (|α| − |k|4 )t ) 2|a|4n

for all x ∈ X and all t > 0. Since |α| > |a|4 , we get lim µ0 (ϕ(x, 0),

|α|n (|α| − |a|4 )t ) = 1, 2|a|4n

lim ν 0 (ϕ(x, 0),

|α|n (|α| − |a|4 )t ) = 0. 2|a|4n

n→∞

n→∞

Therefore, µ(Q(x) − Q0 (x), t) = 1 and ν(Q(x) − Q0 (x), t) = 0 for all x ∈ X and all t > 0, that is, the mapping Q(x) is unique, as desired.  Theorem 3.2. Let X be a linear space and let (Z, µ0 , ν 0 ) be an IFNS. Let ϕ : X × X → Z be a mapping such that, for some 0 < |α| < |a|4 , µ0 (ϕ(ax, 0), t) ≥ µ0 (αϕ(x, 0), t), ν 0 (ϕ(ax, 0), t) ≤ ν 0 (αϕ(x, 0), t). Let (Y, µ, ν) be an intuitionistic fuzzy Banach space and let f : X → Y be a mapping satisfying f (0) = 0 and a ϕ-approximately quartic mapping in the sense that 1 µ(f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) − (a2 − 1)2 (f (x) + f (y)) 2 1 2 − a(a + 1) f (x + y), t) ≥ µ0 (ϕ(x, y), t), 2 1 ν(f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) − (a2 − 1)2 (f (x) + f (y)) 2 1 2 − a(a + 1) f (x + y), t) ≤ ν 0 (ϕ(x, y), t) 2 for all x, y ∈ X and all t > 0. Then there exists a unique quartic mapping Q : X → Y such that |a|4 − |α| µ(Q(x) − f (x), t) ≥ µ0 (ϕ(x, 0), t), 2 |a|4 − |α| ν(Q(x) − f (x), t)νν 0 (ϕ(x, 0), t) (3.9) 2 for all x ∈ X and all t > 0. 584

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Proof. The techniques are similar to that of Theorem 3.1. Hence we present a sketch of the proof. Letting y = 0 in (3.9), we get f (ax) − f (x), t) ≥ µ0 (ϕ(x, 0), t), a4 f (ax) ν( 4 − f (x), t) ≤ ν 0 (ϕ(x, 0), t) a µ(

(3.10)

for all x ∈ X and all t > 0. Replacing x by an x in (3.10), we obtain µ(

f (a(n+1) x) t − f (an x), t) ≥ µ0 (ϕ(x, 0), n ), a4 |α|

ν(

f (a(n+1) x) t − f (an x), t) ≤ ν 0 (ϕ(x, 0), n ) a4 |α|

for all x ∈ X and all t > 0. For positive integers n and m, µ(

t f (a(n+m) x) f (am x) , t) ≥ µ0 (ϕ(x, 0), Pn+m−1 − 4m 4(n+m) a a j=m

ν(

f (a(n+m) x) f (am x) t − , t) ≤ ν 0 (ϕ(x, 0) Pn+m−1 4m 4(n+m) a a j=m

for all x ∈ X and all t > 0. Hence {

f (an x) a4n

),

|α|j a4(j+1) |α|j

),

(3.11)

a4(j+1)

} is a Cauchy sequence in intuitionistic fuzzy Banach

space. Therefore, there is a mapping Q : X → Y defined by Q(x) = limn→∞ (3.11) with m = 0 implies

f (an x) a4n

and hence

|a|4 − |α| t), 2 |a|4 − |α| ν(Q(x) − f (x), t)νν 0 (ϕ(x, 0), t) 2 µ(Q(x) − f (x), t) ≥ µ0 (ϕ(x, 0),

for all x ∈ X and all t > 0. This complete the proof.



4. Intuitionistic fuzzy continuity In this section we establish intuitionistic fuzzy continuity by using continuous approximately quartic mappings. Definition 4.1. Let f : R → X be a mapping, where R is endowed with the Euclidean topology and X is an intuitionistic fuzzy normed space equipped with intuitionistic fuzzy norm (µ, ν). Then f is called intuitionistic fuzzy continuous at a point t0 ∈ R if for all ε > 0 and all 0 < α < 1 there exists δ > 0 such that for each t with 0 < |t − t0 | < δ µ(f (tx) − f (t0 x), ε) ≥ α, ν(f (tx) − f (t0 x), ε) ≤ 1 − α. Theorem 4.1. Let X be a normed space and let (Z, µ0 , ν 0 ) be an IFNS. Let (Y, µ, ν) be an intuitionistic fuzzy Banach space and let f : X → Y be a mapping satisfying f (0) = 0 and an 585

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(r, s)-approximately quartic mapping in the sense that for some r, s and some z0 ∈ Z 1 µ(f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) − (a2 − 1)2 (f (x) + f (y)) 2 1 a(a + 1)2 f (x + y), t) ≥ µ0 ((kxkr + kyks )z0 , t), − 2 1 ν(f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) − (a2 − 1)2 (f (x) + f (y)) 2 1 − a(a + 1)2 f (x + y), t) ≤ ν 0 ((kxkr + kyks )z0 , t) 2 for all x, y ∈ X and all t > 0. If r, s < 4, then there exists a unique quartic mapping Q : X → Y such that (|a|4 − |a|r )t ), 2 (|a|4 − |a|r )t ν(Q(x) − f (x), t) ≤ ν 0 (kxkr z0 , ) (4.1) 2 for all x ∈ X and all t > 0. Furthermore, if for some x ∈ X and all n ∈ N, the mapping H : R → Y defined by H(t) = f (an tx) is intuitionistic fuzzy continuous. Then the mapping t → Q(tx) from R to Y is intuitionistic fuzzy continuous. µ(Q(x) − f (x), t) ≥ µ0 (kxkr z0 ,

Proof. Define ϕ : X × X → Z by ϕ(x, y) = (kxkr + kyks )z0 . By Theorem 3.2, there exists a unique quartic mapping Q satisfying (4.1). We have µ(Q(x) −

f (an x) Q(an x) f (an x) , t) = µ( − , t) a4n a4n a4n

= µ(Q(an x) − f (an x), |a|4n t) ≥ µ0 (|a|4rn kxkr z0 ,

|a|4n (|a|4 − |a|r )t ) 2

|a|4n (|a|4 − |a|r )t ), 2|a|rn f (an x) Q(an x) f (an x) ν(Q(x) − , t) = ν( − , t), a4n a4n a4n ≥ µ0 (kxkr z0 ,

= ν(Q(an x) − f (an x), |a|4n t) ≤ ν 0 (|a|4rn kxkr z0 , ≤ ν 0 (kxkr z0 ,

|a|4n (|a|4 − |a|r )t ), 2

|a|4n (|a|4 − |a|r )t ) 2|a|rn

(4.2)

for all x ∈ X , all t > 0 and n ∈ N. Fix x ∈ X and t0 ∈ R. Given ε > 0 and 0 < α < 1. By (4.2), we obtain µ(Q(tx) −

f (an tx) |a|4n (|a|4 − |a|r )t 0 r , t) ≥ µ (kxk z , ) 0 a4n 2|t|r |a|rn

≥ µ0 (kxkr z0 , µ(Q(tx) −

|a|4n (|a|4 − |a|r )t ), 2|1 + t0 |r |a|rn

f (an tx) |a|4n (|a|4 − |a|r )t 0 r , t) ≥ µ (kxk z , ) 0 a4n 2|t|r |a|rn

≥ µ0 (kxkr z0 ,

|a|4n (|a|4 − |a|r )t ) 2|1 + t0 |r |a|rn 586

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for all t ∈ R and all |t − t0 | < 1. Since r < 3, we have limn→∞ there exists n0 ∈ N such that

|a|4n (|a|4 −|a|r )t 2|1+t0 |r |a|rn

= ∞, and hence

f (an0 tx) ε , ) ≥ α, a4n0 3 f (an0 tx) ε ν(Q(tx) − , )≤1−α a4n0 3 µ(Q(tx) −

for all t ∈ R and all |t − t0 | < 1. By the intuitionistic fuzzy continuity of the mapping s → f (an0 tx), there exists δ < 1 such that for each t with 0 < |t − t0 | < δ, we have f (an0 tx) f (an0 t0 x) ε − , ) ≥ α, a4n0 a4n0 3 n n 0 0 f (a tx) f (a t0 x) ε − , ) ≤ 1 − α. ν( a4n0 a4n0 3 µ(

It follows that f (an0 tx) ε , )∗ a4n0 3 f (an0 t0 x) ε f (an0 tx) f (an0 t0 x) ε − , ) ∗ µ( − Q(t0 x), ) ≥ α µ( a4n0 a4n0 3 a4n0 3 f (an0 tx) ε µ(Q(tx) − Q(t0 x), ε) ≥ µ(Q(tx) − , )∗ a4n0 3 n n n 0 0 0 f (a tx) f (a t0 x) ε f (a t0 x) ε µ( − , ) ∗ µ( − Q(t0 x), ) ≥ α, a4n0 a4n0 3 a4n0 3 f (an0 tx) ε ν(Q(tx) − Q(t0 x), ε) ≤ ν(Q(tx) − , ) a4n0 3 f (an0 tx) f (an0 t0 x) ε f (an0 t0 x) ε ν( − , )  ν( − Q(t x), )≤α 0 4n 4n 4n a 0 a 0 3 a 0 3 f (an0 tx) ε ν(Q(tx) − Q(t0 x), ε) ≤ ν(Q(tx) − , ) a4n0 3 f (an0 tx) f (an0 t0 x) ε f (an0 t0 x) ε ν( − , )  ν( − Q(t0 x), ) ≤ 1 − α 4n 4n 4n 0 0 0 a a 3 a 3 µ(Q(tx) − Q(t0 x), ε) ≥ µ(Q(tx) −

for all t ∈ R and all 0 < |t − t0 | < δ. Therefore, the mapping t → Q(tx) is intuitionistic fuzzy continuous.  Theorem 4.2. Let X be a normed space and let (Z, µ0 , ν 0 ) be an IFNS. Let (Y, µ, ν) be an intuitionistic fuzzy Banach space and let f : X → Y be a mapping satisfying f (0) = 0 and a (r, s)-approximately quartic mapping in the sense that for some r, s and some z0 ∈ Z 1 µ(f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) − (a2 − 1)2 (f (x) + f (y)) 2 1 − a(a + 1)2 f (x + y), t) ≥ µ0 ((kxkr + kyks )z0 , t), 2 1 ν(f (ax + y) + f (x + ay) + a(a − 1)2 f (x − y) − (a2 − 1)2 (f (x) + f (y)) 2 1 − a(a + 1)2 f (x + y), t) ≤ ν 0 ((kxkr + kyks )z0 , t) 2 587

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Intuitionistic fuzzy stability of Euler-Lagrange quartic functional equation

for all x, y ∈ X and all t > 0. If r, s < 4, then there exists a unique quartic mapping Q : X → Y such that (|a|r − |a|4 )t µ(Q(x) − f (x), t) ≥ µ0 (kxkr z0 , ), 2 (|a|r − |a|4 )t ν(Q(x) − f (x), t) ≤ ν 0 (ϕ(x, 0), ) 2 for all x ∈ X and all t > 0. Furthermore, if for some x ∈ X and all n ∈ N, the mapping H : R → Y defined by H(t) = f (an tx) is intuitionistic fuzzy continuous. Then the mapping t → Q(tx) from R to Y is intuitionistic fuzzy continuous. Proof. Define ϕ : X × X → Z by ϕ(x, y) = (kxkr + kyks )z0 for all x ∈ X . Since r > 4, we have α = |a|r > |a|4 . By Theorem 3.1, there exists a unique quartic mapping Q satisfying (4.3). The rest of the proof can be done by the same line as in Theorem 4.1.  Acknowledgments C. Park and D. Y. Shin were 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 (NRF-2010-0021792), respectively. References [1] Abolfathi, M. A., Ebadian, A. and Aghalary, R. Stability of mixed additive-quadratic Jensen type functional equation in non-Archimedean `-fuzzy normed spaces, Ann. Univ. Ferrara Sez. V II Sci. Mat. (to appear). [2] Aoki, T. On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 245–251. [3] Atanassov, K. Intuitionistic fuzzy sets, VII ITKR’S, Session, Sofia, June 1983 (Deposed in Central ScienceTechnical Library of Bulg. Academy of Science, 1697/84)(in Bulgarian). [4] Bag, T. and Samanta, S. K. Finite dimensional fuzzy normed linear space, J. Fuzzy Math. 11 (2003), 687–705. [5] Bag, T. and Samanta, S. K. Fuzzy bounded linear operators, Fuzzy Set Syst. 151 (2005), 513–547. [6] Cheng, S. C. and Mordeson, J. N. Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [7] Ebadian, A. and Zolfaghari, S. Stability of a mixed additive and cubic functional equation in several variables in non-Archimedean spaces, Ann. Univ. Ferrara Sez. V II Sci. Mat. 58 (2012), 291–306. [8] Felbin, C. Finite dimensional fuzzy normed linear space, Fuzzy Set Syst. 48 (1992), 239–248. [9] Gˇ avruta, P. A generalization of the Hyers-Ulam-Rassias stability of the approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [10] Hyers, D. H. On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [11] Hyers, D. H., Isac, G. and Rassias, Th. M. Stability of Functional Equation in Several Variables, Birkh¨ ause, Basel, 1998. [12] Isac, G. and Rassias, Th. M. Stability of ψ-additive mappings: applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [13] Jun, K. and Kim, H. The generalized Hyers-Ulam-Rassias stability of cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878. [14] Jun, K. and Kim, H. On the stability of the Euler-Lagrange type cubic mappings in quasi-Banach spaces, J. Math. Anal. Appl. 33 (2007), 1335–1350. [15] Kaleva, O. and Seikkala, S. On fuzzy metric spaces, Fuzzy Set Syst. 12 (1984), 1–7. [16] Kang, D. On the stability of gneralized quadratic mappings in quasi β-normed space, J. Inequal. Appl. 2010, Article ID 198098 (2010). [17] Katsaras, A. K. Fuzzy topological vector spaces, Fuzzy Set Syst. 12 (1984), 143–154. 588

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO. 3, 2015

On the Shadowing Property of Functional Equations, Sun Young Jang,…………………400 Ulam-Găvruta-Rassias Stability of an Additive Functional Inequality in Fuzzy Normed Modules, Yeol Je Cho, Reza Saadati, and Young-Oh Yang,……………………………………..…409 Almost Ideal Statistical Convergence and Strongly Almost Ideal Lacunary Convergence of Sequences of Fuzzy Numbers with Respect to the Orlicz Functions, Zeng-Tai Gong, Xiao-Xia Liu, and Xue Feng,…………………………………………………………………………418 Some Boundary Value Problems of Fractional Differential Equations with Fractional Impulsive Conditions, Youjun Xu, and Xiaoyou Liu,………………………………………………....426 Volterra Composition Operators From F(p,q,s) to Logarithmic Bloch Spaces, Fang Zhang, and Yongmin Liu,……………………………………………………………………………….444 The (p,q)-Absolutely Summing Operators on the Köthe Bochner Function Spaces. Mona Khandaqji, and Ahmed Al-Rawashdeh,……………………………………………………455 An Estimate for the Four-Dimensional Discrete Derivative Green's Function and Its Applications in FE Superconvergence, Jinghong Liu, and Yinsuo Jia,……………………………….….462 Apollonius Type Additive Functional Equations in C*-Ternary Algebras and JB*-Triples, Yeol Je Cho, Reza Saadati, and Young-Oh Yang,…………………………………………470 On Grüss Type Integral Inequality Involving the Saigo’s Fractional Integral Operators, D. Baleanu, S.D. Purohit, and F. Uçar,…………………………………………………….480 The Value Distribution of Solutions of Some Types of q-Shift Difference Riccati Equations, Yu Xian Chen, and Hong Yan Xu,…………………………………………………………490 Energy Decay Rates for Viscoelastic Wave Equation with Dynamic Boundary Conditions, Jin-Mun Jeong, Jong Yeoul Park, and Yong Han Kang,……………………………………500 Lower order Fractional Monotone Approximation, George A. Anastassiou,……………….518 Modified Quadrature Method with Sidi Transformation for Three Dimensional Axisymmetric Potential Problems with Dirichlet Conditions, Xin Luo, Jin Huang, and Yan-Ying Ma,……526

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO. 3, 2015 (continued) Stability of the Generaliazed Version of Cauchy Type Equation From Functional Operator Viewpoint, Chang Il Kim, Giljun Han, and Seong-A Shim,………………………………540 Inequalities of Simpson Type for Functions Whose Third Derivatives are Extended s-Convex Functions and Applications to Means, Ling Chun, and Feng Qi,………………………….555 Representation of Higher-Order Euler Numbers Using the Solution of Bernoulli Equation, Cheon Seoung Ryoo, Hyuck In Kwon, Jihee Yoon, and Yu Seon Jang,………………….570 Intuitionistic Fuzzy Stability of an Euler-Lagrange Type Quartic Functional Equation, Ali Ebadian, Choonkil Park, and Dong Yun Shin,…………………………………………578

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.4, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Applications of sampling Kantorovich operators to thermographic images for seismic engineering Federico Cluni Danilo Costarelli Anna Maria Minotti Gianluca Vinti ,

,

and

∗

Abstract In this paper, we present some applications of the multivariate sampling Kantorovich operators Sw to seismic engineering. The mathematical theory of these operators, both in the space of continuous functions and in Orlicz spaces, show how it is possible to approximate/reconstruct multivariate signals, such as images. In particular, to obtain applications for thermographic images a mathematical algorithm is developed using MATLAB and matrix calculus. The setting of Orlicz spaces is important since allow us to reconstruct not necessarily continuous signals by means of Sw . The reconstruction of thermographic images of buildings by our sampling Kantorovich algorithm allow us to obtain models for the simulation of the behavior of structures under seismic action. We analyze a real world case study in term of structural analysis and we compare the behavior of the building under seismic action using various models. AMS 2010 Subject Classication: 41A35, 46E30, 47A58,47B38, 94A12 Key words and phrases: Sampling Kantorovich operators, Orlicz spaces, Image Processing, thermographic images, structural analysis.

1

Introduction

The sampling Kantorovich operators have been introduced to approximate and reconstruct not necessarily continuous signals.

In [3], the authors in-

troduced this operators starting from the well-known generalized sampling operators (see e.g. [10, 12, 9, 22, 4]) and replacing, in their denition, the sample values

f (k/w) with w

∫ (k+1)/w k/w

f (u) du.

Clearly, this is the most natu-

ral mathematical modication to obtain operators which can be well-dened also for general measurable, locally integrable functions, not necessarily continuous. Moreover, this situation very often occur in Signal Processing, when ∗

corresponding author, email: [email protected]

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one cannot match exactly the sample at the point

k/w:

this represents the

so-called "time-jitter error. The theory of sampling Kantorovich operators allow us to reduces the time-jitter error, calculating the information in a neighborhood of

k/w

rather that exactly in the node

k/w.

These operators,

as the generalized sampling operators, represent an approximate version of classical sampling series, based on the Whittaker-Kotelnikov-Shannon sampling theorem (see e.g. [1]). Subsequently, the sampling Kantorovich operators have been studied in various settings. In [15, 16] the multivariate version of these operators were introduced. Results concerning the order of approximation are shown in [17]. Extensions to more general contexts are presented in [23, 24, 25, 6]. The multivariate sampling Kantorovich operators considered in this paper are of the form:

(Sw f )(x) :=

∑

[

wn χ(wx − tk ) Ak n

∫

f (u) du , Rkw

k∈Z where

f : Rn → R

] (x ∈ Rn ),

(I)

is a locally integrable function such that the above series

x ∈ Rn . The symbol tk = (tk1 , ..., tkn ) denotes vectors where each (tki )ki ∈Z , i = 1, ..., n is a certain strictly increasing sequence of real numbers with ∆ki = tki+1 − tki > 0. Note that, the sequences (tki )ki ∈Z w are not necessary equally spaced. We denote by Rk the sets: is convergent for every

[ Rkw

:=

] [ ] [ ] tk2 tk2 +1 tkn tkn +1 tk1 tk1 +1 , × , × ... × , , w w w w w w

(II)

w > 0 and Ak = ∆k1 · ∆k2 · ... · ∆kn , k ∈ Zn . Moreover, χ : Rn → R is a kernel satisfying suitable assumptions.

the function

For the operators in (I) we recall some convergence results.

We have

(Sw f )w>0 converges pointwise to f , when f is continuous and (Sw f )w>0 converges uniformly to f , when f is uniformly

that the family and bounded

continuous and bounded. Moreover, to cover the case of not necessarily continuous signal, we study our operators in the general setting of Orlicz spaces

Lφ (Rn ). For functions f belonging to Lφ (Rn ) and generated by the convex φ-function φ, the family of sampling Kantorovich operators is "modularly convergent to f , being the latter the natural concept of convergence in this setting. The latter result, allow us to apply the theory of the sampling Kantorovich operators to approximate and reconstruct images.

In fact, static

gray scale images are characterized by jumps of gray levels mainly concentrated in their contours or edges and this can be translated, from a mathematical point of view, by discontinuities (see e.g. [18]). Here, we introduce and analyze in detail some practical applications of the sampling Kantorovich algorithm to thermographic images, very useful

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for the analysis of buildings in seismic engineering. The thermography is a remote sensing technique, performed by the image acquisition in the infrared. Thermographic images are widely used to make non-invasive investigations of structures, to analyze the story of the building wall, to make diagnosis and monitoring buildings, and to make structural measurements. A further important use, is the application of the texture algorithm for the separation between the bricks and the mortar in masonries images. Through this procedure the civil engineers becomes able to determine the mechanical parameters of the structure under investigation. Unfortunately, the direct application of the texture algorithm to the thermographic images, can produce errors, as an incorrect separation between the bricks and the mortar. Then, we use the sampling Kantorovich operators to process the thermographic images before to apply the texture algorithm. In this way, the result produced by the texture becomes more rened and therefore we can apply structural analysis after the calculation of the various parameters involved. In order to show the feasibility of our applications, we present in detail a real-world case-study.

2

Preliminaries

In this section we recall some preliminaries, notations and denitions.

C(Rn ) (resp. C 0 (Rn )) the space of all uniformly continuous n and bounded (resp. continuous and bounded) functions f : R → R endowed with the usual sup-norm ∥f ∥∞ := supu∈Rn |f (u)|, u = (u1 , ..., un ), and by Cc (Rn ) ⊂ C(Rn ) the subspace of the elements having compact support. n Moreover, M (R ) will denote the linear space of all (Lebesgue) measurable n real functions dened on R . We denote by

We now recall some basic fact concerning Orlicz spaces, see e.g. [20, 19, 21, 8, 7].

+ φ : R+ 0 → R0 is said to be a φ-function if it satises the following assumptions: (i) φ (0) = 0, and φ (u) > 0 for every u > 0; (ii) φ is + continuous and non decreasing on R0 ; (iii) limu→∞ φ(u) = +∞. φ n The functional I : M (R ) → [0, +∞] (see e.g. [19, 7]) dened by ∫ φ (f ∈ M (Rn )) , I [f ] := φ(|f (x)|) dx, The function

Rn

is a modular in

M (Rn )

. The Orlicz space generated by

Lφ (Rn ) := {f ∈ M (Rn ) : I φ [λf ] < +∞, The space

Lφ (Rn )

φ

is given by

for some

λ > 0} .

is a vector space and an important subspace is given by

E φ (Rn ) := {f ∈ M (Rn ) : I φ [λf ] < +∞,

for every

λ > 0} .

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E φ (Rn )

is called the space of all nite elements of

Lφ (Rn ). It is easy to see ⊂ Lφ (Rn ). Clearly,

Cc ⊂ φ n φ n functions belonging to E (R ) and L (R ) are not necessarily continuous. φ n A norm on L (R ), called Luxemburg norm, can be dened by (Rn )

that the following inclusions hold:

E φ (Rn )

∥f ∥φ := inf {λ > 0 : I φ [f /λ] ≤ λ} ,

(f ∈ Lφ (Rn )).

(fw )w>0 ⊂ Lφ (Rn ) is norm convergent to a function f ∈ ∥fw − f ∥φ → 0 for w → +∞, if and only φ if limw→+∞ I [λ(fw − f )] = 0, for every λ > 0. Moreover, an additional We will say that a family of functions

Lφ (Rn ), i.e.,

concept of convergence can be studied in Orlicz spaces: the "modular convergence". The latter induces a topology (modular topology) on the space

Lφ (Rn )

([19, 7]).

(fw )w>0 ⊂ Lφ (Rn ) is modularly limw→+∞ I φ [λ(fw − f )] = 0, for

We will say that a family of functions convergent to a function some

λ > 0.

f ∈ Lφ (Rn )

if

Obviously, norm convergence implies modular convergence,

while the converse implication does not hold in general. The modular and norm convergence are equivalent if and only if the

∆2 -condition, see e.g., [19, 7].

φ-function φ

satises the

Finally, as last basic property of Orlicz spaces,

we recall the following.

Cc (Rn ) is dense in Lφ (Rn ) with respect to the φ n modular topology, i.e., for every f ∈ L (R ) and for every ε > 0 there exists n φ a constant λ > 0 and a function g ∈ Cc (R ) such that I [λ(f − g)] < ε. Lemma 2.1 ([5]). The space

3

The sampling Kantorovich operators

In this section we recall the denition of the operators with which we will work. We will denote by tk = (tk1 , ..., tkn ) a vector where each element (tki )ki ∈Z , i = 1, ..., n is a sequence of real numbers with −∞ < tki < tki+1 < +∞, limki →±∞ tki = ±∞, for every i = 1, ..., n, and such that there exists ∆, δ > 0 for which δ ≤ ∆ki := tki+1 − tki ≤ ∆, for every i = 1, ..., n. Note that, the elements of (tki )ki ∈Z are not necessary equally spaced. In what n follows, we will identify with the symbol Π the sequence (tk )k∈Zn . n A function χ : R → R will be called a kernel if it satises the following properties:

(χ1) χ ∈ L1 (Rn ) (χ2)

for every

and is bounded in a neighborhood of

u ∈ Rn ,

∑

0 ∈ Rn ;

χ(u − tk ) = 1;

k∈Zn

(χ3)

for some

β > 0,

mβ,Πn (χ) = sup

∑

χ(u − tk ) · u − tk β < +∞, 2

u∈Rn k∈Zn

where

∥ · ∥2

denotes the usual Euclidean norm.

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We now recall the denition of the linear multivariate sampling Kantorovich operators introduced in [15]. Dene:

(Sw f )(x) :=

k∈Z where

[

∑

f : Rn → R

wn χ(wx − tk ) Ak n

]

∫

f (u) du , Rkw

(x ∈ Rn ),

(1)

is a locally integrable function such that the above series

x ∈ Rn , where [ ] [ ] [ ] tk1 tk1 +1 tk2 tk2 +1 tkn tkn +1 := , × , × ... × , , w w w w w w

is convergent for every

Rkw w>0

and

Ak = ∆k1 · ∆k2 · ... · ∆kn , k ∈ Zn .

The operators in (1) have been

introduced in [3] in the univariate setting.

(χ1)

Remark 3.1. (a) Under conditions

for the kernel

χ

and

(χ3),

the following properties

can be proved:

m0,Πn (χ) := sup

∑ χ(u − tk ) < +∞,

u∈Rn k∈Zn

and, for every

γ>0 ∑

lim

w→+∞

χ(wu − tk ) = 0,

(2)

∥wu−tk ∥2 >γw

uniformly with respect to

u ∈ Rn ,

(b) By (a), we obtain that

Sw f

see [15].

with

f ∈ L∞ (Rn )

are well-dened. Indeed,

|(Sw f )(x)| ≤ m0,Πn (χ) ∥f ∥∞ < +∞, for every

x ∈ Rn ,

i.e.

Sw : L∞ (Rn ) → L∞ (Rn ).

Remark 3.2. Note that, in the one-dimensional setting, choosing

for every

k ∈ Z,

condition

(χ2)

is equivalent to

{ χ b(k) := where

χ b(v) :=

∫

R χ(u)e

−ivu

tk = k ,

k ∈ Z \ {0} , k = 0,

0, 1,

du, v ∈ R,

denotes the Fourier transform of

χ;

see [11, 3, 15, 17].

4

Convergence results

In this section, we show the main approximation results for the multivariate sampling Kantorovich operators.

In [15], the following approximation

theorem for our operators has been proved.

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Theorem 4.1. Let

f ∈ C 0 (Rn ).

Then, for every

x ∈ Rn ,

lim (Sw f )(x) = f (x).

w→+∞ In particular, if

f ∈ C(R),

then

lim ∥Sw f − f ∥∞ = 0.

w→+∞

Proof. Here we highlight the main points of the proof.

f ∈ C 0 (Rn ) and x ∈ Rn be xed. By the continuity of f we have that for every xed ε > 0 there exists γ > 0 such that |f (x) − f (u)| < ε for every ∥x − u∥2 ≤ γ , u ∈ Rn . Then, by (χ2) we obtain: ∫ ∑ wn |(Sw f )(x) − f (x)| ≤ χ(wx − tk ) |f (u) − f (x)| du Ak Rkw n

Let

k∈Z

= (

∑

∑

+

wn ) χ(wx − tk ) Ak

∫ Rkw

|f (u) − f (x)| du

∥wx−tk ∥2 ≤wγ/2 ∥wx−tk ∥2 >wγ/2 = I1 + I2 .

w For u ∈ Rk and wx − tk ≤ wγ/2 we have ∥u − x∥2 ≤ γ for w > 0 suciently 2 large, then by the continuity of f we obtain I1 ≤ m0,Πn (χ)ε (see Remark 3.1 (a)). Moreover, by the boundedness of f and (2) we obtain I2 ≤ 2∥f ∥∞ ε for w > 0 suciently large, then the rst part of the theorem follows since ε > 0 is arbitrary. The second part of the theorem follows similarly replacing γ > 0 with the parameter of the uniform continuity of f . In order to obtain a modular convergence result, the following normconvergence theorem for the sampling Kantorovich operators (see [15]) can be formulated. Theorem 4.2. Let

φ be a convex φ-function.

For every

f ∈ Cc (Rn ) we have

lim ∥Sw f − f ∥φ = 0.

w→+∞

Now, we recall the following modular continuity property for

Sw ,

useful

to prove the modular convergence for the above operators in Orlicz spaces. Theorem 4.3. Let

φ

be a convex

holds

I φ [λSw f ] ≤ for some

λ > 0.

In particular,

δn

φ-function.

For every

f ∈ Lφ (Rn )

there

∥χ∥1 I φ [λm0,Πn (χ)f ], · m0,Πn (χ)

Sw

maps

Lφ (Rn )

in

Lφ (Rn ).

Now, the main result of this section follows (see [15]).

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Theorem 4.4. Let exists

λ>0

φ

be a convex

φ-function.

For every

f ∈ Lφ (Rn ),

there

such that

lim I φ [λ(Sw f − f )] = 0.

w→+∞

f ∈ Lφ (Rn ) and ε > 0 be xed. By Lemma 2.1, there exists λ > 0 and g ∈ Cc (Rn ) such that I φ [λ(f − g)] < ε. Let now λ > 0 such that 3λ(1 + m0,Πn (χ)) ≤ λ. By the properties of φ and Theorem 4.3, we have

Proof. Let

I φ [λ(Sw f − f )] ≤ I φ [3λ(Sw f − Sw g)] + I φ [3λ(Sw g − g)] + I φ [3λ(f − g)] 1 ≤ ∥χ∥1 I φ [λ(f − g)] + I φ [3λ(Sw g − g)] + I φ [λ(f − g)] m0,Πn (χ) · δ n ( ) 1 < ∥χ∥ + 1 ε + I φ [3λ(Sw g − g)]. 1 m0,Πn (χ) · δ n The assertion follows from Theorem 4.2. The setting of Orlicz spaces allows us to give a unitary approach for the reconstruction since we may obtain convergence results for particular cases of Orlicz spaces. For instance, choosing that

Lφ (Rn ) = Lp (Rn )

and

I φ [f ] =

φ(u) = up , 1 ≤ p < +∞, we have ∥ · ∥p is the usual Lp -norm.

∥f ∥pp , where

Then, from Theorem 4.3 and Theorem 4.4 we obtain the following corollary. Corollary 4.5. For every holds:

f ∈ Lp (Rn ), 1 ≤ p < +∞, the following inequality

∥Sw f ∥p ≤ δ −n/p [m0,Πn (χ)](p−1)/p ∥χ∥1

1/p

∥f ∥p .

Moreover, we have:

lim ∥Sw f − f ∥p = 0.

w→+∞

The corollary above, allows us to reconstruct

Lp -signals

(in

Lp -sense),

therefore signals/images not necessarily continuous. Other examples of Orlicz spaces for which the above theory can be applied can be found e.g., in [20, 19, 7, 3, 15]. The theory of sampling Kantorovich operators in the general setting of Orlicz spaces allows us to obtain, by means of a unied treatment, several applications in many dierent contexts.

5

Examples of special kernels

One important fact in our theory is the choice of the kernels, which inuence the order of approximation that can be achieved by our operators (see e.g. [17] in one-dimensional setting). For instance, one can take into consideration radial kernels, i.e., functions for which the value depends on the Euclidean norm of the argument only. Example of such a kernel can be given, for example, by the Bochner-Riesz

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−(n/2)+α

bα (x) := 2α Γ(α + 1)∥x∥2 B(n/2)+α (∥x∥2 ), for α > (n − 1)/2, Bλ is the Bessel function of order λ and Γ is

kernel, dened as follows

x ∈ Rn ,

where

the Euler function. For more details about this matter, see e.g. [10]. To construct, in general, kernels satisfying all the assumptions

1, 2, 3

(χi ), i =

is not very easy.

For this reason, here we show a procedure useful to construct examples using product of univariate kernels, see e.g.

[10, 15, 16].

In this case, we

tk = k . χi : R → R, χi ∈ L1 (R),

consider the case of uniform sampling scheme, i.e., Denote by

χ1 , ..., χn ,

the univariate functions

satisfying the following assumptions:

mβ,Π1 (χi ) := sup

∑

|χi (u − k)| |u − k|β < +∞,

(3)

u∈R k∈Z

for some

β > 0, χi

is bounded in a neighborhood of the origin and

∑

χi (u − k) = 1,

(4)

k∈Z

∏ u ∈ R, for i = 1, ..., n. Now, setting χ(x) := ni=1 χi (xi ), x = (x1 , ..., xn ) ∈ Rn , it is easy to prove that χ is a multivariate kernel for the operators Sw satisfying all the assumptions of our theory, see e.g., [10, 15]. for every

As a rst example, consider the univariate Fejér's kernel dened by

F (x) :=

1 2 2 sinc

(x)

2 ,

x ∈ R,

where the sinc function is given by

{ sin πx , sinc(x) := πx 1,

x ∈ R \ {0} , x = 0.

F is bounded, belongs to L1 (R) and satises the moment conditions (3) for β = 1, as shown in [11, 3, 15]. Furthermore, taking into account that the Fourier transform of F is given by (see [11]) { 1 − |v/π|, |v| ≤ π, b F (v) := 0, |v| > π, Clearly,

(4)

is fullled. Then, we can dene

F (xi ), x = (x1 , ..., xn ) ∈ Rn ,

the multivariate Fejér's kernel,

we obtain by Remark 3.2 that condition

Fn (x) =

n ∏ i=1

satisfying the condition upon a multivariate kernel. and the bivariate Fejér's kernel The Fejér's kernel

Fn

F (x) · F (y)

The Fejér's kernel

F

are plotted in Figure 1.

is an example of kernel with unbounded support,

then to evaluate our sampling Kantorovich series at any given

x ∈ Rn ,

n need of an innite number of mean values w Rkw

However, if

∫

the function

f

f (u) du.

has compact support, this problem does not arise.

we

In case

of function having unbounded support the innite sampling series must be

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Figure 1:

The Fejér's kernel F (left) and the bivariate Fejér's kernel F(x, y) (right).

Figure 2:

The B-spline M3 (left) and the bivariate B-spline M23 (x, y) (right).

truncated to a nite one, which leads to the so-called truncation error. In order to avoid the truncation error, one can take kernels support.

χ

with bounded

Remarkable examples of kernels with compact support, can be

constructed using the well-known central B-spline of order

∑ 1 Mk (x) := (−1)i (k − 1)! k

i=0

where the function

k ∈ N, dened by

( )( )k−1 k k +x−i , i 2 +

(x)+ := max {x, 0} denotes the positive part of x ∈ R M3 and the bivariate B-spline kernel

(see [3, 23, 15]). The central B-spline

M3 (x)·M3 (y) are plotted in Figure 2. We have that, the Fourier transform of ( ) ck (v) := sincn v , v ∈ R, and then, if we consider the case Mk is given by M 2π of the uniform spaced sampling scheme, condition (4) is satised by Remark

Mk are bounded on R, with compact support [−n/2, n/2], and Mk ∈ L1 (R), for all k ∈ N. Moreover, it easy to deduce that ∏n condition n (3) is fullled for every β > 0, see [3]. Hence Mk (x) := i=1 Mk (xi ), x = (x1 , ..., xn ) ∈ Rn , is the multivariate B-spline kernel of order k ∈ N. 3.2. Clearly, hence

Finally, other important examples of univariate kernels are given by the

(

)

x 2kπα , x ∈ R, with k ∈ N, α ≥ 1, where the normalization coecients ck are given by ck := ( u ) ]−1 [∫ 2k . Since Jk are band-limited to [−1/α, 1/α], i.e., their 2kπα du R sinc Jackson-type kernels, dened by

Jk (x) = ck sinc2k

Fourier transform vanishes outside this interval, condition Remark 3.2 and

(3)

is satised since

Jk (x) =

(χ2) is satised by x → ±∞, k ∈ N.

O(|x|−2k ), as

In similar manner, by the previous procedure we can construct a multivariate version of Jackson kernels. For more details about Jackson-type kernels, and for others useful examples of kernels see e.g. [11, 7, 2, 3].

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6

The sampling Kantorovich algorithm for image reconstruction

In this section, we show applications of the multivariate sampling Kantorovich operators to image reconstruction. First of all, we recall that every bi-dimensional gray scale image

A

can be modeled as a step function

Lp (R2 ), 1 ≤ p < +∞.

is represented by a suitable matrix and

I,

with compact support, belonging to

The denition of

I(x, y) :=

m ∑ m ∑

I

arise naturally as follows:

aij · 1ij (x, y), (x, y) ∈ R2 ,

i=1 j=1

1ij (x, y), i, j = 1, 2, ..., m, are the characteristics functions of the sets (i − 1, i] × (j − 1, j] (i.e. 1ij (x, y) = 1, for (x, y) ∈ (i − 1, i] × (j − 1, j] and 1ij (x, y) = 0 otherwise). The above function I(x, y) is dened in such a way that, to every pixel (i, j) it is associated the corresponding gray level aij . where

We can now consider approximation of the original image by the bivariate sampling Kantorovich operators

(Sw I)w>0

based upon some kernel

χ.

Then, in order to obtain a new image (matrix) that approximates in sense the original one, it is sucient to sample

Sw I

(for some

w > 0)

Lp -

with

a xed sampling rate. In particular, we can reconstruct the approximating images (matrices) taking into consideration dierent sampling rates and this is possible since we know the analytic expression of

Sw I .

If the sampling rate is chosen higher than the original sampling rate, one can get a new image that has a better resolution than the original one's. The above procedure has been implemented by using MATLAB and tools of matrix computation in order to obtain an optimized algorithm based on the multivariate sampling Kantorovich theory. In the next sections, examples of thermographic images reconstructed by the sampling Kantorovich operators will be given to show the main applications of the theory to civil engineering. In particular, a real world case-study is analyzed in term of modal analysis to obtain a model useful to study the response of the building under seismic action.

7

An application of thermography to civil engineering

In the present application, thermographic images will be used to locate the resisting elements and to dene their dimensions, and moreover to investigate the actual texture of the masonry wall, i.e., the arrangement of blocks (bricks and/or stones) and mortar joints.

In general, thermographic images have

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a resolution too low to accurately analyze the texture of the masonries, therefore it has to be increased by means of suitable tools. In order to obtain a consistent separation of the phases, that is a correct identication of the pixel which belong to the blocks and those who belong to mortar joints, the image is converted from gray-scale representation to blackand-white (binary) representation by means of an image texture algorithm, which employs techniques belonging to the eld of digital image processing. The image texture algorithm, described in details in [13], leads to areas of white pixels identied as blocks and areas of black pixels identied as mortar joints. However, the direct application of the image texture algorithm to the thermographic images (see Figure 3(a)), can produce errors, as an incorrect separation between the bricks and the mortar (see Figure 3(b)). Therefore, we can use the sampling Kantorovich operators to process the thermographic images. In particular, here we used the operators Jackson-type kernel with

k = 12

Sw based upon the bivariate w = 40

(see Section 5) and the parameter

(see Figure 3(c)). The application of the image texture algorithm produces a consistent separation of the phases (see Figure 3(d)).

(a)

(b)

(c)

(d)

(a) Original thermographic image (75 × 75 pixel resolution) and (b) its texture; (c) Reconstructed thermographic image (450 × 450 pixel resolution) and (d) its texture.

Figure 3:

In order to perform structural analysis, the mechanical characteristics of an homogeneous material equivalent to the original heterogeneous material are sought. The equivalence is in the sense that, when subjected to the same boundary conditions (b.c.), the overall responses in terms of mean values of stresses and deformations are the same.

The equivalent elastic properties

are estimated by means of the test-window method [14].

8

Case-study

The image texture algorithm described previously has been used to analyze a real-world case-study: a building consisting of two levels and an attic, with a very simple architectural plan, a rectangle with sides 11 m and 11.4 m (see Figure 4). The vertical structural elements consist of masonry walls. At rst level the masonry walls have thickness of 40 cm with blocks made of stones,

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while at second level the masonry is made of squared tu block and it has thickness of about 35 cm. Both surfaces of the walls are plastered. The slab can be assumed to be rigid in the horizontal plane and the building is placed in a medium risk seismic area. According to a preliminary visual survey, the structure does not have strong asymmetries between principal directions and the distribution of the masses are quite uniform.

Figure 4:

Case-study building: plant (left) and main facade (right).

The response of the building is evaluated by means of three dierent models.

In the rst model, only information that can be gathered by the

visual survey have been used, and the characteristics of material has been assume according to Italian Building Code. In the second model information concerning the actual geometry of vertical structural elements acquired by means of thermographic survey have been used; for example, the survey showed that the openings at ground level in the main facade were once larger and have been subsequently partially lled with non-structural masonry, and thus the dimensions of all the masonry walls in the facade have to be reduced.

Eventually, in the third model information about the actual

texture of masonries of ground level (chaotic masonry, Fig. 5(a)) and of second level (periodic masonry, Fig. 5(b)) have been used.

In particular,

the actual textures where established using the reconstructed thermographic images.

(a) Figure 5:

(b)

Texture of (a) ground level and (b) second level masonries.

The main characteristics of the models are reported in Table 1. For the mechanical characteristic the Young's modulus

E

and the shear modulus

12

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G

are shown. In the third model, the following mechanical characteristics

(Young's modulus

E

and Poisson's ratio

ν)

of the constituent phases have

E = 25000 N mm−2 , ν = 0.2, −2 , ν = 0.2, for the mortar of the second level E = 1700 N mm −2 E = 2500 N mm , ν = 0.2.

been used: for the stones of the ground level, for the bricks of both levels

geometry Model #1 Model #2 Model #3 Table 1:

visual survey thermogr. survey thermogr. survey

ground level E N mm−2

3346 3346 7050

second level

G N mm−2

1115 1115 2957

E N mm−2

1620 1620 1996

G N mm−2

540 540 833

Geometry and masonries' mechanical characteristics in the models.

The behavior under seismic actions is estimated by means of modal analysis, using a commercial code based on the Finite Element Method.

The

periods of the rst three modes and the corresponding mass participating ratio for the two principal direction of seismic action are reported in Tab. 2.

y

For all the models, the rst mode is along the the

x

axis, the second is along

axis, and the third is mainly torsional. Anyway, the second mode is

not a pure translational one since the asymmetry in the walls distribution produces also a torsional component.

Periods

Model #1 Model #2 Model #3

Mass partecipating ratio y direction 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd mode mode mode mode mode mode mode mode mode 0.22 0.15 0.13 0.00 0.64 0.13 0.73 0.00 0.00 0.22 0.14 0.13 0.00 0.51 0.27 0.73 0.01 0.00 0.18 0.13 0.11 0.00 0.46 0.26 0.68 0.01 0.00 x direction

Periods and mass participating ratio for two directions of seismic action of the rst three modes . Table 2:

As can be noted, the rst two models have the same period for the fundamental mode in

y

direction; nevertheless, in

x

direction there is a slight

dierence due to the reduced dimensions of masonry walls width discovered by means of thermographic survey. In fact, the reduction of dimensions leads to a decrease of global stiness while the total mass is almost the same. The periods of third model show a reduction of about 20% due to the greater value of equivalent elastic moduli estimated by means of homogenized texture. As already noted, for seismic action in

x direction the structure response is dom-

inated by second mode, with also a signicant contribution from the third mode (which is torsional); for seismic action in

y

the response is dominated

by rst mode. It is also worth noting that Model #3 shows a reduced mass participating ratio for the fundamental mode in each direction, and therefore

13

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a greater number of modes should be considered in the evaluation of seismic response in order to achieve a suitable accuracy.

9

Concluding remarks

The sampling Kantorovich operators and the corresponding MATLAB algorithm for image reconstruction are very useful to enhance the quality of thermographic images of portions of masonry walls. In particular, after the processing by sampling Kantorovich algorithm, the thermographic image has higher denition with respect to the original one and therefore, it was possible to estimate the mechanical characteristics of homogeneous materials equivalent to actual masonries, taking into account the texture (i.e., the arrangement of blocks and mortar joints).

These materials were used

to model the behavior of a case study under seismic action.

This model

has been compared with others constructed by well-know methods, using naked eyes survey and the mechanical parameters for materials taken from the Italian Building Code. Our method based on the processing of thermographic images by sampling Kantorovich operators enhances the quality of the model with respect to that based on visual survey only. In particular the proposed approach allows to overcome some diculties that arise when dealing with the vulnerability analysis of existing structures, which are: i) the knowledge of the actual geometry of the walls (in particular the identication of hidden doors and windows); ii) the identication of the actual texture of the masonry and the distribution of inclusions and mortar joints, and from this iii) the estimation of the elastic characteristics of the masonry. It is noteworthy that, for item i) the engineer has usually limited knowledge, due to the lack of documentation, while for items ii) and iii) he usually use tables proposed in technical manuals and standards which however give large bounds in order to encompass the generality of the real masonries. Instead, the use of reconstruction techniques on thermographic images coupled with homogenization permits to reduce these latter uncertainty on the estimation of the mechanical characteristics of the masonry.

References

[1] L. Angeloni and G. Vinti, Rate of approximation for nonlinear integral operators with applications to signal processing, Dierential and Integral Equations, 18 (8), 855-890, 2005. [2] C. Bardaro, P.L. Butzer, R.L. Stens and G. Vinti, Approximation of the Whittaker Sampling Series in terms of an Average Modulus of Smoothness covering Discontinuous Signals, J. Math. Anal. Appl., 316, 269-306, 2006.

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[3] C. Bardaro, P.L. Butzer, R.L. Stens and G. Vinti, Kantorovich-Type Generalized Sampling Series in the Setting of Orlicz Spaces, Sampling Theory in Signal and Image Processing, 6 (1), 29-52, 2007. [4] C. Bardaro, P.L. Butzer, R.L. Stens and G. Vinti, Prediction by samples from the past with error estimates covering discontinuous signals, IEEE, Transaction on Information Theory, 56 (1), 614-633, 2010. [5] C. Bardaro and I. Mantellini, Modular Approximation by Sequences of Nonlinear Integral Operators in Musielak-Orlicz Spaces, Atti Sem. Mat. Fis. Univ. Modena, special issue dedicated to Professor Calogero Vinti, 46, 403-425, 1998. [6] C. Bardaro and I. Mantellini, On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Optim.. 33 (4), 374-396, 2012. [7] C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, New York, Berlin, 9, 2003. [8] C. Bardaro and G. Vinti, Some Inclusion Theorems for Orlicz and MusielakOrlicz Type Spaces, Annali di Matematica Pura e Applicata, 168, 189-203, 1995. [9] C. Bardaro and G. Vinti, A general approach to the convergence theorems of generalized sampling series, Applicable Analysis, 64, 203-217, 1997. [10] P.L. Butzer, A. Fisher and R.L. Stens, Generalized sampling approximation of multivariate signals: theory and applications, Note di Matematica, 10 (1), 173-191, 1990. [11] P.L. Butzer and R.J. Nessel, Fourier Analysis and Approximation, I, Academic Press, New York-London, 1971. [12] P.L. Butzer and R.L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, (editor R.J. Marks II), Springer-Verlag, New York, 1993. [13] N. Cavalagli, F. Cluni and V. Gusella, Evaluation of a Statistically Equivalent Periodic Unit Cell for a quasi-periodic masonry, International Journal of Solids and Structures, 50, 4226-4240, 2013. [14] F. Cluni and V. Gusella, Homogenization of non-periodic masonry structures, International Journal of Solids and Structures, 41, 1911-1923, 2004. [15] D. Costarelli and G. Vinti, Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces, Bollettino U.M.I., Special volume dedicated to Prof. Giovanni Prodi, 9 (IV), 445-468, 2011. [16] D. Costarelli and G. Vinti, Approximation by Nonlinear Multivariate Sampling Kantorovich Type Operators and Applications to Image Processing, Numerical Functional Analysis and Optimization, 34 (8), 819-844, 2013.

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[17] D. Costarelli and G. Vinti, Order of approximation for sampling Kantorovich operators, in print in: Journal of Integral Equations and Applications, 2014. [18] R. Gonzales and R. Woods, Digital Image Processing, Prentice-Hall NJ - USA, 2002. [19] J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Lecture Notes in Math. 1034, 1983. [20] J. Musielak and W. Orlicz, On modular spaces, Studia Math., 28, 49-65, 1959. [21] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Pure and Appl. Math., Marcel Dekker Inc. New York-Basel-Hong Kong, 1991. [22] G. Vinti, A general approximation result for nonlinear integral operators and applications to signal processing, Applicable Analysis, 79, 217-238, 2001. [23] G. Vinti and L. Zampogni, Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, J. Approx. Theory, 161, 511-528, 2009. [24] G. Vinti and L. Zampogni, A Unifying Approach to Convergence of Linear Sampling Type Operators in Orlicz Spaces, Advances in Dierential Equations, 16 (5-6), 573-600, 2011. [25] G. Vinti, L. Zampogni, Approximation results for a general class of Kantorovich type operators, in print in Advanced Nonlinear Studies (2014).

Authors aliation:

Federico Cluni, Department of Civil and Environmental Engineering, University of Perugia, Via G. Duranti, 93, 06125, Perugia, Italy. Email: [email protected]. 2. Danilo Costarelli, Department of Mathematics and Physics, Section of Mathematics, University of Roma Tre, Largo San Leonardo Murialdo 1, 06146, Rome, Italy. Email: [email protected]. 3. Anna Maria Minotti and Gianluca Vinti, Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli 1, 06123, Perugia, Italy. Emails: [email protected] and [email protected]. 1.

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ON THE STOLARSKY-TYPE INEQUALITY FOR THE CHQOUET INTEGRAL AND THE GENERATED CHOQUET INTEGRAL JEONG GON LEE AND LEE-CHAE JANG

Division of Mathematics and Informational Statistics, and Nanoscale Science and Technology Institute, Wonkwang University, Iksan 570-749, Republic of Korea E-mail : [email protected], Phone:082-63-850-6189 General Education Institute, Konkuk University, Chungju 138-701, Korea E-mail : [email protected], Phone:082-43-840-3591

Abstract. Roman-Fores (2008) have studied Stolarsky-type inequality for the fuzzyintegral with respect to a fuzzy measure. Recently, Daraby(2012) proved the Stolarsky-type inequality for the pseudo-integral with respect to the Lebesgue measure and gave an open problem: ”Does Stolarsky’s inequality hold for the Choquet integral?” In this paper, we prove the Stolarsky-type inequality for the Choquet integral with respect to a fuzzy measure. Furthermore, we investigate the Stolarsky-type inequality for the generated Choquet integral with respect to a fuzzy measure.

1. Introduction Choquet [1], Couse et al. [2], Mihailovic and Pap [10], Murofushi et al. [11], Narukaw et al. [12], Pedrycz et al.[13], Rebille [14], Shieh et al. [17], Torra and Narukawa [20], and Tsai and Lu [21] have studied the Choquet integral with respect to a fuzzy measure, for examples, convergence theorems for the Choquet integral, some properties of the generated Choquet integral, and some applications of the Choquet integral criterion, etc. Furthermore, the authors in [15,16,19] have been studied various inequalities, for examples, Jensen-type inequality, Hardy-type inequality, and give an open problem: Does Stolarsky’s inequality hold for the Choquet integral? Jang et al. [5], Jang and Kwon [6], Jang [7-9], Schjaer-Jacobsen [18] have studied the Choquet integral of measurable interval-valued functions which are used for representing uncertain functions. Roman-Fores [4] also have studied Stolarsky-type inequality for the fuzzyintegral with respect to a fuzzy measure. Recently, Daraby [3] and Flores-Franulic proved the Stolarsky-type inequality for the pseudo-integral with respect to a σ-⊕-decomposable measure. In this paper, we prove a Stolarsky-type inequality for the Choquet integral with respect to a fuzzy measure and for the generated Choquet integral with respect to a fuzzy measure. 1991 Mathematics Subject Classification. 28E10, 28E20, 03E72, 26E50 11B68. Key words and phrases. fuzzy measure, Choquet integral, generated Choquet integral, Stolarsky-type inequality, Lebesgue inetgral, measurable function. 1

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2

The paper is organized in five sections. In section 2, we list definitions and some properties of the Choquet integral with respect to a fuzzy measure and introduce the two types fuzzy Stolarsky’s inequality for the fuzzy integral with respect to a continuous fuzzy measure. In section 3, we prove the Stolarsky-type inequality for the Choquet integral with respect to a fuzzy measure which is the solution of an open problem : Does Stolarsky’s inequality hold for the Choquet integral? In section 4, we prove the Stolarsky-type inequality for the generated Choquet integral with respect to a fuzzy measure which is the generalized solution of an open problem : Does Stolarsky’s inequality hold for the Choquet integral? In section 5, we give a brief summary results and some conclusions. 2. Definitions and Preliminaries In this section, we introduce a fuzzy measure, the fuzzy integral with respect to a fuzzy measure, and the Choquet integral with respect to a fuzzy measure, the generated Choquet integral defined by Mihailovic and Pap [10], and the Stolarsky-type inequality for the fuzzy integral with respect to a continuous fuzzy measure of a measurable nonnegative function. Let ([0, 1], A) be a measurable space and F([0, 1]) be the set of all measurable functions from [0, 1] to [0, 1]. Definition 2.1. ([11-13]) (1) A fuzzy measure µ : A −→ [0, ∞] on a measurable space (X, A) is a real-valued set function satisfying (i) (ii)

µ(∅) = 0 µ(A) ≤ µ(B) whenever A, B ∈ A and A ⊂ B.

(2) A fuzzy measure µ is said to be finite if µ(X) < ∞. (3) A fuzzy measure µ is said to be continuous from below if for any sequence {An } ⊂ A and A ∈ A, such that An ↑ A, then

lim µ(An ) = µ(A).

(1)

n→∞

(4) A fuzzy measure µ is said to be continuous from above if for any sequence {An } ⊂ A and A ∈ A such that µ(A1 ) < ∞ and An ↓ A, then

lim µ(An ) = µ(A).

(2)

n→∞

(5) A fuzzy measure µ is said to be continuous if it is continuous from below and continuous from above.

Definition 2.2. ([4]) (1) Let µ be a fuzzy measure on ([0, 1], A), f ∈ F([0, 1]), and A ∈ A. The fuzzy integral with respect to a fuzzy measure µ of f on A is defined by Z (F ) f dµ = supα≥0 min{α ∧ µf,A (α)}, (3) A

where µf,A (α) = µ (A ∩ {x ∈ X|f (x) ≥ α})

(4)

for all α ∈ [0, ∞). R (2) A measurable function f is said to be integrable if (F ) A f dµ is finite.

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Definition 2.3. ([13]) (1) Let µ be a fuzzy measure on ([0, 1], A), f ∈ F([0, 1]), and A ∈ A. The Choquet integral with respect to a fuzzy measure µ of f is defined by Z Z ∞ (C) f dµ = µf (α)dα (5) A

0

where µf,A (α) = µ (A ∩ {x ∈ X|f (x) ≥ α})

(6)

for all α ∈ [0, ∞) and the integral on the right-hand side is the R Lebesgue integral of µf,A . (2) A measurable function f is said to be integrable if (C) A f dµ is finite. Flores-Franulie et al. [4] proved the two types fuzzy Stolarsky’s inequality for the fuzzy integral with respect to a continuous fuzzy measure. Theorem 2.1. ([4]) (Fuzzy Stolarsky’s inequality: decreasing case) Let a, b > 0. If f ∈ F([0, 1]) is a continuous and strictly decreasing function and µ is the Lebesgue measure on A, then the inequality !Ã ! Ã Z Z Z ³ 1´ ³ 1 ´ ³ 1´ a+b a b (7) dµ ≥ (F ) f x dµ (F ) f x (F ) f x dµ [0,1]

[0,1]

[0,1]

holds.

Theorem 2.2. ([4]) (Fuzzy Stolarsky’s inequality: increasing case) Let a, b > 0. If f ∈ F([0, 1]) is a continuous and strictly increasing function and µ is the Lebesgue measure on A, then the inequality à !à ! Z Z Z ³ 1´ ³ 1 ´ ³ 1´ f x a dµ (8) (F ) f x a+b dµ ≥ (F ) (F ) f x b dµ [0,1]

[0,1]

[0,1]

holds.

3. Stolarsky-type inequality for the Choqeut integral In this section, we consider an open problem: Does Stolarsky’s inequality hold for the Choquet integral? in Daraby [3] and prove a Stolarsky-type inequality for the Choquet integral with respect to a fuzzy measure. Theorem 3.1. Let µ : A −→ [0, ∞) be a continuous fuzzy measure. If f ∈ F([0, 1]) is a continuous function, then we have Z (C) f dµ ∈ [0, 1]. (9) [0,1]

Proof. Since µ is continuous and f is continuous, µf,[0,1] is increasing continuous on [0, 1] and hence we have 0 ≤ µf,[0,1] (α) ≤ 1 for all α ∈ [0, 1]. Thus, we have Z Z 1 (C) f dµ = µf,[0,1] (α)dα ≤ 1. (10) [0,1]

0

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4

That is,

Z (C)

f dµ ∈ [0, 1].

(11)

[0,1]

Theorem 3.2. Let µ : A −→ [0, ∞) be a continuous fuzzy measure and a, b, a + b ∈ (0, ∞). If f ∈ F([0, 1]) is a continuous and increasing function, then the inequality Z Z Z ³ 1´ ³ 1´ ³ 1 ´ f x b dµ (12) f x a dµ(C) (C) f x a+b dµ ≥ (C) [0,1]

[0,1]

[0,1]

holds. 1

1

1

1

Proof. Since a, b, a + b ∈ (0, ∞), x a+b ³> x a´and x³a+b ´> x b for ³ all1 x ´ ∈ [0,³1].1 ´Since f 1 1 is a continuous and increasing function, f x a+b ≥ f x a and f x a+b ≥ f x b for all x ∈ [0, 1]. Hence we have ³ ³ 1´ ´ ³ ³ 1 ´ ´ (13) µ {x|f x a+b } ≥ µ {x|f x a } and

³ ³ 1´ ´ ³ ³ 1 ´ ´ µ {x|f x a+b } ≥ µ {x|f x b } .

By (13) and (14), we have Z (C)

Z ³ 1 ´ f x a+b dµ ≥ (C) Z ³ 1 ´ f x a+b dµ ≥ (C)

Z (C)

³ 1´ f x a dµ

(15)

³ 1´ f x b dµ.

(16)

[0,1]

[0,1]

and

(14)

[0,1]

[0,1]

By Theorem 3.1, (15), and (16), we have à !2 Z Z ³ 1 ´ ³ 1 ´ (C) f x a+b dµ (C) f x a+b dµ ≥ [0,1] Z Z [0,1] ³ ´ 1 a ≥ (C) f x dµ(C)

³ 1´ f x b dµ.

(17)

[0,1]

[0,1]

Let F([0, ∞)) be the set of all measurable function from [0, ∞) to [0, ∞). Theorem 3.3. Let µ : A −→ [0, ∞) be a continuous fuzzy measure and a > 0, b > 0. If f ∈ F([0, ∞)) is a continuous function, and it is increasing on [0, 1] and decreasing on (1, ∞), then the inequality Z Z Z ³ 1´ ³ 1´ ³ 1 ´ f x a dµ(C) f x b dµ (18) (C) f x a+b dµ ≥ (C) [0,1]

[0,1]

[0,1]

holds. Proof. If a > 0, b > 0, then we have 1

1

1

1

x a+b > x a and x a+b > x b for all x ∈ [0, 1]

(19)

and 1

1

1

1

x a+b < x a and x a+b < x b for all x ∈ (1, ∞).

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Since f is increasing on [0, 1] and decreasing on (1, ∞), by (19) and (20), we have ³ 1´ ³ 1´ ³ 1 ´ ³ 1 ´ f x a+b ≥ f x a and f x a+b ≥ f x b for all x ∈ [0, ∞). By (21), we have

and

³ ³ 1´ ´ ³ ³ 1 ´ ´ µ {x|f x a+b } ≥ µ {x|f x a }

(22)

³ ³ 1´ ´ ³ ³ 1 ´ ´ µ {x|f x a+b } ≥ µ {x|f x b } .

(23)

By (22) and (23), we have Z (C)

Z ³ 1 ´ f x a+b dµ ≥ (C)

Z

³

(C)

³ 1´ f x a dµ

(24)

³ 1´ f x b dµ.

(25)

[0,∞)

[0,∞)

and

(21)

f x

1 a+b

Z

´ dµ ≥ (C)

[0,∞)

[0,∞)

By Theorem 3.1, (24), and (25), we have à !2 Z Z ³ 1 ´ ³ 1 ´ (C) f x a+b dµ (C) f x a+b dµ ≥ [0,∞) [0,∞) Z Z ³ 1´ ≥ (C) f x a dµ(C)

³ 1´ f x b dµ.

(26)

[0,∞)

[0,∞)

4. Stolarsky-type inequality for the generated Choquet integral In this section, we consider the generated Choquet integral with respect to a fuzzy measure and investigate Stolarsky-type inequality for the generated Choquet integral. Definition 4.1. (9] Let µ be a fuzzy measure and g : [0, ∞] −→ [0, ∞], g(0) = 0 be an odd, strictly increasing, continuous function. (1) The generated Choquet integral with respect to µ of f ∈ F([0, ∞)) is defined by à ! Z Z (GC) f dµ = g −1 (C) g ◦ f dg ◦ µ (27) [0,∞]

[0,∞)

where g ◦ f is the composition of g and f . R (2) A measurable function f is said to be integrable if (GC) [0,∞) f dµ ∈ [0, ∞).

Theorem 4.1. Let µ : A −→ [0, ∞) be a continuous fuzzy measure, g : [0, ∞] −→ [0, ∞], g(0) = 0 be an odd, strictly increasing, continuous function, and a > 0, b > 0. If f ∈ F([0, ∞)) R is integrable function with (GC) [0,∞) f dµ ∈ [0, 1], and it is increasing on [0, 1] and decreasing ³ 1 ´ R on (1, ∞), and if (GC) [0,∞) f x a+b dµ ∈ (0, 1], then the inequality Z Z Z ³ 1´ ³ 1´ ³ 1 ´ f x a dµ(GC) f x b dµ (28) (GC) f x a+b dµ ≥ (GC) [0,1]

[0,1]

[0,1]

holds.

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Proof. Since a > 0, b > 0 and f is increasing on [0, 1] and decreasing on (1, ∞), by (19) and (20), we have ³ 1 ´ ³ 1´ ³ 1´ ³ 1 ´ f x a+b ≥ f x a and f x a+b ≥ f x b for all x ∈ [0, ∞). (29) Since g is strictly increasing, by (29), we have ³ 1´ ³ 1´ ³ 1 ´ ³ 1 ´ g ◦ f x a+b ≥ g ◦ f x a and g ◦ f x a+b ≥ g ◦ f x b for all x ∈ [0, ∞).

(30)

Note that if µ is a continuous fuzzy measure and g : [0, ∞] −→ [0, ∞], g(0) = 0 be an increasing continuous function, then g ◦ µ is a continuous fuzzy measure. Thus, by (30), we have ³ ³ 1´ ´ ³ ³ 1 ´ ´ (31) g ◦ µ {x|f x a+b } ≥ g ◦ µ {x|f x a } and

³ ³ 1´ ´ ³ ³ 1 ´ ´ g ◦ µ {x|f x a+b } ≥ g ◦ µ {x|f x b } .

By (31) and (32), we have Z Z ³ 1 ´ a+b dg ◦ µ ≥ (C) (C) g◦f x Z

³

(C)

³ 1´ g ◦ f x a dg ◦ µ

(33)

³ 1´ g ◦ f x b dg ◦ µ.

(34)

[0,∞)

[0,∞)

and

(32)

g◦f x

1 a+b

Z

´ dg ◦ µ ≥ (C)

[0,∞)

[0,∞)

We note that if g is an odd, strictly increasing continuous function, then its inverse g −1 is an odd, strictly increasing continuous function. Thus, by (34), we have à ! Z Z ³ 1 ´ ³ 1 ´ −1 (C) g ◦ f x a+b dg ◦ µ (GC) f x a+b dµ = g [0,∞) [0,∞) ! à Z ³ 1´ −1 a ≥ g (C) g ◦ f x dg ◦ µ [0,∞) Z ³ 1´ (35) = (GC) f x a dµ [0,∞)

and

Z (GC)

³ f x

1 a+b

Ã

´ dµ =

g

−1

[0,∞)

Z (C)

à ≥ =

g

³

−1

g◦f x Z

(C) Z

1 a+b

[0,∞)

!

´

dg ◦ µ !

³ 1´ g ◦ f x b dg ◦ µ [0,∞)

(GC)

³ 1´ f x b dµ

(36)

[0,∞)

By Theorem 3.1, (35), and (36), we have à !2 Z Z ³ 1 ´ ³ 1 ´ a+b a+b dµ ≥ (GC) f x dµ (GC) f x [0,∞) Z Z [0,∞) ³ ´ 1 a ≥ (GC) f x dµ(GC) [0,∞)

623

³ 1´ f x b dµ.

(37)

[0,∞)

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7

5. Conclusions This study was to solve an open problem: Does Stolarsky’s inequality for the Choquet integral? Thus, we solved Stolarsky-type inequality for the Choquet integral with respect to a continuous fuzzy integral under some sufficient conditions (see Theorems 3.2 and 3.3). We also proved Stolarsky-type inequality for the generated Choquet integral with respect to a continuous fuzzy integral under some sufficient conditions (see Theorem 4.1). Furthermore, we give another open problem: Does Stolarsky’s inequality for the intervalvalued generated Choquet integral? Acknowledgement: This paper was supported by Konkuk University in 2014. References [1] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131-295. [2] I. Couso, S. Montes, P. Gil, Stochastic convergence, uniform integrability and convergence in mean on fuzzy measure spaces, Fuzzy Sets and Systems 129 (2002) 95-104. [3] B. Daraby, Generalization of the Stolarsky type inequality for pseudo-integrals, Fuzzy Sets and Systems 194 (2012) 90-96. [4] A. Flores-Franulic, H. Roman-Flores, Y. Chalco-Cano, A note on fuzzy integral inequality of Stolarsky type, Applied Mathematics and Computation 196 (2008) 55-59. [5] 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. [6] 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. [7] L.C. Jang, On properties of the Choquet integral of interval-valued functions, Journal of Applied Mathematics 2011 (2011) Article ID 492149, 10pages. [8] L.C. Jang, A note on convergence properties of interval-valued capacity functionals and Choquet integrals, Information Sciences, 183 (2012) 151-158. [9] L.C. Jang, A note on the interval-valued generalized fuzzy integral by means of an interval-representable pseudo-multiplication and their convergence properties, Fuzzy Sets and Systems, 222 (2013) 45-57. [10] B. Mihailovic, E. pap, Asymmetric integral as a limit of generated Choquet integrals based on absolutely monotone real set functions, Fuzzy Sets and Systems 181 (2011) 39-49. [11] T. Murofushi, M. Sugeno, M. Suzaki, Autocontinuity, convergence in measure, and convergence in distribution, Fuzzy Sets and Systems 92(2) (1997) 197-203. [12] Y. Narukawa, M. Murofushi, M. Sugeno, Regular fuzzy measure and representation of comonotonically additive functionsl, Fuzzy Sets and Systems 112 (2000) 177-186. [13] W. Pedrycz, L. Yang, M. Ha, On the fundamental convergence in the (C) mean in problems of information fusion, J. Math. Anal. Appl. 358 (2009) 203-222. [14] Y. Rebille, Decision making over necessity measures through the Choquet criterion, Fuzzy Sets and Systems 157 (2006) 3025-3039. [15] H. Roman-Flores, A. Flores-Franulic, Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Information Sciences 177 (2007) 3192-3201. [16] H. Roman-Flores, A. Flores-Franulic, Y. Chalco-Cano, A Hardy type inequality for fuzzy integrals, Applied Mathematics and Computation 204 (2008) 178-183. [17] J. Shieh, H.H. Wu, H.C. Liu, Applying a complexity-based Choquet integral to evaluate students’ performance, Expert Systems with Applications 36 (2009) 5100-5106. [18] H. Schjaer-Jaconsen, Representation and calculation of economic uncertainties: Intervals, fuzzy numbers, and probabilities, Int. J. Production Economics 78(2002) 91-98. [19] K.B. Stolarsky, From Wythoff ’s Nim to Chebyshev’s inequality , American Mathematical Monthly 98(1991) 669-900. [20] V. Torra, Y. Narukawa, On a comparison between Mahalanobis distance and Choquet integral : The Choquet-Mahalanobis operator, Information Sciences 190(2012) 56-63. [21] H.H. Tsai, I.Y. Lu, The evaluation of service quality using generalized Choquet integral, Information Sciences 176(2006) 640-660.

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Extended Ces´aro operator from weighted Bergman space to Zygmund-type space Yu-Xia Liang School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, P.R. China, [email protected] Ren-Yu Chen ∗ Department of Mathematics, Tianjin University, Tianjin 300072, P.R. China, [email protected] 1

Abstract We discuss the boundedness and compactness of extended ces´ aro operator from weighted Bergman space to Zygmund-type space on the unit ball of C n .

1

Introduction

Let H(Bn ) be the class of all holomorphic functions on Bn , where Bn is the unit ball in the n-dimensional complex space C n . Let dv denote the Lebesegue measure on Bn normalized so that v(Bn ) = 1. For f ∈ H(Bn ), let ρ2m−1 L, (m = 1, 2, ..., m0 , m0 + 1), f0ρ2m < ρ2m L, (m = 1, 2, ..., m0 ). 2m−1 Then the BVP (1.1) has at least 2m0 symmetric positive solutions in K. 0 (C10) There exists {ρi }2m i=1 ⊂ (0, ∞) with ρ1 < ρ2 < Γρ3 < ρ3 < ... < Γρ2m0 −1 < ρ2m0 −1 < ρ2m0 such that

ρ

fΓρ2m−1 > ρ2m−1 L, f0ρ2m < ρ2m L, (m = 1, 2, ..., m0 ). 2m−1 Then the BVP (1.1) has at least 2m0 − 1 symmetric positive solutions in K.

4

An Example

    3 1 1 1 Example 4.1 In BVP (1.1), suppose that T = 0, ∪ , 1 , p(t) = 1, m = 3, α = β = 1, α1 = , ξ1 = and 4 4 2 5 4 η1 = , i.e., 5  4∇ u (t) + f (t, u) = 0, t ∈ (0, 1)T ,           1 1  u(0) − u[4] (0) = u , 2 5         1 4  [4]  ,  u(1) + u (1) = u 2 5

(4.1)

7 676

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where  1   u(t) + t(1 − t),   3       f (t, u) = 5u(t) − 7 + t(1 − t),           − 3 u(t)2 + 6u(t) − 6 + t(1 − t), 4

  3 (t, u) ∈ [0, 1]T × 0, , 2  (t, u) ∈ [0, 1]T ×

(t, u) ∈ [0, 1]T × [2, ∞) .

By simple calculation, we get D = 3, θ(t) = 1 + t, ϕ(t) = 2 − t, ∆ = G(t, s) =

1 3



 3 ,2 , 2

1 ˜ 103 72 9 , Γ= , B = , L= and 20 2 144 103

(1 + t)(2 − s), 0 ≤ t ≤ s ≤ 1, (1 + s)(2 − t), 0 ≤ s ≤ t ≤ 1.

It is clear that conditions (C1) − (C4) are satisfied. Taking ρ1 = ρ1 < Γρ2

and

3 , ρ2 = 4, ρ3 = 9, we can obtain that 2

ρ2 < ρ 3 .

Now, we show that (C5) is satisfied: 3

f02

≤

0.75 < ρ1 L = 1.048,

f24

≥

3 > ρ2 L = 2.796,

f09

≤

6.25 < ρ3 L = 6.291.

Then, (C5) condition of Theorem 3.1 hold. Hence, we get the BVP (4.1) has at least two symmetric positive solutions.

References [1] M. Bohner and A. Peterson, Dynamic Equations on Time Scales:An Introduction with Applications, Birkh¨ auser, Boston, 2001. [2] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨ auser, Boston, 2003. [3] F. Hao, Existence of symmetric positive solutions for m-point boundary value problems for second-order dynamic equations on time scales, Math. Theory Appl. (Changsha) 28 (2008) 65-68. [4] S. Hilger, Ein Masskettenkalk¨ ul mit Anwendug auf zentrumsmanningfaltigkeiten, Ph.D. Thesis, Universitat W¨ urzburg, 1988. [5] I. Y. Karaca, Positive solutions for boundary value problems of second-order functional dynamic equations on time scales, Adv. Difference Equ. Art. ID 829735 (2009). [6] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer, Dordrecht, 1996. [7] K. Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc. 63 (2001) 690-704. [8] R. Ma, Multiple positive solutions for nonlinear m-point boundary value problems, Appl. Math. Comput. 148 (2004) 249-262. [9] Y. Sun and X. Zhang, Existence of symmetric positive solutions for an m-point boundary value problem, Bound. Value Probl. Art. ID 79090 (2007) 14 pp. [10] S. P. Wang, S. W. Lin and C. J. Chyan, Nonlinear two-point boundary value problems on time scales, Math. Comput. Modelling 53 (2011) 985-990. [11] L. Xuan, Existence and multiplicity of symmetric positive solutions for singular second-order m-point boundary value problem, Int. J. Math. Anal. (Ruse) 5 (2011) 1453-1457. [12] I. Yaslan, Multiple positive solutions for nonlinear three-point boundary value problems on time scales, Comput. Math. Appl. 55 (2008) 1861-1869. [13] Q. Yao, Existence and iteration of n symmetric positive solutions for a singular two-point boundary value problem, Comput. Math. Appl. 47 (2004) 1195-1200. [14] C. Yuan and Y. Liu, Multiple positive solutions of a second-order nonlinear semipositone m-point boundary value problem on time scales, Abstr. Appl. Anal. Art. ID 261741 (2010) 19 pp. [15] J. Zhao, F. Geng, J. Zhao and W. Ge, Positive solutions to a new kind Sturm-Liouville-like four-point boundary value problem, Appl. Math. Comput. 217 (2010) 811-819.

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Approximate fuzzy double derivations and fuzzy Lie ∗-double derivations Ali Ebadian1 , Mohammad Ali Abolfathi2 , Choonkil Park3 and Dong Yun Shin4∗ 1,2

Department of Mathematics, Urmia University, P.O. Box 165, Urmia, Iran; 3 Research Institute for Natural

Sciences, Hanyang University, Seoul 133-791, Korea; 4 Department of Mathematics, University of Seoul, Seoul 130-743, Korea. Abstract. Using the fixed point method, we prove the Hyers-Ulam stability of fuzzy double derivations and fuzzy Lie ∗-double derivations on fuzzy Banach algebras, fuzzy C ∗ -algebras and fuzzy Lie C ∗ -algebras associated with the following additive functional equation of n-Apollonius type n X

f (z − xi ) = −

i=1

1 n

X

f (xi + xj ) + nf (z −

1≤i 0 if and only if x = 0; t ) for each c 6= 0; (N 3) N (cx, t) = P (x, |c| (N 4) N (x + y, s + t) ≥ min{N (x, t), N (y, s)}; (N 5) N (x, ·) is a non-decreasing function on 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 linear space. One may regard N (x, t) as the truth value of the statement “the norm of x is less than or equal to the real number t”. Example 1.2. Let (X, k · k) be a normed linear space and α, β > 0. Then ( αt t > 0, x ∈ X, βt+kxk , N (x, t) = 0, t ≤ 0, x ∈ X is a fuzzy norm on X. Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent if there exists x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t > 0. In that 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 fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy normed 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 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 is said to be continuous on X [5]. 679

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Approximate fuzzy double derivations and fuzzy Lie ∗-double derivations

Let X is a Banach algebra. Then an involution on x is a mapping x → x∗ from X into X which satisfies (1) (x∗ )∗ = x for x ∈ X; (2) (αx + βy)∗ = αx∗ + βy ∗ for x, y ∈ X and α, β ∈ C; (3) (xy)∗ = y ∗ x∗ for x, y ∈ X. If, in addition, kx∗ xk = kxk2 for all x ∈ X, then X is a C ∗ -algebra. Definition 1.3. Let X be a ∗-algebra and (X, N ) be a fuzzy normed space. (1) The fuzzy normed space (X, N ) is called a fuzzy normed ∗-algebra if N (xy, ts) ≥ N (x, t).N (y, s),

N (x∗ , t) = N (x, t)

for all x, y ∈ X and all positive real numbers 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 ( t t > 0, x ∈ X, t+kxk , N (x, t) = 0, t ≤ 0, x ∈ X. Then (X, N ) is a fuzzy normed ∗-algebra. Definition 1.5. Let (X, k · k) be a normed ∗-algebra or aC ∗ -algebra and N be a fuzzy norm on X. (1) The fuzzy normed ∗-algebra (X, N ) is called an induced fuzzy normed ∗-algebra. (2) The fuzzy C ∗ -algebra (X, N ) is called an induced fuzzy C ∗ -algebra. Let (X, N ) and (Y, N ) be induced fuzzy normed ∗-algebras. Then a C-linear mapping f : (X, N ) → (Y, N ) is called a fuzzy ∗-homomorphism if f (xy) = f (x)f (y), f (x∗ ) = f (x)∗ and a C-linear mapping f : (X, N ) → (X, N ) is called a fuzzy ∗-derivation if f (xy) = f (x)y + xf (y), f (x∗ ) = f (x)∗ for all x, y ∈ X. Definition 1.6. Let (X, N ) be a fuzzy normed ∗-algebra and g, h : (X, N ) → (X, N ) be Clinear mappings. A C-linear mapping f : (X, N ) → (X, N ) is called a fuzzy *-(g,h)-double derivation if f (xy) = f (x)y + xf (y) + g(x)h(y) + h(x)g(y), f (x∗ ) = f (x)∗ , g(x∗ ) = g(x)∗ , h(x∗ ) = h(x)∗ , for all x, y ∈ X. Example 1.7. Let H : (X, N ) → (X, N ) be a fuzzy ∗-homomorphism. Then H is a fuzzy ∗-( H2 − I)-double derivation where I : (X, N ) → (X, N ) is the identity mapping. Let X be a nonempty 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 for x, y ∈ X; (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 the a fundamental result in fixed point theory. Theorem 1.8. ([25, 39]) Let (X, d) be a complete generalized metric space and J : X → X be a strictly contractive mapping with Lipshitz constant L < 1. Then, for each given x ∈ X, either d(J n x, J n+1 x) = ∞ for all n ≥ 0, or there exists a natural number 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 , y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y. 680

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A. Ebadian, M. A. Abolfathi, C. Park, D. Y. Shin

In 1996, Isac and Rassias [18] 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 [8, 35]). Ebadian and Ghobadipour [9] proved the Hyers-Ulam stability of double derivations on Banach algebras and Lie ∗-double derivations on Lie C ∗ -algebras. Ebadian et al. [10] considered the following additive functional equation of n-Apollonius type n n X 1 X 1 X f (xi + xj ) + nf (z − 2 xi ) (1.2) f (z − xi ) = − n n i=1

i=1

1≤i 0. Since lim

t

m→∞

t+

1 m m λm ϕ(λ x, λ y, 0, 0, 0, 0, 0, 0)

= 1,

f1 (µx + µy) = µf1 (x) + µf1 (y) for all x, y ∈ X and all µ ∈ C. Similarly, we have f2 (µu + µv) = µf2 (u) + µf2 (v) and f3 (µt + µs) = µf3 (t) + µf3 (s) for all x, y ∈ X and all µ ∈ C. Now letting x = y = u = v = t = s = 0 and µ = 1 in (2.2), we have N (zw) − f1 (z)w − zf1 (w) − f2 (z)f3 (w) − f3 (z)f2 (w), t) = N (λm zλm w) − f1 (λm z)λm w − λm zf1 (λm w) − f2 (λm z)f3 (λm w) − f3 (λm z)f2 (λm w), λ2m t) ≥

λ2m t

λ2m t + ϕ(0, 0, λm z, λm w, 0, 0, 0, 0)

for all z, w ∈ X and all t > 0. Since lim

m→∞

t+

t = 1, m z, λm w, 0, 0, 0, 0, ) ϕ(0, 0, λ λ2m 1

f1 (zw) = f1 (z)w + zf1 (w) + f2 (z)f3 (w) + f3 (z)f2 (w) for all x, y ∈ X, that is, f1 is a fuzzy (f2 , f3 )-double derivation on X.



Theorem 2.3. Let ϕ : X n+1 → [0, ∞) be a function such that there exists an L < 1 with  2  n − 1 n2 − 1 n2 − 1 n2 − 1 ϕ z, x , · · · , x ≤ Lϕ(z, x1 , · · · , xn ) (2.3) 1 n n2 n2 n2 n2 for all z, x1 , · · · , xn ∈ X. Let k ∈ J and fk : X → X be mappings satisfying fk (0) = 0 and N (Dµ fk (z, x1 , · · · , xn ), t) ≥ N (∆f1 ,f2 ,f3 (x, y), t) ≥

t , t + ϕ(z, x1 , · · · , xn )

(2.4)

t t + ϕ(x, y, 0, · · · , 0)

(2.5) 2

2

m for all x, y, z, x1 , · · · , xn ∈ X, all µ ∈ T1 and all t > 0. Then dk (x) = N −limm→∞ ( n2n−1 )m fk (( nn−1 2 ) x) exist for each x ∈ X, and define unique C-linear mappings dk : X → X such that

N (fk (x) − dk (x), t) ≥

(n2 − 1)(1 − L)t (n2 − 1)(1 − L)t + nϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

for all x ∈ X and all t > 0. Moreover, d1 is a fuzzy (d2 , d3 )-double derivation on X. 682

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Proof. Let k ∈ J. Consider the set Ω := {u : X → X, u(0) = 0} and introduce the generalized metric t d(u, v) = inf{η ∈ R+ : N (u(x) − v(x), ηt) ≥ , t + ϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

where inf ∅ = +∞. The proof of the fact that (Ω, d) is a complete generalized metric space can be found in [8]. Now we consider the mapping J : Ω → Ω defined by  2  n2 n −1 Ju(x) := 2 u x n −1 n2 for all u ∈ Ω and x ∈ X. Let ε > 0 and u, v ∈ Ω be given such that d(u, v) ≤ ε. Then N (u(x) − v(x), εt) ≥

t t + ϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

for all x ∈ X and all t > 0. Hence n2 n2 − 1 n2 n2 − 1 u( x) − v( x), Lεt) n2 − 1 n2 n2 − 1 n2 n2 − 1 n2 − 1 n2 − 1 x) − v( x), Lεt) = N (u( n2 n2 n2 n2 −1 Lt n2 ≥ n2 − 1 n2 −1 n2 −1 x, 0, · · · , 0) Lt + ϕ( x, 0, · · · , 0, 2 2 n n 2 | n{z }

N (Ju(x) − Jv(x), Lεt) = N (

jth

≥

n2 −1 Lt n2 n2 −1 Lt n2

+

n2 −1 Lϕ(x, 0, · · · n2

, 0, |{z} x , 0, · · · , 0) jth

=

t , t + ϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

and so d(u, v) ≤ ε implies that d(Jg, Jh) ≤ Lε for all u, v ∈ Ω. Letting µ = 1 and z = xj = x for each 1 ≤ p ≤ n with p 6= j, xp = 0 in (2.4) , we have  2  2   n −1 n −1 t N fk (x) − nfk x ,t ≥ (2.6) 2 n n t + ϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

for all x ∈ X and all t > 0. It follows from (2.6) that d(fk , Jfk ) ≤ there exists a mapping dk : X → X such that the following holds: (1) dk is a fixed point of J, that is, dk (

n2 − 1 n2 − 1 x) = dk (x) n2 n2

n . n2 −1

By Theorem 1.8,

(2.7)

for all x ∈ X. The mapping dk is a unique fixed point of J in the set Λ = {v ∈ Ω : d(u, v) < ∞}. This implies that dk is a unique mapping satisfying (2.7) such that there exists η ∈ (0, ∞) 683

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Approximate fuzzy double derivations and fuzzy Lie ∗-double derivations t t+ϕ(x,0,··· ,0,

satisfying N (fk (x) − dk (x), ηt) ≥

x ,0,··· ,0) for all x ∈ X and all t > 0. |{z} jth

(2) d(J m fk , dk ) → 0 as m → ∞. This implies the equality N − lim ( m→∞

n2 m n2 − 1 m ) f (( ) x) = dk (x) k n2 − 1 n2

exists for each x ∈ X. 1 (3) d(fk , dk ) ≤ 1−L d(fk , Jfk ), which implies inequality d(fk , dk ) ≤

1 2 n2 −1 − n n−1 L n

and so

(n2 − 1)(1 − L)t . (n2 − 1)(1 − L)t + nϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0)

N (fk (x) − dk (x), t) ≥

jth

It follows from (2.3) and (2.4) that N (Dµ dk (z, x1 , · · · , xn ), t)   2   n2 m n − 1 m n2 − 1 m n2 − 1 m = lim ( 2 ) N Dµ fk ( ) z, ( ) x1 , · · · , ( ) xn , t m→∞ n − 1 n2 n2 n2 2

≥ lim

m→∞

≥ lim

m→∞

≥ lim

m→∞

m ( nn−1 2 ) t 2

2

2

2

2

2

2

n −1 m n −1 m n −1 m m ( nn−1 2 ) t + ϕ(( n2 ) z, ( n2 ) x1 , · · · , ( n2 ) xn ) t 2

n −1 m n −1 m m t + ( n2n−1 )m ϕ(( nn−1 2 ) z, ( n2 ) x1 , · · · , ( n2 ) xn )

t t+

Lm ϕ(z, x1 , · · ·

, xn )

=1

for all z, x1 , · · · , xn ∈ X, t > 0 and all µ ∈ T1 . Thus n X

µdk (z − xi ) = −

i=1

1 n

X

dk (µxi + µxj ) + ndk (µz −

n k X µxi ) n2 i=1

1≤i 0. Since 2

lim

m→∞

2m t ( nn−1 2 ) 2

2

2m t + ( n −1 )m Lm ϕ(x, y, 0, · · · , 0) ( nn−1 2 ) n2

=1

for all x, y ∈ X and all t > 0, ∆d1 ,d2 ,d3 (x, y) = 0, 684

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A. Ebadian, M. A. Abolfathi, C. Park, D. Y. Shin

for all x, y ∈ X. So d1 (xy) = d1 (x)y + xd1 (y) + d2 (x)d3 (y) + d3 (x)d2 (y) for all x, y ∈ X. Hence d1 is a fuzzy (d2 , d3 )-double derivation on X.



Corollary 2.4. Let X be a normed vector space with norm k · k, δ ≥ 0 and p be a real number with p > 1. Let k ∈ J and fk : X → X be mappings satisfying N (Dµ fk (z, x1 , · · · , xn ), t) ≥

N (∆f1 ,f2 ,f3 (x, y), t) ≥

t P , t + δ(kzkp + ni=1 kxi kp )

(2.8)

t t + δ(kxkp + kykp )

(2.9)

for all x, y, z, x1 , · · · , xn ∈ X, all µ ∈ T1 and all t > 0. Then there exist unique C-linear mappings dk : X → X such that N (fk (x) − dk (x), t) ≥

(n2 − 1)1−p − n2(1−p) t (n2 − 1)1−p − n2(1−p) t + 2nδ(n2 − 1)−p kxkp

for all x ∈ X and all t > 0. Moreover, d1 : X → X is a fuzzy (d2 , d3 )-double derivation on X. Proof. The proof follows from Theorem 2.3 by taking ϕ(z, x1 , · · · , xn ) := δ(kzkp +

n X

kxi kp )

i=1 2

for all x, y, z, x1 , · · · , xn ∈ X. It follows from (2.8) that fk (0) = 0. Choosing L = ( n2n−1 )1−p , we get the desired result.  Theorem 2.5. Let ϕ : X n+1 → [0, ∞) be a function such that there exists an L < 1 such that   n2 n2 n2 n2 ϕ z, x , · · · , x ≤ Lϕ(z, x1 , · · · , xn ) (2.10) 1 n n2 − 1 n2 − 1 n2 − 1 n2 − 1 for all z, x1 , · · · , xn ∈ X. Let k ∈ J and fk : X → Y be mappings satisfying fk (0) = 0, (2.4) 2 n2 m m and (2.5). Then the limit dk (x) = N − limm→∞ ( nn−1 2 ) fk (( n2 −1 ) x) exist for each x ∈ X, and define unique C-linear mappings dk : X → X such that N (fk (x) − dk (x), t) ≥

(n2 − 1)(1 − L)t (n2 − 1)(1 − L)t + nLϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

for all x ∈ X and all t > 0. Moreover, d1 : X → X is a fuzzy (d2 , d3 )-double derivation on X. Proof. Let (Ω, d) be the generalized metric space in the  proof of Theorem 2.3. Consider the n2 −1 n2 linear mapping J : Ω → Ω defined by Ju(x) := n2 u n2 −1 x for all u ∈ Ω and x ∈ X. We can conclude that J is a strictly contractive self-mapping of Ω with the Lipschitz constant L. Let k ∈ J. It follows from (2.6) that   2   n −1 n2 − 1 t N nfk x − fk (x), t ≥ (2.11) 2 n n t + ϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

685

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Approximate fuzzy double derivations and fuzzy Lie ∗-double derivations 2

for all x ∈ X and all t > 0. Replacing x by n2n−1 x in (2.11), we obtain     2 n2 nt n −1 fk x − fk (x), t ≥ N 2 2 n n −1 n2 2 nt + ϕ( n2n−1 x, 0, · · · , 0, 2 x, 0, · · · , 0) −1} |n {z jth

≥

nt nt +

n2 Lϕ(x, 0, · · · n2 −1

, 0, |{z} x , 0, · · · , 0) jth

It follows that d(fk , Jfk ) ≤ nnL 2 −1 . The rest the proof is similar to the proof of Theorem 2.3.



Corollary 2.6. Let X be a normed vector space with norm k · k, δ ≥ 0 and p be a real number with p < 1. Let k ∈ J and fk : X → X be mappings satisfying (2.8) and (2.9). Then there exist unique C-linear mappings dk : X → X such that N (fk (x) − dk (x), t) ≥

(n2 − 1)p−1 − n2(p−1) t (n2 − 1)p−1 − n2(p−1) t + 2δn2p−1 (n2 − 1)−1 kxkp

for all x ∈ X and all t > 0. Moreover, dk : X → X is a fuzzy (d2 , d3 )-double derivation on X. Proof. The proof follows from Theorem 2.5 by taking ϕ(z, x1 , · · · , xn ) := δ(kzkp +

n X

kxi kp )

i=1 2

for all x, y, z, x1 , · · · , xn ∈ X. It follows from (2.8) that fk (0) = 0. Choosing L = ( n2n−1 )p−1 , we get the desired result.  3. Approximate fuzzy ∗-double derivations on fuzzy C ∗ -algebras Throughout this section, we assume that X is a unital C ∗ -algebra with unit e and unitary group U (X) := {u ∈ X : u∗ u = uu∗ = e}, and (X, N ) is an induced fuzzy C ∗ -algebra. Theorem 3.1. Let ϕ : X n+1 → [0, ∞) be a function such that there exists an L < 1 satisfying (2.1). Let k ∈ J and fk : X → X be mappings satisfying fk (0) = 0, (2.4) and N (∆f1 ,f2 ,f3 (u, v), t) ≥

t , t + ϕ(u, v, 0, · · · , 0)

N (fk (u∗ ) − fk (u)∗ , t) ≥

(3.1)

t t + ϕ(u, 0, · · · , 0)

(3.2)

for all u, v ∈ U (X) and all t > 0. Then there exist unique ∗- mappings dk : X → X satisfying N (fk (x) − dk (x), t) ≥

(n2 − 1)(1 − L)t (n2 − 1)(1 − L)t + nϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

for all x ∈ X and all t > 0. Moreover, d1 : X → X is a fuzzy ∗-(d2 , d3 )-double derivation on X. 686

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Proof. Let k ∈ J. By Theorem 2.3,  there are C-linear mappings dk : X → X such that n2 m n2 −1 m N − limm→∞ ( n2 −1 ) fk ( n2 ) x = dk (x) for all x ∈ X. By (2.3) and (3.1),  N

(

 n2 2m n2 − 1 m n2 − 1 m ) ∆ (( ) u, ( ) v), t f1 ,f2 ,f3 n2 − 1 n2 n2 2

≥

2m t ( nn−1 2 ) 2

2

2

2m t + ϕ(( n −1 )m u, ( n −1 )m v, 0, · · · , 0) ( nn−1 2 ) n2 n2 2

≥

2m t ( nn−1 2 ) 2

2

2m t + ( n −1 )m Lm ϕ(u, v, 0, · · · , 0) ( nn−1 2 ) n2

for all u, v ∈ U (X) and all t > 0. Since 2

lim

m→∞

2m t ( nn−1 2 ) 2

2

2m t + ( n −1 )m Lm ϕ(u, v, 0, · · · , 0) ( nn−1 2 ) n2

=1

for all u, v ∈ U (X) and all t > 0, ∆d1 ,d2 ,d3 (u, v) = 0 for all u, v ∈ U (X). Hence d1 (uv) = d1 (u)v + ud1 (v) + d2 (u)d3 (v) + d3 (u)d2 (v)

(3.3)

for all u, v ∈ U (X). The mapping P d1 is C-linear and each x ∈ X is a finite linear combination of unitary elements, that is, x = ni=1 αi ui for αi ∈ C and ui ∈ U (X). It follows from (3.3) that n n X X d1 (xv) = d1 ( αi ui v) = d1 (αi ui v)

=

i=1 n X

i=1

αi (d1 (ui )v + ui d1 (v) + d2 (ui )d3 (v) + d3 (ui )d2 (v))

i=1 n n n n X X X X = d1 ( αi ui )v + αi ui )d1 v + d2 ( αi ui )d3 (v) + d3 ( αi ui )d2 (v) i=1

i=1

i=1

i=1

for all x ∈ X and all v ∈ U (X). So d1 (xv) = d1 (x)v + xd1 (v) + d2 (x)d3 (v) + d3 (x)d2 (v) for all x ∈ A and all v ∈ U (X). Similarly, we can obtain that d1 (xy) = d1 (x)y + xd1 (y) + d2 (x)d3 (y) + d3 (x)d2 (y) for all x, y ∈ X. 687

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By (2.3) and (3.2), N ((

n2 k n2 − 1 m ∗ n2 m n2 − 1 m ∗ ) f (( ) u ) − ( ) f (( ) u) , t) k k n2 − 1 n2 n2 − 1 n2 2

≥

m ( nn−1 2 ) t 2

2

2

n −1 m n −1 m m ( nn−1 2 ) t + ϕ(( n2 ) u, ( n2 ) u, 0, · · · , 0) 2

≥

m ( nn−1 2 ) t 2

2

n −1 m m m ( nn−1 2 ) t + ( n2 ) L ϕ(u, 0, · · · , 0)

for all u ∈ U (X) and all t > 0. Since 2

lim

m→∞

m ( nn−1 2 ) t 2

2

n −1 m m m ( nn−1 2 ) t + ( n2 ) L ϕ(u, 0, · · · , 0)

=1

for all u ∈ U (X) and all t > 0, dk (u∗ ) = dk (u)∗ . The mapping P dk is C-linear and each x ∈ X is a finite linear combination of unitary elements, that is, x = ni=1 αi ui for αi ∈ C and ui ∈ U (X). So ∗

dk (x ) = dk (

n X

αi u∗i )

i=1

=

n X

∗

αi dk (ui ) = dk (

i=1

n X

(αi ui ))∗ = dk (x)∗

i=1

for all x ∈ X. Hence d1 : X → X is a fuzzy ∗-(d2 , d3 )-double derivation on X.



Theorem 3.2. Let ϕ : X n+1 → [0, ∞) be a function such that there exists an L < 1 satisfying (2.3). Let k ∈ J and fk : X → X be mappings satisfying fk (0) = 0, (2.4), (3.1) and (3.2). Then there exist unique ∗-mappings dk : X → X satisfying N (fk (x) − dk (x), t) ≥

(n2

(n2 − 1)(1 − L)t − 1)(1 − L)t + nLϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

for all x ∈ X and all t > 0. Moreover, d1 : X → X is a fuzzy ∗-(d2 , d3 )-double derivation on X. 4. Approximate fuzzy Lie ∗-double derivations on induced fuzzy Lie C ∗ -algebras An induced fuzzy C ∗ -algebra X endowed with the Lie product [x, y] = xy − yx on X, is called an induced fuzzy Lie C ∗ -algebra. Throughout this section, we assume that X is a unital Lie C ∗ -algebra with unit e and unitary group U (X) := {u ∈ X : u∗ u = uu∗ = e}, and (X, N ) is an induced fuzzy Lie C ∗ -algebra. Definition 4.1. Let X be an induced fuzzy Lie C ∗ -algebra. Let k ∈ J and fk : X → X be C-linear mappings. A C-linear mapping f1 is called a fuzzy Lie (f2 , f3 )-double derivation if f1 ([x, y]) = [f1 (x), y] + [x, f1 (y)] + [f2 (x), f3 (y)] + [f3 (x), f2 (y)] for all x, y ∈ X. For given mappings fk : X → X, we define Γf1 ,f2 ,f3 (x, y) := f1 ([x, y]) − [f1 (x), y] − [x, f1 (y)] − [f2 (x), f3 (y)] − [f3 (x), f2 (y)] for all x, y ∈ X. 688

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Theorem 4.2. Let ϕ : X n+1 → [0, ∞) be a function such that there exists an L < 1 satisfying (2.3). Let k ∈ J and fk : X → X be mappings satisfying fk (0) = 0 , (2.4) and t N (Γf1 ,f2 ,f3 (u, v), t) ≥ , (4.1) t + ϕ(u, v, 0, · · · , 0) t (4.2) t + ϕ(u, 0, · · · , 0) for all u, v ∈ U (X) and all t > 0. Then there exist unique C-linear ∗-mappings dk : X → X such that (n2 − 1)(1 − L)t N (fk (x) − dk (x), t) ≥ 2 (n − 1)(1 − L)t + nϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) N (fk (u∗ ) − fk (u)∗ , t) ≥

jth

for all x ∈ X and all t > 0. Moreover, d1 : X → X is a fuzzy Lie ∗-(d2 , d3 )-double derivation on X. Proof. Let k ∈ J. By the same reasoning as in the proofs of Theorems 2.3 and 3.1, we obtain 2 2 m C-linear ∗-mappings dk : X → X such that dk (x) = N − limm→∞ ( n2n−1 )m fk (( nn−1 2 ) x) for each x ∈ X. It follows from (2.3) and (4.1) that N (Γd1 ,d2 ,d3 (u, v), t) = N − lim (( m→∞

n2 − 1 m n2 − 1 m n2 2m ) Γ (( ) u, ( ) v), t) f1 ,f2 ,f3 n2 − 1 n2 n2 2

≥ lim

m→∞

2m t ( nn−1 2 ) 2

2

2

2m t + ϕ(( n −1 )m u, ( n −1 )m v, 0, · · · , 0) ( nn−1 2 ) n2 n2 2

≥ lim

m→∞

2m t ( nn−1 2 ) 2

2

2m t + ( n −1 )m Lm ϕ(u, v, 0, · · · , 0) ( nn−1 2 ) n2

for all u, v ∈ U (X) and all t > 0. Since 2

lim

m→∞

2m t ( nn−1 2 ) 2

2

2m t + ( n −1 )m Lm ϕ(u, v, 0, · · · , 0) ( nn−1 2 ) n2

=1

for all u, v ∈ U (X) and all t > 0, d1 ([u, v]) = [d1 (u), v] + [u, d1 (v)] + [d2 (u), d3 (v)] + [d3 (u), d2 (v)] P for all u, v ∈ U (X). Since d1 : X → X is C-linear and each x ∈ X is x = ni=1 αi ui where ui ∈ U (X) and αi ∈ C, n n X X d1 ([x, v]) = d1 ( [αi ui , v]) = d1 ([αi ui , v]) i=1

=

n X

i=1

αi ([d1 (ui ), v] + [ui , d1 (y)] + [d2 (ui ), d3 (v)] + [d3 (ui ), d2 (v)]

i=1 n n n n X X X X = [d1 ( αi ui ), v] + [( αi ui ), d1 (v)] + [d2 ( αi ui ), d3 (v)] + [d3 ( αi ui ), d2 (v)] i=1

i=1

i=1

i=1

= [d1 (x), v] + [x, d1 (v)] + [d2 (x), d3 (v)] + [d3 (x), d2 (v)] for all x ∈ X and v ∈ U (X). By a similar method, we obtain that d1 [x, y] = [d1 (x), y] + [x, d1 (y)] + [d2 (x), d3 (y)] + [d3 (x), d2 (y)] 689

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for all x, y ∈ X. Thus the C-linear mapping d1 : X → X is a fuzzy Lie ∗-(d2 , d3 )-double derivation on X, as desired.  Corollary 4.3. Let X be a C ∗ -algebra with norm k · k, δ ≥ 0 and p be a real number with p > 1 . Let fk : X → X be mappings satisfying N (Dµ fk (z, x1 , · · · , xn ), t) ≥

t+

δ(kzkp

t P , + ni=1 kxi kp )

(4.3)

  n2 − 1 m t n2 − 1 m ) u, ( ) v), t ≥ N Γf1 ,f2 ,f3 (( , 2 2 2 n mp n n t + 2δ( n−1 2 ) N (fk ((

n2 − 1 m ∗ t n2 − 1 m ∗ u ) − f (( ) ) u) , t) ≥ k 2 2 2 n mp n n t + δ( n−1 2 )

for all z, x1 , · · · , xn ∈ X, all u, v ∈ U (X), all µ ∈ T1 , all t > 0 and m = 0, 1, 2, · · · . Then there exist unique C-linear mappings dk : X → X such that N (fk (x) − dk (x), t) ≥

(n2 − 1)1−p − n2(1−p) t (n2 − 1)1−p − n2(1−p) t + 2nδ(n2 − 1)−p kxkp

for all x ∈ X and all t > 0. Moreover, d1 : X → X is a fuzzy Lie ∗-(d2 , d3 )-double derivation on X. Theorem 4.4. Let ϕ : X n+1 → [0, ∞) be a function such that there exists an L < 1 satisfying (2.10) . Let k ∈ J and fk : X → X be mappings satisfying fk (0) = 0, (2.4), (4.1), and (4.2). Then there exist unique C-linear ∗-mappings fk : X → X such that N (fk (x) − dk (x), t) ≥

(n2 − 1)(1 − L)t (n2 − 1)(1 − L)t + nϕ(x, 0, · · · , 0, |{z} x , 0, · · · , 0) jth

for all x ∈ X and all t > 0. Moreover, d1 : X → X is a fuzzy Lie ∗-(d2 , d3 )-double derivation on X. Proof. The proof is similar to the proofs of Theorems 2.5 and 4.2.



Corollary 4.5. Let X be a C ∗ -algebra with norm k · k, δ ≥ 0 and p be a real number with p < 1. Let k ∈ J and fk : X → X be mappings satisfying (4.3) and   n2 m t n2 m N Γf1 ,f2 ,f3 (( 2 ) u, ( 2 ) v)), t ≥ 2 n −1 n −1 t + 2δ( 2n )mp n −1

N (fk ((

n2 m ∗ n2 m ∗ t ) u ) − f (( ) u) , t) ≥ k 2 2 2 n −1 n −1 t + δ( n2n−1 )mp

for all u, v ∈ U (X), all t > 0 and m = 0, 1, 2, · · · . Then there exist unique C-linear mappings dk : X → X such that N (fk (x) − dk (x), t) ≥

(n2 − 1)p−1 − n2(p−1) t (n2 − 1)p−1 − n2(p−1) t + 2δn2p−1 (n2 − 1)−1 kxkp

for all x ∈ X and all t > 0. Moreover, d1 : X → X is a fuzzy Lie ∗-(d2 , d3 )-double derivation on X. 690

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Acknowledgments C. Park and D. Y. Shin were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299). G. Lu was supported by The PhD Start-up Fund of Science Foundation of Liaoning Province, China (No. 20121055), and (NRF-2010-0021792), respectively. References [1] T. Aoki, On the stability of the lineartransformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 245-251. [2] R. Badora, On approximate ring homomorphism, J. Math. Anal. Appl. 276 (2002), 589-597, doi:10.1016/S0022-247X(02)00293-7. [3] R. Badora, On approximate derivations, Math. Inequal. Appl. 9 (2006), 167-173. [4] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear space, J. Fuzzy Math. 11 (2003), 687-705. [5] T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513-547. [6] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function ring, Duke Math J. 16 (1949), 385-397, doi:10.1215/S0012-7094-49-01639-7. [7] S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429-436. [8] L. C˘ adriu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43-52. [9] A. Ebadian and N. Ghobadipour, A fixed point approach to almost double derivations and Lie ∗-double derivations, Results Math. 63 (2013), 409-423. [10] A. Ebadian, R. Aghalary and M. A. Abolfathi, Approximation of homomorphisms and derivations of additive functional equation of n-Apollonius type in induced fuzzy Lie C ∗ -algebras, J. Adv. Math. 3 (2013), 201-217. [11] M. Eshaghi Gordji and A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Comput. Anal. Appl. 13 (2011), 724–729. [12] M. Eshaghi Gordji, R. Farokhzad Rostami and S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ -algebras, J. Comput. Anal. Appl. 13 (2011), 734–742. [13] M. Eshaghi Gordji, G.H. Kim, J.R. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013) 45–54. [14] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems 48 (1992), 239-248. [15] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of the approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436. [16] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222-224. [17] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equation in Several Variables, Birkh¨ auser, Basel, 1998. [18] G. Isac and Th. M. Rassias, Stability of ψ-additive mappings: applications to nonlinear analysis, Internat. J. Math. Sci. 19 (1996), 219-228. [19] K. Jun and H. Kim, The gegeralized Hyers-Ulam-Rassias stability of cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867-878. [20] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (3) (1984), 1-7. [21] A. K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets and Systems 12 (1984), 143-154. [22] H. Khodaei, R. Khodabakhsh and M. Eshaghi Gordji, Fixed points, Lie ∗-homomorphisms and Lie ∗derivations on Lie C ∗ -algebras, Fixed Point Theory 14 (2013), 387-400. 691

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Approximate fuzzy double derivations and fuzzy Lie ∗-double derivations [23] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 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] 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. 74 (1968), 305-309. [26] M. Mirzavaziri and M. S. Moslehian, Automatic continuity of σ-derivations on C ∗ -algebras, Proc. Amer. Math. Soc. 134 (2006), 3327-3329. [27] M. Mirzavaziri and E. Omidvar Tehrani, δ-Double derivations on C ∗ -algebras, Bull. Iran . Math. Soc. 35 (2009), 141-154. [28] S. A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos, Solitons & Fractals 42 (2009), 29892996. [29] S. A. Mohiuddine, M. Cancan and H. Sevli, Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique, Math. Comput. Modelling 54 (2011), 2403-2409. [30] S. A. Mohiuddine and H. Sevli, Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space, J. Comput. Appl. Math. 235 (2011), 2137-2146. [31] F. Moradlou and M. Eshaghi Gordji, Approximate Jordan derivations on Hilbert C ∗ -modules, Fixed Point Theory 14 (2013), 413-426. [32] F. Moradlou, H. Vaezi and C. Park, Fixed points and stability of an additive functional equation of nApollonius type in C ∗ -algebra, Abst. Appl. Anal. 2008, Article ID 672618 (2008). [33] M. Mursaleen and S. A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos, Solitons & Fractals 42 (2009), 29973005. [34] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Brazil. Math. Soc. 36 (2005), 79-97. [35] C. Park, Fixed points and Hyers-Ulam stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007, Article ID 50175 (2007). [36] C. Park, Fuzzy stability of functional equation associated with inner product spaces. Fuzzy Sets and Systems 160 (2009), 1632-1642. [37] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [38] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [39] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91-96. [40] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. ˇ [41] P. Semrl, The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations Operator Theory 18 (1994), 118-122, doi:10.1007/BF01225216. [42] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [43] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternaty cubic homomorphisms in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [44] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [45] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [46] S. M. Ulam, Problem in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964. [47] C. Wu and J. Fang, Fuzzy generalization of Kolmogoroof ’s theorem, J. Harbin Inst. Technol. 1 (1984), 1-7. [48] J. Z. Xiao and X. H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems. 133 (2003), 389-399. 692

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.4, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

SOME NEW GENERALIZED RESULTS ON OSTROWSKI TYPE INTEGRAL INEQUALITIES WITH APPLICATION 1 A.

QAYYUM, 2 M. SHOAIB, AND 1 IBRAHIMA FAYE

Abstract. The aim of this paper is to establish some new inequalities similar to the Ostrowski’s inequalities which are more generalized than the inequalities of Dragomir and Cerone. The current article obtains bounds for the deviation of a function from a combination of integral means over the end intervals covering the entire interval. Some new purterbed results are obtained. Application for cumulative distribution function is also discussed.

1. Introduction In 1938, Ostrowski [13] established an interesting integral inequality associated with di¤erentiable mappings. This Ostrowski inequality has powerful applications in numerical integration, probability and optimization theory, stochastic, statistics, information and integral operator theory. A number of Ostrowski type inequalities have been derived by Cerone [1], [2] and Cheng [3] with applications in Numerical analysis and Probability. Dragomir et.al [5] combined Ostrowski and Grüss inequality to give a new inequality which they named Ostrowski-Grüss type inequality. Milovanovi´c and Pecari´c [12] gave the …rst generalization of Ostrowski’s inequality. More recent results concerning the generalizations of Ostrowski inequality are given by Liu [11], Hussain [10] and Qayyum [16]. In this paper, we will extend and generalize the results of Cerone [1] and Dragomir et.al [5]-[8] by using a new kernel. Let S (f ; a; b) be de…ned by S (f ; a; b) := f (x)

M (f ; a; b) ;

(1.1)

Zb

(1.2)

where M (f ; a; b) =

1 b

a

f (x) dx

a

is the integral mean of f over [a; b]. The functional S (f ; a; b) represents the deviation of f (x) from its integral mean over [a; b] : Ostrowski [13] proved the following integral inequality: Theorem 1. Let f : [a; b] ! R be continuous on [a; b] and di¤ erentiable on (a; b) ; whose derivative f 0 : (a; b) ! R is bounded on (a; b) ; i.e. kf 0 k1 = 1991 Mathematics Subject Classi…cation. 2010 Mathematics Subject Classi…cation : 26D07; 26D10; 26D15. µ µ Key words and phrases. Ostrowski inequality, Cebyˆ sev-Grüss inequality,Cebyˆ sev functional. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1

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2

A. QAYYUM , 2 M . SHOAIB, AND

supt2[a;b] jf 0 (t)j < 1; then jS (f ; a; b)j

"

b

1

IBRAHIM A FAYE

2

a

+ x

2

2

a+b 2

#

M b

(1.3)

a

for all x 2 [a; b].

In a series of papers, Dragomir et al [5]-[8] proved (1.3) and some of it’s variants for f 0 2 Lp [a; b] when p 1; for Lebesgue norms making use of a peano kernel. If we assume that f 0 2 L1 [a; b] and kf 0 k1 = ess jf 0 (t)j then M in (1.3) may t2[a;b]

be replaced by kf 0 k1 : Dragomir et al [5]-[8] utilizing an integration by parts argument; obtained jS (f ; a; b)j 8 > > > > > > < > > > > > > :

1 b a

1 b a 1 b a

h h

(1.4)

b a 2 2

a+b 2 2

+ x

(x a)q+1 +(b x)q+1 q+1 b a 2

+ x

a+b 2

i q1

i

kf 0 k1 ; f 0 2 L1 [a; b] ;

kf 0 kp ; f 0 2 Lp [a; b] ; p > 1; p1 +

1 q

= 1;

kf 0 k1 ; f 0 2 L1 [a; b] ;

h i q1 1 where f : [a; b] ! R is absolutely continuous on [a; b] and the constants 14 ; q+1 µ and 12 are sharp. In [14], Pachpatte established Cebyˆ sev type inequalities by using Pecari´c’s extension of the Montgomery identity [17]. Cerone [1], proved the following inequality: Lemma 1. Let

f : [a; b] ! R be absolutely continuous function. De…ne 1 (x; ; ) := f (x) [ M (f ; a; x) + M (f ; x; b)] ; +

(1.5)

where j (x; ; )j 8 > > > > > > > > >
; f 0 2 Lp [a; b] ; p > 1; p1 + 1q = 1; > > > > > > > > j : 1 1+ j kf 0 k1 ; f 0 2 L1 [a; b] ; 2 +

where the usual Lp norms kkkp de…ned for a function k 2 Lp [a; b] as follows: kkk1 := ess sup jk (t)j t2[a;b]

and

0 b 1 p1 Z p kkkp := @ jk (t)j dtA ; 1 a

694

p

1:

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With the help of two di¤erent kernels (1.7) and (1.9) given below, we extended the version of Cerone [1] and Dragomir’s result [5]-[8]. Lemma 2. Let P (x; :) : [a; b] ! R; the peano type kernel is given by

p(x; t) =

8
4 2 2 > C 1 B x a > > 0 B C > > @ n o A ( + ) kf k1 > > 2 (b x)2 > b a x+b > +b x + b h 2 > 4 2 > > > 0 > ; f 2 L [a; b] 1 > > > > > > > > 2 o 3 q1 n > q > b a q+1 a b q+1 > x a+h 2 h 2 > q > 7 < 6 (x a) 1 6 7 4 n o 5 (q+1) q1 ( q q+1 q+1 > > + (b x)q b x + h b 2a h a2 b > > > > > > > 0 > > f 2 Lp [a; b] ; p > 1; p1 + 1q = 1; > > > > > > h i 1 0 > > (x a) > > ( + ) h b 2 a (b(x x)+ > a)(b x) > C kf 0 k > > B 1 B C > > @ h i A 2( + ) : > > (x a) (b x) b a : + ( )+h 2 (x a)(b x)

+ )

kf 0 kp ;

Lemma 3. Denote by P (x; :) : [a; b] ! R the kernel is given by

P (x; t) :=

8 > < > :

2( + )(x a)

(t

2

a) ; a

t

x; (1.9)

2( + )(b x)

695

(t

2

b) ; x < t

b;

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A. QAYYUM , 2 M . SHOAIB, AND

1

IBRAHIM A FAYE

Then, j (x; ; )j 1 = [ (x a) (b x)] f 0 (x) f (x) 2( + ) 1 + [ M (f ; a; x) + M (f ; x; b)] ; + 8 h i 00 2 2 kf k > > (x a) + (b x) 6( + 1) ; f 00 2 L1 [a; b] > > > > > > > > h i q1 kf 00 k > > q+1 q+1 > q p 1 q > (x a) + (b x) > 2( + ) ; < (2q+1) q1 > > > f 00 2 Lp [a; b] ; p > 1; p1 + 1q = 1; > > > > > > > > > > ( (x a) + (b x) + j (x a) > > : ; f 00 2 L1 [a; b] :

(b

(1.10)

kf 00 k x)j) 4( + 1)

Using a generalized form of (1.9), we constructed a number of new results for twice di¤erentiable functions. These results are given in Lemma 4 and theorem 2 which are more generalized by (1.8)-(1.10). These generalized inequalities will have applications in approximation theory, probability theory and numerical analysis. We will show in our paper an application of the obtained inequalities for cumulative distribution function. 2. Main Results We will start our main result with this lemma. Lemma 4. Let f : [a; b] ! R be an absolutely continuous mapping. Denote by P (x; :) : [a; b] ! R the kernel P (x; t; h) is given by 8 2 1 1 > a + h b 2a ; a t x; < + x a2 t P (x; t) = (2.1) > : 1 1 b a 2 t b h ; x < t b; +

b x2

2

for all x 2 a + h b 2 a ; b h b 2 a and h 2 [0; 1] ;where ; 2 R are non negative and not both zero. Before we state and prove our main theorem, we will prove the following identity: " Zb 2 1 b a P (x; t)f 00 (t)dt = x a+h (2.2) 2( + ) x a 2 a # 2 b a x b h f 0 (x) b x 2 1 ( + ) x b

x

x

696

a b

x h

a+h b

a 2

b

a 2

f (x)

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1 b a h + 2 x

a

f (a) +

b

x

f (b)

2

1 (b a) h2 f 0 (b) f 0 (a) + 8 b x x a 2 3 Zx Zb 1 4 + f (t) dt + f (t) dt5 : + x a b x +

a

x

Proof. From (2.1), we have

Zb

00

P (x; t)f (t)dt =

1 ( + )x

a

a

Zx

a + h b 2a 2

t

2

f 00 (t)dt

a

+

1 ( + )b

x

Zb

t

h b 2a

b 2

2

f 00 (t)dt:

x

After simpli…cation, we get the required identity (2.2):

We now give our main theorem. Theorem 2. Let

(x; ; )

f : [a; b] ! R be an absolutely continuous mapping. De…ne

:

" 1 = 2( + ) x b

x

x

1 ( + ) x b

x

x

x

a b

h

b

a

b

h

1 b a h + 2 x

2

2

x

a

a+h

a+h b

#

a

f (a) +

a

2

2

(2.3)

f 0 (x)

b

a 2

f (x)

2 a

b

b

x

f (b)

2

1 (b a) h2 f 0 (b) f 0 (a) + 8 b x x a 1 + [ M (f ; a; x) + M (f ; x; b)] ; + +

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6

A. QAYYUM , 2 M . SHOAIB, AND

1

IBRAHIM A FAYE

where M (f ; a; b) is the integral mean de…ned in (1.2), then j (x; ; )j 8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > :

2

x a

4

2 4

f 8 > > > >
> > > : +

+h x)

(b

(b a) 2

(x 2

+h

1 q

00 o 5 kf k1 ; f 6( + )

h a2 b h b 2a

b

2q+1

o 3 q1

o 5 2q+1

00

2 L1 [a; b] ;

kf 00 kp

;

1

2(2q+1) q ( + )

= 1;

hx)

(b a) 2

o 3

3

h b 2a 2q+1

a + h b 2a

x

3

+ h b 2a

b

2 Lp [a; b] ; p > 1; p1 + (x

3

h (b

a) i ( + )

+b x ha) + h (b a) i ( x a

b x

9 > > > > = > > > > ;

)

x a

kf 00 k1 00 4( + ) ; f

2 L1 [a; b]

for all x 2 [a; b] ;where kkk is the usual Lebesgue norm for k 2 L [a; b] with kkk1 := ess sup jk (t)j < 1 t2[a;b]

and

0 b 1 p1 Z p kkkp := @ jk (t)j dtA ; 1

p

1

a

Proof. Taking the modulus of (2.2) and using (2.3) and (1.2), we have j (x; ; )j = Therefore, for f

00

Zb

P (x; t)f (t)dt

a

j (x; ; )j

a

=

a

2 L1 [a; b] we obtain

Now let us observe that Zb

Zb

00

00

kf k1

Zb a

jP (x; t)j jf 00 (t)j dt:

(2.5)

jP (x; t)j dt:

jP (x; t)j dt

2 ( + ) (x

a)

Zx

t

a+h

b

2

a

dt

2

a

+

2 ( + ) (b

x)

Zb

t

b

h

b

a 2

2

dt:

x

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After simple integration, we get Zb a

jP (x; t)j dt 2

1 4 6( + )

=

n

x a

n

b x

3

a + h b 2a

x

3

h b 2a

x

+ h b 2a

3

3

h b 2a

b

o 3

o 5:

Hence the …rst inequality is obtained. j (x; ; )j 2 n x a + h b 2a x a 4 n 3 h b 2a + x b x

3

3

+ h b 2a

3

h b 2a

b

o 3

1 kf 00 k1 : 6( + )

o 5

Further, using Hölder’s integral inequality in (2.5) we have for f

1 p

+

1 q

2 Lp [a; b]

0 b 1 q1 Z q kf 00 kp @ jP (x; t)j dtA :

j (x; ; )j where

00

a

= 1 and p > 1: Now 0

( + )@ 2

6 = 6 4 =

Zb a

Rx

2q (x a)q

"

a+

h b 2a

2q

h b 2a

2q

jP (x; t)j dtA

q

+ 2q (b

q

1 q1

t

a Rb

q

x)q

t

b

x

q

2q (2q+1)(x a)q

a + h b 2a

t

q

+ 2q (2q+1)(b

t

x)q

b

dt dt

1 q

2q+1 2q

h b 2a

3 q1 7 7 5

=xa =bx

# q1

:

Again, after simple integration, we get 0 b 1 q1 Z q ( + ) @ jP (x; t)j dtA a

=

1 2 (2q + 1)

1 q

2 6 6 4

q

(x a)

+ (b

q

n

q

x)q

n

x

a + h b 2a

h a2 b

2q+1

699

2q+1

x

b

h a2 b h b 2a

2q+1

o 3 q1

7 7 o 5 : 2q+1

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A. QAYYUM , 2 M . SHOAIB, AND

1

IBRAHIM A FAYE

Hence the second inequality is obtained as below. j (x; ; )j 1 2 (2q + 1)

Finally, for f

00

1 q

2 6 6 4

q

(x a)q

+ (b

n

q

x)q

n

2q+1

a + h b 2a

x

2q+1

h a2 b

x

h a2 b

2q+1

7 7 kf 00 k : p 5 o 2q+1

h b 2a

b

o 3 q1

2 L1 [a; b], using (2.1), we have the following inequality from (2.5), sup jP (x; t)j kf 00 k1 ;

j (x; ; )j

t2[a;b]

where

( + ) sup jP (x; t)j = t2[a;b]

8 > 1
:

+

a) +

(b 2

+h

1 4j

(b

x)

(b a) 2

(x 2

+h

x)

h (b

h

+

x a

b x

a) + h (b

(b a) 2

h

b x

x a

9 a) ( + ) > = i > ;

a) ( i

)

j

This gives us the last inequality as below.

j (x; ; )j

8 > > > >
> > > : +h (b

a) +

(b 2

+ h (b2 +j (b a) (

a)

hx)

x a

x) )+

h (b

+b x (x h a)

h2 (b a) 2

9 > > > > = kf 00 k 1 : 4( + ) > i > > > ; a j

a) i ( + )

b x

x

This completes the proof of theorem.

Some special cases of Theorem 2 In this section, we will give some useful cases. Remark 1. If we put h = 0 in (2.4), we get (1.10). Remark 2. If we put h = 0 in (1.8), we get Cerone’s result given in (1.6).

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Remark 3. If we put h = 1 in (2.4), we get a new result.

j (x; ; )j 8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > :

n a n(x

2

x

2

1 (x a)q

4

(2.6)

b a 3 2

b x

6 6 4

3

A) +

n (x

+ (x

A)

5

1 6( + )

a b 2q+1 2

2q+1

A)

n

o 3

b a 3 2 o 3

2q+1

8 > > > >
> > > : +

where A =

a) +

o 3 q1

2q+1

a b (x A) + (b 1x)q 2 1 00 ; f 2 Lp [a; b] ; p > 1; p + 1q = 1;

(b h x)

kf 00 k1 ; f

7 7 o 5

1 1

(2q+1) q

(b ia) ( + )

+ b 2a x a + b x x) (xh a) + (b i a) ( + b 2a

b x

x a

j

9 > > > > =

) > > > > ;

00

1 4

2 L1 [a; b] ;

kf 00 kp

kf 00 k1 00 4( + ) ; f

2 L1 [a; b] ;

a+b 2 :

Corollary 1. If we put x = A in above , we get

j (A; ; )j 8 (b a)2 > > 24 > > > > > > h q > > < q (b a)

(2.7)

+

kf 00 k1 ; f

a b 2q+1 2

00

2 L1 [a; b] ;

q q

(b a) 1; p1 + 1q

> ; f 00 2 Lp [a; b] ; p > > > > > > > > n > > : 1 a2 b + + 1

a b 2

701

a b 2q+1 2

= 1;

i q1

o kf 00 k 4

1

1 1

(2q+1) q

;f

00

1 ( + )

kf 00 kp

2 L1 [a; b] :

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A. QAYYUM , 2 M . SHOAIB, AND 1 2

Remark 4. If we put h =

1 2( + )

h

xh a

1 ( + )

x

b x

3a+b 4

x

x a

IBRAHIM A FAYE

in (2.3) and (2.4) we get the following inequality:

3a+b 2 4

x

1

b x

h

x

i

a+3b 2 f 0 4 i a+3b 4i

(x)

f (x) (2.8)

1 b a + 4 h x a f (a) + b x f (b) i 2 (b a) 1 0 0 + + 32 b x f (b) x a f (a) 1 + + [ M (f ; a; x) + M (f ; x; b)]

8 > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > :

2 4

2 4

;f 8 > > > >
1; p1 + 1q = 1;

> > > > : +

q

q

(x (b

x

a+

a) +

1 2

(b hx)

00 5 kf k1 ; f 6( + )

o 3 q1

a b 2q+1 4 o b a 2q+1 4

(b ia) ( + )

+ (b 8 a) x a + b x x) (x h a) + 12 (b ia) ( + (b 8 a) b x x a

Corollary 2. If we put

=

j 21 f b a 32

5

)

00

2 L1 [a; b] ;

kf 00 kp 1

2(2q+1) q ( + )

9 > > > > = > > > > ;

kf 00 k1 00 4( + ) ; f

2 L1 [a; b] :

and x = A in (2.8) we get another result.

a+b 2

[f 0 (b)

+

1 4

[f (a) + f (b)] Rb f 0 (a)] (b a) f (t)dtj

(2.9)

a

8 (b a)2 kf 00 k > 1 > ; f 00 2 L1 [a; b] ; > 192 > > > > > > hn oi q1 > kf 00 kp < 2q+1 2q+1 (b a) (a b) 1 2: 1 16(b a)2 q (2q+1) q > > ; f 00 2 Lp [a; b] ; p > 1; p1 + 1q = 1; > > > > > > > > : (b a)kf 00 k1 ; f 00 2 L1 [a; b] : 16

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Corollary 3. If we put

b a 0 12 f

=

(x)

1 2 3 b a b a 0 + 0 16 f

+ b 2 a @ 31

R

2

1 1 2 3f 1 0 f (a) 3

(b)

Rb

f (t)dt +

a

00

in (2.8) we get another new result.

f (x)

a+3b 4

8 (b a)2 kf 00 k > 1 > ; f > 96 > > > > > > h n > < 1 b a 2q+1 3q

a+3b 4

and x =

a+3b 4

(a) + f (b) 1

(2.10)

f (t)dtA

2 L1 [a; b] ; o

2q+1 a b 2q+1 + a4 b 4 > 1; p1 + 1q = 1;

> > ; f 00 2 Lp [a; b] ; p > > > > > > > > kf 00 k1 : 2 b a 1 ;f 3 + 2 3 8

00

i q1

kf 00 kp

1

(b a)(2q+1) q

2 L1 [a; b] :

Hence, for di¤erent values of h, we can obtain a variety of results. Remark 5. We can write (2.3) in another way. Since

=

M (f ; a; x) + M (f ; x; b) 0b Z @ f (u) du M (f ; a; x) + b x

Zx

a

a

1

f (u) duA :

or

=

M (f ; a; x) + M (f ; x; b) Zx M (f ; a; x) f (u) du + b x b a

=

( +

(x)) M (f ; a; x) +

x

Zb

f (u) du

a

(x) M (f ; a; b) ;

where

b b

a = x

703

(x) :

(2.11)

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Thus, from (2.3), (x; ; ) h 1 2( + )

=

b x

1 ( + )

(2.12)

b

a + h b 2a i f (x) h b 2a

x

x a

x

b x

a + h b 2a i 2 h b 2a f 0 (x)

x

x a

x h

2

b

1 b a h + 2 x

a

f (a) +

b

x

f (b)

2

1 (b a) h2 + 8

+ +

1

b

x

f 0 (b)

x

a

(x) M (f ; a; x) +

+

f 0 (a) (x) M (f ; a; b) :

+

so that for …xed [a; b] ; M (f ; a; b) is also …xed. a+b 2

Corollary 4. If (2.3) and (2.4) is evaluated at x = j (h

1) f

a+b 2

+h2 b 8 a (f 0 (b) 8 > > > > > > > > > > > > < > > > > > > > > > > > > :

(1 2 6 6 4

h) 2

q

(b a)

q

a) (1

then

(f (a) + f (b)) Rb f 0 (a)) + b 1 a f (t) dtj

2

kf 00 k1 ; f

b a 2

(1

h)

2h) + 2h2

00

2q+1

n q 2q+1 + (b 2 a)q h a 2 b ; f 00 2 Lp [a; b] ; p > 1; p1 + (b

=

(2.13)

a

3 (b a) 24

n

and

h 2

2 L1 [a; b] ; h a2 b

b a 2 1 q

(1 = 1;

kf 00 k1 8

;f

00

h)

2q+1

o 3 q1

7 7 o 5 2q+1

kf 00 kp

1

4(2q+1) q

2 L1 [a; b] :

3. Perturbed Results

µ In 1882, Cebyˆ sev [4] gave the following inequality. 1 2 (b a) kf 0 k1 kg 0 k1 (3.1) 12 where f; g : [a; b] ! R are absolutely continuous functions, which has bounded …rst derivatives such that Zb 1 T (f; g) = f (x) g (x) dx (3.2) b a a 10 1 0 Zb Zb 1 @ 1 f (x) dxA @ g (x) dxA b a b a jT (f; g)j

a

= M (f; g; a; b)

a

M (f ; a; b) M (g; a; b) ;

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and k:k1 denotes the norm in L1 [a; b] de…ned as kpk1 = ess sup jP (t)j : t2[a;b]

In 1935, Grüss [9] proved the following inequality: 1 b

a

Zb

Zb

1

f (x) g (x) dx

b

a

a

1 ( 4

f (x) dx

1 b

a

') (

a

Zb

(3.3)

g (x) dx

a

);

provided that f and g are two integrable functions on [a; b] and satisfy the condition: '

f (x)

and

g (x)

; for all x 2 [a; b] :

(3.4)

The constant 41 is best possible. The perturbed version of the results of Theorem µ 2 can be obtained by using Grüss type results involving the Cebyˆ sev functional. T (f; g) = M (f; g; a; b)

M (f ; a; b) M (g; a; b) ;

where M is the integral mean and is de…ned in (1.2). Theorem 3. Let f : [a; b] ! R be an absolutely continuous mapping and ; are non-negative real numbers, then n o 3 2 b a 3 b a 3 x a + h + h 1 2 2 o 5 (3.5) 4 x a n (x; ; ) b a 3 b a 3 ( + ) 6 h + x b h b x

where,

(b

a) N (x)

(b

a) (

1

b

2

a

kf 00 k2

2

(x; ; ) is as given by (2.3) and

then =

1 2

8
> > >
> > > : +

'

a) + +h x)

(b

(b a) 2

(x

hx)

h (b

a) i ( + )

+b x a) + h (b a) h i ( x a

2 + h (b2 a)

b x

x a

inf P (x; t)

)

9 > > > > = > > > > ;

t2[a;b]

2

= where

h2 (b a) 8( + )

x

a

+

b

x

x

a

b

x

'= :

707

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4. An Application to the Cumulative Distribution Function Let X 2 [a; b] be a random variable with the cumulative distributive function

F (x) = Pr (X

x) =

Zx

f (u) du;

a

where f is the probability density function. In particular,

Zb

f (u) du = 1:

a

The following theorem holds. Theorem 4. Let X and F be as above, then

j 12

h

(b

x) x

(b

2

a + h b 2a

(x

a) x

h b 2a

b

h b 2a

x) x

h b 2a

2

i

f 0 (x)

a+ (x a) x b f (x) (4.1) h b 2 a [ (b x) f (a) + (x a) f (b)] 2 2 (b a) 0 0 +h [ (x a) f (b) (b x) f (a)] 8 + [ (b x) (x a)] F (x) + (x a) j o 3 2 n 3 3 (b x) x a + h b 2a + h b 2a n o 5 ; f 00 2 L1 [a; b] ; 4 3 3 (x a) h b 2 a + x b h b 2a

8 > > kf 00 k1 > > > 6 > > > > > > > > > > > > > > > (b x)(x a)kf 00 k > p > > 1 > > 2(2q+1) q >
1; p1 +

> ; ; f 00 > > > > > > > > > > > > > > (b x)(x a)kf 00 k > 1 > > 4 > > > > > > > > : ; f 00 2 L1 [a; b] :

8 > > > >
> > > : +

(x

n 1 q

a + h b 2a

x

h a2 b = 1;

2q+1

a) +

(b 2

(b

+h x)

(b a) 2

(x 2

+h

708

2q+1

x

hx)

(b a) 2

h a2 b

b

h (b

h b 2a

b x

x a

o 3 q1

7 7 o 5 2q+1

a) i ( + )

+b x ha) + h (b a) i ( x a

2q+1

)

:

9 > > > > = > > > > ;

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Proof. From (2:3) ; and by using the de…nition of Probability Density Function, we have

(x; ; )

" 1 = 2( + ) x

:

a

x

a+h

b

2

a 2

b

b a 1 x a+h ( + ) x a 2 1 b a h f (a) + f (b) + 2 x a b x

b

x

x x

x

b b

h

h

b

b

2

a 2

a 2

#

f 0 (x)

f (x)

2

(b a) 1 h2 f 0 (b) f 0 (a) + 8 b x x a 1 [ M (f ; a; x) + M (f ; x; b)] ; + + " 2 1 b a = x a+h 2( + ) x a 2 b x +

1 b a x a+h ( + ) x a 2 b a 1 h f (a) + f (b) + 2 x a b x

b

x

x x

b b

h h

b

a 2

b

a 2

2

#

f 0 (x)

f (x)

2

1 (b a) h2 f 0 (b) f 0 (a) + 8 b x x a 1 (b x) (x a) + F (x) + + (x a) (b x) (b x) +

or ( + ) (x a) (b x) (x; ; ) " # 2 1 (b x) x a + h b 2a f 0 (x) = 2 2 (x a) x b h b 2a

h

b

2 (b +h2 + [ (b

(b x) x a + h b 2a (x a) x b h b 2a a [ (b x) f (a) + (x

(4.2)

f (x) a) f (b)]

2

a) 8

x)

[ (x (x

a) f 0 (b) a)] F (x) +

(b

x) f 0 (a)]

(x

a)

Now using (2.4) and (4.2), we get our required result (4.1).

Putting

=

=

1 2

in Theorem 5 gives the following result.

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Corollary 5. Let X be a random variable, F (x) cumulative distributive function and f is a probability density function. Then h i 2 2 j 14 (b x) x a + h b 2a (x a) x b h b 2a f 0 (x) 1 2

8 > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > :

(b

(b

2

1 (x a)q

6 6 4

;f 8 < :

f (x)

h b 4 a [(b 2 (b +h2 16a) [(x + 21 [(b x)

2 4

a + h b 2a (x a) x b h b 2a x) f (a) + (x a) f (b)] a) f 0 (b) (b x) f 0 (a)] (x a)] F (x) + 21 (x a) j

x) x

x)

(x

n

a) n

x n x

a + h b 2a 3 h b 2a

3

+ x

a + h b 2a

n

(b

a) 1 2

h (b

2q+1

h a2 b

b 1; h

h b 2a

o 3 3

00 o 5 kf k1 ; f 12

2q+1

o 3 q1

7 7 o 5 2q+1

h2 (b a) 1 1 4 h x a + b ix h2 (b a) 1 1 4 b x x a

a) +

+ (a + b 2x) + ; f 2 L1 [a; b] : 00

3

h b 2a

b

2q+1 + (b 1x)q h a 2 b x 00 2 Lp [a; b] ; p > 1; p1 + 1q = 1 2

+ h b 2a

(4.3)

i 9 = (b ;

00

2 L1 [a; b] ;

(b x)(x a) 1

4(2q+1) q

x)(x a)kf 4

00

kf 00 kp

k1

Remark 6. The above result allow the approximation of F (x) in terms of f (x). The approximation of R (x) = 1 F (x) could also be obtained by a simple substitution. R (x) is of importance in reliability theory where f (x) is the probability density function of failure. Remark 7. We put = 0 in (4.1), assuming that 6= 0 to obtain i h 8 9 2 1 > > f 0 (x) x a + h b 2a < = 2 b a (b x) (4.4) x a + h f (x) 2 > > 2 : ; b a 2 (b a) 0 h 2 f (a) h [f (a)] + F (x) 8 8 h n oi kf 00 k 3 3 1 > (b x) x a + h b 2a + h b 2a ; f 00 2 L1 [a; b] ; > > 6 > > > > > > h n oi q1 (b x)(x a)kf 00 k > q 2q+1 2q+1 > p > > (x a)q x a + h b 2a h a2 b 1 > > 2(2q+1) q < ; f 00 2 Lp [a; b] ; p > 1; p1 + 1q = 1; > > > 0 1 > > h2 (b a) > > 00 (x a) h (b a) + > 2 x a > @ A (b x)(x a)kf k1 > 2 > 4 h (b a) > > + (x a) + h (b a) > 2 x a > : ; f 00 2 L1 [a; b] :

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We may replace f by F in any of the equations (4.1);(4.3)and (4.4) so that the bounds are in terms of kf 00 kp ; p 1: Further we note that Zb

F (u) du = uF

a

b (u)ja

Zb

xf (x) dx = b

E (X) :

a

Competing interests: The authors declare that they have no competing interests. Authors’contributions: All authors have contributed equally and signi…cantly in writing this article. References [1] P. Cerone, A new Ostrowski Type Inequality Involving Integral Means Over End Intervals, Tamkang Journal Of Mathematics Volume 33, Number 2, 2002. [2] P. Cerone and S.S. Dragomir, Trapezoidal type rules from an inequalities point of view, Handbook of Analytic-Computational Methods in Applied Mathematics, CRC Press N.Y. (2000). [3] X.L. Cheng, Improvement of some Ostrowski-Grüss type inequalities, Comput. Math. Appl. 42 (2001), 109 114. µ [4] P.L. Cebyˆ sev, Sur less expressions approximatives des integrales de…nies par les autres prises entre les memes limites, Proc. Math. Soc. Charkov 2 (1882) 93–98. [5] S. S. Dragomir and S. Wang, An inequality Ostrowski-Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Computers Math. Applic. 33(1997), 15-22. [6] S. S. Dragomir and S. Wang, A new inequality Ostrowski’s type in L 1 norm and applications to some special means and some numerical quadrature rules, Tamkang J. of Math. 28(1997), 239-244. [7] S. S. Dragomir and S. Wang, A new inequality Ostrowski’s type in L p norm, Indian J. of Math. 40(1998), 299-304. [8] S.S. Dragomir and N. S. Barnett, An Ostrowski type inequality for mappings whose second derivatives are bounded and applications, RGMIA Research Report Collection, V.U.T., 1(1999), 67-76. Rb [9] G. Grüss, Uber das maximum des absoluten Betrages von b 1 a f (x) g (x) dx a

1 (b a)2

Rb

a

Rb f (x) dx g (x) dx Math. Z. 39 (1935) 215–226. a

[10] S. Hussain and A. Qayyum, A generalized Ostrowski-Grüss type inequality for bounded di¤erentiable mappings and its applications. Journal of Inequalities and Applications 2013 2013:1. [11] W. Liu, Y. Jiang and A. Tuna, A uni…ed generalization of some quadrature rules and error bounds, Appl. Math. Comp. 219 (2013), 4765-4774. [12] G.V. Milovanovi´c and J. E. Pecari´c, On generalization of the inequality of A. Ostrowski and some related applications, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. (544-576), 155-158, (1976). [13] A. Ostrowski, Über die Absolutabweichung einer di erentienbaren Funktionen von ihren Integralimittelwert, Comment. Math. Hel. 10(1938), 226-227. µ [14] B.G. Pachpatte, On Cebyˆ sev-Grüss type inequalities via Pecari´c’s extention of the Montgomery identity, J. Inequal. Pure Appl. Math. 7 (1), Art. 108, (2006). µ [15] J.E. Pecari´c, On the Cebyˆ sev inequality, Bul. Sti. Tehn. Inst. Politehn ”Tralan Vuia”Timi¸sora (Romania) 25 (39) (1) (1980) 5–9. [16] A. Qayyum and S. Hussain, A new generalized Ostrowski Grüss type inequality and applications, Applied Mathematics Letters 25 (2012) 1875–1880. [17] D. S. Mitrinovi´c , J. E. Pecari´c and A. M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1994.

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A. QAYYUM , 2 M . SHOAIB, AND

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IBRAHIM A FAYE

1 Department of fundamental and applied sciences, Universiti Teknologi Petronas, Tronoh, Malaysia. E-mail address : [email protected] 2 Department

of Mathematics, University of Hail, 2440, Saudi Arabia E-mail address : [email protected] E-mail address : [email protected]

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A Novel Multistep Generalized Di¤erential Transform Method for Solving Fractional-order Lü Chaotic and Hyperchaotic Systems Mohammed Al-Smadi1 , Asad Freihat2 , Omar Abu Arqub3;a , Nabil Shawagfeh3;4 1

Applied Science Department, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan 2 Pioneer Center for Gifted Students, Ministry of Education, Jerash 26110, Jordan 3 Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan 4 Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan a

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

Abstract. This paper investigates the approximate numerical solutions of the fractionalorder Lü chaotic and hyperchaotic systems based on a multistep generalized di¤erential transform method (MGDTM). This method has the advantage of giving an analytical form of the solution within each time interval which is not possible using purely numerical techniques. In addition, this paper presents a comparative study between a new scheme and the classical Runge-Kutta method to demonstrate the applicability of the MGDTM. Furthermore, numerical results are presented graphically and reveal that the proposed scheme is an e¤ective, simple and convenient method for solving nonlinear fractional-order chaotic systems with less computational and iteration steps. keywords: Chaos, Fractional calculus, Lü system, Di¤erential transform method. AMS Subject Classi…cation: 34C28; 34A08; 74H15 1. Introduction Recently, mathematical modeling of chaotic systems becomes a challenging and interesting for contemporary scientists due to its complex, extremely sensitive, unpredictable dynamical behaviors and potential applications in many scienti…c and engineering …elds such as chaos control [1], electronic circuits [2], secure communication [3], information processing [4], power converters [5], electrical engineering [6], biological and chemical systems [7,8], and so on. Therefore, a wide variety of research activities in the …eld of chaotic systems with complex topological structures as well as in related areas have been proposed by many pioneer researchers, for more details see [9-13] and references therein. In particular, there are some detailed investigations and studies of the Lü system which is transition system that bridges the gap between the Lorenz and Chen systems [10]. The characterization of fractional-order dynamical systems has become a powerful tool to describe memory and hereditary properties of various complex materials, processes and models. Whereas many investigations are devoted to generate new fundamentals which exist

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in the case of fractional-order and disappear when it is reduced to integer. However, along with the development of research on chaos, it was found that many fractional-order di¤erential systems behave chaotically, for instance, fractional-order Chen system [14], fractionalorder Lü system [15], fractional-order Lorenz system [16], fractional-order Chua’s system [17], fractional-order Liu system [18], fractional-order Rössler system [19,20], and so forth. In this paper, the MGDTM is implemented to give approximate numerical solutions for the fractional-order Lü chaotic system, in consideration that the hyperchaotic is chaotic system with two positive Lyapunov exponents. These systems are found to be chaotic in a wide parameter range and have many interesting complex dynamical behaviors. In this point, the chaos and hyperchaos exist in the fractional-order Lü system of order as low as 2.4 and 3.28, respectively. In contrast, the proposed method is just a simple modi…cation of the generalized di¤erential transform method (GDTM), in which it is treated as an algorithm in a sequence of small intervals for …nding accurate approximate solutions to the corresponding systems. Moreover, it provides an accurate numerical solutions over a longer time frame compared to the standard GDTM, and has many advantages over the classical methods. The remainder of this paper is organized as follows. In Section 2, we present basic facts and notations related to the fractional calculus and MGDTM. In Section 3, the MGDTM is applied to the fractional-order Lü chaotic and hyperchaotic systems. In Section 4, Examples and corresponding numerical simulations are shown graphically to illustrate the feasibility and e¤ectiveness of the proposed method. Finally, the conclusions are drawn in Section 5. 2. The multistep generalized di¤erential transform method (MGDTM) In this section, we present some basic de…nitions and properties of the fractional calculus theory and MGDTM which will be used in the remainder of this paper. De…nition .1 A function f (x) (x > 0) is said to be in the space C ( 2 R) if it can be written as f (x) = xp f1 (x) for some p > ; where f1 (x) is continuous in [0; 1), and it is said to be in the space C m if f (m) 2 C ; m 2 N: De…nition .2 The Riemann–Liouville integral operator of order (Ja f )(x) =

1 ( )

Rx

(x

t)

1

with a

f (t) dt; x > a;

a

(Ja0 f )(x) = f (x): Now, for f 2 C ; ;

> 0; a

0 is de…ned as

0; c 2 R;

(1) 1, one can get

>

(Ja Ja f ) (x) = (Ja Ja f ) (x) = (Ja + f ) (x); + Ja x = x ( ) B x a ( ; + 1); x

where B ( ; + 1) is the incomplete beta function which is de…ned as R B ( ; + 1) = 0 t 1 (1 t) dt; 1 P [c(x a)]k Ja ecx = eac (x a) : ( +k+1)

(2)

(3)

k = 0

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The Riemann–Liouville derivative has certain disadvantages when trying to model realworld phenomena with fractional di¤erential equations. Thus, we shall introduce a modi…ed fractional di¤erential operator Da proposed by Caputo in his work on the theory of viscoelasticity. For more details about the fractional calculus theory (see [21-23]). De…nition .3 The Caputo fractional derivative of f (x) of order > 0 with a as Z x f (m) (t) m (m) 1 (Da f )(x) = (Ja f ) (x) = (m ) dt; t) +1 m a (x

for m

1
> < yi;2 (t); yi (t) = .. > . > > : y (t); i;M

t 2 [t0 ; t1 ] ; t 2 [t1 ; t2 ] ; .. . t 2 [tM

4

(9)

1 ; tM ] :

Further, this new scheme of our proposed method is simple for computational performance for all values of h, whereas the obtained solution converges for wide time regions. 3. Solving the fractional-order Lü chaotic systems using the MGDTM In this section, in order to illustrate the performance and e¢ ciency of the MGDTM, we applied the method for solving the fractional-order Lü chaotic and hyperchaotic systems. 3.1.The fractional-order Lü chaotic system For fractional-order Lü system which is the lowest-order chaotic system among all the chaotic systems [26], the integer-order derivatives are replaced by the fractional-order derivatives as follows: D 1 x(t) = a(y x); D 2 y(t) = xz + cy; 3 D z(t) = xy bz;

(10)

where (a; b; c) are the system parameters, (x; y; z) are the state variables, and i ; i = 1; 2; 3; are parameters describing the order of the fractional time-derivatives in the Caputo sense. Applying the MGDT algorithm to (10) yields 8 X(k + 1) = a 1 (Y ( ) X( )); > > > k > P < Y (k + 1) = 2 ( X(l)Z(k l) + cY ( )); (11) l=0 > k > P > > : Z(k + 1) = 3 ( X(l)Y (k l) bZ(k)); l=0

i k+1) ; i = 1; 2; 3; X(k); Y (k) and Z(k) are the di¤erential transformation where i = ( (i ( k+1)+1) of x(t); y(t) and z(t); respectively. The di¤erential transform of the initial conditions are given by X(0) = c1 ; Y (0) = c2 and Z(0) = c3 : In view of the di¤erential inverse transform, the di¤erential transform series solution for the system (10) can be obtained as 8 PN < x(t) = P n=0 X(n)t 1 n ; 2n (12) y(t) = N ; n=0 Y (n)t PN : 3n z(t) = n=0 Z(n)t :

According to the MSGDT method, the series solution for system (10) is suggested by

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8 K P > > > X1 (n)t 1 n ; > > n = 0 > > > K > < P X (n)(t t ) 1 n ; 2 1 x(t) = n = 0 > .. > > . > > > K > P > > XM (n)(t tM 1 ) :

t 2 [0; t1 ]; t 2 [t1 ; t2 ]; 1n

n = 0

8 K P > > > Y1 (n)t 2 n ; > > n = 0 > > > K > < P Y (n)(t t ) 2 n ; 2 1 y(t) = n = 0 > . > . > > > .K > > P > > YM (n)(t tM 1 ) : n = 0

8 K P > > > Z1 (n)t 3 n ; > > n = 0 > > > K > < P Z (n)(t t ) 3 n ; 2 1 z(t) = n = 0 > .. > > . > > > K > P > > ZM (n)(t tM 1 ) : n = 0

5

; t 2 [tM

(13)

1 ; tM ];

t 2 [0; t1 ]; t 2 [t1 ; t2 ]; 2n

; t 2 [tM

(14)

1 ; tM ];

t 2 [0; t1 ]; t 2 [t1 ; t2 ]; 3n

; t 2 [tM

(15)

1 ; tM ];

where Xi (n); Yi (n) and Zi (n); for i = 1; 2; : : : ; M; satisfy the following recurrence relations 8 Xi (k + 1) = a 1 (Y i ( ) X i ( )); > > > k > P < Yi (k + 1) = Xi (l)Z i (k l) + Y i (k)); 2( (16) l=0 > k > P > > : Zi (k + 1) = 3 ( Xi (l)Y i (k l) bZ i (k)); l=0

such that Xi (0) = xi (ti 1 ) = xi 1 (ti 1 ); Yi (0) = yi (ti 1 ) = yi 1 (ti 1 ) and Zi (0) = zi (ti 1 ) = zi 1 (ti 1 ): Finally, starting with X0 (0) = c1 ; Y0 (0) = c2 ; Z0 (0) = c3 and using the recurrence relation given in (16), then the multi-step solution can be obtained as in (13)-(15). 3.2.The fractional-order Lü hyperchaotic system Consider the fractional-order Lü hyperchaotic system [27]. In this system, the integer-order derivatives are replaced by the fractional-order derivatives as follows: D D D D

x(t) 2 y(t) 3 z(t) 4 u(t) 1

= a(y x) + u; = xz + cy; = xy bz; = xz + du;

717

(17)

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where (a; b; c; d) are the system parameters, (x; y; z; u) are the state variables and i ; i = 1; 2; 3; 4; are parameters describing the order of fractional time-derivatives in Caputo sense. In the same manner, applying the MGDT algorithm to system (17) yields 8 X(k + 1) = 1 (a(Y ( ) X( )) + U ( )); > > > k > P > > Y (k + 1) = ( X(l)Z(k l) + cY ( ))); > 2 > < l=0 k P (18) Z(k + 1) = ( X(l)Y (k l) bZ(k)); > 3 > > l=0 > > > k P > > : U (k + 1) = 4 ( X(l)Z(k l) dU ( )); l=0

i k+1) where i = ( (i ( k+1)+1) ; i = 1; 2; 3; 4; X(k); Y (k); Z(k) and U (k) are the di¤erential transformation of x(t); y(t); z(t) and u(t); respectively. The di¤erential transform of the initial conditions are given by X(0) = c1 ; Y (0) = c2 ; Z(0) = c3 and U (0) = c4 : In view of the di¤erential inverse transform, the di¤erential transform series solution for the system (17) can be obtained as 8 N P > > x(t) = X(n)t 1 n ; > > > n=0 > > N > P > > Y (n)t 2 n ; < y(t) = n=0 (19) N P > 3n > > z(t) = Z(n)t ; > > n=0 > > > N P > > : u(t) = U (n)t 4 n

n=0

According to the MSGDT method, the series solution for the system (17) is suggested by 8 K P > > > X1 (n)t 1 n ; t 2 [0; t1 ]; > > n=0 > > > K > < P X (n)(t t ) 1 n ; t 2 [t1 ; t2 ]; 2 1 x(t) = n=0 (20) > . > . > . > > > K > P > > XM (n)(t tM 1 ) 1 n ; t 2 [tM 1 ; tM ]; : n=0

8 K P > > > Y1 (n)t 2 n ; > > n=0 > > > K > < P Y (n)(t t ) 2 n ; 2 1 y(t) = n=0 > .. > > > > K. > > P > > YM (n)(t tM 1 ) : n=0

718

t 2 [0; t1 ];

t 2 [t1 ; t2 ]; 2n

; t 2 [tM

(21)

1 ; tM ];

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8 K P > > > Z1 (n)t 3 n ; > > n=0 > > > K > < P Z (n)(t t ) 3 n ; 2 1 z(t) = n=0 > .. > > > > K. > > P > > ZM (n)(t tM 1 ) :

t 2 [0; t1 ]; t 2 [t1 ; t2 ]; 3n

n=0

8 K P > > > U1 (n)t 4 n ; > > n=0 > > > K > < P U (n)(t t ) 4 n ; 2 1 u(t) = n=0 > .. > > . > > > K > P > > UM (n)(t tM 1 ) : n=0

7

; t 2 [tM

(22)

1 ; tM ];

t 2 [0; t1 ]; t 2 [t1 ; t2 ]; 4n

; t 2 [tM

(23)

1 ; tM ];

where Xi (n); Yi (n); Zi (n) and Ui (n); for i = 1; 2; : : : ; M; satisfy the following recurrence relations 8 X (k + 1) = 1 (a(Yi ( ) Xi ( )) + Ui (k)); > > i Pk < X (l)Zi (k l) + Yi (k)); Yi (k + 1) = 2( Pk l=0 i (24) Zi (k + 1) = 3 ( l=0 Xi (l)Yi (k l) bZi (k)); > > P : Ui (k + 1) = 4 ( kl=0 Xi (l)Zi (k l) + dUi (k)));

such that Xi (0) = xi (ti 1 ) = xi 1 (ti 1 ); Yi (0) = yi (ti 1 ) = yi 1 (ti 1 ); Zi (0) = zi (ti 1 ) = zi 1 (ti 1 ) and Ui (0) = ui (ti 1 ) = ui 1 (ti 1 ): Finally, starting with X0 (0) = c1 ; Y0 (0) = c2 ; Z0 (0) = c3 and U0 (0) = c4 and using the recurrence relation that given in (24), then the multi-step solution can be obtained as in (20)-(23). 4. MGDTM simulation results In this section, some numerical simulations are proposed to show the accuracy of the present method. The method provides immediate and visible symbolic terms of analytic solutions as well as numerical approximate solutions to both Lü chaotic and hyperchaotic systems. Results obtained by the MGDTM are compared with the fourth-order Runge-Kutta (RK4) method and are found to be in good agreement. All computations are performed by Mathematica 0.7. However, the time range studied in this work is [0; 25] and the step size t = 0:025. In this regard, we take the initial condition for Lü chaotic system such as x(0) = 1; y(0) = 1 and z(0) = 1; with parameters a = 36; b = 3 and c = 20 and the initial condition for Lü hyperchaotic system such as x(0) = 5; y(0) = 8; z(0) = 1 and u(0) = 3: Figures 1 and 2 show the phase portrait for the classical Lü chaotic system, when 1 = = 2 3 = 1; using the MGDTM against RK4 methods. While that Figure 3 shows the phase portrait for the classical Lü hyperchaotic system, when 1 = 2 = 3 = 4 = 1: Accordingly, from the graphical results of these …gures, it can be seen the results obtained using the MGDTM match the results of the RK4 method very well, which implies that the present method can predict the behavior of these variables accurately for the region under consideration. In contrast, Figure 4 shows the phase portrait for the fractional Lü 719

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chaotic system using the MGDTM. Obviously, the approximate solutions in Figure 4 depend P continuously on the time-fractional derivative i ; i = 1; P 2; 3; where the e¤ective dimension of (10) is de…ned as the sum of orders 1 + 2 + 3 = . In the meanwhile, we found that the chaos exists in the fractional-order Lü chaotic system with order as low as 2:4. Further, Figures 5 and 6 show that the hyperchaos exists in the fractional-order Lü hyperchaotic system using the MGDTM. This system has a hyperchaotic attractor when d = 1:3 with order as low as 3:28.

(a) The MSGDT method

(b) The RK4 method

Figure 1: Phase plot of chaotic attractor in the x

(a) The MSGDT method

y space, with

1=

2=

3=

1:

(b) The RK4 method

Figure 2: Phase plot of chaotic attractor in the x

720

y

z space, with

1=

2=

3=

1:

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.4, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

(a) The MSGDT method

9

(b) The RK4 method

Figure 3: Phase plot of hyperchaotic attractor in the x

y

(a)

z space, with

1=

2=

3=

4=

1:

(b)

Figure 4: Phase plot of chaotic attractor in the x 1 = 0:79;

2 = 0:81;

y

z space, (a) = 0:8: 3

1=

2=

3=

0:9; (b)

5. Conclusions In this paper, a multi-step generalized di¤erential transform method has been successfully applied to …nd the numerical solutions of the fractional-order Lü chaotic and hyperchaotic systems. This method has the advantage of giving an analytical form of the solution within each time interval which is not possible using purely numerical techniques like the fourthorder Runge-Kutta method (RK4). We conclude that the present method is a highly accurate method in solving a broad array of dynamical problems in fractional calculus due to its consistency used in a longer time frame.

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The reliability of the method and the reduction in the size of computational domain give this method a wider applicability. Many of the results obtained in this paper can be extended to signi…cantly more general classes of linear and nonlinear di¤erential equations of fractional order.

(a) in the x

y

z space

(b) in the x

Figure 5: Phase plot of hyperchaotic attractor, with

(a) in the x

y

z space

1=

(b) in the x

Figure 6: Phase plot of hyperchaotic attractor, with

1=

0:80;

2=

z 2=

z

u space 3=

4

= 0:9:

u space

0:83;

3=

0:81;

4

= 0:84:

References [1] Z.M. Ge and T.N., Lin, Chaos, Chaos Control and Synchronization of ElectroMechanical Gyrostat System, J. Sound Vibr., 259(3) (2003), 585-603.

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[2] K.S. Tang, K.F. Man, G.Q. Zhong and G. Chen, Some new circuit design for chaos generation, “in Chaos in Circuits and Systems, ed. Chen, G. & Ueta, T.”, World Scienti…c, Singapore, (2002), 171-190. [3] M. Feki, An adaptive chaos synchronization scheme applied to secure communication, Chaos Solitons Fractals, (2003), 59–64. [4] Q. Xie and G. Chen, Hybrid Chaos Synchronization and Its Application in Information Processing, Mathematical and Computer Modelling, 35 (2002), 145-163. [5] D.C. Hamill, J.H.B. Deane and D.J. Je¤eries, Modelling of chaotic DC-DC converters by iterated non-linear mappings, IEEE Transactions on Power Electronics, PE-7(1) (1992), 25-36. [6] G. Chen, Controlling Chaos and Bifurcations in Engineering Systems, CRC Press, Boca Raton, FL, 1999. [7] S.K. Dana, P.K. Roy and J. Kurths, Complex Dynamics in Physiological Systems: From Heart to Brain, Springer, NewYork, 2009. [8] B. Peng, V. Petrov, and K. Showalter, Controlling Chemical Chaos, J. Phys. Chem., 95 (1991),4957-4959. [9] J. Lü, G. Chen and D. Cheng, A new chaotic system and beyond: the generalized Lorenz-Like system, Int. J. Bif. and Chaos, 14(5) (2004), 1507-1537. [10] J. Lü, G. Chen, D. Cheng and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Int. J. Bif. and Chaos, 12 (2002), 2917-2926. [11] Z.M. Ge and J.S. Shiue, Nonlinear Dynamics and Control of Chaos for Tachometer, J. Sound Vibr., 253(4) (2002), 773-793. [12] J. Lü, G. Chen and S. Zhang, Dynamical analysis of a new chaotic attractor, Int. J. Bif. and Chaos, 12 (2002), 1001-1015. [13] B.R. Andrievskii and A.L. Fradkov, Control of chaos: methods and applications, II. Applications, Automation and Remote Control, 65(4) (2004), 505-533. [14] W.H. Deng and C.P. Li, Synchronization of Chaotic Fractional Chen System, J. Phys. Soc. Jpn., 74 (2005), 1645-1648. [15] W.H. Deng and C.P. Li, Chaos synchronization of the fractional Lü system, Physica A: Statistical Mechanics and its Applications, 353 (2005), 61-72. [16] L. Chen, Y. Chai and R. Wu, Linear matrix inequality criteria for robust synchronization of uncertain fractional-order chaotic systems, Chaos, 21 (4) (2011), Art. ID 043107. [17] A. Freihat and S. Momani, Application of Multistep Generalized Di¤erential Transform Method for the Solutions of the Fractional-Order Chua’s System, Discrete Dynamics in Nature and Society, 2012 (2012), Art. ID 427393, 12 pages. 723

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[18] A. Freihat and M. AL-Smadi, A new reliable algorithm using the generalized di¤erential transform method for the numeric-analytic solution of fractional-order Liu chaotic and hyperchaotic systems, Pensee Journal, 75(9) (2013), 263-276. [19] C. Li and G. Chen, Chaos and hyperchaos in the fractional-order Rössler equations, Physica A-statistical Mechanics and Its Applications, 341 (2004), 55-61. [20] A. Freihat and S. Momani, Adaptation of Di¤erential Transform Method for the Numeric-Analytic Solution of Fractional-Order Rössler Chaotic and Hyperchaotic Systems, Abstr. Appl. Anal. 2012 (2012), Art. ID 934219, 13 pages. [21] S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993. [22] C.P. Li and W.H. Deng, Remarks on fractional derivatives, Applied Mathematics and Computation, 187 (2007), 777–784. [23] I. Podlubny, Fractional Di¤erential Equations, Academic Press, New York, 1999. [24] Z. Odibat, S. Momani and V.S. Ertürk, Generalized di¤erential transform method: Application to di¤erential equations of fractional order, Appl. Math. Comput., 197 (2008), 467-477. [25] V. S. Ertürk, S. Momani and Z. Odibat, Application of generalized di¤erential transform method to multi-order fractional di¤erential equations, Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 1642-1654. [26] J. Lü, Chaotic dynamics of the fractional order Lü system and its synchronization, Phys. Lett. A, 354 (2006), 05–11. [27] A. Chen, J. Lü and S. Yu, Generating hyperchaotic Lü attractor via state feedback control, Physica A, 364 (2006), 103-110.

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NEW INEQUALITIES OF HERMITE-HADAMARD AND FEJÉR TYPE VIA PREINVEXITY M. A. LATIF 1;2 AND S. S. DRAGOMIR 3;4 Abstract. Several new weighted inequalities connected with Hermite-Hadamard and Fejér type inequalities are established for functions whose derivatives in absolute value are preinvex. The results presented in this paper provide extensions of those given in earlier works.

1. Introduction Many inequalities have been established for convex functions but the most famous is the Hermite-Hadamard inequality, due to its rich geometrical signi…cance and applications, which is stated as (see [23]): Let f : I R ! R be a convex mapping and a; b 2 I with a < b. Then Z b a+b 1 f (a) + f (b) (1.1) f : f (x)dx 2 b a a 2 Both the inequalities hold in reversed direction if f is concave. In [7], Fejér gave a weighted generalization of (1.1) as follows: Z b Z b Z a+b f (a) + f (b) b (1.2) f w(x)dx f (x)w(x)dx w(x)dx; 2 2 a a a where f : [a; b] ! R be a convex function and f : [a; b] ! R is nonnegative, integrable and symmetric about x = a+b 2 . For several results which generalize, improve and extend the inequalities (1.1) and (1.2) we refer the interested reader [5, 6, 8], [10]-[13], [23, 24], [26]-[31]. In recent years, lot of e¤orts have been made by many mathematicians to generalize the classical convexity. These studies include among others the work of Hanson in [9], Ben-Israel and Mond [4], Pini [21], M.A.Noor [18, 19], Yang and Li [33] and Weir [32]. Mond [5], Weir [32] and Noor [18, 19], have studied the basic properties of the preinvex functions and their role in optimization, variational inequalities and equilibrium problems. Hanson in [9], introduced invex functions as a signi…cant generalization of convex functions. Ben-Israel and Mond [4], gave the concept of preinvex function which is special case of invexity. Pini [21], introduced the concept of prequasiinvex functions as a generalization of invex functions. Let us recall some known results concerning invexity and preinvexity. Date : Today. 2000 Mathematics Subject Classi…cation. 26D15, 26D20, 26D07. Key words and phrases. Hermite-Hadamard’s inequality,Fejér’s inequality, invex set, preinvex function, Hölder’s integral inequality. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1

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M . A. LATIF 1;2 AND S. S. DRAGO M IR 3;4

2

Rn is said to be invex if there exists a function

A set K that

:K

K ! R such

x + t (y; x) 2 K; 8x; y 2 K; t 2 [0; 1]: The invex set K is also called a -connected set. De…nition 1. [32] The function f on the invex set K is said to be preinvex with respect to , if f (u + t (v; u))

(1

t) f (u) + tf (v); 8u; v 2 K; t 2 [0; 1]:

The function f is said to be preconcave if and only if

f is preinvex.

It is to be noted that every convex function is preinvex with respect to the map (x; y) = x y but the converse is not true see for instance [32]. In the recent paper, Noor [16] has obtained the following Hermite-Hadamard inequalities for the preinvex functions: Theorem 1. [16] Let f : [a; a + (b; a)] ! (0; 1) be a preinvex function on the interval of the real numbers K (the interior of K) and a, b 2 K with a < a + (b; a). Then the following inequality holds: (1.3)

f

1 (b; a)

2a + (b; a) 2

Z

a+ (b;a)

f (x) dx

a

f (a) + f (b) : 2

For several new results on inequalities for preinvex functions we refer the interested reader to [3, 15, 20, 25] and the references therein. In the present paper we give new inequalities of Hermite-Hadamard and Fejér type for functions whose derivatives are preinvex. Our results generalize those results presented in a very recent papers of M. Z. Sarikaya [25, 28].

2. Main Results The following Lemma is essential in establishing our main results in this section: Lemma 1. Let K R be an open invex subset with respect to : K K ! R and a, b 2 K with a < a + (b; a). Suppose f : K ! R is a di¤ erentiable mapping on K 0 such that f 2 L ([a; a + (b; a)]). If w : [a; a + (b; a)] ! [0; 1) is an integrable mapping, then the following equality holds: (2.1)

1 (b; a)

Z

a+ (b;a)

f (x) w(x)dx

a

Z a+ (b;a) 1 a+ (b; a) w(x)dx 2 a Z 1 0 = (b; a) k(t)f (a + t (b; a)) dt;

1 f (b; a)

0

where k(t) =

8 < :

Rt 0

w (a + s (b; a)) ds;

R1 t

w (a + s (b; a)) ds;

726

t 2 0; 12 t2

1 2; 1

:

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HERM ITE-HADAM ARD AND FEJÉR TYPE INEQUALITIES VIA PREINVEXITY

3

Proof. We observe that (2.2) I =

Z

1

0

k(t)f (a + t (b; a)) dt

0

=

Z

Z

1 2

+

1

0

w (a + s (b; a)) ds f (a + t (b; a)) dt

0

0

Z

t

Z

1 2

1

0

w (a + s (b; a)) ds f (a + t (b; a)) dt = I1 + I2 :

t

By integration by parts, we get

(2.3) I1 =

Z

t

w (a + s (b; a)) ds

0

1 (b; a)

Z

f (a + t (b; a)) (b; a)

1 2

0

1 2

w (a + t (b; a)) f (a + t (b; a)) dt

0

f a + 12 (b; a) = (b; a)

Z

1 2

w (a + t (b; a)) dt

0

Z

1 (b; a)

1 2

w (a + t (b; a)) f (a + t (b; a)) dt:

0

Similarly, we also have

(2.4) I2 =

Z

1

w (a + s (b; a)) ds

t

1 (b; a)

Z

f (a + t (b; a)) (b; a)

1 1 2

1

w (a + t (b; a)) f (a + t (b; a)) dt 1 2

f a + 12 (b; a) = (b; a)

Z

1

w (a + t (b; a)) dt 1 2

1 (b; a)

Z

1

w (a + t (b; a)) f (a + t (b; a)) dt: 1 2

From (2.3) and (2.4), we get

I=

f a + 12 (b; a) (b; a)

Z

1

w (a + t (b; a)) dt

0

1 (b; a)

Z

1

w (a + t (b; a)) f (a + t (b; a)) dt:

0

Using the change of variable x = a + t (b; a) for t 2 [0; 1] and multiplying both sides by (b; a), we get (2.1). This completes the proof of the lemma.

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M . A. LATIF 1;2 AND S. S. DRAGO M IR 3;4

4

Remark 1. If we take w(x) = 1, x 2 [a; a + t (b; a)] in Lemma 1, then (2.1) reduces to 1 (b; a)

(2.5)

Z

a+ (b;a)

f (x) dx + f

a+

a

1 (b; a) 2 = (b; a)

Z

1

0

k(t)f (a + t (b; a)) dt;

0

where k(t) =

8 < t; :

t

t 2 0; 21 1;

1 2; 1

t2

:

Which is one of the results from [25]. Remark 2. If [28, page 379].

(b; a) = b

a in Lemma 1, then (2.1) becomes Lemma 2.1 from

Now using Lemma 1, we prove our results: Theorem 2. Let K R be an open invex subset with respect to : K K ! R and a, b 2 K with a < a + (b; a). Suppose f : K ! R is a di¤ erentiable mapping on K 0 such that f 2 L ([a; a + (b; a)]) and w : [a; a + (b; a)] ! [0; 1) is an integrable 0 mapping and symmetric to a + 21 (b; a). If f is preinvex on K, then we have the following inequality: (2.6) 1 (b; a)

Z

a+ (b;a)

f (x) w(x)dx

a

1 (b; a)

Z

1 f (b; a)

a+ 12 (b;a)

[ (b; a)

2 (x

a

Z a+ (b;a) 1 (b; a) w(x)dx 2 a 1 !0 0 0 f (a) + f (b) A: a)] w(x)dx @ 2 a+

Proof. From Lemma 1 and the preinvexity of f (2.7) 1 (b; a)

Z

a+ (b;a)

f (x) w(x)dx

a

(b; a)

Z

1 2

0

+ (b; a)

Z

Z

0 1

1 2

1 f (b; a)

t

w (a + s (b; a)) ds Z

1

0

a+ h (1

w (a + s (b; a)) ds

t

728

on K, we have

1 (b; a) 2

h

Z

a+ (b;a)

w(x)dx

a

i 0 0 t) f (a) + t f (b) dt (1

i 0 0 t) f (a) + t f (b) dt

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By the change of the order of integration, we have (2.8)

Z

Z

1 2

0

0

Z

Z

Z

i h 0 0 w (a + s (b; a)) (1 t) f (a) + t f (b) dtds 0 s " ! # 2 0 0 1 s2 (1 s) 1 f (a) + f (b) ds: w (a + s (b; a)) 2 8 8 2

= =

i 0 0 t) f (a) + t f (b) dsdt

h w (a + s (b; a)) (1

t

1 2

0

1 2

1 2

Using the change of variable x = a + s (b; a) for s 2 [0; 1], we have from (2.8) that (2.9)

Z

Z

1 2

0

0

0 1 f (a) (b; a)

=

i 0 0 t) f (a) + t f (b) dsdt

h w (a + s (b; a)) (1

t

+

Z

a+ 12 (b;a)

1 2

a

0 1 f (b) (b; a)

Z

2

x a (b; a)

1

a+ 12 (b;a)

a

1 8

1 2

1 8

!

w (x) dx

x a (b; a)

2

!

w (x) dx:

Similarly by change of order of integration and using the fact that w is symmetric to a + 21 (b; a), we obtain (2.10)

Z

1 1 2

=

Z

1

t

Z

1 1 2

h w (a + s (b; a)) (1 Z

w (a + (1

0 1 f (a) (b; a)

=

h s) (b; a)) (1

s

1 2

Z

+

i 0 0 t) f (a) + t f (b) dsdt

1

1 (1 2

1 8

1 2

0 1 f (b) (b; a)

By the change of variable x = a + (1 (2.11)

Z

1 1 2

Z

t

=

1

h w (a + s (b; a)) (1 0 1 f (a) (b; a)

+

Z

2

s) Z

1 1 2

w (a + (1 s2 2

1 8

s) (b; a)) ds

w (a + (1

s) (b; a)) ds

s) (b; a), we get form (2.10) that i 0 0 t) f (a) + t f (b) dsdt

a+ 12 (b;a)

a

0 1 f (b) (b; a)

i 0 0 t) f (a) + t f (b) dtds

Z

1 8

a+ 12 (b;a)

a

1 2 1 2

x a (b; a) 1

2

!

x a (b; a)

w (x) dx 2

1 8

!

w (x) dx:

Substituting (2.9) and (2.11) in (2.7) and simplifying, we get the inequality (2.6). This completes the proof of the theorem.

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6

Corollary 1. If we take w(x) = 1, for x 2 [a; a + (b; a)] in Theorem 2, we get (2.12) Z a+ (b;a) i 0 1 1 (b; a) h 0 f (a) + f (b) : f (x) dx f a + (b; a) (b; a) a 2 8 Which is Theorem 5 from [25]. Remark 3. If f

0

is convex on [a; b], then

(b; a) = b

a+b 2 ,

2, and using the symmetricity of w about 380].

a. Hence from Theorem

we get Theorem 2.3 from [28, page

Theorem 3. Let K R be an open invex subset with respect to : K K ! R and a, b 2 K with a < a + (b; a). Suppose f : K ! R is a di¤ erentiable mapping 0 on K such that f 2 L ([a; a + (b; a)]) and w : [a; a + (b; a)] ! [0; 1) is an

integrable mapping and symmetric to a + K, then we have the following inequality: (2.13) 1 (b; a)

Z

a+ (b;a)

a

(b; a)

where

1 p

+

1 f (b; a)

f (x) w(x)dx

1 q

Z

1

1 2

a+ 12 (b;a)

(b; a). If f

0

q

, q > 1, is preinvex on

Z

1 a+ (b; a) 2

a+ (b;a)

w(x)dx

a

(b; a) 2

! p1

p

(x a) w (x) dx 2 ( (b; a)) a 3 2 0 0 0 q q 1 q1 q q 1 q1 0 0 0 f (a) + 2 f (b) 7 6 2 f (a) + f (b) 6@ A +@ A 7; 5 4 24 24

= 1.

Proof. From Lemma 1 and change of order of integration, we get (2.14) 1 (b; a)

Z

a+ (b;a)

1 f (b; a)

f (x) w(x)dx

a

(b; a) + (b; a)

Z

Z

= (b; a)

Z

1 2

0 1 1 2

Z

a+

t

Z

1 (b; a) 2

a+ (b;a)

w(x)dx

a

0

w (a + s (b; a)) ds

f (a + t (b; a)) dt

w (a + s (b; a)) ds

f (a + t (b; a)) dt

0 1

0

t

Z

1 2

0

Z

1 2

0

w (a + s (b; a)) f (a + t (b; a)) dtds

s

+ (b; a)

Z

1 1 2

Z

s

0

w (a + s (b; a)) f (a + t (b; a)) dtds: 1 2

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7

By the Hölder’s inequality, we have (2.15)

(b; a)

Z

1 2

0

Z

(b; a)

1 2

0

Since f

0

Z

1 2

0

w (a + s (b; a)) f (a + t (b; a)) dtds

s

Z

1 2

! p1

p

Z

w (a + s (b; a)) dtds

s

1 2

0

Z

1 2

! q1

q

0

f (a + t (b; a)) dtds

s

:

q

for q > 1, is preinvex on K, for every a; b 2 K and t 2 [0; 1] we have q

0

q

0

q

0

t) f (a) + t f (b)

(1

f (a + t (b; a))

hence by solving elementary integrals and using the substitution x = a + s (b; a), s 2 [0; 1], we have from (2.15) that (2.16)

(b; a)

Z

1 2

0

Z

1 2

0

w (a + s (b; a)) f (a + t (b; a)) dtds

s

Z

(b; a)

1 2

0

Z

1 2

0

Z

h

1 2

s

1

= (b; a)

2

( (b; a))

(1 Z

Z

1 2

! p1

p

w (a + s (b; a)) dtds

s q

0

t) f (a) + t f (b) a+ 12 (b;a)

a

(b; a) 2

Analogously, using the symmetricity of w about a +

(b; a)

Z

1 1 2

(b; a)

Z

s

! q1

dtds

! p1

p

(x

a) w (x) dx

0

2 f (a) + f (b)

@

(2.17)

qi

0

1 2

0

q

0

24

q 1 q1

A :

(b; a), we also have

0

w (a + s (b; a)) f (a + t (b; a)) dtds 1 2

1 2

( (b; a))

Z

a+ 12 (b;a)

a

(b; a) 2

(x 0 @

! p1

p

a) w (x) dx 0

q

0

f (a) + 2 f (b) 24

q 1 q1

A

Using (2.16) and (2.17) in (2.14), we get the required inequality. This completes the proof of the theorem.

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8

Corollary 2. If the conditions of Theorem 3 are satis…ed and if w(x) = 1, x 2 [a; a + (b; a)], then the following inequality holds: (2.18)

where

Z

1 (b; a)

1 p

+

1 q

a+ (b;a)

1 a+ (b; a) 2

f (x) dx + f

a

(b; a)

1 8

1 p

2 3 0 0 0 q 1 q1 q 1 q1 q q 0 0 0 f (a) + 2 f (b) 6 2 f (a) + f (b) 7 6@ A +@ A 7; 4 5 24 24

= 1.

Corollary 3. [28, Theorem 2.5, page 381] Suppose f : I R ! R is a di¤ erentiable mapping on I , a, b 2 I with a < b. Let w : [a; b] ! [0; 1) be an integrable mapping 0

and symmetric to a+b 2 such that f 2 L ([a; b]). If f then we have the following inequality: 1

(2.19)

b

a

Z

b

1 p

+

1 q

b

a

(b

where

1

f (x) w(x)dx

a)

1 (b

2

a)

a Z

a+b 2

f

b

x a+b 2

Z

0

q

, q > 1, is convex on [a; b],

b

w(x)dx

a

a+b 2

p

w (x) dx

! p1

2 3 0 0 0 q q 1 q1 q q 1 q1 0 0 0 f (a) + 2 f (b) 6 2 f (a) + f (b) 7 6@ A +@ A 7; 4 5 24 24

= 1.

Proof. It follows from Theorem 3 by taking (b; a) = b of w about a+b 2 .

a and using the symmetry

For our next results we need the following Lemma: Lemma 2. Let K R be an open invex subset with respect to : K K ! R and a, b 2 K with a < a + (b; a). Suppose f : K ! R is a di¤ erentiable mapping on K 0 such that f 2 L ([a; a + (b; a)]). If w : [a; a + (b; a)] ! [0; 1) is an integrable mapping, then the following inequality holds: (2.20)

f (a) + f (a + (b; a) 2 (b; a)

Z

a+ (b;a)

Z a+ (b;a) 1 f (x) w(x)dx (b; a) a Z 0 (b; a) 1 p(t)f (a + t (b; a)) dt; 2 0

w(x)dx +

a

= where p(t) =

Z

t

1

w (a + s (b; a)) ds

Z

0

732

t

w (a + s (b; a)) ds; t 2 [0; 1] .

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HERM ITE-HADAM ARD AND FEJÉR TYPE INEQUALITIES VIA PREINVEXITY

9

Proof. It su¢ ces to note that (2.21) J =

Z

1

0

p(t)f (a + t (b; a)) dt

0

Z

=

1

+

t

0

w (a + s (b; a)) ds f (a + t (b; a)) dt

0

0

Z

Z

1

Z

1

0

w (a + s (b; a)) ds f (a + t (b; a)) dt = J1 + J2

t

0

By integration by parts, we get

(2.22) J1 =

Rt 0

1

w (a + s (b; a)) ds f (a + t (b; a))

+

(b; a) 1 (b; a) =

Z

0 1

w (a + t (b; a)) f (a + t (b; a)) dt

0

f (a + (b; a)) (b; a) +

Z

1

w (a + t (b; a)) dt

0

1 (b; a)

Z

1

w (a + t (b; a)) f (a + t (b; a)) dt:

0

Similarly. we also have (2.23) Z 1 Z 1 f (a) 1 J2 = w (a + t (b; a)) dt+ w (a + t (b; a)) f (a + t (b; a)) dt: (b; a) 0 (b; a) 0 Using (2.22) and (2.23) in (2.21), we obtain (2.24) J =

f (a) + f (a + (b; a)) (b; a)

Z

1

w (a + t (b; a)) dt

0

Z

2 (b; a)

+

1

w (a + t (b; a)) f (a + t (b; a)) dt

0

By the change of variable x = a + t (b; a) for t 2 [0; 1] and by multiplying both sides if (2.6) by (b;a) 2 , we get (2.20). This completes the proof of the lemma. Remark 4. If we take w(x) = 1, x 2 [a; a + (b; a)], then we get (2.25)

f (a) + f (a + (b; a) + 2

1 (b; a)

Z

=

a+ (b;a)

f (x) dx

a

(b; a) 2

Z

1

(1

0

2t) f (a + t (b; a)) dt;

0

which is Lemma 2.1 from [3, Page 3]. Theorem 4. Let K R be an open invex subset with respect to : K K ! R and a, b 2 K with a < a + (b; a). Suppose f : K ! R is a di¤ erentiable mapping 0 on K such that f 2 L ([a; a + (b; a)]) and w : [a; a + (b; a)] ! [0; 1) is an

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10

1 2

integrable mapping and symmetric to a + K, then we have the following inequality: (2.26)

f (a) + f (a + (b; a) 2 (b; a)

Z

(b; a). If f

a+ (b;a)

w(x)dx

a

Z

1 2

1 (b; a) 1 p

1

g p (t)dt

0

where g(t) =

Z

0 @

q

0

, q > 1, is preinvex on Z

f (x) w(x)dx

a

0

q

0

f (a) + f (b)

a+(1 t) (b;a)

w (x) dx ; t 2 [0; 1] and

a+t (b;a)

a+ (b;a)

2

q 1 q1

A ;

1 1 + = 1: p q

Proof. From Lemma 2, we get Z Z a+ (b;a) 1 f (a) + f (a + (b; a) a+ (b;a) w(x)dx f (x) w(x)dx (2.27) 2 (b; a) (b; a) a a Z t Z Z 1 0 (b; a) 1 w (a + s (b; a)) ds w (a + s (b; a)) ds f (a + t (b; a)) dt: 2 t 0 0 Since w is symmetric to a + (2.28)

Z

1 2

(b; a), we can write Z

1

w (a + s (b; a)) ds

t

=

Z

w (a + s (b; a)) ds

0

1

Z

w (a + s (b; a)) ds

t

=

t

1 (b; a)

Z

t

w (a + (1

s) (b; a)) ds

0

a+ (b;a)

a+t (b;a)

Z a+(1 t) (b;a) 1 w (x) dx + w (x) dx (b; a) a+ (b;a) 8 R a+(1 t) (b;a) 1 > w (x) dx; t 2 0; 12 < (b;a) a+t (b;a) = R a+t (b;a) > : 1 1 (b;a) a+(1 t) (b;a) w (x) dx; t 2 2 ; 1 :

Using (2.28) in (2.27) we obtain

(2.29)

f (a) + f (a + (b; a) 2 (b; a)

Z

a+ (b;a)

a

Z a+ (b;a) 1 w(x)dx f (x) w(x)dx (b; a) a Z 0 1 1 g(x) f (a + t (b; a)) dt; 2 0

where g(t) =

Z

a+(1 t) (b;a)

a+t (b;a)

w (x) dx ; t 2 [0; 1] :

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11

By Hölder’s inequality, it follows from (2.29) that f (a) + f (a + (b; a) 2 (b; a)

(2.30)

Z

a+ (b;a)

a

Z

1 2

1 p

1 p

g (t)dt

Z

1

Z

a+ (b;a)

f (x) w(x)dx

a 1 q

q

0

f (a + t (b; a)) dt

:

0

0

q

0

1 (b; a)

w(x)dx

is preinvex on K, for every a, b 2 K and t 2 [0; 1], we have

Since f (a + t (b; a))

q

0

f (a + t (b; a))

q

0

q

0

t) f (a) + t f (b)

(1

and hence from (2.30), we get that f (a) + f (a + (b; a) 2 (b; a)

(2.31)

1 2

Z

Z

w(x)dx

a

1 p

1

a+ (b;a)

p

g (t)dt

0

Z

1

0

=

h (1 1 2

0

1 (b; a)

f (a) + f (b) 2 (b a)

(2.32)

Z

(b; a) = b

b

w(x)dx

a

g(t) =

Z

0

t) f (a) + t f (b) Z

1 p

1

g p (t)dt

0

0 @

1 b

a

Z

0

qi

1 q

dt

q

0

f (a) + f (b) 2

q 1 q1

A ;

b

f (x) w(x)dx

a

Z

1 p

1

g p (t)dt

0

0 @

ta+(1 t)b

tb+(1 t)a

f (x) w(x)dx

a in Theorem 4, then we have the inequality:

1 2 where

a+ (b;a)

a

q

which completes the proof of the theorem. Corollary 4. If we take

Z

w (x) dx ; t 2 [0; 1] and

0

q

0

f (a) + f (b) 2

q 1 q1

A ;

1 1 + = 1: p q

Which is Theorem 2.8 from [28, page 383]. Corollary 5. [3, Theorem 2.2, page 4] Under the assumptions of Theorem 4, if we take w (x) = 1, x 2 [a; a + (b; a)]. Then f (a) + f (a + (b; a)) 2

(2.33)

1 (b; a)

Z

a+ (b;a)

f (x) dx

a

(b; a) 1

2 (p + 1) p where

1 p

+

1 q

= 1:

735

0 @

0

q

0

f (a) + f (b) 2

q 1 q1

A ;

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12

Proof. It follows from the fact that Z 1 Z 1 Z p g (t)dt = 0

a+(1 t) (b;a)

dx

a+t (b;a)

0

p

= ( (b; a))

Z

0

p!

dt p

1

p

j1

2tj dt =

( (b; a)) : p+1

Corollary 6. [5] If the conditions of Theorem 4 are ful…lled and if w (x) = 1, x 2 [a; b] and (b; a) = b a, then we have the inequality: f (a) + f (b) 2

(2.34) where

1 p

+

1 q

1 b

a

Z

b

b

f (x) dx

a 1

2 (p + 1) p

a

= 1.

0 @

0

q

0

f (a) + f (b) 2

q 1 q1

A ;

Proof. It follows from Corollary 5. 3. Applications to Special Means In what follows we give certain generalizations of some notions for a positive valued function of a positive variable. De…nition 2. [31]A function M : R2+ ! R+ , is called a Mean function if it has the following properties: (1) (2) (3) (4) (5)

Homogeneity: M (ax; ay) = aM (x; y); for all a > 0, Symmetry : M (x; y) = M (y; x), Re‡exivity : M (x; x) = x, 0 0 0 0 Monotonicity: If x x and y y , then M (x; y) M (x ; y ), Internality: minfx; yg M (x; y) maxfx; yg.

We consider some means for arbitrary positive real numbers , [31]).

(see for instance

(1) The arithmetic mean: + 2

A := A ( ; ) = (2) The geometric mean: G := G ( ; ) = (3) The harmonic mean: H := H ( ; ) =

p 1

2 +

1

(4) The power mean: r

Pr := Pr ( ; ) =

736

+ 2

r

1 r

,r

1

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HERM ITE-HADAM ARD AND FEJÉR TYPE INEQUALITIES VIA PREINVEXITY

(5) The identric mean: I := I ( ; ) =

(

1 e

;

13

6= =

;

(6) The logarithmic mean: L := L ( ; ) = (7) The generalized log-mean:

ln j j

p+1

Lp := Lp ( ; ) =

ln j j

; j j 6= j j

p+1

(p + 1) (

)

;

6= , p 2 Rn f 1; 0g .

It is well known that Lp is monotonic nondecreasing over p 2 R, with L 1 := L and L0 := I. In particular, we have the following inequality H G L I A. Now, let a and b be positive real numbers such that a < b. Consider the function M := M (a; b) : [a; a + (b; a)] [a; a + (b; a)] ! R+ , which is one of the above mentioned means, therefore one can obtain variant inequalities for these means as follows: Setting (b; a) = M (b; a) in (2.12), (2.18) and (2.33), one can obtain the following interesting inequalities involving means: (3.1)

(3.2)

and (3.3)

1 M (b; a)

1 M (b; a)

Z

a+M (b;a)

f (x) dx

1 a + M (b; a) 2

f

a

Z

a

i 0 M (b; a) h 0 f (a) + f (b) ; 8 a+M (b;a)

1 a + M (b; a) 2

f (x) dx + f

M (b; a)

1 8

1 p

3 2 0 0 0 q q 1 q1 q q 1 q1 0 0 0 f (a) + 2 f (b) 7 6 2 f (a) + f (b) 6@ A +@ A 7 5 4 24 24

f (a) + f (a + M (b; a)) 2

1 M (b; a)

Z

a+M (b;a)

f (x) dx

a

0

M (b; a) @ 1 2 (p + 1) p

0

q

0

f (a) + f (b) 2

q 1 q1

A :

Letting M = A, G, H, Pr , I, L, Lp in (3.1), (3.2) and (3.3), we can get the required inequalities, and the details are left to the interested reader. References [1] T. Antczak, Mean value in invexity analysis, Nonl. Anal., 60 (2005), 1473-1484. [2] A. Barani, A.G. Ghazanfari, S.S. Dragomir, Hermite-Hadamard inequality through prequsiinvex functions, RGMIA Research Report Collection, 14(2011), Article 48, 7 pp.

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M . A. LATIF 1;2 AND S. S. DRAGO M IR 3;4

[3] A. Barani, A.G. Ghazanfari, S.S. Dragomir, Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, RGMIA Research Report Collection, 14(2011), Article 64, 11 pp. [4] A. Ben-Israel and B. Mond, What is invexity?, J. Austral. Math. Soc., Ser. B, 28(1986), No. 1, 1-9. [5] S. S. Dragomir, and R. P. Agarwal, Two inequalities for di¤erentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11(5)(1998), 91–95. [6] S. S. Dragomir, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl., 167(1992), 42–56. [7] L. Fejer, Über die Fourierreihen, II. Math. Naturwiss. Anz Ungar. Akad. Wiss., 24(1906), 369–390. (Hungarian). [8] D. -Y. Hwang, Some inequalities for di¤erentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables, Appl. Math. Comp., 217(23)(2011), 9598-9605. [9] M. A. Hanson, On su¢ ciency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545-550. [10] U. S. K¬rmac¬, Inequalities for di¤erentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147(1)(2004), 137-146. [11] U. S. K¬rmac¬ and M. E. Özdemir, On some inequalities for di¤erentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 153(2)(2004), 361-368. [12] K. C. Lee and K. L. Tseng, On a weighted generalization of Hadamard’s inequality for Gconvex functions, Tamsui-Oxford J. Math. Sci., 16(1)(2000), 91–104. [13] A. Lupas, A generalization of Hadamard’s inequality for convex functions, Univ. Beograd. Publ. Elek. Fak. Ser. Mat. Fiz., 544-576(1976), 115–121. [14] S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl. 189 (1995), 901–908. [15] M. Matloka, On some Hadamard-type inequalities for (h1 ; h2 )-preinvex functions on the coordinates. (Submitted) [16] M. Aslam Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, Preprint. [17] M. A. Noor, Variational-like inequalities, Optimization, 30 (1994), 323–330. [18] M. A. Noor, Invex equilibrium problems, J. Math. Anal. Appl., 302 (2005), 463–475. [19] M. A. Noor, Some new classes of nonconvex functions, Nonl. Funct. Anal. Appl.,11(2006),165171 [20] M. A. Noor, On Hadamard integral inequalities involving two log-preinvex functions, J. Inequal. Pure Appl. Math., 8(2007), No. 3, 1-14. [21] R. Pini, Invexity and generalized convexity, Optimization 22 (1991) 513-525. [22] C. E. M. Pearce and J. Peµcari´c, Inequalities for di¤erentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13(2)(2000), 51–55. [23] F. Qi, Z. -L.Wei and Q. Yang, Generalizations and re…nements of Hermite-Hadamard’s inequality, Rocky Mountain J. Math., 35(2005), 235–251. [24] J. Peµcari´c, F. Proschan and Y. L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991. [25] M. Z. Sarikaya, H. Bozkurt and N. Alp, On Hermite-Hadamard type integral inequalities for preinvex and log-preinvex functions, arXiv:1203.4759v1. [26] M. Z. Sar¬kaya and N. Aktan, On the generalization some integral inequalities and their applications Mathematical and Computer Modelling, 54(9-10)(2011), 2175-2182. [27] M. Z. Sarikaya, M. Avci and H. Kavurmaci, On some inequalities of Hermite-Hadamard type for convex functions, ICMS International Conference on Mathematical Science, AIP Conference Proceedings 1309, 852(2010). [28] M. Z. Sarikaya, On new Hermite-Hadamard Fejér type integral inequalities, Stud. Univ. Babe¸s-Bolyai Math. 57(2012), No. 3, 377-386. [29] A. Saglam, M. Z. Sarikaya and H. Y¬ld¬r¬m, Some new inequalities of Hermite-Hadamard’s type, Kyungpook Mathematical Journal, 50(2010), 399-410. [30] C. -L. Wang and X. -H. Wang, On an extension of Hadamard inequality for convex functions, Chin. Ann. Math., 3(1982), 567–570.

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[31] S. -H. Wu , On the weighted generalization of the Hermite-Hadamard inequality and its applications, The Rocky Mountain J. of Math., 39(2009), no. 5, 1741–1749. [32] T. Weir, and B. Mond, Preinvex functions in multiple objective optimization, Journal of Mathematical Analysis and Applications, 136 (1998) 29-38. [33] X. M. Yang and D. Li, On properties of preinvex functions, J. Math. Anal. Appl. 256 (2001), 229-241. 1 College

of Science, Department of Mathematics, University of Hail, Hail 2440, Saudi Arabia,, 2 School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa E-mail address : [email protected] 3 School of Engineering and Science, Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia, , 4 School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa E-mail address : [email protected]

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APPROXIMATE QUADRATIC FORMS IN PARANORMED SPACES JAE-HYEONG BAE AND WON-GIL PARK* Abstract. In this paper, we investigate approximate quadratic forms related to the functional equations f (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w) and f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) + 2f (y, w) in paranormed spaces.

1. Introduction The concept of statistical convergence for sequences of real numbers was introduced by Fast [3] and Steinhaus [15] independently, and since then several generalizations and applications of this notion have been investigated by various authors (see [4, 7, 9, 10, 14]). This notion was defined in normed spaces by Kolk [8]. We recall some basic facts concerning Fr´echet spaces (see [17]). Definition. Let X be a vector space. A paranorm on X is a function P : X → R such that for all x, y ∈ X (i) P (0) = 0; (ii) P (−x) = P (x); (iii) P (x + y) ≤ P (x) + P (y) (triangle inequality); (iv) 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 scalar multiplication). The pair (X, P ) is called a paranormed space if P is a paranorm on X. Note that P (nx) ≤ nP (x) for all n ∈ N and all x ∈ (X, P ). The paranorm P on X is called total if, in addition, P satisfies (v) P (x) = 0 implies x = 0. A Fr´echet space is a total and complete paranormed 2000 Mathematics Subject Classification. Primary 39B52, 39B82. Key words and phrases. Approximate quadratic form, Paranormed space. * Corresponding author. 1

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space. Note that each seminorm P on X is a paranorm, but the converse need not be true. In 1940, S. M. Ulam proposed the general Ulam stability problem (see [16]): Let G1 be a group and let G2 be a metric group with the metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1 then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all x ∈ G1 ? In 1941, this problem was solved by D. H. Hyers [6] in the case of Banach space. Thereafter, we call that type the Hyers-Ulam stability. In 1978, Th. M. Rassias [13] extended the Hyers-Ulam stability by considering variables. It also has been generalized to the function case by P. G˘avruta [5]. In recent, C. Park [11] obtained some stability results in paranormed spaces. Let X and Y be vector spaces. A mapping g : X → Y is called a quadratic mapping [1] if g satisfies the quadratic functional equation g(x + y) + g(x − y) = 2g(x) + 2g(y). The authors [2, 12] considered the following functional equations: (1)

f (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w)

and (2)

f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) + 2f (y, w).

It is easy to show that the functions f (x, y) = ax2 + bxy + cy 2 and f (x, y) = ax2 + by 2 satisfy the functional equations (1) and (2), respectively. From now on, assume that (X, P ) is a Fr´echet space and (Y, ∥ · ∥) is a Banach space. The authors [2, 12] solved the solutions of (1) and (2). In this paper, we investigate approximate quadratic forms related to the equations (1) and (2) in paranormed spaces. 2. Approximate quadratic forms related to (1) Theorem 2.1. Let r, θ be positive real numbers with r > 2, and let f : Y × Y → X be a mapping satisfying f (0, 0) = 0 such that ( ) (3) P f (x+y, z+w)+f (x−y, z−w)−2f (x, z)−2f (y, w) ≤ θ(∥x∥r +∥y∥r +∥z∥r +∥w∥r ) for all x, y, z, w ∈ Y . Then there exists a unique mapping F : Y × Y → X satisfying (1) such that ( ) 2θ (4) P f (x, y) − F (x, y) ≤ r (∥x∥r + ∥y∥r ) 2 −4

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for all x, y ∈ Y . ( ) Proof. Letting x = y and z = w in (3), P f (2x, 2z) − 4f (x, z) ≤ 2θ(∥x∥r + ∥z∥r ) for z x all x, z ∈ Y . Substituting x and z by 2j+1 and 2j+1 in the above inequality, respectively, we have ( x ( (x z) z )) 2θ 1 ≤ r rj (∥x∥r + ∥z∥r ) P f j , j − 4f j+1 , j+1 2 2 2 2 2 2 for all j ∈ N and all x, z ∈ Y . For given integers l, m(0 ≤ l < m), (

(x z ) ( x z )) m−1 ( x ∑ ( (x z) z )) m ≤ P 4j f j , j − 4j+1 f j+1 , j+1 P 4 f l, l − 4 f m, m 2 2 2 2 2 2 2 2 j=l (5)

l

m−1 2θ ∑ 4j ≤ r (∥x∥r + ∥z∥r ) rj 2 j=l 2

for all x, z ∈ Y . By (5), the sequence {4j f ( 2xj , 2zj )} is a Cauchy sequence in X for all x, z ∈ Y . Since X is complete, the sequence {4(j f ( 2xj ,)2zj )} converges for all x, z ∈ Y . Define F : Y × Y → X by F (x, z) := limj→∞ 4j f 2xj , 2zj for all x, z ∈ Y . By (3), we see that ( ) P F (x + y, z + w) + F (x − y, z − w) − 2F (x, z) − 2F (y, w) ( [ ( ( x − y z − w )] [ (x z) ( y w )]) x + y z + w) j j = lim P 4 f , +f , −2·4 f j, j −f j, j j→∞ 2j 2j 2j 2j 2 2 2 2 ([ ( ) ) ( )] [ ( ) ( x+y z+w x−y z−w x z y w )] j ≤ lim 4 P f , +f , −2 f j, j −f j, j j→∞ 2j 2j 2j 2j 2 2 2 2 4j θ ≤ lim jr (∥x∥r + ∥y∥r + ∥z∥r + ∥w∥r ) = 0 j→∞ 2 for all x, y, z, w ∈ Y . Thus F satisfies (1). Setting l = 0 and taking m → ∞ in (5), one can obtain the inequality (4). Let F ′ : Y × Y → X be another mapping satisfying (1) and (4). By [2], there exist four symmetric bi-additive mappings S, T, S ′ , T ′ : Y × Y → X and two bi-additive mappings B, B ′ : Y × Y → X such that F (x, y) = S(x, x) + B(x, y) + T (y, y) and F ′ (x, y) =

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S ′ (x, x) + B ′ (x, y) + T ′ (y, y) for all x, y ∈ Y . Since r > 2, we obtain that P (F (x, y) − F ′ (x, y)) ) ( [ ( ( ( ( (x y) (y y) x) y) y )] x x) ′ x ′ x ′ y n , = P 4 S n, n + B n, n + T n, n − S n, n − B n, n − T 2 2 2 2 2 2 2 2 2 2 2n 2n ( ( ) ( ( ( (x y) (y y) x x) x) y) y) n ′ x ′ x ′ y ≤ 4 P S n, n + B n, n + T n, n − S n, n − B n, n − T , 2 2 2 2 2 2 2 2 2 2 2n 2n [ ( ( ( ( ( x y )) ( x y ))] x y) x y) n ≤ 4 P F n, n − f n, n + P f n, n − G n, n 2 2 2 2 2 2 2 2 2θ 2 ≤ 4n r (∥x∥r + ∥y∥r ) → 0 as n → ∞ nr 2 −42 for all x, y ∈ Y . Hence the mapping F is a unique mapping satisfying (1) and (4), as desired.  Theorem 2.2. Let r be a positive real number with r < 2, and let f : X × X → Y be a mapping satisfying f (0, 0) = 0 such that

(6) f (x+y, z+w)+f (x−y, z−w)−2f (x, z)−2f (y, w) ≤ P (x)r +P (y)r +P (z)r +P (w)r for all x, y, z, w ∈ X. Then there exists a unique mapping F : X × X → Y satisfying (1) such that (7)



f (x, y) − F (x, y) ≤

2 [P (x)r + P (y)r ] 4 − 2r

for all x, y ∈ X. Proof. Letting x = y and z = w and dividing by 4 in (6), we have

1[ ] 1

f (x, z) − f (2x, 2z) ≤ P (x)r + P (z)r 4 2 for all x, z ∈ X. Substituting x and z by 2j x and 2j z, respectively, we obtain that

( j ) 1 ( j+1 ) [ ] j j+1

f 2 x, 2 z − f 2 x, 2 z ≤ 2rj−1 P (x)r + P (z)r

4

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for all j ∈ N and all x, z ∈ X. For given integers l, m(0 ≤ l < m), we obtain



∑ 1 (

1 ( l ) ( m ) m−1 ) ( j+1 ) 1 1 l m j j j+1

f 2 x, 2 z −

f 2 x, 2 z − f 2 x, 2 z f 2 x, 2 z

≤

4j

4l j+1 4m 4 j=l (8)

≤

m−1 ∑ j=l

] 2rj−1 [ P (x)r + P (z)r j 4

{ ( j )} for all x, z ∈ X. By (8), the sequence 41j f 2{ x, 2j(z is a )} Cauchy sequence in Y for all 1 j j x, z ∈ X. Since Y is complete, the sequence 4j f( 2 x, 2 z) converges for all x, z ∈ X. Define F : X × X → Y by F (x, z) := limj→∞ 41j f 2j x, 2j z for all x, z ∈ X. By (6), we see that

F (x + y, z + w) + F (x − y, z − w) − 2F (x, z) − F (y, w)

1[ ( j ) ( j )] ) ( j )] 2[ ( j j j j j

= lim j f 2 (x + y), 2 (z + w) + f 2 (x − y), 2 (z − w) − j f 2 x, 2 z − f 2 y, 2 w

j→∞ 4 4

) ( j )] [ ( j ) ( j )] 1 [ ( j j j j j

≤ lim j f 2 (x + y), 2 (z + w) + f 2 (x − y), 2 (z − w) − 2 f 2 x, 2 z − f 2 y, 2 w

j→∞ 4 ( 2r )j [ ] P (x)r + P (y)r + P (z)r + P (w)r = 0 ≤ lim j→∞ 4 for all x, y, z, w ∈ X. Thus F satisfies (1). Setting l = 0 and taking m → ∞ in (8), one can obtain the inequality (7). Let F ′ : X × X → Y be another mapping satisfying (1) and (7). By [2], there exist four symmetric bi-additive mappings S, T, S ′ , T ′ : X × X → Y and two bi-additive mappings B, B ′ : X × X → Y such that F (x, y) = S(x, x) + B(x, y) + T (y, y) and F ′ (x, y) = S ′ (x, x) + B ′ (x, y) + T ′ (y, y) for all x, y ∈ X. Since 0 < r < 2, we obtain that ∥F (x, y) − F ′ (x, y)∥

1

n n n n n n ′ n n ′ n n ′ n n

= n [S(2 x, 2 x) + B(2 x, 2 y) + T (2 y, 2 y) − S (2 x, 2 x) − B (2 x, 2 y) − T (2 y, 2 y)]

4 1 ∥S(2n x, 2n x) + B(2n x, 2n y) + T (2n y, 2n y) − S ′ (2n x, 2n x) − B ′ (2n x, 2n y) − T ′ (2n y, 2n y)∥ 4n [

] (x y) ( x y ) ( x y )

(x y)

1 ≤ n F , − f n, n + f , − G n, n 4 2n 2n 2 2 2n 2n 2 2 ] 1 2 2 [ ≤ n P (x)r + P (y)r → 0 as n → ∞ r nr 4 4−2 2

=

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for all x, y ∈ X. Hence the mapping F is a unique mapping satisfying (1) and (7), as desired.  3. Approximate quadratic forms related to (2) Theorem 3.1. Let r, θ be positive real numbers with r > 2, and let f : Y × Y → X be a mapping satisfying f (0, 0) = 0 such that ( ) (9) P f (x+y, z−w)+f (x−y, z+w)−2f (x, z)−2f (y, w) ≤ θ(∥x∥r +∥y∥r +∥z∥r +∥w∥r ) for all x, y, z, w ∈ Y . Then there exists a unique mapping F : Y × Y → X satisfying (2) such that ( (x y ) ) 24θ (10) P 4f , − F (x, y) ≤ r r (∥x∥r + ∥y∥r ) 2 2 2 (2 − 4) for all x, y ∈ Y . Proof. Letting y = x and w = −z in (9), we get ( ) (11) P f (2x, 2z) − 2f (x, z) − 2f (x, −z) ≤ 2θ(∥x∥r + ∥z∥r ) for all x, z ∈ Y . Putting x = 0 in the above inequality, we have ( ) P f (0, 2z) − 2f (0, z) − 2f (0, −z) ≤ 2θ∥z∥r for all z ∈ Y . Replacing z by −z in the above inequality, we obtain ( ) P f (0, −2z) − 2f (0, z) − 2f (0, −z) ≤ 2θ∥z∥r for all z ∈ Y . By the above two inequalities, we see that ( ) (12) P f (0, 2z) − f (0, −2z) ≤ 4θ∥z∥r for all z ∈ Y . Taking y = x and w = z in (9), we get ( ) P f (2x, 0) + f (0, 2z) − 4f (x, z) ≤ 2θ(∥x∥r + ∥z∥r ) for all x, z ∈ Y . Replacing z by −z in the above inequality, we have ( ) P f (2x, 0) + f (0, −2z) − 4f (x, −z) ≤ 2θ(∥x∥r + ∥z∥r ) for all x, z ∈ Y . By the above two inequalities, we obtain ( ) P 4f (x, z) − 4f (x, −z) − f (0, 2z) + f (0, −2z) ≤ 4θ(∥x∥r + ∥z∥r ) for all x, z ∈ Y . By (11) and the above inequality, we see that ( ) P 8f (x, z) − 2f (2x, 2z) − f (0, 2z) + f (0, −2z) ≤ 8θ(∥x∥r + ∥z∥r )

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for all x, z ∈ Y . By (12) and the above inequality, we get ( ) P 8f (x, z) − 2f (2x, 2z) ≤ 12θ(∥x∥r + ∥z∥r ) for all x, z ∈ Y . Replacing x and z by x2 and z2 in the above inequality, respectively, we have ( (x z ) ) 12θ P 8f , − 2f (x, z) ≤ r (∥x∥r + ∥z∥r ) 2 2 2 for all x, z ∈ Y . Substituting x and z by 2xj and 2zj , respectively, we obtain ( (x z) ( x z )) 12θ 1 P 2f j , j − 8f j+1 , j+1 ≤ r rj (∥x∥r + ∥z∥r ) 2 2 2 2 2 2 for all j ∈ N and all x, z ∈ Y . Thus we see that ( x ( 4j ( x z ) 4j ( x ( (x z) z )) z )) =P · 2f j , j − · 8f j+1 , j+1 P 4j f j , j − 4j+1 f j+1 , j+1 2 2 2 2 2 2 2 2 2 2 ( ) ( )) j ( j 4 x z x z 4 12θ 1 ≤ P 2f j , j − 8f j+1 , j+1 ≤ · r rj (∥x∥r + ∥z∥r ) 2 2 2 2 2 2 2 2 6θ ( 4 )j = r r (∥x∥r + ∥z∥r ) 2 2 for all j ∈ N and all x, z ∈ Y . For given integers l, m (1 ≤ l < m), we have ( (x z ) ( x z )) m−1 ( x ∑ ( (x z) z )) P 4j f j , j − 4j+1 f j+1 , j+1 P 4l f l , l − 4m f m , m ≤ 2 2 2 2 2 2 2 2 j=l (13)

m−1 6θ ∑ ( 4 )j ≤ r (∥x∥r + ∥z∥r ) 2 j=l 2r

for all x, z ∈ Y . By (13), the sequence {4j f ( 2xj , 2zj )} is a Cauchy sequence in X for all x, z ∈ Y . Since X is complete, the sequence {4(j f ( 2xj ,)2zj )} converges for all x, z ∈ Y . Define F : Y × Y → X by F (x, z) := limj→∞ 4j f 2xj , 2zj for all x, z ∈ Y . By (9), we see that ( ) P F (x + y, z − w) + F (x − y, z + w) − 2F (x, z) − 2F (y, w) ( [ ( ( x − y z + w )] [ (x z) ( y w )]) x + y z − w) j j , , = lim P 4 f +f −2·4 f j, j −f j, j j→∞ 2j 2j 2j 2j 2 2 2 2 ) ([ ( ) ( )] [ ( ) ( x−y z+w x z y w )] x+y z−w j ≤ lim 4 P f , +f , −2 f j, j −f j, j j→∞ 2j 2j 2j 2j 2 2 2 2 j 4θ ≤ lim jr (∥x∥r + ∥y∥r + ∥z∥r + ∥w∥r ) = 0 j→∞ 2

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for all x, y, z, w ∈ Y . Thus F is a mapping satisfying (2). Setting l = 1 and taking m → ∞ in (13), one can obtain the inequality (10). Let F ′ : Y × Y → X be another mapping satisfying (2) and (10). By [12], there exist four symmetric bi-additive mappings S, T, S ′ , T ′ : Y × Y → X such that F (x, y) = S(x, x) + T (y, y) and F ′ (x, y) = S ′ (x, x) + T ′ (y, y) for all x, y ∈ Y . Since r > 2, we obtain that P (F (x, y) − F ′ (x, y)) ) ( [ ( (y y) ( ( x) y )] x x) n ′ x ′ y , = P 4 S n, n + T n, n − S n, n − T 2 2 2 2 2 2 2n 2n ( ( ) ( ( (y y) x x) x) y) n ′ x ′ y ≤ 4 P S n, n + T n, n − S n, n − T , 2 2 2 2 2 2 2n 2n ( ( [ ( ( ( x y )) ( x y ))] x y) x y) n ≤ 4 P F n, n − f n, n + P f n, n − G n, n 2 2 2 2 2 2 2 2 2θ 2 ≤ 4n r (∥x∥r + ∥y∥r ) → 0 as n → ∞ 2 − 4 2nr for all x, y ∈ Y . Hence the mapping F is a unique mapping satisfying (2) and (10), as desired.  Theorem 3.2. Let r be a positive real number with r < 2, and let f : X × X → Y be a mapping satisfying f (0, 0) = 0 such that (14)

f (x + y, z − w) + f (x − y, z + w) − 2f (x, z) − 2f (y, w) ≤ P (x)r + P (y)r + P (z)r + P (w)r for all x, y, z, w ∈ X. Then there exists a unique mapping F : X × X → Y such that

f (x, y) − F (x, y) ≤ 6 [P (x)r + P (y)r ] (15) 4 − 2r for all x, y ∈ X. Proof. Letting y = x and w = −z in (14), we get

1

(16)

f (2x, 2z) − f (x, z) − f (x, −z) ≤ P (x)r + P (z)r 2 for all x, z ∈ X. Putting x = 0 in the above inequality, we have

1



f (0, 2z) − f (0, z) − f (0, −z) ≤ P (z)r 2

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for all z ∈ X. Replacing z by −z in the above inequality, we obtain

1

f (0, −2z) − f (0, z) − f (0, −z) ≤ P (z)r 2 for all z ∈ X. By the above two inequalities, we see that

1

1

(17) f (0, 2z) − f (0, −2z)

≤ P (z)r

4 4 for all z ∈ X. Taking y = x and w = z in (9), we get

1

1 1

f (2x, 0) + f (0, 2z) − f (x, z) ≤ [P (x)r + P (z)r ] 4 4 2 for all x, z ∈ X. Replacing z by −z in the above inequality, we have

1

1 1

f (2x, 0) + f (0, −2z) − f (x, −z)

≤ [P (x)r + P (z)r ] 4 4 2 for all x, z ∈ X. By the above two inequalities, we obtain

1 1

f (x, z) − f (x, −z) − f (0, 2z) + f (0, −2z) ≤ P (x)r + P (z)r 4 4 for all x, z ∈ X. By (16) and the above inequality, we see that

1 1 1

f (x, z) − f (2x, 2z) − f (0, 2z) + f (0, −2z)

≤ P (x)r + P (z)r 4 8 8 for all x, z ∈ X. By (17) and the above inequality, we get

3 1

f (x, z) − f (2x, 2z) ≤ [P (x)r + P (z)r ] 4 2 for all x, z ∈ X. Substituting x and z by 2j x and 2j z, respectively, we have

3 1

j j j+1 j+1

f (2 x, 2 z) − f (2 x, 2 z) ≤ · 2jr [P (x)r + P (z)r ] 4 2 for all nonnegative integers j and all x, z ∈ X. Thus we obtain

3

1 1

j j j+1 j+1

j f (2 x, 2 z) − j+1 f (2 x, 2 z) ≤ · 2(r−2)j [P (x)r + P (z)r ] 4 4 2

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10

JAE-HYEONG BAE AND WON-GIL PARK

for all nonnegative integers j and all x, z ∈ X. For given integers l, m (0 ≤ l < m), we see that

1

m−1

∑ 1 1

1

l l m m j j j+1 j+1

j f (2 x, 2 z) − j+1 f (2 x, 2 z)

l f (2 x, 2 z) − m f (2 x, 2 z) ≤ 4 4 4 4 j=l m−1 3 ∑ (r−2)j 2 [P (x)r + P (z)r ] 2 j=l { } sequence in Y for all for all x, z ∈ X. By (18), the sequence 41j f (2{j x, 2j z) is a Cauchy } 1 j j x, z ∈ X. Since Y is complete, the sequence 4j f (2 x, 2 z) converges for all x, z ∈ X. Define F : X × X → Y by F (x, z) := limj→∞ 41j f (2j x, 2j z) for all x, z ∈ X. By (14), we see that

F (x + y, z − w) + F (x − y, z + w) − 2F (x, z) − 2F (y, w)

1

2 j j j j j j j j

= lim j [f (2 (x + y), 2 (z − w)) + f (2 (x − y), 2 (z + w))] − j [f (2 x, 2 z) − f (2 y, 2 w)]

j→∞ 4 4 1 ≤ lim j ∥f (2j (x + y), 2j (z − w)) + f (2j (x − y), 2j (z + w)) − 2[f (2j x, 2j z) − f (2j y, 2j w)]∥ j→∞ 4 ≤ lim 2(r−2)j [P (x)r + P (y)r + P (z)r + P (w)r ] = 0

(18)

≤

j→∞

for all x, y, z, w ∈ X. Thus F is a mapping satisfying (2). Setting l = 0 and taking m → ∞ in (18), one can obtain the inequality (15). Let F ′ : X × X → Y be another mapping satisfying (2) and (15). By [12], there exist four symmetric bi-additive mappings S, T, S ′ , T ′ : X × X → Y such that F (x, y) = S(x, x) + T (y, y) and F ′ (x, y) = S ′ (x, x) + T ′ (y, y) for all x, y ∈ X. Since 0 < r < 2, we obtain that

1

∥F (x, y) − F ′ (x, y)∥ = n [S(2n x, 2n x) + T (2n y, 2n y) − S ′ (2n x, 2n x) − T ′ (2n y, 2n y)] 4

1 ≤ n S(2n x, 2n x) + T (2n y, 2n y) − S ′ (2n x, 2n x) − T ′ (2n y, 2n y) 4



] 1 [ ≤ n F (2n x, 2n y) − f (2n x, 2n y) + f (2n x, 2n y) − G(2n x, 2n y) 4 ( 1 )n(2−r) 12 ≤ · [P (x)r + P (y)r ] → 0 as n → ∞ r 4−2 2 for all x, y ∈ X. Hence the mapping F is a unique mapping satisfying (2) and (15), as desired. 

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Acknowledgment This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(grant number 2013013682) References [1] J. Acz´el and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989. [2] J.-H. Bae and W.-G. Park, A functional equation originating from quadratic forms, J. Math. Anal. Appl. 326 (2007), 1142–1148. [3] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. [4] J. A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313. [5] P. G˘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. Nat. Acad. Sci. 27 (1941), 222–224. [7] S. Karakus, Statistical convergence on probabilistic nomred spaces, Math. Commun. 12 (2007), 11–23. [8] E. Kolk, The statistical convergence in Banach spaces, Tartu Ul. Toime. 928 (1991), 41–52. [9] M. Mursaleen, λ-statistical convergence, Math. Slovaca. 50 (2000), 111–115. [10] 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. [11] C. Park and J. R. Lee, Functional equations and inequalities in paranormed spaces, J. Inequal. Appl. 2013 (2013), Art. No. 198. [12] W.-G. Park and J.-H. Bae, A functional equation related to quadratic forms without the cross product terms, Honam Math. J. 30 (2008), 219–225. [13] Th. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. ˇ at, On the statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139–150. [14] T. Sal´ [15] H. Steinhaus, Sur la convergence ordinaire et la convergence asymtotique, Colloq. Math. 2 (1951), 73–74. [16] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968, p.63. [17] A. Wilansky, Modern Methods in Topological Vector Space, McGraw-Hill International Book Co., New York, 1978. Jae-Hyeong Bae, Humanitas College, Kyung Hee University, Yongin 446-701, Republic of Korea E-mail address: [email protected] Won-Gil Park, Department of Mathematics Education, College of Education, Mokwon University, Daejeon 302-318, Republic of Korea E-mail address: [email protected]

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Common Fixed Point Theorems for the R-weakly Commuting Mappings in S-metric Spaces Jong Kyu Kim1 , Shaban Sedghi2 and Nabi Shobkolaei3 1 Department of Mathematics Education, Kyungnam University, Changwon Gyeongnam, 631-701, Korea

e-mail: [email protected] 2 Department

of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran emailsedghi [email protected]

3 Department

of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran emailnabi [email protected]

Abstract. In this paper, we prove common fixed point theorems for two mappings under the condition of R-weakly commuting in complete S-metric spaces. A lot of fixed point theorems on ordinary metric spaces are special cases of our main result. Keywords: S-metric space, contractive mapping, R-weakly commuting self-mappings, common fixed points. 2010 AMS Subject Classification: Premary 54H25, Secondary 47H10.

1

Introduction and preliminaries

Recently, Mustafa and Sims [12] introduced a new structure of generalized metric spaces which are called G-metric spaces as generalization of metric space (X, d), to develop and introduce a new fixed point theory for various mappings in this new structure. Some authors [3, 13, 28] have proved some fixed point theorems in these spaces. Fixed point problems of contractive mappings in partically ordered metric spaces have been studied in a number of works (see [2, 4, 7, 10, 11, 23, 24]). Metric spaces are very important in mathematics and applied sciences. So, some authors have tried to give generalizations of metric spaces in several ways. For example, G¨ahler [6] and Dhage [5] introduced the concepts of 2-matric spaces and D-metric spaces, respectively, but some authors pointed out that these attempts are not valid (see [9], [18-22]). Recently, Sedghi et al. introduced D∗ -metric which is a probable modification of the definition of D-metric introduced by Dhage [5] and proved some basic properties in D∗ -metric spaces(see [25-27]). 0 This work was supported by the Basic Science Research Program through the National Research Foundation Grant funded by Ministry of Education of the republic of Korea(2013R1A1A2054617). 0 The corresponding author: [email protected](J.K.Kim).

1

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In this paper, we introduce the concept of S-metric spaces and give some properties of them. And we prove common fixed point theorem for a self-mapping on complete S-metric spaces. In [30], Sedghi et al. have introduced the notion of an S-metric space as follows. Definition 1.1. [30] Let X be a nonempty set. An S-metric on X is a function S : X 3 → [0, ∞) that satisfies the following conditions: (1) S(x, y, z) = 0 if and only if x = y = z, for all x, y, z ∈ X. (2) S(x, y, z) ≤ S(x, x, a) + S(y, y, a) + S(z, z, a), for all x, y, z, a ∈ X. The pair (X, S) is said to be an S-metric space. Example 1.2. [30] Let R be the real line. Then S(x, y, z) = |x − z| + |y − z| for all x, y, z ∈ R is an S-metric on R. This S-metric on R is called the usual S-metric on R. From the definition, we know that the S-metric space is a generalization of a Gmetric space [17] and a D∗ -metric space [26]. That is, every G-metric is a D∗ -metric, and every D∗ -metric is an S-metric, but each converse is not true. Example 1.3. [30] For X = R, let we define d(x, y, z) = |x + y − 2x| + |x + z − 2y| + |y + z − 2x|. Then d is a D∗ -metric, but it is not a G-metric. Example 1.4. [30] For X = Rn , let we define S(x, y, z) = ||y + z − 2x|| + ||y − z||. Then S is an S-metric, but it is not a D∗ -metric. For the fixed point problem in generalized metric spaces, many results have been proved (see [1, 8, 14-16, 29, 31]). In [30], the authors have proved some properties of S-metric spaces. Also, they have been proved some fixed point theorems for a self-map in an S-metric space. Definition 1.5. [30] Let (X, S) be an S-metric space. For r > 0 and x ∈ X, we define the open ball BS (x, r) and the closed ball BS [x, r] with center x and radius r as follows. BS (x, r) = {y ∈ X : S(y, y, x) < r}, BS [x, r] = {y ∈ X : S(y, y, x) ≤ r}. The topology induced by the S-metric is the topology generated by the base of all open balls in X. Definition 1.6. [30] Let (X, S) be an S-metric space. (1) A sequence {xn } ⊂ X is S-convergent to x ∈ X if S(xn , xn , x) → 0 as n → ∞. That is, for all ε > 0, there exists n0 ∈ N such that for all n ≥ n0 , S(xn , xn , x) < ε. We write xn → x for brevity. (2) A sequence {xn } ⊂ X is a Cauchy sequence if S(xn , xn , xm ) → 0 as n, m → ∞. That is, for all ε > 0, there exists n0 ∈ N such that for all n, m ≥ n0 , S(xn , xn , xm ) < ε. 2

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(3) The S-metric space (X, S) is said to be complete if every Cauchy sequence in X is S-convergent. Lemma 1.7. [30] Let (X, S) be an S-metric space, Then, we have S(x, x, y) = S(y, y, x) for all x, y ∈ X. Lemma 1.8. [30] Let (X, S) be an S-metric space. If xn → x and yn → y then S(xn , xn , yn ) → S(x, x, y).

2

Common fixed point results

In this section we give some fixed point results on S-metric spaces. Definition 2.1. Let (X, S1 ) and (Y, S2 ) be two S-metric spaces and T : X → Y be a map. T is called sequentially convergent if {xn } is S1 -convergent in X provided {T xn } is S2 -convergent in Y. Definition 2.2. Let (X, S) be an S-metric space and let f and g be maps from X into itself. The maps f and g are said to be weakly commuting if S(f gx, f gx, gf x) ≤ S(f x, f x, gx) for each x ∈ X. Definition 2.3. Let (X, S) be an S-metric space and let f and g be maps from X into itself. The maps f and g are said to be R-weakly commuting if there exists a positive real number R such that S(f gx, f gx, gf x) ≤ RS(f x, f x, gx) for each x ∈ X. Weak commutativity implies R-weak commutativity in an S−metric space. However, R-weak commutativity implies weak commutativity only when R ≥ 1. Example 2.4. Let X = R2 . Let S be the S−metric on X 3 defined as follows: S(x, y, z) = ||x − y|| + ||y − z||. Then (X, S) is an S−metric space. Define f (x, y) = (x2 , sin y) and g(x, y) = (2x − 1, sin y). Then, we have = = = ≤

S(f g(x, y), f g(x, y), gf (x, y)) S(((2x − 1)2 , sin(siny)), ((2x − 1)2 , sin(siny)), (2x2 − 1, sin(siny))) p (2(x − 1)2 )2 = 2(x − 1)2 2S(f (x, y), f (x, y), g(x, y)) RS(f (x, y), f (x, y), g(x, y)).

Therefore, for R ≥ 2, f and g are R-weakly commuting. But f and g are not weakly commuting. Now we introduce and prove the main theorem in this paper. Theorem 2.5. Let (X, S1 ) be an S-metric space, (Y, S2 ) be a complete S-metric space and let f and g be R-weakly commuting self-mappings of X satisfying the following conditions: (a) f (X) ⊆ g(X); 3

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(b) f or g is continuous; (c) S2 (F f x, F f x, F f y) ≤ kS2 (F gx, F gx, F gy)), where 0 < k < 1 and F : X → Y is one-to-one, continuous and sequentially convergent. Then there exists a unique common fixed point z ∈ X of f and g. proof. Let x0 be an arbitrary point in X. By (a), there exists a point x1 in X such that f x0 = gx1 . Continuing in this process, we can choose xn+1 such that f xn = gxn+1 . Set yn = F f xn . Then S2 (yn , yn , yn+1 )

= ≤ = = ≤ .. . ≤

S2 (F f xn , F f xn , F f xn+1 ) kS2 (F gxn , F gxn , F gxn+1 )) kS2 (F f xn−1 , F f xn−1 , F f xn )) kS2 (yn−1 , yn−1 , yn ) k 2 S2 (yn−2 , yn−2 , yn−1 ) k n S2 (y0 , y0 , y1 ).

Thus for all n < m, we have S2 (yn , yn , ym ) ≤ 2S2 (yn , yn , yn+1 ) + S2 (ym , ym , yn+1 ) = 2S2 (yn , yn , yn+1 ) + S2 (yn+1 , yn+1 , ym ) .. . ≤ 2[k n + · · · + k m−1 ]S2 (y0 , y0 , y1 ) 2k n S2 (y0 , y0 , y1 ). ≤ 1−k Taking the limit as n, m → ∞, we get S2 (yn , yn , ym ) → 0. This means that {yn } is a Cauchy sequence. Since (Y, S2 ) is complete, the sequence {yn } converges to some y ∈ Y. Since F is sequentially convergent, {f xn } converges to some z ∈ X and also from the continuity of F, {F f xn } converges to F z. Note that {yn } converges to y, then yn = F f xn = F gxn+1 → F z = y. Also {gxn } converges to z in X. Let us suppose that the mapping f is continuous. Then limn→∞ f f xn = f z and limn→∞ f gxn = f z. Further, since f and g are R-weakly commuting, we have S1 (f gxn , f gxn , gf xn ) ≤ RS1 (f xn , f xn , gxn ). Taking the limit as n → ∞ in the above inequality, we have S1 (f z, f z, lim gf xn ) n→∞

=

lim S1 (f gxn , f gxn , gf xn )

n→∞

≤ R lim S1 (f xn , f xn , gxn ) n→∞

= RS1 (z, z, z) = 0. Hence, we get limn gf xn = f z. We now prove that z = f z. By (c) S2 (F f z, F f z, F z)

=

lim S2 (F f f xn , F f f xn , F f xn )

n→∞

≤ k lim S2 (F gf xn , F gf xn , F gxn ) n→∞

= kS2 (F f z, F f z, F z)). 4

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By the above inequality, we get F f z = F z. Since F is one-to-one, it follows that f z = z. Since f (X) ⊆ g(X), we can find z1 ∈ X such that z = f z = gz1 . Now, S2 (F f f xn , F f f xn , F f z1 ) ≤ kS2 (F gf xn , F gf xn , F gz1 )). Taking the limit as n → ∞, we get S2 (F f z, F f z, F f z1 )

=

lim S2 (F f f xn , F f f xn , F f z1 )

n→∞

≤ k lim S2 (F gf xn , F gf xn , F gz1 ) n→∞

= kS2 (F f z, F f z, F gz1 )) = 0, which implies that F f z = F f z1 , i.e., z = f z = f z1 = gz1 . Also, we have S1 (f z, f z, gz) = S1 (f gz1 , f gz1 , gf z1 ) ≤ RS1 (f z1 , f z1 , gz1 ) = 0, which implies that f z = gz. Thus z is a common fixed point of f and g. Now, in order to prove uniqueness, let z 0 6= z be another common fixed point of f and g. Then S2 (F z, F z, F z 0 )

S2 (F f z, F f z, F f z 0 ) kS2 (F gz, F gz, F gz 0 ) kS2 (F z, F z, F z 0 ) S2 (F z, F z, F z 0 ),

= ≤ =
1 and that our assumption holds for n − 1. that is, x6n−8

=

x6n−6

=

x6n−4

=

y6n−8

=

y6n−6

=

y6n−4

=

(−1)n−1 x−2 (−1+3x−2 z−1 y0 )n−1 (−1+x−2 z−1 y0 )n−1 (−1+2x−2 z−1 y0 )n−1 , x0 (1+2z−2 y−1 x0 )n−1 (1−z−2 y−1 x0 )n−1 , (1+z−2 y−1 x0 )n−1

−x−1 (1+y−2 x−1 z0 )n (1−2y−2 x−1 z0 )n−1 , (1−y−2 x−1 z0 )n−1 y−2 (1−y−2 x−1 z0 )n−1 (1+y−2 x−1 z0 )n−1 (1−2y−2 x−1 z0 )n−1 ,

x6n−7 = x6n−5 = x6n−3 =

y6n−7 =

(−1)n−1 y0 (−1+x−2 z−1 y0 )n−1 (−1+2x−2 z−1 y0 )n−1 , (−1+3x−2 z−1 y0 )n−1 y−1 (1+z−2 y−1 x0 )n (1−z−2 y−1 x0 )n−1 (1+2z−2 y−1 x0 )n ,

y6n−3 =

761

x−1 (1+y−2 x−1 z0 )n−1 (1−2y−2 x−1 z0 )n−1 , (1−y−2 x−1 z0 )n−1

(−1)n−1 x−2 (−1+3x−2 z−1 y0 )n−1 (−1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n−1 ,

−x0 (1+2z−2 y−1 x0 )n (1−z−2 y−1 x0 )n−1 , (1+z−2 y−1 x0 )n

y−1 (1+z−2 y−1 x0 )n−1 (1−z−2 y−1 x0 )n−1 (1+2z−2 y−1 x0 )n−1 ,

y6n−5 =

y−2 (1−y−2 x−1 z0 )n−1 (1+y−2 x−1 z0 )n (1−2y−2 x−1 z0 )n−1 ,

(−1)n−1 y0 (−1+x−2 z−1 y0 )n−1 (−1+2x−2 z−1 y0 )n , (−1+3x−2 z−1 y0 )n

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and z6n−8

= z−2 , z6n−7 = z−1 , z6n−6 = z0 , z6n−5 =

z6n−4

=

z−1 (−1+x−2 z−1 y0 ) (−1+2x−2 z−1 y0 ) ,

z6n−3 =

z−2 1+z−2 y−1 x0 ,

z0 1−y−2 x−1 z0 .

It follows from Eq.(1) that x6n−2

=

x6n−5 −1+x6n−5 z6n−4 y6n−3

=

 −1+

(−1)n−1 x−2 (−1+3x−2 z−1 y0 )n−1 (−1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n−1 z−1 (−1+x−2 z−1 y0 ) (−1)n−1 y0 (−1+x−2 z−1 y0 )n−1 (−1+2x−2 z−1 y0 )n (−1)n−1 x−2 (−1+3x−2 z−1 y0 )n−1 (−1+3x−2 z−1 y0 )n (−1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n−1 (−1+2x−2 z−1 y0 )

=

(−1)n−1 x−2 (−1+3x−2 z−1 y0 )n−1 1  x−2 z−1 y0 (−1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n−1 −1+ (−1+3x z y

−2 −1 0 )

=

(−1)n−1 x−2 (−1+3x−2 z−1 y0 )n−1 1  x−2 z−1 y0 (−1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n−1 −1+ (−1+3x z y

−2 −1 0 )

= =

(−1)n−1 x−2 (−1+3x−2 z−1 y0 )n−1 (−1+3x−2 z−1 y0 ) (−1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n−1 (1−2x−2 z−1 y0 ) (−1)n x−2 (−1+3x−2 z−1 y0 )n (−1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n

=

 

³

−1+3x−2 z−1 y0 −1+3x−2 z−1 y0



´

(−1)n x−2 (−1+3x−2 z−1 y0 )n (−1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n .

Also, we see from Eq.(1) that y6n−2

=

y6n−5 1+y6n−5 x6n−4 z6n−3

=

y−2 (1−y−2 x−1 z0 )n−1 (1+y−2 x−1 z0 )n (1−2y−2 x−1 z0 )n−1  −x−1 (1+y−2 x−1 z0 )n (1−2y−2 x−1 z0 )n−1 y−2 (1−y−2 x−1 z0 )n−1 z0 1+ 1−y−2 x−1 z0 (1+y−2 x−1 z0 )n (1−2y−2 x−1 z0 )n−1 (1−y−2 x−1 z0 )n−1

=

y−2 (1−y−2 x−1 z0 )n−1 1   (1+y−2 x−1 z0 )n (1−2y−2 x−1 z0 )n−1 1− y−2 x−1 z0 1−y x z −2 −1 0

=

³

1−y−2 x−1 z0 1−y−2 x−1 z0

y−2 (1−y−2 x−1 z0 )n−1 1−y−2 x−1 z0 (1+y−2 x−1 z0 )n (1−2y−2 x−1 z0 )n−1 (1−y−2 x−1 z0 −y−2 x−1 z0 ) .

Then y6n−2 =

´



y−2 (1−y−2 x−1 z0 )n (1+y−2 x−1 z0 )n (1−2y−2 x−1 z0 )n .

Finally from Eq.(1), we see that z6n−2

= = =

z6n−5 1 + z6n−5 y6n−4 x6n−3 z−2 1+z−2 y−1 x0 n −x0 (1+2z−2 y−1 x0 )n (1−z−2 y−1 x0 )n−1 z−2 y−1 (1+z−2 y−1 x0 ) (1+z−2 y−1 x0 )n −2 y−1 x0 (1−z−2 y−1 x0 )n−1 (1+2z−2 y−1 x0 )n

 1+ 1+z

z−2 1   1+z−2 y−1 x0 1− z−2 y−1 x0 1+z y x −2 −1 0

=

z−2 (1+z−2 y−1 x0 −z−2 y−1 x0 )



= z−2 .

Similarly we can prove the other relations. This completes the proof. Lemma 1. Let {yn , zn } be a positive solution of system (1), then every solution {yn , zn } of system (1) is bounded and {yn } converges to zero. Proof: It follows from Eq.(1) that yn+1 =

yn−2 1+zn xn−1 yn−2

≤ yn−2 ,

zn+1 =

zn−2 1+xn yn−1 zn−2

≤ zn−2 .

∞ ∞ Then the subsequences {y3n−2 }∞ n=0 , {y3n−1 }n=0 , {y3n }n=0 ,are decreasing and so are bounded from above by ∞ ∞ M = max{y−2 , y−1 , y0 } and {z3n−2 }n=0 , {z3n−1 }n=0 , {z3n }∞ n=0 ,are decreasing and also, bounded from above by M = max{z−2 , z−1 , z0 }.

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plot of X(n+1)=X(n−2)/−1+X(n−2)Y(n)Z(n−1),Y(n+1)=Y(n−2)/1+Y(n−2)Z(n)X(n−1),Z(n+1)=Z(n−2)/1+Z(n−2)X(n)Y(n−1) 8 X(n) Y(n) Z(n) 7

6

x(n),y(n),Z(n)

5

4

3

2

1

0

−1

0

10

20

30 n

40

50

60

Figure 1.

Lemma 2. It is easy to see that {zn } periodic with period six.

Lemma 3. If x−2 , x−1 , x0 , y−2 , y−1 , y0 , z−2 , z−1 and z0 arbitrary real numbers and let {xn , yn , zn } are solutions of system (1) then the following statements are true:(i) If x−2 = 0, y0 6= 0, z−1 6= 0, then we have x6n−2 = x6n+1 = 0 and y6n = y6n+3 = y0 , z6n+2 = z−1 .

(ii) If x−1 = 0, y−2 6= 0, z0 6= 0, then we have x6n−1 = x6n+2 = 0 and y6n−2 = y6n+1 = y−2 , z6n+3 = z0 .

(iii) If x0 = 0, y−1 6= 0, z−2 6= 0, then we have x6n = x6n+3 = 0 and y6n−1 = y6n+2 = y−1 , z6n+1 = z−2 . z0 .

(iv) If y−2 = 0, x−1 6= 0, z0 6= 0, then we have y6n−2 = y6n+1 = 0 and x6n−1 = x−1 , x6n+2 = −x−1 , z6n+3 =

(v) If y−1 = 0, x0 6= 0, z−2 6= 0, then we have y6n−1 = y6n+2 = 0 and x6n = x0 , x6n+3 = −x0 , z6n+1 = z−2 . (vi) If y0 = 0, x−2 6= 0, z−1 6= 0, then we have y6n = y6n+3 = 0 and x6n−2 = x6n+1 = x−2 , z6n+2 = z−1 .

(vii) If z−2 = 0, x0 6= 0, y−1 6= 0, then we have z6n−2 = z6n+1 = 0 and x6n = x0 , x6n+3 = −x0 , y6n−1 = y6n+2 = y−1 . (viii) If z−1 = 0, x−2 6= 0, y0 6= 0, then we have z6n−1 = z6n+2 = 0 and x6n−2 = x6n+1 = x−2 , y6n = y6n+3 = y0 . (ix) If z0 = 0, x−1 6= 0, y−2 6= 0, then we have z6n = z6n+3 = 0 and x6n−1 = x−1 , x6n+2 = −x−1 , y6n−2 = y6n+1 = y−2 . Proof: The proof follows from the form of the solutions of system (1). Example 1. We consider interesting numerical example for the difference system (1) with the initial conditions x−2 = 0.9, x−1 = 0.4, x0 = 0.3, y−2 = 0.21, y−1 = 0.7, y0 = 0.13, z−2 = 1.9, z−1 = 0.6 and z0 = 1.3. (See Fig. 1). Example 2. See Figure (2) when we put the initial conditions x−2 = 0.9, x−1 = 0.4, x0 = −0.3, y−2 = −0.21, y−1 = 0.7, y0 = 0, z−2 = −1.9, z−1 = 0.6 and z0 = 1.3 for the difference system (1).

XN+1 =

3. THE SECOND SYSTEM: YN −2 YN+1 = −1+YN −2 XN −1 ZN , ZN+1 =

XN −2 −1+XN −2 ZN −1 YN ,

ZN −2 1+ZN −2 YN −1 XN

In this section, we study the solutions of the following system of the difference equations xn+1 =

xn−2 yn−2 zn−2 , yn+1 = , zn+1 = , −1 + xn−2 zn−1 yn −1 + yn−2 xn−1 zn 1 + zn−2 yn−1 xn

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(2)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.4, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

plot of X(n+1)=X(n−2)/−1+X(n−2)Y(n)Z(n−1),Y(n+1)=Y(n−2)/1+Y(n−2)Z(n)X(n−1),Z(n+1)=Z(n−2)/1+Z(n−2)X(n)Y(n−1) 4 X(n) Y(n) Z(n) 3

x(n),y(n),Z(n)

2

1

0

−1

−2

0

5

10

15

20 n

25

30

35

40

Figure 2.

where n ∈ N0 and the initial conditions are arbitrary non zero real numbers such that y−2 x−1 y0 6= 1, x−2 y−1 x0 6= 1. Theorem 3.1. Suppose that {xn , yn , zn } are solutions of system (2). Then the solution of system (2) are given by x3n−2

=

y3n−2

=

y3n

=

(−1)n+1 x−2 (−1+nx−2 z−1 y0 ) ,

(−1)n+1 x−1 (−1+y−2 x−1 z0 ) , (1+(n−1)y−2 x−1 z0 )

x3n−1 =

(−1)n+1 y−2 (1+(n−1)y−2 x−1 z0 ) , (−1+y−2 x−1 z0 )

x3n =

(−1)n x0 (1+nz−2 y−1 x0 ) ,

y3n−1 = (−1)n y−1 (1 + nz−2 y−1 x0 ),

n

(−1) y0 (1−(n+1)x−2 z−1 y0 ) , (1−x−2 z−1 y0 )

z3n−2 =

z−2 (1+nz−2 y−1 x0 ) ,

z−1 (−1+x−2 z−1 y0 ) (−1+(n+1)x−2 z−1 y0 ) ,

z3n−1 =

z3n =

z0 (1+ny−2 x−1 z0 ) .

Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n − 1. that is, x3n−5

=

y3n−5

=

y3n−3

=

(−1)n x−2 (−1+(n−1)x−2 z−1 y0 ) ,

x3n−4 =

(−1)n y−2 (1+(n−2)y−2 x−1 z0 ) , (−1+y−2 x−1 z0 ) (−1)n−1 y0 (1−nx−2 z−1 y0 ) , (1−x−2 z−1 y0 )

z3n−5 =

z−2 (1+(n−1)z−2 y−1 x0 ) ,

(−1)n x−1 (−1+y−2 x−1 z0 ) , (1+(n−2)y−2 x−1 z0 )

x3n−3 =

(−1)n−1 x0 (1+(n−1)z−2 y−1 x0 ) ,

y3n−4 = (−1)n−1 y−1 (1 + (n − 1) z−2 y−1 x0 ),

z3n−4 =

z−1 (−1+x−2 z−1 y0 ) (−1+nx−2 z−1 y0 ) ,

z3n−3 =

z0 (1+(n−1)y−2 x−1 z0 ) .

Now from Eq.(2) it follows that x3n−2

= = =

x3n−5 −1+x3n−5 z3n−4 y3n−3

(−1)n x−2 (−1+(n−1)x−2 z−1 y0 )   z (−1+x−2 z−1 y0 ) (−1)n−1 y0 (1−nx−2 z−1 y0 ) (−1)n x −1+ (−1+(n−1)x −2z y ) −1 (−1+nx−2 z−1 y0 ) (1−x−2 z−1 y0 ) −2 −1 0 n (−1)  x−2 x−2 z−1 y0 (−1+(n−1)x−2 z−1 y0 ) −1− (−1+(n−1)x z

−2 −1 y0 )

764



=

(−1)n x−2 (1−(n−1)x−2 z−1 y0 −x−2 z−1 y0 )

=

(−1)n+1 x−2 (−1+nx−2 z−1 y0 ) ,

M. M. El-Dessoky et al 760-769

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.4, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

y3n−2

= =

y3n−5 −1+y3n−5 x3n−4 z3n−3

=

(−1)n y−2 (1+(n−2)y −2 x−1 z0 )  y

x



z

−2 −1 0 (−1+y−2 x−1 z0 ) −1+ (1+(n−1)y x

−2 −1 z0 )

= =

(−1)n y−2 (1+(n−2)y−2 x−1 z0 ) (−1+y−2 x−1 z0 )  (−1)n x−1 (−1+y−2 x−1 z0 ) z0 (1+(n−2)y−2 x−1 z0 ) (1+(n−1)y−2 x−1 z0 ) −2 −1 0

 (−1)n y−2 (1+(n−2)y−2 x−1 z0 ) −1+ (−1+y x z )

(−1)n y−2 (1+(n−2)y−2 x−1 z0 )(1+(n−1)y−2 x−1 z0 ) (−1+y−2 x−1 z0 )(−(1+(n−1)y−2 x−1 z0 +y−2 x−1 z0 ) (−1)n+1 y−2 (1+(n−1)y−2 x−1 z0 ) , (−1+y−2 x−1 z0 )

and so, z3n−2

=

=

z3n−5 1+z3n−5 y3n−4 x3n−3

=



z−2 (1+(n−1)z−2 y−1 x0 )  z−2 (−1)n−1 x 1+ (1+(n−1)z y x ) (−1)n−1 y−1 (1+(n−1)z−2 y−1 x0 ) (1+(n−1)z y 0 x ) −2 −1 0 −2 −1 0

z−2  z−2 y−1 x0 (1+(n−1)z−2 y−1 x0 ) 1+ (1+(n−1)z y

−2 −1 x0 )



=

z−2 (1+(n−1)z−2 y−1 x0 +z−2 y−1 x0 )

=

z−2 (1+nz−2 y−1 x0 ) .

Also, we can prove the other relation. The proof is complete. Lemma 4. Let {zn } be a positive solution of system (2), then every the sequence {zn } is bounded and converges to zero. Lemma 5. If x−2 , x−1 , x0 , y−2 , y−1 , y0 , z−2 , z−1 and z0 arbitrary real numbers and let {xn , yn , zn } are solutions of system (2) then the following statements are true:(i) If x−2 = 0, y0 6= 0, z−1 6= 0, then we have x3n−2 = 0 and y3n = (−1)n y0 , z3n−1 = z−1 .

(ii) If x−1 = 0, y−2 6= 0, z0 6= 0, then we have x3n−1 = 0 and y3n−2 = (−1)n y−2 , z3n = z0 .

(iii) If x0 = 0, y−1 6= 0, z−2 6= 0, then we have x3n = 0 and y3n−1 = (−1)n y−1 , z3n−2 = z−2 . (iv) If y−2 = 0, x−1 6= 0, z0 6= 0, then we have y3n−2 = 0 and x3n−1 = (−1)n x−1 , z3n = z0 .

(v) If y−1 = 0, x0 6= 0, z−2 6= 0, then we have y3n−1 = 0 and x3n = (−1)n x0 , z3n−2 = z−2 .

(vi) If y0 = 0, x−2 6= 0, z−1 6= 0, then we have y3n = 0 and x3n−2 = (−1)n x−2 , z3n−1 = z−1 .

(vii) If z−2 = 0, x0 6= 0, y−1 6= 0, then we have z3n−2 = 0 and x3n = (−1)n x0 , y3n−1 = (−1)n y−1 .

(viii) If z−1 = 0, x−2 6= 0, y0 6= 0, then we have z3n−1 = 0 and x3n−2 = (−1)n x−2 , y3n = (−1)n y0 .

(ix) If z0 = 0, x−1 6= 0, y−2 6= 0, then we have z3n = 0 and x3n−1 = (−1)n x−1 , y3n−2 = (−1)n y−2 .

Example 3. Figure (3) shows the behavior of the solution of the difference system (2) with the initial conditions x−2 = 0.9, x−1 = 0.4, x0 = 0.3, y−2 = 0.21, y−1 = 0.7, y0 = 0, z−2 = 1.9, z−1 = 0.6 and z0 = 1.3. XN −2 1+XN −2 ZN −1 YN , ZN −2 −1+ZN −2 YN −1 XN

4. THE THIRD SYSTEM: XN+1 = ZN+1 =

YN+1 =

YN −2 1+YN −2 XN −1 ZN ,

In this section, we obtain the form of the solutions of the system of three difference equations xn+1 =

xn−2 yn−2 zn−2 , yn+1 = , zn+1 = , 1 + xn−2 zn−1 yn 1 + yn−2 xn−1 zn −1 + zn−2 yn−1 xn

where n ∈ N0 and the initial conditions are arbitrary nonzero real numbers such that x−2 z−1 y0 6= ±1, 6= z−2 y−1 x0 6= 1, 6=, 12 6= 13 and y−2 x−1 z0 6= ±1, 6= −1 2 . The following theorem is devoted to the expression of the form of the solutions of system (3).

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M. M. El-Dessoky et al 760-769

(3) 1 2,

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.4, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

plot of X(n+1)=X(n−2)/−1+X(n−2)Y(n)Z(n−1),Y(n+1)=Y(n−2)/−1+Y(n−2)Z(n)X(n−1),Z(n+1)=Z(n−2)/1+Z(n−2)X(n)Y(n−1) 5 X(n) Y(n) Z(n) 4

3

x(n),y(n),Z(n)

2

1

0

−1

−2

−3

−4

0

5

10

15

20 n

25

30

35

40

Figure 3.

Theorem 4.1. If {xn , yn , zn } are solutions of difference equation system (3). Then every solution of system (3) are takes the following form for n = 0, 1, 2, ..., x6n−2

=

x6n

=

x6n+2

=

x−2 (−1+x−2 z−1 y0 )n (1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n ,

x6n−1 =

(−1)n x0 (−1+z−2 y−1 x0 )n (−1+2z−2 y−1 x0 )n , (−1+3z−2 y−1 x0 )n (−1)n x−1 (1+y−2 x−1 z0 )n+1 (−1+y−2 x−1 z0 )n (1+2y−2 x−1 z0 )n+1 ,

y6n−2

= y−2 , y6n−1 = y−1 ,

y6n+2

=

y−1 (−1+z−2 y−1 x0 ) (−1+2z−2 y−1 x0 ) ,

(−1)n x−1 (1+y−2 x−1 z0 )n (−1+y−2 x−1 z0 )n (1+2y−2 x−1 z0 )n ,

x6n+1 =

x6n+3 =

(−1)n x0 (−1+2z−2 y−1 x0 )n+1 (−1+z−2 y−1 x0 )n , (−1+3z−2 y−1 x0 )n+1

y6n = y0 , y6n+3 =

x−2 (−1+x−2 z−1 y0 )n (1+x−2 z−1 y0 )n+1 (−1+2x−2 z−1 y0 )n ,

y6n+1 =

y−2 (1+y−2 x−1 z0 ) ,

y0 (1−x−2 z−1 y0 ) ,

and z6n−2

=

z6n

=

z6n+2 =

(−1)n z−2 (−1+3z−2 y−1 x0 )n (−1+z−2 y−1 x0 )n (−1+2z−2 y−1 x0 )n ,

(−1)n z0 (−1+y−2 x−1 z0 )n (1+2y−2 x−1 z0 )n , (1+y−2 x−1 z0 )n

−z−1 (1+x−2 z−1 y0 )n+1 (−1+2x−2 z−1 y0 )n , (−1+x−2 z−1 y0 )n

z−1 (1+x−2 z−1 y0 )n (−1+2x−2 z−1 y0 )n , (−1+x−2 z−1 y0 )n

z6n−1 =

z6n+1 =

z6n+3 =

(−1)n z−2 (−1+3z−2 y−1 x0 )n (−1+z−2 y−1 x0 )n+1 (−1+2z−2 y−1 x0 )n ,

(−1)n+1 z0 (−1+y−2 x−1 z0 )n (1+2y−2 x−1 z0 )n+1 . (1+y−2 x−1 z0 )n+1

Proof: As the proof of Theorem 2.1 and so will be omitted. Lemma 6. Let {xn , yn , zn } be a positive solution of system (3), then the sequences {xn , yn } are bounded and {xn } converges to zero. Lemma 7. It is easy to see that {yn } periodic with period six.

Lemma 8. If x−2 , x−1 , x0 , y−2 , y−1 , y0 , z−2 , z−1 and z0 arbitrary real numbers and let {xn , yn , zn } are solutions of system (3) then the following statements are true:(i) If x−2 = 0, y0 6= 0, z−1 6= 0, then x6n−2 = x6n+1 = 0 and y6n+3 = y0 , z6n−1 = z−1 , z6n+2 = −z−1 .

(ii) If x−1 = 0, y−2 6= 0, z0 6= 0, then x6n−1 = x6n+2 = 0 and y6n+1 = y−2 , z6n = z0 , z6n+3 = −z0 .

(iii) If x0 = 0, y−1 6= 0, z−2 6= 0, then x6n = x6n+3 = 0 and y6n+2 = y−1 , z6n−2 = z−2 , z6n+1 = −z−2 .

(iv) If y−2 = 0, x−1 6= 0, z0 6= 0, then y6n−2 = y6n+1 = 0 and x6n−1 = x6n+2 = x−1 , z6n = z0 , z6n+3 = −z0 .

(v) If y−1 = 0, x0 6= 0, z−2 6= 0, then y6n−1 = y6n+2 = 0 and x6n = x6n+3 = x0 , z6n−2 = z−2 , z6n+1 = −z−2 .

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plot of X(n+1)=X(n−2)/1+X(n−2)Y(n)Z(n−1),Y(n+1)=Y(n−2)/1+Y(n−2)Z(n)X(n−1),Z(n+1)=Z(n−2)/−1+Z(n−2)X(n)Y(n−1) 1 X(n) Y(n) Z(n) 0.8

0.6

x(n),y(n),Z(n)

0.4

0.2

0

−0.2

−0.4

−0.6

0

5

10

15

20

25 n

30

35

40

45

50

Figure 4.

(vi) If y0 = 0, x−2 6= 0, z−1 6= 0, then y6n = y6n+3 = 0 and x6n−2 = x6n+1 = x−2 , z6n−1 = z−1 , z6n+2 = −z−1 . (vii) If z−2 = 0, x0 6= 0, y−1 6= 0, then z6n−2 = z6n+1 = 0 and x6n = x6n+3 = x0 , y6n+2 = y−1 .

(viii) If z−1 = 0, x−2 6= 0, y0 6= 0, then z6n−1 = z6n+2 = 0 and x6n−2 = x6n+1 = x−2 , y6n+3 = y0 .

(ix) If z0 = 0, x−1 6= 0, y−2 6= 0, then z6n = z6n+3 = 0 and x6n−1 = x6n+2 = x−1 , y6n+1 = y−2 .

Example 4. See Figure (4) for an example for the system (3) with the initial values x−2 = 0.9, x−1 = 0.4, x0 = 0.1, y−2 = 0.2, y−1 = 0.7, y0 = 0.16, z−2 = 0.5, z−1 = 0.13 and z0 = 0.3.

XN+1 =

5. THE FOURTH SYSTEM: YN −2 YN+1 = −1+YN −2 XN −1 ZN , ZN +1 =

XN −2 1+XN −2 ZN −1 YN ,

ZN −2 −1+ZN −2 YN −1 XN

In this section, we investigate the solutions of the system of three difference equations xn+1 =

xn−2 yn−2 zn−2 , yn+1 = , zn+1 = , 1 + xn−2 zn−1 yn −1 + yn−2 xn−1 zn −1 + zn−2 yn−1 xn

(4)

where n = 0, 1, 2, ..., and the initial conditions are arbitrary non zero real numbers with z−2 y−1 x0 6= 1 and y−2 x−1 z0 6= 1. Theorem 5.1. Assume that {xn , yn , zn } are solutions of system (4). Then for n = 0, 1, 2, ..., x3n−2

=

y3n−2

=

y3n

=

z3n−2

=

z3n

=

x−2 1+nx−2 z−1 y0 ,

(−1)n+1 y−2 −1+ny−2 x−1 z0 , (−1)n y0 1+nx−2 z−1 y0 ,

x3n−1 = y3n−1 =

x−1 (−1+y−2 x−1 z0 ) (−1+(n+1)y−2 x−1 z0 ) ,

x3n =

(−1)n+1 y−1 (−1+z−2 y−1 x0 ) , (1+(n−1)z−2 y−1 x0 )

x0 1+nz−2 y−1 x0 ,

and (−1)n+1 z−2 (1+(n−1)z−2 y−1 x0 ) , (−1+z−2 y−1 x0 ) (−1)n z0 (−1+(n+1)y−2 x−1 z0 ) . (−1+y−2 x−1 z0 )

z3n−1 = (−1)n z−1 (1 + nx−2 z−1 y0 ),

Proof: As the proof of Theorem 2 and so will be omitted. Lemma 9. Let {xn , yn , zn } be a positive solution of system (4), then the sequences {xn } are bounded and converges to zero.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.4, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

plot of X(n+1)=X(n−2)/1+X(n−2)Y(n)Z(n−1),Y(n+1)=Y(n−2)/−1+Y(n−2)Z(n)X(n−1),Z(n+1)=Z(n−2)/−1+Z(n−2)X(n)Y(n−1) 20 X(n) Y(n) Z(n) 15

10

x(n),y(n),Z(n)

5

0

−5

−10

−15

−20

0

5

10

15

20

25 n

30

35

40

45

50

Figure 5.

Example 5. We assume the initial conditions x−2 = −1.9, x−1 = 1.4, x0 = −1.1, y−2 = 0.2, y−1 = −1.7, y0 = 0.16, z−2 = −1.5, z−1 = 0.13 and z0 = −1.3, for the difference system (4), see Fig. 5.

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. M. Alghamdi, E. M. Elsayed and M. M. Eldessoky, On the Solutions of Some Systems of Second Order Rational Difference Equations, Life Sci J., 10 (3) (2013), 344-351. 2. 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. 3. N. Battaloglu, C. Cinar and I. Yalçınkaya, The dynamics of the difference equation, ARS Combinatoria, 97 (2010), 281-288. 4. C. Cinar and I. Yalçinkaya, On the positive solutions of the difference equation system xn+1 = 1/zn , yn+1 = yn /xn−1 yn−1 , zn+1 = 1/xn−1 , J. Inst. Math. Comp. Sci., 18 (2005), 91-93. 1 5. C. Cinar and I. Yalçinkaya, On the positive solutions of the difference equation system xn+1 = , yn+1 = zn 1 xn , zn+1 = , International Math. J., 5 (2004), 525-527. xn−1 xn−1 6. D. Clark and M. R. S. Kulenovic, A coupled system of rational difference equations, Computers & Mathematics with Applications, 43 (2002), 849—867. 7. S. E. Das and M. Bayram, On a system of rational difference equations, World Applied Sciences Journal, 10(11) (2010), 1306—1312. 8. Q. Din, M. N. Qureshi and A. Qadeer Khan, Dynamics of a fourth-order system of rational difference equations, Advances in Difference Equations 2012, 2012: 215 doi: 10.1186/1687-1847-2012-215. 9. Q. Din, and E. M. Elsayed, Stability analysis of a discrete ecological model, Computational Ecology and Software, 4 (2) (2014), 89-103 10. Q. Din, M. N. Qureshi, and A. Q. Khan, Qualitative behavior of an anti-competitive system of third-order rational difference equations, Computational Ecology and Software, 4(2) (2014), 104-115. 11. A. Q. Khan, Q. Din, M. N. Qureshi, and T. F. Ibrahim, Behavior of an anti-competitive system of fourthorder rational difference equations, Computational Ecology and Software, 4(1) (2014), 35-46.

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12. E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the solutions of a class of difference equations systems, Demonstratio Mathematica, 41(1), 2008, 109-122. 13. M. M. El-Dessoky, E. M. Elsayed and E. O. Alzahrani, On Some System of Three Nonlinear Difference Equations, Life Sci. J., 10(3) ( 2013), 647-657. 14. E. M. Elsayed, Behavior of a rational recursive sequences, Studia Univ. " Babes-Bolyai ", Mathematica, LVI (1) (2011), 27—42. 15. E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 982309, 17 pages. 16. E. M. Elsayed, Solutions of rational difference system of order two, Math. Comput. Mod., 55 (2012), 378—384. 17. E. M. Elsayed and H. El-Metwally, Stability and Solutions for Rational Recursive Sequence of Order Three, Journal of Computational Analysis and Applications, 17 (2) (2014), 305-315. 18. E. M. Elsayed, Behavior and Expression of the Solutions of Some Rational Difference Equations, Journal of Computational Analysis and Applications, 15 (1) (2013), 73-81. 19. E. M. Elsayed and M. M. El-Dessoky, Dynamics and Behavior of a Higher Order Rational Recursive Sequence, Advances in Difference Equations 2012, 2012:69 (28 May 2012). 20. E. M. Elsayed and M. M. El-Dessoky, Dynamics and Global Behavior for a Fourth-Order Rational Difference Equation, Hacettepe Journal of Mathematics and Statistics, 42 (5) (2013), 479—494. 21. E. M. Elsayed, M. M. El-Dessoky and Asim Asiri, Dynamics and Behavior of a Second Order Rational Difference equation, Journal of Computational Analysis and Applications, 16 (4) (2014), 794-807. 22. E. M. Elsayed and H. A. El-Metwally, On the solutions of some nonlinear systems of difference equations, Advances in Difference Equations 2013, 2013:16, doi:10.1186/1687-1847-2013-161, Published: 7 June 2013. 23. E. M. Elsayed, M. M. El-Dessoky and Abdullah Alotaibi, On the Solutions of a General System of Difference Equations, Discrete Dynamics in Nature and Society, Volume 2012, Article ID 892571, 12 pages. 24. L. Keying, Z. Zhongjian, L. Xiaorui and Li Peng, More on three-dimensional systems of rational difference equations, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 178483, 9 pages. 25. A. S. Kurbanli, C. Cinar and I. Yalçınkaya, On the behavior of positive solutions of the system of rational difference equations, Mathematical and Computer Modelling, 53 (2011), 1261-1267. 26. A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations, Advances in Difference Equations 2011, 2011:40 doi:10.1186/1687-1847-2011-40. 27. A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations: xn+1 = xn−1 /xn−1 yn − 1, yn+1 = yn−1 /yn−1 xn − 1, zn+1 = zn−1 /zn−1 yn − 1, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 932362, 12 pages. 28. A. Kurbanli, C. Cinar and M. Erdo˘ gan, On the behavior of solutions of the system of rational difference xn−1 yn−1 xn , yn+1 = , zn+1 = , Applied Mathematics, 2 (2011), equations xn+1 = xn−1 yn − 1 yn−1 xn − 1 zn−1 yn 1031-1038. 29. M. Mansour, M. M. El-Dessoky and E. M. Elsayed, On the solution of rational systems of difference equations, Journal of Computational Analysis and Applications, 15 (5) (2013), 967-976. 30. 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. 31. N. Touafek and E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Mod., 55 (2012), 1987—1997. 32. I. Yalcinkaya, On the global asymptotic stability of a second-order system of difference equations, Discrete Dynamics in Nature and Society, Vol. 2008, Article ID 860152 (2008), 12 pages. 33. I. Yalçınkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria, 95 (2010), 151-159. 34. I. Yalcinkaya and C. Cinar, Global asymptotic stability of two nonlinear difference equations, Fasciculi Mathematici, 43 (2010), 171—180. 35. C. Wang, F. Gong, S. Wang, L. LI and Q. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear difference equation, Adv. Differ. Equ., Volume 2009, Article ID 214309, 8 pages. 36. E. M. E. Zayed and M. A. El-Moneam, On the global asymptotic stability for a rational recursive sequence, Iranian Journal of Science and Technology (IJST Transaction A- Science), (2011), A4: 333-339.

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Generalized Equi-Statistical Convergence of Positive Linear Operators Abdullah Alotaibi1) and M. Mursaleen2) 1) Department of Mathematics, King Abdulaziz University, P.O.Box 80203 Jeddah 21589, Saudi Arabia e-mail : [email protected] 2) Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India e-mail : [email protected]

Abstract The concepts of λ-equi-statistical convergence, λ-statistical pointwise convergence and λ-statistical uniform convergence for sequences of functions were introduced recently by Srivastava, Mursaleen and Khan [Math. Comput. Modelling, 55 (2012) 2040–2051.]. In this paper, we apply the notion of λequi-statistical convergence to prove a Korovkin type approximation theorem by using test functions x 2 x , ( 1+x ) and apply our result for the Bleimann, Butzer and Hahn [4] operators. We also study 1, 1+x

the rate of λ-equi-statistical convergence of a sequence of positive linear operators.

Keywords and phrases: Statistical convergence; λ-Statistical convergence; equi-statistical convergence; λ-equi-statistical convergence; positive linear operators; Korovkin type approximation theorem. AMS subject classification (2000): 41A10, 41A25, 41A36, 40A30, 33C45.

1. Introduction and Preliminaries The following concept of statistical convergence for sequences of real numbers was introduced by Fast [7]. Let K ⊆ N and Kn = {j ∈ K : j ≤ n}, where N is the set of natural numbers . Then the natural density of K is defined by |Kn | n→∞ n

δ(K) := lim

if the limit exists, where |Kn | denotes the cardinality of the set Kn . A sequence x = (xj ) of real numbers is said to be statistically convergent to the number L if, for every  > 0, the set {j : j ∈ N and |xj − L| ≥ } has natural density zero, that is, if, for each  > 0, we have 1 lim {j : j ≤ n and |xj − L| ≥ } = 0. n n 1

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The concept of λ-statistical convergence was studied in [14], [1] and [12] and further applied for deriving approximation theorems in [3], [5], [6], [13] and [15]. Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn + 1 and λ1 = 1. Also let In = [n − λn + 1, n]

and

K ⊆ N.

Then the λ-density of K is defined by 1 {j : n − λn + 1 ≤ j ≤ n and j ∈ K} . n→∞ λn

δλ (K) = lim

Clearly, in the special case when λn = n, the λ-density reduces to the above-defined natural density. The number sequence x = (xj ) is said to be λ-statistically convergent to the number L if, for each  > 0, δλ (K ) = 0, where K = {j : j ∈ In

and |xj − L| > },

that is, if, for each  > 0, 1 {j : j ∈ In n→∞ λn

and |xj − L| > } = 0.

lim

In this case, we write stλ - lim xn = L n→∞

and we denote the set of all λ-statistically convergent sequences by Sλ . The concept of equi-statistical convergence was introduced by Balcerzak et al. [2] and was subsequently applied for deriving approximation theorems in [8], [9] and [10]. In [16], the concepts of λ-equi-statistical convergence, λ-statistical pointwise convergence and λ-statistical uniform convergence for a sequence of real-valued functions were introduced. Further the notion of λ-equi-statistical convergence was used to prove a Korovkin type approximation theorem. In this paper, we prove such type of theorem x x 2 by using the test functions 1, 1+x and ( 1+x ). Let f and fn (n ∈ N) be real-valued functions defined on a subset X of the set N of positive integers. Definition 1.1. A sequence (fn ) of real-valued functions is said to be λ-equi-statistically
convergent to f on X if, for every  > 0, the sequence Sn (, x) n∈N of real-valued functions converges uniformly to the zero function on X, that is, if, for every  > 0, we have lim kSn (, x)kC(X) = 0,

n→∞

2

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where

1 {k : k ∈ In and |fk (x) − f (x)| ≥ } = 0 λn and C(X) denotes the space of all continuous functions on X. In this case, we write Sn (, x) :=

fn

f (λ-equi-stat).

Definition 1.2. A sequence (fn ) is said to be λ-statistically pointwise convergent to f on X if, for every  > 0 and for each x ∈ X, we have 1 lim {k : k ∈ In and | fk (x) − f (x) |≥ } = 0. n→∞ λn In this case, we write fn −→ f (λ-stat). Definition 1.3. A sequence (fn ) is said to be λ-statistically uniform convergent to f on X if (for every  > 0), we have 1 {k : k ∈ In and kfk − f kC(X) ≥ } = 0. n→∞ λn In this case, we write −→ (λ-stat) fn − −→f lim

Definition 1.4 (see [10]). A sequence (fn ) of real-valued functions is said to be equi
statistically convergent to f on X if, for every  > 0, the sequence Pn, (x) n∈N of real-valued functions converges uniformly to the zero function on X, that is, if (for every  > 0) we have lim kPn, (x)kC(X) = 0, n→∞

where Pn, (x) =

1 {k : k 5 n and |fk (x) − f (x)| ≥ } = 0. n

In this case, we write fn

f (equi-stat).

The following implications of the above definitions and concepts are trivial. Lemma 1.1. Each of the following implications holds true: −→ (λ-stat) =⇒ f fn − −→f n

f (λ-equi-stat) =⇒ fn → f (λ-stat).

Furthermore, in general, the reverse implications do not hold true. Example 1.1. Let λ = (λn ) be a sequence as described above and consider the sequence of continuous functions fr : [0, 1] → R (r ∈ N), defined as follows:     1 1  2   (r + 1) x − x+   1 
 r+1 r+1  − x ∈ 0, r+1 x2 + 1 fr (x) =       0 (otherwise). 3

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Then, for every  > 0, we have 1 {r : r ∈ In and |fr (x)| ≥ } λn 1 →0 (n → ∞) ≤ λn 0 (λ-equi-stat). But, since sup |fr (x)| =

uniformly in x. This implies that fr

x∈[0,1]

−→ (λ-stat) does not hold 1 (r ∈ N), we conclude that the following condition: fr − −→0 true. . √ Example 1.2. Let λn = [ n ] and consider the sequence of continuous functions
fr : [0, 1] → R fr (x) = xr (r ∈ N) . If f is the pointwise limit of fr (in the ordinary sense), then fr → f (λ-stat), but the condition: fr

f (equi-stat)

does not hold true. Let us take  = 12 . Then, for all n ∈ N, there exists r > N such that ! r n 1 ,1 , m ∈ [n − [λn ] + 1, n] and x∈ 2 so that

r !m r !n n 1 n 1 1 m |fm (x)| = |x | ≥ ≥ = . 2 2 2

2. Main Result Let C[a, b] be the linear space of all real-valued continuous functions f on [a, b] and let T be a linear operator which maps C[a, b] into itself. We say that T is positive if, for every non-negative f ∈ C[a, b], we have T (f, x) ≥ 0

(x ∈ [a, b]).

We know that C[a, b] is a Banach space with the norm given by kf kC[a,b] := sup |f (x)|

(f ∈ C[a, b]).

x∈[a,b]

The classical Korovkin approximation theorem states as follows [11]. Let {Tn } be a sequence of positive linear operators from C[a, b] into C[a, b]. Then lim kTn (f, x) − f (x)kC[a,b] = 0 n

⇐⇒

(f ∈ C[a, b])

lim kTn (fi , x) − ei (x)kC[a,b] = 0 n

(i = 0, 1, 2),

4

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where ei (x) = xi

(i = 0, 1, 2).

In [16], the following result was proved which is an extension of the result of Karaku¸s et al. [10]. Theorem 2.1. Let X be a compact subset of the set R of real numbers. Also let {Ln } be a sequence of positive linear operators from C(X) into itself. Then, for all f ∈ C(X), Ln (f ) f (λ-equi-stat) on X if and only if Ln (ei ) ei (λ-equi-stat) on X. x and In this paper, we prove such type of theorem by using the test functions 1, 1+x x 2 ). ( 1+x Let K = [0, ∞) and CB (K) denote the space of all bounded and continuous real valued functions on K equipped with norm kf kCB (K) := sup |f (x)|, f ∈ CB (K). x∈K

Let Hω (K) denote the space of all real valued functions f on K such that | f (s) − f (x) |≤ ω(f ; |

x s − | ), 1+s 1+x

where ω is the modulus of continuity, i.e. ω(f ; δ) = sup {|f (s) − f (x)| : |s − x |≤ δ} (δ > 0). s,x∈K

It is to be noted that any function f ∈ Hω (K) is bounded and continuous on K, and a necessary and sufficient condition for f ∈ Hω (K) is that lim ω(f ; δ) = 0.

δ→0

We prove the following theorem. Theorem 2.2. Let {Ln } be a sequence of positive linear operators from Hω (K ) into CB (K ). Then, for all f ∈ Hω (K ), Ln (f )

f (λ-equi-stat)

(2.1)

gi (λ-equi-stat) (i = 0, 1, 2),

(2.2).

if and only if Ln (fi ) with

x x 2 and g2 (x) = ( ). 1+x 1+x Proof. Since each of the functions fi belongs to Hω (K), conditions (3.2) follow immediately. Let g ∈ Hω (K) and x ∈ K be fixed. Then for ε > 0 there exist δ > 0 such s x that | f (s) − f (x) |< ε holds for all s ∈ K satisfying | 1+s − 1+x | < δ. Let g0 (x) = 1, g1 (x) =

K(δ) := {s ∈ K : |

s x − |< δ}. 1+s 1+x 5

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Hence | f (s) − f (x) |=| f (s) − f (x) |χK(δ) (s) + | f (s) − f (x) |χK\K(δ) (s) ≤ ε + 2NχK\K(δ) (s) ,

(2.3)

where χD denotes the characteristic function of the set D and N = kf kCB (K) . Further we get x 2 1 s − ). (2.4) χK\K(δ) (s) ≤ 2 ( δ 1+s 1+x Combining (2.3) and (2.4), we get | f (s) − f (x) |≤ ε +

2N x 2 s − ). ( δ2 1 + s 1 + x

(2.5)

After using the linearity and positivity of operators {Ln }, we get | Ln (f ; x) − f (x) |≤ ε + M { | Ln (g0 ; x) − g0 (x) | + | Ln (g1 ; x) − g1 (x) | + | Ln (g2 ; x) − g2 (x) | + | Ln (g3 ; x) − g3 (x) | },

(2.6)

which implies that |Ln (f ; x) − f (x)| ≤  + B

2 X

|Ln (gi ; x) − gi (x)|,

(2.7)

i=o

where M := ε + N + 4N . Now for a given r > 0, choose  > 0 such that  < r. δ2 Then, for each i = 0, 1, 2, set ψn (x, r) := |{k ∈ In : |Lk (f ; x) − f (x)| ≥ r}| and ψi,n (x, r) := |{k ∈ In : |Lk (gi ; x) − gi (x)| ≥ r− }| for (i = 0, 1, 2), it follows from (2.7) 3K 2 P that ψn (x, r) ≤ ψi,n (x, r). Hence i=o

 2  kψn (., r)kCB (K) X kψi,n (., r)kCB (K) ≤ . λn λ n i=o

(2.8)

Now using the hypothesis (2.2) and the Definition 1.1, the right hand side of (2.8) tends to zero as n → ∞. Therefore, we have kψn (., r)kCB (K) = 0 for every r > 0, n→∞ λn lim

i.e. (2.1) holds. This completes the proof of the theorem. 3. An Application We shall apply our result for the following Bleimann, Butzer and Hahn [4] operators: Let    n X 1 j n k Bn (f ; x) := f x , n (1 + x) k=0 n−k+1 k 6

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be a sequence of positive linear operators from Hω (K) into CB (K), K = [0, ∞) and n ∈ N. Then, for all f ∈ Hω (K), Bn (f ) Since

f (λ-equi-stat).

n   X n k (1 + x) = x , k k=0 n

it is easy to see that n   X 1 n k Bn (f0 ; x) = x n (1 + x) k=0 k

= 1 = g0 (x). Also by simple calculation, we obtain   n x x Bn (f1 ; x) = → = g1 (x), n+1 1+x 1+x and  2   n x x n(n − 1) + Bn (f2 ; x) = (n + 1)2 1 + x (n + 1)2 1 + x x 2 ) = g2 (x). → ( 1+x Therefore Bn (fi ; x) → gi (x) (n → ∞) (i = 0, 1, 2), and cosequently, we have Bn (fi )

gi (λ-equi-stat) (i = 0, 1, 2).

Hence by Theorem 2.2, we have Bn (f )

f (λ-equi-stat).

4. Rate of λ− equistatistical convergence In this section we study the rate of λ-equi-statistical convergence of a sequence of positive linear operators as given in [16]. Definition 4.1. Let (an ) be a positive non-increasing sequence. A sequence (fn ) is equi-statistically convergent to a function f with the rate o(an ) if for every  > 0, lim

n→∞

Λn (x, ) =0 an 7

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uniformly with respect to x ∈ K or equivalently, for every  > 0, kΛn (., )kCB (X) = 0, n→∞ an lim

where Λn (x, ) :=

1 | {k ∈ In :| fk (x) − f (x) |≥ ε} |= 0. λn

In this case, it is denoted by fn − f = o(an )) (λ-equi-stat) on K. We have the following basic lemma. Lemma 4.1. Let (fn ) and (gn ) be sequences of function belonging to C(X). Assume that fn − f = o(an ) ( λ-equi-stat) on X and gn − g = o(bn ) ( λ-equi-stat) on X . Let cn = max{an , bn } . Then the following statement holds: (i) (fn + gn ) − (f + g) = o(cn ) ( λ-equi-stat) on X , (ii) (fn − f )(gn − g) = o(an bn ) ( λ-equi-stat) on X, (iii) µ(fn − f ) = o(an ) ( λ-equi-stat) on X for any real number µ, (iv)

p

|fn − f | = o(an ) ( λ-equi-stat) on X.

We recall that the modulus of continuity of a function f ∈ Hω (K) is defined by ω(f ; δ) = sup {|f (s) − f (x)| : |s − x |≤ δ} (δ > 0). s,x∈K

Now we prove the following result. Theorem 4.2. Let X be a compact subset of the real numbers, and {Ln } be a sequence of positive linear operators from Hω (K ) into CB (K ). Assume that the following conditions hold: (a) Ln (g0 ; x) − g0 = o(an ) ( λ-equi-stat) on K, p (b) ω(f, δn ) = o(bn ) ( λ-equi-stat) on K , where δn (x) = Ln (φ2 ; x)
s x with φ(x) = 1+s − 1+x . Then for all f ∈ Hω (K), we have Ln (f ) − f = o(cn ) (λ-equi-stat) on K, where cn = max{an , bn }.

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Proof. Let f ∈ Hω (K) and x ∈ K. Then it is well known that, |Ln (f ; x) − f (x)| ≤ M |Ln (g0 ; x) − g0 (x)| + (Ln (g0 ; x) +

p Ln (g0 ; x))ω(f, δn ),

where M = kf kHω (K) . This yields that |Ln (f ; x) − f (x)| ≤ M |(Ln (g0 ; x) − g0 (x)| + 2ω(f, δn ) +ω(f, δn )|(Ln (g0 ; x) − g0 (x)| + ω(f, δn )

p |(Ln (g0 ; x) − g0 (x)|.

Now using the conditions (a), (b) and Lemma 4.1 in the above inequality, we get Ln (f )− f = o(cn ) (λ-equi-stat) on K. This completes the proff of the theorem. Acknowledgement This Project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (197/130/1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.

References [1] M.A. Alghamdi and M. Mursaleen, λ-Statistical convergence in paranormed space, Abstract Appl. Anal., Volume 2013, Article ID 264520, 5 pages, http://dx.doi.org/10.1155/2013/264520. [2] M. Balcerzak, K. Dems and A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl., 328 (2007), 715–729. [3] C. Belen, S.A. Mohiuddine, Generalized weighted statistical convergence and application, Applied Math. Comput. 219 (2013) 9821–9826. [4] G. Bleimann, P. L. Butzer and L. Hahn, A Bernstein type operator approximating continuous functions on semiaxis, Indag. Math., 42 (1980) 255–262. [5] N.L. Braha, H.M. Srivastava, S.A. Mohiuddine, A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vall´ee Poussin mean, Appl. Math. Comput. 228 (2014) 162-169. [6] O.H. H. Edely, S.A. Mohiuddine and A. K. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Lett., 23 (2010). 1382–1387. [7] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.

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[8] S. Karaku¸s and K. Demirci, Equi-statistical extension of the Korovkin type approximation theorem, Turkish J. Math., 33 (2009), 159–168; Erratum, Turkish J. Math., 33 (2009), 427–428. [9] S. Karaku¸s and K. Demirci, Equi-statistical σ-convergence of positive linear operators, Comput. Math. Appl., 60 (2010), 2212–2218. [10] S. Karaku¸s, K. Demirci and O. Duman, Equi-statistical convergence of positive linear operators, J. Math. Anal. Appl., 339 (2008), 1065–1072. [11] P.P. Korovkin, Linear Operators and Approximation Theory, [Translated from the Russian edition (1959)], Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. III, Gordon and Breach Publishers, New York; Hindustan Publishing Corporation, Delhi, 1960. [12] S.A. Mohiuddine, A. Alotaibi and M. Mursaleen, A new variant of statistical convergence, Jour. Ineq. Appl., 2013, 2013:309, doi:10.1186/1029-242X-2013-309. [13] 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]. [14] M. Mursaleen, A. Alotaibi and S.A. Mohiuddine, Statistical convergence through de la Vall´ee-Poussin mean in locally solid Riesz spaces, Adv. Difference Equ., 2013, 2013:66, doi:10.1186/1687-1847-2013-66. [15] M. Mursaleen, V. Karakaya, M. Ert¨ urk and F. G¨ ursoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput., 218 (2012) 9132-9137. [16] H.M. Srivastava, M. Mursaleen and Asif Khan, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. Comput. Modelling, 55 (2012) 2040–2051.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO. 4, 2015

Applications of Sampling Kantorovich Operators to Thermographic Images for Seismic Engineering, Federico Cluni, Danilo Costarelli, Anna Maria Minotti, and Gianluca Vinti,…602 On the Stolarsky-Type Inequality for the Chqouet Integral and the Generated Choquet Integral, Jeong Gon Lee, and Lee-Chae Jang,…………………………………………………………618 Extended Cesàro Operator from Weighted Bergman Space to Zygmund-Type Space, Yu-Xia Liang, and Ren-Yu Chen,………………..………………………...........................................625 The Numerical Solution for the Interval-Valued Differential Equations, Li-Xia Pan,………632 Controllability for Semilinear Integrodifferential Control Systems with Unbounded Operators, Jin-Mun Jeong, and Sang-Jin Son,…………………………………………………………...642 Stability of Positive-Additive Functional Equations in Fuzzy C*-Algebras, Afrah A.N. Abdou, Fatma S. Al-Sirehy, and Yeol Je Cho,……………………………………………………….651 Common Fixed Point Theorems for Occasionally Weakly Biased Mappings Under Contractive Conditions in Menger PM-Spaces, Zhaoqi Wu, Chuanxi Zhu, and Jing Wang,…………….661 The Existence of Symmetric Positive Solutions for a Nonlinear Multi-Point Boundary Value Problem on Time Scales, Fatma Tokmak, Ilkay Yaslan Karaca, Tugba Senlik, and Aycan Sinanoglu,……………………………………………………………………………670 Approximate Fuzzy Double Derivations and Fuzzy Lie *-Double Derivations, Ali Ebadian, Mohammad Ali Abolfathi, Choonkil Park, and Dong Yun Shin,…………………………..678 Some New Generalized Results on Ostrowski Type Integral Inequalities with Application, A. Qayyum, M. Shoaib, and Ibrahima Faye,……………………………………………….693 A Novel Multistep Generalized Differential Transform Method for Solving Fractional-order Lü Chaotic and Hyperchaotic Systems, Mohammed Al-Smadi, Asad Freihat, Omar Abu Arqub, and Nabil Shawagfeh,…………………………………………………………………………...713 New Inequalities of Hermite-Hadamard and Fejér Type via Preinvexity, M. A. Latif, and S. S. Dragomir,……………………………………………………………………………….725 Approximate Quadratic Forms in Paranormed Spaces, Jae-Hyeong Bae, and Won-Gil Park,.740

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO. 4, 2015 (continued) Common Fixed Point Theorems for the R-weakly Commuting Mappings in S-metric Spaces, Jong Kyu Kim, Shaban Sedghi, and Nabi Shobkolaei,………………………………………751 On a Solution of System of Three Fractional Difference Equations, M. M. El-Dessoky, and E. M. Elsayed,………………………………………………………………………………………760 Generalized Equi-Statistical Convergence of Positive Linear Operators, Abdullah Alotaibi, and M. Mursaleen,…………………………………………………………………………………770

Volume 19, Number 5 ISSN:1521-1398 PRINT,1572-9206 ONLINE

November 2015

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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 (twelve 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. Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC

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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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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

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27)Svetlozar (Zari) Rachev, Professor of Finance, College of Business,and Director of Quantitative Finance Program, Department of Applied Mathematics & Statistics Stonybrook University 312 Harriman Hall, Stony Brook, NY 117943775 Phone: +1-631-632-1998, Email : [email protected] 28) Alexander G. Ramm Mathematics Department Kansas State University Manhattan, KS 66506-2602 e-mail: [email protected] Inverse and Ill-posed Problems, Scattering Theory, Operator Theory, Theoretical Numerical Analysis, Wave Propagation, Signal Processing and Tomography

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 9) George Cybenko Thayer School of Engineering Dartmouth College 8000 Cummings Hall, Hanover,NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail: [email protected] Approximation Theory and Neural Networks

29) Ervin Y.Rodin Department of Systems Science and Applied Mathematics Washington University,Campus Box 1040 One Brookings Dr.,St.Louis,MO 631304899 314-935-6007 e-mail: [email protected] Systems Theory, Semantic Control, Partial Differential Equations, Calculus of Variations, Optimization

10) Ding-Xuan Zhou Department Of Mathematics City University of Hong Kong

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83 Tat Chee Avenue Kowloon,Hong Kong 852-2788 9708,Fax:852-2788 8561 e-mail: [email protected] Approximation Theory, Spline functions,Wavelets

and Artificial Intelligence, Operations Research, Math.Programming 30) T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece Postal Address: 26 Menelaou St. Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis

11) Sever S.Dragomir School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, MC 8001,AUSTRALIA Tel. +61 3 9688 4437 Fax +61 3 9688 4050 [email protected] Inequalities,Functional Analysis, Numerical Analysis, Approximations, Information Theory, Stochastics.

31) I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3 0651098283

12) Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR-06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

32) Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock,Germany [email protected] Numerical Fourier Analysis, Fourier Analysis,Harmonic Analysis, Signal Analysis, Spectral Methods, Wavelets, Splines, Approximation Theory

13) Saber N.Elaydi Department Of Mathematics Trinity University 715 Stadium Dr. San Antonio,TX 78212-7200 210-736-8246 e-mail: [email protected] Ordinary Differential Equations, Difference Equations

33) Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 P.D.E, Control Theory, Functional Analysis, [email protected]

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

34) Gilbert G.Walter Department Of Mathematical Sciences University of Wisconsin-Milwaukee,Box 413, Milwaukee,WI 53201-0413 414-229-5077 e-mail: [email protected] Distribution Functions, Generalised Functions, Wavelets 35) 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

Lotharstr.65,D-47048 Duisburg,Germany e-mail:[email protected] Fourier Analysis,Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory

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

36) 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

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37) 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

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

38) 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

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 NEW MEMBERS 39)Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] Computational Mathematics

40) Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S.Korea, [email protected] Functional Equations

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

ON THE INTERVAL-VALUED PSEUDO-LAPLACE TRANSFORM BY MEANS OF THE INTERVAL-VALUED PSEUDO-INTEGRAL JEONG GON LEE AND LEE-CHAE JANG

Division of Mathematics and Informational Statistics, and Nanoscale Science and Technology Institute, Wonkwang University, Iksan 570-749, Republic of Korea E-mail : [email protected], Phone:082-63-850-6189 General Education Institute, Konkuk University, Chungju 138-701, Republic of Korea E-mail : [email protected], Phone:082-43-840-3591 Abstract. Pap and Ralevic (1998) introduced the pseudo-Laplace transform and proved the analog of exchange formula. Jang (2013) defined the interval-valued generalized fuzzy integral by using an interval- representable pseudo-operations and Jang (2014) defined the interval- valued g¯-integral. In this paper, by using the concept of the interval-valued pseudo-integral, we define interval-valued pseudo-Laplace transform. Furthermore, we investigate pseudo-exchange formula and inverse of pseudo-Laplace transform.

1. Introduction Many researchers [3,4,15,16,17,18,19,21,22] have studied the theory of the pseudo-integral, for examples, fuzzy integrals, generalized fuzzy integral, and generated pseudo-integral, etc. Aubin [1], Aumann [2], Grbic et al. [5], Grabisch [6], Guo and Zhang [7], Jang [7-14], and Wechselberger [20] have been researching various integrals of measurable multi-valued functions which are used for representing uncertain functions. Pap and Ralevic [17] introduced the pseudo-Laplace transform and proved the analogue of exchange formula. They also discussed inverse of pesudo-Laplace transform. The purpose of this paper is to define the interval-valued pseudo-Laplace transform and investigate some characterizations of them. Furthermore, we also investigate pseudo-exchange formula and inverse of pseudo-Laplace transform. The paper is organized in five sections. In section 2, we list definitions and some properties of the pseudo-integral and the pseudo-Laplace transform by means of the corresponding pseudo-interval. We also discuss the special forms of the pseudo-Laplace transform. In section 3, we define the interval-valued pseudo-Laplace transform and investigate some characterizations of them. Furthermore, we discuss the special forms of the interval-valued pseudo-Laplace transform. In section 4, we prove pseudo-exchange formula of the special forms of intervalvalued pseudo-Laplace transform and discuss inverse of them. 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. σ- -decomposable measure, pseudo-integral, pseudo-Laplace transform, pseudoexchange formular, interval-representable pseudo-operations, interval-valued pseudo-Laplace transform . 1

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2. Definitions and Preliminaries In this section, we introduce a pseudo-addition and a pseudo-multiplication. Let [a, b] be ¯ = [−∞, ∞]. The full a closed (in some cases can be considered semiclosed) subinterval of R order on [a, b] will be denoted by ¹. Definition 2.1. ([3,4,5,12-19,21,22]) (1) The pseudo-addition ⊕ is an operation ⊕ : [a, b] × [a, b] −→ [a, b] which is commutative, non-decreasing with respect to ¹, that is, x ¹ y implies x ⊕ z ¹ y ⊕ z for all z ∈ [a, b], associative, and with a zero (natural) element denoted by 0, that is, for each x ∈ [a, b], 0 ⊕ x = x holds (usually 0 is either a or b). (2) The pseudo-multiplication ¯ is an operation ¯ : [a, b] × [a, b] −→ [a, b] which is commutative, positively non-decreasing with respect to ¹, that is, x ¹ y implies x ¯ z ¹ y ¯ z for all z ∈ [a, b]+, associative, and there exists a unit element 1 ∈ [a, b], that is, for each x ∈ [a, b], 1 ¯ x = x, where [a, b]+ = {x|x ∈ [a, b], 0 ¹ x}. Definition 2.2. ([16,17,21,22]) (1) Let X be a set and M be a σ -algebra of subsets of X. An elementary measurable function e is a mapping e : X −→ [a, b] if it has the following representation : e(x) = ⊕∞ i=1 ai ¯ χAi (x),

(1)

where ai ∈ [a, b], Ai ∈ M for i = 1, 2, · · · , x ∈ X, and pseudo-characteristic function χA of A is defines by ½ 1 for x ∈ A χA (x) = 0 for x6∈A. (2) The pseudo-integral of an elementary function e with respect to the σ−⊕-decomposable measure µ is Z ⊕ (2) e ¯ dµ = ⊕∞ i=1 ai ¯ µ(Ai ) where Ai ∈ M disjoint if ⊕ is not idempotent. (3) The pseudo-integral of a measurable function f : X −→ [a, b] is defined by Z ⊕ Z ⊕ f ¯ dµ = lim en ¯ dµ (3) n→∞

where {en } is the sequence of elementary functions with limn→∞ en = f . R⊕ (4) A measurable function f is said to be integrable if f ¯ dµ is finite. Let (X, +) be a commutative group for X ⊂ R and ([a, b], ⊕, ¯) be a semiring. We also consider the pseudo-Laplace transform of a function. Definition 2.3. ([17]) (1) The semiring B(X, [a, b]) is the set of all bounded (with respect to the order in [a, b]) functions. R (2) The space L1 (X) is the set of Lebesgue integrable functions which satisfy the condition |f (x)|dx < +∞. X (3) The pseudo-character of the group (X, +), X ⊂ R is a map ξ : X → [a, b], of the group (X, +) into semiring ([a, b], ⊕, ¯), with the property ξ(x + y) = ξ(x) ¯ ξ(y), x, y ∈ X.

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(4) The pseudo-Laplace transform L⊕ (f ) of a function f ∈ B(X, [a, b]) is defined by Z ⊕ (L⊕ f )(ξ)(z) = ξ(x, −z) ¯ dmf

(5)

X∩[0,∞)

where ξ is the continuous pseudo-character for z ∈ R for which the right side is meaningful mf (A) = inf x∈A f (x), and dmf = f ¯ mf . Remark 2.4. ([17]) We have the special cases the following forms of the pseudo-Laplace transform for X = R. (1) If ⊕ = max, ¯ = +, and ξ(x, −z) = −xz, then we have (L⊕ 1 f )(z) = supx≥0 (−xz +f (x)) −xz −xz (2) If ⊕ = max, ¯ = ·, and ξ(x, −z) = e , then we have (L⊕ f (x)) 2 f )(z) = supx≥0 (e −1 −1 −xz (3) If x ⊕ y = Rg (g(x) + g(y)), x ¯ y = g (g(x)g(y)), and ξ(x, −z) = e , then we have ∞ −1 (L⊕ ( 0 e−xz g(f (x))dx) 3 f )(z) = g where g : [−∞, ∞] → [−∞, ∞], g(0) = 0 is an odd, strictly increasing, continuous function, f ∈ B(X, [a, b]), and g|f | ∈ L1 (X). R∞ −1 We note that L⊕ (L(g ◦ f )(z)), where (Lf )(z) = 0 e−zx f (x)dx, and that 3 (f )(z) = g R (f ∗ h)(x) = f (x − t)h(t)dt for all x ∈ X. In [17], they proved the following pseudo-exchange formula and inverse of the pseudo-Laplace transform. Theorem 2.1. ([17]) Let f1 , f2 ∈ B(X, [a, b]). Then we have ⊕ ⊕ L⊕ k (f1 ∗k f2 ) = L (f1 ) ¯ L (f2 ), for k = 1, 2, 3

(6)

where (f1 ∗1 f2 )(x) = sup0≤t≤x [f1 (x − t) + f2 (t)], (f1 ∗2 f2 )(x) = sup0≤y≤x (f1 (y)f2 (x − y)), and (f1 ∗3 f2 )(x) = g −1 ((g ◦ f1 ∗ g ◦ f2 )(x)).

Theorem 2.2. ([17]) If f ∈ B(X, [a, b]) and L⊕ k (f ) = Fk for k = 1, 2, 3, and there exists −1 (L⊕ ) (F ), then we have k k −1 (1) (L⊕ (F1 )(x) = inf z≥0 (xz + F (z)), 1) ⊕ −1 (2) (L2 ) (F2 )(x) = inf z≥0 (exz F (z)), and −1 (3) (L⊕ (F3 )(x) = g −1 (L−1 (g ◦ F )(x)), R 3) ∞ where f ∈ B(X, I([a, b])), g|f | ∈ L1 (X), L(f ) = 0 e−zx f (x)dx = F (z) and L−1 (F ) = f (x).

3. The interval-valued pseudo-Laplace transforms In this section, we consider a standard interval-valued addition and a standard intervalvalued pseudo-multiplication (see [12,13]). Let I([a, b]) be the set of all bounded closed intervals in [a, b] as follows : I([a, b]) = {¯ u = [ul , ur ] | ul , ur ∈ [a, b] and ul ¹ ur }.

(7)

We note that the full order on I([a, b]) will be denoted by ¹i . For any a ∈ [a, b], we define a = [a, a]. Obviously, a ∈ I([a, b]). L Definition 3.1. ([12]) (1) An operation : I([a, b])2 −→ I([a, b]) is called a standard interval-valued pseudo-addition if there exist pseudo-additions ⊕l and ⊕r such that x ⊕l y ¹ x ⊕r y for all x, y ∈ [a, b] and such that for each u ¯ = [ul , ur ], v¯ = [vl , vr ] ∈ I([a, b]), M u ¯ v¯ = [ul ⊕l vl , ur ⊕r vr ]. (8)

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L Then ⊕l and ⊕r are J called represents of . (2) An operation : I([a, b])2 −→ I([a, b]) is called a standard interval-valued pseudomultiplication if there exist pseudo-multiplications ¯l and ¯r such that x ¯l y ≤ x ¯r y for all x, y ∈ [a, b] and such that for each u ¯ = [ul , ur ], v¯ = [vl , vr ] ∈ I([a, b]), K u ¯ v¯ = [ul ¯l vl , ur ¯r vr ]. (9) J Then ¯l and ¯r are called represents of .

Definition ¯ = [ul , ur ], v¯ = [vl , vr ] ∈ I([a, b]) and k ∈ [a, b], then we define L 3.2. ([12]) If u (1) u ¯ J v¯ = [ul ⊕l vl , ur ⊕r vr ], (2) kJ u ¯ = [k ¯l vl , k ¯r vr ], (3)¯ u v¯ = [ul ¯l vl , ur ¯r vr ], (4) u ¯ ∨ v¯ = [ul ∨ vl , ur ∨ vr ], (5) u ¯ ∧ v¯ = [ul ∧ vl , ur ∧ vr ], (6) u ¯ ¹i v¯ if and only if ul ¹l vl and ur ¹r vr , (7) u ¯ ≺i v¯ if and only if u ¯ ¹r v¯ and u ¯ 6= v¯, and

Definition 3.3. ([11]) If K ⊂ R and u ¯α = [uα l , vα r ] for all α ∈ K, then we define (1) supα∈K u ¯α = [supα∈K uαl , supα∈K uαr ], and (2) inf α∈K u ¯α = [inf α∈K uαl , inf α∈K uαr ].

Definition 3.4. An ¯ = [µl , µr ] : M −→ I([a, b]) is called the L interval-valued mapping µ interval-valued σ− -decomposable measure µ ¯ = [µl , µr ] if µl and µr are σ−⊕- decomposable measures. L J Let (X, +) be a commutative group for X ⊂ R and (I([a, b]), , ) be a semiring. In [8-14], we see that if dH is the Hausdorff metric on I([a, b]), that is, dH (A, B) = max{sup inf |x − y|, inf sup |x − y|} x∈A y∈B

x∈A x∈B

(10)

then we have dH − lim [un , vn ] = [u, v] if and only if n→∞

lim un = u and lim vn = v.

n→∞

n→∞

(11)

Definition 3.5. (1) The semiring IB(X, I([a, b])) is the set of all bounded (with respect to the pseudo-order ¹i in I([a, b])) interval-valued functions. (2) The space IL1 (X) be the setRof all Lebesgue integrable interval-valued functions f¯ = R [fl , fr ] which satisfy the conditions X |fl (x)|dx < ∞ and X |fr (x)|dx < ∞. (3) The interval-valued pseudo-character of the group (X, +), X ⊂ R is an Linterval-valued J map ξ¯ = [ξl , ξr ] : X → I([a, b]), of the group (X, +) into semiring (I([a, b]), , ), with the property K ¯ + y) = ξ(x) ¯ ¯ ξ(x ξ(y), for all x, y ∈ X. (12)

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Definition 3.6. (1) Let X be a set and M be a σ-algebra of subset of X. A measurable interval-valued elementary function e¯ = [el , er ]is an interval-valued mapping e¯ : X −→ I([a, b]) if it has the following representation : ∞ M K e¯(x) = a¯i χAi (x) (13) i=1

where a¯i = [ai l , ai r ] ∈ I([a, b]), and χAi is a pseudo-characteristic function of Ai for all i = 1, 2, · · · . (2) The interval-valued Lpseudo-integral of an interval-valued elementary function e¯ = [el , er ] with respect to the σ − -decomposable measure µ ¯ is Z  K ∞ M K e¯ d¯ µ= a¯i µ ¯(Ai ). (14) i=1

L where µ ¯ = [µl , µr ], µl , µr are σ − -decomposable measures, and Ai ∈ M disjoint if ⊕l and ⊕r are not idempotent. (3) The interval-valued pseudo-integral of a measurable interval-valued function f¯ = [fl , fr ] : [a, b] −→ I([a, b]) is defined by Z  K Z  K ¯ f d¯ µ = dH − lim e¯n d¯ µ (15) n→∞

where {e¯n } is the sequence of interval-valued elementary functions with dH − limn→∞ e¯ = f¯. R  Jn (4) An interval-valued measurable function f¯ is said to be integrable if f¯ d¯ µ ∈ I([a, b]). By a standard interval-valued addition and a standard interval-valued multiplication, we define the interval-valued pseudo-Laplace transforms as follows. 

Definition 3.7. The interval-valued pseudo-Laplace transform Li (f¯) of an interval-valued function f¯ = [f1 , fr ] ∈ B(X, I([a, b])) is defined by Z   K ¯ ¯ −z) (Li f¯)(ξ)(z) (16) = ξ(x, dmf¯ X∩[0,∞)

where ξ¯ = [ξl , ξr ] is the dH -continuous interval-valued pseudo-character for Jz ∈ R for which the right hand side is meaningful and mf¯(A) = inf x∈A f¯(x), and dmf¯ = f¯ mf¯. We obtain some characterizations of the standard interval-valued pseudo-characters and the interval-valued pseudo-Laplace transforms. L Theorem 3.1. If ⊕l and ⊕r are represents of a standard interval-valued pseudo-addition , J and ¯l and ¯r are represents of a standard interval-valued pseudo-multiplication and if ξ¯ = [ξl , ξr ] is the interval-valued pseudo-character, then ξl and ξr are pseudo-characters. J Proof. Since is a standard interval-valued pseudo-multiplication, for each x, y ∈ X, K ¯ + y) = ξ(x) ¯ ¯ [ξl (x + y), ξr (x + y)] = ξ(x ξ(y) K = [ξl (x), ξr (x)] [ξl (y), ξr (y)] = [ξl (x) ¯l ξl (y), ξr (x) ¯r ξr (y)]. (17) Thus, we have ξl (x + y) = ξl (x) ¯l ξl (y) and ξr (x + y) = ξr (x) ¯r ξr (y). That is, ξl and ξr are pseudo-characters of ¯l and ¯r , respectively. ¤

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Theorem 3.2. If f¯ = [fl , fr ] ∈ B(X, I([a, b])), mf¯(A) = inf x∈A f¯(x) for all A ∈ M, mf¯ = J [mfl , mfr ], and dmf¯ = f¯ mf¯, then we have (1) mf¯(A) = [mfl (A), mfr (A)] for all A ∈ M (2) dmf¯ = [dmfl , dmfr ] where dmfl = fl ¯l mfl and dmfr = fr ¯r mfr . Proof. (1) By Definition 3.3 (2), mf¯ = = (2) Since

J

inf f¯(x) = inf [fl (x), fr (x)]

x∈A

x∈A

[ inf fl (x), inf fr (x)] = [mfl (A), mfr (A)]. x∈A

x∈A

is a standard interval-valued pseudo-multiplication, K K dmf¯ = f¯ mf¯ = [fl , fr ] [mfl , mfr ] = [fl ¯l mfl , fr ¯r mfr ] = [dmfl , dmfr ].

(18)

(19) ¤

We also easily obtain the following theorem without their proof. Theorem 3.3. (1) Let ⊕l and ⊕r are represents of a standard interval-valued pseudo-addition L J , and ¯l and ¯r are represents of a standard interval-valued pseudo-multiplication . Then L we have an interval-valued set function µ ¯ = [µl , µr ] is an interval-valued σ − -decomposable measure if and only if µl and µr are σ − ⊕l -decomposable measure and σ − ⊕r -decomposable measure, respectively. (2) An interval-valued function f¯ = [fl , fr ] is integrable if and only if fl and fr are integrable. L∞ J Theorem 3.4. (1) If e¯(x) = i=1 a¯i χAi (x) is an interval-valued elementary function as in Definition 3.6 (1), then we have ∞ e¯(x) = [⊕l ∞ i=1 ai l ¯l χAi (x), ⊕r i=1 ai r ¯r χA (x)].

(20)

(2) If e¯ L = [el , er ] is an interval-valued elementary function with respect to the intervalvalued σ − -decomposable measure µ ¯ = [µl , µr ] is Z  K Z ⊕l Z ⊕r e¯ d¯ µ=[ el ¯l dµl , er ¯r dµr ]. (21) (3) An interval-valued character ξ¯ : X −→ I([a, b]) is dH -continuous if and only if ξl and ξr are continuous. (4) If f¯ = [fl , fr ] is an integrable, interval- valued function, then we have Z  K Z ⊕l Z ⊕r f¯ d¯ µ=[ fl ¯l dµl , fr ¯r dµr ] (22) L J Proof. (1) Since is a standard interval-valued pseudo-addition and is a standard intervalvalued pseudo-multiplication, e¯(x)

= = =

∞ M

a¯i

K

χAi (x) i=1 K ∞ [⊕l ∞ χAi (x) i=1 ail , ⊕r i=1 air ] ∞ [⊕l ∞ a ¯ χ (x), ⊕ i l A r i=1 l i=1 ai r ¯r i

797

χAi (x)].

(23)

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J L is a standard interval(2) Since is a standard interval-valued pseudo-addition and valued pseudo-multiplication, Z  K ∞ M K e¯ d¯ µ = a¯i µ ¯(Ai ) i=1 K ∞ = [⊕l ∞ [µl (A), µr (A)] i=1 ai l , ⊕r i=1 ai r ] ∞ = [⊕l ∞ a ¯ µ (A), ⊕ a ¯ µ (A)] l l i=1 i l Z ⊕ Z ⊕r r i=1 i r r r l = [ el ¯l dµl , er ¯r dµr ]. (24) (3) By (12), L the proof is trivial. J (4) Since is a standard interval-valued pseudo-addition and is a standard intervalvalued pseudo-multiplication, by (11) Z



f¯

K

Z d¯ µ = dH − lim

n→∞

Z



e¯n

K

d¯ µ

Z ⊕r = dH − lim [ en l ¯l dµl , en r ¯r dµr ] n→∞ Z ⊕l Z ⊕r = [ lim en l ¯l dµl , lim en r ¯r dµr ] n→∞ n→∞ Z ⊕l Z ⊕r = [ fl ¯l dµl , fr ¯r dµr ]. ⊕l

(25) ¤

Theorem 3.5. If ⊕l and ⊕r are represents of a standard interval-valued pseudo-addition L J , and ¯l and ¯r are represents of a standard interval-valued pseudo-multiplication , ¯ and ξ¯ = [ξl , ξr ] is the d -continuous interval-valued pseudo-character, and f = [f , f ] ∈ l r  H B(X, I([a, b])), and Li f¯ is the interval-valued pseudo-Laplace transform, then we have 

¯ (Li f¯)(ξ)(z) = [(L⊕l fl )(ξl )(z), (L⊕r fr )(ξr )(z)]

(26)

for all z ∈ R. L Proof. Since ⊕l and ⊕r are represents of a standard interval-valued pseudo-addition and J ¯l and ¯r are represents of a standard interval-valued pseudo-multiplication , by Theorem 3.1, Theorem 3.2 (2), and Theorem 3.3 (2), and Theorem 3.4 (4), Z   K ¯ −z) (Li f¯)(ξ)(z) = ξ(x, dmf¯ X∩[0,∞)  Z K = [ξl (x, −z), ξr (x, −z)] [dmfl , dmfr ] X∩[0,∞) Z ⊕l Z ⊕r = [ ξl (x, −z) ¯l dmfl , ξr (x, −z) ¯r dmfr ] X∩[0,∞)

=

X∩[0,∞)

[L⊕l (fl )(ξl )(z), L⊕r (fr )(ξr )(z)].

(27) ¤

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4. The interval-valued pseudo-exchange formula and the inverse of interval-valued pseudo-Laplace transform In this section, we study the interval-valued pseudo-exchange formula and the inverse of the interval-valued pseudo-Laplace transform for X = R. Definition 4.1. (1) A standard interval-valued composition ◦s of f¯ = [fl , fr ] and g¯ = [gl , gr ] in B(X, I([a, b])) is defined by ¯ = [fl ◦ hl , fr ◦ hr ], f¯ ◦s h

(28)

where ◦ is the composition of f and h, that is (f ◦ h)(x) = f (h(x)), ∀ ∈ X.

(29)

(2) Let g¯ = [gl , gr ] ∈ B(X, I([a, b])) be an odd, strictly increasing dH -continuous function. If g¯ = [gl , gr ] ∈ B(X, I([a, b])), then the inverse function g¯−1 of g¯ with respect to a standard interval-valued composition ◦s is defined by g¯−1 = [gl−1 , gr−1 ].

TheoremL4.1. (1) If ⊕1l = ⊕1r = max are represents of a standard interval-valued pseudoaddition 1 and J ¯1l = ¯1r = + are represents of a standard interval-valued pseudomultiplication 1 and ξ¯ = [ξl , ξr ] with ξl (x, −z) = ξr (x, −z) = −xz then we have  K (Li1 1 f¯)(z) = sup(−xz f¯(z)) (30) 1

x≥0

where f¯ ∈ B(X, I([a, b])). L (2) If ⊕2l = ⊕2r = max are the represents of a standard interval-valued pseudo-addition interval-valued pseudo- multiplication J2 , and ¯¯2l = ¯2r = · are the represents of a standard −xz , then we have 2 and ξ = [ξl , ξr ] with ξl (x, −z) = ξr (x, −z) = e  K (Li2 2 f¯)(z) = sup exz f¯(z) (31) 2

x≥0

where f¯ ∈ B(X, I([a, b])). (3) Let gl , gr : [−∞, ∞] −→ [−∞, ∞], g(0) = 0 be an odd, strictly increasing, continuous function. If x ⊕l y = gl−1 (gl (x) + gl (y)), x ⊕r y = gr−1 (gr (x) + gr (y)) are the represents L −1 of a standard interval-valued pseudo-addition = 3 , and x ¯3 y = gl (gl (x)gl (y)), x ¯3 yJ −1 gr (gr (x)gr (y)) are the represents of a standard interval-valued pseudo-multiplication , and ξ¯ = [ξl , ξr ] with ξl (x, −z) = ξr (x, −z) = e−xz , then we have 

(Li3 3 f¯)(z) = g¯−1 Li (¯ g ◦s f¯)(z)

(32)

where (Li f¯)(z) = [L(fl )(z), L(fr )(z)]. Proof. (1) By Theorem 3.5, and Definition 3.3 (1), we have 

(Li1 1 f¯)(z) = [L1⊕1 l (fl )(z), L1⊕1 r (fr )(z)] = [sup(−xz + fl (z)), sup(−xz + fr (z))] x≥0

x≥0

=

sup[−xz⊕1 l fl (z), −xz⊕1 r fr (z)] x≥0 K = sup[−xz f¯(z)]. x≥0

1

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(2) By Theorem 3.5 and Definition 3.3(1), we have 

(Li2 2 f¯)(z) = [L2⊕2 l (fl )(z), L2⊕2 r (fr )(z)] = [sup(e−xz + fl (z)), sup(e−xz + fr (z))] x≥0

x≥0

sup[e−xz ⊕2 l fl (z), e−xz ⊕2 r fr (z)] x≥0 K = sup[e−xz f¯(z)].

=

x≥0

(34)

2

(3) By Theorem 3.5 and Definition 4.1, we have 

(Li3 3 f¯)(z)

⊕3 r 3l = [L⊕ (fr )(z)] 3 (fl )(z), L3 = [gl−1 L(gl ◦ fl )(z), gr−1 L(gr ◦ fr )(z)] = g¯−1 [L(gl ◦ fl )(z), L(gr ◦ fr )(z)] = g¯−1 Li (¯ g ◦s f¯)(z).

(35) ¤

Now, we obtain the following pseudo-exchange formula and inverse of interval-valued pseudo-Laplace transform. J L ¯ ¯ Theorem 4.2. Let k be as in Theorem 4.1, for k = 1, 2, 3, and f1 , f2 ∈ k , and B(X, I([a, b])). Then we have   K  Lik k (f¯1 ∗k f¯2 ) = Lik k (f¯1 ) Lk k (f¯2 ) (36) J for k = 1, 2, 3, where (f¯1 ∗1 f¯2 )(x) = sup0≤t≤x [f¯1 (x−t) 1 f¯2 (t)], (f¯1 ∗2 f¯2 )(x) = sup0≤t≤x [f¯1 (x− J ¯ = [fl ∗ hl , fr ∗ hr ]. g ◦s f¯1 ∗I g¯ ◦s f¯2 )(x)) with f¯ ∗I h t) 2 f¯2 (t)], and (f¯1 ∗3 f¯2 )(x) = g¯−1 ((¯ Proof. (1) When k = 1; we see that (f¯1 ∗1 f¯2 )(x)

sup [f¯1 (x − t)

=

0≤t≤x

=

K

f¯2 (t)]

1

sup [f1l (x − t) + f2l (t), f1r (x − t) + f2r (t)]

0≤t≤x

=

[ sup (f1l (x − t) + f2l (t)), sup (f1r (x − t) + f2r (t))]

=

[f1l ∗1 f2l , f1r ∗1 f2r ].

0≤t≤x

0≤t≤x

(37)

By Theorem 4.1(1), (37), and Theorem 2.1, we have 

Li1 1 (f¯1 ∗1 f¯2 ) =

⊕r l [L⊕ 1 (f1l ∗1 f2l ), L1 (f1r ∗1 f2r )] ⊕1

⊕1

⊕

⊕

), L1 1r (f1r ) ¯1r L1 1r (f2r )] = [L1 l (f1l ) ¯1l L1 l (f2K l ⊕1l ⊕1 ⊕1r ⊕ = [L1 (f1l ), L1 (f1r )] [L1 l f2l , L1 1r (f2r )] =



Li1 1 (f¯1 )

K 1

1



Li2 1 (f¯2 ).

(38)

(2) When k = 2; by the some method in the proof of (1), we can obtain the result. (3) When k = 3; we see that (f¯1 ∗3 f¯2 )(x) = g¯−1 (¯ g ◦s f¯1 ∗I g¯ ◦s f¯2 )(x) −1 = g¯ ([gl ◦ f1l , gr ◦ f1r ] ∗I [gl ◦ f2l , gr ◦ f2l ]) = g¯−1 [(gl ◦ f1l ) ∗ (gr ◦ f2l ), (gr ◦ f1r ) ∗ (gr ◦ f2r )] = [¯ g −1 ((gl ◦ f1l ) ∗ (gr ◦ f2l )), gr−1 ((gr ◦ f1r ) ∗ (gr ◦ f2r ))]

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=

[f1l ∗3 f2l , f1r ∗3 f2r ].

(39)

By Theorem 4.1(3), (39), and Theorem 2.1, 

Li3 3 (f¯1 ∗3 f¯2 )

= = =

⊕3

⊕

[L3 l (f1l ∗3 f2l ), L3 3r (f1r ∗3 f2r )]

⊕3 ⊕3 ⊕ ⊕ [L3 l (f1l ) ¯l L3 l (f2l K ), L3 3r (f1r ) ¯r L3 3r (f2r )] ⊕ 3l ⊕ 3l ⊕3r ⊕ 3r [L3 (f1l ), L3 (f1r )] [L3 (f2l ), L3 (f2r )] 3  K 

= Li3 3 (f¯1 )

Li3 3 (f¯2 ).

(40)

3

¤ Finally, we will study the inverse of a standard interval-valued pseudo-Laplace transform. 

Theorem 4.3. If f¯ ∈ B(X, I([a, b])) and Lik k (f¯) = F¯k for k = 1, 2, 3 as in Theorem 4.1,  ⊕k −1 ⊕ −1 and thereexist (Lik k )−1 (F¯k ) = [Lk l (Fkl ), Lk kr (Fkr )], then we have J 1 −1 ¯ (1) (Li ) (F1 )(x) = inf z≥0 (xz 1 F¯1 (z)), 1 J ¯ (2) (L 2 )−1 (F¯2 )(x) = inf z≥0 (exz F2 (z)), and i 2

2

(3) (Li3 3 )−1 (F¯3 )(x) = g¯−1 (L−1 g ◦s F¯3 )(x)). i (¯ 

Proof. (1) Since Li1 1 (f¯) = F¯1 , by Theorem 3.5, [F1l , F1r ]



= F¯1 = Li1 1 (f¯) = [L⊕1l (fl ), L⊕1r (fr )].

⊕1

(41)

⊕

Thus, we have F1l = L1 l (fl ) and F1r = L1 1r (fr ). By Theorem 2.2 (1), we have ⊕1

(L1 l )−1 (F1l )(x) = inf (xz + F1l (z))

(42)

x≥0

and ⊕

(L1 1r )−1 )(F1r )(x) = inf (xz + F1r (z))

(43)

z≥0

By (42) and (43), Definition 3.3 (2), 

(Li1 1 )

−1

⊕1 −1

⊕

−1

(F¯1 ) = [L1 l (F1l ), L2 1r (F1r )] = [ inf (xz + F1l (z)), inf (xz + F1r (z)]) z≥0

z≥0

= [ inf (xz ¯1l F1l (z)), inf (xz ¯1l F1r (z))] z≥0

z≥0

=

inf [xz ¯1l F1l (z), xz ¯1r F1r (z)] z≥0 K = inf [xz F¯1 (z)]. z≥0

(44)

1

(2) By the  same method in the proof of (1), we can obtain the result. (3) Since Li3 3 (f¯) = F¯3 , by Theorem 3.5, 

[F3l , F3r ] = F¯3 = Li3 3 (f¯) = [L⊕3l (fl ), L⊕3r (fr )]

Thus, we have F3l =

⊕3 ⊕ L1 l (fl ) and F3r = L3 3r (fr ). By ⊕3 (L3 l )−1 (F3l )(x) = gl−1 (L−1 (gl

801

(45)

Theorem 2.2 (3), we have ◦ F3l )(x))

(46)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

11

and ⊕

(L3 3r )−1 (F3r )(x) = gr−1 (L−1 (gr ◦ F3r )(x))

(47)

By (46) and (47), Definition 4.1 (2),  −1

L3

3

(F¯3 )

= = = = =

⊕3

⊕

[(L3 l )−1 (F3l ), (L3 3r )−1 (F3r )] [gl−1 (L−1 (gl ◦ F3l ))(x), gr−1 (L−1 (gr ◦ F3r ))(x)] g¯−1 [g −1 (gl ◦ F3l )(x), L−1 (gr ◦ F3r )(x)] g¯−1 L−1 i ([gl ◦ F3l , gr ◦ F3r ])(x) −1 −1 g¯ Li (¯ g ◦s F¯3 )(x).

(48) ¤

Remark 4.2. We find the maximum of the uncertain utility function K K K ¯ (x1 , x2 , · · · , xn ) = f¯1 (x1 ) U f¯2 (x2 ) ··· f¯n (xn ) 1

1

1

on the domain R = {(x1 , x2 , · · · , xn )|x1 + x2 + · · · + xn = x, xi ≥ 0, i = 1, 2, · · · , n}. Such problems economy and operation research. Let f¯(x) = J occurs in J the mathematical J maxR [f¯1 (x1 ) 1 f¯2 (x2L ) 1 · · · J1 f¯n (xn )]. Applying the standard interval-valued pseudo-Laplace transfor (for the case 1 and 1 ) 

F¯ (z) = (L1 1 f¯)(z) = max(−xz

K

x≥0

f¯(x)),

(49)

1

we obtain by interval-valued pseudo-exchange formula 

L1 1 (f¯) =

n K

L

1i=1



1

(f¯i ).

Applying the inverse of a standard pseudo-Laplace transform  K (L1 1 )−1 (F¯ )(x) = min(xz F¯1 (z)), z≥0

(50)

(51)

1

we obtain the solution f (x) = min[xz z≥0

n KK 1

1i=1



L1 1 (fi )(z)].

(52)

5. Conclusions This study was to define the interval-valued pseudo-Laplace transform by means of the L interval-valued pseudo-integral with respect to a σ− -decomposable measure (See Definitions 3.6 and 3.7). By using the method of the standard interval-valued pseudo-operations in [12-14], we also investigated some characterizations of the standard interval-valued pseudo-characters, the interval-valued pseudo-integrals, and the interval-valued pseudo-Laplace transform (See Theorems 3.1, 3.2, 3.3, 3.4, and 3.5.). In particular, we obtained the pseudo-exchange formula for the interval-valued pseudo-Laplace transform and inverse of interval-valued pseudo-Laplace transforms (See Theorems 4.2 and 4.3) and an application (See Remark 4.4).

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JEONG GON LEE AND LEE-CHAE JANG

In the future, we will focus the pseudo-exchange formula for the interval-valued pseudoLaplace transform by means of the interval-valued generalized Choquet integrals and the inverse of interval-valued pseudo-Laplace transforms. Acknowledgement: This paper was supported by Konkuk University in 2014. 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] P. Benvenuti, R. Mesiar, Pseudo-arithmetical operations as a basis for the general measure and integration theory, Information Sciences 160 (2004) 1-11. [4] G. Deschrijver, Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory, Fuzzy Sets and Systems 160 (2009) 3080-3102. [5] T. Grbic, I. Stajner, M. Strboja, An approach to pseudo-integration of set-valued functions, Information Sciences 181 (2011) 2278-2292. [6] M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems 69 (1995) 279-298. [7] C. Guo, D. Zhang, On set-valued measures, Information Science 160 (2004) 13-25. [8] 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. [9] 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. [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, A note on convergence properties of interval-valued capacity functionals and Choquet integrals, Information Sciences 183 (2012) 151-158. [12] L. C. Jang, A note on the interval-valued generalized fuzzy integral by means of an interval-representable pseudo-multiplication and their convergence properties, Fuzzy Sets and Systems 222 (2013) 45-57. [13] L. C. Jang, Some properties of the inteval-valued g¯-integrals and a standard interval-valued g¯-convolution, Journal of Inequalities and Applications 2014 (2014). [14] J. G. Lee, L. C. Jang, On convergence properties of the generalized Choquet integral with respect to a fuzzy measure by means of the generalized Lebesgue integral, Far East J. Math. Sci. 84 (2014) 169-185. [15] R. Mesiar, J. Li, E. Pap, Discrete pseudo-integrals, International Journal of Approximate Reasoning 54 (2013) 357-364. [16] E. Pap, I. Stajner, Generalized pseudo-convolution in the theory of probabilistic metric spaces, information, fuzzy numbers, optimazation, system theory, Fuzzy Sets and Systems 102 (1999) [17] E. Pap, N. Ralevic, Pseudo-Laplace transform, Nonlinear Analysis 33 (1998) 533-550. [18] I. Stajner-Papuga, T. Grbic, M. Dankova, Pseudo-Riemann Stieltjes integral, Information Sciences 180 (2010) 2923-2933. [19] M. Sugeno, T. Murofushi, Pseudo-additive measures and integrals, J. Math. Anal. Appl. 122 (1987) 197-222. [20] K. Wechselberger, The theory of interval-probability as a unifying concept for uncertainty, Int. J. Approximate Reasoning 24 (2000) 149-170. [21] C. Wu, S. Wang, M. Ma, Generalized fuzzy integrals : Part 1. Fundamental concept, Fuzzy Sets and Systems 57 (1993) 219-226. [22] C. Wu, M. Ma, S. Song, S. Zhang, Generalized fuzzy integrals : Part 3. convergence theorems, Fuzzy Sets and Systems 70 (1995) 75-87.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

OPERATORS IDEALS OF GENERALIZED MODULAR SPACES ´ OF CESARO TYPE DEFINED BY WEIGHTED MEANS ˙ S ˙ ¸ EK∗ , VATAN KARAKAYA, AND HARUN POLAT NECIP ¸ IMS

Abstract. In this work, we investigate the ideal of all bounded linear operators between any arbitrary Banach spaces whose sequence of approximation numbers belong to the generalized modular spaces of Ces´ aro type defined by weighted means. Also, we show that the completeness of obtained operator ideals.

1. Introduction At the turn of this century, there has been a considerable interest to study the behavior of the operator ideals using s-numbers and specially approximation numbers. As has been shown in the famous book of Gochberg and Kre˘ın [11], the s-numbers are an important tool in the spectral theory of Hilbert space. We consider s-numbers sequences of operators on Banach spaces in the sense of [4]. He presented the axiomatic theory of s-numbers of operators in Banach spaces. Also, in his work it has been turned out that s-numbers are very powerful tools for estimating eigenvalues of operators in Banach spaces. Many useful operator ideals have been defined by using sequence of s-numbers. The concept of an operator ideal on the class of Banach spaces and the theory of s-numbers of linear bounded operators among Banach spaces was introduced and first studied by [4]. Due to important applications in spectral theory, geometry of Banach spaces, theory of eigenvalue distributions etc., the theory of operator ideals occupies a special importance in functional analysis. Many useful operator ideals have been defined by using sequence of s-numbers. In [14], Faried and Bakery use the de la Val´ee-Poussin means for definition of the operator ideals. By using similar concept, we investigate the ideals of all bounded linear operators generated by the approximation numbers and generalized weighted means defined by [6] and generalized by [16]. 2. Preliminary and Notations By N and R it will be signed the set of all natural numbers and the real numbers, respectively and also by w, it is denoted the space of all real valued sequences. The set B(X, Y ) of all operators from X into Y becomes a Banach space with respect to the so-called operator norm kT k = sup {kT xk : kxk ≤ 1} . The class of all operators between arbitrary Banach spaces is denoted by B. Also if T ∈ B(X, Y ), then we denote null space and range N (T ) and M (T ) respectively. 2000 Mathematics Subject Classification. Primary 46B20, 46B45; Secondary 47B06. Key words and phrases. Approximation numbers; operator ideal; generalized weighted mean. 1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

˙ S ˙ ¸ EK∗ , VATAN KARAKAYA, AND HARUN POLAT NECIP ¸ IMS

2

Let X0 = XN (T ) and Y0 = M (T ). A map which transform every operator T ∈ ∞ B(X, Y ) to a unique sequence (sn (T ))n=0 is called s − f unction, and the number th sn (T ) is called the n s − number of T if the following conditions are satisfied: s1 ) ||T || = s1 (T ) ≥ s2 (T ) ≥ ... ≥ 0 for all T ∈ B(X, Y ) s2 ) sn (T1 +T2 ) ≤ sn (T1 ) +||T2 || for all T1 , T2 ∈ B(X, Y ) s3 ) sn (RST ) ≤ ||R||sn (S)||T || for all T ∈ B(X0 , X), S ∈ B(X, Y ) and R ∈ B(Y, Y0 ) s4 ) sn (λT ) ≤ |λ|sn (T ) for all T∈ B(X, Y ), λ ∈ R 0 ; for r ≥ n s5 ) rank(T ) ≤ n if sn (T ) = 0 for all T ∈ B(X, Y ) s6 ) sr (In ) = 1 ; for r < n where In is the identity map of an n-dimensional normed space. As important examples of s-numbers, we consider approximation numbers αn (T ) (2.1)

αn (T ) = inf {||T − A|| : A ∈ B(X, Y ) and rank(A) ≤ n} ,

We can show that all these s-numbers satisfy the following condition. This is improvement of condition (s2 ) and called additive property: sn+m−1 (T1 + T2 ) ≤ sn (T1 ) + sm (T2 ), for allT1 , T2 ∈ B(X, Y ) and n = 1, 2, ... Many other famous s-numbers are modifications of them. All these numbers express some finite dimensional approximations for the operator T . Geometrically they may be viewed as different widths of the ball T (BX ) as a subset of Y . For details on s-number sequences, we refer [4, 7, 5, 2]. An operator ideal U is a subclass of B such that the components U (X, Y ) = U ∩ B (X, Y ) satisfy the following conditions: O1 ) IK ∈ U , where K denotes the 1-dimensional Banach space, where U ⊂ B, (equally; F (X, Y ) ⊆ U (X, Y ), where F (X, Y ) is the space of all operators of finite rank from the Banach space X into the Banach space Y ), O2 ) If T1 , T2 ∈ U (X, Y ), then λ1 T1 + λ2 T2 ∈ U (X, Y ) for any scalars λ1 and λ2 , O3 ) If V ∈ B(X0 , X), T ∈ U (X, Y ) and R ∈ B(Y, Y0 ) then RT V ∈ U (X0 , Y0 ) (see [3, 1, 15]). The generalized modular spaces of Ces´aro type defined by weighted means is defined by S ¸ im¸sek and Karakaya in [16] as follows: `ρ (u, v; p) = {x ∈ w : ρ(λx) < ∞, for some λ > 0} , where ρ(x) =

∞ X

 pk k X  uk vj |xj | .

k=0

j=0

It can be seen that ρ : `ρ (u, v; p) → [0, ∞] is a modular on `ρ (u, v; p). If p = (pk ) for all k ∈ N, we can write for `ρ (u, v; p) and it’s norm respectively;    pk k ∞   X X  uk vj |xj | < ∞, for some λ > 0 , `ρ (u, v; p) = x ∈ w :   k=0

j=0

o n x kxk = inf λ > 0 : ρ ≤1 , ∀x ∈ `ρ (u, v; p). λ With norm given above, space `ρ (u, v; p) is a Banach space as shown in [16] .One can find detailes about the space `(u, v; p) in list given in ( [9], [6] [6]). Bycombining  1 and special case of uk and vj , we get the following spaces: Firstly, if (uk ) = k+1 vj = 1 for all j, k ∈ N, then the space `ρ (u, v; p) reduces to the space ces(p) (see [12]).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

ON SOME GEOMETRICAL PROPERTIES OF GENERALIZED MODULAR SPACES

(uk ) =



1 Qk



and (vj ) = (qj ) and Qk =

k P

3

qj for all j, k ∈ N, then the space

j=0

`ρ (u, v; p) reduces to the space Nρ (q) ( see [8]). In addition, some related papers on the sequence space can be found in [10, 13, 17, 18]. Also we denote ei = (0, 0, ..., 0, 1, 0, ...) where 1 seats in ith place of ei for all i ∈ N. For any bounded sequence of positive real numbers p = (pk ), we have the following well known inequality: |ak + bk |pk ≤ 2h−1 (|ak |pk + |bk |pk ), where h = sup pk and pk > 1 for all k ∈ N Throughout this paper, the sequence (pk ) is a bounded sequence of positive real numbers with (a1 ) (pk ) is an increasing sequence of positive real numbers, (a2 ) lim sup pk < ∞, k→∞

(a3 )

lim inf pk > 1.

k→∞

Definition 1. [14] A class E of linear sequence spaces is called a special space of sequences (sss) having the following conditions: (i) E is a linear space and en ∈ E, for each n ∈ N, (ii) If x ∈ w, y ∈ E and |xn | ≤ |y n | for all n ∈ N, than x ∈ E, i.e. E is solid, ∞

∞

= (x0 , x0 , x1 , x1 , x2 , x2 , ..., xn , xn , ...) ∈ E, (iii) If (xn )n=0 ∈ E, then x[ n ] 2 n=0 n n where 2 denotes the integral part of 2 . Example 1. a) `p is a special space of sequences for 0 < p < ∞. b) cesp is a special space of sequences for 1 < p < ∞.

Definition 2. UEapp := {UEapp (X, Y ) : X, Y are Banach spaces} where UEapp (X, Y ) = ∞ {T ∈ L(X, Y ) : (αn (T ))n=0 ∈ E}. Definition 3. A class of special sequences(sss) Eρ is called a pre-modular special space of sequences if there exists a function ρ : E −→ [0, ∞), satisfies the following conditions: (i) ρ(x) ≥ 0, ∀x ∈ Eρ and ρ(x) = 0 ⇐⇒ x = θ, where θ is the zero element of E, (ii) there exists a constant ` ≥ 1 such that ρ(λx) ≤ `|λ|ρ(x) for all values of ∀x ∈ E and for any scalar λ, (iii) for some numbers k ≥ 1, we have the inequality ρ(x+y) ≤ k (ρ(x) + ρ(y)) for all x, y ∈ E, (iv) if |xn | ≤ |yn | for all n ∈ N then ρ ((xn )) ≤ ρ ((yn )) ,   (v) for some numbers k0 ≥ 1 we have the inequality ρ ((xn )) ≤ ρ x[ n2 ] ≤ k0 ρ ((xn )) , ∞ ∞ (vi) for each x = (x(i))i=0 ∈ E there exists s ∈ N such that ρ (x(i))i=s < ∞. This means that the set of all finite set is ρ−dense in E, (vii) for any λ > 0 there exists a constant ζ > 0 such that ρ (λ, 0, 0, ...) ≥ ζλρ (1, 0, 0, ...) . It is clear that ρ is continuous at θ. The function ρ defines a metrizable topology in E endowed with this topology is denoted by Eρ . For the arbitrary linear space E, it was provided in the following theorem: Theorem 1. [14] UEapp is an operator ideal if E is a special space of sequences (sss).

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˙ S ˙ ¸ EK∗ , VATAN KARAKAYA, AND HARUN POLAT NECIP ¸ IMS

4

Now we give here the sufficient conditions on the weighted means such that the class of all bounded linear operators between any arbitrary Banach spaces with n-th approximation numbers of the bounded linear operators in the generalized weighted means form an operator ideal. 3. Main Results For the proof of the main theorems, we need the following theorems which includes the notion of special space of sequences. It helps to construction of operators ideals. Theorem 2. U`app is an operator ideal if the conditions (a1 ), (a2 ), (a3 ) are ρ (u,v;p) satisfied. Proof. To prove the theorem, we have to satisfy the three conditions of Definition 1. mentioned above. (1) linearity : Let x, y ∈ `ρ (u, v; p). Since,  pk    pk  pk  ∞ k ∞ k ∞ k X X X X X X    uk vj |xj + yj | ≤ 2h−1  uk vj |xj | + uk vj |yj |  k=0

j=0

k=0

j=0

k=0

j=0

where h = sup pk , then (x + y) ∈ `ρ (u, v; p). Let λ ∈ R, x ∈ `ρ (u, v; p), then   pk  pk ∞ k ∞ k X X X X   uk vj |λxj | ≤ sup|λ|pk uk vj |xj | < ∞. k=0

k

j=0

k=0

j=0

Hence we can get λx ∈ `ρ (u, v; p). To show that em ∈ `ρ (u, v; p) for each m ∈ N, by using condition (a3 ), we have  pk ∞ k ∞ X X X p  ρ(em ) = uk vj |em (j)| = (uk vm ) k < ∞. k=m

j=0

k=m

Hence em ∈ `ρ (u, v; p). ∞ P

∞ P k=0

(2) solidity: Let |xk | ≤ |yk | for each k ∈ N, then k=0 !pk k P uk vj |yj | , since y ∈ `ρ (u, v; p). Thus x ∈ `ρ (u, v; p).

k P

!pk uk vj |xj |

≤

j=0

j=0

(3) x = (xk ) ∈ `ρ (u, v; p), then we have ∞ X k=0

pk  pk  p2k+1  ∞ 2k ∞ 2k+1 k X X X X X    u2k vj |x[ j ] | + u2k+1 vj |x[ j ] | uk vj |x[ j ] | = 2 2 2 j=0

≤

k=0

∞ X k=0

j=0

k=0

j=0

  p2k k X  u2k [(v2j + v2j+1 ) |xj |] + (u2k v2k |xk |) j=0

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ON SOME GEOMETRICAL PROPERTIES OF GENERALIZED MODULAR SPACES

+

∞ X

 u2k+1

≤ 2h−1 

∞ X



k X



k=0

p2k u2k [(v2j + v2j+1 ) |xj |]

∞ X

∞
X h−1

+2h−1

 u2k+1

 u2k+1

k X

≤

k X



p2k  u2k v2j |xj |



j=0

p2k+1 (v2j + v2j+1 ) |xj |

j=0

j=0

k X

k=0

p2k+1 + 2h−1

v2j |xj |

j=0

k=0



  p2k p2k k ∞ k X X X   u2k v2j |xj | + 22h−2 u2k v2j+1 |xj |

k=0

∞ X

+

∞ X k=0

k=0

22h−2 + 2

(v2j + v2j+1 ) |xj |

j=0

+

≤

p2k+1

j=0

k=0



k X

5

∞ X k=0

22h−1 + 32

∞
X h−1 k=0



 u2k+1

j=0

k X

p2k+1 v2j+1 |xj |

j=0

  pk k X  uk vj |xj | < ∞ j=0

∞

∈ `ρ (u, v; p). Consequently, from Theowhere uk → 0(k → ∞). Hence x[ n ] 2 n=0 app rem 1. it follows that U`ρ (u,v;p) is an operator ideal. app Corollary 1. (a) UN is an operator ideal for 1 < p < ∞. ρ (q) app (b) Uces(p) is an operator ideal for 1 < p < ∞.

Theorem 3. The linear space F (X, Y ) is dense in U`app i.e. F (X, Y ) = ρ (u,v;p) U`app . ρ (u,v;p) Proof. We first verify that every finite mapping T ∈ F (X, Y ) belongs to U`app (X, Y ) ρ (u,v;p) i.e. F (X, Y ) ⊂ U`app (X, Y ). Since em ∈ `ρ (u, v; p) for each m ∈ N and ρ (u,v;p) `ρ (u, v; p) is a linear space, then for every finite mapping T ∈ F (X, Y ), the sequence ∞ (αk (T ))n=0 contains only finitely many numbers different from zero. Therefore F (X, Y ) ⊂ U`app (X, Y ). ρ (u,v;p) (X, Y ). Now we have to show that U`app (X, Y ) ⊆ F (X, Y ). Let T ∈ U`app ρ (u,v;p) ρ (u,v;p) ∞ ∞ Thus we get (αk (T ))k=0 ∈ `ρ (u, v; p), and since ρ ((αk (T ))k=0 ) < ∞, for given ∞ ε ∈ (0, 1) then there exists a natural number m > 0 such that ρ ((αk (T ))k=m ) < ε. Therefore we have, for p > 0  pk m+p k X X  (3.1) uk vj |αj (T )| < ε. k=m

j=0

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˙ S ˙ ¸ EK∗ , VATAN KARAKAYA, AND HARUN POLAT NECIP ¸ IMS

6

It means that A ∈ Fm (X, Y ), rank(A) ≤ m. Besides, by using the fact |αj (T )| ≤ | kT − Ak em (j) | , j = m, m + 1, ...

(3.2)

and also from (2.1), (3.1) we have m+p X

pk

(uk vm | kT − Ak em (j) |)

≤ ∞.

k=m

we take m0 ≥ m and from (3.2), we get

d(T, A)

= ρ (αk (T −

∞ A))k=0

=

∞ X k=0

 pk k X  uk vj |αj (T − A)| j=0

  pk   pk m k ∞ k 0 −1 X X X X   ≤ uk vj | kT − Ak em (j) | + uk vj |αj (T − A)| j=0

k=0


0 there exist a constant 0 < ζ < λp0 −1 such that ρ (λ, 0, 0, ...) ≥ ζλρ(1, 0, 0, ...). Theorem 5. Let X be a normed space, Y be a Banach space and the conditions (a1 ), (a2 ) and (a3 ) are satisfied, then U`app (X, Y ) is complete. ρ (u,v;p)

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ON SOME GEOMETRICAL PROPERTIES OF GENERALIZED MODULAR SPACES

7

(X, Y ). Since `ρ (u, v; p) is preProof. Let (Tm ) be a Cauchy sequence in U`app ρ (u,v;p) modular special space of sequences, then by using Definition 3 (vii) and since (X, Y ) ⊆ L(X, Y ), we have U`app ρ (u,v;p) ∞
ρ (αn (Ti − Tj ))n=0 ≥ ρ (α0 (Ti − Tj ), 0, 0, ...) = ρ (α0 kTi − Tj k , 0, 0, ...) ≥ ζ kTi − Tj k ρ(1, 0, 0, ...). Then (Tm ) is also Cauchy sequence in L(X, Y ). Since the space L(X, Y ) is a Banach space, then there exists T ∈ L(X, Y ) such that kTm − T k → 0 as m → ∞ and since ∞ (αn (Tm ))n=0 ∈ `ρ (u, v; p) for all m ∈ N, ρ is continuous at θ and using Definition 3 (iii), we have ρ (αn (T )∞ n=0 )

∞

= ρ (αn (T − Tm + Tm ))n=0  ∞ ∞  ≤ kρ α[ n ] (Tm − T ) + kρ α[ n ] (Tm ) n=0

2

n=0

2

∞

≤ kρ ((kTm − T k)n=0 ) + kρ (αn (Tm ))∞ n=0 < ε. app Hence (αn (T ))∞ n=0 ∈ `ρ (u, v; p) as such T ∈ U`ρ (u,v;p) (X, Y ).

Corollary 2. Let X be a normed space, Y be a Banach space and (pk ) be an increasing sequence of positive real numbers with lim sup(pk ) < ∞ and lim inf(pk ) > 1, k→∞

app (X, Y ) is complete. then Uces (p)

k→∞

Corollary 3. Let X be a normed space, Y be a Banach space and (pk ) be an app increasing sequence of positive real numbers with 1 < p < ∞, then Uces (X, Y ) is p complete. Corollary 4. Let X be a normed space, Y be a Banach space and (pk ) be an app increasing sequence of positive real numbers with 1 < p < ∞, then UN (X, Y ) is ρ (q) complete. References [1]

A. Lima, E. Oja, Ideals of finite rank operators, intersection properties of balls and the approximation property, Stud. Math., 133, 175–186 (1999). [2] A. Pietsch, Eigenvalues and s-numbers, Cambridge University Press, New York (1986). [3] A. Pietsch, Operator Ideals, North-Holland, Amsterdam (1980). [4] A. Pietsch, s-numbers of operators in Banach spaces, Studia Math. T. LI., (1974) 201-223. [5] A. Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann., 247(2)(1980), 149-168. [6] B. Altay, and F. Ba¸sar, Generalization of the sequence space l(p) derived by weighted means, J. Math. Anal. Appl., 330(1), 174-185 (2007). [7] B. Carl, A. Hinrichs, On s-numbers and Weyl inequalities of operators in Banach spaces, Bull. London Math. Soc., 41(2)(2009), 332-340. [8] C.S. Wang, On N¨ orlund sequence space, Tamkang J. Math., 9(1978) 269-274. [9] E. Malkowsky and E. Sava¸s, Matrix transformations between sequence spaces of generalized weighted means, Applied Mathematics and Computation, 147(2)333-345, (2004). [10] H. Polat, V. Karakaya, N. S ¸ im¸sek, Difference sequence spaces derived by using a generalized weighted mean. Appl. Math. Lett. 24 (2011)(5), 608–614. [11] I.Z.Gochberg, M.G. Kre˘ın, Introduction to the theory of linear nonselfadjoint operators, Moscow 1965, engl. Transl. [12] J. S. Shue, Ces´ aro sequence spaces, Tamkang J. Math., 1 (1970) 143-150.

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˙ S ˙ ¸ EK∗ , VATAN KARAKAYA, AND HARUN POLAT NECIP ¸ IMS

[13] M. Karaka¸s, M. Et, V. Karakaya, Some geometric properties of a new difference sequence space involving lacunary sequences. Acta Math. Sci. Ser.B Engl.Ed. 33(2013)(6) 1711-1720. [14] N. Faried, A.A. Bakery, Mappings of type Orlicz and Generalized Ces´ aro sequence space, Journal of Inequalities and Applications, doi:10.1186/1029-242X-2013-186 (2013). [15] N.J. Kalton, Spaces of Compact Operators, Math. Ann., 208(1974), 267-278. [16] N. S ¸ im¸sek, V. Karakaya, On some geometrical properties of generalized modular spaces of Ces´ aro type defined by weighted means, Journal of Inequalities and Applications, Art. ID 932734 (2009). [17] V. Karakaya, Some geometric properties of sequence spaces involving lacunary sequence. Journal of Inequalities and Applications 2007, Art. ID 81028, 8 pp. [18] V. Karakaya, N. S ¸ im¸sek, On some properties of new paranormed sequence space of nonabsolute type. Abstr. Appl. Anal. 2012, Art. ID 921613, 10 pp. ˙ ISTANBUL COMMERCE UNIVERSITY, Faculty of Arts and Sciences, Department ˙ of Mathematics Beyoglu, ISTANBUL-TURKEY E-mail address: [email protected] YILDIZ TECHNICAL UNIVERSITY, Department of Mathematical Engineering, Davut˙ pasa Campus, Esenler, ISTANBUL-TURKEY E-mail address: [email protected] MUS ¸ ALPARSLAN UNIVERSITY, Department of Mathematics, MUS ¸ -TURKEY E-mail address: [email protected]

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Certain subclasses of multivalent uniformly starlike and convex functions involving a linear operator J. Patel1 , A. Ku. Sahoo2 and N.E. Cho3 1

Department of Mathematics, Utkal University, Vani Vihar,Bhubaneswar-751004, India E-mail:[email protected] 2

3

Department of Mathematics, Institute of Technical Education and Research, Jagmohan Nagar, Khandagiri, Bhubaneswar-751030, India E-Mail:[email protected]

Department of Applied Mathematics, Pukyong National University, Pusan 608-737, Republic of Korea E-Mail:[email protected]

Abstract In the present paper, we obtain coefficient inequalities, inclusion relationships involving neighborhoods, subordination results, properties involving modified Hadamard products and also study majorization properties of certain subclasses of multivalent analytic functions, which are defined by means of a certain linear operator. Some useful consequences our main results are mentioned and their relevance with some of the earlier results are also pointed out. 2010 Mathematics Subject Classification: 30C45 Key words and phrases: p−Valent analytic functions, Uniformly starlike and convex functions, Subordination, Majorization, Hadamard product and modified Hadamard product. 1. Introduction and definitions Let Ap denote the class of functions of the form f (z) = z p +

∞ X

ap+k z p+k

(p ∈ N)

(1.1)

k=1

which are analytic and p-valent in the unit disk U = {z ∈ C : |z| < 1}. For convenience, we write A1 = A. For functions f and g analytic in U, we say that f is subordinate to g, written as f ≺ g or f (z) ≺ g(z) (z ∈ U), if there exists a Schwarz function ω, which(by definition) is analytic in U with ω(0) = 0, |ω(z)| < 1 and f (z) = g(ω(z)), z ∈ U. For f ∈ Ap defined by (1.1) and g given by g(z) = z p +

∞ X

bp+k z p+k

(z ∈ U),

k=1

their convolution (or Hadamard product) is defined as (f ? g)(z) = z p +

∞ X

ap+k bp+k z p+k = (g ? f )(z)

(z ∈ U).

k=1

A function f ∈ Ap is said to be β-uniformly p-valent starlike of order α, denoted by U ST (p, α, β), if it satisfies the condition 0  0  zf (z) zf (z) > β − p + α (−p ≤ α < p, β ≥ 0; z ∈ U). (1.3) Re f (z) f (z) Analogously, a function f ∈ Ap is said to be β-uniformly p-valent convex of order α, denoted by U CV (p, α, β), if it satisfies the condition 00   zf (z) zf 00 (z) >β 0 − (p − 1) + α (−p ≤ α < p, β ≥ 0; z ∈ U). (1.4) Re 1 + 0 f (z) f (z) 3 Corresponding

Author

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We note that U ST (1, α, β) = U ST (α, β), U CV (1, α, β) = U CV (α, β), U ST (p, α, 0) = Sp∗ (α) and U CV (p, α, 0) = Cp (α), where U ST (α, β) and U CV (α, β) are the classes of β-uniformly starlike and βuniformly convex functions of order α (−1 ≤ α < 1) studied by Ronning [18, 19] (see also [11,12]). The classes Sp∗ (α), Cp (α) are the well-known classes of p-valently starlike and p-valently convex functions of order α (0 ≤ α < p), respectively. Further, S1∗ (α) = S ∗ (α), C1 (α) = C(α) are the familiar classes of starlike and convex functions of order α (0 ≤ α < 1). We also observe that f ∈ U CV (p, α, β) ⇔ zf 0 /p ∈ U ST (p, α, β)

(−p ≤ α < p).

Let φp be the incomplete beta function defined by φp (a, c; z) = z p +

∞ X (a)k k=1

(c)k

z p+k

(z ∈ U),

where a ∈ R, c ∈ R \ Z− 0 = {0, −1, −2, . . . } and (x)k denote the Pochhammer symbol (or shifted factorial) defined by ( 1, k=0 (x)k = x(x + 1) · · · (x + k − 1), k ∈ N. With the aid of the function φp , we define the linear operator Lp (a, c) : Ap −→ Ap by Lp (a, c)f (z) = φp (a, c; z) ? f (z)

(z ∈ U)

(1.4)

and if f is given by (1.1), then it follows from (1.4) that p

Lp (a, c)f (z) = z +

∞ X (a)k k=1

(c)k

ap+k z p+k

(z ∈ U).

(1.5)

It follows from (1.4) or (1.5) that 0

z (Lp (a, c)f (z)) = aLp (a + 1, c)f (z) − (a − p)Lp (a, c)f (z)

(f ∈ Ap ; z ∈ U).

(1.6)

The operator Lp (a, c) was introduced and studied by Saitoh [22] which yields the operator L(a, c) introduced by Carlson and Shaffer [5] for p = 1. We also note that for f ∈ Ap , (i) Lp (a, a)f (z) = f (z); (ii) Lp (p + 1, p)f (z) = zf 0 (z)/p; (iii) Lp (m + p, 1)f (z) = Dm+p−1 f (z) (m ∈ Z, m > −p), the operator studied by Goel and Sohi [9]. In the case p = 1, Dm f is the familiar Ruscheweyh derivative [20] of f ∈ A. (iv) Lp (p+1, m+p)f (z) = Im,p f (z) (m ∈ Z, m > −p), the extended Noor integral operator considered by Liu and Noor [13]. (λ,p)

(v) Lp (p + 1, p + 1 − λ)f (z) = Ωz f (z) (−∞ < λ < p + 1), the extended fractional differintegral operator considered by Patel and Mishra [16]. Z λ + p z λ+p−1 (vi) Lp (λ + p, λ + p + 1)f (z) = t f (t)dt = Fλ,p (f )(z) (λ > −p; z ∈ U), the generalized zλ 0 Bernardi-Libera-Livingston integral operator (cf., e.g., [6]). With the help of the operator Lp (a, c), we introduce a subclass of Ap as follows: Definition 1. For the fixed parameters A and B with −1 ≤ B < A ≤ 1, we say that a function f ∈ Ap is in the class Sp,j (a, c, β, A, B), if it satisfies the following subordination condition: (j) z (L (a, c)f )(j) (z) (p + 1 − j)(1 + Az) z (Lp (a, c)f ) (z) p , (1.7) − β − (p + 1 − j) ≺ (j−1) (j−1) 1 + Bz (Lp (a, c)f ) (z) (Lp (a, c)f ) (z) where p ∈ N, 1 ≤ j ≤ p, β ≥ 0 and z ∈ U. In view of the definition of the subordination, (1.7) is equivalent to (j) z (L (a, c)f )(j) (z) z (L (a, c)f ) (z) p p − (p + 1 − j) − β − (p + 1 − j) (j−1) (Lp (a, c)f )(j−1) (z) (L (a, c)f ) (z) p < 1. ( ) (j) (j) z (Lp (a, c)f ) (z) z (Lp (a, c)f ) (z) (p + 1 − j)A − B − β − (p + 1 − j) (j−1) (j−1) (Lp (a, c)f ) (z) (Lp (a, c)f ) (z)

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It is easily seen that if f ∈ Sp,j (a, c, β, A, B) with 1 − A − β(1 − B) ≥ 0, then   (p + 1 − j)(1 − A − β(1 − B)) (j−1) ∗ . (Lp (a, c)f ) (z) ∈ Sp (1 − β)(1 − B)

3

(1.9)

for 1 ≤ j ≤ p. Let Tp denote the subclass of Ap consisting of functions of the form f (z) = z p −

∞ X

ap+k z p+k

(ap+k ≥ 0; z ∈ U)

(1.10)

k=1

and for ease of notation, we write T1 = T . Now, we use the operator Lp (a, c) to define a new subclass Tp,j (a, c, β, A, B) as follows: Definition 2. For the fixed parameters A and B with −1 ≤ B < A ≤ 1 (−1 ≤ B < 0), we say that a function f ∈ Tp is in the class Tp,j (a, c, β, A, B), if it satisfies the following subordination condition: (j) z (L (a, c)f )(j) (z) (p + 1 − j)(1 + Az) z (Lp (a, c)f ) (z) p − β − (p + 1 − j) , (1.11) ≺ (j−1) (j−1) 1 + Bz (Lp (a, c)f ) (z) (Lp (a, c)f ) (z) where p ∈ N, 1 ≤ j ≤ p, β ≥ 0 and z ∈ U. We note that the families Sp,j (a, c, β, A, B) and Tp,j (a, c, β, A, B) are of special interest for they contain many well-known as well as many new classes of univalent and multivalent analytic functions. For instance,   2α Example 1. Sp,j a, c, β, 1 − , −1 = Sp,j (a, c, β, α), the class consisting of functions f ∈ p+1−j Ap which satisfy the condition: ) ( (j) z (L (a, c)f )(j) (z) z (Lp (a, c)f ) (z) p >β − (p + 1 − j) + α (0 ≤ α < p + 1 − j; z ∈ U). Re (j−1) (j−1) (Lp (a, c)f ) (z) (Lp (a, c)f ) (z) We note that Sp,j (a, c, β, α) ∩ Tp = Tp,j (a, c, β, α) (0 ≤ α < p + 1 − j).   2α Example 2. (i) Sp,j a, a, β, 1 − p+1−j , −1 = U ST (p, j, β, α), the class of functions f ∈ Ap which satisfy the condition: (j)  (j)  zf (z) zf (z) + α (0 ≤ α < p + 1 − j; z ∈ U). Re > β − (p + 1 − j) (j−1) (j−1) f (z) f (z) and   2α (ii) Sp,j p + 2 − j, p + 1 − j, β, 1 − p+1−j , −1 = U CV (p, j, β, α), the class consisting of functions f ∈ Ap which satisfy the condition: (j+1)   zf zf (j+1) (z) (z) + α (0 ≤ α < p + 1 − j; z ∈ U). > β − (p − j) Re 1 + (j) (j) f (z) f (z) We note that U ST (p, j, 0, α) ∩ Tp = T ∗ (p, α, j)

and U CV (p, j, 0, α) ∩ Tp = C(p, α, j)

are the classes introduced and studied by Aouf [4].   2α , −1 = Sp,j,m (a, c, β, α) (0 ≤ α < p + 1 − j) is the class of Example 3. Sp,j m + p, 1, β, 1 − p+1−j functions in Ap satisfying the condition: ( )
(j) z Dm+p−1 f
(j) (z) (z) z Dm+p−1 f Re > β − (p + 1 − j) + α (z ∈ U), (j−1) (j−1) m+p−1 m+p−1 (D f) (z) (D f) (z)

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where p ∈ N, 1 ≤ j ≤ p, 0 ≤ α < p + 1 − j and m > −p(m ∈ Z).   2α Example 4. Sp,j p + 1, p + 1 − µ, β, 1 − p+1−j , −1 = Sp (α, β, j, µ), the subclass of functions f ∈ Ap satisfying the condition:     (j) (j) (µ,p)    z Ω(µ,p)  z Ω f (z) f (z) z z Re  > β − p +α (j−1)  (µ,p) (j−1)    Ω(µ,p)  Ω f (z) f (z) z z

(z ∈ U),

where p ∈ N, 0 ≤ α < p + 1 − j, 1 ≤ j ≤ p and −∞ < µ < p + 1. For j = 1, this class was studied by Pathak and Sharma [17].   2α Example 5. Sp,j p + 1, p + 1 + λ, β, 1 − p+1−j , −1 = Spλ (α, β, j), the subclass of functions f ∈ Ap satisfying the condition: ( ) (j) z (F (f ))(j) (z) z (Fλ,p (f )) (z) λ,p Re − p > β + α (z ∈ U), (j−1) (j−1) (Fλ,p (f )) (z) (Fλ,p (f )) (z) where p ∈ N, 0 ≤ α < p + 1 − j, 1 ≤ j ≤ p and λ > −p. 2α Example 6. Sp,1 (a, c, β, 1 − , −1) = Sp (a, c, β, α), the class of functions f ∈ Ap satisfying p   aLp (a + 1, c)f (z) aLp (a + 1, c)f (z) − a + p > β − a + α (z ∈ U), Re Lp (a, c)f (z) Lp (a, c)f (z) where 0 ≤ α < p. We note that S1,1 (a, c, β, α) = L(a, c, α, β) was the class introduced and studied by Frasin [8]. Analogous to the classes defined above (i.e., Example 1 to Example 6), we can also define subclasses of multivalent analytic functions with negative coefficients by taking the intersection of these classes with Tp . 2. Coefficient estimates Unless otherwise mentioned, we assume throughout the sequel that p ∈ N, a > 0, c > 0, β ≥ 0, 1 ≤ j ≤ p, −1 ≤ B < A ≤ 1 Next, we give a sufficient condition for functions belonging to the class Sp,j (a, c, β, A, B). Theorem 1. Let the function f be defined by (1.1). If ∞ X δ(p + k, j − 1) {k(1 + β + β|B|) + |(p + 1 − j)A − (p + k + 1 − j)B|)} (a)k

δ(p, j)(A − B)

k=1

(c)k

|ap+k | ≤ 1,

(2.1)

then f ∈ Sp,j (a, c, β, A, B), where ( p(p − 1) . . . (p + 1 − j), δ(p, j) = 1,

j∈N j = 0.

(2.2)

Proof. In view of (1.8), we need to show that for z ∈ ∂U, z (Lp (a, c)f )(j) (z) − (p + 1 − j) (Lp (a, c)f )(j−1) (z) (j) (j−1) (j−1) iθ −β e z (Lp (a, c)f ) (z) − (p + 1 − j) (Lp (a, c)f ) (z) − (p + 1 − j)A (Lp (a, c)f ) (z) (j) (j) (j−1) −B z (Lp (a, c)f ) (z) + βBeiθ z (Lp (a, c)f ) (z) − (p + 1 − j) (Lp (a, c)f ) (z) < 0.

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Substituting the series representation of (Lp (a, c)f ) we deduce that for −π < θ < π.

(j−1)

(j)

(z) and (Lp (a, c)f )

5

(z) in the above expression,

z (Lp (a, c)f )(j) (z) − (p + 1 − j) (Lp (a, c)f )(j−1) (z) (j−1) (j) (j−1) (z) (z) − (p + 1 − j)A (Lp (a, c)f ) −β eiθ z (Lp (a, c)f ) (z) − (p + 1 − j) (Lp (a, c)f ) (j) (j−1) (j) (z) −B z (Lp (a, c)f ) (z) + βBeiθ z (Lp (a, c)f ) (z) − (p + 1 − j) (Lp (a, c)f ) ≤

∞ X

k(1 + β)

k=1

+

∞ X k=1

(p + k)! (a)k p! |ap+k ||z|k − (A − B) (p + k + 1 − j)! (c)k (p − j)!

(p + k)! (p + j − 1)A − (p + k + 1 − j)B (a)k |ap+k ||z|k (p + k + 1 − j)! (c)k +β|B|

∞ X

k

k=1

≤

∞ X k=1

(p + k)! (a)k |ap+k ||z|k (p + k + 1 − j)! (c)k

(a)k |ap+k | δ(p + k, j − 1) k(1 + β + βB) + (p + j − 1)A − (p + k + 1 − j)B (c)k 

−δ(p, j)(A − B) ≤ 0, by applying the hypothesis (2.1). Now, the result follows by using the maximum modulus theorem. Theorem 2. Let the function f be given by (1.10) and −1 ≤ B < 0. Then f ∈ Tp,j (a, c, β, A, B), if and only if ∞ X δ(p + k, j − 1) {k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a)k

δ(p, j)(A − B)

k=1

(c)k

ap+k ≤ 1,

(2.3)

where δ(p, j) is defined by (2.2). The result is sharp. Proof. In view of Theorem 1, we only need to prove the sufficient part of Theorem 2. We assume that f , given by (1.10) belongs to the class Tp,j (a, c, β, A, B). Thus, by (1.11), we deduce that ( (1 − B)Re

z (Lp (a, c)f )

(j)

(z)

)

− (p + 1 − j)(1 − A) (j−1) (Lp (a, c)f ) (z) z (L (a, c)f )(j) (z) p − β(1 − B) − (p + 1 − j) >0 (Lp (a, c)f )(j−1) (z) (j)

Substituting the series expansion of (Lp (a, c)f )

(j−1)

(z) and (Lp (a, c)f )

(z ∈ U).

(2.4)

(z) in (2.4), we get

  P∞ (a)k    ap+k z k   δ(p, j)(A − B) − k=1 δ(p + k, j − 1) {k(1 − B) + (p + 1 − j)(A − B)}  (c)k Re P∞ (a)k       ap+k z k δ(p, j − 1) − k=1 δ(p + k, j − 1)(p + 1 − j) (c)k P∞ (a)k k a z β(1 − B) −kδ(p + k, j − 1) p+k k=1 (c)k > 0 (z ∈ U). − δ(p, j − 1) − P∞ δ(p + k, j − 1)(p + 1 − j) (a)k a k z p+k k=1 (c)k Using the fact that Re(z) ≤ |z| for all z ∈ C and upon choosing the values of z on the real axis, the above

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inequality reduces to   P∞ (a)k  k   δ(p, j)(A − B) − δ(p + k, j − 1) {k(1 − B) + (p + 1 − j)(A − B)} a z p+k   k=1 (c)k Re P∞ (a)k       δ(p, j − 1) − k=1 δ(p + k, j − 1)(p + 1 − j) ap+k z k (c)k   P∞ (a)k     ap+k z k β(1 − B) k=1 −kδ(p + k, j − 1)   (c)k iθ − Re e > 0 (−π < θ < π; z ∈ R ∩ U). P (a)k    k  δ(p, j − 1) − ∞  δ(p + k, j − 1)(p + 1 − j) a z p+k k=1 (c)k (2.5)
Since Re eiθ ≥ −|eiθ | = −1, (2.5) can be rewritten as   P∞ (a)k  k   a z δ(p, j)(A − B) − δ(p + k, j − 1) {k(1 − B) + (p + 1 − j)(A − B)} p+k   k=1 (c)k Re P∞ (a)k       δ(p, j − 1) − k=1 δ(p + k, j − 1)(p + 1 − j) ap+k z k (c)k   P∞ (a)k     ap+k z k β(1 − B) k=1 −kδ(p + k, j − 1)   (c)k − Re > 0 (z ∈ R ∩ U). (2.6)     δ(p, j − 1) − P∞ δ(p + k, j − 1)(p + 1 − j) (a)k a k  z p+k k=1 (c)k Letting r → 1− in (2.6), we get the desired inequality. It is easily verified that the estimate (2.3) is sharp for the function f0 , defined by f0 (z) = z p −

(p + 1 − j)(p + 2 − j)(A − B)c z p+1 (−1 ≤ B < 0; z ∈ U). (p + 1) {(1 + β)(1 − B) + (p + 1 − j)(A − B)} a

(2.7)

Letting A = 1 − {2α/(p + 1 − j)} (0 ≤ α < P + 1 − j) and B = −1 in Theorem 2, we obtain Corollary 1. Let the function f be given by (1.10). Then f ∈ Tp,j (a, c, β, α), if and only if ∞ X δ(p + k, j − 1) {k(1 + β) + (p + 1 − j − α)} (a)k

δ(p, j − 1)(p + 1 − j − α)

k=1

(c)k

ap+k ≤ 1.

The result is sharp. Theorem 3. Let a ≥ c > 0 and −1 ≤ B < 0. If f ∈ Tp,j (a, c, β, A, B), then the unit disk is mapped onto a domain that contains the disk   (p + 1)(1 + β)(1 − B)a + (p + 1 − j)(A − B){(p + 1)(a − c) + (j − 1)c} z ∈ C : |z| < . (p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a The result is sharp. Proof. Let f , defined by (1.10) belongs to the class f ∈ Tp,j (a, c, β, A, B). Then, by virtue of Theorem 2, we get ∞

δ(p + 1, j − 1) {(1 + β)(1 − B) + (p + 1 − j)(A − B)} a X ap+k δ(p, j)(A − B)c k=1

≤

∞ X δ(p + k, j − 1) {k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a)k

δ(p, j)(A − B)(c)k

k=1

ap+k ≤ 1,

which evidently yields ∞ X

ap+k ≤

k=1

=

δ(p, j)(A − B)c δ(p + 1, j − 1) {(1 + β)(1 − B) + (p + 1 − j)(A − B)} a (p + 1 − j)(p + 2 − j)(A − B)c . (p + 1) {(1 + β)(1 − B) + (p + 1 − j)(A − B)} a

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Consequently, for |z| = r < 1, we deduce that ! ∞ X p |f (z)| ≥ r − ap+k rp+1 k=1

≥ rp −

δ(p, j)(A − B)c rp+1 . δ(p + 1, j − 1) {(1 + β)(1 − B) + (p + 1 − j)(A − B)} a

Hence, by letting r → 1− in the above inequality, we get the required result. The result is sharp for the function f0 given by (2.7). This completes the proof of Theorem 3. Remarks 1. A special case of Theorem 2 when j = 1, A = 1 − 2α (0 ≤ α < p) and B = −1 was obtained by Frasin [8, Theorem 1]. Further, if in Theorem 2 with β = 0, A = 1−{2α/(p+1−j)}, B = −1, we set a = c and a = p + 2 − j, c = p + 1 − j, we shall obtain the corresponding results obtained by Aouf [4] for the classes T ∗ (p, α, j) and C(p, α, j), respectively. 2. If, in Theorem 3 with β = 0, A = 1 − {2α/(p + 1 − j)} (0 ≤ α < p + 1 − j) and B = −1, we set a = c and a = p + 2 − j, c = p + 1 − j, we obtain the corresponding results obtained by Aouf [4, Corollary 5 and Corollary 6]. Theorem 4. If a ≥ c > 0 and −1 ≤ B < 0, then Tp,j (a + 1, c, β, A, B) ⊂ Tp,j (a, c, β, C, B), where C=B+

a(1 + β)(1 − B)(A − B) . (a + 1)(1 + β)(1 − B) + (p + 1 − j)(A − B)

The result is sharp. Proof. Suppose f , given by (1.10) belongs to the class Tp,j (a + 1, c, β, A, B). Then by Theorem 2, we obtain ∞ X δ(p + k, j − 1) {k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a + 1)k ap+k ≤ 1, (2.8) δ(p, j)(A − B) (c)k k=1

In view of Theorem 2 and (2.8), we need to find the largest value of C so that {k(1 + β)(1 − B) + (p + 1 − j)(C − B)} a {k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a + k) ≤ C −B A−B each positive integer k ∈ N. Or, equivalently, C≥B+

a(1 + β)(1 − B)(A − B) (a + k)(1 + β)(1 − B) + (p + 1 − j)(A − B)

(k ∈ N).

Since the right side of the above inequality is a decreasing function of k(k ∈ N), putting k = 1 in the above inequality, we get the required result. The result is sharp for the function f0 , defined by f0 (z) = z p −

(p + 1 − j)(p + 2 − j)(A − B)c z p+1 (p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}(a + 1)

(−1 ≤ B < 0; z ∈ U).

Denoting U ST + (p, j, α, β) = U ST (p, j, α, β) ∩ Tp ,

U CV + (p, j, α, β) = U CV (p, j, α, β) ∩ Tp

and taking a = c = p + 1 − j, A = 1 − {2α/(p + 1 − j)} (0 ≤ α < p + 1 − j) and B = −1 in Theorem 4, we get the following result. Corollary 2. We have U CV + (p, j, α, β) ⊂ U ST + (p, j, ρ, β), where ρ=

(p + 1 − j){β(1 + α) + (p + 2 − j)} . (p + 2 − j)(1 + β) + (p + 1 − j − α)

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The result is sharp with the extremal function f0 , given by f0 (z) = z p −

(p + 1 − j)(p + 1 − j − α) p+1 z (p + 1)(p + 2 + β − j − α)

(z ∈ U).

Theorem 5. If 0 ≤ β ≤ (1 − A)/(1 − B) and the function f , defined by (1.1) belongs to the class Sp,j (a, c, β, A, B), then  k  Y δ(p, j − 1)(c)k 2(p + 1 − j)(A − B) (k ∈ N), (2.9) |ap+k | ≤ j − 1) + k! δ(p + k, j − 1)(a)k j=1 (1 − β)(1 − B) where δ(p, j) is defined as in (2.2). Proof. Suppose f , defined by (1.1) belongs to the class Sp,j (a, c, β, A, B). Then, by using (1.9), we deduce that (p + 1 − j){1 − A − β(1 − B)} (j) (j−1) z (Lp (a, c)f ) (z) − (Lp (a, c)f ) (z) (1 − β)(1 − B) (p + 1 − j)(A − B) (j−1) = (Lp (a, c)f ) (z) q(z) (z ∈ U), (2.10) (1 − β)(1 − B) where q(z) = 1 + q1 z + q2 z 2 + . . . is analytic in U and Re{q(z)} > 0 for z ∈ U. Substituting the series (j−1) (j) expansions of (Lp (a, c)f ) , (Lp (a, c)f ) , q in (2.10) and comparing the coefficient of z p+k on both the sides of the resulting equation, we obtain (a)k (p + 1 − j)(A − B) × ap+k = (c)k (1 − β)(1 − B) × {δ(p, j − 1)qk + δ(p + 1, j − 1)qk−1 ap+1 + · · · + δ(p + k − 1, j − 1)q1 ap+k−1 } .

k δ(p + k, j − 1)

(2.11)

Using the fact that |qk | ≤ 2 for k ≥ 1 for Carath´ eodory function q [7] in (2.11), we have 2(p + 1 − j)(A − B)(c)k × k δ(p + k, j − 1)(1 − β)(1 − B)(a)k × {δ(p, j − 1) + δ(p + 1, j − 1)|ap+1 | + · · · + δ(p + k − 1, j − 1)|ap+k−1 |}

|ap+k | ≤

(2.12)

for each k ∈ N. We shall prove the estimate (2.9) by induction on k. For k = 1, (2.12) gives |ap+1 | ≤

2δ(p, j − 1)(p + 1 − j)(A − B)(c)1 δ(p + 1, j − 1)(1 − β)(1 − B)(a)1

so that (2.9) is certainly true. For k = 2, (2.12) yields 2(p + 1 − j)(A − B)(c)2 {δ(p, j − 1) + δ(p + 1, j − 1)|ap+1 |} 2δ(p + 2, j − 1)(1 − β)(1 − B)(a)2   2(p + 1 − j)(A − B) δ(p, j − 1)(c)2 2(p + 1 − j)(A − B) 1+ . ≤ 2! δ(p + 2, j − 1)(a)2 (1 − β)(1 − B) (1 − β)(1 − B)

|ap+2 | ≤

Thus, (2.9) is true for k = 2. We assume that (2.9) holds true for k = n. Then for k = n + 1, we obtain δ(p, j − 1)(c)n+1 |ap+n+1 | ≤ xj × (n + 1)δ(p + n + 1, j − 1)(a)n+1 )   1 1 1 × (1 + xj ) + xj (1 + xj ) + xj (1 + xj )(2 + xj ) + · · · + xj (1 + xj ) · · · (n + xj ) , 2! 3! n!

(2.13)

where xj = 2(p + 1 − j)(A − B)/{(1 − β)(1 − B)}. Since (1 + xj ) +

n 1 1 1 1 Y xj (1 + xj ) + xj (1 + xj )(2 + xj ) + · · · + xj (1 + xj ) · · · (n + xj ) = (j + xj ), 2! 3! n! n! j=1

the estimate (2.13) implies that (2.9) is true for k = n+1. This evidently completes the proof of Theorem 5. Remark. Putting j = 1, a = p + 1, c = p + 1 − µ(0 ≤ µ < 1), A = 1 − (2α/p)(0 ≤ α < p) and B = −1 in Theorem 5, we shall obtain the result due to Pathak and Sharma [17, Theorem 2].

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3. Mazorization problem for the class Sp,j (a, c, β, A, B) Let f and g be analytic functions in the unit disk U. We say that f is majorized by g in U (see [14]) and write f (z)  g(z) (z ∈ U), if there exists a function ϕ, analytic in U such that |ϕ(z)| < 1 and f (z) = ϕ(z)g(z) for all z ∈ U. To establish our result, we need the following lemma. Lemma 1. If a > 0, 1 ≤ j ≤ p, 0 ≤ β < 1 and 0 ≤ aB + (p + 1 − j)(A − B), then aB + (p + 1 − j)(A − B) ≤ aB + (p + 1 − j)(A − B) (θ ∈ R). iθ 1 − βe 1−β

(3.1)

Proof. Since |1 − βeiθ | ≥ (1 − β) for all θ ∈ R, we have for 0 ≤ B < A ≤ 1 aB + (p + 1 − j)(A − B) ≤ aB + (p + 1 − j)(A − B) iθ iθ 1 − βe 1 − βe (p + 1 − j)(A − B) ≤ aB + (θ ∈ R). 1−β This proves the inequality (3.1) for 0 ≤ B < A ≤ 1. Next, we assume that −1 ≤ B < 0 ≤ A ≤ 1. We note that 2 2 {aB(1 − β) + (p + 1 − j)(A − B)} − aB(1 − βeiθ ) + (p + 1 − j)(A − B) = 2β(aB)2 (cos θ − 1) + 2(p + 1 − j)aβ(A − B)(cos θ − 1) = 2aβ B(cos θ − 1) {aB + (p + 1 − j)(A − B)} ≥0 by using the hypothesis and the fact that B < 0 and cos θ ≤ 1 for all θ ∈ R. Thus, the inequality (3.1) holds for −1 ≤ B < 0 and the proof of Lemma 1 is completed. Now, we prove Theorem 6. Let a > 0, 0 ≤ β < 1, aB + (p + 1 − j)(A − B) ≥ 0 and a(1 − β − B) − (p + 1 − j)(A − B) ≥ 0. If the functions f ∈ Ap , g ∈ Sp,j (a, c, β, A, B) and (j−1)

(Lp (a, c)f ) then

(z)  (Lp (a, c)f )

(j−1)

(j−1) (j−1) (z) ≤ (Lp (a + 1, c)f ) (z) (Lp (a + 1, c)f )

(z)

(z ∈ U),

(|z| < r0 = r0 (p, a, j, β, A, B)),

where r0 is the smallest positive root of the equation:     (p + 1 − j)(A − B) (p + 1 − j)(A − B) r3 − (a + 2|B|)r2 − (2 + aB + r + a = 0. aB + 1−β 1−β Proof. Since g ∈ Sp,j (a, c, β, A, B), it follows from (1.7) that ( ) (j) (j) z (Lp (a, c)g) (z) (p + 1 − j)(1 + Az) z (Lp (a, c)g) (z) iθ − βe − (p + 1 − j) ≺ (j−1) (j−1) 1 + Bz (Lp (a, c)g) (z) (Lp (a, c)g) (z)

(3.2)

(3.3)

(z ∈ U)

for −π < θ < π so that   A − βBeiθ (p + 1 − j) 1 + ω(z) (j) z (Lp (a, c)g) (z) 1 − βeiθ = (j−1) 1 + Bω(z) (Lp (a, c)g) (z)

(z ∈ U),

(3.4)

where ω is analytic in U with ω(0) = 0 and |ω(z)| < 1 in U. Also from (1.6), we deduce that (j)

z (Lp (a, c)g)

(j−1)

(z) = a (Lp (a + 1, c)g)

(z) − (a − p + j − 1) (Lp (a, c)g)

820

(j−1)

(z)

(z ∈ U).

(3.5)

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Now, by using (3.5) in (3.4), we get (j−1) (z) (Lp (a, c)g) ≤

a(1 + B|z|) (j−1)   (z) (Lp (a + 1, c)g) (p + 1 − j)(A − B) a − aB + |z| 1−β

(z ∈ U).

(3.6)

From (3.1) and the hypothesis, we have z (Lp (a, c)f )

(j)

(z) = zϕ(z) (Lp (a, c)g)

(j)

(z) + zϕ0 (z) (Lp (a, c)g)

(j−1)

(z ∈ U).

(z)

Using (3.5) in (3.7), we obtain (j−1) (z) a (Lp (a + 1, c)f ) (j−1) (j−1) (z) (z) + |z||ϕ0 (z)| (Lp (a, c)g) ≤ a|ϕ(z)| (Lp (a + 1, c)g)

(z ∈ U).

(3.7)

(3.8)

Using the following inequality [15] |ϕ0 (z)| ≤

1 − |ϕ(z)|2 1 − |z|2

(z ∈ U)

in (3.8) and applying the resulting inequality in (3.6), we deduce that         |z|(1 + B|z|)(1 − |ϕ(z)|2 ) (j−1)     × (z) ≤ |ϕ(z)| + (Lp (a + 1, c)f )   (p + 1 − j)(A − B)    (1 − |z|2 ) a − aB + |z|  1−β (j−1) × (Lp (a + 1, c)g) (z) (z ∈ U), which upon setting |z| = r < 1 and |ϕ(z)| = x (0 ≤ x ≤ 1) leads us to the inequality ψ(x)    × (p + 1 − j)(A − B) 2 (1 − r ) a − aB + r 1−β (j−1) × (Lp (a + 1, c)g) (z) (z ∈ U),

(j−1) (z) ≤ (Lp (a + 1, c)f )

where     (p + 1 − j)(A − B) ψ(x) = −r(1 + |B|r)x2 + (1 − r2 ) a − aB + r x + r(1 + |B|r). 1−β The function ψ takes its maximum value at x = 1 with r0 = r0 (p, a, j, β, A, B), where r0 (p, a, j, β, A, B) is the smallest positive root of the equation (3.2). Further, for 0 ≤ y ≤ r0 (p, a, j, β, A, B), the function     (p + 1 − j)(A − B) 2 2 Φ(x) = −y(1 + |B|y)x + (1 − y ) a − aB + y x + y(1 + |B|y) (3.9) 1−β is increasing on the interval 0 ≤ x ≤ 1 so that     (p + 1 − j)(A − B) Φ(x) ≤ Φ(1) = (1 − y 2 ) a − aB + y 1−β Hence, upon setting x = 1 in (3.9), we conclude that (3.1) holds true for |z| < r0 (p, a, j, β, A, B), where r0 (p, a, j, β, A, B) is the smallest positive root of the equation (3.2). This completes the proof of Theorem 5. Letting A = 1 − {2α/(p + 1 − j)}, B = −1 and a = c = p + 1 − j in Theorem 5, we get the following result which yields the corresponding work of MacGregor [14] for j = 1 and α = 0.

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Corollary 3. Let 1 ≤ j ≤ p, 0 ≤ α ≤ (p + 1 − j)/2 and 0 ≤ β ≤ α/(p + 1 − j). If the functions f ∈ Ap and g ∈ U ST (p, α, β, j) satisfies |f (j−1) (z)|  |g (j−1) (z)| (z ∈ U), then |f (j) (z)| ≤ |g (j) (z)| (|z| < re0 ), where re0 is the smallest positive root of the equation:     2(p + 1 − j − α) p+1−j−α 2 − (p + 1 − j) r − 2 1 + r + (p + 1 − j) = 0. 1−β 1−β

4. Inclusion relatioships involving neighborhoods Following the earlier investigations by Goodman [10], Ruscheweyh [21] and others including Altintas and Owa [2] (see also [1,3]), we define the ε−neighborhoods of a function f ∈ Ap given by (1.1) as follows: ( ) ∞ ∞ X X p p+k Nε (f ) = g ∈ Ap : g(z) = z + bp+k z and (p + k)|bp+k − ap+k | ≤ ε; ε > 0 . (4.1) k=1

k=1

In particular, for the identity function e(z) = z p (z ∈ U), we immediately have ) ( ∞ ∞ X X p+k p (p + k)|bp+k | ≤ ε; ε > 0 . bp+k z and Nε (e) = g ∈ Ap : g(z) = z + k=1

(4.2)

k=1

For convenience, we denote Nε+ (f ) = Nε (f ) ∩ Tp

and Nε+ (e) = Nε (e) ∩ Tp .

Theorem 7. If a ≥ c > 0, 1 ≤ j ≤ p, −1 ≤ B < 0 and ε=

(p + 1)δ(p, j)(1 + β)(1 − B)(A − B)c , δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a

then Tp,j (a, c, β, A, B) ⊂ Nε+ (e). Proof. Suppose f , given by (1.10) belongs to the class Tp,j (a, c, β, A, B). Then by Theorem 2, we have ∞

δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)} a X ap+k ≤ 1, δ(p, j)(A − B) c k=1

so that

∞ X k=1

ap+k ≤

δ(p, j)(A − B)c . δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a

(4.3)

Making use of Theorem 2 again in conjunction with (4.3), we deduce that ∞

δ(p + 1, j − 1)(1 + β)(1 − B) a X (p + k)ap+k c k=1

∞

δ(p + 1, j − 1) a X ≤ δ(p, j)(A − B) + {p(1 + β)(1 − B) − (p + 1 − j)(A − B)} ap+k c k=1

δ(p, j)(A − B) {p(1 + β)(1 − B) − (p + 1 − j)(A − B)} ≤ δ(p, j)(A − B) + (1 + β)(1 − B) + (p + 1 − j)(A − B) δ(p, j)(p + 1)(1 + β)(1 − B)(A − B) ≤ (1 + β)(1 − B) + (p + 1 − j)(A − B)

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that is,

∞ X

(p + k)ap+k ≤

k=1

12

δ(p, j)(p + 1)(A − B)c , δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a

which in view of (4.2) establishes the inclusion relation asserted by Theorem 4. Setting A = 1 − {2α/(p + 1 − j)} and B = −1 in Theorem 7, we have Corollary 4. If a ≥ c > 0, 1 ≤ j ≤ p and ε=

δ(p, j − 1)(p + 1)(p + 1 − j − α)c δ(p + 1, j − 1){(1 + β) + (p + 1 − j − α)}a

(0 ≤ α < p + 1 − j),

then Tp,j (a, c, β, α) ⊂ Nε+ (e). (γ)

Next, we determine the neighborhood for the class Tp,j (a, c, β, A, B) which is defined as follows: (γ)

Definition 3. A function f ∈ Tp is said to be in the class Tp,j (a, c, β, A, B), if there exists a function g ∈ Tp,j (a, c, β, A, B) such that f (z) (4.4) g(z) − 1 < p − γ (0 ≤ γ < p; z ∈ U) Theorem 8. If g ∈ Tp,j (a, c, β, A, B) and ε=

(p − γ) [(p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a − (p + 1 − j)(p + 2 − j)(A − B)c] , {(1 + β)(1 − B) + (p + 1 − j)(A − B)}a

(4.5)

then (γ)

Nε (g) ⊂ Tp,j (a, c, β, A, B). Proof. Suppose that f ∈ Nε (g), where g, defined by g(z) = z p −

∞ X

bp+k z p+k

(z ∈ U).

(4.6)

k=1

belongs to the class Tp,j (a, c, β, A, B). It follows from (3.1) that ∞ X

(p + k)|ap+k − bp+k | ≤ ε

k=1

which implies that

∞ X

|ap+k − bp+k | ≤

k=1

ε . p+1

Since g ∈ Tp,j (a, c, β, A, B), we deduce from (2.3) that ∞ X k=1

bp+k ≤

(p + 1 − j)(p + 2 − j)(A − B)c (p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a

so that P∞ f (z) − bp+k | < k=1 |a Pp+k − 1 ∞ g(z) 1 − k=1 bp+k ε{(1 + β)(1 − B) + (p + 1 − j)(A − B)}a (p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a − (p + 1 − j)(p + 2 − j)(A − B)c = p − γ,

≤

(γ)

provided that ε is given by (4.5). Thus, by (4.4), the function f ∈ Tp,j (a, c, β, A, B). This completes the proof of Theorem 5.

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5. Subordination results To prove our main result, we need the following definition and lemma. ∞ Definition 4. A sequence {bp+k }k=0 of complex numbers is said to be a subordinating factor sequence, if for any analytic convex(univalent) function g(z) = z p +

∞ X

cp+k z p+k

(z ∈ U),

(5.1)

(cp = 1; z ∈ U).

(5.2)

k=1

we have ∞ X

bp+k cp+k z p+k ≺ g(z)

k=0

Wilf [23] gave the following necessary and sufficient condition for a sequence to be a subordinating factor sequence. ∞ Lemma 2. A sequence {bp+k }k=0 of complex numbers is said to be a subordinating factor sequence, if and only if ( ) ∞ X p+k Re 1 + bp+k z > 0 (z ∈ U). (5.3) k=0

Theorem 9. Suppose f , given by (1.10) belongs to the class Tp,j (a, c, β, A, B) and g defined by (4.1) is an analytic convex(univalent) function in U. Then ρ (f ? g)(z) ≺ g(z)

(z ∈ U),

(5.4)

where ρ=

δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a 2 [δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a + δ(p, j)(A − B)c]

(5.5)

Moreover, Re{f (z)} > (−1)p ρ (z ∈ U)

(5.6)

and the subordinating result (5.4) is the best possible for the maximum factor ρ given by (5.5). Proof. We have ∞ X ρ (f ? g)(z) = bp+k z p+k (z ∈ U), k=0

where bp+k

( ρ, k=0 = −ρ ap+k , k = 1.

To prove the subordination (5.4), we need to show that {bp+k }∞ k=0 is a subordinating factor sequence which in view of Lemma 2 is true, if ( Re 1 + 2

∞ X

) bp+k z p+k

>0

(z ∈ U).

k=0

Since δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)}(a)k (c)k

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is an increasing function of k(k ∈ N), we get ( ) ∞ X p+k Re 1 + bp+k z k=0

P∞   p+k p k=1 [δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a] ap+k z = 1 + Re 2ρ z − [δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a + δ(p, j)(A − B)c] δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a ≥1− rp [δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a + δ(p, j)(A − B)c] δ(p, j)(A − B)c − rp × [δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a + δ(p, j)(A − B)c] ∞ X δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a)k × ap+k δ(p, j)(A − B) (c)k k=1

= 1 − rp > 0

(|z| = r < 1),

where we have made use of the assertion (2.4) of Theorem 2. This proves the subordination result (5.4). P∞ Letting g(z) = z p /(1 − z) = z p + k=1 z p+k (z ∈ U) in (5.4), we easily get the inequality (5.5). To prove the sharpness of the constant ρ, we consider the function f0 ∈ Tp,j (a, c, β, A, B) given by f0 (z) = z p −

δ(p, j)(A − B)c z p+1 δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a

(z ∈ U).

Thus, from (5.4), we have δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a zp f0 (z) ≺ (z ∈ U). 2 [δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a + δ(p, j)(A − B)c] 1−z Moreover, it can be easily verified that for the function f0    δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a 1 min Re =− , z∈U 2 [δ(p + 1, j − 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a + δ(p, j)(A − B)c] 2 so that the constant factor ρ in (5.4) cannot be replaced by a larger number. 6. Modified Hadamard product For functions f , defined by (1.10) and g given by (4.6), we define the modified Hadamard product or quasi Hadamard product of f and g by (f ? g)(z) = z p −

∞ X

ap+k bp+k z p+k = (g ? f )(z)

(z ∈ U).

k=1

We now prove e B), Theorem 10. Let a ≥ c > 0. If the functions f, g ∈ Tp,j (a, c, β, A, B), then f ? g ∈ Tp,j (a, c, β, A, where e=B+ A

(p + 1 − j)(p + 2 − j)(1 + β)(1 − B)(A − B)2 c . (p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}2 a − (p + 2 − j){(p + 1 − j)(A − B)}2 c

The result is sharp. Proof. Let the functions f be defined by (1.10) and g be given by (4.6). We need to find the largest e such that value of A ∞ X e − B)} (a)k δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A k=1

e − B) δ(p, j)(A

825

(c)k

ap+k bp+k ≤ 1.

(6.1)

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Since f, g ∈ Tp,j (a, c, β, A, B), by Theorem 2 we get ∞ X δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a)k

δ(p, j)(A − B)

k=1

and

(c)k

∞ X δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a)k

δ(p, j)(A − B)

k=1

(c)k

ap+k ≤ 1

bp+k ≤ 1.

Hence, by using Cauchy-Schwarz inequality, we deduce that ∞ X δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a)k p ap+k bp+k ≤ 1 δ(p, j)(A − B) (c)k

(6.2)

k=1

which implies that p ap+k bp+k ≤

δ(p, j)(A − B) (c)k δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a)k

(k ∈ N).

(6.3)

In view of (6.1) and (6.2), it is sufficient to show that e − B)} δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A ap+k bp+k e − B) δ(p, j)(A δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)} p ≤ ap+k bp+k δ(p, j)(A − B)

(k ∈ N),

or, equivalently e − B){k(1 + β)(1 − B) + (p + 1 − j)(A − B)} p (A ap+k bp+k ≤ e − B)} (A − B){k(1 + β)(1 − B) + (p + 1 − j)(A

(k ∈ N).

In the light of the inequality (6.3), we need to show that δ(p, j)(A − B) (c)k δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (a)k e − B){k(1 + β)(1 − B) + (p + 1 − j)(A − B)} (A ≤ e − B)} (A − B){k(1 + β)(1 − B) + (p + 1 − j)(A

(k ∈ N).

(6.4)

It follows from (6.4) that for k ∈ N e ≥ B+ A kδ(p, j)(1 + β)(1 − B)(A − B)2 (c)k . δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)}2 (a)k − δ(p, j)(p + 1 − j)(A − B)2 (c)k Denoting the right side of the above inequality by ψ(k), we note that ψ is a decreasing function of k (k ∈ N). Therefore, we conclude that e ≥ ψ(1) = B+ A +

(p + 1 − j)(p + 2 − j)(1 + β)(1 − B)(A − B)2 c (p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}2 a − (p + 2 − j){(p + 1 − j)(A − B)}2 c

which evidently completes the proof of Theorem 10. The result is sharp for the functions f0 and g0 defined by f0 (z) = g0 (z) = z p −

(p + 1 − j)(p + 2 − j)(A − B)c z p+1 (p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}a

826

(−1 ≤ B < 0; z ∈ U). (6.5)

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Setting β = 0, A = 1 − {2α/(p + 1 − j)} (0 ≤ α < p + 1 − j − α) and B = −1 in Theorem 10, we get the following result which, in turn, yield the corresponding works for the classes T ∗ (p, α, j) and C(p, α, j) obtained by Aouf [4, Theorem 10 and Corollary 9] when a = c and a = p + 2 − j, c = p + 1 − j respectively. Corollary 5. If a ≥ c > 0 and the functions f, g ∈ Tp,j (a, c, β, α), then the function (f ? g) ∈ Tp,j (a, c, β, κ), where κ = (p + 1 − j) −

(1 + β)(p + 2 − j)(p + 1 − j − α)2 c . (p + 1)(p + 2 + β − j − α)2 a − (p + 2 − j)(p + 1 − j − α)2 c

The result is sharp. Theorem 11. Let a ≥ c > 0 and −1 ≤ B < 0. If the functions f , defined by (1.10) and g given by (4.6) are in the class Tp,j (a, c, β, A, B), then the function h(z) = z p −

∞ X


a2p+k + b2p+k z p+k

(z ∈ U)

k=1

e B), where belongs to the class Tp,j (a, c, β, A, C=

2(p + 1 − j)(p + 2 − j)(1 + β)(1 − B)(A − B)2 c . (p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}2 a − 2(p + 2 − j){(p + 1 − j)(A − B)}2 c

The result is sharp. Proof. By virtue of Theorem 2, we obtain 2 ∞  X δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)}(a)k δ(p, j)(A − B)(c)k

k=1

" ≤

a2p+k

∞ X δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)}(a)k

δ(p, j)(A − B)(c)k

k=1

#2 ap+k

≤ 1.

(6.6)

≤ 1.

(6.7)

Similarly, 2 ∞  X δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)}(a)k δ(p, j)(A − B)(c)k

k=1

" ≤

b2p+k

∞ X δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)}(a)k

k=1

δ(p, j)(A − B)(c)k

#2 bp+k

It follows from (6.6) and (6.7) that 2 ∞  1 X δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)}(a)k (a2p+k + b2p+k ) ≤ 1. 2 δ(p, j)(A − B)(c)k k=1

e such that Thus, we need to find the largest value of A e − B)}(a)k δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A e − B)(c)k δ(p, j)(A  2 1 δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)}(a)k ≤ 2 δ(p, j)(A − B)(c)k

(k ∈ N).

On simplifying the above inequality, we deduce that e ≥ B+ A 2δ(p, j)k(1 + β)(1 − B)(A − B)2 (c)k δ(p + k, j − 1){k(1 + β)(1 − B) + (p + 1 − j)(A − B)}2 (a)k − 2δ(p, j)(p + 1 − j)(A − B)2 (c)k

827

(6.8)

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for k ∈ N. Since the right side of (6.8) is an decreasing function of k (k ∈ N), putting k = 1 in (6.8), we get e≥B+ A

2(p + 1 − j)(p + 2 − j)(1 + β)(1 − B)(A − B)2 c . (p + 1){(1 + β)(1 − B) + (p + 1 − j)(A − B)}2 a − 2(p + 2 − j){(p + 1 − j)(A − B)}2 c

and Theorem 11 follows at once. The result is sharp for the functions f0 and g0 defined by (6.5). Taking β = 0, A = 1 − {2α/(p + 1 − j)} (0 ≤ α < p + 1 − j − α) and B = −1 in Theorem 11, we obtain the following result which yield the corresponding works for the classes T ∗ (p, α, j) and C(p, α, j) due to Aouf [4, Theorem 12 and Corollary 11] for a = c and a = p + 2 − j, c = p + 1 − j respectively. Corollary 6. Let a ≥ c > 0 and β ≥ 0. If the functions f , defined by (1.10) and g given by (3.6) are in the class Tp,j (a, c, β, α), then the function h(z) = z p −

∞ X


a2p+k + b2p+k z p+k

(z ∈ U)

k=1

belongs to the class Tp,j (a, c, β, η), where η = (p + 1 − j) −

2(1 + β)(p + 2 − j)(p + 1 − j − α)2 c . (p + 1)(p + 2 + β − j − α)2 a − 2(p + 2 − j)(p + 1 − j − α)2 c

The result is sharp. Remark. The results obtained in various section can suitably be extended to hold true for the new subclasses of Ap and Tp defined in the introduction. Acknowledgements. This work was supported by a Research Grant of Pukyong National University(2014 year) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007037). References ¨ Ozkan ¨ 1. O. Altinta¸s, O. and H.M. Srivastava, Neighborhoods of a certain family of multivalent functions with negative coefficients, Comput. Math. Appl., 47(2004), 1667-1672. 2. O. Altinta¸s and S. Owa, Neighborhoods of certain analytic functions with negative coefficients, Internat. J. Math. Math. Sci., 19(1996), 797-800. ¨ Ozkan ¨ 3. O. Altinta¸s, O. and H.M. Srivastava, Neighborhoods of a class of analytic functions with negative coefficients, Appl. Math. Lett., 13(3)(2000), 63-67. 4. M.K. Aouf, Some families of p−valent functions with negative coefficients, Acta Math. Univ. Comenianae, LXXVIII (2009), 121-135. 5. B.C. Carlson and D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 159(1984), 737-745. 6. J.H. Choi, M. Saigo and H.M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl., 276(2002), 432-445. 7. P.L. Duren, Univalent Functions, A Series of Comprehensive Studies in Mathematics, Vol. 259, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo, 1983. 8. B.A. Frasin, Subordination results for a class of analytic functions defined by a linear operator, J. Inequal. Pure Appl. Math., 7(4) (2006), Art. 134 [ONLINE:http://jipam.vu.edu.au] 9. R.M. Goel and N.S. Sohi, A new criterion for p-valent functions, Proc. Amer. Math. Soc., 78(1980), 353-357. 10. A.W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8(1957), 598-601.

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11. A.W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl., 155(1991), 364-370. 12. A.W. Goodman, On uniformly convex functions, Ann. Polon. Math., 56(1991), 364-370. 13. J.-L. Liu and K.I. Noor, Some properties of Noor integral operator, J. Natur. Geom., 21(2002), 81-90. 14. T.H. MacGregor, Majorization by univalent functions, Duke Math. J., 34(1967), 95-102. 15. Z. Nehari, Conformal Mapping, McGraw-Hill Book Company, New York, Toronto and Lon- don, 1952. 16. J. Patel and A.K. Mishra, On certain multivalent functions associated with an extended fractional differintegral operator, J. Math. Anal. Appl., 332(2007), 109-122. 17. M. Pathak and P. Sharma, A family of p−valent analytic functions defined by fractional calculus operator, Internat. J. Pure Appl. Math., 87(1)(2013), 129-143. 18. F. Rønning, On starlike functions associated with parabolic regions, Ann. Univ. Mariae CurieSklodowska Sect. A, 45(1991), 117-122. 19. F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118(1) (1993), 189-196. 20. St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109-115. 21. St. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 8(1981), 521-527. 22. H. Saitoh, A linear operator and its application of first order differential subordinations, Math. Japonica, 44(1996), 31-38. 23. H.S. Wilf, Subordinating factor sequence for convex maps of the unit circle, Proc. Amer. Math. Soc., 12(1961), 689-693.

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The characterizations of McShane integral and Henstock integrals for fuzzy-number-valued functions with a small Riemann sum on a small set† a

Muawya Elsheikh Hamida,b , Zeng-Tai Gonga,∗ College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China b School of Management, Ahfad University For Women, Omdurman, Sudan

Abstract In this paper, we shall characterize McShane integral and Henstock integral of fuzzy-numbervalued functions by Riemann-type integral of fuzzy-number-valued functions with a small Riemann sum on a small set, and the results show that McShane integral (Henstock integrals) of fuzzy-number-valued functions could be represented by Riemann integral (McShane integral) with a small Riemann sum on a small set, respectively. Keywords: Fuzzy numbers; fuzzy integrals; small Riemann sum. 1. Introduction Since the concept of fuzzy sets was firstly introduced by Zadeh in 1965[19], it has been studied extensively from many different aspects of the theory and applications, such as fuzzy topology, fuzzy analysis [16], fuzzy decision making and fuzzy logic, information science and so on. Recently, fuzzy integrals of fuzzy-number-valued functions have been studied by many authors from different points of views, including Goetschel [8], Nanda [15], Kaleva [12], Wu [17,18] and other authors [1-7]. For the constructive definition of fuzzy integrals, Goetschel defined and discussed the fuzzy integral of fuzzynumber-valued functions by first taking the sum and then the limit which is known as a Riemann type definition in 1986 [8]. The Mcshane integrals of fuzzy-valued functions were defined and characterized, and it shown that the Mcshane integrals of fuzzy-valued functions are the Riemann-type definitions of (K) integral [6]. In 2001, Wu and Gong defined and discussed the fuzzy Henstock integral for fuzzy-numbervalued functions and several necessary and sufficient conditions of integrability for fuzzy-number-valued functions were given by means of abstract function theory and using a concrete structure into which they embed the fuzzy number space [17]. After that, Hsien-Chung Wu [18] discussed the fuzzy Riemann integral (improper the fuzzy Riemann integral) and its numerical integration by using the probabilistic Monte Carlo method, the Riemann integral (improper the fuzzy Riemann integral) of real-valued functions and its numerical integration respectively, and Barnabbas Bede [1] introduced the quadrature rules of fuzzy Henstock integral on finite interval. In this paper, we shall characterize McShane integral and Henstock integrals of fuzzy-number-valued functions by Riemann integral with a small Riemann sum on a small set, and the results show that McShane integral (Henstock integrals) of fuzzy-number-valued functions could be represented by Riemann integral (McShane integral) with a small Riemann sum on a small set, respectively. The rest of this paper is organized as follows. To make our analysis possible, in Section 2 we shall review the relevant concepts and properties of fuzzy sets and the definition of McShane integral and Henstock integrals for fuzzy-number-valued functions. Section 3 is devoted to discussing McShane integral of fuzzy-number-valued functions by Riemann integral of the fuzzy-number-valued functions with a small Riemann sum on a small set. Section 4 we shall investigate Henstock integrals of fuzzy-number-valued † Supported by the Natural Scientific Fund of China (11461062, 61262022), the Natural Scientific Fund of Gansu Province of China (1208RJZA251),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], [email protected](Muawya Elsheikh).

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functions by McShane integral of fuzzy-number-valued functions with a small Riemann sum on a small set. 2. Preliminaries Fuzzy set u ˜ ∈ E 1 is called a fuzzy number if u ˜ is a normal, convex fuzzy set, upper semi-continuous and supp u = {x ∈ R | u(x) > 0} is compact. Here A¯ denotes the closure of A. We use E 1 to denote the fuzzy number space [8]. Let u ˜, v˜ ∈ E 1 , k ∈ R, the addition and scalar multiplication are defined by [˜ u + v˜]λ = [˜ u]λ + [˜ v ]λ ,

[k˜ u]λ = k[˜ u]λ ,

+ respectively, where [˜ u]λ = {x : u(x) > λ} = [u− λ , uλ ] for any λ ∈ [0, 1]. We use the Hausdorff distance between fuzzy numbers given by D : E 1 × E 1 → [0, +∞) as in [8] − + + D(˜ u, v˜) = sup d([˜ u]λ , [˜ v ]λ ) = sup max{|u− λ − vλ |, |uλ − vλ |}, λ∈[0,1]

λ∈[0,1]

where d is the Hausdorff metric. D(˜ u, v˜) is called the distance between u ˜ and v˜. Furthermore, we write + u, ˜0) = sup max{|u− k˜ ukE 1 = D(˜ λ |, |uλ |}. λ∈[0,1]

Notice that k · kE 1 = D(·, ˜0) doesn’t stands for the norm of E 1 . Definition 2.1 [8]. A fuzzy-number-valued function f˜(x) is said to be fuzzy Riemann integrable on [a, b] ˜ ∈ E 1 such that for every ε > 0, there is a δ > 0 such that for any division if there exists a fuzzy number H T : a = x0 ≤ x1 ≤ ... ≤ xn = b satisfying kT k = maxni=1 {xi − xi−1 } < δ, and for any ξi ∈ [xi−1 , xi ], we have n X ˜ < ε. D( f˜(ξi )(xi − xi−1 ), H) i=1

We write (RH)

Rb a

˜ and f˜ ∈ RH[a, b]. f˜(x)dx = H

Definition 2.2[10, 13]. Let δ : [a, b] → R+ be a positive real-valued function. T = {[xi−1 , xi ]; ξi }ni=1 is said to be a δ-fine division, if the following conditions are satisfied: (1) a = x0 < x1 < x2 < ... < xn = b; (2) ξi ∈ [xi−1 , xi ] ⊂ (ξi − δ(ξi ), ξi + δ(ξi ))(i = 1, 2, · · · , n). If we replace above (2) by (2)’ [xi−1 , xi ] ⊂ (ξi − δ(ξi ), ξi + δ(ξi ))(i = 1, 2, · · · , n), then T = {[xi−1 , xi ]; ξi }ni=1 is said to be a δ-fine (M) division. For brevity, we write T = {[u, v]; ξ}, where [u, v] denotes a typical interval in T and ξ is the associated point of [u, v]. Definition 2.3 [6, 17]. A fuzzy-number-valued function f˜(x) is said to be fuzzy Henstock integrable ˜ ∈ E 1 such that for every ε > 0, (fuzzy McShane integrable) on [a, b] if there exists a fuzzy number H there is a function δ(x) > 0 such that for any δ-fine division (δ-fine (M) division) T = {[xi−1 , xi ]; ξi }ni=1 , we have n X ˜ < ε. D( f˜(ξi )(xi − xi−1 ), H) i=1

Rb

Rb

˜ ((F M ) f˜(x)dx = H) ˜ and f˜ ∈ F H[a, b] (f˜ ∈ F M [a, b]). We write (F H) a f˜(x)dx = H a Obviously, f˜ ∈ RH[a, b] implies that f˜ ∈ F H[a, b] and f˜ ∈ F M [a, b]. Furthermore, from the results of [13] and [6], we have the following Lemma. Lemma 2.1[6, 17]. A fuzzy-number-valued function f˜(x) is fuzzy Henstock integrable (fuzzy McShane integrable) on [a, b] if and only if fλ− (x), fλ+ (x) are (RH) integrable for any λ ∈ [0, 1] uniformly, i.e., for 831

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every ε > 0, there is a function δ(x) > 0 (independent of λ ∈ [0, 1]) such that for any δ-fine division (δ-fine (M) division) T = {[xi−1 , xi ]; ξi }ni=1 , and for any λ ∈ [0, 1] we have |

n X

fλ− (ξi )(xi − xi−1 ) − Hλ− | < ε.

i=1

fλ+ (x) is similar. Here Hλ− and Hλ+ denote respectively the (H) integrals ((M) integrals) of fλ− and fλ+ on [a, b] for any λ ∈ [0, 1]. Lemma 2.2 [7]. Let f˜ : [a, b] → E 1 be a fuzzy-number-valued function, and f˜ ∈ F H[a, b]. Then f˜ ∈ F M [a, b] if and only if kf˜kE 1 is Lebesgue integrable, and Z b Z b ˜ k(F H) f kE 1 ≤ (L) kf˜kE 1 . a

a

Lemma 2.3. Let f˜ : [a, b] → E 1 be a fuzzy-number-valued function and f˜ ∈ F M [a, b]. Then for any ε > 0 there exists a positive integer N such that for every n ≥ N there is a δn −fine division δ-fine (M) division T = {[xi−1 , xi ]; ξi }ni=1 , we have X k fe(ξi )(xi − xi−1 )kE 1 ≤ ε, kf˜kE 1 >n

where the sum is taken over T for which kf˜kE 1 > n. Proof. Since f˜ ∈ F M [a, b], by Lemma 2.2 we know that kf˜kE 1 is Lebesgue integrable. Define a real function ( kf˜kE 1 , if kf˜kE 1 ≤ n, gn (x) = 0, if kf˜kE 1 > n. Then gn (x) is Lebesgue integrable on [a, b] and gn (x) → kf˜kE 1 . Let Gn (a, b) denote the integration of gn (x), and F (a, b) the integration of kf˜kE 1 , then Gn (a, b) → F (a, b). That is to say, for given ε > 0, there is a function δn (x) > 0 such that for any δn -fine division T = {[xi−1 , xi ]; ξi }m i=1 , we have |

m X

gn (ξi )(xi − xi−1 ) − Gn (a, b))| < ε,

i=1

|

m X

kf˜kE 1 (ξi )(xi − xi−1 ) − F (a, b)| < ε.

i=1

Choose N so that whenever n ≥ N |Gn (a, b) − F (a, b)| < ε. Therefore for n ≥ N and δn -fine division T = {[xi−1 , xi ]; ξi }m i=1 , we have X fe(ξi )(xi − xi−1 )kE 1 k kf˜kE 1 >n

X

≤

kfe(ξi )kE 1 (xi − xi−1 )

kf˜kE 1 >n

≤|

X

gn (ξi )(xi − xi−1 ) −

X

≤|

X

gn (ξi )(xi − xi−1 ) − Gn (a, b)| + |Gn (a, b) − F (a, b)|

kf˜kE 1 (ξi )(xi − xi−1 )|

< 3ε. This complete the proof. 832

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3. The characterization of McShane integral of fuzzy-number-valued functions by Riemann integral and a small Riemann sum on a small set Theorem 3.1. Let f˜ : [a, b] → E 1 . Then the following statements are equivalent: (1) fe is fuzzy McShane integrable on [a, b]; (2) for any λ ∈ [0, 1], fλ− and fλ+ are McShane integrable on [a, b] for any λ ∈ [0, 1] uniformly (δ(x) is independent of λ ∈ [0, 1]), and Z b Z b Z b − ˜ [(F M ) f (x)dx]λ = [(M ) fλ (x)dx, (M ) fλ+ (x)dx]; a

a

a

e such that for any  > 0, η > 0, we have (3) there exists a fuzzy number A (i) there exists a constant δ > 0 and a open set G satisfying |G| < η, such that for any δ fine division T : a = x0 ≤ x1 ≤ ... ≤ xn = b satisfying kT k = maxni=1 {xi − xi−1 } < δ, i.e., 0 < xi − xi−1 < δ and ξi ∈ [a, b]\G, we have X e < , D( fe(ξi )(xi − xi−1 ), A) ξi ∈G /

and (ii) there exists a δ(ξi ) > 0, such that for any interval [xi − xi−1 ] satisfying [xi − xi−1 ] ⊂ (ξi − δ(ξi ), ξi + δ(ξi )) ⊂ G, we have X k fe(ξi )(xi − xi−1 )kE 1 < . ξi ∈G

Proof. From the results of [17], (1) and (2) are equivalent, we only prove that (1) and (3) are equivalent. Let fe is fuzzy McShane integrable on [a, b]. By using of the Definition 2.3, there exists a fuzzy number A˜ ∈ E 1 such that for every ε > 0, there is a function δ(x) > 0 such that for any δ-fine (M) division T = {[xi−1 , xi ]; ξi }ni=1 , we have n X ˜ < ε. D( f˜(ξi )(xi − xi−1 ), A) i=1

For any η > 0, due to the results of [14], we could select a δ(ξ) which is measurable and an open set G1 satisfying |G1 | < η2 such that {x|0 < δ(x) < δ0 } ⊂ G1 , and there exists an open set G2 and a natural number N satisfying |G2 | < η2 and {x|kfe(x)kE 1 > N } ⊂ G2 . Let G = G1 ∪ G2 . Then |G| < η. Define ( δ0 , if ξ ∈ / G, δ ∗ (x) = δ(x), such that (x − δ(x), x + δ(x)) ⊂ G, if ξ ∈ G, then according to Lemma 2.3, for all δ ∗ (ξ)-fine Mcshane division, we have X X k fe(ξi )(xi − xi−1 )kE 1 6 kfe(ξi )kE 1 (xi − xi−1 ) ξi ∈G

6

ξi ∈G

X

X

kfe(ξi )kE 1 (xi − xi−1 ) +

kf˜kE 1 >N

kfe(ξi )kE 1 (xi − xi−1 );

ξi ∈G∩{x|kf˜kE 1 ≤N }

<  + N η, and D(

X

e fe(ξi )(xi − xi−1 ), A)

ξi ∈G /

X X e + D( ≤ D( fe(ξi )(xi − xi−1 ), A) fe(ξi )(xi − xi−1 ), ˜0) ξi ∈G

0 there exists a measurable set E ⊂ [a, b] such that µ(E) < η, f and g are Lebesgue integrable on E, and Z Z b (L) f (x)dx = (H) f (x)dx, [a,b]\E

a

Z

Z g(x)dx = (H)

(L)

b

g(x)dx. a

[a,b]\E

Theorem 4.1. Let f˜ : [a, b] → E 1 , and f0− and f0+ be of equi-Riemann tails on E. Then the following statements are equivalent: (1) fe is fuzzy Henstock integrable on [a, b]; (2) for any λ ∈ [0, 1], fλ− and fλ+ are Henstock integrable on [a, b] for any λ ∈ [0, 1] uniformly (δ(x) is independent of λ ∈ [0, 1]), and Z b Z b Z b [(F H) f˜(x)dx]λ = [(H) fλ− (x)dx, (H) fλ+ (x)dx]; a

a

a

e such that the following conditions are satisfied for any given (3) there exists a fuzzy number A  > 0, η > 0: (i) there exists a measurable set E satisfying µ(E) < η, such that f˜(x) is fuzzy McShane integrable on [a, b] \ E; (ii) there is a function δ(x) > 0 such that for any δ fine division T = {[xi−1 , xi ]; ξi }ni=1 and ξi ∈ [a, b]\E, we have X e < , D( fe(ξi )(xi − xi−1 ), A) ξi ∈[a,b]\E

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and k

X

fe(ξi )(xi − xi−1 )kE 1 < .

ξi ∈E

Proof. From the results of [17], (1) and (2) are equivalent, we only prove that (1) and (3) are equivalent. (1) implies (3): If fe is fuzzy McShane integrable on [a, b], then (3) holds, we only define E = [a, b]. Let fe is fuzzy Henstock integrable on [a, b] but it is not fuzzy McShane integrable on [a, b]. By Lemma 2.2, kf˜kE 1 is not Lebesgue integrable on [a, b]. On the other hand, we note that kf˜kE 1 = sup max{|fλ− |, |fλ+ |} = sup max{|fλ−n |, |fλ+n |} ≤ max{|f0− |, |f0+ |}, λn ∈[0,1]

λ∈[0,1]

where {λn } is the set of all rational numbers on [0, 1]. This means that at least one of f0− and f0+ is Henstock integrable on [a, b] but Lebesgue. Since f0− and f0+ are of equi-Riemann tails on E, by Lemma 4.1, for any given  > 0, η > 0 there exists a measurable set E ⊂ [a, b] such that µ(E) < η, f0− and f0+ are Lebesgue integrable on [a, b] \ E. Furthermore, Z b Z − f0− (x)dx| < ε, f0 (x)dx − (H) |(L) a

[a,b]\E

Z |(L) [a,b]\E

f0+ (x)dx − (H)

Z

b

f0+ (x)dx| < ε.

a

Applies the following formula kf˜kE 1 = sup max{|fλ− |, |fλ+ |} = sup max{|fλ−n |, |fλ+n |} ≤ max{|f0− |, |f0+ |} λn ∈[0,1]

λ∈[0,1]

and the mesurability of fλ−n and fλ+n , we have kf˜kE 1 is Lebesgue integrable on [a, b] \ E, where {λn } is the set of all rational numbers on [a, b] \ E. By using Lemma 2.2 again, we obtain that fe is fuzzy McShane integrable on [a, b] \ E. By the Henstock integrability of f0− and f0+ on [a, b], there is a function δ(x) > 0 such that for any δ fine division T = {[xi−1 , xi ]; ξi }ni=1 X k fe(ξi )(xi − xi−1 )kE 1 ξi ∈E

=

sup max{| λ∈[0,1]

X

fλ− (ξi )(xi − xi−1 )|, |

ξi ∈E

≤ max{|

X

f0− (ξi )(xi

X

fλ+ (ξi )(xi − xi−1 )|}

ξi ∈E

− xi−1 )|, |

ξi ∈E

X

f0+ (ξi )(xi

− xi−1 )|}

ξi ∈E

< ε. Since fe is fuzzy McShane integrable on [a, b] \ E, there is a function δ(x) > 0 such that for any δ fine division T = {[xi−1 , xi ]; ξi }ni=1 , we have X e D( fe(ξi )(xi − xi−1 ), A) ξi ∈[a,b]\E

≤ D(

X

e +k fe(ξi )(xi − xi−1 ), A)

X

fe(ξi )(xi − xi−1 )kE 1

ξi ∈E

ξi ∈[a,b]

< 2ε. (3) implies (1):  > 0, η > 0, there exists a measurable set E satisfying µ(E) < η, such that f˜(x) is fuzzy McShane integrable on [a, b] \ E, and there is a function δ(x) > 0 such that for any δ fine division T = {[xi−1 , xi ]; ξi }ni=1 , we have X e < , D( fe(ξi )(xi − xi−1 ), A) ξi ∈[a,b]\E

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and k

X

fe(ξi )(xi − xi−1 )kE 1 < .

ξi ∈E

It follows that D(

X

e fe(ξi )(xi − xi−1 ), A)

ξi ∈[a,b]\E

≤ D(

X

e +k fe(ξi )(xi − xi−1 ), A)

X

fe(ξi )(xi − xi−1 )kE 1

ξi ∈E

ξi ∈[a,b]\E

< 2ε. This prove that fe is fuzzy Henstock integrable on [a, b].

References [1] B. Bede, S.G. Gal, Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems 145 (2004) 359-380. [2] Y.H. Feng, Fuzzy-valued mappings with finite variation, fuzzy-valued measures and fuzzy-valued Lebesgue-Stieltjes integrals, Fuzzy Sets and Systems 121 (2001) 227-236. [3] Z.T. Gong, Y.B. Shao, The controlled convergence theorems for the strong Henstock integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems 160(2009)1528õ1546. [4] Z.T. Gong, L. Wang, The numerical calculus of expectations of fuzzy random variables, Fuzzy Sets and Systems 158 (2007) 722-738. [5] Z.T. Gong, L.L. Wang, The Henstock-Stieltjes integral for fuzzy-number-valued functions, Information Sciences 188 (2012) 276õ297 [6] Z.T. Gong, C.X. Wu, The Mcshane Integral of Fuzzy-Valued Functions, Southeast Asian Bulletin of Mathematics 24(3)(2000) 365-373. [7] Z.T. Gong, Nonabsolute Fuzzy Integrals, Absolute Integrability and Its Absolute Value Inequality, Journal of Mathematical Research and Exposition 28(3)(2008)479õ488. [8] R. Goetschel Jr., W. Voxman. Elementary fuzzy calculus. Fuzzy Sets and Systems 18 (1986) 31-43. [9] R.A. Gordon, Riemann tails and the Lebesgue and Henstock integrals. Real Analysis Exchange 17 (1991-1992) 789-801. [10] R. Henstock, Theory of Integration, Butterworth, London, 1963. [11] W. Pfeffer, W. Yang, A note on conditionally convergent integrals. Real Analysis Exchange 17 (1991-1992) 815-819. [12] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317. [13] P.Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore, 1989. [14] G.Q. Liu, The measurability of δ(x) in Henstock integration. Real Analysis Exchange 13(2) (19871988) 446-450. [15] S. Nanda, On fuzzy integrals, Fuzzy Sets and Systems 32 (1989) 95-101. [16] C.V. Negoita, D. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975. [17] C.X. Wu, Z.T. Gong, On Henstock integral of fuzzy-number-valued functions(I), Fuzzy Sets and Systems 120 (2001) 523-532. [18] Hsing-Chung Wu. The fuzzy Riemann integral and its numerical integration[J]. Fuzzy Sets and Systems, 110 (2000) 1-25. [19] L.A. Zadeh, Fuzzy sets, Information Control 8 (1965) 338-353.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

BARNES-TYPE NARUMI OF THE SECOND KIND AND POISSON-CHARLIER MIXED-TYPE POLYNOMIALS DAE SAN KIM, TAEKYUN KIM, HYUCK IN KWON, AND TOUFIK MANSOUR

Abstract. In this paper, we consider the Barnes-type Narumi of the second kind and PoissonCharlier mixed-type polynomials. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our polynomials and several known families of polynomials.

1. Introduction The aim of this paper is to use umbral calculus to obtain several new and interesting identities of Barnes-type Narumi of the second kind and Poisson-Charlier mixed-type polynomials. Umbral calculus has been used in numerous problems of mathematics and applied mathematics (for instance, see [1, 6, 7, 12, 13, 14, 15, 16]). Di Bucchianico and Loeb [8] give more than five hundred old and new findings related to the study of Sheffer polynomial sequences. For instance, applications of umbral calculus to the physics of gases can be found in [22] and umbral techniques have been used in group theory and quantum mechanics by Biedenharn et al. [2, 3] (for other examples, see [4, 5, 6, 9, 17, 18, 23]). In mathematics, the Narumi polynomials Nn (x) are polynomials introduced by Narumi [19] (also, ∑ n t see [21, Section 4.4]) given by the generating function n≥0 Nn (x) xn! = log(1+t) (1+t)x . We recall here that the Barnes-type Narumi polynomials of the first kind Nn (x|a1 , . . . , ar ) are given by the generating function as r ∏ ∑ (1 + t)aj − 1 tn Nn (x|a1 , . . . , ar ) . (1 + t)x = log(1 + t) n! j=1 n≥0

ˆ Cn (x|a1 , . . . , ar ; b) where the corresponding generating function is Here we consider the polynomials N given by ) r ( n ∏ ∑ (1 + t/b)aj − 1 x ˆ Cn (x|a1 , . . . , ar ; b) t , (1.1) e−t (1 + t/b) = N a (1 + t/b) j log(1 + t/b) n! j=1 n≥0

where r ∈ Z>0 , and a1 , . . . , ar , b ̸= 0. For simple notation, we define ˆ Cn (x) = N ˆ Cn (x|a1 , . . . , ar ; b) and N ˆ Cn;j (x) = N ˆ Cn (x|a1 , . . . , aj−1 , aj+1 , . . . , ar ; b). N ˆn (x) = N ˆn (x|a1 , . . . , ar ) We recall first that the Barnes-type Narumi polynomials of the second kind N are given by the generating function as ) r ( n ∏ ∑ (1 + t)aj − 1 x ˆn (x) t . (1 + t) = N (1 + t)aj log(1 + t) n! j=1 n≥0

2000 Mathematics Subject Classification. 05A40. Key words and phrases. Barnes-type Narumi polynomials, Poisson-Charlier polynomials, Umbral calculus. 1

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DAE SAN KIM, TAEKYUN KIM, HYUCK IN KWON, AND TOUFIK MANSOUR

ˆn = N ˆn (0) are called the Barnes-type Narumi numbers of the second kind. Next, the When x = 0, N Poisson-Charlier polynomial cn (x; b) (b ̸= 0) are defined by the generating function as ∑ tn e−t (1 + t/b)x = cn (x; b) . n! n≥0

In order to study the Barnes-type Narumi polynomials o the second kind, we need the use of the Umbral algebra and Umbral calculus. Let Π be the algebra of polynomials in a single variable x over C and let Π∗ be the vector space of all linear functionals on Π. We denote the action of a linear functional L on a polynomial p(x) by ⟨L|p(x)⟩, and we define the vector space structure on Π∗ by ⟨cL + c′ L′ |p(x)⟩ = c⟨L|p(x)⟩ + c′ ⟨L′ |p(x)⟩, where c, c′ ∈ C (see [10, 11, 20, 21]). The algebra of formal power series in a single variable t is given by     ∑ tk (1.2) H = f (t) = ak | ak ∈ C .   k! k≥0

Each formal power series in the variable t defines a linear functional on Π as (1.3)

⟨f (t)|xn ⟩ = an , for all n ≥ 0, (see [10, 11, 20, 21]).

By (1.2) and (1.3), we have (1.4)

⟨tk |xn ⟩ = n!δn,k , for all n, k ≥ 0, (see [10, 11, 20, 21]),

where δn,k is the Kronecker’s symbol. ∑ n tn n n Let fL (t) = n≥0 ⟨L|x ⟩ n! , thus by (1.4), we have ⟨fL (t)|x ⟩ = ⟨L|x ⟩. Therefore, the map ∗ L 7→ fL (t) defines a vector space isomorphism from Π onto H, which implies that H is thought of as set of both formal power series and linear functionals. We call H the umbral algebra. The umbral calculus is the study of umbral algebra. The smallest integer k for which the coefficient of tk does not vanish in the non-zero power series f (t) is called the order O(f (t)) of f (t) (see [10, 11, 20, 21]). If O(f (t)) = 1 (O(f (t)) = 0), then f (t) is called a delta (invertable) series. Suppose that O(f (t)) = 1 and O(g(t)) = 0, then there exists a unique sequence sn (x) of polynomials such that ⟨g(t)(f (t))k |sn (x)⟩ = n!δn,k , where n, k ≥ 0. The sequence sn (x) is called the Sheffer sequence for (g(t), f (t)) which is denoted by sn (x) ∼ (g(t), f (t)) (see [10, 11, 20, 21]). For f (t) ∈ H and p(x) ∈ Π, we have ⟨eyt |p(x)⟩ = p(y), ⟨f (t)g(t)|p(x)⟩ = ⟨g(t)|f (t)p(x)⟩, and ∑ ∑ tn xn f (t) = ⟨f (t)|xn ⟩ , p(x) = (1.5) ⟨tn |p(x)⟩ , n! n! n≥0

n≥0

(see [10, 11, 20, 21]). From (1.5), we obtain (1.6)

⟨tk |p(x)⟩ = p(k) (0),

⟨1|p(k) (x)⟩ = p(k) (0),

where p(k) (0) denotes the k-th derivative of p(x) with respect to x at x = 0. So, by (1.6), we get that dk tk p(x) = p(k) (x) = dx k p(x), for all k ≥ 0, (see [10, 11, 20, 21]). Let sn (x) ∼ (g(t), f (t)). Then we have ∑ 1 tn y f¯(t) (1.7) e = s (y) , n n! g(f¯(t)) n≥0

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BARNES-TYPE NARUMI AND POISSON-CHARLIER MIXED-TYPE POLYNOMIALS

3

for all y ∈ C, where f¯(t) is the compositional∑inverse of f (t) (see [10, 11, 20, 21]). For sn (x) ∼ n (g(t), f (t)) and rn (x) ∼ (h(t), ℓ(t)), let sn (x) = k=0 cn,k rk (x), then ⟩ ⟨ 1 h(f¯(t)) k n ¯ (1.8) (ℓ(f (t))) |x , cn,k = k! g(f¯(t)) (see [10, 11, 20, 21]). ˆ Cn (x) is the Sheffer sequence for the pair It is immediate from (1.1) to see that N   r aj t ∏ t te ˆ Cn (x) ∼ eb(e −1) (1.9) , b(et − 1) . N aj t − 1 e j=1 ˆ Cn (x) will be called the Barnes-type Narumi of the second kind and PoissonThe polynomials N ˆ Cn = N ˆ Cn (a1 , . . . , ar ; b) = N ˆ Cn (0) are called the Carlier mixed-type polynomials. When x = 0, N Barnes-type Narumi of the second kind and Poisson-Charlier mixed-type numbers, ˆ Cn (x) The aim of the present paper is to present several new identities for the polynomials N by the use of umbral calculus. At first, in the next section, we present explicit formulas for such polynomials. Then, we present several recurrence relations for these polynomials. At the end, we establish a connection between our polynomials and several known families of polynomials.

2. Explicit formulas By the definitions and (1.9), we have ⟨ ⟩ ⟨ ⟩ ) r ( i aj ∑ ∏ t (1 + t/b) − 1 ˆ Ci (y) | xn = e−t ˆ Cn (y) = N N (1 + t/b)y | xn aj log(1 + t/b) i! (1 + t/b) j=1 i≥0 ⟨ ⟩ ) r ( aj ∏ (1 + t/b) − 1 −t y n = e | (1 + t/b) x (1 + t/b)aj log(1 + t/b) j=1 ⟨ ⟩ n ( ) n ( ) ℓ ∑ ∑ ∑ t n 1 ˆ n (−1)n−ℓ ˆ −t −t n−ℓ n ˆℓ (y) = e | N = N (y)⟨e |x ⟩ = Nℓ (y). x ℓ ℓ ℓ b ℓ! ℓ b ℓ bℓ ℓ≥0

ℓ=0

ℓ=0

Also, ⟨

⟩ ⟨ ⟩ ) r ( i aj ∏ t (1 + t/b) − 1 n −t y n ˆ Ci (y) | x ˆ Cn (y) = N N = e (1 + t/b) | x i! (1 + t/b)aj log(1 + t/b) j=1 i≥0 ⟨ r ( ⟩ ) ∏ (1 + t/b)aj − 1 −t y n = | e (1 + t/b) x (1 + t/b)aj log(1 + t/b) j=1 ⟩ ⟨ r ( ) ∑ ∏ tℓ n (1 + t/b)aj − 1 = | cℓ (y; b) x (1 + t/b)aj log(1 + t/b) ℓ! j=1 ℓ≥0 ⟨ ⟩ ( ) n n ( ) i ∑ ∑ ∑ n 1 n ˆi t | xn−ℓ = ˆn−ℓ . = cℓ (y; b) cℓ (y; b)N N ℓ bi i! bn−ℓ ℓ ∑

ℓ=0

i≥0

ℓ=0

Thus, we can state the following identities.

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DAE SAN KIM, TAEKYUN KIM, HYUCK IN KWON, AND TOUFIK MANSOUR

Theorem 1. For all n ≥ 0, ˆ Cn (x) = N

n ( ) ∑ n (−1)n−ℓ

bℓ

ℓ

ℓ=0

ˆℓ (x) = N

n ( ) ˆ ∑ n Nn−ℓ ℓ=0

ℓ

bn−ℓ

cℓ (x; b).

By (1.9), we have t

eb(e

(2.1)

−1)

) r ( ∏ teaj t ˆ Cn (x) ∼ (1, b(et − 1)). N aj t − 1 e j=1

∑n It is well known that (x)n = x(x − 1) · · · (x − n + 1) = k=0 S1 (n, k)xn , where S1 (n, k) is the Stirling number of the first kind, which implies that b−n (x)n ∼ (1, b(et − 1)). Thus ) ) n r ( aj t r ( aj t ∏ ∏ e −1 e − 1 ∑ (−b)ℓ t t ℓ −n −n ˆ N Cn (x) = b (−b(e − 1)) (x)n = b (e − 1)ℓ (x)n . aj t aj t te te ℓ! j=1 j=1 ℓ=0

By the facts that (x)n ∼ (1, e − 1) and (e − 1) (x)n = (n)ℓ (x)n−ℓ , we obtain ) n ( ) r ( aj t ∏ e −1 ∑ −n ℓ n ˆ (−b) (x)n−ℓ N Cn (x) = b teaj t ℓ j=1 t

= b−n

t

r ∏

(

j=1

= b−n

(2.2)

ℓ

ℓ=0

( ) n−ℓ ) n ∑ e −1 ∑ ℓ n (−b) S1 (n − ℓ, m)xm a t j ℓ m=0 te aj t

ℓ=0

( ) ) r ( aj t ∏ n e −1 (−b)ℓ S1 (n − ℓ, m) xm aj t ℓ te m=0 j=1

n n−ℓ ∑ ∑ ℓ=0

( ) m ∑ n tk S1 (n − ℓ, m) Fˆk xm k! ℓ ℓ=0 m=0 k=0 ( ) ( ) n n−ℓ m ∑ ∑ ∑ m ˆ m−k −n ℓ n =b (−b) S1 (n − ℓ, m) Fk x ℓ k ℓ=0 m=0 k=0   ( )( ) n−j n ∑ n−ℓ ∑ ∑ n m  (−b)ℓ S1 (n − ℓ, m)Fˆm−j  xj , = b−n ℓ j m=j j=0 = b−n

n n−ℓ ∑ ∑

(−b)ℓ

ℓ=0

see (2.3), for the definition Fˆk . Clearly, from (2.2), we have ( ) n n−ℓ ∑ ∑ −n ℓ n ˆ N Cn (x) = b (−b) S1 (n − ℓ, m)Fˆm (x). ℓ m=0 ℓ=0

Here, we introduced the Appell numbers Fˆn = Fˆn (a1 , . . . , ar ) by   ) r ( aj t ∏ te Fˆn (a1 , . . . , ar ) ∼  (2.3) , t , aj t − 1 e j=1 so that

) r ( aj t ∏ e −1 j=1

teaj t

ext =

∑ n≥0

n

t Fˆn (x|a1 , . . . , ar ) = n!

840

∑ n≥0

n

t Fˆn (x) . n!

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∏r ( a j t ) ∑ n = n≥0 Fˆn tn! . It is easy to When x = 0, Fˆn (0) will be simply denoted by Fˆn , so that j=1 eteaj−1 t ( ( ) ) ∑n ( n ) ˆ n−m ∏r t Fm x . In [13], the Appell numbers Fn ∼ see that Fˆn (x) = m=0 m a t j=1 e j −1 , t , so that ∏r ( eaj t −1 ) ∑ n = n≥0 Fn tn! . It was shown that j=1 t Fn (x) =

n ∑

∑

i=0 ℓ1 +···+ℓr =i

( )( )∏ r i! n i+r ℓ +1 a j xn−i . (i + r)! i ℓ1 , . . . , ℓr j=1 j

Also, from ) ) r ( aj t r ( aj t ∑ ∏ ∏ tn e − 1 xt e − 1 xt − rj=1 aj t ˆ Fn (x) = e =e e n! j=1 teaj t t j=1 n≥0   ∑ n ( ) r n ∑ ∑ ∑ ∑ (− rj=1 aj )ℓ tℓ ∑ n tn t  (− aj )ℓ Fn−ℓ (x) , = Fn (x) = ℓ! n! ℓ n! j=1 ∑

n≥0

n≥0

ℓ≥0

we see that Fˆn (x) =

ℓ=0

n ( ) r ∑ ∑ n (− aj )ℓ Fn−ℓ (x). ℓ j=1 ℓ=0

Now we invoke the conjugation formula (1.7) with g(f¯(t))−1 = e−t

∏r

(

j=1

(1+t/b)aj −1 (1+t/b)aj log(1+t/b)

) and

f¯(t) = log(1 + t/b): ⟨ ⟨g(f¯(t))−1 f¯(t)j |xn ⟩ =

e

−t

j=1

⟨ e

=

r ( ∏

−t

r ( ∏

j=1

⟨

(1 + t/b)aj − 1 (1 + t/b)aj log(1 + t/b) (1 + t/b)aj − 1 (1 + t/b)aj log(1 + t/b)

⟩

) (log(1 + t/b)) | x j

n

⟩

) | (log(1 + t/b)) x

j n

⟩ ) ∑ (1 + t/b)aj − 1 tℓ n = e | j!S1 (ℓ, j) ℓ x (1 + t/b)aj log(1 + t/b) b ℓ! j=1 ℓ≥j  ⟨ ⟩ ( ) ) n r ( aj ∑ ∏ n (1 + t/b) − 1 b−ℓ S1 (ℓ, j) = j! e−t | xn−ℓ  aj log(1 + t/b) ℓ (1 + t/b) j=1 −t

r ( ∏

ℓ=j

= j!

n ∑

−ℓ

b

ℓ=j

( ) n ˆ S1 (ℓ, j) N Cn−ℓ . ℓ

So we can state the following result. Theorem 2. For all n ≥ 0,   ( ) n−j n ∑ ∑ n ˆ Cn (x) = ˆ Cℓ  xj .  N bℓ−n S1 (ℓ, j) N ℓ j=0 ℓ=j

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By the fact that xn ∼ (1, t) and by (2.1), we obtain ) ( )n r ( ∏ ∑ (n) tk t teaj t b(et −1) n−1 −n ˆ N C (x) = x xn−1 e x = b x Bk n aj t − 1 t − 1) e b(e k! j=1 = b−n x

n−1 ∑( k=0

where

) n−1 ∑ (n − 1) (n) n−1 (n) n−1−k −n Bk x =b Bk xn−k . k k

k≥0

k=0

(n) Bk

is the k-th Bernoulli polynomial of order n. Thus,   ( ) ) n r ( aj t ∑ ∏ t n − 1 e − 1 (n) ˆ Cn (x) = b−n  N Bk e−b(e −1) xn−k  aj t k te j=1 k=0   ( ) ) n−k r ( aj t n ℓ ∏ ∑ ∑ e − 1 n − 1 (−b) (n)  Bk (e− 1)ℓ xn−k  = b−n aj t k te ℓ! j=1 k=0 ℓ=0   ) ( ) n−k r ( aj t n n−k ℓ ∏ ∑ ∑ ∑ n − 1 e − 1 ℓ!(−b) (n)  Bk = b−n S2 (j, ℓ)tj xn−k  aj t k te j! j=1 k=0 ℓ=0 j=ℓ   ) n n−k r ( aj t ∑ ∑ n−k ∑ (n − 1)(n − k ) ∏ e − 1 (n)  = b−n (−b)ℓ S2 (j, ℓ)Bk xn−k−j  aj t k j te j=1 k=0 ℓ=0 j=ℓ

= b−n

] n n−k ∑ ∑ n−k−ℓ ∑ [(n − 1)(n − k ) (n) (−b)ℓ S2 (n − k − j, ℓ)Bk Fˆj (x|a1 , . . . , ar ) k j j=0

k=0 ℓ=0

=b

−n

)( )( ) ] j [( n n−k ∑ ∑ n−k−ℓ ∑ ∑ n−1 n−k j (n) ˆ j−i ℓ (−b) S2 (n − k − j, ℓ)Bk Fi x , k j i j=0 i=0

k=0 ℓ=0

which leads the following result. ˆ Cn (x) is given by Theorem 3. For all n ≥ 0, the polynomial N   n n−m ∑ ∑ n−m−k ∑ n−k−ℓ ∑ (n − 1)(n − k )( j ) (n)  b−n (−b)ℓ S2 (n − k − j, ℓ)Bk Fˆj−m  xm . k j m m=0 j=m k=0

By (1.9), we have

ℓ=0

∏r

(

j=1

ˆ Cn (x + y) = N

teaj t eaj t −1

)

3. Recurrence relations

n ( ) ∑ n j=0

j

t

eb(e

−1)

ˆ Cn (x) = b−n (x)n ∼ (1, b(et − 1)). Thus, N

ˆ Cj (x)bj−n (y)n−j = b−n N

n ( ) ∑ n ˆ N Cj (x)bj (y)n−j . j j=0

ˆ Cn (x) = nN ˆ Cn−1 (x), that is, we have For sn (x) ∼ (g(t), f (t)), f (t)sn (x) = nsn−1 (x). Here b(et − 1)N the following recurrence relation. Theorem 4. We have ˆ Cn (x + 1|a1 , . . . , ar ; b) − N ˆ Cn (x|a1 , . . . , ar ; b) = n N ˆ Cn−1 (x|a1 , . . . , ar ; b). N b

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For sn (x) ∼ (g(t), f (t)), sn+1 (x) = (x − g ′ (t)/g(t)) f ′1(t) sn (x). Thus, ( ) ′ ˆ Cn+1 (x) = b−1 xN ˆ Cn (x − 1) − e−t g (t) N ˆ Cn (x) . N g(t) Observe that  g ′ (t)/g(t) = (log g(t))′ = b(et − 1) + r log t + t

r ∑

aj −

j=1

r ∑

′ log(eaj t − 1)

j=1

r r r ∑ ∑ r− a j ea j t r ∑ t aj − = be + a = be + + j a t t j=1 e j −1 j=1 j=1

∑r

t

aj teaj t j=1 eaj t −1

t

,

where r ∑ aj teaj t r− = eaj t − 1 j=1

∑r j=1

∏

i̸=j (e

ai t

∏r

− 1)(eaj t − 1 − aj teaj t )

i=1 (e

ai t

∑r

− 1)

t+1 + ··· 1 a1 · · · ar j=1 aj t =− =− r 2 a1 a2 · · · ar t + · · ·

∑r j=1

2

aj

t + ···

has order at least 1. Now, by Theorem 2, we obtain

(3.1)

 ∑r r ∑ r − j=1 g ′ (t) ˆ t  N Cn (x) = be + aj g(t) t j=1 ˆ Cn (x + 1) + = bN

r ∑

ˆ Cn (x) + aj N

j=1

aj teaj t eaj t −1

 ˆ Cn (x) N

n n−k ∑ ∑ (n) r− ˆ Cℓ bℓ−n S1 (n − ℓ, k)N ℓ

k=0 ℓ=0

∑r

aj teaj t j=1 eaj t −1

t

xk .

But r−

∑r

aj teaj t j=1 eaj t −1

t



 ) r ( r aj t ∑ xk+1 1 ∑ −eaj t a te j k   x = r− = 1 − −aj t xk+1 aj t − 1 e k + 1 k + 1 e − 1 j=1 j=1   r ∑ (−aj )m Bm tm 1 ∑  xk+1 = 1− k + 1 j=1 m! m≥0 ( ) r k+1 ∑ (k + 1) 1 ∑ k+1 m k+1−m = x − (−aj ) Bm x k + 1 j=1 m m=0 =

) r k+1 ( −1 ∑ ∑ k + 1 (−aj )m Bm xk+1−m . k + 1 j=1 m=1 m

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Thus, by (3.1), we derive g ′ (t) ˆ N Cn (x) g(t) r ∑

ˆ Cn (x + 1) + = bN

ˆ Cn (x) aj N

j=1

−

( )( ) n k+1 1 ˆ Cℓ xk+1−m (−aj )m bℓ−n S1 (n − ℓ, k)Bm N k + 1 ℓ m j=1 m=1

n n−k r k+1 ∑ ∑∑ ∑ k=0 ℓ=0

r ∑

ˆ Cn (x + 1) + = bN

ˆ Cn (x) aj N

j=1

−

( )( ) n k+1 1 ˆ Cℓ xm (−aj )k+1−m bℓ−n S1 (n − ℓ, k)Bk+1−m N k + 1 ℓ m j=1 m=0

n n−k r ∑ k ∑ ∑∑ k=0 ℓ=0

r ∑

ˆ Cn (x + 1) + = bN

ˆ Cn (x) aj N

j=1

−

n ∑

n n−k r ∑ ∑∑

m=0 k=m ℓ=0 j=1

( )( ) n k+1 1 ˆ Cℓ xm . (−aj )k+1−m bℓ−n S1 (n − ℓ, k)Bk+1−m N k+1 ℓ m

Altogether, we obtain the following result. Theorem 5. For n ≥ 0, ˆ Cn+1 (x) + N ˆ Cn (x) = b−1 xN ˆ Cn (x − 1) − b−1 N

r ∑

ˆ Cn (x − 1) aj N

j=1

−

n ∑

n n−k r ∑ ∑∑

m=0 k=m ℓ=0 j=1

( )( ) (−aj )k+1−m bℓ−n−1 n k + 1 ˆ Cℓ (x − 1)m . S1 (n − ℓ, k)Bk+1−m N k+1 ℓ m

Another recurrence can be obtained as follows. For sn (x) ∼ (g(t), f (t)), we have that ∑n−1 (n) ¯ n−ℓ ⟩sℓ (x). By the definitions, we have ℓ=0 ℓ ⟨f (t) | x ⟨f¯(t) | xn−ℓ ⟩ = ⟨log(1 + t/b) | xn−ℓ ⟩ =

d dx sn (x)

=

∑ (−1)m−1 (−1)n−1−ℓ (n − 1 − ℓ)! ⟨tm |xn−ℓ ⟩ = , m mb bn−ℓ

m≥1

which implies

d ˆ dx N Cn (x)

=

∑n−1 (n) (−1)n−1−ℓ (n−1−ℓ)! ˆ N Cℓ (x), and hence the following result. ℓ=0 ℓ bn−ℓ

Theorem 6. For all n ≥ 0, ∑ (−1)n−1−ℓ d ˆ ˆ Cℓ (x). N Cn (x) = n! N dx ℓ!(n − ℓ)bn−ℓ n−1

ℓ=0

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Another recurrence relation can be derived as follows. By definitions, we have ⟨

(3.2)

(3.3)

(3.4)

(3.5)

⟩ ⟨ ⟩ ) r ( i aj ∏ t (1 + t/b) − 1 n −t y n ˆ Cn (y) = ˆ Ci (y) | x N N = e (1 + t/b) | x i! (1 + t/b)aj log(1 + t/b) j=1 i≥0 ⟨ ⟩ ) r ( d −t ∏ (1 + t/b)aj − 1 y n−1 = e (1 + t/b) | x dt (1 + t/b)aj log(1 + t/b) j=1 ⟨ ⟩ ) r ( ∏ (1 + t/b)aj − 1 −t d y n−1 + e (1 + t/b) | x dt j=1 (1 + t/b)aj log(1 + t/b) ⟩ ⟨ ) r ( ∏ d (1 + t/b)aj − 1 y n−1 −t (1 + t/b) | x . + e (1 + t/b)aj log(1 + t/b) dt j=1 ∑

The first term in (3.2), namely (3.3), is ⟨ (3.6)

) r ( d −t ∏ (1 + t/b)aj − 1 e (1 + t/b)y | xn−1 aj log(1 + t/b) dt (1 + t/b) j=1

⟩ ˆ Cn−1 (y). = −N

The third term in (3.2), namely (3.5), is (3.7)

y b

Define L =

⟨ −t

e

r ( ∏ j=1

∑r j=1

(

(1 + t/b)aj − 1 (1 + t/b)aj log(1 + t/b)

aj t/b(1+t/b)aj (1+t/b)aj −1

−

t/b log(1+t/b)

)

d (1 + t/b)y−1 | xn−1 dt

⟩ =

y ˆ N Cn−1 (y − 1). b

) . It is easy to see that

) ∏ ) r ( r ( (1 + t/b)aj − 1 d ∏ (1 + t/b)aj − 1 = dt j=1 (1 + t/b)aj log(1 + t/b) (1 + t/b)aj log(1 + t/b) j=1

(

) ∑r L j=1 aj − , t(1 + t/b) b(1 + t/b)

with L has order at least one. Thus the second term in (3.2), namely (3.4), is ⟨

⟩ ) r ( d ∏ (1 + t/b)aj − 1 y n−1 e (1 + t/b) | x dt j=1 (1 + t/b)aj log(1 + t/b) ⟩ ⟨ ∑r ) r ( ∏ 1 (1 + t/b)aj − 1 j=1 aj ˆ −t y−1 n =− N Cn−1 (y − 1) + (1 + t/b) | Lx e b n (1 + t/b)aj log(1 + t/b) j=1 ∑r j=1 aj ˆ =− N Cn−1 (y − 1) b ⟨ ⟩ ) r r ( aj ai ∏ 1∑ (1 + t/b) log(1 + t/b) (1 + t/b) − 1 t/b + aj e−t (1 + t/b)y−1 | xn n j=1 (1 + t/b)aj − 1 (1 + t/b)ai log(1 + t/b) log(1 + t/b) i=1 ⟨ ⟩ ) r r ( ∏ r∑ (1 + t/b)ai − 1 t/b y−1 n −t − aj e (1 + t/b) | x . n j=1 (1 + t/b)ai log(1 + t/b) log(1 + t/b) i=1 −t

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By the fact that

t/b log(1+t/b)

=

⟨

∑

tℓ ℓ≥0 cℓ bℓ ℓ! ,

we get that

⟩ ) r ( d ∏ (1 + t/b)aj − 1 y n−1 e (1 + t/b) | x dt j=1 (1 + t/b)aj log(1 + t/b) r r n ( ) 1∑ ˆ 1 ∑ ∑ n cℓ ˆ =− N Cn−ℓ;j (y − 1) aj N Cn−1 (y − 1) + aj b j=1 n j=1 ℓ bℓ ℓ=0 n ( ) r ∑ n cℓ ˆ − N Cn−ℓ (y − 1). n ℓ bℓ −t

(3.8)

ℓ=0

By (3.2), (3.6)-(3.8), we obtain the following result. Theorem 7. For n ≥ 1,

∑r x − j=1 aj ˆ ˆ ˆ Cn−1 (x − 1) N Cn (x) = −N Cn−1 (x) + N b   n ( ) r 1 ∑ n cℓ ∑ ˆ r ˆ + aj N Cn−ℓ;j (x − 1) − N Cn−ℓ (x − 1) . ℓ bℓ j=1 n n ℓ=0

Now we compute the following expression in two different ways: ⟩ ⟨ ) r ( aj ∏ (1 + t/b) − 1 (3.9) A = e−t (log(1 + t/b))m | xn . aj log(1 + t/b) (1 + t/b) j=1 On the one hand, it is equal to ⟨ r ( ∏ A = e−t

(3.10)

⟩ ) (1 + t/b)aj − 1 m n | (log(1 + t/b)) x (1 + t/b)aj log(1 + t/b) j=1 ⟨ ⟩ ) ∑ r ( ∏ (1 + t/b)aj − 1 tℓ n −t = m! e S1 (ℓ, m) ℓ x | (1 + t/b)aj log(1 + t/b) b ℓ! j=1 ℓ≥m ⟨ ⟩ ( ) ( ) r n n ∑ ∏ (1 + t/b)aj − 1 −t n−ℓ ℓ = m! S1 (ℓ, m) e |x bℓ (1 + t/b)aj log(1 + t/b) j=1 ℓ=m n (n) ∑ ℓ ˆ Cn−ℓ . = m! S1 (ℓ, m)N bℓ ℓ=m

On the other hand, it is equal to ⟨ ⟩ ) r ( (1 + t/b)aj − 1 d −t ∏ e (3.11) A= (log(1 + t/b))m | xn aj log(1 + t/b) dt (1 + t/b) j=1 ⟨ ⟩ ) r ( aj ∏ d (1 + t/b) − 1 (3.12) + e−t (log(1 + t/b))m | xn dt j=1 (1 + t/b)aj log(1 + t/b) ⟨ ⟩ ) r ( ∏ (1 + t/b)aj − 1 d −t m n (3.13) + e (log(1 + t/b)) | x . (1 + t/b)aj log(1 + t/b) dt j=1

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By (3.9), the right side of (3.11) is equal to −m!

(3.14)

n−1 ∑ ℓ=m

(n−1) ℓ

bℓ

ˆ Cn−1−ℓ . S1 (ℓ, m)N

Also, by (3.9), the expression in (3.13) is equal to ⟨ ⟩ ) r ( ∏ m (1 + t/b)aj − 1 −t −1 m−1 n−1 e (log(1 + t/b)) | (log(1 + t/b)) x b (1 + t/b)aj log(1 + t/b) j=1 ⟨ ⟩ ) r ( n−1 (n−1) ∏ m! ∑ (1 + t/b)aj − 1 −t −1 n−1−ℓ ℓ = S1 (ℓ, m − 1) e (1 + t/b) | x b bℓ (1 + t/b)aj log(1 + t/b) j=1 ℓ=m−1 n−1 (n−1) m! ∑ ℓ ˆ Cn−1−ℓ (−1). (3.15) = S1 (ℓ, m − 1)N b bℓ ℓ=m−1

The term in (3.12) is equal to ⟨ ⟩ ) r ( ∏ 1 (1 + t/b)aj − 1 −t −1 m n e (1 + t/b) L | (log(1 + t/b)) x n (1 + t/b)aj log(1 + t/b) j=1 ⟨ ⟩ ∑r ) r ( ∏ (1 + t/b)aj − 1 j=1 aj −t −1 m n−1 − e , (1 + t/b) | (log(1 + t/b)) x b (1 + t/b)aj log(1 + t/b) j=1 ∑ and by using the fact (log(1 + t/b))m = m! i≥m aj ∑r log(1+t/b) L′ = −r + j=1 aj (1+t/b) , we obtain (1+t/b)aj −1

S1 (i.m)ti bi i!

and

t/b log(1+t/b)

=

∑

cℓ tℓ i≥0 bℓ ℓ!

and defining

⟩ ⟨ ) n (n) r ( ∏ (1 + t/b)aj − 1 m! ∑ i n−i −t −1 S1 (i, m) e (1 + t/b) | Lx n i=m bi (1 + t/b)aj log(1 + t/b) j=1 ⟩ ⟨ ∑r ( ) ) r ( ∑ n−1 ∏ m! j=1 aj n−1 (1 + t/b)aj − 1 −t −1 n−1−i i − S1 (i, m) e (1 + t/b) | x b bi (1 + t/b)aj log(1 + t/b) i=m j=1 ⟨ ⟩ ) n (n) r ( ∏ m! ∑ i (1 + t/b)aj − 1 t/b −t −1 ′ n−i = S1 (i, m) e (1 + t/b) L | x n i=m bi (1 + t/b)aj log(1 + t/b) log(1 + t/b) j=1 ∑r ( ) ∑ n−1 m! j=1 aj n−1 i ˆ Cn−1−i (−1) S1 (i, m)N − b bi i=m   n n−i (n)(n−i) r ∑ m! ∑ ∑ i ℓ ˆ Cn−i−ℓ (−1) ˆ Cn−i−ℓ;j (−1) − rN = cℓ S1 (i, m)  aj N n i=m bℓ+i j=1 ℓ=0 ∑r (n−1) n−1 m! j=1 aj ∑ i ˆ Cn−1−i (−1). (3.16) − S1 (i, m)N i b b i=m Hence, by (3.10), (3.11)-(3.13) and (3.14)-(3.16), we obtain the following result.

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Theorem 8. For all 1 ≤ m ≤ n − 1, ( ) ( ) n−1 n−1 n (n) ∑ ∑ n−1 ∑ n−1 ℓ ℓ ℓ ˆ ˆ ˆ Cn−1−ℓ (−1) S1 (ℓ, m)N Cn−ℓ = − S1 (ℓ, m)N Cn−1−ℓ + S1 (ℓ, m − 1)N bℓ bℓ b1+ℓ ℓ=m ℓ=m−1 ℓ=m   (n)(n−i) r n n−i ∑ c 1 ∑∑ i ℓ ℓ ˆ Cn−i−ℓ;j (−1) − rN ˆ Cn−i−ℓ (−1) + S1 (i, m)  aj N n i=m bℓ+i j=1 ℓ=0

−

r ∑ j=1

aj

n−1 ∑ i=m

(n−1) i

b1+i

ˆ Cn−1−i (−1). S1 (i, m)N

4. Connections with families of polynomials ∑ n −m ˆ Cn (x) = Now let N (x)m . Then by (1.8), (1.9) and b−n (x)n ∼ (1, b(et − 1)), we m=0 cn,m b obtain ⟨ ⟩ 1 1 cn,m = (b(elog(1+t/b) − 1))m | xn m! eb(elog(1+t/b) −1) ∏r log(1+t/b)eaj log(1+t/b) j=1 eaj log(1+t/b) −1 ⟨ ⟩ r ∏ 1 (1 + t/b)aj − 1 −t m n = e |t x m! (1 + t/b)aj log(1 + t/b) j=1 ⟩ ( ) ( )⟨ r ∏ (1 + t/b)aj − 1 n ˆ n −t n−m N Cn−m , e = = |x a j m m (1 + t/b) log(1 + t/b) j=1 which implies the following result. Theorem 9. For all n ≥ 0, ˆ Cn (x) = N

n ( ) ∑ n −m ˆ b N Cn−m (x)m . m m=0

(( t )s ) (s) (s) e −λ ˆ Cn (x) = ∑n cn,m Hm (x|λ) ∼ (x|λ). Then by (1.8), (1.9) and H , t , we Now, let N n m=0 1−λ obtain ⟨ ⟩ ( log(1+t/b) )s 1 1 e −λ cn,m = (log(1 + t/b))m | xn m! eb(elog(1+t/b) −1) ∏r log(1+t/b)eaj log(1+t/b) 1−λ j=1 eaj log(1+t/b) −1 ⟨ ⟩ r ∏ 1 (1 + t/b)aj − 1 −t m s n = e (log(1 + t/b)) | (1 − λ + +t/b) x m!(1 − λ)s (1 + t/b)aj log(1 + t/b) j=1 ⟨ ⟩ () n−m r ∑ sj (1 − λ)s−j ∏ 1 (1 + t/b)aj − 1 −t m n−j = (n)j e (log(1 + t/b)) | x . m!(1 − λ)s j=0 bj (1 + t/b)aj log(1 + t/b) j=1 Thus, by (3.10), we can state the following result. Theorem 10. For all n ≥ 0,   n n−m ∑ ∑ n−m−j ∑ (s)(n − j ) (s) ˆ Cn (x) = ˆ Cℓ  Hm  N (1 − λ)−j bℓ−n (n)j S1 (n − j − ℓ, m)N (x|λ). j ℓ m=0 j=0 ℓ=0

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BARNES-TYPE NARUMI AND POISSON-CHARLIER MIXED-TYPE POLYNOMIALS (s)

By similar arguments, using (1.8), (1.9) and Bn ∼

((

et −1 t

)s

13

) , t , we obtain the following identity.

Theorem 11. For all n ≥ 0,

ˆ Cn (x) = N

n ∑ m=0

 

n−m ∑ n−m−j ∑ ( j=0

ℓ=0

n j

)(

 ) n − j ℓ−n (s) ˆ (s) b (n)j S1 (n − j − ℓ, m)ci N Cℓ  Bm (x). ℓ

References [1] Agapito Ruiz, J., Riordan arrays from an umbral symbolic viewpoint, Bol. Soc. Port. Mat., Special Issue (2012), 5-8 [2] Biedenharn, L.C., Gustafson, R.A., Lohe, M.A., Louck, J.D. and Milne, S.C., Special functions and group theory in theoretical physics. In Special functions: group theoretical aspects and applications, Math. Appl. (1984) 129–162, Reidel, Dordrecht. [3] Biedenharn, L.C., Gustafson, R.A. and Milne, S.C., An umbral calculus for polynomials characterizing U (n) tensor products, Adv. Math. 51 (1984) 36-90. [4] Blasiak, P., Combinatorics of boson normal ordering and some applications, Concepts of Phyiscs 1 (2004) 177–278. [5] Blasiak, P., Dattoli, G., Horzela, A. and Penson, K.A., Representations of monomiality principle with Sheffer-type polynomials and boson normal ordering, Phys. Lett. A 352 (2006) 7–12. [6] Dattoli, G., Levi, D. and Winternitz, P., Heisenberg algebra, umbral calculus and orthogonal polynomials, J. Math. Phys. 49 (2008), no. 5, 053509. [7] Dere, R. and Simsek, Y., Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. 22 (2012), no. 3, 433–438. [8] Di Bucchianico, A. and Loeb, D., A selected survey of umbral calculus, Electronic J. Combina. 2 (2000) #DS3. [9] Gzyl, H., Hamilton Flows and Evolution Semigroups, Research Notes in Mathematics 239, 1990. [10] Kim, D.S. and Kim, T., Applications of umbral calculus associated with p-adic invarient integrals on Zp , Abstract and Appl. Ana. 2012 (2012) Article ID 865721. [11] Kim, D.S. and Kim, T., Some identities of Frobenius-Euler polynomials arising from umbral calculus, Adv. in Difference Equat. 2012 (2012) 196. [12] Kim, D.S., Kim, T., Lee, S.-H., Rim, S.-H., Frobenius-Euler polynomials and umbral calculus in the p-adic case, Adv. Difference Equ. 2012 (2012) 222. [13] Kim, T., Identities involving Laguerre polynomials derived from umbral calculus, Russ. J. Math. Phys. 21 (2014), no. 1, 36–45. [14] Kim, T., Kim, D.S., Mansour, T., Rim, S.-H. and Schork, M., Umbral calculus and Sheffer sequences of polynomials, J. Math. Phys. 54 (2013), no. 8, 083504. [15] Kwa´sniewski, A.K., q-quantum plane, ψ(q)-umbral calculus, and all that, Quantum groups and integrable systems (Prague, 2001), Czechoslovak J. Phys. 51 (2001), no. 12, 1368–1373. [16] Levi, D., Tempesta, P. and Winternitz, P., Umbral calculus, difference equations and the discrete Schr¨ odinger equation, J. Math. Phys. 45 (2004), no. 11, 4077–4105. [17] Morikawa, H., On differential polynomials I, Nagoya Math. J. 148 (1997) 39–72. [18] Morikawa, H., On differential polynomials II, Nagoya Math. J. 148 (1997) 73-112. [19] Narumi, S., On a power series having only a finite number of algebraico logarithmic singularities on its circle of convergence, Tohoku Math. J. 30 (1929) 185–201. [20] Roman, S., More on the umbral calculus, with emphasis on the q-umbral calculus, J. Math. Anal. Appl. 107 (1985) 222–254. [21] Roman, S., The umbral calculus, Dover Publ. Inc. New York, 2005. [22] Wilson, B.G. and Rogers, F.G., Umbral calculus and the theory of multispecies nonideal gases, Phys. A 139 (1986) 359–386. [23] Zachos, C.K., Umbral deformations on discrete space-time, Internat. J. Modern Phys. A 23 (2008), no. 13, 2005– 2014.

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14

DAE SAN KIM, TAEKYUN KIM, HYUCK IN KWON, AND TOUFIK MANSOUR

Department of Mathematics, Sogang University, Seoul 121-742, S. Korea E-mail address: [email protected] Department of Mathematics, Kwangwoon University, Seoul, S. Korea E-mail address: [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, S. Korea E-mail address: [email protected] University of Haifa, Department of Mathematics, 3498838 Haifa, Israel E-mail address: [email protected]

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Local control and approximation properties of a C 2 rational interpolating curve Qinghua Sun∗ School of Mathematics, Shandong University, Jinan, 250100, China

Abstract A new piecewise smooth rational quintic interpolation with a shape parameter is presented based on the function values only. The interpolation has a simple and explicit mathematical representation, and is of C 2 continuity without solving a system of consistency equations for the derivative values at the knots. In order to meet the needs of practical design, a local control method is employed to control the shape of curves. The advantage of the method is that it can be applied to modify the local shape of an interpolating curve by selecting suitable parameter. Also when f (x) ∈ C 2 [a, b], the error estimation formula of the interpolator is obtained. Keywords: C 2 -spline, rational interpolation, local control, error estimates

1

Introduction

In many problems of industrial design and manufacturing, it is usually needed to generate a smooth function. Spline interpolation is a useful and powerful tool to settle the problem, such as the polynomial spline, the rational spline and others [3, 5, 7, 11, 17, 20, 21]. In recent years, the rational spline and its application to shape preserving and shape control have received attention. Since the parameters in the interpolation function are selective according to the control need, the constrained control of the shape becomes possible. In References [4, 6, 13, 14, 15, 16, 18], positivity preserving, monotonic preserving and convexity preserving of the interpolating curves were discussed. In References [8, 9], the region control and convexity control of the interpolating curves were studied. In fact, some practical designs in CAGD need only local control, but there are reports on only a few methods for local control. In Reference [12], a local control method of interval tension using weighted splines was given. In References [1, 2, 10], the local control methods of the rational cubic spline were discussed. For many applications, C 1 smoothness is generally sufficient. However, curvature continuity sometimes is needed and this leads to the need for C 2 or higher order of continuity. These rational interpolations above are also C 2 continuous by solving a system of consistency equations for the derivative values at the knots. In this paper, we will describe a piecewise rational spline function which is of C 2 continuity without solving a systems of consistency equations. The interpolant contains a free parameter αi , and can be used to control the local shape of interpolating curve. ∗

Corresponding author: [email protected]

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SUN: LOCAL CONTROL AND APPROXIMATION PROPERTIES OF A C 2 CURVE This paper is arranged as follows. In Section 2, a piecewise expression of the C 2 rational quintic interpolation with the parameter is described. In Section 3, a local control method of the interpolating curve is developed, including the value control and the derivative control of interpolation function at a point, and the numerical examples show the performance of the method. Section 4 is about the bases and error estimates, the interpolant can be represented by using basis functions clearly, and the error estimate formula is derived when the interpolated function f (x) ∈ C 2 [a, b].

2

Piecewise rational interpolation

Let {(xi , fi ); i = 1, · · · , n} be a given set of data points, a = x1 < x2 < · · · < xn = b be the knot spacing, and di be chosen derivative values at the knots xi . We denote hi = xi+1 −xi , t = (x−xi )/hi for i = 1, 2, · · · , n − 1, and ∆i = fi+1hi−fi . Let αi be a positive parameter. For x ∈ [xi , xi+1 ], i = 1, 2, · · · , n − 1, we construct the interpolant P (x) =

(1 − t)3 fi + t(1 − t)2 (Wi + ϕ(t)) + t2 (1 − t)(Vi + ψ(t)) + t3 fi+1 , (1 − t)2 + t(1 − t)αi + t2

(1)

where both ϕ(t) and ψ(t) are the polynomial functions, and Wi = (αi + 1)fi + hi di , Vi = (αi + 1)fi+1 − hi di+1 . In the special case, when ϕ(t) ≡ 0, ψ(t) ≡ 0, the interpolant P (x) defined by (1) is a piecewise rational cubic interpolation, and which satisfies P (xi ) = fi , P (xi ) = fi+1 , P 0 (xi ) = di , P 0 (xi+1 ) = di+1 . In the follows, we consider the case that both ϕ(t) and ψ(t) are the quadratic polynomials. Let ϕ(t) = (t − t(1 − t)αi )(fi+1 − fi − hi di ), ψ(t) = ((1 − t) − t(1 − t)αi )(fi − fi+1 + hi di+1 ), the interpolation function P (x) defined by (1) is called a piecewise rational quintic interpolation. P (x) is C 1 continuous in interpolating interval [a, b] for any given di , i = 1, 2, · · · , n, and which satisfies P (xi ) = fi , P (xi ) = fi+1 , P 0 (xi ) = di , P 0 (xi+1 ) = di+1 . It is easy to test that when di = f 0 (xi ) and αi → +∞, the interpolation is the well-known standard cubic Hermite interpolation. That is to say, in this special case, the interpolant P (x) defined by (1) will give approximately a standard cubic Hermite interpolation. Further, if the partial derivative values di at the data sites are estimated using the arithmetic mean method: di =

1 (hi−1 ∆i + hi ∆i−1 ), i = 2, 3, · · · , n − 1, hi−1 + hi

852

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SUN: LOCAL CONTROL AND APPROXIMATION PROPERTIES OF A C 2 CURVE then P (x) is C 2 continuous in [a, b], and which satisfies P 00 (xi ) =

2 (∆i − ∆i−1 ), i = 2, 3, · · · , n − 1. hi−1 + hi

It is means that the interpolation function P (x) defined by (1) can be C 2 -continuous without solving a system of consistency equations for the derivative values an the knots. Remark. At the end knots x1 , xn , the derivative values are given as h1 (∆2 − ∆1 ), h1 + h2 hn−1 dn = ∆n−1 + (∆n−1 − ∆n−2 ). hn−1 + hn−2

d1 = ∆ 1 −

(3)

Example 1. The interpolated function f (x) = cos4 (2x/5), x ∈ [0, 1] with interpolating knots at xi = (i − 1)h, i = 1, 2, · · · , 6 and h = 0.2, the values of di at the knots xi are conducted by using (2) and (3). Let P (x) be the interpolation function defined by (1) over interval [0, 1], Q(x) be the interpolation function provided in [8] over interval [0, 0.8]. Since both of the interpolants P (x) and Q(x) only based on function values are local, we consider the interpolation over subinterval [0.2, 0.4] in order to compare their approximation. Let αi = 0.6, βi = 0.8. Figure 1 is the graphs of curves f (x) and P (x), it is obvious that the two curves almost coincide. Figure 2 shows the curve of the error f (t) − P (t). Figure 3 is the graphs of curves f (x) and Q(x), Figure 4 shows the curve of the error f (t) − Q(t). −5

10

x 10

0.99

0.98

5

0.97

0

0.96

0.95

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

−5 0.2

0.4

Figure 1: Graphs of curves f (x) and P (x).

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

Figure 2: Curve of error f (x) − P (x).

From Figure 1 to Figure 4, it is easy to see that the interpolator P (x) defined by (1) approximates f (x) better than that of the interpolator Q(x) defined in [8].

3

Constrained control of the interpolating curve

The shape of interpolating curves on an interpolating interval depends on the interpolating data. Generally speaking, when the interpolating data are given, the shape of the interpolating curve is fixed. For the interpolation function defined by (1), since there is a parameter αi , when the

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SUN: LOCAL CONTROL AND APPROXIMATION PROPERTIES OF A C 2 CURVE

−4

12 0.99

x 10

10

8 0.98

6

4 0.97

2

0 0.96

−2

−4 0.95

0.2

−6

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

−8 0.2

0.4

Figure 3: Graphs of curves f (x) and Q(x).

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

Figure 4: Curve of error f (x) − Q(x).

parameter vary, the interpolation function can be changed for the unchanged interpolating data. On the basis of this, the shape of interpolating curve can be modified by selecting suitable parameter. We consider a method to control the shape of the interpolating curve at a point. Two issues are discussed in the following. The first issue is the value control of interpolation function at a point. Let t be the local co-ordinate for a point x ∈ [xi , xi+1 ]. If the practical design requires the function value of the interpolation at the point x to be equal to a real number M , and M ∈ (min{fi , fi+1 }, max{fi , fi+1 }), this kind of control is called the value control of the interpolation at a point. Denoting M = λfi + (1 − λ)fi+1 , with λ ∈ [0, 1], then λ=

M − fi+1 . fi − fi+1

(4)

Thus, the equation P (x) = M can be written as m1 αi + m2 = 0,

(5)

where m1 = (t(1 − t)3 (1 + 2t) − t(1 − t)λ)(fi − fi+1 ) + t2 (1 − t)2 ((1 − t)di − tdi+1 )hi , m2 = ((1 − t)2 − (1 − 2t + 2t2 )λ)(fi − fi+1 ) + t(1 − t)((1 − t)2 di − t2 di+1 )hi . Obviously, if m1 m2 < 0, then there must exist positive parameter αi satisfying Eq. (5). Thus, we have the following function value control theorem. Theorem 1. Let P (x) be the interpolation function over [xi , xi+1 ] defined by (1), and let x be a point in [xi , xi+1 ], M is a real number, then the sufficient condition for existence of the positive parameter αi to satisfy P (x) = M is that m1 m2 < 0, and the positive parameter αi can be selected by Eq. (5). The second issue is the derivative control of interpolation function at a point. If the practical design requires the first-order derivative of the interpolation function at the point x to be equal to

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a real number N , how this can be achieved. Let P 0 (x) = N,

(6)

if there exist a parameter αi to satisfy this equation, αi is called the solution of the control equation (6). This control is called the derivative control of interpolation function. Eq. (6) can be written as Aαi2 + Bαi + C = 0,

(7)

where A = 6t3 (1 − t)2 (fi − fi+1 ) + t2 (1 − t)2 (N + (t(2 − 3t)di+1 − (1 − 4t + 3t2 )di )hi ), B = 2t2 (1 − t)2 (5 − 6t + 6t2 )(fi − fi+1 ) + 2t(1 − t)(1 − 2t + 2t2 )N +t(1 − t)[t(3 − 7t + 8t2 − 6t3 )di+1 − (1 − t)(2 − 9t + 10t2 − 6t3 )di ]hi , C = 2t(1 − t)(fi − fi+1 ) + (1 − 2t + 2t2 )2 N + t2 (3 − 8t + 8t2 − 4t3 )di+1 hi −(1 − t)2 (1 − 4t + 4t2 − 4t3 )di hi . Eq. (7) is a quadratic algebraic equation of αi , if it exists a positive root, then Eq. (6) holds. This can be stated in the following derivative control theorem. Theorem 2. Let P (x) be the interpolation function over [xi , xi+1 ] defined by (1), and let x be a point in [xi , xi+1 ], N is a real number, then the sufficient condition for existence of the positive parameter αi to satisfy P 0 (x) = N is that Eq. (7) has a positive root. Example 2. Given the interpolating data shown in Table 1, let P (t) be the interpolation function defined by (1) in the interpolating interval [0, 5]. This example will show how the constrained control of the interpolating curves can be achieved by selecting a suitable parameter for unchanged given data. Table 1: Set of the interpolating data. xi

0.0

1.0

2.0

3.0

4.0

5.0

f (xi )

2.0

5.0

3.0

3.0

6.0

4.0

Without loss of generality, we consider the interpolation in [1, 2] only, and all the figures below are in [1, 2]. The values of di at the knots xi are conducted by using (2). Let αi = 0.4, we denote the interpolation function by P1 (x), and it can be shown that P1 (1.2) = 19221 3875 = 4.96026. For the given interpolating data, if the design requires P (1.2) = 4.9, then λ = 0.95 by (4). Thus αi = 95 4 = 23.75 by (5). Denoting the interpolation function by P2 (t). Figure 5 shows the graphs of the interpolating curves P1 (t) and P2 (t), and it can be tested that P2 (1.2) = 4.9. 78251 = −1.08569. If the practical design requires the Also, it can be computed that P10 (1.2) = − 72075 derivative of the interpolation function at the point x = 1.2 to be equal to −1.4, then αi = 18 (−27 + √ 6349) = 6.58508 by Eq. (7). We denote the interpolation function by P3 (x). Figure 6 shows the graphs of the interpolation curves P1 (x) and P3 (t), and it can be tested that P30 (1.2) = −1.4.

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5.5

5.5

5

5

y=P1(x)

y=P1(x)

4.5

4.5

y=P (x)

4

4

2

y=P3(x)

3.5

3

3.5

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

3

2

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Figure 5: Graphs of the curves P1 (x) and P2 (x). Figure 6: Graphs of the curves P1 (x) and P3 (x).

4

The bases and error estimates of the interpolation

In what follows in this paper, we consider the case of equally spaced knots, namely, hi = hj for all i, j ∈ {1, 2, · · · , n − 1}. When di is given by (2) and (3), the interpolation function Pi (x) defined by (1) can be written as follows: P (x) = ω1 (t)fi−1 + ω2 (t)fi + ω3 (t)fi+1 + ω4 (t)fi+2 ,

(8)

where for x ∈ [x1 , x2 ], 1 1 ω1 (t) = 0, ω2 (t) = (2 − 3t + t2 ), ω3 (t) = t(2 − t), ω4 (t) = − t(1 − t); 2 2

(9)

for x ∈ [xi , xi+1 ], i = 2, 3, · · · , n − 2, t(1 − t)3 (1 + tαi ) , 2((1 − t)2 + t2 + t(1 − t)αi ) (1 − t)(2 − 2t + t3 + t(2 − 5t + 3t2 )αi ) ω2 (t) = , 2((1 − t)2 + t2 + t(1 − t)αi ) t(1 − t + 3t2 − t3 + t(1 + 3t − 7t2 + 3t3 )αi ) ω3 (t) = , 2((1 − t)2 + t2 + t(1 − t)αi ) t3 (1 − t)(1 + (1 − t)αi ) ω4 (t) = − ; 2((1 − t)2 + t2 + t(1 − t)αi )

ω1 (t) = −

(10)

and for x ∈ [xn−1 , xn ], 1 1 ω1 (t) = − t(1 − t), ω2 (t) = 1 − t2 , ω3 (t) = t(1 + t), ω4 (t) = 0. 2 2

(11)

The terms ωj (t) (j = 1, 2, 3, 4) are called the basis of the interpolant defined by (8), and which P satisfy 4k=1 ωk (t) = 1, and 4 X

|ωk (t)| = 1 + t − t2 .

(12)

k=1

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We denote M = max{|fk |, k = i − 1, i, i + 1, i + 2}. From (8) and (12), it is easy to see that the following theorem holds. Theorem 3. Let P (x) is the interpolation function defined by (8) in [xi , xi+1 ], then whatever the positive value of the parameter αi might be, the values of P (x) in [xi , xi+1 ] satisfy 5 |P (x)| ≤ M. 4 In order to proceed error estimates of the interpolation function P (x) defined by (8), without loss of generality, it is only necessary to consider the subinterval [xi , xi+1 ]. As f (x) ∈ C 2 [a, b], using the Peano-Kernel Theorem [19] gives the following: Z

f (x) − P (x) =

xi+2

xi−1

f (2) (τ )Rx [(x − τ )+ ] dτ.

(13)

(I) For x ∈ [xi , xi+1 ], i = 2, 3, · · · , n − 2, from (8) and (10), we have   pi (τ ),   

Rx [(x − τ )+ ] =

qi (τ ),

 ri (τ ),   

si (τ ),

xi−1 < τ < xi < x, xi < τ < x, x < τ < xi+1 , xi+1 < τ < xi+2 ,

where pi (τ ) = x − τ − ω2 (t)(xi − τ ) − ω3 (t)(xi+1 − τ ) − ω4 (t)(xi+2 − τ ), qi (τ ) = x − τ − ω3 (t)(xi+1 − τ ) − ω4 (t)(xi+2 − τ ), ri (τ ) = −ω3 (t)(xi+1 − τ ) − ω4 (t)(xi+2 − τ ), si (τ ) = −ω4 (t)(xi+2 − τ ). Now, we consider the properties of the kernel function Rx [(x − τ )+ ] of the variable τ in the interval [xi−1 , xi+2 ]. For all τ ∈ [xi+1 , xi+2 ], it is easy to see that si (τ ) =

t3 (1 − t)(1 + (1 − t)αi )(xi+2 − τ ) ≥ 0. 2((1 − t)2 + t2 + t(1 − t)αi )

For pi (τ ) in [xi−1 , xi ], it is easy to test that pi (xi−1 ) = 0, t(1 − t)3 (1 + tαi )hi ≥ 0. pi (xi ) = 2 − 4t + 4t2 + 2t(1 − t)αi Thus, pi (τ ) ≥ 0 for all τ ∈ [xi−1 , xi ]. For qi (τ ) in [xi , x], we have qi (xi ) = pi (xi ) ≥ 0, t(1 − t)(1 − t + t2 + t(1 + t − 4t2 + 2t3 )αi )hi ≤ 0. qi (x) = − 2 − 4t + 4t2 + 2t(1 − t)αi

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The root of qi (τ ) in [xi , x] can be found, τ ∗ = xi +

t(1 − t)2 (1 + tαi )hi . 2 − 3t + 2t2 + t(2 − t − 3t2 + 2t3 )αi

Similarly, for ri (τ ), we can derive ri (x) = q(x) ≤ 0, t3 (1 − t)(1 + (1 − t)αi )hi ri (xi+1 ) = ≥ 0. 2 − 4t + 4t2 + 2t(1 − t)αi The root of ri (τ ) in [x, xi+1 ] is (1 − t + t2 + t3 + t(1 + t − 3t2 + t3 )αi )hi . 1 − t + 2t2 + t(1 + 2t − 5t2 + 2t3 )αi

τ∗ = xi + From (13), we obtain that Z

|f (x) − P (x)| ≤ kf

(2)

(x)k[

Z

xi

xi−1

pi (τ )dτ +

x

xi

Z

|qi (τ )|dτ +

xi+1

x

Z

|ri (τ )|dτ +

xi+2

xi+1

si (τ )dτ ]

= h2i kf (2) (x)kWi (t), where Wi (t) =

t(1 − t)(1 − t + t2 + t(1 + t − 4t2 + 2t3 )αi )2 . (1 − t + 2t2 + t(1 + 2t − 5t2 + 2t3 )αi )(2 − 3t + 2t2 + t(2 − t − 3t2 + 2t3 )αi )

(14)

(II) For x ∈ [x1 , x2 ], similar to the case (I), we have from (8), (9) and (13) that Z

|f (x) − P (x)| ≤ kf

(2)

(x)k[

x

x1

Z

|p1 (τ )|dτ +

x2

x

Z

|q1 (τ )|dτ +

x3

|r1 (τ )|dτ ]

x2

= h2i kf (2) (x)kW1 (t), where W1 (t) =

t(2 − 3t + t2 ) . 3−t

(15)

(III) For x ∈ [xn−1 , xn ], from (8), (11) and (13), it is easy to derive that Z

|f (x) − P (x)| ≤ kf

(2)

(x)k[

xn−1

xn−2

Z

|pn−1 (τ )|dτ +

x

xn−1

Z

|qn−1 (τ )|dτ +

xn

xn−1

|rn−1 (τ )|dτ ]

= h2i kf (2) (x)kWn−1 (t), where Wn−1 (t) =

t(1 − t2 ) . 2+t

(16)

For the fixed αi , let µi = max Wi (t),

(17)

0≤t≤1

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where Wi (t), i = 1, 2, · · · , n − 1 are defined by (15), (14) and (16), respectively. Based on the analysis above, we have the following theorem. Theorem 4. Let f (x) ∈ C 2 [a, b] and a = x1 < x2 < · · · < xn = b be an equal-knot spacing. For the given positive parameter αi , P (x) is the corresponding rational interpolation function defined by (8). Then for x ∈ [xi , xi+1 ], |f (x) − P (x)| ≤ h2i kf (2) (x)kµi where µi defined by (17). The error coefficient µi is called the optimal error constant. It is evident that the optimal error constant does not depend on the subinterval [xi , xi+1 ]. Since Wi (t) is a continuous function of the variate t in the interval [0, 1], so the coefficient µi is bounded. The following theorem about the optimal error constant µi holds. Theorem 5. For any given positive parameter αi , the optimal error constant µi satisfies µ1 = µn−1 = 0.150644, 9 µi = , i = 2, 3, · · · , n − 2. 64

(18)

Proof. From (15), (16) and (17), it is easy to derive that µ1 = µn−1 = 0.150644. We consider µi , i = 2, 3, · · · , n − 2. (14) can be rewritten as Wi (t) =

k1 + k2 + k3 , k4 + k5 + k6

(19)

where k1 = t(1 − t)(1 − t + t2 )2 , k2 = 2t2 (1 − t)2 (1 − t + t2 )(1 + 2t − 2t2 )αi , k3 = t3 (1 − t)3 (1 + 2t − 2t2 )2 αi2 , k4 = (1 − t + 2t2 )(2 − 3t + 2t2 ), k5 = 2t(1 − t)(2 + t − 5t2 + 8t3 − 4t4 )αi , k6 = t2 (1 − t)2 (1 + 3t − 2t2 )(2 + t − 2t2 )αi2 . For t ∈ (0, 1) and any αi > 0, then ki > 0, i = 1, 2, · · · , 6. Also, we can obtain that k1 (k5 + k6 ) − k4 (k2 + k3 ) = t4 (1 − t)4 (1 − 2t)2 αi2 (2(1 − t + t2 ) + 3t2 (1 − t)2 αi ) ≥ 0. Thus, from (19), we have Wi (t) ≤

k1 t(1 − t)(1 − t + t2 )2 9 = ≤ . 2 2 k4 (2 − 3t + 2t )(1 − t + 2t ) 64

(20)

Further, it is easy to test that k5 (k1 + k3 ) ≥ k2 (k4 + k6 ), it means that Wi (t) ≥

k2 t(1 − t)(1 − t + t2 )(1 + 2t − 2t2 ) = . k5 2 + t − 5t2 + 8t3 − 4t4

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Since max {

t∈[0,1]

t(1 − t)(1 − t + t2 )(1 + 2t − 2t2 ) 9 }= , 2 + t − 5t2 + 8t3 − 4t4 64

from (20), we can see that µi =

9 , i = 2, 3, · · · , n − 2. 64

The proof is completed. 2 Theorem 5 shows that the maximum error of the interpolation function defined by (8) does not depend on the positive parameter αi , it means the interpolation is stable for the parameter.

5

Concluding remarks

In this paper, an explicit representation of a rational quintic interpolator is presented. The interpolation function, which can be represented by the basis functions, contains a free positive parameter αi , and is of C 2 continuity in the interpolating interval without solving a system of consistency equations for the derivative values at the knots. Also, a local control method of the interpolating curve is developed, including the value control and the derivative control. When the given data are not changed, on the basis of this method, the shape of the interpolating curve can be modified merely by selecting suitable parameter. Convergence analysis shows that the interpolator gives a good approximation to the interpolated function, and is stable for the free positive parameter αi . Acknowledgment The author acknowledge supports provided by the National Nature Science Foundation of China (No.61070096) and the Natural Science Foundation of Shandong Province (No.ZR2012FL05).

References [1] F. Bao, Q. Sun, Q. Duan, Point control of the interpolating curve with a rational cubic spline, J. Vis. Commun. Image R. 20(2009)275-280. [2] F. Bao, Q. Sun, J. Pan, Q. Duan, A blending interpolator with value control and minimal strain energy, Comput. & Graph. 34(2010)119-124. [3] C.K. Chui, Multivariate spline, SIAM, 1988. [4] J.C. Clements, Convexity-preserving piecewise rational cubic interpolation, SIAM J. Numer. Anal. 27(1990)1016-1023. [5] C.A. de Boor, Practical guide to splines, Revised Edition, New York: Springer-Verlag, 2001. [6] R. Delbourgo, Accurate C 2 rational interpolants in tension, SIAM J. Numer. Anal. 30(1993)595-607. [7] P. Dierck, B. Tytgat, Generating the B`ezier points of BETA-spline curve, Comput. Aided Geom. Des. 6(1989)279-291. [8] Q. Duan, L. Wang, E.H. Twizell, A new C 2 rational interpolation based on function values and constrained control of the interpolant curves, Appl. Math. Comput. 161(2005)311-322.

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[9] Q. Duan, K. Djidjeli, W.G. Price, E.H. Twizell, Constrained control and approximation properties of a rational interpolating curve, Inform. Sci. 152(2003)181-194. [10] Q. Duan, F. Bao, S. Du, E.H. Twizell, Local control of interpolating rational cubic spline curves, Comput. Aided Des. 41(2009)825-829. [11] G. Farin, Curves and surfaces for computer aided geometric design: A practical guide, Academic press, 1988. [12] T.A. Foley, Local control of interval tension using weighted splines, Comp. Aided Geom. Design. 3(1986)281-294. [13] J.A. Gregory, M. Sarfraz, P.K. Yuen, Interactive curve design using C 2 rational splines, Comput. & Graph. 18(1994)153-159. [14] X. Han, Convexity-preserving piecewise rational quartic interpolation, SIAM J. Numer. Anal. 46(2008)920-929. [15] M.Z. Hussain, M. Sarfraz, Positivity-preserving interpolation of positive data by rational cubics, J. Comput. Appl. Math. 218(2008)446-458. [16] H.T. Nguyen, A. Cuyt, O.S. Celis, Comonotone and coconvex rational interpolation and approximation, Numer. Algor. 58(2011)1-21. [17] G.M. Nielson Rectangular ν-splines, IEEE comp. Graph. Appl. 6(1986)35-40. [18] M. Sarfraz, M.Z. Hussain, M. Hussain, Shape-preserving curve interpolation, Int. J. Comput. Math. 89(2012)35-53. [19] M.H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1973. [20] C. Zhang, W. Wang, J. Wang, X. Li, Local computation of curve interpolation knots with quadratic precision, Comput. Aided Des. 45(2013)853-859. [21] X. Zhao, C. Zhang, B. Yang, P. Li, Adaptive knot placement using a GMM-based continuous optimization algorithm in B-spline curve approximation, Comput. Aided Des. 43(2011)598604.

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¨ CERTAIN NEW GRUSS TYPE INEQUALITIES INVOLVING SAIGO FRACTIONAL q-INTEGRAL OPERATOR Guotao Wang,1 Praveen Agarwal2 and Dumitru Baleanu3 Abstract In the present paper, we aim to investigate a new q-integral inequality of Gr¨ uss type for the Saigo fractional q-integral operator. Some special cases of our main results are also provided. The results presented in this paper improve and extend some recent results.

2010 Mathematics Subject Classification: 26D10; 26D15; 26A33; 05A30. Key words and phrases: Integral inequalities; Gr¨ uss inequality, q-Saigo fractional integral operator.

1

Introduction and preliminaries

Fractional q-calculus, which combines fractional calculus with q-calculus, has the advantage of both. Its origin dates back to the works of Al-Salam [1] and Agarwal [2]. Currently, Fractional q-calculus had been proven to have important applications on many fields, such as physics, quantum theory, theory of relativity, Combinatorics, basic hyper-geometric functions, orthogonal polynomials, mechanics, chemicals and engineering. For some new development on the topic, see book [3] and the papers [4, 5, 6, 7, 8, 9, 10, 11, 12] . In the preset investigation, we assume that f and g be two functions which are defined and integrable on [a, b]. Then the following inequality holds (see also [21], [23, p. 296]): l ≤ f (t) ≤ L,

m ≤ g(t) ≤ M,

(1.1)

Let, for each t ∈ [a, b], l, L, m and, M be real constants satisfying the inequalities (1.1). Then the following Gr¨ uss type inequalities involving Riemann-Liouville fractional integrals holds (see, for example, [27]):  2 tα tα α α α Γq (α + 1) Iq (f g)(t) − Iq (f )(t)Iq (g)(t) ≤ Γq (α + 1) (L − l)(M − m), (1.2) 1

School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, (China) E-mail:[email protected] 2 Department of Mathematics, Anand International College of Engineering, Jaipur-303012, Rajasthan (India) E-mail:[email protected] 3 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, (Saudi Arabia). Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, Ankara, (Turkey). Institute of Space Sciences, P.O. Box MG-23, 76900 Magurele Bucharest, (Romania). E-mail:[email protected]

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Guotao Wang1,∗ , Praveen Agarwal2 and Dumitru Baleanu3,4,5

The inequality (1.2) has various generalizations that have appeared in the literature, for example, e.g. [13, 15, 16, 18, 19, 22, 23, 24, 25, 26] and the references cited therein. Very recently, Zhu et al.[27] gave certain Gr¨ uss type inequalities involving the fractional q-integral operator. Using the same technique, in this paper, we establish certain new Gr¨ uss type fractional q-integral inequalities involving the Saigo fractional q-integral operator due to Garg and Chanchlani[20]. Moreover, we also consider their relevant connections with other known results. Throughout the present paper, we shall investigate a fractional integral over the space Cλ introduced in [17] and defined as follows. Definition 1.1 The space of functions Cλ , λ ∈ R, the set of real numbers, consists of all functions f (t), t > 0, that can be represented in the form f (t) = tp f1 (t) with p > λ and f1 ∈ C[0, ∞), where C[0, ∞) is the set of continuous functions in the interval [0, ∞). Here, we define a new (presumable) fractional integral operator Kqα,β,η associated with the Saigo fractional q-integral operator as follows. Definition 1.2 Let 0 < q < 1, f ∈ Cλ . Then for α > 0, α + η > −1 and β > 1 we define a fractional integral Kqα,β,η as follows:   Γq (1 − β)Γq (α + η + 1) β  α,β,η  Kqα,β,η f (t) = t Iq f (t), (1.3) Γq (η − β + 1) where Iqα,β,η f is the Saigo fractional q-integral operator of order α and is defined in the following. Definition 1.3 Let 0, β and η be real or complex numbers. Then a q-analogue of Saigo’s fractional integral Iqα,β,η is given for | τt | < 1 by (see [20, p. 172, Eq.(2.1)]):
Z ∞ X q α+β ; q m (q −η ; q)m t−β−1 t α,β,η Iq {f (t)} := (q τ /t; q)α−1 Γq (α) 0 (q α ; q)m (q; q)m (1.4) m=0 m (η−β)m m −(m ·q (−1) q 2 ) (τ /t − 1) f (τ ) d τ. q

q

The integral operatorIqα,β,η includes both the q-analogues of the Riemann-Liouville and Erd´elyi-Kober fractional integral operators given by the following relationships:
Iqα {f (t)} : = Iqα,−α,0 {f (t)} Z t tα−1 (q τ /t; q)α−1 f (τ )dq τ = Γq (α) 0

863

(α > 0; 0 < q < 1).

(1.5)

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Certain Gr¨ uss type inequality associated with Saigo fractional q-integral

and
Iqη,α {f (t)} : = Iqα,0,η {f (t)} Z t−η−1 t = (q τ /t; q)α−1 τ η f (τ )dq τ Γq (α) 0 where (a; q)α is the q-shifted factorial. The q-shifted factorial (a; q)n is defined by  1     Y  (a; q)n := n−1 k  1 − a q  

(α > 0; 0 < q < 1),

(1.6)

(n = 0) (1.7)

(n ∈ N),

k=0

where a, q ∈ C and it is assumed that a 6= q −m (m ∈ N0 ). The q-shifted factorial for negative subscript is defined by (a; q)−n :=

(1 −

a q −1 )

1 (1 − a q −2 ) · · · (1 − a q −n )

(n ∈ N0 ) .

(1.8)

We also write (a; q)∞ :=

∞  Y

1 − a qk



(a, q ∈ C; |q| < 1).

(1.9)

k=0

It follows from (1.7), (1.8) and (1.9) that (a; q)n =

(a; q)∞ (a q n ; q)∞

(n ∈ Z),

(1.10)

(α ∈ C; |q| < 1),

(1.11)

which can be extended to n = α ∈ C as follows: (a; q)α =

(a; q)∞ (a q α ; q)∞

where the principal value of q α is taken. For f (t) = tµ in (1.4), we get the known formula [12]: Iqα,β,η {tµ } :=

Γq (µ + 1)Γq (µ − β + η + 1) xµ − β Γq (µ − β + 1)Γq (µ + α + η + 1)

(1.12)

Lemma 1.4(Choi and Agarwal [14]): Let 0 < q < 1 and f : [0, ∞) → R be a continuous function with f (t) ≥ 0 for all t ∈ [0, ∞). Then we have the following inequalities:

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Guotao Wang1,∗ , Praveen Agarwal2 and Dumitru Baleanu3,4,5

4

(i) The Saigo fractional q-integral operator of the function f (t) in (1.4) Iqα,β,η {f (t)} ≥ 0,

(1.13)

for all α > 0 and β, η ∈ R with α + β > 0 and η < 0; (ii) The q-analogue of Riemann-Liouville fractional integral operator of the function f (t) of a order α in (1.5) Iqα {f (t)} ≥ 0,

(1.14)

for all α > 0; (iii) The q-analogue of Erd´elyi-Kober fractional integral operator of the function f (t) in (1.6) Iqη,α {f (t)} ≥ 0,

(1.15)

for all α > 0 and η ∈ R.

Remark 1.5: We can present a large number of special cases of our main fractional integral operator Kqα,β,η . Out of which, here, we give only few ones as follows: (i) For 0, β and η be real or complex numbers and −1, µ−η +β > −1, we have Kqα,β,η (tµ ) =

Γq (1 − β)Γq (α + η + 1)Γq (µ + 1)Γq (µ − β + η + 1) tµ Γq (η − β + 1)Γq (µ − β + 1)Γq (µ + α + η + 1) Γq (η − β + 1) (1.16)

and Kqα,β,η (C) = C

(1.17)

where C is constant. (ii) For 0, and η be real or complex number and −1, µ + η > −1, we have Γq (α + η + 1) Γq (µ + η + 1) µ Kqη,α (tµ ) = t (1.18) Γq (η + 1)Γq (µ + α + η + 1) (iii) For 0and −1, we have Kqα (tµ ) =

Γq (α + 1) µ t Γq (µ + α + 1)

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(1.19)

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2

Gr¨ uss type inequalities for the Saigo fractional qintegral operator

To state the main results in this section, first we establish the following Gr¨ uss type α,β,η lemmas involving the fractional integral operator Kq (1.3), some of which are presumably (new) ones. Lemma 2.1 Let 0 < q < 1, h ∈ Cλ and m, M ∈ R with m ≤ h(t) ≤ M . Then, for all t > 0 we have  2 Kqα,β,η h2 (t) − Kqα,β,η h(t) (2.1)    = M − Kqα,β,η h(t) Kqα,β,η h(t) − m − Kqα,β,η (M − h(t)) (h(t) − m) , for all α > 0, and β, η ∈ R with α + β > 0, and η < 0. Proof. Let h ∈ Cλ and m, M ∈ R; m ≤ h(t) ≤ M , for all t ∈ [0, ∞), then, for any u, v, ∈ [0, ∞), we have (M − h(u)) (h(v) − m) + (M − h(v)) (h(u) − m) − (M − h(u)) (h(u) − m) − (M − h(v)) (h(v) − m) = h2 (u) + h2 (v) − 2h(u)h(v). (2.2) Multiplying both sides of (2.2) by
∞ X q α+β ; q m (q −η ; q)m (η−β)m m t−β−1 (q u/t; q)α−1 ·q (−1)m q −( 2 ) (u/t − 1)m q α Γq (α) (q ; q)m (q; q)m m=0

and taking q-integration of the resulting inequality with respect to u from 0 to t with the aid of Definition 1.2, we get 

   M − Kqα,β,η h(t) (h(v) − m) + (M − h(v)) Kqα,β,η h(t) − m − Kqα,β,η (M − h(t)) (h(t) − m) − (M − h(v)) (h(v) − m)

(2.3)

= Kqα,β,η h2 (t) + h2 (v) − 2Kqα,β,η h(t)h(v). Again multiplying both sides of (2.3) by
∞ X q α+β ; q m (q −η ; q)m (η−β)m m t−β−1 (q v/t; q)α−1 ·q (−1)m q −( 2 ) (v/t − 1)m q α Γq (α) (q ; q)m (q; q)m m=0

and taking q-integration of the resulting inequality with respect to v from 0 to t with the aid of Definition 1.2, we get the required result (2.1). This is complete the proof of the Lemma 2.1.

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Lemma 2.2 Let 0 < q < 1, f, q ∈ Cλ . Then, for all t > 0,we have  2 Kfα,β,η q(t) + Kfγ,δ,ζ q(t) − Kqα,β,η f (t) Kqγ,δ,ζ g(t) − Kqα,β,η g(t) Kqγ,δ,ζ f (t)   ≤ Kqα,β,η f 2 (t) + Kqγ,δ,ζ f 2 (t) − Kqα,β,η f (t) Kqγ,δ,ζ f (t)   − Kqα,β,η g 2 (t) + Kqγ,δ,ζ g 2 (t) − Kqα,β,η g(t) Kqγ,δ,ζ g(t) , (2.4) for all α, γ > 0, and β, η, δ, ζ ∈ R with α + β > 0, γ + δ > 0,, and η, ζ < 0. Proof. Let f and g be two continuous and synchronous functions on [0, ∞). Then, for all u, v ∈ (0, t) with 0 < q < 1, we have A(u, v) = (f (u) − f (v)) (g(u) − g(v)) (u, v ∈ [0, t)).

(2.5)

Multiplying (2.5) by t−β−1 t−δ−1 (q u/t; q)α−1 (q v/t; q)γ−1 (Γq (α)) (Γq (γ))

∞ X ∞ X q α+β ; q (q −η ; q)m q γ+δ ; q

q −ζ ; q n (q γ ; q)n (q; q)n

m

m=0 n=0

(q α ; q)m (q; q)m m


n

n

n · q (η−β)m (−1)m q −( 2 ) q (ζ−δ)n (−1)n q −( 2 ) (u/t − 1)m q (v/t − 1)q . (2.6) Integrating (2.6) twice with respect to u and v from 0 to t, we obtain the following result with the help of (1.3) and (1.4): Z tZ t t−β−1 t−δ−1 (q u/t; q)α−1 (q v/t; q)γ−1 (Γq (α)) (Γq (γ)) 0 0


∞ X ∞ X q α+β ; q m (q −η ; q)m q γ+δ ; q n q −ζ ; q n (q α ; q)m (q; q)m (q γ ; q)n (q; q)n m=0 n=0

m

n

n · q (η−β)m (−1)m q −( 2 ) q (ζ−δ)n (−1)n q −( 2 ) (u/t − 1)m q (v/t − 1)q A(u, v)dudv

= Kqα,β,η f g(t) + Kqγ,δ,ζ f g(t) − Kqα,β,η f (t)Kqγ,δ,ζ g(t) − Kqα,β,η g(t)Kqγ,δ,ζ f (t). (2.7) By making use of the well known Cauchy-Schwarz inequality for double q(.,.,.) integrals [27, pp. 4-5], we obtain the required result (2.4)involving the Kq (1.3), type fractional integral operator. This is complete the proof of Lemma 2.2. Lemma 2.3 Let 0 < q < 1, h ∈ Cλ and m, M ∈ R with m ≤ h(t) ≤ M . Then, for all t > 0 we have Kqα,β,η h2 (t) + Kqγ,δ,ζ h2 (t) − 2Kqα,β,η h(t)Kqγ,δ,ζ h(t)       = M − Kqα,β,η h(t) Kqα,β,η h(t) − m + M − Kqγ,δ,ζ h(t) Kqγ,δ,ζ h(t) − m − Kqα,β,η (M − h(t)) (h(t) − m) − Kqγ,δ,ζ (M − h(t)) (h(t) − m) , (2.8)

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Certain Gr¨ uss type inequality associated with Saigo fractional q-integral

for all α, γ > 0 and β, η, δ, ζ ∈ R with α + β > 0, γ + δ > 0 and η, ζ < 0. Proof. Multiplying both sides of (2.3) by

∞ X q γ+δ ; q n q −ζ ; q n (ζ−δ)n n t−δ−1 ·q (−1)n q −( 2 ) (v/t − 1)nq (q v/t; q)γ−1 γ Γq (γ) (q ; q)n (q; q)n n=0

and taking q-integration of the resulting inequality with respect to v from 0 to t with the aid of Definition 1.2, we get the required result (2.8). This is complete the proof of the Lemma 2.3. Definition 2.4. Two functions f and g are said to be synchronous function on [0, ∞) if A(u, v) = (f (u) − f (v))(g(u) − g(v)) ≥ 0; u, v ∈ [0, ∞). (2.9) Theorem 2.5. Let 0 < q < 1, f, g ∈ Cλ and satisfying the condition (1.1) on [0, ∞). Then the following inequality holds true: 1 α,β,η (2.10) f g(x) − Kqα,β,η f (t)Kqα,β,η g(t) ≤ (L − l)(M − m), K q 4 for all α > 0, and β, η ∈ R with α + β > 0 and η < 0. Proof. Let f and g be two continuous and synchronous functions on [0, ∞). Then, for all u, v ∈ (0, t) with 0 < q < 1, we have A(u, v) = (f (u) − f (v)) (g(u) − g(v)) (u, v ∈ [0, t)).

(2.11)

Multiplying (2.11) by t−2β−2 (q u/t; q)α−1 (q v/t; q)α−1 (Γq (α))2

∞ X ∞ X q α+β ; q m (q −η ; q)m q α+β ; q n (q −η ; q)n (q α ; q)m (q; q)m (q α ; q)n (q; q)n m=0 n=0

m

n

n · q (η−β)m (−1)m q −( 2 ) q (η−β)n (−1)n q −( 2 ) (u/t − 1)m q (v/t − 1)q . (2.12) Integrating (2.12) twice with respect to u and v from 0 to t, we obtain the following result with the help of (1.3) and (1.4) Z tZ t t−2β−2 (q u/t; q)α−1 (q v/t; q)α−1 (Γq (α))2 0 0

∞ X ∞ X (q −η ; q)m q α+β ; q (q −η ; q)n q α+β ; q m=0 n=0

m

n

(q α ; q)m (q; q)m

(q α ; q)n (q; q)n m

n

n · q (η−β)m (−1)m q −( 2 ) q (η−β)n (−1)n q −( 2 ) (u/t − 1)m q (v/t − 1)q .A(u, v)dudv

= 2Kqα,β,η f g(t) − 2Kqα,β,η f (t)Kqα,β,η g(t). (2.13)

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Guotao Wang1,∗ , Praveen Agarwal2 and Dumitru Baleanu3,4,5

8

Making use of the well known Cauchy-Schwarz inequality for linear operator [23, Eq. (1.3), p. 296], we find that  2 α,β,η α,β,η α,β,η Kq f g(t) − Kq f (t)Kq g(t) ≤  (2.14)  2   2   α,β,η 2 α,β,η α,β,η 2 α,β,η . Kq g (t) − Kq g(t) Kq f (t) − Kq f (t) Since (L − f (t)) (f (t) − l) ≥ 0 and (M − g(t)) (g(t) − m) ≥ 0, therefore, we have Kqα,β,η (L − f (t)) (f (t) − l) ≥ 0 and Kqα,β,η (M − g(t)) (g(t) − m) ≥ 0. Thus by using Lemma 2.1, we get  2    Kqα,β,η f 2 (t) − Kqα,β,η f (t) ≤ L − Kqα,β,η f (t) Kqα,β,η f (t) − l

(2.15)

(2.16)

and  2    Kqα,β,η g 2 (t) − Kqα,β,η g(t) ≤ M − Kqα,β,η g(t) Kqα,β,η g(t) − m .

(2.17)

Applying (2.16) and (2.17) in to (2.14), Equation (2.14) reduces to the following form:  2 Kqα,β,η f g(t) − Kqα,β,η f (t)Kqα,β,η g(t) ≤      L − Kqα,β,η f (t) Kqα,β,η f (t) − l M − Kqα,β,η g(t) Kqα,β,η g(t) − m . (2.18) Applying the well known inequality 4ab ≤ (a + b)2 ; a, b ∈ R in the right hand side of the inequality (2.18)and simplifying it, we obtain the required result (2.10). This is complete the proof of Theorem 2.5. Theorem 2.6. Let 0 < q < 1, f and g be two synchronous functions on [0, ∞). Then the following inequality holds: Kqα,β,η f g(t) ≥ Kqα,β,η f (t)Kqα,β,η g(t),

(2.19)

for all α > 0, and β, η ∈ R with α + β > 0 and η < 0. Proof. For the synchronous function f and g, the inequality (2.9) holds for all u, v ∈ [0, ∞). This implies that f (u)g(u) − f (v)g(v) ≥ f (u)g(v) + f (v)g(u).

(2.20)

Following the procedure of the Lemma 2.1 for applying the fractional integral after a little simplification, we arrive at the required result (2.19). This completes the proof of Theorem 2.6. Kqα,β,η ,

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Certain Gr¨ uss type inequality associated with Saigo fractional q-integral

Theorem 2.7. Let 0 < q < 1 and f, g ∈ Cλ . Then, for all t > 0 we have: 2 Kqα,β,η f g(t) + Kqγ,δ,ζ f g(t) − Kqα,β,η f (t)Kqγ,δ,ζ g(t) − Kqα,β,η g(t)Kqγ,δ,ζ f (t) n    o ≤ M − Kqα,β,η f (t) Kqγ,δ,ζ f (t) − m M − Kqγ,δ,ζ f (t) Kqα,β,η f (t) − m n    o × L − Kqα,β,η g(t) Kqγ,δ,ζ g(t) − l L − Kqγ,δ,ζ g(t) Kqα,β,η g(x) − l , (2.21) for all α, γ > 0 and β, η, δ, ζ ∈ R with α + β > 0, γ + δ > 0 and η, ζ < 0.



Proof. Since (L − f (t)) (f (t) − l) ≥ 0 and (M − g(t)) (g(t) − m) ≥ 0, therefore, we have − Kqα,β,η (L − f (t)) (f (t) − l) − Kqγ,δ,ζ (L − f (t)) (f (t) − l) ≥ 0, and

(2.22)

− Kqα,β,η (M − g(t)) (g(t) − m) − Kqγ,δ,ζ (M − g(t)) (g(t) − m) ≥ 0. Thus by using Lemma 2.3 to f and g, then by using Lemma 2.2 and the formula (2.22), we get the desired result (2.21).

3

Special Cases and Concluding Remarks

We conclude our present investigation by remarking further that we can present a large number of special cases of our main inequalities in Theorems 2.5, 2.6 and 2.7. Here we give only three examples: Setting β = 0 in (2.10), (2.19) and β = δ = 0 in (2.21), we obtain an interesting inequalities involving Erd´elyi-Kober fractional integral operator fractional integral operator as follows. Corollary 3.1. Let 0 < q < 1, f, g ∈ Cλ and satisfying the condition (1.1) on [0, ∞). Then the following inequality holds true: η,α Kq f g(x) − Kqη,α f (t)Kqη,α g(t) ≤ 1 (L − l)(M − m), 4

(3.1)

for all α > 0, and η ∈ R with η < 0. Corollary 3.2. Let 0 < q < 1, f and g be two synchronous functions on [0, ∞). Then the following inequality holds: Kqη,α f g(t) ≥ Kqη,α f (t)Kqη,α g(t),

(3.2)

for all α > 0 and η ∈ R with η < 0.

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Guotao Wang1,∗ , Praveen Agarwal2 and Dumitru Baleanu3,4,5

10

Corollary 3.3. Let 0 < q < 1 and f, g ∈ Cλ . Then, for all t > 0 we have:  2 Kqη,α f g(t) + Kqζ,γ f g(t) − Kqη,α f (t)Kqζ,γ g(t) − Kqζ,γ g(t)Kqη,α f (t) n  

o (3.3) ≤ M − Kqη,α f (t) Kqζ,γ f (t) − m M − Kqζ,γ f (t) Kqη,α f (t) − m n     o

× L − Kqη,α g(t) Kqζ,γ g(t) − l L − Kqζ,γ g(t) Kqη,α g(x) − l , for all α, γ > 0 and η, ζ ∈ R with η, ζ < 0. We conclude our present investigation by remarking further that the results obtained here are useful in deriving various fractional integral inequalities involving such relatively more familiar fractional integral operators. For example, if we consider β = −α in (2.10) and β = −α and δ = −γ in (2.21) and making use of the relation (1.5), Theorems 2.5 and 2.7 provide, respectively, the known fractional qintegral inequalities due to Zhu et.al.[27, pp. 6-7, Theorems 1 and 2]. Competing interests The authors declare that they have no competing interests. Authors contributions All authors have equal contributions.

References [1] WA. Al-Salam, Some fractional q-integrals and q-derivatives. Proc Edimb Math Soc 1966;15:13540. [2] R. Agarwal, Certain fractional q-integrals and q-derivatives. Proc Camb Philos Soc 1969;66:36570. [3] MH. Annaby, Z.S. Mansour, q-Fractional calculus and equations. Lecture notes in mathematics, vol. 2056; 2012. [4] F. Jarad, T. Abdeljawad, D. Baleanu, Stability of q-fractional non-autonomous systems. Nonlinear Anal.: Real World Appl 14(2013); 780-784. [5] G. Wu, D. Baleanu, New applications of the variational iteration method - from differential equations to q-fractional difference equations. Adv Differ Eqs.; 2013. 1-21. [6] T. Abdeljawad T, B. Benli, D. Baleanu, A generalized q-Mittag-Leffler function by q-Caputo fractional linear equations. Abstract and applied analysis; 2012.Art ID 546062. [7] T. Abdeljawad, D. Baleanu, Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function, Commun Nonlinear Sci Numer Simul,16(2011), 4682-4688.

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[8] B. Ahmad, S.K. Ntouyas,IK. Purnaras, Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations. Adv Differ Equat 2012;2012:140. [9] R. Almeida, N. Martins, Existence results for fractional q-difference equations of order α ∈ [2, 3] with three-point boundary conditions, Commun Nonlinear Sci Numer Simulat 19(2014), 1675-1685. [10] M. El-Shahed, M. Gaber, M. Al-Yami, The fractional q-differential transformation and its application, Commun Nonlinear Sci Numer Simulat 18(2013) 42-55. [11] J. R. Graef, L. Kong, Positive solutions for a class of higher order boundary value problems with fractional q-derivatives, Applied Mathematics and Computation 218(2012) 9682-9689. [12] G. Anastassiou, q-Fractional Inequalities, CUBO A Mathematical Journal, 13(2011), 61-71. [13] G. Anastassiou, Advances on Fractional Inequalities, Springer Briefs in Mathematics, SpringerLink : Bcher, Springer, 2011. [14] J. Choi and P. Agarwal, Certain inequalities associated with Saigo fractional integrals and their q-analogues, Abstract and Applied Analysis, 2014, Article ID 5792609, 11 pages. [15] Z. Dahmani, O. Mechouar and S. Brahami, Certain inequalities related to the Chebyshev’s functional involving a type Riemann-Liouville operator, Bull. Math. Anal. Appl. 3(4) (2011), 38–44. [16] Z. Dahmani - L. Tabharit - S. Taf, New generalisations of Gr¨ uss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl. 2 (3) (2010),9399. [17] I. H. Dimovski, Operational calculus for a class of differential operators, Compt. Rend. Acad. Bulg. Sci. 19 (1966), 1111-1114. [18] S. S. Dragomir, Some integral inequalities of Gruss type, Indian J. Pure Appl. Math. 31(4)(2000), 397–415. [19] S. S. Dragomir, Some integral inequalities of Gruss type, Indian J. Pur. Appl. Math. 31 (4) (2002), 397-415. [20] M. Garg and L. Chanchlani, q-Analogues of Saigo’s fractional calculus operators, Bull. Math. Anal. Appl. 3(4) (2011), 169–179. ¨ Gr¨ uss, Uber das maximum des absoluten R Rb b 1 f (x)dx a g(x)dx, Math. Z. 39 (1935), 215-226. (b−a)2 a

[21] D.

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Guotao Wang1,∗ , Praveen Agarwal2 and Dumitru Baleanu3,4,5

[22] A. McD. Mercer, An improvement of the Gr¨ uss inequality, J. Inequal. Pure Appl. Math. 6 (4) (2005), 1-4. [23] D. S. Mitrinovi´c - J. E. Peˇcari´c - A. M. Fink, Classical and new inequalities in analysis, Kluwer Academic Publishers, Dordrecht, The Netherlands 1993. [24] B. G. Pachpatte, On Gr¨ uss type integral inequalities, Journal of Inequalities in Pure and Applied Mathematics 3 (1) (2002), 1-5. [25] B. G. Pachpatte, A note on Chebyshev-Gr¨ uss inequalities for differential equations, Tamsui Oxford Journal of Mathematical Sciences 22 (1) (2006), 29-36. [26] G. Wang, P. Agarwal and M. Chand, Certain Gr¨ uss type inequalities involving the generalized fractional integral operator, Journal of Inequalities and Applications (2014), 2014:147. [27] C. Zhu, W. Yang and Q. Zhao, Some new fractional q-integral Gr¨ uss -type inequalities and other inequalities, Journal of Inequalities and Applications (2012), 2012:299.

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ULTRA BESSEL SEQUENCES OF SUBSPACES IN HILBERT SPACES MOHAMMAD REZA ABDOLLAHPOUR1 , AZAM SHEKARI2 , CHOONKIL PARK3 AND DONG YUN SHIN4∗ 1

Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran

2

Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran 3

Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea 4

Department of Mathematics, University of Seoul, Seoul 130-743, Korea [email protected]; [email protected]; [email protected]

Abstract. In this paper, we introduce ultra Bessel sequences of subspaces in Hilbert spaces and we establish some new results about ultra Bessel sequences of subspaces and their perturbation.

1. Introduction Frames for Hilbert spaces were defined by Duffin and Schaeffer [4] in 1952, and were reintroduced and developed in 1986 by Daubechies, Grossmann and Meyer [3]. A sequence {fi }∞ i=1 in Hilbert space H is called a frame for H if there exist two constants A, B > 0 such that ∞ X Akf k2 ≤ |hf, fi i|2 ≤ Bkf k2 (1.1) i=1

for all f ∈ H. The constants A and B are called frame bounds. If A = B then we call the frame {fi }∞ i=1 a tight frame and if A = B = 1 then it is called a Parseval frame. If the right-hand inequality of (1.1) holds for all f ∈ H, then we call {fi }∞ i=1 a Bessel sequence for H. See also [2, 6, 7, 9, 10] for more information on frames and Bessel sequences. In 2008, the concept of ultra Bessel sequences in Hilbert spaces (i.e., Bessel sequences with uniform convergence property) was introduced by Faroughi and Najati [5]. Definition 1.1. Let H0 be an inner product space. The sequence {fi }∞ i=1 ⊆ H0 is called an ultra Bessel sequence in H0 if ∞ X sup |hf, fi i|2 → 0 (1.2) as n → ∞, i.e., the series

kf k=1 i=n 2 i=n |hf, fi i| converges

P∞

uniformly in unit sphere of H0 .

2010 Mathematics Subject Classification. Primary 46C99; 42C15. Keywords: Frame of subspaces; Perturbation; Ultra Bessel sequence of subspaces. ∗ The corresponding author: Dong Yun Shin (email: [email protected].

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M. R. ABDOLLAHPOUR, A. SHEKARI, C. PARK, D. Y. SHIN

In 2004, frame of subspaces as a generalization of ordinary frame was introduced and investigated by Casazza and Kutyniok [1]. Definition 1.2. Let {vi }∞ i=1 be a sequence of weights, i.e., for all i ≥ 1, vi > 0. A family ∞ of closed subspaces {Wi }i=1 of a Hilbert space H is a frame of subspaces with respect to {vi }∞ i=1 for H, if there exist constants 0 < C ≤ D < ∞ such that 2

Ckf k ≤

∞ X

vi2 kπWi (f )k2 ≤ Dkf k2

f ∈ H,

(1.3)

i=1

where πWi is the orthogonal projection of H onto Wi . If v = vi > 0 for all i ≥ 1 in (1.3), then we say that {Wi }∞ i=1 is a frame of subspaces with respect to v for H. If in (1.3) the right hand inequality holds for all f ∈ H, then {Wi }∞ i=1 is called a Bessel ∞ sequence of subspaces with respect to {vi }i=1 with Bessel bound D. Definition 1.3. Let H be a Hilbert space. For each sequence of subspaces {Wi }∞ i=1 of H, we define the set ∞ ∞ X  X  ∞ ⊕Wi 2 = {fi }i=1 |fi ∈ Wi , kfi k2 < ∞ . `

i=1

P

∞ It is clear that i=1 ⊕Wi the inner product given by

 `2

i=1

is a Hilbert space with the point wise operations and with

∞ h{fi }∞ i=1 , {gi }i=1 i

=

∞ X

hfi , gi i.

i=1 ∞ It is proved in [1] that if {Wi }∞ i=1 is a frame of subspaces with respect to {vi }i=1 for H, then the operator ∞ ∞ X  X T : ⊕Wi 2 → H, T ({fi }∞ ) = vi fi i=1 `

i=1

i=1

is bounded and onto and its adjoint is ∞ X  ∗ T :H→ ⊕Wi 2 , i=1

T ∗ (f ) = {vi πWi (f )}∞ i=1 .

`

T∗

The operators T and are called the synthesis and analysis operators for {Wi }∞ i=1 and ∞ {vi }i=1 , respectively. ∞ Also, it is proved in [1] that if {Wi }∞ i=1 is a frame of subspaces with respect to {vi }i=1 , the operator S : H → H,

S(f ) = T T ∗ (f )

is a positive, self-adjoint and invertible operator on H and we have the reconstruction formula f=

∞ X

vi2 S −1 πWi (f ),

f ∈ H.

i=1 ∞ The operator S is called the frame operator for {Wi }∞ i=1 and {vi }i=1 .

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ULTRA BESSEL SEQUENCES OF SUBSPACES IN HILBERT SPACES

2. Ultra Bessel sequence of subspaces In this section we intend to introduce ultra Bessel sequences of subspaces and investigate some of their properties. Definition 2.1. Let H0 be an inner product space. Let {Wi }∞ i=1 be a sequence of closed ∞ subspaces of H0 . Then {Wi }i=1 is called an ultra Bessel sequence of subspaces in H0 , if ∞ X

sup

kf k=1 i=n

as n → ∞, i.e., the series

vi2 kπWi (f )k2 → 0,

P∞

2 2 i=1 vi kπWi (f )k

(2.1)

converges uniformly in the unit sphere of H0 .

It is clear that each ultra Bessel sequence of subspaces is a Bessel sequence of subspaces. Example 2.2. (1) Let {ei }∞ i=1 be an orthonormal basis for some Hilbert space H. We define the subspaces Wi by Wi = span{ei },

i = 1, 2, 3, ....

Then {Wi }∞ i=1 is a Bessel sequence of subspaces with respect to each v > 0. since ∞ X

2

2

v kπWi (f )k =

i=1

∞ X

v 2 |hf, ei i|2 = v 2 kf k2 ,

f ∈ H.

i=1

But {Wi }∞ i=1 is not an ultra Bessel sequence of subspaces with respect to each v > 0. In fact, ∞ X sup v 2 kπWi (f )k2 ≥ v 2 kπWn (en )k2 = v 2 ken k2 = v 2 > 0. kf k=1 i=n

(2) If Wi = span{i−1 ei }, then {Wi }∞ i=1 is an ultra Bessel sequence of subspaces, since sup

∞ X

kf k=1 i=n

2

2

2

v kπWi (f )k = v sup

∞ X

−1

|hf, i

kf k=1 i=n

∞ X 1 ei i| ≤ v → 0, i2 2

2

i=n

as n → ∞. Proposition 2.3. Let {Wi }∞ of closed subspaces in Hilbert space H and i=1 be a sequence P∞ 2 ∞ {vi }i=1 be a sequence of weights such that i=1 vi < ∞. Then {Wi }∞ i=1 is an ultra Bessel sequence of subspaces in H. Proof. For each n ∈ N and each f ∈ H we have ∞ ∞ X X 2 2 2 vi kπWi (f )k ≤ kf k vi2 . i=n

So sup

∞ X

kf k=1 i=n

i=n

vi2 kπWi (f )k2 ≤

∞ X

vi2 → 0,

i=n

as n → ∞.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

M. R. ABDOLLAHPOUR, A. SHEKARI, C. PARK, D. Y. SHIN

The converse of Proposition 2.3 is not true in general. We give an example. Example 2.4. Let {ei }∞ i=1 be an orthonormal basis for H and put Wi = span{ei }. Then {Wi }∞ i=1 is an ultra Bessel sequence of subspaces for H with respect to weights vi := for all i ∈ N. In fact, sup

∞ X 1

kf k=1 i=n

i

2

kπWi (f )k = sup

∞ X 1

kf k=1 i=n i ∞ X

≤ sup

kf k=1

i=n

1 √ i

|hf, ei i|2 ∞

1 |hf, ei i|2  12  X 2 2 |hf, e i| i i2 i=n

∞ X 1  12 ≤ → 0, i2 i=n P∞ 1 P 2 as n → ∞. But this series ∞ i=1 i does not converge. i=1 vi =

Now, we prove the next simple lemma which has an important role in proof of Theorem 2.6. ∞ Lemma 2.5. Let {Wi }∞ i=1 be a Bessel sequence of subspaces with respect to {vi }i=1 in Hilbert space H. Then

sup

∞ X

kf k≤1 i=n

Proof. Let α = supkf k=1

vi2 kπWi (f )k2

≤ sup

P∞

kf k=1 i=n

2 2 i=n vi kπWi (f )k . ∞ X

∞ X

vi2 kπWi (f )k2 ,

n ∈ N.

It is clear that if kf k = 1 or kf k = 0, then

vi2 kπWi (f )k2 ≤ α.

i=n

So we consider 0 < kf k = k < 1. In this case, ∞ ∞ X 1X 2 f vi kπWi (f )k2 = vi2 kπWi ( )k2 ≤ α. k k i=n

Therefore,

P∞

2 2 i=n vi kπWi (f )k

i=n

≤ kα < α, and we conclude that sup

∞ X

kf k≤1 i=n

vi2 kπWi (f )k2 ≤ α. 

Theorem 2.6. Let H and K be Hilbert spaces and T : H → K be a bounded invertible ∞ operator. If {Wi }∞ i=1 is an ultra Bessel sequence of subspaces in H, then {T (Wi )}i=1 also is an ultra Bessel sequence of subspaces in K. Proof. Similar to the proof of Lemma 2.3 of [8] we show that πT Wi = πT Wi (T ∗ )−1 πWi T ∗ . 877

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ULTRA BESSEL SEQUENCES OF SUBSPACES IN HILBERT SPACES

In fact, if f ∈ H then f = πWi f + g, for some g ∈ Wi⊥ . Therefore (T ∗ )−1 f = (T ∗ )−1 πWi f + (T ∗ )−1 g. On the other hand 0 = hw, gi = hT −1 T w, gi = hT w, (T −1 )∗ gi,

w ∈ Wi .

Hence (T −1 )∗ g = (T ∗ )−1 g ∈ (T Wi )⊥ and so for all f ∈ H, πT Wi (T ∗ )−1 f = πT Wi (T ∗ )−1 πWi f + πT Wi (T ∗ )−1 g = πT Wi (T ∗ )−1 πWi f. Now, (2.2) implies that for all f ∈ H, kπT Wi (f )k ≤ k(T ∗ )−1 kkπWi T ∗ f k. If kf k = 1, then by Lemma 2.5 ∞ ∞ X X vi2 kπT (Wi ) (f )k2 ≤ k(T ∗ )−1 k2 vi2 kπWi T ∗ f k2 i=n

i=n

≤ k(T ∗ )−1 k2 kT ∗ k2

∞ X

vi2 kπWi (

i=n

≤ k(T ∗ )−1 k2 kT ∗ k2 sup

∞ X

kgk≤1 i=n ∞ X

≤ k(T ∗ )−1 k2 kT ∗ k2 sup

kgk=1 i=n

T ∗f 2 )k kT ∗ k

vi2 kπWi (g)k2 vi2 kπWi (g)k2 ,

as desired.



Theorem 2.7. Suppose that {Wi }∞ i=1 is a sequence of closed subspaces of Hilbert space H. Let V be a dense subset of the unit sphere of H and ∞ nX o sup vi2 kπWi (f )k2 : f ∈ V → 0, (2.3) i=n

as n → ∞. Then

{Wi }∞ i=1

is an ultra Bessel sequence of subspaces in H.

Proof. Suppose A(n) := sup

∞ nX

o vi2 kπWi (f )k2 : f ∈ H and kf k = 1

i=n

does not tend to zero as n → ∞. Then there exists β > 0 and a subsequence {A(nk )}∞ k=1 of ∞ k k {A(n)}n=1 such that A(nk ) > β for each k ≥ 1. Hence, for some f ∈ H with kf k = 1, ∞ X

vi2 kπWi (f k )k2 > β,

k ≥ 1.

(2.4)

i=nk k k Choose a subspaces {fjk }∞ j=1 of member of V such that fj → f as j → ∞. By (2.3) there exists k0 > 0 such that ∞ X vi2 kπWi (fjk0 )k2 < β, j ≥ 1. i=nk0

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Also, (2.4) implies that there exists l ≥ nk0 such that l X

vi2 kπWi (f k0 )k2 > β.

i=nk0

Since l X

vi2 kπWi (fjk0 )k2

l X

→

i=nk0

vi2 kπWi (f k0 )k2

i=nk0

as j → ∞, for sufficiently large j, we have ∞ X

β>

l X

vi2 kπWi (fjk0 )k2 ≥

i=nk0

vi2 kπWi (fjk0 )k2 > β,

i=nk0

which is a contradiction.



3. Perturbation of Ultra Bessel sequence of subspaces In this section we present some perturbation results for ultra Bessel sequences of subspaces. Proposition 3.1. Let {Wi }∞ i=1 be an ultra Bessel sequence of subspaces in H with respect to ∞ ∞ f {vi }i=1 and {Wi }i=1 be a sequence of closed subspaces of H. If there exist λ ≥ 0, µ < 1 such that

X

X

X  



2 2 (f ) + µ v π (f ) (3.1) (f ) ≤ λ v π vi2 πWi (f ) − πW

,

fi fi i W i Wi i∈J

i∈J

i∈J

fi }∞ is an ultra Bessel sequence of subspaces in for all f ∈ H and all finite J ⊆ N, then {W i=1 ∞ H with respect to {vi }i=1 . fi }∞ is a Bessel sequence of subspaces. Let J be a finite subset Proof. We first prove that {W i=1 of N. By (3.1) we have

X

1 + λ X



2 2 vi πW vi πWi (f ) , f ∈ H.

fi (f ) ≤ 1−µ i∈J

i∈J

{Wi }∞ i=1 .

Let B be the Bessel bound of Then

D X

X E

2 2 v π (f ) = sup v π (f ), g



i Wi i Wi kgk=1

i∈J

i∈J

X D E = sup vi2 πWi (f ), πWi (g) kgk=1

≤ sup kgk=1

i∈J

X

vi2 kπWi (g)k2

1  X 2

i∈J

vi2 kπWi (f )k2

1 2

i∈J

1 √ X 2 2 ≤ B vi kπWi (f )k2 . i∈J

So

X

1 + λ√  X 1 2

vi2 πW B vi2 kπWi (f )k2 ,

fi (f ) ≤ 1−µ i∈J

f ∈ H.

i∈J

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ULTRA BESSEL SEQUENCES OF SUBSPACES IN HILBERT SPACES

P 2 Hence ∞ fi (f ) converges unconditionally for all f ∈ H, and (3.1) holds for all infinite i=1 vi πW subsets of N. Let us consider Γ : H → H,

Γ(f ) =

∞ X

vi2 πW fi (f ).

i=1

Then Γ is a well defined bounded operator with kΓk ≤ ∞ X

1+λ B and 1−µ

2 2 vi2 kπW fi (f )k ≤ kΓkkf k ,

f ∈ H.

i=1

fi }∞ is a Bessel sequence of subspaces for H with respect to {vi }∞ . Let Therefore {W i=1 i=1 f ∈ H and kf k = 1. Similar to above we have ∞ X

2 vi2 kπW fi f k

i=n

∞ ∞

DX E X

2 2 = vi π W vi πW fi (f ), f ≤ fi (f ) i=n

i=n

∞

1 + λ

X 2

≤ vi πWi (f )

1−µ i=n

∞ 1 1 + λ√  X 2 2 → 0, B ≤ vi kπWi (f )k2 1−µ i=n

as n → ∞, and this completes the proof.



Corollary 3.2. Let {Wi }∞ i=1 be an ultra Bessel sequence of subspaces in H with respect to ∞ ∞ f {vi }i=1 , and {Wi }i=1 be a sequence of closed subspaces of H. If there exists λ ≥ 0 such that

X 

X





2 2 v π (f ) − π (f ) ≤ λ v π (f )



, Wi fi i i Wi W i∈J

i∈J

fi }∞ is an ultra Bessel sequence of subspaces in for all f ∈ H and all finite J ⊆ N, then {W i=1 ∞ H with respect to {vi }i=1 Theorem 3.3. Let {Wi }∞ i=1 be an ultra Bessel sequence of subspaces in H with respect to ∞ ∞ f {vi }i=1 and {Wi }i=1 be a sequence of closed subspaces of H. If there exists λ ≥ 0 such that

X

X



fi vi (fi − gi ) ≤ λ vi fi , fi ∈ Wi , gi ∈ W (3.2)

i∈J

i∈J

fi }∞ is an ultra Bessel sequence of subspaces in for all f ∈ H and all finite J ⊆ N, then {W i=1 ∞ H with respect to {vi }i=1 . Proof. Let J be a finite subset of N. By (3.2) we have

X

X



fi . vi gi ≤ (1 + λ) vi fi , fi ∈ Wi , gi ∈ W

i∈J

i∈J

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M. R. ABDOLLAHPOUR, A. SHEKARI, C. PARK, D. Y. SHIN

Let B be the Bessel bound for {Wi }∞ i=1 . We have

X

D X E

vi fi = sup vi fi , g

kgk=1

i∈J

i∈J

XD E = sup fi , vi πWi (g) kgk=1

i∈J

X

≤ sup kgk=1

vi2 kπWi (g)k2

1  X 2

kfi k2

1 2

i∈J

i∈J

1 √ X 2 kfi k2 . ≤ B i∈J

So

X

1 √ X

2 2 kfi k vi gi ≤ (1 + λ) B ,

fi . fi ∈ Wi , gi ∈ W

i∈J

i∈J

P  ∞ ∞ ∈ f v g converges unconditionally for all {g } ⊕ W i i=1 i 2 , and (3.2) holds i=1 i i i=1 ` P  P  ∞ ∞ ∞ ∞ f for all {fi }i=1 ∈ i=1 ⊕Wi 2 and {gi }i=1 ∈ i=1 ⊕Wi 2 . In this case the operator Hence

P∞

`

K:

∞ X

`

fi ⊕W



i=1

`2

K({gi }∞ i=1 ) =

→ H,

√

∞ X

vi gi

i=1

is well defined and bounded with kKk ≤ (1 + λ) B and its adjoint ∞ X  ∗ ∞ fi K : ⊕W → H, K ∗ (f ) = {vi πW fi (f )}i=1 2 `

i=1

will be bounded and

∞ X

2 ∗ 2 vi2 kπW fi (f )k ≤ kK kkf k ,

f ∈ H.

i=1

fi }∞ is a Bessel sequence of subspaces for H with respect to {vi }∞ . Let f ∈ H Therefore {W i=1 i=1 and kf k = 1. Then ∞ ∞ X  X  ∞ ∞ fi , {vi πWi (f )}i=1 ∈ ⊕Wi 2 , {vi πW ⊕W fi (f )}i=1 ∈ 2 i=1

`

i=1

`

and ∞ X i=n

2 vi2 kπW fi (f )k

∞

X

2 ≤ vi πW fi (f ) i=n ∞ ∞

X

  X



2 ≤ vi π W vi2 πWi f fi (f ) − πWi (f ) + i=n

i=n

∞

X

≤ (1 + λ) vi2 πWi (f ) i=n ∞ 1 √ X 2 vi2 kπWi (f )k2 ≤ (1 + λ) B i=n

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ULTRA BESSEL SEQUENCES OF SUBSPACES IN HILBERT SPACES

and this completes the proof.



Proposition 3.4. Let {Wi }∞ i=1 be an ultra Bessel sequence of subspaces in H with respect to ∞ ∞ f {vi }i=1 and {Wi }i=1 be a sequence of closed subspaces of H. If there exist M > 0 such that X X 2 vi2 kπWi (f ) − πW vi2 kπWi (f )k2 , f ∈ H, fi (f )k ≤ i∈J

i∈J

fi }∞ is an ultra Bessel sequence of subspaces in H with respect to all finite J ⊆ N, then {W i=1 {vi }∞ i=1 . Proof. For n ∈ N and for f ∈ H we have ∞ ∞ ∞ X X X 2 2 2 vi2 kπW (f )k ≤ 2 v kπ (f ) − π (f )k + 2 vi2 kπWi (f )k2 Wi fi fi i W i=n

i=n

i=n

≤ 2(M + 1)

∞ X

vi2 kπWi (f )k2 ,

i=n

as desired.

 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), 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) References 1. P. G. Casazza and G. Kutyniok, Frame of subspaces, Contempt. Math. Amer. Math. Soc., Povidence 345 (2004), 87–113. 2. O. Christensen, H. O. Kim and R. Y. Kim, Extensions of Bessel sequences to dual pairs of frames, Appl. Comput. Harmon. Anal. 34 (2013), 224–233. 3. I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271–1283. 4. R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. 5. M. H. Faroughi and A. Najati, Ultra Bessel sequences in Hilbert spaces, Southeast Asian Bull. Math. 32 (2008), 425–436. 6. M. H. Faroughi and M. Rahmani, Bochner pq-frames, J. Inequal. Appl. 2012, Art. No. 2012:196. 7. A. Fereydooni and A. Safapour, Pair frames, Results Math. (in press). 8. P. Gavruta, On the duality of fusion frames, J. Math. Anal. Appl. 333 (2007), 871–879. 9. J.-Z. Li and Y.-C. Zhu, Some equalities and inequalities for g-Bessel sequences in Hilbert spaces, Appl. Math. Lett. 25 (2012), 1601–1607. 10. T. Strohmer and Jr. R. Heath, Grassmanian frames with applications to coding and communications, Appl. Comput. Harmon. Anal. 14 (2003), 257–275.

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On Fixed Point Generalizations to Partial b-metric Spaces Thabet Abeljawad Department of Mathematical Sciences, Prince Sultan University Riyadh, Saudi Arabia 11586 E-mail: [email protected] Kamaleldin Abodayeh Department of Mathematical Sciences, Prince Sultan University Riyadh, Saudi Arabia 11586 E-mail: [email protected] Nabil M. Mlaiki Department of Mathematical Sciences, Prince Sultan University Riyadh, Saudi Arabia 11586 E-mail: [email protected] Abstract We show that many of the recent proved fixed point results in partial b−metric spaces can be obtained by the corresponding ones in b−metric spaces by introducing a simple method. On the other hand, we prove point out that, in general, our method is not applicable for certain partial b-metric fixed point theorems.

1

Introduction and Preliminaries

Starting from the Banach contraction principle huge number of fixed point theorem generalisations have appeared in literature (see [20, 21, 22, 23] and the references therein). Such generalisations required to work on more general metric type spaces such as partial metric spaces (see [14, 15, 16, 17, 18, 24, 25, 26, 27, 28] and the references therein), G−metric spaces [13], cone metric spaces (see [14, 9, 10] and the references therein). On the other hand and after then, several articles have been published where simple methods were presented to reobtain the fixed point theorems in these general metric type spaces by using corresponding ones in metric spaces. However, those simple methods have failed in some cases and hence it remained of interest to work is such spaces. Sometimes, these spaces have been provided 1

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with a partial ordering (see [29] and the references therein). In this article, We show that many of the recent proved fixed point results in partial b−metric spaces can be obtained by the corresponding ones in b−metric spaces by introducing a simple method. On the other hand, we conclude that our method is not applicable for certain partial b-metric fixed point theorems. The details will be presented inside the main result and final conclusions sections. We recall some definitions of partial metric spaces, b−metric spaces and partial b−metric spaces and state some of their properties. A partial metric space (PMS) is a pair (X, p), where p : X × X → R+ and R+ denotes the set of all non negative real numbers, such that (P1) p(x, y) = p(y, x) (symmetry); (P2) If 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)

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. [24] (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 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 T : X → X is said to be continuous at x0 ∈ X, if for every ε > 0, there exists δ > 0 such that T (Bp (x0 , δ)) ⊂ Bp (T (x0 ), ε). Lemma 1.1. [24] (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 ) = lim p(xn , xm ).

n→∞

n→∞

n,m→ ∞

2

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(2)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Definition 2. [5] Let X be a (nonempty) set and s ≥ 1 be a given real number. A function db : X × X → R+ is a b−metric if, for all x, y, z ∈ X, the following conditions hold: • (b1 ) x = y if and only if db (x, y) = 0, • b2 ) db (x, y) = db (y, x), • (b3 ) db (x, y) ≤ s[db (x, z) + db (z, y)]. The pair (X, db ) is called a b−metric space. Example 1. Let (X, d) be a cone metric space over a closed normal cone P with constant s ≥ 1, in a real Banach space (E, k.k). Then db (x, y) = kd(x, y)k, for all x, y ∈ X, defines a b−metric on X. Definition 3. [1] Let X be a (nonempty) set and s ≥ 1 be a given real number. A function pb : X × X → R+ is a partial b−metric if, for all x, y, z ∈ X, the following conditions are satisfied: • (pb1 ) x = y if and only if pb (x, y) = pb (x, x) = pb (y, y), • (pb2 ) pb (x, x) ≤ pb (x, y), • (pb3 ) pb (x, y) = pb (y, x), • (pb4 ) pb (x, y) ≤ s[pb (x, z) + pb (z, y) − pb (z, z)] −

1−s [pb (x, x) 2

+ pb (y, y)].

The pair (X, pb ) is called a partial b−metric space. Since s ≥ 1, from (pb4 ) we have pb (x, y) ≤ s[pb (x, z) + pb (z, y) − pb (z, z)] ≤ s[pb (x, z) + pb (z, y)] − pb (z, z). Proposition 1.2. [1] Every partial b−metric pb defines a b−metric dpb , where dpb (x, y) = 2pb (x, y) − pb (x, x) − pb (y, y), for all x, y ∈ X. For more information about partial b−metric spaces and some fixed point theorems, see [1, 3]. Let (X, , db ) be an ordered b−metric space and let f : X → X be a mapping. Set Ms (x, y) = max{db (x, y), db (x, f x), db (y, f y),

1 [db (x, f y) + db (y, f x)]} 2s

(3)

and Ndb (x, y) = min{db (x, f x), db (x, f y)}.

(4)

Definition 4. [6]A function ψ : [0, ∞) → [0, ∞) is called an alternating distance function if the following properties are satisfied: 3

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• ψ is continuous and nondecreasing, and • ψ(t) = 0 if and only if t = 0. Definition 5. [2] Let (X, db ) be a b−metric space. We say that a mapping f : X → X is an almost generalized (ψ, ϕ)s −contractive mapping if there exist L > 0 and two altering distance functions ψ and φ such that ψ(sdb (f x, f y)) ≤ ψ(Ms (x, y)) − ϕ(Ms (x, y)) + Lψ(Ndb (x, y)),

(5)

for all x, y ∈ X. The following is Theorem 3 in [2]. Theorem 1.3. [2] Let (X, db , ) be a partially ordered complete b-metric space. Let f : X → X be a non-decreasing continuous mapping with respect to . Suppose that f satisfies condition 5, for all comparable elements x, y ∈ X. If there exists x0 ∈ X such that x0  f x0 , then f has a fixed point.

2

Main result

In this section we prove the following two useful lemmas that allow us to prove partial b-metric fixed point results by b−metric ones. Lemma 2.1. Let (X, pb ) be a partial b−metric space. Then the function db : X ×X → [0, ∞) defined by db (x, y) = {

0 if x = y pb (x, y) if x 6= y,

(6)

is a b-metric on X such that τdpb ⊆ τdb . Moreover, (X, db ) is complete if and only if (X, pb ) is 0−complete. Proof. It is clear that db (x, y) = 0 if and only if x = y and db (x, y) = db (y, x) for all x, y ∈ X. To prove the triangle inequality, let x, y, z ∈ X. Then db (x, y) ≤ pb (x, y) ≤ s[pb (x, z) + pb (y, z) − pb (z, z)] and the following cases are observed • if x 6= y and x = z then db (x, y) = pb (x, y) ≤ s[pb (z, z) + pb (z, y) − pb (z, z)] = db (z, y) = db (x, y). • if x 6= y and y = z then db (x, y) = pb (x, y) ≤ s[pb (z, z) + pb (x, z) − pb (z, z)] = db (x, z) = db (x, y). • if x = y then db (x, y) = 0 ≤ s[db (x, z) + db (z, y)]. Thus (x, db ) is a b−metric space. The other parts are similar to those in Proposition 2.1 in [4].

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Let (X, pb ) be a partial metric space and (x, dp ) be its corresponding b−metric space, as defined in Lemma 2.1, and x, y ∈ X. Define Mdb (x, y) = max{dp (x, y), db (x, T x), db (y, T y),

1 [db (x, T y) + db (y, T x)]} 2s

(7)

Mpb (x, y) = max{pb (x, y), pb (x, T x), pb (y, T y),

1 [pb (x, T y) + pb (y, T x)]}. 2s

(8)

and

Lemma 2.2. Mpb (x, y) = Mdb (x, y) for all x, y ∈ X with x 6= y. Proof. First, we show that Mpb (x, y) ≤ Mdb (x, y) for all x, y ∈ X with x 6= y. The cases that Mpb (x, y) = pb (x, y) or Mpb (x, y) = pb (x, T x) or Mpb (x, y) = pb (y, T y) can be easily proved 1 by using that pb (x, x) ≤ p(x, y) as done in Lemma 2.2 in [4]. If Mpb (x, y) = 2s [pb (x, T y) + pb (y, T x)] then the we have the following cases • if x = T y and y = T x, then Mpb (x, y) = Mdb (x, y).

1 [p (x, x) 2s b

+ pb (y, y)] ≤ 1s pb (x, y) ≤ db (x, y) ≤

• if x = T y and y 6= T x, then by the help of triangle inequality in (X, pb ) we have 1 1 Mpb (x, y) = 2s [pb (x, x) + pb (y, T x)] ≤ 2s [spb (x, x) + pb (y, T x)] ≤ 12 [pb (x, y) + pb (x, T x)] 1 which is = 2 [db (x, y) + db (x, T x)] if T x 6= x or ≤ pb (x, y) = db (x, y) if T x = x. In both cases we conclude that Mpb (x, y) ≤ Mdb (x, y). • if x 6= T y and y = T x, then by the help of triangle inequality in (X, pb ) we have 1 1 [pb (x, T y) + pb (y, y)] ≤ 2s [pb (x, T y) + spb (y, y)] ≤ 21 [pb (x, y) + pb (y, T y)] Mpb (x, y) = 2s which is = 21 [db (x, y) + db (y, T y)] if T y 6= y or ≤ pb (x, y) = db (x, y) if T y = y. In both cases we conclude that Mpb (x, y) ≤ Mdb (x, y). The other way inequality ”Mdb (x, y) ≤ Mpb (x, y)” is direct and similar to that in the proof of Lemma 2.2 in [4] or follows by noting that db (x, y) ≤ pb (x, y) for all x, y ∈ X. In closing, assume that x 6= T y and y 6= T x. It is not difficult to see that pb (x, y) = db (x, y) for all x 6= y. Hence, Mpb (x, y) = Mdb (x, y) as desired. Definition 6. Let (X, pb ) be an ordered partial b-metric space. We say that the mapping T : X → X is a generalized (ψ, ϕ)s -weakly contractive mapping if there exist two altering distance functions ψ and ϕ such that ψ(spb (T x, T y)) ≤ ψ(Mpb (x, y)) − ϕ(Mpb (x, y)) for all comparable x, y ∈ X. Next theorem was proved in [1], but we give a much shorter proof using our previous two lemmas. Theorem 2.3. Let (X, , pb ) be a pb -complete ordered partial b-metric space. Let T : X → X be a nondecreasing, with respect to , continuous mapping. Suppose that T is a generalized (ψ, ϕ)s -weakly contractive mapping. If there exists x0 ∈ X such that x0  T x0 , then T has a fixed point. 5

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Proof. Using the facts that db (x, y) ≤ pb (x, y) and the results of Lemma 2.2 we obtain: ψ(sdb (x, y)) ≤ ψ(spb (x, y)) ≤ ψ(Mpb (x, y)) − ϕ(Mpb (x, y)) ≤ ψ(Mpb (x, y)) − ϕ(Mpb (x, y)) + Lψ(Ndb (x, y)) = ψ(Ndb (x, y)) − ϕ(Mdb (x, y)) + Lψ(Ndb (x, y)). Therefore, all the hypothesis of Theorem1.3, are satisfied. Thus, T has a fixed point as required. In our next theorem we prove the results of Theorem 2.3, by replacing the continuity condition. Theorem 2.4. Let (X, , pb ) be a pb -complete ordered partial b-metric space. Let T : X → X be a nondecreasing, with respect to , and whenever {xn } is a nondecreasing sequence in X such that xn → x ∈ X, we have xn  x for all natural numbers n. Suppose that T is a generalized (ψ, ϕ)s -weakly contractive mapping. If there exists x0 ∈ X such that x0  T x0 , then T has a fixed point. Proof. Similarly to the argument in the proof of Theorem 2.3, we have ψ(db (x, y)) ≤ ψ(Mdb (x, y)) − ϕ(Mdb (x, y)) + Lψ(Ndb (x, y)). Hence, all the hypothesis of Theorem 4 in [2], are satisfied. Thus, T has a fixed point as desired. Next, we give the following corollary. Corollary 2.1. Let (X, , pb ) be a pb -complete ordered partial b-metric space. Let T : X → X be a nondecreasing, with respect to , continuous mapping. Suppose there exists k ∈ [0, 1) such that pb (T x, T y) ≤

k pb (x, T x) + pb (y, T y) max{pb (x, y), pb (x, T x), pb (y, T y), } 2 2s

for all comparable elements x, y ∈ X. If there exists x0 ∈ X such that x0  T x0 , then T has a fixed point. Proof. If we take ψ(t) = t and ϕ(t) = (1 − k)t for all t ∈ [0, ∞), then the result follows from Theorem 2.3. Now, we state this trivial corollary. Corollary 2.2. Assume all the hypothesis of Corollary 2.1 without the continuity condition, and suppose that whenever {xn } is a nondecreasing sequence in X such that xn → x ∈ X, we have xn  x for all natural numbers n. Then T has a fixed point in X.

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Remark 1. Notice that the equality Ndb (x, y) = min{db (x, f x), db (x, f y)} ≤ Npb (x, y) = min{pb (x, f x), pb (x, f y)}

(9)

does not hold in general. For example, if f y = x then Ndb (x, y) = 0 6= Npb (x, y) in general. Take R+ with pb (x, y) = max{x, y}. Thus, our method can not be applied to generalise Theorem 1.3 from partially ordered b-metric spaces to partially ordered partial b-metric spaces. It would be interesting to go though this generalisation.

3

Final Conclusion

In the literature of fixed point theory many fixed point theorems have been generalized from the metric space to more general structures of metric spaces such as cone metric spaces, partial metric spaces, cone b-metric spaces, G−metric spaces and partial b−metric spaces. After then, some authors have started to develop easier techniques to reprove these generalisations by using the corresponding fixed point theorems in metric or b−metric spaces (see [4, 7, 8, 9, 10, 11, 12, 13] ). However, those techniques fail to work in some cases and there still some fixed point theorem of special interest in these general metric space structures (see for example [14, 15, 16, 17, 18, 19]). Along with the above, In this article, we have introduced a method to reprove some fixed point theorems in partially ordered b− metric spaces [1] using the corresponding ones in partially ordered b−metric spaces [2] and in Remark 1 we mentioned that the partially ordered b−metric fixed point theorems can not be generalised to partially ordered b−metric spaces by simply applying our method. In the same direction, it would be interesting to generalise the contraction principles in [15] or more generally in [16] from partial metric spaces to partial b−metric spaces.

References [1] Mustafa Z., Roshan J.R., Parvaneh V. and Kadelburg Z., Some common fixed point esults in ordered partial b−metric spaces, Journal of Iequalities and Applications, 2013, 2013:562. [2] Roshan J.R., Parvaneh V., Sedghi S., Shobkol N. and Shatanawi W., Common fixed points of almost generalized (ψ, ϕ)s −contractive mappings in ordered b−metric spaces, Fixed Point Theory Appl. 2013, 159 (2013). [3] Aghajani A., Abbas M. and Roshan J.R., Common fixed point of generalized weak contractive mappings in partially ordered b−metric spaces,Math. slovaca (2012) (in press). [4] Haghi R.H., Rezapour S. and Shahzad N., Be careful on partial metric fixed point results, Toplogy and its Applications, 160 (2013) 450-454. [5] Czerwik S., Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav 1, 5-11 (1993). 7

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[6] Khan M.S., Swaleh M. and Sessa S., Fixed point theorems by alternating distances between the points. Bull. Aust. Math. Soc. 30 1-9 (1984). [7] Samet B., Calogero Vetro and Francesca Vetro, From metric spaces to partial metric spaces, Fixed Point Theory and Applications 2013, 2013:5. [8] Wei-Shih D. and Karapınar E., A note on cone b-metric spacesand its related results: generalisations or equivalence?, Fixed Point Theory and Applications 2013, 2013:210. [9] Du W.S., A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (5), 2259-2261 (2010). [10] Abdeljawad T., Karapınar E., A gap in the paper ”A note on cone metric fixed point theory and its equivalence,[ Nonlinear Anal. 72 (5), 2259-2261], Gazi Univ. J.Sci., 24 (2), 233-234 (2011). [11] Berinde V. and Choban M., Generalized distances and their associate metrics. Impact on fixed point theory, Creat. Math. Inform. 22 (2013) No. 1, 22-32. [12] Haghi R.H., Rezapour S. and Shahzad N., Some fixed point generalizations are not real generalizations, Nonlinear Anal. 74 (2011) 1799-1803 [13] Samet B., Vetro, C. and Vetro F., Remarks on G-Metric Spaces, Int. J. Anal.(2013) Article ID 917158, 6 pages. [14] Abdeljawad T. and Rezapour S., Some Fixed Point Results in TVS-Cone Metric Spaces, Fixed Point Theory, 14(2013), No. 2, 263-268. [15] Ili´c D., Pavlovi´c V. and Rako˘cevi´c V., Some new extensions of Banach’s contraction principle to partial metric spaces, Appl. Math. Lett. 24 (2011), 1326–1330. [16] Abdeljawad T. and Zaidan Y., A φ-contraction principle in partial metric spaces with self-distance terms, Universal Journal of Mathematics and Mathematical Sciences, 2014. [17] Abdeljawad T., Alzabut J.O., Mukheimer E., Zaidan Y., Best proximity points for cyclical contraction mappings with boundedly compact decompositions in partial metric spaces, Journal of Computational analysis and Applications, vol 15 (4), 678-685 (2013). [18] Abdeljawad T., Alzabut J.O., Mukheimer E., Zaidan Y., Banach contraction principle for cyclical mappings on partial metric spaces, Fixed Point Theory and Applications (2012). Fixed Point Theory and Applications, 2012, 2012:154 . [19] Karapınar E., Agarwal R.P., Further fixed point results on G-metric spaces, Fixed Point Theory and Applications 2013, 2013:154. [20] Boyd D.W. and Wong S.W., On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969),458–464. [21] Abdeljawad T., Meir-Keeler α-contractive fixed and common fixed point theorems, Fixed Point Theory and Applications 2013, 2013:19, 10 pages. 8

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[22] Rus I.A., Generalizeed Contractions and Applications, Cluj University Press, ClujNapoca, (2001). [23] Romaguera S., Fixed point theorems for generalized contractions on partial metric spaces, Topology Appl. 159 (2012), 194-199. [24] Matthews S.G., Partial metric topology, in Proceedings of the 11th Summer Conference on General Topology and Applications, vol. 728, pp. 183197, The New York Academy of Sciences, Gorham, Me, USA, August 1995. [25] Abdeljawad T., Karapinar E. and Ta¸s K., A generalized contraction principle with control functions on partial metric spaces, 63 (3) (2012), 716-719 . [26] Abdeljawad T., Fixed points for generalized weakly contractive mappings in partial metric spaces, Math. Comput. Modelling 54 (11-12) (2011), 2923–2927. [27] Matthews S.G., Partial metric topology. Research Report 212. Dept. of Computer Science. University of Warwick, 1992. ´ Raji´c, and S. Radenovi´c, Nonlinear Contractions in [28] Ahmad A.G.B., Fadail Z.M., V.C. 0-Complete Partial Metric Spaces, Abstract and Applied Analysis Volume 2012, Article ID 451239, 12 pages, 2012. [29] Shatanawi W., Samet B. and Abbas M., Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces, Math. Comput. Modelling, doi: 10.1016/j.mcm.2011.08.042.

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Existence of solutions for fractional four point boundary value problems with p-Laplacian operator Erbil Cetin and Fatma Serap Topal Department of Mathematics, Ege University, 35100 Bornova, Izmir-Turkey [email protected] ; [email protected] June 10, 2014 Abstract In this paper, we are concerned with proving the existence of solutions for four point fractional differential equations with p-Laplacian operator. We obtain the existence of at least one solution for the problem applying Schauder fixed-point theorem.

Keywords: Fractional differential equations, p-Laplacian operator, Boundary value problem, Existence of solutions.

1

Introduction Fractional differential equations have been of greet interest due to its numerous applications in

physics, chemistry, engineering, economics and other fields. We can find many applications in viscoelasticity, electrochemistry, systems control theory, electromagnetism, bioscience and diffusion processes etc. There has been a significant development in the study of fractional differential equations in recent years [5, 6, 7, 8, 9, 10, 16]. On the other hand, p-Laplacian problems with two-point, three-point and multi-point boundary value conditions for ordinary differential equations and finite difference equations have been studied extensively, see [17, 18, 19] and the references therein. Besides them, integer order p-Laplacian boundary value problems have been studied to its importance in theory and application of mathematics, physics and so on, see for example [11, 12, 13, 14, 15] and references therein. However, there are few articles dealing with the existence of solutions to boundary value problems for fractional differential equation with p-Laplacian operator. In [3], applying Krasnosel’skii and Leggett-Williams fixed point theorems, the authors obtained some sufficient conditions for the existence of positive solutions for three point boundary value problems D0β+ (φp (D0α+ u))(t) + f (t, u(t)) = 0, t ∈ (0, 1), u(0) = 0, u(1) = au(ξ), D0α+ u(0) = 0. 1 892

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2

In [1], applying the Schauder fixed point theorem, the authors investigated the existence of solutions for a system of nonlinear fractional differential equations with three point boundary conditions Dα u(t) = f (t, v(t), Dp v(t)), t ∈ (0, 1), Dβ v(t) = f (t, u(t), Dq u(t)), t ∈ (0, 1), u(0) = 0, u(1) = γu(η), v(0) = 0, v(1) = γv(η), where 1 < α, β < 2, p, q, γ > 0, 0 < η < 1, α − p ≥ 1, β − q ≥ 1, γη α−1 < 1, γη β−1 < 1, f, g : [0, 1] × R × R → R are given continuous functions. In [4], by using upper and lower solutions method, the authors investigated the existence of positive solutions for nonlocal four point problem. D0β+ (φp (D0α+ u))(t) + f (t, u(t)) = 0, t ∈ (0, 1), u(0) = 0, u(1) = au(ξ), D0α+ u(0) = 0, D0α+ u(1) = bD0α+ u(η) In this paper, we consider the existence of at least one solution for the following four point fractional differential equations with p-Laplacian operator   Dβ+ (φp (Dα+ u))(t) + f (t, u(t), Dγ+ u(t)) = 0, t ∈ (0, 1), 0 0 0  u(0) = 0, Dr1 u(0) − au(ξ) = 0, Dr2 u(1) + bu(η) = 0, Dα u(0) = 0, 0+ 0+ 0+

(1.1)

1 , D0r2+ and D0γ+ are the standard Riemann-Liouville derivative with α ∈ (2, 3], where D0β+ , D0α+ , D0r+

β ∈ (0, 1],r1 , r2 ∈ (0, 1], α − r1 − 2 ≥ 0, α − r2 − 2 ≥ 0, α − γ − 2 ≥ 0, the costants ξ, η, a, b are positive numbers with 0 < ξ ≤ η < 1 and

Γ(α−1) Γ(α−r2 −1)

φp (s) = |s|p−2 s, p > 1 such that (φp )−1 = φq with

1 p

+ bη α−2 > 0, φp is p-Laplacian operator, i.e +

1 q

= 1 and f : [0, 1] × R × R → R is given

continuous function. In section 2, we present some necessary definitions and preliminary results that will be used to prove our main result. In Section 3, we put forward and prove our main result.

2

Preliminaries In this section we collect some preliminary definitions and results that will be used in subsequent

section. Firstly, for convenience of the reader, we give some definitions and fundamental results of fractional calculus theory. Definition 2.1. The Riemann-Liouville fractional integral of order α > 0 of a function y : (0, ∞) → R is given by Iaα+ y(t)

1 = Γ(α)

∫

t

(t − s)α−1 y(s)ds, a

where Γ denotes the gamma function, provided that the right side integral exists.

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3 Definition 2.2. The Riemann-Liouville fractional derivative of order α > 0 of a function y : (0, ∞) → R is given by Daα+ y(t) =

1 Γ(n − α)

(

d dt

)n ∫

t

a

y(s) ds, (t − s)α−n+1

where n = [α] + 1 with [α] denotes the integer part of α and Γ denotes the gamma function, provided that the right side is pointwise defined on (0, ∞). Lemma 2.1. Let α, β > 0, y : (0, ∞) → R is a continuous function and assume that the RiemannLiouville fractional integral and derivative of y exist , then y(t). I0α+ I0β+ y(t) = I0α+β + Lemma 2.2. Let α > 0. If y ∈ C(0, 1) ∩ L(0, 1) possesses a fractional derivative of order α that belongs to C(0, 1) ∩ L(0, 1), then I0α+ D0α+ y(t) = y(t) + c1 tα−1 + c21 tα−2 + ...cn tα−n , for some ci ∈ R, i = 1, 2, ..., n, where n = [α] + 1. For finding a solution of the problem (1.1), we first consider the following fractional differential equation D0α+ u(t) + h(t) = 0, t ∈ (0, 1), 1 u(0) = 0, D0r+ u(0) − au(ξ) = 0, D0r2+ u(1) + bu(η) = 0,

where h ∈ C([0, 1], R). Lemma 2.3. Given h(t) ∈ C(0, 1) and 2 < α < 3. Then the unique solution of   Dα+ u(t) + h(t) = 0, t ∈ (0, 1), 0  u(0) = 0, Dr1 u(0) − au(ξ) = 0, Dr2 u(1) + bu(η) = 0, 0+

(2.2)

0+

is given by

∫ u(t) =

1

G(t, s)h(s)ds, 0

where   G1 (t, s), t ≤ s; G(t, s) =  G (t, s), s ≤ t, 2         G1 (t, s) =

      

Γ(α−1) a dΓ(α) { Γ(α−r2 −1) (t

−

α−1 α−r2 −1 )

(2.3)

+ bη α−2 (t − η)}tα−2 (ξ − s)α−1 +

aξ α−2 dΓ(α−r2 ) (ξ

− t)tα−2 (1 − s)α−r2 −1 +

abξ α−2 dΓ(α) (ξ

− t)tα−2 (η − s)α−1 , s ≤ ξ;

aξ α−2 dΓ(α−r2 ) (ξ

− t)tα−2 (1 − s)α−r2 −1 +

abξ α−2 dΓ(α) (ξ

− t)tα−2 (η − s)α−1 , ξ ≤ s ≤ η;

aξ α−2 dΓ(α−r2 ) (ξ

− t)tα−2 (1 − s)α−r2 −1 ,

894

(2.4)

η ≤ s,

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4

and  Γ(α−1) α−1 a  α−2  (t − η)}tα−2 (ξ − s)α−1  dΓ(α) { Γ(α−r2 −1) (t − α−r2 −1 ) + bη    α−2  aξ α−2 α−2  (ξ − t)tα−2 (1 − s)α−r2 −1 + abξ (η − s)α−1 + dΓ(α−r  dΓ(α) (ξ − t)t  2)     − a (t − s)α−1 , dΓ(α) G2 (t, s) = α−2 α−2 aξ  α−2  dΓ(α−r ) (ξ − t)tα−2 (1 − s)α−r2 −1 + abξ (η − s)α−1  dΓ(α) (ξ − t)t 2     a α−1  ,   − dΓ(α) (t − s)   α−2  aξ  (ξ − t)tα−2 (1 − s)α−r2 −1 − a (t − s)α−1 , dΓ(α−r2 )

s ≤ ξ;

(2.5)

ξ ≤ s ≤ η; η ≤ s.

dΓ(α)

Proof. Applying Lemma 2.2 to reduce the first equation of (2.2) to an equivalent integral equation α u(t) = C1 tα−1 + C2 tα−2 + C3 tα−3 − I+ h(t)

for some C1 , C2 , C3 ∈ R. Hence, the general solution of Eq. (2.2) is u(t) = C1 tα−1 + C2 tα−2 + C3 tα−3 −

1 Γ(α)

∫

t

(t − s)α−1 h(s)ds. 0

It follows from u(0) = 0 that C3 = 0. By using the second boundary condition D0r1+ u(0) − au(ξ) = 0 we have aξ

α−1

C1 + aξ

α−2

1 C2 = a Γ(α)

∫

ξ

(ξ − s)α−1 h(s)ds

(2.6)

0

and by using the third boundary condition D0r2+ u(1) + bu(η) = 0 we have  { ∫ 1 } { } 1  Γ(α−1) Γ(α)  α−1 α−2  + bη C1 + Γ(α−r2 −1) + bη C2 = (1 − s)α−r2 −1 h(s)ds   Γ(α−r2 ) Γ(α − r2 ) 0     

1 +b Γ(α)

∫

(2.7)

η

(η − s)

α−1

h(s)ds.

0

Solving equation (2.6) and (2.7) we find that {( ) ∫ ξ 1 Γ(α) a α−2 C1 = + bη (ξ − s)α−1 h(s)ds d Γ(α − r2 ) Γ(α) 0 )} ( ∫ 1 ∫ η 1 1 α−r2 −1 α−1 α−2 −aξ (1 − s) h(s)ds + b (η − s) h(s)ds Γ(α − r2 ) 0 Γ(α) 0 and C2

=

1 d −

where d = aξ α−2

{

{

( aξ α−1

(

1 Γ(α − r2 )

∫

Γ(α) + bη α−1 Γ(α − r2 )

Γ(α−1) Γ(α−r2 −1) (ξ

−

1

(1 − s)α−r2 −1 h(s)ds +

0

)

α−1 α−r2 −1 )

a Γ(α)

∫

ξ

∫

)

η

(η − s)α−1 h(s)ds 0

(ξ − s)α−1 h(s)ds 0

+ bη α−2 (ξ − η)

895

b Γ(α) }

} and it is easy to see that d < 0.

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Thus the unique solution of (2.2) is given by { }∫ ξ a Γ(α − 1) α−1 α−2 u(t) = (t − ) + bη (t − η) tα−2 (ξ − s)α−1 h(s)ds dΓ(α) Γ(α − r2 − 1) α − r2 − 1 0 ∫ 1 ∫ η aξ α−2 abξ α−2 + (ξ − t) tα−2 (1 − s)α−r2 −1 h(s)ds + (ξ − t) tα−2 (η − s)α−1 h(s)ds Γ(α − r2 ) dΓ(α) 0 0 ∫ t 1 (t − s)α−1 h(s)ds − Γ(α) 0 ∫ 1 = G(t, s)h(s)ds 0

where G(t, s) is defined by (2.3). Let D0α+ u = ϕ, φp (ϕ) = ω. By Lemma 2.2, the solution of initial value problem D0β+ ω(t) + v(t) = 0, t ∈ (0, 1), ω(0) = 0, is given by ω(t) = c1 tβ−1 − I0β+ v(t), t ∈ (0, 1). From the boundary condition and β ∈ (0, 1], it follows that c1 = 0, and so ω(t) = −I0β+ v(t), t ∈ (0, 1). Noting that D0α+ u = ϕ and ϕ = φ−1 p (ω) = φq (ω), we know that the solution of (1.1) satisfies   Dα+ u(t) = −φq (I β+ v)(t), t ∈ (0, 1), 0 0  u(0) = 0, Dr1 u(0) − au(ξ) = 0, Dr2 u(1) + bu(η) = 0. 0+

(2.8)

0+

Thus the solution of the problem (2.8) can be written as

∫ u(t) = 0

1

G(t, s)φq (I0β+ v)(s)ds, t ∈ (0, 1).

Since v(t) > 0 for t ∈ (0, 1), we have φq (I0β+ v)(s) = (I0β+ v)q−1 (s) and so ∫

1

u(t) = 0

G(t, s)(I0β+ v)q−1 (s)ds =

1 (Γ(β))q−1

∫

∫

1 0

s

(s − τ )β−1 v(τ )dτ )q−1 ds.

G(t, s)( 0

So, we have obtained the following lemma. Lemma 2.4. Let f ∈ C([0, 1] × [0, +∞) × R, [0, +∞)). Then BVP (1.1) has a unique solution of the problem (1.1) if and only if u is a soluton of (∫ s )q−1 ∫ 1 1 γ β−1 u(t) = G(t, s) (s − τ ) f (τ, u(τ ), D0+ u(τ ))dτ ds. (Γ(β))q−1 0 0 Proof. The proof can be seen from Lemma 2.3 immediately, so we omit it. Let C(J) denote the space of all continuous functions defined on J = [0, 1]. Let X = {u(t) : u ∈ C(J)} and D0γ+ u ∈ C(J) be a Banach space endowed with the norm ∥u∥ = maxt∈J |u(t)| + maxt∈J |D0γ+ u(t)|.

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6 Let us define an operator T : C(J) → C(J) as (∫ s )q−1 ∫ 1 1 γ β−1 G(t, s) (s − τ ) f (τ, u(τ ), D0+ u(τ ))dτ T u(t) = ds. (Γ(β))q−1 0 0 Because of the continuity of G(t, s) and f, the operator T is continuous. By Lemma 2.4, we know that the fixed point of operator T coincides with the solution of (1.1). The main result of this paper is as follows: Theorem 2.1. Let f ∈ C([0, 1] × [0, +∞) × R, [0, +∞)) and assume that there exists a nonnegative function g(t) ∈ L(0, 1) such that 1

|f (t, x, y)| ≤ g(t) + (ϵ1 |x|ρ1 + ϵ2 |y|ρ2 ) q−1 where ϵ1 , ϵ2 > 0, 0 < ρ1 , ρ2 < 1 and q is a number with

1 p

+

1 q

= 1 such that p > 1 then the problem

(1.1) has a solution. 1

1

Proof. Define a ball as B = {u(t) : ∥u(t)∥ ≤ R, t ∈ [0, 1]}, where R ≥ max{(3Λϵ1 ) 1−ρ1 , (3Λϵ2 ) 1−ρ2 , 3Υ}, } { {∫ 1 max{2q−2 , 1} β β q−1 α q−1 q−1 Υ: = (Γ(β)) max |G(t, s)|(I0+ (g(s))) ds + I0+ (I0+ (g(t))) (Γ(β))q−1 t∈[0,1] 0 { ( ) }} aΓ(α − 1) α−1 q−1 α−2 q−1 q−1 + e(g(ξ)) +ξ ξ+ (g(1)) + b(g(η)) , |d|Γ(α − γ − 1)β q−1 α−γ−1 max{2q−2 , 1} Λ := (Γ(β + 1))q−1 e := (

{

aΓ(α − 1) ε+ |d|Γ(α − γ − 1)Γ(α + 1)

{ eξ

β(q−1)

+ξ

α−2

( ξ+

α−1 α−γ−1

)

}} (1 + bη

β(q−1)

)+1

,

Γ(α − 1) Γ(α − 1) α−1 + bη α−2 ) +( + η) Γ(α − r2 − 1) Γ(α − γ − 1) α − r2 − 1

and

( ε := 1 +

α−1 α − r2 − 1

)

η β(q−1)+2 + (η − ξ)

{ (η − ξ)

( ) 1 } 1 + bη α+β(q−1) + 1. Γ(α − 1) + bη α−2 Γ(α − r2 − 1)

Firstly, we will prove that T : B → B. By using the inequality (a + b)γ ≤ max{2γ−1 , 1}(aγ + bγ ), for a, b ≥ 0, γ > 0, we get (∫ s )q−1 ∫ 1 1 γ β−1 |T u(t)| ≤ |G(t, s)| (s − τ ) |f (τ, u(τ ), D0+ u(τ ))|dτ ds (Γ(β))q−1 0 0 (∫ s )q−1 ∫ 1 1 1 ρ1 ρ2 q−1 β−1 ≤ |G(t, s)| (s − τ ) (g(τ ) + (ϵ |R| + ϵ |R| ) )dτ ds 1 2 (Γ(β))q−1 0 0 (∫ s )q−1 ∫ 1 ∫ s 1 1 β−1 ρ1 ρ2 q−1 β−1 = |G(t, s)| (s − τ ) g(τ )dτ + (ϵ1 |R| + ϵ2 |R| ) ) (s − τ ) dτ ds (Γ(β))q−1 0 0 0 (∫ s )q−1 ∫ 1 1 sβ 1 β−1 ρ1 ρ2 q−1 ) = |G(t, s)| (s − τ ) g(τ )dτ + (ϵ1 |R| + ϵ2 |R| ) ds (Γ(β))q−1 0 β 0 {∫ (∫ s )q−1 1 max{2q−2 , 1} β−1 ≤ |G(t, s)| ds (s − τ ) g(τ )dτ (Γ(β))q−1 0 0 } ∫ (ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) 1 β(q−1) + |G(t, s)|s ds . β q−1 0

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We consider the integral as by using ∫

1

|G(t, s)|sβ(q−1) ds

≤

0

Since |d| = aξ α−2

{

Γ(α−1) α−1 Γ(α−r2 −1) ( α−r2 −1

∫t 0

(t − s)α sβ ds = tα+β+1 Γ(α+1)Γ(β+1) Γ(α+β+2) ,

a Γ(α − 1) α−1 |{ (t − ) |d|Γ(α) Γ(α − r2 − 1) α − r2 − 1 Γ(α)Γ(β(q − 1) + 1) +bη α−2 (t − η)}|tα−2 ξ α+β(q−1) Γ(α + β(q − 1) + 1) α−2 Γ(α − r2 )Γ(β(q − 1) + 1) aξ |ξ − t|tα−2 + |d|Γ(α − r2 ) Γ(α − r2 + β(q − 1) + 1) α−2 Γ(α)Γ(β(q − 1) + 1) abξ |ξ − t|tα−2 η α+β(q−1) + |d|Γ(α) Γ(α + β(q − 1) + 1) 1 α+β(q−1) Γ(α)Γ(β(q − 1) + 1) t . + Γ(α) Γ(α + β(q − 1) + 1) } α−1 − ξ) + bη α−2 (η − ξ) and α−r − ξ > η − ξ, we get 2 −1

{ } } { Γ(α−1) Γ(α−1) α−2 α−2 α−2 . |d| > aξ α−2 Γ(α−r (η − ξ) + bη (η − ξ) + bη = aξ (η − ξ) −1) Γ(α−r −1) 2 2 Also, since 1 < β(q − 1) + 1 ≤ q, 3 < β(q − 1) + α + 1 ≤ 3 + q and 2 < β(q − 1) + α − r2 + 1 < 3 + q then Γ(β(q − 1) + 1) < Γ(β(q − 1) + α + 1), Γ(β(q − 1) + 1) < Γ(β(q − 1) + α − r2 + 1). Together with the above inequalities, we have ( ) β(q−1)+2 ∫1 η α−1 β(q−1) |G(t, s)|s ds < 1 + + α−r −1 (η−ξ) 0 2 ( If we define ε := 1 +

α−1 α−r2 −1

we get |T u(t)| ≤

max{2q−2 , 1} (Γ(β))q−1

1 { } Γ(α−1) (η−ξ) Γ(α−r −1) +bη α−2

( ) 1 + bη α+β(q−1) + 1.

2

)

η

β(q−1)+2

(η−ξ)

{∫

+

{ (η−ξ)

1 Γ(α−1) +bη α−2 Γ(α−r2 −1)

1

|G(t, s)|(Γ(β)I β (g(s)))q−1 ds + 0

}

( ) 1 + bη α+β(q−1) + 1,

} (ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) × ε ds. β q−1

And also, we get |D0γ+ T u(t)|

)q−1 (∫ s ∫ 1 1 γ β−1 D u(τ ))dτ ds}| G(t, s) (s − τ ) f (τ, u(τ ), 0+ (Γ(β))q−1 0 0 { { } γ 1 a Γ(α − 1) α−1 α−2 D + = (t − ) + bη (t − η) (Γ(β))q−1 0 dΓ(α) Γ(α − r2 − 1) α − r2 − 1 )q−1 (∫ s ∫ ξ γ α−1 β−1 α−2 ds t (ξ − s) (s − τ ) f (τ, u(τ ), D0+ u(τ ))dτ

= |D0γ+ {

0

0 1

(∫ s )q−1 aξ (ξ − t) tα−2 (1 − s)α−r2 −1 (s − τ )β−1 f (τ, u(τ ), D0γ+ u(τ ))dτ ds Γ(α − r2 ) 0 0 (∫ s )q−1 ∫ η abξ α−2 γ α−2 α−1 β−1 + (ξ − t) t (η − s) (s − τ ) f (τ, u(τ ), D0+ u(τ ))dτ ds dΓ(α) 0 0 } (∫ s )q−1 ∫ t 1 (t − s)α−1 (s − τ )β−1 f (τ, u(τ ), D0γ+ u(τ ))dτ ds − Γ(α) 0 0 [( ) a Γ(α − 1) α−γ−2 Γ(α − 1) Γ(α − 1) 1 + bη α−2 t = d Γ(α − γ − 1) t q−1 (Γ(β)) Γ(α − r2 − 1) Γ(α − γ − 1) (∫ )q−1 ( )] ξ α−1 γ α β−1 − + η I0+ (ξ − τ ) f (τ, u(τ ), D0+ u(τ ))dτ α − r2 − 1 0 α−2

∫

+

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) (∫ 1 )q−1 α−1 Γ(α − 1) γ β−1 2 t tα−γ−2 I0α−r (1 − τ ) f (τ, u(τ ), D u(τ ))dτ 0+ α−γ−1 Γ(α − γ − 1) + 0 ( ) (∫ )q−1 η α−1 Γ(α − 1) abξ α−2 γ α−γ−2 α β−1 ξ− t t I (η − τ ) f (τ, u(τ ), D0+ u(τ ))dτ + d α−γ−1 Γ(α − γ − 1) 0+ 0 (∫ t )q−1 γ α β−1 − I0+ (t − τ ) f (τ, u(τ ), D0+ u(τ ))dτ 0 { [( ) 1 a Γ(α − 1) Γ(α − 1) Γ(α − 1) α−2 + bη (Γ(β))q−1 |d| Γ(α − γ − 1) Γ(α − r2 − 1) Γ(α − γ − 1) (∫ )q−1 )] ( ξ α−1 γ α β−1 + η I0+ (ξ − τ ) f (τ, u(τ ), D0+ u(τ ))dτ − α − r2 − 1 0 ( ) (∫ 1 )q−1 α−1 Γ(α − 1) aξ α−2 γ β−1 2 ξ− I0α−r (1 − τ ) f (τ, u(τ ), D u(τ + ))dτ + 0 |d| α − γ − 1 Γ(α − γ − 1) + 0 )q−1 ( ) (∫ η abξ α−2 α−1 Γ(α − 1) γ α β−1 + ξ− I (η − τ ) f (τ, u(τ ), D0+ u(τ ))dτ |d| α − γ − 1 Γ(α − γ − 1) 0+ 0 } (∫ )q−1 +

≤

aξ α−2 d

(

ξ−

t

− I0α+

0

(t − τ )β−1 f (τ, u(τ ), D0γ+ u(τ ))dτ

.

By using the properties I0α+ (f + g)(t) = I0α+ f (t) + I0α+ g(t) and if f ≤ g then I0α+ f (t) ≤ I0α+ g(t), we have (∫ I0α+

)q−1

ξ

(ξ − 0

≤

τ )β−1 f (τ, u(τ ), D0γ+ u(τ ))dτ (∫

I0α+

(ξ − τ )

[g(τ ) + (ϵ1 |R|

(ξ − τ )

g(τ )dτ + (ϵ1 |R|

β−1

(∫ ≤

)q−1

ξ

I0α+

+ ϵ2 |R| )

ρ1

ρ2

1 q−1

]dτ

0 ξ β−1

0

≤ max{2q−2 , 1}

   {

ρ1

(∫ I0α+

+ ϵ2 |R| ) ρ2

)q−1

ξ

(ξ − τ )β−1 g(τ )dτ 0

1 q−1

∫

)q−1

ξ

(ξ − τ )

β−1

dτ

0

(

+ I0α+ (ϵ1 |R|ρ1 + ϵ2 |R|ρ2 )

1 q−1

 )q−1  dτ



}

( )q−1 1 ξβ ≤ max{2q−2 , 1} I0α+ (Γ(β)I0β+ (g(ξ)))q−1 + (ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) q−1 I0α+ (1) β { ( ) } ξ β(q−1) 1 ≤ max{2q−2 , 1} Γ(β)q−1 I0α+ (I0β+ (g(ξ)))q−1 + (ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) q−1 tα . β Γ(α + 1) If we define e := (

|D0γ+ T u(t)| ≤

Γ(α − 1) Γ(α − 1) α−1 + bη α−2 ) +( + η), we get Γ(α − r2 − 1) Γ(α − γ − 1) α − r2 − 1

max{2q−2 , 1} a Γ(α − 1) { e{Γ(β)q−1 I0α+ (I0β+ (g(ξ)))q−1 (Γ(β))q−1 |d| Γ(α − γ − 1) +((ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) +

ξ β(q−1) tα }} ) β q−1 Γ(α + 1)

aξ α−2 α−1 Γ(α − 1) (ξ + ) {Γ(β)q−1 I0α+ (I0β+ (g(1)))q−1 |d| α − γ − 1 Γ(α − γ − 1) 1 tα +((ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) q−1 ) } β Γ(α + 1)

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+

α−1 Γ(α − 1) abξ α−2 (ξ + ) {Γ(β)q−1 I0α+ (I0β+ (g(η)))q−1 |d| α − γ − 1 Γ(α − γ − 1) +((ϵ1 |R|ρ1 + ϵ2 |R|ρ2 )

tα η β(q−1) ) } β q−1 Γ(α + 1)

+{Γ(β)q−1 I0α+ (I0β+ (g(t)))q−1 + ((ϵ1 |R|ρ1 + ϵ2 |R|ρ2 )I0α+ (

≤

max{2q−2 , 1} a Γ(α − 1) g(ξ) q−1 Γ(β(q − 1) + 1) α+β(q−1) { e{( ) t (Γ(β))q−1 |d| Γ(α − γ − 1) β Γ(α + β(q − 1) + 1) +

+

1 ξ β(q−1) (ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) q−1 }} Γ(α + 1) β

aξ α−2 α−1 Γ(α − 1) g(1) q−1 Γ(β(q − 1) + 1) (ξ + ) {( ) tα−r2 +β(q−1) |d| α − γ − 1 Γ(α − γ − 1) β Γ(α − r2 + β(q − 1) + 1) 1 1 ((ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) q−1 )} + Γ(α + 1) β

+

abξ α−2 α−1 Γ(α − 1) g(η) q−1 Γ(β(q − 1) + 1) α+β(q−1) (ξ + ) {( ) t |d| α − γ − 1 Γ(α − γ − 1) β Γ(α + β(q − 1) + 1) +

1 η β(q−1) ((ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) q−1 )} Γ(α + 1) β

+{Γ(β)q−1 I0α+ (I0β+ (g(t)))q−1 + ((ϵ1 |R|ρ1 + ϵ2 |R|ρ2 )

Since

≤

tβ(q−1) ))} β q−1

1 β q−1

Γ(β(q − 1) + 1) α+β(q−1) t )}. Γ(α + β(q − 1) + 1)

Γ(β(q − 1) + 1) Γ(β(q − 1) + 1) < 1 and < 1, then we have Γ(α + β(q − 1) + 1) Γ(α − r2 + β(q − 1) + 1) { { } β(q−1) max{2q−2 , 1} a Γ(α − 1) g(ξ) q−1 1 ρ1 ρ2 ξ e ( ) + (ϵ |R| + ϵ |R| ) 1 2 (Γ(β))q−1 |d| Γ(α − γ − 1) β Γ(α + 1) β q−1 ( ) { } aξ α−2 α−1 Γ(α − 1) g(1) q−1 1 1 + ξ+ ( ) + (ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) q−1 |d| α − γ − 1 Γ(α − γ − 1) β Γ(α + 1) β ( ) { } α−2 β(q−1) abξ α−1 Γ(α − 1) g(η) q−1 1 ρ1 ρ2 η + ξ+ ( ) + (ϵ1 |R| + ϵ2 |R| ) q−1 |d| α − γ − 1 Γ(α − γ − 1) β Γ(α + 1) β } 1 + Γ(β)q−1 I0α+ (I0β+ (g(t)))q−1 + (ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) q−1 β

{ ( ) } aΓ(α − 1) α−1 q−1 α−2 q−1 q−1 ≤ e(g(ξ)) +ξ ξ+ (g(1)) + b(g(η)) |d|Γ(α − γ − 1) α−γ−1 } max{2q−2 , 1} + Γ(β)q−1 I0α+ (I0β+ (g(t)))q−1 + (ϵ1 |R|ρ1 + ϵ2 |R|ρ2 ) (βΓ(β))q−1 { ( ( )( )} ) aΓ(α − 1) α−1 β(α−1) α−2 β(α−1) eξ ξ+ 1 + bη +ξ +1 . |d|Γ(α − γ − 1)Γ(α + 1) α−γ−1 max{2q−2 , 1} (βΓ(β))q−1

{

Thus we get ∥T u(t)∥

=

max |T u(t)| + max |D0γ+ T u(t)|

t∈[0,1]

≤ Υ + (ϵ1 |R|

900

t∈[0,1]

ρ1

+ ϵ2 |R|ρ2 ) Λ.

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1

Since R ≥ max{(3Λϵ1 ) 1−ρ1 , (3Λϵ2 ) 1−ρ2 , 3Υ} then we have ∥T u(t)∥

≤

R R R + + = R. 3 3 3

Hence, we conclude that ∥T u(t)∥ ≤ R. Since T u(t) and D0γ+ T u(t) are continuous operator on [0, 1], therefore T : B → B. Secondly, we will show that T is completely continuous operator. For that we fix M := maxt∈[0,1] |f (t, u(t), D0γ+ )|. Thus we get ∫ s q−1 ( )q−1 M β q−1 (s − τ )β−1 |f (τ, u(τ ), Dγ+ u(τ ))|dτ = |Γ(β)I (f )| ≤ sq−1 . 0 β 0

For t1 , t2 ∈ [0, 1] such that t1 < t2 and u ∈ B, we have |T u(t2 ) − T u(t1 )|

≤

1 (Γ(β))q−1

∫

(∫

1

|G(t2 , s)| 0

)q−1

s

(s − τ ) 0

β−1

|f (τ, u(τ ), D0γ+ u(τ ))|dτ

ds

(∫ s )q−1 ∫ 1 1 γ β−1 |G(t , s)| (s − τ ) |f (τ, u(τ ))|dτ ds u(τ ), D 1 0+ (Γ(β))q−1 0 0 )q−1 {( ) ( Γ(α − 1) ξq M a α−1 α−2 ≤ Γ(q)ξ + bη |tα−1 − tα−1 | 1 βΓ(β) |d| Γ(α − r2 − 1) Γ(α + q) 2 ( ) Γ(α − 1) ξq + + bη α−2 |tα−2 − tα−2 | 1 Γ(α − r2 ) Γ(α + q) 2 1 1 |tα−2 − tα−2 |tα−1 − tα−1 |+ | + 1 1 Γ(α − r2 + q) 2 ξΓ(α − r2 + q) 2 } bη α+q−1 α−2 bη α+q−1 α−1 α−2 α−1 |t |t − t1 | + − t1 | + Γ(α + q) 2 ξΓ(α − r2 + q) 2 ∫ )q−1 (∫ s 1 t2 ds (t2 − s)α−1 (s − τ )β−1 f (τ, u(τ ), D0γ+ u(τ ))dτ + Γ(α) 0 0 (∫ s )q−1 ∫ t1 − (t1 − s)α−1 (s − τ )β−1 f (τ, u(τ ), D0γ+ u(τ ))dτ ds 0 0 )q−1 {( ) ( a Γ(α − 1) ξq M Γ(q)ξ α−1 + bη α−2 |tα−1 − tα−1 | ≤ 1 βΓ(β) |d| Γ(α − r2 − 1) Γ(α + q) 2 ( ) Γ(α − 1) ξq + + bη α−2 |tα−2 − tα−2 | 1 Γ(α − r2 ) Γ(α + q) 2 1 1 |tα−2 − tα−2 |tα−1 − tα−1 + |+ | 1 1 Γ(α − r2 + q) 2 ξΓ(α − r2 + q) 2 } bη α+q−1 bη α+q−1 α−2 |t2 − tα−2 |+ |tα−1 − tα−1 | + 1 1 Γ(α + q) ξΓ(α − r2 + q) 2 ( )q−1 { } M + |(t2 − c)α−1 − (t1 − c)α−1 | + |t2 − t1 | , βΓ(α) −

and |D0γ+ T u(t2 ) − D0γ+ T u(t1 )| ( )q−1 { ( β )q−1 [ ( ) a M 1 ξ Γ(α) Γ(α − 1) ≤ + bη α−2 |tα−γ−1 − tα−γ−1 | 2 1 |d| Γ(β) Γ(α + 1) β Γ(α − γ) Γ(α − r2 − 1) ( ) ] Γ(α − 1) α−1 α−γ−2 α−γ−2 − t1 | + + η |t2 Γ(α − γ − 1) α − r2 − 1

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11 Γ(α − 1) Γ(α) ξ α−1 ξ α−2 |tα−γ−2 |tα−γ−1 − t1α−γ−1 | − tα−γ−2 |+ 2 1 Γ(α − r2 + 1) Γ(α − γ − 1) ξΓ(α − r2 + 1) Γ(α − γ) 2 } ( β )q−1 ( β )q−1 Γ(α − 1) η η bξ α−1 bξ α−2 α−γ−2 α−γ−1 α−γ−2 α−γ−1 |t2 |t2 − t1 − t1 + |+ | Γ(α + 1) Γ(α − γ − 1) β Γ(α − 1) β )q−1 ( { } M |(t2 − c)α−1 − (t1 − c)α−1 | + |t2 − t1 | , + βΓ(α)

+

where c ∈ (0, t1 ). Since the function tα−1 , tα−2 , tα−γ−1 , tα−γ−2 and (t−c)α−1 are uniformly continuous on [0, 1], therefore it follows from the above estimates that T B is an equicontinuous set. Also, it is uniformly bounded as T B ⊂ B. Thus, we conclude that T is a completely continuous operator. Hence, by Schauder fixed point theorem, there exists a solutions of (1.1). This completes the proof. Example 2.1. Consider the four point boundary value problem  1 5 1  D 3+ (φ3 (D 2+ u))(t) + f (t, u(t), D 4+ u(t)) = 0, t ∈ (0, 1), 0 0 0  u(0) = 0, D 13 u(0) − 4 u( 3 ) = 0, D 15 u(1) + 2u( 3 ) = 0, D 52 u(0) = 0, 5 8 5 0+ 0+ 0+

(2.9)

1

1

and f (t, u, D 4 (u)) = a + (t + 21 )4 [u2ρ1 (t) + (D 4 u(t))2ρ2 ] where a is constant different from 0 and 0 < ρ1 , ρ2 < 1. Obviously, it follows by Theorem 2.1 that there exists a solution of 2.9.

References [1] Bashir Ahmad, Juan J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three point bounary conditions, Computers and Mathematics with Applications, 58(2009) 1838-1843. [2] Zhi Wei Lv, Existence result for m-point boundary value problems of nonlinear fractional differential equations with p-Laplacian operator, Advances in Difference Equations, (2014),16 pp. [3] Wang, J, Xiang, H, Liu, Z, Positive solutions for three-point boundary value problems of nonlinear fractional differential equations with p-Laplacian. Far East J Appl Math., 37(2009) 33-47. [4] Wang, J, Xiang, H, Liu, Z, Upper and lower solutions method for a class of singular fractional boundary value problems with p-Laplacian operator. Abst Appl Anal. (2010 ),12 pp [5] Glockle, WG, Nonnenmacher, A fractional calculus approach of self-similar protein dynamics. Biophys J. 68(1995), 46-53 [6] Hilfer, R: Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000) [7] Metzler, F, Schick, W, Kilian, HG, Nonnenmacher, TF: Relaxation in filled polymers: a fractional calculus approach. J Chem Phys. 103(1995), 7180-7186 [8] Podlubny, I, Fractional Differential Equations. Academic Press, San Diego (1999) [9] Podlubny, I, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract Calc Appl Anal. 5(2002), 367-386 [10] Kilbas, AA, Srivastava, HM, Trujillo, JJ, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier, Amsterdam (2006) [11] Yang, C, Yan, J, Positive solutions for third-order Sturm-Liouville boundary value problems with p-Laplacian. Comput Math Appl. 59(2010), 2059-2066 [12] Anderson, DR, Avery, RI, Henderson, J, Existence of solutions for a one-dimensional p-Laplacian on time scales. J Diff Equ Appl. 10(2004), 889-896

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[13] Goodrich, CS, The existence of a positive solution to a second-order delta-nabla p-Laplacian BVP on a time scale. Appl Math Lett. 25(2012), 157-162 [14] Graef, JR, Kong, L, First-order singular boundary value problems with p-Laplacian on time scales. J Diff Equ Appl. 17(2011), 831-839 [15] Goodrich, CS, Existence of a positive solution to a first-order p-Laplacian BVP on a time scale. Nonlinear Anal. 74(2011), 1926-1936 [16] Agarwa, RP, Ahmad, B, Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput Math Appl. 62(2011), 1200-1214 [17] D. Ji, W. Ge, Existence of multiple positive solutions for Sturm-Liouville-like four-point boundary value problem with p-Laplacian Nonlinear Anal., 68 (2008), 2638-2646. [18] H. Lian, W. Ge, Positive solutions for a four-point boundary value problem with the p-Laplacian Nonlinear Anal., 68 (2008), 3493-3503. [19] H. Pang, H. Lian, W. Ge, Multiple positive solutions for second-order four-point boundary value problem Comput. Math. Appl., 54 (2007), 1267-1275.

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Differential subordinations obtained with some new integral operators

E-mail:

1

Georgia Irina Oros1 , Gheorghe Oros2 , Radu Diaconu3 1,2 Department of Mathematics, University of Oradea Str. Universit˘a¸tii, No.1, 410087 Oradea, Romania 3 Department of Mathematics, University of Pitesti Str. Targul din Vale, no.1, 110040 - Pitesti, Romania georgia oros [email protected], 2 gh [email protected], 3 [email protected]

Abstract In this paper we define some new integral operators on classes of analytic functions and we study certain differential subordinations obtained by using those operators.

Keywords: Analytic function, univalent function, differential subordination, dominant, best dominant. 2000 Mathematical Subject Classification: 30C80, 30C20, 30C45, 30A40.

1

Introduction and preliminaries

The concept of differential subordination was introduced in [1], [2] and developed in [3], by S.S. Miller and P.T. Mocanu. Denote by U the unit disc of the complex plane U = {z ∈ C : |z| < 1}. Let H(U ) be the space of holomorphic functions in U and let An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U }, with A1 = A, and S = {f ∈ A : f is univalent in U } be the class of holomorphic and univalent functions in the open unit disc U , with conditions f (0) = 0, f 0 (0) = 1, that is the holomorphic and univalent functions with the following power series development f (z) = z + a2 z 2 + . . . , z ∈ U. For a ∈ C and n ∈ N∗ , we denote by H[a, n] = {f ∈ H(U ) : f (z) = a + an z n + an+1 n+1 + . . . , z ∈ U }. Let A(p) denote the subclass Pz∞ of the functions f ∈ H(U ) of the form f (z) = z p + k=p+1 ak z k , p ∈ N and set A = A(1). Denote by   zf 00 (z) + 1 > 0, z ∈ U the class of normalized convex functions in U . K = f ∈ A : Re 0 f (z) If f and g are analytic functions in U , then we say that f is subordinate to g, written f ≺ g, if there is a function w analytic in U , with w(0) = 0, |w(z)| < 1, for all z ∈ U such that f (z) = g(w(z)) for z ∈ U . If g is univalent then f ≺ g if and only if f (0) = 0 and f (U ) ⊆ g(U ). Let ψ : C3 × U → C and let h be univalent in U . If p is analytic in U and satisfies the differential subordination (i) ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z), z ∈ U , then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply, a dominant, if p ≺ q for all p satisfying (i). A dominant qe that satisfies qe ≺ q for all dominants q of (i) is said to be the best dominant of (i). (Note that the best dominant is unique up to a rotation of U .) To prove our main results, we need the following lemmas. Lemma A. (Hallenbeck and Ruscheweyh [3, Th. 3.1.6, p. 71]) Let h be a convex function with h(0) = a, 1 and let γ ∈ C∗ be a complex number with Re γ ≥ 0. If p ∈ H[a, n] and p(z) + zp0 (z) ≺ h(z), z ∈ U then γ γ Z z γ n p(z) ≺ q(z) ≺ h(z), z ∈ U where q(z) = γ h(t)t n −1 dt, z ∈ U. zn 0 Lemma B. [3, Th. 3.44, p. 132] Let the function q be univalent in the unit disc U and let θ and ϕ be analytic in a domain D containing q(U ), with ϕ(w) 6= 0, when w ∈ q(U ). Set Q(z) = zq 0 (z) · ϕ[q(z)] and h(z) = θ[q(z)] + Q(z). 1

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Suppose that (i) Q(z) is starlike univalent in U , zh0 (z) (ii) Re > 0, for z ∈ U . Q(z) If p is analytic with p(0) = q(0), p(U ) ⊆ D and θ[p(z)] + zp0 (z) · ϕ[p(z)] ≺ θ[q(z)] + zq 0 (z) · ϕ[z(q)], then p(z) ≺ q(z), z ∈ U and q(z) is the best dominant.

2

Main results We introduce the following new integral operators: Definition 1. For f ∈ An , n ∈ N∗ , m ∈ N, γ ∈ C, let Lγ be the integral operator given by Lγ : An → An , L0γ f (z) = f (z), Z γ+1 z 0 Lγ (t)tγ−1 dt, . . . = zγ 0 Z γ + 1 z m−1 m Lγ f (z) = Lγ f (t) · tγ−1 dt. zγ 0

L1γ f (z)

By using Definition 1, we can prove the following properties for this integral operator: Property 1. For f ∈ An , n ∈ N∗ , m ∈ N, γ ∈ C, we have (2.1)

Lm γ f (z) = z +

∞ X (γ + 1)m · ak z k , (γ + k)m

z ∈ U.

k=n+1

Property 2. For f ∈ An , n ∈ N∗ , m ∈ N, γ ∈ C, we obtain (2.2)

0 m−1 z · [Lm f (z) − γLm γ f (z)] = (γ + 1)Lγ γ f (z),

z ∈ U.

m Remark 1. For γ = 1, n ∈ N∗ , m ∈ N, γ ∈ C, we obtain the integral operator Lm : An → An , 1 := I given by

I 0 f (z) = f (z), Z 2 z 0 I f (t)dt, . . . I 1 f (z) = z 0 Z z 2 I m f (z) = I m−1 f (t)dt. z 0 In this case, (2.1) and (2.2) become: (2.1)0

∞ X

I m f (z) = z +

k=n+1

2m ak z k (k + 1)m

and (2.2)0

z · [I m f (z)]0 = 2 · I m−1 f (z) − I m f (z),

z ∈ U.

Remark 2. For γ = 0, n = 1, m ∈ N, f ∈ A, we get the integral operator Lm 0 := Is : A → A, introduced by G.St. S˘al˘ agean in [4], given by Is0 f (z) = f (z) Z z Is1 f (z) = f (t) · t−1 dt, . . . Z0 z m Is f (z) = Ism−1 f (t) · t−1 dt. 0

In this case, (2.1) and (2.2) become: (2.1)00

Ism f (z) = z +

∞ X 1 ak z k , km

z∈U

k=2

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and z · [Ism f (z)]0 = Ism−1 f (z),

(2.2)00

z ∈ U.

Remark 3. For n ∈ N∗ , m ∈ N, γ ∈ C, f ∈ H[a, n], we obtain the integral operator Lγ := Lγ,a : H[a, n] → H[a, n] given by L0γ,a f (z) = f (z), Z z γ 1 L0 f (t) · tγ−1 dt, . . . Lγ,a f (z) = γ z 0 γ Z z γ m Lm−1 f (t) · tγ−1 dt. Lγ,a f (z) = γ z 0 γ In this case, (2.1) and (2.2) become: (2.3)

Lm γ,a f (z) = a +

∞ X k=n

γm ak z k , (γ + k)m

z∈U

and (2.4)

0 m−1 m z · [Lm γ,a f (z)] = γLγ,a f (z) − γLγ,a f (z),

z ∈ U.

Remark 4. For a = 0, n ∈ N∗ , m ∈ N, γ ∈ C, f ∈ H[0, n], we have the integral operator Lγ := Gγ : H[0, n] → H[0, n], given by G0γ f (z) = f (z), Z γ+n z 0 Gγ f (t) · tγ−1 dt, . . . = zγ 0 Z γ + n z m−1 Gm f (z) = Gγ f (t)tγ−1 dt. γ zγ 0

G1γ f (z)

In this case, (2.1) and (2.2) become: (2.5)

n Gm γ f (z) = an z +

∞ X (γ + n)n ak z k , (γ + k)m

z∈U

k=n+1

and (2.6)

0 m−1 z · [Gm f (z) − γGm γ f (z)] = (γ + n)Gγ γ f (z),

z ∈ U.

Remark 5. For a ∈ C, n ∈ N∗ , m ∈ N, γ = 1, we get the integral operator Lγ,a := T : H[a, n] → H[a, n], given by T 0 f (z) = f (z) Z 1 z 0 T 1 f (z) = T f (t)dt, . . . z 0 Z 1 z m−1 T m f (z) = T f (t)dt. z 0 In this case, (2.1) and (2.2) become: (2.7)

m

T f (z) = a +

∞ X k=n

1 · ak z k , (k + 1)m

z∈U

and (2.8)

[z · tm f (z)]0 = T m−1 f (z),

z ∈ U.

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Definition 2. For p ∈ N, m ∈ N, f ∈ A(p), let H be the integral operator given by H : A(p) → A(p), H 0 f (z) = f (z) Z p+1 z 0 1 H f (t)dt, . . . H f (z) = z Z0 z p+1 H m f (z) = H m−1 f (t)dt. z 0 Property 3. For f ∈ A(p), m ∈ N, p ∈ N, we have (2.9)

Hm f (z) = z p +

∞ X (p + 1)m ak z k , (k + 1)m

z ∈ U.

k=p+1

Property 4. For f ∈ A(p), m ∈ N, p ∈ N, we have (2.10)

[z · H m f (z)]0 = (p + 1)H m−1 f (z),

z ∈ U.

Remark 6. If p = 0, f ∈ H[1, n], then H m := T m , the integral operator seen in Remark 5. Remark 7. If p = 1, f ∈ A, then H m := I m , the integral operator shown in Remark 1. Remark 8. These operators are also studied in [5], [6] and [1]. We next give certain differential subordinations obtained by using the integral operators we have defined here. 1 + (2α − 1)z Theorem 1. Let h(z) = , be convex in U , with h(0) = 1, 0 ≤ α < 1. 1+z Assume γ ∈ C, Re γ > 0 and that f ∈ H[a, n], satisfies the differential subordination (2.11)

1 z · [T m−1 f (z)]0 + T m−1 f (z) ≺ h(z). γ

Then [z · T m−1 f (z)]0 ≺ q(z), where q is given by (2.12)

q(z) = (2α − 1) + (2 − 2α)

γ σ(z) · , n z γ/n

and z

Z (2.13)

σ(z) = 0

γ

t n −1 dt. 1+t

The function q is convex and is the best dominant. Proof. By using the property (2.8) of the integral operator T seen in Definition 1, Remark 5, the differential subordination (2.11) becomes: (2.14)

1 z · [zT m f (z)]00 + [z · T m f (z)]0 ≺ h(z), γ

z ∈ U.

We let (2.15)

p(z) = [z · T m f (z)]0 = a +

∞ X k=n

1 ak z k , (k + 1)m−1

p ∈ H[a, n].

Using the notation (2.15) in (2.14), the differential subordination (2.11) becomes 1 0 zp (z) ≺ g(z), z ∈ U. γ Z z Z z γ 1 + (2α − 1)t γ/n−1 γ By using Lemma A, we have p(z) ≺ q(z) = t dt = [(2α−1)tγ/n−1 + 1+t nz γ/n 0 nz γ/n 0 2 − 2α γ −1 γ σ(z) t n ]dt = 2α − 1 + (2 − 2α) · · γ/n , where σ is given by (2.13).  1+t n z Theorem 2. Let q be univalent in U , with q(0) = 0 and q(z) 6= 0, for all z ∈ U , and suppose that

(2.16)

p(z) +

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(i) Re q(z)  > 0, 00  zq (z) zq 0 (z) (ii) Re 1 + 0 − > 0, z ∈ U . q (z) q(z) 0 2 m 00 z[Gm γ f (z)] + z [Gγ f (z)] Let n ∈ N∗ , γ ∈ C, 6= 0, z ∈ U, and (γ + n)Gm−1 f (z) − γGm γ γ f (z) (2.17)

(γ + n)Gm−1 f (z) − γGm γ γ f (z) +

0 2 m 00 z[Gm γ f (z)] + z [Gγ f (z)]

(γ +

n)Gm−1 f (z) γ

− γGm γ f (z)

≺ q(z) +

zq 0 (z) . q(z)

Then z · [Gm−1 f (z)]0 ≺ q(z) and q(z) is the best dominant. γ Proof. By using (2.6) we have m 0 (γ + n)Gm+1 f (z) − γGm γ γ f (z) = z · [Gγ f (z)] .

(2.18) We let

0 p(z) = z · [Gm γ f (z)] .

(2.19) Using (2.5) in (2.19) we obtain " (2.20)

# ∞ ∞ X X (γ + 1)m k (γ + 1)m k−1 kz a kz zk . = z + p(z) = z · 1 + k (γ + k)m (γ + k)m k=n+1

k=n+1

Since p(0) = 0, we obtain that p ∈ H[0, n]. Differentiating (2.19) and after a short calculus, we obtain 0 2 m 00 zp0 (z) = z[Gm γ f (z)] + z [Gγ f (z)] .

(2.21)

Using (2.18), (2.19) and (2.21), differential subordination (2.17) becomes (2.22)

p(z) +

zp0 (z) zq 0 (z) ≺ q(z) + , p(z) q(z)

z ∈ U.

In order to prove the theorem, we shall use Lemma B. For that, we show that the necessary conditions are satisfied. Let the function θ : C → C and ϕ : C → C with (2.23)

θ(w) = w

and (2.24)

ϕ(w) =

1 , w

ϕ(w) 6= 0.

We check the conditions from the hypothesis of Lemma B. Using (2.24), we have Q(z) = zq 0 (z) · ϕ(q(z)) =

(2.25)

zq 0 (z) . q(z)

Differentiating (2.25) and after a short calculus, we obtain zQ0 (z) zq 00 (z) zq 0 (z) =1+ 0 − . Q(z) q (z) q(z)

(2.26) From (ii) we have (2.27)

zQ0 (z) Re = Re Q(z)



zq 00 (z) zq 0 (z) 1+ 0 − q (z) q(z)

 > 0,

z ∈ U,

hence function Q is starlike. 5

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Using (2.23) we have (2.28)

h(z) = θ[q(z)] + Q(z) = q(z) + Q(z).

Differentiating (2.28) and after a short calculus, we obtain

zh0 (z) zQ0 (z) = q(z) + . From (i) and (2.27) Q(z) Q(z)

we have   zh0 (z) zQ0 (z) Re = Re q(z) + > 0. Q(z) Q(z)

(2.29)

1 1 , θ[q(z)] = q(z), ϕ(q(z)) = , and p(z) q(z) differential subordination (2.17) becomes θ[p(z)] + zp0 (z)ϕ(p(z)) ≺ θ[q(z)] + zq 0 (z) × (q(z)). 0 Using Lemma B, we obtain p(z) ≺ q(z) i.e. z[Gm  γ f (z)] ≺ q(z), and q is the best dominant. Theorem 3. Let q be univalent in U , with q(0) = 1 and q(z) = 6 0 for all z ∈ U and suppose that   zq 00 (z) zq 0 (z) (j) Re 1 + 0 + > 0, z ∈ U . q (z) q(z) Let f ∈ A(p), p ∈ N. If the differential subordination Using (2.23) and (2.24), we get θ[p(z)] = p(z), ϕ(p(z)) =

(2.30)

1+

zH m f (z)[H m f (z)]0 − p[H m f (z)]2 ≺ 1 + zq(z)q 0 (z) z 2p

H m f (z) is satisfied, then ≺ q(z) and q(z) is the best dominant. zp Proof. We let (2.31)

p(z) = zp +

Using (2.9) in (2.31) we have p(z) =

H m f (z) , zp

z ∈ U.

∞ X (p + 1)m ak z k (k + 1)m

k=p+1

zp

= 1+

∞ X (p + 1)m ak z k−p . Since p(0) = (k + 1)m

k=p+1

1, we get that p ∈ H[1, k − p]. Differentiating (2.31), we have [H m f (z)]0 = pz p−1 p(z) + z p p0 (z).

(2.32) Using (2.31) and (2.32), we have (2.33)

1+

zH m f (z)[H m f (z)]0 − p[H m f (z)]2 = 1 + zp(z)p0 (z). z 2p

Using (2.33), differential subordination (2.30) becomes (2.34)

1 + zp(z)p0 (z) ≺ 1 + zq(z)q 0 (z).

In order to prove the theorem, we shall use Lemma B. For that, we show that the necessary conditions are satisfied. Let the functions θ : C → C and ϕ : C → C with (2.35)

θ[w] = 1

and (2.36)

ϕ[w] = w.

Then (2.37)

Q(z) = zq 0 (z)ϕ[q(z)] = zq 0 (z)q(z)

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and (2.38)

h(z) = θ[q(z)] + Q(z) = 1 + Q(z). zQ0 (z) zq 0 (z) zq 00 (z) =1+ + 0 , z ∈ U. Using (j), we have Q(z) q(z) q (z)   zq 0 (z) zq 00 (z) zQ0 (z) = Re 1 + + 0 > 0, z ∈ U. Re Q(z) q(z) q (z)

Differentiating (2.37) we have

(2.39)

Differentiating (2.38), we have h0 (z) = Q0 (z), and using (2.39) we obtain: (2.40)

Re

zh0 (z) zQ0 (z) = Re > 0. Q(z) Q(z)

From (2.35) and (2.36), differential subordination (2.34) can be written as follows: (2.41)

θ[p(z)] + zp0 (z)ϕ(p(z)) ≺ θ[q(z)] + zq 0 (z)ϕ(q(z)).

H m f (z) ≺ q(z), z ∈ U and q(z) is the best dominant. zp Example 1. Let m ∈ N, p ∈ N, q(z) = 1 + z, z ∈ U , with     zq 00 (z) zq 0 (z) z 1 + 2z Re 1 + 0 + = Re 1 + = Re > 0, z ∈ U. q (z) q(z) 1+z 1+z Using Lemma B, we have p(z) ≺ q(z), i.e.

zH m f (z)[H m f (z)]0 − p[H m f (z)]2 H m f (z) 2 ≺ 1 + z + z then ≺ 1 + z, z ∈ U z 2p zp and q(z) = 1 + z is the best dominant. Remark 9. An open problem is the study of starlikeness, convexity and subordination preserving through those operators. If f ∈ A(p), p ∈ N, and 1 +

References [1] L. Andrei, Strong differential subordinations and superordinations using a new integral operator, Analele Universitatii Oradea, Fasc. Matematica, Tom XXI (2014), Issue No. 1, 191-196. [2] S. S. Miller, P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65(1978), 298-305. [3] S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Michig. Math. J., 28(1981), 157-171. [4] S. S. Miller, P. T. Mocanu, Differential subordinations. Theory and applications, Marcel Dekker, Inc., New York, Basel, 2000. [5] Gh. Oros, R. Sendrutiu, A. Venter, L. Andrei, Strong Differential Superordination and Sandwich Theorem, Journal of Computational Analysis and Applications, Vol. 15, No. 8, 2013, 1490-1495. [6] Gh. Oros, A. Venter, R. Sendrutiu, L. Andrei, Strong Differential Subordinations and Superordinations and Sandwich Theorem, Journal of Computational Analysis and Applications, Vol. 15, No. 8, 2013, 1496-1501. [7] G.St. S˘ al˘ agean, Subclasses of univalent functions, Lecture Notes in Math. (Springer-Verlag), 1013(1983), 362-372.

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Preconditioned SAOR iterative methods for H -matrices linear systems Xue-Zhong Wang, Lei Ma∗ School of Mathematics and Statistics, Hexi University, Zhangye, Gansu, 734000 P.R. China

Abstract: In this paper, we consider a preconditioned SAOR iterative method for solving linear system A x = b . We prove its convergence and give more comparison results of the spectral radius for the case when A is an H -matrix or a strictly diagonally dominant matrix. Numerical example are also given to illustrate our method. Key words: H -matrix; SAOR 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 two 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. (1.2) can be written as x k +1 = T x k + c , k = 0, 1, · · · , where T = M −1 N , c = M −1 b . Let A = D − L − U , where D , −L and −U are diagonal, strictly lower and strictly upper triangular parts of A, respectively. We split A into A = M − N = P − Q , here M=

1 1 (D − r L ), N = [(1 − ω)D + (ω − r )L + ωU ] ω ω

and

1 1 (D − r U ), Q = [(1 − ω)D + (ω − r )U + ωL ]. ω ω Then, the iteration matrix of the SAOR method is given by F =

Lr ω = F −1Q M −1 N , where ω and r are real parameters with ω , 0. Transforming the original systems (1.1) into the preconditioned form PAx = P b,

(1.3)

M p x k +1 = Np x k + P b , k = 0, 1, · · · ,

(1.4)

then, we can define the basic iterative scheme

where P A = M p − Np , and M p is nonsingular. Thus (1.4) can also be written as x k +1 = T x k + c , k = 0, 1, · · · , ∗ Lei

Ma Email address:

[email protected] (Lei Ma)

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Preconditioned SAOR iterative methods for H -matrices where T = M p−1 Np , c = M p−1 P b . In the literature, various authors have suggested different models of (I +S )-type preconditioner [1-8] for linear systems (1.1). These preconditioners have reasonable effectiveness and low construction cost. For example. In this paper, we present the preconditioner of (I + S )-type with the following form      P = I +S =     

α1 − a 1m

1 1 ..

 α2 − a 2s

    ,    

. ..

αr − a r u

. ..

.

αn − a n t

(1.5)

1

where m , s , u, t , 1. We consider the preconditioned SAOR iterative method for solving linear systems with preconditioner P . Let P A = (I + S )(D − L − U ) = D˜ − L˜ − U˜ , where D˜ , −L˜ and −U˜ are diagonally, strictly lower and strictly upper triangular parts of P A, respectively. We split ˜ − N˜ = P˜ − Q˜ , here P A into P A = M ˜ = M

1 1 (D˜ − r L˜ ), N˜ = [(1 − ω)D˜ + (ω − r )L˜ + ωU˜ ] ω ω

and

1 1 (D˜ − r U˜ ), Q˜ = [(1 − ω)D˜ + (ω − r )U˜ + ωL˜ ]. ω ω If D˜ is nonsingular, then (D˜ − r L˜ )−1 and (D˜ − r U˜ )−1 exist, it is possible to define the SAOR iteration matrix for P A. Namely ˜ −1 N˜ , L˜r ω = F˜ −1Q˜ M F˜ =

Our work in the presentation are to prove Convergence of the SAOR method applied to H -matrix with preconditioner P . Also more comparison results of the spectral radius for the case when A is a nonnegative H -matrix are given. Numerical example shows that the results are valid. 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 a n-dimensional vector x , 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 if 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 on special matrices introduced previously are given to be used in this paper. Lemma 2.1 [5]. 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 A x > 0. (c) A −1 ≥ 0. Lemma 2.2 [5]. Let A and B be two n × n matrices with 0 ≤ B ≤ A. Then, ρ(B ) ≤ ρ(A). Lemma 2.3 [8]. If A is an H −matrix, then |A −1 | ≤ 〈A〉−1 .

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Xue-Zhong Wang, Lei Ma Lemma 2.4 [9]. Suppose that A 1 = M 1 − N1 and A 2 = M 2 − N2 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 , ∥ M 2−1 N2 ∥ x ≤∥ M 1−1 N1 ∥ x . (2.1) In particular, if M 1−1 N1 has a positive perron vector, then ρ(M 2−1 N2 ) ≤ ρ(M 1−1 N1 ).

(2.2)

Moreover if x is a Perron vector of M 1−1 N1 and strictly inequality holds in (2.1), then strictly inequality holds in (2.2). Lemma 2.5 [17]. Let A −1 ≥ 0. If the splitting A = M −N = F −Q are weak regular, then ρ(T ) < 1 and the unique splitting A = B − C induced by T is weak regular. Where B = F (M + P − A)−1 M and C = B − A. Lemma 2.6 [5]. Matrix A is a strictly diagonally dominant (SDD) matrix, if ∑ |a i i | > |a i j |, i = 1, 2, . . . , n . j ,i

3. Convergence results (〈A〉x )

For the convenience, let t i = 2xm −(〈A〉xi )m , i = 1, 2, · · · , n . Now we give main results as follows: Theorem 3.1 Let A be an H −matrix with unit diagonally elements. Assume that there exists a positive vector x = (x1 , x2 , · · · , xn )T , such that 〈A〉x > 0. If a i m − t i ≤ αi ≤ a i m + t i ,

i = 1, 2, · · · , n,

(3.1)

then P A is an H −matrix.
T Proof. Let (P A)i j = a i j + (αi − a i m )a m j , i , j = 1, 2, · · · , n , m = r, s , · · · , t , and x = x1, x2, · · · , xn , then (〈P A〉 x )i

= ≥

(αi − a i m ) a m i | xi − |a i m + (αi − a i m )| xm |1 +∑ a i j + (αi − a i m ) a m j x j −

j ,i ,m ∑ a i j x j xi − |αi − a i m | |a mi | xi − |a i m | xm − |αi − a i m | xm − j , i ,m ∑ |αi − a i m | a m j x j . − j ,i ,m

Case 1. a i m ≤ αi ≤ a i m + t i (〈P A〉 x )i

=

∑ a i j x j xi − (αi − a i m ) |a mi | xi − |a i m | xm − (αi − a i m )xm − j ,i ,m ∑ (αi − a i m ) a m j x j − j ,i ,m ∑ a i j x j − (αi − a i m ) |a mi | xi − (αi − a i m )xm xi − |a i m | xm − j , i ,m ∑ − (αi − a i m ) a m j x j j ,i ,m ∑ a m j x j ) (〈A〉 x )i + (αi − a i m ) (−xm − ∑ j ,m a m j x j + xm − 2xm ) (〈A〉 x )i + (αi − a i m ) (−

= >

(〈A〉 x )i + (αi − a i m ) [(〈A〉 x )m − 2xm ] 0.

≥

=

=

j ,m

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Preconditioned SAOR iterative methods for H -matrices Case 2. a i m − t i ≤ αi ≤ a i m (〈P A〉 x )i

=

∑ a i j x j xi + (αi − a i m ) |a mi | xi − |a i m | xm + (αi − a i m ) xm − j ,i ,m ∑ (αi − a i m ) a m j x j + j ,i ,m ∑ a m j x j − xm + 2xm ) (〈A〉 x ) + (αi − a i m ) (

= >

(〈A〉 x )i + (αi − a i m ) [2xm − (〈A〉 x )m ] 0.

≥

i

j ,m

Therefore, 〈P A〉 is an M -matrix, and P A is an H -matrix. Theorem 3.2. If A is an nonsingular H -matrix with unit diagonally elements, and 0 < r ≤ ω ≤ 1, and ω , 0. If a i m − t i ≤ αi ≤ a i m + t i ,

i = 1, 2, · · · , n.

Then ρ(L˜r ω ) < 1. ˜ 〉 − |N˜ | = Proof From Theorem 3.1, we know 〈P A〉 is an M -matrix, and 〈P A〉 = |D˜ | − |L˜ | − |U˜ |. Let 〈P A〉 = 〈M 〈P˜ 〉 − |Q˜ |, where 1 1 ˜ 〉 = (|D˜ | − r |L˜ |), |N˜ | = [(1 − ω)|D˜ | + (ω − r )|L˜ | + |ωU˜ |] 〈M ω ω and 1 1 〈P˜ 〉 = (|D˜ | − r |U˜ |), |Q˜ | = [(1 − ω)|D˜ | + (ω − r )|U˜ | + ω|L˜ |]. ω ω Then the SAOR iteration matrix for 〈P A〉 is as follows. ˜ 〉−1 |N˜ |. L¨r ω = 〈P˜ 〉−1 |Q˜ |〈M ˜ 〉 are two M −matrices, which follow from 〈P A〉 is an M -matrix, by Lemma 2.3, we know Obviously, 〈P˜ 〉 and 〈M −1 −1 ˜ ˜ ˜ −1 | ≤ 〈M ˜ 〉−1 . Thus that |P | ≤ 〈P 〉 and |M |L˜r ω |

=

˜ −1 N˜ | ≤ |P˜ −1 ||Q˜ ||M ˜ −1 ||N˜ | ≤ 〈P˜ 〉−1 |Q˜ |〈M ˜ 〉−1 |N˜ | = ρ(L¨r ω ). |P˜ −1Q˜ M

˜ 〉−|N˜ | = 〈P˜ 〉−|Q˜ | are two weak regular splitting, by Lemma 2.5 ρ(L¨r ω ) < 1. Together with Lemma Since 〈P A〉 = 〈M 2.2, we have ρ(L˜r ω ) ≤ ρ(|L˜r ω |) ≤ ρ(L¨r ω ) < 1. Remark 3.1. Theorem 3.2 shows that presented preconditioned SAOR method is convergent by employing preconditioner P for H -matrices linear systems. 4. Comparison results of spectral radius In what follows we will give some comparison results on the spectral radius of preconditioned SAOR iteration matrices with preconditioner P . ˆ − Nˆ = Fˆ − Qˆ , where Let 〈A〉 = M ˆ = M

1 1 (|D | − r |L |), Nˆ = [(1 − ω)|D | + (ω − r )|L | + ω|U |] ω ω

and

1 1 (|D | − r |U |), Qˆ = [(1 − ω)|D | + (ω − r )|U | + ω|L |]. ω ω Then the SAOR iteration matrix for 〈A〉 is as follows. Fˆ =

ˆ −1 Nˆ . Lˆr ω = Fˆ −1Qˆ M

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Xue-Zhong Wang, Lei Ma ˆ − Nˆ = Fˆ − Qˆ are two weak regular splitting, from Lemma 2.5, the unique splitting It is easy to show 〈A〉 = M ˆ ˆ ˆ ˆ + Fˆ − 〈A〉)−1 M ˆ and Cˆ = Bˆ − 〈A〉. 〈A〉 = B − C induced by Lr ω is weak regular splitting. Where Bˆ = Fˆ (M ˜ ˜ ˜ ˜ From 〈P A〉 = 〈M 〉 − |N | = 〈F 〉 − |Q | are two weak regular splitting, similar to the above analysis, we have the ˜ 〉 + 〈F˜ 〉 − 〈P A〉)−1 〈M ˜〉 unique splitting 〈P A〉 = B˜ − C˜ induced by L¨r ω is weak regular splitting. Where B˜ = 〈F˜ 〉(〈M ˜ ˜ and C = B − 〈P A〉. Theorem 4.1. Let A be a nonnegative SDD matrix with unit diagonally elements and a m j , 0, 0 < r ≤ ω ≤ 1. If a i m − si ≤ αi ≤ a i m − wi , i = 1, 2, · · · , n. 2a (〈A〉x ) (〈A〉x ) Then ρ(L¨r ω ) ≤ ρ(Lˆr ω ), where s = min{ 2xm −(〈A〉xi )m , a m2 j } and w = min{ 2xm −(〈A〉xi )m , a mi jj }. Proof Since 〈A〉 and 〈P A〉 are M -matrices, and 〈A〉 = Bˆ − Cˆ and 〈P A〉 = B˜ − C˜ are two weak regular splittings. From ˜ 〉 + 〈F˜ 〉 − 〈P A〉)−1 〈M ˜〉 B˜ = 〈F˜ 〉(〈M

we have

˜ 〉−1 (〈M ˜ 〉 + 〈F˜ 〉 − 〈P A〉)〈F˜ 〉−1 . B˜ −1 = 〈M

Since 〈P A〉 = (I + |S |)〈A〉. By simple calculation, we have B˜ −1 ≥ Bˆ −1 ≥ 0. Let x = 〈A〉−1 e > 0, then (〈P A〉 − 〈A〉)x = (I + |S |)e > 0. Since B˜ −1 ≥ Bˆ −1 ≥ 0, we get B˜ −1 〈P A〉x = (I − B˜ −1 C˜ )x ≥ Bˆ −1 〈A〉x = (I − Bˆ −1 Cˆ )x . Thus, it follows that

||B˜ −1 C˜ || x ≤ ||Bˆ −1 Cˆ || x ,

as 〈P A〉 = B˜ − C˜ is a weak regular splitting, there exists a positive perron vectors y , thus, by Lemma 2.4, the following inequality holds: ρ(B˜ −1 C˜ ) ≤ ρ(Bˆ −1 Cˆ ) ie, ρ(L¨r ω ) ≤ ρ(Lˆr ω ). Combining the above Theorems, we obtain the following conclusion: Theorem 4.2. Let A be a nonnegative SDD with unit diagonally elements and a m j , 0, 0 < r ≤ ω ≤ 1. If a i m − si ≤ αi ≤ a i m − wi ,

i = 1, 2, · · · , n.

Then, ρ(L˜r ω ) ≤ ρ(L¨r ω ) ≤ ρ(Lˆr ω ) < 1. Theorem 4.3. Let A be a nonnegative H -matrix with unit diagonally elements and a m j , 0, 0 < r ≤ ω ≤ 1. If a i m − si ≤ αi ≤ a i m − wi ,

i = 1, 2, · · · , n.

Then ρ(L¨r ω ) ≤ ρ(Lˆr ω ). Proof It is well known that each H-matrix A there exists a nonsingular diagonal matrix V such that AV is an SDD matrix. By simple calculation, we have iterative matrices of SAOR for A and AV are similar. Since similar matrices have the same eigenvalues, it follows that ρ(L¨r ω ) ≤ ρ(Lˆr ω ). Theorem 4.4. Let A be a nonnegative H -matrix with unit diagonally elements and a m j , 0, 0 < r ≤ ω ≤ 1. If a i m − si ≤ αi ≤ a i m − wi ,

i = 1, 2, · · · , n.

Then, ρ(L˜r ω ) ≤ ρ(L¨r ω ) ≤ ρ(Lˆr ω ) < 1. Proof The proof is similar to the proof of Theorem 4.3, so omitted.

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Preconditioned SAOR iterative methods for H -matrices 5. Example. For randomly generated nonsingular H -matrices for n = 50, 100, 200, 1000. Let r = 〈A〉−1 e , where e = (1, 1, . . . , 1)T . Taking ki = mi n j ∈ { j | max |a i j |} for i , j in (1.5) and αi , (i = 1, 2, . . . , n) meet the inequality in j

(3.1). We have determined the spectral radius of the iteration matrices of SAOR method mentioned previously with preconditioner P . For P , we make two group of experiments. In figure 1, we test the relation between r and ρ, when n = 10, ω = 1, where “+”,“∗”,“·”and “◦”denote the spectral radius of 〈P A〉, 〈A〉, A and P A, respectively. In Table 1, the meaning of notations ρ(L˜ω ), ρ(L¨ω ), ρ(Lˆω ) and ρ(Lω ) denote the spectral radius of P A, 〈P A〉, 〈A〉 and A, respectively. Figure 1: The relation between r and ρ, when n = 10, ω = 1 0.95

0.9

ρ

0.85

0.8

0.75

0.7

0

0.1

0.2

0.3

0.4

0.5 r

0.6

0.7

0.8

0.9

1

From Figure 1 and Table 1, we can conclude that the rate of convergence of the preconditioned SAOR method with preconditioner P is faster than others, which further illustrate the Theorem 4.2 is true. Table 1: Comparison of spectral radius with preconditioner P

ω,r

n

ρ(L˜ω )

ρ(L¨ω )

ω=0.8

5

0.6309

r =0.6

10 50 5

0.8542 0.9488 0.7087

50 200 10 50 100 500

ω=0.6 r =0.5 ω=1 r =0.8

ρ(Lˆω )

ρ(Lω )

0.6899

0.7899

0.7293

0.8755 0.9499 0.7301

0.9210 0.9665 0.7715

0.8638 0.9501 0.7520

0.9454 0.9804 0.7320

0.9659 0.9857 0.7615

0.9805 0.9948 0.7658

0.9721 0.9758 0.7644

0.9204 0.9411 0.9713

0.9508 0.9538 0.9806

0.9819 0.9799 0.9926

0.9606 0.9642 0.9893

References [1] M. J. Wu, L. Wang, and Y. Z. Song, Preconditioned AOR iterative method for linear systems, Appli. Numer. Math., 57(5-7), 672-685 (2007).

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Xue-Zhong Wang, Lei Ma [2] D. J. Evansa, M. M. Martinsb, and M. E. Trigoc, The AOR iterative method for new preconditioned linear systems, J. Compu. Appl.Math., 132, 461-46 (2001). [3] X.Z. Wang, Block preconditioned AOR methods for H-matrices linear systems, J. Comput. Anal. Appl., 15, 714-721 (2013). [4] X. Z. Wang, and T. Z. Huang, Comparison results on preconditioned SOR-type iterative methods for Z -matrices linear systems, J. Comput. Appl. Math., 206, 726-732 (2007). [5] T. Z. Huang, X. Z. Wang, and Y. D. Fu, Improving Jacobi methods for nonnegative H-matrices linear systems, Applied Mathematics and Computation. 186, 1542-1550 (2007). [6] Y. Zhang, and T. Z. Huang, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl., 50, 1587-1602 (2005). [7] D. Noutsos, and M. Tzoumas, On optimal improvements of classical iterative schemes for Z -matrices, J. Comput. Appl. Math., 188, 89-106 (2006). [8] A. Hadjidimos, D. Noutsos, and M. Tzoumas, More on modifications and improvements of classical iterative schemes for Z-matrices, Lin. Alg. Appl., 364, 253-279 (2003). [9] A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA. (1994) [10] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ. (1981) [11] M. Benzi, D.B. Szyld, Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods, Numer. Math. 76: 309-321 (1997).

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO. 5, 2015

On the Interval-Valued Pseudo-Laplace Transform by Means of the Interval-Valued PseudoIntegral, Jeong Gon Lee, and Lee-Chae Jang,….…………………………………………792 Operators Ideals of Generalized Modular Spaces of Cesàro Type Defined By Weighted Means, Necip Şimşek, Vatan Karakaya, and Harun Polat,………………………………………..804 Certain Subclasses of Multivalent Uniformly Starlike and Convex Functions Involving a Linear Operator, J. Patel, A. Ku. Sahoo, and N.E. Cho,………………………………………….812 The Characterizations of McShane Integral and Henstock Integrals for Fuzzy-Nnumber-Valued Functions with a Small Riemann Sum on a Small Set, Muawya Elsheikh Hamid, and Zeng-Tai Gong,………………………………………………………………………………………830 Barnes-Type Narumi of the Second Kind and Poisson-Charlier Mixed-Type Polynomials, Dae San Kim, Taekyun Kim, Hyuck In Kwon, and Toufik Mansour,……………………837 Local Control and Approximation of a C2 Rational Interpolating Curve, Qinghua Sun,…851 Certain New Grüss Type Inequalities Involving Saigo Fractional q-Integral Operator, Guotao Wang, Praveen Agarwal, and Dumitru Baleanu,………………………………….862 Ultra Bessel Sequences of Subspaces in Hilbert Spaces, Mohammad Reza Abdollahpour, Azam Shekari, Choonkil Park, and Dong Yun Shin,………………………………………874 On Fixed Point Generalizations to Partial b-metric Spaces, Thabet Abeljawad, Kamaleldin Abodayeh, and Nabil M. Mlaiki,……………………………………………………………883 Existence of Solutions for Fractional Four Point Boundary Value Problems with p-Laplacian Operator, Erbil Cetin, and Fatma Serap Topal,……………………………………………...892 Differential Subordinations Obtained with Some New Integral Operators, Georgia Irina Oros, Gheorghe Oros, and Radu Diaconu,………………………………………………………..904 Preconditioned SAOR Iterative Methods for H-Matrices Linear Systems, Xue-Zhong Wang, and Lei Ma,………………………………………………………………………………………911

Volume 19, Number 6 ISSN:1521-1398 PRINT,1572-9206 ONLINE

December 2015

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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36) 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

37) 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

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

38) 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

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 NEW MEMBERS 39)Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] Computational Mathematics

40) Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S.Korea, [email protected] Functional Equations

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Common Fixed Point of Mappings Satisfying Almost Generalized Contractive Condition in Partially Ordered G-Metric Spaces M. Abbas1 , Jong Kyu Kim2 and Talat Nazir3 1 Department of Mathematics, Lahore University of Management Sciences, 54792-Lahore, PAKISTAN. e-mail: [email protected] 2 Department of Mathematics Education, Kyungnam University, Changwon Gyeongnam, 631-701, Korea e-mail: [email protected] 3 Department of Mathematics, Lahore University of Management Sciences, 54792-Lahore, PAKISTAN.

e-mail: [email protected]

Abstract: In this paper, we proved some common fixed point theorems of mappings satisfying almost generalized contractive condition in complete partially ordered G−metric spaces. These results extend, unify and generalize several well-known corresponding results in the iterature. Keywords: Common fixed point, partially ordered set, G−metric space. 2010 AMS Subject Classification: 47H10.

1

Introduction and Preliminaries

The study of fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Mustafa and Sims [14] generalized the concept of a metric space. Based on the notion of generalized metric spaces, Mustafa et al.([13]-[17]) obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades [1] initiated the study of a common fixed point theory in generalized metric spaces (see [11]). Abbas et al. [2] and Chugh et al. [8] obtained some fixed point results for maps satisfying property P in G−metric spaces. Recently, Shatanawi [21] proved some fixed point results for self-mapping in a complete G− metric space under some contractive conditions related to a nondecreasing map φ : R+ → R+ with lim φn (t) = 0 for all t ≥ 0. Saadati et al. [20] proved some n→∞ fixed point results for contractive mappings in partially ordered G−metric spaces. Existence of fixed points in ordered metric spaces was first investigated in 2004 by Ran and Reurings [19] (see also, [18]). Several mathematicians then obtained fixed 0 The

corresponding author: [email protected](J.K.Kim).

1

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point and common fixed point theorems in partially ordered metric spaces ([4], [10], [12]). The notion of almost contractive condition was introduced by Berinde ([5]-[7]). ´ c et al. [9] obtained common fixed point theorems of almost generalized Recently, Ciri´ contractive mappings in ordered metric spaces. The aim of this paper is to define almost generalized contractive condition for two mappings in the setting of generalized metric spaces and to obtain common fixed point theorems for such mappings in partially ordered complete G−metric spaces. Consistent with Mustafa and Sims [14], the following definitions and results will be needed in the sequel. Definition 1.1. Let X be a nonempty set. A mapping G : X × X × X → R+ is said to be a G−metric on X if (a) G(x, y, z) = 0 if x = y = z; (b) 0 < G(x, y, z) for all x, y, z ∈ X, with x 6= y; (c) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X, with y 6= z; (d) G(x, y, z) = G(x, z, y) = G(y, z, x) = · · · , (symmetry in all three variables); and (e) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X. Then (X, G) is called a G−metric space. Definition 1.2.

A sequence {xn } in a G− metric space X is said to be

(a) G−Cauchy if for every ε > 0, there is an n0 ∈ N (the set of all natural numbers), such that for all n, m, l ≥ n0 , G(xn , xm , xl ) < ε, (b) G−Convergent if for any ε > 0, there is an x ∈ X and an n0 ∈ N, such that for all n, m ≥ n0 , G(xn , xm , x) < ε. A G−metric space on X is said to be G−complete if every G−Cauchy sequence in X is G−convergent in X. It is known that {xn } is G−converget to x ∈ X if and only if G(xm , xn , x) → 0 as n, m → ∞. Proposition 1.3. [14] equivalent:

Let X be a G−metric space. Then the followings are

(1) The sequence {xn } is G−convergent to x. (2) G(xn , xn , x) → 0 as n → ∞. (3) G(xn , x, x) → 0 as n → ∞. (4) G(xn , xm , x) → 0 as n, m → ∞. Proposition 1.4. [14] equivalent:

Let X be a G−metric space. Then the followings are

(1) The sequence {xn } is G−Cauchy. (2) For every ε > 0, there exists n0 ∈ N, such that for all n, m ≥ n0 , G(xn , xm , xm ) < ε; that is, G(xn , xm , xm ) → 0 as n, m → ∞. 2

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Definition 1.5.

A G−metric on X is said to be symmetric if G(x, y, y) = G(y, x, x)

for all x, y ∈ X. Proposition 1.6. Every G−metric on X defines a metric dG on X by dG (x, y) = G(x, y, y) + G(y, x, x),

(1.1)

for all x, y ∈ X. For a symmetric G− metric, we obtain dG (x, y) = 2G(x, y, y),

(1.2)

for all x, y ∈ X. However, if G is not symmetric, then we have 3 G(x, y, y) ≤ dG (x, y) ≤ 3G(x, y, y), 2 for all x, y ∈ X.

2

Common Fixed Point Theorems

Now, we obtain common fixed point theorems of mappings satisfying almost generalized contractive condition in partially ordered complete generalized metric spaces. Definition 2.1. [3] Let (X, ) be a partially ordered set. Two maps f, g : X → X are said to be weakly increasing if f x  gf x and gx  f gx for all x ∈ X. Theorem 2.2. Let (X, ) be a partially ordered set equipped with a complete metric G and let f, g : X → X be two weakly increasing mappings for which there exist δ ∈ (0, 1) and some L ≥ 0 such that  G(f x, gy, gy) ≤ δM (x, y, y) + L min G(x, f x, f x), G(y, gy, gy), G(x, gy, gy), G(y, f x, f x) , (2.1) for all comparable x, y ∈ X, where M (x, y, y)

= a1 G(x, y, y) + a2 G(x, f x, f x) + a3 G(y, gy, gy)   +a4 G(x, gy, gy) + G(y, f x, f x)

and ai > 0 for i = {1, 2, 3, 4} with a1 +a2 +a3 +2a4 < 1. If either f or g is continuous or for a nondecreasing sequence {xn } in X which is G−convergent to z in X implies that xn  z for all n ∈ N. Then f and g have a common fixed point. Proof. First, we show that if f or g has a fixed point, then it is a common fixed point of f and g. Indeed, let u be a fixed point of f . Now assume that G(u, gu, gu) > 0, then from (2.1) with x = y = u, we obtain G(u, gu, gu)

= G(f u, gu, gu)  ≤ δM (u, u, u) + L min G(u, f u, f u),

(2.2)

G(u, gu, gu), G(u, gu, gu), G(u, f u, f u) = δM (u, u, u), 3

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where M (u, u, u)

= a1 G(u, u, u) + a2 G(u, f u, f u) + a3 G(u, gu, gu)   +a4 G(u, gu, gu) + G(u, f u, f u) = (a3 + a4 )G(u, gu, gu),

which implies that G(u, gu, gu) ≤ δ(a3 + a4 )G(u, gu, gu) < G(u, gu, gu), a contradiction. Therefore G(u, gu, gu) = 0 and so u is a common fixed point of f and g. Similarly, if u is a fixed point of g, then it is also fixed point of f. Now let x0 be an arbitrary point of X. If f x0 = x0 , then the proof is finished. We assume that f x0 6= x0 . Define a sequence {xn } in X as follows: x1 x2

= f x0  gf x0 = gx1 = x2 , = gx1  f gx1 = f x2 = x3

and continuing this process, we have x1  x2  · · ·  xn  xn+1  · · · . We may assume that G(x2n , x2n+1 , x2n+1 ) > 0, for every n ∈ N. If not, then x2n = x2n+1 for some n. Since x2n and x2n+1 are comparable, using (2.1), we obtain

= ≤ = =

G(x2n+1 , x2n+2 , x2n+2 ) G(f x2n , gx2n+1 , gx2n+1 )  δM (x2n , x2n+1 , x2n+1 ) + L min G(x2n , f x2n , f x2n ), G(x2n+1 , gx2n+1 , gx2n+1 ), G(x2n , gx2n+1 , gx2n+1 ), G(x2n+1 , f x2n , f x2n )  δM (x2n , x2n+1 , x2n+1 ) + L min G(x2n , x2n+1 , x2n+1 ), G(x2n+1 , x2n+2 , x2n+2 ), G(x2n , x2n+2 , x2n+2 ), G(x2n+1 , x2n+1 , x2n+1 ) δM (x2n , x2n+1 , x2n+1 ), (2.3)

where M (x2n , x2n+1 , x2n+1 ) = a1 G(x2n , x2n+1 , x2n+1 ) + a2 G(x2n , f x2n , f x2n ) + a3 G(x2n+1 , gx2n+1 , gx2n+1 )   +a4 G(x2n , gx2n+1 , gx2n+1 ) + G(x2n+1 , f x2n , f x2n ) = a1 G(x2n , x2n+1 , xn+1 ) + a2 G(x2n , x2n+1 , x2n+1 ) + a3 G(x2n+1 , x2n+2 , x2n+2 )   +a4 G(x2n , x2n+2 , x2n+2 ) + G(x2n+1 , x2n+1 , x2n+1 ) = a3 G(x2n+1 , x2n+2 , x2n+2 ) + a4 G(x2n , x2n+2 , x2n+2 ) = (a3 + a4 )G(x2n+1 , x2n+2 , x2n+2 ). Hence G(x2n+1 , x2n+2 , x2n+2 ) ≤ δ(a3 + a4 )G(x2n+1 , x2n+2 , x2n+2 ), implies that G(x2n+1 , x2n+2 , x2n+2 ) = 0 as δ(a3 + a4 ) < 1 and hence x2n+1 = x2n+2 . Following the similar arguments, we obtain x2n+2 = x2n+3 and hence x2n is a common fixed point of f and g. 4

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Now, take G(x2n , x2n+1 , x2n+1 ) > 0 for n = 0, 1, 2, 3, · · · . Since x2n and x2n+1 are comparable, therefore

= ≤ = =

G(x2n+1 , x2n+2 , x2n+2 ) G(f x2n , gx2n+1 , gx2n+1 )  δM (x2n , x2n+1 , x2n+1 ) + L min G(x2n , f x2n , f x2n ), G(x2n+1 , gx2n+1 , gx2n+1 ), G(x2n , gx2n+1 , gx2n+1 ), G(x2n+1 , f x2n , f x2n )  δM (x2n , x2n+1 , x2n+1 ) + L min G(x2n , x2n+1 , x2n+1 ), G(x2n+1 , x2n+2 , x2n+2 ), G(x2n , x2n+2 , x2n+2 ), G(x2n+1 , x2n+1 , x2n+1 ) δM (x2n , x2n+1 , x2n+1 ), (2.4)

where

= = ≤ =

M (x2n , x2n+1 , x2n+1 ) a1 G(x2n , x2n+1 , x2n+1 ) + a2 G(x2n , f x2n , f x2n ) + a3 G(x2n+1 , gx2n+1 , gx2n+1 )   +a4 G(x2n , gx2n+1 , gx2n+1 ) + G(x2n+1 , f x2n , f x2n ) a1 G(x2n , x2n+1 , x2n+1 ) + a2 G(x2n , x2n+1 , x2n+1 ) + a3 G(x2n+1 , x2n+2 , x2n+2 )   +a4 G(x2n , x2n+2 , x2n+2 ) + G(x2n+1 , x2n+1 , x2n+1 ) (a1 + a2 )G(x2n , x2n+1 , x2n+1 ) + a3 G(x2n+1 , x2n+2 , x2n+2 )   +a4 G(x2n , x2n+1 , x2n+1 ) + G(x2n+1 , x2n+2 , x2n+2 ) (a1 + a2 + a4 )G(x2n , x2n+1 , x2n+1 ) + (a3 + a4 )G(x2n+1 , x2n+2 , x2n+2 ).

Now if G(x2n+1 , x2n+2 , x2n+2 ) ≥ G(x2n , x2n+1 , x2n+1 ) for some n = 0, 1, 2, ..., then M (x2n , x2n+1 , x2n+1 ) ≤ G(x2n+1 , x2n+2 , x2n+2 ) and from (2.4), we have G(x2n+1 , x2n+2 , x2n+2 ) ≤ δM (x2n , x2n+1 , x2n+1 ) ≤ δG(x2n+1 , x2n+2 , x2n+2 ), a contradiction. Therefore, for all n ≥ 0, G(x2n+1 , x2n+2 , x2n+2 ) ≤ G(x2n , x2n+1 , x2n+1 ). Similarly, we have G(x2n , x2n+1 , x2n+1 ) ≤ G(x2n−1 , x2n , x2n ) for all n ≥ 0. Hence for all n ≥ 0 G(xn+1 , xn+2 , xn+2 ) ≤ δG(xn , xn+1 , xn+1 ). Similarly, repetition of the above process n times gives G(xn+1 , xn+2 , xn+2 ) ≤ δG(xn , xn+1 , xn+1 ) ≤ δ 2 G(xn−1 , xn , xn ) ≤ ··· ≤ δ n+1 G(x0 , x1 , x1 ). For m, n ∈ N with m > n ≥ 0, using property (e) of G−metric, we have G(xn , xm , xm ) ≤ G(xn , xn+1 , xn+1 ) + G(xn+1 , xn+2 , xn+2 ) + · · · + G(xm−1 , xm , xm ) ≤ (δ n + δ n+1 + · · · + δ m−1 )G(x0 , x1 , x1 ) δn ≤ G(x0 , x1 , x1 ), 1 − δn 5

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and so G(xn , xm , xm ) → 0 as n, m → ∞. It follows that {xn } is a G−Cauchy sequence and by the completeness of X, there exists z ∈ X such that {xn } converges to z as n → ∞. If f is continuous, then it is clear that f z = z. Using similar arguments to those given above, we conclude that z is a common fixed point of f and g. Consider the case, when f is not continuous. Since {x2n } is a nondecreasing sequence with x2n → z in X implies x2n  z for all n ∈ N. Now from (2.1) G(x2n+1 , gz, gz) = G(f x2n , gz, gz) ≤ δM (x2n , z, z) + L min{G(x2n , f x2n , f x2n ), G(z, gz, gz), G(x2n , gz, gz), G(z, f x2n , f x2n )} = δM (x2n , z, z) + L min{G(x2n , x2n+1 , x2n+1 ), G(z, gz, gz), G(x2n , gz, gz), G(z, f x2n+1 , f x2n+1 )},

(2.5)

where M (x2n , z, z) = a1 G(x2n , z, z) + a2 G(x2n , f x2n , f x2n ) + a3 G(z, gz, gz) +a4 [G(x2n , gz, gz) + G(z, f x2n , f x2n )] = a1 G(x2n , z, z) + a2 G(x2n , x2n+1 , x2n+1 ) + a3 G(z, gz, gz) +a4 [G(x2n , gz, gz) + G(z, x2n+1 , x2n+1 )], which on taking limit as n → ∞, implies that lim M (x2n , z, z) = (a3 +a4 )G(z, gz, gz). n→∞

From (2.5) G(z, gz, gz) = lim G(x2n+1 , gz, gz) ≤ δ(a3 + a4 )G(z, gz, gz). n→∞

Hence, we have G(z, gz, gz) = 0, so z = gz and z is common fixed point of f and g. Corollary 2.3. Let (X, ) be a partially ordered set equipped with a complete metric G and let f, g : X → X be two weakly increasing mappings for which there exist δ ∈ (0, 1) and some L ≥ 0 such that  G(f x, gy, gy) ≤ δG(x, y, y) + L min G(x, f x, f x), G(y, gy, gy), G(x, gy, gy), G(y, f x, f x) , (2.6) for all comparable x, y ∈ X. If either f or g is continuous or for a nondecreasing sequence {xn } with xn → z in X implies that xn  z for all n ∈ N. Then f and g have a common fixed point. Proof. 2.2.

As inequality (2.6) is a special case of (2.1), the result follows from Theorem

Theorem 2.4. Let (X, ) be a partially ordered set equipped with a complete metric G and let f, g : X → X be two weakly increasing mappings for which there exist δ ∈ (0, 1) and some L ≥ 0 such that  G(f x, gy, gy) ≤ δM (x, y, y) + L min G(x, f x, f x), (2.7) G(y, gy, gy), G(x, gy, gy), G(y, f x, f x) , for all comparable x, y ∈ X, where M (x, y, y)

=

 max G(x, y, y), G(x, f x, f x), G(y, gy, gy), [G(x, gy, gy) + G(y, f x, f x)]/2 . 6

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If either f or g is continuous or for a nondecreasing sequence {xn } with xn → z in X implies that xn  z for all n ∈ N. Then f and g have a common fixed point. Proof. First we show that if f or g has a fixed point, then it is a common fixed point of f and g. Indeed, let u be a fixed point of f and assume that G(u, gu, gu) > 0, then from (2.7) with x = y = u, we have G(u, gu, gu)

= G(f u, gu, gu)  ≤ δM (u, u, u) + L min G(u, f u, f u), G(u, gu, gu), G(u, gu, gu), G(u, f u, f u) = δM (u, u, u),

(2.8)

where M (u, u, u)

 max G(u, u, u), G(u, f u, f u), G(u, gu, gu), [G(u, gu, gu) + G(u, f u, f u)]/2  = max 0, G(u, gu, gu), G(u, gu, gu)/2 = G(u, gu, gu), =

that is, G(u, gu, gu) ≤ δG(u, gu, gu) < G(u, gu, gu), a contradiction. Hence G(u, gu, gu) = 0 and so u is a common fixed point of f and g. Similarly, if u is a fixed point of g, then it is also fixed point of f. Now let x0 be an arbitrary point of X. If f x0 = x0 , then the proof is finished, so we assume that f x0 6= x0 . Define a sequence {xn } in X as follows: x1 x2

= f x0  gf x0 = gx1 = x2 , = gx1  f gx1 = f x2 = x3

and continuing this process we have x1  x2  · · ·  xn  xn+1  · · ·. Assume that G(x2n , x2n+1 , x2n+1 ) > 0, for every n ∈ N. If not, then x2n = x2n+1 for some n. Since x2n and x2n+1 are comparable using (2.7) we obtain

= ≤ = =

G(x2n+1 , x2n+2 , x2n+2 ) G(f x2n , gx2n+1 , gx2n+1 )  δM (x2n , x2n+1 , x2n+1 ) + L min G(x2n , f x2n , f x2n ), G(x2n+1 , gx2n+1 , gx2n+1 ), G(x2n , gx2n+1 , gx2n+1 ), G(x2n+1 , f x2n , f x2n )  δM (x2n , x2n+1 , x2n+1 ) + L min G(x2n , x2n+1 , x2n+1 ), G(x2n+1 , x2n+2 , x2n+2 ), G(x2n , x2n+2 , x2n+2 ), G(x2n+1 , x2n+1 , x2n+1 ) δM (x2n , x2n+1 , x2n+1 ), (2.9)

where M (x2n , x2n+1 , x2n+1 ) 7

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 max G(x2n , x2n+1 , x2n+1 ), G(x2n , f x2n , f x2n ), G(x2n+1 , gx2n+1 , gx2n+1 ), [G(x2n , gx2n+1 , gx2n+1 ) + G(x2n+1 , f x2n , f x2n )]/2  = max G(x2n , x2n+1 , xn+1 ), G(x2n , x2n+1 , x2n+1 ), G(x2n+1 , x2n+2 , x2n+2 ), [G(x2n , x2n+2 , x2n+2 ) + G(x2n+1 , x2n+1 , x2n+1 )]/2  = max 0, G(x2n+1 , x2n+2 , x2n+2 ), G(x2n+1 , x2n+2 , x2n+2 )/2 = G(x2n+1 , x2n+2 , x2n+2 ). =

Therefore, G(x2n+1 , x2n+2 , x2n+2 ) ≤ δG(x2n+1 , x2n+2 , x2n+2 ), implies that G(x2n+1 , x2n+2 , x2n+2 ) = 0 and x2n+1 = x2n+2 . Following the similar arguments, we obtain x2n+2 = x2n+3 and hence x2n becomes a common fixed point of f and g. Suppose that G(x2n , x2n+1 , x2n+1 ) > 0 for n = 0, 1, 2, 3, · · · . Since x2n and x2n+1 are comparable, using (2.7) we obtain G(x2n+1 , x2n+2 , x2n+2 ) G(f x2n , gx2n+1 , gx2n+1 )  δM (x2n , x2n+1 , x2n+1 ) + L min G(x2n , f x2n , f x2n ), G(x2n+1 , gx2n+1 , gx2n+1 ), G(x2n , gx2n+1 , gx2n+1 ), G(x2n+1 , f x2n , f x2n )  δM (x2n , x2n+1 , x2n+1 ) + L min G(x2n , x2n+1 , x2n+1 ), G(x2n+1 , x2n+2 , x2n+2 ), G(x2n , x2n+2 , x2n+2 ), G(x2n+1 , x2n+1 , x2n+1 ) δM (x2n , x2n+1 , x2n+1 ), (2.10)

= ≤ = = where

= = ≤ =

M (x2n , x2n+1 , x2n+1 )  max G(x2n , x2n+1 , x2n+1 ), G(x2n , f x2n , f x2n ), G(x2n+1 , gx2n+1 , gx2n+1 ), [G(x2n , gx2n+1 , gx2n+1 ) + G(x2n+1 , f x2n , f x2n )]/2  max G(x2n , x2n+1 , x2n+1 ), G(x2n , x2n+1 , x2n+1 ), G(x2n+1 , x2n+2 , x2n+2 ), [G(x2n , x2n+2 , x2n+2 ) + G(x2n+1 , x2n+1 , x2n+1 )]/2  max G(x2n , x2n+1 , x2n+1 ), G(x2n+1 , x2n+2 , x2n+2 ), [G(x2n , x2n+1 , x2n+1 ) + G(x2n+1 , x2n+2 , x2n+2 )]/2  max G(x2n , x2n+1 , x2n+1 ), G(x2n+1 , x2n+2 , x2n+2 ) .

Now, if G(x2n+1 , x2n+2 , x2n+2 ) ≥ G(x2n , x2n+1 , x2n+1 ) for some n = 0, 1, 2, · · · , then M (x2n , x2n+1 , x2n+1 ) ≤ G(x2n+1 , x2n+2 , x2n+2 ) and from (2.10), we have G(x2n+1 , x2n+2 , x2n+2 ) ≤ δM (x2n , x2n+1 , x2n+1 ) ≤ δG(x2n+1 , x2n+2 , x2n+2 ), a contradiction. Therefore, for all n ≥ 0, G(x2n+1 , x2n+2 , x2n+2 ) ≤ G(x2n , x2n+1 , x2n+1 ). Similarly, we have G(x2n , x2n+1 , x2n+1 ) ≤ G(x2n−1 , x2n , x2n ) for all n ≥ 0. Hence for all n ≥ 0, G(xn+1 , xn+2 , xn+2 ) ≤ δG(xn , xn+1 , xn+1 ). 8

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Following similar argument to those in Theorem 2.2, it follows that {xn } is a G−Cauchy sequence and by the G−completeness of X, there exists u ∈ X such that {xn } is G−convergent to u as n → ∞. If f is continuous, then it is clear that f u = u and hence u is common fixed point of f and g. Now consider the case when f is not continuous. Since {x2n } is a nondecreasing sequence with x2n → u in X implies x2n  u for all n ∈ N. Now from (2.7) G(x2n+1 , gu, gu)  = G(f x2n , gu, gu) ≤ δM (x2n , u, u) + L min G(x2n , f x2n , f x2n ), G(u, gu, gu), G(x2n , gu, gu), G(u, f x2n , f x2n )  = δM (x2n , u, u) + L min G(x2n , x2n+1 , x2n+1 ), G(u, gu, gu), G(x2n , gu, gu), G(u, f x2n+1 , f x2n+1 ) ,

(2.11)

where M (x2n , u, u)  = max G(x2n , u, u), G(x2n , f x2n , f x2n ), G(u, gu, gu), [G(x2n , gu, gu) + G(u, f x2n , f x2n )]/2  = max G(x2n , u, u), G(x2n , x2n+1 , x2n+1 ), G(u, gu, gu), [G(x2n , gu, gu) + G(u, x2n+1 , x2n+1 )]/2 , which implies that lim M (x2n , u, u) = G(u, gu, gu). From (2.11), we obtain that n→∞

G(u, gu, gu) = lim G(x2n+1 , gu, gu) ≤ δG(u, gu, gu), n→∞

and G(u, gu, gu) = 0 and so u = gu and u is a common fixed point of f and g. Corollary 2.5. Let (X, ) be a partially ordered set equipped with a complete metric G and let f, g : X → X be two weakly increasing mappings for which there exist δ ∈ (0, 1) and some L ≥ 0 such that G(f x, gy, gy)   G(x, f x, f x) + G(y, gy, gy) G(x, gy, gy) + G(y, f x, f x) , ≤ δ max G(x, y, y), 2 2  +L min G(x, f x, f x), G(y, gy, gy), G(x, gy, gy), G(y, f x, f x) , (2.12) for all comparable x, y ∈ X. If either f or g is continuous or for a nondecreasing sequence {xn } with xn → z in X implies that xn  z for all n ∈ N. Then f and g have a common fixed point. Proof. As inequality (2.12) is a special case of (2.7), the result follows from Theorem 2.4. Corollary 2.6. Let (X, ) be a partially ordered set equipped with a complete metric G and let f, g : X → X be two weakly increasing mappings for which there exist δ ∈ (0, 1) and some L ≥ 0 such that G(f x, gy, gy)   G(x, f x, f x) + G(y, gy, gy) G(x, gy, gy) + G(y, f x, f x) , , ≤ δ max 2 2

(2.13)

9

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for all comparable x, y ∈ X. If either f or g is continuous or for a nondecreasing sequence {xn } with xn → z in X implies that xn  z for all n ∈ N. Then f and g have a common fixed point. Proof. As inequality (2.13) is a special case of (2.7), the result follows from Theorem 2.4. Remark 2.7. 1. If L = 0 in Theorem 2.2, then [15, Theorem 2.1 and 2.9] and [17, Theorem 2.1-2.5, Corollary 2.6-2.8] are extended to two mappings in partially ordered G−metric spaces. 2. If L = 0 in Theorem 2.4, then [9, Theorem 2.1-2.2], [15, Theorem 2.3 and 2.9], [16, Theorem 2.1 and 2.6] and [17, Corollary 2.7-2.8] are extended to the common fixed point results in partially ordered G−metric spaces. Also, Theorem 2.4 extends and generalizes the results [9, Theorem 2.3, 2.6, Corollary 2.4-2.5] to the common fixed point result in partially ordered G−metric spaces. 3. Corollary 2.3 extends [9, Corollary 2.5] to the partially ordered G− metric spaces. Also, if L = 0 in Corollary 2.3, then the results [15, Theorem 2.9] and [17, Corollary 2.7-2.8] are extended to the common fixed point results in partially ordered G−metric spaces. 4. If L = 0 in Corollary 2.5, then [17, Corollary 2.8] are extended to two mappings in partially ordered G− metric spaces. Also Corollary 2.5 extends [9, Corollary 2.4-2.5] to partially ordered G−metric spaces. 5. Corollary 2.6 extends [17, Corollary 2.8] to the common fixed point result in partially ordered G−metric spaces. Acknologements: This work was supported by the Basic Science Research Program through the National Research Foundation Grant funded by Ministry of Education of the republic of Korea(2013R1A1A2054617).

References [1] M. Abbas and B. E. Rhoades, Common fixed point results for non-commuting mappings without continuity in generalized metric spaces, Appl. Math. and Computation, 215(2009), 262-269. [2] M. Abbas, T. Nazir and S. Radenovi´c, Some periodic point results in generalized metric spaces, Appl. Math. Comput., 217(2010), 4094-4099. [3] I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory and Appl., (2010), Article ID 621492, 17 pages. [4] A. Amini-Harandi and H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. TMA, 72(2010), 2238-2242. [5] V. Berinde, Iterative Approximation of Fixed Points, 2nd Ed., Springer Verlag, Berlin Heidelberg New York, 2007.

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[6] V. Berinde and M. Pacurar, Fixed points and continuity of almost contractions, Fixed Point Theory, 9(2008), 23-34. ´ c-type almost con[7] V. Berinde, Some remarks on a fixed point theorem for Ciri´ tractions, Carpathian J. Math., 25(2009), 157-162. [8] R. Chugh, T. Kadian, A. Rani and B. E. Rhoades, Property P in G− metric Spaces, Fixed Point Theory and Appl., (2010), Article ID 401684, 12 pages. ´ c, M. Abbas, R. Saadati and N. Hussain, Common fixed points of almost [9] Lj. Ciri´ generalized contractive mappings in ordered metric spaces, Appl. Math. and Computation, 217(2011), 5784-5789. [10] J. Harjani and K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. TMA, 71(2009), 3403 341. [11] J.K. Kim, S. Sedghi and N. Shobe, Common fixed point theorems for a class of weakly compatible mappings in D-metric spaces, East Asian Mathematical Journal, 25(2009), 107-117. [12] J.K. Kim and S. Chandok, Coupled common fixed point theorems for generalized nonlinear contraction mappings with the mixed monotone property in partially ordered metric spaces, Fixed Point Theory and Appl., 2013:317 (2013), 17 pages. [13] Z. Mustafa and B. Sims, Some remarks concerning D− metric spaces, Proc. Int. Conf. on Fixed Point Theory Appl., Valencia (Spain), July (2003), 189-198. [14] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. of Nonlinear and Convex Analysis, 7(2006), 289-297. [15] Z. Mustafa, H. Obiedat and F. Awawdeh, Some fixed point theorem for mapping on complete G− metric spaces, Fixed Point Theory Appl., (2008), Article ID 189870, 12 pages. [16] Z. Mustafa and B. Sims, Fixed point theorems for contractive mapping in complete G− metric spaces, Fixed Point Theory Appl., (2009), Article ID 917175, 10 pages. [17] Z. Mustafa, W. Shatanawi and M. Bataineh, Existance of Fixed point Results in G− metric spaces, Inter. J. Mathematics and Math. Sciences., vol. 2009, Article ID 283028, 10 pages. [18] J. J. Nieto and R. R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22(2005), 223-239. [19] A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc., 132(2004), 1435-1443. [20] R. Saadati, S. M. Vaezpour, P. Vetro, and B. E. Rhoades, Fixed point theorems in generalized partially ordered G− metric spaces, Mathematical and Computer Modelling, 52(2010), 797-801. [21] W. Shatanawi, Fixed point theory for contractive mappings satisfying Φ−maps in G− metric spaces, Fixed Point Theory Appl., (2010), Article ID 181650, 9 pages.

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ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN MATRIX NORMED SPACES AFSHAN BATOOL, CHOONKIL PARK, AND DONG YUN SHIN∗ Abstract. In [22], Kim et al. introduced and investigated the following additive ρ-functional inequalities kf (x + y + z) − f (x) − f (y) − f (z)k (0.1)

   

x+y + z − f (x) − f (y) − 2f (z) ≤

,

ρ 2f 2





2f x + y + z − f (x) − f (y) − 2f (z) (0.2)

2

   

x+y+z ≤ − f (x) − f (y) − f (z)

ρ 2f

2 in complex Banach spaces. We prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in complex matrix normed spaces and prove the Hyers-Ulam stability of additive ρ-functional equations associated with the additive ρ-functional inequalities (0.1) and (0.2) in complex matrix normed spaces.

1. Introduction and preliminaries The abstract characterization given for linear spaces of bounded Hilbert space operators in terms of matricially normed spaces [29] implies that quotients, mapping spaces and various tensor products of operator spaces may again be regarded as operator spaces. Owing in part to this result, the theory of operator spaces is having an increasingly significant effect on operator algebra theory (see [5]). The proof given in [29] appealed to the theory of ordered operator spaces [2]. Effros and Ruan [6] showed that one can give a purely metric proof of this important theorem by using a technique of Pisier [26] and Haagerup [11] (as modified in [4]). We will use the following notations: Mn (X) is the set of all n × n-matrices in X; ej ∈ M1,n (C) is that j-th component is 1 and the other components are zero; Eij ∈ Mn (C) is that (i, j)-component is 1 and the other components are zero; Eij ⊗ x ∈ Mn (X) is that (i, j)-component is x and the other components are zero; 2010 Mathematics Subject Classification. Primary 47L25, 47H10, 39B82, 39B72, 46L07, 39B52, 39B62. Key words and phrases. Hyers-Ulam stability; additive ρ-functional equation; additive ρ-functional inequality; matrix normed space. ∗ Corresponding author: Dong Yun Shin (email: [email protected]).

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For x ∈ Mn (X), y ∈ Mk (X), x⊕y =

x 0 0 y

!

.

Note that (X, {k · kn }) is a matrix normed space if and only if (Mn (X), k · kn ) is a normed space for each positive integer n and kAxBkk ≤ kAkkBkkxkn holds for A ∈ Mk,n (C), x = (xij ) ∈ Mn (X) and B ∈ Mn,k (C), and that (X, {k · kn }) is a matrix Banach space if and only if X is a Banach space and (X, {k · kn }) is a matrix normed space. A matrix normed space (X, {k · kn }) is called an L∞ -matrix normed space if kx ⊕ ykn+k = max{kxkn , kykk } holds for all x ∈ Mn (X) and all y ∈ Mk (X). Let E, F be vector spaces. For a given mapping h : E → F and a given positive integer n, define hn : Mn (E) → Mn (F ) by hn ([xij ]) = [h(xij )] for all [xij ] ∈ Mn (E). Lemma 1.1. ([25]) Let (X, {k.kn }) be a matrix normed space. (1) k Ekl ⊗ x kn =k x k f or x ∈ X. (2) k xkl k≤k [xij ] k n ≤

n P

k xij k f or [xij ] ∈ Mn (X).

i,j=1

(3) limn→∞ xn = x if and only if limn→∞ xnij = xij f or xn = [xnij ], x = [xij ] ∈ Mk (X). The stability problem of functional equations originated from a question of Ulam [32] concerning the stability of group homomorphisms. Hyers [12] 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 Rassias [27] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [8] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. See [3, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 30, 31] for more information on functional equations and their stability. In [9], Gil´anyi showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (xy −1 )k ≤ kf (xy)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (xy) + f (xy −1 ). See also [28]. Gil´anyi [10] and Fechner [7] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [24] proved the Hyers-Ulam stability of additive functional inequalities. Kim et al. [22] solved the additive ρ-functional inequalities (0.1) and (0.2) in complex normed spaces and proved the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces In Section 2, we prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in complex matrix Banach spaces. We moreover prove the Hyers-Ulam stability of an additive ρ-functional equation associated with the additive ρ-functional inequality (0.1) in complex matrix Banach spaces. In Section 3, we prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.2) in complex matrix Banach spaces. We moreover prove the Hyers-Ulam stability

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of an additive ρ-functional equation associated with the addtive ρ-functional inequality (0.2) in complex matrix Banach spaces. Throughout this paper, let (X, {k · kn }) be a matrix normed space and (Y, {k · kn }) a matrix Banach space. Let ρ be a fixed complex number with |ρ| < 1. 2. Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in matrix normed spaces In this section, we prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in complex matrix normed spaces. Theorem 2.1. Let r > 1 and be nonnegative real numbers, and let f : X → Y be a mapping such that kfn ([xij + yij + zij ]) − fn ([xij ]) − fn ([yij ]) − fn ([zij ])kn

≤

ρ 2fn

+

n X

[xij ] + [yij ] + [zij ] 2

!

(2.1)

!

− fn ([xij ]) − fn ([yij ]) − 2fn ([zij ])

n

θ (k xij kr + k yij kr + k zij kr )

i,j=1

for all x = [xij ], y = [yij ], z = [zij ] ∈ Mn (X). Then there exists a unique additive mapping h : X → Y such that kfn ([xij ]) − hn ([xij ])kn ≤

n X i,j=1

2θ k xij kr −2

(2.2)

2r

for all x = [xij ] ∈ Mn (X). Proof. Let n = 1 in (2.1). Then we obtain kf (a + b + c) − f (a) − f (b) − f (c)k

! !

a+b

≤ ρ 2f + c − f (a) − f (b) − 2f (c)

2 +θ (kakr + kbkr + kckr ) for all a, b, c ∈ X. By [22, Theorem 2.3], there is a unique additive mapping h : X → Y such that kf (a) − h(a)k ≤

2θ kakr 2r − 2

for all a ∈ X. By Lemma 1.1, kfn ([xij ]) − hn ([xij ])kn ≤

n X i,j=1

2θ k xij kr −2

2r

for all x = [xij ] ∈ Mn (X). Thus the mapping h : X → Y is a unique additive mapping satisfying (2.2). 

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Theorem 2.2. Let r < 1 and θ be positive real numbers, and let f : X →Y be a mapping satisfying (2.1). Then there exists a unique additive mapping h : X → Y such that kfn ([xij ]) − hn ([xij ])kn ≤

n X

2θ k xij kr r 2 − 2 i,j=1

(2.3)

for all x = [xij ] ∈ Mn (X). Proof. By [22, Theorem 2.4], there is a unique additive mapping h : X → Y such that 2θ kf (a) − h(a)k ≤ kakr 2 − 2r for all a ∈ X. By Lemma 1.1, n X 2θ k xij kr kfn ([xij ]) − hn ([xij ])kn ≤ r 2 − 2 i,j=1 for all x = [xij ] ∈ Mn (X). Thus the mapping h : X → Y is a unique additive mapping satisfying (2.3).  By the triangle inequality, we have kfn ([xij + yij + zij ]) − fn ([xij ]) − fn ([yij ]) − fn ([zij ])kn

− ρ 2fn

!

!

[xij ] + [yij ]

+ [zij ] − fn ([xij ]) − fn ([yij ]) − 2fn ([zij ])

2 n ≤ kfn ([xij + yij + zij ]) − fn ([xij ]) − fn ([yij ]) − fn ([zij ]) ! !

[xij ] + [yij ]

−ρ 2fn + [zij ] − fn ([xij ]) − fn ([yij ]) − 2fn ([zij ]) .

2 n

As corollaries of Theorems 2.1 and 2.2, we obtain the Hyers-Ulam stability results for the additive ρ-functional equations associated with the additive-ρ-functional inequality (0.1) in complex matrix Banach spaces. Corollary 2.3. Let r > 1 and be nonnegative real numbers, and let f : X → Y be a mapping such that kfn ([xij + yij + zij ]) − fn ([xij ]) − fn ([yij ]) − fn ([zij ]) (2.4) ! !

[xij ] + [yij ]

−ρ 2fn + [zij ] − fn ([xij ]) − fn ([yij ]) − 2fn ([zij ])

2 n ≤

n X

θ (k xij kr + k yij kr + k zij kr )

i,j=1

for all x = [xij ], y = [yij ], z = [zij ] ∈ Mn (X). Then there exists a unique additive mapping h : X → Y satisying (2.2). Corollary 2.4. Let r < 1 and be positive real numbers, and let f : X → Y be a mapping satisfying (2.4). Then there exists a unique additive mapping h : X → Y satisfying (2.3). Remark 2.5. If ρ is a real number such that −1 < ρ < 1 and Y is a real Banach space, then all the assertions in this section remain valid.

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Also, we prove the Hyers-Ulam stability of the following additive ρ-functional inequality





x+y

2f + z − f (x) − f (y) − 2f (z)

2 ≤ kρ (f (x + y + z) − f (x) − f (y) − f (z)) k in complex matrix Banach spaces. The proof is similar to the proofs of Theorems 2.1 and 2.2 except for direction of the inequality, and so we will omit it. 3. Hyers-Ulam stability of the additive ρ-functional inequality (0.2) in matrix normed spaces In this section, we prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.2) in complex matrix normed spaces. Theorem 3.1. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that

!

[xij ] + [yij ]

+ [zij ] − fn ([xij ]) − fn ([yij ]) − 2fn ([zij ])

2fn

2 n

! !

[x ] + [y ] + [z ] ij ij ij

≤ ρ 2fn − fn ([xij ]) − fn ([yij ]) − fn ([zij ])

2

(3.1)

n

+

n X

θ (kxij kr + kyij kr + kzij kr )

i.j=1

for all x = [xij ], y = [yij ], z = [zij ] ∈ Mn (X). Then there exists a unique additive mapping h : X → Y satisfying n X

2r−1 θ k xij kr kfn ([xij ]) − hn ([xij ])kn ≤ r i,j=1 (1 − |ρ|)(2 − 2)

(3.2)

Proof. Let n = 1 in (3.1). Then we get

!

a+b

2f + c − f (a) − f (b) − 2f (c)

2

! !

a+b+c

≤

ρ 2f − f (a) − f (b) − f (c)

+ θ (kakr + kbkr + kckr ) 2

for all a, b, c ∈ X. By [22, Theorem 3.3], there is a unique additive mapping h : X → Y such that kf (a) − h(a)k ≤

2r−1 θ kakr (1 − |ρ|)(2r − 2)

for all a ∈ X. By Lemma 1.1, kfn ([xij ]) − hn ([xij ])kn ≤

n X

2r−1 θ k xij kr r − 2) (1 − |ρ|)(2 i,j=1

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for all x = [xij ] ∈ Mn (X). Thus the mapping h : X → Y is a unique additive mapping satisfying (3.2).  Theorem 3.2. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying (3.1). Then there exists a unique mapping h : X → Y such that kfn ([xij ]) − hn ([xij ])kn ≤

n X

2r θ k xij kr r) (1 − |ρ|)(2 − 2 i,j=1

(3.3)

for all x = [xij ] ∈ Mn (X) Proof. By [22, Theorem 3.4], there is a unique additive mapping h : X → Y such that kf (a) − h(a)k ≤

2r θ kakr r (1 − |ρ|)(2 − 2 )

for all a ∈ X. By Lemma 1.1, kfn ([xij ]) − hn ([xij ])kn ≤

n X

2r θ k xij kr r) (1 − |ρ|)(2 − 2 i,j=1

for all x = [xij ] ∈ Mn (X). Thus the mapping h : X → Y is a unique additive mapping satisfying (3.3).  By the triangle inequality, we have



2fn



!

[xij ] + [yij ]

+ [zij ] − fn ([xij ]) − fn ([yij ]) − 2fn ([zij ])

2 n

! !

[xij ] + [yij ] + [zij ]

−

ρ 2fn − fn ([xij ]) − fn ([yij ]) − fn ([zij ])

2 n

!

[x ] + [y ] ij ij ≤

2fn + [zij ] − fn ([xij ]) − fn ([yij ]) − 2fn ([zij ])

2 ! !

[xij ] + [yij ] + [zij ] −ρ 2fn − fn ([xij ]) − fn ([yij ]) − fn ([zij ])

.

2 n

As corollaries of Theorems 3.1 and 3.2, we obtain the Hyers-Ulam stability results for the additive ρ-functional equations associated with the additive-ρ-functional inequality (0.2) in complex matrix Banach spaces. Corollary 3.3. Let r > 1 and be nonnegative real numbers, and let f : X → Y be a mapping such that



2fn

!

[xij ] + [yij ] + [zij ] − fn ([xij ]) − fn ([yij ]) − 2fn ([zij ]) 2 !

−ρ 2fn ≤

n X

(3.4) !



[xij ] + [yij ] + [zij ] − fn ([xij ]) − fn ([yij ]) − fn ([zij ]) 2

n

θ (k xij kr + k yij kr + k zij kr )

i,j=1

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ρ-FUNCTIONAL INEQUALITIES IN MATRIX NORMED SPACES

for all x = [xij ], y = [yij ], z = [zij ] ∈ Mn (X). Then there exists a unique additive mapping h : X → Y satisying (3.2). Corollary 3.4. Let r < 1 and be positive real numbers, and let f : X → Y be a mapping satisfying (3.4). Then there exists a unique additive mapping h : X → Y satisfying (3.3). Remark 3.5. If ρ is a real number such that −1 < ρ < 1 and Y is a real Banach space, then all the assertions in this section remain valid. Also, we prove the Hyers-Ulam stability of the following additive ρ-functional inequality





x+y+z

2f − f (x) − f (y) − f (z)

2    

x+y ≤

ρ 2f + z − f (x) − f (y) − 2f (z)

2 in complex matrix Banach spaces. The proof is similar to the proofs of Theorems 3.1 and 3.2 except for direction of the inequality, and so we will omit it. 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) References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] M.-D. Choi and E. Effros, Injectivity and operator spaces, J. Funct. Anal. 24 (1977), 156–209. [3] A. Ebadian, S. Zolfaghari, S. Ostadbashi and C. Park, Approximately quintic and sextic mappings in matrix non-Archimedean random normed spaces, Sylwan 158 (2014), No. 5 (3), 248–260. [4] E. Effros, On multilinear completely bounded module maps, Contemp. Math. 62, Amer. Math. Soc.. Providence, RI, 1987, pp. 479–501. [5] E. Effros and Z.-J. Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (1990), 163–187. [6] E. Effros and Z.-J. Ruan, On the abstract characterization of operator spaces, Proc. Amer. Math. Soc. 119 (1993), 579–584. [7] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [8] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [9] A. Gil´anyi, Eine zur Parallelogrammgleichung ¨ aquivalente Ungleichung, Aequationes Math. 62 (2001), 303–309. [10] A. Gil´anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [11] U. Haagerup, Decomp. of completely bounded maps, unpublished manuscript. [12] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [13] M. Kim, Y. Kim, G. A. Anastassiou and C. Park, An additive functional inequality in matrix normed modules over a C ∗ -algebra, J. Comput. Anal. Appl. 17 (2014), 329–335.

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[14] M. Kim, S. Lee, G. A. Anastassiou and C. Park, Functional equations in matrix normed modules, J. Comput. Anal. Appl. 17 (2014), 336–342. [15] J. Lee, S. Lee and C. Park, Fixed points and stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras, J. Comput. Anal. Appl. 15 (2013), 692–698. [16] J. Lee, C. Park, Y. Cho and D. Shin, Orthogonal stability of a cubic-quartic functional equation in non-Archimedean spaces, J. Comput. Anal. Appl. 15 (2013), 572–583. [17] L. Li, G. Lu, C. Park and D. Shin, Additive functional inequalities in generalized quasi-Banach spaces, J. Comput. Anal. Appl. 15 (2013), 1165–1175. [18] G. Lu, Y. Jiang and C. Park, Additive functional equation in Fr´echet spaces, J. Comput. Anal. Appl. 15 (2013), 369–373. [19] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [20] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [21] J. Kim, C. Park, R. Saadati and D. Shin, Completeness of fuzzy normed spaces and fuzzy stability of quintic functional equations, Sylwan 158 (2014), No. 5 (3), 27–41. [22] M. Kim, M. Kim, Y. Kim, S. Lee and C. Park, 3-Variable additive ρ-functional inequalities and equations, Results Math. (in press). [23] C. Park, Functional inequalities in β-homogeneous normed spaces, Sylwan 158 (2014), No. 5 (2), 2–10. [24] C. Park, Y. Cho and M. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, J. Inequal. Appl. 2007 (2007), Article ID 41820, 13 pages. [25] C. Park, D. Shin and J. Lee, An additive functional inequality in matrix normed spaces, Math. Inequal. Appl. 16 (2013), 1009–2022. [26] G. Pisier, Grothendieck’s Theorem for non-commutative C ∗ -algebras with an appendix on Grothendieck’s constants, J. Funct. Anal. 29 (1978), 397–415. [27] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [28] J. R¨atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. [29] Z.-J. Ruan, Subspaces of C ∗ -algebras, J. Funct. Anal. 76 (1988), 217–230. [30] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [31] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125– 134. [32] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Afshan Batool Department of Mathematics, Quaid-i-Azam University Islamabad, Pakistan E-mail address: [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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New iterative algorithms for coupled matrix equations Lingling Lv ∗†, Lei Zhang†

‡

September 18, 2014

Abstract Iterative algorithms are presented for solving coupled matrix equations including the well-known coupled Sylvester matrix equations and the discrete-time coupled Markovian jump Lyapunov matrix equations as special cases. The proposed methods remove the restriction that the considered matrix equation has a unique solution, which is required in the existing gradient based iterative algorithms and identification principle based methods. Keywords: Coupled matrix equations; iterative algorithms; least squares solutions.

1

Introduction

For convenience, in this paper we use I[m, n] to denote the set {m, m + 1, · · · , n} for two integers m and n, m ≤ n. In systems and control theory, some coupled Sylvester matrix equations play very vital roles. For example, pairs of generalized Sylvester equations (A1 X − Y B1 , A2 X − Y B2 ) = (C1 , C2 ) are encountered in perturbation analysis of generalized eigenspaces of matrix pencils [1]. In the analysis of discrete-time jump linear systems with Markovian transitions, the following coupled discrete Markovian jump Lyapunov (CDMJL) equation is encountered ([2] [3] [4]) ) (N ∑ (1) pil Pl Ai + Qi , Qi > 0, i ∈ I[1, N ] Pi = ATi l=1

where Pi , i ∈ I[1, N ] are the matrices to be determined. Due to their wide applications, they have attracted considerable attention of many researchers. It was pointed out in [4] that existence of positive definite solutions for the CDMJL equation (1) is related to the spectral radius of an augmented matrix being less than one. A simple explicit solution for the CDMJL equation (1) was given in [4] in terms of matrix inversions by convert them into matrix-vector equations. However, computational difficulties arise when the dimensions of the coefficient matrices are high. To eliminate the high dimensionality problems, a parallel iterative scheme for solving the CDMJL equation was proposed in [3]. Under the condition that all the subsystems are Schur stable, this method was proven to converge to the exact solution if the initial condition is chosen as zero. In [5], the restriction of zero initial conditions of the method in [3] was removed, and a new iteration method was provided by using the solutions of discrete-time Lyapunov equations as its intermediate steps. Very recently, from an optimization point of view a gradient-based algorithm was developed in [6] to solve the CDMJL equation (1). Alternative ways are to transform the matrix equations into forms for which solutions may be readily computed. In this area, Chu gave a numerical algorithm for solving the coupled Sylvester equations in [7]. ∗ Lingling Lv is with Electric Power College, North China University of Water Resources and Electric Power, Zhengzhou, 450045, People’s Republic of China. † This work is supported by the National Natural Science Foundation of China (No. U1204605, 11226239, 61402149), Excellent-Young-Backbone Teacher Project in high school of Henan Province (No. 2013GGJS-087), Scientific Research Key Project Fund of the Education Department of Henan Province(No. 12B120007). ‡ Lei Zhang is with Computer and Information Engineering College, Henan University, Kaifeng, 475004, P. R. China. Email: [email protected]. Corresponding author.

1

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In [8], generalized Schur methods for solving the coupled Sylvester matrix equations was proposed. Recently, some iterative algorithms for solving the pairs of generalized matrix equations were proposed in [9] and [10] by applying the hierarchical identification principle, which has been used to construct iterative algorithms for noncoupled matrix equations in [11]. Different from the methods in [7] and [8], these algorithms do not require matrix transformation, and thus can be realized in terms of original coefficient matrices. Very recently, from an optimization point of view gradient based iteration was constructed in [12] to solve the general coupled matrix equation. A significant characteristic of the method in [12] is that a necessary and sufficient condition guaranteeing the convergence of the algorithm can be explicitly obtained. However, the methods in [10], [12] are only suitable for solving the coupled matrix equation having unique solutions. In addition, it should be pointed out that complex matrix equations with the conjugate of the unknown matrices are attracting more and more researchers. In [13], the so-called Sylvester-polynomial-conjugate matrix equation was investigated, and an explicit solution was given. In [14], an iterative algorithm was given for coupled Sylvester-conjugate matrix equations by using hierarchical identification principle. In [15], an iterative algorithm was constructed for a kind of coupled Sylvester-conjugate matrix equations by using a real inner product as tools. In this paper, we investigate iterative algorithms for solving general coupled matrix equations. The proposed methods do not require that the concerned matrix equations have unique solutions. Compared with the method of obtaining least squares solutions in [9], the method in the current paper removes the restriction that some coefficient matrices must have full column or row ranks, and does not involve matrix inversions. Throughout this paper, the symbol tr(A) is used to denote the trace of A. For the space Rm×n we define an inner product as ⟨A, B⟩ = tr(B T A) for all A, B ∈ Rm×n . The norm of a matrix A generated by this inner product is, obviously, a Frobenius norm, and denoted by ∥A∥. In addition, the symbol ∥A∥2 denotes the spectral norm of matrix A. For two matrices M and N , M ⊗ N is their Kronecker product. For a matrix [ ] X = x1 x2 · · · xn ∈ Rm×n , vec(X) is the column stretching operation of X, and defined as ]T [ . vec(X) = xT1 xT2 · · · xTn A well-known property of Kronecker product is, for matrices M , N and X with appropriate dimension ( ) vec(M XN ) = N T ⊗ M vec(X).

2

The case of two unknown matrices

In this section, we consider the following coupled Sylvester matrix equation { A1 XB1 + C1 Y D1 = E1 A2 XB2 + C2 Y D2 = E2

(2)

where Ai ∈ Rmi ×r , Bi ∈ Rs×ni , Ci ∈ Rmi ×p , Di ∈ Rq×ni , and Ei ∈ Rmi ×ni , i = 1, 2, are known matrices, and X ∈ Rr×s and Y ∈ Rp×q are the matrices to be determined. Obviously, this coupled matrix equation includes the pair of generalized Sylvester matrix equation in [1], [9] as a special case. It should be mentioned that the methods in [1], [9] require the unknown matrices X and Y to have the same dimension. In this paper, the algorithm to be developed removes such restriction. In this paper, we solve it in the least squares sense. That is, we search matrix pair (X, Y ) to minimize the following index function J(X, Y ) =

1 1 2 2 ∥E1 − A1 XB1 − C1 Y D1 ∥ + ∥E2 − A2 XB2 − C2 Y D2 ∥ . 2 2

It is easy to obtain that ∂J ∂X ∂J ∂Y

= AT1 (E1 − A1 XB1 − C1 Y D1 ) B1T + AT2 (E2 − A2 XB2 − C2 Y D2 ) B2T , = C1T (E1 − A1 XB1 − C1 Y D1 ) D1T + C2T (E2 − A2 XB2 − C2 Y D2 ) D2T . 2

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(3)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Obviously, a least squares solution (X∗ , Y∗ ) satisfies ∂J ∂J = 0, = 0. ∂X X=X∗ ∂Y Y =Y∗ In addition, denote R(k) =

∂J ∂J , S(k) = . ∂X X=X(k) ∂Y Y =Y (k)

With the above prelimilaries, the iteration method to solve least squares solutions of the coupled matrix equation (2) can be constructed as the following algorithm. Algorithm 1 (Iterative algorithm for least squares solutions of (2)) 1. Given initial values X(0) and Y (0), calculate ¯ R(0) = ¯ S(0) = R(0) = S(0) = P (0) = Q(0) = k

:

E1 − A1 X(0)B1 − C1 Y (0)D1 ; E2 − A2 X(0)B2 − C2 Y (0)D2 ; T T ¯ T ¯ AT1 R(0)B 1 + A2 S(0)B2 ; T T ¯ T ¯ C1T R(0)D 1 + C2 S(0)D2 ; −R(0); −S(0); = 0;

2. If ∥R(k)∥ ≤ ε, ∥S(k)∥ ≤ ε, then stop; else, k := k + 1; 3. Set k := k + 1. Calculate α(k) = X(k + 1) = Y (k + 1) = ¯ + 1) = R(k ¯ + 1) = S(k R(k + 1) =

[ ] [ ] tr P T (k)R(k) + tr QT (k)S(k)

2;

2

∥A1 P (k)B1 + C1 Q(k)D1 ∥ + ∥A2 P (k)B2 + C2 Q(k)D2 ∥ X(k) + α(k)P (k) ∈ Rr×s ; Y (k) + α(k)Q(k) ∈ Rp×q ;

S(k + 1)

=

E1 − A1 X(k + 1)B1 − C1 Y (k + 1)D1 ∈ Rm1 ×n1 ; E2 − A2 X(k + 1)B2 − C2 Y (k + 1)D2 ∈ Rm2 ×n2 ; ¯ + 1)B2T ∈ Rr×s ; ¯ + 1)B1T + AT2 S(k AT1 R(k ¯ + 1)D2T ∈ Rp×q ; ¯ + 1)D1T + C2T S(k C1T R(k

P (k + 1)

=

−R(k + 1) +

Q(k + 1)

=

−S(k + 1) +

2

∥R(k + 1)∥ + ∥S(k + 1)∥ 2

∥R(k)∥ + ∥S(k)∥ 2

∥R(k)∥ + ∥S(k)∥

P (k) ∈ Rr×s ;

2

∥R(k + 1)∥ + ∥S(k + 1)∥ 2

2

2

2

Q(k) ∈ Rp×q ;

4. Go to Step 2. In the rest of this subsection, we analyze the convergence property of the proposed algorithm. Firstly, we give two lemmas, whose proofs are presented in the Appendix. Lemma 1 For the sequences {R(i)}, {S(i)}, {P (i)} and {Q(i)} generated by Algorithm 1, the following relation holds: [ ] [ ] tr RT (k + 1)P (k) + tr S T (k + 1)Q(k) = 0 for k ≥ 0.

3

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Lemma 2 For the sequences {R (i)}, {S(i)}, {P (i)} and {Q(i)} generated by Algorithm 1, the following relation holds: ( )2 2 2 ∑ ∥R(k)∥ + ∥S(k)∥ < ∞. 2 2 ∥P (k)∥ + ∥Q(k)∥ k≥0 With these two lemmas, the following result on the convergence property of Algorithm 1 can be obtained. Theorem 1 The sequences {R(i)} and {S(i)} generated by Algorithm 1 satisfy lim ∥R(k)∥ = 0, lim ∥S(k)∥ = 0.

k→∞

k→∞

Thus, X(k) and Y (k) given by Algorithm 1 converge to a least squares solution of (2). Proof. By using Lemma 1 and the expressions of P (k + 1) and Q(k + 1) in Algorithm 1, one has 2

2

∥P (k + 1)∥ + ∥Q(k + 1)∥

2

2

2 2 2 2



∥R(k + 1)∥ + ∥S(k + 1)∥ ∥R(k + 1)∥ + ∥S(k + 1)∥



P (k) + −S(k + 1) + Q(k) = −R(k + 1) +

2 2 2 2



∥R(k)∥ + ∥S(k)∥ ∥R(k)∥ + ∥S(k)∥ ( )2 2 2 ( ) ∥R(k + 1)∥ + ∥S(k + 1)∥ 2 2 2 2 = + ∥Q(k)∥ ∥P (k)∥ + ∥R(k + 1)∥ + ∥S(k + 1)∥ . 2 2 ∥R(k)∥ + ∥S(k)∥ Denote

2

tk = (

∥P (k)∥ + ∥Q(k)∥ 2

∥R(k)∥ + ∥S(k)∥

2

2

)2 .

Then the preceding relation can be equivalently written as tk+1 = tk +

1

2.

2

(4)

∥R(k + 1)∥ + ∥S(k + 1)∥

We now proceed by contradiction and assume that ( ) 2 2 lim ∥R(k)∥ + ∥S(k)∥ ̸= 0.

(5)

k→∞

This relation implies that there exists a constant δ > 0 such that 2

2

∥R(k)∥ + ∥S(k)∥ > δ for all k ≥ 0. It follows from (4) and (5) that tk+1 < t0 +

k+1 . δ

This implies that 1 tk+1 Thus one has

δ . δt0 + k + 1

∞ ∞ ∑ ∑ 1 δ ≥ = ∞. tk δt0 + k

k=1

However, it follows from Lemma 2 that

≥

k=1

∞ ∑ 1 < ∞. tk

k=1

This gives a contradiction. So the conclusion of this theorem is true. 4

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3

General cases

In this section, we will extend the iterative methods in Section 2 to solve more general coupled Sylvester matrix equations of the form Ai1 X1 Bi1 + Ai2 X2 Bi2 + · · · + Aip Xp Bip = Ei , i ∈ I[1, N ],

(6)

where Aij ∈ Rmi ×rj , Bij ∈ Rsj ×ni , Ei ∈ Rmi ×ni , i ∈ I[1, N ], j ∈ I[1, p] are known matrices and Xj ∈ Rrj ×sj , j ∈ I[1, p] are matrices to be determined. Such type of matrix matrix equations include the coupled matrix equation (2) as a special case. When N = 2 and p = 2, the matrix equation (6) becomes the coupled matrix equations (2). Similar to the idea of Section 2, we solve the least squares solutions. That is, we search Xj , j ∈ I[1, p] such that the index function

2

p N ∑

1 ∑

J (Xi , i ∈ I[1, N ]) = Ei − Aij Xj Bij (7)

2 i=1

j=1 is minimized. It is easy to obtain that

(

∑ ∂J = ATij ∂Xj i=1 N

Ei −

)

p ∑

Aiω Xω Biω

T Bij j ∈ I[1, p].

ω=1

A least squares solution (X1∗ , X2∗ , · · · , Xp∗ ) should satisfy ∂J = 0, j ∈ I[1, p]. ∂Xj Xj =Xj∗ By generalizing Algorithm 1 for the coupled Sylvester matrix equation (2) in Section 2.2, we give the following iteration method to solve the least squares solution group of the coupled matrix equation (6). Algorithm 2 (Iterative method for least squares solutions of (6)) 1. Given initial values Xj (0), j ∈ I[1, p], calculate ¯ i (0) R

Ei −

=

p ∑

Aij Xj (0) Bij , i ∈ I[1, N ];

j=1 N ∑

T ¯ i (0) Bij , j ∈ I[1, p]; ATij R

Rj (0)

=

Pj (0)

=

−Rj (0) ;

k

:

= 0;

i=1

2. If ∥Rj (k)∥ ≤ ε, j ∈ I[1, p], then stop; else, k := k + 1; 3. Set k := k + 1, calculate α(k)

[ ] tr PjT (k) Rj (k)

2 ; ∑N

∑p

A P (k) B

ij i=1 j=1 ij j ∑p

=

j=1

= Xj (k) + α(k)Pj (k) ∈ Rrj ×sj , j ∈ I[1, p]; p ∑ ¯ i (k + 1) = Ei − R Aij Xj (k + 1) Bij , i ∈ I[1, N ];

Xj (k + 1)

j=1

Rj (k + 1)

=

N ∑

¯ i (k + 1) B T , j ∈ I[1, p]; ATij R ij

i=1

Pj (k + 1)

∑p

i=1 ∥Ri (k + 1)∥ ∑ p 2 i=1 ∥Ri (k)∥

= −Rj (k + 1) +

2

Pj (k) , j ∈ I[1, p].

5

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4. Go to Step 2. Similar to the conclusions in Subsection 2.2, on the convergence property of this algorithm one has the following results. The proofs are all omitted. Lemma 3 For the sequences {Ri (k)}, {Pi (k)}, j ∈ I[1, p] generated by Algorithm 2, the following relation holds: p ∑ [ ] tr RjT (k + 1)Pj (k) = 0 j=1

for k ≥ 0. Lemma 4 For the sequences {Ri (k)}, {Pi (k)}, j ∈ I[1, p] generated by Algorithm 2, the following relation holds: (∑ )2 p 2 ∑ j=1 ∥Ri (k)∥ < ∞. ∑p 2 j=1 ∥Pi (k)∥ k≥0 Theorem 2 The sequences {Ri (k)}, j ∈ I[1, p] generated by Algorithm 2 satisfy lim ∥Rj (k)∥ = 0, j ∈ I[1, p].

k→∞

Thus, Xj (k), j ∈ I[1, p] given by Algorithm 1 converge to a least squares solution of (6). Remark 1 The problem of solving least squares solutions of coupled matrix equations has been investigated in [9]. By using the hierarchical identification principle, an infinite iterative algorithm is constructed to solve the coupled linear matrix equation (6). However, the method in [9] require that some coefficient matrices must have full column or row ranks. In addition, the matrix inversion is required in each iteration, which may turn out to be numerically expensive. The proposed Algorithm 2 removes such restrictions, and thus is expected to have advantage over some existing methods. Remark 2 The coupled matrix equations with the form of (6) have been investigated in [9], [10], [12]. In [9], [10] the considered coupled matrix equations are required to satisfy that p = N , and that all the unknown matrices Xi , i ∈ I[1, N ] have the same dimension. While in [12], it is required that N ∑

m i ni =

i=1

p ∑

rj sj .

j=1

In this paer, the proposed Algorithm 2 removes these restrictions. Remark 3 Different from the methods in [10], [12], an exact solution of the coupled matrix equation can be obtained in finite iteration steps in the absence of round-off errors. It should also be mentioned that the methods in [10], [12] are only suitable for solving the coupled matrix equation (6) with unique solutions. Remark 4 In [10] and [12], in order to guarantee the convergence of the proposed algorithms one must choose an appropriate step-size or convergence factor. In general, such convergence factor can be obtained by some complex computation. Different from these methods, the iterative algorithm given in this paper does not involve the problem of parameters choice, and thus is easier to carry out. Remark 5 Due to round-off errors, Algorithm 2 may not terminate in a finite number of steps. In this case, one can use the algorithm to obtain an approximate solution in sufficient steps.

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4

A special case

In this section, we consider the following matrix equation, which is a special case of the coupled matrix equation (6) Ai XBi = Ei , i ∈ I[1, N ] (8) where Ai ∈ Rmi ×r , Bi ∈ Rs×ni , and Ei ∈ Rmi ×ni , i ∈ I[1, N ] are the known coefficient matrices, and X ∈ Rr×s is the matrix to be determined. Similarly, we search matrix X to minimize the index function 1∑ 2 ∥Ei − Ai XBi ∥ . 2 i=1 N

J (X) =

(9)

One easily obtain that ∑ ∂J = ATi (Ei − Ai XBi ) BiT . ∂X i=1 N

The iterative method to obtain the least squares solution of the matrix equation (8) is constructed as the following algorithm. Algorithm 3 (Finite iterative algorithm for (8)) 1. Given initial values X, calculate Ei − Ai X (0) Bi , i ∈ I[1, N ];

¯ i (0) R

=

R (0)

=

P (0) k

= −R (0) ; : = 0;

N ∑

¯ i (0) BiT ; ATi R

i=1

2. If ∥R(k)∥ ≤ ε, then stop; else, k := k + 1; 3. Set k := k + 1, calculate α(k) X (k + 1) ¯ i (k + 1) R R (k + 1)

=

[ ] tr P T (k) R (k) ; ∑N 2 i=1 ∥Ai P (k) Bi ∥

= X (k) + α(k)P (k) ∈ Rr×s ; = Ei − Ai X (k + 1) Bi , i ∈ I[1, N ]; =

N ∑

¯ i (k + 1) BiT ; ATi R

i=1

P (k + 1)

= −R (k + 1) +

∥R (k + 1)∥ ∥R (k)∥

2

2

P (k) .

4. Go to Step 2. For the convergence properties one has the following results. The proofs are all omitted. Lemma 5 For the sequences {R(k)} and {P (k)} generated by Algorithm 3, the following relation holds: [ ] tr RT (k + 1)P (k) = 0 for k ≥ 0.

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Lemma 6 For the sequences {R(k)} and {Pi (k)} generated by Algorithm 3, the following relation holds: ∑ ∥R(k)∥4 2

k≥0

∥P (k)∥

< ∞.

Theorem 3 The sequences {R(k)} generated by Algorithm 3 satisfy lim ∥R(k)∥ = 0.

k→∞

Thus, X(k), j ∈ I[1, p] given by Algorithm 3 converge to a least squares solution of (8).

5

A numerical example

In this section, we give an example to illustrate the effectiveness of the proposed methods. This example has been used in [9]. The matrix equation is in the form of (2) with [ ] [ ] 2 1 1 0 A1 = , B1 = C1 = B2 = C2 = , −1 2 0 1 [ ] [ ] 1 −0.2 13.2 10.6 D1 = , E1 = , 0.2 1 0.6 8.4 [ ] [ ] [ ] −2 −0.5 −1 −3 −9.5 −18 A2 = , D2 = , E2 = . 0.5 2 2 −4 16 3.5 The solution of X and Y to this equation is [ 4 X= 3

3 4

]

[ ,Y =

2 1 −2 3

] .

Taking X(0) = Y (0) = 10−6 1(2), we apply Algorithm 1 to compute X(k) and Y (k). The iterative solutions of the matrix equation is shown in Table 1. Table 1. The iteration solution by Algorithm 1 k 1 2 3 4 5 6 7 8 Solution

x11 2.583305 4.087500 3.873979 3.678258 4.057452 4.002255 4.000130 4.000000 4

x12 2.473221 1.691281 2.146630 2.509525 2.909192 3.005361 2.999937 3.000000 3

x21 2.502577 3.804495 3.395190 3.149014 2.870312 2.999156 3.000034 3.000000 3

x22 2.123399 2.798320 3.986429 4.126780 4.005162 3.982074 4.000004 4.000000 4

y12 3.648917 3.067254 2.384152 2.016496 2.042451 1.991317 1.999854 2.000000 2

y22 3.240872 1.091186 1.057442 1.250040 1.029262 1.005123 1.000127 1.000000 1

y21 −1.349385 −1.916194 −2.671572 −2.208587 −2.056753 −2.020698 −1.999934 −2.000000 −2

y22 1.297523 2.593294 2.215298 3.192737 3.048898 3.006351 2.999947 3.000000 3

We compare our proposed methods with those given in [10] and [12]. For this aim, as in [10] and [12] we define the relative iteration error as √ 2 2 ∥X(k) − X∥ + ∥Y (k) − Y ∥ δ(k) = . 2 2 ∥X∥ + ∥Y ∥ The results computed by using these methods are plotted in Fig. 1. It is seen that the convergence performances of the proposed algorithms in this paper are better than those by using the methods given in [10] and [12].

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1 ********* Ding and Chen (SIAM 06),µ=0.04 . . . . . . Zhou, Duan and Li (Systems & Control Letters),µ=0.0513 .−−.−−.−−.− Algorithm 1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20

30

40

50

60

Figure 1: Comparison of different iterative algorithms

6

Conclusions

In this paper, iterative algorithms have been developed for solving a class of coupled linear matrix equations which have wide applications in linear system theory. For the case when the considered coupled matrix equation is consistent, the proposed iterative algorithm is proven to converge the exact solution in finite iteration steps in the absence of round-off errors. For the case when the equation is inconsistent, an iterative algorithm is also proposed to give a least squares solution. Compared with some existing methods, the proposed methods in this paper have some advantage. For example, they do not require that some coefficient matrices have full column or row ranks; they are suitable for solving the coupled linear matrix equations having more than one solution.

7

Appendix

This appendix provides the proofs of some lemmas.

7.1

Proof of Lemma 1

From the expressions of X(k + 1) and Y (k + 1) in Algorithm 2, one has

R(k + 1)

= AT1 (E1 − A1 X(k + 1)B1 − C1 Y (k + 1)D1 ) B1T +AT2 (E2 − A2 X(k + 1)B2 − C2 Y (k + 1)D2 ) B2T = AT1 (E1 − A1 X(k)B1 − C1 Y (k)D1 − α(k)A1 P (k)B1 − α(k)C1 Q(k)D1 ) B1T +AT2 (E2 − A2 X(k)B2 − C2 Y (k)D2 − α(k)A2 P (k)B2 − α(k)C2 Q(k)D2 ) B2T = R(k) − α(k)AT1 (A1 P (k)B1 + C1 Q(k)D1 ) B1T −α(k)AT2 (A2 P (k)B2 + C2 Q(k)D2 ) B2T

and S(k + 1)

=

S(k) − α(k)C1T (A1 P (k)B1 + C1 Q(k)D1 ) D1T −α(k)C2T (A2 P (k)B2 + C2 Q(k)D2 ) D2T . 9

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In view of the expression of α(k), it is easily obtained that [ ] [ ] tr RT (k + 1)P (k) + tr S T (k + 1)Q(k) [ ] [ ] [ ( ) ] = tr RT (k)P (k) + tr S T (k)Q(k) − α(k)tr B1 B1T P T (k)AT1 + D1T QT (k)C1T A1 P (k) [ ( ) ] −α(k)tr B2 B2T P T (k)AT2 + D2T QT (k)C2T A2 P (k) [ ( ) ] −α(k)tr D1 B1T P T (k)AT1 + D1T QT (k)C1T C1 Q(k) [ ( ) ] −α(k)tr D2 B2T P T (k)AT2 + D2T QT (k)C2T C2 Q(k) [ ] [ ] = tr RT (k)P (k) + tr S T (k)Q(k) [ ] 2 2 −α(k) ∥A1 P (k)B1 + C1 Q(k)D1 ∥ + ∥A2 P (k)B2 + C2 Q(k)D2 ∥ = 0. The proof is thus completed.

7.2

Proof of Lemma 2

Before giving the proof of this lemma, we need a simple conclusion. Lemma 7 For the sequences {P (k)}, {Q(k)}, {R(k)}, {S(k)} generated by Algorithm 2, for k ≥ 0 the following relation holds: [ ] [ ] 2 2 tr RT (k)P (k) + tr S T (k)Q(k) = − ∥R(k)∥ − ∥S(k)∥ . (10) Proof. Obviously, the relation (10) holds for k = 0. In addition, from the expressions of P (k + 1) and Q(k + 1) in Algorithm 2 one has for k ≥ 0 [ ] [ ] tr RT (k + 1) P (k + 1) + tr S T (k + 1) Q (k + 1) 2

= − ∥R (k + 1)∥ − ∥S (k + 1)∥ +

2

2 2 ] [ ]] ∥R (k + 1)∥ + ∥S (k + 1)∥ [ [ T tr R (k + 1) P (k) + tr S T (k + 1) Q (k) . 2 2 ∥R (k)∥ + ∥S (k)∥

By applying Lemma 1, it is known that [ ] [ ] 2 2 tr RT (k + 1) P (k + 1) + tr S T (k + 1) Q (k + 1) = − ∥R (k + 1)∥ − ∥S (k + 1)∥ . With the above argument, the proof is thus completed. Now we start the proof Lemma 2. Firstly, we denote

[ T

B1 ⊗ A1 π=

B2T ⊗ A2

D1T ⊗ C1 D2T ⊗ C2

] 2

.

(11)

2

By using Kronecker product, one has 2

=

= = ≤ =

2

∥A1 P (k)B1 + C1 Q(k)D1 ∥ + ∥A2 P (k)B2 + C2 Q(k)D2 ∥

( T

) ( )

B1 ⊗ A1 vec (P (k)) + D1T ⊗ C1 vec (Q (k)) 2

(

2 ) ( ) + B2T ⊗ A2 vec (P (k)) + D2T ⊗ C2 vec (Q (k))

( T

) ( )

B1 ⊗ A1 vec (P (k)) + D1T ⊗ C1 vec (Q (k)) 2

( T

) ( T )

B2 ⊗ A2 vec (P (k)) + D2 ⊗ C2 vec (Q (k))

[ T ][ ] 2

B1 ⊗ A1 D1T ⊗ C1 vec (P (k))

B2T ⊗ A2 D2T ⊗ C2 vec (Q (k))

[ T ] [ ]

B1 ⊗ A1 D1T ⊗ C1 2 vec (P (k)) 2



B2T ⊗ A2 D2T ⊗ C2 vec (Q (k)) 2 ( ) 2 2 π ∥P (k)∥ + ∥Q(k)∥ . 10

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With the notation of (11), one has 2

2

∥A1 P (k)B1 + C1 Q(k)D1 ∥ + ∥A2 P (k)B2 + C2 Q(k)D2 ∥ ( ) 2 2 ≤ π ∥P (k)∥ + ∥Q(k)∥ .

(12)

Applying the expressions in Algorithm 1 and Lemma 7, for k ≥ 0 one has

=

=

=

=

=

J (X(k + 1), Y (k + 1))

1 ¯ (k) − α(k) [A1 P (k)B1 + C1 Q(k)D1 ] 2

R 2

2 1 + S¯ (k) − α(k) [A2 P (k)B2 + C2 Q(k)D2 ] 2

) [ T ] 1 ( ¯ (k) (A1 P (k)B1 + C1 Q(k)D1 ) ¯ (k) 2 + S¯ (k) 2 − α(k)tr R

R 2 [ ] −α(k)tr S¯T (k) (A2 P (k)B2 + C2 Q(k)D2 ) 1 1 2 2 + α2 (k) ∥A1 P (k)B1 + C1 Q(k)D1 ∥ + α2 (k) ∥A2 P (k)B2 + C2 Q(k)D2 ∥ 2 2 [ ] [ T ] T T ¯ T ¯ J (X(k), Y (k)) − α(k)tr P T (k) AT1 R(k)B 1 − α(k)tr Q (k) C1 R(k)D1 [ ] [ T ] T T ¯ T ¯ −α(k)tr P T (k) AT2 S(k)B 2 − α(k)tr Q (k) C2 S(k)D2 1 1 2 2 + α2 (k) ∥A1 P (k)B1 + C1 Q(k)D1 ∥ + α2 (k) ∥A2 P (k)B2 + C2 Q(k)D2 ∥ 2 2 [ ] [ ] J (X(k), Y (k)) − α(k)tr P T (k) R (k) − α(k)tr QT (k) S (k) [ [ ] [ ]] 1 + α (k) tr P T (k) R (k) + tr QT (k) S (k) 2 [ [ ] [ ]] 1 J (X(k), Y (k)) − α (k) tr P T (k) R (k) + tr QT (k) S (k) . 2

Again using the expression of α(k) one has

= ≤

J (X(k + 1), Y (k + 1)) − J (X(k), Y (k)) ( )2 2 2 ∥R(k)∥ + ∥S(k)∥ 1 − 2 ∥A1 P (k)B1 + C1 Q(k)D1 ∥2 + ∥A2 P (k)B2 + C2 Q(k)D2 ∥2 0.

(13)

This implies that {J(X(k), Y (k))} is a descent sequence. Therefore, for all k ≥ 1 there holds J(X(k), Y (k)) ≤ J(X(0), Y (0)). Then one has ∞ ∑

[J(X(k), Y (k)) − J(X(k + 1), Y (k + 1))]

k=0

= J(X(0), Y (0)) − lim J(X(k), Y (k)) k→∞

< ∞.

(14)

In addition, combining (12) with (13) gives (

2

2

∥R(k)∥ + ∥S(k)∥ 2

)2

2

∥P (k)∥ + ∥Q(k)∥ ≤ 2π (J(X(k), Y (k)) − J(X(k + 1), Y (k + 1))) . This relation, together with (29), shows that the conclusion of this lemma is true.

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References [1] J. Demmel, B. K¨agstr¨om. Computing stable eigendecompositions of matrix pencils. Linear Alg. Appl., 1987, 88/89: 139-186. [2] A. G. Wu, G. R. Duan. New iterative algorithms for solving coupled Markovian jump Lyapunov equations. IEEE Transactions on Automatic Control, doi: 10.1109/TAC.2014.2326273. [3] I. Borno, Z. Gajic. Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems. Computers Math. Applic., 1995, 30(7): 1-4. [4] O.L.V. Costa, M.D. Fragoso. Stability results for discrete-time linear systems with Markovian jumping parameters. Journal of Mathematical Analysis and Applications, 1993, 179(1): 154–178. [5] Q. Wang, J. Lam, Y. Wei, T. Chen. Iterative solutions of coupled discrete Markovian jump Lyapunov equations. Computers and Mathematics with Applications, 2008, 55(4): 843-850. [6] B. Zhou, J. Lam, G. R. Duan. Convergence of gradient-based iterative solution of coupled Markovian jump Lyapunov equations. Computers and Mathematics with Applications, 2008, 56: 3070-3078. [7] K. E. Chu. The solution of the matrix equations AXB −CXD = E and (Y A−DZ, Y C −BZ) = (E, F ). Linear Algebra Appl., 1987, 93: 93–105. [8] B. K¨agstr¨ om, L. Westin. Generalized Schur methods with condition estimators for sloving the generalized Sylvester equation. IEEE Transactions on Automatic Control, 1989, 34(7): 745-751. [9] F. Ding, T. Chen. Iterative least-squares solutions of coupled Sylvester matrix equations. Systems and Control Letters, 2005, 54: 95-107. [10] F. Ding, T. Chen. On iterative solutions of general coupled matrix equations. SIAM Journal on Control and Optimization, 2006, 44(6): 2269-2284. [11] F. Ding, T. Chen. Gradient based iterative algorithms for solving a class of matrix equations. IEEE Transactions on Automatic Control, 2005, 50(8): 1216-1221. [12] B. Zhou, G. R. Duan, Z. Y. Li. Gradient based iterative algorithm for solving coupled matrix equations. Systems & Control Letters, 2009, 58: 327-333. [13] A. G. Wu, G. Feng, W. Liu, G. R. Duan. The complete solution to the Sylvester-polynomial-conjugate matrix equations. Mathematical and Computer Modelling, 2011, 53: 2044-2056. [14] A. G. Wu, G. Feng, G. R. Duan, W. J. Wu. Iterative solutions to coupled Sylvester-conjugate matrix equations. Computers and Mathematics with Applications, 2010, 60: 54-66. [15] A. G. Wu, B. Li, Y. Zhang, G. R. Duan. Finite iterative solutions to coupled Sylvester-conjugate matrix equations. Applied Mathematical Modelling, 2011, 35: 1065-1080. [16] Z. Peng, Y. Peng. An efficient iterative method for solving the matrix equation AXB + CY D = E. Numer. Linear Algebra Appl., 2006, 13: 473–485.

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ESSENTIAL NORM OF WEIGHTED COMPOSITION OPERATOR FROM HARDY SPACE TO WEIGHTED BERGMAN SPACE ON THE UNIT BALL CHANG-JIN WANG SCHOOL OF SCIENCE, JIMEI UNIVERSITY, XIAMEN FUJIAN 361021, P.R. CHINA. [email protected] YU-XIA LIANG∗ SCHOOL OF MATHEMATICAL SCIENCES, TIANJIN NORMAL UNIVERSITY, TIANJIN 300387, P.R. CHINA. [email protected]

Abstract. In this paper, we estimate the essential norm of a weighted composition operator from Hardy space to weighted Bergman space on the unit ball in the complex N-dimensional Euclidean space. In our results, we use Carleson type measures and certain integral operators to estimate the essential norm.

1. Introduction Throughout this paper, let N ≥ 1 be a fixed integer and B denote the unit ball of the complex N-dimensional Euclidean space C N . Denote H(B) the class of all holomorphic functions on B and S(B) the collection of all holomorphic self-maps of B. Let dν denote the normalized Lebesgue measure on B. For each α > −1, we set cα = Γ(N + α + 1)/{Γ(N + 1)Γ(α + 1)} and dνα (z) = cα (1 − |z|2 )α dν(z), z ∈ B. It is obvious that να (B) = 1. For 0 < p < ∞ and α > −1, the weighted Bergman space Apα = Apα (B) is defined by Z Apα = {f ∈ H(B) : kf kpApα = |f (z)|p dνα (z) < ∞}. B

It is well-known that the weighted Bergman space Apα is a Banach space under the norm k.kApα . For 1 ≤ q < ∞, the Hardy space H q = H q (B) is Z q q H = {f ∈ H(B) : kf kH q = sup |f (rζ)|q dσ(ζ) < ∞}, 0 0), for some constant C > 0. Then there exists a constant K > 0 such that Z |f ∗ |p dµ ≤ Kkf kpH q B

for all f ∈ H . Here, the notation f denotes the function defined on B by f ∗ = f in B and f ∗ = limr→1 fr a.e. [σ] on ∂B. q

∗

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Wang and Liang: Essential norm of weighted composition operator P∞ P For f ∈ Hq , recall that an f ∈ H(B) has the homogeneous expansion f (z) = k=0 |r|=k c(r)z r , rN . For the homowhere r = (r1 , ..., rN ) is a multi-index, |r| = r1 + ... + rN and z r = z1r1 ...zN geneous expansion of f and any integer n ≥ 1, let ∞ X X

Rn f (z) =

c(r)z r , and Kn = I − Rn ,

k=n |r|=k

where If = f is the identity operator. Lemma 2. [8, Corollary 3.4] If 1 < q < ∞, then Rn converges to 0 pointwise in Hq as n → ∞. Moreover, sup{kRn k : n ≥ 1} < ∞. Lemma 3. [8, Lemma 3.5] Let 1 < q < ∞. For each f ∈ H q and n ≥ 1, |Rn f (z)| ≤ kf kH q

∞ X Γ(N + k) k |z| . k!Γ(N )

k=n

By a similar proof of [2, Lemma 3.16], we can obtain the following lemma. Lemma 4. Let 1 < p < ∞. If uCϕ is bounded from H q into Apα , then kuCϕ ke,H q →Apα ≤ lim inf kuCϕ Rn kH q →Apα . n→∞

3. Main Results Before proving our main results, we need to prepare the notation about the measure. Let 1 < q < ∞ and α > −1. For ϕ ∈ S(B) and u ∈ Apα , we define a finite positive Borel measure µϕ,u on B by Z µϕ,u (E) = |u|q dνα , ϕ−1 (E)

for all Borel sets E of B. By a change of variables formula from measure theory, we can verify Z Z gdµϕ,u = |u|q (g ◦ ϕ)dνα , B

B

for each nonnegative measurable function g on B. Theorem 1. Let 1 < q ≤ p < ∞, and α > −1. Suppose that uCϕ is bounded from H q into Apα . Then pN/q  Z 1 − |w|2 dνα (z) kuCϕ kpe,H q →Apα  lim sup |u(z)|p |1 − hϕ(z), wi|2 |w|→1− B  lim sup sup t→0+

ζ∈∂B

µϕ,u (S(ζ, t)) . tpN/q

(3.1)

Proof. The lower estimates. For each w ∈ B, define the function fw on B by  N/q 1 − |w|2 fw (z) = . (1 − hz, wi)2

(3.2)

By [11, Theorem 1.12] we obtain that kfw kqH q

Z =

sup 0 4−N t−N for all z ∈ S(ζ, t), the inequality (3.13) shows that Z

µϕ,u (S(ζ, t)) tpN/q


0 which depend only on p, q, α and N such that µj (j = 1, 2) also satisfies the Carleson measure condition µj (S(ζ, t)) ≤ C(Mj + ε)tpN/q

(3.16)

for all ζ ∈ ∂B and t > 0. By (3.14) and (3.15) we obtain that (3.16) is true for all 0 < t ≤ tj . Hence, we assume that t > tj . For a finite cover {Q(wk , tj /3)}, where wk ∈ Q(ζ, t) of the set Q(ζ, t) = {z ∈ ∂B : |1 − hz, ζi| ≤ t}, the covering property implies that there exists a disjoint subcollection Γ of {Q(wk , tj /3)} satisfying Q(ζ, t) ⊂ ∪Γ Q(wk , tj ).

(3.17)

Moreover, we obtain card (Γ) ≤ C(t/tj )N . By the notation (3.12), it follows that µj (S(ζ, t))

≤ µj (D(ζ, t)) ≤

X

µj (D(wk , tj ))

Γ

≤

X

pN/q

µj (S(wk , 2tj )) ≤ C(t/tj )N (Mj + ε)tj

Γ (p/q−1)N

= C(Mj + ε)tN tj

963

≤ C(Mj + ε)tpN/q ,

(3.18)

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Wang and Liang: Essential norm of weighted composition operator where the constant C depends only on p, q, α and the dimension N . Now, taking a function f ∈ H q with kf kH q ≤ 1, it follows that Z p kuCϕ Rn f kApα = |u(z)|p |Rn f (ϕ(z))|p dνα (z) ZB = |Rn f (z)|p dµϕ,u (z) B ! Z Z |Rn f (z)|p dµϕ,u (z)

+

= B\(1−tj )B

Z =

(1−tj )B

|Rn f (z)|p dµj (z) +

B

Z

|Rn f (z)|p dµϕ,u (z)

(3.19)

(1−tj )B

for integers n ≥ 1. Combining (3.16) with lemma 1 it follows that Z |Rn f (z)|p dµj (z) ≤ C(Mj + ε)kRn f kpH q ≤ C sup kRn kp (Mj + ε).

(3.20)

n≥1

B

On the other hand, by lemma 3, we have Z |Rn f (z)|p dµϕ,u (z) ≤ kf kpH q (1−tj )B

∞ X Γ(N + k) |1 − tj |k k!Γ(N )

!p kukpApα .

(3.21)

k=n

By the boundedness of uCϕ and takingPf (z) = 1 ∈ H q , it is evident that u ∈ Apα . Furthermore, the convergence of the series (Γ(N + k))/(k!Γ(N ))|1 − tj |k implies ∞ X Γ(N + k) |1 − tj |k → 0, n → ∞. k!Γ(N )

(3.22)

k=n

Thus combining (3.21) and (3.22) it is clear that Z |Rn f (z)|p dµϕ,u (z) → 0, n → ∞.

(3.23)

(1−tj )B

Combining (3.19), (3.20) and (3.23) with lemma 4, it follows that kuCϕ kpe,H q →Apα ≤ lim inf kuCϕ Rn kpH q →Apα ≤ C sup kRn kp (Mj + ε). n→∞

(3.24)

n≥1

Since supn≥1 kRn k < ∞ from lemma 2. Moreover, since ε > 0 is arbitrary, we conclude that  pN/q Z 1 − |w|2 p p kuCϕ ke,H q →Apα ≤ C lim sup |u(z)| dνα (z) (3.25) |1 − hϕ(z), wi|2 |w|→1− B and kuCϕ kpe,H q →Apα ≤ C lim sup sup

t→0+ ζ∈∂B

µϕ,u (S(ζ, t)) , tpN/q

completing the upper estimates. This finishes the proof.

(3.26) 

The following corollary is an immediate consequence of the above theorem. Corollary 1. Let 1 < q ≤ p < ∞, and α > −1. Suppose that uCϕ is bounded from H q into Apα . The following conditions are equivalent: (a) uCϕ : H q → Apα is compact; (b) u and ϕ satisfy pN/q  Z 1 − |w|2 p lim sup |u(z)| dνα (z) = 0; |1 − hϕ(z), wi|2 |w|→1− B

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Wang and Liang: Essential norm of weighted composition operator (c) u and ϕ satisfy lim sup sup t→0+

ζ∈∂B

µϕ,u (S(ζ, t)) = 0. tpN/q

References [1] B.D. MacCluer, Compact composition operators on Hp (BN ), Michigan Math. J. 32 (1985) 237-248. [2] C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995. [3] J.H. Shapiro, The essential norm of a composition operator, Ann. of Math. 125 (2) (1987) 375-404. [4] W. Rudin, Function Theory in the Unit Ball of C n , Crundlehren Math. Wiss. 241, Spring-Verlag, New-York Berlin 1980. [5] S. Ueki, Essential norms of weighted composition operators between weighted Bergman spaces of the ball, Acta Sci. Math. 74 (2008) 827-841. [6] J.H. Shapiro, Composition operators and classical function theory, Spriger-Verlag, 1993. [7] S. Stevi´ c, S. Ueki, Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball, Appl. Math. Comput. 215 (2010) 3526-3533. [8] S. Ueki, L. Luo, Compact weighted composition operators and multiplication operators between Hardy spaces, Abstr. Appl. Anal. 2008, Article ID 196498, 12 pages. [9] Z.H. Zhou, Y.X. Liang and H.G. Zeng, Essential norms of weighted composition operators from weighted Bergman space to mixed-norm space on the unit ball, Acta Math. Sin. (Engl. Ser.) 29 (3) (2013) 547-556. [10] K.H. Zhu, Operator Theory in Function Spaces, Marcel Dekker. Inc, New York, 1990. [11] K.H. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics 226, Springer, New York, 2005.

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ON THE q-ANALOGUE OF λ-DAEHEE POLYNOMIALS JIN-WOO PARK

Abstract. Recently, Daehee polynomials and λ-Daehee polynomials are introduced in [3, 4]. In this paper, we study the q-analogue of λ-Daehee polynomials of first kind or second kind. In addition, we investigate their properties arising from the p-adic q-integral equations.

1. Introduction Let d be fixed positive integer and 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 completions of algebraic closure of Qp . the p-adic norm is defined |p|p = p1 We set [ N ∗ Xd = lim Zdp Z, X = (a + dpZp ) , ← − N

0 0, for t ∈ Iq and |f (t, x(t)) − f (t, y(t))| ≤ φ(t)|x(t) − y(t)|,

for t ∈ Iq and for all x, y ∈ R;

(H2 ) there exists a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) and a function p ∈ C(Iq , R+ ) such that |g(t, x)| ≤ p(t)ψ(kxk) for each (t, x) ∈ Iq × R; (H3 ) There exists a constant r > 0 such that 2F0 kpkψ(r) 1+q r> , where 2kφk 1− kpkψ(r) 1+q

2kφk kpkψ(r) < 1, and F0 = sup |f (t, 0)|. 1+q t∈[0,1]

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B. AHMAD, S. K. NTOUYAS AND A. ALSAEDI

Then the boundary value problem (1) has at least one solution on Iq . Proof. Set X = C(Iq , R) and define a subset S of X defined by S = {x ∈ X : kxk ≤ r}, where r is satisfies the inequality (7). Clearly S is closed, convex and bounded subset of the Banach space X. By Lemma 2.2 the boundary value problem (1) is equivalent to the equation  Z t Z 1 (1 − qs)g(s, x(s))dq s , t ∈ Iq . (8) (t − qs)g(s, x(s))dq s − t x(t) = f (t, x(t)) 0

0

Define two operators A : X → X by Ax(t) = f (t, x(t)), t ∈ Iq ,

(9)

and B : S → X by Z Bx(t) =

t

Z

1

(t − qs)g(s, x(s))dq s − t 0

(1 − qs)g(s, x(s))dq s,

t ∈ Iq .

(10)

0

Then x = AxBx. We shall show that the operators A and B satisfy all the conditions of Lemma 3.1. For the sake of convenience, we break the proof into a sequence of steps. Step 1. We first show that A is a Lipschitz on X, i.e. (a) of Lemma 3.1 holds. Let x, y ∈ X. Then by (H1 ) we have |Ax(t) − Ay(t)| = |f (t, x(t)) − f (t, y(t))| ≤ φ(t)|x(t) − y(t)| ≤ kφkkx − yk, for all t ∈ Iq . Taking the supremum over t we obtain kAx − Ayk ≤ kφkkx − yk for all x, y ∈ X. So A is a Lipschitz on X with Lipschitz constant kφk. Step 2. Now we show that the multi-valued operator B is completely continuous on S, i.e. (b) of Lemma 3.1 holds. First we show that B is continuous on S. Let {xn } be a sequence in S coverging to a pont x ∈ S. Then by Lebesque dominated convergence theorem, Z t Z 1 lim Bxn (t) = lim (t − qs)g(s, xn (s))dq s − lim t (1 − qs)g(s, xn (s))dq s n→∞

n→∞

Z

n→∞

0

t

Z (t − qs) lim g(s, xn (s))dq s − t

=

n→∞

0

Z

Z (t − qs)g(s, x(s))dq s − t

=

(1 − qs) lim g(s, xn (s))dq s 0

t

0

0

1 n→∞

1

(1 − qs)g(s, x(s))dq s 0

= Bx(t), for all t ∈ Iq . This shows that B is continuous os S. It is enough to show that B(S) is a uniformly bounded and equicontinuous set in X. First we note that Z t Z 1 |Bx(t)| = (t − qs)g(s, x(s))dq s − t (1 − qs)g(s, x(s))dq s 0 0 Z t Z 1 (t − qs)|g(s, x(s))|dq s + (1 − qs)|g(s, x(s))|dq s ≤ 0

0

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Hybrid Boundary value problems of q-difference equations and inclusions Z ≤ 2

1

(1 − qs)p(s)ψ(kxk)dq s ≤ 2kpkψ(kxk) 0

1 , 1+q

for all t ∈ Iq . Taking supremum over t ∈ Iq , we get kBxk ≤

2 kpkψ(r), 1+q

for all x ∈ S. This shows that B is uniformly bounded on S. Next we show that B is an equicontinuous set in X. Let t1 , t2 ∈ Iq with t1 < t2 . Then we have Z t1 Z t2 (t1 − qs)g(s, x(s))dq s (t2 − qs)g(s, x(s))dq s − |Bx(t2 ) − Bx(t1 )| ≤ 0

0

Z

1

(1 − qs)g(s, x(s))dq s +(t2 − t1 ) 0 Z Z t1 |t2 − t1 |dq s + kpkψ(r) ≤ kpkψ(r)

t2

|t2 − qs|dq s

t1

0

Z

1

+kpkψ(r)(t2 − t1 )

(1 − qs)dq s. 0

Obviously the right hand side of the above inequality tends to zero independently of x ∈ S as t2 −t1 → 0. Therefore, it follows from the Arzel´ a-Ascoli theorem that B is a completely continuous operator on S. Step 3. Next we show that hypothesis (c) of Lemma 3.1 is satisfied. Let x ∈ X and y ∈ S be arbitrary such that x = AxBy. Then we have Z Z 1 t |x(t)| = |Ax(t)||By(t)| = |f (t, x(t))| (t − qs)g(s, y(s))dq s − t (1 − qs)g(s, y(s))dq s 0 0 Z Z t 1 ≤ [|f (t, x(t)) − f (t, 0)| + |f (t, 0)|] (t − qs)g(s, y(s))dq s − t (1 − qs)g(s, y(s))dq s 0 0 Z 1 ≤ [φ(t)x(t) + F0 ]2 (1 − qs)p(s)ψ(kyk)dq s 0

2 2 ≤ [kφkx(t) + F0 ] kpkψ(kyk) ≤ [kφkx(t) + F0 ] kpkψ(r). 1+q 1+q Taking supremum over t ∈ Iq and solving for kxk, we get 2F0 kpkψ(r) 1+q kxk ≤ ≤ r, 2kφk 1− kpkψ(r) 1+q that is, x ∈ S. Step 4. Now we show that M k < 1, i.e. (d) of Lemma 3.1 holds. This is obvious by (H3 ) since we have M = kB(S)k = sup{kBxk : x ∈ S} ≤ 2khkL1 and k = kφk. Thus all the conditions of Lemma 3.1 are satisfied and hence the operator equation x = AxBx has a solution in S. As a result, the boundary value problem (1) has a solution defined on Iq . This completes the proof.  Example 3.3 Consider the boundary value problem    1 x(t)  2  D1/2 = cos x(t), 0 < t < 1, sin x + 2 4   x(0) = x(1) = 0,

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Bashir Ahmad et al 984-993

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

B. AHMAD, S. K. NTOUYAS AND A. ALSAEDI 1 1 cos x. Then (H1 ) and (H2 ) hold with φ(t) = 1 and p(t) = , ψ(x) = 1 4 4 2kφk 1 respectively. Since kpkψ(r) = < 1, the boundary value problem (11) has a solution. 1+q 3 Let f (t, x) = sin x + 2, g(t, x) =

4

Existence result-the multi-valued case

Let us first outline the basic notions of multi-valued maps and fix our terminology [12, 20]. For a normed space (X, k · k), we define Pb (X) = {Y ∈ P(X) : Y is bounded} and Pcp,c (X) = {Y ∈ P(X) : Y is compact and convex}. A multi-valued map G : X → P(X) is convex (closed) valued if G(x) is convex (closed) for all x ∈ X. The map G is bounded on bounded sets if G(B) = ∪x∈B G(x) is bounded in X for all B ∈ Pb (X) (i.e. supx∈B {sup{|y| : y ∈ G(x)}} < ∞). G is called upper semi-continuous (u.s.c.) on X if for each x0 ∈ X, the set G(x0 ) is a nonempty closed subset of X, and for each open set N of X containing G(x0 ), there exists an open neighborhood N0 of x0 such that G(N0 ) ⊆ N. G is said to be completely continuous if G(B) is relatively compact for every B ∈ Pb (X). If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., xn → x∗ , yn → y∗ , yn ∈ G(xn ) imply y∗ ∈ G(x∗ ). R1 Let L1 (Iq ) denote the space of all functions f defined on Iq such that kxkL1q = 0 |x(t)|dq t < ∞. Definition 4.1 A multivalued map F : Iq × R → P(R) is said to be Carath´eodory (in the sense of q-calculus) if x 7−→ F (t, x) is upper semicontinuous on Iq . Further a Carath´eodory function F is called L1 −Carath´eodory if there exists ϕα ∈ L1 (Iq , R+ ) such that kF (t, x)k = sup{|v| : v ∈ F (t, x)} ≤ ϕα (t) for all kxk ≤ α on Iq for each α > 0. Definition 4.2 A function x ∈ C is a solution of problem (2) if there exists a function v ∈ L1 ([0, 1], R) such that it is continuous at t = 0 and v(t) ∈ F (t, x(t)) on Iq , and Z

t

Z (t − qs)v(s)dq s − t

x(t) = f (t, x(t)) 0

1

 (1 − qs)v(s)dq s ,

t ∈ Iq .

0

For each y ∈ C, define the set of selections of F by SF,y := {v ∈ C : v(t) ∈ F (t, y(t)) on Iq }. The following lemma will be used in the sequel. Lemma 4.3 ([21]) Let X be a Banach space. Let F : J × X → Pcp,c (X) be an L1 − Carath´eodory multivalued map and let Θ be a linear continuous mapping from L1 (J, X) to C(J, X). Then the operator Θ ◦ SF : C(J, X) → Pcp,c (C(J, X)), x 7→ (Θ ◦ SF )(x) = Θ(SF,x ) is a closed graph operator in C(J, X) × C(J, X). The following fixed point theorem due to Dhage [9] is fundamental in the proof of our main result in this section. Lemma 4.4 Let X be a Banach algebra and let A : X → X be a single valued and B : X → Pcp,c (X) be a multi-valued operator satisfying the conditions: (a) A is single-valued Lipschitz with a Lipschitz constant k; (b) B is compact and upper semi-continuous; (c) 2M k < 1, where M = kB(X)k. Then either (i) the operator inclusion x ∈ AxBx has a solution, or (ii) the set E = {u ∈ X|µu ∈ AuBu, µ > 1} is unbounded. Theorem 4.5 Assume that (H1 ) holds. In addition we suppose that

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Hybrid Boundary value problems of q-difference equations and inclusions (A1 ) F : [0, 1] × R → P(R) is L1 -Carath´eodory and has nonempty compact and convex values; (A2 ) 2kφkkϕα kL1q
0 such that kxk ≤ r, ∀x ∈ Q. Now for each h ∈ B1 x, there exists a v ∈ SF,x such that Z t Z 1 h(t) = (t − qs)v(s)dq s − t (1 − qs)v(s)dq s. 0

0

Then for each t ∈ [0, 1], Z t Z t Z 1 Z 1 |B1 x(t)| = (t − qs)v(s)dq s − t (1 − qs)v(s)dq s ≤ (t − qs)g(s)dq s + (1 − qs)g(s)dq s 0

0

0

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B. AHMAD, S. K. NTOUYAS AND A. ALSAEDI Z ≤ 2 0

1

(1 − qs)ϕα (s)dq s = 2kϕα kL1q .

This implies that khk ≤ 2kϕα kL1q and so B1 (X) is uniformly bounded. Next we show that B1 maps bounded sets into equicontinuous sets. Let Q be, as above, a bounded set and h ∈ B1 x for some x ∈ Q. Then there exists a v ∈ SF,x such that t

Z

1

Z

(1 − qs)v(s)dq s.

(t − qs)v(s)dq s − t

h(t) =

0

0

Then for any t1 , t2 ∈ [0, 1] with t1 ≤ t2 we have Z |h(t2 ) − h(t1 )| ≤

t2

Z (t2 − qs)v(s)dq s −

0

Z ≤

0

t1

Z

Z 1 (1 − qs)ϕα (s)dq s (t1 − qs)v(s)dq s + (t2 − t1 ) 0

t2

[(t2 − qs) − (t1 − qs)] ϕα (s)dq s + 0

t1

Z (t2 − qs)ϕα (s)dq s + (t2 − t1 )

1

(1 − qs)ϕα (s)dq s.

t1

0

Obviously the right hand side of the above inequality tends to zero independently of x ∈ Q as t2 − t1 → 0. Therefore it follows by the Arzel´a-Ascoli theorem that B1 : C([0, 1], R) → P(C([0, 1], R)) is completely continuous. In our next step, we show that B1 has a closed graph. Let xn → x∗ , hn ∈ B1 (xn ) and hn → h∗ . Then we need to show that h∗ ∈ B1 . Associated with hn ∈ B1 (xn ), there exists vn ∈ SF,xn such that for each t ∈ [0, 1], Z t Z 1 hn (t) = (t − qs)vn (s)dq s − t (1 − qs)vn (s)dq s. 0

0

Thus it suffices to show that there exists v∗ ∈ SF,x∗ such that for each t ∈ [0, 1], Z

t

Z

1

(t − qs)v∗ (s)dq s − t

h∗ (t) = 0

(1 − qs)v∗ (s)dq s. 0

Let us consider the linear operator Θ : L1 ([0, 1], R) → C([0, 1], R) given by Z f 7→ Θ(v)(t) =

t

Z (t − qs)v(s)dq s − t

0

1

(1 − qs)v(s)dq s. 0

Observe that

Z

Z 1

t

khn (t) − h∗ (t)k = (t − qs)(vn (s) − v∗ (s))dq s − t (1 − qs)(vn (s) − v∗ (s))dq s → 0, as n → ∞.

0

0 Thus, it follows by Lemma 4.3 that Θ ◦ SF is a closed graph operator. Further, we have hn (t) ∈ Θ(SF,xn ). Since xn → x∗ , therefore, we have Z

t

Z (t − qs)v∗ (s)dq s − t

h∗ (t) = 0

1

(1 − qs)v∗ (s)dq s. 0

for some v∗ ∈ SF,x∗ . As a result we have that the operator B1 is compact and upper semicontinuous operator on X. Step 3. Now we show that 2M k < 1, i.e. (c) of Lemma 4.4 holds. This is obvious by (H3 ) since we have M = kB(X)k = sup{|B1 x : x ∈ X} ≤ 2kϕα kL1q and k = kφk. Thus all the conditions of Lemma 4.4 are satisfied and a direct application of it yields that either the conclusion (i) or the conclusion (ii) holds. We show that the conclusion (ii) is not possible.

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Hybrid Boundary value problems of q-difference equations and inclusions

Let u ∈ E be arbitrary. Then we have for λ > 1, λu ∈ A1 u(t)B1 u(t). Then there exists v ∈ SF,x such that for any λ > 1, one has −1

u(t) = λ

t

Z

Z (t − qs)v(s)dq s − t

[f (t, u(t)]

!

1

(1 − qs)v(s)dq s ,

0

0

for all t ∈ [0, 1]. Then we have |u(t)|

t

Z

≤ λ−1 |f (t, u(t)|

Z (t − qs)|v(s)|dq s +

0

!

1

(1 − qs)|v(s)|dq s 0

Z

t

Z 0

0

Z ≤ [kφkkuk + F0 ]2 0

(1 − qs)|v(s)|dq s

(t − qs)|v(s)|dq s +

≤ [|f (t, u(t) − f (t, 0)| + |f (t, 0)|]

!

1

1

(1 − qs)ϕα (s)dq s ≤ [kφkkuk + F0 ]2kϕα kL1q ,

where we have used F0 = supt∈[0,1] |f (t, 0)|. In consequence, we have kuk ≤

2F0 kϕα kL1q 1 − 2kφkkϕα kL1q

.

Thus the condition (ii) of Thorem 4.4 does not hold. Therefore the operator equation A1 xB1 x and consequently problem (2) has a solution on [0, 1]. This completes the preginoof.  Example 4.6 Consider the problem  h i x(t)  D3/2 ∈ F (t, x(t)), 0 < t < 1, (t tan−1 x)/8 + 2  x(0) = x(1) = 0,

(14)

where F : [0, 1] × R → P(R) is a multivalued map given by  t → F (t, x) =

 |x|5 | sin x| 6 , + . 5(|x|5 + 4) 7(| sin x| + 1) 7

For f ∈ F we have  |f | ≤ max

|x|5 | sin x| 6 , + 5(|x|5 + 4) 7(| sin x| + 1) 7

 ≤ 1, x ∈ R.

Thus kF (t, x)k = sup{|y| : y ∈ F (t, x)} ≤ 1 = ϕα (t), x ∈ R. Further, kφk = 1/8, and 2kφkkϕα kL1q =

1 1 < . 4 2

Thus all the conditions of Theorem 4.5 are satisfied and consequently, the problem (14) has a solution on [0, 1]. Acknowledgment. This paper was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU.

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B. AHMAD, S. K. NTOUYAS AND A. ALSAEDI

References [1] B. Ahmad, Boundary value problems for nonlinear third-order q-difference equations, Electron. J. Diff. Equ. Vol. 2011 (2011), No. 94, pp. 1-7. [2] B. Ahmad and S.K. Ntouyas, Boundary value problems for q-difference inclusions, Abstr. Appl. Anal. Vol. 2011, Article ID 292860, 15 pages. [3] B. Ahmad, A. Alsaedi and S.K. Ntouyas, A study of second-order q-difference equations with boundary conditions, Adv. Difference Equ. 2012, 2012:35. [4] B. Ahmad and J.J. Nieto, Basic theory of nonlinear third-order q-difference equations and inclusions, Math. Model. Anal. 18 (2013), 122-135. [5] M. Ammi, E. El Kinani, D. Torres, Existence and uniqueness of solutions to functional integrodifferential fractional equations, Electron. J. Diff. Equ., Vol. 2012 (2012), No. 103, pp. 1-9. [6] M. H. Annaby and Z. S. Mansour, q-Taylor and interpolation series for Jackson q-difference operators, J. Math. Anal. Appl. 344 (2008), 472-483. [7] M.H. Annaby and Z.S. Mansour, q-Fractional Calculus and Equations, Lecture Notes in Mathematics 2056, Springer-Verlag, Berlin, 2012. [8] B. C. Dhage, On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett. 18 (2005), 273-280. [9] B. Dhage, Existence results for neutral functional differential inclusions in Banach algebras, Nonlinear Anal. 64 (2006), 1290-1306. [10] A. Dobrogowska, A. Odzijewicz, Second order q-difference equations solvable by factorization method, J. Comput. Appl. Math. 193 (2006), 319-346. [11] M. El Borai, M. Abbas, On some integro-differential equations of fractional orders involving Carath´eodory nonlinearities, Int. J. Mod. Math. 2 (2007), 41-52. [12] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992. [13] M. El-Shahed and H. A. Hassan, Positive solutions of q-difference equation, Proc. Amer. Math. Soc. 138 (2010), 1733-1738. [14] G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990. [15] F.H. Jackson, On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253-281. [16] F.H. Jackson, On q-definite integrals, Quart. J. 41 (1910), 193-203. [17] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002. [18] S. Sun, Y. Zhao, Z. Han, Y. Li, The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4961-4967. [19] Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62 (2011), 1312-1324 [20] Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, 1997. [21] A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786.

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q−Wallis inequality Mansour Mahmoud King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. [email protected]

. . Abstract In the paper, we present the following double inequality 1 1 p p < Wn,q < , Γq2 (1/2) [n + ν]q2 Γq2 (1/2) [n + µ]q2

1 1 0 , [2k]q [2k − 1]q

0 < q < 1; k = 2, 3, ... .

(10)

Now using the inequality (10), we obtain 1 [2]q [4]q [2n − 2]q ... [2]q [3]q [5]q [2n − 1]q µ ¶ [2]q [4]q ...[2n]q 1 > [2]q [2n]q [3]q [5]q ...[2n − 1]q 1 . > [2]q [2n]q Wn,q

Wn,q >

Then Wn,q > p

1 , [2]q [2n]q

n ∈ N ; 0 < q < 1.

(11)

When q → 1− , we get the following result [3]: Corollary 2.2. For n ∈ N , we have 1 1 √ < Wn < √ . 2 n 2n + 1 4

997

(12)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

The q−gamma function Γq (x) is defined by [7] Γq (z) =

(q; q)∞ (1 − q)1−z ; (q z ; q)∞

z 6= 0, −1, −2, ...,

(13)

where 0 < q < 1. Here we use the q−Pochhammer (q−shifted) symbol [14]: (a; q)0 = 1, Q j (a; q)k = k−1 k = 1, 2, ... . j=0 (1 − aq ); This function is a q−analogue of the ordinary gamma function since we have lim Γq (z) = Γ(z).

q→1

Also, it satisfies the functional equation Γq (z + 1) = [z]q Γq (z);

Γq (1) = 1,

(14)

which is a q−extension of the well-known functional equation Γ(z + 1) = zΓ(z); Γ(1) = 1. In the next reult, we will present a q−analogue of the relation (3). Lemma 2.3. For n ∈ N and 0 < q < 1, we have Wn,q =

Γq2 (n + 1/2) Γq2 (1/2)Γq2 (n + 1)

(15)

Proof. Using the relation (14), we have 1 − q n−1/2 1 − q n−3/2 1 − q 1/2 ... Γq (1/2) 1−q 1−q 1−q = ([1/2]q )n [2n − 1]q1/2 Γq (1/2).

Γq (n + 1/2) =

But [1/2]q = then [2n − 1]q !! = Also,

1 , [2]q1/2

([2]q )n Γq (n + 1/2) . Γq2 (1/2)

(16)

(1 − q n )(1 − q n−1 )...(1 − q) Γq (n + 1) = = [2n]q1/2 . ([1/2]q )n (1 − q 1/2 )n

Then [2n]q !! = ([2]q )n Γq2 (n + 1).

(17)

Using the relations (16) and (17), we obtain our result (15). Our next result will present a general q−analogue of the inequality (4). 5

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Theorem 1. For n ∈ N and 0 < q < 1, we have 1 1 p p < Wn,q < , Γq2 (1/2) [n + ν]q2 Γq2 (1/2) [n + µ]q2 Proof. Let Hn,q,h

1 1 0 0; s ∈ (0, 1); 0 < q < 1

(19)

s ln[1−(1−q)[Γq (s)]1/(s−1) ] and β(q, . Using with the best possible values α(q, s) = ln[(q −q)/(1−s)(1−q)] s) = ln q ln q 2 the inequality (19) at s = 1/2, x = n ∈ N and by replacing q by q , we get · ¸1/2 · ¸1/2 Γq2 (n + 1/2) 1 − q2 1 − q2 < , 0 < q < 1. (20) < 1 − q 2n+2β(q,1/2) Γq2 (n + 1) 1 − q 2n+2α(q,1/2) p Now multiply (20) by [n + h]q2 with h > 0 to obtain

·

1 − q 2n+2h 1 − q 2n+2β(q,1/2)

Then

¸1/2

· ¸1/2 Γq2 (n + 1/2) q 1 − q 2n+2h [n + h]q2 < , < Γq2 (n + 1) 1 − q 2n+2α(q,1/2)

Γq2 (n + 1/2) q [n + h]q2 = 1, lim n→∞ Γq 2 (n + 1)

h > 0; 0 < q < 1. (21)

h > 0; 0 < q < 1.

Hence lim Hn,q,h = 1,

h > 0; 0 < q < 1.

n→∞

Now let Mn,q,h = (1 − q 2n+2h )(1 − q 2n+2 )2 − (1 − q 2n+2h+2 )(1 − q 2n+1 )2 ¡ ¡ ¢ ¡ ¢¢ = (1 − q)q 2n q 2 − q 2n+1 − q 2n+2 + q 2h −1 − q + 2q 2n+2 . If we suppose that h = µ with

(22)

1 0 0, ν≥ . 2 Then

s Hn,q,ν 1 − q 2n+2 = Hn+1,q,ν 1 − q 2n+1

1 − q 2n+2ν >1 1 − q 2n+2ν+2

and hence the {Hn,q,ν }; ν ≥ 1/2; is decreasing sequence and its limit is 1 as n tends to ∞, hence Hn,q,ν > 1

ν ≥ 1/2.

Then we have Wn,q >

2.1

1 p , Γq2 (1/2) [n + ν]q2

ν ≥ 1/2; 0 < q < 1; n ∈ N.

Special cases

1− If we let µ1 = inequality (4). 2− Let µ2 =

1 4

1 4

and ν1 =

and ν2 =

1 2

and take the limit as q → 1− for the inequality (18), we get the

4 Γ22 (1/2)

− 1, then µ2 satisfies the condition of (18) and will prove that

q

ν2 also satisfies it. Using the inequality [2] Γq (x) ≤ 1 iff 1 ≤ x ≤ 2,

00 1+q

and we obtain the following result from the inequality (18): Lemma 2.4. For n ∈ N and 0 < q < 1, we have s· Γq2 (1/2)

1

n+

< Wn,q
0 due to the indefiniteness of Q. Let ξ = (ξ1 , . . . , ξn )T > 0 be such that diag(ξ) − Q is positive definite, where diag(ξ) denotes the diagonal matrix with diagonal elements ξ1 , . . . , ξn . For example, we can choose 2 1039

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ξj > λn , j = 1, . . . , n. Then f (x) has the following D.C. decomposition: xT Qx + cT x = xT diag(ξ)x − xT [diag(ξ) − Q]x + cT x,

(1)

where diag(ξ) − Q is positive definite. We have the Cholesky factorization for the positive definite matrix diag(ξ) − Q: diag(ξ) − Q = U T U,

(2)

where U is an upper triangular matrix. Substituting (2) into (1), we obtain xT Qx + cT x = −xT U T U x + xT diag(ξ)x + cT x

(3)

xT Qx + cT x = −y T y + xT diag(ξ)x + cT x,

(4)

Let y = U x, then

So problem (QIP ) is transformed equivalently into the following problem (Pe)

min

− y T y + xT diag(ξ)x + cT x

s.t. y = U x, Ax ≤ b, l ≤ x ≤ u,

e l≤y≤u e,

xj integer, j = 1, 2, · · · , n, where e l = (e l1 , . . . , e ln )T , u e = (e u1 , . . . , u en )T , e lj and u ej (j = 1, 2, · · · , n) are the optimal value of the following linear program, respectively: e lj := min Uj x

(5)

s.t. Ax ≤ b, l ≤ x ≤ u, and u ej := max Uj x

(6)

s.t. Ax ≤ b, l ≤ x ≤ u, where Uj is the jth row of matrix U , j = 1, . . . , n.

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3

Lagrangian Relaxation

Lagrangian dual method has been a powerful methodology in integer programming. It can often provide tight bounds by exploiting the separable structures of the problem. Problem (Pe) is a mixed integer quadratic programming problem. Let Lj (yj ) = αj yj +βj be the linear underestimation of the concave function −y 2 on [lej , uej ], (j = 1, 2, · · · , n) j

satisfying the following conditions: 2 Lj (lej ) = αj lej + βj = −lej , Lj (uej ) = αj uej + βj = −uej 2 , j = 1, 2, · · · , n.

So we obtain αj = −(lej + uej ), βj = lej uej , j = 1, 2, · · · , n. Hence Lj (yj ) = −(lej + uej )yj + lej uej , j = 1, 2, · · · , n.

(7)

Replacing −yj2 by Lj (yj ) in problem (Pe), we can get a convex integer relaxation of problem (Pe), i.e., (CIP1 )

min

n X

Lj (yj ) + xT diag(ξ)x + cT x

j=1

s.t. y = U x, Ax ≤ b, l ≤ x ≤ u,

e l≤y≤u e,

xj integer, j = 1, 2, · · · , n. For reducing computation, we eliminate y by y = U x in (CIP1 ) and obtain (CIP2 )

min ρ + (U T φ + c)T x + xT diag(ξ)x s.t. Ax ≤ b, l ≤ x ≤ u, , xj integer, j = 1, 2, · · · , n,

where ρ =

Pn

j=1 lj uej

e

and φ = (−le1 − u e1 , · · · , −len − u en )T . Obviously, problem (CIP2 ) is also

a convex integer relaxation of problem (QIP ). Problem (CIP2 ) is separable and we can consider the Lagrangian dual scheme. For ν ∈ Rm + , the dual function of (CIP2 ) is defined as follows: d(ν) =

 min ρ + (U T φ + c)T x + xT diag(ξ)x + ν T (Ax − b)

x∈XI

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= ρ − νT b +

n X

 min ξj x2j + τj (ν)xj | lj ≤ xj ≤ uj , xj integer ,

j=1

= ρ − νT b +

n X

ψj (ν),

j=1

where τ (ν) = (U T φ + c + AT ν)T and  2  tj < lj   ξj lj + τj (ν)lj , ψj (ν) = min{ξj btj c2 + τj (ν)btj c, ξj (btj c + 1)2 + τj (ν)(btj c + 1)}, lj ≤ tj ≤ uj    ξ u2 + τ (ν)u , tj > uj j j j j with tj = −

τj (ν) 2ξj

and btj c being the maximum integer less than or equal to tj .

From the weak duality, d(ν) is always a lower bound of (CIP2 ), further, (QIP ). The dual problem of (CIP2 ) is then to search for the maximum lower bound yielded by d(ν): (D)

max d(ν).

ν∈Rm +

From the above discussion, we obtain the following conclusion Theorem 1

v(D) ≤ v(CIP2 ) ≤ v(QIP ), where v(·) denotes the optimal value of problem

(·).

4

Branch-and-Bound Algorithm

In this section we will give a new branch-and-bound algorithm. The algorithm consists of two key procedures, i.e., partition of region and lower bounding. Lower bounding is discussed in the previous section, so in this section we first restate briefly the method of region partition given by Thoai in [20], and then describe the new algorithm in detail.

4.1

Integral hyper-rectangular bisection

Let P = ha, bi ⊂ Rn be a hyper-rectangle, which has integral bound vectors, that is P = {x ∈ Rn | a ≤ x ≤ b} = {x ∈ Rn | aj ≤ xj ≤ bj , j = 1, 2, · · · , n}, where aj and bj are integers satisfying aj ≤ bj (j = 1, 2, · · · , n). We assume that bi is the branch index at the current iteration. We define two subregions P 1 and P 2 of P as: P 1 = {x ∈ P | abi ≤ xbi ≤ η}, P 2 = {x ∈ P | η + 1 ≤ xbi ≤ bbi }, where η = b(abi + bbi )/2c. It is obviously P 1 ∪ P 2 ⊂ P, P 1 ∩ P 2 = ∅, 5 1042

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(P 1 ∩ Z n ) ∪ (P 2 ∩ Z n ) = P ∩ Z n . Remark 1 For convenience, we call a, b a lower corner vertex and an upper corner vertex of box ha, bi, respectively.

4.2

The Algorithm

We are now in a position to present a new branch-and-bound algorithm for solving problem (QIP ) based on Lagrangian dual bounds and integral hyper-rectangle bisection. The proposed method is a branch-and-bound method. Each subproblem (subbox) corresponds to a node in the search tree. The lower bound on each node is computed by using the Lagrangian dual search scheme. The new subproblems are branched by integral hyperrectangle bisection. The algorithm starts by computing a lower bound on the initial box hl, ui. The initial upper bound is set to be infinity if any initial feasible point is not found. At the k-th iteration, the algorithm maintains a list of subboxes. The subbox with the minimum lower bound is chosen from the list and partitioned into two new subboxes. For each new subbox, a lower bound is calculated by the Lagrangian dual search scheme. If the lower bound is greater than or equal to the upper bound of the optimal value of problem (QIP ), then the new subbox is fathomed. All the subboxes in the list with lower bound greater than or equal to the upper bound are fathomed. The process repeats until the list is empty.

Algorithm A Step 0. (Initialization). Compute the eigenvalues λ1 , . . . , λn of the matrix Q. Set l1 = l, u1 = u, P1 = hl1 , u1 i. Let S 0 be a set containing some feasible points of problem (QIP ). Let the initial lower bound ζ1 be the optimal value of problem (D) on P1 . If S 0 6= ∅, then set the initial upper bound γ1 = min{f (x) | x ∈ S 0 } and choose x1 ∈ S 0 such that f (x1 ) = γ1 ; otherwise, set γ1 = +∞. Set Ω = {P1 }, k := 1. Step 1. (Partition). Partition Pk into two subboxes Pk1 , Pk2 by integral hyper-rectangular bisection. Set i = 1, Ω := Ω\Pk . Step 2. (Bounding).

Solve dual problem (D) on Pki and obtain the optimal value dki .

Set ζ(Pki ) = dki . If ζ(Pki ) < γk , then Ω := Ω ∪ {Pki }, i := i + 1. If i = 2, goto Step 3, otherwise, repeat Step 2. Step 3. (Fathoming). Update S 0 by means of the feasible points yielded by the procedure. Set γk+1 = min {f (x) | x ∈ S 0 } and choose xk+1 ∈ S 0 such that γk+1 = f (xk+1 ). Set Ω := Ω\{P | ζ(P ) ≥ γk+1 }. If Ω = ∅, stop, xk+1 is the optimal solution of problem 6 1043

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(QIP ). Otherwise, set ζk+1 = min {ζ(P ) | P ∈ Ω}. Choose P (denote by Pk+1 ) from Ω such that ζ(Pk+1 ) = ζk+1 . Set k := k + 1, go back to Step 1. Remark 2

We choose the index bi of the edge with the maximum length to branch in

implementing integral hyper-rectangular bisection. The following conclusion holds true clearly due to the boundedness of the feasible region and partition method. Theorem 2

5

Algorithm A terminates after a finite number of iterations.

An Illustrative Example

In this section we demonstrate Algorithm A by a small example. Example 1 min f (x) = x22 − x1 x2 s.t. x1 + 2x2 ≤ 6, 0 ≤ x1 ≤ 5, 0 ≤ x2 ≤ 3, x = (x1 , x2 )T ∈ Z2 . The optimal solution of this example is x∗ = (4, 1)T with f (x∗ ) = −3. The feasible region of the example is shown in Figure 1 and the iterations of the algorithm are described as follows. Iteration 1 Step 0. The eigenvalue λ1 = −0.207, λ2 = 1.207. l = (0, 0)T , u = (5, 3)T . S 0 = {(3, 1)T }, P1 = hl, ui. ζ1 = −10.229, x1 = (3, 1)T , γ1 = f (x1 ) = −2, Ω = {P1 }. Step 1. The edge of the box P1 paralleling x1 -axis is the maximum length edge(its length is 5). By the integral midpoint bηc = b 0+5 2 c = 2, we partition P1 into two new subboxes: P11 = h(0, 0)T , (2, 3)T i, P12 = h(3, 0)T , (5, 3)T i, and remove P1 from Ω: Ω := Ω\{P1 } = ∅. Step 2. Compute the lower bound ζ(P11 ) = −2.307, ζ(P12 ) = −8.893. The new subboxes P11 , P12 are added into Ω due to ζ(P11 ) < γ1 , ζ(P12 ) < γ1 : Ω := Ω ∪ {P11 , P12 } = { P11 , P12 } = {h(0, 0)T , (2, 3)T i, h(3, 0)T , (5, 3)T i}. 7 1044

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3

2.5

x2

2

1.5

1

0.5

0

0

1

2

3

4

5

x

1

Figure 1: Feasible region of Example 1. Step 3. The upper bound γ1 is not updated in this process, i.e., γ2 = γ1 = −2. The new lower bound ζ2 = ζ(P12 ) = −8.893. Take P2 = P12 = h(3, 0)T , (5, 3)T i from Ω with the minimum lower bound. The process of Iteration 1 is illustrated in Figure 2. Iteration 2 Step 1. The edge of the box P2 paralleling x2 -axis is the maximum length edge(its length is 3). By the integral midpoint bηc = b 0+3 2 c = 1, we partition P2 into two new subboxes: P21 = h(3, 0)T , (5, 1)T i, P22 = h(3, 2)T , (5, 3)T i. The subbox P22 is deleted directly since it does not contain any feasible solutions of this example. And remove P2 from Ω: Ω := Ω\{P2 } = {P11 } = {h(0, 0)T , (2, 3)T i}. Step 2. Compute the lower bound ζ(P21 ) = −2.4. Add P21 into Ω due to ζ(P21 ) < γ2 : Ω := Ω ∪ {P21 } = {P11 , P21 } = {h(0, 0)T , (2, 3)T i, h(3, 0)T , (5, 1)T i}. Step 3. In this process we obtained a feasible point (4, 1)T , so we have S 0 = S 0 ∪ {(4, 1)T } = {(3, 1)T , (4, 1)T }. The new upper bound γ3 = min{f (x) | x ∈ S 0 } = −3. Choose x3 = (4, 1)T ∈ S 0 such that f (x3 ) = γ3 = −3. In view of ζ(P11 ) > γ3 , ζ(P21 ) > γ3 , we remove P11 , P21 from Ω. Hence,we obtain Ω = ∅, Therefore the algorithm stops and x3 = (4, 1)T is the optimal solution. The process of Iteration 2 is illustrated in Figure 3. 8 1045

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3

2.5

2

1.5

1

0.5

0

0

1

2

3

4

5

Figure 2: Illustration of Iteration 1 of Algorithm A for Example 1.

3

2.5

2

1.5

1

0.5

0

0

1

2

3

4

5

Figure 3: Illustration of Iteration 2 of Algorithm A for Example 1.

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6

Numerical results and Concluding Remarks

Algorithm A was coded by Fortran 90 and run on a Pentium IV PC with 2.0 GHz and 1.0G RAM. The test problems are generated randomly and the indefinite matrix is generated by the way in [22]. We take l = 1, u = 50. In order to test the efficiency of Algorithm A, we compare Algorithm A with the algorithm in [20] (denoted by Algorithm B). The numerical results are summarized in Table 1 , where the average CPU time and the average number of iterations are obtained by testing 10 problems. The ”fail” means that the corresponding algorithm doesn’t find the solution of the test problem in reasonable time. Table 1: Numerical results AverageCP U T ime (Seconds) Algorithm A Algorithm B

AverageN umberof Iterations Algorithm A Algorithm B

n

m

10

4

0.094

0.087

77

52

20

10

11.856

13.221

283

326

30

10

60.894

33.733

1879

413

40

10

61.991

fail

1515

fail

50

10

615.463

fail

3254

fail

60

9

44.731

fail

275

fail

70

10

80.331

fail

874

fail

80

9

137.225

fail

1135

fail

The numerical results show that Algorithm A is capable of finding an optimal solution to medium-scale indefinite integer quadratic programming in reasonable time. Algorithm A is more efficient than Algorithm B: the scale of problem (QIP ) solved by Algorithm A is larger than Algorithm B. Algorithm A can solve test problems with n up to 80, while the largest size of test problems solved by Algorithm B is 30. The outstanding performance of Algorithm A is largely due to the tightness and efficiency of the Lagrangian bounds.

References [1] Kalantari, B., Bagchi, A.: An algorithm for quadratic zero-one programs, Naval Research Logistics, 37, 527–538 (1990) [2] Chardaire, P., Sutter,A.: A decompo sition method for quadratic zero-one programming, Management Science 41, 704–712 (1995) [3] Helmberg, C., Rendl, F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting plane, Mathematical Programming 82, 291–315 (1998) [4] Billionnet, A., Elloumi, S.: Using a mixed integer quadratic programming solve for the unconstrained quadratic 0-1 problem, Mathematical Programming 109, 55–67 (2007) [5] McCormik, E.J.: Human Factors Engineering, McGraw-Hill: New York (1970) [6] Gallo, G., Hammer, P.L., Simone, B.: Quadratic kanpsack problems, Mathematical Programming 12, 132–149 (1980)

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[7] Billionnet, A., Faye, A., Soutif, E.: A new upper bound for the 0-1 quadratic knapsack problem, European Journal of Operational Reaearch 112, 664–672 (1999) [8] Billionnet, A., Soutif, E.: An exact method based on Lagrangian decomposition for the 0-1 quadratic knapsack problem, European Journal of Operational Reaearch 157, 565–575 (2004) [9] Pisinger, D.: The quadratic knapsack problem–A survey, journal=Discrete Applied Mathematics 155, 623–648 (2007) [10] Adams, W.P., Sheriali, H.D.: Mixed-integer bilinear programming problems , Working paper,Department of Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University (1983) [11] Li, D., Sun, X.L., Wang, F.L.: Convergent Lagrangian and contour-cut method for nonlinear integer programming with a quadratic objective function, SIAM Journal on Optimization 17, 372–400 (2006) [12] Syam, S.S.: A dual ascent method for the portfolio selection problem with multiple constraints and linked proposals, European Journal of Operational Research 108, 196–207 (1998) [13] Krarup, J., Pruzan, P.M.: Cmputer-aided layout design, Mathematical Peogramming Study 9, 75–94 (1978) [14] Witzgall, C.: Mathematiocal methods of site selection for Electric System(EMS), NBS Internal Report (1975) [15] Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut apptoach to global optimization, Mathematical Programming, Ser.B 103, 225–249 (2005) [16] Saxena, A., Bonami, P., Lee, J.: Disjuntive cuts for nonconvex mixed integer quadratically constrained programs, Lecture Notes in Computer Science, 5035, 17–33 (2008) [17] Hansen, P., Jaumard, B., Mathon, V.: Constrained nonlinear 0-1 programming, ORSA Journal on Computing 5, 97–119 (1993) [18] Li, D., Sun, X.L.: Nonlinear Integer Programming, Springer, New York (2006) [19] Erenguc, S.S., Benson, H.P.: An algorithm for indefinite integer quadratic programming, Computers and Mathematics with Applications 21, 99–106 (1991) [20] Thoai, N.V.: Global optimization technique for solving the general quadratic integer programming problem, Computational Optimization and Applications 10, 149–163 (1998) [21] Sun, X.L., Li, J.L., Luo, H.Z.: Convex Relaxation and Lagrangian Decomposition for Indefinite Quadratic Integer Programming, Optimization 59, 627–641 (2010) [22] Le Thi Hoai An, Pham Dinh Tao: Solving a Class of Linearly Constrained Indefinite Quadratic Problems by D.C. Algorithms, Journal of Global Optimization 11, 253–285 (1997)

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A NOTE ON AN INTERVAL-VALUED PSEUDO-CONVOLUTION OF INTERVAL-VALUED FUNCTIONS AND ITS APPLICATIONS LEE-CHAE JANG

General Education Institute, Konkuk University, Chungju 138-701, Korea E-mail : [email protected], Phone:082-43-840-3591

Abstract. Markova-Stupnanova (1999) investigated the idempotent with respect to a pseudo-convolution. Jang (2013) defined the interval-valued generalized fuzzy integral and investigate their properties. In this paper, we define the interval-valued pseudo-integral and an interval-valued pseudo-convolution of interval-valued functions. We also investigate an interval-valued idempotent function with respect to an interval-valued pseudo-convolution. Furthermore, we define the interval-valued pseudo-Laplace transform and discuss its characterizations.

1. Introduction Many researchers [1,2,4,9-15,20,22,23] have studied various integrals of measurable multivalued functions which are used for representing uncertain functions, for examples, the Aumann integral, the fuzzy integral, and the Choquet integral of measurable interval-valued functions in many different mathematical theories and their applications. Benvenuti-Mesiar [3], Deschrijver [4], Fang [6,7], Grbic-Stajner and Papuga-Strboja [8], Mesiar et al. [18], and Wu et al. [24, 25] have studied pseudo-multiplications and the pseudo-integrals. Markova-Stupnanova [16] introduced the pseudo-convolution of functions based on pseudomathematical operations and investigated the idempotent with respect to a pseudo-convolution. Recently, Jang [14] defined the interval-valued generalized fuzzy integral and discuss some properties of them. The purpose of this study is to define the interval-valued pseudointegral and to investigate interval-valued pseudo-convolution of the interval-valued functions and the interval-valued idempotent functions with respect to an the interval-valued pseudoconvolution. The paper is organized in six sections. In section 2, we list definitions and some properties of the pseudo-integral, and a pseudo-convolution of functions. In section 3, we define an intervalvalued pseudo-addition, an interval-valued pseudo-multiplication, the interval-valued pseudointegral, and an interval-valued pseudo-convolution of interval-valued functions. In section 4, we define an interval-valued idempotent function with respect to interval-valued pseudointegrals and investigate some properties of them. In section 5, we define the interval-valued pseudo-Laplace transform and investigate the interval-valued pseudo-exchange formula. In section 6, we give a brief summary results and some conclusions. 1991 Mathematics Subject Classification. 28E10, 28E20, 03E72, 26E50 11B68. Key words and phrases. pseudo-integral, pseudo-addition, pseudo-multiplication, interval-valued function, pseudo-convolution, pseudo-Laplace transform. 1

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2. Definitions and Preliminaries In this section, we introduce a pseudo-addition and a pseudo-multiplication as follows. Definition 2.1. ([4,6,16-19]) (1) A binary operation ⊕ : [0, ∞]2 −→ [0, ∞] is called a pseudoaddition if it satisfies the following axioms: (i) x ⊕ y = y ⊕ x for all x, y ∈ [0, ∞], (ii) x ≤ y =⇒ x ⊕ z ≤ y ⊕ z for all x, y, z ∈ [0, ∞], (iii) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) for all x, y, z ∈ [0, ∞], (iv) ∃0 ∈ [0, ∞] such that x ⊕ 0 = x for all x ∈ [0, ∞], (v) xn −→ x, yn −→ y =⇒ xn ⊕ yn −→ x ⊕ y. (2) A binary operation ¯ : [0, ∞]2 −→ [0, ∞] is called a pseudo-multiplication with respect to ⊕ if it satisfies the following axioms: (i) x ¯ y = y ¯ x for all x, y ∈ [0, ∞], (ii) (x ¯ y) ¯ z = x ¯ (y ¯ z) for all x, y, z ∈ [0, ∞], (iii) ∃1 ∈ [0, ∞] such that x ¯ 1 = x for all x ∈ [0, ∞], (iv) x ¯ (y ⊕ z) = (x ¯ y) ⊕ (x ¯ z) for all x, y, z ∈ [0, ∞], (v) x ¯ 0 = 0 for all x ∈ [0, ∞], (vi) x ≤ y =⇒ x ¯ z ≤ y ¯ z if z ≥ 0 and x ¯ z ≥ y ¯ z if z ≤ 0 for all x, y, z ∈ [0, ∞]. Let F([0, ∞)) be the set of all functions f : [0, ∞) −→ [0, ∞) and ([0, ∞], ⊗) be a continuous semiring, and A be a σ-algebra of [0, ∞). Definition 2.2. ([4, 17, 19]) (1) Let f ∈ F([0, ∞)). The pseudo-integral of f with respect to a sup-measure mψ is defined by Z ⊕ f ¯ dmψ = sup {f (x) ¯ ψ(x)}, (1) x∈A

A

where A ∈ A and a function ψ defines sup-measure mψ , that is, mψ (A) = supu∈A ψ(u). R⊕ (2) f is said to be integrable if A f ¯ dmψ is finite. Let F∗ ([0, ∞)) be the set of all integrable functions. We also consider the pseudo-convolution of functions in F([0, ∞))(see[4,17,19]). Definition 2.3. Let f, h ∈ F([0, ∞)). The pseudo-convolution of f and h is defined by Z ⊕ (f ∗ h)(t) = f (t − u) ¯ dmh (u), ∀ t ∈ [0, ∞)

(2)

[0,t]

where mh (A) = supx∈A h(x) for all A ∈ A. By Definition 2.3, we directly obtain the following basic characterizations. Theorem 2.1. (1) If x ⊕ y = max{x, y} and x ¯ y = x ∧ y for all x, y ∈ [0, ∞), and f, h ∈ F([0, ∞)), then for all t ∈ [0, ∞), f ∗ h = sup f (t − u) ∧ h(u).

(3)

u∈[0,t]

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(2) If x ⊕ y = max{x, y} and x ¯ y = x + y for all x, y ∈ [0, ∞), and f, h ∈ F([0, ∞)), then for all t ∈ [0, ∞), f ∗ h = sup (f (t − u) + h(u)).

(4)

u∈[0,t]

Proof. (1) If x⊕y = max{x, y} and x¯y = x∧y for all x, y ∈ [0, ∞), and f, h ∈ F([0, ∞)), and t ∈ [0, ∞), by Definition 2.2 and Definition 2.3, then we have Z max f ∗h = f (t − u) ∧ dmh [0,t]

=

sup {f (t − u) ∧ h(u)}.

(5)

u∈[0,t]

(2) The proof is similar to the proof of (1). Now, we easily obtain commutativity and associativity of a pseudo-convolution. Theorem 2.2. (1) If x ⊕ y = max{x, y} and x ¯ y = x ∧ y for all x, y ∈ [0, ∞), then we have f ∗h=h∗f

(6)

(f ∗ h) ∗ k = f ∗ (h ∗ k).

(7)

and (2) If x ⊕ y = max{x, y} and x ¯ y = x + y for all x, y ∈ [0, ∞), then we have f ∗h=h∗f

(8)

(f ∗ h) ∗ k = f ∗ (h ∗ k).

(9)

and Proof. (1) By Theorem 2.1 (1) and changing w = t − u, (f ∗ h)(t)

=

sup f (t − u) ∧ h(u) u∈[0,t]

=

sup f (w) ∧ h(t − w) w∈[0,t]

= (h ∗ f )(t)

(10)

By Theorem 2.1 (1) and changing w = u − z, ((f ∗ h) ∗ k)(t) =

sup [(f ∗ h)(t − u) ∧ k(u)] "

u∈[0,t]

=

sup u∈[0,t]

=

sup

#

sup (f (t − u − z) ∧ h(z)) ∧ k(u) z∈[0,t−u]

sup [(f (t − w) ∧ h(w − u)) ∧ k(u)]

u∈[0,t] w∈[u,t]

=

sup

sup [f (t − w) ∧ (h(w − u) ∧ k(u))] " #

u∈[0,t] w∈[u,t]

=

sup f (t − w) ∧ w∈[0,t]

=

sup (h(w − u) ∧ k(u)) u∈[0,w]

sup f (t − w) ∧ (h ∗ k)(w) w∈[0,t]

=

(f ∗ (h ∗ k))(u).

(11)

(2) By Theorem 2.1 (2) and changing w = t − u, (f ∗ h)(t)

=

sup f (t − u) + h(u) u∈[0,t]

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=

sup f (w) + h(t − w) w∈[0,t]

=

(h ∗ f )(t)

(12)

By Theorem 2.1 (2) and changing w = u − z, ((f ∗ h) ∗ k)(t)

=

sup [(f ∗ h)(t − u) + k(u)] "

u∈[0,t]

=

sup u∈[0,t]

=

sup

#

sup (f (t − u − z) + h(z)) + k(u) z∈[0,t−u]

sup [(f (t − w) + h(w − u)) + k(u)]

u∈[0,t] w∈[u,t]

=

sup

sup [f (t − w) + (h(w − u) + k(u))] " #

u∈[0,t] w∈[u,t]

=

sup f (t − w) + w∈[0,t]

=

sup (h(w − u) + k(u)) u∈[0,w]

sup f (t − w) + (h ∗ k)(w) w∈[0,t]

= (f ∗ (h ∗ k))(u).

(13)

3. The interval-representable pseudo-convolutions Let I(Y ) be the set of all closed intervals(for short, intervals) in Y as follows: I(Y ) = {a = [al , ar ]|al , ar ∈ Y and al ≤ ar },

(14)

where Y is [0, ∞) or [0, ∞]. For any a ∈ Y , we define a = [a, a]. Obviously, a ∈ I(Y ) (see [1,2,9-15, 22, 23]). Definition 3.1. ([13-15]) If a = [al , ar ], b = [bl , br ], an = [anl , anr ], aα = [aαl , aαr ] ∈ I(Y ) for all n ∈ N and α ∈ [0, ∞), and k ∈ [0, ∞), then we define arithmetic, maximum, minimum, order, inclusion, superior, inferior 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 6= b, (8) a ⊂ b if and only if bl ≤ al and ar ≤ br , (9) supn an = [supn anl , supn anr ], (10) inf n an = [inf n anl , inf n anr ], (11) supα aα = [supα aαl , supα aαr ], and (12) inf α aα = [inf α aαl , inf α aαr ]. We also consider an interval-valued pseudo-addition and an interval-valued pseudo-multiplication as follows.

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L : I([0, ∞])2 −→ I([0, ∞]) is called an Definition 3.2. ([15])(1) A binary operation interval-valued pseudo-addition if there exist pseudo-additions ⊕l and ⊕r such that x ⊕l y ≤ x ⊕r y for all x, y ∈ [0, ∞], and such that for all a = [al , ar ], b = [bl , br ] ∈ I([0, ∞]), M a b = [al ⊕l bl , ar ⊕r br ] . (15) L Then ⊕l and ⊕r are called J the represents of I . (2) A binary operation : I([0, ∞])2 −→ I([0, ∞]) is called an interval-valued pseudomultiplication if there exist pseudo-multiplications ¯l and ¯r such that x ¯l y ≤ x ¯r y for all x, y ∈ [0, ∞], and such that for all a = [al , ar ], b = [bl , br ] ∈ I([0, ∞]), K a b = [al ¯l bl , ar ¯r br ] . (16) J Then ¯l and ¯r are called the representants of . Let IF([0, ∞)) be the set of all interval-valued functions. By using an interval-valued addition and an interval-valued multiplication, we can define the interval-valued pseudo-integral. Definition 3.3. (1) An interval-valued function f : [0, ∞) → I([0, ∞) \ {∅} is said to be measurable if for any open set O ⊂ [0, ∞), f

−1

(O) = {x ∈ [0, ∞) | f (x) ∩ O 6= ∅} ∈ A. (17) J (2)Let x ⊕ y = max{x, y} for all x, y ∈ [0, ∞) and be an interval-representable pseudomultiplication and f ∈ IF(X). The interval-valued pseudo-integral with respect to an intervalvalued sup-measure mψ is defined by Z ⊕ K K f dmψ = sup {f (x) ψ(x)} (18) x∈A

A

where A ∈ A, ψ = [ψl , ψr ], and ψs defines a sup-measure mψs for s = l, r. (3) f is said to be integrable on A ∈ A if Z ⊕ K f dmψ ∈ A \ {∅}.

(19)

A

Let IF∗ ([0, ∞)) be the set of all integrable interval-valued functions. By Definition 3.3, we directly obtain the following basic characterizations. Theorem 3.1. Let ¯l and J ¯r be pseudo-multiplications on [0, ∞] corresponding to a pseudoaddition ⊕ = max. If is a standard interval-valued pseudo-multiplication, A ∈ A, and f ∈ IF∗ ([0, ∞)), then we have ·Z ⊕ ¸ Z ⊕ Z ⊕ K fl ¯l dmψ1 , fr ¯r dmψr , (20) f dmψ = A

A

A

where ψ = [ψl , ψr ] and ψs defines a sup-measure mψs for s = l, r. Proof. By Definition 3.2 (2) , Theorem 2.1 (1), and Definition 3.1 (11), Z ⊕ K K f (x) dmψ = sup {f (x) ψ(x)} x∈A A K = sup [fl (x), fr (x)] [ψl (x), ψr (x)] x∈A

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=

sup [fl (x) ¯l ψl (x), fr (x) ¯r ψr (x)]

x∈A

=

[sup fl (x) ¯l ψl (x), sup fr (x) ¯r ψr (x)] x∈A ·Z ¸ Z x∈A ⊕ ⊕ = fl ¯l dmψ1 , fr ¯r dmψr .

(21)

A

A

By using the interval-valued pseudo-integral with respect to an interval-valued sup-measure, we define the interval-valued pseudo-convolutions of interval-valued functions as follows. J Definition 3.4. Let ⊕ be a pseudo-addition, be an interval-valued pseudo-multiplication, and f = [fl , fr ], h = [hl , hr ] ∈ IF∗ ([0, ∞)). The interval-valued pseudo-convolution of f and h is defined by Z ⊕ K f (t − u) dmh (u), ∀t ∈ [0, ∞) (f ? h((t) = (22) [0,t]

where hs defines a sup-measure mhs for s = l, r. From Definition 3.4, we can obtain some characterizations of interval-valued pseudo-convolutions of interval-valued functions. Theorem 3.2. If x ⊕ y = max{x, y} for all x, y ∈ [0, ∞), f = [fl , fr ], h = [hl , hr ] ∈ IF∗ ([0, ∞)), and if ¯l and ¯r are the represents of an interval-valued pseudo-multiplication J , then we have

where (fs ∗s hs )(t) =

R⊕ [0,t]

f ? h = [fl ∗l hl , fr ∗r hr ],

(23)

fs (t − u) ¯s dms for s = l, r.

Proof. By Theorem 3.2(1) and Definition 3.1 (11), Z ⊕ K f ?h = f (t − u) dmh (u) [0,t] K h(u) = sup f (t − u) u∈[0,t] K = sup [fl (t − u), fr (t − u)] [hl (u), hr (u)] u∈[0,t]

=

sup [fl (t − u) ¯l hl (u), fr (t − u) ¯r hr (u)] u∈[0,t]

= [ sup fl (t − u) ¯l hl (u), sup fr (t − u) ¯r hr (u)] u∈[0,t]

u∈[0,t]

= [fl ∗l hl , fr ∗r hr ].

(24)

Now, we investigate commutativity and associativity of interval-valued pseudo-convolution. Theorem 3.3. (1) J If x ⊕ y = max{x, y} and x ¯l y = x ¯r y = x ∧ y for all x, t ∈ [0, ∞) are the represents of , then we have f ?h=h?f

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and (f ? h) ? k = f ? (h) ? k). of

(26)

(2) If x ⊕ y = max{x, y} and x ¯l y = x ¯r y = x + y for all x, t ∈ [0, ∞) are the represents J , then we have f ?h=h?f

(27)

and (f ? h) ? k = f ? (h) ? k).

(28) ∗

Proof. (1) Let f = [fl , fr ], h = [hl , hr ], k = [kl , kr ] ∈ IF ([0, ∞)). By Theorem 2.2(1) with ∗l = ∗r = ∗, fl ∗ hl = hl ∗ fl and fr ∗ hr = hr ∗ fr

(29)

(fl ∗ hl ) ∗ kl = fl ∗ (hl ∗ kl ) and (fr ∗ hr ) ∗ kr = fr ∗ (hr ∗ kr ).

(30)

and By Theorem 3.3(1) and (29), f ?h

= [fl ∗ hl , fr ∗ hr ] = [hl ∗ fl , hr ∗ fr ] = h ? f.

(31)

By Theorem 3.3(1) and (30), (f ? h) ? k

= [(fl ∗ hl ) ∗ kl , (fr ∗ hr ) ∗ kr ] = [fl ∗ (hl ∗ kl ), fr ∗ (hr ∗ kr )] = f ? (h ? k).

(32)

(2) By Theorem 2.2(2) and the same method of the proof of (1), we can obtain the result.

4. An interval-valued idempotent function with respect to an interval-valued pseudo-convolution In this section, we consider an idempotent function with respect to a pseudo-convolution ∗ and define an interval-valued idempotent function with respect to an interval-representable pseudo-convolution ? and investigate some properties of an interval-valued idempotent function. Definition 4.1. (1) A function f : [0, ∞) −→ [0, ∞) is called an idempotent function with respect to a pseudo-convolution ∗ if f ∗ f = f . (2) An interval-valued function f : [0, ∞) −→ I([0, ∞)) is called an interval-valued idempotent function with respect to an interval-valued pseudo-convolution ? if f ? f = f .

Theorem 4.1. ([16]) If x ⊕ y = max{x, y} for all x, y ∈ [0, ∞), and ¯ is a pseudomultiplication, then there exists a neutral element e of the pseudo-convolution ∗ based on ([0, ∞], ∨, ¯) and e(0) = 1, e(x) = 0 for all x > 0, where x ¯ 1 = x and x ¯ 0 = 0 for all x ∈ [0, ∞).

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J Theorem 4.2. If x ⊕ y = max{x, y} for all x, y ∈ [0, ∞), and ¯l , ¯r are the represents of then there exists an interval-valued neutral element e of the a extended interval-valued pseudoJ convolution ? based on (I([0, ∞]), ∨, ) and e(0) = [1l , 1r ], e(x) = [0l , 0r ] for all x > 0, where x ¯s 1s = x and x ¯s 0s = 0s for all x ∈ [0, ∞) and for s = l, r. Proof. By Theorem 4.1, there exists a neutral element es of the pseudo-convolution ∗s based on ([0, ∞], ∨, ¯s ) and e(0) = 1s , e(x) = 0s for all x > 0, where x ¯s 1s = x and x ¯s 0s = 0s for all x ∈ [0, ∞) and for s = l, r. Thus, by Theorem 3.2, for any t ∈ [0, ∞), K e(u)] (f ? e)(t) = sup [f (t − u) u∈[0,t]

=

sup [fl (t − u) ¯l el (u), fr (t − u) ¯r er (u)] u∈[0,t]

=

[ sup fl (t − u) ¯l el (u), sup fr (t − u) ¯r er (u)] u∈[0,t]

u∈[0,t]

= [(fl ∗ el )(t), (fr ∗ er )(t)] = [fl (t), fr (t)] = f (t).

(33)

Theorem 4.3. ([21]) Let x ⊕ y = max{x, y} and x ¯ y = min{x, y} for all x, y ∈ [0, ∞]. If f is an idempotent function with respect to a pseudo-convolution ∗, then for each c ∈ [0, ∞], f (c) + f (c) ⊂ f (c) , where f (c) = {x ∈ [0, ∞)|f (x) ≥ c} is so called c-cut of f . Conversely, if for each c ∈ [0, ∞], f (c) + f (c) = f (c) , then f is an idempotent function with respect to a pseudo-convolution ∗.

Theorem 4.4. Let x ⊕ Jy = max{x, y} and x ¯l y = x ¯r y = min{x, y} for all x, y ∈ [0, ∞] are the represents of . If f = [fl , fr ] is an interval-valued idempotent with respect to (c) (c) (c) an interval-valued pseudo-convolution ?, then for each c ∈ [0, ∞], fs + fs ⊂ fs , where (c) fs = {x ∈ [0, ∞)|fs (x) ≥ c} is so called c-cut of fs for s = l, r. (c) (c) (c) Conversely, if for each c ∈ [0, ∞], fs +fs = fs for s = l, r, then f is an interval-valued idempotent function with respect to an interval-valued pseudo-convolution ?. Proof. (=⇒) Since (f ? e)(t) = f (t) for all t ∈ [0, ∞), by Theorem 3.3(1), (fl ∗l el )(t) = fl (t) and (fr ∗r er )(t) = fr (t). (c)

(c)

(34)

(c)

By Theorem 4.3, for each c ∈ [0, ∞], we obtain fs + fs ⊂ fs for s = l, r. (c) (c) (c) (⇐=) Suppose that for each c ∈ [0, ∞], fs + fs = fs for s = l, r. By Theorem 4.3, fs is an idempotent function with respect to a pseudo-convolution ∗s induced by ¯s for s = l, r. Therefore, by Theorem 3.3(1), (f ?1 f )(t)

= [(fl ∗l fl )(t), (fr ∗r fr )(t)] = [fl (t), fr (t)] = f (t).

(35)

That is, f is an interval-valued J idempotent function with respect to an interval-valued pseudoconvolution ? induced by .

5. The interval-valued pseudo-Laplace transform In this section, we introduce the pseudo-Laplace transform and the pseudo-exchange formula as follows.

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Definition 5.1. ([19]) Let ¯ be a pseudo-multiplication on [0, ∞] corresponding to a pseudoaddition ⊕ = max. (1) The pseudo-character of the group ([0, ∞), +) is a continuous map ξ : [0, ∞) −→ [0, ∞) of the group [0, ∞), +) into the semigroup ([0, ∞), ⊕, ¯) with property ξ(x + y) = ξ(x) ¯ ξ(y), for x, y ∈ [0, ∞). ⊕

(36)

∗

(2) The pseudo-Laplace transform L (f ) of a function f ∈ F ([0, ∞) is defined by Z ⊕ (L⊕ f )(ξ)(z) = ξ(x, z) ¯ dmf (x),

(37)

[0,∞)

where ξ is the pseudo-character.

Theorem 5.1. ([19]) Let ¯ = + be a pseudo-multiplication on [0, ∞] corresponding to a pseudo-addition ⊕ = max. If functions f1 , f2 ∈ F∗ ([0, ∞)), then we have L⊕ (f1 ∗ f2 ) = L⊕ (f1 ) ¯ L⊕ (f2 ).

(38)

Now, we define an interval-valued pseudo-character and the interval-valued pseudo-Laplace transform of interval-valued functions in IF∗ ([0, ∞)) as follows. J Definition 5.2. (1) Let ¯l and ¯r be the represents of . The interval-valued pseudoξ : character ξ of the group ([0, ∞), +) is L a continuous map [0, ∞) −→ [0, ∞) of the group J ([0, ∞), +) into the semigroup ([0, ∞), , ) with property K ξ(x + y) = ξ(x) ξ(y), for x, y ∈ [0, ∞). (39) (2) Let ¯lJ and ¯r be pseudo-multiplications on [0, ∞] corresponding to a pseudo-addition ⊕ = max. If is a standard interval-valued pseudo-multiplication and f ∈ IF∗ ([0, ∞)), then the interval-valued pseudo-Laplace transform L⊕ (f ) of a function f ∈ IF∗ ([0, ∞)) is defined by Z ⊕ K (40) ξ(x, z) (L⊕ f )(ξ)(z) = dmf (x) , [0,∞)

where ξ = [ψl , ψr ] is the interval-valued pseudo-character. We discuss the pseudo-exchange formula for the interval-valued pseudo-Laplace transform. as follows: Theorem 5.2. Let ¯l and J ¯r be pseudo-multiplications on [0, ∞] corresponding to ∗a pseudoaddition ⊕ = max. If is an interval-valued pseudo-multiplication and f ∈ IF ([0, ∞)), then we have £ ¤ ⊕ (L⊕ f )(ξ)(z) = L⊕ (41) l (fl ), Lr (fr ) , R ⊕ ξs (x, z) ¯s dmfs (x) for s = l, r. where L⊕ s (fs ) = Proof. By Theorem 3.2(1), we have Z ⊕ K ⊕ ξ(x, z) dmf (x) (L f )(ξ)(z) =

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LEE-CHAE JANG

·Z

Z

⊕

⊕

=

ξl (x, z) ¯l dmfl (x), £ ⊕ ¤ = Ll (fl ), L⊕ r (fr ) ,

¸ ξr (x, z) ¯r dmfr (x) (42)

Theorem 5.3. Let ¯lJ and ¯r be pseudo-multiplications on [0, ∞] corresponding to a pseudoaddition ⊕ = max. If is an interval-valued pseudo-multiplication a nd f1 = [f1l , f1r ], f2 = [f2l , f2r ] ∈ IF∗ ([0, ∞)), then we have K L⊕ (f 2 ), (43) L⊕ (f1 ? f2 ) = L⊕ (f 1 ) where ? is an interval-valued pseudo-convolution. Proof. By Theorem 3.3(1), we have f1 ? f2 = [f1l ∗l f2l , f1r ∗r f2r ].

(44)

By (44), Theorem 5.2, and Theorem 5.1 , we have ¤ £ L⊕ (f1 ? f2 ) = L⊕ (f1l ∗l f2l ), L⊕ (f1r ∗r f2r ) r l £ ¤ = L⊕ (f1l ) ¯l L⊕ L⊕ (f ) ¯ L⊕ (f ) l (f2l ), Kr£ 1r r r 2r ¤ £ l⊕ ¤ ⊕ = Ll (f1l ), L⊕ L⊕ r (f1r ) l (f2l ), Lr (f2r ) K L⊕ (f 2 ), = L⊕ (f 1 )

(45)

6. Conclusions In this paper, we introduced four general notions of an interval-valued pseudo-addition and an interval-valued pseudo-multiplication (see Definition 3.2), an interval-valued pseudointegral(see Definition 3.3(2)), an interval-valued pseudo-convolution(see Definition 3.4), an interval-valued idempotent function(see Definition 4.1(2)), and the interval-valued pseudoLaplace transform(see Definition 5.1(2)). From Definitions 2.1, 2.2, 2.3, and Theorems 2.1, 2.2, we defined an interval-valued pseudo-convolution of interval-valued functions by means of the interval-valued pseudo-integral and obtained some characterizations of them including commutativity and associativity of an interval-valued convolution (see Theorems 3.1, 3.2). In particular, we defined an interval-valued J idempotent function with respect to an intervalvalued pseudo-convolution ? induced by (see Definition 4.1 (2)) and investigated an equivalence relation between the existence of an interval-valued idempotent function and an intervalvalued pseudo-convolution(see Theorem 4.4). We also obtain the pseudo-exchange formula for the standard interval-valued pseudo-Laplace transform(see Theorem 5.3). In the future, we can expect that the interval-valued pseudo-convolutions are used (i) to generalize pseudo-Laplace transform, Hamilton-Jacobi equation on the space of functions, such as in nonlinearity and optimization(see [20]) (ii) to generalize the Skolasky type inequality for the pseudo-integral of functions such as in economics, finance, decision making(see [4]), etc. Acknowledgement: This paper was supported by Konkuk University in 2014.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 19, NO. 6, 2015

Common Fixed Point of Mappings Satisfying Almost Generalized Contractive Condition in Partially Ordered G-Metric Spaces, M. Abbas, Jong Kyu Kim, and Talat Nazir,…………...928 Additive 𝜌-Functional Inequalities in Matrix Normed Spaces, Afshan Batool, Choonkil Park, and Dong Yun Shin,………………………………………………………………………………939 New Iterative Algorithms for Coupled Matrix Equations, Lingling Lv, and Lei Zhang,……947 Essential Norm of Weighted Composition Operator from Hardy Space to Weighted Bergman Space on the Unit Ball, Chang-Jin Wang, and Yu-Xia Liang,………………………………959 On the q-Analogue of 𝜆-Daehee Polynomials, Jin-Woo Park,………………………………966

On Fuzzy B-Algebras Over t-Norm, Jung Mi Ko, and Sun Shin Ahn,…………….………...975 Hybrid Boundary Value Problems of q-Difference Equations and Inclusions, Bashir Ahmad, Sotiris K. Ntouyas, and Ahmed Alsaedi,………………………………………………….…984 q-Wallis Inequality, Mansour Mahmoud,……………………………………………………994 On a 𝜇-Mixed Trigonometric Functional Equation, Driss Zeglami, Brahim Fadli, Choonkil Park, and Themistocles M. Rassias,……………………………………………………………….1003 Barnes-Type Narumi of the First Kind and Barnes-Type Peters of the First Kind Hybrid Polynomials, Dae San Kim, Taekyun Kim, Takao Komatsu, and Jong-Jin Seo,……………1012 A New Algorithm Based on Lagrangian Relaxation for Solving Indefinite Quadratic Integer Programming, Jianling Li, Peng Wang, and Jinbao Jian,……………………………………1038 A Note on an Interval-Valued Pseudo-Convolution of Interval-Valued Functions and Its Applications, Lee-Chae Jang,……………………………………………………………..…1049