JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 23, 2017

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Volume 23, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

July 2017

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 (fourteen 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:

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics

Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

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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.

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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Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:[email protected] Approximation Theory and Neural Networks

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J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected]

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Approximation Theory (Positive Linear Operators)

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

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

Ram N. Mohapatra Department of Mathematics University of Central Florida Orlando, FL 32816-1364 tel.407-823-5080 [email protected] Real and Complex Analysis, Approximation Th., Fourier Analysis, Fuzzy Sets and Systems

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Gaston M. N'Guerekata Department of Mathematics Morgan State University Baltimore, MD 21251, USA tel: 1-443-885-4373 Fax 1-443-885-8216 Gaston.N'[email protected] [email protected] Nonlinear Evolution Equations, Abstract Harmonic Analysis, Fractional Differential Equations, Almost Periodicity & Almost Automorphy

Robert Kozma Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA [email protected] Neural Networks, Reproducing Kernel Hilbert Spaces, Neural Percolation Theory Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston, RI 02881,USA [email protected] Differential and Difference Equations

M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations, Optimization, Signal Analysis

Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional Analysis, [email protected]

Mubenga N. Nkashama Department OF Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 205-934-2154 e-mail: [email protected] Ordinary Differential Equations, Partial Differential Equations

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Vassilis Papanicolaou Department of Mathematics

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National Technical University of Athens Zografou campus, 157 80 Athens, Greece tel:: +30(210) 772 1722 Fax +30(210) 772 1775 [email protected] Partial Differential Equations, Probability

Approximation Theory, Banach spaces, Classical Analysis 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

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

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I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3-065-109-8283 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

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Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional

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Analysis, [email protected]

USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected] Approximation Theory, Chebychev Systems, Wavelet Theory

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

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

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 [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Golub-Kahan-Lanczos based preconditioner for least squares problems in overdetermined and underdetermined cases Liang Zhaoa,b∗, Ting-Zhu Huanga†, Liu Zhub‡, Liang-Jian Denga§ a. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P. R. China b. Department of Mathematical Sciences, Clemson University, Clemson, South Carolina, 29634, U.S.A.

Abstract We present an effective preconditioner for solving least squares problems in full ranked overdetermined and underdetermined cases. The preconditioner, generated from Golub-Kahan-Lanczos method, can approximately replace a few largest singular values by one without altering the rest. This property accelerates the convergence, thereby improves the efficiency of the algorithm for solving the least squares problems with ill-conditioned system matrix which is caused by large singular values. In this paper we focus on the overdetermined and the underdetermined cases. Key words: Least squares problems; Preconditioner; Lanczos bidiagonalization process; Krylov subspace method; Golub-Kahan-Lanczos method AMSC : 65K05; 65F08; 65F10

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Introduction

In this paper, we assume that the least squares problems are in the form as min ∥b − Ax∥2 ,

(1)

∗

E-mail: [email protected] (L. Zhao) Corresponding author. E-mail: [email protected] (T.-Z. Huang) ‡ E-mail: [email protected] (L. Zhu) § E-mail: [email protected] (L.-J. Deng) †

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where Am×n is a full-ranked coefficient matrix which is large and sparse. In the situation that m = n, we can obtain an approximate solution by solving the linear system Ax = b and minimize the residual in the sense of 2-norm. The minimal norm residual method, based on the iterative Krylov methods, is a suitable algorithm to obtain the optimal approximation, and full details can be found in [2]. We have superscript T denoted the transposition of a matrix, and use subscript to indicate the size of matrix. The overdetermined cases min ∥b − Ax∥2 , A ∈ Rm×n , m > n

(2)

and the underdetermined cases min ∥b − Ax∥2 , A ∈ Rm×n , m < n

(3)

are taken into consideration in the following. In this paper, we take the preconditioner as a left preconditioner in both overdetermined and underdetermined cases. To the overdetermined system (2) in least squares problems, we generally translate the corresponding linear system Ax = b, A ∈ Rm×n , m > n,

(4)

into a normal equation by premultipling AT on both sides. R is the set of real number here and in the following. Similarly, we translate the underdetermined system (3) into a normal equation in the same way in the corresponding linear system Ax = b, A ∈ Rm×n , m < n. (5) Thereby we have the normal equation in the following form AT Ax = AT b.

(6)

We notice that the coefficient matrix in (6) is symmetric positive definite, so the normal equation can be solved by the CG method[16]. Thanks to previous researchers, many classic methods, such as CGNE [4] and CGLS[3], can be regarded as an extensions of the CG method and solve least squares problems efficiently. Similarly, the LSQR method[7] is an effective method for solving the least squares problems, so does the LSMR method[15]. For the symmetric positive definition (SPD) matrix, we know the convergence of iterative Krylov methods depends on the condition number κ of the coefficien(A) with t matrix, in other word, the spectral distribution, where κ(A) = λλmax min (A) λmax (A) and λmin (A) denoting the largest and the smallest eigenvalues of A, respectively. To discuss the spectral distribution of AT A in (6), we give the singular value decomposition of the original coefficient matrix A as follow. Notice that all the matrixes in this paper are full ranked. 2

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We have the singular value decomposition of A in this form   σ1   σ2   T ˆ ˆ A = Um×n DVn×n , D =  , . .   . σn

(7)

where Uˆm×n and Vˆn×n are both unitary matrices, σi denotes the singular value that σ1 > σ2 > · · · > σn . From (7), we have T AT A = Vˆn×n D2 Vˆn×n ,

(8)

which can be regarded as the eigenvalue decomposition of the coefficient matrix in the normal equation (6). If we denote Σ = diag{σ12 , σ22 , · · · , σr2 }, where r = min(m, n), it could be easily concluded that the spectral distribution of the coefficient matrix in (6) is Σ. Therefore, the condition numbers of linear systems can be presented as σ2 κ(AT A) = σ12 . To accelerate the convergence, thereby improve the algorithm, we r expect the condition number to be as small as possible. Therefore, removing the smallest eigenvalue from the spectrum of the coefficient matrix is purpose of the preconditioner. Also, we leave the rest unchanged. Such kind of preconditioners and relevant applications can be located in [8], [9] and [10]. Also, when the property of ill-condition is caused by a few largest eigenvalues, we expect a preconditioner, from the similar point of view, to eliminate the largest eigenvalues from the spectrum in order to accelerate the convergence. A preconditioner formed by Lanczos bidiagonalization is formulated to change the largest singular values to one approximately without altering the others, so that the preconditioner change the corresponding eigenvalues in normal equations. In the ill-conditioned overdetermined case and the ill-conditioned underdetermined case, we utilize the preconditioner to speed up the convergence. To illustrate the effects of the preconditioners proposed in this paper, we utilize two methods to solve a series of the least squares problems. Of course, we divide every experiments into two parts, using preconditioner and not using it. In the following sections, the process of Lanczos bidiagonalization will be stated in section 2; the preconditioners for solving overdetermined and underdetermined least squares problems (2) (3) will be defined in section 3; numerical examples are demonstrated in section 4; conclusions are presented in section 5 finally.

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2 2.1

The process of Lanczos bidiagonalization Standard Lanczos bidiagonalization

Lanczos biorthogonalization, which can be located in [6] [4], is an important process in methods like LSQR[7], BiCG[11] and BiCGSTAB[12]. A variation of Lanczos biorthogonalization, formed as   α1  β2 α 2      .. .. AVn = Un+1 B, B =  (9) , . .    βn αn  βn+1 is denoted as Golub-Kahan-Lanczos method [5] , where Vn and Un+1 are both unitary matrices and we assume A is a matrix of size n × n. One characteristic of decomposition (9) is that the lower bidiagonal matrix B shares the same singular values as A’s. Furthermore, we have analyzed and concluded in the previous section that the singular values distribution of A directly reflects the spectral distribution of AT A in problems (6). Hence we expect a preconditioner based on Lanczos bidiagonalization to optimize spectral distributions of system matrices in least squares problems. Some similar preconditioner based on the Golub-KahanLanczos bidiagonalization for square coefficient matrixes has been proposed and applied. For example, inreference[13], the author optimized the spectral distribution of a ill-posed coefficient matrix by a Lanczos-based preconditioner. However, limited by the dimension of the coefficient matrix in overdetermined and underdetermined cases, the algorithm will break down when maximal number of iteration is greater than both row dimension and column dimension. Therefore, in order to be applied to overdetermined and underdetermined cases, the standard form of Golub-Kahan-Lanczos method requires modification. To extend applications of the Lanczos-based preconditioner, we define variants of the preconditioner which can be utilized in overdetermined cases and underdetermined cases, thereby it is available for least squares problems. At first, we give the standard algorithm for Golub-Kahan-Lanczos method as stated in [5]. Algorithm 1 Standard Golub-Kahan-Lanczos bidiagonalization 1. β1 = ∥b∥2 , u1 = βb1 , v0 = 0 2. for i = 1, 2, ..., n 3. pk = AT uk − βk vk−1 4. αk = ∥pk ∥2 5. vk = αpkk 6. qk = Avk − αkuk 7. βk+1 = ∥qk ∥2 qk 8. uk+1 = βk+1 4

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The α′ s and β ′ s generated in the above algorithm are equal to the ones in (9), also rows of V and U in (9) are obtained through Algorithm 1 as vk and uk respectively. Therefore, we could establish the Lanczos bidiagonalization form by a series of iterations performed according to Algorithm 1, when the coefficient matrix A is of size n × n. To define the Lanczos-based preconditioners in overdetermined cases and underdetermined cases, we have to modify algorithm 1, the standard Lanczos bidiagonalization process, in order to accommodate the situations that the coefficient matrices are m-by-n and m ̸= n.

2.2

Modified Lanczos bidiagonalization

The main distinction between the overdetermined, or underdetermined, determined and square cases is the dimension of the coefficient matrix A. As stated before, the matrix B, generated by Lanczos bidiagonalization, and A in (9) share the same singular value distribution. We limit the steps of Lanczos bidiagonalizaion process under the minimal number between m and n where A is m-by-n. We utilize iterative Krylov subspace methods to solve the linear systems (6), with symmetric positive definite coefficient matrices. Therefore we conclude easily that the rank of B can not exceed the minimum of m and n. Then, a restrictive condition should be added to the corresponding Lanczos bidiagonalization process to terminate it in appropriate number of steps. Different from (9), We set a termination rule that the maximal iteration in Golub-Kahan-Lanczos bidiagonalization is less or equal to the minimum between the row dimension and the column dimension toensure that the algorithm will terminate in appropriate number of steps. Following this rule, we have the bidiagonalization decomposition of A in overdetermined situation as   α1  β2 α2      .. .. AVn×n = Um×(n+1) Bn , Bn =  (10) , . .    βn α n  βn+1 and the bidiagonalization decomposition of A in underdetermined situation as   α1   β2 α2     . . . . (11) AVn×m = Um×(m+1) Bm , Bm =  . . .     βm α m βm+1 Considering the computational cost of the Lanczos bidiagonalization process, we try to avoid bidiagonalizing A completely. The preconditioner, mentioned in 5

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the previous section and defined in the next section, is structured for the purpose of changing the largest singular values to one, in order to optimize the condition numbers of normal equation (6). Hence, we stop the Lanczos dibiagonalization process when the current smallest singular value σk , generated in the kth step of Lanczos dibiagonalization process, is much smaller than the largest one σ1 . We set a scalar number δ to be the threshold of termination, i.e, terminates when σk < δσ1 . If the bidiagonalization process stops at the kth step, the bidiagonalization composition is of the form below   α1  β2 α 2      . . . . (12) AVn×k = Um×(k+1) Bk , Bk =  . . .    βk α k  βk+1 M Rezghi set the scalar number δ as the square root of machine precision in [13] while applying it in ill-conditioned systems derived from blurring images. Since δ is a scalar to judge whether we should terminate the Lanczos bidiagonalization process and the Lanczos bidiagonalization process aims to remove the largest singular values, the choice of δ has different effects in different numerical examples. We will present the influence caused the change of δ under different numerical examples and iterative methods in the section of experiments. In general ill-conditioned systems, we need not to set δ so small and some cases will be presented in the 4th section. Here we add the above two restrictive conditions to standard Lanczos bidiagonalization, then we have modified Lanczos bidiagonalization as following. Algorithm 2 Modified Lanczos bidiagonalization 1. β1 = ∥b∥2 , u1 = βb1 , v0 = 0, r = min{m, n}, δ 2. for i = 1, 2, ..., r 3. pk = AT uk − βk vk−1 4. αk = ∥pk ∥2 5. vk = αpkk 6. qk = Avk − αkuk 7. βk+1 = ∥qk ∥2 qk 8. uk+1 = βk+1 9. get singular values of B: σ1 , σ2 , · · · , σi 10. if σi < δσ1 , break down. 11.end In this section, we introduced the standard Lanczos bidiagonalization process in Algorithm 1, and defined the modified Lanczos bidiagonalization process in 6

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Algorithm 2, which is adapted to the overdetermined and the underdetermined situations. A preconditioner based on modified Lanczos bidiagonalization process will be introduced and defined in the next section.

3

Lanczos-based preconditioner for least squares problems

To solve the least squares problems formed as (2) and (3), we solve the corresponding linear systems (4) and (5) instead by translating them into normal equations (6) respectively. If we have the singular value decompositions of A which are structured as (7), and the singular value distributions are scattered and wide, that is the largest singular value is much greater than the smallest one, thereby the condition number of the normal equation (6) will be terribly greater according to analysis of (8). For the purpose of speeding up the convergence, we expect to optimize, or reduce, the condition number of AT A. Since the σ2 condition number of normal equations (6) could be presented as κ(AT A) = σ12 r where σ1 and σr denote the largest and the smallest singular value of A, enlargement or elimination of the smallest singular values, and decrease or elimination of the largest singular values are both effective methods to reduce the condition number. Deflation-based preconditioners, like the deflation preconditioner and the balancing preconditioner[8, 9, 10], have such characteristics and properties to eliminate smallest eigenvalues of system matrix. We do not pay much attention to the preconditioners based on deflation, but the preconditioners functioned for decreasing, or eliminating, the largest ones are what we concern. In the following, all the preconditioners based on Lanczos bidiagonalization are defined for the overdetermined cases (2) and the underdetermined cases (3). First we shall discuss the situation of the underdetermined case. In linear system (5), the coefficient matrix A has the singular value decomposition illustrated as (7). We assume a diagonal matrix Dk = diag{σ1 , σ2 , · · · , σk }, where σi with i = 1, 2, · · · , k, denotes the first k largest singular values of A. The Lanczos bidiagonalization process for underdetermined cases within k steps have been proposed as (12). On the premise that B, which is structured by Lanczos bidiagonalization, shares the same singular values with A, we have the following conclusion that: the Bm derived from (11) has singular value decomposition form as ( ) D ˜ V˜ T , Bm = U(m+1)×(m+1) 0 (m+1)×m m×m 7

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where D in the above equation is equal to the one in (7), with U˜m+1 and V˜m×m both unitary matrices. Similarly, the Bk derived from (12) has singular value decomposition form as ( ) D k Bk = U˜k V˜kT , (13) 0 where Dk has been defined at the beginning in this section, with U˜k and V˜k both unitary matrices. When we consider the underdetermined case (11), some deductions are stated as follow. We use singular value decomposition of B replacing the one in (11) and we have ( ) D ˜ AVn×m = Um×(m+1) U(m+1)×(m+1) V˜ T . 0 (m+1)×m m×m The dimension of matrices are denoted as subscripts in previous sections, and now the subscripts will be omitted for simplification. Then we postmultiply V˜ on both sides and we have ( ) D ˜ ˜ AV V = U U , 0 Here we set V¯ = V V˜ = {¯ v1 , v¯2 , · · · , v¯m } and U¯ = U U˜ = {¯ u1 , u¯2 , · · · , u¯m+1 }. As for equation ( ) D ¯ ¯ AV = U , 0 we regard it as a singular value decomposition of A, similar to (7), approximately. If we set U¯m = {¯ u1 , u¯2 , · · · , u¯m }, the first m columns of U U˜ , we assume that U¯m = Uˆ V¯ = Vˆ where Uˆ and Vˆ are obtained from (7). Now we focus on the formulation (8). If a matrix is structured as P = V¯ D−2 V¯ T , then combining with the previous assumption(V¯ = Vˆ ), it gives that P AT A = = =

V¯ D−2 V¯ T Vˆ D2 Vˆ T V¯ I V¯ T I.

It seems that we could have obtained solution directly through the application of such a preconditioner P . In view of computation, however, it is inadvisable for 8

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the following reasons: 1. the preconditioner P is based on a complete Lanczos bidiagonalization, so this process has expensive computational cost even no less than direct methods.; 2. the V¯ is approximately equal to Vˆ in practical implement, but we give the above deduction just in theory, without the consideration of computational errors. Although we can not utilize the preconditioner P in practical computation, a variant of P based on incomplete Lanczos bidiagonalization is defined as follow to solve underdetermined least squares problems. Here we construct a preconditioner P which is similar with the one mentioned above with merely replacing Bm (from (11)) by Bk (from (12)). After simple deduction, we have ( −2 ) Dk 0 ¯ P =V V¯ T , 0 Im−k where Dk is from (13). We set V¯k = V V˜k is the first k columns of V¯ , where V˜k is obviously the first k columns of V˜ . Hence we set V¯ = [V¯k , V¯m−k ]. Based on the definition of V¯ , we have T I = V¯ V¯ T = V¯k V¯kT + V¯m−k V¯m−k .

Analyzing the above information, it gives that P = = =

T V¯k Dk−2 V¯kT + V¯m−k V¯m−k −2 V V˜k Dk V˜kT V T + (Im×m − V¯k V¯kT ) V (BkT Bk )−1 V T + (Im×m − V V T ).

where V and Bk can both be obtained through Algorithm 2. If we utilize P as a left preconditioner in normal equation (6) for underdetermined cases (5), we have ( ) Ik 0 T ˆ PA A = V Vˆ T , 2 0 Dm−k where Dm−k = diag{σk+1 , σk+2 , · · · , σm } with σi ’s denoting the m − k smallest singular values. According to the statement above, we can conclude that the Laczos-based preconditioner has the property to change k largest singular values of coefficient matrix A, or k largest eigenvalues of the system matrix in normal equation (6) in other word, to one without touching the others. The preconditioner is able to optimize the condition number of normal equation (6) when the ill condition is caused by these large singular values. Since k ≪ m, the computational cost is greatly reduced, so is the computational error. The conclusion, furthermore, is under the premise that the linear system corresponding to least squares problems is underdetermined, so that Punder = V (BkT Bk )−1 V T + (In×n − V V T )

(14)

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could be used as a left-preconditioner in underdetermined least squares problems. Next we consider the overdetermined cases. In the overdetermined cases, we construct a Lanczos-based preconditioner that follows the same strategy as stated in the previous subsection. To solve the overdetermined system (4), we solve the normal equation (6) instead to obtain approximate solution. Considering the decomposition form (8) of AT A, we expect to construct a preconditioner, similar to the underdetermined cases, presented as ) ( −2 D 0 k P = Vˆ Vˆ T . 0 In−k Through an analogical deduction to underdetermined cases, a preconditioner formed as Pover = V (BkT Bk )−1 V T + (In×n − V V T ) (15) can be used as a left-preconditioner in overdetermined least squares problems. Bk and Vn×k can be obtained from Algorithm 2. Furthermore it is not computationally costly because of k ≪ n. From the above discussion, we can see that the forms of the Lanczos-based preconditioners in over- and under- determined cases are the same, although we deduced them in separate ways. Also, such a preconditioner for the linear system with a square coeffcient matrix has the same form. Therefore, we can conclude that we deduce the preconditioners, proposed in this paper, from the point of overdetermined and underdetermined cases and ultimately get a result similar to the one in square problems, which has been proposed in [13]. Of course, the result of this paper can also be regarded as the expansion of the application of the Lanczos-based preconditioner into the overdetermined and underdetermined least squares problems. Now we unify the preconditioner as follow P = V (BkT Bk )−1 V T + (I − V V T ),

(16)

which can be used as a left preconditioner in ordinary linear systems, overdetermined least squares problems and underdetermined least squares problems. The relevant numerical experiments are presented in the following section, from which we can see the effects of Lanczos-based preconditioners.

4

Numerical experiments

In this section, we will take a series of numerical examples to present the effect of the Lanczos-based preconditioner in the least squares problems. At first, we introduce two iterative methods as the basic algorithm for solving these underdetermined and overdetermined problems. Here, we choose an old and classic method as the first one for solving the least squares problems. It is the CGLS 10

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method[3]. In this method, we first transform the least squares problems into symmetric positive definite(SPD) problems by the normal equations then solve it by the CG method[16]. Integrating the above ideas, we have the CGLS method. Now we present the preconditioned CGLS method algorithm 3, where we just consider the situation of left precondition. Algorithm 3 Preconditioned CGLS method 1. select x0 as the initial guess, r0 = b − Ax0 and P as the preconditioner 2. initialization: we set r¯0 = AT r0 , rˆ0 = P r¯0 , f0 = z0 2. for i = 0, 1, 2, ... 3. gi = Afi 4. αi = (ˆ ri , r¯i )/∥gi ∥22 5. xi+1 = xi + αi fi 6. ri+1 = ri − αi gi 8. r¯i+1 = AT ri+1 9. rˆi+1 = P r¯i+1 10. βi = (ˆ ri+1 , r¯i+1 )/(ˆ ri , r¯i ) 11. fi+1 = rˆi+1 + βi fi 12. endfor The second method to solve the least squares problems is the BAGMRES method[14], a variant of the GMRES method[1]. In this method, the least squares problems will be post-multiplied by a matrix B, an arbitrary nonsingular matrix. Now we give the BAGMRES method as Algorithm4.

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Ex. Group and name 1 JGD Forest/TF10 2 JGD Forest/TF11 3 HB/wm3 4 Pajek/Sandi sandi Meszaros/refine 5 6 JGD margulies/flower 4 1

id #rows #cols 1944 99 107 1945 216 236 277 207 260 1520 314 360 1759 29 62 2155 121 129

Nonzeros 622 1607 2948 613 153 386

Problem kind Combinatorial Combinatorial Economic Bipartite graph Linear programming Combinatorial

Table 1: The structures of six test underdetermined problems Algorithm 4 BA-GMRES with k restart 1. select x0 as the initial guess, r0 = B(b − Ax0 ) and ν1 = r0 /∥r0 ∥2 2. for i = 1, 2, ..., m 3. ωi = BAνi 4. for j = 1, 2, ..., i 5. hj,i = (ωi , νj ) 6. ωi = ωi − hj,i νj 7. endfor 8. hi+1,i = ∥ωi ∥2 9. νi+1 = ωi /hi+1,i 10. Compute ym to minimize ∥rˆi ∥2 = ∥∥rˆ0 ∥2 e1 − H i y∥2 11. if ∥ri ∥2 < τ 12. xi = x0 + [ν1 , ..., νi ]yi 13. stop 14. endif 15. endfor 16. set x0 = xk and return to line 2 until convergence In the following numerical experiments, the examples all come from practical applications from [17]. All the required information about the underdetermined and overdetermined cases is contained in Table 1 and Table 2 respectively. They both consist of group, number of rows, columns and nonzero elements and the type of problem of each example. In the next two subsections, we solve the above 12 problems by the PCGLS method and the BAGMRES method combined with the Lanczos-based preconditioners. Then we change the scalar δ, involving the termination rule of the modified Lanczos bidiagonalization, and show its influence on the iterative process. Because the preconditioner is designed to modify the singular values, the distributions of singular values under different scalar δ’s will be presented as well.

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Ex. Group and name id #rows #cols 7 HB/abb313 5 313 176 204 181 8 JGD margulies/cat ears 3 1 2151 9 JGD margulies/cat ears 4 1 2153 377 313 JGD margulies/flower 5 1 2157 211 201 10 JGD margulies/flower 7 1 2159 11 463 393 Pajek/Cities 1457 55 46 12

Nonzeros 1557 542 938 602 1178 1342

Problem kind Least squares Combinatorial Combinatorial Combinatorial Combinatorial Weighted bipartite graph

Table 2: The structures of six test overdetermined problems

4.1

The acceleration of iterative processes

To discuss the acceleration of iterative processes, we refer to the PCGLS method and the BAGMRES method in [14, 4]. For the BAGMRES method, we have the following relation between the initial residual and the one from the kth iteration in underdetermined cases, ∥Brk ∥2 = ∥CAT rk ∥ ≤ 2(

σ1 − σm k ) ∥Br0 ∥2 , σ1 + σm

(17)

where C is a nonsingular matrix, κ(C) is the condition number of matrix C and σ’s denote the singular values of BA. And we have the relation between r0 and rk as √ σ1 − σn k ∥Brk ∥2 = ∥CAT rk ∥ ≤ 2 κ(C)( ) ∥Br0 ∥2 , (18) σ1 + σn where C is a nonsingular matrix, κ(C) is the condition number of matrix C and σ’s denote the singular values of BA. More information of the above conclusion can be found in [14]. Now we give the convergence analysis of the PCGLS method, that is σ1 − σr k ∥ek ∥A ≤ 2( ) ∥e0 ∥A , (19) σ1 + σr where r = min(m, n) and σ’s denoting the singular values of P AT A. Based on equation (17), (18) and (19), it is obvious that we can accelerate the convergence if the gap between the largest singular value of normal equations and the smallest one is narrowed. In this paper, the Lanczos-based preconditioner is just for resetting the largest singular values to one, which can be regarded as shrink of the singular value distribution. Now, the effect of the Lanczos-based preconditioner in underdetermined cases is shown from Figure 1 to Figure 6. In the numerical experiments, we set the tolerance tol = 10−12 , the maximal number of iteration maxi t = 1000 and the restarted number in the BAGMRES method restart = 600. Furthermore, the scalar δ upon which to terminates the

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TF10

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abb313

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Figure 12: Relative residuals vs iterations in itercities

Lanczos bidiagonalization process is 0.05 in the test. We set B = P AT in the preconditioned BAGMRES method and B = AT in the nonpreconditioned BAGMRES method. From Figure 1 to Figure 6, we can see that both the BAGMRES method and the PCGLS method are accelerated by the Lanczos-based preconditioner as we expected. Next we show the iterative process while solving the overdetermined problems. Figure 7 to Figure 12 present the results of experiments with the tolerance tol = 10−12 , the maximal number of iteration max it = 1000 and the restarted number in the BAGMRES method restart = 600. The scalar δ upon which to terminate the Lanczos bidiagonalization process is 0.05 in the test. Similarly, we set B = P AT in the preconditioned BAGMRES method and B = AT in the nonpreconditioned BAGMRES method. In Figure 7 to Figure 12, it is obvious that Lanczos-based preconditioners also accelerate the iterative processes in these overdetermined problems, so we think the proconditioner proposed in this paper is helpful to optimize the structure of coefficient matrix thereby accelerate the convergence. Moreover, all the numerical examples here are derived from practical applications. We believe, therefore, the Lanczos preconditioner has the result as

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The distribution of singular values

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Figure 14: The iterative process of BAGMRE in TF10

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Figure 18: The iterative process of PCGLS in TF11

we expected.

4.2

The influence of the scalar δ

Referring to the illustration above, we have known that the scalar δ is used as a termination rule during the implementation of the Lanczos bidiagonalization process. By the definition of scalar δ, the smaller the δ is, the more large singular values will be replaced by one. It means that we can narrow the distribution of singular values. In the following experiments, we set the scalar δ to three different values and take TF10 and TF11 as the underdetermined examples. We test the distributions of the coefficient matrix of corresponding normal equations, the iterative process of the BAGMRES method and the PCGLS method. The results of TF10 and TF11 with varying scalar δ are presented in Figure13-15 and Figure 16-18 respectively. As for the overdetermined cases, we take abb313 as the first numerical examples. The singular values distribution and iterative processes of this example are illustrated by Figure 19-21. 16

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The distribution of singular values

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Figure 24: The iterative process of PCGLS in cat ears 4 1

Similarly, the singular value distribution and iterative process regarding to different δ of the example cat ears 4 1 are presented in Figure 22-24. In the above twelve figures, we classify the δ into three classes: the large delta, the middle delta and the small delta. The different δ stand for different preconditioners, upon which we denote the corresponding singular value distribution and iterative process by colorful points and lines. Theoretically, the small delta is able to reset most largest singular values while the large delta reset least largest singular values. Furthermore, required data of the experiments is presented in Table 3 and Table 4, in which k stands for the step of the Lanczos bidiagonalization process, iterBAGM RES and iterP CGLS represent the number of iterations of the BAGMRES method and the PCGLS method, respectively. From Figure 13, Figure 16, Figure 19 and Figure 22, we can observe that the preconditioner with smaller δ indeed narrows the singular value distribution better than the ones led by larger δ. However, we fail to replace the largest singular values by one, although the improvement has brought us better convergence that is shown in Figure 14-15, Figure 17-18, Figure 20-21 and Figure 23-24. Through Table 3 and Table 4, we can also find that the number of iterations decreases 17

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Example Nonprec δ = 0.8 δ = 0.3 δ = 0.05 Example Nonprec δ = 0.8 δ = 0.3 δ = 0.05

TF10 k 2 6 22 TF11 k 2 5 24

iterBAGM RES 99 99 97 83

iterP CGLS 333 340 314 250

iterBAGM RES 216 216 216 200

iterP CGLS 1000 995 972 872

Table 3: The information along with the change of scalar δ in underdetermined cases TF10 and TF11 Example Nonprec δ = 0.8 δ = 0.3 δ = 0.05 Example Nonprec δ = 0.8 δ = 0.3 δ = 0.05

abb313 k 2 5 24 cat ears 4 1 k 2 5 40

iterBAGM RES 101 101 98 80

iterP CGLS 165 159 152 112

iterBAGM RES 125 124 121 86

iterP CGLS 136 135 130 90

Table 4: The information along with the change of scalar δ in underdetermined cases abb313 and cat ears 4 1 obviously while the δ decreasing. In small-scale problem, the Lanczos-based preconditioner can reset the largest singular values closer to one than in large-scale problems, which is easy to testify by a simple numerical deduction. We suppose that the reason why the preconditioner fails to reset the largest singular values to one, just decreasing them instead, is the accumulation of calculation errors and the assumption U¯m = Uˆ V¯ = Vˆ . From another experiment, the matrix B constructed in Lanczos bidiagonalization 18

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process has approximately equal singular values with coefficient matrix A. Merely focusing on the numerical value, the gap between the singular values of B and A may be underestimated and even ignored. Nevertheless, the gap will be enlarged when we assume the above equalities without considering the calculation errors. In the above experiments, we can also notice that the different δ influence the iterative process distinctly in different method so the perturbation analysis of the Lanczos-based preconditioner may give us a theoretical explanation of the difference between the theory and the numerical experiment. This supposition is remained to be testified in the future work.

5

Conclusions

To the overdetermined and the underdetermined least squares problems, we choose the BA-GMRES method and the PCGLS method to solve them respectively. Variants of the Lanczos bidiagonalization process are defined in the situation that coefficient matrices are not square, and the algorithm of modified Lanczos bidiagonalization is illustrated as conclusion. When we suffer from the ill-conditioned system matrices, the preconditioners based on modified Lanczos bidiagonalization, P structured for the overdetermined cases and the underdetermined cases respectively, are imposed on iterative Krylov subspace methods to accelerate convergence. Finally we prove our statements with numerical experiments and conclude that the preconditioner defined in this paper is effective to solve least squares problems in overdetermined and underdetermined cases. Acknowledgements. This research is supported by NSFC (61370147, 61170309), 973 Program (2013CB329404).

References [1] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear system, SIAM J. Sci. Statist. Comput., 7(1986), pp. 865-869. [2] H. A. Van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge University Press, Cambridge, UK, 2003. [3] A. Bj¨ orck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996. [4] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003. [5] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. A. Van Der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems, SIAM, PA, USA, 2000.

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[6] G. H. Golub, C. F. Van Loan, Matrix Computations 3rd Edition, Johns Hopkins University Press, Maryland, USA, 1996. [7] C. C. Paige and M. A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8(1982), pp. 43-71. [8] J. Frank and C. Vuik, On the construction of deflation-based preconditioners, SIAM J. Sci. Comput., 23(2001), pp. 442-462. [9] Y. A. Erlangga and R. Nabben, Deflation and balancing preconditioners for Ktylov subspace methods applied to nonsymmetric matrices, SIAM J. Matrix Anal. Appl., 30(2008), pp. 684-699. [10] Y. A. Erlangga and R. Nabben, Multilevel prejection-based nested Krylov iteration for boundary value problems, SIAM J. Sci. Comput., 30(2008), pp. 1572-1595. [11] R. Fletcher, Conjugate gradient methods for indefinite systems, volume 506 of Lecture Notes Math., pages 73-89. Springer-Verlag, Berlin-Heidelberg-New York, 1976. [12] H. A. van der Vorst. Bi-CGSTAB: A fast and soomthly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Statist. Comput., 13(1992), pp. 631-644. [13] Mansoor Rezghi. S. M. Hosseini, Lanczos based preconditioner for discrete ill-posed problems, Computing, 88(2010), pp. 79-96. [14] K. Hayami, J. F. Yin, and T. Ito, Numerical methods for least squares problems, SIAM J. Matrix Anal. Appl., 31(2010), pp. 2400-2430. [15] D. C. L. Fong and M. Saunders, LSMR: An iterative algorithm for spare leastsqaures problems, SIAM J. Sci. Comput., 33(2011), pp. 2950-2971. [16] M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49(1952), pp. 409-435. [17] T. Davis, The University of Florida Sparse Matrix Collection, http://www.cise.ufl.edu/research/sparse/matrices, NA Digest 97(23)(1997).

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Classical Model of Prandtl’s Boundary Layer Theory for Radial Viscous Flow: Application of (G0 /G)− Expansion Method Taha Aziza,b , T. Motsepac , A. Azizd , A. Fatimaa,b and C.M. Khalique1,c a

DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, Johannesburg, South Africa

b

School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

c

International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa d

College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Rawalpindi 46070, Pakistan ([email protected], [email protected], [email protected], [email protected], [email protected],) Abstract In this paper, the exact closed-form solutions of the Prandtl’s boundary layer equation for radial flow models with uniform or vanishing mainstream velocity are derived by using the (G0 /G)−expansion method. Many new exact solutions are found for the boundary layer equation, which are expressed by the hyperbolic, trigonometric and rational functions. The solutions are valid for all values of the parameter β. It is shown that the (G0 /G)−expansion method is effective and can be used for many other nonlinear differential equations of mathematical physics.

Keywords: (G0 /G)−Expansion method; Prandtl’s boundary layer equation; Exact solutions 1

Corresponding author: E-mail : [email protected] (C.M. Khalique); Tel : +27 18 389 2009; Fax : +27 18 389 2052

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1

Introduction

Many real world problems in nonlinear science associated with mechanical, structural, aeronautical, ocean, electrical, and control systems can be summarized as solving nonlinear differential equations which arise from mathematically modelling such problems. Therefore, the study of nonlinear differential equations has been an active area of research for the past few years. Investigating integrability and finding exact solutions to such nonlinear differential equations have extensive applications in many scientific fields such as hydrodynamics, fluid dynamics, general relativity, condensed matter physics, solid-state physics, nonlinear optics, neurodynamics, fibre-optic communication and so on. These exact solutions, if reported are helpful for the numerical analyst to verify the complex numerical codes and are also useful in stability analysis for solving special nonlinear models. In recent years, much attention has been devoted to the development of several powerful and useful methods for finding exact and approximate solutions of nonlinear differential equations. These research methods for solving nonlinear differential equations include the bilinear method and multilinear method [1], classical Lie symmetry method [2], nonclassical Lie group approach [3], Clarkson-Kruskal’s direct method [4], deformation mapping method [5], homogenous balance method [6], Weierstrass elliptic function expansion method [7], F -expansion method [8], transformed rational function method [9], auxiliary equation method [10], sine–cosine method [11], tanh-function method [12], Backlund transformation method [13], simplest equation method [14, 15], exponential function rational expansion method [16] and so forth. Prandtl [17] initiated the concept of a boundary layer in large Reynolds number flows in 1904 and he also showed how the Navier-Stokes equation could be simplified to yield approximate solutions. Prandtl introduced boundary layer theory to understand the flow behavior of a viscous Newtonian fluid near a solid boundary. Prandtl’s boundary layer equations arise in various physical models of fluid mechanics. The equations of the boundary layer theory have been the subject of considerable interest, since they represent an important simplification of the original Navier-Stokes equations. These equations arise in the study of steady flows produced by wall jets, free jets and liquid jets, the flow past a stretching plate/surface, flow

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induced due to a shrinking sheet and so on. These boundary layer equations are usually solved subject to specific boundary conditions depending upon the physical model investigation. Blasius [18] solved the Prandtl’s boundary layer equations for a flat moving plate problem and found a power series solution of the model. Falkner and Skan [19] generalized the Blasius problem by considering the boundary layer flow over an wedge inclined at certain angle. Sakiadis [20] studied the boundary layer flow over a continuously moving rigid surface with a constant speed. Crane [21] was the first one who investigated the boundary layer flow due to a stretching surface and developed the exact solutions of boundary layer equations. Gupta and Gupta [22] extended the Crane’s work and for the first time introduced the concept of heat transfer with the stretching sheet boundary layer flow. Schlichting [23] was the first to apply the boundary layer theory to the steady flow produced by a free two-dimensional jet emerging into a fluid at rest and solved the resulting ordinary differential equation numerically. Later, Bickley [24] solved the differential equation analytically. The concept of the boundary layer to laminar jets is discussed fully in standard texts on boundary layer theory such as by Schlichting [25] and Rosenhead [26]. More recently, the similarity solution of axisymmetric non-Newtonian wall jet with swirl effects was obtained by Kolar [27]. Naz et al. [28] and Mason [29] studied the general boundary layer equations for two-dimensional and radial flows by using the classical Lie group approach and recently Naz et al. [30] provided the similarity solutions of the Prandtl’s boundary layer equations by implementing the non-classical symmetry method. The (G0 /G)−expansion method is a powerful mathematical tool for finding exact solutions of certain nonlinear ordinary differential equations. The (G0 /G)−expansion method was introduced by Wang in [31] for constructing the exact solutions of some nonlinear evolution equations. To express the applicability and effectiveness of the (G0 /G)−expansion method, further research has been accomplished by a diverse group of researchers (see, for example, papers [32 − 34] ). The importance of our present work is to find some new class of exact closed-form solutions of Prandtl’s boundary layer equation for radial flow models with constant or uniform main stream velocity by employing the (G0 /G)−expansion method.

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2

Mathematical model

The Prandtl’s boundary layer equation, for the stream function φ(r, θ), for radial flow with uniform or vanishing mainstream velocity is [26] 1 ∂φ ∂ 2 φ 1 − 2 r ∂θ ∂r∂θ r



∂φ ∂θ

2 −

1 ∂φ ∂ 2 φ ∂ 3φ − ν = 0, r ∂r ∂θ2 ∂θ3

(1)

where (r, θ) denote the cylindrical polar coordinates and ν is the kinematic viscosity. The velocity components u(r, θ) and v(r, θ), in the r and θ directions, are related to stream function φ(r, θ) as u(r, θ) =

1 ∂φ , r ∂θ

v(r, θ) = −

1 ∂φ . r ∂r

(2)

By the use of Lie group theoretic method of infinitesimal transformations [2], the general form of similarity solution for equation (1) is φ(r, θ) = r2−β H(ξ), ξ =

θ , rβ

(3)

where β is the constant determined from further conditions and ξ = θ/rβ is the similarity variable. By the substitution of Eq. (3) into Eq. (1), we obtain the third-order nonlinear ordinary differential equation in H(ξ), viz., d3 H d2 H ν 3 + (2 − β)H 2 + (2β − 1) dξ dξ



dH dξ

2 = 0.

(4)

Equation (4) is the general form of Prandtl’s boundary layer equation for radial flow of a viscous incompressible fluid. The boundary layer equation is usually solved subject to certain boundary conditions depending upon the particular physical model under investigation. Here, we find the exact closed-form solutions of Eq. (4) using the (G0 /G)−expansion method. The paper is organised as follows. In Section 3, we provide a brief summary of the (G0 /G)−expansion method. In Sections 4, we apply this method to solve nonlinear Prandtl’s boundary layer equation for radial flow. Finally, some concluding remarks are presented in Section 5.

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3

A description of the (G0/G)−expansion method

In this section, we present a brief summary of the (G0 /G)−expansion method for solving nonlinear ordinary differential equations. The essence of the (G0 /G)−expansion method is given in the following steps: Step 1: We consider a general form of a nonlinear ordinary differential equation   dU d2 U d3 U P U (z), , , , ... = 0, (5) dz dz 2 dz 3 where U is an unknown function of z and P is a polynomial in U and its various derivatives. Step 2: According to the (G0 /G)−expansion method, one assumes that the solution of ODE (5) can be written as a polynomial in (G0 /G) as follows:  0 i M X G , (6) U (z) = βi G i=0 where G = G(z) satisfies the second-order linear ODE with constant coefficients, namely d2 G dG + λ + µG = 0, (7) dz 2 dz with βi (i = 0, 1, 2, ..., M ), λ and µ being constants to be determined. The integer M is found by considering the homogenous balance between the highest order derivatives and nonlinear terms appearing in ODE (5). Step 3: The positive integer M can be accomplished by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq. (5) as follows: If we define the degree of U (z) as D[U (z)] = M , then the degree of other expressions is defined by  q  d U (z) = M + q, D dz q   q s  d U (z) r D U = M r + s(q + M ). (8) dz q

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Therefore, we can get the value of M in Eq. (6). Step 4: We substitute Eq. (6) into Eq. (5) and then use ODE (7) to collect all terms with same order of (G0 /G) together. The left-hand side of (5) is then converted into polynomial in (G0 /G). Now by equating each coefficient of this polynomial to zero, we obtain a system of algebraic equations for βi , λ and µ. Step 5: Since the three types of general solutions of Eq. (7) are well known, we substitute the values of βi and the general solutions of Eq. (7) into Eq. (6) and obtain three types of solutions of the ODE (5).

4

Application of the (G0/G)−expansion method

In this section, we employ the (G0 /G)−expansion method to obtain solutions of Prandtl’s boundary layer Eq. (4). We assume that the solutions of Eq. (4) are of the form H(ξ) =

M X

 Ai

i=0

G0 (ξ) G(ξ)

i ,

(9)

where G(ξ) satisfies the second-order linear ODE with constant coefficients, viz., dG d2 G +λ + µG = 0 2 dξ dξ

(10)

with λ and µ being constants. The balancing procedure yields M = 1, so the solution of the ODE (4) is of the form  0  G (ξ) H(ξ) = A0 + A1 . (11) G(ξ) Now substituting Eq. (11) into Eq. (4), making use of the ODE (10), collecting all terms with same powers of (G0 /G) and equating each coefficient to zero, yields the

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following system of algebraic equations: 2βA21 µ2 − βA0 A1 λµ − A1 λ2 µν + 2A0 A1 λµ − 2A1 µ2 ν − A21 µ2 = 0, 3βA21 λµ − βA0 A1 λ2 − 2βA0 A1 µ − A1 λ3 ν + 2A0 A1 λ2 − 8A1 λµν + 4A0 A1 µ = 0, βA21 λ2 − 3βA0 A1 λ + 2βA21 µ − 7A1 λ2 ν + A21 λ2 + 6A0 A1 λ − 8A1 µν + 2A21 µ = 0, βA21 λ − 2βA0 A1 − 12A1 λν + 4A21 λ + 4A0 A1 = 0, 3A21 − 6A1 ν = 0. Solving this system of algebraic equations, with the aid of Mathematica, we obtain √ λ = 2 µ, A0 = λν, A1 = 2ν.

(12)

Substituting these values of A0 , A1 and the corresponding solution of ODE (4) into Eq. (11), we obtain the following three types of solutions of Eq. (1): Case 1: When λ2 − 4µ > 0 For this case we obtain the hyperbolic function solution given by   C1 sinh(δξ) + C2 cosh(δξ) λ H(ξ) = λν + 2ν − + δ , 2 C1 cosh(δξ) + C2 sinh(δξ) p where δ = 21 λ2 − 4µ, C1 and C2 are arbitrary constants.

(13)

Reverting back to the original variables (r, θ), the corresponding stream function is given by "

!# θ θ C sinh δ + C cosh δ λ 1 2 rβ
rβ
φ(r, θ) = r2−β λν + 2ν − + δ . (14) 2 C1 cosh δ rθβ + C2 sinh δ rθβ Case 2: When λ2 − 4µ < 0 Here we obtain the trigonometric function solution   λ −C1 sin(ξ) + C2 cos(δξ) H(ξ) = λν + 2ν − +  , 2 C1 cos(ξ) + C2 sin(ξ)

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p where  = 21 4µ − λ2 , C1 and C2 are arbitrary constants. The corresponding stream function is given as "

!# θ θ −C sin  + C cos  λ 1 2 β β r
r
. (16) φ(r, θ) = r2−β λν + 2ν − +  2 C1 cos  rθβ + C2 sin  rθβ Case 3: When λ2 − 4µ = 0 For this case we obtain the rational function solution   λ C2 H(ξ) = λν + 2ν − + . 2 C1 + C2 ξ In the form of stream function, the solution is expressed as !# " C λ 2 , φ(r, θ) = r2−β λν + 2ν − + 2 C1 + C2 rθβ

(17)

(18)

where C1 and C2 are arbitrary constants.

5

Concluding remarks

We have employed the (G0 /G)-expansion method for obtaining exact closed-form solutions of the well-known Prandtl’s boundary layer equation for radial flow models with uniform main stream velocity. The advantage of this method is that in this method, there is no need to apply the initial and boundary conditions at the outset. This method yields a general solution with free parameters which can be identified by the specific conditions. Also the general solutions obtained by (G0 /G)-expansion method are not approximate solutions. Prandtl’s boundary layer equations arise in various physical models of fluid dynamics and thus the exact solutions obtained maybe very useful and significant for the explanation of some practical physical models dealing with Prandtl’s boundary layer theory.

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References [1] M.J. Carroa, L. Grafakos, J.M. Martell and F. Soria, Multilinear extrapolation and applications to the bilinear Hilbert transform, J. Math. Anal. Appl. 357 (2009) 479 − 497. [2] P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, (1993). [3] Z Zhang and Yufu Chen, Classical and nonclassical symmetries analysis for initial value problems, Physics Letters A. 374 (2010) 1117 − 1120. [4] S. Shou-Feng, Clarkson–Kruskal direct similarity approach for differentialdifference equations, Commun. Theor. Phys. 44 (2005) 964 − 966. [5] X. Xie, L.C. Musson and M. Pasquali, An isochoric domain deformation method for computing steady free surface flows with conserved volumes, J. Comput. Phys. 226 (2007) 398 − 413. [6] M. Senthilvelan, On the extended applications of homogenous balance method, Appl. Math. Comput. 123 (2001) 381 − 388. [7] E.A. Saied, R.G. Abd El-Rahman and M.I. Ghonamy, A generalized Weierstrass elliptic function expansion method for solving some nonlinear partial differential equations, Comp. Math. Appl. 58 (2009) 1725 − 1735. [8] A. Ebaida and E.H. Aly, Exact solutions for the transformed reduced Ostrovsky equation via the F -expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions, Wave Motion. 49 (2012) 296 − 308. [9] H. Zhanga and Wen-Xiu Ma, Extended transformed rational function method and applications to complexiton solutions, Appl. Math. Comput. 230 (2014) 509 − 515. [10] Y. Chen and H. Gao, Auxiliary equation method for solving nonlinear Wicktype partial differential equations, Commun. Nonlinear. Sci. Numer. Simulat. 16 (2011) 2421 − 2437. [11] A.M. Wazwaz, A sine–cosine method for handling nonlinear wave equations, Mathematical and Computer Modelling 40 (2004) 499 − 508.

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[12] A.M. Wazwaz, The tanh and the sine–cosine methods for the complex modified KdV and the generalized KdV equations Computers & Mathematics with Applications 49 (2005) 1101 − 1112. [13] M.R. Miura, Backlund transformation, Berlin: Springer-Verlag; (1978). [14] A.N. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Solitons Fractals 24 (2005) 1217 − 1231. 27. [15] I.E. Mhlanga, C.M. Khalique, Exact solutions of the symmetric regularized long wave equation and the Klein-Gordon-Zakharov equations, Abstr Appl Anal 2014, (2014) 1 − 7. [16] H. Xin, The exponential function rational expansion method and exact solutions to nonlinear lattice equations system, Appl. Math. Comput. 217 (2010) 1561 − 1565. [17] L. Prandtl, Uber Flussigkeitsbewegungen bei sehr kleiner Reibung. Verhanglg. III. Intern. Math. Kongr. Heidelberg. (1904) 484 − 491. [18] H. Blasius, Grenzschichten in Flussigkeiten mit kleiner Reibung, Z. Math. u. Phys. 56 (1908) 1 − 37. [19] V.M. Falkner and S.W. Skan, Some approximate solutions of the boundary layer equations, Phil. Mag. 12 (1931) 865 − 896. [20] B.C. Sakiadis, Boundary-layer behavior on continuous solid surface: I. Boundary-layer equations for two-dimensional and axisymmetric flow, J. AIChe. 7 (1961) 26 − 28. [21] L.J. Crane, Flow past stretching plate, Z. Angew. Math. Phys. 21 (1970) 645 − 647. [22] P.S. Gupta and A.S. Gupta, Heat and mass transfer on a stretching sheet with suction and blowing, Can. J. Chem. Eng. 55 (1977) 744 − 746. [23] H. Schlichting, Laminare Strahlausbreitung, Z. Angew. Math. Mech. 13 (1933) 260 − 263.

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On properties of meromorphic solutions for a certain q-difference Painlev´e equation Xiu-Min Zheng1∗, Hong-Yan Xu2 and Hua Wang3

Abstract The main purpose of this paper is to investigate some properties on transcendental meromorphic solutions of a certain q-difference Painev´e equation az + b z + c, f (qz) + f (z) + f ( ) = q f (z) where a, b and c are complex constants such that |a| + |b| 6= 0. We obtain some results on the value distribution of f (z) and ∆q f (z) := f (qz) − f (z) , and the nonexistence of rational solutions, which extend some earlier results by Qi and Yang, Chen et al. Key words: q-difference equation; solution; zero order. Mathematical Subject Classification (2010): 39A 50, 30D 35.

1

Introduction and Main Results

In this paper, we shall assume that readers are familiar with the basic theorems and the standard notations of the Nevanlinna value distribution theory of meromorphic functions such as m(r, f ), N (r, f ), T (r, f ), · · · , (see Hayman [12], Yang [19] and Yi and Yang [20]). We also use S(r, f ) to denote any quantity satisfying S(r, f ) = o(T (r, f )) for all r on a set F ⊂ [1, +∞ of logarithmic density 1, where the logarithmic density of a set F is defined by Z 1 1 dt. lim sup log r r→∞ [1,r]∩F t Throughout this paper, the set F of logarithmic density 1 can be not necessarily the same at each occurrence. A century ago, Painlev´e and his colleagues [15] classified all equations of Painlev´e type of the form w00 (z) = F (z; w; w0 ), where F is rational in w and w0 and (locally) analytic in z. They singled out a list of 50 equations, six of which could not be integrated in terms of known functions. These equations are now known as the differential Painlev´e equations. The first two of these equations are PI and PII : w00 = 6w2 + z,

w00 = 2w2 + zw + α,

where α is a complex constant. ∗ Corresponding

author. E-mail: [email protected].

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Differential Painlev´e equations have been an important research subject in the field of the Mathematics and the Physics since the beginning of last century. They occur in many physical situations—-plasma physics, statistical mechanics, nonlinear waves, and so on. Therefore, Painlev´e equations have attracted much interest as the reduction of solution equations which are solvable by inverse scattering transformations, and so on. In the past 22 years, the discrete Painlev´e equations have become important research problems (see [7]). For example, the discrete PI equation can be expressed by yn+1 + yn−1 =

an + b + c, yn

and the discrete PII equation can be expressed by yn+1 + yn−1 =

(an + b)yn + c , 1 − yn2

where a, b, c are real constants, n ∈ N. In 2006-2007, Halburd and Korhonen used the analogues of Nevanlinna value distribution theory to single out the difference Painlev´e I and II equations from the following form w(z + 1) + w(z − 1) = R(z, w), (1) where R(z, w) is rational in w and meromorphic in z (see [9, 10, 11]). They obtained that if (1) has an admissible meromorphic solution of finite order, then either w satisfies a difference Riccati equation, or (1) can be transformed by a linear change in w to some difference equations, which include the difference Painlev´e I equation az + b + c, w(z)

(2)

(az + b)w(z) + c , 1 − w(z)2

(3)

w(z + 1) + w(z − 1) = and the difference Painlev´e II equation w(z + 1) + w(z − 1) =

where a, b, c are complex constants. Chen et al [4, 5, 16] studied some properties of finite order transcendental meromorphic solutions of (2)-(3), and obtained a lot of interesting results. Recently, there were lots of results about q-difference operators, q-difference equations, and so on (see [2, 6, 8, 18, 21, 22]), by applying the analogue of Logarithmic Derivative Lemma on q-difference operators, which was firstly established by Barnett, Halburd, Korhonen and Morgan [1] in 2007. By comparing these results of differences and qdifferences, we find that the usual shift f (z + c) of a meromorphic function are replaced by the q-difference f (qz), and the difference ∆c f = f (z + c) − f (z) are replaced by ∆q f (z) = f (qz) − f (z), q ∈ C\{0, 1}. In 2015, Qi and Yang [17] investigated the following equations az + b z + c, f (qz) + f ( ) = q f (z) z (az + b)f (z) + c f (qz) + f ( ) = , q 1 − f (z)2

(4) (5)

which can be seen as q-difference analogues of (2) and (3), and obtained some theorems as follows.

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Theorem 1.1 [17, Theorem 1.1]. Let f (z) be a transcendental meromorphic solution with zero order of equation (4), and a, b, c be three constants such that a, b cannot vanish simultaneously. Then, (i) f (z) has infinitely many poles. (ii) If a 6= 0, then f (z) has infinitely many finite values. (iii) If a = 0 and f (z) takes a finite value A finitely often, then A is a solution of 2z 2 − cz − b = 0. Theorem 1.2 [17, Theorem 1.2]. Let a, b, c and |q| 6= 1 be four constants, (i) if a 6= 0, then equation (4) has no rational solution; (ii) if a = 0, then the rational solutions of the equation (4) must satisfy f (z) = P (z) B+ Q(z) , where P (z) and Q(z) are relatively prime polynomials and satisfy deg P < deg Q 2 and 2z − cz − b = 0. Theorem 1.3 [17, Theorem 1.3]. Let a, b, c be constants with ac 6= 0, and f (z) be a transcendental meromorphic solution with zero order of equation (5). Then f (z) has infinitely many poles and infinitely many finite values. Inspired by the above results, we further investigate some properties of transcendental meromorphic solutions of the q-difference Painlev´e equation az + b z + c, f (qz) + f (z) + f ( ) = q f (z)

(6)

which is different from (4) and (5) to some extent, and obtain the following theorems. Theorem 1.4 Let a, b, c be complex constants such that |a| + |b| 6= 0, and f (z) be a zeroorder transcendental meromorphic solution of the q-difference Painlev´e equation (6). (i) If a 6= 0, p(z) is a polynomial of degree k(≥ 0) and |q| 6= 1, then f (z) − p(z) has infinitely many zeros; if a = 0, then the Borel exceptional values of f (z) can only come from the set E = {z| 3z 2 − cz − b = 0}; (ii) f (z) and ∆q f (z) have infinitely many poles, where ∆q f (z) = f (qz) − f (z). Theorem 1.5 Let a, b, c be complex constants such that |a| + |b| 6= 0. (i) If a 6= 0, then (6) has no rational solution. (ii) If a = 0, then (6) has a nonzero constant solution f (z) = B, where B satisfies 3B 2 − cB − b = 0. Furthermore, if c2 + 12b = 0, then (6) has no nonconstant rational solution.

2

Some Lemmas

To prove our results, we require some lemmas as follows. Lemma 2.1 [14, Theorem 2.5] Let f (z) be a transcendental meromorphic solution of order zero of a q-difference equation of the form Uq (z, f )Pq (z, f ) = Qq (z, f ), where Uq (z, f ), Pq (z, f ) and Qq (z, f ) are q-difference polynomials such that the total degree deg Uq (z, f ) = n in f (z) and its q-shifts, whereas deg Qq (z, f ) ≤ n. Moreover, we assume that Uq (z, f ) contains just one term of maximal total degree in f (z) and its q-shifts. Then m(r, Pq (z, f )) = o(T (r, f )), on a set of logarithmic density 1.

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Remark 2.1 The above lemma can be called see as a type of a q-difference analogue of Clunie lemma, recently proved by Barnett et al.; see [1, Theorem 2.1]. Remark 2.2 Here, a q-difference polynomial of f (z) for q ∈ C\{0, 1} is a polynomial in f (z) and finitely many of its q-shifts f (qz), . . . , f (q n z) with meromorphic coefficients in the sense that their Nevanlinna characteristic functions are o(T (r, f )) on a set of logarithmic density 1. Lemma 2.2 [1, Theorem 2.5] Let f (z) be a nonconstant zero-order meromorphic solution of Pq (z, f ) = 0, where Pq (z, f ) is a q-difference polynomial in f (z). If Pq (z, a) 6≡ 0 for slowly moving target a(z), then m(r,

1 ) = o(T (r, f )) f −a

on a set of logarithmic density 1. Lemma 2.3 [21, Theorem 1.1 and 1.3] Let f (z) be a nonconstant zero-order meromorphic function and q ∈ C \ {0}. Then T (r, f (qz)) = (1 + o(1))T (r, f ),

N (r, f (qz)) = (1 + o(1))N (r, f ),

on a set of lower logarithmic density 1. Lemma 2.4 (Valiron-Mohon’ko) ([13]). Let f (z) be a meromorphic function. Then for all irreducible rational functions in f (z), Pm ai (z)f (z)i , R(z, f (z)) = Pni=0 j j=0 bj (z)f (z) with meromorphic coefficients ai (z), bj (z), the characteristic function of R(z, f (z)) satisfies that T (r, R(z, f (z))) = dT (r, f ) + O(Ψ(r)), where d = max{m, n} and Ψ(r) = maxi,j {T (r, ai ), T (r, bj )}.

3

Proof of Theorem 1.4

Suppose that f (z) is a zero-order transcendental meromorphic solution of (6). (i) Ifa 6= 0, and p(z) is a polynomial of degree k(≥ 0). Let p(z) = ak z k +· · ·+a1 z +a0 . Let g(z) = f (z) − p(z). Substituting f (z) = g(z) + p(z) into equation (6), we have z z az + b g(qz) + p(qz) + g(z) + p(z) + g( ) + p( ) = + c. q q g(z) + p(z) It follows that

¸ · z z Pq (z, g) := g(qz) + p(qz) + g(z) + p(z) + g( ) + p( ) [g(z) + p(z)] q q − (az + b) − c [g(z) + p(z)] = 0.

(7)

From (7), we have · ¸ z Pq (z, 0) = p(qz) + p(z) + p( ) p(z) − (az + b) − cp(z). q

45

(8)

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If p(z) ≡ 0, then Pq (z, 0) = −(az + b) 6≡ 0. If k = 0 and p(z) = a0 ≡ α ∈ C \ {0}, then Pq (z, 0) = 3α2 − (az + b) − cα 6≡ 0. If k ≥ 1 and ak is a nonzero constant, then, we have from (8) that ¸ · 1 z Pq (z, 0) = p(qz) + p(z) + p( ) p(z) − (az + b) − cp(z) = (q k + 1 + k )a2k z 2k + · · · . (9) q q Since |q| 6= 1, we have q k + 1 + that

1 qk

6= 0, then Pq (z, 0) 6≡ 0. Thus, we have by Lemma 2.2 1 m(r, ) = S(r, g). g

Then, we get µ N

1 r, f −p

¶

µ =N

1 r, g

¶ = T (r, g) + S(r, g) = T (r, f ) + S(r, f ).

(10)

Since f (z) is transcendental, f (z) − p(z) has infinitely many zeros. If a = 0 and p(z) = β 6∈ E, then we have Pq (z, 0) = 3β 2 − cβ − b 6≡ 0. 1 )= Set g(z) = f (z) − β, by using the same argument as above, we can obtain N (r, f −β T (r, f ) + S(r, f ).. Therefore, we can obtain that the Borel exceptional values of f (z) can only come from the set E = {z|3z 2 − cz − b = 0}. (ii) From (6), we have ¸ · z (11) f (z) f (qz) + f (z) + f ( ) = az + b + cf (z). q

It follows from Lemma 2.1 and (11) that ¶ µ z m r, f (qz) + f (z) + f ( ) = S(r, f ). q

(12)

By applying Lemma 2.4 for (6), we have µ ¶ z T r, f (qz) + f (z) + f ( ) = T (r, f ) + S(r, f ). q

(13)

And by Lemma 2.3 we get µ ¶ µ ¶ z z N r, f (qz) + f (z) + f ( ) ≤ N (r, f (qz)) + N (r, f (z)) + N r, f ( ) q q = 3(1 + o(1))N (r, f )

(14)

on a set of lower logarithmic density 1. Thus, by combining (12)-(14), we have T (r, f ) ≤ 3(1 + o(1))N (r, f ) + S(r, f ).

(15)

Since f (z) is transcendental, f (z) has infinitely many poles. Next, we prove that ∆q f (z) has infinitely many poles. Set z = qw, then we can rewrite (6) as the form f (q 2 w) + f (qw) + f (w) =

46

aqw + b + c. f (qw)

(16)

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Then it follows from (16) that £ ¤ f (qw) f (q 2 w) + f (qw) + f (w) = aqw + b + cf (qw).

(17)

Since ∆q f (w) = f (qw)−f (w), we have f (qw) = ∆q f (w)+f (w) and f (q 2 w) = ∆q f (qw)+ ∆q f (w) + f (w). Substituting them into (17), we get [∆q f (w) + f (w)] [∆q f (qw) + 2∆q f (w) + 3f (w)] = (aqw + b) + c [∆q f (w) + f (w)] , i.e., −3f (w)2 = [∆q f (qw) + 5∆q f (w) − c] f (w) − (aqw + b) + [∆q f (qw) + 2∆q f (w) − c] ∆q f (w).

(18)

Since f (z) is a zero-order transcendental meromorphic function and z = qw, by Lemma 2.3, we get that f (w) is of zero order. Thus, by Lemma 2.3 again, we have that f (w), ∆q f (w), ∆q f (qw) are of zero-order. Then by Lemma 2.3 again, we have N (r, ∆q f (qw)) ≤ N (r, ∆q f (w)) + S(r, f ).

(19)

Thus, from (18) and (19) we have 2N (r, f (w)) =N (r, [∆q f (qw) + 3∆q f (w) − c] f (w) − (aqw + b) + [∆q f (qw) + ∆q f (w) − c] ∆q f (w) ≤N (r, f (w)) + 5N (r, ∆q f (w)) + O(log r) + S(r, f ). That is, N (r, f (w)) ≤ 5N (r, ∆q f (w)) + S(r, f ).

(20)

Then, it follows from (15) and (20) that T (r, f (w)) ≤ 15N (r, ∆q f (w)) + S(r, f ).

(21)

Since f (z) is transcendental, that is, f (w) is transcendental, we have from (21) that ∆q f (w) has infinitely many poles, that is, ∆q f (z) has infinitely many poles. Therefore, we complete the proof of Theorem 1.4.

4

Proof of Theorem 1.5

Sppose that f (z) is a nonzero rational solution of (6), and has poles z1 , z2 , . . . , zk . Then, we let αis1 αisi , i = 1, 2, . . . , k + ··· + (z − zi )si (z − zi ) be the principal parts of f (z) at zi respectively, where αisi 6= 0, . . . , αis1 are constants, Thus, we can write f (z) as the following form f (z) =

k µ X i=1

αisi αis1 + ··· + s i (z − zi ) (z − zi )

¶ + β0 + β1 z + · · · + βm z m ,

(22)

where β0 , β1 , . . . , βm are constants.

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Next, we affirm that βm = · · · = β1 = 0. Suppose that βm 6= 0(m ≥ 1). For sufficiently large z, by (22), we have f (z) = βm z m (1 + o(1)), f (qz) = βm q m z m (1 + o(1)), z f ( ) = βm q −m z m (1 + o(1)). q By (6), we have

(23) (24) (25)

· ¸ z f (qz) + f (z) + f ( ) f (z) = az + b + cf (z). q

(26)

Substituting (23)-(25) into (26), we have 2 2m (1 + q m + q −m )βm z (1 + o(1)) = az + b + cβm z m (1 + o(1)).

Since |q| 6= 1, we have 1 + q m + q −m 6= 0. And since βm 6= 0, we can see the above equation is a contradiction for sufficiently large z. Hence we have β1 = · · · = βm = 0. (i) Suppose that a 6= 0. If β0 6= 0, then for sufficiently large z, by (23)-(25), we have z f (qz) = f (z) = f ( ) = β0 + o(1). q

(27)

Substituting (27) into (26), we conclude that (3β0 + o(1))(β0 + o(1)) = az + b + c(β0 + o(1)), which is a contradiction to the assumption that a 6= 0. Thus, β0 = 0. Then we have β0 = β1 = · · · = βm = 0. Thus, f (z) can be rewritten by (22) as f (z) =

P (z) , R(z)

(28)

where P (z) = pz k + pk−1 z k−1 + · · · + p0 , R(z) = rz t + rt−1 z t−1 + · · · + r0 ,

(29)

where p, pk−1 , . . . , p0 and r, rt−1 , . . . , r0 are constants such that pr 6= 0 and k < t. Then substituting (28) into (6), we have z z z P (qz)P (z)R(z)R( ) + P (z)2 R(qz)R( ) + P ( )P (z)R(qz)R(z) q q q z z 2 = (az + b)R(qz)R(z) R( ) + cP (z)R(qz)R(z)R( ). q q

(30)

Then since k < t, we can see that the degree of the left side of (30) does not exceed 2k + 2t, and the degree of the right side of (30) is equal to 1 + 4t by a 6= 0. Thus, we can get a contradiction. Therefore, we have that (6) has no nonzero rational solution when a 6= 0. (ii) Suppose that a = 0. If f (z) = B is a nonzero constant solution of (6), we can easily get from (6) that B satisfies 3B 2 − cB − b = 0. Now, we prove that (6) has no rational solution if a = 0 and c2 + 12b = 0. Suppose that f (z) is a nonconstant rational solution of (6). Since βm = 0(m ≥ 1), f (z) can be rewritten as the form (28), where

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P (z) and R(z) satisfy (29) with k ≤ t. Suppose that k < t. Substituting (28) into (6), we have z z z P (qz)P (z)R(z)R( ) + P (z)2 R(qz)R( ) + P ( )P (z)R(qz)R(z) q q q z z 2 = bR(qz)R(z) R( ) + cP (z)R(qz)R(z)R( ). (31) q q If k < t, then it follows from (31) that there exists only one term bR(qz)R(z)2 R( zq ) with maximal degree, which is a contradiction. Thus, we have k = t. Then, it follows by (29) and (30) that pq k z k + pk−1 q k−1 z k−1 + · · · + p0 pz z + pk−1 z k−1 + · · · + p0 + rq t z t + rt−1 q t−1 z t−1 + · · · + r0 rz t + rt−1 z t−1 + · · · + r0 pq −k z k + pk−1 q −(k−1) z k−1 + · · · + p0 rq −t z t + rt−1 q −(t−1) z t−1 + · · · + r0 b(rz t + rt−1 z t−1 + · · · + r0 ) = + c. pz k + pk−1 z k−1 + · · · + p0

+

Then it follows from (32) that as z → ∞, where B =

p r

(32)

3B 2 − cB − b = 0,

6= 0. Therefore, f (z) can be rewritten as f (z) = B +

G(z) , H(z)

(33)

where G(z) and H(z) are relatively prime polynomials and satisfy deg G(z) = µ < deg H(z) = ν, B is a constant satisfying 3B 2 − cB − b = 0. Denote G(z) = ξz µ + ξµ−1 z µ−1 + · · · + ξ0 , H(z) = ηz ν + ην−1 z ν−1 + · · · + η0 ,

(34)

where ξ, ξµ−1 , . . . , p0 and η, ην−1 , . . . , η0 are constants such that ξη 6= 0. Substituting (34) into (6) and noting 3B 2 − cB − b = 0, we have z z z (4B − c)G(z)H(qz)H(z)H( ) + BG(qz)H(z)2 H( ) + BG( )H(z)2 H(qz) q q q z z z 2 = − G(qz)G(z)H(z)H( ) − G(z) H(qz)H( ) − G( )G(z)H(z)H(qz). (35) q q q By observing the coefficients and degrees of all terms of the above equation, and combining with ν > µ, we have that the term with maximal degree of (35) is £ ¤ (4B − c) + Bq µ−ν + Bq ν−µ ξη 3 z µ+3ν . Since 3B 2 − cB − b = 0 and c2 + 12b = 0, we have B = 6c . And by |q| 6= 1, we can get that (4B − c) + Bq µ−ν + Bq ν−µ 6= 0. In fact, if (4B − c) + Bq µ−ν + Bq ν−µ = 0, i.e. c . B= µ−ν 4+q + q ν−µ Then, we have

c c = . 4 + q µ−ν + q ν−µ 6 By solving the above equation, we get |q| = 1, a contradiction. Thus, (35) is a contradiction for sufficiently large z. Therefore, if a = 0 and c2 + 12b = 0, then (6) has no nonconstant rational solution.

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Competing interests The authors declare that they have no competing interests.

Authors’ contributions All authors drafted the manuscript, read and approved the final manuscript.

Authors’ information 1

Xiu-Min Zheng, Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi, 330022, P.R. China, E-mail: [email protected]. Corresponding author. 2 Hong-Yan Xu, Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, P.R. China, E-mail: [email protected]. 3 Hua Wang, Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, P.R. China, E-mail: [email protected].

Acknowledgements This work was supported by the National Natural Science Foundation of China (11301233, 11561033), the Natural Science Foundation of Jiangxi Province in China (20151BAB201008, 20151BAB201004), and the Youth Science Foundation of Education Bureau of Jiangxi Province in China (GJJ14644, GJJ14271).

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New approximation of fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces Shin Min Kang1 , Arif Rafiq2, Faisal Ali3 and Young Chel Kwun4,∗

1

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] 2

3

Department of Mathematics and Statistics, Virtual University of Pakistan, Lahore 54000, Pakistan e-mail: [email protected]

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan e-mail: [email protected] 4

Department of Mathematics, Dong-A University, Busan 49315, Korea e-mail: [email protected] Abstract

We prove necessary and sufficient conditions for the strong convergence of the modified two-step iteration process to the fixed point of asymptotically demicontractive mappings in real Banach spaces. 2010 Mathematics Subject Classification: 47J25, 65J15. Key words and phrases: Iteration process, asymptotically demicontractive mapping, Banach space

1

Introduction

Let K be a nonempty subset of a real Banach space X and X ∗ be its dual space. We ∗ denote by J the normalized duality mapping from X into 2X defined by J(x) = {f ∗ ∈ X ∗ : hx, f ∗ i = kxk2 = kf ∗ k2 }, where h·, ·i denotes the generalized duality pairing. If X is strictly convex, then J is single-valued. In the sequel, we shall denote the single-valued duality mapping by j. Let T : K → K be a mapping. ∗

Corresponding author

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Definition 1.1. T is called a k-strictly asymptotically pseudo-contractive mapping with sequence {kn } ⊂ [1, ∞), limn→∞ kn = 1 if for all x, y ∈ K there exists j(x − y) ∈ J(x − y) and a constant k ∈ [0, 1) such that h(I − T n )x − (I − T n )y, j(x − y)i 1 1 ≥ (1 − k) k(I − T n )x − (I − T n )yk2 − (kn2 − 1) kx − yk2 2 2

(1.1)

for all n ∈ N. Definition 1.2. T is called an asymptotically demicontractive mapping with sequence {kn } ⊂ [0, ∞), limn→∞ kn = 1 if F (T ) = {x ∈ K : T x = x} 6= ∅ and for all x ∈ K and x∗ ∈ F (T ), there exists k ∈ [0, 1) and j(x − x∗ ) ∈ J(x − x∗ ) such that hx − T n x, j(x − x∗ )i ≥

1 1 (1 − k) kx − T n xk2 − (kn2 − 1) kx − x∗ k2 2 2

(1.2)

for all n ∈ N. Definition 1.3. T : K → K is called uniformly L-Lipschitizian if there exists a constant L > 0 such that kT n x − T n yk ≤ L kx − yk ,

(1.3)

for all x, y ∈ K and n ∈ N. The classes of k-strictly asymptotically pseudo-contractive and asymptotically demicontractive mappings are introduced by Liu [3]. It is easy to see that a k-strictly asymptotically pseudo-contrative mapping with a non-empty fixed point set F (T ) is asymptotically demicontractive. In Hilbert spaces, it is shown in [3] that (1.1) and (1.2) are equivalent to the following inequalities: kT n x − T n yk ≤ kn2 kx − yk2 + k k(I − T n )x − (I − T n )yk2 and kT n x − T n yk2 ≤ kn2 kx − yk2 + kx − T n xk2 , respectively. By using the modified Mann iteration method [4] introduced by Schu [7], Liu [3] proved a convergence theorem for the iterative approximation of fixed points of k-strictly asymptotically pseudo-contractive mappings and asymptotically demicontractive mappings in Hilbert spaces. Osilike [6] extended the results of Liu [3] about the iterative approximation of fixed points of k-strictly asymptotically demicontractive mappings from Hilbert spaces to much more general real q-uniformly smooth Banach spaces, 1 < q < ∞ and specifically proved the following results. 2 53

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Theorem 1.4. Let q > 1 and X be a real q-uniformly smooth Banach space. Let K be a closed convex and bounded subset of X and T : K → K a completely continuous uniformly L-Lipschitizian asymptotically demicontractive mapping with a sequence kn ⊂ P 2 [1, ∞) satisfying ∞ n=1 (kn − 1) < ∞. Let {αn } and {βn } be real sequences satisfying the conditions (i) 0 ≤ αn , βn ≤ 1, n ≥ 1; 1 −(q−2) −  for all n ≥ 1 and for some q (ii) 0 <  ≤ cq αq−1 n (1 + Lβn ) ≤ 2 q(1 − k)(1 + L)  > 0; and P (iii) ∞ n=1 βn < ∞. Then the sequence {xn } generated from an arbitrary x1 ∈ K by ( yn = (1 − βn )xn + βn T n xn , xn+1 = (1 − αn )xn + αn T n yn ,

n≥1

converges strongly to a fixed point of T . Remark 1.5. For Hilbert spaces, in Theorem 1.4, if we put q = 2, cq = 1 and βn = 0, then Theorems 1 and 2 of Liu [3] follow. Recently Chidume and Mˇaru¸ster [1] made a comprehensive and very useful survey on the main convergence properties of the modified Mann iteration method for the demicontractive mappings. The purpose of this work is to prove necessary and sufficient conditions for the strong convergence of the modified two-step iteration process to the fixed point of asymptotically demicontractive mappings in real Banach spaces. Our results extend and improve the results of Igbokwe [2], Liu [3], Moore and Nnoli [5].

2

Main results

The following results are useful: Lemma 2.1. ([8]) For all %, ς ∈ X and j(% + ς) ∈ J(% + ς), k% + ςk2 ≤ k%k2 + 2Re hς, j(% + ς)i . Lemma 2.2. ([2]) Let X be a normed space and K be a nonempty convex subset of X. Let T : K → K be uniformly L-Lipschitzian mapping and let {tn } and {βn } be the sequences in [0, 1]. For arbitrary %1 ∈ K, generate the sequence {%n } by ( %n+1 = (1 − tn )%n + tn T n ςn , ςn = (1 − βn )%n + βn T n %n ,

n ≥ 1.

Then

k%n − T %n k ≤ k%n − T n %n k + L(1 + L)2 %n−1 − T n−1 %n−1 .

(2.1)

We now prove our main results.

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Lemma 2.3. Let X be a real Banach space and K be a nonempty convex subset of X. Let T : K → K be an uniformly L−Lipschitzian asymptotically demicontractive mapping with a sequence {kn} ⊂ [1, ∞) such that limn→∞ kn = 1. For arbitrary %1 ∈ K, generate the sequence {%n } by ( %n+1 = (1 − tn )%n + tn T n ςn , (2.2) ςn = (1 − βn )%n + βn T n %n , n ≥ 1, where {tn } and {βn } are the sequences in [0, 1] satisfying P (i) ∞ n=1 tn = ∞, (ii) limn→∞ tn = 0 = limn→∞ βn . Then (a) the sequence {%n} is bounded, (b) lim inf n→∞ k%n+1 − T n %n+1 k = 0, (c) lim inf n→∞ k%n − T n %n k = 0, (d) lim inf n→∞ k%n − T %nk = 0. Proof. Since T is asymptotically demicontractive, then h% − T n %, j(% − %∗ )i ≥ and hence

1 1 (1 − k) k% − T n %k2 − (kn2 − 1) k% − %∗ k2 2 2

r

(2 k% − T n %k + (kn2 − 1) k% − %∗ k) k% − %∗ k . 1−k Therefore, by the triangle inequality, r (2 k% − T n %k + (kn2 − 1) k% − %∗ k) k% − %∗ k ∗ n ∗ k% − % k 6 kT % − % k + . 1−k n

k% − T %k ≤

(2.3)

Now we shall prove that lim inf k%n+1 − T n %n+1 k = 0. n→∞

If %n = T %n for all n > m for some m ∈ N, then (2.3) trivially holds, as we have

k%n+1 − T n %n+1 k = k%n+1 − T n T %n+1 k = %n+1 − T n+1 %n+1 =0

for all n ≥ m. Suppose now that there exists the smallest positive integer n0 such that %n0 6= T %n0 . Put a0 := kT n0 %n0 − %∗ k s
2 k%n0 − T n0 %n0 k + (kn2 0 − 1) k%n0 − %∗ k k%n0 − %∗ k + + 1. 1−k Then clearly k%n0 − %∗ k ≤ a0 .

(2.4)

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To prove that lim inf n→∞ k%n+1 − T n %n+1 k = 0, we shall assume, to the contrary, that lim inf n→∞ k%n+1 − T n %n+1 k = 2δ > 0. Then there exists n00 ∈ N such that k%n+1 − T n %n+1 k > δ for all n ≥ n00 . Also, by limn→∞ kn = 1 and (ii), we may suppose that 1 (1 − k)δ 2 tn ≤ min , 1 + 2L 24(1 + L)(1 + 2L)a20   (1 − k)δ 2 1 , , βn ≤ min 1 + L 24L(1 + L)a20 (1 − k)δ 2 kn2 − 1 ≤ 24a20 



, (2.5)

for all n ≥ n00 . We now show that the sequence {%n } is bounded. By induction we shall show that k%n − %∗ k 6 a0

(2.6)

for all n ≥ n00 . It is clear that (2.6) holds for n = n0 . Assume it is true for some n > N := max{n0 , n00 }, that is, k%n − %∗ k ≤ a0 for some n ≥ N . Then k%n − T n %n k ≤ k%n − %∗ k + kT n %n − %∗ k ≤ (1 + L) k%n − %∗ k ≤ (1 + L)a0 , kςn − %∗ k = k(1 − βn )%n + βn T n %n − %∗ k = k%n − %∗ − βn (%n − T n %n )k ≤ k%n − %∗ k + βn k%n − T n %n k ≤ a0 + (1 + L)a0 βn ≤ 2a0 , k%n − T n ςn k ≤ k%n − %∗ k + kT n ςn − %∗ k ≤ k%n − %∗ k + L kςn − %∗ k ≤ (1 + 2L)a0 , and k%n+1 − %∗ k = k(1 − tn )%n + tn T n ςn − %∗ k = k%n − %∗ − tn (%n − T n ςn )k ≤ k%n − %∗ k + tn k%n − T n ςn k

(2.7)

≤ a0 + (1 + 2L)a0 tn ≤ 2a0 . 5 56

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On the other hand, by Lemma 2.1, k%n+1 − %∗ k2 = k(1 − tn )%n + tn T n ςn − %∗ k2 = k%n − %∗ − tn (%n − T n ςn )k2 ≤ k%n − %∗ k2 − 2tn h%n − T n ςn , j(%n+1 − %∗ )i = k%n − %∗ k2 − 2tn h%n+1 − T n %n+1 , j(%n+1 − %∗ )i + 2tn hT n ςn − T n %n+1 , j(%n+1 − %∗ )i + 2tn h%n+1 − %n , j(%n+1 − %∗ )i . Since T is asymptotically demicontractive mapping, we obtain k%n+1 − %∗ k2 ≤ k%n − %∗ k2 − (1 − k)tn k%n+1 − T n %n+1 k2
+ kn2 − 1 tn k%n+1 − %∗ k2

(2.8)

+ 2(1 + L)tn k%n+1 − %n k k%n+1 − %∗ k + 2Ltn kςn − %n k k%n+1 − %∗ k .

Consider the following estimates, kςn − %n k = k(1 − βn )%n + βn T n %n − %n k = βn k%n − T n %n k ≤ (1 + L)a0 tn , and

k%n+1 − %n k = k(1 − tn )%n + tn T n ςn − %n k = tn k%n − T n ςn k ≤ (1 + 2L)a0 tn ,

so that (2.8), takes the form k%n+1 − %∗ k2 ≤ k%n − %∗ k2 − (1 − k)tn k%n+1 − T n %n+1 k2
+ kn2 − 1 tn k%n+1 − %∗ k2 + 2(1 + L)(1 + 2L)a0 t2n k%n+1 − %∗ k + 2L(1 + L)a0 tn βn k%n+1 − %∗ k . Then, by (2.5), k%n+1 − %∗ k2 ≤ k%n − %∗ k2 − (1 − k)δ 2 tn 
 + 4a20 kn2 − 1 + (1 + L)(1 + 2L)tn + L(1 + L)βn tn 1 ≤ k%n − %∗ k2 − (1 − k)δ 2 tn + (1 − k)δ 2 tn 2 and hence

1 k%n+1 − %∗ k2 6 k%n − %∗ k2 − (1 − k)δ 2 tn . 2

(2.9)

Thus k%n+1 − %∗ k ≤ k%n − %∗ k ≤ a0 and so we proved (2.6). Therefore, we proved (a). 6 57

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From (2.9) we have that for every r > N , r r X X 1 tn ≤ (k%n − %∗ k2 − k%n+1 − %∗ k2 ) (1 − k)δ 2 2 n=N

n=N

≤ k%N − %∗ k2 .

P∞ Hence we have n=1 tn < ∞, a contradiction with the condition (i). Therefore, our assumption δ > 0 was wrong. Thus lim inf k%n+1 − T n %n+1 k = 0.

(2.10)

n→∞

Therefore, we proved (b). Now according to Lemma 2.1, substituting % = u + v and ς = −v, we obtain ku + vk2 ≥ kuk2 + 2 hv, j(u)i , which is mainly due to Igbokwe [2]. By (2.2) we have k%n+1 − T n %n+1 k2 = k(1 − tn )%n + tn T n ςn − T n %n+1 k2 = k%n − T n %n − tn (%n − T n ςn ) − (T n %n+1 − T n %n )k2 .

(2.11)

Then by (2.11) we get k%n+1 − T n %n+1 k2 ≥ k%n − T n %n k2 − 2 htn (%n − T n ςn ) + (T n %n+1 − T n %n ) , j(%n − T n %n )i . Thus k%n − T n %n k2 ≤ k%n+1 − T n %n+1 k2 + 2 htn (%n − T n ςn ) + (T n %n+1 − T n %n ) , j(%n − T n %n )i ≤ k%n+1 − T n %n+1 k2

(2.12)

+ 2 ktn (%n − T n ςn ) + (T n %n+1 − T n %n )k k%n − T n %n k . Further, ktn (%n − T n ςn ) + (T n %n+1 − T n %n )k ≤ tn k%n − T n ςn k + kT n %n+1 − T n %n k ≤ (1 + 2L)a0 tn + L k%n+1 − %n k ≤ (1 + 2L)a0 tn + L(1 + 2L)a0 tn = (1 + L) (1 + 2L) a0 tn . Therefore, from (2.12), we get k%n − T n %n k2 ≤ k%n+1 − T n %n+1 k2 + 2 (1 + L)2 (1 + 2L)a20 tn .

(2.13)

From (2.13), (ii) and (b), lim inf k%n − T n %n k = 0.

(2.14)

n→∞

Thus we proved (c). At last, from (2.14) and Lemma 2.2, we obtain (d). This completes the proof. 7 58

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Theorem 2.4. Let X be a real Banach space and K be a nonempty convex subset of X. Let T : K → K be an uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence {kn} ⊂ [1, ∞) such that limn→∞ kn = 1. For arbitrary %1 ∈ K, generate the sequence {%n } by (

%n+1 = (1 − tn )%n + tn T n ςn , ςn = (1 − βn )%n + βn T n %n ,

n ≥ 1,

where {tn } and {βn } are the sequences in [0, 1] satisfying P (i) ∞ n=1 tn = ∞, (ii) limn→∞ tn = 0 = limn→∞ βn . If T is completely continuos, then {%n} converges strongly to some fixed point of T in K. Proof. From Lemma 2.3, lim inf n→∞ k%

0. Therefore, there exists a subse n − T %n k = quence {%nj } of {%n } such that limj→∞ %nj − T %nj = 0. Since {%nj } is bounded and T is completely continuous, then {T %nj } has a subsequence {T %njk }, which converges strongly. strongly. Hence {%njk } converges

Let limk→∞ %njk = p. Then limk→∞ T %njk = T p. Thus we have limk→∞ %njk − T %njk = kp − T pk = 0. Hence p ∈ F (T ). From (2.9) and Lemma 2.3 it follows that limn→∞ k%n − pk = 0. This completes the proof. Remark 2.5. 1. We generalize the results of Liu [3] from Hilbert spaces to more general Banach spaces. Moreover the boundedness assumption on the subset K is removed. P P 2. One can see that, with ∞ the condition ∞ t2n < ∞ is not always n=1 tn = ∞, n=1 P P ∞ 2 true. Let us take tn = √1n . Then obviously ∞ n=1 tn = ∞, but n=1 tn = ∞. Hence the results of Igbokwe [2] are need to be improve. 3. We improve the results of Moore and Nnoli [5] by removing the conditions like lim inf n→∞ d(%n , F (T )) = 0.

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

References [1] C. E. Chidume and S¸. Mˇaru¸ster, Iterative methods for the computation of fixed points of demicontractive mappings, J. Comput. Appl. Math., 234 (2010), 861–882. [2] D. I. Igbokwe, Approximation of fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces, J. Inequal. Pure Appl. Math., 3 (2002), Article 3, 11 pages. [3] Q. H. Liu, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal., 26 (1996), 1835–1842. 8 59

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[4] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. [5] C. Moore and B. V. C. Nnoli, Iterative sequence for asymptotically demicontractive maps in Banach spaces, J. Math. Anal. Appl., 302 (2005), 557–562. [6] M. O. Osilike, Iterative approximations of fixed points of asymptotically demicontractive mappings, Indian J. Pure Appl. Math., 29 (1998), 1291–1300. [7] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158 (1991), 407–413. [8] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127–1138.

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VISCOSITY APPROXIMATION OF SOLUTIONS OF FIXED POINT AND VARIATIONAL INCLUSION PROBLEMS B. A. BIN DEHAISH1 , H. O. BAKODAH1 , A. LATIF2 , X. QIN3,∗ 1

Department of mathematics, Faculty of Science-AL Faisaliah Campus, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia 3 Department of Mathematics, Wuhan University of Technology, Wuhan, China Abstract. In this paper, fixed point and variational inclusion problems are investigated based on a viscosity approximation method. Strong convergence theorems are established without the aid of metric projections in the framework of Hilbert spaces. Keywords: maximal monotone operator; fixed point; proximal point algorithm; zero point. 2010 AMS Subject Classification: 47H05, 65D15, 90C33.

1. Introduction A very common problem in diverse areas of mathematics and physical sciences consist of finding a solution which satisfies certain constraints. This problem is referred to as the convex feasibility problem. It can be described as follows: Suppose C1 , C2 , · · · , Cr , where r is some positive integer, are finitely many nonempty convex closed subset of a Hilbert space H with C = ∩ri=1 6= ∅. The convex feasibility problem is to find a point in C. In the real world, many important problems have reformulations which require finding fixed points of some nonlinear operators, for instance, evolution equations, complementarity problems, mini-max problems, variational inequalities and zero point problems; see [1-13] and the references therein. In this paper, we are concerned with the problem of finding a common solution of fixed point and inclusion problems. Many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as this problem. One of the most popular methods for solving inclusion problems goes back to the work of Browder [14]. The basic ideas is to reduce inclusion problems to fixed point problems of nonlinear operators. In this paper, we study a regularization method for two monotone and a nonexpansive mappings. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a viscosity approximation method is introduced. A strong convergence theorem of common solutions is established. In Section 4, applications of the main results are discussed. 2. Preliminaries In what follows, we always assume that H is a real Hilbert space with inner product h·, ·i and norm k · k. Let C be a nonempty, convex and closed subset of H. Let S : C → C be a mapping. F ix(S) stands for the fixed point set of S; that is, F ix(S) := {x ∈ C : x = Sx}. Recall that S is said to be κ-contractive iff there exists a constant κ ∈ (0, 1) such that ∗Corresponding author. E-mail address: [email protected] 1

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2

B. A. BIN DEHAISH, H. O. BAKODAH, A. LATIF, X. QIN

kSx − Syk ≤ κkx − yk, ∀x, y ∈ C. It is well known that every contractive mapping has a unique fixed point in metric spaces. The Picard iterative algorithm xn+1 = Sxn converge to the fixed point of S. S is said to be nonexpansive iff kSx − Syk ≤ kx − yk, ∀x, y ∈ C. If C is a bounded, closed, and convex subset of H, then F (S) is not empty; see [15] and the references therein. Since the nonexpansivity of S, the Picard iterative algorithm may not converge to fixed points of S. The Mann iterative algorithm is powerful and efficient to study fixed points of nonexpansive mappings. However, in infinite dimensional spaces, the Mann iterative algorithm is only weak convergence. To obtain strong convergence of the Mann iterative algorithm, different regularization methods have been investigated recently; see [16]-[29] and the references therein. Let A : C → H be a mapping. Recall that A is said to be monotone iff hAx−Ay, x−yi ≥ 0, ∀x, y ∈ C. Recall that A is said to be inverse-strongly monotone iff there exists a constant α > 0 such that hAx − Ay, x − yi ≥ αkAx − Ayk2 , ∀x, y ∈ C. For such a case, A is also said to be α-inverse-strongly monotone. It is not hard to see that every inverse-strongly monotone mapping is monotone and continuous. Recall that a set-valued mapping B : H ⇒ H is said to be monotone iff, for all x, y ∈ H, f ∈ Bx and g ∈ By imply hx − y, f − gi ≥ 0. In this paper, we use B −1 (0) to stand for the zero point of B. A monotone mapping B : H ⇒ H is maximal iff the graph Graph(B) of B is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping B is maximal if and only if, for any (x, f ) ∈ H × H, hx − y, f − gi ≥ 0, for all (y, g) ∈ Graph(B) implies f ∈ Bx. For a maximal monotone operator B on H, and r > 0, we may define the single-valued resolvent Jr : H → Dom(B), where Dom(B) denote the domain of B. It is known that Jr is firmly nonexpansive, and B −1 (0) = F (Jr ). In this paper, we study fixed points of nonexpansive mappings and zero points of two monotone mappings based on a viscosity approximation method. Strong convergence theorems are established in the framework of Hilbert spaces. The results obtained in this paper mainly improve the corresponding results in [23]-[29]. In order to prove our main results, we also need the following lemmas. Lemma 2.1 [30] Let {an } be a sequence of nonnegative numbers satisfying the condition an+1 ≤ (1 − tn )an +P tn bn + cn , ∀n ≥ 0, where {tn } is a number sequence in (0, 1) such that limn→∞ tn = 0 and ∞ sequence such that lim supn→∞ bn ≤ 0, n=0 tn = ∞, {bn } is a numberP and {cn } is a positive number sequence such that ∞ n=0 cn < ∞. Then limn→∞ an = 0. Lemma 2.2. [31] Let C be a nonempty convex closed subset of a real Hilbert space H. Let A : C → H be an α-inverse-strongly monotone mapping and let B be a maximal monotone operator on H. Then (A + B)−1 (0) = F (Jr (I − rA)). Lemma 2.3. [32] Let H be a Hilbert space, and A an  maximal monotone operator. For µ λ > 0, µ > 0, and x ∈ E, we have Jλ x = Jµ 1 − λ Jλ x + µλ x , where Jλ = (I + λA)−1 and Jµ = (I + µA)−1 . Lemma 2.4. [14] Let C be a nonempty convex closed subset of a real Hilbert space H. Let T be a nonexpansive mapping on C. Then I − T is demiclosed at origin. 3. Main results Theorem 3.1. Let C be a nonempty convex closed subset of a real Hilbert space H. Let A : C → H be an α-inverse-strongly monotone mapping and let B be a maximal

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SOLUTIONS OF FIXED POINT AND VARIATIONAL INCLUSION PROBLEMS

3

monotone operator on H. Let S be a fixed κ-contraction and let T be a nonexpansive mapping on C. Assume Dom(B) ⊂ C and (A + B)−1 (0) ∩ F ix(T ) 6= ∅. Let {αn } be a real number sequence in (0, 1) and let {rn } be a positive real number sequence in (0, 2α). Let {xn } be a sequence in C in the following process: x0 ∈ C, yn = αn Sxn + (1 − αn )T xn , xn+1 ≈ (I +rn B)−1 (yn −rn Ayn ), ∀n ≥ 0. Let the criterion for the approximate computation P∞ −1 of xn+1 be kxn+1 − (I + rn B) (yn − rn Ayn )k ≤ en , where n=1 en < Assume that the P∞. ∞ control sequences P {αn } and {rn } satisfy the following restrictions: |r n=1 n − rn−1 | < ∞, P∞ ∞ 0 limn→∞ αn = 0, n=0 αn = ∞, n=1 |αn − αn−1 | < ∞, and 0 < r ≤ rn ≤ r < 2α, 0 where r and r are two real numbers. Then {xn } converges strongly to a point x¯ ∈ (A + B)−1 (0) ∩ F ix(T ), where x¯ = P roj(A+B)−1 (0)∩F ix(T ) S x¯. Proof. First, we show that {xn } and {yn } are bounded sequences. Using the restrictions imposed on {rn }, one see that I − rn A is nonexpansive. Indeed, we have k(I − rn A)x − (I − rn A)yk2 ≤ kx − yk2 − rn (2α − rn )kAx − Ayk2 ≤ kx − yk2 . That is, k(I − rn A)x − (I − rn A)yk ≤ kx − yk. Fixing p ∈ (A + B)−1 (0) ∩ F ix(T ), we find that kyn − pk ≤ αn kSxn − pk + (1 − αn )kT xn − pk ≤ αn kSxn − pk + (1 − αn )kxn − pk
≤ 1 − αn (1 − κ) kxn − pk + αn kSp − pk. Hence, we have kxn+1 − pk ≤ ken k + k(I + rn B)−1 (yn − rn Ayn ) − pk ≤ en + k(yn − rn Ayn ) − (I − rn A)pk
kSp − pk ≤ en + 1 − αn (1 − κ) kxn − pk + αn (1 − κ) 1−κ kSp − pk } + en ≤ max{kxn − pk, 1−κ .. . ∞

≤ max{kx0 − pk,

X kSp − pk }+ ei < ∞. 1−κ i=0

This proves that the sequence {xn } is bounded, so is {yn }. Notice that
kyn − yn−1 k ≤ 1 − αn (1 − κ) kxn − xn−1 k + |αn − αn−1 |kSxn−1 − xn−1 k. Setting zn = yn − rn Ayn , one further has kzn − zn−1 k ≤ kyn − yn−1 k + krn − rn−1 kkAyn−1 k
≤ 1 − αn (1 − κ) kxn − xn−1 k + |rn − rn−1 |kAyn−1 k

(3.1)

+ |αn − αn−1 |kSxn−1 − xn−1 k.

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Putting Jrn = (I + rn B)−1 , it follows from Lemma 2.3 that kxn+1 − xn k
rn−1 rn−1 zn + (1 − )Jrn zn k rn rn rn−1 rn−1 ≤ en + en−1 + k(1 − )(Jrn zn − zn−1 ) + (zn − zn−1 )k rn rn |rn − rn−1 | ≤ en + en−1 + kzn − Jrn zn k + kzn − zn−1 k, rn ≤ en + en−1 + kJrn−1 zn−1 − Jrn−1

which implies from (3.1) that kxn+1 − xn k
|rn − rn−1 | kzn − Jrn zn k + 1 − αn (1 − κ) kxn − xn−1 k rn + |rn − rn−1 |kAyn−1 k + |αn − αn−1 |kSxn−1 − xn−1 k
≤ 1 − αn (1 − κ) kxn − xn−1 k + en + en−1  kJrn zn − zn k  + |αn − αn−1 |kSxn−1 − xn−1 k. + |rn − rn−1 | kAyn−1 k + rn

≤ en + en−1 +

From the restrictions imposed on the control sequences, we have ∞    X kJrn zn − zn k  en + en−1 + |rn − rn−1 | kAyn−1 k + + |αn − αn−1 |kSxn−1 − xn−1 k < ∞. rn n=1 Using Lemma 2.1, we find limn→∞ kxn+1 − xn k = 0. Since k · k2 is convex, we have kyn − pk2 ≤ αn kSxn − pk2 + (1 − αn )kxn − pk2 , from which it follows that kxn+1 − pk2 ≤ k(yn − rn Ayn ) − (p − rn Ap)k2 + 2en k(I + rn B)−1 (yn − rn Ayn ) − pk + e2n ≤ kyn − pk2 − rn (2α − rn )kAyn − Apk2 + 2en k(I + rn B)−1 (yn − rn Ayn ) − pk + e2n ≤ αn kSxn − pk2 + (1 − αn )kxn − pk2 − rn (2α − rn )kAyn − Apk2 + 2en k(I + rn B)−1 (yn − rn Ayn ) − pk + e2n . This implies that rn (2α − rn )kAyn − Apk2 ≤ αn kSxn − pk2 + kxn − pk2 − kxn+1 − pk2 + 2en k(I + rn B)−1 (yn − rn Ayn ) − pk + e2n . Hence, we have lim kAyn − Apk = 0.

n→∞

(3.2)

Put λn = (I + rn B)−1 (yn − rn Ayn ). Since (I + rn B)−1 is firmly nonexpansive, one has kλn − pk2 ≤ h(yn − rn Axn ) − (p − rn Ap), λn − pi
1 ≤ kyn − pk2 + kλn − pk2 − kyn − λn − rn (Ayn − Ap)k2 2
1 ≤ kyn − pk2 + kλn − pk2 − kyn − λn k2 + 2rn kλn − yn kkAyn − Apk . 2

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It follows that kxn+1 − pk2 ≤ e2n + αn kSxn − pk2 + kxn − pk2 − kyn − λn k2 + 2rn kλn − yn kkAyn − Apk + 2en kλn − pk. Hence, we have kyn − λn k2 ≤ e2n + αn kSxn − pk2 + kxn − pk2 − kxn+1 − pk2 + 2rn kλn − yn kkAyn − Apk + 2en kλn − pk. Using the restrictions imposed on the control sequences and (3.2), we arrive at lim kyn − λn k = 0.

(3.3)

n→∞

Note that kxn − T xn k ≤ kxn − xn+1 k + kλn − yn k + kyn − T xn k + en . This finds from (3.3) limn→∞ kxn − T xn k = 0. Next, we show that lim suphS x¯ − x¯, yn − x¯i ≤ 0, (3.4) n→∞

where x¯ is the unique fixed point of the mapping P roj(A+B)−1 (0)∩F ix(T ) S. To show this inequality, we choose a subsequence {yni } of {yn } such that lim supn→∞ hS x¯ − x¯, yn − x¯i = limi→∞ hS x¯ − x¯, yni − x¯i ≤ 0, Since {yni } is bounded, there exists a subsequence {ynij } of {yni } which converges weakly to xˆ. Without loss of generality, we assume that yni * xˆ. Since kxn − yn k ≤ kxn − xn+1 k + kλn − yn k + en , one has xni * xˆ. Using Lemma 2.4, n Ayn one has xˆ ∈ F ix(T ). Since yn − rn Ayn ∈ λn + rn Bλn , that is, yn −λnr−r ∈ Bλn . Let n yn −λn µ ∈ Bν. Since B is monotone, we find that h rn − µ − Ayn , λn − νi ≥ 0. Hence, one has 0 ≤ h−Aˆ x − µ, xˆ − νi. This implies that −Aˆ x ∈ B xˆ, that is, xˆ ∈ (A + B)−1 (0). This shows (3.4) holds. Notice that kyn − x¯k2 ≤ αn hSxn − S x¯, yn − x¯i + αn hS x¯ − x¯, yn − x¯i + (1 − αn )kT xn − pkkyn − x¯k
≤ 1 − αn (1 − κ) kxn − x¯kkyn − x¯k + αn hS x¯ − x¯, yn − x¯i. It follows that kyn − x¯k2 ≤ (1 − αn (1 − κ))kxn − x¯k2 + 2αn hS x¯ − x¯, yn − x¯i. Hence, we have kxn+1 − x¯k2 ≤ k(yn − rn Ayn ) − (I − rn A)¯ xk2 + 2en kλn − x¯k + e2n ≤ (1 − αn (1 − κ))kxn − x¯k2 + 2αn hS x¯ − x¯, yn − x¯i + 2en kλn − x¯k + e2n . An application of Lemma 2.1 to the above inequality yields that limn→∞ kxn − x¯k = 0. This completes the proof. 4. Applications Let C be a nonempty closed and convex subset of a Hilbert space H. Let iC be the indicator function of C, that is, iC (x) = ∞, x ∈ / C, iC (x) = 0, x ∈ C. Since iC is a proper lower and semicontinuous convex function on H, the subdifferential ∂iC of iC is maximal monotone. So, we can define the resolvent Jr of ∂iC for r > 0, i.e., Jr := (I + r∂iC )−1 . Letting x = Jr y, we find that y ∈ x + r∂iC x ⇐⇒ y ∈ x + rNC x ⇐⇒ x = P rojC y, where P rojC is the metric projection from H onto C and NC x := {e ∈ H : he, v − xi, ∀v ∈ C}.

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Theorem 4.1. Let C be a nonempty convex closed subset of a real Hilbert space H. Let A : C → H be an α-inverse-strongly monotone mapping and let T : C → C be a nonexpansive mapping. Assume that V I(C, A) ∩ F ix(T ) is not empty. Let S : C → C be a fixed κ-contraction. Let {xn } be a sequence in C in the following process: x0 ∈ C, yn = αn Sxn + (1 − αn )T xn , xn+1 ≈ P rojC (yn − rn Ayn ), ∀n ≥ 0. Let the criterion P∞for the approximate computation of xn+1 be kxn+1 − P rojC (yn − rn Ayn )k ≤ en , where n=1 en < ∞. Assume that P the control sequences P∞ {αn } and {rn } satisfy P∞the following restrictions: ∞ limn→∞ αn = 0, n=0 αn = ∞, n=1 |αn − αn−1 | < ∞, n=1 |rn − rn−1 | < ∞, and 0 < r ≤ rn ≤ r0 < 2α, where r and r0 are two real numbers. Then {xn } converges strongly to a point x¯ ∈ V I(C, A) ∩ F ix(T ), where x¯ = P rojV I(C,A)∩F ix(T ) S x¯. Proof. Putting B = ∂iC in Theorem 3.1, we find that Jrn = P rojC . This finds from Theorem 3.1 the desired conclusion immediately. Next, we consider the problem of finding a solution of a Ky Fan inequality [7], which is known as an equilibrium problem in the terminology of Blum and Oettli; see [33] and the references therein. Let B be a bifunction of C × C into R, where R denotes the set of real numbers. Recall the following equilibrium problem: Find x ∈ C such that B(x, y) ≥ 0,

∀y ∈ C.

(4.1)

To study equilibrium problem (4.1), we may assume that B satisfies the following restrictions: (R-a) (R-b) (R-c) (R-d)

B(y, x) + B(x, y) ≤ 0, ∀x, y ∈ C; B(x, x) = 0, ∀x ∈ C; B(x, y) ≥ lim supt↓0 B(tz + (1 − t)x, y), ∀x, y, z ∈ C, y 7→ B(x, y), ∀x ∈ C, is lower semi-continuous and convex.

The following lemmas can be found in [22] and [33]. Lemma 4.2. Let C be a nonempty convex closed subset of a real Hilbert space H. Let B : C × C → R be a bifunction with (R-a), (R-b), (R-c) and (R-d). Then, for any r > 0 and x ∈ H, there exists z ∈ C such that rB(z, y) + hy − z, z − xi ≥ 0, ∀y ∈ C. Further, define n o Tr x = z ∈ C : rB(z, y) + hy − z, z − xi ≥ 0, ∀y ∈ C (4.2) for all r > 0 and x ∈ H. Then Tr is single-valued and firmly nonexpansive and F (Tr ) = EP (F ) is closed convex. Lemma 4.3. Let C be a nonempty convex closed subset of a real Hilbert space H. Let B be a bifunction from C × C to R with (R-a), (R-b), (R-c) and (R-d). Let AB be a multivalued mapping of H into itself defined by ( {z ∈ H : hy − x, zi ≤ B(x, y), ∀y ∈ C}, x ∈ C, AB x = (4.3) ∅, x∈ / C. Then AB is a maximal monotone operator with domain D(AB ) ⊂ C, EP (B) = A−1 B (0), where F P (B) stands for the solution set of (4.1), and Tr x = (I +rAB )−1 x, ∀x ∈ H, r > 0, where Tr is defined as in (4.2).

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Theorem 4.4. Let C be a nonempty convex closed subset of a real Hilbert space H. Let B : C × C → R be a bifunction with (R-a), (R-b), (R-c) and (R-d). Let T : C → C be a nonexpansive mapping. Assume that EP (B) ∩ F ix(T ) is not empty. Let S : C → C be a fixed κ-contraction and let Trn = (I+rn AB )−1 . Let {xn } be a sequence in C in the following process: x0 ∈ C and xn+1 ≈ Trn (αn Sxn + (1 − αn )T xn ), ∀n ≥ 0, Let the criterion for the P∞approximate computation of xn+1 be kxn+1 − Trn (αn Sxn + (1 − αn )T xn )k ≤ en , where the control sequences n=1 en < ∞. Assume that P P∞ {αn } and {rn } satisfy P∞ the following ∞ restrictions: limn→∞ αn = 0, n=0 αn = ∞, n=1 |αn − αn−1 | < ∞, n=1 |rn − rn−1 | < ∞, and 0 < r ≤ rn ≤ r0 < 2α, where r and r0 are two real numbers. Then {xn } converges strongly to a point x¯ ∈ EP (B) ∩ F ix(T ), where x¯ = P rojEP (B)∩F ix(T ) S x¯. Proof. Putting A = 0 in Theorem 3.1, we find that Jrn = Trn . From Theorem 3.1, we draw the desired conclusion immediately. Recall that a mapping T : C → T is said to be α-strictly pseudocontractive iff there exits a constant α ∈ [0, 1) such that kT x − T yk2 ≤ αk(I − T )x − (I − T )yk2 + kx − yk2 ,

∀x, y ∈ C.

The class of strictly pseudocontractive mappings was first introduced by Browder and Petryshyn [28]. It is known if T is α-strictly pseudocontractive, then I − T is 1−α -inverse 2 strongly monotone. Finally, we consider the problem of common fixed point problems of nonlinear mappings. Theorem 4.5. Let C be a nonempty convex closed subset of a real Hilbert space H. Let T1 be a nonexpansive mapping and let T2 be a α-strictly pseudocontractive mapping on C. Let S be a fixed κ-contraction on C. Let {xn } be a sequence generated in the following manner: x0 ∈ C, yn = αn Sxn + (1 − αn )T1 xn , xn+1 ≈ (1 − rn )yn + rn T2 yn , ∀n ≥ 0, Let the criterion for P∞the approximate computation of xn+1 be kxn+1 − (1 − rn )yn − rn T2 yn k ≤ en , where the control sequences n=1 en < ∞. Assume that P P∞ the following P∞ {αn } and {rn } satisfy ∞ restrictions: limn→∞ αn = 0, n=0 αn = ∞, n=1 |αn − αn−1 | < ∞, n=1 |rn − rn−1 | < ∞, and 0 < r ≤ rn ≤ r0 < 1 − α, where r and r0 are two real numbers. Then {xn } converges strongly to a point x¯ ∈ F ix(T1 ) ∩ F ix(T2 ), where x¯ = P rojF ix(T1 )∩F ix(T2 ) S x¯. Proof. Putting A = I − T2 , we find A is 1−α -inverse strongly monotone. We also have 2 V I(C, A) = F ix(T2 ) and rn T2 yn + (1 − rn )yn = P rojC (yn − rn Ayn ). In view of Theorem 3.1, we obtain the desired result immediately. Acknowledgements This article was supported by the National Natural Science Foundation of China under grant No.11401152. References [1] D.H. Peaceman, H.H. Rachford, The numerical solutions of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math. 3 (1955), 28-41. [2] S.Y. Cho, X. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl. 2014 (2014), Article ID 94. [3] Z.M. Wang, X. Zhang, Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal. 2014 (2014), Article ID 15.

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[4] B.A.B. Dehaish, et al., Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal. 16 (2015), 1321-1336. [5] C. Wu, Strong convergence theorems for common solutions of variational inequality and fixed point problems, Adv. Fixed Point Theory 4 (2014), 229-244. [6] S.Y. Cho, Generalized mixed equilibrium and fixed point problems in a Banach space, J. Nonlinear Sci. Appl. 9 (2016), 1083-1092. [7] K. Fan, A minimax inequality and applications. In Shisha, O. (eds.): Inequality III, Academic Press, New york (1972). [8] X. Qin, S.Y. Cho, L. Wang, Convergence of splitting algorithms for the sum of two accretive operators with applications, Fixed Point Theory Appl. 2014 (2014), Article ID 75. [9] M. Zhang, S.Y. Cho, A monotone projection algorithm for solving fixed points of nonlinear mappings and equilibrium problems, J. Nonlinear Sci. Appl. 9 (2016), 1453-1462. [10] J. Zhao, Strong convergence theorems for equilibrium problems, fixed point problems of asymptotically nonexpansive mappings and a general system of variational inequalities, Nonlinear Funct. Appl. Appl. 16 (2011), 447-464. [11] X. Qin, S.Y. Cho, L. Wang, Iterative algorithms with errors for zero points of m-accretive operators, Fixed Point Theory Appl. 2013 (2013), Article ID 148. [12] X. Qin, S.Y. Cho, J.K. Kim, On the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings, Optimization 61 (2012), 805-821. [13] J.K. Kim, Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-φ-nonexpansive mappings, Fixed Point Theory Appl. 2011 (2011), Article ID 10. [14] F.E. Browder, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Natl. Acad. Sci. USA 56 (1966), 1080-1086. [15] F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure Math. 18 (1976), 78-81. [16] X. Qin, Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Aappl. 329 (2007), 415-424. [17] J.K. Kim, Convergence theorems of iterative sequences for generalized equilibrium problems involving strictly pseudocontractive mappings in Hilbert spaces, J. Comput. Anal. Appl. 18 (2015), 454-471. [18] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000), 46-55. [19] Y. Su, X. Qin, Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators, Nonlinear Anal. 68 (2008), 3657-3664. [20] H. Zegeye, N. Shahzad, A hybrid approximation method for equilibrium, variational inequality and fixed point problems, Nonlinear Anal. 4 (2010), 619-630. [21] Y. Hecai, On weak convergence of an iterative algorithm for common solutions of inclusion problems and fixed point problems in Hilbert spaces, Fixed Point Theory Appl. 2013 (2013), 155. [22] S. Takahashi, W. Takahashi, M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl. 147 (2010), 27-41. [23] H.K. Xu, A regularization method for the proximal point algorithm, J. Global Optim. 36 (2006), 115-125. [24] S. Kamimura, W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory 106 (2000), 226-240. [25] Y. Hao, Zero Theorems of accretive operators, Bull. Malaysian Math. Sci. Soc. 34 (2011) 103-112. [26] Y. Qing, S.Y. Cho, Proximal point algorithms for zero points of nonlinear operators, Fixed Point Theory Appl. 2014 (2014), 42. [27] B.A.B. Dehaish, A. Latif, H.O. Bakodah, X. Qin, A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces, J. Inequal. Appl. 2015 (2015), Article ID 51. [28] H. Iiduka, W. Takahashi,Strong convergence theorems for nonexpansive mappings and inversestrongly monotone mappings, Nonlinear Anal. 61 (2005), 341-350. [29] Y. Qing, On nonexpansive mappings and an inverse-strongly monotone mapping in Hilbert spaces, J. Nonlinear Funct. Anal. 2015 (2015), Article ID 1.

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[30] Z. Xue, H. Zhou, Y.J. Cho, Iterative solutions of nonlinear equations for m-accretive operators in Banach spaces, J. Nonlinear Convex Anal. 1 (2000), 313-320. [31] B.A. Bin Dehaish, A. Latif, H.O. Bakodah, X. Qin, A viscosity splitting algorithm for solving inclusion and equilibrium problems, J. Inequal. Appl. 2015 (2015), 50. [32] V. Barbu, Nonlinear Semigroups and differential equations in Banach space, Noordhoff, 1976. [33] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems. Math. Stud, 63 (1994), 123-145. [34] F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228.

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ON THE STABILITY OF ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES CHOONKIL PARK Abstract. In this paper, we solve the following additive ρ-functional inequalities       t x+y − f (x) − f (y) , t ≥ (0.1) N f (x + y) − f (x) − f (y) − ρ 2f 2 t + ϕ(x, y) and     x+y t N 2f − f (x) − f (y) − ρ (f (x + y) − f (x) − f (y)) , t ≥ (0.2) 2 t + ϕ(x, y) in fuzzy normed spaces, where ρ is a fixed real number with ρ 6= 1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in fuzzy Banach spaces.

1. Introduction and preliminaries Katsaras [10] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [6, 12, 27]. 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 [11]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. We use the definition of fuzzy normed spaces given in [2, 16, 17] to investigate the Hyers-Ulam stability of additive ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [2, 16, 17, 18] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [15, 16]. Definition 1.2. [2, 16, 17, 18] 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. 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; additive ρ-functional inequality; Hyers-Ulam stability. Email: [email protected]; Phone: +82-2-2220-0892; Fax: +82-2-2281-0019.

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Definition 1.3. [2, 16, 17, 18] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. 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]). The stability problem of functional equations originated from a question of Ulam [26] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [8] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [24] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [7] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. approach.   Rassias’ x+y 1 The functional equation f 2 = 2 f (x)+ 12 f (y) is called the Jensen equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [4, 9, 13, 14, 19, 22, 23, 25]). Park [20, 21] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. In Section 2, we solve the additive ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in fuzzy Banach spaces by using the direct method. In Section 3, we solve the additive ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.2) in fuzzy Banach spaces by using the direct method. Throughout this paper, assume that X is a real vector space and (Y, N ) is a fuzzy Banach space. 2. Additive ρ-functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in fuzzy Banach spaces. Let ρ be a real number with ρ 6= 1. We need the following lemma to prove the main results. Lemma 2.1. Let f : X → Y be a mapping satisfying     x+y f (x + y) − f (x) − f (y) = ρ 2f − f (x) − f (y) 2 for all x, y ∈ X. Then f : X → Y is additive.

(2.1)

Proof. Letting x = y = 0 in (2.1), we get −f (0) = 0 and so f (0) = 0. Replacing y by x in (2.1), we get f (2x) − 2f (x) = 0 and so f (2x) = 2f (x) for all x ∈ X. Thus     x+y f (x + y) − f (x) − f (y) = ρ 2f − f (x) − f (y) = ρ(f (x + y) − f (x) − f (y)) 2 and so f (x + y) = f (x) + f (y) for all x, y ∈ X. 

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Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that Φ(x, y) :=

∞ X

2j ϕ



j=1

x y , 2j 2j



0. Then A(x) := N -limn→∞ 2n f defines an additive mapping A : X → Y such that

x 2n




t t + ϕ(x, y)

(2.3)

exists for each x ∈ X and

t

N (f (x) − A(x), t) ≥

t+

(2.4)

1 2 Φ(x, x)

for all x ∈ X and all t > 0. Proof. Letting y = x in (2.3), we get t t + ϕ(x, x)

N (f (2x) − 2f (x), t) ≥

(2.5)

and so 

t x
,t ≥ 2 t + ϕ x2 , x2

 



x



N f (x) − 2f for all x ∈ X. Hence 

l



N 2f

x 2l







x ,t 2m



x 2l



x 2l



m

−2 f 

l

≥ min N 2 f 

 

= min N f ≥ min

   t + 2l  





−2

(2.6)

l+1



− 2f







m−1



x



m



x ,t 2m 



,t ,··· ,N 2 f −2 f 2l+1 2m−1        x t x x t , l ,··· ,N f − 2f , m−1 2l+1 2 2m−1 2m 2 f



t l 2 , · · · x x ϕ 2l+1 , 2l+1

 

t

,

2m−1 t 2m−1

+ϕ

x x 2m , 2m




   t t  , · · · ,
= min  t + 2l ϕ x , x t + 2m−1 ϕ 2xm , 2xm  2l+1 2l+1

t

≥ t+

1 2

Pm

j j=l+1 2 ϕ



x x , 2j 2j



for all nonnegative integers m and l with m > l and all x ∈ X and all t > 0. It follows from (2.2) and (2.6) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by A(x) := N - lim 2n f ( n→∞

x ) 2n

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.4).

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By (2.3), x y x+y −f −f N 2 f n n 2 2 2n         x+y x y t n+1 n n n
−ρ 2 f −2 f −2 f ,2 t ≥ n+1 n n 2 2 2 t + ϕ 2xn , 2yn 

n



 









for all x, y ∈ X, all t > 0 and all n ∈ N. So         x+y x y n N 2 f −f −f n n 2 2 2n         x+y x y n+1 n n −ρ 2 f −2 f −2 f ,t ≥ 2n+1 2n 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞

t 2n t 2n

+ϕ


x y 2n , 2n

t t+2n ϕ( 2xn , 2yn )

=

t t + 2n ϕ


x y 2n , 2n

= 1 for all x, y ∈ X and

all t > 0, x+y A(x + y) − A(x) − A(y) = ρ 2A − A(x) − A(y) 2 for all x, y ∈ X. By Lemma 2.1, the mapping A : X → Y is Cauchy additive, as desired. 









Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying 





N f (x + y) − f (x) − f (y) − ρ 2f ≥

x+y 2







− f (x) − f (y) , t

t t + θ(kxkp + kykp )

(2.7)

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2p − 2)t (2p − 2)t + 2θkxkp

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X, as desired.  Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that Φ(x, y) :=

∞ X 1 j=0

2





ϕ 2j x, 2j y < ∞ j

for all x, y ∈ X. Let f : X → Y be a mapping satisfying (2.3). Then A(x) := N -limn→∞ exists for each x ∈ X and defines an additive mapping A : X → Y such that 1 N (f (x) − A(x), t) ≥ 1 t + 2 Φ(x, x)

1 2n f

(2n x)

for all x ∈ X and all t > 0. Proof. It follows from (2.5) that 1 1 t N f (x) − f (2x), t ≥ 2 2 t + ϕ(x, x) 



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ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES

and so 1 2t t N f (x) − f (2x), t ≥ = 2 2t + ϕ(x, x) t + 12 ϕ(x, x) 



for all x ∈ X and all t > 0. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying (2.7). Then A(x) := N limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2 − 2p )t (2 − 2p )t + 2θkxkp

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X, as desired.  3. Additive ρ-functional inequality (0.2) In this section, we prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.2) in fuzzy Banach spaces. Let ρ be a fuzzy number with ρ 6= 1. Lemma 3.1. Let f : X → Y be a mapping satisfying f (0) = 0 and 

2f

x+y 2



− f (x) − f (y) = ρ (f (x + y) − f (x) − f (y))

(3.1)

for all x, y ∈ X. Then f : X → Y is additive. Proof. Letting y = 0 in (3.1), we get 2f x2 − f (x) = 0 and so f (2x) = 2f (x) for all x ∈ X. Thus   x+y − f (x) − f (y) = ρ(f (x + y) − f (x) − f (y)) f (x + y) − f (x) − f (y) = 2f 2


and so f (x + y) = f (x) + f (y) for all x, y ∈ X.



Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that Φ(x, y) :=

∞ X j=0

2j ϕ



x y , 2j 2j



0. Then A(x) := N -limn→∞ 2n f defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

t t + Φ(x, 0)

x 2n




t (3.3) t + ϕ(x, y)

exists for each x ∈ X and (3.4)

for all x ∈ X and all t > 0.

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Proof. Letting y = 0 in (3.3), we get 

x , t = N 2f 2

 

N f (x) − 2f



x 2



 



− f (x), t ≥

t t + ϕ(x, 0)

(3.5)

for all x ∈ X. Hence 

N 2l f



x 2l



− 2m f



≥ min N 2l f







x 2l



 

x 2l



= min N f ≥ min

 

x ,t 2m 





(3.6)

− 2l+1 f 

− 2f



x







, t , · · · , N 2m−1 f



x



− 2m f



x ,t 2m 



2l+1 2m−1        x t x x t , , · · · , N f − 2f , 2l+1 2l 2m−1 2m 2m−1 

t l 2

 

t 2m−1

, · · · ,   t x  t + ϕ x ,0  + ϕ , 0 m−1 m−1 l l 2 2 2 2     t t , · · · ,    = min x  t + 2l ϕ x , 0 t + 2m−1 ϕ ,0  2m−1

2l

t

≥ t+

Pm−1 j=l

2j ϕ





x ,0 2j

for all nonnegative integers m and l with m > l and all x ∈ X and all t > 0. It follows from (3.2) and (3.6) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by A(x) := N - lim 2n f ( n→∞

x ) 2n

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.6), we get (3.4). By (3.3), x+y x y N 2 f − 2n f − 2n f n+1 n 2 2 2n          x y t x+y n n
−f −f ,2 t ≥ −ρ 2 f n n n 2 2 2 t + ϕ 2xn , 2yn 

n+1













for all x, y ∈ X, all t > 0 and all n ∈ N. So x y x+y N 2 f − 2n f − 2n f n+1 n 2 2 2n          x y x+y − ρ 2n f −f −f ,t ≥ n n 2 2 2n 

n+1













for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞

t 2n t 2n

+ϕ


x y 2n , 2n

t t+2n ϕ( 2xn , 2yn )

=

t t+

2n ϕ


x y 2n , 2n

= 1 for all x, y ∈ X and

all t > 0, x+y 2A 2 



− A(x) − A(y) = ρ (A(x + y) − A(x) − A(y))

for all x, y ∈ X. By Lemma 3.1, the mapping A : X → Y is Cauchy additive, as desired.

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ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES

Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and 

N 2f



x+y 2





− f (x) − f (y) − ρ (f (x + y) − f (x) − f (y)) , t ≥

t (3.7) t + θ(kxkp + kykp )

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2p − 2)t (2p − 2)t + 2p θkxkp

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X, as desired.  Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that Φ(x, y) :=

∞ X 1 j=1

2





ϕ 2j x, 2j y < ∞ j

for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.3). Then A(x) := N limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

t t + Φ(x, 0)

for all x ∈ X and all t > 0. Proof. It follows from (3.5) that t 1 N f (x) − f (2x), 2 2 



≥

t t + ϕ(2x, 0)

and so 1 2t t N f (x) − f (2x), t ≥ = 1 2 2t + ϕ(2x, 0) t + 2 ϕ(2x, 0) 



for all x ∈ X and all t > 0. The rest of the proof is similar to the proof of Theorem 3.2.



Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with the norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.7). Then A(x) := N -limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2 − 2p )t (2 − 2p )t + 2p θkxkp

for all x ∈ X. Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X, as desired. 

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C. PARK

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] I. Chang and Y. Lee, Additive and quadratic type functional equation and its fuzzy stability, Results Math. 63 (2013), 717–730. [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] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [7] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [8] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [9] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [10] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [11] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [12] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [13] G. Lu, Y. Wang and P. Ye n-Jordan ∗-derivations on induced fuzzy C ∗ -algebras, J. Comput. Anal. Appl. 20 (2016), 266–276. [14] D. Mihet and R. Saadati, On the stability of the additive Cauchy functional equation in random normed spaces, Appl. Math. Lett. 24 (2011), 2005–2009. [15] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [16] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [17] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. [18] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791–3798. [19] E. Movahednia, S. M. S. M. Mosadegh, C. Park and D. Shin, Stability of a lattice preserving functional equation on Riesz space: fixed point alternative, J. Comput. Anal. Appl. 21 (2016), 83–89. [20] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [21] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [22] C. Park, Stability of ternary quadratic derivation on ternary Banach algebras: revisited, J. Comput. Anal. Appl. 20 (2016), 21–23. [23] W. Park and J. Bae, Approximate quadratic forms on restricted domains, J. Comput. Anal. Appl. 20 (2016), 388–410. [24] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [25] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [26] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [27] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected]

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On the Di¤erence equation 

+1 =  +

 P

¡

=0  Q

+

=0

¡

M. M. El-Dessoky12 and E. O. Alzahrani1 1 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected] Abstract The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the di¤erence equation  +1 =

 P

¡

=0

+

 Q



 = 0 1 

¡

=0

where the parameters   and  are positive real numbers and the initial conditions ¡  ¡+1   ¡ , 0 are nonnegative real numbers.

Keywords: di¤erence equations, stability, global stability, periodic solutions. Mathematics Subject Classi…cation: 39A10 —————————————————

1

Introduction

Di¤erence equations have always played an important role in the construction and analysis of mathematical models of biology, ecology, probability theory, genetics, number theory, physics, economic process, and so forth. 1

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The study of nonlinear rational di¤erence equations of higher order is of paramount importance, since we still know so little about such equations. Ahmed [1] investigated the global asymptotic stability and the periodic character for the rational di¤erence equation, +1 =

¡     + ¡2

 = 0 1 

=

where the parameters    1  2    are nonnegative real numbers, and   are nonnegative integers such that  ·  and the initial conditions ¡2 , ¡2+1 , , ¡1 , 0 are arbitrary nonnegative real numbers. Wang et al. [2] studied the asymptotic behavior of the solutions of the nonlinear di¤erence equation +1 =

 

 ¡

=0

+

 

=0



 = 0 1 

¡

where the initial conditions ¡  ¡+1   ¡1  0 are positive real numbers,  =  f1     1    g  1     1    are nonnegative integers, and  , ,  are arbitrary positive real numbers. Zayed et al. [3] investigated the boundedness character, the periodic character, the convergence and the global stability of positive solutions of the di¤erence equation +

+1 =

 

 ¡

=0  



 = 0 1 

  ¡

=0

where the coe¢cients    ,  and the initial conditions ¡  ¡+1   ¡1  0 are positive real numbers, while  is a positive integer number. In [4] Ibrahim et al. studied the global behavior of the di¤erence equation +1 =

¡   + ¡



 = 0 1 

=0

where the parameters    and initial conditions are non-negative real numbers, f0  1     g is a set of nonnegative even integers and  is an odd positive integer Hamza et al. [5] studied the global asymptotic stability of the di¤erence equation 

+1 =

 

¡2¡1 = ¡1  + ¡2 =0



 = 0 1 

where    are nonnegative parameters and   are nonnegative integers for   . They discussed the existence of unbounded solutions under certain conditions for  = 0 2

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In [6] El-Metwally investigated the global stability character and the oscillatory of the solutions of the following di¤erence equation 

 

¡2¡1

=

+1 = +

 

=0

 ¡2¡1

 



 = 0 1 

¡2¡1

=0

where     2 (0 1) with the initial conditions 0  ¡1   ¡2  ¡2¡1 2 (0 1). For more results in the direction of this study, see, for example, [1–27] and the papers therein. The aim of this paper to study some qualitative behavior of the positive solutions of a higher order di¤erence equation 

+1 =  +

 

¡

=0  



+

=0

(1)

 = 0 1 

¡

where the parameters    and  are positive real numbers and the initial conditions ¡  ¡+1   ¡ , 0 are nonnegative real numbers.

2

Preliminaries

Let  be some interval of real numbers and let  :  +1 !  be a continuously di¤erentiable function. Then for every set of initial conditions ¡  ¡+1   0 2  the di¤erence equation (2)

+1 =  (  ¡1  ¡ )  = 0 1  has a unique solution f g1 =¡ . De…nition 1 (Equilibrium Point) A point  2  is called an equilibrium point of the di¤erence equation (2) if  =  (   ).

That is,  =  for  ¸ 0 is a solution of the di¤erence equation (2), or equivalently,  is a …xed point of  De…nition 2 (Stability) Let  2 (0 1) be an equilibrium point of the di¤erence equation (2). Then, we have 3

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(i) The equilibrium point  of the di¤erence equation (2) is called locally stable if for every   0 there exists   0 such that for all ¡   ¡1  0 2  with j¡ ¡ j +  + j¡1 ¡ j + j0 ¡ j   we have j ¡ j  

for all  ¸ ¡

(ii) The equilibrium point  of the di¤erence equation (2) is called locally asymptotically stable if  is locally stable solution of Eq.(2) and there exists   0 such that for all ¡   ¡1  0 2  with j¡ ¡ j +  + j¡1 ¡ j + j0 ¡ j   we have lim  = 

!1

(iii) The equilibrium point  of the di¤erence equation (2) is called global attractor if for all ¡   ¡1  0 2  we have lim  = 

!1

(iv) The equilibrium point  of the di¤erence equation (2) is called globally asymptotically stable if  is locally stable, and  is also a global attractor of the di¤erence equation (2). (v) The equilibrium point  of the di¤erence equation (2) is called unstable if  is not locally stable. De…nition 3 (Periodicity) A sequence f g1 =¡ is said to be periodic with period  if + =  for all  ¸ ¡ A sequence f g1 =¡ is said to be periodic with prime period  if  is the smallest positive integer having this property. De…nition 4 The linearized equation of the di¤erence equation (2) about the equilibrium  is the linear di¤erence equation +1 =

 X  (   ) =0

¡

(3)

¡ 

Now, assume that the characteristic equation associated with (3) is () = 0  + 1 ¡1 +  + ¡1  +  = 0 where  =

(4)

 (   )  ¡ 4

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Theorem 1 [1]: Assume that  2   = 1 2   and  is non-negative integer. Then  X j j  1 =1

is a su¢cient condition for the asymptotic stability of the di¤erence equation + + 1 +¡1 +  +   = 0  = 0 1  

3

Change of variables 1 ¡  ¢ +1

By using the change of variables  = following di¤erence equation 

+1 =  +

 

=0  



¡

1+

=0

where  = numbers.

4

 

 , the equation (1) reduces to the



(5)

 = 0 1 

¡

and the initial conditions   ¡1  ¡+1 , ¡ are positive real

Local Stability of the Equilibrium Point

In this section, we study the local stability character of the equilibrium point of Eq.(5). Eq.(5) has equilibrium point and is given by 

 =  +

 

=0  

 ¡

1+

=0

or



 ¡

¡ ¢  (1 ¡ ) 1 + +1 = ( + 1)

Thus  1 = 0 is always an equilibrium point of Eq. (5). If   1 and the only positive equilibrium point  2 of Eq. (5) is given by 2 =

µ

(+1) 1¡

 1; then

1 ¶ +1 ( + 1) ¡1  1¡

Theorem 2 The equilibrium  1 of Eq. (5) is locally asymptotically stable if  + ( + 1)  1 5

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Proof: Let  : (0 1)+1 ¡! (0 1) be a continuous function de…ned by  



=0  

 (  ¡1  ¡2   ¡ ) =  +

¡

(7)



1+

¡

=0

Therefore, it follows that (  ¡1  ¡2   ¡ ) 

(  ¡1  ¡2   ¡ ) ¡1

(  ¡1  ¡2   ¡ ) ¡2



 1+

= + 

 1+

=



=0

 1+

=

 

 

=0

   (  ¡1  ¡2   ¡ ) ¡



 1+

=

 

=0

 

=0



¡ ¡ 



1+

¡ ¡ 



1+



 

=0





¡ ¡ 

1+

1+



¡

¡

=0  

=0

¡

¡

 

=0

 

=0

 

=0

 

¡ ¡ ¡1 

 

2



 

¡

=0

2



 

=2

¡

=0

¡

¡



2



2

¡1  =0

 

=1

¡

¡



 



¡





¡

=3











At  1 = 0 we have (  ¡1  ¡2   ¡ ) = +  (  ¡1  ¡2   ¡ ) (  ¡1  ¡2   ¡ ) =  = =  ¡1 ¡ and the linearized equation of Eq. (5) about  1 = 0 is the equation +1 ¡ ( + )  ¡ ¡ ¡  ¡ ¡ = 0 It follows by Theorem 1 that, Eq. (5) is asymptotically stable if and only if j + j + jj +  + jj  1 and so  +  ( + 1)  1 The proof is complete. 6

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Theorem 3 The equilibrium  1 of Eq. (5) is unstable if  + ( + 1)  1 Theorem 4 The equilibrium  2 of Eq. (5) is stable if  + (1 ¡ ) (1 ¡  ¡ )   Proof: At  2 =

³

(+1) 1¡

¡1

1 ´ +1

 we have

(+1) (+1)  (1+ 1¡ ¡1)¡(+1)( 1¡ ¡1) = + 2 ¡1)  (1+ (+1) 1¡ (+1) (+1)¡1+ ( )¡(+1)( 1¡ ) ( (+1) )(¡(+1)+1¡) =  + 1¡ =  + 1¡ (+1) 2 (+1) 2 ( 1¡ ) ( 1¡ )

=  + (1¡)(1¡¡) (+1) ( (+1) ) 1¡  (1 ¡ ) (1 ¡  ¡ )  =  = = ¡  ( + 1) = +

 ¡1

(¡¡+1¡)

and the linearized equation of Eq. (5) about  2 = ³ +1 ¡  +

(1¡)(1¡¡) (+1)

´

 ¡

(1¡)(1¡¡) ¡ (+1)

³

(+1) 1¡

¡  ¡

¡1

1 ´ +1

 is the equation

(1¡)(1¡¡) ¡ (+1)

= 0

It follows by Theorem A that, Eq.(5) is stable if and only if ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (1 ¡ ) (1 ¡  ¡ ) ¯ ¯ (1 ¡ ) (1 ¡  ¡ ) ¯ (1 ¡ ) (1 ¡  ¡ ) ¯ + ¯+¯ ¯++¯ ¯  1 ¯ ¯ ¯ ¯ ¯ ¯  ( + 1)  ( + 1)  ( + 1)

for  +   1 we get

+

(1 ¡ ) (1 ¡  ¡ )  1 

The proof is complete.

5

Existence of Boundedness Solutions

Here we look at the boundedness nature of solutions of Eq.(5). Theorem 5 Every solution of Eq.(5) is bounded if  +  ( + 1)  1 Proof: Let f g1 =0 be a solution of Eq.(5). It follows from Eq.(5) that 

0 · +1 =  +

 

=0  

¡

1+

  +  ¡

 X

¡  ( +  ( + 1)) 

=0

=0

7

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this equation is locally asymptotically stable if  +  ( + 1)  1 and converges to the equilibrium point . Therefore lim sup · ( +  ( + 1)) . !1

Hence, the solution is bounded. Theorem 6 Every solution of Eq.(5) is unbounded if   1 Proof: Let f g1 =0 be a solution of Eq.(5). Then from Eq.(5) we see that 

+1 =  +

 

=0  

¡

1+

=0

  +  ¡

 X

¡  .

=0

This equation is unbounded because   1 and lim  = 1. Then by using ratio !1

test f g1 =0 is unbounded from above.

6

Global Stability of the Equilibrium Point

In this section we study the global stability of the positive solutions of Equation (1). Theorem 7 The following statements are true (a) If  + ( + 1)  1 then the equilibrium point 1 = 0 is a global attractor of equation (1). 1 ³ ´ +1 (+1) (b) If  +   1 then the equilibrium point  2 = ¡1 is a global 1¡ attractor of equation (1). Proof. (a) From Eq. (7) we can see that the function is increasing of all arguments. Now, we can see that the function  (  ¡1   ¡ ) increasing in   ¡1  ¡+1 and ¡  Then · ¸ ( + 1)  + ¡  ( ¡ 1 ) 1 +  +1 · [ + ( + 1) ¡ ] ( ¡ 0) · ¡ (1 ¡  ¡ ( + 1))  2  0 If  + ( + 1)  1 then  (   ) satis…es the inequality [ (   ) ¡ ] ( ¡  1 )  0 for  1 = 0 According to Theorem 1.10 page 15 in [1], then 1 is a global attractor of Eq. (1). This completes the proof. 8

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(b) If  +   1 then we can see that the function (  ¡1  ¡2   ¡ ) de…ned by Eq. (7) increasing of all arguments. Suppose that ( ) is a solution of the system  = (   )

and

 = (   )

Then from Equation (1), we see that  =  +

( + 1) ( + 1) , and  =  +  +1 1+ 1 + +1

then (1 ¡ ) + (1 ¡ ) +1 = ( + 1) (1 ¡ ) + (1 ¡ )+1 = ( + 1) Subtracting this two equations, we obtain ¡ ¢ (1 ¡ )  +1 ¡ +1 = 0

under the condition  6= 1we see that  =  According to Theorem 1.15 page 18 in [1], we see that  2 is a global attractor of Equation (1).

7

Existence of Periodic Solutions

In this section we investigate the existence of periodic solutions of Eq.(5). Theorem 8 If  is even, then equation (5) has not prime period two solution. Proof: Equation (5) can be expressed that +1 =  +

 ( + ¡1 + ¡2 +  + ¡ )  1 +  ¡1 ¡2 ¡

For  = 2 is even, then   ¡2 ¡4   ¡¡2  ¡ are even and ¡1  ¡3  ¡5   ¡¡3  ¡¡1 are odd. Suppose that exists there distinct positive solutions      of Equation (5). Then  =  +

 (( + 1)  + )  (( + 1)  + ) and  =  +  +1  1+  1 + +1  

Therefore,  ¡  + +1 +1 ¡ +1 +1 =  ( + 1)  +   ¡  + +1 +1 ¡ +1 +1 =  ( + 1)  + 

(7) (8)

9

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By subtracting (8) from (7), we have (1 +  + )( ¡ ) = 0 Since  +  + 1 6= 0, then  = . This is a contradiction. Thus, the proof is completed. Theorem 9 If  is odd, then equation (5) has not prime period two solution. Proof: When  = 2 + 1 is odd, then   ¡2  ¡4   ¡¡3  ¡¡1 are even and ¡1  ¡3 ¡5   ¡¡2  ¡ are odd. First suppose that there exists distinct positive solutions      of Equation (5). Then  =  +

 (( + 1)  + ( + 1) )  1 + +1 +1

 =  +

 (( + 1)  + ( + 1) )  1 + +1  +1

and

Therefore,  ¡  +  +1 +2 ¡  +2 +1 =  ( + 1)  +  ( + 1) 

(9)

 ¡  + +1 +2 ¡ +2  +1 =  ( + 1)  +  ( + 1) 

(10)

Subtracting (10) from (9), we get ¢ ¡ ( ¡ ) ( + 1)+1  +1 + 1 +  = 0

Since +1 6= 0, then  = . This is a contradiction. Thus, the proof is completed.

8

Numerical Examples

For con…rming the results of this paper, we consider numerical examples which represent di¤erent types of solutions to Eq. (5). Example 1. The zero solution of the di¤erence equation (5) is local stability if  = 3  = 02  = 01 and the initial conditions ¡3 = 08 ¡2 = 02 ¡1 = 04 and 0 = 07 (See Fig. 1). 10

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plot of y(n+1)=A y(n)+(r (y(n)+y(n-1)+y(n-2)+y(n-3))/(1+y(n)y(n-1)y(n-2)y(n-3)))) 0.8

0.7

0.6

x(n)

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25 n

30

35

40

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Figure 1. Plot the behavior of the zero solution of equation (5). Example 2. The positive solution of the di¤erence equation (5) is local stability if  = 3  = 06  = 02 and the initial conditions ¡3 = 08 ¡2 = 02 ¡1 = 04 and 0 = 07 (See Fig. 2). plot of y(n+1)=A y(n)+(r (y(n)+y(n-1)+y(n-2)+y(n-3))/(1+y(n)y(n-1)y(n-2)y(n-3)))) 1.2 1.1 1 0.9

x(n)

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

5

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15

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25 n

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35

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Figure 2. Plot the behavior of the positive solution of equation (5). Example 3. The solution of the di¤erence equation (5) is global stability if  = 3  = 002  = 033 and the initial conditions ¡3 = 08 ¡2 = 02 ¡1 = 04 and 0 = 07 (See Fig. 3).

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plot of y(n+1)=A y(n)+(r (y(n)+y(n-1)+y(n-2)+y(n-3))/(1+y(n)y(n-1)y(n-2)y(n-3)))) 1

0.9

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0.5

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0.3

0.2

0

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Figure 3. Plot the behavior of the positive solution of equation (5). Example 4. Figure (4) shows the equation (5) is unbounded when  = 3  = 11  = 01 and the initial conditions ¡3 = 08 ¡2 = 02 ¡1 = 04 and 0 = 07. plot of y(n+1)=A y(n)+(r (y(n)+y(n-1)+y(n-2)+y(n-3))/(1+y(n)y(n-1)y(n-2)y(n-3)))) 15000

x(n)

10000

5000

0

0

10

20

30

40

50 n

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70

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Figure 4. Plot the behavior of the solution of equation (5).

Acknowledgements

This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

References [1] A. M. Ahmed, On the Dynamics of a Higher-Order Rational Di¤erence Equation, Discrete Dyn. Nat. Soc., Vol., 2011 (2011), Article ID 419789, 8 pages. [2] Chang-you Wang, Shu Wang, Wei Wang, Global asymptotic stability of equilibrium point for a family of rational di¤erence equations, Appl. Math. Lett., 24 (2011), 714-718. 12

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[3] E. Zayed,¶M. A. El-Moneam, On the rational recursive sequence +1 = µ M. E.   P P  +  ¡    ¡ , Mathematica Bohemica, 133 (2008), No. 3, 225239.

=0

=0

[4] Ibrahim Yalcinkaya, Alaa E. Hamza, Cengiz Cinar, Global Behavior of a Recursive Sequence, Selçuk J. Appl. Math., Vol., 14 (2013), No. 1, 3-10. [5] A. E. Hamza, R. Khalaf-Allah, On the recursive sequence +1 = Ã !  ¡1 Q Q  ¡2¡1   +  ¡2 , Computers and Mathematics with Applications, =0

=

56 (2008), 1726-1731.

[6] H. El-Metwally, On the Dynamics of a Higher-Order Di¤erence Equation, Discrete Dyn. Nat. Soc., Vol., 2012 (2012), Article ID 263053, 8 pages. [7] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Di¤erence Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [8] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Di¤erence Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. [9] E. A. Grove and G. Ladas, Periodicities in nonlinear di¤erence equations, Vol. 4, Chapman & Hall / CRC Press, 2005. [10] H. El-Metwally, R. Alsaedi, and E. M. Elsayed, The Dynamics of the Solutions of Some Di¤erence Equations, Discrete Dyn. Nat. Soc., Vol., 2012 (2012), Article ID 475038, 8 pages. [11] E. ¶ M. µ A. El-Moneam, ¶ On the rational recursive sequence +1 = µ M. E. Zayed and  P P  +  ¡   +   ¡ , Int. J. Math. & Math. Sci; Volume 2007, =0

=0

Article ID 23618, 12 pages, doi: 10.1155/2007/23618.

[12] A. E. Hamza, R. Khalaf-Allah, Global behavior of higher order di¤erence equation, J. Math. Stat., 3 (2007), 17-20. [13] E. M. E. Zayed, On the dynamics of the nonlinear rational di¤erence equation, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 22, (2015), 61-71. [14] E. M. E. Zayed, M. A. El-Moneam, On the Rational Recursive Sequence +1 =  +¡ ¡ +  , Bulletin of the Iranian Mathematical Society, Vol. 36 (1),  ¡¡ (2010), 103-115. 13

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[15] C. J. Schinas, G. Papaschinopoulos, and G. Stefanidou, On the Recursive Se  quence +1 =  + ¡1  , Adv. Di¤er. Equ., Vol. 2009, (2009), Article ID 327649,  11 page. [16] G. Papaschinopoulos, C. J. Schinas and G. Stefanidou, On the nonautonomous   di¤erence equation +1 =  + ¡1  , Appl. Math. Comput., 217(12), (2011),  5573-5580. [17] E. M. Elsayed and M. M. El-Dessoky, Dynamics and global behavior for a fourthorder rational di¤erence equation, Hacettepe J. Math. and Stat., 42 (2013), (5), 479–494. [18] M. A. Obaid, E. M. Elsayed and M. M. El-Dessoky, Global Attractivity and Periodic Character of Di¤erence Equation of Order Four, Discrete Dyn. Nat. Soc., Vol., 2012, (2012), Article ID 746738, 20 pages. [19] M. A. El-Moneam, On the Dynamics of the Higher Order Nonlinear Rational Di¤erence Equation, Math. Sci. Lett. 3 (2014), (2), 121-129. [20] M. M. El-Dessoky, Dynamics and Behavior of the Higher Order Rational Di¤erence equation, J. Comput. Anal. Appl., Vol., 21 (2016), (4), 743-760. [21] R. Abo-Zeid, Global behavior of a higher order di¤erence equation, Mathematica Slovaca, 64 (2014), (4), 931-940. [22] M. M. El-Dessoky, Qualitative behavior of rational di¤erence equation of big Order, Discrete Dyn. Nat. Soc., Vol., 2013 (2013), Article ID 495838, 6 pages. [23] R. Abo-Zeid, On the oscillation of a third order rational di¤erence equation, J. Egyptian Math. Soc., 23 (2015), 62–66. [24] Lin-Xia Hu, Hong-Ming Xia, Global asymptotic stability of a second order rational di¤erence equation, Appl. Math. Comput., 233 (2014), 377–382. [25] E. M. Elsayed, M. M. El-Dessoky and E. O. Alzahrani, The Form of The Solution and Dynamic of a Rational Recursive Sequence, J. Comput. Anal. Appl., Vol., 17 (2014), (1), 172-186. [26] Yuji Liua and Xingyuan Liu, The existence of periodic solutions of higher order nonlinear periodic di¤erence equations, Math. Meth. Appl. Sci. 36, (2013) 1459– 1470. [27] E. M. Elsayed and M. M. El-Dessoky, Dynamics and Behavior of a Higher Order Rational Recursive Sequence, Adv. Di¤erence Equ., 2012 (2012), 69.

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A Kind of Generalized Fuzzy Integro-differential Equations of Mixed Type and Strong Fuzzy Henstock Integrals∗ Qiang Maa , a

Ya-bin Shaob,∗ , Zi-zhong Chenc

Network Information Management Center, Northwest University for Nationalities, Lanzhou Gansu,730030, P.R. China b

School of Science, Chongqing University of Posts and Telecommunications, Nan’an 400065, Chongqing, P. R. China

c

School of Computer Science, Chongqing University of Posts and Telecommunications, Nan’an 400065, Chongqing, P. R. China

December 13, 2015

Abstract In this paper, we prove the existence theorem of solutions for a kind of discontinuous fuzzy integro-differential equation of mixed type by using the definition of the ω − ACG∗ for a fuzzy-number-valued function and a generalized controlled convergence theorem of strong fuzzy Henstock integrals. Keywords: Fuzzy number; ω − ACG∗ ; Discontinuous fuzzy Integrodifferential equation; Controlled convergence theorem; Strong fuzzy Henstock integrals.

1

INTRODUCTION

The Cauchy problems for fuzzy differential equations have been studied by several authors [11, 9, 12, 16, 17, 18] on the metric space (E n , D) of normal fuzzy convex set with the distance D given by the maximum of the Hausdorff distance between the corresponding level sets. In [16], the author has been proved the Cauchy problem has a uniqueness result if f˜ was continuous and bounded. In [11, 12], the authors presented a uniqueness result when f satisfies a Lipschitz condition. For a general reference to fuzzy differential equations, see a recent ∗ The

authors would like to thank National Natural Science Foundation of China (No.11161041, 61472056, 61262022) and the PhD Research Startup Foundation of Chongqing University of Posts and Telecommunications (No. A2014-90) and Basic and Advanced Research Projects in Chongqing (No.cstc2015jcyjA00015).

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book by Lakshmikantham and Mohapatra [13] and references therein. In 2002, Xue and Fu [26] established solutions to fuzzy differential equations with righthand side functions satisfying Caratheodory conditions on a class of Lipschitz fuzzy sets. However, there are discontinuous systems in which the right-hand side functions f˜ : [a, b]×E n → E n are not integrable in the sense of Kaleva [11] on certain intervals and their solutions are not absolute continuous functions. Recently, Wu and Gong [24, 25] have combined the fuzzy set theory [27] and nonabsolute integration theory [10], and discussed the fuzzy Henstock integrals of fuzzy-numbervalued functions which extended Kaleva[11] integration. In order to complete the theory of fuzzy calculus and to meet the solving need of transferring a fuzzy differential equation into a fuzzy integral equation, Gong and Shao [7, 8] have defined the strong fuzzy Henstock integrals and discussed some of their properties and the controlled convergence theorem. So, in [19, 20, 21, 22, 23], the authors used the strong fuzzy Henstock integrals [8], and deal with the Cauchy problem of discontinuous fuzzy systems. In this paper, according to the idea of [4] and using the concept of generalized differentiability [2], the operator j which is the isometric embedding from (E n , D) onto its range in the Banach space X and the generalized controlled convergence theorems for the strong fuzzy Henstock integrals, we will deal with the Cauchy problem of discontinuous fuzzy integro-differential equations of mixed type as following: {

∫t ∫a ˜ x(s))ds), x′ (t) = f˜(t, x(t), 0 k1 (t, s)˜ g (s, x(s))ds, 0 k2 (t, s)h(s, x(0) = x0 , x0 ∈ E n , t ∈ Ia = [0, a], a ∈ R+

(1)

˜ x will be assumed strong fuzzy Henstock integrable and k1 , k2 are where f˜, g˜, h, real-valued functions. To make our analysis possible, in section 2, we will first recall some basic results of fuzzy numbers. In section 3, we give some definitions of ω − ACG∗ of fuzzy-number-valued function. In addition, we present the concept of strong fuzzy Henstock integral and a generalized controlled convergence theorem for the strong fuzzy Henstock integrals. In section 4, we deal with the Cauchy problem of discontinuous fuzzy integro-differential equation of mixed type. And in section 5, we present some concluding remarks.

2

PRELIMINARIES

Let Pk (Rn ) denote the family of all nonempty compact convex subset of Rn and define the addition and scalar multiplication in Pk (Rn ) as usual. Let A and B be two nonempty bounded subset of Rn . The distance between A and B is defined by the Hausdorff metric [6]: dH (A, B) = max{sup inf ∥ a − b ∥, sup inf ∥ b − a ∥}. b∈B a∈A

a∈A b∈B

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Denote E n = {u : Rn → [0, 1]|u satisfies (1)-(4) below} is a fuzzy number space. where (1)u is normal, i.e. there exists an x0 ∈ Rn such that u(x0 ) = 1, (2)u is fuzzy convex, i.e. u(λx+(1−λ)y) ≥ min{u(x), u(y)} for any x, y ∈ Rn and 0 ≤ λ ≤ 1, (3)u is upper semi-continuous, (4)[u]0 = cl{x ∈ Rn |u(x) > 0} is compact. For 0 < α ≤ 1, denote [u]α = {x ∈ Rn |u(x) ≥ α}. Then from above (1)-(4), it follows that the α-level set [u]α ∈ Pk (Rn ) for all 0 ≤ α < 1. According to Zadeh’s extension principle, we have addition and scalar multiplication in fuzzy number space E n as follows [6]: [u + v]α = [u]α + [v]α ,

[ku]α = k[u]α ,

where u, v ∈ E n and 0 ≤ α ≤ 1. Define D : E n × E n → [0, ∞) D(u, v) = sup{dH ([u]α , [v]α ) : α ∈ [0, 1]}, where d is the Hausdorff metric defined in Pk (Rn ). Then it is easy see that D is a metric in E n . Using the results [5], we know that (1) (E n , D) is a complete metric space, (2) D(u + w, v + w) = D(u, v) for all u, v, w ∈ E n , (3)D(λu, λv) = |λ|D(u, v) for all u, v, w ∈ E n and λ ∈ R. The metric space (E n , D) has a linear structure, it can be imbedded isomorphically as a cone in a Banach space of function u∗ : I × S n−1 −→ R, where S n−1 is the unit sphere in Rn , with an imbedding function u∗ = j(u) defined by u∗ (r, x) = sup < α, x > α∈[u]α

for all < r, x >∈ I × S n−1 . (see [5]) Theorem 1 There exist a real Banach space X such that E n can be imbedding as a convex cone C with vertex 0 into X. Furthermore the following conclusions hold: (1) the imbedding j is isometric, (2) addition in X induces addition in E n , (3) multiplication by nonnegative real number in X induces the corresponding operation in E n , (4) C − C is dense in X, (5) C is closed. A fuzzy-number-valued function f : [a, b] → E n is said to satisfy the condition (H) on [a, b], if for any x1 < x2 ∈ [a, b] there exists u ∈ E n such that f (x2 ) = f (x1 ) + u. We call u is the H-difference of f (x2 ) and f (x1 ), denoted f (x2 ) −H f (x1 ) ([11]).

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For brevity, we always assume that it satisfies the condition (H) when dealing with the operation of subtraction of fuzzy numbers throughout this paper. It is well-known that the H-derivative for fuzzy-number-functions was initially introduced by Puri and Ralescu [17] and it is based in the condition (H) of sets. We note that this definition is fairly strong, because the family of fuzzy-number-valued functions H-differentiable is very restrictive. For example, the fuzzy-number-valued function f : [a, b] → E n defined by f (x) = C · g(x), where C is a fuzzy number, · is the scalar multiplication (in the fuzzy context) and g : [a, b] → R+ , with g ′ (t0 ) < 0, is not H-differentiable in t0 (see [2]). To avoid the above difficulty, in this paper we consider a more general definition of a derivative for fuzzy-number-valued functions enlarging the class of differentiable fuzzy-number-valued functions, which has been introduced in [2] and [3]. Definition 1 ([2]) Let f˜ : (a, b) → E n and x0 ∈ (a, b). We say that f˜ is differentiable at x0 , if there exists an element f˜′ (t0 ) ∈ E n , such that (1) for all h > 0 sufficiently small, there exists f˜(x0 + h) −H f˜(x0 ), f˜(x0 ) −H ˜ f (x0 − h) and the limits (in the metric D) f˜(x0 + h) −H f˜(x0 ) f˜(x0 ) −H f˜(x0 − h) = lim = f˜′ (x0 ) h→0 h→0 h h lim

or (2) for all h > 0 sufficiently small, there exists f˜(x0 ) −H f˜(x0 + h), f˜(x0 − h) −H f˜(x0 ) and the limits f˜(x0 ) −H f˜(x0 + h) f˜(x0 − h) −H f˜(x0 ) = lim = f˜′ (x0 ) h→0 h→0 −h −h lim

or (3) for all h > 0 sufficiently small, there exists f˜(x0 + h) −H f˜(x0 ), f˜(x0 − h) −H f˜(x0 ) and the limits f˜(x0 + h) −H f˜(x0 ) f˜(x0 − h) −H f˜(x0 ) = lim = f˜′ (x0 ) h→0 h→0 h −h lim

or (4) for all h > 0 sufficiently small, there exists f˜(x0 ) −H f˜(x0 + h), f˜(x0 ) −H f˜(x0 − h) and the limits lim

h→0

f˜(x0 ) −H f˜(x0 + h) f˜(x0 ) −H f˜(x0 − h) = lim = f˜′ (x0 ) h→0 −h h

(h and −h at denominators mean

1 h·

and − h1 ·, respectively).

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3

THE STRONG FUZZY HENSTOCK INTEGRAL AND ITS CONTROLLED CONVERGENCE THEOREM

In this section we shall give the definition of the strong Henstock integral for fuzzy-number-valued functions [7, 8] on a finite interval, which is an extension of the usual fuzzy Kaleva integral in [11]. In addition, we define the properties of ω − AC ∗ and ω − ACG∗ for fuzzy-number-valued functions. In particular, we shall prove a controlled convergence theorems for the strong fuzzy Henstock integrals. Definition 2 ([10, 14]) Let δ(x) be a positive function defined on the interval [a, b]. A division P = {[xi−1 , xi ] : ξi } is said to be δ−fine if the following conditions are satisfied: (1) a − x0 < x1 < · · · < xn = b; (2) ξi ∈ [xi−1 , xi ] ⊂ (ξi − δ(ξi ), ξi + δ(ξi )). For brevity, we write P = {[u, v]; ξ} Definition 3 ([7, 8]) A fuzzy-number-valued function f˜ is said to be strong Henstock integrable on [a, b] if there exists a additive fuzzy-number-valued function F˜ on [a, b] such that for every ε > 0 there is a function δ(ξ) > 0 and for any δ-fine division P = {([u, v], ξ)} of [a, b], we have ∑ D(f˜(ξi )(vi − ui ), F˜ ([ui , vi ])) i∈Kn

+

∑

D(f˜(ξj )(vj − uj ), (−1) · F˜ ([uj , vj−1 ]))

j∈In

< ε. where Kn = {i ∈ {1, 2, ··, n} such that F˜ ([xi−1 , xi ]) is a fuzzy number and In = {j ∈ {1, 2, ··, n} such that F˜ ([xj , xj−1 ]) is a fuzzy number. We write f˜ ∈ SF H[a, b]. Definition 4 ([10, 14]) A real-valued function F is strong absolute continuous (F ∈ AC ∗ ) on [a, b] if and only if for every ε > 0 there is a η > 0 such that for every finite or infinite sequence ∑ ∑ of non-overlapping interval {[ai , bi ]}, satisfying i O(F ; [ai , bi ]) < ε, where where O denotes the i |bi − ai | < η, we have oscillation of f over [ai , bi ], i.e., O(f, [ai , bi ]) = sup{|F (x) − F (y)|; x, y ∈ [ai , bi ]}. A real-valued function F is said to be ACG∗ on X if X is the union of a sequence of sets {Xi } such that on each Xi the function F is AC ∗ (Xi ). Definition 5 A fuzzy-number-valued function f defined on X ⊂ [a, b] is said to be weak generalized absolute continuous (f˜ ∈ ω − ACG∗ (X)) if for every λ ∈ [0, 1], the real-valued function fλ− (x) and fλ+ (x) are ACG∗ . 5

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Theorem 2 If f˜ is strong fuzzy Henstock integrable on [a, b], then its primitive F is ω − ACG∗ on [a, b]. Proof. For every ε > 0, there is a function δ(ξ) > 0 such that for any δ-fine partial division P = {[u, v], ξ} in [a, b], we have ∑ D(F ([u, v]), f (ξ)(v − u)) < ε. We assume that δ(ξ) ≤ 1. Let 1 1 i−1 i Xn,i = {x ∈ [a, b] : D(f (x), ˜0) ≤ n, < δ(x) ≤ , x ∈ [a + , a + )} n n−1 n n for n = 2, 3, · · ·, i = 1, 2, · · ·. Fixed Xn,i and let {[ak , bk ]} be any finite sequence of non-overlapping intervals with ak , bk ∈ Xn,i for all k. Then {([ak , bk ], ak )} is a δ-fine partial division of [a, b]. Furthermore, if ak ≤ uk ≤ vk ≤ bk , then {([ak , uk ], ak )}, {([ak , vk ], ak )} are δ-fine partial division of [a, b]. Thus ∑ ∑ ∑ D(F (uk ), F (vk )) ≤ D(F (ak ), F (uk )) + D(F (bk ), F (vk )) ∑ + D(F (ak ), F (bk )) ∑ ∑ ≤ 3ε + D(f (ak )(uk − ak ), ˜0) + D(f (bk )(bk − vk ), ˜0) ∑ ∑ + D(f (ak )(bk − ak ), ˜0) ≤ 3ε + 3n (bk − ak ). Choose η ≤

ε 3n

and

∑

(bk − ak ) < η. Then ∑ O(F, [ak , bk ]) ≤ 3ε + ε.

Therefore, F is ω − AC ∗ (Xn,i ). Consequently, F is ω − ACG∗ on [a, b]. Theorem 3 If there exists a fuzzy-number-valued function F is continuous and ω − ACG∗ on [a, b] such that F ′ (x) = f (x) a.e. in [a, b], then f is strong fuzzy Henstock integrable on [a, b] with primitive F . Proof. Let F be the primitive of f and F ′ (x) = f (x) for x ∈ [a, b] \ S where S is of measure zero. For ξ ∈ [a, b] \ S, given ε > 0 there is a δ(ξ) > 0 such that whenever ξ ∈ [u, v] ⊂ (ξ − δ(ξ), ξ + δ(ξ)) we have D(F ([u, v]), f (ξ)(v − u)) ≤ ε|v − u|. Since F is continuous and ω − ACG∗ on [a, b], there is a sequence of closed sets {Xi } such that ∪i Xi = [a, b] and F is ω − AC ∗ (Xi ) for each i. Let Y1 = X1 , Yi = Xi \ (X1 ∪ X2 · · · ∪Xi−1 ) for i = 1, 2, · · · and Sij denote the set of points x ∈ S ∩ Yi such that j − 1 ≤ D(f, ˜0) < j. Obviously, Sij are pairwise disjointed and their union is the set S. Since F is also ω − AC ∗ (Sij ), there is a ηij < ε2−i−j j −1 such that for any sequence of non-overlapping intervals {Ik } ∑ with at least one endpoint of Ik belonging to Sij and satisfying k |Ik | < ηij 6

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∑ we have k D(F (Ik ), ˜ 0) < ε2−i−j . Again, F (I) denotes F (v) −H F (u) where I = [u, v]. Choose Gij to be the union of a sequence of open intervals such that |Gij | < ηij and Gij ⊃ Sij where |Gij | denotes the total length of Gij . Now for ξ ∈ Sij , put (ξ − δ(ξ), ξ + δ(ξ)) ⊂ Gij . Hence we have defined a positive function δ(ξ). ∑ Take division P = {[u, v]; ξ}. Split the over P into partial ∑ any δ−fine ∑ ¯ S and ξ ∈ S respectively and we obtain sums 1 and 2 in which ξ ∈ ∑ D(f (ξ)(v − u), F ([a, b])) ≤ D(f (ξ)(v − u), F ([a, b])) 1

+

∑

D(F ([a, b]), ˜0) +

2

< ε(b − a) +

∑

∑ 2

ε2−i−j +

D(f (ξ)(v − u), ˜0) ∑

jηij

2

i,j

< ε(b − a) + 2ε. That is to say, f is strong fuzzy Henstock integrable to F on [a, b]. Definition 6 A sequence of fuzzy-number-valued functions {Gn (x)} is said to be weak uniformly ACG∗ (U ω − ACG∗ ) if for every λ ∈ [0, 1], the real-valued + ∗ functions {Gn (x)}− λ and {Gn (x)}λ are U ACG . Theorem 4 (Controlled Convergence theorem) If a sequence of strong fuzzy Henstock integrable {fn } satisfies the following conditions: (1) fn (x) → f (x) almost everywhere ∫ x in [a, b] as n → ∞; (2) the primitives Fn (x) = (SF H) a fn (s)dx of fn are ω −ACG∗ uniformly in n; (3) the primitives Fn (x) are equicontinuous on [a, b], then f (x) is strong fuzzy Henstock integrable on [a, b] and we have ∫ b ∫ b lim (SF H) fn (x)dx = (SF H) f (x)dx. n→∞

a

a

If condition (1) and (2) are replaced by condition (4): (4) g(x) ≤ f (x) ≤ h(x) almost everywhere on [a, b], where g(x) and h(x) are steong fuzzy Henstock integrable. Proof. In view of condition (3), F (x) exist as the limit of Fn (x) and is + ∗ continuous. In fact, for ∀λ ∈ [0, 1], (Fn (x))− λ and (Fn (x))λ is uniformly ACG on [a, b]. By the Controlled Convergence theorem of real valued strong Henstock integral([14] Theorem 7.6), F (x) is continuous. Because Fλ− (x) and Fλ+ (x) is Henstock integrable on [a, b], it follows condition (2) that F is ω − ACG∗ . From theorem 3.2, it remains to show that F ′ (x) = f (x) almost everywhere. Hence we obtain f (x) is strong fuzzy Henstock integrable on [a, b]. ∫x Next, we put G(x) = (SF H) a F (t)dt, in view of condition (3), for ∀λ ∈ [0, 1], we have − − lim (Fn (x))− λ = Gλ (x) = Fλ (x) n→∞

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and + + lim (Fn (x))+ λ = Gλ (x) = Fλ (x).

n→∞

So, let x = b, we have ∫ n→∞

∫

b

lim (SF H)

fn (x)dx = (SF H) a

b

f (x)dx. a

This completes the proof.

4

AN EXISTENCE RESULT OF GENERALIZED FUZZY INTEGRO-DIFFERENTIAL EQUATIONS

By using the Controlled Convergence theorem of strong fuzzy Henstock integral, in this section, we prove a theorem for the existence of solution to the Cauchy problem (1). For any bounded subset A of the Banach space X we denote α(A) the Kuratowski measure of non-compactness of A, i.e the infimum of all ε > 0 such that there exist a finite covering of A by sets of diameter less than ε. For the properties of α we refer to [1] for example. Lemma 1 ([1]) Let H ⊂ C(Iγ , X) be a family of strong equicontinuous functions. Then α(H) = sup α(H(t)) = α(H(Iγ )) t∈Iγ

where α(H) denote the Kuratowski measure of non-compactness in C(Iγ , X) and the function t → α(H(t)) is continuous. Theorem 5 ([1]) Let D be a closed convex subset of X, and let F be a continuous function from D into itself. If for x ∈ D the implication V¯ = con({x} ¯ ∪ F (V )) ⇒ V is relatively compact, then F has a fixed point. Theorem 6 If the fuzzy-number-valued function f˜ : Ia −→ E n is (SF H) integrable, then ∫ f˜(t)dt ∈ |I| · conv f˜(I), I

where conv f˜(I) is the closure of the convex of f˜(I), I is an arbitrary subinterval of Ia , and |I| is the length of I.. Proof. Because of j ◦ f˜ is abstract (SH) integrable in a Banach Space, by using the mean valued theorem of (SH) integrals, we have ∫ (SH) j ◦ f˜(t)dt ∈ |I| · convj ◦ f˜(I) = |I| · j ◦ conv f˜(t). I

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∫ ∫ In additional, there exists (SH) I j ◦ f˜(t)dt = j ◦ I f˜(t)dt. ∫ So, we have j ◦ I f˜(t)dt ∈ |I| · convj ◦ f˜(I). And the set {|I| · conv f˜(I)} is a closed convex set, we have ∫ f˜(t)dt ∈ |I| · conv f˜(I). I

Definition 7 A fuzzy-number-valued function f˜ : Ia ×E n −→ E n is L1 −Carath´ eodory if the following conditions hold: (1) the fuzzy mapping (x, y) ∈ E n × E n is measurable for all t −→ f˜(t, x, y); (2) the fuzzy mapping t ∈ Ia is continuous for all (x, y) −→ f˜(t, x, y). We observer that the problem (1) is equivalent to the integral eqution: ∫ t ∫ z ∫ a ˜ ˜ x(s))ds)dz x(t) = x0 + f (z, x(z), k1 (z, s)˜ g (s, x(s))ds, k2 (z, s)h(s, 0

0

0

or ∫

∫

t

x(t) = x0 +(−1)· 0

∫

z

f˜(z, x(z),

a

˜ x(s))ds)dz. k2 (z, s)h(s,

k1 (z, s)˜ g (s, x(s))ds, 0

0

(2) Now, we define a notion of a solution. Definition 8 A ω − ACG∗ function x : Ia → E n is said to be the generalized solutions of the problem (1) if it satisfies the following conditions: (1) x(0) = x0 ; (2) ∫ t ∫ a ˜ x(s))ds). x′ (t) = f˜(t, x(t), k1 (t, s)˜ g (s, x(s))ds, k2 (t, s)h(s, 0

0

for a. e. t ∈ Ia . Definition 9 A continuous function x : Ia → E n is said to be the solutions of problem (2) if ∫ t ∫ z ∫ a ˜ ˜ x(s))ds)dz x(t) = x0 + f (z, x(z), k1 (z, s)˜ g (s, x(s))ds, k2 (z, s)h(s, 0

0

0

or ∫

∫

t

f˜(z, x(z),

x(t) = x0 +(−1)· 0

∫

z

0

a

˜ x(s))ds)dz. k2 (z, s)h(s,

k1 (z, s)˜ g (s, x(s))ds, 0

for every t ∈ Ia For every fuzzy number x ∈ C(Ia , E n ), we define the norm of x by: H(x, ˜0) = sup D(x, ˜0). t∈Ia

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Let B(p) = {x ∈ C(Ia , E n )|H(x, ˜0) ≤ H(x, ˜0) + p, p > 0}. Obviously, B(p) is closed and convex in E n . Define the operator F : C(Ia , E n ) → C(Ia , E n ) by: ∫ t ∫ z ∫ a ˜ ˜ x(s))ds)dz F (x)(t) = x0 + f (z, x(z), k1 (z, s)˜ g (s, x(s))ds, k2 (z, s)h(s, 0

0

0

where integrals are in the sense of strong fuzzy Henstock integral. Let Γ(p) = {F (x) ∈ C(Ia , E n )|x ∈ B(p)} for each p > 0. Let r(K) be the spectral radius of the integral operator K defined by ∫ c K(u)(t) = k(t, s)u(s)ds, 0

where the kernel k ∈ C(Ia ×Ia , R), u ∈ C(Ia , E n ) and c denotes any fixed valued in Ia . Next, we give the main result in this section. Theorem 7 Suppose that for each ω − ACG∗ function x : Ia → E n , the functions ∫ (·) ∫a ˜ x(s))ds are (SF H) g˜(·, x(·)), f˜(·, x(·)), 0 k1 (·, s)˜ g (s, x(s))ds, and 0 k2 (z, s)h(s, 1 ˜ are fuzzy L −Caratheodory functions. Let k1 , k2 : Ia × integrable, g˜, f˜, and h Ia → R+ be measurable functions such that k1 (t, ·), k2 (t, ·) are continuous. Assume that there exists p0 > 0 and positive constants L, L1 and d1 , such that α(j ◦ g˜(I, X)) ≤ Lα(j ◦ X), I ⊂ Ia , X ⊂ B(p0 ), ˜ X)) ≤ L1 α(j ◦ X), I ⊂ Ia , X ⊂ B(p0 ), α(j ◦ h(I, α(j ◦ f˜(t, A, C, D)) ≤ d1 · max{α(j ◦ A), α(j ◦ C), α(j ◦ D)} A, C, D ⊂ B(p0 ), ˜ X) = {h(t, ˜ x(t))|t ∈ I, x ∈ X} where g˜(I, X) = {˜ g (t, x(t))|t ∈ I, x ∈ X}, h(I, and f˜(t, A, C, D) = {f˜(t, x1 , x2 , x3 )|(x1 , x2 , x3 ) ∈ A × C × D} where α denotes the Kuratowski measure of non-compactness. Moreover, let Γ(p0 ) be equicontinuous, equibounded, and uniformly ω−ACG∗ on Ia . Then, there exists at least on solution of problem (1) on Ic , for some 0 < c ≤ a, such that d1 · c < 1 and d1 · c · L · r(K). Proof. By equicontinuity and equiboundedness of Γ(p0 ) there exists a number c, 0 < c ≤ a such that ∫ t ∫ z ∫ a ˜ x(s))ds)dz, ˜0) H( f˜(z, x(z), k1 (z, s)˜ g (s, x(s))ds, k2 (z, s)h(s, 0 0 0 ∫ t ∫ z ∫ a ˜ x(s))ds)dz, ˜0) = sup D( f˜(z, x(z), k1 (z, s)˜ g (s, x(s))ds, k2 (z, s)h(s, t∈Ic

0

0

0

≤p0 , 10

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where p0 > 0, x ∈ B(p0 ). By the definition of F , we have H(F (x)(t), ˜ 0) ∫ t ∫ z ∫ a ˜ x(s))ds)dz, ˜0) f˜(z, x(z), k1 (z, s)˜ g (s, x(s))ds, k2 (z, s)h(s, =H(x0 + 0 0 0 ∫ t ∫ z ∫ a ˜ ˜ x(s))ds)dz, ˜0) ˜ ≤H(x0 , 0) + H( f (z, x(z), k1 (z, s)˜ g (s, x(s))ds, k2 (z, s)h(s, 0

≤H(x0 , ˜ 0) + p0 ,

0

0

t ∈ Ic , x 0 ∈ E n .

Using Theorem 4, we deduce that the fuzzy-number-valued function F is continuous. Obviously, there exists V ⊂ B such that V = conv({x} ∪ F (V )) for every x ∈ B(p0 ). Next, we will prove that V is relatively compact. In fact, let V (t) = {v(t) ∈ E n |v ∈ V } for t ∈ Ic . Since V ⊂ B(p0 ) and F (V ) ⊂ Γ(p0 ), then V ⊂ V is equicontinuous. By Lemma 1, we get that t → v(t) = α(j◦V (t)) is continuous on Ic . For fixed t ∈ Ic , we divide the interval [0, t] into m parts: 0 = t0 < t1 < · · · < tm = t, where ti = it/m, i = 0, 1, 2 · · · , m. Let V ([ti , ti+1 ]) = {u(s) : u ∈ V, ti ≤ s ≤ ti+1 , i = 1, 2, · · · , m − 1} By Lemma 1 and the continuity of v, there exists si ∈ Ii = [ti , ti+1 ] such that α(j ◦ V ([ti , ti+1 ])) = sup {α(j ◦ V (s))|ti ≤ s ≤ ti+1 } := v(si ). t∈Ic

For fixed z ∈ [0, t], we divide the interval [0, z] into m parts: 0 = z0 < z1 < · · · < zm = z, where zj = jz/m, j = 0, 1, 2 · · · , m. Let V ([zj , zj+1 ]) = {u(s)|u ∈ V, zj ≤ s ≤ zj+1 }, j = 0, 1, 2, · · · , m − 1. By Lemma 1 and the continuity of v, there exists sj ∈ Ij = [zj , zj+1 ] such that α(j ◦ V ([zj , zj+1 ])) = sup {α(j ◦ V (s))|zj ≤ s ≤ zj+1 } := v(sj ). t∈Ic

Furthermore, we divide the interval [0, c] into m parts: 0 = r0 < r1 < · · · < rm = c, where rk = kc/m, k = 0, 1, 2 · · · , m. Let V ([rk , rk+1 ]) = {u(s)|u ∈ V, rk ≤ s ≤ rk+1 }, j = 0, 1, 2, · · · , m − 1. By Lemma 1 and the continuity of v, there exists sk ∈ Ik = [rk , rk+1 ] such that α(j ◦ V ([rk , rk+1 ])) = sup {α(j ◦ V (s))|rk ≤ s ≤ rk+1 } := v(sk ). t∈Ic

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By Theorem 3 and Theorem 4, we have F (x)(t) = x0 +

m−1 ∑ ∫ ti+1 i=0

m−1 ∑ ∫ rk+1 k=0

+

ti

f˜(z, x(z),

m−1 ∑ ∫ zj+1

˜ x(s))ds)dz ∈ x0 k2 (z, s)h(s,

rk

m−1 ∑

m−1 ∑

(ti+1 − ti )conv f˜(Ii , V (Ii ),

(zj+1 − zj )conv(k1 (Ii , Ij )˜ g (Ij , V (Ij ))),

i=0 m−1 ∑

k1 (z, s)˜ g (s, x(s))ds,

zj

j=0

j=0

˜ k , V (Ik ))), (rj+1 − rj )conv(k2 (Ii , Ij )h(I

k=0

where k(I, J) = {k(t, s)|t ∈ I, s ∈ J} and g˜(I, V (I)) = {˜ g (t, x(t))|t ∈ I, x ∈ V }. Using the condition in assumption and the properties of noncompactness α ([1]), we have α(j ◦ F (V )(t)) ≤

m−1 ∑

(ti+1 − ti )convα(j ◦ f˜(Ii , V (Ii ),

i=0

m−1 ∑

g (Ij , V (Ij ))), (zj+1 − zj )conv(k1 (Ii , Ij )˜

j=0

m−1 ∑

˜ k , V (Ik )))) (rj+1 − rj )conv(k2 (Ii , Ij )h(I

k=0

≤

m−1 ∑

m−1 ∑

i=0

j=0

(ti+1 − ti )d1 max{(α(j ◦ V (Ii )), αj ◦ (

αj ◦ (

m−1 ∑

(zj+1 − zj )conv(k1 (Ii , Ij )˜ g (Ij , V (Ij )))),

˜ k , V (Ik )))). (rj+1 − rj )conv(k2 (Ii , Ij )h(I

k=0

We observe that if α(j ◦ V (Ii )) = max{(α(j ◦ V (Ii )), αj ◦ (

m−1 ∑

(zj+1 − zj )conv(k1 (Ii , Ij )˜ g (Ij , V (Ij )))),

j=0 m−1 ∑

˜ k , V (Ik ))))), (rj+1 − rj )conv(k2 (Ii , Ij )h(I

α(j ◦ (

k=0

then α(j ◦ V (t)) = αj ◦ (conv({x(t)} ∪ F (V (t))))α(j ◦ F (V (t))) ≤ d1 · c · α(j ◦ V (t)) for every t ∈ Ic . Because d1 · c < 1, we have α(j ◦ V ) < α(j ◦ V ). This is a contradiction.

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If α(j ◦ (

m−1 ∑

(zj+1 − zj )conv(k1 (Ii , Ij )˜ g (Ij , V (Ij )))))

j=0 m−1 ∑

= max{α(j ◦ V (Ii )), αj ◦ (

(zj+1 − zj )conv(k1 (Ii , Ij )˜ g (Ij , V (Ij )))),

j=0

α(j ◦ (

m−1 ∑

˜ k , V (Ik )))))}, (rj+1 − rj )conv(k2 (Ii , Ij )h(I

k=0

we have α(j ◦ F (V )(t)) ≤ ≤ ≤

m−1 ∑

m−1 ∑

i=0

j=0

(ti+1 − ti ) · d1 ·

(zj+1 − zj )k1 (Ii , Ij )α(j ◦ g˜(Ij , V (Ij )))

m−1 ∑

m−1 ∑

i=0

j=0

(ti+1 − ti ) · d1 · L ·

(zj+1 − zj )k1 (Ii , Ij )α(j ◦ V (Ij ))

m−1 m−1 ∑ c ∑ · (zj+1 − zj )α(j ◦ V (Ij )) k1 (Ii , Ij ). m j=0 i=0

For j = 0, 1, 2, . . . , m−1, there exists qj = 0, 1, 2, . . . , m−1 such that k1 (Ii , Ij ) ≤ k1 (Iqj , Ij ). So, α(j ◦ F (V )(t)) ≤ d1 · c · L ·

m−1 ∑

(zj+1 − zj )k1 (Iqj , Ij )v(sj ),

sj ∈ Ij .

j=0

Hence

α(j ◦ F (V )(t)) ≤ d1 · c · L ·

m−1 ∑

(zj+1 − zj )k1 (Iqj , Ij )(v(sj ) − v(pj ))

j=0

+ d1 · c · L ·

m−1 ∑

(zj+1 − zj )k1 (Iqj , Ij )v(pj ).

j=0

By the continuity of v, we have j ◦ v(sj ) − j ◦ v(pj ) < ε. Therefore, we have ∫ c α(j ◦ F (V )(t)) ≤ d1 · c · L · k1 (t, s)v(s)ds 0

for t ∈ Ic . Since ∫ cV = conv({x}∪F (V )), we have α(j ◦V (t)) ≤ α(j ◦F (V )(t)), so, v(t) ≤ d1 ·c·L· 0 k1 (t, s)v(s)ds. By Gronwalls inequality, we have α(j ◦V (t)) = 0 for t ∈ Ic . By Arzel´ a−Ascoli’s theorem, we have V is relatively. Consequently, 13

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by Theorem 5, F has a fixed point. That is to say that problem (1) have at least solutions. Similary, if α(j ◦ (

m−1 ∑

˜ k , V (Ik ))))) (rj+1 − rj )conv(k2 (Ii , Ij )h(I

k=0 m−1 ∑

= max{α(j ◦ V (Ii )), αj ◦ (

g (Ij , V (Ij )))), (zj+1 − zj )conv(k1 (Ii , Ij )˜

j=0

α(j ◦ (

m−1 ∑

˜ k , V (Ik )))))}, (rj+1 − rj )conv(k2 (Ii , Ij )h(I

k=0

then we have α(j ◦ V (t)) ≤ α(j ◦ F (V )(t)). By Arzel´ a−Ascoli’s theorem, the set V is relatively. By Theorem 5, F has a fixed point which is a solution of the problem (1).

5

CONCLUSIONS

In this paper, we give the definition of the ω − ACG∗ for a fuzzy-number-valued function and a generalized controlled convergence theorem. In addition, we deal with the Cauchy problem of discontinuous fuzzy integro-differential equations of mixed type involving the strong fuzzy Henstock integral in fuzzy number space. The function governing the equations is supposed to be discontinuous with respect to some variables and satisfy nonabsolute fuzzy integrablility. Our result improves the result given in Ref. [11, 2] and [26] (where uniform continuity was required), as well as those referred therein.

References [1] J. Banas, K. Goebel, Measures of Noncompactness in Banach Space, Marcel Dekker, New York, NY, USA, 1980. [2] B. Bede, S. Gal, Generalizations of the differentiability of fuzzy-numbervalued functions with applications to fuzzy differential equation, Fuzzy Sets and Systems 151, 581-599 (2005). [3] Y. Chalco-Cano, H. Roman-Flores, On the new solution of fuzzy differential equations, Chaos, Solition & Fractals 38, 112-119 (2008). [4] T. S. Chew, F. Franciso, on x′ = f (t, x) and Henstock-Kurzwell integrals, Differential and Integral Equations 4, 861-868 (1991). [5] P. Diamond, P. Kloeden, Metric Space of Fuzzy Sets: Theory and Applications, World Scientific, Singapore, 1994. [6] D. Dubois, H. Prade, Towards Fuzzy Differential Calculus, Part 1. Integration of fuzzy mappings, Fuzzy Sets and Syst. 8, 1-17 (1982). 14

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[7] Z. Gong, On the Problem of Characterizing Derivatives for the Fuzzyvalued Functions (II): almost everywhere differentiability and strong Henstock integral, Fuzzy Sets and Syst. 145, 381-393 (2004). [8] Z. Gong, Y. Shao, The Controlled Convergence Theorems for the Strong Henstock Integrals of Fuzzy-Number-Valued Functions, Fuzzy Sets and Syst. 160, 1528-1546 (2009). [9] Z. Gong, Y. Shao, Global Existence and Uniqueness of Solutions for Fuzzy Differential Equations under Dissipative-type Conditions, Computers & Math. with Appl. 56, 2716-2723 (2008). [10] R. Henstock, Theory of Integration. Butterworth, London, (1963) [11] O. Kaleva, Fuzzy Differential Equations, Fuzzy Sets and Syst. 24, 301-319 (1987). [12] O. Kaleva, The Cauchy Problem for Fuzzy Differential Equations, Fuzzy sets and syst. 35, 389-396 (1990). [13] V. Lakshmikantham, R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London, 2003. [14] P. Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore, New Jersey, London, Hongkong, 1989. [15] C.V. Negoita, D. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975. [16] J. J. Nieto, The Cauchy Problem for Continuous Fuzzy Differential Equations, Fuzzy Sets and syst. 102, 259-262 (1999). [17] M. L. Puri, D. A. Ralescu, Differentials of Fuzzy Functions, J. Math. Anal. Appl. 91, 552-558 (1983). [18] S. Seikkala, On the Fuzzy Initial Value Problem, Fuzzy Sets and Syst. 24, 319-330 (1987). [19] Yabin Shao, Zengtai Gong, Discontinuous fuzzy systems and Henstock integrals of fuzzy number valued functions, Lectures Note on Comput. Sci. 7389, 65-72 (2012). [20] Yabin Shao, Guoliang Xue. Discontinuous fuzzy Fredholm integral equations and strong fuzzy Henstock integrals, Artificial Intelligence Research 2, 87-95 (2013). [21] Yabin Shao, Huanhuan Zhang, Fuzzy integral equations and strong fuzzy Henstock integrals, Abstract and Applied Analysis Volume 2014, Article ID 932696, 8 pages. [22] Yabin Shao, Huanhuan Zhang, The strong fuzzy Henstock integrals and discontinuous fuzzy differential equations, Journal of Applied Mathematics Volume 2013, Article ID 419701, 8 pages. [23] Yabin Shao, Huanhuan Zhang, Existence of the solution for discontinuous fuzzy integro-differential equations and strong fuzzy Henstock integrals, Nonlinear Dynamics & Systems Theory 14, 148-161 (2014). 15

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[24] C. Wu, Z. Gong, On Henstock Intergrals of Interval-Valued and FuzzyNumber-Valued Functions, Fuzzy Sets and Systems 115, 377-391 (2000). [25] C. Wu, Z. Gong, On Henstock Intergrals of Fuzzy-valued Functions (I), Fuzzy Sets and Syst. 120, 523-532 (2001). [26] X. Xue, Y. Fu, Caratheodory Solution of Fuzzy Differential Equations, Fuzzy Sets and Syst. 125, 239-243 (2002). [27] L. Zadeh, Fuzzy Sets, Information and Control 3, 338-353 (1965).

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On the Generalized Stieltjes Transform of Fox’s Kernel Function and its Properties in the Space of Generalized Functions Shrideh Khalaf Qasem Al-Omari Department of Applied Sciences; Faculty of Engineering Technology Al-Balqa Applied University; Amman 11134; Jordan [email protected] Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University Eskisehir Yolu 29.km, 06810 Ankara, Turkey Institute of Space Sciences, Magurele-Bucharest, Romania [email protected] Abstract In this paper, a Stieltjes transform enfolding some Fox’s H-function has been investigated on certain class of generalized functions named as Boehmians. By developing two spaces of Boehmians, the extended transform has been inspected and some general properties are also obtained. An inverse problem is also discussed in some detail. Keywords: Fox’s H-function; Stieltjes transform; Laplace transform; Boehmian space; Distribution space.

1

Introduction

The Fox’s H-function is a generalization of the Meijer G-function introduced by Charles Fox [15]. It is de…ned by the compact notation adopted for " # (a ; )   =12   H (!) = H ! b ;  =12 and has an exempli…cation in terms of the Barnes-type integral [2] Z 1  H (!) = | (&) !  d&; 2 i L 

where L is a path in the complex plane, !  = exp f& (log j!j + i arg !)g ; and |  (&) =

a (&) b (&) ; c (&) d (&)

where a (&) c (&)

: :

= =

 Y

1  Y

b

 & ; b (&) :=

 Y

(1

a +

 &)

1

1

b

&

and d (&) :=

+1

 Y

(a +

 &) ;

+1

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2

S. K. Q. Al-Omari and D. Baleanu

with m; p; q 2 N; a ; b 2 C;  ;  2 R+ ; n 2 N0 := N [ f0g satisfying 0 < n < p and 0 < m < q; and C; R+ and N denote, respectively, the sets of complex numbers, positive real numbers and positive integers. We refer to the survey article by Braaksma [2] and the book of Charles Fox [15] for asymptotic behaviour of Fox’s H-functions: Fox’s H-functions being an extreme generalization of the generalized hypergeometric functions  F are utilized for applications in a large variety of problems connected with statistical distribution theory, structures of random variables, generalized distributions, Mathai’s pathway models, versatile integrals, reaction, di¤usion, reaction di¤usion, engineering, communication, fractional di¤erential and integral equations and many areas of theoretical physics and statistical distribution theory as well. Recently, utility and importance of H-functions are realized due to their occurrence as kernels of certain integral transforms. The generalized Stieltjes transform of a function ' (t) of one variable with kernel involving Fox’s H-function is de…ned by [5; (1:3)] # " Z 1 (a1 ; 1 ) ; (1 b1  ¬ 1 12 1; 1) '( )d ; (1) ! H22  (') (!) = (e1 ; 1 ) ; (e2 ; 2 ) ! 0  [!] is the usual notation of the Fox H-function. where H An interesting fact that we …nd it worthwhile to be mentioned here is that the transform under consideration is a modulation of the Laplace transform Z 1 (a1 ; 1 ) 12 (') (!) = '( )d (2) H22 ( !)  (e1 ; 1 ) ; (e2 ; 2 ) 0

that recti…ed after some iterations and an appropriate choice on its parameter. Denote by J the Fréchet space of smooth functions ' de…ned for all (0 < the set f  g of seminorms where p  '( ) < 1 % (log ) ( D )  (') = sup

< 1) by (3)

0< 1

for every choice of k (k 2 N0 ) ;



% (log ) =

; 1 0, m(A) represents the belief measuser that one is willing to commit exactly to A, given a certain piece of evidence. Definition 2.6 ([26]). Let Θ be the frame of discernment and m : 2Θ → [0, 1] be a Mass function. Then a belief function on Θ is defined a mapping Bel : 2Θ → [0, 1], Bel satisfies X Bel(∅) = 0, Bel(Θ) = 1, Bel(A) = m(B) for A ⊆ Θ. B⊆A

Bel(A) can be interpreted as a global belief measure that the hypothesis A is true, and represents the imprecision and uncertainty in the decision-making process. In the case of single hypothesis, Bel(A) = m(A).

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Definition 2.7 ([26]). Let Θ be the frame of discernment. Suppose there are two Mass functions are m1 and m2 over Θ, induced by two independent items of evidences A1 , A2 , · · · , As and B1 , B2 , · · · , Bt , respectively. D-S rule of evidence combination is defined and denoted as follows: P ( 1 m1 (Ai )m2 (Bj ), ∀ A ⊆ Θ, A 6= ∅, 1−K Ai ∩Bj =A m(A) = m1 ⊕ m2 (A) = 0, A = ∅, P where K = m1 (Ai )m2 (Bj ) < 1. Ai ∩Bj =∅

K is called the conflict probability and reflects the extent of the conflict 1 between the evidences. Coefficient 1−K is called normalized factor, its role is to avoid the probability of assigning non-0 to empty set ∅ in the combination. D-S rule of evidence combination can be generalized to multiple Mass functions, the belief measure resulting from the combination of multiply evidences Ai is as follows: m1 ⊕ m2 · · · ⊕ mn (A) =

1 1−K

X Tn

i=1

where K = T

P n i=1

m1 (A1 )m2 (A2 ) · · · mn (An ),

Ai =A,Ai ⊂Θ

m1 (A1 )m2 (A2 ) · · · mn (An ) < 1.

Ai =∅,Ai ⊂Θ

D-S rule of evidence combination can increase belief measure of hypotheses and reduce the uncertain degree to improve reliability. Example 2.8. Let Θ = {A1 , A2 } be the frame of discernment. Suppose there are two Mass functions m1 and m2 over Θ, induced by the independent items of evidences A1 , A2 , given by m1 (A1 ) = 0.3, m1 (A2 ) = 0.4, m1 (Θ) = 0.3, m2 (A1 ) = 0.4, m2 (A2 ) = 0.3, m2 (Θ) = 0.3. Combining the two evidences by D-S rule of evidence combination leads to: 1 )m2 (Θ)+m1 (Θ)m2 (A1 ) m(A1 ) = m1 ⊕ m2 (A1 ) = m1 (A1 )m2 (A1 )+m1 (A = 0.44, 1−K

m1 (A2 )m2 (A2 )+m1 (A2 )m2 (Θ)+m1 (Θ)m2 (A2 ) 1−K 2 (Θ) = 0.12, m(Θ) = m1 ⊕ m2 (Θ) = m1 (Θ)m 1−K where K = m1 (A1 )m2 (A2 ) + m1 (A2 )m2 (A1 ) = 0.25.

m(A2 ) = m1 ⊕ m2 (A2 ) =

3

= 0.44,

An approach to interval-valued intuitionistic fuzzy soft sets in decision making

Recently, research on soft sets based decision making has attracted more and more attention. The works of Roy et al. [10, 25, 5, 2, 11] are fundamental and significant. Later other authors like Qin et al. further studied and proposed an adjustable approach to interval-valued intuitionistic fuzzy soft set based decision making using the level soft sets and reductions . Generally, there does not exist 5

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any unique or uniform criterion for the evaluation of decision alternatives under uncertain condition. However, it is very difficult for decision makers to select suitable level soft sets and discuss reduct intuitionistic fuzzy soft sets. Now we investigate interval-valued intuitionistic fuzzy soft sets based decision making by means of grey relational analysis and D-S theory of evidence. It is divided three phases: First, grey relational analysis is applied to calculate the grey mean relational degree and the uncertain degree of each parameter is obtained. Second, the corresponding Mass function with respect to each parameter is constructed by the uncertain degree of each parameter. Third, we apply D-S rule of evidence combination to aggregate individual alternatives into a collective alternative, by which the candidate alternatives are ranked and the best alternative is obtained. In the following, we consider the decision making problem with m mutually exclusive alternatives xi and n evaluation parameters (or indexes) ej . dij denotes the degree that the alternative xi satisfies the parameter ej . Put Θ = {x1 , x2 , · · · , xm } and A = {e1 , e2 , · · · , en }. Define F : A → IV IF (Θ) by F (ej ) = {(xi , µF (ej ) (xi ), νF (ej ) (xi )) | xi ∈ Θ} (ej ∈ A) where µF (ej ) : U → Int[0, 1] and νF (ej ) : U → Int[0, 1] satisfy 0 6 sup µF (ej ) (xi ) + sup νF (ej ) (xi ) 6 1. Then (F, A) is an interval+ valued intuitionistic fuzzy soft set over Θ. Denote µF (ej ) (xi ) = [µ− ij , µij ], − + νF (ej ) (xi ) = [νij , νij ], aij = (µF (ej ) (xi ), νF (ej ) (xi )). D = (aij )m×n is called an interval-valued intuitionistic fuzzy soft decision matrix induced by (F, A). Here, we see the set of parameters as a item of evidences information. The key to solve decision problems by using D-S theory of evidence is how to obtain the uncertain degree of evidences (or parameters). First, inspired by Xu [12], we define the score function of as follows. Definition 3.1. Suppose that (F, A) is an interval-valued intuitionistic fuzzy soft over Θ. Suppose that D = (aij )m×n is an interval-valued intuitionistic + fuzzy soft decision matrix induced by (F, A). Denote µF (ej ) (xi ) = [µ− ij , µij ], − + νF (ej ) (xi ) = [νij , νij ], aij = (µF (ej ) (xi ), νF (ej ) (xi )). Then score function of dij is defined and denoted as + − + + + − − s(aij ) = (µ− ij + µij − νij − νij )/2 + α(µij + νij − µij − νij )/2.

By Definition 4.1, we can convert dij into real numbers. s(aij ) presents the global degree that the alternative xi holds the parameter ej . Obviously, 0 6 s(aij ) 6 1. α is called a risk factor. For α = 0, > 0, < 0, they imply the attitude of decision makers for risk is neutral, positive, oppose, respectively. Decision makers can select a α value according to their risk preference. In this paper, we pick α = 0. To obtain Mass functions of each alternative with respect to each parameter, we consider score function values may be negative, so we should normalize the

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score function values by the following formula: dij =

s(aij ) − min16i6m s(aij ) , 1 6 i 6 m, 1 6 j 6 n. max16i6m s(aij ) − min16i6m S(aij )

Hence, we can get normalized matrix of score function values D = (dij )m×n . Next, inspired by the paper [12], we define the grey mean relational degree and the uncertain degree of the parameter as follows. Definition 3.2. Let Θ = {x1 , x2 , · · · , xm }, A = {e1 , e2 , · · · , en } and let (F, A) be an intuitionistic fuzzy soft set on Θ. Suppose that D = (dij )m×n is normalized matrix of score function values. For any i, j, denote n

1X dei = dij , n j=1 rij =

4dij = |dij − dei |,

min16j6n min16i6m 4dij + ρ max16j6n max16i6m 4dij , ρ ∈ (0, 1), 4dij + ρ max16j6n max16i6m 4dij DOI(ej ) =

m 1 1 X ( (rij )q ) q (j = 1, 2, · · · , n). m i=1

rij is called the grey mean relational degree between dij and dei . DOI(ej ) is called q order uncertain degree of the parameter ej . ρ aims to expand or compress the range of the grey relational coefficient. Decision makers can select q, ρ values according to different circumstance. To obtain strong distinguishing effectiveness, we pick q = 2, ρ = 0.5 in this paper. We call DOI(ej ) the uncertain degree of ej for short. It is worthy to notice that the method to obtain the uncertain degree varies from different situation in Definition 4.2. General speaking, since a index (or parameter) is specially more matching the mean of the index set than other indexes, it contains more satisfying information for decision making and the uncertain degree of the index information is lower. Then, in this paper we just consider grey mean relational degree between dij and dei . Definition 3.3 ([36]). Let X = (x1 , x2 , · · · , xm ) be a finite difference information sequence, where there exists xik 6= 0 for k = 1, 2, · · · , m and 1 6 ik 6 m. Then the information structure image sequence Y = (y1 , y2 , · · · , ym ) is given by xi yi = P . m i=1

xi

In the normalized matrix of score function values D = (dij )m×n , the information structure image sequence with respect to a parameter ej is denoted by dij f f g f . Then we obtain an informadj = {df m 1j , d2j , d3j , · · · , dmj }, where dij = P i=1

dij

e = (df tion structure image matric D ij )m×n induced by dj (j = 1, 2, · · · , n).

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D-S theory of evidence is a powerful method for combining accumulative evidence of changing prior opinions in the light of new evidences [26]. The primary procedure of combining the known evidences or information with other evidences is to construct suitable Mass functions of evidences. Now, by the uncertain degree of each parameter, we can obtain Mass function of each alternative with respect to each parameter. Theorem 3.4. Let Θ = {x1 , x2 , · · · , xm }, A = {e1 , e2 , · · · , en } and let (F, A) be an intuitionistic fuzzy soft set on Θ. Suppose that D = (dij )m×n is the normalized matrix of score function values and DOI(ej ) is the uncertain degree dij . For any i, j, we define functions mej (j = 1, 2, · · · , n) of ej . Denote df m ij = P i=1

dij

with respect to the parameter ej , it satisfies: mej (xi ) = df ij (1 − DOI(ej )), mej (Θ) = 1 −

m X

mj (i).

i=1

Then mej (j = 1, 2, · · · , n) are Mass functions. In a normalized matrix of score function values D = (dij )m×n , denote mej (xi ), mej (Θ) by mj (i) and mj (m + 1), respectively. mj (i) implies the belief measure that holds the alternative xi with the parameter ej and mj (m + 1) implies the belief measure of the whole uncertainty with parameter ej . Next, using D-S rule of evidence combination to compose mj (j = 1, 2, · · · , n), we get the belief measure of each alternative with all the parameters, by which the candidate alternatives are ranked and thus the best alternative is obtained.

4 4.1

Algorithm Algorithm

Based on the above analysis, the detailed step-wise procedure as an algorithm is given as follows: Input: An interval-value intuitionistic fuzzy soft set (F, A). Output: The optimal decision-making results. Step 1. Input an interval-value intuitionistic fuzzy soft set (F, A) and construct an interval-value intuitionistic fuzzy soft decision matrix induced by (F, A). Step 2. Compute the normalized matrix of score function values (D = (dij )m×n ). Step 3. Compute the mean of all the score function values (dei ) with respect to each alternative. Step 4. Compute the difference information between dij and dei . Step 5. Compute the gray mean relational degree between dij and dei . Step 6. Compute the uncertain degree DOI(ej ) of each parameter ej .

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Step 7. Compute the information structure image sequence df ij with respect to each parameter ej by Definition 3.3. Step 8. Compute Mass function values of the alternative xi and Θ with respect to the parameter ej by Theorem 3.4. Step 9. Compute belief measure of each alternative xi by combining these Mass functions mej (j = 1, 2, · · · , n) respectively by Definition 2.8. Step 10. The optimal decision is to select xk if ck = maxi {Bel(xi )}. k has more than one value then any one of xk may be optimal choices .

4.2

An illustrative example

Suppose that a fund manager in a wealth management wants to invest a company. Suppose that the set of four potential investment companies U = {x1 , x2 , x3 , x4 } which are characterized by a set of parameters A = {e1 , e2 , e3 , e4 }. For i = 1, 2, 3, 4, the parameters ei stand for “risk”, “growth ,“socio-political issues” ,and “environmental impacts”, respectively. The fund manager provide his/her assessment of each investment company on each parameter as an interval-valued intuitionistic fuzzy soft set (F, A). Its tabular representation is shown in Table 2. Table 2: Tabular representation of the interval-valued intuitionistic soft set (F, A) x1 x2 x3 x4

[0.4, [0.4, [0.3, [0.2,

e1 0.5],[0.3,0.4] 0.5],[0.4,0.5] 0.5],[0.4,0.5] 0.4],[0.4,0.5]

[0.4, [0.5, [0.1, [0.6,

e2 0.6],[0.2,0.4] 0.8],[0.1,0.2] 0.3],[0.2,0.4] 0.7],[0.2,0.3]

[0.1, [0.3, [0.7, [0.5,

e3 0.3],[0.5,0.6] 0.6],[0.3,0.4] 0.8],[0.1,0.2] 0.6],[0.2,0.3]

[0.5, [0.6, [0.5, [0.7,

e4 0.7],[0.2,0.3] 0.7],[0.1,0.3] 0.7],[0.1,0.2] 0.8],[0.1,0.2]

Now, we suppose that the four mutually exclusive and exhaustive investment companies consist a frame of discernment, denoted Θ = {x1 , x2 , x3 , x4 }. And we consider the set of parameters A = {e1 , e2 , e3 , e4 } as a set of evidences. Step 1. Construct an interval-valued intuitionistic fuzzy soft decision matrix induced by (F, A) as follows: 

([0.4, 0.5], [0.3, 0.4])  ([0.4, 0.5], [0.4, 0.5])  ([0.3, 0.5], [0.4, 0.5]) ([0.2, 0.4], [0.4, 0.5])

([0.4, 0.6], [0.2, 0.4]) ([0.5, 0.8], [0.1, 0.2]) ([0.1, 0.3], [0.2, 0.4]) ([0.6, 0.7], [0.2, 0.3])

([0.1, 0.3], [0.5, 0.6]) ([0.3, 0.6], [0.3, 0.4]) ([0.7, 0.8], [0.1, 0.2]) ([0.5, 0.6], [0.2, 0.3])

 ([0.5, 0.7], [0.2, 0.3]) ([0.6, 0.7], [0.1, 0.3])  ([0.5, 0.7], [0.1, 0.2])  ([0.7, 0.8], [0.1, 0.2])

Step 2. Compute the normalized matrix of score function values as follows: 

D = (dij )4×4

1.0000  0.6000 = 0.4000 0

0.5000 1.0000 0 0.8333

0 0.4737 1.0000 0.6842

 0 0.4000  0.4000  1.0000

Step 3. Compute the mean of all parameters with respect to each investment company xi as follows: de1 = 0.3750, de2 = 0.6184, de3 = 0.4500, de4 = 0.6294 9

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Step 4. Compute the difference information between dij and dei , and construct the difference matrix as follows: 

0.6250  0.0184 4D =  0.0500 0.6294

0.1250 0.3816 0.4500 0.2039

0.3750 0.1447 0.5500 0.0548

 0.3750 0.2184  0.0500  0.3706

Step 5. Compute the gray mean relational degree between dij and dei based on 4D as follows: 

(rij )4×4

0.3545  1.0000 = 0.9134 0.3528

0.7576 0.4784 0.4356 0.6423

0.4830 0.7251 0.3852 0.9015

 0.4830 0.6248  0.9134  0.4861

Step 6. Compute the uncertain degree of each parameter ej by Definition 3.2 as follows: DOI(e1 ) = 0.3609, DOI(e2 ) = 0.2963, DOI(e3 ) = 0.3279, DOI(e4 ) = 0.3254. Step 7. Compute the information structure image sequence with respect to each parameter and construct the matrix as follows:  0.5000  0.3000 f e D = (dij )4×4 =  0.2000 0

0.2143 0.4286 0 0.3571

0 0.2195 0.4634 0.3171

 0 0.2222  0.2222  0.5556

Step 8. Let 2Θ = {{x1 }, {x2 }, {x3 }, {x4 }, Θ}. Compute Mass function values of xi and Θ with respect to the parameter ej by Theorem 3.4: 

(mj (i))4×4

0.3195  0.1917 = 0.1278 0

0.1508 0.3016 0 0.2513

0 0.1475 0.3115 0.2131

 0 0.1499  0.1499  0.3748

and m1 (5) = 0.3609, m2 (5) = 0.2963, m3 (5) = 0.3279, m4 (5) = 0.3254, 4

1X mj (5) = 0.3276. 4 j=1 Step 9. We combine these Mass functions and compute each belief measure of each candidate xi respectively as follows: Bel({x1 }) = m1 ⊕ m2 ⊕ m3 ⊕ m4 ({x1 }) = 0.1098, Bel({x2 }) = m1 ⊕ m2 ⊕ m3 ⊕ m4 ({x2 }) = 0.3298, Bel({x3 }) = m1 ⊕ m2 ⊕ m3 ⊕ m4 ({x3 }) = 0.1700, Bel({x4 }) = m1 ⊕ m2 ⊕ m3 ⊕ m4 ({x4 }) = 0.3309, Bel({x5 }) = m1 ⊕ m2 ⊕ m3 ⊕ m4 (Θ) = 0.0595. 10

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Then the final rang order is x4 Â x2 Â x3 Â x1 . Step 10. x4 is the optimal investment company for maxi {Bel(xi )} = 0.3309. From the above results, the belief measure of the uncertainty with respect to the whole candidates Θ is declined from 0.3276 to 0.0595, after applying grey relational analysis to construct the corresponding Mass functions for different evidences and then using the rule of evidence combination to compose these information. This implies the above algorithm is effective and practical under uncertainties. It not only allows us to avoid selecting the suitable level soft set, but also helps reducing humanistic and subjective in nature to raise the choices decision level. Moreover, it broadens the application field of the grey system theory and D-S theory of evidence.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

PRODUCT-TYPE OPERATORS FROM WEIGHTED ZYGMUND SPACES TO BLOCH-ORLICZ SPACES YONG YANG AND ZHI-JIE JIANG

Abstract. Let D be the open unit disk in the complex plane C and H(D) the class of all analytic functions on D. Let ϕ be an analytic self-map of D and u ∈ H(D). The boundedness and compactness of the product-type operators Dn Mu Cϕ , Dn Cϕ Mu , Mu Dn Cϕ , Cϕ Dn Mu , Mu Cϕ Dn and Cϕ Mu Dn from weighted Zygmund spaces to Bloch-Orlicz spaces are characterized by constructing some test functions in weighted Zygmund spaces.

1. Introduction Let D = {z ∈ C : |z| < 1} be the open unit disk in the complex plane C and H(D) the class of all analytic functions on D. For α > 0, the weighted Zygmund space Z α consists of all f ∈ H(D) such that bZ α (f ) = sup(1 − |z|2 )α |f 00 (z)| < ∞. z∈D

It is a Banach space with the norm kf kZ α = |f (0)| + |f 0 (0)| + bZ α (f ). If α = 1, then it becomes the famous Zygmund space, usually denoted by Z. For some results of weighted Zygmund spaces and some concrete operators on them, see, for example, [9, 22, 24, 43, 56] and the references therein. Next we introduce the Bloch-Orlicz space which was defined by Ramos Fern´andez in [32]. Let Ψ be a Young’s function, i.e., Ψ is a strictly increasing convex function on [0, +∞) such that Ψ(0) = 0 and limt→+∞ Ψ(t) = +∞. The Bloch-Orlicz space B Ψ consists of all f ∈ H(D) such that sup(1 − |z|2 )Ψ(λ|f 0 (z)|) < ∞ z∈D

for some λ > 0 depending on f . The Minkowski’s functional n f0  o ≤1 kf kΨ = inf k > 0 : SΨ k defines a seminorm for B Ψ , where SΨ (f ) = sup(1 − |z|2 )Ψ(|f (z)|). z∈D

2000 Mathematics Subject Classification. Primary 47B38; Secondary 47B33, 47B37. Key words and phrases. Weighted Zygmund space, Bloch-Orlicz space, Product-type operator, Test function, Boundedness, Compactness. 1

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B Ψ becomes a Banach space with the norm kf kBΨ = |f (0)| + kf kΨ . Ramos Fern´andez in [32] proved that it is isometrically equal to a special µΨ -Bloch space, where 1 µΨ (z) = −1 , z ∈ D. 1 Ψ ( 1−|z| 2) Consequently, a equivalent norm on B Ψ is given by kf kBΨ = |f (0)| + bBΨ (f ), where bBΨ (f ) = sup µΨ (z)|f 0 (z)|. z∈D

Clearly, the quantity bBΨ (f ) is a seminorm on the space B Ψ and a norm on the quotient space B Ψ /P0 , where P0 is the set of all constant functions. The Bloch-Orlicz space generalizes some spaces. For example, if Ψ(t) = tp with p > 0, then B Ψ coincides with the weighted Bloch space B α , where α = 1/p; if Ψ(t) = t log(1 + t), then B Ψ coincides with the Log-Bloch space (see [2]). Let ϕ be an analytic self-map of D and u ∈ H(D). The weighted composition operator Wϕ,u on H(D) is defined by Wϕ,u f (z) = u(z)f (ϕ(z)), z ∈ D. If u ≡ 1, it becomes the composition operator, usually denoted by Cϕ . If ϕ(z) = z, it becomes the multiplication operator, usually denoted by Mu . Since Wϕ,u = Mu Cϕ , it is a product-type operator. For some studies on weighted composition operators, see, for example, [1, 4, 7, 10, 19, 22, 29, 42, 49, 50] and the references therein. Let n ∈ N0 = N ∪ {0}. The nth differentiation operator Dn on H(D) is defined by Dn f (z) = f (n) (z), z ∈ D, where f (0) = f . If n = 1, it is the well-known differentiation operator D. Zhu in [57] introduced the following, so-called, generalized weighted composition operator: n Dϕ,u f (z) = u(z)f (n) (ϕ(z)), z ∈ D. n If n = 0, it becomes the weighted composition operator. Since Dϕ,u = Mu Cϕ Dn , it is also a product-type operator. For generalized weighted composition operators, see, for n example, [3, 28, 47, 53, 54, 59, 60] and the references therein. Before the operator Dϕ,u some other product-type operators were introduced and studied. For example, the next product-type operators

Mu Cϕ D, Cϕ Mu D, Mu DCϕ , Cϕ DMu , DCϕ Mu , DMu Cϕ were studied by Sharma in [34]. They were also studied on weighted Bergman spaces by Stevi´c et al. in [51] and [52]. However, a normally systematic study of product-type operators started by Stevi´c et al. since the publication of papers [21] and [25]. Before that there were a few papers in the topic, e.g., [8]. The publication of paper [21] first attracted some attention in product-type operators DCϕ and Cϕ D (see, e.g., [23, 30, 39, 41] and the references therein). The publication of paper [25] attracted some attention in producttype operators involving integral-type ones (see, e.g., [16, 26, 37, 43, 48] and the references therein). Recently there is a great interest in various product-type operators between two given spaces of holomorphic functions (see, e.g., [11, 12, 17, 31, 33, 36, 38, 40, 45, 57] and the references therein). Before this paper some product-type operators from Zygmund spaces or weighted Zygmund spaces to some other spaces were studied, for example, in [3, 13, 14, 18, 27]. In this paper we consider the following product-type operators: Dn Mu Cϕ , Dn Cϕ Mu , Mu Dn Cϕ , Cϕ Dn Mu , Mu Cϕ Dn , Cϕ Mu Dn .

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The boundedness and compactness of operators in (1) from Zygmund spaces to BlochOrlicz spaces were characterized in [14]. As a continuation and completeness of our work, we consider the same problems for operators in (1) from weighted Zygmund spaces with α 6= 1 to Bloch-Orlicz spaces. Because these operators are more complicated than those above mentioned, we need seek some other test functions in weighted Zygmund spaces to achieve our objective. Let X and Y be Banach spaces. A linear operator L : X → Y is bounded if there exists a positive constant K such that kLf kY ≤ Kkf kX for all f ∈ X. The operator L : X → Y is compact if it maps bounded sets into relatively compact sets. The norm of the operator L : X → Y is defined by kLkX→Y = sup kLf kY . kf kX ≤1

In this paper, the letter C denotes a positive constant which may differ from one occurrence to the other. The notation a . b means that there exists a positive constant C such that a ≤ Cb. When a . b and b . a, we write a  b. 2. Preliminaries and test functions We first state the following result which was essentially proved in [35] and [46]. Lemma 2.1. For α > 0 and f ∈ Z α . Then 2 2 kf kZ and |f 0 (z)| ≤ 1−α kf kZ . (a) For 0 < α < 1, |f (z)| ≤ 1−α e 0 (b) For α = 1, |f (z)| ≤ kf kZ and |f (z)| ≤ kf kZ log 1−|z| 2. kf kZ 1 2 (c) For 1 < α < 2, |f (z)| ≤ (α−1)(2−α) kf kZ α and |f 0 (z)| ≤ α−1 (1−|z|2 )α−1 . e e 0 (d) For α = 2, |f (z)| ≤ 2kf kZ 2 log 1−|z| 2 and |f (z)| ≤ 1−|z|2 kf kZ 2 . α

(e) For α > 2, |f (z)| ≤

kf kZ α 1 (α−1)(α−2) (1−|z|2 )α−2

and |f 0 (z)| ≤

kf kZ α 2 α−1 (1−|z|2 )α−1 .

The following result directly follows from the corresponding result for the Bloch type spaces when a function f is replaced by f 0 (see, e.g., [55]). Lemma 2.2. For each k ∈ N and k ≥ 2, there exists a positive constant Ck independent of f ∈ Z α and z ∈ D such that |f (k) (z)| ≤

Ck kf kZ α . (1 − |z|2 )α+k−2

Let w ∈ D and i ∈ N0 . It is easily shown that the next function is in the space Z α rw,i (z) =

(1 − |w|2 )2+i , z ∈ D. (1 − wz)α+i

The following result provides the needed test functions for the cases 0 < α < 1, 1 < α < 2, α = 2 and α > 2. Lemma 2.3. (a) If 0 < α < 1, then for each fixed k ∈ {2, 3, . . . , n + 1}, there exist constants a0,k , a1,k , . . . , an+1,k such that the function fw,k (z) =

n+1 X

ai,k rw,i (z)

i=0

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satisfies (k)

fw,k (w) =

wk (j) and fw,k (w) = 0 (1 − |w|2 )α+k−2

(2)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. (b) If 1 < α ≤ 2, then for each fixed k ∈ {1, 2, . . . , n + 1}, there exist constants b0,k , b1,k , . . . , bn+1,k such that the function gw,k (z) =

n+1 X

bi,k rw,i (z)

i=0

satisfies (k)

gw,k (w) =

wk (j) and gw,k (w) = 0 (1 − |w|2 )α+k−2

(3)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. (c) If α > 2, then for each fixed k ∈ {0, 1, . . . , n + 1}, there exist constants c0,k , c1,k , . . . , cn+1,k such that the function hw,k (z) =

n+1 X

ci,k rw,i (z)

i=0

satisfies (k)

hw,k (w) =

wk (j) and hw,k (w) = 0 (1 − |w|2 )α+k−2

(4)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. Proof. (a). From a calculation, it follows that (2) is equivalent to the following system  n+1 P   (α + i)ai,k = 0    i=0    n+1 P    (α + i)(α + i + 1)ai,k = 0    i=0   ······· (5) n+1 k−1 P Q   (α + i + j)a = 1  i,k    i=0 j=0    ·······    n+1 n P Q    (α + i + j)ai,k = 0.  i=0 j=0

Hence, we only need to prove that there exist constants a0,k , a1,k , . . . , an+1,k such that the system (5) holds. By Lemma 3 in [47], the determinant of the system (5) equals to Qn+1 j=1 j!, which is different from zero. So there exist constants a0,k , a1,k , . . . , an+1,k such that the system (5) holds. Results (b) and (c) can be proved similarly, so we omit.  Let w ∈ D and

 qw (z) = 1 + log2

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Lemma 2.4. For the function qw , it follows that (k) qw (w) = ck

e wk wk + d log−1 , k (1 − |w|2 )k (1 − |w|2 )k 1 − |w|2

(6)

where ck > 0 for each k ≥ 1, d1 = 0 and dk > 0 for each k ≥ 2. Proof. By a direct computation, we have w e e , log log−1 1 − wz 1 − wz 1 − |w|2

(7)

w2 e w2 e e log +2 log−1 . log−1 2 2 (1 − wz) 1 − wz 1 − |w| (1 − wz)2 1 − |w|2

(8)

0 qw (z) = 2

and 00 qw (z) = 2

Also, from a direct computation, we see that for k ≥ 2 wk e e log log−1 (1 − wz)k 1 − wz 1 − |w|2   wk e + k − 1 + 2(k − 1)! log−1 . (1 − wz)k 1 − |w|2

(k) qw (z) = 2(k − 1)!

(9)

Set ck = 2(k − 1)!, d1 = 0 and dk = k − 1 + 2(k − 1)! for k ≥ 2. Then (6) follows from (7)-(9).  Remark 2.1. Let Xw be the functions in Lemmas 2.3 and 2.4. Then sup kXw kZ α . 1,

(10)

w∈D

and Xw → functions in and rw,i → Lemma 2.4,

0 uniformly on compact subsets of D as |w| → 1. In fact, if Xw are the Lemma 2.3, then this remark follows from the facts that supw∈D krw,i kZ α . 1 0 uniformly on compact subsets of D as |w| → 1; if Xw is the function in then it follows from [44].

Stevi´c in [47] used Fa` a di Bruno’s formula of the following version (f ◦ ϕ)(n) (z) =

n X

f (k) (ϕ(z))Bn,k (ϕ0 (z), . . . , ϕ(n−k+1) (z)),

(11)

k=0

where Bn,k (x1 , ..., xn−k+1 ) is the Bell polynomial. See [15] for the Fa`a di Bruno’s formula. For n ∈ N the sum can go from k = 1 since Bn,0 (ϕ0 (z), ..., ϕ(n−k+1) (z)) = 0, however we will keep the summation since for n = 0 the only existing term B0,0 is equal to 1. From (11) and the Leibniz formula the next lemma follows. Lemma 2.5. Let f , u ∈ H(D) and ϕ be an analytic self-map of D. Then n+1 X X j

(n+1) n+1 u(z)f (ϕ(z)) = f (k) (ϕ(z)) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) . k=0

j=k

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3. Boundedness the product-type operators We first characterize the boundedness of the operator Dn Mu Cϕ : Z α → B Ψ . j the binomial coeffiTheorem 3.1. Let ϕ be an analytic self-map of D, u ∈ H(D), Cn+1 cient and 0 < α < 1. Then the following statements are equivalent.

(a) The operator Dn Mu Cϕ : Z α → B Ψ is bounded. (b) The functions u and ϕ satisfy the following conditions: n+1
X j I0 := sup µΨ (z) Cn+1 u(n+1−j) (z)Bj,0 ϕ0 (z), . . . , ϕ(j+1) (z) < ∞, z∈D

j=0

n+1
X j I1 := sup µΨ (z) Cn+1 u(n+1−j) (z)Bj,1 ϕ0 (z), . . . , ϕ(j) (z) < ∞, z∈D

j=1

and n+1
P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) j=k

Ik := sup

(1 − |ϕ(z)|2 )α+k−2

z∈D

k, we get k n+1 X X
0 (j) i Dn Mu Cϕ hk (z) = hk (ϕ(z)) Cn+1 u(n+1−i) (z)Bi,j (ϕ0 (z), . . . , ϕ(i−j+1) (z)) j=0

=

k X j=0

k · · · (k − j + 1)(ϕ(z))k−j

i=j n+1 X

i Cn+1 u(n+1−i) (z)Bi,j (ϕ0 (z), . . . , ϕ(i−j+1) (z)). (15)

i=j

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From (15), the boundedness of function ϕ and the triangle inequality, by noticing that the coefficient at n+1 X j Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) j=k

is independent of z and finally using hypothesis (14) we easily obtain n+1 X j Lk := sup µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) ≤ CkDn Mu Cϕ k. z∈D

j=k

(16) By induction we see that (16) holds for each k ∈ {0, 1, . . . , n + 1}. For a fixed w ∈ D and a fixed k ∈ {2, 3, . . . , n + 1}, by Lemma 2.3 (a) there exists a function n+1 X fw,k (z) = ai,k rϕ(w),i (z) i=0

such that k

(k)

fw,k (ϕ(w)) =

ϕ(w) (j) and fw,k (ϕ(w)) = 0 (1 − |ϕ(w)|2 )α+k−2

(17)

for each j ∈ {0, 1, . . . , n + 1} \ {k}, and sup kfw,k kZ α ≤ C.

(18)

w∈D

Then by (17), (18) and the boundedness of Dn Mu Cϕ : Z α → B Ψ , we have n+1
P j µΨ (w)|ϕ(w)|k Cn+1 u(n+1−j) (w)Bj,k ϕ0 (w), . . . , ϕ(j−k+1) (w) j=k

Ik (w) := n

≤ kD Mu Cϕ fw,k kBΨ

(1 − |ϕ(w)|2 )α+k−2 ≤ CkDn Mu Cϕ k.

(19)

From (19) we see that

sup Ik (z) ≤ C Dn Mu Cϕ , z∈D

which leads to n+1 P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) sup

j=k

(1 − |ϕ(z)|2 )α+k−2

|ϕ(z)|>1/2

≤ CkDn Mu Cϕ k. (20)

On the other hand, by (16) we have n+1 P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) sup |ϕ(z)|≤1/2

j=k

(1 − |ϕ(z)|2 )α+k−2

≤ CkDn Mu Cϕ k. (21)

Hence from (20) and (21) we obtain Ik ≤ CkDn Mu Cϕ k < ∞.

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(b) ⇒ (a). By Lemmas 2.1, 2.2 and 2.5, for all f ∈ Z α we have
0 sup µΨ (z) (Dn Mu Cϕ f (z) z∈D n+1 n+1 X j
X (k) = sup µΨ (z) f (ϕ(z)) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) z∈D

k=0

≤ sup µΨ (z)

n+1 X

z∈D

≤



j=k

X j (k) n+1 f (ϕ(z)) C

(n+1−j) (z)Bj,k n+1 u


ϕ0 (z), . . . , ϕ(j−k+1) (z)

j=k

k=0

1 (I0 + I1 ) + 1−α

n+1 X

 Ck Ik kf kZ α .

(23)

|(Dn Mu Cϕ f )(0)| ≤ Ckf kZ α .

(24)

k=2

It is clear that

Hence, from (23) and (24) it follows that the operator Dn Mu Cϕ : Z α → B Ψ is bounded. Clearly, if the operator Dn Mu Cϕ : Z α → B Ψ is bounded, then the operator Dn Mu Cϕ : α Z → B Ψ /P0 is also bounded. By the definition of the norm in the quotient spaces, and using the same functions in the proofs of (12), (13) and (22), we obtain Ik ≤ CkDn Mu Cϕ kZ α →BΨ /P0 , for each k ∈ {0, 1, 2, . . . , n + 1}, and then n+1 X

Ik ≤ CkDn Mu Cϕ kZ α →BΨ /P0 .

(25)

k=0

By (23) we have kDn Mu Cϕ kZ α →BΨ /P0 ≤ C

n+1 X

Ik .

(26)

k=0

The asymptotic expression of kDn Mu Cϕ kZ α →BΨ /P0 follows from (25) and (26).



Remark 3.1. In fact, from the fact z k ∈ Z α , in the proof of Theorem 3.1 we have seen that if the operator Dn Mu Cϕ : Z α → B Ψ is bounded, then Lk < ∞ for all α > 0. j Theorem 3.2. Let ϕ be an analytic self-map of D, u ∈ H(D), Cn+1 the binomial coefficient and 1 < α < 2. Then the following statements are equivalent.

(a) The operator Dn Mu Cϕ : Z α → B Ψ is bounded. (b) The functions u and ϕ are such that I0 < ∞ and for each k ∈ {1, 2, . . . , n + 1} n+1
P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) Mk := sup z∈D

j=k

(1 − |ϕ(z)|2 )α+k−2

< ∞.

Moreover, if the operator Dn Mu Cϕ : Z α → B Ψ is bounded, then n+1 X

n

D Mu Cϕ α Ψ  I + Mk . 0 Z →B /P 0

k=1

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Proof. (a) ⇒ (b). Let h0 (z) ≡ 1 ∈ Z α . Then I0 < ∞. For a fixed w ∈ D and each fixed k ∈ {1, 2, . . . , n + 1}, by Lemma 2.3 (b) there exists a function gw,k (z) =

n+1 X

bi,k rϕ(w),i (z)

i=0

such that k

(k)

gw,k (ϕ(w)) =

ϕ(w) (j) and gw,k (ϕ(w)) = 0 (1 − |ϕ(w)|2 )α+k−2

(27)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. Moreover, sup kgw,k kZ α ≤ C.

(28)

w∈D

Then from (27), (28) and the boundedness of Dn Mu Cϕ : Z α → B Ψ , we have n+1
P j Cn+1 u(n+1−j) (w)Bj,k ϕ0 (w), . . . , ϕ(j−k+1) (w) µΨ (w)|ϕ(w)|k j=k

Mk (w) :=

≤ kDn Mu Cϕ gϕ(w),k kBΨ

(1 − |ϕ(w)|2 )α+k−2 ≤ CkDn Mu Cϕ k.

(29)

From (29) we see

sup Mk (z) ≤ C Dn Mu Cϕ ,

(30)

z∈D

and then n+1 P j Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) µΨ (z) j=k

sup

(1 − |ϕ(z)|2 )α+k−2

|ϕ(z)|>1/2

≤ CkDn Mu Cϕ k. (31)

On the other hand, by using the fact Lk < ∞ for each k ∈ {0, 1, . . . , n + 1}, we get n+1 P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) j=k

sup

(1 − |ϕ(z)|2 )α+k−2

|ϕ(z)|≤1/2

≤ CkDn Mu Cϕ k. (32)

Hence from (31) and (32) we see that Mk < ∞ for each k ∈ {1, 2, ..., n + 1}. (b) ⇒ (a). By Lemmas 2.1, 2.2 and 2.5, for all f ∈ Z α we have
0 sup µΨ (z) (Dn Mu Cϕ f (z) z∈D n+1 n+1 X j
X (k) = sup µΨ (z) f (ϕ(z)) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) z∈D

≤ sup µΨ (z) z∈D

≤



k=0 n+1 X k=0

j=k

X j (k) n+1 f (ϕ(z)) C

(n+1−j) (z)Bj,k n+1 u


ϕ0 (z), . . . , ϕ(j−k+1) (z)

j=k n+1

 X I0 2M1 + + Ck Mk kf kZ α . (α − 1)(2 − α) α − 1

(33)

k=2

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It is clear that |(Dn Mu Cϕ f )(0)| ≤ Ckf kZ α .

(34)

Hence from (33) and (34) it follows that the operator Dn Mu Cϕ : Z α → B Ψ is bounded. Similarly is obtained the asymptotic formula of kDn Mu Cϕ kZ α →BΨ /P0 , hence we omit.  j the binomial coeffiTheorem 3.3. Let ϕ be an analytic self-map of D, u ∈ H(D), Cn+1 cient and α = 2. Then the following statements are equivalent.

(a) The operator Dn Mu Cϕ : Z 2 → B Ψ is bounded. (b) The functions u and ϕ satisfy the following conditions: n+1
X j R0 := sup µΨ (z) Cn+1 u(n+1−j) (z)Bj,0 ϕ0 (z), . . . , ϕ(j+1) (z) log z∈D

j=0

e < ∞. 1 − |ϕ(z)|2

and for each k ∈ {1, 2, . . . , n + 1} n+1
P j Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) µΨ (z) j=k

Rk := sup

< ∞.

(1 − |ϕ(z)|2 )k

z∈D

Moreover, if the operator Dn Mu Cϕ : Z 2 → B Ψ is bounded, then n+1 X

n

D Mu Cϕ α Ψ  Rk . Z →B /P0 k=0

Proof. (a) ⇒ (b). By using Lemma 2.3 (b), we can prove that Rk < ∞ for each k ∈ {1, 2, . . . , n + 1}, so we do not give the proof again. For a fixed w ∈ D, by Lemma 2.4 there exists a function sϕ(w) (z) = pϕ(w) (z) +

n+1 X

di rϕ(w),i (z)

i=0

such that sϕ(w) (ϕ(w)) = log

e (j) and sϕ(w) (ϕ(w)) = 0 1 − |ϕ(w)|2

(35)

for each j ∈ {1, 2, . . . , n + 2}, moreover, supw∈D ksϕ(w) kZ 2 ≤ C. Then from these and the boundedness of Dn Mu Cϕ : Z 2 → B Ψ , we have X j n+1
R0 (w) := µΨ (w) Cn+1 u(n+1−j) (w)Bj,0 ϕ0 (w), . . . , ϕ(j+1) (w) log j=0

e 1 − |ϕ(w)|2

n

≤ kD Mu Cϕ sϕ(w) kBΨ ≤ C Dn Mu Cϕ .

(36)

Then from (36) it follows that R0 < ∞. (b) ⇒ (a). From Lemmas 2.1, 2.2 and 2.5, for all f ∈ Z 2 we have
0 sup µΨ (z) (Dn Mu Cϕ f (z) z∈D n+1 n+1 X j
X (k) = sup µΨ (z) f (ϕ(z)) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) z∈D

k=0

j=k

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≤ sup µΨ (z) z∈D

n+1 X

11

X j (k) n+1 f (ϕ(z)) C


(n+1−j) 0 (j−k+1) u (z)B ϕ (z), . . . , ϕ (z) j,k n+1

j=k

k=0

n+1   X ≤ 2R0 + eR1 + Ck Rk kf kZ 2 .

(37)

k=2

It is clear that |(Dn Mu Cϕ f )(0)| ≤ Ckf kZ 2 .

(38)

Hence from (37) and (38) it follows that the operator Dn Mu Cϕ : Z 2 → BΨ is bounded. The asymptotic expression of kDn Mu Cϕ kZ α →BΨ /P0 can be similarly obtained.  Theorem 3.4. Let ϕ be an analytic self-map of D, u ∈ H(D) and α > 2. Then the following statements are equivalent. (a) The operator Dn Mu Cϕ : Z α → B Ψ is bounded. (b) The functions u and ϕ satisfy n+1
P j Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) µΨ (z) Sk := sup z∈D

j=k

< ∞, k = 0, . . . , n + 1.

(1 − |ϕ(z)|2 )α+k−2

Moreover, if the operator Dn Mu Cϕ : Z α → B Ψ is bounded, then n+1 X

n

D Mu Cϕ α Ψ  Sk . Z →B /P0 k=0

Proof. Similarly to the proofs of Theorems 3.1-3.3, this result can be proved.



Remark 3.2. By using the similar methods and techniques, the boundedness of the operators Dn Cϕ Mu , Cϕ Dn Mu , Mu Dn Cϕ , Mu Cϕ Dn and Cϕ Mu Dn from weighted Zygmund spaces to Bloch-Orlicz spaces can be characterized, so we omit. 4. Compactness of the product-type operators The first result is an alternative to Proposition 3.11 in [5], which characterizes the compactness in terms of sequential convergence. So the proof is omitted. Lemma 4.1. Let T ∈ {Dn Mu Cϕ , Dn Cϕ Mu , Mu Dn Cϕ , Cϕ Dn Mu , Mu Cϕ Dn , Cϕ Mu Dn }. Then the bounded operator T : Z α → B Ψ is compact if and only if for every bounded sequence {fj } in Z α such that fj → 0 uniformly on compact subsets of D as j → ∞, it follows that limj→∞ kT fj kBΨ = 0. The following lemma was proved in [46]. Lemma 4.2. (a) If 0 < α < 2 and {fj } is a bounded sequence in Z α which uniformly converges to zero on compact subsets of D as j → ∞, then lim sup |fj (z)| = 0.

j→∞ z∈D

(b) If 0 < α < 1 and {fj } is a bounded sequence in Z α which uniformly converges to zero on compact subsets of D as j → ∞, then lim sup |fj0 (z)| = 0.

j→∞ z∈D

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Now we characterize the compactness of the operator Dn Mu Cϕ : Z α → B Ψ . Theorem 4.1. Let ϕ be an analytic self-map of D, u ∈ H(D) and 0 < α < 1. Then the following statements are equivalent. (a) The operator Dn Mu Cϕ : Z α → B Ψ is compact. (b) The functions u and ϕ satisfy Lk < ∞ for each k ∈ {0, 1, . . . , n + 1}, and for each k ∈ {2, 3, . . . , n + 1} n+1
P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) j=k

lim

= 0.

(1 − |ϕ(z)|2 )α+k−2

|ϕ(z)|→1

Proof. (a) ⇒ (b). Suppose that the operator Dn Mu Cϕ : Z α → B Ψ is compact. Clearly the operator Dn Mu Cϕ : Z α → B Ψ is bounded. By Remark 2.1, Lk < ∞ for each k ∈ {0, 1, . . . , n + 1}. Consider a sequence {ϕ(zi )} in D such that |ϕ(zi )| → 1 as i → ∞. If such a sequence does not exist, then the last condition in (b) obviously holds. Without loss of generality, we may suppose that |ϕ(zi )| > 1/2 for all i ∈ N. For each fixed k ∈ {2, 3, . . . , n + 1}, using this sequence we define the function sequence fi,k (z) = fϕ(zi ),k (z), i ∈ N. Then by Lemma 2.3 (a) we have that supi∈N kfi,k kZ α ≤ C and fi,k → 0 uniformly on every compact subset of D as i → ∞, moreover k

(k)

fi,k (ϕ(zi )) =

ϕ(zi ) (j) and fi,k (ϕ(zi )) = 0 (1 − |ϕ(zi )|2 )α+k−2

(39)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. By Lemma 4.1 and (39), we have n+1 P j Cn+1 u(n+1−j) (zi )Bj,k (ϕ0 (zi ), . . . , ϕ(j−k+1) (zi )) µΨ (zi ) lim

j=k

= 0.

(1 − |ϕ(zi )|2 )α+k−2

i→∞

(40)

(b) ⇒ (a). We first check that Dn Mu Cϕ : Z α → B Ψ is bounded. We observe that the last condition in (b) implies that for every ε > 0, there is an η ∈ (0, 1) such that for all z ∈ K = {z ∈ D : |ϕ(z)| > η} and for each k ∈ {2, 3, . . . , n + 1} n+1 P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) j=k

< ε.

(1 − |ϕ(z)|2 )α+k−2

(41)

From the fact Lk < ∞ for each k ∈ {2, 3, . . . , n + 1}, and (41), we have Ik ≤ ε +

Lk . (1 − η 2 )α+k−2

(42)

From (42) and the fact Lk < ∞, it follows that Dn Mu Cϕ : Z α → B Ψ is bounded. To prove that Dn Mu Cϕ : Z α → B Ψ is compact, by Lemma 4.1 we just need to prove that, if {fi } is a sequence in Z α such that supi∈N kfi kZ α ≤ M and fi → 0 uniformly on any compact subset of D as i → ∞, then lim kDn Mu Cϕ fi kBΨ = 0.

i→∞

For such chosen ε and η, by using (39), Lemma 2.1 and Lemma 2.2, we have

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13


0 sup µΨ (z) (Dn Mu Cϕ fi (z) z∈D n+1 n+1 X j
X (k) = sup µΨ (z) fi (ϕ(z)) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) z∈D

k=0

≤ sup µΨ (z) z∈D

n+1 X

j=k

X j (k) n+1 f (ϕ(z)) C


(n+1−j) 0 (j−k+1) u (z)B ϕ (z), . . . , ϕ (z) j,k n+1

i

j=k

k=0

n+1
X j ≤ sup µΨ (z) Cn+1 u(n+1−j) (z)Bj,0 ϕ0 (z), . . . , ϕ(j+1) (z) fi (ϕ(z)) z∈D

j=0

n+1
X j + sup µΨ (z) Cn+1 u(n+1−j) (z)Bj,1 ϕ0 (z), . . . , ϕ(j) (z) fi0 (ϕ(z)) z∈D

j=1

+ sup + sup z∈K

z∈D\K

n+1 X j X (k) n+1

(n+1−j) 0 (j−k+1) f (ϕ(z)) C u (z)B µΨ (z) ϕ (z), . . . , ϕ (z) j,k n+1 i j=k

k=2

≤ L0 sup fi (ϕ(z)) + L1 sup fi0 (ϕ(z)) + z∈D

z∈D

n+1 X k=2

(k) Lk sup fi (z) + Cε.

(43)

|z|≤η

From (43), Lemma 4.2 and the fact fi → 0 uniformly on compact subsets of D as i → ∞ (k) implies that for each k ∈ N, fi → 0 uniformly on compact subsets of D as i → ∞, we finally get lim sup µΨ (z) (Dn Mu Cϕ fi )0 (z) = 0. (44) i→∞ z∈D

It is clear that lim (Dn Mu Cϕ fi )(0) = 0.

(45)

i→∞

From (44) and (45) we obtain lim kDn Mu Cϕ fi kBΨ = 0.

i→∞

This shows that the operator Dn Mu Cϕ : Z α → B Ψ is compact.



Theorem 4.2. Let ϕ be an analytic self-map of D, u ∈ H(D) and 1 < α < 2. Then the following statements are equivalent. (a) The operator Dn Mu Cϕ : Z α → B Ψ is compact. (b) The functions u and ϕ are such that Lk < ∞ for each k ∈ {0, 1, . . . , n + 1}, and for each k ∈ {1, 2, . . . , n + 1} n+1
P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) lim

|ϕ(z)|→1

j=k

(1 − |ϕ(z)|2 )α+k−2

= 0.

Proof. (a) ⇒ (b). Suppose that the operator Dn Mu Cϕ : Z α → B Ψ is compact. Obviously the operator Dn Mu Cϕ : Z α → B Ψ is bounded. Then Lk < ∞ for each k ∈ {0, 1, . . . , n+1}. Consider a sequence {ϕ(zi )}i∈N in D such that |ϕ(zi )| → 1 as i → ∞. If such a sequence does not exist, then the last condition in (b) obviously holds. Without loss of generality,

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we may suppose that |ϕ(zi )| > 1/2 for all i ∈ N. For each fixed k ∈ {1, 2, . . . , n + 1}, by using this sequence we define the function sequence gi,k (z) = gϕ(zi ),k (z), i ∈ N. Then from Lemma 2.3 (b) we see that supi∈N kgi,k kZ α ≤ C and gi,k → 0 uniformly on every compact subset of D as i → ∞, moreover k

(k) gi,k (ϕ(zi ))

ϕ(zi ) (j) = and gi,k (ϕ(zi )) = 0 (1 − |ϕ(zi )|2 )α+k−2

(46)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. From Lemma 4.1 and (46), for each fixed k ∈ {1, 2, . . . , n + 1} we have n+1 P j Cn+1 u(n+1−j) (zi )Bj,k (ϕ0 (zi ), . . . , ϕ(j−k+1) (zi )) µΨ (zi ) j=k

lim

= 0.

(1 − |ϕ(zi )|2 )α+k−2

i→∞

(47)

(b) ⇒ (a). We first check that Dn Mu Cϕ : Z α → B Ψ is bounded. We observe that the last condition in (b) implies that for every ε > 0, there is an η ∈ (0, 1) such that for all z ∈ K = {z ∈ D : |ϕ(z)| > η} and for each k ∈ {1, 2, . . . , n + 1} n+1 P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) j=k

< ε.

(1 − |ϕ(z)|2 )α+k−2

(48)

From the fact Lk < ∞ for each k ∈ {0, 1, . . . , n + 1}, and (48), we have Mk ≤ ε +

Lk . (1 − η 2 )α+k−2

(49)

From (49) and the fact I0 = L0 < ∞, it follows that Dn Mu Cϕ : Z α → B Ψ is bounded. In order to prove that Dn Mu Cϕ : Z α → B Ψ is compact, by Lemma 4.1 we just need to prove that, if {fi } is a sequence in Z α such that supi∈N kfi kZ α ≤ M and fi → 0 uniformly on any compact subset of D as i → ∞, then limi→∞ kDn Mu Cϕ fi kBΨ = 0. For such chosen ε and η, by using (46), Lemma 2.1 and Lemma 2.2, we have
0 sup µΨ (z) (Dn Mu Cϕ fi (z) z∈D n+1 n+1 X j
X (k) = sup µΨ (z) fi (ϕ(z)) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) z∈D

k=0

≤ sup µΨ (z) z∈D

n+1 X

j=k

X j (k) n+1 f (ϕ(z)) C

n+1 u

i

k=0

(n+1−j)


(z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z)

j=k

n+1
X j ≤ sup µΨ (z) Cn+1 u(n+1−j) (z)Bj,0 ϕ0 (z), . . . , ϕ(j+1) (z) fi (ϕ(z)) z∈D

j=0

+ sup + sup z∈K

z∈D\K

n+1 X (k) X j n+1

(n+1−j) 0 (j−k+1) f (ϕ(z)) µΨ (z) u (z)B ϕ (z), . . . , ϕ (z) C j,k n+1 i

≤ L0 sup fi (ϕ(z)) + z∈D

k=1 n+1 X k=1

j=k

(k) Lk sup fi (z) + Cε.

(50)

|z|≤η

From (50), Lemma 4.2 and the fact fi → 0 uniformly on compact subsets of D as i → ∞ (k) implies that for each k ∈ N, fi → 0 uniformly on compact subsets of D as i → ∞, we

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15

get lim sup µΨ (z) (Dn Mu Cϕ fi )0 (z) = 0.

(51)

lim (Dn Mu Cϕ fi )(0) = 0.

(52)

i→∞ z∈D

It is clear that i→∞

From (51) and (52) we obtain lim kDn Mu Cϕ fi kBΨ = 0.

i→∞

This shows that the operator Dn Mu Cϕ : Z α → B Ψ is compact.



Theorem 4.3. Let ϕ be an analytic self-map of D, u ∈ H(D) and α = 2. Then the following statements are equivalent. (a) The operator Dn Mu Cϕ : Z 2 → B Ψ is compact. (b) The functions u and ϕ are such that Lk < ∞ for each k ∈ {0, 1, . . . , n + 1}, lim

|ϕ(z)|→1

n+1
X j µΨ (z) Cn+1 u(n+1−j) (z)Bj,0 ϕ0 (z), . . . , ϕ(j+1) (z) log j=0

e = 0, 1 − |ϕ(z)|2

and for each k ∈ {1, 2, . . . , n + 1} n+1
P j Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) µΨ (z) j=k

lim

= 0.

(1 − |ϕ(z)|2 )k

|ϕ(z)|→1

Proof. (a) ⇒ (b). Suppose that the operator Dn Mu Cϕ : Z 2 → B Ψ is compact. Clearly the operator Dn Mu Cϕ : Z 2 → BΨ is bounded. Then Lk < ∞ for each k ∈ {0, 1, . . . , n + 1}. Consider a sequence {ϕ(zi )}i∈N in D such that |ϕ(zi )| → 1 as i → ∞. If such a sequence does not exist, then the last two conditions in (b) obviously hold. Without loss of generality, we may suppose that |ϕ(zi )| > 1/2 for all i ∈ N. For each fixed k ∈ {1, 2, . . . , n + 1}, by using this sequence we define the function sequence gi,k (z) = gϕ(zi ),k (z), i ∈ N. Then from Lemma 2.3 (b) we see that supi∈N kgi,k kZ 2 ≤ C and gi,k → 0 uniformly on every compact subset of D as i → ∞, moreover k

(k)

gi,k (ϕ(zi )) =

ϕ(zi ) (j) and gi,k (ϕ(zi )) = 0 (1 − |ϕ(zi )|2 )k

(53)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. From Lemma 4.1 and (53), for each fixed k ∈ {1, 2, . . . , n + 1} we have n+1 P j µΨ (zi ) Cn+1 u(n+1−j) (zi )Bj,k (ϕ0 (zi ), . . . , ϕ(j−k+1) (zi )) lim

j=k

= 0.

(1 − |ϕ(zi )|2 )k

i→∞

(54)

Now consider another function sequence qi (z) = qϕ(zi ) (z). Then by Lemma 2.4 we have k

(k) qi (ϕ(zi ))

k

ϕ(zi ) ϕ(zi ) e = ck + dk log−1 , (1 − |ϕ(zi )|2 )k (1 − |ϕ(zi )|2 )k 1 − |ϕ(zi )|2

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where ck > 0 for each k ≥ 1, d1 = 0 and dk > 0 for each k ≥ 2. Moreover, supi∈N kqi kZ 2 ≤ C, and qi → 0 uniformly on every compact subset of D as i → ∞. From Lemma 4.1, we get lim kDn Mu Cϕ qi kBΨ = 0.

(56)

i→∞

By (55) and the triangle inequality, we have n+1
 X j µΨ (zi ) Cn+1 u(n+1−j) (zi )Bj,0 ϕ0 (zi ), . . . , ϕ(j+1) (zi ) log j=0

≤ kDn Mu Cϕ qi kBΨ +

n+1 X

n+1 P j ck µΨ (zi )|ϕ(zi )|k Cn+1 u(n+1−j) (zi )Bj,k (ϕ0 (zi ), . . . , ϕ(j−k+1) (zi )) j=k

(1 − |ϕ(zi )|2 )k

k=1

+

n+1 X

 e e + log−1 2 2 1 − |ϕ(zi )| 1 − |ϕ(zi )|

n+1 P j dk µΨ (zi )|ϕ(zi )|k Cn+1 u(n+1−j) (zi )Bj,k (ϕ0 (zi ), . . . , ϕ(j−k+1) (zi )) j=k

(1 − |ϕ(zi

k=1

log−1

)|2 )k

e . 1 − |ϕ(zi )|2 (57)

Therefore, taking the limit in (57) as i → ∞, from (54), (56) and the fact e log−1 → 0 as i → ∞, 1 − |ϕ(zi )|2 we get n+1
X j lim µΨ (zi ) Cn+1 u(n+1−j) (zi )Bj,0 ϕ0 (z), . . . , ϕ(j+1) (zi ) log

i→∞

j=0

e = 0. 1 − |ϕ(zi )|2

(b) ⇒ (a). We first check that Dn Mu Cϕ : Z 2 → B Ψ is bounded. We observe that the conditions in (b) imply that for every ε > 0, there is an η ∈ (0, 1), such that for any z ∈ K = {z ∈ D : |ϕ(z)| > η} n+1
X j Cn+1 u(n+1−j) (z)Bj,0 ϕ0 (z), . . . , ϕ(j+1) (z) log µΨ (z) j=0

e λ} (λ ∈ [0, 1)), = {(x) ∈ U : A+ (x) > λ]} (λ ∈ [0, 1)).

Definition 3.4 ([4, 25]). Let A ∈ F (i) (U ) and [α, β] ∈ [I]. Denote A[α,β] = {x ∈ U : A− (x) ≥ α, A+ (x) ≥ β]}, A[α,β]+ = {x ∈ U : A(x) > [α, β]}, A(α,β) = {x ∈ U : A− (x) > α, A+ (x) > β]}. Then A[α,β] (resp.A[α,β]+ , A(α,β) ) is called the [α, β]-level (resp. strong [α, β]level, (α, β)-level) set of A. Obviously, A(α,β) ⊆ A[α,β]+ ⊆ A[α,β] . Proposition 3.5 ([4, 25]). Let A, B ∈ F (i) (U ) and [α, β] ∈ [I]. Then (1) A ⊆ B =⇒ A[α,β]+ ⊆ B[α,β]+ ; (2) (A ∪ B)[α,β]+ ⊇ A[α,β]+ ∪ B[α,β]+ ; (2) (A ∩ B)[α,β]+ = A[α,β]+ ∩ B[α,β]+ .

5

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Let R ∈ F (i) (U × U ). Denote Rλ Rλ Rλ+ +

Rλ R[α,β] R[α,β]+

= {(x, y) ∈ U × U : R− (x, y) ≥ λ} (λ ∈ I), = {(x, y) ∈ U × U : R+ (x, y) ≥ λ]} (λ ∈ I), = {(x, y) ∈ U × U : R− (x, y) > λ} (λ ∈ [0, 1)), = {(x, y) ∈ U × U : R+ (x, y) > λ]} (λ ∈ [0, 1)), = {(x, y) ∈ U × U : R(x, y) ≥ [α, β]} ([α, β] ∈ [I]), = {(x, y) ∈ U × U : R(x, y) > [α, β]} (α < 1, [α, β] ∈ [I]).

Proposition 3.6. Let R be an IVF relation on U . + (1) If R is reflexive, then Rλ , Rλ , Rλ+ , Rλ and R[α,β]+ are reflexive. + (2) If R is transitive, then Rλ , Rλ , Rλ+ , Rλ and R[α,β]+ are transitive. Proof. (1) are obvious. (2) For any x, y, z ∈ U , if (x, y), (y, z) ∈ Rλ , we have R− (x, y) ≥ λ and − R (y, z) ≥ λ. Note that R is transitive. Then R(x, z) ≥ R(x, y) ∧ R(y, z) and so R− (x, z) ≥ R− (x, y) ∧ R− (y, z) ≥ λ. Thus (x, z) ∈ Rλ . Hence Rλ is transitive. + Similarly, We can prove that Rλ , Rλ+ and Rλ are transitive. For any x, y, z ∈ U , if (x, y), (y, z) ∈ R[α,β]+ , we have R(x, y) > [α, β] and R(y, z) > [α, β]. Note that R is transitive. Then R(x, z) ≥ R(x, y) ∧ R(y, z) > [α, β]. and so (x, z) ∈ R[α,β]+ . Hence R[α,β]+ is transitive. Theorem 3.7. Let (U, R) be an IVF approximation space. Then IVF rough approximation operator be represented as follows: for each A ∈ F (i) (U ), S can S − 1−λ (1) (R(A)) = λR (Aλ ) = λR1−λ (Aλ+ ), λ∈I λ∈I S S + + = λR(1−λ) (Aλ ) = λR(1−λ) (Aλ+ ); λ∈I λ∈I S S + (2) (R(A))+ = λR1−λ (Aλ ) = λR1−λ (Aλ ), λ∈I

λ∈I)

S + λR(1−λ)+ (Aλ ); λR(1−λ)+ (Aλ ) = λ∈I S S λ∈I (3) (R(A))− = λRλ (Aλ ) = λRλ+ (Aλ ), λ∈I λ∈I S S = λRλ (Aλ+ ) = λRλ+ (Aλ+ ); λ∈I λ∈I S S (4) (R(A))+ = λRλ (Aλ ) = λRλ+ (Aλ ), λ∈I λ∈I S S + + = λRλ (Aλ ) = λRλ+ (Aλ ); =

S

λ∈I

λ∈I

6

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Proof. (1) For each x ∈ U , by Proposition 2.10, _ [ ( λR1−λ (Aλ ))(x) = {λ ∈ I : x ∈ R1−λ (Aλ )} λ∈I

= =

_ _ _

{λ ∈ I : (R1−λ )s (x) ⊆ Aλ } {λ ∈ I : R+ (x, y) ≥ 1 − λ implies A− (y) ≥ λ}

{λ ∈ I : 1 − R+ (x, y) ≤ λ implies A− (y) ≥ λ} _ ^ = {λ ∈ I : (A− (x) ∨ (1 − R+ (x, y))) ≥ λ} =

= = Then (R(A))− =

S

_

y∈U

{λ ∈ I : (R(A))− (x) ≥ λ}

(R(A))− (x).

λR1−λ (Aλ ).

λ∈I

Similarly, we can prove that [ λR1−λ (Aλ+ ) = (R(A))− = λ∈[0,1)

[

+

λR(1−λ) (Aλ ) =

λ∈(0,1]

[

+

λR(1−λ) (Aλ+ ).

λ∈(0,1)

(2) The proof is similar to (1). (3) For each x ∈ U , by Proposition 2.10, _ [ {λ ∈ I : x ∈ Rλ (Aλ )} ( λ(Rλ (Aλ )))(x) = λ∈I

= =

_ _ _

{λ ∈ I : (Rλ )s (x) ∩ Aλ 6= ∅} {λ ∈ I : ∃y ∈ U, y ∈ Aλ ∩ (Rλ )s (x)}

{λ ∈ I : ∃y ∈ U, A− (y) ∧ R− (x, y) ≥ λ} _ _ = {λ ∈ I : (A− (y) ∧ R− (x, y)) ≥ λ}

=

=

_

y∈U

{λ ∈ I : (R(A))− (x) ≥ λ}

= (R(A))− (x). Then

S

λ(Rλ (Aλ )) = (R(A))− .

λ∈I

Similarly, we can prove that [ (R(A))− = λ(Rλ+ (Aλ )) = λ∈[0,1)

[ λ∈[0,1)

λ(Rλ (Aλ+ )) =

[

λ(Rλ+ (Aλ+ )).

λ∈[0,1)

(4) The proof is similar to (3).

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Theorem 3.8. Let (U, R) be an IVF approximation space. Then IVF rough approximation operator can be represented as follows: for each A ∈ F (i) (U ), [ [ R(A) = ([α, β]R[α,β]+ (A[α,β]+ )) = ([α, β]R[α,β]+ (A[α,β] )) [α,β]∈[I]

[

=

[α,β]∈[I]

([α, β]R[α,β] (A[α,β]+ )).

[α,β]∈[I]

S

Proof. Denote B =

([α, β]R[α,β]+ (A[α,β]+ )). By Proposition 2.10,

[α,β]∈[I]

B − (x) =

_

(α ∧ R[α,β]+ (A[α,β]+ )(x))

α∈I

= = = =

_

_ _ _

{α ∈ I : x ∈ (R[α,β]+ (A[α,β]+ ))} {α ∈ I : (R[α,β]+ )s (x) ∩ A[α,β]+ 6= ∅} {α ∈ I : ∃y ∈ U, R(x, y) > [α, β] and A(y) > [α, β]} {α ∈ I : ∃y ∈ U, A− (y) ∧ R− (x, y) > α and A+ (y) ∧ R+ (x, y)

≥ β or A− (y) ∧ R− (x, y) ≥ α and A+ (y) ∧ R+ (x, y) > β} _ = (A− (y) ∧ R− (x, y)) = (R(A))− (x). y∈U

Then (R(A))− = B − . Similarly, we can prove that (R(A))+ = B + . Hence [ R(A) = B = ([α, β]R[α,β]+ (A[α,β]+ )). [α,β]∈[I]

Similarly, we can prove that [ R(A) = ([α, β]R[α,β]+ (A[α,β] )) = [α,β]∈[I]

3.2

[

([α, β]R[α,β] (A[α,β]+ )).

[α,β]∈[I]

Axiomatic characterizations of IVF rough approximation operators

In an axiomatic approach, rough sets are axiomatized by abstract operators. For the case of IVF rough sets, the primitive notion is the system T S (F (i) (U ), , , c, L, H), where L, H : F (i) (U ) → F (i) (U ) be two IVF set operators. In this subsection, rough approximation operators in the IVF environment are characterized by some axioms. Definition 3.9. Let L, H : F (i) (U ) → F (i) (U ) be two IVF set operators. If (L(A))c = H(Ac ) (A ∈ F (i) (U )), then L, H are called two dual operators. 8

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Remark 3.10. L, H : F (i) (U ) → F (i) (U ) are two dual operators iff (H(A))c = L(Ac ) for each A ∈ F (i) (U ). Theorem 3.11. Let L, H : F (i) (U ) → F (i) (U ) be two dual operators. Then there exists an IVF relation R on U such that L = R and H = R iff L satisfies axioms (AL1) and (AL2), or equivalently, H satisfies axioms (AU 1) and (AU 2): (AL1)

gb] ∪ A) = [a, gb] ∪ L(A) (A ∈ F (i) (U ), [a, b] ∈ [I]), L([a, L(A ∩ B) = R(A) ∩ L(B) (A, B ∈ F (i) (U );

(AL2) (AU 1)

H([a, b]A) = [a, b]H(A) (A ∈ F (i) (U ), [a, b] ∈ [I]),

(AU 2)

H(A ∪ B) = H(A) ∪ H(B) (A, B ∈ F (i) (U )).

Proof. Note that L, H : F (i) (U ) → F (i) (U ) are two dual operators. Then (AL1) and (AL2) are equivalent to (AU 1) and (AU 2). We only need to prove that L = R and H = R iff H satisfies axioms (AU 1) and (AU 2). Necessity. This is obvious. Sufficiency. Assume that the operator H satisfies axioms (AU 1) and (AU 2). Define an IVF relation R on U by R(x, y) = H(y¯1 )(x)

(x, y ∈ U ).

Let A ∈ F (i) (U ). Note that [ [ [ H(A)(x) = H( (A(y)y¯1 ))(x) = ( H(A(y)y¯1 ))(x) = ( (A(y)H(y¯1 )))(x) y∈U

=

_

y∈U

_

(A(y) ∧ H(y¯1 )(x)) =

y∈U

(A(y) ∧ R(x, y)) = R(A)(x)

y∈U

y∈U

for each x ∈ U . Then H(A) = R(A). By Theorem 3.1(3), L(A) = (H(Ac ))c = (R(Ac ))c = R(A). Thus L = R, H = R. Theorem 3.12. Let L, H : F (i) (U ) → F (i) (U ) be two dual operators. Then there exists a reflexive IVF relation R on U such that L = R and H = R iff L satisfies axiom (AL1), (AL2) and (ALR), or equivalently, H satisfies axiom (AU 1), (AU 2) and (AU R): (ALR) L(A) ⊆ A (A ∈ F (i) (U )); (AU R) A ⊆ H(A) (A ∈ F (i) (U )). Proof. This holds by Theorems 3.2(1) and 3.11.

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Theorem 3.13. Let L, H : F (i) (U ) → F (i) (U ) be two dual operators. Then there exists a symmetric IVF relation R on U such that L = R and H = R iff L satisfies axiom (AL1), (AL2) and (ALS), or equivalently, H satisfies axiom (AU 1), (AU 2) and (AU S): (ALS)

L((x¯1 )c )(y) = L((y¯1 )c )(x) (x, y ∈ U );

(ALS)

H(x¯1 )(y) = H(y¯1 )(x) (x, y ∈ U ).

Proof. This hold by Remark 2.9(1) and Theorem 3.11. Theorem 3.14. Let L, H : F (i) (U ) → F (i) (U ) be two dual operators. Then there exists a transitive IVF relation R on U such that L = R and H = R iff L satisfies axiom (AL1), (AL2) and (ALT ), or equivalently, H satisfies axiom (AU 1), (AU 2) and (AU T ): (ALT ) L(A) ⊆ L(L(A)) (A ∈ F (i) (U )); (AU T ) H(H(A)) ⊆ H(A) (A ∈ F (i) (U )). Proof. This holds by Theorems 3.2(2) and 3.11.

4

IVF pseudo-closure operators in IVF approximation spaces

In this section, we investigate IVF pseudo-closure operators in IVF approximation spaces. For each [a, b] ∈ [I], X ∈ P(U ), we define ( [a, b], x ∈ X, ([a, b]X)(x) = ¯0, x ∈ U − X. Denote E (U ) = {[a, b]X : [a, b] ∈ [I], X ∈ P(U )}. Then E (U ) ⊆ F

(i)

(U ).

Definition 4.1. Let τ be an IVF topology on U . Define [ Sτ (A) = clτ ([α, β]A[α,β] } (A ∈ F (i) (U )). [α,β]∈[I]

Then Sτ : F (i) (U ) → F (i) (U ) is called the IVF pseudo-closure operator induced by τ on U . Theorem 4.2 ([25]). Let A ∈ F (i) (U ). Then [ [ [α, β]A[α,β] = [α, β]A(α,β) . A= [α,β]∈[I]

[α,β]∈[I]

10

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Theorems 4.3(5) and 4.4 below illustrate the meaning on IVF pseudo-closure operators. Theorem 4.3. Let τ be an IVF topology on U and let Sτ be the IVF pseudoclosure operator induced by τ on U . Then for any A, B ∈ F (i) (U ), (1) Sτ (˜ 0) = ˜ 0. (2) A ⊆ Sτ (A) ⊆ clτ (A). (3) Sτ (A ∪ B) ⊇ Sτ (A) ∪ Sτ (B). Sτ (A ∩ B) ⊆ Sτ (A) ∩ Sτ (B). (4) A ∈ τ c =⇒ Sτ (A) = A. (5) Sτ coincides with clτ as operators from E (U ) to F (i) (U ). Proof. (1) For any [α, β] ∈ [I] and x ∈ U , since ( [0, 0] ∧ ¯1 = ¯0, [α, β] = ¯0, ([α, β]˜ 0[α,β] )(x) = [α, β] ∧ ˜0[α,β] (x) = [α, β] ∧ ¯0 = ¯0, [α, β] ∈ [I] − {¯0}. we have [α, β]˜ 0[α,β] = ˜ 0. Thus [

Sτ (˜ 0) =

[α,β]∈[I]

(2) By Theorem 4.2, [ A= [α, β]A[α,β] ⊆ [α,β]∈[I]

Sτ (A) =

[

[

clτ ([α, β]˜0[α,β] ) =

clτ (˜0) = ˜0.

[α,β]∈[I]

[

clτ ([α, β]A[α,β] ) = Sτ (A) and

[α,β]∈[I]

clτ ([α, β]A[α,β] ) ⊆ clτ (

[α,β]∈[I]

[

[α, β]A[α,β] ) = clτ (A).

[α,β]∈[I] (i)

(3) For any A, B ∈ F (U ), [α, β] ∈ [I] and x ∈ U put ( ( ¯1, x ∈ B[α,β] , ¯ 1, x ∈ A[α,β] , , D(x) = C(x) = ¯0, x ∈ U − B[α,β] . ¯ 0, x ∈ U − A[α,β] Obviously, ] ] [α, β]A[α,β] = [α, β] ∩ C, [α, β]B[α,β] = [α, β] ∩ D, ] [α, β](A[α,β] ∪ B[α,β] ) = [α, β] ∩ (C ∪ D) and

] [α, β](A[α,β] ∩ B[α,β] ) = [α, β] ∩ (C ∩ D).

We can easily prove that (A ∪ B)[α,β] ⊇ A[α,β] ∪ B[α,β] and (A ∩ B)[α,β] = A[α,β] ∩ B[α,β] . By Proposition 2.6(5), 11

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Sτ (A ∪ B) [ = clτ ([α, β](A ∪ B)[α,β] ) ⊇ [α,β]∈[I]

[

= = ( (

] ] clτ (([α, β] ∩ C) ∪ ([α, β] ∩ D))

[α,β]∈[I]

] ] (clτ ([α, β] ∩ C) ∪ clτ ([α, β] ∩ D))

[

[

] clτ ([α, β] ∩ C) ) ∪ (

[α,β]∈[I]

=

[

] clτ ([α, β] ∩ (C ∪ D)) =

[α,β]∈[I]

=

clτ ([α, β](A[α,β] ∪ B[α,β] ))

[α,β]∈[I]

[α,β]∈[I]

[

[

[

] clτ ([α, β] ∩ D) )

[α,β]∈[I]

[

clτ ([α, β]A[α,β] ) ) ∪ (

[α,β]∈[I]

clτ ([α, β]B[α,β] ) )

[α,β]∈[I]

= Sτ (A) ∪ Sτ (B). By Proposition 2.6(3), Sτ (A ∩ B) \ = clτ ([α, β](A ∩ B)[α,β] ) = [α,β]∈[I]

[

= ⊆

(

] ] clτ (([α, β] ∩ C) ∩ ([α, β] ∩ D))

[α,β]∈[I]

] ] (clτ ([α, β] ∩ C) ∩ clτ ([α, β] ∩ D))

[

[

[

] clτ ([α, β] ∩ C) ) ∩ (

[α,β]∈[I]

=

[

] clτ ([α, β] ∩ (C ∩ D)) =

[α,β]∈[I]

⊆ (

clτ ([α, β](A[α,β] ∩ B[α,β] ))

[α,β]∈[I]

[α,β]∈[I]

[

[

] clτ ([α, β] ∩ D) )

[α,β]∈[I]

[

clτ ([α, β]A[α,β] ) ) ∩ (

[α,β]∈[I]

clτ ([α, β]B[α,β] ) )

[α,β]∈[I]

= Sτ (A) ∩ Sτ (B). (4) By (2) and Proposition 2.6(6), clτ (A) ⊆ S(clτ (A)) ⊆ clτ (clτ (A)) = clτ (A), Note that A ∈ τ c . Then Sτ (A) = Sτ (clτ (A)) = clτ (A) = A. (5) Let A ∈ E (U ). Then there exist [a, b] ∈ [I] and X ∈ P(U ) such that A = [a, b]X. (i) If [a, b] 6= ¯ 0, then for each x ∈ U , ( ( ¯1, ([a, b]X)(x) ≥ [a, b] ¯1, x ∈ X, A[a,b] (x) = ([a, b]X)[a,b] (x) = = ¯0, ([a, b]X)(x) 6≥ [a, b] ¯0, x ∈ U − X. 12

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Thus A[a,b] = X. So Sτ (A) =

[

clτ ([α, β]A[α,β] )

[α,β]∈[I]

⊇ clτ ([a, b]A[a,b] ) = clτ ([a, b]X) = clτ (A). By (2), Sτ (A) ⊆ cl(A). Thus Sτ (A) = clτ (A). (ii) If [a, b] = ¯ 0, then A = ˜0. By (1), Sτ (˜0) = ˜0. Thus Sτ (A) = clτ (A). By (i) and (ii), Sτ coincides with clτ as operators from E (U ) to F (i) (U ).

Theorem 4.4. Let (U, R) be an IVF approximation space. If R is preorder, then R(A) = SτR (A) (A ∈ E (U )). Proof. For each A ∈ E (U ), by Theorems 3.11(3) and 4.3(5), R(A) = clτR (A) = SτR (A).

References [1] C.Cornelis, M.De Cock, E.E.Kerre, Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge, Expert Systems, 20(2003), 260-270. [2] D.Dubois, H.Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17(1990), 191-208. [3] M.H.Ghanim, Pseudo-closure operators in fuzzy topological spaces, Fuzzy sets and systems, 39(1991) 339-346. [4] B.Gorzalczany, Interval-valued fuzzy controller based on verbal modal of object, Fuzzy Sets and Systems, 28(1988), 45-53. [5] Z.Gong, B.Sun, D.Chen, Rough set theory for interval-valued fuzzy information systems, Information Sciences, 178(2008), 1968-1985. [6] L.I.Kuncheva, Fuzzy rough sets: application to feature selection, Fuzzy Sets and Systems, 51(1992), 147-153. [7] H.Lai, D.Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157(2006), 1865-1885. [8] Z.Li, R.Cui, T -similarity of fuzzy relations and related algebraic structures, Fuzzy Sets and Systems, 275(2015), 130-143. 13

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[9] Z.Li, R.Cui, Similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators, Information Sciences, 305(2015), 219-233. [10] Z.Li, T.Xie, The relationship among soft sets, soft rough sets and topologies, Soft Computing, 18(2014), 717-728. [11] Z.Li, T.Xie, Roughness of fuzzy soft sets and related results, International Journal of Computational Intelligence Systems, 8(2015), 278-296. [12] Z.Li, T.Xie, Q.Li, Topological structure of generalized rough sets, Computers and Mathematics with Applications, 63(2012), 1066-1071. [13] Z.Li, N.Xie, G.Wen, Soft coverings and their parameter reductions, Applied Soft Computing, 31(2015), 48-60. [14] T.K.Mondal, S.K.Samanta, Topology of interval-valued fuzzy sets, Indian Journal of Pure and Applied Mathematics, 30(1999), 23-38. [15] S.Nanda, Fuzzy rough sets, Fuzzy Sets and Systems, 45(1992), 157-160. [16] Z.Pawlak, Rough sets, International Journal of Computer and Information Science, 11(1982), 341-356. [17] Z.Pawlak, Rough Sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers, Dordrecht, 1991. [18] Z.Pawlak, A.Skowron, Rudiments of rough sets, Information Sciences, 177(2007), 3-27. [19] Z.Pawlak, A.Skowron, Rough sets: some extensions, Information Sciences, 177(2007), 28-40. [20] Z.Pawlak, A.Skowron, Rough sets and Boolean reasoning, Information Sciences, 177(2007), 41-73. [21] A.M.Radzikowska, E.E.Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and Systems, 126(2002), 137-155. [22] B.Sun, Z.Gong, D.Chen, Fuzzy rough set theory for the interval-valued fuzzy information systems, Information Sciences, 178(2008), 2794-2815. [23] A.Skowron, J.Stepaniuk, Tolerance approximation spaces, Fundamenta Informaticae, 27(1996), 245-253. [24] R.Slowinski, D.Vanderpooten, Similarity relation as a basis for rough approximations. ICS Research Report, 53(1995), 249-250. [25] I.B.Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems, 20(1986), 191-210.

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[26] W.Wu, J.Mi, W.Zhang, Generalized fuzzy rough sets, Information Sciences, 151(2003), 263-282. [27] Y.Y.Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences, 111(1998), 239-259. [28] D.Zheng, R.Cui, Z.Li, On IVF approximating spaces, Journal of Applied Mathematics, Volume 2013, Article ID 494857, 1-9. [29] L.Zhou, W.Wu, W.Zhang, On intuitionistic fuzzy rough sets and their topological structures, International Journal of General Systems, 38(2009), 589616. [30] H.Zhang, W.Zhang, W.Wu, On characterization of generalized intervalvalued fuzzy rough sets on two universes of discourse, International Journal of Approximate Reasoning, 51(2009), 56-70.

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SOME WEIGHTED HERMITE-HADAMARD TYPE INEQUALITIES FOR GEOMETRICALLY–ARITHMETICALLY CONVEX FUNCTIONS ON THE CO-ORDINATES WAJEEHA IRSHAD 1 , M.A.LATIF 2 , AND M. IQBAL BHATTI

3

Abstract. In this paper, the concept of GA-convex functions on the coordinates is introduced. By using a concept of GA-convex functions on the co-ordinates, Hermite-Hadamard type inequalities for this class of functions are settled.

1. Introduction A function f : I inequality

R ! R forenamed as convex in the classical touch [24], if the f ( x + (1

)y)

f (x) + (1

)f (y)

holds for all x, y 2 I and 2 [0; 1]. Indeed, a vast literature has been written on inequalities using classical convexity but one of the most celebrated is the Hermite-Hadamard inequelity. This double inequality is stated as follows: Let f : I R ! R be a function and a, b 2 I with a < b. Then f is convex on [a; b] i¤ Z b a+b 1 f (a) + f (b) f f (x) dx : (1.1) 2 b a a 2 This also reveals that (1.1) can be compulsary as a adequate and su¢ cient condition to function f to be convex on [a; b]. Hermite-Hadmard inequality (1.1) has recieved considerable attention of many reserchers because of its various applications and usefulness in the …eld of mathematical inequalities itself as well as in other areas of mathematics. The inequality (1.1) has been extended to various forms by using various generalizations of the de…nition of classical convex functions and it has also been re…ned under di¤erent hypotheses, see for instance [6, 9, 10, 11, 15, 24, 32] and the references therein. As stated above the classical convexity has been generalized to di¤erent forms and we mention below one of the generalizations of the classical convexity which is known as GA-convexity. De…nition 1. [18, 19] A function f : I R0 = [0; 1) ! R is said to be GA-convex function on I if f x y1 f (x) + (1 )f (y) holds for all x, y 2 I and 2 [0; 1], where x y 1 and f (x) + (1 )f (y) are respectively the weighted geometric mean of two positive numbers x and y and the weighted arithmetic mean of f (x) and f (y). For results on Hermite-Hadamard type inequalities on GA-convex functions and their applications we refere to a recent articles of Latif [15] and Zhang et al. [32]. Date : March 4, 2016. 2000 Mathematics Subject Classi…cation.

Primary: 26A15, 26A51; Scondary 52A30. 1

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2

WAJEEHA IRSHAD

1

, M .A.LATIF

2

, AND M . IQBAL BHATTI

3

The de…nition of classical convexity for functions of of one variables was extended to functions two variables as follows. De…nition 2. [5, 6] Let =: [a; b] [c; d] R2 with a < b and c < d be a bidimensional interval. A mapping f : ! R is said to be convex on if the inequality f ( x + (1

)z; y + (1

holds for all (x; y); (z; w) 2

and

)w)

f (x; y) + (1

)f (z; w)

2 [0; 1].

The De…nition 2 of convex functions on functions by Dragomir in [5].

was modi…ed as co-ordinated convex

De…nition 3. [5] A function f : ! R is said to be convex on the co-ordinates on if the partial mappings fy : [a; b] ! R, fy (u) = f (u; y) and fx : [c; d] ! R, fx (v) = f (x; v) are convex where de…ned for all x 2 [a; b], y 2 [c; d]. Remark 1. [12] It is clear that if a function f : ordinates on . Then f (tx + (1

t)z; sy + (1

tsf (x; y) + t(1

s)w)

s)f (x; w) + s(1

holds for all (t; s) 2 [0; 1]

! R is convex on the co-

t)f (z; y) + (1

t)(1

s)f (z; w);

[0; 1] and x; z 2 [a; b]; y; w 2 [c; d].

It is well-known that every convex mapping f : ! R is convex on the coordinates but converse may not be true (see [5]). The following inequalities of Hermite-Hadamard type for co-ordinated convex functions on the rectangle from the plane R2 were established in [5, Theorem 1, page 778]: Most recently, the notion of co-ordinated convexity has also been generalized in a diverse manner and as a result, the author [14] extended the de…ntion of GA-convex functions of one variable to GA-convex functions of two variables. De…nition 4. [14]A function f : f x z

1

;y w

holds for all (x; y) ; (z; w) 2

1

and

(0; 1)

(0; 1) ! R is GA-convex on

f (x; y) + (1

if

) f (z; w)

2 [0; 1].

A modi…cation in De…nition 4 resulted in the notion of GA-convex functions on the co-ordinates on . De…nition 5. [14] A function f : (0; 1) (0; 1) ! R is said to be GAconvex on the co-ordinates on if the partial mappings fy : [a; b] (0; 1) ! R, fy (u) = f (u; y) and fx : [c; d] (0; 1) ! R, fx (v) = f (x; v) are GA-convex where de…ned for all x 2 [a; b], y 2 [c; d]. The following result holds as a consequence of the de…ntion of GA-convex fuctions on the co-ordinates on . Remark 2. If a function f : co-ordinates on . Then f xt z 1 t ; y s w 1 tf x; y s w1

(0; 1)

(0; 1) ! R is GA-convex on the

s s

+ (1

t) f z; y s w1

t [sf (x; y) + (1

s) f (x; w)] + (1

tsf (x; y) + t(1

s)f (x; w) + s(1

holds for all (t; s) 2 [0; 1]

s

t) [sf (z; y) + (1 t)f (z; y) + (1

s) f (z; w)]

t)(1

s)f (z; w)

[0; 1] and x; z 2 [a; b]; y; w 2 [c; d].

182

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INEQUALITIES FOR GA-CO-ORDINATED FUNCTIONS

3

In [13], some H-H type inequalities for GA-convex functions on the co-ordinates on were also proved for GA-convex functions on the co-ordinates on . For more results on H-H type inequalities for di¤erent generilazations of the de…ntion of of co-ordinated convex functions we refer the reader to [1], [2], [7]-[12], [16], [20]-[23], [27], [28] and closely related articles mentioned therein. The main objective of the present paper is to establish some new weighted HH type inequalities for the class of GA-convex functions on the co-ordinates on a rectangle from the plane in Section 2. 2. Weighted Inequalities for co-ordinated GA-convex functions For the sake of convenience to the reader, we will use the following notations L1 (t) = a

1+t 2

1

b

t 2

; L2 (s) = c

1+s 2

d

1

s

; U1 (t) = a

2

1

t 2

b

1+t 2

; U2 (s) = c

1

s 2

d

1+s 2

:

To obtain our main results, we …rst establish the following weighted identity. Lemma 1. Suppose that f : (0; 1) (0; 1) ! R has second order partial derivatives on and [a; b] [c; d] with a < b and c < d. If h : [a; b] [c; d] ! [0; 1) is twice partially di¤ erentiable mapping and fts 2 L ([a; b] [c; d]), then we have (a; b; c; d; f; h) = h (a; c) f (a; c) +

Z

h (a; d) f (a; d) h (b; c) f (b; c) + h (b; d) f (b; d) Z d Z b hy (a; y) f (a; y) dy hy (b; y) f (b; y) dy hx (x; d) f (x; d) dx

d

c

+

Z

c

b

hx (x; c) f (x; c) dx +

a

=

(ln b

Z

b

a

Z

a

d

hxy (x; y) f (x; y) dydx

c

Z 1Z 1 ln a) (ln d ln c) L1 (t) L2 (s) h (L1 (t) ; L2 (s)) fts (L1 (t) ; L2 (s)) dsdt 4 0 0 Z 1Z 1 + U1 (t) L2 (s) h (U1 (t) ; L2 (s)) fts (U1 (t) ; L2 (s)) dsdt 0

+

Z

0

1

0

+

Z

Z

1

L1 (t) U2 (s) h (L1 (t) ; U2 (s)) fts (L1 (t) ; U2 (s)) dsdt

0

1

0

Z

1

U1 (t) U2 (s) h (U1 (t) ; U2 (s)) fts (U1 (t) ; U2 (s)) dsdt : (2.1)

0 1+t

1

t

1+s

1

s

Proof. By letting x = a 2 b 2 , y = c 2 d 2 and by integration by parts with respect to y and then with respect to x, we have Z Z (ln b ln a) (ln d ln c) 1 1 L1 (t) L2 (s) h (L1 (t) ; L2 (s)) fts (L1 (t) ; L2 (s)) dsdt 4 0 0 Z pab Z pcd p p p p = h (x; y) fxy (x; y) dydx = h ab; cd f ab; cd a

c

p

p h a; cd f a; cd Z

p cd

hy

c

+

Z

a

Z p p h ab; c f ab; c +h (a; c) f (a; c)+

p p ab; y f ab; y dy

p ab

hx (x; c) f (x; c) dx +

Z

a

183

Z

p ab

a p ab

Z

p cd

hy (a; y) f (a; y) dy

c

p p hx x; cd f x; cd dx

p cd

hxy (x; y) f (x; y) dydx: (2.2)

c

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, M .A.LATIF

, AND M . IQBAL BHATTI

3

Similarly, we obtain Z Z ln a) (ln d ln c) 1 1 U1 (t) L2 (s) h (U1 (t) ; L2 (s)) fts (U1 (t) ; L2 (s)) dsdt 4 0 0 p p p p p p = h b; cd f b; cd h (b; c) f (b; c) h ab; cd f ab; cd p Z cd p p ab; c f ab; c hy (b; y) f (b; y) dy +h

(ln b

+

c

p cd

Z

hy

c

+

Z

p p ab; y f ab; y dy

Z

b

p ab

hx (x; c) f (x; c) dx +

b

p ab

Z

b

p ab

Z

p p hx x; cd f x; cd dx

p cd

hxy (x; y) f (x; y) dydx; (2.3)

c

Z Z ln a) (ln d ln c) 1 1 L1 (t) U2 (s) h (L1 (t) ; U2 (s)) fts (L1 (t) ; U2 (s)) dsdt 4 0 0 p p p p p p =h ab; d f ab; d h (a; d) f (a; d) h ab; cd f ab; cd p Z cd p p p p hy + h a; cd f a; cd ab; y f ab; y dy

(ln b

+ +

Z

Z

c

p cd

hy (a; y) f (a; y) dy

c

p ab

Z p p x; cd f x; cd dx +

hx

a

p ab

Z

hx (x; d) f (x; d) dx

a p ab

Z

d

p cd

a

hxy (x; y) f (x; y) dydx (2.4)

and (ln b

ln a) (ln d 4

ln c)

Z

0

1

Z

1

U1 (t) U2 (s) h (U1 (t) ; U2 (s)) fts (U1 (t) ; U2 (s)) dsdt

0

p p p p h ab; d f ab; d h b; cd f b; cd Z d p p p p +h ab; cd f ab; cd p hy (b; y) f (b; y) dy

= h (b; d) f (b; d)

+ +

Z

b

p ab

Z

d

p cd

hx

hy

p p ab; y f ab; y dy

Z b p p x; cd f x; cd dx + p

ab

Z

cd b

p ab Z d

p cd

hx (x; d) f (x; d) dx hxy (x; y) f (x; y) dydx: (2.5)

Adding (2.2)-(2.5), we get the desired identity. This completes the proof of the lemma. Lemma 2. Let u, v > 0, ; k 2 R and 6= 0. Then Z 1 1 1 (u; v; k; ) = (1 kt) u 2 + t v 2 t dt 80 1 1 1 1 u 2 [L(u ;v ) u ] kv 2 > > + v 2 u 2 L (u ; v ) ; < (ln u ln v) = > > : u[1 (1 k)2 ] ; 2k

184

u 6= v; u = v;

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

INEQUALITIES FOR GA-CO-ORDINATED FUNCTIONS

where L (u; v) is the logarithmic mean 8 < L (u; v) = :

v u ln v ln u ;

u 6= v;

u;

u = v:

5

Proof. The proof follows by integration by parts. Now we present some new weighted H-H type inequality for GA-convex functions on a rectangle from R2 . In what follows, we will use the following notation to make our presentation compact. 1

(u; v; z; w; q) = +

u; v; 1;

u; v; 1;

1 z; w; 1; 2

2

(u; v; z; w; q) = +

3

1 2

z; w; 1;

1 2

u; v; 1; 1 2

q

1 2

1 2

z; w; 1; 1 2

1 2

1 q

q

jfts (b; d)j

;

q

1 2

u; v; 1; 1 2

z; w; 1;

1 q

q

;

jfts (b; d)j

q

jfts (a; c)j q

jfts (a; d)j +

1 u; v; 1; 2

1 2

jfts (a; c)j q

z; w; 1;

jfts (b; c)j +

1 2

1 2

u; v; 1;

1 z; w; 1; 2

jfts (a; d)j +

u; v; 1;

q

q

jfts (a; c)j

jfts (a; d)j +

z; w; 1;

z; w; 1; q

1 2

1 u; v; 1; 2

1 2

jfts (b; c)j +

u; v; 1; 1 2

z; w; 1;

jfts (b; c)j +

1 2

(u; v; z; w; q) = +

z; w; 1;

q

u; v; 1;

u; v; 1;

z; w; 1;

1 2

1 2

u; v; 1; 1 2

z; w; 1;

1 2 1 q

q

jfts (b; d)j

and 4

(u; v; z; w; q) = +

u; v; 1;

u; v; 1;

1 2 q

(z; w; 1) jfts (b; c)j +

1 2

z; w; 1;

z; w; 1;

1 2

u; v; 1;

1 2

q

jfts (a; c)j q

jfts (a; d)j + (u; v; 1) 1 2

1 2

z; w; 1;

q

1 q

jfts (b; d)j

:

It is easy to observe that when u = v = z = w = 1, then 9 3 3 1 q q q q jfts (a; c)j + jfts (a; d)j + jfts (b; c)j + jfts (b; d)j 1 (1; 1; 1; 1; q) = 4 4 4 4 q

3 1 9 3 q q q jfts (a; c)j + jfts (a; d)j + fts (b; c) + jfts (b; d)j 2 (1; 1; 1; 1; q) = 4 4 4 4 3 9 1 3 q q q q jfts (a; c)j + jfts (a; d)j + jfts (b; c)j + jfts (b; d)j 3 (1; 1; 1; 1; q) = 4 4 4 4

185

1 q

; 1 q

; 1 q

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WAJEEHA IRSHAD

2

, M .A.LATIF

3

, AND M . IQBAL BHATTI

and 1 q

1 3 3 9 q q q q jfts (a; c)j + jfts (a; d)j + jfts (b; c)j + jfts (b; d)j 4 (1; 1; 1; 1; q) = 4 4 4 4

:

Theorem 1. Let f : (0; 1) (0; 1) ! R be a twice partially di¤ erentiable mapping on and [a; b] [c; d] with a < b and c < d. If h : [a; b] [c; d] ! [0; 1) is a twice partially di¤ erentiable mapping such that fts 2 L ([a; b] [c; d]) q and jfts j is GA-convex on the co-ordinates on [a; b] [c; d] for q 1, then we get hands on: 1 q +1

1 4

j (a; b; c; d; f; h)j (

(ln b

a; b; 0;

+

a; b; 0;

+

1 a; b; 0; 2 +

where khk1 =

ln c) khk1

1 2

1 2

1

c; d; 0;

1 2

1 c; d; 0; 2

1

1 2

1

c; d; 0; 1 2

a; b; 0;

sup

ln a) (ln d

c; d; 0;

h (x; y) and

1 q

1

(a; b; c; d; q)

2

(a; b; c; d; q)

3

(a; b; c; d; q)

1 q

1 q

1 2

1

1 q

4

)

(a; b; c; d; q) ; (2.6)

(u; v; k; ) is de…ned in Lemma 2.

(x;y)2[a;b] [c;d]

Proof. By virtue of Lemma 1, we have j (a; b; c; d; f; h)j (ln b

ln a) (ln d 4 Z +

1

0

+

Z

ln c) khk1

Z

0

Z

1

0

Z

Z

1

L1 (t) L2 (s) jfts (L1 (t) ; L2 (s))j dsdt

0

U1 (t) L2 (s) jfts (U1 (t) ; L2 (s))j dsdt

0

1

1

1

0

L1 (t) U2 (s) jfts (L1 (t) ; U2 (s))j dsdt +

Z

0

1

Z

1

U1 (t) U2 (s) jfts (U1 (t) ; U2 (s))j dsdt : (2.7)

0

Now by using Hölder’s inequality for double integrals and by the GA-convexity of q jfts j on the co-ordinates on [a; b] [c; d] for q 1, we acquire Z 1Z 1 L1 (t) L2 (s) jfts (L1 (t) ; L2 (s))j dsdt 0

0

Z

0

1

Z

1

1

L1 (t) L2 (s) dsdt

Z

1

0

0

1 4

1 q

1 q

Z

1

L1 (t) L2 (s) jfts (L1 (t) ; L2 (s))j dsdt

0

1

1 a; b; 0; 2 q

jfts (a; c)j + 1 c; d; 1; 2

1

q 1 c; d; 0; 2 1 1 c; d; 1; a; b; 1; 2 2

q

jfts (b; c)j +

1 q

q

a; b; 1;

q

jfts (a; d)j +

1 a; b; 1; 2

186

1 2

1 c; d; 1; 2

c; d; 1; a; b; 1;

1 2

1 2 q

jfts (b; d)j

1 q

:

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

INEQUALITIES FOR GA-CO-ORDINATED FUNCTIONS

7

Correspondingly Z

1

Z

1

U1 (t) L2 (s) jfts (U1 (t) ; L2 (s))j dsdt

0

0

1 q

1 4

a; b; 0; q

jfts (a; c)j +

Z

1

Z

q

jfts (b; c)j +

1 2

a; b; 1;

c; d; 1; a; b; 1;

1 c; d; 1; 2

1 2 1 2 q

1 q

jfts (b; d)j

;

1

L1 (t) U2 (s) jfts (L1 (t) ; U2 (s))j dsdt

0

0

q 1 1 a; b; 1; c; d; 0; 2 2 1 1 q c; d; 1; jfts (a; d)j + 2 2

a; b; 1;

1 c; d; 1; 2

1

1

1 2

1 q

1 4

a; b; 0; q

jfts (a; c)j + c; d; 1;

1 2

c; d; 0;

a; b; 1;

1 2

1 2

1 2

1 q

1

a; b; 1; 1 2

c; d; 1;

q

jfts (b; c)j +

a; b; 1;

1 2

1 2 q

jfts (a; d)j + 1 2

c; d; 1;

1 2 1 a; b; 1; 2

c; d; 1;

q

1 q

jfts (b; d)j

;

by similar argument Z

0

1

Z

0

1 4

1

U1 (t) U2 (s) jfts (U1 (t) ; U2 (s))j dsdt 1 q

a; b; 0; q

jfts (a; c)j + c; d; 1;

1 2

1 2

c; d; 0; 1 2

a; b; 1; q

jfts (b; c)j +

1 2

1 q

1

1 2

c; d; 1; a; b; 1;

1 2

a; b; 1;

1 2

q

jfts (a; d)j + 1 2

c; d; 1;

1 2 1 a; b; 1; 2 c; d; 1;

q

jfts (b; d)j

1 q

:

Using the above four inequalities in (2.7) and by resolution, it reveals (2.6) and proof is completed.

Corollary 1. Suppose the assumptions of Theorem 1 are met and if q = 1,then ln a) (ln d ln c) khk1 16 f 1 (a; b; c; d; 1) + 2 (a; b; c; d; 1) + 3 (a; b; c; d; 1) +

j (a; b; c; d; f; h)j

(ln b

187

4

(a; b; c; d; 1)g : (2.8)

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1

WAJEEHA IRSHAD

(ln b

(

1 q +1

1 4

1 ln a) (ln d

a; b; 0;

+

a; b; 0; +

(x; y) 2 [a; b]

[c; d] in

c; d; 0;

1 2

c; d; 0; 1 2

1 2

1

1 2

1

1

(a; b; c; d; q)

1 q

2

(a; b; c; d; q)

3

(a; b; c; d; q)

1 q

)

1 q

1

1 2

c; d; 0;

1 q

1

1 c; d; 0; 2

1 2

a; b; 0;

3

ln c)

1 a; b; 0; 2

+

, AND M . IQBAL BHATTI

1 (ln b ln a)(ln d ln c) ,

Corollary 2. If we consider h (x; y) = Theorem 1, then a; b; c; d; f;

2

, M .A.LATIF

(a; b; c; d; q) : (2.9)

4

Theorem 2. Suppose f : (0; 1) (0; 1) ! R be a twice partially di¤ erentiable mapping on and [a; b] [c; d] with a < b and c < d. Further let h : [a; b] [c; d] ! [0; 1) be a twice partially di¤ erentiable mapping. If fts 2 L ([a; b] [c; d]) q and jfts j is GA-convex on the co-ordinates on [a; b] [c; d] for q > 1, then we have inequality of the form:

aq

+

a

+

a

+

aq

q

where khk1 =

1

q 1

q q

q 1

; bq

; bq

;b

1

q

1+ q1

1 4

j (a; b; c; d; f; h)j (

;b

q 1

(ln b

q 1

q q

1

1 2

cq

1 2

c

; 0;

1

q q

; 0;

1 ; 0; 2

c

1 2

; 0;

cq

sup

ln a) (ln d q

q

1

; dq

1

;d

q q

q

q

1

1

;d

; dq

q 1

h (x; y) and

1 ; 0; 2

1

1

1 2

1

q

q q

; 0;

1 2

1

1

q

q

1

ln c) khk1

; 0;

; 0;

1 2

1 q

1

(1; 1; 1; 1; q)

2

(1; 1; 1; 1; q)

1 q

1 q

3 1

1 q

4

(1; 1; 1; 1; q) )

(1; 1; 1; 1; q) ; (2.10)

(u; v; k; ) is de…ned in Lemma 2.

(x;y)2[a;b] [c;d]

Proof. From Lemma 1, we may write j (a; b; c; d; f; h)j (ln b

ln a) (ln d 4 Z +

1

0

+

Z

ln c) khk1

Z

0

Z

0

1

Z

Z

1

0

L1 (t) L2 (s) jfts (L1 (t) ; L2 (s))j dsdt

U1 (t) L2 (s) jfts (U1 (t) ; L2 (s))j dsdt

0

1

1

1

0

L1 (t) U2 (s) jfts (L1 (t) ; U2 (s))j dsdt

+

Z

0

1

Z

0

1

U1 (t) U2 (s) jfts (U1 (t) ; U2 (s))j dsdt : (2.11)

Now by using Hölder’s inequality for double integrals, Lemma 2 and by the GAq convexity of jfts j on the co-ordinates on [a; b] [c; d] for q > 1, consequently we

188

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INEQUALITIES FOR GA-CO-ORDINATED FUNCTIONS

have Z 1Z

1

L1 (t) L2 (s) jfts (L1 (t) ; L2 (s))j dsdt

0

0

9

Z

1

Z

1

1

q

(L1 (t) L2 (s)) q

1 q

dsdt

1

0

0

Z

Z

1

q 1

; bq

q 1

; 0;

1 2

cq

q 1

q

; dq

q 1

; 0;

1 q

1

1 2

9 3 3 1 q q q q jfts (a; c)j + jfts (a; d)j + jfts (b; c)j + jfts (b; d)j 16 16 16 16 In addition Z 1Z 1 U1 (t) L2 (s) jfts (U1 (t) ; L2 (s))j dsdt

1 q

:

0

0

Z

1

0

Z

1

1

q

(U1 (t) L2 (s))

q

1 q

dsdt

1

0

Z

Z

1

0

a

q q

1

;b

q q

1

1 2

; 0;

1

c

q 1

1

0

Z

1 q

jfts (U1 (t) ; L2 (s))j dsdt

0

q

q

;d

q q

1

1 ; 0; 2

1

1 q

3 1 9 3 q q q q jfts (a; c)j + jfts (a; d)j + jfts (b; c)j + jfts (b; d)j 16 16 16 16 Z

1 q

jfts (L1 (t) ; L2 (s))j dsdt

0

0

aq

1

1 q

;

1

L1 (t) U2 (s) jfts (L1 (t) ; U2 (s))j dsdt

0

Z

1

0

Z

1

1

q

(L1 (t) U2 (s)) q

1 q

dsdt

1

Z

1

0

0

a

q q

1

;b

q q

1

1 ; 0; 2

c

Z

1

q 1

1 q

jfts (L1 (t) ; U2 (s))j dsdt

0

q

q

;d

q q

1

; 0;

1 2

1

1 q

9 1 3 3 q q q q jfts (a; c)j + jfts (a; d)j + jfts (b; c)j + jfts (b; d)j 16 16 16 16

1 q

;

equivalently Z

0

1

Z

1

U1 (t) U2 (s) jfts (U1 (t) ; U2 (s))j dsdt

0

Z

0

1

Z

1

1

q

(U1 (t) U2 (s))

q

1

1 q

dsdt

0

Z

1

0

a

q q

1

;b

q q

1

; 0;

1 2

c

Z

0

q q

1

1

;d

q

1 q

jfts (U1 (t) ; U2 (s))j dsdt q q

1

; 0;

1 2

1

1 q

1 3 3 9 q q q q jfts (a; c)j + jfts (a; d)j + jfts (b; c)j + jfts (b; d)j 16 16 16 16

1 q

:

Using the above four inequalities in (2.11) and simplifying, we get the required inequality (2.10).

189

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10

WAJEEHA IRSHAD

1

2

, M .A.LATIF

, AND M . IQBAL BHATTI

1 (ln b ln a)(ln d ln c) ,

Corollary 3. If we take h (x; y) = Theorem 2, then

(x; y) 2 [a; b]

1 a; b; c; d; f; (ln b ln a) (ln d ln c) ( 1+ q1 q q q q 1 1 1 c q 1 ; d q 1 ; 0; a q 1 ; b q 1 ; 0; 4 2 2 +

aq

+

a

+

a

q q

1

q

q q

;b

; bq

1

1

;b

q q

1

q

; 0;

1

q q

1

1 2

cq

1 ; 0; 2

c

1 2

; 0;

q

c

q q

q q

1

1

; dq

;d

1

q q

q

;d

q

q

1

1

1 1 ; 0; 2

1

1 2

1

; 0;

; 0;

[c; d] in

1 q

1

1

(1; 1; 1; 1; q)

1 q

(1; 1; 1; 1; q)

2 1 q

3 1

1 2

3

1 q

4

(1; 1; 1; 1; q) )

(1; 1; 1; 1; q) : (2.12)

We shall use the following notation for the next theorem and its related corollary. 2

1

1

q

q

(a; b; c; d; q) = ( (q)) q jfts (a; c)j + ( (q)) q jfts (a; d)j 1

q

1

q

q

+ ( (q)) q jfts (b; c)j + jfts (b; d)j ; 2

q

(a; b; c; d; q) = ( (q)) q jfts (a; c)j + jfts (a; d)j 2

1

q

q

+ ( (q)) q jfts (b; c)j + ( (q)) q jfts (b; d)j ; 1

3

2

q

q

(a; b; c; d; q) = ( (q)) q jfts (a; c)j + ( (q)) q jfts (a; d)j q

1

q

1

q

+ jfts (b; c)j + ( (q)) q jfts (b; d)j

and 4

q

(a; b; c; d; q) = jfts (a; c)j + ( (q)) q jfts (a; d)j 1

2

q

q

+ ( (q)) q jfts (b; c)j + ( (q)) q jfts (b; d)j ;

where (q) = 2q+1

1.

Theorem 3. Let f : (0; 1) (0; 1) ! R be a twice partially di¤ erentiable mapping on and [a; b] [c; d] with a < b and c < d. Further let h : [a; b] [c; d] ! [0; 1) is a twice partially di¤ erentiable mapping. If fts 2 L ([a; b] [c; d]) q and jfts j is GA-convex on the co-ordinates on [a; b] [c; d] for q > 1, then the following inequality holds: (ln b

j (a; b; c; d; f; h)j ( a

+

+

aq

+

aq aq

q 1

q

q

q

;b

1

; bq

1

; bq

q

; bq

1

q 1

q q

q

1 ; 0; 2

1

; 0;

1

; 0;

q

; 0;

1

ln a) (ln d 16

1 2

c

1 2

q q

cq

1 2

cq cq

q 1

ln c) khk1 ;d

1

; dq

q

q 1

; dq

; dq

q 1

190

1 ; 0; 2

1

1

; 0;

1 2

1

1

1 2

1

q

1

q

q

q 1

; 0;

1 q+1

; 0; 1 2

2=q

1 q

1

(a; b; c; d; q)

2

(a; b; c; d; q)

1 q

1 q

3 1

1 q

4

(a; b; c; d; q) )

(a; b; c; d; q) ; (2.13)

WAJEEHA IRSHAD et al 181-195

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

INEQUALITIES FOR GA-CO-ORDINATED FUNCTIONS

where khk1 =

sup

h (x; y),

11

(u; v; k; ) is de…ned in Lemma 2.

(x;y)2[a;b] [c;d]

Proof. From Lemma 1, we have j (a; b; c; d; f; h)j

(ln b Z

Z

1

Z

0

+

Z

L1 (t) L2 (s) jfts (L1 (t) ; L2 (s))j dsdt

Z

1

0

ln c) khk1

1

0

0

+

ln a) (ln d 4

1

U1 (t) L2 (s) jfts (U1 (t) ; L2 (s))j dsdt

0

1Z

1

L1 (t) U2 (s) jfts (L1 (t) ; U2 (s))j dsdt

0

+

Z

Z

1

1

U1 (t) U2 (s) jfts (U1 (t) ; U2 (s))j dsdt : (2.14)

0

0

q

Now by using the GA-convexity of jfts j on the co-ordinates on [a; b] [c; d] for q > 1, Lemma 2 together with the Hölder’s inequality for double integrals, we have Z

1

0

Z

1

L1 (t) L2 (s) jfts (L1 (t) ; L2 (s))j dsdt

0

Z

1

Z

1

1+t 1+s 1+t jfts (a; c)j + 2 2 2 0 0 1 s 1 t 1+s 1 t 1 s jfts (a; d)j + jfts (b; c)j + jfts (b; d)j 2 2 2 2 2 ( Z Z 1 Z 1Z 1 1 q1 q q 1 1 q q 1 + t 1 + s (L1 (t) L2 (s)) q 1 dsdt dsdt jfts (a; c)j 2 2 0 0 0 0 Z

+

Z

1

0

1

(L1 (t) L2 (s))

1+t 2

0

q

1 2 Z

jfts (b; c)j + =

a

q q

1

;b

q q

1

0

1 ; 0; 2

q

jfts (a; c)j + 2q+1

dsdt

1

Z

1

1

jfts (a; d)j +

1

1=q

1

1

;d

1

q

jfts (a; d)j + 2q+1

1

1

1 q

1=q

1 q

q

1+s 2 )

dsdt

jfts (b; d)j

1 2q (q + 1) 1

q

t 2

dsdt

1 q

1

1 ; 0; 2

Z

0

q

2

q q

1

0

s

2

q q

q

t

0

c

Z

1 q

q

s

2=q

h

2q+1

2=q

1

q

q

jfts (b; c)j + jfts (b; d)j

i

:

Likewise, we have Z

0

1

Z

0

1

U1 (t) L2 (s) jfts (U1 (t) ; L2 (s))j dsdt a

q q

1 2q (q + 1)

1

;b

q q

2=q

h

1

+ 2q+1

; 0; 2q+1 1

1 2

c 1

2=q

1=q

q q

1

;d

q q

1

1 ; 0; 2

1 q

q

q

jfts (a; c)j + jfts (a; d)j q

jfts (b; c)j + 2q+1

191

1

1

1=q

q

jfts (b; d)j

i

;

WAJEEHA IRSHAD et al 181-195

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12

1

WAJEEHA IRSHAD

Z

1

Z

, AND M . IQBAL BHATTI

3

1

L1 (t) U2 (s) jfts (L1 (t) ; U2 (s))j dsdt

0

0

2

, M .A.LATIF

aq

1

h

2=q

1 q 2 (q + 1)

q

q

; bq

1

2q+1

; 0; 1

1 2

q

cq

1=q

1

; dq

q

q

jfts (a; c)j + 2q+1

1 q

1

1 2

; 0;

1

1=q

1

q

+ jfts (b; c)j + 2q+1

1

q

jfts (a; d)j

1=q

q

jfts (b; d)j

i

;

i

:

and Z

1

0

Z

1

0

U1 (t) U2 (s) jfts (U1 (t) ; U2 (s))j dsdt q

a

q

h

2=q

1 2q (q + 1)

1

;b

q q

1

2q+1

1 2

; 0; 1

1=q

1

; 0;

1 2

jfts (a; c)j + 2q+1

1

q

c

q

1

;d

q q

1

q

1=q

q

+ jfts (b; c)j + 2q+1

1

1 q

q

jfts (a; d)j

1=q

q

jfts (b; d)j

Further employing the above four inequalities in (2.14) and after simpli…cation, we built up the required inequality (2.13).

1 (ln b ln a)(ln d ln c) ,

Corollary 4. If we take h (x; y) = Theorem 3,then

1 ln a) (ln d

a; b; c; d; f; (ln b ( aq

+

where

+

a

+

a

q

a

q 1

1

q q

q

q

;b

q

1

;b

q

;b

q 1

q 1

q

1

q

q

; bq

; 0;

1

1 ; 0; 2 1 2

ln c)

1 2

1

q

; 0;

; 0;

cq 1 2

q

c c c

q q

1 16

1

1

q 1

1

;d

;d

1

1

; 0;

q q

q q

q q

q

;d

q

q q

; dq

1

; 0;

(u; v; k; ) is de…ned in Lemma 2 and

1

1 q

1

1 2

1 1

1 2

1

1 2

(a; b; c; d; q)

1 q

2

1 q

(q) = 2q+1

(a; b; c; d; q)

1 q

3 1

[c; d] in

2=q

1 q+1

1 ; 0; 2

; 0;

(x; y) 2 [a; b]

3

(a; b; c; d; q) )

(a; b; c; d; q) ; (2.15)

1.

Theorem 4. Let f : (0; 1) (0; 1) ! R be a twice partially di¤ erentiable mapping on and [a; b] [c; d] with a < b and c < d. Further let h : [a; b] [c; d] ! [0; 1) is a twice partially di¤ erentiable mapping. If fts 2 L ([a; b] [c; d]) q and jfts j is GA-convex on the co-ordinates on [a; b] [c; d] for q > 1 and q r 0,

192

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INEQUALITIES FOR GA-CO-ORDINATED FUNCTIONS

13

then we attain the following inequality: j (a; b; c; d; f; h)j ( q

r 1

aq

+

a

+

a

+

a

q q

r 1

q q

r 1

q q

r 1

q

r 1

; bq

;b ;b

;b

1 q +1

1 4

q q

r 1

q q

r 1

q q

r 1

(ln b 1 2

; 0;

q

c

1 ; 0; 2

c 1 2

; 0;

r 1

cq

1 2

; 0;

ln a) (ln d

q q

q q

r 1

c

q q

q

r 1

q q

;d

;d r 1

r 1

; dq

q q

;d

r 1

1 2

1

1 ; 0; 2

1

1 2

1

; 0;

r 1

; 0;

q q

r 1

ln c) khk1 1

(ar ; br ; cr ; dr ; q)

2

(ar ; br ; cr ; dr ; q)

3

(ar ; br ; cr ; dr ; q)

1 q

1 q

1 q

1

1 2

; 0;

1 q

4

)

(ar ; br ; cr ; dr ; q) ; (2.16)

where khk1 =

sup

h (x; y) and

(u; v; k; ) is de…ned in Lemma 2.

(x;y)2[a;b] [c;d]

Proof. From Lemma 1, it follows that j (a; b; c; d; f; h)j (ln b

ln a) (ln d 4 Z +

Z

1

Z

0

1

0

L1 (t) L2 (s) jfts (L1 (t) ; L2 (s))j dsdt

U1 (t) L2 (s) jfts (U1 (t) ; L2 (s))j dsdt

0

1

Z

1

0

Z

1

0

+

Z

ln c) khk1

1

0

L1 (t) U2 (s) jfts (L1 (t) ; U2 (s))j dsdt

+

Z

0

1

Z

0

1

U1 (t) U2 (s) jfts (U1 (t) ; U2 (s))j dsdt : (2.17) q

Now by virtue of GA-convexity of jfts j on the co-ordinates on [a; b] [c; d] for q > 1, Lemma 2 and by the Hölder’s inequality for double integrals, we have in hand Z 1Z 1 Z 1Z 1 1 q r q 1 dsdt L1 (t) L2 (s) jfts (L1 (t) ; L2 (s))j dsdt (L1 (t) L2 (s)) 0

0

0

Z

1

0

1 q

1 4

Z

1

a

r 1

0

r

1 q

q

(L1 (t) L2 (s)) jfts (L1 (t) ; L2 (s))j dsdt

0

q q

1 q

;b

q q

r 1

1 ; 0; 2

c

q q

r 1

;d

q q

r 1

1

1 ; 0; 2

1 q

1

(ar ; br ; cr ; dr ; q)

Similarly Z

0

1

Z

Z

1

0

U1 (t) L2 (s) jfts (U1 (t) ; L2 (s))j dsdt Z

1

0

1 4

1 q

a

q q

Z

1

;b

Z

1

(U1 (t) L2 (s))

q q

r 1

1

1 q

dsdt

0

r

1 q

q

(U1 (t) L2 (s)) jfts (U1 (t) ; L2 (s))j dsdt

0

r 1

0

1

q q

r 1

; 0;

1 2

c

q q

193

r 1

;d

q q

r 1

1 ; 0; 2

1

1 q

2

(ar ; br ; cr ; dr ; q)

WAJEEHA IRSHAD et al 181-195

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

14

WAJEEHA IRSHAD

Z

1

Z

Z

1 q

1 4

0

1

Z

0

q

aq

, AND M . IQBAL BHATTI

Z

1

Z

1

1 q

a

q q

r 1

1

q

; bq

Z

r 1

; 0;

1 2

q

cq

1

;b

(L1 (t) U2 (s))

q q

r q

1

1 q

1

1 q

dsdt

0 1 q

q

r 1

q

; dq

r 1

1

1 2

; 0;

Z

0

1

1 q

3

Z

(ar ; br ; cr ; dr ; q)

1

(U1 (t) U2 (s))

q q

r 1

dsdt

0

r

1 q

q

(U1 (t) U2 (s)) jfts (U1 (t) ; U2 (s))j dsdt

0

q q

1

r

U1 (t) U2 (s) jfts (U1 (t) ; U2 (s))j dsdt Z

Z

3

(L1 (t) U2 (s)) jfts (L1 (t) ; U2 (s))j dsdt

0

r 1

1

0

1

0

1 4

2

L1 (t) U2 (s) jfts (L1 (t) ; U2 (s))j dsdt

0

and Z

, M .A.LATIF

1

0

0

1

r 1

; 0;

1 2

c

q q

r 1

;d

q q

r 1

; 0;

1 2

1

1 q

4

(ar ; br ; cr ; dr ; q)

Using the above four inequalities in (2.17) and simplifying, we obtained the required inequality (2.16). References [1] M. Alomari and M. Darus, Co-ordinated s-convex function in the …rst sense with some Hadamard-type inequalities, Int. J. Contemp. Math. Sciences, 3 (32) (2008) 1557-1567. [2] M. Alomari and M. Darus, The Hadamard’s inequality for s-convex function of 2-variables on the co-ordinates, Int. Journal of Math. Analysis, 2 (13) (2008) 629–638. [3] S. S. Dragomir and R. P. Agarwal, Two inequalities for di¤erentiable mappings and applications to special means of real numbers and to Trapezoidal formula, Appl. Math. Lett., 11(5) (1998) 91-95. [4] F. Chen and S. wu, Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions, Journal of Nonlinear Sciences and Applications, Volume 9, Issue 2 (2016), 705-716. [5] S.S. Dragomir, On Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese Journal of Mathematics. 4 (2001) 775-788. [6] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University (2000). [7] X.-Y. Guo, F. Qi and B.-Y. Xi, Some new Hermite-Hadamard type inequalities for geometrically quasi-convex functions on co-ordinates, Journal of Nonlinear Sciences and Applications. 8 (2015) 740-749. [8] D. Y. Hwang, K. L. Tseng and G. S. Yang, Some Hadamard’s inequalities for co-ordinated convex functions in a rectangle from the plane, Taiwanese Journal of Mathematics. 11 (2007) 63-73. [9] J. Hua, B.-Y. Xi, and F. Qi, Hermite-Hadamard type inequalities for geometric-arithmetically s-convex functions, Commun. Korean Math. Soc. 29 (1) (2014) 51–63. [10] A. P. Ji, T. Y. Zhang, F. Qi, Integral inequalities of Hermite-Hadamard type for ( ; m)GA-convex functions, Journal of Function Spaces and Applications, 2013 (2013), Article ID 823856, 8 pages. [11] U. S. Kirmaci, Inequalities for di¤erentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput. 147 (2004) 137-146. [12] M. A. Latif and M. Alomari, Hadamard-type inequalities for product two convex functions on the co-ordinates, Int. Math. Forum, 4(47) (2009) 2327-2338. [13] M. A. Latif and M. Alomari, On the Hadamard-type inequalities for h-convex functions on the co-ordinates, Int. J. of Math. Analysis. 3(33) (2009) 1645-1656. [14] M. A. Latif, Hermite-Hadamard type inequalities for GA-convex functions on the co-ordinates with applications, Proceedings of the Pakistan Academy of Science. (to appear) [15] M. A. Latif, New Hermite–Hadamard type integral inequalities for GA-convex functions with applications, Analysis. 34 (4) 379–389. [16] M. A. Latif and S. S. Dragomir, On Some New inequalities for di¤erentiable co-ordinated convex functions, Journal of Inequalities and Applications 2012, (2012) :28.

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INEQUALITIES FOR GA-CO-ORDINATED FUNCTIONS

15

[17] W. Liu, W. Wen and J. Park, Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals, Journal of Nonlinear Science and Applications, Volume 9, Issue 3(2016), 766-777. [18] C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl. 3 (2) (2000) 155–167. [19] C. P. Niculescu, Convexity according to means, Math. Inequal. Appl. 6 (4) (2003) 571–579. [20] M.E. Özdemir, E. Set and M.Z. Sarikaya, New some Hadamard’s type inequalities for coordinated m-convex and ( ; m)-convex functions, RGMIA, Res. Rep. Coll. 13 (2010), Supplement, Article 4. [21] M. Emin Özdemir, Havva Kavurmac¬, Ahmet Ocak Akdemir and Merve Avc¬, Inequalities for convex and s-convex functions on = [a; b] [c; d], Journal of Inequalities and Applications. 2012 (2012):20. [22] M. Emin Özdemir, M. Amer Latif and Ahmet Ocak Akdemir, On some Hadamard-type inequalities for product of two s-convex functions on the co-ordinates, Journal of Inequalities and Applications. 2012 (2012):21. [23] M. Emin Özdemir, Ahmet Ocak Akdemir and Melült Tunc, On the Hadamard-type inequalities for co-ordinated convex functions, arXiv:1203.4327v1. [24] C. M. E. Pearce and J. E. Peµcari´c, Inequalities for di¤erentiable mappings with applications to special means and quadrature formula, Appl. Math. Lett. 13 (2000) 51-55. [25] J. E. Peµcari´c, F. Proschan and Y. L. Tong, Convex Functions, Partial Ordering and Statistical Applications, Academic Press, New York, 1991. [26] H. Ping Yin and F. Qi, Hermite Hadamard type inequalities for the product of ( ; m)-convex functions Journal of Nonlinear Sciences and Applications, Volume 8, Issue 3(2015), 231-236. [27] M.Z. Sarikaya, E. Set, M.E. Özdemir and S. S. Dragomir, New some Hadamard’s type inequalities for co-ordinated convex functions, arXiv:1005.0700v1 [math.CA]. [28] M.Z. Sarikaya, On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integral Transforms and Special Functions. 25 (2) (2014) 134-147. [29] S.H. Wang and X.M. Liu, Hermite-Hadamard type inequalities for operator s-preinvex functions, Journal of Nonlinear Science and Applications, Volume 8, Issue 6(2015), 1070-1081. [30] Ying Wu, Feng Qi, Shu-Ping Bai, and Zhi-Li Pei, Some new integral inequalities of HermiteHadamard type for ( ,m;P )-convex functions on co-ordinates, Journal of Nonlinear Science and Applications (2016), in press. [31] F. Zafar, H. Kalsoom and N. Hussain, Some inequalities of Hermite-Hadamard type for ntimes di¤erentiable ( ; m)-geometrically convex functions, Journal of Nonlinear Science and Applications, Volume 8, Issue 3(2015), 201-217. [32] T.-Y. Zhang , A.-P. Ji and F. Qi, Some inequalities of Hermite-Hadamard type for GA-Convex functions with applications to means, Le Matematiche 48 (1) (2013) 229–239. 1 Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan E-mail address : [email protected]

School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa E-mail address : [email protected] 3 Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan E-mail address : [email protected]

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196

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO. 1, 2017

Golub-Kahan-Lanczos Based Preconditioner for Least Squares Problems on Overdetermined and Underdetermined Cases, Liang Zhao, Ting-Zhu Huang, Liu Zhu, and Liang-Jian Deng,……11 Classical Model of Prandtl's Boundary Layer Theory for Radial Viscous Flow: Application of (G’/G)-Expansion Method, Taha Aziz, T. Motsepa, A. Aziz, A. Fatima, and C.M. Khalique,31 On Properties of Meromorphic Solutions for a Certain q-Difference Painlevé Equation, Xiu-Min Zheng, Hong-Yan Xu, and Hua Wang,……………………………………………………….42 New Approximation of Fixed Points of Asymptotically Demicontractive Mappings in Arbitrary Banach Spaces, Shin Min Kang, Arif Rafiq, Faisal Ali, and Young Chel Kwun,…………….52 Viscosity Approximation of Solutions of Fixed Point and Variational Inclusion Problems, B. A. Bin Dehaish, H. O. Bakodah, A. Latif, and X. Qin,……………………………………………61 On The Stability of Additive 𝜌𝜌-Functional Inequalities in Fuzzy Normed Spaces, Choonkil Park,…………………………………………………………………………………………….70 On the Difference equation 𝑥𝑥𝑛𝑛+1 = 𝐴𝐴𝑥𝑥𝑛𝑛 +

𝐵𝐵 ∑𝑘𝑘 𝑖𝑖=0 𝑥𝑥𝑛𝑛−𝑖𝑖

𝐶𝐶+𝐷𝐷 ∏𝑘𝑘 𝑖𝑖=0 𝑥𝑥𝑛𝑛−𝑖𝑖

, M.M. El-Dessoky, E.O. Alzahrani,…78

A Kind of Generalized Fuzzy Integro-Differential Equations of Mixed Type and Strong Fuzzy Henstock Integrals, Qiang Ma, Ya-bin Shao, and Zi-zhong Chen,……………………………..92 On the Generalized Stieltjes Transform of Fox’s Kernel Function and its Properties in the Space of Generalized Functions, Shrideh Khalaf Qasem Al-Omari,…………………………………108 Decision Making Based On Interval-Valued Intuitionistic Fuzzy Soft Sets and Its Algorithm, Hongxiang Tang,……………………………………………………………………………….119 Product-Type Operators from Weighted Zygmund Spaces to Bloch-Orlicz Spaces, Yong Yang and Zhi-Jie Jiang,………………………………………………………………………………132 Union Soft p-Ideals and Union Soft Sub-Implicative Ideals in BCI-Algebras, Sun Shin Ahn, Jung Mi Ko, and Keum Sook So,…………………………………………………………………152 On Interval-Valued Fuzzy Rough Approximation Operators, Weidong Tang, Jinzhao Wu, and Meiling Liu,…………………………………………………………………………………….166

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO. 1, 2017 (continued)

Some Weighted Hermite-Hadamard Type Inequalities For Geometrically-Arithmetically Convex Functions On The Co-Ordinates, Wajeeha Irshad, M.A.Latif, and M. Iqbal Bhatti,………….181

Volume 23, Number 2 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Effect of RTI drug efficacy on the HIV dynamics with two cocirculating target cells A. M. Elaiw, N. A. Almuallem and Aatef Hobiny Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Emails: a m [email protected] (A. Elaiw), [email protected] (N. Almuallem) Abstract In this paper, we propose and analyze an HIV dynamics model. The model can be seen as a generalization of many HIV dynamics models presented in the literature since it incorporates (i) two classes of target cells, CD4+ T cells and macrophages, (ii) two types of infected cells, short-lived infected cells and the long-lived chronically infected cells, (iii) intracellular discrete delays, (iv) reverse transcriptase inhibitors (RTIs) drugs with different drug efficacies on CD4+ T cells and macrophages. The incidence rate of infection is represented by a general function. A bifurcation parameter, known as the basic reproduction number, R0 is derived. We established a set of conditions on the general function which are sufficient to determine the global dynamics of the model. Using Lyapunov functionals and LaSalle’s invariance principle, the global asymptotic stability of the two equilibria of the model is obtained. An example is presented and some numerical simulations are conducted in order to illustrate the dynamical behavior. Keywords: Delayed-HIV models; Chronically infected cells; Cocirculating target cells; Immune responses; Lyapunov method.

1

Introduction

Human immunodeficiency virus (HIV) is one of the most dangerous human viruses that destroys the immune system and causes acquired immunodeficiency syndrome (AIDS). During the past decades, several HIV mathematical models have been presented and analyzed (see e.g. [1]-[25]). Global stability of equilibria has become one of the most important features which help us to better understanding of the HIV dynamics. Thus, several researchers have devoted extensive efforts to study the global stability of HIV infection models (see e.g. [7], [8], [9], [11], [25], [14], [15], [16], [17], [22], [23], [19] and [24]). Some of these works assume that HIV infects only the CD4+ T cells ([7], [8], [9], [11], [25], [22], [23], [19] and [24]), while, others assume that HIV infects two types of immune cells, CD4+ T cells and macrophages ([14], [15], [18], [16] and [17]). Callaway and Perelson [3] pointed out that there are two types of infected cells, short-lived infected cells (which produce the most amounts of viruses) and the long-lived chronically infected cells. Moreover, the model presented in [3] incroporates reverse transcriptase inhibitors (RTIs) drugs with different drug efficacies on CD4+ T cells and macrophages. Actually, there exists a time lag between the time the HIV contacts CD4+ T cells or macrophages and the time the production of new infectious HIV particles. Intracellular time delay was first introduced into viral infection model by Herz et al. [5]. Since then, several delayed HIV models have been investigated (see e.g. [6], [7], [8], [9], [11], [25], [14], [17], [18], [22] and [19]). In a very recent work, Elaiw and Almualem [17] have

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A. M. Elaiw et al 209-228

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presented the following delayed HIV model: x˙ 1 (t) = λ1 − d1 x1 − (1 − ε)β¯1 x1 v,

(1)

x˙ 2 (t) = λ2 − d2 x2 − (1 − χε)β¯2 x2 v, y˙ 1 (t) = (1 − q1 )(1 − ε)β¯1 x1 (t − τ1 )v(t − τ1 ) − δ1 y1 , y˙ 2 (t) = (1 − q2 )(1 − χε)β¯2 x2 (t − τ2 )v(t − τ2 ) − δ2 y2 , z˙1 (t) = q1 (1 − ε)β¯1 x1 (t − τ1 )v(t − τ1 ) − a1 z1 , z˙2 (t) = q2 (1 − χε)β¯2 x1 (t − τ1 )v(t − τ1 ) − a2 z2 , 2 X

v(t) ˙ =


Ni δi e−ni κi yi (t − κi ) + Mi ai e−hi ωi zi (t − ωi ) − uv(t)

(2)

i=1

where xi , yi , zi , and v represent the concentrations of uninfected cells, short-lived infected cells, long-lived chronically infected cells and free HIV particles, respectively, where i = 1, for the CD4+ T cells and i = 2, for the macrophages. The birth and death rates of uninfected cells are given by λi and di xi , respectively. Parameter β¯i denotes the infection rate constant. Parameters δi and ai are the death rate constants of the two types of infected cells, and u is the clearance rate of HIV. The uninfected target cells become short-lived infected and long-lived chronically infected cells with fractions (1 − qi ) and qi , respectively, where qi .∈ (0, 1). The average number of free viruses produced in the lifetime of the two types of infected cells are given by Ni and Mi , respectively. Parameter τi represents for the time between viral contact with an uninfected cell of class i, until it becomes infected but not yet producer cells. The loss of the cells during the delay period [t − τi , t] is given by e−mi τi , where mi > 0. The parameters κi and ωi represent the time necessary for producing new infectious viruses from the short-lived and long-lived chronically infected cells, respectively. The factors e−ni κi and e−hi ωi represent the loss of the two types of infected cells during the delay periods [t − κi , t] and [t − ωi , t], where ni > 0 and hi > 0. The immune system has two main responses to viral infections. The first is based on the Cytotoxic T Lymphocyte (CTL) cells which are responsible to attack and kill the infected cells. The second immune response is based on the antibodies that are produced by the B cells. The function of the antibodies is to attack the viruses [1]. In some infections such as in malaria, the CTL immune response is less effective than the antibody immune response [26]. Several mathematical models have been proposed to consider the antibody immune response into the viral infection models (see [27]-[33])). All the models presented in [27]-[33] are based on the assumption that, the virus attacks one class of target cells. Moreover, model (1)-(2) did not consider the immune response. Therefore, our aim in this paper is to propose an HIV dynamics model with humoral immunity. Our model generalize model (1)-(2) by taking into account the humoral immune response. We use Lyapunov functionals and LaSalle’s invariance principle to prove the global stability of all the equilibria of the models.

2

The model In this section, we propose and analyze the following HIV model: x˙ i (t) = λi − di xi (t) − φi (xi (t), v(t)), −mi τi

y˙ i (t) = (1 − qi )e

φi ((t − τi ), v(t − τi )) − δi yi (t),

z˙i (t) = qi e−mi τi φi ((t − τi ), v(t − τi )) − ai zi (t), v(t) ˙ =

2 X

i = 1, 2,

(3)

i = 1, 2,

(4)

i = 1, 2,

(5)


Nyi δi e−ni κi yi (t − κi ) + Mzi ai e−ri ωi zi (t − ωi ) − uv(t) − bv(t)f (w(t)),

(6)

i=1

w(t) ˙ = cv(t) − pw(t).

(7)

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The incidence rate of infection is given by a general function φi (xi , v), where φ1 (x1 , v) = (1 − ε)φ¯1 (x1 , v), and φ2 (x2 , v) = (1 − χε)φ¯2 (x2 , v). In addition, the neutralize rate of viruses is given by a general nonlinear function f (w). Parameter b is the B cells neutralize rate, the antibody response is induced at a rate proportional to the concentration of free viruses. Parameters c and p are the recruited rate and death rate constants of B cells, respectively. All the parameters and variables of the model have the same meanings as given in (1)-(2).

2.1

Initial conditions

The initial conditions for system (3)-(7) take the form x1 (θ) = ϕ1 (θ), y1 (θ) = ϕ3 (θ), z1 (θ) = ϕ5 (θ), x2 (θ) = ϕ2 (θ), y2 (θ) = ϕ4 (θ), z2 (θ) = ϕ6 (θ), v(θ) = ϕ7 (θ), w(θ) = ϕ8 (θ) ϕj (θ) ≥ 0, θ ∈ [−%, 0), ϕj (0) > 0, j = 1, ..., 8,

(8)

where % = max{τ1 , τ2 , κ1 , κ2 , ω1 , ω2 } and (ϕ1 (θ), ϕ2 (θ), ..., ϕ8 (θ)) ∈ C([−%, 0], R8≥0 ), where C is the Banach space of continuous functions mapping the interval [−%, 0] into R8≥0 . By the fundamental theory of functional differential equations [35], system (3)-(7) has a unique solution satisfying the initial conditions (8). Assumption A1 Function φi , is continuously differentiable and satisfies the following: (i) φi (xi , v) > 0, φi (xi , 0) = φi (0, v) = 0, for all xi > 0, v > 0, i = 1, 2, (xi ,0) (xi ,v) (xi ,v) > 0, for any xi > 0, v > 0. Furthermore, ∂φi∂v > 0, ∂φi∂x > 0 for any xi > 0, i = 1, 2. (ii) ∂φi∂v i Assumption A2 The function f (θ) is locally Lipschitz on [0, ∞), and satisfies f (θ) > 0 for all θ > 0 and f (0) = 0, and f (θ) is strictly increasing in [0, ∞).

2.2

Non-negativity and boundedness of solutions

In the following, we establish the non-negativity and boundedness of solutions of system (3)-(7) with initial conditions (8). Proposition 1. Let (x1 (t), x2 (t), y1 (t), y2 (t), z1 (t), z2 (t), v(t), w(t)) be any solution of (3)-(7) satisfying the initial conditions (8), then xi (t), yi (t), zi (t), i = 1, 2, v(t) and w(t) are all non-negative for t ≥ 0 and ultimately bounded. Proof. First, we prove that xi (t) > 0, i = 1, 2, for all t ≥ 0. Assume that xi (t) lose its positivity on some local existence interval [0, l] for some constant l and let t∗i ∈ [0, l] be such that xi (t∗i ) = 0. From Eq. (3) we have x˙ i (t∗i ) = λi > 0. Hence xi (t) > 0 for some t ∈ (t∗i , t∗i + ) , where  > 0 is sufficiently small. This leads to a contradiction and hence xi (t) > 0, for all t ≥ 0. Furthermore, from Eqs. (4)-(7) we have yi (t) = yi (0) e

−δi t

−mi τi

Zt

+ (1 − qi )e

e−δi (t−θ) φ(xi (θ − τi ) , v (θ − τi ))dθ,

i = 1, 2,

0

zi (t) = zi (0) e−ai t + qi e−mi τi

Zt

e−ai (t−θ) φ(xi (θ − τi ) , v (θ − τi ))dθ,

i = 1, 2,

0

Zt

Rt

− (u+bf (w(ζ)))dζ

v (t) = v (0) e

+

0

Rt − (u+bf (w(ζ)))dζ

e

θ

i=1

0

w (t) = w(0)e−pt + c

Zt

2 X (Nyi δi e−ni κi yi (θ − κi ) + Mzi ai e−ri ωi zi (θ − ωi ))dθ,

e−p(t−θ) v(θ)dθ,

0

then yi (t) ≥ 0, zi (t) ≥ 0, i = 1, 2, v(t) ≥ 0 and w(t) ≥ 0, for all t ∈ [0, %]. By a recursive argument, we obtain yi (t) ≥ 0, zi (t) ≥ 0, v(t) ≥ 0 and w(t) ≥ 0, i = 1, 2, for all t ≥ 0.

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Next we show the boundedness of the solutions. From Eq. (3) we have x˙ i (t) ≤ λi − di xi (t), i = 1, 2. This implies that lim supt→∞ xi (t) ≤ λdii , i = 1, 2. Let Ti (t) = e−mi τi xi (t − τi ) + yi (t) + zi (t), i = 1, 2 then T˙i (t) = e−mi τi λi − e−mi τi di xi (t − τi ) − δi yi (t) − ai zi (t)
≤ e−mi τi λi − σi e−mi τi xi (t − τi ) + yi (t) + zi (t) ≤ λi − σi Ti (t), λi . Since xi (t), yi (t) and zi (t) are all σi non-negative, then lim supt→∞ yi (t) ≤ Li , and lim supt→∞ zi (t) ≤ Li for all t ≥ 0. Moreover, where σi = min{di , δi , ai }. Hence, lim supt→∞ Ti (t) ≤ Li , where Li =

v˙ =

2 X


Nyi δi e−ni κi yi (t − κi ) + Mzi ai e−ri ωi zi (t − ωi ) − uv − bv(t)f (w(t))

i=1

≤

2 X


Nyi δi e−ni κi + Mzi ai e−ri ωi Li − uv.

i=1

Then lim supt→∞ v(t) ≤ L3 , for all t ≥ 0, where L3 =

2 X (Ny

δ e i i

−ni κi

+Mzi ai e−ri ωi )Li . u

Furthermore, w˙ =

cL3 p .Therefore,

xi (t), yi (t), zi (t), v(t)

i=1

cv − pw ≤ cL3 − pw, then lim supt→∞ w(t) ≤ L4 , for all t ≥ 0, where L4 = and w(t) are ultimately bounded.

2.3

Equilibria

Let Assumptions A1 (i) and A2 be satisfied, then system (3)-(7) has a disease-free equilibrium E0 = (x01 , x02 , 0, 0, 0, 0, 0, 0), where x0i = λdii , i = 1, 2. The system can also has another positive equilibrium E1 = (˜ x1 , x ˜2 , y˜1 , y˜2 , z˜1 , z˜2 , v˜, w) ˜ which is called endemic equilibrium. The coordinates of the endemic equilibrium, if it exists satisfy the equalities: λi = di x ˜i + φi (˜ xi , v˜), u˜ v=

2 X

δi y˜i = (1 − qi )e−mi τi φi (˜ xi , v˜), ai z˜i = qi e−mi τi φi (˜ xi , v˜),


Nyi δi e−ni κi y˜i + Mzi ai e−ri ωi z˜i − b˜ v f (w), ˜

i=1

w ˜=

c v˜. p

Then the basic infection reproduction number for system (3)-(7) is R0 =

2 2 X X ((1 − qi )Nyi e−ni κi + qi Mzi e−ri ωi )e−mi τi ∂φi (x0i , 0) R0i = . u ∂v i=1 i=1

The term ∂φi (x0i , 0)/∂v represents the maximal average number of target cells of class i that infects by viruses, and R01 denotes the basic infection reproduction number of the HIV dynamics with CD4+ T cells (in the absence of macrophages) and R02 denotes the basic infection reproduction number of the HIV dynamics with macrophages (in the absence of CD4+ T cells), respectively. The parameter R0 determines whether the infection can be established.

2.4

Global stability analysis

In this subsection, we establish a set of conditions which are sufficient for the global stability of the two equilibria of system (3)-(7) employing Lyapunov method and LaSalle’s invariance principle. The following function will be used throughout the paper H(s) = s − 1 − ln s. Assumption A3 The function φi , i = 1, 2 satisfies:  
∂φi (x0i ,0) ∂φi (xi ,0) 0 xi − xi ≤ 0, for xi > 0, (i) − ∂v ∂v (xi ,0) (ii) φi (xi , v) ≤ v ∂φi∂v , for all xi , v > 0. Theorem 1. Let Assumptions A1-A3 be satisfied and R0 ≤ 1, then the disease-free equilibrium E0 of system (3)-(7) is GAS.

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Proof. Define a Lyapunov functional W0 as follows:  Zxi 2 X φi (x0i , v) Nyi e−ni κi Mzi e−ri ωi  0 lim+ W0 = γi xi − xi − ds + yi + zi φi (s, v) γi γi v→0 i=1 x0i

Zτi φi (xi (t − θ), v(t − θ))dθ +

+

e

−ni κi

Nyi δi γi

Zκi yi (t − θ)dθ +

e

−ri ωi

Mzi ai γi

0

0

Zωi

 zi (t − θ)dθ

0

Zw b +v+ f (θ)dθ, c 0

where γi = e−mi τi ((1 − qi )e−ni κi Nyi + qi e−ri ωi Mzi ), i = 1, 2. We calculate (3)-(7) as: 2

X dW0 = γi dt i=1

dW0 dt

along the trajectories of system

  φi (x0i , v) (λi − di xi − φi (xi , v)) 1 − lim v→0+ φi (xi , v)

Nyi e−ni κi ((1 − qi )e−mi τi φi (xi (t − τi ), v(t − τi )) − δi y) γi Mzi e−ri ωi (qi e−mi τi φi (xi (t − τi ), v(t − τi )) − ai zi ) + γi +

+ φi (xi , v) − φi (xi (t − τi ), v(t − τi ))  e−ni κi Nyi δi e−ri ωi Mzi ai + (yi − yi (t − κi )) + (zi − zi (t − ωi )) γi γi +

2 X i=1


b Nyi δi e−ni κi yi (t − κi ) + Mzi ai e−ri ωi zi (t − ωi ) − uv − bvf (w) + f (w)(cv − pw). c

(9)

Collecting terms of Eq. (9) we get 2

   ∂φi (x0i , 0)/∂v ∂φi (x0i , 0)/∂v bp 1− (λi − di xi ) + φi (xi , v) − uv − wf (w) ∂φi (xi , 0)/∂v ∂φi (xi , 0)/∂v c       2 X ∂φi (x0i , 0)/∂v bp ∂φi (x0i , 0)/∂v xi = γi λi 1 − 0 + φi (xi , v) − uv − wf (w) 1− xi ∂φi (xi , 0)/∂v ∂φi (xi , 0)/∂v c i=1     2 2 X X ∂φi (x0i , 0)/∂v bp ∂φi (x0i , 0)/∂v xi = 1− − uv − wf (w). γi λ i 1 − 0 + γi φi (xi , v) x ∂φ (x , 0)/∂v ∂φ (x , 0)/∂v c i i i i i i=1 i=1

X dW0 = γi dt i=1

(10)

Using A3 we get   2 X xi dW0 ≤ γi λi 1 − 0 1− dt xi i=1   2 X xi = γi λi 1 − 0 1− xi i=1

∂φi (x0i , 0)/∂v ∂φi (xi , 0)/∂v



∂φi (x0i , 0)/∂v ∂φi (xi , 0)/∂v



+

2 X ∂φi (x0i , 0) bp γi v − uv − wf (w) ∂v c i=1

+ (R0 − 1)uv −

bp wf (w). c

(11)

0 By using Assumption A2, the last term is less than or equal zero. Therefore, If R0 ≤ 1 then dW dt ≤ 0 for all x1 , x2 , v, w > 0. We note that, the solutions of the system (3)-(7) converge to Γ, the largest invariant subset of  dW0 dW0 0 dt = 0 . From Eq. (11) we have dt = 0 iff xi = xi , i = 1, 2, v = 0 and w = 0. The set Γ is invariant and for any element belongs to Γ satisfies w = 0, v = 0 then v˙ = 0. We can see from Eq. (19) that

2 X


Nyi δi e−ni κi yi (t − κi ) + Mzi ai e−ri ωi zi (t − ωi ) = 0.

i=1

Since yi and zi are non-negative for i = 1, 2, then y1 = y2 = 0 and z1 = z2 = 0. It follows that, xi = x0i , yi = zi = v = w = 0, i = 1,2. From LaSalle’s invariance principle, E0 is GAS.

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dW0 dt

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To establish the global stability of the endemic equilibrium, we need the following condition. Assumption A4 Function φi (xi , v) satisfies the following:    φi (xi , v) v φi (xi , v˜) − 1− ≤ 0, xi , v > 0 φi (xi , v˜) v˜ φi (xi , v) Theorem 2. Let Assumptions A1, A2 and A4 hold true and the endemic equilibrium E1 of system (3)-(7) exists, then E1 is GAS. Proof. We consider the Lyapunov functional W1 as:    Zxi 2 X Nyi e−ni κi yi φi (˜ xi , v˜)  W1 = γi xi − x ˜i − ds + y˜i H φ (s, v ˜ ) γ y˜i i i i=1 x ˜i

Mzi e−ri ωi + z˜i H γi



zi z˜i

Zτi

 + φi (˜ xi , v˜)

 H

φi (xi (t − θ), v(t − θ)) φi (˜ xi , v˜)

 dθ

0

e−ni κi Nyi δi y˜i + γi

Zκi

 H

yi (t − θ) y˜i



e−ri ωi Mzi ai z˜i dθ + γi

0

b + c

Zωi

 H

zi (t − θ) z˜i



 dθ + v˜H

v v˜

0

Zw (f (θ) − f (w))dθ. ˜ w ˜

Calculating

dW1 dt

along the solutions of system (3)-(7) we obtain 2

  φi (˜ xi , v˜) 1− (λi − di xi − φi (xi , v)) φi (xi , v˜)   Nyi e−ni κi y˜i + 1− ((1 − qi )e−mi τi φi (xi (t − τi ), v(t − τi )) − δi yi ) γi yi   Mzi e−ri ωi z˜i + 1− (qi e−mi τi φi (xi (t − τi ), v(t − τi )) − ai zi ) γi zi   φi (xi (t − τi ), v(t − τi )) + φi (xi , v) − φi (xi (t − τi ), v(t − τi )) + φi (˜ xi , v˜) ln φi (xi , v)    −ni κi e Nyi δi y˜i yi yi (t − κi ) yi (t − κi ) + − + ln γi y˜i y˜i yi    −ri ωi e Mzi ai z˜i zi zi (t − ωi ) zi (t − ωi ) + − + ln γi z˜i z˜i zi !   X 2
v˜ −ni κi −ri ωi + 1− Nyi δi e yi (t − κi ) + Mzi ai e zi (t − ωi ) − uv − bvf (w) v i=1

X dW1 = γi dt i=1

b ˜ − pw). + (f (w) − f (w))(cv c

(12)

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Collecting terms of Eq. (12) we get 2

X dW1 = γi dt i=1

  φi (˜ xi , v˜) Nyi e−ni κi δi Mzi e−ri ωi ai φi (˜ xi , v˜) (λi − di xi ) + φi (xi , v) + y˜i + z˜i 1− φi (xi , v˜) φi (xi , v˜) γi γi

(1 − qi )e−mi τi Nyi e−ni κi y˜i φi (xi (t − τi ), v(t − τi )) qi e−mi τi Mzi e−ri ωi z˜i φi (xi (t − τi ), v(t − τi )) − γ y γi zi i i φi (xi (t − τi ), v(t − τi )) + φi (˜ xi , v˜) ln φi (xi , v)     −ni κi yi (t − κi ) e−ri ωi Mzi ai z˜i zi (t − ωi ) e Nyi δi y˜i ln + ln + γi yi γi zi −

−

2 2 X v˜yi (t − κi ) X v˜zi (t − ωi ) Nyi δi e−ni κi − − uv + u˜ v Mzi ai e−ri ωi v v i=1 i=1

+ b˜ v f (w) −

bp bp wf (w) − bvf (w) ˜ + wf (w). ˜ c c

Using the equilibrium conditions for E1 : λi = di x ˜i + φi (˜ xi , v˜), u˜ v=

2 X

xi , v˜) = δi y˜i , (1 − qi )e−mi τi φi (˜


Nyi δi e−ni κi y˜i + Mzi ai e−ri ωi z˜i − b˜ v f (w), ˜

i=1

qi e−mi τi φi (˜ xi , v˜) = ai z˜i , w ˜=

c v˜ p

and the following equality v v uv = u˜ v = v˜ v˜

! 2 2 X vX −ni κi −ri ωi γi φi (˜ xi , v˜) − bvf (w), ˜ (Nyi δi e y˜i + Mzi ai e z˜i ) − b˜ v f (w) ˜ = v˜ i=1 i=1

we obtain       2 X dW1 φi (˜ xi , v˜) xi φi (˜ xi , v˜) γi di x = ˜i 1 − + φi (˜ xi , v˜) 1 − 1− dt x ˜i φi (xi , v˜) φi (xi , v˜) i=1   2Nyi e−ni κi δi φi (xi , v) v 2Mzi e−ri ωi ai − + + φi (˜ xi , v˜) y˜i + z˜i φi (xi , v˜) v˜ γi γi   Nyi e−ni κi δi y˜i y˜i φi (xi (t − τi ), v(t − τi )) v˜yi (t − κi ) + − γi yi φi (˜ xi , v˜) v y˜i   −ri ωi Mzi e ai z˜i z˜i φi (xi (t − τi ), v(t − τi )) v˜zi (t − ωi ) − + γi zi φi (˜ xi , v˜) v˜ zi      −ni κi δi y˜i yi (t − κi ) Nyi e φi (xi (t − τi ), v(t − τi )) + ln + ln γi φi (xi , v) yi      −ri ωi φi (xi (t − τi ), v(t − τi )) zi (t − ωi ) Mz e ai z˜i ln + ln + i γi φi (xi , v) zi bp bp − b˜ v f (w) ˜ + b˜ v f (w) − wf (w) + wf (w). ˜ c c Using the following equalities

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(13)

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   φi (˜ xi , v˜) y˜i φi (xi (t − τi ), v(t − τi )) + ln φi (xi , v˜) yi φi (˜ xi , v˜)     vφi (xi , v˜) v˜yi + ln + ln , v˜φi (xi , v) v y˜i       yi (t − κi ) v˜yi (t − κi ) v y˜i ln = ln + ln , yi v y˜i v˜yi       φi (xi (t − τi ), v(t − τi )) φi (˜ xi , v˜) z˜i φi (xi (t − τi ), v(t − τi )) ln = ln + ln φi (xi , v) φi (xi , v˜) zi φi (˜ xi , v˜)     vφi (xi , v˜) v˜zi + ln + ln , v˜φi (xi , v) ve zi       zi (t − ωi ) v˜zi (t − ωi ) ve zi ln = ln + ln . zi ve zi v˜zi 

ln

φi (xi (t − τi ), v(t − τi )) φi (xi , v)





= ln

Eq. (13) can be rewritten as      2  X dW1 φi (˜ xi , v˜) φi (xi , v) v vφi (xi , v˜) xi = 1− + γi φi (˜ xi , v˜) − −1+ γi d i x ˜i 1 − dt x ˜i φi (xi , v˜) φi (xi , v˜) v˜ v˜φi (xi , v) i=1       φi (˜ xi , v˜) φi (˜ xi , v˜) vφi (xi , v˜) vφi (xi , v˜) − γi φi (˜ xi , v˜) − 1 − ln − γi φi (˜ xi , v˜) − 1 − ln φi (xi , v˜) φi (xi , v˜) v˜φi (xi , v) v˜φi (xi , v)    y˜i φi (xi (t − τi ), v(t − τi )) y˜i φi (xi (t − τi ), v(t − τi )) − 1 − ln − Nyi e−ni κi δi y˜i yi φi (˜ xi , v˜) yi φi (˜ xi , v˜)    v˜yi (t − κi ) v˜yi (t − κi ) − Nyi e−ni κi δi y˜i − 1 − ln v y˜i v y˜i    z˜i φi (xi (t − τi ), v(t − τi )) z˜i φi (xi (t − τi ), v(t − τi )) −ri ωi − 1 − ln − Mzi e ai z˜i zi φi (˜ xi , v˜) zi φi (˜ xi , v˜)    v˜zi (t − ωi ) v˜zi (t − ωi ) bp −Mzi e−ri ωi ai z˜i ˜ (w) − f (w)). ˜ (14) − 1 − ln − (w − w)(f ve zi ve zi c Then Eq. (14) becomes,       2  X xi dW1 φi (˜ xi , v˜) φi (xi , v) v φi (xi , v˜) γi di x ˜i 1 − = − 1− + γi φi (˜ xi , v˜) 1− dt x ˜i φi (xi , v˜) φi (xi , v˜) v˜ φi (xi , v) i=1      φi (˜ xi , v˜) vφi (xi , v˜) − γi φi (˜ xi , v˜) H +H φi (xi , v˜) v˜φi (xi , v)      v˜yi (t − κi ) y˜i φi (xi (t − τi ), v(t − τi )) − Nyi e−ni κi δi y˜i H +H yi φi (˜ xi , v˜) v y˜i      v˜zi (t − ωi ) z˜i φi (xi (t − τi ), v(t − τi )) bp −Mzi e−ri ωi ai z˜i H +H ˜ (w) − f (w)). ˜ − (w − w)(f zi φi (˜ xi , v˜) ve zi c By using Assumption A2, the last term is less than or equal zero. It is easy to see that, if x ˜1 , x ˜2 , y˜1 , y˜2 , z˜1 , z˜2 , v˜ dW1 and w ˜ > 0, then dt ≤ 0 for all x1 , x2 , y1 , y2 , z1 , z2 , v and w > 0. The solutions of the system limit to Γ, the dW1 1 largest invariant subset of { dW ei , v = v˜, w = w ˜ and dt = 0}. It can be seen that dt = 0 if and only if xi = x H = 0 i.e. v˜yi (t − κi ) v˜zi (t − ωi ) = =1 (15) v y˜i ve zi From Eq. (15), we have yi = yei and zi = z˜i . It follows that principle implies the global stability of E1 .

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3

Example and numerical simulations

We introduce the following example: x˙ i (t) = λi − di xi (t) − y˙ i (t) = (1 − qi )e−mi τi z˙i (t) = qi e−mi τi v(t) ˙ =

2 X

βi xki i (t)v(t) , + ρi )(v(t) + ςi )

i = 1, 2,

(xki i (t)

βi xki i (t − τi )v(t − τi ) − δi yi (t), (xki i (t − τi ) + ρi )(v(t − τi ) + ςi )

βi xki i (t − τi )v(t − τi ) − ai zi (t), (xki i (t − τi ) + ρi )(v(t − τi ) + ςi )

(16)

i = 1, 2,

(17)

i = 1, 2,

(18)


Nyi δi e−ni κi yi (t − κi ) + Mzi ai e−ri ωi zi (t − ωi ) − uv(t) − bv(t)w(t),

(19)

i=1

w(t) ˙ = cv(t) − pw(t).

(20)

For this example we have φi (xi , v) =

(xki i

βi xki i v , + ρi )(v + ςi )

f (w) = w

(21)

where ki , ρi , ςi > 0, i = 1, 2. Function φi satisfies the following: ∂φi (xi , v) ki ρ β xki −1 v = ki i i i > 0, for all xi > 0, v > 0, ∂xi (xi + ρi )2 (v + ςi ) ςi βi xki i ∂φi (xi , v) = ki > 0, for all xi > 0, ∂v (xi + ρi )(v + ςi )2 βi xki i ∂φi (xi , 0) = > 0, for all xi > 0, v > 0, ∂v ςi (xki i + ρi ) βi xki i v βi xki i v ∂φi (xi , 0) ≤ =v , for all xi > 0, v > 0, ∂v (xiki + ρi )(v + ςi ) ςi (xki i + ρi )    φi (xi , v) v φi (xi , v˜) −ςi (v − v˜)2 − ≤ 0, for all xi , v > 0. 1− = φi (xi , v˜) v˜ φi (xi , v) v˜(˜ v + ςi )(v + ςi ) φi (xi , v) =

Thus Assumption A1-A4 hold true and Theorems 1 and 2 are applicable. The basic reproduction number in this case is given by R0 =

2 2 X X ((1 − qi )e−ni κi Nyi + qi e−ri ωi Mzi )e−mi τi βi (x0i )ki R0i = . u ςi ((x0i )ki + ρi ) i=1 i=1

Without loss of generality we let, τe = τ1 = τ2 = κ1 = κ2 = ω1 = ω2 . In Table 1, we present the values of some parameters of model (16)-(20). The effect of the drug efficacy ε and time delay τe on the qualitative behavior of the system will be studied below in details. All computations are carried out by MATLAB.

3.1

Evolution of the system state with different initial conditions

We have chosen three different initial conditions as follows: IC1: ϕ1 (θ) = 600, ϕ2 (θ) = 200, ϕ3 (θ) = 1, ϕ4 (θ) = 0.5, ϕ5 (θ) = 1, ϕ6 (θ) = 2, ϕ7 (θ) = 1, ϕ8 (θ) = 0.02, IC2: ϕ1 (θ) = 700, ϕ2 (θ) = 350, ϕ3 (θ) = 2, ϕ4 (θ) = 2, ϕ5 (θ) = 3, ϕ6 (θ) = 5, ϕ7 (θ) = 6, ϕ8 (θ) = 1 IC3: ϕ1 (θ) = 800, ϕ2 (θ) = 500, ϕ3 (θ) = 3.5, ϕ4 (θ) = 3.5, ϕ5 (θ) = 6, ϕ6 (θ) = 8, ϕ7 (θ) = 10, ϕ8 (θ) = 1.4, where θ ∈ [−%, 0). We will fix the delay parameter τe = 0.01 day−1 , and using two sets of the parameter ε to get the following two cases. Case (I): In this case, we choose ε = 0.8 then we get R0 = 0.79 < 1. Figure 1 shows that, the state of the system eventually approach to the infection-free equilibrium E0 = (1000, 600, 0, 0, 0, 0, 0, 0) for the three initial conditions IC1-IC3. This supports the results of Theorem 1 that the infection-free equilibrium E0 is GAS. In

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Table 1: The values of the parameters of model (16)-(20). Parameter λ1 β¯1

Value

Parameter

Value 6 cells mm−3 day−1

8 cells mm−3 day−1

λ2 β¯2

0.01 day−1

−3

10 cells mm

day

−1

5 cells mm−3 day−1

d2

0.01 day−1

δ1

−1

0.5 day

δ2

0.3 day−1

a1

0.3 day−1

a2

0.1 day−1

q1

0.5

q2

0.5

ς2

10 virus mm−3

d1

ς1

−3

10 virus mm

k1

2

k2

2

Ny1

9 virus cells−1

Ny2

4 virus cells−1

Mz1

4 virus cells−1

Mz2

1 virus cells−1

ρ1

0.1 cellsk1 mm−3k1

ρ2

0.1 cellsk1 mm−3k1

m1

1 day−1

m2

1 day−1

n1

1 day−1

n2

1 day−1

r1

1 day−1

r2

1 day−1

χ

0.5

u

1 day−1

p

6 day−1

ε

Varied

b

−3

1 cells mm

day

c

1 day−1

τe

Varied

−1

this case, the virus particles will be cleared from the body. Case (II): In this case, we choose ε = 0 then we calculate R0 = 2.13 > 1. Consequently, the system has two equilibria E0 and E1 , and based on Theorem 2, E1 is GAS. From Figure 1 we can see that, our simulation results are consistent with the theoretical results of Theorem 2. We observe that, the state of the system converge the endemic equilibrium E1 = (571.06, 332.13, 4.25, 4.43, 7.08, 13.28, 11.58, 1.93). for the three initial conditions IC1-IC3. In this case, the infection becomes chronic.

3.2

Effect of the drug efficacy on the dynamical behavior of the system

In this case, we will fix the delay parameter τe = 0.01 day−1 . Figures 2 shows the effect of the parameter ε on the evolution of the uninfected CD4+ T cells and macrophages, short-lived infected cells, long-lived chronically infected cells, free virus particles and B cells. When there is no treatment i.e. ε = 0, the trajectory of the system tends to the endemic equilibrium E1 = (571.06, 332.13, 4.25, 4.43, 7.08, 13.28, 11.58, 1.93). Since E1 exists, then according to Theorem 2, E1 is GAS. We can see from the figures that, our simulation results are consistent with the theoretical results of Theorem 2. We observe that, as the drug efficacy is increased from ε = 0 to ε = 0.8, E1 is still exists and is GAS, moreover, the concentrations of the uninfected CD4+ T cells and macrophages are increasing, while the concentrations of the short-lived infected cells, long-lived chronically infected cells, free virus particles and B cells are decreasing. When ε = 0.98, the basic reproduction number is given by R0 = 0.73 < 1, then according to Theorem 1, the disease-free equilibrium E0 is GAS. We can see that, the concentrations of uninfected CD4+ T cells and macrophages are increasing and converge to their normal values λ1 −3 λ2 , d2 = 600 cells mm−3 , respectively, while the concentrations of short-lived infected cells, d1 = 1000 cells mm long-lived chronically infected cells, free viruses and B cells are decaying and tend to zero. It means that, the numerical results are also compatible with the results of Theorem 1. In this case, the treatment with such drug

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1000

600 Case(I)

Case(I)

950

550

Uninfected CD4 T cells (cell/mm3)

Uninfected macrophages (cell/mm3)

R0≤1

900

850

800

750

700

650

R0≤1

500

450

400

350

300

Case(II) R0>1

R0>1

250

Case(II)

600 200 0

550 0

50

100

150

200 250 300 Time (days)

350

400

450

50

(a) Uninfected CD4+ T cells

150

200 250 300 Time (days)

350

400

450

500

(b) Uninfected macrophages 5

4.5

4.5

4

Case(II)

Case(II)

Short−lived infected macrophages (cell/mm3)

Short−lived infected CD4 Tcells (cell/mm3)

100

500

R0>1

3.5 3 2.5 2 1.5 1

0 0

5

R >1 0

3.5 3 2.5 2 1.5 1 R ≤1

R0≤1

0.5

4

0

Case(I)

10

15

20

Case(I)

0.5 0 0

25

10

20

Time (days)

(c) Short-lived infected CD4+ T cells

30 Time (days)

40

50

60

(d) Short-lived infected macrophages 15

8

7

Case(II)

Chronically infected CD4 T cells (cell/mm3)

3

Chronically infected macrophages (cell/mm )

Case(II)

6

R >1 0

5

4

3

2

1

5

10

15

10

5

R0≤1

R0≤1

0 0

R0>1

Case(I)

20

Case(I) 0 0

25

10

20

30

Time (days)

(e) Chronically infected CD4+ T cells

50

60

70

80

(f) Chronically infected macrophages

12

2

Case(II)

Case(II)

1.8

R0>1

10

R0>1

1.6 1.4

8

3

B cells (cell/mm )

Free virus (virus/mm3)

40 Time (days)

6

4

1.2 1 0.8 0.6

R0≤1

2

0.4

Case(I)

R ≤1

Case(I)

0

0.2

0 0

5

10

15

20

25 30 Time (days)

35

40

45

50 0 0

(g) Free virus

5

10

15

20 Time (days)

25

30

35

(h) B cells

Figure 1: The evolution of the system state in different initial conditions for model (16) -(20).

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efficacy succeeded to eliminate the viruses from the blood.

3.3

Effect of the time delay on the dynamical behavior of the system

In this case, we will fix the drug efficacy ε = 0.2. Figure 3 shows the effect of the parameter τe on the evolution of the state variables of the system. When τe = 0.01, the trajectory of the system tends to the endemic equilibrium E1 = (684.2, 378.23, 3.13, 3.66, 5.2, 10.9, 9.75, 1.62). Then E1 exists and according to Theorem 2 E1 is GAS. It means that, both the numerical and theoretical results of Theorem 2 are consistent. One can see that, as the time delay is increased from τe = 0.01 to τe = 0.7, E1 is still exists and is GAS, in addition, the concentrations of the uninfected CD4+ T cells and macrophages are increased, while the concentrations of the short-lived infected cells, long-lived chronically infected cells, free virus particles and B cells are decreased. When τe = 1, the basic reproduction number is given by R0 = 0.71 < 1, then according to Theorem 1, E0 is GAS. We can see that, the concentrations of uninfected CD4+ T cells and macrophages are increasing and converge to their normal values λ1 −3 λ2 , d2 = 600 cells mm−3 , respectively, while the concentrations of short-lived infected cells, d1 = 1000 cells mm long-lived chronically infected cells, free viruses and B cells are decaying and tend to zero. Figure 3 shows that the numerical results are also compatible with the results of Theorem 1. This shows the effect of time delay on preventing the disease from development.

3.4

Effects of the drug efficacy and the delay on the basic reproduction number:

Figure 4 shows the effect of the parameters ε and τe on the basic reproduction number R0 . We note that, R0 > 1 for small values of ε or τe , and the endemic equilibrium exists and is GAS, while the disease-free equilibrium is unstable. When R0 = 1 (which is a bifurcation point), both disease-free equilibrium and endemic equilibrium coincide and it is GAS. Moreover, as ε or τe is increasing, R0 is decreasing until it becomes less than one, which makes the endemic equilibrium does not exists and the disease-free equilibrium is GAS. From a biological point of view, the intracellular delay plays a similar role as antiviral treatment in eliminating the virus. We observe that, even if there is no treatment i.e. ε = 0, sufficiently large delay suppress viral replication and clear the virus. This give us some suggestions on new drugs to prolong the increase the intracellular delay period.

3.5

Effects of two types of target cells on the dynamics and controls of HIV infection

In this subsection, we show the effects of two types of target cells on the dynamics and controls of HIV infection. We note that if R0 < 1, then it is sure that R01 < 1 and R02 < 1. But if one neglect the presence of the macrophages in the HIV dynamics model, then the HIV model system (16) -(20) will become x˙ 1 (t) = λ1 − d1 x1 (t) − y˙ 1 (t) = (1 − q1 )e−m1 τ1

(1 − ε)β¯1 xk11 (t)v(t) , (xk11 (t) + ρ1 )(v(t) + ς1 ) (1 − ε)β¯1 xk1 (t − τ1 )v(t − τ1 ) 1

(xk11 (t

− τ1 ) + ρ1 )(v(t − τ1 ) + ς1 )

(22) − δ1 y1 (t),

(23)

(1 − ε)β¯1 xk11 (t − τ1 )v(t − τ1 ) − a1 z1 (t), (xk11 (t − τ1 ) + ρ1 )(v(t − τ1 ) + ς1 )

(24)

v(t) ˙ = Ny1 δ1 e−n1 κ1 y1 (t − κ1 ) + Mz1 a1 e−r1 ω1 z1 (t − ω1 ) − uv(t) − bv(t)w(t),

(25)

w(t) ˙ = cv(t) − pw(t).

(26)

z˙1 (t) = q1 e−m1 τ1

The basic reproduction number of model (22)-(26) is given by R01 =

((1 − q1 )e−n1 κ1 Ny1 + q1 e−r1 ω1 Mz1 ) e−m1 τ1 (1 − ε)β¯1 (x01 )k1 . u ς1 ((x01 )k1 + ρ1 )

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600

ε=0.0 ε=0.2 ε=0.4 ε=0.6 ε=0.8 ε=0.98

900

800

Uninfected macrophages (cell/mm3)

Uninfected CD4 T cells (cell/mm3)

1000

700

600

500 0

100

200

300 Time (days)

400

500

ε=0.0 ε=0.2 ε=0.4 ε=0.6 ε=0.8 ε=0.98

550 500 450 400 350 300 0

600

100

5 ε=0.0 ε=0.2 ε=0.4 ε=0.6 ε=0.8 ε=0.98

4 3 2 1 0 0

50

100

150 Time (days)

200

250

300

Chronically infected macrophages (cell/mm3)

Chronically infected CD4 T cells (cell/mm3)

ε=0.0 ε=0.2 ε=0.4 ε=0.6 ε=0.8 ε=0.98

7 6 5 4 3 2 1 100

150 Time (days)

200

600

ε=0.0 ε=0.2 ε=0.4 ε=0.6 ε=0.8 ε=0.98

4

3

2

1

0 0

50

100

150 Time (days)

200

250

300

250

15

ε=0.0 ε=0.2 ε=0.4 ε=0.6 ε=0.8 ε=0.98

10

300

5

0

0

50

100

150

200

250

300

Time (days)

(e) Chronically infected CD4+ T cells

(f) Chronically infected macrophages

15

2.5

ε=0.0 ε=0.2 ε=0.4 ε=0.6 ε=0.8 ε=0.98

10

ε=0.0 ε=0.2 ε=0.4 ε=0.6 ε=0.8 ε=0.98

2 B cells (cell/mm3)

Free virus (virus/mm3)

500

(d) Short-lived infected macrophages

8

50

400

5

(c) Short-lived infected CD4+ T cells

0 0

300 Time (days)

(b) Uninfected macrophages Short−lived infected macrophages (cell/mm3)

Short−lived infected CD4 Tcells (cell/mm3)

(a) Uninfected CD4+ T cells

200

5

1.5 1 0.5

0 0

20

40

60

80

100 120 Time (days)

140

160

180

0 0

200

(g) Free virus

10

20

30

40

50 Time (days)

60

70

80

90

100

(h) B cells

Figure 2: The evolution of the system state with different values of drug efficacy for model (16) -(20).

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600

τe=0.01

950

τe=0.2

900

τe=0.4

Uninfected macrophages (cell/mm3)

Uninfected CD4 T cells (cell/mm3)

1000

τe=0.6

850

τe=0.7 τe=1

800 750 700 650 0

100

200

300 Time (days)

400

500

τe=0.01

550

τe=0.2

500

τe=0.6

τe=0.4 τ =0.7 e

τ =1 e

450

400

350 0

600

100

4

τe=0.01

3.5

τe=0.2

3

τe=0.4

2.5

τe=0.6

2

τe=0.7

1.5

τe=1

1 0.5 0 0

50

100

150 Time (days)

200

250

300

Chronically infected macrophages (cell/mm3)

Chronically infected CD4 T cells (cell/mm3)

τ =0.01 e

τe=0.2

5

τe=0.4

4

τ =0.6 e

τe=0.7

3

τe=1 2 1

100

150 Time (days)

200

250

4

τe=0.2

3

τe=0.6

2

τe=1

τe=0.4 τe=0.7

1

0 0

50

100

150 Time (days)

200

250

300

15

τe=0.01 τe=0.2 τe=0.4 τe=0.6

10

300

τe=0.7 τe=1 5

0 0

50

100

150 Time (days)

200

250

300

(f) Chronically infected macrophages

12

2.5

τe=0.01

10

τ =0.2

8

τe=0.4 τe=0.6 τe=0.7

6

τ =1

4

τe=0.01 τe=0.2

2

e

B cells (cell/mm3)

Free virus (virus/mm3)

600

τe=0.01

(e) Chronically infected CD4+ T cells

e

τe=0.4 τe=0.6

1.5

τe=0.7 1

τe=1

0.5

2 0 0

500

(d) Short-lived infected macrophages

6

50

400

5

(c) Short-lived infected CD4+ T cells

0 0

300 Time (days)

(b) Uninfected macrophages Short−lived infected macrophages (cell/mm3)

Short−lived infected CD4 T cells (cell/mm3)

(a) Uninfected CD4+ T cells

200

20

40

60

80

100 120 Time (days)

140

160

180

0 0

200

(g) Free virus

10

20

30

40

50 Time (days)

60

70

80

90

(h) B cells

Figure 3: The evolution of the system state with different values of delayed for model (16) -(20).

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7

Reproduction numder R0

6

ε=0.0 ε=0.2 ε=0.4 ε=0.6 ε=0.8 ε=0.98

E1 is GAS

5 4 3 2 R =1 0

1 E0 is GAS 0 0

0.5

1

1.5

2

2.5

3

3.5

Delay (days)

Figure 4: Effects of the drug efficacy and delays on the basis reproduction number of model (3)-(7) Now we show that there is a number of parameter values for which R01 ≤ 1, but R0 > 1, and in such cases the solutions of system (22)-(26) tend to E0 (in R5≥0 ) as t → ∞, while those of (16) -(20) tend to E1 (in R8≥0 ) as t → ∞. We calculate the critical drug efficacy for system (16) -(20), E0 is GAS when R0 ≤ 1 i.e.   R0 − 1 crit εcrit ≤ ε < 1, ε = max 0, , 1 1 R01 + χR02 where R0 = R0 |ε=0 and R0i = R0i |ε=0 , i = 1, 2. For system (22)-(26), E0 is GAS when R01 ≤ 1 i.e. εcrit 2

≤ ε < 1,

εcrit 2

  R01 − 1 = max 0, . R01

crit Clearly, εcrit > εcrit ≤ ε ≤ εcrit 1 2 . Then, if one design treatment with drug efficacy ε2 1 , then E0 is GAS for system (22)-(26) but unstable for system (16) -(20). Using the data in Table 1 and τe = 0.01, we have εcrit = 0.93 and εcrit = 0.80. Let us choose ε = 0.88, then R01 |ε=0.88 = 0.62 < 1, but R0 |ε=0.88 = 1.31 > 1. 1 2 Therefore, more accurate treatment can be designed using the model (16) -(20) than those designed using model (22)-(26). Figure 5 shows the effect of two target cells on dynamics and control of HIV infection. We observe that, if we choose ε = 0.88, then the trajectory of model (16) -(20 tends to the infection-free equilibrium E0 = (1000, 0, 0, 0, 0, 0), while the trajectory of model (16) -(20) tends to the endemic equilibrium E1 = (990.54, 573.24, 0.1, 0.4, 0.15, 1.31, 1.04, 0.17).

3.6

Effect of long-lived chronically infected cells on the dynamics and controls of HIV infection

To show the effect of the presence of long-lived chronically infected cells on the dynamics and controls of HIV infection, we write the HIV model without long-lived chronically infected cells as: x˙ i (t) = λi − di xi (t) − y˙ i (t) = v(t) ˙ =

βi xki i (t)v(t) , (xki i (t) + ρi )(v(t) + ςi )

(27)

e−mi τi βi xki i (t − τi )v(t − τi ) − δi yi (t), (xki i (t − τi ) + ρi )(v(t − τi ) + ςi )

(28)

2 X Nyi δi e−ni κi yi (t − κi ) − uv(t) − bv(t)w(t),

(29)

i=1

w(t) ˙ = cv(t) − pw(t).

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950

Short−lived infected CD4 Tcells (cell/mm3)

Uninfected CD4 T cells (cell/mm3)

1000

one compartment, E0 is GAS two compartments, E0 is unstable

900

850 0

50

100

150

200 250 300 Time (days)

350

400

450

0.2 one compartment, E0 is GAS two compartment, E0 is unstable 0.15

0.1

0.05

500

0 0

10

20

30

40

50 60 Time (days)

70

80

90

100

2.5 one compartment, E0 is GAS

one compartment, E0 is GAS

two compartment, E0 is unstable

0.25

two compartment, E0 is unstable

2 3

Free virus (virus/mm )

Chronically infected CD4 T cells (cell/mm3)

(a) Uninfected CD4+ T cells for model (16)-(20) and model (22)- (b) Short-lived CD4+ T cells for model (16) -(20) and ((22)-(26). (26).

0.2 0.15 0.1

1.5

1

0.5 0.05 0 0

0 0 10

20

30

40

50 60 Time (days)

70

80

90

(c) Chronically infected CD4+ T cells for model (16) -(20) and (22)-(26). Short−lived infected macrophages (cell/mm3)

10

20

30

100

40

50 60 Time (days)

70

80

90

(d) Free virus for model (16) -(20)) and (22)-(26).

0.4 one compartment, E0 is GAS

0.35

two compartment, E0 is unstable

0.3 0.25 0.2 0.15 0.1 0.05 0 0

10

20

30

40 50 Time (days)

60

70

80

(e) B cells for model (16) -(20) and (22)-(26).

Figure 5: Effect of two types of target cells on the dynamics and controls of HIV infection

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The basic reproduction number for system (27)-(30) is given by e0 = R

2 2 X X e−ni κi e−mi τi Nyi βi (x0i )ki e0i = , R uςi ((x0i )ki + ρi ) i=1 i=1

e0 = R0 |q =q =0 . Since e−ni κi Ny > e−ri ωi Mz , i = 1, 2, then we have where R 1 2 i i R0 =

=

2 X ((1 − qi )e−ni κi Nyi + qi e−ri ωi Mzi ) e−mi τi βi (x0i )ki u ςi ((x0i )ki + ρi ) i=1 2 2 X e−ni κi e−mi τi Nyi βi (x0i )ki X (e−ni κi Nyi − e−ri ωi Mzi )qi e−mi τi βi (x0i )ki − uςi ((x0i )ki + ρi ) uςi ((x0i )ki + ρi ) i=1 i=1

e0 − =R

2 X (e−ni κi Nyi − e−ri ωi Mzi )qi e−mi τi βi (x0i )ki e0 . 1 and the trajectory tends to the endemic equilibrium with humoral immunity model (27)-(30), R E1 = (990.99, 558.53, 0.17, 1.36, 1.83, 0.3).

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600

Uninfected macrophages (cell/mm3)

Uninfected CD4 T cells (cell/mm3)

1000

980

absence of chronically, E0 is unstable

960

presence of chronically, E0 is GAS 940

920

900 0

100

150

200 250 300 Time (days)

350

400

450

540

0

Short−lived infected macrophages (cell/mm3)

absence of chronically, E0 is unstable presence of chronically, E0 is GAS 0.1

0.05

100

150 Time (days)

200

250

presence of chronically, E0 is GAS

520

50

100

150

350

400

450

500

1.5

absence of chronically, E0 is unstable presence of chronically, E0 is GAS

1

0.5

0 0

300

(c) Short-lived CD4+ T cells for model (16) -(20) and (27)-(30).

200 250 300 Time (days)

(b) Uninfected macrophages for model (16) -(20) and (27)-(30).

0.15

50

absence of chronically, E is unstable

500

0.2

Short−lived infected CD4 Tcells (cell/mm3)

560

500 0 50

(a) Uninfected CD4+ T cells for model (16)-(20) and (27)-(30).

0 0

580

50

100

150 Time (days)

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(d) Short-lived macrophages for model (16) -(20) and (27)-(30).

2

absence of chronically, E0 is unstable

B cells (cell/mm3)

Free virus (virus/mm3)

0.3 1.5

presence of chronically, E0 is GAS 1

0.25

absence of chronically, E0 is unstable presence of chronically, E0 is GAS

0.2 0.15 0.1

0.5

0.05 0 0

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150 Time (days)

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(e) Free virus for model (16) -(20) and (27)-(30).

0 0

50

100

150

200 250 Time (days)

300

350

400

(f) B cells for model (16) -(20) and (27)-(30).

Figure 6: Effect of long-lived chronically infected cells on the dynamics and controls of HIV infection

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4

Acknowledgment

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. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford Uni., Oxford, (2000). [2] A.S. Perelson and P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. [3] D. S. Callaway, and A. S. Perelson, “HIV-1 infection and low steady state viral loads”, Bull. Math. Biol.,vol. 64, pp. 29-64, 2002. [4] M. A. Obaid, Dynamical behaviors of a nonlinear virus infection model with latently infected cells and immune response, Journal of Computational Analysis and Applications, 21(1) (2016), 182-193. [5] V. Herz, S. Bonhoeffer, R. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251. [6] N. M. Dixit, A.S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay, J. Theoret. Biol. 226 (2004) 95-109. [7] C. Connell McCluskey, Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78. [8] P. K. Roy, A. N. Chatterjee, D. Greenhalgh, and Q. J.A. Khan, Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model, Nonlinear Anal. Real World Appl., 14 (2013), 1621-1633. [9] Z. Yuan, and X. Zou, Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays, Math. Biosc. Eng., 10 (2) (2013), 483-498. [10] B. Buonomo, and C. Vargas-De-Le, Global stability for an HIV-1 infection model in cluding an eclipse stage of infected cells, J. Math. Anal. Appl. 385 (2012), 709-720. [11] S. Liu, and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. Biosc. Eng., 7(3) (2010), 675-685. [12] A. M. Elaiw and X. Xia, HIV dynamics: Analysis and robust multirate MPC-based treatment schedules, J. Math. Anal. Appl., 356 (2009), 285-301. [13] M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonlinear Anal. Real World Appl., 17 (2014), 147-160. [14] A. M. Elaiw, I. A. Hassanien, and S. A. Azoz, Global stability of HIV infection models with intracellular delays, Journal of the Korean Mathematical Society, 49(4), (2012), 779-794. [15] A.M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Methods Appl. Sci., 36 (2013), 383-394. [16] A.M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263.

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[17] A. M. Elaiw, and N. A. Almuallem, Global properties of delayed-HIV dynamics models with differential drug efficacy in co-circulating target cells, Appl. Math. Comput., 265 (2015), 1067-1089. [18] A. M. Elaiw, R. M. Abukwaik, and E. O. Alzahrani, Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays, Int. J. Biomath., 77(5) (2014), 25 Pages. [19] B. Li, Y. Chen, X. Lu and S. Liu, A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13, (2016), 135-157. [20] D. Huang, X. Zhang, Y. Guo, and H. Wang, Analysis of an HIV infection model with treatments and delayed immune response, Appl. Math. Model., (in press). [21] Y. Zhao, D. T. Dimitrov, H. Liu, and Y. Kuang, Mathematical insights in evaluating state dependent effectiveness of HIV prevention interventions, Bull. Math. Biol. , vol. 75, pp. 649-675, 2013. [22] C. Monica and M. Pitchaimani, Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays, Nonlinear Anal. Real World Appl., 27 (2016), 55-69. [23] L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4 + T cells, Math. Biosci., 200(1) (2006), 44-57. [24] C. Lv, L. Huang, and Z. Yuan, Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 121-127. [25] R. Xu, Global stability of an HIV-1 infection model with saturation infection and in tracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81. [26] J. A. Deans and S. Cohen, Immunology of malaria, Ann. Rev. Microbiol. 37 (1983), 25-49. [27] M. A . Obaid and A. M. Elaiw, Stability of virus infection models with antibodies and chronically infected cells, Abstr. Appl. Anal, (2014) Article ID 650371. [28] S. Wang and D. Zou, Global stability of in host viral models with humoral immunity and intracellular delays, Appl. Math. Model., 36 (2012), 1313-1322. [29] T. Wang, Z. Hu and F. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, J. Math. Anal. Appl., 411 (2014) 63-74. [30] A. M. Elaiw, A. Alhejelan, Global dynamics of virus infection model with humoral immune response and distributed delays, Journal of Computational Analysis and Applications, 17 (2014), 515-523. [31] A. M. Elaiw and N. H. AlShameani, Stability analysis of general viral infection models with humoral immunity,. J. Nonlinear Sci. Appl., 9 (2016), 684-704. [32] A. M. Elaiw and N. H. AlShameani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Anal. Real World Appl., 26 (2015), 161-190. [33] T. Wang, Z. Hu, F. Liao and W. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulation, 89 (2013), 13-22. [34] R. Larson and B. H. Edwards, Calculus of a single variable, Cengage Learning, Inc., USA, (2010). [35] J.K. Hale and S. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, (1993).

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COMPOSITION OPERATORS ON DIRICHLET-TYPE SPACES LIU YANG, YECHENG SHI∗ A BSTRACT. In this note, motivated by [8], under the conditions of weighted function in [10], we characterize bounded and compact composition operator on Dirichlet-type spaces DK . We also give an equavalent charaterization of composition operator on DK , if the composition operator on DK spaces is Hilbert-Schmidt. Keywords: DK spaces; composition operators; Hilbert-Schmidt

1. I NTRODUCTION Let D be the unit disk in the complex plane C and H(D) be the class of functions analytic in D. Let K : [0, ∞) → [0, ∞) be a right-continuous and nondecreasing function. The Dirichlet-type spaces DK , consists of those functions f ∈ H(D), such that Z 2 2 kf kDK = |f (0)| + |f 0 (z)|2 K(1 − |z|2 )dA(z) < ∞. D α

When K(t) = t , 0 < α < 1, it give the classical Dirichlet-type space Dα . For more informations on Dα and DK spaces, we refer to [1], [3], [12], [19], [25]. Let ϕ be a holomorphic self-map of D. The composition operator Cϕ on DK is defined by Cϕ (f ) = f ◦ ϕ, f ∈ DK . There are many papers study composition operator, we refer to [4], [13], [14], [15], [17], [20], [21], [22], [24], [26]. Recently, Kellay and Lef`evre using Nevanlinna counting function, characterize bounded and compact composition operator on Dirichlet-type space DK under certain conditions in [13]. Later, Pau and P`erez studied the essential norm and closed ranged of composition operator on Dα in [17]. 2000 Mathematics Subject Classification. 30D45, 30D50. The first author is supported by NSF of China (No. 11471202) and supported by the Special Fund of the Shaanxi Provincial Education Department (grant Nr. 2013JK0567), Shaanxi Provincial Natural Science Foundation (grant Nr. 2014JM1018). *Corresponding author. 1

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2

In this paper, motivated by [8], we generalize Theorem 2.2 of [8] to DK spaces. We also give a characterizations of boundedness and compactness of composition operator Cϕ on DK spaces by ϕn . Furthermore, equavalent characterizations of composition operator on DK spaces belong to HilbertSchmidt was gave. Throughout this paper, suppose that K : [0, ∞) → [0, ∞) is a rightcontinuous and nondecreasing function. Satisfying Z 1 ϕK (s) ds < ∞ (1.1) s 0 and Z 1

∞

ϕK (s) ds < ∞, s2

(1.2)

where ϕK (s) = sup K(st)/K(t),

0 < s < ∞.

0≤t≤1

To learn more about weight function K, we refer to [2], [3], [9], [10] and [16]. Throughout this paper, for two functions f and g, f  g means that g . f . g, that is, there are positive constants C1 and C2 depend on K and index s, α, such that C1 g ≤ f ≤ C2 g.

2. AUXILIARY

RESULTS

Before to proof, we need to know some results. The following lemma can be found in Lemma 2.1 of [2]. Lemma 1. Let (1.1) and (1.2) hold for K. If 2 − α2 < s < 1 + c, then Z K (1 − |σa (w)|2 ) K (1 − |σa (z)|2 ) dA(w) . 2 s α (1 − |z|2 )s+α−2 D (1 − |w| ) |1 − wz| for all a, z ∈ D, where σa (z) =

z−a . 1−az

Lemma 2. Suppose that K satisfies (1.1) and (1.2). Then 1+

∞ X 1 n+1 n t  1 (1 − t)2 K(1 − t) K( n+1 ) n=1

for all 0 ≤ t < 1.

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Proof. Without loss of generality, we can assume 1/3 < t < 1, otherwise, it obvious. Make change of variables y = x1 , an easy computation gives 1 ∞ ∞ Z 1 X n+1 n X n tx t  dx 1 1 x3 K(x) K( n+1 ) n=1 n=1 n+1 Z 1 Z ∞ 1 tx yty  dx  dy. 3 K( y1 ) 0 x K(x) 1

Let y =

γ . − ln t

We can deduce that

Z ∞ ∞ X n+1 n 1 γe−γ t  dγ 1 K( n+1 (ln 1t )2 − ln t K( γ1 ln 1t ) ) n=1 Z ∞ γe−γ K(ln 1t ) 1 dγ = (ln 1t )2 K(ln 1t ) − ln t K( γ1 ln 1t ) Z ∞ 1 γe−γ ϕK (γ)dγ. . (1 − t)2 K(1 − t) − ln t By [10], under conditions (1.1) and (1.2), there exists an enough small c > 0 only depending on K such that ϕK (s) . sc , 0 < s ≤ 1 and ϕK (s) . s1−c , s ≥ 1. Therefore, Z ∞ ∞ X 1 n+1 n t . γe−γ ϕK (γ)dγ 1 2 (1 − t) K(1 − t) − ln t K( n+1 ) n=1  Z ∞ Z ∞ 1 −γ 2−c −γ 1+c e γ dγ . e γ dγ + (1 − t)2 K(1 − t) 0 0 1 (Γ(3 − c) + Γ(2 + c)) ,  2 (1 − t) K(1 − t) where Γ(.) is the Gamma function. It follows that ∞ X n+1 n 1 1+ t . . 1 2 (1 − t) K(1 − t) K( ) n+1 n=1

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Conversely, since K is nondecreasing, we deduce that Z ∞ ∞ X γe−γ K(ln 1t ) n+1 n 1 t  dγ 1 1 2 1 1 1 K( ) (ln ) K(ln ) K( ln ) − ln t n+1 t t γ t n=1 Z ∞ −γ γe K(ln 1t ) 1 & dγ (1 − t)2 K(1 − t) ln 2 K( γ1 ln 1t ) Z ∞ 1 γe−γ dγ & (1 − t)2 K(1 − t) ln 2 1 .  2 (1 − t) K(1 − t) 

The proof is completed. The next lemma can be found in Theorem 5 of [23].

Lemma 3. Let (1.2) hold for K. Then for any α > 0 and 0 ≤ β < 1, we have  1−β   Z 1 1 −β 1 1−β 1−β α−1 r (log ) K(log )dr  K . r r α α 0 3. B OUNDEDNESS AND

COMPACTNESS

In this section, motivated by [8], we discuss the boundedness and compactness of compostion operators by a general computation. Theorem 1. Suppose that (1.1) and (1.2) hold for K, s ≥ 0. Suppose ϕ(D) ⊂ D and ϕ ∈ DK . Then Cϕ is bounded on DK if and only if Z (1 − |a|2 )2+2s |ϕ0 (z)|2 sup K(1 − |z|2 )dA(z) < ∞. 2) 4+2s K(1 − |a| |1 − aϕ(z)| a∈D D Proof. Let 1 (1 − |a|2 )1+s Fa (z) = p , s ≥ 0. K(1 − |a|2 ) (1 − az)1+s Using Lemma 1, it is easy to check that Fa ∈ DK . If Cϕ is bounded on DK , then kCϕ (Fa )kDK < ∞, that is, Z (1 − |a|2 )2+2s |ϕ0 (z)|2 sup K(1 − |z|2 )dA(z) < ∞. 2) 4+2s K(1 − |a| |1 − aϕ(z)| a∈D D

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On the other hand, we know that for any pseudohyperbolic discs D(z, r), we have 1 − |w|  1 − |z|  |1 − wz|, for any w ∈ D(z, r) (see [27, page 69]). Let f ∈ DK . Applying sub-mean-property to |f 0 |2 , we have 0

|f 0 (w)|2 dA(w) |1 − wz|2

Z

2

|f (z)| ≤ D(z,r)

|f 0 (w)|2 (1 − |w|2 )2+2s dA(w) |1 − wz|4+2s

Z  D(z,r) 0

|f (w)|2 (1 − |w|2 )2+2s dA(w). |1 − wz|4+2s

Z .

D

Therefore, we get Z

|f 0 (ϕ(z))|2 |ϕ0 (z)|2 K(1 − |z|2 )dA(z)

D

 |f 0 (w)|2 2 2+2s (1 − |w| ) dA(w) |ϕ0 (z)|2 K(1 − |z|2 )dA(z) ≤ 4+2s |1 − wϕ(z)| D D  Z (1 − |w|2 )2+2s |ϕ0 (z)|2 2 ≤ sup K(1 − |z| )dA(z) 2 4+2s w∈D K(1 − |w| ) D |1 − wϕ(z)| Z × |f 0 (w)|2 K(1 − |w|2 )dA(w) < ∞. Z Z

D



The proof is completed.

Theorem 2. Suppose that (1.1) and (1.2) hold for K, s ≥ 0. Suppose ϕ(D) ⊂ D and ϕ ∈ DK . Then Cϕ is compact on DK if and only if (1 − |a|2 )2+2s lim |a|→1 K(1 − |a|2 )

Z D

|ϕ0 (z)|2 K(1 − |z|2 )dA(z) = 0. |1 − aϕ(z)|4+2s

Proof. Let (1 − |w|2 )2+2s G(w) = K(1 − |w|2 )

Z D

|ϕ0 (z)|2 K(1 − |z|2 )dA(z). |1 − wϕ(z)|4+2s

Let {fk }∞ k=1 be a bounded sequence of DK such that fk → 0 weakly. There0 fore, fk → 0 uniformly on compact sets. From the proof of Theorem 1 and

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dominated convergence theorem, when k → ∞, and r → 1, it follows that kCϕ (fk )k2DK − |fk (ϕ(0))|2 Z ≤ |fk0 (w)|2 G(w)K(1 − |w|2 )dA(w) ZD ≤ |fk0 (w)|2 G(w)K(1 − |w|2 )dA(w) rD Z + |fk0 (w)|2 G(w)K(1 − |w|2 )dA(w) → 0. D\rD

Thus, Cϕ is compact. Conversely, if Cϕ is compact, let {ak }∞ k=1 ⊆ D, |ak | → 1, (1 − |ak |2 )1+s 1 Fak (z) = p . 2 K(1 − |ak | ) (1 − ak z)1+s Then, it is easy to verify that Fak → 0 uniformly on compact sets. Thus, kCϕ (Fak )kDK → 0 as k → ∞. The proof is completed.  4. ϕn - TYPE C HARACTERIZATIONS In [24], Wulan, Zheng and Zhu gave an interesting characterizations of compostion operators Cϕ by ϕn . In this section, we are going to give an analogy results on DK spaces. Theorem 3. Let (1.1) and (1.2) hold for K. Suppose ϕ ∈ DK satisfies ϕ(D) ⊂ D and Cϕ : DK → DK . Then (1) If 1 n 2 sup 1 kϕ kDK < ∞, n K( n ) then Cϕ is bounded; (2) If Cϕ is bounded, then sup n

1 n 2 1 kϕ kDK < ∞. nK( n )

Proof. (1). Let a, z ∈ D and s > 0. Since |1 − a ¯ϕ(z)| ≥ 1 − |a||ϕ(z)| and

1 1  . 4+2s 2 (|1 − |a||ϕ(z)|) (|1 − |a| |ϕ(z)|2 )4+2s

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Note that ∞

X Γ(n + 4 + 2s) 1 |a|2n |ϕ(z)|2n ,  (|1 − |a|2 |ϕ(z)|2 )4+2s n!Γ(4 + 2s) n=0 it follows that Z (1 − |a|2 )2+2s |ϕ0 (z)|2 K(1 − |z|2 )dA(z) 2 4+2s K(1 − |a| ) D |1 − aϕ(z)| Z |ϕ0 (z)|2 (1 − |a|2 )2+2s . K(1 − |z|2 )dA(z) K(1 − |a|2 ) D (1 − |a||ϕ(z)|)4+2s Z X ∞ (1 − |a|2 )2+2s Γ(n + 4 + 2s) 2n |a| |ϕ(z)|2n |ϕ0 (z)|2 K(1 − |z|2 )dA(z).  2 K(1 − |a| ) D n=0 n!Γ(4 + 2s) By Stirling formula, we get Γ(n + 4 + 2s) ∼ n3+2s , n → ∞. n!Γ(4 + 2s) Therefore, (1 − |a|2 )2+2s K(1 − |a|2 ) .

(1 − |a|2 )2+2s K(1 − |a|2 )

|ϕ0 (z)|2 K(1 − |z|2 )dA(z) 4+2s D |1 − aϕ(z)| Z ∞ X 3+2s 2n n |a| |ϕ(z)|2n |ϕ0 (z)|2 K(1 − |z|2 )dA(z) Z

n=0 ∞ 2 2+2s X

(1 − |a| ) ≤ K(1 − |a|2 ) ≤

n=0 ∞ 2 2+2s X

(1 − |a| ) K(1 − |a|2 )

D

(n + 1)

3+2s

2n

Z

|a|

|ϕ(z)|2n |ϕ0 (z)|2 K(1 − |z|2 )dA(z)

D

(n + 1)1+2s |a|2n kϕn k2DK

n=0

∞ 2 2+2s X 1 1 n 2 (1 − |a| ) . sup (n + 1)1+2s K( )|a|2n . 1 kϕ kDK 2 K(1 − |a| ) n=0 n n K( n )

Following the proof of Lemma 2, we have ∞ X

1 K(1 − |a|2 ) (n + 1)1+2s K( )|a|2n  . n (1 − |a|2 )2+2s n=0

Thus, Z (1 − |a|2 )2+2s |ϕ0 (z)|2 K(1 − |z|2 )dA(z) K(1 − |a|2 ) D |1 − aϕ(z)|4+2s 1 n 2 . sup 1 kϕ kDK . n K( n )

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Hence, by Theorem 1, we prove (1). (2). Suppose that Cϕ is bounded on DK . Let fn (z) = z n /kz n k2DK . Then, we have kfn k2DK = 1. An easy computation gives, ∞ > kCϕ fn k2DK =

kϕn k2DK 1 & kϕn k2DK . 2 kz n kDK nK( n1 )

The last inequality is deduced by Lemma 3. The proof is completed.



Theorem 4. Let (1.1) and (1.2) hold for K. Suppose ϕ ∈ DK satisfies ϕ(D) ⊂ D and Cϕ : DK → DK . Then (1) If 1 kϕn k2DK = 0, n→∞ K( 1 ) n lim

then Cϕ is compact; (2) If Cϕ is compact, then 1 kϕn k2DK = 0. n→∞ nK( 1 ) n lim

Proof. (1). The proof is similar to (1) of Theorem 3. (2). Let {fn } be a bounded sequence in DK that convergence to 0 weakly. If Cϕ is compact on DK , then kCϕ fn kDK → 0, as n → ∞. Thus, for any z ∈ D, we have fn (ϕ(z)) → 0, n → ∞. Since {z n /kz n kDK , n ≥ 1} is bounded in DK and it converges to 0 pointwise, the compactness of Cϕ on DK implies that kϕn k2DK 1 n 2 = 1 kϕ kDK = 0. 2 n→∞ kz n k nK( n ) DK lim



The proof is completed.

5. H ILBERT-S CHMIDT CLASS Let Hilbert-Schmidt class be the space of all compact operators on Hilbert space with its singular value sequence {λn } ∈ l2 , the 2-summable sequence space (see [27, page 18]). The following theorem give an equavalent charaterizations of composition operator on DK spaces, when it belong to Hilbert-Schmidt class.

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Theorem 5. Let (1.1) and (1.2) hold for K. Suppose ϕ(D) ⊂ D, ϕ ∈ DK and Cϕ is compact. Then Cϕ is Hilbert-Schmidt on DK if and only if Z |ϕ0 (z)|2 K(1 − |z|2 ) dA(z) < ∞. 2 2 2 D (1 − |ϕ(z)| ) K(1 − |ϕ(z)| ) n

Proof. Without loss of generality, we can assume {1} ∪ { √ √z n

}∞ 1 K( n ) n=1

is

an orthonormal basis in DK and ϕ(0) = 0. From Theorem 1.22 of [27], Cϕ is Hilbert-Schmidt on DK if and only if ∞ X DK (ϕn ) < ∞. nK( n1 ) n=1 Applying Lemma 2, we have Z ∞ ∞ X DK (ϕn ) X n = |ϕ2 (z)|n−1 |ϕ0 (z)|2 K(1 − |z|2 )dA(z) 1 1 nK( K( ) ) D n n n=1 n=1 ∞ X n+1 Z |ϕ2 (z)|n |ϕ0 (z)|2 K(1 − |z|2 )dA(z) = 1 K( ) D n+1 n=0 Z K(1 − |z|2 ) |ϕ0 (z)|2 dA(z).  2 2 2 D (1 − |ϕ(z)| ) K(1 − |ϕ(z)| ) The proof is completed.  R EFERENCES [1] A. Aleman, Hilbert spaces of analytic functions between the Hardy space and the Dirichlet space, Proc. Amer. Math. Soc. 115(1992), 97-104. [2] G. Bao, Z. Lou, R. Qian and H. Wulan, Improving multipliers and zero sets in QK spaces, Collect. Math. DOI: 10.1007/s13348-014-0113-z. [3] G. Bao, Z. Lou, R. Qian and H. Wulan, On multipliers of Dirichlet type spaces, preprint. [4] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, Florida, 1955. [5] P. Duren, Theory of H p Spaces, Academic Press, New York, 1970. [6] K. Dyakonov, Division and multiplication by inner functions and embedding theorems for star-invariant subspaces, Amer. J. Math. 115(1993) 881-902. [7] K. Dyakonov, Self-improving behaviour of inner functions as multipliers, J. Funct. Anal. 240(2006), 429-444. [8] O. El-Fallah, K. Kellay, M. Shabankhah and H. Youssfi, Level sets and composition operators on the Dirichlet space, J. Funct. Anal. 260(2011) 1721-1733. [9] M. Essen and H. Wulan, On analytic and meromorphic function and spaces of QK -type, Illionis. J. Math. 46(2002), 1233-1258. [10] M. Essen, H. Wulan and J. Xiao, Several function-theoretic characterizations of M¨obius invariant QK spaces, J. Funct. Anal. 230(2006), 78-115. [11] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.

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10

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[12] R. Kerman and E. Sawyer, Carleson measures and multipliers of Dirichlet-type spaces. Trans. Amer. Math. Soc. 309(1988), 87-98. [13] K. Kellay and P. Lef`evre, Compact composition operators on weighted Hilbert spaces of analytic functions, J. Math. Anal. Appl. 386(2012), 718-727. [14] S. Li and S. Stevi´c, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl. 338(2008), 1282-1295. [15] S. Li and S. Stevi´c, Products of Volterra type operator and composition operator from H ∞ and Bloch spaces to the Zygmund space, J. Math. Anal. Appl. 345(2008), 40-52. [16] Z. Lou and W. Chen, Distances from Bloch functions to QK -type spaces, Integral Equations Operator Theory. 67(2010), 171-181. [17] J. Pau and P. A. P´erez, Composition operators acting on weighted Dirichlet spaces, J. Math. Anal. Appl. 401(2012), 682-694. [18] J. Pel´aez, Inner functions as improving multipliers, J. Funct. Anal. 255(2008), 1403-1418. [19] R. Rochberg and Z. Wu, A new characterization of Dirichlet type spaces and applications, Illinois J. Math. 37(1993), 101-122. [20] S. Stevi´c, Norm of weighted composition operators from Bloch space to Hµ∞ on the unit ball. Ars Combin. 88(2008), 125-127. [21] J. H. Shapiro, The essential norm of a composition operator, Ann. of Math. 125(1987) 375-404. [22] K.-J. Wirths, J. Xiao, Global integral criteria for composition operators, J. Math. Anal. Appl. 269(2002) 702-715. [23] H. Wulan and K. Zhu, Lacunary series in QK spaces, Stud. Math. 178(2007) 217-230. [24] H. Wulan, D. Zheng and K. Zhu, Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc. 137(2009) 3861-3868. [25] J. Zhou and Y. Wu, Decomposition theorems and conjugate pair in DK spaces, to appear. [26] N. Zorboska, Composition operators on weighted Dirichlet spaces, Poc. Amer. Math. Soc. 126(1998) 2013-2023. [27] K. Zhu, Operator Theory in Function Spaces, American Mathematical Society, Providence, RI, 2007. D EPARTMENT OF M ATHEMATICS , S HAANXI X UEQIAN N ORMAL U NIVERSITY, S HAANXI X I ’ AN 710100, P. R. C HINA E-mail address: [email protected] (L. Yang) D EPARTMENT OF M ATHEMATICS AND APPLICATION , S HANWEI VOCATIONAL AND T ECHNICAL C OLLEGE , G UANGDONG S HANWEI 516600, P. R. C HINA E-mail address: [email protected] (Y. Shi)

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On left multidimensional Riemann-Liouville fractional integral George Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we study some important properties of left multidimensional Riemann-Liouville fractional integral operator, such as of continuity and boundedness.

2010 AMS Subject Classi…cation: 26A33. Key Words and Phrases: Riemann-Liouville fractional integral, continuity, boundedness.

1

Motivation

From [1], p. 388 we have Theorem 1 Let r > 0, F 2 L1 (a; b), and Z s r G (s) = (s t)

1

F (t) dt;

a

all s 2 [a; b]. Then G 2 AC ([a; b]) (absolutely continuous functions) for r and G 2 C ([a; b]), only for r 2 (0; 1) :

2

1,

Main Results

We give Theorem 2 Let f 2 L1 ([a; b] [c; d]), Z x1 Z x2 F (x1 ; x2 ) = (x1 t1 ) 1 a1

1; 1

2

(x2

> 0. Consider the function t2 )

2

1

f (t1 ; t2 ) dt1 dt2 ;

(1)

a2

1

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where a1 ; x1 2 [a; b], a2 ; x2 2 [c; d] : a1 x1 , a2 Then F is continuous on [a1 ; b] [a2 ; d] :

x2 :

Proof. (I) Let a1 ; b1 ; b1 2 [a; b] with b1 > b1 > a1 , and a2 ; b2 ; b2 2 [c; d] with b 2 > b 2 > a2 : We observe that F (b1 ; b2 ) F (b1 ; b2 ) = Z b1 Z b2 1 1 (b1 t1 ) 1 (b2 t2 ) 2 f (t1 ; t2 ) dt1 dt2 a1

Z

a2

Z

b1

a1

Z

Z

a1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2 =

(b1

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2

(b1

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2 +

(b1

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2 +

(b1

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2 +

(b1

t1 )

(2)

b2

a2

Z

b1

b1

b2

a2

Z

b1

b2

b2

a1

Z

1

b2

Z

a1

Z

t1 )

a2

b1

Z

(b1

a2

b1

Z

b2

b1

b1

Z

b2 1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2 :

b2

Call I (b1 ; b2 ) =

Z

b1

a1

Z

b2

(b1

t1 )

1

1

(b2

t2 )

2

1

(b1

t1 )

1

1

(b2

t2 )

2

1

dt1 dt2 :

a2

(3)

Thus jF (b1 ; b2 )

I (b1 ; b2 ) + (b1

a1 )

1

(b1

(b1

b1 )

b1 )

F (b1 ; b2 )j

1

(b2

a2 )

2

1 1

1

(b2

(b2

b2 )

2

+

2

b2 ) 2

2

+

(b1

b1 )

1

(b2

1

b2 ) 2

2

kf k1 : (4)

Hence, by (4), it holds :=

lim jF (b1 ; b2 ) (b1 ;b2 )!(b1 ;b2 )

F (b1 ; b2 )j

or (b1 ;b2 )!(b1 ;b2 )

(lim I (b1 ; b2 )) kf k1 =: : (b1 ;b2 )!(b1 ;b2 ) or (b1 ;b2 )!(b1 ;b2 )

(5) 2

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If

1

If

= 1

= 0, proving 2 = 1, then = 1, 2 > 0 we get

I (b1 ; b2 ) = (b1

Z

a1 )

= 0:

b2

(b2

1

2

t2 )

(b2

1

2

t2 )

dt2

a2

!

:

Assume 2 > 1, then 2 1 > 0. Hence by b2 > b2 , then b2 t2 > b2 t2 1 1 1 1 and (b2 t2 ) 2 > (b2 t2 ) 2 and (b2 t2 ) 2 (b2 t2 ) 2 > 0: That is " # a (b2 t2 ) 2 2 (b2 a2 ) 2 I (b1 ; b2 ) = (b1 a1 ) 2

= (b1

(b2

a1 )

2

a2 )

2

b2 )

0,

2

b2

(b2

(6)

(b2

2

a2 )

:

(7)

2

Clearly, then lim I (b1 ; b2 ) = 0:

(8)

b2 !b2 or b2 !b2

Similarly and symmetrically, we obtain that lim I (b1 ; b2 ) = 0;

(9)

b1 !b1 or b1 !b1

for the case of 2 = 1, If 1 = 1, and 0 < I (b1 ; b2 ) = (b1

1 2

> 1. < 1, then

a1 )

Z

1 < 0. Hence

2

b2

(b2

t2 )

2

1

(b2

t2 )

2

1

dt2

a2

(b1

a1 )

(b2

a2 )

2

(b2

a2 )

2

+ (b2

b2 )

!

=

2

:

(10)

2

Clearly, then lim I (b1 ; b2 ) = 0:

b2 !b2 or b2 !b2

(11)

Similarly and symmetrically, we derive that lim I (b1 ; b2 ) = 0;

b1 !b1 or b1 !b1

for the case of

2

= 1, 0
1, then Z b1 Z b2 h (b1 t1 ) 1 I (b1 ; b2 ) = 1

a1 )

(b2

t2 )

2

1

(b1

2

(b2

b2 )

1

1

t1 )

(b2

t2 )

1

2

a2

a1

(b1

1

(b1

b1 )

1

(b2

a2 )

1

2

(b1

1

a1 )

2

i

dt1 dt2 =

(13) a2 )

(b2

1

2

:

2

That is I (b1 ; b2 ) = 0:

lim

(14)

(b1 ;b2 )!(b1 ;b2 ) or (b1 ;b2 )!(b1 ;b2 )

Case now of 0 < 1 ; 2 < 1, then Z b1 Z b2 h 1 (b1 t1 ) 1 (b2 I (b1 ; b2 ) = a1 )

1

(b1

1

1

t1 )

(b2

t2 )

2

(b2

1

2

a2

a1

(b1

1

2

t2 )

(b2

2

a2 )

1

(b1

1

a1 )

(b1

2

1

b1 )

(b2

a2 )

1

i

dt1 dt2 =

b2 )

2

:

2

(15) That is again, when 0
1, 0
1, 0
1 and 0
b1 and b2 > b2 , as symmetric to b1 > b1 and b2 > b2 we treated, it is omitted, a totally similar treatment. (II) The remaining cases are: let a1 ; b1 ; b1 2 [a; b]; a2 ; b2 ; b2 2 [c; d], we can have (II1 ) b1 > b1 and b2 < b2 ; or (II2 ) b1 < b1 and b2 > b2 : Notice that (II1 ) and (II2 ) cases are symmetric, and treated the same way. As such we treat only the case (II1 ). We observe again that F (b1 ; b2 ) Z

b1 Z

a1

Z

b1

a1

Z

b1

a1

b2

F (b1 ; b2 ) =

(b1

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2

(b1

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2 =

(b1

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2 +

a2

Z

b2

(24)

a2

Z

b2

a2

5

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Z

Z

b1

b1

Z

b1

a1

b1

a1

Z

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2

Z

(b1

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2

(b1

t1 )

1

1

(b2

t2 )

2

1

f (t1 ; t2 ) dt1 dt2 =

b2

a2

a1

Z

(b1

a2

b1

Z

b2

Z

b2

b2

b2

(b1

a2

+

Z

b1

b1

Z

1

1

t1 )

Z

(b2

t2 )

2

1

(b1

t1 )

1

1

(b2

t2 )

(b1

t1 )

1

(b2

t2 )

b2

(b1

2

1

1

t1 )

(b2

1

2

t2 )

1

f (t1 ; t2 ) dt1 dt2

1

f (t1 ; t2 ) dt1 dt2 :

f (t1 ; t2 ) dt1 dt2

a2

Z

b1

a1

b2 1

2

(25)

b2

We call I (b1 ; b2 ) :=

Z

b1

a1

Z

b2

(b1

1

1

t1 )

(b2

t2 )

2

1

(b1

t1 )

1

1

(b2

t2 )

2

1

dt1 dt2 :

a2

(26)

Hence, we have

I (b1 ; b2 ) +

(b1

b1 )

jF (b1 ; b2 )

1

(b2

a2 )

1

F (b1 ; b2 )j 2

+

(b1

a1 )

2

1

(b2

b2 )

1

2

kf k1 : (27)

2

Therefore it holds :=

jb1 jb2

lim jF (b1 ; b2 ) b1 j!0; b2 j!0

F (b1 ; b2 )j jb1 jb2

(lim I (b1 ; b2 )) kf k1 =: : b1 j!0; b2 j!0

(28)

We will prove that = 0, hence = 0, in all possible cases. If 1 = 2 = 1, then I (b1 ; b2 ) = 0, hence = 0: If 1 = 1, 2 > 0 we get I (b1 ; b2 ) = (b1

Z

a1 )

b2

(b2

t2 )

2

1

(b2

2

t2 )

1

dt2

a2

Assume

2

> 1, then

2

I (b1 ; b2 ) = (b1

:

(29)

1 > 0. Hence a1 )

Z

b2

(b2

t2 )

2

1

(b2

t2 )

a2

= (b1

!

a1 )

"

(b2

a

t2 ) 2 jb22

(b2 2

a2 )

2

2

1

dt2

!

#

6

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= (b1

(b2

a1 )

2

a2 )

(b2

2

b2 )

(b2

2

a2 )

:

(30)

2

Clearly, then lim I (b1 ; b2 ) = 0; b1 j!0; b2 j!0

jb1 jb2 hence = 0: Let the case now of

= 1,

2

I (b1 ; b2 ) = (b2

a2 )

> 1: Then

1

Z

(31)

b1

(b1

1

1

t1 )

(b1

1

1

t1 )

dt1

a1

= (b2

(b1

a2 )

1

a1 )

(b1

1

b1 )

(b1

!

1

a1 )

:

(32)

1

Then If

= 0: = 1, and 0
1 and 0
0, i = 1; :::; k 2 N. Consider the

i

1

f (t1 ; :::; tk ) dt1 :::dtk ;

(43)

i=1

where ai ; xi 2 [ai ; bi ], ai xi , i = 1; :::; k: Qk Then F is continuous on i=1 [ai ; bi ] :

Remark 4 In the setting of Theorem 3: Consider the left multidimensional Riemann-Liouville fractional integral of order = ( 1 ; :::; k ) : Ia+ f (x) = Qk

i=1

1 ( i)

Z

x1

a1

:::

Z

xk

ak

k Y

ti )

i

1

f (t1 ; :::; tk ) dt1 :::dtk ;

i=1

where a = (a1 ; :::; ak ), x = (x1 ; :::; xk ), ai By Theorem 3 we get that Qk x 2 i=1 [ai ; bi ] :

(xi

xi , i = 1; :::; k: Here

(44) denotes the gamma f unction:

Ia+ f (x) is a continuous function for every

9

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We notice that Ia+ f (x)

Qk

1

i=1

kf k1

= Qk

i=1

Z k Y

( i ) i=1

( i)

Z

ti )

ai

= kf k1 That is Ia+ f (x)

:::

a1

xi

(xi

x1

Z

xk

ak

(xi

ti )

i

Ia+ f

1

dt1 :::dtk

i=1

1

dti

k

k Y (xi

ai ) i ( i + 1)

!

kf k1 :

Ia+ f (a ) = 0; and

1

Y (xi kf k1 = Qk i=1 ( i ) i=1 ! k Y (xi ai ) i : ( i + 1) i=1

i

i=1

In particular we get that

!

k Y

k Y (bi

ai ) i ( i + 1)

i=1

!

kf k1

ai )

i

(45)

i

(46)

(47) !

kf k1 :

(48)

That is Ia+ f is a bounded linear operator, which here is also a positive operator.

References [1] G. Anastassiou, Fractional Di¤ erentiation Inequalities, Springer, New York, 2009.

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Weak closure operations on ideals of BCK-algebras Hashem Bordbar1 , Mohammad Mehdi Zahedi2 , Sun Shin Ahn3,∗ and Young Bae Jun4 1

4

Faculty of Mathematics, Statistics and Computer Science, Shahid Bahonar University, Kerman, Iran 2 Department of Mathematics, Graduate University of Advanced Technology, Mahan-Kerman, Iran 3 Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

The Research Institute of Nature Sciences, Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea

Abstract. Weak closure operation, which is more general form than closure operation, on ideals of BCKalgebras is introduced, and related properties are investigated. Regarding weak closure operation, finite type and (strong) quasi-primeness are considered. Also positive implicative (resp., commutative and implicative) weak closure operations are discussed.

1. Introduction Semi-prime closure operations on ideals of BCK-algebras are introduced in the paper [1], and a finite type of closure operations on ideals of BCK-algebras are discussed in [2]. In this paper, we consider more general form than closure operations on ideals of BCK-algebras. We introduce the notion of weak closure operations on ideals of BCK-algebras. Regarding weak closure operation, we define finite type and (strong) quasi-primeness, and investigate related properties. We also discuss positive implicative (resp., commutative and implicative) weak closure operations, and provide several examples to illustrate notions and properties. 2. Preliminaries

A BCK/BCI-algebra is an important class of logical algebras introduced by K. Is´eki and was extensively investigated by several researchers. An algebra (X; ∗, 0) of type (2, 0) is called a BCI-algebra if it satisfies the following conditions: (I) (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (II) (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0), 0

2010 Mathematics Subject Classification: 06F35; 03G25. Keywords: weak closure operation of finite type; (quasi-prime, positive implicative, commutative, implicative) weak closure operation. ∗ The corresponding author. 0 E-mail: [email protected] (H. Bordbar); zahedi [email protected] (M. M. Zahedi); [email protected] (S. S. Ahn); [email protected] (Y. B. Jun) 0

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Hashem Bordbar, Mohammad Mehdi Zahedi, Sun Shin Ahn and Young Bae Jun

(III) (∀x ∈ X) (x ∗ x = 0), (IV) (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y). If a BCI-algebra X satisfies the following identity: (V) (∀x ∈ X) (0 ∗ x = 0), then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following axioms: (a1) (a2) (a3) (a4)

(∀x ∈ X) (x ∗ 0 = x), (∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x), (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y), (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y)

where x ≤ y if and only if x ∗ y = 0. A subset A of a BCK/BCI-algebra X is called an ideal of X (see [4]) if it satisfies: 0 ∈ A,

(2.1)

(∀x ∈ X) (∀y ∈ A) (x ∗ y ∈ A ⇒ x ∈ A) .

(2.2)

For any subset A of X, the ideal generated by A is defined to be the intersection of all ideals of X containing A, and it is denoted by ⟨A⟩. If A is finite, then we say that ⟨A⟩ is finitely generated ideal of X (see [4]). A subset A of a BCK-algebra X is called a commutative ideal of X (see [4]) if it satisfies (2.1) and (∀x, y ∈ X)(∀z ∈ A) ((x ∗ y) ∗ z ∈ A ⇒ x ∗ (y ∗ (y ∗ x)) ∈ A) .

(2.3)

A subset A of a BCK-algebra X is called a positive implicative ideal of X (see [4]) if it satisfies (2.1) and (∀x, y, z ∈ X) ((x ∗ y) ∗ z ∈ A, y ∗ z ∈ A ⇒ x ∗ z ∈ A) .

(2.4)

A subset A of a BCK-algebra X is called an implicative ideal of X (see [4]) if it satisfies (2.1) and (∀x, y ∈ X)(∀z ∈ A) ((x ∗ (y ∗ y)) ∗ z ∈ A ⇒ x ∈ A) .

(2.5)

Denote by Ipi (X) (resp., Ic (X) and Im (X)) the set of all positive implicative (resp., commutative and implicative) ideals of X. We refer the reader to the books [3, 4] for further information regarding BCK/BCI-algebras. 3. Weak Closure operations In what follows, let X and I(X) be a BCK-algebra and a set of all ideals of X, respectively, unless otherwise specified .

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Weak closure operations on ideals of BCK-algebras

Definition 3.1. A mapping c : I(X) → I(X) is called a weak closure operation on I(X) if the following conditions are valid. (∀A ∈ I(X)) (A ⊆ c(A)) ,

(3.1)

(∀A, B ∈ I(X)) (A ⊆ B ⇒ c(A) ⊆ c(B)) .

(3.2)

If a weak closure operation c : I(X) → I(X) satisfies the condition (∀A ∈ I(X)) (c(c(A)) = c(A)) ,

(3.3)

then we say that c is a closure operation on I(X) (see [2]). In what follows, we use Acl instead of c(A). Example 3.2. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4

0 0 1 2 3 4

1 0 0 2 2 4

2 0 1 0 1 4

3 0 0 0 0 4

4 0 1 2 3 0

We have 8 ideals of X, and they are A0 = {0}, A1 = {0, 1}, A2 = {0, 2}, A3 = {0, 4}, A4 = {0, 1, 4}, A5 = {0, 1, 2, 3}, A6 = {0, 2, 4}, and A7 = X. Define a mapping c : I(X) → I(X) cl cl by Acl 0 = A0 , A1 = A4 , A2 = A5 , c(A3 ) = A6 , and c(A4 ) = c(A5 ) = c(A6 ) = c(A7 ) = A7 . Then c is a weak closure operation on I(X). But it is not a closure operation on I(X) since c(Acl 2 ) = c(A5 ) = A7 . In a BCK-algebra X, let x ∧ y denote the greatest lower bound of x and y. Note that 0 ∧ x = 0 for all x ∈ X. For any element x of X, consider the following condition (∃ y ∈ X \ {0}) (x ∧ y = 0) .

(3.4)

In the following example, we know that there are two kinds of element. One is an element x satisfying the condition (3.4). The other is an element x which does not satisfy the condition (3.4). Example 3.3. Let X = {0, 1, 2, 3, 4} be a set with the following Cayley table. ∗ 0 1 2 3 4

0 0 1 2 3 4

1 0 0 2 2 4

251

2 0 1 0 1 4

3 0 0 0 0 4

4 0 0 0 0 0

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Hashem Bordbar, Mohammad Mehdi Zahedi, Sun Shin Ahn and Young Bae Jun

Then X is a BCK-algebra. We know that 1 and 2 satisfy the condition (3.4), but 3 and 4 do not satisfy the condition (3.4). On the basis of this consideration, we define the zeromeet element in a BCK-algebra. Definition 3.4. An element x of X is called a zeromeet element of X if the condition (3.4) is valid. Otherwise, x is called a non-zeromeet element of X. Denote by Z(X) the set of all zeromeet elements of X, that is, Z(X) = {x ∈ X | x ∧ y = 0 for some nonzero element y ∈ X}. Obviously, 0 ∈ Z(X). We know that 0, 1, 2 ∈ Z(X) and 3, 4 ∈ / Z(X) in Example 3.3. Lemma 3.5. For any x, y ∈ X, if x, y ∈ / Z(X), then x ∧ y ∈ / Z(X), that is, the set X \ Z(X) is closed under the operation ∧. Proof. Let x, y ∈ X \ Z(X) and assume that x ∧ y ∈ Z(X). Then x ∧ (y ∧ a) = (x ∧ y) ∧ a = 0 for some nonzero element a ∈ X. Since x ∈ / Z(X), it follows that y ∧ a = 0 and so that a = 0 since y ∈ / Z(X). This is a contradiction, and thus x ∧ y ∈ / Z(X). □ For any subsets A and B of X, we define A ∧ B := ⟨{a ∧ b | a ∈ A, b ∈ B⟩. We use x ∧ A instead of {x} ∧ A, that is, x ∧ A := ⟨{x ∧ a | a ∈ A}⟩. Definition 3.6. A weak closure operation cl : I(X) → I(X) is said to be quasi-prime if it satisfies: ( ) (∀a ∈ X \ Z(X)) (∀A ∈ I(X)) a ∧ Acl ⊆ (a ∧ A)cl . (3.5) Example 3.7. Consider a BCK-algebra X = {0, 1, 2, 3} with the following Cayley table. ∗ 0 1 2 3 0 0 0 0 0 1 1 0 0 0 2 2 2 0 0 3 3 3 3 0 We know that Z(X) = {0} and there are four ideals in X, that is, A0 = {0}, A1 = {0, 1}, cl cl A2 = {0, 1, 2} and A3 = X. Define a mapping cl : I(X) → I(X) by Acl 0 = A0 , A1 = A2 , A2 = A3 and Acl 3 = A3 . Then “cl” is a weak closure operation on I(X). For 1, 2, 3 ∈ X \ Z(X), we have cl cl 1 ∧ Acl 0 = 1 ∧ A0 = ⟨{0}⟩ = A0 = A0 = (1 ∧ A0 ) , cl cl 1 ∧ Acl 1 = 1 ∧ A2 = ⟨{0, 1}⟩ = A1 ⊆ A2 = A1 = (1 ∧ A1 ) , cl cl 1 ∧ Acl 2 = 1 ∧ A3 = ⟨{0, 1}⟩ = A1 ⊆ A2 = A1 = (1 ∧ A2 ) , cl cl 1 ∧ Acl 3 = 1 ∧ A3 = ⟨{0, 1}⟩ = A1 ⊆ A2 = A1 = (1 ∧ A3 ) ,

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Weak closure operations on ideals of BCK-algebras cl cl 2 ∧ Acl 0 = 2 ∧ A0 = ⟨{0}⟩ = A0 = A0 = (2 ∧ A0 ) , cl cl 2 ∧ Acl 1 = 2 ∧ A2 = ⟨{0, 1, 2}⟩ = A2 = A1 = (2 ∧ A1 ) , cl cl 2 ∧ Acl 2 = 2 ∧ A3 = ⟨{0, 1, 2}⟩ = A2 ⊆ A3 = A2 = (2 ∧ A2 ) , cl cl 2 ∧ Acl 3 = 2 ∧ A3 = ⟨{0, 1, 2}⟩ = A2 ⊆ A3 = A2 = (2 ∧ A3 ) , cl cl 3 ∧ Acl 0 = 3 ∧ A0 = ⟨{0}⟩ = A0 = A0 = (3 ∧ A0 ) , cl cl 3 ∧ Acl 1 = 3 ∧ A2 = ⟨{0, 1, 2}⟩ = A2 = A1 = (3 ∧ A1 ) , cl cl 3 ∧ Acl 2 = 3 ∧ A3 = ⟨{0, 1, 2, 3}⟩ = A3 = A2 = (3 ∧ A2 ) , cl cl 3 ∧ Acl 3 = 3 ∧ A3 = ⟨{0, 1, 2, 3}⟩ = A3 = A3 = (3 ∧ A3 ) , Therefore ”cl” is a quasi-prime weak closure operation on I(X).

Definition 3.8. A weak closure operation cl : I(X) → I(X) is said to be strong quasi-prime if it satisfies: ( ) (∀a ∈ X \ Z(X)) (∀A ∈ I(X)) a ∧ Acl = (a ∧ A)cl . (3.6) Example 3.9. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4

0 0 1 2 3 4

1 0 0 2 3 4

2 0 1 0 3 4

3 0 0 0 0 4

4 0 0 0 0 0

We know that Z(X) = {0, 1, 2} and there are six ideals in X, that is, A0 = {0}, A1 = {0, 1}, A2 = {0, 2}, A3 = {0, 1, 2}, A4 = {0, 1, 2, 3} and A5 = X. Define a mapping cl : I(X) → I(X) cl cl cl cl cl as follows: Acl 0 = A1 , A1 = A2 = A3 , A3 = A4 = A4 and A5 = A5 . Then “cl” is a weak closure operation on I(X). For 3, 4 ∈ X \ Z(X), we have cl cl 3 ∧ Acl 0 = 3 ∧ A1 = ⟨{0, 1}⟩ = A1 = A0 = (3 ∧ A0 ) , cl cl 3 ∧ Acl 1 = 3 ∧ A3 = ⟨{0, 1, 2}⟩ = A3 = A1 = (3 ∧ A1 ) , cl cl 3 ∧ Acl 2 = 3 ∧ A3 = ⟨{0, 1, 2}⟩ = A3 = A2 = (3 ∧ A2 ) , cl cl 3 ∧ Acl 3 = 3 ∧ A4 = ⟨{0, 1, 2, 3}⟩ = A4 = A3 = (3 ∧ A3 ) , cl cl 3 ∧ Acl 4 = 3 ∧ A4 = ⟨{0, 1, 2, 3}⟩ = A4 = A4 = (3 ∧ A4 ) , cl cl 3 ∧ Acl 5 = 3 ∧ A5 = ⟨{0, 1, 2, 3}⟩ = A4 = A4 = (3 ∧ A5 ) , cl cl 4 ∧ Acl 0 = 4 ∧ A1 = ⟨{0, 1}⟩ = A1 = A0 = (4 ∧ A0 ) , cl cl 4 ∧ Acl 1 = 4 ∧ A3 = ⟨{0, 1, 2}⟩ = A3 = A1 = (4 ∧ A1 ) , cl cl 4 ∧ Acl 2 = 4 ∧ A3 = ⟨{0, 1, 2}⟩ = A3 = A2 = (4 ∧ A2 ) , cl cl 4 ∧ Acl 3 = 4 ∧ A4 = ⟨{0, 1, 2, 3}⟩ = A4 = A3 = (4 ∧ A3 ) , cl cl 4 ∧ Acl 4 = 4 ∧ A4 = ⟨{0, 1, 2, 3}⟩ = A4 = A4 = (4 ∧ A4 ) , cl cl 4 ∧ Acl 5 = 4 ∧ A5 = ⟨{0, 1, 2, 3}⟩ = A4 = A4 = (4 ∧ A5 ) . Therefore “cl” is a strong quasi-prime weak closure operation on I(X).

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Hashem Bordbar, Mohammad Mehdi Zahedi, Sun Shin Ahn and Young Bae Jun

Given an ideal A of X and an operation cl : I(X) → I(X) on I(X), we consider the following set: K := ∪{B cl | B ⊆ A, B ∈ If (X)} (3.7) where If (X) is the set of all finitely generated ideals of X. The following example shows that the set K in (3.7) may not be an ideal of X in general. Example 3.10. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4

0 0 1 2 3 4

1 0 0 2 3 4

2 0 0 0 3 3

3 0 0 2 0 2

4 0 0 0 0 0

There are five ideals in X, that is, A0 = {0}, A1 = {0, 1}, A2 = {0, 1, 2}, A3 = {0, 1, 3} and cl cl cl A4 = X. Define a mapping cl : I(X) → I(X) as follows: Acl 0 = A3 , A1 = A2 , A2 = A0 , A3 = A4 and Acl 4 = A3 . For the ideal A2 of X, we have cl cl ∪{B cl | B ⊆ A, B ∈ If (X)} = Acl 0 ∪ A1 ∪ A2 = {0, 1, 2, 3}

which is not an ideal of X. We provide a condition for the set K in (3.7) to be an ideal of X. Theorem 3.11. If cl : I(X) → I(X) is a weak closure operation on I(X), then the set K in (3.7) is an ideal of X for any ideal A of X. Proof. Obviously, 0 ∈ K. Let x, y ∈ X such that x ∗ y ∈ K and y ∈ K. Then there exist Bx , By ∈ If (X) such that Bx ⊆ A, By ⊆ A, x ∗ y ∈ Bxcl and y ∈ Bycl . Since Bx , By ⊆ Bx + By = ⟨Bx ∪ By ⟩, we have x ∗ y ∈ Bxcl ⊆ (Bx + By )cl and y ∈ Bycl ⊆ (Bx + By )cl , which imply that x ∈ (Bx + By )cl . Since Bx , By ∈ If (X), we get Bx + By ∈ If (X) and Bx + By ⊆ A. Therefore x ∈ K, and K is an ideal of X. □ Corollary 3.12. If cl : I(X) → I(X) is a closure operation on I(X), then the set K in (3.7) is an ideal of X for any ideal A of X. Lemma 3.13 ([4]). (Extension property) Let A and B be ideals of X such that A ⊆ B. If A is a positive implicative (resp., commutative and implicative) ideal, then so is B. Using Lemma 3.13 and (3.1), we have the following theorem. Theorem 3.14. Let “cl” be a weak closure operation on I(X). If A is a positive implicative (resp., commutative and implicative) ideal of X, then so is Acl . The following example shows that the converse of Theorem 3.14 is not true in general.

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Example 3.15. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4

0 1 2 3 4 0 0 0 0 0 1 0 1 0 0 2 2 0 0 0 3 3 3 0 0 4 3 4 1 0 There are five ideals in X, that is, A0 = {0}, A1 = {0, 1}, A2 = {0, 2}, A3 = {0, 1, 2} and A4 = X. Define a mapping cl : I(X) → I(X) as follows: (A0 )cl = A0 , (A1 )cl = (A2 )cl = A3 , and (A3 )cl = (A4 )cl = A4 . Then “cl”” is a weak closure operation on I(X). The ideal A2 = {0, 2} is not positive implicative (resp., commutative and implicative) ideal, but (A2 )cl = A3 = {0, 1, 2} is a positive implicative (resp., commutative and implicative) ideal of X. Theorem 3.16. An operation cl : I(X) → I(X) on I(X) defined by ( ) (∀A ∈ I(X)) Acl = ∩{Iλ | Iλ ∈ IΓ (X), A ⊆ Iλ , λ ∈ Λ}

(3.8)

is a weak closure operation on I(X) where IΓ (X) ∈ {Ipi (X), Ic (X), Im (X)} and Λ is any index set. Proof. Obviously, A ⊆ Acl for every A ∈ I(X). Let A, B ∈ I(X) be such that A ⊆ B. Then B cl = ∩{Iλ | Iλ ∈ IΓ (X), B ⊆ Iλ , λ ∈ Λ} ⊇ ∩{Iλ | Iλ ∈ IΓ (X), A ⊆ Iλ , λ ∈ Λ} = Acl , and so “cl” is a weak closure operation on I(X).

□

The following example illustrates Theorem 3.16. Example 3.17. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 0 0 0 1 1 0 2 2 2 3 3 1 4 4 4 There are six ideals in X, that is, A0 = {0}, A4 = {0, 2, 4} and A5 = X. (1) Define a mapping cl1 : I(X) → I(X) by

2 0 1 0 3 4 A1

3 0 0 2 0 4 =

4 0 1 0 3 0 {0, 1, 3}, A2 = {0, 2}, A3 = {0, 1, 2, 3},

Acl1 = ∩{B | A ⊆ B and B ∈ Ipi (X)}.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Hashem Bordbar, Mohammad Mehdi Zahedi, Sun Shin Ahn and Young Bae Jun

Then we have cl1 1 Acl 0 = A1 ∩ A3 ∩ A5 = A1 , A1 = A1 ∩ A3 ∩ A5 = A1 , cl1 cl1 cl1 1 Acl 2 = A3 ∩ A5 = A3 , A3 = A3 ∩ A5 = A3 , A4 = A5 = A5 . We can check that “cl1 ” is a weak closure operation on I(X). (2) We define an operation “cl2 ” on I(X) by Acl2 = ∩{B | A ⊆ B and B ∈ Ic (X)}. Then we have cl2 2 Acl 0 = A2 ∩ A3 ∩ A4 ∩ A5 = A2 , A1 = A3 ∩ A5 = A3 , cl2 2 Acl 2 = A2 ∩ A3 ∩ A4 ∩ A5 = A2 , A3 = A3 ∩ A5 = A3 , cl2 2 Acl 4 = A4 ∩ A5 = A4 , A5 = A5 . It is routine to verify that “cl2 ” is a weak closure operation on I(X). (3) We define an operation “cl3 ” on I(X) by Acl3 = ∩{B | A ⊆ B and B ∈ Im (X)}. Then we have cl3 3 Acl 0 = A3 ∩ A5 = A3 , A1 = A3 ∩ A5 = A3 , cl3 3 Acl 2 = A3 ∩ A5 = A3 , A3 = A3 ∩ A5 = A3 , cl3 3 Acl 4 = A5 , A5 = A5 . It is easy to show that “cl3 ” is weak closure operation on I(X). Let {clλ | λ ∈ Λ} be a collection of operations on I(X). We define the intersection of clλ ’s, denoted by ∩ clλ , as follows: λ∈Λ

∩ clλ : I(X) → I(X), A 7→ ∩ Aclλ .

λ∈Λ

λ∈Λ

Note that if clλ is a weak closure operation on I(X) for all λ ∈ Λ, then ∩ clλ is a weak closure λ∈Λ

operation on I(X) (see [2]). But the following example shows that the union of weak closure operations may not be a weak closure operation. Example 3.18. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 0 2 2 2 0 0 2 3 3 2 1 0 2 4 4 1 4 1 0 There are four ideals in X, that is, A0 = {0}, A1 = {0, 1, 4}, A2 = {0, 2} and A3 = X. Define cl1 cl1 cl1 1 a mapping cl1 : I(X) → I(X) as follows: Acl 0 = A1 , A1 = A3 , A2 = A3 , A3 = A3 . Then “cl1 ” is a weak closure operation on I(X). Also, define a mapping cl2 : I(X) → I(X) as follows: cl2 cl2 cl2 2 Acl 0 = A2 , A1 = A3 , A2 = A3 , A3 = A3 . Then “cl2 ” is a weak closure operation on I(X).

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Weak closure operations on ideals of BCK-algebras

Now if we define “cl3 ” by Acl3 = Acl1 ∪ Acl2 , then “cl3 ” is not a weak closure operation on I(X) because for an ideal A0 of X, we have cl1 cl2 3 Acl 0 = A0 ∪ A0 = A1 ∪ A2 = {0, 1, 2, 4}

which is not an ideal of X. Thus “cl3 ” is not a weak closure operation on I(X). Definition 3.19. Given a (weak) closure operation cl : I(X) → I(X) on I(X), we define a new operation clf : I(X) → I(X) by ( ) (∀A ∈ I(X)) Aclf = ∪{B cl | B ⊆ A, B ∈ If (X)} , (3.9) where If (X) is the set of all finitely generated ideals of X. Definition 3.20. A (weak) closure operation cl on I(X) is said to be of finite type if the following assertion is valid. ( ) (∀A ∈ I(X)) Acl = Aclf . (3.10) Note that every weak closure operation on a finite BCK-algebra is of finite type. Example 3.21. Let X be a BCK-algebra of infinite order. Define an operation “cl′′ on I(X) as follows: { X if A is a maximal ideal or A = X, cl (3.11) A = M otherwise, where M is a maximal ideal of X containing A. We can easily check that “cl” is a weak closure operation. Now let A be a maximal ideal of X which is not finitely generated. Then Aclf = ∪{B cl | B ⊆ A and B ∈ If (X)} ⊆ M ⊊ X = Acl , and thus “cl” is a weak closure operation which is not of finite type. For two operations “cl1 ” and “cl2 ” on I(X), we say that “cl1 ” is weaker than “cl2 ”, denoted by cl1 ≤ cl2 , if Acl1 ⊆ Acl2 for every A ∈ I(X). Theorem 3.22. Given an operation “cl” on I(X), we have (i) If “cl” is a weak closure operation of finite type, then so is “clf ”, and it is largest in the set of weak closure operations which are weaker than “cl”. (ii) If “cl” is a (strong) quasi-prime weak closure operation, then so is “clf ”. Proof. (i) Let “cl” be a weak closure operation of finite type. Then “clf ” is a weak closure operation on I(X) (see [2]). To prove that ”clf ” is of finite type, we should prove that Aclf = A(clf )f for every ideal A of X. Clearly, we have Aclf ⊆ A(clf )f . Suppose that x ∈ A(clf )f . Then there exists a finitely generated ideal B such that B ⊆ A and x ∈ B clf . Since “cl” is a weak closure operation of finite type, we have B cl = B clf . Thus x ∈ B cl , B ⊆ A and B is finitely generated ideal. Therefore x ∈ Aclf and Aclf = A(clf )f which means that “clf ” is a weak closure

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Hashem Bordbar, Mohammad Mehdi Zahedi, Sun Shin Ahn and Young Bae Jun

operation on I(X) of finite type. Now let c be a weak closure operation on I(X) of finite type which is weaker than “cl”. Let A be an ideal of X and a ∈ Ac . Then there exists a finitely generated ideal B of X such that B ⊆ A and a ∈ B c . It follows from c ≤ cl that a ∈ B cl . Therefore a ∈ Aclf , and so c ≤ clf . (ii) Suppose that “cl” be a quasi prime weak closure operation on I(X). To prove that “clf ” is a quasi prime weak closure operation, it is enough to show that a ∧ Aclf ⊆ (a ∧ A)clf . Now let x ∈ a ∧ Aclf = ⟨{a ∧ α | α ∈ Aclf }⟩. Then there exist α1 , α2 , · · · , αn ∈ Aclf such that (· · · ((x ∧ (a ∧ α1 )) ∗ (a ∧ α2 )) ∗ · · · ) ∗ (a ∧ αn ) = 0. Since αi ∈ Aclf = ∪{B cl | B ⊆ A and B ∈ If (X)} for each 1 ≤ i ≤ n, we have αi ∈ Aclf = ∪{B cl | B ⊆ A and B ∈ If (X)}, and so there exists a finitely generated ideal B such that αi ∈ B cl and B ⊆ A. Since αi ∈ B cl , we have a ∧ αi ∈ {a ∧ β | β ∈ B} ⊆ ⟨{a ∧ β | β ∈ B}⟩ = a ∧ B cl , which implies that a ∧ αi ∈ a ∧ B and (· · · ((x ∧ (a ∧ α1 )) ∗ (a ∧ α2 )) ∗ · · · ) ∗ (a ∧ αn ) = 0. This means that x ∈ a ∧ B cl . Since “cl” is a quasi prime weak closure operation on I(X), it follows that x ∈ a ∧ B cl ⊆ (a ∧ B)cl ⊆ (a ∧ A)cl ⊆ (a ∧ A)clf . Therefore x ∈ (a ∧ A)clf and “clf ” is a quasi-prime weak closure operation on I(X). Similarly, we can check that if “cl” is a strong quasi-prime weak closure operation on I(X), then “clf ” is a strong quasi-prime weak closure operation on I(X). □ Definition 3.23. An operation α : I(X) → I(X) is called a positive implicative (resp. commutative and implicative) weak closure operation if the following conditions are valid. (i) For any A, B ∈ Ipi (X) (resp. Ic (X) and Im (X)), A ⊆ Aα ,

(3.12)

A⊆B ⇒ A ⊆B . α

α

(3.13)

(ii) (∀A ∈ / Ipi (X)(resp., Ic (X) and Im (X))) (Aα = A) . Obviously, every positive implicative (resp., commutative and implicative) weak closure operation is a weak closure operation, but the converse is not true in general as seen in the following example. Example 3.24. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table.

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∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 1 2 2 2 0 2 0 3 3 1 3 0 3 4 4 4 4 4 0 There are six ideals in X, that is, A0 = {0}, A1 = {0, 1, 3}, A2 = {0, 2}, A3 = {0, 1, 2, 3}, A4 = {0, 2, 4} and A5 = X. Note that A1 , A3 and A5 are positive implicative ideals and A0 , A2 and A4 are not positive implicative ideals. Define a mapping cl : I(X) → I(X) as follows: cl cl cl cl cl Acl 0 = A0 A1 = A3 , A2 = A2 , A3 = A5 , A4 = A4 and A5 = X. Then “cl” is a positive implicative weak closure operation on I(X). Now we define an operation “cl1 ” on I(X) as follows: cl1 cl1 cl1 cl1 cl1 1 Acl 0 = A1 , A1 = A3 , A2 = A4 , A3 = A5 , A4 = A5 and A5 = X.

Then “cl1 ” is a weak closure operation on I(X), but it is not positive implicative because the 1 ideal A2 is not a positive implicative ideal and Acl 2 = A4 ̸= A2 . Example 3.25. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 0 0 0 2 2 2 0 0 0 3 3 3 3 0 0 4 4 4 4 4 0 There are five ideals in X, that is, A0 = {0}, A1 = {0, 1}, A2 = {0, 1, 2}, A3 = {0, 1, 2, 3} and A4 = X where A3 and A4 are commutative ideals and A0 , A1 and A2 are not commutative ideals. Now define “cl” as follows: cl cl cl cl Acl 0 = A1 , A1 = A2 , A2 = A3 , A3 = A4 and A4 = X

Then “cl” is a weak closure operation on I(X), but it is not commutative since the ideal A2 is not a commutative ideal and Acl 2 = A3 ̸= A2 . Example 3.26. Let X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 0 0 0 2 2 2 0 0 0 3 3 3 3 0 0 4 4 4 4 4 0 Then X is a BCK-algebra with seven ideals A0 = {0}, A1 = {0, 1}, A2 = {0, 1, 2}, A3 = {0, 1, 4} A4 = {0, 1, 2, 3}, A5 = {0, 1, 2, 4} and A6 = X. Note that A2 , A4 , A5 and A6 are implicative

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ideals and A0 , A1 and A3 are not implicative ideals. Now we define an operation define “cl” on I(X) by cl cl cl cl cl cl Acl 0 = A1 , A1 = A2 , A2 = A5 , A3 = A5 , A4 = A6 , A5 = A6 and A6 = X.

Then “cl” is a weak closure operation on I(X), but it is not implicative since the ideal A3 is not an implicative ideal and Acl 3 = A5 ̸= A3 . Given a weak closure operation, we kame a positive implicative weak closure operation. Theorem 3.27. Given A ∈ I(X), let “cl” be a weak closure operation on I(X) and “clpi ” be an operation on I(X) such that cl ≤ clpi and ( ) (i) (∀C ∈ I(X)) A ⊆ C ⇒ C clpi = C cl . ( ) (ii) (∀C ∈ I(X)) C ⊊ A ⇒ C clpi = C . (iii) For any C ∈ I(X), if A and C have no inclusion relation, then C clpi = C. If A is positive implicative (resp., commutative and implicative) ideals of X, then “clpi ” is a positive implicative (resp., commutative and implicative) weak closure operation on I(X). Proof. Let A and C be ideals of X such that A ⊆ C. Suppose that A is a positive implicative (resp., commutative and implicative) ideal of X. Then C is a positive implicative (resp., commutative and implicative) ideal of X by Lemma 3.13. Let A and C be ideals of X such that C ⊆ A. If A is not a positive implicative (resp., commutative and implicative) ideal of X, then C is not a positive implicative (resp., commutative and implicative) ideal of X. Therefore “cl” is a positive implicative (resp., commutative and implicative) weak closure operation on I(X). □ The following examples illustrate Theorem 3.27. Example 3.28. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table, ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 0 0 1 2 2 2 0 0 2 3 3 2 1 0 3 4 4 4 4 4 0 There are six ideals in X, that is, A0 = {0}, A1 = {0, 1}, A2 = {0, 4}, A3 = {0, 1, 2, 3}, A4 = {0, 1, 4} and A5 = X in which A1 , A3 , A4 and A5 are positive implicative ideals and A0 and A2 are not positive implicative ideals. Now define “cl” as follows: cl cl cl cl cl Acl 0 = A0 , A1 = A3 , A2 = A4 , A3 = A3 , A4 = A5 and A5 = X.

Then “cl” is a weak closure operation. Now let A = {0, 4} = A2 which is not a positive implicative ideal. By using Theorem 3.27 we have ”clpi as follows: cl

cl

cl

cl

cl

cl

A0 pi = A0 , A1 pi = A1 , A2 pi = A4 , A3 pi = A3 , A4 pi = A5 and A5 pi = X.

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Weak closure operations on ideals of BCK-algebras cl

Clearly, cl ≤ clpi . But, “clpi ” is not a positive implicative weak closure operation because A2 pi = A4 ̸= A2 . Now let A = {0, 1} = A1 which is a positive implicative ideal. By using Theorem 3.27 we have “clpi ” as follows: cl

cl

cl

cl

cl

cl

A0 pi = A0 , A1 pi = A3 , A2 pi = A2 , A3 pi = A3 , A4 pi = A5 and A5 pi = X. Clearly, cl ≤ clpi and “clpi ” is a positive implicative weak closure operation. Example 3.29. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 0 2 2 2 0 0 0 3 3 2 1 0 0 4 4 4 4 4 0 There are five ideals in X, that is, A0 = {0}, A1 = {0, 1}, A2 = {0, 2}, A3 = {0, 1, 2, 3}, and A4 = X in which A3 and A4 are commutative ideals and A0 , A1 and A2 are not commutative ideals. Now define “cl” as follows: cl cl cl cl Acl 0 = A1 , A1 = A3 , A2 = A3 , A3 = A4 and A4 = X.

Then “cl” is a weak closure operation. Now let A = {0, 1} = A1 which is not a commutative ideal. By using Theorem 3.27 we have ”clc as follows: clc clc clc clc c Acl 0 = A0 , A1 = A3 , A2 = A2 , A3 = A4 and A4 = X. c Clearly, cl ≤ clc . But, “clc ” is not a commutative weak closure operation because Acl 1 = A3 ̸= A1 . Now let A = {0, 1, 2, 3} = A3 which is a commutative ideal. By using Theorem 3.27 we have “clc ” as follows:

clc clc clc clc c Acl 0 = A0 , A1 = A1 , A2 = A2 , A3 = A4 and A4 = X.

Clearly, cl ≤ clc and “clc ” is a commutative weak closure operation. Example 3.30. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 0 2 2 2 0 0 2 3 3 2 1 0 2 4 4 4 4 4 0 There are six ideals in X, that is, A0 = {0}, A1 = {0, 2}, A2 = {0, 1}, A3 = {0, 1, 2, 3}, A4 = {0, 1, 4} and A5 = X in which A2 , A3 , A4 and A5 are implicative ideals and A0 and A1 are not implicative ideals. Now define “cl” as follows: cl cl cl cl cl Acl 0 = A1 , A1 = A3 , A2 = A4 , A3 = A5 , A4 = A4 and A5 = X.

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Hashem Bordbar, Mohammad Mehdi Zahedi, Sun Shin Ahn and Young Bae Jun

Then “cl” is a weak closure operation. Now let A = {0, 2} = A1 which is not an implicative ideal. By using Theorem 3.27 we have “clm ” as follows: m m m m m m Acl = A0 , Acl = A3 , Acl = A2 , Acl = A5 , Acl = A4 and Acl = X. 0 1 2 3 4 5 m Clearly, cl ≤ clm . But, “clm ” is not an implicative weak closure operation because Acl = A3 ̸= 1 A1 . Now let A = {0, 1} = A2 which is an implicative ideal. By using Theorem 3.27 we have “clm ” as follows: m m m m m A0clm = A0 , Acl = A1 , Acl = A4 , Acl = A5 , Acl = A4 and Acl = X. 1 2 3 4 5

Clearly, cl ≤ clm and “clm ” is an implicative weak closure operation. References [1] H. Bordbar and M. M. Zahedi, Semi-prime closure operations on BCK-algebra, Commun. Korean Math. Soc. 30 (2015), no. 5, 385–402. [2] H. Bordbar and M. M. Zahedi, A finite type of closure operations on BCK-algebra, Appl. Math. Inf. Sci. Lett. 4 (2016), no. 2, 1–9. [3] Y. Huang, BCI-algebra, Science Press, Beijing 2006. [4] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co. 1994.

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Communication between relation information systems∗ Funing Lin† Shenggang Li‡ December 23, 2015 Abstract: Communication between information systems is considered as an important issue in granular computing. A relation information system is the generalization of an information system. This paper investigates communication between relation information systems and obtain some invariant characterizations of relation information systems under homomorphism. Keywords: Relation information system; Reduction; Consistent function; Relation mapping; Homomorphism.

1

Introduction

Rough set theory, proposed by Pawlak [17], is an important tool for dealing with fuzzyness and uncertainty of knowledge and has become an active branch of information science. With more than thirty years development, rough set theory 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 [13, 14, 15, 16]. Communication between information systems is a very important topic in the field of artificial intelligence. In mathematics, it can be explained as a mapping between information systems. The approximations and reductions in the original system can be regarded as encoding while the image system is seen as an interpretive system. The concept of homomorphisms as a kind of tool to study relationships between information systems with rough sets was introduced by Grzymala-Busse [1, 2]. A homomorphism can be viewed as a special communication between two information systems. As explained in [23], homomorphisms allow one to translate the information contained in one granular world into the ∗ This work was supported by the National Natural Science Foundation of China (61473181, 11501436). † Corresponding Author, College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062, P.R.China; [email protected]. ‡ College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062, P.R.China; [email protected].

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granularity of another granular world and thus provide a communication mechanism for exchanging information with other granular worlds. Li et al. [5] studied invariant characters of information systems under some homomorphism. Wang et al. [20, 21] introduced the notions of consistent functions, relation mappings and relation information systems which are the generalization of information systems. By using these notions, they proposed the homomorphisms as a mechanism for communicating between relation information systems. Zhu et al. [26] obtained some improved results on communication between relation information systems. Li et al. [12] investigated communication between knowledge bases. It should be pointed out that some other related works investigating information systems through homomorphisms [1, 2, 3, 5, 25] are based on equivalence relations or other particular relations and are quite different from [20, 21, 26]. The purpose of this paper is to investigate some invariant characterizations of relation information systems under homomorphisms.

2

Preliminaries

In this section, we recall some basic concepts on consistent functions, relation mappings and relation information systems. Throughout this paper, U denotes a non-empty finite set called the universe, 2U denotes the family of all subsets of U , 2U ×U denotes the family of all binary relations on U , All mappings are assumed to be surjective. For R ∈ 2U ×U , the successor neighborhood of x ∈ U with respect to R will be denoted by Rs (x), that is, Rs (x) = {y ∈ U : xRy} ([22]). Denote U/R = {Rs (x) : x ∈ U }. If R is an equivalence relation on U , then T ∀ x ∈ U , Rs (x) = [x]R . For R ⊆ 2U ×U , denote ind(R) = R. R∈R

2.1

Consistent functions

Definition 2.1 ([20, 21]). Let U and V be two finite nonempty universes, f : U → V a mapping and R ∈ 2U ×U . Define [x]f = {u ∈ U : f (u) = f (x)}, (x)R = {u ∈ U : Rs (u) = Rs (x)}. Then {[x]f : x ∈ U } and {(x)R : x ∈ U } are two partitions on U . If [x]f ⊆ Rs (u) or [x]f ∩Rs (u) 6= ∅ for any x, u ∈ U , then f is called a type-1 consistent function with respect to R on U . If [x]f ⊆ (x)R for any x ∈ U , then f is called a type-2 consistent function with respect to R on U . Remark 2.2. (1) ∀ x ∈ U , [x]f = f −1 (f (x)). (2) If R is an equivalence relation on U , then ∀ x ∈ U , (x)R = [x]R . (3) If f is type-2 consistent with respect to R on U and f (u) = f (x), then Rs (u) = Rs (x). 2

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Obviously, f is type-1 ⇐⇒ If [x]f ∩ Rs (y) 6= ∅, then [x]f ⊆ Rs (y) ⇐⇒ If [x]f * Rs (y), then [x]f ∩ Rs (y) = ∅, f is type-2 ⇐⇒ If f (x1 ) = f (x2 ), then Rs (x1 ) = Rs (x2 ).

2.2

Relation mappings

Definition 2.3 ([20, 21]). Let f : U → V be a mapping. Define [ fˆ : 2U ×U → 2V ×V , R| → fˆ(R) = ({f (x)} × f (Rs (x))); x∈U

fˆ−1 : 2V ×V → 2U ×U , T | → fˆ−1 (T ) =

[

({f −1 (y)} × f −1 (Ts (y))).

y∈V

Then fˆ and fˆ−1 are called the relation mapping and inverse relation mapping induced by f , respectively. Obviously, y1 fˆ(R)y2 ⇐⇒ ∃ x1 , x2 ∈ U, y1 = f (x1 ), y2 = f (x1 ) and x1 Rx2 , x1 fˆ−1 (T )x2 ⇐⇒ ∃ y1 , y2 ∈ V, y1 = f (x1 ), y2 = f (x1 ) and y1 T y2 . For R ⊆ 2U ×U , denote fˆ(R) = {fˆ(R) | R ∈ R}. Proposition 2.4 ([20]). If f : U → V is both type-1 and type-2 consistent with respect to R ∈ 2U ×U , then fˆ−1 (fˆ(R)) = R.

2.3

Relation information systems

Definition 2.5 ([13]). An information system is a pair (U, A) of non-empty finite sets U and A, where U is a set of objects and A is a set of attributes; each attribute a ∈ A is a function a : U → Va , where Va is the set of values (called domain) of attribute a. If (U, A) is an information system and B ⊆ A, then an equivalence relation (or indiscernibility relation) RB can be defined by (x, y) ∈ RB ⇐⇒ a(x) = a(y), ∀ a ∈ B. Definition 2.6 ([20]). A pair (U, R) is called a relation information system, if R ⊆ 2U ×U . Definition 2.7. Let (U, A) be an information system. Put R = {R{a} : a ∈ A}. Then the pair (U, R) is called the relation information system induced by (U, A). 3

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Definition 2.8 ([20]). Let f : U → V be a mapping and R ⊆ 2U ×U . If f is type-1 (resp. type-2) consistent with respect to R on U for every R ∈ R, then f is called type-1 (resp. type-2) consistent with respect to R on U . Proposition 2.9 ([20]). Let f : U → V be a mapping and R ⊆ 2U ×U . If f is both type-1 and type-2 consistent with respect to R, then fˆ(ind(R)) = ind(fˆ(R)). Proposition 2.10 ([20]). Let f : U → V be a mapping and R ⊆ 2U ×U . If f is both type-1 and type-2 consistent with respect to R, then fˆ−1 (fˆ(ind(R)) = ind(R). Definition 2.11 ([20]). Let f : U → V be a mapping and R ⊆ 2U ×U . Then the pair (V, fˆ(R)) is called an f -induced relation information system of (U, R). Definition 2.12 ([20]). Let (U, R) be a relation information system and (V, fˆ(R)) an f-induced relation information system of (U, R). If f is both type-1 and type2 consistent with respect to R on U , then f is called a homomorphism from (U, R) to (V, fˆ(R)). We write (U, R) ∼f (V, fˆ(R)). We often consider reductions in a relation information system by deleting unrelated or unimportant elements with the requirement of keeping the ability of classification. Definition 2.13 ([20]). Let (U, R) be a relation information system and P ⊆ R. (1) P is called a coordination subfamily of R, if ind(P) = ind(R). (2) R ∈ P is called independent in P, if ind(P − {R}) 6= ind(P); P is called a independent subfamily of R, if ∀ R ∈ P, R is independent in P. (3) P is called a reductions of R, if P is both coordination and independent. In this paper, the set of all coordination subfamilies (resp., all reductions) of R is denoted by co(R) (resp., red(R)). Obviously, P ∈ red(R) ⇔ P ∈ co(R) and ∀ Q ⊂ P, Q 6∈ co(R).

3

Some results on reductions in relation information systems

Proposition 3.1. Let (U, R) be a relation information system. Then red(R) 6= ∅. Proof. Suppose ∀ R ∈ R, R − {R} 6∈ co(R). Then R ∈ red(R). Suppose ∃ R1 ∈ R, R − {R1 } ∈ co(R). Then, we consider R − {R1 }. Again suppose ∀ R ∈ R − {R1 }, (R − {R1 }) − {R} 6∈ co(R). Then R − {R1 } ∈ red(R). Again suppose ∃ R2 ∈ R−{R1 }, (R−{R1 })−{R2 } ∈ co(R). Then, we consider 4

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R − {R1 , R2 }. Repeat this process. Since R is finite, we can find a reductions of R. Thus red(R) 6= ∅. Definition 3.2. Let (U, R) be a relation information system. Put D(x, y) = {R ∈ R|(x, y) 6∈ R} (x, y ∈ U ). Then (1) D(x, y) is called is called the discernibility subfamily of R on x and y. (2) D(R) = (dij )n×n is called the discernibility matrix of R where U = {x1 , x2 , · · ·, xn } and dij = D(xi , xj ) (1 ≤ i, j ≤ n). Example 3.3. Let U = {x1 , x2 , x3 , x4 , x5 , x6 }. We consider the relation information system (U, R) where R = {R1 , R2 , R3 , R4 } and U/R1 = {{x1 , x2 , x5 }, {x3 , x4 , x6 }}, U/R2 = {{x1 , x6 }, {x2 , x3 , x4 , x5 }}, U/R3 = {{x1 , x2 , x5 , x6 }, {x3 , x4 }}, U/R4 ={{x1 , x2 , x5 }, {x3 , x4 , x6 }}. We can obtain the discernibility matrix D(R) as follows:  ∅ {R2 } R R {R2 } {R1 , R4 }  {R2 } ∅ {R , R , R } {R , R , R } ∅ {R 1 3 4 1 3 4 1 , R2 , R4 }   R {R , R , R } ∅ ∅ {R , R , R } {R 1 3 4 1 3 4 2 , R3 }   R {R , R , R } ∅ ∅ {R , R , R } {R 1 3 4 1 3 4 2 , R3 }   {R2 } ∅ {R1 , R3 , R4 } {R1 , R3 , R4 } ∅ {R1 , R2 , R4 } {R1 , R4 } {R1 , R2 , R4 } {R2 , R3 } {R2 , R3 } {R1 , R2 , R4 } ∅

       

Discernibility family can expediently judge coordination families and reductions. Proposition 3.4. Let (U, R) be a relation information system. Then P ∈ co(R) ⇐⇒ If (x, y) 6∈ ind(R), then P ∩ D(x, y) 6= ∅. Proof. (1) “=⇒”. Let (x, y) 6∈ ind(R). Since P ∈ co(R), we have ind(P) = ind(R). Then (x, y) 6∈ ind(P). It follows (x, y) 6∈ P f or some P ∈ P. (x, y) 6∈ P implies P ∈ D(x, y). Then P ∈ P ∩ D(x, y). Thus P ∩ D(x, y) 6= ∅. “⇐=”. Suppose P 6∈ co(R). Then ind(P) 6= ind(R). It follows ind(P) − ind(R) 6= ∅. Pick (x, y) ∈ ind(P) − ind(R). Since (x, y) 6∈ ind(R), we have P ∩ D(x, y) 6= ∅. Note that (x, y) ∈ ind(P). Then ∀ P ∈ P, (x, y) ∈ P . So P 6∈ D(x, y). Thus P ∩ D(x, y) = ∅. This is a contradiction. Thus P ∈ co(R).

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Theorem 3.5. Let (U, R) be a relation information system. Then P ∈ red(R) ⇐⇒ (1) If (x, y) 6∈ ind(R), then P ∩ D(x, y) 6= ∅; (2) ∀ R ∈ P, ∃ (xR , yR ) ∈ ind(R), (P − {R}) ∩ D(xR , yR ) = ∅. Proof. This holds by Proposition 3.4. Definition 3.6. Let (U, R) be a relation information system. Put \ core(R) = P. P∈red(R)

Then core(R) is called the core of R. Moreover, (1) R ∈ R is called necessary, if R ∈ core(R). (2) R ∈ R is called relatively necessary, if R ∈

S

P − core(R).

P∈red(R)

(3) R ∈ R is called unnecessary, if R ∈ R −

S

P.

P∈red(R)

Discernibility family can easily determine the core. Proposition 3.7. Let (U, R) be a relation information system. The following are equivalent: (1) R is necessary; (2) R is independent in R; (3) ∃ x, y ∈ U , D(x, y) = {R}. Proof. (1) =⇒ (2). Suppose that R is not independent in R. Then ind(R − {R}) = ind(R). It follows R − {R} ∈ co(R). Consider R − {R}. By Proposition 3.1, ∃ P ⊆ R − {R}, P ∈ red(R). P ⊆ R − {R} implies R 6∈ P. Then R is not necessary. This is a contradiction. (2) =⇒ (1). Suppose that R is not necessary. Then ∃ P ∈ red(R), R 6∈ P. So P ⊆ R − {R} ⊆ R. It follows ind(P) ⊇ ind(R − {R}) ⊇ ind(R). By P ∈ red(R), ind(P) = ind(R). Then ind(R − {R}) = ind(R). So R is not independent in R. This is a contradiction. (2) =⇒ (3). Since R is independent in R, we have ind(R − {R}) 6= ind(R). Then ind(R − {R}) − ind(R) 6= ∅. Pick (x, y) ∈ ind(R − {R}) − ind(R). Denote R = {R1 , R2 , . . . , Rn }. Then R = Rj for some j ≤ n. So \ \ Ri . (x, y) ∈ Ri − 1≤i≤n,i6=j

1≤i≤n

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It follows (x, y) 6∈ Rj and (x, y) ∈ Ri (i 6= j). Thus D(x, y) = {R}. (3) =⇒ (2). Since ∃ x, y ∈ U , D(x, y) = {R}, we have 0

0

(x, y) 6∈ R, (x, y) ∈ R (R 6= R). Then (x, y) ∈ ind(R − {R}). But (x, y) 6∈ ind(R). Thus ind(R − {R}) 6= ind(R). Hence R is independent in R. Proposition 3.8. Let (U, R) be a relation information system. Denote [ R? = ind(P − {R}). P∈co(R)

Then the following are equivalent. (1) R is unnecessary; (2) ∀ P ∈ co(R), P − {R} ∈ co(R); (3) R? = ind(R); (4) R? ⊆ R. Proof. (1) =⇒ (2). By Proposition 3.1, ∃ Q ⊆ P, Q ∈ red(R). Since R is unnecessary, we have R 6∈ Q. It follows Q ⊆ R − {R}. Then Q ⊆ P ∩ (R − {R}) = P − {R} ⊆ P. We have ind(Q) ⊇ ind(R − {R}) ⊇ ind(P). Note that P ∈ co(R) and Q ∈ red(R). Then ind(P) = ind(R) = ind(Q). Thus ind(P − {R}) = ind(R). This shows P − {R} ∈ co(R). (2) =⇒ (3) =⇒ (4) are obvious. (4) =⇒ (1). Suppose ∃ P ∈ red(R), R ∈ P. Then P − {R} ⊂ P. Since P ∈ red(R), we have P − {R} 6∈ co(R). Then ind(P − {R}) − ind(R) 6= ∅. P ∈ red(R) implies ind(P) = ind(R). Then ind(P − {R}) − ind(P) 6= ∅. Pick (x, y) ∈ ind(P − {R}) − ind(P). Note that ind(P) = ind(P − {R}) ∩ R. Then (x, y) 6∈ R. Since P ∈ co(R) and R? ⊆ R, we have ind(P − {R}) ⊆ R. Then (x, y) ∈ R. This is a contradiction. Thus R is unnecessary. Theorem 3.9. Let (U, R) be a relation information system. Then (1) R is necessary ⇔ R − {R} 6∈ co(R). (2) R is relatively necessary ⇔ R − {R} ∈ co(R) and R? 6⊆ R. (3) R is unnecessary ⇔ R? ⊆ R.

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Proof. This holds by Proposition 3.7 and Proposition 3.8. Example 3.10. In Example 3.3, we have (1) R2 is necessary. (2) R1 and R4 are relatively necessary. (3) R3 is unnecessary. (4) red(R) = {{R1 , R2 }, {R2 , R4 }}, core(R) = {R2 }.

4

Communication between relation information systems

Proposition 4.1. Let (U, R) ∼f (V, fˆ(R)). Then (1) P ∈ co(R) ⇐⇒ fˆ(P) ∈ co(fˆ(R)). (2) co(fˆ(R)) = fˆ(co(R)). Proof. (1) “=⇒”. Since P ∈ co(R), we have ind(P) = ind(R). Then fˆ(ind(P)) = fˆ(ind(R)). By Proposition 2.6,

ind(fˆ(P)) = ind(fˆ(R)).

Thus fˆ(P) ∈ co(fˆ(R)). “⇐=”. Since fˆ(P) ∈ co(fˆ(R)), we have ind(fˆ(P)) = ind(fˆ(R)). By Proposition 2.6, Then

fˆ(ind(P)) = fˆ(ind(R)).

fˆ−1 (fˆ(ind(P))) = fˆ−1 (fˆ(ind(R))).

By Proposition 2.7, ind(P) = ind(R). Thus P ∈ co(R). (2) By (1), fˆ(co(R)) = {fˆ(P)|P ∈ co(R)} = {fˆ(P)|fˆ(P) ∈ co(fˆ(R))} = co(fˆ(R)). Theorem 4.2. Let (U, R) ∼f (V, fˆ(R)). Then (1) P ∈ red(R) ⇐⇒ fˆ(P) ∈ red(fˆ(R)). (2) red(fˆ(R)) = fˆ(red(R)).

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Proof. (1) “=⇒”. Since P ∈ red(R), we have P ∈ co(R). By Proposition 4.1, fˆ(P) ∈ co(fˆ(R)). ∀ T ⊂ fˆ(P). Pick Q ⊆ R, T = fˆ(Q). Then fˆ(Q) ⊂ fˆ(P). By Proposition 2.4, Q = fˆ−1 (fˆ(Q)) ⊆ fˆ−1 (fˆ(P)) = P. Suppose Q = P. Then T = fˆ(Q) = fˆ(P). This is a contradiction. Thus Q ⊂ P. Since P ∈ red(R), we have Q 6∈ co(R). By Proposition 4.1, T = fˆ(Q) 6∈ co(fˆ(R)). Hence fˆ(P) ∈ red(fˆ(R)). “⇐=”. Since fˆ(P) ∈ red(fˆ(R)), we have fˆ(P) ∈ co(fˆ(R)). By Proposition 4.1, P ∈ co(R). ∀ Q ⊂ P, fˆ(Q) ⊆ fˆ(P). Suppose fˆ(Q) = fˆ(P). By Proposition 2.4, Q = fˆ−1 (fˆ(Q)) = fˆ−1 (fˆ(P)) = P. This is a contradiction. Thus fˆ(Q) ⊂ fˆ(P). Since fˆ(P) ∈ red(fˆ(R)), we have fˆ(Q) 6∈ co(fˆ(R)). By Proposition 4.1, Q 6∈ co(R). Hence P ∈ red(R). (2) By (1), fˆ(red(R)) = {fˆ(P)|P ∈ red(R)} = {fˆ(P)|fˆ(P) ∈ red(fˆ(R))} = red(fˆ(R)).

Remark 4.3. Theorem 3.20(1) is Theorem 4.4 in [20]. We just prove this result from another angle. Lemma 4.4. Let (U, R) ∼f (V, fˆ(R)). Then fˆ(R − {R}) = fˆ(R) − {fˆ(R)}. Proof. ∀ S ∈ R − {R}, S 6= R. By Proposition 2.4, fˆ(S) 6= fˆ(R). It follows fˆ(S) ∈ fˆ(R) − {fˆ(R)}. Thus fˆ(R − {R}) ⊆ fˆ(R) − {fˆ(R)}. On the other hand, ∀ T ∈ fˆ(R) − {fˆ(R)}, T = fˆ(S) for some S ∈ R. T ∈ 6 {fˆ(R)} implies fˆ(S) 6= fˆ(R). Then S 6= R. So S ∈ R − {R}. It follows T ∈ fˆ(R − {R}). Thus fˆ(R − {R}) ⊇ fˆ(R) − {fˆ(R)}. Hence fˆ(R − {R}) = fˆ(R) − {fˆ(R)}. 9

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Theorem 4.5. Let (U, R) ∼f (V, fˆ(R)). Then R ∈ core(R) ⇐⇒ fˆ(R) ∈ core(fˆ(R)). Proof. This holds by Theorem 3.9(1), Proposition 4.1(1) and Lemma 4.4. Theorem 4.6. Let (U, R) ∼f (V, fˆ(R)). Then fˆ(core(R)) = core(fˆ(R)). Proof. By Theorem 3.23, fˆ(core(R)) = {fˆ(R)|R ∈ core(R)} = {fˆ(R)|fˆ(R) ∈ core(fˆ(R))} = core(fˆ(R)). Theorem 4.7. Let (U, R) ∼f (V, fˆ(R)). Then R is unnecessary ⇔ fˆ(R) is unnecessary. Proof. “=⇒”. ∀ T ∈ co(fˆ(R)), pick P ⊆ R, T = fˆ(P). Then fˆ(P) ∈ co(fˆ(R)). By Proposition 3.19(1), P ∈ co(R). Since R is unnecessary, by Proposition 3.8, we have P − {R} ∈ co(R). Then ind(P − {R}) = ind(R). By Proposition 2.6 and Lemma 4.4, ind(fˆ(P) − {fˆ(R)}) = ind(fˆ(P − {R}) = fˆ(ind(P − {R})), ind(fˆ(R)) = fˆ(ind(R)). Then ind(T − {fˆ(R)}) = ind(fˆ(R)). This implies T − {fˆ(R)} ∈ co(fˆ(R)). By Proposition 3.8, fˆ(R) is unnecessary. “⇐=”. ∀ P ∈ co(R), by Proposition 4.1(1), fˆ(P) ∈ co(fˆ(R)). Since fˆ(R) is unnecessary, by Proposition 3.8, we have fˆ(P) − {fˆ(R)} ∈ co(fˆ(R)). Then

ind(fˆ(P) − {fˆ(R)}) = ind(fˆ(R)).

By Proposition 2.6 and Lemma 4.4, fˆ(ind(P − {R})) = ind(fˆ(P − {R}) = ind(fˆ(P) − {fˆ(R)}), fˆ(ind(R)) = ind(fˆ(R)). Then fˆ(ind(P − {R})) = fˆ(ind(R)). By Proposition 2.7, ind(P − {R}) = fˆ−1 (fˆ(ind(P − {R}))) = fˆ−1 (fˆ(ind(R))) = ind(R). Then P − {R} ∈ co(R). By Proposition 3.8, R is unnecessary. 10

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Corollary 4.8. Let (U, R) ∼f (V, fˆ(R)). Then R is relatively necessary ⇐⇒ fˆ(R) is relatively necessary. Proof. This hods by Theorem 4.5 and Theorem 4.7. Example 4.9. Let U = {xi |1 ≤ i ≤ 15}. We consider the relation information system (U, R) where R = {R1 , R2 , R3 , R4 }, U/R1 = {{x1 , x2 , x4 , x7 , x8 , x9 , x10 , x11 }, {x3 , x5 , x6 , x12 , x13 , x14 , x15 }}, U/R2 = {{x1 , x4 , x11 , x12 , x13 , x14 , x15 }, {x2 , x3 , x5 , x6 , x7 , x8 , x9 , x10 }}, U/R3 = {{x1 , x2 , x4 , x7 , x8 , x9 , x10 , x11 , x12 , x13 , x14 , x15 }, {x3 , x5 , x6 }}, U/R4 = {{x1 , x2 , x4 , x7 , x8 , x9 , x10 , x11 }, {x3 , x5 , x6 , x12 , x13 , x14 , x15 }}. Let V = {y1 , y2 , y3 , y4 , y5 , y6 }. Define a mapping as follows: x1 , x4 , x11 y1

x2 , x8 y2

x3 , x6 y3

x5 y4

x7 , x9 , x10 y5

x12 , x13 , x14 , x15 . y6

Let (V, fˆ(R)) be the f -induced relation information system of (U, R). It is very easy to verify that f is a homomorphism from (U, R) to (V, fˆ(R)). We have fˆ(R) = {fˆ(R1 ), fˆ(R3 ), fˆ(R3 ), fˆ(R4 )} where V /fˆ(R1 ) = {{y1 , y2 , y5 }, {y3 , y4 , y6 }}, V /fˆ(R2 ) = {{y1 , y6 }, {y2 , y3 , y4 , y5 }}, V /fˆ(R3 ) = {{y1 , y2 , y5 , y6 }, {y3 , y4 }}, V /fˆ(R4 ) = {{y1 , y2 , y5 }, {y3 , y4 , y6 }}. By Example 3.10, red(fˆ(R)) = {{fˆ(R1 ), fˆ(R2 )}, {fˆ(R2 ), fˆ(R4 )}}, core(fˆ(R)) = {fˆ(R2 )}. By Proposition 2.4, Theorem 4.2(2) and Theorem 4.6, red(R) = {{R1 , R2 }, {R2 , R4 }}, core(R) = {R2 }.

5

Conclusions

In this paper, we have investigated the original relation information system and image relation information system, and obtained some invariant characterizations of relation information systems under homomorphism. These results will be significant for establishing a framework of granular computing in knowledge bases and may have potential applications to knowledge discovery, decision making and reasoning about data. In the future, we will consider concrete applications of our results.

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References [1] J.W.Graymala-Busse, Algebraic properties of knowledge representation systems, in: Proceedings of the ACM SIGART International Symposium on Methodologies for Intelligent Systems, Knoxville, 1986, pp. 432-440. [2] J.W.Graymala-Busse Jr., W.A.Sedelow, On rough sets and information system homomorphism, Bulletin of the Polish Academy of Technology Science 36(3-4)(1988) 233-239. [3] Z.Gong, Z.Xiao, Communicating between information systems based on including degrees, International Journal of General Systems 39(2)(2010) 189-206. [4] M.Kryszkiewicz, Comparative study of alternative types of knowledge reductions in inconsistent systems, International Journal of Intelligent Systems 16(2001) 105-120. [5] D.Li, Y.Ma, Invariant characters of information systems under some homomorphisms, Information Sciences 129(2000) 211-220. [6] Z.Li, R.Cui, T -similarity of fuzzy relations and related algebraic structures, Fuzzy Sets and Systems 275(2015) 130-143. [7] Z.Li, R.Cui, Similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators, Information Sciences 305(2015) 219-233. [8] Z.Li, T.Xie, The relationship among soft sets, soft rough sets and topologies, Soft Computing 18(2014) 717-728. [9] Z.Li, T.Xie, Roughness of fuzzy soft sets and related results, International Journal of Computational Intelligence Systems 8(2015) 278-296. [10] Z.Li, T.Xie, Q.Li, Topological structure of generalized rough sets, Computers and Mathematics with Applications 63(2012) 1066-1071. [11] Z.Li, N.Xie, G.Wen, Soft coverings and their parameter reductions, Applied Soft Computing 31(2015) 48-60. [12] Z.Li, Y.Liu, Q.Li, B.Qin, Relationships between knowledge bases and related results, Knowledge and Information Systems, First online: 26 November 2015, DOI 10.1007/s10115-015-0902-z. [13] Z.Pawlak, Rough sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers, Dordrecht, 1991. [14] Z.Pawlak, A.Skowron, Rudiments of rough sets, Information Sciences 177(2007) 3-27.

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[15] Z.Pawlak, A.Skowron, Rough sets: some extensions, Information Sciences 177(2007) 28-40. [16] Z.Pawlak, A.Skowron, Rough sets and Boolean reasoning, Information Sciences 177(2007) 41-73. [17] Z.Pawlak, Rough sets, International Journal of Computer and Information Sciences 11(1982) 341-356. [18] A.Skwwron, C.Rauszer, The discernibility matrices and mappings in information systems, in: R. Slowinski (Ed.), Intelligent decision support, Handbook of applications and advances of the rough sets theory, Kluwer Academic, Dordrecht, 1992, pp. 331-362. [19] W.Wu, M.Zhang, H.Li, J.Mi, Knowledge reductionsion in random information systems via Dempster-Shafer theory of evidence, Information Sciences 174(2005) 143-164. [20] C.Wang, C.Wu, D.Chen, W.Du, Some properties of relation information systems under homomorphisms, Applied Mathematics Letters 21(2008) 940-945. [21] C.Wang, C.Wu, D.Chen, Q.Hu, C.Wu, Communicating between information systems, Information Sciences 178(2008) 3228-3239. [22] Y.Y.Yao, Constructive and algebraic methods of the theory of rough sets, Information Sciences 109(1998) 21-47. [23] W.Pedrycz, G.Vukovich, Granular worlds: representation and communication problems, International Journal of Intelligent Systems 15(11)(2000) 1015-1026. [24] W.Zhang, J.Mi, W.Wu, Knowledge reductionsions in inconsistent information systems, International Journal of Intelligent Systems 18(2003) 9891000. [25] Y.Zhai, K.Qu, On characteristics of information system homomorphisms, Theory of Computing Systems 44(3)(2009) 414-431. [26] P.Zhu, Q.Wen, Some improved results on communication between information systems, Information Sciences 180(2010) 3521-3531.

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Global stability in a discrete Lotka-Volterra competition model Sangmok Chooa , Young-Hee Kim∗,b a

b

Department of Mathematics, University of Ulsan, Ulsan 44610, Korea. Division of General Education-Mathematics, Kwangwoon University, Seoul 01897, Korea

Abstract We consider the Euler difference scheme for two-dimensional Lotka-Volterra competition equations and show that the difference scheme has positive and bounded solutions. In addition, we present sufficient conditions that the solutions of the scheme converge to the equilibrium points of the scheme. The convergence is shown based on the two approaches: first, partition of the domain used for the boundedness of the solutions and second, calculation of the movement of the species started in each partitioned region. Numerical examples are presented to verify the results. Key words: Euler difference scheme, positivity, global stability, competition model

1. Introduction The competition model in the two-dimensional case represents two species which are competing for a common resource; an additional term is included within the logistic prey growth Lotka-Volterra model to incorporate this interspecific competition for some limiting resource. This limiting resource can be anything for which supply is smaller than demand. The classic two-dimensional competition model is given by dy dx = x(t)(r1 − a11 x(t) − a12 y(t)), = y(t)(r2 − a21 x(t) − a22 y(t)), dt dt

(1)

where ri > 0 and aij > 0. Here x(t) and y(t) denote the population sizes or population density in the species x and y at time t; the parameters ri ’s are the intrinsic growth rates for the two species x and y; aii ’s measure the inhibiting effect on the two species; a12 and a21 are the interspecific acting coefficients. The species x in the model (1) acts on y with functional response of type a12 x(t)y(t). However other types of functional responses including Holling types [1–5], BeddingtonDeAngelis type [6–8], Crowley-Martin type [9–11], and Ivlev-type of functional responses [12–14] have been applied to many population models The dynamics of the model (1) is well-known [15–17] ; the solutions of (1) are positive and bounded, and the stability of the system (1) has been studied. There are a number of works on investigating continuous time Lokta-Volterra models, but relatively few theoretical papers are published on their discretized models [18–21]. The author in [22] has ∗

Corresponding author Email addresses: [email protected] (Sangmok Choo), [email protected] (Young-Hee Kim)

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introduced a method to present global stability in discrete Lokta-Volterra predator-prey models for the case that all species coexist at a unique equilibrium. In [23], the authors have shown the global stability of the Euler difference scheme for a three-dimensional predator-prey model using a new approach. As far as we know, there is no theoretical research on the global stability of the discrete-time competition model of (1), so that we consider the Euler difference scheme xn+1 = Fyn (xn ), yn+1 = Gxn (yn ), n ≥ 0

(2)

Fy (x) = x {1 + (r1 − a11 x − a12 y)∆t} , Gx (y) = y {1 + (r2 − a21 x − a22 y)∆t} ,

(3) (4)

with

where ∆t is a time step size, xn = x0 +n∆t and yn = y0 +n∆t with (x0 , y0 ) = (x(0), y(0)). The paper is organized as follows. Section 2 gives the positivity and boundedness of solutions of (2). In Section 3, we partition the domain used for the boundedness of the discrete solutions and find the geometric properties of the movement of the solutions starting in the partitioned regions. Using the properties, we present sufficient conditions that the solutions converge to equilibrium points of (2). In Section 4, some numerical examples are presented to verify our results. 2. Positivity of the discrete solutions In this section, we consider the positivity and boundedness of the solutions of (2). Note that if τ1 and τ2 are positive constants satisfying U1 (τ2 ) =

1 + r2 ∆t − a21 τ1 ∆t 1 + r1 ∆t − a12 τ2 ∆t > 0, U2 (τ1 ) = > 0, 2a11 ∆t 2a22 ∆t

(5)

then Fτ2 (x), Gτ1 (y) are increasing on 0 ≤ x ≤ U1 (τ2 ), 0 ≤ y ≤ U2 (τ1 ).

(6)

For the positivity and boundedness of the solutions (xn , yn ) we assume max{r1 , r2 } < 1/∆t

(7)

and consider constants x∗ and y ∗ such that ∗ ∗ ∗ ∗ −1 r1 a−1 11 ≤ x ≤ U1 (y ), r2 a22 ≤ y ≤ U2 (x ).

(8)

Remark 1. For every point (x∗ , y ∗ ) satisfying     1 + r1 ∆t 1 + r2 ∆t r2 1 + r1 ∆t 1 + r2 ∆t r1 ∗ ∗ ≤ x ≤ min , , ≤ y ≤ min , , (9) a11 4a11 ∆t 2a21 ∆t a22 2a12 ∆t 4a22 ∆t

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the two conditions in (8) hold since 1 + r1 ∆t − a12 min

n

∗

1 + r2 ∆t − a21 min

n

∗

1+r1 ∆t 1+r2 ∆t , 4a22 ∆t 2a12 ∆t

o

∆t

1+r1 ∆t 1+r2 ∆t , 2a21 ∆t 4a11 ∆t

o

∆t

1 + r1 ∆t − a12 y ∆t ≥ 2a11 ∆t 2a11 ∆t   a12 1 + r1 ∆t 1 + r2 ∆t 1 + r1 ∆t − min , = 2a11 ∆t 2a11 2a12 ∆t 4a22 ∆t   1 + r1 ∆t 1 + r1 ∆t a12 1 + r2 ∆t 1 + r1 ∆t = max , − ≥ 4a11 ∆t 2a11 ∆t 2a11 4a22 ∆t 4a11 ∆t   1 + r1 ∆t 1 + r2 ∆t ≥ min , ≥ x∗ 4a11 ∆t 2a21 ∆t

U1 (y ∗ ) =

and 1 + r2 ∆t − a21 x ∆t ≥ 2a22 ∆t 2a22 ∆t   a21 1 + r2 ∆t 1 + r1 ∆t 1 + r2 ∆t − min , = 2a22 ∆t 2a22 4a11 ∆t 2a21 ∆t   1 + r2 ∆t a21 1 + r1 ∆t 1 + r2 ∆t 1 + r2 ∆t = max − , ≥ 2a22 ∆t 2a22 4a11 ∆t 4a22 ∆t 4a22 ∆t   1 + r1 ∆t 1 + r2 ∆t ≥ min , ≥ y∗. 2a12 ∆t 4a22 ∆t

U2 (x∗ ) =

Using x∗ and y ∗ in (8), we can obtain the positivity and boundedness of (xn , yn ). Theorem 1. Let (xn , yn ) be the solution of (2). Assume that (7) and (8) hold. If (x0 , y0 ) ∈ (0, x∗ ) × (0, y ∗ ), then (xn , yn ) ∈ (0, x∗ ) × (0, y ∗ ) for all n. Proof. Using the condition in this theorem and (5), we have 0 < x0 < x∗ ≤ U1 (y ∗ ) < U1 (y0 ), 0 < y0 < y ∗ ≤ U2 (x∗ ) < U2 (x0 ),

(10)

and then the increasing property (6) gives the positivity of x1 and y1 : x1 = Fy0 (x0 ) > Fy0 (0) = 0, y1 = Gx0 (y0 ) > Gx0 (0) = 0.

(11)

Now, we claim that x1 < x∗ and y1 < y ∗ . If r1 − a11 x0 − a12 y0 ≤ 0, then x1 = Fy0 (x0 ) ≤ x0 < x∗ . Otherwise, we get −1 0 < x0 < (r1 − a12 y0 )a−1 = U1 (y0 ), 11 < (1 + r1 ∆t − a12 y0 ∆t)(2a11 ∆t)

where the last inequality is obtained from r1 ∆t < 1 in (7). Hence (6) and (8) imply the boundedness of x1 :
−1 −1 ∗ x1 = Fy0 (x0 ) < Fy0 (r1 − a12 y0 )a−1 (12) 11 = (r1 − a12 y0 )a11 < r1 a11 ≤ x . 3 278

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Similarly if r2 − a21 x0 − a22 y0 ≤ 0, then y1 = Gx0 (y0 ) ≤ y0 < y ∗ . Otherwise, we have −1 0 < y0 < (r2 − a21 x0 )a−1 = U2 (x0 ), 22 < (1 + r2 ∆t − a21 x0 ∆t)(2a22 ∆t)

where the last inequality is obtained from r2 ∆t < 1 in (7). Thus (6) and (8) imply the boundedness of y1 that
∗ −1 −1 (13) y1 = Gx0 (y0 ) < Gx0 (r2 − a21 x0 )a−1 22 = (r2 − a21 x0 )a22 < r2 a22 ≤ y . Hence using (11), (12) and (13), we have that if (x0 , y0 ) ∈ (0, x∗ ) × (0, y ∗ ), then (x1 , y1 ) ∈ (0, x∗ ) × (0, y ∗ ). Therefore, using the mathematical induction, we can obtain the desired result. Remark 2. Due to (9), we can choose sufficiently large values of x∗ and y ∗ when letting ∆t be sufficiently small, so that the area of (0, x∗ ) × (0, y ∗ ) for the initial state (x0 , y0 ) in Theorem 1 can be taken large. 3. Stability of the discrete solutions Let D = (0, x∗ ) × (0, y ∗ ) for x∗ and y ∗ defined in (8). In order to discuss the stability of the Euler scheme (2) for each initial position (x0 , y0 ) contained in D, we partition D into the four regions I = {x ∈ D | f (x) ≥ 0, g(x) > 0}, II = {x ∈ D | f (x) < 0, g(x) ≥ 0}, III = {x ∈ D | f (x) ≤ 0, g(x) < 0}, IV = {x ∈ D | f (x) > 0, g(x) ≤ 0},

(14)

where x = (x, y) and f (x, y) = r1 − a11 x − a12 y, g(x, y) = r2 − a21 x − a22 y.

(15)

Since the location of the regions depends on the x and y-intercepts of the two lines f (x, y) = 0 and g(x, y) = 0, we partition D by using the four categories Ci (1 ≤ i ≤ 4) as in −1 −1 −1 Figure 1; we use the symbol C1 for the two conditions r1 a−1 11 < r2 a21 and r1 a12 < r2 a22 , −1 −1 −1 −1 −1 the symbol C2 for r1 a−1 11 > r2 a21 and r1 a12 > r2 a22 , the symbol C3 for r1 a11 < r2 a21 −1 −1 −1 −1 −1 −1 and r1 a12 > r2 a22 , and finally the symbol C4 for r1 a11 > r2 a21 and r1 a12 < r2 a22 . The magenta circles in Figure 1 denote the stable points of the difference model (2) in the categories, which will be proved. Remark 3. In the case of C1 −1 −1 −1 r1 a−1 11 < r2 a21 , r1 a12 < r2 a22 ,

(16)

the region IV is empty. In order to prove this emptiness, suppose, on the contrary, that there exists (x, y) ∈ IV, which means, from (14), that r1 − a11 x − a12 y > 0, r2 − a21 x − a22 y ≤ 0.

(17)

Eliminating x and y from (17), we have the two inequalities, respectively: −r1 a21 + r2 a11 < (a11 a22 − a12 a21 )y, −r1 a22 + r2 a12 < (a12 a21 − a11 a22 )x.

(18) (19)

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(a)

(b) f =0 g =0

1.5

f =0 g =0 1

y

y

1

III

III

II

0.5

0.5

IV

I 0

0

0.5

1

x

0

1.5

(c)

I 0

0.5

1

x

1.5

(d) f =0 g =0

1.5

f =0 g =0 1

y

y

II

1

IV

III

III

0.5 0.5

I 0

0

I

IV 0.5

x

1

0

1.5

0

II 0.5

x

1

1.5

Figure 1: Two lines f = 0 and g = 0 and regions with stable points. (a) r2 = 3.5, a21 = 3.0, a22 = 2 (b) r2 = 1.5, a21 = 3, a22 = 5 (c) r2 = 1.7, a21 = 3, a22 = 1 (d) r2 = 3.5, a21 = 2.5, a22 = 5

We find a contradiction by using the following three cases: Case 1. Let a11 a22 − a12 a21 = 0. In this case, (18) becomes −r1 a21 + r2 a11 < 0, which contradicts (16). Case 2. Let a11 a22 − a12 a21 < 0. Using the positivity of y, (18) becomes −r1 a21 + r2 a11 < 0, which contradicts (16). Case 3. Let a11 a22 − a12 a21 > 0. Using the positivity of x, (19) becomes −r1 a22 + r2 a12 < 0, which contradicts (16). Therefore it follows from Cases 1, 2 and 3 that the region IV is empty and then D = I ∪ II ∪ III for C1

(20)

as in Figure 1-(a). Similarly we can obtain D = I ∪ III ∪ IV for C2

(21)

as in Figure 1-(b). For convenience, we use the difference equations xn+1 = xn {1 + f (xn , yn )∆t}, yn+1 = yn {1 + g(xn , yn )∆t}

(22) (23)

as well as (2), where f (x, y) and g(x, y) are defined in (15). For the stability we need to assume 1 > ∆t (a11 x∗ + a22 y ∗ + x∗ y ∗ |a12 a21 − a11 a22 |∆t) .

(24)

Lemma 1. Let (xn , yn ) be the solution of (2). Assume that (7), (8) and (24) hold. If (xk , yk ) ∈ I for some k, then (xk+1 , yk+1 ) is not contained in III.

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Proof. The condition (xk , yk ) ∈ I gives g(xk , yk ) > 0.

(25)

Suppose, on the contrary, that (xk+1 , yk+1 ) is contained in III, which means f (xk+1 , yk+1 ) ≤ 0 and g(xk+1 , yk+1 ) < 0. Then (22) and (23) give 0 ≥ f (xk+1 , yk+1 ) = f (xk + xk f (xk , yk )∆t, yk + yk g(xk , yk )∆t) = f (xk , yk ) + (−a11 )xk f (xk , yk )∆t + (−a12 )yk g(xk , yk )∆t

(26)

0 > g(xk+1 , yk+1 ) = g(xk + xk f (xk , yk )∆t, yk + yk g(xk , yk )∆t) = g(xk , yk ) + (−a21 )xk f (xk , yk )∆t + (−a22 )yk g(xk , yk )∆t.

(27)

and

We write (26) and (27) as f (xk , yk )(1 − a11 xk ∆t) ≤ a12 yk g(xk , yk )∆t, g(xk , yk )(1 − a22 yk ∆t) < a21 xk f (xk , yk )∆t.

(28)

Combining (24) and Theorem 1 gives 0 < 1 − a11 x∗ ∆t < 1 − a11 xk ∆t and so (28) implies g(xk , yk )(1 − a22 yk ∆t) < a21 xk ∆t

a12 yk g(xk , yk )∆t . (1 − a11 xk ∆t)

(29)

Using (24) and (25), we can simplify (29) as follows. 1 < ∆t{a11 xk (1 − a22 yk ∆t) + a22 yk + a12 yk a21 xk ∆t} ≤ ∆t{a11 xk + a22 yk + xk yk |a12 a21 − a11 a22 |∆t},

(30)

where the last inequality contradicts (24). Hence (xk+1 , yk+1 ) is not contained in III. Remark 4. Similarly to Lemma 1 under the same assumption, we can obtain that if (xk , yk ) ∈ III for some k, then (xk+1 , yk+1 ) is not contained in I

(31)

as follows. The condition (xk , yk ) ∈ III gives g(xk , yk ) < 0.

(32)

f (xk+1 , yk+1 ) ≥ 0 and g(xk+1 , yk+1 ) > 0.

(33)

Suppose, on the contrary, that

Using (33) instead of f (xk+1 , yk+1 ) ≤ 0 and g(xk+1 , yk+1 ) < 0 in the proof of Lemma 1 and following the proof of Lemma 1 with (32), we have g(xk , yk )(1 − a22 yk ∆t) > a21 xk ∆t

a12 yk g(xk , yk )∆t (1 − a11 xk ∆t)

and then obtain the contradiction (30) due to (32). Therefore we obtain (31). 6 281

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Lemma 2. Let (xn , yn ) be the solution of (2). Assume that (7), (8) and (24) hold. If (xk , yk ) ∈ II for some k, then (xn , yn ) ∈ II for all n ≥ k. Proof. Let (xk , yk ) ∈ II, which implies f (xk , yk ) < 0 ≤ g(xk , yk ) and then xk+1 < xk , yk+1 ≥ yk .

(34)

It follows from Theorem 1, (34) and (10) that 0 < xk+1 < xk < U1 (yk ), 0 < yk ≤ yk+1 < y ∗ < U2 (xk ).

(35)

Using the decreasing function Fy (x) of y and combining (6) with (35), we have xk+2 = Fyk+1 (xk+1 ) ≤ Fyk (xk+1 ) < Fyk (xk ) = xk+1

(36)

f (xk+1 , yk+1 ) < 0.

(37)

and then (22) gives Similarly, the strictly decreasing function Gx (y) of x with (6) and (35) gives yk+2 = Gxk+1 (yk+1 ) > Gxk (yk+1 ) ≥ Gxk (yk ) = yk+1 .

(38)

Substituting (23) into (38) yields g(xk+1 , yk+1 ) > 0, with which (37) gives f (xk+1 , yk+1 ) < 0 < g(xk+1 , yk+1 ). This implies (xk+1 , yk+1 ) ∈ II. Hence if (xk , yk ) ∈ II, then (xk+1 , yk+1 ) ∈ II. Therefore using mathematical induction, we can obtain the desired result. Remark 5. Similarly to Lemma 2 under the same assumption, we can obtain that if (xk , yk ) ∈ IV for some k, then (xn , yn ) ∈ IV for all n ≥ k

(39)

as follows. Let (xk , yk ) ∈ IV, which implies f (xk , yk ) > 0 ≥ g(xk , yk ).

(40)

Then replacing f (xk , yk ) < 0 ≤ g(xk , yk ) in the proof of Lemma 2 with (40) and following the proof of Lemma 2, we have f (xk+1 , yk+1 ) > 0 > g(xk+1 , yk+1 ), which implies (xk+1 , yk+1 ) ∈ IV. Hence mathematical induction gives (39). 7 282

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In the following theorem, we show the global stability of the solutions of (2) for the category C1 as in Figure 1-(a); we present the condition that the species y always outcompetes the species x. Theorem 2. Assume that (7), (8) and (24) hold.
−1 −1 −1 −1 If r1 a−1 is globally stable. 11 < r2 a21 and r1 a12 < r2 a22 , then 0, r2 a22 Proof. The condition in this theorem is corresponding to C1 , so that D is partitioned into the three regions I, II and III due to (20). We claim the global stability for (x0 , y0 ) ∈ I ∪ II ∪ III by using mathematical induction as follows. Case 2-1. Let (x0 , y0 ) ∈ II. Using Lemma 2 and Theorem 1, we have that 0 < xn+1 < xn , 0 < yn ≤ yn+1 < y ∗ ,

(41)

which give the convergence of {xn } and {yn } with limits ω1 and ω2 , respectively. Note that the increasing property of {yn } gives ω2 > 0. In addition, the limit ω1 is zero, which can be obtained by indirect proof. Suppose, on the contrary, that ω1 is nonzero. Taking the limit of (2) and using ωi > 0 (i = 1, 2), we have (a11 a22 − a12 a21 ) (ω1 , ω2 ) = (r1 a22 − r2 a12 , −r1 a21 + r2 a11 ) . (42) Since r1 a22 − r2 a12 < 0 and −r1 a21 + r2 a11 > 0 from the conditions in this theorem, the equality (42) with ωi > 0 gives 0 > a11 a22 − a12 a21 > 0,

(43)

which is a contradiction. Consequently, ω1 is zero. Taking the limit of the second equation in (2) with ω1 = 0 and ω2 > 0, we have ω2 = r2 a−1 22 , which completes the proof for Case 2-1. Case 2-2. Let (x0 , y0 ) ∈ I. This case implies that f (x0 , y0 ) ≥ 0 and g(x0 , y0 ) > 0. We use the following three steps to prove this theorem in this case. Step 1. There exists a positive integer m1 such that (xm1 , ym1 ) ∈ / I. Suppose, on the contrary, that (xn , yn ) ∈ I for all n, which means f (xn , yn ) ≥ 0 and g(xn , yn ) > 0 for all n. Then xn+1 = xn {1 + f (xn , yn )∆t} ≥ xn > 0, yn+1 = yn {1 + g(xn , yn )∆t} > yn > 0 and hence the boundedness of (xn , yn ) in Theorem 1 gives the convergence of the increasing sequences {xn } and {yn }, which have positive limits ω1 and ω2 , respectively. Therefore we have a contradiction by using (42)–(43). Step 2. There exists a positive integer m such that (xm , ym ) ∈ II. Using (x0 , y0 ) ∈ I and Step 1, there exists a positive integer m1 such that (xm1 −1 , ym1 −1 ) ∈ I and (xm1 , ym1 ) ∈ D−I. Since D−I = II ∪ III, we have (xm1 , ym1 ) ∈ II or (xm1 , ym1 ) ∈ III.

(44)

Applying Lemma 1 with (xm1 −1 , ym1 −1 ) ∈ I, it is not true that (xm1 , ym1 ) ∈ III and then (xm1 , ym1 ) ∈ II. Taking m = m1 gives the desired result. 8 283

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Step 3. If (x0 , y0 ) ∈ I, then (xm , ym ) ∈ II for some positive integer m due to Step 2. Therefore the proof for Case 2-1 completes the proof for Case 2-2. Case 2-3. Let (x0 , y0 ) ∈ III. This case implies that f (x0 , y0 ) ≤ 0 and g(x0 , y0 ) < 0. We use the following two steps to prove this theorem in this case. Step 1. If (xn , yn ) ∈ III for all n, then limn→∞ (xn , yn ) = (0, r2 a−1 22 ). Assume that (xn , yn ) ∈ III for all n, which implies f (xn , yn ) ≤ 0, g(xn , yn ) < 0

(45)

for all n. The assumption gives the decreasing property 0 < xn+1 = xn {1 + f (xn , yn )∆t} ≤ xn , 0 < yn+1 = yn {1 + g(xn , yn )∆t} < yn and then Theorem 1 gives the convergence of {xn } and {yn } with the nonnegative limits ω1 and ω2 , respectively. It is only possible that ω1 = 0 and ω2 > 0 as follows. If ω1 > 0 and ω2 > 0, then (42)–(43) give a contradiction. If ω1 > 0 and ω2 = 0, then ω1 = r1 a−1 11 . This is impossible due to the unstability of −1 (r1 a11 , 0) since the linearized system of (2) at (r1 a−1 11 , 0) has the eigenvalue 1 + ∆ta−1 11 (r2 a11 − r1 a21 ) > 1 −1 −1 under the condition a21 a−1 11 < r2 r1 . Therefore {(xn , yn )} cannot have the limit (r1 a11 , 0). If ω1 = 0 and ω2 = 0, then

limn→∞ f (xn , yn ) = r1 > 0, limn→∞ g (xn , yn ) = r2 > 0, which are contradictory to (45). Therefore it remains that ω1 = 0 and ω2 > 0, which gives (ω1 , ω2 ) = (0, r2 a−1 22 ). −1 Step 2. If (xm , ym ) ∈ / III for some m, then limn→∞ (xn , yn ) = (0, r2 a22 ). Since (xm , ym ) ∈ D−III and D−III = I ∪ II, we have (xm , ym ) ∈ I or (xm , ym ) ∈ II. However it is not true that (xm , ym ) ∈ I due to Remark 4 and so we have (xm , ym ) ∈ II. Therefore, following the proof for Case 2-1, we obtain limn→∞ (xn , yn ) = (0, r2 a−1 22 ). Finally, we obtain the desired result from the proofs for Cases 2-1, 2-2 and 2-3. In the following theorem, we show the global stability of (2) for C2 as in Figure 1-(b) and present the condition that the species x always outcompetes the species y. Theorem 3. Assume that (7), (8) and (24) hold.
−1 −1 −1 −1 If r1 a−1 11 > r2 a21 and r1 a12 > r2 a22 , then r1 a11 , 0 is globally stable. Proof. The condition in this theorem is corresponding to C1 and so D is partitioned into the three regions I, III and IV due to (21). We claim the global stability for (x0 , y0 ) ∈ I ∪ III ∪ IV by using mathematical induction as follows. Case 3-1. Let (x0 , y0 ) ∈ IV. In this case, (39) gives (xn , yn ) ∈ IV for all n, with which (22) and (23) give xn < xn+1 and yn+1 ≤ yn . Then Theorem 1 gives 0 < xn < xn+1 < x∗ , 0 < yn+1 ≤ yn ,

(46)

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which imply the convergence of {xn } and {yn } with limits ω1 and ω2 , respectively. The increasing property of {xn } gives ω1 > 0. In addition, the limit ω2 is zero, which can be obtained by indirect proof as in Case 2-1. Suppose, on the contrary, that ω2 is nonzero. Taking the limit of (2) and using −1 the positivity of ω1 and ω2 , we have (42). Applying the conditions r1 a−1 11 > r2 a21 and −1 r1 a−1 12 > r2 a22 to (42) yields the contradiction (43) Consequently, ω2 is zero. Taking the limit of the first equation in (2) with ω1 > 0 and ω2 = 0, we have ω1 = r1 a−1 11 , which completes the proof for Case 3-1. Case 3-2. Let (x0 , y0 ) ∈ I. In this case we have f (x0 , y0 ) ≥ 0 and g(x0 , y0 ) > 0, and use the following three steps. Step 1. There exists a positive integer m1 such that (xm1 , ym1 ) ∈ / I. Suppose, on the contrary, that (xn , yn ) ∈ I for all n, which means f (xn , yn ) ≥ 0 and g(xn , yn ) > 0 for all n. Then xn+1 = xn {1 + f (xn , yn )∆t} ≥ xn > 0, yn+1 = yn {1 + g(xn , yn )∆t} > yn > 0, and hence the boundedness of (xn , yn ) in Theorem 1 gives the convergence of the increasing sequences {xn } and {yn }, which have positive limits ω1 and ω2 , respectively. Therefore we have the contradiction (43) as in Case 3-1. Step 2. There exists a positive integer m such that (xm , ym ) ∈ IV. Using (x0 , y0 ) ∈ I and Step 1, there exists a positive integer m1 such that (xm1 −1 , ym1 −1 ) ∈ I and (xm1 , ym1 ) ∈ D−I for some m1 . Since D−I = III ∪ IV, we have (xm1 , ym1 ) ∈ III or (xm1 , ym1 ) ∈ IV. Applying Lemma 1 with (xm1 −1 , ym1 −1 ) ∈ I, it is not true that (xm1 , ym1 ) ∈ III and then (xm1 , ym1 ) ∈ IV. Taking m = m1 gives (xm , ym ) ∈ IV. Step 3. If (x0 , y0 ) ∈ I, then (xm , ym ) ∈ IV for some positive integer m due to Step 2. Therefore the proof used in Case 3-1 completes the proof for Case 3-2. Case 3-3. Let (x0 , y0 ) ∈ III. In this case we have f (x0 , y0 ) ≤ 0 and g(x0 , y0 ) < 0,and use the following two steps. Step 1. If (xn , yn ) ∈ III for all n, then limn→∞ (xn , yn ) = (r1 a−1 11 , 0). As in Step 1 of Case 2-3 in Theorem 2, {(xn , yn )} has the limit (ω1 , ω2 ). It is only possible that ω1 > 0 and ω2 = 0 as follows. If ω1 > 0 and ω2 > 0, then (46)–(??) give a contradiction. If ω1 = 0 and ω2 > 0, then ω2 = r2 a−1 22 . This is impossible due to the unstability of (0, r2 a−1 ) since the linearized system of (2) at (0, r2 a−1 22 ) has the eigenvalue 22 1 + ∆ta−1 22 (r1 a22 − r2 a12 ) > 1 −1 −1 under the condition a22 a−1 12 > r2 r1 . Therefore {(xn , yn )} cannot have the limit (r1 a11 , 0). If ω1 = 0 and ω2 = 0, then

limn→∞ f (xn , yn ) = r1 > 0, limn→∞ g (xn , yn ) = r2 > 0, which are contradictory to (45). It remains that ω1 > 0 and ω2 = 0, which yields the desired result (ω1 , ω2 ) = (r1 a−1 11 , 0). −1 Step 2. If (xm , ym ) ∈ / III for some m, then limn→∞ (xn , yn ) = (r1 a11 , 0). Since (xm , ym ) ∈ D−III = I ∪ IV, we have 10 285

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(xm , ym ) ∈ I or (xm , ym ) ∈ IV. However it is not true that (xm , ym ) ∈ I due to Remark 4. Therefore, we have (xm , ym ) ∈ IV and then limn→∞ (xn , yn ) = (r1 a−1 11 , 0) by following the proof for Case 3-1. Finally, we obtain the desired result from the proofs for Cases 3-1 and 3-2. In the following theorem, we show the convergence of the solutions of (2) for the category C3 as in Figure 1-(c) and the dependence of the limit on the region in which the initial state is located. From now on, in the case that a11 a22 − a12 a21 6= 0, we use the symbol (θ1 , θ2 ) to mean (θ1 , θ2 ) = (a11 a22 − a12 a21 )−1 (r1 a22 − r2 a12 , −r1 a21 + r2 a11 ) ,

(47)

where (θ1 , θ2 ) satisfies f (θ1 , θ2 ) = g(θ1 , θ2 ) = 0.

(48)

Theorem 4. Let the conditions (7), (8) and (24) hold. Assume that −1 −1 −1 r1 a−1 11 > r2 a21 and r1 a12 < r2 a22 .

(a) If (x0 , y0 ) ∈ II, then limn→∞ (xn , yn ) = (0, r2 a−1 22 ). −1 (b) If (x0 , y0 ) ∈ IV, then limn→∞ (xn , yn ) = (r1 a11 , 0). −1 (c) If (x0 , y0 ) ∈ I ∪ III, then {(xn , yn )} converges with the limit (r1 a−1 11 , 0) or (0, r2 a22 ). Proof. For the proof of (a), let (x0 , y0 ) ∈ II. We have from Lemma 2 and Theorem 1 that 0 < xn+1 < xn , 0 < yn ≤ yn+1 < y ∗ ,

(49)

which gives the convergence of {xn } and {yn } with limits ω1 and ω2 , respectively. The increasing property of {yn } gives ω2 > 0. In addition, the limit ω1 is zero, which can be obtained by indirect proof. Suppose, on the contrary, that ω1 is nonzero. Taking the limit of (2) and using the positivity of ω1 and ω2 , we have (a11 a22 − a12 a21 )ω1 = r1 a22 − r2 a12 . (50) Since (x0 , y0 ) ∈ II, the definition of the region II gives f (x0 , y0 ) < 0 ≤ g(x0 , y0 ).

(51)

(r1 a22 − r2 a12 ) − (a11 a22 − a12 a21 )x0 < 0.

(52)

Solving (51) for x0 , we obtain

−1 The conditions a21 a−1 > a22 a−1 11 > r2 r1 12 in this theorem give

a11 a22 − a12 a21 < 0.

(53)

Applying (53) into both (52) and (50) yields ω1 > x0 .

(54)

Combining (54) with (49), we have that for all n 11 286

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ω1 > x0 > xn , which is contradictory to limn→∞ xn = ω1 . Consequently, ω1 is zero. Taking the limit of the second equation in (2) with ω1 = 0 and ω2 > 0, we have ω2 = r2 a−1 22 , which completes the proof of (a). For the proof of (b), let (x0 , y0 ) ∈ IV. Using (46), we have the convergence of {xn } and {yn } with limits ω1 and ω2 , respectively. The increasing property of {xn } gives ω1 > 0. In addition, the limit ω2 is zero, which can be obtained by indirect proof. Suppose, on the contrary, that ω2 is nonzero. Taking the limit of (2) and using the positivity of ω1 and ω2 , we have (a11 a22 − a12 a21 )ω2 = −r1 a21 + r2 a11 . (55) Since (x0 , y0 ) ∈ IV, the definition of the region IV gives f (x0 , y0 ) > 0 ≥ g(x0 , y0 ).

(56)

(r1 a21 − r2 a11 ) + (a11 a22 − a12 a21 )y0 > 0.

(57)

Solving (56) for y0 , we obtain

Applying (53) into (57) yields ω2 > y0 .

(58)

Combining (58) with (46), we have that for all n ω2 > y0 > yn , which is contradictory to limn→∞ yn = ω2 . Consequently, ω2 is zero. Taking the limit of the first equation in (2) with ω1 > 0 and ω2 = 0, we have ω1 = r1 a−1 11 , which completes the proof of (b). For the proof of (c), we consider the following two cases. Case 4-1. Let (x0 , y0 ) ∈ I. We use the following three steps to obtain the desired result in this case. Step 1. There exists a positive constant m such that (xm , ym ) ∈ / I. Suppose, on the contrary, that (xn , yn ) ∈ I for all n. Then {xn } and {yn } have the positive limits (θ1 , θ2 ) defined in (47) by applying (53) and the approach used in Step1 −1 of Case 2-2 in Theorem 2. However the system (2) under the condition r1 a−1 11 < r2 a21 is unstable at the point (θ1 , θ2 ) since the linearized system at (θ1 , θ2 ) has the eigenvalue 1+∆ta−1 11 (r2 a11 − r1 a21 ) greater than 1. Therefore {xn } and {yn } cannot have the positive limits θ1 and θ2 , respectively, which is contradictory. Step 2. There exists a positive constant m such that (xm , ym ) ∈ II ∪ IV. Since (x0 , y0 ) ∈ I, Step 1 gives the existence of a positive integer m such that (xm−1 , ym−1 ) ∈ I and (xm , ym ) ∈ / I, which implies (xm , ym ) ∈ II ∪ IV due to Lemma 1 and D = I ∪ II ∪ III ∪ IV. Step 3. It follows from (a), (b) and Step 2 in this theorem that (xn , yn ) converges and −1 has the limit (r1 a−1 11 , 0) or (0, r2 a22 ). Case 4-2. Let (x0 , y0 ) ∈ III. We use the following two steps to obtain the desired result in this case. 12 287

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Step 1. If (xn , yn ) ∈ III for all n, then {(xn , yn )} converges with the limit (r1 a−1 11 , 0) or (0, r2 a−1 ). To prove this, note that we have the convergence of {(x , y )} with the limit n n 22 (ω1 , ω2 ) by following the proof of Step 1 of Case 2-3 in Theorem 2. If ω1 > 0 and ω2 > 0, then (ω1 , ω2 ) = (θ1 , θ2 ). This is impossible due to the unstability of (θ1 , θ2 ) since the linearized system of (2) at (θ1 , θ2 ) has the eigenvalue greater than 1:   q 2 1 + 0.5∆t − (a11 θ1 + a22 θ2 ) + (a11 θ1 + a22 θ2 ) + α > 1 −1 since α = 4 (a12 a21 − a11 a22 ) θ1 θ2 > 0 under the condition a21 a−1 > a22 a−1 11 > r2 r1 12 . Therefore it is not possible that ω1 > 0 and ω2 > 0. If ω1 = 0 and ω2 = 0, then we have the contradictions to to (45):

limn→∞ f (xn , yn ) = r1 > 0, limn→∞ g (xn , yn ) = r2 > 0. Therefore the remaining signs of ω1 and ω2 are (+, 0) and (0, +), which give the desired result −1 (ω1 , ω2 ) = (r1 a−1 11 , 0) and (0, r2 a22 ),

respectively, by taking the limit of (2) and using the signs of ω1 and ω2 . Step 2. If (xm , ym ) ∈ / III for some m, then {(xn , yn )} converges with the limit (r1 a−1 11 , 0) or (0, r2 a−1 ). To prove this, we follow the proof used in Step 2 of Case 4-1. 22 Since (x0 , y0 ) ∈ III, using the condition (xm , ym ) ∈ / III for some m, we can assume that there exists a positive constant m1 such that (xm1 −1 , ym1 −1 ) ∈ III and (xm1 , ym1 ) ∈ / III, which implies (xm1 , ym1 ) ∈ II ∪ IV

(59)

due to D = I ∪ II ∪ III ∪ IV and Lemma 1. Therefore, using (59) and (a) and (b) in this −1 theorem, we have that (xn , yn ) converges and has the limit (r1 a−1 11 , 0) or (0, r2 a22 ). Finally, we obtain the desired result from the proofs for Cases 4-1 and 4-2. In the following theorem, we show the global stability of the solutions of (2) for the category C4 as in Figure 1-(d) where each component of the equilibrium point is positive. Theorem 5. Let the conditions (7), (8) and (24) hold. Assume that −1 −1 −1 r1 a−1 11 < r2 a21 and r1 a12 > r2 a22 .

Then for (θ1 , θ2 ) defined in (47) (θ1 , θ2 ) is globally stable. −1 −1 −1 Proof. Note that the conditions r1 a−1 11 < r2 a21 and r1 a12 > r2 a22 in this theorem give

a11 a22 − a12 a21 > 0.

(60)

We prove this theorem by using the four cases and mathematical induction. Case 5-1. Let (x0 , y0 ) ∈ II. Lemma 2 and Theorem 1 give (49). Then we have 13 288

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limn→∞ (xn , yn ) = (ω1 , ω2 ), ω2 > 0 and f (xn , yn ) < 0 ≤ g(xn , yn ).

(61)

Solving (61) for xn as in (51) and (52) and using (60), we have that for all n 0 < θ1 < xn and then ω1 ≥ θ1 > 0. Since ω1 and ω2 are positive, we have (ω1 , ω2 ) = (θ1 , θ2 ). Case 5-2. Let (x0 , y0 ) ∈ IV. Using Remark 5 and Theorem 1, we have 0 < xn < xn+1 < x∗ , 0 < yn+1 ≤ yn

(62)

and limn→∞ (xn , yn ) = (ω1 , ω2 ), ω1 > 0. The inequalities (62) implies f (xn , yn ) > 0 ≥ g(xn , yn ).

(63)

Solving (63) for yn as in (56) and (57), we have that for all n 0 < θ2 < yn and then ω2 ≥ θ2 > 0. Since ω1 and ω2 are positive, we have (ω1 , ω2 ) = (θ1 , θ2 ). Case 5-3. Let (x0 , y0 ) ∈ I. If (xm , ym ) ∈ / I for some m, then (xm , ym ) ∈ D − I = II ∪ III ∪ IV and further (xm , ym ) ∈ II ∪ IV due to Lemma 1. By Case 5-1 and 5-2, we have limn→∞ (xn , yn ) = (θ1 , θ2 ). On the other hand, if (xn , yn ) ∈ I for all n, then we have the positive limits ω1 and ω2 of {xn } and {yn }, respectively, due to the definition of I and Theorem 1. Taking the limit of (2) and using ωi (i = 1, 2), we have (ω1 , ω2 ) = (θ1 , θ2 ). Case 5-4. Let (x0 , y0 ) ∈ III. Replacing I in the proof of Case 5-3 with III, we can obtain (ω1 , ω2 ) = (θ1 , θ2 ). Finally, we obtain the desired result from the proofs for Cases 5-1 to 5-4.

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4. Numerical examples In this section, we provide simulations that illustrate our results in Theorem 2 to −1 Theorem 5 for the difference scheme (2) with ∆t = 0.001 and (x∗ , y ∗ ) = (r1 a−1 11 +50, r2 a22 + 50). The values of parameters used in the following four examples satisfy the three conditions (7), (8) and (24). Example 1. Let (r1 , a11 , a12 , r2 , a21 , a22 ) = (1, 1, 2, 3.5, 3, 2), which satisfies the two −1 −1 −1 conditions r1 a−1 11 < r2 a21 and r1 a12 < r2 a22 in Theorem 2. Then the solutions (xn , yn ) of (2) converge to (0, r2 a−1 22 = 1.75) as displayed in Figure 2-(a). Example 2. Let (r1 , a11 , a12 , r2 , a21 , a22 ) = (1, 1, 1, 1.5, 3, 5), which satisfies the two −1 −1 −1 conditions r1 a−1 11 > r2 a21 and r1 a12 > r2 a22 in Theorem 3. Then the solutions (xn , yn ) of (2) converge to (r1 a−1 11 = 1, 0) as displayed in Figure 2-(b). (a)

(b) f =0 g =0

f =0 g =0 1

1.5

III y

III

II

y

1

IV

0.5

0.5

I I 0

0

0.5

x

1

0

1.5

0

0.5

x

1

1.2

Figure 2: (a) Trajectories for different initial points in the regions I, II, III with r1 = 1, a11 = 1, a12 = 2, r2 = 3.5, a21 = 3, a22 = 2 in the category C1 . (b) Trajectories for different initial points in the regions I, III, IV with r1 = 1, a11 = 1, a12 = 1, r2 = 1.5, a21 = 3, a22 = 5 in the category C2 . The box and circle symbols denote initial and equilibrium points, respectively.

Example 3. Let (r1 , a11 , a12 , r2 , a21 , a22 ) = (1, 1, 1, 1.7, 3, 1), which satisfies the two −1 −1 −1 conditions r1 a−1 11 > r2 a21 and r1 a12 < r2 a22 in Theorem 4. Then as displayed in Figure 3-(a), we obtain the results in Theorem 4. If (x0 , y0 ) ∈ II, then the solutions (xn , yn ) of −1 (2) converge to (0, r2 a−1 22 ) = (0, 1.7). If (x0 , y0 ) ∈ IV, then limn→∞ (xn , yn ) = (r1 a11 , 0) = (1, 0). If (x0 , y0 ) ∈ I ∪ III, then {(xn , yn )} converges with the limit (r1 a−1 11 , 0) = (1, 0) −1 or (0, r2 a22 ) = (0, 1.7). Especially, Figure 3-(a) shows that there exist at least two −1 initial points contained in I converging to (r1 a−1 11 , 0) = (1, 0) and (0, r2 a22 ) = (0, 1.7), respectively. In the region III, the same phenomenon happens. The outcome depends on the initial abundances of the two species. Example 4. Let (r1 , a11 , a12 , r2 , a21 , a22 ) = (1, 1, 1, 3.5, 2.5, 5), which satisfies the two −1 −1 −1 conditions r1 a−1 11 < r2 a21 and r1 a12 > r2 a22 in Theorem 5. Then the solutions xn and yn of (2) converge to (r1 a22 − r2 a12 )(a11 a22 − a12 a21 )−1 = 0.6 and 15 290

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(−r1 a21 + r2 a11 )(a11 a22 − a12 a21 )−1 = 0.4, respectively, as displayed in Figure 3-(b). Although the outcome in Example 3 depends on the initial abundances of the two species, the outcome in Example 4 is independent of the initial abundances. (a)

(b) f =0 g =0

f =0 g =0 1

1.5

III y

II

1

III

IV y 0.5

0.5

IV I

I 0

0

0.5

x

1

0

1.2

0

II 0.5

x

1

1.5

Figure 3: Trajectories for different initial points in the regions I, II, III and IV. The values of the parameters are (a) r1 = 1, a11 = 1, a12 = 1, r2 = 1.7, a21 = 3, a22 = 1 in the category C3 . (b) r1 = 1, a11 = 1, a12 = 1, r2 = 3.5, a21 = 2.5, a22 = 5 in the category C4 . The box and circle symbols denote initial and equilibrium points, respectively.

5. Conclusions and future work In this paper, we have studied the Euler difference scheme for a two-dimensional LotkaVolterra competition model and presented sufficient conditions for the global stability of the fixed points of a discrete competition model with two species. The main idea of our approach is to divide the domain used for the boundedness of solutions of the discrete model and to describe how to trace the trajectories with respect to each partition. Although we have applied our method for the two-dimensional discrete model, this method can be utilized to two-dimensional and other higher dimensional discrete models. Acknowledgments The present research has been conducted by the Research Grant of Kwangwoon University in 2015.

References [1] J. Huang, S. Ruan, and J. Song. Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response. J. Differential Equations, 257(6):1721–1752, 2014. [2] J. Alebraheem and Y. Abu-Hasan. Persistence of predators in a two predators-one prey model with non-periodic solution. Appl. Math. Sci., 6(19):943–956, 2012. 16 291

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[3] Y. Li and D. Xiao. Bifurcations of a predator-prey system of Holling and Leslie types. Chaos Solitons Fractals, 34(2):606–620, 2007. [4] S. Ruan and D. Xiao. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math., 61(4):1445–1472, 2000. [5] S.B. Hsu and T.W. Huang. Global stability for a class of predator-prey systems. SIAM J. Appl. Math., 55(3):763–783, 1995. [6] H.K. Baek and D.S. Kim. Dynamics of a predator-prey system with mixed functional responses. J. Appl. Math., pages Art. ID 536019, 10, 2014. [7] S. Shulin and G. Cuihua. Dynamics of a Beddington-DeAngelis type predator-prey model with impulsive effect. J. Math., pages Art. ID 826857, 11, 2013. [8] S. Liu and E. Beretta. A stage-structured predator-prey model of BeddingtonDeAngelis type. SIAM J. Appl. Math., 66(4):1101–1129, 2006. [9] H. Xiang X.Y. Meng, H.F. Huo and Q.Y. Yin. Stability in a predator-prey model with Crowley-Martin function and stage structure for prey. Appl. Math. Comput., 232:810–819, 2014. [10] X.Y. Zhou and J.G. Cui. Global stability of the viral dynamics with Crowley-Martin functional response. Bull. Korean Math. Soc., 48(3):555–574, 2011. [11] P.H. Crowley and E.K. Martin. Functional responses and interference within and between year classes of a dragonfly population. J. North. Am. Benth. Soc., 8:211– 221, 1989. [12] X. Wang and H. Ma. A Lyapunov function and global stability for a class of predatorprey models. Discrete Dyn. Nat. Soc., pages Art. ID 218785, 8, 2012. [13] H.B. Xiao. Global analysis of ivlevs type predator-prey dynamic systems. Applied Mathematics and Mechanics, 28(4):461–470, 2007. [14] J. Sugie. Two-parameter bifurcation in a predator-prey system of Ivlev type. J. Math. Anal. Appl., 217(2):349–371, 1998. [15] L.J.S. Allen. Introduction to mathematical biology. Pearson/Prentice Hall, 2007. [16] M. Townsend, C.R. Begon and J.D. Harper. Ecology: individuals, populations and communities, 1996. [17] C.E. Gordon. The coexistence of species la coexistencia de especies. Revista chilena de historia natural, 73(1):175–198, 2000. [18] Q. Din. Dynamics of a discrete Lotka-Volterra model. Adv. Difference Equ., pages 2013:95, 13, 2013. [19] D. Blackmore, J. Chen, J. Perez, and M. Savescu. Dynamical properties of discrete lotka–volterra equations. Chaos, Solitons & Fractals, 12(13):2553–2568, 2001. 17 292

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[20] L.-I. Roeger and R. Gelca. Dynamical consistent discrete-time lokta-volterra competition models. Discrete Cont. Dyn. Sys., (Supplement 2009):650–658, 2009. [21] T. Wu. Dynamic behaviors of a discrete two species predator-prey system incorporating harvesting. Discrete Dyn. Nat. Soc., pages Art. ID 429076, 12, 2012. [22] S.M. Choo. Global stability in n-dimensional discrete lotka-volterra predator-prey models. Adv. Difference Equ., pages 2014:11, 17, 2014. [23] Y.-H. Kim and S.M. Choo. A new approach to global stability in discrete lotkavolterra predator-prey models. Discrete Dyn. Nat. Soc., pages Art. ID 674027, 11, 2015.

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Weighted Composition Operators from Bloch spaces into Zygmund spaces∗ Shanli Ye† (School of Sciences, Zhejiang University of Science and Technology, Hangzhou 310023, China) (Department of Mathematics, Fujian Normal University, Fuzhou 350007, China)

Abstract In this paper we characterize the boundedness and compactness of the weighted composition operator from the classical Bloch space β to the Zygmund space Z, and from the little Bloch space β0 to the little Zygmund space Z0 , respectively. Keywords Bloch space, Zygmund space; Weighted composition operator; Boundedness; Compactness 2010 MR Subject Classification 47B38, 30D99, 30H05

1

Introduction

Let D = {z : |z| < 1} be the open unit disk in the complex plane and H(D) denote the set of all analytic functions on D. Let u, φ ∈ H(D), where φ is an analytic self-map of D. Then the well-known weighted composition operator uCφ on H(D) is defined by uCφ (f )(z) = u(z)·(f ◦φ(z)) for f ∈ H(D) and z ∈ D. Weighted composition operators can be regarded as a generalization of multiplication operators and composition operators. In 2001, Ohno and Zhao studied the weighted composition operators on the classical Bloch space β in [14], which has led many researchers to study this operator on other Banach spaces of analytic functions. The boundedness and compactness of it have been studied on various Banach spaces of analytic functions, such as Hardy, Bergman, BMOA, Bloch-type spaces, see, e.g. [2, 4, 8, 18, 27]. In 2006, the boundedness of composition operators on the Zygmund space Z was first studied by Choe, Koo, and Smith in [1]. Later, many researchers have studied composition operators and weighted composition operators acting on the Zygmund space Z. Li and Stevi´c in [9] studied the boundedness and compactness of the generalized composition operators on Zygmund spaces and Bloch type spaces. They in [11] considered the boundedness and compactness of the weighted composition operators from Zygmund spaces to Bloch spaces. Ye and Hu in [22] characterized boundedness and compactness of weighted composition operators on the Zygmund space Z. Esmaeili and Lindstr¨om in [7] studied weighted composition operators from Zygmund type spaces to Bloch type spaces and their essential norms. Sanatpour and Hassanlou in [17] gave the essential norms of this operators between Zygmund-type spaces and Bloch-type spaces. See also ∗ The research was supported by the National Natural Science Foundation of China (Grant No. 11571217) and the Natural Science Foundation of Fujian Province, China(Grant No. 2015J01005). † E-mail: [email protected]

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[5, 15, 16, 19, 20, 21, 23, 24, 25, 26] for corresponding results for weighted composition operators from one Banach space of analytic functions to another. It is well-known that Z ⊂ β. It is more interesting to characterize u, φ such that this operator uCφ has the pull-back properly, that is, uCφ f ∈ Z whenever f ∈ β. In this paper we consider this question. Now we give a detailed definition of these spaces. A function f analytic on the unit disk is said to belong to the Bloch space β if b(f ) = sup{(1 − |z|2 )|f ′ (z)|} < ∞, z∈D

and to the little Bloch space β0 if f ∈ β and lim (1 − |z|2 )|f ′ (z)| = 0.

|z|→1−

It is well known that β is a Banach space under the norm ∥f ∥β = |f (0)| + b(f ), and β0 is a closed subspace of β. The Zygmund space Z consists of all analytic functions f defined on D such that z(f ) = sup{(1 − |z|2 )|f ′′ (z)| : z ∈ D},

0 < α < +∞.

From a theorem of Zygmund (see [29, vol. I, p. 263] or [6, Theorem 5.3]), we see that f ∈ Z if and only if f is continuous in the close unit disk D = {z : |z| ≤ 1} and the boundary function f (eiθ ) such that |f (ei(θ+h) ) + f (ei(θ−h) ) − 2f (eiθ )| sup < ∞. h h>0,θ An analytic function f ∈ H(D) is said to belong to the little Zymund space Z0 consists of all f ∈ Z satisfying lim|z|→1− (1 − |z|2 )|f ′′ (z)| = 0. It can easily proved that Z is a Banach space under the norm ∥f ∥Z = |f (0)| + |f ′ (0)| + z(f ) and the polynomials are norm-dense in closed subspace Z0 of Z. For some other information on this space and some operators on it, see, for example, [9, 10, 11]. Throughout this paper, constants are denoted by C, they are positive and only depending on p, and may differ from one occurrence to the other.

2

Auxiliary results

In order to prove the main results of this paper. we need some auxiliary results. The first part of the following lemma is a well known. Lemma 2.1 Suppose that f ∈ β, then e ∥f ∥β for every z ∈ D; (1 − |z|2 ) 8 (ii) |f ′′ (z)| ≤ b(f ) for every z ∈ D. (1 − |z|2 )2 (i) |f (z)| ≤ log

2

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Proof For any f ∈ β. Fix z ∈ D and let ρ = that

1 |f (z)| = | 2πi ′′

∫ |ξ|=ρ

f ′ (ξ) b(f ) 1 dξ| ≤ 2 (ξ − z) 1 − ρ2 2π

1 + |z| , by the Cauchy integral formula, we obtain 2 ∫

ρ dθ ∥f ∥∞ ρ 8 = ≤ . 2 2 2 2 − z| 1 − ρ ρ − |z| (1 − |z|2 )2

2π

|ρeiθ

0

Hence (ii) holds. Lemma 2.2 [28] Suppose that f ∈ β0 , then |f (z)| = 0; log(e/(1 − |z|2 )) (ii) lim − (1 − |z|2 )2 |f ′′ (z)| = 0.

(i)

lim

|z|→1− |z|→1

Lemma 2.3 Suppose uCφ : β0 → Z0 is a bounded operator, then uCφ : β → Z is a bounded operator. The proof is similar to that of Lemma 2.3 in [21]. The details are omitted. Lemma 2.4 Suppose that uCφ be a bounded operator from β to Z, then uCφ is compact if and only if for any bounded sequence {fn } in β which converges to 0 uniformly on compact subsets of D. We have ∥uCφ (fn )∥Z → 0 , as n → ∞ . The proof is similar to that of Proposition 3.11 in [3] . The details are omitted. Lemma 2.5 Let U ⊂ Z0 . Then U is compact if and only if it is closed, bounded and satisfies lim sup (1 − |z|2 )|f ′′ (z)| = 0.

|z|→1 f ∈U

The proof is similar to that of Lemma 1 in [12], we omit it.

3

Main results

Theorem 3.1 Let u be an analytic function on the unit disc D, and φ an analytic self-map of D. Then uCφ is a bounded operator from the classical space β to the Zygmund space Z if and only if the following are satisfied:

sup (1 − |z|2 )|u′′ (z)| log z∈D

sup z∈D

e < ∞; 1 − |φ(z)|2

(3.1)

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| < ∞; 1 − |φ(z)|2

(3.2)

(1 − |z|2 )|u(z)(φ′ (z))2 | < ∞. (1 − |φ(z)|2 )2

(3.3)

sup z∈D

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Proof Suppose uCφ is bounded from the Bloch space β to the Zygmund space Z. Then we can easily obtain the following results by taking f (z) = 1 and f (z) = z in β respectively: u ∈ Z;

(3.4)

sup (1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z) + φ(z)u′′ (z)| < +∞.

(3.5)

z∈D

By (3.4), (3.5) and the boundedness of the function φ(z), we get K1 = sup (1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| < +∞.

(3.6)

z∈D

Let f (z) = z 2 in β again, in the same way we have sup (1 − |z|2 )|4φ(z)φ′ (z)u′ (z) + φ2 (z)u′′ (z) + 2u(z)(φ(z)φ′′ (z) + (φ′ (z))2 )| < ∞. z∈D

Using these facts and the boundedness of the function φ(z) again, we get K2 = sup (1 − |z|2 )|(φ′ (z))2 u(z)| < +∞.

(3.7)

z∈D

Fix a ∈ D, we take the test functions fa (z) = 3 log

e e e 3 1 (log (log + )2 − )3 e 2 e 1−a ¯z log 1−|a|2 1−a ¯z 1−a ¯z log 1−|a|2

(3.8)

for z ∈ D. By a directly calculation we obtain that fa ∈ β and supa ∥fa ∥β ≤ C < ∞, where C is e ′ ′′ not depended on a. Since fa (a) = 5 log 1−|a| 2 , fa (a) = 0, fa (a) = 0, we have C∥fa ∥β

∥uCφ fa ∥Z ≥ sup (1 − |z|2 )|(uCφ fa )′′ (z)| z∈D ( ) 2 = sup (1 − |z| )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) fa′ (φ(z))

≥

z∈D

+ fa′′ (φ(z))(φ′ (z))2 u(z) + u′′ (z)fa (φ(z))|. Let a = φ(λ), it follows that C∥fa ∥β

( ) ′ ≥ (1 − |λ|2 )α | 2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ) fφ(λ) (φ(λ)) ′′ + fφ(λ) (φ(λ))(φ′ (λ))2 u(λ) + u′′ (λ)fφ(λ) (φ(λ))| e = 5(1 − |λ|2 )α |u′′ (λ) log |. 1 − |φ(λ)|2

Hence (3.1) holds. Next, we will show that (3.2) holds. Fix a ∈ D with |a| > 12 , we take another test functions: ga (z) =

14(1 − |a|2 )3 6(1 − |a|2 )4 8(1 − |a|2 )2 − + (1 − a ¯z)2 (1 − a ¯z)3 (1 − a ¯z)4

(3.9)

for z ∈ D. By a directly calculation we obtain that ga ∈ β and supa ∥ga ∥β ≤ C < ∞, where C is −2¯ a , g ′′ (a) = 0, it follows that for all λ ∈ D not depended on a. Since ga (a) = 0, ga′ (a) = 1 − |a|2 a with |φ(λ)| > 12 , we have C∥ga ∥β

≥ =

∥uCφ ga ∥Z ≥ sup (1 − |z|2 )|(uCφ ga )′′ (z)| z∈D ( ) 2 sup (1 − |z| )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) ga′ (φ(z))

z∈D

+ ga′′ (φ(z))(φ′ (z))2 u(z) + u′′ (z)ga (φ(z))|. 4

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Let a = φ(λ), it follows that C∥ga ∥β

( ) ′ (φ(λ)) ≥ (1 − |λ|2 )| 2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ) gφ(λ) ′′ + gφ(λ) (φ(λ))(φ′ (λ))2 u(λ) + u′′ (λ)gφ(λ) (φ(λ))|

( ) −2φ(λ)2 | = (1 − |λ|2 )| 2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ) 1 − |φ(λ)|2 ( ) 1 (1 − |λ|2 )| 2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ) | . ≥ 2 1 − |φ(λ)|2 For ∀λ ∈ D with |φ(λ)| ≤ 12 , by (3.6), we have ( ) ( ) (1 − |λ|2 )| 2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ) | 4 sup ≤ sup (1 − |λ|2 )| 2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ) | < +∞. 2 1 − |φ(λ)| 3 λ∈D λ∈D Hence (3.2) holds. Finally we will show (3.3) holds. Fix a ∈ D with |a| > 12 , we take the test functions: ha (z) = −

6(1 − |a|2 )3 3(1 − |a|2 )4 3(1 − |a|2 )2 + − (1 − a ¯z)2 (1 − a ¯z)3 (1 − a ¯z)4

(3.10)

for z ∈ D. It is easily proved that sup 12 1

+

sup

sup

2

|φ(w)|≤ 12

≤4

(1 − |w|2 )|u(w)(φ′ (w))2 | (1 − |φ(w)|2 )2

sup (1 − |w|2 )|

|φ(w)|> 12

|u(w)(φ′ (w))2 (φ(w))2 | 16 + sup (1 − |w|2 )|u(w)(φ′ (w))2 | (1 − |φ(w)|2 )2 9 |φ(w)|≤ 1 2

< ∞. Hence (3.3) holds. Conversely, suppose that (3.1), (3.2), and (3.2) hold. For f ∈ β, by Lemma 2.1, we have the

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following inequality: (1 − |z|2 )|(uCφ f )′′ (z)| =

( ) (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) f ′ (φ(z))

+ f ′′ (φ(z))(φ′ (z))2 u(z) + u′′ (z)f (φ(z))| ( ) ≤ (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) f ′ (φ(z))| +

(1 − |z|2 )|f ′′ (φ(z))(φ′ (z))2 u(z)| + (1 − |z|2 )|u′′ (z)f (φ(z))|

≤

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| b(f ) 1 − |φ(z)|2

(1 − |z|2 )|(φ′ (z))2 u(z)| e b(f ) + (1 − |z|2 )|u′′ (z)| log( )∥f ∥β 2 2 (1 − |φ(z)| ) 1 − |φ(z)|2 ≤ C∥f ∥β , +

8

and |u(0)f (φ(0))| + |u′ (0)f (φ(0))| + |u(0)f ′ (φ(0))φ′ (0)| ( e |u(0)φ′ (0)| ) ≤ (|u(0)| + |u′ (0)|) log( ) + ∥f ∥β . 1 − |φ(0)|2 1 − |φ(0)|2 This shows that uCφ is bounded. This completes the proof of Theorem 3.1. Corollary 3.1 Let φ be an analytic self-map of D. Then Cφ is a bounded operator from the Bloch space β to the Zygmund space Z if and only if

sup z∈D

(1 − |z|2 )|(φ′ (z))2 | (1 − |z|2 )|φ′′ (z)| < ∞ and sup < ∞. 2 2 (1 − |φ(z)| ) 1 − |φ(z)|2 z∈D

In the formulation of lemma, we use the notation Mu on H(D) defined by Mu f = uf for f ∈ H(D). Corollary 3.2 The pointwise multiplier Mu : β → Z is a bounded operator if and only if u = 0. Theorem 3.2 Let u be an analytic function on the unit disc D and φ an analytic self-map of D. Then uCφ is a compact operator from β to Z if and only if uCφ is a bounded operator and the following are satisfied:

lim

|φ(z)|→1−

lim

|φ(z)|→1−

(1 − |z|2 )|u′′ (z)| log

e = 0; 1 − |φ(z)|2

(3.11)

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| = 0; 1 − |φ(z)|2

(3.12)

(1 − |z|2 )|u(z)(φ′ (z))2 | = 0. (1 − |φ(z)|2 )2

(3.13)

lim

|φ(z)|→1−

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Proof Suppose that uCφ is compact from β to the Zygmund space Z. Let {zn } be a sequence in D such that |φ(zn )| → 1 as n → ∞. If such a sequence does not exist then (3.11), (3.12) and (3.13) are automatically satisfied. Without loss of generality we may suppose that |φ(zn )| > 21 for all n. We take the test functions fn (z) =

e 8 3 6 e e log2 − 2 log3 + 3 log4 . an an an 1 − |φ(zn )|2 1 − φ(zn )z 1 − φ(zn )z

(3.14)

e . By a directly calculation, we may easily prove that {fn } converges to 1 − |φ(zn )|2 0 uniformly on compact subsets of D and supn ∥fn ∥β ≤ C < ∞. Then {fn } is a bounded sequence in β which converges to 0 uniformly on compact subsets of D. Then limn→∞ ∥uCφ (fn )∥Z = 0 by Lemma 2.4. Note that

where an = log

fn (φ(zn )) = an , fn′ (φ(zn )) = 0, fn′′ (φ(zn )) = 0. It follows that ∥uCφ fn ∥Z ≥ (1 − |zn |2 )|(2u′ (zn )φ′ (zn ) + φ′′ (zn )u(zn ))fn′ (φ(zn )) +u(zn )fn′′ (φ(zn ))(φ′ (zn ))2 + u′′ (zn )fn (φ(zn ))| = (1 − |zn |2 )|u′′ (zn )| log

e . 1 − |φ(zn )|2

Then lim (1 − |zn |2 )|u′′ (zn )| log

n→∞

e = 0. 1 − |φ(zn )|2

Next, let gn (z) =

8(1 − |φ(zn )|2 )2 (1 − φ(zn )z)2

−

14(1 − |φ(zn )|2 )3 (1 − φ(zn )z)3

+

6(1 − |φ(zn )|2 )4 (1 − φ(zn )z)4

.

By a directly calculation we obtain that gn ⇒ 0 (n → ∞) on compact subsets of D and supn ∥gn ∥β ≤ C < ∞. Consequently, {gn } is a bounded sequence in β which converges to 0 uniformly on compact subsets of D. Then limn→∞ ∥uCφ (gn )∥Z = 0 by Lemma 2.4. Note that −2φ(zn ) gn (φ(zn )) ≡ 0, gn′′ (φ(zn )) ≡ 0 and gn′ (φ(zn )) = . 1 − |φ(zn )|2 It follows that ∥uCφ gn ∥Z ≥ (1 − |zn |2 )|(2u′ (zn )φ′ (zn ) + φ′′ (zn )u(zn ))gn′ (φ(zn )) +u(zn )gn′′ (φ(zn ))(φ′ (zn ))2 + u′′ (zn )gn (φ(zn ))| = 2(1 − |zn |2 )|(2u′ (zn )φ′ (zn ) + φ′′ (zn )u(zn ))

φ(zn ) 1 − |φ(zn )|2

|.

(1 − |zn |2 )|2u′ (zn )φ′ (zn ) + φ′′ (zn )u(zn )| = 0. n→∞ 1 − |φ(zn )|2 Finally, let

Then lim

hn (z) = −

3(1 − |φ(zn )|2 )2 (1 − φ(zn )z)2

+

6(1 − |φ(zn )|2 )3 (1 − φ(zn )z)3

−

3(1 − |φ(zn )|2 )4 (1 − φ(zn )z)4

.

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By a directly calculation we obtain that hn ⇒ 0 (n → ∞) on compact subsets of D and supn ∥hn ∥Z ≤ C < ∞. Consequently, {hn } is a bounded sequence in Z which converges to 0 uniformly on compact subsets of D. Then limn→∞ ∥uCφ (hn )∥Z = 0 by Lemma 2.4. Note that −6(φ(zn ))2 hn (φ(zn )) ≡ 0, h′n (φ(zn )) ≡ 0 and h′′n (φ(zn )) = . It follows that (1 − |φ(zn )|2 )2 ∥uCφ hn ∥Z ≥ (1 − |zn |2 )|(2u′ (zn )φ′ (zn ) + φ′′ (zn )u(zn ))h′n (φ(zn )) +u(zn )h′′n (φ(zn ))(φ′ (zn ))2 + u′′ (zn )hn (φ(zn ))| = 6(1 − |zn |2 )|u(zn )(φ′ (zn ))2 |

|φ(zn )|2 . (1 − |φ(zn )|2 )2

|u(zn )(φ′ (zn ))2 | = 0. The proof of the necessary is completed. n→∞ (1 − |φ(zn )|2 )2 Conversely, suppose that (3.11), (3.12), and (3.13) hold. Since uCφ is a bounded operator, by Theorem 3.1, we have Then lim (1 − |zn |2 )

M1 , sup (1 − |z|2 )|u′′ (z)| log z∈D

and M2 , sup z∈D

1 < ∞, 1 − |φ(z)|2

M3 , sup z∈D

(1 − |z|2 )|u(z)(φ′ (z))2 | < ∞, (1 − |φ(z)|2 )2

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| < ∞. 1 − |φ(z)|2

Let {fn } be a bounded sequence in β with ∥fn ∥β ≤ 1 and fn → 0 uniformly on compact subsets of D. We only prove lim ∥uCφ (fn )∥Z = 0 by Lemma 2.4. By the assumption, for any n→∞

ϵ > 0, there is a constant δ, 0 < δ < 1, such that δ < |φ(z)| < 1 implies (1 − |z|2 )|u(z)(φ′ (z))2 | < ϵ, (1 − |φ(z)|2 )2 and

(1 − |z|2 )|u′′ (z)| log

e < ϵ, 1 − |φ(z)|2

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| < ϵ. 1 − |φ(z)|2

Let K = {w ∈ D : |w| ≤ δ}. Noting that K is a compact subset of D, we get that

z(uCφ fn )

=

sup (1 − |z|2 )|(uCφ fn )′′ (z)| z∈D

≤

( ) sup (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) fn′ (φ(z))|

z∈D

+

sup (1 − |z|2 )|fn′′ (φ(z))(φ′ (z))2 u(z)| + sup (1 − |z|2 )|u′′ (z)fn (φ(z))|

z∈D

z∈D

( ) ≤ 10ϵ + sup (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) fn′ (φ(z))| |φ(z)|≤δ

+

sup (1 − |z|2 )|fn′′ (φ(z))(φ′ (z))2 u(z)| + sup (1 − |z|2 )|u′′ (z)fn (φ(z))|

|φ(z)|≤δ

|φ(z)|≤δ

≤ 10ϵ + M2 sup |fn′ (w)| + M3 sup |fn′′ (w)| + M1 sup |fn (w)|. w∈K

w∈K

w∈K

As n → ∞, ∥uCφ fn ∥Z → 0. Hence uCφ is compact. This completes the proof of Theorem 3.2. 8

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Corollary 3.3 Let φ be an analytic self-map of D. Then Cφ is a compact operator from the Bloch space β to the Zygmund space Z if and only if Cφ is bounded,

lim

|φ(z)|→1−

(1 − |z|2 )|(φ′ (z))2 | =0 (1 − |φ(z)|2 )2

and

lim

|φ(z)|→1−

(1 − |z|2 )|φ′′ (z)| = 0. 1 − |φ(z)|2

Theorem 3.3 Let u be an analytic function on the unit disc D, and φ an analytic self-map of D. Then uCφ : β0 → Z0 is a bounded operator if and only if u ∈ Z0 , (3.1), (3.2), and (3.3) hold, and the following are satisfied: lim (1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| = 0.

(3.15)

lim (1 − |z|2 )|u(z)(φ′ (z))2 | = 0;

(3.16)

|z|→1−

|z|→1−

Proof Suppose that uCφ is bounded from the little Bloch space β0 to the little Zygmund type spaces Z0 . Then u = uCφ 1 ∈ Z0 . Also uφ = uCφ z ∈ Z0 , thus (1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z) + φ(z)u′′ (z)| −→ 0 Since |φ| ≤ 1 and u ∈ Z0 , we have

(|z| → 1− ).

lim (1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| = 0. Hence (3.15)

|z|→1−

holds. Similarly, uCφ z 2 ∈ Z0 , then (1 − |z|2 )|4φ(z)φ′ (z)u′ (z) + φ2 (z)u′′ (z) + 2u(z)(φ(z)φ′′ (z) + (φ′ (z))2 )| −→ 0 By (3.15), |φ| ≤ 1 and u ∈ Z0 , we get that

(|z| → 1− ).

lim (1 − |z|2 )|u(z)(φ′ (z))2 | = 0, i. e. that (3.16)

|z|→1−

holds. On the other hand, from Lemma 2.3 and Theorem 3.1, we obtain that (3.1), (3.2), and (3.3) hold. |f (z)| → 0 as |z| → 1− by Conversely, for ∀f ∈ β0 , we have both (1 − |z|2 )2 |f ′′ (z)| → 0 and e ln 1−|z| 2 ϵ Lemma 2.2. Given ϵ > 0 there is a 0 < δ < 1 such that (1−|z|2 )|f ′ (z)| < , (1−|z|2 )2 |f ′′ (z)| < 3M2 ϵ |f (z)| ϵ and < for all z with δ < |z| < 1, where M1 , M2 , M3 are defined in above. e 3M3 log 1−|z| 3M 2 1 If |φ(z)| > δ, it follows that (1 − |z|2 )|(uCφ f )′′ (z)| =

(1 − |z|2 )|[2φ′ (z)u′ (z) + φ′′ (z)u(z)]f ′ (φ(z))

+ f ′′ (φ(z))(φ′ (z))2 u(z) + u′′ (z)f (φ(z))| ≤ (1 − |z|2 )|[2φ′ (z)u′ (z) + φ′′ (z)u(z)]f ′ (φ(z))| +

(1 − |z|2 )|f ′′ (φ(z))(φ′ (z))2 u(z)| + (1 − |z|2 )|u′′ (z)f (φ(z))|

≤ M2 (1 − |φ(z)|2 )|f ′ (φ(z))| + M3 (1 − |φ(z)|2 )2 |f ′′ (φ(z))| + M1
0. Next, noting that the proof of Theorem 3.1 and the fact that the functions given in (3.8) are in β0 and have norms bounded independently of a, we obtain that lim − (1 − |z|2 )|u′′ (z)| log

|z|→1

e = 0. 1 − |φ(z)|2

Similarly, noting that the functions given in (3.9) are in β0 and have norms bounded independently of a, we obtain that lim −

|z|→1

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| =0 1 − |φ(z)|2

(3.20)

for |φ(z)| > 12 . However, if |φ(z)| ≤ 12 , by (3.15), we easily have lim −

|z|→1

≤

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| 1 − |φ(z)|2

4 lim (1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| = 0. 3 |z|→1−

Thus (3.18) holds. Also, the third statement, that (3.19), is proved similarly. We omitted it here. This completes the proof of Theorem 4.2. Corollary 3.5 Let φ be an analytic self-map of D. Then Cφ is a compact operator from β0 to Z0 if and only if (1 − |z|2 )|(φ′ (z))2 | =0 lim − (1 − |φ(z)|2 )2 |z|→1 and lim −

|z|→1

(1 − |z|2 )|φ′′ (z)| = 0. 1 − |φ(z)|2

11

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References [1] Choe B.R., Koo H. , Smith W.: Composition operators on small spaces, Integr. equ. oper. Theory, 56(3), 357-380 (2006) [2] Contreras M.D., Hern´ andez-D´iaz A.G.: Weighted composition operators on Hardy spaces, J. Math. Anal. Appl. 263(1), 224-233(2001) [3] Cowen C.C., Maccluer B.D.: Composition Operator on Spaces of Analytic Functions, CRC Press, Boca Raton, (1995) [4] Cuckovic Z., Zhao R.: Weighted composition operators on the Bergman space, J. London Math. Soc. 70(2), 499511(2004) [5] Cuckovic Z., Zhao R.: Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Illinois J. Math. 51(2), 479-498(2007) [6] Duren P.: Theory of H p Spaces, Academic Press, New York, (1970) [7] Esmaeili K., Lindstr¨ om M., Weighted composition operators between Zygmund type spaces and their essential norms, Integr. Equ. Oper Theory, 75(4), 473-490 (2013) [8] Laitila J.: Weighted composition operators on BMOA, Comput. Methods Funct. Theory, 9(1), 27-46(2009) [9] Li S., Stevi´ c S.: Gerneralized composition operators on the Zygmund spaces and Bloch type spaces, J. Math Anal. Appl. 338(2), 1282-1295(2008) [10] Li S., Stevi´ c S.: Products of Volterra type operator and composition operator from H ∞ and Bloch spaces to Zygmund spaces, J. Math Anal. Appl. 345(1), 40-52(2008) [11] Li S., Stevi´ c S.: Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput. 206(2), 825-831(2008) [12] Madigan K., Matheson A.: Compact composition operators on the Bloch space, Trans, Amer. Math. soc. 347(7), 2679-2687(1995) [13] Madigan K.: Composition operators on analytic Lipschitz spaces, Proc. Amer. Math. Soc. 119(2), 465-473(1993) [14] Ohnoand S., Zhao R.: Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc. 63(2), 177-185(2001) [15] Ohno S., Stroethoff K., Zhao R.: Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33(1), 191-215(2003) [16] Sharma A. K.: Products of multiplication, composition and differentiation between weighted Bergman-Nevanlinna and Bloch-type spaces, Turk. J. Math. 35(2), 275-291(2011) [17] Sanatpour A. H., Hassanlou M.: Essential norms of weighted composition operators between Zygmund-type spaces and Bloch-type spaces, Turk. J. Math. 38(5), 872-882 (2014) [18] Smith W.: Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348(6), 23312348(1996) [19] Stevi´ c S.: Weighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ball, Appl. Math. Comput. 217(5), 1939-1943(2010) [20] Stevi´ c S., Sharma A.K.: Essential norm of composition operators between weighted Hardy spaces, Appl. Math. Comput. 217(13), 6192-6197(2011) [21] Ye S., Hu Q.: Weighted composition operators on the Zygmund space, Abstr. Appl. Anal.2012, Article ID 462482(2012) [22] Ye S., Zhuo Z.: Weighted composition operators from Hardy to Zygmund type spaces, Abstr. Appl. Anal. 2013, Article ID 365286(2013) [23] Ye S.: A weighted composition operators on the logarithmic Bloch space, Bull. Korean Math. Soc. 47(3), 527-540 (2010) [24] Ye S.: A weighted composition operator between different weighted Bloch-type spaces, Acta Math. Sinica (Chin. Ser.), 50(4), 927-942(2007)(in Chinese) [25] Ye S.: Weighted composition operators from F (p, q, s) into logarithmic Bloch space, J. Korean Math. Soc. 45(4), 977-991 (2008) [26] Ye S.: Weighted composition operators between the α-Bloch spaces and the little logarithmic Bloch, J. Comput. Anal. Appl. 11(3), 443-450 (2009) [27] Yoneda R.: The composition operators on weighted Bloch space, Arch. Math. 78(4), 310-317 (2002) [28] Zhu K.: Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23(3), 1143-1177 (1993) [29] Zygmund A.: Trigonometric Series, Cambridge, (1959)

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Approximate homomorphisms and derivations on non-Archimedean Lie JC ∗ -algebras

Javad Shokri1 and Dong Yun Shin2∗ 1 2

Department of Mathematics, Urmia University, P. O. Box 165, Urmia, Iran

Department of Mathematics, University of Seoul, Seoul 02504, Republic of Korea

Abstract. In this paper, by using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms in non-Archimedean Lie JC ∗ -algebras and derivations on non-Archimedean Lie JC ∗ -algebras associated with the following additive mapping: n  X k X k=2

k+1 X

i1 =2 i2 =i1 +1

+f

n X

n X

···

n X

  f

in−k+1 =in−k+1

xi −

i=1,i6=i1 ,··· ,in−k+1

n−k+1 X

xir



r=1

 xi = 2n−1 f (x1 )

i=1

for a fixed positive integer n with n > 2.

1. Introduction In 1896, Hensel [4] introduced a field with a valuation in which does not have the Archimedean property. Let K be a field. A non-Archimedean absolute value onK is a function | · | : K → [0, +∞) such that, for any a, b ∈ K, the following conditions are satisfying (i) |a| > 0 and equality holds if and only if a = 0, (ii) |ab| = |a||b|, (iii) |a + b| 6 max{|a|, |b|} (the strict triangle inequality). Note that |1| = | − 1| = 1 and |n| 6 1 for each integer n. We always assume, in addition, that | · | is non-trival, i.e., there exists an a0 6= 0, 1. A function k.k : X → [0, ∞) is called a non-Archimedean norm if it satisfies the following conditions: (i) kxk = 0 if and only if x = 0; (ii) for any r ∈ K, x ∈ X, krxk = |r|kxk; (iii) the strong triangle inequality holds, namely, kx + yk 6 max{kxk, kyk}

(x, y ∈ X).

Then (X, k · k) is called a non-Archimedean normed space. From the fact that kxn − xm k 6 max{knn − xm k : m 6 j 6 n − 1}

(n > m),

0

2010 Mathematics Subject Classification: 39B52, 39B72, 46L05, 47H10, 46B03. Keywords: Hyers-Ulam stability; additive functional equation; fixed point; non-Archimedean space; homomorphisms in a non-Archimedean Lie JC ∗ -algebras; derivations in a non-Archimedean Lie JC ∗ -algebras. ∗ Corresponding author. 0 E-mail:1 [email protected], 2 [email protected] 0

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holds, a sequence {xn } is Cauchy if and only if {xn −xm } converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent. A non-Archimedean Banach algebra is a complete non-Archimedean algebra A which satisfies kabk 6 kak · kbk for all a, b ∈ A. For more detailed definitions of non-Archimedean Banach algebras, we refer the reader to [15]. If U is a non-Archimedean Banach algebra, then an involution on U is mapping t → t∗ from U into U which satisfies (i) t∗∗ = t for t ∈ U; ¯ ∗; (ii) (αs + βt)∗ = α ¯ s∗ + βt (iii) (st)∗ = t∗ s∗ for all s, t ∈ U. If, in addition, kt∗ tk = ktk2 for t ∈ U, then U is a non-Archimedean C ∗ -algebra. The stability problem of functional equations originated from a question of Ulam [16] concerning the stability of group homomorphisms: Let (G1 , ∗) be a group and let (G2 , ) be a metric group (a metric is defined on a set with group property) with the metric d(., .). Given  > 0, does there exist a δ() > 0 such that if a mapping h : G1 → G2 satisfies the inequality d(h(x ∗ y), h(x) ∗ h(y)) < δ for all x, y ∈ G1 , then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) <  for all x ∈ G1 ? If the answer is affirmative, we would say that the equation of homomorphism h(x ∗ y) = h(x) ∗ h(y) is stable (see also [3, 5, 9, 10, 12, 13, 14]). For explicitly later use, we recall a fundamental result in fixed point theory. Theorem 1.1. (The fixed point alternative theorem [2]) Let (Ω, d) be a complete generalized metric space and J : Ω → Ω be a strictly contractive mapping with Lipschitz constant 0 < L < 1, that is, d(Jx, Jy) 6 Ld(x, y),

x, y ∈ Ω.

Then, for each given x ∈ Ω, either d(J n x, J n+1 x) = ∞,

∀n > 0,

or there exists a positive integer n0 such that (1) (2) (3) (4)

d(J n x, J n+1 x) < ∞, ∀n > n0 ; the sequence {J n x} converges to a fixed point y ∗ of J; y ∗ is the unique fixed point of J in the set ∆ = {y ∈ Ω : d(J n0 x, y) < ∞}; 1 d(y, y ∗ ) 6 1−L d(y, Jy) for all y ∈ ∆.

A non-Archimedean C ∗ -algebra C, endowed with the Lie product [x, y] := xy−yx and endowed 2 with anticommutator product (Jordan product) x◦y := xy+yx on C, is called a non-Archimedean 2 Lie JC ∗ -algebra (see [6, 7, 8]). Jordan algebras as coordinates for Lie algebras were created to illuminate a particular aspect of physics, quantum-mechanical observables, but turned out to have illuminating connections with many areas of mathematics. In this paper, we prove the Hyers-Ulam stability of homomorphisms and derivations in nonArchimedean Lie JC ∗ -algebras associated with the following additive functional equation:

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n X k k+1 X X k=2

n X

···

i1 =2 i2 =i1 +1

in−k+1 =in−k+1

+f

n X

n X

  f

xi −

i=1,i6=i1 ,··· ,in−k+1

n−k+1 X

xir



r=1



xi = 2n−1 f (x1 )

(1.1)

i=1

for a fixed positive integer n with n > 2. 2. Stability of homomorphisms in non-Archimedean Lie JC ∗ -algebras Definition 2.1. [7] Let A and B be non-Archimedean Lie JC ∗ -algebras. A C-linear mapping H : A → B is called a (non-Archimedean Lie JC ∗ -algebra) homomorphism if H satisfies H([x, y]) = [H(x), H(y)], H(x ◦ y) = H(x) ◦ h(y), H(x∗ ) = H(x)∗ for all x, y ∈ A. Throughout this section, assume that A and B are two non-Archimedean Lie JC ∗ -algebras, respectively, with norm k · kA and k · kB . For a given mapping f : A → B, we define Dµ f (x1 , · · · , xn ) :=

n X k k+1 X X k=2

i1 =2 i2 =i1 +1

+f

n X

···

n X

n X

  f

in−k+1 =in−k+1

i=1,i6=i1 ,··· ,in−k+1

µxi −

n−k+1 X

µxir



r=1

 µxi − 2n−1 f (µx1 )

i=1

T1

for all µ ∈ := {λ ∈ C : |λ| = 1} and all x1 , · · · , xn ∈ A. We recall the following needed lemmas in this paper. Lemma 2.2. [11] Let V and W be linear spaces and f : V → W be an additive mapping such that f (µx) = µf (x) for all x ∈ V and µ ∈ T1 . Then the mapping f is C-linear. Lemma 2.3. [7] A mapping f :→ A → B with f (0) = 0 satisfies the functional equation (1.1) if and only if f : A → B is additive. We prove the Hyers-Ulam stability of homomorphisms in non-Archimedean Lie JC ∗ -algebras for the functional equation Dµ f (x1 , · · · , xn ) = 0. Theorem 2.4. Let f : A → B be a mapping for which there are functions ϕ : An → [0, ∞), ψ : A2 → [0, ∞), and η : A → [0, ∞) such that |2| < 1 is far from zero and 1 lim ϕ(2m x1 , 2m x2 , · · · , 2m xn ) = 0, (2.1) m→∞ |2|m lim

m→∞

1 ψ(2m x, 2m y) = 0, |2|2m

308

(2.2)

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1 η(2m x) = 0, m→∞ |2|m

(2.3)

kDµ f (x1 , · · · , xn )kB 6 ϕ(x1 , · · · , xn ),

(2.4)

kf ([x, y]) − [f (x), f (y)]kB 6 ψ(x, y),

(2.5)

kf (x ◦ y) − f (x) ◦ f (y)kB 6 ψ(x, y),

(2.6)

kf (x∗ ) − f (x)∗ kB 6 η(x),

(2.7)

lim

T1 .

for all x, y, x1 , · · · , xn ∈ A and µ ∈ If there exists a constant 0 < L < 1 such that x1 x2 xn ϕ(x1 , x2 , · · · , xn ) 6 αLϕ( 2 , 2 , · · · , 2 ) for all x1 , x2 , · · · , xn ∈ A, where α = |2|n−1 , then there exists a unique homomorphism H : A → B such that kf (x) − H(x)k6

x x L ϕ( , , 0, · · · , 0) 1−L 2 2

(2.8)

for all x ∈ A. Proof. Let µ = 1. Using the following relation n−k X n − k  X n − k  n−k = 2n−k = 1+ k k

(2.9)

k=0

k=1

for all n > k and putting x1 = x2 = x and x3 = x4 = · · · = xn = 0 in (2.4), we obtain α k f (2x) − αf (x)kB 6 ϕ(x, x, 0, · · · , 0) 2 for all x ∈ A. So x x  1 1 k f (2x) − f (x)kB 6 ϕ(x, x, 0, · · · , 0) 6 Lϕ , , 0, · · · , 0 (2.10) 2 α 2 2 for all x ∈ A. Let define Ω := {g : A → B} and introduce a generalized metric on Ω as follows x x  d(g, h) = inf{k ∈ (0, ∞) : kg(x) − h(x)kB < kϕ , , 0, · · · , 0 , ∀x ∈ A}. 2 2 It is easy to show that (Ω, d) is a generalized complete metric space (see [1]). 1 Now we consider the function J : Ω → Ω define by Jg(x) = |2| g(2x) for all x ∈ A and g ∈ Ω. Let for all g, h ∈ Ω and an arbitrary constant k ∈ [0, ∞) with d(x, y) 6 k, we have x x  kg(x) − h(x)kB 6 kϕ , , 0, · · · , 0 2 2 for all x ∈ A. Then we can write  1 k αkL  x x kJg(x) − Jh(x)kB = kg(2x) − h(2x)kB 6 ϕ(x, x, 0, · · · , 0) 6 ϕ , , 0, · · · , 0 |2| |2| |2| 2 2 α for all x ∈ A. So we conclude that d(Jg, Jh) 6 |2| Ld(g, h) for all g, h ∈ Ω. It follows from (2.9) that d(Jf, f ) 6 L, that is, J is a self-function of Ω with the Lipchitz constant L. Therefore, from Theorem 1.1, there eists a fixed point H of J set Ω1 = {h ∈ X : d(f, h) < ∞} such that

H(x) = lim

m→∞

309

1 f (2m x) |2|m

(2.11)

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Homomorphisms in non-Archimedean Lie JC ∗ -algebras

for all x ∈ A, since limm→∞ d(J n f, H) = 0. Also 2H( x2 ) = H(x) for all x ∈ A. Thus H : A → B is the unique fixed point of J in Ω1 such that d(H, f ) 6

1 L d(Jf, f ) 6 , 1−L 1−L

i.e., H satisfies (2.8) for all x ∈ A. It follows from the definition of H, (2.1) and (2.4) that n X k k+1 X X k=2

i1 =2 i2 =i1 +1

+H

n X

n X

···

  H

in−k+1 =in−k+1

n X i=1,i6=i1 ,··· ,in−k+1

xi −

n−k+1 X

x ir



r=1

 xi = 2n−1 H(x1 )

i=1

for all x1 , x2 , · · · , xn ∈ A. Since H(0) = 0, by Lemma 2.3, the mapping H is additive. Put x1 = x and x2 = x3 = · · · = 0 in (2.4). It follows from (2.9) that kf (µx) − µf (x)k 6

1 ϕ(x, 0, · · · , 0) α

(2.12)

for all x ∈ A and all µ ∈ T1 . Also we conclude k

1 1 (f (µ2m x) − µf (2m x))kB 6 ϕ(2m x, 0, · · · , 0) m 2 α|2|m

for all x ∈ A and all µ ∈ T1 . The right hand side of the above inequality tends to zero as m → ∞, and so we obtain 1 1 f (µ2m x) = lim µf (2m x) = µH(x) m m→∞ |2| m→∞ |2|m

H(µx) = lim

for all x ∈ A and all µ ∈ T1 . Hence by Lemma 2.2, the mapping H : A → B is C-linear. It follows from (2.2), (2.5), (2.6) and (2.11) that 1 kf ([2m x, 2m y]) − [f (2m x), f (2m y)]kB |2|2m 1 6 lim ψ(2m x, 2m y) = 0 m→∞ |2|2m

kH([x, y]) − [H(x), H(y)]kB = lim

m→∞

and 1 kf (2m x ◦ 2m y) − f (2m x) ◦ f (2m y)kB |2|2m 1 6 lim ψ(2m x, 2m y) = 0 m→∞ |2|2m

kH(x ◦ y) − H(x) ◦ H(y)kB = lim

m→∞

for all x, y ∈ A. So H([x, y]) = [H(x), H(y) and H(x ◦ y) = H(x) ◦ H(y) for all x, y ∈ A. Similarly, by (2.3), (2.7) and (2.11), we have kH(x∗ ) − H(x)∗ kB = lim

m→∞

1 1 kf (2m x∗ ) − f (2m x)∗ kB 6 lim η(2m x) = 0 m→∞ |2|m |2|m

310

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and so H(x∗ ) = H(x)∗ for all x, y ∈ A. Thus H : A → B is the desired homomorphism satisfying (2.8).  Corollary 2.5. Let r > 1 and θ be nonnegative real number, and let f : A → B be a mapping such that kDµ f (x1 , x2 , · · · , xn )kB 6 θ(k|x1 krA + k|x2 krA + · · · + k|xn krA ), kf ([x, y]) − [f (x), f (y)]kB 6 θ · k|xkrA · k|ykrA , kf (x ◦ y) − f (x) ◦ f (y)kB 6 θ · k|xkrA · k|ykrA , kf (x∗ ) − f (x)∗ kB 6 θ · k|xkrA , for all µ ∈ T1 and x, y, x1 , · · · , xn ∈ A. Then there exists a unique homomorphism H : A → B such that |2|θ kxkrA kf (x) − H(x)kB 6 |2| − |2|r for all x ∈ A. Proof. The proof follows from Theorem 2.4 by taking ϕ(x1 , x2 , · · · , xn ) := θ(k|x1 krA + k|x2 krA + · · · + k|xn krA ), ψ(x, y) := θ · k|xkrA · k|ykrA , η(x) := θ · k|xkrA for all x, y, x1 , · · · , xn ∈ A and L = |2|r−1 .



3. Stability of derivations on non-Archimedean Lie JC ∗ -algebras Definition 3.1. [7] Let A be a non-Archimedean Lie JC ∗ -algebra. A C-linear mapping δ : A → A is called a (non-Archimedean Lie JC ∗ -algebra) derivation if δ satisfies δ([x, y]) = [δ(x), y] + [x, δ(y)], δ(x ◦ y) = δ(x) ◦ y + x ◦ δ(y), δ(x∗ ) = δ(x)∗ for all x ∈ A. Throughout this section, assume that A is a non-Archimedean Lie JC ∗ -algebra with norm k · kA . We prove the Hyers-Ulam stability of derivation on non-Archimedean Lie JC ∗ -algebras for the functional equation Dµ f (x1 , · · · , xn ) = 0. Theorem 3.2. Let f : A → A be a mapping for which there are function ϕ : An → [0, ∞), ψ : A2 → [0, ∞) and η : A → [0, ∞) such that (2.1), (2.2), (2.3). (2.4) and (2.7) hold and kf ([x, y]) − [f (x), y]0[x, f (y)]kA 6 ψ(x, y),

(3.1)

kf (x ◦ y) − f (x) ◦ y − x ◦ f (y)kA 6 ψ(x, y)

(3.2)

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Homomorphisms in non-Archimedean Lie JC ∗ -algebras

for all x, y ∈ A. If there exists a constant 0 < L < 1 such that ϕ(x1 , x2 , · · · , xn ) 6 αLϕ( x21 , x22 , · · · , x2n ) for all x1 , x2 , · · · , xn ∈ A, where α = |2|n−1 , then there exists a unique derivation δ : A → A such that x x  L kf (x) − δ(x)k6 ϕ , , 0, · · · , 0 (3.3) 1−L 2 2 for all x ∈ A. Proof. By the same reasoning as in the proof of Theorem 2.4, there exists a unique C-linear mapping δ : A → A satisfying in the desired inequality (3.3) and the mapping δ : A → A is defined by 1 δ(x) = lim f (2m x) (3.4) m→∞ |2|m for all x ∈ A. It follows from (2.2), (3.1), (3.3) and (3.4) that kδ([x, y]) − [δ(x), y] − [x, δ(y)]kA 1 kf ([2m x, 2m y]) − [f (2m x), 2m y] − [2m x, f (2m y)]kA = lim m→∞ |2|2m 1 ψ(2m x, 2m y) = 0 6 lim m→∞ |2|2m and kδ(x ◦ y) − δ(x) ◦ y − x ◦ δ(y)kA 1 = lim kf (2m x ◦ 2m y) − f (2m x) ◦ 2m y − 2m x ◦ f (2m y)kA m→∞ |2|2m 1 6 lim ψ(2m x, 2m y) = 0 m→∞ |2|2m for all x, y ∈ A. So δ([x, y]) = [δ(x), y] + [x, δ(y)], δ(x ◦ y) = δ(x) ◦ y + x ◦ δ(y) for all x, y ∈ A. Similarly, as in the proof of Theorem 2.4, one can show δ(x∗ ) = δ(x)∗ for all x ∈ A. Therefore, δ : A → A is a non-Archimedean Lie JC ∗ -algebra derivation satisfying (3.4).  Corollary 3.3. Let r > 1 and θ be nonnegative and real number, and let f : A → A be a mapping such that kDµ f (x1 , x2 , · · · , xn )kB 6 θ(kx1 krA + kx2 krA + · · · , kxn krA ), kf ([x, y]) − [f (x), y] − [x, f (y)]kB 6 θ · kxkrA · kykrA , kf (x ◦ y) − f (x) ◦ y − x ◦ f (y)kB 6 θ · kxkrA · kykrA , kf (x∗ ) − f (x)∗ kB 6 θ · kxkrA

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for all µ ∈ T1 and x, y, x1 , · · · , xn ∈ A. Then there exists a unique homomorphism H : A → A such that |2|θ kf (x) − δ(x)kB 6 kxkrA |2| − |2|r for all x ∈ A. Proof. The proof follows from Theorem 3.2 by taking ϕ(x1 , x2 , · · · , xn ) := θ.(kx1 krA + kx2 krA + · · · + kxn krA ), ψ(x, y) := θ.(kxkrA .kykrA ), η(x) := θ.kxkrA for all x, y, x1 , · · · , xn ∈ A and L = |2|r−1 .



References [1] Cˇ adariu and V. Radu, Fixed points and stability Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), Art. No. 1. [2] 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. [3] M. Eshaghi Gordji, A. Rahimi, C. Park and D. Shin, Ternary Jordan bi-homomorphisms in Banach Lie triple systems, J. Comput. Anal. Appl. 21 (2016), 1040–1045. [4] K. Hensel, Uber eine neue Begrundung der Theorie der algebraischen Zahlen. Jahres, Deutsch. Math. Verein 6 (1897), 83–88. [5] E. Movahednia, S. M. S. M. Mosadegh, C. Park and D. Shin, Stability of a lattice preserving functional equation on Riesz space: fixed point alternative, J. Comput. Anal. Appl. 21 (2016), 83–89. [6] C. Park, Lie *-homomorphisms between Lie C ∗ -algebras and Lie ∗-derivations on Lie C ∗ -algebras, J. Math.Anal. Appl., 293 (2004), 419–434. [7] C. Park, Homomorphisms between Lie JC ∗ -algebras and Cauchy-Rassias stability of Lie JC ∗ -algebra derivations, J. Lie Theory 15 (2005), 393–414. [8] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [9] C. Park, Stability of ternary quadratic derivation on ternary Banach algebras: revisited, J. Comput. Anal. Appl. 20 (2016), 21–23. [10] W. Park and J. Bae, Approximate quadratic forms on restricted domains, J. Comput. Anal. Appl. 20 (2016), 388–410. [11] H. Khodaei and Th. M. Rassias, Approximately generalized additive functional in several variables, Int. J. Nonlinear Anal. Appl. 1 (2010), 22–41. [12] J. M. Rassias, On approximation of approximately linear mapping by linear mappings, J. Func. Anal. 46 (1982), 126–130. [13] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [14] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [15] N. Shilkret, Non-Archimedean Banach Algebras, Ph.D. thesis, Polytechnic University, 1968, ProQuest LLC. [16] S. M. Ulam, Problems in Modern Mathematics, Science ed., John Wiley & Sons, New York, 1940.

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ON DISTRIBUTION AND PROBABILITY DENSITY FUNCTIONS OF ORDER STATISTICS ARISING FROM INDEPENDENT BUT NOT NECESSARILY IDENTICALLY DISTRIBUTED RANDOM VECTORS

1,2

M. GÜNGÖR1 and Y. BULUT2 Department of Econometrics, Inonu University, 44280, Malatya, TURKEY. 1 [email protected] and [email protected] ABSTRACT

In this study, joint probability density and distribution functions of any d order statistics of innid continuous random vectors are expressed. Then, some results connecting distributions of order statistics of innid random vectors to that of order statistics of iid random vectors are given. Keywords: Order Statistics, Distribution Function, Probability Density Function, Continuous Random Variable. MSC 2010: 62G30, 62E15.

1. Introduction Several identities and recurrence relations for probability density function (pdf) and distribution function (df) of order statistics of independent and identically distributed (iid) random variables were established by numerous authors including (Arnold et al., 1992; Balasubramanian, Beg, 2003; David, 1981; Reiss, 1989). Furthermore, (Arnold et al., 1992; David, 1981; Gan, Bain, 1995; Khatri, 1962) obtained the probability function (pf) and df of order statistics of iid random variables from a discrete parent. (Corley, 1984) defined a multivariate generalization of classical order statistics for random samples from a continuous multivariate distribution. (Goldie, Maller, 1999) derived expressions for generalized joint densities of order statistics of iid random variables in terms of Radon-Nikodym derivatives with respect to product measures based on df. (Guilbaud, 1982) expressed the probability of the functions of independent but not necessarily identically distributed (innid) random vectors as a linear combination of probabilities of the functions of iid random vectors and thus also for order statistics of random variables. (Cao, West, 1997) obtained recurrence relationships among the distribution functions of order statistics arising from innid random variables. (Vaughan, Venables, 1972) derived the joint pdf

and marginal pdf

of order statistics of innid random variables by means of

permanents. (Balakrishnan, 2007; Bapat, Beg, 1989) obtained the joint pdf and df of order statistics of innid random variables by means of permanents. (Childs, Balakrishnan, 2006) obtained, using multinomial arguments, the pdf of independent

random

variable

to

the 314

original

X r:n+1 (1 ≤ r ≤ n+1) by adding another n

variables

X 1 , X 2 ,..., X n .

Also,

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(Balasubramanian et al.,1994) established the identities satisfied by distributions of order statistics from non-independent non-identical variables through operator methods based on the difference and differential operators. In this paper, joint df and pdf of order statistics from innid continuous random vectors are obtained. As far as we know, these approaches have not been considered in the framework of order statistics from innid continuous random vectors. From now on, subscripts and superscripts are defined in first place in which they are used and these definitions will be valid unless they are redefined. Consider x = ( x (1) , x ( 2) ,..., x (b ) ) and y = ( y (1) , y ( 2) ,..., y (b ) ) , then it can be written as; x ≤ y if x ( v ) ≤ y ( v ) ( v=1, 2, …, b ) and x + y = ( x (1) + y (1) , x ( 2) + y ( 2) ,..., x ( b ) + y (b ) ) . Let ξ i = (ξ i(1) , ξ i( 2) ,..., ξ i( b ) ) (i=1,2,…,n) be n innid continuous random vectors which components of ξi are independent.

X r(:vn) = Z r:n ( ξ1(v ) , ξ 2(v ) ,..., ξ n(v ) )

(1.1)

is stated as rth order statistic of vth components of ξ1 , ξ 2 , …, ξ n . From (1.1), ordered values of vth components of ξ1 , ξ 2 , …, ξ n are expressed as X 1(:nv ) ≤ X 2(:vn) ≤ ... ≤ X n(:vn) .

(1.2)

From (1.2), we can write X r:n = ( X r(:1n) , X r(:2n) ,..., X r(:bn) )

( 1 ≤ r ≤ n ).

Also, x w = ( xw(1) , xw( 2) , ... , xw(b ) ) , xw( v ) ∈ R ( w = 1,2,...,d; d = 1,2,...,n ). Let Fi and f i be df and pdf of ξ i( v ) , respectively. Moreover, X 1(:nv ),s , X 2(:vn),s ,..., X n(:vn),s are order statistics of iid continuous random variables with df F s and pdf f s , respectively, defined by

Fs =

1 ns

∑F

1 ns

∑f

(1.3)

i

i∈s

and fs =

i

.

(1.4)

i∈s

Here, s is a subset of integers {1, 2,…, n} with ns ≥ 1 elements.

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In follows, df and pdf of X r1:n , X r2:n ,..., X rd :n ( 1 ≤ r1 < r2 < ... < rd ≤ n ) are given. Let

X ( v ) = ( X r(1v:n) , X r(2v:n) ,..., X r(dv:n) ) and x ( v ) = ( x1( v ) , x2(v ) ,..., xd( v ) ) . For notational convenience we write

∑∑

and

n ,...,m3 ,m2

∑

n

(−1) ∑ κ

instead of

md ,...,m2 ,m1

n −κ

=1

κn

∑ n! κ

n

and

md =rd

ns =

m3

m2

∑ ... ∑ ∑

in the expressions

m2 =r2 m1 = r1

below, respectively.

2. Distribution function of order statistics from innid random vectors

In this section, df of X r1:n , X r2:n ,..., X rd :n and its results are given. The results connect df of order statistics of innid random vectors to that of order statistics of iid random vectors using (1.3). Now, we give the following theorem for establish joint df of d order statistics of innid continuous random vectors. Theorem 2.1. b

Fr1 , r2 ,..., rd :n ( x1 , x 2 ,..., x d ) = ∏{ v =1

n ,..., m3 , m 2

∑

m d ,..., m 2 , m1

d +1

C ∑∏

mw

∏[ F

P w =1 l = m w−1 +1

jl

( xw( v ) ) − Fj l ( xw( v−)1 )]} ,

d +1

x1 < x 2 < ... < x d , where C = [∏ (mw − mw−1 )!]−1 , m0 = 0 , md +1 = n ,

∑

w=1

(2.1)

denotes sum over all

P

n! permutations ( j1 , j2 ,..., jn ) of (1,2,…,n), F jl ( x0(v ) ) = 0 and F jl ( x (dv+1) ) = 1 . Proof. It can be written Fr1 ,r2 ,..., rd :n ( x1 , x 2 ,..., x d ) = P{X r1:n ≤ x1 , X r2:n ≤ x 2 ,..., X rd :n ≤ x d }

= P{X (1) ≤ x (1) , X ( 2 ) ≤ x ( 2) ,..., X ( b ) ≤ x (b ) } b

= ∏ P{X ( v ) ≤ x ( v ) } v =1 b

= ∏ P{ X r(1v:n) ≤ x1( v ) , X r(2v:n) ≤ x2( v ) ,..., X r(dv:n) ≤ xd( v ) } .

(2.2)

v =1

(2.2) can be expressed as Fr1 , r2 ,..., rd :n ( x1 , x 2 ,..., x d ) = b

v =1



n ,...,m3 ,m2

∏{ ∑

md ,...,m2 ,m1

C

m1

∑  ∏ F P



l =1

jl

n   m2 ( x1(v ) )  [ F jl ( x2(v ) ) − F jl ( x1( v ) )]  ... [1 − F jl ( xd(v ) )]} .   l =m +1  l =m1 +1 d 

∏

∏

Thus, (2.1) is obtained. The approach in Theorem 2.1 can also be adapted to Theorem 2.2 for iid case. 3 316

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Theorem 2.2.

Fr1 , r2 ,..., rd :n ( x1 , x 2 ,..., x d ) =

b

n ,...,m3 ,m2

v =1

md ,...,m2 ,m1

∏{∑∑ ∑

d +1

n!C ∏ [ F s ( xw( v ) ) − F s ( xw( v−)1 )]mw −mw−1 }.

(2.3)

w=1

Proof. (2.2) can be expressed as b

Fr1 , r2 ,..., rd :n ( x1 , x 2 ,..., x d ) =

∏ [∑∑ v =1

P{ X r(1v:n),s ≤ x1( v ) , X r(2v:n),s ≤ x2( v ) ,..., X r(dv:n),s ≤ xd( v ) }] .

(2.4)

(2.3) is obtained from (2.1) and (2.4). We now obtain the following three results for df of order statistics of innid continuous random vectors from the above theorems. Result 2.1.

 m1 1  ∏ F jl ( x1(1) ) ∑ ∑ m ! ( n − m )! m1 = r1 P  l =1 1 1 n

Fr1 :n ( x1(1) ) =

(

n

n

∑∑ ∑  m [ F

=

s

) ∏[1 − F n

l = m1 +1

jl

( x1(1) )]

( x1(1) )]m1 [1 − F s ( x1(1) )]n − m1 .

(2.5)

1

m1 = r1

Proof. In (2.1) and (2.3), if b = 1 , d = 1 , (2.5) is obtained.

In addition, Fr1 :n ( x1(1) ) =

n

1 − m1 )! m ! ( n m1 = r1 1

∑

n

=

m1

∑ ∏F P



1

l =1

m1

∑ m !(n − m )! ∑  ∏ F 1

m1 = r1

where

   

∑ nτ =n−t

1

P



l =1

jl

jl

 n ( x1(1) )  [1 − F j l ( x1(1) )]  l = m1 +1

∏

 n ( x1(1) )  (−1) n − t  nτ = n − t t = m1

∑

n −t

∑ ∏ Fτ ( x l

(1) 1 )

,

l =1

 n − m1   subsets τ = {τ 1 , τ 2 ,..., τ n−t } of { jm1 +1 , jm1+ 2 ,..., jn } . denotes sum over all   n−t 

Result 2.2. F1:n ( x1(1) ) = 1 −

=

1 n!

n

∑∏[1 − F P

jl

( x1(1) )]

l =1

∑∑[1 − (1 − F

s

( x1(1) )) n ] .

(2.6)

Proof. In (2.5), if r1 = 1 , (2.6) is obtained.

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Result 2.3. Fn:n ( x1(1) ) = =

1 n!

n

∑∏ F P

jl

( x1(1) )

l =1

∑∑ [ F

s

( x1(1) )]n .

(2.7)

Proof. In (2.5), if r1 = n , (2.7) is obtained.

3. Probability density function of order statistics from innid random vectors

In this section, pdf of X r1:n , X r2:n ,..., X rd :n and its results are given. The results connect pdf of order statistics of innid random vectors to that of order statistics of iid random vectors using (1.3) and (1.4). Joint pdf of d order statistics of innid continuous random vectors is expressed in the following theorem. Theorem 3.1.  d +1

b

 d [ F jl ( xw( v ) ) − F j l ( xw( v−)1 )]  f ( x ( v ) )},  w =1 l = r +1  w =1 j rw w w−1  

f r1 , r2 ,..., rd :n ( x1 , x 2 ,..., x d ) = ∏ {D ∑  ∏ v =1

P

rw −1

∏

∏

(3.1)

d +1

x1 < x 2 < ... < x d , where D = [∏ ( rw − rw−1 − 1)!]−1 , r0 = 0 and rd +1 = n + 1 . w=1

Proof. Let δ x w = (δ xw(1) , δ xw( 2) ,..., δ xw(b ) ) and δ x ( v ) = (δ x1(v ) , δ x2( v ) ,..., δ xd(v ) ) .

Consider {x 1 < X r1:n ≤ x1 + δ x1 , x 2 < X r2:n ≤ x 2 + δ x 2 ,..., x d < X rd :n ≤ x d + δ x d } .

It can be written P{x 1 < X r1:n ≤ x1 + δ x 1 , x 2 < X r2:n ≤ x 2 + δ x 2 ,..., x d < X rd :n ≤ x d + δ x d }

= P{x (1) < X (1) ≤ x (1) + δ x (1) , x (2) < X (2) ≤ x (2) + δ x (2) ,..., x (b) < X (b) ≤ x (b) + δ x (b) } b

= ∏ P{x (v) < X (v) ≤ x (v) + δ x (v) } v =1 b

= ∏ P{x1(v) < X r(v) ≤ x1(v) + δ x1(v) , x2(v) < X r(v) ≤ x2(v) + δ x2(v) ,..., xd(v) < X r(v) ≤ xd(v) + δ xd(v) }. 1:n 2 :n d :n

(3.2)

v =1

b

Dividing (3.2) by

d

∏∏ δ x

(v) w

and then letting δ x1( v ) , δ x2( v ) ,..., δ xd(v ) tend to zero, we obtain

v =1 w=1

b

f r1 ,r2 ,...,rd :n ( x 1 , x 2 ,..., x d ) = ∏ {D∑ F j1 ( x1( v ) )...F jr1 −1 ( x1( v ) ) f jr1 ( x1( v ) )[ F jr1 +1 ( x2( v ) ) − F jr1 +1 ( x1( v ) )] v =1

P

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...[ F jr −1 ( x2( v ) ) − F jr −1 ( x1( v ) )] f jr ( x2( v ) )... f jr ( xd( v ) )[1 − F jr +1 ( xd( v ) ]...[1 − F jn ( xd( v ) )]}. 2

2

2

d

(3.3)

d

From (3.3), we can write b

f r1 , r2 ,..., rd :n ( x1 , x 2 ,..., x d ) =

 r1 −1   [ F j ( x1( v ) )] f j ( x1( v ) ) l   r1  l =1 

∏ ∑∏ {D

v =1

p

 r2 −1



n

  l = r1+1

 

l = rd +1

.  ∏ [ F jl ( x2( v ) ) −F jl ( x1(v ) )] f j r2 ( x2(v ) )... f jrd ( xd(v ) ) ∏ [1 − F jl ( xd(v ) )]} .

(3.4)

Thus, (3.1) is obtained. Next theorem shows that pdf of d order statistics of innid continuous random vectors can be expressed in terms of pdf of d order statistics of iid continuous random vectors. Theorem 3.2. b

 d +1



v =1

 w =1

 w =1

d

f r1 , r2 ,..., rd :n ( x1 , x 2 ,..., x d ) = ∏ {∑∑ n! D ∏ [ F s ( xw(v ) ) − F s ( xw(v−)1 )]rw − rw−1 −1 ∏ f s ( xw(v ) )} .

(3.5)

Proof. (3.2) can be expressed as b

∏[∑∑ P{x

(v) 1

v =1

s s ≤ xd(v) + δ xd(v)}]. (3.6) < X r(v), ≤ x1(v) + δ x1(v) , x2(v) < X r(2v:n),s ≤ x2(v) + δ x2(v) ,..., xd(v) < X r(v), 1:n d :n b

d

∏∏ δ x

Dividing (3.6) by

(v) w

and then letting δ x1( v ) , δ x2(v ) ,..., δ xd( v ) tend to zero, (3.5) is

v =1 w=1

obtained. The following five results of which first three are belong to pdf of single order statistic and last two are belong to joint pdf of d order statistics of innid continuous random vectors can be written from last two theorems. Result 3.1. f r1 :n ( x1(1) ) = =

  r1 −1  n 1  ∏ F jl ( x1(1) )  ∏ [1 − F jl ( x1(1) )]  f j r ( x1(1) ) ∑   1 (r1 − 1)!(n − r1 )! P  l =1  l = r1 +1  n

∑∑ r  r [ F

s

1

( x1(1) )]r1 −1[1 − F s ( x1(1) )]n − r1 f s ( x1(1) ) .

(3.7)

1

Proof. In (3.1) and (3.5), if b = 1 , d = 1 , (3.7) is obtained. Result 3.2. f1:n ( x1(1) ) = =

1 (n − 1)!



n

∑  ∏[1 − F P

 l =2

∑∑ n[1 − F

s

jl

 ( x1(1) )]  f j1 ( x1(1) ) 

( x1(1) )]n −1 f s ( x1(1) ) .

(3.8)

Proof. In (3.7), if r1 = 1 , (3.8) is obtained.

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Result 3.3. f n:n ( x1(1) ) = =

1 (n − 1)!

 n −1   F j l ( x1(1) )  f j n ( x1(1) )   l =1 

∑∏

∑∑ n[ F

P s

( x1(1) )]n −1 f s ( x1(1) ) .

(3.9)

Proof. In (3.7), if r1 = n , (3.9) is obtained. Result 3.4. f1, n:n ( x1(1) , x2(1) ) =

1 (n − 2)!

 n −1   [ F j ( x2(1) ) − F j ( x1(1) )]  f j ( x1(1) ) f j ( x2(1) ) l l n   1  l =2 

∑∏ P

= ∑∑ n(n − 1)[ F s ( x2(1) ) − F s ( x1(1) )]n − 2 f s ( x1(1) ) f s ( x2(1) ) .

(3.10)

Proof. In (3.1) and (3.5), if b = 1 , d = 2 and r1 = 1 , r2 = n , (3.10) is obtained. Result 3.5. b

f1, 2,..., k :n ( x1 , x 2 ,..., x k ) = ∏{ v =1 b

 n  1 ∑  ∏[1 − Fjl ( xk(v ) )] f j1 ( x1(v ) ) f j2 ( x2(v ) )... f jk ( xk(v ) )} (n − k )! P  l = k +1 

= ∏ {∑∑ v =1

n! (v) (v ) (v) [1 − F s ( xk( v ) )]n−k f s ( x1 ) f s ( x2 )... f s ( xk )} . (3.11) (n − k )!

Proof. In (3.1) and (3.5), if d = k and r1 = 1 , r2 = 2 ,…, rk = k , (3.11) is obtained. References ARNOLD, B.C., BALAKRISHNAN, N., NAGARAJA, H.N. (1992). A first course in order statistics. John Wiley and Sons Inc., New York. BALAKRISHNAN, N. (2007). Permanents, order statistics, outliers and robustness. Revista Matematica Complutense 20, pp. 7-107. BALASUBRAMANIAN, K., BEG, M.I. (2003). On special linear identities for order statistics. Statistics 37, pp. 335-339. BALASUBRAMANIAN, K., BALAKRISHNAN, N., MALIK, H.J. (1994). Identities for order statistics from non-independent non- identical variables. Sankhyā Series B 56, pp. 67-75. BAPAT, R.B., BEG, M.I. (1989). Order statistics for nonidentically distributed variables and permanents. Sankhyā Series A 51, pp. 79-93. CAO, G., WEST, M. (1997). Computing distributions of order statistics. Communications in Statistics - Theory and Methods 26, pp. 755-764. CHILDS, A., BALAKRISHNAN, N. (2006). Relations for order statistics from non-identical logistic random variables and assessment of the effect of multiple outliers on bias of linear estimators. Journal of Statistical Planning and Inference 136, pp. 2227-2253.

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CORLEY, H.W. (1984). Multivariate order statistics. Communications in Statistics - Theory and Methods 13, pp. 1299-1304. DAVID, H.A. (1981). Order statistics. John Wiley and Sons Inc., New York. GAN, G., BAIN, L.J. (1995). Distribution of order statistics for discrete parents with applications to censored sampling. Journal of Statistical Planning and Inference 44, pp. 37-46. GOLDIE, C.M., MALLER, R.A. (1999). Generalized densities of order statistics. Statistica Neerlandica 53, pp. 222-246. GUILBAUD, O. (1982). Functions of non-i.i.d. random vectors expressed as functions of i.i.d. random vectors. Scandinavian Journal of Statistics 9, pp. 229-233. KHATRI, C.G. (1962). Distributions of order statistics for discrete case. Annals of the Institute of Statistical Mathematics 14, pp. 167-171. REISS, R.-D. 1989. Approximate distributions of order statistics. Springer-Verlag, New York. VAUGHAN, R.J., VENABLES, W.N. (1972). Permanent expressions for order statistics densities. Journal of the Royal Statistical Society Series B 34, pp. 308-310.

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Stability of homomorphisms and derivations in non-Archimedean random C ∗ -algebras via fixed point method

Javad Shokri1 and Jung Rye Lee2∗ 1 2

Department of Mathematics, Urmia University, P. O. Box 165, Urmia, Iran

Department of Mathematics, Daejin University, Kyunggi 11159, Republic of Korea

Abstract. In this paper, using the fixed point method, we investigate the Hyers-Ulam stability of homomorphisms in non-Archimedean random C ∗ -algebras and non-Archimedean random Lie JC ∗ -algebras and of derivations on non-Archimedean random C ∗ -algebras and non-Archimedean random Lie JC ∗ -algebras related to the generalized Cauchy-Jensen additive functional equation.

1. Introduction A non-Archimedean field is a field like K equipped is a function | · | : K → [0, +∞) such that |a| = 0 if and only if a = 0, |ab| = |a||b| and |a + b| 6 max{|a|, |b|} for all a, b ∈ K. Note that |1| = | − 1| = 1 and |n| 6 1 for each integer n. By the trivial valuation we mean the mapping | · | taking everything but 0 into 1 and |0| = 0. We always assume, in addition, that | · | is non-trivial, i.e., there exists an a0 6= 0, 1. A function k.k : X → [0, ∞) is called a non-Archimedean norm if it satisfies the following conditions: (i) kxk = 0 if and only if x = 0; (ii) for any r ∈ K, x ∈ X, krxk = |r|kxk; (iii) the strong triangle inequality holds; namely, kx + yk 6 max{kxk, kyk}

(x, y ∈ X).

Then (X, k.k) is called a non-Archimedean normed space. From the fact that kxn − xm k 6 max{knn − xm k : m 6 j 6 n − 1}

(n > m)

holds, a sequence {xn } is Cauchy if and only if {xn −xm } converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent. For any nonzero rational number x, there exists a unique integer nx ∈ Z such that x = ab pnx , where a and b are integers not divisible by p. Then |x|p := p−nx defines a non-Archimedean norm on Q. The completion of Q with respect to the metric d(x, y) = |x − y|p is denoted by Qp , which is called the p-adic number field. 0

2000 Mathematics Subject Classification: Primary 39B52; 39B72; 46L05; 47H10; 46B03. Keywords: Hyers-Ulam stability; additive functional equation; fixed point; non-Archimedean random space; homomorphisms in non-Archimedean random C ∗ -algebras and non-Archimedean random Lie JC ∗ -algebras; derivations on random C ∗ -algebras and non-Archimedean random Lie JC ∗ -algebras. ∗ Corresponding author. 0 E-mail:1 [email protected], 2 [email protected] 0

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A non-Archimedean Banach algebra is a complete non-Archimedean algebra A which satisfies kabk 6 kak · kbk for all a, b ∈ A. For more detailed definitions of non-Archimedean Banach algebras, we refer the reader to [25]. If U is a non-Archimedean Banach algebra, then an involution on U is mapping t → t∗ from U into U which satisfies (i) t∗∗ = t for t ∈ U; ¯ ∗; (ii) (αs + βt)∗ = α ¯ s∗ + βt (iii) (st)∗ = t∗ s∗ for all s, t ∈ U. If, in addition, kt∗ tk = ktk2 for t ∈ U, then U is a non-Archimedean C ∗ -algebra. The stability problem of functional equations originated from a question of Ulam [26] concerning the stability of group homomorphisms: Let (G1 , ∗) be a group and let (G2 , ) be a metric group (a metric is defined on a set with group property) with the metric d(., .). Given  > 0, does there exist a δ() > 0 such that if a mapping h : G1 → G2 satisfies the inequality d(h(x ∗ y), h(x) ∗ h(y)) < δ for all x, y ∈ G1 , then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) <  for all x ∈ G1 ? If the answer is affirmative, we we would say that the equation of homomorphism h(x ∗ y) = h(x) ∗ h(y) is stable (see also [10, 11, 14, 18, 19, 20, 21, 22]). 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, y) 6 d(x, z) + d(z, y) for all x, y, z ∈ X. For explicitly later use, we recall a fundamental result in fixed point theory. Theorem 1.1. [9] Let (Ω, d) be a complete generalized metric space and J : Ω → Ω be a strictly contractive mapping with Lipschitz constant 0 < L < 1. Then for each given x ∈ Ω, either d(J n x, J n+1 x) = ∞ for all nonnegative 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 ∈ Ω : d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) 6 1−L d(y, Jy) for all y ∈ ∆. A C ∗ -algebra C, endowed with the Lie product [x, y] := xy−yx and endowed with anticommu2 xy+yx tator product (Jordan product) x◦y := 2 on C, is called a Lie JC ∗ -algebra (see [15, 16, 17]). Jordan algebras as coordinates for Lie algebras were created to illuminate a particular aspect of physics, quantum-mechanical observables, but turned out to have illuminating connections with many areas of mathematics. In this paper, using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms and derivations in non-Archimedean random C ∗ -algebras and non-Archimedean random Lie JC ∗ -algebras associated with f : X → Y satisfying the following functional equation (see [1]) ! Pm   n n−m X X (n − m + 1) n X j=1 xij + xkl = f (xi ) (1.1) f m n m l=1

1 6 i1 < · · · < im 6 n 1 6 kl (6= ij , ∀j ∈ {1, · · · , m}) 6 n

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for all x1 , · · · , xn ∈ X, where m, n ∈ N are fixed integer with n > 2, 1 6 m 6 n. In particular, Pn it is shown that in the case m = 1, (1.1) yields the Cauchy additive equation f ( P xkl ) = l=1 n Pn j=1 xj )= l=1 f (xi ) and also in the case m = n, (1.1) yields the Jensen additive equation f ( n P n 1 f (x ). Then (1.1) is a generalized form of the Cauchy-Jensen additive equation, and i l=1 n thus every solution of the equation (1.1) may be analogously called general (m, n)-CauchyJensen additive. For each m with 1 6 m 6 n, a mapping f : X → Y satisfies (1.1) for all n > 2 if and only if f (x) − f (0) = A(x) is Cauchy additive, where f (0) = 0 if m < n. In particular, we have f ((n − m + 1)x) = (n − m + 1)f (x) and f (mx) = mf (x) for all x ∈ X. 2. Random spaces In this section, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces as in [2, 3, 6, 7, 8]. Throughout this paper, ∆+ is the space of distribution functions, that is the space of all mapping F : R ∪ {−∞, ∞} → [0, 1] such that F is left-continuous and non-decreasing on R, F (0) = 0 and F (+∞) = 1. And D+ is a subset of ∆+ consisting of all functions F ∈ ∆+ for which l− F (+∞) = 1, where l− f (x) denotes the left limit of the function f at the point x, that is, l− f (x) = limt→x− f (t). The space ∆+ is partially ordered by the usual point-wise ordering of functions, i.e., F 6 G if and only if F (t) 6 G(t) for all t in R. The maximal element for ∆+ in this order is distribution function ε0 given by  0 if t 6 0, ε0 (t) = 1 if t > 0. Definition 2.1. [23] A mapping T : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm norm (briefly, a continuous t-norm) if T satisfies the following conditions: (a) T is commutative and associative; (b) T is continuous; (c) T (a, 1) = a for all a ∈ [0, 1]; (d) T (a, b) 6 T (c, d) whenever a 6 c and b 6 d for all a, b, c, d ∈ [0, 1]. Typical examples of continuous t-norms are TP (a, b) = ab, TM (a, b) = min(a, b) and TL (a, b) = max(a + b − 1, 0) (the Lukasiewicz t-norm). Definition 2.2. [24] A non-Archimedean random normed space (briefly, NA-RN-space) is a triple (X, µ, T ), where X is a vector space, T is a continuous t-norm, and µ is a mapping from X into D+ such that the following conditions hold: (RN 1) µx (t) = ε0 (t) for all t > 0 if and only if x = 0; t ) for all x ∈ X, α 6= 0. (RN 2) µαx (t) = µx ( |α| (RN 3) µx+y (t) > T (µx (t), µy (t)) for all x, y ∈ X and all t > 0. Every normed space (X, k · k) defines a non-Archimedean random normed space (X, µ, TM ), where t µx (t) = t + kxk for all t > 0, and TM is the minimum t-norm. This space is called the induced random normed space.

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Definition 2.3. [12] A non-Archimedean random normed algebra (X, µ, T, T 0 ) is a non-Archimedean random normed space (X, µ, T ) with an algebraic structure such that (RN 4) µxy (t) > T 0 (µx (t), µy (t)) for all x, y ∈ X and all t > 0, in which T 0 is a continuous t-norm. Every non-Archimedean normed algebra (X, k·k) defines a non-Archimedean random normed algebra (X, µ, TM ), where µx (t) =

t t + kxk

for all t > 0 if and only if kxyk 6 kxk kyk + tkxk + tkyk (x, y ∈ X; t > 0). This space is called an induced non-Archimedean random normed algebra. Definition 2.4. Let (X, µ, TM ) and (Y, µ, TM ) be non-Archimedean random normed algebras. (1) An R-linear mapping f : X → Y is called a homomorphism if f (xy) = f (x)f (y) for all x, y ∈ X. (2) An R-linear mapping f : X → Y is called a derivation if f (xy) = f (x)y + xf (y) for all x, y ∈ X. Definition 2.5. Let (U, µ, T ) be a non-Archimedean random Banach algebra. Then an involution on U is mapping u → u∗ from U into U which satisfies (i) u∗∗ = u for u ∈ U; ¯ ∗; (ii) (αu + βv)∗ = α ¯ u∗ + βv ∗ ∗ ∗ (iii) (uv) = v u for all u, v ∈ U. If, in addition, µu∗ u (t) = T 0 (µu (t), µu (t)) for u ∈ U, then U is a non-Archimedean random C ∗ -algebra. Definition 2.6. Let (X, µ, T ) be an N A-RN -space. (1) A sequence {xn } in X is said be convergent to x in X if, for every  > 0 and λ > 0, there exists a positive integer N such that µxn −x () > 1 − λ whenever n > N . (2) A sequence {xn } in X is called a Cauchy sequence if, for every  > 0 and λ > 0, there exists a positive integer N such that µxn −xn+1 () > 1 − λ whenever n > m > N . (3) An RN -space (X, µ, T ) is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X. 3. Stability of homomorphisms and derivations in non-Archimedean random C ∗ -algebras Throughout this section, we suppose that A and B are non-Archimedean random C ∗ -algebras, B respectively, with norms µA . and µ. .

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We use the following abbreviation for a given mapping f : A → B: Dλ f (x1 , · · · , xn ) :=

Pm

X

j=1 λxij

f

m

1 6 i1 < · · · < i m 6 n 1 6 kl (6= ij , ∀j ∈ {1, · · · , m}) 6 n

+

n−m X

! λxkl

−

l=1

(n − m + 1)


Pn

n m

i=1 λf (xi )

n

for all λ ∈ T1 := {µ ∈ C : |µ| = 1} and all x1 , · · · , xn ∈ A. It is well-known that a C-linear mapping H : A → B is called a random homomorphism in non-Archimedean random C ∗ -algebras if H satisfies H(xy) = H(x)H(y) and H(x∗ ) = H(x)∗ for all x, y ∈ A. We prove the Hyers-Ulam stability oh homomorphisms in non-Archimedean random C ∗ algebras for the functional equation Dλ f (x1 , · · · , xn ) = 0. Theorem 3.1. Let f : A → B be a mapping for which there are functions ϕ : An → D
+ , ψ : n A2 → D+ , and η : A → D+ such that |M| = |n − m + 1| < 1 and |N | = |(n − m + 1) m | ϕx1 ,··· ,xn (t), µB f (xy)−f (x)f (y) (t) > ψx,y (t),

(3.2)

µB f (x∗ )−f (x)∗ (t) > ηx (t),

(3.3)

for all λ ∈ T1 := {µ ∈ C : |µ| = 1} and all x1 , · · · , xn , x, y ∈ A and t > 0. If there exists an L < 1 such that ϕMx1 ,··· ,Mxn (|M|Lt) > ϕx1 ,··· ,xn (t), (3.4) ψMx,My (|M|2 Lt) > ψx,y (t),

(3.5)

ηMx (|M|Lt) > ηx (t),

(3.6)

for all x1 , · · · , xn , x, y ∈ A and t > 0, then there exists a unique random homomorphism H : A → B such that µB (3.7) f (x)−H(x) (t) > ϕx,··· ,x ((|N | − |N |L)t) for all x ∈ A and t > 0. Proof. It follows from (3.4), (3.5), (3.6), and L < 1 that lim ϕMm x1 ,··· ,Mm xn (|M|m t) = 1,

(3.8)

lim ψMm x,Mm y (|M|2m t) = 1,

(3.9)

lim ηMm x (|M|m t) = 1,

(3.10)

m→∞

m→∞

m→∞

for all x1 , · · · , xn , x, y ∈ A and t > 0. Now we define Ω := {g : A → B; g(0) = 0} and introduce a generalized metric on Ω as following: d(g, h) = inf{k ∈ (0, ∞) : µB g(x)−h(x) (kt) > ϕx,x,··· ,x (t), ∀x ∈ A, t > 0}

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where inf ∅ = +∞. By the same technique as in the proof of [13, Theorem 3.2], we can show 1 that (Ω, d) is a complete generalized metric space. We define J : Ω → Ω by Jg(x) = M g(Mx) for all x ∈ A and g ∈ Ω. Note that for all g, h ∈ Ω, from (3.4), we have d(g, h) 6 k ⇒ µB g(x)−h(x) (kt) > ϕx,··· ,x (t) ⇒ µB1 M

⇒ µB1 M

1 g(Mx)− M h(Mx) 1 g(Mx)− M h(Mx)

(kt) > ϕMx,··· ,Mx (|M|t) (kLt) > ϕx,··· ,x (t)

⇒ d(Jg, Jh) < kL. Then one can show that d(Jg, Jh) 6 Ld(g, h) for all g, h ∈ Ω and so J is self-function of Ω with the the Lipschitz constant L. Letting λ = 1 and putting x1 = x2 = · · · = xn = x in (3.1), we obtain µB ( n )f ((n−m+1)x)−( n )(n−m+1)f (x) (t) > ϕx,x,··· ,x (t) m

m

for all x ∈ A and t > 0. Then µB f (x)−

1 f (Mx) M

(t) > ϕx,x,··· ,x (|N |t)

for all x ∈ A and t > 0. This implies that d(Jf, f ) 6 |N1 | < ∞. By The fixed point alternative theorem, Theorem 1.1, J has a unique fixed point H : A → B in Ω0 := {h ∈ Ω : d(h, f ) < ∞} such that 1 f (Mm x) (3.11) H(x) = lim m→∞ |M|m for all x ∈ A, since limm→∞ d(J m f, H) = 0. On the other hand, it follows from (3.1), (3.8) and (3.11) that µB Dλ H(x1 ,··· ,xn ) (t) = lim µ m→∞

1 D f (Mm x1 ,··· ,Mm xn ) Mm λ

(t)

> lim ϕMm x1 ,··· ,Mm xn (|M|m t) = 1. m→∞

By a similar method to the above, we can get λH(Mx) = H(λMx) for all λ ∈ T and all x ∈ A. Then by using the same technique as in the proof of [10, Theorem 2.1], we can show that H is C-linear. It follows from (3.2), (3.9) and (3.11) that
2m µB lim µB t H(xy)−H(x)H(y) (t) = m→∞ f (M2m xy)−f (Mm x)f (Mm y) |M|
> lim ψMm x,Mm y |M|2m t = 1 m→∞

for all x, y ∈ A. Therefore, we conclude that H(xy) = H(x)H(y) for all x, y ∈ A. Thus H : A → B is a homomorphism satisfying (3.7). By same method as above, from (3.3),(3.10) and (3.11), we can write B µB H(x∗ )−H(x)∗ (t) = lim µ 1 m→∞

(f (M Mm

m x∗ )−f (Mm x)∗ )

(t)

> lim ηMm x (|M|m t) = 1 m→∞

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for all x ∈ A and all t > 0. Then we conclude that H(x∗ ) = H(x)∗ and the proof is complete, as desired.  Corollary 3.2. Let r > 1 and θ be nonnegative real numbers, and let f : A → B be a mapping such that t µB , Dλ f (x1 ,··· ,xn ) (t) > r t + θ(kx1 kA + kx2 krA + · · · + kxn krA ) t µB , f (xy)−f (x)f (y) (t) > t + θ(kxkrA .kykrA ) t µB f (x∗ )−f (x)∗ (t) > t + θkxkrA for all λ ∈ T1 , all x1 , · · · , xn , x, y ∈ A and t > 0. Then there exists a unique random homomorphism H : A → B such that µB f (x)−H(x) (t) >

(|N | − |N |r )t (|N | − |N |r )t + nθkxkrA

for all x ∈ A and t > 0. Proof. Letting ϕx1 ,··· ,xn (t)

t+

θ(kx1 krA

t , + kx2 krA + · · · + kxn krA )

t , t + θ(kxkrA .kykrA ) t ηx (t) = t + θkxkrA ψx,y (t) =

for all x1 , · · · , xn , x, y ∈ A, L = |N |r−1 and t > 0 in Theorem 3.1, we get the desired result.



In the following theorem, we investigate the Hyers-Ulam stability of derivations on nonArchimedean random C ∗ -algebras for the functional equation Dλ f (x1 , · · · , xn ) = 0. Theorem 3.3. Let f : A → A be a mapping for which there are functions ϕ : An → D+ , ψ : A2 → D+ , satisfying (3.1), (3.3), and η : A → D+ such that |M| < 1 and |N | < 1 are far from zero and µA (3.12) f (xy)−f (x)y−xf (y) (t) > ψx,y (t), for all λ ∈ T1 and all x1 , · · · , xn , x, y ∈ A and t > 0. If there exists an L < 1 such that (3.4), (3.5) and (3.6) hold, then there exists a unique random derivation δ : A → A such that µA f (x)−δ(x) (t) > ϕx,··· ,x ((|N | − |N |L))

(3.13)

for all x ∈A and t > 0. Proof. By the same argument as in the proof of Theorem 3.1, there exists a unique C-linear mapping δ : A × A → A satisfying (3.13). The mapping δ is given by δ(x) = lim

m→∞

1 f (Mm x) |M|m

328

(3.14)

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for all x ∈ A. It follows from (3.12), (3.9) and (3.14) that
2m lim µB t µB δ(xy)−δ(x)y−xδ(y) (t) = m→∞ f (M2m xy)−f (Mm x)Mm y−Mm xf (Mm y) |M|
> lim ψMm x,Mm y |M|2m t = 1 m→∞

for all x, y ∈ A. Therefore, we conclude that δ(xy) = δ(x)y + xδ(y) for all x, y ∈ A. The remainder of the proof is similar to the proof of Theorem 3.1.  4. Stability of homomorphisms and derivations in non-Archimedean random Lie JC ∗ -algebras A non-Archimedean random C ∗ -algebra C, endowed with the Lie product [x, y] := xy−yx 2 and xy+yx endowed with anticommutator product (Jordan product) x ◦ y := 2 on C, is called a nonArchimedean random Lie JC ∗ -algebra. Definition 4.1. Let A and B be non-Archimedean random Lie JC ∗ -algebras. A C-linear mapping H : A → B is called a random Lie JC ∗ -algebra homomorphism if H satisfies H([x, y]) = [H(x), H(y)], H(x ◦ y) = H(x) ◦ H(y), H(x∗ ) = H(x)∗ for all x, y ∈ A. Throughout this section, assume that A and B are two non-Archimedean random Lie JC ∗ B algebras respectively with norm µA . and µ. . In the following theorem, we prove the Hyers-Ulam stability of homomorphisms in nonArchimedean random Lie JC ∗ -algebra for the functional equation Dλ f (x1 , · · · , xn ) = 0. Theorem 4.2. Let f : A → B be a mapping for which there are functions ϕ : An → D+ and ψ : A2 → D+ satisfying (3.1), (3.3) and µB f ([x,y])−[f (x),f (y)] (t) > ψx,y (t),

(4.1)

µB H(x◦y)−H(x)◦H(y) (t) > φx,y (t)

(4.2)

for all λ ∈ T1 , all x, y ∈ A and t > 0. If there exists an L < 1 such that (3.4), (3.5) and (3.6) hold, and also φMx,My (|M|2 Lt) > φx,y (t), (4.3) for all x, y ∈ A and t > 0, then there exists a unique random Lie JC ∗ -algebra homomorphism H : A → B satisfying (3.7). Proof. It follows from (4.3) and L < 1 that lim φMm x,Mm y (|M|2m t) = 1,

m→∞

(4.4)

for all x, y ∈ A and t > 0.

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Approximate homomorphisms and derivations on ...

By the same argument as in the proof of Theorem 3.1, there exists a unique C-linear mapping H : A → B satisfying (3.7). The mapping H is given by f (Mm x) m→∞ |M|m

H(x) = lim

(4.5)

for all x ∈ A. It follows from (3.9), (4.4) and (4.5) that
2m lim µB t µB H([x,y])−[H(x),H(y)] (t) = m→∞ f (M2m [x,y])−[f (Mm x),f (Mm y)] |M|
> lim ψMm x,Mm y |M|2m t = 1 m→∞

and 2m lim µB t µB H(x◦y)−H(x)◦H(y) = m→∞ f (M2m (x◦y))−f (Mx)◦f (My) |M|
> lim φMm x,Mm y |M|2m t = 1




m→∞

for all x, y ∈ A and t > 0, then it is concluded that H([x, y]) = [H(x), H(y)]

;

H(x ◦ y) = H(x) ◦ H(y)

for all x, y ∈ A. Therefore, H : A → B is the unique random Lie JC ∗ -algebra homomorphism satisfying (3.7).  Corollary 4.3. Let r > 1 and θ be nonnegative real numbers, and f : A → B be a mapping such that t , µB Dλ f (x1 ,··· ,xn ) (t) > r t + θ(kx1 kA + · · · + kxn krA ) t µB , f ([x,y])−[f (x),f (y)] > t + θ(kxkrA .kykrA ) t µB f (x8 )−f (x)∗ (t) > t + θ.kxkrA for all λ ∈ T1 , all x1 , · · · , xn , x, y ∈ A and t > 0. Then there exists a unique random Lie JC ∗ -algebra homomorphism H : A → B such that µB f (x)−H(x) >

(|N | − |N |r )t (|N | − |N |r )t + nθkxkrA

for all x ∈ A and t > 0. Proof. By the same reasoning as in the proof of Theorem 4.2 and a technique similar to Corollary 3.2, by putting L = |N |r−1 , the proof will be completed.  Definition 4.4. Let A be a non-Archimedean random Lie JC ∗ -algebra. A C-linear mapping δ : A → A is called a random Lie JC ∗ -algebra derivation if δ satisfies δ([x, y]) = [δ(x), y] + [x, δ(y)], δ(x ◦ y) = δ(x) ◦ y + x ◦ δ(y), δ(x∗ ) = δ(x)∗ for all x, y ∈ A.

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In the following theorem, we prove the Hyers-Ulam stability of derivation on non-Archimedean random Lie JC ∗ -algebras for the functional equation Dλ f (x1 , · · · , xn ) = 0. Theorem 4.5. Let f : A → A be a mapping for which there are functions ϕ : An → D+ and ψ : A2 → D+ such that (3.1) and (3.3) hold and µA f ([x,y])−[f (x),y]−[x,f (y)] (t) > ψx,y (t),

(4.6)

µA f (x◦y)−f (x)◦y−x◦f (y) (t) > φx,y (t)

(4.7)

for all x, y ∈ A. If there exists an L < 1 and (3.4), (3.5), (3.6) and (4.3) hold, then there exists a unique random Lie JC ∗ -algebra derivation δ : A → A such that (3.13) holds. Proof. By the same argument as in the proof of Theorem 4.2, there exists a unique C-linear mapping δ : A → A satisfying (3.13), and is given by f (Mm x) m→∞ |M|m

δ(x) = lim

(4.8)

for all x ∈ A. It follows from (3.9), (4.4) and (4.8) that
2m t µδ([x,y])−[δ(x),y]−[x,δ(y)] (t) = lim µA 2m [x,y])−[f (Mm x),Mm y]−[Mm x,f (Mm y)] |M| f (M m→∞
> lim ψMm x,Mm y |M|2m t = 1 m→∞

and
2m µA lim µA t δ(x◦y)−δ(x)◦y−x◦δ(y) (t) = m→∞ f (M2m (x◦y))−f (Mm x)◦y−x◦f (Mm y) |M|
> lim φMm x,Mm y |M|2m t = 1 m→∞

for all x, y ∈ A and t > 0, and so we conclude that δ([x, y]) = [δ(x), y] + [x, δ(y)],

δ(x ◦ y) = δ(x) ◦ y + x ◦ δ(y)

for all x, y ∈ A. Therefore, δ : A → A is the unique desired random Lie JC ∗ -algebra derivation satisfying (3.13).  References [1] H. Azadi Kenary, Non-Archimedean stability of Cauchy-Jensen type functional equation, J. Nonlinear Anal. Appl. 1 (2011), no. 2, 1–10. [2] H. Azadi Kenary, S. Jang and C. Park, A fixed point approach to the Hyers-Ulam stability of a functional equation in various normed spaces, Fixed Point Theory Appl. 2011 (2011), 2011:67. [3] H. Azadi Kenary, S. Rezaei, S. Talebzadeh and C. Park, Stability for the Jensen equation in C ∗ -algebras: a fixed point alternative approach, Adv. Difference Equ. 2012 (2012), 2012:17. [4] Cˇ adariu and V. Radu, Fixed points and stability Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), Art. No. 1. [5] S. Chang, Y. Cho and S. Kang, Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publishers, New York, 2001. [6] Y. Cho, J. Kang and R. Saadati, Fixed points and stability additive functional equations on the Banach algebras, J. Comput. Anal. Appl. 14 (2010), 1103–1111.

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Approximate homomorphisms and derivations on ... [7] Y. Cho, C. Park and R. Saadati, Functional inequalities in non-Archimedean in Banach spaces, Appl. Math. Lett. 60 (2010), 1994–2002. [8] Y. Cho, R. Saadati and J. Vahidi, Approximation of homomorphisms and derivations on non-Archimedean Lie C ∗ -algebras via fixed point meyhod, Discrete Dyn. Nat. Soc. 2012 (2012), Art. ID 373904. [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, Nearly involution on Banach algebras: a fixed point approach, Fixed Point Theory 14 (2013), 117–123. [11] M. Eshaghi Gordji, A. Rahimi, C. Park and D. Shin, Ternary Jordan bi-homomorphisms in Banach Lie triple systems, J. Comput. Anal. Appl. 21 (2016), 1040–1045. [12] A. K. Mirmostafaee, Perturbation of generalized derivations in fuzzy menger normed algebras. Fuzzy Sets Syst. 195 (2012), 109–117. [13] F. Moradlou and M. Eshaghi Gordji, Approximate Jordan derivation on Hilbert C ∗ -modules, Fixed Point Theory 14 (2013), 413–425. [14] E. Movahednia, S. M. S. M. Mosadegh, C. Park and D. Shin, Stability of a lattice preserving functional equation on Riesz space: fixed point alternative, J. Comput. Anal. Appl. 21 (2016), 83–89. [15] C. Park, Lie *-homomorphisms between Lie C ∗ -algebras and Lie ∗-derivations on Lie C ∗ -algebras, J. Math.Anal. Appl., 293 (2004), 419–434. [16] C. Park, Homomorphisms between Lie JC ∗ -algebras and Cauchy-Rassias stability of Lie JC ∗ -algebra derivations, J. Lie Theory 15 (2005), 393–414. [17] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [18] C. Park, Stability of ternary quadratic derivation on ternary Banach algebras: revisited, J. Comput. Anal. Appl. 20 (2016), 21–23. [19] W. Park and J. Bae, Approximate quadratic forms on restricted domains, J. Comput. Anal. Appl. 20 (2016), 388–410. [20] J. M. Rassias, On approximation of approximately linear mapping by linear mappings, J. Func. Anal. 46 (1982), 126–130. [21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [22] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [23] B. Schweizer and A. Sklar, Probabilistic Metirc Spaces, North-Holand, New York, 1983. [24] A. N. Sherstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280–283. [25] N. Shilkret, Non-Archimedean Banach Algebras, Ph.D. thesis, Polytechnic University, 1968, ProQuest LLC. [26] S. M. Ulam, Problems in Modern Mathematics, Science ed., John Wiley & Sons, New York, 1940.

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ON THE FUZZY STABILITY PROBLEMS OF GENERALIZED SEXTIC MAPPINGS HEEJEONG KOH AND DONGSEUNG KANG∗

Abstract. We introduce a fuzzy anti-β-norm and generalized sextic mapping and then investigate the Hyers-Ulam-Rassias stability in quasi β-Banach space and the fuzzy stability by using a fixed point in fuzzy anti-β Banach space for the generalized sextic function.

1. Introduction The concept of stability problem of a functional equation was first posed by Ulam [33] concerning the stability of group homomorphisms. In the next year, Hyers [14] gave a partial answer to the question of Ulam. Hyers’ theorem was generalized in various directions. The very first author who generalized Hyers’ theorem to the case of unbounded control functions was Aoki [1]. Rassias [28] succeeded in extending the result of Hyers’ theorem by weakening the condition for the Cauchy difference operator CDf (x, y) = f (x + y) − [f (x) + f (y)] to be controlled by ε(||x||p + ||y||p ) . Rassias’ paper [28] has provided a lot of influence in the development of Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. In 1996, Isac and Rassias [16] were first to provide applications of new fixed point theorems for the proof of stability theory of functional equations. By using fixed point methods the stability problems of several functional equations have been extensively investigated by a number of authors; see [6], [7], [25] and [26]. Recently, the stability problem of functional equations was investigated by using shadowing properties; see [20] and [31]. During the last three decades, several stability problems of a large variety of functional equations have been extensively studied and generalized by a number of authors [9], [12], [15], [28], and [2]. In particular, Xu and et al. [37] introduced the sextic functional equation (1.1) f (x + 3y) + f (x − 3y) − 6[f (x + 2y) + f (x − 2y)] + 15[f (x + y) + f (x − y)] = 20f (x) + 720f (y) . In fact, Xu and et al. [37] and Gordji and et al. [13] introduced a quintic mapping and sextic mapping. In this paper, we deal with the following functional equation f (ax + y) + f (ax − y) + f (x + ay) + f (x − ay)

(1.2) 2

2

= a (a + 1)[f (x + y) + f (x − y)] + 2(a2 − 1)(a4 − 1)[f (x) + f (y)] 2000 Mathematics Subject Classification. 39B52. Key words and phrases. Hyers-Ulam-Rassias stability, sextic mapping, quasi-β-mormed space, fixed point, fuzzy anti-normed space, fuzzy anti-β-normed space. * Corresponding author.

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holds for all x, y ∈ X and all a ∈ Z (a 6= 0, ±1) . We will use the following definition to prove Hyers-Ulam-Rassias stability for the generalized sextic functional equation in the quasi β-normed space. Let β be a real number with 0 < β ≤ 1 and K be either R or C . Definition 1.1. Let X be a linear space over a field K . A quasi β-norm || · || is a real-valued function on X satisfying the following statements: (1) ||x|| ≥ 0 for all x ∈ X and ||x|| = 0 if and only if x = 0 . (2) ||λx|| = |λ|β · ||x|| for all λ ∈ K and all x ∈ X . (3) There is a constant K ≥ 1 such that ||x+y|| ≤ K(||x||+||y||) for all x, y ∈ X . The pair (X, || · ||) is called a quasi β-normed space if || · || is a quasi β-norm on X . The smallest possible K is called the modulus of concavity of || · || . A quasi β-Banach space is a complete quasi-β-normed space. A quasi β-norm || · || is called a (β, p)-norm (0 < p ≤ 1) if (3) takes the form ||x + y||p ≤ ||x||p + ||y||p for all x, y ∈ X . In this case, a quasi β-Banach space is called a (β, p)-Banach space; see [5], [29] and [27]. In 1984, Katsaras [18] and Wu and Fang [35] independently introduced a notion of a fuzzy norm. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view; see [3], [11], [19], [36] and [23]. In 2003, Bag and Samanta [3] modified the definition of Cheng and Mordeson [8]. Bag and Samanta [3] introduced the following definition of fuzzy normed spaces. The notion of fuzzy stability of functional equations was given in the paper [24]. Jebril and Samanta [17] introduced a fuzzy anti-norm linear space depending on the idea of fuzzy anti-norm was introduced by Bag and Samanta [4] and investigated their important properties. We will use the definition of fuzzy anti-normed spaces to investigate a fuzzy version of Hyers-Ulam-Rassias stability in the fuzzy anti-normed algebra setting. Definition 1.2. [17] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy anti-norm on X if for all x, y ∈ X and all s, t ∈ R , (aN1) N (x, t) = 1 for t ≤ 0 (aN2) N (x, t) = 0 if and only if x = 0 for all t > 0 t ) for c 6= 0 (aN3) N (cx, t) = N (x, |c| (aN4) N (x + y, s + t) ≤ max{N (x, s) , N (y, t)} (aN5) N (x, t) is a non-increasing function of t ∈ R and limt→∞ N (x, t) = 0 , (aN6) for x 6= 0 , N (x, ·) is continuous on R . The pair (X, N ) is called a fuzzy anti-normed space. The property (aN3) implies that N (−x, t) = N (x, t) for all x ∈ X and t > 0 . It is easy to show that (aN4) is equivalent the following condition: N (x + y, t) ≤ max{N (x, t) , N (y, t)} , for all x, y ∈ X and t ∈ R . Definition 1.3. Let X be a real vector space. A fuzzy anti-norm N : X ×R → [0, 1] is called a fuzzy anti-β-norm on X if (aN3 ) in Definition 1.2 takes the form t (aN30 ) N (cx, t) = N (x, β ) (c 6= 0, 0 < β ≤ 1) . |c| Example 1.4. Let (X, || · ||) be a β-normed space. Define ( ||x|| when t > 0, t ∈ R N (x, t) = t+||x|| 1 when t ≤ 0 ,

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GENERALIZED SEXTIC MAPPINGS

where x ∈ X . We note that N (cx, t) =

||cx|| = t + ||cx||

t |c|β

||x|| t = N (x, β ) , |c| + ||x||

for all x ∈ X and c ∈ R (c 6= 0 , 0 < β ≤ 1) . Then (X, N ) is a fuzzy anti-β-normed space induced by the β-norm || · || . Definition 1.5. Let (X, N ) be a fuzzy anti-β-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) = 0 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 1.6. Let (X, N ) be a fuzzy anti-β-normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all integer d > 0 , we have N (xn+d − xn , t) < ε . It is well-known that every convergent sequence in a fuzzy anti-β-normed vector space is Cauchy. If each Cauchy sequaence is convergent, then the fuzzy anti-βnormed space is said to be fuzzy anti-β complete and the fuzzy anti-β-normed vector space is called a fuzzy anti-β Banach space. Now, we will state the theorem, the alternative of fixed point in a generalized metric space. Definition 1.7. 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 1.8 ( The alternative of fixed point [21], [30] ). Suppose that we are given a complete generalized metric space (X, d) and a strictly contractive mapping J : X → X with Lipschitz constant 0 < 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} is convergent to a fixed point y ∗ of J ; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X|d(J n0 x, y) < ∞} ; (4) d(y, y ∗ ) ≤

1 1−L

d(y, Jy) for all y ∈ Y .

In this paper, we investigate the Hyers-Ulam-Rassias stability in quasi β-normed space and then the fuzzy stability by using a fixed point in fuzzy anti-β Banach space for the generalized sextic function f : X → Y satisfying the equation (1.2). Let us fix some notations which will be used throughout this paper. Let a ∈ Z (a 6= 0 , ±1) . 2. A sextic functional equation In this section let X and Y be real vector spaces and we investigate the general solution of the functional equation (1.2). Before we proceed, we would like to introduce some basic definitions concerning n-additive symmetric mappings and key concepts which are found in [32] and [34]. A function A : X → Y is said to be

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additive if A(x + y) = A(x) + A(y) for all x , y ∈ X . Let n be a positive integer. A function An : X n → Y is called n-additive if it is additive in each of its variables. A function An is said to be symmetric if An (x1 , · · · , xn ) = An (xσ(1) , · · · , xσ(n) ) for every permutation {σ(1) , · · · , σ(n)} of {1 , 2 , · · · , n} . If An (x1 , x2 , · · · , xn ) is an nadditive symmetric map, then An (x) will denote the diagonal An (x , x , · · · , x) and An (rx) = rn An (x) for all x ∈ X and all r ∈ Q . such a function An (x) will be called a monomial function of degree n (assuming An 6≡ 0). Furthermore the resulting function after substitution x1 = x2 = · · · = xs = x and xs+1 = xs+2 = · · · = xn = y in An (x1 , x2 , · · · , xn ) will be denoted by As,n−s (x , y) . Theorem 2.1. A function f : X → Y is a solution of the functional equation (1.2) if and only if f is of the form f (x) = A6 (x) for all x ∈ X , where A6 (x) is the diagonal of the 6-additive symmetric mapping A6 : X 6 → Y . Proof. Assume that f satisfies the functional equation (1.2). Letting x = y = 0 in the equation (1.2), we have 2a2 (2a2 + 1)(a2 − 1)f (0) = 0 , that is, f (0) = 0 . Let y = 0 in the equation (1.2). Then we get f (ax) = a6 f (x)

(2.1)

for all x ∈ X . Putting x = 0 in the equation (1.2), we get   (2.2) (a4 − 1)(a2 − 1) f (y) − f (−y) = 0 for all y ∈ X . Hence we have f (y) = f (−y) , for all y ∈ X . That is, f is even. We can rewrite the functional equation (1.2) in the form 1 1 f (ax + y) − f (ax − y) f (x) − 2 4 2 2(a − 1)(a − 1) 2(a − 1)(a4 − 1) 1 1 − f (x + ay) − f (x − ay) 2(a2 − 1)(a4 − 1) 2(a2 − 1)(a4 − 1) a2 (a2 + 1) a2 (a2 + 1) f (x + y) + f (x − y) + f (y) = 0 + 2(a2 − 1)(a4 − 1) 2(a2 − 1)(a4 − 1) for all x , y ∈ X and an integer a(a 6= 0 , ±1) . By Theorem 3.5 and 3.6 in [34], f is a generalized polynomial function of degree at most 6, that is, f is of the form (2.3)

f (x) = A6 (x) + A5 (x) + A4 (x) + A3 (x) + A2 (x) + A1 (x) + A0 (x)

for all x ∈ X , where A0 (x) = A0 is an arbitrary element of Y , and Ai (x) is the diagonal of the i-additive symmetric mapping Ai : X i → Y for i = 1, 2, 3, 4, 5, 6 . By f (0) = 0 and f (−x) = f (x) for all x ∈ X , we get A0 (x) = A0 = 0 , A5 (x) = 0 , A3 (x) = 0 and A1 (x) = 0 . It follows that f (x) = A6 (x) + A4 (x) + A2 (x) for all x ∈ X . By (2.1) and An (rx) = rn An (x) for all x ∈ X and r ∈ Q , we obtain 2 that A2 (x) = − a2a+1 A4 (x) for all x ∈ X and an integer a (a 6= 0, ±1) . Hence we get A4 (x) = A2 (x) = 0 , for all x ∈ X . Thus we have f (x) = A6 (x) for all x ∈ X . Conversely, assume that f (x) = A6 (x) for all x ∈ X , where A6 (x) is the diagonal of a 6-additive symmetric mapping A6 : X 6 → Y . Note that A6 (qx + ry)

= q 6 A6 (x) + 6q 5 rA5,1 (x, y) + 15q 4 r2 A4,2 (x, y) + 20q 3 r3 A3,3 (x, y) +

15q 2 r4 A2,4 (x, y) + 6qr5 A1,5 (x, y) + r6 A6 (y)

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cs As,t (x, y) = As,t (cx, y) , ct As,t (x, y) = As,t (x, cy) where 1 ≤ s, t ≤ 5 and c ∈ Q . Thus we may conclude that f satisfies the equation (1.2).  We note that a mapping f : X → Y is called generalized sextic if f satisfies the functional equation (1.2). 3. Hyers-Ulam-Rassias stability over a quasi β-Banach space Throughout this section, let X be a real linear space and let Y be a quasi βBanach space with a quasi β-norm || · ||Y . Let K be the modulus of concavity of || · ||Y . We will investigate the Hyers-Ulam-Rassias stability for the functional equation (1.2); see also the paper [10]. For a given mapping f : X → Y and all fixed integer a ( a 6= 0, ±1) , let (3.1)

Da f (x, y) := f (ax + y) + f (ax − y) + f (x + ay) + f (x − ay)     −a2 (a2 +1) f (x+y)+f (x−y) −2(a2 −1)(a4 −1) f (x)+f (y)

for all x, y ∈ X . Theorem 3.1. Suppose that there exists a mapping φ : X 2 → [0, ∞) for which a mapping f : X → Y satisfies f (0) = 0 , ||Da f (x, y)||Y ≤ φ(x, y) j K φ(aj x, aj y) converges for all x, y ∈ X . Then there and the series j=0 |a|6β exists a unique generalized sextic mapping S : X → Y satisfying the equation (1.2) and the inequality ∞ K X  K j φ(aj x, 0) , (3.3) ||f (x) − S(x)||Y ≤ β 6β 2 |a| j=0 |a|6β (3.2)

P∞ 

for all x ∈ X . Proof. By letting y = 0 in inequality (3.2), since f (0) = 0 we have ||Da f (x, 0)||Y

= ||2f (ax) + 2f (x) − 2a2 (a2 + 1)f (x) − 2(a2 − 1)(a4 − 1)f (x)||Y 1 = 2β |a|6β ||f (x) − 6 f (ax)||Y ≤ φ(x, 0) , a

that is, (3.4)

||f (x) −

1 1 f (ax)||Y ≤ β 6β φ(x, 0) , a6 2 |a|

for all x ∈ X . We note that putting x = ax and multiplying |a|16β in the inequality (3.4), we get 1 1 1 1 (3.5) ||f (ax) − 6 f (a2 x)||Y ≤ β 6β φ(ax, 0) , |a|6β a 2 |a| |a|6β for all x ∈ X . Combining two inequalities (3.4) and (3.5), we have  1 2  K  1 (3.6) ||f (x) − 6 f (a2 x)||Y ≤ β 6β φ(x, 0) + 6β φ(ax, 0) , a 2 |a| |a|

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for all x ∈ X . Since K ≥ 1 , inductively using the previous note we have the following inequalities k−1  1 k K X  K j ||f (x) − 6 f (ak x)||Y ≤ β 6β (3.7) φ(aj x, 0) , a 2 |a| j=0 |a|6β for all x ∈ X , k ∈ N and also t−1  1 k  1 t K X  K j φ(aj x, 0) , (3.8) || 6 f (ak x) − 6 f (at x)||Y ≤ β 6β a a 2 |a| |a|6β j=k

for all x ∈ X and k , t ∈ N (k < t) . Sincethe right-hand side of the previous inequality (3.8) tends to 0 as t → ∞ , n

hence { a16 f (an x)} is a Cauchy sequence in the quasi β-Banach space Y . Thus we may define  1 n S(x) = lim f (an x) , n→∞ a6 for all x ∈ X . Since K ≥ 1 , replacing x and y by an x and an y respectively and dividing by |a|6βn in the inequality (3.2) , we have  1 n ||Da f (an x, an y)||Y |a|6β  1 n ||f (an (ax + y)) + f (an (ax − y)) + f (an (x + ay)) + f (an (x − ay)) = |a|6β   −a2 (a2 + 1) f (an (x + y)) + f (an (x − y))   −2(a2 − 1)(a4 − 1) f (an x) + f (an y) ||Y  K n ≤ φ(an x, an y) |a|6β for all x, y ∈ X . By taking n → ∞ , the definition of S implies that S satisfies (1.2) for all x, y ∈ X , that is, S is the generalized sextic mapping. Also, the inequality (3.7) implies the inequality (3.3). Now, it remains to show the uniqueness. Assume that there exists T : X → Y satisfying (1.2) and (3.3). Then  1 n ||T (x) − S(x)||Y = ||T (an x) − S(an x)||Y |a|6β  1 n   ≤ K ||T (an x) − f (an x)||Y + ||f (an x) − S(an x)||Y 6β |a| ∞  X K j 2K 2 φ(aj x, 0) ≤ 2β |a|6β K n j=n |a|6β for all x ∈ X . By letting n → ∞ , we immediately have the uniqueness of S .



Corollary 3.2. Let θ ≥ 0 , p < 6 be a real number and X be a normed linear space with norm || · || . Suppose f : X → Y is a mapping satisfying f (0) = 0 and (3.9)

||Da f (x, y)||Y ≤ θ(||x||p + ||y||p )

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for all x, y ∈ X and all t > 0 . Then S(x) := N- limn→∞ a16n f (an x) exists for each x ∈ X and defines a generalized sextic mapping S : X → Y such that ||f (x) − S(x)||Y ≤

θ K ||x||p 2β (|a|6β − K|a|pβ )

for all x ∈ X and all t > 0 . Proof. The proof follows from Theorem 3.1 by taking φ(x, y) = θ(||x||p + ||y||p ) for all x , y ∈ X .  4. Fuzzy fixed point stability over a Fuzzy Banach space Let us fix some notations which will be used throughout this section. We assume X is a vector space and (Y, N ) is a fuzzy anti-β Banach space. Using fixed point method, we will prove the Hyers-Ulam stability of the functional equation satisfying equation (1.2) in fuzzy anti-β Banach space. Theorem 4.1. Let φ : X 2 → [0, ∞) be a function such that there exists an 0 < L < 1 with L (4.1) φ(x, y) ≤ 6β φ(ax, ay) |a| for all x, y ∈ X . Let f : X → Y be a mapping satisfying f (0) = 0 and N (Da f (x, y), t) ≤

(4.2)

φ(x, y) t + φ(x, y)

  for all x, y ∈ X and all t > 0 . Then S(x) := N- limn→∞ a6n f axn exists for each x ∈ X and defines a generalized sextic mapping S : X → Y such that N (f (x) − S(x), t) ≤

(4.3)

L φ(x, 0) 2β |a|6β (1 − L) t + L φ(x, 0)

for all x ∈ X and all t > 0 . Proof. By letting y = 0 in the inequality (4.2), we have   φ(x, 0) (4.4) N 2f (ax) − 2a6 f (x), t ≤ t + φ(x, 0) for all x ∈ X and all t > 0 . We note that by letting x = xa in the inequality (4.4) we have  x  φ( xa , 0) N 2f (x) − 2a6 f ,t ≤ . a t + φ( xa , 0) The inequality (4.1) implies that L  x t  φ(x, 0) |a|6β N f (x) − a6 f , β ≤ . L a 2 t + |a|6β φ(x, 0)

By putting t =

L |a|6β

t , we have

 L , β 6β t ≤ N f (x) − a f a 2 |a| 

6

x

339

L |a|6β

L |a|6β

φ(x, 0)

t+

L |a|6β

φ(x, 0)

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that is,  x  L φ(x, 0) N f (x) − a6 f , β 6β t ≤ , a 2 |a| t + φ(x, 0)

(4.5)

for all x ∈ X and all t > 0 . We consider the set F := {g : X → X} and the mapping d defined on F × F by   d(g, h) = inf{µ ∈ R+ | N g(x) − h(x), µt ≤

φ(x, 0) , ∀x ∈ X and t > 0} t + φ(x, 0)

where inf ∅ = +∞ , as usual. Then (F, d) is a complete generalized metric space; see [22, Lemma 2.1]. Now let’s consider the linear mapping J : F → F such that x Jg(x) := a6 g a for all x ∈ X . Let g , h ∈ F be given such that d(g , h) = ε . Then   φ(x, 0) N g(x) − h(x), εt ≤ t + φ(x, 0) for all x ∈ X and all t > 0 .    x x  N Jg(x) − Jh(x), Lεt = N a6 g − a6 h , Lεt a a  x x L  φ( xa , 0) = N g −h , εt ≤ L a a |a|6β t + φ( xa , 0) |a|6β ≤

L φ(x, 0) |a|6β L t + |a|L6β φ(x, |a|6β

0)

=

φ(x, 0) t + φ(x, 0)

for all x ∈ X and all t > 0 . d(g, h) = ε implies that d(Jg, Jh) ≤ Lε . Hence we get d(Jg, Jh) ≤ L d(g, h) L for all g, h ∈ F . The inequality (4.5) implies that d(f, Jf ) ≤ 2β |a| 6β . By Theorem 1.8, there exists a mapping S : X → Y such that (1) S is a fixed point of J , that is, x 1 (4.6) S = 6 S(x) a a for all x ∈ X . The mapping S is a unique fixed point of J in the set M = {g ∈ F | d(f, g) < ∞} . This means that S is a unique mapping satisfying the equation (4.6) such that there exists a µ ∈ (0, ∞) satisfying   φ(x, 0) N f (x) − S(x), µt ≤ t + φ(x, 0)

for all x ∈ X and all t > 0 ; (2) d(J n f, S) → 0 as n → ∞ . This implies the following equality x N- lim a6n f n = S(x) n→∞ a for all x ∈ X ;

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(3) d(f, S) ≤

1 1−L

d(f, Jf ) , which implies the inequality

d(f, S) ≤

L L 1 · β 6β = β 6β . 1 − L 2 |a| 2 |a| (1 − L)

It implies that  N f (x) − S(x),

L 2β |a|6β (1

− L)

for all x ∈ X and all t > 0 . By replacing t by   N f (x) − S(x), t ≤

 t ≤

φ(x, 0) t + φ(x, 0)

2β |a|6β (1−L) L

t , we have

Lφ(x, 0) 2β |a|6β (1 − L) t + Lφ(x, 0)

for all x ∈ X and all t > 0 . That is, the inequality (4.3) holds. By letting x = and y = ayn in the inequality (4.2), we have

x an

 x y   φ( axn , ayn ) N a6n Da f n , n , |a|6βn t ≤ a a t + φ( axn , ayn ) for all x, y ∈ X , all t > 0 and all n ∈ N . Replacing t by  x y   N a6n Da f n , n , t ≤ a a

φ( axn , ayn ) t + φ( axn , |a|6βn

y an )

t |a|6βn

≤

,

Ln φ(x, y) t + Ln φ(x, y)

for all x, y ∈ X , all t > 0 and all n ∈ N . Since limn→∞ x, y ∈ X and all t > 0 , we may conclude that   N Da S(x, y), t = 0

Ln φ(x, y) t+Ln φ(x, y)

= 0 for all

for all x, y ∈ X and all t > 0 . Thus the mapping S : X → Y is the generalized sextic mapping.  Corollary 4.2. Let θ ≥ 0 , p > 6 be a real number and X be a normed linear space with norm || · || . Suppose f : X → Y is a mapping satisfying f (0) = 0 and (4.7)

N (Da f (x, y), t) ≤

θ(||x||p + ||y||p ) t + θ(||x||p + ||y||p )

  for all x, y ∈ X and all t > 0 . Then S(x) := N- limn→∞ a6n f axn exists for each x ∈ X and defines a generalized sextic mapping S : X → Y such that N (f (x) − S(x), t) ≤

2β (|a|pβ

θ ||x||p − |a|6β ) t + θ ||x||p

for all x ∈ X and all t > 0 . Proof. The proof follows from Theorem 4.1 by taking φ(x, y) = θ(||x||p + ||y||p ) for all x , y ∈ X and L = |a|(6−p)β . 

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[28] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978),297–300. [29] S. Rolewicz, Metric Linear Spaces, Reidel/PWN-Polish Sci. Publ., Dordrecht,(1984). [30] I.A. Rus, Principles and Appications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian). [31] J. Tabor and J. Tabor, General stability of functional equations of linear type, Journal of Mathematical Analysis and Applications, 328, no. 1(2007), 192–200. [32] P.K. Sahoo, A generalized cubic functional equation, Acta Math. Sinica 21 no. 5 (2005), 1159–1166. [33] S. M. Ulam, Problems in Morden Mathematics, Wiley, New York, USA, 1960. [34] T.Z. Xu, J.M. Rassias and W.X. Xu, A generalized mixed quadratic-quartic functional equation, Bull. Malaysian Math. Scien. Soc. 35 no. 3 (2012), 633–649. [35] C. Wu and J. Fang, Fuzzy generalization of Klomogoroffs theorem, J. Harbin Inst. Technol. 1 (1984), 1–7. [36] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 no. 3 (2003), 389–399. [37] T.Z. Xu, J.M. Rassias, M.J. Rassias and W. X. Xu, A Fixed Point Approach to the Stability of Quintic and Sextic Functional Equations in Quasi-β-Normed Spaces, Journal of Inequalities and Applications 2010. Department of Mathematical Education, Dankook University, 126, Jukjeon, Suji, Yongin, Gyeonggi, South Korea 448-701 , Korea E-mail address: [email protected] (H. Koh) E-mail address: [email protected] (D. Kang)

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Existence and uniqueness of solutions to SFDEs driven by G-Brownian motion with non-Lipschitz conditions ∗

Faiz Faizullah Department of BS and H, College of E and ME, National University of Sciences and Technology (NUST) Pakistan. January 1, 2016

Abstract The main aim of this paper is to study the existence, uniqueness and stability of solution for stochastic functional differential equations driven by G-Brownian motion (in short G-SFDEs). The existence-and-uniqueness theorem is established for G-SFDEs under non-Lipschitz condition and weakened linear growth condition. We have used the Picard approximation scheme, Gronwall’s inequality, Bihari’s inequality and Burkholder-Davis-Gundy (in short BDG) inequalities to develop the existence theory for the above mentioned stochastic dynamical systems. In addition, the mean square stability of solutions for these systems has been obtained. Key words: Existence, uniqueness, stability, G-Brownian motion, stochastic functional differential equations.

1

Introduction

Responding to the contemporary developments in the fields of physics, control engineering, economics, and social sciences, a growing concern has recently been witnessed in both stochastic differential and deterministic models. The applications of functional differential equations have been applied in a number of cases in physical phenomena, such as in the relocation of soil moisture, where the fluid flows through the crack of rocks, and the problem of conduction of heat as well as its share in order fluids is investigated. The idea of G-Brownian motion as well as the associated stochastic differential equations were introduced by Peng [8, 10]. These equations were extended to stochastic functional differential equations, which are driven by G-Brownian motion (in short G-SFDEs) by Ren, Bi and Sakthivel [12]. While Faizullah, developed the existence-and-uniqueness theorem for GSFDEs with Cauchy-Maruyama approximation scheme [3], they used the strong Lipschitz and linear growth conditions to develop the mentioned theory. In this article, we have generalized the existence theory for functional stochastic dynamical systems, driven by G-Brownian motion. We have used non-Lipschitz condition and weak linear growth condition to study the existence, uniqueness and stability theory for G-SFDEs. We have considered the following stochastic dynamical system that ∗

Corresponding author, E-mail: faiz [email protected] ¯

1

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is driven by G-Brownian motion. Let 0 ≤ t ≤ T < ∞. Suppose g : [0, T ] × BC([−θ, 0]; Rn ) → Rn , h : [0, T ] × BC([−θ, 0]; Rn ) → Rn and w : [0, T ] × BC([−θ, 0]; Rn ) → Rn are Borel measurable. Consider stochastic functional differential equation driven by G-Brownian motion of the type dX(t) = g(t, Xt )dt + h(t, Xt )d⟨B, B⟩(t) + w(t, Xt )dB(t),

(1.1)

where X(t) is the value of stochastic process at time t and Xt = {X(t + δ) : −θ ≤ δ ≤ 0, θ > 0} is a BC([−θ, 0]; Rn )-valued stochastic process, which presents the family of bounded continuous Rn -valued functions φ defined on [−θ, 0] having norm ∥φ∥ = sup | φ(δ) | . {⟨B, B⟩(t), t ≥ 0} is −θ≤δ≤0

the quadratic variation process of G-Brownian motion {B(t), t ≥ 0} and g, h, w ∈ MG2 ([−τ, T ]; Rn ). Denote the space of all Ft -adapted process X(t), 0 ≤ t ≤ T , such that ∥ X ∥L2 = sup |X(t)| < ∞ −θ≤t≤T

by L2 . The initial data of equation (1.1) is given as follows Xt0 =ζ = {ζ(δ) : −θ < δ ≤ 0} is F0 − measurable, BC([−θ, 0]; Rn ) − valued random variable such that ζ ∈ MG2 ([−θ, 0]; Rn ) .

(1.2)

The integral form of G-SFDE (1.1) with initial data (1.2) is given by ∫ t ∫ t ∫ t X(t) = ζ(0) + g(s, Xs )ds + h(s, Xs )d⟨B, B⟩(s) + w(s, Xs )dB(s). 0

0

0

The solution of G-SFDE (1.1) with initial data (1.2) is an t ∈ [−θ, T ] such that

Rn

valued stochastic processes X(t),

(i) X(t) is Ft -adapted and continuous for all t ∈ [0, T ]; (ii) g(t, Xt ) ∈ L1 ([o, T ]; Rn ) and h(t, Xt ), w(t, Xt ) ∈ L2 ([0, T ]; Rn ); (iii) X0 = ζ and for each t ∈ [0, T ], dX(t) = g(t, Xt )dt + h(t, Xt )d⟨B, B⟩(t) + w(t, Xt )dB(t) q.s. X(t) is called a unique solution if it is indistinguishable from any other solution Y (t), that is, E[ sup |X(q) − Y (q)|2 ] = 0. −θ≤q≤t

Throughout this paper we assume the following two conditions, known as non-uniform Lipschitz condition and weakened linear growth condition respectively. (Ai ) For all φ, ψ ∈ BC([−θ, 0]; Rd ) and t ∈ [0, T ], |g(t, φ) − g(t, ψ)|2 + |h(t, φ) − |h(t, ψ)|2 + |w(t, φ) − w(t, ψ)|2 ≤ λ(|φ − ψ|2 ),

(1.3)

where λ(.) : R+ → R+ is a non-decreasing and concave function such that λ(0) = 0, λ(v) > 0 for v > 0 and ∫ dv = ∞. (1.4) 0+ λ(v) As λ is concave and λ(0) = 0, there exists two positive constants c and d such that λ(v) ≤ c + dv,

(1.5)

for all v ≥ 0. 2

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(Aii ) For all t ∈ [0, T ], g(t, 0), h(t, 0), w(t, 0) ∈ L2 and |g(t, 0)|2 + |h(t, 0)|2 + |w(t, 0)|2 ≤ K,

(1.6)

where K is a positive constant. We have organized the rest of the paper as follows. In section 2, some well-known basic notions and results are included. In section 3, several important lemmas are developed. In section 4, the existence-and-uniqueness theorem is proved. In section 5, the mean square stability for the solution of G-SFDEs is given.

2

Preliminaries

The main purpose of this section is to give some basic concepts and results, which are used in the subsequent sections of this paper. For more detailed literature of G-expectation, we refer the readers to book [9] and papers [1, 2, 4, 5, 13]. Definition 2.1. Let H be a linear space of real valued functions defined on a nonempty basic space Ω. Then a sub-linear expectation E is a real valued functional on H with the following properties: (i) For all X, Y ∈ H, if X ≤ Y then E[X] ≤ E[Y ]. (ii) For any real constant α, E[α] = α. (iii) For all X, Y ∈ H, E[X + Y ] ≤ E[X] + E[Y ]. (iv) For any θ > 0 E[θX] = θE[X]. Let Cb.Lip (Rl×d ) denotes the set of bounded Lipschitz functions on Rl×d and LpG (ΩT ) = {ϕ(Bt1 , Bt2 , ..., Btl /l ≥ 1, t1 , t2 , ..., tl ∈ [0, T ], ϕ ∈ Cb.Lip (Rl×d ))}. Let ξi ∈ LpG (Ωti ), i = 0, 1, ..., N −1 then MG0 (0, T ) denotes the collection of processes of the following type: For a given partition πT = {t0 , t1 , ..., tN } of [0, T ], ηt (w) =

N −1 ∑

ξi (w)I[ti ,ti+1 ] (t).

i=0

∫T Under the norm ∥η∥ = { 0 E[|ηu |p ]du}1/p , MGp (0, T ), p ≥ 1, is the completion of MG0 (0, T ). For every ηt ∈ MG2,0 (0, T ), the G-Itˆo’s integral I(η) and G-quadratic variation process {⟨B⟩t }t≥0 are respectively given by ∫ I(η) =

T

ηu dBu = 0

∫

⟨B⟩t = Bt2 − 2

N −1 ∑

ξi (Bti+1 − Bti ),

i=0 t

Bu dBu . 0

The following definition and lemmas are borrowed from [7, 11]. 3

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Definition 2.2. A solution X(t) of dynamical system (1.1) with initial data (1.2) is said to be stable in mean square if for all ϵ > 0 there exists δ(ϵ) > 0 such that E|ζ − ξ|2 ≤ δ(ϵ) follows that E|X(t) − Y (t)|2 < ϵ for all t ≥ 0, where Y (t) is an other solution of system (1.1) having initial data ξ ∈ M 2 ([−θ, 0] : Rl ). Lemma 2.3. (H¨ older’s inequality) If and ∫ d

1 q

+

1 r

= 1 for any q, r > 1, g ∈ L2 and h ∈ L2 then gh ∈ L1

∫ gh ≤ (

c

d

∫ |g| ) (

d

1 q

q

c

1

|h|r ) r .

c

Lemma 2.4. (Gronwall’s inequality) Let C ≥ 0, h(t) ≥ 0 and w(t) be a real valued continuous ∫d function on [c, d]. If for all c ≤ t ≤ d, w(t) ≤ C + c h(s)w(s)ds, then w(t) ≤ Ce

∫t c

h(s)ds

,

for all c ≤ t ≤ d. Lemma 2.5. (Bihari’s inequality) Suppose T ≥ 0 and h0 ≥ 0. Assume h(t) and w(t) be continuous functions on [0, T ]. Let λ(.) : R+ → R+ be non-decreasing and concave continuous function such ∫T that λ(v) > 0 for v > 0. If for all 0 ≤ t ≤ T, h(t) ≤ h(0)+ 0 w(s)λ(h(s))ds, then for all 0 ≤ t ≤ T , h(t) ≤ H

−1

∫ (H(h0 ) +

T

w(s)ds), t

such that H(h0 ) + function of H.

∫T t

w(s)ds ∈ Dom(H −1 ) where H(q) =

∫q

1 t λ(s) ds,

q ≥ 0 and H −1 is the inverse

Lemma 2.6. Assume the assumptions of lemma 2.5 are satisfied and for 0 ≤ t ≤ T , w(t) ≥ 0. If ∫T ∫T 1 ds holds, then for for all ϵ > 0, there exists t1 ≥ 0 such that for 0 ≤ h0 ≤ ϵ, t1 w(s)ds ≤ h0 λ(s) each t1 ≤ t ≤ T h(t) ≤ ϵ, holds.

3

Important results

In this section, we show some important lemmas. They will be used in the forth coming existenceand-uniqueness theorem. Let X 0 (t) = ζ(0) for t ∈ [0, T ]. Set X l (0) = ζ for each l = 1, 2, ..., and define the following Picard iterations sequence, ∫ t ∫ t l l−1 X (t) = ζ(0) + g(s, Xs )ds + h(s, Xsl−1 )d⟨B, B⟩(s) 0 0 (3.1) ∫ t l−1 + w(s, Xs )dB(s), t ∈ [0, T ]. 0

First, we show that X l (.) ∈ MG2 ([−θ, T ]; Rn ).

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Lemma 3.1. Let assumptions Ai and Aii hold. Then for all l ≥ 1, sup E|X l (t)|2 ≤ C,

−θ≤t≤T

where C is a positive constant. Proof. Obviously, X 0 (.) ∈ MG2 ([−θ, T ]; Rn ). Using the basic inequality |a + b + c + d|2 ≤ 4|a|2 + 4|b|2 + 4|c|2 + 4|d|2 , equation (3.1) yields ∫ t ∫ t l 2 2 l−1 2 |X (t)| ≤ 4|ζ(0)| + 4| g(s, Xs )ds| + 4| h(s, Xsl−1 )d⟨B, B⟩(s)|2 0 0 ∫ t + 4| w(s, Xsl−1 )dB(s)|2 . 0

Taking G-expectation on both sides, using the Burkholder-Davis-Gundy (BDG) inequalities [6] and H¨ older inequality (lemma 2.3) we have ∫ t l 2 2 E|X (t)| ≤ 4E|ζ(0)| + 4C1 E |g(s, Xsl−1 )|2 ds 0 ∫ t ∫ t + 4C2 E |h(s, Xsl−1 )|2 ds + 4C3 |w(s, Xsl−1 )|2 ds 0 0 ∫ t ≤ 4E|ζ(0)|2 + 8C1 E (|g(s, Xsl−1 ) − g(s, 0)|2 + |g(s, 0)|2 )ds 0 ∫ t + 8C2 E (|h(s, Xsl−1 ) − h(s, 0)|2 + |h(s, 0)|2 )ds 0 ∫ t (|w(s, Xsl−1 ) − w(s, 0)|2 + |w(s, 0)|2 )d(s) + 8C3 0 ∫ t ∫ t 2 2 |g(s, Xsl−1 ) − g(s, 0)|2 ds |g(s, 0)| ds + 8C1 E ≤ 4E|ζ(0)| + 8C1 E 0 0 ∫ t ∫ t 2 + 8C2 E |h(s, 0)| d(s) + 8C2 E |h(s, Xsl−1 ) − h(s, 0)|2 ds 0 0 ∫ t ∫ t |w(s, Xsl−1 ) − w(s, 0)|2 ds |w(s, 0)|2 ds + 8C3 + 8C3 0

0

By assumptions Ai and Aii , the above inequality yields E|X l (t)|2 ≤ 4E|ζ(0)|2 + 8C1 KT + 8C2 KT + 8C3 KT ∫ t ∫ t ∫ t l−1 2 l−1 2 + 8C1 E λ(|Xs )| )ds + 8C2 E λ(|Xs )| )d(s) + 8C3 λ(|Xsl−1 )|2 )d(s) 0 0 0 ∫ t = 4E|ζ(0)|2 + 8KT (C1 + C2 + C3 ) + 8(C1 + C2 + C3 )E λ(|Xsl−1 )|2 )ds 0

≤ 4E|ζ(0)|2 + 8KT (C1 + C2 + C3 ) + 8a(C1 + C2 + C3 )T ∫ t + 8b(C1 + C2 + C3 )E |Xsl−1 )|2 ds 0 ∫ t = K1 + 8b(C1 + C2 + C3 )E |Xsl−1 )|2 ds, 0

5

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where K1 = 4E|ζ(0)|2 + 8C0 KT + 8aC0 T. and C0 = C1 + C2 + C3 . Noting that sup |Xsl |2 ≤ sup

sup |X l (s + u)|2 ≤ sup |X l (q)|2 ≤ |ζ|2 + sup |X l (q)|2 ,

0≤s≤t −θ≤u≤0

0≤s≤t

−θ≤q≤t

0≤q≤t

we have ∫ sup E|X l (q)|2 ≤ E|ζ|2 + K1 + 8b(C1 + C2 + C3 )E

−θ≤q≤t

t

sup |X l−1 (q)|2 ds.

0 −θ≤q≤t

Again noting that for any j ≥ 1 max E|Xsl−1 |2 ≤ E|ζ|2 + max E|X l (q)|2 ,

1≤l≤j

1≤l≤j

we obtain ∫ max

sup E|X (q)| ≤ E|ζ| + K1 + 8b(C1 + C2 + C3 ) l

2

t

2

1≤l≤j −θ≤q≤t

[E|ζ|2 + max 0

∫

≤ E|ζ| + K1 + 8b(C1 + C2 + C3 )T E|ζ| + 2

2

∫ = K2 + 8b(C1 + C2 + C3 )

sup E|X l (q)|2 ]ds

1≤l≤j −θ≤q≤t

t

max

sup E|X l (q)|2 ds

0 1≤l≤j −θ≤q≤t

t

max

sup E|X l (q)|2 ds,

0 1≤l≤j −θ≤q≤t

where K2 = K1 + (1 + 8bC0 T )E|ζ|2 . Now the Gronwall inequality (lemma 2.4) yields max

sup E|X l (t)|2 ≤ C,

1≤l≤j −θ≤q≤t

where C = K2 e8bC0 T , but j is arbitrary, so sup E|X l (t)|2 ≤ C.

−θ≤t≤T

The proof is complete. Lemma 3.2. Under the assumptions Ai and Aii there exists a positive constant C ∗ such that for all l, d ≥ 1, ∫ t E sup |X l+d (s) − X l (s)|2 ≤ Cˆ λ(E sup |X l+d−1 (q) − X l−1 (q)|2 )ds −θ≤s≤t

−θ≤q≤s

0 ∗

≤ C t. Proof. Using the basic inequality |a + b + c|2 ≤ 3|a|2 + 3|b|2 + 3|c|2 , equation (3.1) yields ∫ |X l+d (t) − X l (t)|2 ≤ 3| ∫

t

0 t

+ 3|

∫ [g(s, Xsl+d−1 ) − g(s, Xsl−1 )]ds|2 + 3|

t

[h(s, Xsl+d−1 ) − h(s, Xsl−1 )]d⟨B, B⟩(s)|2

0

[w(s, Xsl+d−1 ) − w(s, Xsl−1 )]dB(s)|2

t0

6

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Taking G-expectation on both sides, using the BDG inequalities [6], Jensen inequality E(λ(x)) ≤ λ(E(x)), Holder inequality and assumptions Ai , Ai it gives ∫ E[ sup |X l+d (s) − X l (s)|2 ] ≤ 3C1 −θ≤s≤t

∫

t

0 t

+ 3C2

λ(E[ sup |X l+d−1 (q) − X l−1 (q)|2 ])ds −θ≤q≤s

λ(E[ sup |X l+d−1 (q) − X l−1 (q)|2 ])ds −θ≤q≤s

0

∫

t

+ 3C3

λ(E[ sup |X l+d−1 (q) − X l−1 (q)|2 ])ds −θ≤q≤s

0

∫

≤ 3(C1 + C2 + C3 )

t

λ(E[ sup |X l+d−1 (q) − X l−1 (q)|2 ])ds.

0

∫ E[ sup |X −θ≤s≤t

l+d

t

(s) − X (s)| ] ≤ Cˆ l

2

−θ≤q≤s

λ(E[ sup |X l+d−1 (q) − X l−1 (q)|2 ])ds, −θ≤q≤s

0

where Cˆ = 3C0 . Finally, using lemma 3.1 it yields ˆ E[ sup |X l+d (s) − X l (s)|2 ] ≤ Cλ(4C)t = C ∗ t, −θ≤s≤t

ˆ where C ∗ = Cλ(4C). The proof is complete.

4

Existence and uniqueness results for G-SFDEs

We introduce the following new notations to prepare a key lemma. Choose T1 ∈ [0, T ] such that for all t ∈ [0, T1 ] ∗ ˆ Cλ(C t) ≤ C ∗ . (4.1) For all l, d ≥ 1, define the following recursive function ϕ1 (t) = C ∗ t.

∫ ϕl+1 (t) = Cˆ

(4.2)

t

λ(ϕl (s))ds, 0

ϕl,d (t) = E[ sup |X l+d (q) − X l (q)|2 ].

(4.3)

−θ≤q≤t

Lemma 4.1. Under the hypothesis Ai and Aii for any d ≥ 1 and all l ≥ 1 there exists a positive T1 ∈ [0, T ] such that 0 ≤ ϕl,d (t) ≤ ϕl (t) ≤ ϕl−1 (t) ≤ ... ≤ ϕ1 (t),

(4.4)

for all t ∈ [0, T1 ].

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Proof. We use mathematical induction to prove the inequality (4.4). Using the definition of function ϕ(.) and lemma 3.2, we have ϕ1,d (t) = E[ sup |X 1+d (q) − X 1 (q)|2 ] ≤ C ∗ t = ϕ1 (t). −θ≤q≤t

ϕ2,d (t) = E[ sup |X 2+d (q) − X 2 (q)|2 ] −θ≤q≤t t

∫ ≤ Cˆ

λ(E[ sup |X 1+d (q) − X 1 (q)|2 ])ds −θ≤q≤t

0

∫

t

≤ Cˆ

λ(ϕ1 (s))ds = ϕ2 (t). 0

Using (4.1), we have ∫

∫

t

ϕ2 (t) = Cˆ

t

λ(ϕ1 (s))ds = 0

∗ ˆ Cλ(C t)ds ≤ C ∗ t = ϕ1 (t).

0

Hence for all t ∈ [0, T1 ], we derive that ϕ2,d (t) ≤ ϕ2 (t) ≤ ϕ1 (t). Next, suppose that the inequality (4.4) holds for some l ≥ 1. We now show that lemma 4.1 is valid for l + 1, as follows ϕl+1,d (t) = E[ sup |X l+d+1 (q) − X l+1 (q)|2 ] −θ≤q≤t t

∫ ≤ Cˆ

λ(E[ sup |X l+d (q) − X l (q)|2 ])ds −θ≤q≤s

0

∫ = Cˆ

t

λ(ϕl,d (s))ds 0

∫ ≤ Cˆ

t

λ(ϕl (s))ds 0

= ϕl+1 (t). Also

∫ ϕl+1 (t) = Cˆ

t

∫ λ(ϕl (s))ds ≤ Cˆ

0

t

λ(ϕl−1 (s))ds = ϕl (s). 0

Hence for all t ∈ [0, T1 ], we derive that ϕl+1,d (t) ≤ ϕl+1 (t) ≤ ϕl (s), that is, lemma 4.1 holds for l + 1. The proof is complete. Theorem 4.2. Let assumptions Ai and Aii hold. Then the stochastic system (1.1) with initial data (1.2) has a unique solution. Proof. We split the whole proof in two steps. First, we show uniqueness and then existence. Let system (1.1) with initial data (1.2) has two solutions X(t) and Y (t). Then we have ∫ t ∫ t |X(t) − Y (t)| ≤ |g(s, Xs ) − g(s, Ys )|ds + |h(s, Xs ) − h(s, Ys )|d⟨B, B⟩(s) 0 0 ∫ t + |w(s, Xs ) − w(s, Ys )|dB(s). 0

8

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Taking G-expectation on both sides and using the basic inequality (a + b + c)2 ≤ 3(a2 + b2 + c2 ), H¨ older inequality and BDG inequalities [6], it follows ∫ t ∫ t E|X(t) − Y (t)|2 ≤ 3C1 E|g(s, Xs ) − g(s, Ys )|2 ds + 3C2 E|h(s, Xs ) − h(s, Ys )|2 ds 0 0 ∫ t + 3C3 E|w(s, Xs ) − w(s, Ys )|ds. 0

Using assumptions Ai and Aii we have ∫ E[ sup |X(q) − Y (q)| ] ≤ 3(C1 + C2 + C3 ) 2

−θ≤q≤t

t

λ(E[ sup |X(q) − Y (q)|2 ])ds, −θ≤q≤s

0

Then lemma 2.5 and lemma 2.6 gives E[sup−θ≤q≤t |X(q) − Y (q)|2 ] = 0, t ∈ [0, T ]. The proof of uniqueness is complete. Next we show existence. We note that on t ∈ [0, T1 ], ϕl (t) is continuous. For l ≥ 1, it is decreasing on t ∈ [0, T1 ]. By dominated convergence theorem, we define the function ϕ(t) as follows ∫ t ∫ t λ(ϕl−1 (s))ds = Cˆ λ(ϕ(s))ds, 0 ≤ t ≤ T1 . ϕ(t) = lim ϕl (t) = lim Cˆ l→∞

l→∞

0

0

So, ∫ ϕ(t) ≤ ϕ(0) + Cˆ

t

λ(ϕ(s))ds. 0

Thus for all 0 ≤ t ≤ T1 , lemma 2.5 and lemma 2.6 follow that ϕ(t) = 0. From lemma 4.1 for all t ∈ [0, T1 ] we get ϕl,d (s) ≤ ϕl (s) → 0 as l → ∞, which yields E|X l+d (t) − X l (t)|2 → 0 as l → ∞. By the property of function λ(.), assumptions Ai , Aii and completeness of L2 , it follows that for all t ∈ [0, T1 ], g(t, Xtl ) → g(t, Xt ), h(t, Xtl ) → h(t, Xt ), w(t, Xtl ) → w(t, Xt ) in L2 as l → ∞. Hence for all t ∈ [0, T1 ], ∫

t

l

lim X (t) = ζ(0) + lim g(s, Xsl−1 )ds l→∞ l→∞ 0 ∫ t ∫ t + lim h(s, Xsl−1 )d⟨B, B⟩(s) + lim w(s, Xsl−1 )dB(s), l→∞ 0

l→∞ 0

that is, ∫ X(t) = ζ(0) +

∫

t

g(s, Xs )ds + 0

∫

t

h(s, Xs )d⟨B, B⟩(s) + 0

t

w(s, Xs )dB(s). 0

Thus X(t) is a unique solution of stochastic system (1.1) with initial data (1.2) on t ∈ [0, T1 ]. Thus by iteration, one can obtain that the system (1.1) has a unique solution on t ∈ [0, T ]. The proof is complete. 9

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5

Dependence of solutions

In this section, we use lemma 2.5 and lemma 2.6 to give continuous dependence of solutions for stochastic system (1.1) with initial data (1.2). Theorem 5.1. Let assumptions Ai and Aii hold. Assume X(t) and Y (t) be two solutions of dynamical system (1.1) with initial data ζ and ξ respectively. If for all ϵ > 0 and t ∈ [0, T ] there exists δ(ϵ) > 0 such that E|ζ − ξ|2 < δ(ϵ), then E|X(t) − Y (t)|2 ≤ ϵ. Proof. Since X(t) and Y (t) are any two solutions of system (1.1). It follows that for any t ∈ [0, T ], ∫

∫

t

X(t) = ζ(0) +

g(s, Xs )ds + 0

∫ Y (t) = ξ(0) +

∫

t

w(s, Xs )dB(s) q.s.

0

∫

t

0

∫

t

g(s, Ys )ds + 0

t

h(s, Xs )d⟨B, B⟩(s) + t

h(s, Ys )d⟨B, B⟩(s) +

w(s, Ys )dB(s) q.s.

0

0

Then ∫

∫

t

X(t) − Y (t) = ζ(0) − ξ(0) + [g(s, Xs ) − g(s, Ys )]ds + 0 ∫ t [w(s, Xs ) − w(s, Ys )]dB(s) q.s. +

t

[h(s, Xs ) − h(s, Ys )]d⟨B, B⟩(s)

0

0

Taking G-expectation on both sides, using the fundamental inequality (a + b + c + d)2 ≤ 4(a2 + b2 + c2 + d2 ), BDG inequalities [6] and H¨ older inequality, it follows ∫ E[ sup |X(r) − Y (r)| ] ≤ 4E|ζ(0) − ξ(0)| + 4(C1 + C2 + C3 ) 2

2

−θ≤r≤t

0

t

λ(E[ sup |X(r) − Y (r)|2 ])ds. −θ≤r≤t

Thus from lemma 2.5 and 2.6 we have E[|X(t) − Y (t)|2 ] ≤ ϵ, for t ∈ [0, T ]. The proof is complete.

6

Acknowledgement

The financial support of NUST research directorate for this research work is acknowledged and deeply appreciated.

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References [1] X. Bai, Y. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with Integral-Lipschitz coefficients, Acta Mathematicae Applicatae Sinica, English Series, 30(3) (2014) 589–610. [2] L. Denis, M. Hu, S. Peng, Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths, Potential Anal., 34 (2010) 139–161. [3] F. Faizullah, Existence of solutions for G-SFDEs with Cauchy-Maruyama approximation scheme. Abstract and Applied Analysis, http://dx.doi.org/10.1155/2014/809431, volume 2014 (2014) 1–8. [4] F. Faizullah, A. Mukhtar, M. A. Rana, A note on stochastic functional differential equations driven by G-Brownian motion with discontinuous drift coefficients, J. Computational Analysis and Applications, 21(5) 2016, 910-919. [5] M. Hu, S. Peng, Extended conditional G-expectations and related stopping times, arXiv:1309.3829v1[math.PR] 16 Sep 2013. [6] F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Processes and thier Applications, 2 (2009) 3356– 3382. [7] X. Mao, Stochastic Differential Equations and their Applications, Horwood Publishing Chichester, 1997. [8] S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Ito’s type, The abel symposium 2005, Abel symposia 2, edit. benth et. al., Springer-vertag. (2006) 541-567. [9] S. Peng, Nonlinear Expectations and Stochastic Calculus under Uncertainty, arXiv:1002.4546v1 [math.PR] (2010). [10] S. Peng, Multi-dimentional G-Brownian motion and related stochastic calculus under Gexpectation, Stochastic Processes and thier Applications, 12 (2008) 2223–2253. [11] Y. Ren, NM Xia, Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput. 210(1) (2009) 72– 79. [12] Y. Ren, Q. Bi, R. Sakthivel, Stochastic functional differential equations with infinite delay driven by G-Brownian motion, Mathematical Methods in the Applied Sciences, 36(13) (2013) 1746–1759. [13] Y. Song, Properties of hitting times for G-martingale and their applications, Stochastic Processes and thier Applications, 8(121) (2011) 1770–1784.

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Approximation of a kind of new Bernstein-B´ezier type operators Mei-Ying Ren1∗ , Xiao-Ming Zeng2∗, Wen-Hui Zhang2 1

School of Mathematics and Computer Science, Wuyi University, Wuyishan 354300, China

2

School of Mathematical Sciences, Xiamen University, Xiamen 361005, Chnia E-mail: [email protected],

[email protected], [email protected]

Abstract. In this paper, a kind of new Bernstein-B´ezier type operators is introduced. The Korovkin type approximation theorem of these operators is investigated. The rates of convergence of these operators are studied by means of modulus of continuity. Then, by using the Ditzian-Totik modulus of smoothness, a direct theorem concerned with an approximation for these operators is also obtained. Keywords: Bernstein-B´ezier type operators; Korovich type approximation theorem; rate of convergence; direct theorem; modulus of smoothness Mathematical subject classification: 41A10, 41A25, 41A36

1. Introduction In view of the B´ezier basis function, which was introduced by B´ezier [1], in 1983, Chang [2] defined the generalized Bernstein-B´ezier polynomials for any α > 0, and a function f defined on [0, 1] as follows: Bn,α (f ; x) =

n X k=0

k α α f ( )[Jn,k (x) − Jn,k+1 (x)], n

(1)

n P where Jn,n+1 (x) = 0, and Jn,k (x) = Pn,i (x), k = 0, 1, ..., n, Pn,i (x) = i=k µ ¶ n xi (1 − x)n−i . Jn,k (x) is the B´ezier basis function of degree n. i Obviously, when α = 1, Bn,α (f ; x) become the well-known Bernstein polynomials Bn (f ; x), and for any x ∈ [0, 1], we have 1 = Jn,0 (x) > Jn,1 (x) > ... > Jn,n (x) = xn , Jn,k (x) − Jn,k+1 (x) = Pn,k (x). During the last ten years, the B´ezier basis function was extensively used for constructing various generalizations of many classical approximation processes. Some B´ezier type operators, which are based on the B´ezier basis function, have been introduced and studied (e.g., see [3-9]). In 2013, Ren [10] introduced generalized Bernstein operators as follows:

En,β (f ; x) = f (0)Pn,0 (x) +

n−1 X

(β)

Pn,k (x)Fn,k (f ) + f (1)Pn,n (x),

(2)

k=1 ∗ Corresponding

authors: Mei-Ying Ren and Xiao-Ming Zeng

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µ where f ∈ C[0, 1], x ∈ [0, 1], Pn,k (x) = (β)

Fn,k (f ) =

Z

1 B(nk, n(n − k))

1

0

n k

¶ xk (1 − x)n−k , k = 0, 1, ..., n, and

k tnk−1 (1 − t)n(n−k)−1 f (βt + (1 − β) )dt, n

(3)

where k = 1, ..., n − 1, β ∈ [0, 1], B(., .) is the beta function. The moments of the operators En,β (f ; x) were obtained as follows (see [10]). Remark For En,β (tj ; x), j = 0, 1, 2, we have (i) En,β (1; x) = 1; (ii) En,β (t; x) = x;

¸ 1 (n − 1)β 2 (iii) En,β (t ; x) = x + + x(1 − x). n (n2 + 1)n 2

·

2

In the present paper, we will study the B´ezier variant of the generalized Bernstein operators En,β (f ; x) given by (2). We introduce Bernstein-B´ezier type operators as follows: (α)

(α)

En,β (f ; x) = f (0)Qn,0 (x) +

n−1 X

(α)

(β)

Qn,k (x)Fn,k (f ) + f (1)Q(α) n,n (x),

(4)

k=1 (α)

α α where f ∈ C[0, 1], x ∈ [0, 1], β ∈ [0, 1], α > 0, Qn,k (x) = Jn,k (x) − Jn,k+1 (x), µ ¶ n P n Jn,n+1 (x) = 0, Jn,k (x) = Pn,i (x), k = 0, 1, ..., n, Pn,i (x) = xi (1 − i i=k (β)

x)n−i , and Fn,k (f ) is defined as above (3). (α)

It is clear that En,β (f ; x) are bounded and positive on C[0,1]. When α = 1, (α)

(α)

En,β (f ; x) become the operators En,β (f ; x). When β = 0, En,β (f ; x) become the generalized Bernstein-B´ezier operators Bn,α (f ; x). The goal of this paper is to study the approximation properties of these operators with the help of the Korovkin type approximation theorem. We also estimate the rates of convergence of these operators by using a modulus of continuity. Then we obtain the direct theorem concerned with an approximation for these operators by means of the Ditzian-Totik modulus of smoothness. In the paper, for f ∈ C[0, 1], we denote kf k = max{|f (x)| : x ∈ [0, 1]}. ω(f, δ) (δ > 0) denotes the usual modulus of continuity of f ∈ C[0, 1].

2. Auxiliary results In the sequel, we shall need the following auxiliary results. Lemma 1 (see [2]) Let α > 0. We have n

1X α Jn,k (x) = x uniformly on [0, 1]; n→∞ n

(i) lim

k=1 n X

1 n→∞ n2

(ii) lim

α kJn,k (x) =

k=1

x2 uniformly on [0, 1]. 2

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Lemma 2 Let α > 0. We have (α)

(i) En,β (1; x) = 1; (α)

(ii) lim En,β (t; x) = x uniformly on [0, 1]; n→∞

(α)

(iii) lim En,β (t2 ; x) = x2 uniformly on [0, 1]. n→∞

(β)

Proof 2

β n2 +1

(β)

By simple calculation, we obtain Fn,k (1) = 1, Fn,k (t) = 2

(i)

+ (1 − n2β+1 ) nk 2 . n P (α) (α) Since Qn,k (x) = 1, by (4) we can get En,β (1; x) = 1.

(ii)

By (4), we have

·

k n

k n,

(β)

Fn,k (t2 ) =

2

k=0

(α)

En,β (t; x) =

n−1 X

(α)

Qn,k (x)

k=1

k + Q(α) n,n (x) n

α α = [Jn,1 (x) − Jn,2 (x)]

n 1 n−1 α α α + ... + [Jn,n−1 (x) − Jn,n (x)] + Jn,n (x) n n n

n

=

1X α Jn,k (x), n k=1

(α)

thus, by Lemma 1 (i), we have lim En,β (t; x) = x uniformly on [0, 1]. (iii)

n→∞

By (4), we have (α)

En,β (t2 ; x) =

n−1 X

(α)

k=1

=

·

Qn,k (x)

¸ k β2 k2 β2 · + (1 − ) + Q(α) n,n (x) n2 + 1 n n2 + 1 n2

n n β2 1X β2 1 X 2 (α) (α) · kQ (x) + (1 − ) · k Qn,k (x) n,k n2 + 1 n n2 + 1 n2 k=1

k=1

n n β2 1X α β2 1 X α = 2 · Jn,k (x) + (1 − 2 )· 2 (2k − 1)Jn,k (x), n +1 n n +1 n k=1

k=1

(α)

thus, by Lemma 1, we have lim En,β (t2 ; x) = x2 uniformly on [0, 1]. n→∞

Lemma 3 (see [11]) For x ∈ [0, 1], k = 0, 1, ..., n, we have ½ αPn,k (x), α ≥ 1; (α) 0 ≤ Qn,k (x) ≤ α Pn,k (x), 0 < α < 1. Lemma 4 (see [12]) For 0 < α < 1, γ > 0, we have n X

γ

α |k − nx|γ Pn,k (x) ≤ (n + 1)1−α (A αγ )α n 2 ,

k=0

where the constant As only depends on s.

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Lemma 5 For α ≥ 1, we have α β2 1 (α) (i) En,β ((t − x)2 ; x) ≤ (1 + )· ; 5 n r4 r 2 α β 1 (α) (ii) En,β (|t − x|; x) ≤ (1 + )· . 4 5 n Proof Let α ≥ 1. (i) By (4), Lemma 3 and Remark 1, we obtain (α)

En,β ((t − x)2 ; x) (α)

= x2 Qn,0 (x) +

n−1 X

(α)

(β)

Qn,k (x)Fn,k ((t − x)2 ) + (1 − x)2 Q(α) n,n (x)

k=1

≤ α[x2 Pn,0 (x) +

n−1 X

(β)

Pn,k (x)Fn,k ((t − x)2 ) + (1 − x)2 Pn,n (x)]

k=1

= αEn,β ((t − x)2 ; x) µ ¶ α n−1 2 = 1+ 2 β x(1 − x). n n +1

(5)

Since max x(1 − x) = 14 , and for any n ∈ N , one can get 0≤x≤1

have (α)

En,β ((t − x)2 ; x) ≤

n−1 n2 +1

≤ 15 , so we

β2 1 α (1 + )· . 4 5 n

(α)

(ii) In view of En,β (1; x) = 1, by the Cauchy-Schwarz inequality, we have (α)

En,β (|t − x|; x) ≤

q q (α) (α) En,β (1; x) En,β ((t − x)2 ; x),

q

(α)

thus, we get En,β (|t − x|; x) ≤

α 4 (1

+

β2 5 )

q ·

1 n.

Lemma 6 For 0 < α < 1, we have (α)

(i) En,β ((t − x)2 ; x) ≤ Mα(β) n−α ; q α (α) (β) (ii) En,β (|t − x|; x) ≤ Mα · n− 2 . (β)

Where the constant Mα

only depends on α, β.

Proof Let 0 < α < 1. (β) (i) In view of (4), Lemma 3 and Fn,k ((t − x)2 ) =

(k−nx)2 n2

+

β2 k n2 +1 ( n

−

k2 n2 ),

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we obtain (α)

En,β ((t − x)2 ; x) n−1 X

(α)

= x2 Qn,0 (x) +

(α)

(β)

Qn,k (x)Fn,k ((t − x)2 ) + (1 − x)2 Q(α) n,n (x)

k=1 α ≤ x2 Pn,0 (x) +

n−1 X

(β)

α α Pn,k (x)Fn,k ((t − x)2 ) + (1 − x)2 Pn,n (x)

k=1

=

n X

· α Pn,k (x)

k=0

¸ β2 k k2 (k − nx)2 + ) ( − n2 n 2 + 1 n n2

n n 1 X k2 β2 X α k 2 α = 2 (k − nx) Pn,k (x) + 2 Pn,k (x)( − 2 ) n n +1 n n k=0

k=0

:= I1 + I2 . −α (A α2 )α ≤ 2(A α2 )α n−α , where the By Lemma 4, we have I1 ≤ n+1 n (n + 1) constant A α2 only depends on α. n n P P α Using the H¨older inequality, we have Pn,k (x) ≤ (n + 1)1−α [ Pn,k (x)]α ,

and ( nk −

k2 n2 )

k=0

k=0

≤ 1, so we have n

I2 ≤

X β2 β2 Pn,k (x)]α = 2 (n + 1)1−α [ (n + 1)1−α ≤ β 2 n−α . 2 n +1 n +1 k=0

(β)

Denote Mα (ii) Since

(α)

(β)

= 2(A α2 )α + β 2 , then we can get En,β ((t − x)2 ; x) ≤ Mα n−α . (α)

En,β (|t − x|; x) ≤ thus, we get (α) En,β (|t

q q (α) (α) En,β (1; x) En,β ((t − x)2 ; x),

q α (β) − x|; x) ≤ Mα · n− 2 .

Lemma 7 F or f ∈ C[0, 1], x ∈ [0, 1] and α > 0, we have (α)

| En,β (f ; x) |≤k f k . Proof

By (4) and Lemma 2 (i), we have (α)

(α)

| En,β (f ; x) |≤ kf kEn,β (1; x) = kf k.

3. Main results (α)

First of all we give the following convergence theorem for the sequence {En,β (f ; x)}. (α)

Theorem 1 Let α > 0. T hen the sequence {En,β (f ; x)} converges to f unif ormly on [0, 1] f or any f ∈ C[0, 1].

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Proof

(α)

Since En,β (f ; x) is bounded and positive on C[0, 1], and by Lemma (α)

2, we have lim kEn,β (ej ; ·) − ej k = 0 for ej (t) = tj , j = 0, 1, 2. So, according n→∞

to the well-known Bohman-korovkin theorem ([13, P.40, Theorem 1.9]), we (α) see that the sequence {En,β (f ; x)} converges to f uniformly on [0,1] for any f ∈ C[0, 1]. (α)

Next we estimate the rates of convergence of the sequence {En,β } by means of the modulus of continuity. Theorem 2 Let f ∈ C[0, 1], x ∈ [0, 1]. Then # α β2 1 (i) when α ≥ 1, we have − fk ≤ 1 + (1 + ) ω(f, √ ); 4 5 n q α (α) (β) (ii) when 0 < α < 1, we have kEn,β (f ; ·) − f k ≤ (1 + Mα )ω(f, n− 2 ). "

r

(α) kEn,β (f ; ·)

(β)

Where the constant Mα Proof

only depends on α, β.

(i) When α ≥ 1, by Lemma 2 (i), we have (α)

|En,β (f ; x) − f (x)| (α)

≤ |f (0) − f (x)|Qn,0 (x) +

n−1 X

(α)

(β)

Qn,k (x)Fn,k (|f (t) − f (x)|) + |f (1) − f (x)|Q(α) n,n (x)

k=1 (α)

≤ ω(f, |0 − x|)Qn,0 (x) +

n−1 X

(α)

(β)

Qn,k (x)Fn,k (ω(f, |t − x|)) + ω(f, |1 − x|)Q(α) n,n (x)

k=1

≤ (1 +

√

n−1

X (α) √ 1 1 (α) (β) Qn,k (x)Fn,k ((1 + n|t − x|)ω(f, √ )) n|0 − x|)ω(f, √ )Qn,0 (x) + n n k=1

√

1 +(1 + n|1 − x|)ω(f, √ )Q(α) (x) n n,n √ 1 1 (α) ≤ ω(f, √ ) + nω(f, √ )En,β (|t − x|; x), n n so, by Lemma 5 (ii), we obtain " (α) |En,β (f ; x)

r

− f (x)| ≤ 1 +

# α β2 1 (1 + ) ω(f, √ ). 4 5 n

The desired result follows immediately. (ii) When 0 < α < 1, by Lemma 2 (i), we have (α)

|En,β (f ; x) − f (x)| (α)

≤ ω(f, |0 − x|)Qn,0 (x) +

n−1 X

(α)

(β)

Qn,k (x)Fn,k (ω(f, |t − x|)) + ω(f, |1 − x|)Q(α) n,n (x)

k=1 α

α

α 2

−α 2

(α)

≤ (1 + n 2 |0 − x|)ω(f, n− 2 )Qn,0 (x) +

n−1 X

(α)

(β)

α

α

Qn,k (x)Fn,k (1 + n 2 |t − x|)ω(f, n− 2 )

k=1

+(1 + n |1 − x|)ω(f, n α

α

)Q(α) n,n (x)

α

(α)

= ω(f, n− 2 ) + n 2 ω(f, n− 2 )En,β (|t − x|; x), 6 360

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so, by Lemma 6 (ii), we obtain q (α) |En,β (f ; x)

(β)

α

Mα )ω(f, n− 2 ).

− f (x)| ≤ (1 +

The desired result follows immediately. Theorem 3 Let f ∈ C 1 [0, 1], x ∈ [0, 1]. Then (i) when α ≥ 1, we have r r α β2 1 (α) 0 (1 + )· |En,β (f ; x) − f (x)| ≤ kf k 4 5 n " # r r r 2 1 α α 1 β β2 0 + ω(f , √ ) 1 + (1 + ) · (1 + )· ; 4 5 4 5 n n (ii) when 0 < α < 1, we have q q q (α) (β) −α (β) (β) 0 0 −α 2 |En,β (f ; x) − f (x)| ≤ kf k Mα n + ω(f , n )(1 + Mα ) Mα n−α . (β)

Where the constant Mα Proof

only depends on α, β.

Let f ∈ C 1 [0, 1]. For any t, x ∈ [0, 1], δ > 0, we have Z t 0 |f (t) − f (x) − f (x)(t − x)| ≤ | |f 0 (u) − f 0 (x)|du| x

≤ ω(f 0 , |t − x|)|t − x| ≤ ω(f 0 , δ)(|t − x| + δ −1 (t − x)2 ), hence, by the Cauchy-Schwarz inequality, we have (α)

|En,β (f (t) − f (x) − f 0 (x)(t − x); x)| ³ ´ (α) (α) ≤ ω(f 0 , δ) En,β (|t − x|; x) + δ −1 En,β ((t − x)2 ; x) ·q (α) ≤ ω(f 0 , δ) En,β (1; x) ¸q q (α) (α) +δ −1 En,β ((t − x)2 ; x) En,β ((t − x)2 ; x). So, we get (α)

|En,β (f ; x) − f (x)| (α)

≤ kf 0 kEn,β (|t − x|; x) · ¸q q (α) (α) 0 −1 2 +ω(f , δ) 1 + δ En,β ((t − x) ; x) En,β ((t − x)2 ; x).

(6)

(i) When α ≥ 1, taking δ = √1n in (6), by Lemma 5 and inequality (6), we obtain the desired result. α (ii)When 0 < α < 1, taking δ = n− 2 in (6), by Lemma 6 and inequality (6), we obtain the desired result. Finally we study the direct theorem concerned with an approximation for (α) the sequence {En,β } by means of the Ditzian-Totik modulus of smoothness. For the following theorem we shall use some notations. 7 361

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For f ∈ C[0, 1], ϕ(x) =

p

ωϕλ (f, t) = sup

0 0. Now we state our following main result. p Theorem 4 Let f ∈ C[0, 1], α ≥ 1, ϕ(x) = x(1 − x), x ∈ [0, 1], 0 ≤ β, λ ≤ 1. Then there exists an absolute constant C > 0 such that (α)

|En,β (f ; x) − f (x)| ≤ Cωϕλ (f, Proof

ϕ1−λ (x) √ ). n

Let g ∈ Wλ , by Lemma 2 (i) and Lemma 7, we have (α)

|En,β (f ; x) − f (x)| (α)

(α)

≤ |En,β (f − g; x)| + |f (x) − g(x)| + |En,β (g; x) − g(x)| (α)

≤ 2kf − gk + |En,β (g; x) − g(x)|. Rt (α) Since g(t) = x g 0 (u)du + g(x), En,β (1; x) = 1, so, we have Z t (α) (α) |En,β (g; x) − g(x)| ≤ |En,β ( |g 0 (u)|du; x)| x Z t (α) λ 0 ≤ kϕ g kEn,β (| ϕ−λ (u)du|; x).

(9)

(10)

By the H¨older inequality, we get Z t Z t p | ϕ−λ (u)du| ≤ |

(11)

x

x

x

1 u(1 − u)

du|λ |t − x|1−λ ,

√ √ also, in view of 1 ≤ u + 1 − u < 2, 0 ≤ u ≤ 1, we have Z t Z t 1 1 1 p du| ≤ | (√ + √ )du| | u 1 −u u(1 − u) x x √ √ √ √ ≤ 2(| t − x| + | 1 − x − 1 − t|) |t − x| |t − x| √ ≤ 2( √ ) √ +√ 1−t+ 1−x t+ x 1 1 ≤ 2|t − x|( √ + √ ) x 1−x ≤ 4|t − x|ϕ−1 (x),

(12)

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thus, by (11) and (12), we obtain Z

t

|

ϕ−λ (u)du| ≤ Cϕ−λ (x)|t − x|.

(13)

x

Also, by (10) and (13), we have (α)

(α)

≤ Ckϕλ g 0 kEn,β (ϕ−λ (x)|t − x|; x)

|En,β (g; x) − g(x)|

(α)

= Ckϕλ g 0 kϕ−λ (x)En,β (|t − x|; x).

(14)

In view of (5) and Lemma 2 (i), by the Cauchy-Schwarz inequality, we have q q (α) (α) (α) En,β (|t − x|; x) ≤ En,β (1; x) En,β ((t − x)2 ; x) s µ ¶ α n−1 2 ≤ 1+ 2 β x(1 − x) n n +1 ϕ(x) ≤ C √ , n

(15)

so, by (14) and (15), we obtain (α)

|En,β (g; x) − g(x)| ≤ Ckϕλ g 0 k

ϕ1−λ (x) √ , n

(16)

thus, by (9) and (16), we have (α)

|En,β (f ; x) − f (x)| ≤ 2kf − gk + Ckϕλ g 0 k

ϕ1−λ (x) √ . n

Then, in view of (17), (7) and (8), we obtain (α)

|En,β (f ; x) − f (x)| ≤ CKϕλ (f,

ϕ1−λ (x) ϕ1−λ (x) √ ) ≤ Cωϕλ (f, √ ), n n

where C is a positive constant, in different places, the value of C may be different.

Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 61572020), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant No. JA12324), and the Natural Science Foundation of Fujian Province of China (Grant No. 2014J01021 and 2013J01017).

References 1. B´ezier, P: Numerical Control: Mathematics and Applications. Wiley, London (1972) 2. Chang GZ: Generalized Bernstein-B´ezier polynomial. J.Computer Math. 1 (4), 322-327 (1983)

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3. Zeng XM, Chen W: On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation. J. Approx. Theory 102 (1), 1-12 (2000) 4. Zeng XM, Gupta, V: Rate of convergence of Baskakov-B´ezier type operators for locally bounded functions. Comput. Math. Appl. 44 (3), 1445-1453 (2002) 5. Deo N: Direct and inverse theorems for Sz´asz-Lupas type operators in simultaneous approximation. Math. Vesnik. 58 (1-2), 19-29 (2006) 6. Guo S., Qi Q. and Liu G: The central approximation theorem for BaskakovB´ezier operators. J. Approx. Theory 147 (1) 112-124 (2007) 7. Liang L., Sun W: On pointwise approximation for a type of generalized Kantorovich-B´ezier operators. J. Southwest Univ. 33 (10), 103-106 (2011) 8. Liu G., Yang X: On the approximation for generalized Szasz-Durrmeyer type operators in the space Lp [0, ∞). J. Inequal. Appl. 2014 (2014). doi:10.1186/1029-242X-2014-447 9. Deng X., Wu G: On approximation of Bernstein-Durrmeyer-B´ezier operators in Orlicz spaces. pure. Appl. Math. 31 (3), 307-317 (2015) 10. Ren MY: Approximation for a Kind of generalized Bernstein Operators. J. Wuyi Univ. 31 (2), 1-4 (2012) 11. Li P., Huang Y: Approximation order generalized Bernstein-B´ezier Polynomials. J. Univ. Sci. Technol. Chn. 15 (1), 15-18 (1985) 12. Li Z: Approximation properties of the Bernstein-Kantorovic-B´ezier Polynomials. Nat. Sci. J. Hunan Norm. Univ. 9 (1), 14-19 (1986) 13. Chen WZ: Operators Approximation Theory. Xiamen University Press, Xiamen (1989) (In Chinese) 14. Ditzian Z., Totik V: Moduli of Smoothness. Springer-Verlag, New-York, Berlin (1987)

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Approximation by complex Stancu type summation-integral operators in compact disks Mei-Ying Ren1∗ , Xiao-Ming Zeng2∗, Wen-Hui Zhang2 1

School of Mathematics and Computer Science, Wuyi University, Wuyishan 354300, China

2

School of Mathematical Sciences, Xiamen University, Xiamen 361005, Chnia E-mail: [email protected],

[email protected], [email protected]

Abstract. In this paper we introduce a class of complex Stancu type summationintegral operators and study the approximation properties of these operators. We obtain a Voronovskaja-type result with quantitative estimate for these operators attached to analytic functions on compact disks. We also study the exact order of approximation. More important, our results show the overconvergence phenomenon for these complex operators. Keywords: complex Stancu type summation-integral operators; Voronovskajatype result; Exact order of approximation; Simultaneous approximation; Overconvergence Mathematical subject classification: 30E10, 41A25 , 41A36

1. Introduction In 1986, some approximation properties of complex Bernstein polynomials in compact disks were initially studied by Lorentz [11]. Very recently, the problem of the approximation of complex operators has been causing great concern, which is becoming a hot topic of research. A Voronovskaja-type result with quantitative estimate for complex Bernstein polynomials in compact disks was obtained by Gal [3]. Also, in [1-2, 4-10, 12-15] similar results for complex Bernstein-Kantorovich polynomials, Bernstein-Stancu polynomials, Kantorovich-Schurer polynomials, Kantorovich-Stancu polynomials, complex Favard-Sz´ asz-Mirakjan operators, complex Beta operators of first kind, complex Baskajov-Stancu operators, complex Bernstein-Durrmeyer operators based on Jacobi weights, complex genuine Durrmeyer Stancu polynomials, complex Schurer-Stancu operators, complex q-Sz´ asz-Mirakjan operators, complex q-Gamma operators, and complex q-Durrmeyer type operators were obtained. The aim of the present article is to obtain approximation results for complex Stancu type summation-integral operators which are defined for f : [0, 1] → C continuous on [0, 1] by n−1

Mn(α,β) (f ; z) := pn,0 (z)f (

X n+α α (α,β) )+ pn,k (z)Ln,k (f ) + pn,n (z)f ( ), (1) n+β n+β k=1

∗ Corresponding

authors: Mei-Ying Ren and Xiao-Ming Zeng

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where α, β are two given real parameters satisfying the condition 0 ≤ α ≤ β, R 1 nk−1 (α,β) 1 z ∈ C, n ∈ N, Ln,k (f ) = B(n(n−k),nk) t (1 − t)n(n−k)−1 f ( nt+α n+β )dt with µ0 ¶ n B(x, y) is Beta function, and pn,k (z) = z k (1 − z)n−k . k Note that, for α = β = 0, these operators become the complex summation(0,0) integral type operators Mn (f ; z) = Mn (f ; z), this case has been investigated in [16].

2. Auxiliary results In the sequel, we shall need the following auxiliary results. Lemma 1 Let em (z) = z m , m ∈ N ∪ {0}, z ∈ C, n ∈ N, 0 ≤ α ≤ β, we (α,β) have Mn (em ; z) is a polynomial of degree less than or equal to min (m, n) and ¶ j m−j m µ X n α m Mn(α,β) (em ; z) = Mn (ej ; z). j (n + β)m j=0 Proof By the definition given by (1) , the proof is easy, here the proof is omitted. Let m = 0, 1, 2, according to [16, Lemma 1] , by simple computation, we have Mn(α,β) (e0 ; z) = 1;

α nz + ; n+β n+β · ¸ n2 n(n − 1) 2 n+1 Mn(α,β) (e2 ; z) = z + z (n + β)2 n2 + 1 n2 + 1 2nαz α2 + + . 2 (n + β) (n + β)2

Mn(α,β) (e1 ; z) =

Lemma 2 Let em (z) = z m , m ∈ N ∪ {0}, z ∈ C, n ∈ N, 0 ≤ α ≤ β, for all (α,β) (em ; z)| ≤ rm . |z| ≤ r, r ≥ 1, we have |Mn Proof The proof follows directly Lemma 1 and [16, Corollary 1]. Lemma 3 Let em (z) = z m , m, n ∈ N, z ∈ C and 0 ≤ α ≤ β, we have Mn(α,β) (em+1 ; z) =

z(1 − z)n2 (M (α,β) (em ; z))0 (n + β)(n2 + m) n (m + n2 z)n + α(n2 + 2m) (α,β) + Mn (em ; z) (n + β)(n2 + m) αm(n + α) − M (α,β) (em−1 ; z). (n + β)2 (n2 + m) n

(2)

Proof Let e (α,β) (f ) := L n,k

1 B(n(n − k), nk)

Z

1

tnk−1 (1 − t)n(n−k)−1 tf (

0

nt + α )dt, n+β

2 366

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b (α,β) (f ) := L n,k

Z

1 B(n(n − k), nk)

En(α,β) (f ; z) :=

n−1 X

1

tnk−1 (1 − t)n(n−k)−1 t2 f (

0

nt + α )dt, n+β

(α,β)

pn,k (z)Ln,k (f ),

k=1

we have α n+α ) + En(α,β) (f ; z) + pn,n (z)f ( ), n+β n+β e (α,β) (em ) = n + β L(α,β) (em+1 ) − α L(α,β) (em ), L n,k n,k n n n,k b (α,β) (em ) = ( n + β )2 L(α,β) (em+2 ) − 2α(n + β) L(α,β) (em+1 ) + ( α )2 L(α,β) (em ). L n,k n,k n,k n,k n n2 n By simple calculation, we obtain Mn(α,β) (f ; z) = pn,0 (z)f (

z(1 − z)p0n,k (z) = (k − nz)pn,k (z), t(1 − t)[tnk−1 (1 − t)n(n−k)−1 ]0 = [nk − 1 − (n2 − 2)t]tnk−1 (1 − t)n(n−k)−1 , it follows that z(1 − z)(En(α,β) (em ; z))0 =

n−1 X

(α,β)

(k − nz) pn,k (z)Ln,k (em )

k=1

=

n−1 X k=1

=

1 n +

n−1 X

1 n

1 pn,k (z) B(n(n − k), nk)

n−1 X

pn,k (z)

k=1

+

m n+β

n−1 X

1

tnk−1 (1 − t)n(n−k)−1 (

0

Z

1

nt + α m ) dt − nzEn(α,β) (em ; z) n+β

[nk − 1 − (n2 − 2)t]tnk−1 (1 − t)n(n−k)−1 (

0

n−1 X

nt + α m ) dt n+β

(α,β)

(α,β) e pn,k (z)L (em ; z), n,k (em ) − nzEn

k=1

Z

1

[nk − 1 − (n2 − 2)t]tnk−1 (1 − t)n(n−k)−1 (

0

1 B(n(n − k), nk)

1 2 = − En(α,β) (em ; z) + n n n−1 X

Z

1 B(n(n − k), nk)

n2 − 2 1 (α,β) En (em ; z) + n n

k=1

=

pn,k (z)

k=1

where n−1 1X n

1 kpn,k (z) B(n(n − k), nk)

Z

1

(t − t2 )[tnk−1 (1 − t)n(n−k)−1 ]0 (

0 (α,β)

e pn,k (z)L n,k (em ) −

k=1

nt + α m ) dt n+β

nt + α m ) dt n+β

n−1 m X e (α,β) (em−1 ) pn,k (z)L n,k n+β k=1

b (α,β) (em−1 ). pn,k (z)L n,k

k=1

So, in conclusion, we have z(1 − z)(En(α,β) (em ; z))0 =

(n + β)(n2 + m) (α,β) En (em+1 ; z) n2 αn2 + mn + 2αm −( + nz)En(α,β) (em ; z) n2 αmn + α2 m (α,β) + 2 E (em−1 ; z), n (n + β) n 3 367

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which implies the recurrence in the statement. (α,β)

(α,β)

Lemma 4 Let n ∈ N, m = 2, 3, · · ·, em (z) = z m , Sn,m (z) := Mn z m , z ∈ C and 0 ≤ α ≤ β, we have

(α,β) Sn,m (z) =

z(1 − z)n2 (M (α,β) (em−1 ; z))0 (n + β)(n2 + m − 1) n (m − 1 + n2 z)n + α(n2 + m − 1) (α,β) + Sn,m−1 (z) (n + β)(n2 + m − 1) α(m − 1) + M (α,β) (em−1 ; z) (n + β)(n2 + m − 1) n α(m − 1)(n + α) − M (α,β) (em−2 ; z) (n + β)2 (n2 + m − 1) n (m − 1 + n2 z)n + α(n2 + m − 1) m−1 + z − zm. (n + β)(n2 + m − 1)

(em ; z) −

(3)

Proof Using the recurrence formula (2), by simple calculation, we can easily get the recurrence (3), the proof is omitted.

3. Main results The first main result is expressed by the following upper estimates. Theorem 1 Let 0 ≤ α ≤ β, 1 ≤ r ≤ R, DR = {z ∈ C : |z| < R}. Sup∞ P pose that f : DR → C is analytic in DR , i.e. f (z) = cm z m for all z ∈ DR . m=0

(i) for all |z| ≤ r and n ∈ N, we have (α,β)

|Mn(α,β) (f ; z) − f (z)| ≤ (α,β)

where Kr

(f ) = (1 + r)

∞ P m=1

(f ) Kr , n+β

|cm |m(m + 1 + α + β)rm−1 < +∞.

(ii) (Simultaneous approximation) If 1 ≤ r < r1 < R are arbitrary fired, then for all |z| ≤ r and n, p ∈ N we have (α,β)

|(Mn(α,β) (f ; z))(p) − f (p) (z)| ≤ (α,β)

where Kr1

Kr1 (f )p!r1 , (n + β)(r1 − r)p+1

(f ) is defined as at the above point (i).

Proof Taking em (z) = z m , by hypothesis that f (z) is analytic in DR , i.e. ∞ P f (z) = cm z m for all z ∈ DR , it is easy for us to obtain m=0

Mn(α,β) (f ; z) =

∞ X

cm Mn(α,β) (em ; z),

m=0

4 368

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therefore, we get |Mn(α,β) (f ; z)

− f (z)| ≤ =

∞ X

|cm | · |Mn(α,β) (em ; z) − em (z)|

m=0 ∞ X

|cm | · |Mn(α,β) (em ; z) − em (z)|,

m=1 (α,β)

as Mn (e0 ; z) = e0 (z) = 1. (α,β) (i) For m ∈ N, taking into account that Mn (em−1 ; z) is a polynomial degree ≤ min(m − 1, n), by the well-known Bernstein inequality and Lemma 2 we get |(Mn(α,β) (em−1 ; z))0 | ≤

m−1 max{|Mn(α,β) (em−1 ; z)| : |z| ≤ r} ≤ (m − 1)rm−2 . r

On the one hand, when m = 1, for |z| ≤ r, by Lemma 1, we have |Mn(α,β) (e1 ; z) − e1 (z)| = |

α 1+r nz + − z| ≤ (2 + α + β). n+β n+β n+β

When m ≥ 2, for n ∈ N, |z| ≤ r, 0 ≤ α ≤ β, in view of |(m − 1 + n2 z)n + α(n2 + m − 1)| ≤ (n + β)(n2 + m − 1)r, using the recurrence formula (3) and the above inequality, we have (α,β) |Mn(α,β) (em ; z) − em (z)| = |Sn,m (z)|

r(1 + r) (α,β) · (m − 1)rm−2 + r|Sn,m−1 (z)| n+β α α m+1+β + rm−1 + rm−2 + (1 + r)rm−1 n+β n+β n+β m−1 (α,β) ≤ (1 + r)rm−1 + r|Sn,m−1 (z)| n+β m+1+β α (1 + r)rm−1 + (1 + r)rm−1 + n+β n+β 2m + α + β (α,β) = r|Sn,m−1 (z)| + (1 + r)rm−1 . n+β

≤

By writing the last inequality, for m = 2, · · · , we easily obtain step by step the following ¶ µ 2(m − 1) + α + β (α,β) (α,β) m−2 (1 + r)r |Mn (em ; z) − em (z)| ≤ r r|Sn,m−2 (z)| + n+β 2m + α + β + (1 + r)rm−1 n+β 2(m − 1 + m) + 2(α + β) (α,β) = r2 |Sn,m−2 (z))| + (1 + r)rm−1 n+β 1+r m(m + 1 + α + β)rm−1 . ≤ ... ≤ n+β In conclusion, for any m, n ∈ N, |z| ≤ r, 0 ≤ α ≤ β, we have (α,β)

|Mn+β (em ; z) − em (z)| ≤

1+r m(m + 1 + α + β)rm−1 , n+β 5

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it follows that |Mn(α,β) (f ; z) − f (z)| ≤

∞ 1+r X |cm |m(m + 1 + α + β)rm−1 . n + β m=1

By assuming that f (z) is analytic in DR , we have f (2) (z) = 1)z m−2 and the series is absolutely convergent in |z| ≤ r, so we get (α,β)

1)rm−2 < +∞, which implies Kr

(f ) = (1 + r)

∞ P m=1

∞ P m=2 ∞ P m=2

cm m(m − |cm |m(m−

|cm |m(m + 1 + α +

β)rm−1 < +∞. (ii) For the simultaneous approximation, denoting by Γ the circle of radius r1 > r and center 0, since for any |z| ≤ r and υ ∈ Γ, we have |υ − z| ≥ r1 − r, by Cauchy’s formulas it follows that for all |z| ≤ r and n ∈ N, we have ¯Z ¯ ¯ M (α,β) (f ; υ) − f (υ) ¯ p! n ¯ ¯ |(Mn(α,β) (f ; z))(p) − f (p) (z)| = dυ ¯ ¯ ¯ 2π ¯ Γ (υ − z)p+1 (α,β)

≤

Kr1 (f ) p! 2πr1 n + β 2π (r1 − r)p+1

=

Kr1 (f ) p!r1 · , n+β (r1 − r)p+1

(α,β)

which proves the theorem. Theorem 2 Let 0 ≤ α ≤ β, R > 1, DR = {z ∈ C : |z| < R}. Suppose ∞ P that f : DR → C is analytic in DR , i.e. f (z) = ck z k for all z ∈ DR . For k=0

any fixed r ∈ [1, R] and all n ∈ N, |z| ≤ r, we have ¯ ¯ ¯ (α,β) α − βz 0 z(1 − z) 00 ¯¯ ¯ Mn (f ; z) − f (z) − f (z) − f (z) ¯ ¯ n+β 2(n + β) (α,β)

≤

(α,β)

Mr,1 (f ) Mr,2 (f ) Mr,2 (f ) + + , n(n + β) (n + β)2 n2

where Mr,2 (f ) = Mr (f ) + Mr,1 (f ), Mr (f ) =

∞ P

|ck |(k − 1)Fk,r rk with Fk,r =

k=2

10k 3 − 30k 2 + 39k − 16 + 4(k − 2)(k − 1)2 (1 + r), Mr,1 (f ) = (α,β)

1)(1 + r)rk−1 , Mr,1 ∞ P k=2

2

|ck |[ k(k−1)(α2

+β 2 r 2 )

(f ) =

∞ P k=2

(4)

∞ P

|ck |(β + 1)k(k −

k=2 (α,β)

|ck |[2k(k − 1)2 α + 2k 3 βr]rk−1 , Mr,2

(f ) =

+ k 2 αβr + k 2 β 2 r2 ]rk−2 .

Proof For all z ∈ DR , we have α − βz 0 z(1 − z) 00 Mn(α,β) (f ; z) − f (z) − f (z) − f (z) n+β 2(n + β) · ¸ (n + 1)z(1 − z) 00 = Mn (f ; z) − f (z) − f (z) 2(n2 + 1) · ¸ α − βz 0 (β + 1)n + (β − 1) (α,β) 00 + Mn (f ; z) − Mn (f ; z) − f (z) + z(1 − z)f (z) n+β 2(n + β)(n2 + 1) := I1 + I2 . 6 370

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By [16, Theorem 2 ], we have |I1 | ≤

Mr (f ) n2 ,

where Mr (f ) =

∞ P

|ck |(k −

k=2

1)Fk,r rk and Fk,r = 10k 3 − 30k 2 + 39k − 16 + 4(k − 2)(k − 1)2 (1 + r). Next let us to estimate |I2 |. (β) Denote Qn,k (z) = k(k−1)((β+1)n+(β−1)) z k−1 (1 − z). By f is analytic in DR , 2(n+β)(n2 +1) ∞ P (α,β) (e1 ; z) = Mn (e1 ; z) + α−βz i.e. f (z) = ck z k for all z ∈ DR , and Mn n+β , we have

k=0

¯ · ¸¯¯ ∞ ¯X α − βz k−1 ¯ ¯ (β) (α,β) |I2 | = ¯ ck Mn (ek ; z) − Mn (ek ; z) − kz + Qn,k (z) ¯ ¯ ¯ n+β k=2 ¯ ¯ ∞ X ¯ ¯ α − βz k−1 (β) ≤ |ck | ¯¯Mn(α,β) (ek ; z) − Mn (ek ; z) − kz + Qn,k (z)¯¯ . n+β k=2

When k ≥ 2, since

nk (n+β)k

−1 = −

k−1 P j=0

µ

k j

¶ nj β k−j , (n+β)k

by Lemma 1, we obtain

α − βz k−1 (β) kz + Qn,k (z) n+β · ¸ k−1 X µ k ¶ nj αk−j nk α − βz k−1 = Mn (ej ; z) + − 1 Mn (ek ; z) − kz k k j (n + β) (n + β) n+β j=0

Mn(α,β) (ek ; z) − Mn (ek ; z) −

(β)

+ Qn,k (z) k−2 X µ k ¶ nj αk−j knk−1 α Mn (ej ; z) + Mn (ek−1 ; z) = k j (n + β) (n + β)k j=0 −

k−1 Xµ j=0

=

k−2 Xµ j=0

k j

k j ¶

¶

nj β k−j α − βz k−1 (β) Mn (ek ; z) − kz + Qn,k (z) (n + β)k n+β

nj αk−j knk−1 α Mn (ej ; z) + [Mn (ek−1 ; z) − ek−1 (z)] k (n + β) (n + β)k

¶ j k−j k−2 µ knk−1 α k−1 X k n β + z − Mn (ek ; z) j (n + β)k (n + β)k j=0 knk−1 β knk−1 β k α − βz k−1 (β) [M (e ; z) − e (z)] − z − kz + Qn,k (z) n k k (n + β)k (n + β)k n+β k−2 X µ k ¶ nj αk−j knk−1 α = M (e ; z) + [Mn (ek−1 ; z) − ek−1 (z)] n j j (n + β)k (n + β)k −

j=0

k−2 Xµ

¶

nj β k−j knk−1 β Mn (ek ; z) − [Mn (ek ; z) − ek (z)] k (n + β) (n + β)k j=0 · ¸ · ¸ 1 nk−1 1 nk−1 (β) k−1 − − kαz + − kβz k + Qn,k (z). n+β (n + β)k n+β (n + β)k −

k j

By the proof of the [16, Theorem 1 ], for any k ∈ N, |z| ≤ r, r ≥ 1, we have |Mn (ek ; z)| ≤ rk , |Mn (ek ; z) − ek | ≤

2k 2 k r , n

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hence, for any k ≥ 2, |z| ≤ r, r ≥ 1, we can get ¯ ¯ ¯ ¯k−2 µ ¶ j k−j ¯ ¯X k n α ¯ Mn (ej ; z)¯¯ ¯ k j (n + β) ¯ ¯ j=0 k−2 X µ k ¶ nj αk−j ≤ rk−2 k j (n + β) j=0 =

k−2 X j=0

k(k − 1) (k − j)(k − j − 1)

µ

k−2 j

¶

α2 nj αk−2−j · rk−2 k−2 (n + β) (n + β)2

≤

k−2 X µ k − 2 ¶ nj αk−2−j k(k − 1) α2 rk−2 · j 2 (n + β)2 j=0 (n + β)k−2

≤

k(k − 1) α2 rk−2 · 2 (n + β)2

and

¯ k−1 ¯ ¯ kn ¯ 2k(k − 1)2 α k−1 α ¯ ¯≤ [M (e ; z) − e (z)] r . n k−1 k−1 ¯ (n + β)k ¯ n(n + β)

Also, using k−2 P

1 nk−1 − = n+β (n + β)k

j=0

µ

k−1 j (n +

¶ nj β k−1−j β)k

≤

(k − 1)β , (n + β)2

thus, for any k ≥ 2, |z| ≤ r, r ≥ 1, we get α − βz k−1 (β) kz + Qn,k (z)| n+β k(k − 1) α2 2k(k − 1)2 α k−1 k(k − 1) β2 k−2 ≤ · r + r + · rk 2 (n + β)2 n(n + β) 2 (n + β)2 2k 3 β k 2 αβ k−1 k 2 β 2 k (β + 1)k(k − 1)(1 + r)rk−1 + rk + r + r + 2 n(n + β) (n + β) (n + β)2 n2 ¤ (β + 1)k(k − 1)(1 + r)rk−1 rk−1 £ 2k(k − 1)2 α + 2k 3 βr + = n(n + β) n2 rk−2 k(k − 1)(α2 + β 2 r2 ) [ + k 2 αβr + k 2 β 2 r2 ]. + (n + β)2 2

|Mn(α,β) (ek ; z) − Mn (ek ; z) −

Hence, we have (α,β)

(α,β)

|I2 | ≤

Mr,1 (f ) Mr,2 (f ) Mr,1 (f ) + + , n(n + β) (n + β)2 n2

where Mr,1 (f ) = (α,β)

Mr,1

(α,β)

Mr,2

∞ X

|ck |(β + 1)k(k − 1)(1 + r)rk−1 ,

k=2 ∞ X

(f ) = (f ) =

k=2 ∞ X k=2

|ck |[2k(k − 1)2 α + 2k 3 βr]rk−1 , |ck |[

k(k − 1)(α2 + β 2 r2 ) + k 2 αβr + k 2 β 2 r2 ]rk−2 . 2

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In conclusion, we obtain ¯ ¯ ¯ (α,β) α − βz 0 z(1 − z) 00 ¯¯ ¯ Mn (f ; z) − f (z) − f (z) − f (z)¯ ¯ n+β 2(n + β) (α,β)

≤ |I1 | + |I2 | ≤

(α,β)

Mr,1 (f ) Mr,2 (f ) Mr,2 (f ) + , + n(n + β) (n + β)2 n2

where Mr,2 (f ) = Mr (f ) + Mr,1 (f ). In the following theorem, we will obtain the exact order in approximation. Theorem 3 Let 0 < α ≤ β, R > 1, DR = {z ∈ C : |z| < R}. Suppose that f : DR → C is analytic in DR . If f is not a polynomial of degree 0 , then for any r ∈ [1, R) we have (α,β)

kMn(α,β) (f ; ·) − f kr ≥

Cr (f ) , n ∈ N, n+β (α,β)

where kf kr = max{|f (z)|; |z| ≤ r} and the constant Cr f , r and α, β but it is independent of n.

(f ) > 0 depends on

Proof Denote e1 (z) = z and Hn(α,β) (f ; z) = Mn(α,β) (f ; z) − f (z) −

α − βz 0 z(1 − z) 00 f (z) − f (z). n+β 2(n + β)

For all z ∈ DR and n ∈ N we have Mn(α,β) (f ; z) − f (z) ½ ¾ 1 z(1 − z) 00 0 (α,β) = (α − βz)f (z) + f (z) + (n + β)Hn (f ; z) . n+β 2 In view of the property: kF +Gkr ≥ |kF kr −kGkr | ≥ kF kr −kGkr , it follows kMn(α,β) (f ; ·) − f kr ≥

1 n+β

½ ¾ e1 (1 − e1 ) 00 k(α − βe1 )f 0 + f kr − (n + β)||Hn(α,β) (f ; ·)||r . 2

Considering the hypothesis that f is not a polynomial of degree 0 in DR , we 1 ) 00 f kr > 0. have k(α − βe1 )f 0 + e1 (1−e 2 Indeed, supposing the contrary, it follows that (α − βz)f 0 (z) +

z(1 − z) 00 f (z) = 0, for all z ∈ Dr . 2

Denoting y(z) = f 0 (z) and looking for the analytic function y(z) under ∞ P the form y(z) = ak z k , after replacement in the differential equation, the k=0

identification of the coefficients method immediately leads to ak = 0, for all S k ∈ N {0}. This implies that y(z) = 0 for all z ∈ Dr and therefore f is constant on Dr , a contradiction with the hypothesis. Using the inequality (4), we get lim (n + β)kHn(α,β) (f ; ·)kr = 0,

n→∞

(5)

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therefore, there exists an index n0 depending only on f , r and α, β, such that for all n ≥ n0 , we have e1 (1 − e1 ) 00 f kr − (n + β)||Hn(α,β) (f ; ·)||r 2 ° ° 1° e1 (1 − e1 ) 00 ° 0 ° ≥ °(α − βe1 )f + f ° ° , 2 2 r

k(α − βe1 )f 0 +

which implies kMn(α,β) (f ; ·)

° ° e1 (1 − e1 ) 00 ° 1 ° 0 ° (α − βe1 )f + f ° − f kr ≥ ° , for all n ≥ n0 . 2n ° 2 r (α,β)

For n ∈ {1, 2, · · ·, n0 − 1}, we have kMn (α,β) Wr,n (f )

= (n +

(α,β) (f ; ·) β)kMn

(f ; ·) − f kr ≥

(α,β) Wr,n (f ) , n+β

where

− f kr > 0. C (α,β) (f )

(α,β)

As a conclusion, we have kMn (f ; ·) − f kr ≥ rn+β , for all n ∈ N, where n (α,β) (α,β) (α,β) Cr(α,β) (f ) =min Wr,1 (f ), Wr,2 (f ), . . . , Wr,n0 −1 (f ), ¾ e1 (1 − e1 ) 00 1 0 k(α − βe1 )f + f kr , 2 2 this complete the proof. Combining Theorem 3 with Theorem 1, we get the following result. Corollary Let 0 ≤ α ≤ β, R > 1, DR = {z ∈ C : |z| < R}. Suppose that f : DR → C is analytic in DR . If f is not a polynomial of degree 0 , then for any r ∈ [1, R) we have kMn(α,β) (f ; ·) − f kr ³

1 , n ∈ N, n+β

where kf kr = max{|f (z)|; |z| ≤ r} and the constants in the equivalence depend on f , r and α, β but it is independent of n. Theorem 4 Let 0 ≤ α ≤ β, R > 1, DR = {z ∈ C : |z| < R}. Suppose that f : DR → C is analytic in DR . Also, let 1 ≤ r < r1 < R and p ∈ N be fixed. If f is not a polynomial of degree ≤ p − 1, then we have k(Mn(α,β) (f ; ·))(p) − f (p) kr ³

1 , n ∈ N, n+β

where kf kr = max{|f (z)|; |z| ≤ r} and the constants in the equivalence depend on f , r, r1 , p, α and β, but it is independent of n. Proof Taking into account that the upper estimate in Theorem 1 , it remains to prove the lower estimate only. Denoting by Γ the circle of radius r1 > r and center 0 , by the Cauchy’s formula, it follows that for all |z| ≤ r and n ∈ N, we have Z (α,β) Mn (f ; v) − f (v) p! (α,β) (p) (p) dv. (Mn (f ; z)) − f (z) = 2πi Γ (v − z)p+1

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(α,β)

Keeping the notation there for Hn

(f ; z), for all n ∈ N, we have

Mn(α,β) (f ; z) − f (z) 1 = n+β

¾ ½ z(1 − z) 00 0 (α,β) (α − βz)f (z) + f (z) + (n + β)Hn (f ; z) . 2

by using Cauchy’s formula, for all v ∈ Γ we get (· ¸(p) z(1 − z) 00 1 (α,β) (p) (p) (α − βz)f 0 (z) + f (z) (Mn (f ; z)) − f (z) = n+β 2 p! + 2πi

Z Γ

(α,β)

(n + β)Hn (f ; v) dv (v − z)p+1

) ,

passing now to k · kr and denoting e1 (z) = z, it follows "°· ¸(p) ° ° ° ° ° e (1 − e ) 1 ° ° ° (α,β) ° 1 1 f 00 (f ; ·))(p) − f (p) ° ≥ ° (α − βe1 )f 0 + ° °(Mn ° ° n+β 2 r

r

° ° # ° p! Z (n + β)H (α,β) (f ; v) ° n ° ° −° dv ° . ° 2πi Γ ° (v − ·)p+1 r

Since for any |z| ≤ r and υ ∈ Γ, we have |υ − z| ≥ r1 − r, so, ° ° (α,β) ° p! Z (n + β)H (α,β) (f ; v) ° (f ; ·)kr1 p! 2πr1 (n + β)kHn n ° ° dv ≤ · , ° ° ° ° 2πi Γ (v − ·)p+1 2π (r1 − r)p+1 r

° ° ° p! R (n+β)Hn(α,β) (f ;v) ° thus, by the inequality (5), we can get limn→∞ ° 2πi dv ° = 0. p+1 (v−·) Γ r Taking into account the function f is analytic in DR , by following exactly the lines in Gal [5], seeing ° ° also the book Gal [6, pp. 77-78 ], we have ° e1 (1−e1 ) 00 (p) ° 0 f ] ° > 0, °[(α − βe1 )f + 2 r In continuation, reasoning exactly as in the proof of Theorem 3, we can get the desired conclusion.

Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 61572020) and the Natural Science Foundation of Fujian Province of China (Grant No. 2014J01021 and 2013J01017).

References 1. Anastassiou, G.A., Gal, S.G.: Approximation by complex Bernstein-Schurer and Kantorovich-Schurer polynomials in compact disks. Comput. Math. Appl. 58(4), 734-743 (2009) 2. Cai Q.B., Li C.H. and Zeng X.M.: Approximation by Complex q-Gamma Operators in Compact Disks. J. Comput. Anal. Appl. 20(6), 1088-1096 (2016)

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3. Gal, S.G.: Voronovskaja’s theorem and iterations for complex Bernstein polynomials in compact disks. Mediterr. J. Math. 5(3), 253-272 (2008) 4. Gal, S.G.: Approximation by complex Bernstein-Kantorovich and StancuKantorovich polynomials and their iterates in compact disks. Rev. Anal. Num´er. Th´eor. Approx. (Cluj) 37(2), 159-168 (2008) 5. Gal, S.G.: Exact orders in simultaneous approximation by complex Bernstein Stancu polynomials. Rev. Anal. Num´er. Th´eor. Approx. (Cluj) 37(1), 47-52 (2008) 6. Gal, S.G.: Approximation by Complex Bernstein and Convolution Type Operators. World Scientific, Singapore (2009). 7. Gal, S.G.: Approximation by complex Bernstein-Stancu polynomials in compact disks. Results Math. 53(3 − 4), 245-256 (2009) 8. Gal, S.G.: Approximation by complex Bernstein-Durrmeyer polynomials with Jacobl weights in conpact disks. Math. Balkanica (N.S.). 24(1), 103-119 (2010) 9. Gal, S.G., Gupta, V.: Approximation by complex Beta operators of first kind in strips of compact disks. Mediterr. J. Math. 10(1), 31-39 (2013) 10. Gal, S.G., Gupta, V., etc.: Approximation by complex Baskakov-Stancu operators in compact disks. Rend. Circ. Mat. Palermo. 61(2), 153-165 (2012) 11. Lorentz, G.G.: Bernstein Polynomials. 2nd ed., Chelsea Publ, New York (1986) 12. Mahmudov, N.I.: Approximation properties of complex q-Sz´ asz-Mirakjan operators in compact disks. Comput. Math. Appl. 60(6), 1784-1791 (2010) 13. Mahmudov, N.I., Gupta, V.: Approximation by genuine Durrmeyer-Stancu polynomials in compact disks. Math. Comput. Model. 55(3), 278-285 (2012) 14. Rec M.Y., Zeng X.M.: Approximation by complex Schurer-Stancu operators in Compact Disks. J. Comput. Amal. Appl. 15(5), 833-843 (2013) 15. Ren M.Y., Zeng X.M.: Exact orders in simultaneous approximation by complex q-Durrmeyer type operators. J. Comput. Amal. Appl. 16(5), 895-905 (2014) 16. Ren, M.Y., Zeng, X.M.: Approximation by a complex summation-integral type operators in compact disks. J. Comput. Anal. Appl. 21(3), 439-450 (2016)

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On right multidimensional Riemann-Liouville fractional integral George Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we study some important properties of right multidimensional Riemann-Liouville fractional integral operator, such as of continuity and boundedness.

2010 AMS Subject Classi…cation: 26A33. Key Words and Phrases: Riemann-Liouville fractional integral, continuity, boundedness.

1

Motivation

From [1] we have Theorem 1 Let r > 0, F 2 L1 (a; b), and Z b r G (s) = (t s)

1

F (t) dt;

s

all s 2 [a; b]. Then G 2 AC ([a; b]) (absolutely continuous functions) for r and G 2 C ([a; b]), only for r 2 (0; 1) :

2

1,

Main Results

We give Theorem 2 Let f 2 L1 ([a; b] [c; d]), Z b 1 Z b2 F (x1 ; x2 ) = (t1 x1 ) 1 x1

1; 1

(t2

2

> 0. Consider the function x2 )

2

1

f (t1 ; t2 ) dt1 dt2 ;

(1)

x2

1

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where x1 ; b1 2 [a; b], x2 ; b2 2 [c; d] : x1 b1 , x2 Then F is continuous on [a; b1 ] [c; b2 ] :

b2 :

Proof. (I) Let a1 ; a1 ; b1 2 [a; b] with a1 < a1 < b1 , and a2 ; a2 ; b2 2 [c; d] with a2 < a2 < b2 : We observe that F (a1 ; a2 ) F (a1 ; a2 ) = Z b1 Z b2 1 1 (t1 a1 ) 1 (t2 a2 ) 2 f (t1 ; t2 ) dt1 dt2 a1

Z

a2

b1

a1

Z

a1

a1

Z

a1

Z

b2

a2

h

Z

Z

b1

a1

a1

a1

Z

1

f (t1 ; t2 ) dt1 dt2 =

(t1

a1 )

1

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2 +

(t1

a1 )

1

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2 +

(t1

a1 )

1

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2 +

(t1

a1 )

1

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2

(t1

a1 )

1

(t2

a2 )

1

f (t1 ; t2 ) dt1 dt2 =

(2)

b2

(3)

b2 1

2

a2

Z

Z

2

a2

Z

(t2

a2 )

2

1

(t1

a1 )

1

1

a2

(t1 (t2

1

a1 )

1

2

a2 )

1

(t2

a2 )

1

2

i

f (t1 ; t2 ) dt1 dt2

f (t1 ; t2 ) dt1 dt2 +

(4)

a2

Z

a1

a1

1

1

a1 )

+

a2 )

a2

b1

(t1

(t2

a2

Z

a1

a1

b1

1

b2

Z

a1

Z

1

a2

a1

Z

a1 )

a2

a1

Z

(t1

a2

b1

Z

b2

a2

b1

Z

Z

a2 1

(t1

a1 )

(t1

a1 )

1

(t2

a2 )

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2 +

1

f (t1 ; t2 ) dt1 dt2 :

a2

Z

b2 1

2

a2

Call I (a1 ; a2 ) = Z

b1 Z

a1

b2

(t1

a1 )

1

1

(t2

a2 )

2

1

(t1

a1 )

1

1

(t2

a2 )

2

1

dt1 dt2 :

a2

(5)

2

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Thus jF (a1 ; a2 ) (b1

I (a1 ; a2 ) + (a1

a1 )

1

(a2

1

1

a1 )

(a1

a1 )

1

(a2

+

(a1

2

+

2

1

a1 )

2

a2 )

1

2

a2 )

F (a1 ; a2 )j

(b2

a2 )

2

1

(a2

2

a2 )

kf k1 :

2

Hence, by the last inequality, it holds :=

lim jF (a1 ; a2 ) (a1 ;a2 )!(a1 ;a2 )

F (a1 ; a2 )j

or (a1 ;a2 )!(a1 ;a2 )

0

1

B C B C B C lim I (a1 ; a2 )C kf k1 =: : B B(a1 ;a2 )!(a1 ;a2 ) C @ A

(6)

or (a1 ;a2 )!(a1 ;a2 )

If

1

If

= 1

= 0, proving 2 = 1, then = 1, 2 > 0 we get I (a1 ; a2 ) = (b1

Z

a1 )

= 0:

b2

(t2

a2 )

2

1

(t2

a2 )

2

1

dt2 :

a2 )

2

1

(t2

a2 )

2

1

dt2

(7)

a2

Assume

2

> 1, then

1 > 0. Hence

2

I (a1 ; a2 ) = (b1

a1 )

Z

b2

(t2

a2

= (b1

a1 )

(b2

a2 )

2

(a2

a2 )

2

2

(b2

a2 )

2

:

(8)

2

Clearly, then lim

I (a1 ; a2 ) = 0:

(a1 ;a2 )!(a1 ;a2 ) or (a1 ;a2 )!(a1 ;a2 )

(9)

Similarly and symmetrically, we obtain that lim

I (a1 ; a2 ) = 0

(a1 ;a2 )!(a1 ;a2 ) or (a1 ;a2 )!(a1 ;a2 )

for the case of

2

= 1,

1

(10)

> 1.

3

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If

1

= 1, and 0
1, then I (a1 ; a2 ) =

Z

b1

a1

Z

b2

(t1

a1 )

1

1

(t2

1

2

a2 )

(t1

1

1

a1 )

(t2

a2 )

1

2

dt1 dt2 =

a2

(b1

1

a1 )

(a1

1

a1 )

(b2

2

a2 )

(a2

1

a2 )

2

2

(b1

1

a1 )

(b2

a2 )

1

2

:

(14)

2

That is I (a1 ; a2 ) = 0: lim (a1 ;a2 )!(a1 ;a2 )

(15)

or (a1 ;a2 )!(a1 ;a2 )

Case now of 0
1, 0
1 and 0
1; 0
a1 and a2 > a2 , as symmetric to the already treated one of a1 < a1 and a2 < a2 , is omitted. (II) The remaining cases are: let a1 ; a1 ; b1 2 [a; b]; a2 ; a2 ; b2 2 [c; d], we can have (II1 ) a1 > a1 and a2 < a2 , (23) or (II2 ) a1 < a1 and a2 > a2 : Notice that the subcases (II1 ) and (II2 ) are symmetric, and treated the same way. As such we treat only the case (II2 ). We observe again that F (a1 ; a2 ) Z

Z

b1

a1

Z

a1

Z

b1

a1

a2 )

2

1

f (t1 ; t2 ) dt1 dt2

(t1

a1 )

1

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2 =

(t1

a1 )

1

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2 +

(t1

a1 )

1

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2

(t1

a1 )

1

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2

(t1

a1 )

1

1

(t2

a2 )

1

f (t1 ; t2 ) dt1 dt2 =

(t2

a2 )

2

(t1

a1 )

(t1

a1 )

b2

Z

a2

b1

Z

b2

2

a2

b2

(t1

(25)

a2

a1

Z

(t2

a2

a1

b1

1

b2

Z

a1

Z

1

a2

b1

Z

a1 )

b2

Z

a1

Z

(t1

a2

a1

Z

(24)

a2

b1

Z

b2

F (a1 ; a2 ) =

1

1

a1 )

1

(t1

1

a1 )

1

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2

a2

+

Z

a1

a1

Z

b1

a1

Z

b2 1

1

(t2

a2 )

(t2

a2 )

2

1

f (t1 ; t2 ) dt1 dt2

(26)

a2

Z

a2 1

1

2

1

f (t1 ; t2 ) dt1 dt2 :

a2

6

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Call Z

b1 Z

I (a1 ; a2 ) := b2

(t1

1

1

a1 )

(t2

1

2

a2 )

(t1

a1 )

1

1

(t2

2

a2 )

1

dt1 dt2 :

a2

a1

(27)

Hence, we have

I (a1 ; a2 ) +

(a1

jF (a1 ; a2 )

1

a1 )

(b2

a2 )

1

F (a1 ; a2 )j 2

(b1

+

a1 )

2

1

(a2

2

a2 )

1

kf k1 : (28)

2

Therefore it holds 0

1

C B B C B lim I (a1 ; a2 )C kf k1 =: : @ja1 a1 j!0; A ja2 a2 j!0 (29) We will prove that = 0, hence = 0, in all possible cases. If 1 = 2 = 1, then I (a1 ; a2 ) = 0, hence = 0: If 1 = 1, 2 > 0 we get Z b2 1 1 I (a1 ; a2 ) = (b1 a1 ) (t2 a2 ) 2 (t2 a2 ) 2 dt2 : (30) :=

lim jF (a1 ; a2 ) ja1 a1 j!0; ja2 a2 j!0

F (a1 ; a2 )j

a2

Assume

2

> 1, then

1 > 0. Hence Z b2 a1 ) (t2 a2 )

2

I (a1 ; a2 ) = (b1

2

1

(t2

a2 )

2

1

dt2

a2

= (b1

a1 )

a2 )

(b2

2

(a2

a2 )

2

2

(b2

a2 )

2

2

:

(31)

2

Clearly, then ja2 hence = 0: Let the case now of

2

I (a1 ; a2 ) = (b2

lim I (a1 ; a2 ) = 0; a2 j!0;

= 1, Z a2 )

1

(32)

> 1: Then

b1

(t1

a1 )

1

1

(t1

a1 )

1

1

dt1

a1

= (b2

a2 )

Z

b1

(t1

a1 )

1

1

(t1

a1 )

1

1

dt1

(33)

a1

= (b2

a2 )

(b1

a1 )

1

(a1

1

a1 ) 1

1

(b1

a1 )

1

:

1

7

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Then If

= 0: 1 = 1, and 0
1, 0 < Z

2

< 1. We observe that

b2

(t1

1

1

a1 )

(t2

a2 )

1

2

(t2

a2 )

2

1

dt1 dt2

a2

(42)

b2

(t2

a2 )

1

2

(t1

a1 )

1

1

(t1

a1 )

1

(b2

a2 )

2

1

dt1 dt2 =

a2

1

a1 )

(a1

1

a1 )

1

(b2

a2 )

1

2

2

+

(a2

a2 )

2

2

2

(43) (b2

+

a2 )

2

(a2

2

a2 )

2

(b1

2

Z

b1

a1

+

Z

b1

a1

(b1

Z

Z

2

b2

(t1

a1 )

1

1

(b2

1

a1 )

(b1

(t2

a2 )

;

1

< 1. In that case it holds

1

2

1

a1 )

1

> 1 and 0
0, i = 1; :::; k 2 N. Consider the

i

1

f (t1 ; :::; tk ) dt1 :::dtk ;

(46)

xk i=1

where ai xi bi bi , i = 1; :::; k: Qk Then F is continuous on i=1 [ai ; bi ] :

Remark 4 In the setting of Theorem 3: Consider the right multidimensional Riemann-Liouville fractional integral of order = ( 1 ; :::; k ) ; i > 0, i = 1; :::; k : 1

Ib f (x) = Qk

i=1

( i)

Z

b1

:::

x1

Z

bk

k Y

(ti

xi )

i

1

f (t1 ; :::; tk ) dt1 :::dtk ;

xk i=1

(47) where ai xi bi bi , i = 1; :::; k; where b = (b1 ; :::; bk ), x = (x1 ; :::; xk ), is the gamma function. By Theorem 3 we get that Ib f is a continuous function for every x 2 Qk i=1 [ai ; bi ] : We notice that ! Z b 1 Z bk Y k 1 i 1 Ib f (x) ::: (ti xi ) dt1 :::dtk kf k1 Qk x1 xk i=1 i=1 ( i ) (48) ! Z bi k Y 1 1 = Qk (ti xi ) i dti kf k1 = ( ) x i i i=1 i=1 ! k k Y Y (bi xi ) i (bi xi ) i 1 kf k1 : (49) kf k1 = Qk ( i + 1) i i=1 ( i ) i=1 i=1

That is

k Y (b

Ib f (x)

i=1

In particular we get

xi ) i ( i + 1)

i

!

kf k1 :

Ib f (b ) = 0; and Ib f

1;

Qk

i=1

[ai ;bi ]

k Y (b

i=1

ai ) i ( i + 1)

i

(50)

(51) !

kf k1 :

(52)

That is Ib f is a bounded linear operator, which here is also a positive operator.

10

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References [1] G. A. Anastassiou, Fractional representation formulae and right fractional inequalities, Mathematical and Computer Modelling, 54 (2011), 3098-3115.

11

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388

 

389

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO. 2, 2017

Effect of RTI Drug Efficacy on the HIV Dynamics with Two Cocirculating Target Cells, A. M. Elaiw, N. A. Almuallem, and Aatef Hobiny,…………………………………………………209 Composition Operators on Dirichlet-Type Spaces, Liu Yang and Yecheng Shi,…………….229 On Left Multidimensional Riemann-Liouville Fractional Integral, George Anastassiou,…….239 Weak Closure Operations on Ideals Of BCK-Algebras, Hashem Bordbar, Mohammad Mehdi Zahedi, Sun Shin Ahn, and Young Bae Jun,…………………………………………………..249 Communication Between Relation Information Systems, Funing Lin and Shenggang Li,……263 Global Stability in a Discrete Lotka-Volterra Competition Model, Sangmok Choo and YoungHee Kim,……………………………………………………………………………………….276 Weighted Composition Operators from Bloch Spaces Into Zygmund Spaces, Shanli Ye,…….294 Approximate Homomorphisms and Derivations on Non-Archimedean Lie JC*-Algebras, Javad Shokri and Dong Yun Shin,…………………………………………………………………….306 On Distribution and Probability Density Functions of Order Statistics Arising From Independent But Not Necessarily Identically Distributed Random Vectors, M. Güngör and Y. Bulut,……314 Stability of Homomorphisms and Derivations in Non-Archimedean Random C*-Algebras via Fixed Point Method, Javad Shokri and Jung Rye Lee,…………………………………………322 On The Fuzzy Stability Problems of Generalized Sextic Mappings, Heejeong Koh and Dongseung Kang,……………………………………………………………………………….333 Existence and Uniqueness of Solutions to SFDEs Driven By G-Brownian Motion with NonLipschitz Conditions, Faiz Faizullah,……………………………………………………….….344 Approximation of a Kind of New Bernstein- Bézier Type Operators, Mei-Ying Ren, Xiao-Ming Zeng, and Wen-Hui Zhang,…………………………………………………………………….355 Approximation by Complex Stancu Type Summation-Integral Operators in Compact Disks, Mei.Ying Ren, Xiao-Ming Zeng, and Wen-Hui Zhang,………………………………………………..................................................................365 On Right Multidimensional Riemann-Liouville Fractional Integral, George Anastassiou,……377

Volume 23, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE

September 2017

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 (fourteen 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.

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $700, Electronic OPEN ACCESS. Individual:Print $350. For any other part of the world add $130 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2017 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

392

Editorial Board Associate Editors of Journal of Computational Analysis and Applications Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Dynamics of a difference equation with maximum Guangwang Su∗

Taixiang Sun

College of Information and Statistics, Guangxi Univresity of Finance and Economics Nanning, Guangxi 530003, P.R. China

Abstract The purpose of this work is to investigate the convergence of the solutions of the following max-type difference equation zn = max{

1 zn−s

,

Pn αn }, n = 0, 1, 2, · · · , zn−t

where s, t ∈ {1, 2, 3, · · · } with s 6= t, αn ∈ (0, 1) is an s-periodic sequence, {Pn }+∞ n=0 is a constant sequence satisfying Pn ∈ (0, 1] for every n ≥ 0. We show that if {zn }+∞ n=−r (r = max{s, t}) is a positive solution of the above equation with the initial conditions z−r , z−r+1 , · · · , z−1 ∈ (0, +∞), then limn−→∞ zn = 1 or {z2sn+k }+∞ n=0 is eventually monotone for every 0 ≤ k ≤ 2s − 1. Further, we show that if Pn is a periodic sequence, s = 1 and t is even, then limn−→∞ zn = 1 or {zn }+∞ n=−t is eventually periodic with period 2. AMS Subject Classification: 39A10; 39A11. Keywords: max-type equation, positive solution, eventual periodicity, monotonicity, periodic sequence.

1. Introduction The max operator arises naturally in certain models in automatic control theory (see [6,7]). In the recent years, there has been a lot of interest in studying the convergence and boundedness of max-type difference equations (see [1,3,5,8-11]). In [2], Chen studied the second order max-type difference equation zn+1 = max{

1 An , }, n = 0, 1, 2, · · · , zn zn−1

(1.1)

and showed that every positive solution of (1.1) is eventually periodic with period 2 when {An }+∞ n=0 is a periodic sequence with period k ≥ 2 and An ∈ (0, 1) for all n ≥ 0. In [4], the authors studied the following non-autonomous max-type difference equation with two delays zn = max{

A fn , β }, n = 0, 1, 2, · · · , α zn−m zn−r

? Project Supported by NNSF of China (11461003) ∗ Corresponding author: E-mail address: [email protected]

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where α, β ∈ R, {An }+∞ n=0 is a sequence of positive real numbers with a finite limit and m, r ∈ N ≡ {1, 2, 3, · · · } with m 6= r. In this paper, we study the periodicity, the boundedness and the convergence of the following max-type difference equation zn = max{

Pn 1 , αn }, n = 0, 1, 2, · · · , zn−s zn−t

(1.2)

where s, t ∈ N with s 6= t, αn ∈ (0, 1) is an s-periodic sequence, {Pn }+∞ n=0 is a constant sequence satisfying Pn ∈ (0, 1] for every n ≥ 0.

2. Some Propositions In the following, suppose that {zn }+∞ n=−r is a positive solution of (1.2). To obtain the main results of this paper, we need the following propositions. Proposition 2.1 (i) zn zn−s ≥ 1 for all n ≥ 0. α

n−s (ii) For any n ≥ r, zn ≤ max{zn−2s , Pn zn−s−t }.

αn (iii) If zn = Pn /zn−t > 1/zn−s for some n ≥ r, then zn > zn−2s . If zn = 1/zn−s for some

n ≥ s, then zn ≤ zn−2s . Proof (i) Since zn ≥ 1/zn−s for any n ≥ 0, we have zn zn−s ≥ 1. (ii) According to (i), we get that for every n ≥ r, zn = max

©

ª Pn z α n zn−2s αn ≤ max{zn−2s , Pn zn−s−t }. , αn n−s−t αn zn−s zn−2s zn−s−t zn−t

αn (iii) If zn = Pn /zn−t > 1/zn−s for some n ≥ r, then by (i) we obtain that αn zn zn−t Pn−s αn−s αn } zn−2s zn−t−s zn−t zn zn ≤ max{ , Pn Pn−s } = . zn−2s zn−2s

1 < zn zn−s = max{

zn

,

Which implies zn > zn−2s . If zn = 1/zn−s for some n ≥ s, then by (i) we obtain that zn =

zn−2s ≤ zn−2s . zn−s zn−2s

The proof is complete. Define Un = max{zn−1 , zn−2 , · · · , zn−s−r } (n ≥ r).

(2.1)

According to Proposition 2.1 (i), we get max{zn−1 , zn−s−1 } ≥ 1, from which it follows Un ≥ 1 for any n ≥ r. Proposition 2.2 (i) Let Un be as in (2.1). Then zn ≤ Un for any n ≥ r and {Un }+∞ n=r is a decreasing sequence. (ii) There exist constants R ≥ R0 > 0 such that R0 ≤ zn ≤ R for any n ≥ −r. αn αn Proof (i) If zn−s−t ≤ 1, then zn−s−t ≤ 1. If zn−s−t ≥ 1, then zn−s−t ≤ zn−s−t . According to

Proposition 2.1 (ii), we have that for any n ≥ r, αn zn ≤ max{zn−2s , zn−s−t } ≤ max{zn−1 , zn−2 , · · · , zn−s−r } = Un .

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Further, we get Un+1 = max{zn , zn−1 , · · · , zn−s−r+1 } ≤ Un . (ii) Let R = max{Ur , zr−1 , · · · , z−r } and R0 = min{1/Ur , zr−1 , · · · , z−r }. Then R0 ≤ zn ≤ R for any n ≥ −r. The proof is complete. Now we assume limn−→∞ Un = U and lim inf n−→∞ Un = u. According to Proposition 2.2 (i), we obtain the following corollary. Corollary 2.3 There exists a sequence 1 < n1 < n2 < · · · < nk < · · · such that znk ≥ U and nk+1 − nk ≤ s + r. Proposition 2.4 The following statements hold: (i) U = lim supn−→∞ zn . αn (ii) Assume that U > 1. Then {n : U ≤ zn = Pn /zn−t } is a finite set. Further, there exists

N ∈ N such that: i) zN +2ks ≥ U and zN +2ks = 1/zN +(2k−1)s for any k ≥ 0, and zN +2ks is decreasing. ii) limk−→∞ zN +(2k−1)s = u = 1/U . Proof (i) According to (2.1), we see that Un is a subsequence of zn . Thus U ≤ lim supn−→∞ zn . Further, since zn ≤ Un for all n ≥ r, we obtain lim sup zn ≤ lim sup Un = U. n−→∞

n−→∞

αn (ii) If {n : U ≤ zn = Pn /zn−t } is an infinite set, then there exists a sequence t < n1 < n2
1. A contradiction. It follows from the above that there exists M ∈ N such that if n ≥ M and zn ≥ U , then zn = 1/zn−s . By Corollary 2.3 we see that there exists a sequence 1 < n1 < n2 < · · · < nk < · · · such that znk ≥ U and limk−→∞ znk = U . Without loss of generality, suppose that nk = 2srk +τ > M with 0 ≤ τ < 2s for all k ∈ N. Then znk = 1/znk −s . Write N = 2sr1 + τ . By Proposition 2.1 (iii), we see that for any k ≥ 0, zN +2ks ≥ U and

1 zN +2ks−s

= zN +2ks ≥ zN +2(k+1)s =

1 zN +2(k+1)s−s

.

Let ik −→ +∞ such that zik −→ u and zik −s −→ u1 . Then 1 1 1 1 1 = lim = lim zN +(2k−1)s ≥ u = lim zik ≥ lim = ≥ , k−→∞ zN +2ks k−→∞ k−→∞ k−→∞ zik −s U u1 U this implies limk−→∞ zN +(2k−1)s = u = 1/U . The proof is complete. Proposition 2.5 Let N, p, q ∈ N with q ≥ 2 such that (i) {zN +2ks }+∞ k=0 is monotone.

α

+2s(p+λ)+t (ii) zN +2s(p+λ)+t = PN +2s(p+λ)+t /zNN+2s(p+λ) > 1/zN +2s(p+λ)+t−s for every λ ∈ {0, q}.

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(iii) zN +2s(p+λ)+t = 1/zN +2s(p+λ)+t−s for every 1 ≤ λ ≤ q − 1. Then zN +2s(p+λ)+t = zN +2s(p+λ+1)+t for every 0 ≤ λ ≤ q − 2. Proof There are two cases to be considered. Case 1 {zN +2sk }+∞ k=0 is decreasing. In this case, we claim that zN +2s(p+λ)+t−s = 1/zN +2s(p+λ−1)+t for any 1 ≤ λ ≤ q − 1. Since, otherwise, if for some 1 ≤ λ ≤ q − 1, zN +2s(p+λ)+t−s =

PN +2s(p+λ)+t−s α

+2s(p+λ)+t−s zNN+2s(p+λ)−s

> 1/zN +2s(p+λ−1)+t ,

then by Proposition 2.1 (iii) it follows that PN +2sp+t αN +2s(p+λ)+t zN +2s(p+λ)

PN +2sp+t α +2sp+t = zN +2sp+t ≥ zN +2s(p+λ−1)+t zNN+2sp

≥

α

1

>

zN +2s(p+λ)+t−s

=

+2s(p+λ)+t−s zNN+2s(p+λ)−s

PN +2s(p+λ)+t−s

.

This implies α

α

+2s(p+λ)+t +2s(p+λ)+t−s 1 ≥ PN +2sp+t PN +2s(p+λ)+t−s > zNN+2s(p+λ) zNN+2s(p+λ)−s ≥ 1.

A contradiction. From the above claim it follows that zN +2s(p+λ)+t =

1 zN +2s(p+λ)+t−s

= zN +2s(p+λ−1)+t ≥ zN +2s(p+λ)+t .

Thus zN +2s(p+λ−1)+t = zN +2s(p+λ)+t for every 1 ≤ λ ≤ q − 1. Case 2 {zN +2ks }+∞ k=0 is increasing. In this case, it follows from Proposition 2.1 (iii) that PN +2s(p+q)+t

≥

α +2s(p+q−1)+t zNN+2s(p+q−1)

=

PN +2s(p+q)+t α

+2s(p+q)+t zNN+2s(p+q)

= zN +2s(p+q)+t > zN +2s(p+q−1)+t α

+2s(p+q−1)+t−s zNN+2s(p+q−1)−s

1

= min{zN +2s(p+q−2)+t , } zN +2s(p+q−1)+t−s PN +2s(p+q−1)+t−s = zN +2s(p+q−2)+t ≥ zN +2s(p+q−1)+t since α

α

+2s(p+q−1)+t +2s(p+q−1)+t−s zNN+2s(p+q−1)−s ≥ 1, PN +2s(p+q)+t PN +2s(p+q−1)+r−s ≤ 1 and zNN+2s(p+q−1)

we have zN +2s(p+q−1)+t = zN +2s(p+q−2)+t . In a similar fashion, we may obtain that zN +2s(p+q−1)+t = zN +2s(p+λ)+t for any 0 ≤ λ ≤ q − 2. The proof is complete. +∞ Proposition 2.6 If there exists N ∈ N such that {zN +2ks }+∞ k=0 is monotone, then {zN +t+2ks }k=0

is eventually monotone. Proof If there exists K ∈ N such that zN +2ks+t = 1/zN +2sk+t−s for all k ≥ K 4 404

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or α

+2ks+t zN +2ks+t = PN +2ks+t /zNN+2ks > 1/zN +2ks+t−s for all k ≥ K,

then by Proposition 2.1 (iii) we obtain that zN +2ks+t ≤ zN +2(k−1)s+t for all k ≥ K (or zN +2ks+t > zN +2(k−1)s+t for all k ≥ K). Thus {zN +t+2ks }+∞ k=K is monotone. If there exists a sequence 1 < p1 < q1 < p2 < q2 < · · · < pk < qk < · · · such that PN +2rs+t 1 for every pi ≤ r < qi zN +2rs+t = αN +2rs+t > zN +2rs+t−s zN +2rs and

1

for every qi ≤ r < pi+1 , zN +2rs+t−s then by Proposition 2.1 (iii) and Proposition 2.5 it follows that zN +2(r−1)s+t < zN +2rs+t for zN +2rs+t =

every pi ≤ r < qi and zN +2(r−1)s+t = zN +2rs+t for every qi ≤ r < pi+1 , this follows that {zN +t+2rs }+∞ r=p1 is increasing. The proof is complete.

3. Main Results In section, we state the main results of this paper. +∞ Theorem 3.1 Let {zn }+∞ n=−r be a positive solution of (1.2). Then limn−→∞ zn = 1 or {z2ns+k }n=0

is eventually monotone for every 0 ≤ k ≤ 2s − 1. Proof If U = lim supn−→∞ zn = 1, then let ik −→ +∞ such that zik −→ u = lim inf n−→∞ zn and zik −s −→ u1 . Thus 1 ≥ u = lim zik ≥ lim k−→∞

k−→∞

1 zik −s

=

1 1 = 1. ≥ u1 U

Which implies limn−→∞ zn = 1. Now assume that U = lim supn−→∞ zn > 1. First we suppose that gcd(s, t) = 1. Then by Proposition 2.4 (iii) we see that there exists N ∈ N such that the following statements hold: (1) zN +2ns zN +(2n−1)s = 1 for any n ≥ 0. (2) zN +2ns is decreasing (n ≥ 0) and limn−→∞ zN +2ns = U. xN +(2n−1)s is increasing (n ≥ 0) and limn−→∞ zN +(2n−1)s = u = 1/U. Using Proposition 2.6 repeatedly, it follows that for every 1 ≤ i ≤ s − 1, {zN +2ns+it }+∞ n=0 and {zN +(2n−1)s+it }+∞ n=0 are eventually monotone. Since gcd(s, t) = 1, it follows that for every j ∈ {0, 1, 2, · · · , s − 1} there exist some 0 ≤ ij ≤ s − 1 and integer λj such that ij t = λj s + j and ij t − +∞ s = (λj − 1)s + j. Thus {zN +2ns+λj s+j }+∞ n=0 and {zN +2ns+(λj −1)s+j }n=0 are eventually monotone

for every j ∈ {0, 1, 2, · · · , s − 1}, which implies that {z2ns+k }+∞ n=0 is eventually monotone for every 0 ≤ k ≤ 2s − 1. If gcd(s, t) = d > 1, then we consider the max-type equation zn = max{

1 zn−ds1

,

Pn }, αn zn−dt1

n = 0, 1, 2, · · · · · · ,

(3.1)

where s = ds1 and t = dt1 with gcd(s1 , t1 ) = 1. Write yni = znd+i for every 0 ≤ i ≤ d − 1 and n = 0, 1, 2, · · · . Then (3.1) reduces to the equations yni = max{

Pnd+i 1 , i }, 0 ≤ i ≤ d − 1, n = 0, 1, 2, · · · . i αnd+i yn−s (y n−t1 ) 1

(3.2)

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By an analogous way as in the above, we obtain that for every 0 ≤ i ≤ d − 1, yni is a solution of equation i yni = max{1/yn−s , 1

Pnd+i }. i (yn−t1 )αnd+i

i Then {y2s }+∞ is eventually monotone for every 0 ≤ k ≤ 2s1 − 1. Thus for every 0 ≤ k ≤ 1 n+k n=0

2s − 1, {z2ns+k }+∞ n=0 is eventually monotone. The proof is complete.

Theorem 3.2 Assume that s = 1, and t is even, and Pn is a periodic sequence. Let {zn }+∞ n=−t be a positive solution of (1.2). Then limn−→∞ zn = 1 or {zn }+∞ n=−t is eventually periodic with period 2. Proof If U = lim supn−→∞ zn = 1, then using arguments similar to ones developed in the proof of Theorem 3.1 we can obtain limn−→∞ zn = 1. Now assume that U = lim supn−→∞ zn > 1. According to Proposition 2.4 (iii) and Theorem 3.1, we see that there exists N ∈ N such that the following statements hold: (1) zN +2n zN +2n−1 = 1 for any n ≥ 0. (2) zN +2n is decreasing (n ≥ 0) and limn−→∞ zN +2n = U. zN +2n−1 is increasing (n ≥ 0) and limn−→∞ zN +2n−1 = u = 1/U. We claim that zN +2n+1 = 1/zN +2n eventually. In fact, if there exist 1 ≤ k1 < k2 < · · · < ki < · · · such that zN +2ki +1 =

PN +2ki +1 αN +2k

+1

zN +2kii+1−t

,

then by taking a subsequence we may assume that PN +2ki +1 and αN +2ki +1 are constant seαN +2k

+1

quences since Pn and αn are periodic sequences. Thus zN +2ki +1 is decreasing since zN +2kii+1−t

is increasing. A contradiction. Which implies that {zn }+∞ n=−t is eventually periodic with period

2. The proof is complete. Example 3.3 Assume that s = 1 and t is odd. Let Pn = P ∈ (0, 1) and αn = α ∈ (0, 1) for any n ≥ 0. Then there exists a positive solution {zn }∞ n=−t of (1.2) which is not eventually periodic such that limn−→∞ zn 6= 1. Proof Choose the initial values z−t , z1−t , · · · , z−1 ∈ (0, +∞) satisfying z−t < z2−t < · · · < z−1 < z−t /P, z−t < P 2/(1−α) , zk−t = 1/zk−t−1 k ∈ {1, 3, · · · , t − 2}. Now we show that z2k−1 < z2k+1 and z2k < z2k−2 for any k ∈ N. α . Which implies By z−1 < z−t /P and z−t < P 2/(1−α) , we have z−1 < z−t /P < P z−t

z1 = z2 = z3 = z4 =

1

P 1 1 }= < = z−2 . α z−1 z−t z−1 z−3 1 P α α max{ , α } = max{z−1 , P z−t } = P z−t > z−1 . z0 z1−t 1 P 1 P 1 1 1 max{ , α } = max{ α , α } = = < = z0 . α z1 z2−t P z−t z2−t P z−t z1 z−1 1 P 1 α α α α max{ , α } = max{z1 , P z2−t = z1 . } = max{P z−t , P z2−t } = P z2−t > z2 z3−t z2 1 P 1 P 1 1 1 = < = z2 . max{ , α } = max{ α , α } = α z3 z4−t P z2−t z4−t P z2−t z3 z1

z0 = max{

,

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Assume that there exists some m ∈ N such that (1) z2k−1 < z2k+1 and z2k+2 < z2k for any (−t + 1)/2 ≤ k ≤ m. α (2) z2k+1 = P z2k−t for any 0 ≤ k ≤ m and z2k+2 z2k+1 = 1 for any (−t + 1)/2 ≤ k ≤ m.

Then z2m+3 = max{

,

P

}= α z2m+2 z2m+3−t α α max{P z2m−t , P z2m+2−t }

α } max{z2m+1 , P z2m+2−t

α α = P z2m+2−t > P z2m−t = z2m+1 . 1 P 1 P = max{ , α } = max{ α , α } z2m+3 z2m+4−t P z2m+2−t z2m+4−t 1 1 1 = < = z2m+2 . = α P z2m+2−t z2m+3 z2m+1

= z2m+4

1

Therefore z2k−1 < z2k+1 and z2k+2 < z2k for any k ≥ (−t + 1)/2, which implies that {zn }∞ n=−t is α not eventually periodic. Since z2n+1 = P z2n−t (n ∈ N), we obtain limn−→∞ zn 6= 1. The proof

is complete.

REFERENCES [1] K. Berenhaut, J. Foley and S. Stevi´c, Boundedness character of positive solution of a max difference equation, J. Differ. Eq. Appl. 12(2006), 1193-1199. [2] Y. Chen, Eventually periodicity of xn+1 = max{1/xn , An /xn−1 } with periodic coefficients, J. Differ. Eq. Appl. 11(2005), 1289-1294. [3] Daniel W. Cranston and Candace M. Kent, On the boundedness of positive solutions of the reciprocal max-type difference equation xn = max{A1n−1 /xn−1 , A2n−1 /xn−2 , · · · , Atn−1 /xn−t } with periodic parameters, Appl. Math. Comput. 221(2013), 144-151. [4] W. Liu, X. Yang and S. Stevi´c, On a class of nonautonomous max-type difference equations, Abstr. Appl. Anal. 2011(2011), 15 pages. [5] W. Liu and S. Stevi´c, Global attractivity of a family of nonautonomous max-type difference equations, Appl. Math. Comput. 218(2012), 6297-6303. [6] A. D. Mishkis, On some problems of the theory of differential equations with deviating argument, Uspekhi Mat. Nauk 32(1977), 173-202. [7] E. P. Popov, Automatic regulation and control (in Russian), Nauka, Moscow, Russia, 1966. [8] B. Qin, T. Sun and H. Xi, Dynamics of the max-type difference equation xn+1 = max{A/xn , xn−k }, J. Comput. Appl. Anal. 14(2012), 856-861. [9] S. Stevi´c, On the recursive sequence xn+1 = max{c, xpn /xpn−1 }, Appl. Math. Letter 21(2008), 791-796. [10] T. Sun, B. Qin, H. Xi and C. Han, Global behavior of the max-type difference equation xn+1 = max{1/xn , An /xn−1 }, Abstr. Appl. Anal. 2009(2009), 10 pages. [11] T. Sun, H. Xi, C. Han and B. Qin, Dynamics of the max-type difference equation xn = max{1/xn−r , An /xn−r }, J. Appl. Math. Comput. 38(2012), 173-180. 7 407

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General properties of concave functions defined by the generalized Srivastava-Attiya operator Hasan BAYRAM and S¸ahsene ALTINKAYA Department of Mathematics, Faculty of Arts and Science, Uludag University, Bursa, Turkey. [email protected] [email protected] January 27, 2016 Abstract In this paper we introduce a class =m,k µ,b C0 (α) of concave functions by using the generalized Srivastava-Attiya operator. Also, we get distortion bounds for this class. Keywords: Hadamard product, concave functions, linear operator, distortion theorem, Hurwitz-Lerch Zeta functions, Srivastava-Attiya operator. 2010, Mathematics Subject Classification: 30C45, 30C50.

1

Introduction

Let A denote the class of analytic functions in the unit disk U = {z ∈ C : |z| < 1} that have the form f (z) = z +

∞ X

an z n .

(1)

n=2

Further, by S we shall denote the class of all functions in A which are univalent in U. The study of operators plays an important role in Geometric Function Theory in Complex Analysis and its related fields. Many derivative and integral operators can be written in terms of convolution of certain analytic functions. For functions ∞ X fj (z) = an,j z n (j = 1, 2) n=0

1

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analytic in U , we define the Hadamard product of f1 and f2 as (f1 ∗ f2 ) (z) =

∞ X

an,1, an,2 z n = (f2 ∗ f1 ) (z)

(z ∈ U ).

(2)

n=0

In terms of the Hadamard product (or convolution), the Dziok-Srivastava linear convolution operator involving the generalized hypergeometric function was introduced and studied systematically by Dziok and Srivastava [9], [10] and (subsequently) by many other authors (see, for details, [11] and [20]). We recall here a general Hurwitz-Lerch Zeta function Φ(z, s, a) defined in [19] by ∞ X zn Φ(z, s, a) := (n + a)s n=2
− a ∈ C \ Z− 0 ; s ∈ C, when |z| < 1; Re(s) > 1 when |z| = 1 where, as usual, Z0 := Z\N, and N := {1, 2, 3, . . .}). Several interesting properties and characteristics of the Hurwitz-Lerch Zeta function Φ(z, s, a) can be found in [8], and the references stated there in (see also [16], [21], [22]). Srivastava and Attiya [21] (also see [4], [12]) introduced and investigated the linear operator. =µb : A → A defined in terms of the Hadamard product by =µb f (z) = (Gµb ∗ f ) (z),


z ∈ U ; b ∈ C\Z− 0 ; µ ∈ C; f ∈ A

(3)

where, for convenience, Gµb (z) := (1 + b)µ [Φ(z, µ, b) − b−µ ]

(z ∈ U ).

(4)

We recall here the following relationships which follow easily by using (1), (3) and (4) µ ∞  X 1+b an z n . (5) =µb f (z) = z + n + b n=2 Motivated essentially by the Srivastava-Attiya operator, Murugusundaramoorthy [17] introduced the generalized integral operator =m,k µ,b given by =m,k µ,b f (z) = z +

∞ X

Cnm (b, µ, k)an z n

(6)

n=2

where

  1 + b µ m!(n + k − 2)! Ψn = = (7) n + b (k − 2)!(n + m − 1)! and (throughout this paper unless otherwise mentioned) the parameters µ, b are constrained as b ∈ C\Z− 0 ; µ ∈ C, k ≥ 2 and m > −1. It is of interest to note that =1,2 is the Srivastava-Attiya operator and =m,k µ,b 0,b is the well-known Choi-SaigoSrivastava operator (see [15]). Suitably specializing the parameters m, k, µ and b in =m,k µ,b f (z) we can get various integral operators introduced by Alexander [1] and Bernardi [5], Libera and Livingston [13], [14]. Cnm (b, µ, k)

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2

Preliminaries

Conformal maps of the unit disk onto convex domains are a classical topic. Recently Avkhadiev and Wirths [2] discovered that conformal maps onto concave domains (the complements of convex closed sets) have some novel properties. A function f : U → C is said to belong to the family C0 (α) if f satisfies the following conditions: • f is analytic in U with the standard normalization f (0) = f 0 (0) − 1 = 0. In addition it satisfies f (1) = ∞. • f maps U conformally onto a set whose complement with respect to C is convex. • The opening angle of f (U ) at ∞ is less than or equal to πα, α ∈ (1, 2]. The class C0 (α) is referred to as the class of concave univalent functions and for a detailed discussion about concave functions, we refer to Avkhadiev et al. [3], Cruz and Pommerenke [7] and references there in. In particular, the inequality   zf 00 (z) 0, where   α+11+z f 00 (z) 2 −1−z 0 . Pf (z) = α−1 2 1−z f (z) Definition 1 Let f (z) ∈ A and α ∈ (1, 2] . Then f (z) ∈ =m,k µ,b C0 (α) if and only if  h i00  m,k = f (z) µ,b 2 α + 1 1 + z  Re −1−zh (z ∈ U ). i0  > 0  α−1 2 1−z m,k =µ,b f (z)

3

Main results

Theorem 2 If f (z) ∈ A satisfies the inequality ∞ X   (α − 1)n + 2n2 |Cnm (b, µ, k)||an | < 3 − α, n=2

for some α ∈ (1, 2], n ∈ N, then f (z) ∈ =m,k µ,b C0 (α).

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Proof. We want to prove that  Re

h

i00 

=m,k µ,b f (z)

2 α + 1 1 + z  −zh i0  > 0.  α−1 2 1−z m,k = f (z) µ,b

By using the fact that 1 1 > ⇔ |w − 1| < 1, w 2 it is enough to show that |w| < 1.   1 2 α+11+z g 0 (z) = −z w α−1 2 1−z g(z) Re

where g(z) = z



=m,k µ,b f (z)

0

and g 0 (z) = 1 +

( =z ∞ X

1+

∞ X

(8)

)

Cnm (b, µ, k)nan z n−1 n=2

Cnm (b, µ, k)n2 an z n−1 .

(9)

(10)

n=2

Using (9) and (10), in (8) we obtain   ∞ P m 2(1−z)z 1+ Cn (b,µ,k)nan z n−1 n=2 .     |w| ≤ α−1 ∞ ∞ P P 2 (α+1)(1+z)z 1+n=2Cnm (b,µ,k)nan zn−1 −2(1−z)z 1+n=2Cnm (b,µ,k)n2 an zn−1 Using triangle inequality and letting z → −1, then   ∞ P 1+ Cnm (b, µ, k)|an |n  α−1 n=2  . |w| < ∞   P m 2 2 1− Cn (b, µ, k)|an |n n=2

The last expression is bounded by 1, if 1+ 1−

∞ P

Cnm (b, µ, k)|an |n

n=2 ∞ P


q2 (x) on [−1, 0)∪ ∈ (0, 1], then between any two consecutive zeros of y1 (x) there is at least one zero of y2 (x). Proof. Let x1 and x2 with x1 < x2 be consecutive zeroes of y1 . Suppose, it possible,that y2 does not have a zero on (x1 , x2 ). Lagrange’s identity (see, [12]) gives y2 L1 y1 − y1 L2 y2 =

d 0 {y y1 − y10 y2 } + {q1 (x) − q2 (x)}y1 y2 dx 2

(2.4)

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Hence d 0 {y y2 − y20 y1 } = {q1 (x) − q2 (x)}y1 y2 dx 1

(2.5)

Case 1. Let x1 ∈ [−1, 0), x2 ∈ (0, 1] and δ > 0. Integrating on both sides of the equation (2.9) over [x1 , 0) and (0, x2 ] and then adding we get 2 1 lim (y10 y2 − y20 y1 )|x0−² + lim (y10 y2 − y20 y1 )|x0+² 1 2 ²1 → 0 ²2 → 0 ²1 > 0 ²2 > 0

0−² Z 1

=

lim ²1 → 0 ²1 > 0

Zx2

{q1 (x) − q2 (x)}y1 y2 dx + x1

lim ²2 → 0 ²2 > 0

{q1 (x) − q2 (x)}y1 y2 dx (2.6) 0+²2

Since y1 (x1 ) = y1 (x2 ) = 0 we get lim W (y1 , y2 ; 0 − ²1 ) − lim W (y1 , y2 ; 0 + ²2 ) − y10 (x1 )y2 (x1 ) + y10 (x2 )y2 (x2 ) ²2 → 0 ²1 → 0 ²2 > 0 ²1 > 0 0−² Z 1

=

lim ²1 → 0 ²1 > 0

Zx2

{q1 (x) − q2 (x)}y1 y2 dx + x1

lim ²2 → 0 ²2 > 0

{q1 (x) − q2 (x)}y1 y2 dx

(2.7)

0+²2

Using the transmission conditions we obtain 0−² Z 1

−y10 (x1 )y2 (x1 )

+

y10 (x2 )y2 (x2 )

=

lim ²1 → 0 ²1 > 0

{q1 (x) − q2 (x)}y1 y2 dx x1

Zx2 +

lim ²2 → 0 ²2 > 0

{q1 (x) − q2 (x)}y1 y2 dx

(2.8)

0+²2

In this case with no restriction we can assume that y1 (x) > 0 and y2 (x) > 0 over (x1 , 0)∪(0, x2 ). These conditions ensure that the integral on the right in (2.8) is positive. On the left, since y1 (x) > 0 by assumption, the function is increasing at the point x1 . Hence y10 (x1 ) > 0(it cannot vanish, because then it would follow from the uniqueness theorem for the solutions of (2.1) that y1 (x) ≡ 0, which is impossible). Similarly, y10 (x2 ) < 0. Thus, the left-hand side of the equation (2.8) is less or equal to zero, which is a contradiction. Case 2. Let x1 ∈ [−1, 0), x2 ∈ (0, 1] and δ < 0. In this case with no restriction it can be assumed that, y1 (x) > 0 over (x1 , 0), y1 (x) < 0 over (0, x2 ), y2 (x) > 0 over (x1 , 0) and y2 (x) < 0 over (0, x2 ). Since y1 (x1 ) = 0 and y1 (x1 ) > 0 over (x1 , 0) y10 (x1 ) > 0. Further, since y2 (x2 ) = 0 and y2 (x2 ) < 0 immediately to left of x2 , y20 (x) < 0. Hence, the left-hand side of (2.8) is is less

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or equal zero, but the right-hand side is positive which shows that (2.8) is impossible. Case 3. Let (x1 , x2 ) ⊂ [−1, 0). Integrating on both sides of the equation (2.5) from x1 to x2 , we get (y10 y2

−

y20 y1 )|xx21

Zx2 = {q1 (x) − q2 (x)}y1 y2 dx

(2.9)

x1

Then with no restriction it can be assumed that y1 (x) > 0 and y2 (x) > 0 over (x1 , x2 ). These conditions ensure that the integral on the right in (2.9) is positive. However, on the left, we have y1 (x1 ) = y1 (x2 ) = 0 with y10 (x1 ) > 0 and y10 (x2 ) < 0. The left-hand side therefore becomes y10 (x2 )y2 (x2 ) − y10 (x1 )y2 (x1 ) ≤ 0 which presents us with a contradiction: right-hand side > 0 and left-hand side < 0. Thus y2 (x) = 0 (at least once) between the zeros of y1 (x). Since the conditions describing y1 (x) are given, we conclude that y2 (x) must change sign between x = x1 and x = x2 . Case 4. Let (x1 , x2 ) ⊂ (0, 1]. This case is totaly similar to the previous case.

3

On the zeros of eigenfunctions

In this section we examine the number of zeros of eigenfunctions. Lemma 3.1. There is an unique solution y(x, λ) of the equation (1.1) satisfying the initial conditions y(x0 , λ) = α(λ), y 0 (x0 , λ) = β(λ)

(3.1)

and the transmission conditions (1.2) where α(λ), β(λ) are given entire functions of λ ∈ C and x0 ∈ [−1, 0)∪(0, 1]. Moreover, y(x, λ) is entire function of λ ∈ C for each fixed x ∈ [−1, 0)∪(0, 1]. Proof. The proof is totally similar to [?] and therefore is omitted. ½ φ1 (x, λ1 ), x ∈ [−1, 0) Theorem 3.2. Let φ(x, λ1 ) = be solution of the equation (1.1), for φ2 (x, λ1 ), x ∈ (0, 1] λ = λ1 satisfying the initial conditions φ1 (−1, λ1 ) = α,

φ01 (−1, λ1 ) = β

(3.2)

and the transmission conditions φ2 (0+ , λ1 ) = ½

1 φ1 (0− , λ1 ), φ02 (0+ , λ1 ) = δφ01 (0− , λ1 ) δ

ϕ1 (x, λ2 ), x ∈ [−1, 0) ϕ2 (x, λ2 ), x ∈ (0, 1] fying the initial conditions and ϕ(x, λ2 ) =

(3.3)

be solution of the equation (1.1), for λ = λ2 satis-

ϕ1 (−1, λ2 ) = α,

ϕ01 (−1, λ2 ) = β

(3.4)

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and the transmission conditions ϕ2 (0+ , λ2 ) =

1 φ2 (0− , λ2 ), ϕ02 (0+ , λ2 1) = δφ01 (0− , λ1 ). δ

(3.5)

where δ, β, δ any real numbers with α2 + β 2 6= 0, δ 6= 0. Suppose that φ(x, λ1 ) has a zeros in [−1, 0) ∪ (0, 1] and let x1 (x1 6= −1) be zero of the function φ(x, λ1 ), nearest to x = −1. If λ2 > λ1 then ϕ(x2 , λ2 ) has at least one zero in [−1, x1 ). Proof. From the well-known Lagrange’s identity (see,for example, [12]) we have d {φ0 ϕ1 − ϕ01 φ1 } = {λ2 − λ1 }φ1 ϕ1 dx 1

(3.6)

d {φ0 ϕ2 − ϕ02 φ2 } = {λ2 − λ1 }φ2 ϕ2 dx 2

(3.7)

in the interval (0, 1).

Case 1. Let x1 > 0 and δ > 0. Integrating on both sides of the equation (3.11) from −1 to x1 , we get 1 1 lim (φ02 ϕ2 − ϕ02 φ2 )|x0+² lim (φ01 ϕ1 − ϕ01 φ1 )|0−² −1 + 2 ²2 → 0 ²1 → 0 ²2 > 0 ²1 > 0

Zx1

0−² Z 1

=

lim {λ2 − λ1 } ²1 → 0 ²1 > 0

φ1 ϕ1 dx + −1

φ2 ϕ2 dx lim {λ2 − λ1 } ²2 → 0 0+²2 ²2 > 0

(3.8)

Since W (φ1 , ϕ1 ; −1) = 0 by (3.2) and (3.4) we get lim W (φ1 , ϕ1 ; 0 − ²1 ) − lim W (φ2 , ϕ2 ; 0 + ²2 ) + φ02 (x1 , λ1 )ϕ2 (x1 , λ2 ) ²2 → 0 ²1 → 0 ²2 > 0 ²1 > 0 0−² Z 1

=

lim {λ2 − λ1 } ²1 → 0 ²1 > 0

φ1 ϕ1 dx + −1

Zx1 lim {λ2 − λ1 } φ2 ϕ2 dx ²2 → 0 0+²2 ²2 > 0

(3.9)

Using the transmission conditions we obtain 0−² Z 1

φ02 (x1 , λ1 )ϕ2 (x1 , λ2 )

=

lim {λ2 − λ1 } ²1 → 0 ²1 > 0

φ1 ϕ1 dx −1

Zx1 +

lim {λ2 − λ1 } φ2 ϕ2 dx ²2 → 0 0+²2 ²2 > 0

(3.10)

5

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With no restriction it can be assumed that φ(x, λ1 ) < 0 and ϕ(x, λ2 ) < 0 in [−1, x1 ). These conditions ensure that the integral on the right in (3.10) is positive. Since φ2 (x1 , λ1 ) = 0 and φ2 (x, λ1 ) > 0 immediately to the left of x1 by assumption, the function is increasing at the point x1 . Hence φ02 (x1 , λ1 ) > 0(it cannot vanish, because then it would follow from the uniqueness theorem for the solutions of (2.1) that φ2 (x, λ1 ) ≡ 0, which is impossible). Thus, the left-hand side of the equation (3.10) is less or equal to zero, but the right-hand side is positive, which is a contradiction. Case 2. Let x1 > 0 and δ < 0. In this case with no restriction it can be assumed that φ(x, λ1 ) > 0 and ϕ(x, λ2 ) < 0 in [−1, 0) but φ(x, λ1 ) < 0 and ϕ(x, λ2 ) > 0 in (0, x1 ]. As in the previous case, these conditions ensure that the integral on the right of (3.10) is negative, but left hand side of (3.10) is positive or is equal to zero, i.e. the equality (3.10)is impossible. Case 3. Let x1 ∈ [−1, 0). Integrating on both sides of the equation (2.5) from a to x1 , we get (φ01 ϕ1

−

1 ϕ01 φ1 )|x−1

Zx1 = {λ2 − λ1 }φ1 ϕ1 dx

(3.11)

−1

Since φ1 (x, λ1 ) = 0 by using the initial conditions φ1 (−1, λ1 ) = 0, φ01 (−1, λ1 ) = 0 we get φ01 (x1 )ϕ1 (x1 )

Zx1 = {λ2 − λ1 }φ1 ϕ1 dx

(3.12)

−1

Let x1 < 0. Without loss of generality, we can put φ(x, λ1 ) > 0 and ϕ(x, λ2 ) > 0 in [−1, x1 ). Since, by assumption, φ1 (x, λ1 ) > 0 and ϕ1 (x, λ2 ) > 0 in [−1, x1 ) and λ2 > λ1 , the right-hand side of the equality (3.12) is positive. However, on the left-hand side, since φ1 (x1 , λ1 ) = 0 and φ1 (x, λ1 ) > 0 immediately to the left of x1 , the function φ1 (x, λ1 ) is decreasing in the vicinity of the point x1 . Therefore, φ01 (x1 , λ1 ) ≤ 0(it cannot vanish, because then it would follow from the uniqueness theorem for the solutions of (1.1) that φ1 (x, λ1 ) ≡ 0, which is impossible). The left-hand side therefore becomes φ01 (x1 , λ1 )ϕ1 (x1 , λ1 ) ≤ 0 which presents us with a contradiction: right-hand side > 0 and left-hand side ≤ 0. The proof is complete. Now we are ready to establish the main result. Theorem 3.3. Let ψ1 (x) and ψ2 (x) be two eigenfunction corresponding to the eigenvalues λ1 and λ2 of the problem (1.1)-(1.3) and let λ2 > λ1 . Then if ψ1 (x) has m zeros in [−1, 0) ∪ (0, 1], ψ2 (x) has not fewer than m zeros in the same two-interval [−1, 0) ∪ (0, 1]. Moreover, n − th zero of ψ2 (x) is less than the n − th zero of ψ1 (x). Proof. Let x01 , x02 , ..., x0m with x01 < x2 0; t N (cx, t) = N (x, |c| ) if c 6= 0 N (x + y, s + t > min{N (x, s), N (y, t)};

0

2010 Mathematics Subject Classification: 47H10; 47L25; 46S40; 39B52; 39B72. Keywords: additive-quadratic functional equation; matrix fuzzy normed space; fixed point; Hyers-Ulam stability. ∗ Corresponding author. 0 E-mail:1 [email protected], 2 [email protected] 0

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(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. To see more properties and examples of fuzzy normed vector spaces, we refer to [19, 20]. Definition 1.2. 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 1.3. Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called a Cauchy sequence if for each ε > 0 and each t > 0, there exists an n0 ∈ N such that for all n > n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well known that every convergent sequence in a fuzzy normed vector space is a Cauchy sequence. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, the sequence {f (xn )} converges f (x0 ). If f : X → Y continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [2]). We will use the following notations: Mn (X) is the set of all n × n-matrices in X; ej ∈ M1,n (C) is that the jth component is 1 and the other components are zero; Eij ∈ Mn (C) is that the (i, j)-component is 1 and the other components are zero; Eij ⊗ x ∈ Mn (X) is that the (i, j)-component is x and the other components are zero. For x ∈ Mn (X), y ∈ Mk (X),   x 0 . x⊗y = 0 y Let (X, k · k) be a normed space. 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 6 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). Throughout this paper, let (X, {k · kn }) be a matrix normed space and (Y, {k · kn }) be a matrix Banach space. We introduce the concept of a matrix fuzzy normed space.

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Definition 1.4. Let (X, N ) be a fuzzy normed space. (1) (X, N ) is called a matrix fuzzy normed space if for each positive integer n, (Mn (X), Nn )  t is a fuzzy normed space and Nk (AxB, t) > Nn x, kAk·kBk for all t > 0, A ∈ Mk,n (R), x = [xij ] ∈ Mn (X) and B ∈ Mn,k (R) with kAk 6= 0.kBk 6= 0. (2) (X, {Nn }) is called a matrix fuzzy Banach space if (X, N ) is a fuzzy Banach space and (X, {Nn }) is a matrix fuzzy normed space. Example 1.5. Let (X, {k · kn }) be a matrix normed space. Let Nn (x, t) := and x = [xij ] ∈ Mn (X). Then Nk (AxB, t) =

t t > = t + kAxBkk t + kAk.kxkn .kBk

t t+kxkn

for all t > 0

t kAk.kBk t kAk.kBk

+ kxkn

for all t > 0, A ∈ Mk,n (R), x = [xij ] ∈ Mn (X) and B ∈ Mn,k (R) with kAk.kBk 6= 0. So, (X, {Nn }) is a matrix fuzzy normed space. 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 product of operator spaces may again be regarded as operator spaces. Owing in part to this result, the theory of operator spaces have an increasingly significant effect on operator algebra theory(see [10]). The proof given in [29] appealed to the theory of ordered operator spaces [7]. Effros and Ruan [11] showed that one can give a purely metric proof of this important theorem by using a technique of Pisier [26] and Effors [9]. The study of stability problems have been formulated by Ulam [31] in 1940: Under what condition does there exist a homomorphism near an approximate homomorphism? In the following year, Hyers [14] answered affirmatively the question of Ulam for Banach spaces, which was stated that if ε > 0 and f : X → Y is a mapping with X a normed space and Y is a Banach space such that kf (x + y) − f (x) − f (y)k 6 ε (1.1) for all x, y ∈ X, then there exists a unique additive map T : X → Y such that kf (x + y) − f (x) − f (y)k 6 ε for all x ∈ X. A generalized version of the theorem of Hyers for approximately linear mappings presented by Rassias [27] in 1978 by considering the case when (1.1) is unbounded. In 2003, Cˇadariu and Radu applied the fixed point method to the investigation of the Jensen functional equation [3]. They could present a short and a simple proof (different of the “direct method”, initiated by Hyers in 1941) for the Hyers-Ulam stability of the Jensen functional equation [3] and forthe quadratic functional equation [4]. See [12, 22, 23, 24, 28, 30] for more information on functional equations. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies

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(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, y) 6 d(x, z) + d(z, y) for all x, y, z ∈ X. We recall a fundamental result in fixed point theory. Theorem 1.6. [8] Let (Ω, d) be a complete generalized metric space and J : Ω → Ω be a strictly contractive mapping with Lipschitz constant 0 < L < 1. Then for each given x ∈ Ω, either d(J n x, J n+1 x) = ∞ for all nonnegative n or there exists a positive integer n0 such that (1) (2) (3) (4)

d(J n x, J n+1 x) < ∞, ∀n > n0 ; the sequence {J n x} converges to a fixed point y ∗ of J; y ∗ is the unique fixed point of J in the set Λ = {y ∈ Ω : d(J n0 x, y) < ∞}; 1 d(y, y ∗ ) 6 1−L d(y, Jy) for all y ∈ Λ.

Definition 1.7. A mapping f : X × X → Y is called additive-quadratic if f satisfies the system of equations  f (x + y, z) = f (x, z) + f (y, z), (1.2) f (x, y + z) + f (x, y − z) = 2f (x, y) + 2f (x, z). When X = Y = R, the function f : R×R → R given by f (x, y) := cxy 2 is a solution of (1.2). In particular, letting x = y, we get a cubic function g : R → R given by g(x) := f (x, x) = cx3 . For a mapping f : X × X → Y , consider the functional equation: f (x + y, z + w) + f (x + y, z − w) = 2f (x, z) + 2f (x, w) + 2f (y, z) + 2f (y, w).

(1.3)

for all x, y, z, w ∈ X. The solution of (1.3) was discussed in [25]. In this paper, by using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.3) in matrix fuzzy normed spaces. 2. Fuzzy stability of the additive-quadratic functional equation (1.3) In this section, using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.3) in matrix fuzzy normed space. We need the following lemma. Lemma 2.1. [17, Lemma 2.1] Let (X, {Nn }) be a matrix fuzzy normed space. (1) Nn (Ekl ⊗ x, t) = N (x, t) for all t > 0 and x ∈ X. P (2) for all [xij ] ∈ Mn (X) and t = ni,j=1 tij , N (xkl , t) > N ([xij ], t) > min{N (xij , tij ) : i, j = 1, 2, · · · , n},    t  N (xkl , t) > N ([xij ], t) > min N xij , 2 : i, j = 1, 2, · · · , n n (3) limn→∞ xn = x if and only if limn→∞ xijn = xij for xn = [xijn ], x = [xij ] ∈ Mk (X)

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Fuzzy stability in matrix fuzzy normed spaces

Proof.

(1) Since Ekl ⊗ x = e∗k xel and ke∗k k = kel k = 1, Nn (Ekl ⊗ x, t) > N (x, t). Since ek (Ekl ⊗ x)e∗l = x, Nn (Ekl ⊗ x, t)6 N (x, t). SoN (Ekl ⊗ x, t)=N(x,t).

t (2) N (xkl , t) = N (ek [xij ]e∗l , t) > Nn [xij ], kek k.ke = Nn ([xij ], t). lk

Nn ([xij ], t) = Nn

n X

 Eij ⊗ xij , t > min{Nn (Eij ⊗ xij , tij ) : i, j = 1, 2, · · · , n}

i,j=1

= min{N (xij , tij ) : i, j = 1, 2, · · · , n}, P where t = ni,j=1 tij . So, Nn ([xij ], t) > min{N (xij , nt2 ) : i, j = 1, 2, · · · , n}. (3) By N (xkl , t) > Nn ([xij ], t) > min{N (xij , nt2 ) : i, j = 1, 2, · · · , n}, we obtain the result. This completes the proof.  For a mapping f : X → Y , define Df : X m → Y and Dfn : Mn (X 4 ) → Mn (Y ) by Df (a, b, c, d) := f (a + b, c + d) + f (a + b, c − d)

Dfn



− 2f (a, c) − 2f (a, d) − 2f (b, c) − 2f (b, d),      [xij ], [yij ], [zij ],[wij ] := fn [xij ] + [yij ], [zij ] + [wij ] + fn [xij ] + [yij ], [zij ] − [wij ]         − 2fn [xij ], [zij ] − 2fn [xij ], [wij ] − 2fn [yij ], [zij ] − 2fn [yij ], [wij ]

for all a, b, c, d ∈ X and all x = [xij ], y = [yij ], z = [zij ], w = [wij ] ∈ Mn (X). Theorem 2.2. Let f : X → Y , with f (x, 0) = 0, be a mapping for which there exists a function ϕ : X 4 → [0, ∞) such that   t Pn (2.1) Nn fn ([xij ], [yij ], [zij ], [wij ]), t > t + i,j=1 ϕ(xij , yij , zij , wij ) for all t > 0 and all x = [xij ], y = [yij ], z = [zij ], w = [wij ] ∈ Mn (X). If there exists an α < 1 such that   a b c d ϕ(a, b, c, d) 6 8αϕ , , , (2.2) 2 2 2 2 for all a, b, c, d ∈ X, then there exists a unique additive-quadratic mapping T : X × X → Y such that Nn (fn ([xij ], [yij ]) − Tn ([xij ], [yij ]) , t) >

8(1 − α)t + n2

8(1 − α)t Pn i,j=1 ϕ(xij , xij , yij , yij )

(2.3)

for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Proof. Putting n = 1 in (2.1), we have N (Df (x, y, z, w), t) >

t t + ϕ(x, y, z, w)

(2.4)

for all t > 0 and x, y, z, w ∈ X.

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Letting x = y and z = w in (2.4), we obtain N (f (2x, 2z) − 8f (x, z), t) > and also

t t + ϕ(x, x, z, z)

(2.5)

 t t 1 > N f (2x, 2z) − f (x, z), 8 8 t + ϕ(x, x, z, z) for all t > 0 and x, z ∈ X. Also it can be written as   1 t t N > f (2x, 2y) − f (x, y), 8 8 t + ϕ(x, x, y, y) 

(2.6)

for all t > 0 and x, y ∈ X. By considering the set of Ω := {g : X → Y }, we introduce the generalized metric on Ω as following:   t + , ∀x, y ∈ X, ∀t > 0 d(g, h) = inf k ∈ R : N (g(x, y) − h(x, y), kt) > t + ϕ(x, x, y, y) where, as usual inf ∅ = +∞. It is easy to show that (Ω, d) is complete (see [5, 18]). Now we define J : Ω → Ω by 1 Jg(x, y) := h(2x, 2y) 8 for all x, y ∈ X. Let g, h ∈ Ω be given such that d(g, h) = c. Then t N (g(x, y) − h(x, y), ct) > t + ϕ(2x, 2x, 2y, 2y)   1 1 c t ⇒N g(2x, 2y) − h(2x, 2y), t > 8 8 8 t + ϕ(2x, 2x, 2y, 2y)   1 c t 1 g(2x, 2y) − h(2x, 2y), t > ⇒N 8 8 8 t + 8αϕ(x, x, y, y)   1 1 t ⇒N g(2x, 2y) − h(2x, 2y), αct > 8 8 t + ϕ(x, x, y, y) ⇒ d(Jg, Jh) 6 αc for all x, y ∈ X. Hence we get that d(Jg, Jh) 6 αd(g, h) for all g, h ∈ Ω. It follows from (2.6) that d(f, Jf ) 6 18 . By Theorem 1.6, there exists a mapping T : X → Y satisfying the following: (1) T is a fixed point of J, i.e., T (2x, 2y) = 8T (x, y) for all x ∈ X. The mapping T is a unique fixed point of J in the set X = {g ∈ Ω : d(f, g) < ∞}. (2) d(J k f, T ) → 0 as k → ∞. This implies the inequality N − limk→∞ 81k f (2k x, 2k y) = T (x, y) for all x, y ∈ X.

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Fuzzy stability in matrix fuzzy normed spaces

(3) d(f, T ) 6 (f, T ) 6

1 1−α d(f, Jf ),

which implies the inequality

1 . 8(1 − α)

(2.7)

By (2.2) and (2.4),   1 t k k k k N Df (2 x, 2 y, 2 z, 2 w) > k k 8 t + ϕ(2 x, 2k y, 2k z, 2k w) > for all x, y, z, w ∈ X and t > 0. Since limk→∞ t > 0,

8k t 8k t + 8k αk ϕ(x, y, z, w)

8k t 8k t+8k αk ϕ(x,y,z,w)

= 1 for all x, y, z, w ∈ X and

N (DT (x, y, z, w), t) = 1 for all x, y, z, w ∈ X and t > 0. Therefore T (x + y, z + w) + T (x + y, z − w) = 2T (x, z) + 2T (x, w) + 2T (y, z) + 2T (y, w). for all x, y, z, w ∈ X. Then, the mapping T : X × X → Y is additive-quadratic. It follows from Lemma 2.1 and (2.7) that       t Nn fn ([xij ], [yij ]) − Tn ([xij ],[yij ]), t > N f (xij , yij ) − T (xij , yij ), 2 : i, j = 1, 2, · · · , n n   8(1 − α)t > min : i, j = 1, 2, · · · , n 8(1 − α)t + n2 ϕ(xij , xij , yij , yij ) 8(1 − α)t Pn > 2 8(1 − α)t + n i,j=1 ϕ(xij , xij , yij , yij ) for all x = [xij ] ∈ Mn (X). Therefore, we conclude that T : X × X → Y is the unique mapping satisfying (2.3).  Corollary 2.3. Let p, θ be positive real numbers p < 1. Let f : X × X → Y , with f (x, 0) = 0, be a mapping satisfying t Pn Nn (Dfn ([xij ], [yij ], [zij ], [wij ]), t) > (2.8) p t + i,j=1 θ(kxij k + kyij kp + kzij kp + kwij kp ) for all x = [xij ], y = [yij ], z = [zij ], w = [wij ] ∈ Mn (X) and t > 0. Then T (x, y) := N − limk→∞ 81k f (2k x, 2k y) exists for each x, y ∈ X and defines an additive-quadratic mapping T : X × X → Y such that   2(2 − 2p )t P Nn fn ([xij ], [yij ]) − Tn ([xij ], [yij ]) , t > 2(2 − 2p )t + n2 ni,j=1 θ (kxij kp + kyij kp ) for all x = [xij ], y = [yij ] ∈ Mn (X) and t > 0. P p p p p Proof. Putting ϕ(a, b, c, z) := θ m i=1 (kak + kbk + kck + kdk ) for all a, b, c, d ∈ X and letting α = 2p−1 in Theorem 2.2, we obtain the desired result. 

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Theorem 2.4. Let f : X × X → Y , with f (x, 0) = 0, be a mapping for which there exists a function ϕ : X 4 → [0, ∞) satisfying (2.1). If there exists an α < 1 such that   a b c d α ϕ 6 ϕ(a, b, c, d) , , , 2 2 2 2 8 for all a, b, c, d ∈ X, then there exists a unique additive-quadratic mapping T : X × X → Y such that   8(1 − α)t P N fn ([xij ], [yij ]) − Tn ([xij ], [yij ]) , t > 2 8(1 − α)t + n α ni,j=1 ϕ(xij , xij , yij , yij ) for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Proof. Let (Ω, d) be the generalized metric space defined in the proof of Theorem 2.2. Here, we define the linear mapping J : Ω → Ω such that x y Jg(x, y) := 8g( , ) 2 2 for all x, y ∈ X. It follows from (2.5) that d(f, Jf ) 6 α8 . Thus α d(f, T ) 6 . 8(1 − α) The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let p, θ be positive real numbers with p > 1. Let f : X × X → Y , with f (x, 0) = 0, be a mapping satisfying (2.8). Then T (x, y) := N − limk→∞ 8k f ( 2xk , 2yk ) exists for all x ∈ X and defines an additive-quadratic mapping T : X × X → Y such that   4(2p − 2)t P Nn fn ([xij ], [yij ]) − Tn ([xij ], [yij ]) , t > 4(2p − 2)t + n2 · 2p ni,j=1 θ (kxij kp + kyij kp ) for all x = [xij ], y = [yij ] ∈ Mn (X) and t > 0. Proof. Putting ϕ(a, b, c, d) := θ(kakp + kbkp + kckp + kdkp ) for all a, b, c, d ∈ X and letting α = 21−p in Theorem 2.4, we get the desired result.  References [1] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687-705. [2] T. Bag and S. K. Samanta, Finite fuzzy bounded linear operators, Fuzzy Sets Syst. 151 (2005), 513-547. [3] L. Cˇ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure. Appl. Math. 4 (2003), no. 1, Art. 4. [4] L. Cˇ adariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat. Inform. 41 (2003), 25-48. [5] L. Cˇ adariu and V. Radu, On the stabilityu of the Cauchy functional equation, Grazer Math. Berichte, 346 (2004), 43-52. [6] S. C. Cheng and J. M.Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcuta Math. Soc.86 (1994), 429-436.

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Fuzzy stability in matrix fuzzy normed spaces [7] M. D. Choi and E. Effors, Injectivity and operator space, J. Func. Anal. 24(1997), 156-209. [8] 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. [9] E. Effors, On multilinear Completely bounded module maps. Contemp. Math. Vol. 62, pp. 479-501. Amer. Math. Soc., Providence, 1987. [10] E. Effros and Z. J. Ruan, On approximation properties for operator spaces, Int. J. Math.1 (1990), 163-187. [11] E. Effros and Z. J. Ruan, On the abstarac characterization of operator spaces, Proc. Amer. Math. Soc. 119 (1993), 579-584. [12] M. Eshaghi Gordji, A. Rahimi, C. Park and D. Shin, Ternary Jordan bi-homomorphisms in Banach Lie triple systems, J. Comput. Anal. Appl. 21 (2016), 1040-1045. [13] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst. 48 (1992), 239-248. [14] D. H. Hyers, On the stability of the linear functional equation. Proc. Natl Acad Sci. USA 27 (1941), 222-224. [15] A. K. Katsaras, Fuzzy topological vector space, Fuzzy Sets Syst. 12 (1987), 143-154. [16] S. V. Krishna and K. K. M. Sarma, Saperation of fuzzy normed linear spaces, Fuzzy Sets Syst. 63 (1994), 207-217. [17] J. Lee, D. Shin and C. Park, Hyers-Ulam stability of functional equations in matrix normed spaces, J. Inequal. Appl. 2013 (2013), Art. ID 2013:22. [18] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567-572. [19] A. K. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. 159 (2008), 730-738. [20] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy approximately cubic mappings. Inf. Sci. 178 (2008), 37913798. [21] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159 (2008), 720-729. [22] E. Movahednia, S. M. S. M. Mosadegh, C. Park and D. Shin, Stability of a lattice preserving functional equation on Riesz space: fixed point alternative, J. Comput. Anal. Appl. 21 (2016), 83-89. [23] C. Park, Stability of ternary quadratic derivation on ternary Banach algebras: revisited, J. Comput. Anal. Appl. 20 (2016), 21-23. [24] W. Park and J. Bae, Approximate quadratic forms on restricted domains, J. Comput. Anal. Appl. 20 (2016), 388-410. [25] W. Park, J, Bae and B. Chung, On an additive-quadratic functional equation and its stability, J. Appl. Math. Computing 18 (2005), 563-572. [26] G. Pisier, Grothendieck’s theorem for non-commutative C ∗ -algebras with an appendix on Grothendieck’s constants, J. Func. Anal. 29 (1978), 397-415. [27] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 27 (1978), 297-300. [28] Th. M. Rassias, New characterization of inner product spaces, Bull. Sci. Math. 108 (1984), 95-99. [29] J. Z. Ruan, Subspaces of C ∗ -algebras, J. Func. Anal. 76 (1988), 217-230. [30] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37-49. [31] S. M. Ulam, A Collocation of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics. Interscience Publ., New York, London, 1940. [32] J. Z. Xiao, X. H. Zhu, Fuzzy normed spaces of operators and its completeness, fuzzy Sets Syst. 133 (2003), 389-399.

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Closed Form Expressions of some systems of Nonlinear Partial Difference Equations Tarek F. Ibrahim1,2 1 Department of Mathematics Faculty of Science, Mansoura University Mansoura, EGYPT 2 Department of Mathematics Faculty of Sciences and Arts (S. A.) King Khalid University Abha, SAUDI ARABIA. [email protected] January 13, 2016 Abstract In this paper we give the closed form expressions of some two dimensional systems of nonlinear rational partial difference equations of second order.We shall use a new method to prove the results by using (odd-even) double mathematical induction. As a direct consequences , we investigate and drive the explicit solutions of some partial difference equations and some (systems of) ordinary difference equations . AMS Subject Classification: 39A10, 39A14. Key Words and Phrases: (partial)difference equations, solutions , double mathematical induction.

1

Introduction

While the study of (ordinary)difference equations has been widely treated in the past , partial difference equations (P∆Es) have not received the same full attention .Both of ordinary and partial difference equations may be found in the study of probability ,dynamics and other branches of mathematical physics .Moreover,partial difference equations arise in applications involving population dynamics with spatial migrations , chemical reactions and finite difference schemes . Indeed Laplace and Lagrange considered the solution of partial difference equations in their studies of dynamics and probability. An example of a partial difference equation is the following well known relation (n−1)

(n) (n−1) = Cm−1 + Cm Cm

, 1 ≤ m < n.

The solution of this equation is the celebrated binomial coefficient function defined by n! (n) Cm = , 0 ≤ m < n. m!(n − m)!

(n) Cm

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2 An another example , the following P∆Es : (n+1)

= sk−1 − nsk

(n+1)

= Sk−1 + kSk

sk Sk

(n)

(n)

(n)

(n)

, 1 ≤ k < n. , 1 ≤ k < n. (n)

The solutions of these P∆Es are the stirling numbers of the first kind sk (n) and the stirling numbers of the second kind Sk respectively . Some authors investigate the closed form solutions for certain Partial difference equations . For instance , Heins [[9] ] considered the solution of the partial difference equation Xn+1,m + Xn−1,m = 2Xn,m+1 under some conditions . In [[3]] Carlitz has studied a solution of the partial difference equation Xn,m − Xn,m−1 − Xn−1,m − Xn,m−2 + 3Xn−1,m−1 − Xn−2,m = 0 He used a power series expansion related to the Fibonacci numbers . For more results about partial difference equations we refer to ([1],[2], [4]-[8],[10],[11]-[15]). In this paper , we studied the closed form solutions of the following systems of partial difference equations αXn,m + βXn,m Xn−2,m−2 Yn−1,m−1 − Xn−2,m−2 = 0 (1) γYn,m + δYn,m Yn−2,m−2 Xn−1,m−1 − Yn−2,m−2 = 0 (2) S where n, m ∈ N0 , N0 = N {0} ,α, β, γ, δ ∈ {1, −1} and the initial values Xn,0 ,Xn,−1 ,X0,m ,X−1,m , Yn,0 ,Yn,−1 ,Y0,m ,and Y−1,m are real numbers . As a direct consequence , we can drive the explicit solutions of a family of partial difference equations in the following form αXn,m + βXn,m Xn−2,m−2 Xn−1,m−1 − Xn−2,m−2 = 0 S where n, m ∈ N0 , N0 = N {0} ,α, β ∈ {1, −1} and the initial values Xn,0 ,Xn,−1 ,X0,m , ,and X−1,m are real numbers . Moreover , we can derive the exact solution for the following systems of ordinary difference equations αXn + βXn Xn−2 Yn−1 − Xn−2 = 0 γYn + δYn Yn−2 Xn−1 − Yn−2 = 0 S where n ∈ N0 , N0 = N {0} ,α, β, γ, δ ∈ {1, −1} and the initial values X0 ,X−1 ,Y0 ,and Y−1 are real numbers .

2

Forms of Solutions

In this section we shall give explicit forms of solutions of the system (1)-(2) for particular values of α, β, γ, δ ∈ {1, −1} . We can rewrite system (1)-(2) in the following form Xn,m =

Xn−2,m−2 α + βXn−2,m−2 Yn−1,m−1

,

Yn,m = 434

Yn−2,m−2 γ + δYn−2,m−2 Xn−1,m−1

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3

2.1

Form of Solutions when (α, β) = (γ, δ) = (1, −1)

In this case we have the system Xn,m =

Xn−2,m−2 1 − Xn−2,m−2 Yn−1,m−1

Theorem 1. initial conditions

,

Yn,m =

Yn−2,m−2 1 − Yn−2,m−2 Xn−1,m−1

(4)

Let {Xn,m , Yn,m }∞ n,m=−k be a solution of system (4) with

Xn,0 , Xn,−1 , X0,m , X−1,m , Yn,0 , Yn,−1 , Y0,m , Y−1,m S where n, m ∈ N0 , N0 = N {0} . Suppose X−1,m−2 Y0,m−1 6= 1 ,Xn−2,−1 Yn−1,0 6= 1 ,Y−1,m−2 X0,m−1 6= 1 ,Yn−2,−1 Xn−1,0 6= 1 . Then, the form of solutions of system (4) ,for n, m ≥ 1 and n ≥ m , are as follows:  m−2 2  Q  −1+(2k+1)Xn−m,0 Yn−m−1,−1  , m even;  Xn−m,0 −1+(2k+2)Xn−m,0 Yn−m−1,−1 k=0 Xn,m = (5) m−1  2 Q  1−(2k)Xn−m−1,−1 Yn−m,0   Xn−m−1,−1 , m odd; k=0

Yn,m =

 m−2 2  Q    Yn−m,0 k=0

−1+(2k+1)Yn−m,0 Xn−m−1,−1 , −1+(2k+2)Yn−m,0 Xn−m−1,−1

m−1  2 Q    Yn−m−1,−1

k=0

Xm,n =

 m−1 2  Q    X−1,n−m−1 k=0

m−2  2 Q    X0,n−m

k=0

Ym,n =

k=0

k=0

m even; (6)

1−(2k)Yn−m−1,−1 Xn−m,0 , 1−(2k+1)Yn−m−1,−1 Xn−m,0

1−(2k)X−1,n−m−1 Y0,n−m , 1−(2k+1)X−1,n−m−1 Y0,n−m

m odd;

m odd; (7)

−1+(2k+1)X0,n−m Y−1,n−m−1 , −1+(2k+2)X0,n−m Y−1,n−m−1

 m−1 2  Q    Y−1,n−m−1 m−2  2 Q    Y0,n−m

1−(2k+1)Xn−m−1,−1 Yn−m,0

1−(2k)Y−1,n−m−1 X0,n−m , 1−(2k+1)Y−1,n−m−1 X0,n−m

m even;

m odd; (8)

−1+(2k+1)Y0,n−m X−1,n−m−1 , −1+(2k+2)Y0,n−m X−1,n−m−1

m even;

Proof. We shall use the principle of (odd-even)double mathematical induction . Firstly , we shall prove that the relations (5)-(8) hold for (n, m) = (1, 1). From equations in system (4)we can see 1−1

X1,1

2 Y X−1,−1 1 − (2k)X−1,−1 Y0,0 = = X−1,−1 1 − X−1,−1 Y0,0 1 − (2k + 1)X−1,−1 Y0,0 k=0

Y1,1

2 Y Y−1,−1 1 − (2k)Y−1,−1 X0,0 = = Y−1,−1 1 − Y−1,−1 X0,0 1 − (2k + 1)Y−1,−1 X0,0 k=0

1−1

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4 Now , we shall prove that the relations (5)-(8)hold for (n, m) = (2, 2). X2,2 =

X0,0 X0,0 1 − X0,0 Y−1,−1 = = X0,0 ( ) Y−1,−1 1 − X0,0 Y1,1 1 − 2X0,0 Y−1,−1 1 − X0,0 ( 1−Y−1,−1 X0,0 ) 2−2

= X0,0

2 Y −1 + (2k + 1)X0,0 Y−1,−1

k=0

Y2,2 =

−1 + (2k + 2)X0,0 Y−1,−1

Y0,0 1 − Y0,0 X−1,−1 Y0,0 = = Y0,0 ( ) X−1,−1 1 − Y0,0 X1,1 1 − 2Y0,0 X−1,−1 1 − Y0,0 ( 1−X−1,−1 Y0,0 ) 2−2

= Y0,0

2 Y −1 + (2k + 1)Y0,0 X−1,−1

k=0

−1 + (2k + 2)Y0,0 X−1,−1

Moreover ,We shall prove that the relations (5)-(8) hold for (n, m) = (1, 2) and (n, m) = (2, 1). 1−1

X1,2

2 Y X−1,0 1 − (2k)X−1,0 Y0,1 = = X−1,0 1 − X−1,0 Y0,1 1 − (2k + 1)X−1,0 Y0,1 k=0

Y1,2

2 Y Y−1,0 1 − (2k)Y−1,0 X0,1 = Y−1,0 = 1 − Y−1,0 X0,1 1 − (2k + 1)Y−1,0 X0,1 k=0

X2,1

2 Y 1 − (2k)X0,−1 Y1,0 X0,−1 = = X0,−1 1 − X0,−1 Y1,0 1 − (2k + 1)X0,−1 Y1,0 k=0

1−1

1−1

Y2,1 =

Y0,−1 1 − Y0,−1 X1,0

Now suppose that the relations (5)-(8) hold for m = 1 and m = 2 with n ∈ N . So we have , Xn,1 = Xn−2,−1

0 Y

Xn−2,−1 1 − (2k)Xn−2,−1 Yn−1,0 = 1 − (2k + 1)Xn−2,−1 Yn−1,0 1 − Xn−2,−1 Yn−1,0 k=0 Yn,1 =

Yn−2,−1 1 − Yn−2,−1 Xn−1,0

Xn,2 = Xn−2,0 ( Yn,2 = Yn−2,0 (

1 − Xn−2,0 Yn−3,−1 ) 1 − 2Xn−2,0 Yn−3,−1

1 − Yn−2,0 Xn−3,−1 ) 1 − 2Yn−2,0 Xn−3,−1 436

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5 Now we try to prove that relations (5)-(8) hold for m = 1 with n + 2. 0

Xn+2,1

2 Y Xn,−1 1 − (2k)Xn,−1 Yn+1,0 = = Xn,−1 1 − Xn,−1 Yn+1,0 1 − (2k + 1)Xn,−1 Yn+1,0 k=0

Yn+2,1

2 Y 1 − (2k)Yn,−1 Xn+1,0 Yn,−1 = = Yn,−1 1 − Yn,−1 Xn+1,0 1 − (2k + 1)Yn,−1 Xn+1,0 k=0

1−1

Now we try to prove that relations (5)-(8) hold for m = 2 with n + 2. Xn,0 Xn,0 = Xn+2,2 = 1 − Xn,0 Yn+1,1 1 − Xn,0 ( Yn−1,−1 ) 1−Yn−1,−1 Xn,0

2−2 2 Y Xn,0 (1 − Yn−1,−1 Xn,0 ) 1 − (2k + 1)Xn,0 Yn−1,−1 = = Xn,0 1 − 2Yn−1,−1 Xn,0 1 − (2k + 2)Xn,0 Yn−1,−1 k=0 2−2

Yn+2,2

2 Y Yn,0 1 − (2k + 1)Yn,0 Xn−1,−1 = Yn,0 = 1 − Yn,0 Xn+1,1 1 − (2k + 2)Yn,0 Xn−1,−1 k=0

Finally , we suppose that relations (5)-(8) hold for n, m ∈ N . We shall prove that relations (5)-(8) hold for n, m + 2 ∈ N . From (4)we have Xn−2,m Xn,m+2 = (9) 1 − Xn−2,m Yn−1,m+1 There are four cases : (1) If n > m + 2 and m even . Xn,m+2 =

Xn−2,m 1 − Xn−2,m Yn−1,m+1 m−2 2

Xn−m−2,0

Q

k=0

=

m−2 2

Q

1 − (Xn−m−2,0

k=0

m

1−(2k+1)Xn−m−2,0 Yn−m−3,−1 )(Yn−m−3,−1 1−(2k+2)Xn−m−2,0 Yn−m−3,−1 m−2 2

Xn−m−2,0 =

1−(2k+1)Xn−m−2,0 Yn−m−3,−1 1−(2k+2)Xn−m−2,0 Yn−m−3,−1

1−

Q

2 Q

k=0

1−(2k)Yn−m−3,−1 Xn−m−2,0 ) 1−(2k+1)Yn−m−3,−1 Xn−m−2,0

1−(2k+1)Xn−m−2,0 Yn−m−3,−1 1−(2k+2)Xn−m−2,0 Yn−m−3,−1

k=0 Xn−m−2,0 Yn−m−3,−1 1−(m+1)Xn−m−2,0 Yn−m−3,−1

m

= Xn−m−2,0

2 Y 1 − (2k + 1)Xn−m−2,0 Yn−m−3,−1

k=0

1 − (2k + 2)Xn−m−2,0 Yn−m−3,−1 437

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6 (2) If n > m + 2 and m odd Xn,m+2 =

Xn−2,m 1 − Xn−2,m Yn−1,m+1 m−1 2

Xn−m−3,−1

Q

k=0

=

m−1 2

1 − (Xn−m−3,−1

Q

k=0

1−(2k)Xn−m−3,−1 Yn−m−2,0 )(Yn−m−2,0 1−(2k+1)Xn−m−3,−1 Yn−m−2,0 m−1 2

Xn−m−3,−1 =

1−(2k)Xn−m−3,−1 Yn−m−2,0 1−(2k+1)Xn−m−3,−1 Yn−m−2,0

1−

Q

k=0

1−(2k+1)Yn−m−2,0 Xn−m−3,−1 ) 1−(2k+2)Yn−m−2,0 Xn−m−3,−1

1−(2k)Xn−m−3,−1 Yn−m−2,0 1−(2k+1)Xn−m−3,−1 Yn−m−2,0

k=0 Xn−m−3,−1 Yn−m−2,0 1−(m+1)Xn−m−3,−1 Yn−m−2,0

m+1 2

= Xn−m−3,−1

Q

m−1 2

Y k=0

1 − (2k)Xn−m−3,−1 Yn−m−2,0 1 − (2k + 1)Xn−m−3,−1 Yn−m−2,0

(3) If n < m + 2 and m even By symmetry ,using (7) and (8), we can prove it like part (1) . (4) If n < m + 2 and m odd By symmetry ,using (7) and (8), we can prove it like part (2) .. Yn,m+2 =

Yn−2,m 1 − Yn−2,m Xn−1,m+1

We can do that by the same way in proving equation (9) Proposition 1. system (4) :

We have the following properties for the solutions of

(1) If m even and Xn−m,0 = 0 , then Xn,m = 0 . (2) If m odd and Xn−m,0 = 0 , then Yn,m = Yn−m−1,−1 . (3) If m even and Yn−m,0 = 0 , then Yn,m = 0 . (4) If m odd and Yn−m,0 = 0 , then Xn,m = Xn−m−1,−1 . (5) If m even and Xn−m−1,−1 = 0 , then Yn,m = Yn−m,0 . (6) If m odd and Xn−m−1,−1 = 0, then Xn,m = 0 . (7) If m even and Yn−m−1,−1 = 0 , then Xn,m = Xn−m,0 . (8) If m odd and Yn−m−1,−1 = 0, then Yn,m = 0 . 438

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7 Proposition 2. system (4) :

We have the following properties for the solutions of

(1) If m even and X0,n−m = 0 , then Xm,n = 0 . (2) If m odd and X0,n−m = 0 , then Ym,n = Y−1,n−m−1 . (3) If m even and Y0,n−m = 0 , then Ym,n = 0 . (4) If m odd and Y0,n−m = 0 , then Xm,n = X−1,n−m−1 . (5) If m even and X−1,n−m−1 = 0 , then Ym,n = Y0,n−m . (6) If m odd and X−1,n−m−1 = 0, then Xm,n = 0 . (7) If m even and Y−1,n−m−1 = 0 , then Xm,n = X0,n−m . (8) If m odd and Y−1,n−m−1 = 0, then Ym,n = 0 . Remark 1. If we take into account the one dimensional case of system (4) we have a partial difference equation in the form Xn,m =

Xn−2,m−2 1 − Xn−2,m−2 Xn−1,m−1

(10)

We can see that the closed form solution of equation(10) is given ,from theorem(1) , by the following corollary . Corollary 2. Let {Xn,m }∞ (10) with initial n,m=−k be a solution of equation S conditions Xn,0 , Xn,−1 , X0,m , X−1,m , where n, m ∈ N0 , N0 = N {0} . Suppose X−1,m−2 X0,m−1 6= 1 ,Xn−2,−1 Xn−1,0 6= 1 . Then, the form of solutions of equation (10) ,for n, m ≥ 1 and n ≥ m , are as follows:

Xn,m =

 m−2 2  Q    Xn−m,0 k=0

−1+(2k+1)Xn−m,0 Xn−m−1,−1 , −1+(2k+2)Xn−m,0 Xn−m−1,−1

m−1  2 Q    Xn−m−1,−1

1−(2k)Xn−m−1,−1 Xn−m,0 , 1−(2k+1)Xn−m−1,−1 Xn−m,0

m odd;

 m−1 2  Q    X−1,n−m−1

1−(2k)X−1,n−m−1 X0,n−m , 1−(2k+1)X−1,n−m−1 X0,n−m

m odd;

k=0

Xm,n =

m even;

k=0

m−2  2 Q    X0,n−m

k=0

Proposition 3. equation (4):

−1+(2k+1)X0,n−m X−1,n−m−1 , −1+(2k+2)X0,n−m X−1,n−m−1

m even;

We have the following properties for the solutions of

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8 (1) If m even and Xn−m,0 = 0 , then Xn,m = 0 . (2) If m odd and Xn−m,0 = 0 , then Xn,m = Xn−m−1,−1 . (3) If m even and Xn−m−1,−1 = 0 , then Xn,m = Xn−m,0 . (4) If m odd and Xn−m−1,−1 = 0, then Xn,m = 0 . (5) If m even and X0,n−m = 0 , then Xm,n = 0 . (6) If m odd and X0,n−m = 0 , then Xm,n = X−1,n−m−1 . (7) If m even and X−1,n−m−1 = 0 , then Xm,n = X0,n−m . (8) If m odd and X−1,n−m−1 = 0, then Xm,n = 0 . Remark 2. If we put n = m in system (4) we have a system of ordinary difference equations in the following form Xn =

Xn−2 , 1 − Xn−2 Yn−1

Yn =

Yn−2 1 − Yn−2 Xn−1

(11)

Corollary 3. Let {Xn , Yn }∞ n=−k be a solution of system (11) with initial conditions X0 , X−1 , Y0 , Y−1 . Suppose X−1 Y0 6= 1 ,and Y−1 X0 6= 1 ,. Then, the form of solutions of system (11) ,for n ≥ 1 are as follows:   n−2 n−2 2 2   Q Q   −1+(2k+1)X0 Y−1 −1+(2k+1)Y0 X−1   , n, even Y , n, even   X0 0 −1+(2k+2)X0 Y−1 −1+(2k+2)Y0 X−1 k=0 k=0 Yn = Xn = n−1 n−1   2 2 Q Q   1−(2k)X−1 Y0 1−(2k)Y−1 X0    Y−1  X−1 , n, odd , n, odd k=0

1−(2k+1)X−1 Y0

k=0

1−(2k+1)Y−1 X0

Remark 3. If we put X = Y in system(11) we get an ordinary difference equation in the form Xn−2 Xn = (12) 1 − Xn−2 Xn−1 We can see that the closed form solution of equation(12) is given ,from corollary(3) , by the following  n−2 2  Q  −1+(2k+1)X0 X−1  , n even;  X0 −1+(2k+2)X0 X−1 k=0 Xn = n−1  2 Q  1−(2k)X−1 X0   X−1 , n odd; k=0

1−(2k+1)X−1 X0

where n ∈ N , and X−1 X0 6= −1 .We can easy see that if n even (or odd) and X0 = 0 then Xn = 0(Xn = X−1 ). Also if n even (or odd) and X−1 = 0 then Xn = X0 (Xn = 0). 440

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9

2.2

Form of Solutions when (α, β) = (1, 1)&(γ, δ) = (1, −1)

In this case we have the system Xn,m =

Xn−2,m−2 1 + Xn−2,m−2 Yn−1,m−1

,

Yn,m =

Yn−2,m−2 1 − Yn−2,m−2 Xn−1,m−1

(13)

Theorem 4. Let {Xn,m , Yn,m }∞ n,m=−k be a solution of system (13) with initial conditions S Xn,0 , Xn,−1 , X0,m , X−1,m , Yn,0 , Yn,−1 , Y0,m , Y−1,m where n, m ∈ N0 , N0 = N {0} . Suppose X−1,m−2 Y0,m−1 6= −1 ,Xn−2,−1 Yn−1,0 6= −1 , Y−1,m−2 X0,m−1 6= 1 ,Yn−2,−1 Xn−1,0 6= 1 . Then, the form of solutions of system (13) ,for n, m ≥ 1 and n ≥ m , are as follows: ( Xn−m−1,−1 m odd; m+1 , (1+X n−m−1,−1 Yn−m,0 ) 2 Xn,m = m m (−1) 2 Xn−m,0 (−1 + Xn−m,0 Yn−m−1,−1 ) 2 , m even;  m+1  (−1) 2 Yn−m−1,−1 m odd; m+1 , Yn,m = (−1+Yn−m−1,−1 Xn−m,0 ) 2 m  Y n−m,0 (1 + Yn−m,0 Xn−m−1,−1 ) 2 , m even; ( X−1,n−m−1 m odd; m+1 , (1+X−1,n−m−1 Y0,n−m ) 2 Xm,n = m m (−1) 2 X0,n−m (−1 + X0,n−m Y−1,n−m−1 ) 2 , m even;  m+1  (−1) 2 Y−1,n−m−1 m odd; m+1 , Ym,n = (−1+Y−1,n−m−1 X0,n−m ) 2 m  Y (1 + Y X ) 2 , m even; 0,n−m

0,n−m

−1,n−m−1

Proof. We can prove the theorem by odd-even double mathematical induction as in theorem (1). Remark 4. We can see that both of proposition (1) and proposition (2) hold for the solutions of system (13) included in theorem(4) . Remark 5. If we put n = m in system (13) we have a system of ordinary difference equations in the following form Xn =

Xn−2 , 1 + Xn−2 Yn−1

Yn =

Yn−2 1 − Yn−2 Xn−1

(14)

We can drive the formulas for solutions from theorem(4) in the following corollary . Corollary 5. Let {Xn , Yn }∞ n=−k be a solution of system (14) with initial conditions X0 , X−1 , Y0 , Y−1 . Suppose X−1 Y0 6= −1 ,and Y−1 X0 6= 1 ,. Then, the form of solutions of system (14) ,for n ≥ 1 are as follows: ( ( n+1 X−1 (−1) 2 Y−1 ; n, odd n+1 n+1 ; n, odd (1+X−1 Y0 ) 2 Xn = Yn = (−1+Y−1 X0 ) 2 n n n (−1) 2 X0 (−1 + X0 Y−1 ) 2 ; n, even Y0 (1 + Y0 X−1 ) 2 ; n, even . 441

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10

2.3

Form of Solutions when (α, β) = (1, 1)&(γ, δ) = (−1, 1)

In this case we have the system Xn−2,m−2 Xn,m = 1 + Xn−2,m−2 Yn−1,m−1

,

Yn,m =

Yn−2,m−2 −1 + Yn−2,m−2 Xn−1,m−1

(15)

Theorem 6. Let {Xn,m , Yn,m }∞ n,m=−k be a solution of system (15) with initial conditions S Xn,0 , Xn,−1 , X0,m , X−1,m , Yn,0 , Yn,−1 , Y0,m , Y−1,m where n, m ∈ N0 , N0 = N {0} . Suppose X−1,m−2 Y0,m−1 6= −1 ,Xn−2,−1 Yn−1,0 6= −1 , Y−1,m−2 X0,m−1 6= 1 ,Yn−2,−1 Xn−1,0 6= 1 . Then, the form of solutions of system (15) ,for n, m ≥ 1 and n ≥ m , are as follows:  m−1 (−1) 4 Xn−m−1,−1   m = 4K + 1; m+3 m−1 ,   (1+Xn−m−1,−1 Yn−m,0 ) 4 (−1+Xn−m−1,−1 Yn−m,0 ) 4   m−2 m  (−1) 4 Xn−m,0 (−1+Xn−m,0 Yn−m−1,−1 ) 2   , m = 4K + 2; m+2 (−1+2Xn−m,0 Yn−m−1,−1 ) 4 Xn,m = m+1  (−1) 4 Xn−m−1,−1  m = 4K + 3;  m+1 m+1 ,   (−1+Xn−m−1,−1 Yn−m,0 ) 4 (1+Xn−m−1,−1 Yn−m,0 ) 4  m m   (−1) 4 Xn−m,0 (−1+Xn−m,0 Yn−m−1,−1 ) 2  , m = 4K + 4; m 4 (−1+2Xn−m,0 Yn−m−1,−1 )

Yn,m

 m−1 m−1 (−1) 4 Yn−m−1,−1 (−1+2Yn−m−1,−1 Xn−m,0 ) 4   , m+1   (−1+Yn−m−1,−1 Xn−m,0 ) 2   m−2 m+2   (−1) 4 Yn−m,0 (−1 + Yn−m,0 Xn−m−1,−1 ) 4   m+2 .(1 + Yn−m,0 Xn−m−1,−1 ) 4 , = m+1 m+1  (−1) 4 Yn−m−1,−1 (−1+2Yn−m−1,−1 Xn−m,0 ) 4  ,  m+1   (−1+Yn−m−1,−1 Xn−m,0 ) 2  m m   4 )4   (−1) Yn−m,0 (−1 + Yn−m,0 Xn−m−1,−1 m .(1 + Yn−m,0 Xn−m−1,−1 ) 4 ,

Xm,n =

                  

Ym,n

m−1 4 X−1,n−m−1 m+3 m−1 (1+X−1,n−m−1 Y0,n−m ) 4 (−1+X−1,n−m−1 Y0,n−m ) 4 m−2 m (−1) 4 X0,n−m (−1+X0,n−m Y−1,n−m−1 ) 2 m+2 (−1+2X0,n−m Y−1,n−m−1 ) 4 m+1 (−1) 4 X−1,n−m−1 m+1 m+1 (−1+X−1,n−m−1 Y0,n−m ) 4 (1+X−1,n−m−1 Y0,n−m ) 4 m m (−1) 4 X0,n−m (−1+X0,n−m Y−1,n−m−1 ) 2 m (−1+2X0,n−m Y−1,n−m−1 ) 4

(−1)

,

,

 m−1 m−1 (−1) 4 Y−1,n−m−1 (−1+2Y−1,n−m−1 X0,n−m ) 4   , m+1   (−1+Y−1,n−m−1 X0,n−m ) 2   m+2 m−2   (−1) 4 Y0,n−m (−1 + Y0,n−m X−1,n−m−1 ) 4   m+2 .(1 + Y0,n−m X−1,n−m−1 ) 4 , = m+1 m+1  4 (−1) Y−1,n−m−1 (−1+2Y−1,n−m−1 X0,n−m ) 4  ,  m+1   (−1+Y−1,n−m−1 X0,n−m ) 2  m m   4 )4   (−1) Y0,n−m (−1 + Y0,n−m X−1,n−m−1 m 4 .(1 + Y0,n−m X−1,n−m−1 ) , 442

m = 4K + 1; m = 4K + 2; m = 4K + 3; m = 4K + 4; , m = 4K + 1; m = 4K + 2; , m = 4K + 3; m = 4K + 4; m = 4K + 1; m = 4K + 2; m = 4K + 3; m = 4K + 4; Ibrahim 433-445

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11 where k = 0, 1, 2, 3...... . Proof. We can prove the theorem by piecewise double mathematical induction as in theorem (1). Proposition 4. system (15) :

We have the following properties for the solutions of

(1) If m even and Xn−m,0 = 0 , then Xn,m = 0 . (2) If m odd and Xn−m,0 = 0 , then Yn,m = ±Yn−m−1,−1 . (3) If m even and Yn−m,0 = 0 , then Yn,m = 0 . (4) If m odd and Yn−m,0 = 0 , then Xn,m = Xn−m−1,−1 . (5) If m even and Xn−m−1,−1 = 0 , then Yn,m = ±Yn−m,0 . (6) If m odd and Xn−m−1,−1 = 0, then Xn,m = 0 . (7) If m even and Yn−m−1,−1 = 0 , then Xn,m = ±Xn−m,0 . (8) If m odd and Yn−m−1,−1 = 0, then Yn,m = 0 . Proposition 5. system (15) :

We have the following properties for the solutions of

(1) If m even and X0,n−m = 0 , then Xm,n = 0 . (2) If m odd and X0,n−m = 0 , then Ym,n = ±Y−1,n−m−1 . (3) If m even and Y0,n−m = 0 , then Ym,n = 0 . (4) If m odd and Y0,n−m = 0 , then Xm,n = X−1,n−m−1 . (5) If m even and X−1,n−m−1 = 0 , then Ym,n = ±Y0,n−m . (6) If m odd and X−1,n−m−1 = 0, then Xm,n = 0 . (7) If m even and Y−1,n−m−1 = 0 , then Xm,n = ±X0,n−m . (8) If m odd and Y−1,n−m−1 = 0, then Ym,n = 0 . Remark 6. If we put n = m in system (15) we have a system of ordinary difference equations in the following form Xn =

Xn−2 , 1 + Xn−2 Yn−1

Yn =

Yn−2 −1 + Yn−2 Xn−1

(16)

We can drive the formulas for solutions from theorem(6) in the following corollary . 443

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12 Corollary 7. Let {Xn , Yn }∞ n=−k be a solution of system (16) with initial conditions X0 , X−1 , Y0 , Y−1 . Suppose X−1 Y0 6= −1 ,and Y−1 X0 6= 1 ,. Then, the form of solutionsof system (16) ,for n ≥ 1 are as follows: n−1 (−1) 4 X−1   n = 4K + 1; n+3 n−1 ,   (1+X−1 Y0 ) 4 (−1+X−1 Y0 ) 4   n−2 n   (−1) 4 X0 (−1+X0 Y−1 ) 2  , n = 4K + 2; n+2 (−1+2X0 Y−1 ) 4 Xn = n+1  (−1) 4 X−1  n = 4K + 3;  n+1 n+1 ,   (−1+X−1 Y0 ) 4 (1+X−1 Y0 ) 4  n n   (−1) 4 X0 (−1+X0 Y−1 ) 2  , n = 4K + 4; n 4 (−1+2X0 Y−1 )

Yn =

            

n−1 n−1 4 Y−1 (−1+2Y−1 X0 ) 4 n+1 (−1+Y−1 X0 ) 2 n−2 n+2 4 0 0 −1 4 n+1 n+1 (−1) 4 Y−1 (−1+2Y−1 X0 ) 4 n+1 (−1+Y−1 X0 ) 2 n n 4 4

(−1)

(−1)

Y (−1 + Y X )

,

n = 4K + 1;

(1 + Y0 X−1 )

n+2 4

,

, n = 4K + 2; n = 4K + 3;

n 4

(−1) Y0 (−1 + Y0 X−1 ) (1 + Y0 X−1 ) ,

n = 4K + 4;

where k = 0, 1, 2, 3...... .

References [1] M. J. Ablowitz and J. F. Ladik , On the Solution of a Class of Nonlinear Partial Difference Equations , Studies in Applied Mathematics , Volume 57, Issue 1,(1977), pages 1-12 . [2] F.G. Boese , Asymptotical stability of partial difference equations with variable coefficients, Journal of Mathematical Analysis and Applications , Volume 276, Issue 2, 15 December 2002, PP 709-722 [3] L. Carlitz, A partial difference equation related to the Fibonacci numbers, Fibonacci Quarterly, 2 ,No3 (1964), pp. 185–196. [4] Sui Sun Cheng , Partial Difference Equations, Taylor & Francis, London, 2003. [5] R.Courant, K. Friedrichs, H. Lewy , On the Partial Difference Equations of Mathematical Physics , IBM Journal of Research and Development Volume:11 , Issue: 2 ,(1967), 215-234 . [6] Wolfgang Dahmen, Charles A. Micchelli, On the Solution of Certain Systems of Partial Difference Equations and Linear Dependence of Translates of Box Splines , Transactions of the American Mathematical Society, Vol. 292, No. 1 (Nov., 1985), pp. 305-320 [7] Bart J. Daly, The Stability Properties of a Coupled Pair of Non-Linear Partial Difference Equations , Mathematics of Computation, Vol. 17, No. 84 (Oct., 1963), pp. 346-360 444

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13 [8] Leopold Flatto, Partial Differential Equations and Difference Equations, Proceedings of the American Mathematical Society, Vol. 16, No. 5 (Oct., 1965), pp. 858-863 [9] Albert E. Heins , On the Solution of Partial Difference Equations , American Journal of Mathematics , Vol. 63, No. 2 (Apr., 1941), pp. 435-442 . [10] Fulton Koehler, Charles M. Braden, An Oscillation Theorem for Solutions of a Class of Partial Difference Equations , Proceedings of the American Mathematical Society, Vol. 10, No. 5 (Oct., 1959), pp. 762-766 [11] Alan C. Newell , Finite Amplitude Instabilities of Partial Difference Equations , SIAM Journal on Applied Mathematics, Vol. 33, No. 1 (Jul., 1977), pp. 133-160 . [12] C. Raymond Adams, Existence Theorems for a Linear Partial Difference Equation of the Intermediate Type , Transactions of the American Mathematical Society, Vol. 28, No. 1 (Jan., 1926), pp. 119-128 [13] I. P. Van den Berg , On the relation between elementary partial difference equations and partial differential equations , Annals of Pure and Applied Logic 92 (3),(1998),235-265 [14] David Young, Iterative Methods for Solving Partial Difference Equations of Elliptic Type , Transactions of the American Mathematical Society, Vol. 76, No. 1 (Jan., 1954), pp. 92-111 [15] Doron Zeilberger, Binary Operations in the Set of Solutions of a Partial Difference Equation , Proceedings of the American Mathematical Society, Vol. 62, No. 2 (Feb., 1977), pp. 242-244

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TWO-DIMENSIONAL CHLODOWSKY VARIANT OF q-BERNSTEIN-SCHURER-STANCU OPERATORS MEHMET ALI ÖZARSLAN AND TUBA VEDI Abstract. In this paper, two-dimensional Chlodowsky variant q-based BernsteinSchurer-Stancu operators are introduced. Korovkin-type approximation theorems in di¤erent function spaces are studied. The error of approximation by using full modulus of continuity and partial modulus of continuities are given. Moreover, we introduce a generalization of our operators and investigate its approximation in more general weighted space.

1. Introduction It was Chlodowsky [3] who introduced the classical Bernstein-Chlodowsky operators as n r n r X r n x x f Cn (f ; x) = bn 1 ; n bn bn r r=0

where the function f is de…ned on [0; 1) and fbn g is a positive increasing sequence bn ! 0 as n ! 1. with bn ! 1 and n In 2008, the q-analogue of Chlodowsky operators were introduced and investigated by Karsl¬and Gupta [8] as Cn (f ; q; x) =

n+p X k=0

f

[k] bn [n]

n+p k

k n+p Yk 1

x bn

qs

1

s=0

x bn

;

0

x

bn

where fbn g has the same property of Bernstein-Chlodowsky operators. On the other hand, the q-Bernstein-Schurer operators were de…ned by Muraru [9], for …xed p 2 N0 and for all x 2 [0; 1], by (1.1)

Bnp (f ; q; x) =

n+p X

f

k=0

[k] [n]

n+p k x k

n+p Yk 1

(1

q s x) .

s=0

Note that the case q ! 1 in (1.1) reduces to the operators considered by Schurer [12]. Then, some properties of the q-Bernstein-Schurer operators were given in [13]. ; In 2013, the q-analogue of Bernstein-Schurer-Stancu operators Sn;p : C [0; 1 + p] ! C [0; 1] were introduced by Agrawal, et al in [4] by (1.2)

( ; Sn;p

)

(f ; q; x) =

n+p X k=0

f

n+p k x k

[k] + [n] +

n+p Yk 1

(1

q s x) ;

s=0

Key words and phrases. q-Bernstein operators, Chlodowsky operators, Chlodowsky variant of q-Bernstein-Schurer-Stancu operators, weighted space, modulus of continuity. 2010 AMS Math. Subject Classi…cation. Primary 41A10, 41A25; Secondary 41A36. 1

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M . A. ÖZARSLAN AND T. VEDI

where and are non-negative numbers which satisfy 0 and also p is a ( ; ) non-negative integer. Notice that, if we choose = = 0 in (1.2), Sn;p (f ; q; x) reduces to the classical q-Bernstein operator [10]. Recently, Chlodowsky variant of q-Bernstein-Schurer-Stancu operators were introduced by the authors in [14] as ( ; (1.3) Cn;p

)

(f ; q; x) :=

n+p X

f

k=0

n+p k

[k] + bn [n] +

x bn

k n+p Yk 1 s=0

1

qs

x bn

;

where n 2 N, p 2 N0 := f0g [ N, 0 ; 0 x bn and 0 < q < 1. If = = p = 0 in (1.3), we get the operators Cn (f ; q; x) and if q ! 1 and = = p = 0 in (1.3), we get the operators Cn (f ; x). In 2009, Büyükyaz¬c¬[1] de…ned the two-dimensional q-Bernstein-Chlodowsky polynomials as ! n X m X [j]qm [k]qn x y qn ;qm e f Bn;m (f ; x; y) = n; k;n;qn k;m;qm m [n]qn [m]qm n m j=0 k=0

n k 1 n k Y where k;n;qn (u) = (1 qns ) and investigated its approximation propu k s=0 erties on the rectangular unbounded domain. On the other hand, Büyükyaz¬c¬ and Sharma [2] de…ned the two-dimensional qBernstein-Chlodowsky-Durrmeyer operators on the rectangular unbounded domain and derived the Korovkin type approximation properties. They also computed the order of convergence by means of the modulus of continuity and then examined the weighted approximation properties for these operators. In the present paper we consider the two dimensional Chlodowsky variant of q( ; ) Bernstein-Schurer-Stancu operators. Some of the results about the operators Cn;p (f ; q; x) de…ned in (1.3) will be useful in our investigations. For instance, the …rst three mo( ; ) ments …rst three moments of the operator Cn;p (f ; q; x) are as follows [14]: ( ; )

Lemma 1.1. Let Cn;p ators are, ( ; )

(i) Cn;p

(1; q; x) = 1;

( ; )

(ii) Cn;p

(f ; q; x) de…ned. Then the …rst few moments of the oper-

(t; q; x) =

[n + p] x + bn ; [n] + 1

( ; )

(iii) C n;p (t2 ; q; x)=

2

([n] + )

+ (2 + 1) [n + p] bn x +

2 2 bn

[n + p

1] [n + p] qx2

.

Before proceeding further let us recall that the some basic de…nitions of q-calculus. The q-integer of k 2 R is [7] [k]q =

1 k

q k = (1

447

q) ; q 6= 1 ; q = 1;

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TWO DIM ENSIONAL CHLODOW SKY VARIANT OF q-BERNSTEIN-SCHURER-STANCU OPERATORS3

the q-factorial is de…ned by [k]q ! =

[k]q [k 1

1]q ::: [1]q ; ;

k = 1; 2; 3; :::; k=0

and q-binomial coe¢ cients are de…ned by n k

= q

[n]q ! [n

k]q ! [k]q !

The organization of the paper as follows: In section two, the two dimensional Chlodowsky variant of q-Bernstein-SchurerStancu operators is established and the …rst few moments of the operator is given. In section three, some Korovkin-type theorems in di¤erent function spaces are studied. In section four, we obtain the order of convergence of the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators by means of the …rst modulus of continuity and partial modulus of continuity. In section …ve, we study the generalization of the two-dimensional Chlodowsky variant of q-Bernstein-Schurer-Stancu operators and seek its approximation properties in more general weighted space. 2. Construction of the operators Let fan g and fbm g be increasing sequences of real numbers satisfying lim an = lim bm = 1:

n!1

m!1

Let, Dan ;bm denotes (2.1)

Dan ;bm = f(x; y) : 0

x

an , 0

y

bm g :

For (x; y) 2 Dan ;bm , we construct the two dimensional Chlodowsky variant of qBernstein-Schurer-Stancu operators as ( ; ) Cn;m;p (f ; qn ; qm ; x; y)

(2.2)

:=

n+p X m+p X k=0 j=0

f

[k]qn + [n]qn +

an ;

[j]qm + [m]qm +

bm

!

k;n;qn

x an

j;m;qm

y bm

n+p k 1

Y n+p (1 qns z). zk k qn s=0 We also let 0 < qn < 1 (n 2 N) for the positivity of the operators. It is easy to ( ; ) show that Cn;p (f ; qn ; qm ; x; y) is a linear and positive operator. Now, we start by giving the following lemma which will be used throughout the paper.

where n 2 N, p 2 N0 := f0g[N, 0

.

k;n;qn (z) =

( ; )

Lemma 2.1. Let Cn;m;p (f ; qn ; qm ; x; y) be given in (2.2). Then the …rst few moments of the operators are, ( ; )

(i) Cn;m;p (1; qn ; qm ; x; y) = 1;

( ; )

(ii) Cn;m;p (t1 ; qn ; qm ; x; y) =

[n + p]qn x + an [n]qn +

448

;

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M . A. ÖZARSLAN AND T. VEDI

( ; )

(iii) Cn;m;p (t2 ; qn ; qm ; x; y) =

[m + p]qm y + bm [m]qm +

( ; )

(iv) Cn;m;p t21 + t22 ; qn ; qm ; x; y

1

=

2

[n]qn +

n [n + p

1

+

2

[m]qm +

1]qn [n + p]qn qn x2 + (2 + 1) [n + p]qn an x +

n [m + p

2 2 an

1]qm [m + p]qm qm y 2 + (2 + 1) [m + p]qm bm y +

o

2 2 bm

o

:

Proof. Using Lemma 1.1 and the linearity of the operators, the proof is easily obtained.

3. Korovkin-type approximation theorems In this section, Korovkin-type approximation theorems are given for the two dimensional Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. For …xed 0 consider the space C which consists of all continuous functions f , satisfying the condition jf (x; y)j Clearly, C

Mf

(x; y) ;

[0; 1) := R2+ and

(x; y) 2 [0; 1)

(x; y) = 1 + x2 + y 2 .

is a linear normed space with the following norm kf k

=

sup 0 x;y0 f ( )kC

n;m!1

1+

=0

where fan g, fbm g, fqn g and fqm ghave the same conditions as in Theorem 3.1. Proof. For all " > 0, there exist su¢ ciently large positive real numbers A and B such that 1 + x2 + y 2

(3.1)

0 is arbitrary, then

"(1 + M ) 00

lim yn;m = 0. This completes the proof.

n;m!1

Now, consider the subspace C 0 of C which is de…ned by C 0 :=

f 2 C : lim

x;y!0

jf (x; y)j =0 . 1 + x2 + y 2

Theorem 3.4. Let the sequences fqn g ; fan g and fbm g satisfy the same properties as in Theorem 3.1. Then for all f 2 C 0 R2+ , we obtain lim kTn;m (f ; qn ; qm ; ; )

f ( )kC = 0:

n;m!1

Proof. For all f 2 C 0 R2+ , observe that [k]qn + [n]qn +

f

jf (x; y)j lim = 0, x;y!1 1 + x2 + y 2

lim

n;m!1

1+

[k]qn + [n]qn +

[j]

+

an ; [m]qm + bm qm

2

an

+

[j]qm + [m]qm +

2

= 0:

bm

Therefore, for all " > 0, we can …nd su¢ ciently large numbers A and B such that jf (x; y)j < " 1 + x2 + y 2

(3.2)

for x > A and y > B and there exists natural numbers n0 and m0 such that (3.3) 0 ! !2 !2 1 [k]qn + [k]qn + [j]qm + [j]qm + f an ; bm < " @1 + an + bm A [n]qn + [m]qm + [n]qn + [m]qm +

for all n > n0 and m > m0 . Hence, for large n and m, we have kTn;m (f ; qn ; qm ; ; ) ; Cn;m

sup (x;y)2DA;B

+

f ( )kC

(f ; qn ; qm ; x; y) 1 + x2 + y 2

sup (x;y)2Dan ;bm nDA;B

f (x; y)

; Cn;m (f ; qn ; qm ; x; y) 1 + x2 + y 2

f (x; y)

0

00 = zn;m + zn;m :

00 By Theorem 3.1 it is su¢ cient to show that zn;m ! 0 as n ! 1. Using (3.2) and (3.3), we get 00 zn;m

"+

sup (x;y)2Dan ;bm nDA;B

"+"

sup

; Cn;m (f ; qn ; qm ; x; y)) 1 + x2 + y 2

tn;m (qn ; qm ; x; y)

(x;y)2Dan ;bm nDA;B

=" 1+

sup (x;y)2Dan ;bm =DA;B

452

!

tn;m (qn ; qm ; x; y)

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; ; C ; (1;qn ;qm ;x;y))+Cn;m (t21 ;qn ;qm ;x;y))+Cn;m (t22 ;qn ;qm ;x;y)) where tn;m (qn ; qm ; x; y) := n;m : 2 2 1+x +y By Lemma 2.1, it is clear that there exist K independent of n and m such that

tn;m (qn ; qm ; x; y)

sup

K.

(x;y)2Dan ;bm =DA;B

Therefore, for n > n0 and m > m0 we have 00 zn;m < (1 + K)":

This completes the proof. 4. Order of convergence In this section, we compute the rate of convergence of the operators in terms of the the full modulus of continuity and partial modulus of continuities. Let f 2 DA;B and x 0. Then the de…nition of the modulus of continuity of f is given by (4.1)

!(f ; ) = p

It is known that for any (4.2)

jf (x1 ; y1 )

max

jf (x1 ; y1 )

(x1 x2 )2 +(y1 y2 )2 x;y2C(DA;B )

> 0 we know that 0q ! (f; ) @

f (x2 ; y2 )j

(x1

f (x2 ; y2 )j:

2

x2 ) + (y1

2

y2 )

and its partial modulus of continuies are de…ned by

Also, for any

! (1) (f ; )

=

! (2) (f ; )

=

max

max

jf (x1 ; y)

f (x2 ; y)j

max

max

jf (x; y1 )

f (x; y2 )j :

0 y A jx1 x2 j 0 x B jy1 y2 j

+ 1A

> 0 we have jf (x1 ; y1 )

f (x2 ; y2 )j

! (1) (f; )

jx1

x2 j

+1 ;

jf (x1 ; y1 )

f (x2 ; y2 )j

! (2) (f; )

jy1

y2 j

+1 :

Theorem 4.1. For any f 2 C(DA;B ), the following inequalities h ( ; ) (4.3) Cn;m;p (f ; qn ; qm ; x; y) f (x; y) 2 ! (1) (f ; m ) + ! (2) (f ; ( ; ) Cn;m;p (f ; qn ; qm ; x; y)

(4.4)

1

f (x; y)

are satis…ed where

2! f ;

q

2 m

+

i

n)

2 n

(4.5) 2 n

:=

1 2

[n]qn + 2

[n + p + 1]qn [n + p]qn qn

[n]qn +

453

A2 + (2 + 1) [n + p]qn an A +

2 2 an

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TWO DIM ENSIONAL CHLODOW SKY VARIANT OF q-BERNSTEIN-SCHURER-STANCU OPERATORS9

and

(4.6) 2 m

1

:=

2

[m]qm + 2

[m + p + 1]qm [m + p]qm qm

[m]q + m

B 2 + (2 + 1) [m + p]qm bm B +

2 2 bm

:

Proof. We directly have, ( ; ) Cn;m;p (f ; qn ; qm ; x; y) f (x; y) " ! n+p X m+p X [k]qn + [j]qm + = f an ; bm [n]qn + [m]qm + j=0

#

f (x; y)

k=0

x y j;m;qm an bm ! " n+p X m+p X [j]qm + [k]qn + an ; bm = f [n]qn + [m]qm + k=0 j=0 # [k]qn + x +f ( an ; y) f (x; y) k;n;qn [n]qn + an k;n;qn

f(

[k]qn + [n]qn +

an ; y)

y bm

j;m;qm

:

By linearity and positivity of the operators, we get ( ; ) Cn;m;p (f ; qn ; qm ; x; y) n+p X m+p X

[k]qn +

f

+

n+p X m+p X

x an f(

k=0 j=0

n+p X m+p X

!

j;m;qm

[k]qn + [n]qn +

(2)

f;

k=0 j=0

+

n+p X m+p X

! (1)

f;

k=0 j=0

=

1

(x; y) +

an ;

[n]qn +

k=0 j=0 k;n;qn

f (x; y)

2

[j]qm +

bm

[m]qm +

!

f(

[k]qn + [n]qn +

an ; y)

y bm

an ; y)

[j]qm + [m]qm + [k]qn + [n]qn +

f (x; y)

bm

an

y

x

!

!

k;n;qn

k;n;qn

k;n;qn

x an

x an x an

j;m;qm

j;m;qm

j;m;qm

y bm

y bm y bm

(x; y) :

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Using Lemma 1.1 and Cauchy-Schwartz inequality, we have 1 (x; y) n+p X m+p X

! (2)

=

k=0 j=0

m+p X

=

! (2)

f;

f;

[j]qm + [m]qm +

bm

[j]qm +

bm [m]qm + j=0 8 2 > m+p < X 1 4 ! (2) (f ; m ) 1 + > m :

y

y !

!

[m]qm +

y bm

j;m;qm

y bm

j;m;qm

[j]qm +

j=0

x an

k;n;qn

bm

y

!2

y bm

j;m;qm

Finally, using Lemma 2.1, we get (4.7)

1

(x; y)

2! (2) (f ;

m)

(x; y)

2! (1) (f ;

n)

31=2 9 > = 5 : > ;

where we choose m as in (4.6). In the same way, we obtain (4.8)

2

where n is given in (4.5). Combining (4.7) and (4.8), we obtain (4.3) . Now, by using linearity and the monotonicity of the operators, and taking into account (4.1), we have ( ; ) Cn;m;p (f ; qn ; qm ; x; y) f (x; y) 0 v u n+p X m+p X B u [k]q + n ! @f ; t an [n] qn + j=0 k=0

n+p X m+p X k=0 j=0

1+

1

f(

[k]qn +

[n]qn +

n+p X m+p X k=0 j=0

(4.9) k;n;qn

x an

an ;

[j]qm +

[m]qm +

!2

x

j;m;qm

+

bm )

v u u [k] + qn !(f ; t an [n]qn +

[j]qm + [m]qm + f (x; y)

!2

x

+

bm

k;n;qn

[j]qm + [m]qm +

y

!2

x an bm

1 C A

j;m;qm

j;m;qm

y

!2

y bm y bm

)

y bm

Using (4.2) and the Cauchy-Schwartz inequality, we get (4.4). Theorem 4.2. Let f (x; y) have continuous partial derivatives @f =@x and @f =@y, let ! 1 (fx ; :) and ! 2 (fy ; :) denote the partial moduli of @f =@x and @f =@y, respectively

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TWO DIM ENSIONAL CHLODOW SKY VARIANT OF q-BERNSTEIN-SCHURER-STANCU OPERATORS 11

on DA;B . Then the inequality ( ; ) Cn;m;p (f ; qn ; qm ; x; y)

[n + p]qn

N

where DA;B .

n

[m + p]qm

1 B+

[m]qm +

and

m

!

an 1 A+ [n]qn +

[n]qn +

+M

f (x; y)

bm [m]qn +

+2 !

n!

+2

m!

are the same as in Theorem 4.1 and

@f ; @x

(1)

(2)

@f @x

n

@f ; @y

N,

:

m

@f @y

M on

Proof. By the mean value theorem, we can write [k]qn +

f

[n]qn +

[k]qn +

=f

[n]qn + [k]qn +

f

= +

an ;

[n]qn + [j]qm + [m]qm +

[m]qm + !

an ; y an ; y

[n]qn +

[k]qn +

[j]qm +

bm

!

f (x; y)

f (x; y) + f

!

[k]qn + [n]qn

!

an bm

[j] + an ; bm + [m] +

! [k]qn + @f ( 1 ; y) @f (x; y) x + an x @x [n]qn + @x ! ! [j]qm + @f (x; y) y bm y + @y [m]qm +

!

@f (x; y) @x

(4.10) @f (x; @y

2)

@f (x; y) @y

for any …xed y 2 [0; B] and x 2 [0; A], where x
: f (x; y) ;

; (x; y) 2 Dan ;bn R2+ nDan ;bn

where (x; y) 2 Dan bm and fan g and fbm g have the same properties of two dimensional of Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. Theorem 5.1. For all continuous functions f satisfying jf (x; y)j f (x;y) x; y 0, and lim w(x;y) = 0, we have

Mf w(x; y),

x;y!1

lim

n;m!1

; Ln;p (f ; qn ; qm ; ; )

f(; )

w

=0

where (x; y) = 1 + x2 + y 2 : Proof. Clearly, ; Ln;p (f ; qn ; qm ; x; y)

=

f (x; y)

n+p X m+p X w(x; y) Gf 1 + x2 + y 2 j=0 k=0

k;n;qn

x an

k;n;qn

[k]qn + [n]qn + x an

an ;

j;m;qm

[j]qm + [m]qm + y bm

bm

!

Gf (x; y) ;

thus ; Ln;p (f ; qn ; qm ; ; )

=

sup x;y2R2+

; Ln;p

f(; )

(f ; qn ; qm ; x; y) w(x; y)

w

f (x; y)

= sup x;y2R2+

jTn;p (Gf ; qn ; qm ; x; y) Gf (x; y)j : 1 + x2 + y 2

Since jf (x; y)j Mf w(x; y), then jGf (x; y)j Mf (x; y) for x; y 0 and Gf (x; y) f (x;y) 2 = 0, we have is continuous function on R+ . Furthermore, from lim w(x;y) x;y!1

lim

x;y!1

Gf (x; y) = 0: (x; y)

Thus, from Theorem 3.4 we get the result. ; Finally, note that, taking w(x; y) = 1+x2 +y 2 , then the operators Ln;p (f ; qn ; qm ; x; y) ; reduces Tn;p (Gf ; qn ; qm ; x; y).

References [1] Büyükyaz¬c¬, I·.,On the approximation properties of two-dimensional q-Bernstein-Chlodowsky polynomials, Mathematical Communications, 2009, 14, 255-269. [2] Büyükyaz¬c¬, I·. and Sharma, H., Approximation properties of two-dimensional q-BernsteinChlodowsky-Durrmeyer operators, Nuerical Func. Anal. and Optimization, 2012, 33 (12), 1351-1371, doi: 10.1080/01630563.2012.674594. [3] Chlodowsky, I., Sur le developpement des fonctions de…nes dans un interval in…ni en series de polynomes de M. S. Bernstein, Compositio Math., 1937, 4, 380-393. [4] Agrawal, P. N., Gupta V., Kumar S. A., On a q-analogue of Bernstein-Schurer-Stancu operators, Applied Mahmematics and Computation, 2013, 219, 7754-7764.

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[5] I·bikli, E., Approximation by Bernstein-Chlodowsky polynomials, Hacettepe Journal of Mathematics and Statics, 2003, 32, 1-5. [6] I·bikli, E., On approximation for functions of two variables on a triangular domain, Roucky Mountain Journal of Mathematics, 2005, 5, 1523-1531. [7] Kac, V., Cheung, P.: Quantum Calculus, 2002, Springer. [8] Karsl¬, H., Gupta, V., Some approximation properties of q Chlodowsky operators, Applied Mathematics and Computation, 2008, 195, 220-229. [9] Muraru, C.V., Note on q-Bernstein-Schurer operators, Babe¸s-Bolyaj Math., 2011, 56, 489495. [10] Phillips, G.M., On Generalized Bernstein polynomials, Numerical analysis, World Sci. Publ., River Edge, 1996, 98, 263-269. [11] Phillips, G. M., Interpolation and Approximation by Polynomials, 2003, Newyork. [12] Schurer, F., Linear Positive Operators in Approximation Theory, Math. Inst., Techn. Univ. Delf Report, 1962. [13] Vedi, T. and Özarslan, M.A., Some Properties of q-Bernstein-Schurer operators, J. Applied Functional Analysis, 2013, 8 (1) 45-53. [14] Vedi, T and Özarslan, M.A., Chlodowky variant of q-Bernstein-Schurer-Stancu operators, J. of Ineq. and Appl., 2014, 10.1186/1029-242X-2014-189. (M.A. Özarslan) Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey E-mail address : [email protected] (T. Vedi) University of Kyrenia, Girne, TRNC, Mersin 10, Turkey E-mail address : [email protected]

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Global stability in stochastic difference equations for predator-prey models Sangmok Chooa , Young-Hee Kim∗,b a

b

Department of Mathematics, University of Ulsan, Ulsan 44610, Korea. Division of General Education-Mathematics, Kwangwoon University, Seoul 01897, Korea

Abstract There are many publications on theoretical analysis of deterministic difference equations and stochastic differential equations. However, relatively few theoretical papers are published to consider the positivity of solutions of discrete-time stochastic difference equations (DSDEs), and no theoretical papers investigate the global stability of nontrivial solutions of DSDEs with nonlinear terms. In this paper, we consider a DSDE model that is a generalization of two-dimensional nonlinear models of stochastic predator-prey interactions, and show the positivity and global stability of the nontrivial solutions by using our new discretized version of the Itˆo formula. In addition, our results are compared with those of continuous-time stochastic differential equations and discrete-time deterministic difference equations. Numerical simulations are introduced to support the results. Key words: Discrete-time stochastic difference equations, Positivity, Global stability.

1. Introduction Many predator-prey models have been studied to describe the dynamics of biological systems in which two species interact, one as a predator and the other as a prey. A classic predator-prey model is given by dx dy = x(r1 − a11 x − a12 y), = y(r2 + a21 x − a22 y), dt dt

(1)

where x(t) and y(t) denote the population density of the prey and predator at time t, respectively. In the model (1), r1 is the intrinsic growth rate of the prey in the absence of the predator, −r2 is the death rate of the predator in the absence of the prey, the coefficients aij (i 6= j) give the strength of the interaction between the two species, and aii (i = 1, 2) measure the inhibiting effect of environment on the two species. In the model (1), the predator consumes the prey with functional response of type a12 x(t)y(t). However the rate of prey capture is saturated when the population of the prey is relatively large. Such phenomena are described by nonlinear functions including Holling types [1–5], Beddington-DeAngelis type [6–8], Crowley-Martin type [9–11], and ∗

Corresponding author Email addresses: [email protected] (Sangmok Choo), [email protected] (Young-Hee Kim)

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Ivlev-type of functional responses [12–14]. Other types of nonlinear functions have been applied to express the Allee effect [15–19], which describes a positive relation between the population density and the per capita growth rate of a species. There have been also models to take into account of diffusion of species ([15] and [20–22]). On the other hand, the population is inevitably affected by environmental noise in nature, so that the reproduction rates can change randomly. In order to be more realistic, stochastic models should be considered. Stochastic differential equation (SDE) models have been increasingly used in a range of application areas, including biology, chemistry, mechanics, economics, and finance. The SDE models have been studied to understand extinction, stochastic permanence and stationary distributions of the stochastic systems. In particular, many authors have taken stochastic perturbation into deterministic predator prey models with Beddington-DeAngelis and Holling types of functional responses [23–33]. For example, putting noise into the deterministic model (1) gives the SDE model dx(t) = x(t){r1 − a11 x(t) − a12 y(t)}dt + σ1 x(t)dW1 (t), dy(t) = y(t){r2 + a21 x(t) − a22 y(t)}dt + σ2 y(t)dW2 (t),

(2)

which is a special model studied in [25] with zero-time delays. Here the positive coefficients σ1 and σ2 measure the intensity of environmental perturbations on the underlying growth rate of the prey and the death rate of the predator, respectively. The processes Wi are independent and real valued Wiener processes on a complete probability space (Ω, F, P). In general, the exact solutions of SDEs are not known, so one has to numerically solve these SDEs. This leads us to consider and analyze discrete-time stochastic difference equations (DSDEs), which can be also viewed as stochastically perturbed versions of deterministic difference equations (DDEs) (see [34], [35] and references therein). There are many publications on estimations of the difference between solutions of SDEs and DSDEs. The global asymptotic stability of the trivial solution of DSDEs has been also widely addressed (see [36], [37], [38] and references therein). However, relatively few theoretical studies consider the positivity of solutions of DSDEs that are scalar equations on a finite time interval (see [39] references therein). In particular, to the best of our knowledge, there is no paper that theoretically deals with the global stability of nontrivial solutions of DSDEs. Therefore, to investigate the positivity and global stability, we consider the DSDE model for (2) n   o Xi−1 X2 i xik+1 = xik 1 + h ri + aij xjk − aij xjk + h0.5 σi ξk+1 , (3) j=1

j=i

where 1 ≤ i ≤ 2, k ≥ 0, xi0 > 0 and 0 < h < 1. Although r1 > 0, r2 < 0 and aij > 0 in the SDE model (2) and the DDE model (3) with σi = 0 (see [34] and [35]), we weaken the conditions on the parameters and use the following conditions in the DSDE model (3): for 1 ≤ i, j ≤ 2 and i 6= j ri ∈ R, aii > 0, aij ≥ 0, σi > 0. (4) i The discrete Wiener processes Wi (tk+1 )−Wi (tk ) are h0.5 ξk+1 with a mutually independent 1 2 ∞ and identically distributed sequence (ξk , ξk )k=1 of the standard normal random variables. The solutions of (3) are defined with respect to a complete, filtered probability space ∞ (Ωh , Fh , {Fk }∞ k=1 , Ph ), where {Fk }k=1 is the natural filtration generated by the stochastic 1 2 ∞ sequence (ξk , ξk )k=1 , i.e., Fk = σ(ξ11 , ξ12 , · · · , ξk1 , ξk2 ) for k ≥ 1. Therefore (x1k , x2k )∞ k=1 is

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adapted to the filtration for any initial vector (x10 , x20 ), which is supposed to be nonrandom. The positivity of solutions of the SDEs (2) is obtained in the infinite time interval [0, ∞) without boundedness of the noises Wi (t) by using the concept of explosion time (see [25] and [40]). However, to the best of our knowledge, there is no method for applying the concept of explosion time to DSDEs. Then for obtaining the positivity of solutions of the DSDE model (3) in the infinite time interval, we restrict the noises to bounded noises, which means that ξki (1 ≤ i ≤ 2, k ≥ 1) are assumed to be doubly truncated standard normal random variables with support [−ς, ς] for a positive constant ς −ς ≤ ξki ≤ ς and the probability density function ( q(x) {Φ(ς) − Φ(−ς)}−1 ψ(x) = 0

(5)

if x ∈ [−ς, ς], if x ∈ / [−ς, ς],

(6)

where q and Φ are the probability density and cumulative distribution functions of the standard normal random variable, respectively. Denoting ης = 2ςq(ς) {Φ(ς) − Φ(−ς)}−1 gives that for 1 ≤ i ≤ 2 and k ≥ 1
E(ξki ) = 0, E (ξki )2 = 1 − ης , (7) in which the positive value ης can be assumed to be sufficiently close to 0. For example, when ς = 20, we have 0 < ης < 10−85 . The truncation constant ς will be first used in (12) for the positivity of the solutions xik of the DSDE model (3). The paper is organized as follows. Section 2 gives the positivity and boundedness of solutions of the model (3). In Section 3, we develop a new discrete Itˆo formula for (3) by using a known discrete Itˆo formula for DSDEs (see [41], [42] and [43]). The new discrete Itˆo formula is the main tool for finding conditions for the global stability of solutions of (3). Section 4 introduces auxiliary equations, the solutions of which are used for the upper bounds of solutions of (3). In Section 5, we present sufficient conditions for extinction and non-extinction of solutions of (3). Our results are compared with those for the DDEs in [35] and the SDEs in [25]. Section 6 gives simulation results to confirm the theoretical analysis obtained in this paper.

2. Positivity and boundedness of solutions of DSDEs In this section, we show the positivity and boundedness of solutions of the DSDE model (3) by applying the approach used in the DDE model (3) with σ1 = σ2 = 0 (see [34] and [35]). Notation 1. For simplicity, we use the symbols a ˜ and a ˆ for every constant a to denote a ˜ = a · h0.5 , a ˆ =a·h and the symbols x1k and x2k for a vector xk = (x1k , x2k ) to denote x1k = x2k , x2k = x1k . 3 464

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Write the model (3) as i i xik+1 = Fk,x i (xk ), k

where
1 1 , (x) = x 1 + rˆ1 − a ˆ11 x − a ˆ12 y + σ ˜1 ξk+1 Fk,y
2 2 Fk,x (y) = y 1 + rˆ2 + a ˆ21 x − a ˆ22 y + σ ˜2 ξk+1 .

(8)

Note that for a vector ζ k = (ζk1 , ζk2 ) of real numbers ζk1 and ζk2 , i i Fk,ζ i (τ ) is strictly increasing on 0 ≤ τ < Vk (ζ k ),

(9)

k

in which  Xi−1 X2 aii )−1 1 + rˆi + Vki (ζ k ) = (2ˆ a ˆij ζkj − j=1

j=i+1

 i . a ˆij ζkj + σ ˜i ξk+1

(10)

Denote that for 1 ≤ i ≤ 2 χi =

a ˆ−1 ii



rˆi +

Xi−1 j=1



a ˆij χj + σ ˜i ς∗ ,

(11)

where ς∗ is a constant satisfying ς∗ > ς,

(12) X2





χi ≤ (2ˆ aii )−1 1 + rˆi − a ˆij χj − σ ˜i ς∗ , j=i+1 Xi−1 a ˆij χj + σ ˜i ς∗ < 1. rˆi + j=1

(13) (14)

The relation (12) will be first used in (69) to find upper solutions of the model (3). The initial condition of the model (3) is assumed to satisfy (x10 , x20 ) ∈ (0, χ1 ) × (0, χ2 ).

(15)

σ1 ς∗ Remark 1. The definition (11) gives that χ1 = rˆ1 +˜ and χ2 = a ˆ−1 r2 + a ˆ21 χ1 + σ ˜2 ς∗ ). 22 (ˆ a ˆ11 Letting h in (3) be small, we can choose ς∗ satisfying the two conditions (13) and (14). For example, let h = 0.0001, ς∗ = 20, r1 = 2, r2 = aij = 1 and σi = 0.1 (1 ≤ i, j ≤ 2). Denoting by Ri and Li the right and left-hand sides of (13) and (14), respectively, gives

(χ1 , R1 , L1 ) = (202, 4699.5, 0.3848), (χ2 , R2 , L2 ) = (403, 4900.5, 0.3518), which show that the conditions (13) and (14) are satisfied. Theorem 1. Let xik be the solutions of (3) and χi be defined in (11). Assume that (5), (12), (13), (14) and (15) hold. Then (x1k , x2k ) ∈ (0, χ1 ) × (0, χ2 ), k ≥ 0. Proof. The proof is divided into the following three steps. Step 1. We prove the positivity: xi1 > 0 for 1 ≤ i ≤ 2. Note that for x0 = (x10 , x20 ) 4 465

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  P 0 < xi0 < χi ≤ (2ˆ aii )−1 1 + rˆi − 2j=i+1 a ˆij χj − σ ˜i ς∗ < V0i (x0 ), where the first two inequalities are obtained from (15), the third from (13) and the last from (10), (15), (5) and (12). Then using (9) with ζ 0 = x0 and (15), we have the positivity i i i xi1 = F0,x i (x0 ) > F0,xi (0) = 0. 0

0

xi1

Step 2. We prove the < χi for 1 ≤ i ≤ 2. Pupper-bound Pproperty: 2 j j i Let ω ∈ Ωh . If rˆi + i−1 a ˆ x − a ˆ x + σ ˜ i ξ1 (ω) ≤ 0, then j=1 ij 0 j=i+1 ij 0 i i i xi1 (ω) = F0,x i (x0 )(ω) ≤ x0 < χi . 0

Otherwise, we have 0
0), φ (x) = 0 (x = 0), we modify the function ϕ used in [37]. Define the function ϕ as follows.  ln |x| (|x| ≥ e−1 ), ϕ (x) = 3 3 −4−1 e4 x4 + e2 x2 − 4−1 7 + 6−1 e6 (x − e−1 ) (x + e−1 ) (|x| ≤ e−1 ). Then φ and ϕ satisfy all the conditions in Lemma 1 with δ = 1 − e−1 . Notation 2. For simplicity, we use the notations Xk−1
E xis E(xik ) = k −1

(20)

s=0

and  2 ˚ a = a · 1 + O(h0.5 ) , aη = a · (1 − ης ), riσ = ri − 0.5σiη 2 for k > 0, 1 ≤ i ≤ 2, constants a and ης in (7). Here σiη is equal to {σi · (1 − ης )}2 .

Remark 5. Since the solutions xik of (3) are positive by Theorem 1, we can take logarithm of (3), which gives      
i F k , E ln xik+1 Fk = E ln xik Fk + E ln 1 + hf + h0.5 gξk+1 (21) where f and g are defined in (19). In order to simplify the equation (21), applying Fk independence of ξk+1 , Fk -measurability of xik and Lemma 1 with Remarks 3 and 4 to the three expectation terms in (21), respectively, we have

1 E(ln xik+1 ) = ln xik + hf − hg 2 · (1 − ης ) + hf O h0.5 + hg 2 O h0.5  2  X2 1 2 Xi−1 j j i ˚ aij xk − aij xk . = ln xk + h ri − σiη + j=1 j=i 2 Taking expectation of (22) and adding the result, we obtain n o Xi−1 X2 j j i i ˚ E(ln xk ) = E(ln x0 ) + k h riσ + aij E(xk ) − aij E(xk ) . j=1

j=i

(22)

(23)

4. Auxiliary equations In order to find upper bounds of xik , we consider the auxiliary equations   Xi−1 i i zk+1 = zki 1 + rˆi + a ˆij zkj − a ˆii zki + σ ˜i ξk+1 , z0i = xi0 j=1

(24)

for 1 ≤ i ≤ 2 and k ≥ 0. Since (24) is the system (3) with a12 = 0, Theorem 1 with (4) gives that for k ≥ 0 (zk1 , zk2 ) ∈ (0, χ1 ) × (0, χ2 ). (25) 7 468

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Let βi be the solutions of the equations riσ +

Xi−1 j=1

aij βj − aii βi = 0

(26)

for 1 ≤ i ≤ 2. Note that (22) and (23) with a12 = 0 become

1 E ln zk+1 = ln zk1 + ˚ h r1σ − a11 zk1 ,


 E ln zk1 = E ln z01 + k˚ h r1σ − a11 E zk1 n Xk−1
o
ha11 β1 − k −1 E zs1 = E ln z01 + k˚ s=0

due to (20) and β1 = a−1 11 r1σ in (26). Similarly, we have

2 h r2σ + a21 zk1 − a22 zk2 , = ln zk2 + ˚ E ln zk+1

 E ln zk2 = E ln z02 + k˚ h r2σ + a21 E(zk1 ) − a22 E(zk2 ) ) ( k−1 X

a r 21 2σ E(zk1 ) − k −1 E zs2 . + = E ln z02 + k˚ ha22 a22 a22 s=0

(27)

(28)

(29)

(30)

Lemma 2. Let zk1 and β1 be the solutions of (24) and (26), respectively. If β1 ≥ 0, then for  > 0 and all sufficiently large k k −1

Xk−1 s=0


E zs1 ≤ β1 + .

Proof. Suppose, on the contrary, that the theorem is false, which means that there exist a constant ε0 > 0 and an infinite increasing sequence {km } satisfying both for all km −1 km

Xkm −1


E zs1 >β1 + ε0

(31)


E zs1 ≤β1 + ε0 .

(32)


limm→∞ E ln zk1m = −∞.

(33)

and for all k with k 6= km k −1

s=0

Xk−1 s=0

Combining (31) and (28), we have

Substituting (33) and the boundedness of zk1 into (27) gives lim ln zk1m −1 = −∞ a.s.

m→∞

and then limm→∞ zk1m −1 = 0 a.s.

(34)

Thus the dominated convergence theorem with (25) leads to limm→∞ E(zk1m −1 ) = 0.

(35)

In order to obtain a contraction we follow the two steps: 8 469

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Step 1. If there exists k = km −1 satisfying (32), then the system of (31) and (32) becomes Xkm −1
E zs1 > km (β1 + ε0 ) , s=0 Xkm −2
E zs1 ≤ (km − 1) (β1 + ε0 ) , s=0

which gives E(zk1m −1 ) > β1 + ε0 ,

(36)

and hence there exist finitely many k satisfying (32) due to (35) and (36). Therefore for all sufficiently large k Xk−1
(37) k −1 E zs1 >β1 + ε0 . s=0

Step 2. As (31) implies (35), the equation (37) implies limk→∞ E(zk1 ) = 0, which is contradictory to (37) due to β1 + ε0 > 0 and so the proof is completed. Lemma 3. Let (zk1 , zk2 ) and (β1 , β2 ) be the solutions of (24) and (26), respectively. (a) Assume r1σ < 0. Then limk→∞ zk1 = 0 a.s. 0 a.s. (i) If r1σ < 0 and r2σ < 0, then limk→∞ zk2 =P k−1 −1 2 −1 (ii) If r1σ < 0 and r2σ ≥ 0, then limk→∞ k s=0 E (zs ) = a22 r2σ . P 1 (b) Assume r1σ ≥ 0. Then limk→∞ k −1 k−1 s=0 E (zs ) = β1 . 0 a.s. (i) If r1σ ≥ 0 and r2σ + a21 β1 < 0, then limk→∞ zk2 =P 2 (ii) If r1σ ≥ 0 and r2σ + a21 β1 ≥ 0, then limk→∞ k −1 k−1 s=0 E (zs ) = β2 . Proof. (a) Since r1σ < 0 is equivalent to β1 = a−1 11 r1σ < 0, it follows from (28) and the positivity of zk1 in (25) that if r1σ < 0, then limk→∞ E (ln zk1 ) = −∞, and further limk→∞ zk1 = 0 a.s.

(38)

as (33) implies (34). (a)-(i) Assume that r1σ < 0 and r2σ < 0. As (34) implies (35), the equation (38) yields limm→∞ E(zk1 ) = 0 and then limk→∞ E(zk1 ) = 0.

(39)

Combining (39) and (30) with r2σ < 0 and using zk2 > 0, we have from (30) that
limk→∞ E ln zk2 = −∞.

(40)

Therefore, as (33) implies (34), the equation (40) gives limk→∞ zk2 = 0 a.s. (a)-(ii) Assume that r1σ < 0 and r2σ ≥ 0. 1 Using (zk2 , a−1 22 r2σ ), (29) and (30) instead of (zk , β1 ), (27) and (28) in the proof of Lemma 2, respectively, and applying (39) to (30), we can obtain that for  > 0 and all sufficiently large k Xk−1
E zs2 ≤ a−1 (41) k −1 22 r2σ + . s=0

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In order to show limk→∞ k −1 and all sufficiently large k

Pk−1 s=0

E (zs2 ) = a−1 22 r2σ , it is enough to prove that for  > 0

−1 a−1 22 r2σ −  ≤ k

Xk−1 s=0


E zs2 .

(42)

Suppose that (42) is false, which means that there exist a constant ε0 > 0 and an infinite increasing sequence {km } satisfying Xkm −1
−1 (43) r − ε > k E zs2 . a−1 0 m 22 2σ s=0

Then the boundedness of

zk2

and (30) imply that for all km

∞ > E ln zk2m > E ln z02 + km˚ ha22 ε0 ,

(44)

which is a contradiction. Therefore (42) is true and so the proof is completed due to (41) and (42). −1 (b) Assume r1σ ≥ 0, which means Pk−1β1 = 1a11 r1σ ≥ 0. −1 In order to show limk→∞ k s=0 E (zs ) = β1 , it is enough to prove that for  > 0 and all sufficiently large k Xk−1
β1 −  ≤ k −1 E zs1 (45) s=0

due to Lemma 2. Suppose that (45) is false, so that there exist a constant ε0 > 0 and an infinite increasing sequence {km } such that Xkm −1
−1 β1 − ε0 > km E zs1 . (46) s=0

Then the boundedness of

zk1

and (28) imply that for all km

∞ > E ln zk1m > E ln z01 + km˚ ha11 ε0 ,

(47)

which is a contradiction. Hence (45) is true and, therefore, Lemma 2 with (45) gives limk→∞ E(zk1 ) = β1 .

(48)

(b)-(i) Assume that r1σ ≥ 0 and r2σ + a21 β1 < 0. Applying (48) to (30) with both r2σ + a21 β1 < 0 and zk2 > 0, we have limk→∞ E(ln zk2 ) = −∞. Therefore, as (33) implies (34), we can obtain limk→∞ zk2 = 0 a.s. (b)-(ii) Assume that r1σ ≥ 0 and r2σ + a21 β1 ≥ 0. Following the proof of Lemma 2, we can obtain that Xk−1
k −1 E zs2 ≤ β2 +  s=0

(49)

for  > 0 and all sufficiently large k by using (zk2 , β2 ), (29) and (30) instead of (zk1 , β1 ), (27) and (28), respectively, and applying (48) and β2 = a−1 22 (r2σ + a21 β1 ) ≥ 0 to (30). Similarly, following the proof of (45), we can obtain that Xk−1
β2 −  ≤ k −1 E zs2 (50) s=0

for  > 0 and all sufficiently large k by replacing (zk1 , β1 ) and (28) with (zk2 , β2 ) and (30), respectively, and applying (48) to (30). Therefore (49) and (50) give the desired result. 10 471

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Remark 6. The equations (28) and (30) can be written as n o Xi−1 h riσ + aij E(zkj ) − aii E(zki ) . E(ln zki ) = E(ln z0i ) + k˚ j=1

(51)

Substituting (26) to (51) yields E(ln zki ) = E(ln z0i ) + k˚ h

hXi−1 j=1

  i aij E(zkj ) − βj − aii E(zki ) − βi .

(52)

Applying Lemma 3-(b) and (b)-(ii) to (52) with the notation (20), we have limk→∞ k −1 E(ln zki ) = 0 P under the condition that min{r1σ , riσ + i−1 j=1 aij βj } ≥ 0 for 1 ≤ i ≤ 2.

(53)

Lemma 4. Let xik and zki be the solutions of (3) and (24), respectively for i = 1, 2. Then for k ≥ 0 0 < xik ≤ zki . Proof. Theorem 1 with Remark 2 gives 0 < xik .

(54)

1 Fk,y (x) is nonincreasing in y for x ≥ 0 and k ≥ 0

(55)

2 Fk,x (y) is nondecreasing in x for y ≥ 0 and k ≥ 0

(56)

Note that and by the definition (8). The proof of this lemma is divided into the following two cases. Case 1. Let i = 1. Using x10 = x20 > 0 and (55), we have 1 1 1 1 x11 = F0,x 1 (x0 ) ≤ F0,0 (x0 ). 0

(57)

It follows from Remark 2, (24), (25), (10) and (13) that 0 < x10 ≤ z01 < χ1 < V01 (0, 0), with which (9) yields 1 1 (z01 ) = z11 . F0,0 (x10 ) ≤ F0,0

(58)

Hence combining (54), (57) and (58) gives 0 < x11 ≤ z11 .

(59)

Assume that for some positive integer k 0 < x1k ≤ zk1 .

(60)

Using (54), (60), (25), (10) and (13), we have x1k > 0, 0 < x1k ≤ zk1 < χ1 < Vk1 (0, 0) 11 472

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and so 1 1 1 1 1 1 1 x1k+1 = Fk,x 1 (xk ) ≤ Fk,0 (xk ) ≤ Fk,0 (zk ) = zk+1 , k

where the first inequality is obtained from (55) and the second inequality from (9). Case 2. Let i = 2. Using x20 = x10 ≤ z01 and 0 < x20 ≤ z02 < χ2 < V02 (0, 0), we have 2 2 2 2 2 2 2 x21 = F0,x 2 (x0 ) ≤ F0,z 1 (x0 ) ≤ F0,z 1 (z0 ) = z1 0

0

(61)

0

due to (56) and (9). Similarly as in Case 1, using mathematical induction and zk2 ≤ χ2 < Vk2 (0, 0) instead of zk1 < χ1 < Vk1 (0, 0) in Case 1, we can obtain the desired result. P Remark 7. If min{r1σ , riσ + i−1 j=1 aij βj } ≥ 0 for 1 ≤ i ≤ 2, then Lemma 4 and (53) imply that for  > 0 and all sufficiently large k k −1 E(ln xik ) ≤ ,

(62)

which will be first used in Theorem 4.

5. Extinction and persistence of the discrete solutions In this section, we present several conditions sufficient for the extinction and persistence (non-extinction) of the solutions xik of (3). Theorem 2. Let xik and βi be the solutions of (3) and (26), respectively for i = 1, 2. (a) If r1σ < 0, then limk→∞ x1k = 0 a.s. (b) If r1σ < 0 and r2σ < 0, then limk→∞ x2k = 0 a.s. Proof. The proof is followed by combining Lemma 3-(a) and (a)-(i) with Lemma 4. Remark 8. Since r1σ = 0 gives β1 = a−1 11 r1σ = 0, we obtain that P 1 if r1σ = 0, then limk→∞ k −1 k−1 s=0 E (xs ) = 0 by combining Lemma 3-(b) with Lemma 4. Similarly, Lemma 3-(b)-(ii) gives P 2 if r1σ = r2σ = 0, then limk→∞ k −1 k−1 s=0 E (xs ) = 0 since β2 = a−1 22 (r2σ + a21 β1 ) = 0. 2 Remark 9. By Theorem 2-(a), we find that if r1 < 12 σ1η , then the prey population will be extinct in the future, no matter whether the predator exists. It implies that environmental noise plays a very important role in the biological system.

In order to establish the sufficient condition for the extinction of the predator and the persistence of the prey, we will use the following Lemma 5 as well as Lemma 3-(b). Using Lemmas 4 and 3-(b) with β1 = a−1 11 r1σ we obtain that if r1σ > 0, then lim k −1

Xk−1

k→∞

s=0


E x1s ≤ a−1 11 r1σ .

(63)

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For finding a lower function of x1k , we consider the solution uk, of the equation 1 ), u0, = x1N , uk+1, = uk, (1 + rˆ1 − a ˆ11 uk, − a ˆ12  + σ ˜1 ξN  +k+1

(64)

in which  satisfies that for some positive integer N and all k ≥ N 0 < x2k ≤ , rˆ1 − a ˆ12  + σ ˜1 ς∗ < 1, a ˆ12  + σ ˜1 ς < σ ˜ 1 ς∗ ,

(65) (66) (67)

where (65) is possible under the conditions r1σ > 0 and r2σ + a21 β1 < 0 due to Lemmas 4 and 3-(b)-(i). The inequalities (66) and (67) are possible by (14) and (12), respectively. Lemma 5. Assume that r1σ > 0 and r2σ + a21 β1 < 0. Let  and N satisfy (65)–(67). Let x1k and uk, be the solutions of (3) and (64), respectively. Then (a) 0 < uk, < χ1 for k ≥ 0. (b) uk, ≤ x1N +k for k ≥ 0. P −1 (c) If r1σ − a12  > 0, then limk→∞ k −1 k−1 s=0 E (us, ) = a11 (r1σ − a12 ). Proof. (a)We proceed by induction on k. Since (64) and Theorem 1 with Remark 2 give u0, = x1N , 0 < x1N < χ1 , the statement (a) is true for k = 0. Assume that for a nonnegative integer k 0 < uk, < χ1 .

(68)

Now, in the case of k + 1, the proof of (a) is divided into the following two steps. Step 1. We prove the positivity of uk+1, . Denoting
1 Uk = (2ˆ a11 )−1 1 + rˆ1 − a ˆ12  + σ ˜1 ξN  +k+1 gives that for k ≥ 0 0 < χ1 < (2ˆ a11 )−1 (1 + rˆ1 − σ ˜1 ς∗ ) < Uk ,

(69)

where the second inequality is obtained from (13) and the last from (67), (12) and (5). Letting
1 Gk (x) = x 1 + rˆ1 − a ˆ11 x − a ˆ12  + σ ˜1 ξN ,  +k+1 we have Gk (x) is strictly increasing on 0 ≤ x < Uk .

(70)

Applying (68) and (69) to (70), we have the desired positivity. Step 2. We prove that χ1 is an upper bound of uk+1, . 1 Let ω ∈ Ωh . If rˆ1 − a ˆ11 uk, (ω) − a ˆ12  + σ ˜1 ξN (ω) ≤ 0, then  +k+1 uk+1, (ω) = Gk (uk, )(ω) ≤ uk, (ω) < χ1 ,

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in which (68) gives the last inequality. Otherwise, we have 0 < uk, (ω) < ∆k (ω) with
1 . ˆ1 − a ˆ12  + σ ˜1 ξN ∆k = a ˆ−1 11 r  +k+1 Since ∆k < Uk by (66), we have 0 < uk, (ω) < ∆k (ω) < Uk (ω) and then (70) gives uk+1, (ω) = Gk (uk, )(ω) < Gk (∆k )(ω) = ∆k (ω) < χ1 , where the last inequality is obtained from (11), (12) and (5). (b)We proceed by induction on k. The statement (b) is true for k = 0 due to (64). Assume that for a nonnegative integer k uk, ≤ x1N +k .

(71)

It follows from (a) in this theorem, (71), Theorem 1, Remark 2 and (69) that 0 < uk, ≤ x1N +k < χ1 < Uk and then uk+1, = Gk (uk, ) ≤ Gk (x1N +k ) = FN1  +k, (x1N +k )

(72)

due to (70). Combining (55) and (65) also gives FN1  +k, (x1N +k ) ≤ FN1  +k,x2

N +k

(x1N +k ) = x1N +k+1 .

(73)

Therefore, (72) and (73) give the desired result. (c) Let γ1 = a−1 11 (r1σ − a12 ). Note that E (ln uk+1, ) = ln uk, + ˚ h (r1σ − a11 uk, − a12 ) ,  E (ln uk, ) = E (ln u0, ) + k˚ h r1σ − a12  − a11 E (uk, ) n o Xk−1 = E (ln u0, ) + k˚ ha11 γ1 − k −1 E (us, ) s=0

(74)

(75)

as in (27) and (28). Following the proof of Lemma 2, we can obtain that k −1

Xk−1 s=0

E (us, ) ≤ γ1 + 0

(76)

for 0 > 0 and all sufficiently large k by replacing (27), (28) and (zk1 , r1σ , β1 ) with (74), (75) and (uk, , r1σ − a12 , γ1 ), respectively. Similarly, replacing (28) and (zk1 , β1 ) in (45)–(47) with (75) and (uk, , γ1 ), respectively, we can obtain that for 0 > 0 and all sufficiently large k P γ1 − 0 ≤ k −1 k−1 s=0 E (us, ), with which (76) gives the desired result. Theorem 3. Let xik and β1 be the solutions of (3) and (26), respectively for i = 1, 2. If r1σ ≥ 0 and r2σ + a21 β1 < 0, then lim E(x1k ) = β1 and lim x2k = 0 a.s. k→∞

k→∞

Proof. It follows from Lemma 3-(b)-(i), Lemma 4, Theorem 1 and Remark 2 that 14 475

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limk→∞ x2k = 0 a.s. Using Lemma 5-(a) and Lemma 4, we obtain that for  > 0 and all sufficiently large k 1 0 < uk, ≤ x1N +k ≤ zN .  +k

(77)

Lemma 5-(c) and Lemma 3-(b) give
1 −1 limk→∞ E (uk, ) = a−1 11 (r1σ − a12 ) , limk→∞ E zk = a11 r1σ ,

(78)

where the first and second equalities are valid under the conditions r1σ − a12  > 0 and r1σ ≥ 0, respectively. Therefore using (77), (78) and Remark 8, we obtain the desired result. Remark 10. By Theorems 2 and 3, we find that the value r1σ is the threshold between the extinction and persistence for the prey population. In addition, although the prey population converges to a non-extinction state in the mean when r1σ > 0 and r2σ +a21 β1 < 0, the predators dies out when the diffusion coefficient σ2 is large enough and then −r2σ = −r2 + 0.5 {σ2 · (1 − ης )}2 becomes too large. Remark 11. We can establish one condition for the extinction of the prey and the persistence of the predator as follows. Lemmas 4 and 3-(a)-(ii) yield if r1σ < 0 and r2σ ≥ 0, then lim k −1 k→∞

Xk−1 s=0


E x2s ≤ a−1 22 r2σ .

(79)

For finding a lower function of x2k , we consider the solution vk, of the equation 2 vk+1, = vk, (1 + rˆ2 − a ˆ21  − a ˆ22 vk, + σ ˜2 ξN ), v0, = x2N ,  +k+1

(80)

in which  satisfies that for some positive integer N and all k ≥ N 0 < x1k ≤ , rˆ2 − a ˆ21  + σ ˜2 ς∗ < 1, a ˆ21  + σ ˜2 ς < σ ˜2 ς∗ .

(81) (82) (83)

The inequality (81) is possible under the condition r1σ < 0 due to Lemma 3-(a). Replacing (64)–(67), r1σ > 0, r2σ + a21 β1 < 0 and (uk, , r1 , a11 , a12 , ξ 1 ) in the proof of Lemma 5 with (80)–(83), r1σ < 0, r2σ > 0 and (vk, , r2 , a22 , a21 , ξ 2 ), we can obtain that vk, ≤ x2N +k , lim k −1 k→∞

Xk−1 s=0

E (vs, ) = a−1 22 (r2σ − a21 ) ,

(84)

if r2σ − a21  > 0. Therefore (79) and (84) give the desired result:

a.s. if r1σ < 0 and r2σ > 0, then limk→∞ x1k , E(x2k ) = 0, a−1 22 r2σ

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Now, it remains to establish one condition for persistence of the prey and the predator. Define the matrix A and the constants Di as       a11 a12 r1σ D1 A= , =A , (85) −a21 a22 r2σ D2 which give  |A| = a11 a22 + a12 a21 > 0,

D1 D2

 =A

−1

    a22 r1σ − a12 r2σ r1σ −1 ≥0 = |A| a11 (r2σ + a21 β1 ) r2σ

under the conditions r1σ ≥ a−1 22 a12 r2σ and r2σ + a21 β1 ≥ 0. Using (85), the system (23) can be written as the matrix equation       1 E (ln x1k ) E (ln x10 ) D − E (x ) 1 k = + k˚ hA E (ln x2k ) E (ln x20 ) D2 − E (x2k )

(86)

(87)

and multiplying the matrix |A|A−1 to (87), we have  a22 E(ln x1k ) − a12 E(ln x2k ) = C1 + k˚ h|A| D1 − E (x1k ) ,  a21 E(ln x1 ) + a11 E(ln x2 ) = C2 + k˚ h|A| D2 − E (x2 ) , k

k

k

(88) (89)

where C1 = a22 E(ln x10 ) − a12 E(ln x20 ) and C2 = a21 E(ln x10 ) + a11 E(ln x20 ). Lemma 6. Let x1k and β1 be the solutions of (3) and (26), respectively. If r1σ ≥ a−1 22 a12 r2σ and r2σ + a21 β1 ≥ 0, then for  > 0 and all sufficiently large k E(x1k ) ≤ D1 + ,

(90)

where D1 is defined in (85). Proof. Suppose that (90) is false, which means that there exist a constant 0 > 0 and an infinite increasing sequence {km } satisfying both for all km −1 km

Xkm −1

and for all k with k 6= km k −1

s=0

Xk−1 s=0


E x1s >D1 + 0 ,

(91)


E x1s ≤D1 + 0 .

(92)

Replace (zk1 , β1 ), (31), (32), (28) and (27) in the proof of Lemma 2 with (x1k , D1 ), (91), (92), (88) and (22), respectively, where we apply (22) with i = 1. Then using the boundedness of x1k and following the proof for (37), we can obtain that for all sufficiently large k k −1

Xk−1 s=0


E x1s >D1 + 0 .

(93)

Combining (93) and (88) gives a22 E(ln x1k ) − a12 E(ln x2k ) < C1 + k˚ h|A|(−0 ).

(94)

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Applying Theorem 1 to (22) with i = 2, we obtain supk≥0 E(ln x2k ) < ∞ and then (94) yields
limk→∞ E ln x1k = −∞.

(95)

Substituting (95) into (22) with i = 1 and using the boundedness of x1k , we obtain limk→∞ ln x1k = −∞ a.s., which implies limk→∞ x1k = 0 a.s. Hence the dominated convergence theorem with Theorem 1 leads to limk→∞ E(x1k ) = 0, which is contradictory to (93) due to D1 + ε0 > 0. This completes the proof. Remark 12. The equation (90) with (87) gives that for  > 0 and all sufficiently large k  2 E(ln x2k ) ≤ E(ln x20 ) + k˚ ha22 a−1 (96) 22 a21  + D2 − E(xk ) . Following the proof of Lemma 6 with (96), we can obtain that 2 −1 0 if r1σ ≥ a−1 22 a12 r2σ and r2σ + a21 β1 ≥ 0, then E(xk ) ≤ a22 a21  + D2 + 

(97)

for 0 > 0 and all sufficiently large k by replacing (x1k , D1 ) and (88) in the proof of Lemma 6 with (x2k , a−1 22 a21  + D2 ) and (96), respectively. Theorem 4. Let xik and βi be the solutions of (3) and (26), respectively for i = 1, 2. i If r1σ ≥ a−1 22 a12 r2σ and r2σ + a21 β1 ≥ 0, then limk→∞ E(xk ) = Di ,

where Di are defined in (85). Proof. Substituting (62) into (89) gives that for 0 > 0 and all sufficiently large k 0 ≥ D2 − E(x2k ).

(98)

limk→∞ E(x2k ) = D2 .

(99)

Combining (98) and (97), we have

Applying (99) to (89) with (62) yields limk→∞ k −1 E(ln x1k ) = limk→∞ k −1 E(ln x2k ) = 0, with which (88) gives the desired result lim E(x1k ) = D1 . k→∞

Remark 13. Let (xk , yk ) be the solutions of DDEs (3) with σ1 = σ2 = 0 in [35]. −1 (i) If r1 > 0, r2 < 0 and r2 + a21 a−1 11 r1 ≤ 0, then limk→∞ (xk , yk ) = (a11 r1 , 0).

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(ii) If r1 > 0, r2 < 0 and r2 + a21 a−1 11 r1 > 0, then limk→∞ (xk , yk ) = (Dx , Dy ), where (Dx , Dy ) is equal to (D1 , D2 ) with σ1 = σ2 = 0. Note that the sign of r2 in the DDE model is fixed to r2 < 0. Adding the noise to the DDEs, we have from Theorems 3 and 4 that

−1 2 1 = a r , 0 a.s. ), x (i)0 If r1σ ≥ 0 and r2σ + a21 a−1 r < 0, then lim E(x 1σ 1σ k→∞ 11 11 k k
−1 −1 0 1 2 (ii) If r1σ ≥ a22 a12 r2σ and r2σ + a21 a11 r1σ ≥ 0, then limk→∞ E(xk ), E(xk ) = (D1 , D2 ). Hence we demonstrate that the solutions of the DDEs and the DSDEs with small noise have similar asymptotic behavior by comparing (i), (ii) and (i)0 , (ii)0 , respectively. In −1 0 addition, when comparing r2 + a21 a−1 11 r1 > 0 in (ii) and r2σ + a21 a11 r1σ < 0 in (i) , we understand the effect of strong noise, which changes the behavior of the predator population from non-extinction into extinction. Therefore the main difference between the deterministic and stochastic models is that large stochastic perturbation may result in the extinction of the predator population. Remark 14. Let (x, y) be the solutions of the SDE model (2), which is a special model in [25] with zero time delays. Note that the sign of r2 in the SDE model is also negative. (i) If r1 − 0.5σ12 < 0 and r2 − 0.5σ22 < 0, then limt→∞ (x(t), y(t)) = (0, 0) a.s. 2 (ii) If r1 − 0.5σ12 > 0, r2 − 0.5σ22 < 0 and (r2 − 0.5σ22 ) + a21 a−1 11 (r1 − 0.5σ1 ) < 0, then x is stable in the mean and y goes to extinction: Rt limt→∞ t−1 0 x(s)ds = a−1 11 r1σ , limt→∞ y(t) = 0 a.s. 2 (iii) If r2 − 0.5σ22 < 0 and (r2 − 0.5σ22 ) + a21 a−1 11 (r1 − 0.5σ1 ) > 0, then both x and y are stable in the mean:  R  R −1 t −1 t x(s)ds, t y(s)ds = (D1 , D2 ) a.s. limt→∞ t 0 0

Since r2 < 0 in the SDE model (2), the sign of r2 − 0.5σ22 in (2) is also negative, which is the reason why the condition r2 − 0.5σ22 < 0 is assumed in (i)–(iii). The three results, (i), (ii) and (iii) in this remark, are corresponding to Theorem 2-(b), (i)0 and (ii)0 in Remark 13, respectively. Hence, when replacing the stability of (x(t), y(t)) in the mean with the
stability of E(x1k ), E(x2k ) , we demonstrate that the sufficient conditions for the almost sure global stability of the SDE model (2) also suffice to give the same global stability of the DSDE model (3). In this case, note that there is no constraint on the sign of r2 in the DSDE model. Therefore we show that the DSDE model (3) is a good discrete model for the corresponding SDE model (2).

6. Numerical examples In this section, we provide some simulations that illustrate the results in Theorems 1, 2, 3 and 4 with truncation constants (ς, ς∗ ) = (19.9, 20) in (5) and (12). In this case, we have 0 < ης < 10−85 , so that we can ignore the effect of the term ης when using the values of parameters in the following three examples, where the conditions (12)–(14) are satisfied. In Figures 1, 2 and 3, the DSDE model (3) is simulated 1000 times at each time kh for calculating the expectation values E (xk ) and E (yk ), where xk and yk denote the 18 479

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solutions x1k and x2k , respectively. We compare our results for the DSDE model (3) with the results for the DDE model in [35], which is the model (3) with σ1 = σ2 = 0. Example 1. Let h = 0.0001, r1 = 0.8, r2 = −0.1, a11 = 0.4, a12 = 0.001, a21 = 0.1, a22 = 0.3, σ12 = 2.5 and σ22 = 0.1. Since r1 > 0, r2 < 0 and r2 + a21 a−1 11 r1 > 0, the solutions xk and yk of the DDE model converge to the positive numbers Dx and Dy in Remark 13-(ii), respectively, as displayed in Figure 1-(a). However, since riσ < 0 (i = 1, 2), the noises have a large effect on the convergence and, as a result, the solutions of the stochastically perturbed model (3) go to extinction, which are shown in Figures 1-(b) and (c), as in Theorem 2-(a) and (b), respectively. Therefore Figures 1 demonstrates the important role of noise. (a) 2.0

(b) 1,0

xk yk

(c)

xk yk

×10 −4 E (xk ) E (yk )

5 1

0.1 0.3

0

0 0

50

100

0

50

100

50

75

100

Figure 1: All the x-axes denote time kh. (a) Curves of the solutions of the DDE model. (b) Two realizations of the solutions xk and yk of the DSDE model, which converge to zero. (c) Expectation values of the solutions xk and yk of the DSDE model, which converge to zero in the mean.

Example 2. Let h = 0.001, r1 = 2, r2 = −2, a11 = 1.0, a12 = 0.4, a21 = a22 = 0.3, σ12 = 0.2 and σ22 = 4. Figure 2-(a) shows that the solutions xk and yk of the DDE model converge to a−1 11 r1 and 0, respectively, as in Remark 13-(i) when r1 > 0, r2 < 0 and r2 + a21 a−1 r ≤ 0. The noises satisfy both r1σ > 0 and r2σ + a21 a−1 11 1 11 r1σ < 0, which are the conditions in Theorem 3. Then Figures 2-(b), (c) and (d) show that the stochastically P perturbed model (3) behaves similarly to the DDE model in the sense that k −1 k−1 i=0 E(xi ) and yk converge to a−1 r and 0, respectively, which confirms Theorem 3. 11 1σ (a)

(b)

xk yk

(d)

xk

×10 −4

5

1 k

k−1 P

E (xi ) −

i=0

r 1 −0.5σ 12 a1 1

E (yk )

2

2 1

0 0

50

(c)

1

100 yk −5

0.5 0

0 0

5

10

0

2.5

5

500

750

1000

Figure 2: All the x-axes denote time kh. (a) Curves of the solutions of the DDE model. Curves in (b) and (c) are realizations of the solutions xk and yk of the DSDE model, respectively. (d) Convergence of average of expectation values of xk to non-zero and convergence of yk to zero in the mean.

σ12

Example 3. Let h = 0.001, r1 = 2.0, r2 = −0.1, a11 = a12 = 0.4, a21 = 1, a22 = 0.3 and = σ22 = 0.02, which give that r1 > 0, r2 < 0 and r2 + a21 a−1 11 r1 > 0. Thus Figure 3-(a) 19 480

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shows that the solutions xk and yk of the DDE model converge to Dx and Dy in Remark 13-(ii), respectively, as displayed in Figure 1-(a) in Example 1. However, the condition r1σ > 0 is different from that in Example 1. Realizations of the solutions of the DSDE −1 model are given in Figures 3-(b) and (c). Since r1σ > a−1 22 a12 r2σ and r2σ + a21 a11 r1σ > 0, Figure 3-(d) shows that the DSDE Pk−1model behaves similarly to the DDE model in the sense Pk−1 −1 −1 that k i=0 E(yi ) converge to positive D1 and D2 , respectively, i=0 E(xi ) and k which demonstrate Theorem 4. (a)

xk yk

(b)

xk

(d)

×10 −4 1 k

1.2 1 k

0.5

3.7

0

50

(c)

0

5

2.5 10 0

E(xi ) − D21

i=0 k−1 P

E(yi ) − D22

i=0

100 yk

5 0

3.7 1.2

k−1 P

−5 50

100 1000

1500

2000

Figure 3: All the x-axes denote time kh. (a) Curves of the solutions of the DDE model. Curves in (b) and (c) are realizations of the solutions xk and yk of the DSDE model, respectively. The symbols D21 and D22 in (d) denote D1 and D2 defined in (85).

7. Conclusion In this paper, we have considered a system of discrete-time stochastic difference equations for predator-prey interactions and established sufficient conditions for extinction and non-extinction of the two species. Our results show that if the positive equilibrium point of the deterministic difference system is globally stable, then the stochastic difference model will preserve the nice property in mean provided that the noise is sufficiently small. It is shown, however, that large noise can change the behavior of the predator population from non-extinction into extinction. Our new discrete Itˆo formula has played an important role in the two-dimensional DSDE model. In addition we can apply the new formula for the n-dimensional DSDE model o n   Xi−1 Xn i aij xjk − aij xjk + h0.5 σi ξk+1 xik+1 = xik 1 + h ri + j=1

j=i

for 1 ≤ i ≤ n and k ≥ 0. Therefore it is a further study to establish sufficient conditions for the extinction and non-extinction of the n species.

Appendix A.1. The proof of Lemma 1 By Taylor expansion, ϕ(1 + x) = ϕ(1) + ϕ0 (1)x + 2−1 ϕ00 (1)x2 + 6−1 ϕ000 (θ)x3

(100)

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with θ lying between 1 and x. Let x = hf + h0.5 gξ. Since f, g are G-measurable and ξ is G-independent with E(ξ) = 0, we have
E ( x| G) = E (hf | G) + E h0.5 gξ G = hf + h0.5 gE(ξ) = hf (101) and further



E x2 G = E (hf )2 G + E 2hf h0.5 gξ G + E hg 2 ξ 2 G = (hf )2 + hg 2 · (1 − µ) ≤ hf M5 hε + hg 2 · (1 − µ)

(102)

due to E(ξ 2 ) = 1 − µ and (18). Using Lemma 1-(ii) gives

E 6−1 ϕ000 (θ)x3 G ≤ 6−1 M3 E x3 G

(103)

and expanding x3 = (hf + h0.5 gξ)3 yields
 E x3 G ≤ hf (hf )2 + 3hg 2 · (1 − µ) + hg 2 M1 h0.5 g ≤ hf (M5 hε )2 {1 + 3(1 − µ)} + hg 2 M1 M4 hε

(104)

because of (18) and (16). Inserting (101)–(104) into (100), we have
E ϕ(1 + x) G (105) 0 −1 00 2 ε 2 ε = ϕ(1) + ϕ (1)hf + 2 ϕ (1)hg · (1 − µ) + hf O1 (h ) + hg O2 (h ) , (106) in which the two big O notations denote O1 (hε ) = 2−1 ϕ00 (1)M5 hε + 6−1 M3 (M5 hε )2 {1 + 3(1 − µ)} , O2 (hε ) = M1 M5 hε . Now it remains to show


E φ 1 + hf + h0.5 gξ − ϕ 1 + hf + h0.5 gξ G = hg 2 O (hε ) . Let c1 = 1 + hf and c2 = h0.5 g. Then the disintegration formula for conditional expectations with respect to G gives       √ √ E φ 1 + hf + hgξ − ϕ 1 + hf + hgξ G Z = {φ (c1 + c2 x) − ϕ (c1 + c2 x)} p(x) dx (107) R

due to Lemma 1-(iii) and the fact that f, g are G-measurable, ξ is G -independent, φ is almost everywhere continuous and ϕ is also continuous (see Theorem 5.4 in [44] for the disintegration formula). Let Uδ = [1 − δ, 1 + δ] and s = c1 + c2 x. Then (107) becomes   Z s − c1 ds {φ (s) − ϕ (s)} p (108) c2 |c2 | R−Uδ because of Lemma 1-(i). Here p is the probabilty density function of ξ. 21 482

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Lemma 1-(iii) gives that Z   s − c1 ds {φ (s) − ϕ (s)} p c2 |c2 | R−Uδ      Z s − c1 1 ds sup p ≤ |φ(s) − ϕ(s)| |c2 | s∈U c2 |c2 | / δ R−Uδ     s − c1 1 ≤ M4 |c2 |2 sup p c2 |c2 |3 s∈U / δ     s − 1 − hf 1 2 = M4 hg sup p . h0.5 g |h0.5 g|3 s∈U / δ Since there exists some δ0 such that for s ∈ / Uδ and all sufficiently small h > 0 |s − 1 − hf | > |s − 1| − h|f | > δ − M5 h > δ0 > 0,

(109)

letting y = (s − 1 − hf )/(h0.5 g) yields |y| = and further

|s − 1 − hf | δ0 > 0.5 h |g| M5 hε

(110)

    s − 1 − hf 1 p (y) |y|3 sup p = sup 3. h0.5 g |h0.5 g|3 s∈U / δ s∈U / δ |s − 1 − hf |

Hence it follows from (17), (109) and (110) that M5 ε |y|−1 p (y) |y|3 sup 3 < M2 sup 3 < M2 2 h , δ0 s∈U / δ |s − 1 − hf | s∈U / δ |s − 1 − hf | which gives Z

 {φ (s) − ϕ (s)} p

R−Uδ

s − c1 c2



ds M5 < hg 2 · M4 M2 2 hε . |c2 | δ0

(111)

Therefore using (105), (108) and (111), we obtain the desired result.

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments This work was supported by the 2014 Research Fund of University of Ulsan.

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WEIGHTED SUPERPOSITION OPERATORS FROM ZYGMUND SPACES TO µ-BLOCH SPACES ZHI JIE JIANG, TING WANG, JUAN LIU, TING LUO, TING SONG

Abstract. Let D = {z ∈ C : |z| < 1} be the open unit disk in the complex plane C and H(D) the space of all analytic functions on D. Let ϕ be an entire function on C and u ∈ H(D). The boundedness and compactness of the operators Su,ϕ : f 7→ u · ϕ ◦ f from Zygmund spaces to µ-Bloch spaces are characterized.

1. Introduction Let D = {z ∈ C : |z| < 1} be the open unit disk in the complex plane C, H(D) the space of all analytic functions on D and H ∞ (D) the space of bounded analytic functions. Let ϕ be a complex-valued function on C and u ∈ H(D). We introduce a class of nonlinear operators by Su,ϕ f = u · ϕ ◦ f, f ∈ H(D). This operator can be regarded as a generalization of the superposition operator Sϕ f = ϕ◦f and the multiplication operator Mu f = u · f . Suppose that X and Y are two metric spaces of analytic functions on D. Note that if X contains the linear functions and Sϕ maps X into Y , then ϕ must be an entire function. In recent years, the following natural questions of the superposition operators are considered. (a) When does ϕ induce a superposition operator from X into Y ? (b) When is a superposition operator from X into Y bounded? (c) When is a superposition operator from X into Y compact? Although analogous concepts also make sense in the context of real-valued functions and their theory has a long history (see [2]), the study of such natural questions on analytic function spaces has only begun fairly recently. The operators Sϕ that map Bergman spaces into area Nevanlinna classes were characterized in [6], which have been extended by other authors to some other analytic function spaces, where it is remarkable the works of Vukoti´c et. al. in [1], [4] and [5]. It must be mentioned that the authors of [4] gave a very interesting geometric construction of simple connected domain in several analytic function spaces. This technique has been used by many authors; in particular, Xu used it to study the superposition operators from α-Bloch spaces into β-Bloch spaces in [20] and Xiong used it to characterize the superposition operators from Qp spaces into α-Bloch spaces with 0 < α < 1 in [18]. It should be noted that quite recently, Castillo et.al. and Ramos Fern´ andez have studied the superposition operators from Bloch-Orlicz spaces into α-Bloch spaces and between weighted Banach spaces of analytic functions in [7] and [14], respectively. In this paper we characterize the boundedness and compactness of the operators Su,ϕ from weighted Zygmund spaces to µ-Bloch spaces. We also consider the superposition operators from weighted Zygmund spaces to weighted Bloch spaces. Now we present the needed spaces and some facts. The Zygmund space Z consists of all f ∈ H(D) such that sup(1 − |z|2 )|f 00 (z)| < ∞. z∈D

2000 Mathematics Subject Classification. Primary 47H38; Secondary 46E15, 47B38. Key words and phrases. Weighted Zygmund spaces, µ-Bloch spaces, superposition operators. 1

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With the norm kf kZ = |f (0)| + |f 0 (0)| + sup(1 − |z|2 )|f 00 (z)|, z∈D

it is a Banach space. By Zygmund’s theorem (see Theorem 5.3 in [9]), we know that f ∈ Z if and only if f is continuous on D and |f (ei(θ+h) ) + f (ei(θ−h) ) − 2f (eiθ )| < ∞. h h>0,θ∈R sup

In closed subspaces of Z, the little Zygmund space Z0 is usually considered, which is defined by  Z0 = f ∈ Z : lim (1 − |z|2 )|f 00 (z)| = 0 . |z|→1

Let α ∈ (0, ∞). The weighted Zygmund space Zα consists of all f ∈ H(D) such that sup(1 − |z|2 )α |f 00 (z)| < +∞. z∈D

With the norm kf kZα = |f (0)| + |f 0 (0)| + sup(1 − |z|2 )α |f 00 (z)|, z∈D

Zα is also a Banach space. For the weighted Zygmund spaces and the operators from them into some other spaces, see, e.g., [10], [12] and [15]. Suppose that µ is a positive continuous radial function on D (that is, µ(z) = µ(|z|)) and decreasing on [0, 1) with limr→1 µ(r) = 0. Let µ be a weight. The µ-Bloch space Bµ consists of all f ∈ H(D) such that supz∈D µ(z)|f 0 (z)| < ∞. With kf kBµ = |f (0)| + sup µ(z)|f 0 (z)|, z∈D 2

Bµ is a Banach space. When µ(z) = 1 − |z| , the space Bµ is just Bloch space and denoted by B; while when µ(z) = (1 − |z|2 )α with α > 0, the space Bµ becomes the weighted Bloch space Bα . The µ-Bloch spaces appear in the literature in a natural way when one considers properties of some operators in certain spaces of analytic functions; for example, 2 , Attele in [3] proved that the Hankel operator on Bergman if µ(z) = (1 − |z|) log 1−|z| spaces induced by a function f is bounded if and only if f ∈ Bµ . The logarithmic Bloch type space has been defined and studied in [16]. Recently, the Bloch-Orlicz spaces have been introduced by Ramos-Fernandez in [13]. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation a ' b means that there is a positive constant C such that a/C ≤ b ≤ Ca. 2. The operator Su,ϕ : Z → Bµ First we enumerate several useful lemmas. The first one below is well-known. Lemma 2.1 There is a positive constant Cα depending only on α such that for any z ∈ D and f ∈ Zα (i)  C kf kZα , 0 < α < 2,  α f (z) ≤ Cα kf kZα log 2 2 , α = 2, 1−|z|   2 2−α Cα kf kZα (1 − |z| ) , α > 2. (ii)  C kf kZα , 0  α f (z) ≤ Cα kf kZα log 2 2 , 1−|z|   Cα kf kZα (1 − |z|2 )1−α ,

488

0 < α < 1, α = 1, α > 1.

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WEIGHTED SUPERPOSITION OPERATORS

3

√ Let a ∈ D and 1/ 2 < |a| < 1, define  f (z) = (z − 1) 1 + log

 1 2 +1 1−z

and 1 −1 f (¯ az)  log . a ¯ 1 − |a|2 The function ga is called the test function with the following property (see [11]). ga (z) =

Lemma 2.2 The function ga belongs to Z and kga kZ ' 1. The following result can be found in [17]. Lemma 2.3 Let α ∈ (0, 1]. Then for every bounded sequence uniformly on every compact subset of D as n → ∞, we have (i) if α = 1, then lim sup fn (z) = 0. n→∞ z∈D (ii) if 0 < α < 1, then lim sup fn0 (z) = 0.

 fn in Zα and fn → 0

n→∞ z∈D

The next result is often used in dealing with the compactness of operators on analytic function spaces. Since the proof is standard (see Proposition 3.11 in [8]), it is omitted . Lemma 2.4 Let u ∈ H(D) and ϕ an entire function. Then the bounded operator Su,ϕ : Zα → Bµ is compact if and only if for any bounded sequence {fn } in Zα such that fn → 0 uniformly on every compact subset of D as n → ∞, it follows that limn→∞ kSu,ϕ fn kBµ = 0. Now we characterize the boundedness of the operator Su,ϕ : Z → Bµ . Theorem 2.1 Let u ∈ H(D) and ϕ an entire function with ϕ0 (0) 6= 0. Then the operator Su,ϕ : Z → Bµ is bounded if and only if u ∈ Bµ and L := sup µ(z)|u(z)| log z∈D

2 < ∞. 1 − |z|2

Proof. Suppose that the operator Su,ϕ : Z → Bµ is bounded. By taking f1 the constant function, we obtain u ∈ Bµ . Since operator Su,ϕ : Z → Bµ is bounded, for the function f2 = ga there exists a positive constant C such that ∞ > CkSu,ϕ k ≥ kSu,ϕ f2 kBµ ≥ µ(a) (Su,ϕ f2 )0 (a) = µ(a) u0 (a)ϕ(f2 (a)) + u(a)ϕ0 (f2 (a))f20 (a)
≥ µ(a) u(a) ϕ0 (f2 (a)) f20 (a) − u0 (a) ϕ(f2 (a)) . From this, we get µ(a) u0 (a) ϕ(f2 (a)) + CkSu,ϕ k ≥ µ(a) u(a) ϕ0 (f2 (a)) f20 (a) . Set M = Cα kf2 kZ and M1 = max |ϕ(z)|. By Lemma 2.1 (i), we have |z|=M

M1 kukBµ + CkSu,ϕ k ≥ µ(a) u0 (a) ϕ(f2 (a)) + CkSu,ϕ k ≥ µ(a) u(a) ϕ0 (f2 (a)) f20 (a) 1 1 − |a|2 1 2 , ≥ µ(a) u(a) ϕ0 (ga (a)) log 2 1 − |a|2 √ where we have used that when |a| > 1/ 2, = µ(a) u(a) ϕ0 (ga (a)) log

log

1 1 2 ≥ log . 1 − |a|2 2 1 − |a|2

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It is easy to see that ga (a) → 0 as |a| → 1. Therefore from this and the fact that lim ϕ0 (ga (a)) = |ϕ0 (0)| 6= 0, |a|→1

we obtain sup

µ(z) u(z) log

1/2 0, there is a δ > 0 such that 2 µ(z) u(z) log 0 and kfn kZ ≤ M0 and fn → 0 uniformly on every compact subset of D as n → ∞. By the Cauchy integral formula and an easy calculation, it is clear that {fn0 } also uniformly converges to zero on every compact subset of D as n → ∞. Let M = max |ϕ0 (z)|. By Lemma 2.1 and Lemma 2.3 (i), we have |z|=Cα M0

kSu,ϕ fn kBµ

= u(0)ϕ(fn (0)) + sup µ(z) (Su,ϕ fn )0 (z) z∈D = u(0)ϕ(fn (0)) + sup µ(z) u0 (z)ϕ(fn (z)) + u(z)ϕ0 (fn (z))fn0 (z) z∈D ≤ u(0)ϕ(fn (0)) + sup µ(z) u0 (z) ϕ(fn (z)) + sup µ(z) u(z) ϕ0 (fn (z)) fn0 (z) z∈D z∈D ≤ u(0)ϕ(fn (0)) + kukBµ sup ϕ(fn (z)) + sup µ(z) u(z) ϕ0 (fn (z)) fn0 (z) z∈D

|z|≤δ

+ sup µ(z) u(z) ϕ0 (fn (z)) fn0 (z) δ 0, f ∈ Zα and kf kZα ≤ M . Set M1 =

max |ϕ0 (z)|.

|z|=Cα M

Then we have (1 − |z|2 )β (Sϕ f )0 (z) = (1 − |z|2 )β ϕ0 (f (z)) f 0 (z) ≤ Cα M M1 (1 − |z|2 )β < ∞. This means that the operator Sϕ : Zα → Bβ is bounded. Now we prove (ii). Suppose that kfn kZα ≤ M and {fn } uniformly converges to zero on every compact subset of D as n → ∞, then kSϕ fn kBβ = |ϕ(fn (0))| + sup(1 − |z|2 )β (Sϕ fn )0 (z) z∈D = |ϕ(fn (0))| + sup(1 − |z|2 )β ϕ0 (fn (z)) fn0 (z) z∈D

≤ |ϕ(fn (0))| + M1 sup |fn0 (z)|, z∈D

where M1 =

0

max |ϕ (z)|. By ϕ(0) = 0 and Lemma 2.3 (ii), we know that lim kSϕ fn kBβ = n→∞

|z|=Cα M

0. By Lemma 2.4, the operator Sϕ : Zα → Bβ is compact.



When α = 1, from Theorem 2.1 and Theorem 2.2 we can obtain characterizations of the boundedness and compactness of the operator Sϕ : Z → Bβ . It is unnecessary to go into details here. Theorem 3.2 Let α ∈ (1, 2) and ϕ an entire function. We have the following assertions: (1) If α ≤ 1 + β, then (i) the operator Sϕ : Zα → Bβ is bounded, and (ii) when ϕ(0) = 0, the operator Sϕ : Zα → Bβ is compact. (2) If α > 1 + β, then the operator Sϕ : Zα → Bβ is bounded if and only if ϕ is a constant function. Proof. We first prove the assertion (i) of (1). Let M > 0, f ∈ Zα and kf kZα ≤ M . Set M1 = max |ϕ0 (z)|. Then we have |z|=Cα M

(1 − |z|2 )β (Sϕ f )0 (z) = (1 − |z|2 )β ϕ0 (f (z)) f 0 (z) ≤ CM M1 (1 − |z|2 )1−α+β < ∞. This shows that the operator Sϕ : Zα → Bβ is bounded. As the proof of Theorem 3.1 (ii), the assertion (ii) follows. Note that we have the relation Zα = Bα−1 . By this and Theorem 4 in [5], the assertion (2) is true.  Theorem 3.3 Let α = 2 and ϕ an entire function. (1) When β > 1, (i) the operator Sϕ : Zα → Bβ is bounded if and only if ϕ is a polynomial of degree s ≤ 1, and (ii) the operator Sϕ : Zα → Bβ is compact. (2) When β = 1, (i) the operator Sϕ : Zα → Bβ is bounded if and only if ϕ is a linear function, and (ii) the operator Sϕ : Zα → Bβ is compact. (3) When 0 < β < 1, the operator Sϕ : Zα → Bβ is bounded if and only ϕ is a constant function.

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ZHI JIE JIANG, TING WANG, JUAN LIU, TING LUO, TING SONG

Proof. By Theorem 7 of [5], the assertions (i) of (1) and (i) of (2) hold. Also from Theorem 4 of [5], the assertion (3) follows. Now we want to prove the assertion (ii) of (1). Let the operator Sϕ : Zα → Bβ be compact. From the assertion (i) of (1), we know that, if ϕ is not a constant function, then ϕ(z) = az + b with a 6= 0. Therefore, it is enough to show that Sϕ : Zα → Bβ is compact when ϕ(z) = az. At this time, Sϕ is just the multiplication operator Ma defined by Ma f = a · f . Thus, by Theorem 3.1 of [19], we know that Ma : Zα → Bβ is compact. Similar to the proof of the assertion (ii) of (1), the assertion (ii) of (2) is right.  Theorem 3.4 Let α > 2, β > 1 and ϕ an entire function. (1) The operator Sϕ : Zα → Bβ is bounded if and only if (i) when α > β, ϕ is a constant. (ii) when α = β, ϕ is a linear function. β−1 . (iii) when α < β, ϕ is a polynomial of degree s ≤ α−2 (2) The operator Sϕ : Zα → Bβ is compact if and only if ϕ is a polynomial of degree β−1 . s < α−2 Proof. Note that when α > 2, it follows that Zα = Bα−1 = Hα−2 , where Hα−2 is called the weighted Banach space of analytic functions defined by Hα−2 = {f ∈ H(D) : (1 − |z|2 )α−2 |f (z)| < ∞}. Then (1) and (2) follow from Theorem 4.2 of [14] and Proposition 3.1 of [4].  Acknowledgments. This work is supported by the Key Fund Project of Sichuan Provincial Department of Education (No.15ZA0221), the Cultivation Project of Sichuan University of Science and Engineering (No.2015PY04) and the innovation foundation for the university students (No.cx20141202).

References ´ [1] V. Alvarez, A. M´ arquez, D. Vukoti´ c, Superposition operators between the Bloch spaces and Bergman spaces, Ark. Math., 42 (2004), 205-216. [2] J. Appell, P. P. Zabrejko, Nonlinear superposition operators, Cambridge University Press, Cambridge, 1990. [3] K. Attele, Toeplitz and Hankel operators on Bergman spaces, Hokkaido Math. J., 21 (1992), 279-293. [4] J. Bonet, D. Vukoti´ c, Superposition operators between weighted Banach spaces of analytic functions of controlled growth, Monatsh Math, 170 (3) (2013), 311-323. [5] S. M. Buckley, D. Vukoti´ c, Univalent interpolation in Besov spaces and superposition into Bergman spaces, Potential Anal, 29 (2008), 1-16. [6] G. A. C´ amere, J. Gim´ enez, The nonlinear superposition operator acting Bergman spaces, Compos. Math., 93 (1994), 23-35. [7] R. E. Castillo, J. C. Ramos-Fern´ andez, M. Salazar, Bounded superposition operators between BlochOrlicz and α-Bloch spaces, Appl. Math. Comput., 218 (2011), 3441-3450. [8] C. C. Cowen, B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, 1995. [9] P. L. Duren, Theory of H p spaces, Academic Press, New York, NY, USA, 1970. [10] Z. J. Jiang, On a class of opertors from weighted Bergman spaces to some spaces of analytic functions, Taiwanese Journal of Mathematics, 15 (5) (2011), 2095-2121. [11] S. Li, S. Stevi´ c, Integral-type operators from Bloch-type spaces to Zygmund-type spaces, Appl. Math. Comput., 215 (2009), 464-473. [12] S. Li, S. Stevi´ c, Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces, Appl. Math. Comput., 217 (2010), 3144-3154. [13] J. C. Ramos-Fern´ andez, Composition operators on Bloch-Orlicz type spaces, Appl. Math. Comput., 217 (2010), 3392-3402. [14] J. C. Ramos-Fern´ andez, Bounded superposition operators between weighted Banach spaces of analytic functions, Appl. Math. Comput., 219 (2013), 4942-4949. [15] S. Stevi´ c, On an integral operator from the Zygmund space to the Bloch-type space on the unit ball, Glasg. J. Math. 51 (2009), 275-287. [16] S. Stevi´ c, On new Bloch-type spaces, Appl. Math. Comput., 215 (2009), 841-849.

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[17] S. Stevi´ c, On an integral-type operator from Zygmund-type spaces to mixed-norm spaces on the unit ball, Abstr. Appl. Anal. 2010 (2010), 7 pages. Article ID 198608. [18] C. Xiong, Superposition operators between Qp spaces and Bloch-type spaces, Complex Var. Theory Appl., 50(12) (2005), 935-938. [19] H. M. Xu, T. S. Liu, Weighted composition operators between Bloch-type spaces on the polydisks, Chinese Annals of Mathematics, 26A(1) (2005), 61-72. [20] W. Xu, Superposition operators on Bloch-type spaces, Comput. Methods. Funct. Theory., 7(2)(2007), 501-507. Zhi Jie Jiang, School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China E-mail address: [email protected]

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Dynamical Analysis Of The Rational Difference Equation xn+1 =

αxn−3 A+Bxn−1 xn−3

E. M. Elsayed1.2 , Malek Ghazel3 , and A. E. Matouk3 1 Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 3 Mathematics Department, Faculty of Science, University of Hail, Hail 2440, Saudi Arabia E-mails: [email protected], [email protected], [email protected]. ABSTRACT αxn−3 This article is concerned with the following rational difference equation xn+1 = A+Bx with the initial n−1 xn−3 conditions, x−3 = d, x−2 = c, x−1 = b, and x0 = a are arbitrary real numbers, α, A and B are arbitrary constants. A detailed analytical study of the convergence of the solutions including their dependence on parameters and initial conditions is investigated. The local stability and global attractivity of the difference equation’s equilibrium points are discussed. The existence of periodic solutions in the proposed difference equation is also verified analytically. Moreover, numerical simulations are carried out to verify the correctness of the analytical results.

Keywords: Difference equations, Recursive sequences, Analytical study, Infinite products, Convergence, Periodic solution. Mathematics Subject Classification: 39A10 ––––––––––––––––––––––

1. INTRODUCTION Difference equations arise from the study of the evolution of natural phenomena. The applications of difference equations are rapidly increasing to various fields such as economics [1], [12]-[14], mathematical, biology [15]-[16] physics and engineering [7]. Indeed, difference equations represent chief tools of investigating the qualitative behaviors of dynamical systems [33]. Consequently, studying the solutions of difference equations and its qualitative behaviors have become focal topics for research [1]-[36]. In recent years, difference equations have been investigated by many authors. For some results: In [3], Aloqeili n−1 xn−k . Cinar [5] obtained the solution of the difference found the solution of the difference equation xn+1 = dxb−cx n−s axn−1 equation xn+1 = 1+bxn xn−1 . In [9], Elabbasy et al. discussed the solution and the periodicity character of the bxn . difference equations xn+1 = axn − cxn −dx n−1 In this paper, we study to the following sequence defined recursively by αxn−3 xn+1 = , A + Bxn−1 xn−3

(1)

with the initial data: x−3 = d, x−2 = c, x−1 = b, and x0 = a. Note first that, if α = 0, then for all n ∈ N, xn = 0. Then we will consider that α 6= 0. Although we can (by dividing the numerator and denominator by α) obtain a more simply form of such sequences, we will keep them in order to study of the behaviors with respect to α. Note also that, if one or more of the initial data a, b, c and d is zero, then it will be seen that one or more of the subsequences of (xn )n modulo 4 vanish, so that we will suppose that abcd 6= 0. The cases A = 0 and B = 0 are a trivial, therefore we will assume that A 6= 0 and B 6=Q0. Finally, we will consider the convention: if (ap )p is a sequence of complex numbers, and n > m, in Z, then m p=n ap = 1.

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2. DEFINITIONS AND PRELIMINARIES. A difference equation of order k is an equation of the form xn+1 = F (xn , xn−1 , ..., xn−(k−1) ), n = 0, 1, ...,

(2)

where F is a function that maps on some set I k into I. A solution of Eq. (2) is a sequence xn that satisfies Eq. (2) for all n ≥ 0. With each solution xn of the Eq. (1), we associate the vector of initial conditions v0 (x) = (x0 , x−1 , ..., x−k+1 ) ∈ I k . P0 The norm of the vector u ∈ I k will be defined as kuk = i=−k+1 |ui |. Definition 1. (Equilibrium point) A point x ¯ ∈ R is called an equilibrium point of Eq. (2), if

x ¯ = F (¯ x, x ¯, ..., x ¯). x) = (¯ x, x ¯, ..., x ¯). Let x ¯ ∈ R be an equilibrium point of Eq. (2), and denote by v(¯ x) ∈ I k the vector v(¯ Suppose that the function F is continuously differentiable in some open neighborhood of an equilibrium point x ¯. Consider the linearized equation of Eq. (2) about the equilibrium point x ¯: yn+1 = q0 yn + q1 yn−1 + ... + qk−1 yn−(k−1) , where qi =

∂F x, x ¯, ..., x ¯), ∂xi (¯

(3)

i = 0, 1, ..., k − 1, and the characteristic equation of Eq. (3) about x ¯: λk − q0 λk−1 − ... − qk−2 λ − qk−1 = 0.

(4)

Definition 2. 1. When all the roots of Eq. (4) have absolute value less than one, then the equilibrium point of Eq. (2) is locally asymptotically stable. 2. If at least a root of Eq. (4) have absolute value greater than one, then the equilibrium point of Eq. (2) is unstable. Definition 3. 1. An equilibrium point x ¯ of Eq. (2) is called hyperbolic if no root of Eq. (4) has absolute value equal one. 2. If there exists a root of Eq. (4) with absolute value equal to one, then the equilibrium point x ¯ is called nonhyperbolic. 3. An equilibrium point x ¯ of Eq. (2) is called saddle if there exists a root of Eq. (4) has absolute value less than one. and another root of Eq. (4) greater than one. 4. An equilibrium point x ¯ of Eq. (2) is called a repeller if all roots of Eq. (4) has absolute value greater than one. ¯ or simply nonoscillatory if there exists N ≥ −k 5. A solution xn of Eq. (2) is called nonoscillatory about x ¯, ∀n ≥ N or xn ≤ x ¯, ∀n ≥ N . Otherwise, the solution xn is called oscillatory about such that either xn ≥ x x ¯, or simply oscillatory. 6. A solution xn of Eq. (2) is called periodic with period p if there exists an integer p, such that xn+p = xn ,

∀n ≥ −k.

(5)

A solution is called periodic with prime period p if p is the smallest positive integer for which Eq. (5) holds.

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3. ANALYTICAL EXPRESSIONS OF (XN )N The following Theorem gives an analytical expression of the sequence (xn )n . Theorem 1. Let (xn )n be the sequence given by (1) and the initial data that follow, then For all n ≥ 2 n−2 Y³

A2p+2 + Bbd

dαn x4n−3 =

p=0

A2p+1 + Bbd

p=0

x4n−1 =

Ai α2p+1−i

i=0

n−1 Y³

bαn

2p+1 X

2p X

Ai α2p−i

i=0

n−1 Y³

A2p+1 + Bbd

p=0

n−1 Y³

2p+2

A

+ Bbd

p=0

2p X

´

´

n−2 Y³

A2p+2 + Bac

cαn ,

p=0

x4n−2 =

p=0

´

i=0 2p+1 ´ X i 2p+1−i

aαn ,

x4n =

Aα

A2p+1 + Bac

p=0

2p+2

A

+ Bac

p=0

i=0

2p X

Ai α2p−i

i=0

n−1 Y³

n−1 Y³

Ai α2p+1−i

i=0

n−1 Y³

A2p+1 + Bac

Ai α2p−i

2p+1 X

2p X

´

Ai α2p−i

´

i=0 2p+1 ´ X i 2p+1−i

´

.

.

(6)

(7)

Aα

i=0

Proof. By induction, we prove the result for x4n−3 . Take n ≥ 2, and assume that the results hold for the step n, then prove the result for the step n + 1, we get: x4(n+1)−3

=

αx4n−3 A + Bx4n−1 x4n−3 dαn+1

=

n−1 Y³

A2p+2 + Bbd

p=0

n−1 Y³

A2p+1 + Bbd

p=0

2p X

i=0 n−1 Y³

2p+1

A

+ Bbd

p=0

Hence, we obtain

A2p+2 + Bbd

2p X i=0

dαn+1 x4(n+1)−3 =

i=0

´

i=0

p=0

n−1 Y³

Ai α2p+1−i

2n−1 ´h ³ ´ i X Ai α2p−i A A2n + Bbd Ai α2n−1−i + Bbdα2n

dαn+1

=

2p+1 X

2p+1 X

Ai α2p+1−i

i=0

´

2n ³X ´³ ´´ Ai α2p−i A2n+1 + Bbd Ai α2n−i + α2n

.

i=1

n−1 Y³

A2p+2 + Bbd

p=0

2p+1 X

Ai α2p+1−i

i=0

2p n ³ ´ Y X 2p+1 + Bbd Ai α2p−i A

p=0

´

.

i=0

Similarly, the expression for x4n−2 , x4n−1 , x4n can be easily proved.

Notation. If we denote by (Pn )n the sequence of two variables polynomials defined for every n ∈ N, x and y as, Pn (x, y) = (A − α + Bxy)An − Bxyαn . The following Corollary gives a simplified analytic expression when A 6= α. Corollary 1. Consider the sequence (xn )n defined by the Eq. (1) for A 6= α, the subsequences can be written as: n−2 n−2 Y Y P2p+2 (b, d) P2p+2 (a, c) dαn (A − α) cαn (A − α) x4n−3 =

p=0

n−1 Y

,

P2p+1 (b, d)

p=0

x4n−2 =

p=0

n−1 Y

,

P2p+1 (a, c)

p=0

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bαn

n−1 Y p=0

x4n−1 =

n−1 Y

aαn

P2p+1 (b, d) ,

and

n−1 Y

P2p+1 (a, c)

p=0

x4n =

n−1 Y

P2p+2 (b, d)

p=0

.

P2p+2 (a, c)

p=0

Proof. It is sufficient to use the binomial identity xp+1 −y p+1 = (x−y) of the subsequences defined by Eq. (6) and (7).

Pp

k=0

xk y p−k in the analytical expression

Corollary 2. Consider the sequence (xn )n defined by the Eq. (1). For A = α 6= 0, the sequence can be expressed in Gamma form as A A + n)Γ( + 1) 2Bbd Bbd , A A + 1)Γ( + 2n) BbΓ2 ( 2Bbd Bbd

A22n−2 Γ2 ( x4n−3 =

A A + n)Γ( ) 2Bac Bac = , A A + 1)Γ( + 2n) BaΓ2 ( 2Bac Bac A22n−2 Γ2 (

x4n−2

A A + 2n + 1)Γ2 ( + 1) Bbd 2Bbd x4n−1 = , A A + 1)Γ2 ( + n + 1) 22n Γ( Bbd 2Bbd where Γ is the Euler’s Gamma function. bΓ(

A A + 2n + 1)Γ2 ( + 1) Bac 2Bac = , A A + 1)Γ2 ( + n + 1) 22n Γ( Bac 2Bac aΓ(

x4n

Proof. Using Eq. (6) we have: dAn x4n−3

n−2 Y³

A2p+2 + Bbd

p=0 n−1 Y³

=

A2p+1

i=0

2p+1

A

dA

2p+1 X

+ Bbd

p=0 n−2 Y

2p X i=0

2p

A

´ ³ A + 2p + 2 Bbd Bbd p=0

´

´

,

h n−1 ´i2 Y ³ A A +p 2 2Bbd p=1

2n−1 ´ = ´ ³ A Y ³ A + 2p + 1 +p Bbd Bb Bbd Bbd p=0 p=1 ´ ³ A A A22n−2 Γ2 + n Γ( + 1) 2Bbd Bbd = ´ ³ A ´ . ³ A + 2n Γ2 +1 Bb Γ Bbd 2Bbd Similarly, one can prove the other relations. This ended the proof.

=

n−1 Y

Remark 1. 1. A common hypothesis in the study of rational difference equations is the choice of positive coefficients and initial data. Therefore, all the solutions will be automatically well defined. It is, in general a problem of great difficulty to determine the good set of initial conditions without finding the analytical expression of the considered sequence. 2. According to the Corollaries 1 and 2, the good set G of the sequence (xn )n is given as (a) When A 6= α, G=

½ ½ −(A − α)An , (a, b, c, d) ∈ R4 such that bd, ac ∈ R − B(An − αn )

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(b) When A = α, G = {(a, b, c, d) ∈ R4 such that

A A Bbd , Bac

∈ / 2Z− }.

3. If we choose for example α = A = B, we obtain the expression of the general term which can be written and in gamma form as x4n−3 =

1 1 + n)Γ( bd ) 22n−2 Γ2 ( 2bd , 1 1 bΓ2 ( 2bd + 1)Γ( bd + 2n)

x4n−2 =

1 1 + 2n + 1)Γ2 ( 2bd + 1) bΓ( bd , 1 1 2n 2 2 Γ( bd + 1)Γ ( 2bd + n + 1)

x4n−1 =

x4n =

1 1 + n)Γ( ac ) 22n−2 Γ2 ( 2ac , 1 1 aΓ2 ( 2ac + 1)Γ( ac + 2n)

1 1 + 2n + 1)Γ2 ( 2ac + 1) aΓ( ac . 1 1 2n 2 2 Γ( ac + 1)Γ ( 2ac + n + 1)

In the following section we will study the convergence of sequence (xn )n . This will depend evidently on the parameters α, A, B and the initial data.

4. CONVERGENCE OF SOLUTIONS OF EQ. (1) Consider the function F defined on R4 as: F (u0 , u1 , u2 , u3 ) = written as xn+1 = F (xn , xn−1 , xn−2 , xn−3 ).

αu3 A+Bu1 u3 .

Using the function F , Eq. (1) can be

Theorem 2. The following statements are true: (1) For B(A − α) ≥ 0, Eq.(1) has a unique equilibrium point x = 0, then (a) If A = α, the equilibrium point is nonhyperbolic. (b) If

A α

> 1, the equilibrium point is locally asymptotically stable.

(2) For B(A − α) < 0, then

(a) The Eq. (1) has exactly three equilibrium points which are x1 = 0, x2 = (b) If 0 < A < α, then

q

α−A B ,

q x3 = − α−A B .

(8)

(i) The equilibrium point x1 = 0 is a repeller. (ii) The equilibrium points x2 , x3 are hyperbolic. Proof. (1) For B(A − α) ≥ 0, x ¯ is an equilibrium point is equivalent to x ¯=

α¯ x A + Bx ¯2

⇒ Bx ¯3 + (A − α)¯ x=0 ⇒ x ¯(B x ¯2 + A − α) = 0.

This shows clearly that if B(A − α) ≥ 0, x = 0 is the unique equilibrium point of Eq. (1). ∂F α (0, 0, 0, 0), then q0 = q1 = q2 = 0 and q3 = − A , the characteristic equation of the linearized equation qi = ∂u i associated with Eq. (1) is then all real roots have absolute value equal to one, so the equilibrium points is nonhyperbolic.¯ x is an equilibrium point is equivalent to λ4 −

α = 0. A

(9)

(a) Suppose that A = α, then all real roots have absolute value equal to one, so the equilibrium points is nonhyperbolic. (b) Suppose that A α > 1, so all the roots of Eq. (9) have absolute value less than one, according the linearized stability Theorem, the equilibrium point x = 0 is locally asymptotically stable. (2) For B(A − α) < 0, the equation x ¯(B x ¯2 + A − α) = 0 has exactly three solutions which are the equilibrium points in Eqs. (8).

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α (a) The characteristic equation about x1 = 0 is λ4 − A = 0, since 0 < A < α then all roots has absolute value greater than one and x1 = 0 is repeller. q 2 A A (b) The characteristic equation about x2 is λ4 + α−A λ − = 0. The real roots of this equation are A α α q A and − α , they are less than one, so the equilibrium point x2 is hyperbolic. The proof for x3 can be similarly obtained.

As it is expected, the convergence of (xn )n depends on the parameters α, A, B, and the initial data. We will distinguish the following cases: (i) Case | A α | > 1. Theorem 3. Assume that | A α| > 1

(1) If (A − α + Bbd)(A − α + Bac) 6= 0, then every solution of Eq. (1) converges toward zero.

q (2) If A − α + Bbd = A − α + Bac = 0, then the solution of Eq. (1) converges iff a = b = c = d = ± α−A B .

(3) If (A − α + Bbd)(A − α + Bac) = 0 but not both terms of the product are zero, then every solution of Eq. (1). Proof. (1) Suppose that (A − α + Bbd)(A − α + Bac) 6= 0, then Corollary 1 implies that ³ ´ dαn (A−α)

x4n−3

=

Tn−2

Tn−1 p=0

A2p+2 (A−α+Bbd)−Bbdα2p+2

p=0

³

A2p+1 (A−α+Bbd)−Bbdα2p+1

dαn (A−α)An−1

=

Tn−2

(A−α+Bbd)A2n−1

Denote by β =

Bbd A−α+Bbd

p=0

³

Tn−2 p=0

Bbd α 2p+2 1− A−α+Bbd (A )

³

´

Bbd α 2p+1 1− A−α+Bbd (A )

and by (Up )p the sequence defined as Up = x4n−3

´

´.

α 2p+2 1−β( A ) α 2p+1 , 1−β( A )

we get

n−2 α n Y ) (A − α) d( A ³ ´ = Up . α 2n−1 (A − α + Bbd) 1 − β( A ) p=0

We have either: for p ∈ N big enough, Up > 1 or for p ∈ N big enough, 0 < Up < 1. Using Taylor expansion of the Up , we obtain α α α α α Up = (1 − β( )2p+2 )(1 + β( )2p+1 + o( )2p+1 ) = 1 + β( )2p+1 + o( )2p+1 , A A A A A α 2p+1 ) which is the general term of a convergent infinite product. then Up is equivalent to 1 + β( A We can easily deduce that (x4n−3 )n converges toward zero. same discussion can be obtained for the other subsequences.

(2) If A − α + Bbd = A − α + Bac = 0, then by the proof of (1), the subsequences (x4n−3 )n and (x4n−1 )n are constants: x4n−3 = d and x4n−1 = b, also the subsequences x4n−2 = c and x4n = a. Thus every solution of Eq. (1) converges to a real number l if and only if a = b = c = d = l. (3) Consider for instance the case A − α + Bbd = 0 and A − α + Bac 6= 0, by (2), the subsequences (x4n−3 )n and (x4n−1 )n are constants x4n−3 = d and x4n−1 = b, in other hand and also by the proof of case (1), the subsequences (x4n−2 )n and (x4n )n converge to zero, then the sequence (xn )n diverges. The proof is completed. (ii) Case | A α | = 1.

Theorem 4. Assume that | A α | = 1. We distinguish two subcases, A = α and A = −α.

(1) If A = α, and let sequence (xn )n be the sequence given by the formula (1), then the sequence (xn )n converges toward zero.

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(2) If A = −α, and let sequence (xn )n be the sequence given by the formula (1), then we have x4n−1 = b c dx4n−3 , x4n−2 = ax4n and the sequence (xn )n is divergent. Proof. (1) For A = α, let δ the parameter δ = x4n−3 =

A Bbd .

In the proof of Corollary 2, we find that

A Bb(δ+1)

n−1 Y³ p=1

Denote by (Wp )p the sequence defined as Wp =

δ 2p +1 δ+1 2p +1

δ ´ 2p + 1 . δ+1 2p + 1

, then we get:

For p big enough, we have 0 < Wp < 1. The Taylor expansion for Wp gives: δ 1 1 1 )(1 − δ+1 Wp = (1 + 2p 2p + o( p )) = 1 − 2p + o( p ), which is a general term of divergent infinite product. Since for p big enough, 0 < Wp < 1, then limn→∞ Πn−1 p=1 Wp = 0. So, we get limn→∞ x4n−3 = 0. Similarly, one can easily prove that the other subsequences converge to zero, therefore the sequence (xn )n converges to zero. (2) To prove the second part, we replace α by (−A) in the expression of x4n−3 of Eq. (6), we obtain ³ ´ Tn−2

d(−A)n

x4n−3

=

p=0

Tn−1 p=0

A2p+1 A+Bbd

³

A2p A+Bbd

S2p

S2p+1 k=0

k k=0 (−1)

(−1)k

´

=

d , (−1−δ−1 )n

.

In other hand, If we replace α by (−A) in the first term of Eq. (7), we obtain ³ 2p ´ Q A (A+Bbd) x4n−1 = b(−A)n n−1 = b(−1 − δ −1 )n . p=0 A2p+2

Thus x4n−1 =

b dx4n−3 ,

hence

(a) If |1 + δ −1 | > 1, then the subsequence (x4n−3 )n converges to zero, so (|x4n−1 |)n goes to infinity.

(b) If |1 + δ −1 | < 1, then the subsequence (|x4n−3 |)n goes to infinity. This completed the proof.

(iii) Case | A α | < 1. Theorem 5. Let (xn )n be the sequence given by the formula (1), then For | A α | < 1, then the subsequences (x4n−3 )n , (x4n−1 )n , (x4n−2 )n and (x4n )n converge. Proof. We need to prove that (x4n−3 )n converges. Using Corollary (1), we obtain ³ ´ dαn (A−α)

x4n−3

=

Tn−2

Tn−1 p=0

p=0

³

A2p+2 (A−α+Bbd)−Bbdα2p+2

A2p+1 (A−α+Bbd)−Bbdα2p+1

´

=

α−A

Bb(1

− γλ2n−1 )

Qn−2 p=0

Vp ,

2p+2

1−γλ , λ= A where γ = A−α+Bbd Bbd α and (Vp )p is the sequence defined by Vp = 1−γλ2p+1 . For p ∈ N big enough, we have two cases; either Vp > 1 or 0 < Vp < 1. Applying the transformation of infinite product of positive terms to infinite series, and assuming p0 to be big enough, we get

x4n−3 =

It is clear that the sequence order gives

³

p0 ´ ´ ³ n−2 ³Y X α−A V ln(V ) . exp p p Bb(1 − γλ2n−1 ) p=0 p=p0 +1

α−A Bb(1−γλ2n−1 )

Vp =

´

n

1−γλ2p+2 1−γλ2p+1

converges toward

α−A Bb .

The Taylor expansion of Vp to the first

= 1 + γ(1 − λ)λ2p+1 + o(λ2p+1 ).

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So ln(Vp ) is equivalent to γ(1 − λ)λ2p+1 , which is the general term of a convergent infinite series, then the sequence (x4n−3 )n is convergent. Similarly, one can prove that the other subsequences are convergent. Remark 2. (Commentary on the convergence of (xn )n in the case | A α | < 1). Suppose that | A | < 1, according to Theorem 5, the subsequences (x4n−3 )n , (x4n−1 )n , (x4n−2 )n and (x4n )n α converge, denote by: l3 , l2 , l1 and l0 their limits respectively. The subsequences (x4n−3 )n and (x4n−1 )n are related by the equations: x4(n+1)−3 = x4(n+1)−1 =

αx4n−3 A+Bx4n−1 x4n−3 ,

(10)

αx4n−1 A+Bx4(n+1)−3 x4n−1 .

(11) ⎧ ⎪ ⎨l3 = 0, αl3 Passing to the limit as n goes to infinity in Eq. (10), we obtain l3 = A+Bl3 l1 , then (S1 ) : or ⎪ ⎩ l3 6= 0 and l1 = α−A Bl3 . ⎧ ⎪ ⎨l1 = 0, αl1 Passing to the limit as n goes to infinity in Eq. (11), we obtain l1 = A+Bl , then (S ) : or 2 3 l1 ⎪ ⎩ l1 6= 0 and l3 = α−A Bl1 . Combining systems (S1 ) and (S2 ), since α 6= A, we obtain ⎧ ⎪ l3 = l1 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨or ⎧ ⎪ ⎨l3 = ⎪ ⎪ ⎪ l1 6= 0, l3 6= 0 and (S) : and ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ l1 =

α−A Bl1 , α−A Bl3 .

Q The proposition l3 = l1 = 0 contradicts the fact that the infinite product p≥0 Vp converges, in fact if Qn Pn limn→∞ p=0 Vp = 0, then limn→∞ p=p0 ln(Vp ) = −∞, and this is absurd. Hence the only possibility is that ⎧ α−A ⎪ ⎨l3 = Bl1 , l1 6= 0, l3 6= 0 and (S) : and ⎪ ⎩ l1 = α−A Bl3 . ∗ One can easily see that (S) is equivalent to l3 = α−A Bl1 . Let f be the function defined on R as f (x) = have f of = Id and, l1 and l3 are related by f (l1 ) = l3 . r α−A α−A =x⇔x=∓ . f (x) = x ⇔ Bx B

α−A Bx ,

we

Hence: f has fixed points if and only if α−A B > 0. The numerical example (Figure 4) given in the end of this paper confirm that even we chose α−A > 0 and B |A | < 1, l and l may be different, which implies the sequence (x ) may converge or diverge. 1 3 n n α Finally based on the preview discussion of all preview cases, The following Theorem is now proved. Theorem 6. (Boundedness of (xn )n ). The Eq. (1) has an unbounded solutions if and only if A = −α.

5. PERIODICITY CHARACTER OF SOLUTIONS OF EQ. (1) In the sequel, we need the following lemma, which describes sufficient conditions for Eq. (1) to have a periodic solution. Lemma 1. Let (xn )n≥−3 be a solution of Eq. (1) and the initial data that follow. Suppose that there are real numbers l3 , l2 , l1 , l0 such that limn→∞ x4n−j = lj for j = 0, ..., 3.

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Let (yn )n≥−3 be the period-4 sequence such that y−j = lj , for all j = 0, ..., 3, then the sequence (yn )n≥−3 is a period-4 solution of Eq. (1). The periodicity results are given by the following Theorem Theorem 7. Let (xn )n≥−3 be a solution of Eq. (1) and the initial data that follow, then (1) For | A α | > 1,

(a) If (A − α + Bbd)(A − α + Bac) 6= 0, then then Eq. (1) has no periodic solutions.

(b) If A − α + Bbd = A − α + Bac = 0, then the solution of Eq. (1) is a periodic-4 solution.

(c) If either A − α + Bbd or A − α + Bac equals zero but not both of them, then Eq. (1) has a periodic-4 solution. (2) For | A α | = 1, Eq. (1) has no periodic solutions. (3) For | A α | < 1, Eq. (1) has periodic-4 solutions.

Proof. (1) Suppose that | A α | > 1,

(a) If (A − α + Bbd)(A − α + Bac) 6= 0, then by Theorem 3, every solution of Eq. (1) converges to zero, hence, the solutions are not allowed to be periodic (since the solutions are not identically zero). (b) If A − α + Bbd = A − α + Bac = 0, then by Theorem 3, the subsequences of (xn )n (x4n−j )n , j = 0, .., 3 are constants: x4n−3 = d, x4n−2 = c, x4n−1 = b and x4n = a, and the sequence d, c, b, a, d, c, b, a... is a periodic-4 solution of Eq. (1). (c) Consider for instance the case A − α + Bbd = 0 and A − α + Bac 6= 0, by the proof of Theorem 3, the subsequences (x4n−3 )n and (x4n−1 )n are constants and equal d and b respectively. Also according to the proof of Theorem 3, the subsequences (x4n−2 )n and (x4n )n converge to zero. Applying Lemma 1, the sequence d, 0, b, 0, d, 0, b, 0, ... is a periodic-4 solution of Eq. (1). (2) The case A = α is similar to (1) (a). If A = −α, then every solution of Eq. (1) is unbounded, so Eq. (1) has no periodic solutions.

(3) If | A α | < 1, then by Theorem 5, there are real numbers l3 , l2 , l1 and l0 , such that limn→∞ x4n−j = lj for all j = 0, .., 3. Applying Lemma 1, the sequence l3 , l2 , l1 , l0 , l3 , l2 , l1 , l0 ... is a periodic-4 solution of Eq. (1). This completes the proof. Remark 3. (1) Note that if | A α | > 1, A − α + Bbd = A − α + Bac = 0, a = c, b = d, then Eq. (1) has periodic-2 solution a, b, a, b, .... (2) If | A α | < 1, A − α + Bbd = A − α + Bac = 0, then, by the proof of Theorem 7, we deduce that the values of the limits of the subsequences are l3 = d, l2 = c, l1 = b and l0 = a.

6. NUMERICAL SIMULATION Example 1. Figure (1) illustrates the case | A α | > 1, (A − α + Bbd)(A − α + Bac) 6= 0, we choose a = 2, b = −3, c = 2, d = −2, B = 2, A = 1.1 and α = 1. We notice that the solution is oscillating about zero with a decreasing amplitude. In fact, according to Theorem 3, the solution has to converge to zero. Example 2. In order illustrate the case | A α | > 1, A − α + Bbd = A − α + Bac = 0, we choose a = c = 2, b = d = −2, B = −3, A = 13 and α = 1. Figure (2) depicts that the obtained solution is a 2-prime periodic solution. This is coherent with Remark 3. Example 3. The case | A α | > 1, A − α + Bbd = 0 and A − α + Bac 6= 0 is illustrated in figure (3), in which we set a = c = 1, b = d = −2, B = −2, A = 9 and α = 1. The subsequences (x4n−3 )n and (x4n−1 )n are constants (x4n−3 )n = d and (x4n−1 )n = b, and the subsequences (x4n−2 )n and (x4n )n converge to zero. by Lemma 1, the sequence d, 0, b, 0, d, 0, b, 0, ... is a periodic-4 solution of Eq. (1). Example 4. Figure (4) illustrates the case | A α | < 1, we choose a = −1, b = 0.5, c = −0.2, d = 0.8, B = 1, A = 0.5 and α = 1. the subsequences (x4n−3 )n , (x4n−1 )n , (x4n−2 )n and (x4n )n converge.

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Example 5. To illustrate the case A = α, we choose a = 0.1, b = 0.2, c = 0.3, d = −0.4, B = 1, α = 0.5 and A = 0.5. We notice in the figure (5), that the solution converges to zero (which is coherent to Theorem 4 part (1)), and the Eq. (1) has no periodic solutions (which is coherent to Theorem 7 part (2)). Example 6. In figure (6) (case A = −α), we choose a = 0.2, b = 0.3, c = 0.1, d = −0.3, B = 2, α = −0.4 and A = 0.4. We notice that the solution is oscillating about zero with an increasing amplitude and the solution is unbounded, which is coherent to Theorem 4 part (2).

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Conclusion αxn−3 In this work, some dynamical behaviors of the rational difference equation xn+1 = A+Bx with the n−1 xn−3 initial conditions, x−3 = d, x−2 = c, x−1 = b, and x0 = a are arbitrary real numbers, A and B are arbitrary constants, have been investigated. A detailed analytical study of the convergence of the solutions including their dependence on parameters and initial conditions has been illustrated. The local stability and global attractivity of the difference equation’s equilibrium points have been demonstrated. The existence of periodic solutions in the proposed difference equation has also been shown analytically. Finally, numerical simulations have been carried out to match the analytical results.

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Figure 5.

Figure 6.

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.

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bxn cxn −dxn−1 ,

[10] H. El-Metwally and E. M. Elsayed, Qualitative Behavior of some Rational Difference Equations, Journal of Computational Analysis and Applications, 20 (2) (2016), 226-236. [11] M. A. El-Moneam, On the dynamics of the solutions of the rational recursive sequences, Brit. J. Math. Comput. Sci., 5 (5) (2015), 654—665. [12] A. A. Elsadany, Competition analysis of a triopoly game with bounded rationality, Chaos Solitons and Fractals, 45 (11) (2012), 1343-1348.

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axn−k , b+cxp n

Fasciculi Mathematici,

bxn [36] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence xn+1 = axn − cxn −dx , Comm. n−k Appl. Nonlin. Anal., 15 (2), (2008), 47-57.

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QUADRATIC ρ-FUNCTIONAL EQUATIONS JUNG RYE LEE1 , CHOONKIL PARK2∗ , AND DONG YUN SHIN3∗ Abstract. In this paper, we solve the quadratic ρ-functional equations f (x + y) + f (x − y) − 2f (x) − 2f (y) (0.1)       x−y x+y + 2f − f (x) − f (y) , = ρ 2f 2 2 where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ 6= −1, 2, and     x+y x−y 2f + 2f − f (x) − f (y) (0.2) 2 2 = ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y)), whereρ is a fixed non-Archimedean number or a fixed real or complex number with ρ 6= −1, 12 . Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional equations (0.1) and (0.2) in non-Archimedean Banach spaces and in Banach spaces.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [25] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [13] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [22] for linear mappings by considering an unbounded Cauchy difference. Gajda [11] following the same approach as in Rassias [22], gave an affirmative solution to this question for p > 1. It was shown by Gajda ˇ [11], as well as by Rassias and Semrl [21] that one cannot prove a Rassias’ type theorem when ˇ p = 1. The counterexamples of Gajda [11], as well as of Rassias and Semrl [21] have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings, cf. G˘avruta [12], who among others studied the Hyers-Ulam stability of functional equations (cf. the books of Czerwik [8, 9], Hyers, Isac and Th.M. Rassias [14]). The hyperstability of the Cauchy equation was proved by Brzdek [4]. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [24] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. See [1, 5, 6, 10, 16, 17, 18, 19, 20, 23] for more 2010 Mathematics Subject Classification. Primary 46S10, 39B62, 39B52, 47S10, 12J25. Key words and phrases. Hyers-Ulam stability; non-Archimedean normed space; quadratic ρ-functional equation. ∗ Corresponding authors.

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functional equations. The survey on the Hyers-Ulam stability of functional equations was given by Brillouet-Bulluot, Brzdek and Cieplinski [3]. The functional equation     x+y x−y 2f +2 = f (x) + f (y) 2 2 is called a Jensen type quadratic equation. A valuation is a function | · | from a field K into [0, ∞) such that 0 is the unique element having the 0 valuation, |rs| = |r| · |s| and the triangle inequality holds, i.e., |r + s| ≤ |r| + |s|,

∀r, s ∈ K.

A field K is called a valued field if K carries a valuation. The usual absolute values of R and C are examples of valuations. Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|},

∀r, s ∈ K,

then the function | · | is called a non-Archimedean valuation, and the field is called a nonArchimedean field. Clearly |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N. A trivial example of a non-Archimedean valuation is the function | · | taking everything except for 0 into 1 and |0| = 0. Throughout this paper, we assume that the base field is a non-Archimedean field, hence call it simply a field. Definition 1.1. ([15]) Let X be a vector space over a field K with a non-Archimedean valuation | · |. A function k · k : X → [0, ∞) is said to be a non-Archimedean norm if it satisfies the following conditions: (i) kxk = 0 if and only if x = 0; (ii) krxk = |r|kxk (r ∈ K, x ∈ X); (iii) the strong triangle inequality kx + yk ≤ max{kxk, kyk},

∀x, y ∈ X

holds. Then (X, k · k) is called a non-Archimedean normed space. In Section 2, we solve the quadratic functional equation (0.1) in vector spaces and prove the Hyers-Ulam stability of the quadratic functional equation (0.1) in non-Archimedean Banach spaces. In Section 3, we solve the quadratic functional equation (0.2) in vector spaces and prove the Hyers-Ulam stability of the quadratic functional equation (0.2) in non-Archimedean Banach spaces. In Section 4, we prove the Hyers-Ulam stability of the quadratic functional equation (0.1) in Banach spaces. In Section 5, we prove the Hyers-Ulam stability of the quadratic functional equation (0.2) in Banach spaces. 2. Quadratic ρ-functional equation (0.1) in non-Archimedean Banach spaces Throughout Sections 2 and 3, assume that X is a non-Archimedean normed space and that Y is a non-Archimedean Banach space. Let |2| = 6 1 and let ρ be a fixed non-Archimedean number with ρ 6= −1, 2. Lemma 2.1. Let X and Y be vector spaces. A mapping f : X → Y satisfies f (x + y) + f (x − y) − 2f (x) − 2f (y) = 0

509

(2.1)

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for all x, y ∈ X if and only if the mapping f : X → Y satisfies 

2f

x+y 2





+ 2f

x−y 2



− f (x) − f (y) = 0

(2.2)

for all x, y ∈ X. Proof. Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get f (0) = 0. Letting y = x in (2.1), we get f (2x) − 4f (x) = 0 and so f (2x) = 4f (x) for all x ∈ X. Thus
f x2 = 14 f (x) for all x ∈ X. So f : X → Y satisfies (2.2). Assume that f : X → Y satisfies (2.2). Letting x = y = 0 in (2.2), we get f (0)
= 0. Letting y = 0 in (2.2), we get 4f x2 = f (x) for all x ∈ X. and so f (2x) = 4f (x) for all x ∈ X. So f : X → Y satisfies (2.1).  We solve the quadratic ρ-functional equation (0.1) in vector spaces. Lemma 2.2. Let X and Y be vector spaces. If a mapping f : X → Y satisfies f (x + y) + f (x − y) − 2f (x) − 2f (y)       x−y x+y + 2f − f (x) − f (y) = ρ 2f 2 2 for all x, y ∈ X, then f : X → Y is quadratic.

(2.3)

Proof. Assume that f : X → Y satisfies (2.3). Letting x = y = 0 in (2.3), we get −2f (0) = 2ρf (0). So f (0) = 0. Letting y = x in (2.3), we get f (2x) − 4f (x) = 0 and so f (2x) = 4f (x) for all x ∈ X. Thus x 2

 

f

1 = f (x) 4

(2.4)

for all x ∈ X. It follows from (2.3) and (2.4) that f (x + y) + f (x − y) − 2f (x) − 2f (y)       x+y x−y = ρ 2f + 2f − f (x) − f (y) 2 2 ρ = (f (x + y) + f (x − y) − 2f (x) − 2f (y)) 2 and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X.



We prove the Hyers-Ulam stability of the quadratic ρ-functional equation (2.3) in nonArchimedean Banach spaces. Theorem 2.3. Let ϕ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0 and lim |4|j ϕ

j→∞



x y , 2j 2j

510



= 0,

(2.5)

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kf (x + y) + f (x − y) − 2f (x) − 2f (y)       x+y x−y −ρ 2f + 2f − f (x) − f (y) k ≤ ϕ(x, y) 2 2 for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y such that    x x j−1 kf (x) − h(x)k ≤ sup |4| ϕ j , j 2 2 j∈N

(2.6)

(2.7)

for all x ∈ X. Proof. Letting y = x in (2.6), we get kf (2x) − 4f (x)k ≤ ϕ(x, x)

(2.8)

for all x ∈ X. So

   

f (x) − 4f x ≤ ϕ x , x

2 2 2

for all x ∈ X. Hence

   

l x

4 f x − 4m f (2.9)

2l 2m

         

l

x x x x l+1 m

, · · · , 4m−1 f

4 f ≤ max − 4 f − 4 f

2l 2l+1 2m−1 2m

 

       

x x x x

, · · · , |4|m−1 f

≤ max |4|l f − 4f − 4f

2l 2l+1 2m−1 2m    x x ≤ sup |4|j ϕ j+1 , j+1 2 2 j∈{l,l+1,··· } for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.9) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping h : X → Y by x h(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.9), we get (2.7). It follows from (2.5) and (2.6) that kh(x + y) + h(x − y) − 2h(x) − 2h(y)       x+y x−y −ρ 2h + 2h − h(x) − h(y) k 2 2

       

x+y x−y x y n = lim |4| f +f − 2f − 2f n n n n→∞ 2 2 2 2n         

x+y x−y x y

−ρ 2f + 2f −f −f

n+1 n+1 n n 2 2 2 2   x y ≤ lim |4|n ϕ n , n = 0 n→∞ 2 2 for all x, y ∈ X. So       x+y x−y h(x + y) + h(x − y) − 2h(x) − 2h(y) = ρ 2h + 2h − h(x) − h(y) 2 2 for all x, y ∈ X. By Lemma 2.2, the mapping h : X → Y is quadratic.

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Now, let T : X → Y be another quadratic mapping satisfying (2.7). Then we have

q kh(x) − T (x)k =

4 h

x 2q



x 2q



− 4q T



x

2q 



x

, 4q T

q 2    x x ≤ sup |4|q+j−1 ϕ q+j , q+j , 2 2 j∈N 

q ≤ max

4 h





− 4q f





x 2q



− 4q f



x

2q

 

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that h(x) = T (x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping h : X → Y is a unique quadratic mapping satisfying (2.7).  Corollary 2.4. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that kf (x + y) + f (x − y) − 2f (x) − 2f (y)       x−y x+y + 2f − f (x) − f (y) k ≤ θ(kxkr + kykr ) −ρ 2f 2 2

(2.10)

for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y such that kf (x) − h(x)k ≤

2θ kxkr |2|r

for all x ∈ X. Theorem 2.5. Let ϕ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (2.6) and 

lim

j→∞

1 ϕ(2j−1 x, 2j−1 y) = 0 |4|j 

for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y such that 1 kf (x) − h(x)k ≤ sup ϕ(2j−1 x, 2j−1 x) j j∈N |4| 



(2.11)

for all x ∈ X. Proof. It follows from (2.8) that



f (x) − 1 f (2x) ≤ 1 ϕ(x, x)

4 |4|

for all x ∈ X. Hence

1

f (2l x) − 1 f (2m x) (2.12)

4l

m 4

     

1  l 

1 l+1

, · · · , 1 f 2m−1 x − 1 f (2m x) ≤ max f 2 x − f 2 x

4l

l+1 m−1 m 4 4 4

 



    

1 1

f 2l x − 1 f 2l+1 x , · · · ,

f 2m−1 x − 1 f (2m x) ≤ max



l m−1 |4| 4 |4| 4   1 ≤ sup ϕ(2j x, 2j x) j+1 |4| j∈{l,l+1,··· }

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for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.12) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping h : X → Y by h(x) := lim

n→∞

1 f (2n x) 4n

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.12), we get (2.11). The rest of the proof is similar to the proof of Theorem 2.3.  Corollary 2.6. Let r > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (2.10). Then there exists a unique quadratic mapping h : X → Y such that kf (x) − h(x)k ≤

2θ kxkr |4|

for all x ∈ X. 3. Quadratic ρ-functional equation (0.2) in non-Archimedean Banach spaces Let |2| 6= 1 and let ρ be a fixed non-Archimedean number with ρ 6= −1, 21 . We solve the quadratic ρ-functional equation (0.2) in vector spaces. Lemma 3.1. Let X and Y be vector spaces. If a mapping f : X → Y satisfies 

2f

x−y x+y + 2f − f (x) − f (y) 2 2 = ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y)) 





(3.1)

for all x, y ∈ X, then f : X → Y is quadratic. Proof. Assume that f : X → Y satisfies (3.1). Letting x = y = 0 in (3.1), we get 2f (0) = −2ρf (0). So f (0) = 0. Letting y = 0 in (3.1), we get x 2

 

4f

− f (x) = 0

(3.2)

and so f x2 = 14 f (x) for all x ∈ X. It follows from (3.1) and (3.2) that


1 (f (x + y) + f (x − y) − 2f (x) − 2f (y)) 2     x+y x−y = 2f + 2f − f (x) − f (y) 2 2 = ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y)) and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X.



We prove the Hyers-Ulam stability of the quadratic ρ-functional equation (3.1) in nonArchimedean Banach spaces.

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Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0 and    x y j lim |4| ϕ j , j = 0, j→∞ 2 2     x−y x+y + 2f − f (x) − f (y) (3.3) k2f 2 2 −ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) k ≤ ϕ(x, y) for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y such that    x j−1 kf (x) − h(x)k ≤ sup |4| ϕ j−1 , 0 2 j∈N

(3.4)

for all x ∈ X. Proof. Letting y = 0 in (3.3), we get

 



4f x − f (x) ≤ ϕ(x, 0)

2

(3.5)

for all x ∈ X. So

   

l x

4 f x − 4m f

(3.6)

2l 2m

         

l

x x x x l+1 m

, · · · , 4m−1 f

≤ max 4 f − 4 f − 4 f

2l 2l+1 2m−1 2m

 

       

x x x x

, · · · , |4|m−1 f

≤ max |4|l f − 4f − 4f

2l 2l+1 2m−1 2m    x ≤ sup |4|j ϕ j , 0 2 j∈{l,l+1,··· } for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.6) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping h : X → Y by x h(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.6), we get (3.4). The rest of the proof is similar to the proof of Theorem 2.3.  Corollary 3.3. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that     x+y x−y + 2f − f (x) − f (y) (3.7) k2f 2 2 −ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k ≤ θ(kxkr + kykr ) for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y such that kf (x) − h(x)k ≤ θkxkr for all x ∈ X. Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (3.3) and   1 j j lim ϕ(2 x, 2 y) =0 j→∞ |4|j

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for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y such that   1 j kf (x) − h(x)k ≤ sup ϕ(2 x, 0) j j∈N |4|

(3.8)

for all x ∈ X. Proof. It follows from (3.5) that



f (x) − 1 f (2x) ≤ 1 ϕ(2x, 0)

4 |4|

for all x ∈ X. Hence



1

f (2l x) − 1 f (2m x) (3.9)

4l 4m

     

1  l 

1 l+1

, · · · , 1 f 2m−1 x − 1 f (2m x) ≤ max f 2 x − f 2 x

4l

l+1 m−1 m 4 4 4

 



    

1 1

f 2l x − 1 f 2l+1 x , · · · ,

f 2m−1 x − 1 f (2m x) ≤ max



l m−1 |4| 4 |4| 4   1 ϕ(2j x, 0) ≤ sup j j∈{l+1,l+2,··· } |4| for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.9) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping h : X → Y by 1 f (2n x) 4n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.9), we get (3.8). The rest of the proof is similar to the proof of Theorems 2.3.  h(x) := lim

n→∞

Corollary 3.5. Let r > 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying (3.7). Then there exists a unique quadratic mapping h : X → Y such that kf (x) − h(x)k ≤

|2|r θ kxkr |4|

for all x ∈ X. 4. Quadratic ρ-functional equation (0.1) in Banach spaces Throughout Sections 4 and 5, assume that X is a normed space and that Y is a Banach space. Let ρ be a fixed real or complex number with ρ 6= −1, 2. We prove the Hyers-Ulam stability of the quadratic ρ-functional equation (2.3) in Banach spaces. Theorem 4.1. Let ϕ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0 and ∞ X

x y Ψ(x, y) := 4 ϕ j, j 2 2 j=1 j





< ∞,

(4.1)

kf (x + y) + f (x − y) − 2f (x) − 2f (y)       x+y x−y −ρ 2f + 2f − f (x) − f (y) k ≤ ϕ(x, y) 2 2

515

(4.2)

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for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that 1 kf (x) − Q(x)k ≤ Ψ(x, x) 4

(4.3)

for all x ∈ X. Proof. Letting y = x in (4.2), we get kf (2x) − 4f (x)k ≤ ϕ(x, x)

(4.4)

for all x ∈ X. So

   

f (x) − 4f x ≤ ϕ x , y

2 2 2

for all x ∈ X. Hence

   

l x

4 f x − 4m f

≤

2l 2m

≤

m−1 X j=l m−1 X

4j f

j





4 ϕ

j=l

x 2j x



− 4j+1 f x



x



2j+1



, 2j+1 2j+1

(4.5)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (4.5) that the sequence {4k f ( 2xk )} is Cauchy for all x ∈ X. Since Y is a Banach space, the sequence {4k f ( 2xk )} converges. So one can define the mapping Q : X → Y by k

Q(x) := lim 4 f k→∞



x 2k



for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (4.5), we get (4.3). Now, let T : X → Y be another quadratic mapping satisfying (4.3). Then we have

   

q x x q

kQ(x) − T (x)k = 4 Q q − 4 T 2 2q

       

q

q x x x x q q



≤ 4 Q q − 4 f − 4 f + 4 T

2 2q 2q 2q   q

≤

4 x x Ψ q, q 2 2 2

,

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. It follows from (4.1) and (4.2) that kQ(x + y) + Q(x − y) − 2Q(x) − 2Q(y)       x+y x−y −ρ 2Q + 2Q − Q(x) − Q(y) k 2 2

       

x+y x−y x y + f − 2f − 2f = lim 4n f

n→∞ 2n 2n 2n 2n         

x+y x−y x y

−ρ 2f + 2f − f − f 2n+1 2n+1 2n 2n   x y ≤ lim 4n ϕ n , n = 0 n→∞ 2 2

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for all x, y ∈ X. So 



Q(x + y) + Q(x − y) − 2Q(x) − 2Q(y) = ρ 2Q

x+y 2





+ 2Q

x−y 2





− Q(x) − Q(y)

for all x, y ∈ X. By Lemma 2.2, the mapping Q : X → Y is quadratic.



Corollary 4.2. Let r > 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that kf (x + y) + f (x − y) − 2f (x) − 2f (y)       x−y x+y + 2f − f (x) − f (y) k ≤ θ(kxkr + kykr ) −ρ 2f 2 2

(4.6)

for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

2θ kxkr −4

2r

for all x ∈ X. Theorem 4.3. Let ϕ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (4.2) and Ψ(x, y) :=

∞ X 1

4j j=0

ϕ(2j x, 2j y) < ∞

for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that 1 kf (x) − Q(x)k ≤ Ψ(x, x) 4

(4.7)

for all x ∈ X. Proof. It follows from (4.4) that



f (x) − 1 f (2x) ≤ 1 ϕ(x, x)

4 4

for all x ∈ X. Hence

1

f (2l x) − 1 f (2m x) ≤

4l

m 4

≤

m−1 X

j=l

   

1 f 2j x − 1 f 2j+1 x

4j

j+1 4

m−1 X j=l

1 ϕ(2j x, 2j x) 4j+1

(4.8)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (4.8) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping Q : X → Y by 1 f (2n x) n→∞ 4n

Q(x) := lim

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (4.8), we get (4.7). The rest of the proof is similar to the proof of Theorem 4.1. 

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Corollary 4.4. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying (4.6). Then there exists a unique quadratic mapping Q : X → Y such that 2θ kf (x) − Q(x)k ≤ kxkr 4 − 2r for all x ∈ X. 5. Quadratic ρ-functional equation (0.2) in Banach spaces Let ρ be a fixed real or complex number with ρ 6= −1, 12 . In this section, we prove the Hyers-Ulam stability of the quadratic ρ-functional equation (3.1) in Banach spaces. Theorem 5.1. Let ϕ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0 and   ∞ X x y j Ψ(x, y) := 4 ϕ j , j < ∞, 2 2 j=0 x+y x−y + 2f − f (x) − f (y) 2 2 −ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) k ≤ ϕ(x, y) 







k2f

(5.1)

for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤ Ψ(x, 0)

(5.2)

for all x ∈ X. Proof. Letting y = 0 in (5.1), we get

   



f (x) − 4f x = 4f x − f (x) ≤ ϕ(x, 0)



2 2 for all x ∈ X. So

   

l x

4 f x − 4m f

≤

2l 2m

≤

m−1 X

j=l m−1 X j=l

4j f



x 2j



−4

x 4 ϕ j,0 2 j

j+1

(5.3)



f

x



j+1 2





(5.4)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (5.4) that the sequence {4k f ( 2xk )} is Cauchy for all x ∈ X. Since Y is a Banach space, the sequence {4k f ( 2xk )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4 f k→∞ 2k for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (5.4), we get (5.2). The rest of the proof is similar to the proof of Theorem 4.1.  k





Corollary 5.2. Let r > 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and     x+y x−y + 2f − f (x) − f (y) (5.5) k2f 2 2 −ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k ≤ θ(kxkr + kykr )

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for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that 2r θ kf (x) − Q(x)k ≤ r kxkr 2 −4 for all x ∈ X. Theorem 5.3. Let ϕ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (5.1) and Ψ(x, y) :=

∞ X 1

4j j=1

ϕ(2j x, 2j y) < ∞

for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤ Ψ(x, 0)

(5.6)

for all x ∈ X. Proof. It follows from (5.3) that



f (x) − 1 f (2x) ≤ 1 ϕ(2x, 0)

4 4

for all x ∈ X. Hence

1

f (2l x) − 1 f (2m x) ≤

4l

m 4 ≤

m   X

1  j 

f 2 x − 1 f 2j+1 x

4j

j+1 4 j=l+1 m X

1 ϕ(2j x, 0) j 4 j=l+1

(5.7)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (5.7) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping Q : X → Y by 1 f (2n x) 4n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (5.7), we get (5.6). The rest of the proof is similar to the proof of Theorem 4.1.  Q(x) := lim

n→∞

Corollary 5.4. Let r < 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (5.5). Then there exists a unique quadartic mapping Q : X → Y such that 2r θ kf (x) − Q(x)k ≤ kxkr 4 − 2r for all x ∈ X. References [1] S. Alizadeh and F. Moradlou, Approximate a quadratic mapping in multi-Banach spaces, a fixed point approach, Int. J. Nonlinear Anal. Appl. 7 (2016), no. 1, 63–75. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] N. Brillouet-Belluot, J. Brzdek and K. Cieplinski, On some recent developments in Ulam’s type stability, Abs. Appl. Anal. 2012, Art. ID 716935 (2012). [4] J. Brzdek, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar. 141 (2013), 58–67.

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[5] L. S. Chadli, S. Melliani, A. Moujahid and M. Elomari, Generalized solution of sine-Gordon equation, Int. J. Nonlinear Anal. Appl. 7 (2016), no. 1, 87–92. [6] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [7] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [8] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong, 2002. [9] S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Florida, 2003. [10] M. Eshaghi Gordji, A. Rahimi, C. Park and D. Shin, Ternary Jordan bi-homomorphisms in Banach Lie triple systems, J. Comput. Anal. Appl. 21 (2016), 1040–1045. [11] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [12] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [13] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [14] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [15] M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.–TMA 69 (2008), 3405–3408. [16] E. Movahednia, S. M. S. M. Mosadegh, C. Park and D. Shin, Stability of a lattice preserving functional equation on Riesz space: fixed point alternative, J. Comput. Anal. Appl. 21 (2016), 83–89. [17] C. Park, Stability of ternary quadratic derivation on ternary Banach algebras: revisited, J. Comput. Anal. Appl. 20 (2016), 21–23. [18] C. Park, S. Kim, J. Lee and D. Shin, Quadratic ρ-functional inequalities in β-homogeneous normed spaces, Int. J. Nonlinear Anal. Appl. 6 (2015), no. 2, 21–26. [19] C. Park and A. Najati, Generalized additive functional inequalities in Banach algebras, Int. J. Nonlinear Anal. Appl. 1 (2010), no. 2, 54–62. [20] W. Park and J. Bae, Approximate quadratic forms on restricted domains, J. Comput. Anal. Appl. 20 (2016), 388–410. ˇ [21] Th.M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325–338. [22] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [23] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [24] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [25] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. 1

Daejin University Kyunggi 11159 Republic of Korea E-mail address: [email protected] 2

Research Institute for Natural Sciences Hanyang University Seoul 04763 Republic of Korea E-mail address: [email protected] 3

University of Seoul Seoul 02504 Republic of Korea E-mail address: [email protected]

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ON MODIFIED DEGENERATE GENOCCHI POLYNOMIALS AND NUMBERS HYUCK IN KWON, LEE-CHAE JANG, DAE SAN KIM, JONG-JIN SEO

Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail : [email protected] Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea E-mail : [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail : [email protected] Department of Applied Mathematics, Pukyung National University, Busan 608-737, Republic of Korea E-mail : [email protected]

Abstract. In this paper, we consider the modified partially degenerate Genocchi polynomials and investigate some properties of these polynomials. From these properties, we give some new and interesting identities of them.

1. Introduction The Genocchi polynomials are defined by the generating function ∞ tn 2t xt X e = G (x) (see [2, 3, 7, 9, 12, 14, 17, 19, 27, 28]). n et + 1 n! n=0 When x = 0, Gn = Gn (0) are called the Genocchi numbers. From (1), we see that   ∞ X tn 2t Gn (x) = ext t+1 n! e n=0 ! ∞ ! ∞ m X X tl mt = Gl x l! m! m=0 l=0 !   ∞ n X X n tn = Gl xn−l . l n! n=0

(1)

(2)

l=0

1991 Mathematics Subject Classification. 05A10, 11B68, 11S80, 05A19. Key words and phrases. Euler polynomials, Genocchi polynomials, degenerate Genocchi polynomials, modified degenerate Genocchi polynomials. 1

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Thus, by comparing the coefficients on both sides of (2), we get n   X n Gn (x) = Gl xn−l . l

(3)

l=0

From (1), we can derive the following equation: ∞ X n=0

Gn (x)

−2t −(1−x)t tn = − −t e n! e +1 =

∞ X

n−1

(−1)

n=0

tn Gn (1 − x) . n!

(4)

By comparing the coefficients on both sides of (4), we get Gn (x) = (−1)n−1 Gn (1 − x).

(5)

The gamma and beta function are defined by the following definite integrals: for (α > 0, β > 0), Z ∞ e−t tα−1 dt (6) Γ(α) = 0

and 1

Z

tα−1 (1 − t)β−1 dt

B(α, β) = 0

Z = 0

∞

tα−1 dt (1 + t)α+β

(7)

(see [15,23,24]).Thus by (6) and (7), we get Γ(α + 1) = αΓ(α),

B(α, β) =

Γ(α)Γ(β) . Γ(α + β)

(8)

The classical Genocchi numbers, a sequence of integers introduced by Angelo Genocchi (1817-1889), have been studied in various context in such diverse areas of mathematics and physics as number theory, combinatorics, complex analysis, topology, and quantum physics. In recent years, Genocchi polynomials and numbers have received considerable attention and many researchers have worked on them, their extensions and their connections with some combinatorial counting. The degenerate Bernoulli polynomials, the rst degenerate version of well-known families of polynomials, were introduced by Carlitz and rediscovered by Ustinov under the name of Korobov polynomials of the second kind. On the other hand, Korobov polynomials (of the rst kind) are the degenerate version of the Bernoulli polynomials of the second kind. Recently, many researchers began to study various kinds of degenerate versions of the familiar polynomials like Bernoulli, Euler, Genocchi, falling factorial and Bell polynomials by using generating functions, umbral calculus, and p-adic integrals. The goal of this paper is to introduce the modified degenerate Genocchi polynomials and numbers, a degenerate version of the classical Genocchi polynomials and numbers, in order to study their properties and obtain several new and interesting identities involving them. More precisely, we give some properties, explicit formulas, several identities, a connection with Genocchi polynomials, and some integral formulas. Here they were named as the modified degenerate Genocchi polynomials, since there existed what are called the degenerate Genocchi polynomials whose definition is slightly different from ours (see [1, 4-6, 8, 11-16, 18, 20, 21, 22-26, 28]).

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2. Modified degenerate Genocchi polynomials First, we note that t log(1 + λ). λ→0 λ λ→0 λ→0 From (1) and (9), we define the modified degenerate Genocchi polynomials as ∞ X tx 2t tn λ = (1 + λ) gn,λ (x) t n! (1 + λ) λ + 1 n=0 t

et = lim (1 + λ) λ ,

t

t = loget = lim log(1 + λ) λ = lim

(9)

(10)

When x = 0, gn,λ = gn,λ (0) are called the modified degenerate Genocchi numbers. From (10), we get    t 2t 2t = (1 + λ) λ + 1 t (1 + λ) λ + 1 t 2t 2t (1 + λ) λ + = t t (1 + λ) λ + 1 (1 + λ) λ + 1 ∞ ∞ X tn X tn = gn,λ (1) + gn,λ n! n=0 n! n=0 ∞ n X t = (gn,λ (1) + gn,λ ) . (11) n! n=0 By comparing the coefficients on both sides of (11), we get  g0,λ = 0 gn,λ (1) + gn,λ = 2δ1,n ,

(12)

where δ1,n is the Kronecker delta. From (10), we note that ! ∞  ! ∞ ∞ m X X log(1 + λ) m X tm tn mt gm,λ x gn,λ (x) = n! m! λ m! m=0 m=0 n=0 !     ∞ n m X X n log(1 + λ) tn gn−m,λ = xm . m λ n! n=0 m=0

(13)

Thus, by comparing the coefficients on both sides of (13), we obtain the following theorem. Theorem 2.1. Let n ∈ N ∪ {0}. Then we have m  n   X n log(1 + λ) xm . gn,λ (x) = gn−m,λ m λ m=0 From (10), we derive the following equation: ∞ X −(1−x)t −2t tn (1 + λ) λ gn,λ (x) = − −t n! (1 + λ) λ + 1 n=0 ∞ X tn = (−1)n−1 gn,λ (1 − x) . n! n=0

(14)

(15)

By comparing the coefficients on both sides of (15), gn,λ (x) = (−1)n−1 gn,λ (1 − x)

523

(n ≥ 0).

(16)

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By (10), we see that d gn,λ (x) = gn−1,λ (x) dx



log(1 + λ) λ

 n

(n ≥ 1).

(17)

From (17), we get gn+1,λ (1) − gn+1,λ n+1

Z

1

d gn+1,λ (x) dx dx n + 1  Z0 1 log(1 + λ) = gn,λ (x) dx. (n ≥ 1). λ 0 =

(18)

By (18), we obtain the following theorem. Theorem 2.2. Let n ∈ N ∪ {0}. Then we have   Z 1 gn+1,λ (1) − gn+1,λ log(1 + λ) = gn,λ (x) dx. n+1 λ 0

(19)

We note that the Stirling numbers of the first kind are defined as n X (x)n = S1 (n, l)xl , (n ≥ 0), (see [1, 4 − 6, 8, 11 − 16, 18, 20, 21, 22 − 26, 28]).

(20)

l=0

where (x)n = x(x − 1) · · · (x − n + 1)(n ≥ 1), and (x)0 = 1. By (10), we see that tx 2t (1 + λ) λ t (1 + λ) λ + 1! !   ∞ ∞ X X tk 1 tx m λ = gk,λ k! m! λ m m=0 k=0 ! ∞ m  l ! m ∞ X XX tx tk λ = gk,λ S1 (m, l) k! λ m! k=0 l=0 ! m=0 ! ! ∞ ∞ ∞   m k X X X x lλ tl t S1 (m, l) l! = gk,λ k! λ m! l! l=0 m=l k=0 !   ∞ ∞ n   n m X XX n x lλ t gn−l,λ S1 (m, l) = l! l λ m! n! n=0

(21)

l=0 m=l

From (21), we obtain the following theorem. Theorem 2.3. Let n ∈ N ∪ {0}. Then we have n X ∞    x l λm X n gn,λ (x) = gn−l,λ S1 (m, l) l!. l λ m!

(22)

l=0 m=l

Let d be an odd integer. Then we see that d−1 X lt (−1)l (1 + λ) λ l=0  d   t 2t λ 1 − −(1 + λ) t 1 + (1 + λ) λ

2t =

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= = = = =

  dt 2t 1 + (1 + λ) λ t 1 + (1 + λ) λ dt 2t 2t λ t + t (1 + λ) 1 + (1 + λ) λ 1 + (1 + λ) λ ∞ ∞ X tn X tn gn,λ + gn,λ (d) n! n=1 n! n=1 ∞ n X t (gn,λ + gn,λ (d)) n! n=1  ∞ X gn+1,λ + gn+1,λ (d)  tn t . n+1 n! n=0

(23)

Also, we see that 2

d−1 X lt (−1)l (1 + λ) λ l=0

=

2

d−1 X ∞ X l=0

=

∞ X n=0

2

(−1)l

n=0 d−1 X

l

(−1)

l=0



log(1 + λ) λ

n



log(1 + λ) λ

n

!

tn n!

!

tn . n!

ln l

n

(24)

From (23) and (24), we obtain the following theorem. Theorem 2.4. Let n ∈ N ∪ {0}. Then we have  n d−1 X log(1 + λ) g+1n,λ + gn+1,λ (d) (−1)l ln = . 2 λ n+1

(25)

l=0

From (10) and (14), we note that  m Z 1 Z 1 n   X n log(1 + λ) gn−m,λ y n gn,λ (x + y)dy = y n+m dy m λ 0 0 m=0  m n   X n gn−m,λ (x) log(1 + λ) = λ m n+m+1 m=0

(26)

By (16), we get Z

1 n

n−1

Z

1

y gn,λ (x + y)dy = (−1) y n gn,λ (1 − (x + y))dy 0 0 m Z 1  n   X n log(1 + λ) y n (1 − y)m dy = (−1)n−1 gn−m (−x) m λ 0 m=0  m n   X n log(1 + λ) = (−1)m gn−m,λ (1 + x) B(n + 1, m + 1) m λ m=0  m  −1 n   X n gn−m,λ (1 + x) log(1 + λ) n+m = (−1)m m n+m+1 λ m m=0

(27)

By (26) and (27), we obtain the following theorem.

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Theorem 2.5. For n ∈ N, we have  m n   X n gn−m,λ (x) log(1 + λ)

=

m=0 n  X m=0

λ m n+m+1  m  −1  n+m n gn−m,λ (1 + x) log(1 + λ) (−1)m n+m+1 λ m m

(28)

From (17), we note that Z 1 y n gn,λ (x + y)dy 0 Z n log(1 + λ) 1 n+1 gn,λ (x + 1) = y gn−1,λ (x + y)dy − n+1 n+1 λ 0 gn,λ (x + 1) gn−1,λ (x + 1) n log(1 + λ) = − n+1 n + 1 n + 2 λ Z 2 1 n(n − 1) log(1 + λ) +(−1)2 y n+2 gn−2,λ (x + y)dy (n + 1)(n + 2) λ 0 gn,λ (x + 1) gn−1,λ (x + 1) n log(1 + λ) = − n+1 n+1 n+2 λ  2 n(n − 1) (log(1 + λ) 2 gn−2,λ (x + 1) +(−1) n+1 (n + 2)(n + 3) λ  3 Z 1 n(n − 1)(n − 2) (log(1 + λ) +(−1)3 y n+3 gn−3,λ (x + y)dy (29) (n + 1)(n + 2)(n + 3) λ 0 By continuing this process, we have Z 1 gn,λ (x + 1) y n gn,λ (x + y)dy = n+1 0 (30)  m n−1 X n(n − 1) · · · (n − m + 1) (log(1 + λ) m (−1) + gn−m,λ (x + 1) (n + 1)(n + 2) · · · (n + m + 1) λ m=1

Therefore by (26) and (30), we obtain the following theorem. Theorem 2.6. For n ∈ N, we have n   n−1 X X n gn−m,λ (x) = (−1)m m n + m + 1 m=0 m=0

n m
n+m m




gn−m,λ (x + 1) n+m+1



(log(1 + λ) λ

m (31)

Taking x = 0, From (16) and (31), we obtain the following corollary. Corollary 2.7. For n ∈ N, we have n   n−1 X X n gn−m,λ = (−1)m m n + m + 1 m=0 m=0

n m
n+m m




gn−m,λ (1) n+m+1



(log(1 + λ) λ

m (32)

For n ∈ N, we observe that Z 1 y n gn,λ (x + y)dy 0

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= = = = = =

Z 1 λ gn+1,λ (x + 1) λ n y n−1 gn+1,λ (x + y)dy − log(1 + λ) n+1 log(1 + λ) n + 1 0   Z 1 λ gn+1,λ (x + 1) n n−1 n y (−1) gn+1,λ (1 − (x + y))dy − log(1 + λ) n+1 n+1 0 !  Z 1 n+1  λ gn+1,λ (x + 1) n X n+1 n n−1 l − gn+1−l,λ (−x)(−1) y (1 − y) dy log(1 + λ) n+1 n+1 l 0 l=0 !  n+1  gn+1,λ (x + 1) n X n+1 λ n − gn+1−l,λ (−x)(−1) B(n, l + 1) log(1 + λ) n+1 n+1 l l=0 ! n+1 n+1
λ gn+1,λ (x + 1) 1 X l n −
(−1) gn+1−l,λ (−x) n+l log(1 + λ) n+1 n+1 l l=0 ! n+1 n+1
λ gn+1,λ (x + 1) 1 X l l −
(−1) gn+1−l,λ (1 + x) n+l log(1 + λ) n+1 n+1 l l=0

(33) Therefore, by (30) and (33), we obtain the following theorem. Theorem 2.8. For n ∈ N, we have
 l+1 n−1 n X gn−l,λ (1 + x) log(1 + λ) l l (−1) n+l
n+l+1 λ l l=0
n+1 n+1 X 1 gn+1,λ (x + 1) l l − (−1) n+l
gn+1−l,λ (1 + x) = n+1 n+1 l

(34)

l=0

Replacing λ by e − 1 and t by (e − 1)t in (10), we get ∞ X

Gn (x)

n=0

tn n!

2t xt e +1 ∞ X tn = gn,e−1 (x)(e − 1)n−1 , n! n=0 =

et

(35)

where Gn (x) are the Genocchi polynomials. By comparing both sides of (35), we obtain the following theorem. Theorem 2.9. For n ∈ N ∪ {0}, we have Gn (x) = gn,e−1 (x)(e − 1)n−1 . By (12) and (18), we get Z 1 gn,λ (x)dx = 0

= =

(36)

Z 1 λ d (n + 1)−1 gn+1,λ (x)dx log(1 + λ) dx 0 λ (n + 1)−1 (gn+1,λ (1) − gn+1,λ (0)) log(1 + λ) (−2)λ (n + 1)−1 gn+1,λ log(1 + λ)

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where n ∈ N. Also, we have Z 1 gn,λ (x)gm,λ (x)dx 0

Z 1 λ d 1 λ 1 gn+1,λ (x) gm,λ (x)dx gn+1,λ (x)gm,λ (x) |10 − log(1 + λ) n + 1 log(1 + λ) n + 1 0 dx 1 λ (gn+1,λ (1)gm,λ (1) − gn+1,λ (0)gm,λ (0) = log(1 + λ) n + 1 Z 1 1 log(1 + λ) λ m gn+1,λ (x)gm−1,λ (x)dx − log(1 + λ) n + 1 λ 0 Z 1 m = − gn+1,λ (x)gm−1,λ (x)dx n+1 0 Z 1 m(m − 1) 2 = (−1) gn+2,λ (x)gm−2,λ (x)dx (38) (n + 1)(n + 2) 0 =

By continuing this process, we obtain the following theorem. Theorem 2.10. For m, n ∈ N, we have Z 1 gn,λ (x)gm,λ (x)dx 0

=

m−2

(−1)

m(m − 1) · · · 3 (n + 1)(n + 2) · · · (n + m − 2)

Z

1

gn+m−2,λ (x)g2,λ (x)dx.

(39)

0

Now, we have Z

1

gn+m−2,λ (x)g2,λ (x)dx Z 1 2 gn+m−1,λ (x)g1,λ (x)dx = − n+m−1 0 2 gn+m,λ (x) λ = − |1 n + m − 1 n + m log(1 + λ) 0 λ −2gn+m,λ 2 . = − n + m − 1 log(1 + λ) n + m 0

(40)

By (41), we obtain the following theorem. Theorem 2.11. For m, n ∈ N, we have Z 1 gn,λ (x)gm,λ (x)dx 0  −1 n+m λ m = (−1) 2 gn+m,λ . m log(1 + λ)

(41)

References [1] A. Adelberg, A finite difference approach to degenerate Bernoulli and Stirling polynomials, Discrete Math. 140(1-3) (1995) 1-21. [2] E. Aˇ g y¨ uz, M. Acikgoz,S. Araci, A symmetric identity on the q -Genocchi polynomials of higher-order under third dihedral group D3 , Proc. Jangjeon Math. Soc. 18 (2) (2015) 177-187.

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[3] A. Bayad, T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials., Adv. Stud. Contemp. Math. (Kyungshang) 20(2) (2010) 247-253. [4] A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys. 18(2) (2011) 133-143. [5] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979) 51-88. [6] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) 7 (1956) 28-33. [7] S. Gaboury, R. Tremblay, B.-J. Fugre, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17(1) (2014) 115-123. [8] F. T. Howard, Explicit formulas for degenerate Bernoulli numbers, Discrete Math. 162(1-3) (1996) 175-185. [9] J. Y. Kang, C. S. Ryoo, Interpolation function of the multiple twisted q -Genocchi polynomials with weak weight α, Adv. Stud. Contemp. Math. (Kyungshang) 23(3) (2013) 491-500. [10] B. M. Kim, L. C. Jang, A note on the von Staudt-Clausen’s theorem for the weighted q -Genocchi numbers, Adv. Difference Equ. 2015 (2015) 2015:4. [11] D. S. Kim, T. Kim, Some identities of degenerate Euler polynomials arising from p-adic fermionic integrals on Zp , Integral Transforms Spec. Funct. 26(4) (2015) 295-302. [12] D. S. Kim, T. Kim, Some identities of degenerate special polynomials, Open Math. 13 (2015) 380-389. [13] D. S. Kim, T. Kim, D. V. Dolgy, Degenerate q -Euler polynomials, Adv. Difference Equ. 2015 (2015)2015:246. [14] T. Kim, Note on the Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 17(2) (2008) 131-136. [15] T. Kim, Some properties on the integral of the product of several Euler polynomials, Quaest. Math. 38(4) (2015) 553-562. [16] T. Kim, Degenerate Euler zeta function, Russ. J. Math. Phys. 22(4) (2015) 469-472. [17] T. Kim, On the multiple q -Genocchi and Euler numbers, Russ. J. Math. Phys. 15(4) (2008) 481-486. [18] T. Kim, T. Mansour, Umbral calculus associated with Frobenius-type Eulerian polynomials, Russ. J. Math. Phys. 21(4) (2008) 484-493. [19] T. Kim, New approach to q -Euler, Genocchi numbers and their interpolation functions, Adv. Stud. Contemp. Math. (Kyungshang) 18(2) (2009) 105-112. [20] T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10(3) (2003) 261-267. [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 , Russ. Adv. Stud. Contemp. Math. (Kyungshang) 23(1) (2013) 5-11. [22] T. Kim, H. I. Kwon, J.J. Seo, On the degenerate Frobenius-Euler polynomials, Global J. Pure and Applied Math. 11(4) (2015) 2077-2084. [23] T. Kim, J.W. Park, J.J. Seo, A note on λ-zeta function, Global J. Pure Appl. Math. 11(5) (2015) 3501-3506. [24] J.K. Kwon, A note on weighted Boole polynomials, Global J. Pure Appl. Math. 11(4) (2015) 2055-2063. [25] J. K. Kwon, S.-H. Rim, J.W. Park, Degenerate Daehee polynomials associated with p-adic invariant integral on Zp , Global J. Pure Appl. Math. 11(4) (2015) 1747-1754. [26] G. D. Liu, Degenerate Bernoulli numbers and polynomials of higher order, (Chinese) J. Math. (Wuhan) 25(3) (2005) 283-288. ¨ [27] N. I. Mahmudov, A. Akkeles, A. Oneren, On two dimensional q -Bernoulli and q -Genocchi polynomials: properties and location of zeros, J. Comput. Anal. Appl. 18(5) (2015) 834-843. ¨ [28] B. Yilmaz Yasar, M. A. Ozarslan, Frobenius-Euler and Frobenius-Genocchi polynomials and their differential equations, New Trends Math. Sci. 3(2) (2015) 172-180.

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Hesitant fuzzy implicative filters in BE-algebras Jeong Soon Han1 , Sun Shin Ahn2,∗ 1

Department of Applied Mathematics, Hanyang University, Ahnsan, 15588, Korea

2

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

Abstract. The notion of hesitant fuzzy implicative filter of a BE-algebra is introduced and related properties are investigated. We provide conditions for a hesitant fuzzy filter to be a hesitant fuzzy implicative filter. Also, as a generalization of hesitant fuzzy implicative filter, we consider the hesitant fuzzy n-fold implicative filter. Characterizations of hesitant fuzzy n-fold implicative filter are discussed.

1. Introduction In 2007, Kim and Kim [5] introduced the notion of a BE-algebra, and investigated several properties. In [1], Ahn and So introduced the notion of ideals in BE-algebras. They gave several descriptions of ideals in BE-algebras. Song et al. [8] considered the fuzzification of ideals in BEalgebras. They introduced the notion of fuzzy ideals in BE-algebras, and investigated related properties. They gave characterizations of a fuzzy ideal in BE-algebras. The notions of Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy sets and fuzzy multisets etc. are a generalization of fuzzy sets. As another generalization of fuzzy sets, Torra [9] introduced the notion of hesitant fuzzy sets which are a very useful to express peoples hesitancy in daily life. The hesitant fuzzy set is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. Also, hesitant fuzzy set theory is used in decision making problem etc. (see [11, 12, 13, 14, 15]), and is applied to residuated lattices and M T L-algebras (see [4, 6]). In this paper, we introduce the notion of hesitant fuzzy implicative filter of a BE-algebra, and investigate some properties of it. We consider characterizations of hesitant fuzzy implicative filter of a BE-algebra. We provide conditions for a hesitant fuzzy filter to be a hesitant fuzzy implicative filter. Also, as a generalization of hesitant fuzzy implicative filter, we consider the hesitant fuzzy n-fold implicative filter. We discuss characterizations of hesitant fuzzy n-fold implicative filter. 2. Preliminaries

0

2010 Mathematics Subject Classification: 06F35; 03G25; 06D72. Keywords: filter; implicative filter; hesitant fuzzy (implicative) filter. The corresponding author. Tel: +82 2 2260 3410, Fax: +82 2 2266 3409 0 E-mail: [email protected] (J. S. Han); [email protected] (S. S. Ahn) 0

∗

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Jeong Soon Han and Sun Shin Ahn

By a BE-algebra ([5]) we mean a system (X; ∗, 1) of type (2, 0) which the following axioms hold: (2.1) (2.2) (2.3) (2.4)

(∀x ∈ X) (x ∗ x = 1), (∀x ∈ X) (x ∗ 1 = 1), (∀x ∈ X) (1 ∗ x = x), (∀x, y, z ∈ X) (x ∗ (y ∗ z) = y ∗ (x ∗ z) (exchange).

We introduce a relation “ ≤ ” on X by x ≤ y if and only if x ∗ y = 1. A BE-algebra (X; ∗, 1) is said to be transitive ([5]) if it satisfies: for any x, y, z ∈ X, y ∗ z ≤ (x ∗ y) ∗ (x ∗ z). A BE-algebra (X; ∗, 1) is said to be self distributive ([5]) if it satisfies: for any x, yz ∈ X, x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z). Note that every self distributive BE-algebra is transitive, but the converse is not true in general ([5]). Every self distributive BE-algebra (X; ∗, 1) satisfies the following properties: (2.5) (∀x, y, z ∈ X) (x ≤ y ⇒ z ∗ x ≤ z ∗ y and y ∗ z ≤ x ∗ z), (2.6) (∀x, y ∈ X) (x ∗ (x ∗ y) = x ∗ y), (2.7) (∀x, y, z ∈ X) (x ∗ y ≤ (z ∗ x) ∗ (z ∗ y)), Definition 2.1.([5]) Let (X; ∗, 1) be a BE-algebra and let F be a non-empty subset of X. Then F is a filter of X if (F1) 1 ∈ F ; (F2) (∀x, y ∈ X)(x ∗ y, x ∈ F ⇒ y ∈ F ). F is an implicative filter of X if it satisfies (F1) and (F3) (∀x, y, z ∈ X)(x ∗ (y ∗ z), x ∗ y ∈ F ⇒ x ∗ z ∈ F ). Definition 2.2.([9]) Let E be a reference set. A hesitant fuzzy set on E is defined in terms of a function that when applied to E returns a subset of [0, 1], which can be viewed as the following mathematical representation: HE := {(e, hE (e))|e ∈ E} where hE : E → P([0, 1]). Definition 2.3. Given a non-empty subset A of X, a hesitant fuzzy set HX := {(x, hX (x))|x ∈ X} on satisfying the following condition: hX (x) = ∅ for all x ∈ /A

(2.8)

is called a hesitant fuzzy set related to A (briefly, A-hesitant fuzzy set) on X, and is represented by HA := {(x, hA (x)) | x ∈ X}, where hA is a mapping from X to P([0, 1]) with hA (x) = ∅ for all x ∈ / A.

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For a hesitant set HX := {(x, hX (x)) | x ∈ X} of a BE-algebra X and a subset γ of [0, 1], the hesitant fuzzy γ-inclusive set of HX , denoted by HX (γ), is defined to be the set HX (γ) := {x ∈ X|γ ⊆ hX (x)}. For any hesitant fuzzy set HX = {(x, hX (x)|x ∈ X} and GX = {(x, gX (x))|x ∈ X}, we call e X , if hX (x) ⊆ gX (x) for all x ∈ X. The HX a hesitant fuzzy subset of GX , denoted by HX ⊆G e GX , is defined to be the hesitant fuzzy hesitant fuzzy union of HX and GX , denoted by HX ∪ e gX )(x) = hX (x) ∪ gX (x) for all x ∈ X. The hesitant fuzzy intersection of HX and GX , set (hX ∪ e GX , is defined to be the hesitant fuzzy set (hX ∩ e gX )(x) = hX (x) ∩ gX (x) for all denoted by HX ∩ x ∈ X.

3. Hesitant fuzzy implicative filters Definition 3.1.([3]) Given a non-empty subset (subalgebra as much as possible) A of X, let HA := {(x, hA (x)) | x ∈ X} be an A-hesitant fuzzy set on X. Then HA := {(x, hA (x)) | x ∈ X} is called a hesitant fuzzy filter of X related to A (briefly, A-hesitant fuzzy filter of X) if it satisfies the following condition: (∀x ∈ A) (hA (x) ⊆ hA (1)) ,

(3.1)

(∀x, y ∈ A) (hA (x ∗ y) ∩ hA (x) ⊆ hA (y)) .

(3.2)

An A-hesitant fuzzy filter of X with A = X is called a hesitant fuzzy filter of X. Proposition 3.2.([3]) Let HA := {(x, hA (x))|x ∈ X} be an A-hesitant fuzzy filter of X where A is a subalgebra of X. Then the following assertions are valid. (i) (∀x, y ∈ A)(x ≤ y ⇒ hA (x) ⊆ hA (y)), (ii) (∀x, y, z ∈ A)(hA (x ∗ (y ∗ z)) ∩ hA (y) ⊆ hA (x ∗ z)), (iii) (∀a, x ∈ A)(hA (a) ⊆ hA ((a ∗ x) ∗ x). Definition 3.3. Given a non-empty subset (subalgebra as much as possible) A of X, let HA := {(x, hA (x)) | x ∈ X} be an A-hesitant fuzzy set on X. Then HA := {(x, hA (x)) | x ∈ X} is called a hesitant fuzzy implicative filter of X related to A (briefly, A-hesitant fuzzy implicative filter of X) if it satisfies (3.1) and (∀x, y, z ∈ A) (hA (x ∗ (y ∗ z)) ∩ hA (x ∗ y) ⊆ hA (x ∗ z)) .

(3.3)

An A-hesitant fuzzy implicative filter of X with A = X is called a hesitant fuzzy implicative filter of X.

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Example 3.4. Let X = {1, a, b, c, d, 0} be a BE-algebra with the following Cayley table: ∗ 1 a b c d 0

1 1 1 1 1 1 1

a a 1 1 a 1 1

b b a 1 b a 1

c c c c 1 1 1

d d c c 1 1 1

0 0 d c a a 1

For a subalgebra A = {1, a, b} of X, let HA := {(x, hA (x)) | x ∈ X} be an A-hesitant fuzzy set on X defined by { } HA = (1, [0, 1]), (a, (0, 12 ]), (b, (0, 12 )), (c, (0, 14 )), (d, ∅), (0, ∅) It is easy to check that HA is an A-hesitant fuzzy implicative filter of X. Proposition 3.5. Every A-hesitant fuzzy implicative filter of a BE-algebra X is an A-hesitant fuzzy filter of X. Proof. Let HA := {(x, hA (x)) | x ∈ X} be an A-hesitant fuzzy implicative filter of X. It follows from (2.4) and (3.3) that hA (y ∗ (x ∗ z)) ∩ hA (x ∗ y) =hA (x ∗ (y ∗ z)) ∩ hA (x ∗ y) ⊆hA (x ∗ z)

(3.4)

for any x, y, z ∈ X. Setting x := 1 in (3.4), we have hA (y ∗ z) ∩ hA (y) ⊆ hA (z). Therefore HA := {(x, hA (x)) | x ∈ X} is an A-hesitant fuzzy filter of X. □ The converse of Proposition may not be true in general (see Example 3.6). Example 3.6. Let X = {1, a, b, c, d, 0} be a BE-algebra as in Example 3.4. Let HX := {(x, hX (x)) | x ∈ X} be a hesitant fuzzy set on X defined as follows: { γ2 if x = 1 hX : X → P([0, 1]), x 7→ γ1 if x ∈ {a, b, c, d, 0}, where γ1 and γ2 are subsets of [0, 1] with γ1 ⊊ γ2 . It is easy to check that HX is a hesitant fuzzy filter of X. But it is not a hesitant fuzzy implicative filter of X, since hX (d ∗ (a ∗ 0)) ∩ hX (d ∗ a) = γ2 ⊈ γ1 = hX (d ∗ 0). We provide conditions for a hesitant fuzzy filter to be a hesitant fuzzy implicative filter. Proposition 3.7. Let X be a self distributive BE-algebra. Let HX := {(x, hX (x)) | x ∈ X} be a hesitant fuzzy filter of X satisfying (∀x, y, z ∈ X)(hX (x ∗ (y ∗ (y ∗ z))) ∩ hX (y ∗ x)) ⊆ hX (y ∗ z).

(3.5)

Then HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy implicative filter of X.

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Proof. Since x ∗ (y ∗ z) = y ∗ (x ∗ z) ≤ (x ∗ y) ∗ (x ∗ (x ∗ z)) = x ∗ (y ∗ (x ∗ z)) = y ∗ (x ∗ (x ∗ z)) for all x, y ∈ X, we have hX (x ∗ (y ∗ z)) ⊆ hX (y ∗ (x ∗ (x ∗ z))) by Proposition 3.2(i). It follows from (3.5) that hX (x ∗ z) ⊇ hX (y ∗ (x ∗ (x ∗ z)) ∩ hX (x ∗ y) ⊇ hX (x ∗ (y ∗ z)) ∩ hX (x ∗ y). Thus HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy implicative filter of X. □ Theorem 3.8. Let X be a transitive BE-algebra. For any hesitant fuzzy filter HX := {(x, hX (x)) | x ∈ X} of X, the following are equivalent: (i) HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy implicative filter, (ii) (∀x, y ∈ X) (hX (x ∗ (x ∗ y)) ⊆ hX (x ∗ y)) , (iii) (∀x, y, z ∈ X) (hX (x ∗ (y ∗ z)) ⊆ hX ((x ∗ y) ∗ (x ∗ z))). Proof. (i)⇒(ii) Assume that HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy implicative filter of X. Setting z := y, y := x in (3.3), we get hX (x ∗ y) ⊇hX (x ∗ (x ∗ y)) ∩ hX (x ∗ x) =hX (x ∗ (x ∗ y)) ∩ hX (1) =hX (x ∗ (x ∗ y)). Hence (ii) holds. (ii)⇒(iii) Suppose that (ii) holds. Since x ∗ (y ∗ z) ≤ x ∗ ((x ∗ y) ∗ (x ∗ z)) = x ∗ (x ∗ ((x ∗ y) ∗ z)), by Proposition 3.2(i) we have hX (x ∗ ((x ∗ y) ∗ (x ∗ z))) = hX (x ∗ (x ∗ ((x ∗ y) ∗ z))) ⊇ hX (x ∗ (y ∗ z)). It follows from (ii) that hX ((x ∗ y) ∗ (x ∗ z)) =hX (x ∗ ((x ∗ y) ∗ z)) ⊇hX (x ∗ (x ∗ ((x ∗ y) ∗ z))) ⊇hX (x ∗ (y ∗ z)). Thus (iii) holds. (iii)⇒(ii) Assume that (iii) holds. By (3.2) and (iii), we have hX (x ∗ z) ⊇hX ((x ∗ y) ∗ (x ∗ z)) ∩ hX (x ∗ y) ⊇hX (x ∗ (y ∗ z)) ∩ hX (x ∗ y). Therefore HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy implicative filter of X.

□

Theorem 3.9. Let X be a self distributive BE-algebra. Then the hesitant fuzzy set HX := {(x, hX (x)) | x ∈ X} of X is a hesitant fuzzy implicative filter of X if and only if it is a hesitant fuzzy filter of X. Proof. By Proposition 3.5, every hesitant fuzzy implicative filter of X is a hesitant fuzzy filter of X.

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Jeong Soon Han and Sun Shin Ahn

Conversely, assume that HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy filter of X. For any x, y, z ∈ X, by (3.2) we have hX (x ∗ z) ⊇hX ((x ∗ y) ∗ (x ∗ z)) ∩ hX (x ∗ y) =hX (x ∗ (y ∗ z)) ∩ hX (x ∗ y). Hence HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy implicative filter of X.

□

For any element x and y of a BE-algebra X and positive integer n, let xn ∗ y denote x ∗ (· · · ∗ (x ∗ (x ∗ y)) · · · ) in which x occurs n times, and x0 ∗ y = 1. Definition 3.10. Let X be a BE-algebra and let HX := {(x, hX (x)) | x ∈ X} be a hesitant fuzzy set on X. Then HX := {(x, hX (x)) | x ∈ X} is called a hesitant fuzzy n-fold implicative filter of X if it satisfies (3.1) and (3.6) (∀x, y, z ∈ X) (hX (xn ∗ (y ∗ z)) ∩ hX (xn ∗ y)) ⊆ hX (xn ∗ z)) . Note that a hesitant fuzzy 1-fold implicative filter of X is a hesitant fuzzy implicative filter of X. Example 3.11. Let X := {1, a, b, c, d, 0} Cayley table: ∗ 1 1 1 a 1 b 1 c 1 d 1 0 1

is a transitive BE-algebra ([11]) with the following a a 1 a a 1 1

b b b 1 1 1 1

c c c b 1 b 1

d d b a a 1 1

0 0 c d a b 1

Let HX := {(x, hX (x)) | x ∈ X} be a hesitant fuzzy set on X defined as follows: { γ2 if x ∈ {1, b, c} hX : X → P([0, 1]), x 7→ γ1 if x ∈ {a, d, 0}, where γ1 and γ2 are subsets of U with γ1 ⊊ γ2 . It is easy to check that HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy n-fold implicative filter of X. Theorem 3.12. Every hesitant n-fold fuzzy implicative filter of X is a hesitant fuzzy filter of X. Proof. Taking x := 1 in (3.6) and (2.3), we have hX (z) ⊇ hX (y ∗ z) ∩ hX (y). Hence HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy filter of X. □ The converse of Theorem 3.12 may not be not true in general (see Example 3.13).

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Example 3.13. Let X := {1, a, b, c, d, 0} be a BE-algebra as in Example 3.11. Let HX be a hesitant fuzzy set on X defined as follows: { γ2 if x = 1 hX : X → P([0, 1]), x 7→ γ1 if x ∈ {a, b, c, d, 0}, where γ1 and γ2 are subsets of U with γ1 ⊊ γ2 . It is easy to check that HX is a hesitant fuzzy filter of X. But it is not a hesitant fuzzy 1-fold implicative filter of X, since hX (d ∗ c) = hX (b) = γ1 ⊉ γ2 = hX (1) = hX (d ∗ (b ∗ c)) ∩ hX (d ∗ b). Theorem 3.14. Let X be a transitive BE-algebra. For any hesitant fuzzy filter HX := {(x, hX (x)) | x ∈ X} of X, the following are equivalent: (i) HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy n-fold implicative filter, (ii) (∀x, y ∈ X) (hX (xn+1 ∗ y) ⊆ hX (xn ∗ y)) , (iii) (∀x, y, z ∈ X) (hX (xn ∗ (y ∗ z)) ⊆ hX ((xn ∗ y) ∗ (xn ∗ z))). Proof. (i)⇒(ii) Assume that HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy n-fold implicative filter of X. Setting z := y, y := x in (3.6), we have hX (xn ∗ y) ⊇hX (xn ∗ (x ∗ y)) ∩ hX (xn ∗ x) =hX (xn+1 ∗ y) ∩ hX (1) =hX (xn+1 ∗ y). Hence (ii) holds. (ii)⇒(iii) Suppose that (ii) holds. Since xn ∗ (y ∗ z) ≤ xn ∗ ((xn ∗ y) ∗ (xn ∗ z)), we have hX (xn ∗ ((xn ∗y)∗(xn ∗z))) ⊇ hX (xn ∗(y ∗z)). Since xn+1 ∗(xn−1 ∗((xn ∗y)∗z)) = xn ∗(xn ∗((xn ∗y)∗z))) = xn ∗ ((xn ∗ y)) ∗ (xn ∗ z)) and using (ii), we have hX (xn+1 ∗ (xn−2 ∗ ((xn ∗ y) ∗ z)) =hX (xn ∗ (xn−1 ∗ ((xn ∗ y) ∗ z)) ⊇hX (xn+1 ∗ (xn−1 ∗ ((xn ∗ y) ∗ z))) =hX (xn ∗ ((xn ∗ y) ∗ (xn ∗ z)))

(3.7)

⊇hX (xn ∗ (y ∗ z)). By (ii) and (3.7), we have hX (xn+1 ∗ (xn−3 ∗ ((xn ∗ y) ∗ z))) =hX (xn ∗ (xn−2 ∗ ((xn ∗ y) ∗ z))) ⊇hX (xn+1 ∗ (xn−2 ∗ ((xn ∗ y) ∗ z))) ⊇hX (xn ∗ (y ∗ z)). Continuing this process, we conclude that hX ((xn ∗ y) ∗ (xn ∗ z)) =hX (xn ∗ ((xn ∗ y) ∗ z)) ⊇hX (xn ∗ (y ∗ z)).

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Jeong Soon Han and Sun Shin Ahn

(iii)⇒(i) Let x, y, z ∈ X. It follows from (iii) that hX (xn ∗ z) ⊇hX ((xn ∗ y) ∗ (xn ∗ z)) ∩ hX (xn ∗ y) ⊇hX ((xn ∗ (y ∗ z)) ∩ hX (xn ∗ y). Hence HX := {(x, hX (x)) | x ∈ X} is a hesitant n-fold fuzzy implicative filter

□

Definition 3.15. Let n be a positive integer. A BE-algebra X is said to be n-fold implicative if it satisfies the equality xn+1 ∗ y = xn ∗ y for all x, y ∈ X. Corollary 3.16. In an n-fold implicative BE-algebra, the notion of hesitant fuzzy filters and hesitant fuzzy n-fold implicative filters coincide. □

Proof. Straightforward.

Theorem 3.17. A hesitant fuzzy set HX := {(x, hX (x)) | x ∈ X} of a BE-algebra X is a hesitant fuzzy implicative filter of X if and only if the hesitant fuzzy γ-inclusive set HX (γ) is an implicative filter of X for all γ ∈ P([0, 1]) with HX (γ) ̸= ∅. The filter HX (γ) in Theorem 3.17 is called the γ-inclusive filter of X. Proof. Assume that HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy implicative filter of X. Let x, y, z ∈ X and γ ∈ P([0, 1]) be such that x ∗ (y ∗ z) ∈ HX (γ) and x ∗ y ∈ HX (γ). Then γ ⊆ hX (x ∗ (y ∗ z)) and γ ⊆ hX (x ∗ y). Using (3.1) and (3.3), we have γ ⊆ hX (1) and γ ⊆ hX (x ∗ (y ∗ z) ∩ hX (x ∗ y) ⊆ hX (x ∗ z) for x, y, z ∈ X. Hence 1 ∈ HX (γ) and x ∗ z ∈ HX (γ). Thus HX (γ) is an implicative filter of X. Conversely, suppose that HX (γ) is an implicative filter of X for all γ ∈ P([0, 1]) with HX (γ) ̸= ∅. For any x ∈ X, let hX (x) = γ. Since HX (γ) is an implicative filter of X, we have 1 ∈ HX (γ) and so hX (x) = γ ⊆ hX (1). For any x, y, z ∈ X, let hX (x ∗ (y ∗ z)) = γx∗(y∗z) and hX (x ∗ y) = γx∗y . Take γ = γx∗(y∗z) ∩ γx∗y . Then x ∗ (y ∗ z) ∈ HX (γ) and x ∗ y ∈ HX (γ) which imply that x ∗ z ∈ HX (γ). Hence hX (x ∗ z) ⊇ γ = γx∗(y∗z) ∩ γx∗y = hX (x ∗ (y ∗ z)) ∩ hX (x ∗ y). Thus HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy implicative filter of X.

□

Theorem 3.18. Every hesitant fuzzy implicative filter of a BE-algebra can be represented as a hesitant fuzzy γ-inclusive set of a hesitant fuzzy implicative filter. Proof. Let F be an implicative filter of a BE-algebra X. For a subset γ of [0, 1], define a hesitant fuzzy set HX := {(x, hX (x)) | x ∈ X} of X by { γ if x ∈ F, hX : X → P([0, 1]), x 7→ ∅ if x ∈ / F. Obviously, F = HX (γ). We now prove that HX is a hesitant fuzzy implicative filter of X. Since 1 ∈ F = HX (γ), we have hX (1) = γ ⊇ hX (x) for all x ∈ X. Let x, y, z ∈ X. If x∗(y∗z), x∗y ∈ F, then

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x∗z ∈ F because F is an implicative filter of X. Hence hX (x∗(y ∗z)) = hX (x∗y) = hX (x∗z) = γ, and so hX (x∗(y ∗z))∩hX (x∗y) ⊆ hX (x∗z). If x∗(y ∗z) ∈ F and x∗y ∈ / F, then hX (x∗(y ∗z)) = γ and hX (x ∗ y) = ∅ which imply that hX (x ∗ (y ∗ z)) ∩ hX (x ∗ y) = γ ∩ ∅ = ∅ ⊆ hX (x ∗ z). Similarly, if x ∗ (y ∗ z) ∈ / F and x ∗ y ∈ F, then hX (x ∗ (y ∗ z)) ∩ hX (x ∗ y) ⊆ hX (x ∗ z). Obviously, if x ∗ (y ∗ z) ∈ / F and x ∗ y ∈ / F, then hX (x ∗ (y ∗ z)) ∩ hX (x ∗ y) ⊆ hX (x ∗ z). Therefore HX := {(x, hX (x)) | x ∈ X} is a hesitant fuzzy implicative filter of X. □ a,b For two elements a and b of X, consider a hesitant fuzzy set HX := {(x, ha,b X x)) | x ∈ X} where

{ ha,b X

: X → P([0, 1]), x 7→

γ1 if a ∗ (b ∗ x) = 1, γ2 otherwise,

where γ1 and γ2 are subsets of X with γ2 ⊊ γ1 . a,b There exist a, b ∈ X such that HX is not a hesitant fuzzy implicative filter of X (see Example 3.19). Example 3.19. Consider the BE-algebra X = {1, a, b, c, d, 0} which is given in Example 3.4. 1,a Then HX is not a hesitant fuzzy implicative filter of X since 1,a 1,a h1,a X (1 ∗ (a ∗ b)) ∩ hX (1 ∗ a) = γ1 ⊈ hX (1 ∗ b) = γ2 .

a,b Now we provide a condition for the hesitant fuzzy set HX to be a hesitant fuzzy implicative filter of X for all a, b ∈ X. a,b Theorem 3.20. If X is a self distributive BE-algebra, then the hesitant fuzzy set HX is a hesitant fuzzy implicative filter of X for all a, b ∈ X. a,b Proof. Let a, b ∈ X. Obviously, ha,b X (1) ⊇ hX (x) for all x ∈ X. Let x, y, z ∈ X be such that a,b a ∗ (b ∗ (x ∗ (y ∗ z))) ̸= 1 or a ∗ (b ∗ (x ∗ y)) ̸= 1. Then ha,b X (x ∗ (y ∗ z)) = γ2 or hX (x ∗ y) = γ2 . Hence

a,b a,b ha,b X (x ∗ (y ∗ z)) ∩ hX (x ∗ y) = γ2 ⊆ hX (x ∗ z).

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Jeong Soon Han and Sun Shin Ahn

Assume that a ∗ (b ∗ (x ∗ (y ∗ z))) = 1 and a ∗ (b ∗ (x ∗ y)) = 1. Then 1 = a ∗ (b ∗ (x ∗ (y ∗ z))) = a ∗ (b ∗ ((x ∗ y) ∗ (x ∗ z))) = a ∗ ((b ∗ (x ∗ y)) ∗ (b ∗ (x ∗ z))) = (a ∗ (b ∗ (x ∗ y))) ∗ (a ∗ (b ∗ (x ∗ z))) = 1 ∗ (a ∗ (b ∗ (x ∗ z))) = a ∗ (b ∗ (x ∗ z)), a,b a,b a,b and so ha,b X (x ∗ (y ∗ z))∩ hX (x ∗y) = γ1 = hX (x ∗ z). Therefore HX is a hesitant fuzzy implicative filter of X for all a, b ∈ X. □

Theorem 3.21. If HX and GX are hesitant fuzzy implicative filters of a BE-algebra X, then ˜ GX of HX and GX is a hesitant fuzzy implicative filter of X. the hesitant fuzzy intersection HX ∩ Proof. For any x ∈ X, we have ˜ gX ) (1) = hX (1) ∩ gX (1) ⊇ hX (x) ∩ gX (x) = (hX ∩ ˜ gX )(x). (hX ∩ Let x, y, z ∈ X. Then ˜ gX )(x ∗ z) = hX (x ∗ z) ∩ gX (x ∗ z) (hX ∩ ⊇ (hX (x ∗ (y ∗ z)) ∩ hX (x ∗ y)) ∩ (gX (x ∗ (y ∗ z)) ∩ gX (x ∗ y)) = (hX (x ∗ (y ∗ z)) ∩ gX (x ∗ (y ∗ z))) ∩ (hX (x ∗ y) ∩ gX (x ∗ y)) ˜ gX ) (x ∗ (y ∗ z)) ∩ (hX ∩ ˜ gX ) (x ∗ y). = (hX ∩ ˜ GX is a hesitant fuzzy implicative filter of X. Hence HX ∩

□

The hesitant fuzzy union of hesitant fuzzy implicative filters of a BE-algebra X may not be a hesitant fuzzy implicative filter of X as the following example. Example 3.22. Let X = {1, a, b, c, d} is a BE-algebra with the following Cayley table ([5]): ∗ 1 a b c d

1 1 1 1 1 1

a a 1 a 1 1

b b b 1 b 1

c c c c 1 1

d d d c b 1

Let HX and GX be hesitant fuzzy sets of X defined, respectively, as follows: { γ3 if x ∈ {1, b} hX : X → P([0, 1]), x 7→ γ1 if x ∈ {a, c, d}

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and { gX : X → P(U ), x 7→

γ4 if x ∈ {1, a, c} γ2 if x ∈ {b, d}

where γ1 , γ2 , γ3 , and γ4 are subsets of [0, 1] with γ1 ⊊ γ2 ⊊ γ3 ⊊ γ4 . It is easy to check that ˜ GX is not a hesitant fuzzy HX and GX are hesitant fuzzy implicative filters of X. But HX ∪ implicative filter of X, since ˜ gX )(1 ∗ (c ∗ d))∩(hX ∪ ˜ gX )(1 ∗ c) = (hX ∪ ˜ gX )(b) ∩ (hX ∪ ˜ gX )(c) (hX ∪ = (hX (b) ∪ gX (b)) ∩ (hX (c) ∪ gX (c)) = γ3 ∩ γ4 = γ3 ⊈ γ2 = γ1 ∪ γ2 ˜ gX )(1 ∗ d). = hX (1 ∗ d) ∪ gX (1 ∗ d) = (hX ∪ Let HX be a hesitant fuzzy set set of a BE-algebra X. For any a, b ∈ X and k ∈ N, consider the set { ( ) } hX [ak ; b] := x ∈ X | hX ak ∗ (b ∗ x) = hX (1) where hX (ak ∗ x) = hX (a ∗ (a ∗ (· · · ∗ (a ∗ (a ∗ x)) · · · ))) in which a appears k-times. Note that a, b, 1 ∈ hX [ak ; b] for all a, b ∈ X and k ∈ N. Proposition 3.23. Let HX be a hesitant fuzzy set of a BE-algebra X such that the condition (3.1) and hX (x ∗ y) = hX (x) ∪ hX (y) for all x, y ∈ X. For any a, b ∈ X and k ∈ N, if x ∈ hX [ak ; b], then y ∗ x ∈ hX [ak ; b] for all y ∈ X. Proof. Assume that x ∈ hX [ak ; b]. Then hX (ak ∗ (b ∗ x)) = hX (1), and so hX (ak ∗ (b ∗ (y ∗ x))) = hX (ak ∗ (y ∗ (b ∗ x))) = hX (y ∗ (ak ∗ (b ∗ x))) = hX (y) ∪ hX (ak ∗ (b ∗ x)) = hX (y) ∪ hX (1) = hX (1) for all y ∈ X by the exchange property of the operation ∗. Hence y∗x ∈ hX [ak ; b] for all y ∈ X. □ Proposition 3.24. For any hesitant fuzzy set HX of a BE-algebra X, let a ∈ X satisfy the following condition a ∗ x = 1 for all x ∈ X. Then hX [ak ; b] = X = hX [bk ; a] for all b ∈ X and k ∈ N.

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Jeong Soon Han and Sun Shin Ahn

Proof. For any x ∈ X, we have hX (ak ∗ (b ∗ x)) = hX (ak−1 ∗ (a ∗ (b ∗ x))) = hX (ak−1 ∗ (b ∗ (a ∗ x))) = hX (ak−1 ∗ (b ∗ 1)) = hX (1), and so x ∈ hX [ak ; b]. Similarly, x ∈ hX [bk ; a].

□

Proposition 3.25. Let X be a self distributive BE-algebra and let HX be an order-preserving soft set of X with the property (3.1). If b ≤ c in X, then hX [ak ; c] ⊆ hX [ak ; b] for all a ∈ X and k ∈ N. Proof. Let a, b, c, ∈ X be such that b ≤ c. For any k ∈ N, if x ∈ hX [ak ; c], then hX (1) = hX (ak ∗ (c ∗ x)) = hX (c ∗ (ak ∗ x)) ⊆ hX (b ∗ (ak ∗ x)) = hX (ak ∗ (b ∗ x)) by (2.5), Proposition 3.2(i) and (2.4), and so hX (ak ∗ (b ∗ x)) = hX (1). Thus x ∈ hX [ak ; b], which completes the proof. □ The following example shows that there exists a hesitant fuzzy set HX of X, a, b ∈ X and k ∈ N such that hX [ak ; b] is not a filter of X. Example 3.26. Let X = {1, a, b, c} is a BE-algebra with the following Cayley table: ∗ 1 a b c

1 1 1 1 1

a a 1 1 a

b b a 1 a

c c a a 1

Let HX be a hesitant fuzzy set of X U defined as follows: { γ2 if x = 1 hX : X → P([0, 1]), x 7→ γ1 if x ∈ {a, b, c}, where γ1 and γ2 are subsets of U with γ1 ⊊ γ2 . Then it is a hesitant fuzzy set of X. But hX [c; b] = {x ∈ X|hX (c ∗ (b ∗ x)) = hX (1)} = {1, a, b} is not an implicative filter, since 1 ∗ (a ∗ c) = a ∈ hX [c; b], 1 ∗ a = a ∈ hX [c; b] and 1 ∗ c = c ∈ / hX [c; b]. We provide conditions for a set hX [ak ; b] to be an implicative filter. Theorem 3.27. Let HX be a hesitant fuzzy set of a self distributive BE-algebra X. If hX is injective, then hX [ak ; b] is an implicative filter of X for all a, b ∈ X and k ∈ N.

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Proof. Assume that X is a self distributive BE-algebra and hX is injective. Obviously, 1 ∈ hX [ak ; b]. Let a, b, x, y, z ∈ X and k ∈ N be such that x ∗ (y ∗ z) ∈ hX [ak ; b] and x ∗ y ∈ hX [ak ; b]. Then hX (ak ∗ (b ∗ (x ∗ (y ∗ z)))) = hX (1) which implies that ak ∗ (b ∗ (x ∗ (y ∗ z))) = 1 since hX is injective. Since X is a self distributive BE-algebra, we have hX (1) = hX (ak ∗ (b ∗ (x ∗ (y ∗ z)))) = hX (ak−1 ∗ (a ∗ (b ∗ (x ∗ (y ∗ z))))) = hX (ak−1 ∗ (a ∗ ((b ∗ (x ∗ y)) ∗ (b ∗ (x ∗ z))))) = ··· = hX ((ak ∗ (b ∗ (x ∗ y))) ∗ (ak ∗ (b ∗ (x ∗ z)))) = hX (1 ∗ (ak ∗ (b ∗ (x ∗ z)))) = hX (ak ∗ (b ∗ (x ∗ z))), which implies that x∗z ∈ hX [ak ; b]. Therefore hX [ak ; b] is an implicative filter of X for all a, b ∈ X and k ∈ N. □ Theorem 3.28. Let HX be a hesitant fuzzy set of a self distributive B-algebra X satisfying the condition (3.1) and (∀x, y ∈ X) (hX (x ∗ y) = hX (x) ∩ hX (y)) .

(3.8)

Then hX [ak ; b] is an implicative filter of X for all a, b ∈ X and k ∈ N. Proof. Let a, b ∈ X and k ∈ N. Obviously, 1 ∈ hX [ak ; b]. Let x, y, z ∈ X be such that x ∗ (y ∗ z) ∈ ( ) hX [ak ; b] and x ∗ y ∈ hX [ak ; b]. Then hX ak ∗ (b ∗ (x ∗ (y ∗ z))) = hX (1), which implies from (3.8) and (3.1) that hX (1) = hX (ak ∗ (b ∗ (x ∗ (y ∗ z)))) = hX (ak−1 ∗ (a ∗ (b ∗ (x ∗ (y ∗ z))))) = hX (ak−1 ∗ (a ∗ ((b ∗ (x ∗ y)) ∗ (b ∗ (x ∗ z))))) = ··· = hX ((ak ∗ (b ∗ (x ∗ y))) ∗ (ak ∗ (b ∗ (x ∗ z)))) = hX (ak ∗ (b ∗ (x ∗ y))) ∩ hX (ak ∗ (b ∗ (x ∗ z))) = hX (1) ∩ hX (ak ∗ (b ∗ (x ∗ z))) = hX (ak ∗ (b ∗ (x ∗ z))). Hence x ∗ z ∈ hX [ak ; b] and therefore hX [ak ; b] is an implicative filter of X for all a, b ∈ X and k ∈ N. □

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Jeong Soon Han and Sun Shin Ahn

References [1] S. S. Ahn and K. S. So, On ideals and upper sets in BE-algerbas, Sci. Math. Jpn. 68 (2008), 279–285 . [2] Y. B. Jun and S. S. Ahn, Hesitant fuzzy soft theory applied to BCK/BCI-algebras, J. Comput. Anal. Appl. 20 (2016), no.4, 635–646. [3] Y. B. Jun and S. S. Ahn, On hesitant fuzzy filters in BE-algebras, J. Comput. Anal. Appl. (to appear). [4] Y. B. Jun and S. Z. Song, Hesitant fuzzy set theory applied to filters in M T L-algebras, Honam Math. J. 36 (2014), no.4, 813–830. [5] H. S. Kim and Y. H. Kim, On BE-algerbas, Sci. Math. Jpn. 66 (2007), no. 1, 113–116. [6] G. Muhiuddin and Y. B. Jun, Hesitant fuzzy filters and hesitant fuzzy G-filters in residuated lattices, J. Appl. Math. (submitted). [7] Rosa M. Rodriguez, Luis Martinez and Francisco Herrera, Hesitant fuzzy linguistic term sets for decision making, IEEE Trans. Fuzzy Syst. 20(1) (2012), 109–119. [8] S. Z. Song, Y. B. Jun and K. J. Lee, Fuzzy ideals in BE-algebras, Bull. Malays. Math. Sci. Soc. 33 (2010), 147-153. [9] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010), 529–539. [10] V. Torra and Y. Narukawa, On hesitant fuzzy sets and decision, in: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, 1378-1382. [11] F. Q. Wang, X. Li and X. H. Chen, Hesitant fuzzy soft set and its applications in multicriteria decision making, J. Appl. Math. Volume 2014, Article ID 643785, 10 pages. [12] G. Wei, Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowledge-Based Systems 31 (2012), 176–182. [13] M. Xia and Z. S. Xu, Hesitant fuzzy information aggregation in decision making, Internat. J. Approx. Reason. 52(3) (2011), 395–407. [14] Z. S. Xu and M. Xia, Distance and similarity measures for hesitant fuzzy sets, Inform. Sci. 181(11) (2011), 2128–2138. [15] Z. S. Xu and M. Xia, On distance and correlation measures of hesitant fuzzy information, Int. J. Intell. Syst. 26(5) (2011), 410–425.

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A new quadratic functional equation version and its stability and superstability Shahrokh Farhadabadi1 , Jung Rye Lee2∗ and Choonkil Park3∗ 1

Young Researchers and Elite Club, Parand Brunch, Islamic Azad University, Parand, Iran Javad Shokri1 and Jung Rye Lee2∗ 1

Department of Mathematics, Urmia University, P. O. Box 165, Urmia, Iran 2 Department of Mathematics, Daejin University, Kyunggi 11159, Korea 3 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea e-mail: shahrokh [email protected]; [email protected]; [email protected] Abstract. Let X and Y be vector spaces. It is shown that a mapping f : X → Y satisfies the functional equation f



x+y+z 2



x−y−z y−x−z +f 2 2 = f (x) + f (y) + f (z)

+f









+f



z−x−y 2

 (0.1)

if and only if f : X → Y is a quadratic mapping. Furthermore, we prove the superstability and the Hyers-Ulam stability for the quadratic functional equation (0.1) by using a direct method. Keywords: Hyers-Ulam stability; quadratic functional equation; fixed point method; quadratic functional inequality; orthogonality space.

1. Introduction and preliminaries Studying functional equations focusing on their approximate and exact solutions, conduces to one of the most substantial significant study brunches in functional equations, what we would call “the theory of stability of functional equations”. This theory specifically analyzes relationships between approximate and exact solutions of functional equations. Actually a functional equation is considered to be stable, if one can find an exact solution for any approximate solution of that certain functional equation. Another related and close term in this area is superstability, which has a similar nature and concept to the stability problem. As a matter of fact, superstability for a given functional equation occurs when any approximate solution is an exact solution too. In such this situation the functional equation is called superstable. In 1940, the most preliminary form of stability problems was proposed by Ulam [40]. He gave a talk and asked the following: “when and under what conditions does an exact solution of a functional equation near an approximately solution of that exist?” In 1941, this question that today is considered as the source of the stability theory, was formulated and solved by Hyers [14] for the Cauchy’s functional equation in Banach spaces. Then the result of Hyers was generalized by Aoki [1] for additive mappings and by Rassias [32] for linear mappings by considering an unbounded Cauchy difference. In 1994, G˘ avrut¸a [13] 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. For more epochal information and various aspects about the stability of functional equations theory, we refer the 0

∗

2010 Mathematics Subject Classification: 39B52. Corresponding authors.

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S. Farhadabadi, J. Lee, C. Park reader to the monographs [15, 28, 33, 35], which also include many interesting results concerning the stability of different functional equations in many various spaces. Consider the quadratic functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y).

(1.1)

The function f (x) = cx2 is a solution for the quadratic functional equation and obviously every satisfied function in this equation is said to be a quadratic function. A stability problem for this equation was first proved by Skof [39] and then was generalized by Cholewa [9], Czerwik [7, 8] and others [2, 4, 30, 31, 33, 34]. Moreover, there are some other different types of quadratic functional equations that their stability problems have been investigated by many authors. We refer the readers to the papers [3, 5, 6, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 36, 37, 38, 41]. This paper is organized as follows: In Section 2, we consider the superstability of the quadratic functional equation (0.1) and in Sections 3 and 4, we prove two types of stability problems for the quadratic functional equation (0.1). 2. Superstability of the functional equation (0.1) To commence proving the superstability of the quadratic functional equation (0.1), we first solve it and then will give a superstability theorem. Proposition 2.1. Let X and Y be vector spaces. A mapping f : X → Y satisfies (0.1) if and only if the mapping f : X → Y is a quadratic mapping. Proof. Sufficiency. Assume that f : X → Y satisfies (0.1). Letting x = y = z = 0 in (0.1), we have 4f (0) = 3f (0). So f (0) = 0. Letting y = z = 0 in (0.1), we get x x + 2f − = f (x), 2    2 x x 2f − + 2f = f (−x) 2 2 for all x ∈ X , which imply that f (x) = f (−x) for all x ∈ X . 2f

It follows from (2.1) that 4f

x 2




 





(2.1)

= f (x) and so f (2x) = 4f (x) for all x ∈ X .

Putting z = 0 in (0.1), we see that 1 1 f (x + y) + f (x − y) = f (x) + f (y) 2 2 for all x, y ∈ X , which means that f : X → Y is a quadratic mapping. Necessity. Assume that f : X → Y is quadratic. By (1.1), one can easily get f (0) = 0, f (x) = f (−x) and f (2x) = 4f (x) for all x ∈ X . So by applying (1.1), we obtain f



x+y+z 2



x−y−z y−x−z z−x−y +f +f 2 2 h    i h   2  i x x y+z y−z = 2f + 2f + 2f − + 2f 2 2  2   2  x y+z+y−z y+z−y+z = 4f +f +f 2 2 2 = f (x) + f (y) + f (z)

+f











for all x, y, z ∈ X , which is the functional equation (0.1) and the proof is complete.

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Stability and Superstability for a new quadratic functional equation Theorem 2.2. Let X , Y be normed spaces with norms k · kX and k · kY , respectively. Let δ be a nonnegative real number and ϕ : X 3 → [0, ∞) be a function with ϕ(0, 0, 0) = 0,

ϕ(x, y, 3x + y) = 0

for all x, y ∈ X . Suppose that f : X → Y is a mapping such that



     x−y−z y−x−z

x+y+z

+f +f − f (y) − f (z)

f 2 2 2 Y

  z−x−y

≤ f (x) − f

+ δ · ϕ(x, y, z) 2

(2.2)

Y

for all x, y, z ∈ X . Then f is a quadratic mapping. Proof. Putting x = y = z = 0 in (2.2), we get



f (0) ≤ 0 + δ · ϕ(0, 0, 0) = 0. Y Y So f (0) = 0. Replacing x, y, z by 0, x, x in (2.2), respectively, we obtain





f (−x) − f (x) ≤ 0 + δ · ϕ(0, x, x) = 0. Y Y So f (x) = f (−x) for all x ∈ X . Replacing x, y and z by x, −3x and 0, and then by 2x, −3x and 3x in (2.2), respectively, we have [f (x) − f (3x)] + 2f (2x) = 0, 2[f (x) − f (3x)] + f (4x) = 0, which result that f (2x) = 4f (x) and f (3x) = 9f (x) for all x ∈ X . Letting x = v − u, y = 2u − v and z = 2v − u and then x = u + v, y = −3v and z = 3u in (2.2), respectively, we get the equalities f (2u − v) + f (2v − u)

=

f (u) + f (v) + f (2u − 2v),

f (2u − v) + f (2v − u)

=

f (3u) + f (3v) − f (2u + 2v).

Thus f (u) + f (v) + 4f (u − v) = 9f (u) + 9f (v) − 4f (u + v), which is simplified to f (u + v) + f (u − v) = 2f (u) + 2f (v) for all u, v ∈ X . So f is quadratic.



Theorem 2.2 covers several other cases for ϕ : X 3 → [0, ∞). For example, we can define ϕ satisfying the mentioned conditions with ϕ(x, y, z) := kykX − k3x − zkX or ϕ(x, y, z) := k3x + y − zkX . In addition, to make a simpler result, one can put δ = 0.

3. Hyers-Ulam stability of the functional equation (0.1): Type A In this section, we prove the Hyers-Ulam stability of the quadratic functional equation (0.1). We will suppose that X is a normed space and Y is a complete normed space with norms k · kX and k · kY , respectively.

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S. Farhadabadi, J. Lee, C. Park Theorem 3.1. Let ϕ : X 3 → [0, ∞) be a function with ϕ(0, 0, 0) = 0 and the following condition holds: kxkX ≤ kx0 kX , kykX ≤ ky 0 kX , kzkX ≤ kz 0 kX ,

( if

or or

ϕ(x, y, z) ≤ ϕ(x0 , y 0 , z 0 )

=⇒

(3.1)

for all x, y, z, x0 , y 0 , z 0 ∈ X . Denote by φ a function such that φ(x, y, z) :=

∞ X

22n ϕ

x y z , , 2n 2n 2n



n=0



l. From (3.6), it follows that m−1

         X 2s+1  x x x  x x 1 x x

m l l m−l f ≤ 2 ϕ s, s, s − f

4 f m − 4 f l = 4 4

m−l l l

2

2

Y

2

2

547

2

Y

s=l

2

2

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Stability and Superstability for a new quadratic functional equation for all x ∈ X , in which by (3.2) the right-hand side tends to zero as m, l → ∞. This clarifies that the sequence n
o 4n f

x 2n

is Cauchy in the complete space Y and therefore convergent in it. So we can define for all x ∈ X ,

the mapping Q : X → Y by x . n→∞ 2n Now passing the limit n → ∞ in (3.6) and then using (3.2), we obtain (3.4). Q(x) := lim 4n f





To end the proof, we show that Q is a unique quadratic mapping. It follows from (3.3) that

      x−y−z y−x−z

x+y+z +Q +Q − Q(y) − Q(z)

Q 2 2 2

      Y     x+y+z x−y−z y−x−z y z n = lim 4 f + f + f − f − f

n→∞ 2n+1 2n+1 2n+1 2n 2n Y

      x z−x−y x y z

n n n ≤ lim 4 f −4 f

+ lim 4 ϕ n , n , n n→∞ n→∞ 2n 2n+1 2 2 2 Y for all x, y, z ∈ X , in which by (3.2), the second term of the right-hand side tends to zero as n → ∞, and therefore we obtain



       x−y−z y−x−z z−x−y

x+y+z

+Q +Q − Q(y) − Q(z) ≤ Q(x) − Q

Q

2 2 2 2 Y Y for all x, y, z ∈ X . Now by applying Theorem 2.2 (with δ = 0), we conclude that Q is a quadratic mapping. Let Q0 : X → Y be another quadratic mapping satisfying (3.4). Then we have

 

     

x x x x n n 0

Q(x) − Q0 (x) Q ≤ 4 − f + 4 Q − f



Y 2n 2n Y 2n 2n Y 2 · 4n · 2φ

≤



x x x , , 2n 2n 2n



=4

∞ X

22s+1 ϕ



s=n

x x x , , 2n 2n 2n



for all x ∈ X . By (3.2), the right-hand side tends to zero as n → ∞, and thus Q(x) = Q0 (x) for all x ∈ X . This means the uniqueness of Q : X → Y and so the proof is complete.



Theorem 3.2. Let ϕ : X 3 → [0, ∞) be a function satisfying ϕ(0, 0, 0) = 0 and (3.1). Denote by φ a function such that φ(x, y, z) :=

∞ X 1 n=1

ϕ 2n x, 2n y, 2n z < ∞




22n

(3.7)

for all x, y, z ∈ X . Suppose that f : X → Y is an even mapping satisfying (3.3). Then there exists a unique quadratic mapping Q : X → Y satisfying (3.4). Proof. As in the proof of Theorem 3.1, we can first get the inequality (3.5), and then by replacing x by 2x in (3.5), we obtain

1

1

f (2x) − f (x) ≤ ϕ (2x, 2x, 2x) 4

2

Y

for all x ∈ X . Using the induction method, we get n

X

1

n

n f (2 x) − f (x) ≤

4

Y

s=1

1 ϕ (2s x, 2s x, 2s x) 22s−1

(3.8)

for all n ≥ 1 and all x ∈ X .

n Now by othe same method which was done in the proof of Theorem 3.1, we have the Cauchy sequence 1 f (2n x) 4n

, and then the mapping Q : X → Y defined by Q(x) := lim n→∞

1 f (2n x) 4n

for all x ∈ X .

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S. Farhadabadi, J. Lee, C. Park And finally we can conclude the inequality (3.4) by (3.7) and (3.8). The rest of the proof is similar to the proof of Theorem 3.1.



Corollary 3.3. Let δ be a nonnegative real number and p1 , p2 , p3 be positive real numbers such that p1 , p2 , p3 > 2 or p1 , p2 , p3 < 2. Let f : X → Y be an even mapping satisfying



     x−y−z y−x−z

x+y+z

+f +f − f (y) − f (z)

f 2 2 2 Y

 
z−x−y

p3 p2 p1 ≤ f (x) − f

+ δ kxkX + kykX + kzkX 2 Y for all x, y, z ∈ X . Then there exists a unique quadratic mapping Q : X → Y such that 3 pi +1 X

f (x) − Q(x) ≤ δkxkpXi 2 Y 2pi − 4 i=1

for all x ∈ X . Proof. Defining ϕ(x, y, z) = δ kxkpX1 + kykpX2 + kzkpX3




and applying Theorem 3.1 for the case p1 , p2 , p3 > 2,

and Theorem 3.2 for the case p1 , p2 , p3 < 2, we get the desired results.



Corollary 3.4. Let δ be a nonnegative real number and p1 , p2 , p3 be positive real numbers such that p1 +p2 +p3 6= 2. Let f : X → Y be an even mapping satisfying



     x−y−z y−x−z

x+y+z

+f +f − f (y) − f (z)

f 2 2 2 Y

 
z−x−y

p1 p2 p3 ≤ f (x) − f

+ δ kxkX · kykX · kzkX 2 Y for all x, y, z ∈ X . Then there exists a unique quadratic mapping Q : X → Y such that p1 +p2 +p3 +1

p1 +p2 +p3

f (x) − Q(x) ≤ 2 δkxkX Y 2p1 +p2 +p3 − 4

for all x ∈ X . Proof. Defining ϕ(x, y, z) = δ kxkpX1 · kykpX2 · kzkpX3 and applying Theorem 3.1 for the case p1 + p2 + p3 > 2,




and Theorem 3.2 for the case p1 + p2 + p3 < 2, we get the desired results.



4. Hyers-Ulam stability of the functional equation (0.1): Type B In this section, we bring another type of stability theorems for the quadratic functional equation (0.1) which is more prevalent in considering stability problems rather than the given type in the previous section. First of all, for convenience, we define for a given mapping f : X → Y, the difference operator:         x−y−z y−x−z z−x−y x+y+z +f +f +f Df (x, y, z) =: f 2 2 2 2 − f (x) − f (y) − f (z) for all x, y, z ∈ X . Theorem 4.1. Let ϕ : X 3 → [0, ∞) be a function satisfying ϕ(0, 0, 0) = 0 and (3.1). Denote by φ a function such that φ(x, y, z) :=

∞  n X 9n 2 n=0

4

ϕ n

3

x, n

2n 2n y, z 2,

and Theorem 4.2 for the case p1 , p2 , p3 < 2, we get the desired results.



Corollary 4.4. Let δ be a nonnegative real number and p1 , p2 , p3 be positive real numbers such that p1 +p2 +p3 6= 2. Let f : X → Y be an even mapping satisfying



Df (x, y, z) ≤ δ(kxkp1 · kykp2 · kzkp3 ) X X X Y for all x, y, z ∈ X . Then there exists a unique quadratic mapping Q : X → Y such that

2p1 +p2 +p3 −2 p1 +p2 +p3

f (x) − Q(x) ≤ θkxkX Y p +p 2 1 2 +p3 3p1 +p2 +p3 − 9 4 for all x ∈ X . Proof. Defining ϕ(x, y, z) = δ kxkpX1 · kykpX2 · kzkpX3 and applying Theorem 4.1 for the case p1 + p2 + p3 > 2,




and Theorem 4.2 for the case p1 + p2 + p3 < 2, we get the desired results.



This paper is just a start for the quadratic functional equation (0.1). Actually this functional equation and its stability problems can be studied more in various mathematical structures and spaces. Such this studied approach can cause to have a deeper description of this equation’s unknown properties which will probably be more interesting and remarkable. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] C. Borelli, G.L. Forti, On a general Hyers-Ulam stability result, Int. J. Math. Math. Sci. 18 (1995), 229–236. [3] J. Bae, K. Jun, On the generalized Hyers-Ulam-Rassias stability of an n-dimensional quadratic functional equation, J. Math. Anal. Appl. 258 (2001), 183–193. [4] J. Bae, W. Park, On the generalized Hyers-Ulam-Rassias stability in Banach modules over a C ∗ -algebra, J. Math. Anal. Appl. 294 (2004), 196–205. [5] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [6] A. Chahbi, N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [7] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [8] S. Czerwik, The stability of the quadratic functional equation, in: Th.M. Rassias, J. Tabor (Eds.), Stability of Mappings of Hyers-Ulam Type, Hadronic Press, Florida, 1994, 81–91. [9] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [10] I. Chang, H. Kim, On the Hyers-Ulam stability of quadratic functional equations, JIPAM. J. Inequal. Pure Appl. Math. 3, (2002), Art. 33. [11] M. Eshaghi Gordji, G. Kim, J. Lee, C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 45–54. [12] G. Z. Eskandani, P. Gˇ avruta, Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces, J. Nonlinear Sci. Appl. 5 (2012), 459–465. [13] P. G˘ avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [14] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl Acad. Sci. U.S.A. 27 (1941), 222–224. [15] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [16] S. Jung, On the Hyers-Ulam-Rassias stability of a quadratic functional equation, J. Math. Anal. Appl. 232 (1999), 384–393.

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Some New Results on Preconditioned Generalized Mixed-Type Splitting Iterative Methods Guangbin Wang∗†, Fuping Tan‡ and Yuncui Zhang§

Abstract In this paper, we present three preconditioned generalized mixed-type splitting (GMTS) methods for solving the weighted linear least square problem. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the preconditioned GMTS methods converge faster than the GMTS method whenever the GMTS method is convergent. Finally, we give two numerical examples to confirm our theoretical results.

Keywords: parison.

Preconditioning, GMTS method, linear system, convergence, com-

2000 AMS Classification: 65F10.

1. Introduction We consider the following weighted least squares problem (1.1)

min (Ax − b) W −1 (Ax − b) , T

x∈Rn

where A ∈ Rn×n is nonsigular, b ∈ Rn , W ∈ Rn×n is a symmetric positive definite matrix, see [1,4,9]. In order to solve it, one has to solve a nonsingular linear system as (1.2)

Hy = f,

where (1.3)

T

H=A W

−1

( A=

I −B L

U I −C

) ∈ Rn×n

is an invertible matrix with B = (bij )p×p , C = (cij )q×q , L = (lij )q×p , U = (uij )p×q , p + q = n and f = AT W −1 b ∈ Rn , see [1,4]. Throughout the paper, we consider the following decomposition for the matrix H, ∗

Department of Mathematics, Qingdao Agricultural University, Qingdao, China Email: [email protected] † Corresponding Author. ‡ Department of Mathematics, Shanghai University, Shanghai, China Email: [email protected] § Department of Mathematics, Qingdao University of Science and Technology, Qingdao, China Email: [email protected]

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ˆ −L ˆ−U ˆ , in which H=D ( ) ( ) I 0 0 0 ˆ= ˆ= (1.4) D , L , 0 I −L 0

( ˆ= U

B 0

−U C

) .

In [1], authors established a generalized AOR(GAOR) method to solve systems of linear equations (1.2). In paper [2, 3] , authors studied the preconditioned GAOR methods. In [4], authors presented a generalized mixed-type splitting (GMTS) iterative method which is generalized GAOR method. And they studied the preconditioned generalized mixed-type splitting iterative methods to solve (1.2). They showed that the preconditioned GMTS methods converge faster than the GMTS method, whenever the GMTS method is convergent. In this paper, we propose three new preconditioners and give the comparison theorems between the preconditioned and original methods. These results show that the preconditioned GMTS methods converge faster than the GMTS method whenever the GMTS method is convergent. And we prove that in the case that the GMTS method is convergent, using the third preconditioned GMTS method leads to the better convergence rate than the first and the second preconditioned GMTS methods. In Section 4, we give two examples to confirm our theoretical results. And we know that the preconditioned GMTS methods with preconditioners in this paper have the better converge rate than the preconditioned GMTS method with preconditioner P ∗ .

2. Preliminaries 2.1 Definition [5] A ∈ Rn×n is called a Z-matrix if aij ≤ 0 for i, j = 1, 2, ..., n (i ̸= j). 2.2 Definition [5] Let A be a Z-matrix with positive diagonal elements. Then the matrix A is called an M-matrix if A is nonsingular and A−1 ≥ 0. 2.3 Definition [6] The splitting A = M − N is called (1) a regular splitting of A if M −1 ≥ 0 and N ≥ 0; (2) a nonnegative splitting of A if M −1 ≥ 0, M −1 N ≥ 0 and N M −1 ≥ 0; (3) a weak nonnegative splitting of A if M −1 ≥ 0 and either M −1 N ≥ 0 (the first type) or N M −1 ≥ 0 (the second type); (4) a convergent splitting of A if ρ(M −1 N ) < 1. 2.1. Lemma. [4] Let A be a Z-matrix. Moreover, suppose that A = M − N is a weak nonnegative splitting of the first type. Then ρ(M −1 N ) < 1 if and only if A is an M-matrix. 2.2. Lemma. [7] Let A = M − N be a regular splitting of A. Then ρ(M −1 N ) < 1 if and only if A is nonsingular and A−1 is nonnegative. 2.3. Lemma. [8] Let matrix A = (aij )n×n be given such that (1) aij ≤ 0 for i, j = 1, 2, ..., n (i ̸= j), (2) A is nonsingular, (3) A−1 ≥ 0. Then, (1) aii > 0 for i = 1, 2, ..., n, i.e., A is an M-matrix, (2) ρ(B) < 1 where B = I − D−1 A, where D = diag{a11 , ..., ann }.

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2.4. Lemma. [6] Let A = M1 −N1 = M2 −N2 be two convergent weak nonnegative splittings of A, where A−1 ≥ 0, if M1−1 ≥ M2−1 then ρ(M1−1 N1 ) ≤ ρ(M2−1 N2 ).

3. Comparison results Consider the linear system (1.2), the generalized mixed-type splitting (GMTS) iterative method is given as follows: ˆ + D1 + L1 − L)y ˆ (k+1) = (D1 + L1 + U ˆ )y (k) + f (3.1) (D ˆ L ˆ and U ˆ are defined by (1.4), and D1 is an auxiliary nonnegative block where D, diagonal matrix, L1 is an auxiliary strictly nonnegative block lower triangular ˆ and 0 ≤ L1 ≤ L. ˆ Evidently, the iteration matrix of matrix such that 0 ≤ D1 ≤ D the GMTS iterative method is given as follow: ˆ + D1 + L1 − L) ˆ −1 (D1 + L1 + U ˆ ). T = (D In this paper, we propose the new preconditioners as follows, ( ) I + Si 0 (3.2) Pi∗ = , i = 1, 2, 3 0 I + Vi where



0 b21 .. .

0 ··· 0 ··· .. . . . . 0 ··· 0 ···

0 0 .. .

0 0 .. .



      S1 =  ,    bp−1,1 0 0  bp1 0 0  0 b12 · · · b1,p−1  b21 0 ··· 0   .. . . . .. .. S3 =  . . .   bp−1,1 0 ··· 0 bp1 0 ··· 0   0 0 ··· 0 0  c21 0 ··· 0 0     .. .. . . .. ..  , V1 =  .  . . . .    cq−1,1 0 · · · 0 0  cq1 0 ··· 0 0  0 c12 · · · c1,q−1  c21 0 ··· 0   .. . . . .. .. V3 =  . . .   cq−1,1 0 ··· 0 cq1 0 ··· 0 Then Pi∗ H can be expressed by ( ) I − Bi∗ Ui∗ ∗ Pi H = , L∗i I − Ci∗

    S2 =    b1p 0 .. . 0 0



0 b12 0 0 .. .. . . 0 0 0 0

··· ··· .. .

b1,p−1 0 .. .

b1p 0 .. .

··· ···

0 0

0 0

0 0 .. .

··· ··· .. .

c1,q−1 0 .. .

c1q 0 .. .

    ,  

   ,   

c12 0 .. .

   V2 =    0 0 0 0  c1q 0   ..  . .   0  0

··· 0 ··· 0

0 0

    ,  

where Bi∗ = B − Si (I − B), Ci∗ = C − Vi (I − C), L∗i = (I + Vi )L, Ui∗ = (I + Si )U .

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Let us consider the corresponding splitting for the preconditioned GMTS method, ¯ i = P ∗H = M ¯i − N ¯i , where that is the generalized mixed-type splitting for the H i ¯i = D ˆ i∗ + D ¯1 + L ¯1 − L ˆ ∗i , M and ˆ i∗ = D

(

I 0

0 I

)

ˆ ∗i = , L

(

¯i = D ¯1 + L ¯1 + U ˆi∗ N 0 −L∗i

0 0

)

ˆi∗ = , U

(

Bi∗ 0

−Ui∗ Ci∗

) , i = 1, 2, 3,

¯ 1 is an auxiliary nonnegative block diagonal matrix with 0 ≤ D ¯1 ≤ D ˆ ∗, L ¯ 1 is an D i ¯ ˆ∗. auxiliary strictly nonnegative block lower triangular matrix with 0 ≤ L1 ≤ L i The iteration matrix of the preconditioned GMTS method is ˆ∗ + D ¯1 + L ¯1 − L ˆ ∗ )−1 (D ¯1 + L ¯1 + U ˆ ∗ ). Ti∗ = (D i i i 3.1. Lemma. [4] Assume that L ≤ 0, U ≤ 0, B ≥ 0, C ≥ 0 and H in (1.2) is irreducible. If D1 is nonsingular, then the iteration matrix of the GMTS method is irreducible. 3.2. Lemma. [4] Assume that L ≤ 0, U ≤ 0, B ≥ 0, C ≥ 0, then the corresponding splitting of GMTS method is a regular splitting for the matrix H. Similar to the proof of Lemma 3.2, we can prove the following lemma. 3.3. Lemma. Assume that L ≤ 0, U ≤ 0, B ≥ 0, C ≥ 0, then the corresponding splitting of PGMTS method is a regular splitting for the matrix Pi∗ H (i = 1, 2, 3). 3.4. Theorem. Let H be an M-matrix, then Pi∗ H (i = 1, 2, 3) is an M-matrix. Proof. Consider the following splitting for H, H = M1 − N1 , ˆ∗ + U ˆ ∗ ), N1 = (P1∗()−1 (L where M1 ( = (P1∗ )−1 , ) ) ∗ B1 −U1∗ 0 0 ∗ ˆ ˆ∗ = . , U = and L 0 C1∗ −L∗1 0 ˆ∗ + U ˆ ∗ and M −1 ≥ 0. ThenH = M1 −N1 is a weak We can see that M1−1 N1 = L 1 nonnegative splitting of the first type. By the assumption H is an M-matrix, hence ˆ∗ − U ˆ ∗, Lemma 2.1 implies that ρ(M1−1 N1 ) < 1. Let us assume that P1∗ H = I − L −1 ∗ ∗ ˆ ˆ using the fact that ρ(L + U ) = ρ(M1 N1 ) < 1, by Lemma 2.2 and Lemma 2.3, it is easy to know that P1∗ H is an M-matrix. The similar results can be gotten when i = 2, 3.  Now, we will show that in the case that the GMTS method converges, the preconditioned GMTS methods converge faster. 3.5. Theorem. Let T and T1∗ be the iteration matrices of the GMTS and the preconditioned GMTS methods, respectively, assume that the matrix H is irreˆ ¯1 ≤ D ˆ ∗ , 0 ≤ L1 ≤ L, ˆ ducible, L ≤ 0, U ≤ 0, B ≥ 0, C ≥ 0, 0 ≤ D1 ≤ D, 0≤D 1 ∗ ¯ ˆ 0 ≤ L1 ≤ L1 , bi,1 > 0, cj,1 > 0, for some i ∈ {2, 3, ..., p}, j ∈ {2, 3, ..., q}. If ¯ 1 ≤ D1 and L ¯ 1 ≤ L1 , then ρ(T ∗ ) ≤ ρ(T ). ρ(T ) < 1, D 1 Proof. As the matrix H is irreducible, so the P1∗ H is irreducible. And by Lemma 3.1, we know that T and T1∗ are irreducible. Consider the GMTS splitting for the ˆ + D1 + L1 − L, ˆ N = D1 + L1 + U ˆ. matrix H = M − N , where M = D

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Obviously, H = M −N is a regular splitting, and by the assumption ρ(M −1 N ) < 1, we can get that H is an M-matrix. From Theorem 3.4, we know that P1∗ H is ¯1 = M ¯ 1 −N ¯1 is a regular also an M-matrix. Thus, from Lemma 3.3, we know that H −1 ¯ ∗ ¯ splitting. Therefore, as H is an M-matrix, we can get ρ(T1 ) = ρ(M1 N1 ) < 1. Now, we define the following splitting for the matrix H, H = M1∗ − N1∗ , in ¯1 , N ∗ = (I + S¯1 )−1 N¯1 and which M1∗ = (I + S¯1 )−1 M 1 ( ) S 0 1 S¯1 = . 0 V1 Consider the iteration matrix of the GMTS method T = M −1 N , it is easy to see that ) ( ∗ D11 − D11 0 ¯1 = M −M , ∗ L21 + L − L∗21 − L∗1 D22 − D22 where

(

) ( ∗ ) D11 0 D11 0 ˆ ¯ ˆ 1∗ , D1 = ≤ D, D1 = ≤D ∗ 0 D22 0 D22 ( ) ( ) 0 0 0 0 ˆ1. ˆ and L ¯1 = ≤L L1 = ≤L L21 0 L∗21 0 It is known that L∗1 = (I + V1 )L, hence L∗1 − L = V1 L ≤ 0. ¯ 1 ≤ M , so M ¯ −1 ≥ M −1 . Consequently, By computations, we know that M 1 ¯ −1 ≤ M ¯ −1 (I + S¯1 ) = (M1∗ )−1 . M −1 ≤ M 1 1

From Lemma 2.4, we deduce that ¯ −1 N ¯1 ) = ρ((M1∗ )−1 N1∗ ) ≤ ρ(M −1 N ), ρ(M 1 so ρ(T1∗ ) ≤ ρ(T ).



Similar to the proof of Theorem 3.5, we can get the following two theorems. 3.6. Theorem. Let T and T2∗ be the iteration matrices of the GMTS and the preconditioned GMTS methods, respectively. Assume that the matrix H is irreˆ ˆ ¯2 ≤ D ˆ ∗ , 0 ≤ L2 ≤ L, ducible, L ≤ 0, U ≤ 0, B ≥ 0, C ≥ 0, 0 ≤ D2 ≤ D, 0≤D 2 ∗ ¯ ˆ 0 ≤ L2 ≤ L2 , b1,i > 0, c1,j > 0, for some i ∈ {2, 3, ..., p}, j ∈ {2, 3, ..., q}. If ¯ 2 ≤ D2 and L ¯ 2 ≤ L2 , then ρ(T ∗ ) ≤ ρ(T ). ρ(T ) < 1, D 2 3.7. Theorem. Let T and T3∗ be the iteration matrices of the GMTS and the preconditioned GMTS methods, respectively. Assume that the matrix H is irreducible, ˆ 0≤L ¯3 ≤ ˆ 0≤D ¯3 ≤ D ˆ ∗ , 0 ≤ L3 ≤ L, L ≤ 0, U ≤ 0, B ≥ 0, C ≥ 0, 0 ≤ D3 ≤ D, 3 ∗ ˆ L3 , bi,1 > 0, cj,1 > 0, b1,i > 0, c1,j > 0, for some i ∈ {2, 3, ..., p}, j ∈ {2, 3, ..., q}. If ¯ 3 ≤ D3 and L ¯ 3 ≤ L3 , then ρ(T ∗ ) ≤ ρ(T ). ρ(T ) < 1, D 3 Now, we prove that in the case that the GMTS method is convergent, using the third preconditioned GMTS method leads to the better convergence rate than the first and the second preconditioned GMTS methods. 3.8. Theorem. Suppose that the matrix H is irreducible, L ≤ 0, U ≤ 0, B ≥ 0, C ≥ 0, bi,1 > 0, cj,1 > 0, b1,i > 0, c1,j > 0, for some i ∈ {2, 3, ..., p}, j ∈

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{2, 3, ..., q}, the auxiliary block diagonal matrices are chosen as αi I and the auxiliary block lower triangular matrices as βi L∗i for i = 1, 3, 0 ≤ α3 ≤ α1 ≤ 1, 0 ≤ β1 ≤ β3 ≤ 1. Then ρ(T3∗ ) ≤ ρ(T1∗ ) if ρ(T ) < 1. Proof. By the assumption ρ(T ) < 1, and according the Lemma 2.1, H is an Mfi − N ei , i = 1, 3 where matrix. Assume that Pi∗ H = M ( ) ( ∗ ) i i I + D11 0 Bi + D11 −Ui∗ f e Mi = , Ni = , i i Li21 + L∗i I + D22 Li21 Ci∗ + D22 i i and Li21 = −βi L∗i , D11 = αi Ip , D22 = αi Iq for i = 1, 3. Now, we define the following splitting for the matrix H, i.e. H = Mi − Ni (i = fi and Ni = (I + Sei )−1 N ei , 1, 3) such that ( Mi = (I +)Sei )−1 M Si 0 where Sei = . 0 Vi Since

L121 − L321 = −β1 L∗1 + β3 L∗3 ≥ β1 L∗3 − β1 L∗1 = −β1 (L∗1 − L∗3 ), so L121 − L321 + L∗1 − L∗3 ≥ (1 − β1 )(L∗1 − L∗3 ), ) 3 1 0 − D11 D11 3 1 ∗ ∗ 3 1 ( L21 − L21 + L1 − L3 D22 − D22) (α1 − α3 )Ip 0 , ≥ (1 − β1 )(L∗1 − L∗3 ) (α1 − α3 )Iq f1 ≥ M f3 . as L∗1 − L∗3 = (V1 − V3 )L ≥ 0, then M −1 −1 f ≥ 0, M f ≥ 0, hence M f−1 ≤ M f−1 and Notice that M (

then

f1 − M f3 M

1

M1−1

=

3

1

3

f−1 (I + Se1 ) =M 1 f−1 + M f−1 Se1 =M 1 1 f−1 + M f−1 (Se1 − Se3 ) + M f−1 Se3 ≤M 3 1 1 −1 −1 f +M f Se3 ≤M 3 3 f−1 (I + Se3 ) = M −1 . =M 3

3

Since H is an M-matrix, Lemma 2.4 implies that ρ(M3−1 N3 ) ≤ ρ(M1−1 N1 ). ei for i = 1, 3, we can conclude that f−1 N According Mi−1 Ni = M i ρ(T3∗ ) ≤ ρ(T1∗ ).  Similar to the proof of Theorem 3.8, we can get the following theorem. 3.9. Theorem. Suppose that the matrix H is irreducible, L ≤ 0, U ≤ 0, B ≥ 0, C ≥ 0, bi,1 > 0, cj,1 > 0, b1,i > 0, c1,j > 0, for some i ∈ {2, 3, ..., p}, j ∈ {2, 3, ..., q}, the auxiliary block diagonal matrices are chosen as αi I and the auxiliary block lower triangular matrices as βi L∗i for i = 2, 3, 0 ≤ α3 ≤ α2 ≤ 1, 0 ≤ β2 ≤ β3 ≤ 1. Then ρ(T3∗ ) ≤ ρ(T2∗ ) if ρ(T ) < 1.

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4. Examples 4.1 Example Consider ( ) I −B U H= , L I −C where B = (bij )p×p , C = (cij )(n−p)×(n−p) , L = (lij )(n−p)×p and U = (uij )p×(n−p) with 1 bii = , i = 1, 2, · · · , p, 10 × (i + 1) 1 1 − , i < j, i = 1, 2, · · · , p − 1, j = 2, · · · , p, 30 30 × j + i 1 1 bij = − , i > j, i = 2, · · · , p, j = 1, 2, · · · , p − 1, 30 30 × (i − j + 1) + i 1 , i = 1, 2, · · · , n − p, cii = 10 × (p + i + 1) 1 1 cij = − , i < j, i = 1, 2, · · · , n−p−1, j = 2, · · · , n−p, 30 30 × (p + j) + p + i 1 1 cij = − , i > j, i = 2, · · · , n−p, j = 1, 2, · · · , n−p−1, 30 30 × (i − j + 1) + p + i 1 1 lij = − , i = 1, 2, · · · , n − p, j = 1, 2, · · · , p, 30 × (p + i − j + 1) + p + i 30 1 1 uij = − , i = 1, 2, · · · , p, j = 1, 2, · · · , n − p. 30 × (p + j) + i 30 In the experiments, the auxiliary matrices are chosen such that 1 1 γ c γ b∗ D1 = 0.5( −1)I, D1 = 0.5( −1)I, L1 = 0.5(1− )L i , L1 = 0.5(1− )Li . ω ω ω ω From Table 1, we see that these results accord with Theorems 3.5 - 3.9. bij =

Table 1. The spectral radii of the GMTS and preconditioned GMTS iteration matrices

n 10 20 20 25 30 30

ω 0.9 0.8 0.8 0.8 0.9 0.9

r p ρ(T ) 0.8 5 0.2352 0.6 5 0.5736 0.6 10 0.5551 0.6 8 0.7164 0.7 10 0.8680 0.7 20 0.8676

ρ(T1∗ ) 0.2156 0.5609 0.5413 0.7074 0.8635 0.8630

ρ(T2∗ ) 0.2140 0.5605 0.5404 0.7070 0.8633 0.8627

ρ(T3∗ ) 0.2048 0.5568 0.5334 0.7033 0.8613 0.8605

In [4], the authors considered the following preconditioner (4.1)

P∗ =

(

I +S 0

0 I +V

) ,

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where



0 0 .. .

bp1 α

0 ··· 0 ···

0 0 .. .

0 0 .. .

··· ··· .. .

cq1 β

0 0

··· ···

   S=   0 

··· ··· .. .

0 0 .. .

   V =   0

0 0 .. .

0 0 .. .



   ,  0 0  0 0  0 0 0 0   .. ..  . . .    0 0 0 0

Table 2. The spectral radii of the preconditioned GMTS iteration matrices

n 10 20 20 25 30 30

ω 0.9 0.8 0.8 0.8 0.9 0.9

r p 0.8 5 0.6 5 0.6 10 0.6 8 0.7 10 0.7 20

α=β 3 2 2 3 2 2

ρ(T ∗ ) 0.2335 0.5729 0.5542 0.7161 0.8678 0.8673

ρ(T1∗ ) 0.2156 0.5609 0.5413 0.7074 0.8635 0.8630

ρ(T2∗ ) 0.2140 0.5605 0.5404 0.7070 0.8633 0.8627

ρ(T3∗ ) 0.2048 0.5568 0.5334 0.7033 0.8613 0.8605

Here, T ∗ is the GMTS iteration matrix for solving P ∗ Hy = P ∗ f . From Table 2, we see that the preconditioned GMTS methods with preconditioners in this paper have better converge rates than the preconditioned GMTS method with preconditioner P ∗ . 4.2 Example The coefficient matrix H in Equation (1.2) is given by ( ) I −B U H= , L I −C where



b11

 1 4 B=  0 1 4

1 4

0 1 4

0

0 1 4

0 1 4

 0 0   1 , 4 0

 C=

c11 1 4 1 4

1 4

0 1 4

0 1 4

 ,

0

 − 14 0 0 0 0  0 − 14 0  . 0 − 41 0  , U =  L= 0  0 0 − 14  1 0 −4 0 0 − 14 0 0 Table 3 displays the spectral radii of the corresponding iteration matrices with ω = 0.9, γ = 0.8 and different values of b11 and c11 . From Table 3, we can see that ρ(Ti∗ ) ≤ ρ(T ) for i = 1, 2, 3 and ρ(T3∗ ) ≤ ρ(Ti∗ ) for i = 1, 2 when ρ(T ) < 1. These numerical results are in accordance with the theoretical results given in Theorems 3.5- 3.9. 

− 14

− 14



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Table 3. The spectral radii of the GMTS and preconditioned GMTS iteration matrices

b11 c11 0 0 0 0.3 0.2 0.2 0.2 0.5 0.5 0.5

ρ(T ) 0.6804 0.7657 0.7614 0.8860 0.9553

ρ(T1∗ ) 0.6303 0.7253 0.7186 0.8677 0.9483

ρ(T2∗ ) 0.6381 0.7323 0.7265 0.8713 0.9499

ρ(T3∗ ) 0.6140 0.7071 0.6987 0.8596 0.9453

5. Conclusion In this paper, we propose three new preconditioners and give comparison theorems between the preconditioned and original methods. These results show that the preconditioned GMTS methods converge faster than the GMTS method whenever the GMTS method is convergent. Finally, we give two examples to confirm our theoretical results.

6. Acknowledgments The authors would like to thank the referees for their valuable comments and suggestions, which greatly improved the original version of this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 11401333) and the Natural Science Foundation of Shandong Province (No. ZR2012AL09).

References [1] J.Y.Yuan and X.Q. Jin, Convergence of the generalized AOR method, Applied Mathematics and Computation 99, 35-46, 1999. [2] X. X. Zhou, Y. Z. Song, L. Wang and Q. S. Liu, Preconditioned GAOR methods for solving weighted linear least squares problems, Journal of Computational and Applied Mathematics 224, 242-249, 2009. [3] J. H. Yun, Comparison results on the preconditioned GAOR method for generalized least squares problems, International Journal of Computer Mathematics 89, 2094-2105, 2012. [4] F. Beik, N. Shams, Preconditioned generalized mixed-type splitting iterative method for solving weighted least squares problems, International Journal of Computer Mathematics 91, 944-963, 2014. [5] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. [6] Z. I. Woznicki, Basic comparison theorems for weak and weaker matrix splitting, Electron. J. Linear Algebra 8, 53-59, 2001. [7] R.S.Varga, Matrix Iterative Analysis, in: Springer Series in Computational Mathematics, vol. 27, Springer-Verlag, Berlin, 2000. [8] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2nd edition, Philadelphia, 2003. [9] S. Searle, G. Casella, C. McCulloch, Variance Components, Wiley/Intersicence, New York, 1992.

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A Linear Adaptive time-stepping Method for Solving Vibration Problems with Damping Terms Jianguo Huang

1

and Huashan Sheng

School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University Shanghai 200240, China Abstract A linear adaptive time-stepping method is devised for linear or nonlinear damping vibration analysis, which has wide applications in civil engineering. In the time direction, the underlying problem is discretized by a linear C 0 -continuous discontinuous Galerkin method combined with the technique of linearization. By means of the energy method, some optimal a posteriori error estimates are established for linear vibration problems. Motivated by these estimates, we design an adaptive time-stepping strategy for actual computation. Numerical results are performed to illustrate the efficiency of the adaptive method. Keywords. Time-stepping method, Vibration, Damping, A posteriori error analysis, Adaptive algorithm

1

Introduction

This paper aims to design and analyze an adaptive time-stepping method for solving the following problem: For any real number T > 0, find u : [0, T ] → Rd (with d the spatial dimension) such that { ( ) Mu′′ (t) + F t, u(t), u′ (t) = 0, 0 < t < T, (1.1) ′ u(0) = u0 , u (0) = v0 , where (·)′ and (·)′′ denote respectively the first and second order derivatives in time; M is a given (d × d) matrix and F is a given vector-valued function from [0, T ] × Rd × Rd into Rd ; u0 and v0 are two given vectors in Rd . The above problem is frequently encountered in structure analysis of dynamical transient response (cf. [5]). Concretely speaking, the mathematical models for structure analysis are described by a system of second-order linear/nonlinear evolution equations, which give rise to the problem (1.1), after spatial discretization by finite element methods, finite difference methods or spectral methods (cf. [2, 9, 11, 16, 17, 21, 22]). When the vector-valued function F is linear with respect to u and u′ , there are various numerical methods for solving the problem (1.1). The most widely used may be classified as modal superposition (cf. [6, 14]) and direct-time integration methods including the RungeKutta, central difference, Houbolt, Newmark-β and Wilson-θ methods (see [11] and the references therein for details). The space-time finite element method (cf. [7, 12, 13]) is another widely developed approach for solving second order time evolution equations. One typical way is using the time-discontinuous Galerkin (TDG) method (cf. [7,15]) in the time direction for the displacement and velocity fields together, but it has the disadvantage that an ill-conditioned (4 × 4) block system must be solved at each time step, which is time consuming. To overcome this difficulty, some linear C 0 -continuous time-stepping methods were used in [18], where only the primal variables are involved and only a (1×1) block system 1

Corresponding author. E-mail address: [email protected]. The work of this author was partly supported by NSFC (Grant no. 11571237).

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should be solved at each time step. Moreover, an adaptive method was proposed in [18] for solving second order abstract evolution equations, where the optimal a posteriori error estimates are established, which, in conjunction with the error equidistribution strategy and some ideas implied in the Runge-Kutta-Felberg method, leads to an adaptive time-stepping method. In this paper, we intend to use some ideas in [18] to develop an adaptive time-stepping method for solving the problem (1.1). In the time direction, the problem (1.1) is discretized by a linear C 0 -continuous discontinuous Galerkin method combined with the technique of linearization (including three linearization methods). Then, by means of the energy method, some optimal a posteriori error estimates are established for linear vibration problems via some ideas in [18]. It deserves to emphasize that the mathematical argument developed here is greatly simplified by using the Lagrange basis functions instead of the Legendre polynomials. Motivated by these estimates, we construct a posteriori error estimates for nonlinear problems, based on which we design an adaptive time-stepping strategy for actual computation. Numerical results are performed to illustrate the efficiency of the adaptive method. The rest of this paper is organized as follows. In Section 2, we present a time-stepping finite element method for the problem (1.1), and the detailed implementation of the previous method is also developed for actual computation. In Section 3, a posteriori error analysis is established in detail for linear vibration problems. In Section 4, we propose an adaptive algorithm based on some a posteriori error estimates. A series of numerical results are performed in the final section.

2

A linear time-stepping finite element method

2.1

The formulation of a linear time-stepping finite element method

Throughout this paper, we assume that Problem (1.1) has a unique solution and the matrix M is symmetric positive definite. We use a standard time-stepping method to discretize Problem (1.1) (cf. [10, 18, 19]). To this end, we first partition the time interval I := (0, T ) with the nodes 0 = t0 < t1 < · · · < tN = T, to get the following subintervals: Jn = (tn−1 , tn ],

kn = tn − tn−1 ,

1 ≤ n ≤ N.

Define { V1 =

¯ v|Jn (t) = v : I¯ → Rd ; v ∈ C(I),

¯ v|Jn (t) = v : I¯ → Rd ; v ∈ C 1 (I),

tj wj , wj ∈ Rd , 1 ≤ n ≤ N ,

2 ∑

} tj wj , wj ∈ Rd , 1 ≤ n ≤ N ,

j=0

{ Hq =

}

j=0

{ W2 =

1 ∑

v : I¯ → L (I); v|Jn (t) = 2

q ∑

}

t wj , wj ∈ R , 1 ≤ n ≤ N , j

d

q = 0, 1.

j=0

Let V1 (Jn ) and W2 (Jn ) be the restrictions of V1 and W2 to Jn , respectively. Similarly, denote by Hq (Jn ) the restriction of Hq to Jn . Thus, our time-stepping method for (1.1) is 2

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to find U ∈ V1 such that ∫ ⟨ ⟩ ⟨ ( ) ⟩) (⟨ ′′ ⟩  n−1 n−1 n−1  ˙+ ˙− −U U , w′ M + F t, U, U′ , w′ dt + U , w ˙+ Jn 0

 U = u , 0

˙ 0− U

w ∈ V1 (Jn ),

= v0 ,

M

= 0, (2.1)

1 ≤ n ≤ N,

where ⟨a, b⟩ := bT a,

⟨a, b⟩A := bT Aa,

n−1 w ˙± := lim w′ (tn−1 ± s), s→0+

2.2

a, b ∈ Rd , A ∈ Rd×d ,

(2.2)

wn−1 := w(tn−1 ).

Implementation of the time-stepping method

Since U ∈ V1 , we have by a direct manipulation that, for any t ∈ Jn , ˙ n− , U′ (t) = U

˙ n− , U(t) = Un−1 + (t − tn−1 )U

U′′ (t) = 0.

(2.3)

To implement the method (2.1) in actual computation, we require to linearize the nonlinear function F(t, U, U′ ) with respect to U. As shown in Figure 1, for a given function g(t), its linearization over Jn are usually the interpolants given by IL g(t) = g(tn−1 ) + (t − tn−1 )g′ (tn−1 ) or

IR g(t) = g(tn−1 ) + (t − tn−1 )g′ (tn ),

g(t )

g

g

IL g

g(t ) IR g

t t n 1

t ∈ Jn .

Jn

t

tn

Jn

t n 1

(a) IL

tn

(b) IR

Figure 1: Diagrams of the (local) interpolate operators IL and IR . Note that the function F = F(t, U, U′ ) is discontinuous at the interior node tn . Recalling the expression (2.3), we have by the direct computation that the right limit of F at t = tn−1 can be expressed as n−1 ˙ n ˙ n−1 Fn−1 = F(tn−1 , Un−1 , U , U− ). + + ) = F(tn−1 , U

(2.4)

Using the chain rule for differentiation and (2.3), we find that, at t = tn , the left limit of the full derivative of F(t, U, U′ ) with respect to t is given as follows: ˙ n− = ∂F (tn , Un , U ˙ n− ) + ∂F (tn , Un , U ˙ n− )U ˙ n− + F ∂t ∂U ∂F ˙ n− ) + ∂F (tn , Un , U ˙ n− )U ˙ n− = (tn , Un , U ∂t ∂U ∂F ∂F ˙ n =: + U . ∂t tn ∂U tn − −

Similarly, we have

∂F ˙ n− )0 (tn , Un , U ∂U′

−

∂F ∂F n−1 ˙ ˙ n. F+ := + U ∂t tn−1 ∂U tn−1 − +

+

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With these results in mind, we have by the definitions of the interpolation operators IL and IR that ( ˙ n−1 Left side Scheme : F t, U(t), U′ (t)) ≈ IL F = Fn−1 + (t − tn−1 )F (2.5) + + , ( n−1 ′ n ˙ −. Right side Scheme : F t, U(t), U (t)) ≈ IR F = F+ + (t − tn−1 )F (2.6) Now, inserting (2.3) and (2.6) into the first equation of (2.1) and taking w ˙ to be w∗ or (t − tn−1 )w∗ , where w∗ is any constant vector in Rd , we find that the method (2.1) is ˙ n− }N such that equivalent to finding {U n=0 ) ( 2 1 kn ∂F ∂F n 2 + kn Fn−1 = MU ˙ − + kn ˙ n−1 U 1 ≤ n ≤ N. (2.7) M+ + − , 2 ∂U tn 2 ∂t tn − − ∂F ∂F Note that the quantities ∂t tn , ∂U tn and Fn−1 are all the functions of the unknown vector + − − ˙ n− , so the above scheme is implicit. However, if we use the linearization formulation (2.5) U instead of (2.6), then the system (2.1) reduces to ) ( 1 ∂F kn2 ∂F n 2 ˙ − + kn ˙ n−1 U + kn Fn−1 = MU 1 ≤ n ≤ N. (2.8) M+ + − , 2 ∂U tn−1 2 ∂t tn−1 +

+

It is noted that in most vibration problems, it suffices for us to deal with the linear damping case, indicating that the function F is linear with respect to the independent variable u′ . ∂F ˙ n− , the In this case, since the quantities ∂F and ∂U in (2.8) do not depend on U ∂t tn−1 tn−1 + + ˙ n− . Hence, we can work system (2.8) is essentially a linear system of the unknown vector U ˙ n− with much less computational cost, compared to the method (2.7). out U In order to balance the efficiency and stability of the time-stepping method, it is very natural to split the nonlinear term F into two parts FL and FR , which correspond to the non-stiff and the stiff terms of the original system (1.1), respectively. Then, it is better for us to use IL FL + IR FR to approximate F in (2.1). In other words, we have ( n−1 ) ˙ ˙n Semi − side Scheme : F ≈ IL FL + IR FR = Fn−1 + (t − tn−1 ) F (2.9) + L+ + FR− . It is noted that for the linear damping system, the semi-side scheme also yields a linear ˙ n− . system for getting the unknown vector U Now, let us present the solution process of the method (1.1) in detail. Once we obtain ˙ n− by solving the system (2.7) or (2.8). Then the function U over Jn U in Jn−1 , we can get U ˙ n− for all t ∈ Jn . is completely determined using the formulation U(t) = Un−1 + (t − tn−1 )U On implementing this computation recursively, we can thereby determine the function U completely. In the last part of this subsection, we give the solution process explicitly for the vibration analysis related to linear transient dynamic response. At this moment, we can reformulate the problem (1.1) as follows. For any real number T > 0, find u : [0, T ] → Rd such that  ′′ ′ 0 < t < T,   Mu + Cu + Ku = f , u(0) = u0 , (2.10)   ′ u (0) = v0 , where C and K are the (d × d) damping and stiffness matrices of the dynamic system, respectively. We assume that C and K are symmetric and semi-definite. Observing that ( ) F t, u(t), u′ (t) = Cu′ + Ku − f , 4

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we have from the variational formulation (2.1) that ( 2 ) kn ˙ n− = MU ˙ n−1 K + kn C + M U − kn KUn−1 + f n , − 2 ∫ where f n := Jn f dt.

3

1 ≤ n ≤ N,

(2.11)

A posteriori error analysis for linear problems

For the numerical method (2.1) for the linear vibration problem (2.10), following the similar arguments leading to Theorem 2.5 in [18], we can derive some stability estimates to the numerical solution U and then establish the required a priori error estimates. Another way to derive such estimates is to use the mathematical argument due to [24]. Since the objective of this article is to develop efficient adaptive time stepping method for the linear vibration problem (2.10) and the generalized problem (1.1), we will focus on in this section a posteriori error analysis for the problem (2.10) discretized by the method (2.1). Motivated by such an analysis, we will heuristically mention in the next section some error estimators for the nonlinear problem (1.1) and then devise the corresponding adaptive time stepping method.

3.1

Reconstruction

As shown in [18], in order to get efficient a posteriori error estimates for the method e from the approximate solution (2.1), we require to construct a higher order reconstruction U U. So let us first recall such a reconstruction given in [18]. Introduce an invertible linear e := Ie2 w ∈ operator Ie2 : V1 → W2 as follows. With any w ∈ V1 we associate an element w e Jn ∈ W2 defined by locally interpolating w in each subinterval Jn (1 ≤ n ≤ N ), i.e., w| W2 (Jn ) is uniquely determined by n−1 e e n−1 ) + kn w w(t) = w(t ˙− Φ0 (

t − tn−1 t − tn−1 n ) + kn w ˙− Φ1 ( ), kn kn

1 ≤ n ≤ N,

(3.1)

e e ′ (0) = w′ (0). In (3.1), the definition of Φ0 , Φ1 are and the initial values w(0) = w(0), w given as 1 1 (3.2) Φ0 (ξ) = − ξ 2 + ξ, Φ1 (ξ) = ξ 2 . 2 2 e a time reconstruction of w, as shown in Figure 2. It is easy to check by the We call w

w(t)

w(t) e w(t)

t

Figure 2: Diagram of Ie2 w. above construction that

n e ′ (tn ) = w w ˙− ,

1 ≤ n ≤ N.

(3.3)

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Thus, for an approximate solution U , the reconstructed function we hope to find is e ∈ W2 , defined by U t − tn−1 e e n−1 ) + kn U ˙ n−1 ˙ n− Φ1 ( t − tn−1 ), U(t) = U(t ) + kn U 1 ≤ n ≤ N. (3.4) − Φ0 ( kn kn By a direct computation we have e ′′ (t) = 1 (U ˙n −U ˙ n−1 U − ), kn −

1 ≤ n ≤ N.

(3.5)

Observing that the function U(t) can be rewritten as t − tn−1 ˙ n−1 ˙ n− Φ1 ( t − tn−1 ), ) + kn U U(t) = U(tn−1 ) + kn U + Φ0 ( kn kn

t ∈ Jn ,

subtracting which from (3.4) we know t − tn−1 e e n−1 + kn (U ˙ n−1 ˙ n−1 U(t) − U(t) = Un−1 − U −U ), + − )Φ0 ( kn

t ∈ Jn .

(3.6)

Hence, e n = Un−1 − U e n−1 + 1 k 2 U e ′′ , Un − U 2 n i.e.,

n 1 ∑ 2 e ′′ n e km U |Jm , U −U = 2 n

t ∈ Jn ,

t ∈ Jn .

(3.7)

m=1

Moreover, by integration by parts and (3.3), it follows that ∫ ∫ n−1 ′′ ′ e ˙ n−1 ˙ n−1 ⟨U , w ⟩M dt = ⟨U′′ , w′ ⟩M dt + ⟨U −U ˙+ ⟩M , + − ,w Jn

w ∈ V1 (Jn ),

Jn

and use the variational equation in (2.1) we further have ∫ e ′′ , w′ ⟩M + ⟨CU′ + KU − f , w′ ⟩) dt = 0, (⟨U

w ∈ V1

1 ≤ n ≤ N,

Jn

i.e.,

e ′′ + P0 (CU′ + KU − f ) = 0, MU

t ∈ Jn ,

(3.8)

where Pq (q = 0, 1) stands for the (local) orthogonal projection operator on to Hq (Jn ) (cf. [1]), defined by ∫ ⟨Pq v − v, w⟩dt = 0, w ∈ Hq (Jn ). (3.9) L2

Jn

3.2

Error estimates

Let ∥ · ∥, ∥ · ∥M , ∥ · ∥C and ∥ · ∥K be the norms (or seminorms) over Rd , defined by the inner products (2.2), respectively. We further define ∥v∥L∞ = ess sup ∥v(t)∥M , M (G) t∈G

∥v∥L∞ −1 (G) = ess sup ∥v(t)∥M−1 , M

(3.10)

t∈G

where M−1 is the inverse of the matrix M. We assume that for the given function f , the linear problem (2.10) has a unique solution satisfying that u ∈ C([0, T ]; Rd ) ∩ C 1 ([0, T ]; Rd ). e and R e be the residual of U e given by Let e e := u − U e e ′′ (t) + CU e ′ (t) + KU(t) e R(t) := M−1 (MU − f (t)),

t ∈ Jn , 1 ≤ n ≤ N.

(3.11)

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e be the Theorem 3.1 Let u and U be the solution of (2.10) and (2.1), respectively. Let U reconstruction of U by (3.1). Then for any t ∈ [0, T ], there holds ∫ t ′ e e max ∥(u − U) (τ )∥M ≤ 2 ∥R(s)∥ (3.12) M ds, 0≤τ ≤t

0

e is given by (3.11). where R Proof. Subtracting (3.11) from (2.10) gives e Me e′′ (t) + Ce e′ (t) + Ke e(t) = −MR(t).

(3.13)

Then, we test (3.13) by e e′ and integrate over t ∈ [0, τ ] to get ∫ τ ( ′′ ) ⟨e e (s), e e′ (s)⟩M + ⟨e e′ (s), e e′ (s)⟩C + ⟨e e(s), e e′ (s)⟩K ds 0 ∫ τ e e = ⟨−R(s), e′ (s)⟩M ds.

(3.14)

0

Moreover, using integration by parts and noting that e e(0) = e e′ (0) = 0, we arrive at ∫ τ ∫ τ 1 1 ′ e e ∥e e (τ )∥2M + ∥e e′ (s)∥2C ds + ∥e e(τ )∥2K = ⟨−R(s), e′ (s)⟩M ds, τ ∈ [0, t]. (3.15) 2 2 0 0 Hence, it follows from (3.15) and the Cauchy-Schwarz inequality that ∫ τ 1 e e ( max ∥e e′ (τ )∥M )2 ≤ max |⟨R(s), e′ (s)⟩M | ds 0≤τ ≤t 0 2 0≤τ ≤t ∫ t ∫ t ′ ′ e e ≤ |⟨R(s), e e (s)⟩M | ds ≤ max ∥e e (τ )∥M ∥R(s)∥ M ds, 0≤τ ≤t

0

which readily yields

∫

′

max ∥e e (τ )∥M ≤ 2

0≤τ ≤t

t

0

e ∥R(s)∥ M ds,

(3.16)

0

as required. Now, we proceed with the efficiency of the above a posteriori error estimates. Lemma 3.1 For t ∈ Jn , 1 ≤ n ≤ N , 1 ˙ n− . U(t) − P0 U(t) = (t − tn−1 − kn )U 2

(3.17)

e ′ ∥L∞ (J ) = kn ∥U e ′′ ∥L∞ (J ) . ∥(U − U) M n M n

(3.18)

Moreover, for 1 ≤ n ≤ N ,

Furthermore, there holds ∫ t n ( ∑ 2 3 e e (3) ∥L∞ (J ) + tk 2 ∥KU e ′′ ∥L∞ (J ) 2 ∥R(s)∥ km ∥KU M ds ≤ m m m M−1 M−1 3 0 m−1

1 2 2 e ′′ ∥L∞ (J ) + km ∥KU′ ∥L∞ −1 (Jm ) + km ∥CU m M M−1 2∫ ) +2 ∥f (s) − P0 f (s)∥M−1 ds .

(3.19)

Jm

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Proof. First of all, recalling the definition of (Local) L2 projection (3.9), we can deduce that ∫ ∫ ( n−1 ) 1 1 ˙ n− ds U(s)ds = U + (s − tn−1 )U P0 U(t) = kn Jn kn Jn 1 ˙n = Un−1 + kn U t ∈ Jn , −, 2 so

1 ˙ n− , U(t) − P0 U(t) = (t − tn−1 − kn )U t ∈ Jn . (3.20) 2 On the other hand, differentiating (3.6) with respect to the variable t directly yields e ′ (t) = −(t − tn−1 )U e ′′ , (U − U)

t ∈ Jn ,

(3.21)

which implies (3.18). Moreover, we have by (3.8) and (3.11) that ( ′ ) e = K(U e − P0 U) + C U e − P0 (U′ ) − (f − P0 f ). MR Write

(3.22)

e − P0 U) = K(U e − U) + K(U − P0 U), K(U

and owing to the fact thatP0 (U′ ) = U′ we know ( ′ ) e − P0 (U′ ) = C(U e − U)′ . C U Hence, the equation (3.22) can be reformulated as e e − U)(s) + K(U − P0 U)(s) + C(U e − U)′ (s) − (f − P0 f )(s), MR(s) = K(U which, in conjunction with (3.6), (3.20) and (3.21), yields the estimate (3.19). Now, let us continue to discuss the lower and upper a posteriori error bound for the method (2.1). Theorem 3.2 (lower and upper bounds) Let u and U be the solution of (2.10) and e be the reconstruction of U by (3.1). Then for t ∈ [0, T ], 1 ≤ n ≤ (2.1), respectively. Let U N, 2 e ′′ ∞ e ′ (τ )∥M max km ∥U ∥LM (Jm ) ≤ ∥(u − U)′ ∥L∞ + max ∥(u − U) M (0,t) 0≤τ ≤t ∫ t e ′′ ∥L∞ (J ) + 4 e ≤ max km ∥U ∥R(s)∥ M ds, M m

1≤m≤n

1≤m≤n

(3.23)

0

e is given by (3.11). where the a posteriori term R Proof. Using the triangle inequality and (3.18), we obtain 2 e ′′ ∞ e ′ ∥L∞ (0,t) max km ∥U ∥LM (Jm ) = ∥(U − U) M

1≤m≤n

e ′ (τ )∥M , ≤ ∥(u − U)′ ∥L∞ + max ∥(u − U) M (0,t) 0≤τ ≤t

(3.24)

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which implies the left side estimate of (3.23). Again, by the triangle inequality, (3.18) and (3.16), we have e ′ (τ )∥M + ∥(U − U) e ′ (τ )∥L∞ (0,t) ∥(u − U)′ ∥L∞ ≤ max ∥(u − U) M (0,t) M 0≤τ ≤t ∫ t e ′′ ∥L∞ (J ) + 2 e ≤ max km ∥U ∥R(s)∥ M ds. M m 1≤m≤n

(3.25)

0

This together with (3.16) and (3.24) yields e ′ (τ )∥M + ∥(u − U)′ (τ )∥L∞ (0,t) max ∥(u − U) M ∫ t e ′′ ∥L∞ (J ) + 4 e ≤ max km ∥U ∥R(s)∥ M ds, M m

0≤τ ≤t

1≤m≤n

0

which leads to the right side estimate of (3.23).

4

An adaptive algorithm

Motivated by Theorem 3.2 (cf. the estimate (3.25)), we are tempted to introduce a posteriori error estimator of the time-stepping method (2.1) for solving even a nonlinear problem (1.1) heuristically. That means, let ∫ T e e ′′ ∥L∞ (J ) + 2 ∥R(s)∥ (4.1) η := max kn ∥U M ds, M n 1≤n≤N

0

e is the residual of a nonlinear problem, defined by where R ( ) ( ) −1 ′′ ′ e e e e R(t) = M MU (t) + F t, U(t), U (t) , t ∈ Jn , 1 ≤ n ≤ N. Then the quantity η may be viewed as a posteriori error estimator for the method (2.1). Until now, it is beyond our power to develop reliability and efficiency estimates for such an estimator. Based on the above error estimator, using the error equidistribution strategy as used in [4, 20], we can construct the error indicator corresponding to the subinterval Jn as Θ := 2 max { Θ1 , Θ2 } , where e ′′ ∥L∞ (J ) , Θ1 := kn ∥U M n

T Θ2 := 2 kn

∫

(4.2) e ∥R(s)∥ M ds.

Jn

The magnitude of Θ affects the choice of kn , the length of the subinterval Jn . ˙ n− at each Next, let us study how to compute the quantities Θ1 and Θ2 after we get U time step by (2.1). First of all, from (3.5) and the definition of Θ1 , we have ˙ n− − U ˙ n−1 Θ1 = ∥U − ∥M .

(4.3)

e For deriving Θ2 , we should obtain R(t) in advance. It follows from (3.4) that e e n−1 ) + kn U ˙ n−1 ˙n U(t) = U(t − Φ0 (ξ) + kn U− Φ1 (ξ), e ′ (t) = U e ′′ (t) = 1 (−U ˙ n−1 ˙n ˙ n−1 ˙ n− ), U U +U − (1 − ξ) + U− ξ, − kn

(4.4)

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where ξ = (t − tn−1 )/kn and Φ0 , Φ1 are defined as in (3.2). Furthermore, in actual computation, we will use the Gaussian quadrature formula (cf. [23]) to evaluate Θ2 numerically. In other words, for t ∈ Jn , 1 ≤ n ≤ N , ∫ Jn

e ∥R(t)∥ M dt ≈

Ng ∑

e n−1 + kn ζj )∥M , kn ωj ∥R(t

(4.5)

j=1

where ζj and ωj (1 ≤ j ≤ Ng ) are the Gaussian quadrature points and weights on reference interval [0, 1], respectively. Remark 4.1 Let us discuss the cost of computing Θ2 briefly. It is evident that the cost is taken in numerical integration by Gaussian quadrature formula (4.5). Since the quadrature method is highly accurate, very few nodes are enough for actual computation (with the e number ≤ 10). Next, we have to evaluate ∥R(·)∥ M at the quadrature nodes, the main cost of which corresponds to numerical solution of a linear system with M as a coefficient matrix. Generally speaking, the mass matrix M is a well-conditioned symmetric positive definite matrix, so the linear system can be solved by the conjugate gradient method very efficiently. According to the above analysis, we find that the cost for computing Θ2 is inexpensive. With the help of the previous preparations and using some ideas implied in the RungeKutta-Felberg method (cf. [23]), we are ready to present the following Algorithm 1 to compute the numerical solution of the problem (1.1) by using the adaptive time-stepping strategy. Algorithm 1 Adaptive Time Stepping Method Given a tolerance ϵ, a parameter δ ∈ (0, 1), and the max (min) time step size kmax (kmin ) by user ˙ 0− = v0 · Step 0: Initialize n = 1, t0 = 0, k1 = kmax , U0 = u0 , U · WHILE tn−1 < T ˙ n−1 · Step 1: Given tn−1 , kn , Un−1 , U − ˙ n by (2.7) · 1(a): Get the numerical solution Un , U − n e by (3.4) · 1(b): Get the approximation U · 1(c): Evaluate Θ1 by (4.3) e · 1(d): Get R(t) at Gaussian quadrature points by (4.4) and (3.11) · 1(e): Summation to get the value of Θ2 by (4.5) · 1(f ): Get Θ by (4.2) · Step 2: If δϵ ≤ Θ ≤ ϵ, kn+1 = kn , go to Step 5 · Step 3: If Θ < δϵ, kn+1 = min{2kn , kmax }, go to Step 5 · Step 4: If Θ > ϵ, kn = max{kn /2, kmin }, go to Step 1 · Step 5: Let tn = tn−1 + kn , n = n + 1, go to loop condition judgment · END WHILE

Remark 4.2 Similar to the Runge-Kutta-Felberg method (cf. [23]), the parameter δ ∈ (0, 1) in Algorithm 1 is used to determine how to enlarge the step size during the computation process (see Step 3 in Algorithm 1). The choice of δ is very technical. If δ is chosen too small, the over-refined meshes would be used in time, deteriorating the efficiency of Algorithm 1. If it is chosen too large, the algorithm would enlarge the step size more frequently, increasing the extra computational cost remarkably. From our numerical experience, it’s better to choose δ such that 1/32 ≤ δ ≤ 1/2. 10

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5 5.1

Numerical experiments Efficiency of the estimators

Example 5.1 (Nonlinear lumped mass system) For illustrating the effectiveness of the a posteriori error estimates developed in the previous sections, we first study the vibration of a multi-structure model, a similar one as given in [3]. As shown in Figure 3, the structure consists of two rigid elements (vehicles) with lumped masses equal to m1 and m2 , respectively; these elements are connected with each other by soften, classical and harden springs with linear damping. And the restoring force of these springs are given as follows: Harden Spring

Soften Spring

u1

ks m1 c1 Damping

kh

u2

kc

m2 c2

Classical Spring

Damping

Figure 3: Example: 5.1: The nonlinear dynamic system. Classical Spring (kc ) :

fc = −κ1 u,

Softening Spring (ks ) :

fs = −κ2 tanh(u),

Hardening Spring (kh ) :

fh = −κ3 u(1 + κ4 u ).

(5.1) (5.2) 2

(5.3)

In our actual computation, we choose m1 = m2 = 1, and choose the spring stiffness as κ1 = κ2 = κ3 = 1. The damping coefficients are taken as c1 = c2 = 1. Hence by d Alembert’s principle, we can get the following system of nonlinear dynamic equations, ( ) ( ) ( ) ) ( )′′ ( ′ u1 (t) c1 u1 (t) + fs (u1 (t) ) + fc (u1 (t) ) − fc (u2 (t) ) − f1 (t) , (5.4) = c2 u′2 (t) + fh u2 (t) + fc u2 (t) − fc u1 (t) − f2 (t) u2 (t) where f1 and f2 are the external forces. We choose T = 1 and the exact solution to be u(t) = (u1 , u2 )T = (sin(πt), sin(2πt))T , so the force term f can be computed by the equations (5.4). We solve the solution of the dynamical system by the method (2.1) combined with the right side scheme (2.7). In our numerical computation, for a given natural number N , we adopt the uniform partition in time with the mesh size k = T /N , 1 ≤ n ≤ N . To show the computational performance of our method, define Ed = max ∥(u − U)′ (τ )∥M , 0≤τ ≤T

e ′ (τ )∥M , Etd = max ∥(u − U) 0≤τ ≤T

0

e ′′ ∥L∞ (J ) , ε2 = max kn ∥U M n

ε3 = η = 2ε1 + ε2 .

0≤n≤N

Effld =

e )∥M , Et = max ∥(u − U)(τ 0≤τ ≤T ∫ T e ε1 = 2 ∥R(s)∥ M ds ,

ε2 , Ed + Etd

Effud =

ε3 . Ed + Etd

In Figure 4(a) we present the values of Et and ε1 as well as their orders (which are 1). In Figure 4(b) we give the estimates of the reconstruction solution Et and Etd as well as 11

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10

1

10

2

Estimate

10

10

3

10

1

Lower and upper bound

2

Error

10

3

4

1 1

10

11

12

4

3 Effld Effud 2

1

1

4

10

5

Et Etd

Ed ε1

13

14

15

16

10

5

10

11

12

13

14

15

16

0 9

10

11

log2 (1/k)

log2 (1/k)

(a) Order of Ed and ε1

(b) Order of Et and Etd

12

13

14

15

16

17

log2 (1/k)

(c) Lower and Upper bound

Figure 4: Example 5.1. Numerical results corresponding to estimators in Theorem 3.12 and Theorem 3.2. their orders. Moreover, we present the values of these effectivity indices in Figure 4(c), from which we can observe that 0.77 ≈ Effld < 1 < Effud ≈ 3.98. Therefore, our a posteriori error estimator (4.1) is rather efficient.

5.2

Efficiency of the adaptive algorithm

Example 5.2 (Nonlinear Klein-Gordon equation) In order to test the effectiveness of our adaptive Algorithm 1, we consider the nonlinear Klein-Gordon equations (cf. [8]), utt (x, t) − ∆u(x, t) + βut (x, t) + u2 (x, t) = f (x, t), equipped with the homogeneous Dirichlet boundary condition and the initial conditions. After the discretization by P1 conforming element in the space direction, we obtain the following system of nonlinear ODEs, { Mu′′ (t) + Cu′ (t) + Ku(t) + Mu2 (t) = f (t), 0 < t < T, (5.5) ′ u(0) = u0 , u (0) = v0 , where u is the vector representation of ∑ the finite element solution uh in terms of the shape basis functions {φi }, i.e., uh (x, t) = M i=1 {u(t)}i φi (x). The mass matrix M, the ∫ stiff matrix K, the by [M]ij = Ω φj φi dΩ, ∫ damping matrix C and the∫ force F are defined respectively ∫ [K]ij = Ω ∇φj · ∇φi dΩ, [C]ij = β Ω φj φi dΩ and {f }i = Ω f (t)φi dΩ. In the numerical computation, we choose the damping coefficient β = 0.05 and the terminal time T = 1.0. Consider the 1-dim case of the above problem with the force f given such that the exact solution is u(x, t) = e−t/2 x(1 − x) sin((1.5π + actan(500(2t − 1))x),

0 < x < 1,

which varies rapidly around t = 0.5. After the discretization in space direction with a fine uniform mesh h = 1/5000, we solve the semi-discrete problem by using Algorithm 1 combined with the semi-side scheme (2.9) with F split into FR := Cu′ + Ku − f and FL := Mu2 , so that we only require to solve a linear system at each time subinterval. When implementing Algorithm 1 in this example, we set the related parameters by ϵ = 2.5e − 1, δ = 1/2, kmax = 1e − 1 and kmin = 2e − 4. To show the efficiency of Algorithm 1, we also carry out the numerical simulation using the uniform time stepping method with the same number of subintervals as for the adaptive method. The numerical solution obtained by the uniform time stepping method with k = kmin /100 is used as a reference solution. 12

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−1

Time step size

10

kn −2

10

−3

10

−4

10

0

0.5 t

1

(a) Time steps of Alg. 1

(b) Numerical solution of Alg. 1

(c) Numerical solution with uniform stepsize

(d) Reference solution

Figure 5: Example 5.2. Comparison of numerical results. From Figure 5(a) we can see the time step size becomes extremely small around t = 0.5 in order to capture the rapid change of the solution, and the step size will become large automatically when the solution varies slowly, which illustrates the efficiency of Algorithm 1. The numerical results with Algorithm 1 and the uniform time stepping method, and the reference solution are shown in Figures 5(b), 5(c) and 5(d), respectively, from which we may find that the adaptive method can approximate the exact solution very well even if it varies rapidly, but the uniform time stepping method fails. We mention further that for the adaptive method in this example, the total CPU time used is approximately 147.1 s, while the one for computing Θ is only 7.4 s, only covers a very small amount of the total time.

Acknowledgments The authors would like to thank an anonymous referee for valuable suggestions and comments leading to great improvement of the early version of the paper.

References [1] Akrivis G, Makridakis C, Nochetto R H, Galerkin and runge–kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence. Numer. Math., 2011, 118(3):429– 456. [2] Aksu G, Ali R, Free vibration analysis of stiffened plates using finite difference method. J. Sound Vib., 1976, 48(1):15–25.

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[3] Arora V, Direct structural damping identification method using complex frfs. J. Sound Vib., 2015, 339:304–323. [4] Chen Z, Feng J, An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comput., 2004, 73(247):1167–1193. [5] Clough R, Penzien J, Dynamics of Structures (Third Edition). Computers & Structures, Inc., 2003. [6] Farhat C, Wilson E, Modal superposition dynamic analysis on concurrent multiprocessors. Eng. Computation, 1986, 3(4):305–311. [7] French D A, A space-time finite element method for the wave equation. Comput. Methods Appl. Mech. Engrg., 1993, 107(1):145–157. [8] Grundland A, Infeld E, A family of nonlinear klein–gordon equations and their solutions. J. Math. Phys., 1992, 33(7):2498–2503. [9] Heyliger P, Reddy J, A higher order beam finite element for bending and vibration problems. J. Sound Vib., 1988, 126(2):309–326. [10] Huang J, Lai J, Tang T, An adaptive time-stepping method with efficient error control for second-order evolution problems. Sci. China Math., 2013, 56(12):2753–2771. [11] Hughes T J, The finite element method: linear static and dynamic finite element analysis. Dover Publications, INC, New York, 2010. [12] Hughes T J, Hulbert G M, Space-time finite element methods for elastodynamics: formulations and error estimates. Comput. Methods Appl. Mech. Engrg., 1988, 66(3):339–363. [13] Hulbert G M, Hughes T J, Space-time finite element methods for second-order hyperbolic equations. Comput. Methods Appl. Mech. Engrg., 1990, 84(3):327–348. [14] Itoh T, Damped vibration mode superposition method for dynamic response analysis. Eartho. Eng. Struct. D., 1973, 2(1):47–57. [15] Johnson, C, Discontinuous galerkin finite element methods for second order hyperbolic problems. Comput. Methods Appl. Mech. Engrg., 1993, 107(1):117–129. [16] Komatitsch D, Barnes C, Tromp J, Simulation of anisotropic wave propagation based upon a spectral element method. Geophysics, 2000, 65(4):1251–1260. [17] Krawczuk M, Palacz M, Ostachowicz W, The dynamic analysis of a cracked timoshenko beam by the spectral element method. J. Sound Vib., 2003, 264(5):1139–1153. [18] Lai J, Huang J, An adaptive linear time-stepping algorithm for second-order linear evolution problems. Int. J. Numer. Anal. Mod., 2015, 12(2):230-253. [19] Lai J, Huang J, and Shi Z, Vibration analysis for elastic multi-beam structures by the c0continuous time-stepping finite element method. Int. J. Numer. Meth. Bio., 2010, 26(2):205– 233. [20] Nochetto R H, Savar´e G, Verdi C, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Commun. Pure Appl. Math., 2000, 53(5):525–589. [21] Numayr K S, Haddad R H, Haddad M A, Free vibration of composite plates using the finite difference method. Thin Wall. Struct., 2004, 42(3):399–414. [22] Saravanos D A, Heyliger P R, Hopkins D A, Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates. Int. J. Solids Struct., 1997, 34(3):359–378. [23] Stoer J, Bulirsch R, Introduction to Numerical Analysis (Third Edition), Springer, Berlin, 2002. [24] Walkington N J, Combined DG-CG time stepping for wave equations, SIAM J. Numer. Anal., 2014, 52(3):1398–1417.

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A fractional Means inequality George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we produce an interesting fractional means scalar inequality.

2010 AMS Subject Classi…cation: 26A33, 26D10, 26D15, 26D20. Key Words and Phrases: Means inequality, fractional derivative. We make Remark 1 Let > 0, n := d e (d e ceiling of the number), f ( ; y) 2 AC n ([a; b]), n 1 8 y 2 [c; d] (it means @ @xnf ( 1;y) 2 AC ([a; b]), 8 y 2 [c; d]). Then the left Caputo partial fractional derivative with respect to x, is given by (see [1], p. 270) Z x n @ a f (x; y) 1 n 1 @ f (t; y) = dt; (1) (x t) @x (n ) a @xn 8 y 2 [c; d], and it exists almost everywhere for x in [a; b], denotes the gamma function. Then, we get the left Caputo fractional Taylor formula ([2], p. 54) f (x; y) =

n X1 k=0

@ k f (a; y) (x @xk

1 ( )

k

a) +

Z

x

(x

1

t)

a

@ a f (t; y) dt; @x

(2)

8 x 2 [a; b], for each y 2 [c; d] : Rx 1 @ a f (t;y) Above a (x t) dt 2 AC n ([a; b]), 8 y 2 [c; d] : @x n

1

) Let now f (x; ) 2 AC n ([c; d]), 8 x 2 [a; b] (it means @ @ynf (x; 2 AC ([c; d]), 1 8 x 2 [a; b]). Then the left Caputo partial fractional derivative with respect to y, is given by Z y n @ c f (x; y) 1 n 1 @ f (x; s) = (y s) ds; (3) @y (n ) c @y n

8 x 2 [a; b], and it exists almost everywhere for y in [c; d]. 1

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Then, we get the left Caputo fractional Taylor formula f (x; y) =

n X1 k=0

@ k f (x; c) (y @y k

1 ( )

k

c) +

Z

y

(y

s)

c

1

@ c f (x; s) ds; @y

(4)

8 y 2 [c; d], for each x 2 [a; b] : Ry 1 @ c f (x;s) Above c (y s) ds 2 AC n ([c; d]), 8 x 2 [a; b] : @y Assume @ k f (a; y) = 0, for k = 1; :::; n @xk we get f (x; y)

f (a; y) =

1 ( )

Z

1; 8 y 2 [c; d] ;

x

(x

t)

1

a

(5)

@ a f (t; y) dt: @x

(6)

Additionally assume f (a; y) = 0, 8 y 2 [c; d], then Z x 1 1 @ a f (t; y) f (x; y) = (x t) dt; ( ) a @x

(7)

8 y 2 [c; d] ; 8 x 2 [a; b] : Assume @ k f (x; c) = 0, for k = 1; :::; n @y k we get f (x; y)

f (x; c) =

1 ( )

Z

1; 8 x 2 [a; b] ;

y

c

(y

s)

1

(8)

@ c f (x; s) ds; @y

(9)

8 y 2 [c; d] ; 8 x 2 [a; b] : Additionally assume that f (x; c) = 0, 8 x 2 [a; b], then Z y 1 1 @ c f (x; s) f (x; y) = (y s) ds; ( ) c @y

(10)

8 y 2 [c; d] ; 8 x 2 [a; b] : Assuming (5) and (8), we get 2f (x; y) 1 ( )

Z

a

x

(x

t)

1

f (a; y) f (x; c) = Z y @ a f (t; y) 1 @ c f (x; s) dt + (y s) ds ; @x @y c

(11)

8 x 2 [a; b] ; 8 y 2 [c; d] : Additionally assume that f (a; y) = 0, 8 y 2 [c; d], and f (x; c) = 0, 8 x 2 [a; b], we obtain Z x Z y 1 1 @ a f (t; y) 1 @ c f (x; s) f (x; y) = (x t) dt + (y s) ds ; 2 ( ) @x @y a c (12) 2

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8 x 2 [a; b] ; 8 y 2 [c; d] : We can rewrite (11) as follows: f (a; y) + f (x; c) = 2 Z y @ a f (t; y) 1 @ c f (x; s) (y s) dt + ds ; @x @y c

f (x; y) 1 2 ( )

Z

x

(x

1

t)

a

(13)

8 x 2 [a; b] ; 8 y 2 [c; d] : If 0 < < 1, then n = 1, and (13) is valid without (5) and (8), which in this case are void conditions. Call f (a; y) + f (x; c) f (x; y) := f (x; y) : (14) 2 Assume f 2 C ([a; b] Z

b

a

[c; d]), then

Z

d

f (x; y) dxdy =

c

Z

b

a

(b

a)

Rd c

f (a; y) dy + (d 2

Z

d

f (x; y) dxdy

c

c)

Rb a

f (x; c) dx

!

:

(15)

Hence it holds (b

1 a) (d

c)

Z

b

a

Z

d

f (x; y) dxdy =

(b

c

0

1

@ (d

c)

Rd c

f (a; y) dy +

1 a) (d

1 (b a)

2

Rb

c)

Z

b

a

Assume now that

@ a f (x; y) @ c f (x; y) ; 2 C ([a; b] @x @y

d

f (x; y) dxdy

c

f (x; c) dx

a

Z

1

A:

(16)

[c; d])

(17)

Clearly, it holds 1 2 ( )

Z

x

(x

1

t)

a

1 2 ( )

j f (x; y)j Z y @ a f (t; y) dt + (y @x c

(x

1 2 ( + 1)

a)

(b

@ af @x a)

+

(y

s)

c)

@ cf @y

1

@ af @x

+ (d 1

1

c)

@ c f (x; s) ds @y (18) 1

@ cf @y

: 1

3

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That is 1 2 ( + 1)

j f (x; y)j

(b

@ af @x

a)

Hence 1 a) (d

(b

1 a) (d

(b

c)

c)

Z

b

a

b

c)

1

@ cf @y

=: : (19) 1

d

f (x; y) dxdy

c

a

Z

Z

+ (d

Z

d

j f (x; y)j dxdy

c

:

We have derived: Theorem 2 Let > 0, n := d e, f ( ; y) 2 AC n ([a; b]), 8 y 2 [c; d]; and k f (a;y) n = 0, for k = 1; :::; n 1; f (x; ) 2 AC ([c; d]), 8 x 2 [a; b]. Assume @ @x k @ k f (x;c) @y k

8 y 2 [c; d] ; and

assume f 2 C ([a; b]

(b

1 a) (d

c)

Z

a

b

= 0, for k = 1; :::; n

[c; d]) and

Z

@

d

f (x; y) dxdy

c

1 2 ( + 1)

(b

a)

a f (x;y)

@x

0

;

@ (b

@ af @x

@

1; 8 x 2 [a; b] : Furthermore,

c f (x;y)

@y 1 a)

Rb a

f (x; c) dx +

+ (d 1

2 C ([a; b]

[c; d]) : Then

1 (d c)

2 c)

@ cf @y

Rd c

:

f (a; y) dy

1 A

(20)

1

References [1] G.A. Anastassiou, Fractional Di¤ erentiation Inequalities, Springer, New York, 2009. [2] K. Diethelm, The Analysis of Fractional Di¤ erential Equations, Springer, New York, 2010.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO. 3, 2017

Dynamics of a Difference Equation with Maximum, Taixiang Sun and Guangwang Su,……401 General Properties of Concave Functions Defined by the Generalized Srivastava-Attiya Operator, Hasan Bayram and Sahsene Altinkaya,………………………………………….…408 On the zeros of Eigenfunctions of Discontinuous Sturm-Liouville Problems, K. Aydemir and O. Sh. Mukhtarov,…………………………………………………………………………………417 Fuzzy Stability of an Additive-Quadratic Functional Equation in Matrix Fuzzy Normed Spaces, Javad Shokri and Choonkil Park,………………………………………………………………424 Closed Form Expressions of Some Systems of Nonlinear Partial Difference Equations, Tarek F. Ibrahim,…………………………………………………………………………………………433 Two-Dimensional Chlodowsky Variant of q-Bernstein-Schurer-Stancu Operators, Mehmet Ali Özarslan and Tuba Vedi,………………………………………………………………………..446 Global Stability in Stochastic Difference Equations for Predator-Prey Models, Sangmok Chooa and Young-Hee Kim,…………………………………………………………………………...462 Weighted Superposition Operators from Zygmund Spaces to 𝜇𝜇-Bloch Spaces, Zhi Jie Jiang, Ting Wang, Juan Liu, Ting Luo, and Ting Song,……………………………………………………487 𝛼𝛼𝑥𝑥𝑛𝑛−3

Dynamical Analysis of the Rational Difference Equation 𝑥𝑥𝑛𝑛+1 = 𝐴𝐴+𝐵𝐵𝑥𝑥

𝑛𝑛−1 𝑥𝑥𝑛𝑛−3

, E. M. Elsayed,

Malek Ghazel, and A. E. Matouk,……………………………………………………………...496 Quadratic 𝜌𝜌-Functional Equations, Jung Rye Lee, Choonkil Park, and Dong Yun Shin,……..508

On Modified Degenerate Genocchi Polynomials and Numbers, Hyuck In Kwon, Lee-Chae Jang, Dae San Kim, and Jong-Jin Seo,……………………………………………………………….521 Hesitant Fuzzy Implicative Filters in BE-Algebras, Jeong Soon Han and Sun Shin Ahn,……530 A New Quadratic Functional Equation Version and Its Stability and Superstability, Shahrokh Farhadabadi, Jung Rye Lee, and Choonkil Park,………………………………………………544 Some New Results on Preconditioned Generalized Mixed-Type Splitting Iterative Methods, Guangbin Wang, Fuping Tan, and Yuncui Zhang,……………………………………………..553

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO. 3, 2017 (continued)

A Linear Adaptive time-stepping Method for Solving Vibration Problems with Damping Terms, Jianguo Huang and Huashan Sheng,…………………………………………………………562 A Fractional Means Inequality, George A. Anastassiou,…………………………………….576

Volume 23, Number 4 ISSN:1521-1398 PRINT,1572-9206 ONLINE

October 15, 2017

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 (fifteen 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:

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators. Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities 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. 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 Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics Carlo Bardaro Dipartimento di Matematica e Informatica 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. 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. 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 Luis A. Caffarelli Department of Mathematics The University of Texas at Austin Austin, Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

On A System of Rational Difference Equations Ali GELISKEN ∗ Karamanoglu Mehmetbey University, Kamil Ozdag Science Faculty Department of Mathematics, 70100, Karaman, Turkey In this paper, we investigate behaviors of well-defined solutions of the following system

xn+1 =

A1 yn−(3k−1) , B1 + C1 yn−(3k−1) xn−(2k−1)) yn−(k−1)

yn+1 =

A2 xn−(3k−1) , B2 + C2 xn−(3k−1) yn−(2k−1)) xn−(k−1)

where n ∈ N0 , k ∈ Z+ the coefficients A1 , A2 , B1 , B2 , C1 , C2 and the initial conditions are arbitrary real numbers. Keywords: System of difference equations, Asymptotic behavior, Periodicity, Closed form solution. AMS Classification: 39A10

1

Introduction

There has been a great effort in studying periodic and asymptotic behaviors of solutions of difference equations (see e.g. [3,6,12,15,18,20-23,27,35,45,46]). Also, studying in system of difference equations has increased considerably (see, e.g. [5,7,8,16,17,19,28-30,32-34,37,38,40,43,47]). Ozkan et al. [31] gave the solutions of the systems of the difference equations

∗e

xn+1

=

yn+1

=

zn+1

=

yn−2 , −1 ∓ yn−2 xn−1 yn xn−2 , −1 ∓ xn−2 yn−1 xn xn−2 + yn−2 , n ∈ N0 . −1 ∓ xn−2 yn−1 xn

(1)

mail: [email protected], [email protected]

1

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In [39] it was showed that the system of difference equations, which is an extension of first and second equations of system (1) with respect to coefficients,

xn =

cn yn−3 , an + bn yn−1 xn−2 yn−3 (2)

γn xn−3 yn = , n ∈ N0 , αn + βn xn−1 yn−2 xn−3 where the sequences an , bn , cn , αn , βn , γn , n ∈ N0 , and the initial values xi , yi , i ∈ {1, 2, 3} are real numbers, such that cn 6= 0, γn 6= 0, n ∈ N0 , can be solved in closed form, and for the case when all sequences an , bn , cn , αn , βn , γn , n ∈ N0 are constant it was described the asymptotic behavior of well-defined solutions of the system. In [41] it was showed that an extension of system (2) with respect to indices

xn =

cn yn−(2k−1) , Qk−1 an + bn yn−(2k−1) i=1 yn−(2i−1) xn−2i (3)

γn xn−(2k−1) yn = , Qk−1 αn + βn xn−(2k−1) i=1 xn−(2i−1) yn−2i where an , bn , cn , αn , βn , γn , n ∈ N0 , and the initial conditions xi , yi , i ∈ {1, 2, ....2k − 1} are real numbers, is solved in closed form, and the behavior of its well-defined solutions when all the sequences an , bn , cn , αn , βn , γn are constant was described. Related rational difference equations are studied, e.g. in [1,2,4,9-11,13,14,24-26,31,36,42,44,48]. In this paper we consider an other extension of system (2)

xn+1 =

A1 yn−(3k−1) , B1 + C1 yn−(3k−1) xn−(2k−1)) yn−(k−1) (4)

yn+1

A2 xn−(3k−1) , = B2 + C2 xn−(3k−1) yn−(2k−1)) xn−(k−1)

where n ∈ N0 , k is a positive integer, the initial conditions and the coefficients A1 , A2 , B1 , B2 , C1 , C2 are arbitrary real numbers. We will consider only welldefined solutions, that is, B1 + C1 yn−(3k−1) xn−(2k−1)) yn−(k−1) 6= 0 and B2 + C2 xn−(3k−1) yn−(2k−1)) xn−(k−1) 6= 0, n = 0, 1, 2, ....

2

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2 2.1

Special Cases The case A1 = 0 or A2 = 0

If A1 = 0, we obtain directly xn = 0 for n > 0. By using this, we get yn = 0 for n > 3k. If A2 = 0, we obtain directly yn = 0 for n > 0. By using this, we get xn = 0 for n > 3k. From now on both of A1 and A2 will be considered a non-zero real numbers. System (4) is equivalent to the following system yn−(3k−1) , xn+1 = b1 + c1 yn−(3k−1) xn−(2k−1)) yn−(k−1) (5) yn+1

xn−(3k−1) = , b2 + c2 xn−(3k−1) yn−(2k−1)) xn−(k−1)

i where n ∈ N0 , bi = B Ai and ci = (5) instead of system (4).

2.2

Ci Ai , i

= 1, 2. So, we will consider system

The case b1 = 0 or b2 = 0

If b1 = 0, from the first equation of system (5), we have xn−2k yn−k xn = c11 , n > 0. Using this, we obtain directly yn = αxn−3k , n ≥ k, where α = b2 cc11+c2 . From this and by the change of variables yn yn−3k zn = , wn = , n ≥ k, (6) xn−3k xn system (5) can be transformed into the system wn+1 = c − cb2 zn−(k−1) , zn+1 = α, n ≥ k − 1,

(7)

c1 c2 .

where c = The solutions are obtained easily as zn = wn = α, n ≥ k. This means every solution of system (5) is periodic with 6k periods, not necessarily prime period, such that xn = xn−6k , yn = yn−6k , n ≥ 4k. If b2 = 0, we get immediately yn−2k xn−k yn = c12 , n > 0. From the first equation in system (5) and using this, we obtain xn = βyn−3k , n ≥ k, where β = b1 cc22+c1 . The change of variables un =

xn xn−3k , tn = , n ≥ k, yn−3k yn

(8)

reduces system (5) to the system tn+1 = c¯ − c¯b1 un−(k−1) , un+1 = β, n ≥ 2k − 1,

(9)

c2 c1 .

The solutions of this system tn = un = β, n ≥ 2k − 1, are where c¯ = obtained easily. So, every solution of system (5) is periodic with 6k periods, not necessarily prime period, such that xn = xn−6k , yn = yn−6k , n ≥ 4k. Assume that b1 = 0 and b2 = 0. We have xn−2k yn−k xn = c11 , yn−2k xn−k yn = c2 c1 1 c2 , n > 0. Then, we get immediately xn = c1 yn−3k , yn = c2 yn−3k , n > k. Thus, we can write xn = xn−6k , yn = yn−6k , n > 4k. 3

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2.3

The case c1 = 0 or c2 = 0

If c1 = 0, we have xn = variables

1 b1 yn−3k ,

vn =

n > 0. From this and using the change of

1 , n > 0, xn+3k yn−k xn

(10)

the second equation of system (5) implies the linear equation vn+1 = b1 b2 vn−(2k−1) + b1 2 c2 , n = 0, 1, 2, ....

(11)

We can rewrite the equation (11) in the form of v2kn+m = b1 b2 v2k(n−1)+m + b21 c2 ,

(12)

where n ∈ N0 , m = 1, 2, ..., k. Considering the solution of a nonhomogeneous first order difference equation, we can give the solution of the equation (12) such that n+1

n

v2kn+m = (b1 b2 ) vm−2k + b21 c2

1 − (b1 b2 ) 1 − b1 b2

, n ≥ 0.

(13)

when b1 b2 6= 1. If b1 b2 = 1, the solution of the equation (12) can be written as v2kn+m = vm−2k + (n + 1) b21 c2 , n ≥ 0.

(14)

From (10), we have x2kn+3k+m = Considering xn =

1 b1 yn−3k ,

x6kn+3k+m = x−3k+m

x6kn+5k+m = x−k+m

x6kn+7k+m = xk+m

v2k(n−1)+m v2kn+m x2kn−3k+m .

we obtain the solutions of system (5) as

n n Y Y v6kr−2k+m v6kr−2k+m , y6kn+m = b1 x−3k+m , v v6kr+m 6kr+m r=0 r=0 (15)

n Y

v6kr+m

v r=0 6kr+2k+m

, y6kn++2k+m = b1 x−k+m

n Y

v6kr+m , v 6kr+2k+m r=0 (16)

n n Y Y v6kr+2k+m v6kr+2k+m , y6kn+4k+m = b1 xk+m , (17) v v 6kr+4k+m r=0 r=0 6kr+4k+m

n ≥ 0 and m = 1, 2, ..., 2k. Suppose that c2 = 0. Then, we have yn = using the change of variable

1 b2 xn−3k ,

n > 0. From this and

4

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1 , n > 0, yn+3k yn+k yn−k the first equation of system (5) implies the linear equation un =

(18)

un+1 = b1 b2 un−(2k−1) + b22 c1 , n ≥ 0.

(19)

By similar processes just as we did, we can rewrite the equation (19) as u2kn+m = b1 b2 u2k(n−1)+m + b22 c1 , n ≥ 0,

(20)

where m = 1, 2, ..., k. We obtain the solution of the equation (20) n+1

1 − (b1 b2 ) , n ≥ 0, 1 − b1 b2 when b1 b2 = 6 1. When b1 b2 = 1, the solution of the equation (20) n

u2kn+m = (b1 b2 ) um−2k + b22 c1

(21)

u2kn+m = um−2k + (n + 1) b22 c1 , n ≥ 0.

(22)

From (18), we have yn+3k =

1 un yn+k yn−k , n

> 0,

and y2kn+3k+m = Considering yn =

1 b2 xn−3k ,

x6kn+m = b2 y−3k+m

n > 0, we obtain the solutions of system (5) as

n n Y Y u6kr−2k+m u6kr−2k+m , y6kn+3k+m = y−3k+m , u u6kr+m 6kr+m r=0 r=0 (23)

x6kn++2k+m = b2 y−k+m

x6kn+4k+m = b2 yk+m

u2k(n−1)+m u2kn+m y2kn−3k+m .

n Y

u6kr+m

r=0

u6kr+2k+m

, y6kn+5k+m = y−k+m

n Y r=0

u6kr+m

, u6kr+2k+m (24)

n n Y Y u6kr+2k+m u6kr+2k+m , y6kn+7k+m = yk+m , (25) u u r=0 6kr+4k+m r=0 6kr+4k+m

n ≥ 0 and m = 1, 2, ..., 2k. Suppose that both c1 and c2 are equal to zero. We get immediately xn+1 = 1 y , yn+1 = b12 xn−(3k−1) , n ≥ 0. From this result, we obtain xn+1 = n−(3k−1) b1 1 1 1 b1 b2 xn−(6k−1) , yn+1 = b1 b2 yn−(6k−1) , n ≥ 3k. So, we have x6kn+3k+m = b1 b2 n+1  x6kn−3k+m , y6kn+3k+m = b11b2 y6kn−3k+m and from this x6kn+3k+m = b11b2  n+1 x−3k+m , y6kn+3k+m = b11b2 y−3k+m , n ≥ 0, m = 1, 2, ..., 6k. 5

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3

Main Case

In this section, we will need the following results, given in the reference [16], in the proofs of our results. Consider the first order Riccati difference equation a + bxn , n = 0, 1, ..., (26) c + dxn where the parameters and the initial condition x0 are arbitrary real numbers. xn+1 =

Theorem 1 The followings are true: 1) Eq.(26) has a prime period-2 solution if and only if b + c = 0. 2) Suppose b + c = 0. Then every solution {xn } of Eq. (26) with x0 6= 0 is periodic with period 2. Theorem 2 Assume that d 6= 0, bc − ad 6= 0, b + c 6= 0 and R = Then the forbidden set F of Eq.(26) is given as follows:  n  o n λ1 λ2 −λ2 λn c 1 − : n ≥ 1 . F = b+c n n d λ −λ d 2

bc−ad (b+c)2

< 14 .

1

For any well-defined solution {xn } of Eq. (26), we have  n+1  c1 λ1 −c2 λn+1 2 xn = b+c − dc , n n d c1 λ −c2 λ 1

for n = 0, 1, ..., where λ1 = c2 =

√ 1− 1−4R , 2

2

λ2 =

√ 1+ 1−4R , c1 2

=

λ2 (b+c)−(dx0 +c) (λ2 −λ1 )(b+c)

and

(dx0 +c)−λ1 (b+c) (λ2 −λ1 )(b+c) .

Corollary 1 Assume that the conditions in Theorem2 hold. Let {xn } be a well-defined solution of Eq. (26). Then limn→∞ xn =

λ2 (b+c)−c . d

Theorem 3 Assume that d 6= 0, bc − ad 6= 0, b + c 6= 0 and R = Then the forbidden set F of Eq.(26) is given as follows: n o F = n(b−c)−(b+c) :n≥1 . 2dn

bc−ad (b+c)2

= 14 .

For any well-defined solution {xn } of Eq. (26), we have   (b+c)+(n+1)(2dx0 +(c−b)) − dc , xn = b+c d 2(b+c)+2n(2dx0 +(c−b)) for n = 0, 1, .... Corollary 2 Assume that the conditions in Theorem3 hold. Let {xn } be a well-defined solution of Eq.(26). Then 6

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limn→∞ xn =

b−c 2d .

Now we consider the system (5) with b1 , b2 , c1 , c2 parameters and the initial conditions are non-zero real numbers. By the change of variables (6), the system (5) reduces to wn−(k−1) 1 , wn+1 = − γ1 zn−(k−1) , n ≥ k, 1 β α wn−(k−1) − γ2 c2 b1 c1 c2 c1 b2 c2 , γ2 = c1 , α = b2 c1 +c2 , β = b1 c2 +c1 . We can rewrite

zn+1 = where γ1 = (27) such that

zn+1 = 

1 β

1 αβ

− γ1 zn−(2k−1)  , wn+1 = − γ2 − γα1 zn−(2k−1)



1 αβ

(27) the system

 − γ1 wn−(2k−1) − 1 α wn−(2k−1)

γ2 β

− γ2

,

(28)

n ≥ 2k. Each of the equation in (28)is a 2kth order Riccati difference equation. Furthermore, the equations in (28) can be rewritten such that

z2kn+1+i

=

1 β



1 αβ

− γ1 z2k(n−1)+1+i  , − γ2 − γα1 z2k(n−1)+1+i (29)

 w2kn+1+i

=

1 αβ



− γ1 w2k(n−1)+1+i − 1 α w2k(n−1)+1+i

− γ2

γ2 β

,

n > 0, i = 0, 1, ..., (2k −1). Note that the equations in (29) are first order Riccati difference equation in variables z2kn+i , w2kn+i , for i = 1, 2, ..., 2k. Theorem 4 Assume that b1 b2 = −1 and {xn , yn } is a well-defined solution of system (5). Then, x2k(n−2)+1+i x2k(n−3)+1+i , x2kn+1+i = x2k(n−5)+1+i y2k(n−2)+1+i y2k(n−3)+1+i y2kn+1+i = , y2k(n−5)+1+i for n ≥ 4, i = 0, 1, ..., (2k − 1). Proof 1 Consider system (29) and suppose that b1 b2 = −1. Then, we have 1 − γ1 − γ2 αβ

= = = =

1 c1 b2 − c1 c2 c2 b2 c1 +c2 b1 c2 +c1 2 b1 b2 c1 c2 + c1 b2 + c22 b1

−

c2 b1 c1

+ c1 c2 − c21 b2 − c22 b1

c1 c2 c1 c2 (b1 b2 + 1) c1 c2 0. 7

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So, from Theorem1(2) we conclude that every solution of each equation in system (29) is periodic with period 4k, that is, z2kn+1+i = z2k(n−2)+1+i , w2kn+1+i = w2k(n−2)+1+i ,

(30)

for n ≥ 2, i = 0, 1, ..., (2k − 1). From (6), we have

zn xn = zn−3k wn xn−6k , yn = wn−3k yn−6k , (31) for n ≥ 4k. System (31) can be written such that

x2kn+1+i =

z2kn+1+i−3k w2kn+1+i x2kn+1+i−6k , y2kn+1+i

=

z2kn+1+i w2kn+1+i−3k y2kn+1+i−6k

(32)

for n ≥ 2, i = 0, 1, ..., (2k − 1). From (6), (30) and (32), we get

x2kn+1+i

x2kn+1+i

=

z2k(n−2)+1+i−3k x2kn+1+i−6k w2k(n−2)+1+i

=

y2k(n−2)+1+i−3k x2k(n−2)+1+i−6k y2k(n−2)+1+i−3k x2k(n−2)+1+i

=

x2k(n−2)+1+i x2k(n−3)+1+i x2k(n−5)+1+i

x2kn+1+i−6k

(33)

and similarly

y2kn+1+i =

y2k(n−2)+1+i y2k(n−3)+1+i (34) y2k(n−5)+1+i

for n ≥ 4, i = 0, 1, ..., (2k − 1). Theorem 5 Assume that {xn , yn } is a well-defined solution of system (5). Then the followings are true: i) Assume that b1 b2 = 1. Then every solution converges to a periodic solution with period 6k. ii) Assume that b1 b2 6= 1. Then, a) If b1 b2 < −1 or b1 b2 > 1, then yn xn = limn→∞ yn−6k = limn→∞ xn−6k

b2 c1 +c2 b2 c2 (b1 b2 c1 +c2 b1 +c1 ) .

8

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b) If −1 < b1 b2 < 1, then every solution converges to a periodic solution with period 6k. Proof 2 i) Consider system (29) with γ1 = c1c2b2 , γ2 = Suppose that b1 b2 = 1. Then, we have −γ1



1 αβ

 − γ2 − β1 (− γα1 )

(−γ1 +

1 αβ

− γ2 ) 2

c2 b1 c1 , α

=

c1 b2 c1 +c2 , β

=

c2 b1 c2 +c1 .

γ1 γ2 1 (−γ1 + αβ − γ2 ) 2

=

c1 b2 c2 b1 c2 c1

=

 − c1c2b2 +

1

c1 c2 b2 c1 +c2 b1 c2 +c1

−

c2 b1 c1

2

(35)

b1 b2

=

(b1 b2 + 1) 1 . 4

=

2

Similarly, it can be seen that     γ2 1 1 − γ (−γ ) − − 1 2 αβ β α γ1 γ2 1 = 1 = .  2 2 4 ( − γ − γ ) 1 1 2 αβ αβ − γ1 − γ2

(36)

So, from (31), (32) and Theorem3, we obtain lim

n→∞

x2kn+1+i z2kn+1+i−3k = lim n→∞ w2kn+1+i x2kn+1+i−6k 1 −γ1 −( αβ −γ2 ) γ 2(− α1 ) =1 = 1 ( αβ −γ1 )−(−γ2 ) 1 2( α )

and lim

n→∞

y2kn+1+i y2kn+1+i−6k

= lim

n→∞

z2kn+1+i w2kn+1+i−3k

1 −γ1 −( αβ −γ2 )

=

2(−

(

γ1 α

) = 1, )−(−γ2 ) 1 2( α )

1 αβ −γ1

i = 0, 1, ..., (2k−1). Thus, we have limn→∞ xn = limn→∞ xn−6k and limn→∞ yn = limn→∞ yn−6k . So, theproof of (i) is finished. 9

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ii)a) Assume that b1 b2 < −1 or b1 b2 > 1. From (35) and (36), we get that 
1 1 1 −γ1 αβ − γ2 − β1 − γα1 ) 2 < 4 1 −γ2 ) (−γ1 + αβ (36)

and 

1 αβ

   − γ1 (−γ2 ) − − γβ2 α1 )

1 1 2 < 4. 1 −γ1 −γ2 ) ( αβ So, from (31), (32) and Theorem2, we obtain that

lim

n→∞ v u u −γ1 u 1+t1−4

z2kn+1+i−3k x2kn+1+i = lim n→∞ w2kn+1+i x2kn+1+i−6k

γ1 1 −γ 1 ( αβ 2 )− β (− α ) (−γ1 + 1 −γ2 )2 αβ 2

1 1 −γ2 )−( αβ −γ2 ) (−γ1 + αβ ( ) v u γ2 1 1 −γ u u ( αβ 1 )(−γ2 )−(− β ) α 1+u t1−4 2 1 −γ −γ ( αβ 1 2 ) 1 −γ1 −γ2 )−(−γ2 ) ( αβ 2 γ − α1

=

1 α

b b −1 1+ b1 b2 +1 1 2

2

=

− c1c2b2



 (b1 b2 + 1) − b1 b2 + 1 + b b −1 1+ b1 b2 +1 1 2

2

(b1 b2 + 1) +

c1 b2 c2

c2 b1 c1

 (37)



=

b2 c1 + c2 b2 c2 (b1 b2 c1 + c2 b1 + c1 )

and

limn→∞

y2kn+1+i y2kn+1+i−6k

= limn→∞

z2kn+1+i w2kn+1+i−3k

=(b2 c1 + c2 ) b2 c2 (b1 b2 c11+c2 b1 +c1 ) , i = 0, 1, ..., (2k − 1). Thus, we have limn→∞

xn xn−6k

= limn→∞

yn yn−6k

=

b2 c1 +c2 b2 c2 (b1 b2 c1 +c2 b1 +c1 ) .

b) Assume that −1 < b1 b2 < 1. From (35) and (36), we get that -γ1



1 αβ


− γ2 − β1 (− γα1 ) (−γ

1

1 2 1 + αβ −γ2 )


500 the matrix may or may not be singular.

3.3

Iterative algorithm and its convergence

An iterative algorithm and its convergence are described in this section. 3.3.1

Iterative algorithm based on basis function

The iterative algorithm based on basis function of the subdivision scheme (2.2) are as defined in the following three steps. First step: Initial approximation The initial approximation is important because the numerical solution depends on the initial approximation. We define the process for finding the initial approximation as follows: Let initial approximate solution Z 0 be the solution of the following linear system BZ 0 = F 0 where

(3.25)

 0 F = (0, 0, 0, 0, 0, 0, y ′ (a), y(a), f0 , f1 , f2 , · · · , fN , y(b), y ′ (b), 0, 0, 0, 0, 0, 0)T ,     fi = h4 f (xi , Li , D), i = 0, 1, 2, · · · N ) ( Li = y(0) + ih y(b)−y(a)  b−a    D = y(b) − y(a).

(3.26)

F 0 is the initial linear approximation of the non-linear vector R(z). Second step: Numerical solution The numerical solutions Z ∗ of the nonlinear system are obtained by using the simple iterative scheme BZ (m+1) = R(Z m ),

m = 0, 1, 2, 3, · · ·

(3.27)

Third step: Stopping condition The above iterative processes will terminate when the following condition is satisfied ||z (m) − z (m−1) || ≤ tol

(3.28)

where tolerance is supposed value i.e. tol = 10−6 . The convergence of the above iterative algorithm is guaranteed by the following proposition. Theorem 3. The successive solutions {Z (m) } generated by the iterative algorithm (3.27) linearly converges to the solution Z ∗ of the non-linear solution of the system (3.20) provided that the M0 and M1 are Lipschitz constants and step size h is small. i.e. ) (

−1

B ≤ M0 h4 + 4994220330463 M1 h3 . (3.29) 1460471061420

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Proof. Let Z ∗ and Z (m) be the solutions of the nonlinear system (3.20). Then by definition, for small h we have BZ ∗ = R(Z ∗ ), BZ m+1 = R(Z m ).

(3.30) (3.31)

Let the error vector be defined as e(k) = Z k − Z ∗ at kth iteration which satisfies BZ (m+1) − BZ ∗ = R(Z k ) − R(Z ∗ ), B(Z (m+1) − Z ∗ ) = R(Z k ) − R(Z ∗ ), Be(k+1) = R(Z k ) − R(Z ∗ ).

(3.32)

For i = 0, 1, 2, · · · , N = (F (Z k ) − F (Z ∗ ))i .

(k+1)

D 4 ei

By mean value theorem, which is stated as “If a function f (x, y, z) is continuously differentiable in an open set of R3 containing points (x1 , y1 , z1 ) and (x2 , y2 , z2 ) and the line segment connecting them, then an equation ′

′

′

f (x2 , y2 , z2 ) − f (x1 , y1 , z1 ) = fx (r, s, t)(x2 − x1 ) + fy (r, s, t)(y2 − y1 ) + fz (r, s, t)(z2 − z1 ) is valid for the interior point (a, b, c) of the segment.”, we have (k+1)

D 4 ei

= f (xi , Zi , Z ′(k) ) − f (xi , Zi , Z ′(∗) ). (k)

(∗)

The above equation can be written as (by using mean value theorem) (k+1)

D4 ei

= fx∗ (xi − xi ) + fy∗ (Zi

(k)

− Zi ) + fy∗′ (Z ′(k) − Z ′(∗) ) (∗)

by using the definition of error vector, we have (k+1)

= fy∗ e(k) + fy∗′ e′(k) ,

(k+1)

= fy∗ e(k) + fy∗′ D1 e(k)

D4 ei

D4 ei

where D4 and D1 are the derivative difference operators defined as D1 fi =

1 [1575(fi−8 − fi+8 ) + 1474560(fi−7 − fi+7 ) 2920942122840h +315738080(fi−6 − fi+6 ) + 1397587968(fi−5 − fi+5 ) −43588613880(fi−4 − fi+4 ) + 311679549440(fi−3 − fi+3 ) −1336741045920(fi−2 − fi+2 ) + 4824847319040(fi−1 − fi+1 )]

D4 fi =

1 [392875(fi+8 − fi−8 ) + 45977600(fi+7 − fi−7 ) 183768238080h4 −1296269280(fi+6 − fi−6 ) + 5912719360(fi+5 − fi−5 ) +1180083476(fi+4 − fi−4 ) − 86261280768(fi+3 − fi−3 ) +332951715808(fi+2 − fi−2 ) − 677767008256(fi+1 − fi−1 ) +850467338370fi ] .

This implies (k+1)

D4 ei

= h4 fy∗ e(k) + h3 fy∗′ D1 e(k) .

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i = 0, −1, −2, · · · , −8, we have

Since ei = eN −i = 0,

(k+1)

Bei

= h4 fy∗ e(k) + h3 fy∗′ D1 e(k)

This can be written as (k+1)

ei

= B −1 (h4 fy∗ e(k) + h3 fy∗′ D1 e(k) ).

By taking norm on both sides, we get (k+1)

∥ei

∥ = ∥B −1 (h4 fy∗ e(k) + h3 fy∗′ D1 e(k) )∥.

This implies (k+1)

∥ei

∥ = ∥B −1 ∥∥(h4 fy∗ e(k) + h3 fy∗′ D1 e(k) )∥.

By using the definition of Lipschitz condition, we get ∥e(k+1) ∥ ≤ h4 M0 (b − a)∥B −1 ∥∥ek ∥ + h3 M1 ∥D1 ∥∥e(k) ∥. This implies (k+1) ( ) ∥ei ∥ ≤ ∥B −1 ∥ h4 M0 (b − a) + h3 M1 ∥D1 ∥ , ∥e(k) ∥

which is equivalent to (k+1)

∥ei ∥ ≈ h3 M1 ∥B −1 ∥∥D1 ∥ ≤ hM1 ∥B −1 ∥∥D1 ∥, (k) ∥e ∥ i-e (k+1)

∥ei ∥ ≈ hM1 ∥B −1 ∥∥D1 ∥. ∥e(k) ∥ The results follows immediately from this inequality and the following fact ∥D1 ∥ =

4994220330463 . 1460471061420

(3.33)

A simple approximation of condition by omitting the quatric term is h≤

−1 1460471061420 −1 M1 B −1 . 4994220330463

(3.34)

This complete the proof.

4

Error Estimation

From the approximation properties of the basis function ϕ(x), it is shown that the collocation method (3.1) with nonic precision treatments at the end points has at least power of approximation O(h3 ). Here we present our main results for error estimation. Proof of these results are similar to the proof of Proposition [14, 8]. Theorem 4. Suppose the exact solution y(x) ∈ C 4 [0, 1] and {zi } are obtained by (3.20) then absolute error by interpolating collocation algorithm is ||err(x)||∞ = ||Z (l) (x) − y (l) (x)||∞ = O(h3−l ), l = 0, 1, 2, 3. where l denotes the order of derivative.

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Proof. Since the order of approximation of subdivision scheme (2.2) is ten so by direct calculation (fourth left eigenvector), we can find derivative of smooth function y(x) as y iv (xj ) =

24 {392875y(xj − 8h) + 45977600y(xj − 7h) 183768238080h4

−1296269280y(xj − 6h) + 5912719360y(xj − 5h) + 1180083476y(xj − 4h) −86261280786y(xj − 3h) + 332951715808y(xj − 2h) − 677767008256y(xj − h) +850467338370y(xj ) − 677767008256y(xj + h) + 332951715808y(xj + 2h) −86261280786y(xj + 3h) + 1180083476y(xj + 4h) + 5912719360y(xj + 5h) −1296269280y(xj + 6h) + 45977600y(xj + 7h) + 392875y(xj + 8h)} + O(h10 ). This can be written as yjiv =

24 {392875yj−8 + 45977600yj−7 − 1296269280yj−6 183768238080h4

+5912719360yj−5 + 1180083476yj−4 − 86261280786yj−3 + 332951715808yj−2 −677767008256yj−1 + 850467338370yj − 677767008256yj+1 + 332951715808yj+2 −86261280786yj+3 + 1180083476yj+4 + 5912719360yj+5 − 1296269280yj+6 +45977600yj+7 + 392875yj+8 } + O(h10 ).

(4.1)

Similarly, we have Zjiv =

24 {392875zj−8 + 45977600zj−7 − 1296269280zj−6 183768238080h4

+5912719360zj−5 + 1180083476zj−4 − 86261280786zj−3 + 332951715808zj−2 −677767008256zj−1 + 850467338370zj − 677767008256zj+1 + 332951715808zj+2 −86261280786zj+3 + 1180083476zj+4 + 5912719360zj+5 − 1296269280zj+6 +45977600zj+7 + 392875zj+8 } + O(h10 ).

(4.2)

If we define error function e(x) = Z(x) − y(x) and error vectors at the nodes by e(xj ) = Z(xj ) − y(xj + jh), −8 ≤ j ≤ N + 8, or equivalently ej = Zj − yj , −8 ≤ j ≤ N + 8, This implies  ′ e = Zj′ − yj′ ,    j′′ ej = Zj′′ − yj′′ , ′′′ ′′′ e′′′  j = Zj − yj   iv iv ej = Zj − yjiv .

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By subtracting (4.2) from (4.1), we get 24 {392875(yj−8 − zj−8 ) + 45977600(yj−7 − zj−7 ) 183768238080h4

yjiv − Zjiv =

−1296269280(yj−6 − zj−6 ) + 5912719360(yj−5 − zj−5 ) + 1180083476(yj−4 − zj−4 ) −86261280786(yj−3 − zj−3 ) + 332951715808(yj−2 − zj−2 ) − 677767008256(yj−1 − zj−1 ) +850467338370(yj − zj ) − 677767008256(yj+1 − zj+1 ) + 332951715808(yj+2 − zj+2 ) −86261280786(yj+3 − zj+3 ) + 1180083476(yj+4 − zj+4 ) + 5912719360(yj+5 − zj+5 ) −1296269280(yj+6 − zj+6 ) + 45977600(yj+7 − zj+7 ) + 392875(yj+8 − zj+8 )} + O(h10 ).

This implies eiv j =

24 {392875ej−8 + 45977600ej−7 − 1296269280ej−6 183768238080h4

+5912719360ej−5 + 1180083476ej−4 − 86261280786ej−3 + 332951715808ej−2 −677767008256ej−1 + 850467338370ej − 677767008256ej+1 + 332951715808ej+2 −86261280786ej+3 + 1180083476ej+4 + 5912719360ej+5 − 1296269280ej+6 +45977600ej+7 + 392875ej+8 } + O(h10 ).

(4.4)

From (1.1), (3.1), (4.3) and by assuming the tenth order boundary treatments at the end points, we have ′

0≤i≤N

eiv j = aj ej + bj ej , and

  max {|ek |}O(h10 ), ej =

0≤k≤7



−8 ≤ i ≤ 0

{|ek |}O(h ),

max

(4.6)

N ≤i≤N +8

10

N −3≤k≤N

(4.5)

where j = 0, 1, · · · N ′

aj = fy (tj , yj∗ , yj∗ ),

′

bj = fy′ (tj , yj∗ , yj∗ ),

and yj∗ = yj + θj ej ,

′

′

yj∗ = yj′ + θj ej ,

0 ≤ θj ≤ 1.

Using the results (4.4) and [1575(zi−8 − zi+8 ) + 1474560(zi−7 − zi+7 ) + 315738080(zi−6 − zi+6 ) + 1397587968 (zi−5 − zi+5 ) − 43588613880(zi−4 − zi+4 ) + 311679549440(zi−3 − zi+3 ) − 1336741045920 ′

(zi−2 − zi+2 ) + 4824847319040(zi−1 − zi+1 )] = 2920942122840hZ + O(h10 ),

(4.7)

It can be conclude that relation (4.5)and (4.6) is equivalent to (B + O(h8 ) − O(h4 ) − D1 O(h3 ))E = O(h10 )∥E∥, where E = (e−8 , e−7 , · · · , e7 , e8 ). Hence for small h, the coefficient matrix B + O(h), will be invertible, thus using the standard result from algebra and effect of ∥B −1 ∥ , we have the following estimate ∥E∥ ≤

∥B −1 ∥ O(h10 ) = O(h3 ). 1 − O(h)

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Results and Discussions

In this section, we test the proposed method on some nonlinear problems. Numerical results for each of the problems are presented in the tables. These values are very close to the true solutions and the values of the errors are also given in the table. Example 1. Consider the following non-linear boundary value problem [1] y iv − 6 exp(−4y) = −12(1 + x)−4 ,

(5.1)

with boundary conditions y(0) = 0, y ′ (0) = 1, y(1) = ln(2) = y ′ (1) = 0.5. The exact solution of the problem (5.1) is y = ln(1 + x). Using the collocation method described in Section 3 for N = 10, h = 10−1 and tol = 10−6 with tenth order boundary treatment at end points. The numerical results are obtained after third iteration with the condition (3.28). The obtained numerical results for this problem are presented in Table 1. The maximum absolute error obtained by the proposed method is 1.78 × 10−3 . The graphical comparison between exact and approximate solutions is shown in Figure 2.

Table 1: Numerical results of Example 1 xi 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Analytic solution Yi 0 0.0953101798 0.1823215568 0.2623642645 0.3364722366 0.4054651081 0.4700036292 0.5306282511 0.5877866649 0.6418538862 0.6931471806

Approximate solution Zi 0 0.0950147533 0.1814496227 0.2609546573 0.3347370220 0.4036840381 0.4684459279 0.5294932609 0.5871580370 0.6416636708 0.6931471806

Error = ||Yi − Zi ||∞ 0 0.0002954265 0.0008719341 0.0014096072 0.0017352146 0.0017810699 0.0015577013 0.0011349902 0.0006286279 0.0001902154 0

Example 2. Consider the non-linear boundary value problem [1] y (iv) = y 2 − x10 + 4x9 − 4x8 − 4x7 + 8x6 − 4x4 + 120x − 48

(5.2)

subject to the boundary conditions y(0) = y ′ (0) = 0, y(1) = y ′ (1) = 1. Using the collocation method described in Section 3 for N = 10, h = 10−1 and tol = 10−6 with tenth order boundary treatment at end points. The numerical results are obtained after third iteration with the condition (3.28). The obtained numerical results for this problem are presented in Table 2. The maximum absolute error obtained by the proposed method is 1.73 × 10−2 . The graphical comparison between exact and approximate solutions is shown in Figure 3.

6

Conclusion

This study has presented a numerical approach based on subdivision collocation algorithm for solving the numerical solution of nonlinear fourth order boundary value problems. The proposed iterative method

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Figure 2: Comparison of the analytic and approximate solution of Example 1. Table 2: Numerical results of Example 2 xi 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Analytic solution Yi 0 0.01981 0.07712 0.16623 0.27904 0.40625 0.53856 0.66787 0.78848 0.89829 1.00000

Approximate solution Zi 0 0.0202195 0.0796952 0.1728732 0.2905995 0.4219208 0.5558846 0.6833406 0.7987412 0.9019417 1.0000000

Error = ||Yi − Zi ||∞ 0 0.0004095 0.0025752 0.0066432 0.0115595 0.0156708 0.0173246 0.0154706 0.0102612 0.0036517 0

has been applied on different nonlinear fourth order boundary value problems. Numerical results show that the accuracy of the approximate solution is O(h3 ). We have also observed that the accuracy of the solution can be improved by choosing different subdivision schemes with the proper adjustment of boundary conditions.

Acknowledgement This work is supported by NRPU (P. No. 3183) and Indigenous Ph.D. Scholarship Scheme of HEC Pakistan.

Competing interests The authors declare that they have no competing interests.

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Figure 3: Comparison of the analytic and approximate solution of Example 2.

Author´ s contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

References [1] R. P., Agarwal, Boundary value problems for higher order differential equations, World Scientific, Singapore, 1986. [2] M. Abbas , A. A. Majid, A. I. M. Ismail and A. Rashid, Numerical Method Using Cubic B-Spline for a Strongly Coupled Reaction-Diffusion System, PLOS ONE, 9 (1), e83265, 2014. [3] S. M. Zin, M. Abbas, A. A. Majid, A. I. M. Ismail, A New Trigonometric Spline Approach to Numerical Solution of Generalized Nonlinear Klien-Gordon Equation, PLOS ONE, 9(5), e95774, 2014 [4] M. Abbas, A. A. Majid, A. I. M. Ismail, and A. Rashid, The application of cubic trigonometric Bspline to the numerical solution of the hyperbolic problems, Applied Mathematics and Computation, 239, 7488, 2014 [5] M. Abbas, A. A. Majid, A. I. M. Ismail, A. Rashid, Numerical method using Cubic Trigonometric BSpline Technique for Non-Classical Diffusion Problem, Abstract and applied Analysis Volume 2014, Article ID 849682, 10 pages [6] S. M. Zin, A. A. Majid, A. I. Md. Ismail, M. Abbas, Application of Hybrid Cubic B-Spline Collocation Approach for Solving a Generalized Nonlinear Klien-Gordon Equation, Mathematical Problems in Engineering, Volume 2014, Article ID 108560, 10 pages. [7] R. Qu, Curve and surface interpolation by subdivision algorithms, Computer Aided Drafting Design and Manufacturing, 4(2): 28-39, 1994. [8] R. Qu and R. P. Agarwal, Solving two point boundary value problems by interpolatory subdivision algorithms, International Journal of Computer Mathematics, 60: 279-294, 1996. [9] R. Qu and R. P. Agarwal, An iterative scheme for solving nonlinear two point boundary value problems, International Journal of Computer Mathematics, 64:3-4, 285-302, 1997.

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[10] R. Qu and R. P. Agarwal, A collocation method for solving a class of singular nonlinear two-point boundary value problems, Journal of Computational and Applied Mathematics, 83: 147-163, 1997. [11] R. Qu, A new approach to numerical differentiation and integration, Mathematical and Computer Modelling, 24: 55-68, 1996. [12] C. Deng and W. Ma, A unified interpolatory subdivision schemes for quadrilateral meshes, ACM Transactions on Graphics, Volume 32(3), Article No. 23, 2013. [13] G. Deslauriers and S. Dubuc, Symmetric iterative interpolation processes, Constructive Approximation, 5: 49-68, 1989. [14] G. Mustafa and S. T. Ejaz, Numerical solution of two point boundary value problems by interpolating subdivision schemes, Abstract and Applied Analysis, Article ID 721314, 2014. [15] G. Strang, Linear algebra and its applications, fourth edition Cengage Learning India Private Limited, ISBN-10:81-315-0172-8, 2011.

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On Stability of Quintic Functional Equations in Random Normed Spaces Afrah A.N. Abdou1 , Y. J. Cho1,2,∗ , Liaqat A. Khan1 and S. S. Kim3,∗ 1

Department of Mathematics, King Abdulaziz University Jeddah 21589, Saudi Arabia E-mail: [email protected]; [email protected] 2

Department of Mathematics Education and the RINS Gyeongsang National University Jinju 660-701, Korea E-mail: [email protected] 3

Department of Mathematics, Dongeui University Busan 614-714, Korea E-mail: [email protected]

Abstract. In this paper, using the direct and fixed point methods, we investigate the generalized Hyers-Ulam stability of the quintic functional equation: 2f (2x + y) + 2f (2x − y) + f (x + 2y) + f (x − 2y) = 20[f (x + y) + f (x − y)] + 90f (x) in random normed spaces under the minimum t-norm. 1. Introduction A classical question in stability of functional equations is as follows: Under what conditions, is it true that a mapping which approximately satisfies a functional equation (ξ) must be somehow close to an exact solution of (ξ)? We say the functional equation (ξ) is stable if any approximate solution of (ξ) is near to a true solution of (ξ). The study of stability problem for functional equations is related to a question of Ulam [15] concerning the stability of group homomorphisms. The famous Ulam stability problem was partially solved by Hyers [9] for linear functional equation of Banach spaces. Subsequently, the result of Hyers theorem was generalized by Aoki [2] for additive mappings and by Rassias [12] for linear mappings by considering an unbounded Cauchy difference. C˘adariu and Radu [3] applied the fixed point method to investigation of the Jensen functional equation. They could present a short and a simple proof (different from the direct method initiated by Hyers in 1941) for the generalized Hyers-Ulam stability of Jensen functional equation and for quadratic functional equation. Their methods are a powerful tool for studying the stability of several functional equations. 0

2000 Mathematics Subject Classification: 39B52, 39B72, 47H09, 47H47. Keywords: Generalized Hyers-Ulam stability, quintic functional equation, random normed spaces, fixed point theorem. 0 *The corresponding author. 0

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On Stability of Quintic Functional Equations

On the other hand, the theory of random normed spaces (briefly, RN -spaces) is important as a generalization of deterministic result of normed spaces and also in the study of random operator equations. The notion of an RN -space corresponds to the situations when we do not know exactly the norm of the point and we know only probabilities of passible values of this norm. The RN spaces may provide us the appropriate tools to study the geometry of nuclear physics and have usefully application in quantum particle physics. A number of papers and research monographs have been published on generalizations of the stability of different functional equations in RN spaces [5, 6, 10, 11, 16]. In the sequel, we use the definitions and notations of a random normed space as in [1, 13, 14]. A function F : R ∪ {−∞, +∞} → [0, 1] is called a distribution function if it is nondecreasing and left-continuous, with F (0) = 0 and F (+∞) = 1. The class of all probability distribution functions F with F (0) = 0 is denoted by Λ. D+ is a subset of Λ consisting of all functions F ∈ Λ for which F (+∞) = 1, where l− F (x) = limt→x− F (t). For any a ≥ 0, ϵa is the element of D+ , which is defined by { 0, if t ≤ a, ϵa (t) = 1, if t > a. Definition 1.1. ([13]) A function T : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm (briefly, a t-norm) if T satisfies the following conditions: (1) T is commutative and associative; (2) T is continuous; (3) T (a, 1) = a for all a ∈ [0, 1]; (4) T (a, b) ≤ T (c, d) whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. Three typical examples of continuous t-norms are as follows: TM (a, b) = min{a, b},

TP (a, b) = ab,

TL (a, b) = max{a + b − 1, 0}.

n Recall that, if T is a t-norm and {xn } is a sequence of numbers in [0, 1], then Ti=1 xi is defined n−1 n 1 recurrently by Ti=1 xi = x1 and Ti=1 xi = T (Ti=1 xi , xn ) = T (x1 , · · · , xn ) for each n ≥ 2 and ∞ ∞ Ti=n xn is defined as Ti=1 xn+i ([8]).

Definition 1.2. ([14]) Let X be a real linear space, µ be a mapping from X into D+ (for any x ∈ X, µ(x) is denoted by µx ) and T be a continuous t-norm. The triple (X, µ, T ) is called a random normed space (briefly RN -space) if µ satisfies the following conditions: (RN1) µx (t) = ϵo (t) for all t > 0 if and only if x = 0; t (RN2) µαx (t) = µx ( |α| ) for all x ∈ X, α ̸= 0 and all t ≥ 0; (RN3) µx+y (t + s) ≥ T (µx (t), µy (s)) for all x, y ∈ X and all t, s ≥ 0. Example 1.1. Every normed space (X, ∥ · ∥) defines a RN -space (X, µ, TM ), where µx (t) =

t t + ∥x∥

for all t > 0 and TM is the minimum t-norm. This space is called the induced random normed space.

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Definition 1.3. Let (X, µ, T ) be a RN -space. (1) A sequence {xn } in X is said to be convergent to a point x ∈ X if, for all t > 0 and λ > 0, there exists a positive integer N such that µxn −x (t) > 1 − λ whenever n ≥ N . In this case, x is called the limit of the sequence {xn } and we denote it by limn→∞ µxn −x = 1. (2) A sequence {xn } in X is called a Cauchy sequence if, for all t > 0 and λ > 0, there exists a positive integer N such that µxn −xm (t) > 1 − λ whenever n ≥ m ≥ N . (3) The RN -space (X, µ, T ) is said to be complete if every Cauchy sequence in X is convergent to a point in X. Theorem 1.4. ([13]) If (X, µ, T ) is a RN -space and {xn } is a sequence of X such that xn → x, then limn→∞ µxn (t) = µx (t) almost everywhere. Recently, Cho et. al. [4] was introduced and proved the Hyers-Ulam-Rassias stability of the following quintic functional equations 2f (2x + y) + 2f (2x − y) + f (x + 2y) + f (x − 2y) = 20[f (x + y) + f (x − y)] + 90f (x)

(1.1)

for fixed k ∈ Z+ with k ≥ 3 in quasi-β-normed spaces. Remark 1.1. (1) If we put x = y = 0 in the equation (1.1), then f (0) = 0. (2) f (2n x) = 25n f (x) for all x ∈ X and n ∈ Z+ . (3) f is an odd mapping. Throughout this paper, let X be a real linear space, (Z, µ′ , TM ) be an RN -space and (Y, µ, TM ) be a complete RN -space. For any mapping f : X → Y , we define Df (x, y) = 2f (2x + y) + 2f (2x − y) + f (x + 2y) + f (x − 2y) − 20[f (x + y) + f (x − y)] − 90f (x) for all x, y ∈ X. In this paper, using the direct and fixed point methods, we investigate the generalized Hyers-Ulam stability of the quintic functional equation: 2f (2x + y) + 2f (2x − y) + f (x + 2y) + f (x − 2y) = 20[f (x + y) + f (x − y)] + 90f (x) in random normed spaces under the minimum t-norm. 2. Random stability of the functional equation (1.1) In this section, we investigate the generalized Hyers-Ulam stability problem of the quintic functional equation (1.1) in RN -spaces in the sense of Scherstnev under the minimum t-norm TM . Theorem 2.1. Let ϕ : X 2 → Z be a function such that, for some 0 < α < 25 , µ′ϕ(2x,2y) (t) ≥ µ′αϕ(x,y) (t)

626

(2.1)

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4

and limn→∞ µ′ϕ(2n x,2n y) (25n t) = 1 for all x, y ∈ X and t > 0. If f : X → Y is a mapping with f (0) = 0 such that µDf (x,y) (t) ≥ µ′ϕ(x,y) (t) (2.2) for all x, y ∈ X and t > 0, then there exists a unique quintic mapping Q : X → Y such that ( ) µf (x)−Q(x) (t) ≥ µ′ϕ(x,0) 22 (25 − α)t (2.3) for all x ∈ X and t > 0. Proof. Letting y = 0 in (2.2), we get µ f (2x) −f (x) (t) ≥ µ′ϕ(x,0) (128t)

(2.4)

25

for all x ∈ X and t > 0. Replacing x by 2n x in (2.4), we get (( 25 )n ) 128t µ f (2n+1 x) − f (2n x) (t) ≥ µ′ϕ(x,0) α 25n 25(n+1) ( j+1 ) n ∑ n−1 f (2 x) f (2j x) x) for all x ∈ X and t > 0. Since f (2 − f (x) = − , 5n 5j 5(j+1) j=0 2 2 2 µ f (2n x) −f (x) 25n

( n−1 ∑ 1 ( α )j ) ′ ′ t ≥ TM n−1 j=0 (µϕ(x,0) (t)) = µϕ(x,0) (t) 5 128 2 j=0

for all x ∈ X and t > 0. Substituting x by 2m x in (2.5), we get ( t µ f (2n+m x) − f (2m x) (t) ≥ µ′ϕ(x,0) ∑n+m−1 25(n+m)

25m

j=m

(2.5)

) (2.6)

( 2α5 )j n

x) for all x ∈ X and m, n ∈ Z with n > m ≥ 0. Since α < k 3 , the sequence { f (2 25n } is a Cauchy sequence in the complete RN -space (Y, µ, TM ) and so it converges to some point Q(x) ∈ Y . Fix x ∈ X and put m = 0 in (2.6). Then we get ( ) 128t ′ µ f (2n x) −f (x) (t) ≥ µϕ(x,0) ∑n−1 α , j 25n j=0 ( 25 )

and so, for any δ > 0, µQ(x)−f (x) (δ + t) ( ) ≥ TM µQ(x)− f (2n x) (δ), µ f (2n x) −f (x) (t) 25n 25n ( ( )) 128t ′ ≥ TM µQ(x)− f (2n x) (δ), µϕ(x,0) ∑n−1 α j 25n j=0 ( 25 )

(2.7)

for all x ∈ X and t > 0. Taking the limit as n → ∞ in (2.7), we get ( ) µQ(x)−f (x) (δ + t) ≥ µ′ϕ(x,0) 22 (25 − α)t

(2.8)

Since δ is arbitrary, by taking δ → 0 in (2.8), we have ( ) µQ(x)−f (x) (t) ≥ µ′ϕ(x,0) 22 (25 − α)t

(2.9)

for all x ∈ X and t > 0. Therefore, we conclude that the condition (2.3) holds. Also, replacing x and y by 2n x and 2n y in (2.2), respectively, we have µ Df (2n x,2n y) (t) ≥ µ′ϕ(2n x,2n y) (25n t) 25n

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for all x, y ∈ X and t > 0. It follows from limn→∞ µ′ϕ(2n x,2n y) (25n t) = 1 that Q satisfies the equation (1.1), which implies that Q is a quintic mapping. To prove the uniqueness of the quintic mapping Q, let us assume that there exists another e : X → Y which satisfies (2.3). Fix x ∈ X. Then Q(2n x) = 25n Q(x) and Q(2 e n x) = mapping Q + 5n e 2 Q(x) for all n ∈ Z . Thus it follows from (2.3) that µQ(x)−Q(x) (t) e = µ Q(2n x) − Q(2 e n x) (t) 25n 25n (t) ( t )) ( (2.10) ≥ TM µ Q(2n x) − f (2n x) , µ f (2n x) − Q(2 e n x) 2 2 5n 5n 25n 25n 2 2 ( ( 25 )n ) ′ 2 5 ≥ µϕ(x,0) 2 (2 − α) t . α ( ( 5 )n ) (t) = 1 for all t > 0. Thus the quintic Since limn→∞ 22 (25 − α) 2α t = ∞, we have µQ(x)−Q(x) e mapping Q is unique. This completes the proof.  Theorem 2.2. Let ϕ : X 2 → Z be a function such that, for some 25 < α, µ′ϕ( x , y ) (t) ≥ µ′ϕ(x,y) (αt) 2

(2.11)

2

and limn→∞ µ′25n ϕ( xn , yn ) (t) = 1 for all x, y ∈ X and t > 0. If f : X → Y is a mapping with 2 2 f (0) = 0 which satisfies (2.2), then there exists a unique cubic mapping Q : X → Y such that ( ) (2.12) µf (x)−Q(x) (t) ≥ µ′ϕ(x,0) 22 (α − 25 )t for all x ∈ X and t > 0. Proof. It follows from (2.2) that

( ) µf (x)−25 f ( x2 ) (t) ≥ µ′ϕ(x,0) 22 αt

(2.13)

for all x ∈ X. Applying the triangle inequality and (2.13), we have   µf (x)−25n f ( 2xn ) (t) ≥ µ′ϕ(x,0) 



2

2 αt  ∑n+m−1 ( 25 )j  j=m

(2.14)

α

for all x ∈ X and m, n ∈ Z with n > m ≥ 0. Then the sequence {25n f ( 2xn )} is a Cauchy sequence in the complete RN -space (Y, µ, TM ) and so it converges to some point Q(x) ∈ Y . We can define a mapping Q : X → Y by (x) Q(x) = lim 25n f n n→∞ 2 for all x ∈ X. Then the mapping Q satisfies (1.1) and (2.12). The remaining assertion follows the similar proof method in Theorem 2.1. This complete the proof.  Corollary 2.3. Let θ be a nonnegative real number and z0 be a fixed unit point of Z. If f : X → Y is a mapping with f (0) = 0 which satisfies µDf (x,y) (t) ≥ µ′θz0 (t)

(2.15)

for all x, y ∈ X and t > 0, then there exists a unique quintic mapping C : X → Y such that ( ) µf (x)−Q(x) (t) ≥ µ′θz0 124t (2.16)

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for all x ∈ X and t > 0. Proof. Let ϕ : X 2 → Z be defined by ϕ(x, y) = θz0 . Then, the proof follows from Theorem 2.1 by α = 1. This completes the proof.  Corollary 2.4. Let p, q ∈ R be positive real numbers with p, q < 5 and z0 be a fixed unit point of Z. If f : X → Y is a mapping with f (0) = 0 which satisfies µDf (x,y) (t) ≥ µ′(∥x∥p +∥y∥q )z0 (t)

(2.17)

for all x, y ∈ X and t > 0, then there exists a unique quintic mapping Q : X → Y such that ( ) µf (x)−Q(x) (t) ≥ µ′∥x∥p z0 22 (25 − 2p )t (2.18) for all x ∈ X and t > 0. Proof. Let ϕ : X 2 → Z be defined by ϕ(x, y) = (∥x∥p + ∥y∥q )z0 . Then the proof follows from Theorem 2.1 by α = 2p . This completes the proof.  Now, we give an example to illustrate that the quintic functional equation (1.1) is not stable for r = 5 in Corollary 2.4 Example 2.1. Let ϕ : R → R be defined by { x5 , for |x| < 1, ϕ(x) = 1, otherwise. Consider the function f : R → R defined by f (x) =

∞ ∑ ϕ(2n x) 25n n=0

for all x ∈ R. Then f satisfies the functional inequality |2f (2x + y) + 2f (2x − y) + f (x + 2y) + f (x − 2y) − 20[f (x + y) + f (x − y)] − 90f (x)| ) (2.19) 136 · 322 ( 5 ≤ |x| + |y|5 31 for all x, y ∈ X, but there do not exist a quintic mapping Q : R → R and a constant d > 0 such that |f (x) − Q(x)| ≤ d|x|5 for all x ∈ R. In fact, it is clear that f is bounded by 1 trivial. If |x|5 + |y|5 ≥ 32 , then

32 31

on R. If |x|5 + |y|5 = 0, then (2.19) is

) 136 · 322 ( 5 136 · 32 ≤ |x| + |y|5 . 31 31 1 Now, suppose that 0 < |x|5 + |y|5 < 32 . Then there exists a positive integer k ∈ Z + such that |Df (x, y)| ≤

1 32k+2 and so

≤ |x|5 + |y|5
d + |c|. If x is in (0, 2−m ), then 2n x ∈ (0, 1) for n = 0, 1, · · · , m. For this x, we have ∞ m ∑ ϕ(2n ) ∑ (2n x)5 f (x) = ≥ = (m + 1)x5 > (d + |c|)|x|5 , 5n 5n 2 2 n=0 n=0

which contradiction (2.20). Remark 2.1. In Corollary 2.4, if we assume that ϕ(x, y) = ∥x∥r ∥y∥r z0 or ϕ(x, y) = (∥x∥r ∥y∥s + ∥x∥r+s + ∥y∥r+s )z0 , then we have Ulam-Gavuta-Rassias product stability and JMRassias mixed product-sum stability, respectively. Next, we apply a fixed point method for the generalized Hyer-Ulam stability of the functional equation (1.1) in RN -spaces. The following Theorem will be used in the proof of Theorem 2.6.

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8

Theorem 2.5. ([7]) Suppose that (Ω, d) is a complete generalized metric space and J : Ω → Ω is a strictly contractive mapping with Lipshitz constant L < 1. Then, for each x ∈ Ω, either d(J n x, J n+1 x) = ∞ for all nonnegative integers 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} is convergent to a fixed point y∗ of J; (3) y∗ is the unique fixed point of J in the set Λ = {y ∈ Ω : d(J n0 x, y) < ∞}; 1 (4) d(y, y∗) ≤ 1−L d(y, Jy) for all y ∈ Λ. Theorem 2.6. Let ϕ : X 2 → D+ be a function such that, for some 0 < α < 25 , µ′ϕ(x,y) (t) ≤ µ′ϕ(2x,2y) (αt)

(2.21)

for all x, y ∈ X and t > 0. If f : X → Y is a mapping with f (0) = 0 such that µD(x,y) (t) ≥ µ′ϕ(x,y) (t)

(2.22)

for all x, y ∈ X and t > 0, then there exists a unique quintic mapping Q : X → Y such that ( ) µf (x)−Q(x) (t) ≥ µ′ϕ(x,y) 22 (25 − α)t (2.23) for all x ∈ X and t > 0. Proof. It follows from (2.22) that µf (x)− f (2x) (t) ≥ µ′ϕ(x,0) (128t)

(2.24)

25

for all x ∈ X and t > 0. Let Ω = {g : X → Y, g(x) = 0} and the mapping d defined on Ω by d(g, h) = inf{c ∈ [0, ∞) : µg(x)−h(x) (ct) ≥ µ′ϕ(x,0) (t), ∀x ∈ X} where, as usual, inf ∅ = −∞. Then (Ω, d) is a generalized complete metric space (see [10]). Now, let us consider the mapping J : Ω → Ω defined by 1 Jg(x) = 5 g(2x) 2 for all g ∈ Ω and x ∈ X. Let g, h in Ω and c ∈ [0, ∞) be an arbitrary constant with d(g, h) < c. Then µg(x)−h(x) (ct) ≥ µ′ϕ(x,0) (t) for all x ∈ X and t > 0 and so ( αct ) µJg(x)−Jh(x) = µg(2x)−h(2x) (αct) ≥ µ′ϕ(x,0) (t) (2.25) 25 for all x ∈ X and t > 0. Hence we have αc α d(Jg, Jh) ≤ 5 ≤ 5 d(g, h) 2 2 for all g, h ∈ Ω. Then J is a contractive mapping on Ω with the Lipschitz constant L = 2α5 < 1. Thus it follows from Theorem 2.5 that there exists a mapping Q : X → Y , which is a unique fixed point of J in the set Ω1 = {g ∈ Ω : d(f, g) < ∞}, such that f (2n x) n→∞ 25n n for all x ∈ X since limn→∞ d(J f, Q) = 0. Also, from µf (x)− f (2x) (t) ≥ µ′ϕ(x,0) (128t), it follows Q(x) = lim

that d(f, Jf ) ≤

1 128 .

25

Therefore, using Theorem 2.5 again, we get d(f, Q) ≤

1 1 d(f, Jf ) ≤ 2 5 . 1−L 2 (2 − α)

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Afrah A.N. Abdou, Y. J. Cho, Liaqat A. Khan and S. S. Kim This means that

9

( ) µf (x)−Q(x) (t) ≥ µ′ϕ(x,0) 22 (25 − α)t

for all x ∈ X and t > 0. Also, replacing x and y by 2n x and 2n y in (2.22), respectively, we have (( 25 )n ) µDQ(x,y) (t) ≥ lim µ′ϕ(2n x,2n y) (25n t) = lim µ′ϕ(x,y) t =1 n→∞ n→∞ α for all x, y ∈ X and t > 0. By (RN1), the mapping Q is quintic. To prove the uniqueness, let us assume that there exists a quintic mapping Q′ : X → Y which satisfies (2.23). Then Q′ is a fixed point of J in Ω1 . However, it follows from Theorem 2.5 that J has only one fixed point in Ω1 . Hence Q = Q′ . This completes the proof.  Theorem 2.7. Let ϕ : X 2 → D+ be a function such that, for some 0 < 25 < α, µ′ϕ(x,y) (t) ≤ µ′ϕ( x , y ) (αt) 2

(2.26)

2

for all x, y ∈ X and t > 0. If f : X → Y is a mapping with f (0) = 0 which satisfies (2.22), then there exists a unique quintic mapping Q : X → Y such that ( ) µf (x)−Q(x) (t) ≥ µ′ϕ(x,0) 22 (α − 25 )t (2.27) for all x ∈ X and t > 0. Proof. By a modification in the proofs of Theorem 2.2 and 2.6, we can easily obtain the desired results. This completes the proof.  Now, we present a corollary that is an application of Theorem 2.6 and 2.7 in the classical case. Corollary 2.8. Let X be a Banach space, ϵ and p be positive real numbers with p ̸= 5. Assume that f : X → X is a mapping with f (0) = 0 which satisfies ∥Df (x, y)∥ ≤ ϵ(∥x∥p + ∥y∥p ) for all x, y ∈ X. Then there exists a unique quintic mapping Q : X → Y such that ∥Q(x) − f (x)∥ ≤

ϵ∥x∥p 22 |25 − 2p |

for all x ∈ X and t > 0. Proof. Define µ : X × R → R by

{ µx (t) =

t t+∥x∥ ,

0,

if t > 0, otherwise

for all x ∈ X and t ∈ R. Then (X, µ, TM ) is a complete RN -space. Denote ϕ : X × X → R by ϕ(x, y) = ϵ(∥x∥p + ∥y∥p ) for all x, y ∈ X and t > 0. It follows from ∥Df (x, y)∥ ≤ θ(∥x∥p + ∥y∥p ) that µDf (x,y) (t) ≥ µ′ϕ(x,y) (t)

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10

On Stability of Quintic Functional Equations

for all x, y ∈ X and t > 0, where µ′ : R × R → R given by { t , if t > 0, ′ µx (t) = t+|x| 0, otherwise, is a random norm on R. Then all the conditions of Theorems 2.6 and 2.7 hold and so there exists a unique quintic mapping Q : X → X such that t = µQ(x)−f (x) (t) t + ∥Q(x) − f (x)∥ ( ) ≥ µ′ϕ(x,0) 22 |25 − α|t =

22 |25 − α|t . 22 |25 − α|t + ϵ∥x∥p

Therefore, we obtain the desired result, where α = 2p . This completes the proof.



Acknowledgments This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant no. (18-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100).

References [1] C. Alsina, B. Schweizer, A. Sklar, On the definition of a probabilitic normed spaces, Equal. Math. 46(1993), 91–98. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan. 2 (1950), 64–66. [3] L. C˘adariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), No. 1, Art. 4. [4] I.G. Cho, D.S. Kang, H.J. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Ineq. Appl. 2010, Art. ID 368981, 9 pp. [5] Y.J. Cho, C. Park, TM. Rassias, R. Saadati, Stability of Functional Equations in Banach Alegbras, Springer Optimization and Its Application, Springer New York, 2015. [6] Y.J. Cho, TM. Rassias, R. Saadati, Stability of Functional Equations in Random Normed Spaces, Springer Optimization and Its Application 86, Springer New York, 2013. [7] J.B. Dias, B. Margolis, A fixed point theorem of the alternative for contrations on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [8] O. Hadˇzi´c, E. Pap, M. Budincevi´c, Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces, Kybernetika 38(2002), 363–381. [9] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [10] D. Mihet¸, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343(2008), 567–572. [11] J.M. Rassias, R. Saadati, G. Sadeghi, J. Vahidi, On nonlinear stability in various random normed spaces, J. Inequal. Appl. 2011, 2011:62.

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[12] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [13] B. Schweizer, A. Skar, Probability Metric Spaces, North-Holland Series in Probability and Applied Math. New York, USA 1983. [14] A.N. Sherstnev, On the notion of s random normed spaces, Dokl. Akad. Nauk SSSR 149, 280–283 (in Russian). [15] S.M. Ulam, Problems in Modern Mathematics, Science Editions, John Wiley & Sons, New York, USA, 1940. [16] T.Z. Xu, J.M. Rassias, W.X. Xu, On stability of a general mixed additive-cubic functional equation in random normed spaces, J. Inequal. Appl. 2010, Art. ID 328473, 16 pp. [17] T.Z. Xu, J.M. Rassias, M.J. Rassias, W.X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spaces, J. Inequal. Appl. 2010, Art. ID 423231, 23 pp.

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Generalized composition operators on Zygmund type spaces and Bloch type spaces Juntao Du and Xiangling Zhu∗ Abstract. In this paper, we investigate the boundedness and compactness of generalized composition operators on Zygmund type spaces and Bloch type spaces with normal weight. MSC 2000: 47B33, 30H10. Keywords: Generalized composition operator, Bloch type space, Zygmund type space.

1 Introduction Let k be a positive continuous function on [0, 1). k is called normal, if there exist positive numbers a and b, 0 < a < b, and δ ∈ [0, 1) such that(see [12]), k(r) k(r) is decreasing on [δ, 1) and lim = 0; r→1 (1 − r)a (1 − r)a

(1)

k(r) k(r) is increasing on [δ, 1) and lim = ∞. r→1 (1 − r)b (1 − r)b

(2)

Let D be the open unit disk in the complex plane C and H(D) the space of all analytic functions on D. Let ω be normal on [0, 1). An f ∈ H(D) is said to belong to the Bloch type space, denoted by Bω , if k f kBω = | f (0)| + sup ω(|z|)| f ′ (z)| < ∞. z∈D

It is easy to see that Bω is a Banach space with the norm k · kBω . When ω(t) = 1 − t2 , we get the Bloch space, denoted by B = B(D). See [19] for more information of the Bloch space. Suppose µ is normal on [0, 1). The Zygmund type space, denoted by Zµ , is the space of all f ∈ H(D) such that k f kZµ = | f (0)| + | f ′ (0)| + sup µ(|z|)| f ′′ (z)| < ∞. z∈D

It is also easy to see that Zµ is a Banach space with the norm k·kZµ . When µ(t) = 1 −t2 , we get the Zygmund space (see [2, 8]). 1

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Throughout the paper, S (D) denotes the set of analytic self-map of D. Associated with ϕ ∈ S (D) is the composition operator Cϕ , which is defined by (Cϕ f )(z) = f (ϕ(z)), f ∈ H(D). We refer the books [1, 19] for the theory of composition operators. Composition operators mapping into the Bloch space on D were studied in, for example, [1, 4, 11, 14, 15, 18]. See [5, 6, 9, 10] for some results of the composition operator mapping into the Zygmund space. Motivated by the fact that weighted composition operators naturally come from isometries of some function spaces, for ϕ ∈ S (D) and g ∈ H(D), Li and Stevi´c [9] defined the generalized composition operator, denoted by Cϕg , as follows. Z z g Cϕ f (z) = f ′ (ϕ(ξ))g(ξ)dξ, f ∈ H(D), z ∈ D. 0

They characterized the boundedness and compactness of Cϕg on the Zygmund space and the Bloch space in [9]. See, for example, [7, 13, 16] for the study of the operator Cϕg . In this paper, motivated by [9], we investigate the boundedness and compactness g of the generalized composition operator Cϕ on Zygmund type spaces and Bloch type spaces with normal weight. In this paper, constants are denoted by C, they are positive and may differ from one occurrence to the next. We say that A . B if there exists a constant C such that A ≤ CB. The symbol A ≈ B means that A . B . A.

2 Proof of main results In this section, we give some auxiliary results which will be used in proving the main results of this paper. They are incorporated in the lemmas which follow. Lemma 1. [3] Suppose µ is normal on [0, 1). Then there exists µ∗ ∈ H(D), such that (i) For any t ∈ [0, 1), µ∗ (t) ∈ R+ , µ∗ (t) is increasing on [0, 1); (ii) inf µ(t)µ∗ (t) > 0; t∈[0,1)

sup µ(|z|)|µ∗ (z)| < ∞. z∈D

In the rest of the paper, we will always use µ∗ to denote the analytic function related to µ in Lemma 1. By a calculation, we get the following lemma. Lemma 2. Suppose µ is normal on [0, 1). Then the following statements hold. (i) There exists a δ ∈ (0, 1), such that µ is decreasing on [δ, 1), lim µ(t) = 0. t→1

(ii) For all α > 1, β ∈ (0, 1), when t ∈ (0, 1), s ∈ (β, 1), Z sα Z s 1 1 1 α µ(t) ≈ µ(t ) ≈ , dt ≈ dt. µ∗ (t) µ(t) 0 0 µ( t) R z R |z| (iii) For any z ∈ D, 0 µ∗ (η)dη . 0 µ∗ (t)dt. If |η| ≤ |z|, µ(|z|)|µ∗ (η)| < C. 2

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Proof. (i). By the definition of normal function, there exist positive numbers a and b, µ(t) a 0 < a < b, and δ ∈ [0, 1) such that (1) and (2) hold. Since µ(t) = (1−t) a (1 − t) , we see that µ is decreasing on [δ, 1) and lim µ(t) = 0. t→1

1−t (ii). From lim 1−t α = t→1

1 α

1

> 0, for any t ∈ [δ α , 1),

µ(t) = 1> µ(tα )

µ(t) (1−t)b µ(tα ) (1−tα )b

(1 − t)b (1 − t)b > > C. (1 − tα )b (1 − tα )b

So when t ∈ (0, 1), µ(t) ≈ µ(tα ). By Lemma 1, when t ∈ (0, 1), µ(t) ≈ µ∗1(t) is obvious. When s ∈ (β, 1), Z sα Z βα Z sa Z s α−1 1 αt 1 1 dt = dt + dt = C + dt α µ(t) µ(t) 0 β µ(t ) 0 βα µ(t) Z β Z s Z s 1 1 1 ≈ dt + dt = dt. µ(t) µ(t) µ(t) 0 β 0 (iii). Since µ∗ is analytic, we see that (iii) holds. The proof is completed.



Lemma 3.[17] Suppose µ is normal on [0, 1). Then for all z ∈ D and f ∈ Bµ , | f (z)| < Gµ (z)k f kBµ , where Gµ (z) = 1 +

Z

|z|

0

1 dt. µ(t)

Remark 1. From the definitions of Zµ and Bµ , for all z ∈ D and f ∈ Zµ , | f ′ (z)| ≤ Gµ (z)k f ′ kBµ ≤ Gµ (z)k f kZµ . R1 1 Lemma 4. [17] Suppose that µ is normal on [0, 1) such that 0 µ(t) dt < ∞. If { fn } is bounded in Bµ and converges to 0 uniformly on compact subsets of D, then lim sup | fn (z)| = 0.

n→∞ z∈D

The relationship between Zygmund type spaces and Bloch type spaces was established as follows. Lemma 5. Suppose that µ is normal on [0, 1). Let µ+ (t) = (1 − t)µ(t). Then (i) µ+ is normal on [0, 1), lim Gµ+ (z) = ∞. |z|→1

(ii) Bµ = Zµ+ and k · kBµ ≈ k · kZµ+ . Proof. (i) Obviously, µ+ is normal on [0, 1). Since µ is normal,there exist positive numbers a and b, 0 < a < b, and δ ∈ [0, 1) such that (1) and (2) holds. Then Z 1 Z 1 Z 1 1 (1 − t)a (1 − δ)a 1 1 dt > dt > dt = +∞, a+1 µ(t) µ (t) (1 − t) µ(δ) (1 − t)1+a + 0 δ δ 3

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as desired. (ii) First we prove that Zµ+ ⊆ Bµ . For all f ∈ Zµ+ , we have Z |z| Z z µ(|z|)| f ′ (z) − f ′ (0)| = µ(|z|) f ′′ (η)dη ≤ k f kZµ+ 0

If |z| ≤ δ, Z

R |z|

0

µ(|z|) dt. µ(t)(1 − t)

(3)

µ(|z|) µ(t)(1−t) dt

< C. If |z| > δ, ! Z δ Z |z| µ(|z|) µ(|z|) µ(|z|) dt = dt + dt µ(t)(1 − t) 0 µ(t)(1 − t) δ µ(t)(1 − t)   Z |z| µ(|z|) a   (1−|z|)a (1 − |z|)   dt ≤ C +  µ(t) (1 − t)a+1  δ (1−t)a ! Z |z| (1 − |z|)a ≤ C+ dt ≤ C. a+1 δ (1 − t)

0

|z|

0

From Lemma 2, µ(t) is bounded on [0, 1). By (3), µ(|z|)| f ′ (z)| ≤ Ck f kZµ+ + µ(|z|)| f ′ (0)| . k f kZµ+ + | f ′ (0)| ≤ 2k f kZµ+ . Therefore k f kBµ . k f kZµ+ and Zµ+ ⊆ Bµ . Next we prove that Bµ ⊆ Zµ+ . For any f ∈ Bµ , by Cauchy’s formula, | f ′′ (z)| ≤ If |z| ≤ δ,

µ(|z|) µ( 1+|z| 2 )

2k f kBµ 2 2 . max | f ′ (η)| ≤ max | f ′ (η)| ≤ 1+|z| 1−|z| 1+|z| 1 − |z| |η−z|= 2 1 − |z| |η|= 2 µ( 2 )(1 − |z|)

< C is obvious. When δ < |z| < 1, µ(|z|) µ( 1+|z| 2 )

µ(|z|) (1−|z|)b

= 2b

< 2b .

µ( 1+|z| 2 ) b (1− 1+|z| 2 )

So k f kZµ+ . k f kBµ and hence Bµ ⊆ Zµ+ . The proof is completed.



To study the compactness, we need the following lemma, which can be proved in a standard way (see, for example, Proposition 3.11 in [1]). Lemma 6. Suppose that g ∈ H(D), ϕ ∈ S (D), X, Y are Bloch type spaces or Zygmund g g type spaces. If Cϕ : X → Y is bounded, then Cϕ : X → Y is a compact operator if and only if whenever { fn } is bounded in X and fn → 0 uniformly on compact subsets of D , lim kCϕg fn kY = 0. n→∞

3 The boundness and compactness of Cϕg : Zµ → Zω (Bω ) Theorem 1. Suppose g ∈ H(D), ϕ ∈ S (D) , ω and µ are normal on [0, 1). Then Cϕg : Zµ → Zω is bounded if and only if sup ω(|z|)|g′ (z)|Gµ (ϕ(z)) < ∞

and

z∈D

sup z∈D

ω(|z|)|ϕ′ (z)g(z)| < ∞. µ(|ϕ(z)|)

(4)

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Proof. Suppose that (4) holds. For any f ∈ Zµ , by Lemma 3 and Remark 1, we have sup ω(|z|)|(Cϕg f )′′ (z)| ≤ sup ω(|z|) f ′′ (ϕ(z))ϕ′ (z)g(z) + sup ω(|z|) f ′ (ϕ(z))g′ (z) z∈D

z∈D

z∈D

′

ω(|z|)|ϕ (z)g(z)| k f kZµ + sup ω(|z|)|g′(z)|Gµ (ϕ(z))k f kZµ µ(|ϕ(z)|) z∈D

≤ sup z∈D

12 , let a = ϕ(ξ) and pa (z) =

az

Z

0

t2

Z  

0

2 Z   µ∗ (η)dη dt −

0

qa (z) = R |a| 0

Then p′a (z)

p′′a (z)

Z  = a 

3

(az)2

0

2 2

= 4a zµ∗ (a z )

az

Z

0

(az)2

pa (z)

Z   

0

t3 |a|2

2  µ∗ (η)dη dt, (7)

.

µ∗ (η)dη

2 Z (az)3 2   |a|2   µ∗ (η)dη , µ∗ (η)dη − a  0 6a4 z2 (az)3 µ∗ (η)dη − µ ∗ |a|2 |a|2

!Z

(az)3 |a|2

µ∗ (η)dη.

0

By Lemmas 1 and 2,

So

Z Z (az)3 |a|2 Z |a| (az)2 µ∗ (η)dη µ(|z|)|p′′a (z)| . µ∗ (η)dη + µ∗ (η)dη . 0 0 0 kqa kZµ = qa (0) + q′a (0) + sup µ(|z|)|q′′a (z)| < C.

(8)

z∈D

Hence, when |ϕ(ξ)| > 12 , ω(|ξ|)|ϕ′ (ξ)g(ξ)| g g g ≈ ω(|ξ|)|(Cϕ qa )′′ (ξ)| ≤ kCϕ qa kZω < kqa kZµ kCϕ k < ∞. µ(|ϕ(ξ)|)

(9)

5

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From (6) and (9), we see that the second inequality in (4) holds. R az R η Let fa (z) = 0 0 µ∗ (w)dwdη. Then fa′ (z) = a

Z

az

0

µ∗ (w)dw, fa′′ (z) = a2 µ∗ (az), k fa kZµ ≤ C.

By Lemma 2, when |ϕ(ξ)| > 12 , Z |a| 1 ′ dt ≈ω(|ξ|) (Cϕg fa )′′ (ξ) − fa′′ (ϕ(ξ))ϕ′ (ξ)g(ξ) ω(|ξ|)|g (ξ)| µ(t) 0 g ≤kCϕ fa kZω + sup ω(|ξ|)µ∗ (|ϕ(ξ)|2 )|ϕ′ (ξ)g(ξ)|

(10)

ξ∈D

.k fa kZµ kCϕg k + sup ξ∈D

ω(|ξ|)|ϕ′ (ξ)g(ξ)| . µ(|ϕ(ξ)|)

From (6) and (10), we see that the first inequality in (4) holds. The proof is completed.  Theorem 2. Suppose g ∈ H(D), ϕ ∈ S (D) , ω and µ are normal on [0, 1) such that Cϕg : Zµ → Zω is bounded. Then the following statements hold: (i) When lim Gµ (z) < ∞, Cϕg : Zµ → Zω is compact if and only if |z|→1

ω(|z|)|ϕ′ (z)g(z)| = 0. |ϕ(z)|→1 µ(|ϕ(z)|)

(11)

lim

g

(ii) When lim Gµ (z) = ∞, Cϕ : Zµ → Zω is compact if and only if |z|→1

lim ω(|z|)|g′ (z)|Gµ (ϕ(z)) = 0

|ϕ(z)|→1

and

ω(|z|)|ϕ′ (z)g(z)| = 0. |ϕ(z)|→1 µ(|ϕ(z)|) lim

(12)

Proof. Because Cϕg : Zµ → Zω is bounded, (5) holds. (i). Suppose (11) holds. For any ε > 0, there is a δ ∈ (0, 1), such that ω(|z|)|ϕ′ (z)g(z)| < ε, when |ϕ(z)| > δ. µ(|ϕ(z)|)

(13)

Let { fn } ⊂ Zµ be bounded and converge to 0 uniformly on compact subsets of D. By Lemma 4 and Cauchy estimate, lim sup | fn′ (z)| = 0, and lim sup | fn′′ (z)| = 0.

n→∞ z∈D

n→∞ |z|≤δ

(14)

From Remark 1, (5) and sup k fn kZµ < ∞, n∈N g kCϕ fn kZω

=

g |(Cϕ fn )′ (0)| +

sup ω(|z|) fn′′ (ϕ(z))ϕ′ (z)g(z) + fn′ (ϕ(z))g′ (z) z∈D

.

ω(|z|)|ϕ′ (z)g(z)| + sup | fn′ (z)|. µ(|ϕ(z)|) z∈D |ϕ(z)|>δ

| fn′ (ϕ(0))| + sup | fn′′ (ϕ(z))| + sup |ϕ(z)|≤δ

6

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By (13) and (14), lim kCϕg fn kZµ = 0. Using Lemma 6, we see that Cϕg : Zµ → Zω is n→∞ compact. g Conversely, assume that Cϕ : Zµ → Zω is compact. Suppose {zn } ⊂ D is a sequence such that lim |ϕ(zn )| = 1. Let an = ϕ(zn ) and n→∞

rn (z) = µ(|an |)

Z

0

an z Z t

µ2∗ (η)dηdt.

0

From Lemma 2, {rn } is bounded in Zµ and rn (z) → 0 uniformly on compact subsets of D when n → ∞. By Lemmas 4 and 6, we have lim sup |rn′ (z)| = 0 and lim kCϕg rn kZω = 0.

n→∞ z∈D

(15)

n→∞

Using Lemma 2, (5) and (15), ω(|zn |)|ϕ′ (zn )g(zn )| n→∞ µ(|ϕ(zn )|) lim

lim ω(|zn |) (Cϕg rn )′′ (zn ) − rn′ (an )g′ (zn ) n→∞ g ≤ lim kCϕ rn kZω + lim ω(|zn |) rn′ (an )g′ (zn ) = 0, ≈

n→∞

n→∞

which implies that

′ (z)g(z)| lim ω(|z|)|ϕ µ(|ϕ(z)|) |ϕ(z)|→1

= 0.

(ii). Suppose (12) holds. For any ε > 0, there is a δ ∈ (0, 1), such that ω(|z|)|g′ (z)|Gµ (ϕ(z)) < ε

and

ω(|z|)|ϕ′ (z)g(z)| < ε, µ(|ϕ(z)|)

(16)

when |ϕ(z)| > δ. Let { fn } be a bounded sequence in Zµ and converges to 0 uniformly on compact subsets of D. By Cauchy estimate, lim sup | fn′ (ϕ(w))| = 0, lim sup | fn′′ (ϕ(w))| = 0.

n→∞ |ϕ(w)|≤δ

(17)

n→∞ |ϕ(w)|≤δ

From Lemma 3, Remark 1 and (5), kCϕg fn kZω

=

|(Cϕg fn )′ (0)| + sup ω(|z|) fn′′ (ϕ(z))ϕ′ (z)g(z) + fn′ (ϕ(z))g′ (z) z∈D

.

| fn′ (ϕ(0))| + sup | fn′′ (ϕ(z))| + sup | fn′ (ϕ(z))| + |ϕ(z)|≤δ

|ϕ(z)|≤δ

ω(|z|)|ϕ′ (z)g(z)| sup k fn kZµ + sup ω(|z|)|g′(z)|Gµ (ϕ(z))k fn kZµ µ(|ϕ(z)|) |ϕ(z)|>δ |ϕ(z)|>δ By (16) and (17), we see that lim kCϕg fn kZµ = 0. From Lemma 6, Cϕg : Zµ → Zω is n→∞ compact. g Conversely, suppose that Cϕ : Zµ → Zω is compact. Let {zn } ⊂ D be a sequence such that lim |ϕ(zn )| = 1. Let an = ϕ(zn ) and qn = qan , where qa is defined in (7). By n→∞

(8), {qn } is bounded in Zµ . Obviously, qn (z) → 0 uniformly on compact subsets of D. By Lemma 6, lim kCϕg qn kZω = 0. By (9), n→∞

ω(|zn |)|ϕ′ (zn )g(zn )| g g ≈ lim ω(|zn |)|(Cϕ qn )′′ (zn )| ≤ lim kCϕ qn kZω = 0, n→∞ n→∞ n→∞ µ(|ϕ(zn )|) lim

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ω(|z|)|ϕ′ (z)g(z)| µ(|ϕ(z)|) |ϕ(z)|→1

which implies that lim

= 0.

Let kn (z) = Then kn′ (z) =

an

0

2 an z µ (s)ds ∗ 0

R

R |an | 0

2

R an z R t

µ∗ (s)ds

µ (s)ds 0 ∗ R |an | µ∗ (s)ds 0

dt (18)

.

R an z 2 µ∗ (s)ds 2(a ) µ (a z) n ∗ n 0 , kn′′ (z) = . R |an | µ∗ (s)ds 0

By Lemma 2, {kn } is bounded in Zµ and kn → 0 uniformly on compact subsets of D. From Lemma 6, lim kCϕg kn kZω = 0. By Lemma 2, n→∞

′

lim ω(|zn |)|g (zn )|

n→∞

|ϕ(zn )|

µ∗ (s)ds ′ + 2 lim ϕ (zn )g(zn )ω(|zn |)µ∗ (|ϕ(zn )|2 ) 0

lim kCϕg kn kZω

.

Z

n→∞

n→∞

ω(|zn |)|ϕ′ (zn )g(zn )| = 0, lim n→∞ µ(|ϕ(zn )|)

.

which implies that lim ω(|z|)|g′ (z)|Gµ (ϕ(z)) = 0. The proof is completed.



|ϕ(z)|→1

Theorem 3. Suppose g ∈ H(D), ϕ ∈ S (D), ω and µ are normal on [0, 1). Then the following statements are equivalent. (i) Cϕg : Zµ → Bω is bounded. (ii) sup ω(|z|)|g(z)|Gµ(ϕ(z)) < ∞. z∈D

(iii) sup ω+ (|z|)|g′(z)|Gµ (ϕ(z)) < ∞

′

(z)g(z)| sup ω+ (|z|)|ϕ < ∞. µ(|ϕ(z)|)

and

z∈D

z∈D

Proof. (ii)⇒(i). Suppose that (ii) holds. For any f ∈ Zµ , using Remark 1, kCϕg f kBω

=

sup ω(|z|)|g(z) f ′ (ϕ(z))| ≤ sup ω(|z|)|g(z)|Gµ(ϕ(z))k f kZµ . k f kZµ < ∞. z∈D

z∈D

g

So Cϕ : Zµ → Bω is bounded. (ii)⇒(i). Suppose Cϕg : Zµ → Bω is bounded. Then sup ω(|z|)|g(z)| = kCϕg zkBω < ∞.

(19)

z∈D

R az R t For all η ∈ D, let ua (z) = 0 0 µ∗ (s)dsdt, where a = ϕ(η). By Lemma 2, supη∈D kua kZµ < ∞. Thus supη∈D kCϕg ua kBω < ∞. When |ϕ(η)| > 12 , ω(|η|)|g(η)|

Z

0

|ϕ(η)|

1 g g ds ≈ ω(|η|)|(Cϕ ua )′ (η)| ≤ kCϕ ua kBω < C. µ(s)

(20)

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By (19) and (20), sup ω(|η|)|g(η)|Gµ(ϕ(η)) < ∞. η∈D

By Lemma 5 and Theorem 1, (i)⇔ (iii). The proof is completed.



Theorem 4. Suppose g ∈ H(D), ϕ ∈ S (D) , ω and µ are normal on [0, 1) such that g Cϕ : Zµ → Bω is bounded. Then the following statements hold. g

(i) If lim Gµ (z) < ∞, Cϕ : Zµ → Bω is compact. |z|→1

(ii) if lim Gµ (z) = ∞, then the following statements are equivalent. |z|→1

(a) Cϕg : Zµ → Bω is compact. (b) (c)

lim ω(|z|)|g(z)|Gµ(ϕ(z)) = 0.

|ϕ(z)|→1

lim ω+ (|z|)|g′(z)|Gµ (ϕ(z)) = 0 and

|ϕ(z)|→1

lim

|ϕ(z)|→1

ω+ (|z|)|ϕ′ (z)g(z)| µ(|ϕ(z)|)

= 0.

g

Proof. Since Cϕ : Zµ → Bω is bounded, (19) holds. (i). Suppose { fn } is bounded in Zµ and fn → 0 uniformly on compact subsets of D. Then { fn′ } is also bounded in Bµ and fn′ → 0 uniformly on compact subsets of D. From Lemma 4, lim sup | fn′ (z)| = 0. Using (19), n→∞ z∈D

lim sup ω(|z|)|(Cϕg fn )′ (z)| = lim sup ω(|z|)|g(z) fn′ (ϕ(z))| . lim sup | fn′ (ϕ(z))| = 0.

n→∞ z∈D

n→∞ z∈D

g

n→∞ z∈D

g

Thus lim kCϕ fn kBω = 0. By Lemma 6, Cϕ : Zµ → Bω is compact. n→∞ (ii). (b)⇒(a). Assume that lim ω(|z|)|g(z)|Gµ(ϕ(z)) = 0. Then for any ε > 0, |ϕ(z)|→1

there exists a δ ∈ (0, 1), such that ω(|z|)|g(z)|Gµ(ϕ(z)) < ε, when δ < |ϕ(z)| < 1. Suppose that { fn } is bounded in Zµ and converges to 0 uniformly on compact subsets of D. Then fn′ → 0 uniformly on compact subsets of D. By (19) and Remark 1, g

kCϕ fn kBω

=

sup ω(|z|)|g(z) fn′ (ϕ(z))| z∈D

≤

sup ω(|z|)|g(z) fn′ (ϕ(z))| +

|ϕ(z)|≤δ

.

sup | fn′ (ϕ(z))| +

|ϕ(z)|≤δ

.

sup ω(|z|)|g(z) fn′ (ϕ(z))| δ 0 ( ) 1 and k ∈ 0, 1/(3 · 24(p−1) ) p . Obviously, Dn (Xtn ; Xtn ) − D(Xtn ) ≡ 0 fn (t, Xtn ; Xtn ) − f (t, Xtn ) ≡ 0 gn (t, Xtn ; Xtn ) − g(t, Xtn ) ≡ 0. This shows that the condition (1.9) is satisfied. Thus theorem 2.4 is obtained. Example 2: In particular, we linearize the equation (3.1) by: for n = 0, 1, ..., (3.4) (3.5)

Dn (X; Xtn ) := (X − Xtn ) · φn + D(Xtn ) fn (t, X; Xtn ) := (X − Xtn ) · ψn + f (t, Xtn )

(3.6)

gn (t, X; Xtn ) := (X − Xtn ) · θn + g(t, Xtn ),

where θn = (θ1n , θ2n , ..., θmn ) and φn , ψn , θin (i = 1, 2, ..., m) are scalar sequences. We can easily see that all conditions of Theorem 2.4 are satisfied. So Theorem 2.4 succeed. Example 3: More specifically, we assume that ξ n = ξ a.s. and φn = ψn = θn = 0 in equation (3.4) for all n = 0, 1, ..., then we obtain the Picard iteration. Of course, in this case, Theorem 2.4 is successful. Therefore, the Picard iteration is a special Z-algorithm.

Acknowledgements This work was supported in part by the National Natural Science Foundation of China under grant #11171306 and #11071065, and sponsored by the Scientific Project of Zhejiang Provincial Science Technology Department under grant #2011C33012.

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THE NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

11

References [1] H. Bao, J. Cao, Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput. 215 (2009) 1732-1743. [2] S. Jankovi´ c, Iterative procedure for solving stochastic differential equations, Mathematica Balkanica, New Series, 1 (1987) 64-71. [3] S. Jankovi´ c, Some special iterative procedures for solving stochastic differential equations of Ito type, Mathematica Balkanica, New Series, 3 (1989) 44-50. [4] S. Jankovi´ c, M. Vasilova, M. Krsti´ c, Some analytic approximations for neutral stochastic functional differential equations, Appl. Math. Comput. 217 (2010) 3615-3623. [5] V.B. Kolmanovskii, V.R. Nosov, Stability and periodic modes of control systems with aftereffect, Nauka, Moscow, 1981. [6] X. Mao, Stochastic Differential Equations and Applications, second ed., Horwood, Chichester, UK, 2007. [7] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications Sixth Edition, Springer-Verlag, 2003. [8] F. Wei, K. Wang, The existence and uniqueness of the solutions for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl. 331 (2007) 516-531. [9] S. Zhou, M. Xue, The existence and uniqueness of the solutions for neutral stochastic functional differential equations with infinite delay, Math. Appl. 21 (2008) 75-83. [10] R. Zuber, About one algorithm for solving first order differential equations (I), Zastosow. Math. 8 (1966) 351-363. [11] R. Zuber, About one algorithm for solving first order differential equations (II), Zastosow. Math. 11 (1966) 85-97.

Conflict of interest: The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Department of Mathematics, Zhejiang University of Science & Technology,, Hangzhou 310023, P. R. China E-mail address: [email protected]; [email protected]

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An improved generalized parameterized inexact Uzawa method for singular saddle point problems ∗ Li-Tao Zhang†, Li-Min Shi College of Science, Zhengzhou University of Aeronautics, Zhengzhou, Henan, 450015, P. R. China

Abstract In this paper, based on the generalized parameterized inexact Uzawa method (GPIU) presented by Zhang and Wang [Applied Mathematics and Computation, 219(2013) 4225-4231], we introduce and study an improved generalized parameterized inexact Uzawa method (IGPIU) for singular saddle point problems. Moreover, theoretical analysis shows that the semi-convergence of the IGPIU method can be guaranteed by suitable choices of the iteration parameters. Finally, numerical experiments are carried out, which show that the improved generalized parameterized inexact Uzawa method (IGPIU) with appropriate parameters improve the convergence of iteration method efficiently when solving singular saddle point problems from the classic incompressible steady state Stokes problems. Key words: Krylov subspace methods; Generalized saddle point matrices; Minimal polynomial; Preconditioners. MSC : 65F10; 65F15

1

Introduction

Consider a singular saddle point problem ( ) ( )( ) ( ) x A B x p A ≡ = ≡ b, y −B T 0 y −q

(1)

∗

This research of this author is supported by NSFC Tianyuan Mathematics Youth Fund (11226337), NSFC(11501525,11471098,61203179,61202098,61170309,91130024,61272544, 61472462 and 11171039), Science Technology Innovation Talents in Universities of Henan Province(16HASTIT040), Aeronautical Science Foundation of China (2013ZD55006), Project of Youth Backbone Teachers of Colleges and Universities in Henan Province(2013GGJS-142,2015GGJS-179), ZZIA Innovation team fund (2014TD02), Major project of development foundation of science and technology of CAEP (2012A0202008), Defense Industrial Technology Development Program, China Postdoctoral Science Foundation (2014M552001), Basic and Advanced Technological Research Project of Henan Province (152300410126), Henan Province Postdoctoral Science Foundation (2013031), Natural Science Foundation of Zhengzhou City (141PQYJS560), Research on Innovation Ability Evaluation Index System and Evaluation Model (142400411268). † E-mail: [email protected]. Tel Numbers:+8615238682150.

1

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where A ∈ R ,B ∈ R , m ≥ n. The matrix A is symmetric positive matrix and the matrix and B is a rank-deficient matrix. Systems of the form (1) arise in a variety of scientific and engineering applications and have attracted a lot of research, see [1-7] for a comprehensive survey. When A is symmetric positive definite and B is of full column rank, we refer the reader to [2,7-18] for many efficient iterative methods and [19] for a survey. For large, sparse or structure matrices, iterative method is an attractive option. In particular, Krylov subspace methods apply techniques that involve orthogonal projections onto subspaces of the form { K(A, b) ≡ span b, Ab, A2 b, ..., An−1 b, ...}. The conjugate gradient method (CG), minimum residual method (MINRES) and generalized minimal residual method (GMRES) are common Krylov subspace methods. The CG method is used for symmetric, positive definite matrices, MINRES for symmetric and possibly indefinite matrices and GMRES for unsymmetric matrices [20]. Generally speaking, the matrix B is full column rank, but not always. If B is rankdeficient, how to effectively solve the singular saddle point problem (1) is important in both scientific computing and engineering applications. For solving the rank-deficient saddle point problems, Ma and Zheng et al. [17,21] presented the parameterized Uzawa method. Bai et al. [22-23] studied the PHSS iteration method. Fischer et al. [24] considered the preconditioned minimum residual (PMINRES) method. Wu et al. [7] discussed the preconditioned conjugate gradient (PCG) method. Zhang and Wang [17] introduced the generalized parameterized inexact Uzawa (GPIU) method. In this paper, based on the generalized parameterized inexact Uzawa (GPIU) method presented by Zhang and Wang [17], we introduce and study an improved generalized parameterized inexact Uzawa method (IGPIU) for singular saddle point problems (1). Similar to the proving process of section 3 in [17], theoretical analysis shows that the semi-convergence of IGPIU method can be guaranteed by suitable choices of the iteration parameters. Finally, one numerical example presented shows correctness and availability of our theory about the improved generalized parameterized inexact Uzawa method (IGPIU) for singular saddle point problems. This paper is organized as follows. In Section 2, we will present the improved generalized parameterized inexact uzawa method (IGPIU) for singular saddle point problems (1). The semi-convergence of the IGPIU method are discussed in Section 3. Moreover, our methods are the generalization of known literature. Some numerical examples are given to demonstrate the efficiency of the IGPIU method in Section 4. Finally, conclusions are made in Section 5.

2

An improved generalized parameterized inexact uzawa method (IGPIU)

Recently, for singular saddle point problems (1), Zhang and Wang [17] make the following splitting ( ) A B A := = M − N, −B T 0 (

where M=

) ( ) P 0 P − A −B ,N = −B T + Q1 Q2 Q1 Q2 2

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC P ∈ Rm×m and Q2 ∈ Rn×n are prescribed symmetric positive definite matrices and Q1 n×m

∈

R

is an arbitrary matrix. To construct the improved generalized parameterized inexact Uzawa method (IGPIU), if we can add one parameter in the above splitting, then we may change the parameter to improve the performance of presented method. Hence, we propose the following splitting ( ) A B A := = L − U, −B T 0 (

where L=

) ( ) P 0 P − A −B ,U = −B T + Q1 ωQ2 Q1 ωQ2

P ∈ Rm×m and Q2 ∈ Rn×n are prescribed symmetric positive definite matrices and Q1 ∈ Rn×m is an arbitrary matrix. Based the generalized parameterized inexact Uzawa (GPIU) iteration method presented by Zhang and Wang [17], we consider an improved generalized parameterized inexact uzawa method (IGPIU) for solving the singular saddle point (1). ( ) ( (k+1) ) ( ) ( (k) ) ( ) x P − A −B x p P 0 = + , (2) −q −B T + Q1 ωQ2 Q1 ωQ2 y (k+1) y (k) or equivalently, {

[ ] x(k+1) = x(k) + P −1 p[ − Ax(k) − By](k) , [ (k+1) ] B T x(k+1) − q − ω1 Q−1 − x(k) . y (k+1) = y (k) + ω1 Q−1 2 2 Q1 x

(3)

The iteration matrix of the IGPIU method (2) or (3) is given by ( T =

P 0 −B T + Q1 ωQ2

)−1 ( ) P − A −B = I − L−1 A. Q1 ωQ2

(4)

The IGPIU method: Let P ∈ Rm×m and Q2 ∈ Rn×n be prescribed symmetric positive definite matrices and Q1 ∈ Rn×m be an arbitrary matrix. Given initial vector x(0) ∈ Rm and y (0) ∈ Rn and the relaxation parameter ω with ω ̸= 0. For k = 0, 1, 2, ... until the iteration T T sequence {(x(k) , y (k) )T } is convergent, compute [ ] { (k+1) x = x(k) + P −1 p[ − Ax(k) − By](k) , [ (k+1) ] (5) 1 −1 T (k+1) (k) y (k+1) = y (k) + ω1 Q−1 B x − q − Q Q x − x . 1 2 ω 2 Remark 2.1. It is obvious that when choosing ω = 1, then the IGPIU method reduces to the GPIU method [17]. Hence, we may change the parameter to improve the performance of presented method.

3

The semi-convergence of IGPIU method

In this section, we discuss the semi-convergence of the IGPIU method for solving the singular saddle point problem (1). We first reveal some basic concepts and notations. Denote σ(A) and ρ(A) as the spectrum and spectral radius of the matrix A, respectively. The rank and index of the matrix A are denoted by rank (A) and index (A), respectively. 3

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Assume that the matrix A can be split into A = M − N with M nonsingular. Then we can construct a splitting iteration method: x(k+1) = T x(k) + M−1 c, k = 0, 1, 2, ...

(6)

where T = M−1 N is the iteration matrix. It is well known that any of the following three conditions is necessary and sufficient for guaranteeing the semi-convergence of the iteration method (6) for the singular linear systems AX = c (see [17,22]): (a) The spectral radius of the iteration matrix T is equal to one , i.e., ρ(T ) = 1; (b) The elementary divisor associated with λ ∈ σ(A) is linear when λ = 1, i.e., rank ((I − T )2 ) =rank (I − T ), or equivalently, index (I − T ) = 1; (c) If λ ∈ σ(T ) with |λ| = 1, then λ = 1, i.e., V(T ) ≡ max{|λ| : λ ∈ σ(T ), λ ̸= 1} < 1. When iteration scheme (6) is semi-convergent, V(T ) is said to be the semi-convergence factor. As usual, the splitting A = M − N and the corresponding iteration matrix T are called as semi-convergent if the iteration (6) is semi-convergent. Next we study the semi-convergence of the IGPIU iteration (5). To get the semi-convergence conditions, the following lemmas are used. Lemma 3.1. [25] Consider the quadratic equation x2 − δx + η = 0, where δ and η are real numbers. Both roots of the equation are less than one in modulus if and only if |η| < 1 and |δ| < 1 + η. Lemma 3.2. [17] Let P ∈ Rm×m and Q2 ∈ Rn×n be symmetric positive definite and B ∈ Rm×n be of column rank-deficient, with m ≥ n. Suppose that λ is an eigenvalue of the iteration matrix T and (uT , v T )T ∈ Rm+n is the corresponding eigenvector. Then λ = 1 if and only if u = 0. Theorem 3.3. Let P ∈ Rm×m and Q2 ∈ Rn×n be symmetric positive definite and B ∈ Rm×n be of column rank-deficient, with m ≥ n. Suppose that λ ̸= 1 is an eigenvalue of the iteration matrix T and (uT , v T )T ∈ Rm+n is the corresponding eigenvector. Then λ satisfies the following quadratic equation: λ2 +

β + γ − 2ωα − τ α + τ − ωβ λ+ = 0, α α

where T u∗ P u u∗ Au u∗ BQ−1 u∗ BQ−1 2 B u 2 Q1 u > 0, β = > 0, γ = ≥ 0, τ = . ∗ ∗ ∗ ∗ uu uu uu uu Proof. Firstly, since λ ̸= 1, we know u ̸= 0 from Lemma 3.2. By (4) we have ( )( ) ( )( ) P − A −B u P 0 u =λ . T Q1 ωQ2 v −B + Q1 ωQ2 v

α=

or equivalently

{

[(1 [ − λ)P − A] u T=] Bv, (1 − λ)Q1 + λB u = ω(λ − 1)Q2 v.

(7)

(8)

Because Q2 is symmetric positive definite and λ ̸= 1, from the second equation in (8), we −1 T λ obtain that v = ω1 (−Q−1 2 Q1 + λ−1 Q2 B )u, which together with the first equation in (8), result in −1 −1 T ωλ2 P u + λω(Au + BQ−1 2 B u − 2ωP u − BQ2 Q1 u) + ω(P u + BQ2 Q1 u − ωAu) = 0. (9)

4

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Since u ̸= 0, by left multiplying u and with the positive definiteness of P (u P u ̸= 0), we have (10) λ + α+τα−ωβ = 0. λ2 + β+γ−2ωα−τ α Thus, the proof is completed. Theorem 3.4. Assume that A ∈ Rm×m is symmetric positive definite, B ∈ Rm×n is rankdeficient, P ∈ Rm×m and Q2 ∈ Rn×n are symmetric positive definite and Q1 ∈ Rn×m is an arbitrary matrix such that BQ−1 2 Q1 is symmetric. Then σ(T ) < 1 holds if and only if one of the following conditions hold: ω > 0, τ < ωβ, 0
0, τ < ωβ, 0
0, τ < ωβ, 0
1, 2α > β or 0 < ω < 1, 2α < β, then (1 + ω)α + τ − 1+ω β > 2α + τ − β. Hence, under these conditions, 2 the range of γ is wider and we will have more space of parameters range. Remark 3.2. It is obvious that when choosing ω = 1, P = 1ξ A, Q1 = 0, and Q2 = ζ1 Q, Q is an approximate matrix to the Schur complement B T A−1 B, then the IGPIU method reduces to the PIU method in [10,17]. 5

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Remark 3.3. Some choices of the parameter matrices P, Q1 and Q2 are given in Table 1 [17]. When choosing different parameter matrices P, Q1 and Q2 , we may immediately obtain a series of iterative methods for solving singular saddle problem (1). Table 1: Some choices of the parameter matrices P, Q1 and Q2 . Case P Q1 Q2 1 1 I A 0 I ξ ζ n 1 1 diag(A) 0 I II ξ ζ n 1 1 III tridiag(A) 0 I ξ ζ n 1 θ T 1 IV A −ζ B I ξ ζ n 1 θ T 1 T ˆ P −1 B, ˆ B ˜ T B) ˜ V A −ζ B diag(B ξ ζ 1 1 T T T −1 ˜ ˆ B ˜ B) ˆ P B, VI A −θQ2 B tridiag(B ξ ζ

4

Numerical examples

In this section, we give numerical experiments to demonstrate the conclusions drawn above. The numerical experiments were done by using MATLAB 7.1 and the matrix of the numerical experiments were generated by IFISS software. In all our runs we used as a zero initial guess and stopped the iteration when the relative residual had been reduced by at least seven orders of magnitude (i.e, when ∥b − Axk ∥2 ≤ 10−7 ∥b∥2 ). We consider the classic incompressible steady state Stokes problems: { −∆u + gradp = f, in Ω, −divu = 0, in Ω, with suitable boundary condition on ∂Ω. It is known that many discretization schemes for the above Stokes problems will lead to generalized saddle point problems of the form (1). Here, we get the test problem (leak-lid driven cavity) by using IFISS software written by David Silvester, Howard Elman and Alison Ramage. We take a finite element subdivision based on 32 × 32 uniform grids of square elements. The mixed finite element used is the bilinear-constant velocity-pressure: Q1 − P0 pair with stabilization. Q1 − P0 finite element subdivision is shown in Figure 1. The stabilization parameter is chosen to 14 . We get the (1,1) block A of the coefficient matrix corresponding to the discretization of the conservative term. Since the matrix B produced by the software is rank deficient, so A is singular matrix. In our experiment, we choose uniform grids 8 × 8, 16 × 16. In Tables 2, when choosing different parameters, we show iteration counts, relative residual and computing time about the GPIU and the IGPIU methods for solving singular saddle problem (1), where IT, RES and CPU are the iteration numbers, relative residual and computing time about the GPIU and the IGPIU methods, respectively. Moreover, we also show the corresponding reduction of residual 2-norm and eigenvalues distributions about two methods for different parameters. Figures 2 ∼ 5 show the reduction of residual 2-norm with Case I, II, III and IV of Table 2. Figures 6 ∼ 9 show the eigenvalues distributions with Case I, II, III and IV of Table 2. Figures 10 and 11 show the reduction of residual 2-norm with uniform grids 16 × 16 and Cases I, II. Figures 12 and 13 show the eigenvalues distributions with uniform grids 16 × 16 and Cases I, II. 6

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Remark 3.1. From Table 2, Figures 2 ∼ 5, 10 and 11 , it is very easy to get that the IGPIU method is in general better than the GPIU method when choosing suitable parameters. By numerical experiments for many times, we can find that, when 0.75 ≤ ω ≤ 1.05 the IGPIU method is very efficient. For Case II, when ω = 1.05 the IGPIU method is little efficient. Hence, we suggest that, the selection range of the parameters may be 0.75 ≤ ω ≤ 1. Remark 3.2. From Figures 6 ∼ 9,12 and 13, we may find that the eigenvalue distribution about the GPIU method has the same spectral clustering compared with the IGPIU method when choosing suitable parameters. Q1−P0 finite element subdivision 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

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Table 2: numerical results of different parameters about GPIU and IGPIU methods for solving singular saddle problem (1). Here, uniform grids are 8 × 8. Case (ξ, θ, ζ, ω) IT RES CPU −8 Case I (0.8, 0, 10, 1) 124 9.6146 × 10 1.776 (0.8, 0, 10, 0.85) 100 9.9265 × 10−8 1.437 (0.8, 0, 10, 0.75) 103 9.9106 × 10−8 1.468 (0.8, 0, 10, 1.05) 79 9.8442 × 10−8 1.156 Case II (0.8, 0, 10, 1) 451 8.8682 × 10−8 0.813 (0.8, 0, 10, 0.85) 435 9.6783 × 10−8 0.812 (0.8, 0, 10, 0.75) 439 9.6727 × 10−8 0.797 (0.8, 0, 10, 1.05) 462 7.7405 × 10−8 0.844 Case III (0.8, 0, 10, 1) 375 7.2718 × 10−8 2.797 (0.8, 0, 10, 0.85) 356 9.7272 × 10−8 2.672 (0.8, 0, 10, 0.75) 354 8.3278 × 10−8 2.703 (0.8, 0, 10, 1.05) 373 7.3365 × 10−8 2.829 Case IV (0.8, 0, 10, 1) 141 9.3439 × 10−8 1.984 (0.8, 0, 10, 0.85) 124 9.3478 × 10−8 1.734 (0.8, 0, 10, 0.75) 118 9.91 × 10−8 1.625 −8 (0.8, 0, 10, 1.05) 139 9.913 × 10 1.907

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Figure 2: Reduction of residual 2-norm with Case I of Table 2. The left figure shows that the first line parameters (GPIU) of Case I compare with the second line parameters (IGPIU) of Case I; The middle figure shows that the first line parameters (GPIU) of Case I compare with the third line parameters (IGPIU) of Case I; The right figure shows that the first line parameters (GPIU) of Case I compare with the forth line parameters (IGPIU) of Case I.

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Figure 3: Reduction of residual 2-norm with Case II of Table 2. The left figure shows that the first line parameters (GPIU) of Case II compare with the second line parameters (IGPIU) of Case II; The middle figure shows that the first line parameters (GPIU) of Case II compare with the third line parameters (IGPIU) of Case II; The right figure shows that the first line parameters (GPIU) of Case II compare with the forth line parameters (IGPIU) of Case II.

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Figure 4: Reduction of residual 2-norm with Case III of Table 2. The left figure shows that the first line parameters (GPIU) of Case III compare with the second line parameters (IGPIU) of Case III; The middle figure shows that the first line parameters (GPIU) of Case III compare with the third line parameters (IGPIU) of Case III; The right figure shows that the first line parameters (GPIU) of Case III compare with the forth line parameters (IGPIU) of Case III.

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Figure 5: Reduction of residual 2-norm with Case IV of Table 2. The left figure shows that the first line parameters (GPIU) of Case IV compare with the second line parameters (IGPIU) of Case IV; The middle figure shows that the first line parameters (GPIU) of Case IV compare with the third line parameters (IGPIU) of Case IV; The right figure shows that the first line parameters (GPIU) of Case IV compare with the forth line parameters (IGPIU) of Case IV.

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Figure 10: Reduction of residual 2-norm with uniform grids 16 × 16 and Case I. The left figure shows that the parameters (1, 0, 10, 1) (GPIU) of Case I compare with the parameters (1, 0, 10, 0.85) (IGPIU) of Case I; The middle figure shows that parameters (1, 0, 10, 1) (GPIU) of Case I compare with the parameters (1, 0, 10, 0.75) (IGPIU) of Case I; The right figure shows that the parameters (1, 0, 10, 1) (GPIU) of Case I compare with the parameters (1, 0, 10, 1.05) (IGPIU) of Case I.

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Figure 11: Reduction of residual 2-norm with uniform grids 16 × 16 and Case II. The left figure shows that the parameters (1, 0, 10, 1) (GPIU) of Case II compare with the parameters (1, 0, 10, 0.85) (IGPIU) of Case II; The middle figure shows that parameters (1, 0, 10, 1) (GPIU) of Case II compare with the parameters (1, 0, 10, 0.75) (IGPIU) of Case II; The right figure shows that the parameters (1, 0, 10, 1) (GPIU) of Case II compare with the parameters (1, 0, 10, 1.05) (IGPIU) of Case II.

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5

Conclusions

Based on the generalized parameterized inexact Uzawa method (GPIU) presented by Zhang and wang [17], we introduce and study an improved generalized parameterized inexact Uzawa method (IGPIU) for singular saddle point problems (1). Moreover, theoretical analysis shows that the semi-convergence of IGPIU method can be guaranteed by suitable choices of the iteration parameters. Finally, numerical experiments are carried out, which show that the IGPIU method is in general better than the GPIU method when choosing suitable parameters. Moreover, we also may find that the eigenvalue distribution about GPIU method has the same spectral clustering compared with IGPIU method when choosing suitable parameters.

6

Acknowledgements

This research of this author is supported by NSFC Tianyuan Mathematics Youth Fund (11226337), NSFC(11501525,11471098,61203179,61202098,61170309,91130024,61272544, 61472462 and 11171039), Science Technology Innovation Talents in Universities of Henan Province(16HASTIT040), Aeronautical Science Foundation of China (2013ZD55006), Project of Youth Backbone Teachers of Colleges and Universities in Henan Province(2013GGJS142,2015GGJS-179), ZZIA Innovation team fund (2014TD02), Major project of development foundation of science and technology of CAEP (2012A0202008), Defense Industrial Technology Development Program, China Postdoctoral Science Foundation (2014M552001), Basic and Advanced Technological Research Project of Henan Province (152300410126), Henan Province Postdoctoral Science Foundation (2013031), Natural Science Foundation of Zhengzhou City (141PQYJS560).

References [1] Z.Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput. 75 (2006) 791-815. [2] Z.Z. Bai, G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 27 (2007) 1-23. [3] J.T. Betts, Practical Methods For Optimal Control Using Nonlinear Programming, SIAM, Philadelphia, PA, 2001. [4] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York and London, 1991. [5] H.C. Elman, A. Ramage, D.J. Silvester, Algorithm 866: IFISS, a MatLab toolbox for modelling incompressible flow, ACM Trans. Math. Softw. 33 (2007) 1-18. [6] H.C. Elman, D.J. Silvester, A.J. Wathen, Performance and analysis of saddle point preconditioners for the discrete steady-state NavierCStokes equations, Numer. Math. 90 (2002) 665-688. [7] X. Wu, B.P.B. Silva, J.Y. Yuan, Conjugate gradient method for rank deficient saddle point problems, Numer. Algor. 35 (2004) 139-154.

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[8] Z.Z. Bai, G.Q. Li, Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal. 23 (2003) 561-580. [9] Z.Z. Bai, G.H. Golub, J.Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math. 98 (2004) 1-32. [10] Z.Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1-38. [11] Z.Z. Bai, Z.Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900-2932. [12] Z.Z. Bai, Z.Q. Wang, Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems, J. Comput. Appl. Math. 187 (2006) 202-226. [13] J.H. Bramble, J.E. Pasciak, A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997) 1072-1092. [14] H.C. Elman, G.H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (1994) 1645-1661. [15] G.H. Golub, X. Wu, J.Y. Yuan, SOR-like methods for augmented systems, BIT Numer. Math. 41 (2001) 71-85. [16] Z.H. Huang, T.Z. Huang, Spectral properties of the preconditioned AHSS iteration method for generalized saddle point problems, Comput. Appl. Math. 29 (2010) 269-295. [17] G.F. Zhang, S.S. Wang, A generalization of parameterized inexact Uzawa method for singular saddle point problems, Appl. Math. Comput. 219 (2013) 4225-4231. [18] Y.Y. Zhou, G.F. Zhang, A generalization of parameterized inexact Uzawa method for generalized saddle point problems, Appl. Math. Comput. 215 (2009) 599-607. [19] M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numer. 14 (2005) 1-137. [20] H. A. Van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2003. [21] H.F. Ma, N.M. Zhang, A note on block-diagonally preconditioned PIU methods for singular saddle point problems, Int. J. Comput. Math. 88 (2011) 808-817. [22] Z.Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing 89 (2010) 171-197. [23] Z.Z. Bai, L. Wang, J.-Y. Yuan, Weak convergence theory of quasi-nonnegative splittings for singular matrices, Appl. Numer. Math. 47 (2003) 75-89. [24] B. Fischer, R. Ramage, D.J. Silvester, A.J. Wathen, Minimum residual methods for augmented systems, BIT Numer. Math. 38 (1998) 527-543. [25] D.M. Young, Iterative Solution for Large Linear Systems, Academic Press, New York, 1971.

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IDENTITIES INVOLVING BESSEL POLYNOMIALS ARISING FROM LINEAR DIFFERENTIAL EQUATIONS TAEKYUN KIM AND DAE SAN KIM

Abstract. In this paper, we study linear differential equations arising from Bessel polynomials and their applications. From these linear differential equations, we give some new and explicit identities for Bessel polynomials.

1. Introduction As is well known, the Bessel differential equation is given by
d2 y dy (1.1) x2 2 + x + x2 − α2 y = 0, (see [17]) . dx dx for an arbitrary complex number α. The Bessel functions of the first kind Jα (x) are defined by the solution of (1.1). For n ∈ Z, Jn (x) are sometimes also called cylinder function or cylindrical harmonics. It is known that ∞ l X (−1)  x 2l+n (1.2) Jn (x) = , (see [1, 16, 17]) . l! (n + l)! 2 l=0

The generating function of Bessel functions is given by ∞ X x 1 (1.3) e 2 (t− t ) = Jn (x) tn , n=−∞

and Jn (x) can be also represented by the contour integral as ˛ x 1 1 (1.4) Jn (x) = e 2 (t− t ) t−n−1 dt, (see [17]) , 2πi where the contour encloses the origin and is traversed in a counterclockwise direction. The Bessel polynomials are defined by the solution of the differential equation d2 y dy − n (n + 1) y = 0, (see [1–6, 15, 16]) . + 2 (x + 1) 2 dx dx Indeed, the solutions of (1.5) are given by n X (n + k)!  x k (1.6) yn (x) = (n − k)!k! 2 k=0 r   2 1 1 = e x K−n− 12 , (see [1, 15–17]) , πx x

(1.5)

x2

2010 Mathematics Subject Classification. 05A19, 33C10, 34A30. Key words and phrases. Bessel polynomials, linear differential equation. 1

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TAEKYUN KIM AND DAE SAN KIM

where


ν ˆ ∞ Γ ν + 12 (2z) cos t √ Kν (z) = dt. ν+ 1 π 2 0 (t + z 2 ) 2 We note that yn (x) are very similar to the modified spherical Bessel function of the second kind. The first few are given as y0 (x) = 1,

y1 (x) = x + 1,

y2 (x) = 3x2 + 3x + 1,

y3 (x) = 15x3 + 15x2 + 6x + 1, y4 (x) = 105x4 + 105x3 + 45x2 + 10x + 1, Carlitz reverse Bessel polynomials are defined by   1 n (1.7) pn (x) = x yn−1 , (n ∈ N ∪ {0}) , x

....

(see [4, 15]) .

These polynomials are also given by the generating function as √ 1−2t)

ex(1−

(1.8)

=

∞ X

pn (x)

n=0

tn . n!

The explicit formulas for them are (1.9)

pn (x) =

n X k=1

(2n − k − 1)! xk 2n−k (k − 1)! (n − k)!

= (2n − 3)!!x 1 F1 (1 − n; 2 − 2n; 2x) ,

(see [1, 15, 16]) ,

where   n (n − 2) · · · 5 · 3 · 1 n!! = n (n − 2) · · · 6 · 4 · 2   1

if n > 0 odd, if n > 0 even, if n = −1, 0,

and 1 F1

a a (a + 1) z 2 (a; b; z) = 1 + z + + ··· b b (b + 1) 2! ∞ X a (a + 1) · · · (a + k − 1) z k = b (b + 1) · · · (b + k − 1) k! k=0 ˆ 1 Γ (b) b−a−1 = ezt ta−1 (1 − t) dt. Γ (b − a) Γ (a) 0

The first few polynomials are p1 (x) = x, p2 (x) = x2 + x, p3 (x) = x3 + 3x2 + 3x, p4 (x) = x4 + 6x3 + 15x2 + 15x, · · · . Recently, several authors have studied non-linear differential equations related to special polynomials (see [7–14]). The reverse Bessel polynomials are used in the design of Bessel electronic filters.

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IDENTITIES INVOLVING BESSEL POLYNOMIALS

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In this paper, we consider linear differential equations arising from Carlitz reverse Bessel polynomials and give some new and explicit identities for Bessel polynomials. 2. Identities involving Bessel polynomials arising from linear differential equations Let us put √ 1−2t)

F = F (t, x) = ex(1−

(2.1)

.

Thus, by (2.1), we get d −1 F (t, x) = x (1 − 2t) 2 F, dt

F (1) =

(2.2)

F (2) =

(2.3)

dF (1)  dt

− 32

= x (1 − 2t) F (3) =

(2.4)

d (2) F dt

= 3x (1 − 2t)

− 52

−1

+ x2 (1 − 2t)

+ 3x2 (1 − 2t)

−2



F,

− 23

+ x3 (1 − 2t)



F,

and (2.5) F (4) =

dF (3)  dt

− 72

= 15x (1 − 2t)

−3

+ 15x2 (1 − 2t)

− 25

+ 6x3 (1 − 2t)

−2

+ x4 (1 − 2t)



F.

Continuing this process, we set  N d (N ) F = (2.6) F (t, x) dt =

2N −1 X

− 2i

ai−N (N, x) (1 − 2t)

! F,

i=N

where N = 1, 2, 3, . . . . From (2.6), we note that (2.7) F (N +1) d (N ) F dt !   2N −1 X i − 2i −1 = ai−N (N, x) − (1 − 2t) (−2) F 2 =

i=N

+

2N −1 X

− 2i

ai−N (N, x) (1 − 2t)

F (1)

i=N

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TAEKYUN KIM AND DAE SAN KIM 2N −1 X

=

− i+2 2

!

iai−N (N, x) (1 − 2t)

F

i=N

+

2N −1 X

− 2i

!

ai−N (N, x) (1 − 2t)

− 12

x (1 − 2t)

F

i=N

=

2N −1 X

− i+2 2

iai−N (N, x) (1 − 2t)

! F+

i=N

2N −1 X

− i+1 2

xai−N (N, x) (1 − 2t)

! F

i=N

n − N +1 − 2N +1 = xa0 (N, x) (1 − 2t) 2 + (2N − 1) aN −1 (N, x) (1 − 2t) 2 ) 2N −1 X − i+1 2 F. + ((i − 1) ai−N −1 (N, x) + xai−N (N, x)) (1 − 2t) i=N +1

By replacing N by N + 1 in (2.6), we get (2.8)

F

(N +1)

2N +1 X

=

ai−N −1 (N + 1, x) (1 − 2t)

− 2i

! F

i=N +1 2N X

=

ai−N (N + 1, x) (1 − 2t)

− i+1 2

! F.

i=N

By comparing the coefficients on both sides (2.7) and (2.8), we have (2.9)

a0 (N + 1, x) = xa0 (N, x) , aN (N + 1, x) = (2N − 1) aN −1 (N, x) ,

(2.10) and (2.11)

ai−N (N + 1, x) = (i − 1) ai−N −1 (N, x) + xai−N (N, x) ,

where N + 1 ≤ i ≤ 2N − 1. From (2.2) and (2.6), we can derive the following equation (2.11): (2.12)

x (1 − 2t)

− 21

− 21

F = F (1) = a0 (1, x) (1 − 2t)

F.

Thus, by (2.12), we have (2.13)

a0 (1, x) = x.

From (2.9), we note that (2.14) a0 (N + 1, x) = xa0 (N, x) = x2 a0 (N − 1, x) = · · · = xN a0 (1, x) = xN +1 , and, by (2.10), we see (2.15)

aN (N + 1, x) = (2N − 1) aN −1 (N, x) = (2N − 1) (2N − 3) aN −2 (N − 1, x) .. . = (2N − 1) (2N − 3) · · · 3 · 1a0 (1, x) = (2N − 1)!!x.

The matrix (ai (j, x))0≤i≤N −1,1≤j≤N is given by

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IDENTITIES INVOLVING BESSEL POLYNOMIALS

 1

2

3

4

···

5

N 

0  x x2 x3 x4 · · · xN  1 1!!x  3!!x 2   5!!x 3 ..  ..  . .  0 N −1 (2N − 3)!!x

           

From (2.11), we obtain (2.16)

a1 (N + 1, x) = N a0 (N, x) + xa1 (N, x) = N a0 (N, x) + x (N − 1) a0 (N − 1, x) + x2 a1 (N − 1, x) .. . =

N −2 X

xi (N − i) a0 (N − i, x) + xN −1 a1 (2, x)

i=0

=

N −2 X

xi (N − i) a0 (N − i, x) + xN −1 x

i=0

=

N −1 X

xi (N − i) a0 (N − i, x) ,

i=0

(2.17)

a2 (N + 1, x) = (N + 1) a1 (N, x) + xa2 (N, x) = (N + 1) a1 (N, x) + xN a1 (N − 1, x) + x2 a2 (N − 1, x) .. . =

N −3 X

xi (N + 1 − i) a1 (N − i, x) + xN −2 a2 (3, x)

i=0

=

N −3 X

xi (N + 1 − i) a1 (N − i, x) + 3xN −2 a1 (2, x)

i=0

=

N −2 X

xi (N + 1 − i) a1 (N − i, x) ,

i=0

and (2.18)

a3 (N + 1, x) = (N + 2) a2 (N, x) + xa3 (N, x) = (N + 2) a2 (N, x) + x (N + 1) a2 (N − 1, x) + x2 a3 (N − 1, x)

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TAEKYUN KIM AND DAE SAN KIM

.. . =

N −4 X

xi (N − i + 2) a2 (N − i, x) + 5xN −3 a2 (3, x)

i=0

=

N −3 X

xi (N − i + 2) a2 (N − i, x) .

i=0

Continuing this process, we get (2.19)

aj (N + 1, x) =

N −j X

xi (N − i + j − 1) aj−1 (N − i, x) ,

i=0

where j = 1, 2, . . . , N − 1. Now, we give explicit expressions for aj (N + 1, x) (j = 1, 2, . . . , N − 1) . From (2.14) and (2.16), we can easily derive the following equation: (2.20)

a1 (N + 1, x) =

N −1 X

xi1 (N − i1 ) a0 (N − i1 , x)

i1 =0

= xN

N −1 X

(N − i1 ) .

i1 =0

By (2.17), (2.18) and (2.19), we get (2.21)

a2 (N + 1, x) =

N −2 X

xi2 (N − i2 + 1) a1 (N − i2 , x)

i2 =0

= xN −1

N −2 N −2−i X 2 X i2 =0

(2.22)

a3 (N + 1, x) =

N −3 X

(N − i2 + 1) (N − i2 − i1 − 1) ,

i1 =0

xi3 (N − i3 + 2) a2 (N − i3 , x)

i3 =0

= xN −2

N −3 N −3−i X X 3 N −3−i X3 −i2 i3 =0

i2 =0

(N − i3 + 2) (N − i3 − i2 )

i1 =0

× (N − i3 − i2 − i1 − 2) , and (2.23)

a4 (N + 1, x) =

N −4 X

xi4 (N − i4 + 3) a3 (N − i4 , x)

i4 =0

=x

N −3

N −4 N −4−i X X 4 i4 =0

×

i3 =0

N −4−i 4 −i3 −i2 X4 −i3 N −4−i X i2 =0

(N − i4 + 3) (N − i4 − i3 + 1)

i1 =0

× (N − i4 − i3 − i2 − 1) (N − i4 − i3 − i2 − i1 − 3) .

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Continuing this process, we get (2.24) aj (N + 1, x) =x

N −j+1

N −j N −j−i X Xj

N −j−ij −···−i2

X

j Y

i1 =0

k=1

···

ij =0 ij−1 =0

(N − ij − · · · − ik − (j − (2k − 1))) .

Therefore, we obtain the following theorem. Theorem 1. For N ∈ N, the linear differential equations !  N 2N −1 X d − 2i (N ) F = F F (t, x) = ai−N (N, x) (1 − 2t) dt has a solution F = F (t, x) = e N

, where

aN −1 (N, x) = (2n − 3)!!x,

a0 (N, x) = x , aj (N, x) = x

i=N √ x(1− 1−2t)

N −j

NX −j−1 N −j−1−i X j ij =0

j Y

×

N −j−1−ij −···−i2

X

···

i1 =0

ij−1 =0

! (N − ij − ij−1 − · · · − ik − (j − (2k − 2))) .

k=1

Recall the the reverse Bessel polynomials pk (x) are given by the generating function as √ F = F (t, x) = ex(1− 1−2t) (2.25) =

∞ X

tk . k!

pk (x)

k=0

Thus, by (2.25), we get F

(2.26)

(N )

N d = F (t, x) dt ∞ X tk−N = pk (x) (k)N k! 

= =

k=N ∞ X

pk+N (x) (k + N )N

k=0 ∞ X

pk+N (x)

k=0

tk (k + N )!

tk . k!

On the other hand, by Theorem 1, we get (2.27)

F

(N )

=

2N −1 X

ai−N (N, x) (1 − 2t)

− 2i

! F

i=N

=

2N −1 X i=N

 ∞  l X i (−2t) ai−N (N, x) − 2 l l! l=0

690

!

∞ X

tm pm (x) m! m=0

!

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TAEKYUN KIM AND DAE SAN KIM

=

∞ X k=0

(2N −1 X i=N

)  k    X tk k l i ai−N (N, x) + l − 1 pk−l (x) . 2 2 k! l l l=0

Therefore, by (2.26) and (2.27), we obtain the following theorem. Theorem 2. For k ∈ N ∪ {0}, and N ∈ N, we have  2N −1 k    X X k l i pk+N (x) = + l − 1 pk−l (x) , ai−N (N, x) 2 2 l l i=N

l=0

where (x)n = x (x − 1) (x − 2) · · · (x − n + 1), (n ≥ 1), and (x)0 = 1. References 1. W. A. Al-Salam and L. Carlitz, Bernoulli numbers and Bessel polynomials, Duke Math. J. 26 (1959), 437–445. MR 0105516 (21 #4256) 2. M. J. Atia and S. Chneguir, The exceptional Bessel polynomials, Integral Transforms Spec. Funct. 25 (2014), no. 6, 470–480. MR 3172058 3. G. Bevilacqua, V. Biancalana, Y. Dancheva, T. Mansour, and L. Moi, A new class of sum rules for products of Bessel functions, J. Math. Phys. 52 (2011), no. 3, 033508, 9. MR 2814858 (2012c:33018) 4. R. P. Boas, Book Review: Bessel polynomials, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 5, 799–800. MR 1567180 5. L. Carlitz, A note on the Bessel polynomials, Duke Math. J. 24 (1957), 151–162. MR 0085360 (19,27d) 6. P. Duan and J. Du, Riemann-Hilbert characterization for main Bessel polynomials with varying large negative parameters, Acta Math. Sci. Ser. B Engl. Ed. 34 (2014), no. 2, 557–567. MR 3174101 7. L.-C. Jang and B. M. Kim, On identities between sums of Euler numbers and Genocchi numbers of higher-order, J. Comput. Anal. Appl. 20 (2016), 1240– 1247. 8. D. Kang, J. Jeong, S.-J. Lee, and S.-H. Rim, A note on the Bernoulli polynomials arising from a non-linear differential equation, Proc. Jangjeon Math. Soc. 16 (2013), no. 1, 37–43. MR 3059283 9. D. S. Kim and T. Kim, A note on non-linaer Changhee differential equations, Russ. J. Math. Phys., (to appear). 10. T. Kim, Identities involving Laguerre polynomials derived from umbral calculus, Russ. J. Math. Phys. 21 (2014), no. 1, 36–45. MR 3182545 11. T. Kim, Identities involving Frobenius-Euler polynomials arising from nonlinear differential equations, J. Number Theory 132 (2012), no. 12, 2854–2865. MR 2965196 12. T. Kim, D. S. Kim, T. Mansour, S.-H. Rim, and M. Schork, Umbral calculus and Sheffer sequences of polynomials, J. Math. Phys. 54 (2013), no. 8, 083504, 15. MR 3135486 13. T. Kim and T. Mansour, Umbral calculus associated with Frobenius-type Eulerian polynomials, Russ. J. Math. Phys. 21 (2014), no. 4, 484–493. MR 3284958 14. J.-W. Park, On the q-analogue of λ-Daehee polynomials, J. Comput. Anal. Appl. 19 (2015), no. 6, 966–974. MR 3309750 15. S. Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. MR 741185 (87c:05015)

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16. H. M. Srivastava, S.-D. Lin, S.-J. Liu, and H.-C. Lu, Integral representations for the Lagrange polynomials, Shively’s pseudo-Laguerre polynomials, and the generalized Bessel polynomials, Russ. J. Math. Phys. 19 (2012), no. 1, 121–130. MR 2892608 17. D. G. Zill and W. S. Wright, Advanced Engineering Mathematics, Jones & Bartlett Publishers, 2009. Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected]

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On existence and comparison results for solutions to stochastic functional differential equations in the G-framework ∗

Faiz Faizullah , Matloob-Ur-Rehman1 , Muhammad Shahzad1 , M. Ikhlaq Chohan2 ∗ Department of BS and H, College of E and ME, National University of Sciences and Technology (NUST) Pakistan. 1 Department of Mathematics, Hazara University, Mansehra, Pakistan. 2 Department of Business Administration and Accounting, Al-Buraimi University College, Oman. October 5, 2016

Abstract With the advancement in stochastic calculus, stochastic differential equations have now become very common in different fields such as engineering, population dynamics, physics, system sciences, ecological sciences, medicine and financial mathematics. In several stochastic dynamic systems, one assumes that the future state of the system does not depend on its past states. However, under close analysis, it becomes evident that most realistic models would contain some of the past states of the system, and one would require stochastic functional differential equations in order to study such systems. This paper presents the existence theory for stochastic functional differential equations in the G-framework (in short G-SFDEs). The comparison theorem has been developed in a bid to obtain the required results. It is ascertained that the G-SFDEs, whose coefficients may be discontinuous functions, have more than one continuous and bounded solutions. Key words: Existence, G-Brownian motion, Stochastic functional differential equations, discontinuous coefficients.

1

Introduction

In the last twenty years, the greater requirement for tools and procedure of stochastic calculus has been recorded in different scientific fields. In the study of financial markets, it has acquired the state of an essential element, projected in dynamic phenomena of routine changes in share and stock prices. Stochastic calculus has its applications in engineering, as well as in filtering and control theory, and even in physics, when it deals with the effect of random changes on different physical phenomena. In Biology, its main usage is in modeling the achievement of stochastic changes in reproduction on populations processes. The idea of G-Brownian motion, which is a new ∗

Corresponding author, E-mail: faiz [email protected]/faiz [email protected] ¯ ¯

1

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stochastic process, was given by a Chinese mathematician Shige Peng in 2006 [12]. This theory opened a new era in stochastic calculus and financial mathematics. This type of motion has a newer construction as it does not depend on a specific probability space. This motion explains the ancient Brownian motion in an extraordinary way. In the framework of a sublinear expectation (called as G-expectation), he established the associated Itˆo’s calculus. During his research on stochastic calculus, Peng set up the existence and uniqueness of solutions for stochastic differential equations driven by G-Brownian motion in short (G-SDEs) with Lipschitz continuous coefficients [12, 13]. Then F. Gao generalized the associated Itˆo’s calculus and the existence theory of GSDEs with Lipschitz continuity condition using the concept of G-capacity and quasi-sure analysis [6]. Y. Ren and L. Hu proved the existence and uniqueness of solutions for G-SDEs under the Carath´eodory conditions, while later on, X. Bai and Y. Lin extended the theory for G-SDEs to the integral Lipschitz conditions [1]. In the G-frame, stochastic functional differential equations were introduced by Ren, Bi and Sakthivel [14]. Then studied by Faizullah[4]. He used the CauchyMaruyama approximation scheme to establish the existence-and-uniqueness theorem for SFDEs in the G-frame with linear growth condition as well as Lipschitz continuity condition [4]. In a different manner, this paper explores the existence theory for SFDEs in the G-frame, whose coefficients may not be continuous. This is the generalization of the previous work by Faizullah, Mukhtar and Rana [5]. We consider stochastic functional differential equations in the G-framework of the following type dY (t) = κ(t, Yt )dt + λ(t, Yt )d⟨B, B⟩(t) + λ(t, Yt )dB(t), 0 ≤ t ≤ T. (1.1) Recall that Yt = {Y (t + θ) : −δ ≤ θ ≤ 0, δ > 0} is a bounded continuous stochastic process from [−τ, 0] to R where at time t, the value of stochastic process is denoted by Y (t) [4]. Also, Yt indicates the collection of continuous bounded real-valued functions ψ defined on [−δ, 0] with norm ∥ψ∥ = sup | ψ(θ) | . Let κ, λ and µ are Borel measurable functions from [0, T ] × BC([−τ, 0]; R) −δ≤θ≤0

to R. We define the initial data of equation (1.1) as follows; Yt0 =ζ = {ζ(θ) : −τ < θ ≤ 0} is F0 − measurable, BC([−τ, 0]; R) − valued random variable so that ζ ∈ MG2 ([−τ, 0]; R) .

(1.2)

The integral form of problem (1.1) is given as the following ∫ t ∫ t ∫ t Y (t) = ζ(0) + κ(s, Ys )ds + λ(s, Ys )d⟨B, B⟩(s) + µ(s, Ys )dB(s). 0

0

0

MG2 ([−τ, T ]; R)

The G-SFDE (1.1) admit at most solution Y (t) ∈ if all its coefficients gratify the linear growth condition as well as Lipschitz condition. [4, 14]. On the other hand, in this article we assume that the coefficients κ and λ may be discontinuous functions. The solution to problem 1.1 with initial data 1.2 is a real valued stochastic process Y (t), t ∈ [−τ, T ] if it holds the following characteristics (a) For every t ∈ [0, T ], Y (t) is Ft -adapted as well as path-wise continuous. (b) κ(t, Yt ), λ(t, Yt ) ∈ L1 ([o, T ]; R) and µ(t, Yt ) ∈ L2 ([o, T ]; R); (c) Y0 = ζ and dY (t) = κ(t, Yt )dt + λ(t, Yt )d⟨B, B⟩(t) + µ(t, Yt )dB(t) q.s. for each t ∈ [0, T ]. The rest of the paper is organized as follows. Some basic definitions and notions are given in the subsequent section. Section 3 presents an important results known as the comparison theorem. The final section develops the existence theorem with possible discontinuous coefficients. 2

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2

Preliminary Concerns

This section presents some basic notions and results, which are used in forthcoming research work of this paper [2, 3, 6, 13].

2.1

Sublinear Expectation

Suppose that Ω (sample space) is a grand set and H be a family of linear and real valued functions described on Ω. Suppose that H fulfil k ∈ H, for any constant k and |Y | ∈ H if Y ∈ H. H containing the stochastic variables. Definition 2.1. A functional E, where E : H → R, is known as a G-expectation or sublinear expectation if (1) E is monotonic, that is, if Y ≥ Z for all Y, Z ∈ H ⇒ E[Y ] ≥ E[Z]. (2) E is constant conserving, that is, E[k] = k k ∈ H. (3) E is sub-additive, that is, if E[Y + Z] ≤ E[Y ] + E[Z], for each Y, Z ∈ H. (4) E is positive homogeneous, that is, E[bY ] = b[y] for b ≥ 0. the space given by triple (Ω, H, E) is said to be sublinear expectation space. And E is nonlinear expectation if it satisfies the above two conditions. Sublinear expectation is also able to state the supremum of linear expectation Definition 2.2. G-Brownian motion A d-dimensional process (Bt )t≥0 , define on (Ω, Cl,lip (H), E), is known as G-Browmain motion, if the following conditions are hold. (1) B0 (w) = 0. (2) The increment Bt+r − Bt is G-normally distributed for any t, r ≥ 0 . (3) Bt+r − Bt is independent from Bt1 , Bt2 , ........Btn for any n ∈ N, t, r ≥ 0 and 0 ≤ t1 ≤ t2 ≤ , ........ ≤ tn ≤ t.

2.2

Ito’s integral of G-Brownian motion

Definition 2.3. If T ∈ R+ , a partition π T of the interval [0, T ] is πT = {t0 , t1 , ......, tN }, since ρ(πT ) = max{|tϵ+1 − tϵ | : ϵ = 0, 1, ......N − 1}, where 0 = t0 ≤ t1 ≤, ...... ≤ tN = T,

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N N we customize πTN = {tN 0 , t1 , ......tN } to represent a sequence of partition of [0, T ] since

lim ρ(πTN ) = 0.

N →∞

Let p ≥ 1. Suppose the following sort of processes of a partition πT = {t0 , tπ1 , ......, tN }. We take, ηt (ω) =

N −1 ∑

ξm (ω)I[tm ,tm+1 ) (t)

m=0

where ξm ∈ LpG (ωtm ), for all m = 0, 1, 2, .....N − 1. The group of these process is represented by MGp,0 (0, T ). Definition 2.4. Let η ∈ MG1,0 (0, T ) with ηt (ω) =

N −1 ∑

ξm (ω)I[tm ,tm+1 ) (t)

m=0

it can be written as,

∫

T

ηt (ω)dt = 0

N −1 ∑

ξm (ω)(tm+1 − tm )

m=0

Definition 2.5. For every p ≥ 1, we represent by MGp (0, T ) the completion of MGp,0 (0, T ) under the norm ∫ T 1/p |ηt |p dt]} , ∥η∥M p (0,T ) = {E[ G

where for 1 ≤ p ≤ q,

MGp (0, T )

⊃

0

MGq (0, T ).

Definition 2.6. For every η ∈ MGp (0, T ) of the arrangement ηt (w) =

N −1 ∑

ξϵ (w)I[tϵ , tϵ+1 )(t),

ϵ=0

it can be written as,

∫

T

I(η) =

ηt dBt = 0

N −1 ∑

ξϵ (Btϵ+1 − Btϵ ).

ϵ=0

Lemma 2.7. Let a function I : MG2,0 (0, T ) → L2G (ΩT ), then it can be continuously extended to I : MG2 (0, T ) → L2G (ΩT ). Moreover, ∫ T E[ ηt dBt ] = 0, ∫

0

T

∫ ηt dBt ) ] ≤ σ E[ 2

2

E[( 0

T

ηt2 dt].

0

4

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2.3

(Peng’s quadratic variation process ⟨B⟩t )

Definition 2.8. A 1-dimensional G-quadratic variation process is introduced as follows. Let πtN , N = 1, 2, ...., be a sequence of the partition [0, T ] then Bt2

=

=

N −1 ∑

2 2 (BtN − BtN ) ϵ ϵ+1

ϵ=0 N −1 ∑

2BtNϵ (BtNi+ϵ − BtNϵ ) +

N −1 ∑

(BtNϵ+1 − BtNϵ )2 .

ϵ=0

ϵ=0

Taking limit µ(πtN ) → 0 N −1 ∑

∫ 2BtNϵ (BtNϵ+1 − BtNϵ )

converges to

2

t

Bs dBs , 0

ϵ=0

∫

and we have ⟨B⟩t =

Bt2

−2

t

Bs dBs . 0

Definition 2.9. Let P be a (weakly compact) collection of probability measures P defined on (Ω, B(Ω)) then the capacity cˆ(.) associated to P is defined by cˆ(B) = sup P (B), P ∈P

B ∈ B(Ω),

where B(Ω) is the Borel σ-algebra of Ω. A set B is said to be polar if its capacity is zero, that is, cˆ(B) = 0 and a statement holds quasi-surely in short (q.s.) if it holds except on a polar set.

3

An important result

In this section, we establish an important result known as comparison theorem. First, we assume two stochastic functional integral equations given as follows. ∫ t ∫ t ∫ t Y (t) = ζ1 (0) + κ1 (s, Ys )ds + λ1 (s, Ys )d⟨B, B⟩(s) + µ(s, Ys )dB(s), t ∈ [0, T ], (3.1) t0

∫

t0

∫

t

Y (t) = ζ2 (0) +

∫

t

κ2 (s, Ys )ds + t0

t0 t

λ2 (s, Ys )d⟨B, B⟩(s) + t0

µ(s, Ys )dB(s),

t ∈ [0, T ].

(3.2)

t0

Theorem 3.1. Let Y 1 and Y 2 are the respective unique solutions of equations (3.1) and (3.2). Suppose that κ1 (s, Ys ) ≤ κ2 (s, Ys ) and λ1 (s, Ys ) ≤ λ2 (s, Ys ) are componentwise for every t ∈ [t0 , T ], y ∈ BC([−τ, 0]; Rd ) and ζ 1 ≤ ζ 2 . Also, let the coefficients κ1 , λ1 or κ2 , λ2 are increasing functions. Then for every t > 0, Y 1 ≤ Y 2 q.s.

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Proof. Suppose that κ2 and λ2 are increasing and consider the problem ∫ t ∫ t Z(t) = ζ2 (0) + κ2 (s, max{Ys 1 , Zs })ds + λ2 (s, max{Ys 1 , Ys })d⟨B, B⟩(s) ∫

t0 t

t0

(3.3)

µ(s, max{Ys , Zs })dB(s), t0 ≤ t ≤ T, 1

+ t0

where the function x → max{y, z} satisfies the growth condition | max{y, z}| ≤ |y| + |z| and the Lipschitz condition with constant one. It follows that all coefficients of the above equation 3.3 gratify the growth condition as well as Lipschitz condition. Thus problem 3.3 admit the only one solution say Z(t). Now one has to show that Z(t) ≥ Ys1 q.s. First define stopping times δ1 and δ2 as follows. More details on stopping times can be found in [9, 10, 11]. δ1 = inf{t ∈ [t0 , T ] : Ys1 − Z(t) > 0} where δ1 < T, δ2 = inf{t ∈ [τ1 , T ] : Ys1 − Z(t) < 0}. Contrary assume that (δ1 , δ2 ) ⊂ [t0 , T ] be an arbitrary interval, such that Z(δ1 ) = Y 1 (δ1 ) = ζ ∗ (0) and Z(t) ≤ Y 1 (t) for every t ∈ (δ1 , δ2 ). Then, ∫ t ∫ t 1 ∗ 1 Z(t) − Y (t) = ζ (0) + κ2 (s, max{Ys , Zs })ds + λ2 (s, max{Ys 1 , Zs })d⟨B, B⟩(s) ∫

δ1 t

∗

µ(s, max{Ys , Zs })dB(s) − ζ (0) − 1

+ −

δ1

δ1 ∫ t

∫ λ1 (s, Ys )d⟨B, B⟩(s) −

∫

t

κ1 (s, Ys 1 )ds δ1

t

1

µ(s, Ys 1 )dB(s), t ∈ (δ1 , δ2 ).

δ1

δ1

∫ Z(t) − Y (t) =

t

1

∫

[κ2 (s, max{Ys 1 , Zs }) − κ1 (s, Ys 1 )]ds

δ1 t

[λ2 (s, max{Ys 1 , Zs }) − λ1 (s, Ys 1 )]d⟨B, B⟩(s)

+ δ1 t

∫ +

[µ(s, max{Ys 1 , Zs }) − µ(s, Ys 1 )]dB(s), t ∈ (δ1 , δ2 ).

δ1

But the assumption Z(t) ≤ Y 1 (t) gives max[Y 1 , Z] = Y 1 . So, we have ∫ t 1 [κ2 (s, Ys 1 ) − κ1 (s, Ys 1 )]ds Z(t) − Y (t) = ∫

δ1 t

[λ2 (s, Ys 1 ) − λ1 (s, Ys 1 )]d⟨B, B⟩(s)

+ δ1 ∫ t

+

[µ(s, Ys 1 ) − µ(s, Ys 1 )]dB(s),

δ1

which gives Z(t) ≥ Y 1 (t) because κ2 (t, y) ≥ κ1 (t, y) and λ2 (t, y) ≥ λ1 (t, y). This gives contradiction. So, the supposition Z(t) ≤ Y 1 (t) for every t ∈ (δ1 , δ2 ) is not true. Thus Z(t) ≥ Y 1 (t) q.s. and hence max{Y 1 , Z} = Z. It follows that Z = Y 2 ≥ Y 1 because problem (3.3) admit a single solution Y 2 . The proof is complete. 6

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4

Existence of solutions to SFDEs in the G-framework

Next, we assume that the coefficients κ and λ are not continuous. However, they are increasing, left continuous and κ(t, y) ≥ 0, λ(t, y) ≥ 0 for every (t, y) ∈ [0, T ] × BC([−δ, 0]; R). Assume a sequence of problems given as follows. ∫ t ∫ t ∫ t l l−1 l Y (t) = ζ(0) + κ(s, Ys )ds + λ(s, Ys )d⟨B, B⟩(s) + µ(s, Ysl )dB(s), t ∈ [0, T ], (4.1) 0

0

0

where Y 0 = Lt , Lt is the unique solution of the equation given by ∫ t Lt = ζ + µ(s, Ls )dB(s),

(4.2)

0

where t ∈ [0, T ]. By our supposition κ(t, y) ≥ 0, λ(t, y) ≥ 0 and comparison result we obtain Y 1 ≥ Lt . Thus, one can see that the sequence {Y l : l ≥ 1} is increasing. In the following lemma we show that Y l is bounded. Lemma 4.1. Let Y l (t) denotes a solution of equation (4.1). Then ( ) E

sup |Y l (s)|2

−δ≤s≤T

≤ K,

where K = C6 eC5 T , C6 = E[∥ζ∥] + C4 , C5 = 4(C1 + C2 + C3 ), C4 = 4[E|ζ|2 + C1 T + C2 T + C3 T ], C1 , C2 and C3 are positive constants. Proof. Define the following stopping time, for any l ≥ 1 δm = T ∧ inf {t ∈ [t0 , T ] : ∥Ytl ∥ ≥ m}. We get δm ↑ T and define Y l,m (t) = Y l (t ∧ δm ) for t ∈ (−τ, T ). Next we proceed as follows. ∫ Y

l,m

(t) = ζ(0) + 0

∫

t

κ(s, Ysl−1,m )I[o,δm ] ds

∫

t

+ 0

λ(s, Ysl,m )I[o,δm ] d⟨B, B⟩s

+ 0

t

µ(s, Ysl,m )I[o,δm ] dBs .

∫ t ∫ t λ(s, Ysl,m )I[0,δm ] d⟨B, B⟩s κ(s, Ysl−1,m )I[0,δm ] ds + |Y l,m (t)|2 = |ζ(0) + 0 0 ∫ t + µ(s, Ysl,m )I[0,δm ] dBs |2 0 ∫ t ∫ t 2 l−1,m 2 ≤ 4|ζ(0)| + 4| κ(s, Ys )I[0,δm ] ds| + 4| λ(s, Ysl,m )I[0,δm ] d⟨B, B⟩s |2 0 0 ∫ t + 4| µ(s, Ysl,m )I[0,δm ] dBs |2 0

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By taking G-expectation on both sides, using the linear growth condition and Burkholder-DavisGundy inequalities [6, 13] we proceed as follows ∫ t ∫ t E[|Y l,m (t)|2 ] ≤ 4E|ζ(0)|2 + 4C1 [1 + E|Ysl−1,m |2 ]ds + 4C2 [1 + E|Ysl,m |2 ]ds| 0 0 ∫ t + 4C3 [1 + E|Ysl,m |2 ]ds 0 ∫ t ∫ t ∫ t ∫ t 2 l−1,m 2 ≤ 4E|ζ(0)| + 4C1 ds + 4C1 E|Ys | ds + 4C2 dt + 4C2 E|Ysl,m |2 ds 0 0 0 0 ∫ t ∫ t + 4C3 ds + 4C3 E|Ysl,m |2 ds 0 0 ∫ t ∫ t 2 l−1,m 2 = 4E|ξ(0)| + 4C1 T + 4C1 E|Ys | ds + 4C2 T + 4C2 E|Ysl,m |2 ds 0 0 ∫ t + 4C3 T + 4C3 E|Ysl,m |2 ds. 0

For any j ∈ N we get, ∫ max E[|Y

1≤l≤j

l,m

(t)| ] ≤ C4 +4C1 2

∫

t

max

0 1≤l≤j

E|Ysl−1,m |2 ds+4C2

∫

t

max

0 1≤l≤j

E|Ysl,m |2 ds+4C3

t

max E|Ysl,m |2 ds,

0 1≤l≤j

where C4 = 4[E|ζ|2 + C1 T + C2 T + C3 T ]. Hence by Doob’s martingale inequality we get for any l, m ∈ N ∫ t l,m 2 E|Ysl,m |2 ds, (4.3) E[ sup |Y (s)| ] ≤ C4 + C5 0≤s≤t

0

where C5 = 4(C1 + C2 + C3 ). One can observe the fact [11], sup |Y l,m (v)|2 ≤ ∥ζ∥ + sup |Y l,m (s)|2 ,

−δ≤s≤t

0≤s≤t

and hence 4.3 gives ∫ E[ sup |Y −δ≤s≤t

l,m

(s)| ] ≤ E[∥ζ∥] + C4 + C5

t

2

∫ ≤ C6 + C5

E|Ysl,m |2 ds

0 t

E[ sup |Y l,m (q)|2 ]ds,

0

−δ≤q≤s

where C6 = E[∥ζ∥] + C4 . Finally, taking m → ∞ and by the Gronwall’s inequality we get, E[ sup |Y l (s)|2 ] ≤ C6 eC5 t . −δ≤s≤t

Letting t = T we have E[ sup |Y l (s)|2 ] ≤ K, −δ≤s≤T

where K = C6 eC5 T . Hence, the proof stands completed. 8

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Theorem 4.2. Let the coefficients κ(t, y) and λ(t, y) are increasing in the second variable y and left continuous. For all (t, y) ∈ [0, T ] × BC([−τ, 0]; R), κ(t, y) ≥ 0 and λ(t, y) ≥ 0. Then there exists at least one solution Y (t) ∈ MG2 ([−τ, T ]; R) to problem (1.1). Proof. Theorem 3.1 follows that the sequence {Y l } is increasing. On the other hand, Lemma 4.1 shows that {Y l } is a bounded sequence in the norm L2 . Thus dominated convergence theorem yields that Y n converges in L2 . Let Y be the limit of Y l . Then for almost all w, we have κ(t, Y l (t)) → κ(t, Y (t)) as l → ∞, λ(t, Y l (t)) → λ(t, Y (t)) as l → ∞. Also |κ(t, Y l (t))| ≤ K(1 + sup |Y l (t)|) ∈ L1 ([t0 , T ]), l

|λ(t, Y (t))| ≤ K(1 + sup |Y l (t)|) ∈ L1 ([t0 , T ]). l

l

Since ⟨B⟩ is continuous, so, for uniformly in t and almost all w ∫ t ∫ t l κ(s, Y (s))ds → κ(s, Y (s))ds, l → ∞, 0 0 ∫ t ∫ t l λ(s, Y (s))⟨B, B⟩(s) → λ(s, Y (s))⟨B, B⟩(s), l → ∞. 0

0

Since G-integral is continuous we get, ∫ t ∫ t l µ(s, Y (s))dB(s) → 0 (q.s), l → ∞. sup µ(s, Y (s))dB(s) − 0≤t≤T

0

0

Obviously, the sequence Y l converges uniformly to Y in t, hence Y is continuous. Taking limits l → ∞ on both sides of equation (4.1), we obtain that Y is the solution to G-SFDE (1.1) with initial condition (1.2).

5

Acknowledgment

The financial support of NUST research directorate for this research work is acknowledged and deeply appreciated.

References [1] X. Bai, Y. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with Integral-Lipschitz coefficients, Acta Mathematicae Applicatae Sinica, English Series, 30(3), 589–610 (2014). [2] L. Denis, M. Hu, S. Peng, Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths, Potential Anal., 34, 139-161(2010). 9

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[3] F. Faizullah, A note on the Caratheodory approximation scheme for stochastic differential equations under G-Brownian motion, Zeitschrift fr Naturforschung A, 67a, 699-704 (2012). [4] F. Faizullah, Existence of solutions for G-SFDEs with Cauchy-Maruyama approximation scheme, Abstract and Applied Analysis, http://dx.doi.org/10.1155/2014/809431, (2014). [5] F. Faizullah, A. Mukhtar, M. A. Rana, A note on stochastic functional differential equations driven by G-Brownian motion with discontinuous drift coefficients, J. Computational Analysis and Applications, 21(5), 910-919 (2016). [6] F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Processes and thier Applications, 2, 3356–3382 (2009). [7] N. Halidias, Y. Ren, An existence theorem for stochastic functional differential equations with delays under weak conditions, Statistics and Probability Letters, 78, 2864-2867 (2008). [8] N. Halidias, P. Kloeden, A note on strong solutions for stochastic differential equations with discontinuous drift coefficient, J. Appl. Math. Stoch. Anal., 78, 1-6 (2006). [9] M. Hu, S. Peng, Extended conditional G-expectations and related stopping times. arXiv:1309.3829v1[math.PR], (2013). [10] X. Li, S. Peng, Stopping times and related Ito’s calculus with G-Brownian motion, Stochastic Processes and thier Applications, 121, 1492-1508 (2011). [11] X. Mao, Stochastic differential equations and their applications. Horwood Publishing Chichester, (1997). [12] S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Ito’s type. The abel symposium 2005, Abel symposia 2, edit. benth et. al., Springer-vertag, 541-567 (2006). [13] S. Peng, Multi-dimentional G-Brownian motion and related stochastic calculus under Gexpectation, Stochastic Processes and thier Applications, 12, 2223-2253 (2008). [14] Y. Ren, Q. Bi, R. Sakthivel, Stochastic functional differential equations with infinite delay driven by G-Brownian motion, Mathematical Methods in the Applied Sciences, 36(13), 17461759 (2013).

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Interval-valued intuitionistic fuzzy Choquet integral operators based on Archimedean t-norm and their calculations† San-Fu Wanga,b,∗ School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, Gansu, P.R. China b School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R. China

a

Abstract: It is necessary to assume additivity and independent among decision making criteria for traditional multiple decision making (MDM) in which the weights given by decision makers based on a additive measure. However, most criteria have inter-dependent or interactive characteristics in the real decision making problems. Furthermore, with respect to multiple attribute group decision making (MAGDM) problems in which the attribute weights and the expert weights take the form of real numbers and the attribute values take the form of interval-valued intuitionistic sets, we propose interval-valued intuitionistic fuzzy Choquet integral operators based on Archimedean t-norm and discuss their calculations in this paper. First, we introduce some concepts of fuzzy measure, interval-valued intuitionistic sets and Archimedean t-norm. Then, the representations and transformations of Archimedean t-norm and Archimedean t-conorm are obtained, and the operational rules of interval-valued intuitionistic fuzzy sets based on Archimedean t-norm are presented under intuitionistic fuzzy environment. Finally, as fuzzy Choquet integral operators, some aggregating of interval-valued intuitionistic fuzzy sets based on Archimedean t-norm are given. Keywords: Intuitionistic sets; Fuzzy Choquet integral operators; Archimedean t-norm. 1. Introduction Multiple attribute decision making (MADM) problem is an important research topic in decision theory. Because the objects are fuzzy and uncertain, the attributes involved in decision problems are not always expressed as real numbers, and some better suited to be denoted by fuzzy numbers, such as interval numbers, triangular fuzzy numbers, trapezoidal fuzzy numbers, linguistic numbers on uncertain linguistic variables, and intuitionistic fuzzy numbers. Because Zadeh initially proposed the basic model of fuzzy decision making based on the theory of fuzzy mathematics, fuzzy MADM has been receiving more and more attention. We also notice that the main technologies in multiple attribute decision making, whether the situation is certain or vague, are how to define and calculate the aggregation operators proposed in the practice. The fuzzy set (FS) theory proposed by Zadeh [1] was a very good tool to research the fuzzy MADM problems, the fuzzy set is used to character the fuzziness just by membership degree. Different from fuzzy set, there is another parameter: non-membership degree in intuitionistic fuzzy set (IFS) which is proposed by Atanassov [2, 3]. Clearly, the IFS can describe and character the fuzzy essence of the objective world more accurately [2] than the fuzzy set, and has received more and more attention since its appearance. Later, Atanassov and Gargov [4, 5] further introduced the interval-valued intuitionistic fuzzy set (IVIFS), which is a generalization of the IFS. The fundamental characteristic of the IVIFS is that the values of its membership function and non-membership function are interval numbers rather real numbers. Base on Archimedean t-conorm and t-norm [6, 7], and the aggregation functions for the classical fuzzy sets (FSs), Beliakov et al. gave some operations about intuitionistic fuzzy sets, proposed two general concepts for constructing other types of aggregation operators for intuitionistic fuzzy sets (IFSs) † ∗

This work was supported by the Key Subjects Construction of Tianshui Normal University. Corresponding Author:San-Fu Wang. Tel.: +8613893853838. E-mail addresses: [email protected] 703

San-Fu Wang 703-712

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

San-Fu Wang: Interval-valued intuitionistic fuzzy Choquet integral operators based on...

extending the existing methods and showed that the operators obtained by using the Lukasiewicz t-norm are consistent with the ones on ordinary FSs. We can find above aggregation operators are all based on different relationships of the aggregated arguments, which can provide more choices for the decision makers. As an aggregation function, it is well-known that Choquet integral [8] based on non-additive fuzzy measure, is a kind of non-additive and non-linear integral, and has been successfully used for handling information fusion and decision making problems (MCDM). The main characteristic of this aggregation function is that it is able to flexibly describe the relative importance of decision criteria as well as their interactions. There are many works on the Choquet integral of single-valued functions, set-valued functions and studied their mathematical properties. It is of interest to combine the Choquet integral and the IFS theory or MCDM under intuitionistic fuzzy environment, because, by doing this, we cannot only deals with the imprecise and uncertain decision information but also efficiently take into account the various interactions among the decision criteria. The intuitionistic fuzzy-valued Choquet integral, the combination of the Choquet integral and the IFS theory, can also act an aggregation tool employed in MCDM as well as other multicriteria analysis field. In this paper, we propose the interval-valued intuitionistic fuzzy Choquet integral operators based on Archimedean t-norm and discuss their calculations. First, we introduced some concepts of fuzzy measure and interval-valued intuitionistic sets based on Archimedean t-norm. Then, interval-valued intuitionistic weighted average(geometric) operator based on Archimedean t-norm, interval-valued intuitionistic ordered weighted average operator based on Archimedean t-norm are developed. The rest of this study is organized as follows. In section 2, we recall the definitions of intuitionistic fuzzy set Archimedean t-norm and Choquet integral. In section 3, the representations and transformations of Archimedean t-norm and Archimedean t-conorm are proposed and inveastigated, and some of its properties are investigated in detail by means of the representation theorem. In section 4, the operational rules of interval-valued intuitionistic fuzzy sets based on Archimedean t-norm is presented under intuitionistic fuzzy environment. In section 5, an aggregating of interval-valued intuitionistic fuzzy sets based on Archimedean t-norm are defined and discussed.

!

2. Definitions and preliminaries A fuzzy measure on X is a set function µ : P (X) → [0, 1] such that (i) µ(∅) = 0, µ(X) = 1; (ii) A, B ⊆ X, A ⊆ B implies µ(A) 6 µ(B). Definition 2.1. Let A, B ∈ P (X), A ∩ B = ∅. If fuzzy measure g satisfies the following conditions: g(A ∪ B) = g(A) + g(B) + λg(A)g(B) and λ ∈ (−1, ∞). Especially if λ = 0, then g is an additive measure, which means there is no interaction between coalitions A and B . S Let X = {x1 , x2 , ...xn } be a attribute index set, if i, j = 1, 2, ..., n and i 6= j, xi ∩ xj = ∅, ni=1 xi = X, then  1 Qn λ 6= 0, λ ( i=1 [1 + λg(xi )] − 1) g(X) = P (1) n g(x ) λ = 0, i i=1 From Eq. (1), for the A ∈ P (X) , g can be expressed by  1 Q λ ( i∈A [1 + λg(xi )] − 1) g(X) = P i∈A g(xi )

λ 6= 0, λ = 0,

(2)

For xi , g(xi ) is called a fuzzy measure function, and itQindicates the importance degree of xi . From g(X) = 1, we know λ is determined by λ + 1 = ni=1 (1 + λg(xi )). Definition 2.2. Let f be a positive real-valued function on X, the discrete Choquet integral of f with respect to a fuzzy measure µ on X is defined as Cµ (f (x(1) ), ..., f (x(n) )) =

n X i=1 704

f (x(i) )[µ(A(i) ) − µ(A(i+1) )] San-Fu Wang 703-712

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

San-Fu Wang: Interval-valued intuitionistic fuzzy Choquet integral operators based on...

where (·) indicates a permutation on X such that f (x(1) ) 6 · · · 6 f (x(n) ). A(i) = (i, . . . , n), and A(n+1) = ∅. A function T : [0, 1] × [0, 1] → [0, 1] is called a t-norm if it satisfies the following four conditions [8,9]: 1) T (1, x) = x, for all x. 2) T (x, y) = T (y, x), for all x and y. 3) T (x, T (y, z)) = T (T (x, y), z), for all x, y and z. 0 0 0 0 0 0 4) x 6 x , y 6 y implies T (x, y) 6 T (x , y ), x, y, x , y ∈ [0, 1]. A function S : [0, 1] × [0, 1] → [0, 1] is called a t-conorm if it satisfies the following four conditions[8,9]: 1) S(0, x) = x, for all x. 2) S(x, y) = S(y, x), for all x and y. 3) S(x, S(y, z)) = S(S(x, y), z), for all x, y and z. 0 0 0 0 0 0 4) x 6 x , y 6 y implies S(x, y) 6 S(x , y ), x, y, x , y ∈ [0, 1]. Definition 2.3 [8,9]. A t-norm function T (x, y) is called Archimedean t-norm if it is continuous and T (x, x) < x for all x ∈ [0, 1]. An Archimedean t-norm is called strictly Archimedean t-norm if it is strictly increasing in each variable for x, y ∈ (0, 1). A t-conorm function S(x, y) is called Archimedean t-conorm if it is continuous and S(x, x) > x for all x ∈ [0, 1]. An Archimedean t-conorm is called strictly Archimedean t-conorm if it is strictly increasing in each variable for x, y ∈ (0, 1). Definition 2.4. Let X be in a given domain. Then, A = {hx, µA (x), νA (x)i|x ∈ X} is called an interval-valued intuitionistic fuzzy set (IV IF S), where µA : X → I ⊂ [0, 1], νA : X → J ⊂ [0, 1] and I, J are closed intervals in [0, 1], the following condition is met: sup µA (x) + sup νA (x) 6 1 , x ∈ X. The intervals µA (x) and νA (x) represent, respectively, the membership degree and non-membership degree of the element x on X. Thus for each x, µA (x) and νA (x) are closed intervals and their lower and upper end points are, U L U respectively, denoted by µL A (x), µA (x),νA (x),νA (x). We can denote by U L U A = {hx, [µL A (x), µA (x)], [νA (x), νA (x)]]i|x ∈ X}, U L L where 0 6 µU A (x) + νA (x) 6 1, x ∈ X, µA (x) > 0 and νA (x) > 0. L U L U Simply, we write A = h[µA (x), µA (x)], [νA (x), νA (x)]. For each element x, we can compute its hesitation interval of x as: L U U L L πA (x) = [πA (x), πA (x)] = [1 − νA (x) − µU A (x), 1 − νA (x) − µA (x)].

3. The representations and transformations of Archimedean t-norm and Archimedean tconorm

§

Definition 3.1. A mapping N : [0, 1] → [0, 1] is called negation operator if N is decreasing and N (0) = 1 , N (1) = 0. Especially, we have (i) If N (x) = 1 − x, it is called standard negation operator. (ii) ∀x ∈ [0, 1], if N (N (x)) = x, then it is called cyclotron negation operator. Obviously, cyclotron negation operator is continuous and strictly increasing. (iii) For each negation operator, T and S are dual with respect to N (x) if and only if T (N (x), N (y)) = N (S(x, y)). It is well known [9] that a strict Archimedean t-norm is expressed via its additive generator g as T (x, y) = g −1 (g(x) + g(y)), and similarly, applied to its dual t-conorm S(x, y) = h−1 (h(x) + h(y)) with h(t) = g(N (t)). We notice that an additive generator of a continuous Archimedean t-norm is a strictly decreasing function g : [0, 1] → [0, +∞) such that g(1) = 0. If we assign specific forms to the function g, then some well-known t-conorms and t-norms can be obtained. Let me emphasize that the results (1-4) were shown in [11], however, considering that the representation of the negation operator is always restricted by the policy mak- ers’ historical knowledge, perceptual judgement and other factors in the game playing, benefit groups’ voting or decision making process, we could define the negation operator by means of the fuzzy logic non-portal operators in this paper and calculate Archimedean t-norm and Archimedean t-conorm as results (5-8) as follows. 705

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San-Fu Wang: Interval-valued intuitionistic fuzzy Choquet integral operators based on...

Theorem 3.1 Let T (x, y) be Archimedean t-norm and S(x, y) its dual Archimedean t-conorm. Then we have the following statements: If N (x) = 1 − x, i.e. h(t) = g(1 − t)), then the following are valid: (1) Let g(t) = − log t, then h(t) = − log(1 − t), g −1 (t) = exp−t , h−1 (t) = 1 − exp−t , and Algebraic t-conorm and t-norm [10] are obtained as follows: T A (x, y) = x · y, S A (x, y) = x + y − xy. 2−(1−t) −1 (2) Let g(t) = log( 2−t t ), then h(t) = log( 1−t ), g (t) = Einstein t-conorm and t-norm [10]:

T E (x, y) =

2 −1 expt +1 , h (t)

= 1−

2 expt +1 ,

and we get

xy x+y , S E (x, y) = . 1 + (1 − x)(1 − y) 1 + xy

) , γ > 0, then we have h(t) = log( γ+(1−γ)(1−t) ), g −1 (t) = (3) Let g(t) = log( γ+(1−γ)t t 1−t γ −1 h (t) = 1 − expt +γ−1 , and Hamacher t-conorm and t-norm [10] are obtained as follows: TγH (x, y) =

γ expt +γ−1 ,

xy , γ > 0, γ + (1 − γ)(x + y − xy)

x + y − xy − (1 − γ)xy , γ > 0. 1 − (1 − γ)xy Especially, if γ = 1, then Hamacher t-conorm and t-norm reduce to the Algebraic t-conorm and t-norm respectively; if γ = 1, then Hamacher t-conorm and t-norm reduce to the Einstein t-conorm and t-norm respectively. SγH (x, y) =

t

(4) Let g(t) = t log( γ−1+exp ) expt

log γ

log( γγ−1 t −1 )

, γ > 1, then h(t) =

γ−1 ), g −1 (t) log( γ 1−t −1

=

log( γ−1+exp ) expt log γ

, h−1 (t) = 1 −

, and we have Frank t-conorm and t-norm [10] as follows: TγF (x, y) = logγ (1 + SγF (x, y) = 1 − logγ (1 +

(γ x − 1)(γ y − 1) ) , γ > 1, γ−1 (γ 1−x − 1)(γ 1−y − 1) ) , γ > 1. γ−1

Especially, if γ → 1, then we have lim g(t) = lim log(

γ→1

γ→1

γ−1 1 ) = lim log( t−1 ) = − log t. t γ→1 γ −1 tγ −1

which indicates that limγ→1 SγF (x, y) = SγA (x, y) and limγ→1 TγF (x, y) = TγA (x, y). If N (x) = 1 − x2 , i.e. h(t) = g(1 − t2 ) then the following are also valid: p (5) Let g(t) = − log t, then h(t) = − log(1 − t2 ), g −1 (t) = exp−t , h−1 (t) = 1 − exp−t , and Algebraic t-conorm and t-norm [10] are obtained as follows: p T2A (x, y) = xy , S2A (x, y) = 1 − (1 − x2 )(1 − y 2 ). q t exp −1 1+t2 2 −1 (t) = −1 (t) = (6) Let g(t) = log( 2−t ), then we have h(t) = log , g , h t expt +1 expt +1 , and we 1−t2 get Einstein t-conorm and t-norm [10] are obtained as follows: s xy x2 + y 2 E E T2 (x, y) = , S2 (x, y) = / 1 + (1 − x)(1 − y) 1 + x2 y 2 2

) (7) Let g(t) = log( γ+(1−γ)t ) , γ > 0, then we have h(t) = log( γ+(1−γ)(1−t ), g −1 (t) = expt γ+γ−1 , t 1−t2 q h−1 (t) = 1 − expt γ+γ−1 , and Hamacher t-conorm and t-norm [10] are obtained as follows: H T2γ (x, y) =

xy , γ > 0, γ + (1 − γ)(x + y − xy) 706

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San-Fu Wang: Interval-valued intuitionistic fuzzy Choquet integral operators based on...

s H S2γ (x, y)

=

x2 + y 2 − x2 y 2 − (1 − γ)x2 y 2 , γ > 0. 1 − (1 − γ)x2 y 2

Especially, if γ = 1, then Hamacher t-conorm and t-norm reduce to the Algebraic t-conorm and t-norm respectively; if γ = 1, then Hamacher t-conorm and t-norm reduce to the Einstein t-conorm and t-norm respectively. t

γ−1 (8) Let g(t) = log( γγ−1 ), g −1 (t) = t −1 ) , γ > 1, then h(t) = log( 1−t2 γ −1 r t

log( γ−1+exp ) expt log γ

,

) log( γ−1+exp t

exp , h−1 (t) = 1 − log γ and we have Frank t-conorm and t-norm [10] as follows:

F T2γ (x, y) = logγ (1 +

(γ x − 1)(γ y − 1) ) , γ > 1, γ−1

s F S2γ (x, y)

=

(γ 1−x2 − 1)(γ 1−y2 − 1) 1 − logγ (1 + ) , γ > 1. γ−1

Especially, if γ → 1, then we have lim g(t) = lim log(

γ→1

γ→1

1 γ−1 ) = lim log( t−1 ) = − log t. γ→1 γt − 1 tγ −1

which indicates that limγ→1 SγF (x, y) = SγA (x, y) and limγ→1 TγF (x, y) = TγA (x, y). 4. The operational rules of interval-valued intuitionistic fuzzy sets based on Archimedean t-norm Definition 4.1. Let α ei = h[µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2) be two interval-valued intuitionistic fuzzy sets, T (x, y) Archimedean t-norm and S(x, y) its dual Archimedean t-conorm, and λ > 0. We can define the operational rules about α e1 and α e2 based on Archimedean t-norm as follows (1) α e1 ⊕ α e2 = h[S(µL (α1 ), µL (α2 )), S(µU (α1 ), µU (α2 ))], [T (ν L (α1 ), ν L (α2 )), T (ν U (α1 ), ν U (α2 ))]i = h[h−1 (h(µL (α1 )) + h(µL (α2 ))), h−1 (h(µU (α1 )) + h(µU (α2 )))], [g −1 (g(ν L (α1 )) + g(ν L (α2 ))), g −1 (g(ν U (α1 )) + g(ν U (α2 )))]i; (2) α e1 ⊗ α e2 = h[T (µL (α1 ), µL (α2 )), T (µU (α1 ), µU (α2 ))], [S(ν L (α1 ), ν L (α2 )), S(ν U (α1 ), ν U (α2 ))]i = h[g −1 (g(µL (α1 )) + g(µL (α2 ))), g −1 (g(µU (α1 )) + g(µU (α2 )))], [h−1 (h(ν L (α1 )) + h(ν L (α2 ))), h−1 (h(ν U (α1 )) + h(ν U (α2 )))]i; (3) λe α1 = h[h−1 (λh(µL (α1 ))), h−1 (λh(µU (α1 )))], [g −1 (λg(ν L (α1 ))), g −1 (λg(ν U (α1 )))]i; (4) α e1λ = h[g −1 (λg(µL (α1 ))), g −1 (λg(µU (α1 )))], [h−1 (λh(ν L (α1 ))), h−1 (λh(ν U (α1 )))]i. Obviously, the above operational result is still an the operational rules of interval-valued intuitionistic fuzzy sets based on Archimedean t-norm. According Theorem 3.1 and Definition 4.1, we have Theorem 4.1 and Theorem 4.2, the operational rules of interval-valued intuitionistic fuzzy sets based on Archimedean t-norm are obtained as follows. Theorem 4.1. Let α ei = h[µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2) be two interval-valued fuzzy intuitionistic sets, T (x, y) Archimedean t-norm and S(x, y) its dual Archimedean t-conorm, N (x) = 1−x. Then the following operational rules based on Archimedean t-norm are hold (1) If g(t) = − log t, then [9] (i) α e1 ⊕ α e2 = h[µL (α1 ) + µL (α2 ) − µL (α1 )µL (α2 ), µU (α1 ) + µU (α2 ) − µU (α1 )µU (α2 )], [ν L (α1 )ν L (α2 ), ν U (α1 )ν U (α2 )]i; (ii) α e1 ⊗ α e2 = h[µL (α1 )µL (α2 ), µU (α1 )µU (α2 )], [ν L (α1 ) + ν L (α2 ) − ν L (α1 )ν L (α2 ), ν U (α1 ) + ν U (α2 ) − ν U (α1 )ν U (α2 )]i; (iii) λe α1 = h[sλ×θ(α1 ) , sλ×τ (α1 ) ], [1 − (1 − µL (α1 ))λ , 1 − (1 − µU (α1 ))λ ], [(ν L (α1 ))λ , (ν U (α1 ))λ ]i; λ (iv) α e1 = h[s(θ(α1 ))λ , s(τ (α1 ))λ ], [(µL (α1 ))λ , (µU (α1 ))λ ], [1 − (1 − ν L (α1 ))λ , 1 − (1 − ν U (α1 ))λ ]i. (2) If g(t) = log( 2−t t ), then

µ

L

L

U

U

L

L

U

U

µ (α1 )+µ (α2 ) µ (α1 )+µ (α2 ) ν (α1 )ν (α2 ) ν (α1 )ν (α2 ) (i) α e1 ⊕ α e2 = h[ 1+µ L (α )µL (α ) , 1+µU (α )µU (α ) ], [ 1+(1−ν L (α ))(1−ν L (α )) , 1+(1−ν U (α ))(1−ν U (α )) ]i; 1 2 1 2 1 2 1 2

707

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

San-Fu Wang: Interval-valued intuitionistic fuzzy Choquet integral operators based on...

L

L

U

U

L

L

U

U

µ (α1 )µ (α2 ) µ (α1 )µ (α2 ) ν (α1 )+ν (α2 ) ν (α1 )+ν (α2 ) (ii) α e1 ⊗ α e2 = h[ 1+(1−µ L (α ))(1−µL (α )) , 1+(1−µU (α ))(1−µU (α )) ], [ 1+ν L (α )ν L (α ) , 1+ν U (α )ν U (α ) ]i; 1 2 1 2 1 2 1 2 L

λ

L

λ

U

λ

U

λ

λ 2(ν U (α1 ))λ 1 )) λ +ν L (α )λ , (2−ν U (α ))λ +ν U (α )λ ]i; )) 1 1 1 1 1 1 1 2(µU (α1 ))λ (1+ν L (α1 ))λ −(1−ν L (α1 ))λ (1+ν U (α1 ))λ −(1−ν U (α1 ))λ ], [ , ]i. (2−µU (α1 ))λ +µU (α1 )λ (1+ν L (α1 ))λ +(1−ν L (α1 ))λ (1+ν U (α1 ))λ +(1−ν U (α1 ))λ

(1+µ (α1 )) −(1−µ (α1 )) (1+µ (α1 )) −(1−µ (α1 )) 2(ν (iii) λe α1 = h[ (1+µ L (α ))λ +(1−µL (α ))λ , (1+µU (α ))λ +(1−µU (α ))λ ], [ (2−ν L (α 1

L

λ

(α1 )) , (iv) α e1λ = h[ (2−µL2(µ (α1 ))λ +µL (α1 )λ

L (α

(3) If g(t) = log( γ+(1−γ)t ) , γ > 0, then t

L (α )µL (α )−(1−γ)µL (α )µL (α ) µU (α )+µU (α )−µU (α )µU (α )−(1−γ)µU (α )µU (α ) 1 2 1 2 1 2 1 2 1 2 , ], 1−(1−γ)µL (α1 )µL (α2 ) 1−(1−γ)µU (α1 )µU (α2 ) L L U U (α1 )ν (α2 ) ν (α1 )ν (α2 ) [ γ+(1−γ)(ν L (αν1 )+ν L (α )−ν L (α )ν L (α )) , γ+(1−γ)(ν U (α )+ν U (α )−ν U (α )ν U (α )) ]i; 2 1 2 1 2 1 2 µL (α1 )µL (α2 ) µU (α1 )µU (α2 ) (ii) α e1 ⊗ α e2 = h[ γ+(1−γ)(µL (α1 )+µL (α2 )−µL (α1 )µL (α2 )) , γ+(1−γ)(µU (α1 )+µU (α2 )−µU (α1 )µU (α2 )) ], L L L (α )ν L (α )−(1−γ)ν L (α )ν L (α ) ν U (α )+ν U (α )−ν U (α )ν U (α )−(1−γ)ν U (α )ν U (α ) 1 2 1 2 1 2 1 2 1 2 [ ν (α1 )+ν (α2 )−ν , ]i; 1−(1−γ)ν L (α1 )ν L (α2 ) 1−(1−γ)ν U (α1 )ν U (α2 ) (1+(γ−1)µU (α1 ))λ −(1−µU (α1 ))λ (1+(γ−1)µL (α1 ))λ −(1−µL (α1 ))λ (iii) λe α1 = h[ (1+(γ−1)µL (α1 ))λ +(γ−1)(1−µL (α1 ))λ , (1+(γ−1)µU (α1 ))λ +(γ−1)(1−µU (α1 ))λ ], L (α ))λ γ(ν U (α1 ))λ 1 [ (1+(γ−1)(1−ν Lγ(ν , ]i; (α1 )))λ +(γ−1)(ν L (α1 ))λ (1+(γ−1)(1−ν U (α1 )))λ +(γ−1)(ν U (α1 ))λ L (α ))λ U (α ))λ γ(µ γ(µ 1 1 (iv) α e1λ = h[ (1+(γ−1)(1−µL (α )))λ +(γ−1)(µL (α ))λ , (1+(γ−1)(1−µU (α )))λ +(γ−1)(µU (α ))λ ], 1 1 1 1 (1+(γ−1)ν L (α1 ))λ −(1−ν L (α1 ))λ (1+(γ−1)ν U (α1 ))λ −(1−ν U (α1 ))λ [ (1+(γ−1)ν , L (α ))λ +(γ−1)(1−ν L (α ))λ (1+(γ−1)ν U (α ))λ +(γ−1)(1−ν U (α ))λ ]i. 1 1 1 1 (4) If g(t) = log( γγ−1 t −1 ) , γ > 1, then U U 1−µL (α1 ) −1)(γ 1−µL (α2 ) −1) (γ 1−µ (α1 ) −1)(γ 1−µ (α2 ) −1) ), 1 − log (1 + )], (i) α e1 ⊕ α e2 = h[1 − logγ (1 + (γ γ γ−1 γ−1 L (α ) L (α ) U (α ) U (α ) ν ν ν ν 1 −1)(γ 2 −1) 1 −1)(γ 2 −1) [logγ (1 + (γ ), logγ (1 + (γ )]i; γ−1 γ−1 L L U U µ (α1 ) −1)(γ µ (α2 ) −1) µ (α1 ) −1)(γ µ (α2 ) −1) (ii) α e1 ⊗ α e2 = h[logγ (1 + (γ ), logγ (1 + (γ )], γ−1 γ−1 U (α ) U L (α ) L (α ) 1−ν 1−ν 1−ν 1 1 2 (γ −1)(γ 1−ν (α2 ) −1) −1)(γ −1) (1 + ), 1 − log )]i; [1 − logγ (1 + (γ γ γ−1 γ−1 L (α ) U (α ) 1−µ λ 1−µ λ 1 −1) 1 −1) (iii) λe α1 = h[1 − logγ (1 + (γ (γ−1)λ−1 ), 1 − logγ (1 + (γ (γ−1)λ−1 )],

(i) α e1 ⊕ α e2 = h[ µ

L (α

1 )+µ

L (α

2 )−µ

U

L

[logγ (1 +

(γ ν (α1 ) −1)λ ), logγ (1 (γ−1)λ−1

(iv) α e1λ = h[logγ (1 +

(γ µ (α1 ) −1)λ ), logγ (1 (γ−1)λ−1

+

(γ ν (α1 ) −1)λ )]i; (γ−1)λ−1

+

(γ µ (α1 ) −1)λ )], (γ−1)λ−1

L

U

L

[1 − logγ (1 +

U

1−ν (α1 ) −1)λ (γ 1−ν (α1 ) −1)λ ), 1 − logγ (1 + (γ (γ−1)λ−1 )]i. (γ−1)λ−1 L U L U h[µ (αi ), µ (αi )], [ν (αi ), ν (αi )]i (i = 1, 2)

Theorem 4.2. Let α ei = be be two interval-valued intuitionistic fuzzy sets , T (x, y) Archimedean t-norm and S(x, y) its dual Archimedean t-conorm, N (x) = 1 − x. Then the following operational rules based on Archimedean t-norm valid: (1) If g(t) = − log t, then (i) α e1 ⊕ α e2 = p p h[ (µL (α1 ))2 + (µL (α2 ))2 − (µL (α1 ))2 (µL (α2 ))2 , (µU (α1 ))2 + (µU (α2 ))2 − (µU (α1 ))2 (µU (α2 ))2 ], [ν L (α1 )ν L (α2 ), ν U (α1 )ν U (α2 )]i; (ii) α e1 ⊗ α e2 = h[µL (α1 )µL (α2 ), µU (α1 )µUp (α2 )], p 2 − (ν L (α ))2 (ν L (α ))2 , [ (ν L (α1 ))2 + (ν L (αp )) (ν U (α1 ))2 + (ν U (α2 ))2 − (ν U (α1 ))2 (ν U (α2 ))2 ]i; 2 1 2p L 2 λ λ ], [(ν L (α ))λ , (ν U (α ))λ ]i; (iii) λe α1 = h[ 1 − (1 − (µ (α1 )) )p, 1 − (1 − (µU (α1 ))2 )p 1 1 λ L λ U λ L 2 λ U 2 (iv) α e1 = h[(µ (α1 )) , (µ (α1 )) ], [ 1 − (1 − (ν (α1 )) ) , 1 − (1 − (ν (α1 )) )λ ]i. (2) If g(t) = log( 2−t ), then t q q U (µL (α1 ))2 +(µL (α2 ))2 (µ (α1 ))2 +(µU (α2 ))2 (i) α e1 ⊕ α e2 = h[ 1+(µ , L (α ))2 (µL (α ))2 U (α ))2 (µU (α ))2 ], 1+(µ 1 2 1 2 L

L

U

U

ν (α1 )ν (α2 ) ν (α1 )ν (α2 ) [ 1+(1−ν L (α ))(1−ν L (α )) , 1+(1−ν U (α ))(1−ν U (α )) ]i; 1 2 1 2 L

L

U

U

µ (α1 )µ (α2 ) µ (α1 )µ (α2 ) (ii) α e1 ⊗ α e2 = h[ 1+(1−µ L (α ))(1−µL (α )) , 1+(1−µU (α ))(1−µU (α )) ], 1 2q 1 2 q L (ν (α1 ))2 +(ν L (α2 ))2 (ν U (α1 ))2 +(ν U (α2 ))2 [ 1+(ν , ]i; L (α ))2 (ν L (α ))2 1+(ν U (α1 ))2 (ν U (α2 ))2 1 2 q q U (α ))2 )λ −(1−(µU (α ))2 )λ (1+(µL (α1 ))2 )λ −(1−(µL (α1 ))2 )λ 1 1 (iii) λe α1 = h[ (1+(µL (α ))2 )λ +(1−(µL (α ))2 )λ , (1+(µ ], (1+(µU (α ))2 )λ +(1−(µU (α ))2 )λ 1

1

1

1

2(ν L (α1 ))λ 2(ν U (α1 ))λ [ (2−ν L (α λ L λ , (2−ν U (α ))λ +(ν U (α ))λ ]i; 1 )) +(ν (α1 )) 1 1

708

San-Fu Wang 703-712

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

San-Fu Wang: Interval-valued intuitionistic fuzzy Choquet integral operators based on...

L

λ

U

λ

2(µ (α1 )) 2(µ (α1 )) (iv) α e1λ = h[ (2−µL (α λ L λ , (2−µU (α ))λ +(µU (α ))λ ], 1 )) +(µ (α1 )) 1 1 q q (1+(ν L (α1 ))2 )λ −(1−(ν L (α1 ))2 )λ (1+(ν U (α1 ))2 )λ −(1−(ν U (α1 ))2 )λ [ (1+(ν L (α1 ))2 )λ +(1−(ν L (α1 ))2 )λ , (1+(ν U (α1 ))2 )λ +(1−(ν U (α1 ))2 )λ ]i.

(3) If g(t) = log( γ+(1−γ)t ) , γ > 0, then t q L 2 L 2 −(µL (α ))2 (µL (α ))2 −(1−γ)(µL (α ))2 (µL (α ))2 1 1 1 2 (i) α e1 ⊕ α e2 = h[ (µ (α1 )) +(µ (α2 )) 1−(1−γ)(µ , L (α ))2 (µL (α ))2 1 2 q U 2 U 2 U 2 U 2 U 2 U 2 (µ (α1 )) +(µ (α2 )) −(µ (α1 )) (µ (α1 )) −(1−γ)(µ (α1 )) (µ (α2 )) ], 1−(1−γ)(µU (α1 ))2 (µU (α2 ))2 L

U

L

U

ν (α1 )ν (α2 ) (α1 )ν (α2 ) [ γ+(1−γ)(ν L (αν1 )+ν L (α )−ν L (α )ν L (α )) , γ+(1−γ)(ν U (α )+ν U (α )−ν U (α )ν U (α )) ]i; 2 1 2 1 2 1 2 L

L

U

U

(α1 )µ (α2 ) µ (α1 )µ (α2 ) (ii) α e1 ⊗ α e2 = h[ γ+(1−γ)(µL (αµ1 )+µ L (α )−µL (α )µL (α )) , γ+(1−γ)(µU (α )+µU (α )−µU (α )µU (α )) ], 2 1 2 1 2 1 2 q L (ν (α1 ))2 +(ν L (α2 ))2 −(ν L (α1 ))2 (ν L (α1 ))2 −(1−γ)(ν L (α1 ))2 (ν L (α2 ))2 , [ 1−(1−γ)(ν L (α1 ))2 (ν L (α2 ))2 q U (ν (α1 ))2 +(ν U (α2 ))2 −(ν U (α1 ))2 (ν U (α1 ))2 −(1−γ)(ν U (α1 ))2 (ν U (α2 ))2 ]i; 1−(1−γ)(ν U (α1 ))2 (ν U (α2 ))2 q q (1+(γ−1)(µL (α1 ))2 )λ −(1−(µL (α1 ))2 )λ (1+(γ−1)(µU (α1 ))2 )λ −(1−(µU (α1 ))2 )λ (iii) λe α1 = h[ (1+(γ−1)(µ ], L (α ))2 )λ +(γ−1)(1−(µL (α ))2 )λ , (1+(γ−1)(µU (α ))2 )λ +(γ−1)(1−(µU (α ))2 )λ 1

1

1

1

L (α ))λ γ(ν U (α1 ))λ 1 [ (1+(γ−1)(1−ν Lγ(ν , ]i; (α1 )))λ +(γ−1)(ν L (α1 ))λ (1+(γ−1)(1−ν U (α1 )))λ +(γ−1)(ν U (α1 ))λ L (α ))λ U (α ))λ γ(µ γ(µ (iv) α e1λ = h[ (1+(γ−1)(1−µL (α )))λ1 +(γ−1)(µL (α ))λ , (1+(γ−1)(1−µU (α )))λ1 +(γ−1)(µU (α ))λ ], 1 1 1 1 q q (1+(γ−1)(ν L (α1 ))2 )λ −(1−(ν L (α1 ))2 )λ (1+(γ−1)(ν U (α1 ))2 )λ −(1−(ν U (α1 ))2 )λ [ (1+(γ−1)(ν L (α1 ))2 )λ +(γ−1)(1−(ν L (α1 ))2 )λ , (1+(γ−1)(ν U (α1 ))2 )λ +(γ−1)(1−(ν U (α1 ))2 )λ ]i. If g(t) = log( γγ−1 t −1 ) , γ > 1, then

(4) e1 ⊕ α e2 = r(i) α h[ 1 − logγ (1 +

L (α ))2 L 2 1 −1)(γ 1−(µ (α2 )) −1)

(γ 1−(µ

[logγ (1 +

γ−1 (γ 1−ν

L (α ) L 1 −1)(γ 1−ν (α2 ) −1)

γ−1

(ii) α e1 ⊗ α e2 = h[logγ (1 + r [ 1 − logγ (1 + r

), logγ (1 +

γ−1 γ−1 L

2

(γ 1−(µ (α1 )) −1)λ ), (γ−1)λ−1

+

(γ µ (α1 ) −1)λ ), logγ (1 (γ−1)λ−1 L

[ 1 − logγ (1 +

2

U (α ))2 U 2 1 −1)(γ 1−(µ (α2 )) −1)

γ−1

U (α ) U 1 −1)(γ 1−ν (α2 ) −1)

γ−1 γ−1

)],

U 2 U 2 (γ 1−(ν (α1 )) −1)(γ 1−(ν (α2 )) −1)

γ−1

U

1 − logγ (1 +

)]i;

U U (γ 1−µ (α1 ) −1)(γ 1−µ (α2 ) −1)

1 − logγ (1 +

),

r

)],

)]]i;

2

(γ 1−(µ (α1 )) −1)λ )], (γ−1)λ−1

U

L

L

(γ 1−ν

(γ 1−(µ

), logγ (1 + r

L 2 L 2 (γ 1−(ν (α1 )) −1)(γ 1−(ν (α2 )) −1)

(γ ν (α1 ) −1)λ ), logγ (1 (γ−1)λ−1

(iv) α e1λ = h[logγ (1 + r

1 − logγ (1 +

),

L L (γ 1−µ (α1 ) −1)(γ 1−µ (α2 ) −1)

(iii) λe α1 = h[ 1 − logγ (1 + [logγ (1 +

r

(γ ν (α1 ) −1)λ )]i; (γ−1)λ−1 µU (α1 )

λ

−1) + (γ (γ−1)λ−1 )], r

(γ 1−(ν (α1 )) −1)λ ), (γ−1)λ−1

1 − logγ (1 +

U

2

(γ 1−(ν (α1 )) −1)λ )]i. (γ−1)λ−1

Theorem 4.3. Let α ei (i = 1, 2) be be two interval-valued intuitionistic fuzzy sets, T (x, y) Archimedean t-norm and S(x, y) its dual Archimedean t-conorm, N (x) = 1 − x. We can easily prove the the following statements: (1) α e1 ⊕ α e2 = α e2 ⊕ α e1 ; (2) α e1 ⊗ α e2 = α e2 ⊗ α e1 ; (3) λ(e α1 ⊕ α e2 ) = λe α1 ⊕ λe α2 , λ > 0; (4) λ1 α e 1 ⊕ λ2 α e1 = (λ1 + λ2 )e α1 , λ1 , λ2 > 0; λ1 λ2 λ +λ 1 2 (5) α e1 ⊗ α e1 = (e α1 ) , λ1 , λ2 > 0; (6) α e1λ ⊗ α e2λ = (e α1 ⊗ α e2 )λ , λ > 0. According Theorem 3.1 and Definition 4.1, Theorem 4.3 is easy to prove. 5. Aggregating of interval-valued intuitionistic fuzzy sets based on Archimedean t-norm Definition 5.1. Let α e1 = h[µL (α1 ), µU (α1 )], [ν L (α1 ), ν U (α1 )]i be an interval-valued fuzzy intuitionistic sets. An expected value E(e α1 ) of α e1 can be represented as follows E(e α1 ) =

µL (α1 ) + µU (α1 ) ν L (α1 ) + ν U (α1 ) 1 ×( +1− ) 2 2 2 709

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San-Fu Wang: Interval-valued intuitionistic fuzzy Choquet integral operators based on...

= (µL (α1 ) + µU (α1 ) + 2 − ν L (α1 ) − ν U (α1 ))/4. An accuracy function H(e α1 ) can be represented as follows H(e α1 ) = (

µL (α1 ) + µU (α1 ) ν L (α1 ) + ν U (α1 ) + ) 2 2

= (µL (α1 ) + µU (α1 ) + ν L (α1 ) + ν U (α1 ))/4. L U h[µ (αi ), µ (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2) be two interval-valued

Let α ei = fuzzy intuitionistic sets. Then (1) If E(e α1 ) > E(e α2 ), then α e1  α e2 . (2) If E(e α1 ) = E(e α2 ), then: If H(e α1 ) > H(e α2 ), then α e1  α e2 . If H(e α1 ) = H(e α2 ), then α e1 = α e2 . Based on the the above operational rules, we propose weighted average (geometric) operator, ordered weighted average (geometric) operator and hybrid average (geometric) operator for interval-valued intuitionistic fuzzy sets based on Archimedean t-norm in this part. Definition 5.2. Let α ei = h[µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2, . . . , n) be a collection of interval-valued intuitionistic fuzzy sets, T (x, y) Archimedean t-norm and S(x, y) its dual Archimedean t-conorm, N (x) = 1 − x. We define interval-valued intuitionistic fuzzy weighted average operator based on Archimedean t-norm as follows: AT S − IV IF W A : Ωn → Ω, AT S − IV IF W Aµ (e α1 , α e2 , . . . , α en ) =

n X

µj α ej ,

j=1

Specifically, if µ = ( n1 , n1 , . . . , n1 ), then AT S − IV IF W A operator degenerates interval-valued intuitionistic fuzzy arithmetic average operator based on Archimedean t-norm (AT S − IV IF AA)

µ

1 (e α1 ⊕ α e2 ⊕ . . . ⊕ α en ). n Similarly, we could define interval-valued intuitionistic fuzzy weighted geometric average operator based on Archimedean t-norm, AT S − IV IF W GA : Ωn → Ω, as follows AT S − IV IF AA(e α1 , α e2 , . . . , α en ) =

AT S − IV IF W GAµ (e α1 , α e2 , . . . , α en ) =

n Y

(e αj )µj ,

j=1

Specifically, if µ = ( n1 , n1 , . . . , n1 ), then AT S − IV IF W GA operator degenerates interval-valued intuitionistic fuzzy arithmetic geometric average operator based on Archimedean t-norm (AT S −IV IF GA)

µ

1

AT S − IV IF GA(e α1 , α e2 , . . . , α en ) = (e α1 ⊗ α e2 ⊗ . . . ⊗ α en ) n . where Ω is the set of all interval-valued fuzzy intuitionistic sets, and µ = (µ1 , µ2 , . . . , µn )T is the weighted vector of α ej (j = 1, 2, . . . , n), µ is a fuzzy measure on X with µj ∈ [0, 1], µj = µ(A(j) ) − µ(A(j+1) ), and Pn µ = 1, A(j) = (j, . . . , n) with A(n+1) = ∅. j=1 j Theorem 5.1. Let α ei = h[µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2, . . . , n) be a collection of interval-valued intuitionistic fuzzy sets, T (x, y) Archimedean t-norm and S(x, y) its dual Archimedean t-conorm, N (x) = 1 − x. Then, the result aggregated by Definition 5.1 is still an intuitionistic fuzzy set, and (i)AT S − IV IF W Aµ (e αP e2 , . . . , α en ) = 1, α P P P h[h−1 ( nj=1 µj h(µL (αj ))), h−1 ( nj=1 µj h(µU (αj )))], [g −1 ( nj=1 µj g(ν L (αj ))), g −1 ( nj=1 µj g(ν U (αj )))]i. (ii)AT S − IV IF W GAµP (e α1 , α e2 , . . . , α en ) = P P P h[g −1 ( nj=1 µj g(µL (αj ))), g −1 ( nj=1 µj g(µU (αj )))], [h−1 ( nj=1 µj h(ν L (αj ))), h−1 ( nj=1 µj h(ν U (αj )))]i, where Pn µ = (µ1 , µ2 , . . . , µn ) is a fuzzy measure on X with µj ∈ [0, 1], µj = µ(A(j) ) − µ(A(j+1) ), and e1 6 α e2 6 · · · 6 j=1 µj = 1, the parentheses used for indices represent a permutation on X such that α α en , A(j) = (j, ..., n), A(n+1) = ∅. Theorem 5.1 can be proven by mathematical induction. The steps in the proof are as follows: 710

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San-Fu Wang: Interval-valued intuitionistic fuzzy Choquet integral operators based on...

Proof. We only prove that (i) holds. the proof of (ii) is similar. (1) When n = 1, obviously, it is right. (2) When n = 2, µ1 α e1 = h[h−1 (µ1 h(µL (α1 ))), h−1 (µ1 h(µU (α1 )))], [g −1 (µ1 g(ν L (α1 ))), g −1 (ν1 g(ν U (α1 )))]i. µ2 α e2 = h[h−1 (µ2 h(µL (α2 ))), h−1 (µ2 h(µU (α2 )))], [g −1 (µ2 g(ν L (α2 ))), g −1 (ν2 g(ν U (α2 )))]i. AT S − AT S − IV IF W Aµ (e α1 , α e2 ) = µ1 α e1 ⊕ µ2 α e2 = [h−1 (µ1 h(µL (α1 ))), h−1 (µ1 h(µU (α1 )))], [g −1 (µ1 g(ν L (α1 ))), g −1 (ν1 g(ν U (α1 )))]i ⊕h[h−1 (µ2 h(µL (α2 ))), h−1 (µ2 h(µU (α2 )))], [g −1 (µ2 g(ν L (α2 ))), g −1 (ν2 g(ν U (α2 )))]i = h[h−1 (h(h−1 (µ1 h(µL (α1 )))) + h(h−1 (µ2 h(µL (α2 ))))), h−1 (h(h−1 (µ1 h(µU (α1 )))) + h(h−1 (µ2 h(µU (α2 )))))], [g −1 (g(g −1 (µ1 g(ν L (α1 )))) + g(g −1 (µ2 g(ν L (α2 ))))), g −1 (g(g −1 (µ1 g(ν U (α1 )))) + g(g −1 (µ2 g(ν U (α2 )))))]i = P P P P h[h−1 ( 2j=1 µj h(µL (αj ))), h−1 ( 2j=1 µj h(µU (αj )))], [g −1 ( 2j=1 µj g(ν L (αj ))), g −1 ( 2j=1 µj g(ν U (αj )))]i. Therefore, when n = 2, the conclusion is right. (3) Suppose when n = k, the conclusion is right, i.e. AT S − IV IF W Aµ (e α1 , α e2 , . . . , α e )= Pk Pkk P P −1 L −1 h[h ( j=1 µj h(µ (αj ))), h ( j=1 µj h(µU (αj )))], [g −1 ( kj=1 µj g(ν L (αj ))), g −1 ( kj=1 µj g(ν U (αj )))]i. Then, when n = k + 1, AT S − IV IU LW Aµ (e α1 , α e2 , . . . , α e ,α e )= P P Pkk k+1 U Pk L −1 −1 h[h ( j=1 µj h(µ (αj ))), h ( j=1 µj h(µ (αj )))], [g −1 ( kj=1 µj g(ν L (αj ))), g −1 ( kj=1 µj g(ν U (αj )))]i ⊕h[h−1 (µk+1 h(µL (αk+1 ))), h−1 (µk+1 h(µU (αk+1 )))], [g −1 (µk+1 g(ν L (αk+1 ))), g −1 (νk+1 g(ν U (αk+1 )))]i = P h[h−1 (h(h−1 ( kj=1 µj h(µL (αj )))) + h(h−1 (µk+1 h(µL (αk+1 ))))), P h−1 (h(h−1 ( kj=1 µj h(µU (αj )))) + h(h−1 (µk+1 h(µU (αk+1 )))))], P [g −1 (g(g −1 ( kj=1 µj g(ν L (αj )))) + g(g −1 (µk+1 g(ν L (αk+1 ))))), P g −1 (g(g −1 ( kj=1 µj g(ν U (αj )))) + g(g −1 (µk+1 g(ν U (αk+1 )))))]i = Pk+1 Pk+1 Pk+1 P U L −1 U −1 L −1 h[h−1 ( k+1 j=1 µj h(µ (αj ))), h ( j=1 µj h(µ (αj )))], [g ( j=1 µj g(ν (αj ))), g ( j=1 µj g(ν (αj )))]i. So, when n = k + 1, the conclusion is right, too. According to steps (1), (2) and (3), we can conclude the conclusion is right for all n. 6. Conclusions The main technologies in multiple attribute decision making, whether the situation is certain or vague, are how to define and calculate aggregation operators proposed in the practice. In this study we only discussed and investigated the operational rules of interval-valued intuitionistic fuzzy sets based on Archimedean t-norm, and the aggregating of interval-valued intuitionistic fuzzy sets based on Archimed -ean t-norm. In order to do this we also obtained the representations and transformations of Archimedean t-norm and Archimedean t-conorm. Based on these operators proposed in this note, we could make multiple attribute group decision making problems easily. Limited to the length of this paper it can not be discussed. However, it will be our main work in the future.

References [1] L.A. Zadeh, Fuzzy ses, Information and Control 8 (1965) 338-356. [2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96. [3] K.T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989) 37-46. [4] K.T. Atanassov, G.Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 3 (1989) 343-349. [5] K.T. Atanassov, Operators over interval-valued intuitionistic fuzzy sets, Fuzzy Sets and System 64 (1994) 159-174. [6] G. Klir, B. Yuan, Fuzzy sets and fuzzy logic: theory and applications. NJ: Prentice Hall, Upper Saddle River, 1995. [7] H.T. Nguyen, E.A. Walker, A first course in fuzzy logic. Boca Raton, Florida: CRC Press, 1997. [8] G. Choquet, Theory of capacities, Annales de l’institut Fourier 5 (1953) 131-295. 711

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[9] E.P. Klement, R. Mesiar(Eds.), Logical, algebraic, analytic, and probabilistic aspects of triangular norms. New York: Elsevier, 2005. [10] G. Beliakov, H. Bustince, D.P. Goswami, U.K. Calvo, Aggregation Functions: A guide for Practitioners. Springer, Heidelberg Berlin, New York, 2007. [11] M.M. Xia, Z.S. Xu, B. Zhu, Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm, Knowledge-Based Systems 31 (2012) 78-88

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Approximate bi-homomorphisms and bi-derivations in intuitionistic fuzzy ternary normed algebras

Javad Shokri1 , 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 04763, Korea 3

Department of Mathematics, University of Seoul, Seoul 02504, Korea

Abstract. In this paper, we generalize the concept of homomorphisms and derivations in intuitionistic fuzzy normed algebras for 2-dimensional functional equations. Furthermore, we investigate the Hyers-Ulam stability bi-homomorphisms and bi-derivations in intuitionistic fuzzy ternary normed algebras concerning a 2-dimensional bi-additive functional equation.

1. Introduction and preliminaries We say a functional equation (ζ) is stable if any function g satisfying the equation (ζ) approximately is near to true solution of (ζ). Also, we say that a functional equation is superstable if every approximately solution is an exact solution of it. The stability problem of functional equations originated from a question of Ulam [37] in 1940, concerning the stability of group homomorphisms. We are looking for situations when the homomorphisms are stable, i.e., if a mapping is almost a homomorphism, then there exists a true homomorphism near it. The case of approximately additive mappings was solved by Hyers [11] under the assumption that G1 and G2 are Banach spaces. In 1978, a generalized version of the theorem of Hyers for approximately linear mappings was given by Rassias [28]. In 1991, Gajda [8] answered the question for the case p > 1, which was raised by Rassias. For more information on functional equations, see [18, 25, 26, 27, 32, 34, 35]. Fuzzy set theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. This new theory was introduced by Zadeh [38], in 1965 and since then a large number of research papers have appeared by using the concept of fuzzy set/numbers and fuzzification of many classical theories has also been made. It has also very useful application in various fields, e.g. population dynamics [5], chaos control [7], computer programming [9], nonlinear dynamical systems [10], fuzzy physics [12], fuzzy topology [31], fuzzy stability [13, 14, 15, 16, 24], nonlinear operators [20], statistical convergence [21, 23], etc. The concept of intuitionistic fuzzy normed spaces, initially has been introduced by Saadati and Park [29]. In [30], by modifying the separation condition and strengthening some conditions in the definition of Saadati and Park, Saadati et al. have obtained a modified case of intuitionistic fuzzy normed spaces. Many authors have considered the intuitionistic fuzzy normed linear spaces, and intuitionistic fuzzy 2-normed spaces(see [3, 4, 6, 19]). 0

2010 Mathematics Subject Classification: 39B52; 46S40; 26E50. Keywords: Hyers-Ulam stability; fuzzy ternary Banach space, intuitionistic fuzzy normed algebra; biadditive functional equation. ∗ Corresponding authors. 0 E-mail:1 [email protected]; 2 [email protected]; 3 [email protected] 0

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J. Shokri, C. Park, D. Shin

Let X be a real linear space. A function N : X × R → [0, 1] (the so-called fuzzy subset) is said to be a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N 1) N (x, c) = 0 for c 6 0; (N 2) x = 0 if and only if N (x, c) = 1 for all c > 0; t ) if c 6= 0; (N 3) N (cx, t) = N (x, |c| (N 4) N (x + y, s + t) > min{N (x, s), N (y, t)}; (N 5) N (x, .) is a non-decreasing function on R and limt→∞ N (x, t) = 1; (N 6) 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. The stability problem for a 2-dimensional bi-additive functional equation was proved by Bae and Park [1] for mappings f : X × X → Y , where X is a real normed space and Y is a Banach space. In this paper, we determine some stability results of bi-homomorphism and bi-derivation concerning the 2-dimensional bi-additive functional equation f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) − 2f (y, w)

(1.1)

in intuitionistic fuzzy ternary normed algebras. It has been discussed that f (x, y) = ax2 + by 2 is a solution of (1.1) (see [2]). We recall some notations and basic definitions used in this paper. We use the definition of intuitionistic fuzzy normed spaces given in [17, 22, 29] to investigate some stability results for the functional equation (1.1) in the intuitionistic fuzzy normed vector space setting. Definition 1.1. ([33]) A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-norm if it satisfies the following conditions: (a) is commutative and associative; (b) is continuous; (c) a ∗ 1 = a for all a ∈ [0, 1]; (d) a ∗ b 6 c ∗ d whenever a 6 c and b 6 d for all a, b, c, d ∈ [0, 1]. Definition 1.2. ([33]) A binary operation  : [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-conorm if it satisfies the following conditions: (a) is commutative and associative; (b) is continuous; (c) a  0 = a for all a ∈ [0, 1]; (d) a  b 6 c  d whenever a 6 c and b 6 d for all a, b, c, d ∈ [0, 1]. Using the continuous t-norm and t-conorm, Saadati and Park [29] have introduced the concept of intuitionistic fuzzy normed space. Definition 1.3. ([22, 29]) The five-tuple (X, µ, ν, ∗, ) is said to be an intuitionistic fuzzy normed space (for short, IFNS) 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) µ(x, t)+ν(x, t) 6 1, (ii) µ(x, t) > 0, (iii) µ(x, t) = 1 if and only if x = 0, (iv) µ(αx, t) = t ) for each α 6= 0, (v) µ(x, t) ∗ µ(y, s) 6 µ(x + y, t + s), (vi) µ(x, .) : (0, ∞) → [0, 1] is µ(x, |α| continuous, (vii) limt→∞ µ(x, t) = 1 and limt→0 µ(x, t) = 0, (viii) ν(x, t) < 1, (ix) ν(x, t) = 0 t if and only if x = 0, (x) ν(αx, t) = ν(x, |α| ) for each α 6= 0, (xi) ν(x, t)ν(y, s) > ν(x+y, t+s), (xii) ν(x, .) : (0, 1) → [0, 1] is continuous, (xiii) limt→∞ ν(x, t) = 0 and limt→0 ν(x, t) = 1.

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Approximate bi-homomorphisms and bi-derivations

Definition 1.4. Let (X, µ, ν, ∗, ) be an IFNS. A sequence {xn } is said to be intuitionistic fuzzy convergent to L ∈ X if limk→∞ µ(xk − L, t) = 1 and limk→∞ ν(xk − L, t) = 0 for all t > 0. In this case we write xk → L as k → ∞. A sequence {xn } is said to be intuitionistic fuzzy Cauchy sequence if limk→∞ µ(xk+p − xk , t) = 1 and limk→∞ ν(xk+p − xk , t) = 0 for all p ∈ N and all t > 0. Then IFNS (X, µ, ν, ∗, ) is said to be complete if every intuitionistic fuzzy Cauchy sequence in (X, µ, ν, ∗, ) is intuitionistic fuzzy convergent in (X, µ, ν, ∗, ) and (X, µ, ν, ∗, ) is also called an intuitionistic fuzzy Banach space. The concepts of convergent sequence and Cauchy sequence in an intuitionistic fuzzy normed space are studied in [29]. Definition 1.5. Let X be a ternary algebra with [·, ·, ·] and (X, µ, ν, ∗, ) be an IFNS. (1) The intuitionistic fuzzy normed space (X, µ, ν, ∗, ) is called an intuitionistic fuzzy ternary normed algebra if µ([x, y, z], stu) > µ(x, s) ∗ µ(y, t) ∗ µ(z, u) ν([x, y, z], stu) > ν(x, s) ∗ ν(y, t) ∗ ν(z, u) for all x, y, z ∈ X and s, t, u > 0. (2) A complete intuitionistic fuzzy ternary normed algebra is called an intuitionistic fuzzy ternary Banach algebra. Definition 1.6. Let X be a ternary normed (Banach) algebra and (Y, µ, ν) an intuitionistic fuzzy ternary Banach algebra. (1) A bi-additive mapping H : X × X → Y is called a ternary bi-homomorphism if H([x, y, z], [w, w, w]) = [H(x, w), H(y, w), H(z, w)], H([x, x, x], [y, z, w]) = [H(x, y), H(x, z), H(x, w)] for all x, y, z, w ∈ X. (2) A bi-additive mapping δ : X × X → X is called a ternary bi-derivation if δ([x, y, z], w) = [δ(x, w), y, z] + [x, δ(y, w), z] + [x, y, δ(z, w)], δ(x, [y, z, w]) = [δ(x, y), z, w] + [y, δ(x, z), w] + [y, z, δ(x, w)] for all x, y, z, w ∈ X. 2. Bi-homomorphisms in intuitionistic fuzzy ternary normed algebras We begin with a Hyers-Ulam type theorem in intuitionistic fuzzy ternary normed algebras to approximate bi-homomorphism associated to the functional equation (1.1). For notational convenience, given a function f : X × X → Y , we define the difference operator Dq f (x, y, z, w) = f (x + y, z − w) + f (x − y, z + w) − 2f (x, z) + 2f (y, w) Lemma 2.1. ([36, Theorem 3.1]) Let X be a linear space and let (Z, µ0 , ν 0 ) be an IFNS. Let ϕ : X 4 → Z be a mapping such that, for some 0 < α < 4.  0 µ (ϕ(2x, 2y, 2z, 2w), t) > µ0 (αϕ(x, y, z, w), t), (2.1) ν 0 (ϕ(2x, 2y, 2z, 2w), t) 6 ν 0 (αϕ(x, y, z, w), t), for all x, y, z, w ∈ X and all t > 0. Let (Y, µ, ν) be an intuitionistic fuzzy Banach space and let f : X × X → Y be a mapping satisfying f (0, 0) = 0 and  µ(Dq f (x, y, z, w), t) > µ0 (ϕ(x, y, z, w), t), (2.2) ν(Dq f (x, y, z, w), t) 6 ν 0 (ϕ(x, y, z, w), t)

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for all x, y, z, w ∈ X and all t > 0. Then there exists a unique bi-additive mapping H : X × X → Y satisfying (1.1) such that     µ H(x, y) − f (x, y), t            (4−α) (4−α) (4−α)   > ∗∞ µ0 ϕ(x, x, y, −y), 8 t ∗∞ µ0 ϕ(x, −x, y, y), 8 t ∗∞ µ0 ϕ(0, x, 0, y), 8 t ,     ν H(x, y) − f (x, y), t             6 ∞ ν 0 ϕ(x, x, y, −y), (4−α) t ∞ ν 0 ϕ(x, −x, y, y), (4−α) t ∞ ν 0 ϕ(0, x, 0, y), (4−α) t 8

8

8

(2.3) for all x, y, z, w ∈ X and all t > 0, where ∗∞ a := a ∗ a ∗ · · · and ∞ a := a  a  · · · for all a ∈ [0, 1]. Theorem 2.2. Let X be a ternary algebra and let (Z, µ0 , ν) be an IFNS. Let ϕ : X 4 → Z be a mapping satisfying (2.1). Let (Y, µ, ν) be an intuitionistic fuzzy ternary Banach algebra and let f : X × X → Y be a mapping satisfying f (0, 0) = 0, (2.2) and    µ(f ([x, y, z], [w, w, w]) − [f (x, y), f (y, w), f (z, w)], t)   +µ(f ([x, x, x], [y, z, w]) − [f (x, y), f (x, z), f (x, w)], t) > µ0 (ϕ(x, y, z, w), t), (2.4)  ν(f ([x, y, z], [w, w, w]) − [f (x, y), f (y, w), f (z, w)], t)    +ν(f ([x, x, x], [y, z, w]) − [f (x, y), f (x, z), f (x, w)], t) 6 ν 0 (ϕ(x, y, z, w), t) for all x, y, z, w ∈ X and all t > 0. Then there exists a unique bi-homomorphism H : X × X → Y satisfying (1.1) and (2.3). Proof. In Lemma 2.1, the mapping H : X ×X → Y was defined by H(x, y) = limn→∞ for all x, z ∈ X. From (2.4) and definition of H, it follows that

f (2n x,2n y) 4n

µ(H([x, y, z], [w, w, w]) − [H(x, y), H(y, w), H(z, w)], t) + µ(H([x, x, x], [y, z, w]) − [H(x, y), H(x, z), H(x, w)], t)  f ([2n x, 2n y, 2n z], [2n w, 2n w, 2n w]) h f (2n x, 2n w) f (2n y, 2n w) f (2n z, 2n w) i  =µ − , , ,t 64n 4n 4n 4n  f ([2n x, 2n x, 2n x], [2n x, 2n y, 2n z]) h f (2n x, 2n y) f (2n x, 2n z) f (2n x, 2n w) i  +µ − , , ,t 64n 4n 4n 4n 43n > µ0 (ϕ(2n x, 2n y, 2n z, 2n w), 43n t) > µ0 (ϕ(x, y, z, w), n t) → 1 α as n → ∞ for all x, y, z, w ∈ X and all t > 0, and similarly ν(H([x, y, z], [w, w, w]) − [H(x, y), H(y, w), H(z, w)], t) + ν(H([x, x, x], [y, z, w]) − [H(x, y), H(x, z), H(x, w)], t) 6 0 for all x, y, z, w ∈ X and all t > 0. So we conclude that H([x, y, z], [w, w, w]) = [H(x, w), H(y, w), H(z, w)], H([x, x, x], [y, z, w]) = [H(x, y), H(x, z), H(x, w)] for all x, y, z, w ∈ X.



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Approximate bi-homomorphisms and bi-derivations

Corollary 2.3. Let p be a nonnegative real number with p < 2, X be a ternary normed algebra with norm k.k, (Z, µ0 , ν 0 ) be an intuitionistic fuzzy ternary normed algebra, (Y, µ, ν) be a complete intuitionistic fuzzy ternary normed algebra, and let z0 ∈ Z. If f : X → Y is a mapping satisfying f (0, 0) = 0 and   µ(f ([x, y, z], [w, w, w]) − [f (x, y), f (y, w), f (z, w)], t)     +µ(f ([x, x, x], [y, z, w]) − [f (x, y), f (x, z), f (x, w)], t)     > µ0 ((kxkp + kykp + kzkp + kwkp )z0 , t) (2.5)  ν(f ([x, y, z], [w, w, w]) − [f (x, y), f (y, w), f (z, w)], t)      +µ(f ([x, x, x], [y, z, w]) − [f (x, y), f (x, z), f (x, w)], t)    6 ν 0 ((kxkp + kykp + kzkp + kwkp )z , t) 0 and

µ(Dq f (x, y), t) > µ0 ((kxkp + kykp + kzkp + kwkp )z0 , t) (2.6) ν(Dq f (x, y), t) 6 ν 0 ((kxkp + kykp + kzkp + kwkp )z0 , t) for all x, y, z, w ∈ X and t > 0, then there exists a unique bi-homomorphism H : X × X → Y such that      p p )t 0 (kxk + kyk)z , (4−2 )t  µ(H(x, y) − f (x, y), t) > ∗2 µ0 (kxk + kyk)z0 , (4−2 ∗ µ 0 16 8     p  ν(H(x, y) − f (x, y), t) 6 ∗2 ν 0 (kxk + kyk)z , (4−2 )t ∗ ν 0 (kxk + kyk)z , (4−2p )t 0 0 16 8 

for all x, y ∈ X and t > 0. Lemma 2.4. ([36, Theorem 3.3]) Let X be a linear space and let (Z, µ0 , ν 0 ) be an IFNS. Let ϕ : X × X × X × X → Z be a mapping such that, for some α > 4,       µ0 ϕ x2 , y2 , z2 , w2 , t > µ0 (ϕ(x, y, z, w), αt),     (2.7)  ν 0 ϕ x , y , z , w , t 6 ν 0 (ϕ(x, y, z, w), αt), 2 2 2 2 for all x, y, z, w ∈ X and all t > 0. Let (Y, µ, ν) be an intuitionistic fuzzy Banach space and let f : X × X → Y be a ϕ-approximately bi-additive mapping in the sense of (2.2) and (2.4) with f (0, 0) = 0. Then there exists a unique bi-additive mapping H : X × X → Y such that  (α − 4)  (α − 4)  ∞ 0  t ∗ µ ϕ(x, −x, y, y), t µ(H(x, y) − f (x, y), t) > ∗∞ µ0 ϕ(x, x, y, −y), 8 8  (α − 4)  ∗∞ µ0 ϕ(0, x, 0, y), t (2.8) 8 and  (α − 4)  ∞ 0  (α − 4)  µ(H(x, y) − f (x, y), t) 6 ∞ ν 0 ϕ(x, x, y, −y), t  ν ϕ(x, −x, y, y), t 8 8  (α − 4)  ∞ ν 0 ϕ(0, x, 0, y), t (2.9) 8 for all x, y ∈ X and all t > 0. Theorem 2.5. Let X be a ternary algebra and let (Z, µ0 , ν 0 ) be an IFNS. Let ϕ : X × X × X × X → Z be a mapping satisfying (2.7). Let (Y, µ, ν) be an intuitionistic fuzzy ternary Banach algebra and let f : X × X → Y be a ϕ-approximately bi-additive mapping in the sense of (2.2) and (2.4) with f (0, 0) = 0. Then there exists a unique bi-homomorphism H : X × X → Y satisfying (2.8) and (2.9).

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Proof. The proof is similar to the proof of Theorem 2.2.



Corollary 2.6. Let p be a nonnegative real number with p > 2, X be a ternary normed algebra with norm k.k, (Z, µ0 , ν 0 ) be an intuitionistic fuzzy ternary normed algebra, (Y, µ, ν) be a complete intuitionistic fuzzy ternary normed algebra, and let z0 ∈ Z. If f : X → Y is a mapping satisfying f (0, 0) = 0, (2.5) and (2.6). then there exists a unique bi-homomorphism H : X × X → Y such that      p −4)t p  µ(H(x, y) − f (x, y), t) > ∗2 µ0 (kxk + kyk)z0 , (2 16 ∗ µ0 (kxk + kyk)z0 , (2 −4)t 8     p −4)t p −4)t (2 (2  ν(H(x, y) − f (x, y), t) 6 ∗2 ν 0 (kxk + kyk)z , 0 (kxk + kyk)z , ∗ ν 0 0 16 8 for all x, y ∈ X and t > 0. 3. Bi-derivations on intuitionistic fuzzy ternary normed algebras In this section, we investigate generalized Hyers-Ulam stability of bi-derivations on intuitionistic fuzzy ternary normed algebrasfor the functional equation (1.1). Theorem 3.1. Let X be an intuitionistic fuzzy ternary Banach algebra and let (Z, µ0 , ν 0 ) be an IFNS. Let f : X × X → X be a mapping with f (0, 0) = 0 for which there exists a mapping ϕ : X × X × X × X → Z such that, for some 0 < α < 4 satisfying (2.1), (2.2) and   µ(f ([x, y, z], w) − [f (x, w), y, z] − [x, f (y, w), z] − [x, y, f (z, w)], t)     +µ(f (x, [y, z, w]) − [f (x, y), z, w] − [y, f (x, z), w] − [y, z, f (x, w)], t)     > µ0 (ϕ(x, y, z, w), t), (3.1)  ν(f ([x, y, z], w) − [f (x, w), y, z] − [x, f (y, w), z] − [x, y, f (z, w)], t)      +ν(f (x, [y, z, w]) − [f (x, y), z, w] − [y, f (x, z), w] − [y, z, f (x, w)], t)    6 ν 0 (ϕ(x, y, z, w), t) for all x, y, z, w ∈ X and all t > 0. Then there exists a unique bi-derivation δ : X × X → X satisfying (1.1) such that     µ δ(x, y) − f (x, y), t            (4−α) (4−α) (4−α)   > ∗∞ µ0 ϕ(x, x, y, −y), 8 t ∗∞ µ0 ϕ(x, −x, y, y), 8 t ∗∞ µ0 ϕ(0, x, 0, y), 8 t ,     ν δ(x, y) − f (x, y), t             6 ∞ ν 0 ϕ(x, x, y, −y), (4−α) t ∞ ν 0 ϕ(x, −x, y, y), (4−α) t ∞ ν 0 ϕ(0, x, 0, y), (4−α) t 8

8

8

(3.2) for all x, y, z, w ∈ X and all t > 0, where ∗∞ a := a ∗ a ∗ · · · and ∞ a := a  a  · · · for all a ∈ [0, 1]. Proof. By the same argument as in the proof of Theorem 2.2, there exists a unique bi-additive mapping δ : X × X → X satisfying (3.2). The mapping δ is given by 1 f (2n x, 2n y) n→∞ 4

δ(x, y) = lim

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Approximate bi-homomorphisms and bi-derivations

for all x, y ∈ X. It follows from (3.1) that µ(δ([x, y, z], w) − [δ(x, w), y, z] − [x, δ(y, w), z] − [x, y, δ(z, w)], t) + µ(δ(x, [y, z, w]) − [δ(x, y), z, w] − [y, δ(x, z), w] − [y, z, δ(x, w)], t)  1 h1 i = µ 3n f (23n [x, y, z], 23n w) − n f (2n x, 2n w), y, z 4 4 h 1 i h i  1 n n − x, n f (2 x, 2 w), z − x, y, n f (2n z, 2n w) , t 4 4  1 h1 i + µ 3n f (23n x, 23n [y, z, w]) − n f (2n x, 2n y), z, w 4 4 h 1 i h i  1 n n − y, n f (2 x, 2 z), w − y, z, n f (2n x, 2n w) , t 4 4  1 1 = µ 3n f ([2n x, 2n y, 2n z], 23n w) − 3n [f (2n x, 23n w), 2n y, 2n z] 4 4  1 n 1 − 3n [2 x, f (2n y, 23n w), 2n z] − 3n [2n x, 2n y, f (2n z, 23n w)], t 4 4  1 1 3n n n n + µ 3n f (2 x, [2 y, 2 z, 2 w]) − 3n [f (23n x, 2n y), 2n z, 2n w] 4 4  1 1 − 3n [2n y, f (23n x, 2n z), 2n w] − 3n [2n y, 2n z, f (23n x, 2n w)], t 4 4 6 µ0 (ϕ(2n x, 2n y, 2n z, 23n w), 43n t) + µ0 (ϕ(23n x, 2n y, 2n z, 2n w), 43n t))  43n t  6 2µ0 ϕ(x, y, z, w), 3n −→ 1 α as n → ∞ for all x, y, z, w ∈ A. Similarly, we obtain ν(δ([x, y, z], w) − [δ(x, w), y, z] − [x, δ(y, w), z] − [x, y, δ(z, w)], t) + ν(δ(x, [y, z, w]) − [δ(x, y), z, w] − [y, δ(x, z), w] − [y, z, δ(x, w)], t) = 0 for all x, y, z, w ∈ A. Thus δ([x, y, z], w) = [δ(x, w), y, z] + [x, δ(y, w), z] + [x, y, δ(z, w)], δ(x, [y, z, w]) = [δ(x, y), z, w] + [y, δ(x, z), w] + [y, z, δ(x, w)] for all x, y, z, w ∈ A. So we conclude that δ is a unique bi-derivation satisfying (3.2).



Corollary 3.2. Let p be a nonnegative real number with p < 2, (Z, µ0 , ν 0 ) be an intuitionistic fuzzy ternary normed algebra, (X, µ, ν) be a complete intuitionistic fuzzy ternary Banach algebra, and let z0 ∈ Z. If f : X → X is a mapping with f (0, 0) = 0 such that   µ(f ([x, y, z], w) − [f (x, w), y, z] − [x, f (y, w), z] − [x, y, f (z, w)], t)     +µ(f (x, [y, z, w]) − [f (x, y), z, w] − [y, f (x, z), w] − [y, z, f (x, w)], t)     > µ0 ((kxkp + kykp + kzkp + kwkp )z0 , t) (3.3)  ν(f ([x, y, z], w) − [f (x, w), y, z] − [x, f (y, w), z] − [x, y, f (z, w)], t)      +ν(f (x, [y, z, w]) − [f (x, y), z, w] − [y, f (x, z), w] − [y, z, f (x, w)], t)    > ν 0 ((kxkp + kykp + kzkp + kwkp )z0 , t)

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and

µ(Dq f (x, y), t) > µ0 ((kxkp + kykp + kzkp + kwkp )z0 , t) (3.4) ν(Dq f (x, y), t) 6 ν 0 ((kxkp + kykp + kzkp + kwkp )z0 , t) for all x, y, z, w ∈ X and t > 0, then there exists a unique bi-derivation δ : X × X → X such that      p )t p 0 (kxk + kyk)z , (4−2 )t  µ(δ(x, y) − f (x, y), t) > ∗2 µ0 (kxk + kyk)z0 , (4−2 ∗ µ 0 16 8     p  ν(δ(x, y) − f (x, y), t) 6 ∗2 ν 0 (kxk + kyk)z , (4−2 )t ∗ ν 0 (kxk + kyk)z , (4−2p )t 0 0 16 8 

for all x, y ∈ X and t > 0. Theorem 3.3. Let X be an intuitionistic fuzzy ternary Banach algebra and let (Z < µ0 , ν, ) be an IFNS. Let f : X × X → Y be a mapping with f (0, 0) = 0 for which there exists a mapping ϕ : X × X × X × X → Z satisfying (2.1), (2.7) and (3.1) for some α > 4. Then there exists a unique bi-derivation δ : X × X → X such that  (α − 4)  (α − 4)  ∞ 0  t ∗ µ ϕ(x, −x, y, y), t µ(δ(x, y) − f (x, y), t) > ∗∞ µ0 ϕ(x, x, y, −y), 8 8  (α − 4)  t ∗∞ µ ϕ(0, x, 0, y), 8 and  (α − 4)  ∞ 0  (α − 4)  µ(δ(x, y) − f (x, y), t) 6 ∞ ν 0 ϕ(x, x, y, −y), t  ν ϕ(x, −x, y, y), t 8 8  (α − 4)  ∞ ν 0 ϕ(0, x, 0, y), t 8 for all x, y ∈ X. Proof. The proof is similar to the proof of Theorems 2.5 and 3.1.



(Z, µ0 , ν 0 )

Corollary 3.4. Let p be a nonnegative real number with p > 2, be an intuitionistic fuzzy ternary normed algebra, (X, µ, ν) be a complete intuitionistic fuzzy ternary Banach algebra and let z0 ∈ Z. If f : X → Y is a mapping satisfying f (0, 0) = 0, (3.3) and (3.4), then there exists a unique bi-derivation δ : X × X → X such that      p −4)t p  µ(δ(x, y) − f (x, y), t) > ∗2 µ0 (kxk + kyk)z0 , (2 16 ∗ µ0 (kxk + kyk)z0 , (2 −4)t 8     p p  ν(δ(x, y) − f (x, y), t) 6 ∗2 ν 0 (kxk + kyk)z , (2 −4)t ∗ ν 0 (kxk + kyk)z , (2 −4)t 0 0 16 8 for all x, y ∈ X and t > 0. References [1] J. Bae and W. Park, A functional equation originating from quadratic forms, J. Math. Anal. Appl., 326 (2007), 1142-1148. [2] J. Bae and W. Park, Approximate bi-homomorphisms and bi-derivations in C ∗ -ternery algebras, Bull. Korean Math. Soc. 47 (2010), 195-209. [3] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst. 151 (2005), 513-547. [4] T. Bag and S.K. Samanta, Some fixed point theorems on fuzzy normed linear spaces, Inform. Sci. 177 (2007), 3271-3289. [5] L. C. Barros, R. C. Bassanezi and P. A. Tonelli, Fuzzy modelling in population dynamics, Ecol. Model, 128 (2000), 27-33.

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Approximate bi-homomorphisms and bi-derivations [6] J.X. Fang, On I-topology generated by fuzzy norm, Fuzzy Sets Syst. 157 (2006), 2739-2750. [7] A. L. Fradkol and R. J. Evans, Control of chaos: Methods and applications in engineering, Choas Solitons Fractals 29 (2005), 33-56. [8] Z. Gajda, On the stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431-434. [9] R. Giles, A computer program for fuzzy reasoning, Fuzzy Set Syst. 4 (1980), 221-234. [10] L. Hong and J. Q. Sun, Bifurcations of fuzzy nonlinear dynomical systems, Commun. Nonlinear Sci. Numer. Simul. 1 (2006), 1-12. [11] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. [12] J. Madore, Fuzzy physics, Ann. Phys. 219 (1992), 187-198. [13] D. Mihet, The fixed point method for fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 160 (2009), 1663-1667. [14] A.K. Mirmostafaee, A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces, Fuzzy Sets Syst. 160 (2009), 1653-1662. [15] M. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 159 (2008), 730-738. [16] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791-3798. [17] S.A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos Solitons Fractals 42 (2009), 2989-2996. [18] E. Movahednia, S. M. S. M. Mosadegh, C. Park, D. Shin, Stability of a lattice preserving functional equation on Riesz space: fixed point alternative, J. Comput. Anal. Appl. 21 (2016), 83–89. [19] M. Mursaleen and Q.M.D. Lohani, Intuitionistic fuzzy 2-normed space and some related concepts, Chaos Solitons Fractals 42 (2009), 224-234. [20] M. Mursaleen and S.A. Mohiuddine, Nonlinear operators between intuitionistic fuzzy normed spaces and Frechet derivative, Chaos Solitons Fractals 42 (2009), 1010-1015. [21] M. Mursaleen and S.A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals 41 (2009), 2414-2421. [22] M. Mursaleen and S.A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals 42 (2009), 2997-3005. [23] M. Mursaleen, S.A. Mohiuddine and O.H.H. Edely, On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comp. Math. Appl. 59 (2010), 603-611. [24] C. Park, Fuzzy stability of a functional equation associated with inner product spaces, Fuzzy Sets Syst. 160 (2009), 1632-1642. [25] C. Park, Additive ρ-functional inequalities, J. Nonlinear Sci. Appl. 7 (2014), 296–310. [26] C. Park, Stability of ternary quadratic derivation on ternary Banach algebras: revisited, J. Comput. Anal. Appl. 20 (2016), 21–23. [27] W. Park, J. Bae, Approximate quadratic forms on restricted domains, J. Comput. Anal. Appl. 20 (2016), 388–410. [28] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. [29] R. Saadati and J. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitons Fractals 27 (2006), 331-344. [30] R. Saadati, S. Sedghi and N. Shobe, Modified intuitionistic fuzzy metric spaces and some fixed point theorems, Chaos Solitons Fractals 38 (2008), 36-47. [31] R. Saadati, S. M. Vaezpour and Y. J. Cho, Quicksort algorithem: application of a fixed point theorem in intuitionistic fuzzy quasi-metric space at a domain of words, J. Comput. Appl. Math. 228 (2009), 219-225. [32] S. Schin, D. Ki, J. Chang, M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37-49. [33] B. Schweize and A. Sklar, Satistical metric spaces, Pacific J. Math. 10 (1960), 314-334.

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J. Shokri, C. Park, D. Shin [34] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964-973. [35] D. Shin, C. Park, 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. [36] J. Shokri, Approximate bi-additive mappings in intuitionistic fuzzy normed spaces, Thai J. Math. (in press). [37] S. M. Ulam, Problem in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964. [38] L. A. Zadeh, Fuzzy sets, Inform. Control. 8 (1965), 338-353.

722

Javad Shokri et al 713-722

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

ON NEW REFINEMENTS AND APPLICATIONS OF EFFICIENT QUADRATURE RULES USING N-TIMES DIFFERENTIABLE MAPPINGS 1;2 A.

QAYYUM, 3 M. SHOAIB, AND 1 I. FAYE

Abstract. In this paper, new e¢ cient quadrature rules are established using a newly developed special type of kernel for n-times di¤erentiable mappings, having …ve steps. Some previous inequalities are also recaptured as special cases of our main inequalities. At the end, e¢ ciency of the newly developed quadrature rules are discussed.

1. Introduction In 1938, Ostrowski [13] …rst announced his inequality for di¤erent di¤erentiable mappings, which is given below: Theorem 1. Let f : I R ! R be a di¤ erentiable mapping on I (I is the interior of I) and let a; b 2 I with a < b: If f 0 : (a; b) ! R is bounded on (a; b) i.e. kf 0 k1 = sup jf 00 (t)j < 1; then t2[a;b]

1

f (x)

b

a

Zb

f (t)dt

a

for all x 2 [a; b]. The constant a smaller one.

1 4

"

x 1 + 4 (b

# a+b 2 2 (b 2

a) kf 0 k1 ;

a)

(1.1)

is sharp in the sense that it can not be replaced by

In 1976, Milovanovic et. al [11], proved a generalization of Ostrowski’s inequality for n-time di¤erentiable mappings. Up till now, a large number of research papers and books have been written on inequalities and their applications (see for instance [2]-[5], [8] and [14]-[16]). In many practical problems, it is important to bound one quantity by another quantity. The classical inequalities like Ostrowski are very helpful for this purpose. Ostrowski type inequalities have immediate applications in numerical integration, optimization theory, statistics, and integral operator theory. We indicate another inequality called Grüss inequality [11] which is stated as the integral inequality that establishes a connection between the integral of the product of two functions and the product of the integrals, which is given below. Theorem 2. Let f; g : [a; b] ! R be integrable functions such that ' and g(x) ; for some constants '; ; ; and x 2 [a; b]. Then 1 b

a

Zb

f (x)g(x)dx

1 b

a

a

1 ( 4

Zb a

')(

f (x)dx:

1 b

a

Zb

g(x)dx

f (x)

(1.2)

a

):

2000 Mathematics Subject Classi…cation. Primary 05C38, 15A15; Secondary 05A15, 15A18. Key words and phrases. Ostrowski inequality, Grüss inequality, Quadrature formula,Numerical Integration, peano kerenel. 1

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1;2

2

A. QAYYUM , 3 M . SHOAIB, AND

1

I. FAYE

Dragomir et. al [4] combined Ostrowski and Grüss inequality to give a new inequality which they named Ostrowski-Grüss type inequality. Dragomir [3], Liu [6], Alomari [1] and Liu et. al [8] established some companions of ostrowski type integral inequalities. Recently, Liu [7] proved the following companions of ostrowski type inequalities for 3-step kernels. Theorem 3. Let f : [a; b] ! R be di¤ erentiable in (a; b) : If f 0 2 L1 [a; b] ; and ; we have f 0 (x) , for all x 2 [a; b], then for all x 2 a; a+b 2 f (x) + f (a + b 2

x)

Zb

1 b

a

f (t) dt

b

a 4

+ x

3a + b 4

(S

) (1.3)

a

and f (x) + f (a + b 2

x)

1 b

a

Zb

(1.4)

f (t) dt

a

b

a 4

+ x

3a + b 4

(

S) :

More recently, Qayyum et. al [9]-[10] proved companions of Ostrowski inequality for 5-step linear and quadratic kernels but in this paper, we establish our results for 5-step kernel for n-times di¤erentiable mappings. In this paper, new ontrowski inequalities are extended. Using these inequalities, some e¢ cient quadrature rules are established. Some previous inequalities are also recaptured as special cases of our main inequalities. At the end, e¢ ciency of the newly developed quadrature rules are discussed.

2. Derivation of Ostrowski inequalities using 5-step kernel We will start our work by introducing a new Peano kernel de…ned by P (x; :) : [a; b] ! R 8 1 n a) ; t 2 a; a+x ; > n! (t 2 > n > 3a+b a+x 1 > t ; t 2 ; x ; < n! 4 2 1 a+b n Pn (x; t) = (2.1) t ; t 2 (x; a + b x] ; n! 2 > n 1 a+3b a+2b x > > n! t ; t 2 a + b x; ; > 4 2 : 1 n a+2b x (t b) ; t 2 ; b ; n! 2 for all x 2 a; a+b : 2 The following lemma is the main tool to prove the main results.

Lemma 1. Let f : [a; b] ! R be an n-times di¤ erentiable function such that f (n 1) (x) for n 2 N is absolutely continuous on [a; b] then 1 b

a

Zb

Pn (x; t)f (n) (t)dt

a

=

n X1 k=0

n+k+1

( 1) (k + 1)!

"

1 2k+1

(2.2) (

(x

k+1

a)

724

x

a+b 2

k+1

)

f (k)

a+x 2

A. QAYYUM et al 723-739

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

ON NEW REFINEM ENTS AND APPLICATIONS OF EFFICIENT QUADRATURE RULES

+

(

x

k+1

+ ( 1)

n

( 1) + b a

Zb

(

a+b 2

x k+1

a+b 2

x

k+1

1 2

+

k+1

3a + b 4 ( x

)

f (k) (x) k+1

3a + b 4

x

a+b 2

k+1

k+1 k+1

(x

3

a)

)

f

)

(k)

f (k) (a + b a + 2b 2

x) #

x

f (t)dt;

a

for all x 2 a; a+b 2 . Proof. The proof of (2.2) is established using mathematical induction. Take n = 1; Zb L:H:S of (2.2) = P1 (x; t)f 0 (t)dt:

(2.3)

a

After integrating by parts, we get 1 b

a

Zb

P1 (x; t)f 0 (t)dt

(2.4)

a

1 = f 4

a+x 2

+ f (x) + f (a + b

a + 2b 2

x) + f

x

1 b

a

Zb

f (t)dt:

a

We have L:H:S =

Zb

P1 (x; t)f 0 (t)dt:

a

Equation (2.3), is identical to the R:H:S of (2.2). Assume that (2.2) is true for n. Zb

Pn+1 (x; t)f (n+1) (t)dt

a

n n+k+2 X ( 1) = (k + 1)! k=0 ( 3a + b + x 4 ( k+1

+ ( 1)

+

1 2

+ ( 1)

x

k+1

n+1

"

Zb

(

1 2k+1

(

(x

k+1

a+b 2

x

k+1

a+b 2

)

f (k) (x)

a)

a+b 2

x

x

a+b 2

k+1

x

k+1

3a + b 4

k+1

(x

k+1

a)

k+1

)

f

)

(k)

k+1

)

f (k)

f (k) (a + b a + 2b 2

a+x 2

x) x

#

f (t)dt;

a

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

1;2

4

A. QAYYUM , 3 M . SHOAIB, AND

1

I. FAYE

where 8 > > > > > > >
> > > > > > :

;

After integeration by parts, we get Zb

Pn+1 (x; t)f (n+1) (t)dt

a

" 1 1 = (x (n + 1)! 2n+1 1 2n+1 x n

+ ( 1)

1 6 4 n!

f (n) (a + b

n

(t

a) f

(n)

(t)dt +

a

a+b Z x

+

a+2b 2

a+b 2

t

2n+1

a+b 2

x

n+1

( 1)

1 2

x

x

n+1

a) n+1

f (n)

n+1

f (n) (x) + 3a + b 4

a + 2b 2

a

n+1

x

a + 2b 2

f (n)

x)

x

)#

n

3a + b 4

f (n) (t)dt

a+2b 2

x

Z

f (n) (t)dt +

3a + b 4

t

n

f (n) (t)dt

a+b x

n+1

a)

f (n)

a+x 2

a+x 2 1 2

3a + b 4

+ x n+1

+ ( 1)

n+1

x

x

a+b 2

n+1

a+b 2

n+1

f (n)

f (n) (x) n+1

f (n) (a + b a + 2b 2

x)

x

n+1

f (n) (a + b

n+1

(x

f (n) (a + b

7 n b) f (n) (t)dt5

(t

a+b 2

x

n+1

f (n)

2

t

n

3

" 1 1 = (x (n + 1)! 2n+1 1

a+b 2

a+x 2

x

Zb

Zx

f (n) (x)

n+1

x

x) +

a+x

Z2

x

a+b 2

n+1

3a + b 4 2

n+1

+ ( 1) x

n+1

3a + b 4

+ x

n+1

1 2

f (n) (x) +

x

a+x 2

a+x 2

f (n)

n+1

a+b 2 (

+

f (n)

n+1

a+b 2

x

n+1

a)

f

(n)

a + 2b 2

x) x

726

#

Zb

Pn (x; t)f (n) (t)dt

a

A. QAYYUM et al 723-739

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

ON NEW REFINEM ENTS AND APPLICATIONS OF EFFICIENT QUADRATURE RULES

5

"( ) n+1 1 a+b 1 (n) a + x n+1 = f (x a) x n+1 (n + 1)! 2 2 2 ( ) n+1 n+1 a+b 3a + b x + x f (n) (x) 4 2 ) ( n+1 n+1 3a + b a+b n+1 (n) x ( 1) f (a + b x) + x 2 4 ( ) # n+1 n+1 a+b 1 a + 2b x n+1 + x (x a) f (n) 2 2 2

+

n X1 k=0

+

(

n+k+2

( 1) (k + 1)! x

+

1 2

n+1

+ ( 1)

Zb

(

2k+1

a+b 2

x

k+1

1

(

a)

x

a+b 2

k+1

x

k+1

)

f (k) (x)

3a + b 4

k+1

a+b 2

x

x

a+b 2

k+1

(x

k+1

3a + b 4 (

k+1

+ ( 1)

"

(x

k+1

a)

k+1

)

f

)

(k)

k+1

)

a+x 2

f (k)

f (k) (a + b a + 2b 2

x) x

#

f (t)dt

a

n n+k+2 X ( 1) = (k + 1)! k=0 ( 3a + b + x 4 ( k+1

+ ( 1)

+

1 2

x

k+1

n+1

+ ( 1)

"

Zb

(

1 2k+1

(

(x

k+1

a+b 2

k+1

a+b 2

x

)

f (k) (x)

a)

a+b 2

x

x

a+b 2

k+1

x

k+1

3a + b 4

k+1

(x

k+1

a)

k+1

)

f

)

(k)

k+1

)

f (k)

f (k) (a + b a + 2b 2

a+x 2

x) x

#

f (t)dt:

a

This completes the proof of lemma 1. Now we will present our results by imposing three di¤erent conditions on f and f (n+1) .

(n)

3. Case A: When f (n) 2 L1 [a; b] Theorem 4. Let f : [a; b] ! R be an n-times di¤ erentiable function on (a; b), f (n 1) is absolutely continuous on [a; b] and f (n) (t) , 8 t 2 [a; b] ; then for

727

A. QAYYUM et al 723-739

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

1;2

6

A. QAYYUM , 3 M . SHOAIB, AND

all x 2 a; a+b ; we have 2 n X1 k=0

+

(

n+k+1

( 1) (k + 1)!

k+1

+ ( 1) +

1 2k+1

(

(x

x

a+b 2

k+1

k+1

a+b 2

)

k+1

)

a+x 2

f (k)

k+1

k+1

a)

)

f

)

f (k) (a + b a + 2b 2

(k)

x) x

b n Z ( 1) f (n 1) (b) f (n 1) (a) 1 + f (t)dt 2 b a (n + 1)! (b a) a " 1 3a + b n+1 n+1 n 1 ( 1) (x a) + (1 + ( 1) ) x n+1 2 4 # ! n+1 n+1 ( 1) a+b 1 n+1 + + + ( 1) 1 x 2n+1 2n+1 2

(x) (b

a) (S

(3.1)

f (k) (x)

k+1

(x

I. FAYE

a+b 2

3a + b 4

x

a+b 2

x

k+1

a)

x

x

k+1

(

k+1

3a + b 4 (

x

1 2

"

1

#

n+1

)

and n X1 k=0

+

(

n+k+1

( 1) (k + 1)! x

+

1 2

1 2k+1

k+1

(x

a+b 2 x

k+1

x

a+b 2

k+1

a)

x

x

(

(

k+1

3a + b 4 (

k+1

+ ( 1)

"

k+1

a+b 2

)

k+1

(x

k+1

a)

)

a+x 2

f (k)

k+1

)

f

)

f (k) (a + b a + 2b 2

(k)

x) x

b n Z ( 1) f (n 1) (b) f (n 1) (a) 1 + f (t)dt 2 b a (n + 1)! (b a) a " 1 3a + b n+1 n+1 n 1 ( 1) (x a) + (1 + ( 1) ) x n+1 2 4 # ! n+1 n+1 1 ( 1) a+b n+1 + + + ( 1) 1 x 2n+1 2n+1 2

(x) (b

a) (

(3.2)

f (k) (x)

3a + b 4

x

k+1

a+b 2

#

n+1

S) ;

where S=

f

(n 1)

(b) b

728

f a

(n 1)

(a)

;

(3.3)

A. QAYYUM et al 723-739

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

ON NEW REFINEM ENTS AND APPLICATIONS OF EFFICIENT QUADRATURE RULES

(x) = max 1 n!

(

1 n!

a+b 2

x

x

n

a

1 (x) ; b a n!

2 n

1 (x) ; b a 4n!

(x) ; b a ) (x) (x) ; b a b a

2

a+b 2

x

2

3a + b 4

x

7

and ( " 1 3a + b 1 n n+1 (1 + ( 1) ) (x a) + x (x) = (n + 1)! 2n+1 4 ! # n+1 n+1 1 ( 1) a+b n+1 + + + ( 1) 1 x : 2n+1 2n+1 2

n+1

)

Proof. Let 1 b

a

Zb

Pn (x; t)dt

(3.4)

a

" ( 1 1 n (1 + ( 1) ) (x = n+1 (b a) (n + 1)! 2 ! n+1 1 ( 1) n+1 + + + ( 1) 1 x 2n+1 2n+1

n+1

a)

+ x n+1

a+b 2

#

n+1

3a + b 4

)

:

Using (3.4), we get 1 b

a

Zb

Pn (x; t)f

(n)

a

=

n X1 k=0

+

(

n+k+1

( 1) (k + 1)!

x

+ ( 1) +

1 2

k+1

1 2k+1

a+b 2 x

2

(b

a)

(

Zb a

a+b 2

k+1

x

x

a+b 2

)

f (k) (x)

3a + b 4

k+1

(x

Zb

f (n) (t)dt

a

a)

k+1

a+b 2

Pn (x; t)dt

k+1

(x

x

x

(

(t)dt

k+1

3a + b 4 (

k+1

"

1

k+1

a)

k+1

)

f

)

(k)

k+1

)

f (k)

f (k) (a + b a + 2b 2

b n Z ( 1) f (n 1) (b) f (n 1) (a) 1 + f (t)dt 2 b a (n + 1)! (b a) a " 1 3a + b n n+1 n (1 + ( 1) ) (x a) + (1 + ( 1) ) x 2n+1 4 ! # n+1 n+1 1 ( 1) a+b n+1 + + + ( 1) 1 x : 2n+1 2n+1 2

729

(3.5) a+x 2

x) x

#

n+1

A. QAYYUM et al 723-739

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

1;2

8

A. QAYYUM , 3 M . SHOAIB, AND

1

I. FAYE

Denote the L.H.S of (3.5) by Rn (x) : If C 2 R is an arbitrary constant, then we have 2 3 Zb Zb 1 1 Rn (x) = f (n) (t) C 4Pn (x; t) P (n) (x; s)ds5 dt: (3.6) b a b a a

a

Furthermore, we have 1

jRn (x)j

b

max Pn (x; t) a t2[a;b]

1 b

a

Zb

P

(n)

Zb

(x; s)ds

f (n) (t)

C dt: (3.7)

a

a

Now Pn (x; t)

= max where

8 < :

1 b

a

Zb

P (n) (x; s)ds

(3.8)

a

1 n! 1 n!

x

x a n 2 a+b n 2

(x) b a (x) b a

;

1 n!

;

1 4n!

3a+b 2 4 a+b 2 2

x x

(x) b a (x) b a

; ;

(x) b a

9 = ;

" ( 1 3a + b 1 n n+1 (x) = (1 + ( 1) ) (x a) + x (n + 1)! 2n+1 4 ! # n+1 n+1 1 ( 1) a+b n+1 + + + ( 1) 1 x : 2n+1 2n+1 2

= (x) ;

n+1

)

We also have Zb

f (n) (t)

dt = (S

Zb

f (n) (t)

dt = (

) (b

a) ;

(3.9)

S) (b

a) :

(3.10)

a

a

Using (3.4) to (3.10) and using C = (3.2).

and C =

in (3.7), we can obtain (3.1) and

Remark 1. If we substitute n = 2 in (3.1) and (3.2), we get Qayyum et. al result proved in [9]: Corollary 1. Substitution of x = a in (3.1) and (3.2) gives ( ! ) n k X1 ( 1)n+k+1 (b a)k+1 1 ( 1) k (k) (k) ( 1) f (a) + 1 + k+1 + k+1 f (b) (k + 1)! 2k+1 2 4 k=0

(3.11)

n

+

( 1) b a

Zb

(b

n 1

f (t)dt

a) f (n 2n+1 (n + 1)!

a) (S

)

1)

(b)

f (n

1)

n

(a) (1 + ( 1) )

a

(a) (b

730

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

ON NEW REFINEM ENTS AND APPLICATIONS OF EFFICIENT QUADRATURE RULES

and n X1 k=0

n+k+1

(

k

( 1) f

(k)

(a) +

1+

k

1 2k+1

( 1) + k+1 4

!

f

(k)

)

(b)

(3.12) Zb

n

+

k+1

( 1) (b a) (k + 1)! 2k+1

9

( 1) b a

n 1

(b

a) f (n 2n+1 (n + 1)!

f (t)dt

1)

(b)

f (n

1)

n

(a) (1 + ( 1) )

a

(a) (b

a) (

S) : a+b 2

Corollary 2. Substitution of x = n X1 k=0

n+k+1

k+1

( 1) (b a) (k + 1)! 4k+1 k

+ ( 1) f

3a + b 4

f (k)

Zb

n

a + 3b 4

(k)

in (3.1) and (3.2) gives

( 1) + b a

k

f (k)

a+b 2

(3.13)

k

f (k)

a+b 2

(3.14)

+ 1 + ( 1)

f (t)dt

a

n 1

f (n

1)

f (n

(b)

a+b 2

(b

1)

2 (b a) n (1 + ( 1) ) 4n+1 (n + 1)!

(a)

a) (S

)

and n X1 ( 1)n+k+1 (b a)k+1 (k + 1)! 4k+1

3a + b 4

f (k)

k=0

k

+ ( 1) f

Zb

n

a + 3b 4

(k)

( 1) + b a

+ 1 + ( 1)

f (t)dt

a

n 1

f (n

1)

f (n

(b)

a+b 2

(b

1)

2 (b a) n (1 + ( 1) ) 4n+1 (n + 1)!

(a)

a) (

S) :

in (3.1) and (3.2) gives Corollary 3. Substitution of x = 3a+b 4 2 k n 1 + ( 1) X1 ( 1)n+k+1 (b a)k+1 7a + b k 4 + ( 1) f (k) f (k) (k + 1)! 4k+1 2k+1 8 k=0

+f

a + 3b 4

(k)

+

1 2k+1

k

1 + ( 1)

f

a + 7b 8

(k)

n

( 1) + b a

Zb

3a + b 4 (3.15)

f (t)dt

a

n 1

f (n

1)

(b)

3a + b 4

f (n (b

1)

(a)

a) (S

1 (b a) (n + 1)! 4n+1

n

n

1 + ( 1) +

1 ( 1) + n 2 2n

)

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1

I. FAYE

and n X1 k=0

n+k+1

3a + b 4

k

+ ( 1) f (k)

+

k+1

( 1) (b a) (k + 1)! 4k+1

1 2k+1 f (n

2k+1

(b)

f (n

(b

a)

1)

(a)

2

3a + b 4

(b

n o k 1 + ( 1) f (k)

n

a + 7b 8

( 1) + b a

(3.16)

Zb

f (t)dt

a

n

n+1

1 (b a) (n + 1)! 4n+1

a) (

7a + b 8

a + 3b 4

+ f (k)

n o k 1 + ( 1) f (k) 1)

1

n

1 + ( 1) +

1 ( 1) + n 2 2n

S) :

4. Case B: When f (n+1) 2 L2 [a; b] Theorem 5. Let f : [a; b] ! R be an n-times di¤ erentiable function on (a; b), f (n+1) 2 L2 [a; b] ; then for all x 2 a; a+b ; we have 2 " ( ) n k+1 X1 ( 1)n+k+1 f (k) a+x a+b k+1 2 (x a) x (4.1) (k + 1)! 2k+1 2 k=0 ( ) k+1 k+1 3a + b a+b + x x f (k) (x) 4 2 ( ) k+1 k+1 a+b 3a + b k+1 + ( 1) x x f (k) (a + b x) 2 4 ( ) # k+1 k+1 a + 2b x 1 a+b k+1 (k) x + (x a) f 2 2 2 n

( 1) + b a "

Zb

f (n

f (t)dt

a

n

(1 + ( 1) )

(

1 2n+1

(x

1)

(b)

f (n

(b

2

1)

(a)

a) n+1

a)

+ x

1 (n + 1)! 3a + b 4

n+1

)

! # n+1 n+1 ( 1) a+b n+1 + + + ( 1) 1 x 2n+1 2n+1 2 " ( 2n+1 2n+1 b a (n+1) 1 (x a) 3a + b f + 2 x 2 22n 4 2 (n!) (2n + 1) ) 2n+1 1 a+b +2 x 22n 2 ( ! n+1 n+1 1 (x a) 3a + b n (1 + ( 1) ) + x (b a) (n + 1)! 2n+1 4 ) 3 21 n+1 2 n 1 ( 1) a+b n 5 : + ( 1) 1 x 2n+1 2n+1 2 1

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ON NEW REFINEM ENTS AND APPLICATIONS OF EFFICIENT QUADRATURE RULES 11

Proof. Substitute C = f Inequality, then we get jRn (x)j

; in Rn (x) given in (3.5) and use the Cauchy (4.2)

Zb

1 b

a+b 2

(n)

a

f (n) (t)

a+b 2

f (n)

1

P (n) (x; t)

b

a

a

b

a

P (n) (x; s)ds dt

a

2 b Z 4

1

Zb

f (n) (t)

2

a+b 2

f (n)

dt5

a

2 0 Zb 6 @ (n) P (x; t) 4

1 b

a

a

3 21 3 21

12

Zb

7 P (n) (x; s)dsA dt5 :

a

Use the Diaz-Metcalf inequality [12] or [17], to get Zb

f (n) (t)

2

a+b 2

f (n)

2

(b

dt

a)

f (n+1)

2

2 2

:

a

Therefore, using the above relations, we obtain (4.1). Corollary 4. Substitution of x = a in (4.1) gives ( ! ) n k X1 ( 1)n+k+1 (b a)k+1 1 ( 1) k (k) ( 1) f (a) + 1 + k+1 + k+1 f (k) (b) (k + 1)! 2k+1 2 4 k=0

n

( 1) + b a

Zb

n 1

(b

a) f (n 2n+1 (n + 1)!

f (t)dt

1)

(b)

f (n

1)

n

(a) (1 + ( 1) )

a

b

a

f

(n+1) 2

"

2

22n (n!) (2n + 1)

k=0

n+k+1

k+1

( 1) (b a) (k + 1)! 4k+1 k

+ 1 + ( 1) n

( 1) + b a

Zb

f (k)

a

1 b

f

f (k)

a+b 2

# 21

:

in (4.1) gives 3a + b 4 k

+ ( 1) f (k)

(4.3) a + 3b 4 n 1

f (n

f (t)dt

a

b

a+b 2

2n+1

(1 + ( 1) ) (b a) 22n+2 (n + 1)!

a)

Corollary 5. Substitution of x = n X1

n 2

2n+1

(b

(n+1)

1 a (n + 1)!

2

(

"

1)

(b)

1

f (n

1)

4

2

(n!) (2n + 1) 42n+1

2 (b

n+1

a) 4n+1

(a)

2 (b a) n (1 + ( 1) ) 4n+1 (n + 1)!

(b

a)

2n+1

)2 3 21 n (1 + ( 1) ) 5 :

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Corollary 6. Substitution of x = n X1 k=0

n+k+1

( 1) (b a) (k + 1)! 4k+1 k

+ ( 1) f (k)

+

k+1

1

3a + b 4 k

2k+1

1 + ( 1)

f

3a+b 4

7a + b 8

(4.4)

a + 3b 4

Zb

n

a + 7b 8

(k)

I. FAYE

in (4.1) gives

o 2n k 1 + ( 1) 4 f (k) 2k+1

+ f (k)

1

( 1) + b a

f (n

f (t)dt

1)

(b)

f (n

1)

(a)

a

n

n 1

1 (b a) (n + 1)! 4n+1 b

a

1 b

f

(n+1) 2

n

1 + ( 1) + "

1 2

1 ( 1) + n 2 2n

(b

(n!) (2n + 1)

2n+1

a)

4 22n+1

42n+1

n+1

1 (b a) a (n + 1)! 4n+1

1

n

(1 + ( 1) ) 2 +

5. Case C: When f

2n+1

(n)

2 2

# 21

:

2 L2 [a; b] :

Theorem 6. Let f : [a; b] ! R be an n-times di¤ erentiable function on (a; b), with f (n) 2 L2 [a; b]. Then, we have " ( ) n k+1 X1 ( 1)n+k+1 a+x 1 a+b k+1 (x a) x f (k) (5.1) (k + 1)! 2k+1 2 2 k=0 ( ) k+1 k+1 3a + b a+b + x x f (k) (x) 4 2 ) ( k+1 k+1 a+b 3a + b k+1 x x f (k) (a + b x) + ( 1) 2 4 ( ) # k+1 k+1 1 a+b a + 2b x k+1 (k) + x (x a) f 2 2 2 b n Z f (n 1) (b) f (n 1) (a) 1 ( 1) + f (t)dt 2 b a (n + 1)! (b a) a " n+1 1 3a + b n+1 n+1 n+1 1 ( 1) (x a) + 1 ( 1) x 2n+1 4 # ! n+1 n+1 1 ( 1) a+b n+1 + + + ( 1) 1 x 2n+1 2n+1 2 q " ( 2n+1 2n+1 f (n) 1 (x a) 3a + b + 2 x 2 b a 22n 4 (n!) (2n + 1) ) 2n+1 1 a+b + 2 x 22n 2

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ON NEW REFINEM ENTS AND APPLICATIONS OF EFFICIENT QUADRATURE RULES 13

1 b

n+1

(x

a)

n

2n+1

( 1) 2n+1

n

( 1)

1

n+1

a+b 2

x

for all x 2 a; a+b 2 , where

n+1

3a + b 4

n

(1 + ( 1) ) + (1 + ( 1) ) x

n

1 2n+1

+

(

1 a (n + 1)!

)2 3 21 5 ;

f (n)

(5.2)

= f (n)

f (n

2

1)

2

(b) f (n b a

1)

(a)

2

2

= f (n)

k 2 (b

2

a) ;

where S is as de…ned in Theorem 4. Proof. Let Rn (x) is de…ned as in (3.5). If we choose C =

1 b a

and use the Cauchy inequality and (3.5), then we get jRn (x)j 1 b

a

Zb

f

(n)

1

(t)

b

a

a

1 b

a

Zb

1 b

a

a

=

q

a

f b

(n)

a

1 +2 22n 1 b

"

a

Zb

P (n) (x; s)ds dt

a

7 f (n) (s)dsA dt5 3 12

12

1 b

a

1

0 @

Zb a

12 3 12 7 P (n) (x; t)dtA 5

(

1 (x 2 (n!) (2n + 1) 22n ) 2n+1 a+b x 2

1 a (n + 1)!

b

3 12

12

a

q

1

f (n) (s)ds in (3.6)

a

7 P (n) (x; s)dsA dt5

2 Zb 6 2 4 (Pn (x; t))

f (n) b

a

Zb a

a

Zb a

1 b

(s)ds Pn (x; t)

a

2 0 Zb 6 @ (n) f (t) 4

2 0 Zb 6 @ Pn (x; t) 4

f

(n)

Rb

2n+1

a)

1 n (1 + ( 1) ) (x 2n+1

3a + b + (1 + ( 1) ) x 4 3 ) 12 n+1 2 a+b 5 : x 2 n

n+1

+

+2 x

3a + b 4

2n+1

n+1

a) 1

2n+1

n+1

( 1) + 2n+1

n+1

+ ( 1)

!

1

Hence theorem is completed.

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A. QAYYUM , 3 M . SHOAIB, AND

1

I. FAYE

Corollary 7. Substitution of x = a in (5.1) gives n X1 k=0

n+k+1

(

k

( 1) f

(k)

(a) +

1+

k

( 1) + k+1 4

1 2k+1

!

f

(k)

)

(b)

(5.3) n

+

k+1

( 1) (b a) (k + 1)! 2k+1

( 1) b a

Zb

n 1

(b

f (t)dt

2n+1

a) f (n (n + 1)!

1)

f (n

(b)

1)

n

(a) (1 + ( 1) )

a

q

"

(n)

f

2n+1

(b

1

a)

2 (n!) (2n + 1) 43n+1 )2 3 12 ( n+1 1 (b a) 1 n (1 + ( 1) ) 5 : b a (n + 1)! 2n+1

b

a

a+b 2

Corollary 8. Substitution of x = n X1 k=0

k+1

n+k+1

(b a) ( 1) (k + 1)! 4k+1 k

+ ( 1) f

3a + b 4

f (k) n

a + 3b 4

(k)

in (5.1) gives

( 1) + b a

Zb

k

+ 1 + ( 1)

f (k)

a+b 2

(5.4)

f (t)dt

a

n 1

f (n

q

f b

1)

f (n

(b)

(n)

a

"

1)

2 (b a) n (1 + ( 1) ) 4n+1 (n + 1)!

(a)

2n+1

(b

a)

n X1 k=0

3a+b 4

n+k+1

k+1

( 1) (b a) (k + 1)! 4k+1 3a + b 4

k

+ ( 1) f (k)

+

1+

2

22n (n!) (2n + 1)

Corollary 9. Substitution of x =

1 2k+1

k

1 + ( 1)

1

f

n 2

a) (1 + ( 1) ) 2n+1 4 (n + 1)!

22n+1

# 12

:

in (5.1) gives 2 4

+ f (k) (k)

2n+1

(b

k

1 + ( 1) 2k+1

7a + b 8

f (k)

(5.5)

a + 3b 4

a + 7b 8

n

( 1) + b a

Zb

f (t)dt

a

n 1

f (n

1)

(b)

f (n

1)

1 (b a) (n + 1)! 4n+1

(a) :

n

n

1 + ( 1) +

1 ( 1) + n 2 2n

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ON NEW REFINEM ENTS AND APPLICATIONS OF EFFICIENT QUADRATURE RULES 15

q

"

(n)

f

a

2 (b

2n+1

1 +1 2 42n+1 22n (n!) (2n + 1) )2 3 12 ( n+1 1 1 1 (b a) n 5 : (1 + ( 1) ) 1 + n b a (n + 1)! 4n+1 2 b

1

a)

Remark 2. By choosing n = 1 in case A, B and C, we get all results obtained in [10]. Remark 3. By choosing n = 2 in case A, B and C, we get all results obtained in [9]. 6. Derivation of Numerical Quadrature Rules We propose some new quadrature rules involving higher order derivatives of the function f . In fact, the following new quadrature rules can be obtained while investigating error bounds using theorem 5.

Qn;1 (f ) :=

Zb

f (t)dt

a

k+2 h i a) k f (k) (a) + ( 1) f (k) (b) k+1 2 (k + 1)! k=0 h i (b a)n n (1 + ( 1) ) ; + f (n 1) (b) f (n 1) (a) n+1 2 (n + 1)! n X1

(b

Qn;2 (f ) :=

Zb

f (t)dt

a

n X1

k+2

Qn;3 (f ) :=

a)

k

( 1) 3a + b f (k) 4k+1 (k + 1)! 4 k=0 n o a+b a + 3b k k + 1 + ( 1) f (k) + ( 1) f (k) 2 4 n 2 (b a) n (( 1) + 1) ; + f (n 1) (b) f (n 1) (a) 4n+1 (n + 1)! Zb

(b

f (t)dt

a

n X1 k=0

k

k+2

( 1) (b a) (k + 1)! 4k+1 k

+ ( 1) f (k) h + f (n

1)

(b)

3a + b 4 f (n

1)

1 k 1 + ( 1) 2k+1

f (k)

7a + b 8

a + 3b 4 n (b a) n (( 1) + 1) 4n+1 (n + 1)!

+ f (k)

a + 7b 8

+ f (k)

i (a)

1 +1 : 2n

Performance of the e¢ cient quadrature rules

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1;2

16

Method 1.

R1

A. QAYYUM , 3 M . SHOAIB, AND

n : Qn;1 (f )

f1 (x)dx

n:

Qn;2 (f )

1

I. FAYE

n:

Qn;3 (f )

2: 2.83333

2: 2.83333

2: 2.83333

0

0

0

f2 (x)dx

6: 0.30117

4: 0.301172

4: 0.30117

Error:

1.1381 10

f3 (x)dx

6: 0.909328

Exact Value 2.83333

0

2.

R1

Error:

0

3.

R1

6

0

4.

R1

Error: f4 (x)dx

6

2.33999 10

5: 0.793022

0

5.

R1

Error: f5 (x)dx

11: 1.46266

0

6.

6

8.63182 10

6

Error:

5.8789 10

R1

f6 (x)dx

11: 1.31384

R1

f7 (x)dx

0

7.

Error

6: 1.34146

0

8.

R1

Error:

6

7.37624 10

1.4808 10

f8 (x)dx

6

3.08726 10

4: 0.909324 6

7.13925 10

4: 0.793031 2.9641 10

7

4: 0.909327

4: 0.793031

9: 0.62977

1.31383 6

1.73918 10 4: 1.34147

7

5.42574 10

6

5.20247 10

4: 1.34137

7

1.46265

6: 1.31383 6

5: 0.629762

0.793031

7

1.33626 10

6: 1.31383

0.909331

6

3.21638 10

6: 1.46266 6

2.13363 10

6

1.38925 10

7: 1.46265 2.29707 10

0.301169

1.34147 7

2.44601 10

4: 0.629774

0.629769

0

Error:

1.18074 10

6

6.3567 10

6

5.647 10

6

Table: f1 (x) = x2 + x + 2; x

f3 (x) = e sin x , 2

f5 (x) = ex , f7 (x) = x + cos x,

f2 (x) = x sin x;

(6.1)

2

f4 (x) = x + sin x; f6 (x) = ex cos (ex 2

2x) ;

f8 (x) = log x + 2 sin log x2 + 2

:

From the above table, we observe that all three quadrature rules show exact value of the integral of f1 for n = 2: For any polynomial of degree k; n = k + 1 will give exact value of the integral f1 . Acceptable error estimates can be obtained for smaller values of n to save computational time. The integral of f5 , Qn;3 (f ) report an error of the order of 10 6 for n = 6 while the other two quadrature rules give a similar error for n = 7 and n = 11. Similarly for all other functions Qn;3 (f ) report errors of the order of 10 6 or 10 7 for relatively smaller values of n as compared to the other two quadrature rules. Speci…cally, Qn;3 (f ) give an excellent estimate for the integrals of f5 and f8 at n = 6 and n = 4 respectively. In general Qn;3 (f ) gave better results as compared to the rest of the quadrature rules for much smaller values of n. Therefore we can conclude that overall Qn;3 (f ) is computationally more e¢ cient both in terms of error approximation, simplicity, and time. As a rough estimate we integrated log x2 + 2 sin log x2 + 2 using the built in algorithms of Mathematica 10.0 which took 26.30 seconds to give its approximate answer. To obtain similar approximation for the integral of f8 ; Qn;3 (f ) took less than a second.

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ON NEW REFINEM ENTS AND APPLICATIONS OF EFFICIENT QUADRATURE RULES 17

Based on this analysis, we can conjecture that Qn;3 (f ) is the most e¢ cient quadrature rule, while Qn;2 (f ) comes second in terms of performance. It should be noted that if desired the value of n can be adjusted to improve the error bounds or decrease computational time. References [1] M.W. Alomari, A companion of ostrowski’s inequality for mappings whose …rst derivatives are bounded and applications in numerical integration, Kragujevac Journal of Mathematics. (2012); 36: 77 - 82. [2] N. S. Barnett, S. S. Dragomir and I. Gomma, A companion for the Ostrowski and the generalized trapezoid inequalities, Journal of Mathematical and Computer Modelling, (2009); 50: 179-187. [3] S. S. Dragomir, Some companions of Ostrowski’s inequality for absolutely continuous functions and applications, Bulletin of the Korean Mathematical Society. (2005); 40(2): 213-230. [4] S. S. Dragomir and S. Wang, An inequality of Ostrowski-Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Computers and Mathematics with Applications, (1997); 33(11): 15-20. [5] A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, Journal of Approximation Theory, (2002); 115( 2): 260-288. [6] Z. Liu, Some companions of an Ostrowski type inequality and applications, Journal of Inequalities in Pure and Applied.Mathematics, (2009); 10-12. [7] W. Liu, New Bounds for the Companion of Ostrowski’s Inequality and Applications, Filomat, (2014); 28: 167-178. [8] W. Liu, Y. Zhu and J. Park, Some companions of perturbed Ostrowski-type inequalities based on the quadratic kernel function with three sections and applications, Journal of Inequalities and Applications, (2013); 226. [9] A. Qayyum, M. Shoaib and I. Faye, Companion of Ostrowski-type inequality based on 5-step quadratic kernel and applications, Journal of Nonlinear Sciences and Applications, 9 (2016); 537-552. [10] A. Qayyum, M. Shoaib and I. Faye, A companion of Ostrowski Type Integral Inequality using a 5-step kernel With Some Applications, (Accepted), Filomat, (2016). [11] D. S. Mitrinvi´c, J. E. Pecari´c and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, (1993). [12] D. S. Mitrinovi´c, J. E. Pecari´c and A. M. Fink, Inequalities involving functions and their integrals and derivatives, Mathematics and its Applications. (East European Series), Kluwer Acadamic Publications Dordrecht, (1991); 53. [13] A. Ostrowski, Über die Absolutabweichung einer di¤ erentienbaren Funktionen von ihren Integralimittelwert, Comment. Math. Hel. (1938); 10: 226-227. [14] S. Hussain and A. Qayyum, A generalized Ostrowski-Grüss type inequality for bounded differentiable mappings and its applications, Journal of Inequalities and Applications (2013) ; 2013:1. [15] A. Qayyum, M. Shoaib and I. Faye, Some new generalized results on ostrowski type integral inequalities with application, Journal of computational analysis and applications, vol. 19, No.4, (2015). [16] A. Qayyum and S. Hussain, A new generalized Ostrowski Gruss type inequality and applications, Applied Mathematics Letters, 25 (2012);1875-1880. [17] N. UJevi´c, New bounds for the …rst inequality of Ostrowski-Grüss type and applications, Computers and Mathematics with Applications, (2003); 46: 421-427. 1 Department

of Fundamental and Applied Sciences, 32610 Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia., 2 Department of Mathematics, University of Ha’il, Saudi Arabia., E-mail address : [email protected] Higher Colleges of Technology, Abu Dhabi Men’s College, P.O. Box 25035, Abu Dhabi, United Arab Emirates. E-mail address : [email protected] Department of Fundamental and Applied Sciences, 32610 Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia. E-mail address : [email protected]

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Duality in multiobjective nonlinear programming under generalized second order (F, b, φ, ρ, θ)− univex functions Falleh R. Al-Solamy, Meraj Ali Khan

Abstract In the present paper, second order duality for multiobjective nonlinear programming are investigated under the second order generalized (F, b, φ, ρ, θ)− univex functions. The weak, strong and converse duality theorems are proved. Further, we also illustrated an example of (F, b, φ, ρ, θ)− univex functions. Results obtained in this paper extend some previously known results of multiobjective nonlinear programming in the literature. Keywords: Duality, Multiobjective programming, Univex functions Mathematics Subject Classification (2000): 90C32, 49K35, 49N15

1

Introduction

In recent years, the concept of convexity and generalized convexity is well known in optimization theory and plays a central role in mathematical economics, management science and optimization theory. Therefore, the research on convexity and generalized convexity is one of the most important aspects in mathematical programming. In particular, the concept of generalized (F, ρ)− convexity introduced by Preda [8]. In [9, 13], the concept of V − ρ-invexity and (F, α, ρ, d)− convexity were introduced respectively. Zhang and Mond [12] extended the class of (F, ρ)− convex functions to second oder (F, ρ)− convex functions and obtained the duality results for Mangasarian type, Mond-Weir type and general Mond-Weir type multiobjective dual problems. Motivated by Liang et al. [13] and Aghezzaf [2], I. Ahmad and Z. Husain [5] introduced second order (F, α, ρ, d)− convex functions and their generalization and they investigate weak, strong and strict converse duality theorems for second order Mond Weir type Multiobjective dual. Bector et al. [15] generalized the notion of convex function to univex functions. Rueda et al. [16] obtained optimality and duality results for several mathematical programs by combining the concepts of type I and univex functions. Mishra [8] obtained optimality results and saddle point results for multiobjective programs under generalized type I univex functions. Recently, Zalmai [14] introduced the notion of second order (F, b, φ, ρ, θ)− univex functions and he called these functions (F, b, φ, ρ, θ)−sounivex functions, these function generalize the second order (F, α, ρ, d)−convex functions defined by Ahmad and Husain [5]. The concept of second order duality in nonlinear programming problems was first introduced by Mangasarian [11]. One significant practical application of second order dual over first order is that it may provide tighter bounds for value of objective function because there ae more parameters involved, several researchers [1, 4, 7, 21] considered second order dual models for multiobjective 1 740

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programming. In this paper , we formulate second order dual model and investigate weak, strong and strict converse duality theorems under (F, b, φ, ρ, θ)− sounivexity assumptions. Further, an example have been constructed, which shows the existence of (F, b, φ, ρ, θ)− sounivex functions.

2

Notations and Preliminaries We consider the following multiobjective nonlinear programming problem: (P) subject to

Minimize g(x)
0,

f (x), x ∈ X,

k

(1) m

where f = (f1 , f2 , . . . , fk ) : X → R , g = (g1 , g2 , . . . gm ) : X → R are assumed to be twice differentiable function over X, an open subset of Rn . Definition 2.1. A function F : X × X × Rn → R, where X ⊆ Rn is said to be sublinear in its third argument, if ∀ x, x ¯ ∈ X, (i) F(x, x ¯; a1 + a2 ) ≤ F (x, x ¯; a1 ) + F (x, x ¯; a2 ), ∀ a1 , a2 ∈ Rn , (ii) F(x, x ¯; αa) = αF (x, x ¯; a), ∀ α ∈ R+ , a ∈ Rn . Definition 2.2. A point x ¯ ∈ S is said to efficient solution of (P), if there exists no other feasible point x such that f (x) ≤ f (¯ x) for each x, x ¯ ∈ X. Let u ∈ Rn and assume that the function f : X → R is twice differentiable at u. Definition 2.3. [14] The function f is said to be (strictly) (F, b, φ, ρ, θ)− sounivex at u if there exist functions b : X × X → (0, ∞), φ : R → R, ρ : X × X → R, θ : X × X → Rn , and a sublinear function F (x, u, .) : Rn → R such that for each x ∈ X(x = u) and p ∈ Rn , 1 φ(f (x) − f (u) + pt ∇2 f (u)p)(>)  F (x, u; b(x, u)[∇f (u) + ∇2 f (u)p]) 2 +ρ(x, u) θ(x, u) 2 , where . 2 is a norm on Rn . A twice differentiable vector function f : X → Rk is said to be (F, b, φ, ρ, θ)− sounivex at u, if each of its components fi is (F, b, φ, ρ, θ)−sounivex at u. Now we define generalized (F, b, φ, ρ, θ)− sounivex functions Definition 2.4. A twice differentiable function f, over X is said to be (F, b, φ, ρ, θ)− pseudo sounivex at u if there exist functions b : X × X → (0, ∞), φ : R → R, ρ : X × X → R, θ : X × X → Rn , and a sublinear function F (x, u, .) : Rn → R such that for each x ∈ X(x = u) and p ∈ Rn , 1 φ(f (x) − f (u) + pt ∇2 f (u)p) < 0 2 ⇒ F (x, u; b(x, u)[∇f (u) + ∇2 f (u)p]) < −ρ(x, u) θ(x, u) 2 . 2 741

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A twice differentiable vector function f : X → Rk is said to be (F, b, φ, ρ, θ)− pseudo sounivex at u, if each of its components fi is (F, b, φ, ρ, θ)−pseudo sounivex at u. Definition 2.5. A twice differentiable function f, over X is said to be strictly (F, b, φ, ρ, θ)− pseudo sounivex at u if there exist functions b : X × X → (0, ∞), φ : R → R, ρ : X × X → R, θ : X × X → Rn , and a sublinear function F (x, u, .) : Rn → R such that for each x ∈ X(x = u) and p ∈ Rn , F (x, u; b(x, u)[∇f (u) + ∇2 f (u)p])  −ρ(x, u) θ(x, u) 2 1 ⇒ φ(f (x) − f (u) + pt ∇2 f (u)p) > 0, 2 or equivalently

1 φ(f (x) − f (u) + pt ∇2 f (u)p)
0 2

⇒ F (x, u; b(x, u)[∇f (u) + ∇2 f (u)p]) < −ρ(x, u) θ(x, u) 2 . A twice differentiable vector function f : X → Rk is said to be strictly (F, b, φ, ρ, θ)− pseudo sounivex at u, if each of its components is strictly fi is (F, b, φ, ρ, θ)−pseudo sounivex at u. Definition 2.6. A twice differentiable function f, over X is said to be (F, b, φ, ρ, θ)− quasi sounivex at u if there exist functions b : X × X → (0, ∞), φ : R → R, ρ : X × X → R, θ : X × X → Rn , and a sublinear function F (x, u, .) : Rn → R such that for each x ∈ X(x = u) and p ∈ Rn , 1 φ(f (x) − f (u) + pt ∇2 f (u)p)
0 2 ⇒ F (x, u; b(x, u)[∇f (u) + ∇2 f (u)p])
−ρ(x, u) θ(x, u) 2 . A twice differentiable vector function f : X → Rk is said to be (F, b, φ, ρ, θ)− pseudo sounivex at u, if each of its components fi is (F, b, φ, ρ, θ)−quasi sounivex at u. Definition 2.7. A twice differentiable function f, over X is said to be strong (F, b, φ, ρ, θ)− pseudo sounivex at u if there exist functions b : X × X → (0, ∞), φ : R → R, ρ : X × X → R, θ : X × X → Rn , and a sublinear function F (x, u, .) : Rn → R such that for each x ∈ X(x = u) and p ∈ Rn , 1 φ(f (x) − f (u) + pt ∇2 f (u)p) ≤ 0 2 ⇒ F (x, u; b(x, u)[∇f (u) + ∇2 f (u)p]) ≤ −ρ(x, u) θ(x, u) 2 A twice differentiable vector function f : X → Rk is said to be strong (F, b, φ, ρ, θ)− pseudo sounivex at u, if each of its components fi is strong (F, b, φ, ρ, θ)− pseudo sounivex at u. Every (F, b, φ, ρ, θ)− sounivex function need not to be second order (F, α, ρ, d)− convex, definded in [5]. To show this, consider the following example.

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Example 2.1. Let f : X = (0, ∞) → R be defined as f (x) = −x2 − x. Let and sublinear function is φ(t) = −t, b(x, u) = x − u, ρ = −10, θ(x, u) = u+2 2 defined as F (x, u, a) = a(x − u) + x F (x, u; b(x, u)[∇f (u) + ∇2 f (u)p]) = −(x2 − u2 )(2u + 1 + 2p) + x − 10

u+2 2 , 2

at u = 0, F (x, 0; b(x, 0)[∇f (0) + ∇2 f (0)p]) = −x2 (1 + 2p) + x − 10 and

1 f (x) − f (u) + pt ∇2 f (u)p = −x2 − x + u2 + u − p2 , 2

at u = 0

1 φ(f (x) − f (0) + pt ∇2 f (0)p) = x2 + x + p2 , 2 and it is easy to see that 1 φ(f (x) − f (0) + pt ∇2 f (0)p) − F (x, 0; b(x, 0)[∇f (0) + ∇2 f (0)p]) 2 = x2 + p2 + x2 (1 + 2p) + 10 ≥ 0

for all x ∈ R and −1 ≤ p < ∞, so the function is (F, b, ρ, φ, θ)− sounivex at x = 0, but at p = −1, x = 10 1 (f (x) − f (0) + pt ∇2 f (0)p) − F (x, 0; b(x, 0)[∇f (0) + ∇2 f (0)p]) < 0 2 Hence, the function is not (F, α, ρ, d)−convex at x = 0. Now we have following Kuhn-Tucker type necessary conditions, which will be useful to prove the strong duality theorem. Theorem 2.1. (Kuhn-Tucker type necessary conditions) Assume that x∗ is an efficient solution for (P) at which the Kuhn-Tucker constraint qualification is satisfied. Then there exist λ∗ ∈ Rk and y ∗ ∈ Rm , such that λ∗t ∇f (x∗ ) + y∗t ∇g(x∗ ) = 0, y ∗t ∇g(x∗ ) = 0, y ∗  0, λ∗ ≥ 0.

3

Second order Mond-Weir type duality

In this section, we consider the following Mond-Weir second order dual associated with multiobjective problem (P) and establish weak, strong and strict converse duality theorems under generalized (F, b, ρ, φ, θ)− sounivexity

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(MD) Maximize

1 f (u) − pt ∇2 f (u)p 2

Subject to ∇λt f (u) + ∇2 λt f (u)p + ∇λt g(u) + ∇2 λt g(u)p = 0,

(2)

1 y t g(u) − pt ∇2 yt g(u)p  0, 2 y  0,

(3) (4)

λ ≥ 0,

(5)

where λ is a k−dimensional vector, and y is an m−dimensional vector. Theorem 2 (weak duality) Suppose that for all feasible solutions x in (P) and all feasible solutions (u, y, λ, p) in MD (i) yt g(0) is (F, b, φ, ρ, θ)−quasi sounivex at u, (ii) λ > 0, and f (.) is strong (F, b1 , φ, ρ1 , θ)− pseudo sounivex at u with b−1 ρ + b−1 1 ρ1 λ  0, (iii) u ≤ 0 ⇒ φ(u) ≤ 0 and v
0 ⇒ φ(v)
0, for all u, v ∈ Rn . Then the following can not hold 1 f (x) ≤ f (u) − pt ∇2 f (u)p. 2

(6)

Proof. Now suppose contrary to the result that (6) holds, i.e., 1 f (x) ≤ f (u) − pt ∇2 f (u)p, 2 or

1 f (x) − f (u) + pt ∇2 f (u)p ≤ 0, 2 then by assumption (iii) 1 φ(f (x) − f (u) + pt ∇2 f (u)p) ≤ 0, 2

(7)

which by virtue of assumption (ii) leads F (x, u, b1 (x, u){∇f (u) + ∇2 f (u)p}) ≤ −ρ1 θ(x, u) 2 .

(8)

On multiplying (8) by λ > 0 and using sublinearity of F with b1 (x, u) > 0, we have 2 F (x, u, ∇λt f (u) + ∇2 λt f (u)p) < −b−1 (9) 1 (x, u)ρ1 λ θ(x, u) . The first dual constraint and sublinearity of F give F (x, u; ∇y t g(u) + ∇2 yt g(u)p)  −F (x, u; ∇λt f (u) + ∇2 λt f (u)p). Applying (9) in above inequality, we have 2 F (x, u; ∇y t g(u) + ∇2 yt g(u)) > b−1 1 (x, u)ρ1 λ θ(x, u) .

(10)

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Let x be any feasible solution in (P) and (u, y, λ, p) be any feasible solution in (MD). Then we have 1 yt g(x)
0
y t g(u) − pt ∇2 y t g(u)p), 2

(11)

by assumption (iii), (11) yields 1 φ(y t g(x) − y t g(u) + pt ∇2 y t g(u)p)
0. 2

(12)

Using (F, b, φ, ρ, θ)-quasi sounivexity of y t g(.), we have F (x, u; b(x, u){∇y t g(u) + ∇2 y t g(u)p)})
−ρ θ(x, u) 2 .

(13)

Since b(x, u) > 0, the above inequality with the sublinearity of F give F (x, u; ∇y t g(u) + ∇2 yt g(u)p)
−b−1 ρ θ(x, u) 2 .

(14)

Now using the assumption b−1 ρ + b−1 1 ρ1 λ  0, the above inequality yields 2 F (x, u; ∇yt g(u) + ∇2 y t g(u)p)
b−1 1 ρ1 λ θ(x, u) .

(15)

Which contradict (10), hence (6) can not hold. Theorem 3 (Strong duality). Let x ¯ be an efficient solution of (P) at which the Kuhn-Tucker constraint qualification is satisfied. Then there exist y¯ ∈ Rm ¯ ∈ Rk , such that (¯ ¯ p¯ = 0) is a feasible for (MD) and the correand λ x, y¯, λ, sponding values of (P) and (MD) are equal. If in addition, the assumptions of weak duality (Theorem 2) hold for all feasible solutions of (P) and (MD), then ¯ p¯ = 0) is an efficient solution of (MD). (¯ x, y¯, λ, Proof. Since x ¯ is an efficient solution of (P) at which the Kuhn-Tucker constraint qualification is satisfied, then by Theorem 1, there exist y¯ ∈ Rm and ¯ ∈ Rk , such that λ ¯ t ∇f (¯ λ x) + y¯t ∇g(¯ x) = 0, y¯t ∇g(¯ x) = 0, y¯  0, ¯ ≥ 0. λ ¯ p¯ = 0) is feasible for (MD) and the corresponding values of Therefore (¯ x, y¯, λ, (P) and (MD) are equal. The efficiency of this feasible solution for (MD) thus follows from weak duality (Theorem 2). ¯ p¯) be the efficient soluTheorem 4 (Strict converse duality) Let x ¯ and (¯ u, y¯, λ, tion of (P) and (MD), respectively such that 1 ¯ t f (¯ ¯ t f (¯ ¯ t f (¯ u)¯ p. λ x) = λ u) − p¯t ∇2 λ 2

(16)

Suppose 6 745

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(i) yt g(.) is (F, b, φ, ρ, θ)−quasi sounivex at u¯, ¯ t f (.) be (F, b1 , φ, ρ1 , θ)− pseudo sounivex at u (ii) λ ¯ with b−1 ρ + b−1 1 ρ1 λ  0, (iii) u
0 ⇒ φ(u)
0 and v < 0 ⇒ φ(v) < 0, for all u, v ∈ Rn . Then x¯ = u¯, that is u ¯ is an efficient solution. Proof. We assume that x ¯ = u ¯ and reach a contradiction, since x ¯ and ¯ p) are respectively the feasible solution of (P) and (MD), we have (¯ u, y¯, λ, 1 u)¯ p
0. y¯t g(¯ x) − y¯t g(¯ u) + p¯t ∇2 y¯t g(¯ 2 Using the assumption (iii), we have

(17)

1 u)¯ p)
0. φ(¯ y t g(¯ x) − y¯t g(¯ u) + p¯t ∇2 y¯t g(¯ 2 By (F, b, φ, ρ, θ)−quasi sounivexity of y¯t g(.) at u ¯, we get

(18)

F (¯ x, u¯; b(¯ x, u ¯){∇¯ y t g(¯ u) + ∇2 y¯t g(¯ u)¯ p)})
−ρ θ(¯ x, u ¯) 2 .

(19)

Since b(¯ x, u ¯) > 0, the inequality (19) along with the sublinearity of F, imply F (¯ x, u¯; ∇¯ y t g(¯ u) + ∇2 y¯t g(¯ u)¯ p)
−b−1 (x, u)ρ θ(¯ x, u ¯) 2 .

(20)

The first dual constraint and sublinearity of F imply ¯ t f (¯ F (¯ x, u ¯; ∇λ u) + ∇2 y¯t f (¯ u)¯ p)  −F (¯ x, u ¯, ∇¯ yt g(¯ u) + ∇2 y¯t g(¯ u)¯ p). Applying (20) and b−1 ρ + b−1 1 ρ  0 in above inequality, we get ¯ t f (¯ F (¯ x, u ¯; ∇ λ u) + ∇2 y¯t f (¯ u)¯ p)  −b−1 x, u ¯)ρ θ(¯ x, u¯) 2 . 1 (¯

(21)

Suppose (16) does not holds, then we have 1 ¯ t f (¯ ¯ t f (¯ ¯ t f (¯ u)¯ p, λ x) < λ u) − p¯t ∇2 λ 2 now using assumption (iii) 1 ¯ t f (¯ ¯ t f (¯ ¯ t f (¯ u)¯ p) < 0. φ(λ x) − λ u) + p¯t ∇2 λ 2 Now by the assumption (ii), the above inequality gives ¯ t f (¯ F (¯ x, u ¯; b1 (¯ x, u ¯)(∇λ u) + ∇2 y¯t f (¯ u)¯ p)) < −ρ θ(¯ x, u ¯) 2 , or ¯ t f (¯ F (¯ x, u ¯; ∇ λ u) + ∇2 y¯t f (¯ u)¯ p) < −b−1 x, u ¯)ρ θ(¯ x, u¯) 2 . 1 (¯

(22)

Which contradict (21). Hence result.

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4

Conclusion

In this paper anew concept of generalized invex functions is introduced. Under this generalized invexity we establish weak, strong and converse duality theorems. These duality relations lead to duality in nonlinear fractional programming problems.

5

Authors contributions

Both the authors contributed equally to writing of this paper and the final manuscript is read and approved by the authors.

6

Competing interests The author declare that they have no competing interests.

7

Acknowledgments

This project was funded by the Deanship of Scientific Research(DSR), King Abdulaziz University, Jeddah, under grant No. (G-1436-130-242). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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[9] H. Kuk, G. M. Lee, D. S. Kim, Nonsmooth multiobjective programs with V − ρ−invexity, Ind. J. Pure. Appl. Math. 29(2)(1998)405-412. [10] C. R. Bector, S. Chandra, S. Gupta, S. K. Suneja, Univex sets, functions and univex nonlinear programming, in: Lecture Notes in Economics and Mathematical system, vol. 405, Springer Verlag, Berlin, 1994, pp. 1-8. [11] O. L. Mangasarian, Second and higher order duality in nonlinear programming, J. Math. Anal. Appl., 51 (1975), 607-620. [12] J. Zhang, B. Mond, Second order duality for multiobjective nonlinear programming involving generalized convexity, in: B. M. Glower, B. D. Cravan, D. Ralph (Eds.), Proceeding of the Optimization Miniconference III, University of Ballarat, (1997), 79-95 [13] Z. A. Liang, H. X. Huang, P. M. Pardalos, Efficiency conditions and duality for a class of multiobjective programming problems, Journal of Global Optimization, 27(2003), 1-25. [14] G. J. Zalmai, Second order functions and generalized duality models for multiobjective programming problems containing arbitrary norms, J. Korean Math. Soc., 50(4)(2013), 727-753. [15] C. R. Bector, S. K. Suneja, S. Gupta, Univex functions and univex nonlinear programming, Proceeding of the administrative sciences Association of Canada (1992)115-124. [16] N. G. Rueda, M. A. Hanson, C. Singh, Optimality and duality with generalized convexity, J. Optimization Theory and Applications 86(1995)491-500. [17] S. K. Mishra, S. Y. Wang, K. K. Lai, Nondifferentiable multiobjective programming under generalized d−univexity, Eur. J. Oper. Res. 160(2005)218226. [18] S. K. Mishra, On multiple-objective optimization with generalized univexity, J. Math. Anal. and Appl. 224(1998), 131-148. [19] M. A. hanson, Second order invexity and duality in mathematical programming, Opsearch 30(1993), 311-320. [20] N. G. Rueda, M. A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. anal. Appl. 130(1988)375-385. [21] Z. Husain, I. Ahmad and Sarita Sharma, Second order duality for minimax fractional programming, Optimization Letters, 3 no. 2 (2009), 277-286. Author’s addresses: Falleh R. Al-Solamy Department of Mathematics King Abdulaziz University P.O. Box 80015, Jeddah 21589, Kingdom of Saudi Arabia

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E-mail:[email protected] Meraj Ali Khan Department of Mathematics, University of Tabuk, Tabuk Kingdom of Saudi Arabia E-mail:[email protected]

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STABILITY OF FRACTIONAL DIFFERENTIAL EQUATION WITH BOUNDARY CONDITIONS S. RAJAN1 , P. MUNIYAPPAN2∗ , CHOONKIL PARK3∗ , SUNGSIK YUN4∗ , AND JUNG RYE LEE5∗ Abstract. In this paper, we prove the Hyers-Ulam stability of a fractional differential equation of order α ∈ (1, 2) with certain boundary conditions.

1. Introduction The recent concentric area in the research world of mathematics is fractional differential equations. The concept of fractional derivative is not new and is very much as old as classical differential equations. In recent years, many authors disscussed and proved the existence results of fractional differential equations using various methods. For example, one can refer the monographs of Kilbas et al. [10], Miller and Ross [14], Podulbny [20], Diethelm et al. [4, 5], Benchora [2] and so on. Obviously, the differential equations of fractional order has been proved to be a valuable tool in the modeling of many phenomena in various fields of science and engineering. Indeed, one can find many applications in electromagnetic, control, electrochemistry, etc. (see [6, 7]). At the same instance, the stability concept is more devoloped in the research world of mathematics, particularly in functional equations. But the analysis of stability concepts of fractional differential equations has been very slow and there are only countable number of works. In 2009, Li [12], first proposed the Mittag-Leffler stability and in 2010 [13], the fractional Lyapunovs second method. In the next year, Li and Zhang [11] have been given a brief overview on the stability of the fractional differential equations. However, there are only few works available on the local stability and Mittag-Leffler stability for fractional differential equations and very rare works on the Ulam stability of fractional differential equations. In 2011, Wang [24] carried out a pioneering work on the Hyers-Ulam stability and data dependence for fractional differential equations with Caputo derivative. Wang [25] proved the Hyers-Ulam stability of fractional differential equation of order 0 < α < 1 via a generalized fixed point approach, by adopting some part idea of Wang et al. [24], Cadariu and Radu [3] and Jung [9] in the next year. Particularly, there are very rare works on the Hyers-Ulam stability of fractional differential equations with boundary conditions. Recently, Rabha [8], Muniyappan and Rajan [16] had given Ulam stabilities with boundary conditions in the interval (0, 1). For more information on functional equations and their stability problems, see [15, 17, 18, 19, 21, 22, 23]. 2010 Mathematics Subject Classification. Primary 34A08, 34K10, 34K20. Key words and phrases. Hyers-Ulam stability; fractional differential equation; boundary condition. ∗ Corresponding authors.

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In this paper, the Hyers-Ulam stability of the following fractional boundary value problem is proved. C

Dα y(t) = F (t, y(t)),

1 0. Thus (3.7) implies that d(Λg0 , g0 ) < ∞. Therefore, according to Theorem 2.8, there exists a continuous function y0 : I → R such that Λn g0 → y0 in (X, d) and Λy0 = y0 , that is, y0 satisfies (3.4) for every t ∈ I. we will now verify that {g ∈ X/d(g0 , g) < ∞} = X. For any g ∈ X, since g and g0 are bounded on I and mint∈I ϕ(t) > 0, there exists a constant 0 < Cg < ∞ such that |g0 (t) − g(t)| ≤ Cg ϕ(t) Hence, we have d(g0 , g) < ∞ for all g ∈ X, that is {g ∈ X/d(g0 , g) < ∞} = X. Hence in view of Theorem 2.8, we conclude that y0 is the unique continuous function with the property (3.4). On the other hand, it follows from (3.2) that α −ϕ(t) ≤c Da+ y(t) − F (t, y(t)) ≤ ϕ(t)

for all t ∈ I. If we integrate each term in the above inequality and substitute the boundary conditions, then we obtain   Z t Z T 1 t t t α−1 α−1 |y(t) − (t − s) F (s, y(s))ds − (T − s) F (s, y(s))ds − − 1 y0 + yT | Γ(α) 0 T Γ(α) 0 T T Z T 1 (T − s)α−1 ϕ(t)ds ≤ Γ(α) 0 for all t ∈ I. Thus by (3.3) and (3.8) we get |y(t) − (Λy)(t)| ≤ Kϕ(t) for each t ∈ I, which implies that d(y, Λy) ≤ K.

(3.10)

Finally, Theorem 2.8 and (3.10) imply that d(y, y0 ) ≤

K 1 d(y, Λy) ≤ . 1 − KP L 1 − KP L 

Now, we will prove the Hyers-Ulam stability of the (1.1) with boundary condition (1.2) Theorem 3.2. Let I = [0, T ] be a closed interval. Let r > 0 be a positive constant with 0 ≤ t, T ≤ r and let F : I × R → R be a continuous function which satisfies a Lipschitz condition LP rα (3.1) for all t ∈ I and y, z ∈ R, where L is a constant with 0 < Γ(α+1) < 1. If a continuously differentiable function y : I → R satisfying the differential inequality c α Da+ y(t) − F (t, y(t)) ≤  (3.11) for all t ∈ I and for some  ≥ 0, then there exists a unique continuous function y0 : I → R satisfying (3.4) and rα |y(t) − y0 (t)| ≤  (3.12) Γ(α + 1) − LP rα for all t ∈ I.

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Proof. First, we define a set X of all continuous functions f : I → R by X = {f : I → R|f is continuous} and introduce a generalized complete metric on X as follows d(f, g) = inf {C ∈ [0, ∞]| |f (t) − g(t)| ≤ C

for all

t ∈ I}

Define an operator Λ : X → X by   Z t Z T t t 1 t α−1 α−1 (t−s) F (s, y(s))ds− (T −s) F (s, y(s))ds− (Λf ) (t) = − 1 y0 + yT Γ(α) 0 T Γ(α) 0 T T for all f ∈ X. We now assert that Λ is strictly contractive on X. For all f, g ∈ X, let Cf g ∈ [0, ∞] be an arbitrary constant with d(f, g) ≤ Cf g , that is, let us assume that |f (t) − g(t)| ≤ Cf g (3.13) for any t ∈ I. Moreover, it follows from (3.1), (3.8) and (3.13) that Z t 1 |(Λf )t − (Λg)t| ≤ (t − s)α−1 |F (s, f (s)) − F (s, g(s))| ds Γ(α) 0 Z T t + (T − s)α−1 |F (s, f (s)) − F (s, g(s))| ds T Γ(α) 0 Z t L (t − s)α−1 |f (s) − g(s)| ds ≤ Γ(α) 0 Z T tL + (T − s)α−1 |f (s) − g(s)| ds T Γ(α) 0  α  r trα ≤ LCf g + αΓ(α) T αΓ(α)   LCf g rα t + T ≤ Γ(α + 1) T α LP Cf g r ≤ Γ(α + 1)
t for all t ∈ I, where P = 1 + T , that is d (Λf, Λg) ≤

LP rα Cf g . Γ(α + 1)

Thus it follows that

LP rα d (f, g) Γ(α + 1) LP rα for all f, g ∈ X, and we note that 0 < Γ(α+1) < 1. Analogously to the proof of Theorem 3.1, we can show that each g0 ∈ X satisfies the property d(Λg0 , g0 ) < ∞. Therefore, Theorem 2.8 implies that there exists a continuous function y0 : I → R such that Λn g0 → y0 in (X, d) as n → ∞, and such that y0 = Λy0 , that is, y0 satisfies the equation (3.4) for all t ∈ I. d (Λf, Λg) ≤

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FRACTIONAL DIFFERENTIAL EQUATION WITH BOUNDARY CONDITIONS

If g ∈ X, then g0 and g are continuous functions defined on a compact interval I. Hence, there exists a constant C > 0 with |g0 (t) − g(t)| ≤ C for all t ∈ I. This implies that d(g0 , g) < ∞ for every g ∈ X, or equivalently, {g ∈ X|d(g0 , g) < ∞} = X. Therefore, according to Theorem 2.8, y0 is a unique continuous function with property (3.4). Furtheremore, it follows from (3.11) that α − ≤c Da+ y(t) − F (t, y(t)) ≤ 

for all t ∈ I. If we integrate each term of the above inequality and appling the boundary conditions, then we have rα |(Λy) (t) − y(t)| ≤  Γ(α + 1) α

r . for all t ∈ I, that is, it holds that d (Λy, y) ≤ Γ(α+1) It now follows from Theorem 2.8 that rα 1 , d (Λy, y) ≤ d(y, y0 ) ≤ α LP r Γ(α + 1) − LP rα 1 − Γ(α+1)

which implies the validity of (3.12) for each t ∈ I

(3.14) 

Acknowledgments S. Yun was supported by Hanshin University Research Grant. References 1. R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010), 973-1033. 2. M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3(2008), 1-12. 3. L. Cadariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (1) (2003). Art. No. 4. 4. K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010. 5. K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229-248. 6. K. Diethelm, A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. pp. 217-224, Springer-Verlag, Heidelberg, 1999. 7. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. 8. R.W. Ibrahim, Stability of fractional differential equations, Internat. J. Math. Comput. Sci. Eng. 7 (2013), 212-217. 9. S. Jung, A fixed point approach to the stability of differential equations y 0 (t) = F (x, y), Bull. Malays. Math. Sci. Soc. 33 (2010), 47-56. 10. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Science, Amsterdam, 2006. 11. C.P. Li, F.R. Zhang, A survey on the stability of fractional differential equations, Eur. Phys. J. Special Topics 193 (2011), no. 27, 27-47. 12. Y. Li, Y. Chen, I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica J. IFAC 45 (2009), 1965-1969. 13. Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59 (2010), 1810-1821. 14. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons Inc., New York 1993.

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15. E. Movahednia, S. M. S. M. Mosadegh, C. Park, D. Shin, Stability of a lattice preserving functional equation on Riesz space: fixed point alternative, J. Comput. Anal. Appl. 21 (2016), 83–89. 16. P. Muniyappan, S. Rajan, Hyers-Ulam-Rassias stability of fractional differential equations, Internat. J. Pure Appl. Math. 102 (2015), 631-642. 17. C. Park, Additive ρ-functional inequalities, J. Nonlinear Sci. Appl. 7 (2014), 296–310. 18. C. Park, Stability of ternary quadratic derivation on ternary Banach algebras: revisited, J. Comput. Anal. Appl. 20 (2016), 21–23. 19. W. Park, J. Bae, Approximate quadratic forms on restricted domains, J. Comput. Anal. Appl. 20 (2016), 388–410. 20. I. Podlubny, Fractional Differential Equations, Academic Press, London, 1999. 21. S. Schin, D. Ki, J. Chang, M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37-49. 22. D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964-973. 23. D. Shin, C. Park, 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. 24. J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 63 (2011), Art. No. 63. 25. J. Wang, L. Lv, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul, 17 (2012), 2530-2538. 1

Department of Mathematics, Erode Arts and Science College, Erode, Tamilnaddu, India E-mail address: [email protected] 2

Department of Mathematics, Adhiyamaan College of Engineering, Hosur, Tamilnadu, India E-mail address: [email protected] 3

Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] 4

Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Korea E-mail address: [email protected] 4

Department of Mathematics, Daejin University, Kyunggi 11159, Korea E-mail address: [email protected]

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Bernstein-Stancu type operators which preserve polynomials Young Chel Kwun1 , Ana-Maria Acu2, Arif Rafiq3,∗, Voichit¸a Adriana Radu4 , Faisal Ali5 and Shin Min Kang6,∗

1

2

Lucian Blaga University of Sibiu, Department of Mathematics and Informatics, Str. Dr. I. Ratiu, No.5-7, RO-550012 Sibiu, Romania e-mail: [email protected] 3

4

Department of Mathematics, Dong-A University, Busan 49315, Korea e-mail: [email protected]

Department of Mathematics and Statistics, Virtual University of Pakistan, Lahore 54000, Pakistan e-mail: [email protected]

Babes-Bolyai University, FSEGA, Department of Statistics Forecasts Mathematics, Str. Teodor Mihali, No.58-60, RO-400591 Cluj-Napoca, Romania e-mail: [email protected] 5

6

Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan e-mail: [email protected]

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] Abstract In the last years there is an increasing interest in modifying linear operators so that the new versions reproduce some basic functions. This idea motivated us to modify the sequence of linear Bernstein Stancu type operators. Using numerical examples we show that these operators present a better degree of approximation than the original ones. In this note the modified Bernstein Stancu operators are studied in regard to uniform convergence and global smoothness preservation. 2010 Mathematics Subject Classification: Primary 41A36; Secondary 41A25 Key words and phrases: Bernstein-Stancu operator, rate of convergence, moduli of continuity

∗

Corresponding authors

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1

Y. C. Kwun, A. M. Acu, A. Rafiq, V. A. Radu, F. Ali and S. M. Kang

Introduction

In 1912 in Bernstein’s constructive proof of the Weierstrass approximation theorem [3] were introduced the classical Bernstein operators Bn : C[0, 1] → C[0, 1], defined by Bn (f ; x) =

n X k=0

  k , pn,k (x)f n

  n k where pn,k (x) = x (1 − x)n−k . k

Lemma 1.1. The Bernstein operators verify the following identities (i) Bn (e0 ; x) = 1, (ii) Bn (e1 ; x) = x, (iii) Bn (e2 ; x) = nx (1 + xn − x), where ei (t) = ti , i = 0, 1, . . . . In the last years there is an increasing interest in modifying linear operators so that the new versions reproduce some basic functions. King [12] consider for the first time this kind of modification for the Bernstein operators and proved that the modified operators reproduce the functions ei (x) = xi for i = 0, 2 and approximate each continuous function on [0, 1] with an order of approximation at least as good as that of the classic Bernstein whenever 0 ≤ x < 31 . Using the same type of technique introduced by King or new methods many authors published new results in regard with this subject. C´ardenas-Morales et al. [4] extended this result considering a family of sequences of operators Bn,α that preserve e0 and e2 +αe1 with α ∈ [0, ∞). Gonska et al. [11] studied the sequence Vnτ : C[0, 1] → C[0, 1] defined by Vnτ f := (Bn f ) ◦ (Bn τ )−1 ◦ τ, where τ is a continuous strictly increasing function defined on [0, 1] with τ (0) = 0 and +αe1 τ (1) = 1. Note that if τ = e21+α , then Vnτ = Bn,α and the operators Vnτ preserve e0 and τ . In [5], the authors inspired by the above ideas consider the sequence of linear Bernstein-type operators defined for f ∈ C[0, 1] by Bn (f ◦ τ −1 ) ◦ τ, τ being any function that is continuously differentiable ∞ times on [0, 1] such that τ (0) = 0, τ (1) = 1 and τ 0 (x) > 0 for x ∈ [0, 1]. Note that the Korovkin set {1, e1, e2 } is generalized to {1, τ, τ 2} and these operators present a better degree of approximation than Bn . Since the modified operators present a better degree of approximation than the original ones leads to an interesting area of research, so that generalized Bernstein-Durrmeyer operators and their approximation properties were studied in [1] and [6]. Also, the modified Szasz operators were considered recently in [2].

2

Bernstein-Stancu operators

In 1968, Stancu [15] proposed the sequence of positive linear operators Sn : C[0, 1] → C[0, 1] depending on a non-negative parameter α given by Sn (f ; x)

  n X k = f p n,k (x), x ∈ [0, 1], n

(2.1)

k=0

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Bernstein-Stancu type operators where p n,k (x)

3

  [k,−α] n x (1 − x)[n−k,−α] = k 1[n,−α]

and t[n,h] := t(t − h) · · · (t − n − 1h) is the nth factorial power of t with increment h. For α = 0 these operators reduce to the classical Bernstein operators. The values of the test function by Bernstein-Stancu operators were given by Stancu [15] as follows Lemma 2.1. If x ∈ [0, 1], then (i) Sn (e0 ; x) = 1, (ii)Sn(e1 ; x) = x,   x(1−x) 1 (iii)Sn(e2 ; x) = 1+α + x(x + α) . n Recently, in [13] Micl˘au¸s proposed a new technique to obtain the values of the test function, without using properties of Bernstein operators. It is well known the following form of Bernstein operators using the divided difference    n X k! n 1 k Bn (f ; x) = 0, , . . ., ; f xk . (2.2) nk k n n k=0

Starting with the form (2.2) of the Bernstein operators, the following Stancu type operators are constructed in [7]-[8]: Cn : C[0, 1] → Πn ,     n X k! n 1 k Cn (f ; x) = m 0, , ..., ; f xk , k,n nk k n n k=0

f ∈ C[0, 1],

(2.3)

where the real numbers (mk,n )∞ k=0 are selected in order to preserve some important properties of Bernstein operators and Πn is the linear space of all real polynomials of degree ≤ n. n )k Let m0,n = 1, limn→∞ m1,n = 1 and mk,n = (ak! , an ∈ (0, 1]. For this special case of ∞ real sequence (mk,n )k=0 the Bernstein-Stancu operators Cn were written in the Bernstein basis as follows (see [7], Theorem 10): Cn (f ; x) =

n X

pn,k (x)Ck,n [f ],

(2.4)

k=0

where

k     1 X k j Ck,n [f ] = f (an )j (1 − an )k−j . k! j n j=0

We remark that an ∈ (0, 1] leads to Cn linear positive operators. The coefficients Ck,n [f ] can be written as follows Ck,n [f ] =

k X

p k,j (an )f

j=0

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Therefore, Ck,n [f ] =

Sk (f˜; an ),

where f˜(t) = f

  k t . n

Lemma 2.2. ([7]) The Bernstein-Stancu operators Cn verify the following identities (i) Cn (e0 ; x) = 1, (ii) Cn (e1 ; x) = an x,
2 an n an + 1−a (iii) Cn (e2 ; x) = x2 + x(1−x) n 2 n − (2 + an ) x . Let µn,m (x) = Cn ((t − x)m ; x) =

n X

pn,k (x)

k X

p k,j (an )

j=0

k=0



j −x n

m

n, m ∈ N,

,

be the central moment operators. Lemma 2.3. ([7]) The central moment operators verify
x(1−x) an n (i) µn,2 (x) = n an + x2 (1 − an ) 2−a 2 + 2n ,  6(a )
12(a )
  7(a )
(ii) µn,4 (x) = x4 + nn4 3 n3 − n3n 2 n2 + 6ann x3 + nn4 2 n2 −


+ ann3 x + (ann4)4 n4 − 4(ann3 )3 n3 + 6(ann2 )2 n2 − 4an .

4an n2

 2 x

In [7], Cleciu obtained the following Voronovskaya type theorem: Theorem 2.4. ([7]) Suppose that x0 ∈ [0, 1] and f 00 (x0 ) exists. If an ∈ (0, 1), limn→∞ an = 1 and L := limn→∞ n(1 − an ) exists, then   x0 (1 − x0 ) 00 x20 00 0 lim n [f (x0 ) − Cn (f ; x0 )] = − f (x0 ) + x0 f (x0 ) − f (x0 ) L. n→∞ 2 4

3

Modified Bernstein-Stancu operators

In this section we deal with Bernstein-Stancu type generalization of (2.4). We investigate its sharp preserving and convergence properties. We define the modified Bernstein-Stancu operators as follows: Cnτ (f ; x)

=

n X k=0

pτn,k (x)

k X

pk,j (an )

j=0

f ◦τ

−1




  j , n

(3.1)


where pτn,k (x) = nk τ (x)k (1 − τ (x))n−k and τ is any function that is continuously differentiable ∞ times on [0, 1] such that τ (0) = 0, τ (1) = 1 and τ 0 (x) > 0 for x ∈ [0, 1]. Note that these operators are positive and linear and for the case τ (x) = x, these operators (3.1) reduce to the Bernstein-Stancu operators defined by Cleciu [7]-[8].

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5

Lemma 3.1. The modified operators Cnτ verify (i) Cnτ e0 = 1, (ii) Cnτ τ = an τ,
2 ) 1−an an (iii) Cnτ τ 2 = τ 2 + τ (1−τ n an + 2 n − (2 + an ) τ . Let

µτn,m (x) = Cnτ ((τ (t) − τ (x))m ; x) =

n X

pτn,k (x)

k X

p k,j (an )

j=0

k=0



j − τ (x) n

m

n, m ∈ N,

,

be the central moment operators. Lemma 3.2. The central moment operators verify (i) µτn,0 (x) = 1, (ii) µτn,1 (x) = (an − 1)τ (x),


τ (x)(1−τ (x)) an n an + τ (x)2 (1 − an ) 2−a n 2 + 2n ,  6(a )
12(a )
 = τ (x)4 + nn4 3 n3 − n3n 2 n2 + 6ann τ (x)3 

 2 + an τ (x) + (an )4 n n + 7(ann4 )2 n2 − 4a τ (x) 2 3 4 4 n n n

− 4(ann3 )3 n3 + 6(ann2 )2 n2 − 4an .

(iii) µτn,2 (x) =

(iv) µn,4 (x)

Lemma 3.3. For all n ∈ N we have

2 µτn,2 (x) ≤ δn,τ (x) 2 where δn,τ (x) :=

an 2 n ϕτ (x) +

for all x ∈ [0, 1],

(1 − an ) and ϕ2τ (x) := τ (x)(1 − τ (x)).

Proof. We have |µτn,2 (x)|

  τ (x)(1 − τ (x))an 2 an 2 − an = + + τ (x)(1 − an ) n 2 2n an 2 (x). ≤ ϕ2τ (x) + (1 − an ) = δn,τ n

Lemma 3.4. If f ∈ C[0, 1], then kCnτ f k ≤ kf k, where k · k is the uniform norm on C[0, 1]. Proof. From the definition of the operator Cnτ and using Lemma 3.1 it follows |Cnτ (f ; x)|

≤

n X k=0

pτn,k (x)

k X j=0



p k,j (an )

≤ kf ◦ τ −1 kCnτ (e0 ; x) = kf k.

762

f ◦τ

−1




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Theorem 3.5. Let f ∈ C[0, 1], an ∈ (0, 1] and limn→∞ an = 1. Then Cnτ f converges to f as n tends to infinity, uniformly on [0, 1].  Proof. Using the well known Korovkin theorem and Lemma 3.1 and the fact that e0 , τ, τ 2 is an extended complete Tchebychev system on [0, 1] it follows the uniform convergence of the operators Cnτ . Let ω be the usual modulus of continuity of f ∈ C[0, 1] which is defined as ω(f ; δ) = sup

sup

|h|≤δ x,x+h∈[0,1]

|f (x + h) − f (x)|.

Proposition 3.6. Let f ∈ C[0, 1] with modulus of continuity ω(f, ·). Then |Cnτ (f ; x) −

f (x)| ≤



µτn,2 (x) 1+ δ2



ω(f, δ)

for δ > 0 and x ∈ [0, 1]. 2

Example 3.7. If we choose τ (x) = x 2+x , we have τ (x)(1 − τ (x)) ≤ x(1 − x) for all x ∈ [0, 1/2] and this inequality leads to µτn,2 (x) ≤ µn,2 (x). Therefore, the modified operators Cnτ presents an order of approximation better than Cn in that interval. Example 3.8. Now using a graphical example we try to illustrate these approximation 2 processes. Let f (x) = sin(9x), τ (x) = x 2+x and an = 1/2. For n = 20, the approximation to the function f by Cn and Cnτ is shown in the Figure 1.

Figure 1. Approximation process by Cn and Cnτ 2

Example 3.9. Let us take f (x) = log(x + 1), τ (x) = x 2+x and an = 12 . In the Table 1 we computed the error of approximation for Cn and Cnτ at the point x0 = 0.8.

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7

Table 1. Error of approximation for Cn and Cnτ n 5 10 15 20 25 30 35 40 45 50

|Cn (f; x0 ) − f(x0 )| 0.2800807097 0.2762200954 0.2749367804 0.2742959594 0.2739117553 0.2736557443 0.2734729425 0.2733358757 0.2732292893 0.2731440335

|Cnτ (f; x0 ) − f(x0 )| 0.2613318434 0.2502212648 0.2465367167 0.2447029941 0.2436063038 0.2428768564 0.2423567117 0.2419671158 0.2416644116 0.2414224523

From the above results it follows that Cnτ converge faster than Cn to the function f (x) = log(x + 1) at the point x0 = 0.8. Also, the approximation to the function f by Cn and Cnτ is shown in the Figure 2.

Figure 2. Approximation process by Cn and Cnτ

4

Voronovskaya type theorem

Let Ln : C[0, 1] → C[0, 1], n ≥ 1, be a positive linear operator and Ln e0 = e0 . Acar et al. [1] defined a general operator Kn : C[0, 1] → C[0, 1] by
Kn g := Ln (g ◦ τ −1 ) ◦ τ, n ≥ 1.

The authors obtained the following Voronovskaya type formula for the modified operators Kn . Theorem 4.1. ([1]) Let f ∈ C[0, 1] with f 00 (x) finite for x ∈ [0, 1]. If there exists α, β ∈ C[0, 1] such that lim n(Ln (f, x) − f (x)) = α(x)f 00 (x) + β(x)f 0 (x),

n→∞

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then we have α(τ (t)) lim n (Kn (g, t) − g(t)) = 0 2 g 00 (t) + n→∞ τ (t)



β(τ (t)) α(τ (t))τ 00 (t) − τ 0 (t) τ 0 (t)3



g 0 (t)

for g ∈ C[0, 1] with g 00(x) finite for x ∈ [0, 1]. Using Theorem 2.4 and Theorem 4.1 we obtain a Voronovskaya type theorem for Cnτ . Theorem 4.2. Let f ∈ C 2 [0, 1]. If an ∈ (0, 1), limn→∞ an = 1 and L := limn→∞ n(1−an ) exists, then   α(τ (x)) 00 β(τ (x)) α(τ (x))τ 00(x) τ lim n (Cn (f, x) − f (x)) = 0 2 f (x) + − f 0 (x) n→∞ τ (x) τ 0 (x) τ 0 (x)3 uniformly on [0, 1] with α(x) = − x(1−x) − 2

5

x2 4 L

and β(x) = xL.

Local Approximation

Let  W 2 [0, 1] = g ∈ C[0, 1] : g 0 ∈ C[0, 1] .

For f ∈ C[0, 1] and δ > 0, the Peetre’s K-functional [14] is defined by K2 (f ; δ) =

inf

g∈W 2 [0,1]



kf − gk + δkgkW 2[0,1] ,

where kf kW 2 [0,1] = kf k + kf 0 k + kf 00 k. Throughout this paper we assume that inf x∈[0,1] τ 0 (x) ≥ a, a ∈ R+ . Theorem 5.1. Let an ∈ (0, 1) and limn→∞ an = 1 and f ∈ C[0, 1]. For the operator Cnτ (f ; ·), there exists absolute constant C > 0 such that  
1 τ 2 |Cn (f ; x) − f (x)| ≤ CK2 f ; δn,τ (x) + ω f ; (1 − an )τ (x) . a Proof. Let g ∈ W 2 [0, 1] and t ∈ [0, 1]. Then by Taylor’s expansion, we get
g(t) = g ◦ τ −1 (τ (t))

0 = g ◦ τ −1 (τ (x)) + g ◦ τ −1 (τ (x)) (τ (t) − τ (x)) Z τ (t)
00 + (τ (t) − u) g ◦ τ −1 (u)du.

(5.1)

τ (x)

If we consider the change of variable u = τ (y), it follows Z

τ (t) τ (x)

(τ (t) − u) g ◦ τ −1


00

(u)du =

Z

x

t

(τ (t) − τ (y)) g ◦ τ −1

765


00

(τ (y)) τ 0 (y)dy,

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Bernstein-Stancu type operators but g ◦ τ −1 therefore Z τ (t) τ (x)

(τ (y)) =


00

(u)du

(τ (t) − u) g ◦ τ −1

t

=

Z

τ (t)

=

Z

x


00

g 00 (y) dy − (τ (t) − τ (y)) 0 τ (y)

τ (x)

(τ (t) − u)

Z

(τ (t) − τ (y))

2 du

[τ 0 (τ −1 (u))]

1 g 00(y)τ 0 (y) − g 0 (y)τ 00(y) · , τ 0 (y) (τ 0 (y))2

t

x

g 00 (τ −1 (u))

9

−

Z

g 0 (y)τ 00 (y) dy (τ 0 (y))2

τ (t) τ (x)

(τ (t) − u)

g 0 (τ −1 (u))τ 00 (τ −1 (u)) [τ 0 (τ −1 (u))]3

du.

(5.2)

From (5.1) and (5.2) we can write g(t) = g(x) + g ◦ τ −

Z

τ (t)

τ (x)


−1 0

(τ (x)) (τ (t) − τ (x)) +

(τ (t) − u)

Z

g 0 (τ −1 (u))τ 00 (τ −1 (u)) [τ 0 (τ −1 (u))]3

τ (t)

τ (x)

(τ (t) − u)

g 00 (τ −1 (u)) du [τ 0 (τ −1 (u))]2 (5.3)

du.

We define
C˜nτ (f ; x) = Cnτ (f ; x) + f (x) − f ◦ τ −1 (an τ (x)).

From Lemma 3.1 it follows

C˜nτ (e0 ; x) = 1 and C˜nτ (τ ; x) = Cnτ (τ ; x) + τ (x) − an τ (x) = τ (x).

Now applying C˜nτ to both side of the relation (5.3) we can write  Z τ (t)  g 00(τ −1 (u)) τ C˜nτ (g; x) = g(x) + Cn (τ (t) − u) du [τ 0 (τ −1 (u))]2 τ (x) Z an τ (x) g 00(τ −1 (u)) − (an τ (x) − u) du [τ 0 (τ −1 (u))]2 τ (x)  Z τ (t)  g 0 (τ −1 (u))τ 00(τ −1 (u)) τ − Cn du (τ (t) − u) [τ 0 (τ −1 (u))]3 τ (x) Z an τ (x) g 0 (τ −1 (u))τ 00(τ −1 (u)) du. + (an τ (x) − u) [τ 0 (τ −1 (u))]3 τ (x)

Since inf x∈[0,1] τ 0 (x) ≥ a, a ∈ R+ and τ is strictly increasing on the interval (0, 1), we obtain  00  1 kg k kg 0 k · kτ 00 k ˜τ τ + Cn (τ ; x) − g(x) ≤ µn,2 (x) 2 a2 a3  00  1 kg 0 k · kτ 00 k 2 kg k + (an τ (x) − τ (x)) + 2 a2 a3  00   kg k kg 0k · kτ 00 k 1 2 2 2 ≤ δ (x) + τ (x)(1 − an ) + 2 n,τ a2 a3  00 0 00  kg k kg k · kτ k 2 ≤ δn,τ (x) + . a2 a3

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By Lemma 3.4, it follows
|C˜nτ (g; x)| ≤ |Cnτ (g; x)| + |g(x)| + | g ◦ τ −1 (an τ (x))| ≤ 3kgk. For f ∈ C[0, 1] and g ∈ W2 [0, 1], we can write |Cnτ (f ; x) − f (x)|
= C˜nτ (f ; x) − f (x) + f ◦ τ −1 (an τ (x)) − f (x)

≤ |C˜nτ (f − g; x)| + |C˜nτ (g; x) − g(x)| + |g(x) − f (x)|

+ f ◦ τ −1 (an τ (x)) − f ◦ τ −1 (τ (x))

2 (x) 2 (x)
δn,τ δn,τ 00 kg k + kτ 00 k kg 0k + ω f ◦ τ −1 ; (1 − an )τ (x) . 2 3 a a  00 Let C := max 4, a12 , kτa3 k . Then

≤ 4kf − gk +


2 (x)kgkW 2[0,1] + ω f ◦ τ −1 ; (1 − an )τ (x) . |Cnτ (f ; x) − f (x)| ≤ C kf − gk + δn,τ



Using the following result (see [1]) ω f ◦ τ −1 ; t ≤ ω f ; at , the theorem is proved.

To describe our next result, we recall the definitions of the Ditzian-Totik first order p modulus of smoothness and the K-functional [9]. Let ϕτ (x) := τ (x)(1 − τ (x)) and f ∈ C[0, 1]. The first order modulus of smoothness is given by       hϕτ (x) hϕτ (x) hϕτ (x) ωϕτ (f ; t) = sup −f x− ∈ [0, 1] . (5.4) f x + ,x± 2 2 2 0 0),

g∈Wϕτ [0,1]

(5.5)

where Wϕτ [0, 1] = {g : g ∈ AC[0, 1], kϕτ g 0 k < ∞} and AC[0, 1] is the class of all absolutely continuous functions on [0, 1]. It is well known ([9], p.11 ) that there exists a constant C > 0 such that Kϕτ (f ; t) ≤ Cωϕτ (f ; t).

(5.6)

Now, we establish a direct approximation theorem by means of Ditzian-Totik modulus of smoothness. p Theorem 5.2. Let f ∈ C[0, 1] and ϕτ (x) = τ (x)(1 − τ (x)), then for every x ∈ (0, 1), we have   δ (x) n,τ τ ˜ ϕτ f ; |Cn (f ; x) − f (x)| ≤ Cω , ϕτ (x) where C˜ is a constant independent of n and x.

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11

Proof. Using the representation

g(t) = g ◦ τ −1 (τ (t)) = g ◦ τ −1 (τ (x)) +

Z

τ (t)

τ (x)


0 g ◦ τ −1 (u)du,

we get |Cnτ (g; x) − But,

Z g(x)| = Cnτ

τ (t) τ (x)

g◦τ


−1 0

 (u)du .

(5.7)

Z Z Z t τ (t) t g 0 (y) 0 (y)
g ϕ (y) τ −1 0 0 0 g◦τ (u)du = τ (y)dy = · 0 τ (y)dy 0 τ (x) x τ (y) x ϕτ (y) τ (y) Z kϕτ g 0 k t τ 0 (y) ≤ dy , a x ϕτ (y)

and Z

t x

(5.8)

Z   τ 0 (y) t 1 1 0 p dy ≤ +p τ (y)dy ϕτ (y) τ (y) 1 − τ (y) x p p p
p ≤ 2 τ (t) − τ (x) + 1 − τ (t) − 1 − τ (x)   1 1 p p = 2 |τ (t) − τ (x)| p +p τ (t) + τ (x) 1 − τ (t) + 1 − τ (x)   1 1 < 2|τ (t) − τ (x)| p +p τ (x) 1 − τ (x) √ 2 2|τ (t) − τ (x)| ≤ . (5.9) ϕτ (x)

From relations (5.7)-(5.9) and using Cauchy-Schwarz inequality, we obtain √ kϕτ g 0 k τ |Cnτ (g; x) − g(x)| ≤ 2 2 C (|τ (t) − τ (x)|; x) aϕτ (x) n √ kϕτ g 0 k  τ
1/2 ≤2 2 Cn (τ (t) − τ (x))2 ; x aϕτ (x) √ kϕτ g 0 k ≤2 2 δn,τ (x). aϕτ (x)

(5.10)

Using Lemma 3.4 and (5.10) it follows |Cnτ (f ; x) − f (x)| ≤ |Cnτ (f − g; x)| + |f (x) − g(x)| + |Cnτ (g; x) − g(x)|   δn,τ (x) 0 ≤ C kf − gk + kϕτ g k , ϕτ (x)  √ where C = max 2, 2 a 2 . Taking infimum on the right hand side of the above inequality over all g ∈ Wϕτ [0, 1], we get   δn,τ (x) τ |Cn (f ; x) − f (x)| ≤ CKϕτ f ; . ϕτ (x) Using the relation (5.6) this theorem is proven.

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Acknowledgment This work was supported by the Dong-A University research fund.

References [1] T. Acar, A. Aral and I. Ra¸sa, Modified Bernstein-Durrmeyer operators, Gen. Math., 22(1) (2014), 27–41. [2] A. Aral, D. Inoan and I. Ra¸sa, On the generalized Szasz-Mirakyan Operators, Results Math., 65(3-4) (2014), 441–452. [3] S. Bernstein, D´emonstration du th´eor`eme de Weierstrass fond´ee sur le calcul de probabilit´es, Commun. Kharkov Math. Soc., 13 (1912), 1–2. [4] D. C´ardenas-Morales, P. Garrancho and F. J. Mu˜ noz-Delgado, Sharpe preserving approximation by Bernstein-type operators which fix polynomials, Appl. Math. Comput., 182(2) (2006), 1615–1622. [5] D. C´ardenas-Morales, P. Garrancho and I. Ra¸sa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 62(1) (2011), 158–163. [6] D. C´ardenas-Morales, P. Garrancho and I. Ra¸sa, Approximation properties of Bernstein-Durrmeyer type operators, Appl. Math. Comput., 232 (2014), 1–8. [7] V. A. Cleciu, Bernstein-Stancu operators, Studia Univ. Babe¸s-Bolyai, Mathematica, 52(4) (2007), 53–65. [8] V. A. Cleciu, Approximation properties of a class of Bernstein-Stancu type operators, in Numerical Analysis and Approximation Theory, 171–178, Casa Cˇart¸ii de ¸stiint¸ˇa, Cluj-Napoca, 2006. [9] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer, New York, 1987. [10] H. Gonska, P. Pit¸ul and I. Ra¸sa, On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators, in Numerical Analysis and Approximation Theory, 55–80, Casa Cˇart¸ii de ¸stiint¸ˇa, Cluj-Napoca, 2006. [11] H. Gonska, P. Pit¸ul and I. Ra¸sa, General King-type operators, Results Math., 53(3-4) (2009), 279–286. [12] J. P. King, Positive linear operators which preserve x2 , Acta Math. Hungar., 99 (2003), 203–208. [13] D. Micl˘au¸s, The revision of some results for Bernstein-Stancu type operators, Carpathian J. Math., 28(2) (2012), 289–300.

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13

[14] J. Peetre, A Theory of Interpolation of Normed Spaces, Notas de Matem´atica, No. 39, Rio de Janeiro, 1968. [15] D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl., 13 (1968), 1173–1194.

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A NOTE ON HERMITE POLYNOMIALS TAEKYUN KIM AND DAE SAN KIM

Abstract. In this paper, we consider linear differential equations satisfied by the generating function for Hermite polynomials and derive some new identities involving those polynomials.

1. Introduction The Hermite polynomials form a Sheffer sequence and are given by the generating function ∞ X 2 Hn (x) n (1.1) e2xt−t = t , (see [1–8, 10, 13, 14]) . n! n=0 By using Taylor series, we get  n  2 ∂ e(2xt−t ) Hn (x) = ∂t t=0   n  ∂ x2 −(x−t)2 = e e ∂t  n t=0  2 ∂ n x2 e−(x−t) = (−1) e ∂x t=0 n 2 n x2 d e−x , (n ≥ 0) , (see [1–15, 18]) . = (−1) e dxn The Hermite polynomials can be represented by the Contour integral as follows: ˛ 2 n! e−t +2tz t−n−1 dt, (1.2) Hn (z) = 2πi where the Contour encloses the origin and is traversed in a counterclockwise direction (see [2, 8, 11, 13]). The probabilists’ Hermite polynomials are given by the generating function n 2 x2 n x d (1.3) Hn∗ (x) = (−1) e 2 e− 2 n  dx n d = x− · 1, (see [10]) . dx The physicists’ Hermite polynomials are also given by n 2 d 2 n (1.4) Hn (x) = (−1) ex e−x n dx  n d = 2x − · 1 (see [20]) . dx 2010 Mathematics Subject Classification. 05A19, 11B83, 33C45, 34A30. Key words and phrases. Hermite polynomials, linear differential equation. 1

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TAEKYUN KIM AND DAE SAN KIM

Thus, by (1.3) and (1.4), we get √  n (1.5) Hn (x) = 2 2 Hn∗ 2x ,

Hn∗

−n 2

(x) = 2

 Hn

x √ 2

 ,

where n ≥ 0 (see [9, 11, 12, 15, 18]). The first several Hermite polynomials are H0 (x) = 1, H1 (x) = 2x, H2 (x) = 4x2 −2, H3 (x) = 8x2 −12x, H4 (x) = 16x4 −48x2 +12, H5 (x) = 32x5 −160x3 +120x, H6 (x) = 64x6 − 480x4 + 720x2 − 120, ... The probabilists’ Hermite polynomials are solutions of the differential equation:  x 2 0 1 2 e− 2 u0 + λe− 2 x u = 0, where λ is a constant, with the boundary conditions that u should be polynomially bounded at infinity. The generating function of the probabilists’ Hermite polynomials is given by ∞ X t2 tn Hn∗ (x) , (see [12, 15, 18]) . (1.6) ext− 2 = n! n=0 (ν)

The Hermite polynomials Hn (x) of variance ν form an Appell sequence and are defined by the generating function ∞ (ν) X H (x) k

(1.7)

k=0

k!

tk = ext−

νt2 2

,

(see [12]) .

Thus, by (1.7), we get  m  X 2m + 1 (2m − 2l)!  ν m−l (ν) 2m+1 H2l+1 (x) , (1.8) x = (m − l)! 2 2l + 1 l=0

and (1.9)

x

2m

=

 m  X 2m (2m − 2l)!  ν m−l l=0

2l

(m − l)!

2

(ν)

H2l (x) ,

(see [12]) .

The Hermite polynomials have been studied in probability, combinatorics, numerical analysis, finite element methods, physics and system theory (see [1–15, 18]). Recently, Kim has studied nonlinear differential equations arising from FrobeniusEuler numbers and polynomials. In this paper, we consider linear differential equations arising from Hermite polynomials of variance ν and give some new and explicit identities for those polynomials. 2. Hermite polynomials of variance ν Let (2.1)

F = F (t : x, ν) = ext−

νt2 2

.

From (2.1), we note that (2.2)

F (1) =

d F (t : x, ν) dt

= (x − νt) ext−

νt2 2

= (x − νt) F,

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A NOTE ON HERMITE POLYNOMIALS

3

 d (1)  2 F = −ν + (x − tν) F, dt   d 3 = F (2) = −3ν (x − νt) + (x − νt) F, dt

(2.3)

F (2) =

(2.4)

F (3)

and  d (3)  2 2 4 F = 3ν − 6ν (x − νt) + (x − νt) F. dt Continuing this process, we set  N d (2.6) F (t : x, ν) F (N ) = dt ! N X i = ai (N, ν) (x − νt) F, (2.5)

F (4) =

i=0

where N ∈ N ∪ {0}. From (2.6), we have (2.7)

d (N ) F dt N X i−1 = ai (N, ν) i (x − νt) (−ν) F

F (N +1) =

i=0

+

N X

i

ai (N, ν) (x − νt) F (1) .

i=0

By (2.2) and (2.7), we easily get n N +1 N (2.8) F (N +1) = −νa1 (N, ν) + aN (N, ν) (x − νt) + aN −1 (N, ν) (x − νt) ) N −1 X i + (− (i + 1) νai+1 (N, ν) + ai−1 (N, ν)) (x − νt) F. i=1

By replacing N by (N + 1) in (2.6), we get (2.9)

F

(N +1)

=

N +1 X

! i

ai (N + 1, ν) (x − νt)

F.

i=0

From (2.8) and (2.9), we can derive the following equations: (2.10)

a0 (N + 1, ν) = −νa1 (N, ν) ,

(2.11)

aN (N + 1, ν) = aN −1 (N, ν) ,

(2.12)

aN +1 (N + 1, ν) = aN (N, ν)

and (2.13)

ai (N + 1, ν) = − (i + 1) νai+1 (N, ν) + ai−1 (N, ν) ,

where 1 ≤ i ≤ N − 1. It is not difficult to show that (2.14)

F = F (0) = a0 (0, ν) F.

Thus, by (2.14), we get (2.15)

a0 (0, ν) = 1.

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From (2.2) and (2.6), we note that (x − νt) F = F (1) = (a0 (1, ν) + a1 (1, ν) (x − νt)) F.

(2.16)

Thus, by comparing the coefficients on both sides of (2.16), we get (2.17)

a0 (1, ν) = 0,

a1 (1, ν) = 1.

From (2.11), (2.12), (2.15) and (2.17), we have aN (N + 1, ν) = aN −1 (N, ν) = · · · = a0 (1, ν) = 0,

(2.18) and

aN +1 (N + 1, ν) = aN (N, ν) = · · · = a1 (1, ν) = 1.

(2.19)

Therefore, we obtain the following theorem. Theorem 1. The linear differential equations  N d (N ) F (t : x, ν) F = dt ! N X i = ai (N, ν) (x − νt) F,

(N ∈ N ∪ {0})

i=0

has a solution F = F (t : x, ν) = ext−

νt2 2

, where

a0 (N, ν) = −νa1 (N − 1, ν) , aN −1 (N, ν) = aN −2 (N − 1, ν) = · · · = a1 (2, ν) = a0 (1, ν) = 0, aN (N, ν) = aN −1 (N − 1, ν) = · · · = a1 (1, ν) = a0 (0, ν) = 1, and ai (N, ν) = − (i + 1) νai+1 (N − 1, ν) + ai−1 (N − 1, ν) ,

(1 ≤ i ≤ N − 2) .

Example. (1) N = 3, i = 1. By (2.13), we get a1 (3, ν) = −2νa2 (2, ν) + a0 (2, ν) = −2ν − ν = −3ν. (2) N = 4, 1 ≤ i ≤ 2. By (2.13), we have a1 (4, ν) = 0,

a2 (4, ν) = −6ν.

(3) N = 5, 1 ≤ i ≤ 3. By (2.13), we get a1 (5, ν) = 15ν 2 ,

a2 (5, ν) = 0,

a3 (5, ν) = −10ν.

(4) N = 6, 1 ≤ i ≤ 4. From (2.13), we have a1 (6, ν) = 0,

a2 (6, ν) = 45ν 2 ,

a3 (6, ν) = 0,

a4 (6, ν) = −15ν.

Thus, we obtain the following result.

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5

Remark. The matrix (ai (j, ν))0≤i,j≤6 is given by 3 4 5 6  0 1 2 2 0 −15ν 3 0  1 0 −ν 0 3ν  1 0 −3ν 0 15ν 2 0 1   2 1 0 −6ν 0 45ν 2  1 0 −10ν 0 3   1 0 −15ν 4  1 0 0 5 1 6

      .       

From (1.7), we note that F = F (t : x, ν) = ext−

(2.20)

∞ X

=

(ν)

Hk (x)

k=0

νt2 2

tk . k!

Thus, by (2.20), we get F (N ) =

(2.21)

= = =

N



d dt ∞ X

F (t : x, ν) (ν)

Hk (x) (k)N

k=N ∞ X

(ν)

k=0 ∞ X

(ν)

tk−N k!

Hk+N (x) (k + N )N

tk (n + k)!

tk . k!

Hk+N (x)

k=0

By Theorem 1, we easily get (2.22) F (N ) =

N X

! i

ai (N, ν) (x − νt)

F

i=0

=

N X

ai (N, ν)

(N ∞ X X k=0

=

(i)m x

ai (N, ν)

i=0

 ∞ X N X k=0

∞

i−m

m

(−ν)

m=0

i=0

=

∞ X



i=0

l=0

k   X k l=0

ai (N, ν)

tl tm X (ν) Hl (x) m! l!

l

) (i)k−l (−ν)

k X l=max{0,k−i}

k−l

x

i+l−k

(ν) Hl

(x)

tk k!

    tk k k−l i+l−k (ν) (i)k−l (−ν) x Hl (x) .  k! l

Therefore, by (2.21) and (2.22), we obtain the following theorem.

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Theorem 2. For k, N ∈ N ∪ {0}, we have (ν)

Hk+N (x) =

N X

k X

ai (N, ν)

i=0

l=max{0,k−i}

  k k−l i+l−k (ν) (i)k−l (−ν) x Hl (x) . l

It is easy to show that 

(ν)

(2.23)

x−ν

Hk+1 (x) =

∂ ∂x



(ν)

Hk (x) .

Thus, by (2.23), we have 

(ν)

(2.24)

x−ν

Hk+N (x) =

∂ ∂x

N

(ν)

Hk (x) ,

(N ∈ N ∪ {0}) .

From Theorem 2, we note that N  ∂ (ν) (2.25) Hk (x) x−ν ∂x =

N X

ai (N, ν)

i=0 ∂ ∂x x

k X l=max{0,k−i}

  k k−l i+l−k (ν) (i)k−l (−ν) x Hl (x) , l

∂ x ∂x

where − = identity. Now, we observe explicit determination of ai (j, ν). From (2.12) and (2.13), we can derive the following equations: (2.26) (2.27)

aN (N, ν) = 1, aN −2 (N, ν) = − (N − 1) νaN −1 (N − 1, ν) + aN −3 (N − 1, ν) = − (N − 1) νaN −1 (N − 1, ν) − (N − 2) νaN −2 (N − 2, ν) +aN −4 (N − 2, ν) .. . = − (N − 1) νaN −1 (N − 1, ν) − (N − 2) νaN −2 (N − 2, ν) − · · · − 2νa2 (2, ν) + a0 (2, ν) = − (N − 1) νaN −1 (N − 1, ν) − (N − 2) νaN −2 (N − 2, ν) − · · · − 2νa2 (2, ν) − νa1 (1, ν) = −ν

N −1 X

iai (i, ν) ,

i=1

(2.28)

aN −4 (N, ν) = − (N − 3) νaN −3 (N − 1, ν) + aN −5 (N − 1, ν) = − (N − 3) νaN −3 (N − 1, ν) − (N − 4) νaN −4 (N − 2, ν) +aN −6 (N − 2, ν) .. . = − (N − 3) νaN −3 (N − 1, ν) − (N − 4) νaN −4 (N − 2, ν)

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− · · · − 2νa2 (4, ν) + a0 (4, ν) = − (N − 3) νaN −3 (N − 1, ν) − (N − 4) νaN −4 (N − 2, ν) − · · · − 2νa2 (4, ν) − νa1 (3, ν) = −ν

N −3 X

iai (i + 2, ν) ,

i=0

and (2.29)

aN −6 (N, ν) = − (N − 5) νaN −5 (N − 1, ν) + aN −7 (N − 1, ν) = − (N − 5) νaN −5 (N − 1, ν) − (N − 6) νaN −6 (N − 2, ν) +aN −8 (N − 2, ν) .. . = − (N − 5) νaN −5 (N − 1, ν) − (N − 6) νaN −6 (N − 2, ν) − · · · − 2νa2 (6, ν) − νa1 (5, ν) = −ν

N −5 X

iai (i + 4, ν) .

i=1

Continuing in this fashion, for l with 1 ≤ l ≤ aN −2l (N, ν) = −ν

(2.30)

N −2l+1 X

 N −1  , 2

iai (i + 2l − 2, ν) .

i=1

By (2.26), (2.27), (2.28), (2.29) and (2.30), we get aN −2 (N, ν) = −ν

(2.31)

N −1 X

i1 ,

i1 =1

aN −4 (N, ν) = −ν

(2.32)

N −3 X

i2 ai2 (i2 + 2, ν)

i2 =1 2

= (−ν)

N −3 iX 2 +1 X

i2 i1 ,

i2 =1 i1 =1

aN −6 (N, ν) = −ν

(2.33)

N −5 X

i3 ai3 (i3 + 4, ν)

i3 =1 3

= (−ν)

N −5 iX 3 +1 iX 2 +1 X

i3 i2 i1 ,

i3 =1 i2 =1 i1 =1

and (2.34)

l

aN −2l (N, ν) = (−ν)

N −2l+1 l +1 X iX il =1

where 1 ≤ l ≤

il−1 =1

···

iX 2 +1

il · il−1 · · · i1 ,

i1 =1

 N −1  . 2

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By (2.11) and (2.13), we easily get (2.35) aN −1 (N, ν) = aN −2 (N − 1, ν) = aN −3 (N − 2, ν) = · · · = a0 (1, ν) = 0, (2.36) aN −3 (N, ν) = − (N − 2) νaN −2 (N − 1, ν) + aN −4 (N − 1, ν) = aN −4 (N − 1, ν) .. . = a0 (3, ν) = −νa1 (2, ν) = −νa0 (1, ν) = 0, (2.37) aN −5 (N, ν) = − (N − 4) νaN −4 (N − 1, ν) + aN −6 (N − 1, ν) = aN −6 (N − 1, ν) .. . = a0 (5, ν) = −νa1 (4, ν) = 0, (2.38) aN −7 (N, ν) = − (N − 6) νaN −6 (N − 1, ν) + aN −8 (N − 1, ν) .. . = a0 (7, ν) = −νa1 (6, ν) = 0, and    N . (2.39) aN −(2l−1) (N, ν) = 0, 1≤l≤ 2 Therefore, we obtain the following theorem. Theorem 3. For N ∈ N ∪ {0}, we have aN −2l (N, ν) = (−ν)

l

N −2l+1 l +1 X iX il =1

where 1 ≤ l ≤ Also,

il−1 =1

···

iX 2 +1

il il−1 · · · i1 ,

i1 =1

 N −1  . 2  aN −(2l−1) (N, ν) = 0,

if 1 ≤ l ≤

 N . 2

References 1. P. Barry, Eulerian polynomials as moments, via exponential Riordan arrays, J. Integer Seq. 14 (2011), no. 9, Article 11.9.5, 14. MR 2859989 2. L. Carlitz, The Product of Two Eulerian Polynomials, Math. Mag. 36 (1963), no. 1, 37–41. MR 1571266 3. G. Ferraro, Euler’s treatises on infinitesimal analysis: Introductio in analysin infinitorum, institutiones calculi differentialis, institutionum calculi integralis, Euler reconsidered, Kendrick Press, Heber City, UT, 2007, pp. 39–101. MR 2384378 4. D. Foata, Eulerian polynomials: from Euler’s time to the present, The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, 2010, pp. 253–273. MR 2744266 (2012f:01013)

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Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO. 4, 2017

On A System of Rational Difference Equations, Ali Gelisken,………………………593 A Numerical Approach Based on Subdivision Schemes for Solving Non-Linear Fourth Order Boundary Value Problems, Ghulam Mustafa, Muhammad Abbas, Syeda Tehmina Ejaz, Ahmad Izani Md Ismail, and Faheem Khan,………………………………………………….607 On Stability of Quintic Functional Equations in Random Normed Spaces, Afrah A.N. Abdou, Y. J. Cho, Liaqat A. Khan, and S. S. Kim,……………………………………………624 Generalized composition operators on Zygmund type spaces and Bloch type spaces, Juntao Du and Xiangling Zhu,……………………………………………………………………635 Convergence and Error Estimates for the Series Solutions of Higher-Order Differential Equations, Junchi Ma, Songxin Liang, Xiaolong Zhang, and Li Zou,………………..647 On the Generalized Z-Algorithm for the Neutral Stochastic Functional Differential Equations with Infinite Delay, Xiangxing Tao and Songyan Zhang,…………………………….660 An Improved Generalized Parameterized Inexact Uzawa Method for Singular Saddle Point Problems, Li-Tao Zhang and Li-Min Shi,……………………………………………..671 Identities Involving Bessel Polynomials Arising From Linear Differential Equations, Taekyun Kim and Dae San Kim,…………………………………………………………………684 On Existence and Comparison Results for Solutions to Stochastic Functional Differential Equations in the G-Framework, Faiz Faizullah, Matloob-Ur-Rehman, Muhammad Shahzad, and M. Ikhlaq Chohan,……………………………………………………………………..693 Interval-Valued Intuitionistic Fuzzy Choquet Integral Operators Based On Archimedean t-Norm and Their Calculations, San-Fu Wang,………………………………………………..703 Approximate Bi-Homomorphisms and Bi-Derivations in Intuitionistic Fuzzy Ternary Normed Algebras, Javad Shokri, Choonkil Park, and Dong Yun Shin,…………………………713 On New Refinements and Applications of Efficient Quadrature Rules Using n-Times Differentiable Mappings, A. Qayyum, M. Shoaib, and I. Faye,……………………….723 Duality in Multiobjective Nonlinear Programming Under Generalized Second Order (F,b,𝜙,𝜌,𝜃) − Univex Functions, Falleh R. Al-Solamy and Meraj Ali Khan,………………………740

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO. 4, 2017 (continued) Stability of Fractional Differential Equation With Boundary Conditions, S. Rajan, P. Muniyappan, Choonkil Park, Sungsik Yun, and Jung Rye Lee,………………………750 Bernstein-Stancu Type Operators Which Preserve Polynomials, Young Chel Kwun, Ana-Maria Acu, Arif Rafiq, Voichita Adriana Radu, Faisal Ali, and Shin Min Kang,………………758 A Note on Hermite Polynomials, Taekyun Kim and Dae San Kim,……………………...771

Volume 23, Number 5 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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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 (fifteen 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:

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Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $750, Electronic OPEN ACCESS. Individual:Print $380. For any other part of the world add $140 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2017 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics

Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

Carlo Bardaro Dipartimento di Matematica e Informatica 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.

Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities

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.

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.

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

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

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

Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

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Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design

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

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

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.

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

Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

Tian-Xiao He Department of Mathematics and Computer Science P.O. Box 2900, Illinois Wesleyan University Bloomington, IL 61702-2900, USA Tel (309)556-3089 Fax (309)556-3864 [email protected] Approximations, Wavelet, Integration Theory, Numerical Analysis, Analytic Combinatorics

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

Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected] Approximation Theory (Positive Linear Operators)

J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected] Partial Differential Equations, Semigroups of Operators H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048

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Los Angeles, CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory, Splines, Wavelets, Neural Networks

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

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

On new 2-convergent di¤erence BK-spaces Sinan ERCAN and Çi¼gdem A. BEKTA¸S Department of Mathematics, Firat University, 23119, Elaz¬¼g-TURKEY [email protected]/[email protected] (2  ¢)

2

and 0 (  ¢) which are  -spaces of non-absolute type and we prove that these spaces are linearly isomorphic to the spaces  and 0  respectively. Moreover,

In this paper, we introduce the spaces

we give some inclusion relations and compute the ¡, ¡ and ¡duals of these spaces. We also determine 2 2 the Schauder basis of the (  ¢) and 0 (  ¢) Lastly we give some matrix transformations between of these spaces and others.

2010 Mathematics Subject Classi…cation: 46A45, 46B20 Key words: 2 -convergence,  -spaces, ¡ ¡ and ¡ duals, matrix

mappings, di¤erence se-

quence spaces

1

Introduction

A sequence space is de…ned to be a linear space of real or complex sequences. Let  denote the spaces of all complex sequences. If  2 , then we simply write  = ( ) instead of  = ( )1 =0 . Let  be a sequence space. If  is a Banach space and   :  !    () = 

( = 1 2 )

is a continuous for all ,  is called a ¡space. We shall write 1   and 0 for the sequence spaces of all bounded, convergent and null sequences, respectively, which are ¡spaces with the norm given by kk1 = sup j j for all  2 N For a sequence space  the matrix domain  of an in…nite matrix  de…ned by  = f = ( ) 2  :  2 g

(1)

which is a sequence space. We denote the collection of all …nite subsets of N by F. M. Mursaleen and A. K. Noman [9] introduced the sequence spaces 1   and 0 as the sets of all  ¡   ¡  and  ¡  sequences as follows;

where ¤ () =

1 

 P

=0

1

= f 2  : sup j¤ ()j  1g



= f 2  : lim ¤ () g

0

= f 2  : lim ¤ () = 0g



!1

!1

( ¡ ¡1 )    2 N Also they generalized  and 0 spaces de…ning  (¢),

0 (¢) spaces using the di¤erence operator. They studied some properties of these spaces in [8]. N. L. Braha and F. Ba¸sar introduced the in…nite matrix  () = f ()g1 =0 such as; ( 2 ¢  ¢  0 ·  · ;  () = 0  for all   2 N and they de…ned  (1 )   () and  (0 ) spaces in [11] as follows; ½ ¾  (1 ) =  2  : sup j( ) j  1   n o  () =  2  : 9 2 C 3 lim ( ) =    n o  (0 ) =  2  : lim ( ) = 0 

where ( ) =

1 ¢

 P

=0

¡ 2 ¢ ¢   . They examined some properties of these spaces. In literature,

some authors have constructed new sequence spaces by using matrix domain of in…nite matrix and have introduced some topological properties. (see [2], [4], [12]) 1

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2

The sequence spaces (2  ¢) and 0 (2  ¢)

In this section, we de…ne the sequence spaces (2  ¢) and 0 (2  ¢) as follows; n o (2  ¢) =  2  : lim ¤2 ()  !1

n o 0 (2  ¢) =  2  : lim ¤2 () = 0 !1

where ¤2 () =

1 ¢

 ¡ ¢ P ¢2  ( ¡ ¡1 ) for all   2 N ¢ denotes the di¤erence operator. i.e.,

=0

¢0  =  , ¢ =  ¡ ¡1 , ¢2  =  ¡ 2¡1 + ¡2 and ¢ =  ¡ ¡1 .  = ( )1 =0 is a strictly increasing sequence of positive reals tending to in…nity, that is 0  0  1  and  ! 1 as  ! 1 and +1 ¸ 2 for all  2 N. Here and in sequel, we use the convention that any term with a negative subscript is¡ equal ¢ to naught. e.g. ¡1 = ¡2 = 0 and ¡1 = 0 On the other hand, we de…ne the matrix ¤2 = 2 for all   2 N by 8 2 ¢ ( ¡+1 ) > ;    < ¢ 2 ¢2   = (2)  =  ¢ ; > : 0;   

The equality can be eaisly seen from

¤2 () =

 1 X¡ 2 ¢ ¢  ( ¡ ¡1 ) ¢ =0

(3)

for all   2 N and every  = ( ) 2  Then it leads us together with (1) to the fact that ¡ ¢ ¡ ¢ 0 2  ¢ = (0 )¤2 and  2  ¢ = ()¤2 .

(4)

2 The matrix ¤2 = 2 is a triangle, i.e., 2 6= 0 and  ) for all   2 N. Further, ¡ 2 ¢ ©= 0¡ ( ¢ª for any sequence  = ( ) we de…ne the sequence   =  2 as the ¤2 -transform of , i.e., ¡ ¢  2 = ¤2 () and so we have that

X ¢2 ( ¡ +1 ) ¡ ¢ ¡1 ¢2   2 =  +  ¢ ¢ =0

(5)

for  2 N Here and in what follows, the summation running from 0 to  ¡ 1 is equal to zero when  = 0 Also it can be written from (3) with (5) for  2 N such as;  ¡ ¢ X ¢2   2 = ( ¡ ¡1 )  ¢ =0

Theorem 1 0 (2  ¢) and (2  ¢) are BK-spaces with the norm kk(0 )

¤2

= kk()

¤2

¯ ¯ = sup ¯¤2 ()¯  

Proof. We know that  and 0 are ¡spaces with their natural norms from [6]. (4) holds and ¤2 = 2 is a triangle matrix and from Theorem 4312 of Wilansky [1], we derive that 0 (2  ¢) and (2  ¢) are ¡spaces. This completes the proof. Remark 2 The absolute property does not hold on the 0 (2  ¢) and (2  ¢) spaces. For instance, if we take jj = (j j) we hold kk() 2 6= kjjk() 2 Thus, the space 0 (2  ¢) and (2  ¢) are BK-space ¤ ¤ of non-absolute type. Theorem 3 The sequence spaces 0 (2  ¢) and (2  ¢) of non-absolute type are linearly isomorphic to the spaces 0 and  respectively, that is 0 (2  ¢) » = 0 and (2  ¢) » =  2

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Proof: We only consider 0 (2  ¢) » = 0 and others will prove similarly. To prove the theorem we must show the existence of linear bijection operator between 0 (2  ¢)¡and ¢ 0  Hence, let de…ne the linear operator with the notation (5), from 0 (2  ¢) and 0 by  !  2 =   ¡ ¢ Then   =  2 = ¤2 () 2 0 for every  2 0 (2  ¢) Also, the linearity of  is clear. Further, it is trivial that  = 0 whenever   = 0. Hence  ©is injective. ¡ ¢ª Let  = ( ) 2 0 and de…ne the sequence  =  2 by   X ¡ ¢ X ¢  2 = (¡1)¡ 2   ¢  =0 =¡1

and we have

(6)

 X ¡ ¢ ¡ ¢ ¡ ¢  2 ¡ ¡1 2 = (¡1)   ¢2  =¡1

Thus, for every  2 N we have by (5) that ¤2 () =

 1 X [¢ (  ¡ ¡1 ¡1 )] =  ¢ =0

This shows that ¤2 () =  and since  2 0  we obtain that ¤2 () 2 0  Thus we deduce that  2 0 (2  ¢) and   =  Hence  is surjective. Further, we have for every  2 0 (2  ¢) that ° ¡ ¢° k k0 = k k1 = ° 2 ° 2

1

° ° = °¤2 ()°

1

= kk(

0 )¤2

which means that 0 (  ¢) and 0 are linearly isomorphic.

3

Some inclusion relations

¡ ¢ ¡ ¢ Theorem 4 The inclusion 0 2  ¢ ½  2  ¢ strictly holds.

¡ ¢ ¡ ¢ Proof. 0 2  ¢ ½  2  ¢ is clear. To show strict, consider the sequence  = ( ) de…ned by  =  + 1 for all  2 N Then we obtain that ¤2 () =

 1 X¡ 2 ¢ ¢  ( ¡ ¡1 ) = 1; ( 2 N) ¢ =0

¡ ¢ ¡ ¢ for  2 N which shows that ¤2 () 2  ¡ 0  Thus, the sequence  is in  2  ¢ but not in 0 2  ¢  ¡ ¢ ¡ ¢ Hence the inclusion 0 2  ¢ ½  2  ¢ is strict and this completes the proof.

¡ ¢ Theorem 5 The inclusion  ½ 0 2  ¢ strictly holds.

¡ ¢ Proof. Let  2  Then, ¤2 () 2 0  This shows that  2 0 2  ¢  Hence, the inclusion  ½ ¢ p ¡ 0 2  ¢ holds. Then, consider the sequence  = ( ) de…ned by  =  + 1 for  2 N It is trivial ¡ ¢ that  2   On the other hand, it can easily be seen that ¤2 () 2 0 and  2 0 2  ¢ Consequently, ¡ ¢ ¡ ¢ the sequence  is in 0 2  ¢ but not in  We therefore deduce that the inclusion  ½ 0 2  ¢ is strict. ¡ ¢ ¡ ¢ Corollary 6 0 ½ 0 2  ¢ and  ½  2  ¢ strictly hold.

¡ ¢ Theorem 7 Although the spaces 1 and 0 2  ¢ overlap, the space 1 does not include the space ¡ 2 ¢ 0   ¢ 

¡ ¢ Proof. It can be seen from the sequence  which was de…ned in Theorem 5, is in 0 2  ¢ but not in 1  3

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Lemma 8  2 (1 : 0 ) if and only if lim

P 

j j = 0

¡ ¢ Theorem 9 The inclusion 1 ½ 0 2  ¢ strictly holds if and only if  2  (0 ) where the sequence  = ( ) is de…ned by ¯ ¯ ¯ ¢2 +1 ¯¯ ; ( 2 N)   = ¯¯1 ¡ 2 ¢ ¡1 ¯ ¡ ¢ Proof. Let 1 ½ 0 2  ¢ . Then, we obtain that ¤2 () 2 0 for every  2 1 and the matrix ¡ ¢ ¤2 = 2 is in the class (1 : 0 )  It follows by Lemma 8 X¯ 2 ¯ ¯ ¯ = 0 lim (7) 



¡ ¢ From de…nition of ¤ = 2 given in (2) we have 2

X¯ 2 ¯ ¯ ¯ =  

From (7)

¡1 ¢¯ ¢2  1 X ¯¯¡ 2 ¢  ¡ ¢2 ¡1 ¯ +  ¢ ¢

lim 

and lim 

We have

¢2  =0 ¢

¡1 ¯ ¢¡1 1 X ¯¯ 2 ¢ ( ¡ +1 )¯ = ¢ ¢

¢¡1 ¢

(9)

¡1 ¯ 1 X ¯¯ 2 ¢ ( ¡ +1 )¯ = 0 ¢ =0

=0

and since lim

(8)

=0

"

= 1 by (9); we have from (10) that lim 

(10)

¡1 X¡ ¢ 1 ¢2   ¢¡1 =0

#

¡1 1 X¡ 2 ¢ ¢   = 0 ¢ =0

(11)

which shows that  = ( ) 2  (0 ). Conversely, let  = ( ) 2  (0 )  Then we have that (11) holds. Also we obtain that  ¯ 1 X ¯¯ 2 ¢ ( ¡ +1 )¯ = ¢

¡1 1 X 2 ¢   ¢

=0

=0

¡1 X 1 ¢2    ¢¡1

·

=0

This and (11) provides (10). On the other hand, we have that ¯ 2 ¯ ¯ ¯ ¯ ¢  ¡ 0 ¯ ¯ ¯ ¯ ¯ = ¯ 2¡1 ¡ ( + ¡2 ¡ 0 ) ¯ ¯ ¢ ¯ ¯ ¯ ¢ ¯ ¯ ¯ 1 ¡1 ¯ X ¯ ¯ = ¯ ¢2 ( ¡ +1 )¯ ¯ ¢ ¯ =0

·

1 ¢

From (10), we derive that

¡1 X =0

¯ 2 ¯ ¯¢ ( ¡ +1 )¯ 

¢2   ¢2  ¡ 0 = lim = 0  ¢  ¢ This provides (9). Hence, ¡we obtain from (8) that (7) holds. From Lemma 8 ¤2 2 (1 : 0 )  ¢ 2 Hence, the inclusion 1 ½ 0   ¢ holds. This inclusion is strict from Theorem 7. The proof is completed. lim

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Corollary 10 If lim

4

¢2 +1 ¢2 

¡ ¢ = 1, then the inclusion 1 ½ 0 2  ¢ is strict.

¡ ¢ ¡ ¢ The bases for the spaces  2 ¢ and 0 2  ¢

If a normed sequence space  contains a sequence ( ) with the property that for every  2  there is a unique sequence ( ) of scalars such that lim k ¡ (0 0 + 1 1 +  +   )k = 0 

P Then ( ) is called a Schauder basis (or brie‡y basis) for  The series P  which has the sum  is then called the expansion of  with respect to ( ) and written as  =    

¡ ¢ ¡ ¢ Theorem 11 De…ne the sequence () 2 2 0 2  ¢ for every …xed  2 N and by () 

2

 ¢     ¢2 

( )=   

¢ ¢2 +1 ; ¢ ¢2  ;

¡

0;

    =    

n o1 ¡ ¢ () ¡ ¢ () The sequence  2 is a Schauder basis for the space 0 2  ¢ and every  2 =0 ¡ ¢ 0 2  ¢ has a unique representation of the form =

X 

¡ ¢ ¡ ¢  2 () 2

n o ¡ ¢ (0) ¡ ¢ (1) ¡ ¢ () The sequence   2   2   is a Schauder basis for the space  2  ¢ and every ¡ ¢  2  2  ¢ has a unique representation of the form  =  +

X £ ¡ 2¢ ¤ ¡ 2¢    ¡  ()  

¡ ¢ where  2 = ¤2 () for all  2 N and the sequence  = ( ) is de…ned by  =  + 1 ¡ ¢ ¡ ¢ Corollary 12 The di¤erence sequence spaces  2  ¢ and 0 2  ¢ are seperable.

5

¡ ¢ ¡ ¢ The ¡ ¡ and ¡duals of the spaces  2  ¢ and 0 2  ¢

In this section, we introduce determining the ¡ ¡ and ¡ duals of the ¡ and ¢prove the ¡ theorems ¢ di¤erence sequence spaces  2  ¢ and 0 2  ¢ of non-absolute type. For arbitrary sequence spaces  and  ,the set  (  ) de…ned by  (  ) = f = ( ) 2  :  = (  ) 2  for all  = ( ) 2 g

(12)

is called the multipier space of  and  With the notation of (12); the ¡ ¡ and ¡duals of a sequence space  which are respectively denoted by      and   are de…ned by   =  ( 1 )    =  ( ) and   =  ( )  Now, we may begin with lemmas which are given in [10]. We are needed them in proving theorems. Lemma 13  2 (0 : 1 ) = ( : 1 ) if and only if ¯ ¯ ¯ X ¯¯ X ¯ sup  ¯  1 ¯ ¯ ¯ 2F  2

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Lemma 14  2 (0 : ) if and only if lim  exists for each  2 N

(13)



X

sup 



j j  1

(14)

Lemma 15  2 ( : ) if and only if (13) and (14) hold, and X lim  exists. 

(15)



Lemma 16  2 (0 : 1 ) = ( : 1 ) if and only if (14) holds. Lemma 17  2 (1 : ) if and only if (13) holds and X X j j = j j . lim !1





¡ ¢ ¡ ¢ Theorem 18 The ¡dual of the space  2  ¢ and 0 2  ¢ is the set ¯ ¯ ( ) X ¯¯ X ¡ 2 ¢¯¯ 1 =  = ( ) 2  : sup   ¯  1 ; ¯ ¯ 2F  ¯ 2

where the matrix 

2

³ 2´ =  is de…ned via the sequence  = ( ) by 2

  =

     

³

¢ ¢2 

¢ ¢2 +1 ¢ ¢2   ;

¡

´

 ;   

 =  0;    ¡ 2 ¢ Proof. We prove the theorem for the space 0   ¢  Let  = ( ) 2  Then, we obtain the equality   X X 2 ¢   = (¡1)¡ 2   =  () ; ( 2 N)  (16) ¢  =0 =¡1 ¡ ¢ ¡ ¢ Thus, we observe by (16) that  = (  ) 2 1 whenever  = ( ) 2 0 2  ¢ or  2  ¢ if and 2 only if    2 1¡whenever  ¡= ( )¢2 0 or  This means that the sequence  = ( ) is in the ¡dual ¢ of the spaces 0 2  ¢ or  2  ¢ if and only if  2 (0 : 1 ) = ( : 1 )  We therefore obtain by © ¡ ¢ª © ¡ 2 ¢ª Lemma 13 with   instead of  that  2 0 2  ¢ =   ¢ if and only if ¯ ¯ X ¯¯ X 2 ¯¯  sup  ¯  1 ¯ ¯ 2F  ¯     

2

© ¡ ¢ª © ¡ 2 ¢ª Which leads us to the consequence that 0 2  ¢ =   ¢ = 1  This concludes proof.

Theorem 19 De…ne the sets

8 9 1 < = X  exists for each  2 N 2 =  = ( ) 2  : : ; =

3 =

4 =

(

 = ( ) 2  : sup

½

2N

¡1 X =0

j ()j  1

)

¯ ¯ ¾ ¯ ¢ ¯ ¯  = ( ) 2  : sup ¯ 2  ¯¯  1 2N ¢  

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5 =

(

 = ( ) 2  :

where

X

( + 1)  converges.



)

0

1  X  1 1   () = ¢ @ +( ¡ )  A ¢ ¢ ¢+1 =+1

© ¡ ¢ª © ¡ ¢ª for    Then  2  ¢ = 3 \ 4 \ 5 and 0 2  ¢ = 2 \ 3 \ 4  Proof. We have from (6) that 2 0 13     X X X X ¢ 4 @   = (¡1)¡ 2  A5  ¢  =¡1 =0 =0 =0 2 3 µ ¶ X ¡1  X  1 1 ¢ = ¢ 4 2 + ¡ 2  5  + 2   2 ¢  ¢ ¢  ¢    +1 =0 =+1 =

¡1 X

 ()  +

=0

=  () ;

(17)

¢   ¢2 

( 2 N)

where the matrix  = ( ) 

8 <  () ; ¢  ; = 2 : ¢  0;

    =    

(  2 N) 

¡ ¢ Then we derive that  = (  ) 2  whenever  = ( ) 2 0 2  ¢ if and only if   2  whenever © ¡ ¢ª  = ( ) 2 0  This means that  = ( ) 2 0 2  ¢ if and only if  2 (0 : )  Therefore, by using Lemma 14, we obtain from (13) and (14) that 1 X =

 exists for each  2 N sup 

¡1 X =0

sup 

(18)

j ()j  1

(19)

¢   1 ¢2 

(20)

© ¡ ¢ª Hence we conclude that 0 2  ¢ = 2 \ 3 \ 4 . We can derive from Lemma 15 and 16 that © ¡ 2 ¢ª  = ( ) 2    ¢ if and only if  2 ( : )  Therefore, we have from (13) and (14) that (18), (19) and (20) hold. It can be seen that the equality  X

( + 1)  =

=0

¡1 X

 () +

=0

¢  ; ( 2 N) ¢2 

holds, which can be written as follows;  X

( + 1)  =

=0

X 

 ;

( 2 N) 

Consequently, we have from (15) that f( + 1)  g 2  © ¡ ¢ª Hence (18) is redundant. We conclude that  2  ¢ = 3 \ 4 \ 5  7

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Theorem 20

© ¡ 2 ¢ª © ¡ 2 ¢ª 0   ¢ =   ¢ = 3 \ 4 

Proof. It can be proved similarly as the proof of the Theorem 19 with Lemma 16 instead of Lemma 14.

6

Some matrix transformations

¡ ¢ ¡ ¢ In this section, we state some matrix classes of matrix mappings on the 0 2  ¢ and  2  ¢  Let   2  be connected by the relation  = ¤2 () like given in (5). For an in…nite matrix  = ( ), we have by (17)  X

  =

=0

where

¡1 X

 ()  +

=0

2

µ

  () = ¢ 4 + ¢

¢   ¢2 

1 1 ¡ ¢ ¢+1

¶ X 

=+1

(21) 3

 5 

¡ ¢ ¡ ¡ 2 ¢¢ Let  2  2  ¢ and  = ( )1 for all  2 N By passing limits in (21) as  ! 1 =0 2    ¢ X X   =   +  



X

=



where  = lim!1  and  = lim!1

³

 ( ¡ ) + 

¢  ¢2  

´

à X 

 + 

!

(22)

for all  2 N Let consider following conditions; ¯ ¯ ¯ X ¯¯ X ¯ sup  ¯  1 (23) ¯ ¯ ¯  2F  2

sup

¡1 X

j ()j  1

(24)

f( + 1)  g1 =0 2 

(25)



=0

¢  =   ¢2  X j j  1

lim 

(26) (27)



sup 

X 

j j  1

(28)

sup j j  1

(29)



1 X

 

(30)

¾1

(31)

=

½

¢  ¢2 

=0

2 1 

lim  = 

(32)

lim  =    X lim  = 

(33)





(34)



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lim  = 0

(35)

lim  = 0  X lim  = 0

(36)





(37)



Using Theorem 19 and the results given in [10] with (21) and (22), we derive the following result: Theorem 21 ¡ ¡ ¢ ¢ () Let 1 ·   1 Then  2  2  ¢ :  if and only if (23), (24), (25), (26) and (27). ¡ ¡ ¢ ¢ ()  2  2  ¢ :  if and only if (25), (26), (28), (29). ¡ ¡ ¢ ¢ () Let 1 ·   1 Then  2 0 2  ¢ :  if and only if (23), (24), (30) and (31). ¡ ¡ 2 ¢ ¢ ()  2 0   ¢ : 1 if and only if (28), (30) and (31). ¡ ¡ ¢ ¢ ()  2  2  ¢ :  if and only if (25), (26), (28), (32), (33) and (34). ¡ ¡ 2 ¢ ¢ ( )  2    ¢ : 0 if and only if (25), (26), (28), (35), (36) and (37). ¡ ¡ 2 ¢ ¢ ()  2 0   ¢ :  if and only if (28), (30), (31) and (33). ¡ ¡ ¢ ¢ ()  2 0 2  ¢ : 0 if and only if (28), (30), (31) and (36).

Acknowledgements We thank the reviewer for his/her careful reading and useful comments which improved the presentation of the paper. Disclosure Statement The authors declares to have no competing interests.

References [1] A. Wilansky, Summability Through Functional Analysis, in: North-Holland Mathematics Studies, Elsevier Science Publishers, Amsterdam, New York, 1984 . [2] A. H. Ganie, N. A. Sheikh, On some new sequence spaces of non-absolute type and matrix transformations, Journal of Egyptian Math. Society, 21, (2013) ,108-114. [3] B. Choudhary, S. Nanda, Functional Analysis with Applications, John Wiley & Sons Inc., New Delhi, 1989. [4] Ç. Asma, R. Çolak, On the Köthe-Toeplitz duals of some generalized sets of di¤erence sequences, Demonstratio Math., 33 (2000), 797-803. [5] E. Malkowsky, S. D. Parashar, Matrix transformations in space of bounded and convergent difference sequence of order . Analysis, 17, (1997), 87-97. [6] F. Ba¸sar, Summability Theory and Its Applications, Bentham Science Publishers, 2011, ISBN: 978-1-60805-252-3. [7] I. J. Maddox, Elements of Functional Analysis, 2nd ed., The University Press, Cambridge, 1988. [8] M. Mursaleen, A. K. Noman, On some new di¤erence sequence spaces of non-absolute type, Math.Comput. Mod., 52 (2010), 603-617. [9] M. Mursaleen, A. K. Noman, On the spaces of ¡convergent sequences and bounded sequences, Thai J. Math, Volume 8, Number 2, 2010, 311-329. [10] M. Stieglitz and H. Tietz, Matrix transformationen von folgenraumen. Eine ergebnisübersicht, Mathematische Zeitschrift, vol. 154, no.1, pp. 1-16, 1977. [11] N. L. Braha, F. Ba¸sar, On the domain of the triangle  () on the spaces of null, convergent and bounded sequences, Abstract and Applied Analysis, Volume 2013, Article ID 476363. [12] S. Ercan, Ç. A. Bekta¸s, On some sequence spaces of non–absolute type, Kragujevac Journal of Mathematics, 38, (2014) 195-202.

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Stable cubic sets G. Muhiuddin1 , Sun Shin Ahn2,∗ , Chang Su Kim3 and Young Bae Jun3 1

2 3

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

The Research Institute of Natural Science, Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea

Abstract. The notions of (almost) stable cubic set, stable element, evaluative set and stable degree are introduced, and related properties are investigated. Regarding internal (external) cubic sets and the complement of cubic set, their (almost) stableness and unstableness are discussed. Regarding the P-union, R-union, P-intersection and R-intersection of cubic sets, their (almost) stableness and unstableness are investigated.

1. Introduction Fuzzy sets are initiated by Zadeh [14]. In [15], Zadeh made an extension of the concept of a fuzzy set by an interval-valued fuzzy set, i.e., a fuzzy set with an interval-valued membership function. In traditional fuzzy logic, to represent, e.g., the expert’s degree of certainty in different statements, numbers from the interval [0, 1] are used. It is often difficult for an expert to exactly quantify his or her certainty; therefore, instead of a real number, it is more adequate to represent this degree of certainty by an interval or even by a fuzzy set. In the first case, we get an intervalvalued fuzzy set. In the second case, we get a second-order fuzzy set. Interval-valued fuzzy sets have been actively used in real-life applications. For example, Sambuc [8] in Medical diagnosis in thyroidian pathology, Kohout [7] also in Medicine, in a system CLINAID, Gorzalczany [10] in Approximate reasoning, Turksen [10, 11] in Interval-valued logic, in preferences modelling [12], etc. These works and others show the importance of these sets. Using a fuzzy set and an intervalvalued fuzzy set, Jun et al. [4] introduced a new notion, called a (internal, external) cubic set, and investigated several properties. They dealt with P-union, P-intersection, R-union and Rintersection of cubic sets, and investigated several related properties. Cubic set theory is applied to CI-algebras (see [1]), B-algebras (see [9]), BCK/BCI-algebras (see [5, 6]), KU-Algebras (see [2, 13]), and semigroups (see [3]). In this paper, we introduce the notions of (almost) stable cubic set, stable element, evaluative set and stable degree. We investigate related properties. Regarding internal (external) cubic sets and the complement of cubic set, we investigate their (almost) stableness and unstableness. 0

2010 Mathematics Subject Classification: 03E72, 08A72. Keywords: (almost) stable cubic set, stable element, evaluate set, stable degree. ∗ The corresponding author. Tel.: +82 2 2260 3410, Fax: +82 2 2266 3409 (S. S. Ahn). 0 E-mail: [email protected] (G. Muhiuddin); [email protected] (S. S. Ahn); [email protected] (C. S. Kim); [email protected] (Y. B. Jun). 0

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Regarding the P-union, R-union, P-intersection and R-intersection of cubic sets, we deal with their (almost) stableness and unstableness. 2. Preliminaries A fuzzy set in a set X is defined to be a function λ : X → [0, 1]. Denote by I X the collection of all fuzzy sets in a set X. Define a relation ≤ on I X as follows: (∀λ, µ ∈ I X ) (λ ≤ µ ⇐⇒ (∀x ∈ X)(λ(x) ≤ µ(x))). The join (∨) and meet (∧) of λ and µ are defined by (λ ∨ µ)(x) = max{λ(x), µ(x)}, and (λ ∧ µ)(x) = min{λ(x), µ(x)}, respectively, for all x ∈ X. The complement of λ, denoted by λc , is defined by (∀x ∈ X) (λc (x) = 1 − λ(x)). For a family ( {λi | i)∈ Λ} of fuzzy sets in X, we ∨ define the join (∨) and meet (∧) operations as follows: λi (x) = sup{λi (x) | i ∈ Λ}, i∈Λ ( ) ∧ λi (x) = inf{λi (x) | i ∈ Λ}, respectively, for all x ∈ X. i∈Λ

Let D[0, 1] be the set of all closed subintervals of the unit interval [0, 1]. The elements of D[0, 1] are generally denoted by capital letters M, N, · · · , and note that M = [M − , M + ], where M − and M + are the lower and the upper end points respectively. Especially, we denote 0 = [0, 0], 1 = [1, 1], and a = [a, a] for every a ∈ (0, 1). We also note that (i) (∀M, N ∈ D[0, 1]) (M = N ⇔ M − = N − , M + = N + ). (ii) (∀M, N ∈ D[0, 1]) (M ≤ N ⇔ M − ≤ N − , M + ≤ N + ). For every M ∈ D[0, 1], the complement of M, denoted by M c , is defined by M c = 1 − M = [1 − M + , 1 − M − ]. Let X be a nonempty set. A function A : X → D[0, 1] is called an interval-valued fuzzy set (briefly, an IVF set) in X. For each x ∈ X, A(x) is a closed interval whose lower and upper end points are denoted by A(x)− and A(x)+ , respectively. For any [a, b] ∈ D[0, 1], the IVF set whose gb]. Denote by DX the collection of all value is the interval [a, b] for all x ∈ X is denoted by [a, interval-valued fuzzy sets in a set X. In particular, for any a ∈ [0, 1], the IVF set whose value is a = [a, a] for all x ∈ X is denoted by simply a ˜. X For every A, B ∈ D , we define A = B ⇔ (∀x ∈ X) (A(x)− = B(x)− , A(x)+ = B(x)+ ), A ⊆ B ⇔ (∀x ∈ X) (A(x)− ≤ B(x)− , A(x)+ ≤ B(x)+ ). The complement Ac of A is defined by (∀x ∈ X) (Ac (x)− = 1 − A(x)+ , Ac (x)+ = 1 − A(x)− ) . ∪ For a family {Ai | i ∈ Λ} of IVF sets where Λ is an index set, the union G = Ai and the i∈Λ

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intersection F =

∩ i∈Λ

Ai are defined by ( ) (∀x ∈ X) G(x)− = sup Ai (x)− , G(x)+ = sup Ai (x)+ , (

i∈Λ

i∈Λ

) (∀x ∈ X) F (x) = inf Ai (x) , F (x) = inf Ai (x)+ , −

−

i∈Λ

+

i∈Λ

respectively. Definition 2.1 ([4]). Let X be a nonempty set. By a cubic set in X we mean a structure A = {⟨x, A(x), λ(x)⟩ | x ∈ X} in which A is an IVF set in X and λ is a fuzzy set in X. A cubic set A = {⟨x, A(x), λ(x)⟩ | x ∈ X} is simply denoted by A = ⟨A, λ⟩. Note that a cubic set is a generalization of an intuitionistic fuzzy set. Definition 2.2 ([4]). Let X be a nonempty set. A cubic set A = ⟨A, λ⟩ in X is said to be an internal cubic set (briefly, ICS) if A(x)− ≤ λ(x) ≤ A(x)+ for all x ∈ X. Definition 2.3 ([4]). Let X be a nonempty set. A cubic set A = ⟨A, λ⟩ in X is said to be an external cubic set (briefly, ECS) if λ(x) ̸∈ (A(x)− , A(x)+ ) for all x ∈ X. Theorem 2.4 ([4]). Let A = ⟨A, λ⟩ be a cubic set in X. If A is both an ICS and an ECS, then (∀x ∈ X) (λ(x) ∈ U (A) ∪ L(A)) where U (A) = {A(x)+ | x ∈ X} and L(A) = {A(x)− | x ∈ X}. Definition 2.5 ([4]). Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be cubic sets in X. Then we define (a) (Equality) A = B ⇔ A = B and λ = ν. (b) (P-order) A ⊑ B ⇔ A ⊆ B and λ ≤ ν. (c) (R-order) A ⋐ B ⇔ A ⊆ B and λ ≥ ν. Definition 2.6 ([4]). Let A = ⟨A, λ⟩, B = ⟨B, µ⟩ and Ai = {⟨x, Ai (x), λi (x)⟩ | x ∈ X}, i ∈ Λ, be cubic sets in X for i ∈ Λ. The complement, P-union, P-intersection, R-union and R-intersection are defined as follows; (a) (Complement) A c = {⟨x, Ac (x), 1 − λ(x)⟩ | x ∈ X}. (b) (P-union) A ⊔ B = {⟨x, (A ∪ B)(x), (λ ∨ ν)(x)⟩ | x ∈ X} and ∪ ∨ ⊔Ai = {⟨x, ( Ai )(x), ( λi )(x)⟩ | x ∈ X} for i ∈ Λ. (c) (P-intersection) A ⊓ B = {⟨x, (A ∩ B)(x), (λ ∧ ν)(x)⟩ | x ∈ X} and ∩ ∧ ⊓Ai = {⟨x, ( Ai )(x), ( λi )(x)⟩ | x ∈ X} for i ∈ Λ. (d) (R-union) A ⋓ B = {⟨x, (A ∪ B)(x), (λ ∧ ν)(x)⟩ | x ∈ X} and ∪ ∧ ⋓Ai = {⟨x, ( Ai )(x), ( λi )(x)⟩ | x ∈ X} for i ∈ Λ. (e) (R-intersection) A ⋒ B = {⟨x, (A ∩ B)(x), (λ ∨ ν)(x)⟩ | x ∈ X} and ∩ ∨ ⋒Ai = {⟨x, ( Ai )(x), ( λi )(x)⟩ | x ∈ X} for i ∈ Λ.

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3. (Almost) stable cubic sets In what follows, let X denote a nonempty set unless otherwise specified. Definition 3.1. Let A = ⟨A, λ⟩ be a cubic set in X. Then the evaluative set of A = ⟨A, λ⟩ is defined to be a structure EA = {(x, EA (x)) | x ∈ X}

(3.1)

where EA (x) = ⟨l(EA (x)), r(EA (x))⟩ with l(EA (x)) = λ(x) − A(x)− and r(EA (x)) = A(x)+ − λ(x) which are called the left evaluative point and the right evaluative point, respectively, of A = ⟨A, λ⟩ at x ∈ X. We say that EA (x) is the evaluative point of A = ⟨A, λ⟩ at x ∈ X. Example 3.2. Let A = {⟨x, A(x), λ(x)⟩ | x ∈ I} be a cubic set in I = [0, 1]. (1) If A(x) = [0.3, 0.7] and λ(x) = 0.4 for all x ∈ I, then EA = {(x, ⟨0.1, 0.3⟩) | x ∈ I}. (2) If A(x) = [0.3, 0.7] and λ(x) = 0.2 for all x ∈ I, then EA = {(x, ⟨−0.1, 0.5⟩) | x ∈ I}. (3) If A(x) = [0.3, 0.7] and λ(x) = 0.8 for all x ∈ I, then EA = {(x, ⟨0.5, −0.1⟩) | x ∈ I}. Example 3.3. Let B = {⟨x, B(x), µ(x)⟩ | x ∈ I} be a cubic set in I = [0, 1] with B(x) = [ x4 , 1− x4 ] {( ) } x , 1 − 7x ⟩ | x ∈ I , and so the evaluative point of B at 12 ∈ I and µ(x) = x3 . Then EB = x, ⟨ 12 12 1 17 is EB ( 12 ) = ⟨ 24 , 24 ⟩. Example 3.4. Let A = {⟨x, A(x), λ(x)⟩ | x ∈ I} be a cubic set in X = {0, a, b, c} which is defined by Table 1. Table 1. Tabular representation of the cubic set A X

A(x)

λ(x)

0 a b c

[ 18 , 78 ] [ 14 , 34 ] [ 83 , 58 ] [ 12 , 12 ]

7 8 3 8 1 4 5 8

= 0.875 = 0.375 = 0.250 = 0.625

Then every evaluative point of A at each x ∈ X is EA (0) = ⟨ 34 , 0⟩, EA (a) = ⟨ 18 , 38 ⟩, EA (b) = ⟨− 81 , 83 ⟩, and EA (c) = ⟨ 18 , − 18 ⟩, respectively. Hence the evaluative set of A is EA = {(0, ⟨ 34 , 0⟩), (a, ⟨ 18 , 38 ⟩), (b, ⟨− 18 , 38 ⟩), (c, ⟨ 18 , − 81 ⟩)}. Definition 3.5. Let A = ⟨A, λ⟩ be a cubic set in X with the evaluative set EA = {(x, EA (x)) | x ∈ X} . An element a ∈ X is called a stable element of A = ⟨A, λ⟩ in X if it satisfies: l(EA (a)) = λ(a) − A(a)− ≥ 0, r(EA (a)) = A(a)+ − λ(a) ≥ 0. Otherwise, we say that a is an unstable element of A = ⟨A, λ⟩ in X. The set of all stable elements of A = ⟨A, λ⟩ in X is called the stable cut of

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A = ⟨A, λ⟩ in X and is denoted by SA . The set of all unstable elements of A = ⟨A, λ⟩ in X is called the unstable cut of A = ⟨A, λ⟩ in X and is denoted by UA . We say that A = ⟨A, λ⟩ is a stable cubic set if SA = X. Otherwise, A = ⟨A, λ⟩ is called an unstable cubic set. It is clear that X = SA ∪ UA , SA = {x ∈ X | l(EA (x)) ≥ 0, r(EA (x)) ≥ 0} and UA = {x ∈ X | l(EA (x)) < 0} ∪ {x ∈ X | r(EA (x)) < 0}. Example 3.6. Let A = ⟨A, λ⟩ be a cubic set in X = {0, a, b, c} given by Table 2. Table 2. Tabular representation of the cubic set A X

A(x)

λ(x)

0 a b c

[0.2, 0.3] [0.2, 0.3] [0.7, 0.8] [0.3, 0.7]

0.10 0.25 0.75 0.80

Then a and b are stable elements of A in X, and 0 and c are unstable elements of A in X. Hence SA = {a, b} and UA = {0, c}. Example 3.7. (1) Let A = ⟨A, λ⟩ be a cubic set in X = {a, b, c} defined by Table 3. Table 3. Tabular representation of the cubic set A X

A(x)

λ(x)

a b c

[0.1, 0.6] [0.6, 0.9] [0.1, 0.9]

0.5 0.7 0.6

It is routine to verify that A = ⟨A, λ⟩ is a stable cubic set. (2) Let B = ⟨B, µ⟩ be a cubic set in X = {a, b, c} defined by Table 4. Table 4. Tabular representation of the cubic set B X

B(x)

µ(x)

a b c

[0.1, 0.3] [0.6, 0.9] [0.1, 0.9]

0.5 0.7 0.6

Then B is an unstable cubic set since EB (a) = (0.5 − 0.1, 0.3 − 05) = (0.4, −0.2). Theorem 3.8. Every ICS is a stable cubic set.

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□

Proof. Straightforward. The following example shows that every ECS would be stable or unstable. Example 3.9. (1) Let A = ⟨A, λ⟩ be an ECS in X = {a, b, c} given by Table 5. Table 5. Tabular representation of the cubic set A X

A(x)

λ(x)

a b c

[0.1, 0.6] [0.6, 0.9] [0.1, 0.9]

0.6 0.5 0.1

Then A is unstable because EA (b) = (0.5 − 0.6, 0.9 − 0.5) = (−0.1, 0.4). (2) Let B = ⟨B, µ⟩ be an ECS in X = {a, b, c} defined by Table 6. Table 6. Tabular representation of the cubic set B X

B(x)

µ(x)

a b c

[0.1, 0.3] [0.6, 0.9] [0.1, 0.9]

0.1 0.9 0.1

Then B is stable since EB (a) = (0, 0.2), EB (b) = (0.3, 0), and EB (c) = (0, 0.8). We provide a condition for an ECS to be a stable cubic set. Theorem 3.10. If an ECS A = ⟨A, λ⟩ in X satisfies the following condition ( ) (∀x ∈ X) A − (x) = λ(x) or A + (x) = λ(x) ,

(3.2)

then A = ⟨A, λ⟩ is a stable cubic set. □

Proof. Straightforward.

Corollary 3.11. Let A = ⟨A, λ⟩ be a cubic set in X. If A is both an ICS and an ECS, then A is stable. □

Proof. Straightforward. Theorem 3.12. The complement of a stable cubic set is also stable.

Proof. Let A = ⟨A, λ⟩ be a stable cubic set in X. Then X = SA = {x ∈ X | l(EA (x)) ≥ 0, r(EA (x)) ≥ 0}. Hence λ(x) − A(x)− ≥ 0 and A(x)+ − λ(x) ≥ 0 for all x ∈ X. It follows that l(EA c (x)) = (1 − λ(x)) − (1 − A(x)+ ) = A(x)+ − λ(x)) ≥ 0 and r(EA c (x)) = (1 − A(x)− ) − (1 − λ(x)) = λ(x) − A(x)− ≥ 0. Therefore A c = ⟨Ac , λc ⟩ is a stable cubic set. □

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Theorem 3.13. The complement of an unstable cubic set is also unstable. Proof. Let A = ⟨A, λ⟩ be an unstable cubic set in X. Then UA = {x ∈ X | l(EA (x)) < 0} ∪ {x ∈ X | r(EA (x)) < 0} ̸= ∅, and so there exist x ∈ X such that λ(x) − A(x)− < 0 or A(x)+ − λ(x) < 0. It follows that l(EA c (x)) = (1 − λ(x)) − (1 − A(x)+ ) = A(x)+ − λ(x)) < 0 or r(EA c (x)) = (1 − A(x)− ) − (1 − λ(x)) = λ(x) − A(x)− < 0. Hence UA c ̸= ∅, and therefore A c = ⟨Ac , λc ⟩ is an unstable cubic set in X. □ The following example illustrates Theorem 3.13. Example 3.14. Note that the cubic set B = ⟨B, µ⟩ in Example 3.7(2) is unstable, and its complement is represented by Table 7. Table 7. Tabular representation of the cubic set B c X

B c (x)

µc (x)

a b c

[0.7, 0.9] [0.1, 0.4] [0.1, 0.9]

0.5 0.3 0.4

Then B c = ⟨B c , µc ⟩ is unstable since a ∈ UBc . Theorem 3.15. The P-union and P-intersection of two stable cubic sets in X are stable cubic sets in X. Proof. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be stable cubic sets in X. Then SA = {x ∈ X | l(EA (x)) ≥ 0, r(EA (x)) ≥ 0} = X and SB = {x ∈ X | l(EB (x)) ≥ 0, r(EB (x)) ≥ 0} = X. It follows that λ(x) − A(x)− ≥ 0, A(x)+ − λ(x) ≥ 0 for all x ∈ X and µ(x) − B(x)− ≥ 0, B(x)+ − µ(x) ≥ 0 for all x ∈ X. Assume that λ(x) ≥ µ(x) and consider four cases: (i) (ii) (iii) (iv)

A(x)− A(x)− A(x)− A(x)−

≥ B(x)− ≥ B(x)− ≤ B(x)− ≤ B(x)−

and and and and

A(x)+ A(x)+ A(x)+ A(x)+

≥ B(x)+ , ≤ B(x)+ , ≥ B(x)+ , ≤ B(x)+ .

The first case implies that max{λ(x), µ(x)} = λ(x) ≥ A(x)− = max{A(x)− , B(x)− } and max{λ(x), µ(x)} = λ(x) ≤ A(x)+ = max{A(x)+ , B(x)+ }. It follows that λ(x) − A(x)− ≥ 0 and A(x)+ − λ(x) ≥ 0. From the second case, we have max{λ(x), µ(x)} = λ(x) ≥ A(x)− = max{A(x)− , B(x)− } and max{λ(x), µ(x)} = λ(x) ≤ B(x)+ = max{A(x)+ , B(x)+ }. Hence λ(x) − A(x)− ≥ 0 and B(x)+ − λ(x) ≥ A(x)+ − λ(x) ≥ 0. The third case induces max{λ(x), µ(x)} = λ(x) ≥ µ(x) ≥ B(x)− = max{A(x)− , B(x)− } and max{λ(x), µ(x)} = λ(x) ≤ A(x)+ = max{A(x)+ , B(x)+ }, and so λ(x) − B(x)− ≥ µ(x) − B(x)− ≥ 0 and A(x)+ − λ(x) ≥ 0. For the final case, we get max{λ(x), µ(x)} = λ(x) ≥ µ(x) ≥ B(x)− = max{A(x)− , B(x)− } and max{λ(x), µ(x)} =

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λ(x) ≤ A(x)+ ≤ B(x) = max{A(x)+ , B(x)+ }. Thus λ(x) − B(x)− ≥ µ(x) − B(x)− ≥ 0 and B(x)+ − λ(x) ≥ 0. In the case of µ(x) ≥ λ(x), we can obtain the same results in a similar way. Therefore A ⊔ B is a stable cubic set in X. By the similar method, we know that A ⊓ B is a stable cubic set in X. □ The following example shows that the R-union and the R-intersection of two stable cubic sets in X may not be stable in X. Example 3.16. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be cubic sets in X = {a, b, c} defined by Tables 8 and 9, respectively. Table 8. Tabular representation of the cubic set A X

A(x)

λ(x)

a b c

[0.2, 0.3] [0.7, 0.8] [0.3, 0.7]

0.20 0.75 0.60

Table 9. Tabular representation of the cubic set B X

B(x)

µ(x)

a b c

[0.1, 0.3] [0.6, 0.9] [0.1, 0.9]

0.15 0.70 0.80

Then A ⋓ B = {⟨a, [0.2, 0.3], 0.15⟩, ⟨b, [0.7, 0.9], 0.7⟩, ⟨c, [0.3, 0.9], 0.6⟩} and A ⋒ B = {⟨a, [0.1, 0.3], 0.2⟩, ⟨b, [0.6, 0.8], 0.75⟩, ⟨c, [0.1, 0.7], 0.8⟩}. Hence we know that EA ⋓B (a) = ⟨−0.05, 0.15⟩ and EA ⋒B (c) = ⟨0.7, −0.1⟩. Thus A ⋓ B and A ⋒ B are unstable. Now, we provide conditions for the R-union (resp. R-intersection) of two ICSs to be stable. Theorem 3.17. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be ICSs in X such that ( ) (∀x ∈ X) max{A(x)− , B(x)− } ≤ (λ ∧ µ)(x) .

(3.3)

Then the R-union of A and B is a stable cubic set in X.

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Proof. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be ICSs in X. Then A(x)− ≤ λ(x) ≤ A(x)+ and B(x)− ≤ µ(x) ≤ B(x)+ for all x ∈ X. It follows from (3.3) that max{A(x)− , B(x)− } ≤ (λ ∧ µ)(x) ≤ max{A(x)+ , B(x)+ } for all x ∈ X. Hence the R-union of A and B is an ICS, and so it is stable by Theorem 3.8. □ Theorem 3.18. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be ICSs in X such that ( ) (∀x ∈ X) max{A(x)+ , B(x)+ } ≤ (λ ∨ µ)(x) .

(3.4)

Then the R-intersection of A and B is a stable cubic set in X. □

Proof. The proof is by the similar method to Theorem 3.17.

Theorem 3.19. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be ECSs in X such that A ∗ = ⟨A, µ⟩ and B ∗ = ⟨B, λ⟩ are ICSs in X. Then the P-union A ⊔ B and the P-intersection A ⊓ B of A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are stable in X. □

Proof. It is straightforward by Theorems 3.20 and 3.21 in [4] and Theorem 3.8.

Definition 3.20. Let A = ⟨A, λ⟩ be a cubic set with the evaluative set EA = {(x, EA (x)) | x ∈ X} in X. Then the stable degree of A in X is denoted by SDA and is defined by ( ) ∑ ∑ SDA = l(EA (x)), r(EA (x)) . (3.5) x∈X

x∈X

Definition 3.21. A cubic set A = ⟨A, λ⟩ with the evaluative set EA = {(x, EA (x)) | x ∈ X} in ∑ l(EA (x)) ≥ 0 X is said to be almost stable if there exists the stable degree SDA in which x∈X ∑ r(EA (x)) ≥ 0. and x∈X

Example 3.22. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be cubic sets in X = {a, b, c} defined by Tables 10 and 11, respectively. Table 10. Tabular representation of the cubic set A X

A(x)

λ(x)

a b c

[0.2, 0.3] [0.7, 0.8] [0.3, 0.7]

0.2 0.9 0.6

Then EA = {(a, ⟨0, 0.1⟩), (b, ⟨0.2, −0.1⟩), (c, ⟨0.3, 0.1⟩)} and

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Table 11. Tabular representation of the cubic set B X

B(x)

µ(x)

a b c

[0.2, 0.3] [0.6, 0.9] [0.1, 0.9]

0.9 0.7 1

EB = {(a, ⟨0.7, −0.6⟩), (b, ⟨0.1, 0.2⟩), (c, ⟨0.9, −0.1⟩)}. Thus SDA = (0 + 0.2 + 0.3, 0.1 − 0.1 + 0.1) = (0.5, 0.1) and so A is almost stable. But B is not almost stable since SDB = (0.7 + 0.1 + 0.9, −0.6 + 0.2 − 0.1) = (1.7, −0.5). Theorem 3.23. Every stable cubic set A = ⟨A, λ⟩ in X is almost stable. □

Proof. Straightforward.

In Example 3.22, the almost stable cubic set A = ⟨A, λ⟩ is not stable. This shows that the converse of Theorem 3.23 is not true in general. Combining Theorems 3.8, 3.10, 3.15, 3.19 and 3.23, we know that (1) (2) (3) (4)

Every ICS is almost stable. Every ESC satisfying the condition (3.2) is almost stable. The P-union and P-intersection of two stable cubic sets is almost stable. If A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are ECSs in X such that A ∗ = ⟨A, µ⟩ and B ∗ = ⟨B, λ⟩ are ICSs in X, then the P-union and the P-intersection of A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are almost stable in X.

Proposition 3.24. If A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are cubic sets in X, then either ( ) (∀x ∈ X) max{λ(x), µ(x)} − max{A(x)− , B(x)− } ≤ λ(x) − A(x)−

(3.6)

or ( ) (∀x ∈ X) max{λ(x), µ(x)} − max{A(x)− , B(x)− } ≤ µ(x) − B(x)− .

(3.7)

Proof. For each x ∈ X, we consider the four cases as follows: (1) (2) (3) (4)

max{λ(x), µ(x)} = λ(x) max{λ(x), µ(x)} = λ(x) max{λ(x), µ(x)} = µ(x) max{λ(x), µ(x)} = µ(x)

and and and and

max{A(x)− , B(x)− } = A(x)− . max{A(x)− , B(x)− } = B(x)− . max{A(x)− , B(x)− } = A(x)− . max{A(x)− , B(x)− } = B(x)− .

First two cases induce the inequality (3.6), and the inequality (3.7) is induced by the last two cases. □

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Proposition 3.25. If A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are cubic sets in X, then either ( ) (∀x ∈ X) max{A(x)+ , B(x)+ } − max{λ(x), µ(x)} ≤ A(x)+ − λ(x) or

( ) (∀x ∈ X) max{A(x)+ , B(x)+ } − max{λ(x), µ(x)} ≤ B(x)+ − µ(x) .

(3.8)

(3.9) □

Proof. It is similar to the proof of Proposition 3.24.

In the following example, we know that the P-union and the R-union of almost stable cubic sets may not be almost stable. Example 3.26. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be cubic sets in X = {a, b, c} defined by Tables 12 and 13, respectively. Table 12. Tabular representation of the cubic set A X

A(x)

λ(x)

a b c

[1.0, 1.0] [0.5, 1.0] [0.6, 1.0]

0.7 0.7 0.7

Table 13. Tabular representation of the cubic set B X

B(x)

µ(x)

a b c

[0.5, 1.0] [1.0, 1.0] [0.6, 1.0]

0.7 0.7 0.7

Then A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are almost stable cubic sets in X because ∑ ∑ ∑ ∑ r(EA (x)) = 0.9, l(EB (x)) = 0, and r(EB (x)) = 0.9. l(EA (x)) = 0, x∈X

x∈X

x∈X

x∈X

But the P-union A ⊔ B and the R-union A ⋓ B of A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are not almost ∑ ∑ stable because l(EA ⊔B (x)) = (max{λ(x), µ(x)} − max{A(x)− , B(x)− }) = −0.5 ̸≥ 0 and x∈X ∑ x∈X ∑ l(EA ⋓B (x)) = (min{λ(x), µ(x)} − max{A(x)− , B(x)− }) = −0.5 ̸≥ 0. x∈X

x∈X

We now provide conditions for the P-union of almost stable cubic sets to be almost stable. Theorem 3.27. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be almost stable cubic sets in X such that ( ∑ ) ∑ (|λ(x) − µ(x)| − A(x)− ) ≥ 0, (|A(x)+ − B(x)+ | − λ(x)) ≥ 0 . (∀x ∈ X) (3.10) x∈X

x∈X

Then the P-union A ⊔ B = ⟨A ∪ B, λ ∨ µ⟩ of A = ⟨A, λ⟩ and B = ⟨B, µ⟩ is almost stable in X.

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Proof. Assume that A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are almost stable in X. Then there exist stable degrees SDA and SDB , respectively, such that ∑ ∑ ∑ ∑ l(EA (x)) = (λ(x) − A(x)− ) ≥ 0, r(EA (x)) = (A(x)+ − λ(x)) ≥ 0, x∈X

∑

x∈X

l(EB (x)) =

∑

x∈X

(µ(x) − B(x)− ) ≥ 0, and

x∈X

x∈X

∑

Now, we have to show that

∑ x∈X

l(EA ⊔B (x)) ≥ 0 and

x∈X

x∈X

r(EB (x)) = ∑

∑

(B(x)+ − µ(x)) ≥ 0.

x∈X

r(EA ⊔B (x)) ≥ 0 in the stable degree

x∈X

SDA ⊔B of A ⊔ B. Using (3.10), we have ∑ ∑( ) l(EA ⊔B (x)) = (λ ∨ µ)(x) − (A ∪ B)(x)− x∈X

=

∑(

x∈X

max{λ(x), µ(x)} − max{A(x)− , B(x)− }

)

x∈X

=

∑ ( |λ(x)−µ(x)|+λ(x)+µ(x) 2

−

|A(x)− −B(x)− |+A(x)− +B(x)− 2

)

x∈X

=

∑ ( |λ(x)−µ(x)|−|A(x)− −B(x)− |+λ(x)−A(x)− +µ(x)−B(x)− ) 2

x∈X

=

1 2

=

1 2

∑(

x∈X

∑(

|λ(x) − µ(x)| − |A(x)− − B(x)− | + λ(x) − A(x)− + µ(x) − B(x)− ) |λ(x) − µ(x)| − |A(x)− − B(x)− |

x∈X

+ ≥

1 2

)

∑(

1 2

∑(

∑( ) ) λ(x) − A(x)− + 12 µ(x) − B(x)−

x∈X

x∈X

∑( ∑( ) ) ) |λ(x) − µ(x)| − A(x)− + 21 λ(x) − A(x)− + 12 µ(x) − B(x)−

x∈X

x∈X

x∈X

≥ 0. Similarly, we have

∑

r(EA ⊔B (x)) ≥ 0. Therefore A ⊔ B = ⟨A ∪ B, λ ∨ µ⟩ is almost stable in

x∈X

□

X. Theorem 3.28. The complement of an almost stable cubic set is also almost stable.

Proof. Let A = ⟨A, λ⟩ be an almost stable cubic set in X. Then there exists a stable degree SDA such that ∑ ∑ ∑ ∑ l(EA (x)) = (λ(x) − A(x)− ) ≥ 0, and r(EA (x)) = (A(x)+ − λ(x)) ≥ 0. x∈X

x∈X

x∈X

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∑ ∑ ∑ It follows that l(EA c (x)) = ((1 − λ(x)) − (1 − A(x)+ )) = (A(x)+ − λ(x)) ≥ 0 and x∈X∑ x∈X x∈X ∑ ∑ r(EA c (x)) = ((1 − A(x)− ) − (1 − λ(x))) = (λ(x) − A(x)− ) ≥ 0. Therefore A c = x∈X c

x∈X

x∈X

⟨A , λc ⟩ is almost stable.

□

We now provide conditions for the R-union of almost stable cubic sets to be almost stable. Theorem 3.29. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be almost stable cubic sets in X such that ( ) ∑ ∑ ∑ − − − (µ(x) − B(x)− ) (3.11) (|λ(x) − µ(x)| + |A(x) − B(x) |) ≤ λ(x) − A(x) + x∈X

and

∑

x∈X

(|λ(x) − µ(x)| + |A(x)+ − B(x)+ |) ≥

x∈X

∑

x∈X

∑

(λ(x) − A(x)+ ) +

x∈X

x∈X

(µ(x) − B(x)+ ) (3.12)

for all x ∈ X. Then the R-union A ⋓ B = ⟨A ∪ B, λ ∧ µ⟩ is almost stable in X. Proof. Assume that A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are almost stable in X. Then there exist stable degrees SDA and SDB , respectively, such that ∑ ∑ ∑ ∑ l(EA (x)) = (λ(x) − A(x)− ) ≥ 0, r(EA (x)) = (A(x)+ − λ(x)) ≥ 0, x∈X

∑

x∈X

l(EB (x)) =

x∈X

∑

x∈X

(µ(x) − B(x)− ) ≥ 0, and

x∈X

∑

x∈X

r(EB (x)) =

x∈X

∑

(B(x)+ − µ(x)) ≥ 0.

x∈X

It follows from (3.11) that ∑ ∑( ) l(EA ⋓B (x)) = (λ ∧ µ)(x) − (A ∪ B)(x)− x∈X

=

∑(

x∈X

min{λ(x), µ(x)} − max{A(x)− , B(x)− }

)

x∈X

=

∑ ( −|λ(x)−µ(x)|+λ(x)+µ(x) 2

−

|A(x)− −B(x)− |+A(x)− +B(x)− 2

)

x∈X

=

∑ ( −|λ(x)−µ(x)|−|A(x)− −B(x)− |+λ(x)−A(x)− +µ(x)−B(x)− ) 2

x∈X

= − 12

∑(

) |λ(x) − µ(x)| + |A(x)− − B(x)− |

x∈X

+ ( ≥ − 12

1 2

∑( x∈X

1 2

∑( x∈X

x∈X

∑( ) ∑( ) λ(x) − A(x)− + µ(x) − B(x)−

x∈X

+

∑( ) ) λ(x) − A(x)− + 12 µ(x) − B(x)−

λ(x) − A(x)

) −

)

x∈X

∑( ) + 12 µ(x) − B(x)− = 0. x∈X

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Using (3.12), we have ∑

r(EA ⋓B (x)) =

x∈X

=

∑(

∑(

(A ∪ B)(x)+ − (λ ∧ µ)(x)

x∈X

max{A(x)− , B(x)− } − min{λ(x), µ(x)}

)

)

x∈X

=

∑ ( |A(x)+ −B(x)+ |+A(x)+ +B(x)+ 2

x∈X

=

1 2

∑(

−

−|λ(x)−µ(x)|+λ(x)+µ(x) 2

) |λ(x) − µ(x)| + |A(x)+ − B(x)+ |

x∈X

(

−

1 2

)

∑(

) ∑( ) λ(x) − A(x)+ + µ(x) − B(x)+

x∈X

) ≥ 0.

x∈X

□

Hence A ⋓ B = ⟨A ∪ B, λ ∧ µ⟩ is almost stable in X.

The following examples show that the P-intersection and the R-intersection of almost stable cubic sets may not be almost stable. Example 3.30. (1) Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be cubic sets in X = {a, b, c} defined by Tables 14 and 15, respectively. Table 14. Tabular representation of the cubic set A X

A(x)

λ(x)

a b c

[0.7, 1.0] [0.5, 1.0] [0.6, 1.0]

0.4 0.8 0.7

Table 15. Tabular representation of the cubic set B X

B(x)

µ(x)

a b c

[0.5, 1.0] [0.6, 1.0] [0.7, 1.0]

0.8 0.7 0.4

Then A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are almost stable cubic sets in X because ∑ ∑ ∑ ∑ l(EA (x)) = 0.1, r(EA (x)) = 1.1, l(EB (x)) = 0.1, and r(EB (x)) = 1.1. x∈X

x∈X

x∈X

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But the P-intersection A ⊓ B of A = ⟨A, λ⟩ and B = ⟨B, µ⟩ is not almost stable because ∑ ∑( ) l(EA ⊓B (x)) = min{λ(x), µ(x)} − min{A(x)− , B(x)− } = −0.1 ̸≥ 0. x∈X

x∈X

(2) Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be cubic sets in X = {a, b, c} defined by Tables 16 and 17, respectively. Table 16. Tabular representation of the cubic set A X

A(x)

λ(x)

a b c

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

0.8 0.5 0.5

Table 17. Tabular representation of the cubic set B X

B(x)

µ(x)

a b c

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

0.6 0.7 0.5

Then A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are almost stable cubic sets in X because ∑ ∑ ∑ ∑ r(EB (x)) = 0. l(EB (x)) = 1.2, and r(EA (x)) = 0, l(EA (x)) = 1.2, x∈X

x∈X

x∈X

x∈X

But the R-intersection A ⋒ B of A = ⟨A, λ⟩ and B = ⟨B, µ⟩ is not almost stable since ∑ ∑( ) r(EA ⋒B (x)) = min{A(x)+ , B(x)+ } − max{λ(x), µ(x)} = −0.2 ̸≥ 0. x∈X

x∈X

We now provide conditions for the P-intersection and the R-intersection of almost stable cubic sets to be almost stable. Theorem 3.31. Let A = ⟨A, λ⟩ and B = ⟨B, µ⟩ be almost stable cubic sets in X. (i) Assume that the following condition is valid.  ∑  (|A(x)− − B(x)− | − |λ(x) − µ(x)|) ≥ 0, . ∑ (∀x ∈ X)  x∈X (|λ(x) − µ(x)| − |A(x)+ − B(x)+ |) ≥ 0

(3.13)

x∈X

Then the P-intersection A ⊓ B = ⟨A ∩ B, λ ∧ µ⟩ of A = ⟨A, λ⟩ and B = ⟨B, µ⟩ is almost stable in X.

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(ii) If A = ⟨A, λ⟩ and B = ⟨B, µ⟩ satisfy the following condition ( (∀x ∈ X)

∑(

) ) |λ(x) − µ(x)| + |A(x)+ − B(x)+ | = 0 ,

(3.14)

x∈X

then the R-intersection A ⋒ B = ⟨A ∩ B, λ ∨ µ⟩ of A = ⟨A, λ⟩ and B = ⟨B, µ⟩ is almost stable in X. Proof. Since A = ⟨A, λ⟩ and B = ⟨B, µ⟩ are almost stable in X, there exist stable degrees SDA and SDB , respectively, such that ∑ r(EA (x)) = (A(x)+ − λ(x)) ≥ 0, x∈X x∈X x∈X ∑ x∈X ∑ ∑ ∑ (B(x)+ − µ(x)) ≥ 0. r(EB (x)) = (µ(x) − B(x)− ) ≥ 0, and l(EB (x)) = ∑

∑

l(EA (x)) =

(λ(x) − A(x)− ) ≥ 0,

∑

∑

(i) We have to show that

l(EA ⊓B (x)) ≥ 0 and

SDA ⊓B of A ⊓ B. Using (3.13), we have ∑(

l(EA ⊓B (x)) =

x∈X

=

∑(

∑

r(EA ⊓B (x)) ≥ 0 in the stable degree

x∈X

x∈X

∑

x∈X

x∈X

x∈X

x∈X

(λ ∧ µ)(x) − (A ∩ B)(x)−

x∈X

min{λ(x), µ(x)} − min{A(x)− , B(x)− }

)

)

x∈X

=

∑ ( −|λ(x)−µ(x)|+λ(x)+µ(x) 2

+

|A(x)− −B(x)− |−A(x)− −B(x)− 2

)

x∈X

=

∑ ( −|λ(x)−µ(x)|+|A(x)− −B(x)− |+λ(x)−A(x)− +µ(x)−B(x)− ) 2

x∈X

=

1 2

∑(

−|λ(x) − µ(x)| + |A(x)− − B(x)− | + λ(x) − A(x)− + µ(x) − B(x)−

x∈X

=

1 2

∑(

|A(x)− − B(x)− | − |λ(x) − µ(x)|

x∈X

+

1 2

∑( x∈X

Similarly, we have

∑

)

)

∑( ) ) (λ(x) − (A(x)− + 12 (µ(x)) − B(x)− ) ≥ 0. x∈X

r(EA ⊓B (x)) ≥ 0. Therefore A ⊓ B = ⟨A ∩ B, λ ∧ µ⟩ is almost stable in X.

x∈X

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(ii) We have

∑

l(EA ⋒B (x)) =

x∈X

=

∑(

∑(

(λ ∨ µ)(x) − (A ∩ B)(x)−

x∈X

max{λ(x), µ(x)} − min{A(x)− , B(x)− }

)

)

x∈X

=

∑ ( |λ(x)−µ(x)|+λ(x)+µ(x) 2

+

|A(x)− −B(x)− |−A(x)− −B(x)− 2

)

x∈X

=

∑ ( |λ(x)−µ(x)|+|A(x)− −B(x)− |+λ(x)−A(x)− +µ(x)−B(x)− ) 2

x∈X

=

∑(

1 2

) |λ(x) − µ(x)| + |A(x)− − B(x)− |

x∈X 1 2

+ ( ≥

1 2

∑(

∑( ) ) λ(x) − A(x)− + 12 µ(x) − B(x)−

x∈X

∑(

x∈X

λ(x) − A(x)

) −

x∈X

Using (3.14), we have ∑

=

∑(

) µ(x) − B(x)−

) ≥ 0.

x∈X

∑(

r(EA ⋒B (x)) =

x∈X

+

∑(

) (A ∩ B)(x)+ − (λ ∨ µ)(x)

x∈X

) min{A(x)+ , B(x)+ } − max{λ(x), µ(x)}

x∈X

=

∑ ( −|A(x)+ −B(x)+ |+A(x)+ +B(x)+ 2

x∈X

=

1 2

∑(

( +

=

1 2

|λ(x)−µ(x)|+λ(x)+µ(x) 2

)

) −|λ(x) − µ(x)| − |A(x)+ − B(x)+ |

x∈X

(

−

1 2

∑(

)

A(x)+ − λ(x) +

x∈X

∑( x∈X

∑(

) B(x)+ − µ(x)

x∈X

) ∑( ) A(x)+ − λ(x) + B(x)+ − µ(x)

)

) ≥ 0.

x∈X

□

Hence A ⋒ B = ⟨A ∩ B, λ ∨ µ⟩ is almost stable in X. References

[1] S. S. Ahn. Y. H. Kim and J. M. Ko, Cubic subalgebras and filters of CI-algebras, Honam Math. J. 36(1) (2014) 43–54. [2] M. Akram, N. Yaqoob and M. Gulistan, Cubic KU-subalgebras, Int. J. Pure Appl. Math. 89(5) (2013) 659– 665.

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[3] Y. B. Jun and A. Khan, Cubic ideals in semigroups, Honam Math. J. 35(4) (2013) 607–623. [4] Y. B. Jun, C. S. Kim and K. O. Yang, Cubic sets, Ann. Fuzzy Math. Infom. 4(1) (2012) 83–98. [5] Y. B. Jun and K. J. Lee, Closed cubic ideals and cubic ◦-subalgebras in BCK/BCI-algebras, Appl. Math. Sci. 4(68) (2010) 3395–3402. [6] Y. B. Jun, K. J. Lee and M. S. Kang, Cubic structures applied to ideals of BCI-algebras, Comput. Math. Appl. 62 (2011) 3334–3342. [7] L. J. Kohout and W. Bandler, Fuzzy interval inference utilizing the checklist paradigm and BK-relational products, in: R.B. Kearfort et al. (Eds.), Applications of Interval Computations, Kluwer, Dordrecht, 1996, pp. 291–335. [8] R. Sambuc, Functions Φ-Flous, Application `a l’aide au Diagnostic en Pathologie Thyroidienne, Th`ese de Doctorat en M´edecine, Marseille, 1975. [9] T. Senapati, C. S. Kim, M. Bhowmik and M. Pal, Cubic subalgebras and cubic closed ideals of B-algebras, Fuzzy Inf. Eng. 7 (2015) 129–149. [10] I. B. Turksen, Interval-valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20 (1986) 191–210. [11] I. B. Turksen, Interval-valued fuzzy sets and compensatory AND, Fuzzy Sets and Systems 51 (1992) 295–307. [12] I. B. Turksen, Interval-valued strict preference with Zadeh triples, Fuzzy Sets and Systems 78 (1996) 183–195. [13] N. Yaqoob, S. M. Mostafa and M. A. Ansari, On cubic KU-ideals of KU-algebras, ISRN Algebra 2013, Art. ID 935905, 10 pp. [14] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. [15] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inform. Sci. 8 (1975) 199–249.

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SOME IDENTITIES OF CHEBYSHEV POLYNOMIALS ARISING FROM NON-LINEAR DIFFERENTIAL EQUATIONS TAEKYUN KIM, DAE SAN KIM, JONG-JIN SEO, AND DMITRY V. DOLGY

Abstract. In this paper, we investigate some properties of Chebyshev polynomials arising from non-linear differential equations. From our investigation, we derive some new and interesting identities on Chebyshev polynomials.

1. Introduction As is well known, the Chebyshev polynomials of the first kind, Tn (x), (n ≥ 0), are defined by the generating function (1.1)

∞ ∑ 1 − t2 tn = Tn (x) , 2 1 − 2xt + t n! n=0

(see [1, 3, 5, 8, 17, 21]) .

The higher-order Chebyshev polynomials are given by the generating function ( )α ∑ ∞ 1 − t2 (1.2) = Tn(α) (x) tn , 1 − 2xt + t2 n=0 and Chebyshev polynomials of the second kind are denoted by Un and given by generating function (1.3)

∞ ∑ 1 = Un (x) tn , 1 − 2xt + t2 n=0

(see [1, 7, 12, 17]) .

The higher-order Chebyshev polynomials of the second kind are also defined by ( )α ∑ ∞ 1 (1.4) = Un(α) (x) tn . 1 − 2xt + t2 n=0 The Chebyshev polynomials of the third kind are defined by the generating function ∞ ∑ 1−t (1.5) = Vn (x) tn , (see [1, 7, 8, 17]) . 1 − 2xt + t2 n=0 and the higher-order Chebyshev polynomials of the third kind are also given by the generating function )α ∑ ( ∞ 1−t = Vn(α) (x) tn . (1.6) 1 − 2xt + t2 n=0 2010 Mathematics Subject Classification. 05A19, 33C45, 34A34. Key words and phrases. Chebyshev polynomials of the first kind, Chebyshev polynomials of the second kind, Chebyshev polynomials of the third kind, Chebyshev polynomials of the fourth kind, non-linear differential equation. 1

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Finally, we introduce the Chebyshev polynomials of the fourth kind defined by the generating function ∞ ∑ 1+t = Wn (x) tn . 1 − 2xt + t2 n=0

(1.7)

The higher-order Chebyshev polynomials of the fourth kind are defined by ( )α ∑ ∞ 1+t (1.8) = Wn(α) (x) tn . 1 − 2xt + t2 n=0 It is well known that the Legendre polynomials are defined by the generating function ∞ ∑ 1 √ (1.9) = pn (x) tn , (see [2, 20]) . 1 − 2xt + t2 n=0 Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial nodes (see [19]). The Chebyshev polynomials of the first kind and of the second kind are solutions of the following Chebyshev differential equations ( ) (1.10) 1 − x2 y ′′ − xy ′ + n2 y = 0, and (1.11)

(

) 1 − x2 y ′′ − 3xy ′ + n (n + 2) y = 0.

These equations are special cases of the Strum-Liouville differential equation (see [1–3]). The Chebyshev polynomials of the first kind can be defined by the contour integral ( ) ˛ 1 − t2 1 (1.12) Tn (z) = t−n−1 dt, 4πi 1 − 2tz + t2 where the contour encloses the origin and is traversed in a counterclockwise direction (see [1, 19, 21]). The formula for Tn (x) is given by (1.13)

[ n2 ] ( ) ∑ ( )m n xn−2m x2 − 1 . Tn (x) = 2m m=0

From (1.3), we note that (1.14)

∞ ( )−2 ∑ 2 (x − t) 1 − 2xt + t2 = nUn (x) tn−1 . n=0

Thus, by (1.14), we get (1.15)

∞ ∑ ( )( ) 2 2 −2 = nUn (x) tn . 2xt − 2t 1 − 2xt + t n=0

From (1.3) and (1.15), we can derive the following equation: ( ) ( ) 2xt − 2t2 + 1 − 2xt + t2 1 − t2 (1.16) = 2 2 (1 − 2xt + t2 ) (1 − 2xt + t2 )

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=

∞ ∑

3

(n + 1) Un (x) tn .

n=0

Note that (1.17)

1 − t2 2

(1 − 2xt + t2 ) )( ) ( 1 1 − t2 = 1 − 2xt + t2 1 − 2xt + t2 (∞ )( ∞ ) ∑ ∑ l m = Tl (x) t Um (x) t m=0

l=0

=

( n ∞ ∑ ∑ n=0

)

Tl (x) Un−l (x) tn .

l=0

From (1.16) and (1.17), we have 1 ∑ Tl (x) Un−l (x) . n+1 n

Un (x) =

l=0

The Chebyshev polynomials have been studied by many authors in the several areas (see [1–21]). In [11], Kim-Kim studied non-linear differential equations arising from Changhee polynomials and numbers related to Chebyshev poynomials. In this paper, we study non-linear differential equations arising from Chebyshev polynomials and give some new and explicit formulas for those polynomials. 2. Differential equations arising from Chebyshev polynomials and their applications Let (2.1)

F = F (t, x) =

1 . 1 − 2tx + t2

Then, by (1.1), we get (2.2)

d F (t, x) = 2 (x − t) F 2 . dt

F (1) =

From (2.2), we note that (2.3)

2F 2 = (x − t)

−1

F (1) .

By using (2.3) and (2.2), we obtain the following equations: −3

(2.4)

22 · 2F 3 = (x − t)

(2.5)

23 · 2 · 3F 4 = 3 (x − t)

F (1) + (x − t)

−5

−2

F (2) , −4

F (1) + 3 (x − t)

F (2) + (x − t)

−3

F (3)

and (2.6)

24 · 2 · 3 · 4F 5 = 3 · 5 (x − t)

−6

−5

+ (3 · 2) (x − t)

822

−6

F (2)

−4

F (4) ,

F (1) + 3 · 5 (x − t) F (3) + (x − t)

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where

( F

N

= F × ··· × F | {z }

and F

(N )

=

N −times

d dt

)N F (t, x) .

Continuing this process, we set 2N N !F N +1 =

(2.7)

N ∑

i−2N

ai (N ) (x − t)

F (i) ,

i=1

where N ∈ N. From (2.7), we note that 2N N !F N (N + 1) F (1)

(2.8) =

N ∑

i−2N −1

ai (N ) (2N − i) (x − t)

F (i) +

i=1

N ∑

ai (N ) (x − t)

i−2N

F (i+1) .

i=1

By (2.2) and (2.8), we get

( ) 2N N ! (N + 1) F N 2 (x − t) F 2

(2.9) =

N ∑

i−2N −1

ai (N ) (2N − i) (x − t)

F (i)

i=1

+

N ∑

ai (N ) (x − t)

i−2N

F (i+1) .

i=1

Thus, from (2.9), we have 2N +1 (N + 1)!F N +2

(2.10) =

N ∑

i−2(N +1)

ai (N ) (2N − i) (x − t)

F (i)

i=1

+

N +1 ∑

i−2(N +1)

ai−1 (N ) (x − t)

F (i) .

i=2

On the other hand, by replacing N by N + 1, in (2.7), we get (2.11)

2N +1 (N + 1)!F N +2 =

N +1 ∑

i−2(N +1)

ai (N + 1) (x − t)

F (i) .

i=1

Comparing the coefficients on both sides of (2.10) and (2.11), we have (2.12) (2.13)

a1 (N + 1) = (2N − 1) a1 (N ) , aN +1 (N + 1) = aN (N ) ,

and (2.14)

ai (N + 1) = ai−1 (N ) + (2N − i) ai (N ) ,

(2 ≤ i ≤ N ) .

Moreover, by (2.4) and (2.7), we get (2.15)

−1

2F 2 = (x − t)

−1

F (1) = a1 (1) (x − t)

F (1) .

By comparing the coefficients on both sides of (2.15), we get (2.16)

a1 (1) = 1.

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Now, by (2.12) and (2.16), we have a1 (N + 1) = (2N − 1) a1 (N )

(2.17)

= (2N − 1) (2N − 3) a1 (N − 1) = (2N − 1) (2N − 3) (2N − 5) a1 (N − 2) .. . = (2N − 1) (2N − 3) (2N − 5) · · · 1 · a1 (1) = (2N − 1)!!, where (2N − 1)!! is Arfken’s double factorial. From (2.13), we easily note that aN +1 (N + 1) = aN (N ) = · · · = a1 (1) = 1.

(2.18)

For 2 ≤ i ≤ N , from (2.14), we can derive the following equation: (2.19) ai (N + 1) = ai−1 (N ) + (2N − i) ai (N ) = ai−1 (N ) + (2N − i) ai−1 (N − 1) + (2N − i) (2N − 2 − i) ai (N − 1) .. . =

N −i ∑

(k−1 ∏

k=0

l=0

) (2 (N − l) − i) ai−1 (N − k) +

N −i ∏

(2 (N − l) − i) ai (i)

l=0

) ( ) ( i i N −i+1 ai−1 (N − k) + 2 N− = 2 N− 2 k 2 N −i+1 k=0 ( ) N∑ −i+1 i = 2k N − ai−1 (N − k) , 2 k N −i ∑

k

k=0

where (x)n = x (x − 1) · · · (x − n + 1), (n ≥ 1) and (x)0 = 1. As the above is also valid for i = N + 1, by (2.19), we get (2.20)

ai (N + 1) =

N∑ +1−i k=0

( ) i 2k N − ai−1 (N − k) , 2 k

where 2 ≤ i ≤ N + 1. Now, we give an explicit expression for ai (N + 1). From (2.17) and (2.20), we can derive the following equations: (2.21)

a2 (N + 1) =

N −1 ∑

2k1

) ( 2 N− a1 (N − k1 ) 2 k1

2k1

( ) 2 N− (2 (N − k1 − 1) − 1)!!, 2 k1

k1 =0

=

N −1 ∑ k1 =0

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(2.22) N −2 ∑

a3 (N + 1) =

(

k2 =0 N −2 N −2−k ∑ ∑ 2

=

k2 =0

N −3 ∑

a2 (N − k2 ) k2

2k1 +k2

k1 =0

and (2.23) a4 (N + 1) =

)

3 2

N−

2k2

( 2k3

)

4 2

N−

k3 =0

) ( ) ( 4 3 N − k2 − (2 (N − 2 − k1 − k2 ) − 1)!!, N− 2 k2 2 k1

a3 (N − k3 ) k3

N −3 N −3−k ∑ ∑ 3 N −3−k ∑3 −k2

=

k3 =0

k2 =0

( ) ( ) ( ) 4 5 6 N− N − k3 − N − k3 − k2 − 2 k3 2 k2 2 k1

2k1 +k2 +k3

k1 =0

× (2 (N − 3 − k1 − k2 − k3 ) − 1)!!. Thus, we see that, for 2 ≤ i ≤ N + 1, (2.24) ai (N + 1) =

N∑ −i+1 N −i+1−k ∑ i−1 ki−1 =0

×

i ∏



N −i+1−ki−1 −···−k2

∑

···

ki−2 =0

N −

j=2

2

k1 =0

i−1 ∑ l=j



∑i−1 j=1

kj

 

2i − j  kl − 2

2 N − i + 1 −

i−1 ∑





kj  − 1!!.

j=1

kj−1

Therefore, we obtain the following theorem. Theorem 1. The nonlinear differential equations 2N N !F N +1 =

N ∑

ai (N ) (x − t)

i−2N

F (i) ,

(N ∈ N)

i=1

has a solution F = F (t, x) =

1 1−2tx+t2 ,

where

a1 (N ) = (2N − 3)!!, ai (N ) =

N −i N −i−k ∑ ∑i−1 ki−1 =0

×

i ∏ j=2



N −i−ki−1 −···−k2

···

ki−2 =0

N −

∑

∑i−1

2

j=1

kj

k1 =0

i−1 ∑ l=j

    i−1 ∑ 2i + 2 − j  2 N − i − kl − kj  − 1!! 2 kj−1 j=1

(2 ≤ i ≤ N ). From (1.3) and (1.9), we note that (2.25)

∞ ∑

Un (x) tn

n=0

=

1 1 − 2xt + t2

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(

)2 1 = √ 1 − 2xt + t2 (∞ )( ∞ ) ∑ ∑ l m = pl (x) t pm (x) t l=0 ( n ∞ ∑ ∑

=

n=0

m=0

)

pl (x) pn−l (x) tn .

l=0

Thus, from (2.25), we have n ∑

Un (x) =

pl (x) pn−l (x) .

l=0

From (1.4), we obtain 2N N !F N +1 = 2N N !

(2.26)

∞ ∑

Un(N +1) (x) tn .

n=0

On the other hand, by Theorem 1, we get (2.27) 2N N !F N +1 =

N ∑

ai (N ) (x − t)

i=1

i−2N

F (i)

) )( ∞ ) ∞ ( ∑ ∑ 2N + m − i − 1 i−2N −m m l Ui+l (x) (l + i)i t x t = ai (N ) m m=0 i=1 l=0 } { n ( N ∞ ∑ 2N + n − l − i − 1) ∑ ∑ xi−2N −n+l Ul+i (x) (l + i)i tn = ai (N ) n − l n=0 i=1 l=0 {N } ) ∞ n ( ∑ ∑ ∑ 2N + n − l − i − 1 i+l−2N −n = ai (N ) x Ui+l (x) (l + i)i tn . n − l n=0 i=1 (

N ∑

l=0

Comparing the coefficients on the both sides of (2.26) and (2.27), we obtain the following theorem. Theorem 2. For N ∈ N, and n ∈ N ∪ {0}, the following identity holds. Un(N +1) (x) =

) N n ( ∑ 2N + n − l − i − 1 1 ∑ a (N ) Ul+i (x) xi+l−2N −n (l + i)i . i 2N N ! i=1 n−l l=0

The higher-order Legendre polynomials are given by the generating function )α ∑ ( ∞ 1 n √ = p(α) (2.28) n (x) t . 1 − 2xt + t2 n=0 Thus, by 1.4 and (2.27), we get (2.29)

∞ ∑

Un(α) (x) tn

n=0

(

=

1 1 − 2xt + t2

)α

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(

)2α 1 = √ 1 − 2xt + t2 (∞ )( ∞ ) ∑ (α) ∑ l (α) m = pl (x) t pm (x) t =

l=0 ( n ∞ ∑ ∑ n=0

m=0 (α) pl

(α) (x) pn−l

)

(x) tn .

l=0

From (2.29), we note that Un(α)

(2.30)

(x) =

n ∑

(α)

pl

(α)

(x) pn−l (x) .

l=0

Therefore, we obtian the following corollaries. Corollary 3. For N ∈ N and n ∈ N ∪ {0}, we have n ∑

(N +1) (N +1) pn−l

pl

(x)

l=0

=

) N n ( ∑ 2N + n − l − i − 1 1 ∑ a (N ) Ul+i (x) (l + i)i xi+l−2N −n . i 2N N ! i=1 n−l l=0

Corollary 4. For N ∈ N and n ∈ N, we have Un(N +1) (x)

) N n ∑ l+i ( ∑ 1 ∑ 2N + n − l − i − 1 i+l−2N −n = N ai (N ) x (l + i)i (x) pl+i−j (x) . 2 N ! i=1 n−l j=0 l=0

By (1.6), we get 2N N !F N +1

(2.31)

( )N +1 1−t −N −1 = 2N N ! (1 − t) 1 − 2xt + t2 ) ( ∞ ( )( ∞ ∑ N + m) ∑ (N +1) N l m = 2 N! t Vl (x) t m m=0 l=0 ( n ( ) ∞ ∑ ∑ N + n − l) (N +1) N = 2 N! Vl (x) tn . n − l n=0 l=0

On the other hand, by Theorem 1, we have (2.32)

N

2 N !F

N +1

=

N ∑

ai (N ) (x − t)

i−2N

ai (N ) (x − t)

i−2N

i=1

=

N ∑

F (i) (

i=1

d dt

)i (

1−t 1 · 1 − t 1 − xt + t2

) .

From Leibniz formula, we note that ( )i ( ) d 1−t 1 (2.33) · dt 1 − 2xt + t2 1 − t

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( ) (( ) ) )i−l i ( ) ( l ∑ i d 1 d 1−t = l dt 1−t dt 1 − 2xt + t2 l=0 ( )l ( ) i ( ) ∑ i d 1−t −i+l−1 = (i − l)! (1 − t) l dt 1 − 2xt + t2 l=0 ) ∞ ∞ ( i ( ) ∑ ∑ i−l+s s∑ i t Vp+l (x) (p + l)l tp (i − l)! = s l p=0 s=0 l=0

) ∞ i ∞ ( ∑ i! ∑ i − l + s s ∑ = t Vp+l (x) (p + l)l tp . l! s=0 s p=0 l=0

By (2.32) and (2.33), we get 2N N !F N +1 {N i ( )( ) ∞ ∑ ∑∑ i! ∑ 2N + m − i − 1 i − l + s = ai (N ) l! m+s+p=n m s n=0 i=1 l=0 } × (p + l)l xi−2N −m Vp+l (x) tn .

(2.34)

Therefore, by (2.31) and (2.34), we obtain the following theorem. Theorem 5. For N ∈ N and n ∈ N ∪ {0}, we have the following identity: ) n ( ∑ N +n−l (N +1) Vl (x) n−l l=0 ( )( ) N i 1 ∑∑ i! ∑ 2N + m − i − 1 i − l + s = N ai (N ) (p + l)l 2 N ! i=1 l! m+s+p=n m s l=0

i−2N −m

×x

Vp+l (x) .

From (1.8), we note that (2.35)

2N N !F N +1

( )N +1 1+t −N −1 = 2N N ! (1 + t) 1 − 2xt + t2 ( ∞ ( )( ∞ ) ∑ N + m) ∑ (N +1) m m N l = 2 N! (−1) t Wl (x) t m m=0 l=0 ( n ) ( ) ∞ ∑ ∑ N + n − l (N +1) n−l = 2N N ! (−1) Wl (x) tn . n − l n=0 l=0

On the other hand, by Theorem 1, we get (2.36)

N

2 N !F

N +1

=

N ∑

( ai (N ) (x − t)

i−2N

i=1

d dt

)i {

1+t 1 · 1 + t 1 − 2xt + t2

} .

Now, we observe that ( )i {( )( )} d 1 1+t (2.37) dt 1+t 1 − 2xt + t2

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)i−l+1 ( )l ( ) 1 d 1+t l 1+t dt 1 − 2xt + t2 l=0 ) i ( ) ∞ ( ∞ ∑ ∑ ∑ i i−l+s i−l s = (−1) (i − l)! (−1) ts Wp+l (x) (p + l)l tp . l s s=0 p=0 =

i ( ) ∑ i

(

(−1)

i−l

(i − l)!

l=0

From (2.36) and (2.37), we have (2.38)

2N N !F N +1 {N ( ) ∞ i ∑ ∑ ∑ ∑ s 2N + m − i − 1 i−l i! = (−1) ai (N ) (−1) l! m+s+p=n m n=0 i=1 l=0 ( ) } i−l+s × (p + l)l xi−2N −m Wp+l (x) tn . s

Therefore, by (2.35) and (2.38), we obtain the following theorem. Theorem 6. For N ∈ N and n ∈ N ∪ {0}, the following identity is valid: ( ) n ∑ (N +1) n−l N + n − l (−1) Wl (x) n−l l=0 ( ) N i i! ∑ 1 ∑∑ i−l s 2N + m − i − 1 (−1) ai (N ) (−1) = N 2 N ! i=1 l! m+s+p=n m l=0 ( ) i−l+s × (p + l)l xi−2N −m Wp+l (x) . s From (1.1), we have (2.39) 2N N !F N +1 ( )N +1 1 1 − t2 N = 2 N! · 1 − t2 1 − 2xt + t2 )N +1 ( )N +1 ( )N +1 ( 1 1 1 − t2 = 2N N ! 1−t 1+t 1 − 2xt + t2 (∞ ( ) ( )( ∞ ) ) ∞ ( ∑ N + l) ∑ ∑ m+N m m N l (N +1) p = 2 N! t (−1) t (x) t Tp l m m=0 p=0 l=0   ∞ ∑ ∑ (N + l)(m + N ) m  = 2N N ! (−1) Tp(N +1) (x) tn . l m n=0 l+m+p=n

On the other hand, by Theorem 1, we get (2.40)

2N N !F N +1 =

N ∑

ai (N ) (x − t)

i−2N

F (i)

i=1

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1∑ i−2N ai (N ) (x − t) 2 i=1 N

=

(

d dt

)i {(

1 1 + 1−t 1+t

)

11

1 − t2 1 − 2xt + t2

} .

From Leibniz formula, we note that the following equations: ( )i {( ) ( )} d 1 1 − t2 (2.41) · dt 1−t 1 − 2xt + t2 ) ∞ ( ) i ∞ ( ∑ ∑ i+s−l s∑ i t Tp+l (x) (p + l)l tp , = (i − l)! s l p=0 s=0 l=0

and

(

(2.42)

)} 1 − t2 1 − 2xt + t2 ) i ( ) ∞ ( ∞ ∑ ∑ ∑ i i−l+s i−l s = (i − l)! (−1) (−1) ts Tp+l (x) (p + l)l tp . l s s=0 p=0 d dt

)i {(

1 1+t

)(

l=0

By (2.40), (2.41), and (2.42), we obtain (2.43) 2N N !F N +1 ) ∞ N i ( ) ∞ ( ∑ ∑ 1∑ i i+s−l s∑ i−2N = ai (N ) (x − t) (i − l)! t Tp+l (x) (p + l)l tp 2 i=1 l s s=0 l=0 k=0 ) i ( ) N ∞ ( ∑ ∑ ∑ i 1 i−l+s i−l i−2N s (i − l)! (−1) + ai (N ) (x − t) (−1) ts 2 i=1 l s s=0 l=0

×

∞ ∑

Tp+l (x) (p + l)l tp

p=0

( )( ) ∞ N i 2N + m − i − 1 i + s − l 1 ∑∑∑ i! ∑ (p + l)l = ai (N ) 2 n=0 i=1 l! m+s+p=n m s l=0

∞ N i 1 ∑∑∑ i! i−l ai (N ) (−1) 2 n=0 i=1 l! l=0 ( )( ) ∑ 2N + m − i − 1 i+s−l s × (−1) (p + l)l xi−2N −m Tp+l (x) tn . m s m+s+p=n

×xi−2N −m Tp+l (x) tn +

Therefore, by (2.39) and (2.43), we obtain the following theorem. Theorem 7. For n ∈ N ∪ {0} and N ∈ N, we have the following identity ∑ (N + s)(m + N ) m N +1 (−1) Tp(N +1) (x) 2 N! s m s+m+p=n =

N ∑ i ∑ i=1 l=0

( )( ) i! ∑ 2N + m − i − 1 i + s − l ai (N ) (p + l)l l! m+s+p=n m s

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TAEKYUN KIM, DAE SAN KIM, JONG-JIN SEO, AND DMITRY V. DOLGY

×xi−2N −m Tp+l (x) +

N ∑ i ∑ i=1 l=0

ai (N )

( ) ∑ i! s 2N + m − i − 1 i−l (−1) (−1) m l! m+s+p=n

( ) i+s−l × (p + l)l xi−2N −m Tp+l (x) . s

Acknowledgements. This paper is supported by grant NO 14-11-00022 of Russian Scientific Fund. References 1. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642 (29 #4914) 2. V. M. Babich, To the problem on the asymtotics with respect to the indices of the associated Legendre function, Russ. J. Maht. Phys. 9 (2002), no. 1, 6–13. 3. L. Carlitz, Some arithmetic properties of the Chebyshev polynomials, Arch. Math. (Basel) (1965). 4. Y. V. Darevskaya, On some algebraic properties of the Chebyshev polynomials, Usekhi Mat. Nauk. 58 (2003), no. 1, 181–182, :translation in Russian Math. Surveys 58 (2003), no. 1., 175–176. 5. B. G. Gabdulkhaev and L. B. Ermolaeva, Interpolation over extreme points of Chebyshev polynomials and its applications, Izv. Vyssh. Uchebn. Zaved. Mat. (2005), no. 5, 22–41, translation in Russian Math. (Iz. VUZ) 49 (2005), no. 5, 1937 (2006). MR 2186867 (2006g:41013) 6. K. Hejranfar and M. Hajihassanpour, Chebyshev collocation spectral lattice boltzmann method for simulation of low-speed flows, Phys. Rev. E 91 (2015), 013301. 7. D. S. Kim, D. V. Dolgy, T. Kim, and S.-H. Rim, Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials, Proc. Jangjeon Math. Soc. 15 (2012), no. 4, 361–370. MR 3050107 8. D. S. Kim, T. Kim, and S.-H. Lee, Some identities for Bernoulli polynomials involving Chebyshev polynomials, J. Comput. Anal. Appl. 16 (2014), no. 1, 172–180. MR 3156166 9. P. Kim, J. Kim, W. Jung, and S. Bu, An error embedded method based on generalized Chebyshev polynomials, J. Comput. Phys. 306 (2016), 55–72. MR 3432341 10. T. Kim, Identities involving Laguerre polynomials derived from umbral calculus, Russ. J. Math. Phys. 21 (2014), 36–452. MR 3182545 11. T. Kim, D.S. Kim, A note on nonlinear Changhee differential equations, Russ. J. Math. Phys. 23 (2016), no. 1, 1–5. 12. Y. Liu, Adomian decomposition method with second kind Chebyshev polynomials, Proc. Jangjeon Math. Soc. 12 (2009), no. 1, 57–67. MR 2542048 (2010g:65102) 13. V. D. Lyakhovsky, Chebyshev polynomials for a three-dimensional algebra, Teoret. Mat. Fiz. 185 (2015), no. 1, 118–126. MR 3438608 14. T. Mansour, Adjoint polynomials of bridge-path and bridge-cycle graphs and Chebyshev polynomials, Discrete Math. 311 (2011), no. 16, 1778–1785. MR 2806041 (2012f:05145)

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SOME IDENTITIES OF CHEBYSHEV POLYNOMIALS

13

15. N. N. Osipov and N. S. Sazhin, An extremal property of Chebyshev polynomials, Russian J. Numer. Anal. Math. Modelling 23 (2008), no. 1, 89–95. MR 2384894 (2009f:33011) 16. S. O’Sullivan, A class of high-order Runge-Kutta-Chebyshev stability polynomials, J. Comput. Phys. 300 (2015), 665–678. 17. Steven Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. MR 741185 (87c:05015) 18. W. Siyi, Some new identities of Chebyshev polynomials and their applications, Adv. Difference Equ. (2015), 2015:355. MR 3425412 19. B. Spain and M. G. Smith, Functions of mathematical physics, (1970). 20. H. M. Srivastava, Shy-Der Lin, Shuoh-Jung Liu, and Han-Chun Lu, Integral representations for the Lagrange polynomials, Shively’s pseudo-Laguerre polynomials, and the generalized Bessel polynomials, Russ. J. Math. Phys. 19 (2012), no. 1, 121–130. MR 2892608 21. D. G. Zill and W. S. Wright, Advanced Engineering Mathematics, Jones&Bartlett Publishers, 2009. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China, Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected] Department of Applied Mathematics, Pukyong National University, Pusan, Republic of Korea E-mail address: [email protected] School of Natural Sciences, Far Eastern Federal University, Vladivostok, Russia E-mail address: [email protected]

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Blowup singularity for a degenerate and singular parabolic equation with nonlocal boundary ∗ Dengming Liu1†and Jie Ma2 1. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. China 2. College of Science, Henan University of Engineering, Xinzheng, Henan 451191, P. R. China

Abstract In this paper, we are interested in the blowup behavior of the solution to a degenerate and singular parabolic equation Z l ut = (xα ux )x + up dx − kuq , (x, t) ∈ (0, l) × (0, +∞) 0

with nonlocal boundary condition Z l u (0, t) = f (x) u (x, t)dx,

Z u (l, t) =

0

l

g (x) u (x, t)dx,

t ∈ (0, +∞) ,

0

where p, q ∈ [1, ∞ ), α ∈ [0, 1 ) and k ∈ (0, ∞). In view of comparison principle, we investigate the conditions on the global existence and blowup of the solutions. Moreover, under some suitable hypotheses, we discuss the global blowup and the uniform blowup profile of the blowup solution. Keywords: Degenerate and singular parabolic equation; Nonlocal boundary; Global existence; Blowup singularity Mathematics Subject Classification(2000) : 35K50, 35K55, 35K65

1

Introduction

The main purpose of this paper is to deal with the blowup singularity of the following degenerate and singular parabolic equation with nonlocal source and nonlocal boundary condition  Rl   ut = (xα ux )x + 0 up dx − kuq , (x, t) ∈ (0, l) × (0, +∞) ,     R   u (0, t) = l f (x) u (x, t)dx, t ∈ (0, +∞) , 0 (1.1) R l   u (l, t) = g (x) u (x, t)dx, t ∈ (0, +∞) ,  0      u (x, 0) = u (x) ≥ 0, x ∈ [0, l] , 0

∗ This work is supported by National Natural Science Foundation of China (11426099, 11526076, 11571102), Scientific Research Fund of Hunan Provincial Education Department (14B067, 15A062) † Corresponding Author: [email protected]

1

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where 0 ≤ α < 1, p, q ≥ 1, k > 0, the weight functions f (x) and g (x) in the boundary condition are nonnegative continuous on [0, l] and not identically zero, and the initial value u0 (x) ∈ C 2+δ (0, l) ∩ C [0, l] with 0 < δ < 1, and satisfies the compatibility conditions. It is obvious that the equation in problem (1.1) is singular and degenerate because the coefficients of ux and uxx may tend to ∞ and 0 as x → 0. This type equation in problem (1.1) can be viewed as a model which describes the conduction of heat related to the geometric shape of the body (see [1] and the references therein for more details of the physical background). On the other hand, lots of physical phenomena were formulated into nonlocal mathematical models, for example, Day [4, 5] derived a heat equation with nonlocal boundary in the study of the heat conduction with thermoelastcity. From then on, a lot of mathematicians devoted to studying the blowup behavior of the solutions of various parabolic problems with nonlocal boundary conditions (see [6, 7, 8, 9, 10, 11, 13, 15, 16, 21]). The blowup phenomenon related to problem (1.1) attracted extensive attention of mathematicians in the past several decades (see [2, 3, 12, 18, 20, 22, 23]), but most of them considered the problems with null Dirichlet boundary conditions. Inspired by the works mentioned above, we consider problem (1.1), and our main attention is focused on evaluating the effects of the weighted nonlocal boundary, the nonlocal source and absorption term on the asymptotic blowup behavior of the solution u (x, t) of problem (1.1). Compared with [3] and [18], we need more skills to handle the difficulties, which are produced by the degeneration and singularity of problem (1.1), and the appearance of the nonlinear nonlocal boundary condition. Before stating our main results, for the sake of convenience, we denote (Z ) Z l

N = max

l

f (x)dx, 0

g (x)dx , 0

and let λ1 be the first eigenvalue and ζ1 (x) be the corresponding eigenfunction of the following eigenvalue problem − (xα ζx )x = λ1 ζ, 0 < x < l; ζ (0) = ζ (l) = 0. (1.2) Indeed, from [3, 14], we know that the principle eigenvalue λ1 of the eigenvalue problem (1.2) is the first zero of ! √ 2 λ 2−α J 1−α l 2 = 0, 2−α 2−α and ζ1 (x) can be expressed in an explicit form as follows  √  1−α 2 λ1 2−α , ζ1 (x) = ax 2 J 1−α x 2 2−α 2−α

(1.3)

1−α where J 1−α is Bessel function of the first kind of order 2−α , and a is an appropriate positive parameter 2−α such that kζ1 (x)kL1 ([0,l]) = 1. Furthermore, we know easily that ζ1 (x) is a positive smooth function in (0, l), and in light of d ϑ Jϑ (τ ) = Jϑ (τ ) − Jϑ+1 (τ ) , dτ 2 we can deduce that, for x ∈ (0, l), √    √  √  p 1+α 1−2α d a (1 − α) λ1 2−α 2 λ1 2−α λ1 2−α ζ1 (x) = 1+ x 2 x− 2 J 1−α x 2 − a λ1 x 2 J 3−2α x 2 . 2−α 2−α dx 2 2−α 2−α 2−α

2

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And hence, by making use of Jϑ (τ ) →

 τ ϑ 1 as τ → 0, Γ (ϑ + 1) 2

where Γ (·) is the Gamma function, we find that lim ζ1 (x) = 0

x→0+

and a (1 − α) d   lim ζ1 (x) = + x→0 dx 2Γ 3−2α 2−α

 √  1−α 2 λ1 2−α , 2−α

which imply that d ζ1 (x) < ∞. dx x∈[0,l]

sup ζ1 (x) < ∞ and sup x∈[0,l]

(1.4)

The main results of this paper are stated as follows. Theorem 1.1. Assume that q > p ≥ 1, then all the solutions of problem (1.1) exist globally. Theorem 1.2. Assume that p > q ≥ 1, then problem (1.1) admits blowup solutions as well as global solutions. More precisely,
1 (i) if N ≤ 1, then the solution exists globally provided that u0 (x) ≤ kl p−q ; (ii) if N > 1, then the solution of problem (1.1) blows up in finite time provided that u0 (x) > η1 , where η1 > 1 is an appropriate constant; (iii) there is a suitable positive small constant η2 such that the solution u (x, t) of problem (1.1) blows up in finite time for any f (x) and g (x) provided that   1 l x1−α − x2−α , u0 (x) > η2−ξ 2−α 2−α where ξ >

1 p−1 .

Theorem 1.3. Assume that p = q > 1. The solution u (x, t) of problem (1.1) exists globally provided that N < 1 and u0 (x) ≤ 1 N , where 1 is given by (3.13). For any nonnegative weight functions f (x) and g (x), the solution u (x, t) of problem (1.1) blows up in finite time provided that the initial value u0 (x) is sufficiently large. Remark 1.1. If p = q = 1, one can show that problem (1.1) has no blowup solution. The remaining part is devote to discussing the global blowup and the uniform blowup profile of the blowup solution, to this end, we assume that p > q ≥ 1 (or p = q > 1), N ≤ 1 and u0 (x) is large enough in some suitable sense. Moreover, we assume that u0 (x) satisfies extra Z l α (x u0x )x + up0 dx − kuq0 ≥ 0 for x ∈ (0, l) , (1.5) 0

(xα u0x )x ≤ 0 in (0, l) , and

" lim

x→0+

α

(x u0x )x +

Z 0

#

l

up0 dx

−

kuq0

" α

= lim− (x u0x )x + x→l

(1.6) Z

#

l

up0 dx

−

kuq0

= 0.

(1.7)

0

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Theorem 1.4. Assume that p > q ≥ 1 and N ≤ 1. Suppose that hypotheses (1.5), (1.6) and (1.7) hold. Then − 1 u (x, t) ∼ [l (p − 1) (T − t)] p−1 a.e. in (0, l) as t → T, where T is the blowup time. Corollary 1.1. Under the assumptions of Theorem 1.4, we see that the blowup set of the blowup solution u (x, t) of problem (1.1) is the whole interval (0, l). Theorem 1.5. Assume that p = q > 1, N ≤ 1 and 0 < k < l. Suppose that hypotheses (1.5), (1.6) and (1.7) hold. Then − 1 u (x, t) ∼ [l (p − 1) (T − t)] p−1 a.e. in (0, l) as t → T, where T is the blowup time. Corollary 1.2. Under the assumptions of Theorem 1.5, we know that the blowup set of the blowup solution u (x, t) of problem (1.1) is the whole interval (0, l). The rest of this paper is organized as follows. In Section 2, we shall state the comparison principle and local existence theorem for problem (1.1). In section 3, we shall concern with the conditions on the global existence of solution and prove Theorems 1.1, 1.2 and 1.3. In section 4, we shall estimate the uniform blowup profile and give the proofs of Theorems 1.4 and 1.5.

2

Comparison principle and local existence

In this section, we will establish a suitable comparison principle for problem (1.1) and state the existence and uniqueness result on the local solution. For the sake of simplify, we denote IT = (0, l) × (0, T ) and I T = [0, l] × [0, T ). First, we give the definitions of the super-solution and sub-solution to problem (1.1). Definition 2.1. A nonnegative function u (x, t) is called a super-solution of problem (1.1) if u (x, t) ∈
C 2,1 (IT ) ∩ C I T satisfies  Rl   ut ≥ (xα ux )x + 0 up dx − kuq , (x, t) ∈ IT ,     R   u (0, t) ≥ l f (x) u (x, t)dx, t ∈ (0, T ) , 0 Rl   u (l, t) ≥ 0 g (x) u (x, t)dx, t ∈ (0, T ) ,       u (x, 0) ≥ u (x) , x ∈ [0, l] . 0

(2.1)


Similarly, u (x, t) ∈ C 2,1 (IT ) ∩ C I T is called a sub-solution of problem (1.1) if it satisfies all the reversed inequalities in (2.1). We say that u (x, t) is a solution of problem (1.1) if it is both a sub-solution and a super-solution of problem (1.1). Now, by using the similar arguments as those in [6] (or [10]), we give directly the following maximum principle.

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Lemma 2.1. Let ω (x, t) ∈ C 2,1 (IT ) ∩ C I T satisfy  R   ωt − (xα ωx )x ≥ θ1 (x, t) ω + l θ1 (x, t) ω (x, t) dx, (x, t) ∈ IT ,  0   Rl ω (0, t) ≥ 0 θ3 (x) ω (x, t)dx, t ∈ (0, T ) ,    R   ω (l, t) ≥ l θ4 (x) ω (x, t)dx, t ∈ (0, T ) , 0

(2.2)

where θi (x, t), i = 1, 2, 3, 4, are bounded functions, θ2 (x, t) is nonnegative for (x, t) ∈ IT , θ3 (x) and θ4 (x) are nonnegative, nontrivial in (0, l). Then ω (x, 0) > 0 in [0, l] implies that ω (x, t) > 0 for (x, t) ∈ IT . Moreover, if one of the following conditions holds,o(i) θ3 (x) = θ4 (x) ≡ 0 for x ∈ (0, l); (ii) θ3 (x), θ4 (x) ≥ 0 nR Rl l for x ∈ (0, l) and max 0 θ3 (x) dx, 0 θ4 (x) dx ≤ 1, then ω (x, 0) ≥ 0 in [0, l] leads to ω (x, t) ≥ 0 for (x, t) ∈ IT . Based on the idea of [10], we can establish the comparison principle for problem (1.1) as follows, which is the main tool of establishing the conditions on the global existence and blowup of the solution. Proposition 2.1 (Comparison principle). Let u (x, t) and u (x, t) be a nonnegative super-solution and sub-solution of problem (1.1), respectively. Then u (x, t) ≥ u (x, t) holds in I T if u (x, 0) ≥ u (x, 0) for x ∈ [0, l]. Next, we state the result on the existence and uniqueness of the local solution of problem (1.1) at the end of this section. Theorem 2.1 (Local existence and uniqueness). Assume that (1.5) holds, then there exists a small positive
real number T such that problem (1.1) admits a unique nonnegative solution u(x, t) ∈ C I T ∩ C 2,1 (IT ). Remark 2.1. We can get the proof of Theorem 2.1 by using regularization method and Schauder’s fixed point theorem. For more details, we refer the readers to [3, 23].

3

Global existence of solution

The main goal of this section is to discuss the global existence and blowup property of the solution u (x, t) to the problem (1.1). To this end, by Proposition 2.1, we only need to construct some suitable global super-solutions (or blowup sub-solutions). Proof of Theorem 1.1. Let T be any positive number and u1 (x, t) be defined as u1 (x, t) =

χ2 eχ3 t χ1 ζ1 (x) + 1

where χ1 is large enough such that Z l 0

1 dx ≤ max 1 + χ1 ζ1 (x)



 max f (x) , max g (x) ,

x∈[0,l]

x∈[0,l]

and χ2 = max

 

"

max (u (x) + 1) (χ1 ζ1 (x) + 1) , max x∈[0,l] 0 x∈[0,l]

(χ1 ζ1 (x) + 1) k

q

Z 0

l

 1 # q−p  1 , p dx  (1 + χ1 ζ1 (x))

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dζ1 (x) 2 . χ3 = λ1 + max 2 x∈[0,l] (χ1 ζ1 (x) + 1) dx By the direct calculation, one has Z l P u1 : = u1t − (xα u1x )x − up1 dx + kuq1 0 " 2 !# d λ1 χ1 ζ1 (x) 2xα χ21 ζ1 (x) = u1 χ3 − + 1 + χ1 ζ1 (x) (χ1 ζ1 (x) + 1)2 dx  q Z l
χ2 eχ3 t 1 χ3 t p +k − χ2 e p dx 1 + χ1 ζ1 (x) (1 + χ 1 ζ1 (x)) 0 ≥ 0, 2lα χ21

(3.1)

and

max (u0 (x) + 1) (1 + χ1 ζ1 (x)) χ2 x∈[0,l] ≥ > u0 (x) . (3.2) 1 + χ1 ζ1 (x) 1 + χ1 ζ1 (x) On the other hand, we can verify that Z l Z l Z l 1 f (x) χ2 eχ3 t u1 (0, t) = χ2 eχ3 t ≥ χ2 eχ3 t max f (x) dx ≥ dx = f (x) u1 (x, t) dx, x∈[0,l] 0 1 + χ1 ζ1 (x) 0 1 + χ1 ζ1 (x) 0 (3.3) u1 (x, 0) =

and Z u1 (l, t) ≥

l

(3.4)

g (x) u1 (x, t) dx. 0

Combining now from (3.1) to (3.4), we know that u1 (x, t) is a global super-solution of (1.1) in IT and the solution u (x, t) of (1.1) exists globally by Proposition 2.1. The proof of Theorem 1.1 is complete.
1 Proof of Theorem 1.2. (i) If p > q and N > 1, then it is easy to check that u2 (x) = kl p−q is a global
1 super-solution of problem (1.1) provided that u0 (x) ≤ kl p−q . (ii) Consider the following ordinary differential equation    v 0 (t) = lv p − kv q , t > 0, 1 1 1 (3.5)   v 1 (0) = v 10 . From p > q ≥ 1, it follows that v q1 ≤ v p1 + 1, and hence, we have lv p1 − kv q1 ≥ (l − k) v p1 − k, which tells us that the solution v 1 (t) of (3.5) is a super-solution of the following problem    v 0 (t) = (l − k) v p − k, t > 0, 2 2

(3.6)

  v 2 (0) = v 10 provided l > k. Noticing that (l − k) v p2 is convex, then there exists η1 > 1 such that (l − k) v p2 ≥ 2k holds for v 2 ≥ η1 . It follows easily that if v 2 (0) = v 10 > η1 , then v 2 (t) is increasing on its interval of the existence and l−k p v 2 0 (t) ≥ v . (3.7) 2 2 6

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From the above inequality it follows that v 2 (t) → ∞ as t →

2 , (l − k) (p − 1) v p−1 10

(3.8)

which leads to that v 1 (t) will become infinite in a finite time. Recalling that N > 1, then v 1 (t) is a blowup sub-solution of problem (1.1) when u0 (x) ≥ v 10 > η, so the solution u (x, t) of problem (1.1) blows up in finite time for sufficiently large initial value. (iii) Let v (x, t) be the solution of the following auxiliary problem  Rl    vt = (xα vx )x + 0 v p (x, t)dx − kv q , 0 < x < l, t > 0,   (3.9) v (0, t) = v (l, t) = 0, t > 0,      v (x, 0) = u0 (x) , 0 < x < l, then v (x, t) is a sub-solution of problem (1.1). Set   l 1 −ξ −ξ x1−α − x2−α := (η2 − t) µ (x) , v 3 (x, t) = (η2 − t) 2−α 2−α where η2 and ξ > 0 will be chosen later. Calculating directly, we have Z l α P v 3 : = v 3t − (x v 3x )x − v p3 (x, t)dx + kv q3 0 " −ξp

= (η2 − t)

ξp−ξ−1

ξ (η2 − t)

ξ(p−1)

µ (x) + (η2 − t)

ξ(p−q)

+ k (η2 − t)

q

Z

µ (x) −

#

l p

µ (x) dx . 0

Since p > q ≥ 1, we can take ξ large enough such that ξp − ξ − 1 > 0, then we have P v 3 ≤ 0 with η2 − t small enough, which implies that v 3 (x, t) is a blowup sub-solution to problem (3.9) provided that v (x, 0) = u0 (x) > µ (x) η2−ξ . And hence, Proposition 2.1 tells us that the solution u (x, t) of problem (1.1) blows up in finite time for large initial value. The proof of Theorem 1.2 is completed. Proof of Theorem 1.3. For any given constant 3−α

0 ∈

0,

(1 − N ) (2 − α) l2−α (1 − α)

! ,

1−α

(3.10)

let σ (x) be the unique positive solution of the following ordinary differential equation    − (xα σx ) = 0 , 0 < x < l, x

(3.11)

  σ (0) = σ (l) = N . In fact, we can solve the explicit expression of σ (x) as follows l0 1−α 0 2−α x − x + N, 2−α 2−α Moreover, according to N < 1, we can verify that σ (x) =

x ∈ [0, l] .

1−α

0 < min σ (x) = N < max σ (x) = N + x∈[0,l]

x∈[0,l]

0 l2−α (1 − α)

3−α

(2 − α)

< 1.

(3.12)

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Define u3 (x, t) = 1 σ (x) , where

1 =

   

0 kN p −l

1  p−1

if kN p − l > 0,

,

  any fixed positive constant,

(3.13)

if kN p − l ≤ 0.

Calculating directly, one has Z

P u3 : = u3t − (xα u3x )x − Z

l

up3 dx + kup3

0

l

σ p dx + kp1 σ p 0 p  p p p ≥ 0 1 − l1 max σ (x) + k1 min σ (x)

= 0 1 −

p1

x∈[0,l]

> 0 1 −

p1

(3.14)

x∈[0,l]

p

(kN − l)

≥ 0. Meanwhile, we can prove that Z u3 (0, t) = 1 N ≥

l

l

Z 1 f (x) dx >

Z 1 σ (x) f (x) dx =

0

0

and Z u3 (l, t) >

l

u3 (x, t) f (x) dx

(3.15)

0

l

u3 (x, t) f (x) dx.

(3.16)

0

Then u3 (x, t) is a global super-solution of problem (1.1) if u0 (x) ≤ 1 N , and hence, we obtain our global existence result by Proposition 2.1. The proof of blowup conclusion in this case is similar to the arguments of (iii) in Theorem 1.2, we omit the details here. The proof of Theorem 1.3 is completed.

4

Global blowup set and uniform blowup profile

This section is mainly about the global blowup and the uniform blowup profile of the blowup solution for problem (1.1). Throughout this section, we assume that p > q ≥ 1 (or p = q > 1), N ≤ 1 and u0 (x) is large enough in some suitable sense. From Theorems 1.2 and 1.3, it follows that the solution u (x, t) of problem (1.1) blows up in finite. For convenience, we denote T the blowup time. From the assumptions on the initial value u0 (x) and (1.5), (1.6) and (1.7), we can find a sufficiently small positive constant ε1 and a nonnegative function w0ε (x) such that (1) w0ε ∈ C 2+δ (ε, l − ε) ∩ C [ε, l − ε] with δ ∈ (0, 1) and ε ∈ (0, ε1 ]. R l−ε R l−ε (2) w0ε (ε) = ε f (x) w0ε (x) dx and w0ε (l − ε) = ε g (x) w0ε (x) dx. (3) w0ε (x) < u0 (x) for x ∈ (ε, 2ε) ∪ (l − 2ε, l − ε), and w0ε (x) = u0 (x) for x ∈ [2ε, l − 2ε]. (4) (xα w0εx )x ≤ 0 for x ∈ (ε, l − ε). 8

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(5) (xα w0εx )x +

R l−ε

p q w0ε dx − kw0ε ≥ 0 for ε ∈ (0, ε1 ] and x ∈ (ε, l − ε).

ε

(6) w0ε is non-increasing with respect to ε in (0, ε1 ]. Moreover " # " Z Z l−ε q p α α (x w0εx )x + lim (x w0εx )x + w0ε dx − kw0ε = lim x→ε+

x→(l−ε)−

ε

#

l−ε p dx w0ε

−

q kw0ε

= 0.

ε

It is obvious that lim w0ε (x) = u0 (x) .

ε→0+

Now, we consider the following regularized problem  R l−ε   wεt = (xα wεx )x + ε wεp dx − kwεq , (x, t) ∈ (ε, l − ε) × (0, +∞) ,     R   wε (ε, t) = l−ε f (x) wε (x, t)dx, t ∈ (0, +∞) , ε R l−ε   wε (l − ε, t) = ε g (x) wε (x, t)dx, t ∈ (0, +∞) ,       w (x, 0) = w (x) , x ∈ [0, l] . ε 0ε

(4.1)

Then it is not difficult to show that there exists a unique solution wε (x, t) for problem (4.1). In addition, from the arguments of Section 2 in [23], it follows that lim wε (x, t) = u (x, t) ,

ε→0+

where u (x, t) is the solution of problem (1.1). Lemma 4.1. Suppose that hypotheses (1.5), (1.6) and (1.7) hold, and assume that p ≥ q > 1 and N ≤ 1. Then (xα ux )x ≤ 0 holds for (x, t) ∈ IT . Proof. Taking η = (xα wεx )x , then from (4.1), we have ( " # ) Z l−ε

ηt =

xα (xα wεx )x +

wεp dx − kwεq

ε

2

= (xα ηx )x − kqwεq−1 η − kq (q − 1) wεq−2 |wεx |

x

(4.2)

x

holds for any (x, t) ∈ (ε, l − ε) × (0, T ), which tells us that ηt − (xα ηx )x + kqwεq−1 η ≤ 0.

(4.3)

On the other hand, for any t ∈ (0, T ), we have l−ε

Z

l−ε

Z f (x) wεt (x, t) dx −

η (ε, t) = ε

α

=

f (x) (x wεx )x +

ε

l−ε

Z

wεp

−

f (x) wε (x, t) dx !

wεp

(x, t) dx −

kwεq

dx

ε

Z

!q

l−ε

(x, t) dx + k

(4.4)

f (x) wε (x, t) dx ε

l−ε

=

ε

l−ε

Z

ε

Z

(x, t) dx + k

ε l−ε

Z

!q

l−ε

Z

wεp

Z

f (x) dx − 1

f (x) η (x, t) dx + ε

!Z

l−ε

ε

"Z −k

l−ε

wεp (x, t) dx

ε

l−ε

f (x) wεq (x, t) dx −

Z

l−ε

!q # f (x) wε (x, t) dx

ε

.

ε

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It follows from Jensen’s inequality that Z

l−ε

f (x) wεq dx −

!q

l−ε

Z

f (x) wε (x, t) dx ε

ε

Z ≥

R l−ε

l−ε

f (x) wε (x, t) dx R l−ε f (x) dx ε

ε

f (x) dx ε

!q

!q

l−ε

Z −

f (x) wε (x, t) ε

≥ 0. Exploiting the above inequality and the assumption N ≤ 1 to (4.4), we can claim that l−ε

Z η (ε, t) ≤

f (x) η (x, t) dx,

t ∈ (0, T ) .

(4.5)

ε

By the analogous arguments, one can also show that Z

l−ε

η (l − ε, t) ≤

g (x) η (x, t) dx

(4.6)

ε

holds for all t ∈ (0, T ). Moreover, noticing that η (x, 0) = (xα w0εx )x ≤ 0 holds for x ∈ (ε, l − ε). Then, maximum principle tells us that η (x, t) = (xα wεx )x ≤ 0 holds for all (x, t) ∈ (ε, l − ε) × (0, T ). In addition, by the arbitrariness of ε, we know that (xα ux )x ≤ 0 holds in IT . The proof of Lemma 4.1 is complete. In what follows, for the sake of simplicity, we denote Z ψ (t) =

l

Z

p

u (x, t) dx and Ψ (t) = 0

t

ψ (τ ) dτ. 0

Lemma 4.2. Assume that (1.5), (1.6) and (1.7) hold, p > q ≥ 1 and N ≤ 1, then there exists a positive constant C such that   Z t C sup (Ψ (t) − u (x, t)) ≤ 2 1 + Z (t) + Ψ (τ ) dτ d x∈Kd 0 in [0, l] × [ T2 , T ), where Z (t) = o (Ψ (t)) as t → T, and Kd = {x ∈ (0, l) : dist (x, 0) ≥ d, dist (x, l) ≥ d} ⊂ (0, l) . Proof. Put Z

l

(Ψ (t) − u (x, t)) ζ1 (x) dx,

F (t) =

(4.7)

0

10

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where ζ1 (x) is given by (1.3). Taking the derivative of F (t) with respect to t, we arrive at Z l 0 F (t) = (ψ (t) − ut ) ζ1 (x) dx 0

Z = 0

l

(− (xα ux )x + kuq ) ζ1 (x) dx Z

l

= λ1

Z

l

uq (x, t) ζ1 (x) dx

u (x, t) ζ1 (x) dx + k 0

0

α

+ l ζ1x |x=l Z ≤ λ1

(4.8)

l

Z

g (x) u (x, t) dx 0

l

Z

l

uq (x, t) ζ1 (x) dx

u (x, t) ζ1 (x) dx + k 0

0 l

Z

uq (x, t) ζ1 (x) dx.

= −λ1 F (t) + λ1 Ψ (t) + k 0

On the other hand, it follows from Lemma 4.1 that ut ≤ ψ (t) − kuq , which implies that − max u0 (x) ≤ Ψ (t) − u (x, t) .

(4.9)

x∈[0,l]

Then (4.9) and (4.8) lead to F0 (t) ≤ λ1 max u0 (x) + λ1 Ψ (t) + k

Z

x∈[0,l]

l

uq (x, t) ζ1 (x) dx.

0

Integrating above inequality over from 0 to t, one has !   Z t Z tZ l q F (t) ≤ max λ1 , k max ζ1 (x) , F (0) + λ1 T max u0 (x) 1+ Ψ (τ ) dτ + u (x, τ ) dxdτ . x∈[0,l]

x∈[0,l]

0

0

0

(4.10) Further, since p > q ≥ 1, H¨ older’s inequality implies that Z tZ

l q

u (y, τ ) dydτ ≤ (lT ) 0

Z tZ

p−q p

! pq

l p

u (y, τ ) dydτ

0

0

:= Z (t) .

(4.11)

0

It is not difficult to verify that Z (t) = o (Ψ (t)) as t → T.

(4.12)

Combining (4.13), (4.11) with (4.12), we see that    Z t F (t) ≤ max λ1 , k max ζ1 (x) , F (0) + λ1 T max u0 (x) 1 + Z (t) + Ψ (τ ) dτ . x∈[0,l]

x∈[0,l]

(4.13)

0

Now, by Lemma 4.5 in [17], we can claim that   Z t C sup (Ψ (t) − u (x, t)) ≤ 2 1 + Ψ (τ ) dτ + o (Ψ (t)) d x∈Kd 0
 holds for (x, t) ∈ [0, l] × T2 , T , where C is an appropriate positive constant. The proof of Lemma 4.2 is complete. 11

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In view of Lemma 4.2, and by a slight variant of the proof of Lemma 4.5 in [17], we have the following Lemma. Lemma 4.3. Assume that (1.6) and (1.7) hold, p > q ≥ 1 and N ≤ 1, then lim sup |u (·, t)| = +∞

(4.14)

lim Ψ (t) = +∞

(4.15)

t→T [0,l]

is equivalent to t→T

Moreover, if (4.14) or (4.15) is fulfilled, then lim

t→T

|u (·, t)|∞ u (x, t) = lim =1 t→T Ψ (t) Ψ (t)

(4.16)

uniformly on any compact subset of (0, l). Next, we give the proofs of Theorems 1.4 and Theorem 1.5, respectively. Proof of Theorem 1.4. It follows from (4.16) that up (x, t) ∼ Ψp (t) ,

t → T.

By the Lebesgue’s dominated convergence theorem, we have Z l 0 Ψ (t) = ψ (t) = up (x, t) dx ∼ lΨp (t) ,

t → T.

0

Therefore, by integrating the above equality, we can claim that 1 − p−1

Ψ (t) ∼ (l (p − 1) (T − t))

.

(4.17)

t → T,

(4.18)

Combining (4.16) with (4.17), we find that u (x, t) ∼ (l (p − 1) (T − t))

1 − p−1

,

which means that 1

1 − p−1

1

lim (T − t) p−1 u (x, t) = lim (T − t) p−1 |u (·, t)|∞ = (l (p − 1))

t→T

t→T

.

The proof of Theorem 1.4 is complete. Proof of Theorem 1.5. Denote  p Z l Z t ϕ (t) = up (y, t) dy − k max u (x, t) and Φ (t) = g (τ ) dτ. x∈[0,l]

0

0

Similar to Lemma 4.3, we can get lim

t→T

|u (·, t)|∞ u (x, t) = lim = 1, t→T Φ (t) Φ (t)

(4.19)

uniformly on any compact subset of (0, l). Since, the remaining arguments are the same as those in the proof of Theorem 1.4, we omit it here. The proof of Theorem 1.5 is complete. 12

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Acknowledgements The authors are sincerely grateful to professor Chunlai Mu of Chongqing University for his encouragements and discussions.

Competing interests The authors declare that they have no competing interests.

References [1] C. Y. Chan and C. S. Chen, A numerical method for semilinear singular parabolic quenching problem, Quart. Appl. Math. 47 (1989), 45-57. [2] Y. P. Chen, Q. L. Liu and C. H. Xie, Blow-up for degenerate parabolic equation with nonlocal source, Proc. Amer. Math. Soc. 132 (2003), 135-145. [3] Y. P. Chen, Q. L. Lin and C. H. Xie, The blow-up properties for a degenerate semilinear oparabolic equation with nonlocal source, Appl. Math. J. Chinese Unvi. Ser. B 17 (2002), 413-424. [4] W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 40 (1983), 468-475. [5] W. A. Day, Heat Conduction within Linear Thermoelasticity, Springer-Verlag, New York, USA, 1985. [6] K. Deng, Comparison principle for some nonlocal problems, Quart. Appl. Math. 50 (1992), 517-522. [7] A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44 (1986), 401-407. [8] A. Gladkov and K. I. Kim, Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition, J. Math. Anal. Appl. 338 (2008), 264-273. [9] F. Liang, Global existence and blow-up for a degenerate reaction-diffusion system with nonlinear localized sources and nonlocal boundary conditions, J. Korean Math. Soc. 53 (2016), 27-43. [10] J. Liang, H. Y. Wang and T. J. Xiao, On a comparison principle for delay coupled systems with nonlocal and nonlinear boundary conditions, Nonlinear Anal. 71 (2009), e359-e365. [11] D. M. Liu, Blow-up for a degenerate and singular parabolic equation with nonlocal boundary condition, J. Nonlinear Sci. Appl. 9 (2016), 208-218. [12] Q. L. Liu, Y. P. Chen and C. H. Xie, Blow-up for a degenerate parabolic equation with a nonlocal source, J. Math. Anal. Appl. 285 (2003), 487-505. [13] Z. G. Lin and Y. R. Liu, Uniform blow-up profiles for diffusion equations with nonlocal source and nonlocal boundary, Acta Math. Sci. Ser. B Engl. Ed. 24 (2004), 443-450. [14] N. W. Mclachlan, Bessel Functions For Engineers, 2nd Edition, Oxford at the Clarendon Press, London, (1955). [15] C. L. Mu, D. M. Liu and Y. S. Mi, Blow-up for a degenerate parabolic equation with nonlocal source and nonlocal boundary, Appl. Anal. 90 (2011), 1373-1389. [16] C. V. Pao, Asymptotic behavior of solutions of reaction diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. (88) (1998), 225-238. [17] P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations. 153 (1999), 374-406. [18] M. X. Wang and Y. M. Wang, Properties of positive solutions for nonlocal reaction diffusion problems, Math. Methods Appl. Sci. 19 (1996), 1141-1156. [19] Y. L. Wang, C. L. Mu and Z. Y. Xiang, Blow up of solutions to a porous medium equation with nonlocal boundary condition, Appl. Math. Comput. 192 (2007), 579-585.

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[20] L. Yan, X. J. Song and C. L. Mu, Global existence and blow-up for weekly coupled degenerate and singular equations with nonlocal sources, Appl. Anal. 94 (2015), 1624-1648. [21] H. M. Yin, On a class of parabolic equations with nonlocal boundary conditions, J. Math. Anal. Appl. 333 (2007), 1138-1152. [22] J. Zhou, Blowup for a degenerate and singular parabolic equation with nonlocal source and absorption, Glasgow Math. J. 52 (2010), 209-225. [23] J. Zhou, Global existence and blowup for a degenerate and singular parabolic system with nonlocal source and absorptions, Z. Angew. Math. Phys. 65 (2014), 449-469. [24] J. Zhou and D. Yang, Blowup for a degenerate and singular parabolic equation with nonlocal source and nonlocal boundary, Appl. Math. Comput. 256 (2015), 881-884.

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Approximation properties of Kantorovich-type q-Bernstein-Stancu-Schurer operators Qing-Bo Caia,b,∗ a

School of Information Science and Engineering, Xiamen University Xiamen 361005, P. R. China

b

School of Mathematics and Computer Science, Quanzhou Normal University Quanzhou 362000, P. R. China E-mail: [email protected]

Abstract. In this paper, we introduce a Kantorovich-type Bernstein-Stancu-Schurer α,β based on the concept of q-integers. We investigate statistical apoperators Kn,p,q proximation properties and establish a local approximation theorem, we also give a convergence theorem for the Lipschitz continuous functions. Finally, we give some graphics to illustrate the convergence properties of operators to some functions. 2000 Mathematics Subject Classification: 41A10, 41A25, 41A36. Key words and phrases: q-integers, Bernstein-Stancu-Schurer operators, A-statistical convergence, rate of convergence, Lipschitz continuous functions.

1

Introduction

¨ In 2013, Ozarslan and Vedi [7] introduced the q-Bernstein-Schurer-Kantorovich operators as follows: # "  Z 1  n+p n+p−r−1 X n+p Y 1 + (q − 1)[r]q [r]q p xr + t dq t Kn (f ; q; x) = (1 − q s x) f [n + 1]q [n + 1]q r 0 s=o r=0 q

for any real number 0 < q < 1, fixed p ∈ N0 and f ∈ C[0, p + 1]. They gave the Korovkintype approximation theorem, obtained the rate of convergence of the operators and so on. In 2014, Ren and Zeng [8] introduced two kinds of Kantorovich-type q-Bernstein-Stancu operators based on q-Jackson integral and Riemann-type q-integral respectively and got some approximation properties. In 2015, Acu [1] introduced and studied q analogue of Stancu-Schurer-Kantorovich operators. They proved a convergence theorem, established the rate of convergence, obtained a Voronovskaya type result and so on, they constructed ∗

Corresponding author.

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Q. -B. CAI

the operators as follows: α,β Kn,p (f ; x)

n+p X

=

k=0

"

n+p k

# k

x (1 −

x)n+p−k q

Z

1

 f

0

q

[k]q + q k t + α [n + 1]q + β

 dq t.

In 2015, Agrawal, Finta and Kumar [2] introduced a new Kantorovich-type generalization of the q-Bernstein-Schurer operators, they gave the basic convergence theorem, obtained the local direct results, estimated the rate of convergence and so on. The operators are defined as Z [k+1]q n+p X [n+1]q −k Kn,p (f ; q; x) = [n + 1]q bn+p,k (q; x)q f (t)dR (1) q t, [k] q [n+1]q

k=0

where bn+p,k (q; x) is defined by " bn+p,k (q; x) =

n+p k

# xk (1 − x)n+p−k . q

(2)

q

Motivated by above investigations, it seems there have no papers mentioned about the Stancu-type of the operators defined in (1). In present paper, we will introduce the ^ α,β (f ; x) which will be defined Kantorovich-type q-Bernstein-Stancu-Schurer operators Kn,p,q in (4). We will investigate statistical approximation properties, establish a local approximation theorem and give a convergence theorem for the Lipschitz continuous functions. Furthermore, we will give some graphics to illustrate the convergence properties of operators to some functions. Before introducing the operators, we mention certain definitions based on q-integers, detail can be found in [5, 6]. For any fixed real number 0 < q ≤ 1 and each nonnegative integer k, we denote q-integers by [k]q , where ( 1−q k 1−q , q 6= 1; [k]q = k, q = 1. Also q-factorial and q-binomial coefficients are defined as follows: ( " # [n]q ! [k]q [k − 1]q ...[1]q , k = 1, 2, ...; n , [k]q ! = , = [k]q ![n − k]q ! 1, k = 0, k

(n ≥ k ≥ 0).

q

For x ∈ [0, 1] and n ∈ N0 , we recall that ( (1 − x)nq =

Qn−1 j=0

1−

qj x




1, n = 0;
. n−1 = (1 − x)(1 − qx)... 1 − q x , n = 1, 2, ...

The Riemann-type q-integral is defined by Z a

b

f (t)dR q t = (1 − q)(b − a)

∞ X


f a + (b − a)q j q j ,

(3)

j=0

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APPROXIMATION PROPERTIES OF KANTOROVICH-TYPE q-BERNSTEIN-STANCU-SCHURER OPERATORS where the real numbers a, b and q satisfy that 0 ≤ a < b and 0 < q < 1. For f ∈ C(I), I = [0, 1 + p], p ∈ N0 , 0 ≤ α ≤ β, q ∈ (0, 1) and n ∈ N, we introduce the Kantorovich-type q-Bernstein-Stancu-Schurer operators as follows: n+p

X ^ α,β Kn,p,q (f ; x) = ([n + 1]q + β) bn+p,k (q; x)q −k k=0

Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

f (t)dR q t,

(4)

where bn+p,k (q; x) is defined by (2).

2

Auxiliary Results In order to obtain the approximation properties, We need the following lemmas:

Lemma 2.1. Using the definition (3), we easily get Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

dR qt= tdR qt=

qk , [n + 1]q + β

(5)

([k]q + α)q k q 2k + , 2 ([n + 1]q + β) [2]q ([n + 1]q + β)2

t2 dR qt=

(6)

2q 2k ([k]q + α) q k ([k]q + α)2 q 3k + + . (7) 3 3 ([n + 1]q + β) [2]q ([n + 1]q + β) [3]q ([n + 1]q + β)3

Lemma 2.2. (See [2], Lemma 2.1) The following equalities hold n+p X k=0 n+p X

bn+p,k (q; x)q k = 1 − (1 − q)[n + p]q x,

(8)


bn+p,k (q; x)q 2k = 1 − 1 − q 2 [n + p]q x + q(1 − q)2 [n + p]q [n + p − 1]q x2 . (9)

k=0

Lemma 2.3. For the Kantorovich-type q-Bernstein-Stancu-Schurer operators (4), we have ^ α,β Kn,p,q (1; x) = 1, (10) 2q[n + p]q x + 1 + [2]q α ^ α,β Kn,p,q (t; x) = , (11) [2]q ([n + 1]q + β)

q 2 [3]q + 3q 4 [n + p]q [n + p − 1]q 2 [2]q [3]q α2 + 2[3]q α + [2]q ^ α,β 2 Kn,p,q t ; x = x + [2]q [3]q ([n + 1]q + β)2 [2]q [3]q ([n + 1]q + β)2
4q[3]q α + 3q + 5q 2 + 4q 3 [n + p]q + x. (12) [2]q [3]q ([n + 1]q + β)2 Proof. (10) is easily obtained from (4) and (5). Using (4), (6) and (8), we have ^ α,β Kn,p,q (t; x)

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Q. -B. CAI n+p X

=

 bn+p,q (k; x)

k=0

[k]q + α qk + [n + 1]q + β [2]q ([n + 1]q + β)



n+p [n + p]q X [k]q 1 − (1 − q)[n + p]q x α + bn+p,q (k; x) + [n + 1]q + β [n + p]q [n + 1]q + β [2]q ([n + 1]q + β) k=0 " # n+p−1 X 1 − (1 − q)[n + p]q x [n + p]q n+p−1 xk+1 (1 − x)n+p−k−1 + q [n + 1]q + β [2]q ([n + 1]q + β) k

=

=

k=0

q

α + [n + 1]q + β [n + p]q (1 − q)[n + p]q 1 + [2]q α = x− x+ . [n + 1]q + β [2]([n + 1]q + β) [2]q ([n + 1]q + β) Thus, (11) is proved. Finally, from (4) and (7), we have
^ α,β Kn,p,q t2 ; x =

n+p X

bn+p,q (k; x)

k=0

[k]2q + 2α[k]q + α2 2q k ([k]q + α) q 2k + + ([n + 1]q + β)2 [2]q ([n + 1]q + β)2 [3]q ([n + 1]q + β)2

! ,

since [k]2q = [k]q [k − 1]q + q k−1 [k]q , and from lemma 2.2, we have
^ α,β Kn,p,q t2 ; x =

n+p X

k=0 n+p X

+

k=0

+

n+p

X [k]q [k − 1]q 2α[k]q bn+p,k (q; x) + bn+p,k (q; x) 2 ([n + 1]q + β) ([n + 1]q + β)2 k=0

n+p

X q k−1 [k]q 2q k [k]q α2 bn+p,k (q; x) b (q; x) + + n+p,k ([n + 1]q + β)2 ([n + 1]q + β)2 [2]q ([n + 1]q + β)2

2α [2]q ([n + 1]q + β)2

k=0

n+p X k=0

bn+p,k (q; x)q k +

n+p X

bn+p,k (q; x)

k=0

q 2k [3]([n + 1]q + β)2

[n + p]q [n + p − 1]q x2 2α[n + p]q x [n + p]q x (1 − q)[n + p]q [n + p − 1]q x2 = + + − ([n + 1]q + β)2 ([n + 1]q + β)2 ([n + 1]q + β)2 ([n + 1]q + β)2 2q[n + p]q x 2q(1 − q)[n + p]q [n + p − 1]q x2 α2 + + − ([n + 1]q + β)2 [2]q ([n + 1]q + β)2 [2]q ([n + 1]q + β)2
2α (1 − (1 − q)[n + p]q x) 1 − 1 − q 2 [n + p]q x + q(1 − q)2 [n + p]q [n + p − 1]q x2 + + [2]q ([n + 1]q + β)2 [3]q ([n + 1]q + β)2 [n + p]q [n + p − 1]q 2 (2[2]q α + [2]q + 2q)[n + p]q [2]q [3]q α2 + 2[3]q α + [2]q = x + x + ([n + 1]q + β)2 [2]q ([n + 1]q + β)2 [2]q [3]q ([n + 1]q + β)2
(1 − q) (1 − q + 4q[3]q ) [n + p]q [n + p − 1]q 2 (1 − q) 2α[3]q + [2]2q [n + p]q − x − x [2]q [3]q ([n + 1]q + β)2 [2]q [3]q ([n + 1]q + β)2

q 2 [3]q + 3q 4 [n + p]q [n + p − 1]q 2 4q[3]q α + 3q + 5q 2 + 4q 3 [n + p]q = x + x [2]q [3]q ([n + 1]q + β)2 [2]q [3]q ([n + 1]q + β)2 850

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APPROXIMATION PROPERTIES OF KANTOROVICH-TYPE q-BERNSTEIN-STANCU-SCHURER OPERATORS

+

[2]q [3]q α2 + 2[3]q α + [2]q . [2]q [3]q ([n + 1]q + β)2

Thus, (12) is proved. Remark 2.4. From lemma 2.3, it is observed that for α = β = 0, we get the moments for the operators defined in (1), which are the corresponding results of lemma 2.1 in [2]. Lemma 2.5. Using lemma 2.3 and easily computations, we have   1 + [2]q α 2q[n + p]q . ^ α,β = Aα,β (13) Kn,p,q (t − x; x) = −1 x+ n,p,q (x), [2]q ([n + 1]q + β) [2]q ([n + 1]q + β) " #

q 2 [3]q + 3q 4 [n + p]q [n + p − 1]q 4q[n + p]q ^ α,β 2 Kn,p,q (t − x) ; x ≤ x2 +1− [2]q [3]q ([n + 1]q + β)2 [2]q ([n + 1]q + β)
4q[3]q α + 3q + 5q 2 + 4q 3 [n + p]q . α,β [2]q [3]q α2 + 2[3]q α + [2]q + + x = Bn,p,q (x). (14) [2]q [3]q ([n + 1]q + β)2 [2]q [3]q ([n + 1]q + β)2

3

Statistical approximation properties

In this section, we present the statistical approximation properties of the operator ^ α,β by using the Korovkin-type statistical approximation theorem proved in [4]. Kn,p,q Let K be a subset of N, the set of all natural numbers. The density of K is defined P by δ(K) := limn n1 nk=1 χK (k) provided the limit exists, where χK is the characteristic function of K. A sequence x := {xn } is called statistically convergent to a number L if, for every ε > 0, δ{n ∈ N : |xn − L| ≥ ε} = 0. Let A := (ajn ), j, n = 1, 2, ... be an infinite summability matrix. For a given sequence x := {xn }, the A−transform of x, denoted by P Ax := ((Ax)j ), is given by (Ax)j = ∞ k=1 ajn xn provided the series converges for each j. We say that A is regular if limn (Ax)j = L whenever lim x = L. Assume that A is a non-negative regular summability matrix. A sequence x = {xn } is called A-statistically P convergent to L provided that for every ε > 0, limj n:|xn −L|≥ε ajn = 0. We denote this limit by stA − limn xn = L. For A = C1 , the Ces` aro matrix of order one, A-statistical convergence reduces to statistical convergence. It is easy to see that every convergent sequence is statistically convergent but not conversely. We consider a sequence q := {qn } for 0 < qn < 1 satisfying stA − lim qn = 1, n

(15)

If ei = ti , t ∈ R+ , i = 0, 1, 2, ... stands for the ith monomial, then we have Theorem 3.1. Let A = (ank ) be a non-negative regular summability matrix and q := {qn } be a sequence satisfying (15), then for all f ∈ C(I), x ∈ [0, 1], we have ^ α,β stA − lim Kn,p,q f − f = 0. (16) n

C(I)

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Q. -B. CAI

Proof. Obviously ^ α,β stA − lim Kn,p,qn (e0 ) − e0 n By (13), we have ^ α,β Kn,p,qn (e1 ; x) − e1 (x) ≤

= 0.

1 + [2]qn α 2qn [n + p]qn − 1 + . [2]qn ([n + 1]qn + β) [2]qn ([n + 1]qn + β)

Now for a given ε > 0, let us define the following sets: ) (  ^ α,β ≥ ε , U1 := k U := k : Kn,p,qk (e1 ) − e1 C(I)

 U2 :=

(17)

C(I)

1 + [2]qk α ε k: ≥ [2]qk ([n + 1]qk + β) 2

 ε 2qk [n + p]qk − 1 ≥ , : [2]qk ([n + 1]qk + β) 2

 .

Then one can see that U ⊆ U1 ∪ U2 , so we have ( )  ^ α,β ≤ δ k ≤ n : δ k ≤ n : Kn,p,qk (e1 ) − e1 C(I)

 ε 2qk [n + p]qk − 1 ≥ [2]qk ([n + 1]qk + β) 2   1 + [2]qk α ε +δ k ≤ n : , ≥ [2]qk ([n + 1]qk + β) 2

since stA − lim qn = 1, we have n

[n + p]qn − 1 = 0, stA − lim n [n + 1]q + β

stA − lim n

n

1 + [2]qn α = 0, [2]qn ([n + 1]qn + β)

which implies that the right-hand side of the above inequality is zero, thus we have ^ α,β = 0. (18) stA − lim Kn,p,qn (e1 ) − e1 n

C(I)

Finally, by (10) and (12), we get ^ α,β Kn,p,qn (e2 ; x) − e2 (x)

q 2 [3] + 3q 4 [n + p] [n + p − 1] 4qn [3]qn α + 3qn + 5qn2 + 4qn3 [n + p]qn qn qn n qn n ≤ − 1 + [2]qn [3]qn ([n + 1]qn + β)2 [2]qn [3]qn ([n + 1]qn + β)2 +

[2]qn [3]qn α2 + 2[3]qn α + [2]qn . = αn + βn + γn . [2]qn [3]qn ([n + 1]qn + β)2

Since stA − lim qn = 1, one can see that n

stA − lim αn = stA − lim βn = stA − lim γn = 0. n

n

n

(19)

For ε > 0, we define the following four sets ( ) n n ^ εo εo α,β V := k : Kn,p,q (e ) − e ≥ ε , V := k : α ≥ , V := k : β ≥ , 2 2 1 2 k k k 3 3 C(I) 852

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APPROXIMATION PROPERTIES OF KANTOROVICH-TYPE q-BERNSTEIN-STANCU-SCHURER OPERATORS n εo V3 := k : γk ≥ . 3 Hence, from (19) we obtain the right-hand side of the above inequality is zero, so we have ( ) ^ α,β δ k ≤ n : Kn,p,qk (e2 ) − e2 ≥ ε = 0, C(I)

thus

^ α,β stA − lim Kn,p,qn (e2 ) − e2 n

= 0.

(20)

C(I)

Combining (17), (18) and (20), theorem 3.1 follows from the Korovkin-type statistical approximation theorem established in [4], the proof is completed.

4

Local approximation properties

Let f ∈ C(I), endowed with the norm ||f || = supx∈I |f (x)|. The Peetre’s K−functional is defined by  K2 (f ; δ) = inf ||f − g|| + δ||g 00 || , g∈C 2

where δ > 0 and C 2 = {g ∈ C(I) : g 0 , g 00 ∈ C(I)} . By [3, p.177, Theorem 2.4], there exits an absolute constant C > 0 such that √ K2 (f ; δ) ≤ Cω2 (f ; δ), (21) where ω2 (f ; δ) = sup

|f (x + 2h) − 2f (x + h) + f (x)|

sup

0 1. Theorem 2.1. (l∞ − l1 spaces). Let p > 1 and X = l∞ − l1 which is R2 endowed with the norm ( kxk∞ , if x1 x2 ≥ 0, kxk = kxk1 , if x1 x2 ≤ 0. Then (p)

CN J (l∞ − l1 ) =

1 (1 + t0 )p + 1 = , p−1 2p−1 (1 + tp0 ) 2 (1 − tp−1 0 )

(2.1)

where t0 ∈ (0, 1) is the unique solution of the equation (1 + t)p−1 − tp−1 − tp−1 (1 + t)p−1 = 0.

(2.2)

Proof. Firstly we shall show that kx + tykp + kx − tykp ≤ 1 + (1 + t)p for any x, y ∈ SX and every t ∈ [0, 1]. By Minkowski inequality, for any α, β ∈ [0, 1] and any x1 , x2 , y1 , y2 ∈ BX with x = αX1 + (1 − α)x2 , y = βy2 + (1 − β)y2 , we have kx + tykp + kx − tykp = kα(x1 + ty) + (1 − α)(x2 + ty)kp + kα(x1 − ty) + (1 − α)(x2 − tykp ≤ αkx1 + tykp + (1 − α)kx2 + tykp + αkx1 − tykp + (1 − α)kx2 − tykp = α[kβ(x1 + ty1 ) + (1 − β)(x1 + ty2 )kp + kβ(x1 − ty1 ) + (1 − β)(x1 − ty2 )kp ] +(1 − α)[kβ(x2 + ty1 ) + (1 − β)(x2 + ty2 )kp + kβ(x2 − ty1 ) + (1 − β)(x2 − ty2 )kp ] ≤ αβ[kx1 + ty1 kp + kx1 − ty1 kp ] + α(1 − β)[kx1 + ty2 kp + kx1 − ty2 kp ] +(1 − α)β[kx2 + ty1 kp + kx2 − ty1 kp ] + (1 − α)(1 − β)[kx2 + ty2 kp + kx2 − ty2 kp ] Hence, we only need to prove kx + tykp + kx − tykp ≤ 1 + (1 + t)p for any x, y ∈ ex(BX ) and every t ∈ [0, 1]. Since ex(BX ) = {(1, 0), (0, 1), (1, 1), (−1, 0), (−1, −1), (0, −1)} and we can change x into −x or y into −y. So we may assume that x, y = (0, 1), (1, 0) or (1, 1). Obviously, for these x, y we easily have kx + tykp + kx − tykp ≤ 1 + (1 + t)p for every t ∈ [0, 1]. Therefore,

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3

(p)

CN J (l∞ − l1 ) ≤ sup { t∈[0,1]

Let f (t) =

(1+t)p +1 1+tp ,

(1 + t)p + 1 }. 2p−1 (1 + tp )

then p(1 + t)p−1 t p−1 [1 − tp−1 − ( ) ]. p 2 (1 + t ) 1+t

0

f (t) =

t p−1 1 on [0, 1]. Whence Defining h(t) = 1 − tp−1 − ( 1+t ) , we have h(t) is decreasing from 1 to − 2p−1 there exists an unique t0 ∈ (0, 1) such that h(t0 ) = 0. Therefore, (p)

CN J (l∞ − l1 ) ≤

(1 + t0 )p + 1 . 2p−1 (1 + tp0 )

On the other hand, by taking x0 = (1, 0), y0 = (t0 , t0 ), we have (p)

CN J (l∞ − l1 ) ≥

(1 + t0 )p + 1 . 2p−1 (1 + tp0 )

Hence, (p)

CN J (l∞ − l1 ) =

(1 + t0 )p + 1 , 2p−1 (1 + tp0 )

t p−1 where t0 ∈ (0, 1) is the unique solution of 1 − tp−1 = ( 1+t ) . From (2.2), we also have

tp−1 0

p

(1 + t0 ) + 1 = (1 + t0 )

1 − tp−1 0

+1=

1 + tp0 1 − tp−1 0

.

Therefore (2.1) is obtained. Corollary 2.2. For X = l∞ − l1 , we have 1

(3)

CN2J (X) = √

q 2−

√

2 2+1−

p

√ ≈ 1.5077. 5+4 2

(2.3)

and

p √ √ 3+2 2+ 5+4 2 = ≈ 1.1366. 8 Proof. (1) For p = 23 , (2.2) is equivalent to t4 + 1 − 2t3 − 2t − 5t2 = 0. that is 1 1 t2 + 2 − 2(t + ) = 5. t t √ (3) CN J (X)

√

(2.4)

√

Hence, we can get t = 2 2+1−2 5+4 2 and (2.3) is valid by(2.1). (2) For p = 3, (2.2) is equivalent to t2 = (1 + t)2 (1 − t2 ). Letting t = u − 1, we have u4 + 1 − 2u3 − 2u + u2 = 0. that is u2 + √

Hence, u =

√ 2+1+

(3) CN J (X)

2

√ 2 2−1

√

and t =

1 1 − 2(u + ) = −1. 2 u u √ √ 2 2−1

2−1+ 2

. Therefore

p √ √ 1 1 3+2 2+ 5+4 2 p √ = = = ≈ 1.1366. √ 4(1 − t2 ) 8 2 − 2( 2 − 1) 2 2 − 1

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4

Theorem 2.3. (lq − l1 spaces). If p ≥ q > 1. Let X = R2 endowed with the norm ( kxkq , if x1 x2 ≥ 0 kxk = kxk1 , if x1 x2 ≤ 0

,

then p

(p)

CN J (lq − l1 ) = 1 + 2 q

−p

.

In order to prove this theorem, firstly we give the following lemma. Lemma2.4. Let a, b, c, d ≥ 0 and p ≥ q > 1 such that aq + bq = 1 and cq + dq = 1. If 0 ≤ t ≤ 1, a ≥ ct and b ≤ dt, then p

p

[(a + ct)q + (b + dt)q ] q + (a − ct + dt − b)p ≤ (1 + t)p + (1 + tq ) q . Proof. Clearly, 0 ≤ a − ct + dt − b ≤ 1 + t. So we will consider the following two cases. 1 Case I. if 0 ≤ a − ct + dt − b ≤ (1 + tq ) q , then p

[(a + ct)q + (b + dt)q ] q + (a − ct + dt − b)p 1

1

p

≤ [(aq + bq ) q + t(cq + dq ) q ]p + (1 + tq ) q p = (1 + t)p + (1 + tq ) q . 1

Case II. if (1 + tq ) q ≤ a − ct + dt − b ≤ 1 + t, then 1

[(a + ct)q + (b + dt)q ] q + (a − ct + dt − b) 1

1

≤ (aq + dq tq ) q + (cq tq + bq ) q + a − ct + dt − b 1

≤ (1 + tq ) q + ct + b + a − ct + dt − b 1

≤ (1 + tq ) q + 1 + t. So, 1

1

[(a + ct)q + (b + dt)q ] q ≤ (1 + tq ) q + 1 + t − (a − ct + dt − b). Thus, p

[(a + ct)q + (b + dt)q ] q + (a − ct + dt − b)p 1

≤ [(1 + tq ) q + 1 + t − (a − ct + dt − b)]p + (a − ct + dt − b)p ≤ max

1

1 u∈[(1+tq ) q ,1+t]

[(1 + tq ) q + 1 + t − u]p + up p

= (1 + t)p + (1 + tq ) q . Proof of Theorem 2.3 Note that ex(BX ) = {(x1 , x2 ) : xq1 + xq2 = 1, x1 x2 ≥ 0}. Now we prove that p

kx + tykp + kx − tykp ≤ (1 + t)p + (1 + tq ) q , holds for any x, y ∈ ex(BX ) and any t ∈ [0, 1].

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Case I. If (a − ct)(b − dt) ≥ 0. By Minkowski inequality, we have kx + tykp + kx − tykp = kx + tykpq + kx − tykpq p p = [(a + ct)q + (b + dt)q ] q + [|a − ct|q + |b − dt|q ] q 1

1

≤ [(aq + bq ) q + (cq tq + dq tq ) q ]p + 1 ≤ (1 + t)p + 1 p ≤ (1 + t)p + (1 + tq ) q . Case II. If (a − ct)(b − dt) ≤ 0. By Lemma2.4, we have that kx + tykp + kx − tykp = kx + tykpq + kx − tykp1 p = [(a + ct)q + (b + dt)q ] q + (a − ct + dt − b)p p ≤ (1 + t)p + (1 + tq ) q . p

Therefore, kx + tykp + kx − tykp ≤ (1 + t)p + (1 + tq ) q is also valid for any x, y ∈ SX . Hence , p

(p) CN J (lq

(1 + t)p + (1 + tq ) q − l1 ) ≤ . 2p−1 (1 + tp )

On the other hand, for every t ∈ [0, 1], taking x0 = (1, 0), y0 = (0, 1), we have (p)

CN J (lq − l1 ) p p 0 k +kx0 −ty0 k ≥ kx0 +ty2p−1 (1+tp ) p

=

(1+t)p +(1+tq ) q 2p−1 (1+tp )

.

Hence,

p

(p) CN J (lq

(1 + t)p + (1 + tq ) q . − l1 ) = max 2p−1 (1 + tp ) t∈[0,1]

p

We let f (t) =

(1+t)p +(1+tq ) q 1+tp

, so p

f (t) =

p{(1 + tq ) q

−1

(tq−1 − tp−1 ) + (1 + t)p−1 (1 − tp−1 )} ≥ 0. (1 + tp )2 That imply f (t) is not decreasing. Hence, 0

(p)

CN J (lq − l1 ) = 21−p maxt∈[0,1] f (t) p −p = 21−p f (1) = 1 + 2 q . Lemma2.6. Let p > 1 and

1 p

+

1 q

= 1, then 1− pq

(p)

CN J (X) = 2 and

(p)

(q)

p

CN J (X ∗ ) q

(p)

CN J (X) = CN J (X ∗∗ ), where X ∗ is the dual of X. 1 Proof. Let lp (X) = {(x1 , x2 ) : k(x1 , x2 )k = (kx1 kp + kx2 kp ) p } and define the operator A : lp (X) → p (p) (q) lp (X) by (x1 , x2 ) 7→ (x1 + x2 , x1 − x2 ). Then we easily have CN J (X) = kAk . Similarly, CN J (X ∗ ) = 2p−1

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6 kA∗ kq . 2q−1

(p)

So CN J (X) = 2

1− pq

p

(q)

(q)

CN J (X ∗ ) q by kAk = kA∗ k, and hence CN J (X ∗ ) = 2

(p)

1− pq

(p)

q

CN J (X ∗∗ ) p .

(p)

Therefore, we have CN J (X) = CN J (X ∗∗ ). (p) The relationship between the constant CN J (X) and the uniformly normal structure of X as follows: Theorem 2.7. The Banach space X has uniformly normal structure if any one of the following conditions is valid  q 2p−3 p−1   1+ 1+2 p−1 (p) 3−log2 3 (i)CN J (X) < for some p ∈ 1, 2−log2 3 ; 22p−3 1 3−q 2

(q)

(ii)CN J (X ∗ ) < 1+(1+22 ) for some q > 1, where p−1 + q −1 = 1. (p) Proof. According to CN J (X) < 2, we have X is uniformly non-square, so we only need to prove X has weak normal structure. Assume that X has no weak normal structure. Then it is well known (see[5])that for any ε > 0 there exists z1 , z2 , z3 ∈ SX and g1 , g2 , g3 ∈ SX ∗ satisfying the following statements: (i) for all i 6= j, we have |kzi − zj k − 1| < ε, |gi (zj )| < ε, (ii) gi (zj ) = 1 for i = 1, 2, 3, (iii) kz3 − (z2 + z1 )k ≥ kz2 + z1 k − ε. Let us fix ε > 0 as small as needed. Then, we can find z1 , z2 , z3 ∈ SX and g1 , g2 , g3 ∈ SX ∗ satisfying the above   qproperties. 1+

(1)Taking α =

1+2

2p−3 p−1

22p−3

p−1

. We will consider the following two cases:

Case I. If kz2 + z1 k ≤ α. Then, kg1 +g2 kq +kg2 −g1 kq 2q−1 (kg2 kq +kg1 kq ) z +z z −z [(g1 +g2 )( 2 α 1 )]q +[(g2 −g1 )( kz2 −z1 k )]q

≥

( 2−2ε )q +( 2−2ε )q α 1+ε q 2

2q

2

1

≥ 1−ε q q = ( 1−ε α ) + ( 1+ε ) . Case II. If kz2 + z1 k > α. Then, the contains two sub-cases: (i) If kz3 − z2 + z1 k ≤ α. Then, kg1 +g3 kq +kg3 −g1 kq 2q−1 (kg3 kq +kg1 kq ) z −z +z z −z [(g1 +g3 )( 3 α2 1 )]q +[(g3 −g1 )( kz3 −z1 k )]q

≥ ≥ =

2q ( 2−4ε )q +( 2−2ε )q α 1+ε 2q q + ( 1−ε )q . ( 1−2ε ) α 1+ε

3

1

(ii) If kz3 − z2 + z1 k > α. Then, kz3 −z2 +z1 kp +kz3 −z2 −z1 kp 2p−1 (kz3 −z2 kp +kz1 kp ) p +(kz +z k−ε)p 2 1 ≥ α2p−1 [(1+ε)p +1] p α +(α−ε)p ≥ 2p−1 . [(1+ε)p +1]

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7

Letting ε → 0, and by lemma2.6 we have  1− pq

(p)

CN J (X) ≥ min{2

(

αp

p 1 + 1) q , p−1 } = q α 2

q 1+

2p−3

1 + 2 p−1 22p−3

p−1 ,

which contradicts to the hypothesis(i). 1

(2)Taking α =

1+(1+23−q ) 2 2

. By the proof of (1), we have 1

1 αp q 1 αq 1 + (1 + 23−q ) 2 1− q ≥ min{ q + 1, 2 p ( p−1 ) p } = min{ q + 1, q−1 } = , α 2 α 2 2 which contradicts to the hypothesis(ii). (q) CN J (X ∗ )

References [1] Cui, Y., Huang, W., Generalized von Neumann-Jordan constant and its relationship to the fixed point property, Fixed Point Theory and Applications,(2015). [2] Yang, C., Wang, F., On a new geometric constant related to the von Neumann-Jordan constant, J. Math.Anal.Appl., 324(1) (2006), 555-565. [3] Clarkson,J.A., The von Neumann-Jordan constant for the Lebegue space, Ann. of Math., 38 (1937), 114-115. [4] Kato, M., Maligranda,L., Takahashi,Y., On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces, J.Math. Anal.Appl., 144 (2001), 275-295. [5] Macu˜ n´an-Navarro, Eva M., Banach spaces properties sufficient for normal structure, J.Math. Anal.Appl., 337 (2008), 197-218. [6] Yang,C., A note of Jordan-von Neumann constant and James constant, J. Math.Anal.Appl., 398(2) (2009), 92-102. [7] Dhompongsa,S.,Piraisangjun P.,Saejung, S., On a generalized Jordan-von Neumann constants and uniform normal structure, Bull.Austral.Math.Soc., 67 (2003),225-240. [8] Alonso, J.Llorens-Fuster,E., Geometric mean and triangles inscribed in a semicircle in Banach spaces, J.Math. Anal.Appl., 340 (2008),1271-1283. [9] Alonso,J.,martin P.,Papini,P.L., Wheeling around von Neumann-Jordan constant in Banach spaces, Studia Math, 188(2) (2008),135-150. [10] Kato, M., Takahashi,Y., On the von Neumann-Jordan constant for Banach spaces, J. Inequal.Appl., 2 (1998),302-306.

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Discrete dynamical systems in soft topological spaces ∗ Wenqing Fu,

Hu Zhao

Abstract In this paper the iteration of soft continuous functions is investigated and their discrete dynamical systems in soft topological spaces are defined. Some basic concepts related to discrete dynamical systems (such as soft ω-limit set, soft invariant set, soft periodic point, soft nonwandering point, and soft recurrent point) are introduced into soft topological spaces. Soft topological mixing and soft topological transitivity are also studied. At last, soft topological entropy is defined and several properties of it are discussed. Keywords Soft point, Soft ω-limit set, Soft nonwandering point, Soft topological mixing, Soft topological transitivity, Soft topological entropy

1

Introduction and preliminaries The real world is too complex for our immediate and direct understanding, so we create

models which are simplifications of the real word. In 1999, Molodtsov

[1]

introduced the con-

cept of soft set which gives a new approach to modeling uncertainties. And he also discussed the application of soft set theory in many fields, such as: operations analysis, game theory, the smoothness of function, and so on[2] . Maji et al.[3] and Ali et al.[4] defined some operators of soft sets. Beyond these theoretical works of soft set, research works on its applications in various fields are progressing rapidly, and great progress has been achieved, including soft set theory in abstract algebras[5−10] , decision making, data analysis, information system, and so on[11−14] . The application of soft set theory in algebraic structures was introduced by Akta¸s and C ¸ aˇgman[5] , they defined the notion of soft groups and progressed some basic properties. Jun[6,7] investigated soft BCK/BCI-algebras and its application in ideal theory. Dudek et al.[8] discussed soft ideals in BCC-algebras. Zhang[9] studied intuitionistic fuzzy soft rings. Feng et al.[10] worked on soft semirings, soft ideals and idealistic soft semirings. Maji et al.[11] first applied soft sets to solve the decision making problem that is based on the concept of knowledge reduction in the theory of rough sets[12] . Based on the analysis of the rough set model on a tolerance relation and the fuzzy rough set, two types of fuzzy rough sets models on tolerance relations are constructed and researched by Xu et al.[13] . Chen et al.[14] presented a ∗ Corresponding Author: Wenqing Fu is with the School of Science, Xi’an Technological University, Xi’an 710032, China. E-mail: palace [email protected] Hu Zhao is with Xi’an Polytechnic University, Xi’an 710048, China E-mail: [email protected]

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new definition of soft set parametrization reduction so as to improve the soft set based decision making in [11]. Yang[15] combined the multi-fuzzy set and soft set, from which they obtained a new soft set model named multi-fuzzy soft set, and applied it to decision making. Soft set

theory is also be used in topology. Shabir and Naz’s work[16] on soft topological spaces defined over an initial universe with a fixed set of parameters. The notions of soft open set, soft closed set, soft closure, soft interior point, soft neighborhood of a point, and soft separation axioms (such as soft Ti -space for i = 1, 2, 3, 4, soft normal space, and soft regular space) were also introduced and their basic properties were investigated. Min[17] pointed out some mistakes of [16] and investigated some properties of the soft separation axioms defined in [15]. Zorlutuna etc.[18] introduced some new concepts in soft topological spaces (such as soft point, interior point, interior, neighborhood, continuity, and compactness). Motivated by Chen etc.[19] and Liu[20] , this paper will investigate iteration of soft continuous functions and their discrete dynamical systems in soft topological spaces. Some basic concepts on dynamical systems (such as soft ω-limit set, soft invariant set, soft periodic point, soft nonwandering point, and soft recurrent point) are introduced in soft topological spaces, soft topological mixing, soft topological transitivity, soft topological entropy and its several properties are studied. As a result, some conclusions of discrete dynamical systems in ordinary topological spaces are generalized. Now we give some definitions and results to be used in this paper. Definition 1[1]

A soft set on a set X is a triple (M, E, X), where M : E −→ 2X (the

set of all subsets of X) is a mapping. The set of all soft sets on X is denoted by S(X, E). Roughly speaking, a soft set on a set X is just a family {Me }e∈E of subsets of X; it can be looked to be a subset of X if E is a singleton. Let (M, E, X), (N, E, X) ∈ S(X, E). If M (e) ⊆ N (e) (∀e ∈ E), then (M, E, X) is called e e a soft subset of (N, E, X), denoted by (M, E, X)⊆(N, E, X). If (M, E, X)⊆(N, E, X) and

e (M, E, X)⊇(N, E, X), then (M, E, X) and (N, E, X) are said to be soft equal, denoted by (M, E, X) = (N, E, X).

e : E −→ 2X as A(e) e = A (∀e ∈ E), Remark 1[16] (1) Let X be a set, and A ∈ 2X . Define A

e E, X) ∈ S(X, E); we use A e to denote this soft set (particularly, we use x then (A, e to denote g the soft set {x}). (2) Let X be a set, and (M, E, X) ∈ S(X, E). Then (M ′ , E, X) ∈ S(X, E), where M ′ :

E −→ 2X is defined as M ′ (e) = X − M (e) (∀e ∈ E). Sometimes we use (M, E, X)′ to replace (M ′ , E, X). (3) Let X be a set, {(Hj , E, X)}j∈J ⊆ S(X, E). Then (M, E, X), (N, E, X) ∈ S(X, E), S T called the union (denoted as e (Hj , E, X)) and intersection (denoted as e (Hj , E, X)) j∈J

j∈J

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC of the family {(Hj , E, X)}j∈J respectively, where

M (e) =

[

Hj (e) (∀e ∈ E)

\

Hj (e) (∀e ∈ E).

j∈J

and N (e) =

j∈J

(4) Let X be a set, (H, E, X) ∈ S(X, E), and x ∈ X. Write x ∈ (H, E, X) if x ∈ H(e) (∀e ∈ E), and x 6∈ (H, E, X) if x 6∈ H(e) for some e ∈ E. (5) Let X be a set. The difference of the two soft sets (M, E, X) and (N, E, X) is a soft set (H, E, X) over X (usually, denoted by (M, E, X) − (N, E, X)) which is defined by H(e) = M (e) − N (e) (∀e ∈ E). (6) Let X be a set, and (M, E, X), (N, E, X) ∈ S(X, E). Then e (N, E, X))′ = (M, E, X)′ ∩ e (N, E, X)′ ; (i) ((M, E, X)∪ (ii)

e (N, E, X))′ = (M, E, X)′ ∪ e (N, E, X)′ . ((M, E, X)∩

Definition 2[18] (1) A soft set (M, E, X) ∈ S(X, E) is called elementary (or a soft point

e denoted by eM ) if M (e) 6= ∅ for some e ∈ E and M (e′ ) = ∅ for all e′ ∈ E − {e}. in X,

e and (N, E, X) is a soft set. If M (e) ⊆ N (e), then eM is (2) Let eM be a soft point in X,

e (N, E, X). said to be in (N, E, X), denoted by eM ∈

Definition 3[17] Let X and Y be two sets, E and F be two nonempty parameter sets,

and f : E −→ F and g : X −→ Y are mappings. For each (M, E, X) ∈ S(X, E), define (f, g)(M, E, X) = (g→ (M ), f (E), Y ), where g→ (M )(α) =

[

g(M (e)) (∀α ∈ F ).

f (e)=α

Then we obtain a mapping (f, g) : S(X, E) −→ S(Y, F ). For each (N, F, Y ) ∈ S(Y, F ), define (f, g)−1 (N, F, Y ) = (g−1 ◦ N ◦ f, f −1 (F ), X), where (g−1 ◦ N ◦ f )(e) = g−1 (N (f (e))) (∀e ∈ f −1 (F )). Then we obtain another mapping (f, g)−1 : S(Y, F ) −→ S(X, E). Definition 4[16] (1) Let X be a set, and T ⊆ S(X, E) satisfies 869

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(i) ∅ and X ∈ T ;

(ii) T is closed under arbitrary unions;

(ii) T is closed under finite intersections. Then T is called a soft topology on X, and (X, T , E) is called a soft topological space. The members of T are called soft open sets, members of T ′ = {(M ′ , E, X) | (M, E, X) ∈ T } are called soft closed sets. (2) Let (X, T , E) be a soft topological space, and Y be a non-empty subset of X. Then TY = {(MY , E, X) | (M, E, X) ∈ T } is a soft topology on Y , it is called the soft relative topology on Y , and (Y, TY , E) is called a soft subspace of (X, T , E), where

Example 1

e (M, E, X) (∀(M, E, X) ∈ T ). (MY , E, X) = Ye ∩

(1) Let X = {x1 , x2 , x3 } be a 3-element set, E = {e1 , e2 } be a 2-element

set, and e T = {(Mi , E, X) | i = 1, 2, · · · , 6} ∪ {e ∅, X},

where (Mi , E, X) (i = 1, 2, · · · , 6) are defined as follows:  {x2 }, if e = e1 ; M1 (e) = {x1 }, if e = e2 . M2 (e) = M3 (e) = M4 (e) = M5 (e) = M6 (e) =







 

{x1 }, if e = e1 ; {x3 }, if e = e2 . {x3 }, if e = e1 ; {x2 }, if e = e2 .

{x2 , x3 }, if e = e1 ; {x1 , x2 }, if e = e2 . {x1 , x2 }, if e = e1 ; {x1 , x3 }, if e = e2 . {x1 , x3 }, if e = e1 ; {x2 , x3 }, if e = e2 .

Then T is a soft topology on X and hence (X, T , E) is a soft topological space. (2) Let X = R (the set of all real numbers), E = {e1 , e2 } be a 2-element set, J = {A ⊆ X | X − A is a finite subset of X} ∪ {∅, X} (i.e. the finite complement topology on X), and T = {(M, E, X) | M (e1 ), M (e2 ) ∈ J }. Then T is a soft topology on X and hence (X, T , E) is a soft topological space. 870

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC (3) Let X = R, E = {e1 , e2 } be a 2-element set, J be the ordinary topology on X (i.e. J

is the topology on X generated by the basis B = {(a, b) | a, b ∈ R, a < b}), and T = {(M, E, X) | M (e1 ), M (e2 ) ∈ J }. Then T is a soft topology on X and hence (X, T , E) is a soft topological space. (4) Let X = [0, 1], E = {e1 , e2 } be a 2-element set, J be the ordinary topology on X (i.e. J is the topology on [0, 1] generated by the basis B = {(a, b) | a ∈ [0, 1), b ∈ (0, 1], a < b}), and T = {(M, E, X) | M (e1 ), M (e2 ) ∈ J }. Then T is a soft topology on X and hence (X, T , E) is a soft topological space. e Remark 2 (1)[16] Let (X, T , E) be a soft topological space, eM is a soft point in X,

(N, E, X) ∈ S(X, E). If there exists a (A, E, X) ∈ T such that e e (A, E, X)⊆(N, eM ∈ E, X),

then (N, E, X) is called a neighborhood of eM . e ∈ T ′ , and T ′ is closed under the operations of arbitrary ∅, X (2) It can be easily seen that e

intersections and finite unions. It can be also seen that (N, E, X) ∈ T ′ if and only if e (N, E, X) 6= e ((A, E, X) − eM )∩ ∅

e and any neighborhood (A, E, X) of eM . for anyeM ∈ X

(3)[16] Let(X, T , E) be a soft topological space, and (M, E, X) ∈ S(X, E). Then (M, E, X) =

\ f

e {(N, E, X) | (M, E, X)⊆(N, E, X),

(N, E, X) ∈ TX′ }

is called the closure of (M, E, X). Clearly, (M, E, X) ∈ S(X, E) is a soft closed set of (X, T , E) if and only if (M, E, X) = (M, E, X). (4)[16] Let (X, T , E) be a soft topological space over X, then T e = {M (e) | (M, E, X) ∈ T } is a topology on X (e ∈ E). (5) If E is a single point set, then a soft topological space (X, T , E) can be seen as a common topological space. Definition 5 Let (X, TX , E) and (Y, TY , E) be soft topological spaces. A soft function (f, g) : S(X, E) −→ S(Y, E) is said to be a soft continuous function from (X, TX , E) to (Y, TY , E) if (f, g)−1 (N, E, Y ) ∈ TX (∀(N, E, Y ) ∈ TY ). 871

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC Remark 3 Let (X, TX , E) and (Y, TY , E) be soft topological spaces, and

(idE , g) : S(X, E) −→ S(Y, E) be a soft continuous function from (X, TX , E) to (Y, TY , E). Then g : X −→ Y is a continuous function from (X, TXe ) to (Y, TYe ) (∀e ∈ E). Definition 6[18] (1) Let (X, T , E) be a soft topological space, (P, E, X) ∈ S(X, E), and A ⊆ T . If

[ f

A = (P, E, X),

then A is called an soft open cover of (P, E, X). (2) Let (X, T , E) be a soft topological space, and (P, E, X) ∈ S(X, E). (P, E, X) is said e is compact, then to be soft compact if every open soft cover of it has a finite subcover. If X (X, T , E) is called a soft compact topological space.

Theorem 1[18] Let (X, T , E) be a soft compact topological space, then each soft closed

e subset (P, E, X) is a soft compact subset of X.

Theorem 2 Let (X, TX , E) and (Y, TY , E) be soft topological spaces, and (idE , g) : S(X, E) −→ S(Y, E)

is a soft function. Then the following conditions are equivalent: (1) (idE , g) is a soft continuous function from (X, TX , E) to (Y, TY , E). (2) (idE , g)−1 (N, E, Y ) ∈ TX′

(∀ (N, E, Y ) ∈ TY′ ).

e E , g)(M, E, X) (∀(M, E, X) ∈ S(X, E)). (3) (idE , g)(M, E, X) ⊆(id

e E , g)−1 (P, E, Y ) (∀(P, E, Y ) ∈ S(Y, E)). (4) (idE , g)−1 (P, E, Y )⊇(id

Proof

2

Straightforward. 

Discrete dynamical systems in soft topological spaces Let X be a topological space, and g : X −→ X a continuous mapping, then the family

{gn }n∈N defines a (discrete) semi-dynamical system in topological space X, where N stands for the set of all nonnegative integers. In addition, if g is a homeomorphism (i.e. g is a one-to-one correspondence and both g and its inverse mapping g−1 are continuous), then we can define g−n by g−n = (g−1 )n (∀n ∈ N ), then {gn }n∈Z defines a discrete dynamical system in topological space X, where Z stands for the set of all integers. Let (X, T , E) be a soft topological space and (idE , g) : S(X, E) −→ S(X, E) be a soft continuous function from (X, T , E) to (X, T , E). It can be seen from definition 3 that (gn )→ = (g→ )n , 872

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC so we can define the n-th iterate of (idE , g) (n ∈ N ) as follows:

(idE , g)n = (idE , g) ◦ (idE , g)n−1 = (idE ◦ idE , g ◦ gn−1 ) = (idE , gn ), (idE , g)0 = (idE , g0 ) = (idE , idX ), where idE (resp. idX ) denotes the identity mapping of E (resp., X) onto itself. Then the family {(idE , g)n }n∈N defines a (discrete) semi-dynamical system in soft topological space (X, T , E), where N stands for the set of all nonnegative integers. If g is a one-to-one correspondence and both (idE , g) and its inverse mapping (idE , g)−1 are continuous, it can be seen from definition 3 that (g← )n = (gn )← (∀n ∈ N − {0}) and ((gn )← )m = (g← )nm (∀n ∈ N − {0}, ∀m ∈ N ). Let (idE , g)−n = (idE , g−n ) = (idE , (gn )−1 ) (∀n ∈ N ), then {(idE , g)n }n∈Z defines a discrete dynamical system in soft topological space, and it is denoted by (X, (idE , g)). If (X, T , E) is a soft compact topological space, then (X, (idE , g)) is called a soft compact discrete topological dynamical system. It is easy to show that (idE , g)n (eM ) (∀n ∈ Z) is a soft point when eM is a soft point. Example 2 Let us consider the soft topological space in Example 1(1). Define g : X −→ X as follows: g(x1 ) = x2 , g(x2 ) = x3 , g(x3 ) = x1 . We will verify that both (idE , g) and its inverse mapping (idE , g)−1 are continuous. In fact, (idE , g)−1 (M1 , E, X) = (g−1 ◦ M1 ◦ idE , E, X), where

g−1 ◦ M1 ◦ idE (e) = g−1 ((M1 )(e)) g−1 ({x2 }), if e = e1 ; = −1  g ({x1 }), if e = e2 . {x1 }, if e = e1 ; = {x3 }, if e = e2 . = M2 (e)

Thus (idE , g)−1 (M1 , E, X) = (M2 , E, X) ∈ T . (idE , g)−1 (M2 , E, X) = (g−1 ◦ M2 ◦ idE , E, X),

873

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC where g−1 ◦ M2 ◦ idE (e) = g−1 ((M2 )(e)) g−1 ({x1 }), if e = e1 ; = −1  g ({x3 }), if e = e2 .

{x3 }, if e = e1 ; {x2 }, if e = e2 . = M3 (e) =

Thus (idE , g)−1 (M2 , E, X) = (M3 , E, X) ∈ T . (idE , g)−1 (M3 , E, X) = (g−1 ◦ M3 ◦ idE , E, X), where

g−1 ◦ M3 ◦ idE (e) = g−1 ((M3 )(e)) g−1 ({x3 }), if e = e1 ; = −1  g ({x2 }), if e = e2 . {x2 }, if e = e1 ; = {x1 }, if e = e2 . = M1 (e)

Thus (idE , g)−1 (M3 , E, X) = (M1 , E, X) ∈ T . (idE , g)−1 (M4 , E, X) = (g−1 ◦ M4 ◦ idE , E, X), where

g−1 ◦ M4 ◦ idE (e) =  g−1 ((M4 )(e)) g−1 ({x2 , x3 }), if e = e1 ; = −1  g ({x1 , x2 }), if e = e2 . {x1 , x2 }, if e = e1 ; = {x3 , x1 }, if e = e2 . = M5 (e)

Thus (idE , g)−1 (M4 , E, X) = (M5 , E, X) ∈ T . (idE , g)−1 (M5 , E, X) = (g−1 ◦ M5 ◦ idE , E, X), where

g−1 ◦ M5 ◦ idE (e) =  g−1 ((M5 )(e)) g−1 ({x1 , x2 }), if e = e1 ; = −1  g ({x3 , x1 }), if e = e2 . {x3 , x1 }, if e = e1 ; = {x2 , x3 }, if e = e2 . = M6 (e)

Thus (idE , g)−1 (M5 , E, X) = (M6 , E, X) ∈ T . (idE , g)−1 (M6 , E, X) = (g−1 ◦ M6 ◦ idE , E, X), where

g−1 ◦ M6 ◦ idE (e) =  g−1 ((M6 )(e)) g−1 ({x1 , x3 }), if e = e1 ; = −1  g ({x2 , x3 }), if e = e2 . {x3 , x2 }, if e = e1 ; = {x1 , x2 }, if e = e2 . = M4 (e) 874

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC Thus (idE , g)−1 (M 6 , E, X) = (M4 , E, X) ∈ T . It is easy to see that

(idE , g)−1 (e ∅) = e ∅∈T

and

Therefore, (idE , g) is continuous.

e =X e ∈T. (idE , g)−1 (X)

From the above, it is easy to see that (idE , g)−1 = (idE , g−1 ), since for any (M, E, X) ∈ T , (idE , g−1 )(M, E, X) = ((g−1 )→ (M ), E, X), where (g−1 )→ (M )(e) = g−1 (M )(e) = g−1 ◦ M ◦ idE (e). Thus for any (M, E, X) ∈ T , ((idE , g)−1 )−1 (M, E, X) = (idE , g−1 )−1 (M, E, X) = ((g−1 )−1 ◦ M ◦ idE , E, X) = (g ◦ M ◦ idE , E, X) Hence ((idE , g)−1 )−1 (M1 , E, X) = (g ◦ M1 ◦ idE , E, X), where

g ◦ M1 ◦ idE (e) = g((M  1 )(e)) g({x2 }), =  g({x1 }), {x3 }, if = {x2 }, if = M3 (e)

if e = e1 ; if e = e2 . e = e1 ; e = e2 .

Thus ((idE , g)−1 )−1 (M1 , E, X) = (M3 , E, X) ∈ T . ((idE , g)−1 )−1 (M2 , E, X) = (g ◦ M2 ◦ idE , E, X), where

g ◦ M2 ◦ idE (e) = g((M  2 )(e)) g({x1 }), =  g({x3 }), {x2 }, if = {x1 }, if = M1 (e)

if e = e1 ; if e = e2 . e = e1 ; e = e2 .

Thus ((idE , g)−1 )−1 (M2 , E, X) = (M1 , E, X) ∈ T . ((idE , g)−1 )−1 (M3 , E, X) = (g ◦ M3 ◦ idE , E, X), 875

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where

g ◦ M3 ◦ idE (e) = g((M  3 )(e)) g({x3 }), =  g({x2 }), {x1 }, if = {x3 }, if = M2 (e)

if e = e1 ; if e = e2 . e = e1 ; e = e2 .

Thus ((idE , g)−1 )−1 (M3 , E, X) = (M2 , E, X) ∈ T . ((idE , g)−1 )−1 (M4 , E, X) = (g ◦ M4 ◦ idE , E, X), where

g ◦ M4 ◦ idE (e) = g((M  4 )(e)) g({x2 , x3 }), =  g({x1 , x2 }), {x1 , x3 }, if = {x2 , x3 }, if = M6 (e)

if e = e1 ; if e = e2 . e = e1 ; e = e2 .

Thus ((idE , g)−1 )−1 (M4 , E, X) = (M6 , E, X) ∈ T . ((idE , g)−1 )−1 (M5 , E, X) = (g ◦ M5 ◦ idE , E, X), where

g ◦ M5 ◦ idE (e) = g((M  5 )(e)) g({x1 , x2 }), =  g({x3 , x1 }), {x2 , x3 }, if = {x1 , x2 }, if = M4 (e)

if e = e1 ; if e = e2 . e = e1 ; e = e2 .

Thus ((idE , g)−1 )−1 (M5 , E, X) = (M4 , E, X) ∈ T . ((idE , g)−1 )−1 (M6 , E, X) = (g ◦ M6 ◦ idE , E, X), where

g ◦ M6 ◦ idE (e) = g((M  6 )(e)) g({x1 , x3 }), =  g({x2 , x3 }), {x2 , x1 }, if = {x3 , x1 }, if = M5 (e)

if e = e1 ; if e = e2 . e = e1 ; e = e2 .

Thus ((idE , g)−1 )−1 (M6 , E, X) = (M5 , E, X) ∈ T . It is easy to see that

and

((idE , g)−1 )−1 (e ∅) = e ∅∈T e =X e ∈T. ((idE , g)−1 )−1 (X)

Therefore, (idE , g)−1 is continuous. Hence, (X, (idE , g)) is a soft topological dynamical system. 876

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Example 3 Let us consider the soft topological space in Example 1(2). Let g : X −→ X

be an arbitrary one-to-one respondence on X. Then for any (M, E, X) ∈ T , (idE , g)−1 (M, E, X) = (g−1 ◦ M ◦ idE , E, X), where g−1 ◦ M ◦ idE (e) = g−1 (M (e)) (∀e ∈ E), the complement X − g−1 ◦ M ◦ idE (e) is still a finite subset of X since g is an one-to-one respondence, thus (idE , g)−1 (M, E, X) ∈ T . Therefore, (idE , g) is continuous. On the other hand, for any (M, E, X) ∈ T , ((idE , g)−1 )−1 (M, E, X) = (idE , g−1 )−1 (M, E, X) = ((g−1 )−1 ◦ M ◦ idE , E, X) = (g ◦ M ◦ idE , E, X) where g ◦ M ◦ idE (e) = g(M (e)) (∀e ∈ E), the complement X − g ◦ M ◦ idE (e) is still a finite subset of X since g is an one-to-one respondence, thus (idE , g)(M, E, X) ∈ T . Therefore, (idE , g)−1 is continuous. Hence, (X, (idE , g)) is a soft topological dynamical system. Example 4 Let us consider the soft topological space in Example 1(3). Define g : X −→ X as follows: g(x) = x + 1 (∀x ∈ X). Then for every (a, b) ∈ B, g(a, b) = (a + 1, b + 1), and g−1 (a, b) = (a − 1, b − 1), thus g(B) = g−1 (B) = B. Denote the topology on X generated by g(B) and g−1 (B) by g(J ) and g−1 (J ). Then g(J ) = g−1 (J ) = J . For any (M, E, X) ∈ T , (idE , g)−1 (M, E, X) = (g−1 ◦ M ◦ idE , E, X), where g−1 ◦ M ◦ idE (e) = g−1 (M (e)) (∀e ∈ E), since M (e) ∈ J , we have g−1 (M (e)) ∈ g−1 (J ) = J , thus (idE , g)−1 (M, E, X) ∈ T . Therefore, (idE , g) is continuous. On the other hand, for any (M, E, X) ∈ T , ((idE , g)−1 )−1 (M, E, X) = (idE , g−1 )−1 (M, E, X) = ((g−1 )−1 ◦ M ◦ idE , E, X) = (g ◦ M ◦ idE , E, X) 877

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where

g ◦ M ◦ idE (e) = g(M (e)) (∀e ∈ E), since M (e) ∈ J , we have g(M (e)) ∈ g(J ) = J , thus (idE , g)(M, E, X) ∈ T . Therefore, (idE , g)−1 is continuous. Hence, (X, (idE , g)) is a soft topological dynamical system. Example 5 Let us consider the soft topological space in Example 1(4). Define g : X −→ X as follows: g(x) = For every (a, b) ∈ B,

(

x ∈ [0, 12 ];

2x,

2 − 2x, x ∈ ( 12 , 1].

 a b b ≤ 12 ; ( , ),    2 2 2−b 1 g−1 (a, b) = ( 2−a 2 , 2 ), a ≥ 2 ;    a 2−b ( 2 , 2 ), a < 21 < b.

Thus g−1 (B) ⊆ B. Let g−1 (J ) be the topology on X generated by g−1 (B), then g−1 (J ) ⊆ J . For any (M, E, X) ∈ T , (idE , g)−1 (M, E, X) = (g−1 ◦ M ◦ idE , E, X), where g−1 ◦ M ◦ idE (e) = g−1 (M (e)) (∀e ∈ E), since M (e) ∈ J , we have g−1 (M (e)) ∈ g−1 (J ) ⊆ J , thus (idE , g)−1 (M, E, X) ∈ T . Therefore, (idE , g) is continuous. Hence, (X, (idE , g)) is a semi-soft topological dynamical system. Definition 7

Let (X, (idE , g)) be a soft discrete topological dynamical system and

e is a soft point. Define several soft sets as follows: eM ∈ X

Orb(idE ,g) (eM ) = {(idE , g)n (eM ) | n ∈ Z},

n Orb+ (idE ,g) (eM ) = {(idE , g) (eM ) | n ∈ N − {0}}. −n Orb− (eM ) | n ∈ N − {0}}. (idE ,g) (eM ) = {(idE , g) − Then we call Orb(idE ,g) (eM ) (resp., Orb+ (idE ,g) (eM ), Orb(idE ,g) (eM )) the soft orbit (resp., soft

positive semi-orbit, soft negative semi-orbit) of the soft dynamical system of (idE , g). e if (idE , g)n (eM ) = eM for some n ∈ N − {0}, then eM is called a soft Let eM ∈ X,

periodic point of (idE , g), the smallest one of such integers is referred to as the soft period of eM . In particular, if (idE , g)(eM ) = eM , then eM is called a soft fixed point of (idE , g). Let P er(idE , g) (resp. F ix(idE , g)) be the set of all soft periodic points (resp. all soft fixed points) of (idE , g). Then F ix(idE , g) ⊆ P er(idE , g). e be a soft point, then the soft set Definition 8 Let eM ∈ X ω(eM ) =

\ f

n∈N −{0}

[ f

{(idE , g)k (eM ) | k ≥ n},

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is called a soft ω-limit set.

Obviously ω(eM ) is a soft closed set of (X, T , E). If the soft topological space (X, T , E) is soft compact, then ω(eM ) 6= e ∅ by Theorem 7.4 in [20].

Definition 9 Let (X, (idE , g)) be a soft discrete topological dynamical system, and

e a soft point. eM ∈ X

(1) If for each soft open neighborhood (N, E, X) of eM , there exists an n ∈ N − {0} such

that (idE , g)n (eM )

e (N, E, X), then eM is called a soft recurrent points of (idE , g). The set of all soft recurrent ∈ points of (idE , g) is denoted by Rec(idE , g). Clearly, P er(idE , g) ⊆ Rec(idE , g).

(2) If for each soft open neighborhood (N, E, X) of eM , there exists an n ∈ N − {0} such

that e (N, E, X) 6= e (idE , g)−n (N, E, X)∩ ∅.

Then eM is called a soft nonwandering point of (idE , g). The set of all soft nonwandering points of (idE , g) is denoted by Ω(idE , g), i.e., e | eM be a soft nonwandering Ω(idE , g) = {eM ∈ X

point of (idE , g)}.

e − Ω(idE , g) is called a soft wandering point. Each soft point of X

Definition 10 Let (idE , g) be a soft continuous function from (X, T , E) to (X, T , E). (1) (idE , g) is called soft topological mixing if, for any pair (M, E, X) and (N, E, X) ∈ T

of nonempty soft open sets of (X, T , E), there exists an n ∈ N − {0} such that e (N, E, X) 6= e (idE , g)n (M, E, X)∩ ∅.

e such (2) (idE , g) is called soft topological transitivity if there exists a soft point eM ∈ X

e e (i.e. Orb(id ,g) (eM ) = X). that Orb(idE ,g) (eM ) is dense in X E

e (3) A soft set (N, E, X) is said to be soft invariant of (idE , g) if (idE , g)(N, E, X)⊆(N, E, X)

(i.e. g(N (e)) ⊆ N (e) for each e ∈ E).

Theorem 3 Let (X, T , E) be a soft topological space, and (idE , g) : S(X, E) −→ S(X, E) be a soft continuous function from (X, T , E) to (X, T , E). Then e and Rec(idE , g) (1) Ω(idE , g) is a soft closed set of X,

e ⊆Ω(id E , g).

(2) Orb(idE ,g) (eM ), ω(eM ), P er(idE , g), F ix(idE , g) and Ω(idE , g) are invariant of (idE , g). (3) Ω((idE , g)m ) is an invariant and closed soft set, and e Ω((idE , g)m )⊆Ω(id E , g) (m ∈ N − {0}).

e is a soft nonwandering point if one of the following conditions (4) Each soft point eM ∈ X

is satisfied:

(i) (idE , g) is soft topological mixing, g is a one-to-one correspondence, and both (idE , g) and its inverse mapping (idE , g)−1 are continuous 879

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Proof (1) Suppose that a soft point eM is not a soft wandering point of (idE , g), then

there exists some soft open neighborhood (N, E, X) and some n ∈ N − {0} such that e (N, E, X) = e ∅. (idE , g)−n (N, E, X)∩

So all the soft points in (N, E, X) are not soft wandering points of (idE , g), it follows that e Ω(idE , g) be a soft closed set of X.

Now let soft point eM ∈ Rec(idE , g), then for each soft open neighborhood (N, E, X) of

e eM , there exists some n ∈ N − {0} such that (idE , g)n (eM )⊆(N, E, X), so for any e ∈ E,

gn (M (e)) ⊆ N (e), thus M (e) ⊆ g−n (N (e)), it implies that

then

e (idE , g−n )(N, E, X) = (idE , g)−n (N, E, X), eM ∈

hence

e (N, E, X), e (idE , g)−n (N, E, X)∩ eM ∈ e Rec(idE , g)⊆Ω(id E , g).

(2) We only show that ω(eM ) and Ω(idE , g) are invariant sets of (idE , g). Firstly, we have (idE , g)(ω(eM )) S T = (idE , g)( e n∈N −{0} e {(idE , g)k (eM ) | k ≥ n}) T S e k e e ⊆ n∈N −{0} (idE , g) {(idE , g) (eM ) | k ≥ n}) e ⊆

e ⊆

T e

T e

n∈N −{0}

n∈N −{0}

S e {(idE , g)k+1 (eM ) | k ≥ n})

S e {(idE , g)k (eM ) | k ≥ n} = ω(eM )

e Ω(idE , g) and (N, E, X) a soft open neighborhood of soft point Now let soft point eM ∈

(idE , g)(eM ), we can obtain that (idE , g)−1 (N, E, X) is a soft open neighborhood of soft

point eM since (idE , g) is a soft continuous function, then there exists some n ∈ N − {0} such that

So

Therefore

Hence

e (N, E, X)) (idE , g)−1 ((idE , g)−n (N, E, X))∩

e (idE , g)−1 (N, E, X) = (idE , g)−n ((idE , g)−1 (N, E, X))∩ 6= e ∅ e (N, E, X) 6= e (idE , g)−n (N, E, X))∩ ∅. e Ω(idE , g), (idE , g)(eM )∈ e (idE , g)(Ω(idE , g))⊆Ω(id E , g). 880

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(3) Straightforward.

e be a soft point and (N, E, X) ∈ T be a soft open neighborhood eX (4) Let (i) hold, eM ∈

of eM . Because (idE , g) is soft topological mixing, there exists some n ∈ N − {0} such that e (M, E, X) 6= e (idE , g)n (M, E, X)∩ ∅.

Then

e (M, E, X) 6= e (idE , g)−n (M, E, X)∩ ∅

since g is a one-to-one correspondence and both (idE , g) and its inverse mapping (idE , g)−1 are continuous. Thus eM ∈ Ω(idE , g). Let (ii) hold. Then e e = P er(idE , g)⊆Rec(id e e e X. X E , g)⊆Ω(id E , g) = Ω(idE , g)⊆

e 2 Therefore Ω(idE , g) = X.

Remark 4 If g is a one-to-one correspondence, both (idE , g) : S(X, E) −→ S(X, E)

and its inverse mapping (idE , g)−1 : S(X, E) −→ S(X, E) are continuous, and (M, E, X) ∈ S(X, E). Then e (M, E, X) 6= e (idE , g)n (M, E, X)∩ ∅

if and only if

e (M, E, X) 6= e (idE , g)−n (M, E, X)∩ ∅ (∀n ∈ N − {0}).

So Ω(idE , g) = Ω(idE , g)−1 .

Definition 11 Let (X, TX , E) and (Y, TY , E) be soft topological spaces, (idE , g) : S(X, E) −→ S(X, E) be a soft continuous function from (X, TX , E) to (X, TX , E), (idE , f ) : S(Y, E) −→ S(Y, E)) be a soft continuous function from (Y, TY , E) to (Y, TY , E)). If there exists a soft continuous function (idE , h) : S(X, E) −→ S(Y, E) from (X, TX , E) to (Y, TY , E) such that (idE , h) ◦ (idE , f ) = (idE , g) ◦ (idE , h) (i.e. (idE , h ◦ f ) = (idE , g ◦ h) ), then (idE , h) is said to be soft topology semi-conjugate from (idE , g) to (idE , f ). If g is a one-to-one correspondence and both (idE , g) and its inverse 881

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(idE , g) to (idE , f ). Here, we denote (idE , g) ∼ = (idE , f ). (idE , g) - S(X, E) S(X, E) (idE , h)

(idE , h) ?

S(Y, E)

(idE , f )

? - S(Y, E)

fig.1 Remark 5 (1) ∼ = is an equivalence relation. (2) If (idE , h) is a soft topological conjugate mapping from (idE , g) to (idE , f ), then for e and n ∈ N − {0}, we have each soft point eM ∈ X

(idE , h)((idE , f )n (eM )) = (idE , gn )((idE , h)(eM )),

it follows that (idE , h)(Orb(idE ,g) (eM )) = Orb(idE ,g) ((idE , h)(eM )), and it is easy to show that (idE , h)(ω(eM )) = ω((idE , h)(eM )); (idE , h)(P er(idE , g)) = P er(idE , f ); (idE , h)(F ix(idE , g)) = F ix(idE , f ); (idE , h)(Rec(idE , g)) = Rec(idE , f ); (idE , h)(Ω(idE , g)) = Ω(idE , f ).

3

Soft topological entropy In this section, the definition of soft topological entropy will be given and some fundamental

properties of the soft topological entropy will be studied. Definition 12 Let (X, (idE , g)) be a soft compact discrete topological dynamical system, e Denote the smallest cardinality of all subcovers (for X) e of and α be a soft open cover of X.

α by NXe (α), i.e.,

  [ f e NXe (α) = min |β| | β ⊆ α and X = β .

e is compact soft set, N e (α) is a positive integer. Let H e (α) = log N e (α). Since X X X X e Define their join by Let α and β be two soft open covers of X.

b β = {(P, E, X)∩ e (Q, E, X) | (P, E, X) ∈ α, (Q, E, X) ∈ β}. α∪ 882

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b β is a soft open cover of X. It is well known that β is called a refinement Clearly, the join α∪ of α (denoted by α ≺ β) if for each (Q, E, X) ∈ β, there exists a (P, E, X) ∈ α such that

e (Q, E, X)⊆(P, E, X).

Theorem 4 Let(X, (idE , g)) be a soft compact discrete topological dynamical system, α

e Then the following hold. and β be two soft open covers of X. (1) HXe (α) ≥ 0.

(2) if β ≺ α,then HXe (α) ≤ HXe (β). S (3) H e (α b β) ≤ H e (α) + H e (β). X

(4) HXe ((idE

X −1 , g) (α))

X

= HXe (α).

Proof we only prove (4). Let NXe (α) = n, then any subcover of α containing less than n

e Let elements of α would not cover X.

{(P1 , E, X), (P2 , E, X), · · · , (Pn , E, X)}

e of α with a cardinality n, since (idE , g) is continuous, be a subcover (for X) {(idE , g)−1 (P1 , E, X), (idE , g)−1 (P2 , E, X), · · · , (idE , g)−1 (Pn , E, X)} e of (idE , g)−1 (α). By (idE , g)(X) e =X e we can know X e = is a subcover (for (idE , g)−1 (X))

e so (idE , g)−1 (X),

{(idE , g)−1 (P1 , E, X), (idE , g)−1 (P2 , E, X), · · · , (idE , g)−1 (Pn , E, X)}

e of (idE , g)−1 (α). Therefore, is a finite open subcover (for X)

NXe ((idE , g)−1 (α)) ≤ n = NXe (α)

which implies HXe ((idE , g)−1 (α)) ≤ HXe (α).

Now, suppose that NXe ((idE , g)−1 (α)) = m. Let

{(idE , g)−1 (Q1 , E, X), (idE , g)−1 (Q2 , E, X), · · · , (idE , g)−1 (Qm , E, X)}

e of (idE , g)−1 (α). Therefore, be a finite open subcover (for X)

[m e = f {(idE , g)−1 (Qi , E, X)}. X i=1

e = X, e then Since (idE , g)(X)

e {(idE , g)−1 (Qi , E, X)} e = (idE , g)(X e) = S X i=1 Sm = e i=1 {(idE , g)(idE , g)−1 ((Qi , E, X))} Sn = e i=1 {(Qi , E, X)}. m

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So

{(Qi , E, X) | i = 1, 2, · · · , m} e of α, Hence, m ≥ N e (α), i.e., is a finite open subcover (for X) X NXe ((idE , g)−1 (α)) ≥ NXe (α)

which implies

By the above, we can get that

HXe ((idE , g)−1 (α)) ≥ HXe (α). HXe ((idE , g)−1 (α)) = HXe (α). 

Theorem 5 Let(X, (idE , g)) be a soft compact discrete topological dynamical system, α e Then the limit be a soft open cover of X.

[ 1 c n−1 HXe ( {(idE , g)−k (α)}) k=1 n→∞ n lim

exists. Proof. Let

[ c n−1 an = HXe ( {(idE , g)−k (α)}). k=1

We only need to show that

an+p ≤ an + ap (∀n, p ∈ N − {0}). From theorem 2.7(3) and (4), we have Sn+p−1   an+p = HXe b k=0 (idE , g)−k (α) Sn−1   = HXe ( b k=0 (idE , g)−k (α)

 n+p−1   S −k b S b (idE , g) (α) ) Sn−1  k=n  = HXe ( b k=0 (idE , g)−k (α)

Thus an+p ≤ an + ap . 

  S b (idE , g)−n S b p−1 (idE , g)−k (α) ) k=0 Sn−1   −k b ≤ HXe (id , g) (α) E k=0 Sp−1   +HXe b k=0 (idE , g)−k (α) .

Definition 13 Let (X, (idE , g)) be a soft compact discrete topological dynamical system, e Then let α be a soft open cover of X.

e ) = lim 1 H e Ent((idE , g), α, X n→∞ n X 884

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−k

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC e relative is called the soft topological entropy of (idE , g) on X to α, and

e) | Ent(idE , g) = sup{Ent((idE , g), α, X α

e α is a soft open cover of X}

is called the soft topological entropy of (idE , g).

e is a soft compact subset of X, e then the By Theorem 1, each soft closed subset of X

following theorem holds.

Theorem 6 Let (X, (idE , g)) be a soft compact discrete topological dynamical syse (A1 , E, X) and (A2 , E, X) be two closed soft sets, and tem, α be a soft open cover of X,

e 2 , E, X), Then (A1 , E, X)⊆(A (1)

Ent((idE , g), α, (A1 , E, X)) ≤ Ent((idE , g), α, (A2 , E, X)). (2) Ent((idE , g), (A1 , E, X)) ≤ Ent((idE , g), (A2 , E, X)). Proof. (1) Let

[ c n−1 N(A2 ,E,X)( {(idE , g)−k (α)}) = s. k=0

Then there exists a soft open subcover

{(P1 , E, X), (P2 , E, X), · · · , (Ps , E, X)} of

[ c n−1 k=0

{(idE , g)−k (α)}

e 2 , E, X), we have for (A2 , E, X). Since (A1 , E, X) ⊆(A

{(P1 , E, X), (P2 , E, X), · · · , (Ps , E, X)}

is also a subcover of

[ c n−1 k=0

for (A1 , E, X), and hence

{(idE , g)−k (α)}

[ c n−1 N(A1 ,E,X) ( {(idE , g)−k (α)}) ≤ s k=0

[ c n−1 {(idE , g)−k (α)}). = N(A2 ,E,X) ( k=0

So

[ c n−1 H(A1 ,E,X) ( {(idE , g)−k (α)}) k=0

[ c n−1 {(idE , g)−k (α)}). ≤ H(A2 ,E,X)( k=0

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Then

Ent((idE , g), α, (A1 , E, X)) ≤ Ent((idE , g), α, (A2 , E, X)). (2) Ent((idE , g), (A1 , E, X)) = sup{Ent((idE , g), α, (A1 , E, X)) | α is a soft open α

e cover of X}

≤ sup{Ent((idE , g), α, (A2 , E, X)) | α is a soft open α

= Ent((idE , g), (A2 , E, X)). 

e cover of X}

Theorem 7 Let (X, (idE , g)) be a soft compact discrete topological dynamical system, e Then Ent(idE , idX ) = 0. and α be a soft open cover of X. Proof Straightforward.

Theorem 8 Ent(idE , gm ) ≥ m · Ent(idE , g) ( ∀m ∈ N − {0}).

Proof As ((gn )← )m = (g← )nm (∀n ∈ N − {0}, ∀m ∈ N ), we have

S b n−1 {(idE , gm )−s S b m−1 {(idE , g)−t (α)}} t=0

= Hence

t=0

S b mn−1 {(idE , g)−s (α)} s=0

[ [ c n−1 c m−1 HXe ( {(idE , gm )−s {(idE , g)−t (α))}} t=0

t=0

[mn−1 

= HXe (

Denote

β= Then

c

s=0

[ c m−1 s=0

(idE , g)−s (α) ).

{(idE , g)−s (α)}.

e Ent(idE , gm ) = Ent(idE , g)m ≥ Ent((idE , g)m , β, X) Sn−1  Sm−1 = lim n1 HXe b t=0 {(idE , gm )−s b t=0 {(idE , g)−t (α))}} n→∞

=

Hence,

lim m ·

n→∞

S 1 −s b mn−1 e ( s=0 {(idE , g) (α)}) mn HX

e ). = m · Ent((idE , g), α, X

e) Ent(idE , gm ) ≥ m · sup Ent((idE , g), α, X α

= m · Ent(idE , g). 

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Conclusion

In this paper, the discrete dynamical systems in soft topological spaces are defined, and simple examples are also given. Some basic concepts (such as soft ω-limit set, soft invariant set, soft periodic point, soft nonwandering point, and soft recurrent point) of the discrete dynamical system are introduced into soft topological spaces. Soft topological mixing and soft topological transitivity are also studied. At last, soft topological entropy is defined and several properties of it are discussed.

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

Acknowledgement This work was supported by the International Science and Technology Cooperation Foundation of China (Grant No. 2012DFA11270), the National Natural Science Foundation of China (Grant No. 11371292, 11071151), and Shaanxi Provincial Natural Science Foundation (Grant No. 2014JM1018).

References [1] D. Molodtsov, “Soft set theory — First results”, Computers and Mathematics with Applications, vol. 37, no. 4-5, pp. 19-31, 1999. [2] D. Molodtsov, The theory of soft sets, Moscow: URSS Publisher, 2004(in Russian). [3] P. K. Maji, R. Bismas, and A. R. Roy, “Soft set theory”, Computers and Mathematics with Applications, vol. 45, no. 4, pp. 555-562, 2003. [4] M. I. Ali, F. Feng, X. Y. Liu, W. K. Win, and M. Shabir, “On some new operations in soft set theory”, Computers and Mathematics with Applications, vol. 57, no. 9, pp. 1547-1553, 2009. [5] H. Akta¸s, and N. C ¸ aˇgman, “Soft sets and soft groups”, Information Sciences, vol. 177, no. 13, pp. 2726-2735, 2007. [6] Y.B. Jun, “Soft BCK/BCI-algebras”, Computers and Mathematics with Applications, vol. 56, pp. 1408 C 1413, 2008. [7] Y. B. Jun, and C. H. Park, “Applications of soft sets in ideal theory of BCK/BCIalgrbras”, Information Sciences, vol. 178, pp. 2466-2475, 2008. [8] W. A. Dudek, Y. B. Jun, and Z. Stojakovic, “On fuzzy ideals in BCC-algebras”, Fuzzy Sets and Systems, vol. 123, no. 2, pp.251-258, 2001. [9] Z. M. Zhang. “Intuitionistic fuzzy soft rings”, International Journal of Fuzzy Systems, vol. 14, no. 3, pp. 420-433, 2012. [10] F. Feng, Y. B. Jun, and X. Zhao, “Soft semirings”, Computers and Mathematics with Applications, vol. 56, no. 10, pp. 2621-2628, 2008.

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[11] P. K. Maji, A. R. Roy, and R. Bismas, “Soft set theory”, Computers and Mathematics with Applications, vol. 44, pp. 1077-1083, 2002.

[12] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic, Boston, MA, 1991. [13] W. H. Xu, Q. R. Wang, and X. T. Zhang, “Multi-granulation fuzzy rough sets in a fuzzy tolerance approximation space”, International Journal of Fuzzy Systems, vol. 13, no. 4, pp. 246-259, 2011. [14] D. Chen, E.C.C. Tsang, D. S. Yeung, and X. Wang, “The parametrization reduction of soft sets and its applications”, Computers and Mathematics with Applications, vol. 49, pp. 757-763, 2005. [15] Y. Yang, X. Tan, and C. C. Meng, “The multi-fuzzy soft set and its application in decision making”, Applied Mathematical Modelling, vol. 37, pp. 4915-4923, 2013. [16] M. Shabir, and M. Naz, “On soft topological spaces”, Computers and Mathematics with Applications, vol. 61, pp. 1786-1799, 2011. [17] W. K. Min, “A note on soft topological spaces”, Computers and Mathematics with Applications, vol. 62, pp. 3524-3528, 2011. ˙ Zorlutuna, M. Akdag, W. K. Min, and S. Atmaca, “Remarks on soft topological [18] I. spaces”, Annals of Fuzzy Mathematics and Informatics(in press). [19] L. Chen, H. Kou, M. K. Luo, W. N. Zhang, “Discrete dynamical systems in L-topological spaces”, Fuzzy Sets and Systems, vol. 156, pp. 25-42, 2005. [20] L. Liu, Y. G. Wang, and G. Wei, “Topological entropy of continuous functions on topological spaces”, Chaos, Solitons and Fractals, vol. 39, pp. 417-427, 2009.

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FUNCTIONAL INEQUALITIES IN VECTOR BANACH SPACE GANG LU, JUN XIE, YUANFENG JIN∗ , AND QI LIU Abstract. In this paper, we prove that the generalized Hyers-Ulam stability of the additive functional inequality kf (ax + by + cz) + f (bx + ay + bz) + f (cx + cy + az)k ≤ k(a + b + c)f (x + y + z)k in vector Banach space, where a 6= b 6= c ∈ R are fixed points with 3 > |a + b + c|.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [32] concerning the stability of group homomorphisms. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [24] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [24] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [9] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. The stability problems for several functional equations or inequations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2]–[8],[10], [12]–[16], [22]–[25],[26]-[31],[34]). We recall some basic facts concerning generalized norm. Definition 1.1 (see [15]). Let E be a real vector space. A generalized norm for E is a mapping k · kG : E → Rk+ denoted by kxkG = (α1 (x), α2 (x), α3 (x), · · · , αk (x)) such that (a) kxkG ≥ 0, that is, αi (x) ≥ 0 for all i = 1, 2, · · · , k; (b) kxkG = 0 if and only if x = 0, that is, αi (x) = 0 for all i, if and only if x = 0; (c) kλxkG = |λ|kxkG ,that is,αi (λx) = |λ|αi (x); (d) kx + ykG ≤ kxkG + kykG , which means, αi (x + y) ≤ αi (x) + αi (y); 2010 Mathematics Subject Classification. Primary 39B62, 39B52, 46B25. Key words and phrases. additive functional inequaties; Hyers-Ulam stability; vector Banach space ∗ Corresponding author:[email protected] (Y.Jin). 1

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Example 1.2. In R2 , kxkG = (|x1 |, |x2 |). Definition 1.3. Let (X, k·kG ) be a general normed linear space. Let xn be a sequence in X. Then xn is said to be convergent if there exists x ∈ X such that limn→∞ αi (xn −x) = 0 for all i = 1, 2 · · · , k. In that case, x is called the limit of the sequence xn and we denote it by G-limn→∞ xn = x. Definition 1.4. A sequence xn in X is called Cauchy if for each  > 0 and each a > 0 there exists n0 such that for all n ≥ n0 and all p > 0, we have kxn+p − xn kG ≤ , that is, αi (xn+p − xn ) ≤ . It is known that every convergent sequence in the general normed space is Cauchy. If each Cauchy sequence is convergent, then the the general normed space is said to be complete and the general normed space is called a vector Banach space. 2. HYers-Ulam Stability In vector Banach Space From now on , Let X be a normed linear space and Y a vector Banach space. This paper,we prove that the generalized Hyers-Ulam stability of the additive functional inequality kf (ax + by + cz) + f (bx + cy + bz) + f (cx + ay + az)kG ≤ k(a + b + c)f (x + y + z)kG in the vector Banach space, where a 6= b 6= c ∈ R are fixed points with 3 > |a + b + c|. Lemma 2.1. Let f : X → Y be a mapping. If it satisfies kf (ax + by + cz) + f (bx + cy + bz) + f (cx + ay + az)kG

(2.1)

≤ k(a + b + c)f (x + y + z)kG

for all x, y, z ∈ X and a, b, c are fixed real numbers with 3 > |a + b + c|. Then f is additive. Proof. Letting x = y = z = 0 in (2.1) for all x, y, z ∈ X , we get k3f (0)kG ≤ k(a + b + c)f (0)kG

(2.2)

for a, b, c ∈ R. For any i = 1, 2, · · · , k, αi (3f (0)) ≤ αi ((a + b + c)f (0)) we get 3αi (f (0)) ≤ |a + b + c|αi (f (0)), Thus f (0) = 0. Letting x = 0 and Replacing z by −y in (2.1), we get kf ((b − c)y) + f ((c − b)y)kG ≤ k(a + b + c)f (0)kG = |a + b + c|αi (f (0)) = 0

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3

and so f (−x) = −f (x) for all x ∈ X . Replacing x by −y − z in (2.1), we have kf ((b − a)y + (c − a)z) + f ((a − b)y) + f ((a − c)z)kG ≤ 0 for all y, z ∈ X . Then we can obtain f (x + y) = f (x) + f (y) for all x, y ∈ X .



Theorem 2.2. Let f : X → Y be a mapping with f (0) = 0. If there is a function ϕ : X 3 → [0, ∞) such that kf (ax + by + cz) + f (bx + cy + bz) + f (cx + ay + az)kG ≤ k(a + b + c)f (x + y + z)kG + (ϕ(x, y, z), ϕ(x, y, z), · · · , ϕ(x, y, z)) | {z }

(2.3)

k

and ϕ(x, e y, z) :=

∞ X
1 ϕ (−2)j x, (−2)j x, (−2)j x < ∞ j 2 j=0

(2.4)

for all x, y, z ∈ X and a 6= b 6= c ∈ R are fixed points with 3 > |a + b + c|, then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)kG        (2.5) b + c − 2a 1 1 b + c − 2a 1 1  ≤ ϕ e x, x, x , · · · , ϕ e x, x, x  (a − b)(a − c) a − b a − c (a − b)(a − c) a − b a − c  | {z } k

for all x ∈ X . Proof. Letting x = −y − z in (2.3), we get kf ((b − a)y + (c − a)z) + f ((a − b)y) + f ((a − c)z)kG ≤ (ϕ(−y − z, y, z), · · · , ϕ(−y − z, y, z)) {z } |

(2.6)

k

for all y, z ∈ X . x Letting y = b−a ,z =

y c−a

in (2.6), we get

kf (x + y) + f (−x) + f (−y)kG      x y x y x y x y (2.7) ≤ ϕ + , , ,··· ,ϕ + , , a−b a−c b−a c−a a−b a−c b−a c−a | {z } k

for all x, z ∈ X .

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Letting x = y in (2.7) we get k2f (−x) + f (2x)kG    2a − b − c 1 1 ≤ ϕ x, x, x ,··· , (a − b)(a − c) b − a c − a   1 1 2a − b − c x, x, x ϕ (a − b)(a − c) b − a c − a for all x ∈ X . Thus



f (−2x)

f (x) −

−2 G    b + c − 2a 1 1 1 ϕ x, x, x ,··· , ≤ 2 (a − b)(a − c) a − b a − c   b + c − 2a 1 1 ϕ x, x, x (a − b)(a − c) a − b a − c for all x ∈ X . Hence one may have the following formula for positive integers m, l with m > l,

1


1 l m

(−2)l f (−2) x − (−2)m f ((−2) x) G    m−1 X 1 (−2)i (b + c − 2a) (−2)i (−2)i x, x, x ,··· , ≤ ϕ 2i (a − b)(a − c) a−b a−c i=l   (−2)i (b + c − 2a) (−2)i (−2)i ϕ x, x, x (a − b)(a − c) a−b a−c for all x ∈ X . That is,  
1 1 l m αi f (−2) x − f ((−2) x) (−2)l (−2)m m−1 X 1  (−2)i (b + c − 2a) (−2)i (−2)i  ≤ ϕ x, x, x 2i (a − b)(a − c) a−b a−c i=l for all x ∈ mathcalX. It follows from (2.8) that the sequence

n

f ((−2)k x) (−2)k

(2.8)

o

sequence for all x ∈ X . Since Y is a generalized norm space, the sequence converges. So one may define the mapping A : X → Y by   f ((−2)k x) A(x) := G − lim , ∀x ∈ X . k→∞ (−2)k

is a Cauchy n o k f ((−2) x) (−2)k

Taking m = 0 and letting l tend to ∞ in (2.8), we have the inequality (2.5).

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It follows from (2.3) that kA(ax + by + cz) + A(bx + ay + bz) + A(cx + cy + az)kG 1 f ((−2)k (ax + by + cz)) + f ((−2)k (bx + ay + bz)) = lim k k→∞ (−2)

+f ((−2)k (cx + cy + az)) G

1 (a + b + c)f ((−2)k (x + y + z)) (2.9) ≤ lim G k→∞ (−2)k   1 ϕ((−2)k x, (−2)k y, (−2)k z), · · · , ϕ((−2)k x, (−2)k y, (−2)k z) + lim | {z } k→∞ (−2)k k

≤ k(a + b + c)A(x + y + z)kG for all x, y, z ∈ X . One see that A satisfies the inequality (2.1) and so it is additive by Lemma (2.1). Now, we show that the uniqueness of A. Let T : X → Y be another additive mapping satisfying (2.5). Then one has

1

1 k k

kA(x) − T (x)kG =

(−2)k A (−2) x − (−2)k T (−2) x G

1 ≤ k A (−2)k x − f (−2)k x G

2


+ T (−2)k x − f (−2)k x G      1  (b + c − 2a)(−2)k (−2)k (−2)k  ≤2 k ϕ e x, x, x , · · ·  2  (a − b)(a − c) a−b a−c | {z } k

which tends to zero as k → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X.  Theorem 2.3. Let f : X → Y be a mapping with f (0) = 0. If there is a function ϕ : X 3 → [0, ∞) satisfying (2.3) such that   ∞ X x y z j ϕ(x, e y, z) := 2ϕ , , 0, there is a positive integer N such that G(αn , αm , αl ) < ε for all n, m, l > N. Definition 5. [20] A metric space (X, G) is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in (X, G) is G-convergent in X. Definition 6. [9] Let (X, ≼) be a partially ordered set, T : X ×X → X. Then T is said to have mixed-monotone property if T (x, y) is monotone non-decreasing in x and monotone non-increasing in y. That is., for all x, y ∈ X Definition 7. [17] Let (X, ≼) be a partially ordered set, T : X × X → X and g : X → X. We say T is the g-mixed monotone property if T is monotone gnondecreasing in its first argument and monotone g-non-increasing in its second argument. That is., for all x, y ∈ X x1 , x2 ∈ X, gx1 ≼ gx2 ⇒ T (x1 , y) ≼ T (x2 , y), y1 , y2 ∈ X, gy1 ≼ gy2 ⇒ T (x, y1 ) ≽ T (x, y2 ). Definition 8. [9] Let T : X × X → X be a map such that T (x, y) = x and T (y, x) = y then the pair (x, y) ∈ X × X is called a coupled fixed point of T . It is clear that (x, y)is a coupled fixed point if and only if (y, x) is such. Definition 9. [17] Let T : X × X → X and g : X → X be two map such that T (x, y) = gx and T (y, x) = gy then the pair (x, y) ∈ X × X is called a coupled coincidence point of T and g. Definition 10. [17] Two maps T : X × X → X and g : X → X are said to be commutative if g(T (x, y)) = T (gx, gy). Chakrababati [10] proved the following results. Theorem 1. [10]Let (X, ≼) be a partially ordered set and let (X, G) be a G−complete G-metric space. Suppose T : X×X −→ X be a continuous mapping on X having the mixed monotone property. Suppose for all (x, y), (u, v), (w, z) ∈ X × X with (x, y) ≼ (u, v) ≼ (w, z) holds G(T (x, y), T (u, v), T (w, z)) G (x, T (x, y), T (x, y)) G (u, T (u, v), T (u, v)) G (w, T (w, z), T (w, z)) G2 (x, u, w) + βG(x, u, w),

≤α

3

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where 8α + β < 1. If there exist x0 , y0 ∈ X such that x0 ≼ T (x0 , y0 ) and y0 ≽ T (y0 , x0 ), then T has a coupled fixed point (x∗ , y∗ ) ∈ X. Theorem 2. [10] Let (X, ≼) be a partially ordered set and let (X, G) be a G−complete G-metric space. Suppose T : X × X −→ X and g : X −→ X be a continues mappings on X such that T has the mixed g-monotone property. Suppose that T (X ×X) ⊆ g(X), g commute with T and for (x, y), (u, v), (w, z) ∈ X × X with (x, y) ≼ (u, v) ≼ (w, z) and gx ≼ gu ≼ gw or gy ≽ gu ≽ gz holds G(T (x, y), T (u, v), T (w, z)) G (gx, T (x, y), T (x, y)) G (gu, T (u, v), T (u, v)) G (gw, T (w, z), T (w, z)) G2 (gx, gu, gw) + βG(gx, gu, gw),

≤α

where 8α + β < 1. If there exist x0 , y0 ∈ X such that gx0 ≼ T (x0 , y0 ) and gy0 ≽ T (y0 , x0 ) then T and g have a coupled coincidence point (x∗ , y∗ ) ∈ X ×X, that is., (x∗ , y∗ ) satisfies gx∗ = T (x∗ , y∗ ), gy∗ = T (y∗ , x∗ ).

2

Main Results

In our main results we used the following two classes. ψ ∈ Ψ if and only if ψ : [0, ∞) → [0, ∞), ψ is continuous and non-decreasing function such that ψ (t) = 0 if and only if t = 0. ϕ ∈ Φ if and only if ϕ : [0, ∞) → [0, ∞), ψ is a lower semi continuous and non-decreasing function such that ϕ (t) = 0 if and only if t = 0. Also, for more details of G-metric spaces see ([1]-[4], [7], [16], [18], [21], [26]-[28]). Remark 1. It is worth to noticing that both results in [10] without the conditions G2 (x, u, w) ̸= 0 that is., G(gx, gu, gw) ̸= 0 are not correct. Now, we announce the first our result. Theorem 3. Let (X, ≼) be a partially ordered set and let (X, G) be a G−complete symmetric G-metric space. Suppose T : X × X −→ X be a continuous mapping having the mixed monotone property and satisfying ψ(G(T (x, y), T (u, v), T (w, z)) ≤ ψ(M (x, u, w, y, v, z)) − ϕ(M (x, u, w, y, v, z)), (2.1) for all x, y, z, u, v, w ∈ X with G(x, u, w) ̸= 0 and (x, y) ≼ (u, v) ≼ (w, z) or (x, y) ≽ (u, v) ≽ (w, z), where M (x, u, w, y, v, z) { [G(x, T (x, y), T (x, y)G(u, T (u, v), T (u, v)G(w, T (w, z), T (w, z)] = max , G2 (x, u, w) } G(x, u, w) , ψ ∈ Ψ and ϕ ∈ Φ. If there exist x0 , y0 ∈ X such that x0 ≼ T (x0 , y0 ) and y0 ≽ T (y0 , x0 ). Then T has a coupled fixed point (x∗ , y∗ ) ∈ X. 4

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Proof. Suppose that there exist x0 , y0 ∈ X such that x0 ≼ T (x0 , y0 ) and y0 ≽ T (y0 , x0 ) . Further, define xn+1 = T (xn , yn ) and yn+1 = T (yn , xn ) . Using the mixed monotone property and the mathematical induction we obtain that xn ≼ xn+1 and yn ≽ yn+1 for all n ∈ N (very known method). Consider now ψ (G (xn+1 , xn , xn )) = ψ (G (T (xn , yn ) , T (xn−1 , yn−1 ) , T (xn−1 , yn−1 ))) . Using (2.1) we have that ψ (G (xn+1 , xn , xn )) ≤ψ (M (xn , xn−1 , xn−1 , yn , yn−1 , yn−1 )) − ϕ (M (xn , xn−1 , xn−1 , yn , yn−1 , yn−1 ))

(2.2)

where M (xn , xn−1 , xn−1 , yn , yn−1 , yn−1 ) { G (xn , T (xn , yn ) , T (xn , yn )) G2 (xn−1 , T (xn−1 , yn−1 ) T (xn−1 , yn−1 )) = max , G2 (xn , xn−1 , xn−1 ) } G (xn , xn−1 , xn−1 ) . Let Gn = G(xn , xn−1 , xn−1 ) then, M (xn , xn−1 , xn−1 , yn , yn−1 , yn−1 ) = max{Gn+1 , Gn }. Further we show that Gn is non-incresing. Suppose their exist n0 such that Gn0 +1 > Gn0 then from (2.2) ψ(Gn0 +1 ) ≤ ψ(Gn0 +1 ) − ϕ(Gn0 +1 ). Which implies that ϕ(Gn0 +1 ) ≤ 0. A contradiction. Hence Gn ≥ Gn+1 for all n ≥ 1. Since {Gn } is a non-increasing sequence of positive real numbers there exists r ≥ 0 such that lim Gn = r. (2.3) n→∞

We shall show that r = 0. Suppose r > 0 then applying limit in (2.2) and using (2.3), we have ψ(r) ≤ ψ(r) − ϕ(r) < ψ(r). We obtain a contradiction. Therefore r = 0 that is., lim Gn = 0.

n→∞

(2.4)

Now, we show that {xn } is a G-Cauchy sequence. Suppose that, {xn } is not G-Cauchy. Then, there exist ϵ > 0 and subsequences {xn(k) } and {xm(k) } of {xn } with n(k) > m(k) > k such that, G(xm(k) , xm(k) , xn(k) ) ≥ ϵ, ∀ k ∈ N.

(2.5)

Furthermore, corresponding to m(k) one can choose n(k) such that, it is the smallest integer with n(k) > m(k) satisfying (2.5) then, G(xm(k) , xm(k) , xn(k)−1 ) < ϵ, ∀ k ∈ N

(2.6)

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Now ϵ ≤ G(xm(k) , xm(k) , xn(k) ) = G(xn(k) , xm(k) , xm(k) ) ≤ G(xm(k) , xm(k) , xn(k)−1 ) + G(xn(k)−1 , xn(k)−1 , xn(k) ), Taking limit k → ∞ and using (2.4) we get lim G(xm(k) , xm(k) , xn(k) ) = ϵ.

(2.7)

k→∞

Now G(xm(k)−1 , xm(k)−1 , xn(k)−1 ) = G(xn(k)−1 , xm(k)−1 , xm(k)−1 ) ≤ G(xn(k)−1 , xn(k) , xn(k) ) + G(xn(k) , xm(k)−1 , xm(k)−1 ) ≤ G(xn(k)−1 , xn(k) , xn(k) ) + G(xn(k) , xm(k) , xm(k) ) + G(xm(k) , xm(k)−1 , xm(k)−1 ), (2.8) and G(xn(k) , xm(k) , xm(k) ) ≤ G(xn(k) , xm(k)−1 , xm(k)−1 ) + G(xm(k)−1 , xm(k) , xm(k) ) ≤ G(xn(k) , xn(k)−1 , xn(k)−1 ) + G(xn(k)−1 , xm(k)−1 , xm(k)−1 ) + G(xm(k)−1 , xm(k) , xm(k) ),

(2.9)

Using limit k → ∞ in (2.8) and (2.9) and using (2.4) and (2.7) we get lim G(xm(k)−1 , xm(k)−1 xn(k)−1 ) = ϵ.

(2.10)

k→∞

Consider ( ) ψ G(xm(k) , xm(k) , xn(k) ( ) ≤ ψ M (xm(k)−1 , xm(k)−1 , xn(k)−1 , ym(k)−1 , ym(k)−1 , yn(k)−1 ) ( ) − ϕ M (xm(k)−1 , xm(k)−1 , xn(k)−1 , ym(k)−1 , ym(k)−1 , yn(k)−1 ) ,

(2.11)

where M (xm(k)−1 , xm(k)−1 , xn(k)−1 , ym(k)−1 , ym(k)−1 , yn(k)−1 ) { [[G(xm(k)−1 , xm(k) , xm(k) )]2 G(xn(k)−1 , xn(k) , xn(k) ] = max , G(xm(k)−1 , xm(k)−1 , xn(k)−1 )2 } G(xm(k)−1 , xm(k)−1 , xn(k)−1 ) .

(2.12) (2.13)

Applying limit k → ∞ in (2.13), using (2.7), (2.10) and (2.4) we get lim M (xm(k)−1 , xm(k)−1 , xn(k)−1 , ym(k)−1 , ym(k)−1 , yn(k)−1 ) = ϵ.

k→∞

(2.14)

Taking limit of (2.11) using (2.7), (2.14) and lower semi continuity of ϕ we have ψ(ϵ) ≤ ψ(ϵ) − ϕ(ϵ) < ψ(ϵ), 6

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which is contradiction. So ϵ = 0. Therefore xn is a G-Cauchy sequence. Similarly by the same argument we can show that yn is a G-Cauchy sequence. By completeness of X, there is x∗ , y∗ ∈ X such that xn → x∗ and yn → y∗ as n → ∞. Now we have to show that (x∗ , y∗ ) is a coupled fixed point of T . Since T is continuous on X and G is also continuous in each of its variable, so G(T (x∗ , y∗ ), x∗ , x∗ ) = G( lim T (xn , yn ), x∗ , x∗ ) = G(x∗ , x∗ , x∗ ) = 0. n→∞

Hence, we proved that T (x∗ , y∗ ) = x∗ Similarly by the same argument we obtain that T (y∗ , x∗ ) = y∗ . So (x∗ , y∗ ) is a coupled fixed point of T . Theorem 4. Suppose that the conditions of Theorem 3 are valid. In addition suppose that for each (x, y), (u, v) ∈ X × X exists (w, z) ∈ X × X which is comparable to (x, y) and (u, v). Then coupled fixed point of T is unique. ′

′

Proof. Suppose that (x∗ , y∗ ), (x , y ) ∈ X × X are two coupled fixed points. Case 1 If (x∗ , y∗ ), (x′ , y′ ) are comparable then from (2.1) ′

′

′

′

′

′

′

′

ψ(G(T (x∗ , y∗ ), T (x , y ), T (x , y ) ≤ψ(M (x∗ , x , x , y∗ , y , y ) ′

′

′

′

− ϕ(M (x∗ , x , x , y∗ , y , y ),

(2.15)

where ′

′

′

′

M (x∗ , x , x , y∗ , y , y ) { } ′ ′ ′ ′ ′ ′ ′ G(x∗ , T (x∗ , y∗ ), T (x∗ , y∗ ))[G(x , T (x , y ), T (x , y )]2 ] = max , G(x , x , x ) ∗ G(x∗ , x′ , x′ ) { } ′ ′ ′ ′ ′ G(x∗ , x∗ , x∗ )[G(x , x , x ]2 = max , G(x∗ , x , x ) . ′ ′ G(x∗ , x , x ) Which implies that ′

′

′

′

′

′

M (x∗ , x , x , y∗ , y , y ) = G(x∗ , x , x ). From (2.15) we have ′

′

′

′

′

′

′

′

ψ(G(x∗ , x , x ) = ψ(G(T (x∗ , y∗ ), T (x , y ), T (x , y ) < ϕ(G(x∗ , x , x ), ′

which is contradiction. Hence we must have x∗ = x . Similarly we can easily ′ show that y∗ = y so couple fixed point is unique. Case 2 ′ ′ If (x∗ , y∗ ), (x , y ) are not comparable by Theorem 3 there is a (u, v) ∈ X ×X ′ ′ comparable to (x∗ , y∗ ) and (x , y ) if there is m0 ∈ N such that T m0 (u, v) = (x∗ , y∗ ), then T m0 +1 (u, v) = T (x∗ , y∗ ) = x∗ , in last we get T m (u, v) = x∗ for m ≥ m0 this mean T m (u, v) → x∗ for m → ∞ if there is no such m0 then for any m ≥ 1 ψ(G(T m (u, v), x∗ , x∗ ) = ψ(G(T m (u, v), T m (x∗ , y∗ ), T m (x∗ , y∗ ) ≤ ψ(M (u, x∗ , x∗ , v, y∗ , y∗ ) − ϕ(M (u, x∗ , x∗ , v, y∗ , y∗ ), (2.16) 7

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where M (u, x∗ , x∗ , v, y∗ , y∗ ) { G(T m−1 (u, v), T m (u, v), T m (u, v))[G(T m−1 (x∗ , y∗ ), T m (x∗ , y∗ ), T m (x∗ , y∗ ))]2 = max , G(T m−1 (u, v), T m−1 (x∗ , y∗ ), T m−1 (x∗ , y∗ )) } G(T m−1 (u, v), T m−1 (x∗ , y∗ ), T m−1 (x∗ , y∗ )) { } G(T m−1 (u, v), x∗ , x∗ )[G(x∗ , x∗ , x∗ )]2 m−1 , G(T = max (u, v), x , x ) . ∗ ∗ G(T m−1 (u, v), x∗ , x∗ ) Which implies that M (u, x∗ , x∗ , v, y∗ , y∗ ) = G(T m−1 (u, v), x∗ , x∗ ). Putting M in (2.16), we have ψ(G(T m (u, v), x∗ , x∗ ) ≤ ψ(G(T m−1 (u, v), x∗ , x∗ )) ϕ(G(T m−1 (u, v), x∗ , x∗ )).

(2.17)

This implies that ψ(G(T m (u, v), x∗ , x∗ ) < ψ(G(T m−1 (u, v), x∗ , x∗ ), since ψ is non-decreasing therefore, G(T m (u, v), x∗ , x∗ ) < G(T m−1 (u, v), x∗ , x∗ ) that is, {G(T m (u, v), x∗ , x∗ )} is a decreasing sequence of positive real numbers. Therefore, there is an α1 such that {G(T m (u, v), x∗ , x∗ )} → α1 . We shall show that α1 = 0. Suppose, to the contrary, that α1 > 0. Taking the limit in equation (2.17) we get contradiction. So α1 =0. Implies G(T m (u, v), x∗ , x∗ )=0, that is., ′ T m (u, v) = x∗ . Similarly we can show that T m (u, v) = y∗ , (T m (u, v) = x and ′ (T m (u, v) = y . Hence the coupled fixed point is unique. The next result is the generalization of Theorem 3. Because the proof is similar, then it is omitted. Theorem 5. Let (X, ≼) be a partially ordered set and let (X, G) be a G−complete symmetric G-metric space. Suppose that T : X × X −→ X and g : X −→ X are a continues mappings such that T has the g−mixed monotone property. Suppose that T (X × X) ⊆ g(X), g commute with T and satisfying ψ(G(T (x, y), T (u, v), T (w, z)) ≤ ψ(M (x, u, w, y, v, z) − ϕ(M (x, u, w, y, v, z), (2.18) for all x, y, z, u, v, w ∈ X with G(gx, gu, gw) ̸= 0 and (gx, gy) ≼ (gu, gv) ≼ (gw, gz) or (gx, gy) ≽ (gu, gv) ≽ (gw, gz), where M (x, u, w, y, v, z) { G (gx, T (x, y), T (x, y)) G (gu, T (u, v), T (u, v)) G (gw, T (w, z), T (w, z)) , = max G2 (gx, gu, gw) } G(gx, gu, gw) ,

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ψ ∈ Ψ and ϕ ∈ Φ. If there exist x0 , y0 ∈ X such that gx0 ≼ T (x0 , y0 ) and gy0 ≽ T (y0 , x0 ) then T and g have a coupled coincidence point (x∗ , y∗ ) ∈ X ×X, that is., (x∗ , y∗ ) satisfies gx∗ = T (x∗ , y∗ ), gy∗ = T (y∗ , x∗ ). Corollary 1. Let (X, G) be a partially ordered set and let (X, G) be a G−complete symmetric G-metric space. Suppose that T : X × X −→ X and g : X −→ X are a continues mappings such that T has the g−mixed monotone property. Suppose that T (X × X) ⊆ g(X), g commute with T and for 0 < k < 1 satisfying G(T (x, y), T (u, v), T (w, z)) ≤ k(M (x, u, w, y, v, z), for all x, y, z, u, v, w ∈ X with G(gx, gu, gw) ̸= 0 and (gx, gy) ≼ (gu, gv) ≼ (gw, gz) or (gx, gy) ≽ (gu, gv) ≽ (gw, gz), where M (x, u, w, y, v, z) { G (gx, T (x, y), T (x, y)) G (gu, T (u, v), T (u, v)) G (gw, T (w, z), T (w, z)) , = max G2 (gx, gu, gw) } G(gx, gu, gw) . If there exist x0 , y0 ∈ X such that gx0 ≼ T (x0 , y0 ) and gy0 ≽ T (y0 , x0 ) then T and g have a coupled coincidence point (x∗ , y∗ ) ∈ X × X, that is., (x∗ , y∗ ) satisfies gx∗ = T (x∗ , y∗ ), gy∗ = T (y∗ , x∗ ). Proof. The proof follows by taking ψ(t) = t, ϕ(t) = (1 − k)t where 0 < k < 1 in Theorem 5. 1 Remark 2. For 0 < α < 18 , 0 < β < 16 and for all x, y, z, u, v, w ∈ X with G(gx, gu, gw) ̸= 0 and (gx, gy) ≼ (gu, gv) ≼ (gw, gz) or (gx, gy) ≽ (gu, gv) ≽ (gw, gz) we have

G(T (x, y), T (u, v), T (w, z)) [G(gx, T (x, y), T (x, y)G(gu, T (u, v), T (u, v)G(gw, T (w, z), T (w, z)] G(gx, gu, gw)2 + βG(gx, gu, gw), { G (gx, T (x, y), T (x, y)) G (gu, T (u, v), T (u, v)) G (gw, T (w, z), T (w, z)) ≤(α + β) max , G2 (gx, gu, gw) } G(gx, gu, gw) .

≤α

where k = α +β < 1. Clearly, the relation 0 < 8α +β < 1 implies that Corollary 1 is the generalization of Theorem 2. Therefore Theorem 5 is the generalization of Theorem 2. Now we give example which satisfying Theorem 5 but does not Theorem 2. Example 2. Let X = [0, 1] and consider the natural ordered relation in X, defined G : X × X × X → R+ by { 0, if x = y = z, G(x, y, z) = max{x, y, z}. 9

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Then (X, G) is G−complete symmetric G-metric space. Let T : X ×X → X, g : X → X, ϕ : [0, ∞) → [0, ∞) and ψ : [0, ∞) → [0, ∞) define by, { x3 −y3 , if x ≥ y, 4 T (x, y) = 0, if x < y, g(x) = x2 , ϕ(t) =

t t , ψ(t) = . 2 4

We discuss the following cases. Case 1. (x, y) = (0, 0), (u, v) = (0, 0), (w, z) = (1, 0) it is clear that (gx, gy) ≼ (gu, gv) ≼ (gw, gz) or (gx, gy) ≽ (gu, gv) ≽ (gw, gz) and ψ(G(T (0, 0), T (0, 0), T (1, 0)) ≤ ψ(M (0, 0, 1, 0, 0, 0) − ϕ(M (0, 0, 1, 0, 0, 0), where G (T (0, 0), T (0, 1), T (0, 1)) = 1 and M (0, 1, 1, 1, 1, 1) = 1. Case 2. (x, y) = (0, 1), (u, v) = (1, 1), (w, z) = (1, 1) it is clear that (gx, gy) ≼ (gu, gv) ≼ (gw, gz)or (gx, gy) ≽ (gu, gv) ≽ (gw, gz) and ψ(G(T (0, 1), T (1, 1), T (1, 1)) ≤ ψ(M (0, 1, 1, 1, 1, 1) − ϕ(M (0, 1, 1, 1, 1, 1), where G (T (0, 1), T (1, 1), T (1, 1)) = 0 and M (0, 1, 1, 1, 1, 1) = 1. Case 3. (x, y) = (0, 0), (u, v) = (1, 0), (w, z) = (1, 0) it is clear that (gx, gy) ≼ (gu, gv) ≼ (gw, gz) or (gx, gy) ≽ (gu, gv) ≽ (gw, gz) and ψ(G(T (0, 0), T (1, 0), T (1, 0)) ≤ ψ(M (0, 1, 1, 0, 0, 0) − ϕ(M (0, 1, 1, 0, 0, 0), where G (T (0, 0), T (1, 0), T (1, 0)) = 41 and M (0, 1, 1, 0, 0, 0) = 1. Case 4. (x, y) = (0, 1), (u, v) = (1, 1), (w, z) = (1, 1) again it is clear that (gx, gy) ≼ (gu, gv) ≼ (gw, gz) or (gx, gy) ≽ (gu, gv) ≽ (gw, gz)and ψ(G(T (0, 1), T (1, 1), T (1, 1)) ≤ ψ(M (0, 1, 1, 1, 1, 1) − ϕ(M (0, 1, 1, 1, 1, 1), where G (T (0, 1), T (1, 1), T (1, 1)) = 0 and M (0, 1, 1, 1, 1, 1) = 1. Case 5. (x, y) = (u, v) = (0, 1), (w, z) = (1, 1) also it is clear that (gx, gy) ≼ (gu, gv) ≼ (gw, gz) or (gx, gy) ≽ (gu, gv) ≽ (gw, gz) and ψ(G(T (0, 1), T (0, 1), T (1, 1)) ≤ ψ(M (0, 1, 1, 1, 1, 1) − ϕ(M (0, 1, 1, 1, 1, 1), where G (T (0, 1), T (0, 1), T (1, 1)) = 0 and M (0, 0, 1, 1, 1, 1) = 1. Clearly for (gx, gy) ≼ (gu, gv) ≼ (gw, gz) or (gx, gy) ≽ (gu, gv) ≽ (gw, gz)all the conditions of Theorem 5 hold. So (0, 0) is the unique common coupled fixed point of T and g. On the other side if we taking in the Case 3 α = β = 61 then Theorem 2 fail to satisfy. Acknowledgments. The first author were supported in part by the Serbian Ministry of Science and Technological Developments (Project: Methods of Numerical and Nonlinear Analysis with Applications, grant number #174002)

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TRIANGULAR NORMS BASED ON INTUITIONISTIC FUZZY BCK -SUBMODULES L. B. Badhurays1, S. A. Bashammakh2 and N. O. Alshehri3

Abstract: We introduce the concept of intuitionistic fuzzy BCK-submodules of a BCK-module with respect to a t-norm and a s-norm and present some basic properties. Keywords : Intuitionistic fuzzy BCK-submodules, Triangular Norms, (Imaginable) Intuitionistic (T, S)-fuzzy BCK-submodules.

1. Introduction The theory of fuzzy sets proposed by Zadeh [11] in 1965, and later on several researchers worked in this field. As a natural advancement of these research works we get one of the interesting generalizations of the theory of fuzzy sets that is the theory of intuitionstic fuzzy sets propounded by Atanassov [1, 2]. In 1966 Imai and Iseki [5] proposed the concept of BCK -algebra. Xi [10] applied the concept of fuzzy set to BCK -algebras. Also Bakhshi [3] in 2011 introduced the concept of fuzzy BCK -submodule of BCK -module and gave some related results. Recently, Badhurays and Bashammakh [4] considered the intuitionistic fuzzification of the concept of BCK -submodules in a BCK -module and investigated some properties of such BCK -modules. In this paper, we are going to introduce the notion of intuitionistic (T,S )-fuzzy BCK -submodules by using triangular norms, say T and S, and investigate several properties. We obtain some results on level sets of an intuitionistic (T,S )-fuzzy BCK -submodule by using the concept of level sets and triangular norms. For the notations and terminology not given in this paper, the reader is referred to Atanassov [1, 2] (1986, 1994), Jun [8] (2001), Janiˆs [6] (2010), and Zadeh [11] (1965). 2. Preliminaries First we present the fundamental definitions. Definition 2.1. (Imai and Iseki [5]) a BCK -algebra is a set X with a binary operation ∗ and a constant 0 satisfying the following axioms : (BCK1) ((x ∗ y) ? (x ∗ z)) ? (z ∗ y) = 0 1Department of mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi

Arabia. E-mail address: [email protected] 2Department of mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi

Arabia. E-mail address: [email protected] 3Department of mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi

Arabia. E-mail address: [email protected] 1

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(BCK2) (x ∗ (x ∗ y)) ∗ y = 0, (BCK3) x ∗ x = 0, (BCK4) 0 ∗ x = 0, (BCK5) x ∗ y = 0 and y ∗ x = 0 imply that x = y, for all x, y, z ∈ X. A partial ordering ”≤” is defined on X by x ≤ y iff x ∗ y = 0. Definition 2.2. (Zadeh [11]) By a fuzzy set µ in a nonempty set X we mean a function µ : X 7−→ [0, 1], and the complement of µ denoted by µ ¯ is the fuzzy set in X given by µ ¯ (x) = 1 − µ(x) for all x ∈ X. Definition 2.3. (Atanassov [1]) An intuitionistic fuzzy set (IFS) in a universe X is an object of the form A = {(x, µA (x), λA (x))|x ∈ X}, where the functions µ : X 7−→ [0, 1] and λ : X 7−→ [0, 1] denote the degree of membership (namely µA (x)) and the degree of non-membership (namely λA (x)) of each element x ∈ X to the set A respectively, and 0 ≤ µA (x)) + λA (x) ≤ 1 for all x ∈ X. For the sake of simplicity, we shall use the symbol A = (µA (x), λA (x)) for the IFS A = {(x, µA (x), λA (x))|x ∈ X} Definition 2.4. (Atanassov [1]) Let X be a non-empty set and A = (µA (x), λA (x)), B = (µB (x), λB (x)) be IFS ’s of X. Then (1) A ⊂ B iff µA (x) < µB (x) and λA (x) > λB (x) for all x ∈ X. (2) A = B iff µA (x) = µB (x) and λA (x) = λB (x) for all x ∈ X (3) AC = (λA , µA ). (4) A ∩ B = {x, min{µA (x), µB (x)}, max{λA (x), λB (x)} : x ∈ X}. (5) A ∪ B = {x, max{µA (x), µB (x)}, min{λA (x), λB (x)} : x ∈ X}. (6) 2A = {(x, µA (x), µ¯A (x))|x ∈ X}. (7) 3A = {(x, λ¯A (x), λA (x))|x ∈ X}. Definition 2.5. (Atanassov [1]) Let A = (µA (x), λA (x)) be an intuitionistic fuzzy set in M and let α ∈ [0, 1]. Then the sets U (µA , α) = {x ∈ M : µA (x) ≥ α}, L(λA , α) = {x ∈ M : λA (x) ≤ α} are called a µ-level α-cut and a λ-level α-cut of A, respectively. Theorem 2.1. (Bakhshi [3]) Let X be a bounded implicative BCK-algebra. Then (X, +, 0) is an X-module where ” + ” is defined as x + y = (x ? y) ∨ (y ? x) and xy = x ∧ y. Theorem 2.2. (Bakhshi [3]) A subset A of a BCK-module M is a BCK-submodule of M iff a − b, xa ∈ A, for every a, b ∈ A and x ∈ X. Definition 2.6. (Bakhshi [3]) A fuzzy subset A of M is said to be a fuzzy BCK submodule if for all m, m1 , m2 ∈ M and x ∈ X, the following axioms hold : (1) A(m1 + m2 ) ≥ min{A(m1 ), A(m2 )}

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(2) A(m) = A(−m) (3) A(xm) ≥ A(m) Definition 2.7. (Badhurays and Bashammakh [4]) An intuitionistic fuzzy subset A = (µA (x), λA (x)) of M is said to be an intuitionistic fuzzy BCK -submodule of M if for all m, m1 , m2 ∈ M and x ∈ X, the following axioms hold : (1) µA (m1 + m2 ) ≥ min{µA (m1 ), µA (m2 )}, λA (m1 + m2 ) ≤ max{λA (m1 ), λA (m2 )}. (2) µA (m) = µA (−m), λA (m) = λA (−m), (3) µA (xm) ≥ µA (m), λA (xm) ≤ λA (m). Definition 2.8 (Klir and Yuan [9]) a triangular norm (or t-norm) T is a mapping T : [0, 1] × [0, 1] 7−→ [0, 1], which satisfies the following axioms for every x, y, z, ∈ [0, 1]: (T1) (T2) (T3) (T4)

T (x, 1) = x (boundary condition); y ≤ z implies T (x, y) ≤ T (x, z) (monotonicity); T (x, y) = T (y, x) (commutativity); T (x, T (y, z)) = T (T (x, y), z) (associativity).

Definition 2.9. (Klir and Yuan [9]) a triangular conorm (or t-conorm) S is a mapping S : [0, 1] × [0, 1] 7−→ [0, 1], which satisfies the following axioms for every x, y, z, ∈ [0, 1] : (S1) (S2) (S3) (S4)

S(x, 0) = x (boundary condition); y ≤ z implies S(x, y) ≤ S(x, z) (monotonicity); S(x, y) = S(y, x) (commutativity); S(x, S(y, z)) = S(S(x, y), z) (associativity).

Both t-norm and s-norm are called triangular norms. For all α, β ∈ [0, 1], It is clear that T (α, β) ≤ min{α, β} ≤ max{α, β} ≤ S(α, β). Definition 2.10. ( Jun and Hong [7]) For a t-norm T and a s-norm S, we use the symbols ∆T and ∆S as the sets : ∆T = {a ∈ [0, 1]|T (a, a) = a}, ∆S = {a ∈ [0, 1]|S(a, a) = a}, respectively. Definition 2.11. (Jun and Hong [7]) We say that the intuitionistic fuzzy set A = (µA (x), λA (x)) in M satisfies the imaginable property if Im(µA ) ⊆ ∆T and Im(λA ) ⊆ ∆S . Definition 2.12. (Klir and Yuan [9]) The norms T and S are called dual if and only if D1) T¯(x, y) = S(¯ x, y¯), ¯ y) = T (¯ D2) S(x, x, y¯) for all x, y ∈ [0, 1]

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A few t-norms which are frequently encountered are Tl , Tm , and Tw defined by Tl (a, b) = max{a + b − 1, 0} (Lukasiewicz), Tm (a, b) = min{a, b} (minimum) and Tw (a, b) := {

min{a, b} if a = 1 or b = 1, , 0 otherwise (weak).

A few s-norms which are frequently encountered are Sl ,Sm , and Sw defined by Sl (a, b) = min{a + b, 1} (Lukasiewicz), Sm (a, b) = max{a, b} (maximum) andæ Sw (a, b) := {

max{a, b} if a = 0 or b = 0, , 1 otherwise (strong).

3. Intuitionistic (T, S)-fuzzy BCK -submodules Throughout this paper, M is a BCK -module and T is a t-norm and S is a snorm unless otherwise specified. we can extend the concept of the intuitionistic fuzzy BCK -submodules of M to the concept of intuitionistic (T, S)-fuzzy BCK submodules in the following way: Definition 3.1. Let T be a t-norm and S be a s-norm on [0, 1]. An intuitionistic fuzzy set A = (µA , λA ) in M is called an intuitionistic fuzzy BCK -submodule of M with respect to t-norm and s-norm (briefly, intuitionistic (T, S)-fuzzy BCKsubmodule of M ) if it satisfies the following conditions for all m, m1 , m2 ∈ M : (1) µA (m1 + m2 ) ≥ T {µA (m1 ), µA (m2 )}, λA (m1 + m2 ) ≤ S{λA (m1 ), λA (m2 )}. (2) µA (m) = µA (−m), λA (m) = λA (−m), (3) µA (xm) ≥ µA (m), λA (xm) ≤ λA (m). Example 3.2.

Let X = {0, 1, 2, 3} and consider the following table: * 0 1 2 3

0 0 1 2 3

1 0 0 2 2

2 0 1 0 1

3 0 0 0 0

Then (X, ∗) is a BCK -module over itself. Define a fuzzy set µA : M 7−→ [0, 1] by µ(0) = 0.5, µ(m) = 0.3, m ∈ M and λA : M 7−→ [0, 1] by λA (0) = 0.3, λA (m) = 0.5, m ∈ M . Let Tl : [0, 1] × [0, 1] 7−→ [0, 1] be a function defined by Tl (a, b) = max(a + b − 1, 0) for all a, b ∈ [0, 1] and let Sl : [0, 1] × [0, 1] 7−→ [0, 1] be a function defined by Sl (a, b) = min(a + b, 1) for all a, b ∈ [0, 1]. Then Tl is a t-norm and Sl is a s-norm. By routine calculations, we know that A = (µA (x), λA (x)) is an intuitionistic (Tl , Sl )-fuzzy BCK -submodule of M . Theorem 3.3. An intuitionistic fuzzy subset A of M is an intuitionistic (T, S)fuzzy BCK-submodule of M if and only if (1) µA (m1 − m2 ) ≥ T {µA (m1 ), µA (m2 )}, λA (m1 − m2 ) ≤ S{λA (m1 ), λA (m2 )}. (2) µA (xm) ≥ µA (m), λA (xm) ≤ λA (m).

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proof. Let A be an intuitionistic (T, S)-fuzzy BCK -submodule of M , then µA (m1 − m2 ) = µA (m1 + (−m2 )) ≥ T (µA (m1 ), µA (−m2 )) = T (µA (m1 ), µA (m2 )), Similarly, λA (m1 − m2 ) ≤ S(λA (m1 ), λA (m2 )). Condition 2 is hold by definition. Conversely suppose A satisfies 1 and 2. Then we have by 2 µA (−m) = µA ((−1).m) ≥ µA (m), and µA (m) = µA ((−1).(−1).m) ≥ µA (−m). Thus A (m) = A (−m). Similarly, λA (m) = λA (−m). Also we have µA (m1 + m2 ) = µA (m1 − (−m2 )) ≥ T (µA (m1 ), µA (−m2 )) ≥ T (µA (m1 ), µA (m2 )) Similarly, λA (m1 + m2 ) ≤ S(λA (m1 ), λA (m2 )). Thus A is an intuitionistic (T, S)-fuzzy BCK -submodule of M . Proposition 3.4. Let T and S be dual norms. If A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK-submodule of M , then so is 2A = (µA , µA ). Proof. For all m1 , m2 ∈ M , we have T (µA (m1 ), µA (m2 )) ≤ µA (m1 + m2 ) and so T (1 − µA (m1 ), 1 − µA (m2 )) ≤ 1 − µA (m1 + m2 ) hence 1 − T (1 − µA (m1 ), 1 − µA (m)) ≥ 1 − (1 − µA (m1 + m2 ) which implies T (1 − µA (m1 ), 1 − µA (m2 )) ≥ µA (m1 + m2 ) since T and S are dual, we get S(µA (m1 ), µA (m2 )) ≥ µA (m1 + m2 ) , Moreover µA (m) = µA (−m) imply that 1 − µA (m) = 1 − µA (−m), Thus µA (m) = µA (−m). Now, let m ∈ M and x ∈ X, since µA is T -fuzzy BCK submodule of M , we have µA (x.m) ≥ µA (m). Hence 1 − µA (x.m) ≤ 1 − µA (m) which implies µA (xm) ≤ µA (m). Therefore 2A = (µA , µA ) is an intuitionistic (T, S) - fuzzy BCK -submodule of M .

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Proposition 3.5. Let T and S be dual norms. If A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK- submodule of M , then so is 3A = (λA , λA ). Proof. For all m1 , m2 ∈ M , we have S(λA (m1 ), λA (m2 )) ≥ λA (m1 + m2 ) and so S(1 − λA (m1 ), 1 − λA (m2 )) ≥ 1 − λA (m1 + m2 ) hence 1 − S(1 − λA (m1 ), 1 − λA (m2 )) ≤ 1 − (1 − λA (m1 + m2 )) which implies 1 − S(λA (m1 ), λA (m2 )) ≤ λA (m1 + m2 ) since T and S are dual 1 − T (λA (m1 ), λA (m2 )) ≤ λA (m1 + m2 ) that is T (λA (m1 ), λA (m2 )) ≤ λA (m1 + m2 ). Moreover λA (m) = λA (−m) imply that 1 − λA (m) = 1 − λA (−m), Thus λA (m) = λA (−m). Now, let m ∈ M and x ∈ X, since λA is T -fuzzy BCK -submodule of M we have λA (x.m) ≤ λA (m). Hence 1 − λA (x.m) ≥ 1 − λA (m) which implies λA (xm) ≥ λA (m). Therefore 3A = (λA , λA ) is an intuitionistic (T, S) - fuzzy BCK -submodule of M . Combining the above two Propositions it is not difficult to verify that the following theorem is valid. Theorem 3.6. Let T and S be dual norms. Then A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK-submodule of M if and only if 2A and 3A are intuitionistic (T, S)-fuzzy BCK-submodule of M . Corollary 3.7. Let T and S be dual norms. Then A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK-submodule of M if and only if µA and λA are T -fuzzy BCK-submodule of M . From corollary 3.7 we immediately obtain the following result. Theorem 3.8. An intuitionistic fuzzy set A = (µA , λA ) is an intuitionistic (Tm , Sm )- fuzzy BCK- submodule of M if and only if the fuzzy sets µA and λA are fuzzy BCK-submodule of M . Theorem 3.9. An intuitionistic fuzzy set A = (µA , λA ) is an intuitionistic (Tm , Sm )-fuzzy BCK- submodule of M if and only if 2A = (µA , µ ¯ A ) and 3A = (λA , λA ) are intuitionistic (Tm , Sm )-fuzzy BCK- submodule of M . Proof. Let A = (µA , λA ) be an intuitionistic (Tm , Sm )-fuzzy BCK-submodule of M . By Theorem 3.8, we get µA = µA and λA are fuzzy BCK -submodule of M .

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Therefore 2A = (µA , µA ) and 3A = (λA , λA ) are intuitionistic (Tm , Sm )-fuzzy BCK -submodule of M . Conversely, assume that A = (µA , λA ) and 2A = (µA , µA ) and 3A = (λA , λA ) are intuitionistic (Tm , Sm )-fuzzy BCK submodule of M . Then the fuzzy sets µA and λA are fuzzy BCK -submodule of M . Therefore A = (µA , λA ) is an intuitionistic (Tm , Sm )-fuzzy BCK - submodule of M . Definition 3.10. An intutionistic (T, S)-fuzzy BCK -submodule of M is called an imaginable intuitionistic (T, S)-fuzzy BCK -submodule of M if it satisfies the imaginable property. Proposition 3.11. Every imaginable intuitionistic (T, S)-fuzzy BCK-submodule of M is an intuitionistic fuzzy BCK-submodule of M . Proof. Let A = (µA , λA ) be an imaginable intuitionistic (T, S)-fuzzy BCK -submodule of M . Then µA (m1 + m2 ) ≥ T (µA (m1 ), µA (m2 )) and λA (m1 + m2 ) ≤ S(λA (m1 ), λA (m2 )) for all m1 , m2 ∈ M . Since A = (µA , λA ) is imaginable, we have min{µA (m1 ), µA (m2 )} = T (min{µA (m1 ), µA (m2 )}, min{µA (m), µA (m2 )}) ≤ T (µA (m1 ), µA (m2 )) ≤ min{µA (m1 ), µA (m2 )}, and max{λA (m1 ), λA (m2 )} = S(max{λA (m1 ), λA (m2 )}, max{λA (m), λA (m2 )}) ≥ S(λA (m1 ), λA (m2 )) ≥ max λA (m1 ), λA (m2 ). It follows that µA (m1 − m2 ) ≥ T (µA (m1 ), µA (m2 )) = min{µA (m1 ), µA (m2 )}, and λA (m1 − m2 ) ≤ S(λA (m1 ), λA (m2 )) = max{λA (m1 ), λA (m2 )}. Now let x ∈ X and m ∈ M . Since A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK -submodule of M , we have µA (xm) ≥ µA (m) , λA (xm) ≤ λA (m). Therefore A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK -submodule of M .

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Note that every intuitionistic fuzzy BCK-submodule is an intuitionistic (T, S)fuzzy BCK -submodule but the converse is not true as seen in the following Example. Example 3.12. We consider the BCK -module M which is given in Example 3.2. Define an intuitionistic fuzzy set A = (µA , λA ) in M    0.2 if m = 1  0.5 if m = 1 0.3 if m = 2, 3 ; λA (m) = 0.3 if m = 2, 3 µA (m) =   0.1 if m = 0 0.5 if m = 0 Then A = (µA , λA ) is an intuitionistic (Tw , Sw )-fuzzy BCK -submodule of M , but it is not an intuitionistic fuzzy BCK -submodule of M since µA (2 + 3) = µA (1) = 0.2 < 0.3 = min(µA (2), µA (3)). Proposition 3.13. If an intuitionistic fuzzy set A = (µA , λA ) in M is an imaginable intuitionistic (T, S)-fuzzy BCK-submodule of M , then for all m ∈ M , µA (0) ≥ µA (m) and λA (0) ≤ λA (m) . Proof. From Definition 3.1 (3) it follows that µA (0) = µA (0.m) ≥ µA (m) and λA (0) = λA (0.m) ≤ λA (m) for all m ∈ M . Theorem 3.14. If A = (µA , λA ) is an imaginable intuitionistic (T, S)-fuzzy BCK-submodule of M , then the set H = {m ∈ M |µ(m) = µ(0)} and K = {m ∈ M |λA (m) = λA (0)} are BCK-submodule of M . Proof. Assume that A = (µA , λA ) is an imaginable intuitionistic (T, S)-fuzzy BCK submodule of M , and let m1 , m2 ∈ M . Since A = (µA , λA ) is an imaginable intuitionistic (T, S)-fuzzy BCK -submodule of M , we have µA (m1 − m2 ) ≥ T (µA (m), µA (m)) = T (µA (0), µA (0)) = µA (0) for all m1 , m2 ∈ M , Using Lemma Proposition 3.11., we get µA (m1 − m2 ) = µA (0). Hence m1 − m2 ∈ H. Now let x ∈ Xand m ∈ M . Since A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK -submodule of M , we have µA (x.m) ≥ µA (m) = µA (0). Using Lemma Proposition 3.11., we get µA (x.m) = µA (0) and so x.m ∈ H. Therefore H is a BCK -submodule of M . By similar method, we get K is a BCK submodule of M . Definition 3.15. Let A = (µA , λA ) be an intuitionistic fuzzy set in BCK -submodule M and let α, β ∈ [0, 1] with α + β ≤ 1. Then the set A(α,β) := {m ∈ M |µA (m) ≥ α, λA (m) ≤ β} is called an (α, β)-level set of A = (µA , λA ). Theorem 3.16. Let A = (µA , λA ) be an intuitionistic fuzzy set in M such that

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A(α,β) is a BCK-submodule of M , for all (α, β) ∈ [0, 1] with α + β ≤ 1. Then A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK-submodule of M . Proof. Let m1 , m2 , m ∈ M and x ∈ X be such that A(m1 ) = (α1 , β1 ), A(m2 ) = (α2 , β2 ) where αi + βi ≤ 1 for i = 1, 2. Then m1 , m2 ∈ A(min(α1 ,α2 ),max(β1 ,β2 )) , and so m1 − m2 ∈ Amin(α1 ,α2 ),max(β1 ,β2 )) . Hence µA (m1 − m2 ) ≥ min(α1 , α2 ) ≥ T (α1 , α2 ), and λA (m1 − m2 )) ≤ max(β1 , β2 ) ≤ S(β1 , β2 ). 0

Also, if we put s = A (m), t0 = A (m) where s0 + t0 ≤ 1. Then m ∈ A(s0 ,t0 ) . Since A(s0 ,t0 ) is a BCK - submodule of M , we have xm ∈ A(s0 ,t0 ) . It follows that µA (xm) ≥ s0 = µA (m) and λA (xm) ≤ t0 = λA (m) Therefore A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK -submodule of M . The following Example shows that the converse of Theorem 3.16 is not true. Example 3.17. We consider the intuitionistic (Tw , Sw )-fuzzy BCK -submodule A of M which is given in Example 3.2. Then A(0.3,0.5) = {2, 3, 0} is not BCK submodule of M since 2 + 3 = 1 ∈ / A(0.3,0.5) Theorem 3.18. If A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK-submodule of M , then A(1,0) is either empty or a BCK-submodule of M . Proof. Let m1 , m2 ∈ A(1,0) . Then µA (m1 ) ≥ 1 , µA (m2 ) ≥ 1 , λA (m1 ) ≤ 0 and λA (m2 ) ≤ 0. It follows from Definitions 2.10 and Theorem 3.3 that µA (m1 − m2 ) ≥ T (µA (m1 ), µA (m2 )) ≥ T (1, 1) = 1 and λA (m1 − m2 ) ≤ S(λA (m1 ), λA (m2 )) ≤ S(0, 0) = 0, so m1 − m2 ∈ A(1,0) . Let m ∈ A(1,0) and x ∈ X. Then µA (xm) ≥ µA (m) ≥ 1 and λA (xm) ≤ λA (m) ≤ 0, so xm ∈ A(1,0) . As a generalization of Theorem 3.18, we get the following Theorem. Theorem 3.19. If A = (µA , λA ) is an imaginable intuitionistic (T, S)-fuzzy BCK-submodule of M , then A(α,β) is either empty or a BCK-submodule of M for all α ∈ ∆T and β ∈ ∆S . with α + β ≤ 1.

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Proof. Let m1 , m2 ∈ A(α,β) where α ∈ ∆T , β ∈ ∆S and α + β ≤ 1. Then µA (m1 − m2 ) ≥ T (µA (m1 ), µA (m2 )) ≥ T (α, α) = α and λA (m1 − m2 ) ≤ S(λA (m1 ), λA (m2 )) ≤ S(β, β) = β, and so m1 − m2 ∈ A(α,β) . Let m ∈ A(α,β) and x ∈ X. Then µA (xm) ≥ µA (m) ≥ α and λA (xm) ≤ λA (m) ≤ β, so xm ∈ A(α,β) .Hence A(α,β) is a BCK -submodule of M . Proposition 3.20. (Bakhshi [3]) A fuzzy set in M is a fuzzy BCK-submodule of M if and only if the non-empty U (µ, α), α ∈ [0, 1] is a BCK-submodule of M . By the above Proposition , we get the following result. Corollary 3.21. If A = (µA , λA ) is an imaginable intuitionistic fuzzy set in M . Then A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK-submodule of M if and only if the non-empty sets U (µ, α) and L(λ, α) are BCK-submodules of M , for every (α, β) ∈ [0, 1]. From corollary 3.21 we immediately obtain the following Theorem. Theorem 3.22. Let T be the minimum t-norm and let S the maximum s-norm dual of T . Then an intuitionistic fuzzy set A = (µA , λA ) of M is is an intuitionistic (T, S)-fuzzy BCK-submodule of M if and only if A(α,β) := {m ∈ M |µA (m) ≥ α, λA (m) ≤ β} is a BCK-submodule of M , where (α, β) ∈ [0, 1]. Proposition 3.23. Let S be a non-empty subset of a BCK-module M . Then an intuitionistic fuzzy set A = (µA , λA ) defined by   1 if m ∈ S, 0 if m ∈ S, µA (m) = , λA (m) = α otherwise. β otherwise. where 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and α + β ≤ 1 is an intuitionistic (T, S)-fuzzy BCK -submodule of M if and only if S is a BCK-submodule of M . Proof. Let S be a BCK -submodule of M . Let m1 , m ∈ M . If m1 , m2 ∈ S,

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then m1 − m2 ∈ S, and so µA (m1 − m2 ) = 1 ≥ 1 = T (1, 1) = T (µA (m1 ), µA (m2 )) and λA (m1 − m2 ) = 0 = S(0, 0) = S(λA (m1 ), λA (m2 )) For m1 ∈ S , m2 ∈ / S , we have µA (m1 − m2 ) = α ≥ α = T (1, α) = T (µA (m1 ), µA (m2 )) and λA (m1 − m2 ) = β ≤ β = S(0, β) = S(λA (m1 ), λA (m2 )) Similarly, for the case m1 ∈ / S , m2 ∈ S , we have µA (m1 − m2 ) ≥ T (µA (m1 ), µA (m2 )) and λA (m1 − m2 ) ≤ S(λA (m1 ), λA (m2 )). For m1 ∈ / S , m2 ∈ / S, µA (m1 − m2 ) ≥ α = T (1, α) ≥ T (α, α) = T (µA (m1 ), µA (m2 )), and λA (m1 − m2 ) ≤ β = S(0, β) ≤ S(β, β) = S(λA (m1 ), λA (m2 )). Thus for all cases, µA (m1 − m2 ) ≥ T (µA (m1 ), µA (m2 )) and λA (m1 − m2 ) ≤ S(λA (m1 ), λA (m2 )). Next, let m ∈ M and x ∈ X, Then, if m ∈ S then xm ∈ S and so, µA (xm) = 1 ≥ 1 = µA (m) and

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λA (xm) = 0 ≤ 0 = λA (m). If m ∈ / S, then µA (xm) ≥ α = A (m) and λA (xm) ≤ β = λA (m). Therefore µA (xm) ≥ µA (m) and λA (xm) ≤ λA (m). Thus A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK -submodule of M . Conversely, we assume A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK -submodule of M . Let m1 , m2 ∈ S, x ∈ X. Then, µA (m1 − m2 ) ≥ T (µA (m1 ), µA (m2 )) = T (1, 1) = 1, hence µA (m1 − m2 ) = 1. Thus m1 − m2 ∈ S. Also, µA (xm) ≥ µA (m) = 1 implies µA (xm) = 1 implies xm ∈ S. Hence, S is a BCK -submodule of M . Corollary 3.24. Let S be a non-empy subset of a BCK-module M and let χs be the characteristic function of S. Then A = (χs , χcs ) is an intutionistic (T, S)fuzzy BCK-submodule of M if and only if S is a BCK-submodule of M . Definition 3.25. (Janiˆs [6]) Let A = (µA , λA ) be an intuitionistic fuzzy set of X and let T be a t-norm. Then AT,α is a subset of X defined by AT,α = {x ∈ X|T (µA (x), 1 − λA (x)) ≥ α}, for every α ∈ [0, 1] Theorem 3.26. Let T and S be dual norms. If A = (µA , λA ) is an intuitionistic (T, S)-fuzzy BCK-submodule of M . Then AT,1 = {m ∈ M |T (µA (m), 1 − λA (m)) = 1} is a BCK-submodule of M . Proof. Let m1 , m2 ∈ AT,1 . Then, T (µA (m1 − m2 ), 1 − A (m1 − m2 )) ≥ T (T (µA (m1 ), A (m2 )), 1 − S(A (m1 ), A (m2 ))) = T (T (µA (m2 ), (µA (m1 )), T (1 − λA (m1 ), 1 − λA (m2 ))) = T (µA (m2 ), T (µA (m1 ), T (1 − λA (m1 ), 1 − λA (m2 )))) = T (µA (m2 ), T (T (µA (m1 ), 1 − λA (m1 )), 1 − λA (m2 ))) = T (µA (m2 ), T (1 − λA (m2 ), T (µA (m1 ), 1 − λA (m1 )))) = T (T (µA (m2 ), 1 − λA (m2 )), T (µA (m1 ), 1 − λA (m1 ))) = T (1, 1) = 1 Thus, we have T (µA (m1 − m2 ), 1 − λA (m1 − m2 )) = 1 Therefore m1 − m2 ∈ AT,1 . Also, let x ∈ X and m ∈ AT,1 . Then T (µA (m), 1 − λA (m)) = 1. Further, T (µA (xm), 1−λA (xm)) ≥ T (µA (m), 1−λA (m)) = 1. Therefore xm ∈ AT,1 . Hence, AT,1 is a is a BCK -submodule of M .

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For any tringular norm T , the level set AT,α of an intuitionistic (T, S)-fuzzy BCK submodule of M is not necessarily to be a BCK -submodule of M . However, if T is the minimum tringular norm, then all level sets AT,α of an intuitionistic (T, S)fuzzy BCK -submodule of M are BCK -submodules of M . Theorem 3.27. Let A = (µA , λA ) be an intuitionistic (Tm , Sm )-fuzzy BCKsubmodule of M such that Tm , Sm are dual. Then for every α ∈ [0, 1], ATm ,α = {m ∈ M |T (µA (m), 1 − λA (m)) ≥ α} is a BCK-submodule of M . Proof. Let A = (µA (x), λA (x)) is an intuitionistic (Tm , Sm )-fuzzy BCK -submodule of M . Let m1 , m2 ∈ AT, . Then, Tm (µA (m1 − m2 ), 1 − A (m1 − m2 )) ≥ Tm (Tm (µA (m1 ), µA (m2 )), 1 − Sm (λA (m1 ), λA (m2 ))) = Tm (Tm (µA (m2 ), (µA (m1 )), Tm (1 − λA (m1 ), 1 − λA (m2 ))) = Tm (µA (m2 ), Tm (µA (m1 ), Tm (1 − λA (m1 ), 1 − λA (m2 )))) = Tm (µA (m2 ), Tm (Tm (µA (m1 ), 1 − λA (m1 )), 1 − λA (m2 ))) = Tm (µA (m2 ), Tm (1 − λA (m2 ), Tm (µA (m1 ), 1 − λA (m1 )))) = Tm (Tm (µA (m2 ), 1 − λA (m2 )), Tm (µA (m1 ), 1 − λA (m1 ))) ≥ Tm (α, α) = α Thus, we have Tm (µA (m1 − m2 ), 1 − A (m1 − m2 )) ≥ α Therefore, m1 − m2 ∈ ATm ,α . Also, let x ∈ X and m ∈ ATm ,α . Then Tm (µA (m), 1 − λA (m)) ≥ α Further, Tm (µA (xm), 1 − λA (xm)) ≥ Tm (µA (m), 1 − λA (m)) ≥ α Therefore we have Tm (µA (xm), 1 − λA (xm)) ≥ α. Hence xm ∈ ATm ,α . Thus ATm ,α is a is a BCK-submodule of M . Definition 3.28. Let A = (µA , λA ) be an intuitionistic fuzzy set of X, let T and S be dual norms. Then AT,S,α is a subset of X defined by AT,S,1 = {x ∈ XT (µA (x), S(µA (x), λA (x))) ≥ α} for every α ∈ [0, 1]. Theorem 3.29. Let A = (µA , λA ) be an intuitionistic (T, S)-fuzzy BCK-submodule of M , then AT,S,1 = {m ∈ M |T (µA (m), S(µA (m), λA (m))) = 1} is a BCK-submodule of M . Proof. Let A = (µA , λA ) be an intuitionistic (T, S)-fuzzy BCK -submodule of M . Let m1 , m2 ∈ AT,S,1 , then T (µA (m1 ), S(µA (m1 ), λA (m1 ))) = 1

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and T (µA (m2 ), S(µA (m2 ), λA (m2 ))) = 1. Therefore µA (m1 ) ≥ 1 and µA (m2 ) ≥ 1 which mean that µA (m1 ) = 1 and µA (m2 ) = 1. From monotonicity of T , we have, T (µA (m1 − m2 ), S(µA (m1 − m2 ), λA (m1 − m2 ))) ≥ T (T (µA (m1 − m2 )), T (µA (m1 − m2 ))) ≥ T (T (µA (m), µA (m)), T (µA (m), µA (m))) = T (T (1, 1), T (1, 1)) = T (1, 1) = 1 Therefore, T (µA (m1 − m2 ), S(µA (m1 − m2 ), λA (m1 − m2 ))) = 1 implies m1 , m2 ∈ AT,S,1 . Also, let x ∈ X and m ∈ AT,S,1 . Then, T (µA (m), S(µA (m), λA (m))) = 1. which impliese µA (m) = 1. Now, T (µA (xm), S(µA (xm), λA (xm))) ≥ T (µA (xm), µA (xm)) ≥ T (µA (m), µA (m)) = T (1, 1) = 1 Thus, we have, T (µA (xm), S(µA (xm), λA (xm))) = 1. Therefore, xm ∈ AT,S,1 . Hence, AT,S,1 is a BCK-submodule of M . Theorem 3.30. Let A = (µA , λA ) be an intuitionistic (Tm , Sm )-fuzzy BCKsubmodule of M such that Tm , Sm are dual. Then for every α ∈ [0, 1], AT,S,α = {m ∈ M |T (µA (m), S(µA (m), λA (m))) ≥ α} is a BCK-submodule of M. Proof. Let A = (µA , λA ) is an intuitionistic (Tm , Sm )-fuzzy BCK -submodule of M . Let m1 , m2 ∈ AT,S,α , then Tm (µA (m1 ), Sm (µA (m1 ), λA (m1 ))) ≥ α and Tm (µA (m2 ), Sm (µA (m2 ), λA (m2 ))) ≥ α. Therefore µA (m1 ) ≥ α and µA (m2 ≥ α. Due monotonicity of Tm , we have, Tm (µA (m1 − m2 ), Sm (µA (m1 − m2 ), λA (m1 − m2 ))) ≥ Tm (µA (m1 − m2 )), (µA (m1 − m2 ))) = µA (m1 − m2 ) ≥ Tm (µA (m1 ), µA (m2 )) ≥ Tm (α, α) =α Therefore, Tm (µA (m1 − m2 ), Sm (µA (m1 − m2 ), λA (m1 − m2 ))) ≥ α and hence m1 − m2 ∈ ATm ,Sm ,α . Also, let m ∈ ATm ,Sm ,α and x ∈ X. Then, Tm (µA (m), Sm (µA (m), λA (m))) ≥ α.

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which impliese µA (m) ≥ α. From monotonicity of Tm , we have, Tm (µA (xm), Sm (µA (xm), λA (xm)) ≥ Tm (µA (xm), µA (xm)) = µA (xm) ≥ µA (m) ≥α Thus Tm (µA (xm), Sm (µA (xm), λA (xm)) ≥ α. Therefore, xm ∈ ATm ,Sm ,α . Hence, ATm ,Sm ,α is a BCK -submodule of M .

4. Conclusion One of the generalizations of fuzzy BCK -submodules, namely, intuitionistic (T,S )-fuzzy BCK - submodules was defined and some properties of intuitionistic (T,S )-fuzzy BCK -submodules are investigated. Also, some related results on level sets of an intuitionistic (T,S )-fuzzy BCK -submodule are investigated. These investigations of generalized fuzzy on BCK -modules could be enable us to discuss further study in this field. References [1] K.T.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1),(1986), 87-96. [2] K.T.Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems 61 (1994), 137-142. [3] M.Bakhshi, Fuzzy set theory applied to BCK -modules. Pushpa Publishing House 8 (2011), 61-87. [4] L.B.Badhurays , S.A.Bashammakh, Intuitionistic fuzzy BCK -submodules, The Journal of Fuzzy Mathematics, (2015). (accepted) [5] Imai, K.Iˆseki, On axiom systems of propositional calculi, XIV , Proc. Japan Academy, 42 (1996) , 19-22. [6] V. Janiˆs , t-Norm based cuts of intuitionistic fuzzy sets, Information Sciences, 180 (7), (2010), 1134-1137. Soochow Journal of Mathematics 27.1 (2001): 83-88. [7] Y.B.Jun , S.M.Hong , On imaginable T -fuzzy subalgebras and imaginable T -fuzzy closed ideals in BCH -algebras, International Journal of Mathematicsand Mathematical Sciences 27 (2001), 269-287. [8] Y.B.Jun, M.A.Ozturk , E.H.Roh Triangular normed fuzzy subalgebras of BCK -algebras, Scientiae Mathematicae Japonicae 61(2005),3 : 451-458. [9] G.J.Klir, B.Yuan, Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall of India, Englewood Cliffs(2008). [10] O.G.Xi, Fuzzy BCK -algebras, Math Japon 36 (1991), 935-942. [11] L.A.Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338-353.

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On strongly almost generalized difference lacunary ideal convergent sequences of fuzzy numbers S. A. Mohiuddine1 and B. Hazarika2 1

Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2

Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791112, Arunachal Pradesh, India 1 Email: [email protected]; 2 bh [email protected]

Abstract The purpose of this paper is to introduce some new sequence spaces of fuzzy numbers defined by lacunary ideal convergence using generalized difference matrix and Orlicz functions. We also study some algebraic and topological properties of these classes of sequences. Moreover, some illustrative examples are given in support of our results. Keywords and phrases: Ideal convergence; fuzzy number; difference sequence; Orlicz function; lacunary sequence. AMS subject classification (2010): 40A05; 40C05; 40G15; 06B99.

1

Introduction and preliminaries

The concept of ideal convergence is the dual (equivelant) to the notion of filter convergence introduced by Cartan [4]. The filter convergence is a generalization of the classical notion of convergence of sequences of real or complex numbers and it has been an important tool in the study of functional analysis. Nowadays many authors studied this notion from various aspects and applied this notion to various problems arising in the convergence theory. Kostyrko et al. [13] and Nuray and Ruckle [23] independently studied in detalis about the notion of ideal convergence which is based upon the structure of the admissible ideal I of subsets N of natural numbers. Later on it was further investigated by many authors, e.g. Tripathy and Hazarika [26], Mursaleen and Mohiuddine [22] and references therein. Let S be a non-empty set. Then a non empty class I ⊆ P (S) is said to be an ideal on S if and only if (i) φ ∈ I; (ii) I is additive; (iii) hereditary. An ideal I ⊆ P (S) is said to be non trivial if I 6= φ and S∈ / I. A non-empty family of sets F ⊆ P (S) is said to be a filter on S if and only if (i) φ ∈ / F (ii) for each A, B ∈ F we have A ∩ B ∈ F ; (iii) for each A ∈ F and each B ⊃ A, we have B ∈ F . For each ideal I, there is a filter F (I) corresponding to I i.e. F (I) = {K ⊆ S : K c ∈ I}, where K c = S − K. We say that a non-trivial ideal I ⊆ P (S) is an admissible ideal on S if and only if it contains all singletons, i.e. if it contains {{s} : s ∈ S}. Recall that a sequence x = (xk ) of points in R is said to be I-convergent to the number ` (denoted by I- lim xk = `) if for every ε > 0, the set {k ∈ N : |xk − `| ≥ ε} ∈ I. We used the standard notation θ = (kr ) to denote the lacunary sequence, where θ is a sequence of positive integers such that k0 = 0, 0 < kr < kr+1 and hr := kr − kr−1 → ∞ as r → ∞. The intervals kr determined by θ will be denoted by Jr = (kr−1 , kr ] and the ratio kr−1 (r 6= 1) by qr (see [8]). The notion of lacunary ideal convergence for sequences of real numbers and fuzzy numbers, respectively, has been defined and studied in [27] and [9]. Let I ⊂ 2N be a non-trivial ideal. A real sequence

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x = (xk ) is said to be lacunary I-convergent to L ∈ R, in symbol we shall write Iθ - lim x = L, if for every ε > 0, the set ( ) 1 X |xk − L| ≥ ε ∈ I. r∈N: hr k∈Jr

Throughout the paper we use w to denotes the set of all real sequences x = (xk ). The difference sequence spaces have been introduced by Kızmaz [12] by using the difference operator ∆ as follows: Z(∆) = {(xk ) ∈ w : ∆xk ∈ Z},

for Z = `∞ , c, c0 and ∆xk = ∆1 xk = xk − xk+1 for all k ∈ N, where the standard notations `∞ , c and c0 are used to denote the set of bounded, convergent and null sequences, respectively. Later this idea was generalized by Et and C ¸ olak [6] by considering ∆n instead of ∆, where (∆n xk ) = ∆1 (∆n−1 xk ) for n ≥ 2 and all k ∈ N. In case of n = 0 we obtain xk . Tripathy et al. [28] presented another generalization of difference sequence spaces by introducing the operator ∆nm and is given by ∆nm x = n−1 n (∆nm xk ) = (∆n−1 m xk − ∆m xk+m ) so that ∆m xk has the following binomial representation:   n X ν n n ∆ m xk = (−1) xk+mν , ν ν=0 for all k ∈ N. If we take n = 1, then Z(∆nm ) is reduced to Z(∆m ) which was introduced by Tripathy and Esi [25], in this case the operator ∆m x is given by ∆m x = (∆m xk ) = (xk − xk+m ) for all k, m ∈ N. The choice of m = 1 in the definition of Z(∆nm ) gives us the difference sequence spaces introduced by Et and Colak [6]. Ba¸sar and Altay [1] introduced the generalized difference matrix B(r, s) = (bnk (r, s)) by   if k = n;  r, bnk (r, s) = s, if k = n − 1;   0, if 0 ≤ k < n − 1 or k > n.

for all k, n ∈ N and all non-zero real numbers r, s. The generalized difference matrix B n of order n has been recently defined by Ba¸sarir and Kayik¸ci [2] and its binomial representation is given by n   X n n−ν ν n B xk = r s xk−ν , ν ν=0

for all n ∈ N and r, s ∈ R − {0}. Another generalization of above difference matrix was given by Ba¸sarir n−1 n−1 n n n 0 et al. [3] as B(m) , where B(m) x = (B(m) xk ) = (rB(m) xk + sB(m) xk−m ) and B(m) xk = xk for all k ∈ N, which is equivalent to the following binomial representation: n   X n n−ν ν n B(m) xk = r s xk−mν . ν ν=0 In [24], Orlicz introduced functions nowadays called Orlicz functions and constructed the sequence space (LM ). Krasnoselskii and Rutitsky further investigated the Orlicz space in [14]. Some recent related work we refer to Mohiuddine et al. [19, 20]. A function M : [0, ∞) → [0, ∞) is said to be an Orlicz function if it is non-decreasing, continuous, convex with M (0) = 0, M (x) > 0 as x > 0 and M (x) → ∞ as x → ∞ (see [24]). It is well known that if M is a convex function and M (0) = 0, then M (λx) ≤ λM (x) for all λ ∈ (0, 1). An Orlicz function M is said to be satisfy ∆2 -condition for all values of u, if there exists a constant K > 0 such that M (Lu) ≤ KLM (u) for all values of L > 1 (see, Krasnoselskii and Rutitsky [14]).

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Lindenstrauss and Tzafriri [16] introduced the sequence space `M by using the notion of Orlicz function by `M =

(

(xk ) ∈ w :

∞ X

M

k=1



|xk | ρ



)

< ∞, for some ρ > 0 .

and proved that this space is a Banach space with the norm ( )   ∞ X |xk | ||x|| = inf ρ > 0 : M ≤1 . ρ k=1

Every space `M contains a subspace isomorphic to the classical sequence space `p for some 1 ≤ p < ∞. The space `p , 1 ≤ p < ∞ is itself an Orlicz sequence space with M (t) = |t|p. A sequence space E is said to be (i) normal (or solid) if (αk xk ) ∈ E whenever (xk ) ∈ E and for all sequence (αk ) of scalars with |αk | ≤ 1 for all k ∈ N, (ii) symmetric if (xπ(k) ) ∈ E, whenever (xk ) ∈ E, where π is a permutation of N. Let E be a sequence space and K = {k1 < k2 < ...} ⊆ N. A sequence space of the form λE K = {(xkn ) ∈ w : (kn ) ∈ E} is called a K-step space of E. A canonical preimage of a sequence (xkn ) ∈ λE K is a sequence (yk ) ∈ w and is defined by ( xk , if k ∈ K yk = 0, otherwise. E A canonical preimage of a step space λE K is a set of canonical pre-images of all elements in λK . We say that E is monotone if E contains the canonical pre-image of all its step spaces. Note that every normal space is monotone (see [11], pp. 53). A sequence x = (xk ) ∈ `∞ (the space of bounded sequences) is said to be almost convergent, denoted

by b c, if all of its Banach limits coincide. Lorentz [17] introduced this sequence space as follows:   c = x ∈ `∞ : lim tjk (x) exists uniformly in j , b k

where

tjk (x) = It is clear that tjk (x) =

    

xj + xj+1 + ... + xj+k . k +1 1 k

k P

xj+i

for k ≥ 1;

i=1

xj

for k = 0.

Zadeh [29] introduced the concept of fuzzy set theory and its applications can be found in many branches of mathematical and engineering sciences including management science, control engineering, computer science, artificial intelligence. Matloka [18] introduced the bounded and convergent sequences of fuzzy numbers and proved that every convergent sequence of fuzzy numbers is bounded. Later, various classes of sequences of fuzzy numbers have been defined and studied by Colak et al. [5], Et et al. [7], Mursaleen and Ba¸sarir [21], Hazarika [10] and references therein. Now recalling some notions of fuzzy numbers which we will used to prove our main results. Throughout F the paper we used w F , `F and cF ∞, c 0 to denote the set of all, bounded, convergent and null sequence spaces of fuzzy numbers, respectively. A fuzzy number X is a fuzzy subset of the real line R i.e., a mapping X : R → J(= [0, 1]) associating each real number t with its grade of membership X(t). A fuzzy number X is said to be (i) upper-semi continuous if for each ε > 0, X −1 ([0, a + ε)) for all a ∈ [0, 1] is

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open in the usual topology of R, (ii) convex if X(t) ≥ X(s) ∧ X(r) = min{X(s), X(r)} for s < t < r (iii) normal if there exists t0 ∈ R such that X(t0 ) = 1. We used the notation X α to denotes α-level set of a fuzzy number X, 0 < α ≤ 1 and is given by X α = {t ∈ R : X(t) ≥ α}. The set of all normal, convex and upper semi-continuous fuzzy number with compact support will be denoted by R(J) and the fuzzy number we mean that the number belongs to R(J). We used the symbol D to denote the set of all closed and bounded intervals X = [x1 , x2 ] on R. For any two sets X, Y ∈ D, we define X ≤ Y if and only if x1 ≤ y1 and x2 ≤ y2 . A metric d on D is given by d(X, Y ) = max{|x1 − y1 |, |x2 − y2 |}. It is easy to see that (D, d) is a complete metric space. Also, the relation ≤ is a partial order on D. The absolute value |X| of X ∈ R(J) is given by ( max{X(t), X(−t)}, if t > 0, |X|(t) = 0, if t < 0. ¯ Suppose that d¯ : R(J)×R(J) → R is a mapping such that d(X, Y ) = sup0≤α≤1 d(X α , Y α ). Then (R(J), d) is a complete metric space. We define X ≤ Y if and only if X α ≤ Y α , for all α ∈ J. By ¯0 and ¯1 we denotes the additive and multiplicative identities in R(J), respectively. A sequence u = (uk ) of fuzzy numbers is said to be (i) bounded if the set {uk : k ∈ N} of fuzzy numbers is bounded, (ii) convergent to a fuzzy number u0 if for every ε > 0 , there exists k0 ∈ N such ¯ k , u0 ) < ε, for all k ≥ n0 , (iii) I-convergent (see [15]) if there exists a fuzzy number u0 such that that d(u ¯ k , u0 ) ≥ ε} ∈ I. We write I-lim uk = u0 , (iv) I-bounded if there exists for each ε > 0, the set {k ∈ N : d(u ¯ k , ¯0) ≥ K} ∈ I. K > 0 such that the set {k ∈ N : d(u

2

Main results

Throughout the article we assume that I is an admissible ideal of N. In this section, we introduce the following definitions. We introduce some new strongly almost ideal convergent sequence spaces using the n generalized difference matrix B(m) and Orlicz function M . Let us consider a sequence p = (pk ) of positive real numbers and let m, n be any nonnegative integers. For some ρ > 0, we define the following sequence spaces. ( ( 1 IF n F [w b0 (M, θ, B(m) , p)] = (uk ) ∈ w : r ∈ N : hr

×

X

k∈Jr

[w b

IF

n (M, θ, B(m) , p)]

=

(

n d(tjk (B(m) uk ), 0)

"

M

F

(uk ) ∈ w :

ρ (

!#pk

" 1 X r∈N: M hr k∈Jr

)

≥ ε ∈ I, uniformly in j ∈ N

n d(tjk (B(m) uk ), u0 )

ρ

!#pk

)

≥ε

)

∈ I,

)

uniformly in j ∈ N and for some u0 ∈ R(J) F n [w b∞ (M, θ, B(m) , p)]

=

(

(uk ) ∈ w F : sup r

1 hr

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×

X

k∈Jr IF n [w b∞ (M, θ, B(m) , p)]

=

(

"

M

n d(tjk (B(m) uk ), 0)

ρ (

!#pk

1 hr !#pk

< ∞, uniformly in j ∈ N

)

(uk ) ∈ w F : ∃K > 0 s.t. r ∈ N :

×

X

k∈Jr

"

M

n uk ), 0) d(tjk (B(m)

ρ

≥K

)

)

∈ I, uniformly in j ∈ N .

Particular cases: n n (i) If p = (pk ) = 1 for all k ∈ N, we denote [w b0IF (M, θ, B(m) , p)] = [w b0IF (M, θ, B(m) )], IF n IF n F n F n [w b (M, θ, B(m) , p)] = [w b (M, θ, B(m) )], [w b∞ (M, θ, B(m) , p)] = [w b∞ (M, θ, B(m) )] and IF n IF n [w b∞ (M, θ, B(m) , p)] = [w b∞ (M, θ, B(m) )].

n n n (ii) If M (x) = x, we denote [w b0IF (M, θ, B(m) , p)] = [w b0IF (θ, B(m) , p)], [w bIF (M, θ, B(m) , p)] = IF n F n F n IF n [w b (θ, B(m) , p)], [w b∞(M, θ, B(m) , p)] = [w b∞(θ, B(m) , p)] and [w b∞ (M, θ, B(m) , p)] = IF n [w b∞ (θ, B(m) , p)].

n n n (iii) If θ = (2r ), we denote [w b0IF (M, θ, B(m) , p)] = [w b0IF (M, B(m) , p)], [w b IF (M, θ, B(m) , p)] = n F n [w bIF (M, B(m) , p)], [w b∞ (M, θ, B(m) , p)] IF n [w b∞ (M, B(m) , p)].

=

F n [w b∞ (M, B(m) , p)]

and

IF n [w b∞ (M, θ, B(m) , p)]

=

Throughout the manuscript, we will used the following well-known inequality. Suppose that p = (pk ) is a sequence of positive real numbers with 0 < pk ≤ supk pk = H, D = max{1, 2H−1 }. Then |ak + bk |pk ≤ D(|ak |pk + |bk |pk ) for all k ∈ N and ak , bk ∈ C. Also |a|pk ≤ max{1, |a|H } for all a ∈ C. Now we are ready to give our main results as follows.

Theorem 2.1. Let p = (pk ) be a bounded sequence of positive real numbers. The spaces IF n IF n F n IF n [w b0 (M, θ, B(m) , p)], [w b (M, θ, B(m) , p)], [w b∞(M, θ, B(m) , p)], and [w b∞ (M, θ, B(m) , p)] are closed with respect to addition and scalar multiplication.

n Proof. We prove the result only for the space [w bIF (M, θ, B(m) , p)]. The others can be treated similarly. IF n Let u = (uk ) and v = (vk ) be two elements of [w b (M, θ, B(m) , p)] and α1 , α2 be scalars. Let ε > 0 be given. Then there exist positive numbers ρ1 , ρ2 such that ( " !#pk ) n d(tjk (B(m) uk ), u0 ) 1 X ε P = r∈N: M ≥ ∈I (uniformly in j ∈ N) hr ρ1 2 k∈Jr

and Q=

(

" 1 X r∈N: M hr k∈Jr

n d(tjk (B(m) vk ), v0 )

ρ2

!#pk

ε ≥ 2

)

∈I

(uniformly in j ∈ N).

Let ρ3 = max(2|α1|ρ1 , 2|α2|ρ2 ). Since M is non-decreasing and convex function, we have " !#pk n d(tjk (B(m) (α1 uk + α2 vk )), α1 u0 + α2 v0 ) 1 X M hr ρ3 k∈Jr

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" 1 X ≤ M hr

n uk ), u0 ) α1 d(tjk (B(m)

" 1 X ≤ M hr

n uk ), u0 ) d(tjk (B(m)

ρ3

k∈Jr

ρ1

k∈Jr

!#pk

!#pk

n vk ), v0 ) α2 d(tjk (B(m)

" 1 X + M hr

ρ3

k∈Jr

n vk ), v0 ) d(tjk (B(m)

" 1 X M + hr

ρ2

k∈Jr

!#pk

!#pk

,

uniformly in j. Therefore, we have ) ( !#pk " n d(tjk (B(m) (α1 uk + α2 vk )), α1 u0 + α2 v0 ) 1 X ≥ ε ⊆ P ∪ Q ∈ I. r∈N: M hr ρ3 k∈Jr

n b IF (M, θ, B(m) , p)]. This completes the proof. uniformly in j. This yields (α1 u + α2 v) ∈ [w

Theorem 2.2. Let M1 and M2 be two Orlicz functions. Then n n (i) [Z(M2 , θ, B(m) , p)] ⊆ [Z(M1 M2 , θ, B(m) , p)].

n n n (ii) [Z(M1 , θ, B(m) , p)] ∩ Z(M2 , θ, B(m) , p)] ⊆ [Z(M1 + M2 , θ, B(m) , p)], IF F where Z = w b0IF , w bIF , w b∞ ,w b∞ .

n Proof. (i) Let u = (uk ) ∈ [w bIF (M2 , θ, B(m) , p)] and let ε > 0 be given. For some ρ > 0, we have

(

" 1 X r∈N: M2 hr

!#pk

n uk ), u0 ) d(tjk (B(m)

ρ

k∈Jr

)

≥ε

∈ I,

(2.1)

uniformly in j ∈ N. Choose λ with 0 < λ < 1 such that M1 (t) < ε for 0 ≤ t ≤ λ. We define vk =

n uk ), u0 ) d(tjk (B(m)

ρ

and consider lim

k∈N;0≤vk ≤λ

[M1 (vk )]pk =

lim

k∈N;vk ≤λ

[M1 (vk )]pk +

lim

[M1 (vk )]pk .

k∈N;vk >λ

Therefore, one obtains lim

k∈N;vk ≤λ

[M1 (vk )]pk ≤ [M1 (2)]H

lim

[vk ]pk ,

(H = sup pk ).

k∈N;vk ≤λ

(2.2)

k

For the second summation (i.e. vk > λ), we go through the following procedure. We have vk
λ

[vk ]pk .

(2.3)

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It follows from (2.1), (2.2) and (2.3) that n (uk ) ∈ [w bIF (M1 .M2 , θ, B(m) , p)].

n n Hence, [w bIF (M2 , θ, B(m) , p)] ⊆ [w bIF (M1 .M2 , θ, B(m) , p)].

n n (ii) Let (uk ) ∈ [w bIF (M1 , θ, B(m) , p)] ∩ [w bIF (M2 , θ, B(m) , p)]. Let ε > 0 be given. Then there exists ρ > 0 such that !#pk " ( ) n d(tjk (B(m) uk ), u0 ) 1 X M1 r∈N: ≥ε ∈I (uniformly in j ∈ N) hr ρ k∈Jr

and

(

n d(tjk (B(m) uk ), u0 )

" 1 X r∈N: M2 hr

ρ

k∈Jr

!#pk

)

≥ε

(uniformly in j ∈ N).

∈I

The rest of the proof follows from the following relation: ( " !#pk ) n d(tjk (B(m) uk ), u0 ) 1 X r∈N: (M1 + M2 ) ≥ε hr ρ k∈Jr

⊆

(

" 1 X r∈N: M1 hr

[

n d(tjk (B(m) uk ), u0 )

ρ

k∈Jr

(

" 1 X r∈N: M2 hr

!#pk

n d(tjk (B(m) uk ), u0 )

ρ

k∈Jr

)

≥ε

!#pk

)

≥ε .

Note that if we take M1 (x) = M (x) and M2 (x) = x for all x ∈ [0, ∞) in the above theorem, then we obtain the following corollary: n n IF F Corollary 2.3. One has [Z(θ, B(m) , p)] ⊆ [Z(M, θ, B(m) , p)], where Z = w b0IF , w bIF , w b∞ ,w b∞ .

As in classical theory, the following is easy to prove.

Theorem 2.4.

n n (a) If M1 (x) ≤ M2 (x) for all x ∈ [0, ∞), then [Z(M1 , θ, B(m) , p)] ⊆ [Z(M2 , θ, B(m) , p)]

F for Z = w b0IF , w b IF and w b∞ .

n1 n2 F (b) If n1 < n2 then [Z(θ, B(m) , p)] ⊆ [Z(θ, B(m) , p)] for Z = w b0IF , w bIF and w b∞ .

Theorem 2.5. Let M be an Orlicz function. Then

n n F n [w b0IF (M, θ, B(m) , p)] ⊂ [w bIF (M, θ, B(m) , p)] ⊂ [w b∞ (M, θ, B(m) , p)]

and the inclusions are proper.

n Proof. Suppose that (uk ) ∈ [w bIF (M, θ, B(m) , p)]. Let ε > 0 be given. Then there exists ρ > 0 such that

(

" 1 X r∈N: M hr

n d(tjk (B(m) uk ), u0 )

ρ

k∈Jr

931

!#pk

)

≥ε

∈ I.

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Clearly, M

n d(tjk (B(m) uk ), 0)

ρ

!

1 ≤ M 2

n d(tjk (B(m) uk ), u0 )

ρ

!

1 + M 2



d(u0 , 0) ρ



.

F n Taking supremum over k on both sides of above inequalities implies that (uk ) ∈ [w b∞ (M, θ, B(m) , p)]. n F n Thus, we have [w bIF (M, θ, B(m) , p)] ⊂ [w b∞ (M, θ, B(m) , p)].

n n The inclusion [w b0IF (M, θ, B(m) , p)] ⊂ [w bIF (M, θ, B(m) , p)] is obvious. We now show that the inclusion is strict in the above theorem by constructing the following illustrative example.

Example 2.1. Suppose that θ = (2r ) and M (x) = x for all x ∈ [0, ∞). Suppose also that r = 1, s = −1, n = 1, m = 2. Let us define the sequence (uk ) of fuzzy numbers by  6  if − k6 ≤ t ≤ 0;  kt + 1 uk (t) = − k6 t + 1 if 0 < t ≤ k6 ;   0 , otherwise,

1 uk are where k = 2i (i = 1, 2, 3, ...), otherwise uk (t) = 0. For α ∈ (0, 1], the α-level sets of uk and B(m)

α

[uk ] =

(

[ k6 (α − 1), k6 (1 − α)] if

k = 2i , i = 1, 2, 3, ...

[0, 0]

otherwise

and α 1 [B(2) uk ]

=

(

,

[ 31 (α − 1), 13 (1 − α)] for

k = 2i

[0, 0]

otherwise .

,

Pj 1 1 α 1 It is easy to prove that − ¯31 < [Tj ]α < ¯13 for α ∈ (0, 1], where [Tj ]α = [tj,k (B(2) uk )]α = [ j+1 i=1 B(2) uk ] . Because ( 1 1 [ (α − 1), 13 (1 − α)] for k = 2i ; j ≥ 1 α 1+ 1j 3 1 [tj,k (B(2) uk )] = [0, 0] , otherwise and α 1 [tj,k (B(2) uk )]

=

(

[ 31 (α − 1), 13 (1 − α)] if [0, 0] ,

j=0 otherwise .

Thus (Tj ) is I-bounded but not I-convergent.



n−1 n Theorem 2.6. The inclusions [Z(M, θ, B(m) , p)] ⊆ [Z(M, θ, B(m) , p)] are strict for n ≥ 1. In geni n eral [Z(M, θ, B(m) , p)] ⊆ [Z(M, θ, B(m) , p)] (i = 1, 2, ..., n − 1) and the inclusion is strict, where IF IF IF F Z=w b0 , w b ,w b∞ , w b∞ . n−1 Proof. Suppose that u = (uk ) ∈ [w b0IF (M, θ, B(m) , p)]. Let ε > 0 be given. Then there exists ρ > 0 such

that

(

" 1 X r∈N: M hr

n−1 d(tjk (B(m) uk ), 0)

ρ

k∈Jr

!#pk

)

≥ε

∈ I.

Since M is non-decreasing and convex it follows that " !#pk n d(tjk (B(m) uk ), 0) M 2ρ

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n−1 n−1 uk+1 ), 0) uk ), tjk (B(m) d(tjk (B(m)

"

≤ M

2ρ n−1 uk ), 0) d(tjk (B(m)

"

1 ≤D M 2 "

ρ n−1 d(tjk (B(m) uk ), 0)

≤ DK M where K = max{1, (


1 H }. 2

ρ

n−1 uk+1 ), 0) d(tjk (B(m)

"

1 +D M 2 !#pk "

ρ

+ DK M

!#pk

n−1 d(tjk (B(m) uk+1 ), 0)

ρ

!#pk

,

Therefore we have

" 1 X r∈N: M hr

n d(tjk (B(m) uk ), 0)

2ρ

k∈Jr

⊆

!#pk

!#pk

(

!#pk

" 1 X r ∈ N : DK M hr

≥ε

n−1 uk ), 0) d(tjk (B(m)

ρ

k∈Jr

[

(

)

" 1 X M r ∈ N : DK hr

ρ

k∈Jr

" 1 X r∈N: M hr

n uk ), 0) d(tjk (B(m)

2ρ

k∈Jr

)

≥ε

n−1 d(tjk (B(m) uk+1 ), 0)

i.e., (

!#pk

!#pk

)

≥ε

!#pk

)

≥ε ,

∈ I.

n b0IF (M, θ, B(m) , p)]. Hence, (uk ) ∈ [w We now show that the inclusion is strict in the above theorem (Theorem 2.6) by constructing the

following illustrative example.

Example 2.2. Let θ = (2r ) and M (x) = x for all x ∈ [0, ∞) Suppose also that r = 1, s = −1, n = 2, m = 2 and pk = 1 for all k ∈ N. We now define the  t   − k 2−1 + 1 uk (t) = − k 2t+1 + 1   0

sequence (uk ) of fuzzy numbers by , if k 2 − 1 ≤ t ≤ 0; , if 0 < t ≤ k 2 + 1; , otherwise.

1 2 For α ∈ (0, 1], the α-level sets of uk , B(2) uk and B(2) uk are as follow:

[uk ]α = [(1 − α)(k 2 − 1), (1 − α)(k 2 + 1)], and 1 [B(2) uk ]α = [(1 − α)(4k − 6), (1 − α)(4k − 2)], 2 [B(2) uk ]α = [4(1 − α), 12(1 − α)]. 1 2 It is easy to verified that the sequence [B(2) uk ]α is not I-convergent but [B(2) uk ]α is I-convergent.



n n Theorem 2.7. Let 0 < pk ≤ qk < ∞ for each k. Then [Z(M, θ, B(m) , p)] ⊆ [Z(M, θ, B(m) , q)] for

Z=w b0IF and w bIF .

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n Proof. Let (uk ) ∈ [w b0IF (M, θ, B(m) , p). Then there exists a number ρ > 0 such that

(

" 1 X r∈N: M hr

n d(tjk (B(m) uk ), 0)

ρ

k∈Jr

!#pk

)

≥ε

∈I

(uniformly in j ∈ N).

For sufficiently large k, since pk ≤ qk for each k, therefore we obtain ) ( !#qk " n uk ), 0) d(tjk (B(m) 1 X ≥ε r∈N: M hr ρ k∈Jr

⊆

(

" 1 X r∈N: M hr

n uk ), 0) d(tjk (B(m)

ρ

k∈Jr

!#pk

)

≥ε

∈ I,

n uniformly in j ∈ N, i.e. (uk ) ∈ [w b0IF (M, θ, B(m) , q)].

n n Similarly, we can show that ]w b IF (M, θ, B(m) , p)] ⊆ [w bIF (M, θ, B(m) , q)].

n n , p)] ⊆ [Z(M, θ, B(m) )] for Z = w b0IF Corollary 2.8. (a) Let 0 < inf k pk ≤ pk ≤ 1. Then [Z(M, θ, B(m) IF and w b . n n (b) Let 1 ≤ pk ≤ supk pk < ∞. Then [Z(M, θ, B(m) )] ⊆ [Z(M, θ, B(m) , p)] for Z = w b0IF and w bIF .

n Theorem 2.9. If I is an admissible ideal and I 6= If , then the sequence spaces [w b0IF (M, θ, B(m) , p)] and IF n w b (M, θ, B(m) , p)] are neither normal nor monotone, where If denotes the class of all finite subsets of

N.

Proof. To prove our result, we construct the following example. Example 2.3. Suppose that M (x) = x for all x ∈ [0, ∞) and r = 1, s = −1, n = 1, m = 1. Consider that I = Iδ , where Iδ = {A ⊂ N : asymptotic density of A (in symbol, δ(A)) = 0} and note that Iδ is an ideal of N, and pk = 1 for all k ∈ N. We now define the sequence (uk ) of fuzzy numbers by    1 + t − k , if t ∈ [k − 1, k]; uk (t) = 1 − t + k , if t ∈ [k, k + 1];   0 , otherwise. Let us define

αk =

(

1 , if k is odd; 0 , if k is even.

n n Thus (αk uk ) ∈ / [w b0IF (M, θ, B(m) , p)] and w bIF (M, θ, B(m) , p)]. Therefore, we conclude that the spaces IF n IF n [w b0 (M, θ, B(m) , p)] and w b (M, θ, B(m) , p)] are not normal and hence these spaces are not mono-

tone.



n , p)] is not Theorem 2.10. If I is an admissible ideal and I 6= If , then the sequence space [Z(M, θ, B(m) IF IF symmetric, where Z = w b0 , w b .

n Proof. We shall prove the result only for the space [w bIF (M, θ, B(m) , p)] with the help of the following example. For other space, the proof is similar so we omitted.

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Example 2.4. Suppose that M (x) = x for all x ∈ [0, ∞) and r = 1, s = −1, n = 1, m = 1. Let I = Iδ and pk = 1 for all k ∈ N. We now define the sequence (uk ) of fuzzy numbers by   , if t ∈ [4k − 1, 4k];  t − 4k + 1 uk (t) = −t + 4k + 1 , if t ∈ [4k, 4k + 1];   0 , otherwise. n Thus, we have (uk ) ∈ [w bIF (M, θ, B(m) , p)]. But the rearrangement (vk ) of (uk ) defined as

vk = {u1 , u4 , u2 , u9 , u3 , u16, u5 , u25, u6 , ...}.

n n This implies that (vk ) ∈ / [w bIF (M, θ, B(m) , p)]. Hence [w bIF (M, θ, B(m) , p)] is not symmetric.

3



Acknowledgement

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (292-130-1436-G). The authors, therefore, acknowledge with thanks DSR for technical and financial support.

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The Catalan Numbers: a Generalization, an Exponential Representation, and some Properties Feng Qi1,2,3,†

Xiao-Ting Shi3 1

Mansour Mahmoud4

Fang-Fang Liu3

Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

2

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China 3

Department of Mathematics, College of Science,

Tianjin Polytechnic University, Tianjin City, 300387, China 4

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt †

Corresponding author: [email protected], [email protected]

Abstract In the paper, the authors establish an exponential representation for a function involving the gamma function and originating from investigation of the Catalan numbers in combinatorics, find necessary and sufficient conditions for the function to be logarithmically completely monotonic, introduce a generalization of the Catalan numbers, derive an exponential representation for the generalization, and present some properties of the generalization. 2010 Mathematics Subject Classification: Primary 11R33; Secondary 11B75, 11B83, 11S23, 26A48, 33B15, 44A20. Key words and phrases: exponential representation; necessary and sufficient condition; logarithmically completely monotonic function; gamma function; Catalan number; generalization; property; Catalan–Qi function.

1

Introduction

It is known [4, 21, 22] that, in combinatorics, the Catalan numbers Cn for n ≥ 0 form a sequence of natural numbers that occur in tree enumeration problems such as “In how many ways can a regular n-gon be divided into n−2 triangles if different orientations are counted separately?” whose solution is the Catalan number Cn−2 . Explicit formulas of Cn for n ≥ 0 include     1 2n 2n (2n − 1)!! 1 2n 4n Γ(n + 1/2) Cn = = = = 2 F1 (1 − n, −n; 2; 1) = √ , (1) n+1 n (n + 1)! n n−1 π Γ(n + 2) R∞ where Γ(z) = 0 tz−1 e−t d t for 0 is the classical Euler gamma function and p Fq (a1 , . . . , ap ; b1 , . . . , bq ; z)

=

∞ X (a1 )n · · · (ap )n z n (b1 )n · · · (bq )n n! n=0

(2)

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is the generalized hypergeometric series defined for ai ∈ C and bi ∈ C \ {0, −1, −2, . . . }, for positive Qn−1 integers p, q ∈ N, and in terms of the rising factorials (x)n = k=0 (x + k). The asymptotic form for the Catalan function Cx is   1 9 1 145 1 4x − + + ··· , Cx ∼ √ 8 x5/2 128 x7/2 π x3/2 see [3, 4, 21, 22, 24]. Recently, among other things, the formula     n−1 n k n n X Y 1 X 2n X k! 2n − k − 1 n2 ` k [2(n − k) − 1]!! Cn = (−1) (−1) (` − 2m) = n! 2k ` m=0 n! 2k 2(n − k) k=0

`=0

k=0

was found in [18, Theorem 3]. For more information on the Catalan numbers Cn , please refer to two monographs [2, 3] and references cited therein. In the paper [20], motivated by the explicit expression (1), the authors established an integral representation of the Catalan function Cx for x ≥ 0. Theorem 1.1 ([20, Theorem 1]). For x ≥ 0, we have  Z ∞  
e3/2 4x (x + 1/2)x 1 1 1 1 −t/2 −2t −xt Cx = √ − + e −e e dt . exp t et − 1 t 2 π (x + 2)x+3/2 0

(3)

Recall from [8, Chapter XIII], [19, Chapter 1], and [25, Chapter IV] that an infinitely differentiable function f is said to be completely monotonic on an interval I if it satisfies 0 ≤ (−1)k f (k) (x) < ∞ on I for all k ≥ 0. Recall from [11] that an infinitely differentiable and positive function f is said to be logarithmically completely monotonic on an interval I if 0 ≤ (−1)k [ln f (x)](k) < ∞ hold on I for all k ∈ N. For more information on logarithmically completely monotonic functions, please refer to [14, 19]. The formula (3) can be rearranged as √   Z ∞ 
π (x + 2)x+3/2 1 1 1 1 ln 3/2 x − + C e−t/2 − e−2t e−xt d t. (4) = x t x t e −1 t 2 e 4 (x + 1/2) 0
Since the function 1t et1−1 − 1t + 21 is positive on (0, ∞), the right-hand side of (4) is a completely monotonic function on (0, ∞). This means that the function (x + 2)x+3/2 Cx 4x (x + 1/2)x

(5)

is logarithmically completely monotonic on (0, ∞). Because any logarithmically completely monotonic function must be completely monotonic, see [14, Eq. (1.4)] and references therein, the function (5) is also completely monotonic on (0, ∞). By virtue of (1), the function (5) can be rewritten as (x + 2)x+3/2 Γ(x + 1/2) , (x + 1/2)x Γ(x + 2)

x > 0.

(6)

Hence, the logarithmically complete monotonicity of (5) implies the logarithmically complete monotonicity of (6). The function (6) is the special case F1/2,2 (x) of the general function Fa,b (x) =

Γ(x + a) (x + b)x+b−a , (x + a)x Γ(x + b)

a, b ∈ R,

a 6= b

x > − min{a, b}.

(7)

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We notice that the function Fa,b (x) does not appear in the expository and survey articles [9, 14] and plenty of references therein. Therefore, it is significant to naturally pose an open problem below. Open Problem 1.1 ([20, Open Problem 1]). What are the necessary and sufficient conditions on a, b ∈ R such that the function Fa,b (x) defined by (7) is (logarithmically) completely monotonic in x ∈ (− min{a, b}, ∞)? This problem was answered in [6, Theorem 2] as follows. Theorem 1.2 ([6, Theorem 2]). The sufficient conditions on a, b such that the function [Fa,b (x)]±1 defined by (7) is logarithmically completely monotonic in x ∈ (− min{a, b}, ∞) are (a, b) ∈ D± (a, b), where   1 . D± (a, b) = {(a, b) : a ≷ b, a ≥ 1} ∪ (a, b) : a ≶ b, a ≤ 2 The necessary conditions on a, b for the function [Fa,b (x)]±1 to be logarithmically completely monotonic in x ∈ (− min{a, b}, ∞) are a(a − b) R a−b 2 . The aims of this paper are to establish an exponential representation for the function Fa,b (x), to find necessary and sufficient conditions on a, b for [Fa,b (x)]±1 to be logarithmically completely monotonic on [0, ∞), to introduce a generalization of the Catalan numbers Cn , and to derive an exponential representation for the generalization of Cn . The first main result in this paper can be stated as the following theorem. Theorem 1.3. For a, b > 0, the function Fa,b (x) defined by (7) has the exponential representation    Z ∞ 
1 1 1 −bt −at −xt Fa,b (x) = exp b − a + a+ − e − e e d t (8) t t 1 − e−t 0 on [0, ∞) and the function [Fa,b (x)]±1 is logarithmically completely monotonic on [0, ∞) if and only if (a, b) ∈ D± (a, b). Comparing (3) with (8) hints and stimulates us to consider the three-variable function  z Γ(b) b Γ(z + a) , 1; b b − a −1 n=1  X    ∞ ∞ X x2n b 2 xn 1 b C(a, b; n) = 1 F2 a; , b; x ; C(a, b; n) = 1 F1 a; b; x . (2n)! 2 4a n! a n=0 n=0 C(a, b; z + 1) =

Remark 1.2. When a = and (12).

1 2

and b = 2, the formulas in Theorem 1.5 become those listed in (11)


b 2 Remark 1.3. The last two formulas in Theorem 1.5 show that the functions 1 F2 a; 12 , b; 4a x and
b 1 F1 a; b; a x can be regarded as the generating functions of the Catalan–Qi numbers C(a, b; n).

2

Proofs of Theorems 1.3 to 1.5

We are now start out to prove Theorem 1.3 by two approaches and to prove Theorems 1.4 and 1.5. First proof of Theorem 1.3. Taking the logarithm of Fa,b (x) gives ln Fa,b (x) = ln Γ(x + a) − x ln(x + a) − ln Γ(x + b) + (x + b − a) ln(x + b) , fa (x) − fa (x + b − a). Differentiating twice with respect to the variable x of fa (x) yields fa0 (x) = ψ(x + a) − ln(x + a) +

a −1 x+a

and fa00 (x) = ψ 0 (x + a) −

1 a − . x + a (x + a)2

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By virtue of the formulas ψ

(n)

n+1

Z

(z) = (−1)

0

∞

tn e−zt d t and 1 − e−t

Γ(z) = k

z

Z

∞

tz−1 e−kt d t

0

for 0, 0, and n ∈ N in [1, p. 260, 6.4.1] and [1, p. 255, 6.1.1], we obtain  Z ∞ 1 1 1 a − fa00 (x − a) = ψ 0 (x) − − 2 = − a te−xt d t. x x 1 − e−t t 0 Accordingly, we have ∞

   1 1 − − a t e−(x+a)t − e−(x+b)t d t −t 1−e t 0  Z ∞
1 1 − − a t e−at − e−bt e−xt d t. = −t 1−e t 0

[ln Fa,b (x)]00 = fa00 (x) − fa00 (x + b − a) =

Z

(13)

The famous Bernstein-Widder theorem, [25, p. 161, Theorem 12b], states that aRnecessary and ∞ sufficient condition for f (x) to be completely monotonic on (0, ∞) is that f (x) = 0 e−xt d µ(t), where µ is a positive measure on [0, ∞) such that the above integral converges on (0, ∞). Hence, in order to find necessary and sufficient conditions on a, b such that the function [ln Fa,b (x)]00 is completely monotonic on (0, ∞), it is necessary and sufficient to discuss the positivity or negativity of the function  
1 1 − − a t e−at − e−bt (14) −t 1−e t on (0, ∞). It is clear that the factor e−at − e−bt is positive (or negative, respectively) if and only if b > a (or b < a, respectively). Since the function 1−e1 −t − 1t = et1−1 − 1t + 1 is strictly increasing on (0, ∞)

and has the limits limt→0+ 1−e1 −t − 1t = 12 and limt→∞ 1−e1 −t − 1t = 1, see [5, 15] and references therein, the factor 1−e1 −t − 1t − a is positive (or negative, respectively) on (0, ∞) if and only if a ≤ 12 (or a ≥ 1, respectively). Consequently, the function (14) is 1 2

1. positive if and only if either b > a and a ≤ 2. negative if and only if either b < a and a ≤

1 2

or b < a and a ≥ 1, or b > a and a ≥ 1.

As a result, the function ±[ln Fa,b (x)]00 is completely monotonic on (0, ∞) if and only if (a, b) ∈ D± (a, b). By a straightforward computation, we see that   x+b a(b − a) 0 lim [ln Fa,b (x)] = lim ψ(x + a) − ψ(x + b) + ln + =0 (15) x→∞ x→∞ x + a (x + a)(x + b) for all a, b ∈ R. This implies that, if and only if (a, b) ∈ D± (a, b), the first logarithmic derivative satisfies [ln Fa,b (x)]0 ≶ 0. By the definition of logarithmically completely monotonic functions, we conclude that, if and only if (a, b) ∈ D± (a, b), the function [Fa,b (x)]±1 is logarithmically completely monotonic on (0, ∞). 5

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Integrating from u to ∞ with respect to x on the very ends of (13) and considering the limit (15) give  Z ∞
1 1 − e−at − e−bt e−ut d t. −[ln Fa,b (u)]0 = − a −t 1−e t 0 Further integrating with respect to u from x to ∞ on both sides of the above equality and employing the limit limx→∞ Fa,b (x) = eb−a reveal that Z ln Fa,b (x) = b − a + 0

∞

 
1 1 1 − − a e−at − e−bt e−xt d t. −t t 1−e t

The first proof of Theorem 1.3 is thus complete. Second proof of Theorem 1.3. As did in the proof of [20, Theorem 1], employing the formula  Z ∞  √
1 1 1 1 −zt − + dt e ln Γ(z) = ln 2π z z−1/2 e−z + t et − 1 t 2 0 in [23, (3.22)] and utilizing ln ab =

R∞ 0

e−au −e−bu u

d u in [1, p. 230, 5.1.32] yield

   −xt Z ∞
1 1 1 1 x+a e ln Fa,b (x) = b − a + a − + − + t e−at − e−bt d t ln 2 x+b 2 t e − 1 t 0   Z ∞ −xt  −xt Z ∞

1 e 1 1 1 e −bt −at =b−a+ a− e −e dt + − + t e−at − e−bt d t 2 t 2 t e − 1 t 0 0  Z ∞ 
1 1 1 1 1 a− − + − t e−bt − e−at e−xt d t =b−a+ t 2 2 t e − 1 0  Z ∞ 
1 1 1 a+ − e−bt − e−at e−xt d t. =b−a+ −t t t 1 − e 0 The rest of the second proof is the same as in the first proof after the equation (13). The second proof of Theorem 1.3 is complete. Proof of Theorem 1.4. This follows from straightforwardly combining (7) and (8) with (9). Proof of Theorem 1.5. It is easy to see that C(a, b; z + 1) =

 z+1  z Γ(b) b b z + a Γ(b) b Γ(z + a) b z+a Γ(z + a + 1) = = C(a, b; z). Γ(a) a Γ(z + b + 1) a z + b Γ(a) a Γ(z + b) a z+b

Consequently, when taking z = n − 1,  2 b n+a−1 b n+a−1n+a−2 C(a, b; n) = C(a, b; n − 1) = C(a, b; n − 2) a n+b−1 a n+b−1 n+b−2  n  n n−1 Y a+k b n+a−1n+a−2 a+1a b = ··· = ··· C(a, b; 0) = . a n+b−1 n+b−2 b+1 b a b+k k=0

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By (9), it follows that ∞  n X a n=1

b

C(a, b; n) =

∞ Γ(b) X Γ(n + a) Γ(b) Γ(a + 1)Γ(b − a − 1) a = = . Γ(a) n=1 Γ(n + b) Γ(a) Γ(b)Γ(b − a) b−a−1

The last two formulas in Theorem 1.5 can be straightforwardly derived from the definition (2) of the generalized hypergeometric series. The proof of Theorem 1.5 is complete. Remark 2.1. This paper is a companion of the articles [6, 7, 12, 13, 16, 18, 20] and the preprints [10, 18] and is a revised version of the preprint [17].

References [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington, 1972. [2] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974. [3] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics—A Foundation for Computer Science, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. [4] T. Koshy, Catalan Numbers with Applications, Oxford University Press, Oxford, 2009. [5] A.-Q. Liu, G.-F. Li, B.-N. Guo, and F. Qi, Monotonicity and logarithmic concavity of two functions involving exponential function, Internat. J. Math. Ed. Sci. Tech., 39 (2008), no. 5, 686–691; Available online at http://dx.doi.org/10.1080/00207390801986841. [6] F.-F. Liu, X.-T. Shi, and F. Qi, A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function, Glob. J. Math. Anal., 3 (2015), no. 4, 140–144; Available online at http://dx.doi.org/10.14419/gjma.v3i4.5187. [7] M. Mahmoud and F. Qi, Three identities of the Catalan–Qi numbers, Mathematics, 4 (2016), no. 2, Article 35, 7 pages; Available online at http://dx.doi.org/10.3390/math4020035. [8] D. S. Mitrinovi´c, J. E. Peˇcari´c, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993; Available online at http:// dx.doi.org/10.1007/978-94-017-1043-5. [9] F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl., 2010 (2010), Article ID 493058, 84 pages; Available online at http://dx.doi.org/10.1155/2010/493058. [10] F. Qi, Some properties and generalizations of the Catalan, Fuss, and Fuss–Catalan numbers, ResearchGate Research, (2015), available online at http://dx.doi.org/10.13140/RG.2.1. 1778.3128. [11] F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl., 296 (2004), 603–607; Available online at http://dx.doi.org/10.1016/j.jmaa. 2004.04.026. 7

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[12] F. Qi and B.-N. Guo, Logarithmically complete monotonicity of a function related to the Catalan–Qi function, Acta Univ. Sapientiae Math., 8 (2016), no. 1, 93–102; Available online at http://dx.doi.org/10.1515/ausm-2016-0006. [13] F. Qi and B.-N. Guo, Logarithmically complete monotonicity of Catalan–Qi function related to Catalan numbers, Cogent Math., (2016), 3:1179379, 6 pages; Available online at http: //dx.doi.org/10.1080/23311835.2016.1179379. [14] F. Qi and W.-H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions, J. Appl. Anal. Comput., 5 (2015), no. 4, 626–634; Available online at http://dx.doi.org/10.11948/2015049. [15] F. Qi, Q.-M. Luo, and B.-N. Guo, The function (bx − ax )/x: Ratio’s properties, In: Analytic Number Theory, Approximation Theory, and Special Functions, G. V. Milovanovi´c and M. Th. Rassias (Eds), Springer, 2014, pp. 485–494; Available online at http://dx.doi.org/10.1007/ 978-1-4939-0258-3_16. [16] F. Qi, M. Mahmoud, X.-T. Shi, and F.-F. Liu, Some properties of the Catalan–Qi function related to the Catalan numbers, SpringerPlus, (2016), 5:1126, 20 pages; Available online at http://dx.doi.org/10.1186/s40064-016-2793-1. [17] F. Qi, X.-T. Shi, and F.-F. Liu, An exponential representation for a function involving the gamma function and originating from the Catalan numbers, ResearchGate Research, (2015), available online at http://dx.doi.org/10.13140/RG.2.1.1086.4486. [18] F. Qi, X.-T. Shi, F.-F. Liu, and D. V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, J. Appl. Anal. Comput., (2017), in press; ResearchGate Technical Report, (2015), available online at http://dx.doi.org/10.13140/ RG.2.1.3230.1927. [19] R. L. Schilling, R. Song, and Z. Vondraˇcek, Bernstein Functions—Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012; Available online at http://dx.doi.org/10.1515/9783110269338. [20] X.-T. Shi, F.-F. Liu, and F. Qi, An integral representation of the Catalan numbers, Glob. J. Math. Anal., 3 (2015), no. 3, 130–133; http://dx.doi.org/10.14419/gjma.v3i3.5055. [21] R. P. Stanley, Catalan Numbers, Cambridge University Press, New York, 2015; Available online at http://dx.doi.org/10.1017/CBO9781139871495. [22] R. Stanley and E. W. Weisstein, Catalan number, From MathWorld–A Wolfram Web Resource; Available online at http://mathworld.wolfram.com/CatalanNumber.html. [23] N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996; Available online at http://dx.doi.org/10.1002/9781118032572. [24] I. Vardi, Computational Recreations in Mathematica, Addison-Wesley, Redwood City, CA, 1991. [25] D. V. Widder, The Laplace Transform, Princeton Mathematical Series 6, Princeton University Press, Princeton, N. J., 1941.

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Semiring structures based on meet and plus ideals in lower BCK-semilattices Hashem Bordbar1 , Sun Shin Ahn2,∗ , Mohammad Mehdi Zahedi3 , Young Bae Jun4 1

4

Faculty of Mathematics, Statistics and Computer Science, Shahid Bahonar University, Kerman, Iran 2 Department of Mathematics Education, Dongguk University, Seoul 04620, Korea 3 Department of Mathematics, Graduate University of Advanced Technology, Mahan-Kerman, Iran

Department of Mathematics Education (and RINS), Gyeongsang National University, Jinju 52828, Korea

Abstract. The notion of the meet set based on two subsets of a lower BCK-semilattice X is introduced, and related properties are investigated. Conditions for the meet set to be a (positive implicative, commutative, implicative) ideal are discussed. The meet ideal based on subsets, and the plus ideal of two subsets in a lower BCK-semilattice X are also introduced, and related properties are investigated. Using meet operation and addition, the semiring structure is induced.

1. Introduction Ideal theory has an important role in the development BCK/BCI-algebras (see [1, 3, 4]). It was shown in [5] that if X is a BCK-algebra then (X, ≤) is a poset, and moreover if X is a commutative BCK-algebra, i.e., x ∗ (x ∗ y) = y ∗ (y ∗ x) holds in X, then (X, ≤) is a lower semilattice. Palasi´ nski [7] discussed properties of certain ideals in BCK-algebras which are lower semilattices. In this paper, we introduce the notion of the meet set based on two subsets of a lower BCKsemilattice X and we discuss conditions for the meet set to be a (positive implicative, commutative, implicative) ideal. We also introduced the meet ideal based on subsets, and the plus ideal of two subsets in a lower BCK-semilattice X. We investigate several related properties, and we induce the semiring structure by using meet operation and addition. 2. Prliminaries

A BCK/BCI-algebra is an important class of logical algebras introduced by K. Is´eki and was extensively investigated by several researchers. An algebra (X; ∗, 0) of type (2, 0) is called a BCI-algebra if it satisfies the following conditions 0

2010 Mathematics Subject Classification: 06F35, 03G25. Keywords: Lower BCK-semilattice; meet set; meet ideal; plus ideal; meet operation; addition; semiring. ∗ The corresponding author. Tel.: +82 2 2260 3410, Fax: +82 2 2266 3409 (S. S. Ahn). 0 E-mail: [email protected] (H. Bordbar); [email protected] (S. S. Ahn); zahedi [email protected] (M. M. Zahedi); [email protected] (Y. B. Jun). 0

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(I) (II) (III) (IV)

(∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0), (∀x ∈ X) (x ∗ x = 0), (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y).

If a BCI-algebra X satisfies the following identity (V) (∀x ∈ X) (0 ∗ x = 0), then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following conditions (a1) (a2) (a3) (a4)

(∀x ∈ X) (x ∗ 0 = x), (∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x), (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y), (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y)

where x ≤ y if and only if x ∗ y = 0. A BCK-algebra X is called a lower BCK-semilattice (see [6]) if X is a lower semilattice with respect to the BCK-order. A subset A of a BCK/BCI-algebra X is called an ideal of X (see [6]) if it satisfies 0 ∈ A,

(2.1)

(∀x ∈ X) (∀y ∈ A) (x ∗ y ∈ A ⇒ x ∈ A) .

(2.2)

Note that every ideal A of a BCK/BCI-algebra X satisfies the following implication (see [6]). (∀x, y ∈ X) (x ≤ y, y ∈ A ⇒ x ∈ A) .

(2.3)

For any subset A of X, the ideal generated by A is defined to be the intersection of all ideals of X containing A, and it is denoted by ⟨A⟩. If A is finite, then we say that ⟨A⟩ is finitely generated ideal of X (see [6]). A subset A of a BCK-algebra X is called a commutative ideal of X (see [6]) if it satisfies (2.1) and (∀x, y ∈ X)(∀z ∈ A) ((x ∗ y) ∗ z ∈ A ⇒ x ∗ (y ∗ (y ∗ x)) ∈ A) .

(2.4)

A subset A of a BCK-algebra X is called a positive implicative ideal of X (see [6]) if it satisfies (2.1) and (∀x, y, z ∈ X) ((x ∗ y) ∗ z ∈ A, y ∗ z ∈ A ⇒ x ∗ z ∈ A) .

(2.5)

A subset A of a BCK-algebra X is called an implicative ideal of X (see [6]) if it satisfies (2.1) and (∀x, y ∈ X)(∀z ∈ A) ((x ∗ (y ∗ y)) ∗ z ∈ A ⇒ x ∈ A) .

(2.6)

A proper ideal P of a lower BCK-semilattice X is said to be prime if it satisfies (∀a, b ∈ X) (a ∧ b ∈ P ⇒ a ∈ P or b ∈ P ) .

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Semiring structures based on meet and plus ideals in lower BCK-semilattices

We refer the reader to the books [2, 6] for further information regarding BCK/BCI-algebras. 3. Meet and plus ideals In what follows, let X be a lower BCK-semilattice unless otherwise specified. nonempty subsets A and B of X, we consider the set K := {a ∧ b | a ∈ A, b ∈ B}

For any

where a ∧ b is the greatest lower bound of a and b. We say that K is the meet set based on A and B. Note that A ∩ B ⊆ K, but the reverse inclusion is not true as seen in the following example. Example 3.1. (1) Consider a lower BCK-semilattice X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 1 2 2 2 0 2 0 3 3 1 3 0 3 4 4 4 4 4 0 For A = {2, 3} and B = {1, 4}, we have K := {a ∧ b | a ∈ A, b ∈ B} = {0, 1, 2} ⊈ A ∩ B. (2) Consider a lower BCK-semilattice X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 0 1 1 2 2 1 0 2 2 3 3 3 3 0 3 4 4 4 4 4 0 For subsets A = {1, 2, 3} and B = {1, 3, 4} of X, we have K := {a ∧ b | a ∈ A, b ∈ B} = {0, 1, 3} ⊈ {1, 3} = A ∩ B. The following example shows that the set K := {a ∧ b | a ∈ A, b ∈ B} may not be an ideal of X for some subsets A and B of X. Example 3.2. Let X = {0, 1, 2, 3, 4} be a lower BCK-semilattice in Example 3.1(1). For A = {2, 3} and B = {1, 4}, we have {a ∧ b | a ∈ A, b ∈ B} = {0, 1, 2}, which is not an ideal of X. We provide conditions for the meet set K := {a ∧ b | a ∈ A, b ∈ B} based on A and B to be an ideal.

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Theorem 3.3. If A and B are ideals of X, then so is the meet set K := {a ∧ b | a ∈ A, b ∈ B} based on A and B. Proof. Obviously, 0 ∈ K. Let x ∈ K and y ∗ x ∈ K for x, y ∈ X. Then x = a ∧ b and y ∗ x = a′ ∧ b′ where a, a′ ∈ A and b, b′ ∈ B. Since a ∧ b ≤ a and A is an ideal, we have x = a ∧ b ∈ A. Similarly, we have y ∗ x = a′ ∧ b′ ≤ a′ ∈ A. Since A is an ideal of X, it follows that y ∈ A. By the similar way, we get y ∈ B. Therefore, y = y ∧ y ∈ {a ∧ b | a ∈ A, b ∈ B} = K □

and K is an ideal of X. Lemma 3.4 ([6]). For an ideal A of a BCK-algebra X, the following are equivalent. (i) A is positive implicative. (ii) (∀x, y ∈ X) ((x ∗ y) ∗ y ∈ A ⇒ x ∗ y ∈ A). Lemma 3.5 ([6]). For an ideal A of a BCK-algebra X, the following are equivalent. (i) A is commutative. (ii) (∀x, y ∈ X) (x ∗ y ∈ A ⇒ x ∗ (y ∗ (y ∗ x)) ∈ A).

Lemma 3.6 ([6]). Let A be an ideal of a BCK-algebra X. Then A is implicative if and only if A is both positive implicative and commutative. Theorem 3.7. If A and B are positive implicative (resp., commutative, implicative) ideals of X, then so is the meet set K := {a ∧ b | a ∈ A, b ∈ B} based on A and B. Proof. Assume that A and B are positive implicative ideals of X. Then A and B are ideals of X, and so the set K := {a ∧ b | a ∈ A, b ∈ B} is an ideal of X by Theorem 3.3. Let (x ∗ y) ∗ y ∈ K for every x, y ∈ X. Then (x ∗ y) ∗ y = a ∧ b for some a ∈ A and b ∈ B. Since a ∧ b ≤ a and A is an ideal, we have (x ∗ y) ∗ y ∈ A. Similarly, (x ∗ y) ∗ y ∈ B. Since A and B are positive implicative ideals, it follows from Lemma 3.4 that x ∗ y ∈ A and x ∗ y ∈ B. Therefore x ∗ y = (x ∗ y) ∧ (x ∗ y) ∈ {a ∧ b | a ∈ A, b ∈ B} = K, and so K is a positive implicative ideal of X by Lemma 3.4. Now suppose that A and B are commutative ideals of X. Then A and B are ideals of X, and so the set K := {a ∧ b | a ∈ A, b ∈ B} is an ideal of X by Theorem 3.3. Let x ∗ y ∈ K for every x, y ∈ X. Then x ∗ y = a ∧ b for some a ∈ A and b ∈ B. Since a ∧ b ≤ a and a ∧ b ≤ b, it follows

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that x ∗ y ∈ A ∩ B. Since A and B are commutative, we have x ∗ (y ∗ (y ∗ x)) ∈ A ∩ B by Lemma 3.5. Hence x ∗ (y ∗ (y ∗ x)) = (x ∗ (y ∗ (y ∗ x))) ∧ (x ∗ (y ∗ (y ∗ x))) ∈ {a ∧ b | a ∈ A, b ∈ B} = K, and therefore K is a commutative ideal if X. Now, if A and B are implicative ideals of X, then they are both positive implicative and commutative by Lemma 3.6. Thus K is both a positive implicative ideal and a commutative ideal of X, and so it is an implicative ideal of X. □ Given two nonempty subsets A and B of X, we consider the ideal of X generated by the meet set K := {a ∧ b | a ∈ A, b ∈ B} based on A and B. Definition 3.8. For any nonempty subsets A and B of X, we denote A ∧ B := ⟨{a ∧ b | a ∈ A, b ∈ B}⟩ which is called the meet ideal of X generated by A and B. In this case, we say that the operation “∧” is a meet operation. If A = {a}, then {a} ∧ B is denoted by a ∧ B. Also, if B = {b}, then A ∧ {b} is denoted by A ∧ b. Obviously, A ∧ B = B ∧ A for any nonempty subsets A and B of X. If A and B are ideals of X, then A ∧ B = {a ∧ b | a ∈ A, b ∈ B}. Example 3.9. For two subsets A = {2, 3} and B = {1, 4} of X in Example 3.1, the meet ideal of X generated by A and B is A ∧ B = ⟨{0, 1, 2}⟩ = {0, 1, 2, 3}. For any nonempty subsets A, B and C of X, we have A ⊆ B, A ⊆ C ⇒ A ⊆ B ∧ C.

(3.1)

The following example shows that there are subsets A, B and C of X such that A ⊆ B and A ⊆ C, but B ∧ C ⊈ A. Example 3.10. Consider a lower BCK-semilattice X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 1 2 2 2 0 2 0 3 3 3 3 0 3 4 4 4 4 4 0 For subsets A = {0, 1}, B = {0, 1, 2, 3} and C = {0, 1, 2, 4} of X, we have

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B ∧ C = ⟨{b ∧ c | b ∈ B, c ∈ C}⟩ = {0, 1, 2} ⊈ {0, 1} = A. Proposition 3.11. If A, B and C are ideals of X, then A ∧ {0} = {0}.

(3.2)

A ∧ B = A ∩ B.

(3.3)

(A ∧ B) ∧ C = A ∧ (B ∧ C) = {a ∧ b ∧ c | a ∈ A, b ∈ B, c ∈ C}.

(3.4)

Proof. It is clear that A ∧ {0} = {0}. Using (3.1), we have A ∩ B ⊆ A ∧ B. Let x ∈ A ∧ B. Then there exist a ∈ A and b ∈ B such that x = a ∧ b. Since a ∧ b ≤ a and a ∧ b ≤ b, we have x ∈ A ∩ B by (2.3). Hence A ∧ B = A ∩ B. The result (3.4) is straightforward. □ Corollary 3.12. If A, B and C are ideals of X, then the condition (3.1) is valid. By Proposition 3.11, we know that for ideals A1 , A2 , · · · , An of X n ∧ Ai := A1 ∧ A2 ∧ · · · ∧ An i=1

= {a1 ∧ a2 ∧ · · · ∧ an | a1 ∈ A1 , a2 ∈ A2 , · · · , an ∈ An } =

n ∩

(3.5)

Ai .

i=1

For any nonempty subsets A and B of X, denote by A + B the ideal generated by A ∪ B, and is called the plus ideal of A and B. The operation “+” is called the addition. Obviously, A, B ⊆ A + B, A + {0} = A and A + B = B + A. Example 3.13. Consider a lower BCK-semilattice X table. ∗ 0 1 2 3 0 0 0 0 0 1 1 0 0 1 2 2 2 0 2 3 3 3 3 0 4 4 4 4 4 For subsets A = {1, 3} and B = {2} of X, we have

= {0, 1, 2, 3, 4} with the following Cayley 4 0 0 2 3 0

A + B = ⟨A ∪ B⟩ = {0, 1, 2, 3}, which is a plus ideal of X. Proposition 3.14. For any nonempty subsets A and B of X, we have A ∧ B ⊆ A + B. Proof. If x ∈ A ∧ B, then there exists z1 , z2 , · · · , zn ∈ {a ∧ b | a ∈ A, b ∈ B} such that (· · · ((x ∗ z1 ) ∗ z2 ) ∗ · · · ) ∗ zn = 0.

950

(3.6)

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Semiring structures based on meet and plus ideals in lower BCK-semilattices

For each i ∈ {1, 2, · · · , n}, we have zi = ai ∧ bi where ai ∈ A and bi ∈ B. Thus ai ∧ bi ≤ ai ∈ A ⊆ A ∪ B ⊆ A + B, and so zi ∈ A + B for all i ∈ {1, 2, · · · , n}. Since 0 ∈ A + B, it follows from (3.6) and (2.2) that x ∈ A + B. Hence A ∧ B ⊆ A + B. □ Given two nonempty subsets A and B of X, we note that every ideal I of X is represented by the meet ideal based on some A and B, and every ideal J of X is represented by the plus ideal of A and B. But we know that they are different, that is, I ̸= J in general as seen in the following example. Example 3.15. Consider a lower BCK-semilattice X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 0 1 0 2 2 2 0 2 2 3 3 3 3 0 3 4 4 4 4 4 0 For two subsets A = {1} and B = {2, 3} of X, the ideal I = {0, 1} is represented by the meet ideal based on A and B as follows I = ⟨A ∧ B⟩ = ⟨{0, 1}⟩ = {0, 1}. Also the ideal J = {0, 1, 2, 3} is represented by the plus ideal of A and B as follows: J = A + B = ⟨A ∪ B⟩ = ⟨{1, 2, 3}⟩ = {0, 1, 2, 3}. We know that I ̸= J. The following example shows that the reverse inclusion in Proposition 3.14 is not true in general. Example 3.16. Consider a lower BCK-semilattice X = {0, 1, 2, 3, 4} which is given in Example 3.13. For subsets A = {1, 2} and B = {1, 3} of X, we have A ∧ B = ⟨{0, 1}⟩ = {0, 1} and A + B = ⟨{1, 2, 3}⟩ = {0, 1, 2, 3}. Thus A + B ⊈ A ∧ B. For any nonempty subsets A, B and C of X, consider the following condition. A ⊆ C, B ⊆ C ⇒ A + B ⊆ C.

(3.7)

The following example shows that the condition (3.7) is not valid in general.

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Example 3.17. Consider a lower BCK-semilattice X = {0, 1, 2, 3, 4} with the following Cayley table. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 0 1 1 2 2 2 0 2 2 3 3 3 3 0 3 4 4 4 4 4 0 For subsets A = {1, 3}, B = {2, 3} and C = {1, 2, 3} of X, we have A + B = ⟨A ∪ B⟩ = {0, 1, 2, 3} ⊈ C. We provide conditions for the implication (3.7) to be hold. Proposition 3.18. If A and B are nonempty subsets of X and C is an ideal of X, then the implication (3.7) is valid. Proof. Let A and B be subsets of X and C be an ideal of X such that A ⊆ C and B ⊆ C. If x ∈ A + B, then (· · · ((x ∗ z1 ) ∗ z2 ) ∗ · · · ) ∗ zn = 0 (3.8) for some z1 , z2 , · · · , zn ∈ A ∪ B. It follows that zi ∈ C for all i = 1, 2, · · · , n and 0 ∈ C. Since C is an ideal of X, it follows from (3.8) and (2.2) that x ∈ C. Therefore A + B ⊆ C. □ Let A be an ideal of a BCI-algebra X and S be a subset of X with a nilpotent element. Then x ∈ ⟨A ∪ S⟩ if and only if (· · · ((x ∗ s1 ) ∗ s2 ) ∗ · · · ) ∗ sn ∈ A for some s1 , s2 , · · · , sn ∈ S (see [2]). Since every element of a BCK-algebra is nilpotent, we can apply the result above to BCK-algebras as follows. Lemma 3.19. Let A an ideal of a BCK-algebra X. For any subset S of X, we have x ∈ ⟨A ∪ S⟩ if and only if (· · · ((x ∗ s1 ) ∗ s2 ) ∗ · · · ) ∗ sn ∈ A for some s1 , s2 , · · · , sn ∈ S. Lemma 3.20 ([2]). Let X be a commutative BCK-algebra and x, y, z ∈ X. Then (x ∧ y) ∗ (x ∧ z) = (x ∧ y) ∗ z. Theorem 3.21. For any ideals A, B and C of a commutative BCK-algebra X, we have A ∧ (B + C) = (A ∧ B) + (A ∧ C) and (B + C) ∧ A = (B ∧ A) + (C ∧ A). Proof. Note that A ∧ B ⊆ A and A ∧ B ⊆ B ⊆ B + C. It follows from (3.1) that A ∧ B ⊆ A ∧ (B + C). Similarly A ∧ C ⊆ A ∧ (B + C), and thus

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(A ∧ B) + (A ∧ C) ⊆ A ∧ (B + C) by Proposition 3.18. Now let x ∈ A ∧ (B + C). Then x = a ∧ z for some a ∈ A and z ∈ B + C = ⟨B ∪ C⟩. It follows from Lemma 3.19 that there exist c1 , c2 , · · · , cn ∈ C such that (· · · ((z ∗ c1 ) ∗ c2 ) ∗ · · · ) ∗ cn ∈ B.

(3.9)

Note that a ∧ c1 , a ∧ c2 , · · · , a ∧ cn ∈ A ∧ C. Using Lemma 3.20 and (a3) induces ((a ∧ z) ∗ (a ∧ c1 )) ∗ (a ∧ c2 ) = ((a ∧ z) ∗ c1 ) ∗ (a ∧ c2 ) = ((a ∧ z) ∗ (a ∧ c2 )) ∗ c1 = ((a ∧ z) ∗ c2 ) ∗ c1 = ((a ∧ z) ∗ c1 ) ∗ c2 which implies from Lemma 3.20 and (a3) again that (((a ∧ z) ∗ (a ∧ c1 )) ∗ (a ∧ c2 )) ∗ (a ∧ c3 ) = (((a ∧ z) ∗ c1 ) ∗ c2 ) ∗ (a ∧ c3 ) = (((a ∧ z) ∗ (a ∧ c3 )) ∗ c1 ) ∗ c2 = (((a ∧ z) ∗ c3 ) ∗ c1 ) ∗ c2 = (((a ∧ z) ∗ c1 ) ∗ c2 ) ∗ c3 . By the mathematical induction, we conclude that (· · · (((a ∧ z) ∗ (a ∧ c1 )) ∗ (a ∧ c2 )) ∗ · · · ) ∗ (a ∧ cn ) = (· · · (((a ∧ z) ∗ c1 ) ∗ c2 ) ∗ · · · ) ∗ cn .

(3.10)

The inequality a ∧ z ≤ z implies from (a2) that (· · · (((a ∧ z) ∗ c1 ) ∗ c2 ) ∗ · · · ) ∗ cn ≤ (· · · ((z ∗ c1 ) ∗ c2 ) ∗ · · · ) ∗ cn .

(3.11)

Since (· · · ((z ∗ c1 )) ∗ c2 ) ∗ · · · ) ∗ cn ∈ B and B is an ideal, it follows from (2.3) that (· · · (((a ∧ z) ∗ c1 ) ∗ c2 ) ∗ · · · ) ∗ cn ∈ B.

(3.12)

Note that (· · · (((a ∧ z) ∗ c1 ) ∗ c2 ) ∗ · · · ) ∗ cn ≤ a ∧ z ≤ a and a ∈ A, and so (· · · (((a ∧ z) ∗ c1 ) ∗ c2 ) ∗ · · · ) ∗ cn ∈ A.

(3.13)

Combining (3.10), (3.12) and (3.13), we have (· · · (((a ∧ z) ∗ (a ∧ c1 )) ∗ (a ∧ c2 )) ∗ · · · ) ∗ (a ∧ cn ) ∈ A ∧ B.

(3.14)

Since a ∧ c1 , a ∧ c2 , · · · , a ∧ cn ∈ A ∧ C, it follows from Lemma 3.20 that x = a ∧ z ∈ ⟨(A ∧ B) ∪ (A ∧ C)⟩ = (A ∧ B) + (A ∧ C).

(3.15)

Consequently A∧(B+C) = (A∧B)+(A∧C). Similarly we have (B+C)∧A = (B∧A)+(C∧A). □

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Through our discussion above, we make a semiring as follows. Theorem 3.22. Let I(X) be the set of all ideals of a commutative BCK-algebra X. Then (I(X), +, ∧) is a semiring, that is, two operations + and ∧ are associative on I(X) such that (i) addition + is a commutative operation, (ii) there exist {0} ∈ I(X) such that A + {0} = A and A ∧ {0} = {0} ∧ A = {0} for each A ∈ I(X), and (iii) the meet operation ∧ distributes over addition (+) both from the left and from the right. References [1] [2] [3] [4] [5] [6] [7]

M. Aslam and A. B. Thaheem, On certain ideals in BCK-algebras, Math. Japon. 36 (1991), no. 5, 895–906. Y. Huang, BCI-algebra, Science Press, Beijing 2006. K. Iseki, On some ideals in BCK-algebras, Mathematics Seminar Notes 3 (1975), 65–70. K. Iseki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japon. 21 (1976), 451–466. K. Iseki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978), 1–26. J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co., Seoul 1994. M. Palasi´ nski, Ideals in BCK-algebras which are lower semilattices, Polish Acad. Sci. Inst. Philos. Sociol. Bull. Sect. Logic 10 (1981), no. 1, 48–51.

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The solutions of some types of q-shift difference differential equations ∗ Hua Wang Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, P.R. China

Abstract In this paper, we investigate some properties of solutions of some types of q-shift difference differential equations. In addition, we also generalize the Rellich-Wittichtype theorem about differential equations to the case of q-shift difference differential equations. Moreover, we give some example to show the existence and growth of some q-shift difference differential equations. Key words: q-shift; difference differential equation; zero order. Mathematical Subject Classification (2010): 39A 50, 30D 35.

1

Introduction and Some Results

The main purpose of this paper is to investigate some properties of solutions of some q-shift difference differential equations by using Nevanlinna theory in the fields of complex analysis. Thus, we firstly assume that readers are familiar with the basic results and the notations of the Nevanlinna value distribution theory of meromorphic functions such as m(r, f ), N (r, f ), T (r, f ), · · · , (see Hayman [15], Yang [33] and Yi and Yang [34]). For a meromorphic function f , we use S(r, f ) to denote any quantity satisfying S(r, f ) = o(T (r, f )) for all r outside a possible exceptional set of finite logarithmic measure, S(f ) denotes the family of all meromorphic function a(z) such that T (r, a) = S(r, f ) = o(T (r, f )), where r → ∞ outside of a possible exceptional set of finite logarithmic measure. Besides, we use S1 (r, f ) to denote any quantity satisfying S1 (r, f ) = o(T (r, f )) for all r on a set F of logarithmic density 1, the logarithmic density of a set F is defined by Z 1 1 dt. lim sup r→∞ log r [1,r]∩F t For convenience, we claim that the set F of logarithmic density can be not necessarily the same at each occurrence. ∗ The author was supported by the NSF of China (11561033), the Natural Science Foundation of Jiangxi Province in China (20151BAB201008), and the Foundation of Education Department of Jiangxi Province (GJJ150902) of China.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

About forty years ago, F. Rellich, H. Wittich and I. Laine investigated the existence or growth of solutions of some differential equations (see [17, 18, 20, 22]) and obtained the following results. Theorem 1.1 (see [17, Rellich]). Let the differential equation be the following form w0 (z) = f (w),

(1)

If f (w) is transcendental meromorphic function of w, then equation (1) has no nonconstant entire solution. Theorem 1.2 (see [26, Wittich]). Let X Φ(z, w) = a(i) (z)wi0 (w0 )i1 · · · (w(n) )in be differential polynomial, with coefficients a(i) (z) are polynomials of z. If the right-hand side of the differential equation Φ(z, w) = f (w), (2) f (w) is the transcendental meromorphic function of w, then equation (2) has no nonconstant entire solution. Remark 1.1 H. Wittich [26] studied the more general differential equation than equation (1). Later, Yanagihara and Shimomura extended the above type theorem to the case of difference equations (see [25, 31, 32]), and obtained the following two results Theorem 1.3 (see [25, Shimomura]). For any non-constant polynomial P (w), the difference equation w(z + 1) = P (w(z)) has a non-trivial entire solution. Theorem 1.4 (see [31, Yanagihara]). For any non-constant rational function R(w), the difference equation w(z + 1) = R(w(z)) has a non-trivial meromorphic solution in the complex plane. After theirs work, by using Nevanlinna theory in complex difference equations (see [1, 3, 7, 8, 11, 12, 14]), many mathematicians have done a lot of researches in difference equations, difference product and q-difference in the complex plane C, there were a number of articles (including [5, 13, 16, 19, 24, 36]) focused on the existence and growth of solutions of difference equations. In addition, K. Liu, H.Y. Xu and X. G. Qi investigated some properties of complex q-shift difference equations [23, 24, 28]. Inspired by these papers, the purpose of this paper is to study the above Rellich-Wittich-type theorem of q-shift difference differential equation.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Definition 1.1 We call the equation as q-shift difference differential equation if a equation contains the q-shift term f (z +c), q-difference term f (qz) and differential term f 0 (z) of one function f (z) at the same time. We consider the q-shift difference differential equation of the form Ω(z, w) :=

X

aJ (z)

n  Y

w(j) (qj z + cj )ij



= Ps [f (w)],

(3)

j=1

J

where aJ (z) are polynomials of z and qj , cj ∈ C \ {0}, Pm [f ] is a polynomial of f of degree m, Pm [f ] = dm (z)f m + dm−1 (z)f m−1 + · · · + d0 (z), and dm (z), . . . , d0 (z) are polynomials of z, and obtain the following results. Theorem 1.5 For equation (3), if s ≥ 1 and f is a transcendental meromorphic function, then equation (3) has no non-constant transcendental entire solution with zero order. Theorem 1.6 Under the assumptions of Theorem 1.5, the q-shift difference differential equation n X Y Ps [f (w)] , (w(j) (qj z + cj ))ij = aJ (z) Q t [f (w)] j=1 J

has no non-constant transcendental entire solution with zero order, where s ≥ 1, and Ps [f ] and Qt [f ] are irreducible polynomials in f . In 2012, Beardon [4] studied entire solutions of the generalized functional equation f (qz) = qf (z)f 0 (z),

f (0) = 0,

(4)

where q is a non-zero complex number. Beardon [4] obtained the main theorem as follows. Theorem 1.7 [4]. Any transcendental solution f of equation (4) is of the form f (z) = z + z(bz p + · · · ), where p is a positive integer, b 6= 0 and q ∈ Kp . In particular, if q 6∈ K, then the only formal solutions of (4) are O and I, where K, Kp , O and I were stated as in [4]. In 2013, Zhang [35] further the growth of solutions of equation (4) and obtained the following theorem Theorem 1.8 [35, Theorem 1.1]. Suppose that f is a transcendental solution of (4) for q ∈ K, then we have log 2 , ρ(f ) ≤ log |q| where ρ(f ) = lim sup r→+∞

log T (r, f ) , log r

where K is stated as in Theorem 1.7.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Inspired by the ideas of Xu [27, 30] and Beardon [4], we investigate the growth of solutions of some q-shift difference differential equations and obtain the following results. Theorem 1.9 Suppose that f is a solution of f (qz + c) = ηf (z)f 0 (z),

(5)

where q, c, η ∈ C \ {0} and |q| > 1. If f is a transcendental entire function, then we have ρ(f ) ≤

log 2 . log |q|

Furthermore, if f is a polynomial, then f is a polynomial of degree 1, that is, f (z) = a1 z + a0 , where qc q a1 = , a0 = . η η(1 + q) The following example shows that equation (5) had a transcendental entire solution. Example 1.1 Let q = 2, c = 2π and η = 2. Then f (z) = sin z satisfies equation f (2z + 2π) = 2f (z)f 0 (z), and ρ(f ) = 1 =

log 2 . log 2

We also investigate the existence and growth of solutions of equation (5) when the constant η in equation (5) is replaced by a function, and obtain the following result. Theorem 1.10 Let f be a transcendental solution of equation f (qz + c)n = R(z)f (z)[f (j) (z)]s ,

(6)

where q, c, ∈ C and |q| > 1, n, j, s are positive integers and R(z) is rational function in z. If f is an entire function, then n ≤ s + 1 and ρ(f ) ≤

log(s + 1) − log n . log |q|

Furthermore, if n = 1 and f is a meromorphic function with infinitely many poles, then we have log(s + 1) log(sj + s + 1) ≤ µ(f ) ≤ ρ(f ) ≤ . log |q| log |q| The following example shows that equation (6) has transcendental entire and meromorphic solutions. Example 1.2 Let q = 2, c = 2πi, n = 1 and s = 1, then f (z) = zez satisfies system f (2z + 2πi) =

2z + 2πi f (z)f 0 (z). z(z + 1)

and ρ(f ) = 1 ≤

958

log 2 . log 2

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Example 1.3 Let q = 2, c = πi, n = 1 and s = 1, then f (z) = f (2z + 2πi) = and

e2z z2

satisfies equation

z5 f (z)f 0 (z), (2z − 2)(2z + 2πi)2

log 2 log 3 = 1 ≤ µ(f ) = ρ(f ) = 1 ≤ . log 2 log 2

Theorem 1.11 Let f be a transcendental solution of the equation f (qz + c)n = ϕ(z)f (z)[f (j) (z)]s ,

(7)

where q, c, ∈ C and |q| > 1, n, j, s are positive integers and ϕ(z) is a small function with respect of f . If f is a meromorphic function with N (r, f ) = S(r, f ), then n < s + 1 and f satisfies log(s + 1) − log n ρ(f ) ≤ . log |q| Furthermore, if n = 1 and f has infinitely many poles, and the number of distinct common poles of f and ϕ1 is finite, then we have ρ(f ) =

log(s + 1) . log |q|

The following example shows that equation (7) has transcendental meromorphic solution f with the order ρ(f ) = log(s+1) log |q| . √ 2 1 Example 1.4 Let n = j = s = 1 and q = 2, c = 2√ , then f (z) = ez satisfies 2 equation 1 1 z 1 e 8 e f (z)f 0 (z). f (2z + √ ) = 2z 2 2 Thus, ϕ(z) =

1 81 z 2z e e

with T (r, ϕ) = S(r, f ) and the order of f (z) satisfies ρ(f ) = 2 =

2

log 2 − log 1 . 1 2 log 2

Some Lemmas

Lemma 2.1 (Valiron-Mohon’ko). [18] Let f (z) be a meromorphic function. Then for all irreducible rational functions in f , Pm ai (z)f (z)i R(z, f (z)) = Pni=0 , j j=0 bj (z)f (z) with meromorphic coefficients ai (z), bj (z), the characteristic function of R(z, f (z)) satisfies T (r, R(z, f (z))) = dT (r, f ) + O(Ψ(r)), where d = max{m, n} and Ψ(r) = maxi,j {T (r, ai ), T (r, bj )}.

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Lemma 2.2 (see [23]). Let f (z) be a nonconstant zero-order meromorphic function and q ∈ C \ {0}. Then   f (qz + η) m r, = S1 (r, f ). f (z) Lemma 2.3 (see [28]). Let f (z) be a transcendental meromorphic function of zero order and q, η be two nonzero complex constants. Then T (r, f (qz + η)) = T (r, f (z)) + S1 (r, f ), N (r, f (qz + η)) ≤ N (r, f ) + S1 (r, f ). Lemma 2.4 (see [34, p.37] or [33]). Let f (z) be a nonconstant meromorphic function in the complex plane and l be a positive integer. Then N (r, f (l) ) = N (r, f ) + lN (r, f ), T (r, f (l) ) ≤ T (r, f ) + lN (r, f ) + S(r, f ). Lemma 2.5 Let q, c ∈ C \ {0} and f (z) be a nonconstant meromorphic function with zero order. Then for any positive finite integer k, we have   f (k) (qz + c) = S1 (r, f ), m r, f (z) and

  m r, f (k) (qz + c) ≤ m(r, f ) + S1 (r, f ).

Proof: It follows from Lemma 2.2 that       f (k) (qz + c) f (k) (qz + c) f (qz + c) m r, ≤ m r, + m r, = S1 (r, f ). f (z) f (qz + c) f (z) Moreover, we have     f (k) (qz + c) m r, f (k) (qz + c) = m r, f (z) ≤ m(r, f ) + S1 (r, f ). f (z) This completes the proof of Lemma 2.5.

2

Lemma 2.6 (see [11]). Let Φ : (1, ∞) → (0, ∞) be a monotone increasing function, and let f be a nonconstant meromorphic function. If for some real constant α ∈ (0, 1), there exist real constants K1 > 0 and K2 ≥ 1 such that T (r, f ) ≤ K1 Φ(αr) + K2 T (αr, f ) + S(αr, f ), then the order of growth of f satisfies ρ(f ) ≤

log K2 log Φ(r) + lim sup . − log α log r r→+∞

Lemma 2.7 (see [9]). Let f (z) be a transcendental meromorphic function and p(z) = pk z k + pk−1 z k−1 + · · · + p1 z + p0 be a complex polynomial of degree k > 0. For given 0 < δ < |pk |, let λ = |pk | + δ, µ = |pk | − δ, then for given ε > 0 and for r large enough, (1 − ε)T (µrk , f ) ≤ T (r, f ◦ p) ≤ (1 + ε)T (λrk , f ).

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Lemma 2.8 (see [2, 10] or [6]). Let g : (0, +∞) → R, h : (0, +∞) → R be monotone increasing functions such that g(r) ≤ h(r) outside of an exceptional set E with finite linear measure, or g(r) ≤ h(r), r 6∈ H ∪ (0, 1], where H ⊂ (1, ∞) is a set of finite logarithmic measure. Then, for any α > 1, there exists r0 such that g(r) ≤ h(αr) for all r ≥ r0 .

3

Proofs of Theorems 1.5 and 1.6

3.1

The proof of Theorem 1.5

Suppose that w be non-constant entire solution of equation (3) with zero order. Let E1 = {z : |w(z)| > 1} and E2 = {z : |w(z)| ≤ 1}, then we have  0 i1  0 in1 X w (q z + c ) w (q z + c ) 1 1 n n 1 1 aJ (z)(w(z))λi ··· |Ω(z, w)| = w(z) w(z) J  0 0 P w (qn1 z+cn1 ) in1  w (q1 z+c1 ) i1  |w(z)|λ if z ∈ E1 , ··· , J |aJ (z)| w(z) w(z) ≤ i i 0 1 n P w (qn1 z+cn1 ) 1 w0 (q1 z+c1 )   if z ∈ E2 , ··· , J |aJ (z)| w(z) w(z) where λ = max{λi }, λi = i1 + · · · + in1 . It follows from Lemma 2.2 and Lemma 2.5 that Z Z  1 m(r, Ω(z, w)) = + log+ |Ω(z, w)|dθ ≤ λm(r, w) + S1 (r, w). 2π E2 E1 And since w(z) is a non-constant entire function, we have N (r, w) = 0. Thus, we have N (r, Ω(z, w)) = 0 and T (r, Ω) = m(r, Ω) ≤ λm(r, w) + S1 (r, w) = λT (r, w) + S1 (r, w).

(8)

Since Ps [f (w)] is a polynomial of f (w), we can take a complex constant α such that Ps [f (w)] − α = [f (w) − α1 ] · · · [f (w) − αs ], where α1 , . . . , αs are complex constants, and there at least exists a constant β ∈ {α1 , . . . , αs }, which is not a Picard exceptional value of f (w). Let ξj , j = 1, 2, . . . , p be the zeros of f (w) − β, where p is an any positive integer with p ≥ 1. Then it follows p X j=1

N (r,

1 1 1 ) ≤ N (r, ) ≤ N (r, ). w − ξj f (w) − β Ps [f (w)] − α

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Thus, by using the second main theorem and (8), (9), we can get that (p − 2)T (r, w) ≤

p X

N (r,

j=1

1 ) + S(r, w) w − ξj

1 ) + S(r, w) Ps [f (w)] − α ≤ T (r, Ps [f (w)]) + S(r, w) ≤ T (r, Ω(z, w)) + S(r, w) ≤ λT (r, w) + S1 (r, w).

≤ N (r,

(10)

It follows from (8) and (10) that (p − 2 − λ)T (r, w) ≤ S1 (r, w).

(11)

Since w is transcendental and p is arbitrary, we can get a contradiction with (11). Hence, we complete the proof of Theorem 1.5.

3.2

The proof of Theorem 1.6

By using the same argument as in Theorem 1.5, and applying Lemma 2.1, we can prove the conclusion of Theorem 1.6 easily.

4

The proof of Theorem 1.9

Suppose that f is a solution of (5). If f is a polynomial of degree m ≥ 1, let f (z) = am z m + am−1 z m−1 + · · · + a0 , where am , . . . , a0 are complex constants. From (5), we have am (qz + c)m + am−1 (qz + c)m−1 + · · · + a0 =η(am z m + am−1 z m−1 + · · · + a0 )[mam z m−1 + (m − 1)am−1 z m−1 + · · · + a1 ].

(12)

By computing the degree of two sides in z in (12), we can get that m = 2m − 1, that is, m = 1. Thus, f (z) can be rewritten as f (z) = a1 z + a0 . It follows a1 (qz + c) + a0 = η(a1 z + a0 )a1 , that is, a1 q = ηa21 , q η,

a1 c + a0 = ηa1 a0 .

qc η(1+q) .

a0 = Thus, we have a1 = If f is a transcendental entire function, from Lemma 2.4, we have T (r, f (qz + c)) ≤ 2T (r, f ) + S(r, f ) ≤ 2(1 + ε)T (βr, f ),

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for sufficiently large r and any given β > 1, ε > 0. By Lemma 2.7 and (13), for θ = |q| − δ(0 < δ < |q|, 0 < θ < 1), i = 1, 2 and sufficiently larger r, we get (1 − ε)T (θr, f ) ≤ 2(1 + ε)T (βr, f ), outside of a possible exceptional set E of finite linear measure. From Lemma 2.8, for any given γ > 1 and sufficiently large r, we obtain (1 − ε)T (θr, f ) ≤ 2(1 + ε)T (γβr, f ).

(14)

that is, (1 − ε) T (r, f ) ≤ T 2(1 + ε)



βγ r, f θ

 .

(15)

Since |q| > 1, we can choose δ > 0 such that θ > 1, and let ε → 0, δ → 0, β → 1, γ → 1, and for sufficiently large r, by Lemma 2.6, we have ρ(f ) ≤

log 2 . log |q|

Thus, this completes the proof of Theorem 1.9.

5 5.1

Proofs of Theorems 1.10 and 1.11 The Proof of Theorem 1.10

Since R(z) is a rational function, then we have T (r, R(z)) = O(log r). If f is a transcendental entire function, similar to the argument as in Theorem 1.9, we can get n ρ(f ) ≤ log(s+1)−log easily. log |q| If f is a meromorphic function, by Lemma 2.1 and Lemma 2.4, it follows from (6) that sj + s + 1 T (r, f (qz + c)) ≤ T (r, f (z)) + S(r, f ). n Since |q| > 1, by Lemma 2.7 and using the same argument as in Theorem 1.9, we have n ρ(f ) ≤ log(sj+s+1)−log . log |q| Suppose that n = 1. Since R(z) is a rational function, we can choose a sufficiently large constant R(> 0) such that R(z) has no zeros or poles in {z ∈ C : |z| > R}. Since f has infinitely many poles, we can choose a pole z0 of f of multiplicity τ ≥ 1 satisfying |z0 | > R. Thus, it follows that the right side of the equation (6) has a pole of multiplicity τ1 = (s + 1)τ + sj at z0 , and f has a pole of multiplicity τ1 at qz0 + c. Replacing z by qz0 + c in equation (6), we have that f has a pole of multiplicity τ2 = (s + 1)τ1 + sj at q 2 z0 + qc + c. We proceed to follow the step above. Since R(z) has no zeros or poles in {z ∈ C : |z| > R} and f has infinitely many poles again, we may construct poles ζk = q k z0 + q k−1 c + · · · + c,k ∈ N+ of f of multiplicity τk satisfying τk = (s + 1)τk−1 + sj = (s + 1)k τ + sj[(s + 1)k−1 + · · · + 1],

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as k → ∞, k ∈ N. Since |q| > 1, then |ζk | → ∞ as k → ∞, for sufficiently large k, we have τ (s + 1)k ≤ (τ + j)(s + 1)k − j = τk ≤ τ + τ1 + · · · + τk ≤ n(|ζk |, f ) k

k−1

≤ n(|q| |z0 | + |C|(|q|

(16)

+ · · · + |q| + 1), f ).

Thus, for each sufficiently large r, there exists a k ∈ N+ such that r ∈ [|q|k |z0 | + |C|

k−1 X

|q|i , |q|(k+1) |z0 | + |C|

i=0

k X

|q|i ),

i=0

that is, k>

log r − log(|z0 | +

|c| |q|−1 )

− log

|c| |q|−1

− log |q| .

log |q|

(17)

Thus, it follows from (17) that log r

n(r, f ) ≥ τ (s + 1)k ≥ K1 (s + 1) log |q| , where − log(|z0 |+

K1 = τ (s + 1)

|c| |c| )−log −log |q| |q|−1 |q|−1 log |q|

(18)

.

Since for all r ≥ r0 , log r

K1 (s + 1) log |q| ≤ n(r, f ) ≤

1 1 N (2r, f ) ≤ T (2r, f ), log 2 log 2

it follows from (18) that ρ(f ) ≥ µ(f ) ≥

log(s + 1) . log |q|

Thus, this completes the proof of Theorem 1.10.

5.2

The proof of Theorem 1.11

By using the same argument as in Theorem 1.10, we can prove the conclusion of Theorem 1.11 easily.

Competing interests The authors declare that they have no competing interests.

Author’s contributions HW and HYX completed the main part of this article. All authors read and approved the final manuscript.

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References [1] L. J. Ai, C. F. Yi, The growth for solutions of a class of higher order linear differential equations with meromorphic coefficients, J. Jiangxi Norm. Univ. Nat. Sci. 38 (3) (2014), 250-253. [2] S. Bank, A general theorem concerning the growth of solutions of first-order algebraic differential equations, Compositio Math. 25 (1972), 61-70. [3] D. C. Barnett, R. G. Halburd, R. J. Korhonen and W. Morgan, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edin. Sect. A Math. 137 (2007), 457-474. [4] A.F. Beardon, Entire solutions of f (kz) = kf (z)f 0 (z), Comput. Methods Funct. Theory 12(1) (2012), 273-278. [5] Z. X. Chen, Value distribution of meromorphic solutions of certain difference Painlev equations, J. Math. Anal. Appl. 364 (2010), 556-566. [6] Z. X. Chen, Z. B. Huang, X. M. Zheng, On properties of difference polynomials, Acta Math. Scientia 31B(2) (2011): 627-633. [7] Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of f (z + η) and difference equations in the complex plane, Ramanujan J. 16 (2008), 105-129. [8] H. L. Gan, The zeros and fixed points of difference of entire functions, J. Jiangxi Norm. Univ. Nat. Sci. 38 (5) (2015), 519-521. [9] R. Goldstein, Some results on factorization of meromorphic functions, J London Math. Soc. 4(2) (1971): 357-364. [10] G. G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), 415-429. [11] G. G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo and D. Yang, Meromorphic solutions of generalized Schr¨ oder equations, Aequationes Math. 63 (2002), 110-135. [12] R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), 477487. [13] R. G. Halburd and R. J. Korhonen, Finite order solutions and the discrete Painlev´e equations, Proc. London Math. Soc. 94 (2007), 443-474. [14] R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), 463-478. [15] W. K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964. [16] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and K. Hohge, Complex difference equations of Malmquist type, Comput. Methods Funct. Theory 1 (2001): 27C39. [17] Y. Z. He, On the algebroid function solutions of differential equations, Acta Mathematica Sinica, 24 (1981), 464-471. [18] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993. [19] I. Laine and C. C. Yang, Clunie theorems for difference and q-difference polynomials, J. London Math. Soc. 76 (2) (2007), 556-566. [20] I. Laine, On the behaviour of the solutions of some first order differential equations, Ann. Acad. Sci. Fenn., 1971. [21] H. C. Li, L. Y. Gao, Meromorphic solutions of a type of system of complex differentialdifference equations, Acta Math. Sic. 35B (1) (2015), 195-206. [22] L. W. Liao, The new developments in the research of nonlinear complex differential equations, J. Jiangxi Norm. Univ. Nat. Sci. 39 (2015), 331C339.

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[23] K. Liu and X. G. Qi, Meromorphic solutions of q-shift difference equations, Ann. Polon. Math. 101 (2011), 215-225. [24] X. G. Qi and L. Z. Yang, Properties of meromorphic solutions of q-difference equations, Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 59, pp. 1-9. [25] S. Shimomura, Entire solutins of a polynomial difference equation, J Fac Sci Univ Tokyo Sect IA Math, 28 (1981), 253-266. [26] H. Wittich, Neuere Untersuchungen u ¨ber eindeutige analytische Funktionen, Springer, Berlin-G¨ ottingen-Heidelberg, 1955. [27] H. Y. Xu, B. X. Liu, K. Z. Tang, Some properties of meromorphic solutions of systems of complex q-shift difference equations, Abstract and Applied Analysis, 2013 (2013), Art. 680956, 6 pages. [28] H. Y. Xu and J. Tu,Some properties of meromorphic solutions of q-shift difference equations, Journal of Mathematical Study 45 (2012), 124-132. [29] H. Y. Xu, J. L. Wang, H. Wang, The existence of meromorphic solutions of some types of systems of complex functional equations, Discrete Dynamics in Nature and Society, 2015 (2015), Art. 723025, 10 pages. [30] H. Y. Xu, Z. X. Xuan, Growth of the solutions of some q-difference differential equations, Advancesin Difference Equations 2015 (2015): 172. [31] N. Yanagihara, Meromorphic solutionas of some difference equations, Funkcialaj Ekvacioj, 23 (1980), 309-326. [32] N. Yanagihara, Meromorphic solutionas of some difference equations of the nth order, Arch Ration Mech Anal, 91 (1983), 19-192. [33] L. Yang, Value distribution theory, Springer-Verlag. Berlin(1993). [34] H. X. Yi and C. C. Yang, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, Dordrecht, 2003; Chinese original: Science Press, Beijing, 1995. [35] G. W. Zhang, On a question of Beardon, Journal of Inequalities and Applications 2013, (2013), Art. 331, 1-6. [36] J. L. Zhang, R. J. Korhonen, On the Nevanlinna characteristic of f (qz) and its applications, J. Math. Anal. Appl. 369 (2010), 537-544.

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Numerical method for solving inequality constrained matrix operator minimization problemI Jiao-fen Lia , Tao Lia , Xue-lin Zhou∗,b , Xiao-fan Lva a

School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, P.R. China b Academic Affairs Office, Guilin University of Electronic Technology, Guilin 541004, P.R. China

Abstract In this paper, we considered a matrix inequality constrained linear matrix operator minimization problems with a particular structure, some of whose reduced versions can be applicable to image restoration. We present an efficient iteration method to solve this problem. The approach belongs to the category of PowellHestense-Rockafellar augmented Lagrangian method, and combines a nonmonotone projected gradient type method to minimize the augmented Lagrangian function at each iteration. Several propositions and one theorem on the convergence of the proposed algorithm were established. Numerical experiments are performed to illustrate the feasibility and efficiency of the proposed algorithm, including when the algorithm is tested with randomly generated data and on image restoration problems with some special symmetry pattern images. Key words: matrix equation, matrix minimization problem, matrix inequality, augmented lagrangian method, image restoration. 2000 MSC: 65F30, 65H15, 15A24

1. Introduction Let m, n, l1 , s1 , l2 , s2 be positive integers. Let A(X; A1 , · · · , A p ) be a linear mapping from Rm×n onto R and G(X; E1 , · · · , Eq ) be a linear mapping from Rm×n onto Rl2 ×s2 , where Ai (i = 1, . . . , p) and E j ( j = 1, . . . , q) with suitable sizes are the parameter matrices. In this paper we are interested in solving the following constrained matrix minimization problem

2  1

 minimize

A X; A1 , · · · , A p − C

2 (1.1) subject to  X∈S  L ≤ G X; E1 , · · · , Eq ≤ U. l1 ×s1

where k · k denotes the Frobenius norm, the symbol ≥ means nonnegative, the set S ⊆ Rm×n shows the constraint, C ∈ Rl1 ×s1 and L, U ∈ Rl2 ×s2 are given matrices. In general, S ⊆ Rm×n is a linear space I

Research supported by National Natural Science Foundation of China(11301107,11261014,11561015,51268006), Natural Science Foundation of Guangxi Province (2016GXNSFAA380011,2016GXNSFFA380003). ∗ Corresponding author. Email addresses: [email protected] (Jiao-fen Li), [email protected] (Xue-lin Zhou) Preprint submitted to Journal of Computational Analysis and Applications September 23, 2016

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possessing special structures, such as symmetry/skew-symmetry, centrosymmetry/centro skew-symmetry, mirror-symmetry/mirror-skew-symmetry, P-commuting symmetry/ skew-symmetry with respect to a given symmetric matrix P, Toeplitz matrix and so on. It is obvious that the linear operator equation in (1.1) is quite general and includes several linear matrix equations such as the Lyapunov and Sylvester matrix equations which are shown in Table 1. For an instant, the Lyapunov matrix equation AT1 XA2 + AT2 XA1 = −C is equivalent to the linear operator equation in (1.1), if we define the operator A as: A : X → AT1 XA2 + AT2 XA1 . Table 1: One-sided and two-sided Lyapunov and Sylvester matrix equations.

Name Continuous-time (CT) Lyapunov Generalized continuous-time (CT) Lyapunov Generalized discrete-time (CT) Lyapunov Continuous-time (CT) Sylvester Discrete-time (DT) Sylvester Generalized Sylvester

Matrix equation A1 X + XAT1 + BBT = 0 AT1 XA2 + AT2 XA1 = −C AT1 XA1 + AT2 XA2 = −C A1 X + XA2 = C A1 XAT2 + X = C A1 XAT2 + A3 XAT4 = C

Throughout we always assume that the matrix operator inequality in model (1.1) is consistent with these given matrices E j , L, U and unknown X ∈ S, then we known that the solution set of Problem (1.1) is nonempty. The interest that we have in this problem stems from the following reasons. Firstly, by using the vec operator vec(.) and the Kronecker produc ⊗, the model (1.1) can be equivalently rewritten as the convex linearly constrained quadratic programming(LCQP) in the vector-form 1 T x Qx + gT x + c 2 subject to l ≤ Gx ≤ u,

(1.2)

1 Q = PT M T MP, g = −PT M T vec(C), c = vec(C)T vec(C) 2

(1.3)

minimize f (x) =

where

and Px = vec(X), l = vec(L), u = vec(U).

(1.4) p {Ai }i=1

q {E j } j=1

The matrices M and G are the Kronecker product of the parameter matrices and which satisfies vec(A(X; A1 , . . . , A p )) = Mvec(X) and vec(G(X; E1 , . . . , Eq )) = Gvec(X), respectively. Specifically, in (1.3)-(1.4), P is the matrix that characterizes the elements X ∈ S by vec(X) = Px in terms of its independent parameter vector x of X[18]. In theory, the model (1.2) can be solved by some classical optimization methods, such as interior point method, active set method, trust region method, Newton method, and other available methods. In particular, Delbos F. in [2] considered the vector LCQP(1.2) by using an augmented Lagrangian method and given a global linear convergence of the proposed algorithm. However, using this transformation will on the one hand destroy the original structure of the unknown matrix X ∈ S if the linear 2

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subspace S has some special symmetrical structure. On the other hand, using this transformation will result in a coefficient matrix in large scale, and then increase computational complexity and storage requirement. Indeed, taking l = m = n = s = p = q = 200 in (1.1), then the matrices Q and G in the transformed model (1.2) have sizes of about 40000 × 40000. For these reasons, it cannot be a practicable method for solving Problem (1.1) by the vec operator and the Kronecker produc if the system scale is large. In this paper we will consider directly from the perspective of matrices. Secondly, various simplified versions of Problem (1.1) have been studied extensively. If we drop the matrix inequality constraint, then Problem (1.1) is reduced to the minimization problem with special structures. Methods proposed for solving such problems can be broadly classified into two classes, including factorization techniques for small size problems, based on the special structure of the linear subspace S that produce a low-dimensional problems that are then solved using direct methods[3, 4, 5, 6, 7, 8, 9, 10, 11], and iterative schemes, for large-scale problems, based on Krylov subspace-type methods, such as the wellknown Jacobi and Gauss-Seidel iterations[12, 13], the conjugate gradient-type methods[14, 15] and the least squares QR(LSQR) methods[16, 17, 18] and so on. On the other hand, if we simplify the general matrix inequality constraint in (1.1) into the nonnegative constraint X ≥ 0 or the bound constraint L ≤ X ≤ U, then the similar problem has been studied with Dykstra’s alternating projection algorithm[19, 20] and spectral projection gradient method[21]. In particular, Problem (1.1) can be regarded as a natural generalization of the problems in [21, 22, 23]. The authors in [21] considered the following constrained minimization problem q

X

2 Minimize

subject to X ∈ Ω = {X ∈ Rm×n : L ≤ X ≤ U}. (1.5) Ai XBi − C

i=1

They propose a globalized variants projected gradient method and apply the left and right preconditioning strategies to solve (1.5). While the authors in [22, 23] devoted to solve the matrix equation AX = B or minimize kAX − Bk with special structures under the constraint CXD ≥ E, respectively. The problems considered in [22] and [23] can be transformed into least nonnegative correction problems based on the fact that close-form optimal solutions of AX = B or minimizing kAX − Bk with special structures can be readily derived, and then some fixed point-like algorithms can be applied to solve these transformed problems. However, all these previous ideas show difficulties when dealing with the Problem (1.1), due to the generalization of the objection function and the matrix operator inequality, so that either the projection onto the set {X ∈ Rm×n |L ≤ G(X) ≤ U} is not available, or a close-form optimal solution of minimizing the objection function in (1.1) with X ∈ S is not tractable. Thirdly, we consider the application of the model (1.1) in image restoration. In fact, the authors in [21, 24] consider the problem of image restoration, combined with a Tikhonov regularization term, as a convex constrained minimization problem by use a Kronecker decomposition of the blurring matrix and the Tikhonov regularization matrix. And then they propose and show the effectiveness of their approaches, a globalized variants projected gradient method [21] and a conditional gradient-type method[21], to restore some blurred and highly noisy images. However, in this paper, we are only concerned with the restoration problems with some special symmetric pattern images, which have not yet studied in [21, 24]. Moreover, to the best of our knowledge, this class of image restoration problems have received little attention in the other literature. The main difficult is due to the fact that the restore image should preserve the same special symmetric structure with the original images. In this paper we undertake some significant attempts in this field. In this paper, we will propose and study an algorithm in the framework of the classic Powell-HestenesRockafellar augmented Lagrangian method, first suggested by Hestenes [25] and Powell [26], and developed by E.G. Birgin [27, 28] for solving Problem (1.1). The classic PHR-AL method is a fundamental and 3

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effective approach in inequality-constrained optimization. The algorithm effectively combines a nonmonotone projected gradient type method to minimize the augmented Lagrangian function at each iteration. We will give several propositions and one theorem on the convergence of the proposed algorithm, and apply it to solving Problem (1.1) with randomly generated data and comparing it with existing methods. We also apply our approach, combined with a Tikhonov regularization term, to restore some blurred and highly noisy symmetric pattern images. Throughout this paper, we use the following notations. Let ei be the ith column of the identity matrix Ik and S k = (ek , ek−1 , . . . , e1 ), i.e., the kth backward identity matrix. Let 0 be the zero matrix of suitable size and PS be the Euclidean projection onto set S.We write εk ↓ 0 to indicate that εk is a (not necessarily decreasing) sequence of non-negative numbers that tends to zero. We denote N = {0, 1, 2, . . .}. For A = (ai j ) ∈ Rm×n , A+ (or A− ) be the matrix with the (i, j)-entry equals to max{0, ai j }(or min{0, ai j }), respectively. For A, B ∈ Rm×n , {A, B}− denotes a matrix with the i jth entry being equal to min{ai j , bi j }, hA, Bi = trace(BT A) denotes the inner product of matrices A and B. Then Rm×n is a Hilbert inner product space and the norm generated is the Frobenius norm k · k. For any linear operator L form Rm×n onto Rl1 ×s1 , there is another operator called the adjoint of L, written LT : Rl1 ×s1 → Rm×n . What defines the adjoint is that for any two matrices X ∈ Rm×n and Y ∈ Rl1 ×s1 , hL(X), Yi = hX, LT (Y)i. The rest of this paper is organized as follows. In section 2, we will briefly characterize the application of model (1.1) in image restoration. Based on the classic augmented Lagrangian method, in section 3 we propose, analyze and test an algorithm for solving the inequality-constrained matrix minimization problem (1.1). Some numerical results are reported in section 4 to verify the efficiency of the proposed algorithm. Numerical tests on the proposed algorithm applied to some special image restoration problems are also reported in this section. 2. The application of model (1.1) in image restoration For completeness, in this section we briefly characterize how to apply the model (1.1) into image restoration and we refer to [21, 24] for detailed description. Consider solving the following model in image restoration with Tikhonov regularization: 1 λ2 kHx − gk22 + kT xk22 , l≤x≤u 2 2 min

(2.6)

where k · k2 is the 2-norm. In image restoration, H will be the blurring operator, g the observed image, T the regularization operator, λ the regularization parameter, and x the restored image to be sought. The constraints represent the dynamic range of the image. The minimizer of (2.6) can be computed by the following linear system Hλ x = H T g,

where Hλ = H T H + λ2 T T T.

(2.7)

In some practical problems in image restoration, often the system (2.7) may not be consistent due to measurement errors in the data matrices, and hence it is useful to consider the following minimization problem with constraints

2 1 min

Hλ x − H T g

2 . (2.8) l≤x≤u 2 4

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Here we assume that the matrices H and T can be separated as Kronecker product of matrices with a smaller size, i.e., H = H1 ⊗ H2 and T = T 1 ⊗ T 2 . In the case of nonseparable, one can still obtain an approximation solution of H1 and H2 by solving the Kronecker product approximation problem (KPA) of the form (H1 , H2 ) = argminHˆ 1 ,Hˆ 2 kH − Hˆ 1 ⊗ Hˆ 2 k[29]. Then, (2.8) can be written as min

L≤X≤U

2 o 1

n T T T T T 2

(H1 H1 ) ⊗ (H2 H2 ) + λ (T 1 T 1 ) ⊗ (T 2 T 2 ) vec(X) − (H1 ⊗ H2 ) vec(G)

, 2

(2.9)

where X, G, L and U are the matrices such that vec(X) = x, vec(G) = g, vec(L) = l and vec(U) = u. If some special symmetry pattern images are considered, by using some properties of the Kronecker product, (2.9) is then written as

2 1

A1 XB1 + λ2 A2 XB2 − C

min (2.10) 2 subject to L ≤ X ≤ U, X ∈ S, with A1 = H2 T H2 , B1 = H1 T H1 , A2 = T 2 T T 2 , B2 = T 1 T T 1 and C = H2 T GH1 and S is the matrix set whose elements have the same symmetry structure with the original images. The parameter λ in (2.10) is a scalar need to be determined, and its optimal value can be obtained by the classical Generalized cross-validation (GCV) method[21, 24], which is chosen to minimize the GCV function defined by GCV(λ) =

kH xˆλ − gk22 {trace(I − HHλ−1 H T )}2

=

k(I − HHλ−1 H T )gk22 {trace(I − HHλ−1 H T )}2

,

where Hλ = H T H + λ2 T T T . Then, the method proposed for solving Problem (1.1) could be applied directly to the model (2.10) by considering the linear matrix operators A(X) = A1 XB1 + λ2 A2 XB2 and G(X) = X. 3. Augmented Lagrangian method for solving Problem (1.1) In this section we propose a matrix-form iteration method, in the framework of the classic PowellHestense-Rockafellar augmented Lagrangian(PHR-AL) method, to compute the solution of Problem (1.1). We then prove some convergence results for the proposed algorithm at the end of this section. For convenience, the two linear matrix operators will be simply denote by A(X) and G(X) in the following discussion. Lemma 1. Assume x∗ is a local minimizer of the quadratic program mins f (x) = x∈R

1 T x Mx + gT x + c subject to Gx ≥ b, 2

then there exists a vector y∗ such that Mx∗ + g − GT y∗ = 0, Gx∗ ≥ b, hy∗ , Gx∗ − bi = 0, y∗ ≥ 0. Theorem 1. Matrix X ∗ ∈ Rm×n is a solution of Problem (1.1) if and only if there exists nonnegative matrices Y1∗ , Y2∗ ∈ Rl2 ×s2 such that the following conditions are satisfied: n o 
 PS AT A(X ∗ ) − C − GT (Y1∗ − Y2∗ ) = 0       G(X ∗ ) − L ≥ 0    (3.11)  U − G(X ∗ ) ≥ 0    ∗ ∗   hY , G(X ) − Li = 0     hY1∗ , U − G(X ∗ )i = 0. 2

5

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Proof. Assume that there are nonnegative matrices Y1∗ , Y2∗ ∈ Rl2 ×s2 such that the conditions (3.11) are satisfied. Let

2 1   f (X) =

A X − C

2 and e f (X) = f (X) + hY1∗ , L − G(X)i + hY2∗ , G(X) − Ui. e ∈ S, we have Then for any W e e f (X ∗ + W) E E D D 1 e −U e + Y ∗ , G(X ∗ + W) e − Ck2 + Y ∗ , L − G(X ∗ + W) = 2 kA(X ∗ + W) 2 1 E D E D e e 2 + A(W), e A(X ∗ ) − C − Y ∗ − Y ∗ , G(W) = e f (X ∗ ) + 12 kA(W)k 2 1 E D e AT (A(X ∗ ) − C) − GT (Y ∗ − Y ∗ ) e 2 + W, = e f (X ∗ ) + 21 kA(W)k 2 1 E  D e 2 + 1 W, e PS AT (A(X ∗ ) − C) − GT (Y ∗ − Y ∗ ) = e f (X ∗ ) + 12 kA(W)k 2 2 1 e 2 = e f (X ∗ ) + 21 kA(W)k ∗ ≥ e f (X ). This implies that X ∗ is a global minimizer of the function e f (X). Since hY1∗ , G(X ∗ )−Li = 0, hY2∗ , U −G(X ∗ )i = ∗ 0 and e f (X) ≥ e f (X ) for all X ∈ S, we have D E D E D E D E f (X) ≥ f (X ∗ ) + Y1∗ , L − G(X ∗ ) + Y2∗ , G(X ∗ ) − U − Y1∗ , L − G(X) − Y2∗ , G(X) − U D E D E = f (X ∗ ) − Y1∗ , L − G(X) − Y2∗ , G(X) − U . Hence, we have from Y1∗ ≥ 0 and Y2∗ ≥ 0 that f (X) ≥ f (X ∗ ) for all X ∈ S with G(X) − L ≥ 0 and U − G(X) ≥ 0. Hence X ∗ is a solution to Problem (1.1). Conversely, assuming that X ∗ is a solution to Problem (1.1), then X ∗ certainly satisfies the KarushKuhn-Tucker conditions of Problem (1.1). That is, there exists a nonnegative matrix Y ∗ such that satisfies conditions (3.11). We now define the following Powell-Hestenes-Rockafellar(PHR) Augmented Lagrangian function 1 ρ

 Z1 

2 ρ

 Z2 

2 Lρ (X, Z1 , Z2 ) = kA(X) − Ck2 +

L − G(X) + + G(X) − U + (3.12)

, 2 2 ρ + 2 ρ + where Z1 ≥ 0 and Z2 ≥ 0 are the Lagrangian multiplier matrices and ρ > 0 is the penalty parameter. Clearly, the partial derivative of function Lρ (X, Z1 , Z2 ) with respect to X is given by !     Z1  Z2  T T ∇X Lρ (X, Z1 , Z2 ) = A A(X) − C − ρG L − G(X) + − G(X) − U + . ρ + ρ + The augmented Lagrangian method proposed by E.G. Birgin et al in in [27, 28] (with necessary modifications) to solve Problem (1.1) can be described as follows: Algorithm PHR-AL. (The PHR-AL method for solving Problem (1.1).) 1. Input coefficient matrices Ai , Bi (i = 1, . . . p) in the linear operator A and matrices Ei , E j (i = 1, . . . q) in the linear operator G. Input matrices C, L, U and a large parameter matrix Zmax > 0. Input γ > 1, 1 1 r ∈ (0, 1), ρ1 > 0, a small tolerance ε > 0 and tolerance εk ↓ 0. Choose initial matrices Z 1 and Z 2 1 1 with 0 ≤ Z 1 , Z 2 ≤ Zmax . Set k ← 1. 6

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2. Compute X k as an approximate stationary point of k

k

minimize Lρk (X, Z 1 , Z 2 )

(3.13)

subject to X ∈ S.

n

k k o That is, compute X k such that

PS ∇Lρk (X k , Z 1 , Z 2 )

< εk . 3. Define  k    k Z1k = Z 1 + ρk (L − G(X k )) , Z2k = Z 2 + ρk (G(X k ) − U) . +

+

4. If k = 1 or  

{G(X k ) − L, Z k }

2 +

{U − G(X k ), Z k }

2 1/2 − − 2 1 

2 1/2

2 , ≤ r

{G(X k−1 ) − L, Z1k−1 }−

+

{U − G(X k−1 ), Z2k−1 }−

(3.14)

define ρk+1 = ρk . Else, define ρk+1 = γρk . 5. If 

n o 2



1/2

PS ∇Lρ (X k , Z k , Z k )

+

G(X k ) − L, Z k

2 +

U − G(X k ), Z k

2 < ε, 1 2 k 1 − 2 − then stop. k+1

k+1

k+1

k+1

k+1

k+1

6. Update Z 1 and Z 2 with 0 ≤ Z 1 , Z 2 ≤ Zmax in such a way that (Z 1 )i j = (Z1k )i j and (Z 2 )i j = (Z2k )i j if 0 ≤ (Z1k )i j , (Z2k )i j ≤ (Zmax )i j , i = 1, 2, . . . , p, j = 1, 2, . . . , q. 7. Set k ← k + 1 and go to step 2. Problem (3.13) in Algorithm PHR-AL is a linear constrained matrix minimization problem. It is certainly solvable for all the known matrices and the scalar ρk . Here we will use the spectral projected gradient (SPG) method to compute the approximation stationary point X k of problem (3.13). The SPG method is a nonmonotone projected gradient type method for minimizing general smooth functions on convex sets[27]. The SPG method is simple, easy to code, and does not require matrix factorizations. Moreover, it overcomes the traditional slowness of the gradient method by incorporating a spectral step length and a nonmonotone globalization strategy. The main steps of SPG algorithm (with necessary modifications) to compute an approximate stationary point of problem (3.13) can be described as follows: Algorithm SPG. (Compute an approximate stationary point of problem (3.13)) k

k

γ ∈ (0, 1), 0 < σ1 < 1. Input matrices Z 1 and Z 2 ; an integer M > 1, parameters αmin > 0, αmax > αmin , e σ2 < 1 and α1 ∈ [αmin , αmax ]. Choose an initial matrix X1 ∈ S and let i ← 1.

n k k o 2. If

PS ∇Lρk (Xi , Z 1 , Z 2 )

< εk , stop. (In this case, Xi is an approximate stationary point of problem (3.13).) n k k
o 3. Compute dXi = −αi PS ∇Lρk Xi , Z 1 , Z 2 . Let λ = 1. 4. Compute Xˇ = Xi + λdXi . 5. If

ˇ Z k1 , Z k2 ) ≤ Lρk (X,

max

1≤ j≤min{i,M}

  k k k k Lρk (Xi− j , Z 1 , Z 2 ) + e γλ dXi , ∇Lρk (Xi , Z 1 , Z 2 ) ,

(3.15)

ˇ si = Xi+1 − Xi , yi = ∇Lρk (Xi+1 , Z k1 , Z k2 ) − ∇Lρk (Xi , Z k1 , Z k2 ). Then goto step 6. define λi = λ, Xi+1 = X, If (3.15) does not hold, define λnew ∈ [σ1 λ, σ2 λ], Let λ = λnew and goto step 4. 7

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6. Compute bi = hsi , yi i. If bi ≤ 0, let αi = αmax , otherwise, compute ai = hsi , si i, αi = min{αmax , max{αmin , ai /bi }}. 7. Let i ← i + 1 and goto step2. In the practical implementation of Algorithm PHR-AL, similarly to [27], we take the parameters γ = 5, 1 r = 0.5, ρ1 = 1, and the large matrix Zmax with all elements equal to 1010 . The initial matrices Z 1 and 1 1 1 Z 2 are chosen as Z 1 = Z 2 = 0. For the implementation of Algorithm SPG, similarly to [30], we take the parameters M = 10, γ = 10−4 , αmin = 10−30 , αmax = 1030 , σ1 = 0.1, σ2 = 0.9, λnew = (σ1 λ + σ2 λ)/2 and α0 = 1. The initial matrix X1 is chosen as the (k − 1)th approximate solution of Algorithm PHR-AL. Lemma 2. Assume that X ∗ is limit point of a sequence generated by Algorithm PHR-AL and the sequence ρk is bounded, then we have L ≤ G(X ∗ ) ≤ U. Proof. Let K be an infinite subset of N such that lim X k = X ∗ . Since lim ρk = ∞ when (3.14) does not hold, k→∞

k∈K

the boundedness of ρk implies that there exists k0 ∈ N such that (3.14) takes place for all k ≥ k0 . Therefore,



 lim

G(X k ) − L, Z1k −

= 0 and lim

U − G(X k ), Z2k −

= 0. k∈K

k∈K

Note that Z1k ≥ 0 and Z2k ≥ 0 for all k ∈ N, we have lim L − G(X k )




k∈K

+

= 0 and

lim G(X k ) − U k∈K




+

= 0,

that is, G(X ∗ ) − L ≥ 0 and U − G(X ∗ ) ≥ 0. Lemma 3. Assume that X ∗ is limit point of a sequence generated by Algorithm PHR-AL, then X ∗ is a first-order stationary point of the problem

2

2  1 

∗ )) ∗ ) − U)

(3.16) minimize (L − G(X + (G(X subject to X ∈ S.

+ + 2 In other words, X ∗ ∈ S satisfies n  o PS GT (L − G(X ∗ ))+ − (G(X ∗ ) − U)+ = 0. Proof. Let K be an infinite subset of N such that lim X k = X ∗ . Consider first the case in which the sequence k∈K

ρk is bounded. By the proof of Lemma 2, we have that


lim

L − G(X k ) +

= 0 and


lim

G(X k ) − U +

= 0.

Note that

    

GT L − G(X ∗ )

≤

L − G(X ∗ )

+ +

    

GT G(X ∗ ) − U

≤

G(X ∗ ) − U

, + +

k∈K

we have that

and

k∈K





lim

GT L − G(X k ) + − G(X k ) − U +

= 0. k∈K

Since

Xk

∈ S for all k, this implies the desired result in the case that {ρk } is bounded. 8

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0

Assume now that {ρk } is not bounded. Therefore there exists an infinite sequence of indices K ⊂ K

n k k o such that lim0 ρk = ∞. Note that εk ↓ 0 and

PS ∇Lρk (X k , Z 1 , Z 2 )

< εk , we have k∈K

  

k k          

     Z Z  T 

k ) − C − ρ GT  k) − U + 2  k) + 1    

= 0. A A(X lim

PS  − G(X L − G(X    k     

k∈K ρk + ρk +  Therefore we have

  

k k    .   

   Z Z  T 

k ) − C ρ − GT  k) − U + 2  k) + 1  

= 0. A A(X lim

PS  − G(X L − G(X    k    

k∈K ρk + ρk +  k

k

Since {X k }, {Z 1 } and {Z 2 } are bounded, we obtain

n   o

PS GT L − G(X ∗ )
− G(X ∗ ) − U


= 0. + + This implies that X ∗ is a stationary point of (3.16). Theorem 2. Assume that X ∗ is limit point of a sequence generated by Algorithm PHR-AL and the sequence {ρk } is bounded, then X ∗ is a solution to Problem (1.1). Proof. Let K be an infinite subset of N such that k

∗

k

∗

lim X k = X ∗ , lim ρk = ρ∗ , lim Z 1 = Z 1 and lim Z 2 = Z 2 . k∈K

k∈K

k∈K

k∈K

By Lemma 2, we have L ≤ G(X ∗ ) ≤ U. Since

n o

PS ∇Lρ (X k , Z k , Z k )

< εk 1 2 k holds for all εk ↓ 0, we have Let then

n o

PS ∇Lρ∗ (X ∗ , Z ∗ , Z ∗ )

= 0. 1 2

  ∗ Y1∗ = ρ∗ L − G(X ∗ ) + Z 1 /ρ∗ Y1∗

≥ 0 and

Y2∗

+

and

(3.17)

  ∗ Y2∗ = ρ∗ G(X ∗ ) − U + Z 2 /ρ∗ , +

≥ 0, and, from (3.17), we have n  o  PS AT A(X ∗ ) − C − GT (Y ∗ − Z ∗ ) = 0.

Since {ρk } is bounded, then there exists k0 ∈ N such that (3.14) takes place for all k ≥ k0 . Hence, we have lim {G(X k ) − L, Z1k }− = {G(X ∗ ) − L, Z1∗ }− = 0

k→∞

and lim {U − G(X k ), Z2k }− = {U − G(X ∗ ), Z2∗ }− = 0,

k→∞

which imply that hG(X ∗ ) − L, Z1∗ i = 0 and hU − G(X ∗ ), Z2∗ i = 0. By the definition of Z1k , Z2k and Y1∗ , Y2∗ we know that (Z1∗ )i j > 0 if and only if (Y1∗ )i j > 0 and (Z2∗ )i j > 0 if and only if (Y2∗ )i j > 0 (i = 1, 2, . . . , l2 , j = 1, 2, . . . , s2 ). So we have hG(X ∗ ) − L, Y1∗ i = 0 and hU − G(X ∗ ), Y2∗ i = 0. Hence X ∗ satisfies conditions (3.11). By Theorem 1, we know that X ∗ is a solution to Problem (1.1). 9

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4. Numerical examples In this section, we first report some numerical results when Algorithm PHR-AL is implemented to solve Problem (1.1) with random data, and then we illustrate the applicability when the algorithm is applied to solve the model (2.10) in image restoration. All the tested algorithms were coded by MATLAB 7.8 (R2009a) and all our computational experiments were run on a personal computer with an Intel(R) Core i3 processor at 2.13 GHz with 2.00 GB of memory. 4.1. Tested with random data In this example, we test the two linear operators as A(X) = A1 XB1 + A2 XB2 and G(X) = E1 XF1 , and S as the set of all real m × n rectangular centrosymmetric matrices[31]. Example 1. Given the matrices A1 , B1 , A2 , B2 , E1 , F1 , C, L and U in Matlab style as follows: A1 = randn(l1 , m), B1 = randn(n, s1 ), A2 = randn(l1 , m), B2 = randn(n, s1 ), E1 = rand(l2 , m), F1 = rand(n, s2 ), C = A1 XB1 + A2 XB2 , L = E1 XF1 − 10 ∗ ones(l2 , s2 ), U = E1 XF1 + 10 ∗ ones(l2 , s2 ), where X = Z + S m ZS n with Z = rand(m, n). Matrices L, U and C are chosen in this way to guarantee that Problem (1.1) is solvable. Note that the Algorithm PHR-AL involve an outer iteration and an inner iteration, the convergence stopping criterion of the outer iterations are all set to be ε = 10−8 , and the small tolerance εk in the inner iterations is set to ( 0.1εk−1 if εk−1 > ε, 0 ε0 = 10 and εk = (4.18) εk−1 if εk−1 < ε. The largest number of the inner iteration is set to be 200. We consider the following two cases to be tested: (a) l1 ≥ m and s1 ≥ n and (b) l1 < m and s1 < n. Table 2: Numerical results for the case (a) l1 ≥ m and s1 ≥ n in Example 1.

l1 , m, n, s1 , l2 , s2 10,10,10,10,10,10 30,18,20,30,25,30 50,50,50,50,50,50 80,60,70,100,80,80 100,100,100,100,100,100 150,100,100,150,120,120 150,150,150,150,150,150 200,180,180,200,150,150 250,250,250,250,200,200

CPU 0.1248 0.3588 3.4476 4.0404 13.3537 10.1401 44.2263 53.3367 161.7106

kX ∗ −Xk kXk

5.1294×10−11 3.4006×10−13 1.2540×10−12 6.7827×10−14 6.7580×10−14 4.8226×10−15 4.7307×10−14 1.2976×10−14 1.1052×10−13

For case l1 ≥ m and s1 ≥ n, Problem (1.1) has unique solution and the true solution is X. Therefore in Table 2, we report the mean computing time in seconds and the mean relative error based on their average values of 10 repeated tests with randomly generated matrices A1 , B1 , A2 , B2 , E1 and F1 for each problem ∗ size. Here the relative error is defined as Re = kX −Xk , where X ∗ is the estimated solution. kXk For case l < n and s < n, as Problem(1.1) has multiple solutions, the algorithm is not guaranteed to converge to the solution X, it is not meaningful to record the relative errors. In this case, we report the mean 10

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Table 3: Numerical results for the case (b)l1 < m and s1 < n in Example 1.

l1 , m, n, s1 , l2 , s2

6,10,10,6,10,10 15,30,25,15,20,20 30,60,75,35,50,50 50,120,125,65,80,80 50,200,200,50,100,100 70,150,150,70,120,120 100,200,200,100,150,150 100,300,300,100,180,180

CPU 0.1560 0.7644 4.2432 17.6749 43.9299 44.9595 132.8817 348.3970

kA1 XB1 + A2 XB2 − Ck 9.3373×10−9 1.4437×10−9 3.2598×10−10 2.6637×10−10 7.1933×10−11 1.7993×10−10 1.7718×10−10 5.1966×10−11

computing time in seconds and the mean residual kA1 XB1 + A2 XB2 − Ck (see Table 3) based on 10 repeated tests with randomly generated matrices A, B, E and F for each problem size in each test. 4.2. Application to image restoration with some special symmetry pattern images In this subsection, we test the efficiency when Algorithm PHR-AL is applied to solve the model (2.10) in image restoration. We only focus on some special symmetry pattern images. The original image is denoted b in each example and it consists of m × n grayscale pixel values in the range [0, d] with d = 255 is the by X b denotes the vector obtained by stacking the maximum possible pixel value of the image. Let xˆ = vec(X) b and H represents the blurring matrix. The vector gˆ = H xˆ represents the associated blurred columns of X and noise-free image. In our tests, similarly to [24], we generated a blurred and noisy image g by g = gˆ + n0 × σ xˆ × 10−

S NR 20

,

where n0 is a random vector noise with a zero mean and a variance equal to one, and SNR is the signal to noise ratio defined by  σ2  S NR = 10 log10 2xˆ , σn where σ2xˆ and σ2n are the variance of the noise and the original image, respectively. The performance of the Algorithm PHR-AL and its comparison are evaluated by the peak signal-to-noise ratio (PSNR) in decible (dB):  d2 mn   d2 mn  PS NR(X) = 10 log10 = 10 log . 10 b − Xk2 k xˆ − xk22 kX In all the tests, the largest number of the involved inner iteration(Algorithm SPG) in the Algorithm PHR-AL is set to be 20. The algorithm started with the degraded images and terminated when the relative difference between the successive iterates of the restored image satisfy Rerror =

kX k+1 − X k k ≤ 0.5 × 10−2 . kX k k

Example 2. In the first example, we consider the”butterfly” original image of size 192 × 254 and is shown on the left side of Figure 1. The original image has perfectly mirror-symmetry[32], that is, the pixel value 11

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Original image

Original image

Figure 1: Original images. Left: ”Butterfly”(mirror-symmetric). Right: ”PlayCard-K-Heart” (centro-symmetric).

b can be expressed as X b = (XL , XL S n ), where XL is the left half of the matrix X. b Actually, we have matrix X b − PS (X)k b = 0, where S is the set of all real 192 × 254 column mirror-symmetry matrices and kX X + X S X S + X  L R n L n R PS (X) = , , ∀X ∈ R192×254 2 2 where XR is the left half and the right half of X. The blurring matrix H is chosen to be H = H1 ⊗ H2 ∈ 2 2 192×192 254×254 R192 ×254 , where H1 = [h(1) and H2 = [h(2) are the Toeplitz matrices whose entries ij ] ∈ R ij ] ∈ R are given by    ( 1 (i− j)2  1  √ , |i − j| ≤ r, exp − , |i − j| ≤ r,  σ 2π (2) (1) 2 2σ and hi j = 2r−1 hi j =    0, 0, otherwise. otherwise In this example we choose the band r = 3 and the variance σ = 0.4. A random Gaussian noise, with S NR = 15dB, was added to produce a blurred and noisy image G with PS NR(G) = 8.1411. The blurred and noisy image is shown on the left side of Figure 4. The restoration of the image from the degraded image is obtained by solving the minimization problem (2.10) using the PHR-AL algorithm. The regularization 2 2 matrix T is chosen to be T = T 1 ⊗ T 2 ∈ R192 ×254 , where T 1 = I192 and T 2 is the tridiagonal matrix, of size 254 × 254, generated by vector (1, 2, 1). The optimal value of the parameter λ = 0.015 was obtained by using the GCV method. The corresponding GCV curve is plotted on the right side of Figure 2. The restored image obtained by using Algorithm PHR-AL is given on the left of Figure 4, the relative error was Re(X) = 1.2521 × 10−1 with PS NR(X) = 21.0231, and the iterations are terminated after 3 iterations with a cpu time of 13.9309 s. Table 1 reports on more results for three levels of noise corresponding to different S NR = 5, 10, 15 and to different values of σ = 0.35, 0.55, 0.85 given in the definition of the blurring matrices H1 and H2 in Example 2. Example 3. In the second example, the original image is the ”PlayCard-K-Heart” image of size 628 × 423 and is shown on the right side of Figure 1. The original image is centrosymmetric, that is, the pixel value 12

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Table 4: Results for Example 3. σ

0.35

0.55

0.85

S NR(dB) 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25

λopt 0.036 0.025 0.017 0.011 0.007 0.036 0.025 0.018 0.012 0.008 0.035 0.026 0.019 0.014 0.010

PS NR(G)(dB) 5.3075 6.0042 6.4097 6.6397 6.7709 8.3142 9.3290 9.9286 10.2655 10.4569 8.4387 9.4712 10.0763 10.4170 10.6154

PS NR(X)(dB) 19.6357 20.9344 21.3394 21.6077 21.9395 18.7410 21.1153 21.8547 21.9397 21.1417 18.5387 20.7428 20.9952 20.5296 20.7946

Re(X) 1.4690×10−1 1.2650×10−1 1.2073×10−1 1.1706×10−1 1.1267×10−1 1.6284×10−1 1.2389×10−1 1.1378×10−1 1.1267×10−1 1.2351×10−1 1.6667×10−1 1.2932×10−1 1.2561×10−1 1.3253×10−1 1.2855×10−1

GCV function, minimum at λ=0.017

CPU-times(s) 23.4002 17.8621 18.0337 18.3145 19.5781 29.6090 40.3419 38.4386 28.2830 18.8137 38.4542 39.1875 27.9086 12.9949 18.8137

GCV function, minimum at λ=0.023

0.06

0.045 0.04

0.05 0.035 0.03 GCV(λ)

GCV(λ)

0.04

0.03

0.02

0.025 0.02 0.015 0.01

0.01 0.005 0

0

0.02

0.04

λ

0.06

0.08

0

0.1

0

0.02

0.04

λ

0.06

0.08

0.1

Figure 2: The GCV curve for the Example 2 with the optimal value of λ = 0.017 (left) and the GCV curve for the Example 3 with the optimal value of λ = 0.023.

Restored image with λ=0.017

Blurred and noisy image

Figure 3: The blurred and noisy image (left) with PS NR(G) = 8.1411, r = 3 and σ = 0.45 and the restored image (right) with PS NR(X) = 21.0231 and Re(X) = 1.2521 × 10−1 .

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b satisfies X b = S 628 XS b 423 . Actually, we have kX−P b S (X)k b = 0, where S is the set of all real 628×423 matrix X 1 rectangle centrosymmetry matrices and PS = 2 (X + S 628 XS 423 ) for any X ∈ R628×423 . The blurring matrix 2 2 H is chosen to be H = H1 ⊗ H2 ∈ R256 ×256 , where H1 = I628 is the identity matrix and H2 = [h(2) i j ] is the Toeplitz matrices of dimension 423 × 423 given by ( 1 , |i − j| ≤ r, (2) hi j = 2r−1 0, otherwise. The blurring matrix H models a uniform blur. The regularization matrix T is chosen to be T = T 1 ⊗ T 2 ∈ Restored image with λ=0.023

Blurred and noisy image

Figure 4: The blurred and noisy image (left) with PS NR(G) = 8.0481, r = 3 and σ = 0.45 and the restored image (right) with PS NR(X) = 20.1459 and Re(X) = 1.5784 × 10−1 . 2

2

R256 ×256 , where T 1 and T 2 are similar to the ones given in Example 2. In this example we set r = 3 and a random Gaussian noise, with S NR = 15dB, was added to produce a blurred and noisy image G with PS NR(G) = 8.0481. The obtained image is shown on the middle of Figure 2. The optimal value of the parameter λ = 0.023 was obtained by using the GCV method. The corresponding GCV curve is plotted on the right side of Figure 2. The restored image obtained by using our proposed Algorithm PHR-AL is also denoted by X and it is given on the right side of Figure 4. The relative error was Re(X) = 1.5784 × 10−1 with the PS NR(X) = 20.1459. The iterations are terminated after 5 iterations with a cpu time of 86.9699s. 5. Conclusion In this paper, we consider solving a class of inequality constrained matrix-form minimization problems, whose various simplified versions have been studied extensively. These matrix-form minimization problems problem can be transformed into the convex linearly constrained quadratic programming in the vector-form by using the vec operator vec(.) and the Kronecker produc ⊗. However, using this transformation will destroy the preindicated linear structure of the unknown matrix and will increase computational complexity and storage requirement. In this paper we will consider the problem from a general point of view and 14

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directly from the perspective of matrices. We propose, analyze and test a matrix-form iteration algorithm framework with the augmented Lagrangian method for solving this problem and its reduced versions which are applicable in image restoration. The numerical results, including when the algorithm is tested with some randomly generated data and on some image restoration problems with special symmetry pattern images, illustrate the effectiveness of the proposed algorithm. References [1] Y.Y. Qiu, A.D. Wang. Solving linearly constrained matrix least squares problem by LSQR. Applied Mathematics and Computation 236 (2014) 273-286. [2] F. Delbos, J.C. Gilbert. Global Linear Convergence of an Augmented Lagrangian Algorithm to Solve Convex Quadratic Optimization Problems. Journal of Convex Analysis, 12 (2005) 45-69. [3] N.J. Higham. The symmetric procrustes problem, BIT Numerical Mathematics, 28 (1988) 133-143. [4] L. E. Andersson, T. Elfving. A Constrained Procrustes Problem, SIAM Journal on Matrix Analysis and Applications, 18 (2006) 124-139. [5] F.J. Henk Don. On the symmetric solution of a linear matrix equation, Linear Algebra and Its Applications, 93 (1987) 1-7. [6] A.P. Liao, Y. Lei. Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraint. Numerica Linear Algebra and its Applications, 14 (2007) 425-444. [7] Z.J. Bai. The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation. SIAM Journal on Matrix Analysis and Applications, 26 (2005) 1100-1114. [8] W.F. Trench. Minimization problems for (R, S )-symmetric and (R, S )-skew symmetric matrices, Linear Algebra and its Applications, 389 (2004) 23-31. [9] Z.Y. Peng, X.Y. Hu. The reflexive and anti-reflexive solutions of the matrix equation AX = B, Linear Algebra and its Applications, 375 (2003) 147-155. [10] D.S. Cvetkovi´c-Ili´ıc The reflexive solutions of the matrix equation AXB = C, Computers Mathematics with Applications, 51 (2006) 897-902. [11] C.J. Meng, X.Y. Hu, L. Zhang. The skew symmetric orthogonal solutions of the matrix equation AX = B, Linear Algebra and its Applications, 402 (2005) 303-318. [12] F. Ding, T.W. Chen. Iterative least squares solutions of coupled Sylvester matrix equations. Systems Control Letters, 54 (2005) 95-107. [13] J. Ding, Y.J. Liu, F. Ding. Iterative solutions to matrix equations of the form Ai XBi = Fi . Computers Mathematics with Applications, 59 (2010) 3500-3507. [14] Y. Lei, A.P. Liao. A minimal residual algorithm for the inconsistent matrix equation AXB = C over symmetric matrices. Applied Mathematics and Computation, 188 (2007) 499-513. [15] M. Dehghan, M. Hajarian, On the generalized reexive and anti-reexive solutions to a system of matrix equations, Linear Algebra and its Applications, 437 (2012) 2793-2812. [16] Z.Y. Peng. Solutions of symmetry-constrained least-squares problems, Numerical Linear Algebra with Applications, 15 (2008) 373-389. [17] S.K. Li, T.Z. Huang. LSQR iterative method for generalized coupled Sylvester matrix equations, Applied Mathematical Modelling, 36 (2012) 3545-3554. [18] Y.Y. Qiu, A.D. Wang Solving linearly constrained matrix least squares problem by LSQR, Applied Mathematics and Computation 236 (2014) 273-286. [19] R. Escalante, M. Raydan. Dykstra’s algorithm for constrained least-squares rectangular matrix problems, Computers Mathematics with Applications, 6 (1998) 73-79. [20] J.F. Li, X.Y. Hu, L. Zhang. Dykstra’s algorithm for constrained least-squares doubly symmetric matrix problems, Theoretical Computer Science, 411 (2010) 2818-2826. [21] A. Bouhamidi, K. Jbilou, M. Raydan. Convex constrained optimization for large-scale generalized Sylvester equations, Computational Optimization and Applications, 48 (2011) 233-253. [22] Z.Y. Peng, L. Wang, J.J. Peng. The solutions of matrix equation AX = B over a matrix inequality constraint, SIAM Journal on Matrix Analysis and Applications, 33 (2012) 554-568. [23] J.F. Li, W. Li, Z.Y. Peng. A hybrid algorithm for solving minimization problem over (R, S )-symmetric matrices with the matrix inequality constraint, Linear and Multilinear Algebra, 5 (2015) 1049-1072. [24] A. Bouhamidi, R. Enkhbat, K. Jbilou. Conditional gradient Tikhonov method for a convex optimization problem in image restoration, Journal of Computational and Applied Mathematics, 255 (2014) 580-592. [25] M. R. Hestenes. Multiplier and gradient methods, Journal of Optimization Theory and Applications, 4 (1969) 303-320.

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[26] M.J.D. Powell. A method for nonlinear constraints in minimization problems, in Optimization, R. Fletcher, ed., Academic Press, New York, 1969, 283-298. [27] E.G. Birgin, J.M. Mart´ınez. Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization, Computational Optimization and Applications, 51 (2012) 941-965. [28] E.G. Birgin, D. Fernandez, J.M. Martnez. The boundedness of penalty parameters in an augmented Lagrangian method with constrained subproblems, Optimization Methods and Software, 27 (2012) 1001-1024. [29] A. Bouhamidi, K. Jbilou. A Kronecker approximation with a convex constrained optimization method for blind image restoration, Optimization Letters, 6 (2012) 1251-1264. [30] E.G. Birgin, J.M. Mart´ınez, M. Raydan. Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization, 10 (2000) 1196-1211. [31] J.R. Weaver. Centrosymmetric(cross symmetric)matrices, their properties, eigenvalues, and eigevectors. American Mathematical Monthly, 92 (1985) 711-717. [32] G.L. Li, Z.H. Feng, Mirrorsymetric matrices, their basic properties, and an application on odd/even-mode decomposition of symmetric multiconductor transmission lines, SIAM Journal on Matrix Analysis and Applications, 24 (2002) 78-90.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO. 5, 2017

On new 𝜆2-Convergent Difference BK-Spaces, Sinan Ercan and Çiğdem A. Bektaş,……793

Stable Cubic Sets, G. Muhiuddin, Sun Shin Ahn, Chang Su Kim, and Young Bae Jun,…802

Some Identities of Chebyshev Polynomials Arising From Non-Linear Differential Equations, Taekyun Kim, Dae San Kim, Jong-Jin Seo, and Dmitry V. Dolgy,………………………820 Blowup Singularity for a Degenerate and Singular Parabolic Equation with Nonlocal Boundary, Dengming Liu and Jie Ma,…………………………………………………………………833 Approximation Properties of Kantorovich-Type q-Bernstein-Stancu-Schurer Operators, Qing-Bo Cai,………………………………………………………………………………………847 (𝑝)

On the Generalized von Neumann-Jordan Constant 𝐶𝑁𝑁 (𝑋), Changsen Yang Wang tianyu,860

Discrete Dynamical Systems in Soft Topological Spaces, Wenqing Fu and Hu Zhao,……867

Functional Inequalities in Vector Banach Space, Gang Lu, Jun Xie, Yuanfeng Jin, and Qi Liu,………………………………………………………………………………………889 Coupled Fixed Point Theorems for Generalized (ψ, ϕ)−Weak Contraction in Partially Ordered G-Metric Spaces, Branislav Popović, Muhammad Shoaib, and Muhammad Sarwar,……897 Triangular Norms Based on Intuitionistic Fuzzy BCK-Submodules, L.B. Badhurays, S.A. Bashammakh, and N. O. Alshehri,…………………………………………………………910 On Strongly Almost Generalized Difference Lacunary Ideal Convergent Sequences of Fuzzy Numbers, S. A. Mohiuddine and B. Hazarika,……………………………………………925 The Catalan Numbers: a Generalization, an Exponential Representation, and some Properties, Feng Qi, Xiao-Ting Shi, Mansour Mahmoud, and Fang-Fang Liu,………………………937 Semiring Structures Based On Meet and Plus Ideals in Lower BCK-Semilattices, Hashem Bordbar, Sun Shin Ahn, Mohammad Mehdi Zahedi, and Young Bae Jun,………………945 The Solutions of Some Types of q-Shift Difference Differential Equations, Hua Wang,…955 Numerical Method For Solving Inequality Constrained Matrix Operator Minimization Problem, Jiao-fen Li, Tao Li, Xue-lin Zhou, and Xiao-fan Lv,………………………………………967

Volume 23, Number 6 ISSN:1521-1398 PRINT,1572-9206 ONLINE

November 15, 2017

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 (fifteen 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,[email protected], Madison,WI,USA.

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $750, Electronic OPEN ACCESS. Individual:Print $380. For any other part of the world add $140 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2017 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics Carlo Bardaro Dipartimento di Matematica e Informatica 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.

Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities

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.

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.

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

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

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

Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

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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

Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design 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

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.

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

Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

Tian-Xiao He Department of Mathematics and Computer Science P.O. Box 2900, Illinois Wesleyan University Bloomington, IL 61702-2900, USA Tel (309)556-3089 Fax (309)556-3864 [email protected] Approximations, Wavelet, Integration Theory, Numerical Analysis, Analytic Combinatorics

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

Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected] Approximation Theory (Positive Linear Operators)

J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected] Partial Differential Equations, Semigroups of Operators H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048

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

Los Angeles, CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory, Splines, Wavelets, Neural Networks Ram N. Mohapatra Department of Mathematics University of Central Florida Orlando, FL 32816-1364 tel.407-823-5080 [email protected] Real and Complex Analysis, Approximation Th., Fourier Analysis, Fuzzy Sets and Systems

Jong Kyu Kim Department of Mathematics Kyungnam University Masan Kyungnam,631-701,Korea Tel 82-(55)-249-2211 Fax 82-(55)-243-8609 [email protected] Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, ODE, PDE, Functional Equations.

Gaston M. N'Guerekata Department of Mathematics Morgan State University Baltimore, MD 21251, USA tel: 1-443-885-4373 Fax 1-443-885-8216 Gaston.N'[email protected] [email protected] Nonlinear Evolution Equations, Abstract Harmonic Analysis, Fractional Differential Equations, Almost Periodicity & Almost Automorphy

Robert Kozma Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA [email protected] Neural Networks, Reproducing Kernel Hilbert Spaces, Neural Percolation Theory Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston, RI 02881,USA [email protected] Differential and Difference Equations

M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations, Optimization, Signal Analysis

Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional Analysis, [email protected]

Mubenga N. Nkashama Department OF Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 205-934-2154 e-mail: [email protected] Ordinary Differential Equations, Partial Differential Equations

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

Vassilis Papanicolaou Department of Mathematics National Technical University of Athens Zografou campus, 157 80 Athens, Greece

Hrushikesh N. Mhaskar Department Of Mathematics California State University

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tel:: +30(210) 772 1722 Fax +30(210) 772 1775 [email protected] Partial Differential Equations, Probability

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Some properties on non-admissible and admissible functions sharing some sets in the unit disc ∗ Feng-Lin Zhou Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

Abstract In this paper, we deal with the uniqueness problem of two non-admissible functions sharing some values and sets in the unit disc, and also investigate the problem on an admissible function and a non-admissible function sharing some values and sets. Some theorems of this paper improve the results given by Fang. In addition, the results in this paper analogous version of the uniqueness theorems of meromorphic functions sharing some sets on the whole complex plane which given by Yi and Cao. Key words: uniqueness; meromorphic function; admissible; non-admissible. Mathematical Subject Classification (2010): Primary 30D 35.

1

Introduction and main results

We should assume that reader is familiar with the basic results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions (see Hayman [6] , Yang [14] and Yi and Yang [18]). For a meromorphic function f , we use S(r, f ) to denote any quantity satisfying S(r, f ) = o(T (r, f )) for all r outside a possible exceptional set of finite logarithmic measure, and b := C S{∞} to denote the extended complex plane, use C to denote the open complex plane, C and D = {z : |z| < 1} to denote the unit disc. R. Nevanlinna [10] proved the following well-known theorems. Theorem 1.1 (see [10]) If f and g are two non-constant meromorphic functions that share five distinct values a1 , a2 , a3 , a4 , a5 IM in C, then f (z) ≡ g(z). After this work, the uniqueness of meromorphic functions with shared sets and values attracted many investigations (see [18]). Moreover, the uniqueness theory of meromorphic functions is an important subject in the value distribution theory. In this paper, we mainly investigate the uniqueness of meromorphic functions with slow growth sharing some sets in the unit disc. We firstly introduce the following basic notations and definitions of meromorphic functions in D(see [2, 4, 7, 12, 8, 13, 22]). Definition 1.1 (see [12]). 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. ∗ This work was supported by the NSF of China (11561033), the Natural Science Foundation of Jiangxi Province in China (20151BAB201008), and the Foundation of Education Department of Jiangxi of China (GJJ150902).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Definition 1.2 (see [12]). 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 . The Second Main Theorem for admissible functions (see [12, Theorem 3]) is very important in studying the uniqueness of two admissible functions in the unit disc D, which was proved by in 2005. Theorem 1.2 (see [12, 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, 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 where E ⊂ (0, 1) is a possibly occurring exceptional set with E 1−r < ∞. If the order of f is   1 finite, the remainder S(r, f ) is a O log 1−r without any exceptional set. In 2005, Titzhoff [12] also obtained the five values theorem for admissible functions in the unit disc D as follows. Theorem 1.3 (see [5, 12]). If two admissible functions f, g share five distinct values, then f ≡ g. From Theorem 1.2(see [12, Theorem 3]), we can easily obtain a lot of theorems similar to meromorphic functions in the complex plane. In 1999, Fang [5] investigated the uniqueness of admissible functions sharing two sets and three sets and obtained a series of theorems. In 2015, Xu, Yang and Cao [15] investigated the problem on shared values of admissible function and nonadmissible function, and obtained some interesting results. Inspired by Xu, Yang and Cao [15] and Fang[5], we further study the problem on shared-sets of admissible function and non-admissible function in the unit disc. The following theorem also plays a very important role in studies non-admissible functions sharing some sets in this paper. Theorem 1.4 (see [12, 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 (q − 2)T (r, f ) ≤ N r, + S(r, f ). + log f − a 1 − r j j=1 1 in Theorem 1.4 does not necRemark 1.1 In contrast to admissible functions, the term log 1−r essarily enter the remainder S(r, f ) because the non-admissible function f may have T (r, f ) =   1 O log 1−r .

 Remark 1.2 We can see that S(r, f ) = o log exception set when 0 < D(f ) < ∞.

1 1−r



holds in Theorem 1.4 without a possible

The following lemma for non-admissible functions in the unit disc is used in this paper.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Lemma 1.1 (see [15]). 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 1 1 kj 1 (q − 2)T (r, f ) < N kj ) r, + N r, k +1 f − aj k +1 f − aj j=1 j j=1 j + log

1 + S(r, f ), 1−r

and  q − 2 −

q X j=1

   q X kj 1 1 1  T (r, f ) ≤ N kj ) r, + log + S(r, f ), kj + 1 k +1 f − aj 1−r j=1 j

1 where nk) (r, f −a ) is used to denote the zeros of f − a in |z| ≤ r, whose multiplicities are no 1 greater than k and are counted only once, N k) (r, f −a ) is the corresponding counting functions, and kj kj +1

1.2.

1 1 = 1, N kj ) (r, f −a ) = N (r, f −a ) and j j

1 kj +1

= 0 if kj = ∞, S(r, f ) is stated as in Theorem

The main purpose of this paper is to deal with the problem of two non-admissible functions sharing some sets, and an admissible function sharing some sets with an non-admissible function. Section 2, the uniqueness of two non-admissible functions sharing some sets in D are investigated and some results showed that the number and weight of sharing sets is related with the index of inadmissibility of functions in D. In section 3, the problem of an admissible function and a nonadmissible function sharing some sets is studied, and one of those results shows that admissible function and non-admissible function can share at most five distinct values with reduced weighted 1.

2

The uniqueness and sharing sets of non-admissible functions in the unit disc

b and X ⊆ C. Define Let S be a set of distinct elements in C [ E(S, D, f ) = {z ∈ D|fa (z) = 0, counting

multiplicities},

a∈S

E(S, D, f ) =

[

{z ∈ D|fa (z) = 0,

ignoring

multiplicities},

a∈S

where fa (z) = f (z) − a if a ∈ C and f∞ (z) = 1/f (z). For two non-constant meromorphic functions f, 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 mulb we say f and g tiplicities) in D if E(S, D, f ) = E(S, D, g). In particular, as S = {a} and a ∈ C, share the value a CM in D if E(a, D, f ) = E(a, D, g), and we say f and g share the value a IM in D if E(a, D, f ) = E(a, D, g). We use E k) (a, D, f ) to denote the set of zeros of f − a in D, with multiplicities no greater than k, in which each zero counted only once. We say that f (z) and g(z) share the value a with reduced weight k in D, if E k) (a, D, f ) = E k) (a, D, g). If D = C, we have the simple notation as before, E(S, f ), E(S, f ), E k) (a, f ) and so on(see [18]). b with respect to a meromorphic function f on the unit disc D is defined The deficiency of a ∈ C by 1 1 m(r, f −a ) N (r, f −a ) δ(a, f ) = δ(0, f − a) = lim inf = 1 − lim sup , T (r, f ) T (r, f ) r→1− r→1−

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Feng-Lin Zhou 995-1007

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

and 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 theorems. The first theorem can be called five values theorem of non-admissible functions. Theorem 2.1 Let f1 and f2 be two non-admissible meromorphic functions in the unit disc D satisfying 1 < D(f1 ), D(f2 ) < ∞, and f1 , f2 share aj (j = 1, 2, 3, 4, 5) IM . Then f1 (z) ≡ f2 (z). Remark 2.1 From Theorem 2.1, we can get that f1 (z) ≡ f2 (z) if f1 , f2 share five distinct values and D(f1 ), D(f2 ) > 1. However, the conclusion holds in Theorem 1.3 under the condition which f1 , f2 are admissible functions, that is, D(f1 ) = ∞, and D(f2 ) = ∞. Thus, we can see that Theorem 2.1 is a greatly improvement of Theorem 1.3. In order to prove Theorem 2.1, we will prove the following general results of two non-admissible functions sharing some sets. Theorem 2.2 Let f1 and f2 be two non-admissible meromorphic functions in the unit disc D satisfying 0 < D(f1 ), D(f2 ) < ∞. Suppose that Sj = {aj , aj + b, . . . , aj + (l − 1)b}, j = 1, 2, . . . , q, i h io nh 1 1 , , where [x] denotes the largest with b 6= 0, Si ∩ Sj = ∅, (i 6= j) and q > 2 + max D(f D(f2 ) 1) integer less than or equal to x. Let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying k1 ≥ k2 ≥ · · · ≥ kq

(1)

and E kj ) (Sj , D, f1 ) = E kj ) (Sj , D, f2 ),

(j = 1, 2, . . . , q).

(2)

Furthermore, let Θ(fi ) =

X

Θ(0, fi − a) −

a

q X l−1 X

Θ(0, fi − (aj + sb)), (i = 1, 2),

j=1 s=0

and Pm−1 Pl−1 A1

j=1

=

s=0

km + 1 +

q X l−1 X kj + δ(0, f1 − (aj + sb)) + kj + 1 j=m s=0

(lm − 3l + 1)km (2l − 1)kn − + Θ(f1 ) − 2, km + 1 kn + 1

Pn−1 Pl−1 A2

δ(0, f1 − (aj + sb))

j=1

=

s=0

δ(0, f2 − (aj + sb)) kn + 1

+

q X l−1 X kj + δ(0, f2 − (aj + sb)) + kj + 1 j=n s=0

(ln − 3l + 1)kn (2l − 1)km − + Θ(f2 ) − 2, kn + 1 km + 1

where m and n are positive integers in {1, 2, . . . , q} and a is an arbitrary complex number or ∞. If min{A1 , A2 } ≥

2 , D(f1 ) + D(f2 )

and

max{A1 , A2 } >

2 . D(f1 ) + D(f2 )

(3)

Then f1 (z) ≡ f2 (z).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

By letting l = 1, q = 5 and k1 = k2 = · · · = k5 = ∞ in Theorem 2.2, we can get Theorem 2.1 easily. Now, we start to prove Theorem 2.2 as follows. Proof of Theorem 2.2: Suppose that f1 (z) 6≡ f2 (z). From the second fundamental theorem in the unit disc (Theorem 1.4) we have    X  p q X l−1 X 1 1 N r, N r, + (ql + p − 2)T (r, f1 ) < f1 − (aj + sb) f1 − dk j=1 s=0 k=1

1 + log + S(r, f1 ). 1−r By definition we have  N

1 r, f1 − d k

 < (1 − Θ(0, f1 − dk )) T (r, f1 ) + S(r, f1 ).

From Lemma 1.1 and the definition of deficiency, it follows that for s ∈ {0, 1, . . . , l − 1}   1 N r, f1 − (aj + sb)     kj 1 1 1 ≤ + N r, N kj ) r, kj + 1 f1 − (aj + sb) kj + 1 f1 − (aj + sb)   kj 1 1 < N k ) r, + (1 − δ(0, f1 − (aj + sb))) T (r, f1 ) kj + 1 j f1 − (aj + sb) kj + 1 +S(r, f1 ). Thus, we obtain (ql + p − 2)T (r, f1 ) ( p ) q X l−1 X X < (1 − Θ(0, f1 − dk )) T (r, f1 ) +

kj 1 ) N kj ) (r, k + 1 f − (a j 1 j + sb) j=1 s=0 k=1   q X l−1 X  1 1 (1 − δ(0, f1 − (aj + sb))) T (r, f1 ) + log + S(r, f1 ). +   kj + 1 1−r j=1 s=0

b Since Θ(0, f − a) ≥ 0 for any meromorphic function f and any complex number a ∈ C. Without loss of generality, we assume that there exist infinitely many d such that Θ(0, f1 − d) > 0 and d 6∈ {aj + sb : j = 1, 2, . . . , q and s = 0, 1, . . . , l − 1}. We denote them by dk (k = 1, 2, . . . , ∞). p Obviously, Θ(f1 ) = Σ∞ k=1 Θ(0, f1 −dk ). Thus there exits a p such that Σk=1 Θ(0, f1 −dk ) > Θ(f1 )−ε holds for any given ε (> 0). Noting that 1≥

k1 k2 kq 1 ≥ ≥ ··· ≥ ≥ , k1 + 1 k2 + 1 kq + 1 2

we can deduce that (ql + p − 2)T (r, f1 )   q l−1 km X X 1 N kj ) r, km + 1 j=1 s=0 f1 − (aj + sb)    l−1  m−1  XX kj km + − (1 − δ(0, f1 − (aj + sb))) T (r, f1 )   kj + 1 km + 1 j=1 s=0   q X l−1 X 1 − δ(0, f1 − (aj + sb))  1 + T (r, f1 ) + log ,   kj + 1 1−r j=1 s=0

< (p − Θ(f1 ) + ε) T (r, f1 ) +

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Feng-Lin Zhou 995-1007

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

namely, 

 q X l−1 X 1 l(m − 1)km km + B1 − ε T (r, f1 ) < N kj ) (r, ) km + 1 k + 1 f − (a 1 j + sb) j=1 s=0 m + log

1 , 1−r

where Pm−1 Pl−1 B1 =

j=1

s=0

δ(0, f1 − (aj + sb))

+

km + 1

q X l−1 X kj + δ(0, f1 − (aj + sb)) + Θ(f1 ) − 2. kj + 1 j=m s=0

By a similar discussion as above, we also have 

   q X l−1 X l(n − 1)kn 1 1 kn + B2 − ε T (r, f2 ) < N kj ) r, + log , kn + 1 k + 1 f − (a + sb) 1 − r 2 j j=1 s=0 n

where Pn−1 Pl−1 j=1

B2 =

s=0

δ(0, f2 − (aj + sb)) kn + 1

+

q X l−1 X kj + δ(0, f2 − (aj + sb)) + Θ(f2 ) − 2. kj + 1 j=n s=0

Hence 


2 + D(f and Theorem 1.4, we get a contradiction. 1) Similarly, we have f2 (z) − f1 (z) 6≡ sb, s = 1, 2, . . . , l − 1. By condition (2) and the first fundamental theorem, we have q X l−1 X

 N kj ) r,

j=1 s=0

 ≤N

1 r, f1 − f2

1 f1 − (aj + sb)  +

l−1 X

 N

s=1



1 r, f1 − f2 − sb

 +

l−1 X



1 r, f2 − f1 − sb





1 r, f2 − f1 − sb



N

s=1

≤ (2l − 1)(T (r, f1 ) + T (r, f2 )) + O(1). and q X l−1 X

 N kj ) r,

j=1 s=0

 ≤N

1 r, f1 − f2

1 f2 − (aj + sb)  +

l−1 X s=1

 N



1 r, f1 − f2 − sb

 +

l−1 X s=1

N

≤ (2l − 1)(T (r, f1 ) + T (r, f2 )) + O(1).

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Therefore, from the above discussion we obtain     l(n − 1)kn l(m − 1)km + B1 − ε T (r, f1 ) + + B2 − ε T (r, f2 ) km + 1 kn + 1   km kn 1 < (2l − 1) + (T (r, f1 ) + T (r, f2 )) + 2 log , km + 1 kn + 1 1−r namely, 1 (A1 − ε) T (r, f1 ) + (A2 − ε) T (r, f2 ) ≤ 2 log . (4) 1−r     1 1 , S(r, f2 ) = o log 1−r . And from Since 0 < D(f1 ), D(f2 ) < ∞, we have S(r, f1 ) = o log 1−r the definition of index, for any ε satisfying   2 0 < 2ε < min D(f1 ), D(f2 ), max{A1 , A2 } − , (5) D(f1 ) + D(f2 ) there exists a sequence {rt } → 1− such that T (rt , f1 ) > (D(f1 ) − ε) log

1 , 1 − rt

T (rt , f2 ) > (D(f2 ) − ε) log

1 , 1 − rt

(6)

for all t → ∞. From (4)-(6), we have   1 1 < o log [(D(f1 ) − ε)(A1 − ε) + (D(f2 ) − ε)(A2 − ε) − 2] log . 1 − rt 1 − rt

(7)

From (7) and ε being arbitrary, the above inequality contradicts to (3). Therefore, the proof of Theorem 2.2 is completed. We can get the following corollaries from Theorem 2.2. Corollary 2.1 Let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1), and let f1 and f2 be two non-admissible meromorphic functions in the unit disc D satisfying 0 < D(f1 ), D(f2 ) < ∞ and (2). Suppose that Sj = {aj , aj + b, . . . , aj + (l − 1)b}, j = 1, 2, . . . , q, nh i h io 1 1 with b 6= 0, Si ∩ Sj = ∅, (i 6= j) and q > 2 + max D(f , , where [x] denotes the largest D(f2 ) 1) integer less than or equal to x. If q X l−1 X j=3 s=0

(2 − 2l)k3 2 kj + >2+ . kj + 1 k3 + 1 D(f1 ) + D(f2 )

Then f1 (z) ≡ f2 (z). Proof: Let m = n = 3. Noting that Θ(fi ) ≥ 0 and δ(0, fi − (aj + sb)) ≥ 0 for j = 1, 2, . . . , q and i = 1, 2, one can deduce from Theorem 2.2 that Corollary 2.1 follows. 2 The following corollary is an analog of a result due to H.-X. Yi (Theorem 10.7 in [18], see also [21]) on C. Corollary 2.2 Let f1 and f2 be two non-admissible meromorphic functions in the unit disc D satisfying 0 < D(f1 ), D(f2 ) < ∞. Suppose that Sj = {aj , aj + b, . . . , aj + (l − 1)b},

j = 1, 2, . . . , q,

with b 6= 0, Si ∩ Sj = ∅, (i 6= j) and      2 1 1 q > max 4 + , 2 + max , . (D(f1 ) + D(f2 ))l D(f1 ) D(f2 ) If E(Sj , D, f1 ) = E(Sj , D, f2 ), (j = 1, 2, . . . , q). Then f1 (z) ≡ f2 (z).

1001

Feng-Lin Zhou 995-1007

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Proof: Let k1 = k2 = . . . = kq = ∞. One can deduce from Corollary 2.1 that Corollary 2.2 follows immediately. 2 Let l = 1. Then it is easily derived the following corollary from Corollary 2.1, which is an analog of the Corollary of Theorem 3.15 in [18]. b and kj (j = 1, 2, . . . , q) Corollary 2.3 Let aj (j = 1, 2, . . . , q) be q distinct complex numbers in C, be positive integers or ∞ satisfying (1), and let f1 and f2 be two non-admissible meromorphic functions in the unit disc D satisfying 0 < D(f1 ), D(f2 ) < ∞ and E kj ) (aj , D, f1 ) = E kj ) (aj , D, f2 ). Set D := min{D(f1 ), D(f2 )}. Then (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). We now state another main theorem. Theorem 2.3 Let f1 and f2 be two non-admissible meromorphic functions in the unit disc D satisfying 0 < D(f1 ), D(f2 ) < ∞. Suppose that Sj = {c + aj , c + aj w, . . . , c + aj wl−1 },

j = 1, 2, . . . , q,

with aj 6= 0, (j = 1, 2, . . . , q), w = exp( 2πi l ), Si ∩Sj = ∅, (i 6= j) and q > 2+max Let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1), and E kj ) (Sj , D, f1 ) = E kj ) (Sj , D, f2 ),

nh

1 D(f1 )

i h io 1 , D(f . ) 2

(j = 1, 2, . . . , q).

(8)

Furthermore, let Θ(fi ) =

X

Θ(0, fi − a) −

a

q X l−1 X

Θ(0, fi − (c + aj ws )), (i = 1, 2),

j=1 s=0

and Pm−1 Pl−1 A3

j=1

=

s=0

km + 1 +

+

q X l−1 X kj + δ(0, f1 − (c + aj ws )) kj + 1 j=m s=0

lkn l(m − 2)km − + Θ(f1 ) − 2, km + 1 kn + 1

Pn−1 Pl−1 A4

δ(0, f1 − (c + aj ws ))

j=1

=

s=0

δ(0, f2 − (c + aj ws )) kn + 1

+

+

q X l−1 X kj + δ(0, f2 − (c + aj ws )) kj + 1 j=n s=0

l(n − 2)kn lkm − + Θ(f2 ) − 2, kn + 1 km + 1

where m and n are positive integers in {1, 2, . . . , q} and a is an arbitrary complex number or ∞. If min{A3 , A4 } ≥

2 , D(f1 ) + D(f2 )

and

max{A3 , A4 } >

2 . D(f1 ) + D(f2 )

(9)

Then (f1 (z) − c)l ≡ (f1 (z) − c)l .

1002

Feng-Lin Zhou 995-1007

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Proof: We assume that (f1 (z) − c)l 6≡ (f2 (z) − c)l . Without loss of generality, we assume that there exist infinitely many d such that Θ(0, f1 − d) > 0 and d 6∈ {c + aj ws : j P = 1, 2, . . . , q and s = ∞ 0, 1, . . . , l − 1}. We denote them by d (k = 1, 2, . . . , ∞). Obviously, Θ(f ) = 1 k=1 Θ(0, f1 − dk ). Pp k Thus there exits a p such that k=1 Θ(0, f1 − dk ) > Θ(f1 ) − ε holds for any given ε (> 0). Using a similar discussion as in the proof of Theorem 2.2, we obtain     l(n − 1)kn l(m − 1)km + B3 − ε T (r, f1 ) + + B4 − ε T (r, f2 ) km + 1 kn + 1
(D(f1 ) − ε) log

1 , 1 − rt

T (rt , f2 ) > (D(f2 ) − ε) log

1 , 1 − rt

(12)

for all t → ∞. From (10)-(12), we have   1 1 [(D(f1 ) − ε)(A3 − ε) + (D(f2 ) − ε)(A4 − ε) − 2] log < o log . 1 − rt 1 − rt

(13)

From (13) and ε being arbitrary, the above inequality contradicts to (9). Therefore, the proof of Theorem 2.3 is completed. We have an analog of a result due to H.-X. Yi (Theorem 10.8 in [18], see also [21]).

2

Corollary 2.4 let f1 and f2 be two non-admissible meromorphic functions in the unit disc D satisfying 0 < D(f1 ), D(f2 ) < ∞. Suppose that Sj = {c + aj , c + aj w, . . . , c + aj wl−1 }, with aj 6= 0, (j = 1, 2, . . . , q), q > 2 +

2 l

+

2 D(f1 )+D(f2 ) ,

j = 1, 2, . . . , q,

w = exp( 2πi l ), Si ∩ Sj = ∅, (i 6= j). If

E(Sj , D, f1 ) = E(Sj , D, f2 ) for j = 1, 2, . . . , q, then (f1 (z) − c)l ≡ (f2 (z) − c)l . Proof: Let m = n = 1 and k1 = k2 = . . . = ∞. Noting that Θ(fi ) ≥ 0 and δ(0, fi − (aj + sb)) ≥ 0 for j = 1, 2, . . . , q and i = 1, 2, Then Corollary 2.4 follows immediately from Theorem 2.2. 2

3

The problem of sharing sets of admissible function and non-admissible function in the unit disc

We now show that an admissible function can share sufficiently many sets 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 positive integers or ∞ satisfying (1). Then E kj ) (aj , D, f1 ) = E kj ) (aj , D, f2 ), and

(j = 1, 2, . . . , q).

q X

(m − 1)km kj + −2>0 k + 1 km + 1 j=m+1 j do not hold at same time. Theorem 3.2 If f1 is admissible and f2 is a non-admissible satisfying limr→1− T (r, f2 ) = ∞. Suppose that Sj = {c + aj , c + aj w, . . . , c + aj wl−1 }, j = 1, 2, . . . , q, with aj 6= 0, (j = 1, 2, . . . , q), w = exp( 2πi l ), Si ∩ Sj = ∅, (i 6= j). Then E(Sj , D, f1 ) = E(Sj , D, f2 ) for j = 1, 2, . . . , q, and q > 1 + 2l can not hold at the same time. To prove the above theorems, we require the following lemmas. Lemma 3.1 (see [12, 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).

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Feng-Lin Zhou 995-1007

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Lemma 3.2 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 ∞ m satisfying (1). Set A5 = B1 + [(m−3)l+1]k . Then (2) and A5 > 0 do not hold at same time, where km +1 B1 , Sj (j = 1, 2, . . . , q) are stated as in Theorem 2.1. Proof: Suppose that (2) and A5 > 0 can hold at the same time. Since f1 (z) is an admissible function, using the same argument as in Theorem 2.2 and from Theorem 1.2 and Lemma 1.1, for any ε(0 < 2ε < A5 ), we have 

 q X l−1 X 1 (m − 1)lkm km N kj ) (r, + B1 − ε T (r, f1 ) < ) + S(r, f1 ), km + 1 k +1 f1 − (aj + sb) j=1 s=0 m

where B1 is stated as in Section 2. Since f1 is admissible and f2 is non-admissible, we can get that f1 (z) 6≡ f2 (z). Thus, by condition (2) and the first fundamental theorem, we have q X l−1 X

 N kj ) r,

j=1 s=0

1 f1 − (aj + sb)



 ≤N

+

1 r, f1 − f2

l−1 X

N

s=1

 r,

 +

l−1 X

 N

s=1

1 f2 − f1 − sb

1 r, f1 − f2 − sb





≤(2l − 1)(T (r, f1 ) + T (r, f2 )) + O(1). From the two above inequality, we get   (2l − 1)km [(m − 3)l + 1]km + B1 − ε T (r, f1 ) ≤ T (r, f2 ). km + 1 km + 1 Since 0 < ε < A5 , we have

[(m−3)l+1]km km +1

(14)

+ B1 − ε > 0. From (14), we have

T (r, f1 ) ≤

1 (2l − 1)km T (r, f2 ). A5 − ε km + 1

(15)

m From Lemma 3.1, (15) and A51−ε (2l−1)k > 0, we can get that each S(r, f1 ) is also an S(r, f2 ). km +1 Since f1 (z) is admissible and f2 (z) is non-admissible, we can get T (r, f2 ) = S(r, f1 ). Thus, we have T (r, f2 ) = S(r, f1 ) = S(r, f2 ) = o(T (r, f2 )).

This is a contradiction. Hence, we can get that (2) and A5 > 0 do not hold at the same time. 2 Lemma 3.3 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 ∞ m satisfying (1). Set A6 = B3 + (m−2)lk km +1 . Then (8) and A6 > 0 do not hold at same time, where B3 , Sj (j = 1, 2, . . . , q) are stated as in Theorem 2.3. Proof: Suppose that (8) and A6 > 0 can hold at the same time. Since f1 (z) is an admissible function, using the same argument as in Theorem 2.3 and from Theorem 1.1 and Lemma 1.1, for any ε(0 < ε < A6 ), we have 

 q X l−1 X 1 (m − 1)lkm km + B3 − ε T (r, f1 ) < N kj ) (r, ) + S(r, f1 ), km + 1 k + 1 f − (c + aj ws ) 1 j=1 s=0 m

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Feng-Lin Zhou 995-1007

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

where B3 is stated as in Section 2. From the assumptions of Lemma 3.3, we can get that (f1 (z) − c)l 6≡ (f2 (z) − c)l . Thus, by condition (8) and the first fundamental theorem, we have q X l−1 X

N kj ) (r,

j=1 s=0

1 1 ) ) < N (r, s l f1 − (c + aj w ) (f1 − c) − (f2 − c)l ≤ l(T (r, f1 ) + T (r, f2 )) + O(1).

From the two above inequality, we get   (m − 2)lkm lkm + B3 − ε T (r, f1 ) ≤ T (r, f2 ). km + 1 km + 1 Since 0 < ε < A6 , we have

(m−2)lkm km +1

(16)

+ B3 − ε > 0. From (16), we have

T (r, f1 ) ≤

1 (2l − 1)km T (r, f2 ). A5 − ε km + 1

(17)

m From Lemma 3.1, (17) and A61−ε klk > 0, we can get that each S(r, f1 ) is also an S(r, f2 ). Since m +1 f1 (z) is admissible and f2 (z) is non-admissible, we can get T (r, f2 ) = S(r, f1 ). Thus, we have

T (r, f2 ) = S(r, f1 ) = S(r, f2 ) = o(T (r, f2 )). This is a contradiction. Hence, we can get that (8) and A6 > 0 do not hold at the same time. Thus, the proof of Lemma 3.3 is completed. 2 Proof of Theorem 3.1: Let l = 1, and since Θ(fi ) ≥ 0 (i = 1, 2) and δ(0, f1 − aj ) ≥ 0 (j = 1, 2, . . . , q), the assertion follows from Lemma 3.2. Proof of Theorem 3.2: Let k1 = k2 = · · · = kq = ∞, and since Θ(fi ) ≥ 0 (i = 1, 2) and δ(0, f1 − aj ) ≥ 0 (j = 1, 2, . . . , q), the assertion follows from Lemma 3.3. It is very interesting to consider distinct small functions instead of distinct complex numbers (see [9, 11, 17],etc). Thus it may be interesting to consider the following questions: Question 3.1 What condition on two non-admissible functions in the unit disc D sharing small functions will guarantee that the two non-admissible functions are identical? Question 3.2 How many small functions can an admissible function and non-admissible function in the unit disc D share at most?

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 general position, 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.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[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] L. W. Liao, The new developments in the research of nonlinear complex differential equations, J Jiangxi Norm. Univ. Nat. Sci. 39 (2015), 331C339. [9] Y. H. Li, J. Y. Qiao, On the uniqueness of meromorphic functions concerning small functions, Sci. China Ser. A 29 (1999), 891-900. [10] 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). [11] D. D. Thai, T. V. Tan, Meromorphic functions sharing small functions as targets, Internat. J. Math. 16 (4) (2005), 437-451. [12] F. Titzhoff, Slowly growing functions sharing values, Fiz. Mat. Fak. Moksl. Semin. Darb. 8(2005), 143-164. [13] J. Tu, J. S. Wei, H. Y. Xu, The order and type of meromorphic functions and analytic functions of [p, q] − ϕ(r) order in the unit disc, J Jiangxi Norm. Univ. Nat. Sci. 39 (2) (2015), 207-210. [14] H. Y. Xu, T. B. Cao, Uniqueness of two analytic functions sharing four values in an angular domain, Ann. Polon. Math. 99 (2010), 55-65. [15] H. Y. Xu, L. Z. Yang, T. B. Cao, The admissible function and non-admissible function in the unit disc, Journal of Computational Analysis and Applications, 19 (2015), 144-155. [16] L. Yang, Value distribution theory and its new application, Springer/Science Press, Berlin/Beijing, 1993/1982. [17] 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. [18] H. X. Yi, C. C. Yang, Uniqueness theory of meromorphic functions, Science Press/ Kluwer. Beijing, 2003. [19] H. X. Yi, The multiple values of meromorphic functions and uniqueness, Chinese Ann. Math. Ser. A 10 (4) (1989), 421-427. [20] H. X. Yi, On one problem of uniqueness of meromorphic functions concerning small functions, Proc. Amer. Math. Soc. 130 (2001), 1689-1697. [21] H. X. Yi, On the uniqueness of meromorphic functions, Acta Math. Sinica (Chin. Ser.) 31 (4) (1988), 570-576. [22] M. L. Zhan, X. M. Zheng, The value distribution of differential polynomials generated by solutions of linear differential equations with meromorphic coefficients in the unit disc, J Jiangxi Norm. Univ. Nat. Sci. 38 (6) (2014), 506-511.

1007

Feng-Lin Zhou 995-1007

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

THE FIXED POINT ALTERNATIVE TO THE STABILITY OF AN ADDITIVE (α, β)-FUNCTIONAL EQUATION SUNGSIK YUN1 , CHOONKIL PARK2∗ , AND HEE SIK KIMK3∗ Abstract. In this paper, we solve the additive (α, β)-functional equation f (x) + f (y) + 2f (z) = αf (β(x + y + 2z)),

(0.1)

where α, β are fixed real or complex numbers with α 6= 4 and αβ = 1. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (α, β)-functional equation (0.1) in Banach spaces.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [24] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [9] 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 [18] 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 [5, 7, 14, 15, 20, 21, 19, 22, 23, 19, 25] for more information on functional equations. We recall a fundamental result in fixed point theory. Theorem 1.1. [2, 6] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−α In 1996, G. Isac and Th.M. Rassias [10] 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 2010 Mathematics Subject Classification. Primary 39B52, 39B62, 47H10. Key words and phrases. Hyers-Ulam stability; additive (α, β)-functional equation; fixed point method; direct method; Banach space. ∗ Corresponding authors.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

S. YUN, C. PARK, AND H. KIM

functional equations have been extensively investigated by a number of authors (see [3, 4, 12, 13, 16, 17]). In Section 2, we solve the additive (α, β)-functional equation (0.1) in vector spaces and prove the Hyers-Ulam stability of the additive (α, β)-functional equation (0.1) in Banach spaces by using the fixed point method. In Section 3, we prove the Hyers-Ulam stability of the additive (α, β)-functional equation (0.1) in Banach spaces by using the direct method. Throughout this paper, assume that X is a normed space and that Y is a Banach space. Let α, β be fixed real or complex numbers with α 6= 4 and αβ = 1. 2. Additive (α, β)-functional equation (0.1) in Banach spaces I We solve the additive (α, β)-functional equation (0.1) in vector spaces. Lemma 2.1. Let X and Y be vector spaces. If a mapping f : X → Y satisfies f (x) + f (y) + 2f (z) = αf (β(x + y + 2z)) (2.1) for all x, y, z ∈ X, then f : X → Y is additive. Proof. Assume that f : X → Y satisfies (2.1). Letting x = y = z = 0 in (2.1), we get 4f (0) = αf (0). So f (0) = 0. Letting y = −x and z = 0 in (2.1), we get f (x) + f (−x) = 0 and so f (−x) = −f (x) for all x ∈ X. Letting x = −2z and y = 0 in (2.1), we get f (−2z)+2f (z) = 0 and so f (2z) = 2f (z) for all z ∈ X. Thus   x 1 f = f (x) 2 2 for all x ∈ X. Letting z = − x+y in (2.1), we get 2   x+y f (x) + f (y) − f (x + y) = f (x) + f (y) + 2f − =0 2 and so f (x + y) = f (x) + f (y) for all x, y ∈ X.  Using the fixed point method, we prove the Hyers-Ulam stability of the additive (α, β)-functional equation (2.1) in Banach spaces. Theorem 2.2. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 with   x y z L ϕ , , ≤ ϕ (x, y, z) (2.2) 2 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and kf (x) + f (y) + 2f (z) − αf (β(x + y + 2z))k ≤ ϕ(x, y, z) (2.3) for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that L kf (x) − A(x)k ≤ (ϕ (x, x, −x) + ϕ (2x, 0, −x)) (2.4) 2(1 − L) for all x ∈ X.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

ADDITIVE (α, β)-FUNCTIONAL EQUATION

Proof. Letting y = x and z = −x in (2.3), we get k2f (x) + 2f (−x)k ≤ ϕ(x, x, −x)

(2.5)

for all x ∈ X. Replacing x by 2x and letting y = 0 and z = −x in (2.3), we get kf (2x) + 2f (−x)k ≤ ϕ(2x, 0, −x)

(2.6)

for all x ∈ X. It follows from (2.5) and (2.6) that kf (2x) − 2f (x)k ≤ ϕ(x, x, −x) + ϕ(2x, 0, −x)

(2.7)

for all x ∈ X. Consider the set S := {h : X → Y, h(0) = 0} and introduce the generalized metric on S: d(g, h) = inf {µ ∈ R+ : kg(x) − h(x)k ≤ µ(ϕ (x, x, −x) + ϕ (2x, 0, −x)), ∀x ∈ X} , where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [11]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then kg(x) − h(x)k ≤ ε(ϕ (x, x, −x) + ϕ (2x, 0, −x)) for all x ∈ X. Hence

        

x x

x x x x kJg(x) − Jh(x)k =

2g , , − − 2h ≤ 2ε ϕ + ϕ x, 0, − 2 2 2 2 2 2 L ≤ 2ε (ϕ (x, x, −x) + ϕ (2x, 0, −x)) = Lε(ϕ (x, x, −x) + ϕ (2x, 0, −x)) 2 for all x ∈ X. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.7) that

     

x

x x x x

f (x) − 2f , , − ≤ ϕ + ϕ x, 0, −

2

2 2

≤

2

2

L (ϕ(x, x, −x) + ϕ(2x, 0, −x)) 2

for all x ∈ X. So d(f, Jf ) ≤ L2 . By Theorem 1.1, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e.,   x (2.8) A (x) = 2A 2 for all x ∈ X. The mapping A is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}.

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This implies that A is a unique mapping satisfying (2.8) such that there exists a µ ∈ (0, ∞) satisfying kf (x) − A(x)k ≤ µ(ϕ (x, x, −x) + ϕ (2x, 0, −x)) for all x ∈ X; (2) d(J l f, A) → 0 as l → ∞. This implies the equality x lim 2 f n l→∞ 2 n

for all x ∈ X; (3) d(f, A) ≤

1 d(f, Jf ), 1−L





= A(x)

which implies

kf (x) − A(x)k ≤

L (ϕ (x, x, −x) + ϕ (2x, 0, −x)) 2(1 − L)

for all x ∈ X. It follows from (2.2) and (2.3) that kA(x) + A(y) + 2A(z) − αA (β(x + y + 2z))k

        

x y z x + y + 2z

= lim 2n

f n + f n + 2f n − αf β

n→∞ 2 2 2 2n   x y z ≤ lim 2n ϕ n , n , n = 0 n→∞ 2 2 2 for all x, y, z ∈ X. So A(x) + A(y) + 2A(z) − αA (β(x + y + 2z)) = 0 for all x, y, z ∈ X. By Lemma 2.1, the mapping A : X → Y is additive.



Corollary 2.3. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying kf (x) + f (y) + 2f (z) − αf (β(x + y + 2z))k ≤ θ(kxkr + kykr + kzkr ) (2.9) for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that 2r + 4 kf (x) − A(x)k ≤ r θkxkr 2 −2 for all x ∈ X. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y, z) = θ(kxkr +kykr +kzkr )  for all x, y, z ∈ X. Then we can choose L = 21−r and we get the desired result. Theorem 2.4. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 with   x y z ϕ (x, y, z) ≤ 2Lϕ , , 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.3). Then there exists a unique additive mapping A : X → Y such that 1 kf (x) − A(x)k ≤ (ϕ (x, x, −x) + ϕ (2x, 0, −x)) 2(1 − L) for all x ∈ X.

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ADDITIVE (α, β)-FUNCTIONAL EQUATION

Proof. It follows from (2.7) that



1 1

f (x) − f (2x) ≤ (ϕ (x, x, −x) + ϕ (2x, 0, −x))

2 2 for all x ∈ X. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (2.9). Then there exists a unique additive mapping A : X → Y such that 4 + 2r kf (x) − A(x)k ≤ θkxkr 2 − 2r for all x ∈ X. Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y, z) = θ(kxkr +kykr +kzkr ) for all x, y, z ∈ X. Then we can choose L = 2r−1 and we get desired result.  3. Additive (α, β)-functional equation (0.1) in Banach spaces II In this section, using the direct method, we prove the Hyers-Ulam stability of the additive (α, β)-functional equation (2.1) in Banach spaces. Theorem 3.1. Let ϕ : X 3 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0 and ∞ X

x y z 2 ϕ j, j, j Ψ(x, y, z) := 2 2 2 j=1 

j



< ∞,

kf (x) + f (y) + 2f (z) − αf (β(x + y + 2z))k ≤ ϕ(x, y, z) (3.1) for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that 1 (3.2) kf (x) − A(x)k ≤ (Ψ(x, x, −x) + Ψ(2x, 0, −x)) 2 for all x ∈ X. Proof. It follows from (2.7) that

     

x

x x x x

f (x) − 2f ≤ϕ , ,− + ϕ x, 0, −

2 2 2 2 2 for all x ∈ X. Hence

        X

l x x

m−1 x x

m j+1

2 f

2j f − 2 f ≤ − 2 f

2l 2m 2j 2j+1 j=l ≤

m−1 X j=l

j

2ϕ



x

x

x

, ,− 2j+1 2j+1 2j+1

1012



x x + 2 ϕ j , 0, − j+1 2 2 j





(3.3)

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for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.3) that the sequence {2k f ( 2xk )} is Cauchy for all x ∈ X. Since Y is a Banach space, the sequence {2k f ( 2xk )} converges. So one can define the mapping A : X → Y by x A(x) := lim 2 f k k→∞ 2 k





for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.3), we get (3.2). Now, let T : X → Y be another additive mapping satisfying (3.2). Then we have

   

q x x

q

kA(x) − T (x)k = 2 A q − 2 T 2 2q 

     

q x x x



q x

q q

≤ 2 A q − 2 f q + 2 T − 2 f 2 2 2q 2q    

≤ 2q Ψ

x x x , q,− q q 2 2 2

+ 2q Ψ

2x x , 0, − q , q 2 2

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of A. The rest of the proof is similar to the proof of Theorem 2.2.  Corollary 3.2. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying (2.9). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

2r + 4 θkxkr r 2 −2

for all x ∈ X. Proof. The proof follows from Theorem 3.1 by taking ϕ(x, y, z) = θ(kxkr +kykr +kzkr ) for all x, y, z ∈ X.  Theorem 3.3. Let ϕ : X 3 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (3.1) and Ψ(x, y, z) :=

∞ X 1 j j=0 2

ϕ(2j x, 2j y, 2j z) < ∞

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that 1 kf (x) − A(x)k ≤ (Ψ(x, x, −x) + Ψ(2x, 0, −x)) 2

(3.4)

for all x ∈ X. Proof. It follows from (2.7) that



1 1

f (x) − f (2x) ≤ (ϕ (x, x, −x) + ϕ (2x, 0, −x))

2

2

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ADDITIVE (α, β)-FUNCTIONAL EQUATION

for all x ∈ X. Hence

m−1 X

1

1  j 

1 1  j+1 

m

f (2l x) −

f 2 x − ≤ f (2 x) f 2 x

l

j

2 2m 2j+1 j=l 2 ≤

m−1 X j=l

1 2j+1

j

j

j

ϕ(2 x, 2 x, −2 x) +

1 2j+1

j+1

ϕ(2

j

x, 0, −2 x)



(3.5)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.5) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by 1 A(x) := n→∞ lim n f (2n x) 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.5), we get (3.4). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.1.  Corollary 3.4. Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (2.9). Then there exists a unique additive mapping A : X → Y such that 4 + 2r kf (x) − A(x)k ≤ θkxkr 2 − 2r for all x ∈ X. Proof. The proof follows from Theorem 3.3 by taking ϕ(x, y, z) = θ(kxkr +kykr +kzkr ) for all x, y, z ∈ X.  Acknowledgments This research was supported by Hanshin University Research Grant. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] L. C˘adariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [3] L. C˘adariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [4] L. C˘adariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008). [5] A. Chahbi, N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [6] J. Diaz, 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. [7] G. Z. Eskandani, P. Gˇ avruta, Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces, J. Nonlinear Sci. Appl. 5 (2012), 459–465. [8] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [9] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.

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[10] G. Isac, Th. M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [11] D. Mihet¸, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [12] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007, Art. ID 50175 (2007). [13] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. 2008, Art. ID 493751 (2008). [14] C. Park, Orthogonal stability of a cubic-quartic functional equation, J. Nonlinear Sci. Appl. 5 (2012), 28–36. [15] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [16] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [17] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [18] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [19] K. Ravi, E. Thandapani, B. V. Senthil Kumar, Solution and stability of a reciprocal type functional equation in several variables, J. Nonlinear Sci. Appl. 7 (2014), 18–27. [20] S. Shagholi, M. Bavand Savadkouhi, M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [21] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [22] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [23] D. Shin, C. Park, 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. [24] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [25] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. 1

Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Republic of Korea E-mail address: [email protected] 2

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, t Republic of Korea E-mail address: [email protected] 3

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected]

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The approximation problem of Dirichlet series with regular growth ∗ Hong-Yan Xua , Yin-Ying Kongb†, and Hua Wangc a

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

b

School of Mathematics and Statistics, Guangdong University of Finance and Economics, Guangzhou, Guangdong 510320, China

c

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

Abstract By introducing the concept of βU -order functions, we study the error in approximating Dirichlet series of infinite order in the half plane by Dirichlet polynomials. Some necessary and sufficient conditions on the error and regular growth of finite βU -order of these functions have been obtained. Key words: β-order, βU -order, Regular growth, Dirichlet series. 2010 Mathematics Subject Classification: 30B50, 30D15.

1

Introduction and basic notes Consider Dirichlet series f (s) =

∞ X

an eλn s ,

s = σ + it,

(1)

n=1

where 0 ≤ λ1 < λ2 < · · · < λn < · · · , λn → ∞ as n → ∞;

(2)

s = σ + it (σ, t are real variables); an are nonzero complex numbers and lim sup(λn+1 − λn ) = h < +∞,

(3)

n→+∞

lim sup n→+∞

log+ |an | = 0, λn

(4)

∗ The first author was supported by The Natural Science Foundation of China(11561033, 11301233), the Natural Science Foundation of Jiangxi Province in China (20151BAB201008), and the Foundation of Education Department of Jiangxi of China (GJJ150902). The second author holds the Project Supported by Guangdong Natural Science Foundation(2015A030313628) and The Training plan for Outstanding Young Teachers in Higher Education of Guangdong(Yqgdufe1405). † Corresponding author

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then from (2) and (3), by using the similar method in [19] or [15], we can get lim sup n→∞

n = E < +∞, λn

lim sup n→∞

log n = 0. λn

(5)

Then the abscissas of convergence and absolutely convergence is 0, that is, f (s) is an analytic function in the left half plane H = {s = σ + it : σ < 0, t ∈ R}. We denote D to be the class of all functions f (s) satisfying (2)-(4) and analytic in Res < 0, denote Dα to be the class of all functions f (s) satisfying (2)-(3) and analytic in Re ≤ α where −∞ < α < +∞. Thus, if −∞ < α < 0 and f (s) ∈ D, then f (s) ∈ Dα ; if 0 < α < +∞ and f (s) ∈ Dα , then f (s) ∈ D. We denote Πk to be the class of all exponential polynomial of degree almost k, that is,   k X  Πk = bj eλj s : (b1 , b2 , . . . , bk ) ∈ Ck .   j=1

For f (s) ∈ D, M (σ, f ) =

max

−∞ 1. From β(x) ∈ F , we have limx→∞ log M (x) = ∞. Then from the Cauchy mean value theorem, there exists ξ(log M (x) < ξ < β(x) log M (x)) satisfying β 0 (ξ) β(ϕ(x) log M (x)) − β(log M (x)) = = ξβ 0 (ξ), log(ϕ(x) log M (x)) − log log M (x) (log ξ)0 that is, β(ϕ(x) log M (x)) = β(log M (x)) + log ϕ(x)ξβ 0 (ξ).

(9)

Since xβ 0 (x) = o(1) as x → +∞ and lim supx→∞ loglogϕ(x) x = %, (0 ≤ % < ∞), by (9), we can get the conclusion of Lemma 2.1. Case 2. If ϕ(x) is a constant. By using the same argument as in Case 1, we can prove that Lemma 2.1 is true. Thus, this completes the proof of Lemma 2.1. 2 The following lemma plays an important role to deal with the growth of Dirichlet series, which shows the relation between M (σ, f ) and m(σ, f ) of such functions. Lemma 2.2 ([19]). If Dirichlet series (1) satisfies (2) (3), then for any given ε ∈ (0, 1) and for σ(< 0) sufficiently reaching 0, we have m(σ, f ) ≤ M (σ, f ) ≤ K(ε)

1 m((1 − ε)σ, f ), −σ

where K(ε) is a constant depending on ε and (3). Lemma 2.3 If f (s) ∈ Dα (−∞ < α < +∞), then for any positive integer n ∈ N+ := N\{0}, we have |an |eαλn ≤ K2 En−1 (f, α), where K2 > 1 is a real constant.

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Proof: From the definition of En (f, α), there exists p(s) ∈ Πn−1 such that ||f − p||α ≤ K2 En−1 (f, α).

(10)

Since f (s) ∈ Dα and from [19, P.16], for any real numbers t0 , ϑ(6= 0), we have R

1 R→+∞ R

Z

1 = lim R→∞ R

Z

lim

an e

(11)

f (α + it)e−λn it dt.

(12)

t0

and αλn

eϑit dt = 0

R

t0

From (11), for any real number x 6= 0, we have 1 R→∞ R

R

Z

ex(α+it) dt = 0.

lim

(13)

t0

Thus, from (12) and (13), for any p1 (s) ∈ Πn−1 , we have an e

αλn

1 = lim R→∞ R

Z

R

[f (α + it) − p1 (α + it)]e−λn it dt,

t0

that is, |an |eαλn ≤ ||f − p1 ||α .

(14) 2

From (10) and (14), we can prove the conclusion of Lemma 2.3.

3

The proof of Theorem 1.4

We prove the conclusions of Theorem 1.4 by using the properties of two functions β(x) and U2 (x), this method is different from the previous method to some extent. We first prove ” ⇐= ” of Theorem 1.4. Suppose that β(λn )

lim sup Ψn (f, α, λn ) = lim sup n→∞

n→∞

log U2



λn log+ [En−1 (f,α)e−αλn ]

 = T.

(15)

Let An = En−1 (f, α)e−αλn , n = 1, 2, . . . , then for any positive real number τ > 0, for sufficiently large n, we have    λn , λn < γ (T + τ ) log U2 log+ An where γ(x) is the inverse functions of β(x). Let V2 (x) and U2 (x) be two reciprocally inverse functions, then we have   V2 exp

    −1 1 λn 1 + β(λn ) < , log An ≤ λn V2 exp β(λn ) . T +τ T +τ log+ An

Thus, we have +

λn σ

log (An e

 ) ≤ λn





V2 exp

! −1 1 β(λn ) +σ . T +τ

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For any fixed and sufficiently small σ < 0, set  

 1 1    , + G = γ (T + τ ) log U2  1 −σ −σ log U 2 −σ

that is,    1 1 1   = V2 exp + β(G) . (17) 1 −σ −σ log U T +τ 2 −σ  n o 1 If λn ≤ G, for sufficiently large n, let V2 exp T +τ β(λn ) ≥ 1, from σ < 0,(16),(17) and the definition of U2 (x), we have !    −1 1 + λn σ log An e ≤G V2 exp +σ β(λn ) T +τ    1 1    ≤ G = γ (T + τ ) log U2  + 1 −σ −σ log U 2 −σ     1 ≤ γ (T + τ ) log (1 + o(1))U2 . (18) −σ If λn > G, from (16) and (17), we have +

λn σ

log An e

! −1 1 β(G) +σ ≤ λn V2 exp T +τ   −1 1  1    + σ + ≤ λn   < 0. 1 −σ −σ log U 2 −σ 





(19)

For sufficiently large n, from (18) and (19), we have     1 log+ An eλn σ ≤ γ (T + τ ) log (1 + o(1))U2 −σ Since An = En−1 e−αλn and τ is arbitrary, by Lemma 2.1,Lemma 2.3 and Theorem 1.3, we can get lim sup σ→0−

β(log+ M (σ, f )) ≤ T. 1 log U2 ( −σ )

Suppose that lim sup σ→0+

β(log+ M (σ, f )) = η < T. 1 log U2 ( −σ )

Thus, there exists any real number ε(0 < ε < η2 ), for any positive integer n and any sufficient small σ < 0, from Lemma 2.2, we have   1 + λn σ log |an |e ≤ log M (σ, f ) ≤ γ (T − 2ε) log U2 ( ) . (20) −σ From (15), there exists a subsequence {λn(p) }, for sufficiently large p, we have ! λn(p) β(λn(p) ) > (T − ε) log U2 . log+ An(p)

1022

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Take a sequence {σp } satisfying  log+ An(p) 1 γ (η − 2ε) log U2 ( ) = λ −σp 1 + log U2 ( log+n(p) A 

n(p)

.

(22)

)

From (20) and (22), we get   log+ An(p) 1 log An(p) + λn(p) σp ≤ γ (η − 2ε) log U2 ( ) = λ −σp 1 + log U2 ( log+n(p) A +

n(p)

, )

that is, 

 λn(p) 1 1 1 + ≤ + λ −σp log An(p) log U2 ( log+n(p) A

n(p)

)

.

Thus, we have   λn(p) 1 1 1 + U2 ( ) ≤ U2  + λ −σp log An(p) log U2 ( log+n(p) A

n(p)

  ≤

)

1+o(1) U2

λn(p)

!

log+ An(p)

.

(23)

From (22) and (23), we have λn(p)

!  λn(p) 1 γ (T − 2ε) log U2 ( ) ) = 1 + log U2 ( + σp log+ An(p) log An(p) ! ! λn(p) λn(p) λn(p) = γ (η − 2ε)(1 + o(1)) log U2 ( + ) 1 + log U2 ( + ) . log+ An(p) log An(p) log An(p) λn(p)



λn(p) (1 + log+ An(p) λn(p) γ(η − 2ε)(1 + o(1)) log U2 ( log+ A ) such n(p)

Thus, from the Cauchy mean value theorem, there exists a real number ξ between λ

λ

)γ(η − 2ε)(1 + o(1)) log U2 ( log+n(p) ) and log U2 ( log+n(p) An(p) An(p) that ! !!
λn(p) λn(p) λn(p) β λn(p) = β 1 + log U2 ( + ) γ (η − 2ε)(1 + o(1)) log U2 ( + ) log+ An(p) log An(p) log An(p) !! λn(p) = β γ (T − 2ε)(1 + o(1)) log U2 ( + ) log An(p) !! λn(p) λn(p) + log 1 + log U2 ( + ξβ 0 (ξ), ) log+ An(p) log An(p) Since log lim



λn(p) log+ An(p)

p→∞



λ

1 + log U2 ( log+n(p) A

n(p)

λ

log U2 ( log+n(p) A

n(p)

)

 = 0,

)

then for sufficiently large p, we have
β λn(p) = (η − 2ε)(1 + o(1)) log U2 (

λn(p) log+ An(p)

) + K2 ξβ 0 (ξ) log U2 (

λn(p) log+ An(p)

),

(24)

where K2 is a constant.

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From (21),(24) and η < T , we can get a contradiction. Thus, we can get lim sup σ→0−

β(log+ M (σ, f )) = T. 1 log U2 ( −σ )

Hence, the sufficiency of Theorem 1.4 is completed. We can prove the necessity of Theorem 1.4 by using the similar argument as in the proof of the sufficiency of Theorem 1.4. Thus, the proof of Theorem 1.4 is completed.

4

The Proof of Theorem 1.5

We will consider two steps as follows: Step one: We first prove the sufficiency of Theorem 1.5. From the conditions of Theorem 1.5, for any ε(> 0), there exists a subsequence {λn(p) } such that !! β(λn(p) ) λn(p) = 1, (25) , lim λn(p) ≥ γ (T − ε) log U2 p→∞ β(λn(p+1) ) log+ An(p) that is, λn(p) log+ An(p)

  ≤ V2 exp

 −1   1 1 β(λn(p) ) , log+ An(p) ≥ λn(p) V2 exp β(λn(p) ) . T −ε T −ε

Take the sequence {σp } satisfying !! 1 1 , λn(p) = γ (T − ε) log U2 + 1 −σp σp log U2 ( −σ ) p    1 1 1 = V exp β(λ ) . + 2 n(p) 1 −σp T −ε ) σp log U2 ( −σ p

(26)

For any sufficiently small σ < 0 and −∞ < α < σ < 0, we have En−1 (f, α) ≤ ||f − pn−1 ||α ≤

∞ X

∞ X

|ak |eλk α ≤ M (σ, f )

eλn (α−σ) ,

(27)

k=n

k=n

Pn−1 where pn−1 (s) = k=1 ak eλk s . From (3), we take 0 < h0 < h satisfying λn+1 − λn ≥ h0 for any integer n ≥ 1. Thus, for sufficiently small σ < 0 such that σ ≥ α2 , from (27) we have En−1 (f, α) ≤ M (σ, f )eλn (α−σ)

∞ X

e(λk −λn )(α−σ)

k=n α

0

≤ M (σ, f )eλn (α−σ) e− 2 h n

∞ X

0

α

e2h k

k=n λn (α−σ)

= M (σ, f )e



1−e

α 0 2h

−1

.

Then for sufficiently small σ < 0 and −∞ < α < σ < 0, we have M (σ, f ) ≥ K3 En−1 (f, α)e−λn (α−σ) = K3 An eλn σ ,

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(28)

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α

0

where K3 = 1 − e 2 h . For sufficiently small σ < 0, we take σp ≤ σ < σp+1 , from (25),(26) and (28), we have log+ M (σ, f ) ≥ log+ An(p) + λn(p) σp + O(1) (29) ! −1   1 ≥ λn(p) V2 exp β(λn(p) ) + σp + O(1) T −ε !! 1 −σp 1 + + O(1) ≥ γ (T − ε) log U2 1 1 −σp σp log U2 ( −σ log U ( −σ ) )−1 2 p p !! 1 −σp 1 ≥ (1 + o(1))γ (T − ε) log U2 + 1 1 −σp+1 )−1 σp+1 log U2 ( −σp+1 ) log U2 ( −σ p !! 1 −σ 1 . + ≥ (1 + o(1))γ (T − ε) log U2 1 1 −σ σ log U2 ( −σ ) )−1 log U2 ( −σ Set

    1 1 1 1 + = r, r 1 + = R, R 1 + = R0 , 1 −σ σ log U2 ( −σ log U2 (r) log U2 (R) )

1 by using a simple calculation, we can get R0 ≥ −σ . Thus, from the definitions of U2 (x) (ii), we can get log U2 (r) = 1. (30) lim sup 1 − ) log U2 ( −σ σ→0

Since log lim sup σ→0−

−σ 1 log U2 ( −σ )−1

1 log U2 ( −σ )

= 0,

and from Lemma 2.1, (29) and (30), we have lim sup σ→0−

β(log+ M (σ, f )) = T. 1 log U2 ( −σ )

Step two: The necessity of the Theorem 1.5 will be proved as follows. From Theorem 1.4, we can get that the right hand of (7) is verified. Next, we will prove that (8) also holds. We take a positive decreasing sequence {εi }(0 < εi < T ),εi → 0(i → ∞). Set     β(λn )  > T − εi ,  Fi = n : Ψn (f, α, λn ) = (31)   log U2 logλ+nAn it follows that ∀i, Fi 6= Φ and Fi ⊂ Fi−1 . For each i, we arrange the n(∈ Fi ) in an increasing sequence {n(i) (p)}∞ p=1 , then we consider the two cases in the following. Case 1. Suppose that limν→+∞

β(λn(i) (p+1) ) β(λn(i) (p) )

= 1 for any i. Then there exists Ni ∈ Fi (i ∈ N+ ),

when n(i) (p) ≥ Ni , we have
β λn(i) (p+1)
≤ 1 + εi . β λn(i) (p)

(32)

Note Fi+1 ⊂ Fi , take Ni+1 > Ni , denote Fi0 the subset of Fi Fi0 = {n ∈ Fi : Ni ≤ n ≤ Ni+1 }, thus the elements of Fi0 satisfy (31) and (32).

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S∞ Therefore let F = i=1 Fi0 and arrange the n(∈ Ei0 ) in an increasing sequence {nν }. Thus, the necessity of Theorem 1.5 is proved.   Case 2. If there exists i ∈ N+ satisfying limν→+∞ λn(i) (p) , we get limν→+∞

β(λn(i) (p+1) ) β(λn(i) (p) )

β λn(i) (p+1)   β λn(i) (p)

6= 1, then since λn(i) (p+1) >

> 1. Hence there exists {n(i) (pk )} ⊆ {n(i) (p)} (still marked

with {n(i) (p)}) and positive real constant τ > 0, it follows that
β λn(i) (p+1)
≥ 1 + τ. β λn(i) (p) Let n0 (1) = n(i) (1), n0 (2) = n(i) (3), · · · , n0 (p) = n(i) (2p − 1), · · · n00 (1) = n(i) (1), n00 (2) = n(i) (4), · · · , n00 (p) = n(i) (2p), · · · where {n0 (p)}, {n00 (p)} are two increasing positive integer sequences, and n00 (p) < n0 (p + 1),

β(λn00 (p) ) > (1 + τ )β(λn0 (p) ),

ν = 1, 2, · · · .

From (31), for any sufficiently large p, when n ∈ 6 Fi satisfies n0 (p) < n < n00 (p), there exists a positive real number δ > 0 such that     λn λn 1 λn ≤ γ (T − δ) log U2 ( + ) , ≥ V2 exp{ β(λn )} . (33) T −δ log An log+ An Thus we have

 log+ An eσλn < λn 

Set

 1

  + σ . 1 V2 exp{ T −δ β(λn )}



(34)



 1 1    , G = γ (T − δ) log U2  + 1 −σ −σ log U 2 −σ

that is,    1 1 1   = V2 exp + β(G) . 1 −σ −σ log U T −δ 2 −σ If λn ≥ G, from (34) and (35), we have  log+ An eσλn ≤ λn 

(35)

 1 V2



If λn < G, from (34) and (35), we have 

 + σ  < 0.

(36)

1 exp{ T −δ β(λn )}



log+ |an |eσλn < G = γ (T − δ) log U2 

 1 1    . + 1 −σ −σ log U 2 −σ

(37)

Choose the sequence {σp } satisfying    σp = − V2 exp

1 β(λn00 (p) ) T −δ

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(38)

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from  thenassumptions oof the necessity of Theorem 1.5, there exists an integer N2 ∈ N+ such that 1 V2 exp T −δ β(λn ) ≥ 1. Then for n ≥ N2 , we have   V2 exp

log+ An eσp λn < λn

1 β(λn ) T −δ

!

−1 + σp

.

When n ≥ n00 (p), it follows λn ≥ λn00 (p) , and from (38), we have +

log An e

σp λn

 < λn



V2 exp

! −1 1 + σp = 0. β(λn00 (p) ) T −δ

(39)

For sufficiently large ν, we have λn0 (p) ≥ λn as N2 ≤ n ≤ n0 (p), and +

log An e

Since λn0 (p) < γ



σp λ n



1 00 1+τ β(λn (p) )

+

σν λn

log An e

≤ λn0 (p)

 V2 exp{

! −1 1 + σp . β(λn )} T −δ

and σp < 0, from the definition of σp , N2 , we can get 

≤γ

    T −δ 1 1 β(λn00 (p) ) ≤ γ log U2 . 1+τ 1+τ −σp

Thus, from (36), (37), (39) and (40), we have   log+ An eσp λn ≤ γ (T − δ) log U2 

(40)

 1 1    , as n > N2 . + 1 −σ −σ log U 2 −σ

By Lemma 2.2, we have lim −

σp →0

β(log+ m(σp , f ))  ≤ T − δ < T.  1 log U2 −σ p

(41)

From (41), Theorem 1.3, we can get a contradiction with the following equality lim−

σ→0

β(log+ M (σ, f ))   = T. 1 log U2 −σ

Thus, the proof of Theorem 1.5 is completed by Step one and Step two.

References [1] A. Akanksha, G. Srivastava, Spaces of vector-valued Dirichlet series in a half plane. Frontiers of Mathematics in China, 2014, 9(6): 1239-1252. [2] Z. C. Cheng, G. X. Wu, S. T. Song, A probability approximatiions of belief function based on fusion of the properties of information entropy, J. Jiangxi Norm. Unive. Nat. Sci. 38 (2014), 534-538. [3] P. V. Filevich, M. N. Sheremeta, Regularly Increasing Entire Dirichlet Series, Mathematical Notes 74 (2003), 110-122; Translated from Matematicheskie Zametki 74 (2003), 118-131.

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[4] Z. S. Gao, The growth of entire functions represented by Dirichlet series, Acta Mathematica Sinica 42 (1999), 741-748(in Chinese). [5] Z. Q. Gao, G. T. Deng, M¨ untz-type theorem on the segments emerging from the origin, J. Approx. Theory 151 (2) (2008), 181-185. [6] P. Gong, L.P. Xiao, The growth of solutions of a class of higher order complex differential equations, J. Jiangxi Norm. Unive. Nat. Sci. 38 (2014), 512-516. [7] Z. D. Gu, D. C. Sun,The growth of Dirichlet series, Czechoslovak Mathematical Journal, 62(1), 29-38, 2012. [8] Y. Y. Huo, Y. Y. Kong. On the Generalized Order of Dirichlet Series, Acta Mathematica Scientia. 2015,35B(1):133-139 [9] Q. Y. Jin, G. T. Deng, D. C. Sun, Julia lines of general random dirichlet series, Czechoslovak Mathematical Journal, 62(4), 919-936, 2012. [10] Y. Y. Kong, H. L. Gan, On orders and types of Dirichlet series of slow growth, Turk J. Math. 34 (2010), 1-11. [11] Y. Y. Kong, On some q-order and q-types of Dirichlet-Hadamard function, Acta Mathematica Sinica 52A (6) (2009), 1165-1172(in Chinese). [12] M. S. Liu, The regular growth of Dirichlet series of finite order in the half plane, J. Sys. Sci. and Math. Scis. 22(2) (2002), 229-238. [13] D. C. Sun, Z. S. Gao, The growth of Dirichlet series in the half plane, Acta Mathematica Scientia 22A(4) (2002), 557-563. [14] W. J. Tang, Y. Q. Cui, H. Q. Xu, H. Y. Xu, On some q-order and q-type of Taylor-Hadamard product function, J. Jiangxi Norm. Unive. Nat. Sci. 40 (2016), 276-279. [15] G. Valiron, Entire function and Borel’s directions, Proc. Nat. Acad. Sci. USA. 20 (1934), 211-215. [16] H. Wang, H. Y. Xu, The approximation and growth problem of Dirichlet series of infinite order, J. Comput. Anal. Appl. 16 (2014), 251-263. [17] H. Y. Xu, C. F. Yi, The approximation problem of Dirichlet series of finite order in the half plane, Acta Mathematica Sinica 53 (3) (2010), 617-624(in Chinese). [18] H. Y. Xu, C. F. Yi, The growth and approoximation of Dirichlet series of infinite order, Advances in Mathematics 42 (1) (2013), 81-88(in Chinese). [19] J. R. Yu, X. Q. Ding, F. J. Tian, On The Distribution of Values of Dirichlet Series And Random Dirichlet Series, Wuhan: Press in Wuhan University, 2004.

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On special fuzzy differential subordinations using multiplier transformation Alina Alb Lupa¸s Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected] Abstract In the present paper we establish several fuzzy differential subordinations regardind the operator I (m, λ, l), P∞  1+λ(j−1)+l m given by I (m, λ, l) : A → A, I (m, λ, l) f (z) = z + j=2 aj z j and A = {f ∈ H(U ), f (z) = l+1 z + a2 z 2 + . . . , z ∈ U } is the class of normalized analytic functions. A certain fuzzy class, denoted by δ SIF (m, λ, l) , of analytic functions in the open unit disc is introduced by means of this operator. By making use of the concept of fuzzy differential subordination we will derive various properties and characteristics of δ (m, λ, l) . Also, several fuzzy differential subordinations are established regarding the operator the class SIF I (m, λ, l).

Keywords: fuzzy differential subordination, convex function, fuzzy best dominant, differential operator. 2000 Mathematical Subject Classification: 30C45, 30A20.

1

Introduction

S.S. Miller and P.T. Mocanu have introduced [10], [11] and developed [12] in the one complex variable functions theory the admissible functions method known as ”the differential subordination method” . The application of this method allows to one obtain some special results and to prove easily some classical results from this domain. G.I. Oros and Gh.Oros [13], [14] wanted to launch a new research direction in mathematics that combines the notions from the complex functions domain with the fuzzy sets theory. In the same way as mentioned, we can justify that by knowing the properties of a differential expression on a fuzzy set for a function one can be determined the properties of that function on a given fuzzy set. We have analyzed the case of one complex functions, leaving as ”open problem” the case of real functions. We are aware that this new research alternative can be realized only through the joint effort of researchers from both domains. The ”open problem” statement leaves open the interpretation of some notions from the fuzzy sets theory such that each one interpret them personally according to their scientific concerns, making this theory more attractive. The notion of fuzzy subordination was introduced in [13]. In [14] the authors have defined the notion of fuzzy differential subordination. In this paper we will study fuzzy differential subordinations obtained with the differential operator studied in [3] using the methods from [4], [5]. Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and H(U ) the space of holomorphic functions in U . Let An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U } with A1 = A and H[a, n] = {f ∈ H(U ) : f (z) = a + an z n + an+1 z n+1 n + . . . , z ∈ U 00} for a ∈ C and n ∈oN. (z) Denote by K = f ∈ A : Re zff 0 (z) + 1 > 0, z ∈ U , the class of normalized convex functions in U . In order to use the concept of fuzzy differential subordination, we remember the following definitions: Definition 1.1 [9] A pair (A, FA ), where FA : X → [0, 1] and A = {x ∈ X : 0 < FA (x) ≤ 1} is called fuzzy subset of X. The set A is called the support of the fuzzy set (A, FA ) and FA is called the membership function of the fuzzy set (A, FA ). One can also denote A = supp(A, FA ).

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Remark 1.1 In the development work
we use the following notations for fuzzy sets: Ff (D) (f (z)) =supp f (D) , Ff (D) ·
= {z ∈ D : 0 < Ff (D) f (z) ≤ 1}, Fg(D) (g (z)) =supp g (D)
, Fg(D) · = {z ∈ D : 0 < Fg(D) g (z) ≤ 1}, p (U ) =supp p (U ) , Fp(U ) ·
= {z ∈ U : 0 < Fp(U ) (p (z)) ≤ 1}, q (U ) =supp q (U ) , Fq(U ) ·
= {z ∈ U : 0 < Fq(U ) (q (z)) ≤ 1}, h (U ) =supp h (U ) , Fh(U ) · = {z ∈ U : 0 < Fh(U ) (h (z)) ≤ 1}. We give a new definition of membershippfunction on complex numbers set using the module notion of a complex number z = x + iy, x, y ∈ R, |z| = x2 + y 2 ≥ 0. Example 1.1 Let F : C → R+ a function such that FC (z) = |F (z)|, ∀ z ∈ C. Denote by FC (C) = {z ∈ C : 0 < F (z) ≤ 1} = {z ∈ C : 0 < |F (z)| ≤ 1} =supp(C, FC ) the fuzzy subset of the complex numbers set. Remark 1.2 We call the subset FC (C) = {z ∈ C : 0 < |F (z)| ≤ 1} = UF (0, 1) the fuzzy unit disk. p x2 + y 2 ≥ 0. A fuzzy subset of the comExample 1.2 Let F : C → R+ , F (z) = 2−|z| 2+|z| , where |z| = plex numbers set is A = {z ∈ C : 0 < FA (z) ≤ 1} =supp(A, FA ) = {z ∈ C : |z| < 2}, where FA (z) =  F (z) , z ∈ {|z| ≤ 2} 0, z ∈ C − {|z| ≤ 2}. We show that the fuzzy subset is nonempty. Indeed, for z = 0, FA (0) = F (0) = 1, so z = 0 ∈ A. More we see that the fuzzy subset A contains all the complex numbers with the properties |z| < 2 and all the complex numbers for which |z| > 2 not belong to A, i.e. supp(A, FA ) = {z ∈ C : x2 + y 2 < 4}. Remark 1.3 The membership functions can be defined otherwise and we propose that each choose how to define according to their research. Definition 1.2 ([13]) Let D ⊂ C, z0 ∈ D be a fixed point and let the functions f, g ∈ H (D). The function f is said to be fuzzy subordinate to g and write f ≺F g or f (z) ≺F g (z), if are satisfied the conditions: 1) f (z0 ) = g (z0 ) , 2) Ff (D) f (z) ≤ Fg(D) g (z), z ∈ D. Definition 1.3 ([14, Definition 2.2]) Let ψ : C3 × U → C and h univalent in U , with ψ (a, 0; 0) = h (0) = a. If p is analytic in U , with p (0) = a and satisfies the (second-order) fuzzy differential subordination Fψ(C3 ×U ) ψ(p(z), zp0 (z) , z 2 p00 (z); z) ≤ Fh(U ) h(z),

z ∈ U,

(1.1)

then p is called a fuzzy solution of the fuzzy differential subordination. The univalent function q is called a fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination, or more simple a fuzzy dominant, if Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U , for all p satisfying (1.1). A fuzzy dominant qe that satisfies Fqe(U ) q˜(z) ≤ Fq(U ) q(z), z ∈ U , for all fuzzy dominants q of (1.1) is said to be the fuzzy best dominant of (1.1). Rz Lemma Corollary 2.6g.2, p. 66]) Let h ∈ A and L [f ] (z) = G (z) = z1 0 h (t) dt, z ∈ U. If  00 1.1 ([12,  (z) 1 Re zh h0 (z) + 1 > − 2 , z ∈ U, then L (f ) = G ∈ K. Lemma 1.2 ([15]) Let h be a convex function with h(0) = a, and let γ ∈ C∗ be a complex number with Re γ ≥ 0. If p ∈ H[a, n] with p (0) = a, ψ : C2 × U → C, ψ (p (z) , zp0 (z) ; z) = p (z) + γ1 zp0 (z) an analytic function in U   and Fψ(C2 ×U ) p(z) + γ1 zp0 (z) ≤ Fh(U ) h(z), i.e. p(z) + γ1 zp0 (z) ≺F h(z), z ∈ U, then Fp(U ) p(z) ≤ Fg(U ) g(z) ≤ Rz Fh(U ) h(z), i.e. p(z) ≺F g(z) ≺F h(z), z ∈ U, where g(z) = nzγγ/n 0 h(t)tγ/n−1 dt, z ∈ U. The function q is convex and is the fuzzy best dominant. Lemma 1.3 ([15]) Let g be a convex function in U and let h(z) = g(z)+nαzg 0 (z), z ∈ U, where α > 0 and n is a positive integer. If p(z) = g(0)+pn z n +pn+1 z n+1 +. . . , z ∈ U, is holomorphic in U and Fp(U ) (p(z) + αzp0 (z)) ≤ Fh(U ) h(z), i.e. p(z) + αzp0 (z) ≺F h(z), z ∈ U, then Fp(U ) p(z) ≤ Fg(U ) g(z), i.e. p(z) ≺F g(z), z ∈ U, and this result is sharp. We will study the following differential operator, known as multiplier transformation.

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Definition 1.4 For f ∈ A = {f ∈ H(U ) : f (z) = z + a2 z 2 + . . . , z ∈ U }, m ∈ N∪ {0}, operator  λ, l ≥ 0,the m P∞ λ(j−1)+l+1 aj z j . I (m, λ, l) f (z) is defined by the following infinite series I (m, λ, l) f (z) = z + j=n+1 l+1 Remark 1.4 It follows from the above definition that (l + 1) I (m + 1, λ, l) f (z) = [l + 1 − λ] I (m, λ, l) f (z) + 0 λz (I (m, λ, l) f (z)) , z ∈ U. Remark 1.5 For l = 0, λ ≥ 0, the operator Dλm = I (m, λ, 0) was introduced and studied by Al-Oboudi [2], which is reduced to the S˘ al˘ agean differential operator [16] for λ = 1. The operator I (m, 1, l) was studied by Cho and Srivastava [8] and Cho and Kim [7]. The operator I (m, 1, 1) was studied by Uralegaddi and Somanatha [17] and the operator I (α, λ, 0) was introduced by Acu and Owa [1]. C˘ ata¸s [6] has studied the operator Ip (m, λ, l) which generalizes the operator I (m, λ, l) .

2

Main results

Using the operator I (m, λ, l) we define the class SIFδ (m, λ, l) and we study fuzzy subordinations.
Definition 2.1 Let f (D) =supp f (D) , Ff (D) = {z ∈ D : 0 < Ff (D) f (z) ≤ 1}, where Ff (D) · is the membership function of the fuzzy set f (D) asociated to the function f . The membership function of the fuzzy set (µf ) (D) asociated to the function µf coincide with the membership function of the fuzzy set f (D) asociated to the fuction f , i.e. F(µf )(D) ((µf ) (z)) = Ff (D) f (z), z ∈ D. The membership function of the fuzzy set (f + g) (D) asociated to the function f + g coincide with the half of the sum of the membership functions of the fuzzy sets f (D), respectively g (D), asociated to the function f , F f (z)+Fg(D) g(z) , z ∈ D. respectively g, i.e. F(f +g)(D) ((f + g) (z)) = f (D) 2 Remark 2.1 F(f +g)(D) ((f + g) (z)) can be defined in other ways. Remark 2.2 Since 0 < Ff (D) f (z) ≤ 1 and 0 < Fg(D) g (z) ≤ 1, it is evidently that 0 < F(f +g)(D) ((f + g) (z)) ≤ 1, z ∈ D. Definition 2.2 Let δ ∈ (0, 1], λ, l ≥ 0 and m ∈ N. A function f ∈ A is said to be in the class SIFδ (m, λ, l) if 0 it satisfies the inequality F(I(m,λ,l)f )0 (U ) (I (m, λ, l) f (z)) > δ, z ∈ U. Theorem 2.1 The set SIFδ (m, λ, l) is convex. P∞ Proof. Let the functions fj (z) = z + j=2 ajk z j , k = 1, 2, z ∈ U, be in the class SIFδ (m, λ, l). It is sufficient to show that the function h (z) = η1 f1 (z) + η2 f2 (z) is in the class SIFδ (m, λ, l) with η1 and η2 nonnegative such that η1 + η2 = 1. 0 We have h0 (z) = (µ1 f1 + µ2 f2 ) (z) = µ1 f10 (z) + µ2 f20 (z), z ∈ U , and 0 0 0 0 (I (m, λ, l) h (z)) = (I (m, λ, l) (µ1 f1 + µ2 f2 ) (z)) = µ1 (I (m, λ, l) f1 (z)) + µ2 (I (m, λ, l) f2 (z)) . From Definition 2.1 we obtain that 0 0 F(I(m,λ,l)h)0 (U ) (I (m, λ, l) h (z)) = F(I(m,λ,l)(µ1 f1 +µ2 f2 ))0 (U ) (I (m, λ, l) (µ1 f1 + µ2 f2 ) (z)) = 0 0
F(I(m,λ,l)(µ1 f1 +µ2 f2 ))0 (U ) µ1 (I (m, λ, l) f1 (z)) + µ2 (I (m, λ, l) f2 (z)) = F(µ1 I(m,λ,l)f1 )0 (U ) (µ1 (I(m,λ,l)f1 (z))0 )+F(µ2 I(m,λ,l)f2 )0 (U ) (µ2 (I(m,λ,l)f2 (z))0 ) = 2

F(I(m,λ,l)f1 )0 (U ) (I(m,λ,l)f1 (z))0 +F(I(m,λ,l)f2 )0 (U ) (I(m,λ,l)f2 (z))0 . 2 0 δ Since f1 , f2 ∈ SIF (m, λ, l) we have δ < F(I(m,λ,l)f1 )0 (U ) (I (m, λ, l) f1 (z)) 0 δ < F(I(m,λ,l)f2 )0 (U ) (I (m, λ, l) f2 (z)) ≤ 1, z ∈ U . F (I(m,λ,l)f1 (z))0 +F(I(m,λ,l)f2 )0 (U ) (I(m,λ,l)f2 (z))0 0 Therefore δ < (I(m,λ,l)f1 ) (U ) ≤1 2 0 δ 0 δ < F(I(m,λ,l)h) (U ) (I (m, λ, l) h (z)) ≤ 1, which means that h ∈ SIF (m, λ, l)

≤ 1 and

and we obtain that and SIFδ (m, λ, l) is convex. 1+z We highlight a fuzzy subset obtained using a convex function. Let the function h (z) = 1−z , z ∈ U . After  00  zh (z) 1+z a short calculation we obtain that Re h0 (z) + 1 = Re 1−z > 0, so h ∈ K and h (U ) = {z ∈ C : Rez > 0}. We define the membership
function for the set h (U ) as Fh(U ) (h (z)) = Reh (z), z ∈ U and we have Fh(U ) h (z) =supp h (U ) , Fh(u) = {z ∈ C : 0 < Fh(U ) (h (z)) ≤ 1} = {z ∈ U : 0 < Rez ≤ 1}. Remark 2.3 In this case the membership function can be defined otherwise too and we recommend that those interested to make it in accordance with their scientific concern.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Theorem 2.2 Let g be a convex function in U and let h (z) = g (z) + Rz c t f (t) dt, z ∈ U, then f ∈ SIFδ (m, λ, l) and G (z) = Ic (f ) (z) = zc+2 c+1 0

1 0 c+2 zg

0

(z) , where z ∈ U, c > 0. If

0

F(I(m,λ,l)f )0 (U ) (I (m, λ, l) f (z)) ≤ Fh(U ) h (z) , i.e. (I (m, λ, l) f (z)) ≺F h (z) , 0

z ∈ U,

(2.1)

0

implies F(I(m,λ,l)G)0 (U ) (I (m, λ, l) G (z)) ≤ Fg(U ) g (z), i.e. (I (m, λ, l) G (z)) ≺F g (z), z ∈ U, and this result is sharp. Proof. We obtain that z

c+1

Z G (z) = (c + 2)

z

tc f (t) dt.

(2.2)

0

Differentiating (2.2), with respect to z, we have (c + 1) G (z) + zG0 (z) = (c + 2) f (z) and 0

(c + 1) I (m, λ, l) G (z) + z (I (m, λ, l) G (z)) = (c + 2) I (m, λ, l) f (z) ,

z ∈ U.

(2.3)

Differentiating (2.3) we have 0

(I (m, λ, l) G (z)) +

1 00 0 z (I (m, λ, l) G (z)) = (I (m, λ, l) f (z)) , z ∈ U. c+2

(2.4)

Using (2.4), the fuzzy differential subordination (2.1) becomes     1 1 0 00 0 FI(m,λ,l)G(U ) (I (m, λ, l) G (z)) + z (I (m, λ, l) G (z)) ≤ Fg(U ) g (z) + zg (z) . c+2 c+2 If we denote

(2.5)

0

p (z) = (I (m, λ, l) G (z)) , z ∈ U,

(2.6)

then p ∈ H [1, 1] .  Replacing (2.6) in (2.5) we obtain Fp(U ) p (z) +

1 0 c+2 zp

  (z) ≤ Fg(U ) g (z) +

1 0 c+2 zg

 (z) , z ∈ U. 0

Using Lemma 1.3 we have Fp(U ) p (z) ≤ Fg(U ) g (z) , z ∈ U, i.e. F(I(m,λ,l)G)0 (U ) (I (m, λ, l) G (z)) ≤ Fg(U ) g (z), 0 z ∈ U, and g is the fuzzy best dominant. We have obtained that (Lm α G (z)) ≺F g (z), z ∈ U.
1 3−2z 1 0 00 Example 2.1 If f ∈ SIF1 1, 12 , 12 , then f 0 (z) + 31 zf 00 (z) ≺F 3(1−z) 2 implies G (z) + 3 zG (z) ≺F 1−z , where R z G (z) = z32 0 tf (t) dt. Theorem 2.3 Let h (z) = z ∈ U, then

1+(2β−1)z , 1+z

β ∈ [0, 1) and c > 0. If λ, l ≥ 0, m ∈ N and Ic (f ) (z) =

c+2 z c+1

Rz 0

tc f (t) dt,

h i ∗ Ic SIFβ (m, λ, l) ⊂ SIFβ (m, λ, l) , where β ∗ = 2β − 1 + (c + 2) (2 − 2β)

(2.7)

R1

tc+1 dt. 0 t+1

Proof. The function h is convex and using the same steps  as in the proof of Theorem 2.2 we get from  1 zp0 (z) ≤ fh(U ) h (z) , where p (z) is defined in (2.6). the hypothesis of Theorem 2.3 that Fp(U ) p (z) + c+2 0

Using Lemma 1.2 we deduce that Fp(U ) p (z) ≤ Fg(U ) g (z) ≤ Fh(U ) h (z) , i.e. F(I(m,λ,l)G)0 (U ) (I (m, λ, l) G (z)) ≤ R z c+1 1+(2β−1)t R z tc+1 Fg(U ) g (z) ≤ Fh(U ) h (z) , where g (z) = zc+2 t dt = 2β −1+ (c+2)(2−2β) dt. Since g is convex c+2 1+t z c+2 0 0 t+1 and g (U ) is symmetric with respect to the real axis, we deduce 0

FI(m,λ,l)G(U ) (I (m, λ, l) G (z)) ≥ min Fg(U ) g (z) = Fg(U ) g (1) |z|=1

and β ∗ = g (1) = 2β − 1 + (c + 2) (2 − 2β) From (2.8) we deduce inclusion (2.7).

(2.8)

R1

tc+1 dt. 0 t+1

Theorem 2.4 Let g be a convex function, g(0) = 1 and let h be the function h(z) = g(z) + zg 0 (z), z ∈ U. If λ, l ≥ 0, m ∈ N, f ∈ A and satisfies the fuzzy differential subordination 0

0

F(I(m,λ,l)f )0 (U ) (I (m, λ, l) f (z)) ≤ Fh(U ) h (z) , i.e. (I (m, λ, l) f (z)) ≺F h(z), z ∈ U, (z) then FI(m,λ,l)f (U ) I(m,λ,l)f ≤ Fg(U ) g(z), i.e. z

I(m,λ,l)f (z) z

1032

≺F g(z), z ∈ U, and this result is sharp.

Alina Alb Lupas 1029-1035

(2.9)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

m

1+λ(j−1)+l z+ ∞ ) aj zj (z) j=2 ( l+1 Proof. Consider p(z) = I(m,λ,l)f = = 1 + p1 z + p2 z 2 + ..., z ∈ U. We deduce z z that p ∈ H[1, 1]. 0 Let I (m, λ, l) f (z) = zp(z), for z ∈ U. Differentiating we obtain (I (m, λ, l) f (z)) = p(z) + zp0 (z), z ∈ U. 0 0 Then (2.9) becomes Fp(U ) (p(z) + zp (z)) ≤ Fh(U ) h(z) = Fg(U ) (g(z) + zg (z)) , z ∈ U. (z) By using Lemma 1.3, we have Fp(U ) p(z) ≤ Fg(U ) g(z), z ∈ U, i.e. F(I(m,λ,l)f )0 (U ) I(m,λ,l)f ≤ Fg(U ) g(z), z (z) z ∈ U.We obtain that I(m,λ,l)f ≺ g(z), z ∈ U, and this result is sharp. F z   00 (z) Theorem 2.5 Let h be an holomorphic function which satisfies the inequality Re 1 + zh > − 12 , z ∈ U, 0 h (z) and h(0) = 1. If λ, l ≥ 0, m ∈ N, f ∈ A and satisfies the fuzzy differential subordination

P

0

0

F(I(m,λ,l)f )0 (U ) (I (m, λ, l) f (z)) ≤ Fh(U ) h (z) , i.e. (I (m, λ, l) f (z)) ≺F h(z),

z ∈ U,

(2.10)

Rz (z) (z) then FI(m,λ,l)f (U ) I(m,λ,l)f ≤ Fq(U ) q(z), i.e. I(m,λ,l)f ≺F q(z), z ∈ U, where q(z) = z1 0 h(t)dt. The z z function q is convex and it is the fuzzy best dominant.   00 (z) (z) , z ∈ U, p ∈ H[1, 1]. Since Re 1 + zh > − 12 , z ∈ U, from Lemma 1.1, Proof. Let p(z) = I(m,λ,l)f z h0 (z) R z we obtain that q (z) = z1 0 h(t)dt is a convex function and verifies the differential equation asscociated to the fuzzy differential subordination (2.10) q (z) + zq 0 (z) = h (z), therefore it is the fuzzy best dominant. 0 Differentiating, we obtain (I (m, λ, l) f (z)) = p(z)+zp0 (z), z ∈ U and (2.10) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z), z ∈ U. (z) ≤ Fq(U ) q(z), z ∈ U. Using Lemma 1.3, we have Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U, i.e. FI(m,λ,l)f (U ) I(m,λ,l)f z I(m,λ,l)f (z) We have obtained that ≺F q(z), z ∈ U. z Corollary 2.6 Let h(z) = 1+(2β−1)z a convex function in U , 0 ≤ β < 1. If λ, l ≥ 0, m ∈ N, f ∈ A and verifies 1+z the fuzzy differential subordination 0

0

F(I(m,λ,l)f )0 (U ) (I (m, λ, l) f (z)) ≤ Fh(U ) h(z), i.e. (I (m, λ, l) f (z)) ≺F h(z),

z ∈ U,

(2.11)

(z) (z) ≤ Fq(U ) q(z), i.e. I(m,λ,l)f ≺F q(z), z ∈ U, where q is given by q(z) = 2β − 1 + then FI(m,λ,l)f (U ) I(m,λ,l)f z z 2(1−β) ln (1 + z) , z ∈ U. The function q is convex and it is the fuzzy best dominant. z

Proof. We have h (z) = 1+(2β−1)z with h (0) = 1, h0 (z) = −2(1−β) and h00 (z) = 1+z (1+z)2  00      (z) 1−ρ cos θ−iρ sin θ 1−ρ2 1 1−z Re zh h0 (z) + 1 = Re 1+z = Re 1+ρ cos θ+iρ sin θ = 1+2ρ cos θ+ρ2 > 0 > − 2 .

4(1−β) , (1+z)3

therefore

(z) , the fuzzy Following the same steps as in the proof of Theorem 2.5 and considering p(z) = I(m,λ,l)f z 0 differential subordination (2.11) becomes FI(m,λ,l)f (U ) (p(z) + zp (z)) ≤ Fh(U ) h(z), z ∈ U. (z) By using Lemma 1.2 for γ = 1 and n = 1, we have Fp(U ) p(z) ≤ Fq(U ) q(z), i.e., FI(m,λ,l)f (U ) I(m,λ,l)f ≤ z R R 2(1−β) 1 z 1 z 1+(2β−1)t Fq(U ) q (z) and q (z) = z 0 h (t) dt = z 0 dt = 2β − 1 + ln (1 + z) , z ∈ U. 1+t z −2 4 00 Example 2.2 Let h (z) = 1−z with h (0) = 1, h0 (z) = (1+z) . 2 and h (z) = (1+z)3  00  1+z     2 (z) cos θ−iρ sin θ 1 1−z Since Re zh = Re 1−ρ = 1+2ρ1−ρ h0 (z) + 1 = Re 1+z 1+ρ cos θ+iρ sin θ cos θ+ρ2 > 0 > − 2 , the function h is convex in U . Let f (z) = z + z 2 , z ∈ U . For n = 1, m = 1, l = 2, λ = 1, we obtain I (1, 1, 2) f (z) = 23 f (z) + 31 zf 0 (z) = Rz 0 (z) 2 ln(1+z) z + 34 z 2 . Then (I (1, 1, 2) f (z)) = 1 + 83 z and I(1,1,2)f = 1 + 43 z. We have q (z) = z1 0 1−t . z 1+t dt = −1 + z

Using Theorem 2.5 we obtain 1 + 38 z ≺F

1−z 1+z ,

z ∈ U, induce 1 + 34 z ≺F −1 +

2 ln(1+z) , z

z ∈ U.

Theorem 2.7 Let g be a convex function such that g (0) = 1 and let h be the function h (z) = g (z) + zg 0 (z), z ∈ U . If λ, l ≥ 0, m ∈ N, f ∈ A and the fuzzy differential subordination  0  0 zI (m + 1, λ, l) f (z) zI (m + 1, λ, l) f (z) FI(m,λ,l)f (U ) ≤ Fh(U ) h (z) , i.e. ≺F h (z) , z ∈ U (2.12) I (m, λ, l) f (z) I (m, λ, l) f (z) (z) holds, then FI(m,λ,l)f (U ) I(m+1,λ,l)f I(m,λ,l)f (z) ≤ Fg(U ) g (z), i.e.

1033

I(m+1,λ,l)f (z) I(m,λ,l)f (z)

≺F g (z), z ∈ U, and this result is sharp.

Alina Alb Lupas 1029-1035

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

(z) 0 Proof. Consider p(z) = I(m+1,λ,l)f I(m,λ,l)f (z) . We have p (z) =  0 (z) obtain p (z) + z · p0 (z) = zI(m+1,λ,l)f . I(m,λ,l)f (z)

(I(m+1,λ,l)f (z))0 I(m,λ,l)f (z)

− p (z) ·

(I(m+1,λ,l)f (z))0 I(m,λ,l)f (z)

and we

Relation (2.12) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z) = Fg(U ) (g(z) + zg 0 (z)) , z ∈ U. By using Lemma (z) 1.3, we have Fp(U ) p(z) ≤ Fg(U ) g(z), z ∈ U, i.e. FI(m,λ,l)f (U ) I(m+1,λ,l)f I(m,λ,l)f (z) ≤ Fg(U ) g(z), z ∈ U. We obtain that I(m+1,λ,l)f (z) I(m,λ,l)f (z)

≺F g (z), z ∈ U.

Theorem 2.8 Let g be a convex function such that g(0) = 1 and let h be the function h(z) = g(z) + zg 0 (z), z ∈ U. If λ, l ≥ 0, m ∈ N, f ∈ A and the fuzzy
differential subordination
l+1 I (m, λ, l) f (z) ≤ Fh(U ) h(z), i.e. FI(m,λ,l)f (U ) l+1 λ I (m + 1, λ, l) f (z) + 2 − λ   l+1 l+1 I (m + 1, λ, l) f (z) + 2 − I (m, λ, l) f (z) ≺F h(z), z ∈ U (2.13) λ λ holds, then FI(m,λ,l)f (U ) [I (m, λ, l) f (z)]0 ≤ Fg(U ) g(z), i.e. [I (m, λ, l) f (z)]0 ≺F g(z), z ∈ U. This result is sharp. 0

Proof. Let p(z) = (I (m, λ, l) f (z)) . We deduce that p ∈ H[1, 1]. We obtain p (z) + z · p0 (z) = 0 (z) = I (m, λ, l) f (z) + z (I (m, λ, l) f (z)) = I (m, λ, l) f (z) + (l+1)I(m+1,λ,l)f (z)−(l+1−λ)I(m,λ,l)f λ
l+1 l+1 I (m, λ, l) f (z) . λ I (m + 1, λ, l) f (z) + 2 − λ The fuzzy differential subordination becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z) = Fg(U ) (g(z) + zg 0 (z)) . By 0 using Lemma 1.3, we have Fp(U ) p(z) ≤ Fg(U ) g(z), z ∈ U, i.e. FI(m,λ,l)f (U ) (I (m, λ, l) f (z)) ≤ Fg(U ) g(z), z ∈ U, and this result is sharp. h i 00 (z) Theorem 2.9 Let h be an holomorphic function which satisfies the inequality Re 1 + zh > − 12 , z ∈ U, 0 h (z) and h (0) = 1. If λ, l ≥ 0, m ∈ N, f ∈ A and satisfies the fuzzy
differential subordination
l+1 FI(m,λ,l)f (U ) l+1 I (m + 1, λ, l) f (z) + 2 − I (m, λ, l) f (z) ≤ Fh(U ) h(z), i.e. λ λ   l+1 l+1 I (m + 1, λ, l) f (z) + 2 − I (m, λ, l) f (z) ≺F h(z), z ∈ U, (2.14) λ λ 0

0

then FI(m,λ,l)f (U ) (I (m, λ, l) f (z)) ≤ Fq(U ) q(z), i.e. (I (m, λ, l) f (z)) ≺F q(z), z ∈ U, where q is given by Rz q(z) = z1 0 h(t)dt. The function q is convex and it is the fuzzy best dominant.   Rz 00 (z) > − 12 , z ∈ U, from Lemma 1.1, we obtain that q (z) = z1 0 h(t)dt is a Proof. Since Re 1 + zh h0 (z) convex function and verifies the differential equation asscociated to the fuzzy differential subordination (2.14) q (z) + zq 0 (z) = h (z), therefore it is the fuzzy best dominant.
0 l+1 I (m, λ, l) f (z) , Considering p (z) = (I (m, λ, l) f (z)) , we obtain p(z)+zp0 (z) = l+1 λ I (m + 1, λ, l) f (z)+ 2 − λ z ∈ U. Then (2.14) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z), z ∈ U. 0 Since p ∈ H[1, 1], using Lemma 1.3, we deduce Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U, i.e. FI(m,λ,l)f (U ) (I (m, λ, l) f (z)) ≤ 0 Fq(U ) q(z), z ∈ U. We have obtained that (I (m, λ, l) f (z)) ≺F q(z), z ∈ U. Corollary 2.10 Let h(z) = 1+(2β−1)z be a convex function in U , where 0 ≤ β < 1.If λ, l ≥ 0, m ∈ N, f ∈ A 1+z

l+1 and satisfies the differential subordination FI(m,λ,l)f (U ) l+1 I (m, λ, l) f (z) ≤ λ I (m + 1, λ, l) f (z) + 2 − λ Fh(U ) h(z), i.e.   l+1 l+1 I (m + 1, λ, l) f (z) + 2 − I (m, λ, l) f (z) ≺F h(z), z ∈ U, (2.15) λ λ 0

0

then FI(m,λ,l)f (U ) (I (m, λ, l) f (z)) ≤ Fq(U ) q(z), i.e. (I (m, λ, l) f (z)) ≺F q(z), z ∈ U, where q is given by q(z) = 2β − 1 + 2(1 − β) ln(1+z) , for z ∈ U. The function q is convex and it is the fuzzy best dominant. z 0

Proof. Following the same steps as in the proof of Theorem 2.8 and considering p(z) = (I (m, λ, l) f (z)) , the fuzzy differential subordination (2.15) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z), z ∈ U. 0 By using Lemma 1.2 for γ = 1 and n = 1, we have Fp(U ) p(z) ≤ Fq(U ) q(z), i.e., FI(m,λ,l)f (U ) (I (m, λ, l) f (z)) ≤ R R z z 0 Fq(U ) q(z), i.e. (I (m, λ, l) f (z)) ≺F q(z), z ∈ U, and q(z) = z1 0 h(t)dt = z1 0 1+(2β−1)t dt = 2β − 1 + 2(1 − 1+t 1 β) z ln(z + 1), z ∈ U.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

 00  zh (z) 1 Example 2.3 Let h (z) = 1−z 1+z a convex function in U with h (0) = 1 and Re h0 (z) + 1 > − 2 (see Example 2.2). Let f (z) = z + z 2 , z ∈ U . For n = 1, m = 1, l = 2, λ = 1, we obtain I (1, 1, 2) f (z) = 23 f (z) + 31 zf 0 (z) =
0 l+1 I (m, λ, l) f (z) = z + 34 z 2 and (I (1, 1, 2) f (z)) = 1 + 83 z. We obtain also l+1 λ I (m + 1, λ, l) f (z) + 2 − λ 0 3I (2, 1, 2) f (z) − I (1, 1, 2) f (z) = 2z + 4z 2 , where I (2, 1, 2) f (z) = 32 I (1, 1, 2) f (z) + z3 (I (1, 1, 2) f (z)) = R z 2 ln(1+z) 1 1−t 2 3z + 16 . 3 z . We have q (z) = z 0 1+t dt = −1 + z Using Theorem 2.9 we obtain 2z + 4z 2 ≺F

1−z 1+z ,

z ∈ U, induce 1 + 83 z ≺F −1 +

2 ln(1+z) , z

z ∈ U.

References [1] M. Acu, S. Owa, Note on a class of starlike functions, RIMS, Kyoto, 2006. [2] F.M. Al-Oboudi, On univalent functions defined by a generalized S˘ al˘ agean operator, Ind. J. Math. Math. Sci., 2004, no.25-28, 1429-1436. [3] A. Alb Lupa¸s, A special comprehensive class of analytic functions defined by multiplier transformation, Journal of Computational Analysis and Applications, Vol. 12, No. 2, 2010, 387-395. [4] A. Alb Lupa¸s, Gh. Oros, On special fuzzy differential subordinations using S˘ al˘ agean and Ruscheweyh operators, Applied Mathematics and Computation, Volume 261, 2015, 119-127. [5] Alina Alb Lupa¸s, A Note on Special Fuzzy Differential Subordinations Using Generalized Salagean Operator and Ruscheweyh Derivative, Journal of Computational Analysis and Applications, Vol. 15, No. 8, 2013, 1476-1483. [6] A. C˘ata¸s, On certain class of p-valent functions defined by new multiplier transformations, Proceedings Book of the International Symposium on Geometric Function Theory and Applications, August 20-24, 2007, TC Istanbul Kultur University, Turkey, 241-250. [7] N.E. Cho, T.H. Kim, Multiplier transformations and strongly close-to-convex functions, Bull. Korean Math. Soc., 40 (3) (2003), 399-410. [8] N.E. Cho, H.M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modelling, 37 (1-2) (2003), 39-49. [9] S.Gh. Gal, A. I. Ban, Elemente de matematic˘ a fuzzy, Oradea, 1996. [10] S.S. Miller, P.T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65(1978), 298-305. [11] S.S. Miller, P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 32(1985), 157-171. [12] S.S. Miller, P.T. Mocanu, Differential Subordinations. Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 225, Marcel Dekker Inc., New York, Basel, 2000. [13] G.I. Oros, Gh. Oros, The notion of subordination in fuzzy sets theory, General Mathematics, vol. 19, No. 4 (2011), 97-103. [14] G.I. Oros, Gh. Oros, Fuzzy differential subordinations, Acta Universitatis Apulensis, No. 30/2012, pp. 55-64. [15] G.I. Oros, Gh. Oros, Dominant and best dominant for fuzzy differential subordinations, Stud. Univ. BabesBolyai Math. 57(2012), No. 2, 239-248. [16] G. St. S˘al˘ agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372. [17] B.A. Uralegaddi, C. Somanatha, Certain classes of univalent functions, Current topics in analytic function theory, World Sci. Publishing, River Edge, N.J., (1992), 371-374.

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Alina Alb Lupas 1029-1035

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

On some differential sandwich theorems involving a multiplier transformation and Ruscheweyh derivative Alb Lupa¸s Alina Department of Mathematics and Computer Science, Faculty of Science University of Oradea 1 Universitatii street, 410087 Oradea, Romania [email protected] Abstract m,n In this paper we obtain some subordination and superordination results for the operator IRλ.l and we m,n establish differential sandwich-type theorems. The operator IRλ,l is defined as the Hadamard product of the multiplier transformation I (m, λ, l) and Ruscheweyh derivative Rn .

Keywords: analytic functions, differential operator, differential subordination, differential superordination. 2010 Mathematical Subject Classification: 30C45.

1

Introduction

Consider H (U ) the class of analytic function in the open unit disc of the complex plane U = {z ∈ C : |z| < 1}, H (a, n) the subclass of H (U ) consisting of functions of the form f (z) = a + an z n + an+1 z n+1 + . . . and An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U } with A = A1 . Next we remind the definition of differential subordination and superordination. Let the functions f and g be analytic in U . The function f is subordinate to g, written f ≺ g, if there exists a Schwarz function w, analytic in U , with w(0) = 0 and |w(z)| < 1, for all z ∈ U, such that f (z) = g(w(z)), for all z ∈ U . In particular, if the function g is univalent in U , the above subordination is equivalent to f (0) = g(0) and f (U ) ⊂ g(U ). Let ψ : C3 × U → C and h be an univalent function in U . If p is analytic in U and satisfies the second order differential subordination ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z), for z ∈ U, (1.1) 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 (1.1). A dominant qe that satisfies qe ≺ q for all dominants q of (1.1) is said to be the best dominant of (1.1). The best dominant is unique up to a rotation of U .
Let ψ : C2 × U → C and h analytic in U . If p and ψ p (z) , zp0 (z) , z 2 p00 (z) ; z are univalent and if p satisfies the second order differential superordination h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z) ; z),

z ∈ U,

(1.2)

then p is a solution of the differential superordination (1.2) (if f is subordinate to F , then F is called to be superordinate to f ). An analytic function q is called a subordinant if q ≺ p for all p satisfying (1.2). An univalent subordinant qe that satisfies q ≺ qe for all subordinants q of (1.2) is said to be the best subordinant. Miller and Mocanu [6] obtained conditions h, q and ψ for which the following implication holds h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z) ; z) ⇒ q (z) ≺P p (z) . P∞ ∞ For two functions f (z) = z + j=2 aj z j and g(z) = z + j=2 bj z j analytic in the open unit disc U , the Hadamard product P∞ (or convolution) of f (z) and g (z), written as (f ∗ g) (z) is defined by f (z) ∗ g (z) = (f ∗ g) (z) = z + j=2 aj bj z j . We need the following differential operators. Definition 1.1 [5] For f ∈ A, m ∈ N∪ {0}, λ, l ≥ 0, the multiplier transformation I (m, λ, l) f (z) is defined by m P∞ the following infinite series I (m, λ, l) f (z) := z + j=2 1+λ(j−1)+l aj z j . 1+l

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Alina Alb Lupas 1036-1042

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

0

Remark 1.1 We have (l + 1) I (m + 1, λ, l) f (z) = (l + 1 − λ) I (m, λ, l) f (z) + λz (I (m, λ, l) f (z)) ,

z ∈ U.

Remark 1.2 For l = 0, λ ≥ 0, the operator Dλm = I (m, λ, 0) was introduced and studied by Al-Oboudi , which reduced to the S˘ al˘ agean differential operator S m = I (m, 1, 0) for λ = 1. Definition 1.2 (Ruscheweyh [8]) For f ∈ A and n ∈ N, the Ruscheweyh derivative Rn is defined by Rn : A → A, R0 f (z) = f (z) , R1 f (z) = zf 0 (z) , ... 0 (n + 1) Rn+1 f (z) = z (Rn f (z)) + nRn f (z) , z ∈ U. P∞ P∞ j Remark 1.3 If f ∈ A, f (z) = z + j=2 aj z j , then Rn f (z) = z + j=2 (n+j−1)! n!(j−1)! aj z for z ∈ U . m,n Definition 1.3 ([2]) Let λ, l ≥ 0 and n, m ∈ N. Denote by IRλ,l : A → A the operator given by the m,n Hadamard product of the multiplier transformation I (m, λ, l) and the Ruscheweyh derivative Rn , IRλ,l f (z) = (I (m, λ, l) ∗ Rn ) f (z) , for any z ∈ U and each nonnegative integers m, n. m P∞ P∞  (n+j−1)! 2 j m,n Remark 1.4 If f ∈ A and f (z) = z + j=2 aj z j , then IRλ,l f (z) = z + j=2 1+λ(j−1)+l l+1 n!(j−1)! aj z , z ∈ U.

Using simple computation we obtain the following relation. Proposition 1.1 [1]For m, n ∈ N and λ ≥ 0 we have m+1,n IRλ,l f (z) =

 0 1 + l − λ m,n λ m,n IRλ,l f (z) + z IRλ,l f (z) l+1 l+1

(1.3)

Definition 1.4 [7] Denote by Q the set of all functions f that are analytic and injective on U \E (f ), where E (f ) = {ζ ∈ ∂U : lim f (z) = ∞}, and are such that f 0 (ζ) 6= 0 for ζ ∈ ∂U \E (f ). z→ζ

Lemma 1.1 [7] Let the function q be univalent in the unit disc U and 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). 0 (z) Suppose that Q is starlike univalent in U and Re zh > 0 for z ∈ U . If p is analytic with p (0) = q (0), Q(z) p (U ) ⊆ D and θ (p (z)) + zp0 (z) φ (p (z)) ≺ θ (q (z)) + zq 0 (z) φ (q (z)) , then p (z) ≺ q (z) and q is the best dominant. Lemma 1.2 [4] Let the function q be convexunivalent  in the open unit disc U and ν and φ be analytic in a ν 0 (q(z)) domain D containing q (U ). Suppose that Re φ(q(z)) > 0 for z ∈ U and 2. ψ (z) = zq 0 (z) φ (q (z)) is starlike univalent in U . If p (z) ∈ H [q (0) , 1] ∩ Q, with p (U ) ⊆ D and ν (p (z)) + zp0 (z) φ (p (z)) is univalent in U and ν (q (z)) + zq 0 (z) φ (q (z)) ≺ ν (p (z)) + zp0 (z) φ (p (z)) , then q (z) ≺ p (z) and q is the best subordinant.

2

Main results We intend to find sufficient conditions for certain normalized analytic functions f such that q1 (z) ≺

m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l

≺ q2 (z) , z ∈ U, 0 < δ ≤ 1, where q1 and q2 are given univalent functions. m+1,n z δ IRλ,l f (z)

1+δ ∈ H (U ) and let the function q (z) be analytic and univalent in U such that m,n f (z)) (IRλ,l 0 (z) is starlike univalent in U . Let q (z) 6= 0, for all z ∈ U . Suppose that zqq(z)

Theorem 2.1 Let

 Re

ξ 2µ 2 q 00 (z) q 0 (z) q (z) + q (z) + 1 + z 0 −z β β q (z) q (z)

 > 0,

(2.1)

for α, ξ, β, µ ∈ C, β 6= 0, z ∈ U and m,n ψλ,l (α, ξ, µ, β; z) := α + β

m+2,n f (z) (l + 1) (l + 1) IRλ,l +β − m+1,n λ λ IRλ,l f (z)

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Alina Alb Lupas 1036-1042

(2.2)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

β

(l + 1) (1 + δ) λ

 2 m+1,n 2δ z IR f (z) (z) z (z) λ,l + ξ 1+δ + µ  2+2δ . m,n m,n (z) IRλ,l f (z) IRλ,l f (z)

m+1,n IRλ,l f m+1,n IRλ,l f

δ

m+1,n IRλ,l f

If q satisfies the following subordination 2

m,n ψλ,l (α, β, µ; z) ≺ α + ξq (z) + µ (q (z)) + β

for α, ξ, β, µ ∈ C, β 6= 0, then

m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l

Proof. Consider p (z) := m+1,n δ−1 IRλ,l f (z) δ(1+l) z 1+δ m,n λ (IRλ,l f (z))

+

1+δ

m+2,n δ−1 IRλ,l f (z) l+1 z λ (IRm,n f (z))1+δ λ,l

(2.3)

≺ q (z), and q is the best dominant.

m+1,n z δ IRλ,l f (z) m,n f (z)) (IRλ,l

zq 0 (z) , q (z)

, z ∈ U , z 6= 0, f ∈ A. Differentiating we obtain p0 (z) = 2

−

m+1,n δ−1 f (z)) (IRλ,l (l+1)(1+δ) z 2+δ m,n λ (IR f (z))

.

λ,l

By using the identity (1.3), we obtain m+2,n m+1,n f (z) (l + 1) (1 + δ) IRλ,l f (z) zp0 (z) δ (l + 1) l + 1 IRλ,l − . = + m+1,n m+1,n p (z) λ λ IRλ,l λ f (z) IRλ,l f (z)

(2.4)

β By setting θ (w) := α + ξw + µw2 and φ (w) := w , it can be easily verified that θ is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. 0 0 2 (z) (z) Also, by letting Q (z) = zq 0 (z) φ (q (z)) = β zqq(z) and h (z) = θ (q (z))+Q (z) = α+ξq (z)+µ (q (z)) +β zqq(z) , we find that Q (z) is starlike univalent in U .  0 2 0 00 0 (z) (z) (z) (z) ξ 2µ 2 and zh + βz qq(z) − βz qq(z) We get h0 (z) = ξq 0 (z) + 2µq (z) q 0 (z) + β qq(z) Q(z) = β q (z) + β q (z) + 1 + 00

0

(z) (z) z qq(z) − z qq(z) .

α

zh0 (z) Q(z)

q 00 (z) 2µ 2 β q (z) + 1 + z q(z) − 0 2 (z) By using (2.4), we obtain α + ξp (z) + µ (p (z)) + β zpp(z) = m+2,n m+1,n m+1,n δ IR IR f (z) f (z) z IRλ,l f (z) (l+1)(1+δ) λ,l λ,l + β (l+1) + β (l+1) +ξ 1+δ m,n λ λ IRm+1,n f (z) − β λ IRm+1,n f (z) (IR f (z))

So we deduce that Re





λ,l

= Re



ξ β q (z)

+

λ,l

 0 (z) z qq(z) > 0.

+µ

m+1,n z 2δ (IRλ,l f (z)) 2+2δ

2

m,n f (z)) (IRλ,l 0 2 (z) ≺ α + ξq (z) + µ (q (z)) + β zqq(z) .

.

λ,l

0

2

(z) By using (2.3), we have α + ξp (z) + µ (p (z)) + β zpp(z)

Appying Lemma 1.1, we obtain p (z) ≺ q (z), z ∈ U, i.e.

m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l

≺ q (z), z ∈ U and q is the best

dominant. m,n 1+Az Corollary 2.2 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.1) holds. If f ∈ A and ψλ,l (α, β, µ; z) ≺ α + ξ 1+Bz +  2 β(A−B)z m,n 1+Az µ 1+Bz + (1+Az)(1+Bz) , for α, β, µ, ξ ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where ψλ,l is defined in (2.2), then m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l

≺

1+Az 1+Bz ,

Proof. For q (z) =

and

1+Az 1+Bz

1+Az 1+Bz ,

is the best dominant.

−1 ≤ B < A ≤ 1 in Theorem 2.1 we get the corollary.

m,n Corollary 2.3 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.1) holds. If f ∈ A and ψλ,l (α, β, µ; z) ≺ α +  γ  2γ m,n 2βγz 1+z 1+z ξ 1−z + µ 1−z + 1−z is defined in (2.2), then 2 , for α, β, µ, ξ ∈ C, 0 < γ ≤ 1, β 6= 0, where ψλ,l     m+1,n γ γ z δ IRλ,l f (z) 1+z 1+z , and 1−z is the best dominant. 1+δ ≺ m,n 1−z f (z)) (IRλ,l  γ 1+z Proof. Corollary follows by using Theorem 2.1 for q (z) = 1−z , 0 < γ ≤ 1. 0

(z) Theorem 2.4 Let q be analytic and univalent in U such that q (z) 6= 0 and zqq(z) be starlike univalent in U . Assume that   ξ 2µ 2 0 0 Re q (z) q (z) + q (z) q (z) > 0, for ξ, β, µ ∈ C, β 6= 0. (2.5) β β

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Alina Alb Lupas 1036-1042

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

If f ∈ A,

m+1,n z δ IRλ,l f (z)

m,n m,n ∈ H [q (0) , 1] ∩ Q and ψλ,l (α, β, µ; z) is univalent in U , where ψλ,l (α, β, µ; z) is as

1+δ

m,n f (z)) (IRλ,l defined in (2.2), then

2

α + ξq (z) + µ (q (z)) + implies q (z) ≺

m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l

βzq 0 (z) m,n ≺ ψλ,l (α, β, µ; z) q (z)

(2.6)

, z ∈ U, and q is the best subordinant. m+1,n z δ IRλ,l f (z)

1+δ , z ∈ U , z 6= 0, f ∈ A. m,n f (z)) (IRλ,l β By setting ν (w) := α + ξw + µw2 and φ (w) := w it can be easily verified that ν is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}.    0  0 0 (q(z)) ν (q(z)) ξ 2µ 2 0 0 = Re Since νφ(q(z)) = q (z)q(z)[ξ+2µq(z)] , it follows that Re q (z) q (z) + q (z) q (z) > 0, for β φ(q(z)) β β α, β, µ ∈ C, µ 6= 0. 0 2 2 (z) βzp0 (z) By using (2.4) and (2.6) we get α + ξq (z) + µ (q (z)) + βzq q(z) ≺ α + ξp (z) + µ (p (z)) + p(z) . Applying

Proof. Consider p (z) :=

Lemma 1.2, we obtain q (z) ≺ p (z) =

m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l

, z ∈ U, and q is the best subordinant. z δ IRm+1,n f (z)

λ,l Corollary 2.5 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.5) holds. If f ∈ A, 1+δ ∈ H [q (0) , 1] ∩ Q and m,n f (z)) (IRλ,l 2  β(A−B)z m,n 1+Az 1+Az α + ξ 1+Bz + µ 1+Bz + (1+Az)(1+Bz) ≺ ψλ,l (α, β, µ; z) , for α, β, ξ, µ ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where

m,n ψλ,l is defined in (2.2), then

Proof. For q (z) =

1+Az 1+Bz ,

1+Az 1+Bz

m+1,n z δ IRλ,l f (z)

≺

m,n f (z)) (IRλ,l

1+δ

, and

1+Az 1+Bz

is the best subordinant.

−1 ≤ B < A ≤ 1 in Theorem 2.4 we get the corollary. z δ IRm+1,n f (z)

λ,l Corollary 2.6 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.5) holds. If f ∈ A, 1+δ ∈ H [q (0) , 1] ∩ Q and m,n f (z)) (IRλ,l  γ  2γ m,n m,n 2βγz 1+z 1+z α + ξ 1−z is defined + µ 1−z + 1−z 2 ≺ ψλ,l (α, β, µ; z) , for α, β, µ, ξ ∈ C, β 6= 0, 0 < γ ≤ 1, where ψλ,l γ  γ  m+1,n δ z IR f (z) λ,l 1+z 1+z ≺ is the best subordinant. in (2.2), then 1−z 1+δ , and m,n 1−z f (z)) (IRλ,l  γ 1+z Proof. For q (z) = 1−z , 0 < γ ≤ 1 in Theorem 2.4 we get the corollary. Combining Theorem 2.1 and Theorem 2.4, we state the following sandwich theorem.

Theorem 2.7 Let q1 and q2 be analytic and univalent in U such that q1 (z) 6= 0 and q2 (z) 6= 0, for all zq 0 (z) zq 0 (z) z ∈ U , with q11(z) and q22(z) being starlike univalent. Suppose that q1 satisfies (2.1) and q2 satisfies (2.5). m+1,n z δ IRλ,l f (z)

m,n 1+δ ∈ H [q (0) , 1] ∩ Q and ψλ,l (α, β, µ; z) is as defined in (2.2) univalent in U , then m,n f (z)) (IRλ,l βzq10 (z) βzq20 (z) 2 2 m,n α + ξq1 (z) + µ (q1 (z)) + q1 (z) ≺ ψλ,l , for α, β, µ, ξ ∈ C, (α, β, µ; z) ≺ α + ξq2 (z) + µ (q2 (z)) + q2 (z)

If f ∈ A,

β 6= 0, implies q1 (z) ≺

m+1,n z δ IRλ,l f (z) m,n f (z)) (IRλ,l

1+δ

≺ q2 (z), and q1 and q2 are respectively the best subordinant and the best

dominant. For q1 (z) =

1+A1 z 1+B1 z ,

q2 (z) =

1+A2 z 1+B2 z ,

where −1 ≤ B2 < B1 < A1 < A2 ≤ 1, we have the following corollary. z δ IRm+1,n f (z)

λ,l Corollary 2.8 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.1) and (2.5) hold. If f ∈ A, ∈ 1+δ m,n f (z)) (IRλ,l  2  2 β(A1 −B1 )z m,n 1+A1 z 1+A1 z 1+A2 z 1+A2 z H [q (0) , 1] ∩ Q and α + ξ 1+B + µ 1+B + (1+A ≺ ψλ,l (α, β, µ; z) ≺ α + ξ 1+B + µ 1+B + 1z 1z 1 z)(1+B1 z) 2z 2z

β(A2 −B2 )z (1+A2 z)(1+B2 z) , for α, β, µ, ξ m+1,n z δ IRλ,l f (z) 1+A1 z then 1+B ≺ 1+δ m,n 1z (IR f (z)) λ,l

m,n ∈ C, β 6= 0, −1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1, where ψλ,l is defined in (2.2),

≺

1+A2 z 1+B2 z ,

hence

1+A1 z 1+B1 z

and

1+A2 z 1+B2 z

are the best subordinant and the best dominant,

respectively.

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Alina Alb Lupas 1036-1042

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

For q1 (z) =



1+z 1−z

γ1

, q2 (z) =



1+z 1−z

γ2

, where 0 < γ1 < γ2 ≤ 1, we have the following corollary. z δ IRm+1,n f (z)

λ,l Corollary 2.9 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.1) and (2.5) hold. If f ∈ A, 1+δ ∈ H [q (0) , 1]∩ m,n f (z)) (IRλ,l γ1  2γ1 γ2  2γ2   m,n 1+z 1+z 1+z 1+z 1z 2z +µ 1−z + 2βγ +µ 1−z + 2βγ Q and α+ξ 1−z 1−z 2 ≺ ψλ,l (α, β, µ; z) ≺ α+ξ 1−z 1−z 2 , for α, β, µ, ξ ∈  γ1  γ2 m+1,n z δ IRλ,l f (z) m,n 1+z 1+z C, β 6= 0, 0 < γ1 < γ2 ≤ 1, where ψλ,l is defined in (2.2), then 1−z ≺ , hence 1+δ ≺ m,n 1−z (IRλ,l f (z)) γ1 γ2   1+z 1+z and 1−z are the best subordinant and the best dominant, respectively. 1−z

Changing the functions θ and φ we obtain the following results. m+1,n z δ IRλ,l f (z)

1+δ ∈ H (U ) , f ∈ A, z ∈ U , m, n ∈ N, λ, l ≥ 0 and let the function q (z) be m,n f (z)) (IRλ,l convex and univalent in U such that q (0) = 1, z ∈ U . Assume that   α+β q 00 (z) Re +z 0 > 0, (2.7) β q (z)

Theorem 2.10 Let

for α, β ∈ C, β 6= 0, z ∈ U, and m,n ψλ,l (α, β; z) :=

  δ m+1,n m+2,n δ f (z) f (z) β (l + 1) z IRλ,l βδ (l + 1) z IRλ,l + α +  1+δ  1+δ λ λ m,n m,n IRλ,l f (z) IRλ,l f (z)

(2.8)

 2 m+1,n δ f (z) β (1 + δ) (l + 1) z IRλ,l −  2+δ . λ m,n IRλ,l f (z) If q satisfies the following subordination m,n ψλ,l (α, β; z) ≺ αq (z) + βzq 0 (z) ,

for α, β ∈ C, β 6= 0, z ∈ U, then Proof. Consider p (z) :=

m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l

m+1,n z δ IRλ,l f (z) m,n f (z)) (IRλ,l

Differentiating we get p0 (z) =

1+δ

(2.9)

≺ q (z), z ∈ U, and q is the best dominant.

, z ∈ U , z 6= 0, f ∈ A. The function p is analytic in U and p (0) = 1.

m+1,n δ−1 IRλ,l f (z) δ(1+l) z 1+δ m,n λ (IRλ,l f (z))

+

m+2,n δ−1 IRλ,l f (z) l+1 z λ (IRm,n f (z))1+δ λ,l

2

−

m+1,n δ−1 f (z)) (IRλ,l (l+1)(1+δ) z 2+δ m,n λ (IR f (z))

.

λ,l

By using the identity (1.3), we get  2 m+1,n m+2,n m+1,n δ δ z δ IRλ,l f (z) z IR f (z) z IR f (z) l + 1 δ (1 + l) (l + 1) (1 + δ) λ,l λ,l zp0 (z) =  1+δ +  1+δ −  2+δ . λ λ λ m,n m,n m,n IRλ,l f (z) IRλ,l f (z) IRλ,l f (z)

(2.10)

By setting θ (w) := αw and φ (w) := β, it can be easily verified that θ is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. Also, by letting Q (z) = zq 0 (z) φ (q (z)) = βzq 0 (z) , we find that (z) univalent in U.  Q  is starlike  00 zh0 (z) (z) α+β 0 Let h (z) = θ (q (z)) + Q (z) = αq (z) + βzq (z). We have Re Q(z) = Re β + z qq0 (z) > 0.   m+2,n m+1,n δ δ z IRλ,l f (z) z IRλ,l f (z) By using (2.10), we obtain αp (z) + βzp0 (z) = β(l+1) α + βδ(l+1) 1+δ + 1+δ − m,n m,n λ λ f (z)) f (z)) (IRλ,l (IRλ,l 2 m+1,n δ f (z)) β(1+δ)(l+1) z (IRλ,l 0 0 2+δ . By using (2.9), we have αp (z) + βzp (z) ≺ αq (z) + βzq (z) . From Lemma 1.1, we m,n λ (IRλ,l f (z)) have p (z) ≺ q (z), z ∈ U, i.e.

m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l

≺ q (z), z ∈ U, and q is the best dominant.

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1+Az 1+Bz , z ∈ U, −1 ≤ B < 1+Az + β(A−B)z , for α, β α 1+Bz (1+Bz)2

Corollary 2.11 Let q (z) = f ∈ A and (2.8), then

m,n ψλ,l

(α, β; z) ≺

m+1,n z δ IRλ,l f (z)

≺

1+δ

m,n f (z)) (IRλ,l

Proof. For q (z) =

1+Az 1+Bz ,

1+Az 1+Bz ,

and

1+Az 1+Bz

A ≤ 1, m, n ∈ N, λ, l ≥ 0. Assume that (2.7) holds. If m,n ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where ψλ,l is defined in

is the best dominant.

−1 ≤ B < A ≤ 1, in Theorem 2.10 we get the corollary.

γ  m,n 1+z , m, n ∈ N, λ, l ≥ 0. Assume that (2.7) holds. If f ∈ A and ψλ,l (α, β; z) ≺ Corollary 2.12 Let q (z) = 1−z γ γ   m+1,n δ z IRλ,l f (z) m,n 2βγz 1+z 1+z α 1−z + 1−z , for α, β ∈ C, 0 < γ ≤ 1, β 6= 0, where ψλ,l is defined in (2.8), then 2 1+δ ≺ m,n 1−z IRλ,l f (z)) (  γ  γ 1+z 1+z , and 1−z is the best dominant. 1−z Proof. Corollary follows by using Theorem 2.10 for q (z) =



1+z 1−z

γ

, 0 < γ ≤ 1.

Theorem 2.13 Let q be convex and univalent in U such that q (0) = 1. Assume that   α 0 Re q (z) > 0, for α, β ∈ C, β 6= 0. β If f ∈ A,

m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l in (2.8), then

(2.11)

m,n m,n ∈ H [q (0) , 1] ∩ Q and ψλ,l (α, β; z) is univalent in U , where ψλ,l (α, β; z) is as defined m,n αq (z) + βzq 0 (z) ≺ ψλ,l (α, β; z)

implies q (z) ≺

m+1,n z δ IRλ,l f (z)

(2.12)

, δ ∈ C, δ 6= 0, z ∈ U, and q is the best subordinant.

1+δ

m,n f (z)) (IRλ,l

m+1,n z δ IRλ,l f (z)

1+δ , z ∈ U , z 6= 0, f ∈ A. The function p is analytic in U and p (0) = 1. m,n f (z)) (IRλ,l By setting ν (w) := αw and φ (w) := β it can be easily verified that ν is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}.  0    0 (q(z)) ν (q(z)) α 0 0 Since νφ(q(z)) =α q (z), it follows that Re = Re q (z) > 0, for α, β ∈ C, β 6= 0. β φ(q(z)) β

Proof. Consider p (z) :=

Now, by using (2.12) we obtain αq (z) + βzq 0 (z) ≺ αp (z) + βzp0 (z) , z ∈ U. From Lemma 1.2, we have q (z) ≺ p (z) =

m+1,n z δ IRλ,l f (z) 1+δ

m,n f (z)) (IRλ,l

, z ∈ U, and q is the best subordinant. 1+Az 1+Bz ,

Corollary 2.14 Let q (z) = m+1,n z IRλ,l f (z) 1+δ m,n IRλ,l f (z) δ

holds. If f ∈ A,

(

)

−1 ≤ B < A ≤ 1, z ∈ U, m, n ∈ N, λ, l ≥ 0. Assume that (2.11)

1+Az ∈ H [q (0) , 1] ∩ Q, and α 1+Bz +

m,n −1 ≤ B < A ≤ 1, where ψλ,l is defined in (2.8), then

Proof. For q (z) =

1+Az 1+Bz ,

1+Az 1+Bz

≺

β(A−B)z (1+Bz)2

m,n ≺ ψλ,l (α, β; z) , for α, β ∈ C, β 6= 0,

m+1,n z δ IRλ,l f (z) m,n f (z)) (IRλ,l

1+δ

, and

1+Az 1+Bz

is the best subordinant.

−1 ≤ B < A ≤ 1, in Theorem 2.13 we get the corollary.

 γ m+1,n z δ IRλ,l f (z) 1+z Corollary 2.15 Let q (z) = 1−z , m, n ∈ N, λ, l ≥ 0. Assume that (2.11) holds. If f ∈ A, 1+δ ∈ m,n IRλ,l f (z)) (  γ  γ m,n m,n 2βγz 1+z 1+z H [q (0) , 1] ∩ Q and α 1−z + 1−z ≺ ψλ,l (α, β; z) , for α, β ∈ C, 0 < γ ≤ 1, β 6= 0, where ψλ,l is 2 1−z  γ   m+1,n δ γ z IRλ,l f (z) 1+z 1+z defined in (2.8), then 1−z ≺ is the best subordinant. 1+δ , and m,n 1−z f (z)) (IRλ,l  γ 1+z Proof. Corollary follows by using Theorem 2.13 for q (z) = 1−z , 0 < γ ≤ 1. Combining Theorem 2.10 and Theorem 2.13, we state the following sandwich theorem.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Theorem 2.16 Let q1 and q2 be convex and univalent in U such that q1 (z) 6= 0 and q2 (z) 6= 0, for all m+1,n z δ IRλ,l f (z)

1+δ ∈ H [q (0) , 1] ∩ Q , and m,n f (z)) (IRλ,l m,n m,n ψλ,l (α, β; z) is as defined in (2.8) univalent in U , then αq1 (z) + βzq10 (z) ≺ ψλ,l (α, β; z) ≺ αq2 (z) + βzq20 (z) ,

z ∈ U . Suppose that q1 satisfies (2.7) and q2 satisfies (2.11). If f ∈ A,

for α, β ∈ C, β 6= 0, implies q1 (z) ≺

m+1,n z δ IRλ,l f (z) m,n f (z)) (IRλ,l

1+δ

≺ q2 (z), z ∈ U, and q1 and q2 are respectively the best

subordinant and the best dominant. For q1 (z) =

1+A1 z 1+B1 z ,

q2 (z) =

1+A2 z 1+B2 z ,

where −1 ≤ B2 < B1 < A1 < A2 ≤ 1, we have the following corollary. 1+A1 z 1+B1 z β(A1 −B1 )z ≺ (1+B1 z)2

Corollary 2.17 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.7) and (2.11) hold for q1 (z) = 1+A2 z 1+B2 z ,

and q2 (z) =

m,n 1+A1 z ∈ H [q (0) , 1] ∩ Q and α 1+B + ψλ,l (α, β; z) 1z ( ) m,n 2 −B2 )z + β(A , z ∈ U, for α, β ∈ C, β = 6 0, −1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1, where ψλ,l is defined in (1+B z)2

respectively. If f ∈ A,

1+A2 z ≺ α 1+B 2z

m+1,n z δ IRλ,l f (z) 1+δ m,n IRλ,l f (z)

2

m+1,n z δ IRλ,l f (z)

1+A2 z 1+A1 z 1+A2 z 1+δ ≺ 1+B z , z ∈ U, hence 1+B z and 1+B z are the best subordinant and the m,n 2 1 2 f (z)) (IRλ,l best dominant, respectively.  γ1 γ2  1+z 1+z For q1 (z) = 1−z , q2 (z) = 1−z , where 0 < γ1 < γ2 ≤ 1, we have the following corollary.

(2.2), then

1+A1 z 1+B1 z

≺

γ1  1+z and q2 (z) = Corollary 2.18 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.7) and (2.11) hold for q1 (z) = 1−z  γ2  γ1  γ1 m+1,n δ z IR f (z) m,n λ,l 1+z 1+z 1+z 1z , respectively. If f ∈ A, + 2βγ ≺ ψλ,l (α, β; z) 1+δ ∈ H [q (0) , 1]∩Q and α m,n 1−z 1−z 1−z 2 1−z IRλ,l f (z)) (   γ2 γ2 m,n 1+z 1+z 2z ≺ α 1−z + 2βγ , z ∈ U, for α, β ∈ C, β 6= 0, 0 < γ1 < γ2 ≤ 1, where ψλ,l is defined in (2.2), 1−z 2 1−z  γ1 γ2  γ1  γ2  m+1,n δ z IR f (z) λ,l 1+z 1+z 1+z 1+z ≺ , z ∈ U, hence 1−z and 1−z are the best subordinant and then 1−z 1+δ ≺ m,n 1−z f (z)) (IRλ,l the best dominant, respectively.

References [1] A. Alb Lupas, Differential Sandwich Theorems using a multiplier transformation and Ruscheweyh derivative, Advances in Mathematics: Scientific Journal 4 (2015), no.2, 195-207. [2] A. Alb Lupas, About some differential sandwich theorems using a multiplier transformation and Ruscheweyh derivative, Journal of Computational Analysis and Applications, Vol. 21, No.7 (2016), 1218-1224. [3] F.M. Al-Oboudi, On univalent functions defined by a generalized S˘ al˘ agean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [4] T. Bulboac˘ a, Classes of first order differential superordinations, Demonstratio Math., Vol. 35, No. 2, 287-292. [5] A. C˘ata¸s, On certain class of p-valent functions defined by new multiplier transformations, Proceedings Book of the International Symposium on Geometric Function Theory and Applications, August 20-24, 2007, TC Istanbul Kultur University, Turkey, 241-250. [6] S.S. Miller, P.T. Mocanu, Subordinants of Differential Superordinations, Complex Variables, vol. 48, no. 10, 815-826, October, 2003. [7] S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Dekker Inc., New York, 2000. [8] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [9] G. St. S˘al˘ agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

FUZZY STABILITY OF A CLASS OF ADDITIVE-QUADRATIC FUNCTIONAL EQUATIONS CHANG IL KIM AND GILJUN HAN∗

Abstract. In this paper, we consider the following functional equation af (x + y) + bf (x − y) + cf (y − x) = (a + b)f (x) + cf (−x) + (a + c)f (y) + bf (−y) for a fixed real numbers a, b, c with a = b + c and a 6= 0. We study the fuzzy version of the generalized Hyers-Ulam stability for it in the sense of Mirmostafaee and Moslehian.

1. Introduction and preliminaries In 1940, Ulam proposed the following stability problem (cf. [20]): “Let G1 be a group and G2 a metric group with the metric d. Given a constant δ > 0, does there exists 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 there exists a unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” In the next year, Hyers [11] gave a partial solution of Ulam, s problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki [1] for additive mappings, and by Rassias [19] for linear mappings, to consider the stability problem with unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians ([5], [6], [7], [10], [18]). Recently, the stability in fuzzy spaces has been extensively studied ([3], [12], [15], [16], [17]). The concept of fuzzy norm on a linear space was introduced by Katsaras [14] in 1984. Later, Cheng and Mordeson [4] gave a new definition of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [13]. In 2008, for the first time, Mirmostafaee and Moslehian [16], [17] used the definition of a fuzzy norm in [2] to obtain a fuzzy version of stability for the Cauchy functional equation (1.1)

f (x + y) = f (x) + f (y)

and the quadratic functional equation (1.2)

f (x + y) + f (x − y) = 2f (x) + 2f (y).

2010 Mathematics Subject Classification. 39B52, 39B72, 46S40. Key words and phrases. additive-quadratic mapping, fuzzy almost quadratic-additive mapping, fuzzy normed space. * Corresponding author. 1

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2

CHANG IL KIM AND GILJUN HAN

We call a solution of (1.1) an additive mapping and a solution of (1.2) is called a quadratic mapping. Also, f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y) = 0 is called Drygas functional equation(see [8], [9] for detail.). It is easy to see that the function f (x) = px2 + qx is a solution of Drygas functional equation and so we can expect that a solution of Drygas functional equation is an additive-quadratic mapping. Now, we consider the following functional equation (1.3)

af (x + y) + bf (x − y) + cf (y − x) = (a + b)f (x) + cf (−x) + (a + c)f (y) + bf (−y)

for fixed real numbers a, b, c with a = b + c and a 6= 0 and show the generalized Hyers-Ulam stability of (1.3) in a fuzzy sense [18]. Definition 1.1. Let X be a real vector space. A function N : X × R −→ [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1) N (x, t) = 0 for t ≤ 0; (N2) x = 0 if and only if N (x, t) = 1 for all t > 0; t ) 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 nondecreasing function of R and limt→∞ N (x, t) = 1; (N6) for any x 6= 0, N (x, ·) is continuous on R. In this case, the pair (X, N ) is called a fuzzy normed space. Let (X, N ) be a fuzzy normed space. A sequence {xn } in X is said to be convergent in (X, N ) 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 } in (X, N ) and one denotes it by N − limn→∞ xn = x. A sequence {xn } in X is said to be Cauchy if for any  > 0, there is an m ∈ N such that for any n ≥ m and any positive integer p, N (xn+p − xn , t) > 1 −  for all t > 0. It is well known that every convergent sequence in a fuzzy normed space is Cauchy. A fuzzy normed space is said to be complete if each Cauchy sequence in it is convergent and a complete fuzzy normed space is called a fuzzy Banach space. 2. Solutions and the Generalized Hyers-Ulam stability of (1.3) In this section, we investigate solutions of (1.3) and prove the generalized HyersUlam stability of (1.3) in fuzzy Banach spaces. Throughout this section, we assume that (X, N ) is a fuzzy normed space and (Y, N 0 ) is a fuzzy Banach space. In Theorem 2.3, it can be concluded that any solution of (1.3) is additive-quadratic. We start with the following lemma. Lemma 2.1. Let f : X −→ Y be an odd mapping satisfying (1.3). Then f is an additive mapping. Proof. Since a 6= 0, f (0) = 0. Since f is an odd mapping, the functional equation (1.3) can be written by (2.1)

af (x + y) + (b − c)f (x − y) = (a + b − c)f (x) + (a − b + c)f (y)

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FUZZY STABILITY OF A CLASS OF ADDITIVE-QUADRATIC FUNCTIONAL EQUATIONS 3

for all x, y ∈ X. Interchanging x and y in (2.1), we have (2.2)

af (x + y) − (b − c)f (x − y) = (a + b − c)f (y) + (a − b + c)f (x)

for all x, y ∈ X. By (2.1) and (2.2), af (x + y) = af (x) + af (y) for all x, y ∈ X and since a 6= 0, f is additive.



Lemma 2.2. Let f : X −→ Y be an even mapping satisfying (1.3). Then f is a quadratic mapping. Proof. Since a 6= 0, f (0) = 0. Since f is an even mapping, the functional equation (1.3) can be written by (2.3)

af (x + y) + (b + c)f (x − y) = (a + b + c)f (x) + (a + b + c)f (y)

for all x, y ∈ X. Letting y = −y in (2.3), we have (2.4)

af (x − y) + (b + c)f (x + y) = (a + b + c)f (x) + (a + b + c)f (y)

for all x, y ∈ X. Since a = b + c, by (2.3) and (2.4), we have 2af (x − y) + 2af (x + y) = 4af (x) + 4af (y) for all x, y ∈ X and since a 6= 0, f is a quadratic mapping.



Combining Lemma 2.1 and Lemma 2.2, we have the following theorem. Theorem 2.3. Let f : X −→ Y be a mapping. If f satisfies (1.3), then f is an additive-quadratic mapping. For any mapping f : X −→ Y , we define the difference operator Df : X 2 −→ Y by Df (x, y) = af (x+y)+bf (x−y)+cf (y−x)−(a+b)f (x)−cf (−x)−(a+c)f (y)−bf (−y) for all x, y ∈ X. For a given q > 0, the mapping f is said to be a fuzzy q-almost additive-quadratic mapping if (2.5)

N 0 (Df (x, y), t + s) ≥ min{N (x, tq ), N (y, sq )}

for all x, y ∈ X and all positive real numbers t, s. Theorem 2.4. Let q be a positive real number with q 6= 1, 21 and f : X −→ Y a fuzzy q-almost additive-quadratic mapping. Then there exists a unique additivequadratic mapping F : X −→ Y such that (2.6)  sups 1  N (F (x) − f (x), t) ≥ sups 0 and by (N2), f (0) = 0. Case 1. Let q > 1 and define a mapping Jn f : X −→ Y by

Jn f (x) =

f (2n x) + f (−2n x) f (2n x) − f (−2n x) + 2 · 4n 2 · 2n

for all x ∈ X and all positive integer n. Then we have

Jn f (x) − Jn+1 f (x) (2.7) =

2n+1 − 1 2n+1 + 1 Df (−2n x, −2n x) − Df (2n x, 2n x) n+1 a·2·4 a · 2 · 4n+1

for all x ∈ X and all positive integer n. By (2.5), (2.7), (N3), and (N4), we have (2.8) N 0 (Jm f (x) − Jm+n f (x),

m+n−1 X i=m

2pi p t ) |a| · 2i

m+n−1 X

m+n−1 X

i=m

i=m

= N 0(

[Ji f (x) − Ji+1 f (x)],

2pi p t ) |a| · 2i

2pi p t ) | m ≤ i ≤ m + n − 1} ≥ min{N 0 (Ji f (x) − Ji+1 f (x), |a| · 2i 2i+1 − 1 2i+1 + 1 2pi p i i i i ≥ min{N 0 ( Df (−2 x, −2 x) − Df (2 x, 2 x), t ) | a · 2 · 4i+1 a · 2 · 4i+1 |a| · 2i m ≤ i ≤ m + n − 1} 2i+1 + 1 (2i+1 + 1)2pi p i i Df (2 x, 2 x), t ), a · 2 · 4i+1 |a| · 4i+1 2i+1 − 1 (2i+1 − 1)2pi p i i N 0( Df (−2 x, −2 x), t )} | m ≤ i ≤ m + n − 1} a · 2 · 4i+1 |a| · 4i+1

≥ min{min{N 0 (

≥ min{min{N 0 (Df (2i x, 2i x), 2pi+1 tp ), N 0 (Df (−2i x, −2i x), 2pi+1 tp )}|m ≤ i ≤ m + n − 1} ≥ min{min{N (2i x, 2i t), N (−2i x, 2i t)} | m ≤ i ≤ m + n − 1} = N (x, t) for all x ∈ X, all t > 0, and all positive integers m, n. Let  > 0 be given. Since limt−→∞ N (x, t) = 1, there is a t1 such that N (x, t1 ) > 1 − . Let t2 > t1 . Since P∞ 2pn p p < 1, n=0 |a|·2 n t2 is convergent. Let s > 0. Then there is a positive integer k Pm+n−1 2pi p such that i=m |a|·2i t2 < s for m, n > k and so by (2.8), we have

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FUZZY STABILITY OF A CLASS OF ADDITIVE-QUADRATIC FUNCTIONAL EQUATIONS 5

N 0 (Jm f (x) − Jm+n f (x), s) ≥ N 0 (Jm f (x) − Jm+n f (x),

m+n−1 X i=m

2pi p t ) |a| · 2i 2

≥ N (x, t2 ) ≥1− for all x ∈ X. Hence {Jn f (x)} is a Cauchy sequence in (Y, N 0 ). Since (Y, N 0 ) is a fuzzy Banach space, we can define a mapping F : X −→ Y by F (x) = N 0 − lim Jn f (x) n→∞

for all x ∈ X. Letting m = 0 in (2.8), we have (2.9)

tq N 0 (f (x) − Jn f (x), t) ≥ N (x, Pn−1 2pi ) [ i=0 |a|·2i ]q

for all x ∈ X, all positive integer n, and all t > 0. By (N4), we have N 0 (DF (x, y), t) t t ), N 0 (b[F − Jn f ](x − y), ), 14 14 t t N 0 (c[F − Jn f ](y − x), ), N 0 ((a + b)[F − Jn f ](x), ) (2.10) 14 14 t t − N 0 (c[F − Jn f ](−x), ), N 0 ((a + c)[F − Jn f ](y), ) 14 14 t t 0 0 − N (b[F − Jn f ](−y), ), N (Jn Df (x, y), )} 14 2 for all x, y ∈ X and all positive integer n. The first seven terms on the right-hand of (2.10) tend to 1 as n → ∞ and by (N4), we have t N 0 (Jn Df (x, y), ) 2 n n Df (−2n x, −2n y) t 0 Df (2 x, 2 y) t (2.11) ≥ min{N 0 ( , ), N ( , ), 2 · 4n 8 2 · 4n 8 n n n n Df (−2 x, −2 y) t Df (2 x, 2 y) t N 0( , ), N 0 ( , )} 2 · 2n 8 2 · 2n 8 for all x, y ∈ X, all positive integer n and all t > 0. By (N3) and (2.5), we have ≥ min{N 0 (a[F − Jn f ](x + y),

Df (±2n x, ±2n y) t , ) 2 · 4n 8 = N 0 (Df (±2n x, ±2n y, 4n−1 t)) N 0(

(2.12)

≥ min{N (2n x, 2q(2n−3) tq ), N (2n y, 2q(2n−3) tq )} ≥ min{N (x, 2(2q−1)n−3q tq ), N (y, 2(2q−1)n−3q tq )} for all x, y ∈ X, all positive integer n, and all t > 0. Since q > 1, by (2.11) and (2.12), we have t lim N 0 (Jn Df (x, y), ) = 1 n→∞ 2

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and so by (2.10), N 0 (DF (x, y), t) = 0 for all x, y ∈ X and all t > 0. By (N2), DF (x, y) = 0 for all x, y ∈ X and by Theorem 2.3, F is additive-quadraic. Now we will show that (2.6) holds. Let x ∈ X, t > 0, s > 0 with 0 < s < t and 0 <  < 1. Since F (x) = N 0 − limn→∞ Jn f (x), there is a positive integer n such that N 0 (F (x) − Jn f (x), t − s) ≥ 1 −  and so by (2.9), N 0 (F (x) − f (x), t) ≥ min{N 0 (F (x) − Jn f (x), t − s), N 0 (Jn f (x) − f (x), s)} sq ≥ min{1 − , N (x, Pn−1 2pi )} [ i=0 |a|·2i ]q ≥ min{1 − , N (x, (1 − 2p−1 )q sq |a|q )}. and so we have (2.6). To prove the uniqueness of F , let F1 : X −→ Y be another additive-quadratic mapping satisfying (2.6). Then F (x) − F1 (x) = Jn F (x) − Jn F1 (x) for all x ∈ X and all positive integer n. Hence by (N4), (N5), and (2.6), we have N 0 (F (x) − F1 (x), t) = N 0 (Jn F (x) − Jn F1 (x), t) t t ≥ min{N 0 (Jn F (x) − Jn f (x), ), N 0 (Jn F1 (x) − Jn f (x), )} 2 2 n n F (2n x) − f (2n x) t 0 F (−2 x) − f (−2 x) t ≥ min{N 0 ( ), N ( , , ), 2 · 4n 8 2 · 4n 8 n n n n 0 F (2 x) − f (2 x) t 0 F (−2 x) − f (−2 x) t N( , ), N ( , ), 2 · 2n 8 2 · 2n 8 n n n n 0 F1 (−2 x) − f (−2 x) t 0 F1 (2 x) − f (2 x) t , ), N ( , ), N( 2 · 4n 8 2 · 4n 8 n n n n F (2 x) − f (2 x) t F (−2 x) − f (−2 x) t 1 1 N 0( , ), N 0 ( , )} 2 · 2n 8 2 · 2n 8 ≥ sup{N (2n x, (1 − 2p−1 )q 2(n−3)q sq |a|q )} s 0, and all positive integers m, n. Let  > 0 be given. Since limt−→∞ N (x, t) = 1, there is a t1 such that N (x, t1 ) > 1 − . Let t2 > t1 . Since P∞ 2pn+1 21−p(n+1)+n p 1 < p < 2, n=0 [ |a|·4 ]t2 is convergent. Let s > 0. Then there is a n+1 + |a| Pm+n−1 2pi+1 1−p(i+1)+i positive integer n such that i=m [ |a|·4i+1 + 2 |a| ]tp2 < s for m, n > k and

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so by (2.14), we have N 0 (Jm f (x) − Jm+n f (x), s) ≥ N 0 (Jm f (x) − Jm+n f (x),

m+n−1 X

[

i=m

2pi+1 21−p(i+1)+i p + ]t2 ) i+1 |a| · 4 |a|

≥ N (x, t2 ) ≥1− for all x ∈ X. Hence {Jn f (x)} is a Cauchy sequence in (Y, N 0 ). Since (Y, N 0 ) is a fuzzy Banach space, we can define a mapping F : X −→ Y by F (x) = N 0 − lim Jn f (x) n→∞

for all x ∈ X. Letting m = 0 in (2.14), we have (2.15)

tq N 0 (f (x) − Jn f (x), t) ≥ N (x, Pn−1 2pi+1 [ i=0 ( |a|·4i+1 +

21−p(i+1)+i q )] |a|

)

for all x ∈ X, all positive integer n, and all t > 0. By (N4), we have N 0 (DF (x, y), t) t t ), N 0 (b[F − Jn f ](x − y), ), 14 14 t t N 0 (c[F − Jn f ](y − x), ), N 0 ((a + b)[F − Jn f ](x), ) (2.16) 14 14 t t − N 0 (c[F − Jn f ](−x), ), N 0 ((a + c)[F − Jn f ](y), ) 14 14 t t − N 0 (b[F − Jn f ](−y), ), N 0 (Jn Df (x, y), )} 14 2 for all x, y ∈ X and all positive integer n. The first seven terms on the right-hand of (2.16) tend to 1 as n → ∞ and by (N4), we have t N 0 (Jn Df (x, y), ) 2 n n Df (−2n x, −2n y) t 0 Df (2 x, 2 y) t (2.17) , ), N ( , ), ≥ min{N 0 ( 2 · 4n 8 2 · 4n 8 t t 0 n−1 −n −n 0 n−1 −n N (2 Df (2 x, 2 y), ), N (2 Df (−2 x, −2−n y), )} 8 8 for all x, y ∈ X, all positive integer n and all t > 0. By (N3) and (2.5), we have ≥ min{N 0 (a[F − Jn f ](x + y),

(2.18)

Df (±2n x, ±2n y) t , ) 2 · 4n 8 ≥ min{N (x, 2(2q−1)n−3q tq ), N (y, 2(2q−1)n−3q tq )} N 0(

and (2.19)

t N 0 (2n−1 Df (±2−n x, ±2−n y), ) 8 ≥ min{N (x, 2(1−q)n−3q) tq ), N (y, 2(1−q)n−3q) tq )}

for all x, y ∈ X, all positive integer n, and all t > 0. Since (2.18), and (2.19), we have t lim N 0 (Jn Df (x, y), ) = 1 n→∞ 2

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FUZZY STABILITY OF A CLASS OF ADDITIVE-QUADRATIC FUNCTIONAL EQUATIONS 9

and so by (2.16), N 0 (DF (x, y), t) = 0 for all x, y ∈ X and all t > 0. By (N2), DF (x, y) = 0 for all x, y ∈ X and by Theorem 2.3, F is additive-quadratic. Now we will show that (2.6) holds. Let x ∈ X, t > 0, s > 0 with 0 < s < t and 0 <  < 1. Since F (x) = N 0 − limn→∞ Jn f (x), there is a positive integer n such that N 0 (F (x) − Jn f (x), t − s) ≥ 1 −  and so by (2.15), N 0 (F (x) − f (x), t) ≥ min{N 0 (F (x) − Jn f (x), t − s), N 0 (Jn f (x) − f (x), s)} sq )} ≥ min{1 − , N (x, Pn−1 2pi+1 1−p(i+1)+i [ i=0 ( |a|·4i+1 + 2 |a| )]q ≥ min{1 − , N (x, (2p−1 − 1)q (2 − 2p−1 )q |a|q sq )}. and so we have (2.6). To prove the uniqueness of F , let F1 : X −→ Y be another additive-quadratic mapping satisfying (2.6). Then F (x) − Jn F (x) = F1 (x) − Jn F1 (x) for all x ∈ X and all positive integer n. Hence by (N4), (N5), and (2.6), we have N 0 (F (x) − F1 (x), t) = N 0 (Jn F (x) − Jn F1 (x), t) t t ≥ min{N 0 (Jn F (x) − Jn f (x), ), N 0 (Jn F1 (x) − Jn f (x), )} 2 2 F (2n x) − f (2n x) t F (−2n x) − f (−2n x) t ≥ min{N 0 ( , ), N 0 ( , ), n 2·4 8 2 · 4n 8 t t 0 n−1 −n −n 0 n−1 −n N (2 [F (2 x) − f (2 x)], ), N (2 [F (−2 x) − f (−2−n x)], ), 8 8 n n F1 (2n x) − f (2n x) t 0 F1 (−2 x) − f (−2 x) t N 0( ), N ( , , ), 2 · 4n 8 2 · 4n 8 t t N 0 (2n−1 [F1 (2−n x) − f (2−n x)], ), N 0 (2n−1 [F1 (−2−n x) − f (−2−n x)], )} 8 8 ≥ sup{N (±2n x, (2p−1 − 1)q (2 − 2p−1 )q 4(n−1)q |a|q sq )} s 0. Hence F = F1 . Case 3. Let 0 < q < Jn f (x) = 2

2n−1

1 2

1 2

< q < 1, N 0 (F (x) −

and define a mapping Jn f : X −→ Y by

[f (2−n x) + f (−2−n x)] + 2n−1 [f (2−n x) − f (−2−n x)]

for all x ∈ X and all positive integer n. Then we have (2.20) Jn f (x) − Jn+1 f (x) =

22n−1 + 2n−1 22n−1 − 2n−1 Df (2−(n+1) x, 2−(n+1) x) + Df (−2−(n+1) x, −2−(n+1) x) a a

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CHANG IL KIM AND GILJUN HAN

for all x ∈ X and all positive integer n. By (2.5), (2.20), (N3), and (N4), we have N 0 (Jm f (x) − Jm+n f (x),

m+n−1 X i=m

21−p(i+1)+2i p t ) |a|

m+n−1 X

m+n−1 X

i=m

i=m

= N 0(

[Ji f (x) − Ji+1 f (x)],

21−p(i+1)+2i p t ) |a|

21−p(i+1)+2i p ≥ min{N 0 (Ji f (x) − Ji+1 f (x), t ) | m ≤ i ≤ m + n − 1} |a| 22i−1 + 2i−1 Df (2−(i+1) x, 2−(i+1) x) ≥ min{N 0 ( a 22i−1 − 2i−1 21−p(i+1)+2i p + Df (−2−(i+1) x, −2−(i+1) x), t )} | m ≤ i ≤ m + n − 1} a |a| 22i−1 + 2i−1 22i−1 + 2i−1 1−p(i+1) p ≥ min{min{N 0 ( Df (2−(i+1) x, 2−(i+1) x), 2 t ), a |a| 22i−1 − 2i−1 1−p(i+1) p 22i−1 − 2i−1 t )} Df (−2−(i+1) x, −2−(i+1) x), 2 N 0( a |a| | m ≤ i ≤ m + n − 1} ≥ min{min{N 0 (Df (2−(i+1) x, 2−(i+1) x), 21−p(i+1) tp ), N 0 (Df (−2−(i+1) x, −2−(i+1) x), 21−p(i+1) tp )} | m ≤ i ≤ m + n − 1} ≥ min{min{N (2−(i+1) x, 2−(i+1) t), N (−2−(i+1) x, 2−(i+1) t)} | m ≤ i ≤ m + n − 1} = N (x, t) for all x ∈ X, all t > 0, and all positive integers m, n. Similar to Case 1. and Case 2., there is a unique cubic mapping C : X −→ Y with (2.6).  We can use Theorem 2.4 to get a classical result in the framework of normed spaces. For example, it is well known that for any normed space (X, || · ||), the mapping NX : X × R −→ [0, 1], defined by ( 0, if t < kxk NX (x, t) = 1, if t ≥ kxk a fuzzy norm on X. In [15], [16] and [17], some examples are provided for the fuzzy norm NX . Here using the fuzzy norm NX , we have the following corollary. Corollary 2.5. Let f : X −→ Y be a mapping such that f (0) = 0 and kDf (x, y)k ≤ kxkp + kykp

(2.21)

for a fixed positive number p such that p 6= 1, 2. Then there exists a unique additivequadratic mapping F : X −→ Y such that the inequality

kF (x) − f (x)k ≤

 1 p   (1−2p−1 )|a| kxk ,

1 kxkp , (2p−1 −1)(2−2(p−1) )|a|   1 p (2p−1 −2)|a| kxk ,

if 1 < p if 1 < p < 2 if 2 < p

holds for all x ∈ X.

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FUZZY STABILITY OF A CLASS OF ADDITIVE-QUADRATIC FUNCTIONAL EQUATIONS11

Proof. By the definition of NY , we have ( 0, if s + t ≤ kDf (x, y)k NY (Df (x, y), s + t) = 1, if s + t ≥ kDf (x, y)k. for all x, y ∈ X and all s, t ∈ R. Now, we claim that NY (Df (x, y), s + t) ≥ min{NX (x, sq ), NX (y, tq )} for all x, y ∈ X and s, t > 0. If NY (Df (x, y), s + t) = 1, then it is trivial. Suppose that NY (Df (x, y), s + t) = 0. Then s + t ≤ kDf (x, y)k and by (2.21), either s ≤ kxkp or t ≤ kykp . Hence either NX (x, sq ) = 0 or NX (y, tq ) = 0 and thus f is a fuzzy q-almost additive-quadratic mapping. By Theorem 2.4, we have the results.  The condition p 6= 1, 2 in Corollary 2.5 is indispensable. The following example shows that the inequality (2.21) is not stable for p = 1, 2, especially in the case of b = 2 and c = −1. We will give the proof when p = 1, and the proof when p = 2 is (−x) (−x) similar. For any f : X −→ Y , let fo (x) = f (x)−f and fe (x) = f (x)+f . 2 2 Example 2.6. Define mappings t, s : R −→ R by   if |x| < 1 x, t(x) = −1, if x ≤ −1   1, if 1 ≤ x, ( x2 , if |x| < 1 s(x) = 1, ortherwise and a mapping f : R −→ R by f (x) =

∞ X t(2n x) s(2n x) [ n + ] 2 4n n=0

We will show that there is a positive integer M such that (2.22)

|D2 f (x, y)| ≤ M (|x| + |y|)

for all x, y ∈ R, where D2 g(x, y) = g(x + y) + 2g(x − y) − g(y − x) − 3g(x) + g(−x) − 2g(−y). But there do not exist an additive-quadratic mapping F : R −→ R and a nonnegative constant K such that (2.23)

|F (x) − f (x)| ≤ K|x|2

for all x ∈ R. Proof. Note that so (x) = 0, to (x) = t(x), and |fo (x)| ≤ 2 for all x ∈ R. First, suppose that 21 ≤ |x| + |y|. Then |D2 fo (x, y)| ≤ 40(|x| + |y|). Now suppose that 1 2 > |x| + |y|. Then there is a non-negative integer m such that 1 1 ≤ |x| + |y| < m+1 2m+2 2

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and so 2m |x| < 12 , 2m |y| < 21 . Hence {2m (x ± y), 2m x, 2m y} ⊆ (−1, 1) and so for any n = 0, 1, 2, · · ·, m, D2 t0 (2n x, 2n y) = 0 for all x, y ∈ X. Thus D2 fo (x, y) =

∞ ∞ X X 40 1 1 n n D t(2 x, 2 y) = D t(2n x, 2n y) ≤ m+2 ≤ 40(|x|+|y|). n 2 n 2 2 2 2 n=m+1 n=0

Note that te (x) = 0, se (x) = s(x), and |fe (x)| ≤ 34 for all x ∈ R. First, suppose that 14 ≤ |x| + |y|. Then |D2 fe (x, y)| ≤ 128 3 (|x| + |y|) for all x, y ∈ R. Now suppose that 14 > |x| + |y|. Then there is a non-negative integer k such that   21 1 1 < k+1 . ≤ |x| + |y| 2k+2 2 Hence {2k (x ± y), 2k x, 2k y} ⊆ (−1, 1) and so for any n = 0, 1, 2, · · ·, m, D2 se (2n x, 2n y) = 0. Hence D2 fe (x, y) =

∞ ∞ X X 8 1 1 1 n n D s (2 x, 2 y) = D s (2n x, 2n y) ≤ · 2k . n 2 e n 2 e 4 4 3 2 n=0 n=k+1

and so we have 

D2 fe (x, y)

 12

≤4

 8  12  3

 21 |x| + |y| .

Thus we have D2 fe (x, y) ≤

128 (|x| + |y|). 3

and so we have (2.22). Suppose that there exist an additive mapping A : R −→ R, a quadratic mapping Q : R −→ R, and a non-negative constant K such that A + Q satisfies (2.23). Since |f (x)| ≤ 10 3 , by (2.23), we have 10 A(x) 10 − K|x|2 ≤ + Q(x) ≤ + K|x|2 3n n 3n for all x ∈ X and all positive integers n and so |Q(x)| ≤ K|x|2 for all x ∈ X. Hence by (2.23), we have |f − A(x)| ≤ 2K|x|2 for all x ∈ X. Since fo , A are odd and fe is even, i 1h (2.24) |fe (x)| ≤ |fe (x) + fo (x) − A(x)| + |fe (−x) + fo (−x) − A(−x)| ≤ 4K|x|2 2 for all x ∈ X. Take a positive integer l such that l > 4K, and pick x ∈ R with 0 < 2l x < 1. Then fe (x) =

∞ l−1 X s(2n x) X s(2n x) ≥ ≥ lx2 > 4Kx2 n n 4 4 n=0 n=0

which contradicts to (2.24).



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FUZZY STABILITY OF A CLASS OF ADDITIVE-QUADRATIC FUNCTIONAL EQUATIONS13

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. 3(2003), 687705. [3] I. S. Chang and Y. H. Lee, Additive and quadratic type functional equation and its fuzzy stability, Results in Mathematics 63(2013), 717-730. [4] S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86(1994), 429436. [5] P. W. Cholewa, Remarkes on the stability of functional equations, Aequationes Math., 27(1984), 76-86. [6] K. Cieplinski, Applications of fixed point theorems to the hyers-ulam stability of functional equation-A survey, Ann. Funct. Anal. 3 (2012), no. 1, 151-164. [7] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62(1992), 59-64. [8] H. Drygas, Quasi-inner products and their applications A. K. Gupta (ed.), Advances in Multivariate Statistical Analysis, 13-30, Reidel Publ. Co., 1987. [9] V. A. Faiziev, P. K. Sahoo, On the stability of Drygas functional equation on groups, Banach Journal of Mathematical Analysis, 01/2007; 1(2007), 43-55. [10] P. Gˇ avruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [11] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. [12] H. M. Kim, J. M. Rassias, and J. Lee, Fuzzy approximation of Euler-Lagrange quadratic mappings, Journal of inequalities and Applications, 2013(2013), 1-15. [13] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11(1975), 326334. [14] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst, 12(1984), 143154. [15] A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 159(2008), 730738. [16] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost quadratic functions, Results Math. 52 (2008), 161177. [17] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159(2008), 720729. [18] M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bulletin of the Brazilian Mathematical Society, vol. 37, no. 3, pp. 361376, 2006. [19] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [20] S. M. Ulam, A collection of mathematical problems, Interscience Publisher, New York, 1964. Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 448-701, Korea E-mail address: kci206@@hanmail.net Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 448-701, Korea E-mail address: [email protected]

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Exact controllability for fuzzy differential equations using extremal solutions Jin Hee Jeong∗ Department of Environmental Engineering, Dong-A University, Busan 604-714, South Korea [email protected] Jeong Soon Kim, Hae Eun Youm Department of Mathematics, Dong-A University, Busan 604-714, South Korea [email protected](J.S. Kim), [email protected](H.E. Youm) Jin Han Park† Department of Applied Mathematics, Pukyong National University, Busan 608-737, South Korea [email protected]

Abstract In this paper, we devoted study exact controllability for fuzzy differential equations with the control function in credibility spaces. Moreover we study exact controllability for every solutions of fuzzy differential equations. The result is obtained by using extremal solutions.

1

Introduction

The theory of controlled processes is one of the most recent mathematical concepts to enable very important applications in modern engineering. However, actual systems subject to control do not admit a strictly deterministic analysis in view of various random factors that influence their behavior. The theory of controlled processes takes the random nature of a systems behavior into account. Many researchers have studied controlled processes in a credibility space. Arapostathis et al. [1] studied the controllability properties of the class of stochastic differential systems characterized by a linear controlled diffusion perturbed by a ∗ This

study was supported by research funds from Dong-A University. author: [email protected] (J.H. Park)

† Corresponding

1

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smooth, bounded, and uniformly Lipschitz nonlinearity. Kwun et al. [8] proved the approximate controllability for fuzzy differential equations driven by Liu process. Lee et al. [10] examined the exact controllability for abstract fuzzy differential equations in a credibility space. Recently, Kwun et al. [14] studied the existence of extremal solutions for fuzzy differential equations driven by Liu process. Kwun et al. [6, 7] have studied the existence of extremal solutions for fuzzy differential equations in a n-dimensional fuzzy vector space. In this paper, using the extremal solutions, we study the exact controllability for every solutions of fuzzy differential equations in credibility space. We consider the following fuzzy differential equation: { dx(t, θ) = f (t, x(t, θ))dCt + Bu(t), t ∈ [0, T ], (1) x(0) = x0 ∈ EN , where the state function x(t, θ) takes values in X(⊂ EN ) and another bounded space Y (⊂ EN ). EN is the set of all upper semi-continuously convex fuzzy numbers on R, (Θ, P, Cr) is credibility space, the state function x : [0, T ] × (Θ, P, Cr) → X is a fuzzy process, f : [0, T ] × X → X is a regular fuzzy function, u : [0, T ] × (Θ, P, Cr) → Y is a control function, B is a linear bounded operator from Y to X. Ct is a standard Liu process, x0 ∈ EN is an initial value.

2

Preliminaries

In this section, we give basic definitions, terminologies, notations and lemmas which are most relevant to our investigated and are needed in later section. All undefined concepts and notions used here are standard. A fuzzy set of Rn is a function u : Rn → [0, 1]. For each fuzzy set u, we denote by [u]α = {x ∈ Rn : u(x) ≥ α} for any α ∈ [0, 1], its α-level set. Let u, v be fuzzy sets of Rn . It is well known that [u]α = [v]α for each α ∈ [0, 1] implies u = v. Let E n denote the collection of all fuzzy sets of Rn that satisfies the following conditions: (1) u is normal, i.e., there exists an x0 ∈ Rn such that u(x0 ) = 1; (2) u is fuzzy convex, i.e., u(λx + (1 − λ)y) ≥ min{u(x), u(y)} for any x, y ∈ Rn , 0 ≤ λ ≤ 1; (3) u(x) is upper semi-continuous, i.e., u(x0 ) ≥ limk→∞ u(xk ) for any xk ∈ Rn (k = 0, 1, 2, . . .), xk → x0 ; (4) [u]0 is compact. Definition 2.1. [17] The complete metric DL on EN is defined by DL (u, v) = sup dL ([u]α , [v]α ) 0 0, for all t ∈ [0, T ]. Now we assume the following hypotheses: (H1) For L1 , L2 > 0, x0 ∈ EN , ( ) ( ) dL [U (t)x0 ]α , [x0 ]α ≤ L1 , dL [S(t)x0 ]α , [x0 ]α ≤ L2 . (H2) For x(·), y(·) ∈ C([0, T ]×(Θ, P, Cr ), EN ), t ∈ [0, T ], there exist positive numbers m1 , m2 such that ( ) dL [G(t, x)]α , [G(t, y)]α ≤ m1 dL ([x]α , [y]α ), ( ) dL [F (t, x)]α , [F (t, y)]α ≤ m2 dL ([x]α , [y]α ) and F (0, X{0} (0)) ≡ 0, G(0, X{0} (0)) ( ≡ 0.

) (H3) For L3 > 0, x0 ∈ EN , dL [x0 ]α , [X{0} (0)]α ≤ L3 .

(H4) For ε > 0, (L1 + cm1 KL3 T )ecm1 KT ≤ ε. (H5) For ε > 0, (L2 + dm2 KL3 T )edm2 KT ≤ ε. (H6) Let a, b be, respectively, lower solution and upper solution of equation (1)(u ≡ 0), then [a, b] is convex. We define the controllability concept for a fuzzy differential equation. Definition 3.1. The equation (1) is said to be controllable on [0,T], if for every x0 ∈ EN there exists a control ut ∈ Y such that every solutions x(·) of (1) satisfies a.s. θ, xT = x1 ∈ X (i.e., [xT ]α = [x1 ]α ). Definition 3.2. Define the fuzzy mappings P1 : Pe(R) → X and P2 : e P (R) → X by { ∫T U α (T − s)Bvs ds, v ⊂ Γu , 0 P1α (v) = 0, otherwise, { ∫T S α (T − s)Bvs ds, v ⊂ Γu , 0 P2α (v) = 0, otherwise, where Pe(R) is a nonempty fuzzy subset of R and Γu is the closure of support α α u. Then there exist P1i , P2i (i = l, r) such that ∫ T α 1 P1l (vl ) = Ulα (T − s)B(vs )l ds, (vs )l ∈ [(us )α l , (us ) ], 0

∫

T

Urα (T − s)B(vs )r ds, (vs )r ∈ [(us )1 , (us )α r ],

α P1r (vr ) = 0

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∫ α P2l (vl )

T 1 Slα (T − s)B(vs )l ds, (vs )l ∈ [(us )α l , (us ) ],

= 0

∫

T

Srα (T − s)B(vs )r ds, (vs )r ∈ [(us )1 , (us )α r ].

α P2r (vr ) = 0

α eα eα α We assume that Pe1l , P1r , P2l and Pe2r are bijective mappings.

By Definition 3.2, we can introduce α-level set of us is α [us ]α = [(us )α l , (us )r ] ∫ T } 1 [ eα −1 { 1 α α α α (x )l − Ul (T )(x0 )l − Ulα (T − s)Gα = (P1l ) l (s, (xs )l )dCs 2 0 ∫ T { } α α α α α α −1 (x1 )α − S (T )(x ) − S (T − s)F (s, (x ) )dC , +(Pe2l ) 0 s s l l l l l l 0 T

∫ { α α α −1 (x1 )α − U (T )(x ) − (Pe1r ) 0 r r r

α Urα (T − s)Gα r (s, (xs )r )dCs

}

0

{

α α α −1 ) (x1 )α +(Pe2r r − Sr (T )(x0 )r −

∫

T

Srα (T − s)Frα (s, (xs )α r )dCs

}] .

0

Theorem 3.1. If Lemma 2.3 and hypotheses (H1)-(H5) are satisfied, then the equation (4) is controllable on [0, T ]. Proof By Definition 3.2 and above us , substitute the control into the equation (4) yields α-level of xT . α

∫

[

T

U (T − s)G(s, xs )dCs +

[xT ] = U (T )x0 + 0

[

∫

T

]α U (T − s)Bus ds

0

∫

∫

T

Ulα (T − s)B

0

×

0

∫ T } 1 [ eα −1 { 1 α α α α (P1l ) (x )l − Ulα (T )(x0 )α − U (T − s)G (s, (x ) )dC s l s l l l 2 0 ∫ T { }] α −1 α α +(Pe2l ) (x1 )α Slα (T − s)Flα (s, (xs )α ds, l − Sl (T )(x0 )l − l )dCs 0

∫ Urα (T )(x0 )α r

[

∫

T

Urα (T

+

−

α s)Gα r (s, (xs )r )dCs

0

×

T

α Ulα (T − s)Gα l (s, (xs )l )dCs +

= Ulα (T )(x0 )α l +

T

Urα (T − s)B

+ 0

∫ T } 1 [ eα −1 { 1 α α α α (P1r ) (x )r − Urα (T )(x0 )α − U (T − s)G (s, (x ) )dC s s r r r r 2 0 ∫ T }] ] { α −1 α α α α α +(Pe2r ) (x1 )α − S (T )(x ) − S (T − s)F (s, (x ) )dC ds 0 s s r r r r r r 0

∫

T α Ulα (T − s)Gα l (s, (xs )l )dCs

= Ulα (T )(x0 )α l + 0

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∫ T } 1 α [ eα −1 { 1 α α α α (x )l − Ul (T )(x0 )l − Ulα (T − s)Gα + P1l (P1l ) l (s, (xs )l )dCs 2 0 ∫ T { }] α α α α α α −1 (x1 )α − S (T )(x ) − S (T − s)F (s, (x ) )dC , +(Pe2l ) 0 s s l l l l l l ∫

0 T α Urα (T − s)Gα r (s, (xs )r )dCs

Urα (T )(x0 )α r + 0

∫ T } 1 α [ eα −1 { 1 α α + P1r (x )r − Urα (T )(x0 )α − Urα (T − s)Gα (P1r ) r r (s, (xs )r )dCs 2 0 ∫ T }]] { α α α −1 Srα (T − s)Frα (s, (xs )α (x1 )α +(Pe2r ) r )dCs r − Sr (T )(x0 )r − 0 1 α 1 α = [(x1 )α l , (x )r ] = [x ] .

Hence this control ut satisfy a.s. θ, xT = x1 . Also, using this control, we shall show that the nonlinear operator Φ1 defined by ∫ t ∫ t (Φ1 x)t = U (t)x0 + U (t − s)G(s, xs )dCs + U (t − s)B 0

0

∫ T { } 1[ × Pe1−1 x1 − U (T )x0 − U (T − τ )G(τ, xτ )dCτ 2 0 ∫ T }] { S(T − τ )F (τ, xτ )dCτ ds, +Pe2−1 x1 − S(T )x0 − 0

where the fuzzy mappings (Pe1 )−1 satisfy above statements. Form hypothesis (H2) and Lemma 2.3, for any given θ with Cr{θ} > 0, x(·), y(·) ∈ C([0, T ] × (Θ, P, Cr), EN ), we have ( ) dL [(Φ1 x)t ]α , [(Φ1 y)t ]α ∫ t ([ = dL U (t)x0 + U (t − s)G(s, xs )dCs 0

∫

t

∫

t

∫ T } 1 [ e−1 { 1 + U (t − s)B P1 x − U (T )x0 − U (T − τ )G(τ, xτ )dCτ 2 0 0 ∫ T }] ]α { +Pe2−1 x1 − S(T )x0 − S(T − τ )F (τ, xτ )dCτ ds , 0 ∫ t [ U (t)x0 + U (t − s)G(s, ys )dCs 0

∫ T } 1 [ e−1 { 1 P1 x − U (T )x0 − U (T − τ )G(τ, yτ )dCτ 2 0 0 ∫ T { }] ]α ) −1 +Pe2 x1 − S(T )x0 − S(T − τ )F (τ, yτ )dCτ ds 0 ([ ∫ t ]α [ ∫ t ]α ) ≤ dL U (t − s)G(s, xs )dCs , U (t − s)G(s, ys )dCs U (t − s)B

+

0

0

8

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([ ∫

∫ T } 1 [ e−1 { 1 U (T − τ )G(τ, xτ )dCτ U (t − s)B P1 x − U (T )x0 − 2 0 ∫ T { }] ]α S(T − τ )F (τ, xτ )dCτ ds , +Pe2−1 x1 − S(T )x0 −

t

+dL 0

0

∫ T } { 1[ U (T − τ )G(τ, yτ )dCτ U (t − s)B Pe1−1 x1 − U (T )x0 − 2 0 0 ∫ T { }] ]α ) S(T − τ )F (τ, yτ )dCτ ds +Pe2−1 x1 − S(T )x0 − 0 ([ ∫ t ]α [ ∫ t ]α ) ≤ dL U (t − s)G(s, xs )dCs , U (t − s)G(s, ys )dCs ∫

t

0

+dL

([ 1 2

0

∫ { −1 1 e P1 P1 x − U (T )x0 −

T

U (T − τ )G(τ, xτ )dCτ

}

0

∫ T { }]α 1 + P1 Pe2−1 x1 − S(T )x0 − S(T − τ )F (τ, xτ )dCτ , 2 0 ∫ T [1 { } P1 Pe1−1 x1 − U (T )x0 − U (T − τ )G(τ, yτ )dCτ 2 0 ∫ T { }]α ) 1 S(T − τ )F (τ, yτ )dCτ + P1 Pe2−1 x1 − S(T )x0 − 2 0 ]α [ ∫ T ]α ) ([ ∫ t U (t − s)G(s, ys )dCs ≤ dL U (t − s)G(s, xs )dCs , 0

0

([ ∫

T

+dL 0

∫

t

≤ cm1 K

]α [ ∫ t ]α ) U (T − s)G(s, xs )dCs , U (T − s)G(s, ys )dCs

(

0 T

∫

)

dL [xs ]α , [ys ]α ds + cm1 K 0

( ) dL [xs ]α , [ys ]α ds.

0

Therefore, by Lemma 2.1, we get ) E H1 (Φ1 x, Φ1 y) ( ( )) = E sup DL (Φ1 x)t , (Φ1 y)t (

( =E

t∈[0,T ]

sup

( )) sup dL [(Φ1 x)t ]α , [(Φ1 y)t ]α

t∈[0,T ] 0 |x ∈ U } = {< x, [µ− A (x), µA (x)], [γA (x), γA (x)] > |x ∈ U }, + − + + + where µA (x) = [µ− A (x), µA (x)] and γA (x) = [γA (x), γA (x)] satisfy 0 ≤ µA (x) + γA (x) ≤ 1 for all x ∈ U, and are, respectively, called the degree of membership and the degree of non-membership of the element x ∈ U to A. Let IV IF (U ) denotes the family of all interval-valued intuitionistic fuzzy sets on U .

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3

Construction of generalized interval-valued intuitionistic fuzzy soft rough sets

In this section, we will present the concept of generalized IVIF soft rough sets by using the IVIF soft relation defined by us. Definition 3.1 ( [14]) Let U be an initial universe set and E be a universe set of parameters. A pair (F, E) is called an IVIF soft set over U if F : E → IV IF (U ), where IV IF (U ) is the set of all IVIF subsets of U. In the following, an IVIF soft relation will be presented, which is important for us to construct generalized IVIF soft rough sets. Definition 3.2 Let (F, E) be an IVIF soft set over U . Then an IVIF subset of U × E called an IVIF soft relation from U to E is uniquely defined by R = {< (u, x), µR (u, x), γR (u, x) > |(u, x) ∈ U × E}, where µR : U × E → Int[0, 1] and γR : U × E → Int[0, 1], for all (u, x) ∈ U × E such + − + that µR (u, x) = [µ− R (u, x), µR (u, x)] and γR (u, x) = [γR (u, x), γR (u, x)], which satisfy the + + condition 0 ≤ µR (u, x) + γR (u, x) ≤ 1. + − + Remark 3.3 In Definition 3.2, if µ− R (u, x) = µR (u, x) and γR (u, x) = γR (u, x), namely, µR : U × E → [0, 1] and γR : U × E → [0, 1], for all (u, x) ∈ U × E such that 0 ≤ µR (u, x) + γR (u, x) ≤ 1, then R is referred to as an intuitionistic fuzzy soft relation on U × E. If R is an intuitionistic fuzzy soft relation on U × E and µR (u, x) + γR (u, x) = 1, then R is degenerated to a fuzzy soft relation [8] in Definition 2.4. Hence, among fuzzy soft relation, intuitionistic fuzzy soft relation [42] and IVIF soft relation, the IVIF soft relation is the most generalized one. That is, the IVIF soft relation has included fuzzy soft relation and intuitionistic fuzzy soft relation.

Let U = {u1 , u2 , · · · , um } and E = {x1 , x2 , · · · , xn }. Then the IVIF soft relation R from U to E can be presented by a table as in the following form

··· ··· ··· .. .

xn (µR (u1 , xn ), γR (u1 , xn )) (µR (u2 , xn ), γR (u2 , xn )) .. .

um (µR (um , x1 ), γR (um , x1 )) (µR (um , x2 ), γR (um , x2 )) · · ·

(µR (um , xn ), γR (um , xn ))

R u1 u2 .. .

x1 (µR (u1 , x1 ), γR (u1 , x1 )) (µR (u2 , x1 ), γR (u2 , x1 )) .. .

x2 (µR (u1 , x2 ), γR (u1 , x2 )) (µR (u2 , x2 ), γR (u2 , x2 )) .. .

From the above form and the definition of IVIF soft set, we know that every IVIF soft set (F, E) is uniquely characterized by the IVIF soft relation, namely they are mutual determined. It means that an IVIF soft set (F, E) is formally equal to IVIF soft relation.

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Therefore, we shall identify any IVIF soft set with IVIF soft relation and view these two concepts as interchangeable. Now, any discussion regard to IVIF soft set could be converted into analysis about IVIF soft relation, which will bring great convenience for our future researches. In this case, according to the definition of IVIF soft relation, we can construct generalized IVIF soft rough sets as follows. Definition 3.4 Let U be an initial universe set and E be a universe set of parameters. For an arbitrary IVIF soft relation R over U × E, the pair (U, E, R) is called an IVIF soft approximation space. For any A ∈ IV IF (E), we define the upper and lower soft approximations of A with respect to (U, E, R), denoted by R(A) and R(A), respectively, as follows: R(A) = {< u, µR(A) (u), γR(A) (u) > |u ∈ U }, (1) R(A) = {< u, µR(A) (u), γR(A) (u) > |u ∈ U }.

(2)

where W + W − (µR (u, x) ∧ µ+ µR(A) (u) = [ (µR (u, x) ∧ µ− A (x))], A (x)), x∈E x∈E V V − + + − (x))], γR(A) (u) = [ (u, x) ∨ γA (γR (x)), (γR (u, x) ∨ γA x∈E x∈E V + V − µR(A) (u) = [ (γR (u, x) ∨ µ+ (γR (u, x) ∨ µ− A (x))], A (x)), x∈E x∈E W + W − + − γR(A) (u) = [ (x))]. (µR (u, x) ∧ γA (x)), (µR (u, x) ∧ γA x∈E

x∈E

The pair (R(A), R(A)) is referred to as a generalized IVIF soft rough set of A with respect to (U, E, R). + + + By µ+ R (u, x) + γR (u, x) ≤ 1 and µA (x) + γA (x) ≤ 1, it can be easily verified that R(A) and R(A) ∈ IV IF (U ). So we call R, R : IV IF (E) → IV IF (U ) generalized upper and lower IVIF soft rough approximation operators, respectively.

Remark 3.5 If R is an intuitionistic fuzzy soft relation on U × E, then generalized IVIF soft rough approximation operators R(A) and R(A) in Definition 3.4 degenerate to the following forms: R(A) = {< u, µR(A) (u), γR(A) (u) > |u ∈ U }, R(A) = {< u, µR(A) (u), γR(A) (u) > |u ∈ U }. where W V µR(A) (u) = (µR (u, x) ∧ µA (x)), γR(A) (u) = (γR (u, x) ∨ γA (x)), x∈E

x∈E

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µR(A) (u) =

V

(γR (u, x) ∨ µA (x)), γR(A) (u) =

x∈E

W

(µR (u, x) ∧ γA (x)).

x∈E

In that case, the pair (R(A), R(A)) is generated into a generalized IF soft rough set of A with respect to (U, E, R) proposed by Zhang et al. [42]. That is, generalized IVIF soft rough set in Definition 4.4 includes generalized IF soft rough set [42] as a special case. Remark 3.6 If R is a fuzzy soft relation on U × E and A ∈ F (E), then generalized IVIF soft rough approximation operators R(A) and R(A) degenerate to the following forms: R(A) = {< u, µR(A) (u) > |u ∈ U }, R(A) = {< u, µR(A) (u) > |u ∈ U }. where µR(A) (u) =

W

[µR (u, x) ∧ µA (x)], µR(A) (u) =

x∈E

V

[(1 − µR (u, x)) ∨ µA (x)].

x∈E

In that case, generalized IVIF soft rough approximation operators R(A) and R(A) are identical with the soft fuzzy rough approximation operators defined by Sun [23]. That is, generalized IVIF soft rough approximation operators in Definition 4.4 are an extension of the soft fuzzy rough approximation operators defined by Sun [23]. In order to better understand the concept of generalized IVIF soft rough approximation operators, let us consider the following example. Example 3.7 Suppose that U = {u1 , u2 , u3 , u4 , u5 } is the set of five houses under consideration of a decision maker to purchase. Let E be a parameter set, where E = {e1 , e2 , e3 , e4 }={expensive; beautiful; size; location}. Mr. X wants to buy the house which qualifies with the parameters of E to the utmost extent from available houses in U . Assume that Mr. X describes the “attractiveness of the houses” by constructing an IVIF soft relation R from U to E. And it is presented by a table as in the following form.

R u1 u2 u3 u4 u5

e1 ([0.7, 0.8], [0.2, 0.2]) ([0.1, 0.2], [0.4, 0.6]) ([0.5, 0.6], [0.2, 0.4]) ([0.1, 0.3], [0.2, 0.6]) ([0.8, 0.9], [0.0, 0.1])

e2 ([0.3, 0.4], [0.2, 0.5]) ([0.6, 0.7], [0.1, 0.2]) ([0.3, 0.6], [0.2, 0.3]) ([0.5, 0.7], [0.1, 0.2]) ([0.3, 0.5], [0.4, 0.5])

e3 ([0.1, 0.1], [0.7, 0.8]) ([0.2, 0.3], [0.5, 0.7]) ([0.5, 0.7], [0.1, 0.3]) ([0.1, 0.4], [0.3, 0.5]) ([0.6, 0.8], [0.1, 0.2])

e4 ([0.3, 0.4], [0.1, 0.3]) ([0.3, 0.6], [0.2, 0.3]) ([0.1, 0.8], [0.1, 0.2]) ([0.2, 0.3], [0.5, 0.7]) ([0.4, 0.6], [0.1, 0.4])

We can see that the precise evaluation for each object on each parameter is unknown while the lower and upper limits of such an evaluation are given. For example, we can not present the precise membership degree and non-membership degree of how beautiful house u2 is, however, house u2 is at least beautiful on the membership degree of 0.6 and it is at most beautiful on the membership degree of 0.7; house u2 is not at least beautiful on 1076

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the non-membership degree of 0.1 and it is not at most beautiful on the non-membership degree of 0.2. Now give an IVIF subset A over the parameter set E as follows:

A = { < e1 , [0.7, 0.8], [0.1, 0.2] >, < e2 , [0.5, 0.7], [0.2, 0.3] >, < e3 , [0.4, 0.6], [0.1, 0.3] >, < e4 , [0.2, 0.6], [0.3, 0.4] >}. By Equations (1) and (2), we have µR(A) (u1 ) = [0.7, 0.8], γR(A) (u1 ) = [0.2, 0.2], µR(A) (u2 ) = [0.5, 0.7], γR(A) (u2 ) = [0.2, 0.3], µR(A) (u3 ) = [0.5, 0.6], γR(A) (u3 ) = [0.1, 0.3], µR(A) (u4 ) = [0.5, 0.7], γR(A) (u4 ) = [0.2, 0.3], µR(A) (u5 ) = [0.7, 0.8], γR(A) (u5 ) = [0.1, 0.2]; µR(A) (u1 ) = [0.2, 0.6], γR(A) (u1 ) = [0.3, 0.4], µR(A) (u2 ) = [0.2, 0.6], γR(A) (u2 ) = [0.3, 0.4], µR(A) (u3 ) = [0.2, 0.6], γR(A) (u3 ) = [0.2, 0.4], µR(A) (u4 ) = [0.4, 0.6], γR(A) (u4 ) = [0.2, 0.3], µR(A) (u5 ) = [0.2, 0.6], γR(A) (u5 ) = [0.3, 0.4]. Thus R(A) = { < u1 , [0.7, 0.8], [0.2, 0.2] >, < u2 , [0.5, 0.7], [0.2, 0.3] >, < u3 , [0.5, 0.6], [0.1, 0.3] >, < u4 , [0.5, 0.7], [0.2, 0.3] >, < u5 , [0.7, 0.8], [0.1, 0.2] >} and R(A) = { < u1 , [0.2, 0.6], [0.3, 0.4] >, < u2 , [0.2, 0.6], [0.3, 0.4] >, < u3 , [0.2, 0.6], [0.2, 0.4] >, < u4 , [0.4, 0.6], [0.2, 0.3] >, < u5 , [0.2, 0.6], [0.3, 0.4] >}. In what follows, we investigate the properties of generalized IVIF soft rough approximation operators. Theorem 3.8 Let (U, E, R) be an IVIF soft approximation space. Then the generalized upper and lower IVIF soft rough approximation operators R(A) and R(A) satisfy the following properties: ∀A, B ∈ IV IF (E), (IVIFSL1) R(A) =∼ R(∼ A), (IVIFSU1) R(A) =∼ R(∼ A); (IVIFSL2) R(A ∩ B) = R(A) ∩ R(B), (IVIFSU2) R(A ∪ B) = R(A) ∪ R(B); (IVIFSL3) A ⊆ B ⇒ R(A) ⊆ R(B), (IVIFSU3) A ⊆ B ⇒ R(A) ⊆ R(B); (IVIFSL4) R(A ∪ B) ⊇ R(A) ∪ R(B), (IVIFSU4) R(A ∩ B) ⊆ R(A) ∩ R(B);

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Proof. We only prove the properties of the lower IVIF soft rough approximation operator R(A). The upper IVIF soft rough approximation operator R(A) can be proved similarly. (IVIFSL1) By Definition 3.4, then we have ∼ R(∼ A) = {< u, γR(∼A) (u), µR(∼A) (u) > |u ∈ U } _ _ − + = {< u, [ (µ− (u, x) ∧ γ (x)), (µ+ R ∼A R (u, x) ∧ γ∼A (x))], x∈E

^

[

x∈E

− (γR (u, x)

∨

µ− ∼A (x)),

x∈E

^ x∈E

= {< u, [

_

(µ− R (u, x)

∧

µ− A (x)),

x∈E

^

[

+ (γR (u, x) ∨ µ+ ∼A (x))] > |u ∈ U }

_

+ (µ+ R (u, x) ∧ µA (x))],

x∈E

− (u, x) (γR

∨

^

− (x)), γA

+ (u, x) (γR

+ (x))] > |u ∈ U } ∨ γA

x∈E

x∈E

= {< u, µR(A) (u), γR(A) (u) > |u ∈ U } = R(A). (IVIFSL2) By virtue of Equation (2), we have R(A ∩ B) = {< u, µR(A∩B) (u), γR(A∩B) (u) > |u ∈ U } ^ _ = {< u, (γR (u, x) ∨ µA∩B (x)), (µR (u, x) ∧ γA∩B (x)) > |u ∈ U } x∈E

= {< u, [

^

x∈E − (u, x) (γR

∨

(µ− A (x)

[

+ + (u, x) ∨ (µ+ (γR A (x) ∧ µB (x)))],

x∈E

x∈E

_

^

∧ µ− B (x))),

− − (µ− R (u, x) ∧ (γA (x) ∨ γB (x))),

_

+ + (µ+ R (u, x) ∧ (γA (x) ∨ γB (x)))] > |u ∈ U }

x∈E − − + {< u, [µR(A) (u) ∧ µR(B) (u), µR(A) (u) ∧ µ+ R(B) (u)], − − + + [γR(A) (u) ∨ γR(B) (u), γR(A) (u) ∨ γR(B) (u)] > |u ∈ U } x∈E

=

= {< u, µR(A) (u) ∧ µR(B) (u), γR(A) (u) ∨ γR(B) (u) > |u ∈ U } = R(A) ∩ R(B). (IVIFSL3) It can be easily verified by Definition 3.4. (IVIFSL4) By (IVIFSL3), it is straightforward.

2

In Theorem 3.8, properties (IVIFSL1) and (IVIFSU1) show that the generalized upper lower IVIF soft rough approximation operators R and R are dual to each other. Inspired by the concept of cut sets of IF sets in [44, 45], we first present the concept of cut sets of IVIF sets before investigating the representing method of the generalized IVIF soft rough approximation operators. Definition 3.9 Let A = {< x, µA (x), γA (x) > |x ∈ U } ∈ IV IF (U ), and (α, β) ∈ L, where α = [α1 , α2 ], β = [β1 , β2 ] ∈ Int[0, 1] with α2 + β2 ≤ 1. The (α, β)-level cut set of A, 1078

Yanping He et al 1070-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 23, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

denoted by Aβα , is defined as follows: Aβα = {x ∈ U |µA (x) ≥LI α, γA (x) ≤LI β} + + − = {x ∈ U |µ− A (x) ≥ α1 , µA (x) ≥ α2 , γA (x) ≤ β1 , γA (x) ≤ β2 }. + Aα = {x ∈ U |µA (x) ≥LI α} = {x ∈ U |µ− A (x) ≥ α1 , µA (x) ≥ α2 },

and + Aα+ = {x ∈ U |µA (x) >LI α} = {x ∈ U |µ− A (x) > α1 , µA (x) > α2 }

are, respectively, called the α-level cut set and the strong α-level cut set of membership generated by A. Meanwhile, + − (x) ≤ β2 } (x) ≤ β1 , γA Aβ = {x ∈ U |γA (x) ≤LI β} = {x ∈ U |γA

and + − (x) < β2 } (x) < β1 , γA Aβ+ = {x ∈ U |γA (x)