Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 5827483, 8 pages
http://dx.doi.org/10.1155/2016/5827483

Research Article

Discrete Weighted Pseudo Almost Automorphic Solutions of Nonautonomous Difference Equations

Xiaoqing Wen¹ and Hongwei Yin1,2

¹School of Sciences, Nanchang University, Nanchang 330031, China
²Numerical Simulation and High-Performance Computing Laboratory, School of Sciences, Nanchang University, Nanchang 330031, China

Received 6 December 2015; Accepted 24 March 2016

Academic Editor: Silvia Romanelli

Copyright © 2016 Xiaoqing Wen and Hongwei Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce the concept of a discrete weighted pseudo almost automorphic function and prove some basic results. Further, we investigate the nonautonomous linear and semilinear difference equations and obtain the weighted pseudo almost automorphic solutions of both these kinds of difference equations, respectively. Our results generalize the ones by Lizama and Mesquita (2013).

1. Introduction

The almost automorphic functions that were introduced initially by Bochner [1] are significant and related to some aspects of differential geometry. From then on, the almost automorphic solutions for deterministic differential systems were extensively investigated [2–5]. One can refer to the monograph [6]. Later, the concept of the pseudo almost automorphic function was proposed by N’Guérékata in [6], which seems to be more generalization of the pseudo almost periodic function [7, 8]. In 2009, Blot et al. introduced the concept of the weighted pseudo almost automorphic function [9], which generalized the one of the weighted pseudo almost periodicity proposed in [10]. At present, there are many literatures to study the weighted pseudo almost automorphic function in other types of dynamics. For example, Chang et al. in [11] obtained the weighted pseudo almost automorphic solution for fractional differential equations. Besides, the weighted pseudo almost automorphic solution for stochastic partial differential equations has been discussed in [12, 13].

However, as far as we know, no author has studied the weighted pseudo almost automorphic solution for the nonautonomous difference equation as follows:

where is specified later. Now, few authors investigated the properties of the solution to system (1). In 2013, Lizama and Mesquita have obtained almost automorphic solutions of  (1); see [14]. But they did not deeply study the weighted pseudo almost automorphic solution of (1). This paper mainly discusses this problem and generalizes the result in [14].

The plan of the paper is as follows. In Section 2, some necessary definitions and results are stated. In Section 3, we discuss the weighted pseudo almost automorphic solutions of the linear nonautonomous difference equation. In Section 4, we obtain the weighted pseudo almost automorphic solutions of semilinear nonautonomous difference equation.

2. Preliminaries

In this section, we present some basic concepts and results. Let be any Banach space with the norm . is the space of all bounded sequences from to with the sup norm. To begin this paper, we recall some primary definitions of almost automorphy.

Definition 1. Consider
(i) Let be a (real or complex) Banach space. A function is said to be discrete almost automorphic if for every integer sequence there exists a subsequence such that

is well defined for each and

for each .
(ii) A function is said to be discrete almost automorphic if is almost automorphic in for any .

Denote by (resp., ) the set of all discrete almost automorphic function , (resp., ). With the sup norm , this space is Banach spaces.

Theorem 2 (see [14]). If are discrete almost automorphic functions in for each in , then the following assertions are true:

(1) is discrete almost automorphic in for each in .(2) is discrete almost automorphic in for each in , where is an arbitrary scalar.(3) for each in .(4), for each in .

The following result can be found in [15, Theorem 2.10].

Theorem 3. Let be discrete almost automorphic in for each in , satisfying a Lipschitz condition in uniformly in ; that is,

Suppose is discrete almost automorphic; then the function defined by is discrete almost automorphic.

Let be the set of all functions which are positive and locally summable over . For a given , set

for each . Define

Obviously, . Now for define

Similarly, we define as the collection of all functions is bounded and is bounded for each , and

uniformly in .

Definition 4. A bounded-sequence set is said to be translation invariant if, for any , for .

For simplicity, denote . We are now ready to introduce the spaces and of discrete weighted pseudo almost automorphic functions:

Lemma 5. Assume that is an almost automorphic function. Fix , and write

Then, there exist such that

Proof. It is analogous to Lemma  2.1.1 in [16].

Lemma 6. If with and , where , then .

Proof. If this is not true, then there are such that

Let be as in Lemma 5 and write

For with , we put

where is defined in Lemma 5. Using Lemma 5 gives that

For any ,

Thus, one yields

This is a contradiction and this proof is completed.

Theorem 7. The decomposition of a discrete weighted pseudo almost automorphic function defined on is unique for any .

Proof. Assume that and and then . Since and , we deduce that by Lemma 6. Consequently, ; that is, , which proves the uniqueness of the decomposition of .

Theorem 8. If , then is a Banach space.

Proof. Assume that is a Cauchy sequence in . We can write uniquely . In view of Lemma 6, we see that , from which we deduce that is a Cauchy sequence in the Banach space . So, is also a Cauchy sequence in the Banach space . We can deduce that , , and finally .

Lemma 9. Let . Then , where if and only if, for any ,

where

for .

Proof.  
Necessity. For any , we have

Noticing , we thus know that

which implies

Sufficiency. From , we know there exists a number such that for all . Since , for any there exists a number such that, for ,

This gives

Thus . This proof is finished.

Theorem 10. Suppose , , and there exists a number such that, for any ,

Then, for any , one has .

Proof. Since , there exist two functions and such that . Also, since , there exist and such that . Hence,

Let and . Now, we are in a position to prove and .
First, for any ,

Since , from Lemma 6, it follows that

so that

Thus,

for all . In view of and from Theorem 3, it therefore follows that .
Second, in order to prove , we only need to prove , and . In fact, according to

and , we have

Thus, .
Since , the set for and some are finite. Its elements are denoted by . Hence, we have

Further, we have

Notice . Then, by virtue of Lemma 9, we obtain

for , which consequently gives

Therefore, according to Lemma 9 again, we know . From the above discussion, we obtain that , and this completes proof.

3. Weighted Pseudo Almost Automorphic Solutions of Nonautonomous Line Difference Equation

In this section, consider the following nonautonomous linear difference equation:

where are given nonsingular matrices with elements is given vector function, and is an unknown vector with components . Its associated homogeneous linear difference equation is given by

Definition 11. The matrix which satisfies (38) and is called principal fundamental matrix. One will denote by .

From [14], the following holds:

Definition 12 (see [14]). A matrix function is said to be discrete almost automorphic if each entry of the matrix is discrete almost automorphic function; that is,

is well defined for each and

for each .

Lemma 13 (see [14]). Suppose is discrete almost automorphic and a nonsingular matrix on . Also, suppose that the set is bounded. Then is almost automorphic on ; that is, for every sequence of integer numbers , there exists a subsequence such that

is well defined for each and

for each .

Definition 14. Let be the principal fundamental matrix of (38). System (38) is said to possess an exponential dichotomy if there exist a projection , which commutes with , and positive constants , , , and such that, for all , one has

Specially, for , system (44) is said to possess an ordinary dichotomy.

According to Theorem  2.12 in [14], if system (38) possesses an exponential dichotomy and the function is bounded, then system (37) has a bounded solution which is given by

where is a fundamental matrix of (38).

We start by proving a result concerning existence of a discrete weighted pseudo almost automorphic solution of (37).

Theorem 15. Suppose is discrete almost automorphic and nonsingular matrix and the set is bounded. Also, suppose the function , where , and (38) admits an exponential dichotomy with positive constants , , , and . Then system (37) has a discrete weighted pseudo almost automorphic solution.

Proof. Since is a discrete weighted pseudo almost automorphic function, we have , where and , which is unique according to Theorem 7. In the help of (45), the solution of the system (37) is

Let

By using proof method of Theorem 3.1 in [14], one can know that in (47) is almost automorphic on . Thus, we only need to prove that . Next, we pass to prove

In fact, we have

From the well-known Tonelli theorem, it follows that

Since and , we can know that, for any ,

as . Define two functions as follows:

for any fixed . Obviously, the sequences are measurable on and

Besides, in view of (52), one knows that

Then by the Lebesgue dominated convergence theorem, we have

Thus, one obtains that

and this proof is completed.

4. Weighted Pseudo Almost Automorphic Solutions for Semilinear Nonautonomous Difference Equation

In this section, we consider the following difference equation:

where is described as in (37); besides is a given vector function.

Definition 16. One says that is solution of (58), if it satisfies

for every .

Now, the following result shows the existence and uniqueness of a weighted pseudo almost automorphic solution of (58).

Theorem 17. Suppose is discrete almost automorphic and nonsingular matrix and the set is bounded. Also, assume that (38) admits an exponential dichotomy with positive constants , , , and . Assume the function , satisfying the following condition: there exists a constant such that

for every , and . Then system (58) has a discrete weighted pseudo almost automorphic solution.

Proof. Since the function satisfies the global Lipschitz-type condition, we obtain by Theorem 10 that is in for any . Define an operator as follows:

for all and is the fundamental matrix of (38). By Theorem 15, one can know the operator . Thus we only need to prove that the operator is contraction in . In fact,

Since , we obtain that is a contraction. Then, has a unique fixed point. Thus, system (58) has a unique solution which is discrete weighted pseudo almost automorphic one.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by Grants 11563005 and 61563033 from the National Natural Science Foundation of China and 20151BAB212011 and 20151BAB201021 from the Natural Science Foundation of Jiangxi Province.

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