Journal of Applied Mathematics

Volume 2014 (2014), Article ID 984946, 9 pages

http://dx.doi.org/10.1155/2014/984946

Research Article

Discrete Weighted Pseudo-Almost Automorphy and Applications

Zhinan Xia

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China

Received 12 September 2013; Accepted 27 January 2014; Published 3 March 2014

Academic Editor: Francisco J. Marcellán

Copyright © 2014 Zhinan Xia. 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 deal with discrete weighted pseudo almost automorphy which extends some classical concepts and systematically explore its properties in Banach space including a composition result. As an application, we establish some sufficient criteria for the existence and uniqueness of the discrete weighted pseudo almost automorphic solutions to the Volterra difference equations of convolution type and also to nonautonomous semilinear difference equations. Some examples are presented to illustrate the main findings.

1. Introduction

The notation of (continuous) almost automorphy, introduced by Bochner [1], is related to and more general than (continuous) almost periodicity. Since then, this concept has been attracting the attention of many researchers and the interest in this topic still increases. There is a lager literature on this topic. We refer to the recent books [2, 3], where the authors gave an important overview about the theory of (continuous) almost automorphic functions and their applications to differential equations. Recently, a new more general type of almost automorphy called (continuous) weighted pseudo almost automorphy is proposed by Blot et al. [4], which generalizes various extensions of (continuous) almost automorphy; one can refer to [5] for more details.

Discrete almost automorphic functions, a class of functions which are more general than discrete almost periodic ones, were considered in [6] in connection with the study of (continuous) almost automorphic bounded mild solutions of differential equations; see also [7–9]. Similar to (continuous) almost automorphic functions, discrete almost automorphic functions have made important applications to differential equations in Banach space. The range of applications of discrete almost automorphic functions include first order nonlinear difference equations [10], Volterra difference equations [11–13], nonautonomous difference equations [14, 15], and nonlinear stochastic difference equations [16]. On the other hand, recently, the concept of discrete weighted pseudo almost automorphic functions, which generalizes the notion of discrete almost automorphic functions, is introduced by Abbas in [17] and some basic properties of these functions are explored.

In this paper, we conduct further studies on discrete weighted pseudo almost automorphic functions; the main idea consists of enlarging the weighted ergodic space, with the help of two weighted functions, extending some results of [17]. We systematically explore its properties in Banach space including completeness, translation invariance, and composition results. As an application, the existence and uniqueness of the discrete weighted pseudo almost automorphic solutions to the Volterra difference equations of convolution type and nonautonomous semilinear difference equations are investigated. To the best of our knowledge, discrete weighted pseudo almost automorphy of Volterra difference equations and nonautonomous semilinear difference equations are an untreated topic and this is the main motivation of this paper.

The paper is organized as follows. In Section 2, some notations and preliminary results are presented. In Section 3, we propose a new class of functions called discrete weighted pseudo almost automorphic functions with the help of two weighted functions, explore its properties, and establish the composition theorem. Section 4 is divided into the existence and uniqueness of discrete weighted pseudo almost automorphic solutions to the Volterra difference equations of convolution type and nonautonomous semilinear difference equations, respectively. In Section 5, we provide some examples to illustrate our main results.

2. Preliminaries and Basic Results

Let , be two Banach spaces and , , , , and stand for the set of natural numbers, integers, nonnegative integers, real numbers, nonnegative real numbers, and complex numbers, respectively. Let be bounded linear operator; denotes the point spectrum of . stands for open balls with center and radius less than . Let ; if , we call that is a summable function.

In order to facilitate the discussion below, we further introduce the following notations.

(i).(ii). (iii): the Banach space of bounded linear operators from to endowed with the operator topology. In particular, we write when .(iv): the set of all functions satisfying that , such that

for all and with .

Next, we recall the so-called Matkowski's fixed point theorem [18] and exponential dichotomy on [19, 20] which will be used in the sequel.

Theorem 1 (Matkowski’s fixed point theorem [18]). Let be a complete metric space and let be a map such that

where is a nondecreasing function such that for all . Then has a unique fixed point .

Given a sequence of invertible operators, define

where Id is the identity operator in .

For the first order difference equation

Definition 2 (see [19]). Equation (4) is said to have an exponential dichotomy if there exist projections for all and positive constants such that

(i),(ii),(iii),

where is the complementary projection of .

Finally, we recall the concept of discrete almost automorphic function.

Definition 3 (see [6]). 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 .

Remark 4. (i) If is (continuous) almost automorphic function in , then is discrete almost automorphic.(ii) If the convergence in Definition 3 is uniform on , then we get discrete almost periodicity, so discrete almost automorphy is more general than discrete almost periodicity.(iii) Example of discrete almost automorphic functions which are not discrete almost periodicity was first constructed by Veech [21]. Note that the function

is discrete almost automorphic function but not discrete almost periodic (see [10] for more details).

Throughout the paper, we denote the set of discrete almost automorphic functions. Note that if , then is relative compact in and is a bounded function.

Definition 5 (see [6]). A function is said to be discrete almost automorphic in for each , if for every integer sequence , there exists a subsequence such that is well defined for each , and for each , . We denoted by the spaces of all discrete almost automorphic in for each .

Lemma 6 (see [11]). Let ; then if .

3. Discrete Weighted Pseudo Almost Automorphy

Let denote the collection of functions (weights) . For and , set

Denote

Definition 7. Let . is said to be equivalent to (i.e., ) if .

It is trivial to show that “” is a binary equivalence relation on . The equivalence class of a given weight which is denoted by . It is clear that .

For , define the weighted ergodic space [17]

Particularly, for , define

Clearly, when , coincide with ; that is, , this fact suggests that are more interesting when and are not necessarily equivalent. So are general and richer than .

Definition 8. Let . A function is called discrete weighted pseudo almost automorphic if it can be expressed as , where and . The set of such functions is denoted by .

Remark 9. If , coincide with the discrete weighted pseudo almost automorphic functions defined in [17].

Throughout the rest of the paper, we denote by the set of all the functions satisfying that there exists an unbounded set such that for all ,

Next, we show some properties of the space .

Similar to the proof of [22], one has the following.

Lemma 10. Let , then

(i)for each , one has

(ii) is translation invariant; that is, for each if .(iii) is translation invariant.(iv)If , where , , then

(v) is a Banach space under the supremum norm; that is,

Lemma 11. If and , then .

The proof is straightforward and is therefore omitted.

Similarly, define

Definition 12. Let . A function is said to be discrete weighted pseudo almost automorphic in for each , if it can be decomposed as , where and . Denote by the set of such functions.

We will establish composition theorem for discrete weighted pseudo almost automorphic functions.

Lemma 13. Let be bounded and , then if and only if for any ,

where .

Proof. Sufficiency. It is clear that and by Lemma 10. It follows from (15) that ; there exists such that

then

so

That is, .

Necessity. Suppose the contrary, that there exists , such that does not converge to as . That is, there exists , such that for each ,

Then

which contradicts the fact that

Thus (15) holds.

Remark 14. If and , the result of Lemma 13 is obtained by [23, Lemma 2.9].

Lemma 15. Let and be compact; if , then , where

Proof. Since , , there exists such that

for all and with . Note that is compact; for the above , there exists such that

Since , for the above , there exists such that for all ,

For each , there exists such that ; hence

which implies that

Thus is bounded and

which means that .

Theorem 16. Assume that , ; then if .

Proof. Let

where , , and . The function can be decomposed as

Set

By Lemma 10, , it is not difficult to see that ; hence by Lemma 6.

We claim that . In fact, since , , such that

for all and ; hence

Since , by Lemma 13, one has

which implies that

By Lemma 13, .

Since , then . Let ; by Lemma 15, one has

so

That is . Then , . Hence .

Corollary 17. Let , and satisfies the Lipschitz condition

then if .

The following Lemma is the essential property to study the existence of solutions of Volterra difference equations of convolution type.

Lemma 18. Let be a summable function; if , , then , where

Proof. Note that

hence is bounded.

Let , where , ; then

where

The almost automorphy of follows from [10].

Next, we show that . In fact,

Since , for each by Lemma 10, then

by Lebesgue dominated convergence theorem, one has

which implies that . The proof is completed.

Remark 19. If , the results of Corollary 17 and Lemma 18 are obtained by [17, Theorem 2.8] and [17, Theorem 2.9], respectively.

4. Applications in Difference Equations

As an application, the main goal of this section is to establish some sufficient criteria for the existence and uniqueness of solution to the Volterra difference equations and nonautonomous semilinear difference equations.

4.1. Volterra Difference Equation

This subsection is devoted to establish some sufficient criteria for the existence and uniqueness of solutions of (46).

Consider the Volterra difference equations of convolution type

where , is a summable function; and . Its associated homogeneous linear equation is given by

where , is a summable function.

For a given , let be the solution of the difference equation

In this case, is called the fundamental solution to (47) generated by . We define the set

By [13], if , the solution of (47) is given by

To establish our results, we introduce the following condition.

(H₁), .(H₂), .(H₃₁)There exists a constant such that

(H₃₂)There exists a linear nondecreasing function and satisfies

Theorem 20. Assume that , , and hold and ; then (46) has a unique solution which is given by

Proof. Similar to the proof of [13], it can be shown that a solution of (53) is the solution of (46).

Define the operator by

Since and holds, by Lemma 11, Corollary 17. By Lemma 18, . Hence is well-defined.

For ,

By the Banach contraction mapping principle, has a unique fixed point , which is the unique solution to (46).

Theorem 21. Assume that , , and hold; then (46) has a unique solution if as for each .

Proof. Define the operator as in (54), so is well defined. For , one has

Since as for each , by Matkowski fixed point theorem (Theorem 1), has a unique fixed point , which is the unique solution to (46).

4.2. Nonautonomous Semilinear Difference Equations

In this subsection, consider the following nonautonomous semilinear difference equations

Its associated homogeneous linear difference equation is given by

To establish our results, we introduce the following condition.

(A₁).(A₂), .(A₃)There exists a constant such that

(A₄)Equation (58) admits an exponential dichotomy on with positive constants .

Theorem 22. Assume that hold and ; then (57) has a unique solution which is given by

Proof. Similar to the proof of [14, 15], it can be shown that given by (60) is the solution of (57).

Define an operator as follows:

Since and holds, by Corollary 17. Let , where , ; then

where

Similar to the proof of [14, 15], .

Next, we show that . In fact, let

where

Then,

Since , for each by Lemma 10; then

by Lebesgue dominated convergence theorem, one has

hence . Similarly, one can prove . So is well defined.

For , by the exponential dichotomy and Lipschitz condition, one has

hence is a contraction. By the Banach contraction mapping principle, has a unique fixed point , which is the unique solution to (57). The proof is completed.

5. Examples

In this section, we provide some examples to illustrate our main results.

Example 1. For , where , after a calculation using in (48) the unilateral- transform, that , , and define

Consider the following difference equation:

where , , . It is easy to see that , , and hold with . By Theorem 20, if , then (71) has a unique solution .

Example 2. Consider the system

where is a nonsingular matrix such that , , and there exists a constant such that

Since , the system

admits an exponential dichotomy with positive constants [19] and holds with . By Theorem 22, If we suppose that , then (72) has a unique discrete weighted pseudo almost automorphic solution.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This material is based upon work funded by Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ13A010015.

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