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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 240735, 8 pages
http://dx.doi.org/10.1155/2012/240735
Research Article

Unique Existence Theorem of Solution of Almost Periodic Differential Equations on Time Scales

Department of Mathematics, Anyang Normal University, Anyang, Henan 455000, China

Received 26 October 2011; Accepted 10 December 2011

Academic Editor: Binggen Zhang

Copyright © 2012 Meng Hu and Lili Wang. 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

By using the theory of calculus on time scales and -matrix theory, the unique existence theorem of solution of almost periodic differential equations on almost periodic time scales is established. The result can be used to a large of dynamic systems.

1. Introduction

It is well known that in Celestial mechanics, almost periodic solutions and stable solutions to differential equations or difference equations are intimately related. In the same way, stable electronic circuits, ecological systems, neural networks, and so on exhibit almost periodic behavior. A vast amount of researches have been directed toward studying these phenomena, we refer the readers to [15] and the references therein.

Also, the theory of calculus on time scales (see [6] and references cited therein) was initiated by Stefan Hilger in his Ph.D. thesis in 1988 [7] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has received much attention since his foundational work (see, e.g., [812]). Therefore, it is practicable to study that on time scales which can unify the continuous and discrete situations.

Recently, the conceptions of almost periodic time scales and almost periodic functions on almost periodic time scales have been established, one can see [8]. Consider the following almost periodic system: where is an almost periodic time scale, is an almost periodic matrix function, is an almost periodic vector function. The authors in [8] only proved the existence of almost periodic solution for system (1.1) (see Lemma  2.13 in [8]), but the uniqueness has not been considered. However, the unique existence theorem of solution usually plays an important role in applications, so, the theories need to be explored.

The main purpose of this paper is by using the theory of calculus on time scales and -matrix theory to establish the unique existence theorem of solution of system (1.1).

2. Preliminaries

The basic theories of calculus on time scales, one can see [6]. In order to obtain the unique existence theorem of solution of system (1.1), we first make the following preparations.

Lemma 2.1 (see [6]). Let be a function with Let be differentiable with Then, implies

Theorem 2.2. If the following conditions satisfy:(1)where , and is a constant;(2) is an -matrix; then there exists constants and , such that the solutions of inequality (1) satisfy

Proof. Assume that where
By condition (1), then
By condition (2), there exist constants and such that Choose , such that
If , choose , such that
For any , let
From (2.11), for any , we have that is,
For , by (2.12), we can get Let , then
Together with (2.9), (2.15), and (2.17), by Lemma 2.1, we can get Let , then The proof is completed.

Definition 2.3. The almost periodic solution of (1.1) is said to be exponentially stable, if there exists a positive such that for any , there exists such that for any solution satisfying where , is the initial condition.

3. Unique Existence Theorem

In this section, we will establish the unique existence theorem of solution of system (1.1) based on the theory of calculus on time scales and -matrix theory. The conceptions of almost periodic time scales and almost periodic functions on almost periodic time scales, one can see [8].

Definition 3.1 (see [8]). Let , and let be an rd-continuous matrix on , the linear system is said to admit an exponential dichotomy on if there exist positive constants , projection , and the fundamental solution matrix of (3.1), satisfying where is a matrix norm on .

Lemma 3.2 (see [8]). If the linear system (3.1) admits exponential dichotomy, then system (1.1) has a bounded solution as follows: where is the fundamental solution matrix of (3.1).

Let .

Lemma 3.3. Assume that the conditions of Lemma 3.2 hold, if is an -matrix, then the almost periodic solution of system (1.1) is globally exponentially stable and unique.

Proof. According to Lemma 3.2, we know that (1.1) has an almost periodic solution . Suppose that be an arbitrary solution of (1.1). Then, system (1.1) can be written as
Assume that the initial condition of (1.1) is , then the initial condition of (3.4) is .
Let , the upper right derivative along the solutions of system (3.4) is as follows: where is an -matrix with all its elements are zeros.
Since is an -matrix, according to Theorem 2.2, then there exist constants , for any , where .
Then, there exists a positive number , such that where .
From Definition 2.3, the almost periodic solution is globally exponentially stable. Thus, the almost periodic solution of system (1.1) is globally exponentially stable.
In (3.7), let , then , so, we can get . Hence, the almost periodic system (1.1) has a unique almost periodic solution. The proof is completed.

Together with Lemmas 3.2 and 3.3, we can get the following theorem.

Theorem 3.4. If the linear system (3.1) admits an exponential dichotomy, is an -matrix, then system (1.1) has a unique almost periodic solution , and where is the fundamental solution matrix of (3.1).

Lemma 3.5 (see [8]). Let be an almost periodic function on , where , , for all , and then the linear system admits an exponential dichotomy on .

Corollary 3.6. In system (1.1), if , and , then system (1.1) has a unique almost periodic solution , and where is the fundamental solution matrix of (3.1).

Proof. Obviously, is an -matrix, since , and . By Lemma 3.5, the linear system (3.1) admits an exponential dichotomy. According to Theorem 3.4, it is easy to see that the almost periodic system (1.1) has exactly one almost periodic solution. The proof is completed.

Remark 3.7. As an application, consider system (1.1) in paper [8], by using fixed-point theorem, the authors in [8] proved (1.1) has a unique almost periodic solution, one can see Theorem  3.2 in [8] for more detail. However, from the proof of Theorem  3.2 in paper [8], one can see that (3.5) is a solution of system (3.4), but the uniqueness cannot be determined, so, the proof of Theorem  3.2 in paper [8] is questionable. Our results obtained in this paper can solve the problem. By Corollary 3.6, one can get system (3.4) has exactly one solution as (3.5) in [8], then by the same method in [8], under the conditions of Theorem  3.2, system (1.1) has a unique almost periodic solution. Also, the results can be used to other neural networks.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 61073065.

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