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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 784198, 11 pages
http://dx.doi.org/10.1155/2013/784198
Research Article

Almost Periodic Solutions for Second Order Dynamic Equations on Time Scales

1School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China
2College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, China

Received 25 July 2013; Accepted 3 September 2013

Academic Editor: Mustafa R. S. Kulenovic

Copyright © 2013 Yan Wang and Lan Li. 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 firstly introduce the concept and the properties of almost periodic functions on time scales, which generalizes the concept of almost periodic functions on time scales and the concept of -almost periodic functions. Secondly, we consider the existence and uniqueness of almost periodic solutions for second order dynamic equations on time scales by Schauder’s fixed point theorem and contracting mapping principle. At last, we obtain alternative theorems for second order dynamic equations on time scales.

1. Introduction

The theory of dynamic equations on time scales was first introduced by Hilger [1]. The study of dynamic equations on time scales helps to avoid studying results twice, once for differential equations and once for difference equations. In recent years, the theory of first order and second order dynamic equations on time scales has been studied, and some important results have been presented in [26]. However, to the best of our knowledge, there are no results on the existence of almost periodic solutions for the second order dynamic equations on time scales. The aim of this paper is to consider the existence of almost periodic solutions for second order dynamic equations on time scales.

The concept of almost periodicity was first introduced by Bohr [7] and later generalized by Bochner, Fink, N’Guérékata, and Shen and Yi and others (see [811]). Recently, Guan and Wang [12] and Li and Wang [13, 14] developed the theory of almost periodic functions on time scales, which do not only unify the almost periodic functions on and the almost periodic sequences on but also extend to nontrivial time scales, for example, -difference equations.

The existence and uniqueness solutions for second order dynamic equations have become important in recent years in mathematical models and they rise in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. The existence of oscillatory and nonoscillatory solutions for second order equations has been studied in [1518] (and the references therein).

This paper is concerned with the second order dynamic equation as follows: where is almost periodic in uniformly. Such a type of equation appears in many problems of applications, such as Brillouin focusing systems [19, 20], nonlinear elasticity [21], and Ermakov-Pinney equations [22, 23].

In this paper, we consider the existence of almost periodic solutions (1) and present alternative theorems for second order dynamic equations. In order to do this, we introduce a new concept called almost periodicity, which generalizes the concept of almost periodic functions on time scales and the concept of -almost periodic functions on introduced by Adamczak [24], Bugajewski and N’Guérékata [25]. The study of almost periodic solutions on time scales has tremendous potential for applications in mathematical models of real processes and phenomena.

This paper is organized as follows. In Section 2, we recall some properties on time scales. In Section 3, we give the concept and properties of almost periodicity, and uniformly almost periodicity on time scales. The most important part of this paper are Sections 4 and 5. In Section 4, by exponential dichotomy on time scales and fixed point theorems, the existence and uniqueness theorems of almost periodic solutions for second order dynamic equations are obtained. In Section 5, we show the alternative theorems for second order dynamic equations on time scales by topological degree method.

2. Preliminaries on Time Scales

Suppose that is an arbitrary time scale, that is, a nonempty closed subset of . Set , , resp.) , resp.). First, we recall some results of time scales in [5, 6].

Definition 1 (see [5, 6]). For , the forward jump operator , the graininess function , the backward jump operator , and the backwards graininess function are given by respectively. Set and , where denotes the empty set.
If , then the point is called right-scattered, while if , then the point is termed left-scattered. If and , then the point is called right-dense, while if and , then is called left-dense. Set

Definition 2 (see [5, 6]). (i) Let be a function. Then is called -differentiable at if the limit provided that this limit exists as a finite number , and is called the -derivative of at and we denote it by .
(ii) If , then the -integral is defined by

Definition 3 (see [5, 6]). A function is called rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point. The set of rd-continuous functions is denoted by .
Obviously, . If is -differential, then is continuous. If , then is -integral.

Definition 4 (see [5, 6]). Suppose that . Then (i) for all ; (ii) for all .

Definition 5 (see [5, 6]). Suppose that ; then where
The following lemmas are concerned with the properties of .

Lemma 6 (see [6, 26]). Suppose that , . Then (i) , ; (ii) ; (iii) ; (iv) ; (v) ; (vi) ; (vii) ; (viii) ; (ix) let be a constant, and let the graininess function be uniformly bounded with . Then for ,

Lemma 7 (see [5]). Let be function. Then (i) ; (ii) .

3. Almost Periodic Functions

In this section, we will introduce a new concept called almost periodic and uniformly almost periodic on time scales. First, we recall the conceptions and the properties of almost periodic functions and uniformly almost periodic functions in the works [1214].

Definition 8 (see [12]). Suppose that is a time scale.(i)A real number is called a translation invariant for , if The notation denotes the set of all translation invariants of . (ii)If there exist with , then is called two-way translation invariant time scale or almost periodic time scale.

There exist lots of two-way translation invariant time scales. For example, , , and so on.

Lemma 9 (see [12]). If is a two-way translation invariant time scale, then , .

Thus, follows from being a two-way translation invariant time scale.

Lemma 10. Suppose that is a two-way translation invariant time scale. Then there exists a such that for all .

Proof. Since is a two-way translation invariant time scale, there is a with such that for all . Thus and for all . Furthermore, for all .

If is a two-way translation invariant time scale, set where with denoting the set of all bounded functions in .

Definition 11 (see [12]). Suppose that is a two-way translation invariant time scale. A function is called (Bohr) almost periodic if for every there exists a such that every interval of length contains at least one satisfying

Definition 12 (see [13]). Suppose that is a two-way translation invariant time scale. A function is called (Bochner) almost periodic if any sequence ; there exists a subsequence such that converges uniformly for .

In [14], Li and Wang show that Definition 11 is equivalent to Definition 12. In the following, the notation denotes the set of all almost periodic functions in .

Lemma 13 (see [13]). Suppose that and is a constant. Then (i) ; (ii) is bounded on ; (iii) is uniformly continuous on ; (iv) is almost periodic if and only if is bounded on .

Definition 14. A function is called almost periodic in uniformly for all if is almost periodic for all in each compact subset of .

The notation denotes the set of all uniformly almost periodic functions in .

Similar to the proofs in [14], Lemmas 15 and 16 hold.

Lemma 15. Suppose that and is a constant. Then (i) ; (ii) for each compact subset , is bounded and uniformly continuous on ; (iii) if and only if is bounded on with being compact.

Lemma 16. Let and . Then .

Lemma 17. Let with for . Then if and only if for .

Proof. If , then for each , there exists a such that every interval of length contains a implying that Thus for , ; that is, for .
If for , then for each sequence , there exists a subsequence such that converges uniformly as . Furthermore, there exists a subsequence such that converges uniformly as . Inductively, there exists a subsequence such that converges uniformly as for . Moreover, converges uniformly as which implies that .

Set where . Then is a Banach space, where denotes the set of all bounded functions in , and .

Now we introduce a new concept called almost periodic.

Definition 18. Suppose that is a two-way translation invariant time scale. A function is called (Bohr) almost periodic if for every there exists a such that every interval of length contains at least one satisfying

Definition 19. Suppose that is a two-way translation invariant time scale. A function is called (Bochner) almost periodic if any sequence ; there exists a subsequence such that converges uniformly for .

Theorem 20. Let . Then is (Bohr) almost periodic if and only if are (Bohr) almost periodic.

Proof. If , then for each , there is a such that for every interval of length contains a satisfying which implies are (Bohr) almost periodic.
If are almost periodic, by Lemma 17, . That is, for each , there is a such that for every interval of length contains a satisfying Thus, Therefore, .

If is (Bochner) almost periodic, the following theorem also holds.

Theorem 21. Let . Then is (Bochner) almost periodic if and only if are (Bochner) almost periodic.

Since is (Bohr) almost periodic if and only if is (Bochner) almost periodic, according to Theorems 20 and 21, Definition 18 is equivalent to Definition 19. In the following, the notations , denotes the set of all almost periodic functions in . Obviously, , and

According to Definition 19, we obtain the following theorem.

Theorem 22. Suppose that and is a constant. Then , where .

Theorem 23. Let be a sequence of almost periodic functions. If holds uniformly on , then .

Proof. Noting that converge uniformly, thus for any , there exists a such that
Since , there exists a such that every interval of length contains a such that Then which yields that .

Therefore, is a Banach space with the norm .

Similar to the proof of Lemma 13, Theorem 24 holds.

Theorem 24. Let . If is bounded, then .

Theorem 25. Suppose that , . Then .

Proof. Since , then for each sequence there exists a subsequence and function such that holds uniformly.
By Theorem 25 and Theorem  3.20 in [14], with being hull of and is compact. Thus there exists such that holds uniformly.
Set . Then, by the uniform continuity of , (22), and (23). Therefore .

4. Almost Periodic Solutions

In this section, we will consider the existence and uniqueness of almost periodic solutions of (1). To investigate (1), we will consider the following auxiliary equation: where . Obviously, follows from .

Lemma 26 (see [13, 14]). Let be a -matrix in , where is identity matrix. If the linear system admits an exponential dichotomy on , that is, there exist a projection on and positive constants and for such that then the system has a bounded solution as follows: Moreover, if and are almost periodic functions, then (28) admits a unique almost periodic solution as (29).

We will rewrite (25) by the following lemma.

Lemma 27. Suppose that with is chosen. If is the solution of (25), then is the solution of vector equation where However, if is the solution of (30), then is the solution of (25).

Lemma 28. Suppose that is a two-way translation invariant time scale. If for , then the homogeneous equation admits an exponential dichotomy on . Furthermore, the solution of (32) with the initial value is

Proof. For , the matrix satisfies Thus by Theorem  5.1 in [27], (32) admits an exponential dichotomy on .
According to Theorem  5.35 in [5], we have where is the solution of Thus (33) holds.

Lemma 29. Suppose that with . Then if and only if .

Proof. If holds, from Theorem 20 (or Theorem 21), . According to Lemmas 13 and 17, we have .
If holds, then . Thus and .

Lemma 30. Suppose that is a two-way translation invariant time scale. Assume that (26) admits an exponential dichotomy on , with a projection on and positive constants and for such that (27) holds. If is bounded, where , then for any fixed , the families of functions and are equicontinuous on and , respectively.

Proof. The proof of equicontinuity of on is similar to that of . Thus, we only prove the equi-continuity of .
Since is the fundamental solutions of (26), the function is -differential on . Moreover, for fixed and chosen, we have which implies that . By being -differential -a.e. for and Theorem  4.1 in [28], is absolutely continuous; thus the family is equi-continuous on .

Theorem 31. Suppose that there is a such that for all . Assume takes bounded sets into bounded sets. Then (25) admits a almost periodic solution.

Proof. Set the operator by where is a fundamental solution matrix of (32). From Theorem 25 and Lemma 26, the operator is well defined, is bounded, and the solution of (30) is equivalent to the fixed point of .
Since , then for each compact subset , is bounded and uniformly continuous on . Thus, for each and compact subset , there is a compact subset such that , with implies that . Then which implies that is continuous.
In the following, we will show that is equi-continuous. If and there exists a such that , then there exists such that Without loss of generality, we suppose that , and . Let , and let . We note that
From Lemma 6, we have Thus, there is a such that implies that .
In the following, we will prove there exists a such that implies that and . We will prove that by the following cases: (i); (ii); (iii).
Suppose that case (i) holds. By Theorem  4.1 in [28] is absolutely continuous; that is, there exists a such that implies that Then
If , then there exist such that and . By the equi-continuity of , there exists a such that implies that for given we have Thus,
If , then Similar to the proof above, we can show that there is a such that implies that
Note that where Since for is decreasing and , then is decreasing by Theorem  2.1 in [29] and therefore uniformly bounded on . Thus is bounded as follows. By the boundedness of , , and absolute continuity of , there is a such that implies that
Set ; then for .
Therefore, for ,
As the method in the above proof, we can show that for there is a such that if or with , the inequality (52) holds. Thus, is equi-continuous.
According to Arzela-Ascoli theorem and the continuity of , is completely continuous. Therefore, by Schauder’s fixed point theorem there exists a fixed point of such that . Furthermore, is almost periodic solution of (25).

Theorem 32. Suppose that (H1) is Lipschitz continuous in uniformly on ; that is, there is a such that for , and , , , ;(H2) there exists with such that Then (25) admits a unique almost periodic solution.

Proof. Let , . Note Set as (38). Then which implies that is continuous. And by (H2), is a contraction. Therefore, by contraction mapping principle, there exists a unique fixed point such that . By Lemma 27, implies that is the solution of (25).

Set . Then yields that .

Theorem 33. Suppose that there is a such that for all . Assume takes bounded sets into bounded sets. Then (1) admits a almost periodic solution.

Theorem 34. Let be a two-way translation invariant time scale. Suppose that (H1) is Lipschitz continuous in uniformly on ; that is, there is a such that for , and ;(H2) there exists with such that Then (25) admits a unique almost periodic solution.

5. Alternative Theorem for Second Order Dynamic Equations on Time Scales

In this section, we consider the following equation:

Theorem 35. Assume the following conditions hold: (i) take bounded sets into bounded sets;(ii) there exists an open bounded subset such that every possible solution of (59) satisfies ;(iii) . Then (59) has an almost periodic solution with .

Proof. Let , and let . Then yields that .
For all , set Obviously, is a map from to .
As the proof of Theorem 31, for all is completely continuous. Thus that is, (59) has an almost periodic solution with .

Theorem 36. Consider the following problems: where and takes bounded sets into bounded sets. Assume the following conditions hold.(i) There exists an open bounded subset such that every possible solution of (63) satisfies . (ii) There exists a unique almost periodic solution of (62) such that . Then (63) has an almost periodic solution with .

Proof. Set where is the almost periodic solution of (62). As the proof of Theorem 31, for all is completely continuous. Note (see [30]) Therefore, that is, (63) has an almost periodic solution with .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author’s work was supported NSFC (11026168, 11201046) and the Fundamental Research Funds for the Central Universities in China (DUT12LK32, DUT13JS02). The second author’s work was supported by NSFC (11126070, 11201309) and Natural Science Foundation of SZU (201111). The authors thank the referee and the editor for their valuable suggestions and comments, which improved the presentation of this paper.

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