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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 341520, 22 pages
http://dx.doi.org/10.1155/2011/341520
Research Article

Uniformly Almost Periodic Functions and Almost Periodic Solutions to Dynamic Equations on Time Scales

Department of Mathematics, Yunnan University, Yunnan, Kunming 650091, China

Received 29 April 2011; Revised 27 July 2011; Accepted 25 August 2011

Academic Editor: Detlev Buchholz

Copyright © 2011 Yongkun Li and Chao 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

Firstly, we propose a concept of uniformly almost periodic functions on almost periodic time scales and investigate some basic properties of them. When time scale or , our definition of the uniformly almost periodic functions is equivalent to the classical definitions of uniformly almost periodic functions and the uniformly almost periodic sequences, respectively. Then, based on these, we study the existence and uniqueness of almost periodic solutions and derive some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations. Finally, as an application of our results, we study the existence of almost periodic solutions for an almost periodic nonlinear dynamic equations on time scales.

1. Introduction

In recent years, researches in many fields on time scales have received much attention. The theory of calculus on time scales (see [1, 2] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 [3] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his fundamental work. It has been created in order to unify the study of differential and difference equations. Many papers have been published on the theory of dynamic equations on time scales [410]. Also, the existence of almost periodic, asymptotically almost periodic, and pseudo-almost periodic solutions is among the most attractive topics in qualitative theory of differential equations and difference equations due to their applications, especially in biology, economics and physics [1129]. However, there are no concepts of almost periodic functions on time scales so that it is impossible for us to study almost periodic solutions for dynamic equations on time scales.

Motivated by the above, our main purpose of this paper is firstly to propose a concept of uniformly almost periodic functions on time scales and investigate some basic properties of them. Then we study the existence and uniqueness of almost periodic solutions to linear dynamic equations on almost time scales. Finally, as an application of our results, we study the existence of almost periodic solutions for almost periodic nonlinear dynamic equations on time scales.

The organization of this paper is as follows. In Section 2, we introduce some notations and definitions and state some preliminary results needed in the later sections. In Section 3, we propose the concept of uniformly almost periodic functions on almost periodic time scales and investigate the basic properties of uniformly almost periodic functions on almost periodic time scales. In Section 4, we study the existence and uniqueness of almost periodic solutions and derive some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations on time scales. In Section 5, as an application of our results, we study the existence of almost periodic solutions for almost periodic nonlinear dynamic equations on time scales.

2. Preliminaries

In this section, we will first recall some basic definitions lemmas which are used in what follows.

Let be a nonempty closed subset (time scale) of . The forward and backward jump operators and the graininess : are defined, respectively, by

A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum , then ; otherwise . If has a right-scattered minimum , then ; otherwise .

A function is right-dense continuous provided it is continuous at right-dense point in and its left-side limits exist at left-dense points in . If is continuous at each right-dense point and each left-dense point, then is said to be a continuous function on .

For and , we define the delta derivative of , to be the number (if it exists) with the property that, for a given , there exists a neighborhood of such that for all .

Let be right-dense continuous; if , then we define the delta integral by

A function is called regressive provided for all . The set of all regressive and rd-continuous functions will be denoted by . We define the set .

An -matrix-valued function on a time scale is called regressive provided and the class of all such regressive and rd-continuous functions is denoted, similar to the above scalar case, by .

If is a regressive function, then the generalized exponential function is defined by for all , with the cylinder transformation

Definition 2.1 (see [1, 2]). Let be two regressive functions; define

Lemma 2.2 (see [1, 2]). Assume that are two regressive functions, then(i)  and  ;(ii) ;(iii) ;(iv) ;(v) ;(vi)if   ,  then    .

Lemma 2.3 (see [1, 2]). Assume that  is a function sequence on such that (i)  is a uniformly bounded on ;(ii)  is a uniformly bounded on .Then, there is a subsequence of which converges uniformly on where is an arbitrary compact subset of .

3. Uniformly Almost Periodic Functions

Let be a given time scale, and is a complete metric space with the metric (distance) defined by For a given , the -neighborhood of a given point is the set of all points such that .

Throughout this paper, denotes or , denotes an open set in or , and denotes an arbitrary compact subset of .

Definition 3.1. is called continuous at if, and only if for any , there exists such that, for any , is called continuous on provided that it is continuous for every .

Definition 3.2. is called uniformly continuous on if, for any , there exists such that, for any with it is implied that
Similar to the finite covering theorem in functional analysis (see [30]), one can easily show that the following.

Lemma 3.3. Let be a closed interval. If , where is an index set, and for every , is an open set in , then there exist , such that .

Also, one can easily prove the following two lemmas.

Lemma 3.4. If converges uniformly to on and each is continuous on , then is continuous on , , and converges uniformly to .

Lemma 3.5. If a sequence converges to on and, for each , has continuous derivative and converges uniformly to , then , and converges to uniformly on .

By using Lemma 3.3, one can easily show that the following.

Lemma 3.6. Let , then is uniformly continuous on .

Definition 3.7. A time scale is called an almost periodic time scale if
Obviously, if , then and if is an almost periodic time scale, then and

Example 3.8. If , where , then . Hence, it is an almost periodic time scale. Obviously, if , then . If , then .

Definition 3.9. Let be an almost periodic time scale. A function is called an almost periodic function if the -translation set of is a relatively dense set in for all ; that is, for any given , there exists a constant such that each interval of length contains a such that is called the -translation number of and is called the inclusion length of .

Definition 3.10. Let be an almost periodic time scale. A function is called an almost periodic function in uniformly for if the -translation set of is a relatively dense set in for all and for each compact subset of ; that is, for any given and each compact subset of , there exists a constant such that each interval of length contains a such that is called the -translation number of and is called the inclusion length of .

Obviously, an almost periodic function can be regarded as a special case of a uniformly almost periodic function. So, in the following, we mainly discuss the basic properties of uniformly almost periodic functions. The basic properties of almost periodic functions can be derived directly from the corresponding ones of uniformly almost periodic functions.

Remark 3.11. If , then in this case, Definitions 3.4 and 3.5 are equivalent to the definitions of the almost periodic functions and the uniformly almost periodic functions in [16], respectively. If , then , in this case, Definitions 3.4 and 3.5 are equivalent to the definitions of the almost periodic functions and the uniformly almost periodic sequences in [17, 18].

For convenience, we denote is almost periodic} and introduce some notations: let and be two sequences. Then, means that is a subsequence of , , and and are common subsequences of and , respectively, which means that and for some given function .

We will introduce the translation operator which means that and is written only when the limit exists. The mode of convergence, for example, pointwise, uniform, and so forth, will be specified at each use of the symbol.

Similar to the proof of Theorem 1.13 in [16], one can show that.

Theorem 3.12. Let be almost periodic in uniformly for , then it is uniformly continuous and bounded on .

Theorem 3.13. Let be almost periodic in uniformly for , then, for any given sequence , there exist a subsequence and such that holds uniformly on and is almost periodic in uniformly for .

Proof. For any and , let be an inclusion length of . For any given subsequence , we denote , where , and . (In fact, for any interval with length of , there exists , thus, we can choose a proper interval with length of such that , from the definition of , it is easy to see that .) Therefore, there exists a subsequence such that as , .
Also, it follows from Theorem 3.12 that is uniformly continuous on . Hence, there exists so that , for , implies
Since is a convergent sequence, there exists so that implies . Now, one can take such that common with then; for any integers , we have that is, Hence, we can obtain Thus, we can take sequences , and such that, for any integers , and all , the following holds: For all sequences , we can take a sequence then, it is easy to see that for any integers with and all the following holds: Therefore, converges uniformly on that is, holds uniformly on , where .
Next, we will prove that is continuous on . If this is not true,then there must exist such that is not continuous at this point. Then there exist and sequences , where as , and Let ; obviously, is a compact subset of . Hence, there exists positive integer so that implies According to the uniform continuity of on , for sufficiently large , we have From (3.17)–(3.18), we get this contradicts (3.16). Therefore, is continuous on .
Finally, for any compact set and given , one can take ; then, for all , the following holds: Let , for all ; we have which implies that is relatively dense. Therefore, is almost periodic in uniformly for . This completes the proof.

Theorem 3.14. Let ; if, for any sequence , there exists such that exists uniformly on , then is almost periodic in uniformly for .

Proof. For contradiction, if this is not true, then there exist and such that, for any sufficiently large , we can find an interval with length of and there is no -translation numbers of in this interval; that is, every point in this interval is not in .
One can take a number and find an interval with , where such that there is no -translation numbers of in this interval. Next, taking , obviously, , so ; then, one can find an interval with , where such that there is no -translation numbers of in this interval. Next, taking , obviously, . One can repeat these processes, again and again one can find , such that . Hence, for any , without loss of generality, let , for ; we have Therefore, there is no uniformly convergent subsequence of for ; this is a contradiction. Thus, is almost periodic in uniformly for . This completes the proof.

From Theorems 3.13 and 3.14, we can obtain the following equivalent definition of uniformly almost periodic functions.

Definition 3.15. Let ; if, for any given sequence , there exists a subsequence such that exists uniformly on , then is called an almost periodic function in uniformly for .

Similar to the proof of Theorem 2.11 in [16], one can prove that the following.

Theorem 3.16. If is almost periodic in uniformly for and is almost periodic with , then is almost periodic.

Definition 3.17. Let ; there exits such that exists uniformly on } is called the hull of .

Similar to the proofs of Theorems 1.6 and 1.8 in [19], one can prove the following two theorems, respectively.

Theorem 3.18. is compact if and only if is almost periodic in uniformly for .

Theorem 3.19. If is almost periodic in uniformly for , then for any .

From Definition 3.17 and Theorem 3.19, one can easily show that.

Theorem 3.20. If is almost periodic in uniformly for , then, for any , is almost periodic in uniformly for .

Theorem 3.21. If is almost periodic in uniformly for , then, for any , there exists a positive constant and for any , there exist a constant and such that and .

Proof. Since is uniformly continuous on , for any , there exists so that implies where .
We take , and , where is the inclusion length of .
For any , consider an interval , take and we have Hence, for all , we have . Therefore, for any , So, we let , then . This completes the proof.

Theorem 3.22. If are almost periodic in uniformly for , then, for any , is a nonempty relatively dense set in .

Proof. Since are almost periodic in uniformly for , they are uniformly continuous on . For any given , one can take , and are inclusion lengths of , respectively.
According to Theorem 3.18, we can take Hence, we can find -translation numbers of and : and , respectively, where , are integers, and .
Let , then can only be taken from a finite number set . When , denote the -translation numbers of and by , , respectively, that is, , and we take .
For any , on the interval , we can take -translation numbers of and : and , respectively; there must exist some integer such that Set then , and, for any , we have Therefore, there exists at least a on any interval with the length such that . The proof is complete.

According to Definition 3.10, one can easily prove the following.

Theorem 3.23. If is almost periodic in uniformly for , then, for any , functions are almost periodic in uniformly for .

Theorem 3.24. If are almost periodic in uniformly for , then are almost periodic in uniformly for and if , then are almost periodic in uniformly for .

Proof. From Theorem 3.22, for any , is a nonempty relatively dense set. It is easy to see that if , then . Hence, Therefore, is a relatively dense set, so is almost periodic in uniformly for .
On the other hand, denote , take , then, for all , we have Therefore, is a relatively dense set, so is almost periodic in uniformly for .
Faunally, denote and take , then, for all we have that is, . Hence, is almost periodic in uniformly for , so is almost periodic in uniformly for . The proof is complete.

Theorem 3.25. If is almost periodic in uniformly for , then is also continuous on and almost periodic in uniformly for .

Proof. Let be uniformly almost periodic, then, for any sequence , there exists a subsequence such that exists uniformly on , where is any compact set in . Consequently, exists uniformly on . In view of Theorem 3.14, this shows that is uniformly almost periodic.

Corollary 3.26. If is an almost periodic function, then is an almost periodic function on .

Theorem 3.27. If are almost periodic in for and the sequence uniformly converges to on , then is almost periodic in uniformly for .

Proof. For any , there exists sufficiently large such that, for all , Take , then, for all , we have that is, . Therefore, is also a relatively dense set; is almost periodic in uniformly for . This completes the proof.

Theorem 3.28. If is almost periodic in uniformly for , denote , then is almost periodic in uniformly for if and only if is bounded on .

Proof. If is almost periodic in uniformly for , then it is easy to see that is bounded on .
If is bounded, without loss of generality, then we can assume that is a real-valued function. Denote for any , there exist and such that Let be an inclusion length of , where . For any , take . Denote , so , for all , that is Since in any interval with length , there exist such that
Now, we denote ; in the following, we will prove that if , then . In fact, for any , one can take such that so for , we have That is, for , we have , so is almost periodic in uniformly for . The proof is complete.

Theorem 3.29. If is almost periodic in uniformly for and is uniformly continuous on the value field of , then is almost periodic in uniformly for .

Proof. In fact, since is uniformly continuous on the value field of and is almost periodic in uniformly for , there exists a real sequence such that holds uniformly on . Hence, is almost periodic in uniformly for . The proof is complete.

Similar to the proof of Theorem 1.17 in [19], one can easily get the following.

Theorem 3.30. A function is almost periodic in uniformly for if and only if, for every pair of sequences , there exist common subsequences such that

Definition 3.31. If every element of matrix function , where is almost periodic in uniformly for , then is called almost periodic in uniformly for .

If we use matrix norm: , then the definition above can be rewritten as.

Definition 3.32. A matrix function is almost periodic in uniformly for if and only if, for any , the translation set is a relatively dense set.

Theorem 3.33. Definition 3.31 is equivalent to Definition 3.32.

Proof. In fact, if is almost periodic in uniformly for , by Definition 3.31, then every element is almost periodic in uniformly for . Thus, for any , there exists nonempty relatively dense set . For any , we have
On the other hand, if, for any , is a relatively dense set, then, for any and , we have Hence, every element is almost periodic in uniformly for that is, is almost periodic in uniformly for . The proof is complete.

Definition 3.34. A continuous matrix function is called normal if, for any sequence , thenthere exists subsequence such that exists uniformly on .

Theorem 3.35. A continuous matrix function is normal if and only if is almost periodic in uniformly for .

Proof. If is normal, then every element satisfies Definition 3.15, so is almost periodic.
On the other hand, if is almost periodic in uniformly for , by Definition 3.31, for any sequence , thenthere exists subsequence such that exists uniformly on . Hence, there exists , such that exists uniformly on ; we can repeat this step times, then we can get a series of subsequences satisfying such that exist uniformly on . Therefore, there exists subsequence such that exists uniformly on ; that is, is normal. The proof is complete.

4. Almost Periodic Dynamic Equations on Almost Periodic Time Scales

Consider the following nonlinear dynamic equation where ; let is a bounded solution to (4.1)}.

Definition 4.1. If , then exists is called the least-value of solutions to (4.1). If there exists such that , then is called a minimum norm solution to (4.1), where .

Similar to the proof of Theorem 5.1 in [19], one can easily get the following.

Lemma 4.2. If is bounded on and (4.1) has a bounded solution such that and , then (4.1) must have a minimum norm solution.

Lemma 4.3. If is almost periodic in uniformly for , and (4.1) has a bounded solution on , then (4.1) has a bounded solution on and .

Proof. In fact, we may take such that and holds uniformly on . For any fixed , consider the interval and . It is easy to see that, for sufficiently large, is defined on and is a solution to and is uniformly bounded and equicontinuous on . Then let be a sequence which goes to , according to Lemma 2.3, there must exist such that holds uniformly on any compact subset of and, for all , we have . Since , by Lemma 3.5, is a solution to (4.1). This completes the proof.

Lemma 4.4. Let be almost periodic in uniformly for . If (4.1) has a minimum norm solution, then, for any , the following equation: has the same least-value of solutions as that to (4.1).

Proof. Let be the minimum norm solution to (4.1) and is the least-value. Since , there exists a sequence such that holds uniformly on . From Lemma 2.3, there exists such that holds uniformly on any compact subset of . By Lemma 3.5, is a solution to (4.2). For , we have ; thus, . Since and , we have ; thus, . Therefore, . The proof is complete.

From the process of the proof of Lemma 4.4, one can easily get the following.

Lemma 4.5. If is a minimum norm solution to (4.1) and there exists a sequence such that exists uniformly on ,   furthermore, if there exists a subsequence such that holds uniformly on any compact set of , then is a minimum norm solution to (4.2).

Lemma 4.6. Let be almost periodic in uniformly for and, for every , (4.2) has a unique minimum norm solution; then theses minimum norm solutions are almost periodic on .

Proof. For a fixed , (4.2) has the unique minimum norm solution . Since is almost periodic in uniformly for , we have that for any sequences , there exist common subsequences such that holds uniformly on and , hold uniformly on . It follows from Lemmas 4.4 and 4.5 that and are minimum norm solutions to the following equation: Since the minimum norm solution is unique, we have . Therefore, is almost periodic. The proof is complete.

We will now discuss the linear almost periodic dynamic equation on an almost periodic time scale : and its associated homogeneous equation where is an almost periodic matrix function and is an almost periodic vector function.

Definition 4.7. If , thenwe say that is a homogeneous equation in the hull of (4.4).

Definition 4.8. If and , then we say that is an equation in the hull of (4.4).

Definition 4.9 (see [31]). Let be rd-continuous matrix function on ; the linear system is said to admit an exponential dichotomy on if there exist positive constants ,  projection , and the fundamental solution matrix of (4.8), satisfying

Similar to the proof of Theorem 5.7 (Favard's Theorem) in [19], one can obtain that the following.

Lemma 4.10. If is an almost periodic matrix function and is an almost periodic solution of the homogeneous linear almost periodic dynamic equation , then or .

Similar to the proof of Theorems 6.3 and 5.8 in [19], one can easily get.

Lemma 4.11. Suppose that (4.5) has an almost periodic solution and . If (4.4) has bounded solution on , then (4.4) has an almost periodic solution.

Lemma 4.12. If every bounded solution of a homogeneous equation in the hull of (4.4) is almost periodic, then all bounded solutions of (4.4) are almost periodic.

Proof. According to Lemma 4.10, we know that every nontrivial bounded solution of equations in the hull of (4.4) satisfies . From Lemma 4.11, it follows that if (4.4) has bounded solutions on , then (4.4) must have an almost periodic solution . If is an arbitrary bounded solution of (4.4), then is a bounded solution of its associated homogeneous equation (4.5) and it is almost periodic. Thus, is almost periodic. This completes the proof.

Lemma 4.13. If a homogeneous equation in the hull of (4.4) has the unique bounded solution , then (4.4) has a unique almost periodic solution.

Proof. Let be two bounded solutions to (4.4), then is a solution of a homogeneous equation in the hull of (4.4), since , we have that . Thus, by Lemma 4.12, (4.4) has a unique almost periodic solution. This completes the proof.

Similar to the proof of Lemma 7.4 in [16], one can easily prove that the following.

Lemma 4.14. Let be a projection and a differentiable invertible matrix such that is bounded on . Then, there exists a differentiable matrix such that for all and are bounded on . In fact, there is an of the form , where commutes with .

Similar to the proof of Lemma 7.5, in [19], one can easily get.

Lemma 4.15. If (4.5) has an exponential dichotomy and is the fundamental solution matrix of (4.5), non-singular, then has an exponential dichotomy with the same projection if and only if .

Similar to the proof of Theorem 7.6 in [19], we can easily obtain.

Lemma 4.16. Suppose that is an almost periodic matrix function and (4.5) has an exponential dichotomy, then for every , (4.6) has an exponential dichotomy with the same projection and the same constants .

Lemma 4.17. If the homogeneous equation (4.5) has an exponential dichotomy, then (4.5) has only one bounded solution .

Proof. Let be the fundamental solution matrix to (4.5). For any sequence , denote . Since the homogeneous equation (4.5) has an exponential dichotomy, it is easy to see that there exists a constant such that and , where . Therefore, by Lemma 2.3, there exists such that converges uniformly on any compact subset of and exists uniformly on . So, is almost periodic. Since the homogeneous equation (4.5) has an exponential dichotomy, , from Lemma 4.10, . This completes the proof.

Lemma 4.18. If the homogeneous equation (4.5) has an exponential dichotomy, then all equations in the hull of (4.5) have only one bounded solution .

Proof. By Lemma 4.16, all equations in the hull of (4.5) have an exponential dichotomy; according to Lemma 4.17, all equations in the hull of (4.5) have only one bounded solution . This completes the proof.

Similar to the proof of Theorem 7.7 in [19], we have the following.

Theorem 4.19. Let be an almost periodic matrix function and be an almost periodic vector function. If (4.5) admits an exponential dichotomy, then (4.4) has a unique almost periodic solution: where