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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 186361, 10 pages
http://dx.doi.org/10.1155/2012/186361
Research Article

Existence of Almost Periodic Solutions to th-Order Neutral Differential Equations with Piecewise Constant Arguments

Department of Mathematics, Huizhou University, Huizhou 516007, China

Received 12 November 2011; Accepted 12 December 2011

Academic Editor: SimeonΒ Reich

Copyright Β© 2012 Rong-Kun Zhuang. 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 present some conditions for the existence and uniqueness of almost periodic solutions of th-order neutral differential equations with piecewise constant arguments of the form , here is the greatest integer function, and are nonzero constants, is a positive integer, and is almost periodic.

1. Introduction

In this paper we study certain functional differential equations of neutral delay type with piecewise constant arguments of the form here is the greatest integer function, and are nonzero constants, is a positive integer, and is almost periodic. Throughout this paper, we use the following notations: is the set of reals; the set of positive reals; the set of integers; that is, ; the set of positive integers; denotes the set of complex numbers. A function is called a solution of (1.1) if the following conditions are satisfied:(i) is continuous on ;(ii)the th-order derivative of exists on except possibly at the points , , where one-sided th-order derivatives of exist;(iii) satisfies (1.1) on each interval with integer .

Differential equations with piecewise constant arguments are usually referred to as a hybrid system, and could model certain harmonic oscillators with almost periodic forcing. For some excellent works in this field we refer the reader to [1–5] and references therein, and for a survey of work on differential equations with piecewise constant arguments we refer the reader to [6].

In paper [1, 2], Yuan and Li and He, respectively, studied the existence of almost periodic solutions for second-order equations involving the argument in the unknown function. In paper [3], Seifert intensively studied the special case of (1.1) for and by using different methods. However, to the best of our knowledge, there are no results regarding the existence of almost periodic solutions for th-order neutral differential equations with piecewise constant arguments as (1.1) up to now.

Motivated by the ideas of Yuan [1] and Seifert [3], in this paper we will investigate the existence of almost periodic solutions to (1.1). Both the cases when and are considered.

2. The Main Results

We begin with some definitions, which can be found (or simply deduced from the theory) in any book, say [7], on almost periodic functions.

Definition 2.1. A set is said to be relatively dense if there exists such that for all .

Definition 2.2. A bounded continuous function (resp., ) is said to be almost periodic if the -translation set of is relatively dense for each . We denote the set of all such function by (resp., ).

Definition 2.3. A sequence (resp., ), , , denoted by , is called an almost periodic sequence if the -translation set of is relatively dense for each , here is any convenient norm in (resp., ). We denote the set of all such sequences by (resp., ).

Proposition 2.4. (resp., ) if and only if (resp., ), .

Proposition 2.5. Suppose that , . Then the sets and are relatively dense.

Now one rewrites (1.1) as the following equivalent system Let be solutions of system (2.3) on , for , , using we obtain and using this with we obtain Continuing this way, and, at last, we get Since must be continuous at , using these equations we get for , where

Lemma 2.6. If , then sequences .

Proof. We typically consider and , we have From Definition 2.3, it follows that is an almost periodic sequence. In a manner similar to the proof just completed, we know that are also almost periodic sequences. This completes the proof of the lemma.

Lemma 2.7. The system of difference equations has solutions on ; these are in fact uniquely determined by .

Proof. It is easy to check that , are uniquely determined in term of , for . For , uniquely determines , uniquely determines uniquely determines , and thus since , uniquely determines . So , are determined. Continuing in this way, we establish the lemma.

Lemma 2.8. For any solution , , of system (2.10), there exists a solution , , of (2.3) such that , , .

Proof . Define for , . It can easily be verified that is continuous on ; we omit the details.
Define , , where is continuous, and , ; Continuing this way, we can define for . Similarly, define continuing in this way is defined for , and so is defined for all .
Next, define , , and by the appropriate one-sided derivative of at . It is easy to see that are continuous on , and for ; we omit the details.

Next we express system (2.7) in terms of an equivalent system in give by where

Lemma 2.9. Suppose that all eigenvalues of are simple (denoted by ) and , . Then system (2.14) has a unique almost periodic solution.

Proof. From our hypotheses, there exists a nonsingular matrix such that , where and are the distinct eigenvalues of . Define , then (2.14) becomes where .
For the sake of simplicity, we consider first the case . Define where , . Clearly is almost periodic, since , and is. For , we have this shows that .
If , , in a manner similar to the proof just completed for , we know that , , and so . It follows easily that then and our lemma follows.
Assume now . Now define As before, the fact that follows easily from the fact that . So in every possible case, we see that each component , , of is almost periodic and so .
The uniqueness of this almost periodic solution of (2.14) follows from the uniqueness of the solution of (2.16) since , and the uniqueness of of (2.16) follows, since if were a solution of (2.16) distinct from , would also be almost periodic and solve . But by our condition on , it follows that each component of must become unbounded either as or as , and that is impossible, since it must be almost periodic. This proves the lemma.

Lemma 2.10. Suppose that conditions of Lemma 2.9 hold, is as defined in the proof of Lemma 2.8 with the unique first components of the almost periodic solution of (2.14) given by Lemma 2.9, then is almost periodic.

Proof. For , It follows from definition that is almost periodic.

Theorem 2.11. Suppose that and all eigenvalues of in (2.14) are simple (denoted by ) and satisfy , . Then (1.1) has a unique almost periodic solution , which can, in fact be determined explicitly in terms of as defined in the proof of Lemma 2.8.

Proof. Consider the following.Case 1 (). For each define as follows: here is as defined in the proof of Lemma 2.8, and where is the first component of the solution of (2.14) given by Lemma 2.9. Let , then from (2.21) we get It follows that If , , and so for such ,
Let , we get
Since and are uniformly continuous on , it follows that is equicontinuous on each interval , and by the Ascoli-ArzelΓ‘ Theorem, there exists a subsequence, which we again denote by , and a function such that uniformly on , and by a familiar diagonalization procedure, can find a subsequence, again denoted by which is such that for each . From (2.27) it follows that and so is almost periodic since is almost periodic in for each , and . From (2.21), letting , we get , and since solves (1.1), does also. The uniqueness of as an almost periodic solution of (1.1) follows from the uniqueness of the almost periodic solution of (2.14) given by Lemma 2.9, which determines the uniqueness of , and therefore from (2.21) the uniqueness of .
Case 2 (). Rewriting (2.24) as we deduce in a similar manner that
The remainder of the proof is similar to that of Case 1, we omit the details.

If , the system of difference equations (2.10) of Lemma 2.7 now becomes and system (2.14) reduces to where and , . Then we have the following theorem.

Theorem 2.12. Let and , if all eigenvalues of in (2.32) are simple (denoted by ) and satisfy , then (1.1) has a unique almost periodic solution .

Proof. System (2.32) has a solution on since is nonsingular because . The rest of the proof follows in the same way as the proof of Theorem 2.11 and is omitted.

Funding

This paper was supported by NNSF of China and NSF of Guangdong Province (1015160150100003).

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