Abstract and Applied Analysis

Volume 2014 (2014), Article ID 782018, 11 pages

http://dx.doi.org/10.1155/2014/782018

Research Article

Existence of Almost Periodic Solutions for Impulsive Neutral Functional Differential Equations

Junwei Liu¹ and Chuanyi Zhang²

¹College of Science, Yanshan University, Qinhuangdao 066004, China

²Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 8 May 2014; Accepted 20 July 2014; Published 12 August 2014

Academic Editor: Robert A. Van Gorder

Copyright © 2014 Junwei Liu and Chuanyi Zhang. 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.

The existence of piecewise almost periodic solutions for impulsive neutral functional differential equations in Banach space is investigated. Our results are based on Krasnoselskii’s fixed-point theorem combined with an exponentially stable strongly continuous operator semigroup. An example is given to illustrate the theory.

1. Introduction

In this paper, we study the existence of piecewise almost periodic solutions for a class of abstract impulsive neutral functional differential equations with unbounded delay modeled in the form

where is the infinitesimal generator of an exponentially stable strongly continuous semigroup of linear operators on a Banach space , the history , belongs to an abstract phase space defined axiomatically, are appropriate functions, is a discrete set of real numbers such that when , and the symbol represents the jump of the function at , which is defined by .

The existence of solutions to impulsive differential equations is one of the most attracting topics in the qualitative theory of impulsive differential equations due to their applications in mechanical, electrical engineering, ecology, biology, and others; see, for instance, [1–6] and the references therein. Some recent contributions on mild solutions to impulsive neutral functional differential equations have been established in [7–14]. On the other hand, the existence of almost periodic solutions for impulsive differential equations has been investigated by many authors; see, for example, [2–5, 15, 16]. However, the existence of almost periodic solutions for the impulsive neutral functional differential equations in the form (1) is an untreated topic in the literature and this fact is the motivation of the present work.

The paper is organized as follows: in Section 2, we recall some notations, concepts, and useful lemmas which are used in this paper. In Section 3, some criteria ensuring the existence of almost periodic solutions for impulsive neutral functional differential equations are obtained. In Section 4, we give an application.

2. Preliminaries

Let be a Banach space. is the infinitesimal generator of a strongly continuous semigroup of linear operators on the Banach space and , are positive constants such that for . Let ; it is possible to define the fractional power , , as a closed linear operator with its domain . We denote by a Banach space between and endowed with the norm , ; the following properties hold.

Lemma 1 (see [17, 18]). Let ; then is continuously embedded into with bounded ; that is,

Moreover, the function is continuous in the uniform operator topology on and there exists such that for every .

Throughout this paper, let be the set consisting of all real sequences such that . For , let be the space formed by all piecewise continuous functions such that is continuous at for any and for all ; let be the space formed by all piecewise continuous functions such that, for any , is continuous at for any and for all and for any , is continuous at .

Definition 2. A number is called an -translation number of the function if

for all which satisfies , for all . Denote by the set of all -translation numbers of .

Definition 3 (see [1]). A function is said to be piecewise almost periodic if the following conditions are fulfilled.

(1), , , are equipotentially almost periodic; that is, for any , there exists a relatively dense set of almost periods that are common to all the sequences .(2)For any , there exists a positive number such that if the points and belong to a same interval of continuity of and , then .(3)For every ,   is a relatively dense set in .

We denote by the space of all piecewise almost periodic functions. endowed with the uniform convergence topology is a Banach space.

Definition 4. is said to be piecewise almost periodic in uniformly in if for each compact set , is uniformly bounded and, given , there exists a relatively dense set such that

for all , , and , for all . Denote by the set of all such functions.

Lemma 5 (see [15]). Let ; then the range of , , is a relatively compact subset of .

Lemma 6. Suppose that and is uniformly continuous on each compact subset uniformly for ; that is, for every , there exists such that and implies that for all . Then for any .

Proof. Since , by Lemma 5, is a relatively compact subset of . Because is uniformly continuous on each compact subset uniformly for , then for any , there exist a number , such that

where , , and , . By piecewise almost periodic of and , there exists a relatively dense set of such that the following conditions hold:

for and , , , . Note that

We have

We deduce from (5) and (6) that the following formula holds:

That is, is piecewise almost periodic. The proof is complete.

Since the uniform continuity is weaker than the Lipschitz continuity, we obtain the following lemma as an immediate consequence of the previous lemma.

Lemma 7. Let and is Lipschitz; that is, there is a positive number such that

for all and ; if for any , then .

In this paper, we assume that the phase space is a linear space formed by functions mapping into endowed with a norm and satisfying the following conditions.

(1)If , then for all and , where is a constant independent of and .(2)The space is complete.(3)If is a uniformly bounded sequence in formed by functions with compact support and in the compact open topology, then and , as .

Lemma 8 (see [1, 15]). Assume that , the sequence is almost periodic in , and , , , are equipotentially almost periodic. Then for each , there are relatively dense sets of and of such that the following conditions hold.

(i) for all , , , and .(ii) for all and .(iii)For every , there exists at least one number such that

Definition 9. A bounded function is a mild solution of (1) if the following holds: , the function is integrable, and for any , ,

Since for all , let ; then we have , and the above formula can be replaced by

In fact, for ,

so that

In order to obtain our results, we need to introduce some additional notations. Let be a continuous function such that for all and as . We consider the space

Endowed with the norm , it is a Banach space.

We recall here the following compactness criterion in these spaces, which we can refer to [7, 19–23].

Lemma 10. A set is a relatively compact set if and only if

(1) uniformly for ;(2) is relatively compact in for every ;(3)the set is equicontinuous on each interval .

Theorem 11 (Krasnoselskii’s fixed-point theorem [24]). Let be a closed convex nonempty subset of a Banach space ; suppose that and map into such that

(i), (ii) is completely continuous,(iii) is a contraction with constant .

Then there is a with .

3. Main Results

In this section, we discuss the existence of piecewise almost periodic solutions for impulsive neutral functional differential equation (1). To begin, let us list the following hypotheses.

(A1)The operator is the infinitesimal generator of an exponentially stable strongly continuous semigroup of linear operators ; that is, there exist constants , such that for . Moreover, is compact for  .(A2) is uniformly continuous in uniformly in ; is almost periodic in uniformly in and is a uniformly continuous function in for all . For every , , . Moreover, there exist a number , such that , where , .(A3), , and there exist a number such that

 for all , , .(A4)Let be uniformly bounded in and uniformly convergent in each compact set of ; then is relatively compact in .

Theorem 12. Suppose that conditions (A1)–(A4) hold; then (1) has a piecewise almost periodic solution provided that .

Proof. Let

Note that is a closed convex set of . By (A3) and Lemma 1, we have

and we infer that is integrable on .

Define the operator on by

In order to prove that has a fixed point in , we introduce the decomposition , where

Our proof will be split into the following three steps.

Step  1. We claim that .

For any , by (A3) and Lemma 1, we have

In order to estimate the last part of the second term on the right hand side of the above formula, we assume that, for every , there exists , such that ,

so

Then,

from , and we obtain that

By (A2) and Lemma 6, , and, by (A3) and Lemma 7, . By (A2) and [1, Lemma 37], is almost periodic. From [1, Theorem 73], for the two almost periodic functions , , there exists a relatively dense set of their common -translation numbers. Then by Lemma 8, for every , there exist relatively dense sets of and of such that for q ,

where , , . So for , we know that

so

That is,

so we have

so

Combining (26), (30), and (33), it follows that (for all ).

Step  2. is a contraction.

Let ; by (A3) and Lemma 1, we have

Therefore,

Since , it follows that is a contraction.

Step  3. is completely continuous.

Claim   1. is continuous.

Let , in as ; by Lemma 5, we may find a compact subset such that for all , ; here we assume . By (A2), for the given , there exist such that , , implies that

For the above , there exists such that

for and ; then

for and . Hence,

for and , from which it follows that is continuous.

Claim  2. is a relatively compact subset of for each .

For any , let

where   is uniformly bounded in and is compact, so is relatively compact in . Moreover,

so is a relatively compact subset of for each .

Claim  3.   is equicontinuous on each interval .

Let ,

Moreover,

By (A1), for the given , there exists , such that if , then

So

Similarly,

So that, for , for any , there exists a positive number , ; if ,

That is, is equicontinuous on each interval .

Since and satisfies the conditions of Lemma 10, is completely continuous.

By Krasnoselskii’s fixed-point theorem (Theorem 11), we know that has a fixed point ; that is, (1) has a piecewise almost periodic solution . The proof is complete.

Note that the condition of uniformly continuous is weaker than that of Lipschitz continuous, so if assumption (A2) is replaced by the following assumption:

(A2′) is almost periodic in uniformly in and

for all .

We can get the almost periodic solution of (1) by means of contraction mapping principle.

Corollary 13. Suppose that conditions (A1), (A2′), and (A3)-(A4) hold; (1) has a piecewise almost periodic solution provided that

Proof. As the same discussion as Step 2 of Theorem 12, we can prove that is a contraction, and it remains to show that is a contraction. By Step 1 of Theorem 12, . For , ,

The proof is complete.

4. Application

Consider the following impulsive neutral differential system:

where , , , are equipotentially almost periodic such that . The system (51) arises, for example, in control systems described by abstract retarded functional differential equations with feedback control governed by proportional integrodifferential law; see [25, 26] for details.

Let and be the infinitesimal generator of an analytic semigroup which is compact for and given by with domain . The semigroup is defined for by

where is an orthonormal basis of ; then , . For , , ,   is the Banach space endowed with the norm between and .

In this paper, we assume , .

To study the system (51), we make the following assumptions.

(i) The functions are Lebesgue measurable, and

(ii) The functions , are continuous, and the sequence of functions is almost periodic.

Under these conditions, we define by

We assume further that satisfy the following condition.

(iii) and are uniformly continuous in uniformly in .

Under the above assumptions, we can rewrite (51) as the abstract form (1) and verify that the assumptions of Theorem 12 hold; then we can get the next result, which is a consequence of Theorem 12.

Proposition 14. Assume that the previous conditions are verified, if

The system (51) has a piecewise almost periodic solution.

Conflict of Interests

The authors declare that they have no conflict of interests.

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