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Abstract and Applied Analysis
Volume 2008, Article ID 939062, 19 pages
http://dx.doi.org/10.1155/2008/939062
Research Article

Existence of Periodic Solutions for Integrodifferential Impulsive Periodic System on Banach Space

1College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
2College of Science, Guizhou University, Guiyang, Guizhou 550025, China

Received 5 July 2008; Accepted 28 October 2008

Academic Editor: Agacik Zafer

Copyright © 2008 JinRong Wang et al. 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

This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, the -periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. By virtue of the generalized new Gronwall lemma with impulse and -norm, the estimate on the PC-mild solutions is derived. Showing the continuity and compactness of the Poincaré operator, we utilize Horn's fixed point theorem to prove the existence of -periodic PC-mild solutions when the PC-mild solutions are bounded and ultimate bounded. This extends the study of periodic solutions of integrodifferential periodic system without impulse to integrodifferential periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.

1. Introduction

It is well known that impulsive periodic motion is a very important and special phenomenon not only in natural science but also in social science such as climate, food supplement, insecticide population, and sustainable development. Periodic system with applications on finite dimensional spaces has been extensively studied. Particularly, impulsive periodic systems on finite dimensional spaces are considered and some important results (such as the existence and stability of periodic solution, the relationship between bounded solution and periodic solution, robustness by perturbation) are obtained (see [14]).

Since the end of the last century, many authors including us pay great attention to impulsive systems on infinite dimensional spaces. Ahmed investigated optimal control problems of a system governed by an impulsive system (see [58]). We also gave a series of results for semilinear (integrodifferential, strongly nonlinear) impulsive systems and optimal control problems (see [912]). Particulary, Benchohra et al. give many interesting recent results for various classes of differential equations with impulses on finite and infinite dimensional Banach spaces (see [13]).

Although, there are some papers on periodic solution for periodic system on infinite dimensional spaces (see [1424]) and some results discussing integrodifferential system on finite Banach space and infinite Banach space (see [19, 25]), to our knowledge, integrodifferential impulsive periodic systems on infinite dimensional spaces (with unbounded operator) have not been extensively investigated. Recently, we discuss the impulsive periodic system and integrodifferential impulsive system on infinite dimensional spaces. For linear impulsive evolution operator is constructed and -periodic -mild solution is introduced. Existence of periodic solutions and alternative theorem, criteria of Massera type, and asymptotical stability and robustness by perturbation are established (see [2628]). For integrodifferential impulsive system, existence of -mild solutions and optimal controls are presented (see [9]).

Herein, we go on studying the following integrodifferential impulsive periodic system:on infinite dimensional Banach space , where , , , , , is a fixed positive number, and denotes the number of impulsive points between and . The operator is the infinitesimal generator of a -semigroup on , is a measurable function from to and is -periodic in , is a continuous function from to and is -periodic in and and , . This paper is mainly concerned with the existence of periodic solutions for integrodifferential impulsive periodic system on infinite dimensional Banach space .

In this paper, we use Horn's fixed point theorem to obtain the existence of periodic solutions for integrodifferential impulsive periodic system (1.1). First, by virtue of impulsive evolution operator corresponding to linear homogeneous impulsive system, we construct a new Poincaré operator for integrodifferential impulsive periodic system (1.1), then overcome some difficulties to show the continuity and compactness of Poincaré operator which are very important. By a new generalized Gronwall inequality with impulse and -norm given by us, the estimate of -mild solutions is established. Therefore, the existence of -periodic -mild solutions for impulsive integrodifferential periodic system is shown. This extends the study of periodic solutions of integrodifferential periodic system without impulse to integrodifferential periodic system with impulse on general Banach spaces.

This paper is organized as follows. In Section 2, some results of linear impulsive periodic system and properties of impulsive periodic evolution operator corresponding to homogeneous linear impulsive periodic system are recalled. In Section 3, the new generalized Gronwall inequality with impulse and -norm is established and the existence of -mild solutions for integrodifferential impulsive system is presented. In Section 4, the -periodic -mild solutions for integrodifferential impulsive periodic system (1.1) are introduced. We construct the suitable Poincaré operator and give the relation between -periodic -mild solution and the fixed point of the Poincaré operator . After showing the continuity and compactness of the Poincaré operator , we can use Horn's fixed point theorem to establish the existence of -periodic -mild solutions for integrodifferential impulsive periodic system when the -mild solutions are bounded and ultimate bounded. At last, an example is given to demonstrate the applicability of our result.

2. Preliminaries

In order to study the integrodifferential impulsive periodic system, we first recall some results about linear impulsive periodic system here. Let be a Banach space. denotes the space of linear operators in , and denotes the space of bounded linear operators in . is the Banach space with the usual supremum norm. Define , where denotes the number of impulsive points between 0 and . We introduce is continuous at , is continuous from left and has right-hand limits at and SetIt can be seen that a Banach space is endowed with the norm , .

Firstly, we consider the homogeneous linear impulsive periodic system

We introduce the following assumption.

Assumption [H1] [H1.1] is the infinitesimal generator of a -semigroup on with domain .[H1.2]There exists such that .[H1.3]For each , , and .

In order to study system (2.2), we need to consider the associated Cauchy problem

If and is an invariant subspace of , using [29, Theorem 5.2.2, page 144], step by step, one can verify that the Cauchy problem (2.3) has a unique classical solution represented by where given byThe operator is called impulsive evolution operator associated with .

Now we introduce the -mild solution of Cauchy problem (2.3) and -periodic -mild solution of the system (2.2).

Definition 2.1. For every , the function given by is said to be the -mild solution of the Cauchy problem (2.3).

Definition 2.2. A function is said to be a -periodic -mild solution of system (2.2) if it is a -mild solution of Cauchy problem (2.3) corresponding to some and for .

The following lemma gives the properties of the impulsive evolution operator associated with which are widely used in the sequel.

Lemma 2.3 (see [26, Lemma 1]). Impulsive evolution operator has the following properties.
(1)For , , that is, , where .(2)For , , .(3)For and , (4)For and , .(5)If is a compact semigroup in , then is a compact operator for .

Here, we note that system (2.2) has a -periodic -mild solution if and only if has a fixed point. The impulsive evolution operator can be used to reduce the existence of -periodic -mild solutions for linear impulsive periodic system to the existence of fixed points for an operator equation. This implies that we can build up the new framework to study the periodic -mild solutions for integrodifferential impulsive periodic system on Banach space.

Consider the nonhomogeneous linear impulsive periodic systemand the associated Cauchy problemwhere , and .

Now we introduce the -mild solution of Cauchy problem (2.7) and -periodic -mild solution of system (2.6).

Definition 2.4. A function , for finite interval , is said to be a -mild solution of the Cauchy problem (2.6) corresponding to the initial value and input if is given by

Definition 2.5. A function is said to be a -periodic -mild solution of system (2.6) if it is a -mild solution of Cauchy problem (2.7) corresponding to some and for .

3. The Generalized Gronwall Inequality and Existence of Solutions

In order to derive an estimate of the -mild solutions, we introduce the following generalized Gronwall inequality with impulse and -norm.

Lemma 3.1. Let and let it satisfy the following inequality:where , , are constants, and
Then one has

Proof. (i) For , , let andThen
Using (3.5), we obtainDefinethen we getSince , we then have
For , by (3.9), we obtainfurther,thus,
(ii) For , we only need to defineSimilar to the proof in (i), one can obtainCombining (i) and (ii), one can complete the proof.

Now, we consider the following integrodifferential impulsive periodic system:and the associated Cauchy problem

By virtue of the expression of the -mild solution of the Cauchy problem (2.7), we can introduce the -mild solution of the Cauchy problem (3.16).

Definition 3.2. A function is said to be a -mild solution of the Cauchy problem (3.16) corresponding to the initial value if satisfies the following integral equation:
Now, we introduce the -periodic -mild solution of system (3.15).

Definition 3.3. A function is said to be a -periodic -mild solution of system (3.15) if it is a -mild solution of Cauchy problem (3.16) corresponding to some and for .

Assumption [H2] [H2.1] is measurable for and for any , , satisfying there exists a positive constant such that[H2.2]There exists a positive constant such that [H2.3] is -periodic in t, that is,[H2.4] is continuous for and for any , satisfying there exists a positive constant such that[H2.5]There exists a positive constant such that[H2.6] are -periodic in and , that is,[H2.7]For each and there exists such that .

It is not difficult to verify the following results.

Lemma 3.4. Under the assumptions [H2.4] and [H2.5], one has the following properties: (1)(2)for all and (3)for

Now we present the existence of -mild solution for system (3.16).

Theorem 3.5. Assumptions [H1.1], [H2.1], [H2.2], [H2.4], and [H2.5] hold, and for each , , are fixed. Then system (3.16) has a unique -mild solution given by

Proof. A similar result is given by Wei et al. in [9]. Thus, we only sketch the proof here. In order to make the process clear we divide it into three steps.
Step 1. We consider the following general integrodifferential equation without impulseIn order to obtain the local existence of mild solution for system (3.27), we only need to set up the framework for using the contraction mapping theorem. Consider the ball given bywhere would be chosen and , . is a closed convex set. Define a map on given byUnder the assumptions [H1.1], [H2.1], [H2.2], [H2.4], [H2.5], and Lemma 3.1, one can verify that map is a contraction map on with chosen . This means that system (3.27) has a unique mild solution given byAgain, using Lemma 3.1, we can obtain the a priori estimate of the mild solutions for system (3.27) and present the global existence of mild solutions.Step 2. For , consider the Cauchy problem By Step 1, Cauchy problem (3.31) also has a unique -mild solution:Step 3. Combining all solutions on (), one can obtain the -mild solution of the Cauchy problem (3.16) given by This completes the proof.

4. Poincaré Operator and Existence of Periodic Solutions

To establish the periodic solutions for the system (3.15), we define a Poincaré operator from to as follows:where denotes the -mild solution of the Cauchy problem (3.16) corresponding to the initial value , then examine whether has a fixed point.

We first note that a fixed point of the Poincaré operator gives rise to a periodic solution.

Lemma 4.1. System (3.15) has a -periodic -mild solution if and only if has a fixed point.

Proof. Suppose , then . This implies that is a fixed point of . On the other hand, if , , then for the -mild solution of the Cauchy problem (3.16) corresponding to the initial value , we can define , then . Now, for , we can use the properties (2), (3), and (4) of Lemma 2.3 and assumptions [H1.2], [H1.3], [H2.3], [H2.6], and [H2.7] to arrive at This implies that is a -mild solution of Cauchy problem (3.16) with initial value . Thus the uniqueness implies that , so that is a -periodic.

Second, we show that the Poincaré operator defined by (4.1) is a continuous operator.

Lemma 4.2. Assumptions [H1.1], [H2.1], [H2.2], [H2.4], and [H2.5] hold. Then the operator is a continuous operator of on .

Proof. Let , , where is a bounded subset of . Suppose and are the -mild solutions of Cauchy problem (3.16) corresponding to the initial value and respectively, given byThus, by assumption [H2.2], property (1) of Lemma 2.3, and property (3) of Lemma 3.4, we obtain
By Lemma 3.1, one can verify that there exist constants and such thatLet , then , which imply that they are locally bounded.
By assumption [H2.1], property (1) of Lemma 2.3, and property (2) of Lemma 3.4, we obtainBy Lemma 3.1 again, one can verify that there exists a constant such thatwhich implies thatHence, is a continuous operator of on .

In the sequel, we need to prove the compactness of operator , so we make the following assumption.

Assumption [H3]. The semigroup is a compact semigroup on .
Now, we are ready to prove the compactness of operator defined by (4.1).

Lemma 4.3. Assumptions [H1.1], [H2.1], [H2.2], [H2.4], [H2.5], and [H3] hold. Then the operator is a compact operator.

Proof. We only need to verify that takes a bounded set into a precompact set in .
Let be a bounded subset of . Define .
For , define
Next, we show that is precompact in . In fact, for fixed, we haveThis implies that the set is totally bounded.
By virtue of the compactness of and property (5) of Lemma 2.3, is a compact operator. Thus, is precompact in .
On the other hand, for arbitrary ,Thus, combined with (4.1), we haveIt is showing that the set can be approximated to an arbitrary degree of accuracy by a precompact set . Hence itself is a precompact set in , that is, takes a bounded set into a precompact set in . As a result, is a compact operator.

After showing the continuity and compactness of operator , we can follow and derive periodic -mild solutions for system (3.15). In the sequel, we define the following definitions. The following definitions are standard, we state them here for convenient references. Note that the uniform boundedness and uniform ultimate boundedness are not required to obtain the periodic -mild solutions here, so we only define the (locally) boundedness and ultimate boundedness.

Definition 4.4. One says that -mild solutions of Cauchy problem (3.16) are bounded if for each , there is a such that implies for .

Definition 4.5. One says that -mild solutions of Cauchy problem (3.16) are locally bounded if for each and , there is a such that implies for .

Definition 4.6. One says that -mild solutions of Cauchy problem (3.16) are ultimate bounded if there is a bound such that for each , there is a such that and imply .

We also need the following results as a reference.

Lemma 4.7 (see [21, Theorem 3.1]). Local boundedness and ultimate boundedness imply boundedness and ultimate boundedness.

Lemma 4.8 (Horn's fixed point theorem [20, Lemma 3.1]). Let be convex subsets of Banach space , with and compact subsets and open relative to . Let : be a continuous map such that for some integer , one hasthen has a fixed point in .

With these preparations, we can prove our main result in this paper.

Theorem 4.9. Let assumptions [H1], [H2], and [H3] hold. If the -mild solutions of Cauchy problem (3.16) are ultimate bounded, then system (3.15) has a -periodic -mild solution.

Proof. By Theorem 3.5, (4.4) in Lemma 4.2, and Definition 4.5, the Cauchy problem (3.16) corresponding to the initial value has a -mild solution which is locally bounded. From ultimate boundedness and Lemma 4.7, is bounded. Next, let be the bound in the definition of ultimate boundedness. Then by boundedness, there is a such that implies for . Furthermore, there is a such that implies for . Now, using ultimate boundedness again, there is a positive integer such that implies for .
Define , then From (4.2) in Lemma 4.1, we obtain Thus, Suppose there exists integer such that By induction we arrive atThus, we obtain
It comes from Lemma 4.3 that on is compact. Now letwhere conv is the convex hull of the set defined by conv and cl. denotes the closure. Then we see that are convex subsets of with , compact subsets and open relative to , and from (4.16) one hasWe see that : is a continuous map continuous from Lemma 4.2. Consequently, from Horn's fixed point theorem, we know that the operator has a fixed point . By Lemma 4.1, we know that the -mild solution of the Cauchy problem (3.16) corresponding to the initial value is just -periodic. Therefore, is a -periodic -mild solution of system (3.15). This proves the theorem.

At last, an example is given to illustrate our theory. Consider the following problem:where is a bounded domain and .

Define , , and for . Then, generates a compact semigroup on . Define , , , and

Thus problem (4.19) can be rewritten asIf the -mild solutions of Cauchy problem (4.21) are ultimate bounded, then all the assumptions in Theorem 4.9 are met, and our results can be used to system (4.19), that is, problem (4.19) has a -periodic -mild solution , where

Acknowledgments

This work is supported by Natural Science Foundation of Guizhou Province Education Department (no. 2007008) and Guizhou Province (no. 20052001). This work is also supported by the undergraduate carve out project of Department of Guiyang Science and Technology ([2008] no. 15-2).

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