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Mathematical Problems in Engineering
Volume 2008, Article ID 183489, 15 pages
http://dx.doi.org/10.1155/2008/183489
Research Article

Periodic Solutions of Semilinear Impulsive Periodic System with Time-Varying Generating Operators 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 17 June 2008; Accepted 19 October 2008

Academic Editor: Fernando Lobo Pereira

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

A class of semilinear impulsive periodic systems with time-varying generating operators on Banach space is considered. Using impulsive periodic evolution operator given by us, the -periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of Poincaré operator and using a new generalized Gronwall inequality with mixed type integral operators given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of -periodic PC-mild solutions. Our method is an innovation and it is much different from methods of other papers. 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. No autonomous periodic systems with applications on finite dimensional spaces have been extensively studied. Particularly, no autonomous impulsive periodic systems on finite dimensional spaces are considered and some important results (such as the existence and stability of periodic solutions, the relationship between bounded solution and periodic solution, and robustness by perturbation) are obtained (see [15]).

Since the end of last century, many authors including us pay great attention on impulsive systems with time-varying generating operators on infinite dimensional spaces. Particulary, Dr. Ahmed investigated optimal control problems of system governed by artificial heart model, uncertain systems, impulsive system with time-varying generating operators, access control mechanism model, computer network traffic controllers model, and active queue management (AQM) system (see [614]). We also gave a series of results for semilinear (strongly nonlinear) impulsive systems with time-varying generating operators and optimal control problems (see [1518]).

Although, there are some papers on periodic solutions of periodic system with time-varying generating operators on infinite dimensional spaces (see [1922]), to our knowledge, nonlinear impulsive periodic systems with time-varying generating operators on infinite dimensional (with unbounded operator) have not been extensively investigated. Recently, we consider impulsive periodic system on infinite dimensional spaces. For linear impulsive evolution operator is constructed and -periodic -mild solution is introduced. The existence of periodic solutions and alternative theorem, criteria of Massera type, as well as asymptotical stability and robustness by perturbation are established (see [2325]).

Herein, we go on studying the semilinear impulsive periodic system with time-varying generating operatorsin the parabolic case on infinite dimensional Banach space , where is a family of closed densely defined linear unbounded operators on and the resolvent of the unbounded operator is compact. Time sequence , , , , , is a fixed positive number and denoted the number of impulsive points between and . is a measurable function from to and is -periodic in , , . This paper is mainly concerned with the existence of periodic solution for semilinear impulsive periodic system with time-varying generating operators on infinite dimensional Banach space .

In this paper, we use Leray-Schauder fixed point theorem to obtain the existence of periodic solutions for semilinear impulsive periodic system with time-varying generating operators (1.1). First, by virtue of impulsive evolution operator corresponding to linear homogeneous impulsive system with time-varying generating operators, we construct a new Poincaré operator for semilinear impulsive periodic system with time-varying generating operators (1.1), then overcome some difficulties to show the compactness of Poincaré operator which is very important. By a new generalized Gronwall's inequality with mixed-type integral operators given by us, the estimate of fixed point set is established. Therefore, the existence of -periodic -mild solutions for semilinear impulsive periodic system with time-varying generating operators is shown.

In order to obtain the existence of periodic solutions, many authors use Horn's fixed point theorem or Banach fixed point theorem. In [26, 27], by virtue of Horn's fixed point theorem and Banach fixed point theorem, respectively, we also obtain the existence of periodic solutions for impulsive periodic systems. However, the conditions for Horn's fixed point theorem are not easy to be verified sometimes and the conditions for Banach's fixed point theorem are too strong. Here, a new way to show the existence of periodic solutions is given by us, which is much different from our previous works, and other related results in the literature. In addition, the conditions are easier to be verified and more weak compared with some related papers (see [20, 26]). Of course, the new generalized Gronwall's inequality with mixed-type integral operators given by us which can be used in other problems have played an essential role in the study of nonlinear problems on infinite dimensional spaces.

This paper is organized as follows. In Section 2, some results of linear impulsive periodic system with time-varying generating operators and properties of impulsive periodic evolution operator corresponding to homogeneous linear impulsive periodic system with time-varying generating operators are recalled. In Section 3, first, the new generalized Gronwall's inequality with mixed-type integral operator is shown and the -periodic -mild solution for semilinear impulsive periodic system with time-varying generating operators (1.1) is introduced. We construct the suitable Poincaré operator and give the relation between -periodic -mild solution and the fixed point of . After showing the compactness of the Poincaré operator and obtaining the boundedness of the fixed point set by virtue of the generalized Gronwall's inequality, we can use Leray-Schauder fixed point theorem to establish the existence of -periodic -mild solutions for semilinear impulsive periodic system with time-varying generating operators. At last, an example is given to demonstrate the applicability of our result.

2. Linear impulsive periodic system with time-varying generating operators

In order to study the semilinear impulse periodic system with time-varying generating operators, we first recall some results about linear impulse periodic system with time-varying generating operators here. Let be a Banach space. denotes the space of linear operators in ; 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 . We introduce is continuous at , is continuous from left and has right-hand limits at and SetIt can be seen that endowed with the norm , is a Banach space.

Consider the following homogeneous linear impulsive periodic system with time-varying generating operators:on Banach space , where , is a family of closed densely defined linear unbounded operators on satisfying the following assumption.

Assumption (see [28, page 158]). For one has
(P1)The domain is independent of and is dense in .(P2)For , the resolvent exists for all with , and there is a constant independent of and such that(P3)There exist constants and such that

Lemma 2.1 (see [28, page 159]). Under the Assumption A1, the Cauchy problemhas a unique evolution system in satisfying the following properties:
(1) for ;(2) for ;(3) for , ;(4)For , : and is strongly differentiable in . The derivative and it is strongly continuous on . Moreover,(5)For every and is differentiable with respect to on and, for each , the Cauchy problem (Eq.1) has a unique classical solution given by

In addition to Assumption A1, we introduce the following assumptions.

Assumption 2. There exits such that for .

Assumption 3. For , the resolvent is compact.

Then, we have the following lemma.

Lemma 2.2. Assumptions A1, A2, and A3 hold. Then evolution system in also satisfying the following two properties:
(6) for ;(7) is compact operator for .

In order to introduce an impulsive evolution operator and give it's properties, we need the following assumption.

Assumption 4. For each , , there exists such that

Consider the following Cauchy's problem

For every , is an invariant subspace of , using Lemma 2.1, step by step, one can verify that the Cauchy problem (2.8) has a unique classical solution represented by where given by The operator is called impulsive evolution operator associated with .

The following lemma on the properties of the impulsive evolution operator associated with are widely used in this paper.

Lemma 2.3 (see [24, Lemma 1]). Assumptions A1, A2, A3, and B hold. Impulsive evolution operator has the following properties.
(1)For , , that is, , where .(2)For , , .(3)For and , (4)For and , .(5) is 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 with time-varying generating operators 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 the semilinear impulsive periodic system with time-varying generating operators on Banach space.

Now we introduce the -mild solution of Cauchy's problem (2.8) and -periodic -mild solution of the system (2.2).

Definition 2.4. For every , the function given by is said to be the -mild solution of the Cauchy problem (2.8).

Deffinition 2.2. A function is said to be a -periodic -mild solution of system (2.2) if it is a -mild solution of Cauchy's problem (2.8) corresponding to some and for .

Secondly, we recall the following nonhomogeneous linear impulsive periodic system with time-varying generating operatorswhere , for and satisfies the following assumption.

Assumption 5. For each and , there exists such that .

In order to study system (2.10), we need to consider the following Cauchy problemand introduce the -mild solution of Cauchy's problem (2.11) and -periodic -mild solution of system (2.10).

Definition 2.6. A function , for finite interval , is said to be a -mild solution of the Cauchy problem (2.10) corresponding to the initial value and input if is given by

Definition 2.7. A function is said to be a -periodic -mild solution of system (2.10) if it satisfies the expression (Eq.2) and for .

3. Periodic solutions of semilinear impulsive periodic system with time-varying generating operators

In order to use Leray-Schauder theorem to show the existence of periodic solutions, we need the following generalized Gronwall's inequality with mixed-type integral operator which is much different from the classical Gronwall's inequality and can be used in other problems (such as discussion on integral-differential equation of mixed type, see [15]). It will play an essential role in the study of nonlinear problems on infinite dimensional spaces.

Lemma 3.1. Let , , , , . If satisfiesthen there exists a constant such that

Proof. LetThen,By Gronwall's inequality, we obtainThus,Therefore,This completes the proof.

Now, we consider the following semilinear impulsive periodic system with time-varying generating operatorsand introduce Poincaré operator and study the -periodic -mild solution of system (3.8).

In order to study the system (3.8), we first consider Cauchy's problemBy virtue of the expression of the -mild solution of the Cauchy problem (2.11), we can introduce the -mild solution of the Cauchy problem (3.9).

Definition 3.2. A function is said to be a -mild solution of the Cauchy problem (3.9) corresponding to the initial value if satisfies the following integral equation:

Now, we introduce the -periodic -mild solution of system (3.8).

Definition 3.3. A function is said to be a -periodic -mild solution of system (3.8) if it is a -mild solution of Cauchy's problem (3.9) corresponding to some and for .

In order to prove the existence of the -mild solution of Cauchy's problem (3.9), we need the following assumption.

Assumption 6. (F1): is measurable for and for any satisfying there exists a positive constant such that
(F2): There exists a positive constant such that
(F3): is -periodic in . That is, , .

Then, we have the following theorem.

Theorem 3.4. Assumptions A1, F(F1), and F(F2) hold. Cauchy's problem (3.9) has a unique -mild solution given by

Proof. Under the Assumptions A1, F(F1), and F(F2), using the similar method of Theorem 5.3.3 (see [28, page 169]), Cauchy's problem has a unique mild solution
In general, for , Cauchy's problemhas a unique -mild solution
Combining all of solutions on (), one can obtain the -mild solution of the Cauchy problem (3.9) given by

In order to study the periodic solutions of the system (3.8), we define Poincaré operator from to as following:where denote the -mild solution of the Cauchy problem (3.9) corresponding to the initial value . We note that a fixed point of gives rise to a periodic solution.

Lemma 3.5. System (3.8) 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 (3.9) corresponding to the initial value , we can define , then . Now, for , we can use the (2), (3), and (4) of Lemma 2.3 and Assumptions A2, B, C, F(F3) to obtainThis implies that is a -mild solution of Cauchy's problem (3.9) with initial value . Thus, the uniqueness implies that , so that is a -periodic.

Next, we show that defined by (3.19) is a continuous and compact operator.

Lemma 3.6. Assumptions A1, A3, F(F1), and F(F2) hold. Then, is a continuous and compact operator.

Proof. (1) Show that is a continuous operator on .
Let , , where is a bounded subset of . Suppose and are the -mild solutions of Cauchy's problem (3.9) corresponding to the initial value and , respectively, given byThus, we obtainBy Gronwall's inequality with impulse [5, Lemma 1.7.1], one can verify that there exist constants and such thatLet , then , which imply that they are locally bounded. By Assumption F(F1), we obtainBy Gronwall's inequality with impulse [5, Lemma 1.7.1] again, one can verify that there exists constant such thatwhich implies thatHence, is a continuous operator on .
(2) Verifies 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 bounded.
By (5) of Lemma 2.3, is a compact operator. Thus, is precompact in .
On the other hand, for arbitrary ,Thus, combined with (3.19), we haveIt is showing that the set can be approximated to an arbitrary degree of accuracy by a precompact set . Hence, itself is precompact set in . That is, takes a bounded set into a precompact set in . As a result, is a compact operator.

In order to use Leary-Schauder fixed pointed theorem to examine that the operator has a fixed point, we have to make the Assumption F(F2) a little strong as following.

(F2′): there exist constant and such that

Now, we can give the main results in this paper.

Theorem 3.7. Assumptions A1, A2, A3, B, C, F(F1), F(F2′), and F(F3) hold. Then system (3.8) has a -periodic -mild solution on .

Proof. By (5) of Lemma 2.3, is a compact operator on infinite dimensional space . Thus, , . Then, there exists such thatIn fact, define , , andIt is obvious that . Thus, there exist and such thatIf not, there exits such that . We can assert that unless . Thus, for which is a contradiction with , .
By Theorem 3.4, for fixed , the Cauchy problem (3.9) corresponding to the initial value has the -mild solution . By Lemma 3.6, the operator defined by (3.19), is compact.
According to Leray-Schauder fixed point theory, it suffices to show that the set is a bounded subset of . In fact, let , we have
By Assumption F(F2′),
For , we obtainBy Lemma 3.1, there exists such thatThis implies that for all .
Thus, by Leray-Schauder fixed point theory, there exits such that . By Lemma 3.5, we know that the -mild solution of Cauchy's problem (3.9) corresponding to the initial value , is just -periodic. Therefore is a -periodic -mild solution of system (3.8).

4. Application

In this section, an example is given to illustrate our theory. Consider the following problem:where is bounded domain and .

Define , , and for , which satisfies Assumptions A1, A2, and A3. Define , , and

It is obvious that , which satisfy the Assumption B.

For any satisfying ,Meanwhile,These imply that Assumptions F(F1), F(F2′), and F(F3) hold. It comes fromthat , . In fact, if , then cannot be compact in which is a contradiction with which is compact operator for . Thus problem (4.1) can be rewritten asIt satisfies all the assumptions given in Theorem 3.7, our results can be used to problem (4.1). That is, problem (4.1) has a -periodic -mild solution , where

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 10661044) and Natural Science Foundation of Guizhou Province Education Department (no. 2007008). This work is also supported by undergraduate carve out project of Department of Guiyang City Science and Technology ([2008] no.152).

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