JinRong Wang,¹ X. Xiang,^1,2 W. Wei,² and Qian Chen²

Abstract

This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, the T0-periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of Poincaré operator and using a new generalized Gronwall's inequality with impulse, mixed type integral operators and B-norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of T0-periodic PC-mild solutions. Our method 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. 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, and robustness by perturbation) are obtained (see [1–4]).

Since the end of last century, many researchers pay great attention to impulsive systems on infinite-dimensional spaces. Particulary, Ahmed et al. investigated optimal control problems of system governed by impulsive system (see [5–8]). Many authors including us also gave a series of results for semilinear (integrodifferential, strongly nonlinear) impulsive systems and optimal control problems (see [9–20]).

Although, there are some papers on periodic solution for periodic system on infinite-dimensional spaces (see [12, 21–23]) and some results discussing integrodifferential system on finite Banach space and infinite Banach space (see [11, 13]). 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. Linear impulsive evolution operator is constructed and -periodic -mild solution is introduced. The existence of periodic solutions, alternative theorem (criteria of Massera type), asymptotical stability, and robustness by perturbation is established (see [24–26]). For semilinear impulsive periodic system, a suitable Poincaré operator is constructed which verifies its compactness and continuity. By virtue of a generalized Gronwall inequality with mixed integral operator and impulse given by us, the estimate of the -mild solutions is derived. Some fixed point theorems such as Banach fixed point theorem and Horn fixed point theorem are applied to obtain the existence of periodic -mild solutions, respectively (see [27, 28]). For integrodifferential impulsive system, the existence of -mild solutions and optimal controls is presented (see [15]).

Herein, we go on studying the following integrodifferential impulsive periodic system(1.1)on infinite-dimensional Banach space , where ; , ; , ; is a fixed positive number; and denoted the number of impulsive points between and . The operator is the infinitesimal generator of a -semigroup on ; is a -periodic, with respect to , Carathédory function;   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 Leray-Schauder 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 we overcome some difficulties to show the compactness of Poincaré operator which is very important. By a new generalized Gronwall inequality with impulse, mixed-type integral operators, and -norm given by us, the estimate of fixed point set is established. Therefore, the existence of -periodic -mild solutions for impulsive integrodifferential periodic system is shown.

In order to obtain the existence of periodic solutions, many authors use Horn fixed point theorem or Banach fixed point theorem. However, the conditions for Horn fixed point theorem are not easy to be verified sometimes and the conditions for Banach fixed point theorem are too strong. Our method is much different from others', and we give a new way to show the existence of periodic solutions. In addition, the new generalized Gronwall inequality with impulse, mixed-type integral operator, and -norm 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 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, mixed-type integral operator, and -norm are established. In Section 4, the -periodic -mild solution for integrodifferential impulsive periodic system (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 inequality, we can use Leray-Schauder fixed point theorem to establish the existence of -periodic -mild solutions for integrodifferential impulsive periodic system. At last, an example is given to demonstrate the applicability of our result.

2. Linear Impulsive Periodic System

In order to study the integrodifferential impulse periodic system, we first recall some results about linear impulse periodic system 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 to be continuous at ; is continuous from left and has right-hand limits at and Set(2.1)It can be seen that endowed with the norm , is a Banach space.

Firstly, we consider homogeneous linear impulsive periodic system(2.2)

We introduce the following 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(2.3)

If and is an invariant subspace of , using Theorem 5.2.2, (see [29, page 144]), step by step, one can verify that the Cauchy problem (2.3) has a unique classical solution represented by where(2.4)given by(2.5)The 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 sequel.

Lemma 2.3 (see [24, 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 nonhomogeneous linear impulsive periodic system(2.6)and the associated Cauchy problem(2.7)where , 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(2.8)

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

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

We first introduce the following generalized Gronwall inequality with impulse and -norm.

Lemma 3.1. Let and satisfy the following inequality: (3.1) where , are constants, and . Then (3.2)

Proof. (i) For , , let and(3.3)Then(3.4)

Using (3.4), we obtain(3.5)Define(3.6)we get(3.7)Since , we then have(3.8)

For , by (3.8), we obtain(3.9)further,(3.10)thus,(3.11)

(ii) For , we only need to define(3.12)Similar to the proof in (i), one can obtain(3.13)Combining (i) and (ii), one can complete the proof.

Using Gronwall's inequality with impulse and -norm, we can obtain the following new generalized Gronwall Lemma.

Lemma 3.2. Let satisfy the following inequality: (3.14) where , , are constants. Then there exists a constant such that (3.15)

Proof. By Lemma 3.1, we obtain that(3.16)where (3.17)

Define(3.18)then is a monotone increasing function and(3.19)

Consider(3.20)Integrating from to , we obtain(3.21)that is,(3.22)

On the other hand,(3.23)Now, we observe that(3.24)As a result, we get(3.25)Letting(3.26)we have and . Moreover,(3.27)Hence, there exists enough large such that for arbitrary . Meanwhile, . Thus, .

As a result, we obtain(3.28)

4. Periodic Solutions of Integrodifferential Impulsive Periodic System

In this section, we consider the following integrodifferential impulsive periodic system:(4.1)and the associated Cauchy problem(4.2)

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 (4.2).

Definition 4.1. A function is said to be a -mild solution of the Cauchy problem (4.2) corresponding to the initial value if satisfies the following integral equation:(4.3)

Now, we introduce the -periodic -mild solution of system (4.1).

Definition 4.2. A function is said to be a -periodic -mild solution of system (4.1) if it is a -mild solution of Cauchy problem (4.2) corresponding to some and for .

Assumption [H2] includes the following.

[H2.1] satisfies the following. (i)For each , is measurable.(ii)For each there exists such that, for almost all and all , , , , we have(4.4)

[H2.2] There exists a positive constant such that(4.5)

[H2.3] is -periodic in , that is,

[H2.4] Let . The function is continuous for each , there exists such that, for each and each with , we have(4.6)

[H2.5] There exists a positive constant such that(4.7)

[H2.6] are  -periodic in and , that is, and(4.8)

[H2.7] For each and , there exists such that .

Lemma 4.3. Under assumptions [H2.4] and [H2.5], one has the following properties: (1) . (2) For all and , (4.9) (3) For (4.10)

Proof. (1) Since is continuous in its variables and satisfies linear growth conditions, one can verify that maps to .

(2) Let , , we have(4.11)

(3) For ,(4.12)

Now we present the existence of -mild solution for system (4.2).

Theorem 4.4. Assumptions [H1.1], [H2.1], [H2.4], and [H2.5] hold. Then system (4.2) has a unique -mild solution given by (4.13)

Proof. A similar result is given by Wei et al. [15]. 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 impulse(4.14)In order to obtain the local existence of mild solution for system (4.14), we only need to set up the framework for use of the contraction mapping theorem. Consider the ball given by(4.15)where would be chosen, and , . is a closed convex set. Define a map on given by(4.16)Under 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 (4.14) has a unique mild solution given by(4.17)Again, using Lemma 3.1, we can obtain the a priori estimate of the mild solutions for system (4.14) and present the global existence of mild solutions.Step 2. For , consider the Cauchy problem(4.18)By Step 1, Cauchy problem (4.18) also has a unique -mild solution(4.19)Step 3. Combining all of the solutions on (), one can obtain the -mild solution of Cauchy problem (4.2) given by(4.20)This completes the proof.

To establish the periodic solutions for system (4.1), we define a Poincaré operator from to as follows:(4.21)where denote the -mild solution of Cauchy problem (4.2) corresponding to the initial value , then we examine whether has a fixed point.

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

Lemma 4.5. System (4.1) 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 Cauchy problem (4.2) corresponding to the initial value , we can define , then . Now, for , we can use (2), (3), and (4) of Lemma 2.3 and assumptions [H1.2], [H1.3], [H2.3], [H2.6], and [H2.7] to arrive at(4.22)This implies that is a -mild solution of Cauchy problem (4.2) with initial value . Thus the uniqueness implies that , so that is a -periodic.

Next, we show that defined by (4.21) is a continuous and compact operator.

Lemma 4.6. Suppose that is a compact semigroup in . Then the operator is a continuous and compact operator.

Proof. (1) Show that is a continuous operator on .

Let , where is a bounded subset of . Suppose that and are the -mild solutions of Cauchy problem (4.2) corresponding to the initial values   and respectively, given by(4.23)Thus, we obtain(4.24)where and .

By Lemma 3.1, one can verify that there exist constants and such that(4.25)Let , then which imply that they are locally bounded.

By assumptions [H2.1], [H2.2], [H2.4], [H2.5], and (2) of Lemma 4.3, we obtain(4.26)By Lemma 3.1 again, one can verify that there exists constant such that(4.27)which implies that(4.28)Hence, is a continuous operator on .

(2) Verify that takes a bounded set into a precompact set in .

Let is a bounded subset of . Define .

For , define(4.29)

Next, we show that is precompact in . In fact, for fixed, we have(4.30)This implies that the set is totally bounded.

By virtue of which is a compact semigroup and (5) of Lemma 2.3, is a compact operator. Thus, is precompact in .

On the other hand, for arbitrary ,(4.31)Thus, having this combined with (4.21), we have(4.32)It is shown 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.

In order to use Leray-Schauder fixed pointed theorem to examine whether the operator has a fixed point, we have to make assumptions [H2.2] and [H2.5] a little stronger as follows.

[H2.] There exists constant and such that(4.33)

[H2.] There exists a positive constant and such that(4.34)

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

Theorem 4.7. Assumptions [H1], [H2.1], [H2. ], [H2.3], [H2.4], [H2. ], [H2.6], and [H2.7] hold. Suppose that is a compact semigroup in . Then system (4.1) has a -periodic -mild solution on .

Proof. By virtue of which is a compact semigroup and (5) of Lemma 2.3, is a compact operator on infinite-dimensional space . Thus, , . Then, there exists such that for . In fact, define , , and and . It is obvious that . Thus, there exist and such that(4.35)If not, there exists   such that . We can assert that unless . Thus, for ,(4.36)which is a contradiction with , .

By Theorem 4.4, for fixed , Cauchy problem (4.2) corresponding to the initial value has the -mild solution . By Lemma 4.6, the operator defined by (4.21), 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(4.37)

By assumptions [H2.] and [H2.],(4.38)

For , we obtain(4.39)By Lemma 3.2, there exists such that(4.40)This implies that for all .

Thus, by Leray-Schauder fixed pointed theory, there exists such that . By Lemma 4.5, we know that the -mild solution of Cauchy problem (4.2) corresponding to the initial value is just -periodic. Therefore is a -periodic -mild solution of system (4.1).

5. Application

In this section, an example is given to illustrate our theory. Consider the following problem:(5.1)where is bounded domain and .

Define , , and for . Then, generates a compact semigroup . Define , , where , , and(5.2)

Thus problem (5.1) can be rewritten as(5.3)

It satisfies all the assumptions given in Theorem 4.7, our results can be used to problem (5.1). That is, problem (5.1) has a -periodic -mild solution , where(5.4)

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 partially supported by undergraduate carve-out project of Department of Guiyang City Science and Technology.

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