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

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. By virtue of the generalized new Gronwall lemma with impulse and 𝐵-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 𝑇₀-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 [1–4]).

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 [5–8]). We also gave a series of results for semilinear (integrodifferential, strongly nonlinear) impulsive systems and optimal control problems (see [9–12]). 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 [14–24]) 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 𝑇₀-periodic 𝑃𝐶-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 [26–28]). For integrodifferential impulsive system, existence of 𝑃𝐶-mild solutions and optimal controls are presented (see [9]).

Herein, we go on studying the following integrodifferential impulsive periodic system:𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥,∫^𝑡₀𝑔(𝑡,𝑠,𝑥)𝑑𝑠),𝑡≠𝜏_𝑘,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡)+𝑐_𝑘,𝑡=𝜏_𝑘, (1.1)on infinite dimensional Banach space 𝑋, where 0=𝜏₀<𝜏₁<𝜏₂<⋯<𝜏_𝑘⋯, lim_𝑘→∞𝜏_𝑘=∞, 𝜏_𝑘+𝛿=𝜏_𝑘+𝑇₀, Δ𝑥(𝜏_𝑘)=𝑥(𝜏⁺_𝑘)−𝑥(𝜏⁻_𝑘), 𝑘∈ℤ⁺₀, 𝑇₀ is a fixed positive number, and 𝛿∈ℕ denotes the number of impulsive points between 0 and 𝑇₀. The operator 𝐴 is the infinitesimal generator of a 𝐶₀-semigroup {𝑇(𝑡),𝑡≥0} on 𝑋, 𝑓 is a measurable function from [0,∞)×𝑋×𝑋 to 𝑋 and is 𝑇₀-periodic in 𝑡, 𝑔 is a continuous function from [0,∞)×[0,∞)×𝑋 to 𝑋 and is 𝑇₀-periodic in 𝑡 and 𝑠 and 𝐵_𝑘+𝛿=𝐵_𝑘, 𝑐_𝑘+𝛿=𝑐_𝑘. This paper is mainly concerned with the existence of periodic solutions for integrodifferential impulsive periodic system on infinite dimensional Banach space 𝑋.

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 𝑃 for integrodifferential impulsive periodic system (1.1), then overcome some difficulties to show the continuity and compactness of Poincaré operator 𝑃 which are very important. By a new generalized Gronwall inequality with impulse and 𝐵-norm given by us, the estimate of 𝑃𝐶-mild solutions is established. Therefore, the existence of 𝑇₀-periodic 𝑃𝐶-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 𝐵-norm is established and the existence of 𝑃𝐶-mild solutions for integrodifferential impulsive system is presented. In Section 4, the 𝑇₀-periodic 𝑃𝐶-mild solutions for integrodifferential impulsive periodic system (1.1) are introduced. We construct the suitable Poincaré operator 𝑃 and give the relation between 𝑇₀-periodic 𝑃𝐶-mild solution and the fixed point of the Poincaré operator 𝑃. After showing the continuity and compactness of the Poincaré operator 𝑃, we can use Horn's fixed point theorem to establish the existence of 𝑇₀-periodic 𝑃𝐶-mild solutions for integrodifferential impulsive periodic system when the 𝑃𝐶-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 𝑋 be a Banach space. £(𝑋) denotes the space of linear operators in 𝑋, and £_𝑏(𝑋) denotes the space of bounded linear operators in 𝑋. £_𝑏(𝑋) is the Banach space with the usual supremum norm. Define ^0𝑥0𝐷^02𝑑𝑐={𝜏₁,…,𝜏_𝛿}⊂[0,𝑇₀], where 𝛿∈ℕ denotes the number of impulsive points between 0 and 𝑇₀. We introduce 𝑃𝐶([0,𝑇₀];𝑋)≡{𝑥∶[0,𝑇₀]→𝑋∣𝑥 is continuous at 𝑡∈[0,𝑇₀]⧵𝐷^{0𝑥002𝑑𝑐}, 𝑥 is continuous from left and has right-hand limits at 𝑡∈𝐷^{0𝑥002𝑑𝑐}} and 𝑃𝐶¹([0,𝑇₀];𝑋)≡{𝑥∈𝑃𝐶([0,𝑇₀];𝑋)∣𝑥̇∈𝑃𝐶([0,𝑇₀];𝑋)}. Set‖𝑥‖_𝑃𝐶=max{sup_{𝑡∈[0,𝑇0]}‖𝑥(𝑡+0)‖,sup_{𝑡∈[0,𝑇0]}‖𝑥(𝑡−0)‖},‖𝑥‖_𝑃𝐶¹=‖𝑥‖_𝑃𝐶+‖𝑥̇‖_𝑃𝐶. (2.1)It can be seen that a Banach space is endowed with the norm ‖⋅‖_𝑃𝐶(‖⋅‖_𝑃𝐶¹), 𝑃𝐶([0,𝑇₀];𝑋)(𝑃𝐶¹([0,𝑇₀];𝑋)).

Firstly, we consider the homogeneous linear impulsive periodic system𝑥̇(𝑡)=𝐴𝑥(𝑡),𝑡≠𝜏_𝑘,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡),𝑡=𝜏_𝑘. (2.2)

We introduce the following assumption.

Assumption [H1] [H1.1]𝐴 is the infinitesimal generator of a 𝐶₀-semigroup {𝑇(𝑡),𝑡≥0} on 𝑋 with domain 𝐷(𝐴).[H1.2]There exists 𝛿∈𝑁 such that 𝜏_𝑘+𝛿=𝜏_𝑘+𝑇₀.[H1.3]For each 𝑘∈ℤ⁺₀, 𝛿∈𝑁, 𝐵_𝑘∈£_𝑏(𝑋) and 𝐵_𝑘+𝛿=𝐵_𝑘.

In order to study system (2.2), we need to consider the associated Cauchy problem𝑥̇(𝑡)=𝐴𝑥(𝑡),𝑡∈[0,𝑇₀]⧵𝐷^{0𝑥002𝑑𝑐},Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘),𝑘=1,2,…,𝛿,𝑥(0)=𝑥. (2.3)

If 𝑥∈𝐷(𝐴) and 𝐷(𝐴) is an invariant subspace of 𝐵_𝑘, 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 𝑥∈𝑃𝐶¹([0,𝑇₀];𝑋) represented by 𝑥(𝑡)=𝑆(𝑡,0)𝑥 where 𝑆(⋅,⋅):Δ={(𝑡,𝜃)∈[0,𝑇₀]×[0,𝑇₀]∣0≤𝜃≤𝑡≤𝑇₀}→£_𝑏(𝑋) (2.4)given by𝑆(𝑡,𝜃)=𝑇(𝑡−𝜃),𝜏_𝑘−1≤𝜃≤𝑡≤𝜏_𝑘,𝑇(𝑡−𝜏⁺_𝑘)(𝐼+𝐵_𝑘)𝑇(𝜏_𝑘−𝜃),𝜏_𝑘−1≤𝜃<𝜏_𝑘<𝑡≤𝜏_𝑘+1,𝑇(𝑡−𝜏⁺_𝑘)[_{𝜃<𝜏𝑗<𝑡}(𝐼+𝐵_𝑗)𝑇(𝜏_𝑗−𝜏⁺_𝑗−1)](𝐼+𝐵_𝑖)𝑇(𝜏_𝑖𝜏−𝜃),_𝑖−1≤𝜃<𝜏_𝑖≤⋯<𝜏_𝑘<𝑡≤𝜏_𝑘+1. (2.5)The operator {𝑆(𝑡,𝜃),(𝑡,𝜃)∈Δ} is called impulsive evolution operator associated with {𝐵_𝑘;𝜏_𝑘}^∞_𝑘=1.

Now we introduce the 𝑃𝐶-mild solution of Cauchy problem (2.3) and 𝑇₀-periodic 𝑃𝐶-mild solution of the system (2.2).

Definition 2.1. For every 𝑥∈𝑋, the function 𝑥∈𝑃𝐶([0,𝑇₀];𝑋) given by 𝑥(𝑡)=𝑆(𝑡,0)𝑥 is said to be the 𝑃𝐶-mild solution of the Cauchy problem (2.3).

Definition 2.2. A function 𝑥∈𝑃𝐶([0,+∞);𝑋) is said to be a 𝑇₀-periodic 𝑃𝐶-mild solution of system (2.2) if it is a 𝑃𝐶-mild solution of Cauchy problem (2.3) corresponding to some 𝑥 and 𝑥(𝑡+𝑇₀)=𝑥(𝑡) for 𝑡≥0.

The following lemma gives the properties of the impulsive evolution operator {𝑆(𝑡,𝜃),(𝑡,𝜃)∈Δ} associated with {𝐵_𝑘;𝜏_𝑘}^∞_𝑘=1 which are widely used in the sequel.

Lemma 2.3 (see [26, Lemma 1]). Impulsive evolution operator {𝑆(𝑡,𝜃),(𝑡,𝜃)∈Δ} has the following properties.
(1)For 0≤𝜃≤𝑡≤𝑇₀, 𝑆(𝑡,𝜃)∈£_𝑏(𝑋), that is, sup_{0≤𝜃≤𝑡≤𝑇0}‖𝑆(𝑡,𝜃)‖≤𝑀_𝑇0, where 𝑀_𝑇0>0.(2)For 0≤𝜃<𝑟<𝑡≤𝑇₀, 𝑟≠𝜏_𝑘, 𝑆(𝑡,𝜃)=𝑆(𝑡,𝑟)𝑆(𝑟,𝜃).(3)For 0≤𝜃≤𝑡≤𝑇₀ and 𝑁∈𝑍⁺₀, 𝑆(𝑡+𝑁𝑇₀,𝜃+𝑁𝑇₀)=𝑆(𝑡,𝜃).(4)For 0≤𝑡≤𝑇₀ and 𝑁∈𝑍⁺₀, 𝑆(𝑁𝑇₀+𝑡,0)=𝑆(𝑡,0)[𝑆(𝑇₀,0)]^𝑁.(5)If {𝑇(𝑡),𝑡>0} is a compact semigroup in 𝑋, then 𝑆(𝑡,𝜃) is a compact operator for 0≤𝜃<𝑡≤𝑇₀.

Here, we note that system (2.2) has a 𝑇₀-periodic 𝑃𝐶-mild solution 𝑥 if and only if 𝑆(𝑇₀,0) has a fixed point. The impulsive evolution operator {𝑆(𝑡,𝜃),(𝑡,𝜃)∈Δ} can be used to reduce the existence of 𝑇₀-periodic 𝑃𝐶-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 𝑃𝐶-mild solutions for integrodifferential impulsive periodic system on Banach space.

Consider the nonhomogeneous linear impulsive periodic system𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡),𝑡≠𝜏_𝑘,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡)+𝑐_𝑘,𝑡=𝜏_𝑘, (2.6)and the associated Cauchy problem𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡),𝑡∈[0,𝑇₀]⧵𝐷^{0𝑥002𝑑𝑐},Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘)+𝑐_𝑘,𝑘=1,2,…,𝛿,𝑥(0)=𝑥, (2.7)where 𝑓∈𝐿¹([0,𝑇₀];𝑋), 𝑓(𝑡+𝑇₀)=𝑓(𝑡), and 𝑐_𝑘+𝛿=𝑐_𝑘.

Now we introduce the 𝑃𝐶-mild solution of Cauchy problem (2.7) and 𝑇₀-periodic 𝑃𝐶-mild solution of system (2.6).

Definition 2.4. A function 𝑥∈𝑃𝐶([0,𝑇₀];𝑋), for finite interval [0,𝑇₀], is said to be a 𝑃𝐶-mild solution of the Cauchy problem (2.6) corresponding to the initial value 𝑥∈𝑋 and input 𝑓∈𝐿¹([0,𝑇₀];𝑋) if 𝑥 is given by𝑥(𝑡)=𝑆(𝑡,0)𝑥+^𝑡₀𝑆(𝑡,𝜃)𝑓(𝜃)𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝑆(𝑡,𝜏⁺_𝑘)𝑐_𝑘. (2.8)

Definition 2.5. A function 𝑥∈𝑃𝐶([0,+∞);𝑋) is said to be a 𝑇₀-periodic 𝑃𝐶-mild solution of system (2.6) if it is a 𝑃𝐶-mild solution of Cauchy problem (2.7) corresponding to some 𝑥 and 𝑥(𝑡+𝑇₀)=𝑥(𝑡) for 𝑡≥0.

3. The Generalized Gronwall Inequality and Existence of Solutions

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

Lemma 3.1. Let 𝑥∈𝑃𝐶([0,∞),𝑋) and let it satisfy the following inequality:‖𝑥(𝑡)‖≤𝑎+𝑏^𝑡₀‖𝑥(𝜃)‖^𝜆1𝑑𝜃+𝑐^𝑡₀‖𝑥_𝜃‖^𝜆2_𝐵𝑑𝜃, (3.1)where 𝑎, 𝑏, 𝑐≥0,0≤𝜆₁,𝜆₂≤1 are constants, and‖𝑥_𝜃‖_𝐵=max{sup_{0≤𝜉≤𝜃}‖𝑥(𝜉+0)‖,sup_{0≤𝜉≤𝜃}‖𝑥(𝜉−0)‖}. (3.2)
Then one has‖𝑥(𝑡)‖≤(𝑎+1)𝑒^{(𝑏+𝑐)𝑡}. (3.3)

Proof. (i) For 0≤𝜆₁, 𝜆₂<1, let 𝜆=max{𝜆₁,𝜆₂}∈[0,1) and𝑦(𝑡)=1,‖𝑥(𝑡)‖≤1,𝑥(𝑡),‖𝑥(𝑡)‖>1. (3.4)Then‖𝑥(𝑡)‖≤‖𝑦(𝑡)‖≤(𝑎+1)+𝑏^𝑡₀‖𝑦(𝜃)‖^𝜆𝑑𝜃+𝑐^𝑡₀‖𝑦_𝜃‖^𝜆_𝐵𝑑𝜃,∀𝑡∈[0,𝑇₀], (3.5)
Using (3.5), we obtain‖𝑦_𝑡‖^𝜆_𝐵≤(𝑎+1)+(𝑏+𝑐)^𝑡₀‖𝑦_𝜃‖^𝜆_𝐵𝑑𝜃. (3.6)Define𝑢(𝑡)=(𝑎+1)+(𝑏+𝑐)^𝑡₀‖𝑦_𝜃‖^𝜆_𝐵𝑑𝜃, (3.7)then we get𝑢̇(𝑡)=(𝑏+𝑐)‖𝑦_𝑡‖^𝜆_𝐵,𝑡≠𝜏_𝑘,𝑢(0)=𝑎+1,𝑢(𝜏_𝑘+0)=𝑢(𝜏_𝑘). (3.8)Since ‖𝑦_𝑡‖^𝜆_𝐵≤𝑢(𝑡), we then have𝑢̇(𝑡)≤(𝑏+𝑐)𝑢(𝑡),𝑡≠𝜏_𝑘,𝑢(0)=𝑎+1,𝑢(𝜏_𝑘+0)=𝑢(𝜏_𝑘). (3.9)
For 𝑡∈(𝜏_𝑘,𝜏_𝑘+1], by (3.9), we obtain𝑢(𝑡)≤𝑢(𝜏_𝑘+0)𝑒^{(𝑏+𝑐)(𝑡−𝜏𝑘)}=𝑢(𝜏_𝑘)𝑒^{(𝑏+𝑐)(𝑡−𝜏𝑘)}, (3.10)further,𝑢(𝑡)≤(𝑎+1)𝑒^{(𝑏+𝑐)𝑡}, (3.11)thus,‖𝑥(𝑡)‖≤‖𝑦_𝑡‖_𝐵≤(𝑎+1)𝑒^{(𝑏+𝑐)𝑡}. (3.12)
(ii) For 𝜆₁=𝜆₂=1, we only need to define𝑢₁(𝑡)=𝑎+(𝑏+𝑐)^𝑡₀‖𝑥_𝜃‖_𝐵𝑑𝜃. (3.13)Similar to the proof in (i), one can obtain‖𝑥(𝑡)‖≤‖𝑥_𝑡‖_𝐵≤𝑎𝑒^{(𝑏+𝑐)𝑡}. (3.14)Combining (i) and (ii), one can complete the proof.

Now, we consider the following integrodifferential impulsive periodic system:𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥,∫^𝑡₀𝑔(𝑡,𝑠,𝑥)𝑑𝑠),𝑡≠𝜏_𝑘,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡)+𝑐_𝑘,𝑡=𝜏_𝑘, (3.15)and the associated Cauchy problem𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥,∫^𝑡₀𝑔(𝑡,𝑠,𝑥)𝑑𝑠),𝑡∈[0,𝑇₀]⧵𝐷^{0𝑥002𝑑𝑐},Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘)+𝑐_𝑘,𝑘=1,2,…,𝛿,𝑥(0)=𝑥. (3.16)

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

Definition 3.2. A function 𝑥∈𝑃𝐶([0,𝑇₀];𝑋) is said to be a 𝑃𝐶-mild solution of the Cauchy problem (3.16) corresponding to the initial value 𝑥∈𝑋 if 𝑥 satisfies the following integral equation:𝑥(𝑡)=𝑆(𝑡,0)𝑥+^𝑡₀𝑆(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠))𝑑𝑠)𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝑆(𝑡,𝜏⁺_𝑘)𝑐_𝑘,for𝑡∈[0,𝑇₀]. (3.17)
Now, we introduce the 𝑇₀-periodic 𝑃𝐶-mild solution of system (3.15).

Definition 3.3. A function 𝑥∈𝑃𝐶([0,+∞);𝑋) is said to be a 𝑇₀-periodic 𝑃𝐶-mild solution of system (3.15) if it is a 𝑃𝐶-mild solution of Cauchy problem (3.16) corresponding to some 𝑥 and 𝑥(𝑡+𝑇₀)=𝑥(𝑡) for 𝑡≥0.

Assumption [H2] [H2.1]𝑓∶[0,∞)×𝑋×𝑋→𝑋 is measurable for 𝑡≥0, and for any 𝑥₁, 𝑥₂, 𝑦₁,𝑦₂∈𝑋 satisfying ‖𝑥₁‖,‖𝑥₂‖,‖𝑦₁‖,‖𝑦₂‖≤𝜌 there exists a positive constant 𝐿_𝑓(𝜌)>0 such that‖𝑓(𝑡,𝑥₁,𝑦₁)−𝑓(𝑡,𝑥₂,𝑦₂)‖≤𝐿_𝑓(𝜌)(‖𝑥₁−𝑥₂‖+‖𝑦₁−𝑦₂‖). (3.18)[H2.2]There exists a positive constant 𝑀_𝑓>0 such that ‖𝑓(𝑡,𝑥,𝑦)‖≤𝑀_𝑓(1+‖𝑥‖+‖𝑦‖),∀𝑥,𝑦∈𝑋. (3.19)[H2.3]𝑓(𝑡,𝑥,𝑦) is 𝑇₀-periodic in t, that is,𝑓(𝑡+𝑇₀,𝑥,𝑦)=𝑓(𝑡,𝑥,𝑦),𝑡≥0. (3.20)[H2.4]𝑔∶[0,∞)×[0,∞)×𝑋→𝑋 is continuous for 𝑡≥𝑠≥0 and for any 𝑥, 𝑦∈𝑋 satisfying ‖𝑥‖,‖𝑦‖≤𝜌 there exists a positive constant 𝐿_𝑔(𝜌)>0 such that‖𝑔(𝑡,𝑠,𝑥)−𝑔(𝑡,𝑠,𝑦)‖≤𝐿_𝑔(𝜌)‖𝑥−𝑦‖. (3.21)[H2.5]There exists a positive constant 𝑀_𝑔>0 such that‖𝑔(𝑡,𝑠,𝑥)‖≤𝑀_𝑔(1+‖𝑥‖),∀𝑥∈𝑋. (3.22)[H2.6]𝑔(𝑡,𝑠,𝑥) are 𝑇₀-periodic in 𝑡 and 𝑠, that is,𝑔(𝑡+𝑇₀,𝑠+𝑇₀∫,𝑥)=𝑔(𝑡,𝑠,𝑥),𝑡≥𝑠≥0,_𝑇00𝑔(𝑡,𝑠,𝑥)𝑑𝑠=0,𝑡≥𝑠≥0. (3.23)[H2.7]For each 𝑘∈ℤ⁺₀ and 𝑐_𝑘∈𝑋 there exists 𝛿∈ℕ such that 𝑐_𝑘+𝛿=𝑐_𝑘.

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)∫^⋅₀𝑔(⋅,𝑠,𝑥(𝑠))𝑑𝑠∶𝑃𝐶([0,𝑇₀];𝑋)→𝑃𝐶([0,𝑇₀];𝑋);(2)for all 𝑥₁,𝑥₂∈𝑃𝐶([0,𝑇₀];𝑋) and ‖𝑥₁‖_{𝑃𝐶([0,𝑇0];𝑋)},‖𝑥₂‖_{𝑃𝐶([0,𝑇0];𝑋)}≤𝜌,‖^𝑡₀𝑔(𝑡,𝑠,𝑥₁(𝑠))𝑑𝑠−^𝑡₀𝑔(𝑡,𝑠,𝑥₂(𝑠))𝑑𝑠‖≤𝐿_𝑔(𝜌)𝑇₀‖(𝑥₁)_𝑡−(𝑥₂)_𝑡‖_𝐵; (3.24)(3)for 𝑥∈𝑃𝐶([0,𝑇₀];𝑋),‖^𝑡₀𝑔(𝑡,𝑠,𝑥(𝑠))𝑑𝑠‖≤𝑀_𝑔𝑇₀(1+‖𝑥_𝑡‖_𝐵). (3.25)

Now we present the existence of 𝑃𝐶-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 𝑘∈ℤ⁺₀, 𝐵_𝑘∈£_𝑏(𝑋), 𝑐_𝑘∈𝑋 are fixed. Then system (3.16) has a unique 𝑃𝐶-mild solution given by𝑥(𝑡,𝑥)=𝑆(𝑡,0)𝑥+^𝑡₀𝑆(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠))𝑑𝑠)𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝑆(𝑡,𝜏⁺_𝑘)𝑐_𝑘. (3.26)

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 impulse𝑥^⋅(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥,∫^𝑡₀𝑔(𝑡,𝑠,𝑥)𝑑𝑠),𝑡∈[𝑠,𝜏],𝑥(𝑠)=𝑥∈𝑋. (3.27)In 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 by𝔅={𝑥∈𝐶([𝑠,𝑡₁];𝑋)∣‖𝑥(𝑡)−𝑥‖≤1,𝑠≤𝑡≤𝑡₁}, (3.28)where 𝑡₁ would be chosen and ‖𝑥(𝑡)‖≤1+‖𝑥‖=𝜌, 𝑠≤𝑡≤𝑡₁. 𝔅⊆𝐶([𝑠,𝑡₁],𝑋) is a closed convex set. Define a map 𝐐 on 𝔅 given by(𝐐𝑥)(𝑡)=𝑇(𝑡)𝑥+^𝑡_𝑠𝑇(𝑡−𝜃)𝑓(𝜃,𝑥(𝜃),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠))𝑑𝑠)𝑑𝜃. (3.29)Under the assumptions [H1.1], [H2.1], [H2.2], [H2.4], [H2.5], and Lemma 3.1, one can verify that map 𝐐 is a contraction map on 𝔅 with chosen 𝑡₁>0. This means that system (3.27) has a unique mild solution 𝑥∈𝐶([𝑠,𝑡₁];𝑋) given by𝑥(𝑡)=𝑇(𝑡)𝑥+^𝑡_𝑠𝑇(𝑡−𝜃)𝑓(𝜃,𝑥(𝜃),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠))𝑑𝑠)𝑑𝜃on[𝑠,𝑡₁]. (3.30)Again, 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 𝑡∈(𝜏_𝑘,𝜏_𝑘+1], consider the Cauchy problem𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥,∫^𝑡₀𝑔(𝑡,𝑠,𝑥)𝑑𝑠),𝑡∈(𝜏_𝑘,𝜏_𝑘+1],𝑥(𝜏_𝑘)=𝑥_𝑘≡(𝐼+𝐵_𝑘)𝑥(𝜏_𝑘)+𝑐_𝑘∈𝑋. (3.31) By Step 1, Cauchy problem (3.31) also has a unique 𝑃𝐶-mild solution:𝑥(𝑡)=𝑇(𝑡−𝜏_𝑘)𝑥_𝑘+^𝑡_𝜏𝑘𝑇(𝑡−𝜃)𝑓(𝜃,𝑥(𝜃),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠))𝑑𝑠)𝑑𝜃. (3.32)Step 3. Combining all solutions on (𝜏_𝑘,𝜏_𝑘+1] (𝑘=1,…,𝛿), one can obtain the 𝑃𝐶-mild solution of the Cauchy problem (3.16) given by𝑥(𝑡,𝑥)=𝑆(𝑡,0)𝑥+^𝑡₀𝑆(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠))𝑑𝑠)𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝑆(𝑡,𝜏⁺_𝑘)𝑐_𝑘. (3.33) 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 𝑋 to 𝑋 as follows:𝑃(𝑥)=𝑥(𝑇₀,𝑥)=𝑆(𝑇₀,0)𝑥+^𝑇0₀𝑆(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥))𝑑𝑠)𝑑𝜃+_{0≤𝜏𝑘<𝑇0}𝑆(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘, (4.1)where 𝑥(⋅,𝑥) denotes the 𝑃𝐶-mild solution of the Cauchy problem (3.16) corresponding to the initial value 𝑥(0)=𝑥, then examine whether 𝑃 has a fixed point.

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

Lemma 4.1. System (3.15) has a 𝑇₀-periodic 𝑃𝐶-mild solution if and only if 𝑃 has a fixed point.

Proof. Suppose 𝑥(⋅)=𝑥(⋅+𝑇₀), then 𝑥(0)=𝑥(𝑇₀)=𝑃(𝑥(0)). This implies that 𝑥(0) is a fixed point of 𝑃. On the other hand, if 𝑃𝑥₀=𝑥₀, 𝑥₀∈𝑋, then for the 𝑃𝐶-mild solution 𝑥(⋅,𝑥₀) of the Cauchy problem (3.16) corresponding to the initial value 𝑥(0)=𝑥₀, we can define 𝑦(⋅)=𝑥(⋅+𝑇₀,𝑥₀), then 𝑦(0)=𝑥(𝑇₀,𝑥₀)=𝑃𝑥₀=𝑥₀. Now, for 𝑡>0, 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 at𝑦(𝑡)=𝑥(𝑡+𝑇₀,𝑥₀)=𝑆(𝑡+𝑇₀,𝑇₀)𝑆(𝑇₀,0)𝑥₀+^𝑇0₀𝑆(𝑡+𝑇₀,𝑇₀)𝑆(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥₀),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥₀))𝑑𝑠)𝑑𝜃+_{0≤𝜏𝑘<𝑇0}𝑆(𝑡+𝑇₀,𝑇₀)𝑆(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘+^𝑡+𝑇0_𝑇0𝑆(𝑡+𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥₀),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥₀+))𝑑𝑠)𝑑𝜃_{𝑇0≤𝜏𝑘+𝛿<𝑡+𝑇0}𝑆(𝑡+𝑇₀,𝜏⁺_𝑘+𝛿)𝑐_𝑘+𝛿=𝑆(𝑡,0){𝑆(𝑇₀,0)𝑥₀+^𝑇0₀𝑆(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥₀),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥₀))𝑑𝑠)𝑑𝜃+_{0≤𝜏𝑘<𝑇0}𝑆(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘+𝛿}+^𝑡₀𝑆(𝑡+𝑇₀,𝜃+𝑇₀)𝑓(𝜃+𝑇₀,𝑥(𝜃+𝑇₀,𝑥₀),^𝜃+𝑇0₀𝑔(𝜃+𝑇₀,𝑠,𝑥(𝑠,𝑥₀))𝑑𝑠)𝑑𝜃+_{𝑇0≤𝜏𝑘+𝛿<𝑡+𝑇0}𝑆(𝑡+𝑇₀,𝜏⁺_𝑘+𝛿)𝑐_𝑘+𝛿=𝑆(𝑡,0)𝑥(𝑇₀)+^𝑡₀𝑆(𝑡+𝑇₀,𝜃+𝑇₀)𝑓(𝜃+𝑇₀,𝑥(𝜃+𝑇₀,𝑥₀),^𝜃+𝑇0_𝑇0𝑔(𝜃+𝑇₀,𝑠,𝑥(𝑠,𝑥₀+))𝑑𝑠)𝑑𝜃_{𝑇0≤𝜏𝑘+𝛿<𝑡+𝑇0}𝑆(𝑡+𝑇₀,𝜏⁺_𝑘+𝛿)𝑐_𝑘+𝛿=𝑆(𝑡,0)𝑥(𝑇₀)+^𝑡₀𝑆(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃+𝑇₀,𝑥₀),^𝜃₀𝑔(𝜃+𝑇₀,𝑠+𝑇₀,𝑥(𝑠+𝑇₀,𝑥₀))𝑑𝑠)𝑑𝜃+_{𝑇0≤𝜏𝑘+𝛿<𝑡+𝑇0}𝑆(𝑡+𝑇₀,𝜏⁺_𝑘+𝛿)𝑐_𝑘+𝛿=𝑆(𝑡,0)𝑦(0)+^𝑡₀𝑆(𝑡,𝜃)𝑓(𝜃,𝑦(𝜃,𝑦(0)),^𝜃₀𝑔(𝜃,𝑠,𝑦(𝑠,𝑦(0)))𝑑𝑠)𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝑆(𝑡,𝜏⁺_𝑘)𝑐_𝑘=𝑆(𝑡,0)𝑦(0)+^𝑡₀𝑆(𝑡,𝜃)𝑓(𝜃,𝑦(𝜃,𝑦(0)),^𝜃₀+𝑔(𝜃,𝑠,𝑦(𝑠,𝑦(0)))𝑑𝑠)𝑑𝜃_{0≤𝜏𝑘<𝑡}𝑆(𝑡,𝜏⁺_𝑘)𝑐_𝑘. (4.2) This implies that 𝑦(⋅,𝑦(0)) is a 𝑃𝐶-mild solution of Cauchy problem (3.16) with initial value 𝑦(0)=𝑥₀. Thus the uniqueness implies that 𝑥(⋅,𝑥₀)=𝑦(⋅,𝑦(0))=𝑥(⋅+𝑇₀,𝑥₀), so that 𝑥(⋅,𝑥₀) is a 𝑇₀-periodic.

Second, we show that the Poincaré operator 𝑃 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 𝑃 is a continuous operator of 𝑥 on 𝑋.

Proof. Let 𝑥, 𝑦∈Ξ⊂𝑋, where Ξ is a bounded subset of 𝑋. Suppose 𝑥(⋅,𝑥) and 𝑥(⋅,𝑦) are the 𝑃𝐶-mild solutions of Cauchy problem (3.16) corresponding to the initial value 𝑥 and 𝑦∈𝑋, respectively, given by𝑥(𝑡,𝑥)=𝑆(𝑡,0)𝑥+∫^𝑡₀𝑆(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥),∫^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥))𝑑𝑠)𝑑𝜃+∑_{0≤𝜏𝑘<𝑡}𝑆(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘;𝑥(𝑡,𝑦)=𝑆(𝑡,0)𝑦+∫^𝑡₀𝑆(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃,𝑦),∫^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑦))𝑑𝑠)𝑑𝜃+∑_{0≤𝜏𝑘<𝑡}𝑆(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘. (4.3)Thus, by assumption [H2.2], property (1) of Lemma 2.3, and property (3) of Lemma 3.4, we obtain‖𝑥(𝑡,𝑥)‖≤𝑀_𝑇0‖𝑥‖+(1+𝑀_𝑔𝑇₀)𝑀_𝑇0𝑀_𝑓𝑇₀+𝑀_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖+𝑀_𝑇0𝑀_𝑓^𝑡₀‖𝑥(𝜃,𝑥)‖𝑑𝜃+𝑀_𝑇0𝑀_𝑓𝑀_𝑔𝑇₀^𝑡₀‖𝑥_𝜃,𝑥‖_𝐵𝑑𝜃≤𝑎₀+𝑀_𝑇0𝑀_𝑓^𝑡₀‖𝑥(𝜃,𝑥)‖𝑑𝜃+𝑀_𝑇0𝑀_𝑓𝑀_𝑔𝑇₀^𝑡₀‖𝑥_𝜃,𝑥‖_𝐵𝑑𝜃,‖𝑥(𝑡,𝑦)‖≤𝑀_𝑇0‖𝑦‖+(1+𝑀_𝑔𝑇₀)𝑀_𝑇0𝑀_𝑓𝑇₀+𝑀_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖+𝑀_𝑇0𝑀_𝑓^𝑡₀‖𝑥(𝜃,𝑦)‖𝑑𝜃+𝑀_𝑇0𝑀_𝑓𝑀_𝑔𝑇₀^𝑡₀‖𝑥_𝜃,𝑦‖_𝐵𝑑𝜃≤𝑏₀+𝑀_𝑇0𝑀_𝑓^𝑡₀‖𝑥(𝜃,𝑦)‖𝑑𝜃+𝑀_𝑇0𝑀_𝑓𝑀_𝑔𝑇₀^𝑡₀‖𝑥_𝜃,𝑦‖_𝐵𝑑𝜃. (4.4)(4.5)
By Lemma 3.1, one can verify that there exist constants 𝑀^∗₁ and 𝑀^∗₂>0 such that‖𝑥(𝑡,𝑥)‖≤𝑀^∗₁,‖𝑥(𝑡,𝑦)‖≤𝑀^∗₂. (4.6)Let 𝜌=max{𝑀^∗₁,𝑀^∗₂}>0, then ‖𝑥(⋅,𝑥)‖, ‖𝑥(⋅,𝑦)‖≤𝜌 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 obtain‖𝑥(𝑡,𝑥)−𝑥(𝑡,𝑦)‖≤‖𝑆(𝑡,0)‖‖𝑥−+𝑦‖_𝑡0‖𝑆(𝑡,𝜃)‖‖𝑓(𝜃,𝑥(𝜃,𝑥),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥))𝑑𝑠)−𝑓(𝜃,𝑥(𝜃,𝑦),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑦))𝑑𝑠)‖𝑑𝜃≤𝑀_𝑇0‖𝑥−𝑦‖+𝑀_𝑇0𝐿_𝑓(𝜌)^𝑡₀‖𝑥(𝜃,𝑥)−𝑥(𝜃,𝑦)‖𝑑𝜃+𝑀_𝑇0𝐿_𝑓(𝜌)𝐿_𝑔(𝜌)𝑇₀^𝑡₀‖𝑥_𝜃,𝑥−𝑥_𝜃,𝑦‖_𝐵𝑑𝜃. (4.7)By Lemma 3.1 again, one can verify that there exists a constant 𝑀^∗₃>0 such that‖𝑥(𝑡,𝑥)−𝑥(𝑡,𝑦)‖≤𝑀^∗₃𝑀_𝑇0‖𝑥−𝑦‖≡𝐿‖𝑥−𝑦‖,∀𝑡∈[0,𝑇₀], (4.8)which implies that‖𝑃(𝑥)−𝑃(𝑦)‖=‖𝑥(𝑇₀,𝑥)−𝑥(𝑇₀,𝑦)‖≤𝐿‖𝑥−𝑦‖. (4.9)Hence, 𝑃 is a continuous operator of 𝑥 on 𝑋.

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

Assumption [H3]. The semigroup {𝑇(𝑡),𝑡>0} is a compact semigroup on 𝑋.
Now, we are ready to prove the compactness of operator 𝑃 defined by (4.1).

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

Proof. We only need to verify that 𝑃 takes a bounded set into a precompact set in 𝑋.
Let Γ be a bounded subset of 𝑋. Define 𝐾=𝑃Γ={𝑃(𝑥)∈𝑋∣𝑥∈Γ}.
For 0<𝜀≤𝑇₀, define𝐾_𝜀=𝑃_𝜀Γ=𝑆(𝑇₀,𝑇₀−𝜀){𝑥(𝑇₀−𝜀,𝑥)∣𝑥∈Γ}. (4.10)
Next, we show that 𝐾_𝜀 is precompact in 𝑋. In fact, for 𝑥∈Γ fixed, we have‖𝑥(𝑇₀−𝜀,𝑥)‖≤‖𝑆(𝑇₀−𝜀,0)𝑥‖+^𝑇0₀^−𝜀‖𝑆(𝑇₀−𝜀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥))𝑑𝑠)‖𝑑𝜃+_{0≤𝜏𝑘<𝑇0−𝜀}‖𝑆(𝑇₀−𝜀,𝜏⁺_𝑘)𝑐_𝑘‖≤𝑀_𝑇0‖𝑥‖+𝑀_𝑇0𝑀_𝑓𝑇₀(1+𝑀_𝑔𝑇₀)+𝑀_𝑇0𝑀_𝑓^𝑇0₀‖𝑥(𝜃,𝑥)‖𝑑𝜃+𝑀_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖+𝑀_𝑇0𝑀_𝑓𝑀_𝑔𝑇₀^𝑇0₀‖𝑥_𝜃,𝑥‖_𝐵𝑑𝜃≤𝑀_𝑇0‖𝑥‖+𝑀_𝑇0𝑀_𝑓𝑇₀(1+𝑀_𝑔𝑇₀)+(1+𝑀_𝑔𝑇₀)𝑀_𝑇0𝑀_𝑓𝑇₀𝜌+𝑀_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖. (4.11)This implies that the set {𝑥(𝑇₀−𝜀,𝑥)∣𝑥∈Γ} is totally bounded.
By virtue of the compactness of {𝑇(𝑡),𝑡>0} and property (5) of Lemma 2.3, 𝑆(𝑇₀,𝑇₀−𝜀) is a compact operator. Thus, 𝐾_𝜀 is precompact in 𝑋.
On the other hand, for arbitrary 𝑥∈Γ,𝑃_𝜀(𝑥)=𝑆(𝑇₀,0)𝑥+^𝑇0₀^−𝜀𝑆(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥))𝑑𝑠)𝑑𝜃+_{0≤𝜏𝑘<𝑇0−𝜀}𝑆(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘. (4.12)Thus, combined with (4.1), we have‖𝑃_𝜀(𝑥)−𝑃(𝑥)‖≤‖^𝑇0₀^−𝜀𝑆(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,−𝑥))𝑑𝑠)𝑑𝜃_𝑇00𝑆(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥))𝑑𝑠)𝑑𝜃‖+‖_{0≤𝜏𝑘<𝑇0−𝜀}𝑆(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘−_{0≤𝜏𝑘<𝑇0}𝑆(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘‖≤^𝑇0_𝑇0−𝜀‖𝑆(𝑇₀,𝜃)‖‖𝑓(𝜃,𝑥(𝜃,𝑥),^𝜃₀𝑔(𝜃,𝑠,𝑥(𝑠,𝑥))𝑑𝑠)‖𝑑𝜃+𝑀_{𝑇0𝑇0−𝜀≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖≤𝑀_𝑇0𝑀_𝑓(1+𝑀_𝑔𝑇₀)(1+𝜌)𝜀+𝑀_{𝑇0𝑇0−𝜀≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖. (4.13)It is showing that the set 𝐾 can be approximated to an arbitrary degree of accuracy by a precompact set 𝐾_𝜀. Hence 𝐾 itself is a precompact set in 𝑋, that is, 𝑃 takes a bounded set into a precompact set in 𝑋. As a result, 𝑃 is a compact operator.

After showing the continuity and compactness of operator 𝑃, we can follow and derive periodic 𝑃𝐶-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 𝑃𝐶-mild solutions here, so we only define the (locally) boundedness and ultimate boundedness.

Definition 4.4. One says that 𝑃𝐶-mild solutions of Cauchy problem (3.16) are bounded if for each 𝐵₁>0, there is a 𝐵₂>0 such that ‖𝑥‖≤𝐵₁ implies ‖𝑥(𝑡,𝑥)‖≤𝐵₂ for 𝑡≥0.

Definition 4.5. One says that 𝑃𝐶-mild solutions of Cauchy problem (3.16) are locally bounded if for each 𝐵₁>0 and 𝑘₀>0, there is a 𝐵₂>0 such that ‖𝑥‖≤𝐵₁ implies ‖𝑥(𝑡,𝑥)‖≤𝐵₂ for 0≤𝑡≤𝑘₀.

Definition 4.6. One says that 𝑃𝐶-mild solutions of Cauchy problem (3.16) are ultimate bounded if there is a bound 𝐵>0 such that for each 𝐵₃>0, there is a 𝑘>0, such that ‖𝑥‖≤𝐵₃ and 𝑡≥𝑘 imply ‖𝑥(𝑡,𝑥)‖≤𝐵.

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 𝐸₀⊂𝐸₁⊂𝐸₂ be convex subsets of Banach space 𝑋, with 𝐸₀ and 𝐸₂ compact subsets and 𝐸₁ open relative to 𝐸₂. Let 𝑃 : 𝐸₂→𝑋 be a continuous map such that for some integer 𝑚, one has𝑃^𝑗(𝐸₁)⊂𝐸₂𝑃,1≤𝑗≤𝑚−1,_𝑗(𝐸₁)⊂𝐸₀,𝑚≤𝑗≤2𝑚−1, (4.14)then 𝑃 has a fixed point in 𝐸₀.

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

Theorem 4.9. Let assumptions [H1], [H2], and [H3] hold. If the 𝑃𝐶-mild solutions of Cauchy problem (3.16) are ultimate bounded, then system (3.15) has a 𝑇₀-periodic 𝑃𝐶-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 𝑥(0)=𝑥 has a 𝑃𝐶-mild solution 𝑥(⋅,𝑥) which is locally bounded. From ultimate boundedness and Lemma 4.7, 𝑥(⋅,𝑥) is bounded. Next, let 𝐵>0 be the bound in the definition of ultimate boundedness. Then by boundedness, there is a 𝐵₁>𝐵 such that ‖𝑥‖≤𝐵 implies ‖𝑥(𝑡,𝑥)‖≤𝐵₁ for 𝑡≥0. Furthermore, there is a 𝐵₂>𝐵₁ such that ‖𝑥‖≤𝐵₁ implies ‖𝑥(𝑡,𝑥)‖≤𝐵₂ for 𝑡≥0. Now, using ultimate boundedness again, there is a positive integer 𝑚 such that ‖𝑥‖≤𝐵₁ implies ‖𝑥(𝑡,𝑥)‖≤𝐵 for 𝑡≥(𝑚−2)𝑇₀.
Define 𝑦(⋅,𝑦(0))=𝑥(⋅+𝑇₀,𝑥), then 𝑦(0)=𝑥(𝑇₀,𝑥)=𝑃(𝑥). From (4.2) in Lemma 4.1, we obtain 𝑃(𝑦(0))=𝑦(𝑇₀,𝑦(0))=𝑥(2𝑇₀,𝑥). Thus, 𝑃²(𝑥)=𝑃(𝑃(𝑥))=𝑃(𝑦(0))=𝑥(2𝑇₀,𝑥). Suppose there exists integer 𝑚−1 such that 𝑃^𝑚−1(𝑥)=𝑥((𝑚−1)𝑇₀,𝑥). By induction we arrive at𝑃^𝑚(𝑥)=𝑃^𝑚−1(𝑃(𝑥))=𝑃^𝑚−1(𝑦(0))=𝑦((𝑚−1)𝑇₀,𝑦(0))=𝑥(𝑚𝑇₀,𝑥). (4.15)Thus, we obtain‖𝑃^𝑗−1(𝑥)‖=‖𝑥((𝑗−1)𝑇₀,𝑥)‖<𝐵₂,𝑗=1,2,…,𝑚−1,‖𝑥‖<𝐵₁;‖𝑃^𝑗−1(𝑥)‖=‖𝑥((𝑗−1)𝑇₀,𝑥)‖<𝐵,𝑗≥𝑚,‖𝑥‖<𝐵₁. (4.16)
It comes from Lemma 4.3 that 𝑃(𝑥)=𝑥(𝑇₀,𝑥) on 𝑋 is compact. Now let𝐻={𝑥∈𝑋∶‖𝑥‖<𝐵₂},𝐸₂=cl.(conv(𝑃(𝐻))),𝑊={𝑥∈𝑋∶‖𝑥‖<𝐵₁},𝐸₁=𝑊∩𝐸₂,𝐺={𝑥∈𝑋∶‖𝑥‖<𝐵},𝐸₀=cl.(conv(𝑃(𝐺))), (4.17)where conv(𝑌) is the convex hull of the set 𝑌 defined by conv(𝑌)={∑^𝑛_𝑖=1𝜆_𝑖𝑦_𝑖∣𝑛≥1,𝑦_𝑖∈𝑌,𝜆_𝑖≥0,∑^𝑛_𝑖=1𝜆_𝑖=1}, and cl. denotes the closure. Then we see that 𝐸₀⊂𝐸₁⊂𝐸₂ are convex subsets of 𝑋 with 𝐸₀, 𝐸₂ compact subsets and 𝐸₁ open relative to 𝐸₂, and from (4.16) one has𝑃^𝑗(𝐸₁)⊂𝑃^𝑗(𝑊)=𝑃𝑃^𝑗−1(𝑊)⊂𝑃(𝐻)⊂𝐸₂𝑃,𝑗=1,2,…,𝑚−1;_𝑗(𝐸₁)⊂𝑃^𝑗(𝑊)=𝑃𝑃^𝑗−1(𝑊)⊂𝑃(𝐺)⊂𝐸₀,𝑗=𝑚,𝑚+1,…,2𝑚−1. (4.18)We see that 𝑃 : 𝐸₂→𝑋 is a continuous map continuous from Lemma 4.2. Consequently, from Horn's fixed point theorem, we know that the operator 𝑃 has a fixed point 𝑥₀∈𝐸₀⊂𝑋. By Lemma 4.1, we know that the 𝑃𝐶-mild solution 𝑥(⋅,𝑥₀) of the Cauchy problem (3.16) corresponding to the initial value 𝑥(0)=𝑥₀ is just 𝑇₀-periodic. Therefore, 𝑥(⋅,𝑥₀) is a 𝑇₀-periodic 𝑃𝐶-mild solution of system (3.15). This proves the theorem.

At last, an example is given to illustrate our theory. Consider the following problem:𝜕𝜕𝑡𝑥(𝑡,𝑦)=Δ𝑥(𝑡,𝑦)+𝑥(𝑡,𝑦)+^𝑡₀(1+sin(𝑡−𝑠))3𝑥²1(𝑠,𝑦)+2𝑑𝑠,𝑦∈Ω,𝑡≥𝑠,𝑡,𝑠∈(0,2𝜋]⧵{23𝜋,𝜋,2𝜋},Δ𝑥(𝜏_𝑖,𝑦)=𝑥(𝜏⁺_𝑖,𝑦)−𝑥(𝜏⁻_𝑖,𝑦)=0.05𝐼𝑥(𝜏_𝑖,𝑦),𝑖=1,−0.05𝐼𝑥(𝜏_𝑖,𝑦),𝑖=2,0.05𝐼𝑥(𝜏_𝑖,𝑦),𝑖=3,𝑦∈Ω,𝜏_𝑖=𝑖2𝜋,𝑖=1,2,3,𝑥(𝑡,𝑦)|_{𝑦∈𝜕Ω}=0,𝑡>0,𝑥(0,𝑦)=𝑥(2𝜋,𝑦), (4.19)where Ω⊂ℝ³ is a bounded domain and 𝜕Ω∈𝐶³.

Define 𝑋=𝐿₂(Ω), 𝐷(𝐴)=𝐻²(Ω)∩𝐻¹₀(Ω), and 𝐴𝑥=−(𝜕²𝑥/𝜕𝑦²₁+𝜕²𝑥/𝜕𝑦²₂+𝜕²𝑥/𝜕𝑦²₃) for 𝑥∈𝐷(𝐴). Then, 𝐴 generates a compact semigroup {𝑇(𝑡),𝑡≥0} on 𝐿₂(Ω). Define 𝑥(⋅)(𝑦)=𝑥(⋅,𝑦), sin(⋅)(𝑦)=sin(⋅,𝑦), 𝑓(⋅,𝑥(⋅),∫^⋅₀𝑔(⋅,𝑠,𝑥)𝑑𝑠)(𝑦)=𝑥(⋅)(𝑦)+∫^𝑡₀(1+sin(⋅−𝑠))3𝑥²(⋅)+2𝑑𝑠(𝑦), and𝐵_𝑖=0.05𝐼,𝑖=1,−0.05𝐼,𝑖=2,0.05𝐼,𝑖=3. (4.20)

Thus problem (4.19) can be rewritten as𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥,∫^𝑡₀1𝑔(𝑡,𝑠,𝑥)𝑑𝑠),𝑡≥𝑠,𝑡,𝑠∈(0,2𝜋]⧵{23𝜋,𝜋,2𝑖𝜋},Δ𝑥(2𝜋)=𝐵_𝑖𝑖𝑥(2𝜋),𝑖=1,2,3,𝑥(0)=𝑥(2𝜋). (4.21)If the 𝑃𝐶-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 2𝜋-periodic 𝑃𝐶-mild solution 𝑥_2𝜋(⋅,𝑦)∈𝑃𝐶_2𝜋([0+∞);𝐿₂(Ω)), where𝑃𝐶_2𝜋([0,+∞);𝐿₂(Ω))≡{𝑥∈𝑃𝐶([0,+∞);𝐿₂(Ω))∣𝑥(𝑡)=𝑥(𝑡+2𝜋),𝑡≥0}. (4.22)

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).