Periodic Solutions of Semilinear Impulsive Periodic System with Time-Varying Generating Operators on Banach Space

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 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 inequality with mixed type integral operators given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of 𝑇₀-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 [1–5]).

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 [6–14]). We also gave a series of results for semilinear (strongly nonlinear) impulsive systems with time-varying generating operators and optimal control problems (see [15–18]).

Although, there are some papers on periodic solutions of periodic system with time-varying generating operators on infinite dimensional spaces (see [19–22]), 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 𝑇₀-periodic 𝑃𝐶-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 [23–25]).

Herein, we go on studying the semilinear impulsive periodic system with time-varying generating operators𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥),𝑡≠𝜏_𝑘,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡)+𝑐_𝑘,𝑡=𝜏_𝑘, (1.1)in the parabolic case on infinite dimensional Banach space 𝑋, where {𝐴(𝑡),𝑡∈[0,𝑇₀]} is a family of closed densely defined linear unbounded operators on 𝑋 and the resolvent of the unbounded operator 𝐴(𝑡) is compact. Time sequence 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 denoted the number of impulsive points between 0 and 𝑇₀. 𝑓 is a measurable function from [0,∞)×𝑋 to 𝑋 and is 𝑇₀-periodic in 𝑡, 𝐵_𝑘+𝛿=𝐵_𝑘, 𝑐_𝑘+𝛿=𝑐_𝑘. 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 𝑋.

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 𝑃 for semilinear impulsive periodic system with time-varying generating operators (1.1), then overcome some difficulties to show the compactness of Poincaré operator 𝑃 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 {𝑥=𝜆𝑃𝑥,𝜆∈[0,1]} is established. Therefore, the existence of 𝑇₀-periodic 𝑃𝐶-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 𝑇₀-periodic 𝑃𝐶-mild solution for semilinear impulsive periodic system with time-varying generating operators (1.1) is introduced. We construct the suitable Poincaré operator 𝑃 and give the relation between 𝑇₀-periodic 𝑃𝐶-mild solution and the fixed point of 𝑃. After showing the compactness of the Poincaré operator 𝑃 and obtaining the boundedness of the fixed point set {𝑥=𝜆𝑃𝑥,𝜆∈[0,1]} by virtue of the generalized Gronwall's inequality, we can use Leray-Schauder fixed point theorem to establish the existence of 𝑇₀-periodic 𝑃𝐶-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 𝑋 be a Banach space. £(𝑋) denotes the space of linear operators in 𝑋; £_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,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 endowed with the norm ‖⋅‖_𝑃𝐶(‖⋅‖_𝑃𝐶¹), 𝑃𝐶([0,𝑇₀];𝑋)(𝑃𝐶¹([0,𝑇₀];𝑋)) is a Banach space.

Consider the following homogeneous linear impulsive periodic system with time-varying generating operators:𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑡≠𝜏_𝑘,Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘),𝑡=𝜏_𝑘 (2.2)on Banach space 𝑋, where Δ𝑥(𝜏_𝑘)=𝑥(𝜏⁺_𝑘)−𝑥(𝜏⁻_𝑘), {𝐴(𝑡),𝑡∈[0,𝑇₀]} is a family of closed densely defined linear unbounded operators on 𝑋 satisfying the following assumption.

Assumption (see [28, page 158]). For 𝑡∈[0,𝑇₀] one has
(P₁)The domain 𝐷(𝐴(𝑡))=𝐷 is independent of 𝑡 and is dense in 𝑋.(P₂)For 𝑡≥0, the resolvent 𝑅(𝜆,𝐴(𝑡))=(𝜆𝐼−𝐴(𝑡))⁻¹ exists for all 𝜆 with Re𝜆≤0, and there is a constant 𝑀 independent of 𝜆 and 𝑡 such that‖𝑅(𝜆,𝐴(𝑡))‖≤𝑀(1+|𝜆|)⁻¹forRe𝜆≤0. (2.3)(P₃)There exist constants 𝐿>0 and 0<𝛼≤1 such that‖(𝐴(𝑡)−𝐴(𝜃))𝐴⁻¹(𝜏)‖≤𝐿|𝑡−𝜃|^𝛼for𝑡,𝜃,𝜏∈[0,𝑇₀]. (2.4)

Lemma 2.1 (see [28, page 159]). Under the Assumption A1, the Cauchy problem𝑥̇(𝑡)+𝐴(𝑡)𝑥(𝑡)=0,𝑡∈(0,𝑇₀]with𝑥(0)=𝑥₀ (Eq.1)has a unique evolution system {𝑈(𝑡,𝜃)∣0≤𝜃≤𝑡≤𝑇₀} in 𝑋 satisfying the following properties:
(1)𝑈(𝑡,𝜃)∈£_𝑏(𝑋) for 0≤𝜃≤𝑡≤𝑇₀;(2)𝑈(𝑡,𝑟)𝑈(𝑟,𝜃)=𝑈(𝑡,𝜃) for 0≤𝜃≤𝑟≤𝑡≤𝑇₀;(3)𝑈(⋅,⋅)𝑥∈𝐶(Δ,𝑋) for 𝑥∈𝑋, Δ={(𝑡,𝜃)∈[0,𝑇₀]×[0,𝑇₀]∣0≤𝜃≤𝑡≤𝑇₀};(4)For 0≤𝜃<𝑡≤𝑇₀, 𝑈(𝑡,𝜃) : 𝑋→𝐷 and 𝑡→𝑈(𝑡,𝜃) is strongly differentiable in 𝑋. The derivative (𝜕/𝜕𝑡)𝑈(𝑡,𝜃)∈£_𝑏(𝑋) and it is strongly continuous on 0≤𝜃<𝑡≤𝑇₀. Moreover,𝜕𝜕𝑡𝑈(𝑡,𝜃)=−𝐴(𝑡)𝑈(𝑡,𝜃)for0≤𝜃<𝑡≤𝑇₀,‖𝜕𝜕𝑡𝑈(𝑡,𝜃)‖_£𝑏(𝑋)=‖𝐴(𝑡)𝑈(𝑡,𝜃)‖_£𝑏(𝑋)≤𝐶,𝑡−𝜃‖𝐴(𝑡)𝑈(𝑡,𝜃)𝐴(𝜃)⁻¹‖_£𝑏(𝑋)≤𝐶for0≤𝜃≤𝑡≤𝑇₀. (2.5)(5)For every 𝑣∈𝐷 and 𝑡∈(0,𝑇₀],𝑈(𝑡,𝜃)𝑣 is differentiable with respect to 𝜃 on 0≤𝜃≤𝑡≤𝑇₀𝜕𝜕𝜃𝑈(𝑡,𝜃)𝑣=𝑈(𝑡,𝜃)𝐴(𝜃)𝑣, (2.6)and, for each 𝑥₀∈𝑋, the Cauchy problem (Eq.1) has a unique classical solution 𝑥∈𝐶¹([0,𝑇₀];𝑋) given by𝑥(𝑡)=𝑈(𝑡,0)𝑥₀,𝑡∈[0,𝑇₀]. (2.7)

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

Assumption 2. There exits 𝑇₀>0 such that 𝐴(𝑡+𝑇₀)=𝐴(𝑡) for 𝑡∈[0,𝑇₀].

Assumption 3. For 𝑡≥0, the resolvent 𝑅(𝜆,𝐴(𝑡)) is compact.

Then, we have the following lemma.

Lemma 2.2. Assumptions A1, A2, and A3 hold. Then evolution system {𝑈(𝑡,𝜃)∣0≤𝜃≤𝑡≤𝑇₀} in 𝑋 also satisfying the following two properties:
(6)𝑈(𝑡+𝑇₀,𝜃+𝑇₀)=𝑈(𝑡,𝜃) for 0≤𝜃≤𝑡≤𝑇₀;(7)𝑈(𝑡,𝜃) is compact operator for 0≤𝜃<𝑡≤𝑇₀.

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

Assumption 4. For each 𝑘∈ℤ⁺₀, 𝐵_𝑘∈£_𝑏(𝑋), there exists 𝛿∈ℕ such that 𝜏_𝑘+𝛿=𝜏_𝑘+𝑇₀and𝐵_𝑘+𝛿=𝐵_𝑘.

Consider the following Cauchy's problem𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑡∈[0,𝑇₀]⧵𝐷^{0𝑥002𝑑𝑐},Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘),𝑘=1,2,…,𝛿,𝑥(0)=𝑥₀. (2.8)

For every 𝑥₀∈𝑋, 𝐷 is an invariant subspace of 𝐵_𝑘, using Lemma 2.1, step by step, one can verify that the Cauchy problem (2.8) has a unique classical solution 𝑥∈𝑃𝐶¹([0,𝑇₀];𝑋) represented by 𝑥(𝑡)=𝒮(𝑡,0)𝑥₀ where 𝒮(⋅,⋅)∶Δ→£(𝑋) given by 𝒮(𝑡,𝜃)=𝑈(𝑡,𝜃),𝜏_𝑘−1≤𝜃≤𝑡≤𝜏_𝑘,𝑈(𝑡,𝜏⁺_𝑘)(𝐼+𝐵_𝑘)𝑈(𝜏_𝑘,𝜃),𝜏_𝑘−1≤𝜃<𝜏_𝑘<𝑡≤𝜏_𝑘+1,𝑈(𝑡,𝜏⁺_𝑘)[_{𝜃<𝜏𝑗<𝑡}(𝐼+𝐵_𝑗)𝑈(𝜏_𝑗,𝜏⁺_𝑗−1)](𝐼+𝐵_𝑖)𝑈(𝜏_𝑖𝜏,𝜃),_𝑖−1≤𝜃<𝜏_𝑖≤⋯<𝜏_𝑘<𝑡≤𝜏_𝑘+1. (2.9)The operator 𝒮(𝑡,𝜃)((𝑡,𝜃)∈Δ) is called impulsive evolution operator associated with {𝐵_𝑘;𝜏_𝑘}^∞_𝑘=1.

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

Lemma 2.3 (see [24, Lemma 1]). Assumptions A1, A2, A3, and B hold. 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 𝑀∈𝑍⁺₀, 𝒮(𝑀𝑇₀+𝑡,0)=𝒮(𝑡,0)[𝒮(𝑇₀,0)]^𝑀.(5)𝒮(𝑡,𝜃) is 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 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 𝑃𝐶-mild solutions for the semilinear impulsive periodic system with time-varying generating operators on Banach space.

Now we introduce the 𝑃𝐶-mild solution of Cauchy's problem (2.8) and 𝑇₀-periodic 𝑃𝐶-mild solution of the system (2.2).

Definition 2.4. For every 𝑥₀∈𝑋, the function 𝑥∈𝑃𝐶([0,𝑇₀];𝑋) given by 𝑥(𝑡)=𝒮(𝑡,0)𝑥₀ is said to be the 𝑃𝐶-mild solution of the Cauchy problem (2.8).

Deffinition 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's problem (2.8) corresponding to some 𝑥₀ and 𝑥(𝑡+𝑇₀)=𝑥(𝑡) for 𝑡≥0.

Secondly, we recall the following nonhomogeneous linear impulsive periodic system with time-varying generating operators𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑡≠𝜏_𝑘,Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘)+𝑐_𝑘,𝑡=𝜏_𝑘, (2.10)where 𝑓∈𝐿¹([0,𝑇₀];𝑋), 𝑓(𝑡+𝑇₀)=𝑓(𝑡) for 𝑡≥0 and 𝑐_𝑘 satisfies the following assumption.

Assumption 5. For each 𝑘∈ℤ⁺₀ and 𝑐_𝑘∈𝑋, there exists 𝛿∈ℕ such that 𝑐_𝑘+𝛿=𝑐_𝑘.

In order to study system (2.10), we need to consider the following Cauchy problem𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑡∈[0,𝑇₀]⧵𝐷^{0𝑥002𝑑𝑐},Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘)+𝑐_𝑘,𝑘=1,2,…,𝛿,𝑥(0)=𝑥₀, (2.11)and introduce the 𝑃𝐶-mild solution of Cauchy's problem (2.11) and 𝑇₀-periodic 𝑃𝐶-mild solution of system (2.10).

Definition 2.6. A function 𝑥∈𝑃𝐶([0,𝑇₀];𝑋), for finite interval [0,𝑇₀], is said to be a 𝑃𝐶-mild solution of the Cauchy problem (2.10) corresponding to the initial value 𝑥₀∈𝑋 and input 𝑓∈𝐿¹([0,𝑇₀];𝑋) if 𝑥 is given by𝑥(𝑡)=𝒮(𝑡,0)𝑥₀+^𝑡₀𝒮(𝑡,𝜃)𝑓(𝜃)𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝒮(𝑡,𝜏⁺_𝑘)𝑐_𝑘. (Eq.2)

Definition 2.7. A function 𝑥∈𝑃𝐶([0,+∞);𝑋) is said to be a 𝑇₀-periodic 𝑃𝐶-mild solution of system (2.10) if it satisfies the expression (Eq.2) and 𝑥(𝑡+𝑇₀)=𝑥(𝑡) for 𝑡≥0.

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 𝑎≥0, 𝑏≥0, 𝑐≥0, 0≤𝜆₁≤1, 0≤𝜆₂<1. If 𝑥∈𝑃𝐶([0,𝑇₀];𝑋) satisfies‖𝑥(𝑡)‖≤𝑎+𝑏^𝑡₀‖𝑥(𝜃)‖^𝜆1𝑑𝜃+𝑐^𝑇0₀‖𝑥(𝜃)‖^𝜆2𝑑𝜃,∀𝑡∈[0,𝑇₀], (3.1)then there exists a constant 𝑀^∗=𝑀^∗(𝑎,𝑏,𝑐,𝜆₂,𝑇₀)>0 such that‖𝑥(𝑡)‖≤𝑀^∗,∀𝑡∈[0,𝑇₀]. (3.2)

Proof. Let𝑦(𝑡)=‖𝑥(𝑡)‖+1,∀𝑡∈[0,𝑇₀],𝑀=max_{𝑡∈[0,𝑇0]}‖𝑦(𝑡)‖. (3.3)Then,1≤𝑦(𝑡)≤1+𝑎+𝑏^𝑡₀‖𝑦(𝜃)‖𝑑𝜃+𝑐𝑇₀𝑀^𝜆2,∀𝑡∈[0,𝑇₀]. (3.4)By Gronwall's inequality, we obtain𝑦(𝑡)≤(1+𝑎+𝑐𝑇₀𝑀^𝜆2)𝑒^𝑏𝑡,∀𝑡∈[0,𝑇₀]. (3.5)Thus,𝑀≤(1+𝑎+𝑐𝑇₀𝑀^𝜆2)𝑒^𝑏𝑇0≤(1+𝑎+𝑐𝑇₀)𝑒^𝑏𝑇0𝑀^𝜆2. (3.6)Therefore,‖𝑥(𝑡)‖≤𝑀≤((1+𝑎+𝑐𝑇₀)𝑒^𝑏𝑇0)^{1/(1−𝜆2)}≡𝑀^∗(𝑎,𝑏,𝑐,𝜆₂,𝑇₀),∀𝑡∈[0,𝑇₀]. (3.7)This completes the proof.

Now, we consider the following semilinear impulsive periodic system with time-varying generating operators𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥),𝑡≠𝜏_𝑘,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡)+𝑐_𝑘,𝑡=𝜏_𝑘, (3.8)and introduce Poincaré operator and study the 𝑇₀-periodic 𝑃𝐶-mild solution of system (3.8).

In order to study the system (3.8), we first consider Cauchy's problem𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥),𝑡∈[0,𝑇₀]⧵𝐷^{0𝑥002𝑑𝑐},Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘)+𝑐_𝑘,𝑘=1,2,…,𝛿,𝑥(0)=𝑥. (3.9)By virtue of the expression of the 𝑃𝐶-mild solution of the Cauchy problem (2.11), we can introduce the 𝑃𝐶-mild solution of the Cauchy problem (3.9).

Definition 3.2. A function 𝑥∈𝑃𝐶([0,𝑇₀];𝑋) is said to be a 𝑃𝐶-mild solution of the Cauchy problem (3.9) 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))𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝒮(𝑡,𝜏⁺_𝑘)𝑐_𝑘for𝑡∈[0,𝑇₀]. (3.10)

Now, we introduce the 𝑇₀-periodic 𝑃𝐶-mild solution of system (3.8).

Definition 3.3. A function 𝑥∈𝑃𝐶([0,+∞);𝑋) is said to be a 𝑇₀-periodic 𝑃𝐶-mild solution of system (3.8) if it is a 𝑃𝐶-mild solution of Cauchy's problem (3.9) corresponding to some 𝑥 and 𝑥(𝑡+𝑇₀)=𝑥(𝑡) for 𝑡≥0.

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

Assumption 6. (F1): 𝑓∶[0,∞)×𝑋→𝑋 is measurable for 𝑡≥0 and for any 𝑥,𝑦∈𝑋 satisfying ‖𝑥‖,‖𝑦‖≤𝜌 there exists a positive constant 𝐿_𝑓(𝜌)>0 such that‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖≤𝐿_𝑓(𝜌)‖𝑥−𝑦‖. (3.11)
(F2): There exists a positive constant 𝑀_𝑓>0 such that‖𝑓(𝑡,𝑥)‖≤𝑀_𝑓(1+‖𝑥‖)∀𝑥∈𝑋. (3.12)
(F3): 𝑓(𝑡,𝑥) is 𝑇₀-periodic in 𝑡. That is, 𝑓(𝑡+𝑇₀,𝑥)=𝑓(𝑡,𝑥), 𝑡≥0.

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 𝑃𝐶-mild solution given by𝑥(𝑡,𝑥)=𝒮(𝑡,0)𝑥+^𝑡₀𝒮(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥))𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝒮(𝑡,𝜏⁺_𝑘)𝑐_𝑘. (3.13)

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 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥),𝑡∈[𝑠,𝜏],𝑥(𝑠)=𝑥∈𝑋 (3.14)has a unique mild solution𝑥(𝑡)=𝑈(𝑡,𝑠)𝑥+^𝑡_𝑠𝑈(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃))𝑑𝜃. (3.15)
In general, for 𝑡∈(𝜏_𝑘,𝜏_𝑘+1], Cauchy's problem𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥),𝑡∈(𝜏_𝑘,𝜏_𝑘+1],𝑥(𝜏_𝑘)=𝑥_𝑘≡(𝐼+𝐵_𝑘)𝑥(𝜏_𝑘)+𝑐_𝑘∈𝑋 (3.16)has a unique 𝑃𝐶-mild solution𝑥(𝑡)=𝑈(𝑡,𝜏_𝑘)𝑥_𝑘+^𝑡_𝜏𝑘𝑈(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃))𝑑𝜃. (3.17)
Combining all of solutions on [𝜏_𝑘,𝜏_𝑘+1] (𝑘=1,…,𝛿), one can obtain the 𝑃𝐶-mild solution of the Cauchy problem (3.9) given by𝑥(𝑡,𝑥)=𝒮(𝑡,0)𝑥+^𝑡₀𝒮(𝑡,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥))𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝒮(𝑡,𝜏⁺_𝑘)𝑐_𝑘. (3.18)

In order to study the periodic solutions of the system (3.8), we define Poincaré operator from 𝑋 to 𝑋 as following:𝑃(𝑥)=𝑥(𝑇₀,𝑥)=𝒮(𝑇₀,0)𝑥+^𝑇0₀𝒮(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥))𝑑𝜃+_{0≤𝜏𝑘<𝑇0}𝒮(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘, (3.19)where 𝑥(⋅,𝑥) denote the 𝑃𝐶-mild solution of the Cauchy problem (3.9) corresponding to the initial value 𝑥(0)=𝑥. We note that a fixed point of 𝑃 gives rise to a periodic solution.

Lemma 3.5. System (3.8) 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 (3.9) 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 (2), (3), and (4) of Lemma 2.3 and Assumptions A2, B, C, F(F3) to obtain𝑦(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+))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))𝑑𝜃+_{𝑇0≤𝜏𝑘+𝛿<𝑡+𝑇0}𝒮(𝑡+𝑇₀,𝜏⁺_𝑘+𝛿)𝑐_𝑘=𝒮(𝑡,0){𝒮(𝑇₀,0)𝑥₀+^𝑇0₀𝒮(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥₀))𝑑𝜃+_{0≤𝜏𝑘<𝑇0}𝒮(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘}+^𝑡₀𝒮(𝑡+𝑇₀,𝑠+𝑇₀)𝑓(𝑠+𝑇₀,𝑥(𝑠+𝑇₀,𝑥₀))𝑑𝑠+_{0≤𝜏𝑘<𝑡}𝒮(𝑡,𝜏⁺_𝑘)𝑐_𝑘=𝒮(𝑡,0)𝑦(0)+^𝑡₀𝒮(𝑡,𝑠)𝑓(𝑠,𝑦(𝑠,𝑦(0)))𝑑𝑠+_{0≤𝜏𝑘<𝑡}𝒮(𝑡,𝜏⁺_𝑘)𝑐_𝑘. (3.20)This implies that 𝑦(⋅,𝑦(0)) is a 𝑃𝐶-mild solution of Cauchy's problem (3.9) 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.

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

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

Proof. (1) Show that 𝑃 is a continuous operator on 𝑋.
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's problem (3.9) 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))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))𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝒮(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘. (3.21)Thus, we obtain‖𝑥(𝑡,𝑥)‖≤𝑀_𝑇0‖𝑥‖+𝑀_𝑇0𝑀_𝑓𝑇₀+𝑀_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖+𝑀_𝑇0^𝑡₀‖𝑥(𝜃,𝑥)‖𝑑𝜃,‖𝑥(𝑡,𝑦)‖≤𝑀_𝑇0‖𝑦‖+𝑀_𝑇0𝑀_𝑓𝑇₀+𝑀_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖+𝑀_𝑇0^𝑡₀‖𝑥(𝜃,𝑦)‖𝑑𝜃. (3.22)By Gronwall's inequality with impulse [5, Lemma 1.7.1], 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. (3.23)Let 𝜌=max{𝑀^∗₁,𝑀^∗₂}>0, then ‖𝑥(⋅,𝑥)‖, ‖𝑥(⋅,𝑦)‖≤𝜌 which imply that they are locally bounded. By Assumption F(F1), we obtain‖𝑥(𝑡,𝑥)−𝑥(𝑡,𝑦)‖≤‖𝒮(𝑡,0)‖‖𝑥−𝑦‖+^𝑡₀‖𝒮(𝑡,𝜃)‖𝑓(𝜃,𝑥(𝜃,𝑥))−𝑓(𝜃,𝑥(𝜃,𝑦))‖𝑑𝜃≤𝑀_𝑇0‖𝑥−𝑦‖+𝑀_𝑇0𝐿_𝑓(𝜌)^𝑡₀‖𝑥(𝜃,𝑥)−𝑥(𝜃,𝑦)‖𝑑𝜃. (3.24)By Gronwall's inequality with impulse [5, Lemma 1.7.1] again, one can verify that there exists constant 𝑀>0 such that‖𝑥(𝑡,𝑥)−𝑥(𝑡,𝑦)‖≤𝑀𝑀_𝑇0‖𝑥−𝑦‖≡𝐿‖𝑥−𝑦‖,∀𝑡∈[0,𝑇₀], (3.25)which implies that‖𝑃(𝑥)−𝑃(𝑦)‖=‖𝑥(𝑇₀,𝑥)−𝑥(𝑇₀,𝑦)‖≤𝐿‖𝑥−𝑦‖. (3.26)Hence, 𝑃 is a continuous operator on 𝑋.
(2) Verifies 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<𝑡≤𝑇₀, define𝐾_𝜀=𝑃_𝜀Γ=𝒮(𝑇₀,𝑇₀−𝜀){𝑥(𝑇₀−𝜀,𝑥)∣𝑥∈Γ}. (3.27)
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^−𝜀𝒮(𝑇₀−𝜀,𝜃)𝑓(𝜃,𝑥(𝜃,𝑥))𝑑𝜃+_{0≤𝜏𝑘<𝑇0−𝜀}𝒮(𝑇₀−𝜀,𝜏⁺_𝑘)𝑐_𝑘‖≤𝑀_𝑇0‖𝑥‖+𝑀_𝑇0𝑀_𝑓𝑇₀+^𝑇0₀‖𝑥(𝜃,𝑥)‖𝑑𝜃+𝑀_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖≤𝑀_𝑇0‖𝑥‖+𝑀_𝑇0𝑀_𝑓𝑇₀+𝑇₀𝜌+𝑀_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖. (3.28)This implies that the set {𝑥(𝑇₀−𝜀,𝑥)∣𝑥∈Γ} is bounded.
By (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))𝑑𝜃+_{0≤𝜏𝑘<𝑇0−𝜀}𝒮(𝑇₀,𝜏⁺_𝑘)𝑐_𝑘. (3.29)Thus, combined with (3.19), we have‖𝑃_𝜀(𝑥)−𝑃(𝑥)‖≤‖^𝑇0₀^−𝜀𝒮(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃))𝑑𝜃−^𝑇0₀𝒮(𝑇₀,𝜃)𝑓(𝜃,𝑥(𝜃))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))‖𝑑𝜃+𝑀_{𝑇0𝑇0−𝜀≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖≤2𝑀_𝑇0𝑀_𝑓(1+𝜌)𝜀+𝑀_{𝑇0𝑇0−𝜀≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖. (3.30)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 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.

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

(F2′): there exist constant 𝑁_𝑓>0 and 0<𝜆<1 such that‖𝑓(𝑡,𝑥)‖≤𝑁_𝑓(1+‖𝑥‖^𝜆)∀𝑥∈𝑋. (3.31)

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 𝑇₀-periodic 𝑃𝐶-mild solution on [0,+∞).

Proof. By (5) of Lemma 2.3, 𝒮(𝑇₀,0) is a compact operator on infinite dimensional space 𝑋. Thus, 𝒮(𝑇₀,0)≠𝛼𝐼, 𝛼∈ℝ. Then, there exists 𝛽>0 such that‖[𝜎𝒮(𝑇₀,0)−𝐼]𝑥‖≥𝛽‖𝑥‖for𝜎∈[0,1]. (3.32)In fact, define Π_𝜎=𝐼−𝜎𝒮(𝑇₀,0), 𝜎∈[0,1], andΠ_𝜎∶[0,1]→£_𝑏(𝑋),ℎ(𝜎)=‖Π_𝜎‖∶[0,1]→ℝ⁺. (3.33)It is obvious that ℎ∈𝐶([0,1];ℝ⁺). Thus, there exist 𝜎_∗∈[0,1] and 𝛽>0 such thatℎ(𝜎_∗)=min{ℎ(𝜎)∣𝜎∈[0,1]}≥𝛽>0. (3.34)If not, there exits 𝜎∈[0,1] such that ℎ(𝜎)=0. We can assert that 𝜎≠0 unless ℎ(𝜎)=1. Thus, for 𝜎∈(0,1]𝒮(𝑇₀1,0)=𝜎1𝐼where𝜎≥1, (3.35)which is a contradiction with 𝒮(𝑇₀,0)≠𝛼𝐼, 𝛼∈ℝ.
By Theorem 3.4, for fixed 𝑥∈𝑋, the Cauchy problem (3.9) corresponding to the initial value 𝑥(0)=𝑥 has the 𝑃𝐶-mild solution 𝑥(⋅,𝑥). By Lemma 3.6, the operator 𝑃 defined by (3.19), is compact.
According to Leray-Schauder fixed point theory, it suffices to show that the set {𝑥∈𝑋∣𝑥=𝜎𝑃𝑥,𝜎∈[0,1]} is a bounded subset of 𝑋. In fact, let 𝑥∈{𝑥∈𝑋∣𝑥=𝜎𝑃𝑥,𝜎∈[0,1]}, we have𝛽‖𝑥‖≤‖[𝜎𝒮(𝑇₀,0)−𝐼]𝑥‖=𝜎^𝑇0₀‖𝒮(𝑇₀,𝜃)‖‖𝑓(𝜃,𝑥(𝜃,𝑥))‖𝑑𝜃+𝜎_{0≤𝜏𝑘<𝑇0}‖𝒮(𝑇₀,𝜏⁺_𝑘)‖‖𝑐_𝑘‖. (3.36)
By Assumption F(F2′),‖𝜎𝑥‖≤𝛽^𝑇0₀‖𝒮(𝑇₀,𝜃)‖‖𝑓(𝜃,𝑥(𝜃,𝜎𝑥))‖𝑑𝜃+𝛽_{0≤𝜏𝑘<𝑇0}‖𝒮(𝑇₀,𝜏⁺_𝑘)‖‖𝑐_𝑘‖≤𝜎𝛽𝑀_𝑇0(𝑁_𝑓𝑇₀+𝑁_𝑓^𝑇0₀‖𝑥(𝜃,𝑥)‖^𝜆𝑑𝜃+_{0≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖). (3.37)
For 𝑡∈[0,𝑇₀], we obtain‖𝑥(𝑡,𝑥)‖≤𝑀_𝑇0‖𝑥‖+𝑀_𝑇0𝑁_𝑓𝑇₀+𝑀_𝑇0𝑁_𝑓^𝑡₀‖𝑥(𝜃,𝑥)‖^𝜆𝑑𝜃+𝑀_{𝑇00≤𝜏𝑘<𝑡}‖𝑐_𝑘‖≤𝜎𝛽𝑀²_𝑇0(𝑁_𝑓𝑇₀+𝑁_𝑓^𝑇0₀‖𝑥(𝜃,𝑥)‖^𝜆𝑑𝜃+_{0≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖)+𝑀_𝑇0𝑁_𝑓𝑇₀+𝑀_𝑇0𝑁_𝑓^𝑡₀‖𝑥(𝜃,𝑥)‖^𝜆𝑑𝜃+𝑀_{𝑇00≤𝜏𝑘<𝑡}‖𝑐_𝑘‖≤𝑀_𝑇0𝑁_𝑓𝑇₀+𝜎𝛽𝑀²_𝑇0𝑁_𝑓𝑇₀+𝜎𝛽𝑀²_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖+𝑀_{𝑇00≤𝜏𝑘<𝑇0}‖𝑐_𝑘‖+𝑀_𝑇0𝑁_𝑓^𝑡₀‖𝑥(𝜃,𝑥)‖^𝜆𝜎𝑑𝜃+𝛽𝑀²_𝑇0𝑁_𝑓^𝑇0₀‖𝑥(𝜃,𝑥)‖^𝜆𝑑𝜃. (3.38)By Lemma 3.1, there exists 𝑀^∗>0 such that‖𝑥(𝑡,𝑥)‖≤𝑀^∗for𝑡∈[0,𝑇₀]. (3.39)This implies that ‖𝑥(0,𝑥)‖=‖𝑥‖≤𝑀^∗ for all 𝑥∈{𝑥∈𝑋∣𝑥=𝜎𝑃𝑥,𝜎∈[0,1]}.
Thus, by Leray-Schauder fixed point theory, there exits 𝑥₀∈𝑋 such that 𝑃𝑥₀=𝑥₀. By Lemma 3.5, we know that the 𝑃𝐶-mild solution 𝑥(⋅,𝑥₀) of Cauchy's problem (3.9) corresponding to the initial value 𝑥(0)=𝑥₀, is just 𝑇₀-periodic. Therefore 𝑥(⋅,𝑥₀) is a 𝑇₀-periodic 𝑃𝐶-mild solution of system (3.8).

4. Application

In this section, an example is given to illustrate our theory. Consider the following problem:𝜕𝜕𝑡𝑥(𝑡,𝑦)=sin𝑡Δ𝑥(𝑡,𝑦)+3𝑥^2/31(𝑡,𝑦)+2+sin(𝑡,𝑦),𝑦∈Ω,𝑡∈(0,2𝜋]⧵{23𝜋,𝜋,2𝜋},Δ𝑥(𝜏_𝑖,𝑦)=𝑥(𝜏_𝑖+0,𝑦)−𝑥(𝜏_𝑖=−0,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), (4.1)where Ω⊂𝑅³ is bounded domain and 𝜕Ω∈𝐶³.

Define 𝑋=𝐿²(Ω), 𝐷(𝐴)=𝐻²(Ω)⋂𝐻¹₀(Ω), and 𝐴(𝑡)𝑥=−sin𝑡(𝜕²𝑥/𝜕𝑦²₁+𝜕²𝑥/𝜕𝑦²₂+𝜕²𝑥/𝜕𝑦²₃) for 𝑥∈𝐷(𝐴), which satisfies Assumptions A1, A2, and A3. Define 𝑥(⋅)(𝑦)=𝑥(⋅,𝑦), sin(⋅)(𝑦)=sin(⋅,𝑦), 𝑓(⋅,𝑥(⋅))(𝑦)=3𝑥^2/3(⋅,𝑦)+2+sin(⋅,𝑦) and𝐵_𝑖=0.05𝐼,𝑖=3𝑚−2,−0.05𝐼,𝑖=3𝑚−1,𝜏_𝑖=𝑖𝜋2,𝑖,𝑚∈ℕ.0.05𝐼,𝑖=3𝑚. (4.2)

It is obvious that 𝜏_𝑖+3=𝜏_𝑖+2𝜋, 𝐵_𝑖+3=𝐵_𝑖∈£_𝑏(𝐿²(Ω)) which satisfy the Assumption B.

For any 𝑥,𝑧∈𝑋 satisfying 1≤‖𝑥‖,‖𝑧‖≤𝜌,‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑧)‖≤‖3𝑥^2/3+2−3𝑧^2/3≤+2‖(‖𝑥‖^1/3+‖𝑧‖^1/3)‖𝑥^1/3−𝑧^1/3‖3‖𝑥‖^2/3+2+3‖𝑧‖^2/3≤𝜌+2_1/3√2‖𝑥−𝑧‖. (4.3)(1)Meanwhile,‖𝑓(𝑡,𝑥)‖≤‖3𝑥^2/3+2‖+‖sin𝑡‖≤3‖𝑥‖^2/3+2+1≤3(1+‖𝑥‖^2/3),𝑓(⋅+2𝜋,𝑥)=3𝑥^2/3+2+sin(⋅+2𝜋)=3𝑥^2/3+2+sin(⋅)=𝑓(⋅,𝑥). (4.4)These imply that Assumptions F(F1), F(F2′), and F(F3) hold. It comes from𝒮(2𝜋,0)=𝑈(2𝜋,𝜏₃)(𝐼+𝐵₃)𝑈(𝜏₃,𝜏₂)(𝐼+𝐵₂)𝑈(𝜏₂,𝜏₁)(𝐼+𝐵₁)𝑈(𝜏₁,0)=(0.95²⋅1.05)𝑈(2𝜋,0) (4.5)that 𝒮(2𝜋,0)≠𝛼𝐼, 𝛼>1. In fact, if (0.95²⋅1.05)𝑇(2𝜋)=𝛼𝐼, then 𝑈(2𝜋,0)=(𝛼/(0.95²⋅1.05))𝐼 cannot be compact in 𝐿²(Ω), which is a contradiction with 𝑈(𝑡,𝜃)which is compact operator for 0≤𝜃<𝑡≤𝑇₀. Thus problem (4.1) can be rewritten aṡ1𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥),𝑡∈(0,2𝜋]⧵{23𝜋,𝜋,2𝑖𝜋},Δ𝑥(2𝜋)=𝐵_𝑖𝑖𝑥(2𝜋),𝑖=1,2,3,𝑥(0)=𝑥(2𝜋). (4.6)It 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 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}. (4.7)

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.15‐2).