Existence and Global Asymptotical Stability of Periodic Solution for the 𝑇-Periodic Logistic System with Time-Varying Generating Operators and 𝑇₀-Periodic Impulsive Perturbations on Banach Spaces

Abstract

This paper studies the existence and global asymptotical stability of periodic PC-mild solution for the 𝑇-periodic Logistic system with time-varying generating operators and 𝑇₀-periodic impulsive perturbations on Banach spaces. Two sufficient conditions that guarantee the exponential stability of the impulsive evolution operator corresponding to homogenous well-posed 𝑇-periodic system with time-varying generating operators and 𝑇₀-periodic impulsive perturbations are given. It is shown that the system have a unique periodic PC-mild solution which is globally asymptotically stable when 𝑇 and 𝑇₀ are rational dependent and its period must be 𝑛𝑇₀ for some 𝑛∈𝑁. At last, an example is given for demonstration.

1. Introduction

For modeling the dynamics of an ecological system, Cushing [1] pointed out that it is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite naturally exposed (e.g., those due to seasonal effects of wheatear, food supply, mating habits, etc.). The periodic solution theory of dynamic equations has been developed over the last decades. We refer the readers to [2–6] for infinite dimensional cases, and to [1, 7–9] for finite dimensional cases. Especially, there are many results of periodic solutions (such as existence and stability) for impulsive periodic systems on finite dimensional spaces (see [7–9]). There are also some relative results of periodic solutions for periodic systems with time-varying generating operators on infinite dimensional spaces (see [5, 6, 10–12]).

On the other hand, the ecological system is often deeply perturbed by human exploit activities such as planting and harvesting. Usually, these activities are considered continuously by adding some items to [13–15], whereas this is not how things stand. It is often the case that planting and harvesting of the species are seasonal or occur in regular pulses. These perturbations may also naturally be periodic, for example, a fisherman may go fishing at the same time once a day or once a week. Systems with short-term perturbations are often naturally described by impulsive differential equations, which are found in almost every domain of applied sciences. For the basic theory on impulsive differential equations on finite dimensional spaces, the reader can refer to Lakshmikantham's book and Yang's book (see [9, 16]). For the basic theory on impulsive differential equations on infinite dimensional spaces, the reader can refer to Ahmed's paper, Liu's paper, and Xiang's papers (see [17–19]).

In this paper, we will study the following generalized Logistic system with impulsive perturbations:𝜕𝜕𝑡𝑥(𝑡,𝑦)=𝐴(𝑦,𝑡,𝐷)𝑥(𝑡,𝑦)+𝑓(𝑡,𝑦),𝑦∈Ω,𝑡>0,𝑡≠𝜏_𝑘,𝑘∈𝑍⁺₀,𝑥(𝑡,𝑦)=0,𝑦∈𝜕Ω,𝑡>0,Δ𝑥(𝑡,𝑦)=𝐵_𝑘𝑥(𝑡,𝑦)+𝑐_𝑘,𝑦∈Ω,𝑡=𝜋_𝑘,𝑘∈𝑍⁺₀, (1.1) where 𝑥(𝑡,𝑦) denotes the population number of isolated species at time 𝑡 and location 𝑦, Ω is an open-bounded domain in 𝑅², and 𝜕Ω is smooth enough. The operator𝐴(𝑦,𝑡,𝐷)≡_{1≤𝑖,𝑗≤2}𝑎_𝑖𝑗(𝑦,𝑡)𝐷_𝑖𝐷_𝑗+_1≤𝑖≤2𝑏_𝑖(𝑦,𝑡)𝐷_𝑖+𝑐(𝑦,𝑡),(𝑦,𝑡)∈(Ω×[0,+∞)), (1.2)where all the coefficients are smooth functions enough and 𝐷_𝑖 denotes the spatial derivative with respect to 𝑦_𝑖. 𝑓 is related to the periodic change of the resources maintaining the evolution of the population. Time sequence 0=𝜏₀<𝜏₁<⋯<𝜏_𝑘⋯ and 𝜏_𝑘→∞ as 𝑘→∞, Δ𝑥(𝜏_𝑘,𝑦)=𝑥(𝜏⁺_𝑘,𝑦)−𝑥(𝜏_𝑘,𝑦)=𝐵_𝑘𝑥(𝜏_𝑘,𝑦)+𝑐_𝑘 denote mutation of the isolate species at time 𝜏_𝑘.

Suppose the first equation of (1.1) is 𝑇-periodic and the third equation of (1.1) is 𝑇₀-periodic, that is, 𝐴(𝑦,𝑡+𝑇,𝐷)=𝐴(𝑦,𝑡,𝐷), 𝑓(𝑡+𝑇,𝑦)=𝑓(𝑡,𝑦), 𝑡≥0, and 𝑇₀ is the least-positive constant such that there are 𝛿  𝜏_𝑘s in the interval (0,𝑇₀) and 𝜏_𝑘+𝛿=𝜏_𝑘+𝑇₀,𝐵_𝑘+𝛿=𝐵_𝑘,𝑐_𝑘+𝛿=𝑐_𝑘, 𝑘∈𝑍⁺₀.

The first equation of (1.1) describes the variation of the population number 𝑥 of the species in 𝑇-periodically changing environment. The second equation of (1.1) shows that the species is isolated. The third equation of (1.1) reflects the possibility of impulsive effects on the population. As we assumed, these impulsive perturbations are 𝑇₀-periodic. Naturally, this period is distinct from 𝑇, the period of the change of environment. Even when we want to carry out the perturbations according to the period 𝑇, we cannot do it since we do not know 𝑇 exactly. Thus, it is interesting how the dynamics of the first equation of (1.1) are affected by the periodic changing of environment together with the periodic impulsive perturbations.

Define 𝐴(𝑡)(𝑦)=𝐴(𝑦,𝑡,𝐷), 𝑥(⋅)(𝑦)=𝑥(⋅,𝑦), 𝑓(⋅)(𝑦)=𝑓(⋅,𝑦), (1.1) can be abstracted impulsive periodic evolution equations of the form𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑡≠𝜏_𝑘,(𝑇-periodic)Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡)+𝑐_𝑘,𝑡=𝜏_𝑘,(𝑇₀-periodic) (1.3)on the Banach space 𝑋. It is obvious that the investigation of (1.3) cannot only be used to discuss (1.1) but also provide a foundation for study of the general impulsive periodic systems.

Assume that 𝑇-periodic evolution equation of the form 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑥∈𝑋,𝑡>0, is well posed, that is, there exists a 𝑇-periodic evolutionary process {𝑈(𝑡,𝜃),𝑡≥𝜃≥0} which satisfies, among other things, the conditions (1)–(5) of Definition 2.1 which follows. Once the evolution equations 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑥∈𝑋,𝑡>0, and 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑥∈𝑋,𝑡>0, are well posed, the asymptotic behavior of solutions at infinity, such as stability and periodicity, is of particular interest, which has been a central topic discussed for the past decade. We refer the reader to the books [20, 21], and the surveys [22], and the references therein for more complete information on the subject. Because (1.3) can be used to describe more social and natural phenomena, it is naturally needed to study the stability and periodicity of solutions for (1.3).

The aim of this paper is to study the existence and global asymptotical stability of periodic PC-mild solution of (1.3) without compactness condition. We will show that (i) if 𝛾=𝑇/𝑇₀ is rational, that is, 𝑇 and 𝑇₀ are rational dependent, then (1.3) may have a unique periodic PC-mild solution which is globally asymptotically stable and (ii) if 𝑥(𝑡,𝑥) is a periodic PC-mild solution of (1.3) with 𝑥(0)=𝑥, then its period must be 𝑛𝑇₀ for some 𝑛∈𝑁.

This paper is organized as follows. In Section 2, the properties of the impulsive evolution operator are collected, two sufficient conditions that guarantee the exponential stability of the impulsive evolution operator are given. In Section 3, the existence of periodic PC-mild solution which is globally asymptotically stable for (1.3) is obtained. At last, the abstract results are applied to a special case of (1.1). This work not only provides the theory basis for managing some single species but is also fundamental for further discussion on the existence and stability of periodic solution for nonlinear impulsive periodic system with time-varying generating operators on infinite dimensional spaces.

2. Exponential Stability of Impulsive Evolution Operator

Let 𝑋 be a Banach space, £(𝑋) denotes the space of linear operators on 𝑋; £_𝑏(𝑋) denotes the space of bounded linear operators on 𝑋. Let £_𝑏(𝑋) be the Banach space with the usual supremum norm. Denote 𝛾 is rational, let 𝛾=𝑝/𝑞, 𝑝,𝑞∈𝑁, 𝑝, 𝑞 are relatively prime. Denote ^0𝑥0𝑇^02𝑑𝑐=𝑝𝑇₀=𝑞𝑇, ^0𝑥0𝐷^02𝑑𝑐={𝜏₁,…,𝜏_𝑝𝛿}⊂[0,𝑇^{0𝑥002𝑑𝑐}] and define PC([0,𝑇^{0𝑥002𝑑𝑐}];𝑋)≡{𝑥∶[0,𝑇^{0𝑥002𝑑𝑐}]→𝑋∣𝑥 is continuous at 𝑡∈[0,𝑇^{0𝑥002𝑑𝑐}]⧵𝐷^{0𝑥002𝑑𝑐}, 𝑥 is continuous from left and has right-hand limits at 𝑡∈𝐷^{0𝑥002𝑑𝑐}} , and PC¹([0,𝑇^{0𝑥002𝑑𝑐}];𝑋)≡{𝑥∈PC([0,𝑇^{0𝑥002𝑑𝑐}];𝑋)∣𝑥̇∈PC([0,𝑇^{0𝑥002𝑑𝑐}];𝑋)}. Set‖𝑥‖_PC=max{sup_{𝑡∈[0,0𝑥002𝑑𝑐𝑇]}‖𝑥(𝑡+0)‖,sup_{𝑡∈[0,0𝑥002𝑑𝑐𝑇]}‖𝑥(𝑡−0)‖},‖𝑥‖_PC¹=‖𝑥‖_PC+‖𝑥̇‖_PC. (2.1)It can be seen that endowed with the norm ‖⋅‖_PC (‖⋅‖_PC¹), PC([0,𝑇^{0𝑥002𝑑𝑐}];𝑋) (PC¹([0,𝑇^{0𝑥002𝑑𝑐}];𝑋)) is a Banach space.

In order to investigate periodic solution, introduce the following two spaces:𝐿¹_𝑇([0,+∞);𝑋)≡{𝑓∶[0,+∞)→𝑋∣𝑓(𝑡)=𝑓(𝑡+𝑇),^𝑇₀‖𝑓(𝑡)‖𝑑𝑡<∞},PC_{0𝑥002𝑑𝑐𝑇}([0,+∞);𝑋)≡{𝑥∈PC([0,+∞);𝑋)∣𝑥(𝑡)=𝑥(𝑡+𝑇^{0𝑥002𝑑𝑐}),𝑡≥0}. (2.2)

Set‖𝑓‖_𝐿¹_𝑇=^𝑇₀‖𝑓(𝑡)‖𝑑𝑡<∞,‖𝑥‖_{PC𝑇0𝑥002𝑑𝑐}=max{sup_{𝑡∈[0,0𝑥002𝑑𝑐𝑇]}‖𝑥(𝑡+0)‖,sup_{𝑡∈[0,0𝑥002𝑑𝑐𝑇]}‖𝑥(𝑡−0)‖}. (2.3)It can be seen that endowed with the norm ‖⋅‖_𝐿¹_𝑇 (‖⋅‖_{PC𝑇0𝑥002𝑑𝑐}), 𝐿¹_𝑇([0,+∞);𝑋) (PC_{0𝑥002𝑑𝑐𝑇}([0,+∞);𝑋)) is a Banach space.

The following notation will be used throughout the paper, we recall these concepts in the following definitions.

Definition 2.1. A family of bounded linear operators {𝑈(𝑡,𝜃),𝑡≥𝜃≥0} from a Banach space 𝑋 to itself is called strongly continuous evolutionary process if the following conditions (1)–(4) are satisfied:(1)𝑈(𝑡,𝑡)=𝐼, 𝑡≥0,(2)𝑈(𝑡,𝑟)𝑈(𝑟,𝜃)=𝑈(𝑡,𝜃),𝑡≥𝑟≥𝜃≥0,(3)the map (𝑡,𝜃)→𝑈(𝑡,𝜃)𝑥 is continuous for every fixed 𝑥∈𝑋,(4)‖𝑈(𝑡,𝜃)‖≤𝑀𝑒^{𝜔(𝑡−𝜃)} for some 𝑀≥1, 𝜔∈𝑅 independent of 𝑡≥𝜃≥0,
further, if(5)𝑈(𝑡+𝑇,𝜃+𝑇)=𝑈(𝑡,𝜃) for all 𝑡≥𝜃≥0.
Then the strongly continuous evolutionary process {𝑈(𝑡,𝜃),𝑡≥𝜃≥0} is called 𝑇-periodic.

Definition 2.2. The evolutionary process {𝑈(𝑡,𝜃),𝑡≥𝜃≥0} is called exponentially bounded if𝜛(𝑈)=inf{𝜔∈𝑅∶∃𝑀≥1with‖𝑈(𝑡,𝜃)‖≤𝑀𝑒^{𝜔(𝑡−𝜃)}for𝑡≥𝜃≥0}<∞. (2.4)

Definition 2.3. The linear equation𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑥∈𝑋,𝑡≥0 (2.5)is said to be well posed if there exists an evolutionary process {𝑈(𝑡,𝜃),𝑡≥𝜃≥0}, which satisfies the conditions (1)–(4) in Definition 2.1, such that for every 𝜃≥0 and 𝑥∈𝐷(𝐴(𝜃)), the function 𝑥(𝑡)=𝑈(𝑡,𝜃)𝑥 is the uniquely determined classical solution of (2.5) satisfying 𝑥(𝜃)=𝑥.

Definition 2.4. The function 𝑥∈𝑋 is said to be a mild solution to the linear equation (2.5) if and only if𝑥(𝑡)=𝑈(𝑡,𝜃)𝑥(𝜃),𝑡≥𝜃≥0. (2.6)
We introduce the following assumption (H1):(H1.1)𝐴(𝑡)∶𝐷(𝐴(𝑡))→𝑋, 𝑡≥0, is a family of linear unbounded operators on 𝑋, 𝐴(𝑡) is 𝑇-periodic, that is, for 𝑡≥0, 𝐴(𝑡+𝑇)=𝐴(𝑡);(H1.2) the linear equation 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑥∈𝑋,𝑡≥0, is well posed;(H1.3)there exists 𝛿∈𝑁 such that 𝜏_𝑘+𝛿=𝜏_𝑘+𝑇₀;(H1.4)for each 𝑘∈𝑍⁺₀, 𝐵_𝑘∈£_𝑏(𝑋) and 𝐵_𝑘+𝛿=𝐵_𝑘.
By (H1.1) and (H1.2), {𝑈(𝑡,𝜃),𝑡≥𝜃≥0} is 𝑇-periodic strongly continuous evolutionary process.
Under (H1), consider𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑡≠𝜏_𝑘,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡),𝑡=𝜏_𝑘, (2.7)and Cauchy problem𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑡∈[0,𝑇^{0𝑥002𝑑𝑐}]⧵𝐷^{0𝑥002𝑑𝑐},Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘),𝑘=1,2,…,𝑝𝛿,𝑥(0)=𝑥. (2.8)
For every 𝑥∈𝑋, 𝐷(𝐴(𝑡)), 𝑡≥0, is an invariant subspace of 𝐵_𝑘; and step by step, one can verify that the Cauchy problem (2.7) has a unique classical solution 𝑥∈PC¹([0,𝑇^{0𝑥002𝑑𝑐}];𝑋) represented by 𝑥(𝑡)=Φ(𝑡,0)𝑥, whereΦ(⋅,⋅)∶Δ={(𝑡,𝜃)∈[0,𝑇^{0𝑥002𝑑𝑐}]×[0,𝑇^{0𝑥002𝑑𝑐}]∣0≤𝜃≤𝑡≤𝑇^{0𝑥002𝑑𝑐}}→£(𝑋), (2.9)given byΦ(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_𝑖,𝜃),𝜏_𝑖−1≤𝜃<𝜏_𝑖≤⋯<𝜏_𝑘<𝑡≤𝜏_𝑘+1. (2.10)The operator {Φ(𝑡,𝜃),(𝑡,𝜃)∈Δ} is called impulsive evolution operator associated with {𝑈(𝑡,𝜃),(𝑡,𝜃)∈Δ} and {𝐵_𝑘;𝜏_𝑘}^∞_𝑘=1.
The following lemma on the properties of the impulsive evolution operator {Φ(𝑡,𝜃),(𝑡,𝜃)∈Δ} associated with {𝑈(𝑡,𝜃),(𝑡,𝜃)∈Δ} and {𝐵_𝑘;𝜏_𝑘}^∞_𝑘=1 are widely used in this paper.

Lemma 2.5. Assumption (H1) holds. The impulsive evolution operator {Φ(𝑡,𝜃),(𝑡,𝜃)∈Δ} has the following properties:(1) for 0≤𝜃≤𝑡≤𝑇^{0𝑥002𝑑𝑐}, Φ(𝑡,𝜃)∈£_𝑏(𝑋), there exists 𝑀_{0𝑥002𝑑𝑐𝑇}>0 such that sup_{0≤𝜃≤𝑡≤0𝑥002𝑑𝑐𝑇}‖Φ(𝑡,𝜃)‖≤𝑀_{0𝑥002𝑑𝑐𝑇},(2) for 0≤𝜃<𝑟<𝑡≤𝑇^{0𝑥002𝑑𝑐}, 𝑟≠𝜏_𝑘, Φ(𝑡,𝜃)=Φ(𝑡,𝑟)Φ(𝑟,𝜃),(3)for 0≤𝜃≤𝑡≤𝑇^{0𝑥002𝑑𝑐}, 𝑛∈𝑍⁺, Φ(𝑡+𝑛𝑇^{0𝑥002𝑑𝑐},𝜃+𝑛𝑇^{0𝑥002𝑑𝑐})=Φ(𝑡,𝜃),(4)for 0≤𝜃≤𝑡≤𝑇^{0𝑥002𝑑𝑐}, 𝑛∈𝑍⁺, Φ(𝑡+𝑛𝑇^{0𝑥002𝑑𝑐},0)=Φ(𝑡,0)[Φ(𝑇^{0𝑥002𝑑𝑐},0)]^𝑛,(5) for 0≤𝜃<𝑡, there exits 𝑀≥1, 𝜔∈𝑅 such that‖Φ(𝑡,𝜃)‖≤𝑀exp{𝜔(𝑡−𝜃)+_{𝜃≤𝜏𝑛<𝑡}ln(𝑀‖𝐼+𝐵_𝑛‖)}. (2.11)

Proof. For (1)–(4), the reader can see [23, Lemma 2.7]. We only need to verify (5). Without loss of generality, for 𝜏_𝑖−1≤𝜃<𝜏_𝑖≤⋯<𝜏_𝑘<𝑡≤𝜏_𝑘+1,‖Φ(𝑡,𝜃)‖=‖𝑈(𝑡,𝜏⁺_𝑘)‖‖𝐼+𝐵_𝑘‖‖𝑈(𝜏_𝑘,𝜏⁺_𝑘−1)‖⋯‖𝐼+𝐵_𝑖‖‖𝑈(𝜏_𝑖,𝜃)‖≤𝑀𝑒^{𝜔(𝑡−𝜏+𝑘)}[^𝑘_𝑛=𝑖+1‖𝐼+𝐵_𝑛‖𝑀𝑒^{(𝜏𝑛−𝜏+𝑛−1)}]‖𝐼+𝐵_𝑖‖𝑀𝑒^{𝜔(𝜏𝑖−𝜃)}≤𝑀exp{𝜔(𝑡−𝜃)+_{𝜃≤𝜏𝑛<𝑡}ln(𝑀‖𝐼+𝐵_𝑛‖)}. (2.12)This completes the proof.

In order to study the asymptotical properties of periodic solution, it is necessary to discuss the exponential stability of the impulsive evolution operator {Φ(𝑡,𝜃),𝑡≥𝜃≥0}. We first give the definition of exponentially stable for {Φ(𝑡,𝜃),𝑡≥𝜃≥0}.

Definition 2.6. {Φ(𝑡,𝜃),𝑡≥𝜃≥0} is called exponentially stable if there exist 𝐾≥0 and 𝜈>0 such that‖Φ(𝑡,𝜃)‖≤𝐾𝑒^{−𝜈(𝑡−𝜃)},𝑡>𝜃≥0. (2.13)
Assumption (H2): {𝑈(𝑡,𝜃),𝑡≥𝜃≥0} is exponentially stable, that is, there exist 𝐾₀>0 and 𝜈₀>0 such that‖𝑈(𝑡,𝜃)‖≤𝐾₀𝑒^{−𝜈0(𝑡−𝜃)},𝑡>𝜃≥0. (2.14)

Next, two sufficient conditions that guarantee the impulsive evolution operator {Φ(𝑡,𝜃),𝑡≥𝜃≥0} with rational 𝛾 is exponentially stable are given.

Lemma 2.7. Suppose 𝛾 is rational and (H1) and (H2) hold. There exists 0<𝜆<𝜈₀ such that^𝛿_𝑘=1(𝐾₀‖𝐼+𝐵_𝑘‖)^𝛾𝑒^−𝜆𝑇<1. (2.15)
Then {Φ(𝑡,𝜃),𝑡≥𝜃≥0} is exponentially stable.

Proof. Since 𝛾 is rational, let 𝛾=𝑝/𝑞, 𝑝,𝑞∈𝑁 and 𝑝, 𝑞 are relatively prime. Let ^0𝑥0𝑇^02𝑑𝑐=𝑝𝑇₀ (=𝑞𝑇), then (2.7) is ^0𝑥0𝑇^02𝑑𝑐-periodic. Without loss of generality, for 𝜏_𝑖−1≤𝜃<𝜏_𝑖≤⋯<𝜏_𝑘<𝑡≤𝜏_𝑘+1, by (5) of Lemma 2.5, we have‖Φ(𝑡,𝜃)‖≤𝐾₀𝑒^{−(𝜈0−𝜆)(𝑡−𝜃)}(_{𝜃≤𝜏𝑘<𝑡}(𝐾₀‖𝐼+𝐵_𝑘‖)𝑒^{−𝜆(𝑡−𝜃)}). (2.16)Suppose 𝑡∈(𝑛𝑇^{0𝑥002𝑑𝑐},(𝑛+1)𝑇^{0𝑥002𝑑𝑐}] and let 𝑏=max_{𝑠∈[0,0𝑥002𝑑𝑐𝑇]}Π_{0<𝜏𝑘<𝑠}{𝐾₀‖𝐼+𝐵_𝑘‖}. Then_{𝜃≤𝜏𝑘<𝑡}(𝐾₀‖𝐼+𝐵_𝑘‖)𝑒^{−𝜆(𝑡−𝜃)}≤_{0≤𝜏𝑘<𝑛0𝑥002𝑑𝑐𝑇}(𝐾₀‖𝐼+𝐵_𝑘‖)𝑒^{−𝜆𝑛0𝑥002𝑑𝑐𝑇}⋅_{𝑛0𝑥002𝑑𝑐𝑇≤𝜏𝑘<𝑡}(𝐾₀‖𝐼+𝐵_𝑘‖)𝑒^{−𝜆(𝑡−𝑛0𝑥002𝑑𝑐𝑇)}𝑒^𝜆𝜃≤_{0≤𝜏𝑘<𝑛𝑝𝑇0}(𝐾₀‖𝐼+𝐵_𝑘‖)𝑒^{−𝜆𝑛𝑞𝑇}𝑏𝑒^𝜆𝜃≤^𝛿_𝑘=1(𝐾₀‖𝐼+𝐵_𝑘‖)^𝑛𝑝𝑒^{−𝜆𝑛𝑞𝑇}𝑏𝑒^𝜆𝜃=[^𝛿_𝑘=1(𝐾₀‖𝐼+𝐵_𝑘‖)^𝛾𝑒^−𝜆𝑇]^𝑛𝑞𝑏𝑒^𝜆𝜃<𝑏𝑒^𝜆𝜃. (2.17)Let 𝐾=𝐾₀𝑏𝑒^𝜆𝜃>0 and 𝜈=𝜈₀−𝜆>0, then we obtain ‖Φ(𝑡,𝜃)‖≤𝐾𝑒^{−𝜈(𝑡−𝜃)},𝑡>𝜃≥0.

Remark 2.8. An exponentially bounded evolutionary process {𝑈(𝑡,𝜃),𝑡≥𝜃≥0} is exponential stability if and only if for some 1≤𝑝<∞ and all 𝑥∈𝑋 and 𝑡>𝜃≥0, there is a constant 𝜅>0 such that^∞_𝜃‖𝑈(𝑡,𝜃)𝑥‖^𝑝𝑑𝑡≤𝜅^𝑝‖𝑥‖^𝑝. (2.18)

Lemma 2.9. Assumption (H1) holds. Suppose0<𝜇₁=inf_{𝑘=1,2,…,𝛿}(𝜏_𝑘−𝜏_𝑘−1)≤sup_{𝑘=1,2,…,𝛿}(𝜏_𝑘−𝜏_𝑘−1)=𝜇₂<∞. (2.19)If there exists 𝛼>0 such that1𝜔+𝜇ln(𝑀‖𝐼+𝐵_𝑘‖)≤−𝛼<0,𝑘=1,2,…,𝛿, (2.20)where𝜇𝜇=¹𝜇,𝛼+𝜔<0,₂,𝛼+𝜔≥0, (2.21)then {Φ(𝑡,𝜃),𝑡>𝜃≥0} is exponentially stable.

Proof. It comes from (2.20) thatln(𝑀‖𝐼+𝐵_𝑘‖)≤−𝜇(𝛼+𝜔)<0,𝑘=1,2,…,𝛿. (2.22)Further,_{𝜃≤𝜏𝑘<𝑡}ln(𝑀‖𝐼+𝐵_𝑘‖)≤−_{𝜃≤𝜏𝑘<𝑡}𝜇(𝛼+𝜔)=−𝜇(𝛼+𝜔)𝑁(𝜃,𝑡), (2.23)where 𝑁(𝜃,𝑡) is denoted the number of impulsive points in [𝜃,𝑡).
For 𝜏_𝑖−1≤𝜃<𝜏_𝑖≤⋯<𝜏_𝑘<𝑡≤𝜏_𝑘+1, by (2.19), we obtain the following two inequalities:𝑡−𝜃≥(𝜏_𝑘−𝜏_𝑘−1)+⋯+(𝜏_𝑖+1−𝜏_𝑖)≥(𝑁(𝜃,𝑡)−1)𝜇₁,𝑡−𝜃≤(𝜏_𝑘+1−𝜏_𝑘)+(𝜏_𝑘−𝜏_𝑘−1)+⋯+(𝜏_𝑖+1−𝜏_𝑖)+(𝜏_𝑖−𝜏_𝑖−1)≤(𝑁(𝜃,𝑡)+1)𝜇₂. (2.24)This implies that𝜇₁(𝑁(𝜃,𝑡)−1)≤𝑡−𝜃≤𝜇₂(𝑁(𝜃,𝑡)+1), (2.25)that is,1𝜇₂1(𝑡−𝜃)−1≤𝑁(𝜃,𝑡)≤𝜇₁(𝑡−𝜃)+1. (2.26)Then−𝜇(𝛼+𝜔)𝑁(𝜃,𝑡)≤−𝜇₁1(𝛼+𝜔)[𝜇₁(𝑡−𝜃)+1]=−(𝛼+𝜔)(𝑡−𝜃)−𝜇₁(𝛼+𝜔),𝛼+𝜔<0.−𝜇₂1(𝛼+𝜔)[𝜇₂(𝑡−𝜃)−1]=−(𝛼+𝜔)(𝑡−𝜃)+𝜇₂(𝛼+𝜔),𝛼+𝜔≥0.=−(𝛼+𝜔)(𝑡−𝜃)+𝜇|𝛼+𝜔|. (2.27)Thus, we obtain𝜔(𝑡−𝜃)+_{𝜃≤𝜏𝑘<𝑡}ln(𝑀‖𝐼+𝐵_𝑘‖)≤−𝛼(𝑡−𝜃)+𝜇|𝛼+𝜔|. (2.28)By (5) of Lemma 2.5, let 𝐾=𝑀𝑒^{𝜇|𝛼+𝜔|}>0, 𝜈=𝛼>0, ‖Φ(𝑡,𝜃)‖≤𝐾𝑒^{−𝜈(𝑡−𝜃)},𝑡>𝜃≥0.

3. Periodic Solution and Global Asymptotical Stability

Assumption (H3): 𝑓∈𝐿¹_𝑇([0,+∞);𝑋). For each 𝑘∈𝑍⁺₀, 𝛿∈𝑁, and 𝑐_𝑘∈𝑋, 𝑐_𝑘+𝛿=𝑐_𝑘.

Under the (H1) and (H3), consider the following system𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑡≠𝜏_𝑘,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡)+𝑐_𝑘,𝑡=𝜏_𝑘, (3.1)and Cauchy problem𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑡∈[0,𝑇^{0𝑥002𝑑𝑐}]⧵𝐷^{0𝑥002𝑑𝑐},Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘)+𝑐_𝑘,𝑘=1,2,…,𝑝𝛿,𝑥(0)=𝑥. (3.2)

Now we list the PC-mild solution of Cauchy problem (3.2) and ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution of (3.1).

Definition 3.1. For every 𝑥∈𝑋, 𝑓∈𝐿¹([0,+∞);𝑋), the function 𝑥∈PC([0,𝑇^{0𝑥002𝑑𝑐}];𝑋), given by𝑥(𝑡)=Φ(𝑡,0)𝑥+^𝑡₀Φ(𝑡,𝜃)𝑓(𝜃)𝑑𝜃+_{0≤𝜏𝑘<𝑡}Φ(𝑡,𝜏⁺_𝑘)𝑐_𝑘 (3.3)for 𝑡∈[0,𝑇^{0𝑥002𝑑𝑐}], is said to be a PC-mild solution of the Cauchy problem (3.2).

Definition 3.2. A function 𝑥∈PC([0,+∞);𝑋) is said to be a ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution of (3.1) if it is a PC-mild solution of Cauchy problem (3.2) corresponding to some 𝑥 and 𝑥(𝑡+𝑇^{0𝑥002𝑑𝑐})=𝑥(𝑡) for 𝑡≥0.

In the sequel, we show the existence and global asymptotical stability of ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution of (3.1).

Theorem 3.3. Assumptions (H1) and (H3) hold. Suppose ‖Φ(𝑇,0)‖=𝑙<1, then (3.1) has a unique ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution 𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡)≡(𝑃(𝑓,𝑐_𝑘))(𝑡), 𝑡≥0, given by𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡)=Φ(𝑡,0)[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹𝑧+^𝑡₀Φ(𝑡,𝜃)𝑓(𝜃)𝑑𝜃+_{0≤𝜏𝑘<𝑡}Φ(𝑡,𝜏⁺_𝑘)𝑐_𝑘≡(𝑃(𝑓,𝑐_𝑘))(𝑡), (3.4)where𝑧=^{0𝑥002𝑑𝑐𝑇}₀Φ(𝑇^{0𝑥002𝑑𝑐},𝜃)𝑓(𝜃)𝑑𝜃+_{0≤𝜏𝑘<0𝑥002𝑑𝑐𝑇}Φ(𝑇^{0𝑥002𝑑𝑐},𝜏⁺_𝑘)𝑐_𝑘. (3.5)Further, operator𝑃∶𝐿¹_𝑇([0,+∞);𝑋)×𝑋^𝛿→PC_{0𝑥002𝑑𝑐𝑇}([0,+∞);𝑋) (3.6)is a bounded linear operator, that is, there exists ^0𝑥0𝐵^02𝑑𝑐>0 such that‖𝑃(𝑓,𝑐_𝑘)‖_{PC𝑇0𝑥002𝑑𝑐}=‖𝑥_{0𝑥002𝑑𝑐𝑇}‖_{PC𝑇0𝑥002𝑑𝑐}≤𝐵^{0𝑥002𝑑𝑐}(𝑞‖𝑓‖_𝐿¹_𝑇+𝑝^𝛿_𝑘=1‖𝑐_𝑘‖), (3.7)where 𝐵 is independent on 𝑓 and 𝑐_𝑘.

Proof. By (1) and (4) of Lemma 2.5, ‖Φ(𝑇^{0𝑥002𝑑𝑐},0)‖=‖Φ(𝑇,0)‖^𝑞=𝑙^𝑞<1, operator [𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹ exists, is bounded and[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹=^∞_𝑛=0Φ(𝑛𝑇^{0𝑥002𝑑𝑐},0). (3.8)Thus,‖[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹1‖≤1−‖Φ(𝑇^{0𝑥002𝑑𝑐}=1,0)‖1−𝑙^𝑞. (3.9)For𝑧=^{0𝑥002𝑑𝑐𝑇}₀Φ(𝑇^{0𝑥002𝑑𝑐},𝜃)𝑓(𝜃)𝑑𝜃+_{0≤𝜏𝑘<0𝑥002𝑑𝑐𝑇}Φ(𝑇^{0𝑥002𝑑𝑐},𝜏⁺_𝑘)𝑐_𝑘, (3.10)operator equation [𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]𝑥=𝑧 has a unique solution 𝑥=[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹𝑧.
Let 𝑥(0)=𝑥=[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹𝑧, consider the Cauchy problem𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑡∈[0,𝑇^{0𝑥002𝑑𝑐}]⧵𝐷^{0𝑥002𝑑𝑐},Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘)+𝑐_𝑘,𝑘=1,2,…,𝑝𝛿,𝑥(0)=[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹𝑧. (3.11)It has a PC-mild solution 𝑥_{0𝑥002𝑑𝑐𝑇}(⋅) given by𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡)=Φ(𝑡,0)[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹𝑧+^𝑡₀Φ(𝑡,𝜃)𝑓(𝜃)𝑑𝜃+_{0≤𝜏𝑘<𝑡}Φ(𝑡,𝜏⁺_𝑘)𝑐_𝑘≡(𝑃(𝑓,𝑐_𝑘))(𝑡). (3.12)By (2), (3), and (4) of Lemma 2.5, one can easily verify that 𝑥_{0𝑥002𝑑𝑐𝑇}(⋅) is just the unique ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution of (3.1).
Obviously, operator 𝑃∶𝐿¹_𝑇([0,+∞);𝑋)×𝑋^𝛿→PC_{0𝑥002𝑑𝑐𝑇}([0,+∞);𝑋) is linear.
For 𝑡∈[0,𝑇^{0𝑥002𝑑𝑐}],‖𝑃(𝑓,𝑐_𝑘)(𝑡)‖=‖𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡)‖≤(‖Φ(𝑡,0)‖‖[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹‖+1)×(^{0𝑥002𝑑𝑐𝑇}₀‖Φ(𝑇^{0𝑥002𝑑𝑐},𝜃)‖‖𝑓(𝜃)‖𝑑𝜃+_{0≤𝜏𝑘<0𝑥002𝑑𝑐𝑇}‖Φ(𝑇^{0𝑥002𝑑𝑐},𝜏⁺_𝑘)‖‖𝑐_𝑘‖)≤𝑀_{0𝑥002𝑑𝑐𝑇}(𝑀_{0𝑥002𝑑𝑐𝑇}1−𝑙^𝑞+1)(𝑞‖𝑓‖_𝐿¹_𝑇+𝑝^𝛿_𝑘=1‖𝑐_𝑘‖). (3.13)Let ^0𝑥0𝐵^02𝑑𝑐=𝑀_{0𝑥002𝑑𝑐𝑇}(𝑀_{0𝑥002𝑑𝑐𝑇}/(1−𝑙^𝑞)+1), one can obtain the estimation immediately.

Theorem 3.4. Assumptions (H1) and (H3) hold. Suppose {Φ(𝑡,𝜃),𝑡≥𝜃≥0} is exponentially stable, then (3.1) has a unique ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution 𝑥_{0𝑥002𝑑𝑐𝑇}(⋅,𝑥) given by𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡,𝑥)=Φ(𝑡,0)𝑥+^𝑡₀Φ(𝑡,𝜃)𝑓(𝜃)𝑑𝜃+_{0≤𝜏𝑘<𝑡}Φ(𝑡,𝜏⁺_𝑘)𝑐_𝑘, (3.14)where𝑥=[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹𝑧,𝑧=^{0𝑥002𝑑𝑐𝑇}₀Φ(𝑇^{0𝑥002𝑑𝑐},𝜃)𝑓(𝜃)𝑑𝜃+_{0≤𝜏𝑘<0𝑥002𝑑𝑐𝑇}Φ(𝑇^{0𝑥002𝑑𝑐},𝜏⁺_𝑘)𝑐_𝑘, (3.15)and there exists 𝐵>0 such that‖𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡,𝑥)‖≤𝐵(𝑞‖𝑓‖_𝐿¹_𝑇+𝑝^𝛿_𝑘=1‖𝑐_𝑘‖), (3.16)where 𝐵>0 is independent on 𝑓 and 𝑐_𝑘.
Further, for arbitrary 𝑥₀∈𝑋, the PC-mild solution of the Cauchy problem (3.2) corresponding to the initial value 𝑥₀ satisfies the following inequality:‖𝑥(𝑡,𝑥₀)−𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡,𝑥)‖≤𝐵₁𝐵₂𝑒^−𝜈𝑡, (3.17)where 𝑥_{0𝑥002𝑑𝑐𝑇}(⋅,𝑥) is the ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution of (3.1), 𝐵₁>0 is independent on 𝑥₀, 𝑓, and 𝑐_𝑘; 𝐵₂=‖𝑥₀‖+𝑞‖𝑓‖_𝐿¹_𝑇+𝑝∑^𝛿_𝑘=1‖𝑐_𝑘‖.

Proof. Consider the operator series 𝑆=∑^∞_𝑛=0[Φ(𝑇^{0𝑥002𝑑𝑐},0)]^𝑛. By (4) of Lemma 2.5 and the stability of {Φ(⋅,⋅)}, we have‖[Φ(𝑇^{0𝑥002𝑑𝑐},0)]^𝑛‖=‖Φ(𝑛𝑇^{0𝑥002𝑑𝑐},0)‖≤𝐾𝑒^{−𝜈𝑛0𝑥002𝑑𝑐𝑇}→0as𝑛→∞. (3.18)Thus, we obtain‖𝑆‖≤^∞_𝑛=0‖[Φ(𝑇^{0𝑥002𝑑𝑐},0)]^𝑛‖≤^∞_𝑛=0𝐾𝑒^{−𝜈𝑛0𝑥002𝑑𝑐𝑇}. (3.19)Obviously, the series ∑^∞_𝑛=0𝐾𝑒^{−𝜈𝑛0𝑥002𝑑𝑐𝑇} is convergent, thus operator 𝑆∈£_𝑏(𝑋). It comes from[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]𝑆=𝑆[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]=𝐼 (3.20)that𝑆=[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹∈£_𝑏(𝑋). (3.21)Similar proof in Theorem 3.3, it is not difficult to verify that (3.1) has a unique ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution 𝑥_{0𝑥002𝑑𝑐𝑇}(⋅,𝑥) and 𝑥_{0𝑥002𝑑𝑐𝑇}(⋅,𝑥) can be given by (3.14) and (3.15).
Next, verify the estimation (3.16). In fact, for 𝑡∈[0,𝑇^{0𝑥002𝑑𝑐}], we have‖𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡,𝑥)‖≤‖Φ(𝑡,0)‖‖𝑥‖+^𝑡₀‖Φ(𝑡,𝜃)‖‖𝑓(𝜃)‖𝑑𝜃+_{0≤𝜏𝑘<𝑡}‖Φ(𝑡,𝜏⁺_𝑘)‖‖𝑐_𝑘‖. (3.22)On the other hand,‖𝑥‖≤‖[𝐼−Φ(𝑇^{0𝑥002𝑑𝑐},0)]⁻¹‖[^{0𝑥002𝑑𝑐𝑇}₀‖Φ(𝑇^{0𝑥002𝑑𝑐},𝜃)‖‖𝑓(𝜃)‖𝑑𝜃+_{0≤𝜏𝑘<0𝑥002𝑑𝑐𝑇}‖Φ(𝑇^{0𝑥002𝑑𝑐},𝜏⁺_𝑘)‖‖𝑐_𝑘‖]≤‖𝑆‖[^{0𝑥002𝑑𝑐𝑇}₀𝐾𝑒^{−𝜈(0𝑥002𝑑𝑐𝑇−𝜃)}‖𝑓(𝜃)‖𝑑𝜃+𝐾𝑒^{−𝜈(0𝑥002𝑑𝑐𝑇−𝜏+𝑘)}𝑝^𝛿_𝑘=1‖𝑐_𝑘‖]≤𝐾‖𝑆‖(𝑞‖𝑓‖_𝐿¹_𝑇+𝑝^𝛿_𝑘=1‖𝑐_𝑘‖). (3.23)Let 𝐵=𝐾(𝐾‖𝑆‖+1), one can obtain (3.16) immediately.
System (3.1) has a unique ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution 𝑥_{0𝑥002𝑑𝑐𝑇}(⋅,𝑥) given by (3.14) and (3.15). The PC-mild solution of the Cauchy problem (3.2) corresponding to initial value 𝑥₀ can be given by (3.3) replacing 𝑥 with 𝑥₀. Combining with (3.23), we obtain‖𝑥(𝑡,𝑥₀)−𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡,𝑥)‖≤𝐾𝑒^−𝜈𝑡[‖𝑥₀‖+𝐾‖𝑆‖(𝑞‖𝑓‖_𝐿¹_𝑇+𝑝^𝛿_𝑘=1‖𝑐_𝑘‖)]. (3.24)Let 𝐵₁=max{𝐾,𝐾²‖𝑆‖}>0, 𝐵₂=‖𝑥₀‖+𝑞‖𝑓‖_𝐿¹_𝑇+𝑝∑^𝛿_𝑘=1‖𝑐_𝑘‖, one can obtain (3.17) immediately.

Definition 3.5. The ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution 𝑥_{0𝑥002𝑑𝑐𝑇}(⋅,𝑥) of (3.1) is said to be globally asymptotically stable in the sense thatlim_𝑡→+∞‖𝑥(𝑡,𝑥₀)−𝑥_{0𝑥002𝑑𝑐𝑇}(𝑡,𝑥)‖=0, (3.25)where 𝑥(⋅,𝑥₀) is any PC-mild solution of the Cauchy problem (3.2) corresponding to initial value 𝑥₀∈𝑋.

By Theorem 3.4 and Lemma 2.7 (Lemma 2.9), one can obtain the following results.

Corollary 3.6. Assumptions of Lemma 2.7 (Lemma 2.9) and (H3) hold. System (3.1) has a unique ^0𝑥0𝑇^02𝑑𝑐-periodic PC-mild solution 𝑥_{0𝑥002𝑑𝑐𝑇}(⋅,𝑥) which is globally asymptotically stable.

Theorem 3.7. If 𝑥(𝑡,𝑥) is a periodic PC-mild solution of (3.1), then its period must be 𝑛𝑇₀ for some 𝑛∈𝑁.

Proof. Let ^0𝑥0𝑇^02𝑑𝑐 be the period of 𝑥(𝑡)=𝑥(𝑡,𝑥). Then𝑥((𝑇^{0𝑥002𝑑𝑐}+𝑡)±0)=𝑥(𝑡±0),𝑡≥0. (3.26)Clearly, ^0𝑥0𝑇^02𝑑𝑐 is not an impulsive moment, suppose there are 𝑠𝜏_𝑘s in the interval (0,𝑇^{0𝑥002𝑑𝑐}). Let 𝑡=𝜏₁. We have𝑥(𝑇^{0𝑥002𝑑𝑐}+𝜏₁)=𝑥(𝜏₁),𝑥((𝑇^{0𝑥002𝑑𝑐}+𝜏₁)⁺)=𝑥(𝜏⁺₁)=(𝐼+𝐵₁)𝑥(𝜏₁)+𝑐₁=(𝐼+𝐵₁)𝑥(𝑇^{0𝑥002𝑑𝑐}+𝜏₁)+𝑐₁, (3.27)which means that ^0𝑥0𝑇^02𝑑𝑐+𝜏₁ is one of the impulsive moments. Clearly, there is no 𝜏_𝑘s in the interval (𝑇^{0𝑥002𝑑𝑐},𝑇^{0𝑥002𝑑𝑐}+𝜏₁). For otherwise, suppose 𝑡∈(𝑇^{0𝑥002𝑑𝑐},𝑇^{0𝑥002𝑑𝑐}+𝜏₁) is an impulsive moment, then𝑥((𝑇^{0𝑥002𝑑𝑐}+(𝑡−𝑇^{0𝑥002𝑑𝑐}))⁺)=𝑥(𝑡⁺)≠𝑥(𝑡)=𝑥(𝑇^{0𝑥002𝑑𝑐}+(𝑡−𝑇^{0𝑥002𝑑𝑐}))=𝑥(𝑡−𝑇^{0𝑥002𝑑𝑐})=𝑥((𝑡−𝑇^{0𝑥002𝑑𝑐})⁺), (3.28)which is a contradiction. Thus, ^0𝑥0𝑇^02𝑑𝑐+𝜏₁=𝜏_𝑠+1, 𝐵_𝑠+1=𝐵₁, and 𝑐_𝑠+1=𝑐₁. Similarly, we have^0𝑥0𝑇^02𝑑𝑐+𝜏_𝑘=𝜏_𝑠+𝑘,𝐵_𝑠+𝑘=𝐵_𝑘,𝑐_𝑠+𝑘=𝑐_𝑘,𝑘∈𝑍⁺₀. (3.29)Now we can claim that 𝑠=𝑛𝛿 for some 𝑛∈𝑁. Otherwise, suppose 𝑠=𝑛𝛿+𝑗 for some 𝑛∈𝑁∪{0} and 1<𝑗<𝛿. As a consequence, 𝑛𝑇₀<𝑇^{0𝑥002𝑑𝑐}<(𝑛+1)𝑇₀. By (3.29), we have^0𝑥0𝑇^02𝑑𝑐+𝜏_𝑘=𝜏_𝑠+𝑘=𝜏_{𝑛𝛿+𝑗+𝑘}=𝑛𝑇₀+𝜏_𝑗+𝑘,𝐵_𝑘=𝐵_𝑠+𝑘=𝐵_{𝑛𝛿+𝑗+𝑘}=𝐵_𝑗+𝑘,𝑐_𝑘=𝑐_𝑠+𝑘=𝑐_{𝑛𝛿+𝑗+𝑘}=𝑐_𝑗+𝑘,𝑘∈𝑍⁺₀, (3.30)or(𝑇^{0𝑥002𝑑𝑐}−𝑛𝑇₀)+𝜏_𝑘=𝜏_𝑗+𝑘,𝐵_𝑘=𝐵_𝑗+𝑘,𝑐_𝑘=𝑐_𝑗+𝑘,𝑘∈𝑍⁺₀. (3.31)Thus ^0𝑥0𝑇^02𝑑𝑐−𝑛𝑇₀∈(0,𝑇₀) is a period of the impulsive perturbations for the second equation of (3.1), which contradicts to that 𝑇₀ is the least-positive period. Therefore,^0𝑥0𝑇^02𝑑𝑐+𝜏₁=𝜏_𝑠+1=𝜏_𝑛𝛿+1=𝑛𝑇₀+𝜏₁ (3.32)and ^0𝑥0𝑇^02𝑑𝑐=𝑛𝑇₀ for some 𝑛∈𝑁. The proof is complete.

4. Example

Consider a special case of (1.1)𝜕𝜕𝑡𝑥(𝑡,𝑦)=((Δ−𝜆𝐼)+𝑄(𝑡))𝑥(𝑡,𝑦)+sin(𝑡+|𝑦|),|𝑦|=𝑦²₁+𝑦²₂∈Ω,𝑡>0,𝑡≠𝜏_𝑘,𝑘∈𝑍⁺₀,𝑥(𝑡,𝑦)=0,𝑦∈𝜕Ω,𝑡>0,Δ𝑥(𝑡,𝑦)=0.05𝐼𝑥(𝑡,𝑦),𝑘=3𝑚−2,−0.05𝐼𝑥(𝑡,𝑦),𝑘=3𝑚−1,𝑦∈Ω,𝑡>0,𝑡=𝜏_𝑘=𝑘+𝑚−14𝜋,𝑘,𝑚∈𝑍⁺₀.0.05𝐼𝑥(𝑡,𝑦),𝑘=3𝑚, (4.1)where Ω⊂𝑅² is bounded domain and 𝜕Ω∈𝐶², Δ is the Laplace operator in 𝑅², 𝜆 is a parameter. For each 𝑡∈[0,2𝜋], 𝑄(𝑡)∈£_𝑏(𝐿₂(Ω),𝐿₂(Ω)), sup{‖𝑄(𝑡)‖,𝑡∈[0,2𝜋]}<∞, and 𝑄(⋅+2𝜋)=𝑄(⋅).

Set 𝑋=𝐿₂(Ω), for fixed 𝜆≥1, 𝐷(𝐴_𝜆)=𝐻²(Ω)∩𝐻¹₀(Ω). Define operator ℬ_𝜆𝑥=(Δ−𝜆𝐼)𝑥,𝑥∈𝐷(𝐴_𝜆). By [24, Theorem 2.5], ℬ_𝜆 is just the infinitesimal generator of a contraction 𝐶₀-semigroup in 𝐿₂(Ω), that is, ‖𝑇_𝜆(𝑡)‖≤1. Obviously, for 𝜆>1, ℬ_𝜆+𝐼=Δ−(𝜆−1)𝐼 can generate a exponentially stable 𝐶₀-semigroup in 𝐿₂(Ω) and ‖𝑇_𝜆(𝑡)‖≤𝑒^{−(𝜆−1)𝑡}.

Define operator series𝑈_𝜆,0(𝑡,𝜃)=𝑇_𝜆𝑈(𝑡−𝜃),_𝜆,𝑛(𝑡,𝜃)𝑥=∫^𝑡_𝜃𝑇_𝜆(𝑡−𝜏)𝑄(𝜏)𝑈_{𝜆,𝑛−1}(𝜏,𝜃)𝑥𝑑𝜏,0≤𝜃≤𝑡≤2𝜋,𝑛=1,2,…. (4.2)One can easily verify the following results:(i) the evolution operator 𝑈_𝜆(𝑡,𝜃)=∑^∞_𝑛=0‖𝑈_𝜆,𝑛(𝑡,𝜃)‖, 0≤𝜃≤𝑡≤2𝜋 is uniformly convergence;(ii)𝑈_𝜆(𝑡,𝜃) is the unique solution of the integral equation𝑈_𝜆(𝑡,𝜃)𝑥=𝑇_𝜆(𝑡−𝜃)𝑥+^𝑡_𝜃𝑇_𝜆(𝑡−𝜏)𝑄(𝜏)𝑈_𝜆(𝜏,𝜃)𝑥𝑑𝜏; (4.3)(iii)𝑈_𝜆(𝑡,𝜃) satisfies:(1)𝑈_𝜆(𝑡,𝑡)=𝐼, 𝑡≥0,(2)𝑈_𝜆(𝑡,𝜏)𝑈_𝜆(𝜏,𝜃)=𝑈_𝜆(𝑡,𝜃),𝑡≥𝜏≥𝜃≥0,(3) for every fixed 𝑥∈𝑋, (𝑡,𝜃)→𝑈_𝜆(𝑡,𝜃)𝑥 is strongly continuous,(4)‖𝑈_𝜆(𝑡,𝜃)‖≤𝑒^{−(𝜆−1−‖𝑄‖)(𝑡−𝜃)},𝑡≥𝜃≥0,(5)𝑈_𝜆(𝑡+2𝜋,𝜃+2𝜋)=𝑈_𝜆(𝑡,𝜃),𝑡≥𝜃≥0.

Thus, linear equation 𝑥̇(𝑡)=𝐴_𝜆(𝑡)𝑥(𝑡),𝑥∈𝑋,𝑡≥0 is well posed, where 𝐴_𝜆(𝑡)≡Δ−(𝜆−1)𝐼+𝑄(𝑡). Obviously, for 𝜆>1+‖𝑄‖, 𝒜_𝜆(𝑡)≡Δ−(𝜆−1−‖𝑄‖)𝐼+𝑄(𝑡) can determine an exponentially stable 2𝜋-periodic strongly continuous evolutionary process {𝒰_𝜆(⋅,⋅)} in 𝐿₂(Ω).

Define 𝑥(⋅)(𝑦)=𝑥(⋅,𝑦), 𝑓(⋅)(𝑦)=sin(⋅+|𝑦|), then (4.1) can be abstracted𝑥̇(𝑡)=𝒜_𝜆(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑡>0,𝑡≠𝜏_𝑘,𝑘∈𝑍⁺₀,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡),𝑡=𝜏_𝑘,𝑘∈𝑍⁺₀, (4.4)where𝐵_𝑘=0.05𝐼,𝑘=3𝑚−2,−0.05𝐼,𝑘=3𝑚−2,0.05𝐼,𝑘=3𝑚, (4.5)and 𝜏_𝑘=((𝑘+𝑚−1)/4)𝜋, 𝑘,𝑚∈𝑁.

Obviously, 𝒜_𝜆(⋅+2𝜋)=𝒜_𝜆(⋅), 𝜏_𝑘+3=𝜏_𝑘+𝜋, 𝐵_𝑘+3=𝐵_𝑘 and 𝑓(⋅+2𝜋)(𝑦)=sin(⋅+|𝑦|+2𝜋)=sin(⋅+|𝑦|)=𝑓(⋅)(𝑦), that is, 𝑓(⋅+2𝜋)=𝑓(⋅). For 𝜆>1+‖𝑄‖+𝜆, where 𝜆>ln[(1.05)²×0.95]²/2𝜋≈0.0147, by Lemma 2.7, {Φ_𝜆(⋅,⋅)} is exponentially stable. Now, all the assumptions are met in Theorem 3.4. Thus (4.4) has a unique 2𝜋-periodic PC-mild solution 𝑥_2𝜋(⋅,𝑦)∈PC_2𝜋([0+∞);𝐿₂(Ω)) which is globally asymptotically stable.

That is, suppose 𝑥(⋅,𝑦) is the PC-mild solution of the following initial-boundary value problem:𝜕𝜕𝑡𝑥(𝑡,𝑦)=((Δ−𝜆𝐼)+𝑄(𝑡))𝑥(𝑡,𝑦)+sin(𝑡+|𝑦|),|𝑦|=𝑦²₁+𝑦²₂∈Ω,𝑡>0,𝑡≠𝜏_𝑘,𝑘∈𝑍⁺₀,𝑥(𝑡,𝑦)=0,𝑦∈𝜕Ω,𝑡>0,𝑥(0,𝑦)=cos|𝑦|∈𝐿²(Ω),Δ𝑥(𝑡,𝑦)=0.05𝐼𝑥(𝑡,𝑦),𝑘=3𝑚−2,−0.05𝐼𝑥(𝑡,𝑦),𝑘=3𝑚−1,𝑦∈Ω,𝑡>0,𝑡=𝜏_𝑘=𝑘+𝑚−14𝜋,𝑘,𝑚∈𝑍⁺₀.0.05𝐼𝑥(𝑡,𝑦),𝑘=3𝑚, (4.6)Then, for 𝜆>1+‖𝑄‖+𝜆, where 𝜆>0.0147,‖𝑥(𝑡,𝑦)−𝑥_2𝜋(𝑡,𝑦)‖_𝐿²_(Ω)=(_Ω|𝑥(𝑡,𝑦)−𝑥_2𝜋(𝑡,𝑦)|²𝑑𝑦)^1/2→0as𝑡→+∞. (4.7)

From the above discussion, it is not difficult to find that a suitable parameter 𝜆 chosen by human, which will guarantee the model (4.1) has a unique 2𝜋-periodic PC-mild solution which is globally asymptotically stable. That is, we can use a biological approach to maintain the balance of a single, isolated species or eradicate pests. It provides us a reliable method for managing the single and isolated species in the nature.

Acknowledgments

This work is supported by National Natural Science foundation of China (no. 10661044) and Guizhou Province Found (no. 2008008). This work is partially supported by undergraduate carve out project of department of Guiyang City Science and Technology (no. 6-5).