and T0-Periodic Impulsive Perturbations on Banach Spaces

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


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 timevarying 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: x 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, 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 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 T 0periodic, that is, A y, t T, D A y, t, D , f t T, y f t, y , t ≥ 0, and T 0 is the least-positive constant such that there are δ τ k s in the interval 0, T 0 and 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 T 0 -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.
1 can be abstracted impulsive periodic evolution equations of the form 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/T 0 is rational, that is, T and T 0 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 nT 0 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.

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 T pT 0 qT, D {τ 1 , . . ., τ pδ } ⊂ 0, T and define PC 0, T ; X ≡ {x : 0, T → X | x is continuous at t ∈ 0, T \ D, x is continuous from left and has right-hand limits at t ∈ D} , and It can be seen that endowed with the norm In order to investigate periodic solution, introduce the following two spaces: It can be seen that endowed with the norm 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: Then the strongly continuous evolutionary process

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 ; X represented by x t Φ t, 0 x, where given by

2.12
This completes the proof.
Proof.Since γ is rational, let γ p/q, p, q ∈ N and p, q are relatively prime.Let T pT 0 qT , then 2.7 is T -periodic.Without loss of generality, for 2.16 Suppose t ∈ n T, n 1 T and let b max s∈ 0, T Π 0<τ k <s {K 0 I B k }.Then
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 p dt ≤ κ p x p .

2.19
If there exists α > 0 such that Further, where N θ, t is denoted the number of impulsive points in θ, t .

Discrete Dynamics in Nature and Society
For 19 , we obtain the following two inequalities:

2.24
This implies that

2.27
Thus, we obtain

Periodic solution and global asymptotical stability
Assumption H3 : f ∈ L 1 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 system

3.2
Now we list the PC-mild solution of Cauchy problem 3.2 and T -periodic PC-mild solution of 3.1 .Definition 3.1.For every x ∈ X, f ∈ L 1 0, ∞ ; X , the function x ∈ PC 0, T ; X , given by for t ∈ 0, T , 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 T -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 x t for t ≥ 0.
In the sequel, we show the existence and global asymptotical stability of T -periodic PCmild solution of 3.1 .

Theorem 3.3. Assumptions (H1) and (H3) hold. Suppose Φ T, 0
l < 1, then 3.1 has a unique T -periodic PC-mild solution x T t ≡ P f, c k t , t ≥ 0, given by where Further, operator is a bounded linear operator, that is, there exists B > 0 such that where B is independent on f and c k .

3.11
It has a PC-mild solution x T • given by By 2 , 3 , and 4 of Lemma 2.5, one can easily verify that x T • is just the unique T -periodic PC-mild solution of 3.1 .Obviously, operator P :

3.13
Let B M T M T / 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 T -periodic PC-mild solution x T •, x given by and there exists B > 0 such that where B > 0 is independent on f and c k .
Further, for arbitrary x 0 ∈ X, the PC-mild solution of the Cauchy problem 3.2 corresponding to the initial value x 0 satisfies the following inequality: 3.17 Proof.Consider the operator series S ∞ n 0 Φ T, 0 n .By 4 of Lemma 2.5 and the stability of {Φ •, • }, we have 3.18 Thus, we obtain Ke −νn T .

3.19
Obviously, the series Similar proof in Theorem 3.3, it is not difficult to verify that 3.1 has a unique T -periodic PCmild solution x T •, x and x T •, x can be given by 3.14 and 3.15 .
Next, verify the estimation 3.16 .In fact, for t ∈ 0, T , we have

3.22
On the other hand,

3.23
Let B K K S 1 , one can obtain 3.16 immediately.System 3.1 has a unique T -periodic PC-mild solution x T •, x given by 3.14 and 3.15 .The PC-mild solution of the Cauchy problem 3.2 corresponding to initial value x 0 can be given by 3.3 replacing x with x 0 .Combining with 3.23 , we obtain Proof.Let T be the period of x t x t, x .Then x T t ± 0 x t ± 0 , t ≥ 0.

3.26
Clearly, T is not an impulsive moment, suppose there are s τ k s in the interval 0, T .Let t τ 1 .
We have which means that T τ 1 is one of the impulsive moments.Clearly, there is no τ k s in the interval T, T τ 1 .For otherwise, suppose t ∈ T, T τ 1 is an impulsive moment, then which is a contradiction.Thus, T τ 1 τ s 1 , B s 1 B 1 , and c s 1 c 1 .Similarly, we have

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, nT 0 < T < n 1 T 0 .By 3.29 , we have

3.31
Thus T − nT 0 ∈ 0, T 0 is a period of the impulsive perturbations for the second equation of 3.1 , which contradicts to that T 0 is the least-positive period.Therefore, and T nT 0 for some n ∈ N. The proof is complete.

Example
Consider a special case of 1.1 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 equation −→ 0 as t −→ ∞.

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. 4 one can obtain 3.17 immediately.Definition 3.5.The T -periodic PC-mild solution x T •, x of 3.1 is said to be globally asymptotically stable in the sense that If x t, x is a periodic PC-mild solution of 3.1 , then its period must be nT 0 for some n ∈ N.