Existence and Asymptotic Stability of Periodic Solutions for Impulsive Delay Evolution Equations

In this paper, we are devoted to consider the periodic problem for the impulsive evolution equations with delay in Banach space. By using operator semigroups theory and fixed point theorem, we establish some new existence theorems of periodic mild solutions for the equations. In addition, with the aid of an integral inequality with impulsive and delay, we present essential conditions on the nonlinear and impulse functions to guarantee that the equations have an asymptotically stable $\omega$-periodic mild solution.


Introduction
Let X be a real Banach space with norm · , L(X) stand for the Banach space of all bounded linear operators from X to X equipped with its natural topology.
Let r > 0 be a constant, we denote P C([−r, 0], X) as the Banach space of piecewise continuous functions from [−r, 0] to X with finite points of discontinuity where functions are left continuous and have the right limits, with the sup-norm φ P r = sup s∈[−r,0] φ(s) .
In this paper, we consider the periodic problem for the impulsive delay evolution equation in Banach space X u ′ (t) + Au(t) = F (t, u(t), u t ), t ∈ R, t = t i , where A : D(A) ⊂ X → X is a closed linear operator and −A generates a C 0semigroup T (t)(t ≥ 0) in X; F : R × X × P C([−r, 0], X) → X is a nonlinear mapping which is ω-periodic in t; u t ∈ P C([−r, 0], X) is the history function defined by u t (s) = u(t + s) for s ∈ [−r, 0]; p ∈ N denotes the number of impulsive points between 0 and ω, 0 < t 1 < t 2 < · · · < t p < ω < t p+1 are given numbers satisfying represents the jump of the function u at t i , u(t + i ) and u(t − i ) are the right and left limits of u(t) at t i , respectively; I i : X → X(i ∈ Z) are continuous functions satisfying I i+p = I i .
The theory of partial differential equations with delays has extensive physical background and realistic mathematical model, and it has undergone a rapid development in the last fifty years. The evolution equations with delay are more realistic than the equations without delay in describing numerous phenomena observed in nature, hence the numerous properties of their solutions have been studied, see [10,24] and references therein for more comments.
One of the important research directions related to the asymptotic behavior of the solutions for the evolution equations with delay is to find conditions for the existence and stability of ω-periodic solutions in the case that the nonlinear mapping is ω-periodic function in t. In the last few decades, the existence and asymptotic stability of periodic solutions have been investigated by some authors(see [7,25,17,18,19,11,12,9] and the references therein). In [7], under the assumption that the solutions of the associated homotopy equations were uniformly bounded, Burton and Zhang obtained the existence of periodic solutions of an abstract delay evolution equation. In [25], Xiang and Ahmed showed an existence result of periodic solution to the delay evolution equations in Banach spaces under the assumption that the corresponding initial value problem had a priori estimate. In [17,18,19], Liu studied periodic solutions by using bounded solutions or ultimate bounded solutions for delay evolution equations in Banach spaces. In [9], Huy and Dang studied the existence, uniqueness and stability of periodic solutions to a partial functional differential equation in Banach space in the case that the nonlinear function satisfied Lipschitz-type condition. Specially, in [11], Li discussed the existence and asymptotic stability of periodic solutions to the evolution equation with multiple delays in a Hilbert space. By using the analytic semigroups theory and the integral inequality with delays, the author obtained the essential conditions on the nonlinearity F to guarantee that the equation has ω-periodic solutions or an asymptotically stable ω-periodic solution.
On the other hand, it is well known that impulsive evolution equation has an extensive physical, chemical, biological, engineering background and realistic mathematical model, and hence has been emerging as an important area of investigation in the last few decades. Since the end of last century, theory of impulsive evolution equation in Banach space has been largely developed(see [8,1,2,26,13,14]  Although there have been many meaningful results on the delay or impulsive evolution equation periodic problem in Banach space, to our knowledge, these results have relatively large limitation. First of all, the most popular approach is the use of ultimate boundedness of solutions and the compactness of Poincaré map realized through some compact embeddings. However, in some concrete applications, it is difficult to choose an appropriate initial conditions to guarantee the boundedness of the solution. Secondly, we observe that the most popular condition imposed on the nonlinear term F is its Lipschitz-type condition. In fact, for equations arising in complicated reaction-diffusion processes, the nonlinear function F represents the source of material or population, which dependents on time in diversified manners in many contexts. Thus, we may not hope to have the Lipschitz-type condition of F .
Finally, there are few papers to study the asymptotically stable of periodic solutions for the impulsive evolution equations with delay.
Motivated by the papers mentioned above, we consider the periodic problem for impulsive delay evolution equation (1.1) in Banach space. By using periodic extension and fixed point theorem, we study the existence of ω-periodic mild solutions for Eq.(1.1). It is worth mentioning that the assumption of prior boundedness of solutions is not employed and the nonlinear term F satisfies some growth condition, which is weaker than Lipschitz-type condition. On the other hand, by means of an integral inequality with impulsive and delay, we present the asymptotic stability result for Eq.(1.1), which will make up the research in this area blank.
The rest of this paper is organized as follows. In Section 2, we collect some known definitions and notions, and then provide preliminary results which will be used throughout this paper. In Section 3, we apply the operator semigroup theory to find the ω-periodic mild solutions for Eq.(1.1) and in Section 4, by strengthening the condition, we obtain the global asymptotic stability theorems for Eq.(1.1). In the last section, we give an example to illustrate the applicability of abstract results obtained in Section 3 and Section 4.

Preliminaries
Throughout this paper, we assume that X is a Banach space with norm · . Now, we recall some notions and properties of operator semigroups, which are essential for us. For the detailed theory of operator semigroups, we refer to [20].
Assume that A : D(A) ⊂ X → X is a closed linear operator and −A is the infinitesimal generator of a C 0 -semigroup T (t)(t ≥ 0) in X. Then there exist M > 0 and ν ∈ R such that then ν 0 is called the growth exponent of the semigroup T (t)(t ≥ 0). If ν 0 < 0, then T (t)(t ≥ 0) is called an exponentially stable C 0 -semigroup.
If C 0 -semigroup T (t) is continuous in the uniform operator topology for every t > 0 in X, it is well known that ν 0 can also be determined by σ(A) (the resolvent set of A), where −A is the infinitesimal generator of C 0 -semigroup T (t)(t ≥ 0). We know that is compact semigroup (see [23]).
Let J denote the infinite interval [0, +∞), from [20], it follows that when x 0 ∈ D(A) and h ∈ C 1 (J, X), the following initial value problem of the linear evolution equation has a unique classical solution u ∈ C 1 (J, X) ∩ C(J, X 1 ), which can be expressed by Generally, for x 0 ∈ X and h ∈ C(J, X), the function u given by (2.4) belongs to C(J, X) and it is called a mild solution of the linear evolution equation (2.3).
It can be seen that equipped with the norm · P C , P C ω (R, X) (or P C([0, ω], X)) is a Banach space.
Given h ∈ P C ω (R, X), we consider the existence of ω-periodic mild solution for the linear impulsive evolution equation in X in X and ν 0 be a growth index of the semigroup T (t)(t ≥ 0). Then the linear impuisive evolution equation (2.5) exists a unique ω-periodic mild solution u := P h ∈ P C ω (R, X). Furthermore, the operator P : Proof. Firstly, for x 0 ∈ X, we consider the existence of mild solution for the initial value problem of the linear impulsive evolution equation (2.7) Thus, the initial value problem (2.7) exists a unique mild solution u on J i which can be expressed by Iterating successively in the above equation with u(t j ), j = i − 1, i − 2, · · · , 1, we can verify that u satisfies Inversely, we can see that the function u ∈ P C(J, X) defined by (2.9) is a mild solution of the initial value problem (2.6).
Secondly, we demonstrate that the linear impuisive evolution equation (2.5) exists a unique ω-periodic mild solution. It is clear that the ω-periodic mild solution of Eq.(2.5) restricted on J is the mild solution of the initial value problem (2.6) with the initial value For any ν ∈ (0, |ν 0 |), there exists M > 0 such that , it is easy to obtain that (2.12) Therefore, there exists a unique initial value such that the mild solution u of Eq.(2.6) given by (2.9) satisfies the periodic bound- For t ∈ J, by the semigroup properties of T (t), we have Therefore, the ω-periodic extension of u on R is a unique ω-periodic mild solution of Eq.(2.5).
Finally, by (2.9) and (2.13), we obtain thus, it is easy to prove that the solution operator P : P C ω (R, X) → P C ω (R, X) is a bounded linear operator. This completes the proof of Theorem 2.1. ✷

The Existence and Uniqueness
In this section, we discuss the existence of ω-periodic mild solution to Eq. (1.1) for the case that the semigroup T (t)(t ≥ 0) generated by −A is a compact semigroup, which implies that T (t) is a compact operator for any t > 0. Now, we are in a position to state and prove our main results of this section.
for every I k , I k (θ) = θ, and there exist positive constants a k such that then Eq. (1.1) has at least one ω-periodic mild solution u.
Proof Define an operator Q : It is not difficult to prove that Q is continuous on P C ω (R, X). In fact, let {u n } ⊂ P C ω (R, X) be a sequence such that u n → u ∈ P C ω (R, X) as n → ∞, hence, for every t ∈ R, we have u n (t) → u(t) ∈ X and u n,t → u t ∈ P C([−r, 0], X) as n → ∞.
From F : R × X × P C([−r, 0], X) → X is continuous, and I k ∈ C(X, X)(k ∈ Z), it follows that and I k (u n (t k )) → I k (u(t k )), n → ∞. we have which implies that Q : P C ω (R, X) → P C ω (R, X) is continuous, where C = (I − T (ω)) −1 , by the proof of Lemma 2.1, one can obtain For any R > 0, let Note that Ω R is a closed ball in P C ω (R, X) with centre θ and radius r. Now, we show that there is a positive constant R such that Q(Ω R ) ⊂ Ω R . If this were not case, then for any R > 0, there exist u ∈ Ω R and t ∈ R such that (Qu)(t) > R.
Thus, we see by (H1) and (H2) that Dividing on both sides by R and taking the lower limit as R → ∞, we have which contradicts (H3). Hence, there is a positive constant R such that Q(Ω R ) ⊂ Ω R .
In order to show that the operator Q has a fixed point on Ω R , we also introduce Then we will prove that Q 1 is a compact operator and Q 2 is a contraction.
Firstly, we prove that Q 1 is a compact operator. Clearly, Q 1 is continuous and is equicontinuous. For every u ∈ Ω R , by the periodicity of u, we only consider it on It is clear that Thus, we only need to check I i tend to 0 independently of u ∈ Ω R when t 2 − t 1 → 0, i = 1, 2, 3.
From the condition (H1), it follows that there is a constant M ′ > 0 such that Combined this fact with the equicontinuity of the semigroup T (t)(t ≥ 0), we have Hence, Q 1 u(t 2 ) − Q 1 u(t 1 ) tends to 0 independently of u ∈ Ω R as t 2 − t 1 → 0, which means that Q 1 (Ω R ) is equicontinuous.
It remains to show that (Q 1 Ω R )(t) is relatively compact in X for all t ∈ R. To do this, we define a set (Q ε Ω r )(t) by Since the operator T (ε) is compact in X, thus, the set (Q ε Ω R )(t) is relatively compact in X. For any u ∈ Ω R and t ∈ R, from the following inequality which implies that the set (Q 1 Ω R )(t) is relatively compact in X for all t ∈ R.
Thus, the Arzela-Ascoli theorem guarantees that Q 1 is a compact operator.
Secondly, we prove that Q 2 is a contraction. Let u, v ∈ Ω R , by the condition (H2), we have therefore, From the condition (H3), we can deduce Q 2 is a contraction.
Therefore, by the famous Sadovskii fixed point theorem [21], we know that Q has a fixed point u ∈ Ω R , that is, Eq. (1.1) has a ω-periodic mild solution. The proof is completed. ✷ Furthermore, we assume that F satisfies Lipschitz condition, namely, (H1 ′ ) there are positive constants c 1 , c 2 , such that for every t ∈ R, x 0 , x 1 ∈ X and φ, ψ ∈ P C([−r, 0], X) then we can obtain the following result.
Theorem 3.2. Let X be a Banach space, −A generates an exponentially stable Proof From (H1 ′ ) we easily see that (H1) holds. In fact, for any t ∈ R, x ∈ X and φ ∈ P C([−r, 0], X), by the condition (H1 ′ ), Let u, v ∈ P C ω (R, X) be the ω-periodic mild solutions of Eq.(1.1), then they are the fixed points of the operator Q which is defined by (3.1). Hence, which implies that From this and the condition (H3), it follows that u 2 = u 1 . Thus, Eq.(1.1) has only one ω-periodic mild solution. ✷

The Asymptotic Stability
In order prove the asymptotic stability of ω-periodic solutions for Eq. (1.1), we need discuss the existence and uniqueness of the following initial value problem where F : J × X × P C([−r, 0], X) → X is continuous and ϕ ∈ P C([−r, 0], X). Define u is continuous from left and has right limits at t j , j ∈ N .
If there exists u ∈ P C([−r, ∞), X) satisfying u(t) = ϕ(t) for −r ≤ t ≤ 0 and then u is called a mild solution of the initial value problem (4.1).
In order to obtain the results about asymptotic stability, we need the following integral inequality of Gronwall-Bellman type with delay and impulsive.
Proof For t ∈ [−r, t 1 ], the initial value problem (4.1) is in the following form: It is easy to see that Q is well defined, Qu ∈ P C ϕ ([−r, t 1 ], X), and the mild solution Now, we prove that Q has a fixed point in P C ϕ ([−r, t 1 ], X). From the condition (H1 ′ ) and (4.4), it follows that for any u, v ∈ P C ϕ ([−r, t 1 ], X), n = 1, 2, · · ·, where M 1 is the bound of T (t) on [0, t 1 ]. By the contraction principle, one shows that Q has a unique fixed point u 1 in P C ϕ ([−r, t 1 ], X), which means the initial value problem (4.3) has a mild solution and For t ∈ [−r, t 2 ], the initial value problem (4.1) is in the following form: Similar to the proof of (4.3), we can prove that the initial value problem (4.6) has a mild solution u 2 ∈ P C ϕ ([−r, t 2 ], X) Doing this interval by interval, we obtain that there exists u ∈ P C ϕ ([−r, ∞), X) satisfying u(t) = ϕ(t) for −r ≤ t ≤ 0 and which is a mild solution of the the initial value problem (4.1).

Application
In this section, we present one example, which does not aim at generality, but indicates how our abstract results can be applied to concrete problems.
Let X = L 2 (Ω) with the norm · 2 , then X is a Banach space. Define an From [5], we know that −A is a selfadjoint operator in X, and generates an exponentially stable analytic semigroup T p (t)(t ≥ 0), which is contractive in X.