CONDENSING OPERATORS AND PERIODIC SOLUTIONS OF INFINITE DELAY IMPULSIVE EVOLUTION EQUATIONS

. By showing the existence of the ﬁxed point of the condensing operators in the phase space C µ for the Cauchy problem for impulsive evolution equations with inﬁnite delay in a Banach space X :

For the more advanced and complex differential equations, some challenges are met. For example, consider the Cauchy problem for following infinite delay evolution equation x(s) = ϕ(s), s ≤ 0, in a general Banach space (X, · ), where A(t) is a unbounded operator for each t > 0 and F is a continuous function in its variables, x t (s) = x(t + s), s ≤ 0, x t ∈ C((−∞, 0], X) (the space of continuous functions on (−∞, 0] with values in X), and φ : (−∞, 0] → X is a given function. Recently, there have been some significant developments in the study of evolution equations with impulsive conditions, which are the combinations of the traditional initial value problems and the short-term perturbations whose duration can be negligible in comparison with the duration of the process. In particular, in [8,9,10], we present existence results for the periodic solutions of the Cauchy problem for impulsive equations without delay (r = 0) or with finite delay (r > 0): , 0] → X and I i : X → X (i = 1, · · · , n) are given functions. Moreover, in [8,9,10], a series of techniques are developed in order to handle impulsive conditions for equations without delay or with finite delay.
A natural question to ask is what will happen for the periodicity if impulsive conditions are imposed to the problem (1)- (2), that is, what can we obtain for the periodicity of the solutions to the Cauchy problem for following impulsive evolution equations with infinite delay, x(s) = ϕ(s), s ≤ 0, where A(·) and F(·, u, w) are -periodic. As noted in [11] that the infinite delay problem (1)-(2) is itself difficult because most fixed point theorems were not applicable as they would require some compactness which was not obtainable for equations with infinite delay. Moreover, as can be seen from [8,9,10] that impulsive conditions are also difficult to deal with. So now, the combination of the two major difficulties creates a system that is significantly more difficult.
In this paper, with further detailed analysis, we are able to combine the techniques developed in [8,9,10], the treatments in [11], and of course some new ideas in this paper, to attack the problem (3)-(5) and derive fixed points and then periodic solutions.
The procedure is to study the problem (3)-(5) in a phase space C µ , and carefully handle the impulsive conditions and overcome the related difficulties so as to prove that the Poincare operator given by P(ϕ) = x (ϕ) (i.e., units along the unique solution x(ϕ) determined by the initial function ϕ) is a condensing operator with respect to the Kuratowski's measure of non-compactness in C µ . Then as an application, we derive periodic solutions from bounded solutions by using Sadovskii's Fixed Point Theorem. The new results obtained here extend some earlier results in this area for evolution equations without impulsive conditions or without infinite delay.
2. Preliminary results. We start with some basic definitions, settings and results here.
Definition 2.1. Let α be the Kuratowski's measure of non-compactness and let P be a continuous and bounded operator on X. If α(P(B)) < α(B) for any bounded B of E with α(B) > 0, then P is called a condensing operator.
Our purpose of this paper is to derive a condensing operator associated with the problem (3)-(5), so that we need to use the following Sadovskii's Fixed Point Theorem to study the periodic solutions.  (1). F(·, u, w) and A(·) are periodic, that is, (3). For any t ≥ 0 and Reλ ≤ 0, (where we use |·| to denote the operator norm) and (λI −A(t)) −1 is a compact operator. (4). For any s, t, r ∈ [0, ], If Assumption 2.3 holds, then it is known (cf., e.g., [12]) that there exists a unique evolution family U(t, s), 0 ≤ s ≤ t ≤ such that U(t, s) is strongly continuous, and where C > 0 is a constant. Moreover, by Assumption 2.3, we have are finite numbers. Let 0 < Υ < 1, 0 < r < 1 and > 0 be constants. Then there exists an integer m 0 > 1 such that It is easy to see that there exists a function µ on (−∞, 0] such that Let P C((−∞, 0], X) be the space of piecewise continuous functions from (−∞, 0] to X. For the function µ given above, we define and Then C µ is a Banach space.
is a piecewise continuous function with points of discontinuity t i where u is left continuous and has the right limits, and satisfies It is easily seen that mild solutions satisfy the impulsive condition (5). With some additional conditions on the Lipschitzian operators Ii, the proof in [11] can be modified to show that mild solutions do exist on [0, α) for some α > 0. In this paper, we focus our attention to the existence of fixed points of condensing operators for infinite delay impulsive evolution equations and then the existence of periodic solutions, we suppose that for each ϕ ∈ C µ , the problem (3)-(5) has a unique mild solution x(·, ϕ) existing on [0, ∞). Since everything in the whole paper will be clear, we will use "solutions" to mean "mild solutions". Assumption 2.5. (1) F(t, u, w) : + × X × C µ → X is continuous and Lipschitzian in u and in w. Ii, i = 1, 2, · · · , are Lipschitzian and compact.
The following lemma from [8] is also needed here. where Throughout this paper, we write for any real numbers a and b. Proof. First, we prove that if x and y are two solutions of the problem (3)-(5) with initial value ϕ and ψ respectively on (−∞, ], > 0, then where c 1 , c 2 and β i are some constants. Actually, it is not so hard to see that for t ∈ [0, ], By the related Lipschitz continuity, we know that there is a constant M > 0 such that Hence, for s ∈ [0, t], Thus it follows from (16) that Now, Lemma 2.6 implies (15). Let ψ ∈ P 0 be fixed. Then and hence (15) implies that {|x (ϕ)| µ : ϕ ∈ P 0 } is bounded. Therefore the result is true by using the definition of the norm in C µ . The proof ends then.
Based on the result above, for P 0 ⊂ C µ and x(ϕ) the unique (mild) solution of the problem (3)-(5) with ϕ ∈ P 0 , we define where, as in Section 2, Proof. It follows from Theorem 3.1 that Z [0, ] (P 0 ) is bounded in P C([0, ], X). Moreover, in view of the properties of µ and (10), we deduce that for every x r ∈ Z r (P 0 ) (r ∈ [0, ]), Hence, Z r (P 0 ) is bounded in C µ for each r ∈ [0, ]. On the other hand, it follows from the properties of µ and (10) that for any x r , y r ∈ Z r (P 0 ) (r ∈ [0, ]), and for every 0 ≤ τ < r ≤ with r − τ ≥ m0 , we have Thus, by the definition and the basic properties of the Kuratowski's measure of noncompactness, we obtain the desired conclusion. The proof is then complete.

JIN LIANG, JAMES H. LIU AND TI-JUN XIAO
Let s ∈ [l, r] be fixed. For 0 < < l, consider Based on (7) and (8), we conclude that the set {P ϕ(s) : ϕ ∈ P 0 } is precompact in X since U(s, s − ) is compact.
Thus, in view of Theorem 4.1, we see that the problem (3)-(5) has a periodic solution. This completes the proof.