EXISTENCE RESULTS OF SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY AND FRACTIONAL DAMPING∗

In this paper we prove the existence, uniqueness, regularity and continuous dependence of mild solutions for second order impulsive functional differential equations with infinite delay and fractional damping in Banach spaces. We generalize the existence theorem of integer order differential equations to the fractional order case. The results obtained here improve and generalize some known results.


Introduction
Consider the following second order impulsive functional differential equations Cauchy problem with infinite delay and fractional damping: where D α t , D γ t are the Caputo's fractional derivative operator of order α, γ ∈ (0, 1), A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators (C(t)) t∈R on a Banach space X, B : X → X is a bounded linear operator. The history x t , x t : (−∞, 0] → X, x t (θ) = x(t + θ) and x t (θ) = x (t + θ) belong to some abstract phase space B defined axiomatically; 0 = t 0 < t 1 < · · · < t n < t n+1 = b are fixed numbers and the symbol x(t i ) represent the jump of the function x at t i , which is defined by x(t i ) = x(t + i ) − x(t − i ) for i = 1, 2, . . . , n. x (t i ) has the same meaning.
Impulsive differential equations with delay and fractional derivatives have played an important role in describing dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so forth, which has been used for constructing many mathematical models in science and engineering. The theory of fractional differential equations has been extensively studied by many authors [3-5, 7, 15-17, 20-24, 27, 29, 30]. To obtain existence of mild solutions in these papers, usually,the compactness condition on associated family of operators and pulse items, restrictive conditions on a priori estimation are used. For example, authors in [10] studied the existence of mild solutions for a damped impulsive system: x (t i )), i = 1, 2, . . . , n, are used. Recently, authors [1] used the measure of noncompactness without the compactness assumption on associated family of operators, to obtain the existence of mild solutions for the following fractional order integro-differential system: x(t) = ϕ(t), t ∈ (−∞, 0], the restrictive conditions on a priori estimation and measure of noncompactness estimation b q aC q,M µ 2 L 1 (J,R + ) < q, 16aη * < 1, (1.6) are used in [1]. However, to the best of the author's knowledge, fractional derivatives are introduced here for such problems for the first time.
In this paper, using the Kuratowski measure of noncompactness and progressive estimation method, we prove the existence, uniqueness, regularity and continuous dependence of mild solutions for the problem (1.1)-(1.2). The compactness condition of pulse items, some restrictive conditions on a priori estimation and noncompactness measure estimation have been removed to obtain the existence and uniqueness of mild solutions. Our results improve and generalize the corresponding results in [2,10]. Finally, an example of non-compact semigroups is given.
The paper is organized as follows. In section 2 we give some basic concepts and Lemmas. In section 3 we discuss the existence, uniqueness and regularity of mild solutions, in section 4 we discuss the continuous dependence of mild solutions. Our results are based on the properties of equicontinuous semigroups and the ideas and techniques in Xie [28].

Preliminaries
In this paper, X will be a Banach space with norm · and A : D(A) ⊂ X → X is the infinitesimal generator of a strongly continuous cosine family (C(t)) t∈R of bounded linear operators on X and (S(t)) t∈R is the sine function associated with (C(t)) t∈R , which is defined by S(t)x = t 0 C(s)xds, x ∈ X, t ∈ R. We designate by N, N certain constants such that C(t) ≤ N and S(t) ≤ N for every t ∈ J = [0, b]. We refer the reader to [6] for the necessary concepts about cosine functions. As usual we denote by [D(A)] the domain of A endowed with the graph norm x A = x + Ax , x ∈ D(A). Moreover, the notation E stands for the space formed by the vector x ∈ X for which the function C(·)x is of class C 1 . It was proved by Kisyński [18] that the space E endowed with the norm x E = x + sup 0≤t≤b AS(t)x , x ∈ E, is a Banach space. The operator valued function is a strongly continuous group of linear operators on the space E × X generated by the operator A = [ 0 I A 0 ] defined on D(A) × E. It follows from this that AS(t) : E → X is a bounded linear operator and that defines an (E × X)-valued continuous function. Next N 1 =: sup t∈J AS(t) L(E,X) in which L(E, X) stands for the Banach space of bounded linear operators from E into X and we abbreviate this notation to L(X) when E = X.
We say that a function u : [σ, b] → X is a normalized piecewise continuous function on [σ, b] if u is piecewise continuous and left continuous on (σ, b]. We denote by P C([σ, b], X) the space formed by the normalized piecewise continuous functions from [σ, b] into X. In particular, we introduce the space P C formed by all functions u : [0, b] → X such that u is continuous at t = t i , u(t − i ) = u(t i ) and u(t + i ) exists for all i = 1, 2, · · · , n. It is clear that P C endowed with the norm x pc = sup t∈J x(t) is a Banach space. Similarly, We say that x ∈ P C is piecewise smooth if x is continuously differentiable at t = t i , i = 1, 2, · · · , n, and for t = t i , i = 1, 2, · · · , n, there are the right derivative ) and x (t + i ) exist, i = 1, 2, · · · n}. Then P C 1 endowed with the norm u 1 = u pc + u pc is a Banach space. Next, for u ∈ P C 1 we represent by u (t) the left derivative at t > 0 and by u (0) the right derivative at zero.
. For x ∈ P C, we denote by x i , i = 0, 1, . . . , n, the function x i ∈ C(J i , X) given by x i (t) = x(t), t ∈ (t i , t i+1 ] and x i (t i ) = x(t + i ). Moreover, for V ⊂ P C and i = 0, 1, . . . , n, we use the notation V i for V i = { x i : x ∈ V }. By Lemma 1.1 in [10], we know that a set V ⊆ P C is relatively compact if and only if each set V i = { x i : x ∈ V } is relatively compact in C(J i , X) for every i = 0, 1, . . . , n.
In this work we will employ an axiomatic definition of the phase space ß introduced by Hale and Kato [13] which appropriated to treat retarded impulsive differential equations. 13]). The phase space B is a linear space of functions mapping (−∞, 0] into X endowed with a seminorm · B and B satisfies the following axioms: , then for every t ∈ [σ, σ + b) the following conditions hold: is locally bounded and H, K, M are independent of x(·).
(B) The space B is complete.
be a positive function verifying the conditions (g 6 ) and (g 7 ) of [14]. This means that h(·) is Lebesgue integrable on (−∞, −r) and that there exists a non-negative and locally bounded function γ : It follows from the proof of [14] (Th.1.3.8) that B is a phase space which verifies the axioms (A) and (B) of our work. Moreover, when r = 0 this space coincides with C 0 × L 2 (h, X) and the parameters H = 1, In this paper, we denote by α(·) and α pc (·) the Kuratowski measure of noncompactness of X and P C(J, X).
(3) If W ⊂ P C 1 (J, X) is bounded and the elements of W are equicontinuous on each J i (i = 0, 1, · · · , n), then α pc 1 (W ) = max sup t∈J α(W (t)), sup t∈J α(W (t)) , where α pc 1 (·) denotes the Kuratowski measure of noncompactness in space P C 1 (J, X). 25,26]). Suppose that A is the infinitesimal generator of a strongly continuous cosine family (C(t)) t∈R , g : R → X is a continuously differentiable function and [19]). Let Ω be a bounded open subset in a Banach space X and 0 ∈ Ω. Assume that the operator F : Ω → X is continuous and satisfies the following conditions: for any countable set V ⊂ Ω. Then F has a fixed point in Ω.

Existence and uniqueness of mild solutions
We make the following hypotheses.
(H 1 ) f : J × B × X → X satisfies the following conditions: (2) There is an integrable function q(·) : J → R + such that (3) For any bounded set U, V ⊂ P C, there is a constant l > 0 such that , there are constants c i > 0 and d i > 0 such that, for each i = 1, 2, · · · , n, Proof. If x(·) satisfies the equations (3.1) and (3.2), we can decompose x(·) as Then T 1 , T 2 are well defined with values in S(b). In addition, from the axioms of phase space, the Lebesgue dominated convergence theorem,the condition (H 1 ) and (H 2 ), it is easy to show that T = (T 1 , T 2 ) is continuous. It is easy to see that if (u, v) is a fixed point of T , then u + y is a mild solution of the system (1.1)-(1.2). First, we show that the set When t ∈ J 0 = [0, t 1 ], it follows from (3.4), (3.5) and (H 1 )(2) that is the Gamma function, q = sup t∈J q(t) , u s = sup 0≤r≤s u(r) , v s = sup 0≤r≤s v(r) , s ∈ J 0 . Since u s and v s are continuous nondecreasing on J 0 , (3.6) and (3.7) imply that It follows from this and the condition (H 2 ) that When t ∈ J 1 , u 1 , v 1 ∈ C([J 1 , , X). It is similar to (3.6) and (3.7), we get where u 1 s = sup t1≤r≤s u 1 (r) , v 1 s = sup t1≤r≤s v 1 (r) , Q 1 > 0 and Q 1 > 0 are fixed constants. We have by (3.9) and (3.10), (3.11) Using the Gronwall Lemma once again and (3.11), there exists a constant G 1 > 0 such that Similarly, there is a constant G i > 0 such that sup ti≤s≤t u i (s) + sup ti≤s≤t v i (s) ≤ G i , t ∈ J i (i = 2, 3, · · · , n). Let G ≥ G 0 + G 1 + · · · + G n , then (u, v) b ≤ G and Ω 0 is bounded. Second, we verify that the conditions of Lemma 2.3 are satisfied. Let R > G and Then Ω R is a bounded open set and (0, 0) ∈ Ω R . Since R > G, we know that (u, v) = λT (u, v) for any (u, v) ∈ ∂Ω R and λ ∈ (0, 1). Suppose that V ⊂ Ω R is a countable set and V ⊂ co({(0, 0)} ∪ T (V )). Let Then we have (3.12) Since C(t) and S(t) are strongly continuous, it follows from (3.4) and (3.5) that , which together with (3.12) implies that ( V k ) i is equicontinuous on every J i (k = 1, 2, i = 0, 1, · · · , n).
In the following, we verify that the sets V 1 , V 2 are relatively compact in P C. Without loss of generality, we do not distinguish V k | Ji and ( V k ) i , where V k | Ji is the restriction of V k on J i = (t i , t i+1 ] (k = 1, 2, i = 1, 2, · · · , n).
A Banach space X has the Radon-Nikodym property if for each λ-continuous vector measure µ : Σ → X of bounded variation there exists h ∈ L 1 (µ, X) such that µ(B) = B hdλ for all B ∈ Σ.
Theorem 3.5. Assume that the conditions of Theorem 3.2 are satisfied, the space X has Radon-Nikodym property and Bx ∈ L 1 (J, E). If u(·) is a mild solution of the system (1.1)-(1.2) and the following conditions are satisfied: (a) ϕ(0), u ti + I i (u ti , u ti ) ∈ D(A) and ψ(0), u ti + J i (u ti , u ti ) ∈ E for every i = 1, 2, · · · , n; (b) The function f : J × Ω → X (Ω is a open set in B × X) is continuous and satisfies the Carathédodory conditions on E: (i) f (·, u, v) is measurable on J for each (u, v) ∈ Ω, (ii) f (t, ·, ·) is continuous on Ω for t ∈ J a.e., (iii) For each R > 0, there is a integrable function β R : J → R + such that f (t, u, v) E ≤ β R (t) a.e. for t ∈ J and for all (u, v) ∈ Ω such that u B + v ≤ R. Then u(·) is a classical solution of the system (1.1)-(1.2).
Proof. Theorem 3.4 implies that u i (·) is the mild solution of the system (3.20) for each i = 1, 2, · · · , n. Then u i , u i ∈ C(J i , X), and so ( u i ) t , ( u i ) t are continuous on J i .
Since u i (·) is the mild solution of the following Cauchy problem Then u(t) is a strong solution of the system (1.1)-(1.2).
Proof. For any t ∈ (t i , t i+1 ) and ε > 0 such that t + ε ∈ (t i , t i+1 ), we get from (3.21) that where C 1 > 0 is a constant independent of t and ε, and the fact that are Lipschitz continuous on J i has been used (see [12]). Therefore, from the Gronwall Lemma we know that u i (t) is Lipschitz continuous on J i for each i = 1, 2, · · · , n. Since X has Radon-Nikodym property, it follows from Proposition 3.3 [11] that u i (t) is a strong solution of the system (1.1)-(1.2) for each i = 1, 2, · · · , n. A similar argument permit us to prove that u 0 (t) is a strong solution of the system (1.1)-(1.2). The proof is completed.
(ii) For f ∈ X, C(t)f = ∞ n=1 cos(nt)(f ; z n )z n . Moreover, it follow from this expression that S(t)f = ∞ n=1 sin(nt) n (f ; z n )z n , that S(t) is compact for t > 0 and that C(t) = 1 and S(t) = 1 for every t ∈ R.