Second Order Semilinear Volterra-Type Integro-Differential Equations with Non-Instantaneous Impulses

: We consider a non-instantaneous system represented by a second order nonlinear differential equation in a Banach space E . We use the family of linear bounded operators introduced by Kozak, Darbo ﬁxed point method and Kuratowski measure of noncompactness. A new set of sufﬁcient conditions is formulated which guarantees the existence of the solution of the non-instantaneous system. An example is also discussed to illustrate the efﬁciency of the obtained results.

The theory and application of integrodifferential equations are important subjects in applied mathematics, see, for example [1][2][3][4][5][6][7][8] and recent development of the topic, see the monographs of [9]. In recent times there have been an increasing interest in studying the abstract autonomous second order, see for example [10][11][12][13][14]. Useful for the study of abstract second order equations is the existence of an evolution system S(t, s) for the homogenous equation y (t) = A(t)y(t), for t ≥ 0. (2) For this purpose there are many techniques to show the existence of S(t, s) which has been developed by Kozak [15]. In many problems, such as the transverse motion of an extensible beam, the vibration of hinged bars and many other physical phenomena, we deal with the second-order abstract differential equations in the infinite dimensional spaces. On the other hand, recently there exists an extensive literature for the non-autonomous second order see, for example, [16][17][18][19][20][21][22].
The dynamics of many evolving processes are subject to abrupt changes such as shocks, harvesting, and natural disaster. These phenomena involve short term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. Particularly, the theory of instantaneous impulsive equations have wide applications in control, mechanics, electrical engineering, biological and medical fields. Recently, Hernandez et al. [23] use first time not instantaneous impulsive condition for semi-linear abstract differential equation of the form and introduced the concepts of mild and classical solution. Wang and Fečkan have changed the conditions y(t) = g i (t, y(t)) in (3) as follows Of course then y(t + i ) = g i (t, y(t − i )), where y(t + i ) and y(t − i ) represent respectively the right and left limits of y(t) at t = t i . Motivated by above remark, Wang and Fečkan [24] have shown existence, uniqueness and stability of solutions of such general class of impulsive differential equations. To learn more about this kind of problems, we refer [25][26][27][28][29][30][31][32][33][34].
To deal with the above mentioned issues, we investigate necessary and sufficient conditions for the existence of a mild solution of system (1). By virtue of the theory of measure of noncompactness associated with Darbo's and Darbo-Sadovskii's fixed point theorem. This technique was considered by Banas and Goebel [35] and subsequently used in many papers; see, for example, [33,[36][37][38][39].
A brief outline of this paper is given:. Some preliminaries are presented in Section 2. Section 3, we obtain necessary and sufficient conditions for System (1). An Appropriate example is given to illustrate our results.

Basic Definitions and Preliminaries
In this section, we review some basic concepts, notations, and properties needed to establish our main results.
Denote by C(J, E) the space of all continuous E-valued functions on interval J which is a Banach space with the norm y = sup t∈J |y(t)|.
It can be easily proved that PC(J, E) is a Banach space endowed with For a positive number R, let be a bounded set in PC(J, E). L r (J, E) denotes the space of E-valued Bochner functions on [0, a] with the norm B(E) the Banach space of bounded linear operators from E into E. First we recall the concept of the evolution operator S(t, s) for problem (2), introduced by Kozak in [15] and recently used by Henríquez, Poblete and Pozo in [20].
The family is said to be an evolution operator generated by the family {A(t) : t ∈ J} if the following conditions are satisfied [15]: (e 1 ) For each y ∈ E the function S(·, ·)y : J × J → E is of class C 1 and (i) for each t ∈ J, S(t, t) = 0, (ii) for all (t, s) ∈ D and for each y ∈ E, ∂ ∂t S(t, s)y t=s = y, ∂ ∂s S(t, s)y t=s = −y. (e 3 ) For all (t, s) ∈ D, if y ∈ D(A(t)), then ∂ ∂s S(t, s)y ∈ D(A(t)). Moreover, there exist ∂ 3 ∂t 2 ∂s S(t, s)y, S(t, s)y and and for all y ∈ D(A) the function (t, s) −→ A(t) ∂ ∂s S(t, s)y is continuous in D.

Definition 2.
A function f : J × E × E → E is said to be a Carathéodory function if it satisfies: For W, a nonempty subset of E, we denote by W and ConvW the closure and the closed convex hull of W, respectively. Finally, the standard algebraic operations on sets are denoted by aW and Y + W, respectively. Now, we recall some basic definitions and properties about Kuratowski measure of noncompactness that will be used in the proof of our main results. Definition 3. [35] The Kuratowski measure of noncompactness α E (·) defined on bounded set W of Banach space E is Some basic properties of α E (·) are given in the following lemma.

Lemma 1.
Let Y and W be bounded sets of E and a be a real number [35]. The Kuratowski measure of noncompactness satisfies some properties: The map Q : X ⊂ E → E is said to be a α-contraction if there exists a positive constant λ < 1 such that α E (Q(W)) ≤ λα E (W) for any bounded closed subset W ⊂ E.

Lemma 2.
[40] Let E be a Banach space, W ⊂ E be bounded. Then there exists a countable set W 0 ⊂ W, such that

Lemma 3.
[41] Let E be a Banach space, −∞ < a 1 < a 2 < +∞ for constants, and let W = {y n } ⊂ PC([a 1 , a 2 ], E), be a bounded and countable set. Then α E (W(t)) is Lebesgue integral on [a 1 , a 2 ], and Denote by α PC the Kuratowski measure of noncompactness of PC(J, E). Before proving the existence results, we need the following Lemmas. [43] Assume that W is a non-empty, closed and convex subset of a Banach space E and 0 ∈ W. Let Q : W → W be a continuous mapping and α E -contraction. If the set {y ∈ W : y = λQy} is bounded for 0 < λ < 1, then the map Q has at least one fixed point in W.

Theorem 2.
(Darbo-Sadovskii) [35] Assume that W is a non-empty, closed, bounded, and convex subset of a Banach space E. Let Q : W → W be a continuous mapping and α E -contraction. Then the map Q has at least one fixed point in W.

Existence Results
In this section, we discuss the existence of mild solutions for system (1). Firstly, let us propose the definition of the mild solution of system (1).

Definition 4.
A function y ∈ PC(J, E) is said to be a mild solution to the system (1), if it satisfies the following relations: the non-instantaneous conditions and y is the solution of the following integral equations In this manuscript, we list the following hypotheses: (H 1 ) There exist a pair of constants M ≥ 1 and δ > 0, such that (H 2 ) There exists a constantM > 0 such that: (H 3 ) f : J × E × E → E is of Carathéodory type and satisfies: (a) There exist Θ f ∈ L r (J, R + ), r ∈ [1, ∞) and a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) such that: for a.a t ∈ J and each y, z ∈ E.
(b) There exist integrable functions σ, :J → R + , such that: for a.a t ∈ J and W 1 , (a) There exist Θ g ∈ L 1 (J, R + ), and a continuous nondecreasing function ϕ are continuous, and they satisfy the following conditions: (a) there exist positive constants c i , i = 1, · · · , N such that are continuous, and satisfy the following conditions:

Remark 1. From Lemma 5 and (H
Theorem 3. Under the assumptions (H 1 )-(H 7 ), the system (1) has at least one mild solution on J, provided that Proof. Define the mapping Λ : It is obvious that the fixed point of Λ is the mild solution of (1). We shall show that Λ satisfies the assumptions of Theorem 1. The proof will be given in four steps.
Case 3. For each t ∈ (s i , t i ], we have, Then, for all t ∈ J, we have Let us take the right-hand side of the above inequality as µ(t). Then

This implies that
This above inequality implies that there exists a constant L such that β(t) ≤ L, t ∈ J, and hence µ(t) ≤ L, t ∈ J. Since for every t ∈ J, |y(t)| ≤ µ(t), we have y PC ≤ L.
Suppose that (y n ) n∈N is a sequence in B R which converges to y in B R as n → ∞. By the continuity of nonlinear term γ and ζ with respect to the second argument, for each s ∈ J, we have sup s∈J |γ i (s, y n (s)) − γ i (s, y(s))| → 0 as n → ∞, sup s∈J |ζ(s, y n (s)) − ζ(s, y(s))| → 0 as n → ∞.
By the Carathéodory character of nonlinear term f , for each s ∈ J, we have f s, y n (s), Since the function γ i is continuous and → 0, as n → ∞.
For every bounded subset B ⊂ PC(J, E), then we know that there exists a countable set B 1 = {y} ∞ n=1 ⊂ B (see Lemma 2), such that for any t ∈ J, we have Note that B and ΛB are equicontinuous, we can get from Lemma 2, Lemma 3, Lemma 4 and using the assumptions (H 1 )-(H 6 ), we obtain Case 1. For the interval (t i , s i ], we have Then Case 2. For the interval [0, t 1 ], we have Then α E (Λ(B(t))) ≤ 2M( σ L 1 + 2K * a L 1 ))α PC (B(t)). (11) Case 3. For the interval (s i , t i+1 ], we have Then From the above cases (10)- (12), for all t ∈ J, we obtain Thus, we find that Λ is α PC -contraction operator. Applying now theorem 1, we conclude that Λ has a fixed point which is an solution of the system (1).
Next, we present another existence result for the mild solution of the system (1).

Theorem 4.
Assume that hypotheses (H 1 )-(H 6 ) are fulfilled and Then, there exists a mild solution of system (1).
Proof. Following the proof of Theorem 3 we conclude that the map Λ : B R → B R given by Equation (5) is continuous. Next, we show that there exists R > 0 such that Λ(B R ) ⊂ B R . In fact, if it is not true, then for each positive number R , there exists a functiony ∈ B R andť ∈ J such that R ≤ (Λy)(ť) . Therefore for Case 1. Forť ∈ (s i , t i ], andy ∈ B R , we have,

Then
(Λy)(ť) ≤Mc i−1 R +Md i−1 It follows from the Hölder's inequality that Case 2. Forť ∈ [0; t 1 ], andy ∈ B R , we get, It follows from the Hölder's inequality that It follows from the Hölder's inequality that |(Λy)(t)| ≤Md i + Ml i Therefore for allť ∈ J, we have Dividing both sides by R and taking the lim inf as R → +∞, we havẽ which contradicts (13). Hence, the operator Λ transforms the set B R into itself. The proof of Λ : B R → B R is α E -contraction is similar to those in Theorem 3. Therefore, we omit the details. By the Darbo-Sadovskii fixed point theorem 2 we deduce that Λ has a fixed point which is a mild solution of system (1).
It is well known that A is the infinitesimal generator of a C 0 -semigroup and of a strongly continuous cosine function on E, which will be denoted by (C(t)). From [14], for all x ∈ H 2 ([0, π], R), t ∈ R, C(t) B(E) ≤ 1. Define also the operator B : H 1 ([0, π], R) → E by B(t)u(s) = a(t)u (s), where a : [0, 1] → R is a Hölder continuous function.
We consider the following system: From the above discussion, we obtain For each t ∈ J, and W 1 , W 2 ⊂ E, we get We shall show that condition (H 3 ) holds with Moreover By (19), for any t ∈ J and W ⊂ E, we get α E (g(t, s, W)) ≤ 1 8 sup α E (W).
Clearly all the conditions of theorem 3 are satisfied. Hence by the conclusion of Theorem 3, it follows that problem (14) has a solution.