Mild Solutions for Fractional Impulsive Integro-Differential Evolution Equations with Nonlocal Conditions in Banach Spaces

: In this paper, by using the cosine family theory, measure of non-compactness, the Mönch ﬁxed point theorem and the method of estimate step by step, we establish the existence theorems of mild solutions for fractional impulsive integro-differential evolution equations of order 1 < β ≤ 2 with nonlocal conditions in Banach spaces under some weaker conditions. The results obtained herein generalizes and improves some known results. Finally, an example is presented for the demonstration of obtained results.


Introduction
Fractional differential equations, in comparison with classical integer order ones, have apparent advantages in modeling mechanical and electrical properties of various real materials and in some other fields.The theory of fractional differential evolution has been emerging as an important area of investigation in recent years (see [1][2][3][4]).By using semigroup theory, the properties of noncompact measures, the references [5][6][7] studied local and global existence of solutions of the initial value problem for a class of fractional evolution equations of order 0 < α ≤ 1.The references [8][9][10][11][12][13] studied local and global existence of solutions for a class of fractional evolution equations of order 0 < α ≤ 1 with nonlocal conditions.By using semigroup theory, the properties of noncompact measures or Lipschitz conditions, the references [14,15] studied local and global existence and uniqueness of mild solutions for fractional impulisive evolution differential equations of order 0 < α ≤ 1.The references [16][17][18] studied the existence and uniqueness of mild solutions for fractional differential evolution equations of order 1 < α ≤ 2 with nonlocal conditions by using semigroup theory, the properties of noncompact measures or Lipschitz conditions.The references [19][20][21][22][23][24][25] studied the controllability of nonlocal fractional differential evolution equations with nonlocal conditions.
In recent years, many scholars have studied the existence and uniqueness of mild solutions for fractional differential evolution equations by using semigroup theory, the properties of noncompact measures and various fixed point theorems.In [16], the authors studied the existence and uniqueness of mild solutions for semilinear fractional integrodifferential equations with nonlocal conditions of order 1 < β ≤ 2: q(t − s)g(s, x(s))ds, t ∈ J = [0, T], x(0) + m(x) = x 0 ∈ X, x (0) + n(x) = x 1 ∈ X, The reference [17] studied the local and global existence of mild solutions for fractional integro-differential evolution equations of order 1 < β ≤ 2 with nonlocal conditions: where Under two cases where the solution operator is compact and noncompact, respectively, the reference [26] investigated the existence and uniqueness of mild solutions of the following fractional impulsive integro-differential evolution equations of order 1 < β ≤ 2 with nonlocal conditions in Banach space E: x| t=t k denotes the jump of x(t) at t = t k , i.e., k ) represent the right and left limits of x(t) at t = t k ,, respectively.They established the existence results of mild solutions by the(generalized) Darbo fixed point theorem and Schauder fixed point theorem, improving the known results, while the compactness condition is added to the impulse term.
In most of the works mentioned above, the strict conditions on the nonlinearity or impulsive term and the corresponding coefficients are still imposed.Evidently, it is essential and interesting to widen or remove these conditions, which is very helpful for the applications of the problem.The purpose of this paper is to further study the existence of mild solutions of problem (1) in Banach space.Through Mönch fixed point theorem and the method of estimate step by step, under simple conditions and without restrictive ones on the impulsive terms, the existence of a global mild solution for problem (1) is obtained.It should be pointed out that the compactness condition on the impulsive term is removed.
The rest of this paper is organized as follows.In Section 2, we present some notations, definitions and lemmas.In Section 3, we give the the existence theorems of solutions for fractional impulsive integro-differential evolution equations of order 1 < β ≤ 2 with nonlocal conditions.An illustrated example is presented in Section 4.

Preliminaries and Lemmas
Throughout this paper, let E be a real Banach space with the norm For any S ⊂ PC[J, E], we note S(t) = {x(t)|x ∈ S}, t ∈ J.For any R > 0, we denote

Definition 1 ([2]
).The Riemann-Liouville fractional derivative of order β > 0 for function x(t) is defined by (2) ).The Caputo fractional derivative of order β > 0 for function x(t) is defined by ( is called solution operator (or a strongly continuous fractional cosine family of order β) for the problem if the following conditions are satisfied: (i) {C β (t)} t>0 is strongly continuous for t > 0, and {C β (0 , where A: D(A) ⊂ E → E is a closed densely defined linear operator and be known as the infinitesimal generator of C β (t).

Property 4 ([26]). If the solution operator C
In the sequel, let α, α PC denote the Kuratowski measure of noncompactness in E, PC[J, E], , respectively.We first give the following lemmas in order to prove our main results.
Lemma 2 ([20]).If H is a bounded subset of PC[J, E], the element of H is equicontinuous at J k for all k = 1, 2, . .., m, then CoH ⊂ PC[J, E] is bounded and equicontinuous.
where M 1 > 0, M 2 ≥ 0, M 3 ≥ 0 are constants, then m(t) ≡ 0 for any t ∈ J, provided one of following conditions holds Lemma 7 ([29]).Let E be a Banach space, Ω ⊂ E is a bounded open set, θ ∈ Ω, F : Ω → Ω is continuous and satisfy the following conditions: Then F has at least a fixed point on Ω.

Lemma 8 ([26]
). x ∈ PC[J, E] is a solution of the problem (1) if and only if x(t) satisfy the following integral equations:

Main Results
In this section, we are in a position to prove our main results concerning the solutions of fractional impulsive integro-differential evolution Equation (1) in Banach spaces.Now, let us first list the following assumptions for convenience.
(H 1 ) For any R > 0, f is bounded and uniformly continuous on There exists M * > 0 such that for any t ≥ 0,we have (H 6 ) There exist constants L h > 0, L g > 0 such that for any bounded set S ⊂ E satisfying α(h(t, s, S)) ≤ L h α(S), α(g(t, s, S)) ≤ L g α(S).
(H 9 ) There exist non-negative constants L i (i = 1, 2, 3, 4) satisfying one of the following two conditions: (i) < 2, and for any bounded set B i ⊂ E(i = 1, 2, . .., n + 3) and all t ∈ J, We now prove the following main result of this paper.
Theorem 1.Let E be a Banach space, assume that conditions (H 1 ) − (H 7 ) hold, g i (i = 1, 2) is compact.Then (1) has at least a global mild solution on PC[J, E].
Proof.We define F : PC[J, E] → PC[J, E] as follows: As f is uniformly continuous and g 1 , g 2 , I k , Îk are continuous, we can get that F is continuous operator.
Next, let D ⊂ Ω be a countable set and D ⊂ co({θ} ∪ F(D)), where θ is an element of Ω, then D is a relative compact set.
Remark 2. In [26], the authors used the following conditions to the impulse term: through the method of estimate step by step, we remove the compactness condition on the impulsive term and obtain the existence of a mild solution for problem (1).Remark 3. When L 4 = 0, the condition (i) in (H 7 ) is obviously true.In this case, we can get the result of Theorem 1 in [26], while the condition is weaker.

Example
In this section, we give an example to illustrate our main results.

Conclusions
In this paper, by using the Mönch fixed point theorem and the method of estimate step by step, we remove the compactness condition on the impulsive term and obtain the existence of a mild solution for problem (1).The results obtained herein generalize and improve some known results.In our future work, we aim to study the fractional non-autonomous evolution equations with impulses and delay.
(1) t 2 ], especially α(D(t 2 )) = 0, so D is a relative compact set on C[J 1 , E].Similarly, we can proveD is a relative compact set on C[J k , E], (k = 2, 3, ..., m), then D is a relative compact set on C[J, E].By Lemma 7, F has at least a fixed point on Ω, then problem (1) have a solution x * in PC[J, E], that is to say, x * is a global mild solution for problem(1).Let E be a Banach space, assume that conditions (H 1 ) − (H 6 ), (H 8 ), (H 9 ) hold.Then (1) has at least a global mild solution on PC[J, E].