### Abstract

Using the Kuratowski measure of noncompactness and progressive estimation method,
we obtain the existence results of mild solutions for impulsive partial neutral second-order
functional integro-differential equations with infinite delay in Banach spaces. The
compactness condition of the impulsive term, some restrictive conditions on *a priori* estimation and noncompactness measure estimation have been deleted. Our conditions
are simple and our results essentially improve and extend some known results. As applications,
some examples are provided to illustrate the obtained results.

**MSC: **
34K30, 34K40, 35R10, 47D09.

##### Keywords:

impulsive partial neutral functional integro-differential equations; mild solutions; fixed point; Banach spaces### 1 Introduction

Consider the following impulsive partial neutral second-order functional integro-differential
systems with infinite delay in a Banach space *X*:

where *A* is the infinitesimal generator of a strongly continuous cosine function of bounded
linear operators,
*X*. In both cases, the history
*g*, *f*,
*x* at

The study of impulsive functional differential equations is linked to their utility
in simulating processes and phenomena subject to short-time perturbations during their
evolution. The perturbations are performed discretely and their duration is negligible
in comparison with the total duration of the processes and phenomena. Now impulsive
partial neutral functional differential equations have become an important object
of investigation in recent years stimulated by their numerous applications to problems
arising in mechanics, electrical engineering, medicine, biology, ecology, *etc.* With regard to this matter, we refer the reader to [1-12] and references therein. However, in order to obtain the existence of solutions in
these study papers, the compactness condition on the associated family of operators
and the impulsive term, some similar restrictive conditions on *a priori* estimation,

are used. In [13-16], authors used a strict set contraction mapping fixed point theorem without the compactness
assumption on the associated family of operators to obtain the existence results of
system (1) when

improved and generalized some results in [1,7]. However, the compactness condition of the impulsive terms
*a priori* estimation (3), (4) and the restrictive condition on measure of noncompactness estimation

are used in [13-16]. So far we have not seen the existence results of system (2).

In this paper, using the Kuratowski measure of noncompactness and progressive estimation
method, we obtain the existence results of mild solutions of impulsive partial neutral
second-order functional integro-differential systems (1) and (2). The compactness
condition of impulsive terms
*a priori* estimation and measure of noncompactness estimation (3), (4) and (6) have been deleted.
Our conditions are simple and our results essentially improve and extend some corresponding
results in papers [1,2,13,14]. As applications, some examples are provided to illustrate the obtained results.

### 2 Preliminaries

In this paper, *X* is a Banach space with the norm
*A* is the infinitesimal generator of a strongly continuous cosine function of bounded
linear operators,
*X* and
*N*,
*A* endowed with the graph norm
*E* stands for the space formed by the vector
*E* endowed with the norm

is a Banach space. The operator-valued function
*E*-valued continuous function. This is a consequence of the fact that

defines an
*E* into *X*, and we abbreviate this notation to

To describe appropriately our system (1), we say that the function
*u* is piecewise continuous and left continuous on
*X*. In particular, we introduce the space *PC* formed by all functions
*u* is continuous at
*PC* endowed with the norm

For

Then

For system (2), we give the precise meaning of the derivative in (2). We say that
*x* is continuously differentiable at

In this work we employ an axiomatic definition of the phase space ß introduced by Hale and Kato [19] which appropriated to treat retarded impulsive differential equations. For other abstract phase spaces, we can refer to [20,21].

**Definition 2.1**[19]

The phase space ß is a linear space of functions mapping
*X* endowed with a seminorm

(A) If

(i)

(ii)

(iii)
*K* is continuous, *M* is locally bounded and *H*, *K*, *M* are independent of

(B) The space ß is complete.

In this paper we denote by
*X*, by
*PC*.

The following lemma is easy to get.

**Lemma 2.2***If the cosine function family*
*is equicontinuous and*
*then the set*

*is equicontinuous for*

(1) *If*
*is bounded*, *then*
*for any*
*where*

(2) *If**W**is piecewise equicontinuous on*
*then*
*is piecewise continuous for*
*and*

(3) *If*
*is bounded and piecewise equicontinuous*, *then*
*is piecewise continuous for*
*and*

*where*

(4) *If*
*is bounded and the elements of*
*are equicontinuous on each*
*then*

*where*
*denotes the Kuratowski measure of noncompactness in the space*

**Lemma 2.4**[23]

*Let*
*be an integrable function such that*
*Then the function*
*belongs to*
*the function*
*is integrable on*
*for*
*and*

**Lemma 2.5**[24]

*Let*
*If there is*
*such that*
*for*
*and a*.*e*.
*then*
*and*

**Lemma 2.6**[25] (Mónch)

*Let**X**be a Banach space*, Ω *be a bounded open subset in**X**and*
*Assume that the operator*
*is continuous and satisfies the following conditions*:

(1)

(2) *D**is relatively compact if*
*for any countable set*

*Then**F**has a fixed point in*

### 3 Main results

Firstly, we discuss the existence of mild solutions for the impulsive second-order system (1).

**Definition 3.1** A function

For system (1), we make the following hypotheses.

(H_{1}) The functions

(1) For every

(2) There are integrable functions

(3) For any bounded set

where

(H_{2})

(1) For every

(2) There are continuous functions

(3) For any bounded set

(H_{3}) The functions

Let the function

**Theorem 3.2***Suppose that the cosine function family*
*is equicontinuous*,
*satisfy the condition* (H_{1}), (H_{2}) *and* (H_{3}) *are satisfied*. *Then the impulsive second*-*order system* (1) *has at least one mild solution*.

*Proof* Let

Clearly,
*F* is well defined with values in
_{1}), (H_{2}) and (H_{3}), we can show that *F* is continuous (see [5]). It is easy to see that if *x* is a fixed point of *F*, then

Firstly, we show that the set

is bounded. In fact, if

When
_{1}),

Consequently,

By well-known Gronwall’s lemma and (10), there are constants
*x* and
_{3}) that

Nextly, when

Then

where

where
*v* and

It is similar to the proof above, there is a constant
*x* and
*i.e.*,

Lastly, we verify that all the conditions of Lemma 2.6 are satisfied. Let

Then

Nextly, let

It follows from (H_{1})-(H_{3}) and Lemma 2.2 that
*V* is equicontinuous on every

When
_{1})(3), (H_{2})(3) and Lemma 2.5, we have

where

From this and Gronwall’s lemma, we know that
*V* is a relative compact set in

and

When

where

Therefore
*V* is a relative compact set in

Similarly, we can show that *V* is a relative compact set in
*V* is a relative compact set in
*F* has a fixed point in
*x* be a fixed point of *F* on

Nextly, we discuss the existence of mild solutions for the impulsive system (2).

**Definition 3.3** A function

Differentiate (16) to get

Let functions

Clearly,

where

Let

We make the following hypotheses for convenience.

(H_{f})

(1) For every

(2) There is an integrable function

(3) For any bounded set

where

(H_{g})

(1) The function

(2) For every bounded set

(3) For any bounded set

(H_{I}) The functions

**Theorem 3.4***Let the conditions* (H_{f}), (H_{g}) *and* (H_{I}) *be satisfied*, *the cosine function family*
*be equicontinuous and*
*Then system* (2) *has at least one mild solution*.

*Proof* Let the function

and

The product space
_{f}), (H_{g}) and (H_{I}), we can show that

Firstly, we show that the set

is bounded. If

When
_{f})(2), (H_{g})(1), (H_{I}) that

Equations (20) and (21) imply that

Since
_{I}) that

Nextly, when

Then

We have, by (23) and (24),

where

Using Gronwall’s lemma once again and (25), there is a constant

It is similar to the proof above, there are constants

Let

Then

Suppose that

Then we have

It follows from (18), (19) and (H_{g})(2) that

In the following, we verify that the set
*PC*. Without loss of generality, we do not distinguish

When
_{f})(3), (H_{g})(3) and Lemma 2.5, we have

Since

By Gronwall’s lemma and (29), we have

When

Equations (30) and (31) imply that

Consequently,