1. Introduction
In recent years, impulsive differential and partial differential equations have become more important in some mathematical models of real phenomena, especially in control, biological and medical domains. In these models, the investigated simulating processes and phenomena usually are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously in the form of impulses. The theory of impulsive differential equations has seen considerable development, see the monographs of Bainov and Semeonov [
1], Lakshimikantham [
2] and Perestyuk [
3]. Recently, several works reported existence results for mild solutions for impulsive neutral functional differential equations or inclusions, such as ([
4,
5,
6,
7,
8,
9,
10,
11]) and references therein. However, the results obtained there are only in connection with finite delay. Since many systems arising from realistic models heavily depend on histories (
i.e., there is an effect of infinite delay on state equations), there is a real need to discuss partial functional differential systems with infinite delay, where numerous properties of their solutions are studied and detailed bibliographies are given. The literature related to first and second order nonlinear non autonomous neutral impulsive systems with or without state dependent delay is not vast. To the best of our knowledge, this is almost an untreated article in a literature and is one of the main motivations of this paper.
When the delay is infinite, the notion of the phase space plays an important role in the study of both qualitative and quantitative theory. A usual choice is a seminormed space satisfying suitable axioms, introduced by Hale and Kato in [
12]; see also Corduneanu and Lakshmikantham [
13]; Graef [
14] and Baghli and Benchohra [
15,
16]. Unfortunately, we can not find broad literature about the system involving infinite delay with impulse effects. Henderson and Ouahab [
17] discussed existence results for nondensely defined semilinear functional differential inclusions in Frechet spaces. Hernández
et al. [
18] studied existence of solutuions for impulsive partial neutral functional differential equations for first and second order systems with infinite delay. Recently, Arthi and Balachandran [
19] proved controllability of the second order impulsive functional differential equations with state dependent delay using fixed point approach and cosine operator theory. It has been observed that the existence or the controllability results proved by different authors are through an axiomatic defination of the phase space given by Hale and Kato [
12]. However, as remarked by Hino, Murakami, and Naito [
20], it has come to our attention that these axioms for the phase space are not correct for the impulsive system with infinite delay [
21,
22]. This motivated us to generate a new phase space for the existence of a nonautonomous impulsive neutral inclusion with infinite delay. This is another motivation of this paper. To the best of our knowledge, the result proved in this paper is new and are not available in the literature.
On the other hand, researchers have been proving the controllability results using compactness assumption of semigroups and the family of cosine operators. However, as remarked by Triggiani [
23], if
X is an infinite dimensional Banach space, then the linear control system is never exactly controllable on the given interval if either
B is compact or associated semigroup is compact. According to Triggiani [
23], this is a typical case for most control systems governed by parabolic partial differential equations and hence the concept of exact controllability is very limited for many parabolic partial differential equations. Nowadays, researchers are engaged to overcome this problem, refer to ([
19,
21,
22]). Very recently, Chalishajar and Acharya [
22] studied the controllability of second order neutral functional differential inclusion, with infinite delay and impulse effect on unbounded domain, without compactness of the family of cosine operators. Ntouyas and O’Regan [
24] gave some remarks on controllability of evolution equations in Banach paces and proved a result without compactness assumption.
In the last few years, researchers have diverted to fractional impulsive equations due to their extensive applications in various fields. Fečkan
et al. [
25] have discussed the existence of PC-mild solutions for Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving Caputo fractional derivative by utilizing the theory of operators semigroup, probability density functions via impulsive conditions, a new concept on a PC-mild solution is introduced in their paper. We refer to the readers the book of Zhou [
26]. Recently, Fu
et al. [
27] studied the existence of PC-mild solutions for Cauchy and nonlocal problems of impulsive fractional evolution equations for which the impulses are not instantaneous, by using the theory of operator semigroups, probability density functions, and some suitable fixed point theorems.
The rest of this paper is organized as follows: In
Section 2, we introduce the system, recall some basic definitions, and preliminary facts that will be used throughout this paper. The existence theorems for semi linear impulsive neutral evolution inclusions with infinite delay, and their proofs are arranged in
Section 3. Finally, in
Section 4, an example is presented to illustrate the applications of the obtained results.
2. Preliminaries
In this paper, we shall consider the existence of mild solutions for first order impulsive partial neutral functional evolution differential inclusions with infinite delay in a Banach space
E
where
is a multivalued map with nonempty compact values,
is the family of all subsets of
E,
and
are given functions,
are given functions and
is a family of linear closed (not necessarily bounded) operators from
E into
E that generate an evolution system of operators
for
.
,
and
represent right and left limits of
at
respectively. For any continuous function
y and any
, we denote by
the element of
defined by
for
. We assume that the histories
belongs to some abstract phase space
to be specified below.
We present the abstract phase space
Assume that
be a continuous function with
Define,
Here,
endowed with the norm
Then, it is easy to show that is a Banach space.
Lemma 1. Suppose then, for each Moreover,where Proof. For any
, it is easy to see that,
is bounded and measurable on
for
, and
Since
, then
. Moreover,
The proof is complete. ▢
Next, we introduce definitions, notation and preliminary facts from multi-valued analysis, which are useful for the development of this paper (see [
28]).
Let
denote the Banach space of all continuous functions from
into
E with the norm
and let
be the Banach space of measurable functions
, that are Lebesgue integrable with the norm
Let
X be a Frechet space with a family of semi-norms
. Let
, we say that
Y is bounded if for every
, there exists
such that
To
X we associate a sequence of Banach spaces
as follows: for every
, we consider the equivalence relation
defined by :
if and only if
for all
. We denote
the quotient space, the completion of
with respect to
. To every
, we associate a sequence the
of subsets
as follows: for every
, we denote
the equivalence class of
x of subset
and we define
. We denote
and
, respectively, the closure, the interior, and the boundary of
with respect to
in
. We assume that the family of semi-norms
verifies:
Let be a metric space. We use the following notations:
, , .
Consider
, given by
where
. Then,
is a metric space and
is a generalized (complete) metric space (see [
29]).
Definition 1. We say that a family generates a unique linear evolution system for satisfying the following properties:- (1)
where I is the identity operator in E,
- (2)
for ,
- (3)
the space of bounded linear operators on E, where for every and for each , the mapping is continuous.
More details on evolution systems and their properties could be found in the books of Ahmed [
30], Engel and Nagel [
31], and Pazy [
32].
Definition 2. A multivalued map is said to be measurable if for each ,
the function defined byis measurable where d is the metric induced by the normed Banach space X. Definition 3. A function is said to be an -Caratheodory multivalued map if it satisfies:- (i)
is continuous(with respect to the metric ) for almost all ;
- (ii)
is measurable for each ;
- (iii)
for every positive constant k there exists such that
A multivalued map
has convex(closed) values if
is convex (closed) for all
. We say that
G is bounded on bounded sets if
is bounded in
X for each bounded set
B of
X,
i.e.,
Finally, we say that G has fixed point if there exists such that .
For each
, let the set
known as the set of selectors from
F defined by
For more details on multivalued maps, we refer to the books of Aubin and Cellina [
33] and Deimling [
34], Gorniewicz [
35], Hu and Papageorgiou [
36], and Tolstonogov [
37].
Definition 4. A multivalued map is called an admissible contraction with constant if for each there exists such that- (i)
for all .
- (ii)
For every and every , there exists such that
The following nonlinear alternative will be used to prove our main results.
Theorem 2.1 (Nonlinear Alternative of Frigon, [
38,
39]). Let
X be a Frechet space and
U an open neighborhood of the origin in
X and let
be an admissible multivalued contraction. Assume that
N is bounded. Then, one of the following statements holds:
- (C1)
N has a fixed point;
- (C2)
There exists and such that .
3. Existence Results
We consider the space
where
is the restriction of
y to
Let
be the semi-norm in
defined by
To prove our existence results for the impulsive neutral functional differential evolution problem with infinite delay . Firstly, we define the mild solution.
Definition 5. We say that the function is a mild solution of the evolution system if for all and the restriction of to the interval J is continuous and there exists a.e in J such that y satisfies the following integral equation: We need to introduce the following hypotheses, which are assumed hereafter:
- (H1)
There exists a constant
such that
- (H2)
The multifunction
is
-Caratheodory with compact and convex values for each
and there exist a function
and a continuous nondecreasing function
such that
- (H3)
For all
, there exists
such that
for each
and for all
with
and
and
- (H4)
There exists a constant
such that
- (H5)
There exists a constant
such that
- (H6)
There exists a constant
such that
- (H7)
There exists a positive constant
such that
- (H8)
There exists a constant
such that
For every
, let us take here
for the family of semi-norm
. In what follows, we fix
and assume
Theorem 3.1 Suppose that hypotheses
are satisfied. Moreover
with
Then, the impulsive neutral evolution problem has a mild solution.
Proof. We transform the Problem
into a fixed point problem. Consider an operator
defined by
where
. Clearly, the fixed points of the operator
N are mild solutions of the Problem
. We remark also that, for each
, the set
is nonempty since
F has a measurable selection by
(see [
40], Theorem III.6).
For
, we will define the function
by
Then, . For each function, . We can decompose y into .
Let
, we have
Thus
is a Banach space, if we set
with the Bielecki-norm on
defined by
where
and
is a constant. Then
is a Frechet space with family of seminorms
. It is obvious that
y satisfies
if and only if
z satisfies
and
where
.
Let us define a multivalued operator
by
where
for a.e
}. Obviously, the operator inclusion
N has a fixed point is equivalent to the operator inclusion
has one, so it turns to prove that
has a fixed point.
Let
be a possible fixed point of the operator
. Given
, then
z should be solution of the inclusion
for some
and there exists
such that, for each
, we have
Assumption
gives
Set
then we have
Using the inequality Equation
and the nondecreasing character of
ψ, we obtain,
Then
Set
Thus,
We consider the function
μ defined by
Let
be such that
. By the previous inequality, we have
Let us take the right-hand side of the above inequality as
. Then, we have
for all
. From the definition of
v, we have
and
Using the nondecreasing character of
ψ, we get
This implies that for each
and using the condition
, we get
Thus, for every
, there exists a constant
such that
and hence
. Since
, we have
.
Set .
Clearly, Ω is an open subset of
. We shall show that
is a contraction and an admissible operator. First, we prove that
is a contraction. Let
and
. Then, there exists
such that for each
,
From (H3) it follows that,
Hence, there is
such that
Consider
, given by
Since the multivalued operator
is measurable (in [
40], see proposition III, 4), there exists a function
, which is a measurable selection for
. Thus,
and using (A1), we obtain for each
Let us define, for each
Then, for each
and
and using (H1) and (H3)-(H6) and (H8), we get
Using (A1) and (7), we obtain
Therefore,
By an analogous relation, obtained by interchanging the roles of
z and
, it follows that
Thus, the operator
is a contraction for all
.
Now, we shall show that
is an admissible operator. Let
. Set, for every
, the space,
and let us consider the multivalued operator
defined by
where
.
From
, and since
F is multivalued map with compact values, we can prove that for every
and there exists
such that
. Let
and
. Assume that
, then we have
Since
h is arbitrary, we may suppose that
Therefore,
If
z is not in
, then
. Since
is compact, there exists
such that
Then, we have
Thus,
Therefore, is an admissible operator contraction.
From the choice of Ω, there is no such that for some . Then, the statement in Theorem does not hold. This implies that the operator has a fixed point . Then is a fixed point of the operator , which is a mild solution of the Problem .
Hence the proof. ▢
4. Example
As an application of Theorem
, we study the following impulsive neutral differential system:
where
is a continuous function and is uniformly Holder continuous in
t;
and
are continuous functions and
is a multivalued map with compact convex values.
Consider and define by with domain .
Then,
generates an evolution system
satisfying assumption (H1)(see [
41]). We can define respectively that
and
Then, in order to prove the existence of mild solutions of the System
, we suppose the following assumptions:
- (i)
u is Lipschitz with respect to its second argument. Let lip(u) denotes the Lipschitz constant of u.
- (ii)
There exist
and a nondecreasing continuous function
such that
- (iii)
are integrable on .
- (iv)
There exist positive constants
and
such that
By the dominated convergence theorem, one can show that
is a continuous function from
to
E. Moreover, the mapping
g is Lipschitz continuous in its argument. In fact, we have
On the other hand, we have for
and
Since the function
ξ is nondecreasing, it follows that
Proposition: Under the above assumptions, if we assume that condition in Theorem is true, , then the System has a mild solution which is defined in .