Existence of Semi Linear Impulsive Neutral Evolution Inclusions with Infinite Delay in Frechet Spaces

Abstract

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct.

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

$$\begin{array}{ccc}\hfill \frac{d}{dt}\left[y\left(t\right)-g(t,y_t)\right]& \in & A\left(t\right)y\left(t\right)+F(t,y_t)\hfill \\ \hfill t\in J=[0,+\infty ),t& \ne & t_k,k=1,2,\cdots \hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\Delta y|}_{t=t_k}& =& I_k\left(y\left(t_k^{-}\right)\right),k=1,2\cdots \hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill y_0& =& \varphi \in {\mathcal{B}}_h\hfill \end{array}$$

(3)

where $F:J\times {\mathcal{B}}_h\to \mathcal{P}\left(E\right)$ is a multivalued map with nonempty compact values, $\mathcal{P}\left(E\right)$ is the family of all subsets of E, $g:J\times {\mathcal{B}}_h\to E$ and $I_k:E\to E,k=1,2,\cdots$ are given functions, $\varphi \in {\mathcal{B}}_h$ are given functions and ${\left\{A\left(t\right)\right\}}_{0\le t<+\infty }$ is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of operators ${\left\{U(t,s)\right\}}_{(t,s)\in J\times J}$ for $0\le s\le t<+\infty$. ${\Delta y|}_{t=t_k}=y\left(t_k^{+}\right)-y\left(t_k^{-}\right)$, $y\left(t_k^{+}\right)$ and $y\left(t_k^{-}\right)$ represent right and left limits of $y\left(t\right)$ at $t=t_k$ respectively. For any continuous function y and any $t\ge 0$, we denote by $y_t$ the element of ${\mathcal{B}}_h$ defined by $y_t\left(\theta \right)=y(t+\theta )$ for $\theta \in (-\infty ,0]$. We assume that the histories $y_t$ belongs to some abstract phase space ${\mathcal{B}}_h$ to be specified below.

We present the abstract phase space ${\mathcal{B}}_h.$ Assume that $h:(-\infty ,0)\to (0,\infty )$ be a continuous function with $l={\int }_{-\infty }^{0}h\left(s\right)ds<+\infty .$ Define,

$$\begin{array}{ccc}\hfill {\mathcal{B}}_h& :=& \{\varphi :[-\infty ,0]\to X\mathrm{such}\mathrm{that},\mathrm{for}\mathrm{any}r>0,\varphi (\theta )\mathrm{is}\mathrm{bounded}\mathrm{and}\mathrm{measurable}\hfill \\ & & \mathrm{function}\mathrm{on}[-r,0]\mathrm{and}{\int }_{-\infty }^{0}h\left(s\right){\sup }_{s\le \theta \le 0}|\varphi \left(\theta \right)|ds<+\infty \}\hfill \end{array}$$

Here, ${\mathcal{B}}_h$ endowed with the norm

$${\parallel \varphi \parallel }_{{\mathcal{B}}_h}={\int }_{-\infty }^{0}h\left(s\right)\underset{s\le \theta \le 0}{sup}|\varphi \left(\theta \right)|ds,\forall \varphi \in {\mathcal{B}}_h$$

Then, it is easy to show that $({\mathcal{B}}_h,\parallel .{\parallel }_{{\mathcal{B}}_h})$ is a Banach space.

Lemma 1. 

Suppose $y\in {\mathcal{B}}_h;$ then, for each $t\in J,y_t\in {\mathcal{B}}_h.$ Moreover,

$$l|y\left(t\right)|\le \parallel y_t{\parallel }_{{\mathcal{B}}_h}\le l\underset{s\le \theta \le 0}{sup}(|y\left(s\right)|+\parallel y_0{\parallel }_{{\mathcal{B}}_h})$$

where $l:={\int }_{-\infty }^{0}h\left(s\right)ds<+\infty .$

Proof. 

For any $t\in [0,a]$, it is easy to see that, $y_t$ is bounded and measurable on $[-a,0]$ for $a>0$, and

$$\begin{array}{ccc}\hfill \parallel y_t{\parallel }_{{\mathcal{B}}_h}& =& {\int }_{-\infty }^{0}h\left(s\right)\underset{\theta \in [s,0]}{sup}|y_t\left(\theta \right)|ds\hfill \\ & =& {\int }_{-\infty }^{-t}h\left(s\right)\underset{\theta \in [s,0]}{sup}|y(t+\theta )|ds+{\int }_{-t}^{0}h\left(s\right)\underset{\theta \in [s,0]}{sup}|y(t+\theta )|ds\hfill \\ & =& {\int }_{-\infty }^{-t}h\left(s\right)\underset{{\theta }_1\in [t+s,t]}{sup}|y\left({\theta }_1\right)|ds+{\int }_{-t}^{0}h\left(s\right)\underset{{\theta }_1\in [t+s,t]}{sup}|y\left({\theta }_1\right)|ds\hfill \\ & \le & {\int }_{-\infty }^{-t}h\left(s\right)\left[\underset{{\theta }_1\in [t+s,0]}{sup}|y\left({\theta }_1\right)|+\underset{{\theta }_1\in [0,t]}{sup}|y\left({\theta }_1\right)|\right]ds+{\int }_{-t}^{0}h\left(s\right)\underset{{\theta }_1\in [0,t]}{sup}|y\left({\theta }_1\right)|ds\hfill \\ & =& {\int }_{-\infty }^{-t}h\left(s\right)\underset{{\theta }_1\in [t+s,0]}{sup}|y\left({\theta }_1\right)|ds+{\int }_{-\infty }^{0}h\left(s\right)ds.\underset{s\in [0,t]}{sup}|y\left(s\right)|\hfill \\ & \le & {\int }_{-\infty }^{-t}h\left(s\right)\underset{{\theta }_1\in [s,0]}{sup}|y\left({\theta }_1\right)|ds+l.\underset{s\in [0,t]}{sup}|y\left(s\right)|\hfill \\ & \le & {\int }_{-\infty }^{0}h\left(s\right)\underset{{\theta }_1\in [s,0]}{sup}|y\left({\theta }_1\right)|ds+l\underset{s\in [0,t]}{sup}|y\left(s\right)|\hfill \\ & =& {\int }_{-\infty }^{0}h\left(s\right)\underset{{\theta }_1\in [s,0]}{sup}|y_0\left({\theta }_1\right)|ds+l\underset{s\in [0,t]}{sup}|y\left(s\right)|\hfill \\ & =& l\underset{s\in [0,t]}{sup}|y\left(s\right)|+\parallel y_0{\parallel }_{{\mathcal{B}}_h}\hfill \end{array}$$

Since $\varphi \in {\mathcal{B}}_h$, then $y_t\in {\mathcal{B}}_h$. Moreover,

$$\parallel y_t{\parallel }_{{\mathcal{B}}_h}={\int }_{-\infty }^{0}h\left(s\right)\underset{\theta \in [s,0]}{sup}|y_t\left(\theta \right)|ds\ge |y_t\left(\theta \right)|{\int }_{-\infty }^{0}h\left(s\right)ds=l|y\left(t\right)|$$

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 $C\left(\right[0,b],E)$ denote the Banach space of all continuous functions from $[0,b]$ into E with the norm

$${\parallel y\parallel }_{\infty }=sup\{\parallel y\left(t\right)\parallel :0\le t\le b\}$$

and let $L^1([0,\infty ),E)$ be the Banach space of measurable functions $y:[0,\infty )\to E$, that are Lebesgue integrable with the norm

$$\begin{array}{c}\hfill {\parallel y\parallel }_{L^1}={\int }_0^{\infty }\parallel y\left(t\right)\parallel dt\text{for}\text{all}y\in L^1([0,\infty ),E)\end{array}$$

Let X be a Frechet space with a family of semi-norms ${\{\parallel ·\parallel }_n{\}}_{n\in \mathbb{N}}$. Let $Y\subset X$, we say that Y is bounded if for every $n\in \mathbb{N}$, there exists $\overline{M_n}>0$ such that

$$\begin{array}{c}\hfill {\parallel y\parallel }_n\le \overline{M_n}\text{for}\text{all}y\in Y\end{array}$$

To X we associate a sequence of Banach spaces $\left\{\right(X^n{,\parallel ·\parallel }_n\left)\right\}$ as follows: for every $n\in \mathbb{N}$, we consider the equivalence relation ${\sim }_n$ defined by :$x{\sim }_{n}y$ if and only if ${\parallel x-y\parallel }_n=0$ for all $x,y\in X$. We denote $X^n={(X|}_{{\sim }_n}{,\parallel ·\parallel }_n)$ the quotient space, the completion of $X^n$ with respect to ${\parallel ·\parallel }_n$. To every $Y\subset X$, we associate a sequence the $\left\{Y^n\right\}$ of subsets $Y^n\subset X^n$ as follows: for every $x\in X$, we denote ${\left[x\right]}_n$ the equivalence class of x of subset $X^n$ and we define $Y^n=\{{\left[x\right]}_n:x\in Y\}$. We denote $\overline{Y_n},\text{int}\left(Y^n\right)$ and ${\partial }_n{Y}^n$, respectively, the closure, the interior, and the boundary of $Y^n$ with respect to $\parallel ·\parallel$ in $X^n$. We assume that the family of semi-norms $\{\parallel ·{\parallel }_n\}$ verifies:

$$\begin{array}{c}\hfill {\parallel x\parallel }_1\le {\parallel x\parallel }_2\le {\parallel x\parallel }_3\le \cdots \text{for}\text{every}x\in X\end{array}$$

Let $(X,d)$ be a metric space. We use the following notations: ${\mathcal{P}}_{cl}\left(X\right):=\{Y\in \mathcal{P}\left(X\right):Y\text{closed}\}$, ${\mathcal{P}}_b\left(X\right):=\{Y\in \mathcal{P}\left(X\right):Y\text{bounded}\}$ ${\mathcal{P}}_{cv}\left(X\right):=\{Y\in \mathcal{P}\left(X\right):Y\text{convex}\}$, ${\mathcal{P}}_{cp}\left(X\right):=\{Y\in \mathcal{P}\left(X\right):Y\text{compact}\}$.

Consider $H_d:\mathcal{P}\left(X\right)\times \mathcal{P}\left(X\right)\to R^{+}\cup \left\{\infty \right\}$, given by

$$\begin{array}{c}\hfill H_d(\mathcal{A},\mathcal{B}):=max\{\underset{a\in \mathcal{A}}{sup}d(a,\mathcal{B}),\underset{b\in \mathcal{B}}{sup}d(\mathcal{A},b)\}\end{array}$$

where $d(\mathcal{A},b):={inf}_{a\in \mathcal{A}}d(a,b),d(a,\mathcal{B}):={inf}_{b\in \mathcal{B}}d(a,b)$. Then, $({\mathcal{P}}_{b,cl}\left(X\right),H_d)$ is a metric space and $({\mathcal{P}}_{cl}\left(X\right),H_d)$ is a generalized (complete) metric space (see [29]).

Definition 1. 

We say that a family ${\left\{A\left(t\right)\right\}}_{t\ge 0}$ generates a unique linear evolution system ${\left\{U(t,s)\right\}}_{(t,s)\in \Delta }$ for ${\Delta }_1=\{(t,s)\in J\times J:0\le s\le t<+\infty \}$ satisfying the following properties:

(1) 

$U(t,t)=I$ where I is the identity operator in E,

(2) 

$U(t,s)U(s,\tau )=U(t,\tau )$ for $0\le \tau \le s\le t<+\infty$,

(3) 

$U(t,s)\in B\left(E\right)$ the space of bounded linear operators on E, where for every $(t,s)\in {\Delta }_1$ and for each $y\in E$, the mapping $(t,s)\to U(t,s)y$ 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 $G:J\to {\mathcal{P}}_{cl}\left(X\right)$ is said to be measurable if for each $x\in E$, the function $Y:J\to X$ defined by

$$\begin{array}{c}\hfill Y\left(t\right)=d(x,G(t\left)\right)=inf\{|x-z|:z\in G(t\left)\right\}\end{array}$$

is measurable where d is the metric induced by the normed Banach space X.

Definition 3. 

A function $F:J\times {\mathcal{B}}_h\to \mathcal{P}\left(X\right)$ is said to be an $L_{loc}^1$-Caratheodory multivalued map if it satisfies:

(i) 

$x\to F(t,y)$ is continuous(with respect to the metric $H_d$) for almost all $t\in J$;

(ii) 

$t\to F(t,y)$ is measurable for each $y\in {\mathcal{B}}_h$;

(iii) 

for every positive constant k there exists $h_k\in L_{loc}^1(J;R^{+})$ such that

$$\begin{array}{c}\hfill \parallel F(t,y)\parallel \le h_k\left(t\right)forall{\parallel y\parallel }_{{\mathcal{B}}_h}\le kandforalmostallt\in J\end{array}$$

A multivalued map $G:X\to \mathcal{P}\left(X\right)$ has convex(closed) values if $G\left(x\right)$ is convex (closed) for all $x\in X$. We say that G is bounded on bounded sets if $G\left(B\right)$ is bounded in X for each bounded set B of X, i.e.,

$$\begin{array}{c}\hfill \underset{x\in B}{sup}\{sup\{\parallel y\parallel :y\in G\left(x\right)\left\}\right\}<\infty \end{array}$$

Finally, we say that G has fixed point if there exists $x\in X$ such that $x\in G\left(x\right)$.

For each $y\in B_{*}$, let the set $S_{F,y}$ known as the set of selectors from F defined by

$$\begin{array}{c}\hfill S_{F,y}=\{v\in L^1(J;E):v\left(t\right)\in F(t,y_t),\text{a.e}.t\in J\}\end{array}$$

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 $F:X\to \mathcal{P}\left(X\right)$ is called an admissible contraction with constant ${\left\{k_n\right\}}_{n\in \mathbb{N}}$ if for each $n\in \mathbb{N}$ there exists $k_n\in (0,1)$ such that

(i) 

$H_d(F\left(x\right),F\left(y\right))\le k_n{\parallel x-y\parallel }_n$ for all $x,y\in X$.

(ii) 

For every $x\in X$ and every $ϵ\in {(0,\infty )}^n$, there exists $y\in F\left(x\right)$ such that

$$\begin{array}{c}\hfill {\parallel x-y\parallel }_n\le {\parallel x-F\left(x\right)\parallel }_n+{ϵ}_n\mathit{\text{for}}\mathit{\text{every}}n\in \mathbb{N}\end{array}$$

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 $N:\overline{U}\to \mathcal{P}\left(X\right)$ 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 $\lambda \in [0,1)$ and $x\in \partial U$ such that $x\in \lambda N\left(x\right)$.

3. Existence Results

We consider the space

$$\begin{array}{c}\hfill PC=\{y:(-\infty ,\infty )\to E|y\left(t_k^{-}\right)\text{and}y\left(t_k^{-}\right)\text{exist}\text{with}y\left(t_k\right)=y\left(t_k^{-}\right),\\ \hfill y\left(t\right)=\varphi \left(t\right)\text{for}t\in (-\infty ,\infty ),y_k\in C(J_k,E),k=1,2,3,\cdots \}\end{array}$$

where $y_k$ is the restriction of y to $J_k=(t_k,t_k+1],k=1,2,3,\cdots$

Now, we set

$$\begin{array}{c}\hfill B_{*}=\{y:(-\infty ,\infty )\to E:y\in PC\cap {\mathcal{B}}_h\}\\ \hfill B_k=\{y\in B_{*}:\underset{t\in J_k^{*}}{sup}|y\left(t\right)|<\infty \},\text{where}{J}_k^{*}=(-\infty ,t_k]\end{array}$$

Let ${\parallel ·\parallel }_k$ be the semi-norm in $B_k$ defined by

$$\begin{array}{c}\hfill {\parallel y\parallel }_k=\parallel y_0{\parallel }_{{\mathcal{B}}_h}+sup\{|y\left(s\right)|:0\le s\le t_k\},y\in B_k\end{array}$$

To prove our existence results for the impulsive neutral functional differential evolution problem with infinite delay $\left(1\right)-\left(3\right)$. Firstly, we define the mild solution.

Definition 5. 

We say that the function $y(·):(-\infty ,+\infty )\to E$ is a mild solution of the evolution system $\left(1\right)-\left(3\right)$ if $y\left(t\right)=\varphi \left(t\right)$ for all ${t\in (-\infty ,0],\Delta y|}_{t=t_k}=I_k\left(y\left(t_k^{-}\right)\right),k=1,2,\cdots$ and the restriction of $y(·)$ to the interval J is continuous and there exists $f(·)\in L^1(J;E):f\left(t\right)\in F(t,y_t)$ a.e in J such that y satisfies the following integral equation:

$$\begin{array}{ccc}\hfill y\left(t\right)& =& U(t,0)\left[\varphi \left(0\right)-g(0,\varphi )\right]+g(t,y_t)+{\int }_0^{t}U(t,s)A\left(s\right)g(s,y_s)ds\hfill \\ & & +{\int }_0^{t}U(t,s)f\left(s\right)ds+\sum _{0<t_k<t}U(t,t_k)I_k\left(y\left(t_k^{-}\right)\right),foreacht\in [0,+\infty )\hfill \end{array}$$

(4)

We need to introduce the following hypotheses, which are assumed hereafter:

(H1)

There exists a constant $M_1\ge 1$ such that

$${\parallel U(t,s)\parallel }_{B\left(E\right)}\le M_1\text{for}\text{every}(t,s)\in {\Delta }_1$$

(H2)

The multifunction $F:J\times {\mathcal{B}}_h\to \mathcal{P}\left(E\right)$ is $L_{loc}^1$-Caratheodory with compact and convex values for each $u\in {\mathcal{B}}_h$ and there exist a function $p\in L_{loc}^1(J,R^{+})$ and a continuous nondecreasing function $\psi :J\to (0,\infty )$ such that

$$\begin{array}{c}{\parallel F(t,u)\parallel }_{\mathcal{P}\left(E\right)}\le {p\left(t\right)\psi (\parallel u\parallel }_{{\mathcal{B}}_h})\text{for}\text{a.e}t\in J\text{and}\text{each}u\in {\mathcal{B}}_h\end{array}$$

(H3)

For all $r>0$, there exists $l_r\in L_{loc}^1(J;R^{+})$ such that

$$\begin{array}{c}{H}_d(F(t,u)-F(t,v))\le l_r\left(t\right){\parallel u-v\parallel }_{{\mathcal{B}}_h}\end{array}$$

for each $t\in J$ and for all $u,v\in {\mathcal{B}}_h$ with ${\parallel u\parallel }_{{\mathcal{B}}_h}\le r$ and ${\parallel v\parallel }_{{\mathcal{B}}_h}\le r$ and

$$\begin{array}{c}d(0,F(t,0))\le l_r\left(t\right)\text{a.e}t\in J\end{array}$$

(H4)

There exists a constant $M_0>0$ such that

$$\begin{array}{c}\parallel A^{-1}{\left(t\right)\parallel }_{B\left(E\right)}\le M_0\text{for}\text{all}t\in J\end{array}$$

(H5)

There exists a constant $d_k>0,k=1,2,\cdots$ such that

$$\begin{array}{c}\hfill \parallel I_k\left(x\right)-I_k\left(\overline{x}\right)\parallel \le d_k\parallel x-\overline{x}\parallel \text{for}\text{each}k=1,2,\cdots \text{for}\text{all}x,\overline{x}\in E\end{array}$$

(H6)

There exists a constant $0<L<\frac{1}{M_0{K}_n}$ such that

$$\parallel A\left(t\right)g(t,\varphi )\parallel \le {L(\parallel \varphi \parallel }_{{\mathcal{B}}_h}+1)\mathrm{for}\mathrm{all}t\in J,\varphi \in {\mathcal{B}}_h$$

(H7)

There exists a positive constant $c_k,k=1,2,\cdots$ such that

$$\begin{array}{c}\hfill \parallel I_k\left(z\right)\parallel \le c_k\text{for}\text{all}z\in E,\sum _{k=1}^{\infty }{c}_k<\infty \end{array}$$

(H8)

There exists a constant $L_{*}>0$ such that

$$\begin{array}{c}\parallel A\left(s\right)g(s,\varphi )-A\left(\overline{s}\right)g(\overline{s},\overline{\varphi })\parallel \le L_{*}(|s-\overline{s}|+\parallel \varphi -\overline{\varphi }{\parallel }_{{\mathcal{B}}_h})\text{for}\text{all}s,\overline{s}\in J\text{and}\varphi ,\overline{\varphi }\in {\mathcal{B}}_h\end{array}$$

For every $n\in \mathbb{N}$, let us take here $\overline{l_n}\left(t\right)=M_1{K}_n\left[L_{*}+l_n\left(t\right)\right]$ for the family of semi-norm ${\{\parallel ·\parallel }_n{\}}_{n\in \mathbb{N}}$. In what follows, we fix $\tau >1$ and assume

$$\begin{array}{c}\left[M_0{L}_{*}{K}_n+\frac{1}{\tau }+M_1\sum _{k=1}^m{d}_k\right]<1\end{array}$$

Theorem 3.1 

Suppose that hypotheses $\left(\mathrm{H}1\right)-\left(\mathrm{H}8\right)$ are satisfied. Moreover

$$\begin{array}{c}\hfill {\int }_{{\delta }_n}^{+\infty }\frac{ds}{s+\psi \left(s\right)}>\frac{M_1{K}_n}{1-M_{0}L{K}_n}{\int }_0^{n}max(L,p\left(s\right))ds\text{for}\text{each}n\in \mathbb{N}\end{array}$$

(5)

with

$$\begin{array}{ccc}\hfill {\delta }_n& =& (K_n{M}_{1}H+M_n){\parallel \varphi \parallel }_{{\mathcal{B}}_h}+\frac{K_n}{1-M_{0}L{K}_n}[(M_1+1)M_{0}L+M_{1}Ln\hfill \\ & & +M_{0}L[M_1(K_{n}H+1)+M_n]{\parallel \varphi \parallel }_{{\mathcal{B}}_h}+M_1\sum _{k=1}^m{c}_k]\hfill \end{array}$$

Then, the impulsive neutral evolution problem $\left(1\right)-\left(3\right)$ has a mild solution.

Proof. 

We transform the Problem $\left(1\right)-\left(3\right)$ into a fixed point problem. Consider an operator $N:B_{*}\to B_{*}$ defined by

$$\begin{array}{c}\hfill N\left(y\right)=h\in B_{*}:h\left(t\right)=\left\{\begin{array}{c}\varphi \left(t\right)\text{if}t\le 0\hfill \\ U(t,0)[\varphi \left(0\right)-g(0,\varphi )]+g(t,y_t)+{\int }_0^{t}U(t,s)A\left(s\right)g(s,y_s)ds\hfill \\ +{\int }_0^{t}U(t,s)f\left(s\right)ds+\sum _{0<t_k<t}U(t,t_k)I_k\left(y\left(t_k^{-}\right)\right),t\in J\hfill \end{array}\right.\end{array}$$

where $f\in S_{F,y}=\{v\in L^1(J,E):v\left(t\right)\in F(t,y_t)\text{for}\text{a.e}t\in J\}$. Clearly, the fixed points of the operator N are mild solutions of the Problem $\left(1\right)-\left(3\right)$. We remark also that, for each $y\in B_{*}$, the set $S_{F,y}$ is nonempty since F has a measurable selection by $\left(\mathrm{H}2\right)$ (see [40], Theorem III.6).

For $\varphi \in {\mathcal{B}}_h$, we will define the function $x(·):(-\infty ,+\infty )\to E$ by

$$\begin{array}{c}\hfill x\left(t\right)=\left\{\begin{array}{c}\varphi \left(t\right),\text{if}t\in (-\infty ,0]\hfill \\ U(t,0)\varphi \left(0\right),\text{if}t\in J\hfill \end{array}\right.\end{array}$$

Then, $x_0=\varphi$. For each function, $z\in C\left(\right[0,\infty ),E)$. We can decompose y into $y\left(t\right)=z\left(t\right)+x\left(t\right)$.

Let $B_{*}^k=\{z\in B_k:z_0=0\}.\text{Then,}\text{for}\text{any}z\in B_{*}^k$, we have

$$\begin{array}{c}\hfill {\parallel z\parallel }_k=\parallel z_0{\parallel }_{{\mathcal{B}}_h}+sup\{\parallel z\left(s\right)\parallel :0\le s\le t_k\parallel =sup\{\parallel z\left(s\right)\parallel :0\le s\le t_k\}\end{array}$$

Thus $(B_{*}^k,\parallel ·{\parallel }_k)$ is a Banach space, if we set $C_1=\{z\in B_{*};z_0=0\}$ with the Bielecki-norm on $B_{*}^k$ defined by

$$\begin{array}{c}\hfill {\parallel z\parallel }_{B_{*}^k}=max\{\parallel z\left(t\right)\parallel e^{-\tau L_n^{*}\left(t\right)}:t\in [0,t_k]\}\end{array}$$

where $L_n^{*}\left(t\right)={\int }_0^t{\overline{l}}_n\left(s\right)ds,{\overline{l}}_n\left(t\right)=M_1{K}_n{l}_n\left(t\right)$ and $\tau >0$ is a constant. Then $C_1$ is a Frechet space with family of seminorms ${\parallel ·\parallel }_{B_{*}^k}$. It is obvious that y satisfies $\left(4\right)$ if and only if z satisfies $z_0=0$ and

$$\begin{array}{ccc}\hfill z\left(t\right)& =& g(t,z_t+x_t)-U(t,0)g(0,\varphi )+{\int }_0^{t}U(t,s)A\left(s\right)g(s,z_s+x_s)ds+{\int }_0^{t}U(t,s)f\left(s\right)ds\hfill \\ & & +\sum _{0<t_k<t}U(t,t_k)I_k(z\left(t_k^{+}\right)+x\left(t_k^{-}\right))\hfill \end{array}$$

where $f\left(t\right)\in F(t,z_t+x_t)\text{a.e}t\in J$.

Let us define a multivalued operator $\mathcal{F}:C_1\to C_1$ by

$$\begin{array}{ccc}\hfill \mathcal{F}\left(z\right)& =& h\in B_{*}^k:h\left(t\right)=g(t,z_t+x_t)-U(t,0)g(0,\varphi )+{\int }_0^{t}U(t,s)A\left(s\right)g(s,z_s+x_s)ds\hfill \\ & & +{\int }_0^{t}U(t,s)f\left(s\right)ds+\sum _{0<t_k<t}U(t,t_k)I_k(z\left(t_k^{-}\right),x\left(t_k^{-}\right)),t\in J\hfill \end{array}$$

where $f\in S_{F,z}=\{v\in L^1(J,E):v\left(t\right)\in F(t,z_t+x_t)$ for a.e $t\in J$}. Obviously, the operator inclusion N has a fixed point is equivalent to the operator inclusion $\mathcal{F}$ has one, so it turns to prove that $\mathcal{F}$ has a fixed point.

Let $z\in B_{*}^k$ be a possible fixed point of the operator $\mathcal{F}$. Given $n\in \mathbb{N}$, then z should be solution of the inclusion $z\in \lambda \mathcal{F}\left(z\right)$ for some $\lambda \in (0,1)$ and there exists $f\in S_{F,z}\iff f\left(t\right)\in F(t,z_t+x_t)$ such that, for each $t\in [0,n]$, we have

$$\begin{array}{ccc}\hfill \parallel z\left(t\right)\parallel & =& \parallel A^{-1}{\left(t\right)\parallel }_{B\left(E\right)}\parallel A\left(t\right)g(t,z_t+x_t)\parallel +\parallel U(t,0){\parallel }_{B\left(E\right)}\parallel A^{-1}\left(0\right){\parallel }_{B\left(E\right)}\parallel A\left(0\right)g(0,\varphi )\parallel \hfill \\ & & +{\int }_0^t{\parallel U(t,s)\parallel }_{B\left(E\right)}\parallel A\left(s\right)g(s,z_s+x_s)\parallel ds+{\int }_0^t\parallel U(t,s){\parallel }_{B\left(E\right)}\parallel f\left(s\right)\parallel ds\hfill \\ & & +\sum _{0<t_k<t}\parallel U(t,t_k){\parallel }_{B\left(E\right)}\parallel I_k(z\left(t_k^{-}\right)+x\left(t_k^{-}\right))\parallel \hfill \\ & \le & M_{0}L(\parallel z_t+x_t{\parallel }_{{\mathcal{B}}_h}+1)+M_1{M}_0{L(\parallel \varphi \parallel }_{{\mathcal{B}}_h}+1)+M_1{\int }_0^{t}L(\parallel z_s+x_s{\parallel }_{{\mathcal{B}}_h}+1)ds\hfill \\ & & +M_1{\int }_0^{t}p\left(s\right)\psi (\parallel z_s+x_s{\parallel }_{{\mathcal{B}}_h})ds+M_1\sum _{k=1}^m{c}_k\hfill \\ & \le & M_{0}L\parallel z_t+x_t{\parallel }_{{\mathcal{B}}_h}+M_{0}L(1+M_1)+M_{1}Ln+M_1{M}_0{L\parallel \varphi \parallel }_{{\mathcal{B}}_h}+M_1{\int }_0^{t}L{\parallel z_s+x_s\parallel }_{{\mathcal{B}}_h}ds\hfill \\ & & +M_1{\int }_0^{t}p\left(s\right)\psi (\parallel z_s+x_s{\parallel }_{{\mathcal{B}}_h})ds+M_1\sum _{k=1}^m{c}_k\hfill \end{array}$$

Assumption $\left(A1\right)$ gives

$$\begin{array}{ccc}\hfill \parallel z_s+x_s{\parallel }_{{\mathcal{B}}_h}& \le & \parallel z_s{\parallel }_{{\mathcal{B}}_h}+{\parallel x_s\parallel }_{{\mathcal{B}}_h}\hfill \\ & \le & K\left(s\right)\parallel z\left(s\right)\parallel +M\left(s\right)\parallel z_0{\parallel }_{{\mathcal{B}}_h}+K\left(s\right)\parallel x\left(s\right)\parallel +M\left(s\right)\parallel x_0{\parallel }_{{\mathcal{B}}_h}\hfill \\ & \le & K_n\parallel z\left(s\right)\parallel +K_n{\parallel U(s,0)\parallel }_{B\left(E\right)}\parallel \varphi \left(0\right)\parallel +M_n{\parallel \varphi \parallel }_{{\mathcal{B}}_h}\hfill \\ & \le & K_n\parallel z\left(s\right)\parallel +K_n{M}_1\parallel \varphi \left(0\right)\parallel +M_n{\parallel \varphi \parallel }_{{\mathcal{B}}_h}\hfill \\ & \le & K_n\parallel z\left(s\right)\parallel +K_n{M}_1{H\parallel \varphi \parallel }_{{\mathcal{B}}_h}+M_n{\parallel \varphi \parallel }_{{\mathcal{B}}_h}\hfill \\ & \le & K_n\parallel z\left(s\right)\parallel +(K_n{M}_{1}H+M_n){\parallel \varphi \parallel }_{{\mathcal{B}}_h}\hfill \end{array}$$

Set

$$\begin{array}{c}\hfill C_n=(K_n{M}_{1}H+M_n){\parallel \varphi \parallel }_{{\mathcal{B}}_h}\end{array}$$

then we have

$$\begin{array}{c}\hfill \parallel x_s+z_s{\parallel }_{{\mathcal{B}}_h}\le K_n\parallel z\left(s\right)\parallel +C_n\end{array}$$

(6)

Using the inequality Equation $\left(6\right)$ and the nondecreasing character of ψ, we obtain,

$$\begin{array}{ccc}\hfill \parallel z\left(t\right)\parallel & \le & M_{0}L(K_n\parallel z\left(t\right)\parallel +C_n)+M_{0}L(1+M_1)+M_{1}Ln+M_1{M}_{0}L{\parallel \varphi \parallel }_{{\mathcal{B}}_h}\hfill \\ & & +M_1{\int }_0^{t}L(K_n\parallel z\left(s\right)\parallel +C_n)ds+M_1{\int }_0^{t}p\left(s\right)\psi (K_n\parallel z\left(s\right)\parallel +C_n)ds\hfill \\ & & +M_1\sum _{k=1}^m{c}_k\hfill \\ & \le & M_{0}L{K}_n\parallel z\left(t\right)\parallel +M_{0}L(1+M_1)+M_{1}Ln+M_{0}L{C}_n+M_1{M}_{0}L{\parallel \varphi \parallel }_{{\mathcal{B}}_h}\hfill \\ & & +M_1\left[{\int }_0^{t}L(K_n\parallel z\left(s\right)\parallel +C_n)ds+{\int }_0^{t}p\left(s\right)\psi (K_n\parallel z\left(s\right)\parallel +C_n)ds\right]\hfill \\ & & +M_1\sum _{k=1}^m{c}_k\hfill \end{array}$$

Then

$$\begin{array}{ccc}\hfill (1-M_{0}L{K}_n)\parallel z\left(t\right)\parallel & \le & (M_1+1)M_{0}L+MLn+M_{0}L{C}_n+M_1{M}_{0}L{\parallel \varphi \parallel }_{{\mathcal{B}}_h}+M_1\sum _{k=1}^m{c}_k\hfill \\ & & +M_1\left[{\int }_0^{t}L(K_n\parallel z\left(s\right)\parallel +C_n)ds+{\int }_0^{t}p\left(s\right)\psi (K_n\parallel z\left(s\right)\parallel +C_n)ds\right]\hfill \end{array}$$

Set

$$\begin{array}{c}\hfill {\delta }_n=C_n+\frac{K_n}{1-M_{0}L{K}_n}\left[(M_1+1)M_{0}L+MLn+M_{0}L{C}_n+M_1{M}_{0}L{\parallel \varphi \parallel }_{{\mathcal{B}}_h}+M_1\sum _{k=1}^m{c}_k\right]\end{array}$$

Thus,

$$\begin{array}{c}\hfill K_n\parallel z\left(t\right)\parallel +C_n\le {\delta }_n+\frac{M_1{K}_n}{1-M_{0}L{K}_n}\left[{\int }_0^{t}L(K_n\parallel z\left(s\right)\parallel +C_n)ds+{\int }_0^{t}p\left(s\right)\psi (K_n\parallel z\left(s\right)\parallel +C_n)ds\right]\end{array}$$

We consider the function μ defined by

$$\begin{array}{c}\hfill \mu \left(t\right)=sup\{K_n\parallel z\left(s\right)\parallel +C_n:0\le s\le t\},0\le t<+\infty \end{array}$$

Let $t^{*}\in [0,t]$ be such that $\mu \left(t\right)=K_n\parallel z\left(t^{*}\right)\parallel +C_n$. By the previous inequality, we have

$$\begin{array}{c}\hfill \mu \left(t\right)\le {\delta }_n+\frac{M_1{K}_n}{1-M_{0}L{K}_n}\left[{\int }_0^{t}L\mu \left(s\right)ds+{\int }_0^{t}p\left(s\right)\psi \left(\mu \left(s\right)\right)ds\right]\text{for}t\in [0,n]\end{array}$$

Let us take the right-hand side of the above inequality as $v\left(t\right)$. Then, we have $\mu \left(t\right)\le v\left(t\right)$ for all $t\in [0,n]$. From the definition of v, we have $v\left(0\right)={\delta }_n$ and

$$\begin{array}{c}\hfill v^{\prime }\left(t\right)=\frac{M_1{K}_n}{1-M_{0}L{K}_n}[L\mu \left(t\right)+p\left(t\right)\psi \left(\mu \left(t\right)\right)]\text{a.e}t\in [0,n]\end{array}$$

Using the nondecreasing character of ψ, we get

$$\begin{array}{c}\hfill v^{\prime }\left(t\right)=\frac{M_1{K}_n}{1-M_{0}L{K}_n}[Lv\left(t\right)+p\left(t\right)\psi \left(v\left(t\right)\right)]\text{a.e}t\in [0,n]\end{array}$$

This implies that for each $t\in [0,n]$ and using the condition $\left(5\right)$, we get

$$\begin{array}{ccc}\hfill {\int }_{{\delta }_n}^{v\left(t\right)}\frac{ds}{s+\psi \left(s\right)}& \le & \frac{M_1{K}_n}{1-M_{0}L{K}_n}{\int }_0^{t}max(L,p\left(s\right))ds\hfill \\ & \le & \frac{M_1{K}_n}{1-M_{0}L{K}_n}{\int }_0^{n}max(L,p\left(s\right))ds\hfill \\ & <& {\int }_{{\delta }_n}^{+\infty }\frac{ds}{s+\psi \left(s\right)}\hfill \end{array}$$

Thus, for every $t\in [0,n]$, there exists a constant ${\Lambda }_n$ such that $v\left(t\right)\le {\Lambda }_n$ and hence $\mu \left(t\right)\le {\Lambda }_n$. Since ${\parallel z\parallel }_{B_{*}^k}\le \mu \left(t\right)$, we have ${\parallel z\parallel }_{B_{*}^k}\le {\Lambda }_n$.

Set $\Omega =\{z\in B_{*}^k:sup\{\parallel z\left(t\right)\parallel :0\le t\le n\}<{\Lambda }_n+1\text{for}\text{all}n\in \mathbb{N}\}$.

Clearly, Ω is an open subset of $B_{*}^k$. We shall show that $\mathcal{F}:\overline{\Omega }\to \mathcal{P}\left(B_{*}^k\right)$ is a contraction and an admissible operator. First, we prove that $\mathcal{F}$ is a contraction. Let $z,\overline{z}\in B_{*}^k$ and $h\in \mathcal{F}\left(\overline{z}\right)$. Then, there exists $f\left(t\right)\in F(t,z_t+x_t)$ such that for each $t\in [0,n]$,

$$\begin{array}{ccc}\hfill h\left(t\right)& =& g(t,z_t+x_t)-U(t,0)g(0,\varphi )+{\int }_0^{t}U(t,s)A\left(s\right)g(s,z_s+x_s)ds\hfill \\ & & +{\int }_0^{t}U(t,s)f\left(s\right)ds+\sum _{0<t_k<t}U(t,t_k)I_k(z\left(t_k^{-}\right)+x\left(t_k^{-}\right))\hfill \end{array}$$

From (H3) it follows that,

$$\begin{array}{c}\hfill H_d(F(t,z_t+x_t),F(t,{\overline{z}}_t+x_t))\le l_n\left(t\right){\parallel z_t-{\overline{z}}_t\parallel }_{{\mathcal{B}}_h}\end{array}$$

Hence, there is $\rho \in F(t,{\overline{z}}_t+x_t)$ such that

$$\begin{array}{c}\hfill \parallel f\left(t\right)-\rho \parallel \le l_n\left(t\right){\parallel z_t-{\overline{z}}_t\parallel }_{{\mathcal{B}}_h}t\in [0,n]\end{array}$$

Consider ${\Omega }_{*}:[0,n]\to \mathcal{P}\left(E\right)$, given by

$$\begin{array}{c}\hfill {\Omega }_{*}=\{\rho \in E:\parallel f\left(t\right)-\rho \parallel \le l_n\left(t\right)\parallel z_t-{\overline{z}}_t\parallel \}\end{array}$$

Since the multivalued operator $\mathcal{V}\left(t\right)={\Omega }_{*}\left(t\right)\bigcap F(t,{\overline{z}}_t+x_t)$ is measurable (in [40], see proposition III, 4), there exists a function $\overline{f}\left(t\right)$, which is a measurable selection for $\mathcal{V}$. Thus, $\overline{f}\left(t\right)\in F(t,{\overline{z}}_t+x_t)$ and using (A1), we obtain for each $t\in [0,n]$

$$\begin{array}{ccc}\hfill \parallel f\left(t\right)-\overline{f}\left(t\right)\parallel & \le & l_n\left(t\right){\parallel z_t-{\overline{z}}_t\parallel }_{{\mathcal{B}}_h}\hfill \\ & \le & l_n\left(t\right)\left[K\left(t\right)\parallel z\left(t\right)-\overline{z}\left(t\right)\parallel +M\left(t\right)\parallel z_0-{\overline{z}}_0{\parallel }_{{\mathcal{B}}_h}\right]\hfill \\ & \le & l_n\left(t\right)K_n\parallel z\left(t\right)-\overline{z}\left(t\right)\parallel \hfill \end{array}$$

(7)

Let us define, for each $t\in [0,n]$

$$\begin{array}{ccc}\hfill \overline{h}\left(t\right)& =& g(t,{\overline{z}}_t+x_t)-U(t,0)g(0,\varphi )+{\int }_0^{t}U(t,s)A\left(s\right)g(s,{\overline{z}}_s+x_s)ds\hfill \\ & & +{\int }_0^{t}U(t,s)\overline{f}\left(s\right)ds+\sum _{0<t_k<t}U(t,t_k)I_k(\overline{z}\left(t_k^{-}\right)+x\left(t_k^{-}\right))\hfill \end{array}$$

Then, for each $t\in [0,n]$ and $n\in \mathbb{N}$ and using (H1) and (H3)-(H6) and (H8), we get

$$\begin{array}{ccc}\hfill \parallel h\left(t\right)-\overline{h}\left(t\right)\parallel & \le & \parallel g(t,z_t+x_t)-g(t,{\overline{z}}_t+x_t)\parallel \hfill \\ & & +{\int }_0^t\parallel U(t,s)A\left(s\right)\left[g(s,z_s+x_s)+g(s,{\overline{z}}_s+x_s)\right]\parallel ds\hfill \\ & & +{\int }_0^t\parallel U(t,s)\left[f\left(s\right)-\overline{f}\left(s\right)\right]\parallel ds\hfill \\ & & +\sum _{0<t_k<t}\parallel U(t,t_k)\left[I_k(z\left(t_k^{-}\right)+x\left(t_k^{-}\right))-I_k(\overline{z}\left(t_k^{-}\right)+x\left(t_k^{-}\right))\right]\parallel \hfill \\ & \le & \parallel A^{-1}{\left(t\right)\parallel }_{B\left(E\right)}\parallel A\left(t\right)g(t,z_t+x_t)-A\left(t\right)g(t,{\overline{z}}_t+x_t)\parallel \hfill \\ & & +{\int }_0^t{\parallel U(t,s)\parallel }_{B\left(E\right)}\parallel A\left(s\right)g(s,z_s+x_s)-A\left(s\right)g(s,{\overline{z}}_s+x_s)\parallel ds\hfill \\ & & +{\int }_0^t{\parallel U(t,s)\parallel }_{B\left(E\right)}\parallel f\left(s\right)-\overline{f}\left(s\right)\parallel ds\hfill \\ & & +\sum _{0<t_k<t}\parallel U(t,t_k){\parallel }_{B\left(E\right)}\parallel I_k(z\left(t_k^{-}\right)+x\left(t_k^{-}\right))-I_k(\overline{z}\left(t_k^{-}\right)+x\left(t_k^{-}\right))\parallel \hfill \\ & \le & M_0{L}_{*}\parallel z_t-{\overline{z}}_t{\parallel }_{{\mathcal{B}}_h}+{\int }_0^t{M}_1{L}_{*}{\parallel z_s-{\overline{z}}_s\parallel }_{{\mathcal{B}}_h}ds\hfill \\ & & +{\int }_0^t{M}_1\parallel f\left(s\right)-\overline{f}\left(s\right)\parallel ds\hfill \\ & & +M_1\sum _{k=1}^m{d}_k\parallel (z\left(t_k^{-}\right)-\overline{z}\left(t_k^{-}\right))\parallel \hfill \end{array}$$

Using (A1) and (7), we obtain

$$\begin{array}{ccc}\hfill \parallel h\left(t\right)-\overline{h}\left(t\right)\parallel & \le & M_0{L}_{*}K\left(t\right)\parallel z\left(t\right)-\overline{z}\left(t\right)\parallel +{\int }_0^t{M}_1{L}_{*}K\left(s\right)\parallel z\left(s\right)-\overline{z}\left(s\right)\parallel ds\hfill \\ & & +{\int }_0^t{M}_1{l}_n\left(s\right)K_n\parallel z\left(s\right)-\overline{z}\left(s\right)\parallel ds\hfill \\ & & +M_1\sum _{k=1}^m{d}_k\underset{s\in [0,t_k]}{sup}\parallel z\left(s\right)-\overline{z}\left(s\right)\parallel \hfill \\ & \le & M_0{L}_{*}{K}_n\parallel z\left(t\right)-\overline{z}\left(t\right)\parallel +{\int }_0^t{M}_1{K}_n\left[L_{*}+l_n\left(s\right)\right]\parallel z\left(s\right)-\overline{z}\left(s\right)\parallel ds\hfill \\ & & +M_1\sum _{k=1}^m{d}_k\underset{s\in [0,t_k]}{sup}\parallel z\left(s\right)-\overline{z}\left(s\right)\parallel \hfill \\ & \le & M_0{L}_{*}{K}_n\parallel z\left(t\right)-\overline{z}\left(t\right)\parallel +{\int }_0^t{\overline{l}}_n\left(s\right)\parallel z\left(s\right)-\overline{z}\left(s\right)\parallel ds\hfill \\ & & +M_1\sum _{k=1}^m{d}_k\underset{s\in [0,t_k]}{sup}\parallel z\left(s\right)-\overline{z}\left(s\right)\parallel \hfill \\ & \le & M_0{L}_{*}{K}_n\left[e^{\tau L_n^{*}\left(t\right)}\right]\left[e^{-\tau L_n^{*}\left(t\right)}\parallel z\left(t\right)-\overline{z}\left(t\right)\parallel \right]\hfill \\ & & +{\int }_0^t\left[l_n\left(s\right)e^{\tau L_n^{*}\left(s\right)}\right]\left[e^{-\tau L_n^{*}\left(s\right)}\parallel z\left(s\right)-\overline{z}\left(s\right)\parallel \right]ds\hfill \\ & & +M_1\sum _{k=1}^m{d}_k{e}^{\tau L_n^{*}\left(s\right)}{e}^{-\tau L_n^{*}\left(s\right)}\underset{s\in [0,t_k]}{sup}\parallel z\left(s\right)-\overline{z}\left(s\right)\parallel \hfill \\ & \le & M_0{L}_{*}{K}_n{e}^{\tau L_n^{*}\left(t\right)}\parallel z-\overline{z}{\parallel }_{B_{*}^k}+{\int }_0^t{\left[\frac{e^{\tau L_n^{*}\left(s\right)}}{\tau }\right]}^{{}^{\prime }}ds{\parallel z-\overline{z}\parallel }_{B_{*}^k}\hfill \\ & & +M_1\sum _{k=1}^m{d}_k{e}^{\tau L_n^{*}\left(s\right)}{\parallel z-\overline{z}\parallel }_{B_{*}^k}\hfill \\ & \le & M_0{L}_{*}{K}_n{e}^{\tau L_n^{*}\left(t\right)}\parallel z-\overline{z}{\parallel }_{B_{*}^k}+\frac{1}{\tau }{e}^{\tau L_n^{*}\left(t\right)}{\parallel z-\overline{z}\parallel }_{B_{*}^k}\hfill \\ & & +M_1\sum _{k=1}^m{d}_k{e}^{\tau L_n^{*}\left(t\right)}{\parallel z-\overline{z}\parallel }_{B_{*}^k}\hfill \\ & \le & \left[M_0{L}_{*}{K}_n+\frac{1}{\tau }+M_1\sum _{k=1}^m{d}_k\right]e^{\tau L_n^{*}\left(t\right)}{\parallel z-\overline{z}\parallel }_{B_{*}^k}\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \parallel h-\overline{h}{\parallel }_{B_{*}^k}\le \left[M_0{L}_{*}{K}_n+\frac{1}{\tau }+M_1\sum _{k=1}^m{d}_k\right]{\parallel z-\overline{z}\parallel }_{B_{*}^k}\end{array}$$

By an analogous relation, obtained by interchanging the roles of z and $\overline{z}$, it follows that

$$\begin{array}{c}\hfill H_d(\mathcal{F}\left(z\right)-\mathcal{F}\left(\overline{z}\right))\le \left[M_0{L}_{*}{K}_n+\frac{1}{\tau }+M_1\sum _{k=1}^m{d}_k\right]{\parallel z-\overline{z}\parallel }_{\overline{B}}\end{array}$$

Thus, the operator $\mathcal{F}$ is a contraction for all $n\in \mathbb{N}$.

Now, we shall show that $\mathcal{F}$ is an admissible operator. Let $z\in B_{*}^k$. Set, for every $n\in \mathbb{N}$, the space,

$$\begin{array}{c}\hfill B_{**}^k=\{y:(-\infty ,n]\to E:y{|}_{[0,n]}\in PC([0,n],E),y_0\in {\mathcal{B}}_h\}\end{array}$$

and let us consider the multivalued operator $\mathcal{F}:B_{**}^k\to {\mathcal{P}}_{cl}\left(B_{**}^k\right)$ defined by

$$\begin{array}{ccc}\hfill \mathcal{F}\left(z\right)=h\in B_{**}^k:h\left(t\right)& =& g(t,z_t+x_t)-U(t,0)g(0,\varphi )+{\int }_0^{t}U(t,s)A\left(s\right)g(s,z_s+x_s)ds\hfill \\ & & +{\int }_0^{t}U(t,s)f\left(s\right)ds+\sum _{0<t_k<t}U(t,t_k)I_k(z\left(t_k^{-}\right)+x\left(t_k^{-}\right)),t\in [0,n]\hfill \end{array}$$

where $f\in S_{F,y}^n=\{v\in L^1([0,n],E):v\left(t\right)\in F(t,y_t)\text{for}\text{a.e}t\in [0,n]\}$.

From $\left(\mathrm{H}1\right)\mathrm{to}\left(\mathrm{H}7\right)$, and since F is multivalued map with compact values, we can prove that for every $z\in B_{**}^k,\mathcal{F}\left(z\right)\in {\mathcal{P}}_{cl}\left(B_{**}^k\right)$ and there exists $z_{*}\in B_{**}^k$ such that $z_{*}\in \mathcal{F}\left(z_{*}\right)$. Let $h\in B_{**}^k,\overline{y}\in \overline{\mu }$ and $ϵ>0$. Assume that $z_{*}\in \mathcal{F}\left(\overline{z}\right)$, then we have

$$\begin{array}{ccc}\hfill \parallel \overline{z}\left(t\right)-z_{*}\left(t\right)\parallel & \le & \parallel \overline{z}\left(t\right)-h\left(t\right)\parallel +\parallel z_{*}\left(t\right)-h\left(t\right)\parallel \hfill \\ & \le & e^{\tau L_n^{*}\left(t\right)}\parallel \overline{z}\left(t\right)-\mathcal{F}\left(\overline{z}\right){\parallel }_{B_{*}^k}+\parallel z_{*}-h\parallel \hfill \end{array}$$

Since h is arbitrary, we may suppose that

$$\begin{array}{c}\hfill h\in B(z_{*},ϵ)=\{h\in B_{**}^k:\parallel h-z_{*}{\parallel }_{B_{*}^k}\le ϵ\}\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \parallel \overline{z}-z_{*}{\parallel }_n\le {\parallel \overline{z}-\mathcal{F}\left(\overline{z}\right)\parallel }_{B_{*}^k}+ϵ\end{array}$$

If z is not in $\mathcal{F}\left(\overline{z}\right)$, then $\parallel z_{*}-\mathcal{F}\left(\overline{z}\right)\parallel \ne 0$. Since $\mathcal{F}\left(\overline{z}\right)$ is compact, there exists $x\in \mathcal{F}\left(\overline{z}\right)$ such that

$$\begin{array}{c}\hfill \parallel z_{*}-\mathcal{F}\left(\overline{z}\right)\parallel =\parallel z_{*}-x\parallel \end{array}$$

Then, we have

$$\begin{array}{ccc}\hfill \parallel \overline{z}\left(t\right)-z_{*}\left(t\right)\parallel & <& \parallel \overline{z}\left(t\right)-h\left(t\right)\parallel +\parallel x\left(t\right)-h\left(t\right)\parallel \hfill \\ & \le & e^{\tau L_n^{*}}\parallel \overline{z}-\mathcal{F}\left(\overline{z}\right){\parallel }_{B_{*}^k}+\parallel x\left(t\right)-h\left(t\right)\parallel \hfill \end{array}$$

Thus, $\parallel \overline{z}{-x\parallel }_{B_{*}^k}\le {\parallel \overline{z}-\mathcal{F}\left(\overline{z}\right)\parallel }_{B_{*}^k}+ϵ.$

Therefore, $\mathcal{F}$ is an admissible operator contraction.

From the choice of Ω, there is no $z\in \partial \Omega$ such that $z=\lambda \mathcal{F}\left(z\right)$ for some $\lambda \in (0,1)$. Then, the statement $\left(C2\right)$ in Theorem $(2.1)$ does not hold. This implies that the operator $\mathcal{F}$ has a fixed point $z^{*}$. Then $y^{*}\left(t\right)=z^{*}\left(t\right)+x\left(t\right),t\in (-\infty ,+\infty )$ is a fixed point of the operator $\mathbb{N}$, which is a mild solution of the Problem $\left(1\right)-\left(3\right)$.

Hence the proof.   ▢

4. Example

As an application of Theorem $(3.1)$, we study the following impulsive neutral differential system:

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial t}\left[v(t,\xi )-{\int }_{-\infty }^{0}T\left(\theta \right)u(t,v(t+\theta ,\xi ))d\theta \right]& \in & a(t,\xi )\frac{{\partial }^{2}v}{\partial {\xi }^2}(t,\xi )\hfill \\ & & +{\int }_{-\infty }^{0}P\left(\theta \right)Q(t,v(t+\theta ,\xi ))d\theta \hfill \\ \hfill t\in [0,+\infty ),\xi \in [0,\pi ]\\ \hfill v(t,0)=v(t,\pi )=0\\ \hfill z\left(t_k^{+}\right)-z\left(t_k^{-}\right)=I_k\left(z\left(t_k^{-}\right)\right),k=1,2,\cdots \\ \hfill v(\theta ,\xi )=v_0(\theta ,\xi )-\infty <\theta \le 0,\xi \in [0,\pi ]\end{array}$$

(8)

where $a(t,\xi )$ is a continuous function and is uniformly Holder continuous in t; $T,P:(-\infty ,0]\to R;u:(-\infty ,0]\times R\to R$ and $v_0:(-\infty ,0]\times [0,\pi ]\to R$ are continuous functions and $Q:[0,+\infty )\times R\to \mathcal{P}\left(R\right)$ is a multivalued map with compact convex values.

Consider $E=L^2([0,\pi ],R)$ and define $A\left(t\right)$ by $A\left(t\right)w=a(t,\xi )w^{{}^{\prime \prime }}$ with domain $D\left(A\right)=\{w\in E:w,w^{{}^{\prime }}\text{are}\text{absolutely}\text{continuous},w^{{}^{\prime \prime }}\in E,w\left(0\right)=w\left(\pi \right)=0\}$.

Then, $A\left(t\right)$ generates an evolution system $U(t,s)$ satisfying assumption (H1)(see [41]). We can define respectively that

$$\begin{array}{c}\hfill g(t,\varphi )\left(\xi \right)={\int }_{-\infty }^{0}T\left(\theta \right)u(t,\varphi \left(\theta \right)\left(\xi \right)d\theta ,-\infty <\theta \le 0,\xi \in [0,\pi ]\end{array}$$

and

$$\begin{array}{c}\hfill F(t,\varphi )\left(\xi \right)={\int }_{-\infty }^{0}P\left(\theta \right)Q(t,\varphi \left(\theta \right)\left(\xi \right)d\theta ,-\infty <\theta \le 0,\xi \in [0,\pi ]\end{array}$$

Then, in order to prove the existence of mild solutions of the System $\left(8\right)$, 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 $p\in L^1(J,R^{+})$ and a nondecreasing continuous function $\psi :[0,+\infty )\to (0,+\infty )$ such that

$$\begin{array}{c}\hfill |Q(t,x)|\le p\left(t\right)\psi (|x|),\text{for}t\in J,\text{and}x\in R\end{array}$$

(iii)

$T,P$ are integrable on $(-\infty ,0]$.

(iv)

There exist positive constants $c_k$ and $d_k,k=1,2,\cdots$ such that

$$\begin{array}{c}\hfill \parallel I_k\left(x\right)-I_k\left(\overline{x}\right)\parallel \le d_k\parallel x-\overline{x}\parallel ,\\ \hfill \parallel I_k\left(z\right)\parallel \le c_k,\text{for}\text{all}x,\overline{x},z\in E\end{array}$$

By the dominated convergence theorem, one can show that $f\in S_{F,y}$ is a continuous function from ${\mathcal{B}}_h$ to E. Moreover, the mapping g is Lipschitz continuous in its argument. In fact, we have

$$\begin{array}{c}\hfill \parallel g(t,{\varphi }_1)-g(t,{\varphi }_2)\parallel \le M_0{L}_{*}lip\left(u\right){\int }_{-\infty }^0\parallel T\left(\theta \right)\parallel d\theta \parallel {\varphi }_1-{\varphi }_2\parallel ,\text{for}{\varphi }_1,{\varphi }_2\in {\mathcal{B}}_h\end{array}$$

On the other hand, we have for $\varphi \in {\mathcal{B}}_h$ and $\xi \in [0,\pi ]$

$$\begin{array}{c}\hfill \parallel F(t,\varphi )\left(\xi \right)\parallel \le {\int }_{-\infty }^0\parallel p\left(t\right)P\left(\theta \right)\parallel \xi (\parallel \left(\varphi \left(\theta \right)\right)\left(\xi \right)\parallel )d\theta \end{array}$$

Since the function ξ is nondecreasing, it follows that

$$\begin{array}{c}\hfill {\parallel F(t,\varphi )\parallel }_{\mathcal{P}\left(E\right)}\le p\left(t\right){\int }_{-\infty }^0\parallel P\left(\theta \right)\parallel d\theta \xi (\parallel \varphi \parallel ),\text{for}\varphi \in {\mathcal{B}}_h\end{array}$$

Proposition: 

Under the above assumptions, if we assume that condition $\left(5\right)$ in Theorem $(3.1)$ is true, $\varphi \in {\mathcal{B}}_h$, then the System $\left(8\right)$ has a mild solution which is defined in $(-\infty ,+\infty )$.

5. Conclusions

In this manuscript, we have proved the existence result of first order impulsive neutral evolution inclusion with infinite delay in Frechet spaces. Here, we defined a new notion of phase space and proved the result without compactness of an evolution operator, using a recently developed nonlinear alternative for contractive multivalued maps due to Frigon. The same result can be generalized for controllability of an impulsive neutral evolution inclusion with infinite delay of the form [19,21]

$$\begin{array}{ccc}\hfill \frac{d}{dt}\left[y\left(t\right)-g(t,y_t)\right]& \in & A\left(t\right)y\left(t\right)+Bu\left(t\right)+F(t,y_t)\hfill \\ \hfill t\in J=[0,+\infty ),t& \ne & t_k,k=1,2,\cdots \hfill \\ \hfill {\Delta y|}_{t=t_k}& =& I_k\left(y\left(t_k^{-}\right)\right),k=1,2,\cdots \hfill \\ \hfill y_0& =& \varphi \in {\mathcal{B}}_h\hfill \end{array}$$