\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 81, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/81\hfil Existence of solutions]
{Existence of solutions to impulsive fractional partial neutral
stochastic integro-differential inclusions with  state-dependent
delay}

\author[Z. Yan,  H. Zhang\hfil EJDE-2013/81\hfilneg]
{Zuomao Yan, Hongwu Zhang}  % in alphabetical order

\address{Zuomao Yan \newline
Department of Mathematics, Hexi University,
 Zhangye, Gansu 734000, China}
\email{yanzuomao@163.com}

\address{Hongwu Zhang \newline
Department of Mathematics, Hexi University,
 Zhangye, Gansu 734000, China}
\email{zh-hongwu@163.com}

\thanks{Submitted September 25, 2012. Published March 29, 2013.}
\subjclass[2000]{34A37,  60H10, 34K50, 34G25, 26A33}
\keywords{Impulsive stochastic integro-differential
inclusions;  \hfill\break\indent
state-dependent delay; multi-valued map;
fractional neutral integro-differential inclusions}

\begin{abstract}
 We study the existence of mild solutions for a class of impulsive
 fractional partial neutral stochastic integro-differential inclusions
 with  state-dependent delay. We assume that the undelayed part
 generates a solution operator and transform it into an integral equation.
 Sufficient conditions for the existence of solutions are derived by
 using the nonlinear alternative of Leray-Schauder type for multivalued
 maps due to O'Regan and  properties of the solution operator. An example is
 given to illustrate the theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

The study of impulsive functional differential or
integro-differential systems 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 or
integro-differential systems 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 \cite{c4,c5,h2,h3,y2}.
Besides impulsive effects, stochastic effects likewise
exist in real systems. Therefore, impulsive stochastic  differential equations
describing these dynamical systems subject to both impulse and
stochastic  changes have attracted considerable attention.
Particularly,  the papers \cite{a5,h5,l4}
considered the existence of mild
solutions for some impulsive neutral stochastic functional
differential and  integro-differential equations with infinite delay
in Hilbert spaces. As the generalization of classic impulsive
differential equations, impulsive  stochastic  differential
inclusions  in Hilbert spaces have attracted the researchers great
interest. Among them, Ren et al \cite{r1} established the
controllability of impulsive neutral stochastic functional
differential inclusions with infinite delay in an abstract space by
means of the fixed point theorem for discontinuous multi-valued
operators due to Dhage.

 On the other hand, fractional differential equations have gained considerable
importance due to their application in various sciences, such as
physics, mechanics, chemistry, engineering, etc.. In the recent
years, there has been a significant development in ordinary and
partial differential equations involving fractional derivatives; see
the monograph of Kilbas et al  \cite{k1} and the papers 
\cite{a1,a3,b2,l1,l2} and
the references therein. The existence of solutions for fractional
semilinear differential  or integro-differential equations is one of
the theoretical fields that investigated by many authors \cite{a2,e1,y1}.
 Several papers \cite{a4,d3} devoted
to the existence of mild solutions for  abstract fractional
functional differential and integro-differential equations with
state-dependent delay in Banach spaces  by using fixed point
techniques.  Recently, the existence, uniqueness and other
quantitative and qualitative properties of solutions to various
impulsive semilinear fractional differential and integrodifferential
systems have been extensively studied  in Banach spaces. For
example, Balachandran et al \cite{b1}, Chauhan et al \cite{c1},
  Debbouche and Baleanu \cite{d2}, Mophou \cite{m1},  Shu et al \cite{s1}.
However, the deterministic models often fluctuate due to noise,
which is random or at least appears to be so. Therefore, we must
move from deterministic problems to stochastic ones.
  In this paper, we consider the existence of a class of impulsive
fractional partial  neutral
stochastic integro-differential inclusions with  state-dependent
delay  of the form
\begin{gather}
 d D(t,x_t)\in\int_0^t\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}AD(s,x_{s})\,ds\,dt
 + F(t,x_{\rho(t,x_t)})\,dw(t), \label{e1.1}\\
 t\in J=[0,b], t\neq t_k,  k=1,\dots, m,\nonumber \\
x_0=\varphi\in\mathcal{B}, \label{e1.2} \\
 \Delta
x(t_k)=I_k(x_{t_k}), \quad k=1,\dots, m, \label{e1.3}
\end{gather}
 where the state $x(\cdot)$ takes values in a separable real Hilbert
space $H$ with inner product $(\cdot,\cdot)$ and  norm
$\|\cdot\|$, $1 < \alpha < 2$, $A: D(A)\subset H
\to H$ is a linear densely defined operator of sectorial
type on   $H$. The time history $x_t:(-\infty,0]\to H$
given by $ x_t(\theta)=x(t + \theta)$ belongs to some abstract
phase space $ \mathcal{B}$ defined axiomatically;  Let $K$ be
another separable Hilbert space with inner product
$(\cdot,\cdot)_K$  and norm $\| \cdot\|_K$.
Suppose $\{w(t):t\geq0\}$ is a given $K$-valued Brownian motion or
Wiener process with a finite trace nuclear covariance operator $Q >
0$ defined on a complete probability space $(\Omega,\mathcal{F},P)$
equipped with a normal filtration $\{\mathcal{F}_t\}_{t\geq0}$,
which is generated by the Wiener process $w$. We are also employing
the same notation $\|\cdot\|$ for the norm $L(K,H)$,
where $L(K,H)$ denotes the space of all bounded linear operators
from $K$ into $H$. The initial data $\{\varphi(t):-\infty<t\leq 0\}$
is an $\mathcal{F}_0$-adapted, $\mathcal{B}$-valued random
variable independent of the Wiener process $w$ with finite second
moment. $F,G, D(t,\varphi)=\varphi(0)+G(t,\varphi),\varphi \in
\mathcal{B},\rho, I_k(k=1,\dots,m)$, are given functions to be
specified later. Moreover, let $ 0 <t_1 < \dots< t_{m}< b$, are
prefixed points and the symbol $ \Delta
x(t_k)=x(t_k^{+})-x(t_k^{-})$, where $x(t^{-}_k) $ and $
x(t^{+}_k)$ represent the right and left limits of $x(t)$ at
$t =t_k$, respectively.


We notice that the convolution integral in \eqref{e1.1} is known as the
Riemann-Liouville fractional integral (see \cite{c2,c3}).
In \cite{c3}, the
authors established the existence of $S$-asymptotically
$\omega$-periodic solutions for fractional order functional
integro-differential equations with infinite delay. To the best of
our knowledge, the existence of mild solutions for the impulsive
fractional partial neutral stochastic integro-differential
inclusions with state-dependent delay in Hilbert spaces has not been
investigated yet. Motivated by this consideration, in this paper we
will study  this interesting problem, which are natural
generalizations of the concept of mild solution for impulsive
fractional evolution equations
  well known in the theory of
infinite dimensional deterministic systems. Specifically, sufficient
conditions for the existence are given by means of the nonlinear
alternative of Leray-Schauder type for multivalued maps due to
O'Regan  combined with the  solution operator. The known results
appeared in \cite{b1,c1,d2,m1,s1} are generalized to the fractional
stochastic multi-valued settings and the case of infinite delay.

The rest of this paper is organized as follows. In Section 2, we
introduce some notations and necessary preliminaries. In Section 3,
we give our main results. In Section 4, an example is given to
illustrate our results. In the last section, concluding remarks are
given.


\section{Preliminaries}

In this section, we introduce some basic definitions, notation and
lemmas which are used throughout this paper.

Let $(\Omega,\mathcal{F},P)$
  be a complete probability space equipped with
some filtration $\{\mathcal{F}_t\}_{t\geq0}$ satisfying the usual
conditions (i.e., it is right continuous and $\mathcal{F}_0$
contains all $P$-null sets).
 Let $\{e_i\}_{i=1}^{\infty}$ be a complete
orthonormal basis of $K$. Suppose that $\{w(t):t\geq0\}$ is a
cylindrical $K$-valued Wiener process with a finite trace nuclear
covariance operator $Q \geq 0$, denote
$\operatorname{Tr}(Q)=\sum_{i=1}^{\infty}\lambda_i=\lambda< \infty$, which
satisfies that $Qe_i =\lambda_ie_i$. So, actually,
$w(t)=\sum_{i=1}^{\infty}\sqrt{\lambda_i}w_i(t)e_i$, where
$\{w_i(t)\}_{i=1}^{\infty}$ are mutually independent
one-dimensional standard Wiener processes. We assume that
$\mathcal{F}_t=\sigma\{w(s):0\leq s\leq t\}$ is the
$\sigma$-algebra generated by $w$ and $\mathcal{F}_{b}=\mathcal{F}$.


Let $L(K,H)$ denote the space of all bounded linear operators from
$K$ into $H$ equipped with the usual operator norm
$\|\cdot\|_{L(K,H)}$. For $\psi\in L(K,H)$ we define
\[
\| \psi\|_{Q}^2=\operatorname{Tr}(\psi Q\psi^{*})
=\sum_{n=1}^{\infty}\| \sqrt{\lambda_n}\psi
e_n\|^2.
\]
If $\| \psi\|^2_{Q}<\infty$, then $\psi$  is called
a $Q$-Hilbert-Schmidt operator. Let $L_{Q}(K,H)$ denote the space of
all $Q$-Hilbert-Schmidt operators $\psi$. The completion
$L_{Q}(K,H)$ of $L(K,H)$ with respect to the topology induced by the
norm $\| \cdot\|_{Q}$ where $\|\psi\|_{Q}^2=(\psi,\psi)$ is a
 Hilbert space with the above norm topology.

The collection of all strongly measurable,
  square integrable, $H$-valued random variables, denoted by $L_2(\Omega,H)$
   is a Banach space equipped with norm$
\| x(\cdot)\|_{L_2}=(E\| x(\cdot,w)\|^2)^{\frac{1}{2}}$,
 where the expectation, $E$ is defined by $Ex=\int_\Omega x(w)dP$.
Let $C(J,L_2(\Omega,H))$  be the Banach space of all continuous
maps from $J$ into $L_2(\Omega,H)$ satisfying the condition
$\sup_{0\leq t\leq b}E\| x(t)\|^2<\infty$. Let
$L_2^{0}(\Omega,H)$ denote the family of all $\mathcal{F}_0$-measurable,
$H$-valued random variables $x(0)$.

\begin{definition}[\cite{d1}] \label{def2.1} \rm
 We call $S\subset\Omega$ a
$P$-null set if there is $B\in \mathcal{F}$ such that $S\subseteq B$
and $P(B)=0$.
\end{definition}

\begin{definition}[\cite{d1}] \label{def2.2} \rm
 A stochastic process $\{x(t) :t \geq 0\}$ in a real separable Hilbert space
$H$ is a Wiener process if for each $t \geq 0$,
\begin{itemize}
\item[(i)] $x(t)$ has continuous sample paths and independent increments.

\item[(ii)] $x(t) \in L^2( \Omega,H)$ and $E(x(t)) = 0$.

\item[(iii)] $ \operatorname{Cov}(w(t) - w(s)) = (t - s)Q$, where
$Q \in L(K,H)$ is a nonnegative nuclear operator.
\end{itemize}
\end{definition}


\begin{definition}[\cite{d1}] \label{def2.3} \rm Brownian motion is a
continuous adapted real-valued process $x(t), t \geq 0$ such that
\begin{itemize}
\item[(i)] $x(0)= 0$.

\item[(ii)] $x(t) -x(s)$ is independent of $\mathcal{F}_{s}$ for all
$0 \leq s < t$.

\item[(iii)] $x(t) - x(s)$ is $N(0, t - s)$-distributed for all $0 \leq s \leq
t$.
\end{itemize}
\end{definition}

\begin{definition}[\cite{d1}] \label{def2.4} \rm
 Normal filtration $\{\mathcal{F}_t : 0 \leq t \leq b\}$ is a right-continuous,
increasing family of sub $\sigma$-algebras of $\mathcal{F}$.
\end{definition}

\begin{definition}[\cite{d1}] \label{def2.5} \rm
The process $x$ is $\mathcal{F}_0$-adapted if each $x(0)$ is measurable
with respect to $\mathcal{F}_0$.
\end{definition}

We say that a function $x: [\mu,\tau]\to H$ is
 a normalized piecewise continuous function on $[\mu,\tau]$ if $x$ is piecewise
continuous and left continuous on $(\mu,\tau]$. We denote by
$\mathcal{PC}([\mu, \tau],H)$ the space formed by the normalized
piecewise continuous, $\mathcal{F}_t$-adapted measurable processes
from $[\mu,\tau]$ into $H$. In particular, we introduce the space
$\mathcal{PC}$ formed by all $\mathcal{F}_t$-adapted measurable,
$H$-valued stochastic processes $\{x(t):t\in [0,b]\}$ such that $x$
is continuous at $t\neq t_k$, $x(t_k)= x(t_k^{-})$ and $
x(t_k^{+})$ exists for $ k=1,2...,m$. In this paper, we always
assume that $\mathcal{PC}$ is endowed with the norm
\[
\| x\|_{\mathcal{PC}}
=(\sup_{0\leq t\leq b}E\|
x(t)\|^2)^{\frac{1}{2}}.
 \]
Then, we have the following conclusion.

\begin{lemma} \label{lem2.1}
The set $(\mathcal{PC},\|\cdot\|_{\mathcal{PC}}) $ is a Banach space.
\end{lemma}

 \begin{proof}
 Let $\{x_n\}$ be a Cauchy sequence in
 $\mathcal{PC}$, and fix any $\varepsilon > 0$.
There is $n_0\in\mathbb{N}$ such that
for all $n > n_0$ and $p \in\mathbb{N}$
\[
\|x_{n+p}-x_n\|_{\mathcal{PC}}
=(\sup_{0\leq t\leq b} E\|x_{n+p}(t)-x_n(t)\|^2)^{\frac{1}{2}}<\varepsilon
\]
for each $t \in[0,b]$. From the above inequality it follows that the
sequence $x_n(t)$ is a Cauchy
sequence in $L^2(\Omega,H)$; moreover,
 by the completeness of $L^2(\Omega,H)$ with respect to $\|\cdot\|_{L_2}$,
for its limit $x(t) := \lim x_n(t)$, we obtain
\[
E\| x_n(t)-x(t)\|^2<\varepsilon^2
\]
for all $ n > n_0$. Consequently,
$\|x_n-x\|_{\mathcal{PC}}\to0$ as $ n \to\infty$. Next, we need to show
that $x \in \mathcal{PC}$.
In fact, we verify that $x$ is  continuous.  By
\[
x(t + \Delta t) - x(t) = x(t + \Delta t) - x_n(t + \Delta t) +
x_n(t + \Delta t) - x_n(t) + x_n(t) - x(t),
\]
it follows that
\begin{align*}
E\| x(t + \Delta t) - x(t) \|^2
&\leq 3E\| x(t + \Delta t) - x_n(t + \Delta t)\|^2 \\
&\quad+3 E\| x_n(t + \Delta t) - x_n(t)\|^2 +3
E\|  x_n(t) - x(t)\|^2.
\end{align*}
Using the uniform convergence of $x_n$ to $x$ with respect to
$\|\cdot\|_{L_2}$  and the  continuity
of $x_n$, the  continuity of $x$ follows. The proof is
complete.
\end{proof}

To simplify notation, we put $t_0 = 0, t_{m+1} = b$  and for
$x \in \mathcal{PC}$, we denote by $\hat{x}_k\in C([t_k,
t_{k+1}];L_2(\Omega,H))$, $k = 0, 1, \dots , m$, the function given
by
\[
\hat{x}_k(t):=\begin{cases}
x(t) & \text{for }   t\in(t_k,t_{k+1}] ,\\
x(t_k^{+}) & \text{for }   t=t_k.
\end{cases}
\]
Moreover, for $B \subseteq\mathcal{PC}$ we denote by
$\hat{B_k}$, $k = 0, 1,\dots , m$, the set
 $\hat{B_k} = \{\hat{x}_k : x\in B\}$. The notation $B_{r}(x,H)$
stands for the closed ball with center at $x$ and radius $r > 0$ in $H$.


\begin{lemma} \label{lem2.2}
 A set $B \subseteq \mathcal{PC}$ is
relatively compact in $\mathcal{PC}$ if, and only if, the set
$\hat{B_k}$ is relatively compact in $C([t_k,t_{k+1}];L_2(\Omega,H))$,
for every $k = 0, 1,\dots, m$.
\end{lemma}

\begin{proof}
Let $B \subseteq \mathcal{PC}$ be a subset
and $\{x^{(i)}(\cdot)\}$  be any sequence of $B$. Since
$\hat{B_0}$ is a relatively compact subset of
$C([0, t_1];L_2(\Omega,H))$. Then, there exists a subsequence of
$x^{(i)}$, labeled $\{x_1^{(i)}\}\subset B$,
and $x_1 \in C([0, t_1];L_2(\Omega,H))$, such that
$$
x_1^{(i)} \to x_1 \quad \text {in }  C([0, t_1];L_2(\Omega,H)) \quad
\text {as }  i \to\infty .
$$
Similarly, $\hat{B_k}$ is a relatively compact
subset of $C([t_k, t_{k+1}];L_2(\Omega,H))$, for $k=1,2,\dots,m$. 
Then, there exists
a subsequence of $x^{(i)}$, labeled $\{x_k^{(i)}\}\subset B$, such that
$x_k \in C([t_k, t_{k+1}];L_2(\Omega,H))$, and
$$
x_k^{(i)} \to x^{k} \quad \text {in }   C([t_k, t_{k+1}];L_2(\Omega,H))
\text { as } i \to\infty .
$$
 Setting
\[
 x(t)= \begin{cases}
 x_1(t),  & t\in [0,t_1], \\
x_2(t),  & t\in (t_1,t_2], \\
\dots\\
 x_{m}(t),  &  t\in (t_{m},b],
\end{cases}
\]
then
$$
x_{m}^{(i)} \to x\quad \text {in } \mathcal{PC}
 \text{ as }  i \to\infty .
$$
Thus, the set $B$  is  relatively compact.

If set $B \subseteq \mathcal{PC}$ is relatively compact in
$\mathcal{PC}$ and $\{x^{(i)}(\cdot)\}$  be any sequence of $B$.
Then, for each $t\in[0,t_1]$, there exists a subsequence of
$x^{(i)}$, labeled $\{x_1^{(i)}\}\subset B, $ and $x_1 \in
\mathcal{PC}$, such that $x_1^{(i)} \to x_1$ in $
\mathcal{PC}$ as $i \to\infty $. From  the definition of the
set $\hat{B}_0$, we can get
$$
\hat{x}_1^{(i)} \to \hat{x}_1 \quad \text{in }
  C([0,t_1];L_2(\Omega,H)) \text { as } i \to\infty .
$$
Similarly, for each $t\in[t_k,t_{k+1}](k=1,2,\dots,m)$, there
exists a subsequence of $x^{(i)}$, labeled
$\{x_k^{(i)}\}\subset B$ and $x_k \in \mathcal{PC}$, such that
$x_k^{(i)} \to x_k$ in $ \mathcal{PC}$ as $i \to\infty $. From the
definition of the set $\hat{B}_k$, we can get
$$
\hat{x}_k^{(i)} \to \hat{x}_k \quad \text {in }
  C([t_k, t_{k+1}];L_2(\Omega,H)) \text { as } i \to\infty .
$$
Thus, the set $ \hat{B_k}$ is relatively compact in
$C([t_k, t_{k+1}];L_2(\Omega,H))$, for every $k = 0, 1,\dots , m$. The
proof is complete.
\end{proof}

 In this article, we assume that the phase space
$(\mathcal{B},\|\cdot\|_{\mathcal{B}})$ is a
seminormed linear space of $\mathcal{F}_0$-measurable functions
mapping $(-\infty,0]$ into $H$, and satisfying the following
fundamental axioms due to Hale and Kato (see e.g., in \cite{h1}).

\begin{itemize}
\item[(A)] If $x : (-\infty,\sigma+ b]\to H$, $b > 0$,
 is such that $x|_{[\sigma,\sigma+b]}\in C([\sigma,\sigma+b],H)$
  and $x_{\sigma} \in\mathcal{B}$,
  then for every $t \in [\sigma,\sigma+b]$ the following conditions hold:
\begin{itemize}
\item[(i)] $x_t$ is in $\mathcal{B}$;

\item[(ii)]  $\| x(t)\|\leq \tilde{H}\| x_t\|_{\mathcal{B}} $;


\item[(iii)]  $\| x_t\|_{\mathcal{B}}\leq K(t-\sigma)
\sup\{\| x(s)\|: \sigma \leq s \leq
t\}+M(t-\sigma)\| x_{\sigma}
\|_{\mathcal{B}}$, where $\tilde{H} \geq 0$
 is a constant; $K,M : [0,\infty)\to[1,\infty)$, $K$ is
continuous and $M$ is locally bounded, and  $\tilde{H},K,M$ are
independent of $x(\cdot)$.
\end{itemize}
\item[(B)]  For the function $x(\cdot)$ in (A),
the function $t \to x_t$ is continuous from
$[\sigma,\sigma+b]$ into  $\mathcal{B}$.

\item[(C)] The space $\mathcal{B}$ is complete.
\end{itemize}
The next result is a consequence of the phase space
axioms.

\begin{lemma} \label{lem2.3}
 Let $x: (-\infty,b]\to H$ be an
$\mathcal{F}_t$-adapted measurable process such that the
$\mathcal{F}_0$-adapted process $x_0=\varphi(t)\in
L_2^{0}(\Omega, \mathcal{B})$ and $x|_{J}\in \mathcal{PC}(J,H)$,
then
\[
\| x_{s}\|_{\mathcal{B}}
\leq  M_{b} E\| \varphi\|_{\mathcal{B}}+K_{b}
\sup_{0\leq s\leq b}E\| x(s)\|,
\]
where
$K_{b} = \sup\{K(t):0\leq t\leq b\}$, 
$M_{b} = \sup\{M(t):0\leq t\leq b\}$.
\end{lemma}

\begin{proof} For each fixed $x\in H$, we consider the
function $\xi(t)$ defined by 
$\xi(t) = \sup\{\|x_{s}\|_{\mathcal{B}}: 0 \leq s \leq t\}$, 
$0 \leq t \leq b$.
Obviously, $\xi$ is increasing.
This combined with the phase space axioms, we have
\begin{align*}
\xi(t)
&\leq M(t) \| \varphi\|_{\mathcal{B}}+K(t) \sup_{0\leq s\leq t}\| x(s)\|\\
&\leq M_{b} \| \varphi\|_{\mathcal{B}}+K_{b}\| x(t)\|.
\end{align*}
Since $E\| \varphi\|_{\mathcal{B}}<\infty, E\|x(t)\|<\infty$, 
the previous inequality holds.
Consequently
\begin{align*}
E(\xi(t)) 
&\leq E(M_{b} \| \varphi\|_{\mathcal{B}}+K_{b}\| x(t)\|)\\
&\leq M_{b} E\| \varphi\|_{\mathcal{B}}+K_{b}
\sup_{0\leq s\leq b}E\| x(s)\|
\end{align*}
for each $t\in J$. By the definition of $\xi$, we have
\[
\xi(b)=E(\xi(b))\leq M_{b} E\|
\varphi\|_{\mathcal{B}}+K_{b} \sup_{0\leq s\leq
b}E\| x(s)\|,
\]
and $\|x_{s}\|_{\mathcal{B}}\leq\xi(b)$ for each $s\in J$; therefore,
\[
\| x_{s}\|_{\mathcal{B}}\leq M_{b} E\|
\varphi\|_{\mathcal{B}}+K_{b} \sup_{0\leq s\leq
b}E\| x(s)\|.
\]
 The proof is complete.
\end{proof}


Let $\mathcal{P}(H)$ denote all the  nonempty subsets of
$H$. Let $\mathcal{P}_{bd,cl}(H) $, $\mathcal{P}_{cp,cv}(H) $,
$\mathcal{P}_{bd,cl,cv}(H) $,  and $\mathcal{P}_{cd}(H) $
 denote
respectively the family of all nonempty bounded-closed,
compact-convex, bounded-closed-convex and compact-acyclic subsets of  $H$
(see \cite{f1}). For $x \in H $ and $Y , Z \in
\mathcal{P}_{bd,cl}(H)$, we denote by 
$D(x,Y)=\inf\{\| x-y\|:y\in Y\}$ and 
$\tilde{\rho}(Y,Z)=\sup_{a\in Y}D(a,Z)$,
and the Hausdorff metric 
$H_{d}:\mathcal{P}_{bd,cl}(H)\times \mathcal{P}_{bd,cl}(H)\to \mathbb{R}^{+}$ by
$H_{d}(A,B)=\max\{\tilde{\rho}(A,B),\tilde{\rho}(B,A)\}$.


A multi-valued map $ G$ is called upper semicontinuous (u.s.c.) on $H$ if, for each
$x_0\in H$, the set $G(x_0)$ is a nonempty, closed subset of $H$
and if, for each open set $S$ of $H$ containing $ G(x_0)$, there
exists an open neighborhood $S$ of $x_0$ such that $ G(S)
\subseteq V$. $ F$ is said to be completely continuous if $ G(V)$ is
relatively compact, for every bounded subset $V \subseteq H$.

If the multi-valued map $G$ is completely continuous with nonempty
compact values, then  $ G$  is u.s.c. if and only if  $ F$  has a
closed graph, i.e. $x_n\to x_{*}, y_n\to y_{*},
y_n \in  G(x_n)$ imply $y_{*}\in   G(x_{*})$.

A multi-valued map $  G: J\to \mathcal{P}_{bd,cl,cv} (H) $ is 
 measurable if for each $x\in H$,  the function $t \mapsto D(x,
G(t))$ is a measurable function on $J$.

\begin{definition}[\cite{f1}] \label{def2.6} \rm 
Let  $  G: H\to \mathcal{P}_{bd,cl} (H) $ be a multi-valued map. Then $ G$ is called a
multi-valued contraction if there exists a constant $\kappa\in(0,1)$
such that for each $ x, y \in H$ we have
\[
H_{d}( G(x)- G(y))\leq \kappa \| x-y\|.
\]
The constant $\kappa$ is called a contraction constant of $G$.
\end{definition}


 A closed and linear operator $A$ is said to be
sectorial of type $\omega$ if there exist $ 0 < \theta < \pi/2$, 
$M > 0$ and $\omega\in \mathbb{R}$ such that its resolvent exists outside
the sector
$\omega+S_{\theta}:=\{\omega+\lambda:\lambda\in\mathbb{C}|\arg(-\lambda)<\theta\}$
and
$\|(\lambda-A)^{-1}\|\leq\frac{M}{|\lambda-\omega|},\lambda
\notin\omega+S_{\theta}$. To give an operator theoretical
approach we recall the following definition.

\begin{definition}[\cite{c3}] \label{def2.7} \rm
 Let $A$ be a closed and linear
operator with domain $D(A)$ defined on a Hilbert space $H$. We call
$A$ the generator of a solution operator if there exist 
$\omega\in \mathbb{R}$ and a strongly continuous function
$S_{\alpha}:\mathbb{R}^{+} \to L(H)$ such that
$\{\lambda^{\alpha}: \ \text{Re}(\lambda)>\omega\}\subset\rho(A)$
and $\lambda^{\alpha-1}(\lambda^{\alpha}-A)^{-1}x=\int_0^\infty
e^{-\lambda t}S_{\alpha}(t)dt,\text{Re}(\lambda)>\omega,x\in H$. In
this case, $S_{\alpha}(\cdot)$ is called the solution operator
generated by $A$.
\end{definition}

We note that, if $A$ is sectorial of type $\omega$ with $0 < \theta
< \pi(1-\frac{\alpha}{2})$ then $A$ is the generator of a solution
operator given by
\[
S_{\alpha}(t)=\frac{1}{2\pi i}\int_{\Sigma}e^{-\lambda
t}\lambda^{\alpha-1}(\lambda^{\alpha}-A)^{-1}d\lambda,
\]
where $\Sigma$  is a suitable path lying outside the sector
$\omega+S_{\alpha}$.


Cuesta \cite{c3}  proved that, if $A$ is a sectorial operator of type
$\omega< 0$, for some $M > 0$ and $0 < \theta <
\pi(1-\frac{\alpha}{2})$, there is $C> 0$ such that
\begin{equation} \label{e2.1}
\| S_{\alpha}(t)\|\leq \frac{CM}{1+|\omega|t^{\alpha}}, \ t\geq0.
\end{equation}
Moreover, we have the following results.

\begin{lemma}[\cite{c3}] \label{lem 2.4}
 Let  $S_{\alpha}(t)$ be a solution
operator on $H$ with generator $A$. Then, we have
\begin{itemize}
\item[(a)]
 $S_{\alpha}(t)D(A)\subset D(A)$ and
$AS_{\alpha}(t)x=S_{\alpha}(t)Ax$ for all $x \in D(A), t \geq 0;$

\item[(b)] Let $x \in D(A)$ and $t \geq 0$. Then $S_{\alpha}(t)x=x+
\int_0^t\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}A
S_{\alpha}(s)xds;$

\item[(c)]
Let $x \in H$ and $t > 0$. Then
$\int_0^t\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}S_{\alpha}(s)x\,ds\in
D(A)$ and
 $$
S_{\alpha}(t)x
=x+ A\int_0^t\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}S_{\alpha}(s)xds.
$$
\end{itemize}
\end{lemma}

Note that the Laplace transform of the abstract function $f \in
L^2(\mathbb{R}^{+},L(K,H)) $ is defined by
\[
\tilde{f}(\varsigma)=\int_0^\infty e^{-\varsigma t}f(t)dw(t).
\]
Now we consider the  problem
\begin{gather} \label{e2.2}
dx(t)=\int_0^t\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}Ax(s)\,ds\,dt
 +  f(t)dw(t), \quad t> 0, 1 < \alpha < 2, \\
\label{e2.3}
x_0=\varphi\in H.
\end{gather}
Formally applying the Laplace transform, we obtain
\[
\lambda\tilde{x}(\varsigma)-\varphi=\lambda^{1-\alpha}A\tilde{x}(\varsigma)
+\tilde{f}(\lambda)dw(\lambda),
\]
which establishes the  result
\[
\lambda\tilde{x}(\varsigma)=\lambda^{\alpha-1}R(\lambda^{\alpha},A)\varphi
+\lambda^{\alpha-1}R(\lambda^{\alpha},A)\tilde{f}(\lambda)dw(\lambda).
\]
This implies that
\[
x(t)=S_{\alpha}(t)\varphi+\int_0^tS_{\alpha}(t-s)f(s)dw(s).
\]

Let $x : (-\infty, b] \to H$ be a function such that $x, x' \in \mathcal{PC}$. 
If $x$ is a solution of \eqref{e1.1}-\eqref{e1.3}, from the
partial neutral integro-differential inclusions theory, we obtain
\[
x(t)\in S_{\alpha}(t)[\varphi(0)- G(0,\varphi)]
 + G(t,x_t)+\int_0^t S_{\alpha}(t-s)F(s,x_{\rho(s,x_{s})})dw(s),
\quad     t\in [0,t_1].
\]
By using that $x(t_1^{+}) = x(t_1^{-}) + I_k(x_{t_1})$, for
 $t \in(t_1, t_2] $ we have
\begin{align*}
x(t)&\in S_{\alpha}(t-t_1)[x(t_1^{+})- G(t_1,x_{t^{+}_1})] +
G(t,x_t)
+\int_{t_1}^t S_{\alpha}(t-s)F(s,x_{\rho(s,x_{s})})dw(s)\\
&=S_{\alpha}(t-t_1)[x(t_1^{-})+I_1(x_{t_1})-
G(t_1,x_{t^{+}_1})] + G(t,x_t) \\
&\quad +\int_{t_1}^t S_{\alpha}(t-s)F(s,x_{\rho(s,x_{s})})dw(s).
\end{align*}
By repeating the same procedure, we can easily deduce that
\begin{align*}
x(t)&\in S_{\alpha}(t-t_k)[x(t_k^{-})+I_k(x_{t_k})-
G(t_1,x_{t^{+}_k})] + G(t,x_t) \\
&\quad +\int_{t_k}^t S_{\alpha}(t-s)F(s,x_{\rho(s,x_{s})})dw(s)
\end{align*}
holds for any $t \in(t_k,t_{k+1}], k=2,\dots,m$.
This expression motivates the following
definition.



\begin{definition} \label{def2.8} \rm
An $\mathcal{F}_t$-adapted stochastic process $x : (-\infty,b]\to H$  
is called a mild solution of the system \eqref{e1.1}-\eqref{e1.3} 
if $x_0 = \varphi, x_{\rho(s,x_{s})} \in \mathcal{B}$ for every $s \in J$  
and $\Delta x(t_k)=I_k(x_{t_k}), k=1,\dots, m$, the restriction of
$x(\cdot)$ to the interval $(t_k, t_{k+1}]  (k= 0, 1,\dots , m) $
is continuous, and
\[
x(t)\in \begin{cases}
 S_{\alpha}(t)[\varphi(0)- G(0,\varphi)]
 + G(t,x_t)\\
+\int_0^t S_{\alpha}(t-s)F(s,x_{\rho(s,x_{s})})dw(s),
    &t\in [0,t_1], \\[3pt]
 S_{\alpha}(t-t_1)[x(t_1^{-})+I_1(x_{t_1})- G(t_1,x_{t^{+}_1})] + G(t,x_t)\\
+\int_{t_1}^t S_{\alpha}(t-s)F(s,x_{\rho(s,x_{s})})dw(s),  &t\in (t_1,t_2], \\
\dots\\
S_{\alpha}(t-t_{m})[x(t_{m}^{-})+I_{m}(x_{t_{m}})- G(t_{m},x_{t^{+}_{m}})]
+ G(t,x_t)\\
+\int_{t_{m}}^t S_{\alpha}(t-s)F(s,x_{\rho(s,x_{s})})dw(s),
 &t\in (t_{m},b].
\end{cases}
\]
\end{definition}

Now we have a nonlinear alternative of Leray-Schauder type for multivalued maps 
due to  O'Regan.

\begin{lemma}[\cite{o1}] \label{lem2.5}  
 Let $H$ be a Hilbert space with $V$ an open,convex subset of $H$ and $
y\in H$. Suppose
\begin{itemize}
\item[(a)]  $ \Phi :\overline{V}  \to \mathcal{P}_{cd}(H)$ has closed
graph, and

\item[(b)]  $ \Phi :\overline{V}  \to \mathcal{P}_{cd}(H)$ is a
condensing map with $\Phi(\overline{V})$ a subset of a bounded set
in $H$ hold.
\end{itemize}
 Then either
\begin{itemize}
\item[(i)]  $\Phi$ has a fixed point in $\overline{V};$ or

\item[(ii)]    There exist $y \in \partial V$ and $\lambda\in(0,1)$ with
$y \in\lambda \Phi(y)+(1-\lambda)\{y_0\}$.
\end{itemize}
\end{lemma}

\section{Main results}

In this section we shall present and prove our main result. Assume
that $\rho :J \times \mathcal{B}\to (-\infty,b]$ is
continuous. In addition, we make the following hypotheses:
\begin{itemize}
\item[(H1)] The function $t \to \varphi_t$ is continuous from
$\mathcal{R}(\rho^{-})=\{\rho(s,\psi)\leq 0, (s,\psi)\in J\times
\mathcal{B}\}$ into $\mathcal{B}$ and there exists a continuous and
bounded function
$J^{\varphi}:\mathcal{R}(\rho^{-})\to(0,\infty)$ such that
$\| \varphi_t\|_{\mathcal{B}}\leq
J^{\varphi}(t)\| \varphi\|_{\mathcal{B}}$ for each
$t\in \mathcal{R}(\rho^{-})$.

\item[(H2)] The multi-valued map $F:J\times\mathcal{B}\to \mathcal{P}_{bd,cl,cv}(L(K,H));$
for each $t \in J$, the function $F(t, \cdot) :
\mathcal{B}\to \mathcal{P}_{bd,cl,cv}(L(K,H))$ is u.s.c. and
for each $\psi \in \mathcal{B}$, the function $F(\cdot, \psi)$ is
measurable; for each fixed $\psi\in\mathcal{B}$, the set
\[
S_{F,\psi}= \{f
\in  L^2(J,L(K,H)) : f(t)\in F(t,\psi) \quad \text{for a.e }  t \in
J\}
\]
is nonempty.

\item[(H3)] There exists a positive function $l: J\to
\mathbb{R}^{+}$ such that the function $s
\mapsto(\frac{1}{1+|\omega|(t-s)^{\alpha}})^2l(s)$ belongs to
$L^{1}([0, t], \mathbb{R}^{+}),t\in J$, and
\[
\limsup_{\|\psi\|^2_{\mathcal{B}}\to\infty}\frac{\|
F(t,\psi)\|^2}{l(t)\|
\psi\|^2_{\mathcal{B}}}= \gamma
\]
uniformly in $t\in J$ for a nonnegative constant
$\gamma$, where
\[
\| F (t,\psi)\|^2= \sup\{E\| f \|^2:
f\in F (t,\psi)\}.
\]


\item[(H4)]   The function $G:J\times\mathcal{B}\to H$
is continuous and there exist $L,L_1>0$ such that
\begin{gather*}
E\| G(t,\psi_1)- G(t,\psi_2)\|^2\leq L\|
\psi_1-\psi_2\|^2_{\mathcal{B}},  \quad t\in J, \psi_1,\psi_2\in \mathcal{B},
\\
E\| G(t,\psi)\|^2\leq L_1(\|
\psi\|^2_{\mathcal{B}}+1),   \ \ \ t\in J, \psi\in
\mathcal{B},
\end{gather*} 
with 
$4[(CM)^2+1]LK^2_{b}< 1$.

\item[(H5)]   The functions $I_k: \mathcal{B}\to H$ are completely continuous
 and there exist constants $c_k$ such that
\[
\limsup_{\|\psi\|^2_{\mathcal{B}}\to\infty} \frac{E\|
I_k(\psi)\|^2}{\|\psi\|^2_{\mathcal{B}}}= c_k
\]
for every $\psi \in\mathcal{B}$,  $k = 1,\dots ,m$.
\end{itemize}

\begin{remark} \label{rmk3.1} \rm
 Let $\varphi\in \mathcal{B} $ and $t \leq 0$. 
The notation $\varphi_t$ represents the function defined by
$\varphi_t(\tau)=\varphi(t+\theta)$. Consequently, if the function
$x(\cdot)$ in axiom (A) is such that $x_0=\varphi$, then
$x_t=\varphi_t $. We observe that $\varphi_t$ is well-defined
for $t < 0$ since the domain of $\varphi$ is $(-\infty,0]$. We also
note that, in general, $\varphi_t\notin\mathcal{B};$ consider, for
instance, a discontinuous function in $C_{r} \times L^{p}(h,H)$ 
 for $r> 0$ (see \cite{h4}).
\end{remark}

\begin{remark} \label{rmk3.2}\rm
 The condition (H1) is frequently verified
by continuous and bounded functions. In fact, if $\mathcal{B}$
verifies axiom (C$_2$) in the nomenclature of \cite{h4}, then there
exists $\tilde{L}> 0 $ such that 
$\| \varphi\|_{\mathcal{B}}\leq \tilde{L}\sup_{\tau\leq0}\|\varphi(\tau)\|$
for every $\varphi\in\mathcal{B}$ continuous and bounded, see 
 \cite[Proposition 7.1.1]{h4} for details. Consequently,
$$
\| \varphi_t\|_{\mathcal{B}}\leq \tilde{L}\frac{\sup_{\tau\leq0}\varphi(\tau)}
{\| \varphi\|_{\mathcal{B}}},
$$ 
for every continuous and bounded function $\varphi\in\mathcal{B}\setminus\{0\}$ 
and every $t \leq 0$. We also observe that the space $C_{r}\times L^{p}(h,H)$
verifies axiom (C$_2$)  see  \cite[p. 10]{h4} for details.
\end{remark}

\begin{lemma} \label{lem3.1} 
 Let $x:(-\infty,b]\to H$ such
that $x_0=\varphi$ and $x|_{[0,b]}\in \mathcal{PC}(J,H)$. If (H1)
be hold, then
\[
\| x_{s}\|_{\mathcal{B}}\leq (M_{b}+J_0^{\varphi})
\| \varphi\|_{\mathcal{B}}+K_{b}\sup\{\|
x(\theta)\|; \theta\in[0,\max\{0,s\}]\},s\in
\mathcal{R}(\rho^{-})\cup J,
\]
where $J_0^{\varphi}=\sup_{t\in \mathcal{R}(\rho^{-})}J^{\varphi}(t)$.
\end{lemma}

\begin{proof} 
For any $s\in \mathcal{R}(\rho^{-})$, by (H1), we have
\[
\| x_{s}\|_{\mathcal{B}}
\leq\| \varphi_{s}\|_{\mathcal{B}}
\leq J^{\varphi}(s)\| \varphi\|_{\mathcal{B}}\leq
J_0^{\varphi}\| \varphi\|_{\mathcal{B}}.
\]
For any $s\in[0,b]$, $x\in \mathcal{PC}(J,H)$.
Using the phase spaces axioms,  we have
\begin{align*}
\| x_{s}\|_{\mathcal{B}}
&\leq  M(s)\| \varphi\|_{\mathcal{B}}+K(s)
\sup\{\| x(s)\|:0\leq s\leq t\}\\
&\leq M_{b}\| \varphi\|_{\mathcal{B}}+K_{b}
\sup\{\| x(s)\|:0\leq s\leq t\}.
\end{align*}
Then, for $s\in(-\infty,b]$, we have
\[
\| x_{s}\|_{\mathcal{B}}\leq
(M_{b}+J_0^{\varphi})
\| \varphi\|_{\mathcal{B}}+K_{b}\sup\{\|
x(\theta)\|; \theta\in[0,\max\{0,s\}]\},s\in
\mathcal{R}(\rho^{-})\cup J.
\]
The proof is complete.
\end{proof}

\begin{lemma}[\cite{l3}] \label{lem3.2}
 Let $J$ be a compact interval and  $H$ be a Hilbert space.
Let $F$ be a multivalued map satisfying {\rm (H2)} and
$\Gamma$ be a linear continuous operator from $L^2(J, H)$ to 
$C(J, H)$. Then the operator $\Gamma\circ S_{F}: C(J, H)\to
\mathcal{P}_{cp,cv}(C(J, H)) $ is a closed graph in $C(J, H)\times C(J, H )$.
\end{lemma}

\begin{theorem} \label{thm3.1} 
Let {\rm (H1)--(H5)} be satisfied  and 
$x_0 \in L^{0}_2 ( \Omega,H)$, with $\rho(t,\psi) \leq t$ for every
 $(t,\psi) \in J\times \mathcal{B}$. 
Then problem \eqref{e1.1}-\eqref{e1.3}  has at least one mild solution 
on  $J$, provided that
\begin{equation}
\max_{1\leq k\leq m}
\{9(CM)^2[1+2K_{b}^2c_k +2K_{b}^2L_1]+6K_{b}^2L_1\}<1.
\end{equation}
\end{theorem}

 \begin{proof} 
Consider the space 
$\mathcal{BPC}=\{x:(-\infty,b]\to  H; x_0=0, x|_{J}\in\mathcal{PC}\}$
 endowed with the uniform convergence topology and define
 the multi-valued map $\Phi:\mathcal{BPC}\to \mathcal{P}(\mathcal{BPC})$ by
 $\Phi x$ the set of $ h\in \mathcal{BPC}$ such that
\[
h(t)= \begin{cases}
 0, & t\in (-\infty,0], \\[3pt]
 S_{\alpha}(t)[\varphi(0)- G(0,\varphi)]
 + G(t,\bar{x}_t)+\int_0^t S_{\alpha}(t-s)f(s)dw(s),
    &t\in [0,t_1], \\[3pt]
 S_{\alpha}(t-t_1)[\bar{x}(t_1^{-})+I_1(\bar{x}_{t_1})
- G(t_1,\bar{x}_{t^{+}_1})] + G(t,\bar{x}_t)\\
+\int_{t_1}^t S_{\alpha}(t-s)f(s)dw(s),  & t\in (t_1,t_2], \\
\dots\\
 S_{\alpha}(t-t_{m})[\bar{x}(t_{m}^{-})+I_{m}(\bar{x}_{t_{m}})
- G(t_{m},\bar{x}_{t^{+}_{m}})] + G(t,\bar{x}_t)\\
+\int_{t_{m}}^t S_{\alpha}(t-s)f(s)dw(s), &t\in (t_{m},b],
\end{cases}
\]
where $f\in S_{F,\bar{x}_{\rho}}= \{ f \in
 L^2 (L(K, H)) : f (t) \in F(t,\bar{x}_{\rho(s,\bar{x}_t)})\
\text{a.e. }  t \in J \}$ and
$\bar{x}:(-\infty,0]\to H$
is such that $\bar{x}_0=\varphi$ and $\bar{x}=x$ on $J$.
 In what follows, we aim to show that the operator $\Phi$ has a fixed point,
which is a solution of the problem  \eqref{e1.1}-\eqref{e1.3}.

Let $\bar{\varphi}:(-\infty,0)\to H$ be the extension of
$(-\infty,0]$ such that $\bar{\varphi}(\theta)=\varphi(0)=0$ on $J$
and
$J^{\varphi}_0=\sup\{J^{\varphi}(s):s\in\mathcal{R}(\rho^{-})\}$.
We now show that $\Phi$ satisfies all the conditions of Lemma \ref{lem2.5}.
The proof will be given in several steps.
\smallskip

\noindent\textbf{Step 1.} We shall show there exists an open set $V\subseteq
\mathcal{BPC}$   with $x \in\lambda \Phi x$ for
$\lambda\in(0, 1) $ and $ x\notin \partial V$.
Let $\lambda\in (0, 1)$ and let $x\in\lambda \Phi x$, then there
exists an $f\in S_{F,\bar{x}_{\rho}}$ such that
 we have
\[
x(t)=\begin{cases}
 \lambda S_{\alpha}(t)[\varphi(0)- G(0,\varphi)]
 +\lambda G(t,\bar{x}_t)+\lambda\int_0^t S_{\alpha}(t-s)f(s)dw(s),
  &t\in [0,t_1], \\[3pt]
 \lambda S_{\alpha}(t-t_1)[\bar{x}(t_1^{-})+I_1(\bar{x}_{t_1})
- G(t_1,\bar{x}_{t^{+}_1})]  +\lambda G(t,\bar{x}_t)\\
+\lambda\int_{t_1}^t S_{\alpha}(t-s)f(s)dw(s),   &t\in (t_1,t_2], \\
\dots\\
\lambda S_{\alpha}(t-t_{m})[\bar{x}(t_{m}^{-})+I_{m}(\bar{x}_{t_{m}})-
G(t_{m},\bar{x}_{t^{+}_{m}})]
 +\lambda G(t,\bar{x}_t)\\
+\lambda\int_{t_{m}}^t S_{\alpha}(t-s)f(s)dw(s),  & t\in (t_{m},b],
\end{cases}
\]
for some $\lambda\in (0, 1)$. It follows from assumption (H3) that
there exist two nonnegative real numbers $a_1$ and $a_2$ such
that for any $\psi\in \mathcal{B}$ and $t\in J$,
\begin{equation} \label{e3.2}
\| F(t,\psi)\|^2 \leq a_1l(t)+a_2l(t)\|
\psi\|^2_{\mathcal{B}}.
\end{equation}
On the other hand, from  condition (H5), we conclude
that there exist positive constants
$\epsilon_k(k=1,\dots,m),\gamma_1$  such that, for all
$\| \psi\|_{\mathcal{B}}^2>\gamma_1$,
\begin{gather}
E\| I_k(\psi)\|^2 \leq (c_k+\epsilon_k)
\| \psi\|^2_{\mathcal{B}}, \nonumber \\
\max_{1\leq k\leq m} \{9(CM)^2[1+2K_{b}^2(c_k+\epsilon_k)
+2K_{b}^2L_1]+6K_{b}^2L_1\}<1. \label{e3.3}
\end{gather}
Let
\begin{gather*}
F_1 = \{\psi:
 \| \psi\|^2_{\mathcal{B}}\leq  \gamma_1\}, \ \ \ F_2 = \{\psi:
 \| \psi \|^2_{\mathcal{B}}> \gamma_1\},\\
C_1= \max \{ E\| I_k(\psi)\|^2, x\in F_1 \}.
\end{gather*}
 Therefore,
\begin{equation} \label{e3.4}
E\| I_k(\psi)\|^2
\leq C_1+(c_k+\epsilon_k)\|\psi\|_{\mathcal{B}}^2.
\end{equation}
Then, by (H4), \eqref{e3.2}  and \eqref{e3.4}, from the above equation,
for $t \in [0,t_1]$, we have
\begin{align*}
E\| x(t)\|^2
&\leq 3E\| S_{\alpha}(t)[\varphi(0)- G(0,\varphi)]\|^2
 +3E\| G(t,\bar{x}_t)\|^2\\
&\quad+3E\big\|\int_0^t S_{\alpha}(t-s)f(s)dw(s)\big\|^2\\
&\leq   6(CM)^2[E\| \varphi(0)
\|^2+L_1(\| \varphi
\|^2_{\mathcal{B}}+1)]+3L_1(\| \bar{x}_t
\|^2_{\mathcal{B}}+1)\\
 &\quad+3(CM)^2\operatorname{Tr}(Q)\int_0^{t}
 \Big(\frac{1}{1+|\omega|(t-s)^{\alpha}}\Bigr)^2[a_1l(s)+a_2l(s)\|
 \bar{x}_{\rho(s,\bar{x}_{s})}\|^2_{\mathcal{B}}]ds\\
&\leq 6(CM)^2[\tilde{H}^2E\| \varphi
\|^2_{\mathcal{B}}+L_1(\| \varphi
\|^2_{\mathcal{B}}+1)]+3L_1(\| \bar{x}_t
\|^2_{\mathcal{B}}+1)\\
 &\quad+3(CM)^2\operatorname{Tr}(Q)a_1\int_0^{t_1}
 \Big(\frac{1}{1+|\omega|(t_1-s)^{\alpha}}\Bigr)^2l(s)ds\\
 &\quad  +3(CM)^2\operatorname{Tr}(Q)a_2\int_0^{t}
 \Big(\frac{1}{1+|\omega|(t-s)^{\alpha}}\Bigr)^2l(s)\|
 \bar{x}_{\rho(s,\bar{x}_{s})}\|^2_{\mathcal{B}}ds.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$, $k = 1, \dots ,m$, we
have
\begin{align*}
&E\| x(t)\|^2\\
&\leq3E\| S_{\alpha}(t-t_k)[\bar{x}(t_k^{-})+I_k(\bar{x}_{t_k})-
 G(t_k,\bar{x}_{t^{+}_k})]\|^2 +3E\| G(t,\bar{x}_t)\|^2\\
&\quad +3E\big\|\int_{t_k}^t
 S_{\alpha}(t-s)f(s)dw(s)\big\|^2\\
&\leq9(CM)^2[E\| \bar{x}(t_k^{-})
 \|^2+C_1+(c_k+\epsilon_k)\|\bar{x}_{t_k}\|^2_{\mathcal{B}}
 +L_1(\| \bar{x}_{t^{+}_k}\|^2_{\mathcal{B}}+1)]\\
&\quad+3L_1(\| \bar{x}_t
 \|^2_{\mathcal{B}}+1) +3(CM)^2a_1\operatorname{Tr}(Q)\int_{t_k}^{t_{k+1}}
 \Big(\frac{1}{1+|\omega|(t_{k+1}-s)^{\alpha}}\Bigr)^2l(s)ds\\
&\quad+3(CM)^2a_2\operatorname{Tr}(Q)\int_{t_k}^{t}
 \Big(\frac{1}{1+|\omega|(t-s)^{\alpha}}\Bigr)^2l(s)\|
 \bar{x}_{\rho(s,\bar{x}_{s})}\|^2_{\mathcal{B}}ds.
\end{align*}
Then, for all $t \in [0, b]$, we have
\begin{align*}
&E\| x(t)\|^2\\
&\leq \widetilde{M}+9(CM)^2[E\| \bar{x}(t_k^{-})
 \|^2+(c_k+\epsilon_k)\|\bar{x}_{t_k}\|^2_{\mathcal{B}}
 +L_1\| \bar{x}_{t^{+}_k}\|^2_{\mathcal{B}}]\\
&\quad+3L_1\| \bar{x}_t
 \|^2_{\mathcal{B}} +3(CM)^2a_2\operatorname{Tr}(Q)\int_0^{t}
 \Big(\frac{1}{1+|\omega|(t-s)^{\alpha}}\Bigr)^2l(s)\|
 \bar{x}_{\rho(s,\bar{x}_{s})}\|^2_{\mathcal{B}}ds,
\end{align*}
where
\begin{align*}
\widetilde{M}
&=\max\Big\{6(CM)^2[\tilde{H}^2E\| \varphi
\|^2_{\mathcal{B}}+L_1(\| \varphi
\|^2_{\mathcal{B}}+1)]+3L_1\\
 &\quad+3(CM)^2\operatorname{Tr}(Q)a_1\int_0^{t_1}\Big(\frac{1}{1+|\omega|(b-s)^{\alpha}}\Bigr)^2l(s)ds,\
9(CM)^2(C_1 +L_1)\\
&\quad+3L_1 +3(CM)^2a_1\operatorname{Tr}(Q)\int_{t_k}^{t_{k+1}}
\Big(\frac{1}{1+|\omega|(t_{k+1}-s)^{\alpha}}\Bigr)^2l(s)ds\Bigr\}.
\end{align*}
By  Lemmas \ref{lem2.3} and  \ref{lem3.1},  it follows that
$\rho(s,\overline{x}_{s})\leq s, s\in [0,t], t\in [0,b]$ and
 \begin{equation} \label{e3.5}
\| \overline{x}_{\rho(s,\overline{x}_{s})}\|^2_{\mathcal{B}}
\leq 2[(M_{b}+J^{\varphi}_0)E\|
\varphi\|_{\mathcal{B}}]^2 +2K_{b}^2\sup_{0\leq s
\leq b}E\| x(s)\|^2.
\end{equation}
For each $t \in [0,b]$,
 we have
\begin{align*}
E\| x(t)\|{^2}
&\leq  M_{*}
 +\{9(CM)^2[1+2K_{b}^2(c_k+\epsilon_k) +2K_{b}^2L_1]+6K_{b}^2L_1\}
 \sup_{t\in[0,b]}E\| x(t)\|{^2} \\
 &\quad +6(CM)^2a_2K_{b}^2\operatorname{Tr}(Q)\int_0^{t}
\Big(\frac{1}{1+|\omega|(t-s)^{\alpha}}\Bigr)^2l(s)
 \sup_{\tau\in[0,s]}E\| x(\tau)\|{^2} ds,
\end{align*}
where
\begin{gather*}
\begin{aligned}
M_{*}&=\widetilde{M}+9(CM)^2[C_1+(c_k+\epsilon_k)C^{*}
+L_1(C^{*}+1)]+3L_1(C^{*}+1)\\
 &\quad+3(CM)^2\operatorname{Tr}(Q)a_2C^{*}\int_0^{b}
 \Big(\frac{1}{1+|\omega|(b-s)^{\alpha}}\Bigr)^2l(s)ds,
\end{aligned}\\
 C^{*}=2[(M_{b}+J^{\varphi}_0)\|\varphi\|_{\mathcal{B}}]^2.
\end{gather*}
Since $L_{*}=\max_{1\leq k\leq m}
\{9(CM)^2[1+2K_{b}^2(c_k+\epsilon_k)
+2K_{b}^2L_1]+6K_{b}^2L_1\}<1$, we have
\begin{align*}
&\sup_{t\in[0,b]}E\| x(t)\|^2 \\
&\leq\frac{M_{*}}{1-L_{*}}
 +\frac{6(CM)^2a_2K_{b}^2\operatorname{Tr}(Q)}{1-L_{*}}
\int_0^{b}\Big(\frac{1}{1+|\omega|(b-s)^{\alpha}}\Bigr)^2l(s)
 \sup_{\tau\in[0,s]}E\| x(\tau)\|{^2} ds.
\end{align*}
 Applying  Gronwall's inequality in the above
expression, we obtain
\[
\sup_{t\in[0,b]}E\| x(s)\| ^2\leq
\frac{M_{*}}{1-L_{*}} \exp \Big\{\frac{6(CM)^2a_2K_{b}^2\operatorname{Tr}(Q)}{1-L_{*}}
\int_0^{b}\Big(\frac{1}{1+|\omega|(b-s)^{\alpha}}\Bigr)^2l(s)ds\Bigr\}
\]
and, therefore,
\[
\|x\|^2_{\mathcal{PC}}\leq\frac{M_{*}}{1-L_{*}}
\exp \Big\{\frac{6(CM)^2a_2K_{b}^2\operatorname{Tr}(Q)}{1-L_{*}}
\int_0^{b}\Big(\frac{1}{1+|\omega|(b-s)^{\alpha}}\Bigr)^2l(s)ds\Bigr\}<\infty.
\]
Then,  there exists $r^{*}$ such that $\|x\|^2_{\mathcal{PC}}\neq r^{*}$.
 Set
$$
V = \{x\in \mathcal{BPC} :\| x\|^2_{\mathcal{PC}}<r^{*}\}.
 $$
From the choice of $V$, there is no $x\in \partial V $ such that
$x \in \lambda \Phi x $ for $\lambda\in (0, 1)$.
\smallskip

\noindent\textbf{Step 2.} $\Phi$ has a closed graph.
Let $x^{(n)}\to x^{*},  h_n\in \Phi x^{(n)}, x^{(n)}\in
\overline{V}= B_{r^{*}}(0,\mathcal{BPC})$ and $h_n\to
h_{*}$. From Axiom (A), it is easy to see that
$({\overline{x^{(n)}}})_{s}\to \overline{x^{*}}_{s}$
uniformly for $s \in (-\infty, b]$ as $n\to\infty$. We prove
that $ h_{*} \in \Phi {\overline{x^{*}}}$. Now $ h_n \in \Phi
{\overline{x^{(n)}}}$ means that there exists $f_n\in
S_{F,{\overline{x^{(n)}}}_{\rho}}$ such that, for each $t \in
[0,t_1]$,
\[
 h_n(t) =S_{\alpha}(t)[\varphi(0)-
G(0,\varphi)] +G(t,(\overline{x^{(n)}})_t)
+\int_0^tS_{\alpha}(t-s) f_n(s)dw(s),  \quad t\in [0,t_1].
\]
 We must prove that there
exists $f_{*}\in S_{F,{\overline{x^{*}}}_{\rho}}$  such that, for
each $t\in [0,t_1]$,
\[
 h_{*}(t) =S_{\alpha}(t)[\varphi(0)-
G(0,\varphi)] +G(t,(\overline{x^{*}})_t) +\int_0^tS_{\alpha}(t-s)
f_{*}(s)dw(s),  \quad t\in [0,t_1].
\]
  Now, for every $t \in  [0,t_1]$, we have
\begin{align*}
&\Big\| \Big(h_n(t)-S_{\alpha}(t)[\varphi(0)-
G(0,\varphi)]
-G(t,(\overline{x^{(n)}})_t)  -\int_0^tS_{\alpha}(t-s) f_n(s)dw(s)\Bigr)\\
& -\Big(h_{*}(t)-S_{\alpha}(t)[\varphi(0)- G(0,\varphi)]
-G(t,(\overline{x^{*}})_t)  \\
&-\int_0^tS_{\alpha}(t-s)
f_{*}(s)dw(s)\Bigr)\Big\|^2_{\mathcal{PC}}
 \to0 \quad \text{as } n\to\infty.
\end{align*}
Consider the linear continuous operator
 $\Psi:  L([0, t_1 ], H) \to C([0, t_1 ], H)$,
\[
\Psi( f)(t) =\int_0^tS_{\alpha}(t-s)f(s)dw(s).
\]
From Lemma \ref{lem3.2}, it follows that
 $\Psi\circ S_{F}$ is a closed graph operator. Also, from the definition of
$\Psi$, we have that, for every $t \in [0,t_1]$,
\[
h_n(t)-S_{\alpha}(t)[\varphi(0)- G(0,\varphi)]
-G(t,(\overline{x^{(n)}})_t)  -\int_0^tS_{\alpha}(t-s)
f_n(s)dw(s)\Bigr)\in \Gamma(S_{F,{\overline{x^{(n)}}}}).
\]
Since $\overline{x^{(n)}}\to \overline{x^{*}}$, for some
$f_{*}\in S_{F,\overline{x^{*}}_{\rho}}$ it follows that, for every
$t \in [0,t_1]$,
\[
h_{*}(t)-S_{\alpha}(t)[\varphi(0)- G(0,\varphi)]
-G(t,(\overline{x^{*}})_t)=\int_0^t\mathcal{S}_{\alpha}(t-s)f_{*}dw(s).
\]
Similarly, for any $t\in (t_k, t_{k+1}], k = 1, \dots ,m$, we
have
\begin{align*}
 h_n(t)
 &=S_{\alpha}(t-t_k)[\overline{x^{(n)}}(t_k^{-})+I_k(\overline{x^{(n)}}_{t_k})-
  G(t_k,(\overline{x^{(n)}})_{t_k^{+}})]+ G(t,(\overline{x^{(n)}})_t) \\
 &\quad+\int_{t_k}^tS_{\alpha}(t-s) f_n(s)dw(s),  \quad t\in
(t_k, t_{k+1}].
\end{align*}
We must prove that there
exists $f_{*}\in S_{F,{\overline{x^{*}}}_{\rho}}$  such that, for
each $t\in (t_k, t_{k+1}]$,
\begin{align*}
h_{*}(t)
&=S_{\alpha}(t-t_k)[\overline{x^{*}}(t_k^{-})+I_k(\overline{x^{*}}_{t_k})-
  G(t_k,(\overline{x^{*}})_{t_k^{+}})]+ G(t,(\overline{x^{*}})_t) \\
&\quad +\int_{t_k}^tS_{\alpha}(t-s) f_{*}(s)dw(s),  \quad t\in
(t_k, t_{k+1}].
\end{align*}
  Now, for every $t \in  (t_k,t_{k+1}]$,  $k = 1, \dots,m$, we have
\begin{align*}
&\Big\|\Big(h_n(t)-S_{\alpha}(t-t_k)[\overline{x^{(n)}}(t_k^{-})
+I_k(\overline{x^{(n)}}_{t_k})-
  G(t_k,(\overline{x^{(n)}})_{t_k^{+}})]-G(t,(\overline{x^{(n)}})_t) \\
&-\int_{t_k}^tS_{\alpha}(t-s) f_n(s)dw(s)\Bigr)
-\Big(h_{*}(t)-S_{\alpha}(t-t_k)\big[\overline{x^{*}}(t_k^{-})
 +I_k(\overline{x^{*}}_{t_k})\\
&-  G(t_k,(\overline{x^{*}})_{t_k^{+}})\big]
 - G(t,(\overline{x^{*}})_t)
 -\int_{t_k}^tS_{\alpha}(t-s) f_{*}(s)dw(s)\Bigr)\Big\|^2_{\mathcal{PC}}
 \to 0 \quad \text{as }  n\to\infty.
\end{align*}
Consider the linear continuous operator
 $\Psi: L^2((t_k,t_{k+1}],H)\to C((t_k,t_{k+1}] ,H)$, $k = 1, \dots,m$,
\[
\Psi( f)(t)= \int_{t_k}^tS_{\alpha}(t-s) f(s)dw(s).
\]
From Lemma \ref{lem3.2}, it follows that
 $\Psi\circ S_{F}$ is a closed graph operator. Also, from the definition of
$\Psi$, we have that, for every $t \in  (t_k,t_{k+1}],  k = 1,
\dots ,m$,
\[
h_n(t)-S_{\alpha}(t-t_k)[\overline{x^{(n)}}(t_k^{-})
 +I_k(\overline{x^{(n)}}_{t_k})-
 G(t_k,(\overline{x^{(n)}})_{t_k^{+}})]
 -G(t,(\overline{x^{(n)}})_t) \in \Gamma(S_{F,{\overline{x^{(n)}}}_{\rho}}).
\]
Since $\overline{x^{(n)}}\to \overline{x^{*}}$, for some
$f_{*}\in S_{F,\overline{x^{*}}_{\rho}}$ it follows that, for every
$t \in (t_k, t_{k+1}]$, we have
\begin{align*}
&h_{*}(t)-S_{\alpha}(t-t_k)[\overline{x^{*}}(t_k^{-})
 +I_k(\overline{x^{*}}_{t_k})-
  G(t_k,(\overline{x^{*}})_{t_k^{+}})]- G(t,(\overline{x^{*}})_t)\\
&=\int_{t_k}^tS_{\alpha}(t-s)f_{*}(s)dw(s).
\end{align*}
Therefore, $\Phi$ has a closed graph.
\smallskip

\noindent\textbf{Step 3.} 
We show that the operator $\Phi$ condensing.
For this purpose,  we decompose $\Phi$ as $\Phi_1+\Phi_2$, where
the map $\Phi_1: \overline{V}\to
\mathcal{P}(\mathcal{BPC})$ be defined by $\Phi_1x$, the set $
h_1\in \mathcal{BPC}$ such that
\[
h_1(t)=\begin{cases}
  0, &  t\in (-\infty,0], \\
  -S_{\alpha}(t)G(0,\varphi) +G(t,\bar{x}_t), &t\in [0,t_1], \\
  -S_{\alpha}(t-t_1)  G(t_1,\bar{x}_{t_1^{+}})
 +  G(t,\bar{x}_t), &t\in (t_1,t_2],\\
 \dots\\
 -S_{\alpha}(t-t_{m})
  G(t_{m},\bar{x}_{t_{m}^{+}})
 +  G(t,\bar{x}_t),\quad &t\in (t_{m},b],
\end{cases}
\]
and the map $\Phi_2 : \overline{V} \to
\mathcal{P}(\mathcal{BPC})$ be defined by $\Phi_2 x$, the set $
h_2\in \mathcal{BPC}$ such that
\[
h_2(t)=\begin{cases}
 0, &  t\in (-\infty,0], \\
 S_{\alpha}(t)\varphi(0)+\int_0^t\mathcal{S}_{\alpha}(t-s)
f(s)ds,  & t\in [0,t_1], \\
 S_{\alpha}(t-t_1)[\bar{x}(t_1^{-})+I_1(\bar{x}_{t_1})]
+\int_{t_1}^t S_{\alpha}(t-s)f(s)dw(s), &  t\in(t_1,t_2],\\
\dots\\
 S_{\alpha}(t-t_{m})[\bar{x}(t_{m}^{-})+I_{m}(\bar{x}_{t_{m}})]
+\int_{t_{m}}^t S_{\alpha}(t-s)f(s)dw(s),\quad &t\in (t_{m},b].
\end{cases}
\]
We first show that $\Phi_1$ is a contraction while $\Phi_2$ is a
completely continuous operator.

\noindent\textbf{Claim 1.}  $\Phi_1$ is a contraction on
$\mathcal{BPC}$.
Let $t \in [0,t_1]$ and  $v^{*}, v^{**} \in \mathcal{BPC}$. From
(H4),  Lemmas \ref{lem2.3} and \ref{lem3.1}, we have
\begin{align*}
E\|(\Phi_1v^{*})(t)-(\Phi_1v^{**})(t)\|^2 & \leq
E\| G(t,\overline{v^{*}}_t) -G(t,\overline{v^{**}}_t)\|^2\\
& \leq L\| \overline{v^{*}}_t-\overline{v^{**}}_t\|^2_{\mathcal{B}}\\
&\leq 2LK^2_{b} \sup\{\| \overline{v^{*}}(\tau)-\overline{v^{**}}
 (\tau)\|^2, 0 \leq \tau \leq t\}\\
&\leq 2LK^2_{b}  \sup_{s\in[0,b]}\|
\overline{v^{*}}(s)-\overline{v^{**}}(s)\|^2\\
&=2 LK^2_{b} \sup_{s\in[0,b]}\|
v^{*}(s)-v^{**}(s)\|^2 \quad  \text{(since $\bar{v}= v$ on $J$)}\\
 &= 2LK^2_{b}\| v^{*}-v^{**}\|^2_{\mathcal{PC}}.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$,  $k = 1, \dots ,m$, we
have
\begin{align*}
&E\|(\Phi_1v^{*})(t)-(\Phi_1v^{**})(t)\|^2 \\
&\leq 2E\| S_{\alpha}(t-t_k)[-G(t_k,\overline{v^{*}}_{t^{+}_k})
 +G(t_k,\overline{v^{**}}_{t^{+}_k})]\|^2 +2E\|
 G(t,\overline{v^{*}}_t) -G(t,\overline{v^{**}}_t)\|^2\\
&\leq 2(CM)^2L\|\overline{v^{*}}_{t^{+}_k}
 -\overline{v^{**}}_{t^{+}_k}\|^2_{\mathcal{B}}
 +2L\| \overline{v^{*}}_t-\overline{v^{**}}_t\|^2_{\mathcal{B}}\\
&\leq 4((CM)^2+1)LK^2_{b} \sup_{s\in[0,b]}\|
 \overline{v^{*}}(s)-\overline{v^{**}}(s)\|^2\\
&= 4((CM)^2+1)LK^2_{b} \sup_{s\in[0,b]}\|v^{*}(s)-v^{**}(s)\|^2 \quad
\text{(since $\bar{v}= v$ on  $J$)}\\
&= 4[(CM)^2+1)]LK^2_{b}\|v^{*}-v^{**}\|^2_{\mathcal{PC}},
\end{align*}
Thus, for all $t \in [0, b ]$, we have
\[
E\|(\Phi_1v^{*})(t)-(\Phi_1v^{**})(t)\|^2
\leq L_0\|v^{*}-v^{**}\|^2_{\mathcal{PC}}.
\]
Taking supremum over $t$,
\[
\| \Phi_1v^{*}-\Phi_1v^{**}\|^2_{\mathcal{PC}}\leq L_0\|
v^{*}-v^{**}\|^2_{\mathcal{PC}},
\]
where $L_0=4[(CM)^2+1]LK^2_{b}< 1$. Hence, $\Phi_1$ is a
contraction on $\mathcal{BPC}$.


\noindent\textbf{Claim 2.} 
$\Phi_2 $ is convex for each $x \in \overline{V}$.
In fact, if $h^{1}_2, h^2_2$ belong to $\Phi_2x$, then there
exist $f_1, f_2 \in S_{F,\overline{x}_{\rho} }$ such that
\[
h^{i}_2(t) =S_{\alpha}(t)\varphi(0)+ \int_0^tS_{\alpha}(t-s)
f_i(s)dw(s),  \quad t\in [0,t_1], \; i=1,2.
\]
Let $0\leq \lambda\leq1$. For each $t \in [0,t_1]$  we have
\[
(\lambda h^{1}_2+(1-\lambda)h^2_2)(t)=
S_{\alpha}(t)\varphi(0)+\int_0^tS_{\alpha}(t-s)[\lambda
f_1(s)+(1-\lambda)f_2(s)]dw(s).
\]
Similarly, for any $t\in (t_k, t_{k+1}]$, $k = 1, \dots ,m$, we
have
\[
h^{i}_2(t)
=S_{\alpha}(t-t_k)[\bar{x}(t_k^{-})+I_k(\bar{x}_{t_k})]+
\int_{t_k}^tS_{\alpha}(t-s) f_i(s)dw(s), \quad  i=1,2.
\]
Let $0\leq \lambda\leq1$. For each $t \in (t_k, t_{k+1}]$,
$k = 1, \dots ,m$, we have
\begin{align*}
(\lambda h^{1}_2+(1-\lambda)h^2_2)(t)
&=S_{\alpha}(t-t_k)[\bar{x}(t_k^{-})+I_k(\bar{x}_{t_k})]\\
&\quad+\int_{t_k}^tS_{\alpha}(t-s)[\lambda
f_1(s)+(1-\lambda)f_2(s)]dw(s).
\end{align*}
Since $ S_{F,\bar{x}_{\rho} }$ is convex (because $F$ has convex
values) we have
$(\lambda h^{1}_2+(1-\lambda)h^2_2)\in \Phi_2x$.


\noindent\textbf{Claim 3.}  $\Phi_2(\overline{V})$ is completely
continuous.
We begin by showing $\Phi_2(\overline{V})$ is equicontinuous. If
$x \in \overline{V}$, from   Lemmas \ref{lem2.3} and \ref{lem3.1}, it follows
that
 \[
\| \bar{x}_{\rho(s,\bar{x}_{s})}\|^2_{\mathcal{B}}
\leq 2[(M_{b}+J^{\varphi}_0)\|
\varphi\|_{\mathcal{B}}]^2 +2K^2_{b}r^{*}:=r'.
\]
Let $0 < \tau_1 < \tau_2 \leq t_1$. For  each
$x\in \overline{V}$, $ h_2\in \Phi_2x$, there exists
$f\in S_{F,\bar{x}_{\rho} }$, such that
\begin{equation}
 h_2(t)=S_{\alpha}(t)\varphi(0) +\int_0^tS_{\alpha}(t-s) f(s)dw(s).
\end{equation}
Then
\begin{align*}
&E\| h_2(\tau_2)- h_2(\tau_1)\|^2\\
& \leq 4E\|[S_{\alpha}(\tau_2)-S_{\alpha}(\tau_1)] \varphi(0)\|^2
+4E\big\|\int_0^{\tau_1-\varepsilon}[
S_{\alpha}(\tau_2-s)-S_{\alpha}(\tau_1-s)]f(s)dw(s)\big\|^2
\\
&\quad +4E\big\|\int_{\tau_1-\varepsilon}^{\tau_1}
[S_{\alpha}(\tau_2-s)-S_{\alpha}(\tau_1-s)] f(s)dw(s)\big\|^2\\
&\quad +4E\big\| \int_{\tau_1}^{\tau_2}S_{\alpha}
(\tau_2-s) f(s)dw(s)\big\|^2\\
& \leq 4E\|[S_{\alpha}(\tau_2)-S_{\alpha}(\tau_1)]
\varphi(0)\|^2+
4(CM)^2(a_1+a_2r')[1+|\omega|b^{\alpha}]^2\operatorname{Tr}(Q)
\\
&\quad\times\int_0^{\tau_1-\varepsilon}\|
S_{\alpha}(\tau_2-s)-S_{\alpha}(\tau_1-s)\|^2
\Big(\frac{1}{1+|\omega|(\tau_1-\varepsilon-s)^{\alpha}}\Bigr)^2l(s)ds\\
&\quad
+4(CM)^2(a_1+a_2r')\operatorname{Tr}(Q)\int_{\tau_1-\varepsilon}^{\tau_1}
\Big(\frac{1}{1+|\omega|(\tau_2-s)^{\alpha}}\Bigr)^2l(s)ds\\
&\quad
+4(CM)^2(a_1+a_2r')\operatorname{Tr}(Q)\int_{\tau_1-\varepsilon}^{\tau_1}
\Big(\frac{1}{1+|\omega|(\tau_1-s)^{\alpha}}\Bigr)^2l(s)ds\\
 &\quad + 4(CM)^2(a_1+a_2r')\operatorname{Tr}(Q)\int_{\tau_1}^{\tau_2}
\Big(\frac{1}{1+|\omega|(\tau_2-s)^{\alpha}}\Bigr)^2l(s)ds.
\end{align*}
Similarly, for any $\tau_1, \tau_2 \in (t_k, t_{k+1}]$,
$\tau_1 < \tau_2$, $k = 1, \dots ,m$, we have
\begin{equation}
 h_2(t)=S_{\alpha}(t-t_k)[\bar{x}(t_k^{-})+I_k(\bar{x}_{t_k})]
 +\int_{t_k}^tS_{\alpha}(t-s) f(s)dw(s).
\end{equation}
Then
\begin{align*}
&E\| h_2(\tau_2)- h_2(\tau_1)\|^2\\
& \leq 4E\|[S_{\alpha}(\tau_2)-S_{\alpha}(\tau_1)] \varphi(0)\|^2
+4E\big\|\int_{t_k}^{\tau_1-\varepsilon}[
S_{\alpha}(\tau_2-s)-S_{\alpha}(\tau_1-s)]f(s)dw(s)\big\|^2
\\
&\quad +4E\Big\|
\int_{\tau_1-\varepsilon}^{\tau_1}
[S_{\alpha}(\tau_2-s)-S_{\alpha}(\tau_1-s)]f(s)dw(s)\Big\|^2\\
&\quad +4E\big\| \int_{\tau_1}^{\tau_2}S_{\alpha}
(\tau_2-s) f(s)dw(s)\big\|^2\\
& \leq 4E\|[S_{\alpha}(\tau_2)-S_{\alpha}(\tau_1)]
\varphi(0)\|^2+
4(CM)^2(a_1+a_2r')[1+|\omega|b^{\alpha}]^2\operatorname{Tr}(Q)\\
&\quad\times\int_{t_k}^{\tau_1-\varepsilon}\|
S_{\alpha}(\tau_2-s)-S_{\alpha}(\tau_1-s)\|^2
\Big(\frac{1}{1+|\omega|(\tau_1-\varepsilon-s)^{\alpha}}\Bigr)^2l(s)ds\\
&\quad
+4(CM)^2(a_1+a_2r')\operatorname{Tr}(Q)\int_{\tau_1-\varepsilon}^{\tau_1}
\Big(\frac{1}{1+|\omega|(\tau_2-s)^{\alpha}}\Bigr)^2l(s)ds\\
&\quad
+4(CM)^2(a_1+a_2r')\operatorname{Tr}(Q)\int_{\tau_1-\varepsilon}^{\tau_1}
\Big(\frac{1}{1+|\omega|(\tau_1-s)^{\alpha}}\Bigr)^2l(s)ds\\
 &\quad+ 4(CM)^2(a_1+a_2r')\operatorname{Tr}(Q)\int_{\tau_1}^{\tau_2}
\Big(\frac{1}{1+|\omega|(\tau_2-s)^{\alpha}}\Bigr)^2l(s)ds.
\end{align*}
From the above inequalities,
 we see that the right-hand side of
$E\|  h_2(\tau_2)- h_2(\tau_1)\|^2$
 tends to zero independent of $x\in \overline{V}$
as $\tau_2-\tau_1\to0$ with $\varepsilon$ sufficiently
small, since $I_k, k = 1, 2,\dots,m$, are completely continuous
in $H$ and the compactness of $S_{\alpha}(t)$ for $ t> 0$ imply the
continuity in the uniform operator topology. Indeed, the fact of
$S_{\alpha}(\cdot)$  is compact in $H$ since it is generated by the
dense operator $A$. Thus the set $\{\Phi_2x : x\in
\overline{V}\}$ is equicontinuous. The equicontinuities for the
other cases $ \tau_1 < \tau_2 \leq 0$ or $ \tau_1\leq 0 \leq
\tau_2 \leq b$ are very simple.

Next, we prove that $\Phi_2(\overline{V})(t)=\{ h_2(t):
h_2(t)\in\Phi_2(\overline{V})\}$ is relatively compact for every
$t\in [0,b]$.
To this end, we decompose  $\Phi_2$ by  $\Phi_2(\overline{V}) =
\Gamma_1(\overline{V}) + \Gamma_2(\overline{V})$, where the map
$\Gamma_1$ is defined by $\Gamma_1 x, x\in \overline{V}$ the set
$ \tilde{h}_1$ such that
\[
\tilde{h}_1(t)= \begin{cases}
 \int_0^tS_{\alpha}(t-s)f(s)dw(s), &t\in [0,t_1], \\
 \int_{t_k}^tS_{\alpha}(t-s)f(s)dw(s),  &t\in (t_1,t_2],\\
\dots\\
 \int_{t_{m}}^tS_{\alpha}(t-s)f(s)dw(s), &t\in (t_{m},b],
\end{cases}
\]
and the map $\Gamma_2$ is defined by $\Gamma_2 x, x\in
\overline{V}$ the set $ \tilde{h}_2$ such that
\[
\tilde{h}_2(t)= \begin{cases}
  S_{\alpha}(t)\varphi(0), &t\in [0,t_1], \\
S_{\alpha}(t-t_1)[\bar{x}(t_1^{-})+I_1(\bar{x}_{t_1})],
&t\in (t_1,t_2], \\
\dots\\
S_{\alpha}(t-t_{m})[\bar{x}(t_{m}^{-})+I_{m}(\bar{x}_{t_{m}})],
&t\in (t_{m},b].
\end{cases}
\]

We now prove that $\Gamma_1(\overline{V}))(t)=\{\tilde{h}_1(t):
\tilde{h}_1(t)\in\Gamma_1(\overline{V}))\}$ is relatively
compact for every $t\in [0,b]$.
 Let $0<t\leq s \leq t_1$  be fixed and let
$\varepsilon$ be a real number satisfying $0 < \varepsilon < t$. For
$x\in \overline{V}$, we define
\[
\tilde{h}_{1,\varepsilon}(t)=
\int_0^{t-\varepsilon}S_{\alpha}(t-s)f(s)dw(s),
\]
where $f \in S_{F,\bar{x}_{\rho}}$. Using the compactness of
$\mathcal{S}_{\alpha}(t)$ for $t>0$, we deduce that  the set
 $U_{\varepsilon}(t)=\{\tilde{h}_{1,\varepsilon}(t) : x\in \overline{V}\}$
is relatively compact in $H$ for every $\varepsilon, 0<\varepsilon<t$. Moreover,
for every $x\in \overline{V}$ we have
\begin{align*}
E\|\tilde{h}_1(t)-\tilde{h}_{1,\varepsilon}(t)\|^2
& \leq \big\|\int_{t-\varepsilon}^{t}
\mathcal{S}_{\alpha}(t-s)f(s)dw(s)\big\|\\
& \leq (CM)^2(a_1+a_2r')\operatorname{Tr}(Q)\int_{t-\varepsilon}^{t}
\Big(\frac{1}{1+|\omega|(t-s)^{\alpha}}\Bigr)^2l(s)ds.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$ with $ k = 1, \dots ,m$.
 Let $t_k<t\leq s \leq t_{k+1}$  be fixed and let $\varepsilon$
be a real number satisfying $0 < \varepsilon < t$. For $x\in
\overline{V}$, we define
\[
\tilde{h}_{1,\varepsilon}(t)=
\int_{t_k}^{t-\varepsilon} S_{\alpha}(t-s)f(s)dw(s),
\]
where $f \in S_{F,\bar{x}_{\rho}}$. Using the compactness of
$\mathcal{S}_{\alpha}(t)$ for $t >0$, we deduce that  the set
 $U_{\varepsilon}(t)=\{\tilde{h}_{1,\varepsilon}(t) : x\in \overline{V}\}$
 is relatively compact in $H$ for every $\varepsilon, 0<\varepsilon<t$. Moreover,
for every $x\in \overline{V}$ we have
\begin{align*}
E\|\tilde{h}_1(t)-\tilde{h}_{1,\varepsilon}(t)\|^2
& \leq \big\|\int_{t-\varepsilon}^{t}
\mathcal{S}_{\alpha}(t-s)f(s)dw(s)\big\|\\
& \leq(CM)^2(a_1+a_2r')\operatorname{Tr}(Q)\int_{t-\varepsilon}^{t}
\Big(\frac{1}{1+|\omega|(t-s)^{\alpha}}\Bigr)^2l(s)ds.
\end{align*}
The right hand side of the above inequality tends to zero as
$\varepsilon\to0$. Since there are relatively compact sets
arbitrarily close to the set $U(t)=\{\tilde{h}_1(t) : x\in
\overline{V}\}$. Hence the set $U(t)$ is relatively compact  in
$H$. By Arzel\'{a}-Ascoli theorem, we conclude that
$\Gamma_1(\overline{V})$ is completely continuous.

Next, we show that $\Gamma_2(\overline{V})(t)=\{\tilde{h}_2(t):
\tilde{h}_2(t)\in\Gamma_2(\overline{V})\}$ is relatively compact
for every $t\in [0,b]$. For all $t \in [0, t_1]$, since
$\tilde{h}_2(t) = S_{\alpha}(t)\varphi(0)$, by the
$S_{\alpha}(\cdot)$ is compact operator, it follows that
$\{\tilde{h}_2(t) : t \in [0,t_1], x \in \overline{V} \}$ is a
compact subset of $H$. On the other hand, for $t\in (t_k,
t_{k+1}], k= 1, \dots ,m$, and $x \in \overline{V}$, there exists
$r' > 0$ such that
\[
[\widehat{\tilde{h}_2}]_k(t) \in
\begin{cases}
 S_{\alpha}(t-t_k)[\bar{x}(t_k^{-})+I_k(\bar{x}_{t_k})],
    &t\in (t_k,t_{k+1}), \ x \in \overline{V}
_{r''},\\
S_{\alpha}(t_{k+1}-t_k)[\bar{x}(t_k^{-})+I_k(\bar{x}_{t_k})],
   & t=t_{k+1}, \ x \in
\overline{V}_{r''},\\
\bar{x}(t_k^{-})+I_k(\bar{x}_{t_k}),     & t=t_k, \ x
\in \overline{V}_{r''},
\end{cases}
\]
where $\overline{V}_{r''}$ is an open ball of radius
$r''$. From  (H5), it follows that
$[\widehat{\tilde{h}_2}]_k(t)$ is relatively compact in $H$, for
all  $t\in [t_k, t_{k+1}]$, $ k= 1, \dots ,m$. By Lemma \ref{lem2.2}, we
infer that $\Gamma_2(\overline{V})$ is relatively compact.
Moreover, using the compactness of $\{I_k\} (k= 1, \dots ,m)$ and
the continuity of the  operator $S_{\alpha}(t)$, for all $t \in [0,
b]$, $\Gamma_2(\overline{V})$ is completely continuous, and hence
$\Phi_2(\overline{V})$ is completely continuous.


As a consequence of the above steps 1-3, we conclude that
$\Phi=\Phi_1+\Phi_2$ is a condensing map. All of the conditions
of Lemma \ref{lem2.5} are satisfied, we deduce that $\Phi$ has a fixed point
$x\in\mathcal{BPC}, $  which is in turn  a mild solution of the
problem \eqref{e1.1}-\eqref{e1.3}. The proof is complete.
\end{proof}

\begin{remark} \label{rmk3.3}\rm  Note that by the condition
$\rho(s,\overline{x}_{s})\leq s$, $s\in [0,t]$, $t\in [0,b]$ and using
Lemma \ref{lem3.1}, we have
 \[
\| \overline{x}_{\rho(s,\overline{x}_{s})}\|_{\mathcal{B}}
\leq (M_{b}+J^{\varphi}_0)\|
\varphi\|_{\mathcal{B}}+K_{b}\sup\{\|
\bar{x}(s)\|:0\leq s \leq t\}.
\]
By  lemma \ref{lem2.3} this implies that
\[
\| \overline{x}_{\rho(s,\overline{x}_{s})}\|_{\mathcal{B}}
\leq (M_{b}+J^{\varphi}_0)E\|
\varphi\|_{\mathcal{B}} +K_{b}\sup_{0\leq s \leq
b}E\| x(s)\|,
\]
and so \eqref{e3.5} holds.
\end{remark}


\section{Application} 

Consider the following  impulsive fractional partial neutral
stochastic functional integro-differential inclusions of the form
\begin{gather}
\begin{aligned}
 d D(t,z_t)(x)& \in J_t^{\alpha-1}
\Big(\frac{\partial^2}{\partial x^2}-\nu\Big) D(t,z_t)(x)dt\\
&\quad + \int_{-\infty}^t \mu_2(t,s-t,x,z(s-\rho_1(t)\rho_2(\|z(t)\|),x))dw(s),\\
&\quad 0\leq t\leq b,\; 0\leq x\leq \pi, \\
\end{aligned} \label{e4.1} \\
z(t,0)=z(t,\pi)=0, \quad  0\leq t\leq b,
z(\tau,x)=\varphi(\tau,x), \quad  \tau\leq 0,\; 0\leq x\leq \pi,\\
\triangle  z(t_k,x)=\int_{-\infty}^{t_k}\eta_k(s-t_k)z(s,x)ds,
 \quad k=1,2,\dots,m, \label{e4.4}
\end{gather}
where  $1 < \alpha < 2$, $\nu> 0 $ and $\varphi$ is continuous and
$w(t)$ denotes a standard cylindrical Wiener process in $H$ defined
on a stochastic space $(\Omega,\mathcal{F},P)$. In this system,
$$
D(t,z_t)(x)=z(t,x)-\int_{-\infty}^t \mu_1(s-t)z(s,x) ds.
$$
 Let $H=L^2([0,\pi])$ with the norm
$\|\cdot\|$ and define the operator $A: D(A)\subset H \to H$ is 
the operator given by
$A\omega=\omega''-\nu\omega$ with the domain 
$$
D(A)
:=\{\omega \in H :  \omega'' \in H, \omega(0) =\omega(\pi) = 0\}.
$$ 
It is well known that $\Delta x =x''$ is the infinitesimal 
generator of an analytic
semigroup  $T (t), t \geq0$ on $H$. Hence, $A$ is sectorial of type
 $\mu=-\nu< 0$.

Let $r \geq 0, 1 \leq p < 1$ and let $h:(-\infty,-r]\to
\mathbb{R}$ be a nonnegative measurable function which satisfies the
conditions (h-5), (h-6) in the terminology of Hino et al \cite{h4}.
Briefly, this means that $h$ is locally integrable and there is a
non-negative, locally bounded function
 $\gamma$ on $(-\infty,0]$ such that $h(\xi+\tau)\leq\gamma(\xi)h(\tau)$
  for all $\xi\leq0$ and $\theta\in(-\infty,-r)\setminus N_{\xi}$,
  where $N_{\xi}\subseteq(-\infty,-r)$ is a set whose Lebesgue
measure zero. We denote by $\mathcal{PC}_{r} \times L^2(h ,H)$ the
set consists of all classes of functions
$\varphi:(-\infty,0]\to X$  such that
$\varphi_{|_{[-r,0]}}\in\mathcal{PC}([-r,0],H)$, $\varphi(\cdot)$ is
Lebesgue measurable on $(-\infty,-r)$, and $h\|
\varphi\|^{p}$ is Lebesgue integrable on $(-\infty,-r)$. The
seminorm is given by
$$
\| \varphi\|_{\mathcal{B}}=\sup_{-r\leq
\tau\leq0}\|\varphi(\tau)\|+\Big(\int_{-\infty}^{-r}h(\tau)\|
\varphi\|^{p}d\tau\Bigr)^{1/p}.
$$ 
The space
$\mathcal{B}=\mathcal{PC}_{r} \times L^{p}(h,H)$ satisfies axioms
(A)--(C). Moreover, when $r=0 $ and $p =2$, we can take
$\tilde{H}=1$, $M(t)=\gamma(-t)^{1/2}$ and
$K(t)=1+(\int_{-t}^{0}h(\tau)d\tau)^{1/2}$, for $t \geq0$
 (see \cite[Theorem 1.3.8]{h4} for details).

Additionally, we will assume that
\begin{itemize}
\item[(i)] The functions $\rho_i:[0,\infty)\to [0,\infty),i=1,2$, are
 continuous.

\item[(ii)] The functions $\mu_1 : \mathbb{R} \to\mathbb{R}$,
are continuous, and
$l_1=(\int_{-\infty}^0\frac{(\mu_1(s))^2}{h(s)}ds)^{1/2}<\infty$.

\item[(iii)] The function $\mu_2 : \mathbb{R}^{4} \to\mathbb{R}$
is continuous and there exist continuous functions
$b_1,b_2:\mathbb{R} \to\mathbb{R}$ such that
$$
|\mu_2(t,s,x,y)|\leq b_1(t)b_2(s)|y|, \quad (t, s, x,y) \in
\mathbb{R}^{4}
$$ 
with
$l_2=(\int_{-\infty}^0\frac{(b_2(s))^2}{h(s)}ds)^{1/2}<\infty$.

\item[(iv)] The functions $\eta_k : \mathbb{R} \to\mathbb{R},k=1,2,\dots,m $,
are continuous, and
 $L_k=(\int_{-\infty}^0\frac{(\eta_k(s))^2}{h(s)}ds)^{1/2}<\infty$ for every
$k=1,2,\dots,m$.
\end{itemize}

In the sequel, $\mathcal{B}$ will be the phase space
$\mathcal{PC}_0 \times L^2(h,H)$. Set
$\varphi(\theta)(x)=\varphi(\theta,x)\in \mathcal{B}$, defining the
maps $G:[0,b]\times \mathcal{B}\to H $, 
$ F:[0,b]\times \mathcal{B}\to \mathcal{P}(H) $
 by
\begin{gather*}
G(t,\varphi)(x)=\int_{-\infty}^0\mu_1(\theta)\varphi(\theta)(x)d\theta,\\
 D(t,\varphi)(x)=\varphi(0)x+G(t,\varphi)(x), \quad
 J_t^{\alpha-1}G(t)=\int_0^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}G(s)ds,
\\
F(t,\varphi)(x)=\int_{-\infty}^0\mu_2(t,\theta,x,
\varphi(\theta))(x) d\theta, \quad
\rho(t,\varphi)=\rho_1(t)\rho_2(\| \varphi(0)\|).
\end{gather*}
From these definitions, it follows that $G,F $ are bounded linear
operators on $\mathcal{B}$ with 
$\| G\|\leq L_{G}$ and
 $\| F\|\leq L_{F},\| I_k\|\leq L_k$, $k=1,2,\dots,m$,
 where $L_{G}=l_1,L_{F}=\| b_1\|_{\infty}l_2$.
 Then the problem \eqref{e4.1}-\eqref{e4.4} can be written as system
\eqref{e1.1}-\eqref{e1.3}.  Further, we can impose some suitable conditions 
on the above-defined functions to verify the assumptions on
 Theorem \ref{thm3.1}, we
can conclude that system \eqref{e4.1}-\eqref{e4.4} has at least one 
mild solution on $[0,b]$.


\subsection*{Conclusion} 
We have studied the  existence of mild solutions for a class of
impulsive fractional partial neutral stochastic integro-differential
inclusions with  state-dependent delay and  solution operator, which
is new and allow us to develop the
 existence of various  partial fractional
integro-differential inclusions and partial stochastic
integro-differential inclusions. An application is provided to
illustrate the applicability of the new result. The results
presented in this paper extend and improve the corresponding ones
announced by  Chauhan et al \cite{c1},
 Shu et al \cite{s1}, Hu and  Ren \cite{h5}, Lin et al \cite{l4}, and others.

\subsection*{Acknowledgments}
The authors want to thank the anonymous referees and the editor 
for their valuable suggestions and comments.


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\end{document}

\end{document}