††∗ Corresponding author<EMAIL_ADDRESS>
# A Study of nonlocal fractional neutral stochastic integrodifferential
inclusions of order $1<\alpha<2$ with impulses

A. Afreen∗, A. Raheem & A. Khatoon  A. Afreen
Department of Mathematics, Aligarh Muslim University,
Aligarh - 202002, India<EMAIL_ADDRESS>A. Raheem
Department of Mathematics, Aligarh Muslim University,
Aligarh - 202002, India<EMAIL_ADDRESS>A. Khatoon
Department of Mathematics, Aligarh Muslim University,
Aligarh - 202002, India<EMAIL_ADDRESS>
###### Abstract.

This paper considers a class of nonlocal fractional neutral stochastic
integrodifferential inclusions of order $1<\alpha<2$ with impulses in a
Hilbert space. We study the existence of the mild solution for the cases when
the multi-valued map has convex and non-convex values. The results are
obtained by combining fixed-point theorems with the fractional order cosine
family, semigroup theory, and stochastic techniques. A new set of sufficient
conditions is developed to demonstrate the approximate controllability of the
system. Finally, an example is given to illustrate the obtained results.

###### Key words and phrases:

Fractional differential inclusion, Approximate controllability, Fractional
order cosine family, Stochastic system

###### 2010 Mathematics Subject Classification:

34A08, 34K35, 46D09, 47H04, 93B05

##  1\. Introduction

The limitations of using traditional integer-order calculus and standard
differential equations to model specific complex systems and processes are
inadequate for describing various phenomena, including those found in
viscoelastic systems, dielectric polarization, and electromagnetic wave
propagation. This insufficiency is sometimes caused by these events exhibiting
memory-like behavior or other complex features that conventional derivatives
cannot adequately represent. Many researchers have turned to fractional
derivatives as an alternative to these limitations. Fractional derivatives
offer a way to extend the concept of differentiation beyond integer orders.
They are a powerful approach for modeling processes with memory and complex
behaviors in diverse scientific and engineering fields.

In recent decades, the theory of fractional-order differential systems has
attracted considerable attention mainly due to its impressive applicability in
various scientific and engineering domains, such as medical model simulations
and electrical engineering. The nonlocal properties of fractional differential
and integral operators have made them valuable tools. It has been found that
arbitrary-order fractional differential equations provide a more accurate
description of the dynamic response of real-world objects. The importance of a
thorough understanding of fractional calculus has increasingly been recognized
[1, 21, 24].

It is well-established that many phenomena in the physical universe, including
the human cardiovascular system and blood circulation, are influenced by the
abrupt shift in their state after certain intervals. Impulsive differential
equations describing sudden changes are used to explain these events. The
study of impulsive systems is crucial to analyzing more realistic mathematical
models. It has numerous applications in various fields, such as drug diffusion
in the human body, population dynamics, theoretical physics, industrial
robotics, mathematical economy, chemical technology, etc. Ahmed et al. [16]
examined the approximate controllability of noninstantaneous impulsive Hilfer
fractional integrodifferential equations with fractional Brownian motion.
Sivasankar and Udhayakumar [36] studied the approximate controllability of
second-order neutral stochastic delay integrodifferential evolution inclusions
with impulses. Refer to [30, 39, 42] for more details.

Stochastic differential equations (SDEs) are widely used to model real-world
phenomena influenced by random factors or noises. Unlike deterministic models,
which describe how a system evolves based on fixed parameters, stochastic
models account for the inherent randomness in the system. They incorporate the
concept of Brownian motion and stochastic processes to capture the
unpredictable nature of the phenomena being studied.

Stochastic analysis, which deals with SDEs, has applications in various
disciplines. SDEs are extensively used in financial modeling and option
pricing. They provide a framework to account for random market fluctuations
and uncertainty. In physics and engineering, SDEs are used in modeling systems
that exhibit unexpected behavior, such as heat conduction in materials with
memory. They are also applied to describe diffusion processes, particle motion
in fluids, and other phenomena influenced by random forces. Stochastic models
are valuable for understanding biological systems, where random fluctuations
play a significant role. They can help simulate the dynamics of biochemical
reactions, population growth, genetic mutations, and epidemiological
processes. Chemical reactions often involve random events, such as molecular
collisions. Stochastic models are employed to analyze reaction rates, reaction
pathways, and the behavior of chemical systems under uncertainty. Stochastic
models are essential in economic analysis and forecasting. They are used to
model market dynamics, price movements, and the behavior of economic agents
under uncertainty [4, 15, 17, 27, 35].

One of the fundamental concepts of contemporary control theory is the idea of
a dynamical system being controllable. In general, controllability is the
capability of a control system to be directed from an arbitrary initial state
to a likewise arbitrary final state through the permitted set of controls.
Significant consequences for the behavior of linear and nonlinear dynamical
systems are drawn from this idea. Moreover, the approximate controllability
means that the system can be steered to an arbitrarily small neighborhood of
the final state. Approximate controllable systems are more prevalent, and very
often, approximate controllability is completely adequate in applications, see
[2, 5, 6, 19, 20, 34, 40] and the references therein.

It should be pointed out that Sathiyaraj and Balasubraminaim [38] discussed
the controllability of fractional higher-order stochastic integrodifferential
inclusions:

${}^{C}{D}^{q}x(t)\in
A^{C}{D}^{p}x(t)+Bu(t)+F(t,x(t))+\displaystyle\int_{0}^{t}{G}(s,x(s))dw(s),\quad
t\in[0,T],$ $\displaystyle x(0)=x_{0},~{}x^{\prime}(0)=x^{\prime}_{0},\hskip
227.62204pt$

where ${}^{C}{D}^{q}$ and ${}^{C}{D}^{p}$ denote Caputo fractional derivatives
of order $0<p\leq 1<q\leq 2,$ $A$ and $B$ are matrices of dimensions $n\times
n$ and $n\times m,$ respectively, and
$x\in\mathbb{R}^{n},~{}u\in\mathbb{R}^{m}$ are the state and control vectors,
respectively. The results are established using the fixed point techniques for
both the convex and non-convex cases. It is noted that the study is in finite
dimension.

Sivasankar and Udhayakumar [36] studied the existence of solutions and the
approximate controllability of the following integrodifferential evolution
inclusions with infinite delay

$\displaystyle\left\\{\begin{array}[]{lll}d\big{[}x^{\prime}(t)+\sigma(t,x_{t})\big{]}\in\big{[}{A}(t)x(t)+{B}u(t)+F(t,x_{t})\big{]}dt\\\
\hskip 113.81102pt+\displaystyle\int_{0}^{t}e(t-s){G}(s,x_{s})dw(s),\quad
t\in[0,a],~{}t\neq t_{k},~{}k=1,2,\ldots,m,&\\\ x(0)=\phi\in
L^{2}(\Omega,B_{h}),\quad t\in(-\infty,0],\quad x^{\prime}(0)=\zeta,&\\\
\Delta x\rvert_{t=t_{k}}={I}_{k}(x_{t_{k}}),\quad k=1,2,\ldots,m,&\\\ \Delta
x^{\prime}\rvert_{t=t_{k}}={J}_{k}(x_{t_{k}}),\quad
k=1,2,\ldots,m,\end{array}\right.$

where $x(\cdot)$ takes values in a Hilbert space. The results are proved only
for convex multivalued maps using fixed point methods.

Several real-world systems, such as sudden stock price fluctuations brought on
by war, epidemics, market crashes, etc., are subject to stochastic
perturbations with impulsive effects. For these models, the path continuity
supposition could be more plausible. Consequently, we should consider
impulsive effects in stochastic processes when modeling such systems. Due to
the numerous applications of second-order abstract differential equations in
physics and engineering, they have received much more attention. Second-order
differential equations, for instance, can be used to analyze the system of
dynamical buckling of a hinged extensible beam [37, 41]. The integrated
process in continuous time, which can be made stationary, is best described by
second-order stochastic differential equations. For instance, engineers can
use second-order stochastic differential equations to describe mechanical
vibrations or the charge on a capacitor. Researchers have recently obtained
numerous findings about the controllability of second-order stochastic control
systems [4, 28, 31, 36, 40].

Numerous social, physical, biological, and engineering issues are
characterized mainly by differential inclusions; for instance, see [30, 36].
Ahmed and Ragusa [18] examined the nonlocal controllability of Sobolev-type
conformable fractional stochastic evolution inclusions with Clarke
subdifferential. Kavitha et al. [23] studied the approximate controllability
of Sobolev-type fractional neutral differential inclusions of Clarke
subdifferential type. Subsequently, a few authors have reviewed the
controllability of fractional stochastic differential inclusions involving
Hilfer fractional derivatives—for example, Ahmed et al. [12] studied the null
controllability of Hilfer fractional differential inclusions. Ahmed et al.
[13] studied fractional stochastic evolution inclusions with control on the
boundary. Many authors examined the existence and controllability of various
systems with nonlocal conditions as their application results are more
favorable than those of the classical initial condition [26, 28, 29]. Many
authors studied the various controllability of Hilfer fractional differential
systems [10, 11, 14]. To the authors’ knowledge, the existence of mild
solutions of nonlocal fractional neutral stochastic integrodifferential
inclusions of order $1<\alpha<2$ with impulses in Hilbert spaces has not been
addressed for the convex and non-convex case of the multivalued map. This work
aims to address this gap.

In light of the above discussion, we consider the following nonlocal
fractional neutral stochastic integrodifferential inclusions with impulses

$\displaystyle\left\\{\begin{array}[]{lll}{}^{C}{D}^{\alpha}\big{[}\zeta(x)+f_{1}\big{(}x,\zeta(x),\zeta(\nu_{1}(x))\big{)}\big{]}\in\mathcal{A}\zeta(x)+\mathcal{B}u(x)+f_{2}\big{(}x,\zeta(x),\zeta(\nu_{2}(x))\big{)}\\\
\hskip
39.83368pt+\displaystyle\int_{0}^{x}\varrho(x-y)\mathcal{G}\big{(}y,\zeta(y),\zeta(\nu_{3}(y))\big{)}dw(y),\quad
x\in[0,b],~{}x\neq x_{p},~{}p=1,2,\ldots,n,&\\\
\zeta(x)=\phi(x)+h(\zeta)(x),\quad
x\in[-a,0],\quad\zeta^{\prime}(0)=\xi\in\mathcal{H},&\\\
\Delta\zeta(x_{p})=\hat{I}_{p}(\zeta(x_{p})),\quad p=1,2,\ldots,n,&\\\
\Delta\zeta^{\prime}(x_{p})=\hat{J}_{p}(\zeta(x_{p})),\quad
p=1,2,\ldots,n,\par\end{array}\right.$ (1.7)

where $0<a,b<\infty,$ $1<\alpha<2$, ${}^{C}{D}^{\alpha}$ represents fractional
derivative in the Caputo sense and $\zeta(\cdot)$ takes values in a real
separable Hilbert space $\mathcal{H}$ with inner product $(\cdot,\cdot)$ and
norm $\|\cdot\|.$ A densely defined closed linear operator
$\mathcal{A}:\mathcal{D(A)}\subset\mathcal{H}\rightarrow\mathcal{H}$ generates
a strongly continuous cosine family $\\{C(x)\\}_{x\geq 0}$ in $\mathcal{H}$.
$\mathcal{B}$ is bounded linear operator from a real separable Hilbert space
$\mathcal{U}$ to $\mathcal{H},$ and $u(\cdot)\in L^{2}([0,b],\mathcal{U}),$ a
Hilbert space of admissible control functions. Let $\mathcal{W}$ be another
real separable Hilbert space with inner product $(\cdot,\cdot)_{\mathcal{W}}$
and norm $\|\cdot\|_{\mathcal{W}}.$ Let $w$ be a $\mathcal{W}$-valued Wiener
process with a finite trace nuclear covariance operator $Q\geq 0$ on a
complete probability space $(\Omega,\Upsilon,\Upsilon_{x},P),$ where
$\Upsilon_{x}\subset\Upsilon,$ $x\in[0,b]$ is a normal filtration.
$\Upsilon_{x}$ is a right continuous increasing family and $\Upsilon_{0}$
contains all $P$-null sets. Also, let
$L_{2}^{0}=L_{2}\big{(}Q^{1/2}\mathcal{W},\mathcal{H}\big{)},$ the space of
all Hilbert-Schmidt operators from $Q^{1/2}\mathcal{W}$ to $\mathcal{H}$ be a
separable Hilbert space with the norm $\|\psi\|^{2}_{Q}=tr[\psi Q\psi^{*}].$
The maps $f_{1},f_{2}:[0,b]\times\mathcal{H}^{2}\to\mathcal{H},$
$\mathcal{G}:[0,b]\times\mathcal{H}^{2}\to
2^{L(\mathcal{W},\mathcal{H})}\setminus\\{\emptyset\\},$
$\varrho:[0,b]\rightarrow\mathbb{R}$ and
$\hat{I}_{p},\hat{J}_{p}:\mathcal{H}\rightarrow\mathcal{H}$ are appropriate
functions. $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}<x_{n+1}=b$ are impulse points and
$\Delta\zeta(x_{p})=\zeta(x_{p}^{+})-\zeta(x_{p}^{-})$ denotes the jump of a
function $\zeta$ at $x_{p}.$ For $i=1,2,3,$ the functions
$\nu_{i}:[0,b]\rightarrow[-a,b]$ are continuous such that $0\leq\nu_{i}(x)\leq
x,$ $x\in[0,b]$.

The set of all strongly measurable, square integrable $\mathcal{H}$-valued
random variables, denoted by $L_{2}(\Omega,\mathcal{H}),$ is a Banach space
with norm $\|\zeta\|_{L_{2}}=\Big{(}{E}\|\zeta(x)\|^{2}\Big{)}^{\frac{1}{2}}.$
Let $C([0,b],L_{2}(\Omega,\mathcal{H}))$ be the Banach space of all continuous
maps from $[0,b]$ to $L_{2}(\Omega,\mathcal{H})$ satisfying
$\sup\limits_{x\in[0,b]}{E}\|\zeta(x)\|^{2}<\infty.$ A significant subspace is
given by $L_{2}^{0}(\Omega,\mathcal{H})=\\{\Upsilon\in
L_{2}(\Omega,\mathcal{H}):\Upsilon~{}\mbox{is}~{}\Upsilon_{0}\mbox{-measurable}\\}.$

We define the space $\mathcal{PC}_{0}=C([-a,0],\mathcal{H}):$ The space of all
$\Upsilon_{0}$-adapted, $\mathcal{H}$-valued measurable stochastic processes
$\zeta:[-a,0]\rightarrow\mathcal{H},$ which are continuous everywhere, forms a
Banach space with the norm
$\|\zeta\|_{\mathcal{PC}_{0}}=\sup\limits_{x\in[-a,0]}{E}\Big{(}\|\zeta(x)\|^{2}\Big{)}^{\frac{1}{2}}.$

$\mathcal{PC}_{b}=PC([-a,b],\mathcal{H}):$ The space of all
$\Upsilon_{x}$-adapted, $\mathcal{H}$-valued measurable stochastic processes
$\zeta:[-a,b]\rightarrow\mathcal{H},$ which are continuous everywhere except
the points $x_{p}\in(0,b),~{}p=1,2,\ldots,n.$ Moreover, right limits
$\zeta(x_{p}^{+})$ and left limits $\zeta(x_{p}^{-})$ of $\zeta$ exist at
$x_{p},~{}p=1,2,\ldots,n,$ and $\zeta$ is left continuous, i.e.,
$\zeta(x_{p}^{-})=\zeta(x_{p}).$ The space $\mathcal{PC}_{b}$ is a Banach
space equipped with the norm
$\|\zeta\|_{\mathcal{PC}_{b}}=\sup\limits_{x\in[-a,b]}{E}\Big{(}\|\zeta(x)\|^{2}\Big{)}^{\frac{1}{2}}.$
Similarly, $\mathcal{PC}=PC([0,b],\mathcal{H})$ is a Banach space equipped
with the norm
$\|\zeta\|_{\mathcal{PC}}=\sup\limits_{x\in[0,b]}{E}\Big{(}\|\zeta(x)\|^{2}\Big{)}^{\frac{1}{2}}.$
Let $\mathcal{PC}^{1}_{b}$ be the space of all functions
$\zeta\in\mathcal{PC}_{b},$ which are continuously differentiable on
$[0,b]-\\{x_{1},x_{2},\ldots,x_{p}\\},$ and the lateral derivatives

$\zeta^{\prime}_{R}(x)=\lim\limits_{y\rightarrow
0+}\frac{\zeta(x+y)-\zeta(x^{+})}{y}~{}~{}\mbox{and}~{}~{}\zeta^{\prime}_{L}(x)=\lim\limits_{y\rightarrow
0-}\frac{\zeta(x+y)-\zeta(x^{-})}{y}$

are continuous on $[0,b)$ and $(0,b],$ respectively. Clearly, the space
$\mathcal{PC}^{1}_{b}$ is a Banach space with

$\displaystyle\|\zeta\|_{\mathcal{PC}^{1}_{b}}=\max\bigg{\\{}\sup\limits_{x\in[-a,b]}\|\zeta\|_{\mathcal{PC}_{b}},\sup\limits_{x\in[-a,b]}\|\zeta^{\prime}\|_{\mathcal{PC}_{b}}\bigg{\\}}.$

The main contributions of the current work:

  * •

A class of nonlocal fractional neutral stochastic integrodifferential
inclusions of order $1<\alpha<2$ with impulses is introduced.

  * •

We show the existence of mild solution of the system (1.7) for both the convex
and non-convex values of multivalued maps.

  * •

We establish a set of sufficient conditions that demonstrate the approximate
controllability of the system (1.7).

  * •

An example is demonstrated to validate the developed theoretical results.

##  2\. Preliminaries and Assumptions

We use the following notations.
$P(\mathcal{H})=\\{V\in\mathcal{H}:V\neq\emptyset\\}$,
$P_{cl}(\mathcal{H})=\\{V\in P(\mathcal{H}):V~{}\mbox{is closed}\\}$,
$P_{bd}(\mathcal{H})=\\{V\in P(\mathcal{H}):V~{}\mbox{is bounded}\\}$,
$P_{cp}(\mathcal{H})=\\{V\in P(\mathcal{H}):V~{}\mbox{is compact}\\}$,
$P_{cv}(\mathcal{H})=\\{V\in P(\mathcal{H}):V~{}\mbox{is convex}\\}$,
$P_{bd,cl,cv}(\mathcal{H})=\\{V\in P(\mathcal{H}):V~{}\mbox{is bounded, closed
and convex}\\}.$

For the sake of convenience in writing, we always set $r=\frac{\alpha}{2},$
for $\alpha\in(1,2).$

######  Definition 2.1.

[43] An $\Upsilon_{x}$-adapted stochastic process
$\zeta:[-a,b]\rightarrow\mathcal{H}$ is called a mild solution of the system
(1.7) if
$\zeta(x)=\phi(x)+h(\zeta)(x),~{}x\in[-a,0],~{}\zeta^{\prime}(0)=\xi,$
$u(\cdot)\in L^{2}([0,b],\mathcal{U}),$
$\Delta\zeta(x_{p})=\hat{I}_{p}(\zeta(x_{p})),~{}\Delta\zeta^{\prime}(x_{p})=\hat{J}_{p}(\zeta(x_{p})),~{}p=1,2,\ldots,n;$
$\zeta(\cdot)_{[0,b]}\in\mathcal{PC}^{1}_{b}$ and satisfies

$\displaystyle\zeta(x)$ $\displaystyle=$ $\displaystyle
C_{r}(x)[\phi(0)+h(\zeta)]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}$
(2.1)
$\displaystyle\quad-\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy+\int_{0}^{x}P_{r}(x-y)\mathcal{B}u(y)dy$
$\displaystyle\quad+\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy+\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy$
$\displaystyle\quad+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta(x_{p}))+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta(x_{p})),$

where $g\in S_{\mathcal{G},\zeta}=\big{\\{}g\in
L^{2}(L(\mathcal{W},\mathcal{H})):g(x)\in\mathcal{G}\big{(}y,\zeta(y),\zeta(\nu_{3}(y))\big{)}~{}\mbox{a.e.}~{}x\in[0,b]\big{\\}}$
is a selection of $\mathcal{G},$
$C_{r}(x)=\displaystyle\int_{0}^{\infty}M_{r}(\theta)C(x^{r}\theta)d\theta,$
$S_{r}(x)=\displaystyle\int_{0}^{x}C_{r}(s)ds$,
$P_{r}(x)=\displaystyle\int_{0}^{\infty}r\theta
M_{r}(\theta)S(x^{r}\theta)d\theta,$ and $M_{r}(\theta)$ is the Mainardi’s
Wright-type function.

From [3], we introduce the controllability operator as follows:

$\mu_{0}^{b}=\displaystyle\int_{0}^{b}P_{r}(b-s)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-s)ds:\mathcal{H}\to\mathcal{H},$

where $*$ denotes the adjoint. Clearly, $\mu_{0}^{b}\geq 0$ and hence, the
resolvent

$\mathcal{R}\big{(}\delta,-\mu_{0}^{b}\big{)}=(\delta
I+\mu_{0}^{b})^{-1}:\mathcal{H}\to\mathcal{H},$

is well defined for all $\delta>0.$

Consider
$H_{d}:\mathcal{P}(\mathcal{H})\times\mathcal{P}(\mathcal{H})\to\mathbb{R}^{+}\cup\\{\infty\\}$
given by

$\displaystyle H_{d}(R,S)=\max\bigg{\\{}\sup\limits_{a\in
R}d(a,S),\sup\limits_{b\in S}d(R,b)\bigg{\\}},$

where $d(R,b)=\inf\limits_{a\in R}d(a,b).$ Then, $(P_{cl}(\mathcal{H}),H_{d})$
is a generalized metric space.

######  Theorem 2.2.

[32] Let $\mathcal{H}$ be a Hilbert space and $\mathcal{K}:\mathcal{H}\to
P_{bd,cl,cv}(\mathcal{H})$ be an upper semi continuous (u.s.c.) and condensing
map. If

$\mathcal{M}=\\{\zeta\in\mathcal{H}:\lambda\zeta\in\mathcal{K}\zeta~{}\mbox{for
some}~{}\lambda>1\\},$

is bounded, then $\mathcal{K}$ has a fixed point.

######  Lemma 2.3.

[9] Let $(\mathcal{H},d)$ be a complete metric space. If
$\mathcal{K}:\mathcal{H}\to\mathcal{P}_{cl}(\mathcal{H})$ is a contraction,
then $\mathcal{K}$ has a fixed point.

Consider the following assumptions.

  * (A1)

The operator $C_{r}(x),~{}x>0$ is compact. Moreover, there exist constants
$M_{1}\geq 1,M_{2}=M_{1}b,M_{3}=\dfrac{M_{1}b^{r}}{\Gamma(2r)},M_{4}>0$ such
that

$\displaystyle\|C_{r}(x)\|^{2}\leq M_{1},\quad\|S_{r}(x)\|^{2}\leq
M_{2},\quad\|P_{r}(x)\|^{2}\leq M_{3},\quad\|\mathcal{B}\|^{2}\leq M_{4}.$

  * (A2)

The function
$f_{1}:[0,b]\times\mathcal{PC}\times\mathcal{PC}_{0}\to\mathcal{H}$ is
continuous and there exist positive constants $L_{f_{1}},k_{1},k_{2}$ such
that

    * (i)

$E\left\|f_{1}\big{(}x,\zeta_{1}(x),\zeta_{1}(\nu_{1}(x))\big{)}-f_{1}\big{(}x,\zeta_{2}(x),\zeta_{2}(\nu_{1}(x))\big{)}\right\|^{2}\\\
\leq
L_{f_{1}}E\big{(}\|\zeta_{1}(x)-\zeta_{2}(x)\|^{2}+\|\zeta_{1}(\nu_{1}(x))-\zeta_{2}(\nu_{1}(x))\|^{2}\big{)},$

    * (ii)

$E\left\|f_{1}\big{(}x,\zeta(x),\zeta(\nu_{1}(x))\big{)}\right\|^{2}\leq
k_{1}E\big{(}1+\|\zeta(x)\|^{2}+\|\zeta(\nu_{1}(x))\|^{2}\big{)},$

    * (iii)

$E\|f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\|^{2}=k_{2}.$

  * (A3)

The function
$f_{2}:[0,b]\times\mathcal{PC}\times\mathcal{PC}_{0}\to\mathcal{H}$ satisfies
the following

    * (i)

$x\mapsto f_{2}\big{(}x,\zeta(x),\zeta(\nu_{2}(x))\big{)}$ is measurable for
$\zeta(x)\in\mathcal{PC},\zeta(\nu_{2}(x))\in\mathcal{PC}_{0},$

    * (ii)

$\big{(}\zeta(x),\zeta(\nu_{2}(x))\big{)}\mapsto
f_{2}\big{(}x,\zeta(x),\zeta(\nu_{2}(x))\big{)}$ is continuous for almost all
$x\in[0,b],$

    * (iii)

there exists $L_{f_{2}}>0$ such that
$E\left\|f_{2}\big{(}x,\zeta_{1}(x),\zeta_{1}(\nu_{2}(x))\big{)}-f_{2}\big{(}x,\zeta_{2}(x),\zeta_{2}(\nu_{2}(x))\big{)}\right\|^{2}\\\
\leq
L_{f_{2}}E\big{(}\|\zeta_{1}(x)-\zeta_{2}(x)\|^{2}+\|\zeta_{1}(\nu_{2}(x))-\zeta_{2}(\nu_{2}(x))\|^{2}\big{)},$

    * (iv)

for almost every $x\in[0,b],$ $\zeta(x)\in\mathcal{PC}$ and
$\zeta(\nu_{2}(x))\in\mathcal{PC}_{0}$ such that

$\displaystyle
E\left\|f_{2}\big{(}x,\zeta(x),\zeta(\nu_{2}(x))\big{)}\right\|^{2}\leq\Theta(x)\pounds\Big{(}E\|\zeta(x)\|^{2}+E\|\zeta(\nu_{2}(x))\|^{2}\Big{)},$

where $\Theta\in L^{1}([0,b],\mathbb{R}^{+})$ and
$\pounds:\mathbb{R}^{+}\to(0,\infty)$ is continuous and increasing function.

  * (A4)

The real valued function $\varrho$ is continuous on $[0,b]$ and there exists
$\ell>0$ such that $\|\varrho(x)\|^{2}\leq\ell$ for $x\in[0,b].$

  * (A5)

$\mathcal{G}$ is a multivalued map satisfying
$\mathcal{G}:[0,b]\times\mathcal{PC}\times\mathcal{PC}_{0}\to
P_{bd,cl,cv}L(\mathcal{W},\mathcal{H})$ is measurable with respect to $x$ for
each fixed $\zeta(x)\in\mathcal{PC}$ and
$\zeta(\nu_{3}(x))\in\mathcal{PC}_{0},$ u.s.c. with respect to
$\zeta(x)\in\mathcal{PC}$ and $\zeta(\nu_{3}(x))\in\mathcal{PC}_{0}$ for each
$x\in[0,b]$, and for each fixed $\zeta\in L_{2}(L(\mathcal{W},\mathcal{H})),$
the selection

$S_{\mathcal{G},\zeta}=\big{\\{}g\in
L^{2}(L(\mathcal{W},\mathcal{H})):g(x)\in(\hat{\mathcal{G}}\zeta)(x)=\mathcal{G}\big{(}x,\zeta(x),\zeta(\nu_{3}(x))\big{)}~{}\mbox{a.e.}~{}x\in[0,b]\big{\\}}$

is nonempty.

  * (A6)

There exists $\wp\in L^{1}([0,b],\mathbb{R}^{+})$ such that

$\displaystyle\int_{0}^{x}E\|\mathcal{G}\big{(}y,\zeta(y),\zeta(\nu_{3}(y))\big{)}\|^{2}_{L_{2}^{0}}dy$
$\displaystyle=$
$\displaystyle\sup\limits_{x\in[0,b]}\bigg{\\{}\int_{0}^{x}E\|g(y)\|^{2}dy:g\in(\hat{\mathcal{G}}\zeta)(y)\bigg{\\}}$
$\displaystyle\leq$
$\displaystyle\wp(x)\beta\Big{(}E\|\zeta(x)\|^{2}+E\|\zeta(\nu_{3}(x))\|^{2}\Big{)},$

where $\beta:\mathbb{R}^{+}\to(0,\infty)$ is continuous and increasing
function.

  * (A7)

The functions $\hat{I}_{p},\hat{J}_{p}:\mathcal{H}\rightarrow\mathcal{H}$ are
continuous functions and there exist $L_{\hat{I}_{p}},L_{\hat{J}_{p}}>0$ such
that $E\left\|\hat{I}_{p}(\zeta(x_{p}))\right\|^{2}\leq L_{\hat{I}_{p}},\quad
E\left\|\hat{J}_{p}(\zeta(x_{p}))\right\|^{2}\leq L_{\hat{J}_{p}}$ and

$E\left\|\hat{I}_{p}(\zeta_{1}(x_{p}))-\hat{I}_{p}(\zeta_{2}(x_{p}))\right\|^{2}\leq
L_{\hat{I}_{p}}\|\zeta_{1}-\zeta_{2}\|^{2},$
$E\left\|\hat{J}_{p}(\zeta_{1}(x_{p}))-\hat{J}_{p}(\zeta_{2}(x_{p}))\right\|^{2}\leq
L_{\hat{J}_{p}}\|\zeta_{1}-\zeta_{2}\|^{2},$

for any $\zeta_{1},\zeta_{2}\in\mathcal{H}$ and $p=1,2,\ldots,n.$

  * (A8)

$h$ is completely continuous and $L_{h}>0$ such that

    * (i)

$E\left\|h(\zeta)-h(\tilde{\zeta})\right\|^{2}\leq
L_{h}\|\zeta-\tilde{\zeta}\|^{2},$

    * (ii)

$E\left\|h(\zeta)\right\|^{2}\leq L_{h}\big{(}1+\|\zeta\|^{2}),$

for all $\zeta,\tilde{\zeta}\in\mathcal{H}.$

  * (A9)

The resolvent operator $(\delta I+\mu_{0}^{b})^{-1}$ satisfies the following
conditions:

    * (i)

$\big{\|}(\delta
I+\mu_{0}^{b})^{-1}\big{\|}\leq\frac{1}{\delta},\quad\mbox{for
all}~{}\delta\geq 0.$

    * (ii)

$\delta(\delta I+\mu_{0}^{b})^{-1}\rightarrow 0$ in the strong operator
topology as $\delta\rightarrow 0^{+}.$

  * (A10)

The multivalued map
$\mathcal{G}:[0,b]\times\mathcal{PC}\times\mathcal{PC}_{0}\rightarrow
P_{cp}\big{(}L(\mathcal{W},\mathcal{H})\big{)}$ has the property that
$\mathcal{G}\big{(}\cdot,\zeta(y),\zeta(\nu_{3}(y))\big{)}:[0,b]\to
P_{cp}\big{(}L(\mathcal{W},\mathcal{H})\big{)}$ is measurable for each
$\zeta(y)\in\mathcal{PC},~{}\zeta(\nu_{3}(y))\in\mathcal{PC}_{0}.$

  * (A11)

There exists $\hat{\wp}\in L^{1}([0,b],\mathbb{R}^{+})$ such that

$\displaystyle
H_{d}\big{(}(\hat{\mathcal{G}}\zeta_{1})(x),(\hat{\mathcal{G}}\zeta_{2})(x)\big{)}$
$\displaystyle\leq$ $\displaystyle\hat{\wp}(x)\|\zeta_{1}-\zeta_{2}\|^{2},$

and $d\big{(}0,\mathcal{G}(x,0,0)\big{)}\leq\hat{\wp}(x)$ for a.e.
$x\in[0,b].$

From [3], it can be observed that (A9)(ii) satisfies if and only if the
corresponding linear deterministic system of (1.7) is approximately
controllable on $[0,b].$

######  Lemma 2.4.

[35] Let $\chi:[0,b]\times\Omega\rightarrow L_{2}^{0}$ be strongly measurable
mapping such that
$\displaystyle\int\limits_{0}^{b}{E}\|\chi(s)\|_{L_{2}^{0}}^{p}ds<\infty.$
Then

${E}\left\|\int\limits_{0}^{t}\chi(s)dw(s)\right\|^{p}\leq
C_{p}\int\limits_{0}^{t}{E}\|\chi(s)\|_{L_{2}^{0}}^{p}ds,$

for all $0\leq t\leq b$ and $p\geq 2,$ where $C_{p}$ is the constant involving
$p$ and $b.$

######  Lemma 2.5.

[35] For any $\bar{\zeta}_{b}\in L^{2}(\Upsilon_{b},\mathcal{H}),$ there
exists $\varphi\in
L^{2}_{\Upsilon}\big{(}\Omega,L^{2}([0,b],L_{2}^{0})\big{)}$ such that
$\bar{\zeta}_{b}=E\bar{\zeta}_{b}+\displaystyle\int_{0}^{b}\varphi(y)dw(y).$

For any $\delta>0$ and $\bar{\zeta}_{b}\in L^{2}(\Upsilon_{b},\mathcal{H})$
the control function is defined as

$\displaystyle u_{\delta}(x)$ $\displaystyle=$
$\displaystyle\mathcal{B}^{*}P^{*}_{r}(b-x)\bigg{[}\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta)]-S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}\bigg{)}$
$\displaystyle\quad+\int_{0}^{b}\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\varphi(y)dw(y)+\int_{0}^{b}\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}C_{r}(b-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}P_{r}(b-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}P_{r}(b-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy$
$\displaystyle\quad-\sum\limits_{p=1}^{n}\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}C_{r}(b-x_{p})\hat{I}_{p}(\zeta(x_{p}))-\sum\limits_{p=1}^{n}\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}S_{r}(b-x_{p})\hat{J}_{p}(\zeta(x_{p}))\bigg{]}.$

##  3\. Existence of Mild Solutions

######  Theorem 3.1.

(Convex Case) If the hypotheses (A1)-(A8), (A9)(i) are satisfied, then the
system (1.7) has at least one mild solution.

###### Proof.

Transform the system (1.7) such that it becomes a fixed point problem. Let the
multivalued map $\mathcal{K}:\mathcal{PC}^{1}_{b}\rightarrow
P\big{(}{\mathcal{PC}^{1}_{b}}\big{)}$ and $g\in S_{\mathcal{G},\zeta}$
determined by

$\displaystyle\big{(}\mathcal{K}\zeta\big{)}(x)=\left\\{\begin{array}[]{lll}\eta\in\mathcal{PC}^{1}_{b}:\eta(x)=\left\\{\begin{array}[]{lll}\phi(x)+h(\zeta)(x),\quad
x\in[-a,0],&\\\
C_{r}(x)[\phi(0)+h(\zeta)]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}\\\
-\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}u(y)dy\\\
+\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy\\\
+\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy\\\
+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta(x_{p}))\\\
+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta(x_{p})),\quad
x\in[0,b].\end{array}\right.\end{array}\right.$

We verify that $\mathcal{K}$ is completely continuous with bounded, closed and
convex values, and that it is u.s.c. For convenience, we divide the proof into
following steps.
Step 1. $\mathcal{K}\zeta$ is convex for every $\zeta\in\mathcal{PC}^{1}_{b}.$
In fact, if $\eta_{1},\eta_{2}\in\mathcal{K}\zeta,$ then there exist
$g_{1},g_{2}\in S_{\mathcal{G},\zeta}$ such that $x\in[0,b],$ we obtain

$\displaystyle\eta_{i}(x)$ $\displaystyle=$ $\displaystyle
C_{r}(x)[\phi(0)+h(\zeta)]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle-\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)$
$\displaystyle\quad\times\bigg{[}\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta)]-S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}\bigg{)}$
$\displaystyle\quad+\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{\\{}\int_{0}^{b}\varphi(y)dw(y)+\int_{0}^{b}C_{r}(b-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}P_{r}(b-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy-\int_{0}^{b}P_{r}(b-y)\int_{0}^{y}\varrho(y-\tau)g_{i}(\tau)dw(\tau)dy$
$\displaystyle\quad-\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta(x_{p}))-\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta(x_{p}))\bigg{\\}}\bigg{]}ds$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g_{i}(\tau)dw(\tau)dy$
$\displaystyle+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta(x_{p}))+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta(x_{p})),\quad
i=1,2.$

Let $0\leq\lambda\leq 1.$ Then for each $x\in[0,b],$ we have

$\displaystyle\big{(}\lambda\eta_{1}+(1-\lambda)\eta_{2}\big{)}(x)$
$\displaystyle=$ $\displaystyle
C_{r}(x)[\phi(0)+h(\zeta)]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle-\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)\bigg{[}\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta)]$
$\displaystyle\quad-
S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}\bigg{)}+\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{\\{}\int_{0}^{b}\varphi(y)dw(y)$
$\displaystyle\quad+\int_{0}^{b}C_{r}(b-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}P_{r}(b-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}P_{r}(b-y)\int_{0}^{y}\varrho(y-\tau)[\lambda
g_{1}(\tau)+(1-\lambda)g_{2}(\tau)]dw(\tau)dy$
$\displaystyle\quad-\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta(x_{p}))-\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta(x_{p}))\bigg{\\}}\bigg{]}ds$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)[\lambda
g_{1}(\tau)+(1-\lambda)g_{2}(\tau)]dw(\tau)dy$
$\displaystyle+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta(x_{p}))+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta(x_{p})).$

Clearly, $S_{\mathcal{G},\zeta}$ is convex, since $\mathcal{G}$ is convex
values. Therefore, $\lambda g_{1}+(1-\lambda)g_{2}\in S_{\mathcal{G},\zeta}.$
Consequently, $\lambda\eta_{1}+(1-\lambda)\eta_{2}\in\mathcal{K}\zeta$.
Step 2. $\mathcal{K}$ maps bounded sets into bounded sets in
$\mathcal{PC}^{1}_{b}.$

In fact, it is enough to prove that there exists a constant $\epsilon>0$ such
that for any $\eta\in\mathcal{K}\zeta,~{}\zeta\in\mathcal{Z}_{q},$ where

$\displaystyle\mathcal{Z}_{q}=\big{\\{}\zeta\in\mathcal{PC}^{1}_{b}:\|\zeta\|^{2}_{\mathcal{PC}^{1}_{b}}\leq
q,~{}\mbox{where}~{}q=\max\\{q_{1},q_{2}\\}~{}\mbox{such
that}~{}\|\zeta\|^{2}_{\mathcal{PC}_{b}}\leq
q_{1},\|\zeta^{\prime}\|^{2}_{\mathcal{PC}^{1}_{b}}\leq q_{2}\big{\\}},$

one possesses $\|\eta\|^{2}_{\mathcal{PC}^{1}_{b}}\leq\epsilon.$ If
$\eta\in\mathcal{K}\zeta,$ then there exists $g\in S_{\mathcal{G},\zeta}$ such
that for any $x\in[0,b],$ we have

$\displaystyle\eta(x)$ $\displaystyle=$ $\displaystyle
C_{r}(x)[\phi(0)+h(\zeta)]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle-\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}u_{\delta}(y)dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy$
$\displaystyle+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta(x_{p}))+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta(x_{p})).$

Hence, from the hypotheses and Lemma 2.4, it follows that

$\displaystyle E\left\|\eta(x)\right\|^{2}$ $\displaystyle\leq$ $\displaystyle
8E\left\|C_{r}(x)[\phi(0)+h(\zeta)]\right\|^{2}+8E\left\|S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}\right\|^{2}$
$\displaystyle+8E\left\|\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy\right\|^{2}+8E\left\|\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}u_{\delta}(y)dy\right\|^{2}$
$\displaystyle+8E\left\|\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy\right\|^{2}$
$\displaystyle+8E\left\|\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy\right\|^{2}$
$\displaystyle+8E\Bigg{\|}\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta(x_{p}))\Bigg{\|}^{2}+8E\Bigg{\|}\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta(x_{p}))\Bigg{\|}^{2}$
$\displaystyle\leq$ $\displaystyle
16\left\|C_{r}(x)\right\|^{2}E\big{[}\|\phi(0)\|^{2}+\|h(\zeta)\|^{2}\big{]}$
$\displaystyle+16\left\|S_{r}(x)\right\|^{2}E\big{[}\|\xi\|^{2}+\big{\|}f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{\|}^{2}\big{]}$
$\displaystyle+8\displaystyle\int_{0}^{x}\|C_{r}(x-y)\|^{2}E\big{\|}f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}\big{\|}^{2}dy$
$\displaystyle+64\displaystyle\int_{0}^{x}\|P_{r}(x-y)\|^{2}\|\mathcal{B}\|^{2}\|\mathcal{B}^{*}\|^{2}\|P^{*}_{r}(b-y)\|^{2}\bigg{[}\left\|\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\right\|^{2}\bigg{(}2E\|\bar{\zeta}_{b}\|^{2}$
$\displaystyle\quad+2\|C_{r}(b)\|^{2}E\big{[}\|\phi(0)\|^{2}+\|h(\zeta)\|^{2}\big{]}$
$\displaystyle\quad+2\|S_{r}(b)\|^{2}E\big{[}\|\xi\|^{2}+\big{\|}f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{\|}^{2}\big{]}\bigg{)}$
$\displaystyle\quad+\left\|\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\right\|^{2}\bigg{\\{}2C_{2}\int_{0}^{b}E\left\|\varphi(y)\right\|^{2}_{L_{2}^{0}}dy$
$\displaystyle\quad+\int_{0}^{b}\|C_{r}(b-y)\|^{2}E\left\|f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}\right\|^{2}dy$
$\displaystyle\quad+\int_{0}^{b}\|P_{r}(b-y)\|^{2}E\left\|f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}\right\|^{2}dy$
$\displaystyle\quad+C_{2}\int_{0}^{b}\|P_{r}(b-y)\|^{2}\int_{0}^{y}\|\varrho(y-\tau)\|^{2}E\left\|g(\tau)\right\|^{2}d\tau
dy$
$\displaystyle\quad+\sum\limits_{p=1}^{n}\|C_{r}(b-x_{p})\|^{2}E\|\hat{I}_{p}(\zeta(x_{p}))\|^{2}+\sum\limits_{p=1}^{n}\|S_{r}(b-x_{p})\|^{2}E\|\hat{J}_{p}(\zeta(x_{p}))\|^{2}\bigg{\\}}\bigg{]}$
$\displaystyle+8\displaystyle\int_{0}^{x}\|P_{r}(x-y)\|^{2}E\left\|f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}\right\|^{2}dy$
$\displaystyle+8C_{2}\displaystyle\int_{0}^{x}\|P_{r}(x-y)\|^{2}\int_{0}^{y}\|\varrho(y-\tau)\|^{2}E\left\|g(\tau)\right\|^{2}d\tau
dy$
$\displaystyle+8\sum\limits_{0<x_{p}<x}\left\|C_{r}(x-x_{p})\right\|^{2}E\left\|\hat{I}_{p}(\zeta(x_{p}))\right\|^{2}+8\sum\limits_{0<x_{p}<x}\left\|S_{r}(x-x_{p})\right\|^{2}E\left\|\hat{J}_{p}(\zeta(x_{p}))\right\|^{2}$
$\displaystyle\leq$ $\displaystyle
16M_{1}\big{[}E\|\phi(0)\|^{2}+L_{h}(1+q)\big{]}+16M_{2}\big{[}E\|\xi\|^{2}+k_{2}\big{]}+8M_{1}bk_{1}\big{(}1+2q\big{)}$
$\displaystyle+\frac{64}{\delta}M_{3}^{2}M_{4}^{2}b\bigg{[}2E\|\bar{\zeta}_{b}\|^{2}+2M_{1}\big{[}E\|\phi(0)\|^{2}+L_{h}(1+q)\big{]}+2M_{2}\big{[}E\|\xi\|^{2}+k_{2}\big{]}$
$\displaystyle\quad+2C_{2}\int_{0}^{b}E\left\|\varphi(y)\right\|^{2}_{L_{2}^{0}}dy+M_{1}bk_{1}\big{(}1+2q\big{)}+M_{3}\int_{0}^{b}\Theta(y)\pounds(2q)dy$
$\displaystyle\quad+C_{2}M_{3}\ell\beta(2q)\int_{0}^{b}\wp(y)dy+\sum\limits_{p=1}^{n}M_{1}L_{\hat{I}_{p}}+\sum\limits_{p=1}^{n}M_{2}L_{\hat{J}_{p}}\bigg{]}$
$\displaystyle+8M_{3}\pounds(2q)\int_{0}^{x}\Theta(y)dy+8C_{2}M_{3}\ell\beta(2q)\displaystyle\int_{0}^{x}\wp(y)dy+8\sum\limits_{0<x_{p}<x}M_{1}L_{\hat{I}_{p}}$
$\displaystyle+8\sum\limits_{0<x_{p}<x}M_{2}L_{\hat{J}_{p}}$
$\displaystyle\leq$ $\displaystyle\epsilon.$

Then, for each $\eta\in\mathcal{K}\zeta,$ we get
$\|\eta\|^{2}_{\mathcal{PC}^{1}_{b}}\leq\epsilon.$
Step 3. $\mathcal{K}$ maps bounded sets into equicontinuous sets of
$\mathcal{PC}^{1}_{b}.$
For any $\eta\in\mathcal{K}\zeta$ and $\zeta\in\mathcal{Z}_{q},$ there exists
$g\in S_{\mathcal{G},\zeta}$ such that

$\displaystyle\eta(x)$ $\displaystyle=$ $\displaystyle
C_{r}(x)[\phi(0)+h(\zeta)]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle-\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}u_{\delta}(y)dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy$
$\displaystyle+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta(x_{p}))+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta(x_{p})).$

Let $0<x_{1}<x_{2}\leq b,$ then we get
$E\|\eta(x_{2})-\eta(x_{1})\|^{2}$

$\displaystyle\leq$ $\displaystyle
14E\left\|[C_{r}(x_{2})-C_{r}(x_{1})][\phi(0)+h(\zeta)]\right\|^{2}$
$\displaystyle+14E\big{\|}\big{[}S_{r}(x_{2})-S_{r}(x_{1})\big{]}\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}\big{\|}^{2}$
$\displaystyle+14E\left\|\displaystyle\int_{0}^{x_{1}}\big{[}C_{r}(x_{2}-y)-C_{r}(x_{1}-y)\big{]}f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy\right\|^{2}$
$\displaystyle+14E\left\|\displaystyle\int_{x_{1}}^{x_{2}}C_{r}(x_{2}-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy\right\|^{2}$
$\displaystyle+14E\left\|\displaystyle\int_{0}^{x_{1}}\big{[}P_{r}(x_{2}-y)-P_{r}(x_{1}-y)\big{]}\mathcal{B}u_{\delta}(y)dy\right\|^{2}$
$\displaystyle+14E\left\|\displaystyle\int_{x_{1}}^{x_{2}}P_{r}(x_{2}-y)\mathcal{B}u_{\delta}(y)dy\right\|^{2}$
$\displaystyle+14E\left\|\displaystyle\int_{0}^{x_{1}}\big{[}P_{r}(x_{2}-y)-P_{r}(x_{1}-y)\big{]}f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy\right\|^{2}$
$\displaystyle+14E\left\|\displaystyle\int_{x_{1}}^{x_{2}}P_{r}(x_{2}-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy\right\|^{2}$
$\displaystyle+14E\left\|\displaystyle\int_{0}^{x_{1}}\big{[}P_{r}(x_{2}-y)-P_{r}(x_{1}-y)\big{]}\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy\right\|^{2}$
$\displaystyle+14E\left\|\displaystyle\int_{x_{1}}^{x_{2}}P_{r}(x_{2}-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy\right\|^{2}$
$\displaystyle+14E\left\|\sum\limits_{0<x_{p}<x_{1}}\big{[}C_{r}(x_{2}-x_{p})-C_{r}(x_{1}-x_{p})\big{]}\hat{I}_{p}(\zeta(x_{p}))\right\|^{2}$
$\displaystyle+14E\left\|\sum\limits_{x_{1}\leq
x_{p}<x_{2}}C_{r}(x_{2}-x_{p})\hat{I}_{p}(\zeta(x_{p}))\right\|^{2}$
$\displaystyle+14E\left\|\sum\limits_{0<x_{p}<x_{1}}\big{[}S_{r}(x_{2}-x_{p})-S_{r}(x_{1}-x_{p})\big{]}\hat{J}_{p}(\zeta(x_{p}))\right\|^{2}$
$\displaystyle+14E\left\|\sum\limits_{x_{1}\leq
x_{p}<x_{2}}S_{r}(x_{2}-x_{p})\hat{J}_{p}(\zeta(x_{p}))\right\|^{2}$
$\displaystyle\leq$ $\displaystyle
28\left\|C_{r}(x_{2})-C_{r}(x_{1})\right\|^{2}\big{[}E\|\phi(0)\|^{2}+L_{h}(1+q)\big{]}$
$\displaystyle+28\big{\|}S_{r}(x_{2})-S_{r}(x_{1})\big{\|}^{2}\big{[}E\|\xi\|^{2}+k_{2}\big{]}$
$\displaystyle+14\displaystyle\int_{0}^{x_{1}}\left\|C_{r}(x_{2}-y)-C_{r}(x_{1}-y)\right\|^{2}k_{1}\big{(}1+2q\big{)}dy$
$\displaystyle+14\displaystyle\int_{x_{1}}^{x_{2}}M_{1}k_{1}\big{(}1+2q\big{)}dy+\frac{112}{\delta}M_{3}M_{4}^{2}\displaystyle\int_{0}^{x_{1}}\|P_{r}(x_{2}-y)-P_{r}(x_{1}-y)\|^{2}$
$\displaystyle\quad\times\bigg{[}2E\|\bar{\zeta}_{b}\|^{2}+2C_{2}\int_{0}^{b}E\left\|\varphi(y)\right\|^{2}_{L_{2}^{0}}dy+2M_{1}\big{[}E\|\phi(0)\|^{2}+L_{h}(1+q)\big{]}$
$\displaystyle\quad\quad+2M_{2}\big{[}E\|\xi\|^{2}+k_{2}\big{]}+M_{1}bk_{1}\big{(}1+2q\big{)}+M_{3}\int_{0}^{b}\Theta(y)\pounds(2q)dy$
$\displaystyle\quad\quad+C_{2}M_{3}\ell\beta(2q)\int_{0}^{b}\wp(y)dy+\sum\limits_{p=1}^{n}M_{1}L_{\hat{I}_{p}}+\sum\limits_{p=1}^{n}M_{2}L_{\hat{J}_{p}}\bigg{]}dy$
$\displaystyle+\frac{112}{\delta}M_{3}^{2}M_{4}^{2}\displaystyle\int_{x_{1}}^{x_{2}}\bigg{[}2E\|\bar{\zeta}_{b}\|^{2}+2C_{2}\int_{0}^{b}E\left\|\varphi(y)\right\|^{2}_{L_{2}^{0}}dy$
$\displaystyle\quad+2M_{1}\big{[}E\|\phi(0)\|^{2}+L_{h}(1+q)\big{]}+2M_{2}\big{[}E\|\xi\|^{2}+k_{2}\big{]}+M_{1}bk_{1}\big{(}1+2q\big{)}$
$\displaystyle\quad+M_{3}\int_{0}^{b}\Theta(y)\pounds(2q)dy+C_{2}M_{3}\ell\beta(2q)\int_{0}^{b}\wp(y)dy+\sum\limits_{p=1}^{n}M_{1}L_{\hat{I}_{p}}+\sum\limits_{p=1}^{n}M_{2}L_{\hat{J}_{p}}\bigg{]}dy$
$\displaystyle+14\displaystyle\int_{0}^{x_{1}}\left\|P_{r}(x_{2}-y)-P_{r}(x_{1}-y)\right\|^{2}\pounds(2q)\Theta(y)dy$
$\displaystyle+14\displaystyle\int_{x_{1}}^{x_{2}}M_{3}\pounds(2q)\Theta(y)dy$
$\displaystyle+14\displaystyle\int_{0}^{x_{1}}\left\|P_{r}(x_{2}-y)-P_{r}(x_{1}-y)\right\|^{2}C_{2}\ell\beta(2q)\wp(y)dy$
$\displaystyle+14\displaystyle\int_{x_{1}}^{x_{2}}C_{2}M_{3}\ell\beta(2q)\wp(y)dy$
$\displaystyle+14\sum\limits_{0<x_{p}<x_{1}}\left\|C_{r}(x_{2}-x_{p})-C_{r}(x_{1}-x_{p})\right\|^{2}L_{\hat{I}_{p}}$
$\displaystyle+14\sum\limits_{x_{1}\leq
x_{p}<x_{2}}\left\|C_{r}(x_{2}-x_{p})\right\|^{2}L_{\hat{I}_{p}}$
$\displaystyle+14\sum\limits_{0<x_{p}<x_{1}}\left\|S_{r}(x_{2}-x_{p})-S_{r}(x_{1}-x_{p})\right\|^{2}L_{\hat{J}_{p}}$
$\displaystyle+14\sum\limits_{x_{1}\leq
x_{p}<x_{2}}\left\|S_{r}(x_{2}-x_{p})\right\|^{2}L_{\hat{J}_{p}}.$

The right-hand side of the above inequality is independent of
$\zeta\in\mathcal{Z}_{q}$ and $E\|\eta(x_{2})-\eta(x_{1})\|^{2}\to 0$ as
$x_{2}-x_{1}\to 0,$ for all $\zeta\in\mathcal{Z}_{q}.$ Thus, the compactness
of $C_{r}(x)$ and $S_{r}(x)$ for $x>0$ gives the continuity in the uniform
operator topology. Hence, the set $\\{\mathcal{K}(\mathcal{Z}_{q})\\}$ is
equicontinuous on $[0,b].$

Step 4. $\mathcal{K}$ has a closed graph.
Consider $\zeta_{n}\to\zeta_{*},~{}\eta_{n}\in\mathcal{K}\zeta_{n}$ and
$\eta_{n}\to\eta_{*}.$ We prove that
$\eta_{*}\in\mathcal{K}\zeta_{*},~{}\eta_{n}\in\mathcal{K}\zeta_{n}$ means
that there exists $g_{n}\in S_{\mathcal{G},\zeta_{n}}$ such that

$\displaystyle\eta_{n}(x)$ $\displaystyle=$ $\displaystyle
C_{r}(x)[\phi(0)+h(\zeta_{n})]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta_{n}(0),\zeta_{n}(\nu_{1}(0))\big{)}\big{]}$
(3.3)
$\displaystyle-\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{1}(y))\big{)}dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)\bigg{[}\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta_{n})]$
$\displaystyle\quad-
S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta_{n}(0),\zeta_{n}(\nu_{1}(0))\big{)}\big{]}\bigg{)}+\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{\\{}\int_{0}^{b}\varphi(y)dw(y)$
$\displaystyle\quad+\int_{0}^{b}C_{r}(b-y)f_{1}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{1}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}P_{r}(b-y)f_{2}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{2}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}P_{r}(b-y)\int_{0}^{y}\varrho(y-\tau)g_{n}(\tau)dw(\tau)dy$
$\displaystyle\quad-\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta_{n}(x_{p}))-\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta_{n}(x_{p}))\bigg{\\}}\bigg{]}dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{2}(y))\big{)}dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g_{n}(\tau)dw(\tau)dy+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta_{n}(x_{p}))$
$\displaystyle+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta_{n}(x_{p})),\quad
x\in[0,b].$

We need to show that there exists $g_{*}\in S_{\mathcal{G},\zeta_{*}},$ for
every $x\in[0,b]$ such that

$\displaystyle\eta_{*}(x)$ $\displaystyle=$ $\displaystyle
C_{r}(x)[\phi(0)+h(\zeta_{*})]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta_{*}(0),\zeta_{*}(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle-\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{1}(y))\big{)}dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)\bigg{[}\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta_{*})]$
$\displaystyle\quad-
S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta_{*}(0),\zeta_{*}(\nu_{1}(0))\big{)}\big{]}\bigg{)}+\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{\\{}\int_{0}^{b}\varphi(y)dw(y)$
$\displaystyle\quad+\int_{0}^{b}C_{r}(b-y)f_{1}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{1}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}P_{r}(b-y)f_{2}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{2}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}P_{r}(b-y)\int_{0}^{y}\varrho(y-\tau)g_{*}(\tau)dw(\tau)dy$
$\displaystyle\quad-\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta_{*}(x_{p}))-\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta_{*}(x_{p}))\bigg{\\}}\bigg{]}dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{2}(y))\big{)}dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g_{*}(\tau)dw(\tau)dy+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta_{*}(x_{p}))$
$\displaystyle+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta_{*}(x_{p})),\quad
x\in[0,b].$

Clearly, since $\hat{I}_{p},\hat{J}_{p},~{}p=1,2,\ldots,n$ are continuous, we
have

$\displaystyle
E\bigg{\|}\bigg{[}\eta_{n}(x)-C_{r}(x)[\phi(0)+h(\zeta_{n})]-S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta_{n}(0),\zeta_{n}(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle+\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{1}(y))\big{)}dy-\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{2}(y))\big{)}dy$
$\displaystyle-\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta_{n}(x_{p}))-\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta_{n}(x_{p}))$
$\displaystyle-\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)\bigg{\\{}\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta_{n})]$
$\displaystyle-
S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta_{n}(0),\zeta_{n}(\nu_{1}(0))\big{)}\big{]}\bigg{)}+\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{(}\int_{0}^{b}\varphi(y)dw(y)$
$\displaystyle+\int_{0}^{b}C_{r}(b-y)f_{1}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{1}(y))\big{)}dy-\int_{0}^{b}P_{r}(b-y)f_{2}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{2}(y))\big{)}dy$
$\displaystyle-\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta_{n}(x_{p}))-\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta_{n}(x_{p}))\bigg{)}\bigg{\\}}dy\bigg{]}$
$\displaystyle-\bigg{[}\eta_{*}(x)-C_{r}(x)[\phi(0)+h(\zeta_{*})]-S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta_{*}(0),\zeta_{*}(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle+\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{1}(y))\big{)}dy$
$\displaystyle-\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{2}(y))\big{)}dy-\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta_{*}(t_{p}))$
$\displaystyle-\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta_{*}(x_{p}))-\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)\bigg{\\{}\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}$ $\displaystyle-
C_{r}(b)[\phi(0)+h(\zeta_{*})]-S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta_{*}(0),\zeta_{*}(\nu_{1}(0))\big{)}\big{]}\bigg{)}$
$\displaystyle+\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{(}\int_{0}^{b}\varphi(y)dw(y)+\int_{0}^{b}C_{r}(b-y)f_{1}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{1}(y))\big{)}dy$
$\displaystyle-\int_{0}^{b}P_{r}(b-y)f_{2}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{2}(y))\big{)}dy-\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta_{*}(x_{p}))$
$\displaystyle-\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta_{*}(x_{p}))\bigg{)}\bigg{\\}}dy\bigg{]}\bigg{\|}^{2}\to
0~{}\mbox{as}~{}n\to\infty.$

Let the linear operator $\Sigma:L^{2}([0,b],\mathcal{H})\to
C([0,b],\mathcal{H})$

$\displaystyle g\to\big{(}\Sigma g\big{)}(x)$ $\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy$
$\displaystyle-\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{(}\int_{0}^{b}P_{r}(b-y)$
$\displaystyle\quad\times\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy\bigg{)}dy.$

From [36, Lemma 2.7], $\Sigma\circ S_{\mathcal{G}}$ is a closed graph
operator, we have

$\displaystyle\eta_{n}(x)-C_{r}(x)[\phi(0)+h(\zeta_{n})]-S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta_{n}(0),\zeta_{n}(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle+\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{1}(y))\big{)}dy-\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{2}(y))\big{)}dy$
$\displaystyle-\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta_{n}(x_{p}))-\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta_{n}(x_{p}))$
$\displaystyle-\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)\bigg{\\{}\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta_{n})]$
$\displaystyle-
S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta_{n}(0),\zeta_{n}(\nu_{1}(0))\big{)}\big{]}\bigg{)}+\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{(}\int_{0}^{b}\varphi(y)dw(y)$
$\displaystyle+\int_{0}^{b}C_{r}(b-y)f_{1}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{1}(y))\big{)}dy-\int_{0}^{b}P_{r}(b-y)f_{2}\big{(}y,\zeta_{n}(y),\zeta_{n}(\nu_{2}(y))\big{)}dy$
$\displaystyle-\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta_{n}(x_{p}))-\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta_{n}(x_{p}))\bigg{)}\bigg{\\}}dy\in\Sigma\big{(}S_{\mathcal{G},\zeta_{n}}\big{)}.$

Since $\zeta_{n}\to\zeta_{*},$ it follows from [36, Lemma 2.7],

$\displaystyle\eta_{*}(x)-C_{r}(x)[\phi(0)+h(\zeta_{*})]-S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta_{*}(0),\zeta_{*}(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle+\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{1}(y))\big{)}dy-\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{2}(y))\big{)}dy$
$\displaystyle-\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta_{*}(x_{p}))-\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta_{*}(x_{p}))$
$\displaystyle-\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)\bigg{\\{}\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta_{*})]$
$\displaystyle-
S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta_{*}(0),\zeta_{*}(\nu_{1}(0))\big{)}\big{]}\bigg{)}+\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{(}\int_{0}^{b}\varphi(y)dw(y)$
$\displaystyle+\int_{0}^{b}C_{r}(b-y)f_{1}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{1}(y))\big{)}dy-\int_{0}^{b}P_{r}(b-y)f_{2}\big{(}y,\zeta_{*}(y),\zeta_{*}(\nu_{2}(y))\big{)}dy$
$\displaystyle-\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta_{*}(x_{p}))-\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta_{*}(x_{p}))\bigg{)}\bigg{\\}}dy$
$\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g_{*}(\tau)dw(\tau)dy$
$\displaystyle-\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}\mathcal{B}^{*}P^{*}_{r}(b-y)\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\bigg{(}\int_{0}^{b}P_{r}(b-y)\int_{0}^{y}\varrho(y-\tau)g_{*}(\tau)dw(\tau)dy\bigg{)}dy,$

for some $g_{*}\in S_{\mathcal{G},\zeta_{*}}.$ Therefore, $\mathcal{K}$ has a
closed graph.
Hence, $\mathcal{K}$ is completely continuous multivalued map, u.s.c. with
convex closed values. To apply Theorem 2.2, we need the following step.
Step 5. We show that the set $\mathcal{Y}$ is bounded, where

$\displaystyle\mathcal{Y}=\big{\\{}\zeta\in\mathcal{PC}^{1}_{b}:\lambda\zeta\in\mathcal{K}\zeta~{}\mbox{for
some}~{}\lambda>1\big{\\}}.$

Let $\zeta\in\mathcal{Y}.$ In addition, $\lambda\zeta\in\mathcal{K}\zeta$ for
some $\lambda>1.$ So, there exists $g\in S_{\mathcal{G},\zeta}$ such that

$\displaystyle\zeta(x)$ $\displaystyle=$
$\displaystyle\lambda^{-1}C_{r}(x)[\phi(0)+h(\zeta)]+\lambda^{-1}S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle-\lambda^{-1}\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy$
$\displaystyle+\lambda^{-1}\int_{0}^{x}P_{r}(x-y)\mathcal{B}u(y)dy+\lambda^{-1}\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy$
$\displaystyle+\lambda^{-1}\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy+\lambda^{-1}\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta(x_{p}))$
$\displaystyle+\lambda^{-1}\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta(x_{p})).$

Using the hypotheses, we have

$\displaystyle E\left\|\zeta(x)\right\|^{2}$ $\displaystyle\leq$
$\displaystyle
8E\left\|\lambda^{-1}C_{r}(x)[\phi(0)+h(\zeta)]\right\|^{2}+8E\left\|\lambda^{-1}S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta(0),\zeta(\nu_{1}(0))\big{)}\big{]}\right\|^{2}$
$\displaystyle+8E\left\|\lambda^{-1}\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta(y),\zeta(\nu_{1}(y))\big{)}dy\right\|^{2}$
$\displaystyle+8E\left\|\lambda^{-1}\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}u_{\delta}(y)dy\right\|^{2}$
$\displaystyle+8E\left\|\lambda^{-1}\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta(y),\zeta(\nu_{2}(y))\big{)}dy\right\|^{2}$
$\displaystyle+8E\left\|\lambda^{-1}\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy\right\|^{2}$
$\displaystyle+8E\Bigg{\|}\lambda^{-1}\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta(x_{p}))\Bigg{\|}^{2}$
$\displaystyle+8E\Bigg{\|}\lambda^{-1}\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta(x_{p}))\Bigg{\|}^{2}$
$\displaystyle\leq$ $\displaystyle
16M_{1}\big{[}E\|\phi(0)\|^{2}+L_{h}(1+q)\big{]}+16M_{2}\big{[}E\|\xi\|^{2}+k_{2}\big{]}+8M_{1}bk_{1}\big{(}1+2q\big{)}$
$\displaystyle+\frac{64}{\delta}M_{3}^{2}M_{4}^{2}b\bigg{[}2E\|\bar{\zeta}_{b}\|^{2}+2C_{2}\int_{0}^{b}E\left\|\varphi(y)\right\|^{2}_{L_{2}^{0}}dy+2M_{1}\big{[}E\|\phi(0)\|^{2}$
$\displaystyle\quad+L_{h}(1+q)\big{]}+2M_{2}\big{[}E\|\xi\|^{2}+k_{2}\big{]}+M_{1}bk_{1}\big{(}1+2q\big{)}+M_{3}\int_{0}^{b}\Theta(y)\pounds(2q)dy$
$\displaystyle\quad+C_{2}M_{3}\ell\beta(2q)\int_{0}^{b}\wp(y)dy+\sum\limits_{p=1}^{n}M_{1}L_{\hat{I}_{p}}+\sum\limits_{p=1}^{n}M_{2}L_{\hat{J}_{p}}\bigg{]}$
$\displaystyle+8M_{3}\pounds(2q)\int_{0}^{x}\Theta(y)dy+8C_{2}M_{3}\ell\beta(2q)\displaystyle\int_{0}^{x}\wp(y)dy$
$\displaystyle+8\sum\limits_{0<x_{p}<x}M_{1}L_{\hat{I}_{p}}+8\sum\limits_{0<x_{p}<x}M_{2}L_{\hat{J}_{p}}.$

$\implies\mathcal{Y}$ is bounded on $[0,b].$
Hence, it follows from Theorem 2.2 that $\mathcal{K}$ has a fixed point
$\zeta\in\mathcal{PC}^{1}_{b}.$ ∎

######  Theorem 3.2.

(Non-Convex Case) If the hypotheses (A1)-(A4), (A7-A8), (A9)(i), (A10)-(A11)
are satisfied, then the system (1.7) has at least one mild solution, provided
that

$\displaystyle\Lambda\bigg{(}1+\frac{6}{\delta}M_{3}^{2}M_{4}^{2}b\bigg{)}<1,$
(3.4)

where

$\displaystyle\Lambda=14\bigg{[}M_{1}L_{h}+2L_{f_{1}}(M_{2}+M_{1}b)+2M_{3}bL_{f_{2}}+C_{2}\displaystyle\int_{0}^{b}M_{3}\ell\hat{\wp}(y)dy+\sum\limits_{p=1}^{n}M_{1}L_{\hat{I}_{p}}+\sum\limits_{p=1}^{n}M_{2}L_{\hat{J}_{p}}\bigg{]}.$

###### Proof.

Since $S_{\mathcal{G},\zeta}\neq\emptyset$. Therefore, $\mathcal{G}$ has a
nonempty measurable selection. We shall show that the operator $\mathcal{K}$
defined in Theorem 3.1 satisfies the assumption of Lemma 2.3. The proof is as
follows.
Step 1. $\mathcal{K}(\zeta)\in P_{cl}({\mathcal{PC}^{1}_{b}})$ for each
$\zeta\in{\mathcal{PC}^{1}_{b}}.$
Indeed, let $(\eta_{n}),n\geq 0\in\mathcal{K}(\zeta)$ such that
$\eta_{n}\to\eta.$ Then, $\eta\in{\mathcal{PC}^{1}_{b}}$ and there exists
$g_{n}\in S_{\mathcal{G},\zeta_{n}}$ such that for each
$x\in[0,b],\eta_{n}(x)$ is defined in (3.3). Using (A11), we have for a.e.
$x\in[0,b]$

$\|g_{n}(x)\|^{2}\leq\hat{\wp}(x)+\hat{\wp}(x)\|\zeta\|^{2}.$

The Lebesgue dominated convergence theorem implies that

$\|g_{n}-g\|_{L_{2}^{0}}\to 0~{}\mbox{as}~{}n\to\infty.$

Hence, $g\in S_{\mathcal{G},\zeta}.$ Then, for each
$x\in[0,b],~{}\eta_{n}(x)\to\eta(x),$ where

$\displaystyle\eta(x)$ $\displaystyle=$ $\displaystyle
C_{r}(x)[\phi(0)+h(\zeta_{1})]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta_{1}(0),\zeta_{1}(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle-\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta_{1}(y),\zeta_{1}(\nu_{1}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}u_{\delta,1}(y)dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta_{1}(y),\zeta_{1}(\nu_{2}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)dw(\tau)dy$
$\displaystyle+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta_{1}(x_{p}))+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta_{1}(x_{p})).$

So, $\eta\in\mathcal{K}(\zeta).$
Step 2. There exists $\gamma<1$ such that
$H_{d}\big{(}\mathcal{K}(\zeta_{1}),\mathcal{K}(\zeta_{2})\big{)}\leq\gamma\|\zeta_{1}-\zeta_{2}\|_{\mathcal{PC}^{1}_{b}}$
for each $\zeta_{1},\zeta_{2}\in{\mathcal{PC}^{1}_{b}}.$ Let
$\zeta_{1},\zeta_{2}\in{\mathcal{PC}^{1}_{b}}$ and
$\eta\in\mathcal{K}(\zeta).$ Then, there exists $g_{1}\in
S_{\mathcal{G},\zeta_{1}}$ such that $\eta(x)$ is defined as above. From
(A11), it follows that

$H_{d}\big{(}(\hat{\mathcal{G}}\zeta_{1})(x),(\hat{\mathcal{G}}\zeta_{2})(x)\big{)}\leq\hat{\wp}(x)\|\zeta_{1}(x)-\zeta_{2}(x)\|^{2}.$

Hence, there exists $g_{2}\in S_{\mathcal{G},\zeta_{2}}$ such that

$\displaystyle\int_{0}^{x}\|g_{1}(x)-g_{2}(x)\|^{2}dx\leq\hat{\wp}(x)[\|\zeta_{1}(x)-\zeta_{2}(x)\|^{2}],\quad
x\in[0,b].$

Consider the map $S:[0,b]\to P(\mathcal{H})$ defined by

$\displaystyle S(x)=\bigg{\\{}g_{2}(x)|g_{2}:[0,b]\to\mathcal{H}~{}\mbox{is
Lebesgue integrable
and}~{}\displaystyle\int_{0}^{x}\|g_{1}(x)-g_{2}(x)\|^{2}dx$
$\displaystyle\leq\hat{\wp}(x)\|\zeta_{1}(x)-\zeta_{2}(x)\|^{2}\bigg{\\}}.$

Since the multivalued operator $S(x)\cap(\hat{\mathcal{G}}\zeta_{2})(x)$ is
measurable, there exists a function $\bar{g}_{1}(x)$ which is a measurable
selection for $S.$ So, $\bar{g}_{1}(x)\in S_{\mathcal{G},\zeta_{2}},$ and for
each $x\in[0,b],$

$\displaystyle\int_{0}^{x}\|g_{1}(x)-\bar{g}_{1}(x)\|^{2}dx\leq\hat{\wp}(x)\|\zeta_{1}(x)-\zeta_{2}(x)\|^{2}.$

Let us define

$\displaystyle\bar{\eta}(x)$ $\displaystyle=$ $\displaystyle
C_{r}(x)[\phi(0)+h(\zeta_{2})]+S_{r}(x)\big{[}\xi+f_{1}\big{(}0,\zeta_{2}(0),\zeta_{2}(\nu_{1}(0))\big{)}\big{]}$
$\displaystyle-\displaystyle\int_{0}^{x}C_{r}(x-y)f_{1}\big{(}y,\zeta_{2}(y),\zeta_{2}(\nu_{1}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\mathcal{B}u_{\delta,2}(y)dy$
$\displaystyle+\displaystyle\int_{0}^{x}P_{r}(x-y)f_{2}\big{(}y,\zeta_{2}(y),\zeta_{2}(\nu_{2}(y))\big{)}dy+\displaystyle\int_{0}^{x}P_{r}(x-y)\int_{0}^{y}\varrho(y-\tau)\bar{g}(\tau)dw(\tau)dy$
$\displaystyle+\sum\limits_{0<x_{p}<x}C_{r}(x-x_{p})\hat{I}_{p}(\zeta_{2}(x_{p}))+\sum\limits_{0<x_{p}<x}S_{r}(x-x_{p})\hat{J}_{p}(\zeta_{2}(x_{p})).$

Then, for each $x\in[0,b]$, we get
$E\|\eta(x)-\bar{\eta}(x)\|^{2}$

$\displaystyle\leq$ $\displaystyle
14\|C_{r}(x)\|^{2}E\|h(\zeta_{1})-h(\zeta_{2})\|^{2}$
$\displaystyle+14\|S_{r}(x)\|^{2}E\big{\|}f_{1}\big{(}0,\zeta_{1}(0),\zeta_{1}(\nu_{1}(0))\big{)}-f_{1}\big{(}0,\zeta_{2}(0),\zeta_{2}(\nu_{1}(0))\big{)}\big{\|}^{2}$
$\displaystyle+14\displaystyle\int_{0}^{x}\|C_{r}(x-y)\|^{2}E\big{\|}f_{1}\big{(}y,\zeta_{1}(y),\zeta_{1}(\nu_{1}(y))\big{)}-f_{1}\big{(}y,\zeta_{2}(y),\zeta_{2}(\nu_{1}(y))\big{)}\big{\|}^{2}dy$
$\displaystyle+14\displaystyle\int_{0}^{x}\|P_{r}(x-y)\|^{2}\|\mathcal{B}\|^{2}E\|u_{\delta,1}-u_{\delta,2}\|^{2}(y)dy$
$\displaystyle+14\displaystyle\int_{0}^{x}\|P_{r}(x-y)\|^{2}E\big{\|}f_{2}\big{(}y,\zeta_{1}(y),\zeta_{1}(\nu_{2}(y))\big{)}-f_{2}\big{(}y,\zeta_{2}(y),\zeta_{2}(\nu_{2}(y))\big{)}\big{\|}^{2}dy$
$\displaystyle+14C_{2}\displaystyle\int_{0}^{x}\|P_{r}(x-y)\|^{2}\int_{0}^{y}|\varrho(y-\tau)|^{2}\|g(\tau)-\bar{g}(\tau)\|^{2}_{L_{2}^{0}}d\tau
dy$
$\displaystyle+14\sum\limits_{0<x_{p}<x}\|C_{r}(x-x_{p})\|^{2}\|\hat{I}_{p}(\zeta_{1}(x_{p}))-\hat{I}_{p}(\zeta_{2}(x_{p}))\|^{2}$
$\displaystyle+14\sum\limits_{0<x_{p}<x}\|S_{r}(x-x_{p})\|^{2}\|\hat{J}_{p}(\zeta_{1}(x_{p}))-\hat{J}_{p}(\zeta_{2}(x_{p}))\|^{2}$
$\displaystyle\leq$ $\displaystyle
14M_{1}L_{h}\|\zeta_{1}-\zeta_{2}\|^{2}+28M_{2}L_{f_{1}}\|\zeta_{1}-\zeta_{2}\|^{2}+28M_{1}bL_{f_{1}}\|\zeta_{1}-\zeta_{2}\|^{2}$
$\displaystyle+14\displaystyle\int_{0}^{x}\|P_{r}(x-y)\|^{2}\|\mathcal{B}\|^{2}E\|u_{\delta,1}-u_{\delta,2}\|^{2}(y)dy$
$\displaystyle+28M_{3}bL_{f_{2}}\|\zeta_{1}-\zeta_{2}\|^{2}+14C_{2}\displaystyle\int_{0}^{x}M_{3}\ell\hat{\wp}(y)\|\zeta_{1}-\zeta_{2}\|^{2}dy$
$\displaystyle+14\sum\limits_{0<x_{p}<x}M_{1}L_{\hat{I}_{p}}\|\zeta_{1}-\zeta_{2}\|^{2}+14\sum\limits_{0<x_{p}<x}M_{2}L_{\hat{J}_{p}}\|\zeta_{1}-\zeta_{2}\|^{2}$
$\displaystyle\leq$ $\displaystyle
14\bigg{[}M_{1}L_{h}+2L_{f_{1}}(M_{2}+M_{1}b)+2M_{3}bL_{f_{2}}+C_{2}\displaystyle\int_{0}^{x}M_{3}\ell\hat{\wp}(y)dy$
$\displaystyle\quad+\sum\limits_{0<x_{p}<x}M_{1}L_{\hat{I}_{p}}+\sum\limits_{0<x_{p}<x}M_{2}L_{\hat{J}_{p}}\bigg{]}\|\zeta_{1}-\zeta_{2}\|^{2}$
$\displaystyle+\frac{84}{\delta}M_{3}^{2}M_{4}^{2}b\bigg{[}M_{1}L_{h}+2L_{f_{1}}(M_{2}+M_{1}b)+2M_{3}bL_{f_{2}}+C_{2}\displaystyle\int_{0}^{b}M_{3}\ell\hat{\wp}(y)dy$
$\displaystyle\quad+\sum\limits_{p=1}^{n}M_{1}L_{\hat{I}_{p}}+\sum\limits_{p=1}^{n}M_{2}L_{\hat{J}_{p}}\bigg{]}\|\zeta_{1}-\zeta_{2}\|^{2}$
$\displaystyle\leq$ $\displaystyle
2\Lambda\bigg{(}1+\frac{6}{\delta}M_{3}^{2}M_{4}^{2}b\bigg{)}\|\zeta_{1}-\zeta_{2}\|^{2},$

where

$\displaystyle\Lambda=14\bigg{[}M_{1}L_{h}+2L_{f_{1}}(M_{2}+M_{1}b)+2M_{3}bL_{f_{2}}+C_{2}\displaystyle\int_{0}^{b}M_{3}\ell\hat{\wp}(y)dy+\sum\limits_{p=1}^{n}M_{1}L_{\hat{I}_{p}}+\sum\limits_{p=1}^{n}M_{2}L_{\hat{J}_{p}}\bigg{]}.$

From the analogous relation obtained by interchanging the roles of $\zeta_{1}$
and $\zeta_{2}$, it follows that

$\displaystyle
H_{d}\big{(}\mathcal{K}(\zeta_{1}),\mathcal{K}(\zeta_{2})\big{)}$
$\displaystyle\leq$ $\displaystyle
2\Lambda\bigg{(}1+M_{3}^{2}M_{4}^{2}\frac{b^{2}}{\delta}\bigg{)}\|\zeta_{1}-\zeta_{2}\|^{2}.$

From (3.4), $\mathcal{K}$ is a contraction and thus, by Lemma 2.3,
$\mathcal{K}$ has a fixed point which is the solution to (1.7) on $[0,b]$. ∎

######  Theorem 3.3.

If (A1)-(A9) hold, and $f_{1},f_{2}$ and $g$ are uniformly bounded, then the
system (1.7) is approximately controllable on $[0,b]$.

###### Proof.

Let $\zeta^{\delta}(\cdot)\in\mathcal{Z}_{q}$ be a fixed point of the operator
$\mathcal{K}$. From Theorem 3.1, any fixed point of $\mathcal{K}$ is a mild
solution of the system (1.7). This means that there is
$\zeta^{\delta}\in\mathcal{K}(\zeta^{\delta})$ with $g^{\delta}\in
S_{\mathcal{G},\zeta^{\delta}}$ such that

$\displaystyle\zeta^{\delta}(x)$ $\displaystyle=$
$\displaystyle\bar{\zeta}_{b}-\delta\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\bigg{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta^{\delta})]-S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta^{\delta}(0),\zeta^{\delta}(\nu_{1}(0))\big{)}\big{]}\bigg{)}$
$\displaystyle\quad+\int_{0}^{b}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\varphi(y)dw(y)+\int_{0}^{b}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}C_{r}(b-y)f_{1}\big{(}y,\zeta^{\delta}(y),\zeta^{\delta}(\nu_{1}(y))\big{)}dy$
$\displaystyle\quad-\int_{0}^{b}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}P_{r}(b-y)f_{2}\big{(}y,\zeta^{\delta}(y),\zeta^{\delta}(\nu_{2}(y))\big{)}dy$
$\displaystyle\quad-\displaystyle\int_{0}^{b}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}P_{r}(b-y)\int_{0}^{y}\varrho(y-\tau)g^{\delta}(\tau)dw(\tau)dy$
$\displaystyle\quad-\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta^{\delta}(x_{p}))-\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta^{\delta}(x_{p})).$

In view of the unifrom boundedness of $f_{1},f_{2}$ and $g$, there are
subsequences still denoted by
$f_{1}\big{(}y,\zeta^{\delta}(y),\zeta^{\delta}(\nu_{1}(y))\big{)},$
$f_{2}\big{(}y,\zeta^{\delta}(y),\zeta^{\delta}(\nu_{2}(y))\big{)}$ and
$g^{\delta}(\tau)$, which converge weakly to $f_{1}(y),f_{2}(y)$ and
$g(\tau).$ Moreover, the operator $\delta\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\rightarrow 0$ strongly as $\delta\rightarrow 0^{+}$
and $\big{\|}\delta\big{(}\delta I+\mu_{0}^{b}\big{)}^{-1}\big{\|}\leq 1$.
Hence, from the Lebesgue-dominated convergence theorem and the compactness of
$C_{r}(x)$ and $S_{r}(x),x>0,$ yield

$\displaystyle E\left\|\zeta^{\delta}(x)-\bar{\zeta}_{b}\right\|^{2}$
$\displaystyle\leq$ $\displaystyle 10E\Big{\|}\delta\big{(}\delta
I+\mu_{0}^{b}\big{)}^{-1}\Big{(}E\bar{\zeta}_{b}-C_{r}(b)[\phi(0)+h(\zeta^{\delta})]$
$\displaystyle\quad\quad-
S_{r}(b)\big{[}\xi+f_{1}\big{(}0,\zeta^{\delta}(0),\zeta^{\delta}(\nu_{1}(0))\big{)}\big{]}\Big{)}\Big{\|}^{2}$
$\displaystyle+10C_{2}E\bigg{(}\int_{0}^{b}\big{\|}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\varphi(y)\big{\|}_{L_{2}^{0}}^{2}dy\bigg{)}$
$\displaystyle+10E\bigg{(}\int_{0}^{b}\big{\|}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}C_{r}(b-y)\big{[}f_{1}\big{(}y,\zeta^{\delta}(y),\zeta^{\delta}(\nu_{1}(y))\big{)}-f_{1}(y)\big{]}\big{\|}dy\bigg{)}^{2}$
$\displaystyle+10E\bigg{(}\int_{0}^{b}\big{\|}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}C_{r}(b-y)f_{1}(y)\big{\|}dy\bigg{)}^{2}$
$\displaystyle+10E\bigg{(}\int_{0}^{b}\big{\|}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}P_{r}(b-y)\big{[}f_{2}\big{(}y,\zeta^{\delta}(y),\zeta^{\delta}(\nu_{2}(y))\big{)}-f_{2}(y)\big{]}\big{\|}dy\bigg{)}^{2}$
$\displaystyle+10E\bigg{(}\int_{0}^{b}\big{\|}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}P_{r}(b-y)f_{2}(y)\big{\|}dy\bigg{)}^{2}$
$\displaystyle+10C_{2}E\bigg{(}\displaystyle\int_{0}^{b}\Big{\|}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}P_{r}(b-y)\int_{0}^{y}\varrho(y-\tau)\big{[}g^{\delta}(\tau)-g(\tau)\big{]}\Big{\|}d\tau
dy\bigg{)}^{2}$
$\displaystyle+10C_{2}E\bigg{(}\displaystyle\int_{0}^{b}\Big{\|}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}P_{r}(b-y)\int_{0}^{y}\varrho(y-\tau)g(\tau)\Big{\|}d\tau
dy\bigg{)}^{2}$ $\displaystyle+10E\bigg{(}\Big{\|}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\sum\limits_{p=1}^{n}C_{r}(b-x_{p})\hat{I}_{p}(\zeta^{\delta}(x_{p}))\Big{\|}\bigg{)}^{2}$
$\displaystyle+10E\bigg{(}\Big{\|}\delta\big{(}\delta
I+\mu_{y}^{b}\big{)}^{-1}\sum\limits_{p=1}^{n}S_{r}(b-x_{p})\hat{J}_{p}(\zeta^{\delta}(x_{p}))\Big{\|}\bigg{)}^{2}\to
0~{}\mbox{as}~{}\delta\to 0^{+}.$

Thus, $\zeta^{\delta}(x)\to\bar{\zeta}_{b}$ as $\delta\to 0^{+}$ and this
shows that the system (1.7) is approximately controllable. ∎

######  Remark 3.4.

The case when
$f_{1}\big{(}x,\zeta(x),\zeta(\nu_{1}(x))\big{)}=\sigma(x,\zeta_{x}),$
$f_{2}\big{(}x,\zeta(x),\zeta(\nu_{2}(x))\big{)}=F(x,\zeta_{x}),$
$\mathcal{G}\big{(}x,\zeta(x),\zeta(\nu_{3}(x))\big{)}={G}(x,\zeta_{x})$ and
$h(\zeta)(x)=q(\zeta_{x_{1}},\zeta_{x_{2}},\ldots,\zeta_{x_{m}})$ with
integer-order derivative instead of Caputo fractional derivative has been
discussed in [36].

######  Remark 3.5.

The case when $f_{1}\big{(}x,\zeta(x),\zeta(\nu_{1}(x))\big{)}=0,$
$\mathcal{A}\zeta(x)=\mathcal{A}^{C}{D}^{p}\zeta(x),~{}(0<p<1),$
$f_{2}\big{(}x,\zeta(x),\zeta(\nu_{2}(x))\big{)}=F(x,\zeta(x)),$
$\mathcal{G}\big{(}x,\zeta(x),\zeta(\nu_{3}(x))\big{)}={G}(x,\zeta(x))$ has
been studied in finite dimensional space in [38] without nonlocal and
impulsive conditions.

##  4\. Application

Example. Consider the following system

$\displaystyle\left\\{\begin{array}[]{lll}\partial_{x}^{\frac{4}{3}}\big{[}\zeta(x,z)+\tilde{f}_{1}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}\big{]}\in\dfrac{\partial^{2}\zeta(x,z)}{\partial
x^{2}}+\tilde{f}_{2}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}\\\ \hskip
14.22636pt+\mathcal{B}u(x,z)+\displaystyle\int_{0}^{x}|x-s|\tilde{\mathcal{G}}\big{(}s,\zeta(s,z),\zeta(s-1,z))\big{)}dw(s),\quad
z\in[0,\pi],~{}x\in[0,1],&\\\
\zeta(x,z)=\phi(x,z)+h(\zeta),~{}\dfrac{\partial\zeta(x,z)}{\partial
x}\Big{\rvert}_{x=0}=2+z,\quad x\in[-1,0],~{}z\in[0,\pi],&\\\
\Delta\zeta(x,z)\Big{\rvert}_{x=x_{p}}=\displaystyle\int_{-1}^{x_{p}}\beta_{p}(x_{p}-s)\zeta(s,z)ds,\quad
p=1,2,\ldots,n,&\\\ \Delta\dfrac{\partial\zeta(x,z)}{\partial
x}\Big{\rvert}_{x=x_{p}}=\displaystyle\int_{-1}^{x_{p}}\hat{\beta}_{p}(x_{p}-s)\zeta(s,z)ds,\quad
p=1,2,\ldots,n,&\\\ \zeta(x,0)=\zeta(x,\pi)=0,\quad
x\in[0,1],\end{array}\right.$ (4.7)

where $\partial_{x}^{\frac{4}{3}}$ represents fractional partial derivative of
Caputo type and $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}<x_{n+1}=1$ are impulse
points. $w(x)$ represents $\mathcal{W}$-valued standard Wiener process on a
complete probability space $(\Omega,\Upsilon,\Upsilon_{x},P)$ with usual
conditions and $u$ is a control function. $\beta_{p},\hat{\beta}_{p}$ are
continuous and satisfy Lipschitz condition. However, $h$ can be chosen in such
a way that assumption (A8) holds.
Let $\mathcal{H}=\mathcal{W}=\mathcal{L}^{2}[0,\pi].$ Let the operator
$A:\mathcal{H}\rightarrow\mathcal{H}$ be defined by
$A\zeta(x)=\dfrac{\partial^{2}\zeta(x)}{\partial x^{2}}$ with

$D(A)=\big{\\{}\zeta\in\mathcal{H}:\zeta,\zeta^{\prime}~{}\mbox{are absolutely
continuous},~{}\zeta^{\prime\prime}\in\mathcal{H}~{}~{}\mbox{and}~{}\zeta(0)=\zeta(\pi)=0\big{\\}}.$

The operator $A$ has discrete spectrum with normalized eigenvectors
$e_{n}(z)=\sqrt{\frac{2}{\pi}}\sin(nz)$ corresponding to the eigenvalues
$\lambda_{n}=-n^{2},$ where $n\in\mathbb{N}.$ Moreover,
$\\{e_{n}:n\in\mathbb{N}\\}$ forms an orthonormal basis for $\mathcal{H}.$
Thus, we have

$A\zeta=\sum\limits_{n\in\mathbb{N}}{-n^{2}}\langle\zeta,e_{n}\rangle
e_{n},~{}\zeta\in D(A).$

$A$ generates a strongly continuous cosine family given by

$C(x)\zeta=\sum\limits_{n\in\mathbb{N}}\cos(nx)\langle\zeta,e_{n}\rangle
e_{n},$

and that the associated sine family is given by

$S(x)\zeta=\sum\limits_{n\in\mathbb{N}}\frac{1}{n}\sin(nx)\langle\zeta,e_{n}\rangle
e_{n}.$

From the subordinate principle [8, Theorem 3.1], it follows that $A$ generates
a strongly continuous exponentially bounded fractional cosine family
$C_{r}(x),$ such that $C_{r}(0)=I$ and

$\displaystyle C_{r}(x)=\int_{0}^{\infty}\psi_{x,\frac{r}{2}}(s)C(s)ds,\quad
x>0,$ (4.8)

where
$\psi_{x,\frac{r}{2}}(s)=x^{-\frac{r}{2}}\chi_{\frac{r}{2}}(sx^{-\frac{r}{2}})$
and

$\chi_{\rho}(z)=\sum\limits_{n=0}^{\infty}\frac{(-z)^{n}}{n!\Gamma(-\rho
n+1-\rho)},~{}~{}0<\rho<1.$

Define

$\mathcal{U}=\bigg{\\{}u:u=\sum\limits_{n=2}^{\infty}u_{n}e_{n}~{}\Big{\rvert}\sum\limits_{n=2}^{\infty}u_{n}^{2}<\infty\bigg{\\}},$

with the norm
$\|u\|=\bigg{(}\sum\limits_{n=2}^{\infty}u_{n}^{2}\bigg{)}^{1/2}.$
Define the operator $\mathcal{B}:\mathcal{U}\rightarrow\mathcal{H}$ by
$\mathcal{B}u=\big{(}\mathcal{B}u\big{)}(x),$ where
$\mathcal{B}\in\mathcal{L(U,H)}$ such that
$\mathcal{B}u(x)=2u_{2}(x)e_{1}(z)+\sum\limits_{n=2}^{\infty}u_{n}(x)e_{n}(z).$
Next, we verify the hypotheses for the system (4.7) one by one.
Verification of A1:
From (4.8), $C_{r}(x)$ is compact and undoubtedly, $\|C_{r}\|^{2}\leq 1.$
Moreover, there exists $M_{4}>0$ such that $\|\mathcal{B}\|^{2}\leq M_{4}.$
Verification of A2:
Assume that
$f_{1}\big{(}x,\zeta(x),\zeta(\nu_{1}(x))\big{)}(z)=\tilde{f}_{1}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}.$
Define the operator
$\tilde{f}_{1}:[0,1]\times\mathcal{PC}\times\mathcal{PC}_{0}\to\mathcal{H},$
by
$\tilde{f}_{1}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}=\dfrac{2e^{-x}\sin
x}{\sqrt{3}+|\zeta(x,z)|+|\zeta(x-1,z)|}$.
Therefore,

$\displaystyle
E\|\tilde{f}_{1}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}-\tilde{f}_{1}\big{(}x,\tilde{\zeta}(x,z),\tilde{\zeta}(x-1,z)\big{)}\|^{2}$
$\displaystyle\quad=E\bigg{\|}\frac{2e^{-x}\sin
x}{\sqrt{3}+|\zeta(x,z)|+|\zeta(x-1,z)|}-\frac{2e^{-x}\sin
x}{\sqrt{3}+|\tilde{\zeta}(x,z)|+|\tilde{\zeta}(x-1,z)|}\bigg{\|}^{2}$
$\displaystyle\quad\leq
2E\Big{(}\|\zeta(x,z)-\tilde{\zeta}(x,z)\|^{2}+\|\zeta(x-1,z)-\tilde{\zeta}(x-1,z)\|^{2}\Big{)}.$

Verification of A3:
Define
$\tilde{f}_{2}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}=\dfrac{|\zeta(x,z)|+|\zeta(x-1,z)|}{5\pi+|\zeta(x,z)|+|\zeta(x-1,z)|}$.
Thus, there exists $L_{\tilde{f_{2}}}>0$ such that

$\displaystyle
E\|\tilde{f}_{2}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}-\tilde{f}_{2}\big{(}x,\tilde{\zeta}(x,z),\tilde{\zeta}(x-1,z)\big{)}\|^{2}$
$\displaystyle\quad\leq
L_{\tilde{f_{2}}}E\Big{(}\|\zeta(x,z)-\tilde{\zeta}(x,z)\|^{2}+\|\zeta(x-1,z)-\tilde{\zeta}(x-1,z)\|^{2}\Big{)}.$

Verification of A4:
Define $\varrho:[0,1]\to\mathbb{R}$ be such that $\varrho(x-s)=|x-s|.$
Therefore, $\|\varrho(x-s)\|^{2}\leq\ell,$ where $\ell>0$. Verification of A5:
Define

$\displaystyle\tilde{\mathcal{G}}\big{(}x,\zeta(x),\zeta(\nu_{3}(x))\big{)}(z)$
$\displaystyle=$
$\displaystyle\tilde{\mathcal{G}}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}$
$\displaystyle=$
$\displaystyle\big{\\{}g\in\mathcal{H}:g_{1}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}\leq
g\leq g_{2}\big{(}x,\zeta(x,z),\zeta(x-1,z)\big{)}\big{\\}},$

where $g_{1},g_{2}:[0,1]\times\mathcal{PC}\times\mathcal{PC}_{0}\rightarrow
P_{bd,cl,cv}\big{(}L(\mathcal{W},\mathcal{H})\big{)}.$ We assume that for each
$x\in[0,1]$, $g_{1}$ is lower semi continuous and $g_{2}$ is upper semi
continuous.
Verification of A6:
Assume that there exists $\wp\in L^{1}([0,1],\mathbb{R}^{+})$ and
$\Theta:\mathbb{R}^{+}\to(0,\infty)$ is continuous increasing function such
that

$\displaystyle\max\bigg{\\{}\int_{0}^{x}E\|g_{1}\big{(}x,\zeta(x,z),\zeta(x-1,z))\big{)}\|^{2},\int_{0}^{x}E\|g_{2}\big{(}x,\zeta(x,z),\zeta(x-1,z))\big{)}\|^{2}\bigg{\\}}$
$\displaystyle\leq\wp(x)\Theta\Big{(}E\|\zeta(x,z)\|^{2}+E\|\zeta(x-1,z)\|^{2}\Big{)}.\hskip
153.6447pt$ (4.10)

Verification of A7:
From (4.7),

$\Delta\zeta(x,z)\Big{\rvert}_{x=x_{p}}=\displaystyle\int_{-1}^{x_{p}}\beta_{p}(x_{p}-s)\zeta(s,z)ds,\quad
p=1,2,\ldots,n$

and

$\Delta\dfrac{\partial\zeta(x,z)}{\partial
x}\Big{\rvert}_{x=x_{p}}=\displaystyle\int_{-1}^{x_{p}}\hat{\beta}_{p}(x_{p}-s)\zeta(s,z)ds,\quad
p=1,2,\ldots,n,$

and we consider the function
$\hat{I}_{p},\hat{J}_{p}:\mathcal{H}\rightarrow\mathcal{H}$ such that the
required conditions are fulfilled.

If we define $\zeta(x)(z)=\zeta(x,z),$ $u(x)(z)=u(x,z),$ then system (4.7) can
be written as in the abstract form (1.7). If all the hypotheses of Theorem 3.1
are satisfied, then the system (4.7) admits a mild solution.

##  5\.  Conclusion

The existence and approximate controllability results for a class of nonlocal
fractional neutral stochastic integrodifferential inclusions of order
$1<\alpha<2$ with impulses are studied. Sufficient conditions for the
existence of mild solutions are given for the cases when the multivalued map
is convex and non-convex. Fractional calculus, cosine and sine families,
stochastic analysis, and fixed point theorems are all applied to achieve the
desired outcomes. Next, we derived sufficient conditions for the approximate
controllability of the system. Finally, we illustrated the practicality of the
finding using a theoretical scenario.

Second-order differential equations, for instance, can be used to analyze the
system of dynamical buckling of a hinged extensible beam [37, 41]. The
integrated process in continuous time, which can be made stationary, is best
described by second-order stochastic differential equations. For instance,
engineers can make use of second-order stochastic differential inclusions to
describe mechanical vibrations or the charge on a capacitor. On the other
hand, in control problems, we look for a control that steers the system from a
given initial state to a prescribed trajectory or a desired final state.
Moreover, while launching a rocket into space, it may be desirable to have a
precise path along with the preferred destination for cost-effectiveness and
collision avoidance. The efficiency of describing the real-life phenomena by
the fractional differential equation are more accurate than by classical
differential equations due to their nonlocal property. In the future, we plan
to examine optimal controllability for non-instantaneous impulsive stochastic
integrodifferential evolution inclusions of order $1<\alpha<2$.

Acknowledgements: The first and third authors acknowledge UGC, India, for
providing financial support through MANF F.82-27/2019 (SA-III)/ 4453 and
F.82-27/2019 (SA-III)/191620066959, respectively.
Competing Interest: None.

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