Existence results of mild solutions for some stochastic integrodifferential equations with state-dependent delay and noninstantaneous impulses in Hilbert spaces

ABSTRACT

The primary objective of this work is to investigate the existence of mild solutions for some neutral stochastic integrodifferential systems in Hilbert spaces. These systems have a state-dependent delay and noninstantaneous impulses. In order to achieve this objective, the neutral stochastic integrodifferential system that has been proposed with state-dependent delay and noninstantaneous impulses is converted into an equivalent fixed point problem by means of an appropriate integral operator.Together with the theory of stochastic analysis, we develop several existence results that are founded on the theory of resolvent operators in the sense of Grimmer, Hausdorff’s measures of noncompactness, and fixed point theorems.In the last part of this paper, specific examples are used to show how the general conclusions reached in this paper are made.

KEYWORDS:

• Wiener process
• resolvent operator
• stochastic evolution equations
• hausdorff measure of noncompactness
• state dependent delay
• impulsive effects

Public Interest Statement

In physics, automatic control, neural networks, infectious diseases, population growth, and cell production, stochastic differential equations with state-dependent delays arise. Some of these models are quite distinct from one another, and the majority of them do not appear to have an easy construction method. The delay may be defined implicitly by a functional, integral, or differential equation and should frequently be considered as part of the state variables. In most cases, the delay is not given explicitly as a function of what appears to be the natural state variable. Most of the time, the delay is not given as a direct function of what seems to be the natural state variable.

1 Introduction

The theory of state-dependent delay differential equations is the subject of a lot of research because it has real-world applications and because the qualitative theory is different and more complex than the usual ones for differential equations with delay. As a direct result of this fact, there has been a lot of interest in the study of the various properties that are associated with this class of equations, see for instance(Anguraj et al., 2007; Chadha & Pandey, 2015; Das, Pandey, Sukavanam et al., 2016a, 2016b; Dineshkumar et al., 2021; Dos Santos, Cuevas et al., 2011; Dos Santos, Mallika Arjunan et al., 2011; Hernández et al., 2018, 2006; Hernández, Wu, Chadha et al., 2020; Hernández, Wu, Fernandes et al., 2020; Sakthivel & Anandhi, 2010; Sakthivel & Ren, 2013; Suganya, Baleanu et al., 2015; Suganya, Mallika Arjunan et al., 2015; Valliammal et al., 2019; Vijayakumar et al., 2013) and the references therein. On the other hand when mathematics is used to model multiple things that happen in the real world or are the result of human actions, dynamical systems with some randomness are the result. Stochastic differential equations use randomness to give a mathematical description of an event. Over the course of the past few years, stochastic differential equations in both finite and infinite dimensions have garnered a significant amount of attention in a wide variety of research areas. This is due to the fact that they are capable of describing a diverse selection of occurrences in the fields of population dynamics, physics, electrical engineering, ecology, medicine, biology, and other scientific and engineering disciplines. See, (Deng et al., 2018; Y. Guo et al., 2019; Ma et al., 2020; Yuchen Guo et al., 2020) and references therein for a comprehensive introduction to stochastic differential equations and the applications of these equations.

Impulsive differential equation theory can accurately simulate many practical scenarios such as biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, and frequency modulated systems. Impulses of evolution processes formulated by impulsive differential equations are generally abrupt and instantaneous, i.e. the perturbations (impulses) start abruptly and their duration is negligible compared to the process’s duration. In many cases, instant impulses do not adequately reflect evolution’s dynamics. A good example of this would be medication that takes into account an individual’s hemodynamic balance. The process of a drug being absorbed into the body is one that happens gradually and in a continual fashion. As a result, instantaneous impulses are unable to adequately characterize such a process.Hernandez and O’Regan (Hernandez & O’Regan, 2013) give a new example of impulsive behavior that starts at a random time and ends after a certain amount of time.

On the other hand, taking drugs has an effect on memory. Because of this, the newly proposed category of impulsive conditions in (Hernandez & O’Regan, 2013) does not fully or very well explain this phenomenon. When used in this situation, the theory of fractional calculus is a powerful way to explain why this happened. There has been a recent impulse in the scientific community to develop rigorous mathematical models and to access historic information on input. In this approach, the fractional derivative is one such concept that is more effective in complicated models with heredity properties. Several applications can be seen in the modeling of human liver (Baleanu, Jajarmi et al., 2020), hearing loss due to Mumps virus (Tuan et al., 2020), COVID-19 transmission (Baleanu, Mohammadi et al., 2020), etc.

Stochastic differential equations with state-dependent delay have become a popular topic. Several authors have looked at both qualitative and quantitative results for different types of stochastic systems that have state-dependent delays. In (Chaudhary & Pandey, 2019), Chaudhary and Pandey provided a framework for study the mild solution to fractional neutral stochastic integrodifferential systems with state dependent delay and noninstantaneous impulses in Hilbert spaces. Thiagu in (Thiagu, 2021) focused the approximate controllability of second order ($q\in (1,2]$) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. Diop et al. in (Diop, Diop et al., 2022) investigated optimal control problems for a class of stochastic functional integral-differential equations in Hilbert spaces. A class of multi-term time-fractional random integrodifferential equations with state-dependent delay was investigated by Diop et al. in (Diop, Frederico et al., 2022). Yan and Lu in (Yan & Lu, 2018), studied a new class of fractional impulsive stochastic partial integrodifferential control systems with state-dependent delay and their optimal controls in a Hilbert space. Huang et al in (Huang et al., 2018), considered the existence and controllability of mild solutions for a class of second-order neutral impulsive stochastic evolution integrodifferential equations with state-dependent delay in a real separable Hilbert space among. However, no work has been reported in the literature regarding the existence of mild solutions of stochastic integrodifferential equations with state-dependent delay and noninstantaneous impulses in Hilbert spaces. Motivated by these facts, the purpose of this paper is to study the existence results for a class of impulsive neutral stochastic integrodifferential equations with state dependent delay, of the form

(1) $d\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]=\left\{\mathcal{A}\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]+{\displaystyle {\int }_0^t\mathcal{G}\left(t-s\right)\left[\vartheta \left(s\right)-\mathbb{D}\left(s,{\vartheta }_{\rho \left(s,{\vartheta }_s\right)}\right)\right]ds}\right\}dt+\mathbb{F}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)dt+{\displaystyle {\int }_0^t\mathbb{G}\left(t,s,{\vartheta }_{\rho }\left(s,{\vartheta }_s\right)\right)dW(s),t\in {\displaystyle \underset{i=0}{\overset{n}{\cup }}(s_i,t_{i+1}],n}}\in \mathbb{N},\vartheta \left(t\right)={\mathbb{S}}_i\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right),t\in {\displaystyle \underset{i=1}{\overset{n}{\cup }}(t_i,s_i],}\vartheta \left(t\right)=\varphi (t)\in ℬ,t\in (-\infty ,0],$(1)

where, the state $\vartheta (\cdot )$ takes values in a separable real Hilbert space $\left(\mathrm{H},(\cdot ,\cdot ),\parallel \cdot \parallel \right)$.

$\mathcal{A}:\mathcal{D}(\mathcal{A})\subset \mathrm{H}\to \mathrm{H}$ is a closed linear operator which generates a semigroup $(\mathrm{T}(t){)}_{t\ge 0}$, $\mathcal{G}(t)$ is a closed linear operator with domain $\mathcal{D}(\mathcal{G}(t))\subset \mathcal{D}(\mathcal{A})$. Further, the history function ${\vartheta }_t:(-\mathrm{\infty },0]\to \mathrm{H}$ representes the time history of the function $\vartheta$ from $-\mathrm{\infty }$ to the present time $t$ and defined by ${\vartheta }_t(\theta )=\vartheta (t+\theta )$ for $\theta \in (-\mathrm{\infty },0]$, and belongs to the abstract phase space $\mathcal{B}$; $w(t)$ is a brownian motion on a real separable Hilbert space $(\mathrm{K},(\cdot ,\cdot {)}_K,\parallel \cdot {\parallel }_K)$. Let $n\in \mathbb{N},$ $0=t_0=s_0<t_1<s_1<t_2\text{break}<\cdots <t_n<s_n<t_{n+1}<\cdots$, are prefixed numbers.The functions $\mathbb{D}:[0,T]\times \mathcal{B}\to \mathrm{H}$, $\rho :[0,T]\times \mathcal{B}\to \text{break}(-\mathrm{\infty },T]$, $\mathbb{F}:[0,T]\times \mathcal{B}\to \mathrm{H}$, ${\mathbb{S}}_i:[t_i,s_i]\times \mathcal{B}\to \mathrm{H}$ and $\mathbb{G}:\mathrm{\Delta }\times \mathcal{B}\to {\mathcal{L}}_2^0(\mathrm{K},\mathrm{H})$ are appropriate functions to be specified later, where $\mathrm{\Delta }=\{(t,s):0\le s<t\le T\}$.

Three existence results for the system (1) are derived in this paper using resolvent operator theory in the sense of Grimmer and fixed point techniques, which are discussed in detail in the paper. We obtain sufficient conditions to prove the existence and uniqueness of the mild solution of system (1) in the first result by employing the Banach fixed point theorem under Lipschitz conditions on nonlinear terms. The second and third existence results, on the other hand, have demonstrated that mild solutions can be found by applying the Darbo and Darbo-Sadovskii fixed point theorems on nonlinear terms.

The following is how the paper is organized: In Section 2, we give notations, definitions and auxiliary outcomes on stochastic integral with respect to Wiener process (Prato & Zabczyk, 1992), measure of non compactness (Deimling, 1985) and resolvent operators theory for integro-differential equations (Grimmer, 1982). In Section 3, we use fixed point theorems to obtain the existence results for equation (1)(1) $d\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]=\left\{\mathcal{A}\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]+{\displaystyle {\int }_0^t\mathcal{G}\left(t-s\right)\left[\vartheta \left(s\right)-\mathbb{D}\left(s,{\vartheta }_{\rho \left(s,{\vartheta }_s\right)}\right)\right]ds}\right\}dt+\mathbb{F}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)dt+{\displaystyle {\int }_0^t\mathbb{G}\left(t,s,{\vartheta }_{\rho }\left(s,{\vartheta }_s\right)\right)dW(s),t\in {\displaystyle \underset{i=0}{\overset{n}{\cup }}(s_i,t_{i+1}],n}}\in \mathbb{N},\vartheta \left(t\right)={\mathbb{S}}_i\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right),t\in {\displaystyle \underset{i=1}{\overset{n}{\cup }}(t_i,s_i],}\vartheta \left(t\right)=\varphi (t)\in ℬ,t\in (-\infty ,0],$(1) . Section 4 concludes with examples that demonstrate the results that were obtained.

2 Preliminaries

Throughout this paper $\left(\mathrm{H},{⟨\cdot ,\cdot ⟩}_{\mathrm{H}},\parallel \cdot {\parallel }_{\mathrm{H}}\right)$ and $\left(\mathrm{K},{⟨\cdot ,\cdot ⟩}_{\mathrm{K}},\parallel \cdot {\parallel }_{\mathrm{K}}\right)$ are two real separable Hilbert spaces. Let $(\mathrm{\Omega },\mathcal{F},\{{\mathcal{F}}_t{\}}_{t\ge 0},\mathbb{P})$ be a complete probability space with a normal filtration $\{{\mathcal{F}}_t{\}}_{t\ge 0}$ satisfying the usual conditions(i.e. it is increasing and right-continuous, while ${\mathcal{F}}_0$ contains all $\mathbb{P}$-null sets). Let $\left\{w(t):t\ge 0\right\}$ denote a $\mathrm{K}$-valued Wiener process defined on the probability space $(\mathrm{\Omega },\mathcal{F},\{{\mathcal{F}}_t{\}}_{t\ge 0},\mathbb{P})$, with covariance operator $Q$, that is,

$\mathbb{E}({⟨w(t),x⟩}_{\mathrm{K}}{⟨w(s),y⟩}_{\mathrm{K}})=(t\wedge s){⟨Qx,y⟩}_{\mathrm{K}},$

for all $x,y\in \mathrm{K}$, where $Q$ is a positive, self adjoint, trace class operator on $\mathrm{K}$. In particular, we denote by $w(t)$ a $\mathrm{K}-$ valued $Q-$ Wiener process with respect to $\{{\mathcal{F}}_t{\}}_{t\ge 0}$. To define stochastic integrals with respect to the $Q-$ Wiener process $w(t)$, we introduce the subspace ${\mathrm{K}}_0=Q^{\frac{1}{2}}\mathrm{K}$ of $\mathrm{K}$ endowed with the inner product

${⟨u,v⟩}_{{\mathrm{K}}_0}={⟨Q^{\frac{-1}{2}}u,Q^{\frac{-1}{2}}v⟩}_{\mathrm{K}}$

as a Hilbert space. We assume that there exists a complete orthonormal system $\{e_i\}$ in $\mathrm{K}$, a bounded sequence of positive real numbers $\{{\lambda }_i\}$ such that $Q{e}_i={\lambda }_i{e}_i$, $i=1,2,\cdots ,$ and a sequence $\{w_i(t){\}}_{i>1}$ of independent standard Brownian motions such that $w(t)=\sum _{i=1}^{+\mathrm{\infty }}\sqrt{{\lambda }_i}{w}_i(t)e_i$ for $t\ge 0$ and ${\mathcal{F}}_t={\mathcal{F}}_t^w$, where ${\mathcal{F}}_t^w$ is the $\sigma -$ algebra generated by $\left\{w(s):0\le s\le t\right\}$. Let ${\mathcal{L}}_2^0={\mathcal{L}}_2({\mathrm{K}}_0,\mathrm{H})$ be the space of all Hilbert-Schmidt operators from ${\mathrm{K}}_0$ to $\mathrm{H}$. It turns out to be a separable Hilbert space equipped with the norm $\parallel v{\parallel }_{{\mathcal{L}}_2^0}^2=tr\left((v{Q}^{\frac{1}{2}})(v{Q}^{\frac{1}{2}}{)}^{\ast }\right)$ for any $v\in {\mathcal{L}}_2^0$. Obviously, for any bounded operator $v\in {\mathcal{L}}_2^0$, this norm reduces to $\parallel v{\parallel }_{{\mathcal{L}}_2^0}^2=tr\left(vQ{v}^{\ast }\right)=\sum _{i=1}^{+\mathrm{\infty }}\parallel \sqrt{{\lambda }_i}v{e}_i{\parallel }^2$.

Let ${\mathcal{L}}_2(\mathrm{\Omega },\mathcal{F},\mathbb{P};\mathrm{H})\equiv {\mathcal{L}}_2(\mathrm{\Omega };\mathrm{H})$ denote the Banach space of strongly-measurable, square integrable random variables equipped with norm

$\parallel v{\parallel }_{{\mathcal{L}}_2(\mathrm{\Omega },\mathrm{H})}={\left(\mathbb{E}\parallel v{\parallel }_{\mathrm{H}}^2\right)}^{\frac{1}{2}}.$

The phase space $\mathcal{B}$ described by Hale and Kato in (Hale & Kato, 1978) is taken into consideration in order to deal with the infinite delay. Phase space $\mathcal{B}$ must possess the characteristics specified in the lemma below.

Lemma 2.1 (Hino et al., 1991) The abstract phase space is a seminormed linear space $(\mathcal{B},\parallel .\parallel )$ of ${\mathcal{F}}_0$-measurable functions which maps $(-\mathrm{\infty },0]$ into $\mathrm{H}$ and satisfies the following axioms:

1. If $v:(-\mathrm{\infty },T]\to \mathrm{H}$ is a continuous function on $[0,T]$ such that $v{|}_{[0,T]}\in C([0,T],\mathrm{H})$ and $v_0\in \mathcal{B}$, then for every $t\in [0,T]$, the following condition hold:

(a) $v_t$ is in $\mathcal{B}$,

(b)$\parallel v(t)\parallel \le K\parallel v_t{\parallel }_{\mathcal{B}}$,

(c)$\begin{array}{rl}& \parallel v_t{\parallel }_{\mathcal{B}}\le \mathcal{H}(t)\parallel v_0{\parallel }_{\mathcal{B}}+\mathcal{N}(t)\\ & sup\left\{\parallel v(s)\parallel :0\le s\le t\right\}.\end{array}$.Where $K$ is a positive constant, $\mathcal{H}:[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ is locally bounded, $\mathcal{N}:[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ is continuous, $\mathcal{H}$ and $\mathcal{N}$ are independent of $v(\cdot )$.

2. The phase space $\mathcal{B}$ is complete.

3. The function $t\to {\varphi }_t$ is well defined from the set $\mathbb{R}(-)=\left\{\rho (s,\psi ):(s,\psi )\in [0,T]\times \mathcal{B}\right\}$ into $\mathcal{B}$ and there exists a continuous and bounded function

${\mathcal{J}}^{\varphi }:\mathbb{R}(-)\to (0,+\mathrm{\infty })$ such that $\parallel {\varphi }_t{\parallel }_{\mathcal{B}}\le {\mathcal{J}}^{\varphi }(t)\text{break}\parallel \varphi {\parallel }_{\mathcal{B}}$, for every $t\in \mathbb{R}(-)$.

Lemma 2.2 (Hernandez et al., 2006) Let $v:(-\mathrm{\infty },T]\to \mathrm{H}$ be a function such that $v_0=\varphi$ and $v{|}_{[0,T]}\in \mathrm{P}\mathrm{C}([0,T],\mathrm{H})$. Then

$\parallel v_t{\parallel }_{\mathcal{B}}\le ({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })\parallel \varphi {\parallel }_{\mathcal{B}}+{\mathcal{N}}_{T}sup\left\{\parallel v(\varsigma )\parallel :\varsigma \in [0,max\{0,t\}]\right\},t\in \mathbb{R}(-)\cup [0,T]$

where ${\mathcal{J}}^{\varphi }=\underset{t\in \mathbb{R}(-)}{sup}{\mathcal{J}}^{\varphi }(t)$, ${\mathcal{H}}_T=\underset{t\in [0,T]}{sup}\mathcal{H}(t)$, ${\mathcal{N}}_T=\underset{t\in [0,T]}{sup}\mathcal{N}(t)$.

Lemma 2.3 (Prato & Zabczyk, 1992) For any $q\ge 1$ and for arbitrary ${\mathcal{L}}_2^0(\mathrm{K},\mathrm{H})$- valued predictable process $z(\cdot )$,

$\underset{s\in [0,t]}{sup}\mathbb{E}{∥{\int }_0^{s}z(r)dW(r)∥}_{\mathrm{H}}^{2q}\le C_q{\left({\int }_0^t\mathbb{E}\parallel z(r){\parallel }_{{\mathcal{L}}_2^0}^{2}dr\right)}^q,\mathrm{\forall }t\in [0,+\mathrm{\infty }),$

where $C_q=(q(2q-1){)}^q.$

Before going into greater detail about the main findings of this study, we’d like to provide a brief overview of the Hausdorff measure of noncompactness.

Definition 2.1 (Banas & Goebel, 1980) Let $\mathrm{Q}$ be bounded subset of normed space $\mathrm{Y}$. The Hausdorff measure of noncompactness(shortly $MNC$) is defined by

$\beta (\text{Q})=\inf \{ϵ0:\text{Q}hasafinitecoverbyballsofradiuslessthanϵ\}.$

Lemma 2.4 (Banas & Goebel, 1980) The Hausdorff measure verifies the following proprieties: let $\mathrm{Q},{\mathrm{Q}}_1,{\mathrm{Q}}_2$ be bounded subset of a normed space $\mathrm{Y}$.

(a) ${\mathrm{Q}}_1\subset {\mathrm{Q}}_2$ then $\beta ({\mathrm{Q}}_1)\le \beta ({\mathrm{Q}}_2)$ (monotonicity);

(b) $\beta (\mathrm{Q})=\beta (\bar{\mathrm{Q}})$

(c) $\beta ({\mathrm{Q}}_1+{\mathrm{Q}}_2)\le \beta ({\mathrm{Q}}_1)+\beta ({\mathrm{Q}}_2)$,where ${\mathrm{Q}}_1+{\mathrm{Q}}_2=\{x+y:x\in {\mathrm{Q}}_1,y\in {\mathrm{Q}}_2\}$;

(d) $\beta (\lambda \mathrm{Q})\le |\lambda |\beta (\mathrm{Q})$ for any $\lambda \in \mathbb{R}$;

(e) $\beta (\{a\}\cup \mathrm{Q})=\beta (\mathrm{Q})$ for every $a\in \mathrm{Y}$;

(f) $\beta (\mathrm{Q})=0$ if and only if $\mathrm{Q}$ is relatively compact;

(g) For any map $G:\mathcal{D}(G)\subseteq \mathrm{Y}\to \mathrm{Y}$ which is Lipschitz continuous with Lipschitz constant $\alpha$, we have

$\beta (G(\mathrm{Q}))\le \alpha \beta (\mathrm{Q}),foranysubset\mathrm{Q}\subseteq \mathcal{D}(G).$

Definition 2.2 A continuous and bounded map$\mathbb{K}:\mathrm{U}\subset \mathrm{X}\to \mathrm{X}$ is called $\beta$-contraction if there exists a constant $0<\alpha <1$ such that $\beta \left(\mathbb{K}(E)\right)\le \alpha \beta (E)$; for any noncompact bounded subset $E\subset \mathrm{U}$; where $\mathrm{X}$ is a Banach space.

The subsequently fixed point theorems will be employed in the proofs of our outcomes.

Lemma 2.5 (Darbo) If $\mathrm{Q}\subseteq X$ is closed and convex and $0\in \mathrm{Q}$, the continuous map $\mathcal{K}:\mathrm{Q}\to \mathrm{Q}$ is a $\beta$-contraction and the set $\{\vartheta \in \mathrm{Q}:\vartheta =\alpha \mathcal{K}(\vartheta )\}$ is bounded for $0<\alpha <1$, then the map $\mathcal{K}$ has at least one fixed point in $\mathrm{Q}$.

Lemma 2.6 (Darbo-Sadovskii) If $\mathrm{Q}\subseteq X$ be closed, bounded and convex. If the continuous map $\mathcal{K}:\mathrm{Q}\to \mathrm{Q}$ is a $\beta$-contraction, then $\mathcal{K}$ has a fixed point in $D$.

In the sequel, we denote ${\beta }_C$ and ${\beta }_{\mathrm{P}\mathrm{C}}$ to denote the Hausdorff measure of noncompactness of $C([0,T],\mathrm{H})$ and $\mathrm{P}\mathrm{C}([0,T],\mathrm{H})$ respectively.

Next, to be able to access the existence of mild solutions for (1) we need to introduce partial integrodifferential equations and resolvent operators that will be used to develop the main results. Let $\mathbb{X}$ and $\mathbb{M}$ be Banach spaces. We denote by $\mathcal{L}(\mathbb{X},\mathbb{M})$ the Banach space of bounded linear operators from $\mathbb{X}$ to $\mathbb{M}$ endowed with the operator norm, and we abbreviate this notation to $\mathcal{L}(\mathbb{X})$ when $\mathbb{X}=\mathbb{M}$.

In what follows, ${\mathbf{X}}_1$ is a Banach space, $A$ and $\mathcal{G}(t)$ are closed linear operators on ${\mathbf{X}}_1$, ${\mathbf{X}}_2$ is the Banach space $D(A)$ endowed with the graph norm $\parallel z{\parallel }_{{\mathbf{X}}_2}=\parallel Az\parallel +\parallel z\parallel \mathrm{f}\mathrm{o}\mathrm{r}z\in {\mathbf{X}}_2$ and $\mathcal{C}({\mathbb{R}}^{+},{\mathbf{X}}_2)$ denotes the space of continuous functions from ${\mathbb{R}}^{+}$ into ${\mathbf{X}}_2$. For further purposes, let us consider the following system

(2) $\left\{\begin{array}{c}\vartheta { }^{\mathrm{\prime }}(t)=\mathrm{A}\vartheta (t)+{\int }_0^t\mathcal{G}(t-s)\vartheta (s)ds\mathrm{f}\mathrm{o}\mathrm{r}t\in [0,T],\\ \vartheta (0)={\vartheta }_0\in {\mathbf{X}}_1.\end{array}\right.$(2)

Definition 2.3 (Grimmer, 1982) A bounded linear operator valued function $\mathrm{R}(t)\in \mathcal{L}({\mathbf{X}}_1)$, $t\ge 0$ is called the resolvent operator for system (2) if it satisfies the following conditions:

(i) $\mathrm{R}(0)=Id$ and $\parallel \mathrm{R}(t){\parallel }_{\mathcal{L}({\mathbf{X}}_1)}\le \mathcal{D}{e}^{\delta t}$ for some constants $\mathcal{D}$ and $\delta$.

(ii) For each $x\in {\mathbf{X}}_1$, $\mathrm{R}(t)x$ is continuous for $t\ge 0$.

(iii) $\mathrm{R}(t)\in \mathcal{L}({\mathbf{X}}_2)$ for $t\ge 0$. For $x\in {\mathbf{X}}_2$,$\mathrm{R}(\cdot )x\in {\mathcal{C}}^1({\mathbb{R}}_{+},{\mathbf{X}}_1)\cap \mathcal{C}({\mathbb{R}}_{+},{\mathbf{X}}_2)$ and

${\mathrm{R}}^{\prime }(t)x=A\mathrm{R}(t)x+{\int }_0^t\mathcal{G}(t-s)\mathrm{R}(s)xds$

$=\mathrm{R}(t)Ax+{\int }_0^t\mathrm{R}(t-s)\mathcal{G}(s)xds,fort\ge 0.$

In what follows, we suppose the following assumptions.

$({\mathbf{R}}_1)$ $\mathrm{A}$ is the infinitesimal generator of a $C_0$-semigroup $(S(t){)}_{t\ge 0}$ on ${\mathbf{X}}_1$.

$({\mathbf{R}}_2)$ For all $t\ge 0$, $\mathcal{G}(t)$ is a closed linear operator from $D(A)$ to ${\mathbf{X}}_1$ and $\mathcal{G}(t)\in \mathcal{L}({\mathbf{X}}_2,{\mathbf{X}}_1)$. For any $z\in {\mathbf{X}}_2$, the map $t\to \mathcal{G}(t)z$ is bounded, differentiable and the derivative $t\to {\mathcal{G}}^{\prime }(t)z$ is bounded and uniformly continuous on ${\mathbb{R}}^{+}$.

The next theorem gives sufficient conditions ensuring the existence of the resolvent operators for Eq. (2)(2) $\left\{\begin{array}{c}\vartheta { }^{\mathrm{\prime }}(t)=\mathrm{A}\vartheta (t)+{\int }_0^t\mathcal{G}(t-s)\vartheta (s)ds\mathrm{f}\mathrm{o}\mathrm{r}t\in [0,T],\\ \vartheta (0)={\vartheta }_0\in {\mathbf{X}}_1.\end{array}\right.$(2) .

Theorem 2.7 (Grimmer, 1982) Assume that $({\mathbf{R}}_1)$ and $({\mathbf{R}}_2)$ hold. Then, there exists a unique resolvent operator of Eq. (2).

For more details on the theory of resolvent operators, one can see, (Desch et al., 1984; Grimmer, 1982).

Lemma 2.8 (Desch et al., 1984) Let assumptions $({\mathbf{R}}_1)$ and $({\mathbf{R}}_2)$ hold. The corresponding resolvent operator $\mathrm{R}(\cdot )$ is compact for $t>0$ if and only if the $C_0$-semigroup $S(\cdot )$ is compact for $t>0$.

Lemma 2.9 (Liang et al., 2008) Let the assumptions $({\mathbf{R}}_1)$ and $({\mathbf{R}}_2)$ be satisfied. Then there exists a constant $L_0=L_0(T)$ such that

$\parallel \mathrm{R}(t+\mathrm{\ell })-\mathrm{R}(\mathrm{\ell })\mathrm{R}(t){\parallel }_{\mathcal{L}(\mathrm{H})}\le L_0\mathrm{\ell },\mathrm{\forall }0\le \mathrm{\ell }\le t\le T.$

Motivated by Grimmer (Grimmer, 1982), we present the concept of mild solutions for the non-instantaneous impulses, neutral stochastic integrodifferential equation(1).

Definition 2.4 A stochastic process $\vartheta :(-\mathrm{\infty },T]\to \mathrm{H}$ is said to be a mild solution to the problem (1) if

1. $\vartheta (t)$ is measurable and ${\mathcal{F}}_t$-adapted for each $t\ge 0$.

2. $\vartheta (t)$ has Cadlag paths on $t\in [0,T]$, ${\vartheta }_0=\varphi \in \mathcal{B}$ satisfying ${\vartheta }_0\in {\mathcal{L}}_2^0(\mathrm{\Omega },\mathrm{H})$, $v{|}_{[0,T]}\in \mathrm{P}\mathrm{C}([0,T],\mathrm{H})$ such that $\vartheta$ satisfies the following integral equation

(3) $\vartheta (t)=\left\{\begin{array}{cc}\varphi (t),& t\in (-\mathrm{\infty },0];\\ \mathrm{R}(t)\left[\varphi (0)-\mathbb{D}(0,\varphi (0))\right]+\mathbb{D}(t,{\vartheta }_{\rho (t,{\vartheta }_t)})+{\int }_0^t\mathrm{R}(t-s)\mathbb{F}(s,{\vartheta }_{\rho (s,{\vartheta }_s)})ds& \hspace{0.17em}\\ +{\int }_0^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}(s,\varsigma ,{\vartheta }_{\rho (\varsigma ,{\vartheta }_{\varsigma })})dW(\varsigma )\right]ds,& t\in [0,t_1];\\ {\mathbb{S}}_i(t,{\vartheta }_{\rho (t,{\vartheta }_t)}),& t\in \bigcup _{i=1}^n(t_i,s_i];\\ \mathrm{R}(t-s_i)\left[{\mathbb{S}}_i(s_i,{\vartheta }_{\rho (s_i,{\vartheta }_{s_i})})-\mathbb{D}(s_i,{\vartheta }_{\rho (s_i,{\vartheta }_{s_i})})\right]+\mathbb{D}(t,{\vartheta }_{\rho (t,{\vartheta }_t)})& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\mathbb{F}(s,{\vartheta }_{\rho (s,{\vartheta }_s)})ds& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}(s,\varsigma ,{\vartheta }_{\rho (\varsigma ,{\vartheta }_{\varsigma })})dW(\varsigma )\right]ds,& t\in \bigcup _{i=1}^n(s_i,t_{i+1}].\end{array}\right.$(3)

3 Main results

Consider the space ${\mathcal{B}}_T$ of all functions $\vartheta :(-\mathrm{\infty },T]\to \mathrm{H}$ such that ${\vartheta }_0\in \mathcal{B}$ and $\vartheta {|}_{[0,T]}\in \mathrm{P}\mathrm{C}([0,T],\mathrm{H})$ with seminorm $\parallel .{\parallel }_T$ defined by

$\parallel \vartheta {\parallel }_T=\parallel {\vartheta }_0{\parallel }_{\mathcal{B}}+\underset{s\in [0,T]}{sup}{\left(\mathbb{E}(\parallel \vartheta (s){\parallel }^2)\right)}^{\frac{1}{2}},\vartheta \in {\mathcal{B}}_T.$

Now consider the operator $\mathrm{\Psi }:{\mathcal{B}}_T\to {\mathcal{B}}_T$ defined by

(4) $(\mathrm{\Psi }\vartheta )(t)=\left\{\begin{array}{cc}\varphi (t),& t\in (-\mathrm{\infty },0];\\ \mathrm{R}(t)\left[\varphi (0)-\mathbb{D}(0,\varphi (0))\right]+\mathbb{D}(t,{\vartheta }_{\rho (t,{\vartheta }_t)})+{\int }_0^t\mathrm{R}(t-s)\mathbb{F}(s,{\vartheta }_{\rho (s,{\vartheta }_s)})ds& \hspace{0.17em}\\ +{\int }_0^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}(s,\varsigma ,{\vartheta }_{\rho (\varsigma ,{\vartheta }_{\varsigma })})dW(\varsigma )\right]ds,& t\in [0,t_1];\\ {\mathbb{S}}_i(t,{\vartheta }_{\rho (t,{\vartheta }_t)}),& t\in \bigcup _{i=1}^n(t_i,s_i];\\ \mathrm{R}(t-s_i)\left[{\mathbb{S}}_i(s_i,{\vartheta }_{\rho (s_i,{\vartheta }_{s_i})})-\mathbb{D}(s_i,{\vartheta }_{\rho (s_i,{\vartheta }_{s_i})})\right]+\mathbb{D}(t,{\vartheta }_{\rho (t,{\vartheta }_t)})& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\mathbb{F}(s,{\vartheta }_{\rho (s,{\vartheta }_s)})ds& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}(s,\varsigma ,{\vartheta }_{\rho (\varsigma ,{\vartheta }_{\varsigma })})dW(\varsigma )\right]ds,& t\in \bigcup _{i=1}^n(s_i,t_{i+1}].\end{array}\right.$(4)

Let $\bar{\varphi }(\cdot ):(-\mathrm{\infty },T]\to \mathrm{H}$ be the function defined by

$\bar{\varphi }(t)=\left\{\begin{array}{cc}\varphi (t),& t\in (-\mathrm{\infty },0];\\ \mathrm{R}(t)\varphi (0),& t\in [0,T].\end{array}\right.$

Therefore ${\bar{\varphi }}_0=\varphi$.

If $\vartheta$ satisfies Equation $(3)$, we can decompose $\vartheta (t)=\bar{\varphi }(t)+y(t)$, $t\in (-\mathrm{\infty },T]$ if and only if $y_0=0$ and

(5) $y(t)=\left\{\begin{array}{cc}-\mathrm{R}(t)\mathbb{D}(0,\varphi (0))+\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)& \hspace{0.17em}\\ +{\int }_0^t\mathrm{R}(t-s)\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)ds& \hspace{0.17em}\\ +{\int }_0^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds,& t\in [0,t_1];\\ \hspace{0.17em}& \hspace{0.17em}\\ {\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right),& t\in \bigcup _{i=1}^n(t_i,s_i];\\ \mathrm{R}(t-s_i)[{\mathbb{S}}_i(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})})& \hspace{0.17em}\\ -\mathbb{D}(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})})]+\mathbb{D}(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)})& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\mathbb{F}(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)})ds& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })})dW(\varsigma )\right]ds,& t\in \bigcup _{i=1}^n(s_i,t_{i+1}].\end{array}\right.$(5)

Define ${\mathcal{B}}_T^0=\left\{y\in {\mathcal{B}}_T:y_0=0\in \mathcal{B}\right\}$. For any $y\in {\mathcal{B}}_T^0$,

$\parallel y{\parallel }_T=\parallel y_0{\parallel }_{\mathcal{B}}+\underset{s\in [0,T]}{sup}(\mathbb{E}\left(\parallel y(s){\parallel }^2\right){)}^{1/2}=\underset{s\in [0,T]}{sup}(\mathbb{E}\left(\parallel y(s){\parallel }^2\right){)}^{1/2},y\in {\mathcal{B}}_T.$

It can be shown that $\left({\mathcal{B}}_T^0,\parallel \cdot {\parallel }_{{\mathcal{B}}_T^0}\right)$ is a Banach space. For each $r>0$, set

$B_r=\left\{y\in {\mathcal{B}}_T^0:\mathbb{E}\parallel y{\parallel }^2\le r\right\}$

then for each $r$, $B_r$ is a bounded closed convex set in ${\mathcal{B}}_T^0$.

Next, by using Lemma 2.2 and relation $(a+b{)}^2\le 2a^2+2b^2$, we have

(6) $\begin{array}{rl}& \parallel {\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}{\parallel }_{\mathcal{B}}^2\\ & \le 2\left(\parallel {\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}{\parallel }_{\mathcal{B}}^2+\parallel y_{\rho (t,{\bar{\varphi }}_t+y_t)}{\parallel }_{\mathcal{B}}^2\right)\\ & \le 4\left[{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\mathbb{E}\parallel {\bar{\varphi }}_0{\parallel }_{\mathcal{B}}^2+{\mathcal{N}}_T^2\underset{0\le s\le T}{sup}\mathbb{E}\parallel \bar{\varphi }(s){\parallel }^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\mathbb{E}\parallel y_0{\parallel }_{\mathcal{B}}^2+{\mathcal{N}}_T^2\underset{0\le s\le T}{sup}\mathbb{E}\parallel y(s){\parallel }^2\right]\\ & \le 4\left[{\mathcal{N}}_T^2\underset{0\le s\le T}{sup}\mathbb{E}\parallel y(s){\parallel }^2+{\mathcal{N}}_T^2{\tilde{M}}^2\mathbb{E}\parallel \varphi (0){\parallel }^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]\\ & \le 4\left[{\mathcal{N}}_T^{2}r+\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]=\tilde{r},t\in [0,T].\end{array}$(6)

Now, we define the operator $\mathrm{\Pi }:{\mathcal{B}}_T^0\to {\mathcal{B}}_T^0$ by

(7) $\mathrm{\Pi }(y)(t)=\left\{\begin{array}{cc}-\mathrm{R}(t)\mathbb{D}(0,\varphi (0))+\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)& \hspace{0.17em}\\ +{\int }_0^t\mathrm{R}(t-s)\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)ds& \hspace{0.17em}\\ +{\int }_0^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds,& t\in [0,t_1];\\ {\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right),& t\in \bigcup _{i=1}^n(t_i,s_i];\\ \mathrm{R}(t-s_i)[{\mathbb{S}}_i(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})})& \hspace{0.17em}\\ -\mathbb{D}(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})})]+\mathbb{D}(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)})& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\mathbb{F}(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)})ds& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })})dW(\varsigma )\right]ds,& t\in \bigcup _{i=1}^n(s_i,t_{i+1}].\end{array}\right.$(7)

Throughout the paper, $\tilde{M}=\underset{t\in [0,T]}{sup}\parallel \mathrm{R}(t){\parallel }^2$.

3.1 Existence of mild solution under lipschitz conditions

In the following, we give the first existence result for Eq.(1)(1) $d\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]=\left\{\mathcal{A}\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]+{\displaystyle {\int }_0^t\mathcal{G}\left(t-s\right)\left[\vartheta \left(s\right)-\mathbb{D}\left(s,{\vartheta }_{\rho \left(s,{\vartheta }_s\right)}\right)\right]ds}\right\}dt+\mathbb{F}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)dt+{\displaystyle {\int }_0^t\mathbb{G}\left(t,s,{\vartheta }_{\rho }\left(s,{\vartheta }_s\right)\right)dW(s),t\in {\displaystyle \underset{i=0}{\overset{n}{\cup }}(s_i,t_{i+1}],n}}\in \mathbb{N},\vartheta \left(t\right)={\mathbb{S}}_i\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right),t\in {\displaystyle \underset{i=1}{\overset{n}{\cup }}(t_i,s_i],}\vartheta \left(t\right)=\varphi (t)\in ℬ,t\in (-\infty ,0],$(1) under Lipschitz conditions on nonlinear terms. The proof is based on Banach fixed point theorem under the following assumptions:

(I) The functions $\mathbb{D}:[0,T]\times \mathcal{B}\to \mathrm{H}$, $\mathbb{F}:[0,T]\times \mathcal{B}\to \text{break}\mathrm{H}$, $\mathbb{G}:\mathrm{\Delta }\times \mathcal{B}\to {\mathcal{L}}_2^0(\mathrm{K},\mathrm{H})$ are continuous and there exists constant $M_1,M_2,M_3>0$ such that

$\begin{array}{c}\mathbb{E}\parallel \mathbb{D}(t,v_1)-\mathbb{D}(t,v_2){\parallel }_{\mathrm{H}}^2\le M_1\parallel v_1-v_2{\parallel }_{\mathcal{B}}^2,\\ \mathbb{E}\parallel \mathbb{F}(t,v_1)-\mathbb{F}(t,v_2){\parallel }_{\mathrm{H}}^2\le M_2\parallel v_1-v_2{\parallel }_{\mathcal{B}}^2,\\ {\int }_0^t\mathbb{E}\parallel \mathbb{G}(t,s,v_1)-\mathbb{G}(t,s,v_2){\parallel }_{{\mathcal{L}}_2^0}^{2}ds\le M_3\parallel v_1-v_2{\parallel }_{\mathcal{B}}^2\end{array}$

for all $t\in [0,T]$, and $v_1,v_2\in \mathcal{B}$.

(II) The function ${\mathbb{S}}_i:(t_i,s_i]\times \mathcal{B}\to \mathrm{H}$, $i=1,2,\cdots ,n$ are continuous and there exists $L_i>0$, $i=1,2,\cdots ,n$ such that

$\mathbb{E}\parallel {\mathbb{S}}_i(t,v_1)-{\mathbb{S}}_i(t,v_2){\parallel }_{\mathrm{H}}^2\le L_i\parallel v_1-v_2{\parallel }_{\mathcal{B}}^2$

for all $t\in [0,T]$ and $v_1,v_2\in \mathcal{B}$.

Theorem 3.1 Let ${\vartheta }_0\in {\mathcal{L}}_2^0(\mathrm{\Omega },\mathrm{H})$ and suppose that $({\mathbf{R}}_1)$, $({\mathbf{R}}_2)$, $\mathbf{(}\mathbf{I}\mathbf{)}$ and $\mathbf{(}\mathbf{I}\mathbf{I}\mathbf{)}$ are satisfied. Then there exists a unique mild solution of (1) provided that

(8) ${\mathcal{N}}_T^2\underset{1\le i\le n}{sup}\left[(1+{(\tilde{M})}^2)L_i+2M_1+{(\tilde{M})}^2(M_2+M_3)T\right]<\frac{1}{4}.$(8)

Proof. To prove this result, it is enough to prove that the map $\mathrm{\Pi }$ has a unique fixed point. For $y,z\in {\mathcal{B}}_T^0$ and $t\in [0,t_1]$, we have

(9) $\begin{array}{rl}& \mathbb{E}\parallel (\mathrm{\Pi }y)(t)-(\mathrm{\Pi }z)(t){\parallel }_{\mathrm{H}}^2\\ & \le 3\mathbb{E}{∥\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)-\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+z_t)}+z_{\rho (t,{\bar{\varphi }}_t+z_t)}\right)∥}_{\mathrm{H}}^2\\ & +3\mathbb{E}{∥{\int }_0^t\mathrm{R}(t-s)\left[\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)-\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+z_s)}+z_{\rho (s,{\bar{\varphi }}_s+z_s)}\right)\right]ds∥}_{\mathrm{H}}^2\\ & +3\mathbb{E}∥{\int }_0^t\mathrm{R}(t-s)\left[{\int }_0^s\left[\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)\right.\right.\\ & {\left.\left.-\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+z_{\varsigma })}+z_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+z_{\varsigma })}\right)\right]dW(\varsigma )\right]ds∥}_{\mathrm{H}}^2\\ & \le 3M_1\parallel y_{\rho (t,{\bar{\varphi }}_t+y_t)}-z_{\rho (t,{\bar{\varphi }}_t+z_t)}{\parallel }_{\mathcal{B}}^2+(3{\tilde{M}}^2{M}_2){\int }_0^t\parallel y_{\rho (s,{\bar{\varphi }}_s+y_s)}-z_{\rho (s,{\bar{\varphi }}_s+z_s)}{\parallel }_{\mathcal{B}}^{2}ds\\ & +(3{\tilde{M}}^2{M}_3){\int }_0^t\parallel y_{\rho (s,{\bar{\varphi }}_s+y_s)}-z_{\rho (s,{\bar{\varphi }}_s+z_s)}{\parallel }_{\mathcal{B}}^{2}ds\\ & \le 3\left[M_1+{(\tilde{M})}^2(M_2+M_3)t_1\right]\times {\mathcal{N}}_T^2\mathbb{E}\parallel y(t)-z(t){\parallel }_{\mathrm{H}}^2\\ & \le 4{\mathcal{N}}_T^2\left[M_1+{(\tilde{M})}^2(M_2+M_3)t_1\right]\mathbb{E}\parallel y(t)-z(t){\parallel }_{\mathrm{H}}^2.\end{array}$(9)

For $t\in \bigcup _{i=1}^n(t_i,s_i]$, we have

$\begin{array}{rl}& \mathbb{E}\parallel (\mathrm{\Pi }y)(t)-(\mathrm{\Pi }z)(t){\parallel }_{\mathrm{H}}^2\\ & \le \mathbb{E}\parallel {\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)\\ & -{\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+z_t)}+z_{\rho (t,{\bar{\varphi }}_t+z_t)}\right){\parallel }_{\mathrm{H}}^2\\ & \le L_i\parallel y_{\rho (t,{\bar{\varphi }}_t+y_t)}-z_{\rho (t,{\bar{\varphi }}_t+z_t)}{\parallel }_{\mathcal{B}}^2\\ & \le L_i{\mathcal{N}}_T^2\mathbb{E}\parallel y(t)-z(t){\parallel }_{\mathrm{H}}^2.\end{array}$

Similarly for $t\in \bigcup _{i=1}^n(s_i,t_{i+1}]$, we get

$\mathbb{E}\parallel (\mathrm{\Pi }y)(t)-(\mathrm{\Pi }z)(t){\parallel }_{\mathrm{H}}^2\le 4{\mathcal{N}}_T^2\left[{(\tilde{M})}^2{L}_i+2M_1+{(\tilde{M})}^2(M_2+M_3)(t_{i+1}-s_i)\right]\mathbb{E}\parallel y(t)-z(t){\parallel }_{\mathrm{H}}^2.$

Thus for all $t\in [0,T]$, we obtain that

$\mathbb{E}\parallel (\mathrm{\Pi }y)(t)-(\mathrm{\Pi }z)(t){\parallel }_{\mathrm{H}}^2\le 4{\mathcal{N}}_T^2\underset{1\le i\le n}{sup}\left[(1+{(\tilde{M})}^2)L_i+2M_1+{(\tilde{M})}^2(M_2+M_3)T\right]\mathbb{E}\parallel y(t)-z(t){\parallel }_{\mathrm{H}}^2.$

In accordance with condition (3.1), we can conclude that the $\mathrm{\Pi }$ is a contraction map, and that it has a a unique fixed point $y\in {\mathcal{B}}_T^0$ . Since $v(t)=\bar{\varphi }(t)+y(t)$ for $t\in (-\mathrm{\infty },T]$, therefore $v$ is the unique mild solution of Equation (1).

★

3.2 Existence of mild solution under non-Lipschitz conditions

We establish the existence results for Eq.(1)(1) $d\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]=\left\{\mathcal{A}\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]+{\displaystyle {\int }_0^t\mathcal{G}\left(t-s\right)\left[\vartheta \left(s\right)-\mathbb{D}\left(s,{\vartheta }_{\rho \left(s,{\vartheta }_s\right)}\right)\right]ds}\right\}dt+\mathbb{F}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)dt+{\displaystyle {\int }_0^t\mathbb{G}\left(t,s,{\vartheta }_{\rho }\left(s,{\vartheta }_s\right)\right)dW(s),t\in {\displaystyle \underset{i=0}{\overset{n}{\cup }}(s_i,t_{i+1}],n}}\in \mathbb{N},\vartheta \left(t\right)={\mathbb{S}}_i\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right),t\in {\displaystyle \underset{i=1}{\overset{n}{\cup }}(t_i,s_i],}\vartheta \left(t\right)=\varphi (t)\in ℬ,t\in (-\infty ,0],$(1) under the case that the nonlinear terms satisfies non-Lipschitz continuous conditions. We will employ Darbo and Darbo-Sadovskii’s fixed point theorems. We introduce the following assumptions:

(B₀) The resolvent operator $(\mathrm{R}(t){)}_{t\ge 0}$ compact for $t>0$

(B₁) The function $\mathbb{D}:[0,T]\times \mathcal{B}\to \mathrm{H}$ is continuous and there exists a positive constante $M_1$ such that

$\mathbb{E}\parallel \mathbb{D}(t,v){\parallel }_{\mathrm{H}}^2\le M_1\left(1+\parallel v{\parallel }_{\mathcal{B}}^2\right),$

for all $t\in [0,T]$ and $v\in \mathcal{B}$.

(B₂) The function $\mathbb{F}:[0,T]\times \mathcal{B}\to \mathrm{H}$ satisfies the following condition:

(i) $\mathbb{F}$ verifies the Caratheodory’s condition i.e. $\mathbb{F}(t,\cdot ):\mathcal{B}\to \mathrm{H}$ is continuous for all $t\in [0,T]$ and the function $\mathbb{F}(\cdot ,v):[0,T]\to \mathrm{H}$ is strongly measurable for all $v\in \mathcal{B}$.

(ii) There exists a continuous nondecreasing function $n_{\mathbb{F}}:[0,\mathrm{\infty }]\to (0,\mathrm{\infty })$ and a positive integrable function ${\chi }_{\mathbb{F}}\in {\mathcal{L}}^1([0,T],{\mathbb{R}}^{+})$ such that

$\mathbb{E}\parallel \mathbb{F}(t,v){\parallel }_{\mathrm{H}}^2\le {\chi }_{\mathbb{F}}(t)n_{\mathbb{F}}\left(\parallel v{\parallel }_{\mathcal{B}}^2\right).$

(B₃) The function $\mathbb{G}:\mathrm{\Delta }\times \mathcal{B}\to {\mathcal{L}}_2^0(\mathrm{K},\mathrm{H})$ satisfies the following condition

(i) $\mathbb{G}$ verifies the Caratheodory’s condition i.e. $\mathbb{G}(t,s,\cdot ):\mathcal{B}\to {\mathcal{L}}_2^0(\mathrm{K},\mathrm{H})$ is continuous for all $(t,s)\in \mathrm{\Delta }$ and the function $\mathbb{G}(\cdot ,\cdot ,v):\mathrm{\Delta }\to {\mathcal{L}}_2^0(\mathrm{K},\mathrm{H})$ is strongly measurable for all $v\in \mathcal{B}$.

(ii) There exists a continuous nondecreasing function $n_{\mathbb{G}}:[0,\mathrm{\infty }]\to (0,\mathrm{\infty })$ and a positive integrable function ${\chi }_{\mathbb{G}}\in {\mathcal{L}}^1([0,T],{\mathbb{R}}^{+})$ such that

${\int }_0^t\mathbb{E}\parallel \mathbb{G}(t,s,v){\parallel }_{{\mathcal{L}}_2^0(\mathrm{K},\mathrm{H})}^{2}ds\le {\chi }_{\mathbb{G}}(t)n_{\mathbb{G}}\left(\parallel v{\parallel }_{\mathcal{B}}^2\right).$

(B₄) The function ${\mathbb{S}}_i:(t_i,s_i]\times \mathcal{B}\to \mathrm{H}$, $i=1,2,\cdots ,n$ are continuous and there exists $L_i>0$, $i=1,2,\cdots ,n$ such that

$\mathbb{E}\parallel {\mathbb{S}}_i(t,v){\parallel }_{\mathrm{H}}^2\le L_i\left(1+\parallel v{\parallel }_{\mathcal{B}}^2\right),$

for all $t\in (t_i,s_i]$ and $v\in \mathcal{B}$.

Theorem 3.2 Let ${\vartheta }_0\in {\mathcal{L}}_2^0(\mathrm{\Omega },\mathrm{H})$ and suppose that $({\mathbf{R}}_1)$, $({\mathbf{R}}_2)$, $\mathbf{(}\mathbf{I}\mathbf{)}$-$\mathbf{(}\mathbf{I}\mathbf{I}\mathbf{)}$ and $\mathbf{(}{\mathbf{B}}_{\mathbf{0}}\mathbf{)}$-$\mathbf{(}{\mathbf{B}}_{\mathbf{4}}\mathbf{)}$ are satisfied. If ${\int }_1^{\mathrm{\infty }}\frac{ds}{n_{\mathbb{F}}(s)+n_{\mathbb{G}}(s)}=\mathrm{\infty }$ and

(10) $4{\mathcal{N}}_T^2\underset{1\le i\le n}{max}\left[(1+8{\tilde{M}}^2)L_i+4\left(1+2{\tilde{M}}^2\right)M_1\right]<1.$(10)

Then Equation (1)(1) $d\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]=\left\{\mathcal{A}\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]+{\displaystyle {\int }_0^t\mathcal{G}\left(t-s\right)\left[\vartheta \left(s\right)-\mathbb{D}\left(s,{\vartheta }_{\rho \left(s,{\vartheta }_s\right)}\right)\right]ds}\right\}dt+\mathbb{F}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)dt+{\displaystyle {\int }_0^t\mathbb{G}\left(t,s,{\vartheta }_{\rho }\left(s,{\vartheta }_s\right)\right)dW(s),t\in {\displaystyle \underset{i=0}{\overset{n}{\cup }}(s_i,t_{i+1}],n}}\in \mathbb{N},\vartheta \left(t\right)={\mathbb{S}}_i\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right),t\in {\displaystyle \underset{i=1}{\overset{n}{\cup }}(t_i,s_i],}\vartheta \left(t\right)=\varphi (t)\in ℬ,t\in (-\infty ,0],$(1) admits a mild solution on $[0,T]$.

Proof. The following are steps we take to verify the outcome.

Step 1. the set $\left\{y\in \mathrm{P}\mathrm{C}:y=\alpha \mathrm{\Pi }y\mathbf{f}\mathbf{o}\mathbf{r}0<\alpha <1\right\}$ is bounded

Let $\bar{y}$ be a solution of $y=\alpha \mathrm{\Pi }y$ for $0<\alpha <1$. We have

(11) $\underset{0\le s\le t}{sup}\parallel {\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+{\bar{y}}_s)}+{\bar{y}}_{\rho (s,{\bar{\varphi }}_s+{\bar{y}}_s)}\parallel {}_{\mathcal{B}}^2\le 4[{\mathcal{N}}_T^2\mathbb{E}\parallel \bar{y}{\parallel }_t^2+\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2]$(11)

where

$\parallel \bar{y}{\parallel }_t^2=\underset{0\le s\le t}{sup}\parallel \bar{y}(s){\parallel }_{\mathrm{H}}^2.$

Take

(12) $q(t)=4\left[{\mathcal{N}}_T^2\mathbb{E}\parallel \bar{y}{\parallel }_t^2+\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right].$(12)

For $t\in [0,t_1]$, using Lemma 2.3, we have that

$\mathbb{E}\parallel \overline{y}(t){\parallel }_{\text{H}}^2\le 4\mathbb{E}\text{R}(t)\mathbb{D}(0,\varphi (0))2+\mathbb{E}\mathbb{D}\left(t,{\overline{\varphi }}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}+{\overline{y}}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}\right)2+\mathbb{E}{\Vert {\displaystyle {\int }_0^{t}1pt}\text{R}(t-s)F\left(s,{\overline{\varphi }}_{\rho (s,{\overline{\varphi }}_s+{\overline{y}}_s)}+{\overline{y}}_{\rho (s,{\overline{\varphi }}_s+{\overline{y}}_s)}\right)1ptds\Vert }^2+\mathbb{E}{\Vert {\displaystyle {\int }_0^{t}1pt}\text{R}(t-s)\left[{\displaystyle {\int }_0^s\mathbb{G}}\left(s,\varsigma ,{\overline{\varphi }}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+{\overline{y}}_{\varsigma })}+{\overline{y}}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+{\overline{y}}_{\varsigma })}\right)dW(\varsigma )\right]1ptds\Vert }^2\le 4\{M^2\mathbb{E}\parallel \mathbb{D}(0,\varphi (0)){\parallel }^2+\mathbb{E}{\Vert \mathbb{D}\left(t,{\overline{\varphi }}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}+{\overline{y}}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}\right)\Vert }^2+M^{2}1pt{t}_{1}1pt{\displaystyle {\int }_0^t\mathbb{E}}s,{\overline{\varphi }}_{\rho (s,{\overline{\varphi }}_s+{\overline{y}}_s)}+\overline{y}\rho (s,{\overline{\varphi }}_s+{\overline{y}}_s)21ptds+M^{2}1pt{t}_{1}1pt{\displaystyle {\int }_0^t\mathbb{E}}{\Vert {\displaystyle {\int }_0^s\mathbb{G}}\left(s,\varsigma ,{\overline{\varphi }}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+{\overline{y}}_{\varsigma })}+{\overline{y}}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+{\overline{y}}_{\varsigma })}\right)dW(\varsigma )\Vert }_{{ℒ}_2^0}^{2}1ptds\}\le 4\{M^2{M}_1(1+\parallel \varphi {\parallel }_{ℬ}^2)+M_1\left(1+{\Vert {\overline{\varphi }}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}+{\overline{y}}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}\Vert }_{ℬ}^2\right)+M^{2}1ptT{\displaystyle {\int }_0^{t}1pt}{\chi }_{\mathbb{F}}(s)1pt{n}_{\mathbb{F}}(q(s))1ptds+M^{2}1ptT\int 0t{\displaystyle {\int }_0^{s}1pt}\mathbb{E}\mathbb{G}\left(s,\varsigma ,{\overline{\varphi }}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+{\overline{y}}_{\varsigma })}+{\overline{y}}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+{\overline{y}}_{\varsigma })}\right){ℒ}_2^{0}21ptd\varsigma 1ptds\}\le 4\{M^2{M}_1(1+\parallel \varphi {\parallel }_{ℬ}^2)+M_1\left(1+{\Vert {\overline{\varphi }}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}+{\overline{y}}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}\Vert }_{ℬ}^2\right)+M^{2}1ptT{\chi }_{\mathbb{F}}(t)1pt{n}_{\mathbb{F}}\left({\Vert {\overline{\varphi }}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}+{\overline{y}}_{\rho (t,{\overline{\varphi }}_t+{\overline{y}}_t)}\Vert }_{ℬ}^2\right)+M^{2}1ptT{\displaystyle {\int }_0^{t}1pt}{\chi }_{\mathbb{G}}(s)1pt{n}_{\mathbb{G}}(q(s))1ptds\}.$

For $t\in \bigcup _{i=1}^n(t_i,s_i]$, we have

$\mathbb{E}\parallel \bar{y}(t){\parallel }^2\le L_i\left(1+{∥{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+{\bar{y}}_t)}+{\bar{y}}_{\rho (t,{\bar{\varphi }}_t+{\bar{y}}_t)}∥}_{\mathcal{B}}^2\right)\le L_i(1+q(t)).$

Similarly for $t\in \bigcup _{i=1}^n(s_i,t_{i+1}]$, we have

$\begin{array}{rl}& \mathbb{E}\parallel \bar{y}(t){\parallel }^2\\ & \le 4{\left\{\mathbb{E}∥\mathrm{R}(t-s_i)\left[{\mathbb{S}}_i\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+{\bar{y}}_{s_i})}+{\bar{y}}_{\rho (s_i,{\bar{\varphi }}_{s_i}+{\bar{y}}_{s_i})}\right)-\mathbb{D}\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+{\bar{y}}_{s_i})}+{\bar{y}}_{\rho (s_i,{\bar{\varphi }}_{s_i}+{\bar{y}}_{s_i})}\right)\right]∥\right.}^2\\ & +\mathbb{E}{∥\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+{\bar{y}}_t)}+{\bar{y}}_{\rho (t,{\bar{\varphi }}_t+{\bar{y}}_t)}\right)∥}^2\\ & +\mathbb{E}{∥{\int }_{s_i}^t\mathrm{R}(t-s)\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+{\bar{y}}_s)}+{\bar{y}}_{\rho (s,{\bar{\varphi }}_s+{\bar{y}}_s)}\right)ds∥}^2\\ & \left.+\mathbb{E}{∥{\int }_{s_i}^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+{\bar{y}}_{\varsigma })}+{\bar{y}}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+{\bar{y}}_{\varsigma })}\right)dW(\varsigma )\right]ds∥}^2\right\}\end{array}$

$\begin{array}{c}\le 8{\tilde{M}}^2[L_i(1+q(s_i))+M_1(1+q(s_i))]+4M_1(1+q(t))\\ +4{\tilde{M}}^2(t_{i+1}-s_i){\int }_{s_i}^t{\chi }_{\mathbb{F}}(s)n_{\mathbb{F}}(q(s))ds\\ +4{\tilde{M}}^2(t_{i+1}-s_i){\int }_0^t{\chi }_{\mathbb{G}}(s)n_{\mathbb{G}}(q(s))ds.\end{array}$

Therefore for all $t\in [0,T]$, we obtain that

$\begin{array}{rl}& \mathbb{E}\parallel \bar{y}(t){\parallel }^2\le C^{\ast }+L_{i}q(t)+8{\tilde{M}}^2\left[L_{i}q(t)+M_{1}q(t)\right]+4M_{1}q(t)\\ & +4{\tilde{M}}^{2}T{\int }_0^t{\chi }_{\mathbb{F}}(s)n_{\mathbb{F}}(q(s))ds+4{\tilde{M}}^{2}T{\int }_0^t{\chi }_{\mathbb{G}}(s)n_{\mathbb{G}}(q(s))ds,\end{array}$

where

$C^{\ast }=4{\tilde{M}}^2{M}_1\left(1+\parallel \varphi {\parallel }_{\mathcal{B}}^2\right)+(1+8{\tilde{M}}^2)L_i+4\left(2{\tilde{M}}^2+1\right)M_1.$

Putting these values in Equation (12)(12) $q(t)=4\left[{\mathcal{N}}_T^2\mathbb{E}\parallel \bar{y}{\parallel }_t^2+\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right].$(12) , we get

$\begin{array}{rl}& q(t)\le 4\left[{\mathcal{N}}_T^2{C}^{\ast }+\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]\\ & +4{\mathcal{N}}_T^2\left[(1+8{\tilde{M}}^2)L_i+4\left((1+2{\tilde{M}}^2)\right)M_1\right]q(t)\\ & +16{\tilde{M}}^2{\mathcal{N}}_T^{2}T{\int }_0^t{\chi }_{\mathbb{F}}(s)n_{\mathbb{F}}(q(s))ds+16{\tilde{M}}^2{\mathcal{N}}_T^{2}T{\int }_0^t{\chi }_{\mathbb{G}}(s)n_{\mathbb{G}}(q(s))ds.\end{array}$

Since

${\alpha }_1=4{\mathcal{N}}_T^2\underset{1\le i\le N}{max}\left[(1+8{\tilde{M}}^2)L_i+4\left((1+2{\tilde{M}}^2)\right)M_1\right]<1,$

we have

$q(t)\le \frac{{\alpha }_2}{1-{\alpha }_1}+\frac{{\alpha }_2}{1-{\alpha }_1}{\int }_0^t{\chi }_{\mathbb{F}}(s)n_{\mathbb{F}}(q(s))ds+\frac{{\alpha }_3}{1-{\alpha }_1}{\int }_0^t{\chi }_{\mathbb{G}}(s)n_{\mathbb{G}}(q(s))ds$

where ${\alpha }_2=4\left({\mathcal{N}}_T^2{C}^{\ast }+\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right)$, and ${\alpha }_3=16{\tilde{M}}^2{\mathcal{N}}_T^{2}T$.

Let $\alpha (t)=\frac{{\alpha }_2}{1-{\alpha }_1}+\frac{red{\alpha }_2}{1-{\alpha }_1}{\int }_0^t{\chi }_{\mathbb{F}}(s)n_{\mathbb{F}}(q(s))ds+\frac{{\alpha }_3}{1-{\alpha }_1}{\int }_0^t{\chi }_{\mathbb{G}}(s)n_{\mathbb{G}}(q(s))ds$

with $\alpha (0)=\frac{{\alpha }_2}{1-{\alpha }_1}$. Therefore we get

$\begin{array}{rl}& \alpha { }^{\mathrm{\prime }}(t)\le \frac{{\alpha }_3}{1-{\alpha }_1}\left[{\chi }_{\mathbb{F}}(t)n_{\mathbb{F}}(q(t))+{\chi }_{\mathbb{G}}(t)n_{\mathbb{G}}(q(t))\right]\\ & \le \frac{{\alpha }_3}{1-{\alpha }_1}max\left\{{\chi }_{\mathbb{F}}(t),{\chi }_{\mathbb{G}}(t)\right\}\times \left[n_{\mathbb{F}}(q(t))+n_{\mathbb{G}}(q(t))\right]\end{array}$

and from which we derive that

${\int }_{\alpha (0)}^{\alpha (t)}\frac{du}{n_{\mathbb{F}}(u)+n_{\mathbb{G}}(u)}\le {\int }_0^T\frac{{\alpha }_3}{1-{\alpha }_1}max\left\{{\chi }_{\mathbb{F}}(t),{\chi }_{\mathbb{G}}(t)\right\}dt<\mathrm{\infty }.$

This provides that the function $\alpha (t)$ are bounded on $[0,T]$. Therefore, the functions $q(t)$ are bounded and $y(\cdot )$ are bounded on $[0,T]$.

Step 2. $\mathrm{\Pi }:{\mathbb{B}}_T^0\to {\mathbb{B}}_T^0$ is continuous

Let $\{y^{(k)}{\}}_{r\in \mathbb{N}}\subset {\mathcal{B}}_T^0$ be a sequence such that $\{y^{(k)}\}\to y\in {\mathcal{B}}_T^0$ as $k\to \mathrm{\infty }$. Then there exists a number ${\mathrm{\ell }}_0>0$ such that $\parallel y^{(k)}(t)\parallel \le {\mathrm{\ell }}_0$ for all $k$ and a.s. $t\in [0,T]$. Also using Equation (6)(6) $\begin{array}{rl}& \parallel {\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}{\parallel }_{\mathcal{B}}^2\\ & \le 2\left(\parallel {\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}{\parallel }_{\mathcal{B}}^2+\parallel y_{\rho (t,{\bar{\varphi }}_t+y_t)}{\parallel }_{\mathcal{B}}^2\right)\\ & \le 4\left[{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\mathbb{E}\parallel {\bar{\varphi }}_0{\parallel }_{\mathcal{B}}^2+{\mathcal{N}}_T^2\underset{0\le s\le T}{sup}\mathbb{E}\parallel \bar{\varphi }(s){\parallel }^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\mathbb{E}\parallel y_0{\parallel }_{\mathcal{B}}^2+{\mathcal{N}}_T^2\underset{0\le s\le T}{sup}\mathbb{E}\parallel y(s){\parallel }^2\right]\\ & \le 4\left[{\mathcal{N}}_T^2\underset{0\le s\le T}{sup}\mathbb{E}\parallel y(s){\parallel }^2+{\mathcal{N}}_T^2{\tilde{M}}^2\mathbb{E}\parallel \varphi (0){\parallel }^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]\\ & \le 4\left[{\mathcal{N}}_T^{2}r+\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]=\tilde{r},t\in [0,T].\end{array}$(6) , we have

$\parallel {\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}{\parallel }_{\mathcal{B}}^2\le \tilde{r}.$

Now using Lemma 2.1, we obtain that

$\begin{array}{rl}& \parallel y_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}^{(k)}-y_{\rho (t,{\bar{\varphi }}_t+y_t)}{\parallel }_{\mathcal{B}}^2\le 2\left[{(\mathcal{N}(t))}^2\underset{s\in [0,t]}{sup}\left\{\parallel y^{(k)}(s)-y(s){\parallel }^2\right\}+{(\mathcal{H}(t))}^2\parallel y_0^{(k)}-y_0{\parallel }_{\mathcal{B}}^2\right]\\ & =2(\mathcal{N}(t){)}^2\underset{s\in [0,t]}{sup}\left\{\parallel y^{(k)}(s)-y(s){\parallel }^2\right\}\\ & \le 2{\mathcal{N}}_T^2\parallel y^{(k)}-y{\parallel }^2\to 0,\mathrm{a}\mathrm{s}k\to \mathrm{\infty }.\end{array}$

By continuity of functions $\mathbb{F}(t,\cdot )$ for a.e $t\in [0,T]$ and $\mathbb{G}(t,s,\cdot )$ for a.e $(t,s)\in \mathrm{\Delta }$, we get that

$\underset{k\to \mathrm{\infty }}{lim}\mathbb{F}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}+y_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}^{(k)}\right)=\mathbb{F}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right),$

$\underset{k\to \mathrm{\infty }}{lim}\mathbb{G}\left(t,s,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}+y_{\rho (t,s,{\bar{\varphi }}_t+y_t^{(k)})}^{(k)}\right)=\mathbb{G}\left(t,s,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right).$

Thanks to Lebesgue dominated convergence theorem, we have that for $t\in [0,t_1]$

$\begin{array}{rl}& \mathbb{E}\parallel (\mathrm{\Pi }{y}^{(k)})(t)-(\mathrm{\Pi }y)(t){\parallel }^2\\ & \le 3\left\{\mathbb{E}{∥\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}+y_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}^{(k)}\right)-\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)∥}^2\right.\\ & +\mathbb{E}{∥{\int }_0^t\mathrm{R}(t-s)\left[\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s^{(k)})}+y_{\rho (s,{\bar{\varphi }}_s+y_s^{(k)})}^{(k)}\right)-\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)\right]ds∥}^2\\ & +\mathbb{E}∥{\int }_0^t\mathrm{R}(t-s)({\int }_0^s(\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma }^{(k)})}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma }^{(k)})}^{(k)}\right)\\ & \left.-{\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right))dW(\varsigma ))ds∥}^2\right\}\end{array}$

$\begin{array}{rl}& 3\left\{\mathbb{E}{∥\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}+y_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}^{(k)}\right)-\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)∥}^2\right.\\ & +{\tilde{M}}^2{\int }_0^t\mathbb{E}{∥\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s^{(k)})}+y_{\rho (s,{\bar{\varphi }}_s+y_s^{(k)})}^{(k)}\right)-\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)∥}^{2}ds\\ & +{\tilde{M}}^2{\int }_0^t\left({\int }_0^s\mathbb{E}∥\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma }^{(k)})}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma }^{(k)})}^{(k)}\right)\right.\\ & \left.{\left.-\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)\right)∥}_{{\mathcal{L}}_2^0}^{2}d\varsigma ds\right\}\\ & \to 0\mathrm{a}\mathrm{s}k\to \mathrm{\infty }.\end{array}$

For $t\in \bigcup _{i=1}^n(t_i,s_i]$, we have that

$\begin{array}{rl}& \mathbb{E}\parallel (\mathrm{\Pi }{y}^{(k)})(t)-(\mathrm{\Pi }y)(t){\parallel }^2\\ & =\mathbb{E}{∥{\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho }(t,{\bar{\varphi }}_t+y_t^{(k)})+y_{\rho }^{(k)}(t,{\bar{\varphi }}_t+y_t^{(k)})\right)-{\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)∥}^2\\ & \to 0\mathrm{a}\mathrm{s}k\to \mathrm{\infty }.\end{array}$

For $t\in \bigcup _{i=1}^n(s_i,t_{i+1}]$, we also have that

$\begin{array}{rl}& \mathbb{E}{∥(\mathrm{\Pi }{y}^{(k)})(t)-(\mathrm{\Pi }y)(t)∥}^2\\ & \le 4\left\{\mathbb{E}∥\mathrm{R}(t-s_i)\left[{\mathbb{S}}_i\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i}^{(k)})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i}^{(k)})}^{(k)}\right)\right.-{\mathbb{S}}_i\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}\right)\right.\\ & -{\left.\mathbb{D}\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i}^{(k)})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i}^{(k)})}^{(k)}\right)+\mathbb{D}\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}\right)\right]∥}^2\\ & +\mathbb{E}{∥\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}+y_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}^{(k)}\right)-\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)∥}^2\\ & +\mathbb{E}{∥{\int }_{s_i}^t\mathrm{R}(t-s)\left[\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s^{(k)})}+y_{\rho (s,{\bar{\varphi }}_s+y_s^{(k)})}^{(k)}\right)-\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)\right]ds∥}^2\\ & +\mathbb{E}∥{\int }_{s_i}^t\mathrm{R}(t-s)\left({\int }_0^s\left[\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma }^{(k)})}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma }^{(k)})}^{(k)}\right)\right.\right.\\ & \left.-{\left.\left.\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)\right]dW(\varsigma )\right)ds∥}^2\right\}\end{array}$

$\begin{array}{rl}& \le 4{\left|{\tilde{M}}^2\left[\mathbb{E}∥{\mathbb{S}}_i\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i}^{(k)})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i}^{(k)})}^{(k)}\right)-{\mathbb{S}}_i\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}\right)∥\right.\right.}^2\\ & \left.+\mathbb{E}{∥\mathbb{D}\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i}^{(k)})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i}^{(k)})}^{(k)}\right)-\mathbb{D}\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}\right)∥}^2\right]\\ & +\mathbb{E}{∥\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}+y_{\rho (t,{\bar{\varphi }}_t+y_t^{(k)})}^{(k)}\right)-\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)∥}^2\\ & +{\tilde{M}}^2{\int }_{s_i}^t\mathbb{E}{∥\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s^{(k)})}+y_{\rho (s,{\bar{\varphi }}_s+y_s^{(k)})}^{(k)}\right)-\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)∥}^{2}ds\\ & +{\tilde{M}}^2{\int }_{s_i}^t\left({\int }_0^s\mathbb{E}∥\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma }^{(k)})}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma }^{(k)})}^{(k)}\right)\right.\\ & \left.\left.{-\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)∥}_{{\mathcal{L}}_2^0}^{2}d\varsigma \right)ds\right\}\\ & \to 0\mathrm{a}\mathrm{s}k\to \mathrm{\infty }.\end{array}$

It turns out that we can conclude that the nonlinear mapping $\mathrm{\Pi }:{\mathcal{B}}_T^0\to {\mathcal{B}}_T^0$ is continuous.

Step 3. II is a $\beta$-contraction.

To prove this, decompose $\mathrm{\Pi }$ as follows $\mathrm{\Pi }={\mathrm{\Pi }}_1+{\mathrm{\Pi }}_2$ for $t\in [0,T]$, where

$({\mathrm{\Pi }}_{1}y)(t)=\left\{\begin{array}{cc}-\mathrm{R}(t)\mathbb{D}(0,\varphi (0))+\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)& t\in [0,t_1];\\ {\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right),& t\in \bigcup _{i=1}^n(t_i,s_i],\\ \mathrm{R}(t-s_i)[{\mathbb{S}}_i(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})});& \hspace{0.17em}\\ -\mathbb{D}(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})})]+\mathbb{D}(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)})& t\in \bigcup _{i=1}^n(s_i,t_{i+1}];\end{array}\right.$

and$({\mathrm{\Pi }}_{2}y)(t)=\left\{\begin{array}{cc}{\int }_0^t\mathrm{R}(t-s)\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)ds& \hspace{0.17em}\\ +{\int }_0^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds,& t\in [0,t_1];\\ 0,& t\in \bigcup _{i=1}^n(t_i,s_i];\\ {\int }_{s_i}^t\mathrm{R}(t-s)\mathbb{F}(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)})ds& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })})dW(\varsigma )\right]ds,& t\in \bigcup _{i=1}^n(s_i,t_{i+1}].\end{array}\right.$ Claim 1: ${\mathrm{\Pi }}_1$ is Lipschitz continuous.

For $t\in [0,t_1]$, using assumption (A1), Lemmas 2.2 and $u,v\in {\mathcal{B}}_T^0$, we have

$\begin{array}{rl}& \mathbb{E}\parallel ({\mathrm{\Pi }}_{1}u)(t)-({\mathrm{\Pi }}_{1}v)(t){\parallel }^2\le \mathbb{E}{∥\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+u_t)}+u_{\rho (t,{\bar{\varphi }}_t+u_t)}\right)-\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+v_t)}+v_{\rho (t,{\bar{\varphi }}_t+v_t)}\right)∥}^2\\ & \le M_1\parallel u_{\rho (t,{\bar{\varphi }}_t+u_t)}-v_{\rho (t,{\bar{\varphi }}_t+v_t)}{\parallel }^2\\ & \le 2{\mathcal{N}}_T^2{M}_1\underset{t\in [0,t_1]}{sup}\mathbb{E}\parallel u(t)-v(t){\parallel }^2.\end{array}$

For $t\in \bigcup _{i=1}^n(t_i,s_i]$, we have

$\begin{array}{rl}& \mathbb{E}\parallel ({\mathrm{\Pi }}_{1}u)(t)-({\mathrm{\Pi }}_{1}v)(t){\parallel }^2\le \mathbb{E}\parallel {\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+u_t)}+u_{\rho (t,{\bar{\varphi }}_t+u_t)}\right)-{\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+v_t)}+v_{\rho (t,{\bar{\varphi }}_t+v_t)}\right){\parallel }^2\\ & \le L_i\parallel u_{\rho (t,{\bar{\varphi }}_t+u_t)}-v_{\rho (t,{\bar{\varphi }}_t+v_t)}{\parallel }^2\\ & \le {\mathcal{N}}_T^2{L}_i\underset{t\in (t_i,s_i]}{sup}\mathbb{E}\parallel u(t)-v(t){\parallel }^2.\end{array}$

Similarly for $t\in \bigcup _{i=1}^n(s_i,t_{i+1}]$, we have

$\begin{array}{rl}& \mathbb{E}\parallel ({\mathrm{\Pi }}_{1}u)(t)-({\mathrm{\Pi }}_{1}v)(t){\parallel }^2\\ & \le 3\left\{\mathbb{E}{∥\mathrm{R}(t-s_i)\left[{\mathbb{S}}_i\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+u_{s_i})}+u_{\rho (s_i,{\bar{\varphi }}_{s_i}+u_{s_i})}\right)-{\mathbb{S}}_i\left(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+v_{s_i})}+v_{\rho (s_i,{\bar{\varphi }}_{s_i}+v_{s_i})}\right)\right]∥}^2\right.\end{array}$

$\begin{array}{cc}\hspace{0.17em}& +{∥\mathbb{D}(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+u_{s_i})}+u_{\rho (s_i,{\bar{\varphi }}_{s_i}+u_{s_i})})-\mathbb{D}(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+v_{s_i})}+v_{\rho (s_i,{\bar{\varphi }}_{s_i}+v_{s_i})})∥}^2\\ \hspace{0.17em}& \left.+\mathbb{E}{∥\mathbb{D}(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+u_t)}+u_{\rho (t,{\bar{\varphi }}_t+u_t)})-\mathbb{D}(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+v_t)}+v_{\rho (t,{\bar{\varphi }}_t+v_t)})∥}^2\right\}\\ \le & \left.3\left\{{\tilde{M}}^2[L_i\parallel u_{\rho (s_i,{\bar{\varphi }}_{s_i}+u_{s_i})}-v_{\rho (s_i,{\bar{\varphi }}_{s_i}+v_{s_i})}{\parallel }^2+M_1\parallel u_{\rho (s_i,{\bar{\varphi }}_{s_i}+u_{s_i})}-v_{\rho (s_i,{\bar{\varphi }}_{s_i}+v_{s_i})}{\parallel }^2\right.\right]\\ \hspace{0.17em}& \left.+M_1\parallel u_{\rho (t,{\bar{\varphi }}_t+u_t)}-v_{\rho (t,{\bar{\varphi }}_t+v_t)}{\parallel }^2\right\}\\ \le & 3\left\{{\tilde{M}}^2\left[L_i\parallel u_{\rho (s_i,{\bar{\varphi }}_{s_i}+u_{s_i})}-v_{\rho (s_i,{\bar{\varphi }}_{s_i}+v_{s_i})}{\parallel }^2+M_1\parallel u_{\rho (s_i,{\bar{\varphi }}_{s_i}+u_{s_i})}-v_{\rho (s_i,{\bar{\varphi }}_{s_i}+v_{s_i})}{\parallel }^2\right]\right.\\ \hspace{0.17em}& \left.+M_1\parallel u_{\rho (t,{\bar{\varphi }}_t+u_t)}-v_{\rho (t,{\bar{\varphi }}_t+v_t)}{\parallel }^2\right\}\\ \le & 3{\mathcal{N}}_T^2\left\{{\tilde{M}}^2{L}_i+\left(1+{\tilde{M}}^2\right)M_1\}\underset{t\in (s_i,t_{i+1}]}{sup}\mathbb{E}\parallel u(t)-v(t){\parallel }^2\right..\end{array}$

Thus for all $t\in [0,T]$, we have

$\mathbb{E}\parallel ({\mathrm{\Pi }}_{1}u)(t)-({\mathrm{\Pi }}_{1}v)(t){\parallel }^2\le 3{\mathcal{N}}_T^2\left\{(1+{\tilde{M}}^2)L_i+\left(1+{\tilde{M}}^2\right)M_1\right\}\underset{t\in [0,T]}{sup}\mathbb{E}\parallel u(t)-v(t){\parallel }^2.$

Taking the supremum over $t$, we obtain

$\parallel {\mathrm{\Pi }}_{1}u-{\mathrm{\Pi }}_{1}v{\parallel }^2\le \kappa \parallel u-v{\parallel }^2$

where $\kappa =3{\mathcal{N}}_T^2\left\{(1+{\tilde{M}}^2)L_i+\left(1+{\tilde{M}}^2\right)M_1\right\}$. It turns out from (10) that $\kappa <1$ and thus we get ${\mathrm{\Pi }}_1$ is Lipschitz continuous.

Claim 2: $\{{\mathrm{\Pi }}_{2}y,y\in {\mathcal{B}}_T^0\}$ is an equicontinuous family of functions on $[0,T]$.

Let $0<h_1<h_2\le t_1$. For each $y\in {\mathcal{B}}_T^0$, we get that

(13) $\begin{array}{cc}\hspace{0.17em}& \mathbb{E}\parallel ({\mathrm{\Pi }}_{2}y)(h_2)-({\mathrm{\Pi }}_{2}y)(h_1){\parallel }^2\\ \le & 4\left\{\mathbb{E}{∥{\int }_{h_1}^{h_2}\mathrm{R}(h_2-s)\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)ds∥}^2\right.\\ \hspace{0.17em}& +\mathbb{E}{∥{\int }_0^{h_1}\left[\mathrm{R}(h_2-s)-\mathrm{R}(h_1-s)\right]\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)ds∥}^2\\ \hspace{0.17em}& +\mathbb{E}{∥{\int }_{h_1}^{h_2}\mathrm{R}(h_2-s)\left[{\int }_0^s\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds∥}^2\\ \hspace{0.17em}& \left.+\mathbb{E}{∥{\int }_0^{h_1}\left[\mathrm{R}(h_2-s)-\mathrm{R}(h_1-s)\right]\left[{\int }_0^s\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds∥}^2\right\}\\ \le & 4\left\{{\tilde{M}}^2{n}_{\mathbb{F}}(\tilde{r}){\int }_{h_1}^{h_2}{\chi }_{\mathbb{F}}(s)ds+n_{\mathbb{F}}(\tilde{r}){\int }_0^{h_1}\parallel \mathrm{R}(h_2-s)-\mathrm{R}(h_1-s){\parallel }_{\mathcal{L}(\mathrm{H})}^2{\chi }_{\mathbb{F}}(s)ds\right.\\ \hspace{0.17em}& +{\tilde{M}}^2{n}_{\mathbb{G}}(\tilde{r}){\int }_{h_1}^{h_2}{\chi }_{\mathbb{G}}(s)ds+n_{\mathbb{G}}(\tilde{r}){\int }_0^{h_1}\parallel \mathrm{R}(h_2-s)-\mathrm{R}(h_1-s){\parallel }_{\mathcal{L}(\mathrm{H})}^2{\chi }_{\mathbb{G}}(s)ds\end{array}$(13)

Similarly, for any $h_1,h_2\in (s_i,t_{i+1}]$, $h_1<h_2$, $i=1,\cdots ,n$ and each $y\in {\mathcal{B}}_T^0$, we obtain that

(14) $\begin{array}{cc}\hspace{0.17em}& \mathbb{E}\parallel ({\mathrm{\Pi }}_{2}y)(h_2)-({\mathrm{\Pi }}_{2}y)(h_1){\parallel }^2\\ \le & 4\left\{{\tilde{M}}^2{n}_{\mathbb{F}}(\tilde{r}){\int }_{h_1}^{h_2}{\chi }_{\mathbb{F}}(s)ds+n_{\mathbb{F}}(\tilde{r}){\int }_{s_i}^{h_1}\parallel \mathrm{R}(h_2-s)-\mathrm{R}(h_1-s){\parallel }_{\mathcal{L}(\mathrm{H})}^2{\chi }_{\mathbb{F}}(s)ds\right.\\ \hspace{0.17em}& +{\tilde{M}}^2{n}_{\mathbb{G}}(\tilde{r}){\int }_{h_1}^{h_2}{\chi }_{\mathbb{G}}(s)ds+n_{\mathbb{G}}(\tilde{r}){\int }_{s_i}^{h_1}\parallel \mathrm{R}(h_2-s)-\mathrm{R}(h_1-s){\parallel }_{\mathcal{L}(\mathrm{H})}^2{\chi }_{\mathbb{G}}(s)ds.\end{array}$(14)

By the facts that $s\mapsto {\chi }_{\mathbb{F}}(s)$ and $s\mapsto {\chi }_{\mathbb{G}}(s)$ are Lebesgue integrable and the continuity of $(\mathrm{R}(t){)}_{t\ge 0}$ for $t>0$ in the operator norm topology, we derive that the right hand side of (13) and (14) goes to zero as $h_2-h_1\to 0$ independently of $y$. Thus, the set $\{{\mathrm{\Pi }}_{2}y,y\in {\mathcal{B}}_T^0\}$ is equicontinuous.

Claim 3: ${\mathrm{\Pi }}_2$ maps ${\mathcal{B}}_T^0$ onto a precompact set in ${\mathcal{B}}_T^0$.

Let $t\in \bigcup _{i=1}^n(s_i,t_{i+1}]$ be fixed and $\text{isin}$ be a real number satisfying $0\in t$.

Define the operators ${\Pi }_2^{\in },{\tilde{\Pi }}_2^{\in }$ on ${\mathcal{B}}_T^0$ by:

$\begin{array}{c}({\mathrm{\Pi }}_2^{\text{isin}}y)(t)=\mathrm{R}(\text{isin})\underset{0}{\overset{t-\text{isin}}{\int }}\mathrm{R}(t-s-\text{isin})\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)ds\\ +\mathrm{R}(\text{isin})\underset{0}{\overset{t-\text{isin}}{\int }}\mathrm{R}(t-s-\text{isin})\left[\underset{0}{\overset{s}{\int }}\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds\end{array}$

and

$\begin{array}{c}(\stackrel{}{{\mathrm{\Pi }}_2^{\text{isin}}}y)(t)=\underset{0}{\overset{t-\text{isin}}{\int }}\mathrm{R}(t-s)\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)ds\\ +\underset{0}{\overset{t-\text{isin}}{\int }}\mathrm{R}(t-s)\left[\underset{0}{\overset{s}{\int }}\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds.\end{array}$

In view of the compactness of $\mathrm{R}(\text{isin})$, it is clear that the set $\left\{({\mathrm{\Pi }}_2^{\text{isin}}y)(t),y\in {\mathcal{B}}_T^0\right\}$ is precompact for every $\text{isin}\in (0,t)$. Moreover, for each $y\in {\mathcal{B}}_T^0$, by using Lemma 2.9, we have that

$\mathbb{E}\parallel ({\Pi }_2^{\in }1pty)(t)-({\Pi }_2^{\in }y)(t){\parallel }^2\le 2\mathbb{E}\Vert {\displaystyle {\int }_0^{t-\in }\text{R}}(t-s)\mathbb{F}\left(s,{\overline{\varphi }}_{\rho (s,{\overline{\varphi }}_s+y_s)}+y_{\rho (s,{\overline{\varphi }}_s+y_s)}\right)1ptds{-\text{R}(\in ){\displaystyle {\int }_0^{t-\in }\text{R}}(t-s-\in )\mathbb{F}\left(s,{\overline{\varphi }}_{\rho (s,{\overline{\varphi }}_s+y_s)}+y_{\rho (s,{\overline{\varphi }}_s+y_s)}\right)1ptds\Vert }^2+\mathbb{E}\Vert \text{R}(\in ){\displaystyle {\int }_0^{t-\in }\text{R}}(t-s-\in )\left[{\displaystyle {\int }_0^{s}1pt}\mathbb{G}\left(s,\varsigma ,{\overline{\varphi }}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds-{{\displaystyle {\int }_0^{t-\in }\text{R}}(t-s)\left[{\displaystyle {\int }_0^{s}1pt}\mathbb{G}\left(s,\varsigma ,{\overline{\varphi }}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds\Vert }^2\}\le 21pt{n}_{\mathbb{F}}(\tilde{r}){\displaystyle {\int }_0^{t-\in }\parallel }\text{R}(\in )\text{R}(t-s-\in )-\text{R}(t-s){\parallel }_{ℒ(\text{H})}^{2}1pt{\chi }_{\mathbb{F}}(s)1ptds+21pt{n}_{\mathbb{G}}(\tilde{r}){\displaystyle {\int }_0^{t-\in }\parallel }\text{R}(\in )\text{R}(t-s-\in )-\text{R}(t-s){\parallel }_{ℒ(\text{H})}^{2}1pt{\chi }_{\mathbb{G}}(s)1ptds\le 21pt{n}_{\mathbb{F}}(\tilde{r})(L_0\in ){\displaystyle {\int }_0^{t-\in }1pt}{\chi }_{\mathbb{F}}(s)1ptds+21pt{n}_{\mathbb{G}}(\tilde{r})(L_0\in ){\displaystyle {\int }_0^{t-\in }1pt}{\chi }_{\mathbb{G}}(s)1ptds\to 01pt\text{as}1pt\in \to 0.$

Hence, by total boundedness, we derive that the set $\left\{(\stackrel{}{{\mathrm{\Pi }}_2^{\text{isin}}}y)(t),y\in {\mathcal{B}}_T^0\right\}$ is precompact for every $\text{isin}$, $\text{isin}\in (0,t)$. Applying this idea again, we have that

$\mathbb{E}\parallel ({\Pi }_{2}1pty)(t)-({\Pi }_2^{\in }1pty)(t){\parallel }^2\le 2\mathbb{E}\Vert {\displaystyle {\int }_0^t\text{R}}(t-s)\mathbb{F}\left(s,{\overline{\varphi }}_{\rho (s,{\overline{\varphi }}_s+y_s)}+y_{\rho (s,{\overline{\varphi }}_s+y_s)}\right)1ptds{-{\displaystyle {\int }_0^{t-\in }\text{R}}(t-s)\mathbb{F}\left(s,{\overline{\varphi }}_{\rho (s,{\overline{\varphi }}_s+y_s)}+y_{\rho (s,{\overline{\varphi }}_s+y_s)}\right)1ptds\Vert }^2+\mathbb{E}\Vert {\displaystyle {\int }_0^t\text{R}}(t-s)\left[{\displaystyle {\int }_0^{s}1pt}\mathbb{G}\left(s,\varsigma ,{\overline{\varphi }}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds-{{\displaystyle {\int }_0^{t-\in }\text{R}}(t-s)\left[{\displaystyle {\int }_0^{s}1pt}\mathbb{G}\left(s,\varsigma ,{\overline{\varphi }}_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\overline{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds\Vert }^2\}\le 21pt{n}_{\mathbb{F}}(\tilde{r})1pt(M^2){\displaystyle {\int }_{t-\in }^{t}1pt}{\chi }_{\mathbb{F}}(s)1ptds+21pt{n}_{\mathbb{G}}(\tilde{r})(M^2){\displaystyle {\int }_{t-\in }^{t}1pt}{\chi }_{\mathbb{G}}(s)1ptds\to 01pt\text{as}\in \to 0.$

It turns out that there are precompact sets arbitrary close to the set $\{({\mathrm{\Pi }}_{2}y)(t):y\in {\mathcal{B}}_T^0\}$. Hence $\{({\mathrm{\Pi }}_{2}y)(t):y\in {\mathcal{B}}_T^0\}$ is precompact. The case $t\in (0,t_1]$ is similar, so we omit it.

Let $\mathcal{O}$ be an arbitrary bounded subset of ${\mathcal{B}}_T^0$. Then using precompactness of ${\mathrm{\Pi }}_2$, we obtain that

(15) $\begin{array}{rl}& {\beta }_{\mathrm{P}\mathrm{C}}(\mathrm{\Pi }(\mathcal{O}))={\beta }_{\mathrm{P}\mathrm{C}}({\mathrm{\Pi }}_1(\mathcal{O})+{\mathrm{\Pi }}_2(\mathcal{O}))\le {\beta }_{\mathrm{P}\mathrm{C}}({\mathrm{\Pi }}_1(\mathcal{O}))+{\beta }_{\mathrm{P}\mathrm{C}}({\mathrm{\Pi }}_2(\mathcal{O}))\\ & \le {\beta }_{\mathrm{P}\mathrm{C}}({\mathrm{\Pi }}_1(\mathcal{O}))\\ & \le {\kappa }^{\ast }{\beta }_{\mathrm{P}\mathrm{C}}(\mathcal{O}).\end{array}$(15)

Hence $\mathrm{\Pi }$ is $\beta -$ contraction.

Therefore, by the Darbo fixed point theorem, the operator $\mathrm{\Pi }$ has at least one fixed point $y^{\ast }\in {\mathcal{B}}_T^0$. Let $v(t)=\bar{\varphi }(t)+y^{\ast }(t)$, $t\in (-\mathrm{\infty },T]$. Then, $v$ is a fixed point of the operator $\mathrm{\Pi }$, i.e. $v$ is a mild solution of the problem (1).

★

Now, we employ the Darbo-Sadovskii fixed point theorem (Lemma 2.6) to obtain another existence result.

Theorem 3.3 Let ${\vartheta }_0\in {\mathcal{L}}_2^0(\mathrm{\Omega },\mathrm{H})$ and assume that the assumptions $({\mathbf{R}}_1)$, $({\mathbf{R}}_2)$, $\mathbf{(}\mathbf{I}\mathbf{)}-\mathbf{(}\mathbf{I}\mathbf{I}\mathbf{)}$ and $\mathbf{(}{\mathbf{B}}_{\mathbf{0}}\mathbf{)}-\mathbf{(}{\mathbf{B}}_{\mathbf{4}}\mathbf{)}$ are satisfied. If

(16) $\begin{array}{rl}& 4{\mathcal{N}}_T^2\underset{1\le i\le N}{max}\left\{(1+8{\tilde{M}}^2)L_i+4\left(2{\tilde{M}}^2+1\right)M_1+4{\tilde{M}}^{2}T\underset{\varsigma \to \mathrm{\infty }}{limsup}\frac{n_{\mathbb{F}}(\varsigma )}{\varsigma }{\int }_0^T{\chi }_{\mathbb{F}}(s)ds\right.\\ & +\left.4{\tilde{M}}^{2}T\underset{\varsigma \to \mathrm{\infty }}{limsup}\frac{n_{\mathbb{G}}(\varsigma )}{\varsigma }{\int }_0^T{\chi }_{\mathbb{G}}(s)ds\right\}<1.\end{array}$(16)

Then Equation (1)(1) $d\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]=\left\{\mathcal{A}\left[\vartheta \left(t\right)-\mathbb{D}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)\right]+{\displaystyle {\int }_0^t\mathcal{G}\left(t-s\right)\left[\vartheta \left(s\right)-\mathbb{D}\left(s,{\vartheta }_{\rho \left(s,{\vartheta }_s\right)}\right)\right]ds}\right\}dt+\mathbb{F}\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right)dt+{\displaystyle {\int }_0^t\mathbb{G}\left(t,s,{\vartheta }_{\rho }\left(s,{\vartheta }_s\right)\right)dW(s),t\in {\displaystyle \underset{i=0}{\overset{n}{\cup }}(s_i,t_{i+1}],n}}\in \mathbb{N},\vartheta \left(t\right)={\mathbb{S}}_i\left(t,{\vartheta }_{\rho \left(t,{\vartheta }_t\right)}\right),t\in {\displaystyle \underset{i=1}{\overset{n}{\cup }}(t_i,s_i],}\vartheta \left(t\right)=\varphi (t)\in ℬ,t\in (-\infty ,0],$(1) has at least one mild solution on $[0,T]$.

Proof. In view of the proof of Theorem 3.2, it turns out that the nonlinear mapping $\mathrm{\Pi }:{\mathcal{B}}_T^0\to {\mathcal{B}}_T^0$ given by Equ. (7)(7) $\mathrm{\Pi }(y)(t)=\left\{\begin{array}{cc}-\mathrm{R}(t)\mathbb{D}(0,\varphi (0))+\mathbb{D}\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right)& \hspace{0.17em}\\ +{\int }_0^t\mathrm{R}(t-s)\mathbb{F}\left(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)}\right)ds& \hspace{0.17em}\\ +{\int }_0^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}\left(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}\right)dW(\varsigma )\right]ds,& t\in [0,t_1];\\ {\mathbb{S}}_i\left(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)}\right),& t\in \bigcup _{i=1}^n(t_i,s_i];\\ \mathrm{R}(t-s_i)[{\mathbb{S}}_i(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})})& \hspace{0.17em}\\ -\mathbb{D}(s_i,{\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+y_{s_i})})]+\mathbb{D}(t,{\bar{\varphi }}_{\rho (t,{\bar{\varphi }}_t+y_t)}+y_{\rho (t,{\bar{\varphi }}_t+y_t)})& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\mathbb{F}(s,{\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_s)}+y_{\rho (s,{\bar{\varphi }}_s+y_s)})ds& \hspace{0.17em}\\ +{\int }_{s_i}^t\mathrm{R}(t-s)\left[{\int }_0^s\mathbb{G}(s,\varsigma ,{\bar{\varphi }}_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })}+y_{\rho (\varsigma ,{\bar{\varphi }}_{\varsigma }+y_{\varsigma })})dW(\varsigma )\right]ds,& t\in \bigcup _{i=1}^n(s_i,t_{i+1}].\end{array}\right.$(7) is continuous.

In what follows, we demonstrate that there exists $r>0$ such that $\mathrm{\Pi }(B_r)\subset B_r$. We argue by contradiction. Suppose that it is not true, then for any $r>0$, there exist $\tilde{y}\in B_r$ and $\tilde{t}\in [0,T]$ such that $r<\mathbb{E}\parallel (\mathrm{\Pi }\tilde{y})(\tilde{t})$. Therefore

for $\tilde{t}\in [0,t_1]$ and $\tilde{y}\in B_r$, we have that

$\begin{array}{rl}& r<\mathbb{E}\parallel (\mathrm{\Pi }\tilde{y})(\tilde{t})\parallel \le 4\left\{{\tilde{M}}^2{M}_1\left(1+\parallel \varphi {\parallel }_{\mathcal{B}}^2\right)+M_1\left(1+\parallel {\bar{\varphi }}_{\rho (\tilde{t},{\bar{\varphi }}_{\tilde{t}}+{\tilde{y}}_{\tilde{t}})}+y_{\rho (\tilde{t},{\bar{\varphi }}_{\tilde{t}}+{\tilde{y}}_{\tilde{t}})}{\parallel }_{\mathcal{B}}^2\right)\right.\\ & +{\tilde{M}}^2{t}_1{\int }_0^{t_1}{\chi }_{\mathbb{F}}(s)n_{\mathbb{F}}\left(\parallel {\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+{\tilde{y}}_s)}+{\tilde{y}}_{\rho (s,{\bar{\varphi }}_s+{\tilde{y}}_s)}{\parallel }_{\mathcal{B}}^2\right)ds\\ & \left.+{\tilde{M}}^2{t}_1{\int }_0^{t_1}{\chi }_{\mathbb{G}}(s)n_{\mathbb{G}}\left(\parallel {\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+{\tilde{y}}_s)}+{\tilde{y}}_{\rho (s,{\bar{\varphi }}_s+{\tilde{y}}_s)}{\parallel }_{\mathcal{B}}^2\right)ds\right\}.\end{array}$

For $\tilde{t}\in \bigcup _{i=1}^n(t_i,s_i]$ and $\tilde{y}\in B_r$, we have that

$r<\mathbb{E}\parallel (\mathrm{\Pi }\tilde{y})(\tilde{t}){\parallel }^2=L_i\left(1+\parallel {\bar{\varphi }}_{\rho (\tilde{t},{\bar{\varphi }}_{\tilde{t}}+y_{\tilde{t}})}+{\tilde{y}}_{\rho (\tilde{t},{\bar{\varphi }}_{\tilde{t}}+{\tilde{y}}_{\tilde{t}})}{\parallel }_{\mathcal{B}}^2\right).$

Likewise, for $\tilde{t}\in \bigcup _{i=1}^n(s_i,t_{i+1}]$, we get that

$\begin{array}{rl}& r<\mathbb{E}\parallel (\mathrm{\Pi }\tilde{y})(\tilde{t}){\parallel }^2\\ & \le 8{\tilde{M}}^2\left[L_i\left(1+\parallel {\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+{\tilde{y}}_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+{\tilde{y}}_{s_i})}{\parallel }_{\mathcal{B}}^2\right)+M_1\left(1+\parallel {\bar{\varphi }}_{\rho (s_i,{\bar{\varphi }}_{s_i}+{\tilde{y}}_{s_i})}+y_{\rho (s_i,{\bar{\varphi }}_{s_i}+{\tilde{y}}_{s_i})}{\parallel }_{\mathcal{B}}^2\right)\right]\\ & +4M_1\left(1+\parallel {\bar{\varphi }}_{\rho (\tilde{t},{\bar{\varphi }}_{\tilde{t}}+y_{\tilde{t}})}+{\tilde{y}}_{\rho (\tilde{t},{\bar{\varphi }}_{\tilde{t}}+{\tilde{y}}_{\tilde{t}})}{\parallel }_{\mathcal{B}}^2\right)\\ & +{\tilde{M}}^{2}4(t_{i+1}-s_i){\int }_{s_i}^{t_{i+1}}{\chi }_{\mathbb{F}}(s)n_{\mathbb{F}}\left(\parallel {\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_{\tilde{s}})}+{\tilde{y}}_{\rho (s,{\bar{\varphi }}_s+{\tilde{y}}_s)}{\parallel }_{\mathcal{B}}^2\right)ds\\ & +{\tilde{M}}^{2}4(t_{i+1}-s_i){\int }_{s_i}^{t_{i+1}}{\chi }_{\mathbb{G}}(s)n_{\mathbb{G}}\left(\parallel {\bar{\varphi }}_{\rho (s,{\bar{\varphi }}_s+y_{\tilde{s}})}+{\tilde{y}}_{\rho (s,{\bar{\varphi }}_s+{\tilde{y}}_s)}{\parallel }_{\mathcal{B}}^2\right)ds.\end{array}$

From the above estimations, for all $\tilde{t}\in [0,T]$, we have that

$\begin{array}{rl}& r<\mathbb{E}\parallel (\mathrm{\Pi }\tilde{y})(\tilde{t}){\parallel }^2\\ & \le C^{\ast }+L_i\left[4{\mathcal{N}}_T^{2}r+4\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]\\ & +8{\tilde{M}}^2\left\{L_i\left[4{\mathcal{N}}_T^{2}r+4\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]\right.\\ & \left.+8{\tilde{M}}^2{M}_1\left[4{\mathcal{N}}_T^{2}r+4\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]\right\}\\ & +4M_1\left[4{\mathcal{N}}_T^{2}r+4\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]\\ & +4{\tilde{M}}^{2}T{\int }_0^T{\chi }_{\mathbb{F}}(s)n_{\mathbb{F}}\left(4\left[{\mathcal{N}}_T^{2}r+\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]\right)ds\\ & +4{\tilde{M}}^{2}T{\int }_0^T{\chi }_{\mathbb{G}}(s)n_{\mathbb{G}}\left(4\left[{\mathcal{N}}_T^{2}r+\left({\mathcal{N}}_T^2{\tilde{M}}^2{K}^2+{({\mathcal{H}}_T+{\mathcal{J}}^{\varphi })}^2\right)\parallel \varphi {\parallel }_{\mathcal{B}}^2\right]\right)ds.\end{array}$

By dividing both sides by $r$ and letting $r\to \mathrm{\infty }$, we obtain that

$\begin{array}{rl}& 1<4{\mathcal{N}}_T^2\underset{1\le i\le N}{max}\left\{(1+8{\tilde{M}}^2)L_i+4\left(2{\tilde{M}}^2+1\right)M_1+4{\tilde{M}}^{2}T\underset{\varsigma \to \mathrm{\infty }}{limsup}\frac{n_{\mathbb{F}}(\varsigma )}{\varsigma }{\int }_0^T{\chi }_{\mathbb{F}}(s)ds\right.\\ & \left.+4{\tilde{M}}^{2}T\underset{\varsigma \to \mathrm{\infty }}{limsup}\frac{n_{\mathbb{G}}(\varsigma )}{\varsigma }{\int }_0^T{\chi }_{\mathbb{G}}(s)ds\right\}<1\end{array}$

This is a contradiction, therefore there exists a positive constant $r>0$ such that $\mathrm{\Pi }(B_r)\subset B_r$. Arguing likewise the proof of the Theorem 3.2, one can conclude that the system (1) has a mild solution.

★

4 Illustratives Examples

Example 1. We Consider the following neutral stochastic problem

(17) $\left\{\begin{array}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left[z(t,\zeta )-{\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),\zeta \right)}{25}ds\right]=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[z(t,\zeta )-{\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),\zeta \right)}{25}ds\right]\\ +{\int }_0^{t}a(t-s)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[z(s,\zeta )-{\int }_{-\mathrm{\infty }}^s{e}^{2(r-s)}\frac{z\left(r-\rho (\parallel z(s)\parallel ),\zeta \right)}{25}dr\right]ds\\ +{\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),\zeta \right)}{20}ds\\ +{\int }_0^{t}sin(t-s){\int }_{-\mathrm{\infty }}^s{e}^{2(k-s)}\frac{z\left(z-\rho (\parallel z(s)\parallel ),x\right)}{20}\frac{dw(k)}{dt}ds,(t,\zeta )\in \bigcup _{i=0}^n(s_i,t_{i+1}]\times [0,\pi ]\\ z(t,\zeta )={\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),x\right)}{30}ds,(t,\zeta )\in \bigcup _{i=1}^n(t_i,s_i]\times [0,\pi ]\\ z(t,0)=z(t,\pi )=0,t\ge 0\\ z(t,\zeta )=\varphi (t,\zeta ),t\in (-\mathrm{\infty },0],\zeta \in [0,\pi ],\end{array}\right.$(17)

where, $\varphi \in \mathcal{B}$, $0=t_0=s_0<t_1<s_1<t_2<\cdots <t_n<s_n<t_{n+1}=1$ are fixed real numbers, $w(t)$ is a standard cylindrical wiener process defined on a complete probability $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, $a:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$.

Let $\mathrm{H}={\mathbb{L}}^2(0,\pi )$ with the norm $\parallel \cdot \parallel$. Consider the operator $\mathcal{A}u=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\zeta }^2}$ with the domain

$D(\mathcal{A})=\left\{u\in \mathrm{H}|u\mathrm{a}\mathrm{n}\mathrm{d}\frac{\mathrm{\partial }u}{\mathrm{\partial }\zeta }\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s},\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\zeta }^2}\in \mathrm{H},u(0)=u(\zeta )=0\right\}.$

The spectrum of $\mathcal{A}$ consists of the eigenvalues $-n^2$ for $n\in {\mathbb{N}}^{\star }$, with associated eigenvectors $e_n:=\sqrt{\frac{2}{\pi }}sin(n\tilde{x}),(n=1,2,3,\cdots )$. Furthermore, the set $\{e_n:n\in {\mathbb{N}}^{\star }\}$ is an orthogonal basis in $\mathrm{H}$. Then

$\mathcal{A}\tilde{x}=\sum _{n=1}^{\mathrm{\infty }}-n^2⟨\tilde{x},e_n⟩e_n,\tilde{x}\in \mathrm{H}.$

It is well known that $\mathcal{A}$ is the infinitesimal generator of a strongly continuous semigroup $\{\mathrm{T}(t){\}}_{t\ge 0}$ on $\mathrm{H}$, which is compact and is given by

$\mathrm{T}(t)\tilde{x}=\sum _{n=1}^{\mathrm{\infty }}{e}^{-n^{2}t}⟨\tilde{x},e_n⟩e_n,\tilde{x}\in \mathrm{H}.$

Thus $({\mathbf{R}}_1)$ is true.

Let $\mathcal{G}(t):D(\mathcal{A})\subset \mathrm{H}\to \mathrm{H}$, $t\ge 0$, be the operator defined by

$\mathcal{G}(t)(u)=a(t)\mathcal{A}u,t\ge 0,u\in \mathcal{D}(\mathcal{A}).$

Let $p\in [1,\mathrm{\infty })$, $r\in [0,\mathrm{\infty })$ and $h:(-\mathrm{\infty },-r)\to \mathbb{R}$ be a nonnegative borel measurable function with satisfies such that

(i) $h$ is locally integrable function;

(ii) There exists a nonnegative locally bounded function $\rho$ on $(-\mathrm{\infty },0]$ such that $h(\mathrm{\ell }+\tau )\le \rho (\mathrm{\ell })h(\tau )$ for all $\mathrm{\ell }\le 0$ and $\tau \in (-\mathrm{\infty },-r)/N_{\mathrm{\ell }}$, where $N_{\mathrm{\ell }}\subset (-\mathrm{\infty },-r)$ is a set with Lebesgue measure $0$.

More details on the choice of $h$ can be found in (Hino et al., 1991). Now, let $\mathcal{B}=C_r\times L^2(h,\mathrm{H})$ be the space of all functions $\nu :(-\mathrm{\infty },0]\to X$ such that $\nu {|}_{[-r,0]}\in C\left([-r,0],\mathrm{H}\right)$, $\nu (\cdot )$ is Lebesgue measurable on $(-\mathrm{\infty },-r)$ with seminorm

$\parallel \nu {\parallel }_{\mathcal{B}}=\underset{\theta \in [-r,0]}{sup}\parallel \nu (\theta )\parallel +{\left({\int }_{-\mathrm{\infty }}^{-r}h(\theta )\parallel \nu (\theta ){\parallel }^{p}d\theta \right)}^{\frac{1}{p}}.$

Thanks to [38, Theorem 1.3.8], it turns out that $\mathcal{B}$ is a phase space and for $r=0$ and $p=2$, we derive that$\mathcal{B}=C_0\times L^2(h,\mathrm{H})$ with

$K=1,\mathcal{H}(t)=\rho (-t)and\mathcal{N}(t)=1+{\left({\int }_{-t}^{0}h(\theta )d\theta \right)}^{\frac{1}{2}}fort\ge 0.$

Set $\varphi (\theta )(\zeta )=\varphi (\theta ,\zeta )\in \mathcal{B}$ and define

$\vartheta (t)(\zeta )=z(t,\zeta ),$

$\rho (t,\varphi )(\zeta )=\rho (\parallel \varphi (0)\parallel ),$

$\mathbb{D}(t,\varphi )(\zeta )={\int }_{-\mathrm{\infty }}^0{e}^{2s}\frac{\varphi }{25}ds,$

$\mathbb{F}(t,\varphi )(\zeta )={\int }_{-\mathrm{\infty }}^0{e}^{2s}\frac{\varphi }{20}ds,$

${\int }_0^t\mathbb{G}(t,s,\varphi )(\zeta )ds={\int }_0^{t}sin(t-s){\int }_{-\mathrm{\infty }}^0{e}^{2\tau }\frac{\varphi }{20}d\tau ds,$

${\mathbb{S}}_i(t,\varphi )(\zeta )={\int }_{-\mathrm{\infty }}^0{e}^{2s}\frac{\varphi }{30}ds.$

Then Eq. (17(17) $\left\{\begin{array}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left[z(t,\zeta )-{\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),\zeta \right)}{25}ds\right]=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[z(t,\zeta )-{\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),\zeta \right)}{25}ds\right]\\ +{\int }_0^{t}a(t-s)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[z(s,\zeta )-{\int }_{-\mathrm{\infty }}^s{e}^{2(r-s)}\frac{z\left(r-\rho (\parallel z(s)\parallel ),\zeta \right)}{25}dr\right]ds\\ +{\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),\zeta \right)}{20}ds\\ +{\int }_0^{t}sin(t-s){\int }_{-\mathrm{\infty }}^s{e}^{2(k-s)}\frac{z\left(z-\rho (\parallel z(s)\parallel ),x\right)}{20}\frac{dw(k)}{dt}ds,(t,\zeta )\in \bigcup _{i=0}^n(s_i,t_{i+1}]\times [0,\pi ]\\ z(t,\zeta )={\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),x\right)}{30}ds,(t,\zeta )\in \bigcup _{i=1}^n(t_i,s_i]\times [0,\pi ]\\ z(t,0)=z(t,\pi )=0,t\ge 0\\ z(t,\zeta )=\varphi (t,\zeta ),t\in (-\mathrm{\infty },0],\zeta \in [0,\pi ],\end{array}\right.$(17) ), takes the following abstract form

(18) $\left\{\begin{array}{c}d[\vartheta (t)-\mathbb{D}(t,{\vartheta }_{\rho (t,{\vartheta }_t)})]=\left\{\mathcal{A}[\vartheta (t)-\mathbb{D}(t,{\vartheta }_{\rho (t,{\vartheta }_t)})]+{\int }_0^t\mathcal{G}(t-s)[\vartheta (s)-\mathbb{D}(s,{\vartheta }_{\rho (s,{\vartheta }_s)})]ds\right\}dt\\ +\mathbb{F}(t,{\vartheta }_{\rho (t,{\vartheta }_t)})dt+{\int }_0^t\mathbb{G}(t,s,{\vartheta }_{\rho (s,{\vartheta }_s)})dW(s),t\in \bigcup _{i=0}^n(s_i,t_{i+1}],n\in \mathbb{N},\\ \vartheta (t)={\mathbb{S}}_i(t,{\vartheta }_{\rho (t,{\vartheta }_t)}),t\in \bigcup _{i=1}^n(t_i,s_i],\\ \vartheta (t)=\varphi (t)\in \mathcal{B},t\in (-\mathrm{\infty },0].\end{array}\right.$(18)

Moreover, if $a$ is bounded and $C^1$–function such that $a^{\prime }$ is bounded and uniformly continuous, then $({\mathbf{R}}_2)$ is satisfied and hence, by Theorem 2.7, Eq. (2)(2) $\left\{\begin{array}{c}\vartheta { }^{\mathrm{\prime }}(t)=\mathrm{A}\vartheta (t)+{\int }_0^t\mathcal{G}(t-s)\vartheta (s)ds\mathrm{f}\mathrm{o}\mathrm{r}t\in [0,T],\\ \vartheta (0)={\vartheta }_0\in {\mathbf{X}}_1.\end{array}\right.$(2) has a resolvent operator ${\left(\mathrm{R}(t)\right)}_{t\ge 0}$ on $\mathrm{H}$.

Simple computations yield that the functions $\mathbb{D},\mathbb{F},\mathbb{G}$ and ${\mathbb{S}}_i$ satisfies assumptions $\mathbf{(}\mathbf{I}\mathbf{)}-\mathbf{(}\mathbf{I}\mathbf{I}\mathbf{)}$. Hence, from Theorem 3.1, the Equation (17)(17) $\left\{\begin{array}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left[z(t,\zeta )-{\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),\zeta \right)}{25}ds\right]=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[z(t,\zeta )-{\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),\zeta \right)}{25}ds\right]\\ +{\int }_0^{t}a(t-s)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[z(s,\zeta )-{\int }_{-\mathrm{\infty }}^s{e}^{2(r-s)}\frac{z\left(r-\rho (\parallel z(s)\parallel ),\zeta \right)}{25}dr\right]ds\\ +{\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),\zeta \right)}{20}ds\\ +{\int }_0^{t}sin(t-s){\int }_{-\mathrm{\infty }}^s{e}^{2(k-s)}\frac{z\left(z-\rho (\parallel z(s)\parallel ),x\right)}{20}\frac{dw(k)}{dt}ds,(t,\zeta )\in \bigcup _{i=0}^n(s_i,t_{i+1}]\times [0,\pi ]\\ z(t,\zeta )={\int }_{-\mathrm{\infty }}^t{e}^{2(s-t)}\frac{z\left(s-\rho (\parallel z(t)\parallel ),x\right)}{30}ds,(t,\zeta )\in \bigcup _{i=1}^n(t_i,s_i]\times [0,\pi ]\\ z(t,0)=z(t,\pi )=0,t\ge 0\\ z(t,\zeta )=\varphi (t,\zeta ),t\in (-\mathrm{\infty },0],\zeta \in [0,\pi ],\end{array}\right.$(17) has a unique mild solution on $[0,1]$.

Example 2. Consider the following impulsive noninstantaneous stochastic integro-differential equation:

(19) $\left\{\begin{array}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left[y(t,\zeta )-{\int }_{-\mathrm{\infty }}^t{\mathcal{P}}_1\left(t,s-t,x,y(s,\rho (\parallel y(t)\parallel ,\zeta ))\right)ds\right]=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[y(t,\zeta )\right.\\ \left.-{\int }_{-\mathrm{\infty }}^t{\mathcal{P}}_1\left(t,s-t,x,y(s,\rho (\parallel y(t)\parallel ,\zeta ))\right)ds\right]\\ +{\int }_0^{t}a(t-s)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[y(s,\zeta )-{\int }_{-\mathrm{\infty }}^s{\mathcal{P}}_1\left(s,r-s,x,y(r,\rho (\parallel y(s)\parallel ,\zeta ))\right)dr\right]ds\\ +{\int }_{-\mathrm{\infty }}^t{\mathcal{P}}_2\left(t,s-t,x,y(s,\rho (\parallel y(t)\parallel ,\zeta ))\right)ds\\ +{\int }_0^t{\int }_{-\mathrm{\infty }}^s{\mathcal{P}}_3\left(s,z-s,x,\vartheta (z,\rho (\parallel y(s)\parallel ,\zeta ))\right)\frac{dW(z)}{dt}ds,(t,\zeta )\in \bigcup _{i=0}^n(s_i,t_{i+1}]\times [0,\pi ];\\ y(t,\zeta )={\int }_{-\mathrm{\infty }}^t{\mathcal{P}}_4\left(t,s-t,x,y(s,\rho (\parallel y(t)\parallel ,\zeta ))\right)ds,(t,\zeta )\in \bigcup _{i=1}^n(t_i,s_i]\times [0,\pi ];\\ y(t,0)=y(t,\pi )=0,t\ge 0\\ y(t,\zeta )=\varphi (t,\zeta ),t\in (-\mathrm{\infty },0],\end{array}\right.$(19)

where $\varphi \in \mathcal{B}$, $0=t_0=s_0<t_1<s_1<t_2<\cdots <t_n<s_n<t_{n+1}=1$ are fixed real numbers. The function $\rho$ is a continuous function, $a:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$. Moreover, we consider the functions ${\mathcal{P}}_i:{\mathbb{R}}^4\to \mathbb{R}$, $i=1,2,3,4$ are continuous functions and there exists continuous functions ${\epsilon }_i$, ${\kappa }_i:\mathbb{R}\to \mathbb{R}$, $i=1,2,3,4$ such that

(20) $|{\mathcal{P}}_i(t,s,u,v)|\le {\epsilon }_i(t){\kappa }_i(s)|v|,(t,s,u,v)\in {\mathbb{R}}^4,$(20)

with ${\epsilon }_i={\left({\int }_{-\mathrm{\infty }}^0\frac{{({\kappa }_i(s))}^2}{g(s)}ds\right)}^{\frac{1}{2}}<\mathrm{\infty }$. Define the operator $\mathcal{A}$ and phase space $\mathcal{B}=C_r\times L^2(h,\mathrm{H})$ as in Example 1.

Set $\varphi (\tau )(\zeta )=\varphi (\tau ,\zeta )\in \mathcal{B}$, define

$\vartheta (t)(\zeta )=y(t,\zeta ),$

$\rho (t,\varphi )(\zeta )=\rho (\parallel \varphi (0)\parallel ),$

$\mathbb{D}(t,\nu )(\zeta )={\int }_{-\mathrm{\infty }}^0{\mathcal{P}}_1\left(t,s,x,\nu (s)(\zeta )\right)ds,$

$\mathbb{F}(t,\nu )(\zeta )={\int }_{-\mathrm{\infty }}^0{\mathcal{P}}_2\left(t,s,x,\nu (s)(\zeta )\right)ds,$

${\int }_0^t\mathbb{G}(t,s,\nu )(\zeta )ds={\int }_0^t{\int }_{-\mathrm{\infty }}^0{\mathcal{P}}_3\left(s,z,x,\nu (s)(\zeta )\right)dW(s)ds$

${\mathbb{S}}_i(t,\nu )(\zeta )={\int }_{-\mathrm{\infty }}^0{\mathcal{P}}_4\left(t,s,x,\nu (s)(\zeta )\right)ds.$

Then Equation (19)(19) $\left\{\begin{array}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left[y(t,\zeta )-{\int }_{-\mathrm{\infty }}^t{\mathcal{P}}_1\left(t,s-t,x,y(s,\rho (\parallel y(t)\parallel ,\zeta ))\right)ds\right]=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[y(t,\zeta )\right.\\ \left.-{\int }_{-\mathrm{\infty }}^t{\mathcal{P}}_1\left(t,s-t,x,y(s,\rho (\parallel y(t)\parallel ,\zeta ))\right)ds\right]\\ +{\int }_0^{t}a(t-s)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\zeta }^2}\left[y(s,\zeta )-{\int }_{-\mathrm{\infty }}^s{\mathcal{P}}_1\left(s,r-s,x,y(r,\rho (\parallel y(s)\parallel ,\zeta ))\right)dr\right]ds\\ +{\int }_{-\mathrm{\infty }}^t{\mathcal{P}}_2\left(t,s-t,x,y(s,\rho (\parallel y(t)\parallel ,\zeta ))\right)ds\\ +{\int }_0^t{\int }_{-\mathrm{\infty }}^s{\mathcal{P}}_3\left(s,z-s,x,\vartheta (z,\rho (\parallel y(s)\parallel ,\zeta ))\right)\frac{dW(z)}{dt}ds,(t,\zeta )\in \bigcup _{i=0}^n(s_i,t_{i+1}]\times [0,\pi ];\\ y(t,\zeta )={\int }_{-\mathrm{\infty }}^t{\mathcal{P}}_4\left(t,s-t,x,y(s,\rho (\parallel y(t)\parallel ,\zeta ))\right)ds,(t,\zeta )\in \bigcup _{i=1}^n(t_i,s_i]\times [0,\pi ];\\ y(t,0)=y(t,\pi )=0,t\ge 0\\ y(t,\zeta )=\varphi (t,\zeta ),t\in (-\mathrm{\infty },0],\end{array}\right.$(19) can be written in the form of system (1). By using (20), we obtain

$\begin{array}{rl}& \mathbb{E}\parallel \mathbb{F}(t,\nu ){\parallel }^2=\mathbb{E}{\left[{\left({\int }_0^{\pi }|{\int }_{-\mathrm{\infty }}^0{\mathcal{P}}_2\left(t,s,x,\nu (s)(\zeta )\right)ds{|}^{2}dx\right)}^{\frac{1}{2}}\right]}^2\\ & \le \mathbb{E}{\left[{\left({\int }_0^{\pi }{\left({\int }_{-\mathrm{\infty }}^0{\epsilon }_2(t){\kappa }_2(s)|\nu (s)(\zeta )|ds\right)}^{2}dx\right)}^{\frac{1}{2}}\right]}^2\\ & \le \mathbb{E}{\left[\parallel {\epsilon }_2{\parallel }_{\mathrm{\infty }}{\left({\int }_{-\mathrm{\infty }}^0\frac{{({\kappa }_2(s))}^2}{h(s)}ds\right)}^{\frac{1}{2}}{\left({\int }_{-\mathrm{\infty }}^{0}h(s)\parallel \nu (s){\parallel }^{2}ds\right)}^{\frac{1}{2}}\right]}^2\\ & \le {\chi }_{\mathbb{F}}(t)\parallel \nu {\parallel }_{\mathcal{B}}^2.\end{array}$

for all $(t,\nu )\in [0,1]\times \mathcal{B}$, with ${\chi }_{\mathbb{F}}(t)={\left(\parallel {\epsilon }_2{\parallel }_{\mathrm{\infty }}{\mathrm{\Xi }}_2\right)}^2$. Likewise, one can obtain that

$\begin{array}{c}\mathbb{E}\parallel \mathbb{D}(t,\nu ){\parallel }^2\le M_1\parallel \nu {\parallel }_{\mathcal{B}}^2,\\ \mathbb{E}\parallel \mathbb{G}(t,s,\nu ){\parallel }^2\le {\chi }_{\mathbb{G}}(t)\parallel \nu {\parallel }_{\mathcal{B}}^2,\\ \mathbb{E}\parallel {\mathbb{S}}_i(t,\nu ){\parallel }^2\le L_i\parallel \nu {\parallel }_{\mathcal{B}}^2.\end{array}$

for all $(t,\nu )\in [0,1]\times \mathcal{B}$. Therefore all the assumptions $({\mathbf{A}}_1)-({\mathbf{A}}_2)$ and $({\mathbf{B}}_1)-({\mathbf{B}}_2)$ are satisfied and

$n_{\mathbb{F}}(s)=n_{\mathbb{G}}(s)=s,{\int }_1^{\mathrm{\infty }}\frac{ds}{n_{\mathbb{F}}(s)+n_{\mathbb{G}}(s)}=\mathrm{\infty }.$

In addition, if Equations (10)(10) $4{\mathcal{N}}_T^2\underset{1\le i\le n}{max}\left[(1+8{\tilde{M}}^2)L_i+4\left(1+2{\tilde{M}}^2\right)M_1\right]<1.$(10) or (16) holds, then using Theorem 3.2 or Theorem 3.3, we can say that the problem (19) has a mild solution on $[0,1]$.

5 Conclusion

We have obtained various types of existence results for a class of neutral stochastic integro-differential systems with state-dependent delays and noninstantaneous impulses in Hilbert spaces as a result of applying functional analysis and the stochastic analysis method. In future, we plan to improved system (1) through the use of stochastic processes driven by fractional Brownian motions or G-Brownian motions. These processes can be applied to more complex situations in order to improve the overall performance of the system. We also intend to integrate a numerical treatement with different criteria of our outcomes.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Mamadou Niang

Mamadou Niang and Mohamed Didiya arePh.D. student at the Gaston Berger University, Saint-Louis, sénégal. Their main research area is stochastic (hybrid) non- linear differential equations with delay.

Amadou Diop

Amadou Diop received the Ph.D. degree in applied mathematics from the Gaston Berger University, Saint-Louis, sénégal in August 2020. His research interest include stochastic differential equations.

Mamadou Abdoul Diop

Mamadou Abdoul Diop is presently working as a professor, Department of mathematics, University of Gaston Berger . He has vast experience in research and has done several publications in different peer-reviewed and reputed journal. His research interests include nonlinear analysis, control theory and stochastic differential equations.

Amadou Diop received the Ph.D. degree in applied mathematics from the Gaston Berger University, Saint-Louis, sénégal in August 2020. His research interest include stochastic differential equations.