Existence and approximation of solution to stochastic fractional integro-differential equation with impulsive effects

Abstract

In this paper, we study a stochastic fractional integro-differential equation with impulsive effects in separable Hilbert space. Using a finite dimensional subspace, semigroup theory of linear operators and stochastic version of the well-known Banach fixed point theorem is applied to show the existence and uniqueness of an approximate solution. Next, these approximate solutions are shown to form a Cauchy sequence with respect to an appropriate norm, and the limit of this sequence is then a solution of the original problem. Moreover, the convergence of Faedo–Galerkin approximation of solution is shown. In the last, we have given an example to illustrate the applications of the abstract results.

1 Introduction

Stochastic differential equation is an emerging field drawing attention from both theoretical and applied disciplines. In many real world phenomenon, the deterministic models often fluctuate due to random influences or noise, so we have to move from deterministic models to stochastic models. Stochastic differential equation involves randomness into mathematical description of the phenomenon and thus helps to understand more precise description of it, therefore, these differential equations play an important role in option pricing, forecast of the growth of population, electromagnetic theory, heat conduction in material with memory etc. The theory of stochastic differential equation can also be successfully applied to various problems outside mathematics for example in chemistry, economics, epidemiology, mechanics, finance and several fields in engineering. For recent works on the existence results of mild solutions for stochastic integro-differential equations see [7, 11, 12, 20, 33, 36, 39]. There have been good literature available on this field see [13, 18, 29, 30].

The fractional differential equations are the generalization of ordinary differential equations to arbitrary non integer orders. The fact that the fractional derivative(integral) is an operator which includes integer order derivative(integral) as special case, is the reason why in present fractional differential equations becomes very popular and many applications are available. Fractional differential equations also provide an appropriate tool for the description of hereditary properties of various materials and processes. In recent years, fractional differential equations have drawn attention to many researchers and the solutions of fractional differential equations in analytical and numerical sense have been finding out. Fractional differential equations have lots of applications in the field of fluid flow, viscoelasticity, control theory of dynamical systems, electrical networks, probability and statistics, dynamical processes in self-similar and porous structures, electrochemistry of corrosion, optics and signal processing, nonlinear oscillations of earthquake, rheology, Bio-sciences etc. For more details on fractional differential equations and applications, we refer to books [21, 27, 32, 35] and papers [8,9,10, 14, 25, 39, 40].

On the other hand, the differential equations involving impulse effects arise naturally in the description of phenomena that are subjected to sudden changes in their states, such as population dynamics, biological systems, optimal control, chemotherapeutic treatment in medicine, mechanical systems with impact, financial systems. In these models, the processes are characterized by the fact that they undergo abrupt changes of state at certain moments of time between intervals of continuous evolution. The presence of impulses implies that the trajectories of the system do not necessarily preserve the basic properties of the non-impulsive dynamical systems. For the study of impulsive differential equations, we refer to books [5, 24] and papers [9, 14, 16, 17, 20, 33, 37, 38].

The Faedo–Galerkin approximation provide a better tool for numerical approximation of the equation and to study more regular solutions of the equations by imposing higher consistency on the given data. This method may also be appropriate in the variational formulation to find the solutions of the equations under weaker assumptions on the data. Initially, Segal [34], Murakami [26], Heinz and Von Wahl [19] studied about the existence, uniqueness and finite-time blow-up of solutions to the functional Cauchy problem in a separable Hilbert space. Then using the existence results of Heinz and von Wahl [19], Bazley [1, 2] showed the uniform convergence of the approximations to solutions of the semilinear wave equation on any closed subinterval. Using the idea of Bazley [1, 2], Miletta [28] established the existence of the mild solution and proved convergence results to the functional Cauchy problem by using the Faedo–Galerkin approximations in a separable Hilbert space. Bahuguna and Srivastava [3] extended the results of Miletta [28] and considered the Faedo–Galerkin approximations of the solutions to the functional integro-differential equation. Later on, P. Balasubramaniam [6] studied the Faedo–Galerkin approximate solutions for Stochastic semilinear integro-differential equation in Hilbert Space. Recently, Chadha and Pandey [10] discussed the approximation of the solution for neutral fractional differential equation with nonlocal conditions in an arbitrary separable Hilbert space.

Motivated by the above work, the focus of this investigation is the approximation of mild solutions for the following stochastic fractional integro-differential equation with impulsive effects in separable Hilbert space

$$\begin{aligned} {}^c\mathbf D ^\rho [u(t)+G(t,u_t)]+Au(t)=\,&F(t,u_t,u[b(u(t),t)])\nonumber \\&+\int _0^ta(t-s)k(s,u_s)d\omega (s),\,\, 0<t\leqslant T<\infty , t\ne t_j \end{aligned}$$

(1.1)

$$\begin{aligned} \Delta u|_{t=t_j}=\,&I_j(u_{t_j}), \quad j=1,2,\ldots ,r,\,\, r\in \mathbb {N}, \end{aligned}$$

(1.2)

$$\begin{aligned} u(t)=\,&\phi (t),\quad t\in [-\tau ,0],\,\,\tau >0, \end{aligned}$$

(1.3)

where the state $u(\cdot )$ takes values in a separable Hilbert space $(\mathcal {H},\Vert \cdot \Vert ,(\cdot ))$. $^c\mathbf D ^\rho$ denotes the Caputo fractional derivative of order $\rho$, $0<\rho \leqslant 1$. $0<t_1<t_2<\cdots< t_r<T$ are pre fixed numbers. $\Delta u|_{t=t_j}$ denotes the jump of u(t) at $t=t_j$ i.e. $\Delta u|_{t=t_j}=u(t_j^+)-u(t_j^-)$, where $u(t_j^+)$ and $u(t_j^-)$ represent the right and left limits of u(t) at $t = t_j$ respectively. $A:D(A)\subset \mathcal {H}\rightarrow \mathcal {H}$ is a closed, positive definite, self adjoint linear operator with dense domain D(A) such that $-A$ is the infinitesimal generator of an analytic semigroup $\{S(t):t\geqslant 0\}$ in $\mathcal {H}$. The map a is such that $a^2\in L^p_{loc }(0,\infty )$ for some $1< p < \infty$. F, G, I, b and k are suitably defined functions satisfying certain conditions to be stated later. $u_t$ denotes the function defined by $u_t(\nu )=u(t+\nu )$ for $\nu \in [-\tau ,0]$, here $u_t$ represents the time history of the state from the time $t-\tau$ up to the present time t. Let $(\mathcal {K},\Vert \cdot \Vert ,(\cdot ))$ be another separable Hilbert space. $\{\omega (t): t\geqslant 0\}$ is a given $\mathcal {K}$-valued Wiener process defined on a complete probability space $(\Omega , \mathfrak {I},\{\mathfrak {I}_t\}_{t\geqslant 0},\mathbb {P})$ with a finite trace nuclear covariance operator $Q\geqslant 0$. The initial data $u_0$ is an $\mathfrak {I}_0-$ adapted random variable independent of the winer process.

This manuscript develop a continuation and generalization of the existing results in the literature in two ways. First, we study the Faedo–Galerkin approximate solution to the impulsive stochastic fractional integro-differential equation with impulsive effects, to the best of our knowledge this problem has not been discussed earlier in the literature and second, the results in the manuscript constitute impulsive effects and stochastic invariant of some existing results, for instance,[4, 9, 10, 22, 23], which permit us to introduce the noise as well as impulses in the physical models. Further, in past few years, the integro-differential equation with impulsive effects have emerged as a new area of investigation as it describes a kind of system present in the real world, therefore, the stochastic fractional integro-differential equation with impulsive effects deserves a deep study.

2 Preliminaries and assumptions

Let $(\Omega , \mathfrak {I},\{\mathfrak {I}_t\}_{t\geqslant 0},\mathbb {P})$ be a filtered complete probability space such that the filtration $\{\mathfrak {I}_t\}_{t\geqslant 0}$ is a right continuous increasing family and $\mathfrak {I}_0$ contains all $\mathbb {P}$-null sets. $\omega (t)$ is a $\mathcal {K}$-valued Q-Wiener process with respect to $\{\mathfrak {I}_t\}_{t\geqslant 0}$. A $\mathcal {H}$-valued random variable is a $\mathfrak {I}$-measurable function $u(t): \Omega \rightarrow \mathcal {H}$ and the collection of random variables $S = \{u(t):\Omega \rightarrow \mathcal {H}|t\in [0,T] \}$ is called a stochastic process. We assume that there exists complete orthonormal system $e_k$, $k\geqslant 1$ in $\mathcal {K}$, a bounded sequence of nonnegative real numbers $\lambda _k$ such that $Qe_k =\lambda _k e_k$, $k = 1, 2, \ldots ,$ and a sequence $\{\widehat{B}_k\}$ of real valued mutually independent Brownian motions such that

$$\begin{aligned} (\omega (t),e)_\mathcal {K}=\displaystyle \sum _{k=1}^\infty \sqrt{\lambda _k}( e_k,e)\widehat{B}_k(t),\quad e\in \mathcal {K},\,\, t\geqslant 0. \end{aligned}$$

In order to define stochastic integrals with respect to the Q-Wiener process $\omega (t)$, we introduce the subspace $\mathcal {K}_0=Q^{1/2}(\mathcal {K})$ of $\mathcal {K}$ which is endowed with the inner product $(\widetilde{u}, \widetilde{v})_{\mathcal {K}_0}=(Q^{-1/2}\widetilde{u}, Q^{-1/2}\widetilde{v})$ is a Hilbert space. We assume that $\mathfrak {I}_t=\mathfrak {I}_t^\omega$, where $\mathfrak {I}_t^\omega$ is the $\sigma -$algebra generated by ($\omega (s):0\leqslant s\leqslant t)$. Let $\mathcal {L}_2 (\Omega ,\mathfrak {I},\mathbb {P};\mathcal {H})\equiv \mathcal {L}_2(\Omega ;\mathcal {H})$ denote the Banach space of strongly-measurable, square integrable random variables equipped with norm

$$\begin{aligned} \Vert u\Vert _{\mathcal {L}_2(\Omega , \mathcal {H})}=(\mathbb {E}\Vert u\Vert ^2_\mathcal {H})^{1/2}, \end{aligned}$$

where $\mathbb {E}$ is defined as integration with respect to probability measure $\mathbb {P}$. Let $\mathcal {L}_2^0=\mathcal {L}_2(\mathcal {K}_0, \mathcal {H})$ denote the space of all Hilbert-Schmidt operators from $\mathcal {K}_0$ to $\mathcal {H}$ with the norm

$$\begin{aligned} \Vert \phi \Vert ^2_{\mathcal {L}_2^0}=Tr\left( \left( \phi Q^{1/2}\right) \left( \phi Q^{1/2}\right) ^*\right) , \end{aligned}$$

for $\phi \in \mathcal {L}_2^0$. Clearly, for any bounded linear operators $\phi \in \mathcal {L}(\mathcal {K}, \mathcal {H})$, this norm reduces to

$$\begin{aligned} \Vert \phi \Vert ^2_{\mathcal {L}_2^0}=Tr((\phi Q\phi ^*))=\displaystyle \sum _{k=1}^\infty \Vert \sqrt{\lambda _k}\phi e_k\Vert ^2. \end{aligned}$$

Since −A the infinitesimal generator of an analytic semigroup $\{S(t):t\geqslant 0\}$ in $\mathcal {H}$. Therefore there exists constants $C\geqslant 1$ and $\delta \geqslant 0$ such that $\Vert S(t)\Vert \leqslant Ce^{\delta t}$, $t\geqslant 0$. Moreover

$$\begin{aligned} \left\| \frac{d^m}{dt^m}S(t)\right\| \leqslant C_m,\quad t>0,\,\,m=1,2,\ldots \end{aligned}$$

where $C_m$, $m=1,2,\ldots$ are some positive constants. Hence without loss of generality, we might accept S(t) is uniformly bounded by C i.e. $\Vert S(t)\Vert \leqslant C$ and $0\in \rho (-A)$, the resolvent set of −A. Then for $0<\alpha \leqslant 1$, it is possible to define the fractional power $A^\alpha$ as a closed linear operator on its domain $D(A^\alpha )$, being dense in $\mathcal {H}$ and we denote the Banach space $D(A^\alpha )$ by $\mathcal {H}_\alpha$ endowed with the norm

$$\begin{aligned} \Vert u\Vert _\alpha =\Vert A^\alpha u\Vert ,\quad u\in D(A^\alpha ), \end{aligned}$$

which is equivalent to the graph norm of $A^\alpha$.

Lemma 2.1

([31]) Let −A be the infinitesimal generator of an analytic semigroup $\{S(t): t \geqslant 0\}$ such that $\parallel S(t)\parallel \leqslant C$, for $t \geqslant 0$ and $0 \in \rho (-A)$. Then,

• 1. For $\alpha \in (0,1]$, $D(A^\alpha )$ is a Hilbert space.

• 2. The operator $A^\alpha S(t)$ is bounded for every $t > 0$ and

  $$\begin{aligned}&\Vert A S(t)\Vert \leqslant Ct^{-1},\\&\Vert A^\alpha S(t)\Vert \leqslant \frac{C_\alpha }{ t^{\alpha }}. \end{aligned}$$

For more details on fractional power operators see Pazy [31].

Now, we have some basic definitions of fractional calculus.

Definition 2.2

[35] The fractional integral of order $\rho$ for a function $F\in L^1(\mathbb {R}^+)$ is defined by

$$\begin{aligned} I_{t}^\rho F(t) =\frac{1}{\Gamma {(\rho )}}\int _0^t (t-s)^{\rho -1}F(s)ds,\quad t>0,\quad \rho >0. \end{aligned}$$

Definition 2.3

[21] The Caputo fractional derivative of order $\rho$ for a function $F\in C^{m-1}((0,T);\mathcal {H})\cap \mathcal {L}^1((0,T);\mathcal {H})$ is defined by

$$\begin{aligned} {}^c\mathbf D _t^\rho F(t)=\frac{1}{\Gamma (m-\rho )}\int _0^t(t-s)^{m-\rho -1}F^m(s)ds, \end{aligned}$$

where $m-1<\rho <m$, $\,m=[\rho ]+1$ and $[\rho ]$ denotes the integral part of the real number $\rho$.

Lemma 2.4

[13], For any $r\geqslant 1$ and for arbitrary $\mathcal {L}_0^2$-valued predictable process $h(\cdot )$,

$$\begin{aligned} \displaystyle \sup _{s\in [0,t]}\mathbb {E}\bigg \Vert \int _0^sh(l)d\omega (l)\bigg \Vert ^{2r}\leqslant C_r \bigg (\int _0^t\mathbb {E}\Vert h(s)\Vert ^{2}_{\mathcal {L}_2^0}ds\bigg )^r,\quad \forall \, t\in [0, \infty ). \end{aligned}$$

where $C_r=(r(2r-1))^r$.

Let $C_t^\alpha = PC([-\tau , t],\mathcal {H}_\alpha )$ be the Banach space formed by all strongly measurable $\mathcal {H}_\alpha$-valued stochastic processes on $[-\tau ,t]$ such that u is continuous everywhere except for a finite number of points $t_j$ such that u is left continuous at $t_j$ and the right limit $u(t_j^+)$ exists for $j=1,2,\ldots , r$, endowed with the supremum norm,

$$\begin{aligned} \Vert u\Vert _{t,\alpha }=\displaystyle \left( \sup _{s\in [-\tau ,t]}\mathbb {E}\Vert A^\alpha u(s)\Vert ^2\right) ^{1/2}. \end{aligned}$$

Set $C_t^{\alpha -1} = PC([-\tau , t],\mathcal {H}_{\alpha -1})=\{u\in C_t^\alpha :\mathbb {E}\Vert u(x)-u(y)\Vert ^2_{\alpha -1}\leqslant L|x-y|^2,\forall \,x,y\in [-\tau ,t]\}$, where L is a positive constant and $0<\alpha \leqslant 1$.

In order to prove main results we require the following assumptions:

(H1) :

$A:D(A)\subset \mathcal {H}\rightarrow \mathcal {H}$ is a closed, positive definite, self adjoint linear operator with dense domain D(A) such that A has a pure point spectrum $0 < \lambda _0 \leqslant \lambda _1 \leqslant \cdots$ and a corresponding complete orthonormal system $\{\psi _i\}$ so that

$$\begin{aligned} A\psi _i= \lambda _i\psi _i,\quad (\psi _i,\psi _j)=\delta _{ij}, \end{aligned}$$

where $(\cdot ,\cdot )$ is the inner product in $\mathcal {H}$ and $\delta _{ij}$ is the Kronecker delta function i.e. $-A$ is the infinitesimal generator of an analytic semigroup $\{S(t):t\geqslant 0\}$ in $\mathcal {H}$.

(H2) :

Let $U_1\subset \text{ Dom }(G)$ is an open subset of $[0, T] \times C_0^{\alpha -1}$ and for each $(t, u)\in U_1$ there is a neighborhood $V_1 \subset U_1$. There exist positive constants $0< \alpha< \beta < 1$, such that the function $A^\beta G$ is continuous for $(t, u) \in [0, T] \times C_0^{\alpha -1}$ such that

$$\begin{aligned} \mathbb {E}\Vert A^\beta G(t,u)-A^\beta G(s,v)\Vert ^2&\leqslant L_{G} \left\{ |t-s|^2+\mathbb {E}\Vert u-v\Vert _{0,\alpha -1}^2\right\} ,\\ \mathbb {E}\Vert A^\beta G(t,u)\Vert ^2&\leqslant L_{G}, \end{aligned}$$

where (t, u), $(s,v)\in V_1$ and $L_{G}$ is a positive constant.

(H3) :

Let $U_2\subset \text{ Dom }(F)$ is an open subset of $[0, T] \times C_0^{\alpha }\times \mathcal {H}_{\alpha -1}$ and for each $(t, u, v)\in U_2$ there is a neighborhood $V_2 \subset U_2$. The nonlinear map $F:[0, T] \times C_0^{\alpha }\times \mathcal {H}_{\alpha -1}\rightarrow \mathcal {H}$ satisfies

$$\begin{aligned} \mathbb {E}\Vert F(t,u,\widetilde{u})-F(s,v,\widetilde{v})\Vert ^2\leqslant L_f\left[ |t-s|^{2\gamma _1}+\mathbb {E}\Vert u-v\Vert _{0,\alpha }^2+ \mathbb {E}\Vert \widetilde{u}-\widetilde{v}\Vert _{\alpha -1}^2\right] , \end{aligned}$$

where $0<\gamma _1\leqslant 1$, $(t,u,\widetilde{u})\in V_2$, $(s,v,\widetilde{v})\in V_2$ and $L_f$ is a positive constant.

(H4) :

Let $U_3 \in \text{ Dom }(b)$ is an open subset of $\mathcal {H}_{\alpha }\times [0,T]$ and for each $(u, t)\in U_3$ there is a neighborhood $V_3\subset U_3$. The map $b:\mathcal {H}_{\alpha }\times [0,T]\rightarrow [0,T]$ satisfies

$$\begin{aligned} |b(u,t)-b(v,s)|^2\leqslant L_b\left[ \mathbb {E}\Vert u-v\Vert ^2_\alpha +|t-s|^{2\gamma _2}\right] , \end{aligned}$$

where $0<\gamma _2\leqslant 1$, (u, t), $(v,s)\in V_3$, $b(.,0)=0$ and $L_b$ is a positive constant.

(H5) :

The nonlinear map k is defined from $[0, T]\times C_0^{\alpha }$ into $\mathcal {L}_2^0$ and there exists a nonnegative function $L_k \in L^q_{loc}(0,\infty )$, where $1< q < \infty$, $(1/p)+(1/q)=1$, such that

$$\begin{aligned} \mathbb {E}\Vert k(t, u)-k(t, v)\Vert _{\mathcal {L}_2^0}^2\leqslant L_k\mathbb {E}\Vert u-v\Vert _{0,\alpha }^2, \end{aligned}$$

and

$$\begin{aligned} \mathbb {E}\Vert k(t, u)\Vert _{\mathcal {L}_2^0}^2\leqslant L_k. \end{aligned}$$

(H6) :

The functions $I_j:C_0^{\alpha }\rightarrow \mathcal {H}_{\alpha }$, $j=1,2,\ldots ,r$ are continuous and there exists positive constants $L_j$ such that

$$\begin{aligned} \mathbb {E}\Vert I_j(u)-I_j(v)\Vert ^2_\alpha \leqslant L_j\mathbb {E}\Vert u-v\Vert ^2_{0,\alpha }, \end{aligned}$$

and

$$\begin{aligned} \mathbb {E}\Vert I_j(u)\Vert _{\alpha }^2 \leqslant L_j. \end{aligned}$$

Define the function $\widetilde{\phi }$ by

$$\begin{aligned} \widetilde{\phi }(t)=\left\{ \begin{array}{ll} \phi (t), &{} t\in [-\tau ,0]; \\ \phi (0), &{} t\in [0,T_0]. \end{array} \right. \end{aligned}$$

Definition 2.5

[13] A stochastic process $\{u:[-\tau , T_0]\rightarrow \mathcal {H}_{\alpha }\}$, $0<T_0\leqslant T$ is called a mild solution for the system (1.1)–(1.3) if

• 1. u(t) is measurable, $\mathfrak {I}_t$-adapted and has C$\grave{a}$dl$\grave{a}$g paths on $t\in [-\tau , T_0]$.

• 2. $u(t)\in C^\alpha _{T}\cap C^{\alpha -1}_{T}$ and for every $t>s\geqslant 0$, the function $s\rightarrow A Q_\rho (t-s)G(s, u_s)ds$ is integrable such that u satisfies the following stochastic integral equation

  $$\begin{aligned} u(t)=\left\{ \begin{array}{ll} \widetilde{\phi }(t), &{} t\in [-\tau , 0]; \\ S_\rho (t)[\widetilde{\phi }(0) + G(0, \widetilde{\phi }(0))]-G(t, u_t)+\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AG(s, u_s)ds\\ \quad +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[F(s, u_s, u[b(u(s), s)])+\int _0^sa(s-r) k(r, u_r)d\omega (r)]ds\\ \quad +\displaystyle \sum _{0<t_j<t}S_\rho (t-t_j)I_j(u_{t_j}), &{} t\in [0,T_0]. \end{array} \right. \end{aligned}$$

  with initial value $\widetilde{\phi }(t)\in \mathcal {L}_2^0(\Omega , \mathcal {H}_\alpha )$ for all $t\in [-\tau ,0]$, where

  $$\begin{aligned} S_\rho (t)= & {} \int _0^\infty \zeta _\rho (\theta )S(t^\rho \theta )d\theta , \\ Q_\rho (t)= & {} \rho \int _0^\infty \theta \zeta _\rho (\theta )S(t^\rho \theta )d\theta . \end{aligned}$$

  Here $\zeta _\rho (\theta ) = \frac{1}{\rho }\theta ^{1-\frac{1}{\rho }}\times \psi _ \rho (\theta ^{-\frac{1}{\rho }})$ is a probability density function defined on $(0,\infty )$ i.e., $\zeta _\rho (\theta ) \geqslant 0, \int _0^\infty \zeta _\rho (\theta )d\theta = 1$ and

  $$\begin{aligned} \psi _\rho (\theta ) =\frac{1}{\pi }\displaystyle \sum _{n=1}^\infty (-1)^{n-1}\theta ^{-n\rho -1}\frac{\Gamma (n\rho + 1)}{n!} \sin (n\pi \rho ), 0< \theta < \infty . \end{aligned}$$

For more details see [15, 40].

Lemma 2.6

([40]) The operators $\{S_\rho (t), t \geqslant 0\}$ and $\{Q_\rho (t), t \geqslant 0\}$ are bounded linear operators such that

(i) :

$\Vert S_\rho (t)z\Vert \leqslant C\Vert z\Vert , \Vert Q_\rho (t)z\Vert \leqslant \frac{\rho C}{\Gamma (1+\rho )}\Vert z\Vert$ and $\Vert A^\alpha Q_\rho (t)z\Vert \leqslant \frac{\rho C_\alpha \Gamma (2-\alpha )t^{-\rho \alpha }}{\Gamma (1+\rho (1-\alpha )}\Vert z\Vert$, for any $z \in \mathcal {H}$.

(ii) :

The families $\{S_\rho (t) : t \geqslant 0\}$ and $\{Q_\rho (t) : t \geqslant 0\}$ are strongly continuous.

(iii) :

If S(t) is compact, then $S_\rho (t)$ and $Q_\rho (t)$ are compact operators for any $t > 0$.

3 Existance of approximate solutions

Let $\mathcal {H}_n$ denote the finite dimensional subspace of $\mathcal {H}$ spanned by $\{\psi _0, \psi _1,\ldots , \psi _n\}$ and let $P^n : \mathcal {H}\rightarrow \mathcal {H}_n$ be the corresponding projection operator for $n = 0,1,2,\ldots$.

For $R>0$, set

$$\begin{aligned} \mathcal {B}_R=\left\{ u\in C_{T_0}^\alpha \cap C_{T_0}^{\alpha -1}: \Vert u-\widetilde{\phi }\Vert ^2_{T_0,\alpha }\leqslant R\right\} . \end{aligned}$$

Assumptions (H3)–(H4) and $u\in C_{T_0}^\alpha$ imply that $F(s, u_s, u[b(u(s), s)])$ is continuous on $[0, T_0]$. Therefore there exists a positive constant N such that

$$\begin{aligned} \mathbb {E}\Vert F(s, u_s, u[b(u(s), s)])\Vert ^2\leqslant L_f\left[ T_0^{2\gamma _1}+R(1+LL_b)+LL_bT_0^{2\gamma _2}\right] +N_0=N, \end{aligned}$$

where $N_0=\mathbb {E}\Vert F(0, \widetilde{\phi }(0), \widetilde{\phi }(0))\Vert ^2$. Choose $0<T_0<T$ such that

$$\begin{aligned} M(R) =\,&N+\Vert a^2\Vert _{L^p(0,T_0)}\Vert L_k\Vert _{L^q(0,T_0)}, \end{aligned}$$

(3.1)

$$\begin{aligned} \vartheta :=\,&2\Vert A^{\alpha -\beta -1}\Vert ^2L_G+rC^2\displaystyle \sum _{j=1}^rL_j< 1/12, \end{aligned}$$

(3.2)

$$\begin{aligned} \mathbb {E}\Vert (S_\rho (t)-I)A^\alpha [\widetilde{\phi }(0)+G_n(0, \widetilde{\phi }(0))]\Vert ^2+\mathbb {E}\Vert A^{\alpha -\beta }\Vert ^2L_{G}T_0^2(1+L)\leqslant \frac{R}{12}, \end{aligned}$$

(3.3)

$$\begin{aligned}&\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-(\alpha -\beta ))}{\Gamma (1+\rho (\beta -\alpha ))}\bigg )^2\frac{T_0^{2\rho (\beta -\alpha )-1}}{2\rho (\beta -\alpha )-1}L_{G} \nonumber \\&\qquad +\,\bigg (\frac{\rho C_{\alpha } \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))} \bigg )^2M(R)\frac{T_0^{2\rho (1-\alpha )-1}}{2\rho (1-\alpha )-1} +rC^2\displaystyle \sum _{j=1}^rL_j\leqslant \frac{R}{12}, \end{aligned}$$

(3.4)

$$\begin{aligned}&5\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (\beta -\alpha ))}\bigg )^2 \frac{T_0^{2\rho (\beta -\alpha )-1}}{2\rho (\beta -\alpha )-1}L_{G}\Vert A^{-1}\Vert \nonumber \\&\quad +\,5\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))} \bigg )^2 \frac{T_0^{2\rho (1-\alpha )-1}}{2\rho (1-\alpha )-1}\nonumber \\&\quad (L_f(2+LL_b)+M(R))\leqslant 1-\vartheta <1. \end{aligned}$$

(3.5)

We define

$$\begin{aligned} G_n : [0,T_0]\times C_t^{\alpha -1} \rightarrow \mathcal {H}; \, G_n(t, u_t)= & {} G(t, P^nu_t), \\ F_n : [0,T_0]\times C_t^{\alpha }\times \mathcal {H}_{\alpha -1} \rightarrow \mathcal {H}; \,F_n(t, u_t, u[b(u(t), t)])= & {} F(t, P^nu_t, P^nu[(b(P^nu(t), t)]), \\ k_n : [0,T_0]\times C_t^{\alpha }\rightarrow \mathcal {L}_2^0; k_n(t, u_t)= & {} k(t, P^nu_t), \end{aligned}$$

and

$$\begin{aligned} I_{j,n}:C_0^{\alpha }\rightarrow \mathcal {H}_\alpha ; I_{j,n}(u_t)=I_j(P^nu_t). \end{aligned}$$

Now, consider the map $\Phi _n:\mathcal {B}_R\rightarrow \mathcal {B}_R$ given by

$$\begin{aligned} (\Phi _n u)(t)=\left\{ \begin{array}{ll} \widetilde{\phi }(t), \quad t\in [-\tau , 0]; \\ S_\rho (t)[\widetilde{\phi }(0) + G_n(0, \widetilde{\phi }(0))]-G_n(t, u_t)+\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AG_n(s, u_s)ds\\ \quad +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[F_n(s, u_s, u[b(u(s), s)])+\int _0^sa(s-r) k_n(r, u_r)d\omega (r)]ds\\ \quad +\displaystyle \sum _{0<t_j<t}S_\rho (t-t_j)I_j(u_{t_j}), \quad t\in [0,T_0]. \end{array} \right. \nonumber \\ \end{aligned}$$

(3.6)

Theorem 3.1

Let (H1)–(H6) hold and $\widetilde{\phi }(t)\in \mathcal {L}_2^0(\Omega , \mathcal {H}_\alpha )$ for all $t\in [-\tau ,0]$. Then there exists a unique $u_n\in \mathcal {B}_R$ such that $\Phi _nu_n=u_n$ $\forall \, n=0, 1, 2,\ldots$.

Proof

First we show that $\Phi _nu\in C^{\alpha }_{T_0}\cap C^{\alpha -1}_{T_0}$. Clearly $\Phi _n:C^{\alpha }_{T_0}\rightarrow C^{\alpha }_{T_0}$. Therefore we show that $\Phi _nu\in C^{\alpha -1}_{T_0}$ for any $u\in C^{\alpha -1}_{T_0}$. For $u\in C^{\alpha -1}_{T_0}$ and $0<t'<t''<T_0$, using H$\ddot{o}$lder inequality and inequality $\left( \displaystyle \sum \nolimits _{i=1}^na_i\right) ^m\leqslant n^{m-1}\displaystyle \sum \nolimits _{i=1}^na_i^m$, where $a_i$ are nonnegative constants, we have

$$\begin{aligned}&\mathbb {E}\Vert (\Phi _nu)(t'')-(\Phi _nu)(t')\Vert ^2_{\alpha -1} \nonumber \\&\quad \leqslant 6\bigg \{\Vert (S_\rho (t'')-S_\rho (t'))(\widetilde{\phi }(0)+G_n(0, \widetilde{\phi }(0)))\Vert ^2_{\alpha -1}\nonumber \\&\qquad +\mathbb {E}\Vert A^{\alpha -\beta -1}\Vert ^2\Vert (A^\beta G_n(t'',u_{t''})-A^\beta G_n(t',u_{t'}))\Vert ^2\nonumber \\&\qquad +\int _0^{t'} \Vert A^{\alpha -\beta }((t''-s)^{\rho -1}Q_\rho (t''-s)\nonumber \\&\qquad -(t'-s)^{\rho -1}Q_\rho (t'-s))\Vert ^2\mathbb {E}\Vert A^\beta G_n(s, u_s)\Vert ^2ds\nonumber \\&\qquad +\int _{t'}^{t''}\Vert A^{\alpha -\beta }(t''-s)^{\rho -1}Q_\rho (t''-s)\Vert ^2\mathbb {E}\Vert A^\beta G_n(s, u_s)\Vert ^2ds\nonumber \\&\qquad +\int _0^{t'}\Vert A^{\alpha -1}((t''-s)^{\rho -1}Q_\rho (t''-s)-(t'-s)^{\rho -1}Q_\rho (t'-s))\Vert ^2\nonumber \\&\qquad \quad \bigg [\mathbb {E}\Vert F_n(s,u_s, u[b(u(s), s)])\Vert ^2 +\int _0^s|a(s-r)|^2\mathbb {E}\Vert k_n(r, u_{r})\Vert ^2_{\mathcal {L}_2^0}dr\bigg ]ds\nonumber \\&\qquad +\int _{t'}^{t''}\Vert A^{\alpha -1}(t''-s)^{\rho -1}Q_\rho (t''-s)\Vert ^2\bigg [\mathbb {E}\Vert (F_n(s,u_s, b[h(u(s), s)]))\Vert ^2\nonumber \\&\quad \qquad +\int _0^s|a(s-r)|^2\mathbb {E}\Vert k_n(r, u_{r})\Vert ^2_{\mathcal {L}_2^0}dr\bigg ]ds\nonumber \\&\qquad +r\displaystyle \sum _{j=1}^r\mathbb {E}\Vert (S_\rho (t''-t_j)-S_\rho (t'-t_j))I_{j,n}(u_{t_j})\Vert _{\alpha -1}^2\bigg \}\nonumber \\&\quad \leqslant \displaystyle \sum _{i=1}^7 J_i. \end{aligned}$$

(3.7)

For $u\in \mathcal {H}$, we have

$$\begin{aligned}{}[S((t'')^\rho \theta ) - S((t')^\rho \theta )]u =\int _{t'}^{t''}\frac{d}{dt}S(t^\rho \theta )udt =\int _{t'}^{t''}\rho \theta t^{\rho -1}A S(t^\rho \theta )udt. \end{aligned}$$

Therefore

$$\begin{aligned} J_1&\leqslant \,6\left( \int _{0}^{\infty }\zeta _\rho (\theta )\mathbb {E}\Vert S(t'')^\rho \theta \right) - S(t'^\rho \theta )\Vert \Vert A^{\alpha -1}(\widetilde{\phi }(0)+G_n(0, \widetilde{\phi }(0)))\Vert d\theta )^2\nonumber \\&\leqslant \,6\left( \int _{0}^{\infty }\zeta _\rho (\theta )\left[ \int _{t'}^{t''}\mathbb {E} \Vert \frac{d}{dt}S(t^\rho \theta )\Vert dt\right] \Vert (\widetilde{\phi }(0)+G_n(0, \widetilde{\phi }(0)))\Vert _{\alpha -1}d\theta \right) ^2\nonumber \\&\leqslant M_1(t''-t')^2, \end{aligned}$$

(3.8)

where $M_1=6C_1^2\mathbb {E}\Vert (\widetilde{\phi }(0)+G_n(0, \widetilde{\phi }(0)))\Vert ^2_{\alpha -1}$.

$$\begin{aligned} J_2&\leqslant \,6\Vert A^{\alpha -1-\beta }\Vert ^2L_{G}[|t''-t'|^2+\Vert u_{t''}-u_{t'}\Vert ^2_{0,\alpha -1}]\nonumber \\&\leqslant \,6\Vert A^{\alpha -1-\beta }\Vert ^2L_{G}[|t''-t'|^2+\Vert u(t''+\nu )-u(t'+\nu )\Vert ^2_{\alpha -1}] \quad \forall \, \nu \in [-\tau ,0]\nonumber \\&\leqslant \,6\Vert A^{\alpha -1-\beta }\Vert ^2L_{G}(1+L)|t''-t'|^2\nonumber \\&\leqslant M_2(t''-t')^2, \end{aligned}$$

(3.9)

where $M_2=6\Vert A^{\alpha -1-\beta }\Vert ^2L_{G}(1+L)$.

$$\begin{aligned}&6A\left[ (t''-s)^{\rho -1}S_\rho (t''-s)-(t'-s)^{\rho -1}S_\rho (t'-s))\right] \nonumber \\&\quad =6\int _{0}^{\infty }\zeta _\rho (\theta ) \left[ \frac{d}{dt}S((t-s)^\rho \theta )|_{t=t''}-\frac{d}{dt} S((t-s)^\rho \theta )|_{t=t'}\right] d\theta \nonumber \\&\quad =6\int _{0}^{\infty }\zeta _\rho (\theta ) \left[ \int _{t'}^{t''}\frac{d^2}{dt^2}S((t-s)^\rho \theta )dt\right] d\theta \nonumber \\&\quad \leqslant 6C_2(t''-t'). \end{aligned}$$

(3.10)

Therefore

$$\begin{aligned} J_3\leqslant&\,6\int _{0}^{t'}\bigg (\int _{0}^{\infty }\zeta _\rho (\theta ) \bigg [\frac{d}{dt}S((t-s)^\rho \theta )|_{t=t''} -\frac{d}{dt} S((t-s)^\rho \theta )|_{t=t'}\bigg ] d\theta \bigg )^2 \nonumber \\&\Vert A^{\alpha -\beta -1}\Vert ^2\mathbb {E}\Vert A^\beta G_n(s, x_s)\Vert ^2ds\nonumber \\ \leqslant&M_3(t''-t')^2, \end{aligned}$$

(3.11)

$$\begin{aligned} J_4\leqslant&\,6\bigg (\frac{\rho C_{\alpha -\beta }\Gamma (2-(\alpha -\beta ))}{\Gamma (1+\rho (1-\alpha +\beta ))}\bigg )^2\int _{t'}^{t''}(t''-s)^{2\rho (\beta -\alpha +1)-2}\mathbb {E}\Vert A^\beta G_n(s,u_s)\Vert ^2ds\nonumber \\ \leqslant&\,6\bigg (\frac{\rho C_{\alpha -\beta }\Gamma (2-(\alpha -\beta ))}{\Gamma (1+\rho (1-\alpha +\beta ))} \bigg )^2L_{G}\frac{(t''-t')^{2\rho (\beta -\alpha +1)-1}}{2\rho (\beta -\alpha +1)-1}, \end{aligned}$$

(3.12)

$$\begin{aligned} J_5\leqslant&\,6\int _{0}^{t'}\bigg (\int _{0}^{\infty }\zeta _\rho (\theta )\bigg [\int _{t'}^{t''} \frac{d^2}{dt^2}S((t-s)^\rho \theta )dt\bigg ]d\theta \bigg )^2\Vert A^{\alpha -2}\Vert ^2 [N+\Vert a^2\Vert _{L^p}\Vert L_k\Vert _{L^q}]ds\nonumber \\ \leqslant&M_4(t''-t')^2, \end{aligned}$$

(3.13)

$$\begin{aligned} J_6\leqslant&6\bigg (\frac{\rho C_{\alpha -1}\Gamma (3-\alpha )}{\Gamma (1+\rho (2-\alpha ))} \bigg )^2\int _{t'}^{t''}(t''-s)^{2\rho (2-\alpha )-2} [N+\Vert a^2\Vert _{L^p}\Vert L_k\Vert _{L^q}]ds\nonumber \\ \leqslant&6\bigg (\frac{\rho C_{\alpha -1}\Gamma (3-\alpha )}{\Gamma (1+\rho (2-\alpha ))}\bigg )^2M(R)\frac{(t''-t')^{2\rho (2-\alpha )-1}}{2\rho (2-\alpha )-1}, \end{aligned}$$

(3.14)

$$\begin{aligned} J_7\leqslant&M_5(t''-t')^2, \end{aligned}$$

(3.15)

where $M_3=6C_2^2\Vert A^{\alpha -\beta -1}\Vert ^2L_{G}T_0$, $M_4=6C_2^2\Vert A^{\alpha -2}\Vert ^2M(R)T_0$ and $M_5=rC_1^2\Vert A^{-1}\Vert ^2 \sum _{j=1}^rL_j$ are constants. Using (3.7)–(3.15), we have the map $\Phi _n:C_{T_0}^{\alpha -1}\rightarrow C_{T_0}^{\alpha -1}$ is well defined. Now we will prove that $\Phi _n:\mathcal {B}_R\rightarrow \mathcal {B}_R$ i.e. $\Phi _n\in \mathcal {B}_R$ for any $u\in \mathcal {B}_R$

$$\begin{aligned} \mathbb {E}\Vert (\Phi _nu)(t)-\widetilde{\phi }(0)\Vert ^2_\alpha&\leqslant 6\bigg \{\mathbb {E}\Vert (S_\rho (t)-I)A^\alpha [\widetilde{\phi }(0)+G_n(0, \widetilde{\phi }(0))]\Vert ^2\\&\qquad +\mathbb {E}\Vert A^{\alpha -\beta }\Vert ^2\Vert A^\beta G_n(t,u_t)-A^\beta G_n(0,\widetilde{\phi }(0))\Vert ^2\\&\qquad +\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-(\alpha -\beta ))}{\Gamma (1+\rho (\beta -\alpha ))}\bigg )^2\frac{T_0^{2\rho (\beta -\alpha )-1}}{2\rho (\beta -\alpha )-1}L_{G}\\&\qquad +\bigg (\frac{\rho C_{\alpha }\Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2M(R)\frac{T_0^{2\rho (1-\alpha )-1}}{2\rho (1-\alpha )-1}+rC^2\displaystyle \sum _{j=1}^rL_j\bigg \}. \end{aligned}$$

It follows from (3.3) and (3.4) that,

$$\begin{aligned} \mathbb {E}\Vert (\Phi _nu)(t)-\widetilde{\phi }(0)\Vert ^2_\alpha \leqslant R. \end{aligned}$$

Taking supremum over $(0,T_0]$ we obtain that $\Phi _n$ maps $\mathcal {B}_R$ into $\mathcal {B}_R$.

Now we show that $\Phi _n$ is a contraction map. For $u,v\in \mathcal {B}_R$ and $-\tau \leqslant t\leqslant 0$, we have

$$\begin{aligned} \mathbb {E}\Vert (\Phi _nu)(t)-(\Phi _nv)(t)\Vert ^2_\alpha =\mathbb {E}\Vert \widetilde{\phi }(t)-\widetilde{\phi }(t)\Vert ^2_\alpha . \end{aligned}$$

For $t\in ( 0,T_0]$ and $u,v\in \mathcal {B}_R$, we have

$$\begin{aligned}&\mathbb {E}\Vert (\Phi _nu)(t)-(\Phi _nv)(t)\Vert ^2_\alpha \\&\quad \leqslant 5\bigg \{\Vert A^{\alpha -\beta }\Vert ^2\mathbb {E}\Vert A^\beta G_n(t, u_t)-A^\beta G_n(t, v_t)\Vert ^2\\&\qquad \quad +\int _0^t \Vert (t-s)^{\rho -1}Q_\rho (t-s)A^{1+\alpha -\beta }\Vert ^2E\Vert A^\beta G_n(s, u_s)-A^\beta G_n(s, v_s)\Vert ^2ds\\&\qquad \quad + \int ^t_0 \Vert (t-s)^{\rho -1}Q_\rho (t-s)A^\alpha \Vert ^2\mathbb {E}\Vert F_n(s, u_s, u[b(u(s), s)])\\&\qquad \quad -F_n(s, v_s, v[b(v(s), s)])\Vert ^2ds+ \int ^t_0 \Vert (t-s)^{\rho -1}Q_\rho (t-s)A^\alpha \Vert ^2\\&\qquad \quad \bigg (\int _0^s|a(s-r)|^2\mathbb {E}\Vert k(r, u_r)-k(r, v_r)\Vert ^2dr\bigg )ds\\&\qquad \quad +r\displaystyle \sum _{j=1}^r\Vert S_q(t-t_k)(I_{j,n}(u_{t_j})-I_{j,n}(v_{t_j}))\Vert ^2_{\alpha }\bigg \}\\&\quad \leqslant 5\bigg [\Vert A^{\alpha -\beta -1}\Vert ^2L_{G}+\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (\beta -\alpha ))}\bigg )^2 \frac{T_0^{2\rho (\beta -\alpha )-1}}{2\rho (\beta -\alpha )-1}L_{G}\Vert A^{-1}\Vert \\&\qquad \quad +\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2\frac{T_0^{2\rho (1-\alpha )-1}}{2\rho (1-\alpha )-1}(L_f(1+LL_b)+M(R))\\&\quad \qquad +\,rC^2\displaystyle \sum _{j=1}^rL_j\bigg ]\mathbb {E}\Vert u-v\Vert ^2_{T_0,\alpha }. \end{aligned}$$

Using (3.5) and taking supremum over $t\in ( 0,T_0]$, we get

$$\begin{aligned} \mathbb {E}\Vert (\Phi _nu)(t)-(\Phi _nv)(t)\Vert ^2_\alpha <\mathbb {E}\Vert u-v\Vert ^2_{T_0,\alpha }. \end{aligned}$$

Thus $\Phi _n$ is a strict contraction on $\mathcal {B}_R$. Therefore by Banach contraction principle, there exists a unique $u_n\in \mathcal {B}_R$ such that $\Phi _nu_n=u_n$ i.e. $u_n$ satisfies the approximate integral equation

$$\begin{aligned} u_n(t)= \left\{ \begin{array}{ll} \widetilde{\phi }(t), \quad t\in [-\tau , 0]; \\ S_\rho (t)[\widetilde{\phi }(0) + G_n(0, \widetilde{\phi }(0))]-G_n(t, (u_n)_t)+\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AG_n(s, (u_n)_s)ds\\ \quad +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[F_n(s, (u_n)_s, u_n[b(u_n(s), s)])\\ \quad +\int _0^sa(s-r) k_n(r, (u_n)_r)d\omega (r)]ds +\displaystyle \sum \nolimits _{j=1}^rS_\rho (t-t_j)I_{j,n}(u_n)_{t_j}), \quad t\in [0,T_0]. \end{array} \right. \nonumber \\ \end{aligned}$$

(3.16)

$\square$

Lemma 3.2

Let (H1)–(H5) hold and $\widetilde{\phi }(t)\in \mathcal {L}^0_2(\Omega , D(A^\alpha ))$ for all $t\in [-\tau ,0]$. Then there exists a constant $N_{t_0}$, independent of n, such that

$$\begin{aligned} \mathbb {E}\Vert u_n(t)\Vert ^2_{\mu }\leqslant N_{t_0}, \quad 0\leqslant \mu <1,\,\, -\tau \leqslant t\leqslant T. \end{aligned}$$

Moreover, if $\widetilde{\phi }(0)\in \mathcal {L}^0_2(\Omega , D(A))$, there exists a constant $N_{0}$, independent of n, such that

$$\begin{aligned} \mathbb {E}\Vert u_n(t)\Vert ^2_{\mu }\leqslant N_{0}, \quad 0\leqslant \mu <1,\,\, 0\leqslant t\leqslant T. \end{aligned}$$

Proof

For $t\in [-\tau ,0]$, applying $A^\mu$ on both the sides of (3.16) and taking norm, we have

$$\begin{aligned} \mathbb {E}\Vert u_n(t)\Vert _\mu ^2 \leqslant \Vert \widetilde{\phi }(t)\Vert _\mu ^2\leqslant \Vert \widetilde{\phi }\Vert _{0,\mu }^2. \end{aligned}$$

For $t\in [t_0,T]$, on applying $A^\mu$ on both the sides of (3.16) and taking norm, we have

$$\begin{aligned} \mathbb {E}\Vert u_n(t)\Vert _\mu ^2&\leqslant 6\bigg \{\Vert A^\mu S_\rho (t)[\widetilde{\phi }(0) + G_n(0, \widetilde{\phi }(0))]\Vert ^2+\Vert A^{\mu -\beta }\Vert ^2\mathbb {E}\Vert A^\beta G_n(t, (u_n)_t)\Vert ^2\\&\quad \qquad +\int _0^t \Vert (t-s)^{\rho -1}Q_\rho (t-s)A^{1+\mu -\beta }\Vert ^2\mathbb {E}\Vert A^\beta G_n(s, (u_n)_s)\Vert ^2ds \\&\quad \qquad +\int ^t_0 \Vert (t-s)^{\rho -1}Q_\rho (t-s)A^{\mu }\Vert ^2\\&\quad [\mathbb {E}\Vert F_n(s, (u_n)_s, u_n[b(u_n(s), s)])\Vert ^2+\int _0^s|a(s-r)|^2\mathbb {E}\Vert k_n(r, (u_n)_r)\Vert ^2dr]ds\\&\quad +\,r\displaystyle \sum _{j=1}^r|S_\rho (t-t_j)|^2\Vert A^{\mu }I_j(u_n)_{t_j}\Vert ^2\bigg \}\\&\leqslant 6\bigg \{C_\mu ^2 t_0^{-2\rho \mu }\Vert \widetilde{\phi }(0) +G_n(0,\widetilde{\phi }(0))\Vert ^2+\Vert A^{\mu -\beta }\Vert ^2L_{G} \\&\qquad \quad +\bigg (\frac{\rho C_{1+\mu -\beta }\Gamma (1-\mu +\beta )}{\Gamma (1+\rho (\beta -\mu ))} \bigg )^2\frac{T_0^{2\rho (\beta -\mu )-1}}{2\rho (\beta -\mu )-1}L_{G}\\&\qquad \quad +\bigg (\frac{\rho C_\mu \Gamma (2-\mu )}{\Gamma (1+\rho (1-\mu ))}\bigg )^2M(R)\frac{T_0^{2\rho (1-\mu )-1}}{2\rho (1-\mu )-1}+rC^2\displaystyle \sum _{j=1}^rL_j\bigg \}\\&\leqslant N_{t_0}. \end{aligned}$$

From Theorem 3.1, there exists a unique $u_n\in \mathcal {B}_R$ which satisfies (3.16). Now using Part (a) of Theorem 6.13 in Pazy [31] we have $S(t):\mathcal {H}\rightarrow \mathcal {L}_2(\Omega , D(A^\mu ))$ for $t>0$ and $0\leqslant \mu < 1$. Also Theorem 2.4 in Pazy [31] implies that if $u\in \mathcal {L}_2(\Omega , D(A))$ then $S(t)u\in \mathcal {L}_2(\Omega , D(A))$. And Theorem 6.8 in Pazy [31] implies $\mathcal {L}_2(\Omega , D(A))\subseteq \mathcal {L}_2(\Omega , D(A^\mu ))$ for $0\leqslant \mu <1$. Thus if $\widetilde{\phi }(t)\in \mathcal {L}^0_2(\Omega , D(A))$ then $\widetilde{\phi }(t)\in \mathcal {L}^0_2(\Omega , D(A^\mu ))$ for $0\leqslant \mu <1$ and we get,

$$\begin{aligned} \mathbb {E}\Vert u_n(t)\Vert _\mu ^2 \leqslant&6\bigg \{C^2\Vert \widetilde{\phi }(0) + G_n(0, \widetilde{\phi }(0))\Vert ^2_{\mu }+\Vert A^{\mu -\beta }\Vert ^2L_{G} \\&+\bigg (\frac{\rho C_{1+\mu -\beta }\Gamma (1-\mu +\beta )}{\Gamma (1+\rho (\beta -\mu ))}\bigg )^2\frac{T_0^{2\rho (\beta -\mu )-1}}{2\rho (\beta -\mu )-1}L_{G}\\&+\bigg (\frac{\rho C_\mu \Gamma (2-\mu )}{\Gamma (1+\rho (1-\mu ))}\bigg )^2M(R)\frac{T_0^{2\rho (1-\mu )-1}}{2\rho (1-\mu )-1}+rC^2\displaystyle \sum _{j=1}^rL_j\bigg \}\\ \leqslant&N_{0}. \end{aligned}$$

$\square$

4 Convergence of solutions

Theorem 4.1

Let (H1)–(H6) hold and $\widetilde{\phi }(t) \in \mathcal {L}_2^0(\Omega ,D(A^\alpha ))$ for all $t\in [-\tau , 0]$. Then the sequence $\{u_n\}\in \mathcal {B}_R$ is a Cauchy sequence and therefore converges to a function $u\in \mathcal {B}_R$ satisfying (3.16).

Proof

For $0<t_0'<t$, we have

$$\begin{aligned}&\mathbb {E}\Vert u_n(t)-u_m(t)\Vert ^2_\alpha \nonumber \\&\quad \leqslant 6\bigg \{C^2\Vert A^{\alpha -\beta }\Vert ^2 \mathbb {E}\Vert A^\beta G_n(0, \widetilde{\phi }(0))- A^\beta G_m(0, \widetilde{\phi }(0)))\Vert ^2\nonumber \\&\qquad \quad +\Vert A^{\alpha -\beta }\Vert ^2\mathbb {E}\Vert A^\beta G_n(t, (u_n)_t-A^\beta G_m(t, (u_m)_t)\Vert ^2\nonumber \\&\qquad \quad +\bigg (\int _0^{t_0'}+\int _{t_0'}^t\bigg )\Vert (t-s)^{\rho -1}Q_\rho (t-s)A^{1+\alpha -\beta }\Vert ^2\nonumber \\&\qquad \quad \times \mathbb {E}\Vert A^\beta G_n(s,(u_n)_s-A^\beta G_m(s, (u_m)_s)\Vert ^2ds\nonumber \\&\qquad \quad +\bigg (\int _0^{t_0'}+\int _{t_0'}^t\bigg )\Vert (t-s)^{\rho -1}Q_\rho (t-s)A^{\alpha }\Vert ^2\bigg [\mathbb {E}\Vert F_n(s, (u_n)_s, u_n[b(u_n(s), s)])\nonumber \\&\qquad \quad -F_m(s, (u_m)_s, u_m[b(u_m(s), s)])\Vert ^2\nonumber \\&\qquad \quad +\int _0^s|a(s-r)|^2\mathbb {E}\Vert k_n(r, (u_n)_{r})-k_m(r, (u_m)_r)\Vert ^2dr\bigg ]ds\nonumber \\&\qquad \quad +\,r\displaystyle \sum _{j=1}^r|S_\rho (t-t_j)|^2\mathbb {E}\Vert I_{j,n}(u_n)_{t_j})-I_{j,m}(u_m)_{t_j})\Vert ^2_\alpha \bigg \}. \end{aligned}$$

(4.1)

Here

$$\begin{aligned}&C^2\Vert A^{\alpha -\beta }\Vert ^2 \mathbb {E}\Vert A^\beta G_n(0, \widetilde{\phi }(0))- A^\beta G_m(0, \widetilde{\phi }(0)))\Vert ^2 \\&\quad \leqslant C^2\Vert A^{\alpha -\beta }\Vert ^2L_{G}\mathbb {E}\Vert (P^n-P^m)\widetilde{\phi }(0)\Vert ^2_{\alpha -1}\\&\quad \leqslant C^2\Vert A^{\alpha -\beta -1}\Vert ^2L_{G}\mathbb {E}\Vert (P^n-P^m)A^\alpha \widetilde{\phi }(0)\Vert ^2, \end{aligned}$$

and

$$\begin{aligned}&\mathbb {E}\Vert F_n(s, (u_n)_s, u_n[b(u_n(s), s)])-F_m(s, (u_m)_s, u_m[b(u_m(s), s)])\Vert ^2\\&\quad \leqslant 2[\mathbb {E}\Vert F_n(s, (u_n)_s, u_n[b(u_n(s), s)])-F_n(s, (u_m)_s, u_m[b(u_m(s), s)])\Vert ^2\\&\qquad +\mathbb {E}\Vert F_n(s, (u_m)_s, u_m[b(u_m(s), s)])-F_m(s, (u_m)_s, u_m[b(u_m(s), s)])\Vert ^2]\\&\quad \leqslant 2[L_f(1+LL_b)\mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha }+L_f[\mathbb {E}\Vert (P^n-P^m)u_m\Vert ^2_{s,\alpha }\\&\qquad +\Vert A^{-1}\Vert ^2\mathbb {E}\Vert (P^n-P^m)u_m[b(u_m(s),s)]\Vert ^2_\alpha ]. \end{aligned}$$

Also

$$\begin{aligned} \mathbb {E}\Vert (P^n-P^m)u_m\Vert ^2_{s,\alpha }\leqslant \mathbb {E}\Vert A^{\alpha -\mu }(P^n-P^m)A^\mu u_m\Vert ^2_s\leqslant \frac{1}{\lambda _m^{2(\mu -\alpha )}}\mathbb {E}\Vert A^\mu u_m\Vert ^2_s. \end{aligned}$$

Therefore, we have

$$\begin{aligned}&\mathbb {E}\Vert F_n(s, (u_n)_s, u_n[b(u_n(s), s)])-F_m(s, (u_m)_s, u_m[b(u_m(s), s)])\Vert ^2\\&\quad \leqslant 2L_f(1+LL_b)\mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha }+2L_f\bigg [\frac{1}{\lambda _m^{2(\mu -\alpha )}}\mathbb {E}\Vert A^\mu u_m\Vert ^2_s\\&\qquad +\frac{\Vert A^{-1}\Vert ^2}{\lambda _m^{2(\mu -\alpha )}}\mathbb {E}\Vert A^\mu u_m[b(u_m(s), s)]\Vert ^2\bigg ]. \end{aligned}$$

Similarly

$$\begin{aligned}&\mathbb {E}\Vert A^\beta G_n(s, (u_n)_s)-A^\beta G_m(s, (u_m)_s)\Vert ^2\\&\quad \leqslant 2\mathbb {E}\Vert A^\beta G_n(s, (u_n)_s-A^\beta G_n(s, (u_m)_s)\Vert ^2+2\mathbb {E}\Vert A^\beta G_n(s, (u_m)_s-A^\beta G_m(s, (u_m)_s)\Vert ^2\\&\quad \leqslant 2L_{G}\Vert A^{-1}\Vert ^2\left[ \mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha }+ \frac{1}{\lambda _m^{2(\mu -\alpha )}}\mathbb {E}\Vert A^\mu u_m\Vert ^2_s\right] , \end{aligned}$$

and

$$\begin{aligned}&\mathbb {E}\Vert k_n(s, (u_n)_s)-k_m(s, (u_m)_s)\Vert ^2\\&\quad \leqslant 2[\mathbb {E}\Vert k_n(s, (u_n)_s)-k_n(s, (u_m)_s)\Vert ^2+\mathbb {E}\Vert k_n(s, (u_m)_s-k_m(s, (u_m)_s)\Vert ^2]\\&\quad \leqslant 2L_k\left[ \mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha }+ \frac{1}{\lambda _m^{2(\mu -\alpha )}}\mathbb {E}\Vert A^\mu u_m\Vert ^2_s\right] . \end{aligned}$$

The first and third integrals of (4.1) can be estimated as

$$\begin{aligned}&\int _0^{t_0'}\Vert (t-s)^{\rho -1}Q_\rho (t-s)A^{1+\alpha -\beta } \Vert ^2\mathbb {E}\Vert A^\beta G_n(s, (u_n)_s-A^\beta G_m(s, (u_m)_s)\Vert ^2ds\\&\quad \leqslant 2L_{G}\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (-\alpha +\beta ))} \bigg )^2(t_0-t_0')^{2\rho (\beta -\alpha )-2}t_0',\\&\int _0^{t_0'}\Vert (t-s)^{\rho -1}Q_\rho (t-s)A^{\alpha }\Vert ^2\bigg [\mathbb {E}\Vert F_n(s, (u_n)_s, u_n[b(u_n(s), s) ]) \\&\qquad -F_m(s, (u_m)_s, u_m[b(u_m(s), s)])\Vert ^2\\&\qquad +\int _0^s|a(s-r)|^2\mathbb {E}\Vert k_n(r, (u_n)_r)-k_m(r, (u_m)_r)\Vert ^2dr\bigg ]ds\\&\quad \leqslant 2M(R)\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))} \bigg )^2(t_0-t_0')^{2\rho (1-\alpha )-2}t_0'. \end{aligned}$$

Second and fourth integrals of (4.1) can be estimated as

$$\begin{aligned}&\int _{t_0'}^t\Vert (t-s)^{\rho -1}Q_\rho (t-s)A^{1+\alpha -\beta }\Vert ^2\mathbb {E}\Vert A^\beta G_n(s, (u_n)_s-A^\beta G_m(s, (u_m)_s)\Vert ^2ds\\&\quad \leqslant 2\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (-\alpha +\beta ))}\bigg )^2L_{G}\Vert A^{-1}\Vert ^2\bigg (\frac{N_{t_0}T_0^{2\rho (\beta -\alpha )-1}}{\lambda _m^{2(\mu -\alpha )}(2\rho (\beta -\alpha )-1)}\\&\qquad +\int _{t_0'}^t(t-s)^{2\rho (\beta -\alpha )-2}\mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha } ds \bigg ),\\&\int _{t_0'}^t\Vert (t-s)^{\rho -1}Q_\rho (t-s)A^{\alpha }\Vert ^2\bigg [\mathbb {E}\Vert F_n(s, (u_n)_s, u_n[b(u_n(s), s)])\\&\qquad -\,F_m(s, (u_m)_s, u_m[b(u_m(s), s)])\Vert ^2\\&\qquad +\int _0^s|a(s-r)|^2\mathbb {E}\Vert k_n(r, (u_n)_r)-k_m(r, (u_m)_r)\Vert ^2dr\bigg ]ds\\&\quad \leqslant 2\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2\int _{t_0'}^t(t-s)^{2\rho (1-\alpha )-2}\bigg [L_f(1+LL_b)\mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha }\\&\qquad +\,L_f\bigg (\frac{1}{\lambda _m^{2(\mu -\alpha )}}\mathbb {E}\Vert A^\mu u_m\Vert ^2_s +\frac{\Vert A^{-1}\Vert ^2}{\lambda _m^{2(\mu -\alpha )}}\mathbb {E}\Vert A^\mu u_m[b(u_m(s), s)]\Vert ^2\bigg )\\&\qquad +\int _0^s|a(s-r)|^2L_k(\mathbb {E}\Vert u_n-u_m\Vert ^2_{r,\alpha }+\frac{1}{\lambda _m^{2(\mu -\alpha )}}\mathbb {E}\Vert A^\mu u_m\Vert ^2_r)dr \bigg ]ds\\&\quad \leqslant 2\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2\bigg [(L_f(1+\Vert A^{-1}\Vert ^2)+\Vert a^2\Vert _{L^p}\Vert L_k\Vert _{L^q})\frac{N_{t_0}}{\lambda _m^{2(\mu -\alpha )}}\frac{T_0^{2\rho (1-\alpha )-1}}{2\rho (1-\alpha )-1}\\&\qquad +\,(L_f(1+LL_b)+\Vert a^2\Vert _{L^p}\Vert L_k\Vert _{L^q})\int _{t_0'}^t(t-s)^{2\rho (1-\alpha )-2}\mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha } ds\bigg ]. \end{aligned}$$

Thus (4.1) can be estimated as

$$\begin{aligned}&\mathbb {E}\Vert u_n(t)-u_m(t)\Vert ^2_\alpha \nonumber \\&\quad \leqslant 6C^2\Vert A^{\alpha -\beta -1}\Vert ^2L_{G}\mathbb {E}\Vert (P^n-P^m)A^\alpha \widetilde{\phi }(0)\Vert ^2\nonumber \\&\qquad +12\Vert A^{\alpha -\beta -1}\Vert ^2L_{G}\bigg (\mathbb {E}\Vert u_n-u_m\Vert ^2_{t,\alpha }+\frac{N_{t_0}}{\lambda _m^{2(\mu -\alpha )}}\bigg )\nonumber \\&\qquad +12\bigg (L_{G}\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (-\alpha +\beta ))}\bigg )^2(t_0-t_0')^{2\rho (\beta -\alpha )-2}\nonumber \\&\qquad +M(R)\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2(t_0-t_0')^{2\rho (1-\alpha )-2}\bigg )t_0'\nonumber \\&\qquad +\bigg [12\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (-\alpha +\beta ))}\bigg )^2L_{G}\Vert A^{-1}\Vert ^2\frac{T_0^{2\rho (\beta -\alpha )-1}}{(2\rho (\beta -\alpha )-1)}\nonumber \\&\qquad +12\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2(L_f(1+\Vert A^{-1}\Vert ^2) \nonumber \\&\qquad + \Vert a^2\Vert _{L^p}\Vert L_k\Vert _{L^q}) \frac{T_0^{2\rho (1-\alpha )-1}}{2\rho (1-\alpha )-1}\bigg ] \frac{N_{t_0}}{\lambda _m^{2(\mu -\alpha )}}\nonumber \\&\qquad +\int _{t_0'}^t\bigg \{12\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (-\alpha +\beta ))}\bigg )^2L_{G}\Vert A^{-1}\Vert ^2(t-s)^{2\rho (\beta -\alpha )-2}\nonumber \\&\qquad +12\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2(L_f(1+LL_b)+\Vert a^2\Vert _{L^p}\Vert L_k\Vert _{L^q})\nonumber \\&\qquad (t-s)^{2\rho (1-\alpha )-2}\bigg \}\mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha } ds+6rC^2\displaystyle \sum _{j=1}^rL_j\mathbb {E}\Vert u_n-u_m\Vert ^2_{t_j,\alpha }\nonumber \\&\leqslant D_1\mathbb {E}\Vert u_n-u_m\Vert ^2_{t,\alpha }+D_2t_0'+\frac{D_3}{\lambda _m^{2(\mu -\alpha )}}+D_4\int _{t_0'}^t[(t-s)^{2\rho (\beta -\alpha )-2}\nonumber \\&\qquad +(t-s)^{2\rho (1-\alpha )-2}]\mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha } ds+6rC^2\displaystyle \sum _{j=1}^rL_j\mathbb {E}\Vert u_n-u_m\Vert ^2_{t_j,\alpha }, \end{aligned}$$

(4.2)

where

$$\begin{aligned} D_1=&12\Vert A^{\alpha -\beta -1}\Vert ^2L_{G},\\ D_2=&12\bigg (L_{G}\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (-\alpha +\beta ))}\bigg )^2(t_0-t_0')^{2\rho (\beta -\alpha )-2} \\&\quad +M(R)\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2(t_0-t_0')^{2\rho (1-\alpha )-2}\bigg ),\\ D_3=&6C^2N_{t_0}\Vert A^{\alpha -\beta -1}\Vert ^2L_{G}\Vert \widetilde{\phi }\Vert ^2_{0,\mu }+12N_{t_0}\Vert A^{\alpha -\beta -1}\Vert ^2L_{G}\\&+12N_{t_0}\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (-\alpha +\beta ))}\bigg )^2L_{G}\Vert A^{-1}\Vert ^2\frac{T_0^{2\rho (\beta -\alpha )-1}}{(2\rho (\beta -\alpha )-1)}\\&+12N_{t_0}\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2(L_f(1+\Vert A^{-1}\Vert ^2)+\Vert a^2\Vert _{L^p}\Vert L_k\Vert _{L^q}) \frac{T_0^{2\rho (1-\alpha )-1}}{2\rho (1-\alpha )-1},\\ D_4=&12\bigg (\frac{\rho C_{1+\alpha -\beta }\Gamma (1-\alpha +\beta )}{\Gamma (1+\rho (-\alpha +\beta ))} \bigg )^2L_{G}\Vert A^{-1}\Vert ^2 \\&\quad +12\bigg (\frac{\rho C_\alpha \Gamma (2-\alpha )}{\Gamma (1+\rho (1-\alpha ))}\bigg )^2(L_f(1+LL_b) +\Vert a^2\Vert _{L^p}\Vert L_k\Vert _{L^q}). \end{aligned}$$

Replace t by $t+\nu$ in the above inequality, where $\nu \in [t_0'-t,0]$, we get

$$\begin{aligned} \mathbb {E}\Vert u_n(t+\nu )-u_m(t+\nu )\Vert ^2_\alpha&\leqslant D_1\mathbb {E}\Vert u_n-u_m\Vert ^2_{t,\alpha }+D_2t_0'+\frac{D_3}{\lambda _m^{2(\mu -\alpha )}} \\&\quad +D_4\int _{t_0'}^{t+\nu }[(t+\nu -s)^{2\rho (\beta -\alpha )-2}\\&\quad +(t+\nu -s)^{2\rho (1-\alpha )-2}]\mathbb {E}\Vert u_n-u_m\Vert ^2_{s,\alpha } ds \\&\quad +6rC^2\displaystyle \sum _{j=1}^rL_j\mathbb {E}\Vert u_n-u_m\Vert ^2_{t_j,\alpha } \end{aligned}$$

Now put $s-\nu =\gamma$ in the integral term of above inequality and we get

$$\begin{aligned} \mathbb {E}\Vert u_n(t+\nu )-u_m(t+\nu )\Vert ^2_\alpha&\leqslant D_1\mathbb {E}\Vert u_n-u_m\Vert ^2_{t,\alpha }+D_2t_0'+\frac{D_3}{\lambda _m^{2(\mu -\alpha )}} \\&\quad +D_4\int _{t_0'-\nu }^{t}[(t-\gamma )^{2\rho (\beta -\alpha )-2}\\&\quad +(t-\gamma )^{2\rho (1-\alpha )-2}]\mathbb {E}\Vert u_n-u_m\Vert ^2_{\gamma +\nu ,\alpha } d\gamma \\&\quad +6rC^2\displaystyle \sum _{j=1}^rL_j\mathbb {E}\Vert u_n-u_m\Vert ^2_{t_j,\alpha }\\&\leqslant D_1\mathbb {E}\Vert u_n-u_m\Vert ^2_{t,\alpha }+D_2t_0'+\frac{D_3}{\lambda _m^{2(\mu -\alpha )}} \\&\quad +D_4\int _{t_0'}^{t}[(t-\gamma )^{2\rho (\beta -\alpha )-2}\\&\quad +(t-\gamma )^{2\rho (1-\alpha )-2}]\mathbb {E}\Vert u_n-u_m\Vert ^2_{\gamma ,\alpha } d\gamma \\&\quad +6rC^2\displaystyle \sum _{j=1}^rL_j\mathbb {E}\Vert u_n-u_m\Vert ^2_{t_j,\alpha }. \end{aligned}$$

Thus,

$$\begin{aligned} \displaystyle \sup _{t_0'-t\leqslant \nu \leqslant 0}\mathbb {E}\Vert u_n(t+\nu )-u_m(t+\nu )\Vert ^2_\alpha \leqslant&D_1\mathbb {E}\Vert u_n-u_m\Vert ^2_{t,\alpha }+D_2t_0'+\frac{D_3}{\lambda _m^{2(\mu -\alpha )}} \nonumber \\&+D_4\int _{t_0'}^{t}[(t-\gamma )^{2\rho (\beta -\alpha )-2}\nonumber \\&+(t-\gamma )^{2\rho (1-\alpha )-2}]\mathbb {E}\Vert u_n-u_m\Vert ^2_{\gamma ,\alpha } d\gamma \nonumber \\&+6rC^2\displaystyle \sum _{j=1}^rL_j\mathbb {E}\Vert u_n-u_m\Vert ^2_{t_j,\alpha }. \end{aligned}$$

(4.3)

Since $u_n(t+\nu )=\widetilde{\phi }(t+\nu )$ for $t+\nu \leqslant 0$ and $n\geqslant n_0$. Therefore, we have

$$\begin{aligned} \displaystyle \sup _{-\tau -t\leqslant \nu \leqslant 0}\mathbb {E}\Vert u_n(t+\nu )-u_m(t+\nu )\Vert ^2_\alpha&\leqslant \displaystyle \sup _{0\leqslant \nu +t \leqslant t_0'}\mathbb {E}\Vert u_n(t+\nu )-u_m(t+\nu )\Vert ^2_\alpha \nonumber \\&\quad +\displaystyle \sup _{t_0'-t\leqslant \nu \leqslant 0}\mathbb {E}\Vert u_n(t+\nu )-u_m(t+\nu )\Vert ^2_\alpha .\quad \end{aligned}$$

(4.4)

For $t\in (0,t_0']$, we have from (4.2)

$$\begin{aligned} \mathbb {E}\Vert u_n(t+\nu )-u_m(t+\nu )\Vert ^2_\alpha\leqslant & {} D_1\mathbb {E}\Vert u_n-u_m\Vert ^2_{t,\alpha }+D_2t_0'+ \frac{D_5}{\lambda _m^{2(\mu -\alpha )}} \nonumber \\&+\,6rC^2\displaystyle \sum _{j=1}^rL_j\mathbb {E}\Vert u_n-u_m\Vert ^2_{t_j,\alpha }, \end{aligned}$$

(4.5)

where $D_5=6C^2N_{t_0}\Vert A^{\alpha -\beta -1}\Vert ^2L_{G}\Vert \widetilde{\phi }\Vert ^2_{0,\mu }+12N_{t_0}\Vert A^{\alpha -\beta -1}\Vert ^2L_{G}$. Using (4.3) and (4.5) in (4.4), we get

$$\begin{aligned} \displaystyle \sup _{-\tau \leqslant t+\nu \leqslant t}\mathbb {E}\Vert u_n(t+\nu )-u_m(t+\nu )\Vert ^2_\alpha \leqslant&2D_1\mathbb {E}\Vert u_n-u_m\Vert ^2_{t,\alpha }+2D_2t_0'+ \frac{D_3+D_5}{\lambda _m^{2(\mu -\alpha )}} \\&+D_4\int _{t_0'}^{t}[(t-\gamma )^{2\rho (\beta -\alpha )-2} \\&+(t-\gamma )^{2\rho (1-\alpha )-2}]\mathbb {E}\Vert u_n-u_m\Vert ^2_{\gamma ,\alpha } d\gamma \\&+12rC^2\displaystyle \sum _{j=1}^rL_j\mathbb {E}\Vert u_n-u_m\Vert ^2_{t_j,\alpha }. \end{aligned}$$

Since $2D_1+12rC^2\displaystyle \sum _{j=1}^rL_j<1$, we have

$$\begin{aligned} \mathbb {E}\Vert u_n(t)-u_m(t)\Vert ^2_\alpha \leqslant&\frac{1}{\bigg (1-2D_1+12rC^2\displaystyle \sum _{j=1}^rL_j\bigg )}\bigg [2D_2t_0'+\frac{D_3+D_5}{\lambda _m^{2(\mu -\alpha )}} \\&+D_4\int _{t_0'}^{t}[(t-\gamma )^{2\rho (\beta -\alpha )-2} +(t-\gamma )^{2\rho (1-\alpha )-2}]\mathbb {E}\Vert u_n-u_m\Vert ^2_{\gamma ,\alpha } d\gamma \bigg ]. \end{aligned}$$

From Lemma 5.6.7 in Pazy [31], there exists a constant M such that

$$\begin{aligned} \mathbb {E}\Vert u_n(t)-u_m(t)\Vert ^2_\alpha&\leqslant \frac{1}{\bigg (1-2D_1+12rC^2\displaystyle \sum _{j=1}^rL_j\bigg )}\bigg [2D_2t_0'+\frac{D_3+D_5}{\lambda _m^{2(\mu -\alpha )}}\bigg ]M. \end{aligned}$$

Because $t_0'$ is arbitrary and letting $m\rightarrow \infty$, the right hand side may be made as small as desired by taking $t_0'$ sufficiently small and we get the required result. $\square$

Theorem 4.2

Let (H1)–(H6) hold and $\widetilde{\phi }(0)\in \mathcal {L}_2(\Omega , D(A^\alpha ))$ for all $t\in [-\tau , 0]$. Then there exists a unique function $u_n\in \mathcal {B}_R$ and unique $u\in \mathcal {B}_R$ satisfying

$$\begin{aligned} u_n(t)= \left\{ \begin{array}{ll} \widetilde{\phi }(t), \quad t\in [-\tau , 0]; \\ S_\rho (t)[\widetilde{\phi }(0) + G_n(0, \widetilde{\phi }(0))]-G_n(t, (u_n)_t)+\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AG_n(s, (u_n)_s)ds\\ \quad +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[F_n(s, (u_n)_s, u_n[b(u_n(s), s)])\\ \quad +\int _0^sa(s-r) k_n(r, (u_n)_r)d\omega (r)]ds +\displaystyle \sum \nolimits _{j=1}^rS_\rho (t-t_j)I_{j,n}(u_n)_{t_j}), \quad t\in [0,T_0]. \end{array} \right. \nonumber \\ \end{aligned}$$

(4.6)

and

$$\begin{aligned} u(t)= \left\{ \begin{array}{ll} \widetilde{\phi }(t), \quad t\in [-\tau , 0]; \\ S_\rho (t)[\widetilde{\phi }(0) + G(0, \widetilde{\phi }(0))]-G(t, u_t)+\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AG(s, u_s)ds\\ \quad +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[F(s, u_s, u[b(u(s), s)])\\ \quad +\int _0^sa(s-r) k(r, u_r)d\omega (r)]ds +\displaystyle \sum \nolimits _{j=1}^rS_\rho (t-t_j)I_{j,n}u_{t_j}),\quad t\in [0,T_0]. \end{array} \right. \nonumber \\ \end{aligned}$$

(4.7)

such that $u_n\rightarrow u$ in $\mathcal {B}_R$ as $n\rightarrow \infty .$

Proof

Let $\widetilde{\phi }(0)\in \mathcal {L}_2(\Omega , D(A^\alpha ))$ for all $t\in [-\tau , 0]$. Since for $t\in (0,T]$, there exists $u_n\in \mathcal {B}_R$ such that $A^\alpha u_n(t)\rightarrow A^\alpha u(t)\in \mathcal {B}_R$ as $n\rightarrow \infty$ and $u(t)=u_n(t)=\widetilde{\phi }(t)$ for all $t\in [-\tau , 0]$. Also for $t\in [-\tau ,T]$, we have $A^\alpha u_n(t)\rightarrow A^\alpha u(t)$ in $\mathcal {L}_2(\Omega , \mathcal {H})$ as $n\rightarrow \infty$. Furthermore, since for each $u_n\in \mathcal {B}_R$, we have $u\in \mathcal {B}_R$ and for any $0<t_0\leqslant T$,

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty }\displaystyle \sup _{t_0\leqslant t\leqslant T} \mathbb {E}\Vert u_n(t)-u(t)\Vert ^2_\alpha =0. \end{aligned}$$

Also, we have

$$\begin{aligned}&\displaystyle \sup _{s\in [t_0,T]} \mathbb {E}\Vert F_n(s, (u_n)_s, u_n[b(u_n(s), s)])-F(s, u_s, u[b(u(s), s)])\Vert ^2\\&\quad \leqslant 2[L_f(1+LL_b)\mathbb {E}\Vert u_n-u\Vert ^2_{s,\alpha }+L_f(\mathbb {E}\Vert (P^n-I)u_s\Vert ^2_\alpha \\&\qquad +\Vert A^{-1}\Vert ^2\mathbb {E}\Vert (P^n-I)u[b(u(s),s)]\Vert ^2_\alpha )]\rightarrow 0, \end{aligned}$$

and

$$\begin{aligned}&\displaystyle \sup _{s\in [t_0,T]}\mathbb {E}\Vert A^\beta G_n(s, (u_n)_s)-A^\beta G(s, u_s)\Vert ^2 \\&\quad \leqslant 2L_{G}\Vert A^{-1}\Vert ^2[\mathbb {E}\Vert u_n-u\Vert ^2_{s,\alpha }+ \mathbb {E}\Vert (P^n-I)u_s\Vert ^2_\alpha ]\rightarrow 0, \end{aligned}$$

and

$$\begin{aligned} \displaystyle \sup _{s\in [t_0,T]}\mathbb {E}\Vert k_n(s, (u_n)_s-k(s, u_s)\Vert ^2 \leqslant 2L_k[\mathbb {E}\Vert u_n-u\Vert ^2_{s,\alpha }+\mathbb {E}\Vert (P^n-I)u_s\Vert ^2_\alpha ]\rightarrow 0, \end{aligned}$$

as $n\rightarrow \infty$. For $t_0'\in (0,t)$, rewrite Eq.  (3.16) as

$$\begin{aligned} u_n(t) =&S_\rho (t)[\widetilde{\phi }(0) + G_n(0, \widetilde{\phi }(0))]-G_n(t, (u_n)_t) \nonumber \\&+\bigg (\int _0^{t_0'}+\int _{t_0'}^{t}\bigg ) (t-s)^{\rho -1}Q_\rho (t-s)AG_n(s, (u_n)_s)ds\nonumber \\&+\bigg (\int _0^{t_0'}+\int _{t_0'}^{t}\bigg )(t-s)^{\rho -1}Q_\rho (t-s)[F_n(s, (u_n)_s, u_n[b(u_n(s), s)])\nonumber \\&+\int _0^sa(s-r) k_n(r, (u_n)_r)d\omega (r)]ds +\displaystyle \sum _{j=1}^rS_\rho (t-t_j)I_{j,n}(u_n)_{t_j}. \end{aligned}$$

(4.8)

The first and third integrals of (4.8) can be estimated as

$$\begin{aligned}&\bigg \Vert \int _0^{t_0'}(t-s)^{\rho -1}A^{1-\beta }Q_\rho (t-s)A^\beta G_n(s, (u_n)_s)ds\bigg \Vert ^2 \leqslant L_{G}\bigg (\frac{\rho C_{1-\beta }\Gamma (1+\beta )}{\Gamma (1+\rho \beta )}\bigg )^2T_0^{2(\rho \beta -1)}t_0',\\&\bigg \Vert \int _0^{t_0'}(t-s)^{\rho -1}Q_\rho (t-s)[F_n(s, (u_n)_s, u_n[b(u_n(s), s)])+\int _0^sa(s-r) k_n(r, (u_n)_r)d\omega (r)]ds\bigg \Vert \\&\quad \leqslant M(R)\bigg (\frac{\rho C}{\Gamma (1+\rho )}\bigg )^2T_0^{2\rho -2}t_0'. \end{aligned}$$

Thus we conclude that

$$\begin{aligned}&\bigg \Vert u_n(t)-S_\rho (t)[\widetilde{\phi }(0) + G_n(0, \widetilde{\phi }(0))]+G_n(t, (u_n)_t)-\int _{t_0'}^{t}(t-s)^{\rho -1}Q_\rho (t-s)AG_n(s, (u_n)_s)ds\\&\quad -\int _{t_0'}^{t}(t-s)^{\rho -1}Q_\rho (t-s)[F_n(s, (u_n)_s, u_n[b(u_n(s), s)])+\int _0^sa(s-r) k_n(r, (u_n)_r)d\omega (r)]ds\\&\quad -\displaystyle \sum _{j=1}^rS_\rho (t-t_j)I_{j,n}(u_n)_{t_j}\bigg \Vert \leqslant \bigg [L_{G}\bigg (\frac{\rho C_{1-\beta }\Gamma (1+\beta )}{\Gamma (1+\rho \beta )}\bigg )^2T_0^{2(\rho \beta -1)}+M(R)\bigg (\frac{\rho C}{\Gamma (1+\rho )}\bigg )^2T_0^{2\rho -2}\bigg ]t_0'. \end{aligned}$$

Letting $n\rightarrow \infty$ in the above inequality, we get

$$\begin{aligned}&\bigg \Vert u(t)-S_\rho (t)[\widetilde{\phi }(0) + G(0, \widetilde{\phi }(0))]+G(t, u_t)-\int _{t_0'}^{t}(t-s)^{\rho -1}Q_\rho (t-s)AG(s, u_s)ds\\&\quad -\int _{t_0'}^{t}(t-s)^{\rho -1}Q_\rho (t-s)[F(s, u_s, u[b(u(s), s)])+\int _0^sa(s-r) k(r, u_r)d\omega (r)]ds\\&\quad -\displaystyle \sum _{j=1}^rS_\rho (t-t_j)I_{j}(u_{t_j})\bigg \Vert \leqslant \bigg [L_{G}\bigg (\frac{\rho C_{1-\beta }\Gamma (1+\beta )}{\Gamma (1+\rho \beta )}\bigg )^2T_0^{2(\rho \beta -1)}+M(R)\bigg (\frac{\rho C}{\Gamma (1+\rho )}\bigg )^2T_0^{2\rho -2}\bigg ]t_0'. \end{aligned}$$

Since $t_0'$ is arbitrary and hence we conclude that u(t) satisfies Eq. (4.7). $\square$

5 Faedo–Galerkin approximations

For any $0<T_0<T$, we have a unique u satisfying the integral equation

$$\begin{aligned} u(t)= \left\{ \begin{array}{ll} \widetilde{\phi }(t), \quad t\in [-\tau , 0]; \\ S_\rho (t)[\widetilde{\phi }(0) + G(0, \widetilde{\phi }(0))]-G(t, u_t)+\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AG(s, u_s)ds\\ +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[F(s, u_s, u[b(u(s), s)])\\ +\int _0^sa(s-r) k(r, u_r)d\omega (r)]ds +\displaystyle \sum \nolimits _{j=1}^rS_\rho (t-t_j)I_{j,n}u_{t_j}, \quad t\in [0,T_0]. \end{array} \right. \nonumber \\ \end{aligned}$$

(5.1)

Also we have a unique solution $u_n$ of the approximate integral equation

$$\begin{aligned} u_n(t)=\left\{ \begin{array}{ll} \widetilde{\phi }(t), \quad t\in [-\tau , 0]; \\ S_\rho (t)[\widetilde{\phi }(0) + G(0, P^n\widetilde{\phi }(0))]-G(t, P^n(u_n)_t)\\ \quad +\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AG(s, P^n(u_n)_s)ds\\ +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[F(s, P^n(u_n)_s, P^nu_n[b(P^nu_n(s), s)])\\ {+}\int _0^sa(s{-}r) k(r, P^n(u_n)_r)d\omega (r)]ds {+}\displaystyle \sum \nolimits _{j=1}^rS_\rho (t{-}t_j)I_{j,n}(u_n)_{t_j}, \quad t\in [0,T_0]. \end{array} \right. \nonumber \\ \end{aligned}$$

(5.2)

Now, if we project (5.2) onto $H_n$, we get the Faedo–Galerkin approximations $\widehat{u}_n(t)=P^nu_n(t)$ satisfying

$$\begin{aligned} \widehat{u}_n(t)=\left\{ \begin{array}{ll} P^n\widetilde{\phi }(t), \quad t\in [-\tau , 0]; \\ S_\rho (t)P^n[\widetilde{\phi }(0) + G(0, P^n\widetilde{\phi }(0))]-P^nG(t, (\widehat{u}_n)_t)\\ \quad +\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AP^nG(s, (\widehat{u}_n)_s)ds\\ \quad +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[P^nF(s, (\widehat{u}_n)_s, \widehat{u}_n[b(\widehat{u}_n(s), s)])\\ \quad +\int _0^sa(s-r)P^n k(r, (\widehat{u}_n)_t)d\omega (r)]ds\\ \quad +\displaystyle \sum \nolimits _{j=1}^rS_\rho (t-t_j)P^nI_{j}(\widehat{u}_n)_{t_j}), \quad t\in [0,T_0]. \end{array} \right. \end{aligned}$$

(5.3)

Solutions u of (5.1) and $\widehat{u}_n$ of (5.3) have the representations

$$\begin{aligned} u(t)= & {} \displaystyle \sum _{j=0}^\infty \sigma _j(t)u_j, \quad \quad \sigma _j(t)=(u(t), u_j), \quad j=0,1,\ldots ;\nonumber \\ \widehat{u}_n(t)= & {} \displaystyle \sum _{j=0}^n\sigma _j^n(t)u_j,\quad \quad \sigma _j^n(t)=(\widehat{u}_n(t), u_j), \quad j=0,1,\ldots ; \end{aligned}$$

(5.4)

as a consequence of Theorems 3.1 and 4.1, we have the following results.

Corollary 5.1

Let (H1)–(H6) hold. Then

(a) :

if $\widetilde{\phi }(t)\in \mathcal {L}_2^0(\Omega , D(A^\alpha )$ for all $t\in [-\tau , 0]$, then

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty } \displaystyle \sup _{\{n\geqslant m, -\tau \leqslant t\leqslant T_0\}}\mathbb {E}\Vert A^\alpha [\widehat{u}_n(t)-\widehat{u}_m(t)]\Vert ^2=0. \end{aligned}$$

(b) :

if $\widetilde{\phi }(t)\in \mathcal {L}_2^0(\Omega , D(A)$ for all $t\in [-\tau , 0]$, then

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty } \displaystyle \sup _{\{n\geqslant m, -\tau \leqslant t\leqslant T_0\}}\mathbb {E}\Vert A^\alpha [\widehat{u}_n(t)-\widehat{u}_m(t)]\Vert ^2=0. \end{aligned}$$

Theorem 5.2

Let (H1)–(H6) hold and let $\widetilde{\phi }(t)\in \mathcal {L}_2^0(\Omega , D(A^\alpha )$ for all $t\in [-\tau , 0]$. Then, there exist a unique $\widehat{u}_n\in \mathcal {B}_R$ satisfying

$$\begin{aligned} \widehat{u}_n(t)=\left\{ \begin{array}{ll} P^n\widetilde{\phi }(t), \quad t\in [-\tau , 0]; \\ S_\rho (t)P^n[\widetilde{\phi }(0) + G(0, P^n\widetilde{\phi }(0))]-P^nG(t, (\widehat{u}_n)_t)\\ \quad +\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AP^nG(s, (\widehat{u}_n)_s)ds\\ \quad +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[P^nF(s, (\widehat{u}_n)_s,\\ \quad \widehat{u}_n[b(\widehat{u}_n(s), s)])+\int _0^sa(s-r)P^n k(r, (\widehat{u}_n)_t)d\omega (r)]ds\\ \quad +\displaystyle \sum \nolimits _{j=1}^rS_\rho (t-t_j)P^nI_{j}(\widehat{u}_n)_{t_j}), \quad t\in [0,T_0]. \end{array} \right. \end{aligned}$$

(5.5)

and $u\in \mathcal {B}_R$ satisfying

$$\begin{aligned} u(t)=\left\{ \begin{array}{ll} \widetilde{\phi }(t), \quad t\in [-\tau , 0]; \\ S_\rho (t)[\widetilde{\phi }(0) + G(0, \widetilde{\phi }(0))]-G(t, u_t)+\int _0^t (t-s)^{\rho -1}Q_\rho (t-s)AG(s, u_s)ds\\ +\int ^t_0 (t-s)^{\rho -1}Q_\rho (t-s)[F(s, u_s, u[b(u(s), s)])\\ +\int _0^sa(s-r) k(r, u_r)d\omega (r)]ds +\displaystyle \sum _{j=1}^rS_\rho (t-t_j)I_{j,n}u_{t_j}, \quad t\in [0,T_0]. \end{array} \right. \nonumber \\ \end{aligned}$$

(5.6)

such that $\widehat{u}_n(t)\rightarrow u(t)$ as $n\rightarrow \infty$ in $\mathcal {B}_R$.

Proof

We have

$$\begin{aligned} \mathbb {E}\Vert \widehat{u}_n(t)-u(t)\Vert _\alpha ^2&=\mathbb {E}\Vert P^nu_n(t)-P^nu(t)+P^nu(t)-u(t)\Vert _\alpha ^2\\&\leqslant \mathbb {E}\Vert P^n(u_n(t)-u(t))\Vert _\alpha ^2+\mathbb {E}\Vert (P^n-I)u(t)\Vert _\alpha ^2. \end{aligned}$$

By Theorem 4.2, we have

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty } \displaystyle \sup _{t\in [-r,T]}\mathbb {E}\Vert u_n(t)-u(t)\Vert _\alpha ^2=0. \end{aligned}$$

Thus, this completes the proof of the theorem. $\square$

Now for the convergence $\sigma _j^n$ to $\sigma _j$, we have the following convergence result.

Theorem 5.3

Let (H1)–(H6) hold. Then

(a) :

if $\widetilde{\phi }(0)\in \mathcal {L}_2^0(\Omega , D(A^\alpha )$ for all $t\in [-\tau , 0]$, then

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty } \displaystyle \sup _{t\in [-\tau , T_0]}\left[ \displaystyle \sum _{i=0}^n \lambda _i^{2\alpha }\mathbb {E}\Vert \sigma ^n_i(t)-\sigma _i(t)\Vert ^2\right] =0. \end{aligned}$$

(5.7)

(b) :

if $\widetilde{\phi }(0)\in \mathcal {L}_2^0(\Omega , D(A))$, then

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty } \displaystyle \sup _{t\in [-\tau , T_0]}\left[ \displaystyle \sum _{i=0}^n \lambda _i^{2\alpha }\mathbb {E}\Vert \sigma ^n_i(t)-\sigma _i(t)\Vert ^2\right] =0. \end{aligned}$$

(5.8)

Proof

Since

$$\begin{aligned} \mathbb {E}\Vert A^\alpha (\widehat{u}_n(t)-u(t))\Vert ^2&=\displaystyle \sum _{i=0}^\infty \mathbb {E}\Vert A^\alpha (\sigma _i^n(t)-\sigma _i(t))u_i\Vert ^2 \\&=\displaystyle \sum _{i=0}^\infty \lambda _i^{2\alpha }\mathbb {E}\Vert (\sigma _i^n(t)-\sigma _i(t))u_i\Vert ^2. \end{aligned}$$

Therefore

$$\begin{aligned} \mathbb {E}\Vert A^\alpha (\widehat{u}_n(t)-u(t))\Vert ^2\geqslant \displaystyle \sum _{i=0}^n \lambda _i^{2\alpha }\mathbb {E}\Vert (\sigma _i^n(t)-\sigma _i(t))u_i\Vert ^2, \end{aligned}$$

taking $\displaystyle \lim _{n\rightarrow \infty }$ both sides and using Theorem 5.2, we obtain the required result. $\square$

6 Example

Consider the fractional differential equation in the separable Hilbert space $\mathcal {H}=\mathcal {L}^2(0,1)$

$$\begin{aligned} \left\{ \begin{array}{ll} {}^c\mathbf D ^\rho [V(t,x)+G(t, x, \partial _xV(t+\nu ,x))]-\partial _x^2V(t,x)=F_1(x, V(t,x))+ F_2(t, x, V(t+\nu ,x))\\ \quad \quad \quad \quad +\int _0^ta(t-s)K(s, x, \partial _xV(s,x))\partial \omega (s), \quad x\in (0,1),\, t\in (0,\frac{1}{2})\cup (\frac{1}{2},1) \\ \Delta V(t,x)|_{t=1/2}=\frac{2V(1/2,x)^-}{2+V(1/2,x)^-},\\ V(t, 0)=V(t, 1)=0,\\ V(\nu , x)=\frac{1}{N^2}\cdot \frac{V(\nu , x)}{1+ V(\nu , x)},\quad -\tau \leqslant \nu \leqslant 0, \end{array} \right. \nonumber \\ \end{aligned}$$

(6.1)

where

$$\begin{aligned} F_1(x, V(t,x))=\int _0^x W(s, x)V(s, h(t)|V(t, s)|)ds, \end{aligned}$$

and $F_2:\mathbb {R}_+\times [0, 1]\times \mathbb {R}$ is measurable in x, locally H$\ddot{o}$lder continuous in t, locally Lipschitz continuous in V, uniformly in x, $N\in \mathbb {N}$. We also assume that $W\in C^1([0,1]\times [0,1],\mathbb {R})$ and $h:\mathbb {R}_+\rightarrow \mathbb {R}_+$ is locally H$\ddot{o}$lder continuous in t with $h(0)=0$. $K\in \mathcal {L}_Q(\mathcal {K}, \mathcal {H})$ and $a^2\in \mathcal {L}^q(0, \infty )$ and $\omega$ is a standard Winer process.

Define operator A such that

$$\begin{aligned} Au=-u''\quad \text{ with }\quad u\in D(A)=\mathcal {H}_0^1(0, 1)\cap \mathcal {H}^2(0, 1). \end{aligned}$$

(6.2)

Here $-A$ is infinitesimal generator of an analytic semigroup S(t). Moreover, A has a discrete spectrum with the eigenvalues of the form $k^2\pi ^2$ for $k\in \mathbb {N}$, whose corresponding(normalized) eigenfunctions are given by $e_k(x)=\sqrt{2} \sin k\pi x$. Therefore for $u\in D(A)$

$$\begin{aligned} u(x)=\displaystyle \sum _{k\in \mathbb {N}}<u(x),e_k(x)>e_k(x). \end{aligned}$$

Now, for $\alpha =1/2$, $D(A^{1/2})$ (denoted by $\mathcal {H}_{1/2}$) is a Banach space endowed with the norm

$$\begin{aligned} \Vert u\Vert _{1/2}=\Vert A^{1/2}u\Vert \quad \text{ for }\,u\in D(A^{1/2}). \end{aligned}$$

Also, define the space

$$\begin{aligned} C_t^{1/2}=C([-\tau , t], D(A^{1/2})),\quad t\in [0,T], \end{aligned}$$

endowed with the sup norm

$$\begin{aligned} \Vert \varphi \Vert _{t,1/2}=\displaystyle \sup _{-\tau \leqslant \nu \leqslant t}\Vert \varphi (\nu )\Vert _\alpha ,\quad \varphi \in C_t^{1/2}. \end{aligned}$$

Then, we have

$$\begin{aligned} A^{1/2}u(x)=\displaystyle \sum _{k\in \mathbb {N}}k\pi <u(x),e_k(x)>e_k(x)\quad \text{ with } \,u\in D(A^{1/2}). \end{aligned}$$

The Eq. (6.1) can be reformulated as the following abstract stochastic fractional integro-differential equation with impulsive effects in $\mathcal {H}=\mathcal {L}^2(0, 1)$

$$\begin{aligned} {}^c\mathbf D ^\rho [u(t)+G(t,u_t)]+Au(t)dt=&F(t,u_t,u[b(u(t),t)])dt\nonumber \\&+\int _0^ta(t-s)k(s,u_s)d\omega (s), \nonumber \\ \Delta u(t)|_{t=1/2}=&I(u_t),\nonumber \\ u(0)=&\widetilde{\phi }(0), \end{aligned}$$

(6.3)

where $u(t)=V(t, \cdot )$ i.e. $u(t)(x)=V(t, x)$, A is defined in (6.2), the function $G:[0, T] \times D(A^{1/2})\rightarrow \mathcal {H}$ is defined as

$$\begin{aligned} G(t,u_t)(x)=G(t, x, \partial _xV(t+\nu ,x)). \end{aligned}$$

The function $F:[0, T]\times \mathcal {H}_{1/2}\times \mathcal {H}_{-1/2}$ is defined as

$$\begin{aligned} F(t,\widetilde{\phi },\varphi )(x)=F_1(x, \varphi )+ F_2(t, x, \widetilde{\phi }), \end{aligned}$$

where

$$\begin{aligned} F_1(x, \varphi )=\int _0^x W(x, y) \varphi (y)dy,\quad F_2(t, x, \widetilde{\phi })\leqslant Z(t, x)(1+\Vert \widetilde{\phi }\Vert _{\mathcal {H}^2(0, 1)}). \end{aligned}$$

After a simple calculation, we get

$$\begin{aligned} \mathbb {E}\Vert F_1(x, \varphi _1)-F_1(x, \varphi _2)\Vert ^2\leqslant L_\varphi \mathbb {E}\Vert \varphi _1-\varphi _2\Vert ^2, \end{aligned}$$

and

$$\begin{aligned} \mathbb {E}\Vert F_2(t, x, \widetilde{\phi }_1)-F_2(s, x, \widetilde{\phi }_2)\Vert ^2\leqslant L(|t-s|^{2\gamma _1}+\mathbb {E}\Vert \widetilde{\phi }_1-\widetilde{\phi }_2\Vert ^2). \end{aligned}$$

The function $b:H_0^1(0,1)\times \mathbb {R}^+\rightarrow \mathbb {R}^+$ defined by $b(u(t),t)=h(t)|u(t)|$ satisfies

$$\begin{aligned} |b(u,t)|=|h(t)|u(t)||\leqslant \Vert h\Vert _\infty \times \Vert u\Vert _\infty , \quad t\in [0,1] \end{aligned}$$

As h is a H$\ddot{o}$lder continuous function, there exists a positive constant $L_h$ and $\gamma \in (0,1]$ such that

$$\begin{aligned} |h(t)-h(s)|\leqslant L_h|t-s|^{\gamma }, \quad t,s\in [0,1]. \end{aligned}$$

For $u_1$, $u_2\in H_0^1(0,1)$, we have

$$\begin{aligned} |b(u_1,t)-b(u_2,s)|^2&=|h(t)[|u_1|-|u_2|]+(h(t)-h(s))u_2|^2\\&\leqslant \Vert h\Vert ^2_\infty \mathbb {E}\Vert u_1-u_2\Vert ^2_{H_0^1(0,1)}+L_h|t-s|^{2\gamma }\Vert u_2\Vert ^2_\infty \\&\leqslant \max \{\Vert h\Vert ^2_\infty ,L_h\Vert u_2\Vert ^2_\infty \} \left( \mathbb {E}\Vert u_1-u_2\Vert ^2_{H_0^1(0,1)}+|t-s|^{2\gamma }\right) . \end{aligned}$$

For $u_1$, $u_2\in D(A^{1/2})$, we have

$$\begin{aligned} \mathbb {E}\Vert I(u_1)-I(u_2)\Vert _{1/2}^2\leqslant \frac{16\mathbb {E}\Vert u_1-u_2\Vert ^2_{1/2}}{\Vert (2+u_1)(2+u_2)\Vert ^2_{1/2}}. \end{aligned}$$

It can be easily checked that the assumptions (H1)–(H6) are satisfied. Therefore we may use the results established in the earlier sections to obtain approximate solutions and their convergence.