Well-posedness and regularity of some stochastic time-fractional integral equations in Hilbert space

Abstract

In the current work, we deal with a class of stochastic time-fractional integral equations in Hilbert space by studying their well-posedness and regularity. Precisely, we use the celebrity fixed point theorem to prove the well-posedness of the problem by imposing the global Lipschitz and the linear growth conditions. Further, we prove the spatial and temporal regularity by imposing only a regularity condition on the initial value. An important example is considered in order to confirm and support the validity of our theoretical results.

Keywords:

• Integral equations
• Riemann–Liouville integral operator
• cylindrical Wiener process
• fixed point theorem
• spatial regularity
• temporal regularity

2020 MSCs:

• 35B65
• 35R11
• 35R60
• 60H15
• 65M75

1. Introduction

There are many phenomena in applied sciences as anomalous diffusion in some physical processes and dynamical systems with memory in medicine, which cannot be described adequately by the classical differential and integral equations. This fact gives rise of the theory of fractional differential and integral equations. Such kinds of equations became an effective tool to model such phenomena in the past four decades, although the roots of the fractional calculus that extends to the year 1695, see for a short list [1–9].

However, the nondeterministic nature of the most of such phenomena obliges us to incorporate randomness into their mathematical descriptions and provide more realistic mathematical models of them, and this frequently results in stochastic fractional differential and/or integral equations. For example, Arab et al. [10] studied the fractional stochastic Burgers-type equation and proved not only its well-posedness in the Hölder space, but its numerical approximations as well. The authors in Ref. [11] developed the basic theory of fractional calculus and anomalous diffusion from probability's point of view. While Metzler et al. [12] discussed extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

Recently, a few contributions have been considered by some authors in the analysis of stochastic integro-differential equations driven by a fractional Brownian motion. For details, Caraballo et al. [13] studied the asymptotic behaviour for neutral stochastic integro-differential equations driven by a fractional Brownian motion. Arthi et al. [14] proved the existence and exponential stability of a same class of equations with impulses. In Ref. [15], Sathiyara et al. dealt with the contrabillity of fractional higher order stochastic integro-differential systems with fractional Brownian motion. Moreover, a class of fractional stochastic Itö integral or Skorokhod integral equations has been studied by El-borai et al. [16,17], where the well-posedness of such classes has been established and in the recent literature various qualitative behaviors of solutions of Volterra integral equations, ordinary and stochastic differential delay equations of second order and integro-differential equations of first order have been investigated in [18–22] and some new qualitative results have been obtained in these papers.

From these facts and regarding the importance of the stochastic time-fractional integral equations in the description of some anomalous diffusions, our contribution in the current paper will be the study of such class of equations, which is given in the following general form: (1) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{Au(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}\\ & \times {\int }_0^t\frac{F(u(s))}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{G}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(1)

where, $t\in [0,T]$, for fixed T>0, $\alpha \in (\frac{1}{2},1]$, the initial condition $u_0:=u(0)$ is a H-valued ${\mathcal{F}}_0$-measurable random variable, with H be a real Hilbert space, $A:D(A)\subset H\to H$ is a linear closed operator generates a semigroup $(S(t){)}_{t\in [0,T]}$, $F:H\to H$ and $G:H\to H$ are two operators, W is a H-valued cylindrical Wiener process.

In our work, we consider the fractional integrals in the Riemann–Liouville approach.

It is worth mentioning that in Ref. [16], the authors studied an equation similar to (1(1) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{Au(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}\\ & \times {\int }_0^t\frac{F(u(s))}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{G}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(1) ), perturbed by a multiplicative noise, where W is a Wiener process on the real separable Hilbert space K, with covariance operator Q. They proved only the well-posedness of the problem, where the mild solution $(u(t){)}_{t\in [0,T]}$ satisfies $u(t)$ is an $L^2(\mathrm{\Omega },H)$-valued random variable, for any $t\in [0,T]$. In our case, the equation is perturbed by an additive noise, where we prove the well-posedness, with $u(t)$ is an $L^p(\mathrm{\Omega },H)$-valued random variable, for $p\ge 2$. Further, to the best of the authors knowledge, there is no work has been dealt with the regularity of solutions for such class of equations. For this, our main contribution is not only to study the temporal regularity, but to study the spatial regularity as well.

This work is ordered by the following: Section 2 is devoted to prove the well-posedness of the problem. In Section 3 we prove the spatial regularity of the problem. The temporal regularity is postponted to Section 4. Finally, conclusion is presented in Section 6.

We close this introduction by giving the following notations. For $\mathcal{O}$ an operator we mean by $D(\mathcal{O})$ its domain of definition, H is a real Hilbert space endowed with the norm ${|.|}_H$, $\mathcal{L}(H)$ is the space of linear bounded operators defined on H into it self endowed with the norm ${\Vert .\Vert }_{\mathcal{L}(H)}$, HS is the space of Hilbert-Schmidt operators defined from the Hilbert space H into it self, and we indicate its norm by ${\Vert .\Vert }_{HS}$. Let $(\mathrm{\Omega },\mathcal{F},\mathbb{F},\mathbb{P})$ be a filtered probability space, where $\mathbb{F}:=({\mathcal{F}}_t{)}_{t\in [0,T]}$ is a normal filtration and X be a Banach space, $L^p(\mathrm{\Omega },X)$, for $p\ge 2$ is the space of X-valued pth integrable random variables on Ω, its norm is denoted by ${\Vert .\Vert }_{L^p(\mathrm{\Omega },X)}$, $\mathrm{\Lambda }([0,T];H):=\{v\in C([0,T];L^p(\mathrm{\Omega },H)),v\mathrm{is}\mathbb{F}-adapted\}$ is a Banach space equipped with the norm ${\Vert v\Vert }_{\mathrm{\Lambda }}:=\underset{t\in [0,T]}{sup}{\Vert v(t)\Vert }_{L^p(\mathrm{\Omega },H)}$. We use the abbreviation RHS for right hand side.

2. Well-posedness of problem (1)

The current section is devoted to prove the existence and the uniqueness of a mild solution u defined as follows (see Refs. [16,23,24]).

Definition 2.1

Let $u:=(u(t){)}_{t\in [0,T]}$ be an H-valued stochastic process. u is said to be a mild solution of problem (1(1) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{Au(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}\\ & \times {\int }_0^t\frac{F(u(s))}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{G}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(1) ) if

• for all $t\in [0,T]$, $u(t)$ is ${\mathcal{F}}_t$-adapted,

• u satisfies the following equality in H, $\mathbb{P}$-a.s., (2) $\begin{array}{rl}u(t)& ={\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \\ & +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times F(u(s))\mathrm{d}\theta \mathrm{d}s\\ & +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times G\mathrm{d}\theta \mathrm{d}W(s),\mathrm{\forall }t\in [0,T],\end{array}$(2) where ${\xi }_{\alpha }$ is a probability density function defined on $(0,\mathrm{\infty })$.

We need here to impose the following assumptions to prove the well-posedness of problem (1(1) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{Au(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}\\ & \times {\int }_0^t\frac{F(u(s))}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{G}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(1) ). For $p\ge 2$:

${\mathcal{H}}_A$- The linear operator $A:D(A)\subset H\to H$ is an infinitesimal generator of $C_0$-semigroup $(S(t){)}_{t\in [0,T]}$, satisfies: for all $\gamma \ge 0$ and all $\xi >0$, there exist $C_S,C_{\xi }>0$ such that (3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) and (4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ${\mathcal{H}}_F$- The operator $F:H\to H$ (not necessarily linear) satisfies the global Lipschitz and the linear growth conditions, i.e. (5) $\begin{array}{r}{|F(u)-F(v)|}_H\le C_F{|u-v|}_H,\end{array}$(5) and (6) $\begin{array}{r}{|F(u)|}_H\le C_F{\left|u\right|}_H,\end{array}$(6) for some positive constant $C_F$.

We need here to reformulate Assumption ${\mathcal{H}}_F$ in the random context. To this end, let x and y be two H-valued random variables. Then, we have (7) $\begin{array}{rl}{\Vert F(x)-F(y)\Vert }_{L^p(\mathrm{\Omega },H)}^p& =\mathbb{E}{|F(x)-F(y)|}_H^p\le C_F^p\mathbb{E}{|x-y|}_H^p\\ & =C_F^p{\Vert x-y\Vert }_{L^p(\mathrm{\Omega },H)}^p,\end{array}$(7) and (8) $\begin{array}{r}{\Vert F(x)\Vert }_{L^p(\mathrm{\Omega },H)}^p=\mathbb{E}{|F(x)|}_H^p\le C_F^p\mathbb{E}{\left|x\right|}_H^p=C_F^p{\Vert x\Vert }_{L^p(\mathrm{\Omega },H)}^p.\end{array}$(8) ${\mathcal{H}}_G$- The operator $G:H\to H$ is linear and bounded, i.e. ${\Vert G\Vert }_{\mathcal{L}(H)}\le C_G$, for some positive constant $C_G$.

${\mathcal{H}}_{u_0}$- The initial condition $u_0$ is an ${\mathcal{F}}_0$-measurable random variable, satisfies $u_0\in L^p(\mathrm{\Omega },{\mathcal{F}}_0,\mathbb{P};H)$, i.e. ${\Vert u_0\Vert }_{L^p(\mathrm{\Omega },H)}<\mathrm{\infty }$.

Remark 2.1

In the rest of this paper, when we need to use estimations in the random context as it has been proved above for Assumption ${\mathcal{H}}_F$, we will do it without proof in order to avoid the repetitions.

Theorem 2.1

Under the Assumptions ${\mathcal{H}}_A$ (Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) with $\gamma \in [0,1-\frac{1}{2\alpha })$ and Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) with $\xi \in (0,1-\frac{1}{2\alpha })$), ${\mathcal{H}}_F$, ${\mathcal{H}}_G$ and ${\mathcal{H}}_{u_0}$, the problem (1(1) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{Au(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}\\ & \times {\int }_0^t\frac{F(u(s))}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{G}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(1) ) admits a unique mild solution $u\in \mathrm{\Lambda }([0,T];H)$, provided that $C_S{C}_{\alpha ,1}{C}_F{T}^{\alpha }<1,$ for $\alpha \in (\frac{1}{2},1)$ and $p\ge 2$.

To prove this theorem, we need the following useful result.

Lemma 2.1

see [24]

Let $\alpha \in (0,1)$ and $\nu \in (-1,+\mathrm{\infty })$. It is true that ${\int }_0^{\mathrm{\infty }}{\theta }^{\nu }{\xi }_{\alpha }(\theta )\mathrm{d}\theta =\frac{\mathrm{\Gamma }(1+\nu )}{\mathrm{\Gamma }(1+\alpha \nu )}=:C_{\alpha ,\nu },$ where ${\xi }_{\alpha }$ is a probability density function defined on $(0,\mathrm{\infty })$ and Γ is Gamma function.

Proof

Proof of Theorem 2.1

Let $\alpha \in (\frac{1}{2},1)$ and $p\ge 2$. We define the mapping $\psi :\mathrm{\Lambda }([0,T];H)\to \mathrm{\Lambda }([0,T];H)$ as follows (9) $\begin{array}{rl}& \mathrm{\Psi }(u(t))\\ & :={\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}\\ & \times {\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )F(u(s))\mathrm{d}\theta \mathrm{d}s\\ & +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times G\mathrm{d}\theta \mathrm{d}W(s),\mathrm{\forall }t\in [0,T],\end{array}$(9) First we have, (10) $\begin{array}{rl}& {\Vert (\psi (u))(t)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(10) To estimate ${\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}$, we use Assumption ${\mathcal{H}}_A$; Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) (with $\gamma =0$), Assumption ${\mathcal{H}}_{u_0}$ and Lemma 2.1 (with $\nu =0$) to get (11) $\begin{array}{rl}& {\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\int }_0^{\mathrm{\infty }}{\Vert {\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & \le {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert S(t^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert u_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & \le C_S{\Vert u_0\Vert }_{L^p(\mathrm{\Omega },H)}{\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )\mathrm{d}\theta \le C_S{C}_{\alpha ,0}{\Vert u_0\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(11) To estimate the second term in the RHS of (10(10) $\begin{array}{rl}& {\Vert (\psi (u))(t)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(10) ), let $\alpha \in (\frac{1}{2},1)$. Then, by using Assumption ${\mathcal{H}}_A$; Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) (with $\gamma =0$), Lemma 2.1(with $\nu =1$) and Assumption ${\mathcal{H}}_F$, we obtain (12) $\begin{array}{rl}& \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\Vert \theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & {\times F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta ){\Vert S((t-s{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha C_S{\int }_0^t({\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )\mathrm{d}\theta )(t-s{)}^{\alpha -1}\\ & \times {\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s\\ & \le \alpha C_S{C}_{\alpha ,1}{\int }_0^t(t-s{)}^{\alpha -1}{\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s,\\ & \le \alpha C_S{C}_{\alpha ,1}{C}_F{\int }_0^t(t-s{)}^{\alpha -1}{\Vert u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s,\\ & \le \alpha C_S{C}_{\alpha ,1}{C}_F{\Vert u\Vert }_{\mathrm{\Lambda }}{\int }_0^t(t-s{)}^{\alpha -1}\mathrm{d}s,\\ & \le C_S{C}_{\alpha ,1}{C}_F{\Vert u\Vert }_{\mathrm{\Lambda }}{T}^{\alpha }.\end{array}$(12) Now, to deal with the stochastic term in the RHS of (10(10) $\begin{array}{rl}& {\Vert (\psi (u))(t)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(10) ), we use Burkholder–Davis–Gundy inequality (see [25, Proposition 2.12, p.24]) as follows $\begin{array}{rl}& \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}\times G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \alpha C_p(\mathbb{E}({\int }_0^t\Vert {\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {{{\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )G\mathrm{d}\theta \Vert }_{HS}^2\mathrm{d}s)}^{\frac{p}{2}})}^{\frac{1}{p}}\\ & =\alpha C_p\Vert {\int }_0^t\Vert {\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )G\mathrm{d}\theta \\ & \times {\Vert {}_{HS}^2\mathrm{d}s\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}},\end{array}$ where $C_p:=(\frac{p}{2}(p-1){)}^{\frac{1}{2}}(\frac{p}{p-1}{)}^{(\frac{p}{2}-1)}$.

First, we need to estimate $\Vert {\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta ){S((t-s{)}^{\alpha }\theta )G\mathrm{d}\theta \Vert }_{HS}^2$. To do this, let $\alpha \in (\frac{1}{2},1)$ and $\gamma \in (0,1-\frac{1}{2\alpha })$, by using the fact that ${\Vert AB\Vert }_{HS}\le {\Vert A\Vert }_{HS}{\Vert B\Vert }_{\mathcal{L}(H)}$, for every $A\in HS$ and every $B\in \mathcal{L}(H)$, we obtain $\begin{array}{rl}& {\Vert {\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )G\mathrm{d}\theta \Vert }_{HS}^2\\ & \le {\left({\int }_0^{\mathrm{\infty }}{\Vert \theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )G\Vert }_{HS}\mathrm{d}\theta \right)}^2\\ & \le ({\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta ){\Vert S((t-s{)}^{\alpha }\theta )\Vert }_{HS}\\ & \times {\phantom{{\int }_0^{\mathrm{\infty }}}{\Vert G\Vert }_{\mathcal{L}(H)}\mathrm{d}\theta )}^2\\ & =({\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta ){\Vert A^{-\gamma }{A}^{\gamma }S((t-s{)}^{\alpha }\theta )\Vert }_{HS}\\ & \times {\phantom{{\int }_0^{\mathrm{\infty }}}{\Vert G\Vert }_{\mathcal{L}(H)}\mathrm{d}\theta )}^2\\ & \le ({\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta ){\Vert A^{-\gamma }\Vert }_{HS}\\ & \times {\phantom{{\int }_0^{\mathrm{\infty }}}{\Vert A^{\gamma }S((t-s{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert G\Vert }_{\mathcal{L}(H)}\mathrm{d}\theta )}^2.\end{array}$ The use of Assumption ${\mathcal{H}}_A$; Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) (with $\gamma \in (0,1-\frac{1}{2\alpha })$) and Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) (with $\xi =\gamma$), Assumption ${\mathcal{H}}_G$ and Lemma 2.1 (with $\nu =1-\gamma$, which is possible since $1-\gamma >-1$) helps us to get $\begin{array}{rl}& {\Vert {\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )G\mathrm{d}\theta \Vert }_{HS}^2\\ & \le C_S^2{C}_{\gamma }^2{C}_G^2{C}_{\alpha ,1-\gamma }^2(t-s{)}^{2(\alpha (1-\gamma )-1)}.\end{array}$ And so, $\begin{array}{rl}& \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times {\phantom{{\int }_0^{\mathrm{\infty }}}G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \alpha C_p{\Vert {\int }_0^t{C}_S^2{C}_{\gamma }^2{C}_G^2{C}_{\alpha ,1-\gamma }^2(t-s{)}^{2(\alpha (1-\gamma )-1)}\mathrm{d}s\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le \alpha C_p{C}_S{C}_{\gamma }{C}_G{C}_{\alpha ,1-\gamma }{({\int }_0^t(t-s{)}^{2(\alpha (1-\gamma )-1)}\mathrm{d}s)}^{\frac{1}{2}}.\end{array}$ Thanks to the condition $\gamma \in (0,1-\frac{1}{2\alpha })$, we can easily obtain (13) $\begin{array}{rl}& \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times {\phantom{{\int }_0^{\mathrm{\infty }}}G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \alpha C_p{C}_S{C}_{\gamma }{C}_G{C}_{\alpha ,1-\gamma }\frac{T^{\alpha (1-\gamma )-\frac{1}{2}}}{(2\alpha (1-\gamma )-1{)}^{\frac{1}{2}}}.\end{array}$(13) Coming back to Est.(10(10) $\begin{array}{rl}& {\Vert (\psi (u))(t)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(10) ), we replace Est.(11(11) $\begin{array}{rl}& {\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\int }_0^{\mathrm{\infty }}{\Vert {\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & \le {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert S(t^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert u_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & \le C_S{\Vert u_0\Vert }_{L^p(\mathrm{\Omega },H)}{\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )\mathrm{d}\theta \le C_S{C}_{\alpha ,0}{\Vert u_0\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(11) ), Est.(12(12) $\begin{array}{rl}& \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\Vert \theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & {\times F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta ){\Vert S((t-s{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha C_S{\int }_0^t({\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )\mathrm{d}\theta )(t-s{)}^{\alpha -1}\\ & \times {\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s\\ & \le \alpha C_S{C}_{\alpha ,1}{\int }_0^t(t-s{)}^{\alpha -1}{\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s,\\ & \le \alpha C_S{C}_{\alpha ,1}{C}_F{\int }_0^t(t-s{)}^{\alpha -1}{\Vert u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s,\\ & \le \alpha C_S{C}_{\alpha ,1}{C}_F{\Vert u\Vert }_{\mathrm{\Lambda }}{\int }_0^t(t-s{)}^{\alpha -1}\mathrm{d}s,\\ & \le C_S{C}_{\alpha ,1}{C}_F{\Vert u\Vert }_{\mathrm{\Lambda }}{T}^{\alpha }.\end{array}$(12) ) and Est.(13(13) $\begin{array}{rl}& \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times {\phantom{{\int }_0^{\mathrm{\infty }}}G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \alpha C_p{C}_S{C}_{\gamma }{C}_G{C}_{\alpha ,1-\gamma }\frac{T^{\alpha (1-\gamma )-\frac{1}{2}}}{(2\alpha (1-\gamma )-1{)}^{\frac{1}{2}}}.\end{array}$(13) ) in it, to get (14) $\begin{array}{rl}& {\Vert (\psi (u))(t)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le C_S{C}_{\alpha ,0}{\Vert u_0\Vert }_{L^p(\mathrm{\Omega },H)}+C_S{C}_{\alpha ,1}{C}_F{\Vert u\Vert }_{\mathrm{\Lambda }}{T}^{\alpha }\\ & +\le \alpha C_p{C}_S{C}_{\gamma }{C}_G{C}_{\alpha ,1-\gamma }\frac{T^{\alpha (1-\gamma )-\frac{1}{2}}}{(2\alpha (1-\gamma )-1{)}^{\frac{1}{2}}},\end{array}$(14) For all $t\in [0,T]$. And so Ψ is well defined.

Now, we prove that Ψ is a contraction mapping. To do this, let $u,v\in \mathrm{\Lambda }([0,T];H)$. From Est.(9(9) $\begin{array}{rl}& \mathrm{\Psi }(u(t))\\ & :={\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}\\ & \times {\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )F(u(s))\mathrm{d}\theta \mathrm{d}s\\ & +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times G\mathrm{d}\theta \mathrm{d}W(s),\mathrm{\forall }t\in [0,T],\end{array}$(9) ) we infer that (15) $\begin{array}{rl}& {\Vert (\psi (u))(t)-(\psi (v))(t)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times {\phantom{{\int }_0^{\mathrm{\infty }}}(F(u(s))-F(v(s)))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(15) To estimate the term in the RHS of Est.(15(15) $\begin{array}{rl}& {\Vert (\psi (u))(t)-(\psi (v))(t)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times {\phantom{{\int }_0^{\mathrm{\infty }}}(F(u(s))-F(v(s)))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(15) ), we follow the same steps to lead to Est.(12(12) $\begin{array}{rl}& \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {\phantom{{\int }_0^t}\times S((t-s{)}^{\alpha }\theta )F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\Vert \theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & {\times F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta ){\Vert S((t-s{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha C_S{\int }_0^t({\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )\mathrm{d}\theta )(t-s{)}^{\alpha -1}\\ & \times {\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s\\ & \le \alpha C_S{C}_{\alpha ,1}{\int }_0^t(t-s{)}^{\alpha -1}{\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s,\\ & \le \alpha C_S{C}_{\alpha ,1}{C}_F{\int }_0^t(t-s{)}^{\alpha -1}{\Vert u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s,\\ & \le \alpha C_S{C}_{\alpha ,1}{C}_F{\Vert u\Vert }_{\mathrm{\Lambda }}{\int }_0^t(t-s{)}^{\alpha -1}\mathrm{d}s,\\ & \le C_S{C}_{\alpha ,1}{C}_F{\Vert u\Vert }_{\mathrm{\Lambda }}{T}^{\alpha }.\end{array}$(12) ). Thus (16) $\begin{array}{rl}& \Vert \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times {\phantom{{\int }_0^{\mathrm{\infty }}}(F(u(s))-F(v(s)))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le C_S{C}_{\alpha ,1}{C}_F{T}^{\alpha }{\Vert u-v\Vert }_{\mathrm{\Lambda }}.\end{array}$(16) Consequently, (17) $\begin{array}{r}{\Vert \psi (u)-\psi (v)\Vert }_{\mathrm{\Lambda }}\le C_S{C}_{\alpha ,1}{C}_F{T}^{\alpha }{\Vert u-v\Vert }_{\mathrm{\Lambda }}.\end{array}$(17) If $C_S{C}_{\alpha ,1}{C}_F{T}^{\alpha }<1$, then ψ is a contraction, and so it has a unique fixed point that coincides with a unique mild solution of problem (1(1) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{Au(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}\\ & \times {\int }_0^t\frac{F(u(s))}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{G}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(1) ).

3. Spatial regularity of problem (1)

In this section, we study the spatial regularity of the mild solution. Its main result is the following.

Theorem 3.1

Let $u\in \mathrm{\Lambda }([0,T];H)$ be the mild solution of problem (1(1) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{Au(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}\\ & \times {\int }_0^t\frac{F(u(s))}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{G}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(1) ) with an initial condition $u_0$ satisfies ${\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}<\mathrm{\infty }$, for all $\eta \in (0,\frac{1}{2}-\frac{1}{4\alpha })$. According to Assumptions ${\mathcal{H}}_A$, ${\mathcal{H}}_F$ and ${\mathcal{H}}_G$, the solution u satisfies $\begin{array}{r}{\Vert u\Vert }_{C([0,T];L^p(\mathrm{\Omega },D(A^{\eta })))}\le C^{\ast },\end{array}$ for some positive constant $C^{\ast }$.

Before proving this result, we first need to recall the Grönwall lemma (see [26, Lemma 7.1.1] and [25, Lemma A.2]).

Lemma 3.1

Let $f:[0,T]\to \mathbb{R}$ be a positive and continuous function, for a fixed T>0. If there exists $\varrho >0$ such that $\begin{array}{r}f(t)\le C_1+C_2{\int }_0^t(t-s{)}^{\varrho -1}f(s)\mathrm{d}s,\mathrm{\forall }t\in (0,T],\end{array}$ for some two positive constants $C_1$ and $C_2$. Then, there exists $C=C_{(C_2,T,\varrho )}>0$, such that $\begin{array}{r}f(t)\le C_1{C}_{(C_2,T,\varrho )}.\end{array}$

Proof

Proof of Theorem 3.1

Let $p\ge 2$, $\alpha \in (\frac{1}{2},1)$ and $\eta \in (0,\frac{1}{2}-\frac{1}{4\alpha })$ (which is possible thanks to the condition $\alpha \in (\frac{1}{2},1)$). First, from Equation (2(2) $\begin{array}{rl}u(t)& ={\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \\ & +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times F(u(s))\mathrm{d}\theta \mathrm{d}s\\ & +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times G\mathrm{d}\theta \mathrm{d}W(s),\mathrm{\forall }t\in [0,T],\end{array}$(2) ), we have (18) $\begin{array}{rl}& {\Vert A^{\eta }u(t)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )A^{\eta }S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\alpha \Vert {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )A^{\eta }S((t-s{)}^{\alpha }\theta )\\ & {\times \phantom{{\int }_0^{\mathrm{\infty }}}F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\alpha \Vert {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )A^{\eta }S((t-s{)}^{\alpha }\theta )\\ & {\times \phantom{{\int }_0^{\mathrm{\infty }}}G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)},\\ & :=S_1+S_2+S_3,\end{array}$(18) for all $t\in [0,T]$. Then, the use of Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) (with $\gamma =0$) and the application of Lemma 2.1 (with $\nu =0$) lead to (19) $\begin{array}{rl}{S}_1& :={\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )A^{\eta }S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert A^{\eta }S(t^{\alpha }\theta )u_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & ={\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert S(t^{\alpha }\theta )A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & \le {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert S(t^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & \le C_S{C}_{\alpha ,0}{\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(19) Again, the use of Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) (with $\gamma =\eta$), the Assumption ${\mathcal{H}}_F$ and Lemma 2.1 (with $\nu =1-\eta$) yields $\begin{array}{rl}{S}_2& :=\alpha \Vert {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )A^{\eta }S((t-s{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}\times F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(t-s{)}^{\alpha -1}\\ & \times {\Vert A^{\eta }S((t-s{)}^{\alpha }\theta )F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(t-s{)}^{\alpha -1}{\Vert A^{\eta }S((t-s{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha C_S{\int }_0^t{\int }_0^{\mathrm{\infty }}{\theta }^{1-\eta }{\xi }_{\alpha }(\theta )(t-s{)}^{\alpha (1-\eta )-1}\\ & \times {\Vert F(u(s))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha C_S{C}_F{\int }_0^t{\int }_0^{\mathrm{\infty }}{\theta }^{1-\eta }{\xi }_{\alpha }(\theta )(t-s{)}^{\alpha (1-\eta )-1}\\ & \times {\Vert u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}s\\ & \le \alpha C_S{C}_F{C}_{\alpha ,1-\eta }{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}{\Vert u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s.\end{array}$ The fact ${\Vert A\Vert }_{\mathcal{L}(H)}\le {\Vert A\Vert }_{HS}$, $\mathrm{\forall }A\in HS$ and Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) (with $\xi =\eta$) yield (20) $\begin{array}{rl}{S}_2& \le \alpha C_S{C}_F{C}_{\alpha ,1-\eta }{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}\\ & \times {\Vert A^{-\eta }{A}^{\eta }u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s\\ & \le \alpha C_S{C}_F{C}_{\alpha ,1-\eta }{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}{\Vert A^{-\eta }\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert A^{\eta }u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s\\ & \le \alpha C_S{C}_F{C}_{\alpha ,1-\eta }{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}{\Vert A^{-\eta }\Vert }_{HS}\\ & \times {\Vert A^{\eta }u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s\\ & \le \alpha C_S{C}_F{C}_{\alpha ,1-\eta }{C}_{\eta }{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}\\ & \times {\Vert A^{\eta }u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s.\end{array}$(20) An application of Burkholder-Davis-Gundy inequality enables us to estimate $S_3$ as follows $\begin{array}{rl}{S}_3& :=\alpha \Vert {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )A^{\eta }S((t-s{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}\times G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le \alpha C_p\Vert {\int }_0^t\Vert {\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )A^{\eta }S((t-s{)}^{\alpha }\theta )\\ & {{\phantom{{\int }_0^t}\times G\mathrm{d}\theta \Vert }_{HS}^2\mathrm{d}s\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le \alpha C_p\Vert {\int }_0^t({\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {{\phantom{{\int }_0^t}\times {\Vert A^{\eta }S((t-s{)}^{\alpha }\theta )G\Vert }_{HS}\mathrm{d}\theta )}^2\mathrm{d}s\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le \alpha C_p\Vert {\int }_0^t({\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\\ & {{\phantom{{\int }_0^t}\times \Vert A^{\eta }S((t-s{)}^{\alpha }\theta ){\Vert }_{HS}\Vert G{\Vert }_{\mathcal{L}(H)}\mathrm{d}\theta )}^2\mathrm{d}s\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le \alpha C_p\Vert {\int }_0^t({\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )\Vert A^{-\eta }{\Vert }_{HS}\\ & {{\times \phantom{{\int }_0^t}\Vert A^{2\eta }S((t-s{)}^{\alpha }\theta ){\Vert }_{\mathcal{L}(H)}\Vert G{\Vert }_{\mathcal{L}(H)}\mathrm{d}\theta )}^2\mathrm{d}s\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}.\end{array}$ The use of Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) (with $\gamma =2\eta$), Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) (with $\xi =\eta$), Lemma 2.1 (with $\nu =1-2\eta$) and Assumption ${\mathcal{H}}_G$, helps us to get (21) $\begin{array}{rl}{S}_3& \le \alpha C_p{C}_S{C}_{\xi }{C}_{\alpha ,1-2\eta }\Vert {\int }_0^t(t-s{)}^{2\alpha (1-2\eta )-2}\\ & {\phantom{{\int }_0^t}\times {\Vert G\Vert }_{\mathcal{L}(H)}^2\mathrm{d}s\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le \alpha C_p{C}_S{C}_{\xi }{C}_{\alpha ,1-2\eta }{C}_G{({\int }_0^t(t-s{)}^{2\alpha (1-2\eta )-2}\mathrm{d}s)}^{\frac{1}{2}}\\ & \le \alpha C_p{C}_S{C}_{\xi }{C}_{\alpha ,1-2\eta }{C}_G\frac{T^{\alpha (1-2\eta )-\frac{1}{2}}}{(2\alpha (1-2\eta )-1{)}^{\frac{1}{2}}}.\end{array}$(21) By replacing Est.(19(19) $\begin{array}{rl}{S}_1& :={\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )A^{\eta }S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert A^{\eta }S(t^{\alpha }\theta )u_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & ={\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert S(t^{\alpha }\theta )A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & \le {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert S(t^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & \le C_S{C}_{\alpha ,0}{\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$(19) ), Est.(20(20) $\begin{array}{rl}{S}_2& \le \alpha C_S{C}_F{C}_{\alpha ,1-\eta }{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}\\ & \times {\Vert A^{-\eta }{A}^{\eta }u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s\\ & \le \alpha C_S{C}_F{C}_{\alpha ,1-\eta }{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}{\Vert A^{-\eta }\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert A^{\eta }u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s\\ & \le \alpha C_S{C}_F{C}_{\alpha ,1-\eta }{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}{\Vert A^{-\eta }\Vert }_{HS}\\ & \times {\Vert A^{\eta }u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s\\ & \le \alpha C_S{C}_F{C}_{\alpha ,1-\eta }{C}_{\eta }{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}\\ & \times {\Vert A^{\eta }u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s.\end{array}$(20) ) and Est.(21(21) $\begin{array}{rl}{S}_3& \le \alpha C_p{C}_S{C}_{\xi }{C}_{\alpha ,1-2\eta }\Vert {\int }_0^t(t-s{)}^{2\alpha (1-2\eta )-2}\\ & {\phantom{{\int }_0^t}\times {\Vert G\Vert }_{\mathcal{L}(H)}^2\mathrm{d}s\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le \alpha C_p{C}_S{C}_{\xi }{C}_{\alpha ,1-2\eta }{C}_G{({\int }_0^t(t-s{)}^{2\alpha (1-2\eta )-2}\mathrm{d}s)}^{\frac{1}{2}}\\ & \le \alpha C_p{C}_S{C}_{\xi }{C}_{\alpha ,1-2\eta }{C}_G\frac{T^{\alpha (1-2\eta )-\frac{1}{2}}}{(2\alpha (1-2\eta )-1{)}^{\frac{1}{2}}}.\end{array}$(21) ) in Est.(18(18) $\begin{array}{rl}& {\Vert A^{\eta }u(t)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le {\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )A^{\eta }S(t^{\alpha }\theta )u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\alpha \Vert {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )A^{\eta }S((t-s{)}^{\alpha }\theta )\\ & {\times \phantom{{\int }_0^{\mathrm{\infty }}}F(u(s))\mathrm{d}\theta \mathrm{d}s\Vert }_{L^p(\mathrm{\Omega },H)}\\ & +\alpha \Vert {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )A^{\eta }S((t-s{)}^{\alpha }\theta )\\ & {\times \phantom{{\int }_0^{\mathrm{\infty }}}G\mathrm{d}\theta \mathrm{d}W(s)\Vert }_{L^p(\mathrm{\Omega },H)},\\ & :=S_1+S_2+S_3,\end{array}$(18) ), we obtain $\begin{array}{rl}{\Vert A^{\eta }u(t)\Vert }_{L^p(\mathrm{\Omega },H)}& \le C_1+C_2{\int }_0^t(t-s{)}^{\alpha (1-\eta )-1}\\ & \times {\Vert A^{\eta }u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}s.,\end{array}$ where $C_1:=C_S{C}_{\alpha ,0}{\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}+\alpha C_p{C}_S{C}_{\xi }{C}_{\alpha ,1-2\eta }{C}_G\frac{T^{\alpha (1-2\eta )-\frac{1}{2}}}{(2\alpha (1-2\eta )-1{)}^{\frac{1}{2}}}$ and $C_2:=\alpha C_S{C}_F{C}_{\alpha ,1-\eta }{C}_{\eta }$.

An application of Grönwall Lemma 3.1 with $\varrho =\alpha (1-\eta )$ (which is possible thanks to the condition $\eta <\frac{1}{2}-\frac{1}{4\alpha }<1$) yields (22) $\begin{array}{r}{\Vert A^{\eta }u(t)\Vert }_{L^p(\mathrm{\Omega },H)}\le C_1{C}_{(C_2,T,\alpha (1-\eta ))},\text{for all}t\in (0,T].\end{array}$(22) Then, the desired result is obtained.

4. Temporal regularity of problem (1)

We deal with the temporal regularity of the mild solution in this section, where its main result is the following.

Theorem 4.1

According to Assumptions ${\mathcal{H}}_A$, ${\mathcal{H}}_F$ and ${\mathcal{H}}_G$, the mild solution u of (1(1) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{Au(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}\\ & \times {\int }_0^t\frac{F(u(s))}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{G}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(1) ) with an initial condition $u_0$ satisfies ${\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}<\mathrm{\infty }$, for all $\eta \in (0,1-\frac{1}{2\alpha })$, is time Hölder continuous with exponent $\mu :=min\{\alpha \eta ,1-\alpha ,\alpha (1-\eta )-\frac{1}{2}\}$, i.e. $\begin{array}{r}{\Vert u(t)-u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\le C(t-s{)}^{\mu },\end{array}$ for all $0<t_0\le s<t\le T$, with $t_0$ and T be fixed and for some positive constant C.

In order to prove Theorem 4.1, we need the following useful lemma besides Lemma 2.1.

Lemma 4.1

see [24, Lemma 2.5]

Let T>0 be fixed. For any $0\le s<t\le T$ and any $\kappa \in [0,1]$. It is true that $\begin{array}{r}{t}^{\kappa }-s^{\kappa }\le \kappa c^{\kappa -1}(t-s{)}^{\kappa },\end{array}$ where $c\in (0,1)$.

Proof

Proof of Theorem 4.1

Let $p\ge 2$, $\alpha \in (\frac{1}{2},1)$, $t_0>0$, $0<t_0\le s<t\le T$ and $\eta \in (0,1-\frac{1}{2\alpha })$ (which is possible thanks to the condition $\alpha <\frac{1}{2}$). From Equation (2(2) $\begin{array}{rl}u(t)& ={\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )S(t^{\alpha }\theta )u_0\mathrm{d}\theta \\ & +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times F(u(s))\mathrm{d}\theta \mathrm{d}s\\ & +\alpha {\int }_0^t{\int }_0^{\mathrm{\infty }}\theta (t-s{)}^{\alpha -1}{\xi }_{\alpha }(\theta )S((t-s{)}^{\alpha }\theta )\\ & \times G\mathrm{d}\theta \mathrm{d}W(s),\mathrm{\forall }t\in [0,T],\end{array}$(2) ) we have (23) $\begin{array}{r}{\Vert u(t)-u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\le \sum _{i=1}^7{R}_i,\end{array}$(23) where $\begin{array}{rl}{R}_1& :={\Vert {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )(S(t^{\alpha }\theta )-S(s^{\alpha }\theta ))u_0\mathrm{d}\theta \Vert }_{L^p(\mathrm{\Omega },H)},\\ R_2& :=\alpha \Vert {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}(S((t-r{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}-S((s-r{)}^{\alpha }\theta ))F(u(r))\mathrm{d}\theta \mathrm{d}r\Vert }_{L^p(\mathrm{\Omega },H)}\\ R_3& :=\alpha \Vert {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1})\\ & {\phantom{{\int }_0^t}\times S((t-r{)}^{\alpha }\theta )F(u(r))\mathrm{d}\theta \mathrm{d}r\Vert }_{L^p(\mathrm{\Omega },H)}\\ R_4& :=\alpha \Vert {\int }_s^t{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(t-r{)}^{\alpha -1}S((t-r{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}\times F(u(r))\mathrm{d}\theta \mathrm{d}r\Vert }_{L^p(\mathrm{\Omega },H)},\\ R_5& :=\alpha \Vert {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}(S((t-r{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}-S((s-r{)}^{\alpha }\theta ))G\mathrm{d}\theta \mathrm{d}W(r)\Vert }_{L^p(\mathrm{\Omega },H)}\\ R_6& :=\alpha \Vert {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1})\\ & {\phantom{{\int }_0^t}\times S((t-r{)}^{\alpha }\theta )G\mathrm{d}\theta \mathrm{d}W(r)\Vert }_{L^p(\mathrm{\Omega },H)}\end{array}$ and $\begin{array}{rl}{R}_7& :=\alpha \Vert {\int }_s^t{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(t-r{)}^{\alpha -1}S((t-r{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}\times G\mathrm{d}\theta \mathrm{d}W(r)\Vert }_{L^p(\mathrm{\Omega },H)}.\end{array}$ To estimate $R_1$, let $\eta \in (0,1-\frac{1}{2\alpha })$. First, the semigroup property enables us to write $\begin{array}{rl}{R}_1& \le {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert (S(t^{\alpha }\theta )-S(s^{\alpha }\theta ))u_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & ={\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert {\int }_s^t\frac{\mathrm{d}S(r^{\alpha }\theta )}{\mathrm{d}r}{u}_0\mathrm{d}r\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & ={\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\Vert {\int }_s^t\alpha \theta r^{\alpha -1}AS(r^{\alpha }\theta )u_0\mathrm{d}r\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \\ & \le {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\int }_s^t\alpha \theta r^{\alpha -1}{\Vert AS(r^{\alpha }\theta )u_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}r\mathrm{d}\theta .\end{array}$ By using Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) (with $\gamma =1-\eta$) and Lemma 2.1 (with $\nu =\eta$), we obtain $\begin{array}{rl}{R}_1& \le \alpha {\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\int }_s^t\theta r^{\alpha -1}{\Vert A^{1-\eta }S(r^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}r\mathrm{d}\theta \\ & \le \alpha {\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}{\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta ){\int }_s^t\theta r^{\alpha -1}\\ & \times {\Vert A^{1-\eta }S(r^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\mathrm{d}r\mathrm{d}\theta \\ & \le \alpha {\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}{C}_S{\int }_0^{\mathrm{\infty }}{\theta }^{\eta }{\xi }_{\alpha }(\theta ){\int }_s^t{r}^{\alpha \eta -1}\mathrm{d}r\mathrm{d}\theta \\ & =\alpha {\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}{C}_S{C}_{\alpha ,\eta }\frac{t^{\alpha \eta }-s^{\alpha \eta }}{\alpha \eta }.\end{array}$ The use of Lemma 4.1 (with $\kappa =\alpha \eta$), which is possible since $\alpha \eta \in (0,1)$ yields (24) $\begin{array}{r}{R}_1\le C_1(t-s{)}^{\alpha \eta },\end{array}$(24) where $C_1:=\alpha {\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}{C}_S{C}_{\alpha ,\eta }{c}^{\alpha \eta -1}$, with $c\in (0,1)$.

Let $\eta \in (0,1-\frac{1}{2\alpha })$. To estimate $R_2$ we have $\begin{array}{rl}{R}_2& \le \alpha {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}\Vert (S((t-r{)}^{\alpha }\theta )\\ & {-S((s-r{)}^{\alpha }\theta ))F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}r\\ & \le {\alpha }^2{\int }_0^s{\int }_0^{\mathrm{\infty }}{\theta }^2{\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}{\int }_s^t(\zeta -r{)}^{\alpha -1}\\ & \times {\Vert AS((\zeta -r{)}^{\alpha }\theta )F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\zeta \mathrm{d}\theta \mathrm{d}r\\ & \le {\alpha }^2{\int }_0^s{\int }_0^{\mathrm{\infty }}{\theta }^2{\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}{\int }_s^t(\zeta -r{)}^{\alpha -1}\\ & \times {\Vert A^{1+\eta }S((\zeta -r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert A^{-\eta }F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\zeta \mathrm{d}\theta \mathrm{d}r\\ & \le {\alpha }^2{\int }_0^s{\int }_0^{\mathrm{\infty }}{\theta }^2{\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}{\int }_s^t(\zeta -r{)}^{\alpha -1}\\ & \times {\Vert A^{1+\eta }S((\zeta -r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\zeta \mathrm{d}\theta \mathrm{d}r\\ & \le {\alpha }^2{\int }_0^s{\int }_0^{\mathrm{\infty }}{\theta }^2{\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}{\int }_s^t(\zeta -r{)}^{\alpha -1}\\ & \times {\Vert A^{1+\eta }S((\zeta -r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }\Vert }_{HS}\\ & \times {\Vert F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\zeta \mathrm{d}\theta \mathrm{d}r.\end{array}$ The use of Assumption ${\mathcal{H}}_A$ (Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) with $\gamma =1+\eta$ and Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) with $\xi =\eta$), Assumption ${\mathcal{H}}_F$ and Lemma 2.1 (with $\nu =1-\eta$) lead to $\begin{array}{rl}{R}_2& \le {\alpha }^2{C}_S{C}_{\eta }{C}_F{\int }_0^s{\int }_0^{\mathrm{\infty }}{\theta }^{1-\eta }{\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}\\ & \times {\int }_s^t(\zeta -r{)}^{-\alpha \eta -1}{\Vert u(r)\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\zeta \mathrm{d}\theta \mathrm{d}r\\ & \le {\alpha }^2{C}_S{C}_{\eta }{C}_F{C}_{\alpha ,1-\eta }{\Vert u\Vert }_{\mathrm{\Lambda }}{\int }_0^s(s-r{)}^{\alpha -1}\\ & \times {\int }_s^t(\zeta -r{)}^{-\alpha \eta -1}\mathrm{d}\zeta \mathrm{d}r\\ & ={\alpha }^2{C}_S{C}_{\eta }{C}_F{C}_{\alpha ,1-\eta }{\Vert u\Vert }_{\mathrm{\Lambda }}\frac{1}{\alpha \eta }{\int }_0^s(s-r{)}^{\alpha -1}\\ & \times ((s-r{)}^{-\alpha \eta }-(t-r{)}^{-\alpha \eta })\mathrm{d}r\\ & ={\alpha }^2{C}_S{C}_{\eta }{C}_F{C}_{\alpha ,1-\eta }{\Vert u\Vert }_{\mathrm{\Lambda }}\frac{1}{\alpha \eta }{\int }_0^s\\ & \times (s-r{)}^{\alpha -1}\frac{((t-r{)}^{\alpha \eta }-(s-r{)}^{\alpha \eta })}{(t-r{)}^{\alpha \eta }(s-r{)}^{\alpha \eta }}\mathrm{d}r\\ & \le {\alpha }^2{C}_S{C}_{\eta }{C}_F{C}_{\alpha ,1-\eta }{\Vert u\Vert }_{\mathrm{\Lambda }}\frac{1}{\alpha \eta }{t}_0^{-2\alpha \eta }{\int }_0^s(s-r{)}^{\alpha -1}\\ & \times ((t-r{)}^{\alpha \eta }-(s-r{)}^{\alpha \eta })\mathrm{d}r\end{array}$ An application of Lemma 4.1 (with $\kappa =\alpha \eta$) yields (25) $\begin{array}{rl}{R}_2& \le {\alpha }^2{C}_S{C}_{\eta }{C}_F{C}_{\alpha ,1-\eta }{\Vert u\Vert }_{\mathrm{\Lambda }}{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}(t-s{)}^{\alpha \eta }\\ & \times {\int }_0^s(s-r{)}^{\alpha -1}\mathrm{d}r\le C_2(t-s{)}^{\alpha \eta },\end{array}$(25) where $C_2:=\alpha C_S{C}_{\eta }{C}_F{C}_{\alpha ,1-\eta }{\Vert u\Vert }_{\mathrm{\Lambda }}{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}{T}^{\alpha }$ and $c\in (0,1)$.

Let $\eta \in (0,1-\frac{1}{2\alpha })$. We estimate $R_3$ according to Assumption ${\mathcal{H}}_F$, Assumption ${\mathcal{H}}_A$ (Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) with $\gamma =\eta$ and Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) with $\xi =\eta$), Lemma 2.1 (with $\nu =1-\eta$) and Lemma 4.1 (with $\kappa =1-\alpha$), as follows (26) $\begin{array}{rl}{R}_3& \le \alpha {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )((s-r{)}^{\alpha -1}-(t-r{)}^{\alpha -1})\\ & \times {\Vert S((t-r{)}^{\alpha }\theta )F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}r\\ & \le \alpha {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )((s-r{)}^{\alpha -1}-(t-r{)}^{\alpha -1})\\ & \times {\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}r\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{\int }_0^s{\int }_0^{\mathrm{\infty }}{\theta }^{1-\eta }{\xi }_{\alpha }(\theta )\\ & \times ((s-r{)}^{\alpha -1}-(t-r{)}^{\alpha -1})(t-r{)}^{-\alpha \eta }\mathrm{d}\theta \mathrm{d}r\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }{\int }_0^s((s-r{)}^{\alpha -1}-(t-r{)}^{\alpha -1})\\ & \times (t-r{)}^{-\alpha \eta }\mathrm{d}r\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }{\int }_0^s\\ & \times \frac{((t-r{)}^{1-\alpha }-(s-r{)}^{1-\alpha })}{(s-r{)}^{1-\alpha }(t-r{)}^{1-\alpha }}(t-r{)}^{-\alpha \eta }\mathrm{d}r\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }{t}_0^{-2(1-\alpha )}(1-\alpha )\\ & \times c^{-\alpha }(t-s{)}^{1-\alpha }{\int }_0^s(t-r{)}^{-\alpha \eta }\mathrm{d}r\\ & \le C_3(t-s{)}^{1-\alpha },\end{array}$(26) where $C_3:=\alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }{t}_0^{-2(1-\alpha )}(1-\alpha )c^{-\alpha }\frac{T^{1-\alpha \eta }}{1-\alpha \eta }$.

To estimate $R_4$, let $\eta \in (0,1-\frac{1}{2\alpha })$. We use Assumption ${\mathcal{H}}_F$, Assumption ${\mathcal{H}}_A$ (Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) with $\gamma =\eta$ and Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) with $\xi =\eta$) and Lemma 2.1 (with $\nu =1-\eta$) and argue as above. Then, (27) $\begin{array}{rl}{R}_4& \le \alpha {\int }_s^t{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(t-r{)}^{\alpha -1}\\ & \times {\Vert S((t-r{)}^{\alpha }\theta )F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}r,\\ & \le \alpha {\int }_s^t{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(t-r{)}^{\alpha -1}{\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert A^{-\eta }F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}r,\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{\int }_s^t{\int }_0^{\mathrm{\infty }}{\theta }^{1-\eta }{\xi }_{\alpha }(\theta )(t-r{)}^{\alpha -1}\\ & \times (t-r{)}^{-\alpha \eta }\mathrm{d}\theta \mathrm{d}r,\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }{\int }_s^t(t-r{)}^{\alpha (1-\eta )-1}\mathrm{d}r\\ & \le C_4(t-s{)}^{\alpha (1-\eta )},\end{array}$(27) where $C_4:=\alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }\frac{1}{\alpha (1-\eta )}$.

To estimate $R_5$ we first use Burkholder–Davis–Gundy inequality as follows (28) $\begin{array}{rl}{R}_5& :=\alpha \Vert {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}(S((t-r{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}-S((s-r{)}^{\alpha }\theta ))G\mathrm{d}\theta \mathrm{d}W(r)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le C_p\alpha \Vert {\int }_0^s(s-r{)}^{2\alpha -2}\\ & \times ({\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )\Vert (S((t-r{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}\times -S((s-r{)}^{\alpha }\theta ))G{\Vert }_{HS}\mathrm{d}\theta )}^2\mathrm{d}r{\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}},\end{array}$(28) To deal with the term $\Vert (S((t-r{)}^{\alpha }\theta )-S((s-r{)}^{\alpha }\theta )){G\Vert }_{HS}$, let $\eta \in (0,1-\frac{1}{2\alpha })$. The use of Assumption ${\mathcal{H}}_A$ (Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) with $\gamma =1+\eta$ and Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) with $\xi =\eta$) and Lemma 4.1 (with $\kappa =\alpha \eta$) yields $\begin{array}{rl}& {\Vert (S((t-r{)}^{\alpha }\theta )-S((s-r{)}^{\alpha }\theta ))G\Vert }_{HS}\\ & ={\Vert {\int }_s^t\alpha \theta (\zeta -r{)}^{\alpha -1}AS((\zeta -r{)}^{\alpha }\theta )G\mathrm{d}\zeta \Vert }_{HS}\\ & \le \alpha \theta {\int }_s^t(\zeta -r{)}^{\alpha -1}{\Vert A^{1+\eta }S((\zeta -r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert A^{-\eta }G\Vert }_{HS}\mathrm{d}\zeta \\ & \le \alpha \theta {\int }_s^t(\zeta -r{)}^{\alpha -1}{\Vert A^{1+\eta }S((\zeta -r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert A^{-\eta }\Vert }_{HS}{\Vert G\Vert }_{\mathcal{L}(H)}\mathrm{d}\zeta \\ & \le \alpha {\theta }^{-\eta }{C}_S{C}_{\eta }{\Vert G\Vert }_{\mathcal{L}(H)}{\int }_s^t(\zeta -r{)}^{-\alpha \eta -1}\mathrm{d}\zeta \\ & \le \alpha {\theta }^{-\eta }{C}_S{C}_{\eta }{\Vert G\Vert }_{\mathcal{L}(H)}\frac{1}{\alpha \eta }((s-r{)}^{-\alpha \eta }-(t-r{)}^{-\alpha \eta })\\ & \le \alpha {\theta }^{-\eta }{C}_S{C}_{\eta }{\Vert G\Vert }_{\mathcal{L}(H)}\frac{t_0^{-2\alpha \eta }}{\alpha \eta }((t-r{)}^{\alpha \eta }-(s-r{)}^{\alpha \eta })\\ & \le \alpha {\theta }^{-\eta }{C}_S{C}_{\eta }{\Vert G\Vert }_{\mathcal{L}(H)}{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}(t-s{)}^{\alpha \eta }.\end{array}$ Coming back to Est.(28(28) $\begin{array}{rl}{R}_5& :=\alpha \Vert {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(s-r{)}^{\alpha -1}(S((t-r{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}-S((s-r{)}^{\alpha }\theta ))G\mathrm{d}\theta \mathrm{d}W(r)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le C_p\alpha \Vert {\int }_0^s(s-r{)}^{2\alpha -2}\\ & \times ({\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )\Vert (S((t-r{)}^{\alpha }\theta )\\ & {\phantom{{\int }_0^t}\times -S((s-r{)}^{\alpha }\theta ))G{\Vert }_{HS}\mathrm{d}\theta )}^2\mathrm{d}r{\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}},\end{array}$(28) ), an application of Lemma 2.1 (with $\nu =1-\eta$) and the use of Assumption ${\mathcal{H}}_G$ give us (29) $\begin{array}{rl}{R}_5& \le C_p{\alpha }^2{C}_S{C}_{\eta }{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}(t-s{)}^{\alpha \eta }{C}_{\alpha ,1-\eta }\\ & \times \Vert {\int }_0^s(s-r{)}^{2\alpha -2}\Vert G{\Vert {}_{\mathcal{L}(H)}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le C_p{\alpha }^2{C}_S{C}_{\eta }{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}(t-s{)}^{\alpha \eta }{C}_{\alpha ,1-\eta }{C}_G\\ & \times {({\int }_0^s(s-r{)}^{2\alpha -2}\mathrm{d}r)}^{\frac{1}{2}}\\ & \le C_p{\alpha }^2{C}_S{C}_{\eta }{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}(t-s{)}^{\alpha \eta }{C}_{\alpha ,1-\eta }{C}_G\frac{T^{\alpha -\frac{1}{2}}}{(2\alpha -1{)}^{\frac{1}{2}}}\\ & \le C_5(t-s{)}^{\alpha \eta },\end{array}$(29) where $C_5:=C_p{\alpha }^2{C}_S{C}_{\eta }{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}{C}_{\alpha ,1-\eta }{C}_G\frac{T^{\alpha -\frac{1}{2}}}{(2\alpha -1{)}^{\frac{1}{2}}}$.

To estimate $R_6$, we have (30) $\begin{array}{rl}{R}_6& :=\alpha \Vert {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1})\\ & {\phantom{{\int }_0^s}\times S((t-r{)}^{\alpha }\theta )G\mathrm{d}\theta \mathrm{d}W(r)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le C_p\alpha \Vert {\int }_0^s((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1}{)}^2\\ & {\times {({\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )\Vert S((t-r{)}^{\alpha }\theta )G{\Vert }_{HS}\mathrm{d}\theta )}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}.\end{array}$(30) Let $\eta \in (0,1-\frac{1}{2\alpha })$. The use of Assumption ${\mathcal{H}}_A$ (Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) with $\gamma =\eta$ and Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) with $\xi =\eta$) enables us to write (31) $\begin{array}{rl}& {\Vert S((t-r{)}^{\alpha }\theta )G\Vert }_{HS}\\ & \le {\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }G\Vert }_{HS}\\ & \le {\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }\Vert }_{HS}{\Vert G\Vert }_{\mathcal{L}(H)}\\ & \le C_S(t-r{)}^{-\alpha \eta }{\theta }^{-\eta }{C}_{\eta }{\Vert G\Vert }_{\mathcal{L}(H)}.\end{array}$(31) From (30(30) $\begin{array}{rl}{R}_6& :=\alpha \Vert {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1})\\ & {\phantom{{\int }_0^s}\times S((t-r{)}^{\alpha }\theta )G\mathrm{d}\theta \mathrm{d}W(r)\Vert }_{L^p(\mathrm{\Omega },H)}\\ & \le C_p\alpha \Vert {\int }_0^s((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1}{)}^2\\ & {\times {({\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )\Vert S((t-r{)}^{\alpha }\theta )G{\Vert }_{HS}\mathrm{d}\theta )}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}.\end{array}$(30) ) and (31(31) $\begin{array}{rl}& {\Vert S((t-r{)}^{\alpha }\theta )G\Vert }_{HS}\\ & \le {\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }G\Vert }_{HS}\\ & \le {\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }\Vert }_{HS}{\Vert G\Vert }_{\mathcal{L}(H)}\\ & \le C_S(t-r{)}^{-\alpha \eta }{\theta }^{-\eta }{C}_{\eta }{\Vert G\Vert }_{\mathcal{L}(H)}.\end{array}$(31) ) with an application of Lemma 2.1 (with $\nu =1-\eta$), Assumption ${\mathcal{H}}_G$ and Lemma 4.1 (with $\kappa =1-\alpha$), we arrive at (32) $\begin{array}{rl}{R}_6& \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }\Vert {\int }_0^s((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1}{)}^2\\ & {\phantom{{\int }_0^t}\times (t-r{)}^{-2\alpha \eta }{\Vert G\Vert }_{\mathcal{L}(H)}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G({\int }_0^s((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1}{)}^2\\ & {\phantom{{\int }_0^t}\times (t-r{)}^{-2\alpha \eta }\mathrm{d}r)}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G(1-\alpha )c^{-\alpha }{t}_0^{-2(1-\alpha )}(t-s{)}^{1-\alpha }\\ & \times {({\int }_0^s(t-r{)}^{-2\alpha \eta }\mathrm{d}r)}^{\frac{1}{2}}\\ & \le C_6(t-s{)}^{1-\alpha },\end{array}$(32) where $C_6:=C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G(1-\alpha )c^{-\alpha }{t}_0^{-2(1-\alpha )}\frac{T^{\frac{1}{2}-\alpha \eta }}{(1-2\alpha \eta {)}^{\frac{1}{2}}}$.

To deal with $R_7$, we argue as above with the use of Est.(31(31) $\begin{array}{rl}& {\Vert S((t-r{)}^{\alpha }\theta )G\Vert }_{HS}\\ & \le {\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }G\Vert }_{HS}\\ & \le {\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }\Vert }_{HS}{\Vert G\Vert }_{\mathcal{L}(H)}\\ & \le C_S(t-r{)}^{-\alpha \eta }{\theta }^{-\eta }{C}_{\eta }{\Vert G\Vert }_{\mathcal{L}(H)}.\end{array}$(31) ). Then, (33) $\begin{array}{rl}{R}_7& \le C_p\alpha \Vert {\int }_s^t(t-r{)}^{2\alpha -2}({\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )\\ & {{\phantom{{\int }_0^t}\times {\Vert S((t-r{)}^{\alpha }\theta )G\Vert }_{HS}\mathrm{d}\theta )}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }\Vert {\int }_s^t(t-r{)}^{2\alpha -2}({\int }_0^{\mathrm{\infty }}{\theta }^{1-\eta }{\xi }_{\alpha }(\theta )\\ & {{\phantom{{\int }_0^t}\times (t-r{)}^{-\alpha \eta }{\Vert G\Vert }_{\mathcal{L}(H)}\mathrm{d}\theta )}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G{({\int }_s^t(t-r{)}^{2\alpha (1-\eta )-2}\mathrm{d}r)}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G\frac{1}{(2\alpha (1-\eta )-1{)}^{\frac{1}{2}}}\\ & \times (t-s{)}^{\alpha (1-\eta )-\frac{1}{2}}\\ & \le C_7(t-s{)}^{\alpha (1-\eta )-\frac{1}{2}},\end{array}$(33) where $C_7:=C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G\frac{1}{(2\alpha (1-\eta )-1{)}^{\frac{1}{2}}}$.

Finally, we replace Est.(24(24) $\begin{array}{r}{R}_1\le C_1(t-s{)}^{\alpha \eta },\end{array}$(24) ), Est.(25(25) $\begin{array}{rl}{R}_2& \le {\alpha }^2{C}_S{C}_{\eta }{C}_F{C}_{\alpha ,1-\eta }{\Vert u\Vert }_{\mathrm{\Lambda }}{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}(t-s{)}^{\alpha \eta }\\ & \times {\int }_0^s(s-r{)}^{\alpha -1}\mathrm{d}r\le C_2(t-s{)}^{\alpha \eta },\end{array}$(25) ), Est.(26(26) $\begin{array}{rl}{R}_3& \le \alpha {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )((s-r{)}^{\alpha -1}-(t-r{)}^{\alpha -1})\\ & \times {\Vert S((t-r{)}^{\alpha }\theta )F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}r\\ & \le \alpha {\int }_0^s{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )((s-r{)}^{\alpha -1}-(t-r{)}^{\alpha -1})\\ & \times {\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}{\Vert A^{-\eta }F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}r\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{\int }_0^s{\int }_0^{\mathrm{\infty }}{\theta }^{1-\eta }{\xi }_{\alpha }(\theta )\\ & \times ((s-r{)}^{\alpha -1}-(t-r{)}^{\alpha -1})(t-r{)}^{-\alpha \eta }\mathrm{d}\theta \mathrm{d}r\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }{\int }_0^s((s-r{)}^{\alpha -1}-(t-r{)}^{\alpha -1})\\ & \times (t-r{)}^{-\alpha \eta }\mathrm{d}r\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }{\int }_0^s\\ & \times \frac{((t-r{)}^{1-\alpha }-(s-r{)}^{1-\alpha })}{(s-r{)}^{1-\alpha }(t-r{)}^{1-\alpha }}(t-r{)}^{-\alpha \eta }\mathrm{d}r\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }{t}_0^{-2(1-\alpha )}(1-\alpha )\\ & \times c^{-\alpha }(t-s{)}^{1-\alpha }{\int }_0^s(t-r{)}^{-\alpha \eta }\mathrm{d}r\\ & \le C_3(t-s{)}^{1-\alpha },\end{array}$(26) ), Est.(27(27) $\begin{array}{rl}{R}_4& \le \alpha {\int }_s^t{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(t-r{)}^{\alpha -1}\\ & \times {\Vert S((t-r{)}^{\alpha }\theta )F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}r,\\ & \le \alpha {\int }_s^t{\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )(t-r{)}^{\alpha -1}{\Vert A^{\eta }S((t-r{)}^{\alpha }\theta )\Vert }_{\mathcal{L}(H)}\\ & \times {\Vert A^{-\eta }F(u(r))\Vert }_{L^p(\mathrm{\Omega },H)}\mathrm{d}\theta \mathrm{d}r,\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{\int }_s^t{\int }_0^{\mathrm{\infty }}{\theta }^{1-\eta }{\xi }_{\alpha }(\theta )(t-r{)}^{\alpha -1}\\ & \times (t-r{)}^{-\alpha \eta }\mathrm{d}\theta \mathrm{d}r,\\ & \le \alpha C_F{\Vert u\Vert }_{\mathrm{\Lambda }}{C}_S{C}_{\eta }{C}_{\alpha ,1-\eta }{\int }_s^t(t-r{)}^{\alpha (1-\eta )-1}\mathrm{d}r\\ & \le C_4(t-s{)}^{\alpha (1-\eta )},\end{array}$(27) ), Est.(29(29) $\begin{array}{rl}{R}_5& \le C_p{\alpha }^2{C}_S{C}_{\eta }{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}(t-s{)}^{\alpha \eta }{C}_{\alpha ,1-\eta }\\ & \times \Vert {\int }_0^s(s-r{)}^{2\alpha -2}\Vert G{\Vert {}_{\mathcal{L}(H)}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le C_p{\alpha }^2{C}_S{C}_{\eta }{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}(t-s{)}^{\alpha \eta }{C}_{\alpha ,1-\eta }{C}_G\\ & \times {({\int }_0^s(s-r{)}^{2\alpha -2}\mathrm{d}r)}^{\frac{1}{2}}\\ & \le C_p{\alpha }^2{C}_S{C}_{\eta }{t}_0^{-2\alpha \eta }{c}^{\alpha \eta -1}(t-s{)}^{\alpha \eta }{C}_{\alpha ,1-\eta }{C}_G\frac{T^{\alpha -\frac{1}{2}}}{(2\alpha -1{)}^{\frac{1}{2}}}\\ & \le C_5(t-s{)}^{\alpha \eta },\end{array}$(29) ), Est.(32(32) $\begin{array}{rl}{R}_6& \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }\Vert {\int }_0^s((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1}{)}^2\\ & {\phantom{{\int }_0^t}\times (t-r{)}^{-2\alpha \eta }{\Vert G\Vert }_{\mathcal{L}(H)}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G({\int }_0^s((t-r{)}^{\alpha -1}-(s-r{)}^{\alpha -1}{)}^2\\ & {\phantom{{\int }_0^t}\times (t-r{)}^{-2\alpha \eta }\mathrm{d}r)}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G(1-\alpha )c^{-\alpha }{t}_0^{-2(1-\alpha )}(t-s{)}^{1-\alpha }\\ & \times {({\int }_0^s(t-r{)}^{-2\alpha \eta }\mathrm{d}r)}^{\frac{1}{2}}\\ & \le C_6(t-s{)}^{1-\alpha },\end{array}$(32) ) and Est.(33(33) $\begin{array}{rl}{R}_7& \le C_p\alpha \Vert {\int }_s^t(t-r{)}^{2\alpha -2}({\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )\\ & {{\phantom{{\int }_0^t}\times {\Vert S((t-r{)}^{\alpha }\theta )G\Vert }_{HS}\mathrm{d}\theta )}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }\Vert {\int }_s^t(t-r{)}^{2\alpha -2}({\int }_0^{\mathrm{\infty }}{\theta }^{1-\eta }{\xi }_{\alpha }(\theta )\\ & {{\phantom{{\int }_0^t}\times (t-r{)}^{-\alpha \eta }{\Vert G\Vert }_{\mathcal{L}(H)}\mathrm{d}\theta )}^2\mathrm{d}r\Vert }_{L^{\frac{p}{2}}(\mathrm{\Omega },\mathbb{R})}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G{({\int }_s^t(t-r{)}^{2\alpha (1-\eta )-2}\mathrm{d}r)}^{\frac{1}{2}}\\ & \le C_p\alpha C_S{C}_{\eta }{C}_{\alpha ,1-\eta }{C}_G\frac{1}{(2\alpha (1-\eta )-1{)}^{\frac{1}{2}}}\\ & \times (t-s{)}^{\alpha (1-\eta )-\frac{1}{2}}\\ & \le C_7(t-s{)}^{\alpha (1-\eta )-\frac{1}{2}},\end{array}$(33) ) in Est.(23(23) $\begin{array}{r}{\Vert u(t)-u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\le \sum _{i=1}^7{R}_i,\end{array}$(23) ) to get (34) $\begin{array}{r}{\Vert u(t)-u(s)\Vert }_{L^p(\mathrm{\Omega },H)}\le C(t-s{)}^{\mu },\end{array}$(34) where $\mu :=min\{\alpha \eta ,1-\alpha ,\alpha (1-\eta )-\frac{1}{2}\}$, where $\alpha \in (\frac{1}{2},1)$, $\eta \in (0,1-\frac{1}{2\alpha })$, and $C:=\sum _{i=1}^7{C}_i$. By this the proof is completed.

5. Example: stochastic space-time fractional integro-differential equation

We give in this section an important example of problem (1(1) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{Au(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}\\ & \times {\int }_0^t\frac{F(u(s))}{(t-s{)}^{1-\alpha }}\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{G}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(1) ) in order to confirm and support the validity of our theoretical results. Namely; Theorems 2.1, 3.1 and 4.1.

Example 5.1

Let $\acute{\alpha }>\frac{2\alpha }{2\alpha -1}$ where $\alpha \in (\frac{1}{2},1)$. We consider the following stochastic space-time fractional integro-differential equation in the Hilbert space $H:=L^2(0,1)$ with an initial condition $u_0=0$. (35) $\begin{array}{rl}u(t)& =u_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{{\mathcal{A}}_{\acute{\alpha }}u(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s\\ & +\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{u(s)}{(t-s{)}^{1-\alpha }}\mathrm{d}s\\ & +\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{I_H}{(t-s{)}^{1-\alpha }}\mathrm{d}W(s),\end{array}$(35) where

• ${\mathcal{A}}_{\acute{\alpha }}:=(-\mathrm{\Delta }{)}^{\frac{\acute{\alpha }}{2}}=:{\mathcal{A}}^{\frac{\acute{\alpha }}{2}}$ is the fractional Laplacian and $\mathcal{A}:=-\mathrm{\Delta }$ is Laplace operator endowed with the Dirichlet boundary conditions. We know that (see for more details [10,27]), the operator ${\mathcal{A}}_{\acute{\alpha }}:D({\mathcal{A}}_{\acute{\alpha }})\subset L^2(0,1)\to L^2(0,1)$ is linear generates an analytic semigroup $(S_{\acute{\alpha }}(t):=e^{-t{\mathcal{A}}_{\acute{\alpha }}}{)}_{t\in [0,T]}$.

  Further, it satisfies the following estimates.

  Lemma 5.1

  	–	Let $\acute{\alpha }>0$ and $t\in (0,T]$ for T be fixed. For all $\delta \ge 0$, $\mathrm{\exists }{C}_{\delta }>0$ such that (36) $\begin{array}{r}{\Vert {\mathcal{A}}^{\delta }{S}_{\acute{\alpha }}(t)\Vert }_{\mathcal{L}(H)}\le C_{\delta }{t}^{-\frac{2\delta }{\alpha }}.\end{array}$(36)
  	–	For all $\acute{\xi }>\frac{1}{4}$, there exists $C_{\acute{\xi }}:=\frac{1}{{\pi }^{4\acute{\xi }}(4\acute{\xi }-1)}>0$ such that (37) $\begin{array}{r}{\Vert {\mathcal{A}}^{-\acute{\xi }}\Vert }_{HS}\le C_{\acute{\xi }}.\end{array}$(37)

  Proof.

  The first estimate already exists, see, e.g. [10, Lemma 2.3].

  About the second estimate, let $\acute{\xi }>\frac{1}{4}$ and let $(e_j,{\lambda }_j{)}_{j=1}^{+\mathrm{\infty }}$ are the eigenpairs of the operator $\mathcal{A}$, where $e_j(.):=\sqrt{2}sin(j\pi .)$ and ${\lambda }_j:=(j\pi {)}^2$. The use of the definition of the Hilbert-Schmidt norm, with the fact that ${\left|e_j\right|}_H=1$, lead to (38) $\begin{array}{rl}{\Vert {\mathcal{A}}^{-\acute{\xi }}\Vert }_{HS}^2& =\sum _{j\in {\mathbb{N}}_0}{\left|{\mathcal{A}}^{-\acute{\xi }}{e}_j\right|}_H^2=\sum _{j\in {\mathbb{N}}_0}{\left|{\lambda }_j^{-\acute{\xi }}{e}_j\right|}_H^2\\ & =\sum _{j\in {\mathbb{N}}_0}{\lambda }_j^{-2\acute{\xi }}{\left|e_j\right|}_H^2\\ & =\sum _{j\in {\mathbb{N}}_0}{\lambda }_j^{-2\acute{\xi }}{\pi }^{-4\acute{\xi }}\sum _{j\in {\mathbb{N}}_0}{j}^{-4\acute{\xi }}\\ & \le {\pi }^{-4\acute{\xi }}{\int }_1^{\mathrm{\infty }}{x}^{-4\acute{\xi }}\mathrm{d}x.\end{array}$(38) The integral ${\int }_1^{\mathrm{\infty }}{x}^{-4\acute{\xi }}\mathrm{d}x=\frac{1}{4\acute{\xi }-1}$ converges, thanks to $\acute{\xi }>\frac{1}{4}$, and so $\begin{array}{r}{\Vert {\mathcal{A}}^{-\acute{\xi }}\Vert }_{HS}^2\le \frac{1}{{\pi }^{4\acute{\xi }}(4\acute{\xi }-1)}.\end{array}$

  The operator ${\mathcal{A}}_{\acute{\alpha }}$ satisfies Assumption ${\mathcal{H}}_A$; Est.(3(3) $\begin{array}{r}{\Vert A^{\gamma }S(t)\Vert }_{\mathcal{L}(H)}\le C_S{t}^{-\gamma },\text{for all}t\in (0,T],\end{array}$(3) ) (with $\gamma \ge 0$) and Est.(4(4) $\begin{array}{r}{\Vert A^{-\xi }\Vert }_{HS}\le C_{\xi },\end{array}$(4) ) (with $\xi \in (\frac{1}{2\acute{\alpha }},\frac{1}{2}-\frac{1}{4\alpha })$).

  Indeed, let $\gamma \ge 0$ and $\xi \in (\frac{1}{2\acute{\alpha }},\frac{1}{2}-\frac{1}{4\alpha })$ (which is possible thanks to the condition $\acute{\alpha }>\frac{2\alpha }{2\alpha -1}$). From Est.(36(36) $\begin{array}{r}{\Vert {\mathcal{A}}^{\delta }{S}_{\acute{\alpha }}(t)\Vert }_{\mathcal{L}(H)}\le C_{\delta }{t}^{-\frac{2\delta }{\alpha }}.\end{array}$(36) ) (with $\delta =\frac{\acute{\alpha }\gamma }{2}$) and Est.(37(37) $\begin{array}{r}{\Vert {\mathcal{A}}^{-\acute{\xi }}\Vert }_{HS}\le C_{\acute{\xi }}.\end{array}$(37) ) (with $\acute{\xi }=\frac{\acute{\alpha }\xi }{2}>\frac{1}{4}$, which is possible since $\xi >\frac{1}{2\acute{\alpha }}$) respectively, we have $\begin{array}{r}{\Vert {\mathcal{A}}_{\acute{\alpha }}^{\gamma }{S}_{\acute{\alpha }}(t)\Vert }_{\mathcal{L}(H)}={\Vert {\mathcal{A}}^{\frac{\acute{\alpha }\gamma }{2}}{S}_{\acute{\alpha }}(t)\Vert }_{\mathcal{L}(H)}\le C{t}^{-\gamma },\end{array}$ and $\begin{array}{r}{\Vert {\mathcal{A}}_{\acute{\alpha }}^{-\xi }\Vert }_{HS}={\Vert {\mathcal{A}}^{-\frac{\acute{\alpha }\xi }{2}}\Vert }_{HS}\le C_{\xi },\text{ and for some }{C}_{\xi }>0,\end{array}$

• The operator $F:H\to H$ is given by $F(v)=v$, for any $v\in H$. It is easy to check that F satisfies the global Lipschitz and the linear growth conditions imposed in this paper.

  In fact, let $u,v\in H$, we have $\begin{array}{r}{|F(u)-F(v)|}_H={|u-v|}_H,\end{array}$ and $\begin{array}{r}{|F(u)|}_H={\left|u\right|}_H,\end{array}$ Hence, the operator F satisfies Assumption ${\mathcal{H}}_F$.

• The operator $G:H\to H$ is the identity operator $I_H$ defined on the Hilbert space H. It is well known that, $I_H\in \mathcal{L}(H)$. Thus, G satisfies Assumption ${\mathcal{H}}_G$.

6. Conclusion

Stochastic fractional integral (or integro-differential) equations have been used as mathematical models of many physical processes as the anomalous diffusions. Nevertheless, we can find in the literature a few works concerned with this type of equations. Motivated by this fact, we considered in this paper a class of stochastic time-fractional integral equations in a Hilbert space H. We used the fixed point theorem in order to prove the well-posedness of the problem by imposing the global Lipschitz and the linear growth conditions. Moreover, we achieved not only the spatial regularity of the mild solution $u:=(u(t){)}_{t\in [0,T]}$ but the temporal regularity as well. Precisely, by imposing a regularity condition on the initial value (i.e. ${\Vert A^{\eta }{u}_0\Vert }_{L^p(\mathrm{\Omega },H)}<\mathrm{\infty }$), we proved that such solution satisfies $u\in C([0,T];L^p(\mathrm{\Omega },D(A^{\eta })))$, where $\eta \in (0,\frac{1}{2}-\frac{1}{4\alpha })$, $p\ge 2$ and it is time Hölder continous with exponent $\mu :=min\{\alpha \eta ,1-\alpha ,\alpha (1-\eta )-\frac{1}{2}\}$, for $\alpha \in (\frac{1}{2},1)$ and $\eta \in (0,1-\frac{1}{2\alpha })$. In general, it is not easy to solve this kind of equations analytically, for this, the numerical study plays an important role by providing a numerical approximations of the analytic solutions with respect to time, space or to both simultaneously. Motivated by this fact, some numerical approximations for the mild solution of the problem are interesting directions for our future research.

Acknowledgments

The authors would like to thank the reviewers for giving them constructive comments and suggestions which would help them in order to improve the quality of the paper.

Disclosure statement

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