Attractivity for Hilfer fractional stochastic evolution equations

Abstract

This article is devoted to the study of the attractivity of solutions to a class of stochastic evolution equations involving Hilfer fractional derivative. By employing the semigroup theory, fractional calculus and the fixed point technique, we establish new alternative criteria to ensure the existence of globally attractive solutions for the Cauchy problem when the associated semigroup is compact.

1 Introduction

In the past three decades, fractional differential equations received much attention. The growing interest in the subject is due to its extensive applications in diverse fields such as physics, fluid mechanics, viscoelasticity, heat conduction in materials with memory, chemistry and engineering. Much of the work is devoted to the existence and uniqueness of solutions for fractional differential equations; see, for example, Kilbas et al. [10], Miller and Ross [13], Podlubny [14], Zhou [22] and [1, 5, 19, 21, 23, 24] and the references cited therein. Since Hilfer [9] proposed the generalized Riemann–Liouville fractional derivative (Hilfer fractional derivative), there has been shown some interest in studying evolution equations involving Hilfer fractional derivatives (see [2, 4, 7, 8, 18, 20]).

Recently, the stability of fractional-order systems has been discussed by several authors. Abbas et al. [2] investigated the Ulam stability for Hilfer fractional differential inclusions via the weakly Picard operators theory. Chen et al. [6] established the global attractivity for nonlinear fractional differential equations. Losada, et al. [11] studied the attractivity of solutions for a class of multi-term fractional functional differential equations. Rajivganthi, Rihan et al. [15–17] studied the stability of a fractional-order prey–predator system. Moreover, stochastic perturbation is unavoidable in nature and sometimes useful in research, for example, the existence of stochastic perturbation in the mathematical model has been found to be quite effective in preventing the explosion of the population. Hence, it is important and necessary to consider stochastic perturbation into the investigation of fractional differential equations (see [3, 12]. However, it seems that there is less literature related to the stability of Hilfer fractional stochastic evolution equations. Therefore, the attractivity of solutions of Hilfer fractional stochastic evolution equations might be a fascinating and useful problem.

In this work, we study the attractivity of solutions for the following Hilfer fractional stochastic evolution equations:

$$\begin{aligned} \textstyle\begin{cases} D^{\mu ,\beta }_{0^{+}} x(t)=Ax(t)+f(t,x(t))+\sigma (t,x(t)) \frac{d\omega (t)}{dt}, \quad t>0, \\ I^{(1-\mu )(1-\beta )}_{0^{+}}x(0)=x_{0}, \end{cases}\displaystyle \end{aligned}$$

(1.1)

where $D^{\mu ,\beta }_{0^{+}}$ denotes the Hilfer fractional derivative of order μ and type β which will be given in next section, $\frac{1}{2}<\mu <1$, $0\leq \beta <1$, A is the infinitesimal generator of a strongly continuous semigroup $S(t)$, ($t\geq 0$) on a separable Hilbert space X with inner product $\langle \cdot \rangle$, and norm $\Vert \cdot \Vert$. Let $\{\omega ( t )\}_{t \geq 0}$ denote a K-valued Wiener process with a finite trace nuclear covariance operator $Q\geq 0$ defined on the filtered complete probability space $(\varOmega ,\mathcal{F},P)$. The functions f, σ are given functions satisfying some appropriate assumptions. $x_{0}$ is an element of the Hilbert space $L_{2}^{0}(\varOmega ,X)$ which will be specified later.

The objective of this paper is to discuss the attractivity of solutions for Cauchy problem (1.1). In fact, we establish sufficient conditions for the global attractivity of mild solutions for system (1.1) in cases that semigroup associated with A is compact. The obtained results essentially reveal certain characteristics of solutions for Hilfer fractional evolution equations, in contrast to integer-order evolution equations.

The rest of this paper is organized as follows. Section 2 contains some basic notations and essential preliminary results. In Sect. 3, we obtain alternative sufficient conditions for the attractivity of fractional stochastic evolution equations. Some conclusions are presented in Sect. 4.

2 Preliminaries

In this section, we provide some basic definitions, notations, lemmas, properties of semigroup theory and fractional calculus which are used throughout this paper.

Definition 2.1

([10])

The fractional integral of order α with the lower limit 0 for a function f is defined as

$$ I^{\alpha } f(t)=\frac{1}{\varGamma (\alpha )} \int _{0}^{t} \frac{f(s)}{(t-s)^{1-\alpha }}\,ds,\quad t>0, \alpha >0, $$

provided the right side is pointwise defined on $[0,\infty )$, where $\varGamma (\cdot )$ is the gamma function.

Definition 2.2

([10])

The left-sided Riemann–Liouville fractional-order derivative of order α with the low limit 0 for a function $f:[0,+\infty )\rightarrow \mathbb{R}$ is defined as

$$ ^{L}D^{\alpha } f(t)=\frac{1}{\varGamma (n-\alpha )}\frac{d^{n}}{dt^{n}} \int _{0}^{t}\frac{f(s)}{(t-s)^{\alpha +1-n}}\,ds,\quad t>0, n-1< \alpha \leq n. $$

Definition 2.3

([10])

The left-sided Caputo derivative of order $\alpha \in (n-1,n)$, $n\in \mathbb{Z}^{+}$ for a function $f:[a,+\infty )\rightarrow \mathbb{R}$ is defined by

$$ ^{c}D^{\alpha }_{a^{+}} f(t)=\frac{1}{\varGamma (n-\alpha )} \int _{a}^{t} \frac{f^{(n)}(s)}{(t-s)^{\alpha +1-n}}\,ds,\quad t>a, n-1< \alpha < n. $$

Definition 2.4

([9]; Hilfer fractional derivative)

The left-sided Hilfer fractional derivative of order $0<\mu <1$ and $0\leq \beta \leq 1$ of the function $f(t)$ is defined as

$$ D^{\mu ,\beta }_{a^{+}}f(t)=\bigl(I^{\beta (1-\mu )}_{a^{+}}D \bigl(I^{(1-\mu )(1- \beta )}_{a^{+}}f\bigr)\bigr) (t),\quad \text{where } D:= \frac{d}{dt}. $$

Lemma 2.1

([9])

• (i) When$\beta =0$, $0<\mu <1$and$a=0$, the Hilfer fractional derivative reduces to the classical Riemann–Liouville fractional derivative:

  $$ D^{\mu ,0}_{0^{+}}f(t)=\frac{d}{dt}I^{1-\mu }_{0^{+}}f(t)=^{L}D^{\mu }_{0^{+}}f(t). $$

• (ii) For$\beta =1$, $0<\mu <1$and$a=0$, the Hilfer fractional derivative becomes the classical Caputo fractional derivative:

  $$ D^{\mu ,1}_{0^{+}}f(t)=I^{1-\mu }_{0^{+}} \frac{d}{dt}f(t)=^{c}D^{\mu }_{0^{+}}f(t). $$

Lemma 2.2

([10])

For$\sigma \in (0,1]$and$0< a\leq b$, we have$\vert a^{\sigma }-b^{\sigma } \vert \leq (b-a)^{\sigma }$.

For convenience, let $\nu =\beta +\mu -\beta \mu$, by a simple calculation, we show that $0<\nu <1$.

Lemma 2.3

The Cauchy problem (1.1) is equivalent to the integral equation

$$ \begin{aligned}[b] x(t)&= \frac{x_{0}}{\varGamma (\beta (1-\mu )+\mu )}t^{\nu -1} \\ &\quad{} + \frac{1}{\varGamma (\mu )} \int _{0}^{t}(t-s)^{\mu -1}\bigl[Ax(s)+f \bigl(s,x(s)\bigr)\bigr]\,ds \\ &\quad{} + \frac{1}{\varGamma (\mu )} \int _{0}^{t}(t-s)^{\mu -1} \sigma \bigl(s,x(s) \bigr)\,d\omega (s),\quad t>0. \end{aligned} $$

(2.1)

Proof

We omit it and refer the reader to [22].

To define a mild solution of system (1.1), we use the Wright function $M_{\mu }(\theta )$ defined by

$$ M_{\mu }(\theta )=\sum_{n=1}^{\infty } \frac{(-\theta )^{n-1}}{(n-1)\varGamma (1-\mu n)}, \quad 0< \mu < 1, \theta \in \mathbb{C}, $$

and $\int _{0}^{\infty } \theta ^{\tau } M_{\mu }(\theta )\,d \theta = \frac{\varGamma (1+\tau )}{\varGamma (1+\mu \tau )}$, for $\theta \geq 0$.

Denote by $L_{2}(\varOmega ,X)$ the collection of all strongly-measurable, square-integrable, and X-valued random variables, which is a Banach space equipped with the norm $\Vert x(\cdot ) \Vert _{L_{2}(\varOmega , X)} = (E \Vert x(\cdot , \omega ) \Vert ^{2} )^{ \frac{1}{2}}$, where the expectation E is defined by $E(x) = \int _{\varOmega }x(\omega ) \,dP$. An important subspace of $L_{2}(\varOmega , X)$ is $L_{2}^{0}(\varOmega ,X) = \{ x \in L_{2} (\varOmega , X), x\text{ is } \mathcal{F}_{0}\text{-measurable}\}$. Let $C((0,+\infty ) , L_{2}(\varOmega , X))$ be the Banach space of all continuous maps from $(0,+\infty )$ into $L_{2}(\varOmega ,X)$ with $\Vert x \Vert _{C((0,+\infty ) ,L_{2}(\varOmega , X))}=(\sup_{t\in (0,\infty )}E \vert x(t) \vert ^{2})^{ \frac{1}{2}}<\infty$, for each $x\in C((0,+\infty ) ,L_{2}(\varOmega , X))$. Let $C_{0}((0,\infty ),L_{2}(\varOmega ,X))=\{x\in C((0,\infty ),L_{2}( \varOmega ,X)):\lim_{t\rightarrow \infty }E \vert x(t) \vert ^{2}=0\}$ be endowed with the norm $\Vert x \Vert _{0}=(\sup_{t\in (0,\infty )}E \vert x(t) \vert ^{2})^{\frac{1}{2}}< \infty$. Obviously, $C_{0}((0,\infty ),L_{2}(\varOmega ,X))$ is a Banach space. □

Lemma 2.4

By a mild solution of system (1.1) we mean the$\mathcal{F}_{t}$-adapted stochastic progress$x: (0,+\infty )\rightarrow L_{2}(\varOmega ,H)$that satisfies

$$ x(t)=S_{\mu ,\beta }(t)x_{0}+ \int _{0}^{t}T_{\mu }(t-s)f\bigl(s,x(s)\bigr) \,ds + \int _{0}^{t}T_{\mu }(t-s)\sigma \bigl(s,x(s) \bigr)\,d\omega (s), \quad t>0,$$

(2.2)

where$T_{\mu }(t)=t^{\mu -1}P_{\mu }(t)$, $P_{\mu }(t)=\int _{0}^{ \infty }\mu \theta M_{\mu }(\theta )S(t^{\mu }\theta )\,d\theta$, and$S_{\mu ,\beta }(t)=I^{\beta (1-\mu )}_{0^{+}}T_{\mu }(t)$.

Lemma 2.5

([10])

If$a>0$and$b>0$, then

$$ \int _{0}^{t}(t-s)^{a-1}s^{b-1} \,ds= \frac{\varGamma (a)\varGamma (b)}{\varGamma (a+b)}t^{a+b-1}. $$

(2.3)

To formulate some essential propositions, we introduce the assumption:

(H₀):

$S(t)$ is continuous in the uniform operator topology for $t\geq 0$, and $\{S(t)\}_{t\geq 0}$ is uniformly bounded, i.e., there exists $M>1$ such that $\sup_{t\in [0,+\infty )} \vert S(t) \vert < M$.

Remark 2.1

([7])

Under the assumption (H₀), $P_{\mu }(t)$ is continuous in the uniform operator topology for $t>0$.

Remark 2.2

([22])

Under the assumption (H₀), for any fixed $t > 0$, $\{T_{\mu }(t)\}_{t>0}$ and $\{S_{\mu ,\beta }(t)\}_{t>0}$ are linear operators, and, for any $x\in X$,

$$ \bigl\Vert T_{\mu }(t)x \bigr\Vert \leq \frac{Mt^{\mu -1}}{\varGamma (\mu )} \Vert x \Vert , \quad\quad \bigl\Vert S_{\mu ,\beta }(t)x \bigr\Vert \leq \frac{Mt^{\nu -1}}{\varGamma (\beta (1-\mu )+\mu )} \Vert x \Vert , $$

where $\nu =\beta +\mu -\beta \mu$.

Remark 2.3

([7])

Under the assumption (H₀), $\{T_{\mu }(t)\}_{t>0}$ and $\{S_{\mu ,\beta }(t)\}_{t>0}$ are strongly continuous, that is, for any $x\in X$, and $0< t^{\prime }< t^{\prime \prime }\leq b$, we have

$$ \bigl\Vert T_{\mu }\bigl(t^{\prime }\bigr)x-T_{\mu } \bigl(t^{\prime \prime }\bigr)x \bigr\Vert \rightarrow 0 \quad \text{and} \quad \bigl\Vert S_{\mu ,\beta }\bigl(t^{\prime }\bigr)x-S_{\mu ,\beta } \bigl(t^{\prime \prime }\bigr)x \bigr\Vert \rightarrow 0, \quad \text{as } t^{\prime \prime }\rightarrow t^{\prime }. $$

We also need the following generalization of the Ascoli–Arzela theorem.

Lemma 2.6

The set$H \subset C_{0}((0,\infty ),X)$is relatively compact if and only if the following conditions hold:

• (i) for any$b > 0$, the function inHis equicontinuous on$[0, b]$;

• (ii) for any$t \in [0,\infty )$, $H(t) = \{x(t) : x \in H\}$is relatively compact inX;

• (iii) $\lim_{t\rightarrow \infty } \vert x(t) \vert = 0$uniformly for$x \in H$.

Theorem 2.1

([5])

LetSbe a nonempty, closed, convex and bounded subset of the Banach spaceXand let$A : X \rightarrow X$and$B : S \rightarrow X$be two operators such that

• (a) Ais a contraction with constant$L < 1$,

• (b) Bis continuous, BSresides in a compact subset ofX,

• (c) $[x = Ax + By, y \in S] \Rightarrow x \in S$.

Then the operator equation$Ax + Bx = x$has a solution in S.

3 Main results

In this section, we establish the attractivity of solutions for system (1.1).

Since $T_{\mu }(t)=t^{\mu -1}P_{\mu }(t)$, (2.2) takes the form

$$ \begin{aligned}[b] x(t)&=S_{\mu ,\beta }(t)x_{0}+ \int _{0}^{t}(t-s)^{\mu -1}P_{ \mu }(t-s)f \bigl(s,x(s)\bigr)\,ds \\ &\quad{} + \int _{0}^{t}(t-s)^{\mu -1}P_{\mu }(t-s) \sigma \bigl(s,x(s)\bigr)\,d \omega (s), \quad t>0.\end{aligned} $$

(3.1)

In order to establish the attractivity of solutions for system (1.1), we need the following assumptions:

(H₁):

$S(t)$ is a compact operator for each $t>0$.

(H₂):

$f(t, x(t))$ and $\sigma (t, x(t))$ are Lebesgue measurable with respect to t on $(0,\infty )$,

and $f (t, x(t))$, $\sigma (t, x(t))$ are continuous with respect to x on $C((0,\infty ),L_{2}(\varOmega ,X))$.

(H₃):

There exists a constant $\mu _{1}\in (\frac{1}{2},\mu )$ such that

$$ \int _{0}^{h}E \bigl\vert f \bigl(t, x(t)\bigr) \bigr\vert ^{\frac{2}{2\mu _{1}-1}} \,dt < \infty ,\qquad \int _{0}^{h}E \bigl\vert \sigma \bigl(t, x(t) \bigr) \bigr\vert ^{\frac{2}{2\mu _{1}-1}} \,dt < \infty , \quad \text{for all } h < \infty . $$

(H₄):

$E \vert f(t,x) \vert ^{2}\leq Lt^{-\eta }$, $E \vert \sigma (t,x) \vert ^{2}\leq Lt^{-\eta }$ for $t \in (0,\infty )$, $L\geq 0$ and $x\in C((0,\infty ), L_{2}(\varOmega ,X))$, where $2\mu -1<\eta <\min \{1,2\mu +1-2\nu \}$.

By a simple calculation, we can infer that $c\in (-1,0)$ when $c=\frac{\mu -1}{1-\mu _{1}}$.

For any $x\in C((0,\infty ),L_{2}(\varOmega ,X))$, we define an operator F as

$$ \begin{aligned}[b] (Fx) (t)&=(F_{1}x) (t)+(F_{2}x) (t) \\ &=S_{\mu ,\beta }(t)x_{0}+ \int _{0}^{t}(t-s)^{\mu -1}P_{ \mu }(t-s)f \bigl(s,x(s)\bigr)\,ds \\ & \quad{} + \int _{0}^{t}(t-s)^{\mu -1}P_{\mu }(t-s) \sigma \bigl(s,x(s)\bigr)\,d \omega (s),\quad t>0,\end{aligned} $$

(3.2)

where

$$ (F_{1}x) (t)=S_{\mu ,\beta }(t)x_{0}, \quad t>0, $$

(3.3)

and

$$ \begin{aligned}[b] (F_{2}x) (t)&= \int _{0}^{t}(t-s)^{\mu -1}P_{\mu }(t-s)f \bigl(s,x(s)\bigr)\,ds\\&\quad{} + \int _{0}^{t}(t-s)^{\mu -1}P_{\mu }(t-s) \sigma \bigl(s,x(s)\bigr)\, d\omega (s),\quad t>0.\end{aligned} $$

(3.4)

Observe that x is a mild solution of (1.1) if and only there exists a fixed point $x^{*}$ such that the operator equation $x^{*}=F x^{*}=F_{1}x^{*}+F_{2}x^{*}$ holds for $t>0$.

Lemma 3.1

Assume that (H₀) and (H₂)–(H₄) hold. Then the operator$F_{2}$is continuous and for any$b>0$, $F_{2}S_{1}$is equicontinuous on$[0,b)$, where$D_{1}=\{y(t)\mid y(t) \in C((0,\infty ), L_{2}(\varOmega ,X)), E \vert y(t) \vert ^{2} \leq t ^{-\delta}\textit{ for }t \geq T_{1}\}$, $\delta =\frac{1}{2}(1+\eta -2\mu )$and$T_{1}$satisfies the inequality

$$ 3\frac{M^{2} \vert x_{0} \vert ^{2}}{\varGamma ^{2}(\beta (1-\mu )+\mu )}T_{1}^{\nu -1}+3 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ)T_{1}^{- \frac{1}{2}(1+\eta -2\mu )}\leq 1. $$

(3.5)

Proof

Obviously, $D_{1}$ is a nonempty bounded closed and convex subset of $C_{0}((0,\infty ), L_{2}(\varOmega ,X))$. The proof will be completed in three steps.

Step 1. $F_{2}$ maps $D_{1}$ into itself for $t\geq T_{1}$.

For any $t>0$, from (H₄) and Lemma 2.5, we obtain

$$\begin{aligned} E \Vert F_{2} x \Vert ^{2}&\leq 2E \biggl\Vert \int _{0}^{t}(t-s)^{\mu -1}P_{ \mu }(t-s)f \bigl(s,x(s)\bigr)\,ds \biggr\Vert ^{2} \\ & \quad{} +2E \biggl\Vert \int _{0}^{t}(t-s)^{\mu -1}P_{\mu }(t-s) \sigma \bigl(s,x(s)\bigr)\,d \omega (s) \biggr\Vert ^{2} \\ &\leq 2 \biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \int _{0}^{t}(t-s)^{2 \mu -2}E \bigl\Vert f \bigl(s,x(s)\bigr) \bigr\Vert ^{2}\,ds \\ &\quad{}+2TrQ \biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \int _{0}^{t}(t-s)^{2 \mu -2}E \bigl\Vert \sigma \bigl(s,x(s)\bigr) \bigr\Vert ^{2}\,ds \\ &\leq 2 \biggl(\frac{ML}{\varGamma (\mu )}\biggr)^{2}(1+TrQ) \int _{0}^{t}(t-s)^{2 \mu -2}s^{-\eta }\,ds \\ &\leq 2 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ)t^{-(1+ \eta -2\mu )}. \end{aligned}$$

Note that the above inequity is restricted to the integrability of $s^{-\eta }$, which is indeed true for $\eta < 1$.

For $t>T_{1}$, taking into account the fact that $\eta >2\mu -1$ and inequality (3.5), we infer that

$$\begin{aligned}& 2 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ) t^{-\frac{1}{2}(1+\eta -2\mu )} \\& \quad \leq 2 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ) T_{1}^{-\frac{1}{2}(1+\eta -2\mu )}\leq 1. \end{aligned}$$

Then, for $t \geq T_{1}$, we have

$$\begin{aligned} E \bigl\vert F_{2}y(t) \bigr\vert ^{2} \leq & \biggl[2 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ)t^{- \frac{1}{2}(1+\eta -2\mu )} \biggr]t^{-\frac{1}{2}(1+\eta -2\mu )} \\ \leq &t^{-\frac{1}{2}(1+\eta -2\mu )}. \end{aligned}$$

Thus $F_{2}D_{1} \subset D_{1}$ for $t \geq T_{1}$.

Step 2. $F_{2}$ is continuous.

For any $y_{m}(t), y(t) \in D_{1}$, $m = 1, 2,\ldots$ with $\lim_{m\rightarrow \infty } E \vert y_{m}(t)-y(t) \vert ^{2}= 0$, we get $\lim_{m\rightarrow \infty }E \vert y_{m}(t) \vert ^{2}= E \vert y(t) \vert ^{2}$ and $\lim_{m\rightarrow \infty }E \vert f(t, y_{m}(t)) \vert ^{2} = E \vert f(t, y(t)) \vert ^{2}$ for $t \geq T_{1}$. By direct computation, we infer that $(t-s)^{2\mu -2}\in L^{\frac{1}{2-2\mu _{1}}}$ for $t >0$ and $\mu _{1}\in (\frac{1}{2}, \mu )$.

Applying Hölder’s inequality, we have

$$\begin{aligned}& \begin{gathered} E \bigl\vert F_{2}y_{m}(t)-F_{2}y(t) \bigr\vert ^{2} \\ \quad \leq 2 \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{ \mu }(t-s) \bigr\vert ^{2}E \bigl\vert f \bigl(s,y_{m}(s)\bigr)-f\bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ \quad \quad{} +2TrQ \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y_{m}(s)\bigr)- \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ \quad \leq 2\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \int _{0}^{t}(t-s)^{2\mu -2}\bigl(E \bigl\vert f\bigl(s,y_{m}(s)\bigr)-f\bigl(s,y(s)\bigr) \bigr\vert ^{2}\bigr)\,ds \\ \quad \quad{} +2TrQ\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \int _{0}^{t}(t-s)^{2\mu -2}\bigl(E \bigl\vert \sigma \bigl(s,y_{m}(s)\bigr)-\sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\bigr)\,ds \\ \quad \leq 2\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \frac{t^{(1+c)(2-2\mu _{1})}}{(1+c)^{2-2\mu _{1}}}\biggl( \int _{0}^{t} \bigl\vert f\bigl(s,y_{m}(s) \bigr)-f\bigl(s,y(s)\bigr) \bigr\vert ^{ \frac{2}{2\mu _{1}-1}}\,ds\biggr)^{2\mu _{1}-1} \\ \quad \quad{} +2TrQ\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \frac{t^{(1+c)(2-2\mu _{1})}}{(1+c)^{2-2\mu _{1}}} \biggl( \int _{0}^{t} \bigl\vert \sigma \bigl(s,y_{m}(s)\bigr)-\sigma \bigl(s,y(s)\bigr) \bigr\vert ^{\frac{2}{2\mu _{1}-1}}\,ds\biggr)^{2 \mu _{1}-1} \\ \quad \leq 2\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \frac{t^{(1+c)(2-2\mu _{1})}}{(1+c)^{2-2\mu _{1}}}T^{2\mu _{1}-1} \bigl\Vert f\bigl( \cdot ,y_{m}(\cdot )\bigr)-f\bigl(\cdot ,y(\cdot )\bigr) \bigr\Vert _{L^{\frac{1}{2\mu _{1}-1}}} \\ \quad \quad{} + 2TrQ\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \frac{t^{(1+c)(2-2\mu _{1})}}{(1+c)^{2-2\mu _{1}}}T^{2\mu _{1}-1} \bigl\Vert \sigma \bigl(\cdot ,y_{m}(\cdot )\bigr)-\sigma \bigl(\cdot ,y(\cdot )\bigr) \bigr\Vert _{L^{ \frac{1}{2\mu _{1}-1}}}\rightarrow 0 \\ \quad \quad \text{as } m\rightarrow \infty , \end{gathered} \end{aligned}$$

which implies that $F_{2}$ is continuous.

Step 3. $F_{2}D_{1}$ is equicontinuous.

Let $\varepsilon > 0$ be given. Since $\lim_{t\rightarrow \infty }t^{-(1+\eta -2\mu )}= 0$, there exists a $T^{\prime }> T_{1}$ such that for $t>T^{\prime }$

$$ \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)(1+TrQ)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}t^{-(1+ \eta -2\mu )}< \frac{\varepsilon }{8}. $$

Let $T_{1}\leq t^{\prime }< t^{\prime \prime } \leq T^{\prime }$, then

$$\begin{aligned} &E \bigl\vert F_{2}y\bigl(t^{\prime \prime }\bigr)-F_{2}y \bigl(t^{\prime }\bigr) \bigr\vert ^{2} \\ &\quad =2E \biggl\vert \int _{0}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{\mu -1}P_{\mu }\bigl(t^{\prime \prime }-s\bigr)f\bigl(s,y(s) \bigr)\,ds- \int _{0}^{t^{\prime }}\bigl(t^{\prime }-s \bigr)^{\mu -1}P_{\mu }\bigl(t^{\prime }-s\bigr)f\bigl(s,y(s) \bigr)\,ds \biggr\vert ^{2} \\ &\quad\quad{} +2E \biggl\vert \int _{0}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{\mu -1}P_{\mu }\bigl(t^{\prime \prime }-s\bigr)\sigma \bigl(s,y(s)\bigr)\,d \omega (s) \\ &\quad\quad{}- \int _{0}^{t^{\prime }}\bigl(t^{\prime }-s \bigr)^{\mu -1}P_{\mu }\bigl(t^{\prime }-s\bigr) \sigma \bigl(s,y(s)\bigr)\,d\omega (s) \biggr\vert ^{2} \\ &\quad \leq 4 \int _{0}^{t^{\prime }}\bigl(t^{\prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\quad\quad{}+4 \int _{0}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\quad\quad{} +4TrQ \int _{0}^{t^{\prime }}\bigl(t^{\prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\quad\quad{} +4TrQ \int _{0}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\quad \leq 4 \frac{M^{2}L^{2}}{{\varGamma ^{2}(\mu )}}(1+TrQ)\biggl[ \int _{0}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2}s^{-\eta }\,ds+ \int _{0}^{t^{\prime }}\bigl(t^{\prime }-s \bigr)^{2 \mu -2}s^{-\eta }\,ds\biggr] \\ &\quad \leq 4 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ) \bigl({t^{\prime \prime }}^{-(1+ \eta -2\mu )}+{t^{\prime }}^{-(1+\eta -2\mu )} \bigr)\leq \varepsilon . \end{aligned}$$

Moreover, for $0 < t^{\prime } < t^{\prime \prime } \leq T_{1}$, by using Hölder’s inequality and Remark 2.3, we have

$$\begin{aligned} &E \bigl\vert F_{2}y\bigl(t^{\prime \prime }\bigr)-F_{2}y \bigl(t^{\prime }\bigr) \bigr\vert ^{2} \\ &\quad \leq 2E \biggl\vert \int _{0}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{\mu -1}P_{\mu }\bigl(t^{\prime \prime }-s\bigr)f\bigl(s,y(s) \bigr)\,ds- \int _{0}^{t^{\prime }}\bigl(t^{\prime }-s \bigr)^{\mu -1}P_{\mu }\bigl(t^{\prime }-s\bigr)f\bigl(s,y(s) \bigr)\,ds \biggr\vert ^{2} \\ &\quad \quad{} +2E \biggl\vert \int _{0}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{\mu -1}P_{\mu }\bigl(t^{\prime \prime }-s\bigr)\sigma \bigl(s,y(s)\bigr)\,d \omega (s) \\ &\quad\quad{}- \int _{0}^{t^{\prime }}\bigl(t^{\prime }-s \bigr)^{\mu -1}P_{\mu }\bigl(t^{\prime }-s\bigr) \sigma \bigl(s,y(s)\bigr)\,d\omega (s) \biggr\vert ^{2} \\ &\quad \leq 6E \biggl\vert \int _{t^{\prime }}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{\mu -1}P_{\mu }\bigl(t^{\prime \prime }-s\bigr)f\bigl(s,y(s) \bigr)\,ds \biggr\vert ^{2} \\ &\quad\quad{} +6E \biggl\vert \int _{t^{\prime }}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{\gamma -1}P_{\mu }\bigl(t^{\prime \prime }-s\bigr) \sigma \bigl(s,y(s)\bigr)\,d\omega (s) \biggr\vert ^{2} \\ &\quad\quad{} +6E \biggl\vert \int _{0}^{t^{\prime }}\bigl[\bigl(t^{\prime \prime }-s \bigr)^{\mu -1}-\bigl(t^{\prime }-s\bigr)^{\mu -1} \bigr]P_{ \mu }\bigl(t^{\prime \prime }-s\bigr)f\bigl(s,y(s)\bigr)\,ds \biggr\vert ^{2} \\ &\quad\quad{} +6E \biggl\vert \int _{0}^{t^{\prime }}\bigl[\bigl(t^{\prime \prime }-s \bigr)^{\mu -1}-\bigl(t^{\prime }-s\bigr)^{\mu -1} \bigr]P_{ \mu }\bigl(t^{\prime \prime }-s\bigr)\sigma \bigl(s,y(s)\bigr)\,d \omega (s) \biggr\vert ^{2} \\ &\quad\quad{} +6E \biggl\vert \int _{0}^{t^{\prime }}\bigl(t^{\prime }-s \bigr)^{\mu -1}\bigl[P_{\mu }\bigl(t^{\prime \prime }-s \bigr)-P_{\mu }\bigl(t^{\prime }-s\bigr)\bigr]f\bigl(s,y(s)\bigr)\,ds \biggr\vert ^{2} \\ &\quad\quad{} +6E \biggl\vert \int _{0}^{t^{\prime }}\bigl(t^{\prime }-s \bigr)^{\mu -1}\bigl[P_{\mu }\bigl(t^{\prime \prime }-s \bigr)-P_{\mu }\bigl(t^{\prime }-s\bigr)\bigr] \sigma \bigl(s,y(s) \bigr)\,d\omega (s) \biggr\vert ^{2} \\ &\quad \leq 6 \int _{t^{\prime }}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\quad\quad{} +6TrQ \int _{t^{\prime }}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\quad\quad{} +6 \int _{0}^{t^{\prime }} \bigl\vert t^{\prime \prime }-s)^{\mu -1}- \bigl(t^{\prime }-s\bigr)^{\mu -1} \bigr\vert ^{2} \bigl\vert P_{ \mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{2} \,ds \\ &\quad\quad{} +6TrQ \int _{0}^{t^{\prime }} \bigl\vert t^{\prime \prime }-s)^{\mu -1}- \bigl(t^{\prime }-s\bigr)^{\mu -1} \bigr\vert ^{2} \bigl\vert P_{ \mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\quad\quad{} +6 \int _{0}^{t^{\prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime }-s \bigr)-P_{\mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{2} \,ds \\ &\quad\quad{} +6TrQ \int _{0}^{t^{\prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime }-s \bigr)-P_{ \mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\quad =\sum_{i=1}^{6}6I_{i}, \end{aligned}$$

where

$$\begin{aligned}& \begin{aligned} I_{1}&= \int _{t^{\prime }}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\leq \biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2}\biggl( \int _{t^{\prime }}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{ \frac{2\mu -2}{2-2\mu _{1}}}\biggr)^{2-2\mu _{1}}\biggl( \int _{t^{\prime }}^{t^{\prime \prime }} \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{ \frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2\mu _{1}-1}, \end{aligned} \\ & \begin{aligned} I_{2}&=TrQ \int _{t^{\prime }}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\leq \biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2}TrQ\biggl( \int _{t^{\prime }}^{t^{\prime \prime }}\bigl(t^{\prime \prime }-s \bigr)^{ \frac{2\mu -2}{2-2\mu _{1}}}\biggr)^{2-2\mu _{1}}\biggl( \int _{t^{\prime }}^{t^{\prime \prime }} \bigl\vert \sigma \bigl(s,y(s) \bigr) \bigr\vert ^{\frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2\mu _{1}-1}, \end{aligned} \\ & \begin{aligned} I_{3}&= \int _{0}^{t^{\prime }} \bigl\vert \bigl(t^{\prime \prime }-s \bigr)^{\mu -1}-\bigl(t^{\prime }-s\bigr)^{\mu -1} \bigr\vert ^{2} \bigl\vert P_{ \mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\leq \biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2}\frac{1}{(1+c)(2-2\mu _{1})} \bigl[{t^{\prime }}^{(1+c)(2-2 \mu _{1})}-{t^{\prime \prime }}^{(1+c)(2-2\mu _{1})}+ \bigl(t^{\prime \prime }-t^{\prime }\bigr)^{(1+c)(2-2 \mu _{1})}\bigr] \\ &\quad{} \times \biggl( \int _{0}^{t^{\prime }} \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{\frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2 \mu _{1}-1}, \end{aligned} \\ & \begin{aligned} I_{4}&=TrQ \int _{0}^{t^{\prime }} \bigl\vert t^{\prime \prime }-s)^{\mu -1}- \bigl(t^{\prime }-s\bigr)^{\mu -1} \bigr\vert ^{2} \bigl\vert P_{ \mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\leq \biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2}TrQ\frac{1}{(1+c)(2-2\mu _{1})} \bigl[{t^{\prime }}^{(1+c)(2-2 \mu _{1})}-{t^{\prime \prime }}^{(1+c)(2-2\mu _{1})}+ \bigl(t^{\prime \prime }-t^{\prime }\bigr)^{(1+c)(2-2 \mu _{1})}\bigr] \\ &\quad{} \times \biggl( \int _{0}^{t^{\prime }}E \bigl\vert \sigma \bigl(s,y(s) \bigr) \bigr\vert ^{\frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2 \mu _{1}-1}, \end{aligned} \\ & \begin{aligned} I_{5}&= \int _{0}^{t^{\prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime }-s \bigr)-P_{ \mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{2} \,ds \\ &\leq \bigl\vert P_{\mu }\bigl(t^{\prime \prime }-s\bigr)-P_{\mu } \bigl(t^{\prime }-s\bigr) \bigr\vert ^{2} \int _{0}^{t^{\prime }}\bigl(t^{\prime \prime }-s \bigr)^{2 \mu -2}E \bigl\Vert f\bigl(s,y(s)\bigr) \bigr\Vert ^{2}\,ds, \end{aligned} \\ & \begin{aligned} I_{6}&=TrQ \int _{0}^{t^{\prime }}\bigl(t^{\prime \prime }-s \bigr)^{2\mu -2} \bigl\vert P_{\mu }\bigl(t^{\prime }-s \bigr)-P_{ \mu }\bigl(t^{\prime \prime }-s\bigr) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\leq TrQ \bigl\vert P_{\mu }\bigl(t^{\prime \prime }-s \bigr)-P_{\mu }\bigl(t^{\prime }-s\bigr) \bigr\vert ^{2} \int _{0}^{t^{\prime }}\bigl(t^{\prime \prime }-s \bigr)^{2 \mu -2}E \bigl\Vert \sigma \bigl(s,y(s)\bigr) \bigr\Vert ^{2}\,ds. \end{aligned} \end{aligned}$$

By a standard calculation, we have

$$\begin{aligned} I_{1}\leq \frac{M^{2}}{\varGamma ^{2}(\mu )} \frac{(t^{\prime \prime }-t^{\prime })^{(1+c)(2-2\mu _{1})}}{(1+c)^{2-2\mu _{1}}}\biggl( \int _{t^{\prime }}^{t^{\prime \prime }}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{ \frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2\mu _{1}-1}\rightarrow 0, \quad \text{as } t^{\prime \prime }-t^{\prime }\rightarrow 0. \end{aligned}$$

In a similar manner, one can show that $I_{2}\rightarrow 0$ as $t^{\prime \prime }-t^{\prime }\rightarrow 0$. Moveover, from Lemma 2.2 and the fact that $1+c\in (0,1)$, it follows that $I_{3},I_{4}\rightarrow 0$ as $t^{\prime \prime }-t^{\prime }\rightarrow 0$. As for $I_{5}$, $I_{6}$, by the strong continuity of $S(t)$, $(t>0)$, we can infer that $I_{5},I_{6}\rightarrow 0$ as $t^{\prime \prime }-t^{\prime }\rightarrow 0$.

On the other hand, if $T_{1}\leq t^{\prime }< T< t^{\prime \prime }$ and $t^{\prime \prime }-t^{\prime }\rightarrow 0$, then $t^{\prime \prime }\rightarrow T$ and $t^{\prime }\rightarrow T$. Thus one can easily get

$$\begin{aligned} E \bigl\vert F_{2}y\bigl(t^{\prime \prime }\bigr)-F_{2}y \bigl(t^{\prime }\bigr) \bigr\vert ^{2} \leq &2E \bigl\vert F_{2}y\bigl(t^{\prime \prime }\bigr)-F_{2}y(T) \bigr\vert ^{2}+2E \bigl\vert F_{2}y(T)-F_{2}y \bigl(t^{\prime }\bigr) \bigr\vert ^{2} \\ \rightarrow &0, \quad \text{as } t^{\prime \prime }-t^{\prime }\rightarrow 0. \end{aligned}$$

 □

Lemma 3.2

Suppose the assumptions (H₀), (H₂) and (H₄) are satisfied, then a solution of (1.1) is in$D_{1}$for$t \geq T_{1}$.

Proof

It is clear that $x(t)$ is a fixed point of $L_{2}(\varOmega ,X)$ if and only if it is a solution of system (1.1). To prove this assertion, we only need to show that, for each fixed $y \in D_{1}$ and for $\forall x \in C((0,\infty ), L_{2}(\varOmega ,X))$, $x = F_{1}x + F_{2}y \Rightarrow x \in D_{1}$ holds. In fact, if $x = F_{1}x + F_{2}y$, according to (H₄), we can obtain

$$\begin{aligned} E \bigl\vert x(t) \bigr\vert ^{2} \leq &E \bigl\vert F_{1}x(t)+F_{2}y(t) \bigr\vert ^{2} \\ \leq &3\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2\nu -2}E \vert x_{0} \vert ^{2} \\ &{}+3 \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s) \bigr) \bigr\vert ^{2}\,ds \\ &{}+3TrQ \int _{0}^{t}(t-s)^{\mu -1} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ \leq &3\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2\nu -2}E \vert x_{0} \vert ^{2} \\ &{}+3\biggl( \frac{ML}{\varGamma (\mu )} \biggr)^{2}(1+TrQ) \int _{0}^{t}(t-s)^{2\mu -2}s^{- \eta }\,ds \\ =&3\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2\nu -2}E \vert x_{0} \vert ^{2} \\ &{}+3(1+TrQ) \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}t^{-(1+ \eta -2\mu )}. \end{aligned}$$

For $t \geq T_{1}$, from inequality (3.5) and $0 < 2\mu -1 < \eta$, it follows that

$$\begin{aligned}& 3\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{\nu -1}E \vert x_{0} \vert ^{2}+3(1+TrQ) \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}t^{- \frac{1}{2}(1+\eta -2\mu )} \\& \quad \leq 3\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}T_{1}^{\nu -1}E \vert x_{0} \vert ^{2} \\& \quad\quad{} +3(1+TrQ) \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}T_{1}^{- \frac{1}{2}(1+\eta -2\mu )} \\& \quad \leq 1. \end{aligned}$$

In addition, since $\eta <\min \{1,2\mu +1-2\nu \}$ and $\delta =\frac{1}{2}(1+\eta -2\mu )$, we can deduce $\nu -1-\delta -(2\nu -2)>0$, then it follows that

$$\begin{aligned} E \bigl\vert x(t) \bigr\vert ^{2} \leq &3\biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2 \nu -2}E \vert x_{0} \vert ^{2} \\ &{}+3 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ)t^{-(1+ \eta -2\mu )} \\ \leq &3 \biggl[\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{\nu -1}E \vert x_{0} \vert ^{2} \\ &{}+ \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ)t^{- \frac{1}{2}(1+\eta -2\mu )} \biggr]t^{-\delta } \\ \leq &t^{-\delta }, \end{aligned}$$

which implies that $x(t) \in D_{1}$ for $t \geq T_{1}$. □

Lemma 3.3

Assume that (H₁)–(H₃) hold. Then the set$V_{2} (t) = \{(F_{2}x)(t) : x \in D_{1}\}$is relatively compact in$L_{2}(\varOmega ,X)$for any$t \in (0,\infty )$.

Proof

Define $F_{2}^{\lambda ,\delta }x$ as

$$\begin{aligned}& \bigl(F_{2}^{\lambda ,\delta }x\bigr) (t) \\& \quad =S\bigl(\lambda ^{\mu }\theta \bigr) \int _{0}^{t- \lambda } \int _{\delta }^{\infty }\mu \theta M_{\gamma }(\theta )S \bigl((t-s)^{ \gamma }\theta -\lambda ^{\gamma }\theta \bigr) (t-s)^{\mu -1}f\bigl(s,x(s)\bigr)\,d \theta \,ds \\& \quad\quad {}+S\bigl(\lambda ^{\mu }\theta \bigr) \int _{0}^{t-\lambda } \int _{\delta }^{ \infty }\mu \theta M_{\mu }(\theta )S \bigl((t-s)^{\mu }\theta -\lambda ^{ \mu }\theta \bigr) (t-s)^{\mu -1}\sigma \bigl(s,x(s)\bigr)\,d\theta \,d\omega (s). \end{aligned}$$

From the compactness of $S(\lambda ^{\mu }\theta )$, $(\lambda ^{\mu }\theta >0)$, it is easy to see that, for $\forall \lambda \in (0,t)$ and $\forall \delta >0$, the set $V^{\lambda ,\delta }(t)=\{(F_{2}^{\lambda ,\delta }x)(t), x\in D_{1} \}$ is relatively compact in $L_{2}(\varOmega ,X)$. Then we obtain

$$\begin{aligned}& E \bigl\vert F_{2}x(t)-F_{2}^{\lambda ,\delta }x(t) \bigr\vert ^{2} \\& \quad \leq E \biggl\vert \int _{0}^{t} \int _{0}^{\delta }\mu \theta (t-s)^{\mu -1}M_{ \mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr)f\bigl(s,x(s)\bigr)\,d\theta \,ds \\& \quad \quad {} + \int _{0}^{t} \int _{\delta }^{\infty }\mu \theta (t-s)^{\mu -1}M_{ \mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr)f\bigl(s,x(s)\bigr)\,d\theta \,ds \\& \quad \quad {} - \int _{0}^{t-\lambda } \int _{\delta }^{\infty }\mu \theta (t-s)^{ \mu -1}M_{\mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr)f\bigl(s,x(s)\bigr)\,d\theta \,ds \\& \quad \quad {} + \int _{0}^{t} \int _{0}^{\delta }\mu \theta (t-s)^{\mu -1}M_{\mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr)\sigma \bigl(s,x(s)\bigr)\,d \theta \,d\omega (s) \\& \quad \quad {} + \int _{0}^{t} \int _{\delta }^{\infty }\mu \theta (t-s)^{\mu -1}M_{ \mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr)\sigma \bigl(s,x(s)\bigr)\,d\theta \,d\omega (s) \\& \quad \quad {} - \int _{0}^{t-\lambda } \int _{\delta }^{\infty }\mu \theta (t-s)^{ \mu -1}M_{\mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr)\sigma \bigl(s,x(s)\bigr)\,d\theta \,d \omega (s) \biggr\vert ^{2} \\& \quad \leq 4E \biggl\vert \int _{0}^{t} \int _{0}^{\delta }\mu \theta (t-s)^{\mu -1}M_{ \mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr) f\bigl(s,x(s)\bigr)\,d\theta \,ds \biggr\vert ^{2} \\& \quad \quad {} +4E \biggl\vert \int _{t-\lambda }^{t} \int _{\delta }^{\infty }\mu \theta (t-s)^{ \mu -1}M_{\mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr) f\bigl(s,x(s)\bigr)\,d\theta \,ds \biggr\vert ^{2} \\& \quad \quad {} +4TrQE \biggl\vert \int _{0}^{t} \int _{0}^{\delta }\mu \theta (t-s)^{\mu -1}M_{ \mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr) \sigma \bigl(s,x(s)\bigr)\,d \theta \,ds \biggr\vert ^{2} \\& \quad \quad {} +4TrQE \biggl\vert \int _{t-\lambda }^{t} \int _{\delta }^{\infty }\mu \theta (t-s)^{ \mu -1}M_{\mu }( \theta )S\bigl((t-s)^{\mu }\theta \bigr) \sigma \bigl(s,x(s)\bigr)\,d \theta \,ds \biggr\vert ^{2} \\& \quad \leq 4\mu ^{2}M^{2} \frac{t^{(1+c)(2-2\mu _{1})}}{(1+c)^{2-2\mu _{1}}} \biggl( \int _{0}^{t}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{ \frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2\mu _{1}-1} \biggl\vert \int _{0}^{\delta }\theta M_{ \mu }(\theta )\,d \theta \biggr\vert ^{2} \\& \quad \quad {} +4\mu ^{2}M^{2}TrQ \frac{t^{(1+c)(2-2\mu _{1})}}{(1+c)^{2-2\mu _{1}}} \biggl( \int _{0}^{t}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{\frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2\mu _{1}-1} \biggl\vert \int _{0}^{ \delta }\theta M_{\mu }(\theta )\,d \theta \biggr\vert ^{2} \\& \quad \quad {} +4\mu ^{2} M^{2} \frac{\lambda ^{(1+c)(2-2\mu _{1})}}{(1+c)^{2-2\mu _{1}}}\biggl( \int _{t- \lambda }^{t}E \bigl\vert f\bigl(s,y(s)\bigr) \bigr\vert ^{\frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2\mu _{1}-1} \biggl\vert \frac{1}{\varGamma ^{2}(1+\mu )} \\& \quad \quad {} +4\mu ^{2} M^{2}TrQ \frac{\lambda ^{(1+c)(2-2\mu _{1})}}{(1+c)^{2-2\mu _{1}}} \biggl( \int _{t- \lambda }^{t}E \bigl\vert \sigma \bigl(s,y(s) \bigr) \bigr\vert ^{\frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2\mu _{1}-1} \biggr\vert \frac{1}{\varGamma ^{2}(1+\mu )} \\& \quad \rightarrow 0, \quad \text{as } \lambda \rightarrow 0,\delta \rightarrow 0. \end{aligned}$$

Therefore, there do exist relatively compact sets arbitrarily close to the set $\{V(t),t>0\}$. Thus we deduce that the set $\{V(t),t>0\}$ is also relatively compact in $L_{2}(\varOmega ,X)$.

Hence, $\{F_{2}x,x\in D_{1}\}$ is a relatively compact set in $L_{2}(\varOmega ,X)$ by the Arzola–Ascoli theorem. As a consequence of Lemmas 3.1–3.3 and Theorem 2.1, there exists a $y \in D_{1}$ such that $y = F_{1}y + F_{2}y$, that is, H has a fixed point in $D_{1}$ which is a solution of system (1.1) for $t \geq T_{1}$. □

Now, we are well prepared to present our first attractivity result for system (1.1).

Theorem 3.1

Suppose that assumptions (H₁)–(H₄) hold. Then the zero solution of system (1.1) is globally attractive.

Proof

From Lemmas 3.1–3.3 and the properties of $D_{1}$, for $t \geq T_{1}$, we know that the solution of (1.1) does exist which is still in $D_{1}$. Moreover, all functions in $D_{1}$ tend to 0 as $t\rightarrow \infty$. Therefore the solution of system (1.1) tends to zero as $t\rightarrow \infty$. The proof is completed. □

To give our second attractivity result, we require the following hypothesis:

(H₅):

$E \vert f(t,x) \vert ^{2}\leq Lt^{-\eta _{1}}E \vert x(t) \vert ^{2}$, $E \vert \sigma (t,x) \vert ^{2} \leq Lt^{-\eta _{1}}E \vert x(t) \vert ^{2}$, for $\forall t \in (0,\infty )$ and each $x\in C((0,\infty ),L_{2}(\varOmega ,X))$, where $L\geq 0$, $2\mu -1<\eta _{1}<\mu$.

Theorem 3.2

Suppose that assumptions (H₁)–(H₃) and (H₅) hold, then the zero solution of (1.1) is globally attractive.

Proof

Set $D_{2} = \{y(t)\mid y(t) \in C((0,\infty ), L_{2}(\varOmega ,X)), E \vert y(t) \vert ^{2} \leq t^{{-\delta }_{1}}\text{ for } t\geq T_{2}\}$, where $\delta _{1} = 1-\mu$. We choose constant $T_{2}$ large enough such that the following inequality holds:

$$ 3 \frac{M^{2}E \vert x_{0} \vert ^{2}}{\varGamma ^{2}(\beta (1-\mu )+\mu )}T_{2}^{\nu -1} +3 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ)T_{2}^{-(1+ \eta _{1}-2\mu )}\leq 1.$$

(3.6)

One can easily see that operator $F_{1}$ is contraction. In addition, for each fixed $y \in D_{2}$ and for $\forall x \in L_{2}(\varOmega ,X)$, $x = F_{1}x + F_{2}y \Rightarrow x \in D_{2}$ holds. If $x = F_{1}x + F_{2}y$, then the application of (H₅) yields

$$\begin{aligned} E \bigl\vert x(t) \bigr\vert ^{2}& \leq E \bigl\vert F_{1}x(t)+F_{2}y(t) \bigr\vert ^{2} \\ &\leq 3\biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2 \nu -2}E \vert x_{0} \vert ^{2}+3 \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(s) \bigr\vert ^{2}E \bigl\vert f\bigl(s,y(s) \bigr) \bigr\vert ^{2}\,ds \\ &\quad{} +3TrQ \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,y(s)\bigr) \bigr\vert ^{2}\,ds \\ &\leq 3\biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2 \nu -2}E \vert x_{0} \vert ^{2} \\ &\quad{}+3\biggl(\frac{ML}{\varGamma (\mu )} \biggr)^{2}(1+TrQ) \int _{0}^{t}(t-s)^{2 \mu -2}s^{-\eta _{1}}E \bigl\vert y(s) \bigr\vert ^{2}\,ds \\ &\leq 3\biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2 \nu -2}E \vert x_{0} \vert ^{2} \\ &\quad{}+3\biggl(\frac{ML}{\varGamma (\mu )} \biggr)^{2}(1+TrQ) \int _{0}^{t}(t-s)^{2 \mu -2}s^{-\eta _{1}-\delta _{1}}\,ds \\ &\leq 3\biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2 \nu -2}E \vert x_{0} \vert ^{2} \\ &\quad{}+3 \frac{M^{2}L^{2}\varGamma (1-\eta _{1}-\delta _{1})\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta _{1}-\delta _{1})}(1+TrQ)t^{-(1+ \eta _{1}+\delta _{1}-2\mu )}. \end{aligned}$$

(3.7)

Since $\eta _{1} < \mu$ and $\delta _{1} =1-\mu$, we have $-\eta _{1}-\delta _{1}>-1$, therefore $s ^{-\eta _{1}-\delta _{1}}$ in (3.7) is integrable.

Furthermore, for $t>T_{2}$, we have

$$\begin{aligned} \begin{aligned}[b] &3\biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{\nu -1}+3 \frac{M^{2}L^{2}\varGamma (1-\eta _{1}-\delta _{1})\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta _{1}-\delta _{1})}(1+TrQ)t^{- \frac{1}{2}(1+\eta _{1}+\delta _{1}-2\mu )} \\ &\quad \leq 3\biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}T_{2}^{ \nu -1} \\ &\quad\quad{} +3 \frac{M^{2}L^{2}\varGamma (1-\eta _{1}-\delta _{1})\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta _{1}-\delta _{1})}(1+TrQ)T_{2}^{- \frac{1}{2}(1+\eta _{1}+\delta _{1}-2\mu )} \\ &\quad \leq 1. \end{aligned} \end{aligned}$$

(3.8)

From inequality (3.6) and $\nu -1<\delta _{1}$, we have

$$ \begin{aligned}[b] E \bigl\vert x(t) \bigr\vert ^{2}& \leq 3\biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2\nu -2}E \vert x_{0} \vert ^{2} \\ &\quad{} + \frac{M^{2}L^{2}\varGamma (1-\eta _{1}-\delta _{1})\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta _{1}-\delta _{1})}(1+TrQ)t^{-(1+ \eta _{1}+\delta _{1}-2\mu )} \\ &\leq \biggl[3\biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{ \nu -1}E \vert x_{0} \vert ^{2} \\ &\quad{}+ \frac{M^{2}L^{2}\varGamma (1-\eta _{1}-\delta _{1})\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta _{1}-\delta _{1})}(1+TrQ)t^{-(1+ \eta _{1}-2\mu )} \biggr]t^{-\delta _{1}} \\ &\leq t^{-\delta _{1}}, \end{aligned} $$

(3.9)

which implies that $x(t)\in D_{2}$ for $t\geq T_{2}$. Moreover, taking (3.6) and (3.9) into account, we can also get $E \vert F_{2}y(t) \vert ^{2}\leq t^{-\delta _{1}}$, which implies that $F_{2}D_{2} \subset D_{2}$ for $t\geq T_{2}$.

By applying a similar argument to the one used in Lemma 3.1, we can deduce that the operator $F_{2}$ is continuous and $F_{2}D_{2}$ resides in a compact subset of $L_{2}(\varOmega ,X)$ for $t\geq T_{2}$. Using Theorem 2.1 and Krasnoselskii’s theorem, there exists some $y \in D_{2}$ such that $y = F_{1}y + F_{2}y$, that is, F has a fixed point in $D_{2}$ which is indeed a solution of (1.1). Moreover, it is obvious that all functions in $D_{2}$ tend to 0 as $t \rightarrow \infty$, therefore, the solution of (1.1) tends to zero as $t\rightarrow \infty$ which implies the zero solution of (1.1) is globally attractive. This completes the proof. □

Corollary 3.1

Suppose that assumptions (H₁)–(H₃) hold and that

(H₆):

$E \vert f (t, x(t))-f (t, y(t)) \vert ^{2}\leq Lt^{-\eta _{2}}E \vert x(t)-y(t) \vert ^{2}$, $E \vert \sigma (t, x(t))-\sigma (t, y(t)) \vert ^{2}\leq Lt^{-\eta _{2}}E \vert x(t)-y(t) \vert ^{2}$for$t \in (0,\infty )$and$x(t), y(t) \in C((0,\infty ), L_{2}(\varOmega ,X))$, $L \geq 0$and$2\mu -1< \eta _{2} < \mu$, $f (t, 0) \equiv 0$, $\sigma (t, 0) \equiv 0$.

Then the zero solution of system (1.1) is globally attractive.

Proof

From condition (H₆), we have

$$\begin{aligned}& E \bigl\vert f \bigl(t, x(t)\bigr) \bigr\vert ^{2}=E \bigl\vert f \bigl(t, x(t)\bigr)-f (t, 0) \bigr\vert ^{2} \leq Lt^{-\eta _{1}}E \bigl\vert x(t)-0 \bigr\vert ^{2} = Lt^{-\eta _{1}}E \bigl\vert x(t) \bigr\vert ^{2}, \end{aligned}$$

(3.10)

$$\begin{aligned}& E \bigl\vert \sigma \bigl(t, x(t)\bigr) \bigr\vert ^{2}=E \bigl\vert \sigma \bigl(t, x(t)\bigr)-\sigma (t, 0) \bigr\vert ^{2} \leq Lt^{- \eta _{1}}E \bigl\vert x(t)-0 \bigr\vert ^{2} = Lt^{-\eta _{1}}E \bigl\vert x(t) \bigr\vert ^{2}, \end{aligned}$$

(3.11)

which imply that condition (H₅) holds. According to Theorem 3.2, the solution of (1.1) is global attractive. □

Lemma 3.4

Suppose that assumptions (H₀), (H₂) and (H₄) hold. ThenFmapsD̄intoD̄andFis continuous inD̄.

Proof

Since $0 < 2\mu -1< \eta$ and $0<\nu <1$, we can choose a constant $\xi > 0$ sufficiently small such that $\xi <1-\nu$ and $\xi -\eta +2\mu -1< 0$. Let $T > 0$ be sufficiently large such that

$$ \begin{aligned}[b] & 3\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}T^{\xi +2 \nu -2}E \vert x_{0} \vert ^{2} \\ &\quad{} +3 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{\varGamma ^{2}(\mu )\varGamma (2\mu -\eta )}(1+TrQ)T^{ \xi -(1+\eta -2\mu )}< 1. \end{aligned} $$

(3.12)

Define a set $\bar{D} = \{x(t) : x \in C((0,\infty ),L_{2}(\varOmega ,X)), E \vert x(t) \vert ^{2} \leq t^{-\xi }\text{ for }t \geq T\}$, we note that D̄ is a nonempty closed bounded and convex subset of $C_{0}((0,\infty ),L_{2}(\varOmega ,X))$.

We now show that F maps D̄ into D̄. For $x \in \bar{D}$, from (H₂), (H₄) and (3.12), we can deduce that

$$\begin{aligned}& E \bigl\vert (Fx) (t) \bigr\vert ^{2} \\& \quad \leq 3E \bigl\vert S_{\mu ,\beta }(t)x_{0} \bigr\vert ^{2}+3E \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{ \mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad\quad{} +3TrQE \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,d \omega (s) \\& \quad = \biggl(3t^{\xi } \bigl\vert S_{\mu ,\beta }(t)x_{0} \bigr\vert ^{2}+3t^{\xi } \int _{0}^{t}(t-s)^{2 \mu -2} \bigl\vert P_{\mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad\quad{} +3t^{\xi }TrQ \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,d\omega (s) \biggr)t^{-\xi } \\& \quad \leq \biggl(\frac{3M^{2}}{\varGamma ^{2}(\beta (1-\mu )+\mu )}t^{\xi +2 \nu -2} \vert x_{0} \vert \\& \quad\quad{} +3 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{\varGamma ^{2}(\mu )\varGamma (2\mu -\eta )}(1+TrQ)t^{ \xi -(1+\eta -2\mu )} \biggr)t^{-\xi } \\& \quad \leq t^{-\xi }, \quad t>T, \end{aligned}$$

which implies that $F\bar{D}\subset \bar{D}$. We next shall prove that F is continuous in D̄ by considering F on two intervals.

Firstly, for $t>T_{1}>T$, for each $x_{m}\in \bar{D}$, $m=1,2,\ldots$ , with $\lim_{m\rightarrow \infty }E \vert x^{(m)}(t) \vert ^{2}=E \vert x(t) \vert ^{2}$, we have

$$\begin{aligned}& E \bigl\vert F x_{m}(t)-F x(t) \bigr\vert ^{2} \\& \quad \leq 2 \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}E \bigl\vert (f \bigl(s,x^{(m)}(s)\bigr)-f\bigl(s,x(s)\bigr) \bigr\vert ^{2} \,ds \\& \quad\quad{} +2TrQ \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}E \bigl\vert (\sigma \bigl(s,x^{(m)}(s)\bigr)- \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad \leq 2 \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}(E \bigl\vert \bigl(f \bigl(s,x^{(m)}(s)\bigr) \bigr\vert ^{2}+E \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert ^{2}\bigr)\,ds \\& \quad\quad{} +2TrQ \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}(E \bigl\vert \bigl(\sigma \bigl(s,x^{(m)}(s)\bigr) \bigr\vert ^{2}+E \bigl\vert \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\bigr)\,ds \\& \quad \leq 4\biggl(\frac{ML}{\varGamma (\mu )}\biggr)^{2}(1+TrQ) \int _{0}^{t}(t-s)^{2 \mu -2}s^{-\eta }\,ds \\& \quad \leq 4 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{{\varGamma ^{2}(\mu )}\varGamma (2\mu -\eta )}(1+TrQ)T_{1}^{-(1+ \eta -2\mu )}< \varepsilon . \end{aligned}$$

Next, for $0< t\leq T_{1}$, we have

$$\begin{aligned}& E \bigl\vert F x_{m}(t)-F x(t) \bigr\vert ^{2} \\& \quad\leq 2 \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}E \bigl\vert (f \bigl(s,x^{(m)}(s)\bigr)-f\bigl(s,x(s)\bigr) \bigr\vert ^{2} \,ds\,ds \\& \quad\quad{} +2TrQ \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}E \bigl\vert (\sigma \bigl(s,x^{(m)}(s)\bigr)- \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad \leq 2\frac{M^{2}}{\varGamma ^{2}(\mu )} \int _{0}^{t}(t-s)^{2\mu -2}E \bigl\vert (f \bigl(s,x^{(m)}(s)\bigr)-f\bigl(s,x(s)\bigr) \bigr\vert ^{2} \,ds \\& \quad\quad{} +2TrQ\frac{M^{2}}{\varGamma ^{2}(\mu )} \int _{0}^{t}(t-s)^{2\mu -2}E \bigl\vert ( \sigma \bigl(s,x^{(m)}(s)\bigr)-\sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds. \end{aligned}$$

Since $\lim_{m\rightarrow \infty } E \vert f(t, x_{m}(t))-f(t, x(t)) \vert ^{2} = 0$, $\lim_{m\rightarrow \infty } E \vert \sigma (t, x_{m}(t))- \sigma (t, x(t)) \vert ^{2} = 0$ as $m\rightarrow \infty$, it follows from the Lebesgue dominated convergence theorem that $E \vert (Fx_{m})(t)-(Fx)(t) \vert ^{2} \rightarrow 0$ as $m\rightarrow \infty$. The above arguments lead to the fact that the operator F is continuous. □

Lemma 3.5

Suppose that assumptions (H₀), (H₂) and (H₄) hold. Then$\{Fx : x \in \bar{D}\}$is equicontinuous and$\lim_{t\rightarrow \infty } E \vert (Fx)(t) \vert ^{2} = 0$uniformly for$x\in \bar{D}$.

Proof

Since $\nu <1$ and $1+\eta -2\mu >0$, there exists a $T_{1} > T$ such that, for $t > T_{1}$,

$$ \biggl( \frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2\nu -2}E \vert x_{0} \vert ^{2}< \frac{\varepsilon }{6} $$

and

$$\begin{aligned}& \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{\varGamma ^{2}(\mu )\varGamma (2\mu -\eta )}t^{-(1+ \eta -2\mu )}< \frac{\varepsilon }{6},\\& \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{\varGamma ^{2}(\mu )\varGamma (2\mu -\eta )}TrQt^{-(1+ \eta -2\mu )}< \frac{\varepsilon }{6}. \end{aligned}$$

For any $x \in \bar{D}$ and $t_{1}, t_{2} > T_{1}$, we can deduce that

$$\begin{aligned}& E \bigl\vert (Fx) (t_{2})-(Fx) (t_{1}) \bigr\vert ^{2} \\& \quad \leq 6E \bigl\vert S_{\mu ,\beta }(t_{2})x_{0} \bigr\vert ^{2}+6E \bigl\vert S_{\mu ,\beta }(t_{1})x_{0} \bigr\vert ^{2}+6 \int _{0}^{t_{2}}(t_{2}-s)^{2\gamma -2} \bigl\vert P_{\mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert f\bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad\quad{} +6 \int _{0}^{t_{1}}(t_{1}-s)^{2\mu -2} \bigl\vert P_{\mu }(t_{1}-s) \bigr\vert ^{2}E \bigl\vert f\bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad\quad{} +6TrQ \int _{0}^{t_{2}}(t_{2}-s)^{2\mu -2} \bigl\vert P_{\mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad\quad{} +6TrQ \int _{0}^{t_{1}}(t_{1}-s)^{2\mu -2} \bigl\vert P_{\mu }(t_{1}-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad \leq 6\frac{M^{2}}{\varGamma ^{2}(\beta (1-\mu )+\mu )}t_{2}^{2\nu -2}E \vert x_{0} \vert ^{2}+6 \frac{M^{2}}{\varGamma ^{2}(\beta (1-\mu )+\mu )}t_{1}^{2\nu -2}E \vert x_{0} \vert ^{2} \\& \quad\quad{} +6\frac{M^{2}L^{2}}{\varGamma ^{2}(\mu )}(1+TrQ) \int _{0}^{t_{2}}(t_{2}-s)^{2 \mu -2}s^{-\eta } \,ds \\& \quad\quad{} +6\frac{M^{2}L^{2}}{\varGamma ^{2}(\mu )}(1+TrQ) \int _{0}^{t_{1}}(t_{1}-s)^{2 \mu -2}s^{-\eta } \,ds \\& \quad \leq 6\frac{M^{2}}{\varGamma ^{2}(\beta (1-\mu )+\mu )}t_{2}^{2\nu -2}E \vert x_{0} \vert ^{2}+6 \frac{M^{2}}{\varGamma ^{2}(\beta (1-\mu )+\mu )}t_{1}^{2\nu -2}E \vert x_{0} \vert ^{2} \\& \quad\quad{} +6 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{\varGamma ^{2}(\mu )\varGamma (2\mu -\eta )}(1+TrQ)\bigl[t_{2}^{-(1+ \eta -2\mu )}+t_{1}^{-(1+\eta -2\mu )} \bigr]. \end{aligned}$$

Furthermore, for $0 < t_{1} < t_{2} \leq T_{1}$, we have

$$\begin{aligned}& E \bigl\vert (Fx) (t_{2})-(Fx) (t_{1}) \bigr\vert ^{2} \\ & \quad \leq E \biggl\vert S_{\mu ,\beta }(t_{2})x_{0}-S_{\mu ,\beta }(t_{1})x_{0} \\ & \quad\quad{} + \int _{0}^{t_{2}}(t_{2}-s)^{\mu -1}P_{\mu }(t_{2}-s)f \bigl(s,x(s)\bigr)\,ds- \int _{0}^{t_{1}}(t_{1}-s)^{\gamma -1}P_{\mu }(t_{1}-s)f \bigl(s,x(s)\bigr)\,ds \\ & \quad\quad{} + \int _{0}^{t_{2}}(t_{2}-s)^{\mu -1}P_{\mu }(t_{2}-s) \sigma \bigl(s,x(s)\bigr)\,d \omega (s) \\ & \quad\quad{} - \int _{0}^{t_{1}}(t_{1}-s)^{\mu -1}P_{\mu }(t_{1}-s)f \bigl(s,x(s)\bigr)\,d \omega (s) \biggr\vert ^{2} \\ & \quad \leq 7 \bigl\vert S_{\mu ,\beta }(t_{2})-S_{\mu ,\beta }(t_{1}) \bigr\vert ^{2}E \vert x_{0} \vert ^{2} \\ & \quad\quad{} +7 \int _{0}^{t_{1}}\bigl[(t_{2}-s)^{2\mu -2}-(t_{1}-s)^{2\mu -2} \bigr] \bigl\vert P_{ \mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert f\bigl(s,x(s)\bigr) \bigr\vert ^{2} \,ds \\ & \quad\quad{} +7TrQ \int _{0}^{t_{1}}\bigl[(t_{2}-s)^{2\mu -2}-(t_{1}-s)^{2\mu -2} \bigr]^{2} \bigl\vert P_{ \mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\ & \quad\quad{} +7 \int _{0}^{t_{1}}(t_{1}-s)^{2\mu -2} \bigl\vert P_{\mu }(t_{2}-s)-P_{\mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert f\bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\ & \quad\quad{} +7TrQ \int _{0}^{t_{1}}(t_{1}-s)^{2\mu -2} \bigl\vert P_{\mu }(t_{2}-s)-P_{ \mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\ & \quad\quad{} +7 \int _{t_{1}}^{t_{2}}\bigl[(t_{2}-s)^{2\mu -2}-(t_{1}-s)^{2\mu -2} \bigr] \bigl\vert P_{ \mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert f\bigl(s,x(s)\bigr) \bigr\vert ^{2} \,ds \\ & \quad\quad{} +7TrQ \int _{t_{1}}^{t_{2}}\bigl[(t_{2}-s)^{2\mu -2}-(t_{1}-s)^{2\mu -2} \bigr] \bigl\vert P_{ \mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds. \end{aligned}$$

By using the arguments employed in Lemma 3.1, the Lebesgue dominated convergence theorem and Remark 2.3, we get $E \vert (Fx)(t_{2})-(Fx)(t_{1}) \vert ^{2}\rightarrow 0$ as $t_{2} - t_{1}\rightarrow 0$. Therefore, we conclude that the family of functions $\{Fx : x \in \bar{D}\}$ is equicontinuous.

We now verify that $\lim_{t\rightarrow \infty }E \vert (Fx)(t) \vert ^{2}=0$ uniformly for $x\in \bar{D}$. Indeed, by a standard calculation, we have

$$\begin{aligned}& E \bigl\vert (Fx) (t) \bigr\vert ^{2} \\ & \quad\leq 3E \bigl\vert S_{\mu ,\beta }(t)x_{0} \bigr\vert +3E \biggl\vert \int _{0}^{t}(t-s)^{\mu -1}P_{ \mu }(t-s)f \bigl(s,x(s)\bigr)\,ds \biggr\vert ^{2} \\ & \quad\quad{} +3E \biggl\vert \int _{0}^{t}(t-s)^{\mu -1}P_{\mu }(t-s) \sigma \bigl(s,x(s)\bigr)\,d \omega (s) \biggr\vert ^{2} \\ & \quad\leq 3\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2\nu -2}E \vert x_{0} \vert ^{2}+3 \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\ & \quad\quad{} +3TrQ \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t_{2}-s) \bigr\vert ^{2}E \bigl\vert \sigma \bigl(s,x(s)\bigr) \bigr\vert ^{2}\,ds \\ & \quad \leq 3\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2\nu -2}E \vert x_{0} \vert ^{2}+3\biggl( \frac{ML}{\varGamma (\mu )} \biggr)^{2}(1+TrQ) \int _{0}^{t}(t-s)^{2\mu -2}s^{- \eta }\,ds \\ & \quad \leq 3\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{2\nu -2}E \vert x_{0} \vert ^{2}+3 \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{\varGamma ^{2}(\mu )\varGamma (2\mu -\eta )}(1+TrQ)t^{-(1+ \eta -2\mu )} \\ & \quad \leq 3 \biggl[\biggl(\frac{M}{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}t^{\nu -1}E \vert x_{0} \vert ^{2} \\ & \quad\quad{} + \frac{M^{2}L^{2}\varGamma (1-\eta )\varGamma (2\mu -1)}{\varGamma ^{2}(\mu )\varGamma (2\mu -\eta )}(1+TrQ)t^{- \frac{1}{2}(1+\eta -2\mu )} \biggr]t^{-\frac{1}{2}(1+\eta -2\gamma )} \\ & \quad \leq t^{-\frac{1}{2}(1+\eta -2\mu )}\rightarrow 0, \quad \text{as } t \rightarrow \infty . \end{aligned}$$

Thus, we deduce that $\lim_{t\rightarrow \infty }E \vert (Fx)(t) \vert ^{2}=0$ uniformly for $x\in \bar{D}$. □

We now proceed to our third attractivity result for system (1.1).

Theorem 3.3

Suppose that assumptions (H₁)–(H₄) hold. Then the Cauchy problem (1.1) admits at least one attractive solution.

Proof

In order to verify that x is a mild solution of (1.1) in D̄, we need to verify that the operator $Fx=x$ has a fixed point in D̄. By Lemmas 3.4–3.5, it follows that $F : \bar{D} \rightarrow \bar{D}$ is continuous and F maps D̄ into itself. Moreover, $\{Fx : x \in \bar{D}\}$ is equicontinuous and $\lim_{t\rightarrow \infty }E \vert Fx(t) \vert ^{2}= 0$ is uniformly valid for $x \in \bar{D}$. We next show that F is relatively compact. From the definition of $S_{\mu ,\beta }(t)$, we have

$$\begin{aligned} S_{\mu ,\beta }(t)x_{0} =&I^{\beta (1-\mu )}_{0^{+}}T_{\mu }(t)x_{0} \\ =&\frac{1}{\varGamma (\beta (1-\mu ))} \int _{0}^{t}(t-s)^{\beta (1-\mu )-1}T_{ \mu }(s)x_{0} \,ds \\ =&\frac{1}{\varGamma (\beta (1-\mu ))} \int _{0}^{t}(t-s)^{\beta (1-\mu )-1}s^{ \mu -1}P_{\mu }(s)x_{0} \,ds \\ =&\frac{1}{\varGamma (\beta (1-\mu ))} \int _{0}^{t}(t-s)^{\beta (1-\mu )-1}s^{ \mu -1} \int _{0}^{\infty }\mu \theta M_{\mu }(\theta )x_{0}\,d\theta \,ds. \end{aligned}$$

For $\forall \lambda \in (0,t)$ and $\forall \delta >0$, we define

$$\begin{aligned} \bigl(F^{\lambda ,\delta }x\bigr) (t) =&\bigl(F_{1}^{\lambda ,\delta }x \bigr) (t)+\bigl(F_{2}^{ \lambda ,\delta }x\bigr) (t) \\ =&\frac{1}{\varGamma (\beta (1-\mu ))} \int _{0}^{t-\lambda }(t-s)^{ \beta (1-\mu )-1}s^{\mu -1}S \bigl(s^{\mu }\theta \bigr) \int _{\delta }^{\infty } \mu \theta M_{\mu }(\theta )x_{0}\,d\theta \,ds \\ &{} +S\bigl(\lambda ^{\mu }\theta \bigr) \int _{0}^{t-\lambda } \int _{\delta }^{ \infty }\mu \theta M_{\mu }(\theta )S \bigl((t-s)^{\mu }\theta -\lambda ^{ \mu }\theta \bigr) (t-s)^{\mu -1}f\bigl(s,x(s)\bigr)\,d\theta \,ds \\ &{} +S\bigl(\lambda ^{\mu }\theta \bigr) \int _{0}^{t-\lambda } \int _{\delta }^{ \infty }\mu \theta M_{\mu }(\theta )S \bigl((t-s)^{\mu }\theta -\lambda ^{ \mu }\theta \bigr) (t-s)^{\mu -1} \\ &{}\times\sigma \bigl(s,x(s)\bigr)\,d\theta \,d\omega (s). \end{aligned}$$

Since $\{S(t)\}_{t>0}$ is compact, it is easy to see that $\bar{V}_{1}^{\lambda ,\delta }(t)=\{(F_{1}^{\lambda ,\delta }x)(t),x \in \bar{D}\}$ is relatively compact for $t>0$. Next, we verify that $\bar{V}_{1}(t)=\{(F_{1}x)(t),x\in \bar{D}\}$ is relatively compact for $t>0$. Observe that

$$\begin{aligned}& E \bigl\vert (F_{1}(t)-\bigl(F_{1}^{\lambda ,\delta }x \bigr) (t) \bigr\vert ^{2} \\& \quad \leq E \biggl\vert \frac{1}{\varGamma (\beta (1-\mu ))} \int _{0}^{t}(t-s)^{\beta (1- \mu )-1}s^{\mu -1}S \bigl(s^{\mu }\theta \bigr) \int _{0}^{\delta }\mu \theta M_{\mu }(\theta )x_{0}\,d\theta \,ds \\& \quad\quad{} +\frac{1}{\varGamma (\beta (1-\mu ))} \int _{0}^{t}(t-s)^{\beta (1- \mu )-1}s^{\mu -1}S \bigl(s^{\mu }\theta \bigr) \int _{\delta }^{\infty }\mu \theta M_{\mu }(\theta )x_{0}\,d\theta \,ds \\& \quad\quad{}-\frac{1}{\varGamma (\beta (1-\mu ))} \int _{0}^{t-\lambda }(t-s)^{ \beta (1-\mu )-1}s^{\mu -1}S \bigl(s^{\mu }\theta \bigr) \int _{\delta }^{\infty } \mu \theta M_{\mu }(\theta )x_{0}\,d\theta \,ds \biggr\vert ^{2} \\& \quad \leq 2\frac{M}{\varGamma (\beta (1-\mu )+\mu )}E \biggl\vert \int _{0}^{t}(t-s)^{ \beta (1-\mu )-1}s^{\mu -1} \int _{0}^{\delta }\mu \theta M_{\mu }( \theta )x_{0}\,d\theta \,ds \biggr\vert ^{2} \\& \quad\quad{} + 2\frac{M}{\varGamma (\beta (1-\mu )+\mu )}E \biggl\vert \int _{t-\lambda }^{t}(t-s)^{ \beta (1-\mu )-1}s^{\mu -1} \int _{\delta }^{\infty }\mu \theta M_{\mu }( \theta )x_{0}\,d\theta \,ds \biggr\vert ^{2} \\& \quad \leq 2\biggl(\frac{M\mu }{\varGamma (\beta (1-\mu )+\mu )}\biggr)^{2}E \vert x_{0} \vert ^{2}B^{2}\bigl( \beta (1-\mu ),\mu \bigr) \biggl\vert \int _{0}^{\delta }\theta M_{\mu }(\theta )\,d \theta \biggr\vert ^{2} \\& \quad\quad{} +2\biggl(\frac{M\mu }{\varGamma (\beta (1-\mu )+\mu )} \frac{1}{\varGamma (1+\mu )}\biggr)^{2}E \vert x_{0} \vert ^{2} \biggl\vert \int _{t-\lambda }^{t}(t-s)^{ \beta (1-\mu )-1}s^{\mu -1} \,ds \biggr\vert ^{2}\rightarrow 0, \\& \quad \quad \text{as } \lambda ,\delta \rightarrow 0^{+}, \end{aligned}$$

which, together with Lemma 3.3 implies that $\bar{V}(t)=\{(Fx)(t),x\in \bar{D}\}$ is relatively compact for $t>0$. In view of the foregoing arguments, it follows by Schauder’s fixed point theorem that system (1.1) has a mild solution $x \in \bar{D}$ and that $x(t)$ tends to zero as $t\rightarrow \infty$. This completes the proof. □

To end this section, we now present the last result of our paper.

Theorem 3.4

Suppose that assumptions (H₁)–(H₃) and the following assumptions are satisfied:

(H₇):

There exists a strictly decreasing function$\mathcal{A} : (0,\infty ) \rightarrow [0,\infty )$such that,

for all$\psi \in C((0,\infty ),L_{2}(\varOmega ,X))$, and$t>0$,

$$\begin{aligned} \begin{gathered} E \biggl\vert S_{\mu ,\beta }(t)x_{0}+ \int _{0}^{t}(t-s)^{\mu -1}P_{ \mu }(t-s)f \bigl(s,x(s)\bigr)\,ds \\ \quad{} + \int _{0}^{t}(t-s)^{\mu -1}P_{ \mu }(t-s) \sigma \bigl(s,x(s)\bigr)\,d\omega (s) \biggr\vert ^{2}\leq \mathcal{A}(t), \end{gathered} \end{aligned}$$

and$\lim_{t\rightarrow +\infty }\mathcal{A}(t)=0$.

Then the system (1.1) has at least one attractive solution in$C((0,\infty ),L_{2}(\varOmega ,X))$.

Proof

Define a set $D_{3}:= \{u \in C((0,\infty ),L_{2}(\varOmega ,X)) : E \vert u(t) \vert ^{2} \leq \mathcal{A}(t)\text{ for all }t \geq 0\}$. We note that $D_{3}$ is nonempty and convex. We next verify that $D_{3}$ is bounded. Indeed, from (H₇), we have

$$ \sup_{t>0}E \bigl\vert u(t) \bigr\vert ^{2} \leq \mathcal{A}(t), \quad t>0 \text{ for every } u \in C\bigl((0,\infty ),L_{2}(\varOmega ,X)\bigr). $$

For $t>0$, we introduce

$$\begin{aligned} (\varPsi u) (t) =&S_{\mu ,\beta }(t)x_{0}+ \int _{0}^{t}(t-s)^{\mu -1}P_{ \mu }(t-s)f \bigl(s,u(s)\bigr)\,ds\\ &{} + \int _{0}^{t}(t-s)^{\mu -1}P_{\mu }(t-s) \sigma \bigl(s,u(s)\bigr)\,d\omega (s). \end{aligned}$$

Clearly $\varPsi (D_{3}) \subset D_{3}$ by (H₇). We next show continuity of Ψ. Let $\{u_{n}\}_{n\in N}$ be a sequence in $D_{3}$ such that $\lim_{n\rightarrow \infty }u_{n} = u$ is uniformly valid on compact subsets of $(0,\infty )$. By (H₇), we deduce that $\mathcal{A}$ tend to 0 at infinity and $\mathcal{A}$ is strictly decreasing. Thus, for an arbitrary given $\epsilon > 0$, there exists $T > 0$ such that $\mathcal{A}(t)<\epsilon$ for all $t>T$. Moreover, by (H₂), there exists $\delta _{1} > 0$ such that $\Vert u_{n}-u \Vert _{C((0,\infty ),L_{2}(\varOmega ,X))}<\delta _{1}$, which yields

$$ \begin{gathered} \bigl\Vert f\bigl(t,u_{n}(t)\bigr)-f\bigl(t,u(t)\bigr) \bigr\Vert _{C((0,\infty ),L_{2}(\varOmega ,X))}< \epsilon , \\ \bigl\Vert \sigma \bigl(t,u(t)\bigr)-\sigma \bigl(t,v(t)\bigr) \bigr\Vert _{C((0,\infty ),L_{2}( \varOmega ,X))}< \epsilon . \end{gathered} $$

(3.13)

Then there exist $N \in \mathbb{N}$ and $\delta ^{\prime } > 0$ such that $E \vert u_{n}(t)-u(t) \vert ^{2} < \delta ^{\prime }$ is uniformly valid on compact subsets of $(0,\infty )$ for $\forall n > N$ and $\forall t>0$. Since $\Vert u \Vert _{C((0,\infty ),L_{2}(\varOmega ,X))} :=(\sup_{t>0}E \vert u(t) \vert ^{2})^{ \frac{1}{2}}$, we have $\Vert u_{n}-u \Vert _{C((0,\infty ),L_{2}(\varOmega ,X))} := (\sup_{t>0}E \vert u_{n}(t)-u(t) \vert ^{2})^{ \frac{1}{2}}$ for $\forall t > 0$.

Now we prove continuity of Ψ as follows.

For $0 < t < T$, by Hölder’s inequality, (H₃) and (3.13), we obtain

$$\begin{aligned}& E \bigl\vert \varPsi (u_{n})-\varPsi (u) \bigr\vert ^{2} \\& \quad \leq 2 \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}E \bigl\vert (f \bigl(s,u_{n}(s)\bigr)-f\bigl(s,u(s)\bigr) \bigr\vert ^{2} \,ds \\& \quad\quad{} +2 \int _{0}^{t}(t-s)^{2\mu -2} \bigl\vert P_{\mu }(t-s) \bigr\vert ^{2}E \bigl\vert (\sigma \bigl(s,u_{n}(s)\bigr)- \sigma \bigl(s,u(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad \leq 2\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \int _{0}^{t}(t-s)^{2\mu -2}E \bigl\vert (f \bigl(s,u_{n}(s)\bigr)-f\bigl(s,u(s)\bigr) \bigr\vert ^{2} \,ds \\& \quad\quad{} +2TrQ\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \int _{0}^{t}(t-s)^{2\mu -2}E \bigl\vert ( \sigma \bigl(s,u_{n}(s)\bigr)-\sigma \bigl(s,u(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad \leq 2\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \int _{0}^{t}(t-s)^{2\mu -2}\,ds \int _{0}^{t}E \bigl\vert (f\bigl(s,u_{n}(s) \bigr)-f\bigl(s,u(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad\quad{} + 2TrQ\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2} \int _{0}^{t}(t-s)^{2\mu -2}\,ds \int _{0}^{t}E \bigl\vert (\sigma \bigl(s,u_{n}(s)\bigr)-\sigma \bigl(s,u(s)\bigr) \bigr\vert ^{2}\,ds \\& \quad \leq 2\biggl(\frac{M}{\varGamma (\mu )}\biggr)^{2}(1+TrQ) \biggl( \int _{0}^{t}(t-s)^{ \frac{2\mu -2}{2-2\mu _{1}}} \biggr)^{2-2\mu _{1}}\biggl( \int _{0}^{t} \vert \varepsilon \vert ^{\frac{1}{2\mu _{1}-1}}\,ds\biggr)^{2\mu _{1}-1}. \end{aligned}$$

Next, for $0< T < t$, using (H₇) and the S-invariance of Ψ, we have the estimate

$$ E \bigl\vert \varPsi (u_{n}) (t)-\varPsi (u) (t) \bigr\vert ^{2}\leq 2E \bigl\vert \varPsi (u_{n}) (t) \bigr\vert ^{2}+2E \bigl\vert \varPsi (u) (t) \bigr\vert ^{2} \leq 2\mathcal{A}(t) < 2\varepsilon . $$

This leads to the fact that Ψ is continuous.

As argued in Lemma 3.1, we conclude that $\varPsi (D_{3})$ is equicontinuous on each compact interval $[0, T]$ for $\forall T > 0$, which establishes the claim.

Now, we claim that $\varPsi (D_{3})$ is uniformly bounded. Indeed, by the definition of the set $D_{3}$ and the fact that $\mathcal{A}$ is strictly decreasing, we can infer $\sup_{u\in \varPsi (D_{3})}\sup_{t>T}E \vert u(t) \vert ^{2}\leq \mathcal{A}(T-0)$. By (H₇), we obtain $\lim_{T\rightarrow \infty }\sup_{u\in \varPsi (D_{3})} \sup_{t>T}E \vert u(t) \vert ^{2}= 0$. On the other hand, by similar arguments to Theorem 3.3, we can show that $V(t)=\{(\varPsi u)(t),u\in D_{3}\}$ is relatively compact for $t>0$. Consequently, Lemma 2.6 enables us to claim that the family $\varPsi (D_{3})$ is relatively compact in $C((0,\infty ),L_{2}(\varOmega ,X))$. Thus, from Schauder’s fixed point theorem, it shows that the operator Ψ has a fixed point in $D_{3}$. Hence the system (1.1) has at least one attractive solution $u_{0}\in D_{3}$. This completes the proof. □

4 Conclusion

In this paper, we have revealed that a certain class of Hilfer fractional stochastic evolution equations with fast decaying nonlinear term have global attractive solutions, whereas the integer-order evolution equations do not have such attractivity. Our future work will be focused on addressing the attractivity for Hilfer fractional stochastic evolution equations with fractional Brownian motion and Poisson jumps.