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Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion

Abstract

We introduce the investigation of approximate controllability for a new class of nonlocal and noninstantaneous impulsive Hilfer fractional neutral stochastic integrodifferential equations with fractional Brownian motion. An appropriate set of sufficient conditions is derived for the considered system to be approximately controllable. For the main results, we use fractional calculus, stochastic analysis, fractional power of operators and Sadovskii’s fixed point theorem. At the end, an example is also given to show the applicability of our obtained theory.

Introduction

Fractional differential equations have received great attention due to their applications in many important applied fields such as population dynamics, heat conduction in materials with memory, seepage flow in porous media, autonomous mobile robots, fluid dynamics, traffic models, electro magnetic, aeronautics, economics, and diffusion theory; see for instance [13]. Moreover, stochastic perturbation is unavoidable in nature and hence it is important and necessary to consider stochastic effect into the investigation of fractional differential equations. Recently, stochastic fractional differential equations driven by fractional Brownian motion have been considered greatly by research community in various aspects due to its salient features for real world problems (see [410]). The theory of impulsive differential equations and impulsive differential inclusions has wide applications in control, electrical engineering, mechanics and biology [11]. In general, the classical instantaneous impulses cannot describe certain dynamics of evolution processes. For example, when we consider the hemodynamic equilibrium of a person, the introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous processes. In fact, the above situation can be characterized by a new case of impulsive action, which starts at an arbitrary fixed point and stays active on a finite time interval called noninstantaneous impulsive differential equations. Hernández and O’Regan [12] and Pierri et al. [13] introduced some initial value problems for a new class of noninstantaneous impulsive differential equations to describe some certain dynamic change of evolution processes in the pharmacotherapy (as therapy using pharmaceutical drugs). Very recently, Pierri et al. [14] studied the existence of global solutions for a class of impulsive abstract differential equations with non-instantaneous impulses. On the other hand, controllability results for linear and nonlinear integer order differential systems was studied by several authors. The concept of controllability is an important part of mathematical control theory. Generally speaking, controllability means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. Controllability problems for different kinds of dynamical systems have been studied by several authors (see [1518]) and the references therein. Thus, the dynamical systems must be treated by the weaker concept of controllability, namely approximate controllability. Many authors studied the approximate controllability, for example, Sakthivel et al. studied the approximate controllability of nonlinear fractional dynamical systems (see [19]). Sakthivel et al. obtained sufficient conditions for the approximate controllability of fractional nonlinear differential inclusions (see [20]). Sakthivel et al. obtained sufficient conditions for the approximate controllability of fractional stochastic differential inclusions with nonlocal conditions (see [21]). Debbouche and Torres established sufficient conditions for the approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions (see [22]). Ahmed studied the approximate controllability of impulsive neutral stochastic differential equations with fractional Brownian motion in a Hilbert space (see [23]). Muthukumar and Rajivganthi obtained sufficient conditions for the approximate controllability of second-order neutral stochastic differential equations with infinite delay and Poisson jumps (see [24]). Yan and Jia established sufficient conditions for the approximate controllability of partial fractional neutral stochastic functional integrodifferential inclusions with state-dependent delay (see [25]). Yana and Lu studied the approximate controllability of a multi-valued fractional impulsive stochastic partial integrodifferential equation with infinite delay (see [26]). Very recently, a new set of sufficient conditions are established in [27] for the approximate controllability of a class of semilinear Hilfer fractional differential control inclusions in Banach spaces by using the fractional calculus, fixed point technique, semigroup theory and multi-valued analysis. Moreover, up to now no work has been reported yet regarding the approximate controllability results for noninstantaneous impulsive Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion, which motivates the present study. The purpose of this paper is to study the approximate controllability of noninstantaneous impulsive semilinear Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and nonlocal conditions in a Hilbert space of the form

$$\begin{aligned} \textstyle\begin{cases} D^{\nu,\mu}_{0+} [x(t) + F(t,x(t), x(b_{1}(t)),\ldots, x(b_{m}(t)))]+ A x(t) \\ \quad= B u(t)+ \int_{0}^{t} G(s,x(s), x(a_{1}(s)),\ldots,x(a_{k}(s))) \,d\omega(s)\\ \qquad{}+ \sigma(t,x(t), x(c_{1}(t)),\ldots, x(c_{p}(t))) \frac {dB^{H}}{dt},\quad t \in(s_{i},t_{i+1}], i \in[0,m],\\ x(t)=g_{i}(t,x(t)), \quad t \in(t_{i},s_{i}], i \in[1,m],\\ I^{(1-\nu)(1-\mu)}_{0+}x(0)+ \xi(x)= x_{0} , \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(D^{\nu,\mu}_{0+}\) is the Hilfer fractional derivative with \(0 \leq\nu\leq1\), \(0 < \mu< 1\), −A is the infinitesimal generator of an analytic semigroup of bounded linear operators \(S(t)\), \(t \geq0\), on a separable Hilbert space X with inner product \(\langle\cdot,\cdot \rangle\) and norm \(\|\cdot \|\) and the control function \(u(\cdot)\) is given in \(L_{2}(J, U)\), the Hilbert space of admissible control functions with U a Hilbert space, \(J=(0.T]\). The symbol B stands for a bounded linear from U into X, \(t_{i}\), \(s_{i}\) are fixed number satisfying \(0 = s_{0} < t_{1} \leq s_{1} \leq t_{2} < \cdots< s_{m-1} < t_{m} \leq s_{m} \leq t_{m+1} = T \) and \(g_{i}\) is noninstantaneous impulsive function for all \(i=1,2,\ldots,m\). Let K be another separable Hilbert space with inner product \(\langle\cdot ,\cdot\rangle_{K}\) and norm \(\|\cdot \|_{K}\). Suppose \(\{ \omega(t) \}_{t \geq0}\) is Q-Wiener process defined on \((\varOmega, \varUpsilon, \{ \varUpsilon_{t} \}_{t \geq0}, P)\) with values in Hilbert space K and \(\{B^{H}(t)\}_{t \geq0}\) is Q-fractional Brownian motion (fBm) with Hurst parameter \(H \in(\frac{1}{2}, 1)\) defined on \((\varOmega, \varUpsilon , \{\varUpsilon_{t} \}_{t \geq0}, P)\) with values in Hilbert space Y. We are also employing the same notation \(\|\cdot \|\) for the norm in X, K, Y, \(L(K, X)\) and \(L(Y,X)\) where \(L(K,X)\) and \(L(Y,X)\) denote, respectively, the space of all bounded linear operators from K into X and Y into X. The functions F, G, σ, \(g_{i}\) and ξ are given functions to be defined later.

The main contributions of this paper are summarized as follows:

  • The study of approximate controllability of noninstantaneous impulsive semilinear Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and nonlocal conditions described in the form (1.1) is an untreated topic in the literature and this is an additional motivation for writing this paper.

  • Using methods of functional analysis, a set of sufficient conditions are proposed for approximate controllability.

  • The results are established with the use of semigroup theory, fractional calculus and stochastic analysis.

  • The application is demonstrated through an example of stochastic control Hilfer fractional partial differential equation with fractional Brownian motion.

Preliminaries

In this section, some definitions and results are given which will be used throughout this paper.

Definition 2.1

([28])

The left-sided Riemann–Liouville fractional integral of order \(\mu> 0 \) with the lower limit a for a function \(f: [a, \infty)\rightarrow R \) is defined as

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

provided that the right side is pointwise defined on \([a,\infty )\), where \(\varGamma(\cdot)\) is the Gamma function.

Definition 2.2

(Hilfer fractional derivative [29])

The left-sided Hilfer fractional derivative of order \(0\leq\nu\leq1 \) and \(0 < \mu< 1\) of function \(f(t)\) is defined as

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

where \(D:= \frac{d}{dt}\).

Let \((\varOmega, \varUpsilon, P)\) be a complete probability space equipped with a normal filtration \(\varUpsilon_{t}, t \in[0,T]\) where \(\varUpsilon _{t}\) is the σ-algebra generated by random variables \(\{ \omega (s), B^{H}(s), s \in[0,T] \}\) and all P-null sets.

Suppose that \(\{\beta^{H}( t) , t \in[0, T ]\}\) is the one-dimensional fractional Brownian motion with Hurst parameter \(H \in (1/2,1)\). That is, \(\beta^{H}\) is a centered Gaussian process with covariance function \(R_{H}(s,t)= \frac{1}{2}(t^{2H} + s^{2H}- |t - s|^{2H})\) (see [30]).

Consider the Wiener process \(\omega= \omega(t)\), \(t \in[0,T]\) defined by \(\omega(t)= \beta^{H}((K^{*}_{H})^{-1}1_{[0,T]})\) then ω is a Wiener process. Moreover, \(\beta^{H}\) has the following Wiener integral representation:

$$ \beta^{H}(t)= \int_{0}^{t} K_{H} (t,s) \,d \omega(s) $$

where \(K_{H}(t,s)\) is the kernel given by

$$ K_{H}(t,s) = c_{H} s^{\frac{1}{2}- H} \int_{s}^{t} (v-s)^{H- \frac{3}{2}} v^{H-\frac{1}{2}} \,du $$

for \(s< t\), where \(c_{H} = \sqrt{\frac{H(2H-1)}{\beta(2-2H, H - \frac {1}{2})}}\) and

$$ \beta(p,q) = \int_{0}^{1} t^{p-1}(1-t)^{q-1}, \quad p>0, q>0 . $$

We put \(K_{H}(t,s)=0\) if \(t \leq s\).

We will denote by ζ the reproducing kernel Hilbert space of the fBm. In fact ζ is the closure of set of indicator functions \(\{ 1_{[0,T]}, t \in[0,T] \}\) with respect to the scalar product \(\langle 1_{[0,t]}, 1_{[0,s]} \rangle_{\zeta}= R_{H}(t,s)\).

The mapping \(1_{[0,T]} \rightarrow\beta^{H}(t)\) can be extended to an isometry from ζ onto the first Wiener chaos and we will denote by \(\beta^{H}(\varphi)\) the image of φ under this isometry.

We recall that for \(\psi, \varphi \in\zeta\) their scalar product in ζ is given by

$$ \langle\psi, \varphi\rangle_{\zeta}= H(2H-1) \int _{0}^{T} \int_{0}^{T} \psi(s) \varphi(t) \vert t-s \vert ^{2H-2} \,ds \,dt. $$

Let us consider the operator \(K^{*}\) from ζ to \(L_{2}([0,T])\) defined by

$$ \bigl(K^{*}_{H} \varphi\bigr) (s) = \int_{s}^{T} \varphi(t) \frac{\partial K_{H}}{\partial t} (t,s) \,dt. $$

Let \(Q \in L(Y,Y)\) be an operator defined by \(Qe_{n}=\lambda_{n} e_{n}\) with finite trace

$$ \operatorname{Tr} Q= \sum_{n=1}^{\infty} \lambda_{n} < \infty,\quad \lambda_{n} \geq0\ (n=1,2,\ldots), $$

are non-negative real numbers and \(\{e_{n}\}_{n=1}^{\infty}\) is a complete orthonormal basis in Y.

We define the infinite-dimensional fBm on Y with covariance Q as

$$ B^{H}(t)= B_{Q}^{H}(t)= \sum _{n=1}^{\infty}\sqrt {\lambda_{n} } e_{n} \beta_{n}^{H}(t), $$

where \(\beta_{n}^{H}(t)\) are standard fBms mutually independent on \((\varOmega, \varUpsilon,P)\). In order to define Wiener integrals with respect to the Q-fBm, we introduce the space \(L^{0}_{2}:= L^{0}_{2}(Y,X)\) of all Q-Hilbert Schmidt operators \(\psi : Y \rightarrow X\). We recall that \(\psi\in L(Y,X)\) is called a Q-Hilbert–Schmidt operator, if

$$ \Vert \psi \Vert ^{2}_{L^{0}_{2}}:= \sum _{n=1}^{\infty} \Vert \sqrt{\lambda}_{n} \psi e_{n} \Vert ^{2} < \infty $$

and that the space \(L^{0}_{2}\) equipped with the inner product \(\langle \varphi, \psi\rangle_{L_{2}^{0}} = \sum_{n=1}^{\infty}\langle \varphi e_{n}, \psi e_{n} \rangle\) is a separable Hilbert space.

Let \(\phi(s); s \in[0,T]\) be a function with values in \(L_{2}^{0}(Y,X)\), the Wiener integral of ϕ with respect to \(B^{H}\) is defined by

$$\begin{aligned} \int_{0}^{t} \phi(s) \,d B^{H}(s)= \sum _{n=1}^{\infty} \int_{0}^{t} \sqrt{\lambda}_{n} \phi(s) e_{n} \,d \beta^{H}_{n} = \sum _{n=1}^{\infty} \int_{0}^{t} \sqrt{\lambda}_{n} K^{*}(\phi e_{n}) (s) \,d \beta_{n}(s), \end{aligned}$$
(2.1)

where \(\beta_{n}\) is the standard Brownian motion.

Lemma 2.1

(see [4])

If\(\psi: [0,T]\rightarrow L^{0}_{2}(Y,X)\)satisfies\(\int_{0}^{T} \| \psi(s) \|^{2}_{L^{0}_{2}}< \infty\)then the above sum in (2.1) is well defined asX-valued random variable and we have

$$ E \biggl\Vert \int_{0}^{t} \psi(s) \,dB^{H} (s) \biggr\Vert ^{2} \leq2H t^{2H-1} \int_{0}^{t} \bigl\Vert \psi(s) \bigr\Vert ^{2}_{L^{0}_{2} } \,ds. $$

We suppose that \(0 \in\rho(A)\), the resolvent set of A, and \(\| S(t) \|\leq M \) for some constant \(M \geq1\) and every \(t \geq0\). We define the fractional power \(A^{-\gamma}\) by

$$ A^{-\gamma}= \frac{1}{\varGamma(\gamma)} \int _{0}^{\infty}t^{\gamma- 1}S(t) \,dt, \quad \gamma>0. $$

For \(\gamma\in(0,1]\), \(A^{\gamma}\) is a closed linear operator on its domain \(D(A^{\gamma})\). Furthermore, the subspace \(D(A^{\gamma})\) is dense in X. We will introduce the following basic properties of \(A^{\gamma}\).

Theorem 2.1

(see [31])

  1. (1)

    Let\(0 < \gamma\leq1\), then\(X_{\gamma}:= D(A^{\gamma})\)is a Banach space with the norm\(\|x \|_{\gamma}= \|A^{\gamma}x \|\), \(x \in X_{\gamma}\).

  2. (2)

    If\(0 < \beta< \gamma\leq1 \), then\(D(A^{\gamma}) \hookrightarrow D(A^{\beta})\)and the embedding is compact whenever the resolvent operator ofAis compact.

  3. (3)

    For every\(0 < \gamma\leq1\), there exists a positive constant\(C_{\gamma}\)such that

    $$ \bigl\Vert A^{\gamma}S(t) \bigr\Vert \leq\frac{C_{\gamma}}{t^{\gamma}},\quad 0 < t \leq T. $$

The collection of all strongly-measurable, square-integrable, X-valued random variables, denoted by \(L_{2}(\varOmega,X)\), is a Banach space equipped with norm

$$ \bigl\Vert x(\cdot) \bigr\Vert _{L_{2}(\varOmega,X)} = \bigl(E \bigl\Vert x( \cdot,\omega) \bigr\Vert ^{2}\bigr)^{\frac{1}{2}}, $$

where the expectation, E is defined by \(E(x)= \int_{\varOmega}x(\omega ) \,dP\).

Let \(C(J, L_{2}(\varOmega, X))\) be the Banach space of all continuous maps from J into \(L_{2}(\varOmega,X)\) satisfying the condition \(\sup_{t \in J} E \|x(t) \|^{2} < \infty\).

Define \(\bar{C} = \{ x:t^{(1- \nu)(1-\mu)} x(t) \in C(J, L_{2}(\varOmega ,X) )\}\), with norm \(\| \cdot\|_{\bar{C}}\) defined by

$$ \Vert \cdot \Vert _{\bar{C}} =\Bigl( \sup_{t \in J} E \bigl\vert t^{(1-\nu )(1-\mu)} x(t) \bigr\vert ^{2} \Bigr)^{\frac{1}{2}}. $$

Obviously, is a Banach space.

We impose the following conditions on data of the problem:

\((H1)\):

\(F: J \times X^{m+1} \rightarrow X \) is a continuous function, and there exist a constant \(\beta\in(0,1)\) and \(M_{1}, M_{2} > 0\) such that the function \((-A)^{\beta}F\) satisfies the Lipschitz condition:

$$ E \bigl\Vert A^{\beta}F(s_{1},x_{0},x_{1}, \ldots,x_{m})- A^{\beta}F(s_{2},y_{0},y_{1}, \ldots,y_{m}) \bigr\Vert ^{2} \leq M_{1} \Bigl( \vert s_{1} - s_{2} \vert + \max_{i=0,1,\ldots,m} E \Vert x_{i} - y_{i} \Vert ^{2}\Bigr), $$

for \(0 \leq s_{1}\), \(s_{2} \leq b\), \(x_{i}, y_{i} \in X\), \(i=0,1,\ldots ,m\) and the inequality

$$\begin{aligned} E \bigl\Vert A^{\beta}F(t,x_{0},x_{1}, \ldots,x_{m}) \bigr\Vert ^{2} \leq M_{2} \Bigl( \max_{i=0,1,\ldots,m} E \Vert x_{i} \Vert ^{2} + 1\Bigr) \end{aligned}$$
(2.2)

holds for \((t,x_{0},x_{1},\ldots,x_{m}) \in J \times X^{m+1}\).

\((H2)\):

The function \(G:J \times X^{k+1} \rightarrow L(K,X)\) satisfies the following conditions:

\((i)\):

for each \(t \in J\), the function \(G(t,\cdot): X^{k+1} \rightarrow L(K,X)\) is continuous and for each \((x_{0},x_{1},\ldots ,x_{n}) \in X^{n+1}\); the function \(G(\cdot,x_{0},x_{1},\ldots,x_{k}): J \rightarrow L(K,X)\) is \(\varUpsilon_{t}\)-measurable;

\((ii)\):

for each positive number \(q \in N\), there is a positive function \(h_{q}(\cdot): (0,T]\rightarrow R^{+} \) such that

$$ \sup_{ \Vert x_{0} \Vert ^{2},\ldots, \Vert x_{n} \Vert ^{2} \leq q} \int_{0}^{t} E \bigl\Vert G(s,x_{0}, x_{1},\ldots,x_{k}) \bigr\Vert ^{2}_{Q} \,ds \leq h_{q}(t), $$

the function \(s \rightarrow(t -s)^{\mu-1} h_{q}(s) \in L^{1}((0,T], R^{+})\) and there exists a \(\varLambda_{1} > 0 \) such that

$$ \lim_{q\rightarrow\infty} \inf\frac{\int_{0}^{t} (t-s)^{\mu-1} h_{q}(s) \,ds}{q} = \varLambda_{1} < \infty, \quad t \in(0,T]. $$

\((H3)\):

The function \(\sigma:J \times X^{p+1} \rightarrow L^{0}_{2}(Y,X)\) satisfies the following conditions:

\((i)\):

for each \(t \in J\), the function \(\sigma(t,\cdot): X^{p+1} \rightarrow L^{0}_{2}(Y,X)\) is continuous and for each \((x_{0},x_{1},\ldots,x_{p}) \in X^{p+1}\); the function \(\sigma (\cdot,x_{0},x_{1},\ldots,x_{p}): J \rightarrow L^{0}_{2}(Y,X)\) is \(\varUpsilon_{t}\)-measurable;

\((ii)\):

for each positive number \(q \in N\), there is a positive function \(\bar{h}_{q}(\cdot): (0,T]\rightarrow R^{+} \) such that

$$ \sup_{ \Vert x_{0} \Vert ^{2},\ldots, \Vert x_{n} \Vert ^{2} \leq q} E \bigl\Vert \sigma (t,x_{0}, x_{1},\ldots,x_{p}) \bigr\Vert ^{2}_{L^{0}_{2}} \leq\bar{h}_{q}(t), $$

the function \(s \rightarrow(t -s)^{\mu-1} \bar{h}_{q}(s) \in L^{1}((0,T], R^{+})\) and there exists a \(\varLambda_{2} > 0 \) such that

$$ \lim_{q\rightarrow\infty} \inf\frac{\int_{0}^{t} (t-s)^{\mu-1} \bar{h}_{q}(s) \,ds}{q} = \varLambda_{2} < \infty,\quad t \in(0,T], $$

\((H4)\):

The function \(g_{i}:(t_{i},s_{i}] \times X \rightarrow X\) is continuous and satisfies the following two conditions:

\((i)\):

There exists a constant \(M_{3}>0\), such that

$$ E { \bigl\Vert g_{i}(t,x) \bigr\Vert }^{2} \leq M_{3} E { \Vert x \Vert }^{2}, \quad \forall x\in X ; t\in(t_{i},s_{i}], i=1,2,\ldots,m. $$
\((ii)\):

There exists a constant \(M_{6}>0\), such that

$$ E { \bigl\Vert g_{i}(t,x_{1})-g_{i}(t,x_{2}) \bigr\Vert }^{2} \leq M_{6} E { \Vert x_{1}-x_{2} \Vert }^{2}, \quad \forall x_{1},x_{2}\in X ; t\in (t_{i},s_{i}], i=1,2,\ldots,m. $$

\((H5)\):

The function \(\xi:C(J,X)\rightarrow X\) satisfies the following two conditions:

\((i)\):

There exist positive constants \(M_{4}\) and \(M_{5}\) such that

$$ E { \bigl\Vert \xi(x) \bigr\Vert }^{2}\leq M_{4} E { \Vert x \Vert }^{2}+M_{5} \quad \forall x \in X; $$
\((ii)\):

There exists a constant \(M_{7}>0\), such that

$$ E { \bigl\Vert \xi(x_{1})-\xi(x_{2}) \bigr\Vert }^{2} \leq M_{7} E { \Vert x_{1}-x_{2} \Vert }^{2}, \quad \forall x_{1},x_{2}\in X . $$

Theorem 2.2

LetΦbe a condensing operator on a Banach spaceX, that is, Φis continuous and takes bounded sets into bounded sets, and\(\mu(\varPhi(\bar{B})) \leq\mu(\bar{B})\)for every bounded setofXwith\(\mu(\bar{B})>0\). If\(\varPhi(Z) \subset Z\)for a convex, closed and bounded setZofX, thenΦhas a fixed point inX (where\(\mu(\cdot)\)denotes Kuratowski’s measure of noncompactness).

Definition 2.3

An \(\varUpsilon_{t}\)-adapted stochastic process \(x(t): J \rightarrow X\) is said to be a mild solution of problem (1.1) if \(x_{0} \in X\) for each \(s \in[0,T)\) the function \(A P_{\mu}(t-s) F(s, x(s), x(b_{1}(s)),\ldots, x(b_{m}(s)))\) is integrable and the following stochastic integral equation is verified:

$$\begin{aligned}& \begin{aligned} x(t)&=S_{\nu, \mu}(t)\bigl[x_{0} - \xi(x) +F \bigl(0, x(0), x\bigl(b_{1}(0)\bigr),\ldots , x\bigl(b_{m}(0) \bigr)\bigr)\bigr]\\ &\quad {}-F\bigl(t, x(t), x\bigl(b_{1}(t)\bigr),\ldots, x \bigl(b_{m}(t)\bigr)\bigr) \\ &\quad {}- \int_{0}^{t} A P_{\mu}(t-s) F\bigl(s, x(s), x\bigl(b_{1}(s)\bigr),\ldots, x\bigl(b_{m}(s)\bigr)\bigr) \,ds + \int_{0}^{t} P_{\mu}(t-s) B u(s)\,ds \\ &\quad {}+ \int_{0}^{t} P_{\mu}(t-s) \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \\ &\quad {}+ \int_{0}^{t} P_{\mu}(t-s) \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d B^{H}(s), \quad t \in(0,t_{1}], \end{aligned} \\& x(t)= g_{i}\bigl(t,x(t)\bigr), \quad t \in(t_{i},s_{i}], i=1,2, \ldots,m, \\& \begin{aligned} x(t)&=S_{\nu, \mu}(t-s_{i})g_{i} \bigl(s_{i},x(s_{i})\bigr)-F\bigl(t, x(t), x \bigl(b_{1}(t)\bigr),\ldots, x\bigl(b_{m}(t)\bigr)\bigr) \\ &\quad {}- \int_{s_{i}}^{t} A P_{\mu}(t-s) F\bigl(s, x(s), x\bigl(b_{1}(s)\bigr),\ldots, x\bigl(b_{m}(s)\bigr)\bigr) \,ds + \int_{s_{i}}^{t} P_{\mu}(t-s) B u(s)\,ds \\ &\quad {}+ \int_{s_{i}}^{t} P_{\mu}(t-s) \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \\ &\quad {}+ \int_{s_{i}}^{t} P_{\mu}(t-s) \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d B^{H}(s),\\ &\quad t \in(s_{i},t_{i+1}], i=1,2,\ldots,m, \end{aligned} \end{aligned}$$
(2.3)

where

$$ \begin{gathered} S_{\nu, \mu}(t)= I^{\nu(1- \mu)}_{0+} P_{\mu}(t), \\ P_{\mu}(t)= t^{\mu-1} T_{\mu}(t), \\ T_{\mu}(t)= \int_{0}^{\infty}\mu \theta\varPsi_{\mu}(\theta) S \bigl(t^{\mu}\theta\bigr) \,d \theta, \end{gathered} $$
(2.4)

where

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

is a function of Wright-type which satisfies the following inequality \(\int_{0}^{\infty}\theta^{\tau}\varPsi_{\mu}(\theta) \,d \theta= \frac{\varGamma(1+ \tau)}{\varGamma(1+ \mu\tau)}\) for \(\theta\geq0\).

Lemma 2.2

(see [32])

The operator\(S_{\nu, \mu}\)and\(P_{\mu}\)have the following properties.

  1. (i)

    \(\{ P_{\mu}(t): t > 0\}\)is continuous in the uniform operator topology.

  2. (ii)

    For any fixed\(t >0\), \(S_{\nu, \mu}(t)\)and\(P_{\mu}(t)\)are linear and bounded operators, and

    $$ \bigl\Vert P_{\mu}(t) x \bigr\Vert \leq \frac{M t^{\mu-1}}{\varGamma(\mu)} \Vert x \Vert , \qquad \bigl\Vert S_{\nu,\mu}(t) x \bigr\Vert \leq\frac{M t^{(\nu-1)(1-\mu )}}{\varGamma(\nu(1-\mu)+ \mu)} \Vert x \Vert . $$
    (2.5)
  3. (iii)

    \(\{ P_{\mu}(t): t > 0\}\)and\(\{ S_{\nu, \mu} (t): t > 0\}\)are strongly continuous.

Lemma 2.3

For any\(x \in X\), \(\beta\in(0,1)\)and\(\delta\in(0,1]\), we have\(A T_{\mu}(t) x=A^{1-\beta} T_{\mu}(t) A^{\beta}x\), \(0\leq t \leq T\)and

$$ \bigl\Vert A^{\delta}T_{\mu}(t) x \bigr\Vert \leq \frac{\mu C_{\delta} \varGamma(2-\delta) }{t^{\delta\mu} \varGamma(1+\mu (1-\delta))} \Vert x \Vert , \quad 0< t \leq T. $$

In order to study the approximate controllability for the fractional control system (1.1), we introduce the following linear fractional differential system

$$\begin{aligned} \textstyle\begin{cases} D^{\nu,\mu}_{0+} x(t)+ A x(t)= B u(t),\quad t \in(0,T], \\ I^{(1-\nu)(1-\mu)}_{0+}x(0)=x_{0}. \end{cases}\displaystyle \end{aligned}$$
(2.6)

It is convenient at this point to introduce the operators associated with (2.6) as

$$ \varGamma^{T}_{0} = \int_{0}^{T} (T-s)^{\mu-1}T_{\mu}(T-s)BB^{*} T^{*}_{\mu}(T-s) \,ds, $$

and \(R(T,\varGamma_{0}^{T})= (T I + \varGamma_{0}^{T})^{-1}\), \(T>0\), where \(B^{*}\) and \(T^{*}_{\mu}\) denote the adjoint of B and \(T_{\mu}\), respectively.

Let \(x(T;x_{0}, u)\) be the state value of (1.1) at terminal state T, corresponding to the control u and the initial value \(x_{0}\). Denote by \(R(T,x_{0}) = \{ x(T;x_{0},u):u \in L_{2}(J, U) \}\) the reachable set of system (1.1) at terminal time T, its closure in X is denoted by \(\overline{ R(T,x_{0})}\)

Definition 2.4

The system (1.1) is said to be approximately controllable on the interval J if \(\overline{ R(T,x_{0})} = L_{2}(\varOmega, X)\).

Lemma 2.4

(see [20])

The fractional linear control system (2.6) is approximately controllable onJif and only if\(z (z I + \varGamma^{T}_{0})^{-1}\rightarrow0\)as\(z \rightarrow0^{+}\).

Lemma 2.5

For any \(\bar{x}_{T} \in L_{2}(\varOmega, X)\) there exist ψ̄ and \(\bar{\varphi}\in L_{2}(\varOmega; L_{2}(J;L^{0}_{2}))\) such that

$$ \bar{x}_{T}= E \bar{x}_{T} + \int_{0}^{T} \bar{\psi}(s) \,d\omega(s)+ \int_{0}^{T} \bar{\varphi}(s) \,dB^{H}(s). $$

Now for any \(\delta>0\) and \(\bar{x}_{T} \in L_{2}(\varOmega,X)\), we define the control function in the following form

$$\begin{aligned}& \begin{aligned} u^{\delta}(t)&=B^{*} T^{*}_{\mu}(T-t) \bigl(z I + \varGamma^{T}_{0}\bigr)^{-1} \\ &\quad {}\times\biggl\{ E \bar{x}_{T} - S_{\nu, \mu}(T)\bigl[x_{0} -\xi(x)+F \bigl(0, x(0), x\bigl(b_{1}(0)\bigr),\ldots, x\bigl(b_{m}(0) \bigr)\bigr)\bigr] \\ &\quad {}-F\bigl(T, x(T), x\bigl(b_{1}(T)\bigr),\ldots, x \bigl(b_{m}(T)\bigr)\bigr)+ \int_{0}^{T} \bar {\psi}(s) \,d\omega(s)+ \int_{0}^{T} \bar{\varphi}(s) \,dB^{H}(s) \biggr\} \\ &\quad {}-B^{*} T^{*}_{\mu}(T-t) \int_{0}^{T} \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1} P_{\mu}(T-s) A F\bigl(s, x(s), x \bigl(b_{1}(s)\bigr),\ldots, x\bigl(b_{m}(s)\bigr)\bigr)\,ds \\ &\quad {}-B^{*} T^{*}_{\mu}(T-t) \int_{0}^{T} \bigl(z I + \varGamma ^{T}_{0}\bigr)^{-1}P_{\mu}(T-s)\\ &\quad {}\times \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \\ &\quad {}-B^{*} T^{*}_{\mu}(T-t) \int_{0}^{T} \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1} P_{\mu}(T-s)\\ &\quad {}\times \sigma\bigl(s,x(s), x \bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s)\bigr)\bigr) \,dB^{H}(s),\quad t \in(0,t_{1}], \end{aligned} \\ & \\ & \begin{aligned} u^{\delta}(t)&=B^{*} T^{*}_{\mu}(T-t) \bigl(z I + \varGamma^{T}_{0}\bigr)^{-1} \biggl\{ E \bar{x}_{T} - S_{\nu, \mu}(T-s_{i})g_{i} \bigl(s_{i},x(s_{i})\bigr) \\ &\quad {}-F\bigl(T, x(T), x\bigl(b_{1}(T)\bigr),\ldots, x \bigl(b_{m}(T)\bigr)\bigr)+ \int_{0}^{T} \bar {\psi}(s) \,d\omega(s)+ \int_{0}^{T} \bar{\varphi}(s) \,dB^{H}(s) \biggr\} \\ &\quad {}-B^{*} T^{*}_{\mu}(T-t) \int_{s_{i}}^{T} \bigl(z I + \varGamma ^{T}_{0}\bigr)^{-1} P_{\mu}(T-s) A F \bigl(s, x(s), x\bigl(b_{1}(s)\bigr),\ldots, x\bigl(b_{m}(s) \bigr)\bigr)\,ds \\ &\quad {}-B^{*} T^{*}_{\mu}(T-t) \int_{s_{i}}^{T} \bigl(z I + \varGamma ^{T}_{0}\bigr)^{-1}P_{\mu}(T-s) \\ &\quad {}\times\int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \\ &\quad {}-B^{*} T^{*}_{\mu}(T-t) \int_{s_{i}}^{T} \bigl(z I + \varGamma ^{T}_{0}\bigr)^{-1} P_{\mu}(T-s)\\ &\quad {}\times \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,dB^{H}(s),\quad t \in(s_{i},t_{i+1}]. \end{aligned} \end{aligned}$$
(2.7)

Approximate controllability

In this section, we formulate sufficient conditions for the approximate controllability of the system (1.1). For this purpose, we first prove the existence of a mild solution for the system (1.1). Second we shall prove the system (1.1) is approximately controllable under certain assumptions.

Theorem 3.1

If the assumptions\((H1)\)\((H5)\)are satisfied, then the system (1.1) has a mild solution onJ, provided that

$$\begin{aligned}& \biggl[1 +\frac{ M^{4} T^{2\mu} M^{4}_{B}}{z^{2} \mu^{2} \varGamma ^{4}(\mu)} \biggr] \biggl\lbrace \frac{36 M^{2} ( M_{0}^{2} M_{2} + M_{3} + M_{4} )}{\varGamma^{2}(\nu(1-\mu)+\mu)} \\& \quad {}+ 36T^{2(1-\nu)(1-\mu )} \biggl[{M_{0}}^{2} M_{2} + \frac{ M^{2} T^{\mu}\varLambda_{1} \operatorname{Tr}(Q)}{\mu\varGamma^{2}(\mu)} \\& \quad {} + \frac{2H M^{2} \varLambda_{2} T^{2 H+\mu-1}}{\mu\varGamma ^{2}(\mu)}+\frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{2\mu \beta} M_{2} }{ \beta^{2} \varGamma^{2} (1+\mu\beta)} \biggr] \biggr\rbrace + T^{2(1-\nu)(1-\mu)} M_{3} < 1 \end{aligned}$$
(3.1)

and

$$\begin{aligned} \gamma_{1} =&9 \biggl[\frac{M^{2} ({M_{0}}^{2}{M_{1}}+ M_{6} + M_{7})}{\varGamma^{2}(\nu(1-\mu)+\mu)}+T^{2(1-\nu)(1-\mu)}\bigl( M_{6} +{M_{0}}^{2} {M_{1}}\bigr) \\ &{}+\frac{{M_{1}}(C_{1 - \beta})^{2} \varGamma^{2}(1+ \beta) T^{2\mu \beta+2(1-\nu)(1-\mu)}}{\beta^{2} \varGamma^{2} (1+\mu\beta)} \biggr] \end{aligned}$$
(3.2)
$$\begin{aligned} < &1, \end{aligned}$$
(3.3)

where\(M_{0} = \|A^{-\beta} \|\)and\(M_{B} = \|B \|\).

Proof

For any \(\delta> 0 \), consider the map \(\varPhi_{\delta}\) on defined by

$$\begin{aligned} (\varPhi_{\delta}x) (t) =&S_{\nu, \mu}(t)\bigl[x_{0} - \xi(x) +F\bigl(0, x(0), x\bigl(b_{1}(0)\bigr),\ldots, x\bigl(b_{m}(0) \bigr)\bigr)\bigr]\\ &{}-F\bigl(t, x(t), x\bigl(b_{1}(t)\bigr),\ldots, x \bigl(b_{m}(t)\bigr)\bigr) \\ &{}- \int_{0}^{t} A P_{\mu}(t-s) F\bigl(s, x(s), x\bigl(b_{1}(s)\bigr),\ldots, x\bigl(b_{m}(s)\bigr)\bigr) \,ds + \int_{0}^{t} P_{\mu}(t-s) B u^{\delta}(s)\,ds \\ &{}+ \int_{0}^{t} P_{\mu}(t-s) \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \\ &{}+ \int_{0}^{t} P_{\mu}(t-s) \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d B^{H}(s),\quad t \in(0,t_{1}], \\ (\varPhi_{\delta}x) (t) =& g_{i}\bigl(t,x(t)\bigr), t \in(t_{i},s_{i}],\quad i=1,2, \ldots,m, \\ (\varPhi_{\delta}x) (t) =&S_{\nu, \mu}(t-s_{i})g_{i} \bigl(s_{i},x(s_{i})\bigr)-F\bigl(t, x(t), x \bigl(b_{1}(t)\bigr),\ldots, x\bigl(b_{m}(t)\bigr)\bigr) \\ &{}- \int_{s_{i}}^{t} A P_{\mu}(t-s) F\bigl(s, x(s), x\bigl(b_{1}(s)\bigr),\ldots, x\bigl(b_{m}(s)\bigr)\bigr) \,ds + \int_{s_{i}}^{t} P_{\mu}(t-s) B u^{\delta}(s)\,ds \\ &{}+ \int_{s_{i}}^{t} P_{\mu}(t-s) \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \\ &{}+ \int_{s_{i}}^{t} P_{\mu}(t-s) \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d B^{H}(s),\\ & t \in(s_{i},t_{i+1}], i=1,2,\ldots,m. \end{aligned}$$

We shall show that the operator \(\varPhi_{\delta}\) has a fixed point, which then is a solution of system (1.1).

For each positive integer q, set \(B_{q} = \{ x \in\bar{C}, \|x \|^{2}_{\bar{C}} \leq q \}\).

Then, for each q, \(B_{q} \subset\bar{C}\) is clearly a bounded closed convex set in . From Lemma 2.2, Lemma 2.3 and (2.2) together with the Hölder inequality,

$$\begin{aligned}& E \biggl\Vert \int_{0}^{t} A P_{\mu}(t-s) F\bigl(s, x(s), x\bigl(b_{1}(s)\bigr),\ldots, x\bigl(b_{m}(s)\bigr)\bigr) \,ds \biggr\Vert ^{2} \\& \quad \leq E\biggl[ \int_{0}^{t} \bigl\Vert A^{1-\beta} P_{\mu}(t-s) A^{\beta}F\bigl(s, x(s), x\bigl(b_{1}(s) \bigr),\ldots, x\bigl(b_{m}(s)\bigr)\bigr) \bigr\Vert \,ds \biggr]^{2} \\& \quad \leq\frac{\mu^{2} (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta )}{\varGamma^{2} (1+\mu\beta)} \int_{0}^{t} (t-s)^{\mu\beta- 1} \,ds \\& \qquad {}\times \int_{0}^{t} (t-s)^{\mu\beta-1} E \bigl\Vert A^{\beta}F\bigl(s, x(s), x\bigl(b_{1}(s)\bigr),\ldots, x \bigl(b_{m}(s)\bigr)\bigr) \bigr\Vert ^{2} \,ds \\& \quad \leq\frac{\mu(C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{\mu \beta} M_{2} }{\beta\varGamma^{2} (1+\mu\beta)} \int_{0}^{t} (t-s)^{\mu\beta- 1} \Bigl(\max _{i=1,2,\ldots,m} E \Vert x_{i} \Vert ^{2} + 1 \Bigr) \,ds. \end{aligned}$$
(3.4)

It follows that \(A P_{\mu}(t-s) F(s, x(s), x(b_{1}(s)),\ldots, x(b_{m}(s)))\) is integrable on J, by Bochner’s theorem [33] so \(\varPhi_{\delta}\) is well defined on \(B_{q}\).

From \((H2)(ii)\) together with Burkholder Gundy’s inequality, we obtain

$$\begin{aligned}& E \biggl\Vert \int_{0}^{t} P_{\mu}(t-s) \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{n}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \biggr\Vert ^{2} \\& \quad \leq\operatorname{Tr}(Q) \frac{ M^{2} T^{\mu}}{\mu\varGamma^{2}(\mu )} \int_{0}^{t} (t-s)^{\mu-1} \\& \qquad {}\times \biggl(\sup _{ \Vert x_{0} \Vert ^{2},\ldots, \Vert x_{n} \Vert ^{2} \leq q} \int_{0}^{s} E \bigl\Vert G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau )\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr) \bigr\Vert ^{2}_{Q} \,d\tau \biggr) \,ds \\& \quad \leq\operatorname{Tr}(Q) \frac{ M^{2} T^{\mu}}{\mu\varGamma^{2}(\mu )} \int_{0}^{t} (t-s)^{\mu-1}h_{q} (s)\,ds. \end{aligned}$$
(3.5)

Similarly from \((H3)(ii)\) together with Burkholder Gundy’s inequality, we obtain

$$\begin{aligned}& E \biggl\Vert \int_{0}^{t} P_{\mu}(t-s) \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d B^{H}(s) \biggr\Vert ^{2} \\& \quad \leq\frac{2H M^{2} T^{2 H+\mu -1}}{\mu\varGamma^{2}(\mu)} \int_{0}^{t} (t-s)^{\mu-1} \\& \qquad {}\times \sup _{ \Vert x_{0} \Vert ^{2},\ldots, \Vert x_{n} \Vert ^{2} \leq q} E \bigl\Vert \sigma\bigl(s,x(s), x \bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s)\bigr)\bigr) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \\& \quad \leq\frac{2H M^{2} T^{2 H+\mu-1}}{\mu\varGamma^{2}(\mu)} \int _{0}^{t} (t-s)^{\mu-1} \bar{h}_{q} (s)\,ds. \end{aligned}$$
(3.6)

Also, by using \((H1)\)\((H5)\) together with Hölder inequality, we obtain

$$\begin{aligned} E \biggl\Vert \int_{0}^{t} P_{\mu}(t-s) B u^{\delta}(s)\,ds \biggr\Vert ^{2} =& E \biggl\Vert \int _{0}^{t} (t-s)^{\mu-1} T_{\mu}(t-s) B u^{\delta}(s)\,ds \biggr\Vert ^{2} \\ \leq & \frac{ M^{2} T^{\mu}M^{2}_{B}}{\mu\varGamma^{2}(\mu)} \int_{0}^{t} (t-s)^{\mu-1} E \bigl\Vert u^{\delta}(s) \bigr\Vert ^{2} \,ds \end{aligned}$$

where, for \(t \in(0,t_{1}]\),

$$\begin{aligned} &E \bigl\Vert u^{\delta}(s) \bigr\Vert ^{2} \\ &\quad \leq \frac{M^{2}_{B} M^{2}}{z^{2} \varGamma ^{2}(\mu)}\biggl\{ E \Vert \bar{x}_{T} \Vert ^{2}+ \frac{M^{2} T^{2(\nu-1)(1-\mu )}}{\varGamma^{2}(\nu(1-\mu)+ \mu)}\bigl[E \bigl\Vert x(0) \bigr\Vert ^{2}+ M_{4} q + M_{5} + M_{0}^{2} M_{2}(q+1)\bigr] \\ &\qquad {} +M_{0}^{2} M_{2}(q+1)+ \operatorname{Tr}(Q) \int_{0}^{T} E \bigl\Vert \bar{\psi}(s) \bigr\Vert ^{2}_{Q} \,ds+ 2H T^{2H-1} \int_{0}^{T} E \bigl\Vert \bar{\varphi}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \biggr\} \\ &\qquad {}+\frac{M^{2}_{B} M^{2}}{z^{2} \varGamma^{2}(\mu)}\biggl\{ \frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{2\mu\beta} M_{2} }{ \beta^{2} \varGamma^{2} (1+\mu\beta)} ( q + 1) \\ &\qquad {}+\operatorname{Tr}(Q) \frac{ M^{2} T^{\mu}}{\mu\varGamma ^{2}(\mu)} \int_{0}^{T} (T-s)^{\mu-1} h_{q} (s) \,ds + \frac{2H M^{2} T^{2 H+\mu-1}}{\mu\varGamma^{2}(\mu)} \int_{0}^{T} (T-s)^{\mu-1} \bar {h}_{q} (s) \,ds\biggr\} , \end{aligned}$$

and for \(t \in(s_{i},t_{i+1}]\)

$$\begin{aligned} &E \bigl\Vert u^{\delta}(s) \bigr\Vert ^{2} \\ &\quad \leq \frac{M^{2}_{B} M^{2}}{z^{2} \varGamma ^{2}(\mu)}\biggl\{ E \Vert \bar{x}_{T} \Vert ^{2}+ \frac{M^{2} T^{2(\nu-1)(1-\mu )}}{\varGamma^{2}(\nu(1-\mu)+ \mu)} M_{3} q + M_{0}^{2} M_{2}(q+1) \\ &\qquad {}+ \operatorname{Tr}(Q) \int_{0}^{T} E \bigl\Vert \bar{\psi}(s) \bigr\Vert ^{2}_{Q} \,ds+ 2H T^{2H-1} \int_{0}^{T} E \bigl\Vert \bar{\varphi}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \biggr\} \\ &\qquad {}+\frac{M^{2}_{B} M^{2} }{z^{2} \varGamma^{2}(\mu)}\biggl\{ \frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{2\mu\beta} M_{2} }{ \beta^{2} \varGamma^{2} (1+\mu\beta)} ( q + 1) \\ &\qquad {}+\operatorname {Tr}(Q) \frac{ M^{2} T^{\mu}}{\mu\varGamma^{2}(\mu)} \int_{s_{i}}^{T} (T-s)^{\mu-1} h_{q} (s) \,ds\\ &\qquad {} + \frac{2H M^{2} T^{2 H+\mu-1}}{\mu\varGamma ^{2}(\mu)} \int_{s_{i}}^{T} (T-s)^{\mu-1} \bar{h}_{q} (s) \,ds\biggr\} , \end{aligned}$$

thus, we have

$$\begin{aligned}& \begin{gathered}[b] E \biggl\Vert \int_{0}^{t} P_{\mu}(t-s) B u^{\delta}(s)\,ds \biggr\Vert ^{2} \\ \quad \leq \frac { M^{4} T^{2\mu} M^{4}_{B}}{z^{2} \mu^{2} \varGamma^{4}(\mu)} \biggl\{ E \Vert \bar{x}_{T} \Vert ^{2}+ \frac{M^{2} T^{2(\nu-1)(1-\mu)}}{\varGamma^{2}(\nu (1-\mu)+ \mu)}\bigl[E \bigl\Vert x(0) \bigr\Vert ^{2} \\ \qquad {}+ M_{4} q + M_{5}+ M_{0}^{2} M_{2}(q+1)\bigr]+M_{0}^{2} M_{2} (q+1) \\ \qquad {}+ \operatorname{Tr}(Q) \int_{0}^{T} E \bigl\Vert \bar{\psi}(s) \bigr\Vert ^{2}_{Q} \,ds+ 2H T^{2H-1} \int_{0}^{T} E \bigl\Vert \bar{\varphi}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \\ \qquad {}+ \frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{2\mu\beta} M_{2} }{ \beta^{2} \varGamma^{2} (1+\mu\beta)} ( q + 1)\\ \qquad {}+\operatorname {Tr}(Q) \frac{ M^{2} T^{\mu}}{\mu\varGamma^{2}(\mu)} \int_{0}^{T} (T-s)^{\mu-1} h_{q} (s) \,ds \\ \qquad {}+ \frac{2H M^{2} T^{2 H+\mu-1}}{\mu\varGamma^{2}(\mu)} \int_{0}^{T} (T-s)^{\mu-1} \bar{h}_{q} (s) \,ds\biggr\} , \quad t \in(0,t_{1}], \end{gathered} \\& \begin{gathered} E \biggl\Vert \int_{s_{i}}^{t} P_{\mu}(t-s) B u^{\delta}(s)\,ds \biggr\Vert ^{2} \\ \quad \leq \frac{ M^{4} T^{2\mu} M^{4}_{B}}{z^{2} \mu^{2} \varGamma^{4}(\mu)} \biggl\{ E \Vert \bar{x}_{T} \Vert ^{2}+ \frac{M^{2} T^{2(\nu-1)(1-\mu)}}{\varGamma ^{2}(\nu(1-\mu)+ \mu)} M_{3} q \\ \qquad {}+ M_{0}^{2} M_{2} (q+1)+ \operatorname{Tr}(Q) \int_{0}^{T} E \bigl\Vert \bar {\psi}(s) \bigr\Vert ^{2}_{Q} \,ds+ 2H T^{2H-1} \int_{0}^{T} E \bigl\Vert \bar{\varphi}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \\ \qquad {}+ \frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{2\mu\beta} M_{2} }{ \beta^{2} \varGamma^{2} (1+\mu\beta)} ( q + 1)+\operatorname {Tr}(Q) \frac{ M^{2} T^{\mu}}{\mu\varGamma^{2}(\mu)} \int_{s_{i}}^{T} (T-s)^{\mu-1} h_{q} (s) \,ds \\ \qquad {}+ \frac{2H M^{2} T^{2 H+\mu-1}}{\mu\varGamma^{2}(\mu)} \int _{s_{i}}^{T} (T-s)^{\mu-1} \bar{h}_{q} (s) \,ds\biggr\} , \quad t \in (s_{i},t_{i+1}]. \end{gathered} \end{aligned}$$
(3.7)

We claim that there exists a positive number q such that \(\varPhi _{\delta}(B_{q})\subseteq B_{q}\). If it is not true, then, for each positive number q, there is a function \(x_{q}(\cdot) \in B_{q}\), but \(\varPhi_{\delta}(x_{q}) \notin B_{q}\), that is \(\|(\varPhi_{\delta}x_{q}) (t) \|^{2}_{\bar{C}} > q \) for some \(t=t(q) \in J\), where \(t(q)\) denotes that t is dependent of q. However, from \((H4)\)\((H5)\) and Eqs. (2.2), (3.4), (3.5), (3.6) and (3.7), we have for \(t \in(0,t_{1}]\)

$$\begin{aligned}& \Vert \varPhi_{\delta}x_{q} \Vert ^{2}_{\bar{C}} \\& \quad \leq36 \sup_{t \in J} t^{2(1-\nu)(1-\mu)} \biggl\{ E \bigl\Vert S_{\nu, \mu}(t)\bigl[x_{0} + \xi(x)+ F\bigl(0, x(0), x \bigl(b_{1}(0)\bigr),\ldots, x\bigl(b_{m}(0)\bigr)\bigr) \bigr] \bigr\Vert ^{2} \\& \qquad {}+ E \bigl\Vert F\bigl(t, x(t), x\bigl(b_{1}(t)\bigr),\ldots, x \bigl(b_{m}(t)\bigr)\bigr) \bigr\Vert ^{2} \\& \qquad {}+E \biggl\Vert \int _{0}^{t} A P_{\mu}(t-s) F\bigl(t, x(t), x\bigl(b_{1}(t)\bigr),\ldots, x\bigl(b_{m}(t)\bigr)\bigr) \,ds \biggr\Vert ^{2} \\& \qquad {}+E \biggl\Vert \int_{0}^{t} P_{\mu}(t-s) B u^{\delta}(s)\,ds \biggr\Vert ^{2} \\& \qquad {}+E \biggl\Vert \int _{0}^{t} P_{\mu}(t-s) \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau )\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \biggr\Vert ^{2} \\& \qquad {}+ E \biggl\Vert \int_{0}^{t} P_{\mu}(t-s) \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d B^{H}(s) \biggr\Vert ^{2} \biggr\} \\& \quad \leq36 \biggl\{ \frac{ M^{2}}{\varGamma^{2}(\nu(1-\mu)+\mu)} \bigl[E \bigl\Vert x(0) \bigr\Vert ^{2} + M_{4} q + M_{5} + {M_{0}}^{2} M_{2} (q +1)\bigr] \\& \qquad {}+ T^{2(1-\nu)(1-\mu )} {M_{0}}^{2} M_{2} (q + 1) \\& \qquad {}+ \frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta)T^{2\mu\beta+ 2(1-\nu)(1-\mu)} M_{2} (q +1)}{\beta^{2} \varGamma^{2} (1+\mu\beta)} \\& \qquad {} + \operatorname{Tr}(Q) \biggl(\frac{ M^{2} T^{\mu+2(1-\nu)(1-\mu)}}{\mu \varGamma^{2}(\mu)}\biggr)q \frac{1}{q} \int_{0}^{t} (t-s)^{\mu-1} h_{q} (s)\,d s \\& \qquad {}+ \frac{2H M^{2} T^{2 H+\mu-1+ 2(1-\nu)(1-\mu)}}{\mu\varGamma^{2}(\mu )}q \frac{1}{q} \int_{0}^{t} (t-s)^{\mu-1} \bar{h}_{q} (s) \,ds \\& \qquad {}+\frac{ M^{4} T^{2\mu}T^{2(1-\nu)(1-\mu)} M^{4}_{B}}{z^{2} \mu^{2} \varGamma^{4}(\mu)} \biggl\{ E \Vert \bar{x}_{T} \Vert ^{2} \\& \qquad {}+ \frac{M^{2} T^{2(\nu -1)(1-\mu)}}{\varGamma^{2}(\nu(1-\mu)+ \mu)}\bigl[E \bigl\Vert x(0) \bigr\Vert ^{2}+ M_{4} q + M_{5} + M_{0}^{2} M_{2}(q+1)\bigr]+ M_{0}^{2} M_{2}(q+1) \\& \qquad {}+ \operatorname{Tr}(Q) \int_{0}^{T} E \bigl\Vert \bar{\psi}(s) \bigr\Vert ^{2}_{Q} \,ds+ 2H T^{2H-1} \int_{0}^{T} E \bigl\Vert \bar{\varphi}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \\& \qquad {}+ \frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{2\mu\beta} M_{2}}{ \beta^{2} \varGamma^{2} (1+\mu\beta)} ( q + 1) \\& \qquad {}+\operatorname{Tr}(Q) \frac{ M^{2} T^{\mu}}{\mu\varGamma^{2}(\mu)} q \frac{1}{q} \int_{0}^{T} (T-s)^{\mu-1} h_{q} (s) \,ds \\& \qquad {}+ \frac{2H M^{2} T^{2 H+\mu-1}}{\mu\varGamma^{2}(\mu)} q \frac{1}{q} \int_{0}^{T} (T-s)^{\mu-1} \bar{h}_{q} (s) \,ds \biggr\} \biggr\} , \end{aligned}$$
(3.8)

for \(t \in(t_{i},s_{i}]\)

$$ \Vert \varPhi_{\delta}x_{q} \Vert ^{2}_{\bar{C}} \leq\sup_{t \in J} t^{2(1-\nu )(1-\mu)} E \bigl\Vert g\bigl(t,x(t) \bigr) \bigr\Vert ^{2} \leq T^{2(1-\nu)(1-\mu)} M_{3} q , $$
(3.9)

and for \(t \in(s_{i},t_{i+1}]\)

$$\begin{aligned}& \Vert \varPhi_{\delta}x_{q} \Vert ^{2}_{\bar{C}} \\& \quad \leq36 \sup_{t \in J} t^{2(1-\nu)(1-\mu)} \biggl\{ E \bigl\Vert S_{\nu, \mu}(t-s_{i})g_{i}\bigl(s_{i},x(s_{i}) \bigr) \bigr\Vert ^{2} \\& \qquad {}+ E \bigl\Vert F\bigl(t, x(t), x\bigl(b_{1}(t)\bigr), \ldots, x\bigl(b_{m}(t)\bigr)\bigr) \bigr\Vert ^{2} \\& \qquad {}+E \biggl\Vert \int _{s_{i}}^{t} A P_{\mu}(t-s) F\bigl(t, x(t), x\bigl(b_{1}(t)\bigr),\ldots, x\bigl(b_{m}(t)\bigr) \bigr) \,ds \biggr\Vert ^{2} \\& \qquad {}+E \biggl\Vert \int_{s_{i}}^{t} P_{\mu}(t-s) B u^{\delta}(s)\,ds \biggr\Vert ^{2} \\& \qquad {}+E \biggl\Vert \int_{s_{i}}^{t} P_{\mu}(t-s) \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \biggr\Vert ^{2} \\& \qquad {}+ E \biggl\Vert \int_{s_{i}}^{t} P_{\mu}(t-s) \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d B^{H}(s) \biggr\Vert ^{2} \biggr\} \\& \quad \leq36 \biggl\{ \frac{ M^{2}}{\varGamma^{2}(\nu(1-\mu)+\mu)} M_{3} q + T^{2(1-\nu)(1-\mu)} {M_{0}}^{2} M_{2} (q + 1) \\& \qquad {}+ \frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta)T^{2\mu\beta+ 2(1-\nu)(1-\mu)} M_{2} (q +1)}{\beta^{2} \varGamma^{2} (1+\mu\beta)} \\& \qquad {}+ \operatorname{Tr}(Q) \biggl(\frac{ M^{2} T^{\mu+2(1-\nu)(1-\mu)}}{\mu \varGamma^{2}(\mu)} \biggr)q \frac{1}{q} \int_{s_{i}}^{t} (t-s)^{\mu-1} h_{q} (s)\,d s \\& \qquad {}+ \frac{2H M^{2} T^{2 H+\mu-1+ 2(1-\nu)(1-\mu)}}{\mu\varGamma^{2}(\mu )}q \frac{1}{q} \int_{s_{i}}^{t} (t-s)^{\mu-1} \bar{h}_{q} (s) \,ds \\& \qquad {}+\frac{ M^{4} T^{2\mu}T^{2(1-\nu)(1-\mu)} M^{4}_{B}}{z^{2} \mu^{2} \varGamma^{4}(\mu)} \biggl\{ E \Vert \bar{x}_{T} \Vert ^{2}+\frac{ M^{2} T^{2(\nu -1)(1-\mu)}}{\varGamma^{2}(\nu(1-\mu)+\mu)} M_{3} q+ M_{0}^{2} M_{2}(q+1) \\& \qquad {}+ \operatorname{Tr}(Q) \int_{0}^{T} E \bigl\Vert \bar{\psi}(s) \bigr\Vert ^{2}_{Q} \,ds+ 2H T^{2H-1} \int_{0}^{T} E \bigl\Vert \bar{\varphi}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \\& \qquad {}+ \frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{2\mu\beta} M_{2}}{ \beta^{2} \varGamma^{2} (1+\mu\beta)} ( q + 1) \\& \qquad {}+\operatorname{Tr}(Q) \frac{ M^{2} T^{\mu}}{\mu\varGamma^{2}(\mu)} q \frac{1}{q} \int_{s_{i}}^{T} (T-s)^{\mu-1} h_{q} (s) \,ds \\& \qquad {}+ \frac{2H M^{2} T^{2 H+\mu-1}}{\mu\varGamma^{2}(\mu)} q \frac{1}{q} \int _{s_{i}}^{T} (T-s)^{\mu-1} \bar{h}_{q} (s) \,ds \biggr\} \biggr\} , \end{aligned}$$
(3.10)

Combining (3.8), (3.9), (3.10) in the inequality \(q \leq\|(\varPhi x_{q})(t) \|^{2}_{\bar{C}}\) then dividing both sides of the inequality by q and taking the lower limit \(q \rightarrow+ \infty\), we get

$$\begin{aligned}& \biggl[1 +\frac{ M^{4} T^{2\mu} M^{4}_{B}}{z^{2} \mu^{2} \varGamma ^{4}(\mu)} \biggr] \biggl\lbrace \frac{36 M^{2} ( M_{0}^{2} M_{2} + M_{3} + M_{4})}{\varGamma^{2}(\nu(1-\mu)+\mu)}\\& \quad {}+ 36T^{2(1-\nu)(1-\mu )} \biggl[{M_{0}}^{2} M_{2}+ \frac{ M^{2} T^{\mu}\varLambda_{1} \operatorname{Tr}(Q)}{\mu\varGamma^{2}(\mu)} \\& \quad {}+ \frac{2H M^{2} \varLambda_{2} T^{2 H+\mu-1}}{\mu \varGamma^{2}(\mu)}+\frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{2\mu\beta} M_{2} }{ \beta^{2} \varGamma^{2} (1+\mu\beta)} \biggr] \biggr\rbrace + T^{2(1-\nu)(1-\mu)} M_{3} \geq 1. \end{aligned}$$

This contradicts (3.1). Hence for positive q, \(\varPhi_{\delta}(B_{q}) \subseteq B_{q}\).

Next we will show that the operator \(\varPhi_{\delta}\) has a fixed point on \(B_{q}\), which implies that Eq. (1.1) has a mild solution. We decompose \(\varPhi_{\delta}\) as \(\varPhi_{\delta}= \varPhi_{1} + \varPhi _{2}\), where the operators \(\varPhi_{1}\) and \(\varPhi_{2}\) are defined on \(B_{q}\), respectively, by

$$\begin{aligned}& (\varPhi_{1} x) (t) = \textstyle\begin{cases} S_{\nu, \mu}(t)[x_{0} - \xi(x) +F(0, x(0), x(b_{1}(0)),\ldots, x(b_{m}(0)))]\\ \quad {}-F(t, x(t), x(b_{1}(t)),\ldots, x(b_{m}(t)))\\ \quad{}-\int_{0}^{t} A P_{\mu}(t-s) F(s, x(s), x(b_{1}(s)),\ldots, x(b_{m}(s)))\,ds , \quad t \in(0,t_{1}],\\ g_{i}(t,x(t)), \quad t \in(t_{i},s_{i}], i= 1,2,\ldots ,m,\\ S_{\nu, \mu}(t-s_{i})g_{i}(s_{i},x(s_{i}))-F(t, x(t), x(b_{1}(t)),\ldots, x(b_{m}(t)))\\ \quad{}-\int_{s_{i}}^{t} A P_{\mu}(t-s) F(s, x(s), x(b_{1}(s)),\ldots, x(b_{m}(s)))\,ds , \\ \quad t \in(s_{i},t_{i+1}], i = 1,2,\ldots,m, \end{cases}\displaystyle \\& (\varPhi_{2} x) (t)= \textstyle\begin{cases} \int_{s_{i}}^{t} P_{\mu}(t-s) B u(s)\,ds\\ \quad {}+ \int_{s_{i}}^{t} P_{\mu}(t-s) \int_{0}^{s} G(\tau, x(\tau), x(a_{1}(\tau)),\ldots, x(a_{k}(\tau)))\,d\omega(\tau) \,ds\\ \quad{}+ \int_{s_{i}}^{t} P_{\mu}(t-s) \sigma(s,x(s), x(c_{1}(s)),\ldots, x(c_{p}(s))) \,d B^{H}(s),\\ \quad t \in (s_{i},t_{i+1}], i= 0,1,\ldots,m,\\ 0,\quad \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$

for \(t \in J \). We will show that \(\varPhi_{1}\) verifies a contraction condition while \(\varPhi_{2}\) is a compact operator.

To prove that \(\varPhi_{1}\) satisfies a contraction condition, we take \(x_{1}, x_{2} \in B_{q}\), then, for each \(t \in J\) and by condition \((H1)\), \((H4)\) and \((H5)\), we have for \(t \in(0,t_{1}]\)

$$\begin{aligned}& E \bigl\Vert (\varPhi_{1} x_{1}) (t) - (\varPhi_{1} x_{2}) (t) \bigr\Vert ^{2} \\& \quad\leq9 \biggl\{ E \bigl\Vert S_{\nu, \mu}(t) \bigl[ \xi(x_{1})- \xi(x_{2})\bigr] \bigr\Vert ^{2} \\& \qquad{}+ E \bigl\Vert S_{\nu, \mu}(t) \bigl[F\bigl(0, x_{1}(0), x_{1}\bigl(b_{1}(0)\bigr),\ldots , x_{1}\bigl(b_{m}(0)\bigr)\bigr) \\& \qquad {}- F\bigl(0, x_{2}(0), x_{2}\bigl(b_{1}(0)\bigr),\ldots, x_{2}\bigl(b_{m}(0)\bigr)\bigr)\bigr] \bigr\Vert ^{2} \\& \qquad{}+ E \bigl\Vert F\bigl(t, x_{1}(t), x_{1} \bigl(b_{1}(t)\bigr),\ldots, x_{1}\bigl(b_{m}(t) \bigr)\bigr) - F\bigl(t, x_{2}(t), x_{2} \bigl(b_{1}(t)\bigr),\ldots, x_{2}\bigl(b_{m}(t) \bigr)\bigr) \bigr\Vert ^{2} \\& \qquad{}+ E \biggl\Vert \int_{0}^{t} A P_{\mu}(t-s) \bigl[F\bigl(t, x_{1}(t), x_{1}\bigl(b_{1}(t)\bigr),\ldots, x_{1}\bigl(b_{m}(t)\bigr)\bigr) \\& \qquad {} - F\bigl(t, x_{2}(t), x_{2}\bigl(b_{1}(t)\bigr),\ldots, x_{2}\bigl(b_{m}(t)\bigr)\bigr)\bigr] \,ds \biggr\Vert ^{2} \biggr\} \\& \quad\leq9 \biggl[\frac{M^{2} T^{2(\nu-1)(1-\mu )}({M_{0}}^{2}{M_{1}}+ M_{7})}{\varGamma^{2}(\nu(1-\mu)+\mu)} \\& \qquad {}+{M_{0}}^{2} {M_{1}}+\frac{{M_{1}}(C_{1 - \beta})^{2} \varGamma^{2}(1+ \beta) T^{2\mu\beta}}{\beta^{2} \varGamma^{2} (1+\mu\beta)} \biggr] E \bigl\Vert x_{1}(t) - x_{2}(t) \bigr\Vert ^{2}, \end{aligned}$$
(3.11)

for \(t \in(t_{i},s_{i}]\)

$$\begin{aligned}& E \bigl\Vert (\varPhi_{1} x_{1}) (t) - (\varPhi_{1} x_{2}) (t) \bigr\Vert ^{2} \\& \quad \leq E \bigl\Vert g_{i}\bigl(t,x_{1}(t)\bigr)- g_{i}\bigl(t,x_{2}(t)\bigr) \bigr\Vert ^{2} \\& \quad \leq M_{6} E \bigl\Vert x_{1}(t) - x_{2}(t) \bigr\Vert ^{2}, \end{aligned}$$
(3.12)

and for \(t \in(s_{i},t_{i+1}]\)

$$\begin{aligned}& E \bigl\Vert (\varPhi_{1} x_{1}) (t) - (\varPhi_{1} x_{2}) (t) \bigr\Vert ^{2} \\& \quad \leq9 \biggl\{ E \bigl\Vert S_{\nu, \mu}(t-s_{i}) \bigl( g_{i}\bigl(s_{i},x_{1}(s_{i})\bigr)- g_{i}\bigl(s_{i},x_{2}(s_{i})\bigr) \bigr) \bigr\Vert ^{2} \\& \qquad {}+ E \bigl\Vert F\bigl(t, x_{1}(t), x_{1} \bigl(b_{1}(t)\bigr),\ldots, x_{1}\bigl(b_{m}(t) \bigr)\bigr) - F\bigl(t, x_{2}(t), x_{2} \bigl(b_{1}(t)\bigr),\ldots, x_{2}\bigl(b_{m}(t) \bigr)\bigr) \bigr\Vert ^{2} \\& \qquad {}+ E \biggl\Vert \int_{s_{i}}^{t} A P_{\mu}(t-s) \bigl[F\bigl(t, x_{1}(t), x_{1}\bigl(b_{1}(t)\bigr),\ldots, x_{1}\bigl(b_{m}(t)\bigr)\bigr) \\& \qquad {}- F\bigl(t, x_{2}(t), x_{2}\bigl(b_{1}(t)\bigr),\ldots, x_{2}\bigl(b_{m}(t)\bigr)\bigr)\bigr] \,ds \biggr\Vert ^{2} \biggr\} \\& \quad \leq9 \biggl[\frac{ M^{2} T^{2(\nu-1)(1-\mu)}}{\varGamma^{2}(\nu(1-\mu )+\mu)} M_{6} +{M_{0}}^{2} {M_{1}} \\& \qquad {}+\frac{{M_{1}}(C_{1 - \beta})^{2} \varGamma^{2}(1+ \beta) T^{2\mu\beta}}{\beta^{2} \varGamma^{2} (1+\mu \beta)} \biggr] E \bigl\Vert x_{1}(t) - x_{2}(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.13)

Combining (3.11), (3.12), (3.13) and taking \(\sup_{t \in J} t^{2(1-\nu )(1-\mu)}\) for both sides of the inequality, we get

$$ \begin{gathered} \sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \bigl\Vert (\varPhi_{1} x_{1}) (t) - (\varPhi _{1} x_{2}) (t) \bigr\Vert ^{2} \\ \quad\leq9 \biggl[\frac{M^{2} ({M_{0}}^{2}{M_{1}}+ M_{6} + M_{7})}{\varGamma^{2}(\nu(1-\mu)+\mu)}+T^{2(1-\nu)(1-\mu)}\bigl( M_{6} +{M_{0}}^{2} {M_{1}}\bigr) \\ \qquad {}+\frac{{M_{1}}(C_{1 - \beta})^{2} \varGamma^{2}(1+ \beta) T^{2\mu\beta+2(1-\nu)(1-\mu)}}{\beta^{2} \varGamma^{2} (1+\mu\beta)} \biggr] \\ \qquad {}\times\sup_{t \in J} E \bigl\Vert x_{1}(t) - x_{2}(t) \bigr\Vert ^{2} \end{gathered} $$

hence, from the definition of and (3.3) we get

$$ \Vert \varPhi_{1} x_{1} - \varPhi_{1} x_{2} \Vert ^{2}_{\bar{C}} \leq \gamma_{1} \Vert x_{1} - x_{2} \Vert ^{2}_{\bar{C}}. $$

Thus, \(\varPhi_{1}\) is a contraction.

To prove that \(\varPhi_{2}\) is compact, first we prove that \(\varPhi_{2}\) is continuous on \(B_{q}\).

Let \(\{ x_{n} \} \subseteq B_{q}\) with \(x_{n} \rightarrow x\) in \(B_{q}\) and rewrite \(u^{\delta}(t)= u^{\delta}(t,x)\), the control function defined above. Then, for each \(s \in J\), \(x_{n}(s) \rightarrow x(s)\), and by \(H2(i)\) and \(H3(i)\), we have \(G(s, x_{n}(s), x_{n}(a_{1}(s)),\ldots, x_{n}(a_{k}(s))) \rightarrow G(s, x(s), x(a_{1}(s)),\ldots, x(a_{k}(s)))\), as \(n \rightarrow\infty \), and \(\sigma(s, x_{n}(s), x_{n}(c_{1}(s)),\ldots, x_{n}(c_{p}(s))) \rightarrow\sigma(s, x(s), x(c_{1}(s)),\ldots, x(c_{p}(s)))\), as \(n \rightarrow\infty\).

By the dominated convergence theorem, we have

$$\begin{aligned} \Vert \varPhi_{2} x_{n} - \varPhi_{2} x \Vert ^{2}_{\bar{C}} &= \sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \biggl\Vert \int_{s_{i}}^{t} P_{\mu}(t-s) B \bigl(u^{\delta}(s,x_{n}) -u^{\delta}(s,x)\bigr)\,ds \\ &\quad {}+ \int_{s_{i}}^{t} P_{\mu}(t-s) \int_{0}^{s} \bigl(G\bigl(\tau, x_{n}( \tau), x_{n}\bigl(a_{1}(\tau)\bigr),\ldots, x_{n}\bigl(a_{k}(\tau)\bigr)\bigr)\\ &\quad {} -G\bigl(\tau, x(\tau), x\bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr) \bigr)\,d\omega(\tau) \,ds \\ &\quad {}+ \int_{s_{i}}^{t} P_{\mu}(t-s) \bigl(\sigma \bigl(s,x_{n}(s), x_{n}\bigl(c_{1}(s)\bigr), \ldots, x_{n}\bigl(c_{p}(s)\bigr)\bigr)\\ &\quad {}-\sigma\bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s)\bigr)\bigr) \bigr) \,d B^{H}(s) \biggr\Vert ^{2} \\ &\rightarrow0, \end{aligned}$$

as \(n \rightarrow\infty\), which is continuous.

Next we prove that the family \(\{ \varPhi_{2} x: x \in B_{q} \}\) is an equicontinuous family of functions.

To do this, let \(\epsilon> 0\) be small, \(s_{i} < t_{\alpha}< t_{\beta}\leq t_{i+1}\), then

$$\begin{aligned}& E \bigl\Vert (\varPhi_{2} x) (t_{\beta}) - (\varPhi_{2} x) (t_{\alpha}) \bigr\Vert ^{2} \\& \quad \leq E \biggl\Vert \int_{t_{\alpha}}^{t_{\beta}} P_{\mu}(t_{\beta}-s) Bu^{\delta}(s) \,ds \biggr\Vert ^{2} \\& \qquad {}+E \biggl\Vert \int_{s_{i}}^{t_{\alpha}-\epsilon} \bigl( P_{\mu}(t_{\beta}-s) - P_{\mu}(t_{\alpha}-s) \bigr) Bu^{\delta}(s) \,ds \biggr\Vert ^{2}\\& \qquad {}+ E \biggl\Vert \int _{t_{\alpha}-\epsilon}^{t_{\alpha}} \bigl(P_{\mu}(t_{\beta}-s) -P_{\mu}(t_{\alpha}-s) \bigr) Bu^{\delta}(s) \,ds \biggr\Vert ^{2} \\& \qquad {}+ E \biggl\Vert \int_{s_{i}}^{t_{\alpha}-\epsilon} \bigl(P_{\mu}(t_{\beta}-s) -P_{\mu}(t_{\alpha}-s)\bigr)\\& \qquad {}\times \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau )\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \biggr\Vert ^{2} \\& \qquad {}+ E \biggl\Vert \int_{t_{\alpha}-\epsilon}^{t_{\alpha}} \bigl( P_{\mu}(t_{\beta}-s) -P_{\mu}(t_{\alpha}-s) \bigr)\\& \qquad {}\times \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr)\,d\omega(\tau) \,ds \biggr\Vert ^{2} \\& \qquad {}+E \biggl\Vert \int_{t_{\alpha}}^{t_{\beta}} P_{\mu}(t_{\beta}-s) \int _{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau )\bigr) \bigr)\,d\omega(\tau) \,ds \biggr\Vert ^{2} \\& \qquad {}+E \biggl\| \int_{s_{i}}^{t_{\alpha}-\epsilon} \bigl( P_{\mu}(t_{\beta}-s) -P_{\mu}(t_{\alpha}-s) \bigr) \sigma\bigl(s,x(s), x \bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s)\bigr)\bigr)) \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+ E \biggl\| \int_{t_{\alpha}-\epsilon}^{t_{\alpha}} \bigl(P_{\mu}(t_{\beta}-s) -P_{\mu}(t_{\alpha}-s) \bigr) \sigma\bigl(s,x(s), x \bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s)\bigr)\bigr)) \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+ E \biggl\| \int_{t_{\alpha}}^{t_{\beta}} P_{\mu}(t_{\beta}-s) \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x \bigl(c_{p}(s)\bigr)\bigr)) \,dB^{H}(s)\biggr\| ^{2}. \end{aligned}$$

We see that \(E\|(\varPhi_{2} x)(t_{\beta}) - (\varPhi_{2} x)(t_{\alpha})\| ^{2}\) tends to zero independently of \(x \in B_{q}\) as \(t_{\beta}\rightarrow t_{\alpha}\), with ϵ sufficiently small since the compactness of \(S_{\nu, \mu}(t)\) for \(t >0\) (see [28]) implies the continuity in the uniform operator topology. Similarly, we can prove that the function \(\varPhi_{2} x\), \(x \in B_{q}\) are equicontinuous at \(t = 0\). Hence \(\varPhi_{2}\) maps \(B_{q}\) into a family of equicontinuous functions.

It remains to prove that \(V(t) = \{ (\varPhi_{2} x)(t) : x \in B_{q} \}\) is relatively compact in \(B_{q}\). Obviously, by condition \((H3)\), \(V(0)\) is relatively compact in \(B_{q}\).

Let \({s_{i}}< t \leq t_{i+1}\) be fixed, \({s_{i}} < \epsilon< t\), arbitrary \(\rho> 0\), for \(x \in B_{q}\), we define

$$\begin{aligned}& \bigl(\varPhi^{\epsilon, \rho}_{2} x\bigr) (t) \\& \quad = \mu \int_{s_{i}}^{t-\epsilon} \int_{\rho}^{\infty}\theta(t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta\bigr) Bu^{\delta}(s) \,d\theta \,ds \\& \qquad {}+ \mu \int_{s_{i}}^{t-\epsilon} \int_{\rho}^{\infty}\theta (t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta\bigr)\\& \qquad {}\times \int _{0}^{s} G(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \,d \omega(\tau) \,d \theta \,ds \\& \qquad {}+ \mu \int_{s_{i}}^{t-\epsilon} \int_{\rho}^{\infty}\theta (t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta\bigr)\\& \qquad {}\times \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d\theta \,dB^{H}(s) \\& \quad = \mu S\bigl(\epsilon^{\mu}\rho\bigr) \int_{s_{i}}^{t-\epsilon} \int_{\rho}^{\infty}\theta(t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta- \epsilon^{\mu} \rho\bigr) Bu^{\delta}(s) \,d\theta \,ds \\& \qquad {}+ \mu S\bigl(\epsilon^{\mu}\rho\bigr) \int_{s_{i}}^{t-\epsilon} \int_{\rho}^{\infty}\theta(t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta- \epsilon^{\mu} \rho\bigr)\\& \qquad {}\times \int_{0}^{s} G(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \,d \omega(\tau) \,d \theta \,ds \\& \qquad {}+\mu S\bigl(\epsilon^{\mu}\rho\bigr) \int_{s_{i}}^{t-\epsilon} \int_{\rho}^{\infty}\theta(t-s)^{\mu- 1} \theta(t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu} \theta- \epsilon^{\mu}\rho\bigr)\\& \qquad {}\times \sigma \bigl(s,x(s), x \bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s)\bigr)\bigr)\,d \theta \,dB^{H}(s). \end{aligned}$$

Since \(S(\epsilon^{\mu}\rho)\), \(\epsilon^{\mu}\rho> 0\) is a compact operator, the set \(V^{\epsilon,\rho} (t)= \{ (\varPhi _{2}^{\epsilon,\rho} x)(t) : x \in B_{q} \}\) is relatively compact in X for every ϵ, \({s_{i}} <\epsilon< t\) and for all \(\rho>0\).

Moreover, for every \(x \in B_{q}\), we have

$$\begin{aligned}& \bigl\Vert \varPhi_{2} x - \varPhi_{2}^{\epsilon,\rho} x \bigr\Vert ^{2}_{\bar {C}} \\& \quad \leq9 \sup_{t \in J}t^{2(1-\nu)(1-\mu)}\biggl\{ \mu^{2} E \biggl\Vert \int _{s_{i}}^{t} \int_{0}^{\rho}\theta(t-s)^{\mu- 1} \varPsi_{\mu}(\theta ) S\bigl((t-s)^{\mu}\theta\bigr) B u^{\delta}(s) \,d \theta \,ds \biggr\Vert ^{2} \\ & \qquad {}+\mu^{2} E \biggl\Vert \int_{t-\epsilon}^{t} \int_{\rho}^{\infty}\theta (t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta\bigr) B u^{\delta}(s) \,d \theta \,ds \biggr\Vert ^{2} \\ & \qquad {}+ \mu^{2} E \biggl\| \int_{s_{i}}^{t} \int_{0}^{\rho}\theta(t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta\bigr)\\ & \qquad {}\times \int_{0}^{s} G(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \,d \omega(\tau) \,d \theta \,ds \biggr\Vert ^{2} \\ & \qquad {}+\mu^{2} E \biggl\| \int_{t-\epsilon}^{t} \int_{\rho}^{\infty}\theta (t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta\bigr)\\ & \qquad {}\times \int _{0}^{s} G(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \,d \omega(\tau) \,d \theta \,ds \biggr\Vert ^{2} \\ & \qquad {}+ \mu^{2} E \biggl\Vert \int_{s_{i}}^{t} \int_{0}^{\rho}\theta(t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta\bigr)\\ & \qquad {}\times \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d \theta \,dB^{H}(s) \biggr\Vert ^{2} \\ & \qquad {}+\mu^{2} E \biggl\Vert \int_{t-\epsilon}^{t} \int_{0}^{\rho}\theta (t-s)^{\mu- 1} \varPsi_{\mu}(\theta) S\bigl((t-s)^{\mu}\theta\bigr)\\ & \qquad {}\times \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \,d \theta \,dB^{H}(s) \biggr\Vert ^{2} \biggr\} \\ & \quad \leq9\biggl\{ T^{\mu+ 2(1-\nu)(1-\mu)} \mu M^{2} M^{2}_{B} \int _{s_{i}}^{t} (t-s)^{\mu- 1} E \bigl\Vert u^{\delta}(s) \bigr\Vert ^{2} \,ds \biggl( \int _{0}^{\rho}\theta\varPsi_{\mu}(\theta) \,d \theta \biggr)^{2} \\ & \qquad {} +T^{2(1-\nu)(1-\mu)}\mu M^{2} M^{2}_{B} \epsilon^{\mu} \int _{t-\epsilon}^{t} (t-s)^{\mu- 1} E \bigl\Vert u^{\delta}(s) \bigr\Vert ^{2} \,ds \biggl( \int_{\rho}^{\infty}\theta\varPsi_{\mu}(\theta) \,d \theta \biggr)^{2} \\ & \qquad {}+T^{\mu+ 2(1-\nu)(1-\mu)} \mu M^{2} \operatorname{Tr}(Q) \int _{s_{i}}^{t} (t-s)^{\mu- 1}\\ & \qquad {}\times \int_{0}^{s} E \bigl\| G(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr\| ^{2}_{Q} \,d\tau \,ds \biggl( \int_{0}^{\rho}\theta\varPsi_{\mu}(\theta) \,d \theta \biggr)^{2} \\ & \qquad {}+T^{\mu+ 2(1-\nu)(1-\mu)}\mu M^{2} \operatorname{Tr}(Q) \epsilon ^{\mu}\int_{t-\epsilon}^{t} (t-s)^{\mu- 1} \\ & \qquad {}\times \int_{0}^{s} E \bigl\| G(\tau , x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau)\bigr) \bigr\| ^{2}_{Q} \,d\tau \,ds \biggl( \int_{\rho}^{\infty}\theta\varPsi_{\mu}(\theta) \,d \theta \biggr)^{2} \\ & \qquad {}+2T^{2H+ \mu-1+ 2(1-\nu)(1-\mu)} \mu M^{2} \\ & \qquad {}\times\int_{s_{i}}^{t} (t-s)^{\mu- 1} E \bigl\| \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \bigr\| ^{2}_{L^{0}_{2}} \,ds \biggl( \int_{0}^{\rho}\theta\varPsi_{\mu}(\theta ) \,d \theta \biggr)^{2} \\ & \qquad {}+2T^{2H-1 +2(1-\nu)(1-\mu)}\mu M^{2} \epsilon^{\mu} \\ & \qquad {} \times\int _{t-\epsilon}^{t} (t-s)^{\mu- 1} E \bigl\| \sigma \bigl(s,x(s), x\bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s) \bigr)\bigr) \bigr\| ^{2}_{L^{0}_{2}} \,ds \biggl( \int _{\rho}^{\infty}\theta\varPsi_{\mu}(\theta) \,d \theta \biggr)^{2} \biggr\} . \end{aligned}$$

We see that, for each \(x \in B_{q}\), \(\|\varPhi_{2} x - \varPhi _{2}^{\epsilon,\rho}\|^{2}_{\bar{C}}\rightarrow0\) as \(\epsilon \rightarrow0^{+}\), \(\rho\rightarrow0^{+}\). Therefore, there are relative compact sets arbitrarily close to the set \(V(t) = \{ (\varPhi _{2} x)(t): x \in B_{q} \}\), hence the set \(V(t)\) is also relatively compact in \({B_{q}}\).

Thus, by the Ascoli–Arzela theorem \(\varPhi_{2}\) is a compact operator. These arguments enable us to conclude that \(\varPhi_{\delta}= \varPhi_{1} + \varPhi_{2} \) is a condensing map on \(B_{q}\), and by the fixed point theorem of Sadovskii there exists a fixed point \(x(\cdot)\) for \(\varPhi _{\delta}\) on \(B_{q}\). Therefore the system (1.1) has a mild solution. □

Theorem 3.2

Assume that\((H1)\)\((H5)\)are satisfied. Furthermore, if the functionsF, G, andσare uniformly bounded, then the system (1.1) is approximately controllable onJ.

Proof

Let \(x_{\delta}\) be a fixed point of \(\varPhi_{\delta}\). By using the stochastic Fubini theorem, it can be easily seen that

$$\begin{aligned} x_{\delta}(T)&=\bar{x}_{T} - z \bigl(z I + \varGamma^{T}_{0}\bigr)^{-1} \biggl\{ E \bar {x}_{T} - g_{m}\bigl(T,x(T)\bigr) \\ &\quad {}+F\bigl(T, x(T), x\bigl(b_{1}(T)\bigr),\ldots, x \bigl(b_{m}(T)\bigr)\bigr)+ \int_{0}^{T} \bar {\psi}(s) \,d\omega(s)+ \int_{0}^{T} \bar{\varphi}(s) \,dB^{H}(s) \biggr\} \\ &\quad {}- z \int_{s_{m}}^{T} \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1} P_{\mu}(T-s) A F\bigl(s, x(s), x \bigl(b_{1}(s)\bigr),\ldots, x\bigl(b_{m}(s)\bigr)\bigr)\,ds \\ &\quad {}+z \int_{s_{m}}^{T} \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1}P_{\mu}(T-s) \int _{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau )\bigr) \bigr)\,d\omega(\tau) \,ds \\ &\quad {}+z \int_{s_{m}}^{T} \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1} P_{\mu}(T-s) \sigma\bigl(s,x(s), x \bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s)\bigr)\bigr) \,dB^{H}(s). \end{aligned}$$

It follows from the assumption on F, G and σ that there exists \(D > 0\) such that

$$\begin{aligned}& \bigl\Vert F\bigl(s,x_{\delta}(s), x_{\delta}\bigl(b_{1}(s) \bigr),\ldots, x_{\delta}\bigl(b_{m}(s)\bigr)\bigr) \bigr\Vert ^{2} \leq D,\\& \bigl\Vert G\bigl(s,x_{\delta}(s), x_{\delta}\bigl(a_{1}(s)\bigr),\ldots,x_{\delta} \bigl(a_{k}(s)\bigr)\bigr) \bigr\Vert ^{2} \leq D, \\& \bigl\Vert \sigma\bigl(s,x_{\delta}(s), x_{\delta} \bigl(c_{1}(s)\bigr),\ldots, x_{\delta}\bigl(c_{p}(s) \bigr)\bigr) \bigr\Vert ^{2} \leq D. \end{aligned}$$

Consequently, the sequences \(\{ F(s,x_{\delta}(s), x_{\delta}(b_{1}(s)),\ldots, x_{\delta}(b_{m}(s)))\}\), \(\{G(s,x_{\delta}(s), x_{\delta}(a_{1}(s)),\ldots , x_{\delta}(a_{k}(s)))\}\), \(\{\sigma(s,x_{\delta}(s), x_{\delta}(c_{1}(s)),\ldots, x_{\delta}(c_{p}(s)))\}\) are weakly compact in \(L_{2}(J,X)\), \(L_{2}(L_{Q}(K,X))\) and \(L_{2}(L^{0}_{2}(Y,X))\), so, there are subsequences, still denoted by \(\{ F(s,x_{\delta}(s), x_{\delta}(b_{1}(s)),\ldots, x_{\delta}(b_{m}(s)))\}\), \(\{G(s,x_{\delta}(s), x_{\delta}(a_{1}(s)),\ldots ,x_{\delta}(a_{k}(s)))\}\), \(\{\sigma(s,x_{\delta}(s), x_{\delta}(c_{1}(s)),\ldots, x_{\delta}(c_{p}(s)))\}\), that are weakly converge to \(\{ F (s)\}\), \(\{G (s)\}\), \(\{\sigma(s)\}\) in \(L_{2}(J,X)\), \(L_{2}(L_{Q}(K,X))\) and \(L_{2}(L^{0}_{2}(Y,X))\).

From the last equation, we have

$$\begin{aligned}& E \bigl\Vert x_{\delta}(T)-\bar{x}_{T} \bigr\Vert ^{2} \\& \quad \leq9 E \bigl\Vert z \bigl(z I + \varGamma ^{T}_{0} \bigr)^{-1} \bigl\{ E \bar{x}_{T} - g_{m} \bigl(T,x(T)\bigr)\bigr\} \bigr\Vert ^{2} \\& \qquad {}+9 E \bigl\Vert z \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1}F\bigl(T, x(T), x\bigl(b_{1}(T)\bigr),\ldots, x \bigl(b_{m}(T)\bigr)\bigr) \bigr\Vert ^{2}\\& \qquad {}+ 9E \biggl\Vert \int_{0}^{T} z \bigl(z I + \varGamma^{T}_{0}\bigr)^{-1} \bar{\psi}(s) \,d \omega(s) \biggr\Vert ^{2} \\& \qquad {}+ 9E \biggl\Vert \int_{0}^{T} z \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1}\bar{\varphi}(s) \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+9 E \biggl\Vert \int_{s_{m}}^{T} z \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1} P_{\mu}(T-s) A \bigl(F\bigl(s, x(s), x \bigl(b_{1}(s)\bigr),\ldots, x\bigl(b_{m}(s)\bigr) \bigr)-F(s)\bigr)\,ds \biggr\Vert ^{2} \\& \qquad {}+ 9E \biggl\Vert \int_{s_{m}}^{T} z \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1} P_{\mu}(T-s) A F(s)\,ds \biggr\Vert ^{2} \\& \qquad {}+9 E \biggl\Vert \int_{s_{m}}^{T} z \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1}P_{\mu}(T-s)\\& \qquad {}\times \int_{0}^{s} G\bigl(\tau, x(\tau), x \bigl(a_{1}(\tau)\bigr),\ldots, x\bigl(a_{k}(\tau )\bigr) \bigr)-G(\tau)\,d\omega(\tau) \,ds \biggr\Vert ^{2} \\& \qquad {}+9 E \biggl\Vert \int_{s_{m}}^{T} z \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1}P_{\mu}(T-s) \int_{0}^{s} G(\tau) \,d\omega(\tau) \,ds \biggr\Vert ^{2} \\& \qquad {}+ 9 E \biggl\Vert \int_{s_{m}}^{T} z \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1} P_{\mu}(T-s) \sigma\bigl(s,x(s), x \bigl(c_{1}(s)\bigr),\ldots, x\bigl(c_{p}(s)\bigr)\bigr)- \sigma(s)\,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+ 9 E \biggl\Vert \int_{s_{m}}^{T} z \bigl(z I + \varGamma^{T}_{0} \bigr)^{-1} P_{\mu}(T-s)\sigma(s)\,dB^{H}(s) \biggr\Vert ^{2} . \end{aligned}$$

On the other hand, by Lemma 2.4, the operator \(z(z I + \varGamma ^{T}_{0})^{-1} \rightarrow0\) strongly as \(z \rightarrow0^{+}\) for all \({s_{m}} < s \leq T\), and, moreover, \(\| z(z I + \varGamma^{T}_{0})^{-1} \| \leq1\). Thus, by the Lebesgue dominated convergence theorem and the compactness of \(P_{\mu}(t)\) implies that \(E\| x_{\delta}(T) - \bar {x}_{T} \|^{2} \rightarrow0\) as \(z \rightarrow0^{+}\). This proves the approximate controllability of (1.1). □

Application

In this section, we present an example to illustrate our main result.

Let us consider the following stochastic control Hilfer fractional partial differential equation with fractional Brownian motion:

$$\begin{aligned} \textstyle\begin{cases} D_{0+}^{\frac{1}{3}, \frac{3}{5}}[x(t, z)+ \int_{0}^{\pi}a(y, z)x(t,y)\,dy]\\ \quad = \frac{\partial^{2}}{\partial z^{2}}x(t,z) +\eta(t, z) + \int_{0}^{t} 3^{-s}x(s,z) \,d \omega(s)\\ \qquad{}+ \frac{\sin t}{1+\sin t}x(t,z) \frac{dB^{H}(t)}{dt} , \quad t \in(0,\frac{1}{3}]\cup(\frac{2}{3},1] , 0 \leq z \leq\pi,\\ x(t, 0) = x(t, \pi) = 0, \quad t \in(0,1],\\ x(t,z)=\frac{1}{5} e^{-(t-\frac{1}{3})}\frac{ \Vert x(t,z) \Vert }{1+ \Vert x(t,z) \Vert }, \quad t \in(\frac{1}{3},\frac{2}{3}] , 0 \leq z \leq\pi,\\ I_{0+}^{\frac{4}{15}}(x(0,z))+ \sum_{i=1}^{2} c_{i} x(t_{i},z)= x_{0}(z) , \quad 0 \leq z \leq\pi, \end{cases}\displaystyle \end{aligned}$$
(4.1)

where \(D_{0+}^{\frac{1}{3}, \frac{3}{5}}\) is Hilfer fractional derivative of order \(\nu= \frac{1}{3} \), \(\mu=\frac{3}{5}\), ω is a Wiener process and \(B^{H}\) is a fractional Brownian motion with Hurst parameter \(H \in(\frac{1}{2},1)\).

Let \(X=Y=K=U = L_{2}([0, \pi])\) and A be defined by \(Ay = -(\frac {\partial^{2}}{\partial z^{2}})y \) with domain \(D(A) = \{ y \in X: y, \frac{dy}{dz}\text{ are absolutely continuous, and }(\frac{d^{2}}{dz^{2}})y \in X, y(0)= y(\pi)= 0 \}\).

Then −A generates a strongly continuous semigroup \(S(\cdot)\) which is compact, analytic, and self-adjoint. Furthermore, A has a discrete spectrum with eigenvalues \(n^{2}\), \(n \in N\) and the corresponding normalized eigenfunctions are given by

$$ e_{n} = \sqrt{\frac{2}{\pi}} \sin nx, \quad n = 1,2,\ldots. $$

In addition \((e_{n})_{n \in N} \) is a complete orthonormal basis in X. Then

$$ -Ay = \sum_{n=1}^{\infty}n^{2} \langle y, e_{n}\rangle e_{n},\quad y \in D(A). $$

Furthermore, −A is the infinitesimal generator of an analytic semigroup of bounded linear operator, \(\{S(t)\}_{t \geq0}\) on X and is given by

$$ S(t) y = \sum_{n= 1}^{\infty}e^{-n^{2} t} \langle y, e_{n}\rangle_{e_{n}},\quad y \in X, t\geq0, $$

with \(\| S(t)\| \leq e^{-t}\leq1\).

Moreover, the two operators \(S_{\frac{1}{3}, \frac{3}{5}}(t)\) and \(P_{\frac{3}{5}}(t)\) can be defined by

$$\begin{aligned}& S_{\frac{1}{3}, \frac{3}{5}}(t) x=\frac{3}{5\varGamma(\frac{2}{15})} \int _{0}^{t} \int_{0}^{\infty}\theta(t-s)^{\frac{-13}{15}} s^{\frac{-2}{5}} \varPsi_{\frac{3}{5}}(\theta)S\bigl(s^{\frac{3}{5}} \theta\bigr) x \,d \theta \,ds, \\& P_{\frac{3}{5}}(t )x = \frac{3}{5} \int_{0}^{\infty}\theta t^{\frac {-2}{5}} \varPsi_{\frac{3}{5}}(\theta)S\bigl(s^{\frac{3}{5}} \theta\bigr) x \,d \theta. \end{aligned}$$

Clearly,

$$\begin{aligned} \bigl\Vert P_{\frac{3}{5}}(t) \bigr\Vert \leq\frac{ t^{\frac{-2}{5}}}{\varGamma(\frac {3}{5})},\qquad \bigl\Vert S_{\frac{1}{3},\frac{3}{5}}(t) \bigr\Vert \leq\frac{ t^{\frac{-4}{15}}}{\varGamma(\frac{11}{15})}. \end{aligned}$$

In order to define the operator \(Q: Y \rightarrow Y\), we choose a sequence \(\{ \lambda_{n} \}_{n \in N} \subset R^{+}\), set \(Q e_{n} = \lambda_{n} e_{n}\), and assume that

$$ \operatorname{Tr}(Q)= \sum_{n= 1}^{\infty}\sqrt {\lambda_{n}} < \infty. $$

Define the fractional Brownian motion in Y by

$$ B^{H}(t)= \sum_{n= 1}^{\infty}\sqrt{ \lambda_{n}} \beta^{H}(t) e_{n} $$

where \(H \in(\frac{1}{2},1)\) and \(\{\beta^{H}_{n} \}_{n \in N}\) is a sequence of one-dimensional fractional Brownian motions mutually independent.

We also use the following properties:

\((a)\):

If \(y \in D(A)\), then \(Ay = \sum_{n=1}^{\infty}n^{2} \langle y, x_{n} \rangle x_{n} \).

\((b)\):

For each \(y \in X\), \(A^{-1/2}y = \sum_{n=1}^{\infty}\frac {1}{n} \langle y, x_{n} \rangle x_{n}\). In particular, \(\|A^{-1/2} \| ^{2} = 1\).

\((c)\):

The operator \(A^{1/2}\) is given by \(A^{1/2}y = \sum_{n=1}^{\infty}n \langle y, x_{n} \rangle x_{n}\) on the space \(D[A^{1/2}]= \{ y(\cdot) \in X, \sum_{n=1}^{\infty}n \langle y, x_{n} \rangle x_{n} \in X \} \).

We assume the following conditions hold:

\((i)\):

The function a is measurable and

$$ \int_{0}^{\pi} \int_{0}^{\pi}a^{2} (y,z) \,dy \,dz < \infty. $$
\((ii)\):

The function \(\frac{\partial}{\partial z}a(y,z)\) is measurable, \(a(y,0)= a(y, \pi)=0\), and let

$$ N_{1} =\biggl[ \int_{0}^{\pi} \int_{0}^{\pi}\biggl( \frac {\partial}{\partial z} a(y,z) \biggr) ^{2} \,dy \,dz \biggr]^{1/2} < \infty. $$
Define the bounded operator \(B: U\rightarrow X\) by \(Bu(t)(z) = \eta(t, z)\), \(0 \leq z \leq\pi\), \(u \in U\).

We define \(F : (0,1] \times X \rightarrow X\), \(G: (0,1] \times X \rightarrow L(K,X)\), \(\sigma: (0,1] \times X \rightarrow L^{0}_{2}(Y,X)\), \(g_{i}:(t_{i},s_{i}] \times X \rightarrow X\) and \(\xi :C((0,1],X)\rightarrow X\) by \(F(t,x) = Z_{1}(x)\), \(G(s,x)(z)= 3^{-s}x(s,z)\), \(\sigma(t,x)(z)=\frac {\sin t}{1+\sin t}x(t,z)\), \(g_{1}=\frac{1}{5} e^{-(t-\frac{1}{3})}\frac {\|x(t,z)\|}{1+\|x(t,z)\|}\) and \(\xi=\sum_{i=1}^{2} c_{i} x(t_{i},z)\), respectively, where

$$ Z_{1}(x) (z)= \int_{0}^{\pi}a(y,z)x(y) \,dy. $$

Then G, σ, \(g_{1}\) and ξ satisfy \((H2)\)\((H5)\). From \((i)\) it is clear that \(Z_{1}\) is a bounded linear operator on X. Furthermore, \(Z_{1}(x) \in D[A^{1/2}]\), and \(\|A^{1/2} Z_{1} \|^{2} \leq N_{1}\). In fact, from the definition of \(Z_{1}\) and \((ii)\) it follows that

$$ \bigl\langle Z_{1} (x), x_{n} \bigr\rangle = \int_{0}^{\pi}x_{n}(z)\biggl[ \int_{0}^{\pi}a(y,z) x(y)\,dy\biggr] \,dz = \frac{1}{n} \biggl(\frac{2}{\pi}\biggr)^{1/2} \bigl\langle Z_{2}(x), \cos(nx) \bigr\rangle , $$

where \(Z_{2}\) is defined by

$$ Z_{2}(x) (z)= \int_{0}^{\pi}\frac{\partial}{\partial z}a(y,z) x(y) \,dy. $$

From \((ii)\) we know that \(Z_{2}: X \rightarrow X \) is a bounded linear operator with \(\| Z_{2} \|^{2} \leq N_{1}\).

Hence \(\| A^{1/2} Z_{1}(x)\|^{2} = \| Z_{2}(x) \|^{2}\).

If \(u \in L_{2}((0,1],U)\), then \(B=I\), \(B^{*}=I\).

Therefore, with the above choice, the system (4.1) can be written in the abstract form of (1.1). On the other hand the linear system corresponding to (4.1) is approximately controllable. Therefore, all the hypotheses of Theorem 3.1 and Theorem 3.2 are satisfied and

$$\begin{aligned} & \biggl[1 +\frac{ M^{4} T^{2\mu} M^{4}_{B}}{z^{2} \mu^{2} \varGamma ^{4}(\mu)} \biggr] \biggl\lbrace \frac{36 M^{2} ( M_{0}^{2} M_{2} + M_{3} + M_{4} )}{\varGamma^{2}(\nu(1-\mu)+\mu)}\\ &\quad {}+ 36T^{2(1-\nu)(1-\mu )} \biggl[{M_{0}}^{2} M_{2} + \frac{ M^{2} T^{\mu}\varLambda_{1} \operatorname{Tr}(Q)}{\mu\varGamma^{2}(\mu)} \\ &\quad {} + \frac{2H M^{2} \varLambda_{2} T^{2 H+\mu-1}}{\mu\varGamma ^{2}(\mu)}+\frac{ (C_{1-\beta})^{2} \varGamma^{2} (1+ \beta) T^{2\mu \beta} M_{2} }{ \beta^{2} \varGamma^{2} (1+\mu\beta)} \biggr] \biggr\rbrace + T^{2(1-\nu)(1-\mu)} M_{3} < 1 \end{aligned}$$

and

$$\begin{aligned} \gamma_{1}&=9 \biggl[\frac{M^{2} ({M_{0}}^{2}{M_{1}}+ M_{6} + M_{7})}{\varGamma^{2}(\nu(1-\mu)+\mu)}+T^{2(1-\nu)(1-\mu)}\bigl( M_{6} +{M_{0}}^{2} {M_{1}}\bigr) \\ &\quad {}+\frac{{M_{1}}(C_{1 - \beta})^{2} \varGamma^{2}(1+ \beta) T^{2\mu \beta+2(1-\nu)(1-\mu)}}{\beta^{2} \varGamma^{2} (1+\mu\beta)} \biggr]\\ &< 1. \end{aligned}$$

Thus, we can conclude that the Hilfer fractional stochastic partial differential equation with fractional Brownian motion and nonlocal conditions (4.1) is approximately controllable on \((0, 1]\).

Conclusion

In this paper, by using fractional calculus, stochastic analysis, the fractional power of operators and the Sadovskii fixed point theorem, we obtained sufficient conditions for the approximate controllability of a class of noninstantaneous impulsive Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and nonlocal conditions. Also, we provided an example to illustrate our results. In the future we aim to study the existence of mild solutions and controllability of a class of Sobolev-type nonlinear Hilfer fractional stochastic differential inclusions with noninstantaneous impulsive in Hilbert space.

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Acknowledgements

We would like to thank the referees and the editor for their important comments and suggestions, which have significantly improved the paper.

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Correspondence to Hamdy M. Ahmed.

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Ahmed, H.M., El-Borai, M.M., El Bab, A.S.O. et al. Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion. Bound Value Probl 2020, 120 (2020). https://doi.org/10.1186/s13661-020-01418-0

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MSC

  • 93B05
  • 26A33
  • 93C10
  • 60H10
  • 34K40
  • 60G22

Keywords

  • Hilfer fractional neutral stochastic differential equations
  • Nonlocal conditions
  • Approximate controllability
  • Fractional Brownian motion
  • Noninstantaneous impulsive
  • Sadovskii fixed point theorem