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 C̄ 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 C̄. 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 C̄ 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). □