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IMA Journal of Mathematical Control and Information
, Volume Advance Article – Mar 25, 2017

25 pages

/lp/ou_press/approximate-controllability-of-caputo-fractional-neutral-stochastic-YSjVPFqBOy

- Publisher
- Oxford University Press
- Copyright
- © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 0265-0754
- eISSN
- 1471-6887
- D.O.I.
- 10.1093/imamci/dnx014
- Publisher site
- See Article on Publisher Site

Abstract In this article, by the semi-group theory, fractional calculus, stochastic analysis theory and the fixed-point technique, we provide sufficient conditions for the existence of mild solutions and the approximate controllability of Caputo fractional neutral stochastic differential inclusions with state-dependent delay (under the assumption that the corresponding linear system is approximately controllable). An example is provided to illustrate the abstract results. 1. Introduction Fractional differential equations have been proved to be useful tools to study the behaviour of real world problems in various areas, for instance, memory and hereditary properties of various materials and processes, physics, fluid mechanics and in almost every field of science and engineering. There has been a significant development in fractional differential equations in the past decades, see the monographs of Miller & Ross (1993), Kilbas et al. (2006), Lakshmikantham & Vatsala (2008) and Zhou (2014) and the references therein. For the existence of mild solutions and weak solutions for fractional differential equations, see Hernandez et al. (2006), Hernandez et al. (2008), Hernandez et al. (2009), Zhou et al. (2009), Zhou & Jiao (2010a), Zhou & Jiao (2010b), Agarwal et al. (2011), Shu et al. (2011), Shu & Wang (2012), Sakthivel et al. (2013b), Arthi et al. (2014), Zhou & Peng (2017a) and Zhou & Peng (2017b) and the references therein. Control theory is an important topic in mathematics which deals with the design and analysis of control systems and meanwhile it plays an important role in both engineering and sciences. In recent years, controllability problems for various types of nonlinear dynamical systems in infinite dimensional spaces by using different kinds of approaches have been considered in many publications. Two basic concepts of controllability should be distinguished: exact controllability and approximate controllability. Exact controllability of fractional differential equations was studied in Triggiani (1977), Wang & Zhou (2011), Wang & Zhou (2012) and Zhou et al. (2015). However, the concept of exact controllability is usually too strong and indeed has limited applicability in infinite-dimensional spaces while approximate controllability is a weaker concept which is completely adequate in applications. Recently, many authors have paid their attention to the approximate controllability of differential systems under the assumption that the associated linear system is approximately controllable. For example, Sakthivel et al. (2011) studied the approximate controllability of semi-linear fractional differential systems. For more details, we can refer to Mahmudov (2008), Sakthivel & Anandhi (2010), Yan (2012) and Mahmudov & Zorlu (2014). On the other hand, the fractional differential inclusions arise in the mathematical modelling of certain problems in economics, optimal controls, etc. Sakthivel et al. (2013a) formulated a new set of sufficient conditions for the approximate controllability of fractional non-linear differential inclusions and consequently numerous results have been developed. For instance, Debbouche & Torres (2014) derived the approximate controllability of fractional delay dynamic inclusions with non-local control conditions. Yang & Wang (2016) studied the approximate controllability of Riemann–Liouville fractional differential inclusions. For more details, see Vijayakumar et al. (2013) and Yan (2013). It should be pointed out that the deterministic models often fluctuate due to noise, which is random or at least appears to be so. Therefore, we should move from deterministic problems to stochastic ones. It is well known that many control systems arising from realistic models can be described as fractional stochastic differential equations and inclusions (see Agarwal et al., 2011; Balachandran & Sathya, 2011; Cui & Yan, 2013; Sakthivel et al., 2013b; Ren et al., 2014). However, only limited works are available in the existing literature for dealing the approximate controllability of fractional stochastic differential inclusions. So it is essential to extend the approximate controllability to the fractional stochastic differential inclusions. For more details about the approximate controllability of stochastic systems, we refer the reader to Mahmudov (2003), Balasubramaniam & Muthukumar (2009), Sakthivel et al. (2012), Muthukumar & Rajivganthi (2013), Sakthivel et al. (2013a), Guendouzi & Bousmaha (2014), Mahmudov (2014) and Balasubramaniam & Tamilalagan (2015). Recently, by using the $$\alpha$$-revolent operators, Yan & Jia (2015) studied the approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay. Muthukumar & Rajivganthi (2013) established the approximate controllability of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces. However, the approximate controllability of fractional neutral stochastic differential inclusions with state-dependent delay by using probability density function has not been studied yet. Motivated by these facts, this article is concerned with the approximate controllability of the following fractional neutral stochastic differential inclusions with state-dependent delay in the following form {cDα[x(t)−G(t,xt)]∈Ax(t)+h(t,xρ(t,xt))+F(t,xρ(t,xt))dω(t)dt +Bu(t), t∈J=[0,b],x0=ϕ∈B, (1.1) where $$^{c}D^\alpha$$ is Caputo fractional derivative of order $$0<\alpha<1$$, the stochastic process $$x(t)$$ takes values in a real separable Hilbert space $$H$$ with inner product $$(\cdot,\cdot)$$ and the norm $$\|\cdot\|$$. Let $$A$$ be a closed linear operator defined on a dense domain $$D(A)$$ in $$H$$ into $$H$$ that generates an analytic semigroup $$\{S(t)\}_{t\geq0}$$ and there exists a constant $$M>0$$ such that $$\sup_{t\in J}\|S(t)\|\leq M$$. Without the loss of generality, we assume that $$0\in\rho(A)$$, the resolvent set of $$A$$. Then, it is possible to define the fractional power $$A^{\beta}$$ for $$0<\beta\leq1$$ as a closed linear operator on its domain $$D(A^{\beta})$$ with inverse $$A^{-\beta}$$; the control function $$u$$ is given in $$L^{2}_{\mathcal{F}}(J,U)$$, a Hilbert space of admissible control functions, and $$U$$ is a Banach space; $$B:U\rightarrow H$$ is a linear bounded operator. Let $$K$$ be another separable Hilbert space with inner product $$\langle,\cdot,\rangle_{K}$$ and norm $$\|\cdot\|_K$$. $$\{\omega(t)\}_{t\geq0}$$ is a given $$K$$-valued Wiener process with a finite trace nuclear covariance operator $$Q>0$$ defined on the filtered complete probability space $$({\it{\Omega}},\mathcal{F},P)$$ equipped with a normal filtration $$\{\mathcal{F}_t\}_{t\geq0}$$ generated by $$w$$. $$x_t$$ represents the function $$x_t:(-\infty,0]\rightarrow H$$ defined by $$x_t(\theta)=x(t+\theta), -\infty<\theta\leq0$$, which belongs to some abstract phase space $$\mathcal{B}$$ defined axiomatically; $$\mathcal{P}(H)$$ is the family of all non-empty subsets of $$H$$; $G,~h: J\times\mathcal{B}$$\rightarrow H$ are given functions satisfying some assumptions; $$\rho:J\times\mathcal{B}\rightarrow(-\infty,b]$$ is appropriate function and will be specified later; $F: J\times \mathcal{B}$$\rightarrow \mathcal{P}(L(K,H))$ is a bounded closed convex$$-$$valued multi$$-$$valued map, where $$L(K,H)$$ denotes the space of all bounded linear operators from $$K $$ to $$H$$. The rest of the article is organized as follows. In Section 2, we will introduce some useful notations and preliminaries. In Section 3, some sufficient conditions are obtained to ensure the existence of mild solutions for the system. The approximate controllability result is presented in Section 4. In Section 5, an example is presented to illustrate our obtained results. 2. Preliminaries In this section, we introduce some definitions, lemmas and notations. In this article, we use the symbol $$\|\cdot\|$$ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let $$({\it{\Omega}},\mathcal{F},P)$$ be a filtered complete probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and $$\mathcal{F}_0$$ contains all $$P$$-null sets. A $$H$$-valued random variable is a $$\mathcal{F}$$-measurable function $$x(t):{\it{\Omega}}\rightarrow H$$, and a collection of random variables $$S=\{x(t,\omega):{\it{\Omega}}\rightarrow H|t\in J\}$$ is called a stochastic process. Generally, we suppress the dependence on $$\omega\in {\it{\Omega}}$$ and write $$x(t)$$ instead of $$x(t,\omega)$$ and $$x(t):J\rightarrow H$$ in place of $$S$$. Let $$\beta_n(t)(n=1,2,...)$$ be a sequence of real-valued independent one-dimensional standard Brownian motions over $$({\it{\Omega}},\mathcal{F},P).$$ Set $$\omega(t)=\sum\limits_{n=1}^{\infty}\sqrt \lambda_n\beta_n e_n,t\geq0$$, where $$\{e_n\},~(n=1,2,...)$$ is a complete orthonormal basis in $$K$$ and $$\lambda_n\geq 0 ~(n=1,2,...)$$ are non-negative real numbers. Let $$Q\in L(K,H)$$ be an operator defined by $$Q e_k=\lambda_k e_k$$ with $$\mbox{Tr}\, Q=\sum_{n=1}^{\infty}\lambda_n<\infty$$ ($$\mbox{Tr}$$ denotes trace of $$Q$$). The $$K$$-valued stochastic process $$\omega(t),t\geq0$$ is called a $$Q$$-Wiener process. It is assumed that $$\mathcal{F}_t=\sigma(\omega(s): 0\leq s\leq t)$$ is the $$\sigma$$-algebra generated by $$w$$ and $$\mathcal{F}_b=\mathcal{F}$$. Let $$\phi\in L(K,H)$$ and define $$\|\varphi\|_H^2=Tr(\varphi Q\varphi^*)=\sum_{k=1}^{\infty}\|\sqrt \lambda_k\varphi e_k\|^2$$. If $$\|\varphi\|_Q^2<\infty$$, then $$\varphi$$ is called a $$Q$$-Hilbert Schmidt operator. Let $$L_Q(K,H)$$ denote the space of all $$Q$$-Hilbert$$-$$Schmidt operators $$\varphi:K\rightarrow H.$$ The completion $$L_Q(K,H)$$ of $$L(K,H)$$ with respect to the topology induced by the norm $$\|\cdot\|_Q$$, where $$\|\varphi\|_Q=\langle \varphi,\varphi\rangle^{\frac{1}{2}}$$ is a Hilbert space with the above norm topology. The notation $$L_2({\it{\Omega}},H)$$ stands for the space of all $$H$$-valued random variables $$x$$ such that $$\|x\|_{L_2}=(E\|x\|^2)^{\frac{1}{2}}$$, where the expectation $$E$$ is defined by $$Ex=\int_{{\it{\Omega}}}x(\omega)dP$$. It is easy to check that $$L_2({\it{\Omega}},H)$$ is a Hilbert space equipped with the norm $$\|\cdot\|_{L_2}$$. Let $$C(J,L_2({\it{\Omega}},H))$$ stand for the Banach space of all continuous maps from $$J$$ into $$L_2({\it{\Omega}},H)$$ satisfying the condition $$\sup_{t\in J} E\|x(t)\|^2<\infty$$. Further, we denote $$\mathcal{C}=\{x\in C(J,L_2({\it{\Omega}},H))\mid x$$ is $$\mathcal{F}_t-$$adapted$$\}$$, which is also a Banach space endowed with the norm $$\|x\|_{\mathcal{C}}=\sup_{t\in J}(E\|x(t)\|^2)^{\frac{1}{2}}<\infty.$$ Finally, an important subspace of $$L_2({\it{\Omega}},H)$$ is given by $$L_2^0({\it{\Omega}},H)=\{x\in L_2({\it{\Omega}},H):x $$ is $$\mathcal{F}_{0}$$–measurable$$\}$$. We recall some basic definitions and properties of fractional calculus and multi-valued analysis. Definition 2.1 (Zhou, 2014) The fractional integral of order $$\alpha$$ with the lower limit zero for a function $$f$$ is defined as Iαf(t)=1Γ(α)∫0tf(s)(t−s)1−αds, t>0, α>0 provided the right side is pointwise defined on $$[0,\infty)$$, where $${\it{\Gamma}}(\cdot)$$ is the gamma function. Definition 2.2 (Zhou, 2014) Caputo’s derivative of order $$\alpha$$ for a function $$f$$ can be written as cDαf(t)=1Γ(n−α)∫0tfn(s)(t−s)α+1−nds, t>0, n−1<α<n. Lemma 2.1 (Zhou & Jiao, 2010b) (Bochner’s theorem) A measurable function $$H:[0,a]\to E$$ is Bochner integrable if $$|H|$$ is Lebesgue integrable. Lemma 2.2 (Wang & Zhou, 2011) For $$\sigma\in(0,1]$$ and $$0<a\leq b$$, we have $$|a^\sigma-b^\sigma|\leq(b-a)^\sigma$$. Lemma 2.3 (Wang & Zhou, 2011) The operators $$S_{\alpha}$$ and $$T_{\alpha}$$ have the following properties: (i) For any fixed $$t\geq0,S_{\alpha}(t)$$ and $$T_{\alpha}(t)$$ are linear and bounded operators, i.e. for any $$x\in H$$, $$\|S_{\alpha}(t)x\|\leq M\|x\|$$ and $$ \|T_{\alpha}(t)x\|\leq\frac{M}{{\it{\Gamma}}(\alpha)}\|x\|$$. (ii) $$\{S_{\alpha}(t)\}_{t\geq0}$$ and $$\{T_{\alpha}(t)\}_{t\geq0}$$ are strongly continuous. (iii) For any $$t>0$$, $$S_{\alpha}(t)$$ and $$T_{\alpha}(t)$$ are compact, if $$S(t),~t>0$$ is compact. Lemma 2.4 (Zhou & Jiao, 2010a) For any $$x\in D(A^{\beta})$$,$$~\beta\in(0,1),$$ and $$\eta\in(0,1]$$, we have ATα(t)x=A1−βTα(t)Aβx, 0≤t≤b and ‖AηTα(t)‖≤αCηΓ(2−η)tαηΓ(1+α(1−η)) 0<t≤b. Lemma 2.5 (Sakthivel et al., 2013a) For any $\bar{x}_{b}\in L^{2}(\mathcal{F}$$_{b},H)$ , there exists$$~\tilde{\varphi}\in L_{\mathcal{F}}^{2}({\it{\Omega}};L^{2}(0,b;L_2^0))$$ such that $$\bar{x}_{b}=E\bar{x}_{b}+\int_{0}^{b}\tilde{\varphi}(s)d\omega(s)$$. In this article, the phase space $$(\mathcal{B},\|\cdot\|_{\mathcal{B}})$$ will denote a semi-normed linear space of $\mathcal{F}$$_0$ -measurable functions mapping from $$(-\infty,0]$$ into $$H$$ and such that the following axioms hold due to Hale & Kato (1978): (1) If $$x:(-\infty,b+\eta]\rightarrow H,~b\geq0,$$ is such that $$x\mid_{[\eta,\eta+b]}\in \mathcal{C},$$ then, for every $$t\in[\eta,\eta+b]$$, the following conditions hold: (a) $$x_t\in\mathcal{B};$$ (b) $$\|x(t)\|\leq \tilde{H}\|x_t\|_{\mathcal{B}};$$ (c) $$\|x_t\|_{\mathcal{B}}\leq K(t-\eta)\sup\{\|x(s):\eta\leq s\leq t\|\}+M(t-\eta)\|x_\eta\|_{\mathcal{B}}$$; where $$\tilde{H}$$ is a constant, $$K,M:[0,\infty)\rightarrow[1,\infty)$$, $$K$$ is continuous, $$M$$ is locally bounded, $$\tilde{H},H,K,$$ are independent of $$x(\cdot)$$. (2) For the function $$x(\cdot)$$ in $$(1)$$, the function $$t\rightarrow x_t$$ is continuous from $$[\eta,\eta+b]$$ into $$\mathcal{B}$$. (3) The space $$\mathcal{B}$$ is complete. Lemma 2.6 (Arthi et al., 2014) Let $$x:(-\infty,b]\rightarrow H$$ be a function such that $$x_0=\phi,$$ and $$M_b:=\sup\{M(t):0\leq t\leq b\}.$$ Then ‖xs‖B≤(Mb+J0ϕ)‖ϕ‖B+Kbsup{‖x(θ)‖;θ∈[0,max{0,s}]}, s∈R(ρ−)∪J, where $$J_0^{\phi}=\sup\{J^{\phi}(t):t\in\mathcal{R(\rho^{-})} \}$$. Remark 2.1 Let $$\phi\in \mathcal{B}$$ and $$t \leq 0$$. The notation $$\phi_{t}$$ represents the function defined by $$\phi_{t}=\phi(t+\theta)$$. Consequently if the function $$x(\cdot)$$ in the axiom $$(1)$$ is such that $$x_0 =\phi$$, then $$x_t =\phi_{t}$$. We observe that $$\phi_{t}$$ is well defined for $$t < 0$$, since the domain of $$\phi$$ is $$(-\infty,0]$$. Let $$(H,d)$$ be a metricspace. We will use the notations: $$\mathcal{P}(H):=\{Y\in2^H:Y\neq\emptyset\}$$, $\mathcal{P}_{cl}(H):=\{Y\in\mathcal{P}$$(H)\}$ is closed, $\mathcal{P}_{b}(H):=\{Y\in\mathcal{P}$$(H)\}$ is bounded, $\mathcal{P}_{cv}(H):=\{Y\in\mathcal{P}$$(H)\}$ is convex, $ \mathcal{P}_{cp}(H):=\{Y\in\mathcal{P}$$(H)\}$ is compact. Proposition 2.1 (Debbouche & Torres, 2014) (1) A measurable function $$u:J\rightarrow H $$ is Bochner integrable if and only if $$\|u\|$$ is Lebesgue integrable. (2) A multi-valued map $$F:H\rightarrow2^H$$ is said to be convex-valued (closed-valued) if $$F(u)$$ is convex (closed) for all $$u\in H$$; is said to be bounded on bounded sets if $$F(B)=\bigcup\limits_{u\in B}$$ is bounded in $$H$$ for all $$B\in \mathcal{P}_b(x)$$. (3) A map $$F$$ is said to be upper semi-continuous (u.s.c.) on $$H$$ if for each $$u_0\in H$$, the set $$F(u_0)$$ is a non-empty closed subset of $$H$$, and if for each open subset $${\it{\Omega}}$$ of $$H$$ containing $$F(u_0)$$, there exists an open neighbourhood $$\nabla$$ of $$u_0$$ such that $$F(\nabla)\subseteq{\it{\Omega}}$$. (4) A map $$F$$ is said to be completely continuous if $$F(B)$$ is relatively compact for every $$B\in \mathcal{P}_b(H)$$. If the multi-valued map $$F$$ is completely continuous with non-empty compact values, then $$F$$ is u.s.c. if and only if $$F$$ has a closed graph, i.e. $$u_n\rightarrow u, y_n\rightarrow y,y_n\in F(u_0)$$ imply $$y\in F(u)$$. We say that $$F$$ has a fixed point if there is $$u\in H$$ such that $$u\in F(u).$$ (5) A multi-valued map $$F:J\rightarrow \mathcal{P}_{cl}(H)$$ is said to be measurable if for each $$u\in H$$ the function $$y:J\rightarrow R$$ defined by $$y(t)=d(u,F(t))=\inf\{\|u-z\|,z\in F(t)\}$$ is measurable. (6) A multi-valued map $$F:H\rightarrow 2^H$$ is said to be condensing if for any bounded subset $$B\subset H$$ with $$\beta(B)\neq0$$ we have $$\beta(F(B))<\beta(B),$$ where $$\beta(\cdot)$$ denote the Kuraowski measurable of non-compactness defined as follows: $$\beta(B):=\inf\{d>0:B$$ can be covered by a finite number of balls of radius $$d\}$$. For more details on multi-valued maps, see the books of Deimling (1992) and Hu & Papageorgiou (1997). Lemma 2.7 (Guendouzi & Bousmaha, 2014) (Lasota and Opial). Let $$J$$ be a compact real interval, $$ \mathcal{P}_{bd,cl,cv}(H)$$ be the set of all non-empty, bounded, closed and convex subset of $$H$$ and $$F$$ be a $$L^2$$-Carath$$\acute{e}$$odory multi-valued map $$S_{F,x}\neq\emptyset$$ and let $$\gamma$$ be a linear continuous mapping from $$L^2(J,H)$$ to $$C(J,H)$$. Then, the operator Γ∘SF:C(J,H)→Pbd,cl,cv(C(J,H)),x↦(Γ∘SF)(x):=Γ(SF,x) is a closed graph operator in $$C(J,H)\times C(J,H),$$ where $$S_{F,x}$$ is known as the selectors set from $$F,$$ is given by f∈SF,x={f∈L2(L(K,H)):f(t)∈F(t,x), for a.e. t∈J}. Lemma 2.8 (Yan, 2013) Let $$D$$ be a non-empty subset of $$H$$ which is bounded, closed, and convex. Suppose $$F:D\rightarrow2^D\setminus\{\emptyset\}$$ is u.s.c. condensing multi-valued map. If for every $$u\in H,$$$$F(u)$$ is a closed and convex set such that $$F(D)\subset D,$$ then $$F$$ has a fixed point in $$D$$. 3. Existence of mild solutions In this section, we study the existence of mild solutions for the system (1.1). We first present the definition of mild solutions for the system (1.1). Definition 3.1 A stochastic progress $$x(t), t\in J$$ is said to be a mild solution of (1.1) if for every control $$u\in L_{\mathcal{F}}^{2}(J,U)$$, it satisfies the following conditions: (1) $$x_0=\phi,x_{\rho(s,x_s)}\in \mathcal{B}$$ satisfying $$x_0\in L_2^0({\it{\Omega}},H),x|_J\in \mathcal{C}$$; (2) $$x(t), t\geq0$$ is $\mathcal{F}$$_{t}-$ adapted and measurable; (3) $$x(t)$$ is continuous on $$[0, b]$$ almost surely and for each $$s \in [0, t)$$, the function $$(t-s)^{\alpha-1}AT_{\alpha}(t-s)G(s,x_s)$$ is integrable; (4) there exists $$f(s)\in S_{F,x_{\rho}}=\{f\in L^2(J,L_{Q}(K,H)):f(t)\in F(t,x_{\rho})$$ for a.e. $$t\in J\}$$ such that $$x$$ satisfies the following stochastic integral equation: x(t)=Sα(t)[ϕ(0)−G(0,ϕ)]+G(t,xt)+∫0t(t−s)α−1ATα(t−s)G(s,xs)ds+∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs))ds+∫0t(t−s)α−1Tα(t−s)f(s)dω(s)+∫0t(t−s)α−1Tα(t−s)Bu(s)ds, (3.1) where $$S_{\alpha}(t)x=\int_{0}^{\infty}\phi_{\alpha}(\theta)S(t^{\alpha}\theta)xd\theta, T_{\alpha}(t)x=\int_{0}^{\infty}\alpha\theta\phi_{\alpha}(\theta)S(t^{\alpha}\theta)xd\theta, \phi_{\alpha}(\theta)=\frac{1}{\alpha}\theta^{-1-\frac{1}{\alpha}}\psi_{\alpha}(\theta^{\frac{-1}{\alpha}}), \psi_{\alpha}(\theta)=\frac{1}{\pi}\sum\limits_{n=1}^{\infty}(-1)^{n-}\theta^{-\alpha n-1}\frac{\it{\Gamma}(n\alpha+1)}{n!}\sin(n\pi\alpha), \theta\in(0,\infty),$$$$\phi_{\alpha}$$ is the probability density function defined on $$(0,\infty)$$, that is, $$\phi_{\alpha}(\theta)\geq0, \theta\in(0,\infty)$$ and $$\int_0^{\infty}\phi_{\alpha}(\theta)d\theta=1$$. Let $\mathcal{H}$$(b)$ be the space of all functions $$x:(-\infty,b]\rightarrow H$$ such that $$x_0=\phi$$ and the restriction $$x|_{J}\in \mathcal{C}$$ endowed with the sup norm. Set $B_r(0,\mathcal{H}$$(b))=\{x\in\mathcal{H}$$(b),\sup\limits_{0\leq s\leq b}E\|x(s)\|_H^2\leq r\}$. We observe that for every $x\in B_r(0,\mathcal{H}$$(b))$ and for each $$t\in J,$$ ‖xρ(t,xt)‖B2≤2[(Mb+J0ϕ)2‖ϕ‖B2+Kb2r]:=r∗. To prove our results, we always assume that $$\rho:J\times\mathcal{B}\rightarrow(-\infty,b]$$ is continuous and that $$\phi\in\mathcal{B}$$. For convenience, we set 12‖A−β‖2c1Kb2+12c1Kb2K(α,β)b2αβ(αβ)2+84c1Kb2MB4 ×(MΓ(α))41a2b2αα2[‖A−β‖2+K(α,β)b2αβ(αβ)2]=N1, and 12Kb2(MΓ(α))2b2αα2[1+7MB41a2(MΓ(α))4b2αα2]=N2, where $$K(\alpha,\beta):=\displaystyle\frac{\alpha^{2}C_{1-\beta}^{2}\it{\Gamma}^2(1+\beta)}{\it{\Gamma}^2(1+\alpha\beta)}$$ and $$M_B=\|B\|$$. Before stating and proving the main results, we introduce the following hypotheses. (H$$_0$$) Assume the semigroup $$S(t)$$ is compact, and for any $$a>0,$$$$\|a(R(a,\it{\Gamma}_{s}^{b}))\|\leq1$$ for $$0\leq s< b$$; (H$$_1$$) Let $$\mathcal{R}(\rho^{-})=\{\rho(s,\psi)\leq0:(s,\psi)\in J\times\mathcal{B}\}$$. The function $$t\rightarrow\phi_t$$ is well defined from $$\mathcal{R}(\rho^{-})$$ into $$\mathcal{B}$$ and there exists a continuous and bounded function $$J^{\phi}:\mathcal{R}(\rho^{-})\rightarrow(0,\infty)$$ such that $$E\|\phi_t\|_{\mathcal{B}}^{2}\leq J^{\phi}(t)E\|\phi\|_{\mathcal{B}}^{2}$$ for every $$t\in \mathcal{R}(\rho^{-})$$; (H$$_2$$) Function $$G:J\times \mathcal{B}\rightarrow H$$ satisfies the following: (2a) $$G:J\times\mathcal{B}\rightarrow H$$ is continuous for each $$t\in J,$$ and there exist constant $$\beta\in(0,1)$$ and $$c_1,c_2>0$$ such that $$G\in D(A^{\beta})$$ and for any $$\psi\in \mathcal{B},$$$$t\in J$$, the function $$A^{\beta}G(\cdot,\psi)$$ is strongly measurable; (2b) There exist positive constants $$M_G,c_1,c_2$$ such that that for any $$\psi_1, \psi_2\in\mathcal{B},$$$$A^{\beta}G(t,\psi_1)$$ satisfies E‖AβG(t,ψ1)−AβG(t,ψ2)‖H2≤MG‖ψ1−ψ2‖B2,t∈J and E‖AβG(t,ψ1)‖H2≤(c1‖ψ1‖B2+c2),∀ψ∈B,t∈J; (H$$_3$$) Function $$h:J\times \mathcal{B}\rightarrow H$$ satisfies the following: (3a) $$h(t,\cdot):\mathcal{B}\rightarrow H$$ is continuous for each $$t\in J$$, and for each $$\psi\in\mathcal{B},h(\cdot,\psi):J\rightarrow H$$ is strongly measurable; (3b) There exists $$\mu(t)\in L^{1}(J,\mathbb{R}^+)$$ and a continuous non-decreasing function $${{\it{\Omega}}}_{1}:[0,\infty)\to(0,\infty)$$ such that for any $$(t,\psi)\in J\times\mathcal{B},$$ we have E‖h(t,ψ)‖H2≤μ(t)×Ω1(‖ψ‖B2),limr→∞infΩ1(r)rds=Λ<∞; (H$$_4$$) The multi$$-$$valued map $$F:J\times\mathcal{B}\rightarrow \mathcal{P}_{bd,cl,cv}(L(K,H))$$ satisfies the following: (4a) $$F(t,\cdot):\mathcal{B}\rightarrow \mathcal{P}_{bd,cl,cv}(L(K,H))$$ is u.s.c for each $$t\in J$$, and for each $$\psi\in \mathcal{B}$$, the function $$F(\cdot,\psi):J\rightarrow \mathcal{P}_{bd,cl,cv}(L(K,H))$$ is strongly measurable, and the set $$S_{F,\psi}=\{f\in L^2(J,L_{Q}(K,H)):f(t)\in F(t,\psi)$$ for a.e. $$t\in J\}$$ is non-empty; (4b) There exists $$\varphi(t)\in L^{1}(J,\mathbb{R}^+)$$ and a continuous non-decreasing function $${\it{\Omega}}_{2}:[0,\infty)\to(0,\infty)$$ such that for any $$(t,\psi)\in J\times\mathcal{B}$$, we have E‖F(t,ψ)‖H2≤φ(t)×Ω2(‖ψ‖B2),limr→∞infΩ2(r)rds=Υ<∞, where $$E\|F(t,\psi)\|^2_H=\sup\limits_{t\in J}\{E\|f\|^2_H: f\in F(t,\psi)$$ for a.e. $$t\in J \}$$. To prove our results, we introduce two relevant operators: Γτb=∫τb(b−s)α−1Tα(b−s)BB∗Tα∗(b−s)ds,0<τ≤b and R(a,Γτb)=(aI+Γτb)−1,∀a>0, where $$B^{*}$$ denotes the adjoint of $$B$$ and $$T_{\alpha}^{*}$$ is the adjoint of $$T_{\alpha}$$. It is straightforward that the operator $$\it{\Gamma}_{\tau}^b$$ is a linear bounded operator. Now for any $$a > 0$$ and $$\tilde{x}_b\in L^2(\mathcal{F}_b, H)$$, we define the control function uxa(s)=B∗Tα∗(b−s){aI+Γ0b)−1[Ex¯b+∫0bφ~(τ)dω(τ)−Sα(b)[ϕ(0))−G(0,ϕ)]−G(b,xb)]}−B∗Tα∗(b−s)∫0t(aI+Γτb)−1(b−τ)α−1ATα(b−τ)G(τ,xτ)dτ−B∗Tα∗(b−s)∫0t(aI+Γτb)−1(b−τ)α−1Tα(b−τ)h(τ,xρ(τ,xτ))dτ−B∗Tα∗(b−s)∫0t(aI+Γτb)−1(b−τ)α−1Tα(b−τ)f(τ)dω(τ), where $$f\in S_{F,x_{\rho}}=\{f\in L^2(L(K,H)):f(t)\in F(t,x_{\rho(t,x_t)})$$, a.e. $$t\in J\}$$. In order to prove the existence of mild solutions for system (1.1), we transform it into a fixed point problem. We define the multi-valued map $${\it{\Psi}}:\mathcal{H}(b)\rightarrow\mathcal{P}(\mathcal{H}(b))$$ by $${\it{\Psi}} (x)$$, the set of $$\eta\in\mathcal{H}(b)$$ such that η(t):={0,t∈(−∞,0],Sα(t)[ϕ(0)−G(0,ϕ)]+G(t,xt)+∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs))ds+∫0t(t−s)α−1Tα(t−s)f(s)dω(s)+∫0t(t−s)α−1ATα(t−s)G(s,xs)ds+∫0t(t−s)α−1Tα(t−s)Buxa(s)ds,t∈J, (3.2) where $$f\in S_{F,x_{\rho}}=\{f\in L^2(L(K,H)):f(t)\in F(t,x_{\rho(t,x_t)})$$ a.e. $$t\in J\}$$. Now, we define the following operator $${\it{\Psi}}_{1}:\mathcal{H}(b)\rightarrow\mathcal{H}(b)$$ by Ψ1(x)(t)={0,t∈(−∞,0],G(t,xt)+∫0t(t−s)α−1ATα(t−s)G(s,xs)ds,t∈J, (3.3) and multi-valued operator $${\it{\Psi}}_{2}:\mathcal{H}(b)\rightarrow\mathcal{P}(\mathcal{H}(b))$$ by $${\it{\Psi}}_2 (x)$$, the set of $$\bar{\eta}\in\mathcal{H}(b)$$ such that η¯(t)={0,t∈(−∞,0],Sα(t)[ϕ(0)−G(0,ϕ)]+∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs))ds+∫0t(t−s)α−1Tα(t−s)f(s)dω(s)+∫0t(t−s)α−1Tα(t−s)Buxa(s)ds,t∈J. (3.4) Obviously, we have $${\it{\Psi}}={\it{\Psi}}_1+{\it{\Psi}}_2$$. In what follows, we aim to show that the operator $${\it{\Psi}}$$ has a fixed point, which is a mild solution of system (1.1). Theorem 3.1 Suppose that assumptions (H$$_{0}$$)–(H$$_{4}$$) are satisfied, then system (1.1) has at least one mild solution provided that (i) $$N_1+N_2(\sup\limits_{s\in J}\mu(s)\it{\Lambda}+\sup\limits_{s\in J}\varphi(s){\it{\Upsilon}})<1$$, (ii) $$4\|A^{-\beta}\|^2M_GK_b^2+4M_GK_b^2K(\alpha,\beta)\frac{b^{2\alpha\beta}}{(\alpha\beta)^2}<1$$. Proof. We divide the proof into five steps. Step 1. We shall show that there exists a constant $$r=r(a)$$ such that $${\it{\Psi}} (B_r(0,\mathcal{H}(b)))\subset B_r(0,\mathcal{H}(b))$$. In fact, if it is not true, then for each positive constant $$r$$ there exists $$\bar{x}\in B_r(0,\mathcal{H}(b)),$$$$\bar{u}\in L_{\mathcal{F}}^2(J,U)$$ corresponding to $$\bar{x}$$, but $${\it{\Psi}}(\bar{x})\notin B_r(0,\mathcal{H}(b))$$ for some $$t=t(r)\in J$$, i.e. r<E‖(Ψx¯)(t)‖H2≤6∑i=16Ii=6E‖Sα(t)[ϕ(0)−G(0,ϕ)]‖H2+6E‖G(t,x¯t)‖H2+6E‖∫0t(t−s)α−1ATα(t−s)G(s,x¯s)ds‖H2+6E‖∫0t(t−s)α−1Tα(t−s)h(s,x¯ρ(s,x¯s))ds‖H2+6E‖∫0t(t−s)α−1Tα(t−s)f(s)dω(s)‖H2+6E‖∫0t(t−s)α−1Tα(t−s)Bu¯xa(s)ds‖H2. Due to assumption (H$$_2$$), it follows that I1≤2M2E‖ϕ(0)‖H2+2M2‖A−β‖2(2c1[(Mb+J0ϕ)2‖ϕ‖B2+Kb2E‖ϕ(0)‖H2]+c2), and I2=E‖G(t,x¯t)‖H2≤‖A−β‖2E‖AβG(t,x¯t)‖H2≤‖A−β‖2(c1r∗+c2). By a standard calculation, assumption (H$$_2$$) and the Hölder inequality, we can deduce that I3=E‖∫0t(t−s)α−1ATα(t−s)G(s,x¯s)ds‖H2≤K(α,β)∫0t(t−s)αβ−1ds∫0t(t−s)αβ−1E‖AβG(s,x¯s)‖H2ds≤K(α,β)b2αβ(αβ)2(c1r∗+c2). From assumption (H$$_3$$), we derive that I4=E‖∫0t(t−s)α−1Tα(t−s)h(s,x¯ρ(s,x¯s))ds‖H2≤(MΓ(α))2b2αα2sups∈Jμ(s)Ω1(r∗). Similarly, by assumption (H$$_4$$), we obtain I5=E‖∫0t(t−s)α−1Tα(t−s)f(s)dω(s)‖H2≤(MΓ(α))2b2αα2sups∈Jφ(s)Ω2(r∗). Then by the estimates above and (H$$_0$$), for each $$s\in J$$, we can get E‖uxa(s)‖H2 ≤7MB2(MΓ(α))21a2{E‖x¯b‖H2+∫0bE‖φ~(s)‖2ds+2M2‖ϕ(0)‖H2 +2M2‖A−β‖2{c1[2(Mb+J0ϕ)2‖ϕ‖B2+2Kb2‖ϕ(0)‖H2]+c2} +‖A−β‖2(c1r∗+c2)+K(α,β)b2αβ(αβ)2(c1r∗+c2) +(MΓ(α))2b2αα2[sups∈Jμ(s)Ω1(r∗)+sups∈Jφ(s)Ω2(r∗)]} :=Mu. (3.5) Now, by the estimate (3.5), we know that I6=E‖∫0t(t−s)α−1Tα(t−s)Bu¯xa(s)ds‖H2≤MB2(MΓ(α))2b2αα2Mu. From the estimates above, we get r<E‖(Ψx¯)(t)‖H2≤∑i=16Ii=6{2M2E‖ϕ(0)‖H2+2M2‖A−β‖2{2c1[(Mb+J0ϕ)2‖ϕ‖B2+Kb2E‖ϕ(0)‖H2]+c2}}+6‖A−β‖2(c1r∗+c2)+6K(α,β)b2αβ(αβ)2(c1r∗+c2)+6(MΓ(α))2b2αα2sups∈Jμ(s)×Ω1(r∗)+6(MΓ(α))2b2αα2sups∈Jφ(s)Ω2(r∗)+42MB4(MΓ(α))41a2b2αα2×{E‖x¯b‖H2+∫0bE‖φ~(s)‖2ds+2M2‖A−β‖2‖ϕ(0)‖H2+2M2‖A−β‖2{2c1[(Mb+J0ϕ)2×‖ϕ‖B2+Kb2E‖ϕ(0)‖H2]+c2}+‖A−β‖2(c1r∗+c2)+K(α,β)b2αβ(αβ)2(c1r∗+c2)}+42MB4(MΓ(α))61a2b4αα4[sups∈Jμ(s)Ω1(r∗)+sups∈Jφ(s)Ω2(r∗)]=L0+12‖A−β‖2c1Kb2r+12K(α,β)b2αβ(αβ)2c1Kb2r+6(MΓ(α))2b2αα2[sups∈Jμ(s)Ω1(r∗)+sups∈Jφ(s)Ω2(r∗)]+42MB4(MΓ(α))41a2b2αα2{2c1Kb2r[‖A−β‖2+K(α,β)b2αβ(αβ)2]}+42MB4(MΓ(α))61a2b4αα4[sups∈Jμ(s)Ω1(r∗)+sups∈Jφ(s)Ω2(r∗)], where L0=12M2E‖ϕ(0)‖H2+12M2‖A−β‖2{2c1[(Mb+J0ϕ)2‖ϕ‖B2+Kb2E‖ϕ(0)‖H2]+c2}+6‖A−β‖2[2c1(Mb+J0ϕ)2‖ϕ‖B2+c2]+6K(α,β)b2αβ(αβ)2[2c1(Mb+J0ϕ)2‖ϕ‖B2+c2]+42MB4(MΓ(α))41a2b2αα2{E‖x¯b‖H2+∫0bE‖φ~(s)‖2ds+2M2E‖ϕ(0)‖H2+2M2‖A−β‖2×[2c1[(Mb+J0ϕ)2‖ϕ‖B2+Kb2E‖ϕ(0)‖H2]+c2]+‖A−β‖2[2c1(Mb+J0ϕ)2‖ϕ‖B2+c2]}+42MB4(MΓ(α))41a2b2αβ(αβ)2K(α,β)[2c1(Mb+J0ϕ)2‖ϕ‖B2+c2]. Dividing both sides by $$r$$ and taking $$r\rightarrow\infty,$$ we obtain $$N_1+N_2(\sup\limits_{s\in J}\mu(s)\it{\Lambda}+\sup\limits_{s\in J}\varphi(s){\it{\Upsilon}})>1$$, which is a contradiction to our assumption (i) of Theorem 3.1. Thus there exists $$r$$ such that $${\it{\Psi}}$$ maps $$B(r,\mathcal{H}(b))$$ into itself. Step 2. The operator $${\it{\Psi}}(x)$$ is convex for each $$x\in B_r(0,\mathcal{H}(b))$$. In fact, if $$\eta_1,\eta_2$$ belong to $${\it{\Psi}}(x)$$, then there exist $$f_1,f_2\in S_{F,x_{\rho}},$$ such that for each $$t\in J$$, $$i=1,2$$, ηi(t)=Sα(t)[ϕ(0)−G(0,ϕ)]+G(t,xt)+∫0t(t−s)α−1ATα(t−s)G(s,xs)ds+∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs))ds+∫0t(t−s)α−1Tα(t−s)fi(s)dω(s)+∫0t(t−s)α−1Tα(t−s)B{B∗Tα∗(b−s)R(a,Γ0b)×[Ex¯b+∫0bφ~(s)dω(s)−G(b,x(b))−Sα(b)[ϕ(0)−G(0,ϕ)]]−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1ATα(b−τ)G(τ,xτ)dτ−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)h(τ,xρ(τ,xτ))dτ−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)fi(τ)dω(τ)}ds. Let $$0\leq\lambda\leq1$$, then for each $$t\in J$$, we have λη1(t)+(1−λ)η2(t)=Sα(t)[ϕ(0)−G(0,ϕ)]+G(t,xt)+∫0t(t−s)α−1ATα(t−s)G(s,xs)ds+∫0t(t−s)α−1Tα(t−s)[λf1+(1−λ)f2](s)dω(s)+∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs))ds+∫0t(t−s)α−1Tα(t−s)B{B∗Tα∗(b−s)R(a,Γ0b)×[Ex¯b+∫0bφ~(s)dω(s)−G(b,x(b))−Sα(b)[ϕ(0)−G(0,ϕ)]]−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1ATα(b−τ)G(τ,xτ)dτ−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)h(τ,xρ(τ,xτ))dτ−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)[λf1+(1−λ)f2](τ)dω(τ)}ds. Since $$S_{F,\psi}$$ is convex, we have $$\lambda f_1+(1-\lambda)f_2\in S_{F,x_{\rho}}$$, then $$\lambda\eta_1(t)+(1-\lambda)\eta_2(t)\in{\it{\Psi}}(x)$$. Step 3. $${\it{\Psi}}_1$$ is a contraction operator. For any $$x,y\in B_r(0,\mathcal{H}(b)), t\in J$$, we have E‖(Ψ1x)(t)−(Ψ1y)(t)‖H2 ≤2E‖G(t,xt)−G(t,yt)‖H2+2E‖∫0t(t−s)α−1ATα(t−s)[G(s,xs)−G(s,ys)]ds‖H2 ≤‖A−β‖2MG‖xt−yt‖B2+2K(α,β)b2αβ(αβ)2E‖Aβ[G(s,xs)−G(s,ys)]‖H2 ≤4‖A−β‖2MGKb2sup0≤s≤tE‖x(s)−y(s)‖H2 +4MGKb2K(α,β)b2αβ(αβ)2sup0≤r≤sE‖x(r)−y(r)‖H2 ≤[4‖A−β‖2MGKb2+4MGKb2K(α,β)b2αβ(αβ)2]sup0≤s≤tE‖x(s)−y(s)‖H2. It follows that, E‖Ψ1(x)−Ψ1(y)‖H2≤[4‖A−β‖2MGKb2+4MGKb2K(α,β)b2αβ(αβ)2]E‖x−y‖H2. Thus $${\it{\Psi}}_{1}$$ is a contraction operator by our assumption (ii) of Theorem 3.1. Step 4. $${\it{\Psi}}_{2}$$ is completely continuous. We subdivide its proof into three claims. Claim 1. $${\it{\Psi}}_{2}$$ map bounded sets into bounded sets in $B_r(0,\mathcal{H}$$(b))$ . Indeed, it is enough to show that there exists a positive constant $${\it{\Delta}}$$ such that for each $\bar{\eta}\in{\it{\Psi}}_{2}(x), x\in B_r(0,\mathcal{H}$$(b))$, one has $$\sup\limits_{0\leq t\leq b}E\|\bar{\eta}(t)\|_H^2\leq{\it{\Delta}}$$. If $\bar{\eta}\in{\it{\Psi}}_{2}(x), x\in B_r(0,\mathcal{H}$$(b))$, then there exists $$f\in S_{F,x_{\rho}}$$ such that for each $$t\in J$$, η¯(t)=Sα(t)[ϕ(0)−G(0,ϕ)]+∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs))ds+∫0t(t−s)α−1Tα(t−s)f(s)dω(s)+∫0t(t−s)α−1Tα(t−s)Buxa(s)ds. By using Hölder inequality and assumptions (H$$_{0}$$)-(H$$_{4}$$), for each $$t\in J,$$ we have E‖η¯(t)‖H2≤ 4E‖Sα(t)[ϕ(0)−G(0,ϕ)]‖H2+4E‖∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs))ds‖H2 +4E‖∫0t(t−s)α−1Tα(t−s)f(s)dω(s)‖H2+4E‖∫0t(t−s)α−1Tα(t−s)Buxa(s)ds‖H2≤ 8M2E‖ϕ(0)‖H2+8‖A−β‖2{2c1[(Mb+J0ϕ)2‖ϕ‖B2+Kb2E‖ϕ(0)‖H2]+c2} +4(MΓ(α))2b2αα2sups∈Jμ(s)Ω1(r∗)+4(MΓ(α))2b2αα2sups∈Jφ(s)Ω2(r∗)+4MB2(MΓ(α))2b2αα2Mu:=Δ. Then for each $$\bar{\eta}\in{\it{\Psi}}_{2}(x),$$ we have $$\sup\limits_{0\leq t\leq b}E\|\bar{\eta}(t)\|_{H}^2\leq{\it{\Delta}}$$. Claim 2. $${\it{\Psi}}_2$$ maps bounded sets into equicontinuous sets of $$\mathcal{H}(b)$$. For $$0< t_1<t_2\leq b$$, for each $$x\in B_r(0,\mathcal{H}(b)), \bar{\eta}\in{\it{\Psi}}_2(x)$$, there exists $$f\in S_{F,x_{\rho}}$$, such that η¯(t)=Sα(t)[ϕ(0)−G(0,x0)]+∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs))ds+∫0t(t−s)α−1Tα(t−s)f(s)dω(s)+∫0t(t−s)α−1Tα(t−s)Buxa(s)ds. Let $$0<|t_2-t_1|<\delta,t_1+\delta<b, \delta>0$$ and from the compactness of $$\{T_{\alpha}(t)$$, $$t>0\}$$, we know that $$T_{\alpha}(t)$$ is continuous in $$t$$ in the uniform operators topology, thus there exist $$\varepsilon>0$$ small enough such that as $$\delta\rightarrow0^{+}$$, for each $$t_{1},t_{2}\in J$$, we have E‖η¯(t2)−η¯(t1)‖H2≤∑i=113Fi =13E‖(Sα(t2)−Sα(t1))[ϕ(0)+G(0,ϕ)]‖H2 +13E‖∫0t1[(t2−s)α−1−(t1−s)α−1]Tα(t2−s)[h(s,xρ(s,xs))+Buxa(s)]ds‖H2 +13E‖∫0t1[(t2−s)α−1−(t1−s)α−1]Tα(t2−s)f(s)dω(s)‖H2 +13E‖∫t1t2(t2−s)(α−1)Tα(t2−s)[h(s,xρ(s,xs))+Buxa(s)]ds‖H2 +13E‖∫t1t2(t2−s)(α−1)Tα(t2−s)f(s)dω(s)‖H2 +13E‖∫0t1−ε(t1−s)(α−1)[Tα(t2−s)−Tα(t1−s)][h(s,xρ(s,xs))+Buxa(s)]ds‖H2 +13E‖∫0t1−ε(t1−s)(α−1)[Tα(t2−s)−Tα(t1−s)]f(s)dω(s)‖H2 +13E‖∫t1−εt1(t1−s)(α−1)[Tα(t2−s)−Tα(t1−s)][h(s,xρ(s,xs))+Buxa(s)]ds‖H2 +13E‖∫t1−εt1(t1−s)(α−1)[Tα(t2−s)−Tα(t1−s)]f(s)dω(s)‖H2 ≤13‖Sα(t2)−Sα(t1)‖2E‖ϕ(0)+G(0,ϕ)‖H2+13(MΓ(α))2∫0t1[(t2−s)α−1−(t1−s)α−1]ds ×∫0t1[(t2−s)α−1−(t1−s)α−1][sups∈Jμ(s)Ω1(r∗)+sups∈Jφ(s)Ω2(r∗)+MB2sups∈JE‖uxa(s))‖H2]ds +13(MΓ(α))2(t2−t1)2αα2[sups∈Jμ(s)Ω1(r∗)+sups∈Jφ(s)Ω2(r∗)+MB2sups∈JE‖uxa(s)‖H2] +13(MΓ(α))2t12αα2sups∈[0,t1−ε]‖Tα(t2−s)−Tα(t1−s)‖2 ×[sups∈Jμ(s)Ω1(r∗)+sups∈Jφ(s)Ω2(r∗)+MB2sups∈JE‖uxa(s))‖H2] +13(MΓ(α))2ε2αα2[sups∈Jμ(s)Ω1(r∗)+sups∈Jφ(s)Ω2(r∗)+MB2sups∈JE‖uxa(s))‖H2]. The compactness of $$\{S_{\alpha}(t)$$, $$ t>0\}$$; $$\{T_{\alpha}(t)$$, $$ t>0\}$$ implies that $$S_{\alpha}(t)$$ and $$T_{\alpha}(t)$$ are continuous in $$t$$ in the uniform operator topology. Thus we can get $$E\|\bar{\eta}(t_2)-\bar{\eta}(t_1)\|_{H}^{2}\rightarrow0,$$ as $$t_2-t_1\rightarrow 0^{+}$$. For $$t_1=0, 0<t_2\leq b,$$ we can easily prove that $$E\|\bar{\eta}(t_2)-\bar{\eta}(0)\|_{H}^{2}\rightarrow0,$$ as $$t_2\rightarrow t_1=0.$$ Hence the set $\{{\it{\Psi}}_{2}(x):x\in B_r(0,\mathcal{H}$$(b)\}$ is equicontinuous. Claim 3. $V(t)=\{\bar{\eta}(t), \bar{\eta}\in{\it{\Psi}}_2 (B_r(0,\mathcal{H}$$(b))\}$ is relatively compact in $$H$$. Let $$0<t\leq b$$ be fixed, for $$\forall \lambda\in (0,t)$$, and $$\forall \delta>0$$, define an operator η¯λ,δ(t)=∫δ∞ϕα(θ)S(tαθ)(ϕ(0)−G(0,ϕ))dθ+∫0t−λ∫δ∞αθ(t−s)α−1ϕα(θ)S((t−s)αθ)(h(s,xρ(s,xs)+Buxa(s))dθds+∫0t−λ∫δ∞αθ(t−s)α−1ϕα(θ)S((t−s)αθ)f(s)dθdω(s)=S(λαδ)∫δ∞ϕα(θ)S(tαθ−λαδ)(ϕ(0)−G(0,ϕ))dθ+S(λαδ)∫0t−λ∫δ∞αθ(t−s)α−1ϕα(θ)S((t−s)αθ−λαδ)×[h(s,xρ(s,xs)+Buxa(s)]dθds+S(λαδ)∫0t−λ∫δ∞αθ(t−s)α−1ϕα(θ)S(tαθ−λαδ)f(s)dθdω(s). From the compactness of $$S(\lambda^{\alpha}\delta) (\lambda^{\alpha}\delta>0)$$, we obtain that the set $V_{\lambda,\delta}(t)=\{\bar{\eta}^{\lambda,\delta}(z)(t),x\in B_r(0,\mathcal{H}$$(b)\}$ is relatively compact in $$H$$ for $$\forall \lambda>0$$ and $$\forall \delta\in(0,t)$$. By calculating, we have E‖η¯λ,δ(t)−η¯(t)‖H2= 7E‖∫0δϕα(θ)S(tαθ)(ϕ(0)−G(0,ϕ))dθ‖H2 +7E‖∫0t∫0δαθ(t−s)α−1ϕα(θ)S((t−s)αθ)h(s,xρ(s,xs))dθds‖H2 +7E‖∫0t∫0δαθ(t−s)α−1ϕα(θ)S((t−s)αθ)f(s)dθdω(s)‖H2 +7E‖∫0t∫0δαθ(t−s)α−1ϕα(θ)S((t−s)αθ)Buxa(s)dθds‖H2 +7E‖∫t−λt∫δ∞αθ(t−s)α−1ϕα(θ)S((t−s)αθ)h(s,xρ(s,xs))dθds‖H2 +7E‖∫t−λt∫δ∞αθ(t−s)α−1ϕα(θ)S((t−s)αθ)f(s)dθdω(s)‖H2 +7E‖∫t−λt∫δ∞αθ(t−s)α−1ϕα(θ)S((t−s)αθ)Buxa(s)dθds‖H2≤7M2E‖ϕ(0)−G(0,ϕ)‖2‖∫0δϕα(θ)dθ‖H2+7α2M2b2αα2(∫0δθϕα(θ)dθ)2sups∈Jμ(s)Ω1(r∗) +7α2M2b2αα2(∫0δθϕα(θ)dθ)2sups∈Jφ(s)Ω2(r∗)+7α2M2b2αα2MB2Mu(∫0δθϕα(θ)dθ)2 +7α2λ2αα2M2sups∈Jμ(s)Ω1(r∗)(1Γ(1+α))2+7α2λ2αα2M2sups∈Jφ(s)Ω2(r∗)(1Γ(1+α))2 +7α2λ2αα2M2MB2(1Γ(1+α))2Mu. Therefore, as $$\lambda\rightarrow0^{+},\delta\rightarrow0^{+}$$, we can verify that the right-hand side of the above inequality tends to zero. Since there are relatively compact sets arbitrarily close to the set $$V(t)=\{\bar{\eta}(t), \bar{\eta}\in {\it{\Psi}}_2(B_r(0,\mathcal{H}(b))\}$$, hence, $V(t)=\{\bar{\eta}(t), \bar{\eta}\in{\it{\Psi}}_2( B_r(0,\mathcal{H}$$(b))\}$ is relatively compact in $$H.$$ From Steps 2 and 4, we know that $${\it{\Psi}}_2$$ is a completely continuous multi-valued map with compact convex values. Step 5. $${\it{\Psi}}_2$$ has a closed graph. Let $$x^{(n)}\rightarrow x^{*}(n\rightarrow\infty), \bar{\eta}^{(n)}\in{\it{\Psi}}_2(x^{(n)}), x^{(n)}\in B_r(0,\mathcal{H}(b))$$ and $$\bar{\eta}^{(n)}\rightarrow\bar{\eta}^{*} (n\rightarrow\infty).$$ We will prove that $$\bar{\eta}^*\in{\it{\Psi}}_2(x^*),$$ since $$\bar{\eta}^{(n)}\in {\it{\Psi}}_2(x^{(n)}),$$ there exists $$f^{(n)}\in S_{F,x^{(n)}_{\rho}}$$ such that for each $$t\in J$$, η¯(n)(t)=Sα(t)[ϕ(0)−G(0,ϕ)]+∫0t(t−s)α−1Tα(t−s)f(n)(s)dω(s)+∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs(n)))dω(s)+∫0t(t−s)α−1Tα(t−s)B{B∗Tα∗(b−s)R(a,Γ0b)×[Ex¯b+∫0bφ(r)dr−G(b,x(n)(b))−Sα(b)[ϕ(0)−G(0,ϕ)]]−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1ATα(b−τ)G(τ,xτ(n))dτ−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)h(τ,xρ(τ,xτ(n))(n))dτ−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)f(n)(τ)dω(τ)}ds. Now, we investigate the convergence of the sequences $$\{x^{(n)}_{\rho(s,x_s^{(n)})}\}_{n\in N}, s\in J$$. We assume $$x^{(n)}\rightarrow x^{*}$$ in $$H$$. If $$s\in J$$, such that $$\rho(s,x_s)>0$$, then we have ‖xρ(s,xs(n))(n)−xρ(s,xs∗)∗‖B2≤2‖xρ(s,xs(n))(n)−xρ(s,xs(n))∗‖B2+2‖xρ(s,xs(n))∗−xρ(s,xs∗)∗‖B2≤2Kb2E‖x(n)−x∗‖H2+2‖xρ(s,xs(n))∗−xρ(s,xs∗)∗‖B2, which proves that $$x^{(n)}_{\rho(s,x_s^{(n)})}\rightarrow x^{*}_{\rho(s,x_s^{*})}$$ in $$\mathcal{B}$$ as $$n\rightarrow\infty.$$ If $$s\in J$$, such that $$\rho(s,x_s)<0$$, ‖xρ(s,xs(n))(n)−xρ(s,xs∗)∗‖B2=‖ϕρ(s,ρs(n))(n)−ϕρ(s,ρs∗)∗‖B2=0. Combining the pervious arguments, we can prove that $$x^{(n)}_{\rho(s,x_s^{(n)})}\rightarrow\phi$$ for every $$s\in J$$ such that $$\rho(s,x_s^{(n)})=0$$. Thus $$x^{(n)}_{\rho(s,x_s^{(n)})}\rightarrow x^{*}_{\rho(s,x_s^{*})}$$ in $$\mathcal{B}$$ as $$n\rightarrow\infty$$ for each $$s\in J$$. We have to prove that there exists $$f^*\in S_{F,x^{*}_{\rho}}$$, such that for each $$t\in J$$, η¯∗(t)=Sα(t)[ϕ(0)−G(0,ϕ)]+∫0t(t−s)α−1Tα(t−s)f∗(s)dω(s)+∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs∗))dω(s)+∫0t(t−s)α−1Tα(t−s)B{B∗Tα∗(b−s)R(a,Γ0b)×[Ex¯b+∫0bφ(r)dr−G(b,x(b))−Sα(b)[ϕ(0)−G(0,ϕ)]]−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1ATα(b−τ)G(τ,xτ∗)dτ−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)h(τ,xρ(τ,xτ∗)∗)dτ−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)f∗(τ)dω(τ)}ds. Since $$\bar{\eta}^{(n)}\rightarrow\bar{\eta}^*, (n\rightarrow\infty)$$, we can get E‖η¯(n)(t)−Sα(t)[ϕ(0)−G(0,ϕ)]−∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs(n))(n))ds −∫0t(t−s)α−1Tα(t−s)B{B∗Tα∗(b−s)R(a,Γ0b)[Ex¯b+∫0bφ(r)dr −Sα(b)[ϕ(0)−G(0,ϕ)]−G(b,xb(n))] −B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1ATα(b−τ)G(τ,xτ(n))dτ −B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)h(τ,xρ(τ,xτ(n))(n))dτ}ds −(η¯∗(t)−Sα(t)[ϕ(0)−G(0,ϕ)]−∫0t(t−s)α−1Tα(t−s)h(s,xρ(s,xs∗)(n))ds −∫0t(t−s)α−1(b−s)α−1Tα(t−s)B{B∗Tα∗(b−s)R(a,Γ0b)[Ex¯b +∫0bφ(r)dr−Sα(b)[ϕ(0)−G(0,ϕ)]−G(b,xb∗)] −B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1ATα(b−τ)G(τ,xτ∗)dτ −B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)h(τ,xρ(τ,xτ∗)∗)dτ}ds)‖H2→0. Consider the linear continuous operator Γ:L2(J,H)→C(J,H), (Γf)(t)=∫0t(t−s)α−1Tα(t−s)f(s)dω(s)−∫0t(t−s)α−1Tα(t−s)BB∗Tα∗(b−s)×(∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)f(τ)dω(τ))ds. From Lemma 2.7, we know that $$\it{\Gamma}\circ S_{F}$$ is a closed graph operator. Moreover, we have η¯(n)(t)−Sα(t)[ϕ(0)−G(0,ϕ)]−∫0t(t−s)α−1Tα(t−s)B{B∗Tα∗(b−s)×R(a,Γ0b)[Ex¯b+∫0bφ(r)dr−Sα(b)[ϕ(0)−G(0,ϕ)]−G(b,xb(n))]−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1ATα(b−τ)G(τ,xτ(n))dτ−B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)h(τ,xρ(τ,xτ(n))(n))dω(τ)}ds∈Γ(SF,xρ(n)). Since $$x^{(n)}\rightarrow x^*$$, from Lemma 2.7, it follows that η¯∗(t)−Sα(t)[ϕ(0)−G(0,ϕ)]−∫0t(t−s)α−1Tα(t−s)B{B∗Tα∗(b−s) ×R(a,Γ0b)[Ex¯b+∫0bφ(r)dr−Sα(b)[ϕ(0)−G(0,ϕ)]−G(b,xb∗)] −B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1ATα(b−τ)G(τ,xτ∗)dτ −B∗Tα∗(b−s)∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)h(τ,xρ(τ,xτ∗)∗)dτ}ds=∫0t(t−s)α−1Tα(t−s)f∗(s)dω(s)−∫0t(t−s)α−1Tα(t−s)BB∗Tα∗(b−s) ×(∫0bR(a,Γτb)(b−τ)α−1Tα(b−τ)f∗(τ)dω(τ))ds. for some $$f^*\in S_{F,x^*_{\rho}}$$, this show that $$\bar{\eta}^*\in{\it{\Psi}}_2(x^*)$$. Hence $${\it{\Psi}}_2$$ has a closed graph and Steps 1-5 are complete. Since $${\it{\Psi}}_2$$ is a completely continuous multi-valued map with compact, convex value, from the Proposition 2.1 (4), we get that $${\it{\Psi}}_2$$ is u.s.c. On the other hand, $${\it{\Psi}}_1$$ is proved a contraction operator and hence $${\it{\Psi}}={\it{\Psi}}_1+{\it{\Psi}}_2$$ is u.s.c. and condensing. Thus from Lemma 2.8, we know that operator $${\it{\Psi}}$$ has a fixed point in $$B(r,\mathcal{H}(b))$$, which is the mild solution of the system (1.1). The proof is complete. 4. Approximate controllability Consider the stochastic linear system {cDαx(t)∈Ax(t)+(Bu)(t)+σ(t)dω(t)dt, t∈[0,b],x0=ϕ∈B, (4.1) and the deterministic linear system {cDαx(t)∈Ax(t)+(Bv)(t), t∈[0,b],x0=ϕ∈B, (4.2) where $$B:U\rightarrow H$$ is a linear bounded operator, $$u\in L_{2}^{\mathcal{F}}([0,b],U), v\in L_{2}([0,b],U)$$. Definition 4.1 Let $$x_{b}(\phi,u)$$ be the state value of system at terminal time $$b$$ and the corresponding control is $$u,$$ and the initial value is $$\phi$$. Introduce the set R(b,ϕ)={xb(ϕ,u)(0):u(⋅)∈L2(J,u)}, which is called the reachable set of system (1.1) at terminal time $$b$$ and its closure in $$H$$ is denoted by $$\overline{{R}(b,\phi)}$$. The system (1.1) is said to be approximately controllable on the interval $$J$$ if the closure of the reachable set $$\overline{{R}(b,\phi)}=H$$, that is, given an arbitrary $$\varepsilon>0$$, it is possible to steer from the point $$\phi(0)$$ to within a distance $$\varepsilon$$ from all points in the state space $$H$$ at time $$b$$. Lemma 4.1 (Mahmudov, 2014) The control system (4.1) is approximately controllable in $$[s,b]$$ if and only if one of the following conditions holds: (a) $$~{\it{\Gamma}}_{s}^{b}>0$$; (b) $$~a(a I+{\it{\Gamma}}_{s}^{b})^{-1}$$ converges to the zero operator as $$a\rightarrow0^{+}$$ in the strong operator topology; (c) $$~a(a I+{\it{\Gamma}}_{s}^{b})^{-1}$$ converges to the zero operator as $$a\rightarrow0^{+}$$ in the weak operator topology. Remark 4.1 Assume that the linear fractional control system is approximate controllable. We introduce the operators associated with (4.2) as $${\it{\Gamma}}_{0}^{b}=\int_{0}^{b}(b-s)^{(\alpha-1)}T_{\alpha}(b-s)BB^{*}T_{\alpha}^{*}(b-s)ds$$,$$~R(a,{\it{\Gamma}}_{0}^{b})=(a I+{\it{\Gamma}}_{0}^{b})^{-1}$$ for $$a>0.$$ We recall that the linear fractional control system (4.2) is approximately controllable on $$J$$ if and only if $$a (a I+{\it{\Gamma}}_{0}^{b})^{-1}\rightarrow0$$ as $$a\rightarrow0^{+}$$ in the strong operator topology. (see Sakthivel et al., 2013a) Lemma 4.2 The stochastic system (4.1) is approximately controllable in $$[0,b]$$ if and only if the deterministic system (4.2) is approximately controllable in every $$[s,b]$$, $$0\leq s<b$$. Lemma 4.3 The solution of (1.1) corresponding to the $$u_x^a(t,x)$$ satisfies the following identity: x(a)(b)=x¯b−a(aI+Γ0b)−1[Ex¯b+∫0bφ~(r)dr−Sα(b)[ϕ(0)−G(0,ϕ)]−G(b,xb(a))]+a∫0b(aI+Γrb)−1(b−r)α−1ATα(b−r)G(r,xr(a))dr+a∫0b(aI+Γrb)−1(b−r)α−1Tα(b−r)h(r,xρ(r,xr(a))(a))dr+a∫0b(aI+Γrb)−1(b−r)α−1Tα(b−r)f(a)(r)dω(r), where $$f^{(a)}\in S_{F,x^{(a)}_{\rho}}$$. Proof. By using the stochastic Fubini theorem, we can deduce the identity. Here we omit it. □ Theorem 4.1 Assume that assumptions of Theorem 3.1 are satisfied and in addition, the sequences $$\{h(s,x^{(a)}_{\rho(s,x^{(a)}_s)})\},~\{F(s,x^{(a)}_{\rho(s,x_s)})\}$$ and $$\{A^{\beta}G(s,x^{(a)}_{\rho(s,x_s)})\}$$ are all uniformly bounded for each $$s\in J,$$ and the linear system (4.1) is approximately controllable, then the stochastic system (1.1) is approximately controllable on $$J$$. Proof. Under the above assumptions, we know there exists a mild solution corresponding to the control $$u_{x}^a(t,x).$$ Let $$x^{(a)}$$ be a fixed point of $${\it{\Psi}}$$. By our assumption, the sequences $$\{h(s,x^{(a)}_{\rho(s,x^{(a)}_s)})\}$$, $$\{A^{\beta}G(s,x^{(a)}_s)\}$$, $$\{f^{(a)}(s)\}$$ are all uniformly bounded, where $$\{f^{(a)}(s)\}\in S_{F,x^{(a)}_{\rho}}$$. So there exist subsequences still denoted by $$\{h(s,x^{(a)}_{\rho(s,x^{(a)}_s)})\}$$, $$\{A^{\beta}G(s,x^{(a)}_s)\}$$, $$\{f^{(a)}(s)\}$$ converging to, say, $$\{h(s)\}$$, $$\{A^{\beta}G(s)\}$$ in $$H$$ and $$\{f^{*}(s)\}$$ in $$L(K,H)$$, respectively. The compactness of $$\{T_{\alpha}(t),t>0\}$$ implies that {Tα(b−r)f(a)(r)→Tα(b−r)f∗(r),Tα(b−r)h(r,xρ(r,xr(a))(a))→Tα(b−r)h(r),Tα(b−r)AβG(r,xr(a))→Tα(b−r)AβG(r). (4.3) Moreover, by the assumption that system (4.1) is approximately controllable and Lemmas 4.1 and 4.2, we know that for all $0\leq s<b,$$~a(a I+{\it{\Gamma}}_{s}^{b})^{-1}\rightarrow0,$ strongly as $$a\rightarrow0^{+}$$. In addition, $$\|a(a I+{\it{\Gamma}}_{s}^{b})^{-1}\|\leq1$$, for $$\forall~s\in[0,b)$$. Then from the Lebesgue dominated convergence theorem, it follows that as $$a\rightarrow0^{+}$$, E‖x(a)(b)−x¯b‖H2 ≤8E‖a(aI+Γ0b)−1[Ex¯b+∫0bφ~(r)dr−Sα(b)[ϕ(0)−G(0,ϕ)]−G(b,xb(a))]‖2 +8‖A−β‖2‖a(aI+Γrb)−1‖2E[∫0b(b−r)(α−1)Tα(b−r)‖[AβG(r,xr(a))−AβG(r)]‖dr]2 +8‖A−β‖2‖a(aI+Γrb)−1‖2E[∫0b(b−r)(α−1)‖Tα(b−r)AβG(r)‖dr]2 +8‖a(aI+Γrb)−1‖2E[∫0b(b−r)(α−1)‖Tα(b−r)[h(r,xρ(r,xr(a))(a))−h(r)]‖dr]2 +8‖a(aI+Γrb)−1‖2E[∫0b(b−r)(α−1)‖Tα(b−r)h(r)‖dr]2 +8‖a(aI+Γrb)−1‖2E[∫0b(b−r)(α−1)‖Tα(b−r)[f(a)(r)−f∗(r)]‖dω(r)]2 +8‖a(aI+Γrb)−1‖2E[∫0b(b−r)(α−1)‖Tα(b−r)f∗(r)‖dω(r)]2→0. This proves the approximate controllability of system (1.1). The proof is complete. □ 5. An example As an application of our results, we consider the following control system governed by the fractional order neutral stochastic differential inclusion with state-dependent delay of the form {cD34(w(t,z)−∫−∞t∫0πb(t−s,η,z)w(s,z)dηds)∈wzz(t,z)+μ(t,z)+∫−∞th¯(t,t−s,z,w(s−ρ1(t)ρ2(‖w(t)‖),z)ds+∫−∞tF¯(t,t−s,z,w(s−ρ1(t)ρ2(‖w(t)‖),z)dsdβ(t)dt, 0≤z≤π, t∈J=[0,b],w(t,0)=w(t,π)=0, t∈[0,b],w(θ,z)=ϕ(θ,z), θ≤0, z∈[0,π], (5.1) where $$b>0$$, $$\beta(t)$$ is a two-sided and standard one-dimensional Brownian motion defined on the filtered probability space $$({\it{\Omega}},\mathcal{F},P)$$. We choose the space $$H=U=L^{2}([0,\pi])$$ with the norm $$\|\cdot\|$$ and define an operator $$A$$ by $$Av=v^{''}$$ with the domain $$D(A)=\{v\in H:v,v^{'}$$ absolutely continuous, $$v^{''}\in H,v(0)=v(\pi)=0\}$$, then $$A$$ generates a strongly continuous semigroup $$\{S(t)_{t\geq0}\}$$ which is compact, analytic and self-adjoint. Furthermore, $$A$$ has a discrete spectrum, the eigenvalues are $$-n^{2},n\in \mathbb{N}^{+}$$, with corresponding orthogonal eigenvectors $$e_{n}(z)=\sqrt\frac{2}{\pi} \sin(nz)$$. We also need the following properties: (1) for each $$v\in H,S(t)v=\sum_{n=1}^{\infty}e^{-n^{2}t}\langle v,e_{n}\rangle e_{n},$$ in particular, $$S(\cdot)$$ is a uniformly stable semigroup and $$\|S(t)\|\leq e^{-t}$$; (2) for each $$v\in H,~A^{-\frac{1}{2}}v=\Sigma_{n=1}^{\infty}\frac{1}{n}\langle v,v_n\rangle v_n$$ and $$\|A^{-\frac{1}{2}}\|=1$$; (3) the operator $$A^{\frac{1}{2}}$$ is given by $$A^{\frac{1}{2}}v=\Sigma_{n=1}^{\infty}n\langle v,v_n\rangle v_n$$ defined on the space $$D(A^{\frac{1}{2}})=\{v(\cdot)\in H,\Sigma_{n=1}^{\infty}n\langle v,v_n\rangle v_n\in H\}$$. To study this system, we impose the following conditions: (i) functions $$\rho_i:[0,\infty)\rightarrow[0,\infty),~i=1,2$$ are continuous; (ii) functions $$b(s,\eta,z),\frac{\partial b(s,\eta,z)}{\partial z}$$ are measurable, $$b(s,\eta,\pi)=b(s,\eta,0)=0$$ with LG=max{(∫0π∫−∞0∫0π1g(s)(∂ib(s,η,z)∂zi)2)dηdsdz)12:i=0,1}<∞; (iii) functions $$\bar{h}: \mathbb{R}^4\rightarrow \mathbb{R}$$, $$\bar{F}:\mathbb{R}^4\rightarrow \mathbb{R}$$ are continuous and there are continuous functions $$c:\mathbb{R}\rightarrow \mathbb{R},~d:\mathbb{R}\rightarrow \mathbb{R}$$ such that ‖h¯(t,s,u,y)‖≤c(s)‖y‖, ‖F¯(t,s,u,y)‖≤d(s)‖y‖, (t,s,u,y)∈R4, with Lh¯=(∫−∞0(c(s))2g(s)ds)12<∞, LF¯=(∫−∞0(d(s))2g(s)ds)12<∞. Let $$G:J\times \mathcal{B}\rightarrow H,~F:J\times\mathcal{B} \rightarrow\mathcal{P}(H)$$ be the operators respectively defined by x(t)(z)=w(t,z), G(t,ψ)(z)=∫−∞t∫0πb(t−s,η,z)ψ(s,η)dηds,h(t,ψ)(z)=∫−∞0h¯(t,s,z,ψ(s,z)ds, F(t,ψ)(z)=∫−∞0F¯(t,s,z,ψ(s,z)dsdβ(t)dt,ρ(s,φ)=ρ1(s)ρ(‖φ(0)‖)), where $$b,\bar{h},\bar{F}$$ are continuous functions. $$\mathcal{B}=C_0\times L^2(g,H)$$ is the phase space introduced in Hale & Kato (1978). Hence, from the discussion above, we know the operators $$F,h$$ are bounded with $$E\|F\|^2\leq L_{\bar{F}}^2,~E\|h\|^2\leq L_{\bar{h}}^2$$. A straightforward estimation involving (ii) enables us to prove that $$G$$ is $$D(A^{\frac{1}{2}})-$$valued with $$E\|A^{\frac{1}{2}}G\|^2\leq L_G^2$$. Thus, $$F,h,A^{\frac{1}{2}}G$$ are all bounded. Define the bounded linear operator $$B:U \rightarrow H$$ by $$Bu(t)(z) =\mu(t,z),~0\leq z\leq\pi,~u\in U$$ where $$\mu:[0,b]\times[0,\pi]\rightarrow H$$ is continuous. 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