Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces

Approximate controllability of impulsive fractional integro-differential equation with... Abstract In this paper, the problem of approximate controllability for non-linear impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces is investigated. We study the approximate controllability for non-linear impulsive integro-differential systems under the assumption that the corresponding linear control system is approximately controllable. By utilizing the methods of fractional calculus, semigroup theory, fixed-point theorem coupled with solution operator, sufficient conditions are formulated and proved. Finally, an example is provided to illustrate the proposed theory. 1 Introduction Fractional calculus has emerged as an interesting area of research in view of growing applications of its modelling techniques in various fields such as fluid flow, viscoelasticity, control theory of dynamical systems, electrical networks, probability and statistics, dynamical processes in porous structures, electrochemistry of corrosion, optics and signal processing, rheology, hydraulics of dams, potential fields, diffusion problems and waves in liquids and gases etc. Fractional-order operators provide a perfect tool for the description of memory and hereditary properties of various materials and processes, in contrast to their integer-order counterparts. Nowadays, irrational and complex orders also appear in certain studies besides rational orders. The recent works on the topic include the improvement of theoretical aspects and analytic/numerical methods for solving fractional-order differential equations appearing in the mathematical models of applied and scientific problems (for instance, see the studies by Podlubny, 1999; Baleanu et al., 2012; Hilfer, 2000; Kilbas et al., 2006; Lakshmikantham et al., 2009; Miller & Ross, 1993; Tarasov, 2010; Zhou, 2014, 2016, and a series of papers by Zhou et al., 2013a,b, 2015; Zhou & Peng, 2017a,b,c; Zhou & Zhang, 2017, and the references cited therein). However, the development of control theory in the perspective of fractional calculus is at its initial stage and needs to be explored further. Functional differential equations with state-dependent delay are found to be of great interest as such equations appear in the mathematical models associated with real world problems. In particular, the existence and approximate controllability results for mild solutions of certain problems have attracted considerable attention and a great deal of work is underway. For some recent works on fractional-order problems with state-dependent delay, for example, see Agarwal & Andrade (2011), Benchohra & Berhoun (2016), Benchohra et al.,(2012, 2013), Aissani & Benchohra (2014), Kavitha et al. (2012), Carvalho dos Santos et al. (2016), Sakthivel & Yong (2013), Vijayakumar et al. (2013), Yan (2012) and references cited therein. The theory of impulsive differential equation has also become an important area of investigation in recent years. It has been stimulated by numerous applications of these equations in mechanics, electrical engineering, medicine, biology, ecology etc. For further details and examples, we refer the reader to the books by Stamova (2009), Graef et al. (2013), Bainov & Covachev (1995), Benchohra et al. (2006) and the papers by Chang et al. (2008) and Hernandez et al. (2010) and the references cited therein. The concept of controllability is of enormous influence in mathematical control theory and engineering because they have closely related to pole assignment, structural decomposition, observer design etc. In the case of infinite dimensional systems, two basic concept of controllability can be identified which are exact controllability and approximate controllability. Exact controllability enables to steer the system to arbitrary final state while approximate controllability means that system can be steered to arbitrary small neighborhood of final state. The approximate controllability is essentially weaker notion than exact controllability and it gives the possibility of steering the system to states which form the dense subspace in the state space. However, in the case of infinite dimensional systems exact controllability appears rather exceptionally but in the case of finite dimensional systems notions of exact and approximate controllability coincide. The recent work on controllability for different types of fractional differential and integro-differential systems has generated a great deal of interest among scientists. Sakthivel et al. (2013), Ganesh et al. (2013, 2014), Liu et al. (2014) and Guendouzi & Bousmaha (2014) investigated the approximate controllability results for various types of fractional differential equations and inclusions. Wang & Zhou (2011) obtained the existence and controllability results for fractional differential inclusions in Banach spaces under suitable fixed point theorem. Debbouche & Baleanu (2011) discussed the exact controllability for fractional evolution non-local impulsive quasilinear delay integro-differential systems in Banach spaces. Recently, Vijayakumar et al. (2013) and Yan (2012) studied the approximate controllability for fractional-order partial neutral functional integro-differential inclusions with state-dependent delay in Hilbert spaces by applying different types of fixed-point theorem. Ravichandran & Trujillo (2013) addressed the controllability of impulsive fractional integro-differential systems with finite delay with the help of fractional calculus, semigroup concept and fixed-point techniques. In the study of fractional differential systems in the infinite dimensional space, the primary step is to develop a method for finding the mild solution. In the study by Wang et al. (2015), the authors presented and discussed an appropriate idea for constructing the PC-mild solutions for the considered problem. We emphasize that the contraction mapping principle cannot be applied directly to a kind of equations with delay, especially state-dependent delay, due to the presence of not same subscripts in the state-dependent delay terms, which fail to survive upon the application of Lipschitz condition. For example, consider the simple function $$f(t,x_{\varrho (t,x_{t})})$$, where f and ϱ are same as in our main problem (1.1). In case the function f(⋅, ⋅) satisfies the Lipschitz condition with respect to the second variable, it is usually written in the literature as follows: $$ \left\|\,f\left(t,x_{\varrho(t,x_{t})}\right)-f\left(t,y_{\varrho(t,y_{t})}\right)\right\|\le \nu\left\|x_{\varrho(t,x_{t})}-y_{\varrho(t,y_{t})}\right\|, $$ which is impossible as the subscripts are the same. In consequence, the contraction mapping principle is not applicable in this situation. In this paper, we present new techniques to utilize the contraction mapping principle for the problems involving state-dependent delay. This is the main motivation and contribution of this paper. Inspired by the works Wang et al. (2015), Aissani & Benchohra (2014), Ganesh et al. (2014), Sakthivel & Yong (2013), and Andrade et al. (2016), our purpose here is to establish the approximate controllability of mild solutions for the following model problem involving an impulsive fractional integro-differential equation with state-dependent delay: \begin{align} ^{C}D_{t}^{\alpha} x(t)&={\mathscr{A}} x(t)+\mathbb{J}_{t}^{1-\alpha}\left[\mathcal{C}u(t)+f\left(t,x_{\varrho(t,x_{t})}, (\mathscr{G}x)(t), (\mathscr{H}x)(t)\right)\right],\nonumber\\ &\qquad\text{a.e. on}\, {\mathscr{J}}-\{t_{1},t_{2},\cdots,t_{m}\}, \end{align} (1.1) \begin{equation} {\hskip-94pt}\Delta x(t_{k})={\mathcal{I}}_{k}\left(x\left(t_{k}^{-}\right)\right),\quad k=1,2,\cdots,m, \end{equation} (1.2) \begin{equation} {\hskip-133pt}x(t)=\varsigma(t),\quad \varsigma(t)\in{{\mathscr{B}}_h}, \end{equation} (1.3) where $${\mathscr{J}}=[0,T]$$ with T > 0, $$^{C}D_{t}^{\alpha }$$ denote the Caputo fractional derivative of order α ∈ (0, 1), $$ {\mathscr{A}}: D({\mathscr{A}})\subset \mathbb{X}\rightarrow \mathbb{X}$$ is an infinitesimal generator of the solution operator $$ \{\mathbb{S}_{\alpha }(t)\}_{t\ge 0}$$ defined on a Hilbert space $$\mathbb{X}$$ having its norm defined as $$\|\cdot \|_{\mathbb{X}}; D({\mathscr{A}})$$ represents the domain of $${\mathscr{A}}$$; $$\mathcal{C}$$ is a bounded linear operator from U into $$\mathbb{X}$$; the control function $$ u(\cdot )\in L^{2}({\mathscr{J}},U)$$, a Hilbert space of admissible control functions. Further, $$ f: {\mathscr{J}}\times{{\mathscr{B}}_h}\times \mathbb{X}\times \mathbb{X}\rightarrow \mathbb{X}$$ represents a class of $$\mathbb{X}$$-valued functions, $${\mathscr{G}}$$ and $${\mathscr{H}}$$ are defined as $$ ({\mathscr{G}} x)(t)={\int_{0}^{t}}e_{1}\left(t,s,x_{\varrho(s,x_{s})}\right) \textrm{d}s, \quad ({\mathscr{H}} x)(t)={\int_{0}^{T}}e_{2}\left(t,s,x_{\varrho(s,x_{s})}\right) \textrm{d}s, $$ with $$ e_{1}, e_{2}:\mathscr{D}\times{{\mathscr{B}}_h}\rightarrow \mathbb{X}, \varrho :{\mathscr{J}}\times{{\mathscr{B}}_h}\rightarrow{\mathbb{R}}$$ being appropriate functions. Here $$ \mathscr{D}=\{ (t,s)\in{\mathscr{J}}\times{\mathscr{J}}: 0\le s\le t\le b\}$$ and $${\mathcal{I}}_{k}: \mathbb{X} \rightarrow \mathbb{X} \, (k=1,2,\cdots ,m)$$ are bounded functions. Also, the fixed times tk satisfy $$ 0=t_{0}<t_{1}<t_{2}<\cdots <t_{m}<t_{m+1}=T, \Delta x(t_{k})=x(t_{k}^{+})-x(t_{k}^{-}),$$ and $$ x(t_{k}^{+})=\lim _{h\rightarrow 0}x(t_{k}+h), x(t_{k}^{-})=\lim _{h\rightarrow 0}x(t_{k}-h)$$ denote the right and left limits of x at the points tk, respectively. We assume that the histories $$x_{t}:(-\infty ,0]\rightarrow{\mathbb{X}}, x_{t}(\theta )=x(t+\theta )$$ for θ ≤ 0, belongs to an abstract phase space $${{\mathscr{B}}_h}$$ defined axiomatically. To the best of our knowledge, the approximate controllability of fractional integro-differential systems with state-dependent delay and form (1.1)–(1.3) is an untreated topic in the literature and this fact is the main motivation of the present work. The rest of the manuscript is organized as follows. In Section 2, we recall some preliminary concepts, definitions and lemmas that we need in the sequel. The approximate controllability of mild solutions for the model problem (1.1)–(1.3) is discussed via Banach fixed point theorem in Section 3. In Section 4, an example is given to illustrate our main results. 2 Preliminaries In this section, we recall some definitions, lemmas and preparatory facts from functional analysis, solution operator and fractional calculus, which will be used throughout this paper without any further mention. Let $$L(\mathbb{X})$$ denote the Hilbert space of all bounded linear operators from $$ \mathbb{X}$$ into $$\mathbb{X}$$, equipped with the norm $$\|\cdot \|_{L(\mathbb{X})}$$ and $$C({\mathscr{J}},\mathbb{X})$$ be the space of all continuous functions from $$ {\mathscr{J}}$$ into $$\mathbb{X}$$ with norm defined as $$ \|\cdot \|_{C({\mathscr{J}},\mathbb{X})}$$. Notice that we consider the theoretical phase space $${{\mathscr{B}}_h}$$ when the delay is infinite. In this case, we work in the phase spaces $${{\mathscr{B}}_h}$$ as described in the study of Dabas & Chauhan (2013). Consider the space $${\mathscr{B}_T}=\{ x: (-\infty ,T]\rightarrow \mathbb{X}$$ be such that $$x|_{{\mathscr{J}}_{k}}\in C({\mathscr{J}}_{k},\mathbb{X})$$, and we can find $$x(t_{k}^{+})$$ and $$x(t_{k}^{-})$$ with $$ x(t_{k})=x(t_{k}^{-}), \quad x_{0}=\varsigma \in{{\mathscr{B}}_h},\quad k=0,1,2,\dots ,m \}, $$ where $$ x|_{{\mathscr{J}}_{k}}$$ is the constraint of x to $${\mathscr{J}}_{k}=(t_{k},t_{k+1}],\ k=0,1,2,\dots ,m$$. The seminorm $$\|\cdot \|_{{\mathscr{B}_T}}$$ in $${\mathscr{B}_T}$$ is defined by the following: $$ \|x\|_{{\mathscr{B}_T}}=\|\varsigma\|_{{{\mathscr{B}}_h}}+\sup\left\{|x(s)|:s\in [0,T]\right\}, \quad x\in{\mathscr{B}_T}. $$ We assume that the phase space $$({{\mathscr{B}}_h},\|\cdot \|_{{{\mathscr{B}}_h}})$$ is a seminormed linear space of functions from $$(-\infty ,0]$$ into $$\mathbb{X}$$, and satisfies the following basic axioms as suggested in the result in the study by Hale & Kato (1978) (see also Hino et al., 1991). For T > 0, let $$x:(-\infty ,T]\rightarrow \mathbb{X}$$ be a function such that $$ x_{0}=\varsigma \in{{\mathscr{B}}_h}, x\in{\mathscr{B}_T} $$. Then, for every $$t\in{\mathscr{J}}_{k}$$, the following conditions hold: (P1) xt is in $${{\mathscr{B}}_h}$$; (P2) $$ \|x(t)\|_{\mathbb{X}}\le H\|x_{t}\|_{{{\mathscr{B}}_h}};$$ (P3) $$\|x_{t}\|_{{{\mathscr{B}}_h}}\le{\mathscr{D}}_{1}(t)\sup \{\|x(s)\|:0\le s\le t\}+{\mathscr{D}}_{2}(t)\|x_{0}\|_{{{\mathscr{B}}_h}}$$. Here, H is a positive constant, $$ {\mathscr{D}}_{1}(\cdot ):[0,+\infty )\rightarrow [0,+\infty )$$ is continuous, $${\mathscr{D}}_{2}(\cdot ):[0,+\infty )\rightarrow [0,+\infty )$$ is locally bounded and $${\mathscr{D}}_{1}, {\mathscr{D}}_{2}$$ are independent of x(⋅). Next, we present some fundamental definitions and results from fractional calculus. Definition 2.1 (Kilbas et al., 2006) For $$ n-1<\alpha <n,\ n\in \mathbb{N}$$, the Caputo fractional derivative of order α for a function $$ f:[0,+\infty )\rightarrow \mathbb{R}$$ is defined as $$ D_{t}^{\alpha} f(t)=\frac{1}{\Gamma(n-\alpha)}{\int_{0}^{t}}(t-s)^{n-\alpha-1}f^{(n)}(s) \textrm{d}s=\mathbb{J}^{n-\alpha}f^{(n)}(t). $$ If 0 < α < 1, then $$ D_{t}^{\alpha} f(t)=\frac{1}{\Gamma(1-\alpha)}{\int_{0}^{t}}(t-s)^{-\alpha}f^{\prime}(s) \textrm{d}s. $$ Obviously, Caputo’s derivative of a non-zero constant is zero. Definition 2.2 (Dabas & Chauhan, 2013) A two-parameter function of the Mittag–Leffler type is defined by the series representation: $$ E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty} \frac{z^{k}}{\Gamma(\alpha k+\beta)}=\frac{1}{2\pi i}\int_{C} \frac{\mu^{\alpha-\beta}e^{\mu}}{\mu^{\alpha}-z} \;\text{d}\mu,\, \alpha,\beta>0,\, z\in\mathbb{C}, $$ where $$\mathbb{C}$$ is a contour which starts and ends at $$-\infty $$ and encircles the disc $$ \|\mu \|\le |z|^{\frac{1}{\alpha }}$$ counter clockwise. A useful property of the Mittag–Leffler functions involving Laplace integral is $$ \int_{0}^{\infty}e^{-\lambda t}t^{\beta-1}E_{\alpha,\beta}(\omega t^{\alpha})\; \textrm{d}t=\frac{\lambda^{\alpha-\beta}}{\lambda^{\alpha}-\omega},\, \textrm{Re}\ \lambda>\omega^{\frac{1}{\alpha}},\, \omega>0. $$ Remark 2.1 (i) The details about the solution operator and its results are quite standard. Hence we omit these details and we suggest the interested reader to refer to the studies by Wang et al. (2015) and Dabas & Chauhan (2013). (ii) To determine the mild solutions for the problem (1.1)–(1.3), we consider the following Cauchy problem: \begin{cases} {}^{C}D^{\alpha} x(t)={\mathscr{A}} x(t)+\mathbb{J}_{t}^{1-\alpha}f(t),\qquad t\in{\mathscr{J}},\\ x(0)=x_{0}\in\mathbb{X}. \end{cases} As argued in the study by Dabas & Chauhan, (2013), the mild solution of the above Cauchy problem is $$ x(t)= \mathbb{S}_{\alpha}(t)x_{0}+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s)f(s)\; \textrm{d}s, $$ where $$\mathbb{S}_{\alpha }(t)$$ is the solution operator generated by $${\mathscr{A}}$$ and is given by $$ \mathbb{S}_{\alpha}(t)=E_{\alpha,1}({\mathscr{A}} t^{\alpha})=\frac{1}{2\pi i}\int_{\widehat{B}_{r}}e^{\lambda t}\frac{\lambda^{\alpha-1}}{\lambda^{\alpha}-{\mathscr{A}}} \textrm{d}\lambda, $$ where $$ \widehat{B}_{r}$$ denotes the Bromwich path and $$f:{\mathscr{J}}\rightarrow \mathbb{X}$$ is continuous. Let xT(ς;u) be the state value of the problem (1.1)–(1.3) at terminal time T corresponding to the control u and the initial value $$ \varsigma \in{{\mathscr{B}}_h}$$. The set $$\mathscr{R}(T,\varsigma )=\{x_{T}(\varsigma ;u)(0): u(\cdot )\in L^{2}({\mathscr{J}},U)\}$$ is known as the reachable set of the problem (1.1)–(1.3) at terminal time T and its closure in $$\mathbb{X}$$ is denoted by $$\overline{\mathscr{R}(T,\varsigma )}$$. Definition 2.3 The fractional control system (1.1)–(1.3) is said to be approximately controllable on $${\mathscr{J}}$$ if the reachable set $${\mathscr{R}(T,\varsigma )}$$ is dense in $$\mathbb{X}$$, that is, $$\overline{\mathscr{R}(T,\varsigma )}=\mathbb{X}$$. Equivalently, given an arbitrary ϵ > 0, it is possible to steer the system from the point ς at time T to all points in the state space $$\mathbb{X}$$ within a distance ϵ. Assume that the linear fractional differential control model \begin{align} ^{C}D^{\alpha} x(t)={\mathscr{A}} x(t)+\mathbb{J}_{t}^{1-\alpha}(\mathcal{C}u)(t),\qquad t\in{\mathscr{J}}, \end{align} (2.1) \begin{equation} {\hskip-82pt}x_{0}=\varsigma\in{{\mathscr{B}}_h}, \end{equation} (2.2) is approximately controllable. The operator associated with (2.1)–(2.2) as $$ {\Gamma_{0}^{T}}={\int_{0}^{T}} \mathbb{S}_{\alpha}(T-s){\mathcal{C}}{\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-s)\; \textrm{d}s, $$ where $${\mathcal{C}}^{\ast }$$ and $$\mathbb{S}_{\alpha }^{\ast }(t)$$ denote the adjoint of $${\mathcal{C}}$$ and $$\mathbb{S}_{\alpha }(t),$$ respectively. Next, we introduce the following relevant operator $$ \mathcal{R}\left(\gamma,{\Gamma_{0}^{T}}\right)=\left(\gamma I+{\Gamma_{0}^{T}}\right)^{-1}, \quad \textrm{for}\quad \gamma>0. $$ Note that the operator $${\Gamma _{0}^{T}}$$ is a linear bounded operator. Lemma 2.1 (Sakthivel & Yong, 2013) (H0) The linear fractional control system (4.1)–(4.2) is approximately controllable on $${\mathscr{J}}$$ if and only if $$\gamma \mathcal{R}(\gamma ,{\Gamma _{0}^{T}})\rightarrow 0$$ as $$\gamma \rightarrow 0^{+}$$, in the strong operator topology. In accordance with the above discussion, we define the mild solution for the problem (1.1)–(1.3) as follows: Definition 2.4 (Dabas & Chauhan, 2013, Definition 2.20) A function $$ x:(-\infty ,T]\rightarrow \mathbb{X}$$ is said to be a mild solution of the problem if $$u\in L^{2}({\mathscr{J}},U),\ x_{0}=\varsigma \in{{\mathscr{B}}_h}$$ on $$(-\infty ,0]; \Delta x|_{t=t_{k}}={\mathcal{I}}_{k}(x(t_{k}^{-})),\, k=1,2,\dots ,m$$, the constraint of x(⋅) to the interval [0, T) −{t1, t2, ⋯ , tm} is continuous and satisfies the following integral equation: \begin{align} x(t)= \begin{cases} \varsigma(t), &t\in(-\infty,0],\\{\mathbb{S}_{\alpha}}(t)\varsigma(0)+{\sum\limits_{i=1}^{m}} {\mathbb{S}_{\alpha}}(t-{t_{i}}){{\mathcal{I}}_{i}}(x(t_{i}^{-}))+{\int_{0}^{t}} {\mathbb{S}_{\alpha}}(t-s){\mathcal{C}{u}}_{x}(s)ds\\\qquad+{\int_{0}^{t}}{\mathbb{S}_{\alpha}}(t-s)f\left(s,x_{\varrho(s,{x_{s}})}, ({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\right)ds, &t\in{\mathscr{J}}. \end{cases} \end{align} (2.3) 3 Approximate controllability results In this section, we present and prove the approximate controllability of mild solutions for the problem (1.1)–(1.3) by applying Banach’s fixed-point theorem. Let $$\varsigma \in{{\mathscr{B}}_h} $$ be a fixed function. In what follows, we assume that 0 ≤ ϱ(t, ψ) ≤ t for all $$\psi \in{{\mathscr{B}}_h}$$ and assume the following hypotheses: (H1) $$\mathbb{S}_{\alpha }(t)$$ is a compact operator for t > 0 and there exists a constant $$ \widetilde{M}_{S}>0$$ such that $$ \|{\mathbb{S}_\alpha(t)}\|_{L(\mathbb{X})}\le \widetilde{M}_{S}, \quad \textrm{for all}\quad t\in{\mathscr{J}}. $$ (H2) Let $$f:{\mathscr{J}}\times{{\mathscr{B}}_h}\times \mathbb{X}\times \mathbb{X}\rightarrow \mathbb{X}$$ be continuous and there exist constants νi > 0, i = 1, 2, 3 such that \begin{gather*} \left\|\,f(t,\varsigma,x,u)-f(t,\psi,y,v)\right\|_{\mathbb{X}}\le \nu_{1}\|\varsigma-\psi\|_{{{\mathscr{B}}_h}}+\nu_{2}\|x-y\|_{\mathbb{X}}+\nu_{3}\|u-v\|_{\mathbb{X}},\\ t\in{\mathscr{J}},\quad (\varsigma,\psi)\in{{\mathscr{B}}_h^2},\quad x,y,u,v\in\mathbb{X}. \end{gather*} (H3) Let $$ e_{1}:\mathscr{D}\times{{\mathscr{B}}_h}\rightarrow \mathbb{X}$$ be continuous and there exists a constant ξ1 > 0 so that $$ \Bigg\|{\int_{0}^{t}}\big[e_{1}(t,s,\varsigma)-e_{1}(t,s,\psi)\big] \;\text{d}s\Bigg\|_{\mathbb{X}}\le \xi_{1}\|\varsigma-\psi\|_{{{\mathscr{B}}_h}},\quad (t,s)\in \mathscr{D},\quad (\varsigma,\psi)\in{{\mathscr{B}}_h^2}. $$ (H4) Let $$ e_{2}:\mathscr{D}\times{{\mathscr{B}}_h}\rightarrow \mathbb{X}$$ be continuous and there exists a constant ξ2 > 0 so that $$ \Bigg\|{\int_{0}^{t}}\big[e_{2}(t,s,\varsigma)-e_{2}(t,s,\psi)\big] \;\text{d}s\Bigg\|_{\mathbb{X}}\le \xi_{2}\|\varsigma-\psi\|_{{{\mathscr{B}}_h}},\quad (t,s)\in \mathscr{D},\quad (\varsigma,\psi)\in{{\mathscr{B}}_h^2}. $$ (H5) There exists a constant ρ > 0 such that $$ \left\|{\mathcal{I}}_{k}(x)-{\mathcal{I}}_{k}(y)\right\|_{\mathbb{X}}\le \rho\|x-y\|_{\mathbb{X}}, \quad \textrm{for all}\quad x,y\in\mathbb{X},\, k=1,2,\dots,m. $$ (H6) For every q > 0, there exist constants Lf(q) > 0 and ξi(q) > 0 for i = 1, 2 such that \begin{align*} \parallel f(t,x_{t_{2}},x_{2},x_{4})- f(t,x_{t_{1}},x_{1},x_{3})\parallel&\leq \nu(q)\left[|t_{2}-t_{1}|+\|x_{2}-x_{1}\|_{\mathbb{X}}+\|x_{4}-x_{3}\|_{\mathbb{X}}\right],\\ \parallel e_{i}(t,s,x_{t_{2}})- e_{i}(t,s,x_{t_{1}})\parallel&\leq \xi_{i}(q)|t_{2}-t_{1}|, \quad t, t_{1}, t_{2}\in{\mathscr{J}}, \end{align*} for all functions $$x:(-\infty , T]\rightarrow \mathbb{X}$$ such that $$x_{0}=\varsigma \in{{\mathscr{B}}_h}, x:{\mathscr{J}}\rightarrow \mathbb{X}$$ is continuous with $$\max\limits _{0\leq s\leq T}\|x(s)\|\leq q.$$ (H7) The function $$\varrho : {\mathscr{J}}\times{{\mathscr{B}}_h}\rightarrow [0,\infty )$$ is such that (i) the function t↦ϱ(t, ) is continuous for every $$\psi \in{{\mathscr{B}}_h}$$; (ii) there exists a constant Lϱ > 0 such that $$ \left|\varrho(t,{\varphi}_{2})-\varrho(t,{\varphi}_{1})\right|\leq L_{\varrho}\|{\varphi}_{2}-{\varphi}_{1}\|_{{{\mathscr{B}}_h}},\, {\varphi}_{1}, {\varphi}_{2}\in{{\mathscr{B}}_h}\quad \textrm{for all}\quad t\in{\mathscr{J}}. $$ (H8) The function $$f:{\mathscr{J}}\times{{\mathscr{B}}_h}\times \mathbb{X}\times \mathbb{X}\rightarrow \mathbb{X}$$ is continuous and uniformly bounded and there exists an N > 0 such that ∥f(t, ς, x, u)∥≤ N for all $$(t,\varsigma ,x,u)\in{\mathscr{J}}\times{{\mathscr{B}}_h}\times \mathbb{X}\times \mathbb{X}.$$ Remark 3.1 Here, by an example, we shall show that the function f satisfy the hypothesis (H6). Example 3.1 (Andrade et al., 2016, Example 2.1) Let $$h:(-\infty ,0]\rightarrow L(\mathbb{X})$$ be a strongly continuous operator-valued map that satisfies the following conditions: (i) $$\|h\|=\sup \limits _{\theta \leq 0}\|h(\theta )\|<\infty $$ . (ii) There exists Lh > 0 such that ∥h(θ − t2) − h(θ − t1)∥≤ h(θ)Lh|t2 − t1|, for all t2, t1 ≥ 0. We define \begin{align*} f\left(t,\varsigma, \overline{\mathscr{G}}\varsigma, \overline{\mathscr{H}}\varsigma\right)&=\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta+{\int^{t}_{0}}k_{1}(t-s)\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta \;\text{d}s\\&\quad+{\int^{t}_{0}}k_{2}(t-s)\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta\; \textrm{d}s,\, t\ge 0,\, \varsigma\in{{\mathscr{B}}_h}, \end{align*} where $$ \overline{\mathscr{G}}\varsigma={\int^{t}_{0}}k_{1}(t-s)\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta \text{d}s,\quad \overline{\mathscr{H}}\varsigma={\int^{t}_{0}}k_{2}(t-s)\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta \text{d}s.$$ Moreover, for all function $$x:(-\infty , T]\rightarrow \mathbb{X}$$ such that $$x_{0}={\varphi }\in{{\mathscr{B}}_h}, x:[0,T]\rightarrow \mathbb{X}$$ is continuous and $$\max\nolimits _{0\leq s\leq T}\|x(s)\|\leq q$$, we have \begin{align*} &f\left(t,x_{t_{2}},x_{2}, x_{4}\right)-f\left(t,x_{t_{1}}, x_{1}, x_{3}\right)\\[6pt] &\leq \int^{0}_{-\infty}h(\theta)x_{t_{2}}(\theta)\, \textrm{d}\theta+{\int^{t}_{0}} k_{1}(t-s)\int^{0}_{-\infty}h(\theta)x_{t_{2}}(\theta)\text{d}\theta \text{d}s+{\int^{t}_{0}} k_{2}(t-s)\int^{0}_{-\infty}h(\theta)x_{t_{2}}(\theta)\,\textrm{d}\theta \text{d}s\\[6pt] &\quad-\int^{0}_{-\infty}h(\theta)x_{t_{1}}(\theta)\,\textrm{d}\theta-{\int^{t}_{0}} k_{1}(t-s)\int^{0}_{-\infty}h(\theta)x_{t_{1}}(\theta)\,\textrm{d}\theta\, \textrm{d}s-{\int^{t}_{0}} k_{2}(t-s)\int^{0}_{-\infty}h(\theta)x_{t_{1}}(\theta)\,\textrm{d}\theta \text{d}s\\[6pt] &\le\int^{0}_{\!-\infty}h(\theta)x(t_{2}\!+\theta)\,\textrm{d}\theta+ {\int^{t}_{0}} k_{1}(t-s)\int^{0}_{\!-\infty}h(\theta)x(t_{2}\!+\theta)\,\textrm{d}\theta\, \textrm{d}s + {\int^{t}_{0}} k_{2}(t-s)\int^{0}_{\!-\infty}h(\theta)x(t_{2}\!+\theta)\,\textrm{d}\theta \text{d}s\\[6pt] &\quad-\!\int^{0}_{\!-\infty}h(\theta)x(t_{1}\!+\theta)\,\textrm{d}\theta -\!{\int^{t}_{0}}k_{1}(t-s)\!\int^{0}_{\!-\infty}h(\theta)x(t_{1}\!+\theta)\,\textrm{d}\theta \text{d}s-\!{\int^{t}_{0}}k_{2}(t-s)\!\int^{0}_{\!-\infty}h(\theta)x(t_{1}\!+\theta)\,\textrm{d}\theta\text{d} s\\[6pt] &\leq \int^{t_{2}}_{-\infty}h(q-t_{2})\varsigma(q)\,\textrm{d}q-\int^{t_{1}}_{-\infty}h(q-t_{1})\varsigma(q)\,\textrm{d}q+ {\int^{t}_{0}}k_{1}(t-s)\int^{t_{2}}_{-\infty}h(q-t_{2})\varsigma(q)\,\textrm{d}q\,\textrm{d}s\\[6pt] &\quad-{\int^{t}_{0}}k_{1}(t-s)\int^{t_{1}}_{-\infty}h(q-t_{1})\varsigma(q)\,\textrm{d}q\,\textrm{d}s+ {\int^{t}_{0}}k_{2}(t-s)\int^{t_{2}}_{-\infty}h(q-t_{2})\varsigma(q)\,\textrm{d}q\,\textrm{d}s\\[6pt] &\quad-{\int^{t}_{0}}k_{2}(t-s)\int^{t_{1}}_{-\infty}h(q-t_{1})\varsigma(q)\,\textrm{d}q\,\textrm{d}s\\[6pt] &\leq \int^{0}_{-\infty}[h(q-t_{2})-h(q-t_{1})]\varsigma(q)\,\textrm{d}q+{\int^{t}_{0}}k_{1}(t-s)\int^{0}_{-\infty}[h(q-t_{2})-h(q-t_{1})]\varsigma(q)\,\textrm{d}q\,\textrm{d}s \\[6pt] &\quad+{\int^{t}_{0}}k_{2}(t-s)\int^{0}_{-\infty}[h(q-t_{2})-h(q-t_{1})]\varsigma(q)\,\textrm{d}q\,\textrm{d}s+\int^{0}_{-t_{1}}[h(q+t_{1}-t_{2})-h(q)]x(q+t_{1})\,\textrm{d}q\\[6pt] &\quad+{\int^{t}_{0}}k_{1}(t\!-\!s)\!\int^{0}_{-t_{1}}\![h(q\!+\!t_{1}\!-\!t_{2})\!-\!h(q)]x(q+t_{1})\,\textrm{d}q\,\textrm{d}s\\[6pt] &\quad+\!{\int^{t}_{0}}k_{2}(t\!-\!s)\!\int^{0}_{-t_{1}}\![h(q+t_{1}\!-\!t_{2})\!-\!h(q)]x(q+\!t_{1})\, \textrm{d}q\,\textrm{d}s\\[6pt] &\quad+\int^{0}_{t_{1}-t_{2}}h(q)x(q+t_{2})\,\textrm{d}q+{\int^{t}_{0}}k_{1}(t-s)\int^{0}_{t_{1}-t_{2}}h(q)x(q+t_{2})\,\textrm{d}q\,\textrm{d}s\\[6pt] &\quad+{\int^{t}_{0}}k_{2}(t-s)\int^{0}_{t_{1}-t_{2}}h(q)x(q+t_{2})\,\textrm{d}q\,\textrm{d}s, \end{align*} for 0 ≤ t1 ≤ t2. Using (i)−(ii), we obtain \begin{align*} &\big\|f\left(t,x_{t_{2}},x_{2}, x_{4}\right)-f\left(t,x_{t_{1}}, x_{1}, x_{3}\right)\big\|\\[3pt] &\leq L_{h}\bigg[\int^{0}_{-\infty}h(q)\sup\limits_{\theta\in[q,0]}\|\varsigma(\theta)\|\,\textrm{d}q\bigg]|t_{2}-t_{1}|+{\int^{t}_{0}} \|k_{1}(t-s)\| L_{h}\bigg[\int^{0}_{-\infty}h(q)\sup\limits_{\theta\in[q,0]}\|\varsigma(\theta)\|\,\textrm{d}q\bigg]|t_{2}-t_{1}|\,\textrm{d}s\\[3pt] &\quad+{\int^{t}_{0}} \|k_{2}(t-s)\| L_{h}\bigg[\int^{0}_{-\infty}h(q)\sup\limits_{\theta\in[q,0]}\|\varsigma(\theta)\|\,\textrm{d}q\bigg]|t_{2}-t_{1}|\,\textrm{d}s+ L_{h}\bigg[\int^{0}_{-\infty}h(q)\,\textrm{d}q\bigg]q|t_{2}-t_{1}|\\[3pt] &\quad+ {\int^{t}_{0}}\|k_{1}(t-s)\| L_{h}\bigg[\int^{0}_{-\infty}h(q)\,\textrm{d}q\bigg]q|t_{2}-t_{1}|\,\textrm{d}s+ {\int^{t}_{0}}\|k_{2}(t-s)\| L_{h}\bigg[\int^{0}_{-\infty}h(q)\,\textrm{d}q\bigg]q|t_{2}-t_{1}|\,ds\\[3pt] &\quad+\|h\|q|t_{2}-t_{1}|+{\int^{t}_{0}}\|k_{1}(t-s)\|h\|q|t_{2}-t_{1}|\,\textrm{d}s+{\int^{t}_{0}}\|k_{2}(t-s)\|h\|q|t_{2}-t_{1}|\,\textrm{d}s\\[3pt] &\leq \bigg[L_{h}\left(\|\varsigma\|_{{{\mathscr{B}}_h}}+\frac{q}{2}\right)+\|h\|q\bigg]+T\|k_{1}(T-t)\|\bigg[L_{h}\left(\|\varsigma\|_{{{\mathscr{B}}_h}}+\frac{q}{2}\right)+\|h\|q\bigg]\\[3pt] &\quad+T\|k_{2}(T-t)\|\bigg[L_{h}\left(\|\varsigma\|_{{{\mathscr{B}}_h}}+\frac{q}{2}\right)+\|h\|q\bigg]\\[3pt] &\leq \nu(q)(1+T\xi_{1}(q)+T\xi_{2}(q))|t_{2}-t_{1}|, \end{align*} where $$\nu (q)= L_{h}\left (\|\varsigma \|_{{{\mathscr{B}}_h}}+\frac{q}{2}\right )+\|h\|q$$ and ∥ki(T − t)∥ = ξi, i = 1, 2. This shows that f satisfies (H6). Remark3.2 From the study by Andrade et al. (2016, Example 2.2), we observe that (H7) is fulfilled. It will be shown that the system (1.1)–(1.3) is approximately controllable, if for all γ > 0, there exists a function x(⋅) such that \begin{align} x(t)&=\mathbb{S}_{\alpha}(t)\varsigma(0)+\sum_{i=1}^{k} \mathbb{S}_{\alpha}(t-t_{i}){\mathcal{I}}_{i}\left(x\left(t_{i}^{-}\right)\right)+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s){\mathcal{C}}{u}_{x}(s)\,\textrm{d}s \nonumber\\ &\quad+{\int_{0}^{t}}\mathbb{S}_{\alpha}(t-s)f\Big(s,x_{\varrho(s,x_{s})}, ({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\Big)\text{d}s, \end{align} (3.1) \begin{equation} {\hskip-130pt}u_{x}(t)={\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-t)\mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)p(x(\cdot)), \end{equation} (3.2) where \begin{align*} p(x(\cdot))&=x_{T}\!-\mathbb{S}_{\alpha}(T)\varsigma(0)\!-\!\sum_{i=1}^{k}\mathbb{S}_{\alpha}(T-t_{i}){\mathcal{I}}_{i}(x(t_{i}^{-}))-\!{\int_{0}^{T}}\mathbb{S}_{\alpha}(T\!-s)f\bigg(s,x_{\varrho(s,x_{s})},({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\bigg)\text{d}s. \end{align*} Theorem 3.1 Assume that the hypotheses (H0)–(H7) are satisfied. Further, suppose that for all γ > 0 \begin{align} \Omega_{m}&=\!\Bigg(1\!+\!\frac{1}{\gamma}M_{{\mathcal{C}}}^{2}\widetilde{M}_{S}^{2}T\Bigg)\Bigg[m\widetilde{M}_{S}\rho+\widetilde{M}_{S}T{\mathscr{D}}_{1}^{\ast}\bigg(\nu_{1}\!+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\bigg)\Bigg]\!<\!1 \end{align} (3.3) with $$\|M_{{\mathcal{C}}}\|=\|{\mathcal{C}}\|$$, then the system (1.1)–(1.3) has a mild solution on $${\mathscr{J}}$$. Proof. The main aim in this theorem is to find conditions for solvability of the system (3.1) and (3.2) for γ > 0. We show that, using the control u(⋅), the operator $${\Upsilon }:{\mathscr{B}_T}\rightarrow{\mathscr{B}_T}$$ defined by $$({\Upsilon} x)(t)= \begin{cases} \varsigma(t), &t\in(-\infty,0],\\[4pt] \mathbb{S}_{\alpha}(t)\varsigma(0)+\sum\limits_{i=1}^{k} \mathbb{S}_{\alpha}(t-t_{i}){\mathcal{I}}_{i}(x(t_{i}^{-}))+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s){\mathcal{C}}{u}_{x}(s)\;\textrm{d}s\\[4pt]\qquad+{\int_{0}^{t}}\mathbb{S}_{\alpha}(t-s)f\Big(s,x_{\varrho(s,x_{s})}, ({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\Big)\text{d}s, &t\in{\mathscr{J}}. \end{cases} $$ has a fixed point x, which is a mild solution of the system (1.1)–(1.3). For $$\varsigma \in{{\mathscr{B}}_h}$$, we define a function $$y:(-\infty ,T]\rightarrow \mathbb{X}$$ by $$ y(t)= \begin{cases} \varsigma(t),& t\le 0;\\ 0,& t\in{\mathscr{J}}, \end{cases} $$ with y0 = ς. For every function $$z\in C({\mathscr{J}},\mathbb{R})$$ with z(0) = 0, we define $$\overline{z}$$ by $$ \overline{z}(t)= \begin{cases} 0,& t\le 0;\\ z(t),&t\in{\mathscr{J}}. \end{cases} $$ If x(⋅) satisfies (2.3), then we can split it as $$ x(t)=y(t)+\overline{z}(t),\, t\in{\mathscr{J}}$$, which implies that $$ x_{t}=y_{t}+\overline{z}_{t}$$ for each $$ t\in{\mathscr{J}}$$ and the function z(⋅) satisfies \begin{align*} z(t)&=\mathbb{S}_{\alpha}(t)\varsigma(0)+\sum_{i=1}^{k} \mathbb{S}_{\alpha}(t-t_{i}){\mathcal{I}}_{i}\left(y\left(t_{i}^{-}\right)+\overline{z}\left(t_{i}^{-}\right)\right)+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s){\mathcal{C}}{u}_{y+\overline{z}}(s)\;\textrm{d}s\\[4pt] &\quad+{\int_{0}^{t}}\mathbb{S}_{\alpha}(t-s)f\Bigg(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})},{\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau,\\[4pt] &\qquad\qquad\qquad\qquad{\quad\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau\Bigg)\text{d}s,\quad t\in{\mathscr{J}}, \end{align*} where \begin{align*} u_{y+\overline{z}}(t)&= {\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-t)\mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)\\ &\qquad\Bigg[x_{T}-\mathbb{S}_{\alpha}(T)\varsigma(0)-\sum_{i=1}^{k}\mathbb{S}_{\alpha}(T-t_{i}){\mathcal{I}}_{i}\left(y(t_{i}^{-})+\overline{z}(t_{i}^{-})\right) \\ &\quad\qquad-{\int_{0}^{T}}\mathbb{S}_{\alpha}(T-s)f\Bigg(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau, \\ &\quad\qquad\qquad\qquad\qquad\qquad\quad{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau\Bigg)\text{d}s\Bigg], \quad t\in{\mathscr{J}}. \end{align*} Let $${\mathscr{B}_T^{0}}=\{ z\in{\mathscr{B}_T}$$: $$z_{0}=0\in{{\mathscr{B}}_h}\}$$ be equipped with the seminorm $$ \|z\|_{{\mathscr{B}_T^{0}}}=\sup_{t\in{\mathscr{J}}}\|z(t)\|_{\mathbb{X}}+\|z_{0}\|_{{{\mathscr{B}}_h}}=\sup_{t\in{\mathscr{J}}}\|z(t)\|_{\mathbb{X}},\quad z\in{\mathscr{B}_T^{0}}. $$ Clearly $$({\mathscr{B}_T^{0}}, \|\cdot \|_{{\mathscr{B}_T^{0}}})$$ is a Banach space. We define the operator $${\overline{\Upsilon }}:{\mathscr{B}_T^{0}}\rightarrow \mathscr{B}_T^{0}$$ by \begin{align*} ({\overline{\Upsilon}} z)(t)&= \mathbb{S}_{\alpha}(t)\varsigma(0)+\sum_{i=1}^{m} \mathbb{S}_{\alpha}(t-t_{i}){\mathcal{I}}_{i}\left(y\left(t_{i}^{-}\right)+\overline{z}\left(t_{i}^{-}\right)\right)+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s){\mathcal{C}}{u}_{y+\overline{z}}(s)\;\textrm{d}s\\ &\quad+{\int_{0}^{t}}\mathbb{S}_{\alpha}(t-s)f\Bigg(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau,\\ &{\qquad\qquad\qquad\qquad\quad\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau\Bigg)\text{d}s,\quad t\in{\mathscr{J}}. \end{align*} Evidently the operator Υ has a fixed point if and only if $${\overline{\Upsilon }}$$ has a fixed point. In consequence, we have to show that $${\overline{\Upsilon }}$$ has a fixed point. From the preceding arguments, we obtain the following estimates: (i) \begin{align*} &\bigg\|\;f\Big(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau, \\ &\quad{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Big) \\ &\qquad-f\Big(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}+y_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})})\;\textrm{d}\tau, \nonumber \end{align*} \begin{align} &{\int_{0}^{T}}e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\big)\text{d}\tau\Big)\bigg\|_{\mathbb{X}} \nonumber\\ &\qquad\leq \bigg\|f\Big(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau,\nonumber \\ &{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Big) \nonumber\\ &\qquad-f\Big(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau, \nonumber\\ &{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Big)\bigg\|_{\mathbb{X}} \nonumber\\ &\qquad+\bigg\|f\Big(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau, \nonumber\\&{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Big) \nonumber\\ &-f\Big(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}+y_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})})\;\textrm{d}\tau, \nonumber\\ &{\int_{0}^{T}}e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\big)\text{d}\tau\Big)\bigg\|_{\mathbb{X}} \nonumber\\ &\le \nu_{1}\|\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}-\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}-y_{\varrho(s,\overline{z}_{s}+y_{s})}\|_{{{\mathscr{B}}_h}} \nonumber\\ &\quad+\xi_{1}\nu_{2}\|\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}} \nonumber\\ &\quad+\xi_{2}\nu_{3}\|\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}}\nonumber\\ &\quad+\nu(q)\bigg[\|\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}-\overline{z}^{\ast}_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}-y_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}\|_{{{\mathscr{B}}_h}}\nonumber\\ &\quad+{\int^{s}_{0}}\xi_{1}(q)\|\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}-y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}}\text{d}s\nonumber\\ &\quad+{\int^{s}_{0}}\xi_{2}(q)\|\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}-y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}}\text{d}s\bigg] \nonumber\\ &\le \nu_{1}\|\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}-\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}\|_{{{\mathscr{B}}_h}}+\xi_{1}\nu_{2}\|\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}} \nonumber\\ &\quad+\xi_{2}\nu_{3}\|\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}}+2\nu(q) \nonumber\\ &\qquad\bigg[L_{\varrho} \|\overline{z}_{s}-\overline{z}^{\ast}_{s}\|_{{{\mathscr{B}}_h}}+T\xi_{1}(q)L_{\varrho} \|\overline{z}_{s}-\overline{z}^{\ast}_{s}\|_{{{\mathscr{B}}_h}}+T\xi_{2}(q)L_{\varrho} \|\overline{z}_{s}-\overline{z}^{\ast}_{s}\|_{{{\mathscr{B}}_h}}\bigg]\nonumber\\ &\le{\mathscr{D}}_{1}^{\ast}\Big[\nu_{1}+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\Big]\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}, \end{align} (3.4) since, in the view of (P3), we have \begin{align*} \|\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}-\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}\|_{{{\mathscr{B}}_h}}&\le{\mathscr{D}}_{1}^{\ast}\max_{0\le\tau\le\varrho(s,\overline{z}_{s}+y_{s})}\|\overline{z}(\tau)-\overline{z}^{\ast}(\tau)\|+{\mathscr{D}}_{2}^{\ast}\|\overline{z}_{0}-\overline{z}^{\ast}_{0}\|_{{{\mathscr{B}}_h}}\\ &\le{\mathscr{D}}_{1}^{\ast}\max\limits_{0\le\tau\le s} \|\overline{z}(s)-\overline{z}^{\ast}(s)\|_{\mathbb{X}} \le{\mathscr{D}}_{1}^{\ast}\|\overline{z}-\overline{z}^{\ast}\|_{\mathscr{B}_T^{0}} \end{align*} and $$ \|\overline{z}_{s}-\overline{z}^{\ast}_{s}\|_{{{\mathscr{B}}_h}}\le{\mathscr{D}}_{1}^{\ast}\max_{0\le s\le t}\|\|\overline{z}(s)-\overline{z}^{\ast}(s)\|_{\mathbb{X}} \le{\mathscr{D}}_{1}^{\ast}\|\overline{z}-\overline{z}^{\ast}\|_{{\mathscr{B}_T^{0}}}. $$ (ii) By the assumption (H5), we obtain \begin{align*} &\left\|\sum_{i=1}^{k}S_{\alpha}(t-t_{k}){\mathcal{I}}_{i}\left(\overline{z}\left(t_{i}^{-}\right)\right)-\sum_{i=1}^{k}S_{\alpha}(t-t_{i}){\mathcal{I}}_{i}\left(\overline{z}^{\ast}\left(t_{i}^{-}\right)\right)\right\|_{\mathbb{X}}\\ &\quad\le \sum_{i=1}^{k}\left\|S_{\alpha}(t-t_{i})\right\|_{L(\mathbb{X})}\left\|{\mathcal{I}}_{i}\left(\overline{z}\left(t_{i}^{-}\right)\right)-{\mathcal{I}}_{i}\left(\overline{z}^{\ast}\left(t_{i}^{-}\right)\right)\right\|_{\mathbb{X}}\\ &\quad\le k\widetilde{M}_{S}\rho\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}. \end{align*} When k = m, it is obvious that \begin{align} &\left\|\sum_{i=1}^{k}S_{\alpha}(t-t_{k}){\mathcal{I}}_{i}(\overline{z}(t_{i}^{-}))-\sum_{i=1}^{k}S_{\alpha}(t-t_{i}){\mathcal{I}}_{i}(\overline{z}^{\ast}(t_{i}^{-}))\right\|_{\mathbb{X}}\le m\widetilde{M}_{S}\rho\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}. \end{align} (3.5) (iii) From the foregoing arguments, we get \begin{align}&{\int_{0}^{t}}\|\mathbb{S}_{\alpha}(t-s)\|_{L(\mathbb{X})}\nonumber\\ &\qquad\Bigg\|\,f\Bigg(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau, \nonumber\\&\qquad{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Bigg) \nonumber\\&\qquad\ -f\Bigg(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}+y_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\left(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\right)\;\textrm{d}\tau, \nonumber\\&\qquad{\int_{0}^{T}}e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\big)\text{ d}\tau\Bigg)\Bigg\|_{\mathbb{X}}\textrm{d}s \nonumber\\&\quad\le \widetilde{M}_{S} T{\mathscr{D}}_{1}^{\ast}\big[\nu_{1}+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\big]\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}} \end{align} (3.6) and \begin{align} &{\int_{0}^{t}}\left\|\mathbb{S}_{\alpha}(t-\eta){\mathcal{C}}{\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-\eta)\mathcal{R}\left(\gamma,{\Gamma_{0}^{T}}\right)\right\|_{\mathbb{X}}\nonumber\\ &\quad\left[\sum_{i=1}^{k}\|S_{\alpha}(T-t_{i})\|_{L(\mathbb{X})}\|{\mathcal{I}}_{i}(\overline{z}(t_{i}^{-}))-{\mathcal{I}}_{i}(\overline{z}^{\ast}(t_{i}^{-}))\|_{\mathbb{X}}+{\int_{0}^{T}}\|\mathbb{S}_{\alpha}(T-s)\|_{L(\mathbb{X})} \right.\nonumber\\ &\qquad\left\|\,f\left(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{ d}\tau,\right.\right. \nonumber\\ &\qquad\ \left.{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\text{d}\tau\right) \nonumber\\ &\qquad\ -f\left(s,\overline{z}^{\ast}_{\varrho\left(s,\overline{z}^{\ast}_{s}+y_{s}\right)}+y_{\varrho\left(s,\overline{z}^{\ast}_{s}+y_{s}\right)}, {\int_{0}^{s}}e_{1}\left(s,\tau,\overline{z}^{\ast}_{\varrho\left(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau}\right)}+y_{\varrho\left(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau}\right)}\right)\text{d}\tau,\right. \nonumber\\ &\qquad\quad\left.\left. \left.\phantom{\sum_{i=1}^{k}}{\int_{0}^{T}}e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\big)\text{ d}\tau\right)\right\|_{\mathbb{X}}\text{d}s\right]\text{d}\eta \nonumber\\ &\quad \le \Bigg(\frac{1}{\gamma}M_{{\mathcal{C}}}^{2}\widetilde{M}_{S}^{2}T\Bigg)\Bigg[m\widetilde{M}_{S}\rho+\widetilde{M}_{S} T{\mathscr{D}}_{1}^{\ast}\big[\nu_{1}+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\ \quad+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\big]\Bigg]\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}. \end{align} (3.7) Now, we shall prove that $${\overline{\Upsilon }}$$ has a unique fixed point. In fact $$ z,z^{\ast }\in{\mathscr{B}_T^{0}}$$. Then, for all $$ t\in{\mathscr{J}}$$, it follows from estimates (3.4)–(3.6) and (3.7) that \begin{align*} &\|({\overline{\Upsilon}} z)(t)-({\overline{\Upsilon}} z^{\ast})(t)\|_{\mathbb{X}}\\ &\le \Bigg(1+\frac{1}{\gamma}M_{{\mathcal{C}}}^{2}\widetilde{M}_{S}^{2}T\Bigg)\\&\quad(\times)\Bigg[m\widetilde{M}_{S}\rho+\widetilde{M}_{S} T{\mathscr{D}}_{1}^{\ast}\bigg(\nu_{1}+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\bigg)\Bigg]\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}\\ &\le \Omega_{m}\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}. \end{align*} From (3.3), we notice that $${\Omega_m<1,\ \textrm{therefore }\overline{\Upsilon }}$$ is a contraction mapping. Hence, it follows by the contraction mapping principle that $${\overline{\Upsilon }}$$ has a unique fixed point $$z\in{\mathscr{B}_T^{0}}$$, which is a mild solution of the system (1.1)–(1.3). The proof is now completed. Theorem 3.2 Assume the condition in Theorem 3.1 and the assumption (H8) are satisfied, then system (1.1)–(1.3) is approximately controllable on $${\mathscr{J}}$$. Proof. Let xγ(⋅) be a fixed point of $${\overline{\Upsilon }}$$. By Theorem 3.1, unique fixed point of $${\overline{\Upsilon }}$$ is a mild solution of (1.1)–(1.3) under the control $$ x^{\gamma}(t)={\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-t)\mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right){p}(x^{\gamma}) $$ and satisfies the inequality \begin{align} x^{\gamma}(T)=x_{T}+\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right){p}(x^{\gamma}). \end{align} (3.8) Moreover, by assumption on f with Dunford–Pettis theorem, we have that {fγ(s)} is weakly compact in $$L^{1}({\mathscr{J}},\mathbb{X})$$, so there is a sub-sequence, still denoted by {fγ(s)} that converges weakly to say f(s) in $$L^{1}({\mathscr{J}},\mathbb{X})$$. Define \begin{align*} w&=x_{T}-\mathbb{S}_{\alpha}(T)\varsigma(0)-\sum_{i=1}^{k}\mathbb{S}_{\alpha}(T-t_{i}){\mathcal{I}}_{i}\left(x\left(t_{i}^{-}\right)\right)-{\int_{0}^{T}}\mathbb{S}_{\alpha}(T-s)f\bigg(s,x_{\varrho(s,x_{s})},({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\bigg)\textrm{d}s. \end{align*} Now, we have \begin{align} \|{p}(x^{\gamma})-w\|&=\left\|{\int_{0}^{T}} \mathbb{S}_{\alpha}(T-s)[f(s,x^{\gamma}_{\rho(s,x^{\gamma}_{s})}, ({\mathscr{G}} x^{\gamma})(s), ({\mathscr{H}} x^{\gamma})(s) )-f(s)]\;\textrm{d}s\right\|\nonumber \\ &\le \sup_{t\in[0,T]}\left[\left\|{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s)\left[f\left(s,x^{\gamma}_{\rho(s,x^{\gamma}_{s})}, ({\mathscr{G}} x^{\gamma})(s), ({\mathscr{H}} x^{\gamma})(s) \right)-f(s)\right]\textrm{d}s\right\|\right]. \end{align} (3.9) By using infinite-dimensional version of the Ascoli–Arzela theorem, one can show that an operator $$ l(\cdot )\rightarrow \int _{0}^{\cdot } \mathbb{S}_{\alpha }(\cdot -s)l(s)ds: L^{1}({\mathscr{J}},\mathbb{X})\rightarrow C({\mathscr{J}},\mathbb{X})$$ is compact. Consequently, we obtain that $$\|{p}(x^{\gamma })-w\|\rightarrow 0 $$ as $$\gamma \rightarrow 0^{+}$$. Moreover, from (3.8), we obtain \begin{align*} \|x^{\gamma}(T)-x_{T}\|&\le \left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right){p}(x^{\gamma})\right\|\\ &\le \left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)\left({p}(x^{\gamma})-w+w\right)\right\|\\ &\le \left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)(w)\right\|+\left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)\right\|\left\|{p}(x^{\gamma})-w\right\|\\ &\le \left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)(w)\right\|+\left\|{p}(x^{\gamma})-w\right\|. \end{align*} It follows from assumption (H0) and the estimation (3.9) that $$\|x^{\gamma }(T)-x_{T}\|\rightarrow 0$$ as $$\gamma \rightarrow 0^{+}$$. This proves the approximate controllability of the system (1.1)–(1.3). 4 Applications Consider an impulsive fractional integro-differential equation with state-dependent delay given by \begin{align} ^{C}D_{t}^{\alpha} u(t,x)&=\frac{\partial^{2}}{\partial x^{2}}u(t,x)+\frac{1}{\Gamma(1-\alpha)}{\int_{0}^{t}} (t-s)^{-\alpha}\nonumber\\ &\quad\ \Bigg[\mu(s,x)+\int_{-\infty}^{s}\mu_{1}(s,x,\xi-s)u(\xi-\varrho_{1}(s)\varrho_{2}(\|u(s)\|),x)\;\textrm{d}\xi \nonumber\\ &\qquad+{\int_{0}^{s}}\int_{-\infty}^{\eta} k_{1}(\eta-\xi)u(\xi-\varrho_{1}(\xi)\varrho_{2}(\|u(\xi)\|),\xi)\;\textrm{d}\xi\; \textrm{d}\eta\nonumber\\ &\qquad+{\int_{0}^{T}}\int_{-\infty}^{\eta} k_{2}(\eta-\xi)u(\xi-\varrho_{1}(\xi)\varrho_{2}(\|u(\xi)\|),\xi)\;\textrm{d}\xi \text{ d}\eta\Bigg]\textrm{d}s,\nonumber\\ &\qquad\;x\in[0,\pi],\, t\in{\mathscr{J}},\, t\ne t_{k},\, k=1,2,\dots,m, \end{align} (4.1) \begin{equation}{\hskip-135pt}u(t,0)=0=u(t,\pi),\qquad t\ge 0, \end{equation} (4.2) \begin{equation} {\hskip-110pt}u(t,x)=\varsigma(t,x), \quad t \le 0,\quad x\in[0,\pi], \end{equation} (4.3) \begin{equation} {\hskip-25pt}\Delta u(t_{k})(x)=\int_{-\infty}^{t_{k}}q_{k}(t_{k}-s)u(s,x)\textrm{d}s,\quad x\in[0,\pi],\, k=1,2,\dots,m, \end{equation} (4.4) where $$^{C}D_{t}^{\alpha }$$ is Caputo’s fractional derivative of order $$ 0<\alpha <1, \mu :[0,1]\times [0,\pi ]\rightarrow [0,\pi ]$$ is continuous; 0 < t1 < t2 < ⋯ < tn < T are pre-fixed numbers and $$\varsigma \in{{{\mathscr{B}}_h}}$$. We consider $$\mathbb{X}=L^{2}[0,\pi ]$$ with the norm $$ |\cdot |_{L^{2}}$$ and determine the operator $${\mathscr{A}}:D({\mathscr{A}})\subset \mathbb{X}\rightarrow \mathbb{X}$$ satisfying $${\mathscr{A}} w=w^{\prime \prime }$$ with the domain $$ D({\mathscr{A}})=\left\{w\in\mathbb{X}: w,w^{\prime} \quad\textrm{are absolutely continuous},\quad w^{\prime\prime}\in\mathbb{X},\, w(0)=w(\pi)=0\right\}. $$ Then $$ {\mathscr{A}} w=-\sum_{n=1}^{\infty} n^{2}\langle{w,w_{n}\rangle}w_{n},\quad w\in D({\mathscr{A}}), $$ where $$w_{n}(s)=\sqrt{\frac{2}{\pi }}\sin (ns),\, n=1,2,\dots ,$$ is the orthogonal set of eigenvectors of $${\mathscr{A}}$$. It is well known that $${\mathscr{A}}$$ is the infinitesimal generator of an analytic semigroup (T(t))t≥0 in $$\mathbb{X}$$ and is given by $$ T(t)w=\sum_{n=1}^{\infty} e^{-n^{2} t}\langle{w,w_{n}\rangle}w_{n},\quad \textrm{for all}\quad w\in \mathbb{X}, \quad \textrm{and every}\quad t>0. $$ Furthermore, consider an infinite dimensional control space U defined by $$ U=\left\{ v|v=\sum_{n=2}^{\infty} v_{n}w_{n} \quad \textrm{with}\quad \sum_{n=2}^{\infty} {v_{n}^{2}}<\infty\right\}$$ endowed with the norm $$ \|v\|_{U}=\left(\sum_{n=2}^{\infty} {v_{n}^{2}}\right)^{\frac{1}{2}}.$$ Define a continuous linear map $${\mathcal{C}}$$ from U to $$\mathbb{X}$$ as $$ {\mathcal{C}}{v}=2v_{2}w_{1}+ \sum _{n=2}^{\infty } v_{n}w_{n}$$ for $$v= \sum _{n=2}^{\infty } v_{n}w_{n}\in U$$. The subordination concept of solution operator (Bazhlekova, 2001, Theorem 3.1) suggests that $${\mathscr{A}}$$ is the infinitesimal generator of a solution operator $$(\mathbb{S}_{\alpha }(t))_{t\ge 0}$$. Since $$ \mathbb{S}_{\alpha }(t)$$ is strongly continuous on $$[0,\infty )$$, by uniformly bounded theorem, we can find constant $$\widetilde{M}_{S}>0$$ such that $$\|\mathbb{S}_{\alpha }(t)\|_{L(\mathbb{X})}\le \widetilde{M}_{S}$$ for $$t\in{\mathscr{J}}$$. For the phase space, we choose h = e2s, s < 0. Then $$ l=\int _{-\infty }^{0}h(s)\;\textrm{d}s=\frac{1}{2}<\infty $$ for t ≤ 0 and $$ \|\varsigma\|_{{{{\mathscr{B}}_h}}}=\int_{-\infty}^{0}h(s)\sup_{\theta\in[s,0]}\|\varsigma(\theta)\|_{L^{2}}\text{ d}s. $$ Hence, for $$(t,\varsigma )\in [0,T]\times{{\mathscr{B}}_h}$$, we have $$\varsigma (\theta )(x)=\varsigma (\theta ,x),\, (\theta ,x)\in (-\infty ,0]\times [0,\pi ]$$. Set $$ u(t)(x)=u(t,x),\quad \varrho(t,\varsigma)=\varrho_{1}(t)\varrho_{2}(\|\varsigma(0)\|),\quad ({\mathcal{C}}{u})(t)(x)=\mu(t,x),\quad 0\le x\le\pi, $$ where $${\mathcal{C}}:U\rightarrow \mathbb{X}$$ is a bounded linear operator. Thus, we obtain $$ f(t,\varsigma,{\mathscr{G}}\varsigma,\mathscr{H}\varsigma)(x)=\int_{-\infty}^{0}\mu_{1}(t,x,\theta)(\varsigma(\theta)(x))\;\textrm{d}\theta+{\mathscr{G}}\varsigma(x)+\mathscr{H}\varsigma(x), $$ where $$ \mathscr{G}\varsigma(x)={\int_{0}^{t}}\int_{-\infty}^{0}k_{1}(s-\theta)(\varsigma(\theta)(x))\;\textrm{d}\theta\;\textrm{d} s,\quad \mathscr{H}\varsigma(x)={\int_{0}^{T}}\int_{-\infty}^{0}k_{2}(s-\theta)(\varsigma(\theta)(x))\;\textrm{d}\theta\; \textrm{d}s. $$ Here we have assumed that (i) the functions $$\varrho _{i}:[0,\infty )\rightarrow [0,\infty ),\, i=1,2$$ are continuous; (ii) the function μ1(t, x, θ) is continuous in $$[0,T]\times [0,\pi ]\times (-\infty ,0]$$; (iii) the function ki(t − s) is continuous in [0, T] and ki(t − s) ≥ 0, i = 1, 2; (iv) the functions $$q_{k}:\mathbb{R}\rightarrow \mathbb{R},\, k=1,2,\dots ,m$$ are continuous and $$d_{i}=\int _{-\infty }^{0} h(s){q_{i}^{2}}(s)\textrm{d }s<\infty $$ for $$ i=1,2,\dots ,n$$. Thus, with the above choices, the system (4.1);(4.4) can be written in the abstract form of (1.1);(1.3). Further, we can impose some suitable conditions on the above defined functions to verify the assumptions of Theorem 3.2, we can conclude that (4.1)–(4.4) is approximately controllable on $${\mathscr{J}}$$. Conclusion In this paper, we have obtained abstract results concerning the approximate controllability for impulsive fractional integro-differential equation of order α ∈ (0, 1) with state-dependent delay by applying the fractional calculus, contraction mapping principle and semigroup techniques. A new set of sufficient conditions ensuring the controllability of the system (1.1)–(1.3) has been presented. Finally, we demonstrate the application of the obtained results. For the future research, we propose the investigation of approximate controllability for fractional neutral integro-differential equations with state-dependent delay, Poisson jumps and time varying delays. 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Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces

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Abstract

Abstract In this paper, the problem of approximate controllability for non-linear impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces is investigated. We study the approximate controllability for non-linear impulsive integro-differential systems under the assumption that the corresponding linear control system is approximately controllable. By utilizing the methods of fractional calculus, semigroup theory, fixed-point theorem coupled with solution operator, sufficient conditions are formulated and proved. Finally, an example is provided to illustrate the proposed theory. 1 Introduction Fractional calculus has emerged as an interesting area of research in view of growing applications of its modelling techniques in various fields such as fluid flow, viscoelasticity, control theory of dynamical systems, electrical networks, probability and statistics, dynamical processes in porous structures, electrochemistry of corrosion, optics and signal processing, rheology, hydraulics of dams, potential fields, diffusion problems and waves in liquids and gases etc. Fractional-order operators provide a perfect tool for the description of memory and hereditary properties of various materials and processes, in contrast to their integer-order counterparts. Nowadays, irrational and complex orders also appear in certain studies besides rational orders. The recent works on the topic include the improvement of theoretical aspects and analytic/numerical methods for solving fractional-order differential equations appearing in the mathematical models of applied and scientific problems (for instance, see the studies by Podlubny, 1999; Baleanu et al., 2012; Hilfer, 2000; Kilbas et al., 2006; Lakshmikantham et al., 2009; Miller & Ross, 1993; Tarasov, 2010; Zhou, 2014, 2016, and a series of papers by Zhou et al., 2013a,b, 2015; Zhou & Peng, 2017a,b,c; Zhou & Zhang, 2017, and the references cited therein). However, the development of control theory in the perspective of fractional calculus is at its initial stage and needs to be explored further. Functional differential equations with state-dependent delay are found to be of great interest as such equations appear in the mathematical models associated with real world problems. In particular, the existence and approximate controllability results for mild solutions of certain problems have attracted considerable attention and a great deal of work is underway. For some recent works on fractional-order problems with state-dependent delay, for example, see Agarwal & Andrade (2011), Benchohra & Berhoun (2016), Benchohra et al.,(2012, 2013), Aissani & Benchohra (2014), Kavitha et al. (2012), Carvalho dos Santos et al. (2016), Sakthivel & Yong (2013), Vijayakumar et al. (2013), Yan (2012) and references cited therein. The theory of impulsive differential equation has also become an important area of investigation in recent years. It has been stimulated by numerous applications of these equations in mechanics, electrical engineering, medicine, biology, ecology etc. For further details and examples, we refer the reader to the books by Stamova (2009), Graef et al. (2013), Bainov & Covachev (1995), Benchohra et al. (2006) and the papers by Chang et al. (2008) and Hernandez et al. (2010) and the references cited therein. The concept of controllability is of enormous influence in mathematical control theory and engineering because they have closely related to pole assignment, structural decomposition, observer design etc. In the case of infinite dimensional systems, two basic concept of controllability can be identified which are exact controllability and approximate controllability. Exact controllability enables to steer the system to arbitrary final state while approximate controllability means that system can be steered to arbitrary small neighborhood of final state. The approximate controllability is essentially weaker notion than exact controllability and it gives the possibility of steering the system to states which form the dense subspace in the state space. However, in the case of infinite dimensional systems exact controllability appears rather exceptionally but in the case of finite dimensional systems notions of exact and approximate controllability coincide. The recent work on controllability for different types of fractional differential and integro-differential systems has generated a great deal of interest among scientists. Sakthivel et al. (2013), Ganesh et al. (2013, 2014), Liu et al. (2014) and Guendouzi & Bousmaha (2014) investigated the approximate controllability results for various types of fractional differential equations and inclusions. Wang & Zhou (2011) obtained the existence and controllability results for fractional differential inclusions in Banach spaces under suitable fixed point theorem. Debbouche & Baleanu (2011) discussed the exact controllability for fractional evolution non-local impulsive quasilinear delay integro-differential systems in Banach spaces. Recently, Vijayakumar et al. (2013) and Yan (2012) studied the approximate controllability for fractional-order partial neutral functional integro-differential inclusions with state-dependent delay in Hilbert spaces by applying different types of fixed-point theorem. Ravichandran & Trujillo (2013) addressed the controllability of impulsive fractional integro-differential systems with finite delay with the help of fractional calculus, semigroup concept and fixed-point techniques. In the study of fractional differential systems in the infinite dimensional space, the primary step is to develop a method for finding the mild solution. In the study by Wang et al. (2015), the authors presented and discussed an appropriate idea for constructing the PC-mild solutions for the considered problem. We emphasize that the contraction mapping principle cannot be applied directly to a kind of equations with delay, especially state-dependent delay, due to the presence of not same subscripts in the state-dependent delay terms, which fail to survive upon the application of Lipschitz condition. For example, consider the simple function $$f(t,x_{\varrho (t,x_{t})})$$, where f and ϱ are same as in our main problem (1.1). In case the function f(⋅, ⋅) satisfies the Lipschitz condition with respect to the second variable, it is usually written in the literature as follows: $$ \left\|\,f\left(t,x_{\varrho(t,x_{t})}\right)-f\left(t,y_{\varrho(t,y_{t})}\right)\right\|\le \nu\left\|x_{\varrho(t,x_{t})}-y_{\varrho(t,y_{t})}\right\|, $$ which is impossible as the subscripts are the same. In consequence, the contraction mapping principle is not applicable in this situation. In this paper, we present new techniques to utilize the contraction mapping principle for the problems involving state-dependent delay. This is the main motivation and contribution of this paper. Inspired by the works Wang et al. (2015), Aissani & Benchohra (2014), Ganesh et al. (2014), Sakthivel & Yong (2013), and Andrade et al. (2016), our purpose here is to establish the approximate controllability of mild solutions for the following model problem involving an impulsive fractional integro-differential equation with state-dependent delay: \begin{align} ^{C}D_{t}^{\alpha} x(t)&={\mathscr{A}} x(t)+\mathbb{J}_{t}^{1-\alpha}\left[\mathcal{C}u(t)+f\left(t,x_{\varrho(t,x_{t})}, (\mathscr{G}x)(t), (\mathscr{H}x)(t)\right)\right],\nonumber\\ &\qquad\text{a.e. on}\, {\mathscr{J}}-\{t_{1},t_{2},\cdots,t_{m}\}, \end{align} (1.1) \begin{equation} {\hskip-94pt}\Delta x(t_{k})={\mathcal{I}}_{k}\left(x\left(t_{k}^{-}\right)\right),\quad k=1,2,\cdots,m, \end{equation} (1.2) \begin{equation} {\hskip-133pt}x(t)=\varsigma(t),\quad \varsigma(t)\in{{\mathscr{B}}_h}, \end{equation} (1.3) where $${\mathscr{J}}=[0,T]$$ with T > 0, $$^{C}D_{t}^{\alpha }$$ denote the Caputo fractional derivative of order α ∈ (0, 1), $$ {\mathscr{A}}: D({\mathscr{A}})\subset \mathbb{X}\rightarrow \mathbb{X}$$ is an infinitesimal generator of the solution operator $$ \{\mathbb{S}_{\alpha }(t)\}_{t\ge 0}$$ defined on a Hilbert space $$\mathbb{X}$$ having its norm defined as $$\|\cdot \|_{\mathbb{X}}; D({\mathscr{A}})$$ represents the domain of $${\mathscr{A}}$$; $$\mathcal{C}$$ is a bounded linear operator from U into $$\mathbb{X}$$; the control function $$ u(\cdot )\in L^{2}({\mathscr{J}},U)$$, a Hilbert space of admissible control functions. Further, $$ f: {\mathscr{J}}\times{{\mathscr{B}}_h}\times \mathbb{X}\times \mathbb{X}\rightarrow \mathbb{X}$$ represents a class of $$\mathbb{X}$$-valued functions, $${\mathscr{G}}$$ and $${\mathscr{H}}$$ are defined as $$ ({\mathscr{G}} x)(t)={\int_{0}^{t}}e_{1}\left(t,s,x_{\varrho(s,x_{s})}\right) \textrm{d}s, \quad ({\mathscr{H}} x)(t)={\int_{0}^{T}}e_{2}\left(t,s,x_{\varrho(s,x_{s})}\right) \textrm{d}s, $$ with $$ e_{1}, e_{2}:\mathscr{D}\times{{\mathscr{B}}_h}\rightarrow \mathbb{X}, \varrho :{\mathscr{J}}\times{{\mathscr{B}}_h}\rightarrow{\mathbb{R}}$$ being appropriate functions. Here $$ \mathscr{D}=\{ (t,s)\in{\mathscr{J}}\times{\mathscr{J}}: 0\le s\le t\le b\}$$ and $${\mathcal{I}}_{k}: \mathbb{X} \rightarrow \mathbb{X} \, (k=1,2,\cdots ,m)$$ are bounded functions. Also, the fixed times tk satisfy $$ 0=t_{0}<t_{1}<t_{2}<\cdots <t_{m}<t_{m+1}=T, \Delta x(t_{k})=x(t_{k}^{+})-x(t_{k}^{-}),$$ and $$ x(t_{k}^{+})=\lim _{h\rightarrow 0}x(t_{k}+h), x(t_{k}^{-})=\lim _{h\rightarrow 0}x(t_{k}-h)$$ denote the right and left limits of x at the points tk, respectively. We assume that the histories $$x_{t}:(-\infty ,0]\rightarrow{\mathbb{X}}, x_{t}(\theta )=x(t+\theta )$$ for θ ≤ 0, belongs to an abstract phase space $${{\mathscr{B}}_h}$$ defined axiomatically. To the best of our knowledge, the approximate controllability of fractional integro-differential systems with state-dependent delay and form (1.1)–(1.3) is an untreated topic in the literature and this fact is the main motivation of the present work. The rest of the manuscript is organized as follows. In Section 2, we recall some preliminary concepts, definitions and lemmas that we need in the sequel. The approximate controllability of mild solutions for the model problem (1.1)–(1.3) is discussed via Banach fixed point theorem in Section 3. In Section 4, an example is given to illustrate our main results. 2 Preliminaries In this section, we recall some definitions, lemmas and preparatory facts from functional analysis, solution operator and fractional calculus, which will be used throughout this paper without any further mention. Let $$L(\mathbb{X})$$ denote the Hilbert space of all bounded linear operators from $$ \mathbb{X}$$ into $$\mathbb{X}$$, equipped with the norm $$\|\cdot \|_{L(\mathbb{X})}$$ and $$C({\mathscr{J}},\mathbb{X})$$ be the space of all continuous functions from $$ {\mathscr{J}}$$ into $$\mathbb{X}$$ with norm defined as $$ \|\cdot \|_{C({\mathscr{J}},\mathbb{X})}$$. Notice that we consider the theoretical phase space $${{\mathscr{B}}_h}$$ when the delay is infinite. In this case, we work in the phase spaces $${{\mathscr{B}}_h}$$ as described in the study of Dabas & Chauhan (2013). Consider the space $${\mathscr{B}_T}=\{ x: (-\infty ,T]\rightarrow \mathbb{X}$$ be such that $$x|_{{\mathscr{J}}_{k}}\in C({\mathscr{J}}_{k},\mathbb{X})$$, and we can find $$x(t_{k}^{+})$$ and $$x(t_{k}^{-})$$ with $$ x(t_{k})=x(t_{k}^{-}), \quad x_{0}=\varsigma \in{{\mathscr{B}}_h},\quad k=0,1,2,\dots ,m \}, $$ where $$ x|_{{\mathscr{J}}_{k}}$$ is the constraint of x to $${\mathscr{J}}_{k}=(t_{k},t_{k+1}],\ k=0,1,2,\dots ,m$$. The seminorm $$\|\cdot \|_{{\mathscr{B}_T}}$$ in $${\mathscr{B}_T}$$ is defined by the following: $$ \|x\|_{{\mathscr{B}_T}}=\|\varsigma\|_{{{\mathscr{B}}_h}}+\sup\left\{|x(s)|:s\in [0,T]\right\}, \quad x\in{\mathscr{B}_T}. $$ We assume that the phase space $$({{\mathscr{B}}_h},\|\cdot \|_{{{\mathscr{B}}_h}})$$ is a seminormed linear space of functions from $$(-\infty ,0]$$ into $$\mathbb{X}$$, and satisfies the following basic axioms as suggested in the result in the study by Hale & Kato (1978) (see also Hino et al., 1991). For T > 0, let $$x:(-\infty ,T]\rightarrow \mathbb{X}$$ be a function such that $$ x_{0}=\varsigma \in{{\mathscr{B}}_h}, x\in{\mathscr{B}_T} $$. Then, for every $$t\in{\mathscr{J}}_{k}$$, the following conditions hold: (P1) xt is in $${{\mathscr{B}}_h}$$; (P2) $$ \|x(t)\|_{\mathbb{X}}\le H\|x_{t}\|_{{{\mathscr{B}}_h}};$$ (P3) $$\|x_{t}\|_{{{\mathscr{B}}_h}}\le{\mathscr{D}}_{1}(t)\sup \{\|x(s)\|:0\le s\le t\}+{\mathscr{D}}_{2}(t)\|x_{0}\|_{{{\mathscr{B}}_h}}$$. Here, H is a positive constant, $$ {\mathscr{D}}_{1}(\cdot ):[0,+\infty )\rightarrow [0,+\infty )$$ is continuous, $${\mathscr{D}}_{2}(\cdot ):[0,+\infty )\rightarrow [0,+\infty )$$ is locally bounded and $${\mathscr{D}}_{1}, {\mathscr{D}}_{2}$$ are independent of x(⋅). Next, we present some fundamental definitions and results from fractional calculus. Definition 2.1 (Kilbas et al., 2006) For $$ n-1<\alpha <n,\ n\in \mathbb{N}$$, the Caputo fractional derivative of order α for a function $$ f:[0,+\infty )\rightarrow \mathbb{R}$$ is defined as $$ D_{t}^{\alpha} f(t)=\frac{1}{\Gamma(n-\alpha)}{\int_{0}^{t}}(t-s)^{n-\alpha-1}f^{(n)}(s) \textrm{d}s=\mathbb{J}^{n-\alpha}f^{(n)}(t). $$ If 0 < α < 1, then $$ D_{t}^{\alpha} f(t)=\frac{1}{\Gamma(1-\alpha)}{\int_{0}^{t}}(t-s)^{-\alpha}f^{\prime}(s) \textrm{d}s. $$ Obviously, Caputo’s derivative of a non-zero constant is zero. Definition 2.2 (Dabas & Chauhan, 2013) A two-parameter function of the Mittag–Leffler type is defined by the series representation: $$ E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty} \frac{z^{k}}{\Gamma(\alpha k+\beta)}=\frac{1}{2\pi i}\int_{C} \frac{\mu^{\alpha-\beta}e^{\mu}}{\mu^{\alpha}-z} \;\text{d}\mu,\, \alpha,\beta>0,\, z\in\mathbb{C}, $$ where $$\mathbb{C}$$ is a contour which starts and ends at $$-\infty $$ and encircles the disc $$ \|\mu \|\le |z|^{\frac{1}{\alpha }}$$ counter clockwise. A useful property of the Mittag–Leffler functions involving Laplace integral is $$ \int_{0}^{\infty}e^{-\lambda t}t^{\beta-1}E_{\alpha,\beta}(\omega t^{\alpha})\; \textrm{d}t=\frac{\lambda^{\alpha-\beta}}{\lambda^{\alpha}-\omega},\, \textrm{Re}\ \lambda>\omega^{\frac{1}{\alpha}},\, \omega>0. $$ Remark 2.1 (i) The details about the solution operator and its results are quite standard. Hence we omit these details and we suggest the interested reader to refer to the studies by Wang et al. (2015) and Dabas & Chauhan (2013). (ii) To determine the mild solutions for the problem (1.1)–(1.3), we consider the following Cauchy problem: \begin{cases} {}^{C}D^{\alpha} x(t)={\mathscr{A}} x(t)+\mathbb{J}_{t}^{1-\alpha}f(t),\qquad t\in{\mathscr{J}},\\ x(0)=x_{0}\in\mathbb{X}. \end{cases} As argued in the study by Dabas & Chauhan, (2013), the mild solution of the above Cauchy problem is $$ x(t)= \mathbb{S}_{\alpha}(t)x_{0}+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s)f(s)\; \textrm{d}s, $$ where $$\mathbb{S}_{\alpha }(t)$$ is the solution operator generated by $${\mathscr{A}}$$ and is given by $$ \mathbb{S}_{\alpha}(t)=E_{\alpha,1}({\mathscr{A}} t^{\alpha})=\frac{1}{2\pi i}\int_{\widehat{B}_{r}}e^{\lambda t}\frac{\lambda^{\alpha-1}}{\lambda^{\alpha}-{\mathscr{A}}} \textrm{d}\lambda, $$ where $$ \widehat{B}_{r}$$ denotes the Bromwich path and $$f:{\mathscr{J}}\rightarrow \mathbb{X}$$ is continuous. Let xT(ς;u) be the state value of the problem (1.1)–(1.3) at terminal time T corresponding to the control u and the initial value $$ \varsigma \in{{\mathscr{B}}_h}$$. The set $$\mathscr{R}(T,\varsigma )=\{x_{T}(\varsigma ;u)(0): u(\cdot )\in L^{2}({\mathscr{J}},U)\}$$ is known as the reachable set of the problem (1.1)–(1.3) at terminal time T and its closure in $$\mathbb{X}$$ is denoted by $$\overline{\mathscr{R}(T,\varsigma )}$$. Definition 2.3 The fractional control system (1.1)–(1.3) is said to be approximately controllable on $${\mathscr{J}}$$ if the reachable set $${\mathscr{R}(T,\varsigma )}$$ is dense in $$\mathbb{X}$$, that is, $$\overline{\mathscr{R}(T,\varsigma )}=\mathbb{X}$$. Equivalently, given an arbitrary ϵ > 0, it is possible to steer the system from the point ς at time T to all points in the state space $$\mathbb{X}$$ within a distance ϵ. Assume that the linear fractional differential control model \begin{align} ^{C}D^{\alpha} x(t)={\mathscr{A}} x(t)+\mathbb{J}_{t}^{1-\alpha}(\mathcal{C}u)(t),\qquad t\in{\mathscr{J}}, \end{align} (2.1) \begin{equation} {\hskip-82pt}x_{0}=\varsigma\in{{\mathscr{B}}_h}, \end{equation} (2.2) is approximately controllable. The operator associated with (2.1)–(2.2) as $$ {\Gamma_{0}^{T}}={\int_{0}^{T}} \mathbb{S}_{\alpha}(T-s){\mathcal{C}}{\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-s)\; \textrm{d}s, $$ where $${\mathcal{C}}^{\ast }$$ and $$\mathbb{S}_{\alpha }^{\ast }(t)$$ denote the adjoint of $${\mathcal{C}}$$ and $$\mathbb{S}_{\alpha }(t),$$ respectively. Next, we introduce the following relevant operator $$ \mathcal{R}\left(\gamma,{\Gamma_{0}^{T}}\right)=\left(\gamma I+{\Gamma_{0}^{T}}\right)^{-1}, \quad \textrm{for}\quad \gamma>0. $$ Note that the operator $${\Gamma _{0}^{T}}$$ is a linear bounded operator. Lemma 2.1 (Sakthivel & Yong, 2013) (H0) The linear fractional control system (4.1)–(4.2) is approximately controllable on $${\mathscr{J}}$$ if and only if $$\gamma \mathcal{R}(\gamma ,{\Gamma _{0}^{T}})\rightarrow 0$$ as $$\gamma \rightarrow 0^{+}$$, in the strong operator topology. In accordance with the above discussion, we define the mild solution for the problem (1.1)–(1.3) as follows: Definition 2.4 (Dabas & Chauhan, 2013, Definition 2.20) A function $$ x:(-\infty ,T]\rightarrow \mathbb{X}$$ is said to be a mild solution of the problem if $$u\in L^{2}({\mathscr{J}},U),\ x_{0}=\varsigma \in{{\mathscr{B}}_h}$$ on $$(-\infty ,0]; \Delta x|_{t=t_{k}}={\mathcal{I}}_{k}(x(t_{k}^{-})),\, k=1,2,\dots ,m$$, the constraint of x(⋅) to the interval [0, T) −{t1, t2, ⋯ , tm} is continuous and satisfies the following integral equation: \begin{align} x(t)= \begin{cases} \varsigma(t), &t\in(-\infty,0],\\{\mathbb{S}_{\alpha}}(t)\varsigma(0)+{\sum\limits_{i=1}^{m}} {\mathbb{S}_{\alpha}}(t-{t_{i}}){{\mathcal{I}}_{i}}(x(t_{i}^{-}))+{\int_{0}^{t}} {\mathbb{S}_{\alpha}}(t-s){\mathcal{C}{u}}_{x}(s)ds\\\qquad+{\int_{0}^{t}}{\mathbb{S}_{\alpha}}(t-s)f\left(s,x_{\varrho(s,{x_{s}})}, ({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\right)ds, &t\in{\mathscr{J}}. \end{cases} \end{align} (2.3) 3 Approximate controllability results In this section, we present and prove the approximate controllability of mild solutions for the problem (1.1)–(1.3) by applying Banach’s fixed-point theorem. Let $$\varsigma \in{{\mathscr{B}}_h} $$ be a fixed function. In what follows, we assume that 0 ≤ ϱ(t, ψ) ≤ t for all $$\psi \in{{\mathscr{B}}_h}$$ and assume the following hypotheses: (H1) $$\mathbb{S}_{\alpha }(t)$$ is a compact operator for t > 0 and there exists a constant $$ \widetilde{M}_{S}>0$$ such that $$ \|{\mathbb{S}_\alpha(t)}\|_{L(\mathbb{X})}\le \widetilde{M}_{S}, \quad \textrm{for all}\quad t\in{\mathscr{J}}. $$ (H2) Let $$f:{\mathscr{J}}\times{{\mathscr{B}}_h}\times \mathbb{X}\times \mathbb{X}\rightarrow \mathbb{X}$$ be continuous and there exist constants νi > 0, i = 1, 2, 3 such that \begin{gather*} \left\|\,f(t,\varsigma,x,u)-f(t,\psi,y,v)\right\|_{\mathbb{X}}\le \nu_{1}\|\varsigma-\psi\|_{{{\mathscr{B}}_h}}+\nu_{2}\|x-y\|_{\mathbb{X}}+\nu_{3}\|u-v\|_{\mathbb{X}},\\ t\in{\mathscr{J}},\quad (\varsigma,\psi)\in{{\mathscr{B}}_h^2},\quad x,y,u,v\in\mathbb{X}. \end{gather*} (H3) Let $$ e_{1}:\mathscr{D}\times{{\mathscr{B}}_h}\rightarrow \mathbb{X}$$ be continuous and there exists a constant ξ1 > 0 so that $$ \Bigg\|{\int_{0}^{t}}\big[e_{1}(t,s,\varsigma)-e_{1}(t,s,\psi)\big] \;\text{d}s\Bigg\|_{\mathbb{X}}\le \xi_{1}\|\varsigma-\psi\|_{{{\mathscr{B}}_h}},\quad (t,s)\in \mathscr{D},\quad (\varsigma,\psi)\in{{\mathscr{B}}_h^2}. $$ (H4) Let $$ e_{2}:\mathscr{D}\times{{\mathscr{B}}_h}\rightarrow \mathbb{X}$$ be continuous and there exists a constant ξ2 > 0 so that $$ \Bigg\|{\int_{0}^{t}}\big[e_{2}(t,s,\varsigma)-e_{2}(t,s,\psi)\big] \;\text{d}s\Bigg\|_{\mathbb{X}}\le \xi_{2}\|\varsigma-\psi\|_{{{\mathscr{B}}_h}},\quad (t,s)\in \mathscr{D},\quad (\varsigma,\psi)\in{{\mathscr{B}}_h^2}. $$ (H5) There exists a constant ρ > 0 such that $$ \left\|{\mathcal{I}}_{k}(x)-{\mathcal{I}}_{k}(y)\right\|_{\mathbb{X}}\le \rho\|x-y\|_{\mathbb{X}}, \quad \textrm{for all}\quad x,y\in\mathbb{X},\, k=1,2,\dots,m. $$ (H6) For every q > 0, there exist constants Lf(q) > 0 and ξi(q) > 0 for i = 1, 2 such that \begin{align*} \parallel f(t,x_{t_{2}},x_{2},x_{4})- f(t,x_{t_{1}},x_{1},x_{3})\parallel&\leq \nu(q)\left[|t_{2}-t_{1}|+\|x_{2}-x_{1}\|_{\mathbb{X}}+\|x_{4}-x_{3}\|_{\mathbb{X}}\right],\\ \parallel e_{i}(t,s,x_{t_{2}})- e_{i}(t,s,x_{t_{1}})\parallel&\leq \xi_{i}(q)|t_{2}-t_{1}|, \quad t, t_{1}, t_{2}\in{\mathscr{J}}, \end{align*} for all functions $$x:(-\infty , T]\rightarrow \mathbb{X}$$ such that $$x_{0}=\varsigma \in{{\mathscr{B}}_h}, x:{\mathscr{J}}\rightarrow \mathbb{X}$$ is continuous with $$\max\limits _{0\leq s\leq T}\|x(s)\|\leq q.$$ (H7) The function $$\varrho : {\mathscr{J}}\times{{\mathscr{B}}_h}\rightarrow [0,\infty )$$ is such that (i) the function t↦ϱ(t, ) is continuous for every $$\psi \in{{\mathscr{B}}_h}$$; (ii) there exists a constant Lϱ > 0 such that $$ \left|\varrho(t,{\varphi}_{2})-\varrho(t,{\varphi}_{1})\right|\leq L_{\varrho}\|{\varphi}_{2}-{\varphi}_{1}\|_{{{\mathscr{B}}_h}},\, {\varphi}_{1}, {\varphi}_{2}\in{{\mathscr{B}}_h}\quad \textrm{for all}\quad t\in{\mathscr{J}}. $$ (H8) The function $$f:{\mathscr{J}}\times{{\mathscr{B}}_h}\times \mathbb{X}\times \mathbb{X}\rightarrow \mathbb{X}$$ is continuous and uniformly bounded and there exists an N > 0 such that ∥f(t, ς, x, u)∥≤ N for all $$(t,\varsigma ,x,u)\in{\mathscr{J}}\times{{\mathscr{B}}_h}\times \mathbb{X}\times \mathbb{X}.$$ Remark 3.1 Here, by an example, we shall show that the function f satisfy the hypothesis (H6). Example 3.1 (Andrade et al., 2016, Example 2.1) Let $$h:(-\infty ,0]\rightarrow L(\mathbb{X})$$ be a strongly continuous operator-valued map that satisfies the following conditions: (i) $$\|h\|=\sup \limits _{\theta \leq 0}\|h(\theta )\|<\infty $$ . (ii) There exists Lh > 0 such that ∥h(θ − t2) − h(θ − t1)∥≤ h(θ)Lh|t2 − t1|, for all t2, t1 ≥ 0. We define \begin{align*} f\left(t,\varsigma, \overline{\mathscr{G}}\varsigma, \overline{\mathscr{H}}\varsigma\right)&=\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta+{\int^{t}_{0}}k_{1}(t-s)\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta \;\text{d}s\\&\quad+{\int^{t}_{0}}k_{2}(t-s)\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta\; \textrm{d}s,\, t\ge 0,\, \varsigma\in{{\mathscr{B}}_h}, \end{align*} where $$ \overline{\mathscr{G}}\varsigma={\int^{t}_{0}}k_{1}(t-s)\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta \text{d}s,\quad \overline{\mathscr{H}}\varsigma={\int^{t}_{0}}k_{2}(t-s)\int^{0}_{-\infty}h(\theta)\varsigma(\theta)\; \textrm{d}\theta \text{d}s.$$ Moreover, for all function $$x:(-\infty , T]\rightarrow \mathbb{X}$$ such that $$x_{0}={\varphi }\in{{\mathscr{B}}_h}, x:[0,T]\rightarrow \mathbb{X}$$ is continuous and $$\max\nolimits _{0\leq s\leq T}\|x(s)\|\leq q$$, we have \begin{align*} &f\left(t,x_{t_{2}},x_{2}, x_{4}\right)-f\left(t,x_{t_{1}}, x_{1}, x_{3}\right)\\[6pt] &\leq \int^{0}_{-\infty}h(\theta)x_{t_{2}}(\theta)\, \textrm{d}\theta+{\int^{t}_{0}} k_{1}(t-s)\int^{0}_{-\infty}h(\theta)x_{t_{2}}(\theta)\text{d}\theta \text{d}s+{\int^{t}_{0}} k_{2}(t-s)\int^{0}_{-\infty}h(\theta)x_{t_{2}}(\theta)\,\textrm{d}\theta \text{d}s\\[6pt] &\quad-\int^{0}_{-\infty}h(\theta)x_{t_{1}}(\theta)\,\textrm{d}\theta-{\int^{t}_{0}} k_{1}(t-s)\int^{0}_{-\infty}h(\theta)x_{t_{1}}(\theta)\,\textrm{d}\theta\, \textrm{d}s-{\int^{t}_{0}} k_{2}(t-s)\int^{0}_{-\infty}h(\theta)x_{t_{1}}(\theta)\,\textrm{d}\theta \text{d}s\\[6pt] &\le\int^{0}_{\!-\infty}h(\theta)x(t_{2}\!+\theta)\,\textrm{d}\theta+ {\int^{t}_{0}} k_{1}(t-s)\int^{0}_{\!-\infty}h(\theta)x(t_{2}\!+\theta)\,\textrm{d}\theta\, \textrm{d}s + {\int^{t}_{0}} k_{2}(t-s)\int^{0}_{\!-\infty}h(\theta)x(t_{2}\!+\theta)\,\textrm{d}\theta \text{d}s\\[6pt] &\quad-\!\int^{0}_{\!-\infty}h(\theta)x(t_{1}\!+\theta)\,\textrm{d}\theta -\!{\int^{t}_{0}}k_{1}(t-s)\!\int^{0}_{\!-\infty}h(\theta)x(t_{1}\!+\theta)\,\textrm{d}\theta \text{d}s-\!{\int^{t}_{0}}k_{2}(t-s)\!\int^{0}_{\!-\infty}h(\theta)x(t_{1}\!+\theta)\,\textrm{d}\theta\text{d} s\\[6pt] &\leq \int^{t_{2}}_{-\infty}h(q-t_{2})\varsigma(q)\,\textrm{d}q-\int^{t_{1}}_{-\infty}h(q-t_{1})\varsigma(q)\,\textrm{d}q+ {\int^{t}_{0}}k_{1}(t-s)\int^{t_{2}}_{-\infty}h(q-t_{2})\varsigma(q)\,\textrm{d}q\,\textrm{d}s\\[6pt] &\quad-{\int^{t}_{0}}k_{1}(t-s)\int^{t_{1}}_{-\infty}h(q-t_{1})\varsigma(q)\,\textrm{d}q\,\textrm{d}s+ {\int^{t}_{0}}k_{2}(t-s)\int^{t_{2}}_{-\infty}h(q-t_{2})\varsigma(q)\,\textrm{d}q\,\textrm{d}s\\[6pt] &\quad-{\int^{t}_{0}}k_{2}(t-s)\int^{t_{1}}_{-\infty}h(q-t_{1})\varsigma(q)\,\textrm{d}q\,\textrm{d}s\\[6pt] &\leq \int^{0}_{-\infty}[h(q-t_{2})-h(q-t_{1})]\varsigma(q)\,\textrm{d}q+{\int^{t}_{0}}k_{1}(t-s)\int^{0}_{-\infty}[h(q-t_{2})-h(q-t_{1})]\varsigma(q)\,\textrm{d}q\,\textrm{d}s \\[6pt] &\quad+{\int^{t}_{0}}k_{2}(t-s)\int^{0}_{-\infty}[h(q-t_{2})-h(q-t_{1})]\varsigma(q)\,\textrm{d}q\,\textrm{d}s+\int^{0}_{-t_{1}}[h(q+t_{1}-t_{2})-h(q)]x(q+t_{1})\,\textrm{d}q\\[6pt] &\quad+{\int^{t}_{0}}k_{1}(t\!-\!s)\!\int^{0}_{-t_{1}}\![h(q\!+\!t_{1}\!-\!t_{2})\!-\!h(q)]x(q+t_{1})\,\textrm{d}q\,\textrm{d}s\\[6pt] &\quad+\!{\int^{t}_{0}}k_{2}(t\!-\!s)\!\int^{0}_{-t_{1}}\![h(q+t_{1}\!-\!t_{2})\!-\!h(q)]x(q+\!t_{1})\, \textrm{d}q\,\textrm{d}s\\[6pt] &\quad+\int^{0}_{t_{1}-t_{2}}h(q)x(q+t_{2})\,\textrm{d}q+{\int^{t}_{0}}k_{1}(t-s)\int^{0}_{t_{1}-t_{2}}h(q)x(q+t_{2})\,\textrm{d}q\,\textrm{d}s\\[6pt] &\quad+{\int^{t}_{0}}k_{2}(t-s)\int^{0}_{t_{1}-t_{2}}h(q)x(q+t_{2})\,\textrm{d}q\,\textrm{d}s, \end{align*} for 0 ≤ t1 ≤ t2. Using (i)−(ii), we obtain \begin{align*} &\big\|f\left(t,x_{t_{2}},x_{2}, x_{4}\right)-f\left(t,x_{t_{1}}, x_{1}, x_{3}\right)\big\|\\[3pt] &\leq L_{h}\bigg[\int^{0}_{-\infty}h(q)\sup\limits_{\theta\in[q,0]}\|\varsigma(\theta)\|\,\textrm{d}q\bigg]|t_{2}-t_{1}|+{\int^{t}_{0}} \|k_{1}(t-s)\| L_{h}\bigg[\int^{0}_{-\infty}h(q)\sup\limits_{\theta\in[q,0]}\|\varsigma(\theta)\|\,\textrm{d}q\bigg]|t_{2}-t_{1}|\,\textrm{d}s\\[3pt] &\quad+{\int^{t}_{0}} \|k_{2}(t-s)\| L_{h}\bigg[\int^{0}_{-\infty}h(q)\sup\limits_{\theta\in[q,0]}\|\varsigma(\theta)\|\,\textrm{d}q\bigg]|t_{2}-t_{1}|\,\textrm{d}s+ L_{h}\bigg[\int^{0}_{-\infty}h(q)\,\textrm{d}q\bigg]q|t_{2}-t_{1}|\\[3pt] &\quad+ {\int^{t}_{0}}\|k_{1}(t-s)\| L_{h}\bigg[\int^{0}_{-\infty}h(q)\,\textrm{d}q\bigg]q|t_{2}-t_{1}|\,\textrm{d}s+ {\int^{t}_{0}}\|k_{2}(t-s)\| L_{h}\bigg[\int^{0}_{-\infty}h(q)\,\textrm{d}q\bigg]q|t_{2}-t_{1}|\,ds\\[3pt] &\quad+\|h\|q|t_{2}-t_{1}|+{\int^{t}_{0}}\|k_{1}(t-s)\|h\|q|t_{2}-t_{1}|\,\textrm{d}s+{\int^{t}_{0}}\|k_{2}(t-s)\|h\|q|t_{2}-t_{1}|\,\textrm{d}s\\[3pt] &\leq \bigg[L_{h}\left(\|\varsigma\|_{{{\mathscr{B}}_h}}+\frac{q}{2}\right)+\|h\|q\bigg]+T\|k_{1}(T-t)\|\bigg[L_{h}\left(\|\varsigma\|_{{{\mathscr{B}}_h}}+\frac{q}{2}\right)+\|h\|q\bigg]\\[3pt] &\quad+T\|k_{2}(T-t)\|\bigg[L_{h}\left(\|\varsigma\|_{{{\mathscr{B}}_h}}+\frac{q}{2}\right)+\|h\|q\bigg]\\[3pt] &\leq \nu(q)(1+T\xi_{1}(q)+T\xi_{2}(q))|t_{2}-t_{1}|, \end{align*} where $$\nu (q)= L_{h}\left (\|\varsigma \|_{{{\mathscr{B}}_h}}+\frac{q}{2}\right )+\|h\|q$$ and ∥ki(T − t)∥ = ξi, i = 1, 2. This shows that f satisfies (H6). Remark3.2 From the study by Andrade et al. (2016, Example 2.2), we observe that (H7) is fulfilled. It will be shown that the system (1.1)–(1.3) is approximately controllable, if for all γ > 0, there exists a function x(⋅) such that \begin{align} x(t)&=\mathbb{S}_{\alpha}(t)\varsigma(0)+\sum_{i=1}^{k} \mathbb{S}_{\alpha}(t-t_{i}){\mathcal{I}}_{i}\left(x\left(t_{i}^{-}\right)\right)+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s){\mathcal{C}}{u}_{x}(s)\,\textrm{d}s \nonumber\\ &\quad+{\int_{0}^{t}}\mathbb{S}_{\alpha}(t-s)f\Big(s,x_{\varrho(s,x_{s})}, ({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\Big)\text{d}s, \end{align} (3.1) \begin{equation} {\hskip-130pt}u_{x}(t)={\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-t)\mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)p(x(\cdot)), \end{equation} (3.2) where \begin{align*} p(x(\cdot))&=x_{T}\!-\mathbb{S}_{\alpha}(T)\varsigma(0)\!-\!\sum_{i=1}^{k}\mathbb{S}_{\alpha}(T-t_{i}){\mathcal{I}}_{i}(x(t_{i}^{-}))-\!{\int_{0}^{T}}\mathbb{S}_{\alpha}(T\!-s)f\bigg(s,x_{\varrho(s,x_{s})},({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\bigg)\text{d}s. \end{align*} Theorem 3.1 Assume that the hypotheses (H0)–(H7) are satisfied. Further, suppose that for all γ > 0 \begin{align} \Omega_{m}&=\!\Bigg(1\!+\!\frac{1}{\gamma}M_{{\mathcal{C}}}^{2}\widetilde{M}_{S}^{2}T\Bigg)\Bigg[m\widetilde{M}_{S}\rho+\widetilde{M}_{S}T{\mathscr{D}}_{1}^{\ast}\bigg(\nu_{1}\!+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\bigg)\Bigg]\!<\!1 \end{align} (3.3) with $$\|M_{{\mathcal{C}}}\|=\|{\mathcal{C}}\|$$, then the system (1.1)–(1.3) has a mild solution on $${\mathscr{J}}$$. Proof. The main aim in this theorem is to find conditions for solvability of the system (3.1) and (3.2) for γ > 0. We show that, using the control u(⋅), the operator $${\Upsilon }:{\mathscr{B}_T}\rightarrow{\mathscr{B}_T}$$ defined by $$({\Upsilon} x)(t)= \begin{cases} \varsigma(t), &t\in(-\infty,0],\\[4pt] \mathbb{S}_{\alpha}(t)\varsigma(0)+\sum\limits_{i=1}^{k} \mathbb{S}_{\alpha}(t-t_{i}){\mathcal{I}}_{i}(x(t_{i}^{-}))+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s){\mathcal{C}}{u}_{x}(s)\;\textrm{d}s\\[4pt]\qquad+{\int_{0}^{t}}\mathbb{S}_{\alpha}(t-s)f\Big(s,x_{\varrho(s,x_{s})}, ({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\Big)\text{d}s, &t\in{\mathscr{J}}. \end{cases} $$ has a fixed point x, which is a mild solution of the system (1.1)–(1.3). For $$\varsigma \in{{\mathscr{B}}_h}$$, we define a function $$y:(-\infty ,T]\rightarrow \mathbb{X}$$ by $$ y(t)= \begin{cases} \varsigma(t),& t\le 0;\\ 0,& t\in{\mathscr{J}}, \end{cases} $$ with y0 = ς. For every function $$z\in C({\mathscr{J}},\mathbb{R})$$ with z(0) = 0, we define $$\overline{z}$$ by $$ \overline{z}(t)= \begin{cases} 0,& t\le 0;\\ z(t),&t\in{\mathscr{J}}. \end{cases} $$ If x(⋅) satisfies (2.3), then we can split it as $$ x(t)=y(t)+\overline{z}(t),\, t\in{\mathscr{J}}$$, which implies that $$ x_{t}=y_{t}+\overline{z}_{t}$$ for each $$ t\in{\mathscr{J}}$$ and the function z(⋅) satisfies \begin{align*} z(t)&=\mathbb{S}_{\alpha}(t)\varsigma(0)+\sum_{i=1}^{k} \mathbb{S}_{\alpha}(t-t_{i}){\mathcal{I}}_{i}\left(y\left(t_{i}^{-}\right)+\overline{z}\left(t_{i}^{-}\right)\right)+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s){\mathcal{C}}{u}_{y+\overline{z}}(s)\;\textrm{d}s\\[4pt] &\quad+{\int_{0}^{t}}\mathbb{S}_{\alpha}(t-s)f\Bigg(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})},{\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau,\\[4pt] &\qquad\qquad\qquad\qquad{\quad\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau\Bigg)\text{d}s,\quad t\in{\mathscr{J}}, \end{align*} where \begin{align*} u_{y+\overline{z}}(t)&= {\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-t)\mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)\\ &\qquad\Bigg[x_{T}-\mathbb{S}_{\alpha}(T)\varsigma(0)-\sum_{i=1}^{k}\mathbb{S}_{\alpha}(T-t_{i}){\mathcal{I}}_{i}\left(y(t_{i}^{-})+\overline{z}(t_{i}^{-})\right) \\ &\quad\qquad-{\int_{0}^{T}}\mathbb{S}_{\alpha}(T-s)f\Bigg(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau, \\ &\quad\qquad\qquad\qquad\qquad\qquad\quad{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau\Bigg)\text{d}s\Bigg], \quad t\in{\mathscr{J}}. \end{align*} Let $${\mathscr{B}_T^{0}}=\{ z\in{\mathscr{B}_T}$$: $$z_{0}=0\in{{\mathscr{B}}_h}\}$$ be equipped with the seminorm $$ \|z\|_{{\mathscr{B}_T^{0}}}=\sup_{t\in{\mathscr{J}}}\|z(t)\|_{\mathbb{X}}+\|z_{0}\|_{{{\mathscr{B}}_h}}=\sup_{t\in{\mathscr{J}}}\|z(t)\|_{\mathbb{X}},\quad z\in{\mathscr{B}_T^{0}}. $$ Clearly $$({\mathscr{B}_T^{0}}, \|\cdot \|_{{\mathscr{B}_T^{0}}})$$ is a Banach space. We define the operator $${\overline{\Upsilon }}:{\mathscr{B}_T^{0}}\rightarrow \mathscr{B}_T^{0}$$ by \begin{align*} ({\overline{\Upsilon}} z)(t)&= \mathbb{S}_{\alpha}(t)\varsigma(0)+\sum_{i=1}^{m} \mathbb{S}_{\alpha}(t-t_{i}){\mathcal{I}}_{i}\left(y\left(t_{i}^{-}\right)+\overline{z}\left(t_{i}^{-}\right)\right)+{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s){\mathcal{C}}{u}_{y+\overline{z}}(s)\;\textrm{d}s\\ &\quad+{\int_{0}^{t}}\mathbb{S}_{\alpha}(t-s)f\Bigg(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau,\\ &{\qquad\qquad\qquad\qquad\quad\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau\Bigg)\text{d}s,\quad t\in{\mathscr{J}}. \end{align*} Evidently the operator Υ has a fixed point if and only if $${\overline{\Upsilon }}$$ has a fixed point. In consequence, we have to show that $${\overline{\Upsilon }}$$ has a fixed point. From the preceding arguments, we obtain the following estimates: (i) \begin{align*} &\bigg\|\;f\Big(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau, \\ &\quad{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Big) \\ &\qquad-f\Big(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}+y_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})})\;\textrm{d}\tau, \nonumber \end{align*} \begin{align} &{\int_{0}^{T}}e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\big)\text{d}\tau\Big)\bigg\|_{\mathbb{X}} \nonumber\\ &\qquad\leq \bigg\|f\Big(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau,\nonumber \\ &{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Big) \nonumber\\ &\qquad-f\Big(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau, \nonumber\\ &{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Big)\bigg\|_{\mathbb{X}} \nonumber\\ &\qquad+\bigg\|f\Big(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau, \nonumber\\&{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Big) \nonumber\\ &-f\Big(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}+y_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})})\;\textrm{d}\tau, \nonumber\\ &{\int_{0}^{T}}e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\big)\text{d}\tau\Big)\bigg\|_{\mathbb{X}} \nonumber\\ &\le \nu_{1}\|\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}-\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}-y_{\varrho(s,\overline{z}_{s}+y_{s})}\|_{{{\mathscr{B}}_h}} \nonumber\\ &\quad+\xi_{1}\nu_{2}\|\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}} \nonumber\\ &\quad+\xi_{2}\nu_{3}\|\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}}\nonumber\\ &\quad+\nu(q)\bigg[\|\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}-\overline{z}^{\ast}_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}-y_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}\|_{{{\mathscr{B}}_h}}\nonumber\\ &\quad+{\int^{s}_{0}}\xi_{1}(q)\|\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}-y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}}\text{d}s\nonumber\\ &\quad+{\int^{s}_{0}}\xi_{2}(q)\|\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}-y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}}\text{d}s\bigg] \nonumber\\ &\le \nu_{1}\|\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}-\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}\|_{{{\mathscr{B}}_h}}+\xi_{1}\nu_{2}\|\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}} \nonumber\\ &\quad+\xi_{2}\nu_{3}\|\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}-\overline{z}^{\ast}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\|_{{{\mathscr{B}}_h}}+2\nu(q) \nonumber\\ &\qquad\bigg[L_{\varrho} \|\overline{z}_{s}-\overline{z}^{\ast}_{s}\|_{{{\mathscr{B}}_h}}+T\xi_{1}(q)L_{\varrho} \|\overline{z}_{s}-\overline{z}^{\ast}_{s}\|_{{{\mathscr{B}}_h}}+T\xi_{2}(q)L_{\varrho} \|\overline{z}_{s}-\overline{z}^{\ast}_{s}\|_{{{\mathscr{B}}_h}}\bigg]\nonumber\\ &\le{\mathscr{D}}_{1}^{\ast}\Big[\nu_{1}+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\Big]\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}, \end{align} (3.4) since, in the view of (P3), we have \begin{align*} \|\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}-\overline{z}^{\ast}_{\varrho(s,\overline{z}_{s}+y_{s})}\|_{{{\mathscr{B}}_h}}&\le{\mathscr{D}}_{1}^{\ast}\max_{0\le\tau\le\varrho(s,\overline{z}_{s}+y_{s})}\|\overline{z}(\tau)-\overline{z}^{\ast}(\tau)\|+{\mathscr{D}}_{2}^{\ast}\|\overline{z}_{0}-\overline{z}^{\ast}_{0}\|_{{{\mathscr{B}}_h}}\\ &\le{\mathscr{D}}_{1}^{\ast}\max\limits_{0\le\tau\le s} \|\overline{z}(s)-\overline{z}^{\ast}(s)\|_{\mathbb{X}} \le{\mathscr{D}}_{1}^{\ast}\|\overline{z}-\overline{z}^{\ast}\|_{\mathscr{B}_T^{0}} \end{align*} and $$ \|\overline{z}_{s}-\overline{z}^{\ast}_{s}\|_{{{\mathscr{B}}_h}}\le{\mathscr{D}}_{1}^{\ast}\max_{0\le s\le t}\|\|\overline{z}(s)-\overline{z}^{\ast}(s)\|_{\mathbb{X}} \le{\mathscr{D}}_{1}^{\ast}\|\overline{z}-\overline{z}^{\ast}\|_{{\mathscr{B}_T^{0}}}. $$ (ii) By the assumption (H5), we obtain \begin{align*} &\left\|\sum_{i=1}^{k}S_{\alpha}(t-t_{k}){\mathcal{I}}_{i}\left(\overline{z}\left(t_{i}^{-}\right)\right)-\sum_{i=1}^{k}S_{\alpha}(t-t_{i}){\mathcal{I}}_{i}\left(\overline{z}^{\ast}\left(t_{i}^{-}\right)\right)\right\|_{\mathbb{X}}\\ &\quad\le \sum_{i=1}^{k}\left\|S_{\alpha}(t-t_{i})\right\|_{L(\mathbb{X})}\left\|{\mathcal{I}}_{i}\left(\overline{z}\left(t_{i}^{-}\right)\right)-{\mathcal{I}}_{i}\left(\overline{z}^{\ast}\left(t_{i}^{-}\right)\right)\right\|_{\mathbb{X}}\\ &\quad\le k\widetilde{M}_{S}\rho\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}. \end{align*} When k = m, it is obvious that \begin{align} &\left\|\sum_{i=1}^{k}S_{\alpha}(t-t_{k}){\mathcal{I}}_{i}(\overline{z}(t_{i}^{-}))-\sum_{i=1}^{k}S_{\alpha}(t-t_{i}){\mathcal{I}}_{i}(\overline{z}^{\ast}(t_{i}^{-}))\right\|_{\mathbb{X}}\le m\widetilde{M}_{S}\rho\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}. \end{align} (3.5) (iii) From the foregoing arguments, we get \begin{align}&{\int_{0}^{t}}\|\mathbb{S}_{\alpha}(t-s)\|_{L(\mathbb{X})}\nonumber\\ &\qquad\Bigg\|\,f\Bigg(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{d}\tau, \nonumber\\&\qquad{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\textrm{d}\tau\Bigg) \nonumber\\&\qquad\ -f\Bigg(s,\overline{z}^{\ast}_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}+y_{\varrho(s,\overline{z}^{\ast}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\left(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\right)\;\textrm{d}\tau, \nonumber\\&\qquad{\int_{0}^{T}}e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\big)\text{ d}\tau\Bigg)\Bigg\|_{\mathbb{X}}\textrm{d}s \nonumber\\&\quad\le \widetilde{M}_{S} T{\mathscr{D}}_{1}^{\ast}\big[\nu_{1}+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\big]\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}} \end{align} (3.6) and \begin{align} &{\int_{0}^{t}}\left\|\mathbb{S}_{\alpha}(t-\eta){\mathcal{C}}{\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-\eta)\mathcal{R}\left(\gamma,{\Gamma_{0}^{T}}\right)\right\|_{\mathbb{X}}\nonumber\\ &\quad\left[\sum_{i=1}^{k}\|S_{\alpha}(T-t_{i})\|_{L(\mathbb{X})}\|{\mathcal{I}}_{i}(\overline{z}(t_{i}^{-}))-{\mathcal{I}}_{i}(\overline{z}^{\ast}(t_{i}^{-}))\|_{\mathbb{X}}+{\int_{0}^{T}}\|\mathbb{S}_{\alpha}(T-s)\|_{L(\mathbb{X})} \right.\nonumber\\ &\qquad\left\|\,f\left(s,\overline{z}_{\varrho(s,\overline{z}_{s}+y_{s})}+y_{\varrho(s,\overline{z}_{s}+y_{s})}, {\int_{0}^{s}}e_{1}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}\big)\text{ d}\tau,\right.\right. \nonumber\\ &\qquad\ \left.{\int_{0}^{T}} e_{2}\big(s,\tau,\overline{z}_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}_{\tau}+y_{\tau})})\;\text{d}\tau\right) \nonumber\\ &\qquad\ -f\left(s,\overline{z}^{\ast}_{\varrho\left(s,\overline{z}^{\ast}_{s}+y_{s}\right)}+y_{\varrho\left(s,\overline{z}^{\ast}_{s}+y_{s}\right)}, {\int_{0}^{s}}e_{1}\left(s,\tau,\overline{z}^{\ast}_{\varrho\left(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau}\right)}+y_{\varrho\left(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau}\right)}\right)\text{d}\tau,\right. \nonumber\\ &\qquad\quad\left.\left. \left.\phantom{\sum_{i=1}^{k}}{\int_{0}^{T}}e_{2}\big(s,\tau,\overline{z}^{\ast}_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}+y_{\varrho(\tau,\overline{z}^{\ast}_{\tau}+y_{\tau})}\big)\text{ d}\tau\right)\right\|_{\mathbb{X}}\text{d}s\right]\text{d}\eta \nonumber\\ &\quad \le \Bigg(\frac{1}{\gamma}M_{{\mathcal{C}}}^{2}\widetilde{M}_{S}^{2}T\Bigg)\Bigg[m\widetilde{M}_{S}\rho+\widetilde{M}_{S} T{\mathscr{D}}_{1}^{\ast}\big[\nu_{1}+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\ \quad+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\big]\Bigg]\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}. \end{align} (3.7) Now, we shall prove that $${\overline{\Upsilon }}$$ has a unique fixed point. In fact $$ z,z^{\ast }\in{\mathscr{B}_T^{0}}$$. Then, for all $$ t\in{\mathscr{J}}$$, it follows from estimates (3.4)–(3.6) and (3.7) that \begin{align*} &\|({\overline{\Upsilon}} z)(t)-({\overline{\Upsilon}} z^{\ast})(t)\|_{\mathbb{X}}\\ &\le \Bigg(1+\frac{1}{\gamma}M_{{\mathcal{C}}}^{2}\widetilde{M}_{S}^{2}T\Bigg)\\&\quad(\times)\Bigg[m\widetilde{M}_{S}\rho+\widetilde{M}_{S} T{\mathscr{D}}_{1}^{\ast}\bigg(\nu_{1}+\xi_{1}\nu_{2}+\xi_{2}\nu_{3}+2\nu(q)L_{\varrho}[1+T(\xi_{1}(q)+\xi_{2}(q))]\bigg)\Bigg]\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}\\ &\le \Omega_{m}\|z-z^{\ast}\|_{{\mathscr{B}_T^{0}}}. \end{align*} From (3.3), we notice that $${\Omega_m<1,\ \textrm{therefore }\overline{\Upsilon }}$$ is a contraction mapping. Hence, it follows by the contraction mapping principle that $${\overline{\Upsilon }}$$ has a unique fixed point $$z\in{\mathscr{B}_T^{0}}$$, which is a mild solution of the system (1.1)–(1.3). The proof is now completed. Theorem 3.2 Assume the condition in Theorem 3.1 and the assumption (H8) are satisfied, then system (1.1)–(1.3) is approximately controllable on $${\mathscr{J}}$$. Proof. Let xγ(⋅) be a fixed point of $${\overline{\Upsilon }}$$. By Theorem 3.1, unique fixed point of $${\overline{\Upsilon }}$$ is a mild solution of (1.1)–(1.3) under the control $$ x^{\gamma}(t)={\mathcal{C}}^{\ast}\mathbb{S}_{\alpha}^{\ast}(T-t)\mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right){p}(x^{\gamma}) $$ and satisfies the inequality \begin{align} x^{\gamma}(T)=x_{T}+\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right){p}(x^{\gamma}). \end{align} (3.8) Moreover, by assumption on f with Dunford–Pettis theorem, we have that {fγ(s)} is weakly compact in $$L^{1}({\mathscr{J}},\mathbb{X})$$, so there is a sub-sequence, still denoted by {fγ(s)} that converges weakly to say f(s) in $$L^{1}({\mathscr{J}},\mathbb{X})$$. Define \begin{align*} w&=x_{T}-\mathbb{S}_{\alpha}(T)\varsigma(0)-\sum_{i=1}^{k}\mathbb{S}_{\alpha}(T-t_{i}){\mathcal{I}}_{i}\left(x\left(t_{i}^{-}\right)\right)-{\int_{0}^{T}}\mathbb{S}_{\alpha}(T-s)f\bigg(s,x_{\varrho(s,x_{s})},({\mathscr{G}} x)(s), ({\mathscr{H}} x)(s)\bigg)\textrm{d}s. \end{align*} Now, we have \begin{align} \|{p}(x^{\gamma})-w\|&=\left\|{\int_{0}^{T}} \mathbb{S}_{\alpha}(T-s)[f(s,x^{\gamma}_{\rho(s,x^{\gamma}_{s})}, ({\mathscr{G}} x^{\gamma})(s), ({\mathscr{H}} x^{\gamma})(s) )-f(s)]\;\textrm{d}s\right\|\nonumber \\ &\le \sup_{t\in[0,T]}\left[\left\|{\int_{0}^{t}} \mathbb{S}_{\alpha}(t-s)\left[f\left(s,x^{\gamma}_{\rho(s,x^{\gamma}_{s})}, ({\mathscr{G}} x^{\gamma})(s), ({\mathscr{H}} x^{\gamma})(s) \right)-f(s)\right]\textrm{d}s\right\|\right]. \end{align} (3.9) By using infinite-dimensional version of the Ascoli–Arzela theorem, one can show that an operator $$ l(\cdot )\rightarrow \int _{0}^{\cdot } \mathbb{S}_{\alpha }(\cdot -s)l(s)ds: L^{1}({\mathscr{J}},\mathbb{X})\rightarrow C({\mathscr{J}},\mathbb{X})$$ is compact. Consequently, we obtain that $$\|{p}(x^{\gamma })-w\|\rightarrow 0 $$ as $$\gamma \rightarrow 0^{+}$$. Moreover, from (3.8), we obtain \begin{align*} \|x^{\gamma}(T)-x_{T}\|&\le \left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right){p}(x^{\gamma})\right\|\\ &\le \left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)\left({p}(x^{\gamma})-w+w\right)\right\|\\ &\le \left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)(w)\right\|+\left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)\right\|\left\|{p}(x^{\gamma})-w\right\|\\ &\le \left\|\gamma \mathcal{R}\left(\gamma, {\Gamma_{0}^{T}}\right)(w)\right\|+\left\|{p}(x^{\gamma})-w\right\|. \end{align*} It follows from assumption (H0) and the estimation (3.9) that $$\|x^{\gamma }(T)-x_{T}\|\rightarrow 0$$ as $$\gamma \rightarrow 0^{+}$$. This proves the approximate controllability of the system (1.1)–(1.3). 4 Applications Consider an impulsive fractional integro-differential equation with state-dependent delay given by \begin{align} ^{C}D_{t}^{\alpha} u(t,x)&=\frac{\partial^{2}}{\partial x^{2}}u(t,x)+\frac{1}{\Gamma(1-\alpha)}{\int_{0}^{t}} (t-s)^{-\alpha}\nonumber\\ &\quad\ \Bigg[\mu(s,x)+\int_{-\infty}^{s}\mu_{1}(s,x,\xi-s)u(\xi-\varrho_{1}(s)\varrho_{2}(\|u(s)\|),x)\;\textrm{d}\xi \nonumber\\ &\qquad+{\int_{0}^{s}}\int_{-\infty}^{\eta} k_{1}(\eta-\xi)u(\xi-\varrho_{1}(\xi)\varrho_{2}(\|u(\xi)\|),\xi)\;\textrm{d}\xi\; \textrm{d}\eta\nonumber\\ &\qquad+{\int_{0}^{T}}\int_{-\infty}^{\eta} k_{2}(\eta-\xi)u(\xi-\varrho_{1}(\xi)\varrho_{2}(\|u(\xi)\|),\xi)\;\textrm{d}\xi \text{ d}\eta\Bigg]\textrm{d}s,\nonumber\\ &\qquad\;x\in[0,\pi],\, t\in{\mathscr{J}},\, t\ne t_{k},\, k=1,2,\dots,m, \end{align} (4.1) \begin{equation}{\hskip-135pt}u(t,0)=0=u(t,\pi),\qquad t\ge 0, \end{equation} (4.2) \begin{equation} {\hskip-110pt}u(t,x)=\varsigma(t,x), \quad t \le 0,\quad x\in[0,\pi], \end{equation} (4.3) \begin{equation} {\hskip-25pt}\Delta u(t_{k})(x)=\int_{-\infty}^{t_{k}}q_{k}(t_{k}-s)u(s,x)\textrm{d}s,\quad x\in[0,\pi],\, k=1,2,\dots,m, \end{equation} (4.4) where $$^{C}D_{t}^{\alpha }$$ is Caputo’s fractional derivative of order $$ 0<\alpha <1, \mu :[0,1]\times [0,\pi ]\rightarrow [0,\pi ]$$ is continuous; 0 < t1 < t2 < ⋯ < tn < T are pre-fixed numbers and $$\varsigma \in{{{\mathscr{B}}_h}}$$. We consider $$\mathbb{X}=L^{2}[0,\pi ]$$ with the norm $$ |\cdot |_{L^{2}}$$ and determine the operator $${\mathscr{A}}:D({\mathscr{A}})\subset \mathbb{X}\rightarrow \mathbb{X}$$ satisfying $${\mathscr{A}} w=w^{\prime \prime }$$ with the domain $$ D({\mathscr{A}})=\left\{w\in\mathbb{X}: w,w^{\prime} \quad\textrm{are absolutely continuous},\quad w^{\prime\prime}\in\mathbb{X},\, w(0)=w(\pi)=0\right\}. $$ Then $$ {\mathscr{A}} w=-\sum_{n=1}^{\infty} n^{2}\langle{w,w_{n}\rangle}w_{n},\quad w\in D({\mathscr{A}}), $$ where $$w_{n}(s)=\sqrt{\frac{2}{\pi }}\sin (ns),\, n=1,2,\dots ,$$ is the orthogonal set of eigenvectors of $${\mathscr{A}}$$. It is well known that $${\mathscr{A}}$$ is the infinitesimal generator of an analytic semigroup (T(t))t≥0 in $$\mathbb{X}$$ and is given by $$ T(t)w=\sum_{n=1}^{\infty} e^{-n^{2} t}\langle{w,w_{n}\rangle}w_{n},\quad \textrm{for all}\quad w\in \mathbb{X}, \quad \textrm{and every}\quad t>0. $$ Furthermore, consider an infinite dimensional control space U defined by $$ U=\left\{ v|v=\sum_{n=2}^{\infty} v_{n}w_{n} \quad \textrm{with}\quad \sum_{n=2}^{\infty} {v_{n}^{2}}<\infty\right\}$$ endowed with the norm $$ \|v\|_{U}=\left(\sum_{n=2}^{\infty} {v_{n}^{2}}\right)^{\frac{1}{2}}.$$ Define a continuous linear map $${\mathcal{C}}$$ from U to $$\mathbb{X}$$ as $$ {\mathcal{C}}{v}=2v_{2}w_{1}+ \sum _{n=2}^{\infty } v_{n}w_{n}$$ for $$v= \sum _{n=2}^{\infty } v_{n}w_{n}\in U$$. The subordination concept of solution operator (Bazhlekova, 2001, Theorem 3.1) suggests that $${\mathscr{A}}$$ is the infinitesimal generator of a solution operator $$(\mathbb{S}_{\alpha }(t))_{t\ge 0}$$. Since $$ \mathbb{S}_{\alpha }(t)$$ is strongly continuous on $$[0,\infty )$$, by uniformly bounded theorem, we can find constant $$\widetilde{M}_{S}>0$$ such that $$\|\mathbb{S}_{\alpha }(t)\|_{L(\mathbb{X})}\le \widetilde{M}_{S}$$ for $$t\in{\mathscr{J}}$$. For the phase space, we choose h = e2s, s < 0. Then $$ l=\int _{-\infty }^{0}h(s)\;\textrm{d}s=\frac{1}{2}<\infty $$ for t ≤ 0 and $$ \|\varsigma\|_{{{{\mathscr{B}}_h}}}=\int_{-\infty}^{0}h(s)\sup_{\theta\in[s,0]}\|\varsigma(\theta)\|_{L^{2}}\text{ d}s. $$ Hence, for $$(t,\varsigma )\in [0,T]\times{{\mathscr{B}}_h}$$, we have $$\varsigma (\theta )(x)=\varsigma (\theta ,x),\, (\theta ,x)\in (-\infty ,0]\times [0,\pi ]$$. Set $$ u(t)(x)=u(t,x),\quad \varrho(t,\varsigma)=\varrho_{1}(t)\varrho_{2}(\|\varsigma(0)\|),\quad ({\mathcal{C}}{u})(t)(x)=\mu(t,x),\quad 0\le x\le\pi, $$ where $${\mathcal{C}}:U\rightarrow \mathbb{X}$$ is a bounded linear operator. Thus, we obtain $$ f(t,\varsigma,{\mathscr{G}}\varsigma,\mathscr{H}\varsigma)(x)=\int_{-\infty}^{0}\mu_{1}(t,x,\theta)(\varsigma(\theta)(x))\;\textrm{d}\theta+{\mathscr{G}}\varsigma(x)+\mathscr{H}\varsigma(x), $$ where $$ \mathscr{G}\varsigma(x)={\int_{0}^{t}}\int_{-\infty}^{0}k_{1}(s-\theta)(\varsigma(\theta)(x))\;\textrm{d}\theta\;\textrm{d} s,\quad \mathscr{H}\varsigma(x)={\int_{0}^{T}}\int_{-\infty}^{0}k_{2}(s-\theta)(\varsigma(\theta)(x))\;\textrm{d}\theta\; \textrm{d}s. $$ Here we have assumed that (i) the functions $$\varrho _{i}:[0,\infty )\rightarrow [0,\infty ),\, i=1,2$$ are continuous; (ii) the function μ1(t, x, θ) is continuous in $$[0,T]\times [0,\pi ]\times (-\infty ,0]$$; (iii) the function ki(t − s) is continuous in [0, T] and ki(t − s) ≥ 0, i = 1, 2; (iv) the functions $$q_{k}:\mathbb{R}\rightarrow \mathbb{R},\, k=1,2,\dots ,m$$ are continuous and $$d_{i}=\int _{-\infty }^{0} h(s){q_{i}^{2}}(s)\textrm{d }s<\infty $$ for $$ i=1,2,\dots ,n$$. Thus, with the above choices, the system (4.1);(4.4) can be written in the abstract form of (1.1);(1.3). Further, we can impose some suitable conditions on the above defined functions to verify the assumptions of Theorem 3.2, we can conclude that (4.1)–(4.4) is approximately controllable on $${\mathscr{J}}$$. Conclusion In this paper, we have obtained abstract results concerning the approximate controllability for impulsive fractional integro-differential equation of order α ∈ (0, 1) with state-dependent delay by applying the fractional calculus, contraction mapping principle and semigroup techniques. A new set of sufficient conditions ensuring the controllability of the system (1.1)–(1.3) has been presented. Finally, we demonstrate the application of the obtained results. For the future research, we propose the investigation of approximate controllability for fractional neutral integro-differential equations with state-dependent delay, Poisson jumps and time varying delays. 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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Jan 30, 2018

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