IMA Journal of Mathematical Control and Information, Volume Advance Article – Dec 22, 2017

/lp/ou_press/controllability-of-fractional-non-instantaneous-impulsive-differential-xriR5y6s49

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- Oxford University Press
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- © The Author(s) 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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- 0265-0754
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- 1471-6887
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- 10.1093/imamci/dnx055
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Abstract In this paper, we study the controllability for a system governed by fractional non-instantaneous non-linear impulsive differential inclusions in Banach spaces. We adopt a new approach to derive the controllability results under weak conditions by establishing a new version weakly convergent criteria in the piecewise continuous functions spaces. In particular, we emphasize that we do not assume any regularity conditions on the multivalued non-linearity expressed in terms of measures of non-compactness. Moreover, unlike the previous literatures, we also do not restrict that the invertibility of the linear controllability operator satisfies a condition expressed in terms of measures of non-compactness. It allows us to apply the weakly topology theory for weakly sequentially closed graph operator and to obtain the controllability results for both upper weakly sequentially closed and relatively weakly compact types of non-linearity. 1. Introduction It is well known that the issue of controllability plays the fundamental role in the design of engineering control problems. Indeed, the most important property of a controlled system is just controllability. Generally speaking, the problem of controllability is to seek a suitable control function from admissible control set to guarantee the output state arising from the controlled system to achieve the terminal state. Note that many control procedures from mathematical modelling can be formulated into the system governed by semi-linear differential equations or inclusion, the related topic of such kind of systems have been studied extensively, see for example Agarwal et al. (2009) and Balasubramaniam & Ntouyas (2006). During the past decades, fractional differential equations and fractional differential inclusions have gained more and more consideration due to their wide applications in various fields. On the classical existence results for fractional differential equations and inclusions and the applications of fractional calculus, one can see Baleanu et al. (2011); Bajlekova (2001); Hilfer (1999); Kilbas et al. (2006); and some recent references such as Li & Wang (2017), Wang et al. (2017b,c) and Zhou & Peng (2017). The theory of impulsive differential equations and impulsive differential inclusions has been an object of interest because of its wide applications in physics and engineering fields (Ballinger & Liu, 2003). The reason for this applicability arises from the fact that impulsive problems can be regarded as an appropriate model for describing process, which at certain moments change their state rapidly and that cannot be described using the classical differential equations. Many authors were devoted to study mild solutions to instantaneous impulsive differential equations and inclusions, one can see for instance Benchohra et al. (2007), Ravichandran & Trujillo (2013), Shu et al. (2011) and Wang et al. (2015a). However, the action of instantaneous impulsive phenomena seems do not describe some certain dynamics of evolution processes in pharmacotherapy, for example, taking in account development of the hemodynamic equilibrium of a person. In the case of a decompensation, for example, high or low levels of glucose, one can prescribe some intravenous drugs (insulin). The introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous process. Thus, we do not expect to use the standard impulsive conditions to write this process. In fact, the above situation is fallen in a new case of impulsive action, which starts at any arbitrary fixed point and stays active on a finite time interval. To this end, Hernández & O’Regan (2013) and Hernández et al. (2015) introduce the so-called non-instantaneous impulsive differential equations and analyze the existence of mild solutions and classical solutions. Thereafter, Wang et al. (2014b) and Wang & Fečkan (2015c) generalized the model to two general classes of impulsive differential equations, which are more reasonable to show dynamics of evolution processes in pharmacotherapy. We also recommend the reader to the recent literatures Wang et al. (2016a), Wang et al. (2017a) and Wang (2017). In recent years, the existence of mild solutions and controllability problems for various types of integer or fractional differential inclusions in Banach spaces by using different kinds of approach have been considered in many recent publications (see Abada et al., 2009; Benedetti et al., 2014; Debbouche & Baleanu, 2011; Debbouche & Torres, 2013; Debbouche & Torres, 2014, 2015; Feckan et al., 2013; Guo et al., 2004; Henderson & Ouahab, 2010; Kumar & Sukavanam, 2012; Li et al., 2012; Liu & Li, 2015; Machado et al., 2013; Obukhovski & Zecca, 2009; Wang et al., 2015b, 2014a; Yan & Lu, 2016; Zhou et al., 2015) and the references therein. After reviewing some of the above articles, we find that the authors usually assume that the semi-group is compact or transfer the compactness condition into the non-linear part satisfies a condition expressed in terms of the measure of non-compactness. In this paper, by means of weakly topology theory of the state space and avoiding any regularity conditions on the multivalued non-linearity F expressed in terms of measures of non-compactness, we study the controllability of \begin{align} \left\{\begin{array}{l} ^{c}D_{0,t}^{\alpha}v(t)\in F(t,v(t))+B(u(t)),\\[5pt] \mbox{a.e.}\ t\in (\sigma_{i},\tau_{i+1}],\ i=0,1,\cdots,m,\ \alpha\in(0,1),\\[5pt] v(\tau_{i}^{+})=g_{i}(\tau_{i},v(\tau_{i}^{-})),\ i=1,\cdots,m, \\[5pt] v(t)=g_{i}(t,v(\tau_{i}^{-})),\ t\in (\tau_{i}, \sigma_{i}],\ i=1,\cdots,m,\\[5pt] v(0)=v_{0}, \end{array}\right.\end{align} (1) in a real reflexive Banach space E. Here, $$^{c}D_{0,t}^{\alpha }v$$ denotes a generalized Caputo derivative of α-order with zero as its lower limit (p.91, Kilbas et al., 2006), the multivalued non-linearity $$F:[0,b]\times E\rightarrow 2^{E}-\{{{\emptyset }}\}$$ is a given multifunction, the fixed points τi and σi satisfy 0 = σ0 < τ1 < σ1 < τ2 < ⋯ < τm < σm < τm+1 = b, $$v(\tau _{i}^{+})$$ and $$v(\tau _{i}^{-})$$ are the right and left limits of v at the point τi respectively and v0 ∈ E be a fixed point. Moreover, gi : [τi, σi] × E→E is a function for all i = 1, 2, ⋯ , m, the control function u, is given in Lp(J, X), $$p>\frac {1}{\alpha }$$, a Banach space of admissible control functions, with X being a real Banach space and B is a bounded linear operator from X into E. To the best of our knowledge, up to now, a few work has reported on the controllability for non-instantaneous impulsive fractional differential inclusion in Banach spaces even involving compactness condition. Moreover, we don’t assume any conditions on the multivalued function F expressed in terms of measures of non-compactness, and these facts are the main novelty in the present paper. Since our work space is piecewise continuous functions spaces, we have to seek a new sequence weakly converges criteria, which generalize the same result in continuous functions spaces. To obtain the controllability results, we utilize the weakly topology theory for weakly sequentially closed graph operator when the non-linearity satisfies both upper weakly sequentially closed and relatively weakly compactness. The structure of the paper is as follows. In Section 2, we collect some background material about multifunctions and fractional calculus to be used later. In particular, we establish sufficient and necessary conditions to guarantee a sequence in piecewise continuous function spaces is weakly convergence. In Section 3, we prove the main results, controllability results for (1) under the mild conditions via a fixed-point theorem for weakly sequentially closed graph operator. An example is given in the final section to demonstrate the application of our main results. 2. Preliminaries Let J = [0, b], $$P_{cl}(E)=\{\mathfrak {B}\subseteq E:\mathfrak {B}$$ is non-empty, convex and closed}, $$P_{ck}(E)=\{\mathfrak {B}\subseteq E:\mathfrak {B}\ \mbox {is non-empty, convex and compact} \}$$, $$P_{cwk}(E)=\{\mathfrak {B}\subseteq E:\mathfrak {B}$$ is non-empty, convex and weakly compact}, $$conv(\mathfrak {B})$$ (respectively, $$\overline {conv}(\mathfrak {B})$$) be the convex hull (respectively, convex closed hull in E) of a subset $$\mathfrak {B}$$. Let Ew be the space E endowed with the weak topology. For a set D ⊆ E, the symbol $$\overline {D}^{w}$$ denotes the weak closure of D. We recall that any bounded set in a reflexive Banach space is weakly relatively compact. If X is a normed space and $$G:J\rightarrow P_{cl}(X)$$, then the set $$ {S_{G}^{p}}=\{f\in L^{p}(J,X):f(t)\in G(t),\ a.e.\ t\in J\}$$ is called the set of Lebesgue integrable selections of G. To give the concept of mild solution of (1) we consider the set of piecewise continuous functions \begin{align*} PC(J,E)&=\left\{v:J\rightarrow E:v_{|_{J_{i}}}\in C(J_{i},E),\ J_{i}:=(\tau_{i},\tau_{i+1}],\ i=0,1,2,\cdots ,m\ \text{and}\right.\\ &\quad\left.v(t_{i}^{+})\ \text{and}\ v(t_{i}^{-})\ \mbox{exist for each}\ i=1,2,\cdots,m\vphantom{\frac{}{2}}\right\}, \end{align*} which endowed with PC −norm: $$ \|v\|_{PC(J,E)}=\max \{\|v(t)\|:\ t\in J\}. $$ For sake of completeness, we recall some results that we will need in the main section. Lemma 2.1 (see Theorem 2.2, O'Regan, 2000) Let X be a metrizable locally convex linear topological space and let G be a weakly compact, convex subset of X. Suppose that $$R:G\rightarrow P_{cl}\left (G\right )$$ has weakly sequentially closed graph. Then R has a fixed point. The following result is well-known in weakly topology theory. We denote $$\rightharpoonup $$ by weak convergence. Definition 2.2 (see p.924, Bochner & Taylor, 1938) A sequence {xn} of elements of Banach space X is said to converge weakly to x ∈ X if $$\lim _{n\to \infty }\widetilde {T}(x_{n})=\widetilde {T}(x)$$ for each linear functional $$\widetilde {T}$$ defined on X (i.e. $$\widetilde {T}\in X^{*}$$). Lemma 2.3 (see Theorem 4.3, Bochner & Taylor, 1938) A sequence {vn}⊆ C(J, X) converges weakly to an element v ∈ C(J, X) if and only if there is a positive real number L such that, for every $$n\in \mathbb {N}$$ and t ∈ J, ∥vn(t)∥≤ L; and $$v_{n}(t) \rightharpoonup v(t)$$ for each t ∈ J. Remark 2.4 From Lemma 2.3, one can conclude that in order that vn converges weakly to v ∈ C(J, X), we need to check two steps. In the first step, ∥vn∥ is uniformly bounded and in the second step, $$\widetilde {T}(v_{n})\rightarrow \widetilde {T}(v)$$ for each $$n\in \mathbb {N}$$ for bounded and linear functional $$\widetilde {T}: C(J,X)\to \mathbb {R}$$. Next, we give an analogous version for Lemma 2.3 in PC(J, X), which will be used in the sequel. In addition, this result can be also applied to deal with other relative problems. Lemma 2.5 Let E be a Banach space. A sequence {vn} in PC(J, E) weakly converges to an element v ∈ PC(J, E) if and only if There exists a L > 0 such that ∥vn(t)∥≤ L for every $$n\in \mathbb {N}$$ and t ∈ J. For each t ∈ Ji, i = 0, 1, 2, ⋯ , m, $$v_{n}(t)\rightharpoonup v(t)$$. For each i = 0, 1, 2, ⋯ , m, $$v_{n}(\tau _{i}^{+}) \rightharpoonup v(\tau _{i}^{+})$$. Proof. We show the sufficiency. Let $$T:PC(J,E)\rightarrow \mathbb {R}$$ be a linear bounded functional. For any i = 0, 1, 2, ⋯ , m, define $$T_{i}:C(\overline {J_{i}},E)\rightarrow \mathbb {R}$$, $$\overline {J_{i}}:=[\tau _{i},\tau _{i+1}]$$ as follows: let $$f\in C(\overline {J_{i}},E)$$ and define $$f_{i}: J\rightarrow E$$, where \begin{align*} f_{i}(t)= \left\{\begin{array}{l} f(t),\ t\in J_{i},\ i=0,1,\cdots,m, \\[3pt] 0,\ t\notin J_{i}. \end{array}\right. \end{align*} Then, we put Ti(f) := T(fi). Obviously, Ti is linear and bounded. In fact, for any $$f,g\in C(\overline {J_{i}},E)$$ and any $$\alpha ,\beta \in \mathbb {R}$$ we have \begin{align*} T_{i}(\alpha f+\beta g)&=T((\alpha f+\beta g)_{i})=T(\alpha f_{i}+\beta g_{i}) \\ &=\alpha T(f_{i})+\beta T(g_{i})=\alpha T_{i}(f)+\beta T_{i}(g). \end{align*} Next, for any $$f\in C(\overline {J_{i}},E)$$, one obtains \begin{align*} \|T_{i}(f)\|=\|T(f_{i})\|\leq\|T\|\|f_{i}\|_{C(J,E)}=\|T\|\|f\|_{C(\overline{J_{i}},E)}. \end{align*} Now, for any z ∈ PC(J, E), $$z=\sum _{i=0}^{i=m}z_{i}$$, where \begin{align*} z_{i}(t)= \left\{\begin{array}{l} z(t),\ t\in J_{i},\ i=0,1,\cdots,m, \\ 0,\ t\notin J_{i}. \end{array}\right. \end{align*} From the linearity of T, we get \begin{align} T(z)=\sum_{i=0}^{i=m}T_{i}(z_{i})=\sum_{i=0}^{i=m}T_{i\ }((z_{|\overline{J_{i}}})^{\ast}), \end{align} (2) where $$(z_{|\overline {J_{i}}})^{\ast }:\overline {J_{i}}\rightarrow E$$ is given by \begin{align*} (z_{|\overline{J_{i}}})^{\ast }(t)= \left\{\begin{array}{l} z(t),\ t\in J_{i}, \\ z(\tau_{i}^{+}),\ t=\tau_{i}. \end{array}\right. \end{align*} By applying Lemma 2.3, the conditions (i) and (ii) imply that $$ (v_{n|\overline {J_{i}}})^{\ast } \rightharpoonup (v_{|\overline {J_{i}}})^{\ast } $$ weakly in $$C(\overline {J_{i}},E)$$. Further, for any i = 0, 1, 2, ⋯ , m, we obtain \begin{align} \lim\limits_{n\rightarrow\infty}T_{i}\left(\left(v_{n|\overline{J_{i}}}\right)^{\ast}\right) =T_{i}\left(\lim\limits_{n\rightarrow\infty}\left(v_{n|\overline{J_{i}}}\right)^{\ast}\right)=T_{i}\left(v^{\ast}_{|\overline{J_{i}}}\right). \end{align} (3) Therefore, linking (2) and (3), we have \begin{align*}\lim\limits_{n\rightarrow \infty }T(v_{n})=\lim\limits_{n\rightarrow \infty }\sum_{i=0}^{i=m}T_{i}\left(\left(v_{n|\overline{J_{i}}}\right)^{\ast}\right)=\sum_{i=0}^{i=m}T_{i}\left(\left(v_{|\overline{J_{i}}}\right)^{\ast}\right)=T(v).\end{align*} Thus, $$v_{n}\rightharpoonup v$$ weakly in PC(J, E). Now, we show necessity. Assume that $$v_{n}\rightharpoonup v$$ in PC(J, E). For any t ∈ J, consider the following two functions \begin{align*} \delta_{t}:\ PC(J,E)\rightarrow\mathbb{R}, \ \ \delta_{t}(f)=v(f(t)) \end{align*} and \begin{align*} \rho_{t}:\ PC(J,E)\rightarrow\mathbb{R}, \ \ \rho_{t}(f)=v(f(t^{+})). \end{align*} It is easy to see that δt and ρt are linear and bounded. Since $$v_{n}\rightharpoonup v$$ in PC(J, E), then $$\delta _{t}(v_{n})\rightarrow $$δt(v), and $$\rho _{t}(v_{n})\rightarrow $$ρt(v). Hence, we get (ii) and (iii) via Definition 1. Moreover, it is known that any weakly convergent sequence is bounded. Hence (i) is satisfied. The proof is completed.□ In the following lemma, we recall another well-known result, Krein–Simulian theorem. Lemma 2.6 (p.434 Dunford & Schwartz, 1976) The convex hull of a weakly compact set in a Banach space X is weakly compact. Remark 2.7 Every open (closed) set in Ew is open (closed) in E. If G is closed and convex in E, then G is closed in Ew. In fact, let vn$$\rightharpoonup v$$, vn ∈ G. By Mazure’s lemma, there is a sequence of convex combinations of vn, denoted by $$\widetilde {v_{n}}$$, such that $$\widetilde {v_{n}}\rightarrow v$$ in E. Since G is convex, $$\widetilde {v_{n}}\in G$$. From the closedness of G we get v ∈ G. If vn$$\rightharpoonup v$$ and G is a weakly open and v ∈ G, then there is a natural number N such that vn ∈ G, ∀n ≥ N. Indeed, since G is a weakly open and v ∈ G, then from the definition of the weak topology, there is ϵ > 0 and finite number of linear continuous functionals f1 , f2, ⋯ , fm such that v + {z : | fk(z)| < ϵ, ∀ k = 1, 2, ⋯ , m} ⊆ G. From the weak convergence of vn towards v, there is a natural number N such that | fk(vn − v)| < ϵ, ∀ k = 1, 2, ⋯ , m, ∀ n ≥ N, which implies vn − v ∈{z : | fk(z)| < ϵ, ∀ k = 1, 2, ⋯ , m} and hence, vn ∈ G, ∀ n ≥ N. Let us recall some facts about multifunctions (see Aubin & Frankoeska, 1990). Definition 2.8 Let X and Y be two topological spaces. A multifunction $$G: X\longrightarrow P(Y)\backslash \{\mathcal {\emptyset }\}$$ is said to be upper semi-continuous at v0 ∈ X, u.s.c. for short, if for any open V containing G(v0) there exists a neighborhood N(v0) of v0 such that G(v) ⊆ V for all v ∈ N(v0). We say that F is upper semi-continuous if it is so at every v0 ∈ X. In the following lemma, we collect some properties of u.s.c. multifunctions Lemma 2.9 (Aubin & Frankoeska, 1990) Let X, Y be two Hausdorff topological spaces and $$G:X\longrightarrow P(Y)\backslash \{\mathcal {\phi }\}$$. If G is upper semi-continuous with closed values, then the graph of G is closed in X × Y, that is to say, if yn ∈ G(xn); n ≥ 1 and $$(x_{n},y_{n})\rightarrow (x,y)$$ with respect to the product topology on X × Y, then y ∈ G(x); If G is a closed and locally compact (i.e. for any x ∈ X, there is a neighborhood N(x) of x such that {G(z) : z ∈ N(x)} is relatively compact in Y) with closed values, then G is u.s.c.; If F is upper semi-continuous and K is compact subset of X, then G(K) = {G(x) : x ∈ K} is compact in Y. Let l : J → E be a given function and consider \begin{align} \left\{\begin{array}{l} ^{c}D_{0,t}^{\alpha}v(t)=l(t),\text{ a.e. }t\in (\sigma_{i},\tau_{i+1}],\ i=0,1,\cdots,m, \\[3pt] v(\tau_{i}^{+})=g_{i}(\tau_{i},v(\tau_{i}^{-})),\ i=1,\cdots,m, \\[3pt] v(t)=g_{i}(t,v(\tau_{i}^{-})),\ t\in (\tau_{i},\sigma_{i}],\ i=1,\cdots,m, \\[3pt] v(0)=v_{0}. \end{array}\right. \end{align} (4) Similar to the procedure in (Lemma 2.7, Wang et al., 2014b), one can obtain the following result. Since the proof is standard, we omit the proof here. Lemma 2.10 Let $$l\in L^{p}(J,E), p> \frac {1}{\alpha }, \alpha \in (0,1)$$. A function $$v:J\rightarrow E$$ is a solution of \begin{align} v(t)= \left\{\begin{array}{l} v_{0}+\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha -1}l(s)\,\text{d}s,\ t\in [0,\tau_{1}], \\[10pt] g_{i}(t,v(\tau_{i}^{-})),\ t\in (\tau_{i},\sigma_{i}],\ i=1,2,\cdots,m, \\[8pt] g_{i}(\sigma_{i},v(\tau_{i}^{-}))-\displaystyle\frac{1}{\Gamma (\alpha )} \int_{0}^{\sigma_{i}}(\sigma_{i}-s)^{\alpha -1}l(s)\,\text{d}s \\[10pt] +\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha -1}l(s)\,\text{d}s,\ t\in [ \sigma_{i},\tau_{i+1}],\ i=1,2,\cdots,m. \end{array}\right. \end{align} (5) if and only if v is a solution of (4). The concept of PC-mild solutions of (1) is introduced as follows: Definition 2.11 A function v ∈ PC(J, E) is said to be a mild solution for (1) if there exists an integrable selection f from $$S_{F(\cdot ,v(\cdot ))}^{p}$$ such that for each t ∈ J, \begin{align*} v(t)= \left\{\begin{array}{l} v_{0}+\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}(f(s)+B(u(s))),\ t\in [0,\tau_{1}], \\[9pt] g_{i}(t,v(\tau_{i}^{-})),\ t\in (\tau_{i},\sigma_{i}],\ i=1,2,\cdots,m, \\[8pt] g_{i}(\sigma_{i},v(\tau_{i}^{-}))-\displaystyle\frac{1}{\Gamma(\alpha)}\int_{0}^{\sigma_{i}}(\sigma_{i}-s)^{\alpha -1}(f(s)+B(u(s)))\,\text{d}s \\[10pt] +\displaystyle\frac{1}{\Gamma(\alpha )}{\int_{0}^{t}}(t-s)^{\alpha -1}(f(s)+B(u(s)))\,\text{d}s,\ t\in(\sigma_{i},\tau_{i+1}],\ i=1,2,\cdots,m. \end{array}\right. \end{align*} Definition 2.12 The equation (1) is said to be controllable on J if for revery v0, v1 ∈ E, there exists a control function u ∈ Lp(J, X) such that a mild solution of (1) satisfies v(0) = v0 and v(b) = v1. 3. Main results In this section, we study the controllability of the system (1) via the weakly topology theory for weakly sequentially closed graph operator. We assume the following conditions: (H1) Let $$F:J\times E\rightarrow P_{ck}(E)$$ be a multifunction. For every v ∈ PC(J, E), the multifunction $$\cdot \rightarrow F(\cdot ,v)$$ has a measurable selection. (H2) For any natural number n there is a function $$\varphi _{n}\in L^{p}(J,\mathbb {R}^{+})$$ such that $$\sup \limits _{\|v\|\leq n}\|F(t,v)\|\leq \varphi _{n}(t)$$, for a.e. t ∈ J and \begin{align*} \underset{n\rightarrow\infty}{\lim\inf}\frac{\left\|\varphi_{n}\right\|_{L^{p}(J,\mathbb{R}^{+})}}{n}=0. \end{align*} (H3) For almost a.e. t ∈ J that the multifunction $$x\rightarrow F(t,x)$$ is u.s.c. from Xw to Xw. (H4) Let $$g_{i}:[\tau _{i},\sigma _{i}]\times E\rightarrow E,\ i=1,2,\cdots ,m$$. For every i = 1, 2, ⋯ , m, gi(t, ⋅), t ∈ J is continuous from Ew to itself and maps any bounded subset of E to a relatively weakly compact and there exists a positive constant hi such that \begin{align*} \left\| g_{i}(t,v)\right\|\leq h_{i}\left\|v\right\|,\ t\in(\tau_{i},\sigma_{i}],\ v\in E. \end{align*} (HW) Let $$B:X\rightarrow E$$ be a bounded linear operator. The linear bounded operator $$W:L^{p}(J,X)\rightarrow E \ (p>\frac {1}{\alpha })$$ defined by \begin{align*} W(u)=\frac{1}{\Gamma (\alpha )}{\int_{0}^{b}}(b-s)^{\alpha -1}B(u(s)) \text{ d}s -\frac{1}{\Gamma (\alpha )}\int_{0}^{\sigma_{m}}(\sigma_{m}-s)^{\alpha -1}B(u(s)) \text{ d}s, \end{align*} has an invertible $$W^{-1}:E\rightarrow L^{p}(J,X)/Ker(W)$$ and such that there exists a positive constant N such that ∥W−1∥≤ N and ∥B∥≤ N. Concerning on the construction of W−1, we will demonstrate the process in Section 4. For more examples, one can also refer to Feckan et al. (2013) and Wang et al. (2014a). Note that W is well defined. In fact, since $$p>\frac {1}{\alpha }$$, the functions $$\cdot \rightarrow (b-\cdot )^{\alpha -1}$$ and $$\cdot \rightarrow (\sigma _{m}-\cdot )^{\alpha -1}$$ belong to $$L^{\frac {p}{p-1}}([0,b],\mathbb {R}^{+})$$ and $$L^{\frac {p}{p-1}}([0,\sigma _{m}],\mathbb {R}^{+})$$, respectively. Then, by the Hölder’s inequality, for any u ∈ Lp(J, X), we have \begin{align*} \|W(u)\|\leq\frac{2N}{\Gamma(\alpha )}\|u\|_{L^{p}(J,X)}\left(\frac{p-1}{\alpha p-1}\right)^{\frac{p-1}{p}}b^{\alpha -\frac{1}{p}} \leq\frac{2N\eta }{\Gamma(\alpha)}\|u\|_{L^{p}(J,X)}. \end{align*} In view of (H1) for every v ∈ PC(J, E), the multifunction $$t\rightarrow F(t,v)$$ has a measurable selection f and by (H2) this selection belongs to $$S_{F(\cdot ,v(\cdot ))}^{p}$$. So, for any v ∈ PC(J, E) and any $$f\in S_{F(\cdot ,v)}^{p}$$, we can define, using (HW), the control function uv, f ∈ Lp(J, X) by \begin{align} u_{v,f}:&=W^{-1}\left[v_{1}-g_{m}(\sigma_{m},v(\tau_{m}^{-}))+\frac{1}{\Gamma(\alpha)} \int_{0}^{\sigma_{m}}(\sigma_{m}-s)^{\alpha -1}f(s)\text{ d}s\right.\nonumber\\ &\quad\left.-\frac{1}{\Gamma(\alpha)}{\int_{0}^{b}}(b-s)^{\alpha -1}f(s) \ \text{d}s\right]. \end{align} (6) Therefore, we can define a multifunction $$\mathcal R:PC(J,E)\rightarrow 2^{PC(J,E)}$$ as follows: For v ∈ PC(J, E), $$\mathcal R(v)$$ is the set of all the function y ∈ PC(J, E) such that \begin{align} y(t)= \left\{\begin{array}{l} v_{0}+\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}(f(s)+B(u_{v,f}(s))),\ t\in [0,\tau_{1}],\\[8pt] g_{i}(t,v(\tau_{i}^{-})),\ t\in (\tau_{i},\sigma_{i}],\ i=1,2,\cdots,m, \\[8pt] g_{i}(\sigma_{i},v(\tau_{i}^{-}))-\displaystyle\frac{1}{\Gamma(\alpha)}\int_{0}^{\sigma_{i}}(\sigma_{i}-s)^{\alpha -1}(f(s)+B(u_{v,f}(s)))\ \text{d}s \\ +\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}(f(s)+B(u_{v,f}(s)))\ \text{d}s,\\[8pt] \ \ t\in(\sigma_{i},\tau_{i+1}],\ i=1,2,\cdots,m, \end{array}\right. \end{align} (7) where $$f\in S_{F(\cdot ,v)}^{p}$$. In view of (H1) and (H3), the values of $$\mathcal R$$ are non-empty. Clearly, using the control function defined by (6), any fixed point for $$\mathcal R$$ is a mild solution for (1) and v(0) = v0 and v(b) = v1. In fact, if v is a fixed point for $$\mathcal R$$, then from (HW), (6) and (7), one has \begin{align} v(b)= \ &g_{m}(\sigma_{m},v(\tau_{m}^{-}))-\frac{1}{\Gamma (\alpha )} \int_{0}^{\sigma_{m}}(\sigma_{m}-s)^{\alpha -1}f(s)\ \text{d}s\\ &+\frac{1}{\Gamma (\alpha )} {\int_{0}^{b}}(b-s)^{\alpha -1}f(s)\ \text{d}s+W(u_{v,f}) \\ = \ &g_{m}(\sigma_{m},v(\tau_{m}^{-}))-\frac{1}{\Gamma (\alpha )} \int_{0}^{\sigma_{m}}(\sigma_{m}-s)^{\alpha -1}f(s)\ \text{d}s+\frac{1}{\Gamma (\alpha )} {\int_{0}^{b}}(b-s)^{\alpha -1}f(s)\ \text{d}s \\ &+v_{1}-g_{m}(\sigma_{m},v(\tau_{m}^{-}))+\frac{1}{\Gamma (\alpha )} \int_{0}^{\sigma_{m}}(\sigma_{m}-s)^{\alpha -1}f(s)\ \text{d}s\\ &-\frac{1}{\Gamma (\alpha )}{\int_{0}^{b}}(b-s)^{\alpha -1}f(s)\ \text{d}s =v_{1}. \end{align} In the following, we give the first controllability result for our problem. Theorem 3.1 Assume that (H1) − (H4) and (HW) are fulfilled. Then (1) is controllable on J provided that \begin{align} \frac{{2}N^{2}\eta h}{\Gamma (\alpha )}+h<1, \end{align} (8) where $$\eta =b^{\alpha -\frac {1}{p}}\left (\frac {p-1}{p\alpha -1}\right )^{\frac {p-1}{p}}$$ and $$h=\sum _{i=0}^{i=m}h_{i}$$. Proof. Our goal is to prove, by using Lemma 1, that $$\mathcal R$$ has a fixed point. The proof will be given in several steps. Step 1. In this step, we claim that there is a natural number n such that $$\mathcal R(\mathcal B_{n})\subseteq \mathcal B_{n}$$ where $$\mathcal B_{n}=\{v\in PC(J,E):\left \Vert v\right \Vert_{PC(J,E)}\leq n\}$$ and $$\mathcal R(\mathcal B_{n})=\{ y\in \mathcal R(u),u\in \mathcal B_{n}: \|y\|_{PC(J,E)}\leq n\}$$. If not, we suppose that the contrary holds. Then, for any natural number n, there are vn, yn ∈ PC(J, E) with $$y_{n}\in \mathcal R(v_{n}),\left \Vert v_{n}\right \Vert_{PC(J,E)}\leq n$$ and $$\left \Vert y_{n}\right \Vert_{PC(J,E)}>n$$. Then, there is $$\{f_{n}\}_{n\geq 1}\in S_{F(\cdot ,v_{n}(\cdot ))}^{p}$$ such that \begin{align*} y_{n}(t)= \left\{\begin{array}{l} v_{0}+\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}(f_{n}(s)+B(u_{v_{n},f_{n}}(s))\ \text{d}s,\ t\in[0,\tau_{1}], \\[10pt] g_{i}(t,v_{n}(\tau_{i}^{-})),\ t\in (\tau_{i},\sigma_{i}],\ i=1,2,\cdots,m, \\[8pt]\displaystyle g_{i}(\sigma_{i},v_{n}(\tau_{i}^{-}))-\frac{1}{\Gamma (\alpha )}\int_{0}^{\sigma_{i}}(\sigma_{i}-s)^{\alpha -1}(f_{n}(s)+B(u_{v_{n},f_{n}}(s))\ \text{d}s \\[8pt] +\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}(f_{n}(s)+B(u_{v_{n},f_{n}}(s))\ \text{d}s,\ t\in(\sigma_{i},\tau_{i+1}],\ i=1,2,\cdots,m. \end{array}\right. \end{align*} Then, there exists t ∈ [0, τ1], we get by Hölder’s inequality that \begin{align*} n&<\|y_{n}(t)\| \nonumber\\ &\leq\|v_{0}\|+\frac{1}{\Gamma(\alpha)}{\int_{0}^{t}}(t-s)^{\alpha-1}\varphi_{n}(s)\ \text{d}s +\frac{N}{\Gamma(\alpha)}{\int_{0}^{t}}(t-s)^{\alpha-1}\|u_{v_{n},f_{n}}(s)\|\ \text{d}s \nonumber\\ &\leq\|v_{0}\|+\frac{1}{\Gamma(\alpha)}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}b^{\alpha -\frac{1}{p}}\left(\frac{p-1}{p\alpha -1}\right)^{\frac{p-1}{p}} \nonumber\\ &\quad+\frac{N}{\Gamma(\alpha)}\|u_{v_{n},f_{n}}\|_{L^{p}(J,\mathbb{R}^{+})}b^{\alpha -\frac{1}{p}}\left(\frac{p-1}{p\alpha -1}\right)^{\frac{p-1}{p}}. \end{align*} Remark that \begin{align} \|u_{v_{n},f_{n}}\|_{L^{p}(J,\mathbb{R}^{+})} &\leq\|W^{-1}\|\left[\|v_{1}\|+\|g_{m}(\sigma_{m},v(\tau_{m}^{-}))\|+\frac{1}{\Gamma(\alpha )}\int_{0}^{\sigma_{m}}(\sigma_{m}-s)^{\alpha -1}\varphi_{n}(s)\ \text{d}s \nonumber\right.\\ &\quad+\left.\frac{1}{\Gamma(\alpha )}{\int_{0}^{b}}(b-s)^{\alpha-1}\varphi_{n}(s)\ \text{d}s\right]\nonumber\\ &\leq N\left[\|v_{1}\|+h_{m}n+\frac{2}{\Gamma(\alpha)}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}b^{\alpha -\frac{1}{p}}\left(\frac{p-1}{p\alpha -1}\right)^{\frac{p-1}{p}}\right]. \end{align} (9) Then \begin{align} n&<\|v_{0}\|+\frac{1}{\Gamma(\alpha )}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}\eta+\frac{N^{2}}{\Gamma (\alpha )}\eta\left[\|v_{1}\|+h_{m}n+\frac{2}{\Gamma (\alpha )}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}\eta \right]. \end{align} (10) By dividing both side of (10) by n and taking the limit as $$n\rightarrow \infty $$, we obtain \begin{align*} 1<\frac{N^{2}\eta }{\Gamma(\alpha)}h, \end{align*} which contradicts (8). For t ∈ (τi, σi], i = 1, 2, ⋯ , m, then \begin{align} n<\|y(t)\|\leq \|g_{i}(t,v_{n}(\tau_{i}^{-}))\|\leq h_{i}\|v_{n}(\tau_{i}^{-}))\|\leq h_{i}n. \end{align} (11) By dividing both side of (11) by n and taking the limit as $$n\rightarrow \infty $$, we obtain \begin{align*} 1\leq h_{i}<h,\, \end{align*} which contradicts (8) again. Similarly, we get for t ∈ [σi, τi+1], i = 1, 2, ⋯ , m. \begin{align} n<&\|y(t)\| \nonumber \\ \leq &\ h_{i}n+\frac{1}{\Gamma(\alpha )}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}\left[\left(\int_{0}^{\sigma_{i}}(\sigma_{i}-s)^{\frac{p(\alpha-1)}{p-1}}\ \text{d}s\right)^{\frac{p-1}{p}}\nonumber\right.\\ &+\left.\left({\int_{0}^{t}}(t-s)^{\frac{p(\alpha -1)}{p-1}}\ \text{d}s\right)^{\frac{p-1}{p}}\right] +\frac{2N^{2}}{\Gamma(\alpha)}\eta\left[\|v_{1}\|+hn+\frac{2}{\Gamma(\alpha )}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}\eta\right] \nonumber \\ \leq&\ hn+\frac{2}{\Gamma(\alpha)}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}\eta +\frac{2N^{2}}{\Gamma (\alpha )}\eta\left[\|v_{1}\|+hn +\frac{2}{\Gamma (\alpha)}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}\eta\right]. \end{align} (12) By dividing both side of (12) by n, and passing to the limit as $$n\rightarrow \infty $$, we get \begin{align*} 1<h+\frac{2N^{2}\eta h}{\Gamma(\alpha)}, \end{align*} which contradicts (8) again. Thus, there exists a n0 such that $$\mathcal R(\mathcal B_{n_{0}})\subseteq \mathcal B_{n_{0}}$$. Step 2. The restriction of $$\mathcal R$$ on $$\mathcal B_{n_{0}}$$ has a weakly sequentially closed graph. Consider a sequence {vn}n≥1 with $$v_{n}\rightharpoonup v$$ in $$\mathcal B_{n_{0}}$$ and let $$y_{n}\in \mathcal R(v_{n})$$ with $$y_{n}\rightharpoonup y$$ in PC(J, E). We have to show that $$y\in \mathcal R(v)$$. By recalling the definition of $$\mathcal R$$, for any n ≥ 1, there is $$f_{n}\in S_{F(\cdot ,v_{n}(\cdot ))}^{p}$$ such that \begin{align} y_{n}(t)= \left\{\begin{array}{l} v_{0}+\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}(f_{n}(s)+B(u_{v_{n},f_{n}}(s)))\ \text{d}s,\ t\in [0,\tau_{1}], \\[10pt] g_{i}(t,v_{n}(\tau_{i}^{-})),\ t\in (\tau_{i},\sigma_{i}],\ i=1,2,\ldots,m, \\[10pt] g_{i}(\sigma_{i},v_{n}(\tau_{i}^{-}))-\displaystyle\frac{1}{\Gamma (\alpha )}\int_{0}^{\sigma_{i}}(\sigma_{i}-s)^{\alpha -1}(f_{n}(s)+B(u_{v_{n},f_{n}}(s)))\ \text{d}s \\[10pt] +\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}(f_{n}(s)+B(u_{v_{n},f_{n}}(s)))\ \text{d}s,\\[10pt] \ \ t\in(\sigma_{i},\tau_{i+1}],i=1,2,\cdots,m. \end{array}\right. \end{align} (13) Since $$v_{n}\in \mathcal B_{n_{0}}$$, then the set Z = {vn(t) : n ≥ 1, t ∈ J} is contained in $$\mathcal B(0,n_{0})=\{v\in E:\|v\|\leq n_{0}\}$$. Hence, by (H2) there is $$\varphi _{n_{0}}\in L^{p}(J,\mathbb {R}^{+})$$ such that $$ \|f_{n}(t)\|\leq \varphi _{n_{0}}(t),\ \forall \ n\geq 1,\ a.e.\ t\in J. $$ This implies that the sequence {fn : n ≥ 1} is bounded in Lp(J, E). Since $$ p>\frac {1}{\alpha }>1,$$ then Lp(J, E) is reflexive, and hence the set {fn : n ≥ 1} is relatively weakly compact in Lp(J, E). Therefore, there is a subsequence, denoted again by (fn) such that $$ f_{n}\rightharpoonup f\in L^{p}(J,E).$$ Next, let $$\mathcal S_{1}$$ and $$\mathcal S_{2}$$ be two operators from Lp(J, E) to C(J, E) as: \begin{align*} \mathcal S_{1}(f)(t)=\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}f(s)\ \text{d}s, \end{align*} and \begin{align*} \mathcal S_{2}(f)(t)=\,&\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}BW^{-1}\left[\frac{1}{\Gamma (\alpha )}\int_{0}^{\sigma_{m}}(\sigma_{m}-\tau )^{\alpha-1}f(\tau)\ \text{d}\tau\right. \\ &\left.-\frac{1}{\Gamma(\alpha)}{\int_{0}^{b}}(b-\tau )^{\alpha-1}f(\tau)d\tau\right](s)\ \text{d}s. \end{align*} From the linearity of the integral operator and of the operators B and W−1, it is easy to see that $$\mathcal S_{1}$$ and $$\mathcal S_{2}$$ are linear. Furthermore, for t ∈ J, \begin{align} \|\mathcal S_{1}(f)(t)\|\leq \frac{\eta }{\Gamma (\alpha )}\|f\|_{L^{p}(J,E)}. \end{align} (14) Also, by Hölder’s inequality we have \begin{align} \|&\mathcal S_{2}(f)(t)\|\\&\leq\frac{\eta N}{\Gamma(\alpha )}\left\|W^{-1}\left[\frac{1}{\Gamma(\alpha )}\int_{0}^{\sigma_{m}}(\sigma_{m}-\tau)^{\alpha-1}f(\tau)\ \text{d}\tau -\frac{1}{\Gamma(\alpha)}{\int_{0}^{b}}(b-\tau )^{\alpha -1}f(\tau)\ \text{d}\tau\right]\right\| \nonumber\\ &\leq \frac{\eta N}{\Gamma(\alpha )}\|W^{-1}\|\left\|\frac{1}{\Gamma(\alpha)}\int_{0}^{\sigma_{m}}(\sigma_{m}-\tau )^{\alpha -1}f(\tau )\ \text{d}\tau -\frac{1}{\Gamma(\alpha)}{\int_{0}^{b}}(b-\tau)^{\alpha -1}f(\tau)\ \text{d}\tau\right\| \nonumber \\ &\leq \frac{\eta N^{2}}{(\Gamma (\alpha ))^{2}}\int_{0}^{\sigma_{m}}(\sigma_{m}-\tau )^{\alpha -1}\left\|f(\tau)\right\|\ \text{d}\tau+{\int_{0}^{b}}(b-\tau )^{\alpha -1}\left\|f(\tau)\right\|\ \text{d}\tau \nonumber \\ &\leq \frac{2\eta^{2}N^{2}}{(\Gamma (\alpha ))^{2}}\|f\|_{L^{p}(J,E)}.\nonumber \end{align} (15) This shows that $$\mathcal S_{1}$$ and $$\mathcal S_{2}$$ are linear bounded and hence, continuous. Let us claim that $$\mathcal S_{i}(f_{n})\rightharpoonup \mathcal S_{i}(f),\ i=1,2$$, in C(J, E). Let $$v:E\rightarrow \mathbb {R}$$ be a linear continuous operator and t be a fixed point in J. By the linearity and continuity of $$\mathcal S_{i}$$, the operator $$f\rightarrow v(\mathcal S_{i}(f)(t)\ $$is linear and continuous from Lp(J, E) to $$\mathbb {R}$$, and hence $$\mathcal S_{i}(f_{n})(t)\rightharpoonup \mathcal S_{i}(f)(t),i=1,2$$. Hence, in view of Lemma 2.3, (14) and (15), we conclude that $$\mathcal S_{i}(f_{n})\rightharpoonup \mathcal S_{i}(f),i=1,2$$. Furthermore, by (H4), $$g_{i}(t,v_{n}(\tau _{i}^{-}))\rightharpoonup g_{i}(t,v(\tau _{i}^{-})),\ t\in (\tau _{i},\sigma _{i}],\ i=1,2,\cdots ,m$$. Then, by the linearity of the integral operator and of the operators B and W−1 we get \begin{align*} \frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}BW^{-1}[v_{1}-g_{m}(\sigma_{m},v_{n}(\tau_{m}^{-}))](s)\ \text{d}s, \end{align*} converges weakly to \begin{align*} \frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}BW^{-1}[v_{1}-g_{m}(\sigma_{m},v(\tau_{m}^{-}))](s)\ \text{d}s. \end{align*} From the above discussion we obtain $$y_{n}(t)\rightharpoonup w(t)$$, for all t ∈ J, and $$y_{n}(\tau _{i}^{+})\rightharpoonup w(\tau _{i}^{+})$$, i = 0, 1, 2, ⋯ , m, where \begin{align} w(t)= \left\{\begin{array}{l} v_{0}+\displaystyle\frac{1}{\Gamma(\alpha)}{\int_{0}^{t}}(t-s)^{\alpha-1}(h(s)+B(u_{v,f}(s))),\ t\in [0,\tau_{1}], \\[8pt] g_{i}(t,v(\tau_{i}^{-})),\ t\in (\tau_{i},\sigma_{i}],\ i=1,2,\cdots,m, \\[8pt] g_{i}(\sigma_{i},v(\tau_{i}^{-}))-\displaystyle\frac{1}{\Gamma (\alpha )}\int_{0}^{\sigma_{i}}(s_{i}-s)^{\alpha -1}(f(s)+B(u_{v,f}(s)))\ \text{d}s \\[8pt] +\displaystyle\frac{1}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}(f(s)+B(u_{v,f}(s)))\ \text{d}s,\\[8pt] \ \ t\in(\sigma_{i},\tau_{i+1}],\ i=1,2,\cdots,m. \end{array}\right. \end{align} (16) Observe that, by (H4), for any i = 1, 2, ⋯ , m, \begin{align*} y_{n}(\tau_{i}^{+})=g_{i}(\tau_{i},v_{n}(\tau_{i}^{-}))\rightharpoonup g_{i}(\tau_{i},v(\tau_{i}^{-}))=w(\tau_{i}^{+}). \end{align*} Also, by arguing as in (9), (11) and (12), we have for ∈ J, \begin{align*} \|y_{n}(t)\|&\leq\|v_{0}\|+\frac{2}{\Gamma (\alpha )}\|\varphi_{n}|_{L^{p}(J,\mathbb{R}^{+})}\eta +\frac{2N^{2}}{\Gamma(\alpha )}\eta\left[\|v_{1}\|+hn_{0} +\frac{2}{\Gamma(\alpha)}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}\eta\right]+hn_{0}. \end{align*} Then, by Lemma 2.5, $$y_{n}\rightharpoonup w$$ in PC(J, E). By the uniqueness of the weak limit we get y(t) = w(t), t ∈ J. It remains to shows that f(t) ∈ F(t, v(t)), for a.e. t ∈ J. By Mazure’s lemma, there is a sequence, $$\widetilde {f_{n}}$$, of convex combinations of fn such that $$\widetilde {f_{n}}\rightarrow f$$ in Lp(J, E). So, there is a subsequence, denoted again by $$\widetilde {f_{n}}$$, such that $$\widetilde {f_{n}}(t)\rightarrow f(t)$$, for a.e. t ∈ J. Let A be a set of Lebesgue measure zero such that for any t ∈ J − A, $$\widetilde {f_{n}}(t)\rightarrow f(t)$$, fn(t) ∈ F(t, vn(t)) and F(t, ⋅) is upper semi-continuous multifunction from Ew to Ew. Fix t0 ∈ J − A and assume, by contradiction, that f(t0)∉F(t0, v(t0)). Since F(t0, v(t0)) is closed and convex from Hahn Banach theorem, there is a weakly open convex set V such that F(t0, v(t0)) ⊆ V and $$f(t_{0})\notin \overline {V}$$. Because F(t0, ⋅) is upper semi-continuous at v(t0) from Ew to Ew, there is a weakly neighborhood U for v(t0) such that if z ∈ U, then F(t0, z) ⊆ V. The convergence $$v_{n}(t_{0})\rightharpoonup v(t_{0})$$ implies the existence of a natural number N such that vn(t0) ∈ U, ∀ n ≥ N (see Remark 2(ii)). Hence fn(t0) ∈ F(t0, vn(t0)) ⊆ V, ∀ n ≥ N. Since V is convex, $$\widetilde {f_{n}}(t_{0})\in V,\ \forall \ n\geq N$$. This implies that $$f(t_{0})\in \overline {V}$$, which contradicts with the fact that $$f(t_{0})\notin \overline {V}$$. Therefore, f(t) ∈ F(t, v(t)), for a.e. t ∈ J. We remark that a similar result is also reported in Zhou et al. (2015). Step 3. The restriction of $$\mathcal R$$ on $$\mathcal B_{n_{0}}$$ is weakly compact, that is, $$\mathcal R$$ maps any bounded subset of $$\mathcal B_{n_{0}}$$ into relatively weakly compact. Obviously, it suffices to show that $$\mathcal R(\mathcal B_{n_{0}})$$ is relatively weakly compact. Consider a sequence {vn}n≥1 in $$\mathcal B_{n_{0}}$$ and $$y_{n}\in \mathcal R(v_{n})$$. By recalling the definition of $$\mathcal R$$, for any n ≥ 1, there is $$f_{n}\in S_{F(\cdot ,v_{n}(\cdot ))}^{p}$$ such that (13) holds. By arguing as in the previous step, there is a subsequence of {fn}, denoted again by {fn} such that $$f_{n}\rightharpoonup f\in L^{p}(J,E)$$, and $$y_{n}\rightharpoonup w$$, where w is given by (16). Then $$\mathcal R(B_{n_{0}})$$ is relatively weakly compact. Step 4. For any $$v\in \mathcal B_{n_{0}}$$, $$\mathcal R(v)$$ is convex and weakly compact. Since the values of F are convex, it is easy to see that $$\mathcal R(v)$$ is convex. The weak compactness of $$\mathcal R(v)$$ follows from Step 3. Now, set $$W_{n_{0}}=\overline {co}\left(\overline {\mathcal R(\mathcal B_{n_{0}})}^{\,w}\right)$$. From Step 4, $$\left(\overline {\mathcal R(\mathcal B_{n_{0}})}^{\,w}\right)$$ is weakly compact, and, by Lemma 2.6, $$W_{n_{0}}$$ is convex weakly compact. Moreover, from the fact that $$\mathcal B_{n_{0}}$$ is convex and closed we deduce, from Remark 2.7(i), that $$\overline {\mathcal B_{n_{0}}}^{\,w}=\mathcal B_{n_{0}}$$. Then, by Step 1, \begin{align*} \mathcal R(W_{n_{0}})\subseteq \mathcal R\left(\overline{co}(\overline{\mathcal R(\mathcal B_{n_{0}})}^{\,w}\right)\subseteq \mathcal R\left(\overline{co}(\overline{\mathcal B_{n_{0}})}^{\,w}\right)=\mathcal R(\overline{co} (\mathcal B_{n_{0}}))=\mathcal R(\mathcal B_{n_{0}})\subseteq W_{n_{0}}. \end{align*} By applying Lemma 2.1, $$\mathcal R$$ has a fixed point and the proof is completed. □ In the following, we give the second controllability result for (1) under less restrictive growth assumption. (H2)* There exists $$\varphi \in L^{p}(J,\mathbb {R}^{+})$$ such that for any v ∈ E, \begin{align} \|F(t,v)\|\leq\varphi(t)(1+\|v\|),\ for\ a.e.\ t\in J, \end{align} (17) Theorem 3.2 Assume that (H1), (H2)*, (H3), (H4) and (HW) hold. Then (1) is controllable on J provided that \begin{align} h+\frac{2\eta}{\Gamma(\alpha)}\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}+\frac{2N^{2}\eta}{\Gamma(\alpha)}\left[h+\frac{2\eta }{\Gamma(\alpha )}\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}\right]<1. \end{align} (18) Proof. Since the proof is very similar to Theorem 3.1, we only present the main different parts. In fact, we only need to show that there is a natural number n such that $$\mathcal R(\mathcal B_{n})\subseteq \mathcal B_{n}$$, where $$\mathcal R$$ is defined in the above theorem. Suppose the contrary holds. Then, for any natural number n, there are vn, yn ∈ PC(J, E) with $$y_{n}\in \mathcal R(v_{n})$$, ∥vn∥PC(J, E) ≤ n and $$\left \Vert y_{n}\right \Vert _{PC(J,E)}>n$$. Then, there is $$\{f_{n}\}_{n\geq 1}\in S_{F(\cdot ,v_{n}(\cdot ))}^{p}$$ such that (13) holds. Then there exits t ∈ [0, τ1], we get by (17) and Hölder’s inequality \begin{align*} n&<\|y_{n}(t)\|\leq \|v_{0}\|+\ \frac{(1+n)}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha -1}\varphi (s)\ \text{d}s +\frac{N}{\Gamma (\alpha )}{\int_{0}^{t}}(t-s)^{\alpha-1}\|u_{v_{n},f_{n}}(s)\|\ \text{d}s \\ &\leq \|v_{0}\|+\ \frac{(1+n)}{\Gamma (\alpha )}\|\varphi \|_{L^{p}(J,\mathbb{R}^{+})}\eta +\frac{N}{\Gamma (\alpha )}\|u_{v_{n},f_{n}}\|_{L^{p}(J,\mathbb{R}^{+})}\eta. \end{align*} Remark that (9), then \begin{align*} n &<\|v_{0}\|+\frac{(1+n)}{\Gamma (\alpha )}\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}\eta +\frac{N^{2}}{\Gamma(\alpha )}\eta\left[\|v_{1}\|+hn+\frac{2(1+n)}{\Gamma (\alpha )}\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}\eta\right]. \end{align*} By dividing both side of the above inequality by n and taking the limit as $$n\rightarrow \infty $$, we obtain \begin{align*} 1<\frac{\eta}{\Gamma(\alpha )}\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}+\frac{N^{2}\eta }{\Gamma(\alpha)}\left[h+\frac{2\eta}{\Gamma(\alpha) }\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}\right], \end{align*} which contradicts (18). For t ∈ (τi, σi], i = 1, 2, ⋯ , m, the proof is the same as the procedure in the above theorem. Finally, we get for t ∈ [σi, τi+1], i = 1, 2, ⋯ , m. \begin{align*} n <&\|y(t)\| \\ \leq&\ hn+\frac{(1+n)\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}}{\Gamma (\alpha)}\left[\left(\int_{0}^{\sigma_{i}}(\sigma_{i}-s)^{\frac{p(\alpha -1)}{p-1}}\ \text{d}s\right)^{\frac{p-1}{p}}+\left({\int_{0}^{t}}(t-s)^{\frac{p(\alpha -1)}{p-1}}\ \text{d}s\right)^{\frac{p-1}{p}}\right]\\ &+\frac{2N^{2}}{\Gamma (\alpha )}\eta \left[\|v_{1}\|+hn+\frac{2\eta(n+1)}{\Gamma (\alpha )}\|\varphi_{n}\|_{L^{p}(J,\mathbb{R}^{+})}\right] \\ \leq&\ hn+\frac{\ 2(1+n)\eta }{\Gamma (\alpha )}\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}+\frac{2N^{2}\eta }{\Gamma(\alpha )}\left[\|v_{1}\|+hn+\ \frac{2\eta (1+n)}{ \Gamma (\alpha )}\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}\right]. \end{align*} By dividing both side of the above inequality by and passing to the limit as $$n\rightarrow \infty $$, we get \begin{align*} 1<h+\frac{2\eta }{\Gamma (\alpha )}\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}+\frac{2N^{2}\eta }{\Gamma (\alpha )}\left[h+ \frac{2\eta }{\Gamma (\alpha ) }\|\varphi\|_{L^{p}(J,\mathbb{R}^{+})}\right], \end{align*} which contradicts (18). The rest of the proof is very similar to Theorem 3.1, so we omit it here. □ 4. An example Take J = [0, 1], E = X = L2[0, 1] and p = 2. Clearly, E is a separable Hilbert space. For any function $$x:J\rightarrow \ L^{2}[0,1]$$ and any t ∈ [0, 1], we define x(t)(y) := x(t, y), y ∈ J. Let $$F:J\times L^{2}[0,1]\rightarrow 2^{L^{2}[0,1]}-\{{{\emptyset }}\},z\in F(t,x)\Longleftrightarrow z(y)\in P(t,x(t,y))$$, where $$P:J\times \mathbb {R}\rightarrow P_{ck}(\mathbb {R})=\{z\in L^{2}[0,1]$$ is a chosen function such that (H1) − (H3) are verified} and $$g_{i}:[\tau _{i},\sigma _{i}]\times L^{2}[0,1]\rightarrow L^{2}[0,1],\ g_{i}(t,x)=tx,\ i=1,\cdots ,m$$. Obviously, (H4) holds. Consider the following non-instantaneous impulsive fractional control systems: \begin{align} \left\{\begin{array}{l} ^{c}D_{0,t}^{\frac{2}{3}}x(t,y)\in F(t,x(t,y))+B(u(t,y)),\\[5pt] \ \mbox{a.e.}\ t\in(\sigma_{i},\tau_{i+1}],\ i=0,1,\cdots,m,\ y\in J, \\[5pt] x(\tau_{i}^{+},y)=g_{i}(\tau_{i},x(\tau_{i}^{-},y)),\ i=1,\cdots,m,\ y\in J, \\[5pt] x(t,y)=g_{i}(t,x(\tau_{i}^{-},y)),\ t\in(\tau_{i},\sigma_{i}],\ i=1,\cdots,m,\ y\in J,\\[5pt] x(0,y)=\varphi(y),\ y\in J, \end{array}\right. \end{align} (19) where $$\alpha =\frac {2}{3}>\frac {1}{2}=\frac {1}{p}$$, u ∈ L2(J, L2[0, 1]) and φ ∈ L2[0, 1] is a fixed element. Next, let $$B:L^{2}[0,1]\rightarrow L^{2}[0,1],\ B=\gamma I_{d}$$, where Id is the identity operator and γ > 0. Define $$W:L^{2}(J,L^{2}[0,1])\rightarrow L^{2}[0,1]$$ by: \begin{align*} W(u)(\cdot):=\frac{\gamma }{\Gamma (\alpha )}{\int_{0}^{1}}(1-s)^{\frac{-1}{3}}u(s,\cdot)\ \text{d}s -\frac{\gamma}{\Gamma(\alpha)}\int_{0}^{\sigma_{m}}(\sigma_{m}-s)^{\frac{-1}{3}}u(s,\cdot)\ \text{d}s. \end{align*} It is easy to show that Wis linear and bounded. Next, we show that W is surjective. In fact, let x ∈ L2[0, 1] and define $$u:J\rightarrow L^{2}[0,1]$$, \begin{align} u(\cdot)(y)=\delta x(y),\ y\in J, \end{align} (20) where $$\delta =\frac {\Gamma (\alpha )}{\gamma [1-(\sigma _{m})^{\frac {1}{3}}]}$$. Then \begin{align*} W(u)(y)&:=\frac{\gamma\delta }{\Gamma(\alpha)}{\int_{0}^{1}}(1-s)^{\frac{-1}{3}}x(y)\ \text{d}s -\frac{\gamma\delta }{\Gamma(\alpha)}\int_{0}^{\sigma_{m}}(\sigma_{m}-s)^{\frac{-1}{3}}x(y)\ \text{d}s \\ &=\frac{\gamma\delta x(y)}{\Gamma(\alpha)}{\int_{0}^{1}}(1-s)^{\frac{-1}{3}}\ \text{d}s -\frac{\gamma\delta x(y)}{\Gamma(\alpha)}\int_{0}^{\sigma_{m}}(\sigma_{m}-s)^{\frac{-1}{3}}\ \text{d}s \\ &=\frac{\gamma\delta x(y)}{\Gamma(\alpha)}\left[1-(\sigma_{m})^{\frac{1}{3}}\right]=x(y). \end{align*} This shows that W(u) = x, and hence W is surjective. Consequently, W has an invertible $$W^{-1}:L^{2}[0,1]\rightarrow L^{2}(J,L^{2}[0,1])/Ker(W)$$, where W−1(x) = u and u is given by (20). Note that W−1 is linear and \begin{align*} \|W^{-1}(x)\|^{2}&={\int_{0}^{1}}\|u(s)\|^{2}\ \text{d}s \\&={\int_{0}^{1}}\left({\int_{0}^{1}}|u(s)(y)|^{2}\ \text{d}y\right)\ \text{d}s \\ &={\int_{0}^{1}}\left({\int_{0}^{1}}|\delta x(y)|^{2}\ \text{d}y\right)\ \text{d}s\\&=\delta^{2}\|x\|^{2}, \end{align*} which proves that W−1 is bounded and ∥W−1∥≤ δ. From above, by Theorem 3.1, the system (19) is controllable on J. 5. Conclusions Controllability of fractional non-instantaneous non-linear impulsive differential inclusions in Banach spaces have been investigated. By utilizing fixed point approach in the sense of weakly topology for weakly sequentially closed graph operator, sufficient conditions for controllability of such new case of impulsive differential equations are established under both upper weakly sequentially closed and relatively weakly compact types of non-linearity. 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IMA Journal of Mathematical Control and Information – Oxford University Press

**Published: ** Dec 22, 2017

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