\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 98, pp. 1--22.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/98\hfil Multi-strip BVP for fractional differential equations] {Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions} \author[B. Ahmad, S. K. Ntouyas\hfil EJDE-2012/98\hfilneg] {Bashir Ahmad, Sotiris K. Ntouyas} % in alphabetical order \address{Bashir Ahmad \newline Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia} \email{bashir\_qau@yahoo.com} \address{Sotiris K. Ntouyas \newline Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece} \email{sntouyas@uoi.gr} \thanks{Submitted May 20, 2012. Published June 12, 2012.} \subjclass[2000]{26A33, 34A60, 34B10, 34B15} \keywords{Fractional differential inclusions; integral boundary conditions; \hfill\break\indent existence; contraction principle; Krasnoselskii's fixed point theorem; Leray-Schauder degree; \hfill\break\indent Leray-Schauder nonlinear alternative; nonlinear contractions} \begin{abstract} We study boundary value problems of nonlinear fractional differential equations and inclusions of order $q \in (m-1, m]$, $m \ge 2$ with multi-strip boundary conditions. Multi-strip boundary conditions may be regarded as the generalization of multi-point boundary conditions. Our problem is new in the sense that we consider a nonlocal strip condition of the form: $$x(1)=\sum_{i=1}^{n-2}\alpha_i \int^{\eta_i}_{\zeta_i} x(s)ds,$$ which can be viewed as an extension of a multi-point nonlocal boundary condition: $$x(1)=\sum_{i=1}^{n-2}\alpha_i x(\eta_i).$$ In fact, the strip condition corresponds to a continuous distribution of the values of the unknown function on arbitrary finite segments $(\zeta_i,\eta_i)$ of the interval $[0,1]$ and the effect of these strips is accumulated at $x=1$. Such problems occur in the applied fields such as wave propagation and geophysics. Some new existence and uniqueness results are obtained by using a variety of fixed point theorems. Some illustrative examples are also discussed. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In recent years, boundary value problems for nonlinear fractional differential equations have been addressed by several researchers. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes, see \cite{Pod}. These characteristics of the fractional derivatives make the fractional-order models more realistic and practical than the classical integer-order models. As a matter of fact, fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. \cite{d2, Kil, Sag, SKM}. For some recent development on the topic, see \cite{r1, r2, t1, b3, t2, d1} and the references therein. In the first part of this paper, we consider the following nonlinear fractional BVP of an arbitrary order with multi-strip boundary conditions: \label{e1} \begin{gathered} ^cD^qx(t) =f(t,x(t)), \quad 00, provided the integral exists. \end{definition} \begin{lemma} \label{l2} For any \sigma\in C([0,1], \mathbb{R}), the unique solution of the boundary value problem \label{e1a} \begin{gathered} ^cD^qx(t) =\sigma(t), \quad 00, x, y\in \mathbb{R}, \end{itemize} with L < 1/\Lambda, where \Lambda is given by \eqref{e302}. Then the boundary value problem \eqref{e1} has a unique solution. \end{theorem} \begin{proof} Setting \sup_{t \in [0,1]}|f(t,0)|=M and choosing  r \ge \frac{\Lambda M}{1-L\Lambda}, we show that F B_r \subset B_r, where B_r=\{x \in {\mathcal{C}}: \|x\|\le r \}. For x \in B_r, we have \begin{align*} &\|(Fx)(t)\|\\ &\leq \sup_{t \in [0, 1]}\Big\{\int_0^t \frac{(t-s)^{q-1} }{\Gamma(q)}|f(s,x(s))|ds +|\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} |f(s,x(s))|ds \\ &\quad + \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} |f(u,x(u))|du\Big)ds\Big]\Big\} \\ &\leq \sup_{t \in [0,1]}\Big\{\frac{1}{\Gamma(q)}\int_0^t (t-s)^{q-1} (|f(s,x(s))-f(s,0)|+|f(s,0)|)ds \\ &\quad + |\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} (|f(s,x(s))-f(s,0)|+|f(s,0)|)ds \\ &\quad + \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} (|f(u,x(u))-f(u,0)|+|f(u,0)|)du\Big)ds\Big]\Big\} \\ &\leq (Lr+M )\sup_{t \in [0,T]} \Big\{\frac{1}{\Gamma(q)}\int_0^t (t-s)^{q-1} ds \\ &\quad + |\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} ds + \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} du\Big)ds\Big]\Big\} \\ &\leq \frac{(Lr+M)}{\Gamma(q+1)} \Big[1+ |\vartheta| \Big\{1+\sum_{i=1}^{n-2}\frac{\alpha_i (\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big\}\Big] \\ &= \left(Lr+M \right)\Lambda \le r. \end{align*} Now, for x, y \in {\mathcal{C}} and for each t \in [0,1], we obtain \begin{align*} &\|(F x)(t)-(F y)(t)\|\\ &\leq \sup_{t \in [0, 1]}\Big\{\int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} |f(s,x(s))-f(s,y(s))|ds \\ &\quad + |\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} |f(s,x(s))-f(s,y(s))|ds \\ & \quad + \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} |f(u,x(u))-f(u,y(u))|du\Big)ds\Big]\Big\} \\ &\leq L\|x-y\|\sup_{t \in [0, 1]}\Big\{\frac{1}{\Gamma(q)}\int_0^t (t-s)^{q-1} ds \\ &\quad + |\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} ds + \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} du\Big)ds\Big]\Big\}\\ &\leq \frac{L}{\Gamma(q+1)}\Big[1+ |\vartheta| \Big\{1+\sum_{i=1}^{n-2}\frac{\alpha_i (\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big\}\Big]\|x-y\| =L \Lambda \|x-y\|, \end{align*} where \Lambda is given by \eqref{e302}. Observe that \Lambda depends only on the parameters involved in the problem. As L < 1/\Lambda, therefore F is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem).\end{proof} \subsection{Existence result via Krasnoselskii's fixed point theorem} \begin{lemma}[Krasnoselskii's fixed point theorem \cite{MAK}] \label{lk} Let M be a bounded, closed, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that: \begin{itemize} \item[(i)] Ax+By \in M whenever x, y \in M; \item[(ii)] A is compact and continuous; \item[(iii)] B is a contraction mapping. \end{itemize} Then there exists z \in M such that z=Az+Bz. \end{lemma} \begin{theorem}\label{tk} Let f : [0,1]\times \mathbb{R} \to \mathbb{R} be a jointly continuous function satisfying the assumption {\rm (A1)}. Moreover we assume that \begin{itemize} \item[(A2)] |f(t,x)|\le \mu (t), for all (t,x) \in [0,1] \times \mathbb{R}, and \mu \in C([0,1], \mathbb{R}^+). \end{itemize} If $$\label{e4} \frac{L|\vartheta|}{\Gamma(q+1)}\Big( 1+\sum_{i=1}^{n-2}\frac{\alpha_i (\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big)<1,$$ then the boundary value problem \eqref{e1} has at least one solution on [0,1]. \end{theorem} \begin{proof} By the assumption (A2), we can fix \overline{r} \ge \frac{|\vartheta|\|\mu\|}{\Gamma(q+1)}\Big( 1+\sum_{i=1}^{n-2}\frac{\alpha_i (\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big), and consider B_{\overline{r}}=\{x \in {\mathcal{C}}: \|x\|\le \overline{r} \}. We define the operators {\mathcal{P}} and {\mathcal{Q}} on B_{\overline{r}} as \begin{gather*} ({\mathcal{P}} x)(t)= \int_{0}^t\frac{(t-s)^{q-1}}{\Gamma(q)}f(s,u(s))ds, \quad t\in [0,1],\\ \begin{aligned} ({\mathcal{Q}} x)(t) &= - \vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds \\ &\quad -\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f(u,x(u))du\Big)ds\Big],\quad t\in [0,1]. \end{aligned} \end{gather*} For x, y \in B_{\overline{r}}, we find that \|{\mathcal{P}} x+{\mathcal{Q}} y\| \le \frac{|\vartheta|\|\mu\|}{\Gamma(q+1)}\Big( 1+\sum_{i=1}^{n-2}\frac{\alpha_i (\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big) \le \overline{r}. $$Thus, {\mathcal{P}} x+{\mathcal{Q}} y \in B_{\overline{r}}. It follows from the assumption (A1) together with \eqref{e4} that {\mathcal{Q}} is a contraction mapping. Continuity of f implies that the operator {\mathcal{P}} is continuous. Also, {\mathcal{P}} is uniformly bounded on B_{\overline{r}} as$$ \|{\mathcal{P}} x\| \le \frac{\|\mu\|}{\Gamma(q+1)}. $$Now we prove the compactness of the operator {\mathcal{P}}. In view of (A1), we define$$ \sup_{(t,x) \in [0,1] \times B_{\overline{r}}}|f(t,x)|=\overline{f}. Consequently we have \begin{align*} &\|({\mathcal{P}} x)(t_1)-({\mathcal{P}} x)(t_2)\|\\ &= \big\|\frac{1}{\Gamma(q)}\int_0^{t_1}[(t_2-s)^{q-1}-(t_1-s)^{q-1}]f(s,x(s))ds + \int_{t_1}^{t_2}(t_2-s)^{q-1}f(s,x(s))ds\big\|\\ &\leq \frac{\overline{f}}{\Gamma(q+1)}|2(t_2-t_1)^q+t_1^q-t_2^q|, \end{align*} which is independent of x and tends to zero as t_2 \to t_1. Thus, {\mathcal{P}} is relatively compact on B_{\overline{r}}. Hence, by the Arzel\'a-Ascoli Theorem, {\mathcal{P}} is compact on B_{\overline{r}}. Thus all the assumptions of Lemma \ref{lk} are satisfied. So by the conclusion of Lemma \ref{lk}, problem \eqref{e1} has at least one solution on [0,1]. \end{proof} \subsection{Existence result via Leray-Schauder Alternative} \begin{lemma}[Nonlinear alternative for single valued maps \cite{GrDu}] \label{lls}. Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and 0\in U. Suppose that F:\overline{U}\to C is a continuous, compact (that is, F(\overline{U}) is a relatively compact subset of C) map. Then either \begin{itemize} \item[(i)] F has a fixed point in \overline{U}, or \item[(ii)] there is a u\in \partial U (the boundary of U in C) and \lambda\in(0,1) with u=\lambda F(u). \end{itemize} \end{lemma} \begin{theorem}\label{tls} Let f: [0,1]\times \mathbb{R} \to \mathbb{R} be a jointly continuous function. Assume that: \begin{itemize} \item[(A3)] There exist a function p \in L^1([0,1], \mathbb{R}^+), and a nondecreasing function \psi: {\mathbb{R}}^+\to { \mathbb{R}}^+ such that |f(t,x)|\le p (t)\psi(\|x\|), for all (t,x) \in [0,1] \times \mathbb{R}. \item[(A4)] There exists a constant M>0 such that \frac{M}{\frac{\psi(M)}{\Gamma(q)}\big[\{1+|\vartheta|\}\int_0^1 p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s) ds\big]}> 1. \end{itemize} Then the boundary value problem \eqref{e1} has at least one solution on [0,1]. \end{theorem} \begin{proof} Consider the operator F : \mathcal{C} \to \mathcal{C} with  x=F (x), where \begin{align*} (F x)(t)&= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds - \vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f(u,x(u))du\Big)ds\Big], \quad t \in [0,1]. \end{align*} We show that F maps bounded sets into bounded sets in  C([0,1], \mathbb{R}). For a positive number r, let B_r = \{x \in C([0,1], \mathbb{R}): \|x\| \le r \} be a bounded set in C([0,1], \mathbb{R}). Then \begin{align*} |(Fx)(t)| &\leq \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} |f(s,x(s))|ds+ |\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} |f(s,x(s))|ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} |f(u,x(u))|du\Big)ds\Big] \\ &\leq \int_0^t \frac{(t-s)^{q-1} }{\Gamma(q)} p(s)\psi(\|x\|)ds +|\vartheta| t^{m-1}\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} p(s)\psi(\|x\|)ds\\ &\quad + |\vartheta| t^{m-1} \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} p(s)\psi(\|x\|)du\Big)ds\\ &\leq \frac{\psi(\|x\|)}{\Gamma(q)}\int_0^1 (t-s)^{q-1} p(s)ds + \frac{|\vartheta|\psi(\|x\|)}{\Gamma(q)} \int_0^1 (1-s)^{q-1}p(s)ds \\ &\quad + \frac{|\vartheta|\psi(\|x\|)}{ \Gamma(q)}\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s (s-u)^{q-1} p(u)du\Big)ds\\ &\leq \frac{\psi(\|x\|)}{\Gamma(q)}\int_0^1 (t-s)^{q-1} p(s)ds+ \frac{|\vartheta|\psi(\|x\|)}{\Gamma(q)} \int_0^1 (1-s)^{q-1}p(s)ds \\ &\quad + \frac{|\vartheta|\psi(\|x\|)}{ \Gamma(q)}\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s (s-u)^{q-1} p(u)du\Big)ds\\ &\leq \frac{\psi(\|x\|)}{\Gamma(q)}\int_0^1 p(s)ds + \frac{|\vartheta|\psi(\|x\|)}{\Gamma(q)} \int_0^1 p(s)ds \\ &\quad + \frac{|\vartheta|\psi(\|x\|)}{ \Gamma(q)}\sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q-\zeta_i^q}{q} \int_{\zeta_i}^{\eta_i}p(s)ds\\ &= \frac{\psi(\|x\|)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1 p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q -\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s) ds\Big]. \end{align*} Consequently, \|Fx\|\le \frac{\psi(r)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1 p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s) ds\Big]. Next we show that F maps bounded sets into equicontinuous sets of  C([0,1], \mathbb{R}). Let t', t'' \in [0,1] with t'< t'' and x \in B_r, where B_r is a bounded set of C([0,1], \mathbb{R}). Then we obtain \begin{align*} &|(Fx)(t'')-(Fx)(t')|\\ &= \Big| \frac{1}{\Gamma(q)}\int_0^{t''} (t''-s)^{q-1} f(s,x(s))ds -\frac{1}{\Gamma(q)}\int_0^{t'} (t'-s)^{q-1} f(s,x(s))ds \\ &\quad - \vartheta \Big((t'')^{m-1}-(t')^{m-1}\Big)\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds\\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f(u,x(u))du\Big)ds\Big]\Big|\\ &\leq \Big| \frac{1}{\Gamma(q)}\int_0^{t'}[ (t''-s)^{q-1}-(t'-s)^{q-1}] \psi(r)p(s)ds\Big|\\ &\quad +\Big|\frac{1}{\Gamma(q)}\int_{t'}^{t''} (t''-s)^{q-1} \psi(r)p(s)ds\Big|\\ &\quad +\Big|\vartheta \Big((t'')^{m-1}-(t')^{m-1}\Big)\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} \psi(r)p(s)ds\\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} \psi(r)p(u)du\Big)ds\Big]\Big|. \end{align*} Obviously the right hand side of the above inequality tends to zero independently of x \in B_{r'} as t''- t' \to 0. As F satisfies the above assumptions, therefore it follows by the Arzel\'a-Ascoli theorem that F: C([0,1], \mathbb{R}) \to C([0,1], \mathbb{R}) is completely continuous. The result will follow from the Leray-Schauder nonlinear alternative (Lemma \ref{lls}) once we have proved the boundendness of the set of all solutions to equations x=\lambda F x for \lambda\in [0,1]. Let x be a solution. Then, for t\in [0,1], and using the computations in proving that F is bounded, we have \begin{align*} |x(t)| &= |\lambda (F x)(t)| \\ &\leq \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} |f(s,x(s))|ds+ |\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} |f(s,x(s))|ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} |f(u,x(u))|du\Big)ds\Big] \\ &\leq \frac{\psi(\|x\|)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1 p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s) ds\Big]. \end{align*} Consequently, we have \frac{\|x\|}{ \frac{\psi(\|x\|)}{\Gamma(q)}\big[\{1+|\vartheta|\}\int_0^1 p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s) ds\big]}\le 1. $$In view of (A4), there exists M such that \|x\| \ne M. Let us set$$ U = \{x \in C([0,1], \mathbb{R}):\|x\| < M\}. $$Note that the operator F:\overline{U} \to C([0,1], \mathbb{R}) is continuous and completely continuous. From the choice of U, there is no x \in \partial U such that x =\lambda F(x) for some \lambda \in (0,1). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma \ref{lls}), we deduce that F has a fixed point x \in \overline{U} which is a solution of the problem \eqref{e1}. This completes the proof. \end{proof} \subsection{Existence result via Leray-Schauder degree theory} \begin{theorem}\label{tlsd} Let f: [0,1]\times \mathbb{R} \to \mathbb{R}. Assume that there exist constants 0 \le \kappa < \frac{1}{\Lambda}, where \Lambda is given by \eqref{e302} and M>0 such that |f(t,x)| \le \kappa \|x\| +M for all t \in [0,1], x \in \mathbb{R}. Then the boundary value problem \eqref{e1} has at least one solution. \end{theorem} \begin{proof} In view of the fixed point problem \eqref{e301}, we just need to prove the existence of at least one solution x \in \mathbb{R} satisfying \eqref{e301}. Define a suitable ball B_R \subset C[0,1] with radius R>0 as$$ B_R=\{x \in \mathcal{C} : \|x\|0$with the property: $$\|Fx-Fy\|\le \Psi(\|x-y\|), \quad \forall x,y\in E.$$ \end{definition} \begin{lemma}[Boyd and Wong \cite{BW}]\label{BW} Let$E$be a Banach space and let$F: E\to E$be a nonlinear contraction. Then$F$has a unique fixed point in$E$. \end{lemma} \begin{theorem}\label{tnc} Assume that: \begin{itemize} \item[(A5)]$|f(t,x)-f(t,y)|\le h(t)\frac{|x-y|}{H^*+|x-y|}$,$t\in [0,1]$,$x,y\ge 0$, where$h:[0,1]\to \mathbb{R}^+is continuous and \label{e3004} \begin{aligned} H^*&= \int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} h(s)ds + |\vartheta|\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} h(s)ds \\ &\quad +\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} h(u)du \Big)ds\Big]. \end{aligned} \end{itemize} Then the boundary value problem \eqref{e1} has a unique solution. \end{theorem} \begin{proof} We define the operatorF: \mathcal{C}\to \mathcal{C}by \begin{align*} Fx(t) &= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds + \vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds \\ &\quad + \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f(u,x(u))du\Big)ds\Big], \quad t \in [0,1]. \end{align*} Let a continuous nondecreasing function\Psi: \mathbb{R}^+\to \mathbb{R}^+$satisfying$\Psi(0)=0$and$\Psi(\xi)<\xi$for all$\xi>0$be defined by $$\Psi(\xi)=\frac{H^*\xi}{H^*+\xi},\quad \forall \xi\ge 0.$$ Let$x, y\in \mathcal{C}. Then $$|f(s,x(s))-f(s,y(s))|\le \frac{h(s)}{H^*}\Psi(\|x-y\|)$$ so that \begin{align*} &|Fx(t)-Fy(t)| \\ &\leq \int_0^t \frac{(1-s)^{q-1}}{\Gamma(q)} h(s)\frac{|x(s)-y(s)|}{H^*+|x(s)-y(s)|}ds \\ &\quad + |\vartheta| \int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} h(s)\frac{|x(s)-y(s)|}{H^*+|x(s)-y(s)|}ds \\ &\quad + |\vartheta| \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} h(m)\frac{|x(m)-y(m)|}{H^*+|x(m)-y(m)|}dm \Big)ds, \end{align*} fort \in [0,1]$. In view of \eqref{e3004}, it follows that$\|Fx-Fy\|\le \Psi(\|x-y\|)$and hence$F$is a nonlinear contraction. Thus, by Lemma \ref{BW}, the operator$F$has a unique fixed point in$\mathcal{C}$, which in turn is a unique solution of problem \eqref{e1}. \end{proof} \subsection{Examples} For the forthcoming examples, we consider the following boundary conditions: $$\label{bc} x(0)=0, \quad x'(0)=0, \quad x''(0)=0, \quad x(1)=\sum_{i=1}^{3}\alpha_i \int^{\eta_i}_{\zeta_i} x(s)ds,$$ where$\zeta_1=1/16$,$\zeta_2=5/16$,$\zeta_3=9/16$,$\eta_1=1/4$,$\eta_2=1/2$,$\eta_3=3/4$,$\alpha_1=1/3$,$\alpha_2=2/3$,$\alpha_3=1$. \begin{example} \label{examp3.10}\rm Consider the fractional differential equation \label{ex1} ^cD^{7/2}x(t) =L\big( \cos t +\tan^{-1}x(t)\big), \quad 0 0$, there exists $\varphi_{\alpha} \in L^1([0,1],\mathbb{R}^+)$ such that $$\|F (t, x)\| = \sup \{|v| : v \in F (t, x)\} \le \varphi_{\alpha} (t)$$ for all $\|x\| \le \alpha$ and for a. e. $t \in [0,1]$. \end{itemize} \end{definition} For each $y \in C([0,1], \mathbb{R})$, define the set of selections of $F$ by $$S_{F,y} := \{ v \in L^1([0,1],\mathbb{R}) : v (t) \in F (t, y(t)) ~\text{for a.e.} ~t \in [0,1]\}.$$ The consideration of this subsection is based on the following lemmas. \begin{lemma}[Nonlinear alternative for Kakutani maps \cite{GrDu}] \label{NAK} Let $E$ be a Banach space, $C$ a closed convex subset of $E$, $U$ an open subset of $C$ and $0\in U$. Suppose that $F: \overline{U}\to \mathcal{P}_{c,cv}(C)$ is a upper semicontinuous compact map; here $\mathcal{P}_{c,cv}(C)$ denotes the family of nonempty, compact convex subsets of $C$. Then either \begin{itemize} \item[(i)] $F$ has a fixed point in $\overline{U}$, or \item[(ii)] there is a $u\in \partial U$ and $\lambda\in(0,1)$ with $u\in \lambda F(u)$. \end{itemize} \end{lemma} \begin{lemma}[\cite{LaOp}] \label{l1i} Let $X$ be a Banach space. Let $F : [0, 1] \times \mathbb{R} \to \mathcal{P}_{cp,c}(\mathbb{R})$ be an $L^1-$ Carath\'{e}odory multivalued map and let $\Theta$ be a linear continuous mapping from $L^1([0,1],\mathbb{R})$ to $C([0,1],\mathbb{R})$. Then the operator $$\Theta \circ S_F : C([0,1],\mathbb{R}) \to P_{cp,c} (C([0,1],\mathbb{R})), \quad x \mapsto (\Theta \circ S_F) (x) = \Theta( S_{F,x})$$ is a closed graph operator in $C([0,1],\mathbb{R}) \times C([0,1],\mathbb{R})$. \end{lemma} \begin{theorem}\label{tcar} Assume that {\rm (A4)} holds. In addition we assume that: \begin{itemize} \item[(H1)] $F : [0,1] \times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})$ is Carath\'{e}odory and has nonempty compact convex values; \item[(H2)] there exists a continuous nondecreasing function $\psi : [0,\infty) \to (0,\infty)$ and a function $p \in L^1([0,1],\mathbb{R}^+)$ such that $$\|F(t,x)\|_\mathcal{P}:=\sup\{|y|: y \in F(t,x)\}\le p(t)\psi(\|x\|) \quad \text{for each} ~(t,x) \in [0,1] \times \mathbb{R}.$$ \end{itemize} Then the boundary value problem \eqref{e1i} has at least one solution on $[0,1]$. \end{theorem} \begin{proof} Define an operator $\Omega: C([0,1], \mathbb{R})\to \mathcal{P}(C([0,1], \mathbb{R}))$ by \begin{align*} &\Omega(x)\\ &=\Big\{ h \in C([0,1], \mathbb{R}): h(t) = \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big], \; 0\le t\le 1 \Big\}, \end{align*} for $f \in S_{F,x}$. We will show that $\Omega$ satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that $\Omega$ is convex for each $x \in C([0,1], \mathbb{R})$. For that, let $h_1, h_2 \in \Omega(x)$. Then there exist $f_1, f_2 \in S_{F,x}$ such that for each $t \in [0,1]$, we have \begin{align*} h_i(t) &= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f_i(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f_i(s)ds\\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f_i(u)du\Big)ds\Big], \quad i=1,2. \end{align*} Let $0 \le \omega \le 1$. Then, for each $t \in [0,1]$, we have \begin{align*} &[\omega h_1+(1-\omega)h_2](t)\\ &= \int_0^t \frac{(t-s)^{q-1} }{\Gamma(q)} [\omega f_1(s)+(1-\omega)f_2(s)]ds\\ &\quad -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} [\omega f_1(s)+(1-\omega)f_2(s)]ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} [\omega f_1(s)+(1-\omega)f_2(s)]du\Big)ds\Big]. \end{align*} Since $S_{F,x}$ is convex ($F$ has convex values), therefore it follows that $\omega h_1+(1-\omega)h_2 \in \Omega(x)$. Next, we show that $\Omega$ maps bounded sets into bounded sets in $C([0,1], \mathbb{R})$. For a positive number $r$, let $B_r = \{x\in C([0,1], \mathbb{R}): \|x\| \le r \}$ be a bounded set in $C([0,1], \mathbb{R})$. Then, for each $h \in \Omega (x), x \in B_r$, there exists $f \in S_{F,x}$ such that \begin{align*} h(t)&= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds\\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big]. \end{align*} Then, as in Theorem \ref{tls}, \begin{align*} |h(t)| &\leq \int_0^t \frac{(t-s)^{q-1} }{\Gamma(q)} |f(s)|ds +|\vartheta| t^{m-1}\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} |f(s)|ds \\ &\quad + |\vartheta| t^{m-1} \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} |f(u)|du\Big)ds\\ &\leq \frac{\psi(\|x\|)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1 p(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s) ds\Big]. \end{align*} Thus, $$\|h\| \le \frac{\psi(r)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1 p(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i \frac{\eta_1^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s) ds\Big].$$ Now we show that $\Omega$ maps bounded sets into equicontinuous sets of $C([0,1], \mathbb{R})$. Let $t', t'' \in [0,1]$ with $t'< t''$ and $x \in B_r$, where $B_r$ is a bounded set of $C([0,1], \mathbb{R})$. For each $h \in \Omega(x)$, we obtain \begin{align*} |h(t'')-h(t')| &= \Big| \frac{1}{\Gamma(q)}\int_0^{t''} (t''-s)^{q-1} f(s)ds -\frac{1}{\Gamma(q)}\int_0^{t'} (t'-s)^{q-1} f(s)ds \\ &\quad - \vartheta \Big((t'')^{m-1}-(t')^{m-1}\Big) \Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big]\Big| \\ &\leq \Big| \frac{1}{\Gamma(q)}\int_0^{t'}[ (t''-s)^{q-1}-(t'-s)^{q-1}] \psi(r)p(s)ds\Big|\\ &\quad +\Big|\frac{1}{\Gamma(q)}\int_{t'}^{t''} (t''-s)^{q-1} \psi(r)p(s)ds\Big|\\ &\quad +\Big|\vartheta \Big((t'')^{m-1}-(t')^{m-1}\Big)\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} \psi(r)p(s)ds\\ &\quad \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} \psi(r)p(u)du\Big)ds\Big]\Big|. \end{align*} Obviously the right hand side of the above inequality tends to zero independently of $x \in B_{r'}$ as $t''- t' \to 0$. As $\Omega$ satisfies the above three assumptions, therefore it follows by the Arzel\'a-Ascoli theorem that $\Omega: C([0,1], \mathbb{R}) \to {\mathcal{P}}(C([0,1], \mathbb{R}))$ is completely continuous. \\ In our next step, we show that $\Omega$ has a closed graph. Let $x_n \to x_*, h_n \in \Omega (x_n)$ and $h_n \to h_*$. Then we need to show that $h_* \in \Omega (x_*)$. Associated with $h_n \in \Omega(x_n)$, there exists $f_n \in S_{F,x_n}$ such that for each $t \in [0,1]$, \begin{align*} h_n(t) &= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f_n(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f_n(s)ds\\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f_n(u)du\Big)ds\Big]. \end{align*} Thus we have to show that there exists $f_* \in S_{F,x_*}$ such that for each $t \in [0,1]$, \begin{align*} h_*(t) &= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f_*(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f_*(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f_*(u)du\Big)ds\Big]. \end{align*} Let us consider the continuous linear operator $\Theta : L^1([0,1], \mathbb{R}) \to C([0,1], \mathbb{R})$ given by \begin{align*} f \mapsto \Theta(f) &= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big]. \end{align*} Observe that \begin{align*} &\| h_n(t)-h_*(t)\|\\ &= \Big\| \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}(f_n(s)-f_*(s))ds - \vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} (f_n(s)-f_*(s))ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} (f_n(u)-f_*(u))du\Big)ds\Big]\Big\|\to 0, \end{align*} as $n \to \infty$. Thus, it follows by Lemma \ref{l1i} that $\Theta \circ S_F$ is a closed graph operator. Further, we have $h_n(t) \in \Theta(S_{F,x_n})$. Since $x_n \to x_*$, therefore, we have \begin{align*} h_*(t) &= \int_0^t \frac{ (t-s)^{q-1}}{\Gamma(q)} f_*(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f_*(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f_*(u)du\Big)ds\Big], \end{align*} for some $f_* \in S_{F,x_*}$. Finally, we discuss a priori bounds on solutions. Let $x$ be a solution of \eqref{e1i}. Then there exists $f \in L^1([0,1], \mathbb{R})$ with $f \in S_{F,x}$ such that, for $t \in [0,1]$, we have \begin{align*} x(t)&= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big]. \end{align*} In view of (H2), and using the computations in second step above, for each $t \in [0,1]$, we obtain \begin{align*} |x(t)|&\leq \int_0^t \frac{(t-s)^{q-1} }{\Gamma(q)} |f(s)|ds +|\vartheta| t^{m-1}\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} |f(s)|ds \\ &\quad + |\vartheta| t^{m-1} \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} |f(u)|du\Big)ds\\ &\leq \frac{\psi(\|x\|)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1 p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s) ds\Big]. \end{align*} Consequently, $$\frac{\|x\|}{ \frac{\psi(\|x\|)}{\Gamma(q)}\big[\{1+|\vartheta|\}\int_0^1 p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i \frac{\eta_1^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s) ds\big]}\le 1.$$ In view of (A4), there exists $M$ such that $\|x\| \ne M$. Let us set $$U = \{x \in C([0,1], \mathbb{R}) : \|x\| < M\}.$$ Note that the operator $\Omega :\overline{U} \to \mathcal{P}(C([0,1], \mathbb{R}))$ is upper semicontinuous and completely continuous. From the choice of $U$, there is no $x \in \partial U$ such that $x \in \mu \Omega(x)$ for some $\mu \in (0,1)$. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma \ref{NAK}), we deduce that $\Omega$ has a fixed point $x \in \overline{U}$ which is a solution of the problem \eqref{e1i}. This completes the proof. \end{proof} \subsection{The lower semi-continuous case} Here, we study the case when $F$ is not necessarily convex valued. Our strategy to deal with this problems is based on the nonlinear alternative of Leray Schauder type together with the selection theorem of Bressan and Colombo \cite{BrCo} for lower semi-continuous maps with decomposable values. \begin{definition} \label{def4.9} \rm Let $X$ be a nonempty closed subset of a Banach space $E$ and $G: X \to {\mathcal{P}}(E)$ be a multivalued operator with nonempty closed values. $G$ is lower semi-continuous (l.s.c.) if the set $\{y \in X : G(y)\cap B \ne \emptyset\}$ is open for any open set $B$ in $E$. \end{definition} \begin{definition} \label{def4.10} \rm Let $A$ be a subset of $[0,1]\times \mathbb{R}$. $A$ is $\mathcal{L}\otimes \mathcal{B}$ measurable if $A$ belongs to the $\sigma-$algebra generated by all sets of the form $\mathcal{J} \times \mathcal{D}$, where $\mathcal{J}$ is Lebesgue measurable in $[0,1]$ and $\mathcal{D}$ is Borel measurable in $\mathbb{R}$. \end{definition} \begin{definition} \label{def4.11} \rm A subset $\mathcal{A}$ of $L^1([0,1], \mathbb{R})$ is decomposable if for all $x, y \in \mathcal{A}$ and measurable $\mathcal{J} \subset [0,1]=J$, the function $x \chi_{\mathcal{J}}+y \chi_{J-\mathcal{J}} \in \mathcal{A}$, where $\chi_{\mathcal{J}}$ stands for the characteristic function of $\mathcal{J}$. \end{definition} \begin{definition} \label{def4.12} \rm Let $Y$ be a separable metric space and $N : Y \to {\mathcal{P}}(L^1([0,1],\mathbb{R}))$ be a multivalued operator. We say $N$ has a property (BC) if $N$ is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values. \end{definition} Let $F : [0,1] \times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})$ be a multivalued map with nonempty compact values. Define a multivalued operator $\mathcal{F} : C([0,1] \times \mathbb{R}) \to {\mathcal{P}}(L^1([0,1],\mathbb{R}))$ associated with $F$ as $$\mathcal{F}(x)=\{w \in L^1([0,1],\mathbb{R}) : w(t) \in F(t,x(t)) \text{ for a.e. } t \in [0,1]\},$$ which is called the Nemytskii operator associated with $F$. \begin{definition} \label{def4.13} \rm Let $F : [0,1] \times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})$ be a multivalued function with nonempty compact values. We say $F$ is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator $\mathcal{F }$ is lower semi-continuous and has nonempty closed and decomposable values. \end{definition} \begin{lemma}[\cite{BrCo}] \label{l2i} Let $Y$ be a separable metric space and $N : Y \to {\mathcal{P}}(L^1([0,1],\mathbb{R}))$ be a multivalued operator satisfying the property (BC). Then $N$ has a continuous selection, that is, there exists a continuous function (single-valued) $g : Y \to L^1([0,1],\mathbb{R})$ such that $g(x) \in N(x)$ for every $x \in Y$. \end{lemma} \begin{theorem}\label{tbc} Assume that {\rm (H2), (H3)} and the following condition holds: \begin{itemize} \item[(H4)] $F : [0,1] \times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})$ is a nonempty compact-valued multivalued map such that \begin{itemize} \item[(a)] $(t,x) \mapsto F(t,x)$ is $\mathcal{L}\otimes \mathcal{B}$ measurable, \item[(b)] $x \mapsto F(t,x)$ is lower semicontinuous for each $t \in [0,1]$. \end{itemize} \end{itemize} Then the boundary value problem \eqref{e1i} has at least one solution on $[0,1]$. \end{theorem} \begin{proof} It follows from (H2) and (H4) that $F$ is of l.s.c. type. Then, by Lemma \ref{l2i}, there exists a continuous function $f : C([0,1],\mathbb{R}) \to L^1([0,1],\mathbb{R})$ such that $f (x) \in \mathcal{F}(x)$ for all $x \in C([0,1],\mathbb{R})$. Consider the problem \label{et2i} \begin{gathered} ^cD^qx(t)= f(x(t)), \quad 0 0$such that $$H_d(N(x),N(y)) \le \gamma d(x,y) \text{ for each } x, y \in X;$$ \item[(b)] a contraction if and only if it is$\gamma$-Lipschitz with$\gamma < 1$. \end{itemize} \end{definition} \begin{lemma}[Covitz-Nadler, \cite{CoNa}] \label{l3i} Let$(X,d)$be a complete metric space. If$N : X \to P_{cl}(X)$is a contraction, then${\rm Fix }N \ne \emptyset $. \end{lemma} \begin{definition} A measurable multi-valued function$F:[0,1]\to \mathcal{P}(X)$is said to be integrably bounded if there exists a function$h\in L^1([0,1], X)$such that for all$v\in F(t)$,$\|v\|\le h(t)$for a.e.$t\in [0,1]$. \end{definition} \begin{theorem}\label{tcn} Assume that the following conditions hold: \begin{itemize} \item[(H5)]$F : [0,1] \times \mathbb{R} \to P_{cp}(\mathbb{R})$is such that$F(\cdot,x) : [0,1] \to P_{cp}(\mathbb{R})$is measurable for each$x \in \mathbb{R}$; \item[(H6)]$H_d(F(t,x), F(t,\bar{x}))\le m(t)|x-\bar{x}|$for almost all$t \in [0,1]$and$x, \bar{x} \in \mathbb{R}$with$m \in L^1([0,1], \mathbb{R}^+)$and$d(0,F(t,0))\le m(t)$for almost all$t \in [0,1]$. \end{itemize} Then the boundary-value problem \eqref{e1i} has at least one solution on$[0,1]$if $$\frac{1}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1 m(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i \frac{\eta_1^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}m(s) ds\Big] <1.$$ \end{theorem} \begin{proof} We transform the problem \eqref{e1i} into a fixed point problem. Consider the set-valued map$\Omega: C([0,1], \mathbb{R})\to \mathcal{P}(C([0,1], \mathbb{R}))$defined at the beginning of the proof of Theorem \ref{tcar}. It is clear that the fixed point of$\Omega$are solutions of the problem \eqref{e1i}. Note that, by the assumption (H5), since the set-valued map$F(\cdot, x)$is measurable, it admits a measurable selection$f: [0,1]\to \mathbb{R}$(see \cite[Theorem III.6]{CaVa}). Moreover, from assumption (H6) $$|f(t)|\le m(t)+m(t)|x(t)|,$$ i.e.$f(\cdot)\in L^1([0,1], \mathbb{R})$. Therefore the set$S_{F,x}$is nonempty. Also note that since$S_{F,x}\ne \emptyset$, therefore$\Omega(x)\ne \emptyset$for any$x\in C([0,1],\mathbb{R})$. Now we show that the operator$\Omega$satisfies the assumptions of Lemma \ref{l3i}. To show that$\Omega(x) \in P_{cl}((C[0,1],\mathbb{R}))$for each$x \in C([0,1], \mathbb{R})$, let$\{u_n\}_{n \ge 0} \in \Omega(x)$be such that$u_n \to u ~(n \to \infty)$in$C([0,1],\mathbb{R})$. Then$u \in C([0,1],\mathbb{R})$and there exists$v_n \in S_{F,x}$such that, for each$t \in [0,1], we have \begin{align*} u_n(t) &= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} v_n(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} v_n(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} v_n(u)du\Big)ds\Big]. \end{align*} AsF$has compact values, we may pass onto a subsequence (if necessary) to obtain that$v_n$converges to$v$in$L^1 ([0,1],\mathbb{R})$. Thus,$v \in S_{F,x}$and for each$t \in [0,1], \begin{align*} u_n(t) \to u(t) &= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} v(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} v(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} v(u)du\Big)ds\Big]. \end{align*} Hence,u \in \Omega(x)$and$\Omega(x)$is closed. Next we show that$\Omega$is a contraction on$C([0,1],\mathbb{R})$; i.e., there exists$\gamma <1$such that $$H_d(\Omega(x), \Omega(\bar{x}))\le \gamma \|x-\bar{x}\|_{\infty} \quad \text{for each} \quad x, \bar{x} \in C([0,1], \mathbb{R}).$$ Let$x, \bar{x} \in C([0,1], \mathbb{R})$and$h_1 \in \Omega(x)$. Then there exists$v_1(t) \in F(t,x(t))$such that, for each$t \in [0,1], \begin{align*} h_1(t) &= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} v_1(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} v_1(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} v_1(u)du\Big)ds\Big]. \end{align*} By (H6), we have $$H_d(F(t,x), F(t,\bar{x}))\le m(t)|x(t)-\bar{x}(t)|.$$ So, there existsw \in F(t,\bar{x}(t))$such that $$|v_1(t)-w|\le m(t)|x(t)-\bar{x}(t)|, \quad t \in [0,1].$$ Define$U : [0,1] \to \mathcal{P}(\mathbb{R})$by $$U(t)=\{w \in \mathbb{R} : |v_1(t)-w|\le m(t)|x(t)-\bar{x}(t)|\}.$$ Since the multivalued operator$U(t)\cap F(t,\bar{x}(t))$is measurable (\cite[Proposition III.4]{CaVa}), there exists a function$v_2(t)$which is a measurable selection for$U$. So$v_2(t) \in F(t,\bar{x}(t))$and for each$t \in [0,1]$, we have$|v_1(t)-v_2(t)|\le m(t)|x(t)-\bar{x}(t)|$. For each$t \in [0,1], let us define \begin{align*} h_2(t) &= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} v_2(s)ds -\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} v_2(s)ds \\ &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} v_2(u)du\Big)ds\Big]. \end{align*} Thus, \begin{align*} | h_1(t)-h_2(t)| &\leq \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} |v_1(s)-v_2(s)|ds\\ &\quad +|\vartheta| t^{m-1} \int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} |v_1(s)-v_2(s)|ds \\ &\quad +|\vartheta| t^{m-1} \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s \frac{(s-u)^{q-1}}{\Gamma(q)} |v_1(u)-v_2(u)|du\Big)ds \\ &\leq \frac{\|x-\overline{x}\|}{\Gamma(q)} \Big[\{1+|\vartheta|\}\int_0^1 m(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}m(s) ds\Big]. \end{align*} Hence, \begin{align*} \| h_1-h_2\| \le \frac{1}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1 m(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q -\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}m(s) ds\Big]\|x-\overline{x}\|. \end{align*} Analogously, interchanging the roles ofx$and$\overline{x}, we obtain \begin{align*} & H_d(\Omega(x), \Omega(\bar{x})) \\ &\leq \gamma \|x-\bar{x}\|\\ &\leq \frac{1}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1 m(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i \frac{\eta_i^q -\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}m(s) ds\Big]\|x-\overline{x}\|. \end{align*} Since\Omega$is a contraction, it follows by Lemma \ref{l3i} that$\Omega$has a fixed point$x$which is a solution of \eqref{e1i}. 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