\documentclass[amsmath,amssymb,twoside,reqno,12pt]{amsart} \thispagestyle{empty} \pagestyle{myheadings} \markboth{J. R. Graef, N. Guerraiche and S. Hamani}{Fractional differential inclusions} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{notation}{Notation} \allowdisplaybreaks \def\Int{\displaystyle\int} \def\Sum{\displaystyle\sum} \def\Frac{\displaystyle\frac} \def\Min{\displaystyle\min} \def\Sup{\displaystyle\sup} \def\Max{\displaystyle\max} \def\Inf{\displaystyle\inf} \def\Lim{\displaystyle\lim} \def\Cup{\displaystyle\cup} \def\Cap{\displaystyle\cap} \def\R{{\mathbb R}} \def\N{{\mathbb N}} \textheight=8.5in \textwidth=6.5in \oddsidemargin=0.15in \evensidemargin=0.15in \begin{document} \begin{center} {\Large\bf Boundary Value Problems for Fractional Differential Inclusions with Hadamard Type Derivatives in Banach Spaces} \vskip 1cm {\bf John R. Graef, Nassim Guerraiche, and Samira Hamani}\\ Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403-2504, USA\\ e-mail: John-Graef@utc.edu\\[0.3cm] Laboratoire des Math\'ematiques Appliqu\'es et Pures, Universit\'e de Mostaganem\\ B.P. 227, 27000, Mostaganem, ALGERIE\\ e-mail: hamani\_samira@yahoo.fr \\[0.3cm] Laboratoire des Math\'ematiques Appliqu\'es et Pures, Universit\'e de Mostaganem\\ B.P. 227, 27000, Mostaganem, ALGERIE\\ nassim.guerraiche@univ-mosta.dz \\[0.3cm] \vskip0.2cm \end{center} %\begin{abstract} The authors establish sufficient conditions for the existence of solutions to boundary value problems for fractional differential inclusions involving the Hadamard type fractional derivative of order $\alpha \in (1,2]$ in Banach spaces. Their approach uses M\"onch's fixed point theorem and the Kuratowski measure of noncompacteness. %\end{abstract} \\ \noindent {\bf Key words and phrases:} Fractional differential inclusion, Hadamard-type fractional derivative, fractional integral, M\"onch's fixed point theorem, Kuratowski measure of noncompacteness. \\ \noindent {\bf AMS (MOS) Subject Classifications}: 26A33, 34A08, 34A60, 34B15 \section{Introduction} In this paper we are concerned with the existence of solutions to boundary value problems (BVP for short) for fractional order differential inclusions. In particular, we consider the boundary value problem \begin{equation} \label{e1} {}^{H} D^{r}y(t) \in F(t,y(t)), \, \, \, \text{for a.e.} \ t \in J=[1,T], \ 1< r \leq 2, \end{equation} \begin{equation}\label{e2} y(1)= 0, \ y(T)=y_{T}, \end{equation} where ${}^{H} D^{r}$ is the Hadamard fractional derivative, $(E, |\cdot|)$ is a Banach space, ${\mathcal P}(E)$ is the family of all nonempty subsets of $E$, $F:[1,T] \times E \to {\mathcal P}(E)$ is a multivalued map, and $y_{T} \in \R$. Differential equations of fractional order are valuable in modeling phenomena in various fields of science and engineering. They can be found in viscoelasticity, electrochemistry, control, porous media, electromagnetism, etc. The monographs of Hilfer \cite{Hil}, Kilbas {\em et al.} \cite{KST}, Podlubny \cite{Pod}, and Momani {\em et al.} \cite{MoHaAl} are very good sources on the background mathematics and various applications of fractional derivatives. The literature on Hadamard-type fractional differential equations has not undergone as much development as it has for the Caputo and Riemann-Liouville fractional derivatives; see, for example, the papers of Ahmed and Ntouyas \cite{AhNt}, Benhamida, Graef, and Hamani \cite{HG1}, and Thiramanus, Ntouyas, and Tariboon \cite{ThNtTa}. The fractional derivative that Hadamard \cite{Ha} introduced in 1892 differs from other fractional derivatives in the sense that the kernel of the integral in the definition of the Hadamard derivative contains a logarithmic function with an arbitrary exponent. A detailed description of the Hadamard fractional derivative and integral can be found in \cite{BuKiTr,BuKiTr1,BuKiTr2}. In this paper, we present existence results for the problem (\ref{e1})--(\ref{e2}) in the case where the right hand side is convex valued. This result relies on the set-valued analog of M\"{o}nch's fixed point theorem combined with the technique of measure of noncompactness. Recently, this has proved to be a valuable tool in studying fractional differential equations and inclusions in Banach spaces; for additional details, see the papers of Laosta {\em et al.} \cite{LaOp}, Agarwal {\em et al.} \cite{AgBeSe}, and Benchohra {\em et al.} \cite{BeHeSe1, BeHeSe, BeNiSe}. Our results here extend to the multivalued case some previous results in the literature and constitutes what we hope is an interesting contribution to this emerging field. We include an example to illustrate our main results. \section{Preliminaries} This section contains definitions, concepts, lemmas, and preliminary facts that will be used in the remainder of this paper. Let $C(J,E)$ be the Banach space of all continuous functions from $J$ into $E$ with the norm $$ \|y\|_{\infty}=\sup\{|y(t)|: t \in J\}, $$ and let $L^{1}(J,E)$ be the Banach space of Lebesgue integrable functions $y :J \to E$ with the norm $$ \|y\|_{L^{1}}=\Int_{1}^{T} |y(t)| dt. $$ The space $AC^{1}(J,E)$ is the space of functions $y :J \to E$ that are absolutely continuous and have an absolutely continuous first derivative. %Also $C([1-r,1],\R)$ is endowed with the norm %$$ \|\varphi\|_{C}=\sup\{|\varphi(\theta)|: 1-r \leq \theta \leq 1\}. %whose first derivative, $y^{\prime}$, is absolutely continuous.\\ For any Banach space $X$, we set $P_{cl}(X)= \{ Y \in {\mathcal P}(X): \mbox{$Y$ is closed}\}$, $P_{b}(X)=\{Y \in {\mathcal P}(X): \mbox{$Y$ is bounded}\}$, $P_{cp}(X)= \{ Y \in{\mathcal P}(X): \mbox{$Y$ is compact}\}$, and $P_{cp,c}(X)=\{Y \in {\mathcal P}(X): \mbox{$Y$ is compact and convex}\}$. A multivalued map $G:X \to {\mathcal P}(X)$ is {\it convex (closed)} valued if $G(X)$ is convex (closed) for all $x \in X$. We say that $G$ is {\it bounded on bounded sets} if $G(B)=\cup_{x \in B} G(x)$ is bounded in $X$ for all $B \in P_{b}(X)$ (i.e., $\sup_{x \in B}\{\sup\{|y|: y \in G(x)\}\}$ is bounded). The mapping $G$ is {\it upper semi-continuous (u.s.c)} on $X$ if for each $x_{0} \in X$, the set $G(x_{0})$ is a nonempty closed subset of $X$, and for each open set $N$ of $X$ containing $G(x_{0})$, there exists an open neighborhood $N_{0}$ of $x_{0}$ such that $G(N_{0}) \subset N$. A map $G$ is said to be {\it completely continuous} if $G(B)$ is relatively compact for every $B \in P_{b}(X)$. If the multivalued map $G$ is completely continuous with nonempty compact values, then $G$ is u.s.c if and only if $G$ has a closed graph (i.e., $ x_{n} \to x_{*}$, $y_{n} \to y_{*}$, $y_{n} \in G(x_{n})$ imply $y_{*} \in G(x_{*})$). The mapping $G: X \to \mathcal{P}(X)$ has a fixed point if there exists $x \in X$ such that $x \in G(x)$. The set of fixed points of the multivalued operator $G$ will be denoted by $Fix\, G$. A multivalued map $G:J\to P_{cl}(X)$ is said to be measurable if for every $y \in X$, the function $$ t \to d(y,G(t))=inf\{|y-z|: z \in G(t)\} $$ is measurable. \begin{definition} A multivalued map $F:J \times E \to {\mathcal P}(E) $ is said to be Carath\'eodory if: \begin{itemize} \item [(1)] $t \to F(t,u)$ is measurable for each $ u \in E$; \item [(2)] $u \to F(t,u)$ is upper semicontinuous for a.e. $t \in J$. \end{itemize} \end{definition} For each $y \in AC^{1}(J,E)$, define the set of selections of $F$ by $$ S_{F,y}=\{v \in L^{1}(J,E):v(t)\in F(t,y (t)) \text{ a.e. } t \in J\}. $$ Let $(X,d)$ be a metric space induced from the normed space $(X,|\cdot|)$. The function $H_{d}:{\mathcal P}(X) \times {\mathcal P}(X) \to \R_{+} \cup \{\infty\}$ given by \begin{equation*} H_{d}(A,B)=max\{\sup\limits_{a \in A}d(a,B),\sup\limits_{b \in B}d(A,b)\} \end{equation*} is known as the Hausdorff-Pompeiu metric. For more details on multivalued maps, see the books of Aubin and Cellina \cite{AuCe}, Aubin and Frankowska \cite{AuFr}, Castaing and Valadier \cite{CaVa}, and Deimling \cite{De}. For convenience, we first recall the definitions of the Kuratowski measure of noncompacteness and summarize the main properties of this measure. \begin{definition} {\rm (\cite{AkKaPaRoSa, BaGo})} Let $E$ be a Banach space and let $\Omega_{E}$ be the bounded subsets of $E.$ The Kuratowski measure of noncompactness is the map $ \beta \,: \Omega_{E} \to [0,\infty ) $ defined by $$ \beta (B)=\inf \{ \epsilon > 0 \, : B \subset \bigcup_{j=1}^{m}B_j \hbox{ and } diam (B_{j})\leq \epsilon \}\,. $$ \end{definition} {\bf Properties:} The Kuratowski measure of noncompactness satisfies the following properties (for more details see \cite{AkKaPaRoSa, BaGo}): \begin{itemize} \item[(P$_1$)] \quad $\beta (B) = 0$ if and only if $\overline{B}$ is compact ($B$ is relatively compact). \item[(P$_2$)] \quad $ \beta (B) = \beta(\overline{B})$. \item[(P$_3$)] \quad $ A\subset B$ implies $\beta (A)\leq \beta (B)$. \item[(P$_4$)] \quad $ \beta (A+B)\leq \beta (A)+ \beta (B).$ \item[(P$_5$)] \quad $ \beta ( c B)=|c| \beta(B)$, \ $c\in \R. $ \item[(P$_6$)] \quad $ \beta (conv B) = \beta(B).$ \end{itemize} Here $\overline{B}$ and $conv\, B$ denote the closure and the convex hull of the bounded set $B$, respectively. For a given set $V$ of functions $u : J \to E$, we set $$ V(t) =\{u(t)\, : u \in V\}, \, t\in J, $$ and $$ V(J) =\{u(t)\, : u \in V(t), \, t\in J\}. $$ \begin{theorem} \label{th1} {\rm (\cite{He}, \cite[Theorem 1.3]{OrPr})} Let $E$ be a Banach space and let $C$ be a countable subset of $L^{1}(J,E)$ such that there exists $h \in L^{1}(J, \R_{+})$ with $|u(t)| \leq h(t)$ for a.e. $t \in J$ and every $u \in C$. Then the function $\varphi(t) = \beta(C(t))$ belongs to $L^{1}(J, \R_{+})$ and satisfies $$ \beta\left(\left\{\int_{0}^{T} u(s) ds : u \in C \right\}\right) \leq 2 \int_{0}^{T} \beta(C(s)) ds. $$ \end{theorem} \begin{lemma} \label{lem1} {\rm (\cite[Lemma 2.6]{LaOp})} Let $J$ be a compact real interval, $F$ be a Carath\'eodory multivalued map, and let $\theta$ be a linear continuous map from $L^{1}(J,E) \to C(J,E)$. Then the operator $$ \theta \circ S_{F,y}: L^{1}(J,E) \to P_{cp,c}(C(J,E)), \quad y \to (\theta \circ S_{F,y})(y)=\theta(S_{F,y}) $$ is a closed graph operator in $L^{1}(J,E) \times C(J,E)$. \end{lemma} In what follows, $\log(\cdot)=\log_{e}(\cdot)$, and $n=[r]+1$ where $[r]$ denotes the integer part of $r$. \begin{definition} {\rm (\cite{KST})} The Hadamard fractional integral of order $r$ for a function $h\, : [1,+\infty) \to \R $ is defined by $$ I^{r}h(t)= \Frac{1}{\Gamma(r)}\Int_1^t\left(\log \Frac{t}{s}\right)^{r-1}\Frac{h(s)}{s}ds,\,\, r>0, $$ provided the integral exists. \end{definition} \begin{definition} {\rm (\cite{KST})} For a function $h$ on the interval $[1,+\infty)$, the Hadamard fractional derivative of $h$ of order $r$ is defined by \begin{equation*} ({}^{H}D^{r}h)(t) = \Frac{1}{\Gamma(n-r)}\left(t\Frac{d}{dt}\right)^{n}\Int_ 1^t\left(\log \Frac{t}{s}\right)^{n-r-1}\Frac{h(s)}{s}ds, \ n-10$, $B = \{x \in E : \|x\| \leq R\}$, $U = \{x \in C(J,E) : \|x\| \leq R\}$, and assume that: \begin{itemize} \item[(H1)] $F: J \times E \to {\mathcal P}_{cp,p}(E)$ is a Carath\'eodory multi-valued map; \item[(H2)] For each $ R > 0$, there exists a function $p \in L^{1}(J, E)$ such that $$ \|F(t, u)\|_{\mathcal P} = \sup\{|v| : v(t) \in F(t, y)\} \leq p(t) $$ for each $(t, y) \in J \times E$ with $|y| \geq R$, and $$ \liminf_{R\to \infty} \Frac{\int_{0}^{T} p(t)dt}{R}=\delta < \infty; $$ \item[(H3)] There exists a Carath\'eodory function $\psi : J\times [1, 2R] \to \R_{+}$ such that $$ \beta(F(t, M)) \leq \psi(t, \beta(M)) \ \mbox{a.e. $t \in J$ and each $M \subset B$;} $$ \item[(H4)] The function $\varphi = 0$ is the unique solution in $C(J, [1, 2R])$ of the inequality \begin{align} \label{new-801} \varphi(t) &\leq 2\left\{ \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1} \psi(s, \varphi(s))\Frac{ds}{s}\right. \nonumber\\ &\quad + \left.\Frac{(\log t)^{r-1}}{(\log T)^{r-1}}\left[y_T + \Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}\psi(s, \varphi(s))\Frac{ds}{s}\right]\right\} \ \ \mbox{for $t \in J$.} \end{align} \end{itemize} Then the BVP (\ref{e1})--(\ref{e2}) has at least one solution in $C(J, B)$, provided that \begin{equation} \label{new-501} \delta < \Frac{\Gamma(r+1)}{(\log T)^{r}}. \end{equation} \end{theorem} \begin{proof} We wish to transform the problem (\ref{e1})--(\ref{e2}) into a fixed point problem, so consider the multivalued operator \begin{multline*} N(y)=\left\{h \in C(J,\R): h(t) = \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}v(s)\Frac{ds}{s}\right.\\ \quad + \left. \Frac{(\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T}-\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}v(s)\Frac{ds}{s}\right], \ \ v \in S_{F,y}\right\}. \end{multline*} Clearly, from Lemma \ref{l2}, the fixed points of $N$ are solutions to (\ref{e1})--(\ref{e2}). We shall show that $N$ satisfies the assumptions of M\"{o}nch's fixed point theorem. The proof will be given in several steps. First note that $\overline{U} = C(J,B)$. \vskip 0.3cm {\bf Step 1:} {\em $N(y) $ is convex for each $y \in C(J,B)$.} \smallskip Take $h_{1}$, $h_{2} \in N(y)$; then there exist $v_{1}$, $v_{2} \in S_{F,y}$ such that for each $t \in J$, we have \begin{align*} h_{i}(t)&= \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}v_{i}(s)\Frac{ds}{s}\\ &\quad + \Frac{ (\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T}-\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}v_{i}(s)\Frac{ds}{s}\right] \end{align*} for $i=1$, $2$. Let $0 \leq d \leq 1$; then for each $t \in J$, \begin{align*} (dh_{1}+(1-d)h_{2})(t) &= \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}[dv_{1}+(1-d)v_{2}]\Frac{ds}{s}\\ &\quad + \Frac{ (\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T}-\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}[dv_{1}+(1-d)v_{2}]\Frac{ds}{s}\right]. \end{align*} Since $S_{F,y}$ is convex (because $F$ has convex values), we have $$ dh_{1}+(1-d)h_{2} \in N(y). $$ \vskip 0.3cm {\bf Step 2:} {\em $N(M)$ is relatively compact for each compact $M \subset \overline{U}$.} \smallskip Let $ M \subset \overline{U}$ be a compact set and let $\{h_{n}\}$ be any sequence of elements of $N(M)$. We will show that $\{h_{n}\}$ has a convergent subsequence by using the Arzel\`a-Ascoli criterion of compactness in $C(J,B)$. Since $h_{n} \in N(M)$, there exist $y_{n} \in M$ and $v_{n} \in S_{F,y}$ such that \begin{align*} h_{n}(t) &= \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}v_{n}(s)\Frac{ds}{s}\\ &\quad + \Frac{ (\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T}-\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}v_{n}(s)\Frac{ds}{s}\right] \end{align*} for $n \geq 1$. Using Theorem \ref{th1} and the properties of the Kuratowski measure of noncompactness, we have \begin{align} \label{eq1} \beta(\{h_{n}(t)\}) &\leq 2 \left\{ \Frac{1}{\Gamma(r)}\Int_{1}^{t} \beta\left(\left\{ \left(\log \Frac{t}{s}\right)^{r-1}\Frac{v_{n}(s)}{s} : n\geq 1 \right\}\right)ds\right. \nonumber\\ &\quad + \left.\Frac{ (\log t)^{r-1}}{(\log T)^{r-1}} \left[y_{T} + \Frac{1}{\Gamma(r)} \Int_{1}^{T} \beta\left(\left\{\left(\log \Frac{T}{s}\right)^{r-1}\Frac{v_{n}(s)}{s} : n \geq 1\right\}\right)ds\right]\right\}. \end{align} On the other hand, since $M(s)$ is compact in E, the set $\{v_{n}(s) : n \geq 1\}$ is compact. Consequently, $\beta(\{v_{n}(s) : n \geq 1\}) = 0$ for a.e. $s \in J$. Furthermore, $$ \beta\left(\left \{\left(\log \Frac{t}{s}\right)^{r-1} \Frac{v_{n}(s)}{s}\right\}\right) =\left(\log \Frac{t}{s}\right)^{r-1}\Frac{1}{s}\, \beta(\{v_{n}(s) : n \geq 1\}) = 0 $$ and $$ \beta\left(\left\{\left(\log \Frac{T}{s}\right)^{r-1} \Frac{v_{n}(s)}{s}\right\}\right) = \left(\log \Frac{T}{s}\right)^{r-1}\Frac{1}{s}\, \beta(\{v_{n}(s) : n \geq 1\}) = 0 $$ for a.e. $t$, $s \in J$. Hence, from this and (\ref{eq1}), $\{h_{n}(t) : n \geq 1\}$ is relatively compact in $B$ for each $t \in J$. In addition, for each $t_{1}$, $t_{2} \in J$ with $t_{1} < t_{2}$, we have \begin{align*} |h_{n}(t_{2})-h_{n}(t_{1})| &= \left|\Frac{1}{\Gamma(\alpha)} \Int_{1}^{t_{1}}\left[\left(\log \Frac{t_2}{s}\right)^{\alpha-1}- \left(\log \Frac{t_1}{s}\right)^{\alpha-1}\right]\Frac{v_{n}(s)}{s} ds \right. \\ &\quad + \left. \Frac{1}{\Gamma(\alpha)}\Int_{t_{1}}^{t_{2}} \left(\log \Frac{t_{2}}{s}\right)^{\alpha-1}\Frac{v_{n}(s)}{s} ds \right| \\ &\leq \Frac{p(t)}{\Gamma(\alpha)} \Int_{1}^{t_{1}}\left[ \left(\log \Frac{t_2}{s}\right)^{\alpha-1}- \left(\log \Frac{t_1}{s}\right)^{\alpha-1} \right]\Frac{ds}{s} \\ &\quad + \Frac{p(t)}{\Gamma(\alpha)}\Int_{t_{1}}^{t_{2}} \left(\log \Frac{t_{2}}{s}\right)^{\alpha-1}\Frac{ds}{s}. \end{align*} As $t_{1} \to t_{2}$, the right hand side of the above inequality tends to zero. This shows that $\{h_{n} : n \geq 1\}$ is equicontinuous. Consequently, $\{h_{n} : n \geq 1\}$ is relatively compact in $C(J,B)$. \vskip 0.3cm {\bf Step 3:} {\em $N$ has a closed graph.} \smallskip Let $y_{n} \to y_{*}$, $h_{n} \in N(y_{n})$, and $h_{n} \to h_{*}$. We need to show that $h_{*} \in N(y_{*})$. Now $h_{n} \in N(y_{n})$ means that there exists $v_{n} \in S_{F,y}$ such that, for each $t \in J$, \begin{align*} h_{n}(t) &= \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}v_{n}(s)\Frac{ds}{s}\\ &\quad + \Frac{(\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T}-\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}v_{n}(s)\Frac{ds}{s}\right]. \end{align*} Consider the continuous linear operator $\theta : L^{1}(J, E) \to C(J, E)$ defined by \begin{align*} \theta (v) (t) \to h_{n}(t)&= \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}v_{n}(s)\Frac{ds}{s}\\ &\quad + \Frac{(\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T}-\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}v_{n}(s)\Frac{ds}{s}\right]. \end{align*} Clearly, $\|h_{n}(t) - h_*(t)\| \to 0$ as $n \to \infty$. From Lemma \ref{lem1} it follows that $\theta \circ S_{F}$ is a closed graph operator. Moreover, $h_{n}(t) \in \theta(S_{F,y_{n}})$. Since $y_{n} \to y$, Lemma \ref{lem1} implies \begin{equation*} h(t) = \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}v(s)\Frac{ds}{s} + \Frac{ (\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T}-\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}v(s)\Frac{ds}{s}\right]. \end{equation*} \vskip 0.3cm {\bf Step 4:} {\em $M$ is relatively compact in $C(J, B)$.} \smallskip Suppose $M \subset \overline{U}$, $M \subset conv(\{0\}\cup N(M))$, and $\overline{M} = \overline{C}$ for some countable set $C \subset M$. Using an argument similar to the one used in Step 2 shows that $N(M)$ is equicontinuous. Then, since $ M \subset conv(\{0\} \cup N(M))$, we see that $M$ is equicontinuous as well. To apply the Arzel\`a-Ascoli theorem, it remains to show that $M(t)$ is relatively compact in $E$ for each $t \in J$. Since $C \subset M \subset conv(\{0\}\cup N(M))$ and $C$ is countable, we can find a countable set $H = \{h_{n} : n \geq 1\} \subset N(M)$ with $C \subset conv(\{0\} \cup H)$. Then, there exist $y_{n} \in M$ and $v_{n} \in S_{F,y_{n}}$ such that \begin{align*} h_{n}(t) &= \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}v_{n}(s)\Frac{ds}{s}\\ &\quad + \Frac{ (\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T}-\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}v_{n}(s)\Frac{ds}{s}\right]. \end{align*} Since $M \subset C \subset conv(\{0\} \cup H))$, from the properties of the Kuratowski measure of noncompactness, we have $$ \beta(M(t)) \leq (\beta(C(t)) \leq \beta(H(t)) = \beta(\{h_{n}(t) : n \geq 1\}). $$ Using (\ref{eq1}) and the fact that $v_{n}(s) \in M(s)$, we obtain \begin{align*} \beta(M(t)) &\leq 2 \left\{ \Frac{1}{\Gamma(r)}\Int_{1}^{t} \beta\left(\left\{ \left(\log \Frac{t}{s}\right)^{r-1}\Frac{v_{n}(s)}{s} : n \geq 1\right\}\right)ds\right.\\ &\quad + \left.\Frac{ (\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T} + \Frac{1}{\Gamma(r)}\Int_{1}^{T} \beta\left(\left\{\left(\log \Frac{T}{s}\right)^{r-1}\Frac{v_{n}(s)}{s} : n \geq 1\right\}\right)ds\right]\right\}\\ &\leq 2 \left\{ \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}\beta(M(s))\Frac{ds}{s}\right.\\ &\quad + \left.\Frac{ (\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T} + \Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}\beta(M(s))\Frac{ds}{s}\right]\right\} \\ &\leq 2\left\{ \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1} \psi(s,\beta(M(s)))\Frac{ds}{s}\right. \\ &\quad + \left.\Frac{ (\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T} + \Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}\psi(s,\beta(M(s)))\Frac{ds}{s}\right]\right\}. \end{align*} We also have that the function $\varphi$ given by $\varphi(t) = \beta(M(t))$ belongs to $C(J, [1, 2R])$. Consequently, by $(H4)$, $\varphi= 0$; that is, $\beta(M(t)) = 0$ for all $t \in J$. Now, by the Arzel\`a-Ascoli theorem, $M$ is relatively compact in $C(J, B)$. \vskip 0.3cm {\bf Step 5:} Let $h \in N(y)$ with $y \in U$. We claim that $N(U) \subset U$. If this were not the case, then in view of $(H2)$, there exists functions $v \in S_{F,y}$ and $p \in L^{1}(J,E)$ such that \begin{align*} h(t)&= \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}v(s)\Frac{ds}{s}\\ &\quad + \Frac{ (\log t)^{r-1}}{(\log T)^{r-1}}\left[y_{T}-\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}v(s)\Frac{ds}{s}\right], \end{align*} and \begin{align*} R < \|N(y)\|_{\mathcal P} &\leq \Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1}|v(s)|\Frac{ds}{s}\\ &\quad + \Frac{(\log t)^{r-1}}{(\log T)^{r-1}}\left[|y_{T}|+\Frac{1}{\Gamma(r)}\Int_{1}^{T} \left(\log \Frac{T}{s}\right)^{r-1}|v(s)|\Frac{ds}{s}\right]\\ &\leq \Frac{(\log T)^{r}}{\Gamma(r+1)}\Int_{1}^{t} p(s)ds + \Frac{ (\log T)^{r}}{\Gamma(r+1)}\Int_{1}^{T} p(s)ds\\ &\leq 2\Frac{(\log T)^{r}}{\Gamma(r+1)}\Int_{1}^{t} p(s)ds. \end{align*} Dividing both sides by $R$ and taking the $\liminf$ as $R \to \infty$, we have $$ 2\left[\Frac{(\log T)^{r}}{\Gamma(r+1)}\right] \delta \geq 1 $$ which contradicts \eqref{new-501}. Hence, $N(U) \subset U$. As a consequence of Steps 1--5 and M\"onch's Theorem (Theorem \ref{t1} above), $N$ has a fixed point $y \in C(J, B)$ that in turn is a solution of problem (\ref{e1})--(\ref{e2}). \end{proof} \section{An Example} We conclude this paper with an example to illustrate our main result, namely, Theorem \ref{t2} above. Consider the fractional differential inclusion \begin{equation}\label{ex1} {}^{H} D^{\alpha}y(t) \in F(t,y(t)), \ \text{for a.e.} \ t \in J=[1,e], \ 0<\alpha\leq 1, \end{equation} \begin{equation}\label{ex2} y(1)=0, \ y(e)=1. \end{equation} Here, $F:[1,e] \times \R \to {\mathcal P}(\R)$ is a multivalued map satisfying \begin{equation*} F(t,y)=\{v \in \R:f_{1}(t,y) \leq v \leq f_{2}(t,y)\}, \end{equation*} where $f_{1}$, $f_{2}: [1,e] \times \R \to \R$, $f_{1}(t,\cdot)$ is lower semi-continuous (i.e., the set $\{y \in \R : f_{1}(t,y) >\mu$\} is open for each $\mu \in \R$), and $f_{2}(t,\cdot)$ is upper semi-continuous (i.e., the set the set $\{y \in \R : f_{2}(t,y) <\mu$\} is open for each $\mu \in \R$). We assume that there is a function $p \in L^{1}(J, \R)$ such that \begin{multline*} \|F(t, u)\|_{\mathcal P} = \sup\{|v|, v(t) \in F(t, y)\}=\max (|f_{1}(t,y)|,|f_{2}(t,y)|\} \leq p(t), \ t\in [1,e], \ y \in \R. \end{multline*} It is clear that $F$ is compact and convex valued, and is upper semi-continuous. Choose $C(s)$ to be the space of linear functions and choose $\varphi(t) = \beta(C(t))$ such that $$ \beta(u(s))=\Frac{u(s)}{2} $$ with $$ u(s)=as, \ a > 0, \ \Frac{2}{a} \leq s \leq \Frac{4R}{a}. $$ For $(t, y) \in J \times \R$ with $|y| \geq R$, we have $$ \liminf_{R\to \infty} \Frac{\int_{0}^{e} p(t) dt }{R}=\delta < \infty. $$ Finally, we assume that there exists a Carath\'eodory function $\psi : J × [1, 2R] \to \R_{+}$ such that $$ \beta(F(t, M)) \leq \psi(t, \beta(M)) \ a.e. \ t \in J \ \mbox{and each} \ M \subset B = \{x \in \R : |x| \leq R\}, $$ and $\varphi = 0$ is the unique solution in $C(J, [1, 2R])$ of the inequality \begin{align*} \varphi(t) &\leq 2 \left\{\Frac{1}{\Gamma(r)}\Int_{1}^{t} \left(\log \Frac{t}{s}\right)^{r-1} \psi(s, \varphi(s))\Frac{ds}{s}\right. \\ &\quad + \left.(\log t)^{r-1} \left[1+\Frac{1}{\Gamma(r)}\Int_{1}^{e} \left(\log \Frac{e}{s}\right)^{r-1}\psi(s, \varphi(s))\Frac{ds}{s}\right] \right\}. \end{align*} for $t \in J$. 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