\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 233, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/233 fractional differential inclusions\hfil ]
{Boundary-value problems for Riemann-Liouville
fractional differential inclusions in \\
Banach spaces}

\author[S. Hamani, J. Henderson \hfil EJDE-2015/233\hfilneg]
{Samira Hamani, Johnny Henderson}

\address{Samira Hamani \newline
D\'epartement de Math\'ematiques,
Universit\'e de Mostaganem,
B.P. 227, 27000, Mostaganem, Alg\'erie}
\email{hamani\_samira@yahoo.fr}

\address{Johnny Henderson \newline
Department of Mathematics, Baylor University,
Waco, TX  76798-7328, USA}
\email{johnny\_henderson@baylor.edu}

\thanks{Submitted July 10, 2014 Published September 11, 2015.}
\subjclass[2010]{26A33, 34A60}
\keywords{Differential inclusion; Riemann-Liouville fractional derivative; 
\hfill\break\indent fractional integral; Banach space; fixed point}

\begin{abstract}
 In this article, we sudy the existence of solutions of
 boundary-value problems for Riemann-Liouville fractional differential
 inclusions of order $r\in (2,3]$ in a Banach space.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

This article concerns the existence of solutions for boundary-value 
problems (BVP for short), for  fractional order differential
inclusions. We consider the bound\-ary-value problem
\begin{gather}\label{e1}
D^ry(t)\in F(t,y), \quad\text{for a.e. } t\in J=[0,T],\\
\label{e2}
 y(0)=0, \quad  y'(0)=0,\quad y''(T)=0,
\end{gather}
where $ 2<r\leq3$, $D^r$ is the
 Riemann-Liouville fractional derivative, 
$F: J\times E \to\mathcal{P}( E)$  is a multivalued map, $\mathcal{P}( E)$ is
 the family of all nonempty subsets of $E$, and
 $(E, |\cdot|)$ denotes a Banach space.

Differential equations of fractional order have recently proved to be valuable
tools in the modeling of many phenomena in various fields of
science and engineering. Indeed, there are numerous applications
in viscoelasticity, electrochemistry, control, porous media,
electromagnetism, and so on. There has been a significant development in
fractional differential equations in recent years; see the
 monographs of Hilfer \cite{Hil}, Kilbas et al.\ \cite{KiMa,KiSrTr},
Delbosco  et al.\  \cite{DeRo}, Miller et al.\ \cite{MiRo},
Heymans  et al.\  \cite{HePo}, Podlubny\cite{Pod,Pod1},
Kaufman  et al.\  \cite{KaMb}, Karakostas  et al.\  \cite{KaTs},
Momani and Hadid \cite{MoHaAl}, and the papers by Agarwal  et al.\
\cite{AgBeHa,AgBeHa1,AgBeHa2}, Bai  et al.\  \cite{Ba,BaLu}, Benchohra  
et al.\  \cite{ BeDjHa,BeHa1,BeHa2}, and Hamani
 et al.\  \cite{HaBeGr}.

In this article, we present existence results for the problem
\eqref{e1}-\eqref{e2}, when 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 valued tool in solving fractional
differential equation and inclusions in Banach spaces; for details, see
the papers of Lasota  et al.\  \cite{LaOp}, Agarwal  et al.\  \cite{AgBeSe} 
and Benchohra  et al.\  \cite{BeHeSe,BeHeSe1,BeNiSe}.
This result extends to the multivalued
case some previous results in the literature, and constitutes an interesting
contribution to this emerging field.

\section{Preliminaries}

  In this section, we introduce definitions, 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
 $$
\|y\|=\sup\{|y(t)|: 0\le t \le T\}, 
$$ 
and we let
$L^{1}(J,E)$ denote the Banach space of functions
$y:J\to E $ which are Bochner integrable with norm 
$$ 
\|y\|_{L^1}=\int_{0}^{T}|y(t)|dt.
 $$
$AC^{1}(J,E)$ is the space of functions
$y:J \to E$, which are absolutely continuous whose first
derivative, $y'$, is absolutely continuous.

Let $(E,|\cdot|)$ be a Banach space. Let
$ P_{cl}(E)=\{A\in \mathcal{P}(E): A\,\text{ closed} \}$,
$P_{c}(E)=\{A\in \mathcal{P}(E): A \text{ convex} \}$,
$P_{cp,c}(E)=\{A\in \mathcal{P}(E): A  \text{ compact and convex}\}$.
A multivalued map $G:E \to \mathcal{P}(E)$
 has a fixed point if there is $x\in E$ such
that $x\in G(E)$. The fixed point set of the multivalued operator
$G$ will be denoted by $Fix G$. A multivalued map $G:J\to
P_{cl}(\mathbb{R})$ is said to be measurable if for every $y\in \mathbb{R}$, the
function 
$t\mapsto d(y,G(t))=\inf\{|y-z|: z\in G(t) \}$
is measurable.

Let $X,Y $ be two sets, and $N :X \to \mathcal{P}(Y) $ be a set-valued map.
We define the graph of $N$, as
\[
\operatorname{graph}(N)= \{(x,y): x\in X,\; y\in N(X)\}\,.
\]
For more details on multivalued maps see the books of
Deimling \cite{De}, Aubin  et al.\  \cite{AuCe,AuFr} and
 Hu and Papageorgiou \cite{HuPa}.

Let $ R > 0$, and 
$$ 
B =\{ x\in E: |x| \leq R \},\quad 
 U =\{ x\in  C(J,E): \|x\| \leq R \},
$$
Clearly $U$ is a closed subset of $C(J,B)$.

\begin{definition}[\cite{KiSrTr,Pod}] \label{def2.1} \rm 
The fractional (arbitrary) order integral of the function 
$h\in L^1([a,b],\mathbb{R}_+)$ of order $r\in\mathbb{R}_+$ is defined
by 
$$
I^r_ah(t)=\int_a^t\frac{(t-s)^{r-1}}{\Gamma(r)}h(s)\,ds,
$$ 
where $\Gamma$ is the gamma function. When $a=0$, we write
$I^rh(t)=h(t)*\varphi_{r}(t)$, where
$\varphi_{r}(t)=\frac{t^{r-1}}{\Gamma(r)}$ for
$t>0$, and  $\varphi_{r}(t)=0$ for $t\leq 0$, and
$\varphi_{r}\to \delta(t)$\ as $r\to 0$, where $\delta$
is the delta function.
\end{definition}

\begin{definition}[\cite{KiSrTr,Pod}]\rm
For a function $h$ given on the interval $[a,b]$, the $r $
Riemann-Liouville fractional-order derivative of $h$, is defined
by
\[
(D^r_{a+}h)(t)=\frac{1}{\Gamma(n-r)}\big(\frac{d}{dt}\big)^{n}
\int_a^t(t-s)^{n-r-1}h(s)\,ds.
\]
Here $n=[r]+1$ and $[r]$ denotes the integer part of
$r$.
\end{definition}

For convenience, we recall the definition of the Kuratowski measure of
noncompactness.

\begin{definition}[\cite{AkKaPaRoSa, BaGo}] \rm
Let $E$ be a Banach space and let $\Omega_{E}$ be the family of
bounded subsets of $E$. The Kuratowski measure of noncompactness is
the map $ \alpha : \Omega_{E} \to [0,\infty ) $ defined by
\[ 
\alpha (M)=\inf \{ \epsilon > 0  : M \subset \cup_{j=1}^{m}M_j, 
\operatorname{diam}(M_j)\leq \epsilon \}\,,
\]
where $ M \in \Omega_{E}$.
\end{definition}

\noindent\textbf{Properties:}
\begin{itemize}
\item[(1)] $ \alpha (M) = 0 \Leftrightarrow \overline{M}$ is compact
($M$ is relatively compact).
\item[(2)] $ \alpha (M)=\alpha(\overline{M})$.
\item[(3)] $ M_1\subset M_2 \Rightarrow \alpha (M_1)\leq \alpha (M_2)$.
\item[(4)] $ \alpha (M_1+M_2)\leq \alpha (M_1)+\alpha (M_2)$.
\item[(5)] $ \alpha ( cM)=c\alpha(M)$, $c\in \mathbb{R}$. 
\item[(6)] $ \alpha (\operatorname{conv} M)=\alpha(M)$.
\end{itemize}
More properties of $\alpha$ can be found in \cite{AkKaPaRoSa, BaGo}.

\begin{definition} \rm
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 almost all
$t\in J$.
\end{itemize}
\end{definition}

For each $ y \in C(J,E)$, define the set of selections of $F$ by
$$
S_{F,y} =\{ v \in L^{1}(J,E) : v(t) \in F(t,y(t)) \text{ a.e. } t\in J \}.
$$

\begin{theorem}[\cite{He}] \label{thm1}
Let $E$ be a Banach space and $C\subset L^{1}(J,E) $ be countable with
$|u(t)| \leq h(t) $ for a.e. $t\in J$, and every $u \in C$, where
$h \in L^{1}(J,\mathbb{R}_{+})$. Then the function
$\phi(t)= \alpha (C(t))$ belongs to  $L^{1}(J,\mathbb{R}_{+})$ and satisfies
$$
 \alpha \Big( \Big\{ \int_{0}^{T}u(s) \,ds : u\in C \Big\}\Big)
\leq 2 \int_{0}^{T} \alpha(C(s))\,ds. 
$$
\end{theorem}

Let us now recall the set-valued analog of M\"{o}nch's fixed point theorem.

\begin{theorem}[\cite{OrPr}] \label{thm2}
Let $K$ be a closed, convex subset of a Banach space $E,U$ a relatively 
open subset of $K$, and $N:\overline{U} \to\mathcal{P}_c(K)$.
Assume $\operatorname{graph}(N)$ is closed, $N$ maps compact sets into 
relatively compact sets, and that, for some $x_{0}\in U$, the following 
two conditions are satisfied:
\begin{gather}\label{e3}
\parbox{10cm}{
  $M\subset \overline{U}$, $M \subset \operatorname{conv}(x_{0}\cup N(M))$ 
  and $\overline{M}=\overline{U}$ with $C\subset M$ countable imply
that $\overline{M}$ is compact}, \\
\label{e4}
x \in (1-\lambda) x_{0}+\lambda N(x)\quad \text{for all }
x \in \overline{U}\backslash  U, \; \lambda \in (0,1).
\end{gather}
Then there exists $x \in \overline{U}$ with $x\in N(x)$.
\end{theorem}

\begin{lemma}[\cite{LaOp}] \label{lem0}
Let $J$ be a compact real interval. Let $F$ 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
$$ 
\Theta \circ S_{F,y} :C(J,E) \to \mathcal{P}_{cp,c} (C(J,E)),\quad
 y \mapsto (\Theta \circ S_{F,y})(y)= \Theta(S_{F,y})
$$
is a closed graph operator in $C(J,E) \times C(J,E)$.
\end{lemma}

\section{Main results}
Let us start by defining what we mean by a solution of the problem
\eqref{e1}--\eqref{e2}.

\begin{definition}\label{def1} \rm
A function $y\in AC^{2}([0,T],E)$ is said to be a solution of
\eqref{e1}--\eqref{e2} if there exist a function $v \in L^{1}(J,E)$
with $ v(t) \in F(t,y(t)) $, for a.e. $ t\in J$, such that
$D^ry(t)=v(t)$  on $J$, and the  condition $ y(0)=0$,
$y'(0)=0$, $y''(T)=0$ are satisfied.
\end{definition}

For the existence of solutions for the problem
\eqref{e1}--\eqref{e2}, we need the following auxiliary lemma.

\begin{lemma}[\cite{BaLu}] \label{lem1} 
Let $r > 0 $, and $ h\in C(0,T)\cap L^1(0,T)$.  Then 
$$
{I^r}D^rh(t)= h(t)+c_1t^{r-1}+c_2t^{r-2}+\ldots+c_{n}t^{r-n}
$$ 
for some $c_{i}\in \mathbb{R}$, $i=1,\ldots,n$, where $n$ is the smallest integer 
greater than or equal to $r$.
\end{lemma}

\begin{lemma}\label{lem2}  
Let  $2 < r \leq 3$ and let  $ h:[0,T]\to E$ be continuous.  A function $y$
is a solution of the fractional integral equation
\begin{equation}\label{e5}
\begin{aligned}
y(t)&=\frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}h(s)\,ds\\
&\quad - \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}h(s)\,ds.
\end{aligned}
\end{equation}

if and only if  $y$\ is a solution of the fractional  BVP
\begin{gather}\label{e6}
D^ry(t)=h(t), \quad t\in[0,T], \\
\label{e7}
 y(0)=0,\quad y'(0)=0,\quad y''(T)=0.
\end{gather}
\end{lemma}

\begin{proof} 
Assume $y$ satisfies \eqref{e5}.  Then Lemma \ref{lem1}
implies that 
$$ 
y(t)=c_1t^{r-1}+c_2t^{r-2}+c_{3}t^{r-3}+\frac{1}{\Gamma(\alpha)}
\int_{0}^{t}(t-s)^{r-1}h(s)\,ds.
$$ 
From \eqref{e7}, a simple calculation gives 
\begin{gather*}
c_1=-\frac{1}{(r-1)(r-2)\Gamma
(r-3)}\int_{0}^{T}(T-s)^{r-3}h(s)\,ds, \\
c_2=0,quad  c_{3}=0\,.
\end{gather*}
Hence we get equation \eqref{e5}.  Conversely, it is clear that if
$y$ satisfies equation \eqref{e5}, then equations
\eqref{e6}-\eqref{e7} hold. 
\end{proof}

\begin{theorem} Assume the following
hypotheses  hold:
\begin{itemize}
\item[(H1)]  $F: J\times \mathbb{R}\to \mathcal{P}_{cp,c}(\mathbb{R})$ is a
Carath\'eodory  multivalued map.

\item[(H2)] For each $R> 0$, there exists a function
 $p\in L^{1}(J,E) $ and  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| \leq R $, and
$$ 
\lim _{R \to +\infty} \inf \frac{\int_{0}^{T} p(t) dt }{R}=\delta <\infty .
$$


\item[(H3)] There exists a Carath\'eodory function 
$\psi : J \times [0,2R] \to \mathbb{R}_{+} $
such that
$$ 
\alpha (F(t,M))\leq \psi (t,\alpha(M)),\quad \text{a.e. $t \in J$ 
 and each $ M\subset B$}, 
$$
and $\phi \equiv 0$ is the unique solution in $ C(J,[0,2R]) $ of the inequality
\begin{equation}\label{e8}
\begin{aligned}
\phi(t) 
&\leq  2 [ \frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}\varphi(s,\phi(s))\,ds
\\
&\quad - \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}\varphi(s,\phi(s))\,ds],
\end{aligned}
\end{equation}
for $t\in J$.
\end{itemize}
Then the BVP \eqref{e1}--\eqref{e2} has at least one
solution on $C (J,B) $, provided that
\begin{equation}\label{e11}
 \delta <\Big[\frac{T}{\Gamma(r+1)}+
\frac{T^{2}}{(r-1)(r-2)\Gamma(r-1)}\Big].
\end{equation}
\end{theorem}

\begin{proof} 
First we transform  problem \eqref{e1}--\eqref{e2} into 
a fixed point problem. Consider the multivalued  operator
\begin{align*} 
N(y)=\Big\{&h\in C(J,E):
(Ny)(t)=\frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}v(s)\,ds\\
&-\frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}v(s)\,ds,\;
 v \in S_{F,y}\Big\}.
\end{align*}

 Clearly, from Lemma \ref{lem2}, the fixed points of
$N$ are solutions to \eqref{e1}--\eqref{e2}.
We shall show that $N$ satisfies the assumptions of the set-valued analog of
M\"{o}nch's fixed point theorem.  The proof will be given in
several steps.
\smallskip

\noindent\textbf{Step 1:}  $N(y)$ is
convex for each $y\in C(J,E)$.
   Indeed, if $h_1,h_2$ belong to $N(y)$, 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_{0}^{t}(t-s)^{r-1}v_{i}(s)\,ds\\
&\quad - \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}v_{i}(s)\,ds,
\quad i=1,2.
\end{align*}
Let $0\leq d\leq 1$. Then, for each $t\in J$, we have 
\begin{align*}
&(dh_1+(1-d)h_2)(t)\\
&=\frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}
[dv_1(s)+(1-d)v_2(s)]\,ds
\\
&\quad+ \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}[dv_1(s)+(1-d)v_2(s)]\,ds.
\end{align*}
Since $S_{F,y}$ is convex (because $F$ has convex values), we
have $dh_1+(1-d)h_2\in N(y)$.
\smallskip

\noindent\textbf{Step 2:}
$N(M)$ is relatively compact for each compact  $ M \subset \overline{U}$.
Let $M \subset \overline{U}$ be a compact set and let $(h_{n})$ by any
sequence of elements of $N(M)$. We show that $(h_{n})$ has a convergent 
subsequence by using the Arz\'ela-Ascoli criterion of compactness in $C(J,E)$. 
Since $h_{n} \in N(M)$ 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_{0}^{t}(t-s)^{r-1}v_{n}(s)\,ds\\
&\quad - \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}v_{n}(s)\,ds.
\end{align*}
Using Theorem \ref{thm1} and the properties of the measure of
noncompactness of Kuratowski,
we have
\begin{equation} \label{e9}
\begin{aligned}
\alpha (\{h_{n}(t)\}) 
&\leq  2 [ \frac{1}{\Gamma(r)} \int_{0}^{t}\alpha (\{(t-s)^{r-1}v_{n}(s) \})\,ds
\\
&\quad - \frac{t^{\alpha-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}\alpha (\{(T-s)^{r-3} v_{n}(s)\})\,ds].
\end{aligned}
\end{equation}
On the other hand, since $M(s)$ is compact in $E$, the set 
$\{ v_{n}(s): n\geq 1\}$ is compact.
Consequently, $\alpha (\{ v_{n}(s): n\geq 1\})=0 $ for a.e. $s\in J$. 
Furthermore
\begin{gather*}
\alpha (\{(t-s)^{r-1}v_{n}(s) \}) 
= (t-s)^{r-1}\alpha (\{ v_{n}(s): n\geq 1\})=0, \\
\alpha (\{(T-s)^{r-1}v_{n}(s) \}) = (T-s)^{r-1}\alpha (\{ v_{n}(s): n\geq 1\})=0,
\end{gather*}
for a.e. $t,s\in J$. Now \eqref{e9} implies that
$\{ h_{n}(t): n\geq 1\}$ is
relatively compact in $E$, for each $t\in J$.
In addition, for each $t_1$ and $t_2$ from $J$,  $t_1<t_2$, we have
\begin{equation}\label{e10}
\begin{aligned}
&|h_{n}(t_2)-h_{n}(t_1)|\\
&=\Bigl|\frac{1}{\Gamma(r)}\int_0^{t_1}[(t_2-s)^{r-1}
-(t_1-s)^{r-1}]v_{n}(s)\,ds 
 +\frac{1}{\Gamma(r)}\int_{t_1}^{t_2}(t_2-s)^{r-1}v_{n}(s)\,ds \Bigl|\\
&\quad +\frac{(t_2-t_1)^{r-1}}{(r-1)(r-2)\Gamma(r-2)}\int_{0}^{T}
 (T-s)^{r-3}|v_{n}(s)|\,ds\\
& \leq \frac{p(t)}{\Gamma(r)}\int_0^{t_1}[(t_1-s
)^{r-1}-(t_2-s)^{r-1}]\,ds 
+\frac{p(t)}{\Gamma(r)}\int_{t_1}^{t_2}(t_2-s )^{r-1}\,ds \\
&\quad +  \frac{ p(t)(t_2-t_1)^{r-1}}{ (r-1)(r-2)\Gamma(r-2)}\int_{0}^{T}
(T-s)^{r-3}\,ds
\\
&\leq \frac{p(t)
}{\Gamma(r+1)}[(t_2-t_1)^r+t_1^r-t_2^r]
+\frac{p(t)}{\Gamma(r+1)}(t_2-t_1)^r
+\frac{p(t)(t_2-t_1)^{r-1}}{ \Gamma(r-1)} \\
&\leq  \frac{p(t)}{\Gamma(r+1)}(t_2-t_1)^r
+\frac{p(t)}{ \Gamma(r+1)}(t_1^r-t_2^r)
+ \frac{ T^rp(t) (t_2-t_1)^{r-1}}{ \Gamma(r-1)}.
\end{aligned}
\end{equation}
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
$C(J,E)$.
\smallskip

\noindent\textbf{Step 3:} The graph of $N$ is closed.
Let $(y_{n},h_{n}) \in \operatorname{graph}(N)$, $n\geq 1$, with 
$\|y_{n}-y\|$, $\|h_{n}-h\|\to 0$,as 
$n\to \infty$. We must show that $(y,h) \in \operatorname{graph}(N)$.

$(y_{n},h_{n}) \in \operatorname{graph}(N) $ means that 
$h_{n}\in N(y_{n})$, which means that there exists
$v_{n} \in S_{F,y_{n}}$, such that for each $t\in J$,
\begin{align*}
h_{n}(t)
&=\frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}v_{n}(s)\,ds\\
&\quad -\frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}v_{n}(s)\,ds.
\end{align*}
Consider the continuous linear operator
$\Theta : L^{1}(J,E) \to C(J,E)$,
\begin{align*}
\Theta(v)(t)\mapsto h_{n}(t)
&=\frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}v_{n}(s)\,ds\\
&\quad - \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}v_{n}(s)\,ds.
\end{align*}
Clearly,
$\| h_{n}(t) -h(t)\| \to 0$ as as $n \to \infty$.
From Lemma \ref{lem0} 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{lem0} implies
\[
h(t)
=\frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}v(s)\,ds\\
- \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}\int_{0}^{T}(T-s)^{r-3}v(s)\,ds,
\]
for some $v \in S_{F,y}$.
\smallskip

\noindent\textbf{Step 4.}
Suppose $ M\subset\overline{U}$, $M \subset \operatorname{conv}(\{0\}\cup N(M))$,
and $\overline{M}=\overline{C}$ for some countable set $C\subset M$. 
Using an estimation of type \eqref{e10}, we see that $N(M)$ is equicontinuous.  
Then from $M \subset \operatorname{conv}(\{0\}\cup N(M))$, we deduce that 
$M$ is equicontinuous, too. 
To apply the Arz\'ela-Ascoli theorem, it remains to show that $M(t)$ 
is relatively compact in $E$ for each $t\in J$. Since
$C\subset M \subset \operatorname{conv}(\{0\}\cup N(M))$
and $C$  is countable, 
we can find a countable set $H=\{ h_{n}: n\geq 1\} \subset N(M)$ with
$C \subset \operatorname{conv}(\{0\}\cup H)$. 
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_{0}^{t}(t-s)^{r-1}v_{n}(s)\,ds\\
&\quad - \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}v_{n}(s)\,ds.
\end{align*}
From $ M\subset\overline{C} \subset \overline{\rm conv}
(\{0\}\cup H))$, and according to Theorem \ref{thm1}, we have
$$ 
\alpha(M(t)) \leq (\alpha (\overline{C}(t))\leq \alpha (H(t))=
\alpha(\{ h_{n}((t)\,:n\geq 1 \} ). 
$$
Using\eqref{e9}, we obtain
\begin{align*}
\alpha ( M(t)) 
&\leq  2 [ \frac{1}{\Gamma(r)} \int_{0}^{t}\alpha (\{(t-s)^{r-1}v_{n}(s) \})\,ds\\
&\quad  -\frac{t^{\alpha-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}\alpha (\{(T-s)^{r-3} v_{n}(s)\})\,ds].
\end{align*}
Now, since $v_{n}(s) \in M(s)$, we have
\begin{align*}
\alpha (M(t)) 
&\leq  2 [ \frac{1}{\Gamma(r)} \int_{0}^{t}\alpha (\{(t-s)^{r-1}v_{n}(s):
 n\geq 1 \})\,ds\\
&\quad - \frac{t^{\alpha-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}\alpha (\{(T-s)^{r-3} v_{n}(s): n\geq 1\})\,ds].
\end{align*}
Also, since $v_{n}(s) \in M(s)$, we have
\begin{gather*}
\alpha (\{(t-s)^{r-1}v_{n}(s);\, n\geq 1 \}) = (t-s)^{r-1}\alpha (M (s)),\\
\alpha (\{(T-s)^{r-1}v_{n}(s);\, n\geq 1 \}) = (T-s)^{r-1}\alpha (M(s)).
\end{gather*}
It follows that
\begin{align*}
\alpha (M(t)) 
&\leq  2 [ \frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}\alpha (M(s))\,ds\\
&\quad -\frac{t^{\alpha-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}\alpha (M(s))\,ds] \\
&\leq  2 [ \frac{1}{\Gamma(r)} \int_{0}^{t} (t-s)^{r-1}\psi (s,\alpha(M(s)))\,ds\\
&\quad - \frac{t^{\alpha-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T} (T-s)^{r-3}\psi (s,\alpha(M(s)))\,ds].
 \end{align*}
Also, the function $\phi$ given by $\phi(t)=\alpha(M(t)) $ belongs to
$C(J,[0,2R])$. Consequently by (H3), $ \phi \equiv 0$; that is, 
$\alpha(M(t))=0 $ for all $t\in J$. 
Now, by the Arz\'ela-Ascoli theorem, $M$ is relatively compact in $C(J,E)$. 
\smallskip


\noindent\textbf{Step 5.} 
Let $h \in N(y)$ with $y\in \overline{U}$. Since $|y(s)|\leq R $
and by (H2), we have $N(\overline{U}) \subseteq \overline{U}$, 
because if it were not true,
then there exists a function $y \in \overline{U}$, but 
$\|N(y)\|_\mathcal{P}> R $ and
\begin{align*}
h(t)&= \frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}v(s)\,ds\\
&\quad - \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}v(s)\,ds,
\end{align*}
for some $v \in S_{F,y}$. On the other hand, 
\begin{align*}
R &\leq  \|N(y)\|_\mathcal{P} \\ 
& \leq 
\frac{1}{\Gamma(r)} \int_{0}^{t}(t-s)^{r-1}|v(s)|\,ds\\
&\quad - \frac{t^{r-1}}{(r-1)(r-2)\Gamma(r-2)}
\int_{0}^{T}(T-s)^{r-3}|v(s)|\,ds  \\ 
& \leq  \frac{T}{\Gamma(r+1)} \int_{0}^{t} p(s)\,ds\\
&\quad - \frac{T^{2}}{(r-1)(r-2)\Gamma(r-1)}
\int_{0}^{T} p(s)\,ds \\ 
&\leq \big[\frac{T}{\Gamma(r+1)}+ \frac{T^{2}}{(r-1)(r-2)\Gamma(r-1)}\big]
\int_{0}^{T} p(s)\,ds.
\end{align*}
Dividing both sides by $R$ and taking the lower limits as $R \to \infty$, 
we conclude that 
\[
\big[\frac{T}{\Gamma(r+1)}+\frac{T^{2}}{(r-1)(r-2)\Gamma(r-1)}\big] \delta \geq 1
\]
which contradicts \eqref{e11}. Hence $N(\overline{U}) \subseteq \overline{U}$. 

As a consequence of Steps 1-5 and Theorem \ref{thm2}, we conclude that
$N$ has a fixed point $y \in C(J,B)$ which is a solution of  
problem \eqref{e1}-\eqref{e2}. 
\end{proof}

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\end{document}