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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 80, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/80\hfil Impulsive fractional
differential inclusions]
{Existence of solutions to differential inclusions with fractional
order and impulses}

\author[M. Benchohra, S. Hamani, J. J. Nieto, B. A. Slimani\hfil EJDE-2010/80\hfilneg]
{Mouffak Benchohra, Samira Hamani,\\
 Juan Jose Nieto, Boualem Attou Slimani}  % in alphabetical order

\address{Mouffak Benchohra  \newline
Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel-Abb\`es,\\
B.P. 89, 22000, Sidi Bel-Abb\`es, Alg\'erie}
\email{benchohra@univ-sba.dz}

\address{Samira Hamani  \newline
Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel-Abb\`es,\\
B.P. 89, 22000, Sidi Bel-Abb\`es, Alg\'erie}
\email{hamani\_samira@yahoo.fr}

\address{Juan Jose Nieto \newline
Departamento de Analisis Matematico,
Facultad de Matematicas\\
Universidad de Santiago de Compostela,
Santiago de Compostela, Spain}
\email{juanjose.nieto.roig@usc.es}

\address{Boualem Attou Slimani \newline
Facult\'e des Sciences de l'Ing\'enieur,
Universit\'e de Tlemcen,
B.P. 119, 13000, Tlemcen, Alg\'erie}
\email{ba\_slimani@yahoo.fr}

\thanks{Submitted March 11, 2010. Published June 15, 2010.}
\subjclass[2000]{26A33, 34A37}
\keywords{Initial value problem; impulses; differential inclusion;
\hfill\break\indent
 Caputo fractional derivative; fractional integral; existence;
uniqueness; fixed point}

\begin{abstract}
 We establish sufficient conditions for the existence of
 solutions for a class of initial value problem  for impulsive
 fractional differential inclusions involving the Caputo
 fractional derivative. We consider the cases when the
 multivalued nonlinear term takes convex values as well as
 nonconvex values. The topological structure of the set of
 solutions is also considered.
\end{abstract}

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

\section{Introduction}

This article studies the existence and uniqueness of solutions
for the initial value problems (IVP for short), for fractional
order differential inclusions,
\begin{gather}\label{e1}
^{c}D^{\alpha}y(t)\in F(t,y(t)), \quad   t\in J=[0,T],
\; t\neq t_k, \; k=1,\dots ,m, \; 1<\alpha\leq 2,
\\ \label{e2}
\Delta y|_{t=t_k}= I_k(y(t_k^{-})), \quad  k=1,\dots,m,
\\ \label{e3}
\Delta y'|_{t=t_k}= \overline I_k(y(t_k^{-})), \quad
k=1,\dots,m,
\\ \label{e4}
y(0)= y_0, \quad y'(0)=y_1,
\end{gather}
where $^{c}D^{\alpha}$ is the Caputo fractional derivative,
$F: J\times\mathbb{R} \to\mathcal{P}(\mathbb{R})$
is a multivalued map, $(\mathcal{P}(\mathbb{R})$ is
the family of all nonempty subsets of
$\mathbb{R})$, $I_k$ and $\overline I_k :\mathbb{R}\to\mathbb{R}$,
$k=1,\dots,m$, and $ y_0, y_1\in\mathbb{R}$,
$0=t_0<t_1<\dots <t_{m}<t_{m+1}=T$,
$\Delta y|_{t=t_k}=y(t_k^{+})-y(t_k^{-})$,
$\Delta y'|_{t=t_k}=y'(t_k^{+})-y'(t_k^{-})$,
$y(t_k^{+})=\lim_{h\to 0^+}y(t_k+h)$ and
$y(t_k^{-})=\lim_{h\to 0^-}y(t_k+h)$ represent the right and
left limits of $y(t)$ at $t=t_k$, $k=1,\dots,m$.

Differential equations of fractional order have recently proved to
be valuable tools in  modeling  many phenomena in various fields of
science and engineering. Indeed, we can find numerous applications
in viscoelasticity, electrochemistry, control, porous media,
electromagnetic, etc. (see \cite{Ao, DiFr, Hil, Mai, SaAgTe}). There
has been a significant development in fractional differential and
inclusions in recent years; see the monographs of Kilbas \emph{et
al} \cite{KST}, Podlubny \cite{Pod}, Samko \emph{et al}
\cite{SaKiMa} and the papers of Agarwal \emph{et al} \cite{ABH},
Belarbi \emph{et al} \cite{BBHN, BBO}, Benchohra \emph{et al}
\cite{BeHa, BeHaNt, BeHeNtOu1, BeHeNtOu2}, Chang and Nieto
\cite{ChNi}, Diethelm \emph{et al } \cite{DiFr,DiFo}, Furati and
Tatar \cite{FuTa}, Henderson and Ouahab \cite{HeOu}, Kilbas and
Marzan \cite{KiMa}, Mainardi \cite{Mai}, Ouahab \cite{Oua}, and
Zhang \cite{Zha} and the references therein.


Impulsive integer order differential equations have become
important in recent years as mathematical models of phenomena in
both the physical and social sciences. There has a significant
development in impulsive theory especially in the area of impulsive
differential equations with fixed moments; see for instance the
monographs by Benchohra \emph{et al} \cite{BeHeNt1}, Lakshmikantham
\emph{et al} \cite{LBS}, and Samoilenko and Perestyuk \cite{SaPe} and
the papers \cite{AhNi, YaZhNi, ZeWaNi}. To the best knowledge of the
authors, no papers exist in the literature devoted to differential
inclusions with fractional order and impulses. In \cite{BaSi, BeSl}
some classes of of fractional differential
equations with impulses have been considered. 
The aim of this paper is to continue this study.
Thus the results of the present paper initiate this subject.

 This paper is organized as follows. In Section 2 we introduce some
preliminary  results needed in the following sections. In Section 3 we
present an existence result for the problem
\eqref{e1}-\eqref{e4}, when the right hand side is convex valued
using the nonlinear alternative of Leray-Schauder type. In Section 4
two results are given for nonconvex valued right hand side. The
first one is based upon a fixed point theorem for contraction
multivalued maps due to Covitz and Nadler, and the second on the
nonlinear alternative of Leray-Schauder type \cite{GrDu} for
single-valued  maps, combined with a selection theorem due to
Bressan-Colombo \cite{BrCo} for lower semicontinuous
multivalued maps with decomposable values. The topological structure
of the solutions set is considered in Section 5. An example is
presented in the last section. These results extend to the
multivalued case the paper by Benchohra and Slimani \cite{BeSl} and
those considered in  the above cited literature in the absence of
impulsive effect. The present results constitute a contribution to
this emerging field of research.

\section{Preliminaries}

 In this section, we introduce  notation, definitions, and
preliminary facts that will be used in  the
remainder of this paper. Let $C(J,\mathbb{R})$ be the Banach space of
continuous functions from $J$ to $\mathbb{R}$ with the norm
 $$
\|y\|_{\infty}=\sup\{|y(t)|: 0\le t \le T\},
$$
and  let
$L^{1}(J,\mathbb{R})$
 denote the Banach space of
functions $y:J\to \mathbb{R}$ that  are Lebesgue
integrable with the norm
$$
\|y\|_{L^1}=\int_0^{T}|y(t)|dt.
$$
The space $AC^{1}(J,\mathbb{R})$ consists of functions
$y:J \to \mathbb{R}$, which are absolutely continuous, whose first
derivative, $y'$ is absolutely continuous.
 Let $(X,\|\cdot\|)$ be a Banach space. let
 $P_{cl}(X)=\{Y\in \mathcal{P}(X): Y\text{ closed}\}$,
$P_{b}(X)=\{Y\in \mathcal{P}(X): Y\text{ bounded}\}$,
$P_{cp}(X)=\{Y\in \mathcal{P}(X): Y\text{ compact}\}$ and
$P_{cp,c}(X)=\{Y\in \mathcal{P}(X): Y\text{ compact and convex}\}$.
A multivalued map $G:X\to P(X)$ is convex
(closed) valued if $G(x)$ is convex (closed) for all $x\in X$.
$G$ is 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.,
$\sup_{x\in B}\{\sup\{|y|: y\in G(x) \}\}<\infty)$.  $G$ is called
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 if for each open set $N$ of $X$ containing
$G(x_0)$, there exists an open neighborhood $N_0$ of $x_0$ such
that $G(N_0)\subseteq N$.  $G$ is said to be completely continuous
if $G(\mathcal{B})$ is relatively compact for every $\mathcal{B}\in
P_{b}(X)$. 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 $y_{*}\in G(x_{*})$). $G$ has a fixed
point if there is $x\in X$ such that
$x\in G(x)$. The fixed point set of the multivalued operator $G$
will be denoted by $\mathop{\rm Fix} G$. 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.
For more details on multivalued maps see the books of Aubin and
Cellina \cite{AuCe}, Deimling \cite{Dei} and Hu and Papageorgiou
\cite{HuPa}.

\begin{definition} \label{def2.1} \rm
A multivalued map $F: J\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$
is said to be Carath\'eodory if
\begin{itemize}
\item[(i)] $t\mapsto F(t,u)$ is  measurable for each $u\in\mathbb{R}$;
\item[(ii)] $u\mapsto F(t,u)$ is upper semicontinuous for almost
all $t\in J$.
\end{itemize}
For each $y\in PC(J,\mathbb{R})$, define the set of selections of
$F$ by
$$
S_{F,y}=\{v\in L^1(J,\mathbb{R}): v(t)\in F(t,y(t))\text{ a.e. }
 t\in J\}.
$$
\end{definition}

Let $(X,d)$ be a metric space induced from the normed space
$(X, |\cdot |)$.  Consider
$H_{d}:\mathcal{P}(X)\times \mathcal{P}(X)\to\mathbb{R}_{+}
\cup\{\infty\}$
given by
$$
H_{d}(A,B)=\max\big\{\sup_{a\in A}d(a,B),\sup_{b\in
B}d(A,b)\big\},
$$
where $d(A,b)=\inf_{a\in A}d(a,b)$,
$d(a,B)=\inf_{b\in B}d(a,b)$. Then $( P_{b,cl}(X),H_{d})$ is a
metric space and $(P_{cl}(X),H_{d})$ is a generalized metric space
 \cite{Kis}.

\begin{definition} \rm
A multivalued operator $N:X\to  P_{cl}(X)$ is called
\begin{itemize}
\item[(a)] $\gamma$-Lipschitz if and only if there
exists $\gamma>0$ such that
$$
H_d(N(x),N(y))\leq \gamma d(x,y),\quad \text{for each } x, y\in X,
$$
\item[(b)] a contraction if and only if it is
$\gamma$-Lipschitz with $\gamma<1$.
\end{itemize}
\end{definition}

\begin{lemma}[\cite{CoNa}] \label{CN}
Let $(X,d)$ be a complete metric space. If $N: X\to P_{cl}(X)$ is
a contraction, then $\mathop{\rm Fix}N \neq  \emptyset$.
\end{lemma}

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


\begin{definition}[\cite{KST, Pod}] \rm
For a function $h$ given on the interval $[a,b]$, the Caputo
fractional-order derivative of $h$, is defined by
$$
(^{c}D_{a+}^{\alpha}h)(t)=\frac{1}{\Gamma(n-\alpha)}\int_
a^t(t-s)^{n-\alpha-1}h^{(n)}(s)ds,
$$
where $n=[\alpha]+1$.
\end{definition}

Sufficient conditions for  the fractional differential and
fractional integrals to exist are given in \cite{KST}.

\section{The Convex Case}

In this section, we are concerned with the existence of solutions
for the problem \eqref{e1}-\eqref{e4} when the right hand side has
convex values. Initially, we assume that $F$ is a compact and
convex valued  multivalued map.  Consider the
Banach space
\begin{align*}
PC(J,\mathbb{R})
=\big\{&y: J\to \mathbb{R}: y\in C((t_k,t_{k+1}],\mathbb{R}),\;
 k=0,\dots,m+1  \text{ and there exist}\\
 &y(t^{-}_k), y(t^{+}_k), \; k=1,\dots,m  \text{ with }
y(t^{-}_k)=y(t_k)\}.
\end{align*}
with the norm
$$
\|y\|_{PC}=\sup_{t\in J}|y(t)|.
$$
 Set $J':=[0,T]\backslash\{t_1,\dots,t_{m}\}$.

\begin{definition} \rm
A function $y\in PC(J,\mathbb{R})\bigcap\cup_{k=0}^{m}
AC^{1}((t_k,t_{k+1}),\mathbb{R})$
with its $\alpha$-derivative exists on $J'$ is said to be a solution
of \eqref{e1}--\eqref{e4} if there exists a function $v\in
L^{1}([0,T], \mathbb{R})$ such that $v(t)\in F(t,y(t))$ a.e.
$ t\in J$ satisfies the differential equation
$^{c}D^{\alpha}y(t)=v(t)$ on
$J'$,  and conditions
\begin{gather*}
\Delta y|_{t=t_k}= I_k(y(t_k^{-})),\quad k=1,\dots,m,\\
\Delta y'|_{t=t_k}= \overline I_k(y(t_k^{-})), \quad
  k=1,\dots,m,\\
 y(0)= y_0, \quad y'0)=y_1
\end{gather*}
 are satisfied.
\end{definition}

Let $h:[a,b]\to\mathbb{R}$ be a continuous function.
For the existence of solutions for the problem
\eqref{e1}--\eqref{e4}, we need the following auxiliary lemmas.

\begin{lemma}[\cite{Zha}] \label{l1}
Let $\alpha > 0 $, then the differential equation
$$
^{c}D^{\alpha}h(t)=0
$$
has solutions
$h(t)=c_0+c_1(t-a)+c_{2}(t-a)^{2}+\dots+c_{n-1}(t-a)^{n-1}$,
$c_{i}\in \mathbb{R}$,
$i=0,1,2,\dots,n-1$, $n=[\alpha]+1$.
\end{lemma}

\begin{lemma}[\cite{Zha}] \label{l2}
Let $\alpha > 0 $, then
$$
{I^{\alpha}}^{c}D^{\alpha}h(t)=h(t)+
c_0+c_1(t-a)+c_{2}(t-a)^{2}+\dots+c_{n-1}(t-a)^{n-1}
$$
for some $c_{i}\in \mathbb{R}$,
$i=0,1,2,\dots,n-1$, $n=[\alpha]+1$.
\end{lemma}

As a consequence of Lemma \ref{l1} and Lemma \ref{l2} we have the
following result which is useful in what follows.

\begin{lemma}\label{l3}
Let  $1< \alpha\leq 2$ and let $ \rho\in PC(J,\mathbb{R})$.
A function $y$ is a solution of the fractional
integral equation
\begin{equation}\label{e5}
y(t)=\begin{cases}
y_0+y_1t+\frac{1}{\Gamma(\alpha)}\int_0^{t}(t-s)^{\alpha-1}\rho(s)ds
&\text{if }t\in [0,t_1], \\[4pt]
y_0+y_1t+\frac{1}{\Gamma(\alpha)}
 \sum_{i=1}^{k}\int_{t_{i-1}}^{t_{i}}(t_{i}-s)^{\alpha-1}\rho(s)ds\\
+\frac{1}{\Gamma(\alpha-1)}\sum_{i=1}^{k}(t-t_{i})
 \int_{t_{i-1}}^{t_{i}}(t_{i}-s)^{\alpha-2}\rho(s)ds\\
+ \frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}\rho(s)ds\\
+\sum_{i=1}^{k}I_{i}(y(t_{i}^{-}))+\sum_{i=1}^{k}(t-t_{i})\overline
I_{i}(y(t_{i}^{-})),
& \text{if } t\in (t_k,t_{k+1}]\\
& k=1,\dots,m
\end{cases}
\end{equation}
if and only if  $y$ is a solution of the fractional
initial-value problem
\begin{gather}\label{e6}
^{c}D^{\alpha}y(t)= \rho(t), \quad\text{for each }  t\in J', \\
\label{e7}
\Delta y|_{t=t_k}= I_k(y(t_k^{-})), \quad  k=1,\dots,m,\\
\label{e8}
\Delta y'|_{t=t_k}= \overline I_k(y(t_k^{-})), \quad k=1,\dots,m, \\
\label{e9}
y(0)= y_0, \quad y'(0)=y_1.
\end{gather}
\end{lemma}

\begin{proof}
 Assume $y$ satisfies \eqref{e6}-\eqref{e9}. If $t\in
[0,t_1]$ then
$^{c}D^{\alpha}y(t)= \rho(t)$.
Lemma \ref{l2} implies
$$
y(t)=c_0+c_1t+\frac{1}{\Gamma(\alpha)}
\int_0^{t}(t-s)^{\alpha-1}\rho(s)ds.
$$
Hence $c_0=y_0$ and
$c_1=y_1$. Thus
$$
y(t)=y_0+y_1t+\frac{1}{\Gamma(\alpha)}
\int_0^{t}(t-s)^{\alpha-1}\rho(s)ds.
$$
If $t\in (t_1,t_{2}]$ then Lemma \ref{l2} implies
\begin{equation}\label{eqq1}
 y(t)=c_0+c_1(t-t_1)+\frac{1}{\Gamma(\alpha)}
\int_{t_1}^{t}(t-s)^{\alpha-1}\rho(s)ds.
\end{equation}
\begin{align*}
\Delta y|_{t=t_1}
&= y(t_1^+)-y(t_1^-)\\
&= c_0-\Big(y_0+y_1t_1+\frac{1}{\Gamma(\alpha)}
\int_0^{t_1}(t_1-s)^{\alpha-1}\rho(s)ds\Big)\\
&= I_1(y(t_1^{-})).
\end{align*}
Hence
\begin{equation}\label{eqq2}
c_0=y_0+y_1t_1+\frac{1}{\Gamma(\alpha)}
\int_0^{t_1}(t_1-s)^{\alpha-1}\rho(s)ds+I_1(y(t_1^{-})).
\end{equation}
\begin{align*}
\Delta y'|_{t=t_1}
&= y'(t_1^+)-y'(t_1^-)\\
&= c_1-\Big(y_1+\frac{1}{\Gamma(\alpha-1)}
\int_0^{t_1}(t_1-s)^{\alpha-2}\rho(s)ds\Big)\\
&= \overline I_1(y(t_1^{-})),
\end{align*}
 and
\begin{equation}\label{eqq3}
c_1=y_1+\frac{1}{\Gamma(\alpha-1)}
\int_0^{t_1}(t_1-s)^{\alpha-2}\rho(s)ds+\overline
I_1(y(t_1^{-})).
\end{equation}
 Then by \eqref{eqq1}-\eqref{eqq3}, we have
\begin{align*}
y(t)&= y_0+y_1t+\frac{1}{\Gamma(\alpha)}
\int_0^{t_1}(t_1-s)^{\alpha-1}\rho(s)ds\\
&\quad +\frac{(t-t_1)}{\Gamma(\alpha-1)}
\int_0^{t_1}(t_1-s)^{\alpha-2}\rho(s)ds\\
&\quad +I_1(y(t_1^{-}))+(t-t_1)\overline I_1(y(t_1^{-}))
 +\frac{1}{\Gamma(\alpha)}
\int_{t_1}^{t}(t-s)^{\alpha-1}\rho(s)ds.
\end{align*}
If $t\in (t_k,t_{k+1}]$ then again from Lemma \ref{l2} we obtain
\eqref{e5}.

Conversely, assume that $y$ satisfies the impulsive fractional
integral equation \eqref{e5}. If $t\in [0,t_1]$ then
$y(0)=y_0$,
$y'(0)=y_1$ and using the fact that $^{c}D^{\alpha}$ is the left
inverse of $I^{\alpha}$ we get
$$
^{c}D^{\alpha}y(t)= \rho(t), \quad \text{for each }  t\in [0,t_1].
$$
If $t\in [t_k,t_{k+1})$, $k=1,\dots,m$ and using the fact that
$^{c}D^{\alpha}C=0$, where $C$
is a constant, we get
$$
^{c}D^{\alpha}y(t)= \rho(t), \quad \text{for each }  t\in [t_k,t_{k+1}).
$$
Also, we can easily show that
\begin{gather*}
\Delta y|_{t=t_k}= I_k(y(t_k^{-})), \quad  k=1,\dots,m,\\
\Delta y'|_{t=t_k}= \overline I_k(y(t_k^{-})), \quad k=1,\dots,m.
\end{gather*}
Our first result is based on the nonlinear
alternative of Leray-Schauder type for multivalued maps \cite{GrDu}.
We assume the following hypotheses:
\begin{itemize}
\item[(H1)]  $F: J\times \mathbb{R}\to \mathcal{P}_{cp,c}(\mathbb{R})$
is a Carath\'eodory  multi-valued map;

\item[(H2)] there exist  $p\in C(J,\mathbb{R}^+) $ and
$\psi:[0,\infty)\to (0,\infty)$ continuous and  nondecreasing such that
 $$
  \|F(t,u)\|_\mathcal{P}=\sup\{|v|: v\in F(t,u)\}
\leq p(t)\psi(|u|)
$$
for $t\in J$ and $u\in \mathbb{R}$;

\item[(H3)] There exist
$\psi^*, \overline \psi^{*}:[0,\infty)\to (0,\infty)$ continuous
and nondecreasing such that
\begin{gather*}
 |I_k(u)|\leq \psi^*(|u|) \quad\text{for }  u\in \mathbb{R},\\
 |\overline I_k(u)|\leq \overline \psi^*(|u|) \quad \text{for }
  u\in \mathbb{R};
\end{gather*}

\item[(H4)] There exists a number $\overline M>0$ such that
\begin{equation}\label{eq2}
\frac{M}{|y_0|+T|y_1|+a\psi(M) +m\psi^*( M)+mT\overline
\psi^*( M)}>1,
 \end{equation}
 where $p^{0}=\sup\{p(t):  t\in J\}$ and
 $$
 a=\frac{mT^{\alpha}p^{0}}{\Gamma(\alpha+1)}
+\frac{mT^{\alpha}p^{0}}{\Gamma(\alpha)} +
\frac{T^{\alpha}p^{0}}{\Gamma(\alpha+1)}.
$$

\item[(H5)] there exists $l\in L^{1}(J,\mathbb{R}^+)$  such that
\begin{gather*}
  H_d(F(t,u),F(t,\overline u))\leq l(t)|u-\overline u|\quad
\text{for a.e. }  t\in J .\;   u,\overline u\in \mathbb{R},\\
d(0,F(t,0))\leq l(t), \quad \text{a.e. }  t\in J.
\end{gather*}
\end{itemize}

\begin{theorem}\label{t1}
Under Assumptions {\rm (H1)-(H5)}, the
initial-value problem  \eqref{e1}-\eqref{e4} has at least one
 solution on $J$.
\end{theorem}

{\bf Proof.} We transform \eqref{e1}--\eqref{e3} into a fixed point
problem. Consider the multivalued  operator
\begin{align*}
N(y)=\Big\{&h\in PC(J,\mathbb{R}):
h(t)= y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v(s)ds\\
&+\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v(s)ds\\
&+\frac{1}{\Gamma(\alpha)} \int_{t_k}^{t}(t-s)^{\alpha-1}v(s)ds\\
& +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})), \; v \in S_{F,y}.
  \Big\}
\end{align*}
 Clearly, from Lemma \ref{l3},  fixed points of
$N$ are solutions to \eqref{e1}--\eqref{e4}. We shall show that $N$
satisfies the assumptions of the nonlinear alternative of
Leray-Schauder type \cite{GrDu}. The proof will be given in several
steps.

{\bf Step 1:}  $N(y)$ is convex for each
$y\in PC(J,\mathbb{R})$.  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)&=  y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v_{i}(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v_{i}(s)ds\\ &\quad +
\frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}v_{i}(s)ds\\
&\quad +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})) , \ \ 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)\\
&= y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}[dv_1(s)+(1-d)v_{2}(s)]ds
\\&\quad + \frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}[dv_1(s)+(1-d)v_{2}(s)]ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}[dv_1(s)+(1-d)v_{2}(s)]ds\\
&\quad +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})).
\end{align*}
Since $S_{F,y}$ is convex (because $F$ has convex values), we
have
$$
 dh_1+(1-d)h_{2}\in N(y).
$$

{\bf Step 2}: $N$ maps bounded sets into bounded sets
in $PC(J,\mathbb{R})$.
Let $ B_{\eta^*}=\{y\in PC(J,\mathbb{R}): \|y\|_{\infty}\leq
\eta^* \}$ be bounded set in $PC(J,\mathbb{R})$ and $y\in B_{\eta^*}$.
Then for each $h\in N(y)$ and $t\in J$, we have by (H2)-(H3),
\begin{align*}
|h(t)|&\leq
|y_0|+|y_1|T+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}\int_{t_{k-1}}^{t_k}
(t_k-s)^{\alpha-1}|v(s)|ds\\
&\quad + \frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}|v(s)|ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}|v(s)|ds\\
&\quad +\sum_{0<t_k<t}|I_k(y(t_k^{-}))|+\sum_{0<t_k<t}(t-t_k)|\overline
I_k(y(t_k^{-}))|\\
&\leq
|y_0|+|y_1|T+\frac{mT^{\alpha}p^{0}}{\Gamma(\alpha+1)}\psi(\eta^*)
+\frac{T^{\alpha}p^{0}}{\Gamma(\alpha)}\psi(\eta^*)\\
&\quad +
\frac{T^{\alpha}p^{0}}{\Gamma(\alpha+1)}\psi(\eta^*)+m
\psi^*(\eta^*)+ m \overline{\psi}^*(\eta^*).
\end{align*}
Thus
\begin{align*}
\|h\|_{\infty}
&\leq |y_0|+|y_1|T+\frac{mT^{\alpha}p^{0}}{\Gamma(\alpha+1)}
 \psi(\eta^*)
+\frac{T^{\alpha}p^{0}}{\Gamma(\alpha)}\psi(\eta^*)\\
&\quad + \frac{T^{\alpha}p^{0}}{\Gamma(\alpha+1)}\psi(\eta^*)+m
\psi^*(\eta^*)+ m \overline{\psi}^*(\eta^*)
:=\ell.
\end{align*}

  {\bf Step 3}: $N$ maps bounded sets into
equicontinuous sets of $PC(J,\mathbb{R})$.
Let $\tau_1, \tau_{2}\in J$, $\tau_1<\tau_{2}$,  $B_{\eta^*}$ be a
bounded set of $PC(J,\mathbb{R})$ as in Step  2, let
$y\in B_{\eta^*}$ and
$h\in N(y)$, then
\begin{align*}
|h(\tau_{2})-h(\tau_1)|
&\leq |y_1|(\tau_{2}-\tau_1)
 +\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<
\tau_{2}-\tau_1}\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}|v(s)|ds\\
&\quad +\frac{1}{\Gamma(\alpha)}\int_{t_k}^{\tau_1}|(\tau_2-s)^{
\alpha-1}-(\tau_1-s)^{\alpha-1}||v(s)|ds\\ 
&\quad +
\frac{1}{\Gamma(\alpha)}\int_{\tau_1}^{\tau_{2}}|(\tau_2-s)^{
\alpha-1}||v(s)|ds\\ &\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<
\tau_{2}-\tau_1}(\tau_{2}-t_k)\int_{t_{k-1}}^{t_k}
 (t_k-s)^{\alpha-2}|v(s)|ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<\tau_1}(\tau_{2}-\tau_1)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}|v(s)|ds\\
&\quad +\sum_{0<t_k<\tau_{2}-\tau_1}|I_k(y(t_k^{-}))|
+\sum_{0<t_k<\tau_{2}-\tau_1}(\tau_{2}-t_k)|\overline
I_k(y(t_k^{-}))|\\
 &\quad + (\tau_{2}-\tau_1)\sum_{0<t_k<\tau_1}|
 \overline I_k(y(t_k^{-}))|.
\end{align*}
Using (H2), (H3) we can easily prove that the right-hand side of the
above inequality tends to zero independently of $y$ as $\tau_1\to
\tau_{2}$. As a consequence of Steps 1 to 3 together with the
Arzel\'a-Ascoli theorem, we can conclude that $N:PC(J,\mathbb{R})\to
\mathcal{P} (PC(J,\mathbb{R}))$ is completely continuous.

{\bf Step 4:} $N$ has a closed graph.  Let $y_{n}\to y_{*}$,
$h_{n}\in N(y_{n})$  and  $h_{n} \to h_{*}$.
We need to show that $h_{*}\in N(y_{*})$.
 $h_{n}\in N(y_{n})$
means that there exists $v_{n}\in S_{F, y_{n}}$ such that, for
each $t\in J$,
\begin{align*}
h_{n}(t)&=  y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v_{n}(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v_{n}(s)ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}v_{n}(s)ds\\
&\quad +\sum_{0<t_k<t}I_k(y_n(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y_n(t_k^{-})).
\end{align*}
We must show that there exists $v_{*}\in S_{F, y_{*}}$ such
that, for each $t\in J$,
\begin{equation}\label{iie}
\begin{aligned}
 h_{*}(t)&= y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v_{*}(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v_{*}(s)ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}v_{*}(s)ds\\
&\quad +\sum_{0<t_k<t}I_k(y_*(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y_*(t_k^{-})).
\end{aligned}
\end{equation}
Since $F(t,\cdot)$ is upper semicontinuous, then for every
$\varepsilon>0$, there exist $n_0(\epsilon)\geq 0$ such
 that for every $n\geq n_0$, we have
$$
v_n(t)\in F(t,y_n(t))\subset F(t,y_{*}(t))+\varepsilon  B(0,1),
\quad \text{a.e. } t\in J.
$$
 Since $F(\cdot,\cdot)$ has compact values, then there exists a
subsequence $v_{n_m}(\cdot)$ such that
\begin{gather*}
v_{n_m}(\cdot)\to v_*(\cdot) \quad \text{as } m\to\infty,\\
v_*(t)\in F(t,y_*(t)),\quad  \text{a.e. } t\in J.
\end{gather*}
Using the fact that the functions $I_k$ and $\overline I_k$,
$k=1,\dots,m$ are continuous, it can be easily shown that
$h_*$ and $v_*$ satisfy \eqref{iie}.

 {\bf Step 5:} A priori bounds on solutions.
 Let $y\in PC(J,\mathbb{R})$ be such that $y\in\lambda N(y)$ for
$\lambda\in (0,1)$. Then
 there exists $v\in S_{F,y}$ such that, for
 each  $t\in J$,
\begin{align*}
|y(t)|&\leq
|y_0|+|y_1|T+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}\int_{t_{k-1}}^{t_k}
(t_k-s)^{\alpha-1}p(s)\psi(|y(s)|)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2} p(s)\psi(|y(s)|)ds\\&\quad +
\frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}p(s)\psi(|y(s)|)ds\\
&\quad +
\sum_{0<t_k<t}\psi^*(|y(s)|)+\sum_{0<t_k<t}\overline{\psi}^*(|y(s)|)\\
&\leq
|y_0|+|y_1|T+\psi(\|y\|_{\infty})\frac{mT^{\alpha}p^{0}}{\Gamma(\alpha+1)}
+ \psi(\|y\|_{\infty})\frac{mT^{\alpha}p^{0}}{\Gamma(\alpha)}\\
&\quad + \frac{mT^{\alpha}p^{0}}{\Gamma(\alpha+1)}
\psi(\|y\|_{\infty})+m\psi^*(\|y\|_{\infty})
 +m\overline{\psi}^*(\|y\|_{\infty}).
\end{align*}
Thus
$$
\frac{\|y\|_{\infty}}{|y_0|+|y_1|T+a\psi(\|y\|_{\infty})
+m\psi^*(\|y\|_{\infty})+m\overline{\psi}^*(\|y\|_{\infty})}\leq 1.
$$
Then by  (H4), there exists $M$ such that
 $\|y\|_{\infty} \neq M$.
Let
$$
U=\{y\in PC(J,\mathbb{R}): \|y\|_{\infty}<M\}.
$$
The operator $N:\overline U\to \mathcal{P}(PC(J,\mathbb{R}))$
is upper semicontinuous and
completely continuous.  From the choice of $U$, there is no
$y\in \partial U$ such that $y\in \lambda N(y)$ for some
$\lambda\in(0,1)$. As a consequence of the nonlinear alternative
of Leray-Schauder type \cite{GrDu}, we deduce that $N$
has a fixed point $y$ in $\overline U$ which is a solution
of the problem \eqref{e1}--\eqref{e4}.  This completes the proof.
\end{proof}

\section{The nonconvex case}

 This section is devoted to the existence of solutions for the problem
\eqref{e1}-\eqref{e4} with a nonconvex valued right hand side. Our
first result is based on the fixed point theorem for contraction
multivalued map given by Covitz and Nadler \cite{CoNa}, and the
second one on a selection theorem due to Bressan and Colombo
\cite{BrCo} for lower semicontinuous operators with decomposable
values combined with the nonlinear Leray-Schauder alternative. Some
existence results for nonconvex valued differential inclusions can
be found in \cite{AuCe,HuPa}.

For the next theorem, we use the following assumptions:
\begin{itemize}

\item[(H6)] $F: J\times \mathbb{R} \to  P_{cp}(\mathbb{R})$
has the property that $F(\cdot,u): J\to P_{cp}(\mathbb{R})$
is measurable, convex valued and integrable bounded for each
$u\in \mathbb{R}$;

\item[(H7)] There exist  constants $l^*, \overline l^*>0$ such that
\begin{gather*}
|I_k(u)-I_k(\overline u)|\leq l^*|u-\overline u|, \quad
\text{for each } u,  \overline u \in \mathbb{R},  \text{ and }
k=1,\dots,m,\\
 |\overline I_k(u)-\overline
I_k(\overline u)|\leq \overline l^*|u-\overline u|, \quad
\text{for each }  u,  \overline u \in \mathbb{R},  \text{ and }
k=1,\dots,m.
\end{gather*}
\end{itemize}

\begin{theorem}\label{t2}
Assume {\rm (H5)--(H7)}.
 If
\begin{equation}\label{eq1}
\big[\frac{mlT^{\alpha}}{\Gamma(\alpha+1)}+\frac{mlT^{\alpha}}{\Gamma(\alpha)}
+\frac{lT^{\alpha}}{\Gamma(\alpha+1)}+m(l^{*}+T\overline l^{*})
\big]<1,
\end{equation}
where  $l=\sup\{l(t): \ t\in J\}$, then
\eqref{e1}-\eqref{e4} has one solution on $J$.
\end{theorem}

\begin{proof}
 For each $y\in PC(J,\mathbb{R})$, the set $S_{F,y}$ is nonempty
since by (H6), $F$ has  a measurable selection
(see \cite[Theorem III.6]{CaVa}).
We shall show that $N$ satisfies the assumptions of
Lemma \ref{CN}. The proof will be given in two steps.

  {\bf Step 1}:  $N(y)\in  P_{cl}(PC(J,\mathbb{R}))$ for each $y\in
PC(J,\mathbb{R})$.   Indeed, let $(y_{n})_{n\geq 0}\in N(y)$
such that $y_{n}\to \tilde y$ in $ PC(J,\mathbb{R})$. Then,
$\tilde y\in PC(J,\mathbb{R})$ and  there exists
$v_n\in S_{F,y}$ such that,
for each $t\in J$,
 \begin{align*}
 y_{n}(t)&=  y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v_{n}(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v_{n}(s)ds\\ &\quad +
\frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}v_{n}(s)ds\\
&\quad +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})).
\end{align*}
 Using the fact that $F$ has compact values and from (H5),
we may pass to a subsequence if necessary to get that $v_n$
converges weakly to $v$ in $L_{w}^1(J,\mathbb{R})$
(the space endowed with
the weak topology). A standard argument shows that $v_n$ converges
strongly to $v$ and hence $v\in S_{F,y}$. Then, for each $t\in J$,
\begin{align*}
 y_{n}(t) \to  \tilde y(t)
&=  y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v(s)ds\\
&\quad + \frac{1}{\Gamma(\alpha)} \int_{t_k}^{t}(t-s)^{\alpha-1}v(s)ds\\
&\quad +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})).
\end{align*}
 So, $\tilde y\in N(y)$.

 {\bf Step 2}:  There exists $\gamma < 1$ such that
$$
H_d(N(y),N(\overline y))\leq \gamma\|y-\overline y\|_{\infty}
\text{ for each }  y, \overline y\in PC(J,\mathbb{R}).
$$
Let $y, \overline y \in PC(J,\mathbb{R})$ and $h_1\in N(y)$.
Then there
exists $v_1(t)\in F(t,y(t))$ such that for each $t\in J$,
 \begin{align*}
 h_1(t)&= y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v_1(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_{i}}(t_k-s)^{\alpha-2}v_1(s)ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}v_1(s)ds\\
&\quad +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})).
\end{align*}
 From (H5) it follows that
$$
H_d(F(t,y(t)), F(t,\overline y(t)))\leq l(t)|y(t)-\overline y(t)|.
$$
Hence, there exists $w\in F(t,\overline y(t))$ such that
$$
|v_1(t)-w|\leq l(t)|y(t)-\overline y(t)|, \quad t\in J.
$$
Consider $U: J\to \mathcal{P}(\mathbb{R})$ given by
$$
U(t)=\{w\in \mathbb{R}: |v_1(t)-w|\leq
l(t)|y(t)-\overline y(t)|\}.
$$
Since the multivalued operator
$V(t)=U(t)\cap F(t,\overline y(t))$ is measurable
(see  \cite[Proposition III.4]{CaVa}), there exists a function
$v_{2}(t)$ which is
a measurable selection for $V$. So, $v_{2}(t)\in F(t,\overline
y(t))$, and for each $t\in J$,
$$
|v_1(t)-v_{2}(t)|\leq l(t)|y(t)-\overline y(t)|.
$$
 Let us define  for each $t\in J$,
 \begin{align*}
 h_{2}(t)&= y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v_{2}(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v_{2}(s)ds\\ &\quad +
\frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}v_{2}(s)ds\\
&\quad +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})).
\end{align*}
Then for $t\in J$,
\begin{align*}
|h_1(t)-h_{2}(t)|
&\leq \frac{1} {\Gamma(\alpha)}\sum_{0<t_k<t}\int_{t_{k-1}}^{t_k}
(t_k-s)^{\alpha-1}|v_1(s))-v_{2}(s)|ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}|v_1(s))-v_{2}(s)|ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}|v_1(s)-v_{2}(s)|ds\\
&\quad +\sum_{0<t_k<t}|I_k(y(t_k^{-}))-I_k(\overline{y}(t_k^{-}))|+
\sum_{0<t_k<t}|\overline{I}_k(y(t_k^{-}))
 -\overline{I}_k(\overline{y}(t_k^{-}))|\\
&\leq \frac{l}{\Gamma(\alpha)}\sum_{k=1}^{m}\int_{t_{k-1}}^{t_k}
 (t_k-s)^{\alpha-1}|y(s)-\overline{y}(s)|ds\\
&\quad +\frac{l}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}|y(s)-\overline{y}(s)|ds\\
&\quad +\frac{l}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}|y(s)-\overline{y}(s)|ds\\
&\quad +\sum_{k=1}^{m}l^*|y(t_k^{-})-\overline{y}(t_k^{-})|+
\sum_{k=1}^{m}\overline{l}^*|y(t_k^{-})-\overline{y}(t_k^{-})|\\
&\leq \frac{mlT^{\alpha}}{\Gamma(\alpha+1)} \|y-\overline{y}\|_{\infty}+
 \frac{mlT^{\alpha}}{\Gamma(\alpha)}\|y-\overline{y}\|_{\infty}\\
&\quad +\frac{T^{\alpha}l}{\Gamma(\alpha+1)}\|y-\overline{y}\|_{\infty}
+ ml^* \|y-\overline{y}\|_{\infty}+ mT \overline{l}^*
\|y-\overline{y}\|_{\infty}.
\end{align*}
Thus
$$ 
\|h_1-h_{2}\|_{\infty} \leq
\big[\frac{mlT^{\alpha}}{\Gamma(\alpha+1)}
+\frac{mlT^{\alpha}}{\Gamma(\alpha)}
+\frac{lT^{\alpha}}{\Gamma(\alpha+1)}+m(l^{*}+T\overline
l^{*})\big] \|y-\overline y\|_{\infty}.
$$
By an analogous relation, obtained by interchanging the roles
of $y$ and $\overline y$, it follows that
$$
H_d(N(y),N(\overline y)) \leq
\big[\frac{mlT^{\alpha}}{\Gamma(\alpha+1)}+\frac{mlT^{\alpha}}{\Gamma(\alpha)}
+\frac{lT^{\alpha}}{\Gamma(\alpha+1)}+m(l^{*}+T\overline
l^{*})\big]\|y-\overline y\|_{\infty}.
$$
So by \eqref{eq1}, $N$ is a contraction and thus,
by Lemma \ref{CN}, $N$ has a fixed
point $y$ which is solution to \eqref{e1}--\eqref{e4}. The proof
is complete.
\end{proof}

Now we present a result for  problem \eqref{e1}-\eqref{e4} in
the spirit of the nonlinear alternative of Leray-Schauder type
\cite{GrDu} for single-valued
 maps, combined with a selection theorem
due to Bressan-Colombo for lower semicontinuous multivalued maps with
decomposable values. Details on multivalued maps with decomposable
values and their properties can be found in the recent book by
Fryszkowski \cite{Fry}.

Let $A$ be a subset of $[0,T]\times \mathbb{R}$. $A$ is $\mathcal{L}\otimes\mathcal{B}$ measurable if $A$ belongs to the
$\sigma$-algebra generated by all sets of the form $\mathcal{J}\times
D$ where $\mathcal{J}$ is Lebesgue measurable in $[0,T]$ and $D$ is
Borel measurable in $\mathbb{R}$. A subset\ $A$\ of\ $L^1([0,T],\mathbb{R})$\ is
decomposable if for all $u,v\in A$\ and $\mathcal{J}\subset [0,T]$\
measurable, $u\chi_\mathcal{J} +v\chi_{[0,T]-\mathcal{J}}\in A$, where
$\chi$ stands for the characteristic function.

Let $G:X\to\mathcal{P}(X)$  a multivalued operator with  nonempty
closed values. $G$ is lower semi-continuous (l.s.c.) if the set
$\{x\in X: G(x)\cap B\neq \emptyset\}$ is open for any open set
$B$ in  $X$.

\begin{definition} \rm
Let  $Y$ be a separable metric space and let
$N: Y\to\mathcal{P}(L^1([0,T],\mathbb{R}))$ be a multivalued operator.
We say  $N$ has property  (BC) if
\begin{itemize}
\item[(1)] $N$ is lower  semi-continuous (l.s.c.);
\item[(2)] $N$  has nonempty closed and  decomposable values.
\end{itemize}
\end{definition}

Let $F: [0,T]\times \mathbb{R}\to\mathcal{P}(\mathbb{R})$ be a
multivalued map with nonempty compact values. Assign to
$F$  the multivalued operator
$\mathcal{F}: PC([0,T],\mathbb{R})\to\mathcal{P}(L^1([0,T],\mathbb{R}))$
 by letting
$$
 \mathcal{F}(y)=\{w\in L^1([0,T],\mathbb{R}): w(t)\in F(t, y(t))
\text{ for a.e. } t\in[0,T]\}.
$$
The operator $\mathcal{F}$ is called the Niemytzki operator
associated with $F$.

\begin{definition} \rm
Let $F: [0,T]\times \mathbb{R}\to\mathcal{P}(\mathbb{R})$ be
a multivalued function
with nonempty compact values. We say $F$ is of lower
semi-continuous type (l.s.c. type) if its associated Niemytzki
operator $\mathcal{F}$ is lower semi-continuous and has nonempty
closed and decomposable values.
\end{definition}

Next we state a selection theorem  due to Bressan and
Colombo \cite{BrCo}.

\begin{theorem}[\cite{BrCo}]\label{BC}
Let $Y$ be a separable metric space
and let the operator $N: Y\to \mathcal{P}(L^1([0,T],\mathbb{R}))$
be a multivalued satisfying property (BC). Then $N$ has a continuous
selection, i.e. there exists a continuous function (single-valued)
$\tilde g:Y\to L^1([0,1],\mathbb{R})$ such that
$\tilde g(y)\in N(y)$ for every $y\in Y$.
\end{theorem}

Let us introduce the following hypotheses:
\begin{itemize}
\item[(H8)]  $F:[0,T]\times \mathbb{R} \to\mathcal{P}(\mathbb{R})$ is a
nonempty compact valued multivalued map such that:
\begin{itemize}
\item[(a)] $(t,y)\mapsto F(t,y)$\ is $\mathcal{L}\otimes\mathcal{B}$
measurable;
\item[(b)] $y\mapsto F(t,y)$ is lower semi-continuous for a.e.
$t\in[0,T]$;
\end{itemize}
\item[(H9)] for each $q>0$, there exists a function
$h_q\in L^1([0,T],\mathbb{R}^+)$ such that
$\|F(t,y)\|_\mathcal{P}\leq h_q(t)$
for a.e. $t\in[0,T]$ and for $y\in \mathbb{R}$
with $|y|\leq q$.
\end{itemize}
The following lemma is crucial in the proof of our main theorem.

\begin{lemma}\label{FG} \cite{FrGr}.
Let $F: [0,T]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$
 be a multivalued map with
nonempty, compact values. Assume that  {\rm (H8), (H9)} hold.
Then $F$ is of l.s.c.
\end{lemma}

\begin{theorem}\label{t3}
 Suppose that  hypotheses {\rm (H2)-(H4), (H8), (H9)}
are satisfied. Then the problem \eqref{e1}--\eqref{e4} has at
least one solution.
\end{theorem}

\begin{proof}
 (H8) and (H9) imply by Lemma \ref{FG} that $F$ is of
lower semi-continuous type. Then from Theorem \ref{BC} there exists
a continuous function $f: PC([0,T],\mathbb{R})\to L^1([0,T],\mathbb{R})$
such that $f(y)\in\mathcal{F}(y) $ for all $y\in PC([0,T],\mathbb{R})$.
Consider the  problem
\begin{gather}\label{eqq10}
^{c}D^{\alpha}y(t)\in f(y)(t), \quad \text{for a.e. }  t\in J=[0,T], \;
t\neq t_k, \; k=1,\dots,m, \; 1<\alpha\leq 2, \\
\label{eqq11}
\Delta y|_{t=t_k}= I_k(y(t_k^{-})), \quad  k=1,\dots,m, \\
\label{eqq12}
\Delta y'|_{t=t_k}= \overline I_k(y(t_k^{-})), \quad k=1,\dots,m,\\
\label{eqq13}
y(0)= y_0, \quad y'(0)=y_1.
\end{gather}
 Clearly, if $y$ is a solution of
 \eqref{eqq10}--\eqref{eqq13}, then  $y$ is a solution of
 \eqref{e1}-\eqref{e4}. Problem \eqref{eqq10}-\eqref{eqq13}
can be reformulated as a fixed point problem for the operator
$N_1: PC([0,T,\mathbb{R})\to PC([0,T],\mathbb{R})$ defined by
\begin{align*}
 N_1(y)(t)&= y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}f(y)(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}f(y)(s)ds\\ &\quad +
\frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}f(y)(s)ds\\
&\quad +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})).
\end{align*}
Using (H2)-(H4) we can easily show (using similar argument as
in Theorem \ref{t1})
that the operator $N_1$ satisfies all conditions in the
Leray-Schauder alternative.
\end{proof}

\section{Topological structure of the solution set}

In this section, we present a result on the topological structure
of the set of solutions to \eqref{e1}--\eqref{e4}.

\begin{theorem}\label{t4}
Assume that {\rm (H1), (H5)} and the following hypotheses hold:
\begin{itemize}
\item[(H10)] there exists  $p_1\in C(J,\mathbb{R}^+) $  such that
 $ \|F(t,u)\|_\mathcal{P}\leq p_1(t)$ for  $t\in J$ and
$u\in \mathbb{R}$;

\item[(H11)] There exist $ d_1,d_{2} >0$
  such that
\begin{gather*}
  |I_k(u)|\leq d_1 \quad \text{for }   u\in \mathbb{R},\\
  |\overline{I}_k(u)|\leq d_{2} \quad \text{for }  u\in \mathbb{R}.
\end{gather*}
\end{itemize}
Then the solution set of  \eqref{e1}-\eqref{e4} in not empty and
is compact in $PC(J,\mathbb{R})$.
\end{theorem}

\begin{proof}
 Let
$$
S=\{y\in PC(J, \mathbb{R}): y  \text{ is solution of }
\eqref{e1}-\eqref{e4}\}.
$$
 From Theorem \ref{t1}, $S\neq \emptyset$. Now, we prove that
$S$ is compact. Let $(y_n)_{n\in\mathbb{N}}\in S$, then there exists
$v_n\in S_{F,y_n}$ and $t\in J$ such that
\begin{align*}
 y_{n}(t)&= y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v_{n}(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v_{n}(s)ds\\ &\quad +
\frac{1}{\Gamma(\alpha)}
\int_{t_k}^{t}(t-s)^{\alpha-1}v_{n}(s)ds\\
&\quad +\sum_{0<t_k<t}I_k(y_n(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y_n(t_k^{-})).
\end{align*}
 From (H1), (H10) and (H11) we can prove that there exists an $M_1>0$
such that
$\|y_n\|_{\infty}\leq M_1$ for every $n\geq1$.
As in Step 3 in Theorem \ref{t1}, we can easily show that the
set $\{y_n:n\geq 1\}$ is
equicontinuous in $PC(J,\mathbb{R})$, hence by Arz\'ela-Ascoli
Theorem we can conclude that, there exists a subsequence
(denoted again by
$\{y_n\}$) of $\{y_n\}$ such that $y_n$ converges to $y$ in
$PC(J, \mathbb{R})$. We shall show that there exist
$v(.)\in F(.,y(.))$ and $t\in J$ such that
\begin{align*}
y(t)&= y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v(s)ds\\ &\quad +
\frac{1}{\Gamma(\alpha)} \int_{t_k}^{t}(t-s)^{\alpha-1}v(s)ds
 +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})).
\end{align*}
Since $F(t,.)$ is upper semicontinuous, for
every $\varepsilon>0$, there exists $n_0(\epsilon)\geq 0$ such that
for every $n\geq n_0$, we have
$$
v_n(t)\in F(t,y_n(t))\subset F(t,y(t))+\varepsilon B(0,1), \quad
\text{a.e. }  t\in J.
$$
Since $F(.,.)$ has compact values, there exists subsequence
$v_{n_m}(.)$ such that
\begin{gather*}
v_{n_m}(.)\to v(.)\quad \text{as } m\to\infty,\\
v(t)\in F(t,y(t)),\quad \text{a.e. } t\in J.
\end{gather*}
It is clear that
$$
|v_{n_m}(t)|\leq p_1(t),\quad \text{a.e. } t\in J.
$$
By Lebesgue's dominated convergence theorem, we conclude that
$v\in L^{1}(J, \mathbb{R})$ which implies that $v\in S_{F,y}$.
Thus, for $t\in J$, we have
\begin{align*}
 y(t)&= y_0+y_1t+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-1}v(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha-1)}\sum_{0<t_k<t}(t-t_k)
\int_{t_{k-1}}^{t_k}(t_k-s)^{\alpha-2}v(s)ds\\
 &\quad + \frac{1}{\Gamma(\alpha)} \int_{t_k}^{t}(t-s)^{\alpha-1}v(s)ds
 +\sum_{0<t_k<t}I_k(y(t_k^{-}))+\sum_{0<t_k<t}(t-t_k)\overline
I_k(y(t_k^{-})).
\end{align*}
Then $S\in\mathcal{P}_{cp}(PC(J,\mathbb{R}))$.
\end{proof}

\section{An Example}

As an application of the main results, we consider the fractional
differential inclusion
\begin{gather}\label{ex1}
^c D^{\alpha}y(t)\in F(t,y), \quad  \text{a.e. }  t\in J=[0,1],\;
t\neq \frac{1}{2}, \; 1<\alpha\leq 2, \\
\label{ex2}
\Delta y|_{t=\frac{1}{2}}= \frac{1}{3+|y(\frac{1}{2}^{-})|},\\
\label{ex3}
\Delta y|_{t=\frac{1}{2}}= \frac{1}{5+|y(\frac{1}{2}^{-})|}, \\
\label{ex4}
y(0)=0, \quad y'(0)=0.
\end{gather}
We have $T=1$, $m=1$, $t_1=1/2$ and $y_0=y_1=0$. Set
 $$
F(t,y)=\{v\in \mathbb{R}: f_1(t,y)\leq v\leq f_2(t,y)\},
$$
where $f_1,  f_2:J\times \mathbb{R}\to \mathbb{R}$ are given functions,
$$
 I_1(y(t_1))= \frac{1}{3+|y(\frac{1}{2}^{-})|},\quad
 \overline I_1(y(t_1))= \frac{1}{5+|y(\frac{1}{2}^{-})|}.
$$
Then  \eqref{ex1}-\eqref{ex4} takes the form \eqref{e1}-\eqref{e4}.
We assume that for each
 $t\in J$, the function
$f_1(t,\cdot)$ is lower semi-continuous
(i.e, the set $\{y\in  \mathbb{R}: f_1(t,y)>\mu\}$
is open for each $\mu\in \mathbb{R}$), and assume that
 for each $t\in J,  f_2(t,\cdot)$ is upper semi-continuous
(i.e the set $\{y\in \mathbb{R}: f_2(t,y)<\mu\}$ is open for each
$\mu\in \mathbb{R}$). Assume
that there are  $p\in C(J,\mathbb{R}^{+})$ and $\psi:[0,\infty)\to
(0,\infty)$ continuous and nondecreasing such that
$$
\max(|f_1(t,y)|, |f_2(t,y)|)\leq p(t)\psi(|y|),\quad t\in J,
 \text{ and } y\in \mathbb{R}.
$$
Assume there exists a constant $M>0$ such that
$$
\frac{M}{\big(\frac{2p^{0}}{\Gamma(\alpha+1)}
+\frac{p^{0}}{\Gamma(\alpha)}\big)\psi(M)+\frac{8}{15}}>1.
$$
 It is clear that $F$ is compact and convex valued, and it is upper
semi-continuous (see \cite{Dei}). Since all the conditions of
Theorem \ref{t1}  are satisfied, the problem
\eqref{ex1}-\eqref{ex4} has at least one solution $y$ on $J$.


\subsection*{Acknowledgements}
This paper has been completed while the first
author was visiting the Abdus Salam International Centre for
Theoretical Physics in Trieste as regular associate. He likes
to express his gratitude for the provided financial support. The
research of J. J. Nieto was partially supported by Ministerio de
Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta
de Galicia and FEDER, project PGIDIT05PXIC20702PN.

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\end{document}