\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 18, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/18\hfil Existence of convex and non convex solutions]
{Existence of convex and non convex local solutions
 for fractional differential inclusions}

\author[R. W. Ibrahim\hfil EJDE-2009/18\hfilneg]
{Rabha W. Ibrahim}

\address{Rabha W. Ibrahim \newline
School of Mathematical Sciences \\
Faculty of Sciences and Technology\\
UKM, Malaysia}
\email{rabhaibrahim@yahoo.com}

\thanks{Submitted February 20, 2008. Published January 20, 2009.}
\subjclass[2000]{34G10, 26A33, 34A12, 42B05}
\keywords{Fractional calculus; set-valued function; extremal
solution; \hfill\break\indent
maximal solution; minimal solution; differential
inclusions}

\begin{abstract}
 In this paper, we establish the existence theorems for a class of
 fractional differential inclusion of order $n-1 < \alpha \leq n $.
 The study holds in two cases, when the set-valued function has
 convex and non-convex values.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\section{ Introduction}

We study the existence of solutions for a class of nonlinear
differential inclusions of fractional order. The operators are
taken in the Riemann-Liouville  sense and the initial conditions
are specified according to Caputo's suggestion, thus allowing
for interpretation in a physically meaningful way. There are
numerous books focused in this direction, that is concerning the
linear and nonlinear problems involving different types of
fractional derivatives as well as integral (see \cite{r21,r24,r25,r27,r28}).
 El-Sayed and Ibrahim  \cite{r13,r14,r18} gave the concept of the
definite integral of fractional order for set-valued function. As
applications of this type of problem, it arises in the study of
control systems, game theory and programing languages (see
\cite{r2,r3,r20}).

 The Riemann-Liouville fractional operators are defined
as follows; see \cite{r24,r27}:

\begin{definition}  \label{def1.1} \rm
The fractional integral
operator $I^{\alpha}$ of order $\alpha >0 $ of a
 continuous function $f(t)$ is given by
\[
I^{\alpha}_{0}f(t):=I^{\alpha}f(t)=
\frac{1}{\Gamma(\alpha)}\int^{t}_{0} (t-\tau)^{\alpha-1}f(\tau)d
\tau.
\]
We can write
$I^{\alpha}_{0}f(t)=f(t)*\psi_{\alpha}(t)$ where
$\psi_{\alpha}(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}$ for $t>0$
and $\psi_{\alpha}(t)=0$ for $t \leq 0$ and $\psi_{\alpha}(t)
\to \delta(t)$ (the delta function) as $\alpha \to
0$ (see \cite{r24,r27}).
\end{definition}

 \begin{definition}  \label{def1.2} \rm
 The fractional derivatives
$D^{\alpha}$ of order $n-1 <\alpha \leq n$ of the function $f(t)$
is given by
\[
D^{\alpha}_{a}f(t)=\frac{1}{\Gamma(n-\alpha)}\frac{d^{n}}{dt^{n}}\int^{t}_{0}
(t-\tau)^{n-\alpha-1}f(\tau)d \tau.
\]
\end{definition}

This paper concerns the  fractional differential
inclusion
\begin{equation} \label{e1}
\begin{gathered}
D^{\alpha}(u-T_{n-1}[u])(t) \in F(t,u(t),\rho(t)); \quad
n-1 < \alpha \leq n , \; t\in J:=[0,T],\\
u^{(k)}(0)=u^{(k)}_{0} \in \mathbb{R}, \quad  k=0,1,\dots,n-1
\end{gathered}
\end{equation}
where $T_{n-1}[u]$ is the Taylor polynomial of order $(n-1)$ for
$u$, centered at $0$, $\rho: J \to \mathbb{R}$ is a
continuous function and  $F: J \times \mathbb{R}\times \mathbb{R}
\to \mathcal{P}(\mathbb{R})$ is a set-valued function with
nonempty values in $\mathbb{R}$, where $\mathcal{P}(\mathbb{R})$
is the family of all nonempty subsets of $\mathbb{R}$.

 This paper is organized as follows: In Section 2,
we will recall briefly some basic definitions and preliminary
facts from set-valued analysis which will be used later. In
Section 3, we shall establish the existence and uniqueness
solution for the single-valued problem
\begin{equation} \label{e2}
\begin{gathered}
D^{\alpha}(u-T_{n-1}[u])(t) = f(t,u(t),\rho(t)); \quad
n-1 < \alpha \leq n , \; t\in J=[0,T], \\
u^{(k)}(0)=u^{(k)}_{0}, \; k=0,1,\dots,n-1
\end{gathered}
\end{equation}
by using the Schauder fixed point theorem (see \cite{r6}) and the Banach
fixed point theorem (see \cite{r29}) respectively. In Section 4, we shall
study the existence of solution for the set-valued problem \eqref{e1}
when $F$ has a convex as well as non-convex values via the
single-valued problem as well as  fixed point theorems of the
set-valued function. In the first case (convex) a fixed point
theorem due to Martelli \cite{r23} is used. A fixed point theorem
for contraction set-valued functions due to Covitz and Nadler
\cite{r9} is applied in the second one (non-convex).

\section{Preliminaries}

 In this section, we introduce notation,
definitions, and preliminary facts from set-valued analysis which
are used throughout this paper. For further background and details
pertaining to this section we refer the reader to
\cite{r4,r7,r16,r17,r19,r26,r30}.

$\mathcal{B}:=C[J,\mathbb{R}]$ is the Banach space of
all continuous functions from $J$ into $\mathbb{R}$ with the norm
\[
\|u\|=\sup \{|u(t)|: t \in J\}
\]
for each $u \in \mathcal{B}$.
$\mathcal{L}:=L^{1}[J,\mathbb{R}]$ denotes the Banach space of
measurable functions $u: J \to \mathbb{R}$ which are
Lebesgue integrable normed by
\[
\|u\|_{L^{1}}= \int^{T}_{0}|u(t)|dt,
\]
for  $u \in \mathcal{L}.
$ Let $(X,|\cdot |)$ be a normed space,
$\mathcal{P}_{cl}(X)=\{Y \in
\mathcal{P}(X): Y \text{ is closed}\}$,
$\mathcal{P}_{b}(X)=\{Y \in \mathcal{P}(X): Y \text{ is bounded}\}$,
$\mathcal{P}_{cp}(X)=\{Y \in \mathcal{P}(X): Y \text{ is compact}\}$,
$\mathcal{P}_{c}(X)=\{Y \in \mathcal{P}(X): Y \text{ is convex}\}$,
$\mathcal{P}_{cl,c}(X)=\{Y \in \mathcal{P}(X): Y \text{ is closed
 and  convex}\}$,
$\mathcal{P}_{cp,c}(X)=\{Y \in \mathcal{P}(X): Y \text{ is compact
 and  convex}\}$.
  A set-valued function $F: X \to \mathcal{P}(X)$ is called
 \textbf{convex (closed)} valued if
$F(x)$ is convex (closed) for all $x \in X$.
$F$ is called \textbf{bounded} valued on bounded set $B$
if $F(B)= \bigcup _{x \in B}F(x)$ is bounded in $X$ for all
$B \in \mathcal{P}_{b}(X)$ i.e.
$\sup_{x \in B }\{\sup\{|u|: u \in F(x)\}\} < \infty$.
$F$ is called \textbf{upper semi-continuous (u.s.c)} on $X$
if for each $x_{0} \in X$ the set $F(x_{0})$ is
nonempty closed subset of $X$ and if for each open set $N$ of $X$
containing $F(x_{0})$, there exists an open neighborhood $N_{0}$
of $x_{0}$ such that $F(N_{0})\subseteq N$. In other wards $F$ is
u.s.c if the set $F^{-1}(A)=\{x\in X: Fx \subset A \}$ is open
in $X$ for every open set $A$ in $X$. $F$ is called \textbf{lower
semi-continuous (l.s.c)} on $X$ if $A$ is any open subset of $X$
then $F^{-1}(A)=\{x\in X: Fx \cap A \neq \emptyset\}$ is open
in $X$. $F$ is called \textbf{continuous} if it is lower as
well as upper semi-continuous on $X$. $F$ is called
\textbf{compact} if for every $M$ bounded subset of $X$, $F(M)$
is relatively compact. Finally $F$ is called \textbf{completely
continuous} if it is upper semi-continuous and compact on $X$.
The following definitions are used in the sequel.

 \begin{definition}  \label{def2.1}\rm
 A mapping $p:J \times \mathbb{R}\to \mathbb{R}$ is said to be
Carath\'eodory if
\begin{itemize}
\item[(i)] $t \to p(t,u)$ is measurable for each $u \in
\mathbb{R}$,

\item[(ii)] $u \to p(t,u)$ is continuous a.e. for $t
\in J$.

\end{itemize}
A Carath\'eodory function $p(t,u)$ is called
$L^{1}(J,\mathbb{R})$-Carath\'eodory if
\begin{itemize}
\item[(iii)] for each number $r>0$ there exists a function
$h_{r} \in L^{1}(J,\mathbb{R})$ such that $|p(t,u)|\leq h_{r}(t)$
a.e. $t \in J$ for all $u \in \mathbb{R}$ with $|u| \leq r$.
\end{itemize}
 A Carath\'eodory function $p(t,u)$ is called
$L^{1}_{X}(J,\mathbb{R})$-Carath\'eodory if
\begin{itemize}
\item[(iv)] there exists a function $h \in L^{1}(J,\mathbb{R})$
such that $|p(t,u)|\leq h(t)$ a.e $t \in J$ for all $u \in
\mathbb{R}$ where $h$ is called the bounded function of $p$.
\end{itemize}
\end{definition}

 \begin{definition} \label{def2.2}\rm
  A set-valued function $F:J \to \mathcal{P}(\mathbb{R})$ is said to be
\textbf{measurable} if for any $x \in X$, the function $t
\mapsto d(x,F(t))=inf \{|x-u|: u \in F(t) \} $ is measurable.
\end{definition}

 \begin{definition}  \label{def2.3}\rm
  A set-valued function $F:J \times \mathbb{R} \to \mathcal{P}(\mathbb{R})$
is called Carath\'eodory if
\begin{itemize}
\item[(i)] $t \mapsto F(t,x)$ is measurable for each $x \in
\mathbb{R}$, and

\item[(ii)] $x \mapsto F(t,x)$ is u.s.c. for almost $t \in J$.
\end{itemize}
\end{definition}

\begin{definition} \label{def2.4}\rm
  A set-valued function $F:J \times \mathbb{R} \to \mathcal{P}(\mathbb{R})$
is called $L^{1}$-Carath\'eodory  if
\begin{itemize}
\item[(i)] $F$ is Carath\'eodory and

\item[(ii)] For each $r>0$, there exists $h_{r}  \in
L^{1}(J,\mathbb{R})$ such that $\|F(t,u)\|=\sup \{|f|: f \in
F(t,u)\} \leq h_{r}(t)$ for all $|u| \leq r$ and  for a.e. $t \in
J$.
\end{itemize}
\end{definition}

 \begin{definition}[\cite{r11}] \label{def2.5}  \rm
 A set-valued function $F:J \times \mathbb{R} \to \mathcal{P}(\mathbb{R})$
 is called $L^{1}_{X}$-Carath\'eodory
if there exists a function $h \in L^{1}(J,\mathbb{R})$ such that
\[
\|F(t,u)\|=\sup \{|f|: f \in F(t,u)\} \leq h(t), \quad\text{a.e. }
 t \in J
 \]
for all $x \in \mathbb{R}$, and the function $h$ is called a growth
function of $F$ on $J \times \mathbb{R}$.

 Let $A,B \in \mathcal{P}_{cl}(X)$, let $a \in A$ and let
\[
D(a,B)=\inf \{\|a-b\|: b \in B\} \quad\text{and}\quad
\rho(A,B)= \sup \{D(a,B):a \in A\}\,.
\]
The function $H: \mathcal{P}_{cl}(X)\times
\mathcal{P}_{cl,b}(X)\to \mathbb{R}^{+}$ defined by
\[
H(A,B)=\max \{ \rho(A,B),  \rho(B,A)\}
\]
is a metric and is called Hausdorff metric on $X$.
Moreover $(\mathcal{P}_{cl,b}(X),H)$ is
a metric space and $(\mathcal{P}_{cl}(X),H)$ is a complete metric
space (see \cite{r22}).
It is clear that
\[
H(0,C)=sup \{\|c\|:c \in C; C \in \mathcal{P}_{b}(X)\}.
\]
\end{definition}

\begin{definition}  \label{def2.6}\rm
  A set-valued function $F:\mathbb{R} \to \mathcal{P}_{cl}(\mathbb{R})$
is called

\noindent (i) $\gamma$-Lipschitz if  there exists
$\gamma > 0$ such that
\[
H(F(x),F(y)) \leq \gamma \|x-y\|,\quad\text{for  each } x,y \in X
\]
 the constant $\gamma$ is called a Lipschitz constant.

\noindent (ii) a contraction if  it is
$\gamma$-Lipschitz with $\gamma < 1$.
\end{definition}

\begin{definition} \label{def2.7} \rm
 A set-valued function
$F:J \times \mathbb{R} \to \mathcal{P}_{cl}(\mathbb{R})$ is
called

\noindent (i) $\gamma(t)$-Lipschitz if  there exists
$\gamma \in L^{1}(J,\mathbb{R}^{+})$ such that
\[
H(F(t,x),F(t,y)) \leq \gamma(t) \|x-y\|, \quad\text{for  each }
 x,y \in X.
\]

\noindent (ii) a contraction if  it is
$\gamma(t)$-Lipschitz with $\|\gamma\| < 1$.
\end{definition}

 The following remark and lemmas are used in the sequel.

\begin{remark}[\cite{r5}] \label{rmk2.1} \rm
 Let $M \subset X$. If $F:M \to \mathcal{P}(X)$ is closed and
$F(M)$ is relatively compact then $F$ is u.s.c. on $M$.
And if $F:X \to \mathcal{P}(X)$ is closed and compact operator
then $F$ is u.s.c.on $X$.
\end{remark}

\begin{lemma}[\cite{r23}] \label{lem2.1}
 Let $T:X \to \mathcal{P}_{c,cp}(X)$
be a completely continuous set-valued function. If
\[
\varepsilon = \{u \in X: \lambda u \in Tu, \text{ for some } \lambda >
1\}
\]
is a bounded set, then $T$ has a fixed point.
\end{lemma}

\begin{lemma}[\cite{r9}] \label{lem2.2}
 Let $(X,d)$ be a complete metric space. If $G: X \to \mathcal{P}_{cl}(X)$
is a contraction, then $G$ has a fixed point.
\end{lemma}

\section{Single-valued problem}

In this section we prove that the fractional differential equation
\eqref{e2} has a solution
$u(t)$ on $J$. By using some classical results from the fractional
calculus, the following result held (see \cite{r28}).

\begin{lemma} \label{lem3.1.} If the function $f$ is
continuous, then the initial value problem \eqref{e2} is equivalent to
the nonlinear Volterra integral equation of the second kind,
\begin{equation} \label{e3}
u(t)=\sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau,u(\tau),\rho(\tau))d \tau,
\end{equation}
where $n-1 <\alpha \leq n$,  $t\in J=[0,T]$.
In other words, every solution of the Volterra
equation \eqref{e3} is also a solution of the initial value
problem \eqref{e2} and vice versa.
\end{lemma}

 Diethelm and Ford \cite{r10} proved the existence of solutions
for \eqref{e3} in the case $0 < \alpha < 1$.
Let us formulate the following assumption:

\begin{itemize}
\item[(H1)] The function $f$ is $L^{1}_{X}$-Carath\'eodory with
bounded function $h \in L^{1}(J \times \mathbb{R},\mathbb{R}^{+})$; i.e.,
 $|f(t,u,\rho)|\leq h(t,\rho)$ a.e $t \in J$ for all $u \in \mathbb{R}$
such that $\|h\|_{L^{1}} < \infty$.
\end{itemize}

\begin{theorem} \label{thm3.1}
Let the assumption {\rm (H1)} hold.
 Then the fractional differential
equation \eqref{e2} has at least one solution  $u(t)$ on $J$.
\end{theorem}

\begin{proof}
 Define an operator $P$ by
\begin{equation} \label{e4}
(Pu)(t):= \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau,u(\tau),\rho(\tau))d \tau
\end{equation}
then by the assumption of the theorem and the properties of
fractional calculus we obtain
\begin{equation*}
\begin{split}
|(Pu)(t)|& \leq  \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}|u^{(k)}(0)|+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}|f(\tau,u(\tau),\rho(\tau))|d \tau\\
& \leq  \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}|u^{(k)}(0)|+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}h(\tau,\rho) d \tau\\
& \leq  \sum^{n-1}_{k=0}
\frac{T^{k}}{k!}|u^{(k)}(0)|+\frac{\|h\|_{L^{1}}T^{\alpha}}{\Gamma(\alpha+1)}.\\
\end{split}
\end{equation*}
Hence
\[
\|Pu\| \leq \sum^{n-1}_{k=0}
\frac{T^{k}}{k!}|u^{(k)}(0)|+\frac{\|h\|_{L^{1}}
T^{\alpha}}{\Gamma(\alpha+1)}.
\]
 Set $r:=\sum^{n-1}_{k=0} \frac{T^{k}}{k!}|u^{(k)}(0)|
+\frac{\|h\|_{L^{1}}T^{\alpha}}{\Gamma(\alpha+1)}$
 that is $P:B_{r}\to B_{r}$. Then $P$ maps $B_{r}$ into itself.
In fact, $P$ maps the
convex closure of $P[B_{r}]$ into itself. Since $f$  is bounded on
$B_{r}$, thus $ P[B_{r}]$ is equicontinuous and  the Schauder
fixed point theorem shows that $P$ has at least one fixed point
$u \in \mathcal{B}=C[J,\mathbb{R}]$ such that $ Pu=u$, which is
corresponding to the solution of \eqref{e2}.
\end{proof}

 For the uniqueness of solutions, we introduce the
following assumption:
\begin{itemize}
\item[(H2)]
 The function $f$ satisfies that
there exists a function $\ell(t) \in L^{1}(J,\mathbb{R}^{+})$
with, $\|\ell\|_{L^{1}}< \infty$, such that for each $u,v \in
C[J,\mathbb{R}]$ we have
\[
|f(t,u,\rho)-f(t,v,\rho)|\leq \ell(t)\|u-v\|.
\]
\end{itemize}

\begin{theorem} \label{thm3.2}
Let {\rm (H2)} hold. If
$ \|\ell\|_{L^{1}}T^{\alpha}/\Gamma(\alpha+1) < 1 $, then
the fractional differential equation \eqref{e2} has a unique solution
$u(t)$ on $J$.
\end{theorem}

\begin{proof} Using the operator $P$ defined in
 \eqref{e2},  we have
\begin{equation*}
\begin{split}
|(Pu)(t)-(Pv)(t)|
& \leq \frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}|f(\tau,u(\tau),\rho(\tau))- f(\tau,v(\tau),\rho(\tau))|d \tau\\
& \leq \frac{\|u-v\|_{\infty}}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}
\ell(\tau)d \tau\\
& \leq
\frac{\|\ell\|_{L^{1}}T^{\alpha}}{\Gamma(\alpha+1)}\|u-v\|_{\infty}.
\end{split}
\end{equation*}
Hence $P$ is a contraction mapping. Then in virtue of the Banach
fixed point theorem, $P$ has a unique fixed point which is
corresponding to the solution of equation \eqref{e2}.
\end{proof}

\section{Set-valued problem}

In this section we study the existence results for the differential
inclusion \eqref{e1} when the right hand side is convex as well
as non-convex valued. The study
will be taken in view of the single-valued problem
(Theorems \ref{thm3.1}, \ref{thm3.2}) as well as fixed point theorems of
set-valued function. The
definite integral for the set-valued function $F$ of order
$\alpha$ defines as follows:
\[
I^{\alpha}F(t,u(t),\rho(t))=\{\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau,u(\tau),\rho(\tau))d \tau:
f(t,u,\rho)\in S_{F}(u)\},
\]
where
\[
S_{F}(u)=\{f \in L^{1}(J,\mathbb{R}): f(t) \in F(t,u(t),\rho(t))
\text{ a.e. } t\in J \}
\]
denotes the set of selections of $F$.
Let us introduce the following assumption
\begin{itemize}
\item[(H3)]
 The set-valued function  $F: J \times
\mathbb{R} \to \mathcal{P}_{cl,c}(\mathbb{R})$ is
$L^{1}_{X}$-Carath\'eodory with a growth function
$h \in L^{1}(J \times \mathbb{R},\mathbb{R}^{+})$; i.e.,
 $\|F(t,u,\rho)\| \leq h(t,\rho)$ a.e $t \in J$ for all
$u \in \mathbb{R}$ such that $\|h\|_{L^{1}}$.
\end{itemize}

\begin{theorem} \label{thm4.1}
Let {\rm (H3)} hold.
If $F$ is lower semi-continuous (l.s.c). Then the differential
inclusion \eqref{e1} has at least one solution $u(t)$ on $J$.
\end{theorem}

\begin{proof}
This proof depends on the (single-valued problem).
Inclusion \eqref{e1} can reduce to the integral inclusion
\begin{equation} \label{e5}
u(t) \in \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}F(\tau,u(\tau),\rho(\tau))d \tau,
\end{equation}
where
$n-1 <\alpha \leq n$,  $t\in J=[0,T]$.
For each $u(t)$ in $\mathbb{R}$, the set $S_{F}(u)$ is nonempty
since by (H3), $F$ has a non-empty measurable selection (see
\cite{r8}). Thus there exists a function $f(t) \in F$ where $f$ is a
$L^{1}_{X}$-Carath\'eodory function with a bounded
function $h \in L^{1}(J \times \mathbb{R},\mathbb{R}^{+})$ such
that $\|f\| \leq \|h\|$ a.e $t \in J$ for all $u \in \mathbb{R}$.
Hence the assumptions of Theorem \ref{thm3.1} are satisfied then the
inclusion \eqref{e5} has a solution and consequently \eqref{e1}.
\end{proof}

  We define the partial ordering $\leq$ in
$W^{n,1}(J,\mathbb{R})$, the Sobolev class of functions
$u: J \to \mathbb{R}$ for which  $u^{(n-1)}$are absolutely
continuous and $u^{(n)} \in L^{1}(J,\mathbb{R})$ as follows: Let
$u,v \in W^{n,1}(J,\mathbb{R})$ then define
\[
u \leq v \Leftrightarrow u(t) \leq v(t), \quad
\text{for  all } t  \in J.
\]
If $a,b \in W^{n,1}(J,\mathbb{R})$ and $a \leq b$ then we
define an order interval $[a,b] \in W^{n,1}(J,\mathbb{R})$ by
\[
[a,b] := \{u \in W^{n,1}(J,\mathbb{R}): a \leq u \leq b \}.
\]

\begin{definition}[\cite{r1}]  \label{def4.1} \rm
A function $\underline{u}$ is called a lower solution of \eqref{e1}
if there exists an $L^{1}(J,\mathbb{R})$ function
 $f_{1}(t)$ in $F(t,\underline{u}(t),\rho(t))$ a.e. $t \in J$.
 such  that
$\underline{u}^{(n)}(t) \leq f_{1}(t), a.e. t \in J$ and
$\underline{u}^{(k)}(0) \leq \underline{u}^{(k)}_{0}$,
$k=0,1,\dots,n-1$. Similarly a function $\overline{u}$ is called an
upper solution of the problem \eqref{e1} if there exists
an $L^{1}(J,\mathbb{R})$ function $f_{2}(t)$ in
$F(t,\overline{u}(t),\rho(t))$, a.e. $t \in J$ such  that
$\overline{u}^{(n)}(t) \geq f_{2}(t)$, a.e. $t \in J$ and
$\overline{u}^{(k)}(0) \geq \overline{u}^{(k)}_{0}$,
$k=0,1,\dots,n-1$.
\end{definition}

\begin{itemize}
 \item[(H4)] The initial value problem \eqref{e1}  has a
lower solution $\underline{u}$ and an upper solution
$\overline{u}$ with $\underline{u} \leq \overline{u}$.
\end{itemize}

\begin{theorem}[Convex case] \label{thm4.2}
 Let {\rm (H3)-(H4)} hold.
Then the differential inclusion \eqref{e1} has at least one solution
$u(t)$ such that
\[
\underline{u}(t) \leq u(t) \leq \overline{u}(t), \quad
\text{for  all } t \, \in J.
\]
\end{theorem}

\begin{proof} Now we shall show that the
assumptions of Lemma \ref{lem2.1} are satisfied in a suitable Banach space.
Consider the  problem
\begin{equation} \label{e6}
\begin{gathered}
D^{\alpha}(u-T_{n-1}[u])(t) \in F(t,Au(t),\rho(t)),\quad
n-1 <\alpha \leq n,\; t\in J:=[0,T],\\
u^{(k)}(0)=u^{(k)}_{0} \in \mathbb{R}, \quad
n=0,1,\dots,n-1
\end{gathered}
\end{equation}
where $A: C(J,\mathbb{R}) \to C(J,\mathbb{R})$ is the
truncation operator defined by
 \[
(Au)(t)= \begin{cases}
 \underline{u}(t) & \text{if }  u(t)< \underline{u}(t);\\
 u(t) & \text{if }  \underline{u}(t) \leq u(t) \leq  \overline{u}(t);\\
 \overline{u}(t) & \text{if }  \overline{u}(t)< u(t).
\end{cases}
\]
 The problem of the existence of a solution to
\eqref{e1} reduce to finding a solution to the integral inclusion
\begin{equation} \label{e7}
u(t) \in \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}F(\tau,Au(\tau),\rho(t))d \tau,
\end{equation}
where $n-1 <\alpha \leq n$,  $t\in J=[0,T]$.
We study \eqref{e7} in the space of all continuous
real functions on  $J$ endow with a supremun norm. Define a
set-valued function operator $N: C(J,\mathbb{R}) \to
\mathcal{P}(C(J,\mathbb{R})) $ by
\begin{equation} \label{e8}
Nu = \{ u \in C(J,\mathbb{R}): u(t)= \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau)d \tau, f \in \overline{S}_{F}(Au)\}
\end{equation}
where
\[
\overline{S}_{F}(Au)=\{f \in S_{F}(Au): f(t)
 \geq \underline{u}(t) \text{ a.e. } t \in J_{1} \text{ and }
 f(t) \leq \overline{u}(t)\text{ a.e. } t \in J_{2}\}
\]
and
\begin{gather*}
J_{1}=\{t \in J: u(t) < \underline{u}(t) \leq \overline{u}(t)\},\\
J_{2}=\{t \in J: \underline{u}(t)  \leq \overline{u}(t) < u(t)\},\\
J_{3}=\{t \in J: \underline{u}(t) \leq u(t) \leq \overline{u}(t)\}.
\end{gather*}
We shall show that the set-valued operator $N$ satisfies all the
conditions of Lemma \ref{lem2.1}. Firstly, since $F$ is measurable (H3),
then it has a nonempty closed selection set $S_{F}(u)$ (see \cite{r8})
consequently $\overline{S}_{F}(u)$. The proof holds in several
steps.

\noindent \textbf{Step 1:} $N(u)$ is convex subset of
 $C(J,\mathbb{R})$. Let $u_{1}, u_{2} \in N(u)$. Then there exist
 $f_{1}, f_{2} \in \overline{S}_{F}(u)$ satisfy
 \[
u_{i}(t)= \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f_{i}(\tau)d \tau, \quad i=1,2.
\]
Since $F(t,u)$ has convex values, then for $0 \leq \delta \leq 1$ we
obtain
\begin{equation*}
\begin{split}
[\delta f_{1}+(1-\delta)f_{2}](t)&=\delta [\sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f_{1}(\tau)d
\tau]\\
&\quad +(1-\delta)[\sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f_{2}(\tau)d \tau ]\\
&= \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}[\delta f_{1}+(1-\delta)f_{2}](\tau)d \tau.
\end{split}
\end{equation*}
Therefore, $[\delta f_{1}+(1-\delta)f_{2}] \in Nu$
and consequently $N$  has a convex values in $C(J,\mathbb{R})$.

\noindent \textbf{Step 2:} $N(u)$ maps bounded sets into bounded
sets in $C(J,\mathbb{R})$. Let $B$ be bounded set in
$C(J,\mathbb{R})$. Then there exists a real number $r >0$ such
that $\|u\| \leq r$, for all $u \in B$. Now for each $u \in
N$ there exists $f \in \overline{S}_{F}(u)$ such that
\[
u(t)= \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau)d \tau.
\]
Then for each $t \in J$,
\begin{equation*}
\begin{split}
|u(t)|& \leq  \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}|u^{(k)}(0)|+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}|f(\tau)|d \tau\\
& \leq  \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}|u^{(k)}(0)|+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}h(\tau) d \tau \\
&\leq  \sum^{n-1}_{k=0}
\frac{T^{k}}{k!}|u^{(k)}(0)|+\frac{\|h\|_{L^{1}}T^{\alpha}}{\Gamma(\alpha+1)}
\end{split}
\end{equation*}
Implies that $N(B)$ is bounded such that
$\|u(t)\|_{C} \leq \sum^{n-1}_{k=0} \frac{T^{k}}{k!}|u^{(k)}(0)|
+\frac{\|h\|_{L^{1}}T^{\alpha}}{\Gamma(\alpha+1)}:=r$.

  \noindent \textbf{Step 3:} $N(u)$ maps bounded sets into
equicontinuous sets in $C(J,\mathbb{R})$. From above we have for
any $t_{1}, t_{2} \in J $  such that $|t_{1}-t_{2}| \leq \delta,
\,\, \delta > 0$
\begin{equation*}
\begin{split}
|u(t_{1})-u(t_{2})|& =\big|  \sum^{n-1}_{k=0}
\frac{t_{1}^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t_{1}}_{0}
(t_{1}-\tau)^{\alpha-1}f(\tau)d \tau \\
&\quad -\sum^{n-1}_{k=0}
\frac{t_{2}^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t_{2}}_{0}
(t_{2}-\tau)^{\alpha-1}f(\tau)d \tau\big|\\
 & \leq 2 \sum^{n-1}_{k=0}
\frac{T^{k}}{k!}|u^{(k)}(0)|+[\frac{\|h\|_{L^{1}}
}{\Gamma(\alpha+1)}]
(t_{1}^{\alpha}-t_{2}^{\alpha}+2(t_{1}-t_{2})^{\alpha}))\\
& \leq 2 \sum^{n-1}_{k=0}
\frac{T^{k}}{k!}|u^{(k)}(0)|+[\frac{2\|h\|_{L^{1}}
}{\Gamma(\alpha+1)}]|(t_{1}-t_{2})|^{\alpha}\\
& \leq 2[\sum^{n-1}_{k=0}
\frac{T^{k}}{k!}|u^{(k)}(0)|+\frac{\delta^{\alpha} \|h\|_{L^{1}}
}{\Gamma(\alpha+1)}]
\end{split}
\end{equation*} which is independent of $u$ hence $N(B)$ is
equicontinuous set.

 \noindent \textbf{Step 4:} $N(u)$ is u.s.c.
 As an application of the Arzela-Ascoli theorem
yields that $N(B)$ is relatively compact set. Thus $N$ is compact
operator, hence in view of Remark \ref{rmk2.1}, we have that $N$ is u.s.c.

\noindent \textbf{Step 5:} Finally we show that the set
\[
\varepsilon = \{u \in C(J,\mathbb{R}): \lambda u \in Nu \text{ for some }
 \lambda > 1\}
\]
is bounded. Let $u \in \varepsilon$. Then
there exists a $f \in \overline{S}_{F}(u)$ such that
\begin{equation*}
\begin{split}
|u(t)|&  \leq \lambda^{-1} \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}|u^{(k)}(0)|+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}|f(\tau)|d \tau\\
& \leq  \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}|u^{(k)}(0)|+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}h(\tau) d \tau\\
& \leq  \sum^{n-1}_{k=0}
\frac{T^{k}}{k!}|u^{(k)}(0)|+\frac{\|h\|_{L^{1}}T^{\alpha}}{\Gamma(\alpha+1)}.\\
\end{split}
\end{equation*}
 Hence $\varepsilon$ is bounded set. As a consequence of Lemma \ref{lem2.1},
 we deduce that $N$ has a
fixed point which is a solution for $A$. Next we show that $u$ is
a solution for the problem \eqref{e1}. First we show that $u \in
[\underline{u}, \overline{u}]$. Suppose not, then either
$\underline{u} \nleq u$ or $u \nleq \overline{u}$ on $\overline{J}
\subset J$. If $\underline{u} \nleq u$ then for $t_{1} < t_{2} $
we have $\underline{u}(t) > u(t)$ for all $t$ in $(t_{1}, t_{2})
\subset J$. Since $\underline{u}$ is the lower solution of the
problem then for $f \in \overline{S}_{F}(u)$ yields
\begin{equation*}
\begin{split}
u(t)& = \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau)d \tau\\
& \geq \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}\underline{u}^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1} \underline{u}(\tau) d \tau= \underline{u}(t)\\
\end{split}
\end{equation*}
for all $t \in (t_{1}, t_{2})$. This is a
a contradiction. Similarly for $u \nleq \overline{u}$ yields a
contradiction. Hence $\underline{u}(t) \leq u(t) \leq
\overline{u}(t)$,  for all $t \in J$. As a result,
problem \eqref{e1} has a solution $u \in [\underline{u},\overline{u}]$.
\end{proof}

\begin{example} \label{exa4.1} \rm
 Let $J=[0,1]$ denote a closed and bounded interval in $\mathbb{R}$.
Consider $\alpha=1/2$ and
\[
F(t,u,\rho)=\begin{cases}
p(t,\rho), & \text{if }  u <1;\\
[p(t,\rho)\exp(-u^{2}(t)), p(t,\rho)], &\text{if }  u \geq 1.
\end{cases}
\]
in problem \eqref{e1}, subject to the condition $u(0)=1$. It is clear
that $F(t,u,\rho)$ is $L^{1}_{X}$-Carath\'eodory
with a growth function $p \in L^{1}(J\times\mathbb{R},\mathbb{R})$
such that
$\|F(t,u,\rho)\| \leq p(t,\rho)$ a.e $t \in J$ for all
$u \in \mathbb{R}$. Thus we have
\[
\overline{u}(t) = 1+ \frac{1}{\Gamma(1/2)}\int^{t}_{0}
(t-\tau)^{-0.5}p(\tau)d \tau \quad\text{and}\quad
\underline{u}(t)=1.
\]
In view of Theorem \ref{thm4.2}, the problem has a
convex solution $u \in [\underline{u},\overline{u}]$.
\end{example}

To study the existence for the problem \eqref{e1} in
non-convex case by using Theorem \ref{thm3.2} (the existence of the single
valued problem \eqref{e2}) and Lemma \ref{lem2.2} (the fixed point theorem for
set valued functions), we introduce the
following assumptions.
\begin{itemize}

\item[(H5)] $F: J \times \mathbb{R}\times \mathbb{R}
\to \mathcal{P}_{cl}(\mathbb{R})$,
$(t,.) \mapsto F(t,u,\rho)$ is measurable for each $u \in \mathbb{R}$.

\item[(H6)] $F: J \times \mathbb{R}\times \mathbb{R} \to
\mathcal{P}_{cl}(\mathbb{R})$ is $\ell(t)$-Lipschitz; i.e.,
$H(F(t,u,\rho),F(t,v,\rho)) \leq \ell(t) \|u-v\|$.
\end{itemize}

 \begin{theorem}[Non-convex case] \label{thm4.4}
 Let {\rm  (H5-H6)} hold. If
 $ \|\ell\|_{L^{1}}T^{\alpha}/\Gamma(\alpha+1) < 1 $,
then the differential inclusion \eqref{e1} has at least one solution
$u(t)$ on $J$.
\end{theorem}

\begin{proof}  For each $u(t)$ in $\mathbb{R}$, $F$
has a nonempty measurable selection (H5) then the set $S_{F}(u)$
is nonempty (see \cite{r8}). Then there exists a function $f(t) \in F$
such that $f$ is $\ell(t)-Lipschitz$. Thus by the assumption (H6),
we deduce that the conditions of Theorem \ref{thm3.2} hold, which implies
that the inclusion \eqref{e1} has a solution. Hence the proof is complete
in view of the single-valued problem.
\end{proof}

 \begin{theorem}[Non-convex case] \label{thm4.5}
 Let {\rm (H4-H6)} hold. If
 $\|\ell\|_{L^{1}}T^{\alpha}/\Gamma(\alpha+1) < 1$,
then the differential inclusion \eqref{e1} has at least one solution
$u(t)$ on $J$ such that $\underline{u}(t) \leq u(t) \leq
\overline{u}(t)$,   for   all $t  \in J$.
\end{theorem}

\begin{proof}
Define the operator $N$ as in \eqref{e8} then
the proof is done in two steps.

  \noindent \textbf{Step 1:} $N(u) \in
\mathcal{P}_{cl}(\mathcal{B})$ for each $u \in \mathcal{B}:=
C(J,\mathbb{R})$. Let $\{u_{m}\}_{m \geq 0} \in N(u)$ such that
$u_{m} \to \widetilde{u}$ in $\mathcal{B}$. Then
$\widetilde{u} \in \mathcal{B}$ and there exists
$f_{m} \in S_{F}(u)$ such that for $t \in J$
\[
u_{m}(t)=\sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f_{m}(\tau)d \tau.
\]
Using the fact that $F$
has closed values, we get that $f_{m}$ converges to $f$ in
$L^{1}(J,\mathbb{R})$ and hence $f \in S_{F}(u)$. Then for each
$t \in J$,
\[
u_{m}(t) \to \widetilde{u}(t)=\sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau)d \tau.
\]
So $\widetilde{u}  \in N(u)$.

  \noindent \textbf{Step 2:} There exists $\gamma < 1$ such that
\[
H(N(u),N(v)) \leq \gamma\|u-v\|_{\mathcal{B}}, \quad
\text{for  each } u,v  \in  \mathcal{B}.
\]
 Let $ u,v  \in  \Omega$. Then by (H6) there exists $f \in F$
 satisfies
 \[
|f(t,u,\rho)-f(t,v,\rho)|\leq \ell(t)\|u-v\|_{\mathcal{B}}
\]
then for $h_{1}(t)\in N(u)$ where
 \[
h_{1}(t)=\sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau,u(\tau),\rho(\tau))d \tau.
\]
And for $h_{2}(t)\in N(v)$ where
 \[
h_{2}(t)= \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau, v(\tau),\rho(\tau))d \tau
\]
we have
\begin{equation*}
\begin{split}
|h_{1}(t)-h_{2}(t)|& \leq \frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}|f(\tau,u(\tau),\rho(\tau))- f(\tau,v(\tau),\rho(\tau))|d \tau\\
& \leq \frac{\|u-v\|_{\mathcal{B}}}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}
\ell(\tau)d \tau\\
& \leq
\frac{\|\ell\|_{L^{1}}T^{\alpha}}{\Gamma(\alpha+1)}\|u-v\|_{\mathcal{B}}.
\end{split}
\end{equation*}
Let
 \[
\gamma:= [\frac{\|\ell\|_{L^{1}} T^{\alpha}}{\Gamma(\alpha+1)}].
\]
It follows that
\[
H(N(u),N(v)) \leq \gamma\|u-v\|_{\mathcal{B}}, \quad
\text{for  each } u,v \in \mathcal{B} ,
\]
where $\gamma < 1$. Implies that $N$ is a
contraction set-valued mapping. Then in view of Lemma \ref{lem2.3}, $N$
has a fixed point which is corresponding to a solution of
inclusion \eqref{e1}. The same conclusion holds in Theorem \ref{thm4.2}, we obtain
that problem \eqref{e1} has a solution
$u \in [\underline{u}, \overline{u}]$.
\end{proof}

\begin{example} \label{exa4.2}\rm
 Let $J=[0,1]$ denote a closed
and bounded interval in $\mathbb{R}$. Consider
\[
F(t,u,\rho)=\begin{cases}
[0,u l(t)], &\text{if }   1 \leq u \leq 2; \\
1, &\text{if }  u > 2.
\end{cases}
\]
in problem \eqref{e1}, subject to the condition $u(0)=1$. It is clear
that $F$ is $\gamma$-Lipschitzean continuous and bounded function
on $J \times \mathbb{R} $ with bound 1. Thus we have
\[
\overline{u}(t) = 1+ \frac{2}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}l(\tau)d \tau \quad\text{and}\quad
\underline{u}(t)=1
\]
where $\gamma:=2\|l\|/\Gamma(\alpha+1)<1$. In view of
Theorem \ref{thm4.5}, the problem has a non-convex solution
$u \in [\underline{u},\overline{u}]$.
\end{example}

\section{Extremal solutions}

In this section, we establish the
existence of extremal solutions to  \eqref{e1} on ordered
Banach spaces. The cone $K=\{u \in C(J,\mathbb{R}): u(t) \geq 0,
\forall \, t \in J \}$ defines an order relation, $\leq$ in
$C(J,\mathbb{R})$ by $u \leq v \Leftrightarrow u(t) \leq v(t)$,
for all $t \in J $. It is clear that $K$ is normal in
$C(J,\mathbb{R})$ (see \cite{r15}). Let
$S_{1}, S_{2} \in \mathcal{P}(X)$. Then by $S_{1} \leq S_{2}$
we mean $s_{1} \leq s_{2}$ for all $s_{1} \in S_{1}$ and
$s_{2} \in S_{2}$. Thus if
$S_{1} \leq S_{1}$ then it follows that $S_{1} $ is a singleton
set.

 We need to the following definitions and result due to
Dhage.

\begin{definition}  \label{def5.1}\rm
 Let $X$ be an ordered Banach space. A mapping
$T: X \to \mathcal{P}(X)$ is called
\textbf{isotone increasing} if $x,y \in X$ with $x < y$, then
we have that $T(x) \leq T(y)$.
\end{definition}

\begin{definition} \label{def5.2}\rm
A solution $u_{M}(t)$ of \eqref{e1} is said to be
\textbf{maximal solution} if for every
solution $u(t)$ of \eqref{e1}, we have
$u(t)\leq u_{M}(t)$ for all $t \in J$.
A solution $u_{m}(t)$ of  \eqref{e1} is said to
be \textbf{minimal solution} if $u_{m}(t) \leq u(t)$ for all $t
\in J$ where $u(t)$ is any solution of \eqref{e1}.
\end{definition}

\begin{lemma}[\cite{r12}] \label{lem5.1}
Let $[\underline{u},\overline{u}]$ be
an order interval in a Banach space and let $T:
[\underline{u},\overline{u}] \to \mathcal{P}(
[\underline{u},\overline{u}]) $ be a completely continuous and
isotone increasing set-valued. Further if the cone $K$ in $X$ is
normal, then $T$ has a least $u_{*}$ and a greatest fixed point
$v^{*}$ in $[\underline{u},\overline{u}]$. Moreover, the
sequences $\{u_{n}\}$ and  $\{v_{n}\}$ defined by
$u_{n+1} \in Tu_{n}$, $u_{0}=\underline{u}$ and
$v_{n+1} \in Tv_{n}$, $v_{0}=\overline{u}$, converge to $u_{*}$
and $v^{*}$ respectively.
\end{lemma}

 Let us consider the following assumptions:
\begin{itemize}
\item[(H7)] The set-valued function
$F: J \times \mathbb{R} \to \mathcal{P}(\mathbb{R})$ is
Carath\'eodory.

\item[(H8)] $F(t,u(t))$ is nondecreasing in
$u$ a.e. $t \in  J$; i.e., if $u < v$ then $F(t,u) \leq F(t,v)$
 a.e. $t\in J$.
\end{itemize}

\begin{theorem} \label{thm5.1}
Assume {\rm (H4), (H7), (H8)} hold. Then  \eqref{e1} has a
 minimal and a maximal solution on $J$.
\end{theorem}

\begin{proof}
Define an operator
 $H: C(J,\mathbb{R}) \to \mathcal{P}(C(J,\mathbb{R}))$ as
 follows
\begin{equation} \label{e9}
Hu = \big\{ u \in C(J,\mathbb{R}): u(t)= \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau)d \tau, f \in S_{F}(u)\big\}.
\end{equation}
We show that $H$ satisfies the conditions of Lemma \ref{lem5.1}.
Firstly, proceeding as in Theorem \ref{thm4.2}, is proved that $H$ is
completely continuous set-valued operator on
$[\underline{u},\overline{u}]$. Finally, we show that $H$ is
isotone  increasing on $C(J,\mathbb{R})$.
Let $u,v \in C(J,\mathbb{R})$ be such that $u < v$.
Let $\underline{u} \in Hu$  be arbitrary. Then there is a function
$f_{1} \in S_{F}(u)$ such that
\[
\underline{u}(t)=\sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f_{1}(\tau)d \tau.
\]
Since $F$ is
nondecreasing in $u$ we obtain that $S_{F}(u) \leq S_{F}(v)$. As
a result for any $f_{2} \in S_{F}(v)$ we have
\[
\underline{u}(t) \leq \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f_{2}(\tau)d \tau= \overline{u}
\]
for all $t\in J$ and $\overline{u} \in Hv$. This shows that the set-valued
operator $H$ is isotone increasing on $C(J,\mathbb{R})$. And in
particular in $[\underline{u},\overline{u}]$. Since
$\underline{u}$ and $\overline{u}$ are lower and upper solutions
of the problem \eqref{e1} on $J$ we have
\[
\underline{u}(t) \leq \sum^{n-1}_{k=0}
\frac{t^{k}}{k!}u^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}
(t-\tau)^{\alpha-1}f(\tau)d \tau
\]
 for all $f \in S_{F}(\underline{u})$ and so
$\underline{u} \leq H \underline{u}$.
Similarly $\overline{u}\geq H \overline{u}$. Hence we have
\[
\underline{u} \leq H\underline{u}\leq H \overline{u}\leq
\overline{u}.
\]
Since $H$ satisfies all the conditions of Lemma \ref{lem5.1},
yields that $H$ has a least and greatest fixed point
$[\underline{u},\overline{u}]$. This implies that problem \eqref{e1} has
a minimal and maximal solution on $J$.
\end{proof}

\subsection*{Conclusion}
We remark that when $\alpha=n$ in problem
\eqref{e1}, we obtain the existence of solution of the n-th order
differential inclusions studied in \cite{r12}.
Again problem \eqref{e1} has special cases that have been discussed
in \cite{r1}. Further, this work holds
for any kind of fractional operators: Caputo's, Erdelyi-Kober,
Weyl-Riesz, etc.

\subsection*{Acknowledgment}
The author thankful to the anonymous referee for his/her
helpful suggestions for the improvement of this article.

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\end{document}