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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 260, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/260\hfil Fractional differential inclusions]
{Existence of solutions to fractional differential inclusions
 with $p$-Laplacian operator}

\author[A. Yantir, F. S. Topal \hfil EJDE-2014/260\hfilneg]
{Ahmet Yantir, Fatma Serap Topal }  % in alphabetical order

\address{Ahmet Yantir \newline
Department of Mathematics, Yasar University,
Bornova, Izmir, Turkey}
\email{ahmet.yantir@yasar.edu.tr}

\address{Fatma Serap Topal \newline
Department of Mathematics, Ege University,
35100 Bornova, Izmir, Turkey}
\email{f.serap.topal@ege.edu.tr}

\thanks{Submitted April 4, 2014. Published December 17, 2014.}
\subjclass[2000]{34A60, 34B15, 34A08, 26A33, 34A12}
\keywords{Fractional differential equations; $p$-Laplacian operator; 
\hfil\break\indent boundary-value problem; existence of solutions}

\begin{abstract}
 In this article, we prove the existence of solutions for
 three-point fractional differential inclusions with $p$-Laplacian operator.
 We use fixed point theory for set valued upper semi-continuous maps
 for obtaining the solutions.
\end{abstract}

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

\section {Introduction} \label{sec1}

Since fractional derivatives provide an excellent tool for the description of 
the memory and hereditary properties of various materials and processes, 
the differential equations/inclusions of fractional-order  are more suitable to 
describe a model in some real-life problems than integer-order equations 
 \cite{weak}. The most widespread areas whose mathematical models involves 
derivatives of fractional order are  viscoelasticity, electrochemistry, 
control, electromagnetism, aerodynamics, electrodynamics of complex media, 
polymer rheology, and so forth \cite{bagley,sorrenitos,wang}. Because of 
these wide range of application areas, the fractional differential equations 
gain importance and attention day by day. Due to this importance several 
monographs are written. For the detailed information about differential 
equations involving fractional derivatives, we refer to the monographs 
of Kilbas et al \cite{kilbas}, Podlunby \cite{podlubny},
 Lakshimikantham et al \cite{laks} and Samko et al \cite{samko} and the 
references therein.

Besides, integer order $p$-Laplacian boundary-value problems have been studied
in terms of their importance in theory and applications in mathematics, 
physics and so on, see for example, \cite{chai,yang} and the references therein.

By unifying the ideas of fractional differential equations and $p$-laplacian 
operator which are mentioned above,  Liu, Jia and Xiang \cite{liu} 
studied the fractional differential equations with $p$-Laplacian operator 
(for the first time in the literature as the authors claim). 
They studied the existence and uniqueness of solutions of Caputo fractional 
differential equation involving the $p$-Laplacian operator
\[
(\varphi_p(^cD^\alpha x(t)))'=f(t,x(t)),
\]
with the boundary conditions
\[
x(0)=r_0x(1), \quad x'(0)=r_1x'(1), \quad x^{(i)}(0)=0, \quad i=2, 3, 
\ldots [\alpha]-1.
\]
However there are some older studies in this area \cite{chai, wang, wang1}.
Chai \cite{chai} studied the existence and multiplicity of the solutions of
\begin{gather*}
D_{0^+}^\beta(\phi_p(D_{0^+}^\alpha u))(t)+f(t,u(t))=0, \quad 0<t<1,\\
u(0)=0, \quad u(1)+\sigma D_{0^+}^\gamma u(1)=0, D_{0^+}^\alpha u(0)=0
\end{gather*}
by using fixed point theorem on cones. A similar problem with different boundary 
conditions
\begin{gather*}
D_{0^+}^\gamma(\phi_p(D_{0^+}^\alpha u))(t)+f(t,u(t))=0, \quad 0<t<1,\\
u(0)=0, \quad u(1)=au(\xi), \quad D_{0^+}^\alpha u(0)=0, \quad
D_{0^+}^\alpha u(1)=bD_{0^+}^\alpha u(\mu)
\end{gather*}
is studied by Wang and Xiang \cite{wang1}. 
The upper and lower solutions method is used for the existence of solutions. 
In another article, Wang et al \cite{wang} studied the above equation with 
the boundary conditions 
$$
u(0)=0, \quad u(1)=au(\xi), \quad D_{0^+}^\alpha u(0)=0
$$ 
where Krasnoselskii's and Legget-Williams fixed point theorems are used to 
obtain the main results.

On the other hand, realistic problems arising from economics and optimal control 
can be modeled as differential inclusions which are the generalization of 
the concept of ordinary differential equations. Therefore,
differential inclusions have been widely studied by many
authors, see \cite{agarwall,agarwall1,agarwall2,bagley,chai,ibrahim,podlubny} and
the references therein.

The differential inclusions with fractional derivatives have been studied
 by many authors in the literature. Among these studies one of the principle 
one belongs to Chang and Nieto \cite{chang}.  Authors deal with the existence of
solutions for the following fractional differential inclusion
\begin{gather*}
_{0}^{c}D_{t}^{\delta}y(t) \in F(t, y(t))\quad t\in[0, 1], \; \delta \in (1,2)\\
y(0)=\alpha, \quad y(1) = \beta, \quad \alpha, \beta \neq 0,
\end{gather*}
where $_{0}^{c}D_{t}^{\delta}y(t)$ is the Caputo's
derivative and $F:[0, 1] \times \mathbb{R} \to
2^{\mathbb{R}} \backslash \emptyset$.


It is worthwhile to emphasize that the framawork presented in this article has 
the following features which are different from those mentioned above.
 As stated above, in the literature the unification of fractional calculus 
with $p$-laplacian differential equations was studied as well as its 
unification with differential inclusions was also analyzed. 
However, to our knowledge, there are no contributions in the literature 
addressing the unification of fractional calculus, $p$-laplacian operator 
and differential inclusions. Motivated by  this gap the intrinsic feature 
of the present article is to analyze the existence of solutions for the 
following \emph{fractional differential inclusions
with $p$-Laplacian operator}
\begin{gather}
D_{0^{+}}^{\beta}(\varphi_{p}(D_{0^{+}}^{\alpha}u))(t) \in F(t, u(t)), \quad
t\in[0, 1],\label{main}\\
u(0)=0, \quad u(1) = \gamma u(\eta), \quad D_{0^{+}}^{\alpha}u(0) = 0  \label{boundary}
\end{gather}
where $D_{0^{+}}^{\beta}$ and $D_{0^{+}}^{\alpha}$ are the standard
Riemann-Liouville derivatives of order $\alpha$ and $\beta$ with
$\alpha \in (1, 2]$, $\beta \in (0, 1]$. Moreover $\eta \in (0, 1)$
with $1- \gamma \eta^{\alpha-1} > 0$ and $\varphi_p$ is $p$-Laplacian
operator; i.e., $\varphi_p(s)=|s|^{p-2}s$, $p>1$ such that
$(\varphi_p)^{-1}=\varphi_q$ with $\frac{1}{p}+\frac{1}{q}=1$.


Also $F:[0, 1] \times \mathbb{R} \to 2^{\mathbb{R}}
\backslash \emptyset$ is a multi-valued function with compact and
convex values such that $|F(t, u)| = \sup \{|v|: v \in F(t, u)\}$.
By $F(t, u)>0$, we mean $v>0$ for each $v \in F(t, u)$.

By  $u$ being a solution of \eqref{main}-\eqref{boundary}, we mean that
there exists a function $v \in C^{1}([0, 1], \mathbb{R})$ such that
$v(t)\in F(t, u(t))$ on $[0, 1]$ satisfying the equation
\begin{equation} \label{linear}
D_{0^{+}}^{\beta}(\varphi_{p}(D_{0^{+}}^{\alpha}u))(t) = v(t), \ \
t\in[0, 1]
\end{equation}
and the boundary conditions \eqref{boundary}.

This article is organized as follows:  Section \ref{sec2} is devoted
to preliminary definitions of multi-valued maps and a very brief
introduction to fractional calculus. In Section \ref{sec3}, we first
state the fixed point results. Consequently, we prove the existence
results for the integral inclusion corresponding to the main problem
\eqref{main}-\eqref{boundary}. Finally, we illustrate our main
result by an example.

\section{Preliminaries} \label{sec2} 

In this section, we list some preliminary definitions,
notation and results that will be used in the rest of the article.

Let $C(I)$ denote the Banach space of continuous functions from $I$
into $\mathbb{R}$ with the supremum norm
 $\|y\|=\sup_{t\in I}\{|y(t)|\}$.

Let $(X,\|\cdot\|)$ be a Banach space and $C K(X)$ denote the family
of nonempty, closed and convex subsets of $X$. A multi-valued map
$H: X \to C K(X)$ is said to be \emph{upper semi-continuous
(u.s.c.)} provided that $\{u_{k}\}_{k\in \mathbb{N}}$,
$\{v_{k}\}_{k\in \mathbb{N}} \subset X$ with $u_{k} \to u,
v_{k} \to v$ (while $ k\to\infty$) and $v_{k} \in
H(u_{k})$ for all $k \in \mathbb{N}$ imply $v \in H(u)$. Morevover a
multi-valued map $H$ is said to be \emph{completely continuous} if $H(B)$
is relatively compact for every bounded subset $B$ of $X$.
Furthermore, we say that $H$ has a fixed point if there exists $x \in X$
such that $x \in H(X)$.

Throughout this article, we impose the following condition on the
multi-valued function $F$:
\begin{itemize}
\item [(H1)] $F:[0, 1] \times \mathbb{R} \to C
K(\mathbb{R})$ is a multi-valued map such that $F(t, .)$ is u.s.c. 
for all $t\in[0,1]$.
\end{itemize}

\begin{definition}[\cite{kilbas,podlubny}] \rm
 The Riemann-Liouville fractional integral of order $\alpha>0$ of a 
function $y: (a, b] \to \mathbb{R}$ is given by
$$
I_{a^{+}}^{\alpha}y(t) = \frac{1}{\Gamma(\alpha)}\int_{a}^{t}(t-s)^{\alpha-1}y(s) ds,
  \quad t\in (a, b].
$$
\end{definition}

\begin{definition}[\cite{kilbas,podlubny}] \rm
 The Riemann-Liouville fractional derivative of order $\alpha>0$ of a function 
$y: (a, b] \to \mathbb{R}$ is given by
$$
D_{a^{+}}^{\alpha}y(t) = \frac{1}{\Gamma(n-\alpha)}(\frac{d}{dt})^{n}
\int_{a}^{t}\frac{y(s)}{(t-s)^{\alpha-n+1}}ds,  \quad t\in (a, b],
$$
where $n=[\alpha]+1$ and $[\alpha]$ denotes the integer part of
$\alpha$.
\end{definition}

\begin{lemma} \label{lemma2}
 Let $\alpha>0$. If $y \in C(0,1)\cap L(0,1)$ possesses a fractional derivative 
of order $\alpha$ that belongs to $C(0,1)\cap L(0,1)$, then
$$ 
I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}y(t)
= y(t) + c_{1}t^{\alpha-1}+c_{21}t^{\alpha-2}+ \dots + c_{n}t^{\alpha-n},$$
for some $c_{i}\in \mathbb{R}$, $i=1, 2, \dots, n$, where
$n=[\alpha]+1$.
\end{lemma}
To find the form of the a solution of the problem
\eqref{linear}-\eqref{boundary}, we first consider the 
fractional boundary-value problem
\begin{gather*}
D_{0^{+}}^{\alpha}u(t) = \phi(t), \quad t\in[0, 1],\\
u(0)=0, \quad u(1) = \gamma u(\eta), 
\end{gather*}
where $\phi \in C([0, 1], \mathbb{R})$. Ahmad and Nieto \cite{ahmat}
presented the unique solution of the above problem by
$$
u(t)= \int_{0}^{1} G(t, s) \phi(s) ds,
$$ 
where $G(t, s)$ is the Green's function given by
\begin{equation} \label{green}
G(t, s) =  \frac{1}{\Gamma(\alpha)(1-\gamma\eta^{\alpha-1})}
\begin{cases} G_{1}(t, s), & 0\leq t\leq \eta,\\
G_{2}(t, s), & \eta < t\leq 1.\
\end{cases}
\end{equation}
Here $G_1(t,s)$ and $G_2(t,s)$ are given by
\begin{gather*}
G_{1}(t, s) =  \begin{cases}
(t-s)^{\alpha-1}(1- \gamma \eta^{\alpha-1})-t^{\alpha-1}[(1-s)^{\alpha-1}
-\gamma (\eta-s)^{\alpha-1}], & 0\leq s\leq t,\\
-t^{\alpha-1}[(1-s)^{\alpha-1}-\gamma (\eta-s)^{\alpha-1}], & t < s \leq \eta,\\
-(t(1-s))^{\alpha-1}, & \eta < s \leq 1,\
\end{cases}
\\
G_{2}(t, s) =  \begin{cases}
(t-s)^{\alpha-1}(1- \gamma \eta^{\alpha-1})-t^{\alpha-1}[(1-s)^{\alpha-1}
 -\gamma (\eta-s)^{\alpha-1}], & 0\leq s\leq \eta,\\
(t-s)^{\alpha-1}(1- \gamma \eta^{\alpha-1})-(t(1-s))^{\alpha-1},
 & \eta < s \leq t,\\
-(t(1-s))^{\alpha-1}, & t < s \leq 1\
\end{cases}
\end{gather*}
respectively.

Substituting $D_{0^{+}}^{\alpha}u = \phi$, 
$\varphi_{p}(\phi) = \omega$ in \eqref{linear}, we obtain the equation
$D_{0^{+}}^{\beta}\omega(t) = v(t)$.  Lemma \ref{lemma2} implies
that the solution of initial value problem
\[
D_{0^{+}}^{\alpha}\omega(t) = v(t), \quad t\in[0, 1],\\
\omega(0)=0,
\]
is of the form $\omega(t) = c_{1}t^{\beta-1}+I_{0^{+}}^{\beta}v(t)$. 
It follows from the initial condition and
$\beta\in(0, 1]$ that $c_{1} = 0$. Hence 
$$
\omega(t) = I_{0^{+}}^{\beta}v(t), \quad t\in[0, 1].
$$

Taking $D_{0^{+}}^{\alpha}u = \phi$  and 
$\phi = \varphi_{p}^{-1}(\omega) = \varphi_{q}(\omega)$ into account, we
establish that the solution of \eqref{linear}-\eqref{boundary}
satisfies the boundary-value problem
\begin{gather}
D_{0^{+}}^{\alpha}u(t) = \varphi_{q}(I_{0^{+}}^{\beta}v)(t), \quad
  t\in[0, 1],\label{e1}\\
u(0)=0, \quad u(1) = \gamma u(\eta), \quad
D_{0^{+}}^{\alpha}u(0) = 0\label{b1}.
\end{gather}
Using the result of Ahmad and Nieto \cite{ahmat}, we obtain the
solution of the problem \eqref{e1}-\eqref{b1} as
$$
u(t)= \int_{0}^{1} G(t, s) \varphi_{q}(I_{0^{+}}^{\beta}v)(s) ds, \quad t\in[0,1].
$$
Since $v(t)>0$ for $t\in[0,1]$, we have
$\varphi_{q}(I_{0^{+}}^{\beta}v)(s)=(I_{0^{+}}^{\beta}v)^{q-1} (s)$
and therefore
\begin{align*}
u(t)
&= \int_{0}^{1} G(t, s) (I_{0^{+}}^{\beta}v)^{q-1} (s) ds \\ 
&=  \frac{1}{(\Gamma(\beta))^{q-1}}\int_{0}^{1} G(t, s) 
\Big(\int_{0}^{s}(s-\tau)^{\beta-1} v(\tau) d\tau \Big)^{q-1}ds. 
\end{align*}
Hence we have the following lemma.

\begin{lemma} \label{lem2.4}
Let $v(t) > 0$ for $t\in[0,1]$. Then the solution of 
\eqref{linear}-\eqref{boundary} is given by
$$
u(t)= \frac{1}{(\Gamma(\beta))^{q-1}}\int_{0}^{1} G(t, s) 
\Big(\int_{0}^{s}(s-\tau)^{\beta-1} v(\tau) d\tau \Big)^{q-1}ds. 
$$
\end{lemma}

\section {Existence results} \label{sec3} 

This section is devoted to the existence results
regarding solutions for the differential inclusion
\eqref{main}-\eqref{boundary}. For this purpose, we first give an
existence result for the integral inclusion corresponding to
\eqref{main}-\eqref{boundary}. Consequently, we derive an existence
result for the differential inclusion \eqref{main}-\eqref{boundary}.
Finally, we illustrate our result by an example.

Assume that $G: [0, 1] \times [0, 1] \to \mathbb{R}$ is a
single-valued function. By a solution $u$ of the integral inclusion
\begin{equation} \label{intinclusion}
u(t) \in \frac{1}{(\Gamma(\beta))^{q-1}}\int_{0}^{1} G(t,
s)\Big(\int_{0}^{s}(s-\tau)^{\beta-1}F(\tau, u(\tau)) d\tau\Big)^{q-1} ds, \quad
t\in[0,1]
\end{equation}
we mean that there exists a function $v \in C^{1}([0, 1],
\mathbb{R})$ such that $v(t)\in F(t, u(t))$ on $[0, 1]$ satisfying
the integral equation
\[
u(t) = \frac{1}{(\Gamma(\beta))^{q-1}}\int_{0}^{1} G(t,
s)\Big(\int_{0}^{s}(s-\tau)^{\beta-1} v(\tau) d\tau\Big)^{q-1} ds, \quad
t\in[0,1].
\]
The proofs of our main existence results are based on the following two fixed point
results.

\begin{theorem}[\cite{agarwall1}] \label{fpt1} 
 Let $C$ be a nonempty closed convex subset of a Banach space $E$ and 
$H:C \to CK(C)$ an u.s.c. compact map, then $H$ has a fixed point in $C$.
\end{theorem}

\begin{theorem}[\cite{agarwall1}] \label{fpt2} 
 Let $E$ be a Banach space, $U$ an open subset of $E$ and $0 \in U$. 
If $H:\overline{U} \to CK(E)$ is an u.s.c. compact map, then either
\begin{itemize}
\item[(1)] $H$ has a fixed point in $\overline{U}$
or
\item[(2)] there exists $u \in \partial U$ and $\lambda \in (0,
1)$ such that $u \in \lambda H(u)$.
\end{itemize}
\end{theorem}

Our main result is based on the following existence principle.

\begin{lemma}\label{lemma} 
Assume that {\rm (H1)} holds and $G(t, s) :[0, 1] \times [0, 1] \to \mathbb{R}$ 
is continuous. Then we have the following existence results:

(a) For any $r>0$, suppose that there exists a continuous
$h_{r}\in C([0, 1])$ with $|F(t, u)| \leq h_{r}(t)$ for all $t \in
[0, 1]$ and all $|u| \leq r$. If there exists a constant $M$ with
$\| u \| \neq M$ for all solutions $u$ of integral inclusion
 \begin{equation} \label{lemmaeq}
u(t) \in \frac{\lambda}{(\Gamma(\beta))^{q-1}}\int_{0}^{1} G(t,
s)\Big(\int_{0}^{s}(s-\tau)^{\beta-1}F(\tau, u(\tau)) d\tau\Big)^{q-1} ds,
\end{equation}
for each $\lambda \in (0, 1)$, then the inclusion
\eqref{intinclusion} has a solution.

(b) Suppose that there exists a continuous function $h \in C([0, 1])$ with 
$|F(t, u)| \leq h(t)$ for all $t \in [0, 1]$ and all
$u \in \mathbb{R}$, then the inclusion \eqref{intinclusion} has a
solution.
\end{lemma}

\begin{proof}
(a) We define a linear and continuous
operator $H: C([0, 1]) \to C([0, 1]) $ by
$$
H u(t):= \frac{1}{(\Gamma(\beta))^{q-1}}\int_{0}^{1} G(t,
s)\Big(\int_{0}^{s}(s-\tau)^{\beta-1} u(\tau) d\tau\Big)^{q-1} ds, \quad
 t \in [0, 1].
$$
Let
$$
\mathcal{F}(u):= \{v \in  C([0, 1]): v(t) \in F(t, u(t)) \text{ for } t \in
[0, 1] \}. 
$$
Clearly $F: [0, 1]\times \mathbb{R} \to  CK(\mathbb{R})$ implies that 
$\mathcal{F}: C([0, 1])\to CK(C([0, 1]))$.
Note that the integral inclusion \eqref{lemmaeq} is
equivalent to the fixed point problem
$$
u \in \lambda (H \circ \mathcal{F})(u),
$$
where $H \circ \mathcal{F}:C([0, 1])\to CK(C([0,1]))$.

 Let $U:=\{u \in C([0, 1]): \|u\|< M \} $ and 
$E=C([0, 1])$.
Now we apply Theorem \ref{fpt2} to the function $H \circ \mathcal{F}$ for
the existence of solutions for the inclusion \eqref{intinclusion}.

 Assume there exists $u \in \partial U$ and 
$\lambda \in (0, 1)$ with $u \in \lambda (H \circ \mathcal{F})(u)$. Then
$\|u\|=M$ and so the second possibility given in Theorem \ref{fpt2} is
ruled out. Hence, if $H \circ \mathcal{F}: \bar{U}\to CK(E)$
is u.s.c. and compact then Theorem \ref{fpt2} guarantees that 
$H \circ \mathcal{F}$ has a fixed point in $\bar{U}$, i.e. \eqref{intinclusion} 
has a solution.

 First, we show that $H \circ \mathcal{F}: \bar{U}\to
CK(E)$ is u.s.c. For this purpose we let 
$\{u_{k}\}_{k\in\mathbb{N}}, \{w_{k}\}_{k\in\mathbb{N}} \subset \mathbb{R}$ with
$u_{k}\to u_{0}, w_{k}\to w_{0}$ in $C([0, 1])$ as
$k \to \infty$ and $w_{k} \in H \circ \mathcal{F}(u_{k})$
for $k\in \mathbb{N}$. Thus there exist 
$v_{k} \in \mathcal{F}(u_{k})$ with $w_{k}=H v_{k}$. Since $u_{k}\in \bar{U}$
for all $k\in \mathbb{N}$, the condition $|F(t, u)| \leq h_{r}(t)$ for all
 $|u|<r$ with $h_{r} \in C([0, 1])$ guarantees 
(see the proof of \cite[Remark 2.1]{agarwall2}) that there exists a 
compact set $\Omega \subset E$ with
$\{v_{k}\}_{k\in\mathbb{N}}\subseteq \Omega$. Therefore there exists
a convergent subsequence  $\{v_{k_{n}}\}_{n\in\mathbb{N}}$ of
$\{v_{k}\}_{k\in\mathbb{N}}$, such that $v_{k_{n}} \to
v_{0}$ as $n \to \infty$. Now $v_{k_{n}} \to v_{0}$
and $u_{k_{n}} \to u_{0}$ as $n \to \infty$ and
$v_{k_{n}}(t) \in F(t, u_{k_{n}}(t))$ for all $t \in [0, 1]$. Thus,
since $F(t, \cdot)$ is u.s.c. for all $t \in [0, 1]$, we conclude with
$v_{0}(t) \in F(t, u_{0}(t))$ which results $v_{0} \in
\mathcal{F}(u_{0})$. Since $v_{k_{n}}\to v_{0}$ as $n
\to \infty$ and $H$ is continuous, we see that $w_{k_{n}}= H
v_{k_{n}}\to H v_{0}$ as $n \to \infty$, and hence
$w_{0}= H v_{0} \in (H \circ \mathcal{F})(u_{0})$. As a result $H
\circ \mathcal{F}: \bar{U}\to CK(E)$ is u.s.c. The compactness of
$H \circ \mathcal{F}$ follows from the Arzela-Ascoli theorem. Therefore the
proof of part (a) is complete.
\smallskip

(b) Let $H$ and $\mathcal{F}$ be as in part $(a)$. 
Clearly proving the existence a solution for the integral inclusion
\eqref{intinclusion} is equivalent to the fixed point problem
$$
u \in H \circ \mathcal{F}(u).
$$
The argument in part (a) guarantees that $H \circ \mathcal{F}$ is u.s.c. 
and compact and hence the claim follows by
using Theorem \ref{fpt1}. 
\end{proof}

With the help of Lemma \ref{lemma}, we present our existence result
for fractional differential inclusion \eqref{main} with the boundary
conditions \eqref{boundary}.

\begin{theorem} \label{thm3.4}
 Assume that {\rm (H1)} holds, and there exist a continuous nondecreasing 
function $\psi: [0, \infty) \to [0, \infty)$ with $\psi(u) > 0$ for $u > 0$ 
and a function $r: [0, 1] \to [0, \infty)$ such that $|F(t, u)| \leq r(t) \psi(|u|)$ 
for all $u \in \mathbb{R}$ and $t \in [0, 1]$.
If $Q$ defined by
\[
Q:= \frac{1}{(\Gamma(\beta))^{q-1}}\max_{t\in[0, 1]}\Big(\int_{0}^{1}
|G(t, s)|(\int_{0}^{s}(s-\tau)^{\beta-1} r(\tau) d\tau)^{q-1} ds\Big),
\]
where $G(t, s)$ is defined by \eqref{green}, satisfies the property
\[
\sup_{c\in (0,\infty)}\frac{c}{(\psi(c))^{q-1}}> Q,
\]
 then the problem
\eqref{main}-\eqref{boundary} has a solution.
\end{theorem}

\begin{proof}
 Suppose $M > 0$ satisfies $M/(\psi(M))^{q-1}> Q$. Consider
\begin{equation} \label{proof1}
D_{0^{+}}^{\beta}(\varphi_{p}(D_{0^{+}}^{\alpha}u))(t) \in \lambda
F(t, u(t)), \quad t\in[0, 1],
\end{equation}
with the boundary condition \eqref{boundary} and  $\lambda \in (0,1)$.
 Solving the inclusion \eqref{proof1} is equivalent to finding a
function $u \in C([0, 1])$ which satisfies the equation
\eqref{lemmaeq} for $t\in[0, 1]$. Therefore by Lemma \ref{lemma}, it
is enough to show that there exists a constant $M>0$ with
$\|u\| = \max_{t\in[0, 1]}|u(t)|\neq M$ for any solution $u$ of the
inclusion \eqref{lemmaeq}. Let $u$ be any solution of
\eqref{lemmaeq} for $\lambda \in (0, 1)$.

For $t\in[0, 1]$, we have 
\begin{align*} 
|u(t)| &\leq  \frac{1}{(\Gamma(\beta))^{q-1}}\int_{0}^{1}|G(t,
s)|\Big(\int_{0}^{s}(s-\tau)^{\beta-1}|F(\tau, u(\tau))| d \tau\Big)^{q-1}ds
\\ 
&\leq \frac{1}{(\Gamma(\beta))^{q-1}}\int_{0}^{1} 
|G(t, s)|\Big(\int_{0}^{s}(s-\tau)^{\beta-1}|r(\tau)| \psi(|u(\tau)|) d
\tau\Big)^{q-1} ds\\
&\leq  \Big(\frac{\psi(\|u\|)}{\Gamma
(\beta)}\Big)^{q-1}\int_{0}^{1} |G(t, s)|\Big(\int_{0}^{s}(s-\tau)^{\beta-1}
r(\tau)) d \tau\Big)^{q-1} ds.
\end{align*}
Therefore,
$$ 
\frac{\|u\|}{(\psi(\|u\|))^{q-1}} \leq Q.
$$
If $\|u\| = M$, then
$$ 
\frac{M}{(\psi(M))^{q-1}} \leq Q,
$$
which contradicts our assumption. Hence we obtain $\|u\| \neq M$
which is the desired result.
\end{proof}

We illustrate our result with the following example.
Consider the problem
\begin{gather}
D_{0^{+}}^{\frac{1}{2}}(\varphi_{\frac{3}{2}}(D_{0^{+}}^{\frac{4}{3}}u))(t) 
\in F(t, u(t)), \quad t\in[0, 1],\label{ex}\\
u(0)=0, \quad u(1) = \frac{1}{4^{\frac{2}{3}}} u(\frac{1}{4}), \quad
D_{0^{+}}^{\frac{4}{3}}u(0) = 0\label{exboun}
\end{gather}
where $F:[0, 1] \times \mathbb{R} \to 2^{\mathbb{R}}
\backslash \emptyset$ is a multi-valued map defined by
\[
(t, u) \to F(t, u):= \big[\frac{u^{4}}{4(u^{4}+1)} +
\frac{t^{2}+1}{8}, \frac{1}{4}\cos u + \frac{t^{2}+1}{4}\big].
\]
For $v \in F$, we have
\[
|v| \leq \max \big[\frac{u^{4}}{4(u^{4}+1)} + \frac{t^{2}+1}{8},
\frac{1}{4}\cos u + \frac{t^{2}+1}{4}\big] \leq \frac{3}{4}.
\]
Thus, 
$$
|F(t, u)| = \sup \{|v|: v \in F(t, u)\} \leq r(t) \psi(|u|),
\quad  u \in \mathbb{R},
$$ 
with  $r(t)=1$, $\psi(u)= \sqrt{u} + 1$.
Clearly, 
${ \sup_{c \in (0, \infty)}\frac{c}{(\psi(c))^{2}}= 1 > 0,0785502}$ is
satisfied. Hence the problem \eqref{ex}-\eqref{exboun} has a
solution on $[0, 1]$.


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\end{document}