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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 98, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/98\hfil Multi-strip BVP for fractional differential equations]
{Nonlinear fractional differential equations and inclusions 
 of arbitrary order and multi-strip boundary conditions}

\author[B. Ahmad, S. K. Ntouyas\hfil EJDE-2012/98\hfilneg]
{Bashir Ahmad,  Sotiris K. Ntouyas}  % in alphabetical order

\address{Bashir Ahmad  \newline
Department of Mathematics, Faculty of Science,
King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia}
\email{bashir\_qau@yahoo.com}

\address{Sotiris K. Ntouyas \newline
Department of Mathematics,  
University of Ioannina, 451 10 Ioannina, Greece}
\email{sntouyas@uoi.gr}

\thanks{Submitted May 20, 2012. Published June 12, 2012.}
\subjclass[2000]{26A33, 34A60, 34B10, 34B15}
\keywords{Fractional differential inclusions;
   integral boundary conditions; \hfill\break\indent
existence; contraction principle; Krasnoselskii's fixed point theorem;
Leray-Schauder degree; \hfill\break\indent
 Leray-Schauder nonlinear alternative;  nonlinear contractions}

\begin{abstract}
 We study boundary value problems of nonlinear fractional
 differential equations and inclusions of order $q \in (m-1, m]$,
 $m \ge 2$ with multi-strip boundary conditions. Multi-strip boundary
 conditions may be regarded as the generalization of multi-point
 boundary conditions.  Our problem is new in the sense
 that we consider a nonlocal strip condition of the form:
 $$
 x(1)=\sum_{i=1}^{n-2}\alpha_i \int^{\eta_i}_{\zeta_i} x(s)ds,
 $$
 which can be viewed as an extension of a multi-point nonlocal boundary condition:
 $$
 x(1)=\sum_{i=1}^{n-2}\alpha_i x(\eta_i).
 $$
 In fact, the strip condition  corresponds to a continuous distribution
 of the values of the unknown function on  arbitrary finite
 segments $(\zeta_i,\eta_i)$ of the interval $[0,1]$ and the effect
 of these strips is accumulated at $x=1$. Such problems occur in
 the applied fields  such as wave propagation and geophysics. Some
 new existence and uniqueness results are obtained by using a
 variety of fixed point theorems. Some illustrative examples are
 also discussed.
\end{abstract}

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


\section{Introduction}

In recent years, boundary value problems for nonlinear fractional
differential equations have been addressed by several researchers.
Fractional derivatives provide an excellent tool for the
description of memory and hereditary properties of various
materials and processes, see \cite{Pod}. These characteristics of
the fractional derivatives make the fractional-order models more
realistic and practical than the classical integer-order models.
As a matter of fact, fractional differential equations arise in
many engineering and scientific disciplines such as physics,
chemistry, biology, economics, control theory, signal and image
processing, biophysics, blood flow phenomena, aerodynamics,
fitting of experimental data, etc. \cite{d2, Kil, Sag, SKM}. For
some recent development on the topic, see   \cite{r1, r2, t1, b3,
t2, d1} and the references therein.

  In the first part of this paper, we consider the following  nonlinear
fractional BVP of an arbitrary order with
multi-strip boundary conditions:
\begin{equation}\label{e1}
\begin{gathered}
 ^cD^qx(t) =f(t,x(t)), \quad  0<t<1,  \; m-1 < q \le m,\; m\ge2,\; m \in \mathbb{N},
\\
 x(0)=0, \quad x'(0)=0, \quad x''(0)=0,  \dots,x^{(m-2)}(0)=0, \\
  x(1)=\sum_{i=1}^{n-2}\alpha_i
\int^{\eta_i}_{\zeta_i} x(s)ds, \quad 0< \zeta_i < \eta_i <1, \; i=1,
2,\dots, (n-2),
\end{gathered}
\end{equation}
 where $^cD^q$ denotes the Caputo
fractional derivative of order $q$, $f$ is a given continuous
function and $\alpha_i \in \mathbb{R}$ satisfy the condition:
$$
\sum_{i=1}^{n-2}\alpha_i (\eta_i^m-\zeta_i^m) \ne m.
$$

The strip boundary condition in problem \eqref{e1} can be regarded
as a multi-point nonlocal integral boundary condition. Integral
boundary conditions  have various applications in applied sciences
such as blood flow problems, chemical engineering,
thermoelasticity, underground water flow, population dynamics,
etc. For a detailed description of the integral boundary
conditions, we refer the reader to the papers \cite{b1, ch1} and
references therein. Regarding the application of the strip
conditions of fixed size, we know that such conditions appear in
the mathematical modeling of real world problems, for example, see
\cite{b2, b4}. Thus, the present idea of nonlocal strip conditions
will be quite fruitful in modeling the strip problems as one can
choose an arbitrary set of strips of desired size, which can be
fixed according to the requirement by fixing the nonlocal
parameters involved in the problem. Furthermore, these conditions
can be understood in the sense that the controllers at the
end-points of the interval dissipate/absorb energy due to the
sensors of finite lengths (continuous distribution of intermediate
points of arbitrary length:  subsegments of the interval) located
at the intermediate positions of the interval.

Recently nonlocal problems with several  types of
integral boundary conditions have studied in  \cite{n1, n2, n5, n6, n3, n4,
n7, n8}.

We prove some new existence and uniqueness results by using a
variety of fixed point theorems. In Theorem \ref{tb} we prove an
existence and uniqueness result by using Banach's contraction
principle, in Theorem \ref{tk} we prove the existence of a
solution by means of Krasnoselskii's fixed point theorem, while in
Theorem \ref{tls} we prove the existence of a solution via
Leray-Schauder nonlinear alternative.
 The Leray-Schauder degree theory is used in proving the existence result
in Theorem \ref{tlsd}.
In Theorem \ref{tnc} we prove an existence and uniqueness result
by applying a fixed point theorem of Boyd and Wong \cite{BW} for
nonlinear contractions.

 In the second  part of the paper,  we consider a  nonlinear fractional differential inclusion  of an arbitrary order
 with multi-strip boundary conditions:
\begin{equation}\label{e1i}
\begin{gathered}
 ^cD^qx(t) \in F(t,x(t)), \quad  0<t<1,  \; m-1 < q \le m,~ m\ge2,\;
m \in \mathbb{N}, \\
 x(0)=0, \quad x'(0)=0,\quad x''(0)=0, \dots ,x^{(m-2)}(0)=0, \\
  x(1)=\sum_{i=1}^{n-2}\alpha_i
\int^{\eta_i}_{\zeta_i} x(s)ds, \quad 0< \zeta_i < \eta_i <1, \; i=1,
2,\dots, (n-2),
\end{gathered}
\end{equation}
where $^cD^q$ denotes the
Caputo fractional derivative of order $q$,  and $F : [0,1] \times
\mathbb{R} \to {\mathcal{P}}(\mathbb{R})$ is a multivalued map,
${\mathcal{P}}(\mathbb{R})$ is the family of all subsets of
$\mathbb{R}$.

The aim here is to establish  existence results for the problem
\eqref{e1i}, when the right hand side is convex as well as
nonconvex valued.  In the first result (Theorem \ref{tcar}) we
consider the case when the right hand side has convex values, and
prove an existence result via Nonlinear alternative for Kakutani
maps.  In the second  result (Theorem \ref{tbc}), we shall combine
the nonlinear alternative of Leray-Schauder type for single-valued
maps with a selection theorem due to Bressan and Colombo for lower
semicontinuous multivalued maps with nonempty closed and
decomposable values, while in the third result (Theorem
\ref{tcn}), we shall use the fixed point theorem for contraction
multivalued maps due to Covitz and Nadler.

The methods used are standard, however their exposition in the
framework of problems \eqref{e1} and \eqref{e1i} is new.

\section{Preliminaries from fractional calculus}

Let us recall some basic definitions of fractional calculus
\cite{Kil, SKM}.

 \begin{definition} \rm
For function $g : [0,\infty) \to \mathbb{R}$,
at least $n$-times continuously differentiable, the Caputo derivative 
of fractional order $q$ is defined as
 $$
^cD^q g(t)=\frac{1}{\Gamma(n-q)}\int_{0}^t(t-s)^{n-q-1}g^{(n)}(s)ds, \quad
 n-1 < q < n, \; n=[q]+1,
$$
where $[q]$ denotes the integer part of the real number $q$.
\end{definition}

\begin{definition}\rm
The Riemann-Liouville fractional integral of order $q$ is defined as
$$
I^q g(t)=\frac{1}{\Gamma(q)}\int_0^t \frac{g(s)}{(t-s)^{1-q}}ds, \quad q>0,
$$
provided the integral exists.
\end{definition}

\begin{lemma} \label{l2}
For any $\sigma\in C([0,1], \mathbb{R})$, the unique solution of
the boundary value problem
\begin{equation}\label{e1a}
\begin{gathered}
 ^cD^qx(t) =\sigma(t), \quad  0<t<1,  \; m-1 < q \le m,\; m\ge2, \; m \in \mathbb{N}, \\
 x(0)=0, \quad x'(0)=0, \quad x''(0)=0,  \dots ,x^{(m-2)}(0)=0, \\
  x(1)=\sum_{i=1}^{n-2}\alpha_i
\int^{\eta_i}_{\zeta_i} x(s)ds, \quad 0< \zeta_i < \eta_i <1, \; i=1,
2, \dots, (n-2),
\end{gathered}
\end{equation}
is given by
\begin{equation}\label{e202}
\begin{split}
x(t)&= \int_0^t\frac{ (t-s)^{q-1}}{\Gamma(q)}\sigma(s)ds
 -\frac{mt^{m-1}}{\big(m-\sum_{i=1}^{n-2}\alpha_i
(\eta_i^m-\zeta_i^m)\big) }\\
&\quad\times \Big[\int_0^1\frac{(1-s)^{q-1}}{\Gamma(q)} \sigma(s)ds
-\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} \sigma(u)du\Big)ds\Big].
\end{split}
\end{equation}
\end{lemma}

\begin{proof}
 It is well known  \cite{Kil} that the general
solution of the fractional differential equation in \eqref{e1a}
can be written as
\begin{equation}\label{e203} 
x(t)= \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}\sigma(s)ds-c_0-c_1
t-c_2t^2- \dots -c_{m-1} t^{m-1}, 
\end{equation} 
where $c_0, c_1,
c_2, \dots ,c_{m-1}$ are arbitrary constants. Applying the
boundary conditions for the problem \eqref{e1a}, we find that
$c_0=0,\dots, c_{m-2}=0$,
 and
\begin{align*}
c_{m-1}
&  = \frac{m}{\big(m-\sum_{i=1}^{n-2}\alpha_i
(\eta_i^m-\zeta_i^m)\big) }\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds \\
&\quad -\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u,x(u))du\Big)ds\Big].
\end{align*}
Substituting the values of $c_0, c_1, c_2, \dots ,c_{m-1}$  in
\eqref{e203} yields the solution \eqref{e202}.\end{proof}



\section{Existence results - the single-valued  case}

Let ${\mathcal{C}}=C([0,1], \mathbb{R})$ denote the Banach space
of all continuous functions from $[0,1] \to \mathbb{R}$ endowed
with the  norm  defined by $\|x\|= \sup \{|x(t)|, t \in
[0,1]\}$.

In view of  Lemma \ref{l2}, we transform problem \eqref{e1} as
\begin{equation}\label{e301}
x=F (x).
\end{equation}
Here the operator  $F : {\mathcal{C}} \to
{\mathcal{C}}$ is defined by
 \begin{align*}
(F x)(t)
&=   \int_0^t\frac{ (t-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds
- \vartheta t^{m-1}\Big[ \int_0^1\frac{ (1-s)^{q-1}}{\Gamma(q)}
f(s,x(s))ds \\
&\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u,x(u))du\Big)ds\Big], \quad t \in
[0,1],
\end{align*}
where
$$
\vartheta=m\Big(m-\sum_{i=1}^{n-2}\alpha_i (\eta_i^m-\zeta_i^m)\Big)^{-1}.
$$
For convenience, let us set
\begin{equation}\label{e302}
 \Lambda=\frac{1}{\Gamma(q+1)}\Big[1+ |\vartheta| \Big\{1+\sum_{i=1}^{n-2}\frac{\alpha_i
(\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big\}\Big].
 \end{equation}

\subsection{Existence result via Banach's fixed point theorem}

\begin{theorem}\label{tb}
Assume that $f : [0,1]\times \mathbb{R} \to
 \mathbb{R}$  is a jointly continuous function and satisfies the
assumption
\begin{itemize}
\item[(A1)] $|f(t,x)-f(t,y)| \le L |x-y|$, for all
$t \in [0,1]$, $L>0$, $x, y\in \mathbb{R}$,
 \end{itemize}
  with $L  < 1/\Lambda$, where $\Lambda$ is given by
\eqref{e302}. Then the boundary value problem \eqref{e1} has a
unique solution.
\end{theorem}

\begin{proof}
Setting $\sup_{t \in [0,1]}|f(t,0)|=M$ and choosing
$ r \ge \frac{\Lambda M}{1-L\Lambda}$, we show that
$F B_r \subset B_r$, where $B_r=\{x \in {\mathcal{C}}: \|x\|\le r \}$.
For $x \in B_r$, we have
\begin{align*}
&\|(Fx)(t)\|\\
&\leq     \sup_{t \in [0, 1]}\Big\{\int_0^t
 \frac{(t-s)^{q-1} }{\Gamma(q)}|f(s,x(s))|ds 
 +|\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)}
|f(s,x(s))|ds \\ 
&\quad + \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}
|f(u,x(u))|du\Big)ds\Big]\Big\}
\\
 &\leq  \sup_{t \in [0,1]}\Big\{\frac{1}{\Gamma(q)}\int_0^t (t-s)^{q-1}
(|f(s,x(s))-f(s,0)|+|f(s,0)|)ds \\
&\quad + |\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)}
(|f(s,x(s))-f(s,0)|+|f(s,0)|)ds \\ 
&\quad + \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}
(|f(u,x(u))-f(u,0)|+|f(u,0)|)du\Big)ds\Big]\Big\}
\\
 &\leq  (Lr+M )\sup_{t \in [0,T]}
\Big\{\frac{1}{\Gamma(q)}\int_0^t (t-s)^{q-1} ds \\
&\quad + |\vartheta| t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} ds +
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} du\Big)ds\Big]\Big\}
\\
&\leq  \frac{(Lr+M)}{\Gamma(q+1)} \Big[1+ |\vartheta|
 \Big\{1+\sum_{i=1}^{n-2}\frac{\alpha_i
(\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big\}\Big]
\\
&= \left(Lr+M \right)\Lambda \le r.
\end{align*}
Now, for $x, y \in {\mathcal{C}}$ and for each $t \in [0,1]$, we
obtain
\begin{align*}
&\|(F x)(t)-(F y)(t)\|\\
&\leq  \sup_{t \in [0,
1]}\Big\{\int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
|f(s,x(s))-f(s,y(s))|ds \\
 &\quad + |\vartheta|
t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)}
|f(s,x(s))-f(s,y(s))|ds \\
& \quad + \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}
|f(u,x(u))-f(u,y(u))|du\Big)ds\Big]\Big\}
\\
&\leq  L\|x-y\|\sup_{t \in [0,
1]}\Big\{\frac{1}{\Gamma(q)}\int_0^t (t-s)^{q-1} ds \\
&\quad + |\vartheta| t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} ds +
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} du\Big)ds\Big]\Big\}\\
&\leq \frac{L}{\Gamma(q+1)}\Big[1+ |\vartheta|
\Big\{1+\sum_{i=1}^{n-2}\frac{\alpha_i
(\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big\}\Big]\|x-y\| =L
\Lambda \|x-y\|,
\end{align*}
where $\Lambda$ is given by \eqref{e302}. Observe that  $\Lambda$
depends only on the parameters involved in the problem. As  $L <
1/\Lambda$, therefore $F$ is a contraction. Thus, the conclusion
of the theorem follows by the contraction mapping principle
(Banach fixed point theorem).\end{proof}


\subsection{Existence result via  Krasnoselskii's fixed point theorem}

\begin{lemma}[Krasnoselskii's fixed point
theorem \cite{MAK}] \label{lk}
 Let $M$ be a bounded, closed, convex, and nonempty
subset of a Banach space $X$. Let $A, B$ be the operators such
that:
\begin{itemize}
\item[(i)] $Ax+By \in M$ whenever $x, y \in M$;
\item[(ii)] $A$ is compact and continuous;
\item[(iii)] $B$ is a contraction mapping.
\end{itemize}
Then there exists $z \in M$ such that $z=Az+Bz$.
\end{lemma}

\begin{theorem}\label{tk}
Let $f : [0,1]\times \mathbb{R} \to \mathbb{R}$ be a
jointly continuous function satisfying the assumption
 {\rm  (A1)}. Moreover we assume that
\begin{itemize}
\item[(A2)] $|f(t,x)|\le \mu (t)$,  for all $(t,x) \in [0,1]
\times \mathbb{R}$, and $\mu \in C([0,1], \mathbb{R}^+)$.
 \end{itemize}
 If
\begin{equation}\label{e4}
\frac{L|\vartheta|}{\Gamma(q+1)}\Big(
1+\sum_{i=1}^{n-2}\frac{\alpha_i
(\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big)<1,
 \end{equation}
 then the boundary value problem \eqref{e1} has  at least one solution on $[0,1]$.
\end{theorem}

\begin{proof}
By the assumption (A2), we can fix
$$
\overline{r} \ge \frac{|\vartheta|\|\mu\|}{\Gamma(q+1)}\Big(
1+\sum_{i=1}^{n-2}\frac{\alpha_i
(\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big),
$$
and consider
$B_{\overline{r}}=\{x \in {\mathcal{C}}: \|x\|\le \overline{r} \}$. We
define the operators ${\mathcal{P}}$ and ${\mathcal{Q}}$ on
$B_{\overline{r}}$ as
\begin{gather*}
({\mathcal{P}} x)(t)= \int_{0}^t\frac{(t-s)^{q-1}}{\Gamma(q)}f(s,u(s))ds,
\quad t\in [0,1],\\
\begin{aligned}
({\mathcal{Q}} x)(t)
&=   - \vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds \\
&\quad -\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}
f(u,x(u))du\Big)ds\Big],\quad t\in [0,1].
\end{aligned}
\end{gather*}
 For $x, y \in B_{\overline{r}}$, we find that
$$
\|{\mathcal{P}} x+{\mathcal{Q}} y\|
\le  \frac{|\vartheta|\|\mu\|}{\Gamma(q+1)}\Big(
1+\sum_{i=1}^{n-2}\frac{\alpha_i
(\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\Big) \le \overline{r}.
$$
Thus, ${\mathcal{P}} x+{\mathcal{Q}} y \in B_{\overline{r}}$. It follows from
the assumption (A1) together with \eqref{e4} that ${\mathcal{Q}}$
is a contraction mapping. Continuity of $f$ implies that the
operator ${\mathcal{P}}$ is continuous. Also, ${\mathcal{P}}$ is uniformly
bounded on $B_{\overline{r}}$ as
$$
\|{\mathcal{P}} x\| \le \frac{\|\mu\|}{\Gamma(q+1)}.
$$
Now we prove the compactness of the operator ${\mathcal{P}}$. In view
of (A1), we define
$$
\sup_{(t,x) \in [0,1] \times B_{\overline{r}}}|f(t,x)|=\overline{f}.
$$
Consequently we have
\begin{align*}
&\|({\mathcal{P}} x)(t_1)-({\mathcal{P}} x)(t_2)\|\\
&=   \big\|\frac{1}{\Gamma(q)}\int_0^{t_1}[(t_2-s)^{q-1}-(t_1-s)^{q-1}]f(s,x(s))ds
+ \int_{t_1}^{t_2}(t_2-s)^{q-1}f(s,x(s))ds\big\|\\
&\leq  \frac{\overline{f}}{\Gamma(q+1)}|2(t_2-t_1)^q+t_1^q-t_2^q|,
\end{align*}
which is independent of $x$ and tends to zero as $t_2 \to t_1$.
Thus, ${\mathcal{P}}$ is relatively compact on $B_{\overline{r}}$.
Hence, by the Arzel\'a-Ascoli Theorem, ${\mathcal{P}}$ is compact on
$B_{\overline{r}}$. Thus all the assumptions of Lemma \ref{lk}
are satisfied. So by the conclusion of Lemma \ref{lk}, problem
$\eqref{e1}$ has  at least one solution on $[0,1]$.
\end{proof}

 \subsection{Existence result via  Leray-Schauder Alternative}

\begin{lemma}[Nonlinear alternative for single valued maps \cite{GrDu}] \label{lls}.
Let $E$ be a Banach space,
$C$ a closed, convex subset of $E$, $U$ an open subset of $C$ and
$0\in U$. Suppose that $F:\overline{U}\to C$ is a continuous,
compact (that is, $F(\overline{U})$ is a relatively compact subset
of $C$) map. Then either
\begin{itemize}
\item[(i)] $F$ has a fixed point in $\overline{U}$, or \item[(ii)]
there is a $u\in \partial U$ (the boundary of $U$ in $C$) and
$\lambda\in(0,1)$ with $u=\lambda F(u)$.
\end{itemize}
\end{lemma}

\begin{theorem}\label{tls}
Let $f: [0,1]\times \mathbb{R} \to \mathbb{R}$ be a  jointly
continuous function.  Assume that:
\begin{itemize}
 \item[(A3)] There exist a function   $p \in
L^1([0,1], \mathbb{R}^+)$, and a nondecreasing function $\psi:
{\mathbb{R}}^+\to { \mathbb{R}}^+$  such that
 $|f(t,x)|\le p (t)\psi(\|x\|)$,  for all $(t,x) \in [0,1] \times \mathbb{R}$.

 \item[(A4)] There exists a constant $M>0$ such that
$$
\frac{M}{\frac{\psi(M)}{\Gamma(q)}\big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i
\frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\big]}> 1.
$$
 \end{itemize}
  Then the boundary value problem \eqref{e1} has  at least one solution on $[0,1]$.
\end{theorem}

\begin{proof}
Consider the operator  $F :  \mathcal{C} \to  \mathcal{C}$ with
$ x=F (x)$,
 where
  \begin{align*}
(F x)(t)&=  \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds -
\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)}
f(s,x(s))ds \\
&\quad  - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u,x(u))du\Big)ds\Big], \quad t \in
[0,1].
\end{align*}
We show that $F$ maps bounded sets into bounded sets in
$ C([0,1], \mathbb{R})$. For a positive number $r$, let
 $B_r = \{x \in C([0,1], \mathbb{R}): \|x\| \le r \}$ be a bounded set in
$C([0,1], \mathbb{R})$. Then
\begin{align*}
|(Fx)(t)| 
&\leq \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
|f(s,x(s))|ds+ |\vartheta| t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} |f(s,x(s))|ds \\
&\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} |f(u,x(u))|du\Big)ds\Big]
\\
&\leq  \int_0^t \frac{(t-s)^{q-1} }{\Gamma(q)}  p(s)\psi(\|x\|)ds 
+|\vartheta| t^{m-1}\int_0^1  \frac{(1-s)^{q-1}}{\Gamma(q)}  p(s)\psi(\|x\|)ds\\
 &\quad +   |\vartheta| t^{m-1}
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} p(s)\psi(\|x\|)du\Big)ds\\
&\leq    \frac{\psi(\|x\|)}{\Gamma(q)}\int_0^1
(t-s)^{q-1}  p(s)ds + \frac{|\vartheta|\psi(\|x\|)}{\Gamma(q)} 
\int_0^1 (1-s)^{q-1}p(s)ds \\
 &\quad + \frac{|\vartheta|\psi(\|x\|)}{
\Gamma(q)}\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
(s-u)^{q-1} p(u)du\Big)ds\\
&\leq    \frac{\psi(\|x\|)}{\Gamma(q)}\int_0^1
(t-s)^{q-1}  p(s)ds+   \frac{|\vartheta|\psi(\|x\|)}{\Gamma(q)} 
\int_0^1 (1-s)^{q-1}p(s)ds \\
 &\quad + \frac{|\vartheta|\psi(\|x\|)}{
\Gamma(q)}\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
(s-u)^{q-1} p(u)du\Big)ds\\
&\leq  \frac{\psi(\|x\|)}{\Gamma(q)}\int_0^1
  p(s)ds + \frac{|\vartheta|\psi(\|x\|)}{\Gamma(q)} \int_0^1  p(s)ds
   \\
&\quad + \frac{|\vartheta|\psi(\|x\|)}{
\Gamma(q)}\sum_{i=1}^{n-2}\alpha_i   \frac{\eta_i^q-\zeta_i^q}{q}
 \int_{\zeta_i}^{\eta_i}p(s)ds\\
&=  \frac{\psi(\|x\|)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i   \frac{\eta_i^q
-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\Big].
\end{align*}
Consequently,
$$
\|Fx\|\le  \frac{\psi(r)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i
\frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\Big].
$$

Next we show that $F$ maps bounded sets into equicontinuous sets
of $ C([0,1], \mathbb{R})$. Let $t', t'' \in [0,1]$ with $t'< t''$
and  $x \in B_r$, where $B_r$ is a bounded set of $C([0,1],
\mathbb{R})$. Then we obtain
\begin{align*}
&|(Fx)(t'')-(Fx)(t')|\\
&= \Big| \frac{1}{\Gamma(q)}\int_0^{t''} (t''-s)^{q-1} f(s,x(s))ds
-\frac{1}{\Gamma(q)}\int_0^{t'} (t'-s)^{q-1}  f(s,x(s))ds \\
&\quad - \vartheta \Big((t'')^{m-1}-(t')^{m-1}\Big)\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)}  f(s,x(s))ds\\
&\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}  f(u,x(u))du\Big)ds\Big]\Big|\\
&\leq \Big| \frac{1}{\Gamma(q)}\int_0^{t'}[
(t''-s)^{q-1}-(t'-s)^{q-1}] \psi(r)p(s)ds\Big|\\
&\quad +\Big|\frac{1}{\Gamma(q)}\int_{t'}^{t''} (t''-s)^{q-1}
 \psi(r)p(s)ds\Big|\\
&\quad +\Big|\vartheta \Big((t'')^{m-1}-(t')^{m-1}\Big)\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} \psi(r)p(s)ds\\
&\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}
\psi(r)p(u)du\Big)ds\Big]\Big|.
\end{align*}
Obviously the right hand side of the above inequality tends to
zero independently of $x \in B_{r'}$ as $t''- t' \to 0$. As $F$
satisfies the above   assumptions, therefore it follows by the
Arzel\'a-Ascoli  theorem that $F: C([0,1], \mathbb{R}) \to
C([0,1], \mathbb{R})$ is completely continuous.

The result will follow from the Leray-Schauder nonlinear
alternative (Lemma \ref{lls}) once we have proved the boundendness
of the set of all solutions to equations $x=\lambda F x$ for
$\lambda\in [0,1]$.

Let $x$ be a solution. Then, for $t\in [0,1]$, and using the
computations in proving that $F$ is bounded,  we have
\begin{align*}
|x(t)|
&= |\lambda (F x)(t)| \\
&\leq \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} |f(s,x(s))|ds+
|\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)}
|f(s,x(s))|ds \\
&\quad  -
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} |f(u,x(u))|du\Big)ds\Big]
\\
 &\leq  \frac{\psi(\|x\|)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i
\frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\Big].
\end{align*}
Consequently, we have
$$
\frac{\|x\|}{
\frac{\psi(\|x\|)}{\Gamma(q)}\big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i
 \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\big]}\le 1.
$$
In view of (A4), there exists $M$ such that $\|x\| \ne M$. Let
us set
$$
U = \{x \in  C([0,1], \mathbb{R}):\|x\| < M\}.
$$
 Note that the operator $F:\overline{U} \to  C([0,1], \mathbb{R})$ is
 continuous and completely continuous. From the choice of $U$,
there is no $x \in \partial U$ such that $x =\lambda F(x)$ for
some $\lambda \in (0,1)$. Consequently, by the nonlinear
alternative of Leray-Schauder type (Lemma \ref{lls}), we deduce
that $F$ has a fixed point $x \in \overline{U}$ which is a
solution of the problem \eqref{e1}. This completes the
proof.
\end{proof}


\subsection{Existence result via Leray-Schauder degree theory}


\begin{theorem}\label{tlsd} Let $f: [0,1]\times \mathbb{R} \to
\mathbb{R}$. Assume that there exist constants $0 \le \kappa <
\frac{1}{\Lambda}$, where $\Lambda$ is given by \eqref{e302} and
$M>0$ such that $|f(t,x)| \le \kappa \|x\| +M$ for all $t \in
[0,1], x \in \mathbb{R}$. Then the  boundary value problem
\eqref{e1} has at least one solution.
\end{theorem}

\begin{proof}
In view of the fixed point problem \eqref{e301}, we
just need to prove the existence of at least one solution $x \in
\mathbb{R}$ satisfying \eqref{e301}. Define a suitable ball $B_R
\subset C[0,1]$ with radius $R>0$ as
$$
B_R=\{x \in \mathcal{C} : \|x\|<R\},
 $$
where $R$ will be fixed later. Then, it is sufficient to show that
$F : \overline{B}_R \to \mathcal{C}$ satisfies
\begin{equation}\label{e304}
x \ne \lambda F x, \quad  \forall x \in \partial B_R \quad \text{and} \quad
\forall \lambda \in [0,1].
\end{equation}
Let us set
$$
H(\lambda, x)=\lambda F x, \quad x \in \mathcal{C},   \quad \lambda \in [0,1].
$$
Then, by the Arzel\'a-Ascoli Theorem, $h_\lambda(x)=x-H(\lambda,
x)=x- \lambda F x$ is completely continuous. If \eqref{e304} is
true, then the following Leray-Schauder degrees are well defined
and by the homotopy invariance of topological degree, it follows
that
\begin{align*}
\deg(h_\lambda, B_R, 0)
&= \deg(I-\lambda F, B_R, 0)=\deg(h_1, B_R, 0)\\
&= \deg(h_0, B_R, 0) =\deg(I, B_R, 0)=1\ne 0, \quad  0 \in B_R,
\end{align*}
 where  $I$ denotes the unit operator. By the
nonzero property of Leray-Schauder degree,
$h_1(t)=x- \lambda F x=0$ for at least one $x \in B_R$.
To prove \eqref{e304},
we assume that $x = \lambda F x$ for some $\lambda \in [0,1]$ and
for  all $t \in [0,1]$ so that
\begin{align*}
|x(t)| &= |\lambda (F x)(t)| \\
&\leq \int_0^t  \frac{(t-s)^{q-1}}{\Gamma(q)} |f(s,x(s))|ds  +
|\vartheta| t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)}
|f(s,x(s))|ds \\
&\quad+
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} |f(u,x(u))|du\Big)ds\Big]
\\
&\leq  (\kappa \|x\| +M)
\Big\{\frac{1}{\Gamma(q)}\int_0^t (t-s)^{q-1} ds
\\
& \quad + |\vartheta| t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} ds +
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} du\Big)ds\Big]\Big\}\\
 &\leq \frac{(\kappa \|x\| +M)}{\Gamma(q+1)}
\Big[1+ |\vartheta| \big\{1+\sum_{i=1}^{n-2}\frac{\alpha_i
(\eta_i^{q+1}-\zeta_i^{q+1})}{q+1}\big\}\Big]
 \\
 &=  (\kappa \|x\| +M)\Lambda,
\end{align*}
which, on taking norm ($\sup_{t\in[0,1]}|x(t)|=\|x\|$) and solving
for $\|x\|$, yields
$$
\| x \|\le \frac{M \Lambda}{1-\kappa \Lambda}.
$$
Letting $R= \frac{M \Lambda}{1-\kappa \Lambda}+1$,
\eqref{e304} holds. This completes the proof.
\end{proof}

\subsection{Existence result via  nonlinear contractions}

\begin{definition} \rm
Let $E$ be a Banach space and let $F: E\to E$ be a mapping. $F$ is
said to be a nonlinear contraction if there exists a continuous
nondecrasing function $\Psi: \mathbb{R}^+\to \mathbb{R}^+$ such
that $\Psi(0)=0$ and $\Psi(\xi)<\xi$ for all $\xi>0$ with the
property:
$$
\|Fx-Fy\|\le \Psi(\|x-y\|), \quad \forall x,y\in E.
$$
\end{definition}

\begin{lemma}[Boyd and Wong \cite{BW}]\label{BW}
Let $E$ be a Banach space and let $F: E\to E$ be a nonlinear
contraction. Then $F$ has a unique fixed point in $E$.
\end{lemma}

\begin{theorem}\label{tnc}
Assume that:
\begin{itemize}
\item[(A5)]  $|f(t,x)-f(t,y)|\le
h(t)\frac{|x-y|}{H^*+|x-y|}$, $t\in [0,1]$,
$x,y\ge 0$,
where $h:[0,1]\to \mathbb{R}^+$ is continuous and
\begin{equation}\label{e3004}
\begin{aligned}
 H^*&=  \int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} h(s)ds +
|\vartheta|\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} h(s)ds \\
&\quad +\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} h(u)du \Big)ds\Big].
\end{aligned}
\end{equation}
\end{itemize}
 Then the boundary value problem \eqref{e1}  has a
unique solution.
\end{theorem}

\begin{proof}   We define the operator $F:  \mathcal{C}\to \mathcal{C}$ by
\begin{align*}
Fx(t) &=  \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f(s,x(s))ds +
\vartheta t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)}
f(s,x(s))ds \\
&\quad +
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u,x(u))du\Big)ds\Big], \quad t \in [0,1].
\end{align*}

 Let a continuous nondecreasing function $\Psi: \mathbb{R}^+\to \mathbb{R}^+$ satisfying $\Psi(0)=0$ and $\Psi(\xi)<\xi$ for all $\xi>0$ be defined by
$$
\Psi(\xi)=\frac{H^*\xi}{H^*+\xi},\quad \forall \xi\ge 0.
$$
Let $x, y\in \mathcal{C}$. Then
$$
|f(s,x(s))-f(s,y(s))|\le \frac{h(s)}{H^*}\Psi(\|x-y\|)
$$
so that
\begin{align*}
&|Fx(t)-Fy(t)| \\
&\leq  \int_0^t  \frac{(1-s)^{q-1}}{\Gamma(q)}
h(s)\frac{|x(s)-y(s)|}{H^*+|x(s)-y(s)|}ds \\
&\quad + |\vartheta|  \int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)}  h(s)\frac{|x(s)-y(s)|}{H^*+|x(s)-y(s)|}ds  \\
&\quad + |\vartheta|
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}
h(m)\frac{|x(m)-y(m)|}{H^*+|x(m)-y(m)|}dm \Big)ds,
\end{align*}
for $t \in [0,1]$. In view of \eqref{e3004}, it follows that
$\|Fx-Fy\|\le \Psi(\|x-y\|)$ and hence $F$ is a nonlinear
contraction. Thus,  by Lemma \ref{BW}, the operator $F$ has a
unique fixed point in $\mathcal{C}$, which in turn is a unique
solution of problem \eqref{e1}.
\end{proof}

\subsection{Examples}
For the forthcoming examples, we consider the following boundary
conditions:
\begin{equation}\label{bc}
 x(0)=0, \quad x'(0)=0, \quad x''(0)=0, \quad
x(1)=\sum_{i=1}^{3}\alpha_i \int^{\eta_i}_{\zeta_i} x(s)ds,
\end{equation}
where $\zeta_1=1/16$, $\zeta_2=5/16$, $\zeta_3=9/16$,
 $\eta_1=1/4$, $\eta_2=1/2$, $\eta_3=3/4$, $\alpha_1=1/3$, $\alpha_2=2/3$,
$\alpha_3=1$.

\begin{example} \label{examp3.10}\rm
Consider the fractional differential equation
\begin{equation}\label{ex1}
 ^cD^{7/2}x(t) =L\big( \cos t +\tan^{-1}x(t)\big), \quad  0<t<1,
\end{equation}
subject to the strip boundary conditions \eqref{bc}.
\end{example}

Here,  $q=7/2$, $m=4$ and $f(t,x)= L\big( \cos t +\tan^{-1}x(t)\big)$. Clearly
\begin{gather*}  
|f(t,x)-f(t,y)|  \le  L |\tan^{-1}x-\tan^{-1}y | \le L|x-y|,
 \\
\vartheta = m \Big(m-\sum_{i=1}^{n-2}\alpha_i
(\eta_i^m-\zeta_i^m)\Big)^{-1}=1.0675, 
\\
\Lambda = \frac{1}{\Gamma(q+1)}\Big[1+ |\vartheta| \Big
\{1+\sum_{i=1}^{3}\frac{\alpha_i
(\eta_i^{q+1}-\zeta_i^{q+1})}{(q+1)}\Big \}\Big] \approx
\frac{34}{105\sqrt{\pi}}.
\end{gather*}
With $L< \frac{1}{\Lambda} \approx 105\sqrt{\pi}/34$,
all the assumptions of Theorem \ref{tb} are satisfied. Hence,
there exists a unique solution for problem \eqref{ex1}-\eqref{bc}
on $[0,1]$.

\begin{example} \label{e311} \rm
Consider the equation
\begin{equation}\label{ex2}
 ^cD^{7/2}x(t) =\frac{1}{4 \pi}\sin(2 \pi x)+\frac{x^2}{1+x^2}, \quad  0<t<1,
\end{equation}
subject to the strip boundary conditions \eqref{bc}.
\end{example}

Here,  $q=7/2$, $m=4$ and $ f(t,x)=\frac{1}{4 \pi}\sin(2\pi
x)+\frac{x^2}{1+x^2}$. Observe that
$$
|f(t,x)|=|\frac{1}{4 \pi}\sin(2 \pi x)+\frac{x^2}{1+x^2}| 
\le \frac{1}{2}\|x\|+1.
$$
with $\kappa=\frac{1}{2}<\frac{1}{\Lambda} \approx
105\sqrt{\pi}/34$  and $M=1$. Thus, the conclusion of Theorem
\ref{tlsd} applies to  problem \eqref{ex2}-\eqref{bc}. 

\begin{example} \label{examp3.12} \rm
Let us consider the fractional differential equation
\begin{equation}\label{ex3}
 ^cD^{7/2}x(t) =\frac{t|x|}{1+|x|}, \quad  0<t<1,
\end{equation}
subject to the strip boundary conditions \eqref{bc}.
\end{example}
Here,  $q=7/2$, $m=4$ and $f(t,x)=\frac{t|x|}{1+|x|}$.
 We choose $h(t)=(1+t)$ and find that
\begin{align*}
H^*&=   \int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} h(s)ds +
|\vartheta|\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} h(s)ds \\
&\quad +
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} h(u)du \Big)ds\Big]=0.222818.
\end{align*}
Clearly,
\begin{equation*} 
 |f(t,x)-f(t,y)|  = |\frac{t(|x|-|y|)}{1+|x|+|y|+|x||y|}|
 \le \frac{(1+t)|x-y|}{0.222818+|x-y|}. 
\end{equation*}
 Thus, the conclusion of Theorem
\ref{tnc} applies and   problem \eqref{ex3}-\eqref{bc} has a unique solution. 

\section{Existence results - the multi-valued case}

\begin{definition}  \rm
A function $x \in C([0,1], \mathbb{R})$ with its Caputo derivative of order $q$
existing on $[0, 1]$ is a solution of the problem \eqref{e1i} if
there   exists a function $f \in L^1([0,1],\mathbb{R})$ such that
$f(t) \in F(t, x(t))$ a.e. on $[0,1]$ and
\begin{align*}
  x(t)&=   \int_0^t\frac{ (t-s)^{q-1}}{\Gamma(q)}
f(s)ds - \vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds \\
&\quad  -
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big].
\end{align*}
 \end{definition}

 \subsection{The Carath\'eodory case}

In this subsection, we are concerned with the existence of
solutions for the problem \eqref{e1i} when the right hand side has
convex values. We first recall some preliminary facts.
For a normed space 
$(X, \|\cdot\|)$, let 
\begin{gather*}
P_{cl}(X)=\{Y \in {\mathcal{P}}(X) : Y  \text{ is  closed}\},\\ 
P_{b}(X)=\{Y \in {\mathcal{P}}(X) : Y  \text{ is  bounded}\},\\
P_{cp}(X)=\{Y \in {\mathcal{P}}(X) : Y \text{ is  compact}\}, \\ 
P_{cp, c}(X)=\{Y \in {\mathcal{P}}(X) : Y  \text{ is compact and
 convex}\}.
\end{gather*}

 \begin{definition} \label{def4.2}\rm
 A multi-valued map
$G : X \to {\mathcal{P}}(X):$
\begin{itemize}
\item[(i)] is convex (closed) valued if $G(x)$ is convex (closed)
for all $x \in X$;
 
\item[(ii)]  is bounded on bounded sets if
$G(\mathbb{B}) = \cup_{x \in \mathbb{B}}G(x)$ is bounded in $X$
for all $\mathbb{B} \in P_{b}(X)$  (i.e. $\sup_{x \in
\mathbb{B}}\{\sup \{|y| : y \in G(x)\}\} < \infty )$;

\item[(iii)] 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 $\mathcal{N}_0$ of $x_0$ such that
$G(\mathcal{N}_0) \subseteq N$;

\item[(v)]    is said to be completely continuous if $G(\mathbb{B})$ 
is relatively compact for every $\mathbb{B} \in P_{b}(X)$;

\item[(v)]   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 ${\it{Fix}}
G$.
\end{itemize}
\end{definition}

\begin{remark} \label{rmk4.3} \rm
  It is known that, if the multi-valued 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 $y_* \in G(x_*)$.
\end{remark}

\begin{definition} \label{def4.4} \rm
 A multivalued map $G : [0;1] \to P_{cl}(\mathbb{R})$ 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.
 \end{definition}

\begin{definition}\label{def4.5} \rm
 A multivalued map $F : [0,1]
\times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})$ is called
Carath\'{e}odory if
\begin{itemize}
 \item[(i)] $t  \mapsto F(t,x)$ is measurable for each
$x \in \mathbb{R}$;

 \item[(ii)] $x  \mapsto F(t,x)$ is
upper semicontinuous for almost all $t\in [0,1]$;

\end{itemize}
Further a Carath\'{e}odory function $F$ is called
$L^1-$Carath\'{e}odory if
\begin{itemize}
\item[(iii)] for each $\alpha > 0$, there exists
$\varphi_{\alpha} \in L^1([0,1],\mathbb{R}^+)$ such that 
$$
\|F (t, x)\| = \sup \{|v| : v \in F (t, x)\} \le \varphi_{\alpha} (t)
$$
for all  $\|x\| \le \alpha$ and for a. e.
$t \in [0,1]$. 
\end{itemize}
\end{definition}

For each $y \in C([0,1], \mathbb{R})$, define the set of
selections of $F$ by 
$$
S_{F,y} := \{ v \in L^1([0,1],\mathbb{R}) :
v (t) \in F (t, y(t)) ~\text{for a.e.} ~t \in [0,1]\}.
$$
The consideration  of this subsection is based on the following
lemmas.

\begin{lemma}[Nonlinear alternative for Kakutani maps \cite{GrDu}] \label{NAK}
Let $E$ be a Banach space, $C$ a   closed convex subset of $E$, $U$ an open 
subset of  $C$ and  $0\in U$. Suppose that 
$F: \overline{U}\to \mathcal{P}_{c,cv}(C)$ is a upper semicontinuous compact map;
here $\mathcal{P}_{c,cv}(C)$ denotes the family of nonempty, compact
convex subsets of $C$. Then either
\begin{itemize}
\item[(i)] $F$ has a fixed point in $\overline{U}$, or
\item[(ii)] there is a $u\in \partial U$  and $\lambda\in(0,1)$ with $u\in
\lambda F(u)$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{LaOp}] \label{l1i}  
Let $X$ be a Banach space. Let
$F : [0, 1] \times \mathbb{R} \to \mathcal{P}_{cp,c}(\mathbb{R})$
be an $L^1-$ Carath\'{e}odory multivalued map and let $\Theta$ be
a linear continuous mapping from $L^1([0,1],\mathbb{R})$ to
$C([0,1],\mathbb{R})$.  Then the operator 
$$
\Theta \circ S_F : C([0,1],\mathbb{R}) \to P_{cp,c} (C([0,1],\mathbb{R})),
 \quad  x \mapsto (\Theta \circ S_F) (x) = \Theta( S_{F,x})
$$ 
is a closed graph operator in $C([0,1],\mathbb{R}) \times C([0,1],\mathbb{R})$.
\end{lemma}

\begin{theorem}\label{tcar} 
 Assume that {\rm (A4)} holds. In addition we assume that:
\begin{itemize}
\item[(H1)] $F : [0,1] \times \mathbb{R} \to
{\mathcal{P}}(\mathbb{R})$ is Carath\'{e}odory and has nonempty
compact convex values; 

\item[(H2)] there exists a continuous
nondecreasing function $\psi : [0,\infty) \to (0,\infty)$ and a
function $p \in L^1([0,1],\mathbb{R}^+)$ such that
$$
\|F(t,x)\|_\mathcal{P}:=\sup\{|y|: y \in F(t,x)\}\le p(t)\psi(\|x\|) 
\quad  \text{for each} ~(t,x) \in [0,1] \times \mathbb{R}.
$$
\end{itemize} 
Then the boundary value problem \eqref{e1i} has at least one solution on
$[0,1]$.
\end{theorem}

 \begin{proof} 
Define an operator 
$\Omega: C([0,1],  \mathbb{R})\to \mathcal{P}(C([0,1], \mathbb{R}))$ by
\begin{align*}
&\Omega(x)\\
&=\Big\{ h \in C([0,1], \mathbb{R}): 
h(t) =  \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} f(s)ds
 -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds \\
&\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big],
\; 0\le t\le 1 \Big\},
\end{align*}
for $f \in S_{F,x}$. We will show that $\Omega$ satisfies the
assumptions of the nonlinear alternative of Leray-Schauder type.
The proof consists of several steps.  As a first step, we show
that $\Omega$ is convex for each $x \in C([0,1], \mathbb{R})$. For
that, let $h_1, h_2 \in \Omega(x)$. Then there exist
 $f_1, f_2 \in S_{F,x}$ such that for each $t \in [0,1]$, we have
\begin{align*}
  h_i(t) 
&=   \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
f_i(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f_i(s)ds\\
 &\quad  -
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f_i(u)du\Big)ds\Big],   \quad
i=1,2.
\end{align*}
Let $0 \le \omega \le 1$. Then, for each $t \in [0,1]$, we have
\begin{align*}
&[\omega h_1+(1-\omega)h_2](t)\\
&=   \int_0^t \frac{(t-s)^{q-1} }{\Gamma(q)} [\omega f_1(s)+(1-\omega)f_2(s)]ds\\
&\quad -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} [\omega f_1(s)+(1-\omega)f_2(s)]ds \\
 &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}
 [\omega f_1(s)+(1-\omega)f_2(s)]du\Big)ds\Big].
\end{align*}
Since $S_{F,x}$ is convex ($F$ has convex values), therefore it
follows that $\omega h_1+(1-\omega)h_2 \in \Omega(x)$.

Next, we show that $\Omega$ maps bounded sets into bounded sets in
$ C([0,1], \mathbb{R})$. For a positive number $r$, let 
$B_r = \{x\in C([0,1], \mathbb{R}): \|x\| \le r \}$ be a bounded set in
$C([0,1], \mathbb{R})$. Then, for each 
$h \in \Omega (x), x \in B_r$, there exists  $f \in S_{F,x}$ such that
\begin{align*}
  h(t)&=    \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
f(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds\\
 &\quad  - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big].
\end{align*}
Then, as in Theorem \ref{tls},
\begin{align*}
  |h(t)|
&\leq   \int_0^t
\frac{(t-s)^{q-1} }{\Gamma(q)} |f(s)|ds +|\vartheta| t^{m-1}\int_0^1  \frac{(1-s)^{q-1}}{\Gamma(q)}   |f(s)|ds \\
 &\quad +  |\vartheta| t^{m-1}
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} |f(u)|du\Big)ds\\
  &\leq  \frac{\psi(\|x\|)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i   
\frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\Big].
\end{align*}
Thus,
$$
 \|h\| \le \frac{\psi(r)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i   
\frac{\eta_1^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\Big].
$$
Now we show that $\Omega$ maps bounded sets into equicontinuous
sets of $ C([0,1], \mathbb{R})$. Let $t', t'' \in [0,1]$ with
 $t'< t''$ and  $x \in B_r$, where $B_r$ is a bounded set of $C([0,1],
\mathbb{R})$. For each  $h \in \Omega(x)$, we obtain
\begin{align*}
 |h(t'')-h(t')|
&= \Big| \frac{1}{\Gamma(q)}\int_0^{t''} (t''-s)^{q-1} f(s)ds
 -\frac{1}{\Gamma(q)}\int_0^{t'} (t'-s)^{q-1} f(s)ds \\
&\quad - \vartheta \Big((t'')^{m-1}-(t')^{m-1}\Big)
\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds \\
&\quad - 
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big]\Big|
\\
&\leq \Big| \frac{1}{\Gamma(q)}\int_0^{t'}[
(t''-s)^{q-1}-(t'-s)^{q-1}] \psi(r)p(s)ds\Big|\\
&\quad +\Big|\frac{1}{\Gamma(q)}\int_{t'}^{t''} (t''-s)^{q-1}
 \psi(r)p(s)ds\Big|\\
&\quad +\Big|\vartheta \Big((t'')^{m-1}-(t')^{m-1}\Big)\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} \psi(r)p(s)ds\\
&\quad 
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}
\psi(r)p(u)du\Big)ds\Big]\Big|.
\end{align*}
Obviously the right hand side of the above inequality tends to
zero independently of $x \in B_{r'}$ as $t''- t' \to 0$. As
$\Omega$ satisfies the above three assumptions, therefore it
follows by the Arzel\'a-Ascoli  theorem that $\Omega: C([0,1],
\mathbb{R}) \to {\mathcal{P}}(C([0,1], \mathbb{R}))$ is completely
continuous. \\
In our next step, we show that $\Omega$ has a closed graph. Let
$x_n \to x_*, h_n \in \Omega (x_n)$ and $h_n \to  h_*$. Then we
need to show that $h_* \in  \Omega (x_*)$. Associated with $h_n
\in \Omega(x_n)$, there exists $f_n \in S_{F,x_n}$ such that for
each $t \in [0,1]$,
\begin{align*}
  h_n(t)
&=    \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
f_n(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f_n(s)ds\\
 &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f_n(u)du\Big)ds\Big].
\end{align*}
Thus we have to show that there exists $f_* \in  S_{F,x_*}$ such
that for each $t \in [0,1]$,
\begin{align*}
  h_*(t)
&=    \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
f_*(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f_*(s)ds \\
 &\quad  -
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f_*(u)du\Big)ds\Big].
\end{align*}
 Let us consider the continuous linear operator
$\Theta : L^1([0,1], \mathbb{R}) \to C([0,1], \mathbb{R})$ given
by
\begin{align*} f \mapsto \Theta(f)
&=   \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
f(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds \\
 &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big].
\end{align*}
 Observe that
 \begin{align*}
&\| h_n(t)-h_*(t)\|\\
&= \Big\| \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}(f_n(s)-f_*(s))ds 
 - \vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} (f_n(s)-f_*(s))ds \\
 &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)}
(f_n(u)-f_*(u))du\Big)ds\Big]\Big\|\to 0,
\end{align*}
 as  $n \to \infty$.
 Thus, it follows by Lemma \ref{l1i} that $\Theta \circ S_F$ is a closed
graph operator. Further, we have $h_n(t) \in \Theta(S_{F,x_n})$.
Since  $x_n \to x_*$, therefore, we have
\begin{align*}
  h_*(t)
&=    \int_0^t \frac{ (t-s)^{q-1}}{\Gamma(q)}
f_*(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f_*(s)ds \\
 &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f_*(u)du\Big)ds\Big],
\end{align*}
for some $f_* \in  S_{F,x_*}$.

Finally, we discuss a priori bounds on solutions. Let $x$ be a
solution of \eqref{e1i}. Then there exists $f \in L^1([0,1],
\mathbb{R})$ with $f \in S_{F,x}$ such that, for $t \in [0,1]$, we
have
\begin{align*}
  x(t)&=    \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
f(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f(s)ds \\
 &\quad -
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(u)du\Big)ds\Big].
\end{align*}
In view of (H2), and using the computations in second step
above, for each $t \in [0,1]$, we obtain
\begin{align*}
  |x(t)|&\leq    \int_0^t
\frac{(t-s)^{q-1} }{\Gamma(q)} |f(s)|ds
 +|\vartheta| t^{m-1}\int_0^1  \frac{(1-s)^{q-1}}{\Gamma(q)}   |f(s)|ds \\
 &\quad +  |\vartheta| t^{m-1}
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} |f(u)|du\Big)ds\\
&\leq
\frac{\psi(\|x\|)}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i  
 \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\Big].
\end{align*}
Consequently, 
$$
\frac{\|x\|}{
\frac{\psi(\|x\|)}{\Gamma(q)}\big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i  
 \frac{\eta_1^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\big]}\le 1.
$$
In view of (A4), there exists $M$ such that $\|x\| \ne M$. Let us set
$$
U = \{x \in  C([0,1], \mathbb{R}) : \|x\| < M\}.
$$ 
Note that the operator $\Omega :\overline{U} \to \mathcal{P}(C([0,1], \mathbb{R}))$ 
is upper semicontinuous and completely continuous. From the choice of $U$,
there is no $x \in \partial U$ such that $x \in \mu \Omega(x)$ for
some $\mu \in (0,1)$. Consequently, by the nonlinear alternative
of Leray-Schauder type (Lemma \ref{NAK}), we deduce that $\Omega$
has a fixed point $x \in \overline{U}$ which is a solution of the
problem \eqref{e1i}. This completes the proof.
 \end{proof}

\subsection{The lower semi-continuous case}

Here, we study the case when $F$ is not necessarily convex valued.
Our strategy to deal with this problems is based on the nonlinear
alternative of Leray Schauder type together with the selection
theorem of Bressan and Colombo \cite{BrCo} for lower
semi-continuous maps with decomposable values.

\begin{definition} \label{def4.9} \rm
Let $X$ be a nonempty closed subset of a Banach space $E$ and $G:
X \to {\mathcal{P}}(E)$ be a multivalued operator with nonempty
closed values. $G$ is lower semi-continuous (l.s.c.) if the set
$\{y \in X : G(y)\cap B \ne \emptyset\}$ is open for any open set
$B$ in $E$.
\end{definition}

\begin{definition} \label{def4.10} \rm
Let $A$ be a subset of $[0,1]\times \mathbb{R}$.   $A$ is
$\mathcal{L}\otimes \mathcal{B}$ measurable if $A$ belongs to the
$\sigma-$algebra generated by all sets of the form $\mathcal{J}
\times \mathcal{D}$, where $\mathcal{J}$ is Lebesgue measurable in
$[0,1]$ and $\mathcal{D}$ is Borel measurable in $\mathbb{R}$.
\end{definition}

\begin{definition} \label{def4.11} \rm
A subset $\mathcal{A}$ of $L^1([0,1], \mathbb{R})$ is decomposable
if for all $x, y \in  \mathcal{A}$ and measurable $\mathcal{J}
\subset [0,1]=J$, the function $x \chi_{\mathcal{J}}+y
\chi_{J-\mathcal{J}} \in \mathcal{A}$, where $\chi_{\mathcal{J}}$
stands for the characteristic function of $\mathcal{J}$.
\end{definition}


\begin{definition} \label{def4.12} \rm
 Let $Y$ be a separable metric
space and  $N : Y \to {\mathcal{P}}(L^1([0,1],\mathbb{R}))$ be
a multivalued operator. We say $N$ has  a property (BC) if
 $N$ is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable
 values.
 \end{definition}

Let $F : [0,1] \times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})$ be
a multivalued map with nonempty compact values. Define a
multivalued operator $\mathcal{F} : C([0,1] \times \mathbb{R}) \to
{\mathcal{P}}(L^1([0,1],\mathbb{R}))$ associated with $F$ as
$$
\mathcal{F}(x)=\{w \in L^1([0,1],\mathbb{R}) : w(t) \in
F(t,x(t)) \text{ for a.e. }  t \in [0,1]\},
$$ 
which is called the Nemytskii operator associated with $F$.


\begin{definition} \label{def4.13} \rm 
Let $F : [0,1] \times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})$
 be a multivalued function with
nonempty compact values. We say $F$ is of lower semi-continuous
type (l.s.c. type) if its associated Nemytskii operator
$\mathcal{F }$ is lower semi-continuous and has nonempty closed
and decomposable values.
\end{definition}

\begin{lemma}[\cite{BrCo}] \label{l2i}
 Let $Y$ be a separable metric
space and  $N : Y \to {\mathcal{P}}(L^1([0,1],\mathbb{R}))$ be
a multivalued operator satisfying the property (BC). Then $N$ has
a continuous selection,  that is, there exists a continuous
function (single-valued) $g : Y \to L^1([0,1],\mathbb{R})$ such
that $g(x) \in N(x)$ for every $x \in Y$.
\end{lemma}


\begin{theorem}\label{tbc} 
 Assume that {\rm (H2), (H3)} and the following condition holds:
\begin{itemize} 
\item[(H4)] $F : [0,1] \times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})$  
is a nonempty compact-valued multivalued map such that
\begin{itemize} 
\item[(a)] $(t,x) \mapsto F(t,x)$ is  $\mathcal{L}\otimes
\mathcal{B}$ measurable,
 \item[(b)]  $ x \mapsto F(t,x)$ is lower semicontinuous for each $t \in [0,1]$.
\end{itemize}
\end{itemize} 
Then the boundary value problem \eqref{e1i} has at least one solution on $[0,1]$.
\end{theorem}

\begin{proof} 
It follows from (H2) and (H4) that $F$ is of l.s.c. type. 
Then, by Lemma \ref{l2i}, there
exists a continuous function $f : C([0,1],\mathbb{R})  \to
L^1([0,1],\mathbb{R})$ such that $f (x) \in  \mathcal{F}(x)$ for
all $x \in C([0,1],\mathbb{R})$.
Consider the problem
\begin{equation}\label{et2i}
\begin{gathered}
 ^cD^qx(t)= f(x(t)), \quad  0<t<1,  \quad m-1 < q \le m,\; m\ge2,\; 
 m \in \mathbb{N}, \\
 x(0)=0, \quad x'(0)=0,\quad x''(0)=0,\quad \dots ,\quad x^{(m-2)}(0)=0, \\
  x(1)=\sum_{i=1}^{n-2}\alpha_i
\int^{\eta_i}_{\zeta_i} x(s)ds, \quad 0< \zeta_i < \eta_i <1, \; i=1,
2,\dots, (n-2),
\end{gathered}
\end{equation}
in the space $C([0,1], \mathbb{R})$. It is clear that if $x$ is a
solution of the problem \eqref{et2i}, then $x$ is a solution to
the problem \eqref{e1i}. In order to transform the problem
\eqref{et2i} into a fixed point problem, we define the operator
$\overline{\Omega}$ as
\begin{align*}
\overline{\Omega}x(t)
=\Big\{&
\frac{1}{\Gamma(q)}\int_0^t (t-s)^{q-1} f(x(s))ds
-\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} f(x(s))ds  \\
 & - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} f(x(u))du\Big)ds\Big],\; 0\le t\le 1.
\Big\}
\end{align*}
It can easily be shown that $\overline{\Omega}$ is continuous and
completely continuous. The remaining part of the proof is similar
to that of Theorem \ref{tcar}. So we omit it. This completes the
proof.
\end{proof}

\subsection{The Lipschitz case}

 Now we prove  the existence of solutions for the problem \eqref{e1i}
 with a nonconvex valued right hand
side by applying a fixed point theorem for multivalued maps due to
Covitz and Nadler \cite{CoNa}.


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 $d(A,b) = \inf_{a\in A}d(a;b)$ and $d(a,B) = \inf_{b\in B}d(a;b)$. Then
$(P_{b,cl}(X), H_d)$ is a metric space and $(P_{cl}(X), H_d)$ is a
generalized metric space (see \cite{Ki}).

\begin{definition} \label{def4.16} \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 
$$
H_d(N(x),N(y)) \le \gamma d(x,y) \text{ for each }  x, y \in X;
$$
 \item[(b)] a contraction if and only if it is $\gamma$-Lipschitz  with
$\gamma <  1$.
\end{itemize}
\end{definition}

\begin{lemma}[Covitz-Nadler, \cite{CoNa}] \label{l3i}
 Let $(X,d)$ be a complete metric space. If $N : X \to P_{cl}(X)$ 
is a contraction, then ${\rm Fix }N \ne \emptyset $.
\end{lemma}

\begin{definition}
A measurable multi-valued function $F:[0,1]\to \mathcal{P}(X)$ is
said to be integrably bounded if there exists a function 
$h\in L^1([0,1], X)$ such that for all $v\in F(t)$, $\|v\|\le h(t)$ for
a.e. $t\in [0,1]$.
\end{definition}

\begin{theorem}\label{tcn} 
 Assume that the following conditions hold:
\begin{itemize}
 \item[(H5)] $F : [0,1] \times \mathbb{R} \to P_{cp}(\mathbb{R})$ is
 such that $F(\cdot,x) : [0,1] \to P_{cp}(\mathbb{R})$ is measurable 
for each $x \in \mathbb{R}$;

\item[(H6)]  $H_d(F(t,x), F(t,\bar{x}))\le m(t)|x-\bar{x}|$ for almost all 
$t \in [0,1]$ and $x, \bar{x} \in \mathbb{R}$ with
$m \in L^1([0,1], \mathbb{R}^+)$ and $d(0,F(t,0))\le m(t)$ for
almost all $t \in [0,1]$.
 \end{itemize} 
Then the boundary-value problem \eqref{e1i} has at least one solution on
 $[0,1]$ if
$$ 
\frac{1}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1
  m(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i  
 \frac{\eta_1^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}m(s)  ds\Big] <1.
$$
\end{theorem}

\begin{proof} 
We transform the problem \eqref{e1i} into a fixed
point problem. Consider the set-valued map $\Omega: C([0,1],
\mathbb{R})\to \mathcal{P}(C([0,1], \mathbb{R}))$ defined at
the beginning of the proof of Theorem \ref{tcar}. It is clear that
the fixed point of $\Omega$ are solutions of the problem
\eqref{e1i}.

Note that, by the assumption (H5), since the set-valued map
$F(\cdot, x)$ is measurable, it admits a measurable selection 
$f: [0,1]\to \mathbb{R}$ (see \cite[Theorem III.6]{CaVa}).  Moreover,
from assumption (H6)
$$
|f(t)|\le m(t)+m(t)|x(t)|,
$$
i.e. $f(\cdot)\in L^1([0,1], \mathbb{R})$. Therefore the set
$S_{F,x}$ is nonempty. Also note that since $S_{F,x}\ne
\emptyset$, therefore $\Omega(x)\ne \emptyset$ for any $x\in
C([0,1],\mathbb{R})$.

 Now we show that
the operator $\Omega$ satisfies the assumptions of Lemma
\ref{l3i}. To show that $\Omega(x) \in
P_{cl}((C[0,1],\mathbb{R}))$ for each $x \in C([0,1],
\mathbb{R})$, let $\{u_n\}_{n \ge 0} \in \Omega(x)$ be such that
$u_n \to u ~(n \to \infty)$ in $C([0,1],\mathbb{R})$. Then $u \in
C([0,1],\mathbb{R})$ and there exists  $v_n \in S_{F,x}$ such
that, for each $t \in [0,1]$, we have
\begin{align*}
  u_n(t)
&=    \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
v_n(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} v_n(s)ds \\
 &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} v_n(u)du\Big)ds\Big].
\end{align*}
As $F$ has compact values, we may  pass onto a subsequence  (if
necessary) to obtain that $v_n$ converges to $v$ in 
 $L^1 ([0,1],\mathbb{R})$. Thus, $v \in S_{F,x}$ and for each
 $t \in [0,1]$,
\begin{align*}
 u_n(t) \to u(t)
&=    \int_0^t
\frac{(t-s)^{q-1}}{\Gamma(q)} v(s)ds -\vartheta
t^{m-1}\Big[\int_0^1 \frac{(1-s)^{q-1}}{\Gamma(q)} v(s)ds \\
 &\quad - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} v(u)du\Big)ds\Big].
\end{align*}
 Hence, $u \in \Omega(x)$ and $\Omega(x)$ is closed.

Next we show that $\Omega$ is a contraction on $C([0,1],\mathbb{R})$; i.e.,  
there exists  $\gamma <1$ such that
$$
 H_d(\Omega(x), \Omega(\bar{x}))\le \gamma \|x-\bar{x}\|_{\infty} \quad  
\text{for each} \quad x, \bar{x} \in C([0,1], \mathbb{R}). 
$$ 
Let $x, \bar{x} \in C([0,1], \mathbb{R})$ and $h_1 \in \Omega(x)$. 
Then there exists $v_1(t) \in F(t,x(t))$ such that, for each $t \in [0,1]$,
\begin{align*}
  h_1(t)
&=    \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
v_1(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} v_1(s)ds \\
 &\quad  - \sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} v_1(u)du\Big)ds\Big].
\end{align*}

 By (H6), we have 
$$
H_d(F(t,x), F(t,\bar{x}))\le m(t)|x(t)-\bar{x}(t)|.
$$
 So, there exists $w \in F(t,\bar{x}(t))$ such that
$$
|v_1(t)-w|\le m(t)|x(t)-\bar{x}(t)|, \quad t \in [0,1].
$$
Define $U : [0,1] \to \mathcal{P}(\mathbb{R})$ by
$$
U(t)=\{w \in \mathbb{R} : |v_1(t)-w|\le m(t)|x(t)-\bar{x}(t)|\}.
$$
Since the multivalued operator $U(t)\cap F(t,\bar{x}(t))$ is
measurable (\cite[Proposition III.4]{CaVa}), there exists a
function $v_2(t)$ which is  a measurable selection for $U$. So
$v_2(t) \in F(t,\bar{x}(t))$ and for each $t \in [0,1]$, we have
$|v_1(t)-v_2(t)|\le m(t)|x(t)-\bar{x}(t)|$.

For each $t \in [0,1]$, let us define
\begin{align*}
  h_2(t)
&=    \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)}
v_2(s)ds -\vartheta t^{m-1}\Big[\int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} v_2(s)ds \\
 &\quad -
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} v_2(u)du\Big)ds\Big].
\end{align*}
 Thus,
 \begin{align*}
  | h_1(t)-h_2(t)|
&\leq   \int_0^t \frac{(t-s)^{q-1}}{\Gamma(q)} |v_1(s)-v_2(s)|ds\\
&\quad +|\vartheta| t^{m-1} \int_0^1
\frac{(1-s)^{q-1}}{\Gamma(q)} |v_1(s)-v_2(s)|ds \\
 &\quad +|\vartheta| t^{m-1}
\sum_{i=1}^{n-2}\alpha_i\int_{\zeta_i}^{\eta_i}\Big(\int_0^s
\frac{(s-u)^{q-1}}{\Gamma(q)} |v_1(u)-v_2(u)|du\Big)ds \\
&\leq \frac{\|x-\overline{x}\|}{\Gamma(q)}
\Big[\{1+|\vartheta|\}\int_0^1
  m(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i  
 \frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}m(s)  ds\Big].
\end{align*}
 Hence,
\begin{align*}  
\| h_1-h_2\|  \le  \frac{1}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1
  m(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i   \frac{\eta_i^q
  -\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}m(s)  ds\Big]\|x-\overline{x}\|.
\end{align*}
Analogously, interchanging the roles of $x$ and $\overline{x}$, we
obtain
\begin{align*}
& H_d(\Omega(x), \Omega(\bar{x})) \\
&\leq  \gamma \|x-\bar{x}\|\\
&\leq   \frac{1}{\Gamma(q)}\Big[\{1+|\vartheta|\}\int_0^1
  m(s)ds+ |\vartheta| \sum_{i=1}^{n-2}\alpha_i   \frac{\eta_i^q
  -\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}m(s)  ds\Big]\|x-\overline{x}\|.
\end{align*}
Since $\Omega$ is  a contraction, it follows by Lemma \ref{l3i}
that $\Omega$ has a fixed point $x$ which is a solution of
\eqref{e1i}. This completes the proof.
\end{proof}

\subsection{Example}

\begin{example} \rm
Consider the strip fractional boundary value problem
\begin{equation}\label{ex4}
\begin{gathered}
 ^cD^{7/2}x(t) \in F(t,x(t)), \quad  0<t<1,  \\
 x(0)=0, \quad x'(0)=0, \quad x''(0)=0, \quad 
x(1)=\sum_{i=1}^{3}\alpha_i \int^{\eta_i}_{\zeta_i} x(s)ds,
\end{gathered} 
\end{equation}
\end{example}

Here,  $q=7/2$, $m=4$, $\zeta_1=1/16$, $\zeta_2=5/16$, $\zeta_3=9/16$,
 $\eta_1=1/4$, $\eta_2=1/2$, $\eta_3=3/4$, $\alpha_1=1/3$, $\alpha_2=2/3$,
$\alpha_3=1$,
and $F : [0,1] \times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})$ is
a multivalued map given by
$$
x \to F(t,x)=\big[\frac{|x|^3}{|x|^3+3}+3t^3+5,\frac{|x|}{|x|+1}+t+1\big].
 $$
For $f \in F$, we have
$$|f| \le \max\left(\frac{|x|^3}{|x|^3+3}+3t^3+5,\frac{|x|}{|x|+1}+t+1\right)\le 9, \quad  x \in \mathbb{R}.$$
Thus,
$$
\|F(t,x)\|_\mathcal{P}:=\sup\{|y|: y \in F(t,x)\}\le
9=p(t)\psi(\|x\|), \quad  x \in \mathbb{R},
$$ 
with $p(t)=1$, $\psi(\|x\|)=9$.
Further, using the condition
$$ 
\frac{M}{ \frac{\psi(M)}{\Gamma(q)}\big[\{1+|\vartheta|\}\int_0^1
  p(s)ds+ |\vartheta|\sum_{i=1}^{n-2}\alpha_i   
\frac{\eta_i^q-\zeta_i^q}{q}\int_{\zeta_i}^{\eta_i}p(s)  ds\big]}> 1,
$$
we find that $  M > 5.6427$. Clearly, all the conditions of
Theorem \ref{tcar} are satisfied. So there exists at least one
solution of the problem \eqref{ex4} on $[0,1]$.

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\end{document}