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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 58, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/58\hfil Fractional multi-point boundary-value problems]
{Solvability of fractional multi-point boundary-value problems with
$p$-Laplacian operator at resonance}

\author[T. Shen, W. Liu, T. Chen, X. Shen \hfil EJDE-2014/58\hfilneg]
{Tengfei Shen, Wenbin Liu, Taiyong Chen, Xiaohui Shen}  % in alphabetical order

\address{Tengfei Shen \newline
College of Sciences, China University of Mining and Technology,
 Xuzhou 221008, China}
\email{shentengfei1987@126.com}

\address{Wenbin Liu (corresponding author) \newline
College of Sciences, China University of Mining and Technology,
 Xuzhou 221008, China}
\email{wblium@163.com}

\address{Taiyong Chen \newline
College of Sciences, China University of Mining and Technology,
 Xuzhou 221008, China}
\email{taiyongchen@cumt.edu.cn}

\address{Xiaohui Shen \newline
College of Sciences, China University of Mining and Technology,
 Xuzhou 221008, China}
\email{shenxiaohuicool@163.com}

\thanks{Submitted January 13, 2013. Published February 28, 2014.}
\subjclass[2000]{34A08, 34B15}
\keywords{Fractional differential equation;  boundary value problem;
\hfill\break\indent
$p$-Laplacian operator; Coincidence degree theory; Resonance}

\begin{abstract}
 In this article, we  consider the multi-point boundary-value problem
 for nonlinear fractional differential equations with $p$-Laplacian operator:
 \begin{gather*}
 D_{0^+}^\beta  \varphi_p (D_{0^+}^\alpha  u(t))
  = f(t,u(t),D_{0^+}^{\alpha  - 2} u(t),D_{0^+}^{\alpha  - 1} u(t),
   D_{0^+}^\alpha  u(t)),\quad t \in (0,1), \\
 u(0) = u'(0)=D_{0^+}^\alpha  u(0) = 0,\quad
 D_{0^+}^{\alpha  - 1} u(1) = \sum_{i = 1}^m
  {\sigma_i D_{0^+}^{\alpha  - 1} u(\eta_i )} ,
 \end{gather*}
 where $2 < \alpha  \le 3$, $0 < \beta  \le 1$, $3 < \alpha  + \beta  \le 4$,
 $\sum_{i = 1}^m {\sigma_i }  = 1$, $D_{0^+}^\alpha$ is the standard
 Riemann-Liouville fractional derivative. $\varphi_{p}(s)=|s|^{p-2}s$ is
 $p$-Laplacians operator. The existence of solutions for above fractional
 boundary value problem is obtained by using the extension of Mawhin's
 continuation theorem due to Ge, which enrich konwn results.
 An example is given to illustrate the main result.
\end{abstract}

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

\section{Introduction}

In recent years, fractional differential equations play a important role 
in many fields such as physics, engineering, biology, control theory,
etc., see \cite{b1,k2,m1,s1,s2}. 
It has been studied extensively by scholars have obtained many results,
see \cite{b2,b5,j1,k1,l1,s3,z2}.

However, the existence of solutions for fractional boundary value problems 
at resonance is less studied, 
see \cite{b3,b4, c2,h1,w1,z1}.
There are few articles which consider the boundary value problems (BVPs for shorts) 
at resonance for nonlinear fractional differential equation with 
$p$-Laplacian operator. In 2012, Chen, Liu and Hu \cite{c1} considered existence 
of solutions of boundary value problems for a Caputo fractional differential 
equation with $p$-Laplacian operator at resonance by coincidence degree 
theory by Mawhin:
\begin{equation} \label{e1.1}
\begin{gathered}
 D_{0^+}^\beta  \varphi_p (D_{0^+}^\alpha  u(t))
= f(t,u(t),D_{0^+}^\alpha  u(t)),\quad t \in (0,1),\\
 D_{0^+}^\alpha  u(0)=D_{0^+}^\alpha  u(1)=0,
 \end{gathered}
\end{equation}
where $0 < \alpha ,\beta  < 1$, $1 < \alpha  + \beta  \le 2$,
 $D_{0^+}^\alpha$ is a Caputo fractional derivative,
 $ \varphi_p (s) = | s |^{p - 2} s $ is a $p$-Laplacian operator,
$f:[ {0,1} ] \times \mathbb{R}^2  \to \mathbb{R}$
is continuous.

In this article, we study  fractional multi-point boundary value problem 
with $p$-Laplacian operator at resonance by using the extension of 
Mawhin's continuation theorem due to Ge,
\begin{equation} \label{e1.2}
\begin{gathered}
 D_{0^+}^\beta  \varphi_p (D_{0^+}^\alpha  u(t)) = f(t,u(t),
D_{0^+}^{\alpha  - 2} u(t),D_{0^+}^{\alpha  - 1} u(t),
D_{0^+}^\alpha  u(t)),\quad t \in (0,1) ,\\
u(0) = u'(0)=D_{0^+}^\alpha  u(0) = 0,\quad
D_{0^+}^{\alpha  - 1} u(1) = \sum_{i = 1}^m {\sigma_i D_{0^+}^{\alpha  - 1}
u(\eta_i )},
\end{gathered}
\end{equation}
where $2 < \alpha  \le 3$, $0 < \beta  \le 1$, $3 < \alpha  + \beta  \le 4$,
$\eta_i \in (0,1)$, $\sigma_i\in \mathbb{R}$,
$\sum_{i = 1}^m {\sigma_i }  = 1$, $1<m, m\in N$, $\varphi_{p}(s)=|s|^{p-2}s$,
$1<p,1/p+1/q=1$, $\varphi_{p}$ is invertible
and its inverse operator is $\varphi_{q}$, $D_{0^+}^\alpha$ is
 Riemann-Liouville standard fractional derivative,
$f:[ {0,1} ] \times \mathbb{R}^4  \to \mathbb{R}$ is continuous.

There are few articles to investigate fractional multi-point boundary 
value problem with $p$-Laplacian operator at resonance.
By constructing suitable continuous linear projectors and using the 
extension of Mawhin's continuation theorem due to Ge, the existence 
of solutions were obtained. Our paper perfect and generalize some known results.

To investigate the problem, we use the condition
\[
\Delta  = \frac{1}{{\Gamma (\beta  + 1)^{q - 1} 
(q\beta  - \beta  + 1)}}(1 - \sum_{i = 1}^m {\sigma_i
\eta_i^{q\beta  - \beta  + 1} } ) \neq 0.
\]

The rest of this article is organized as follows: 
In Section 2, we give some notations, definitions and Lemmas. 
In Section 3, basing on the extension of Mawhin's continuation theorem 
due to Ge, we establish a theorem of existence result for BVP \eqref{e1.2}.

\section{Preliminaries}

For the convenience of the reader, we present here some basic knowledge
and definitions for fractional calculus theory, that can be found in 
\cite{b2,g1,k2}.

Let $X$ and $Y$ be two Banach spaces with norms 
$\| \cdot \|_X$ and $\| u \|_Y $, respectively. A continuous operator
\[
M|_{\operatorname{dom}M \cap X}: X\cap \operatorname{dom}M \to Y
\]
is said to be quasi-linear if
\begin{itemize}
\item[(i)] $\operatorname{Im}M:=M(X\cap \operatorname{dom}M)$
is a closed subset of $Y$,

\item[(ii)] $\ker M:=\{u\in X\cap \operatorname{dom}M:Mu=0\}$
is is linearly homeomorphic to $\mathbb{R}^{n}$, $n<\infty$.
\end{itemize}

Let $X_1=\ker  M$ and $X_2$ be the complement space of $X_1$ in $X$,
then $X=X_1\oplus X_2$. On the other hand, suppose $Y_1$ is a subspace
of $Y$ and $Y_2$ is the complement space of $Y_1$ in $Y$ so that 
$Y=Y_1\oplus Y_2$. Let $P: X \to X_1$ be a projector and
 $Q: Y \to Y_1$ a semi-projector, and
$\Omega\subset X$ an open and bounded set with origin 
$\theta \in \Omega$. Where $\theta$ is the origin of a linear space.

Suppose $N_{\lambda} : \overline{\Omega} \to Y$, $\lambda\in [0,1]$ 
is a continuous operator. Denote $N_1$ by $N$. Let
$\Sigma_{\lambda}=\{u\in \overline{\Omega}:Mu=N_{\lambda}u\}$. 
$N_{\lambda}$ is said to be $M$-compact in $ \overline{\Omega}$ if
there is a $Y_1\subset Y$ with
$\dim Y_1$ = $\dim X_1$ and an operator 
$R : \overline{\Omega}\times [0,1] \to X$ continuous and compact such that for
$\lambda\in [0,1]$,
\begin{gather} \label{e2.1}
(I-Q)N_{\lambda}(\overline{\Omega})\subset \operatorname{Im}M \subset (I-Q)Y,\\
 \label{e2.2}
QN_{\lambda}x=\theta, \lambda\in(0,1) \Leftrightarrow QNx=\theta, \\
 \label{e2.3}
 R(\cdot,\lambda)\mid_{\Sigma_{\lambda}}=(I-P)\mid_{\Sigma_{\lambda}}
\end{gather}
 and $R(\cdot,0)$ is the zero operator,
\begin{equation} \label{e2.4}
M[P+R(\cdot,\lambda)]=(I-Q)N_{\lambda}.
\end{equation}


\begin{lemma}[\cite{g1}] \label{lem2.1} 
 Let $(X,\| \cdot \|_X)$ and $(Y,\| \cdot \|_Y)$ be two Banach spaces, and 
$\Omega\subset X$ an open and bounded nonempty set. Suppose 
$M : X\cap \operatorname{dom}M \to Y$ is a quasi-linear operator
$N_{\lambda} : \overline{\Omega} \to Y$, $\lambda\in [0,1]$ is $M$-compact 
in $ \overline{\Omega}$.
In addition, if:
\begin{itemize}
\item[(i)] $Mu\neq N_{\lambda}u$ for all 
$(u,\lambda)\in (\operatorname{dom}M \cap \partial\Omega)\times (0,1)$,

\item[(ii)] $QNu\neq0$ for all $u \in \partial\Omega\cap\ker M$,

\item[(iii)] $\deg (JQN,\ker M\cap\Omega,0)\neq 0$,
 where $J:\operatorname{Im}Q\to \ker M$ is a homeomorphism with 
$J(\theta)=\theta$ and $N=N_1$,
\end{itemize}
then the equation $Mu=Nu$ has at least 
one solution in $\operatorname{dom}M\cap\bar{\Omega}$.
\end{lemma}


\begin{definition} \label{def2.1} \rm
 The Riemann-Liouville fractional integral of order $\alpha >0$ of a 
function $u$ is given by
\[
I_{0^+}^\alpha u(t)=\frac{1}{\Gamma(\alpha)}\int_0^t
(t-s)^{\alpha -1}u(s)ds,
\]
provided that the right side integral is pointwise defined on $(0,+\infty)$.
\end{definition}


\begin{definition} \label{def2.2} \rm
The Riemann-Liouville fractional derivative of order $\alpha >0$ of a 
function $u$ is given by
\[
D_{0+}^\alpha u(t)=\frac{1}{\Gamma(n-\alpha)}(\frac{d}{dt})^n
\int_0^t\frac{u(s)}{(t-s)^{\alpha-n+1}}ds,
\]
provided that the right side integral is pointwise defined on $(0,+\infty)$.
\end{definition}

\begin{lemma} \label{lem2.2}
Assume that $u\in C(0,1)\cap L^1(0,1)$ with a fractional derivative of
 order $\alpha>0$ that belongs to $C(0,1)\cap L^1(0,1)$. Then
\[
I_{0+}^\alpha D_{0+}^\alpha u(t)=u(t)+c_1t^{\alpha-1}+c_2t^{\alpha-2}+\dots
+c_Nt^{\alpha-N},
\]
for some $c_i\in \mathbb{R}$, $i=1,2,\dots ,N$, where $N$ is the smallest 
integer grater than or equal to $\alpha$.
\end{lemma}

\begin{lemma} \label{lem2.3}
Assume $u(t) \in C[0,1]$ and 
$0\leq \beta \leq \alpha$, then
$D_{0^+}^\beta  I_{0^+}^\alpha  u(t) = I_{0^+}^{\alpha  - \beta } u(t)$.
 And, for all $\alpha  \ge 0$, $\beta  >  - 1$, we have 
\[
D_{0^+}^\alpha  t^\beta
= \frac{{\Gamma (\beta  + 1)}}{{\Gamma (\beta- \alpha + 1)}}t^{ \beta- \alpha} ,
\]
giving in particular $D_{0^+}^\alpha  t^{\alpha  - m}  = 0$,
$m = 1,2,\dots  ,N$, where $N$ is the smallest integer grater than or equal 
to $\alpha$.
\end{lemma}

In this article, we take 
$X = \{ {u|u,D_{{0^+}}^{\alpha  - 2}u,D_{{0^+}}^{\alpha  - 1}u,
D_{{0^+}}^\alpha u\in C[0,1]} \}$ with the norm
$\| u \|_X  = \max \{ \| u \|_\infty$,
$\| {D_{0^+}^{\alpha-2}  u} \|_\infty  \,
 \| {D_{0^+}^{\alpha  - 1} u} \|_\infty  ,\| {D_{0^+}^\alpha  u} \|_\infty  \}$,
 where $\| u \|_\infty   = \max_{t \in [0,1]} $
$|{u(t)}|$, and $Y = C[0,1]$ with the norm ${\| y \|_Y}={\| y \|_\infty} $.
By means of the linear functional analysis theory, 
it is easy to prove that $X$ and $Y$ are Banach spaces. so, we omit it.

Define the operator $M:\operatorname{dom}M\subset X\to Y$ by
\begin{gather} \label{e2.5}
Mu = D_{0^+}^\beta \varphi_p (D_{0^+}^\alpha  u(t)), \\
\begin{aligned}
\operatorname{dom}M = \Big\{& u \in X :
D_{0^+}^\beta  \varphi_p (D_{0^+}^\alpha  u) \in Y,\,
u(0) =u'(0)= D_{0^+}^\alpha  u(0) = 0, \\
&D_{0^+}^{\alpha  - 1}
u(1) =\sum_{i = 1}^m  {\sigma_i }D_{0^+}^{\alpha  - 1}u(\eta_i )\Big\}.
\end{aligned} \label{e2.6}
\end{gather}
Define the operator $N_{\lambda}:X \to Y$, $\lambda\in [0,1]$, 
\[
N_{\lambda}u(t) = f(t,u(t),D_{{0^+}}^{\alpha  - 2}u(t),
D_{{0^+}}^{\alpha  - 1}u(t),D_{{0^+}}^\alpha u(t)),t \in [ {0,1} ],
\]
then \eqref{e1.2} is equivalent to the operator equation $Mu = Nu$, where $N=N_1$.

\section{Main result}

In this section, we show  existence of solutions for BVP \eqref{e1.2}. 
Let us make some assumptions which will be used throughout this article.
\begin{itemize}
\item[(H1)] There exist nonnegative functions $a,b,c,d,e\in Y$ such that
 \[
| {f(t,u,v,w,z))} | \le a(t) + b(t){| u |^{p - 1}} + c(t){| v |^{p - 1}}
 + d(t){| w |^{p - 1}} + e(t){| z |^{p - 1}},
\]
for all $t \in [0,1]$, $(u,v,w,z) \in {\mathbb{R}^4}$.

\item[(H2)] There exists a constant $A > 0$ such that
\begin{align*}
&\int_0^1 \varphi_q \Big(\frac{1}{{\Gamma (\beta )}}
\int_0^s {(s - \tau )^{\beta  - 1} f(\tau ,u,v,w,z })d\tau \Big)ds\\
& - \sum_{i = 1}^m  {\sigma_i } \int_0^{\eta_i }
\varphi_q (\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1}
f(\tau ,u,v,w,z })d\tau  )ds \ne 0, 
\end{align*}
for all $t\in [0, 1]$, $(u,v,w,z) \in {\mathbb{R}^4}$, $|v|+|w|>A$.


\item[(H3)] There exists a constant $B> 0$ such that
\begin{align*}
0\neq \Lambda
&:= c\frac{1}{\Delta }\Big(\int_0^1 \varphi_q (\frac{1}{{\Gamma (\beta )}}
\int_0^s {(s - \tau )^{\beta  - 1} f(\tau ,c\tau ^{\alpha  - 1},
c\Gamma(\alpha)\tau ,c\Gamma (\alpha ),0)d\tau } )ds \\
&\quad -  \sum_{i = 1}^m  {\sigma_i } \int_0^{\eta_i }
 \varphi_q (\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1}
 f(\tau ,c\tau ^{\alpha  - 1} ,c\Gamma(\alpha)\tau,c\Gamma (\alpha ),0)
 d\tau } )ds\Big),
\end{align*}
for all $|c|>B$, $c\in \mathbb{R}$.
\end{itemize}


\begin{theorem} \label{thm3.1}
Let $f:[0,1]\times\mathbb{R}^4\to\mathbb{R}$ be continuous and the condition 
{\rm (H1)--(H3)} hold.
Then BVP \eqref{e1.2} has at least one solution provided that
\begin{equation} \label{e3.1}
\frac{1}{{\Gamma (\beta  + 1)}}
\Big(\frac{D{\|b \|_\infty}}{{\Gamma {{(\alpha )}^{p - 1}}}} 
+ D{\| c \|_\infty} + D{\| d \|_\infty}+{\| e \|_\infty}\Big) < 1.
\end{equation}
\end{theorem}


\begin{lemma} \label{lem3.1} 
The operator $M:\operatorname{dom}M \cap X\to Y$ is a quasi-linear, and
\begin{gather} \label{e3.2} %\label{kerl}
\ker M=\{u\in X:u(t)=ct^{\alpha-1}, \, \forall t\in [0,1],\, c \in \mathbb{R}\}\\
\begin{aligned}
\operatorname{Im} M=\Big\{& y \in Y:\int_0^1  \varphi_q (\frac{1}{{\Gamma (\beta )}}
\int_0^s {(s - \tau )^{\beta  - 1} y(\tau )d\tau } )ds \label{e3.3} \\
&- \sum_{i = 1}^m{\sigma_i } \int_0^{\eta_i } \varphi_q
 (\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1} y(\tau )d\tau } )ds 
= 0\Big\}
\end{aligned} 
\end{gather}
\end{lemma}

\begin{proof}
 By Lemma \ref{lem2.2} and $D_{0^+}^\beta \varphi_p (D_{0^+}^\alpha  u(t)) = 0$,
we have
\[
D_{0^+}^\alpha u(t)= \varphi_q (c_0 t^{\beta  - 1}).
\]
From condition $D_{0^+}^\alpha  u(0)=0$, we obtain that ${c_0}=0$.
Thus,
\[
u(t)=c_1 t^{\alpha  - 1}  + c_2 t^{\alpha  - 2} + c_3 t^{\alpha  - 3} .
\]
Combined with $u(0)=u'(0)=0$ , we have ${c_2} ={c_3}=0$, 
$u(t) = {c_1}{t^{\alpha  - 1}}$, ${c_1}\in \mathbb{R}$. 
Thus, \eqref{e3.2} is satisfied.

If $y\in\operatorname{Im}M$, then there exists a function 
$u\in \operatorname{dom}M$ such that
\[
y(t)=D_{0^+}^\beta \varphi_p (D_{0^+}^\alpha u(t)).
\]
Then by Lemma \ref{lem2.2} and boundary value condition, we have
\begin{gather*}
u(t) = I_{0^+}^\alpha  \varphi_q (I_{0^+}^\beta y(s)  )
+ c_1 t^{\alpha  - 1} , \\
D_{0^+}^{\alpha  - 1} u(t) = D_{0^+}^{\alpha  - 1} I_{0^+}^\alpha
 \varphi_q (I_{0^+}^\beta  y(s)  ) + c_1 \Gamma (\alpha ).
\end{gather*}
Combing this  with  $\sum_{i = 1}^m  {\sigma_i }  = 1$,
we obtain
\begin{align*}
&\int_0^1 \varphi_q (\frac{1}{{\Gamma (\beta )}}
 \int_0^s {(s - \tau )^{\beta  - 1} y(\tau )d\tau } )ds \\
&- \sum_{i = 1}^m  {\sigma_i } \int_0^{\eta_i }
\varphi_q (\frac{1}{{\Gamma (\beta )}}
\int_0^s {(s - \tau )^{\beta  - 1} y(\tau )d\tau } )ds = 0.
\end{align*}

On the other hand, suppose $y\in Y$ and satisfies \eqref{e3.3}. 
Let $u(t) = I_{0^+}^\alpha  \varphi_q (I_{0^+}^\beta  y(t))$,
then $u\in \operatorname{dom}M$ and 
$Mu(t)=D_{0^+}^\beta \varphi_p (D_{0^+}^\alpha  u(t))=y(t)$. so
$y\in \operatorname{Im}M$ and 
$\operatorname{Im}M:=M(\operatorname{dom}M)$ is a closed subset of $Y$.
Thus, $M$ is a quasi-linear operator.
\end{proof}

\begin{lemma} \label{lem3.2} 
Let $\Omega\subset X$ be an open and bounded set, then $N_{\lambda}$ 
is $M$-compact in $\overline{\Omega}$.
\end{lemma}

\begin{proof}  
Define the continuous projectors $P:X\to X_1$ and  $Q:Y\to Y_1$ by
\begin{gather*}
Pu(t) = \frac{1}{{\Gamma (\alpha )}}D_{0^+}^{\alpha  - 1} u(0)t^{\alpha  - 1} ,
\quad t \in [ 0,1] ,\\
\begin{aligned}
Qy(t) 
&= \varphi_p (\frac{1}{\Delta }(\int_0^1 \varphi_q
 (\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1}
 y(\tau )d\tau } )ds \\
&\quad - \sum_{i = 1}^m {\sigma_i } \int_0^{\eta_i }
\varphi_q (\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1}
y(\tau )d\tau } )ds)), t \in [ 0,1 ].
\end{aligned}
\end{gather*}
Obviously, $X_1=\ker M=\operatorname{Im}P$ and $Y_1=\mbox{ Im}Q$. 
Thus, we have $\dim Y_1$ = $\dim X_1=1$.
 For any $y \in Y$, we have
\begin{align*}
Q^2y=Q(Qy)
&=Qy\varphi_p \Big(\frac{1}{\Delta }\Big(\int_0^1 \varphi_q
(\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1} d\tau } )ds \\
&\quad - \sum_{i = 1}^m {\sigma_i } \int_0^{\eta_i }
\varphi_q \Big(\frac{1}{{\Gamma (\beta )}}
\int_0^s {(s - \tau )^{\beta  - 1} d\tau } \Big)ds\Big)\Big)
=Qy.
\end{align*}
Hence, $Q^{2}=Q$, $Q$ is a semi-projector. Based on the definition of 
$M$ and $Q$, it is easy to see that $\ker Q=\operatorname{Im}M$. 
Let $\Omega\subset X$ be an open and bounded set with $\theta \in \Omega$. 
For each $u\in \overline{\Omega}$, we can get $Q[(I-Q)N_{\lambda}(u)]=0$. 
Thus, $(I-Q)N_{\lambda}(u)\in \operatorname{Im}M=\ker Q $.
Taking any $y\in \mbox{ImM}$ and noting $Qy=0$ , we can get $y\in (I-Q)Y$. 
So \eqref{e2.1} holds. It is easy to verify \eqref{e2.2}.

Define $R:\overline{\Omega}\times [0,1] \to X_2$ by
\[
R(u,\lambda)(t) = \frac{1}{{\Gamma (\alpha )}}
\int_0^t (t - s)^{\alpha  - 1} 
\varphi_q \Big(\frac{1}{\Gamma (\beta )}
\int_0^s (s - \tau )^{\beta  - 1} ((I-Q)N_{\lambda}u(\tau))d\tau \Big)ds .
\]
By the continuity of $f$, it is easy to get that $R(u,\lambda)$ 
is continuous on $\overline{\Omega}\times[0,1]$. Moreover, for all
 $u \in \overline \Omega $, there exists a constant $L > 0$ such 
that $|{I_{0^+}^\beta  (I-Q)N_{\lambda}u(\tau))}|$ $\le L$,
so we can easily obtain that $R(\overline{\Omega},\lambda)$, 
$D_{0^+}^{\alpha  - 2} R(\overline{\Omega},\lambda)$,
$D_{0^+}^{\alpha  - 1} R(\overline{\Omega},\lambda)$ and
$D_{0^+}^{\alpha} R(\overline{\Omega},\lambda)$ are uniformly bounded.
By Arzela-Ascoli theorem, we just need to prove that 
$R:\overline{\Omega}\times [0,1] \to X_2$ is equicontinuous.


For $u \in \overline \Omega$, $0 < t_1  < t_2  \le 1$, 
$2 < \alpha  \le 3$, $0 < \beta  \le 1$, $3< \alpha  + \beta  \le 4$, we have
\begin{align*}
& | {R(u,\lambda)(t_2 ) - R(u,\lambda)(t_1 )} |\\
&=\frac{1}{{\Gamma (\alpha )}}|\int_0^{t_2 } {(t_2  - s)^{\alpha  - 1}\varphi_q (
 {I_{0^+}^\beta  ((I-Q)N_{\lambda}u(\tau))}} )ds \\
&\quad -\int_0^{t_1 } {(t_1  - s)^{\alpha  - 1} \varphi_q ({I_{0^+}^\beta
((I-Q)N_{\lambda}u(\tau))}} )ds| \\
&\leq \frac{{\varphi_q (L)}}{{\Gamma (\alpha )}}(\int_0^{t_1 } ({(t_2  - s)^{\alpha  - 1}  - (t_1  - s)^{\alpha  - 1} }) ds + \int_{t_1 }^{t_2 } {(t_2  - s)^{\alpha  - 1} } ds)\\
&= \frac{{\varphi_q (L)}}{{\Gamma (\alpha  + 1)}}(t_2^\alpha   - t_1^\alpha ),
\end{align*}
\begin{align*}
& | {D_{0^+}^{\alpha  - 2} R(u,\lambda)(t_2 )
 - D_{0^+}^{\alpha  - 2} R(u,\lambda)(t_1 )} |\\
&= |\int_0^{t_2 } {(t - s)\varphi_q (I_{0^+}^\beta
((I-Q)N_{\lambda}u(\tau))} )ds - \int_0^{t_1 } {(t - s)\varphi_q
(I_{0^+}^\beta  ((I-Q)N_{\lambda}u(\tau))} )ds|\\
&\leq \varphi_q (L)(\int_0^{t_1 } {(t_2  - s) - (t_1  - s)} ds
 + \int_{t_1 }^{t_2 } {(t_2  - s)} ds)\\
&= \frac{\varphi_q (L)}{2}(t_2^2  - t_1^2 )
\end{align*}
 and
\begin{align*}
& |{D_{0^+}^{\alpha  - 1} R(u,\lambda)(t_2 ) - D_{0^+}^{\alpha  - 1}
 R(u,\lambda)(t_1 ) }| \\
&=|\int_0^{t_2 } {\varphi_q ( I_{0^+}^\beta ((I-Q)N_{\lambda}u(\tau))} )ds
 - \int_0^{t_1 } {\varphi_q (I_{0^+}^\beta  ((I-Q)N_{\lambda}u(\tau))} )ds| \\
&\leq \varphi_q (L)(t_2 - t_1 ).
\end{align*}

Since $t^\alpha$ is uniformly continuous on $[0, 1]$, it follows that  
$R(\overline{\Omega},\lambda)$, 
$D_{0^+}^{\alpha  - 2} R(\overline{\Omega},\lambda)$ and
$D_{0^+}^{\alpha  - 1} R(\overline{\Omega},\lambda)$ are equicontinuous.
Similarly, we can get ${I_{0^+}^\beta  ((I-Q)N_{\lambda}u(\tau))} \subset C[0,1]$
is equicontinuous, Considering of $\varphi_q (s)$ is uniformly continuous on
$[-L,L]$, we have 
$D_{0^+}^\alpha  R(\overline{\Omega},\lambda)=\varphi_q ( {I_{0^+}^\beta
((I -Q)N_{\lambda}(\overline{\Omega}))})$ is also equicontinuous. So,
 we can obtain that $R:\overline{\Omega}\times [0,1] \to X_2$ is compact.

For each $u\in \Sigma_{\lambda}$, we have 
$D_{0^+}^\beta  \varphi_p (D_{0^+}^\alpha  u(t))
= N_{\lambda}(u(t))\in \operatorname{Im}M$. Thus,
 \begin{align*}
 R(u,\lambda)(t)
  &= \frac{1}{{\Gamma (\alpha )}}\int_0^t {(t - s)^{\alpha  - 1} 
\varphi_q (\frac{1}{\Gamma (\beta )}\int_0^s {(s - \tau )^{\beta  - 1}
((I-Q)N_{\lambda}u(\tau))d\tau )ds} }\\
 &=\frac{1}{{\Gamma (\alpha )}}\int_0^t {(t - s)^{\alpha  - 1} \varphi_q
(\frac{1}{\Gamma (\beta )}\int_0^s {(s - \tau )^{\beta  - 1} 
D_{0^+}^\beta  \varphi_p (D_{0^+}^\alpha  u(\tau))d\tau )ds} },
 \end{align*}
 which together with $u(0) = u'(0)=D_{0^+}^\alpha  u(0) = 0$ yields
 \[
R(u,\lambda)(t)=u(t)-\frac{1}{{\Gamma (\alpha )}}D_{0^+}^{\alpha  - 1}
u(0)t^{\alpha  - 1} =(I-P)u(t).
\]
It is easy to verify that $R(u,0)(t)$ is the zero operator. 
So \eqref{e2.3} holds. Besides, for any $u\in \overline{\Omega}$,
 \begin{align*}
&M[Pu+R(u,\lambda)](t)\\
&=M[\frac{1}{{\Gamma (\alpha )}}\int_0^t {(t - s)^{\alpha  - 1} 
\varphi_q (\frac{1}{\Gamma (\beta )}\int_0^s {(s - \tau )^{\beta  - 1}
((I-Q)N_{\lambda}u(\tau))d\tau )ds} }\\
&\quad +\frac{1}{{\Gamma (\alpha )}}D_{0^+}^{\alpha  - 1} u(0)t^{\alpha  - 1}\\
&=(I-Q)N_{\lambda}u(t),
\end{align*}
 which implies \eqref{e2.4}. So $N_{\lambda}$ is $M$-compact in 
$\overline{\Omega}$.
\end{proof}


\begin{lemma} \label{lem3.3} 
Suppose {\rm (H1), (H2)}  hold, Then the set
\begin{align*}
\Omega_1=\big\{u\in \operatorname{dom}M\setminus \ker M:
 Mu=\lambda Nu,\ \lambda \in (0,1)\}
\end{align*}
is bounded.
\end{lemma}

\begin{proof} By lemma \ref{lem2.2}, for each $u \in \operatorname{dom}M$,
$D_{0^+}^{\alpha  - 1} u \in C[0,1]$, we have
\[
u(t) = I_{0^+}^{\alpha  - 1} D_{0^+}^{\alpha  - 1} u(t)
+ c_1t^{\alpha  - 2}+c_2t^{\alpha  - 3}.
\]
Combining this with $u(0) =u'(0)=0$, we get $c_1=c_2=0$. Thus,
\begin{align*}
\| u\|_\infty  
& = \| {I_{0^+}^{\alpha  - 1} D_{0^+}^{\alpha  - 1} u}
\|_\infty \le | {\frac{1}{{\Gamma (\alpha  - 1)}}
\int_0^t {(t - s)^{\alpha  - 2} ds} } |\| {D_{0^+}^{\alpha  - 1} u} \|_\infty \\
&\le \frac{1}{{\Gamma (\alpha )}}\| {D_{0^+}^{\alpha  - 1} u} \|_\infty .
\end{align*}
Take any $u \in \Omega_1 $, then $Nu \in {\mathop{\rm Im}}M = \ker Q$.
Thus, $QNu = 0$ for all $ t\in [0,1]$.
It follows from (H2) that there exists $t_{0}\in [0,1]$ such that 
$| {D_{0^+}^{\alpha  - 2} u(t_0 )} |+| {D_{0^+}^{\alpha  - 1} u(t_0 )} | \le A$.
Thus
\begin{gather*}
D_{0^+}^{\alpha  - 1} u(t) = D_{0^+}^{\alpha  - 1} u(t_0 )
 + \int_{t_0 }^t {D_{0^+}^\alpha  } u(t)dt,\\
D_{0^+}^{\alpha  - 2} u(t) = D_{0^+}^{\alpha  - 2} u(t_0 )
 + \int_{t_0 }^t {D_{0^+}^{\alpha-1}  } u(t)dt,\\
\| {D_{0^+}^{\alpha  - 1} u} \|_\infty
\le A + \| {D_{0^+}^\alpha  u} \|_\infty, \\
\| {D_{0^+}^{\alpha  - 2} u} \|_\infty
\le A + \| {D_{0^+}^{\alpha-1}  u} \|_\infty
\le 2A+\| {D_{0^+}^\alpha  u} \|_\infty ,  \\
\| u \|_\infty   \le \frac{1}{{\Gamma (\alpha )}}(A
 + \| {D_{0^+}^\alpha  u} \|_\infty  ).
\end{gather*}
Combined with $Mu = \lambda Nu$ and $D_{0^+}^\alpha  u(0) = 0$, we obtain
\[
\varphi_p (D_{0^+}^\alpha  u(t)) = \lambda I_{0^+}^\beta  Nu(t).
\]
From (H1) and $\lambda\in(0,1)$, we have
\begin{align*}
| {\varphi_p (D_{0^+}^\alpha  u(t))} |
&\leq \frac{1}{{\Gamma (\beta )}}\int_0^t {(t - s)^{\beta  - 1} 
 | {f(s,u(s),D_{0^+}^{\alpha  - 2} u(s),D_{0^+}^{\alpha  - 1} u(s),
 D_{0^+}^\alpha  u(s))} |} ds\\
&\le \frac{1}{{\Gamma (\beta )}}\int_0^t {(t - s)^{\beta  - 1} (a(s) 
 + b(s)| {u(s)} |^{p - 1} + c(s)| {D_{0^+}^{\alpha  - 2} u(s)} |^{p - 1} } \\
&\quad+ {d(s)| {D_{0^+}^{\alpha  - 1} u(s)} |^{p - 1} } + {e(s)|
 {D_{0^+}^\alpha  u(s)} |^{p - 1} )}ds\\
&\leq \frac{{1}}{{\Gamma (\beta  + 1)}}(\| a\|_\infty  
 + \| b \|_\infty \| u \|_\infty  ^{p - 1}  
 + \| c \|_\infty\| {D_{0^+}^{\alpha  -2 } u} \|_\infty ^{p - 1}\\
 &\quad +\| d \|_\infty \| {D_{0^+}^{\alpha  -1 } u} \|_\infty  ^{p - 1}
 + \| e \|_\infty \| {D_{0^+}^\alpha  u} \|_{_\infty  }^{p - 1}),
\quad \forall t\in [0,1],
\end{align*}
which together with $| {\varphi_p (D_{0^+}^\alpha  u(t))} |
= | {D_{0^+}^\alpha  u(t)} |^{p - 1} $, and the basic inequality
$(|a|+|b|)^{p}\leq C_{p}(|a|^{p}+|b|^{p})$, where $ C_{p}=2^{p-1}$
 when $p>1$ and where $ C_{p}=1$ when $0<p\leq1$, $a, b\in \mathbb{R}$ 
(see \cite{k3}).
We can get
\begin{align*}
\| {D_{0^+}^\alpha  u} \|_\infty ^{p - 1}
&\leq \frac{{1}}{{\Gamma (\beta  + 1)}}(\| a\|_\infty  + \| b \|_\infty \| u \|_\infty ^{p - 1}  + \| c \|_\infty \| {D_{0^+}^{\alpha  -2 } u} \|_\infty ^{p - 1}\\
&\quad +\| d \|_\infty \| {D_{0^+}^{\alpha  -1 } u} \|_\infty ^{p - 1}
 + \| e \|_\infty \| {D_{0^+}^\alpha  u} \|_\infty ^{p - 1})\\
&\le \frac{{1 }}{{\Gamma (\beta  + 1)}}(\| a \|_\infty  
 + \| b \|_\infty D (\frac{1}{{\Gamma (\alpha )^{p - 1}}}\| {D_{0^+}^\alpha  u}
 \|_\infty^{p - 1} + \frac{A^{p - 1}}{{\Gamma (\alpha )^{p - 1}}})\\
 &\quad + \| c \|_\infty D(2^{p - 1}A^{p - 1} + \| {D_{0^+}^\alpha  u}
 \|_\infty ^{p - 1} )
 + \| d \|_\infty D(A^{p - 1} + \| {D_{0^+}^\alpha  u} \|_\infty ^{p - 1} )\\
 &\quad + \| e \|_\infty \| {D_{0^+}^\alpha  u} \|_\infty ^{p - 1} ).
\end{align*}
where $D=\max \{1,2^{p-2}\}$. From \eqref{e3.1}, we can see that there 
exists a constant $M_1 > 0$ such that
\begin{gather*}
\| {D_{0^+}^\alpha  u} \|_\infty   \le M_1 ,\quad
\| {D_{0^+}^{\alpha  - 1} u} \|_\infty   \le A + M_1 : = M_2 ,\\
\| {D_{0^+}^{\alpha  - 2} u} \|_\infty   \le 2A + M_1 : = M_3,\quad
\| u \|_\infty   \le \frac{1}{{\Gamma (\alpha )}}M_1  
 + \frac{A}{{\Gamma (\alpha )}}: = M_4. 
\end{gather*}
Thus
\begin{align*}
\| u \|_X 
& = \max \big\{ {\| u \|_\infty ,\| {D_{0^+}^{\alpha  - 2} u} \|_\infty  ,
 \| {D_{0^+}^{\alpha  - 1} u} \|_\infty  ,\| {D_{0^+}^\alpha  u} \|_\infty  }
 \big\} \\
&\le \max \{ {M_1 ,M_2 ,M_3,M_4 } \} := M.
\end{align*}
Therefore, $\Omega_1$ is bounded.
\end{proof}

\begin{lemma} \label{lem3.4} Suppose {\rm (H2)} holds, then the set
$\Omega_2=\{ u\in \ker M:  Nu \in \operatorname{Im} M\}$
is bounded.
\end{lemma}

\begin{proof} 
For each $u\in\Omega_2$, we can have that $u(t) = ct^{\alpha-1}$ 
for all $c\in \mathbb{R}$ and $QNu = 0$.
It follow from (H2) that there exists a $t_{0}\in [0,1]$ such that 
$| {D_{0^+}^{\alpha  - 1} u(t_0 )} |+| {D_{0^+}^{\alpha  - 2} u(t_0 )}| \le A$,
which implies $|c| \le \frac{A}{{\Gamma (\alpha )}(1+t_0)}$. 
Therefore, $\Omega_2$ is bounded.
\end{proof}

\begin{lemma} \label{lem3.5} 
Suppose {\rm (H3)} holds, then the set
\[
\Omega_3  = \{ {u \in \ker M: (-1)^{m}\lambda J^{ - 1} u + (1 - \lambda )QNu = 0,\,
\lambda  \in [0,1]}\}
\]
is bounded, where $m=1$ when $\Lambda < 0$ and  $m=2$ when $\Lambda >0$.
\end{lemma}

\begin{proof} 
Case 1, suppose  $\Lambda < 0$, for each $u \in\Omega_{3}$, we can get 
that $u(t) =ct^{\alpha-1}$ for all $c\in \mathbb{R}$. 
We define the isomorphism
$J: \operatorname{Im}Q \to \ker M $ by 
$J(c) = ct^{\alpha  - 1} ,c \in R,t \in [0,1]$. So, we have
\begin{equation} \label{e3.4}
\begin{split}
\lambda c
&=(1 - \lambda )\varphi_{p}\Big(\frac{1}{\Delta }
\Big(\int_0^1 \varphi_q (\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1}
f(\tau ,c\tau ^{\alpha  - 1} ,c\Gamma(\alpha)\tau,c\Gamma (\alpha ),0)d\tau } )ds  \\
&\quad -\sum_{i = 1}^m  {\sigma_i } \int_0^{\eta_i }  \varphi_q
\Big(\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1}
 f(\tau ,c\tau ^{\alpha  - 1},c\Gamma(\alpha)\tau ,c\Gamma (\alpha ),0)d\tau } 
\Big)ds\Big)\Big).
\end{split}
\end{equation}
If $\lambda = 0$, then $|c|\leq B$ because of the first part of (H3). 
If $\lambda\in (0, 1]$, we can also obtain $|c|\leq B$. 
Otherwise, if $|c| > B$, in view of the first part of (H3), one has
\begin{equation} \label{e3.5}
\begin{split}
&c(1 - \lambda )\varphi_{p} \Big(\frac{1}{\Delta }
\Big(\int_0^1  \varphi_q (\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1}
f(\tau ,c\tau ^{\alpha  - 1},c\Gamma(\alpha)\tau ,c\Gamma (\alpha ),0)d\tau } )ds\\
&-\sum_{i = 1}^m  {\sigma_i } \int_0^{\eta_i } \varphi_q
 \Big(\frac{1}{{\Gamma (\beta )}}\int_0^s {(s - \tau )^{\beta  - 1}
 f(\tau ,c\tau ^{\alpha  - 1},c\Gamma(\alpha)\tau ,c\Gamma (\alpha ),0)d\tau } \Big)ds
\Big)\Big) 
\leq 0.
\end{split}
\end{equation}
On the other hand, $\lambda c^{2}>0$ which contradicts to \eqref{e3.4}.  
Therefore, $\Omega_3$  is bounded.

 Case 2, suppose  $\Lambda > 0$, it is similar to case 1 to proof $\Omega_3$
is bounded. So, we omit it.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.1}]
 Assume that $\Omega$ is a bounded open set of $X$ with 
 $\cup_{i = 1}^3 {\overline {\Omega_i } }  \subset \Omega $.
 By Lemma \ref{lem3.2}, we obtain that $N$ is $M$-compact on $\overline{\Omega}$. 
Then by Lemmas \ref{lem3.3} and \ref{lem3.4}, we have
\begin{itemize}
\item[(i)] $Mx \ne N_{\lambda}x$ for each 
$(u,\lambda ) \in (\operatorname{dom}M\backslash \ker M)  \times (0,1)$,

\item[(ii)] $QNu\neq0$, for all $u \in \partial\Omega\cap\ker M$.
\end{itemize}
Thus, we need to prove that (iii) of Lemma \ref{lem2.1} is true,
Let $I$ be the identity operator in the Banach space $X$, and 
$H(u,\lambda ) =  (-1)^{m}\lambda J^{ - 1} (u) + (1 - \lambda )QN(u)$.
According to Lemma \ref{lem3.5} we know that for each 
$u \in \partial \Omega  \cap \ker M$, $H(u,\lambda ) \ne 0$. 
Thus, by the homotopic property of degree, we have
\begin{align*}
\deg (JQN|_{\ker M} ,\Omega  \cap \ker M,0)
&= \deg (H( \cdot ,0),\Omega  \cap \ker M,0)\\
&= \deg (H( \cdot ,1),\Omega  \cap \ker M,0)\\
& = \deg ( \pm I,\Omega  \cap \ker M,0) \ne 0.
\end{align*}
which means (iii) of Lemma \ref{lem2.1} is satisfied. 
Consequently, by Lemma \ref{lem2.1}, the equation $Mu = Nu$ has at least one 
solution in $\operatorname{dom} M \cap {\Omega}$. 
Namely, BVP \eqref{e1.2} have at least one solution in the space $X$.
\end{proof}


\subsection*{Acknowledgements}
This research as supported by grant 11271364 from the NNSF of China.
The authors are grateful to those who give valuable suggestions 
about the original manuscript.

\begin{thebibliography}{00}

\bibitem{b1} J. Bai, X. Feng; 
\emph{Fractional-order anisotropic diffusion for image denoising, 
Computers and Mathematics with
Applications}, IEEE Transactions on Image Processing, 16 (2007), 2492-2502.

\bibitem{b2} Z. Bai, H. L\"u; 
\emph{Positive solutions for boundary value problem of nonlinear fractional 
differential equation}, J. Math. Anal. Appl., 311 (2005), 495-505.

\bibitem{b3} Z. Bai; 
\emph{On solutions of some fractional m-point boundary value
problems at resonance}, Electron. J. Qual. Theory Differ. Equ., 37 (2010), 1-15.

\bibitem{b4} Z. Bai; 
\emph{Solvability of fractional three-point boundary value problems with
nonlinear growth}, Appl. Math. Comput., 218 (2011), 1719-1725.

\bibitem{b5} M. Benchohra, S. Hamani, S. K. Ntouyas;
\emph{Boundary value problems for differential equations with fractional
order and nonlocal conditions}, Nonlinear Anal., 71 (2009), 2391-2396.

\bibitem{c1} T. Chen, W. Liu, Z. Hu;
 \emph{A boundary value problem for fractional differential equation with 
p-Laplacian operator at Resonance}, Nonlinear Analysis, 75 (2012), 3210-3217.

\bibitem{c2} Y. Chen, X. Tang; 
\emph{Solvability of sequential fractional order multi-point boundary value
problems at resonance}, Applied Mathematics and Computation, 218 (2012), 7638-7648.

\bibitem{g1} W. Ge; 
\emph{Boundary Value Problems for Ordinary Nonlinear Differential Equations}, 
Science Press, Beijing, Chinese, 2007.

\bibitem{h1} Z. Hu , W. Liu; 
\emph{Solvability for fractional order boundary value
problem at resonance}, Boundary Value Problem, 20 (2011), 1-10.

\bibitem{j1} H. Jafari, V. D. Gejji; 
\emph{Positive solutions of nonlinear fractional boundary value problems 
using Adomian decomposition method}, Appl. Math. Comput., 180 (2006), 700-706.

\bibitem{k1} E. R. Kaufmann, E. Mboumi; 
\emph{Positive solutions of a boundary value problem for a nonlinear fractional 
differential equation}, Electron. J. Qual. Theory Differ. Equ., 3 (2008), 1-11.

\bibitem{k2} A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; 
\emph{Theory and Applications of Fractional
Differential Equations}, Elsevier B.V, Netherlands, 2006.

\bibitem{k3} J. Kuang; 
\emph{Applied Inequalities}, Shandong Science and Technology Press, 
Shandong, Chinese, 2004.

\bibitem{l1} S. Liang, J. Zhang; 
\emph{Positive solutions for boundary value problems of nonlinear fractional 
differential equation}, Nonlinear Anal., 71 (2009), 5545-5550.

\bibitem{m1} R. Magin;
\emph{Fractional calculus models of complex dynamics in biological tissues},
Computers and Mathematics with Applications, 59 (2010), 1586-1593.

\bibitem{m2} J. Mawhin; 
\emph{Topological degree and boundary value problems for nonlinear differential
equations in topological methods for ordinary differential equations}, Lecture
Notes in Math, 1537 (1993), 74-142.

\bibitem{s1} J. Sabatier, O. P. Agrawal, J. A. T. Machado;
\emph{Advances in fractional calculus: Theoretical developments and applications 
in physics and engineering}, Springer, Dordrecht, 2007.

\bibitem{s2} S. G. Samko, A. A. Kilbas, O. I. Marichev; 
\emph{Fractional Integrals and Derivatives: Theory and Applications},
 Gordon and Breach, New York, NY, USA, 1993.

\bibitem{s3} X. Su; 
\emph{Boundary value problem for a coupled system of nonlinear fractional
 differential equations}, Appl. Math. Lett., 22 (2009), 64-69.

\bibitem{w1} J. Wang; 
\emph{The existence of solutions to boundary value problems of fractional
differential equations at resonance}, Nonlinear Analysis, 74 (2011), 1987-1994.

\bibitem{z1} Y. Zhang, Z. Bai;
\emph{ Existence of positive solutions for nonlinear
fractional three-point boundary value problen at resonance}, Appl.
Math. Comput., 36 (2011), 417-440.

\bibitem{z2} S. Zhang; 
\emph{Positive solutions for boundary-value problems of nonlinear fractional 
differential equations}, Electron. J. Differ. Equ., 36 (2006), 1-12.

\end{thebibliography}

\end{document}