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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 213, pp. 1--14.\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-2012/213\hfil Positive solutions to boundary-value problems]
{Positive solutions to boundary-value problems of p-Laplacian fractional
differential equations with a parameter in the boundary}

\author[Z. Han, H. Lu, S. Sun, D. Yang \hfil EJDE-2012/213\hfilneg]
{Zhenlai Han, Hongling Lu, Shurong Sun, Dianwu Yang }  % in alphabetical order

\address{Zhenlai Han \newline
School of Mathematical Sciences, University of Jinan, Jinan,
Shandong 250022,  China}
\email{hanzhenlai@163.com}

\address{Hongling Lu \newline
 School of Mathematical Sciences, University of Jinan, Jinan,
  Shandong 250022,  China}
\email{lhl4578@126.com}

\address{Shurong Sun \newline
School of Mathematical Sciences, University of Jinan, Jinan,
Shandong 250022, China}
\email{sshrong@163.com}

\address{Dianwu Yang \newline
School of Mathematical Sciences, University of Jinan, Jinan,
Shandong 250022, China}
\email{ss\_yangdw@ujn.edu.cn}


\thanks{Submitted September 5, 2012. Published November 27, 2012.}
\subjclass[2000]{34A08, 34B18, 35J05}
\keywords{Fractional boundary-value problem;
 positive solution; cone; \hfill\break\indent 
Schauder fixed point theorem;  uniqueness; $p$-Laplacian operator}

\begin{abstract}
 In this article, we consider the following boundary-value problem
 of nonlinear fractional differential equation with
 $p$-Laplacian operator
 \begin{gather*}
 D_{0+}^\beta(\phi_p(D_{0+}^\alpha u(t)))+a(t)f(u)=0, \quad 0<t<1, \\
  u(0)=\gamma u(\xi)+\lambda, \quad
  \phi_p(D_{0+}^\alpha u(0))=(\phi_p(D_{0+}^\alpha u(1)))'
  =(\phi_p(D_{0+}^\alpha u(0)))''=0,
 \end{gather*}
 where $0<\alpha\leqslant1$, $2<\beta\leqslant 3$ are real numbers,
 $D_{0+}^\alpha, D_{0+}^\beta$ are the standard Caputo fractional derivatives,
 $\phi_p(s)=|s|^{p-2}s$, $p>1$,
 $\phi_p^{-1}=\phi_q$, $1/p+1/q=1$, $0\leqslant\gamma<1$,
 $0\leqslant\xi\leqslant1$, $\lambda>0$ is a parameter,
 $a:(0,1)\to [0,+\infty)$ and $f:[0,+\infty)\to[0,+\infty)$ are continuous.
 By the properties of Green function and Schauder fixed point theorem,
 several  existence and nonexistence results for positive solutions,
 in terms of the parameter $\lambda$ are obtained.
 The uniqueness of positive solution on the parameter $\lambda$ is also studied.
 Some examples are presented to illustrate the main results.
\end{abstract}

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

\section{Introduction}

  Fractional differential equations have been of great
interest. The motivation for those works stems from both the
intensive development of the theory of fractional calculus itself
and the applications such as economics, engineering and other fields
\cite{Agarwal,Kilbas, machado, Meral, Oldham,Podlubny,Weitzner}.

Recently, much attention has been focused on the study of the
existence and multiplicity of solutions or positive solutions for
boundary-value problems of fractional differential equations by the
use of techniques of nonlinear analysis  (fixed point theorems
\cite{bai, chai,Chen21,Feng17, wang12,Wang14, xu16,Xu20, yang19, 
Zhao15, Zhao18, zhou13},
upper and lower solutions method \cite{Liang,Lu,Wang}, fixed point
index \cite{Dix,Xu}, coincidence theory \cite{Chen22}, Banach
contraction mapping principle \cite{Liu}, etc).

Ma \cite{ma23} considered the boundary-value problem
\begin{gather*}
u''+a(t)f(u)=0, \quad 0<t<1,\\
u(0)=0, \quad u(1)-\alpha u(\eta)=b,
\end{gather*}
where $b, \alpha>0$, $\eta\in(0,1)$, $\alpha\eta<1$ are given.
Under some assumptions, it was shown that there exists $b^*>0$ such that
the boundary-value problem has at least one positive solution for $0<b<b^*$
and no positive solution for $b>b^*$.

Kong et al \cite{kong} studied the boundary-value problem with
 nonhomogeneous three-point boundary condition
\begin{gather*}
(\phi_p (u'))''+a(t)f(u)=0, \quad 0<t<1, \\
u(0)=\xi u(\eta)+\lambda, \quad u'(0)=u'(1)=0,
\end{gather*}
where $\phi_p(s)=|s|^{p-2}s$, $p>1$,
 $\phi_p^{-1}=\phi_q$, $1/p+1/q=1$. $0\leqslant\xi<1$, 
 $0\leqslant\eta\leqslant1$, $\lambda>0$ is a parameter,
 $a\in C(0,1)$ and $f\in C([0,+\infty))$ are nonnegative functions.
Under some assumptions, several existence, nonexistence, and multiplicity 
results for positive solutions in terms of different values of the parameter
 $\lambda$ are derived.

Zhao et al \cite{Zhao25} studied the existence of positive solutions 
for the boundary-value problem of nonlinear fractional differential equations
\begin{gather*}
D_{0+}^\alpha u(t)+\lambda f(u(t))=0, \quad 0<t<1, \\
u(0)=u(1)=u'(0)=0,
\end{gather*}
where $2<\alpha\leqslant3$ is a real number, $D_{0+}^\alpha$ is the standard
Riemann-Liouville fractional derivative, $\lambda$ is a positive parameter, 
and $f: (0,+\infty)\to(0,+\infty)$ is continuous.
 By the properties of the Green function
and Guo-Krasnoselskii fixed point theorem on cones, some sufficient conditions
for the nonexistence and existence of at least one or two positive solutions 
for the boundary-value problem are established.

Chai \cite{chai} investigated the existence and multiplicity
of positive solutions for a class of boundary-value problem of
fractional differential equation with $p$-Laplacian operator
\begin{gather*}
D_{0+}^\beta(\phi_p(D_{0+}^\alpha u(t)))+f(t,u(t),D_{0+}^\rho
u(t))=0,\quad 0<t<1, \\
u(0)=0, u(1)+\sigma D_{0+}^\gamma u(1)=0, \quad D_{0+}^\alpha u(0)=0,
\end{gather*}
where $1<\alpha\leq 2, 0<\gamma\leq 1$, $0<\gamma\leq 1$,
$0\leq\alpha-\gamma-1$, $\sigma$ is a positive constant number,
$D_{0+}^\alpha, D_{0+}^\beta, D_{0+}^\gamma$ are the standard
Riemann-Liouville derivatives. By means of the fixed point theorem
on cones, some existence and multiplicity results of positive
solutions are obtained.

Although the fractional differential equation boundary-value problems have been studied 
by several authors, very little is known in the literature on the existence 
and nonexistence of positive solutions of fractional differential equation
boundary-value problems with $p$-Laplacian operator
when a parameter $\lambda$ is involved in the boundary conditions.
 We also mention that, there is very little known about the uniqueness 
 of the solution of fractional differential equation boundary-value problems
with $p$-Laplacian operator on the parameter $\lambda$. 
Therefore,  to enrich the theoretical knowledge of the above,
in this paper, we investigate the following $p$-Laplacian fractional
differential equation boundary-value problem
\begin{gather} \label{e1.1}
D_{0+}^\beta(\phi_p(D_{0+}^\alpha u(t)))+a(t)f(u)=0, \quad 0<t<1,\\
 \label{e1.2}
u(0)=\gamma u(\xi)+\lambda, \quad 
\phi_p(D_{0+}^\alpha u(0))=(\phi_p(D_{0+}^\alpha u(1)))'
=(\phi_p(D_{0+}^\alpha u(0)))''=0,
\end{gather}
where $0<\alpha\leqslant1$, $2<\beta\leqslant3$ are real numbers,
$D_{0+}^\alpha, D_{0+}^\beta$ are the standard Caputo fractional derivatives,
$\phi_p(s)=|s|^{p-2}s$, $p>1$,
$\phi_p^{-1}=\phi_q$, $1/p+1/q=1$, $0\leqslant\gamma<1$, 
$0\leqslant\xi\leqslant1$, $\lambda>0$ is a parameter,
$a:(0,1)\to [0,+\infty)$ and $f:[0,+\infty)\to[0,+\infty)$ are continuous.
 By the properties of Green function and Schauder fixed point theorem, 
several new existence and nonexistence results for positive
solutions in terms of different values of the parameter $\lambda$ are obtained.
The uniqueness of positive solution is also obtained for fractional differential
equation boundary-value problem \eqref{e1.1} and \eqref{e1.2}.
As applications, examples are presented to illustrate our main results.

The rest of this paper is organized as follows. In Section 2, we
shall introduce some definitions and lemmas to prove our main
results. In Section 3, we investigate the existence of
positive solution for boundary-value problems \eqref{e1.1} and \eqref{e1.2}.
In Section 4, the uniqueness of positive solution on the parameter $\lambda$ 
is studied.
In Section 5, we consider the nonexistence of
positive solution for boundary-value problems \eqref{e1.1} and \eqref{e1.2}. As
applications, examples are presented to illustrate our main results
in Section 3, Section 4 and Section 5, respectively.

\section{Preliminaries and lemmas}

  For the convenience of the reader, we give some
background material from fractional calculus theory to facilitate
analysis of problem \eqref{e1.1} and \eqref{e1.2}. 
These results can be found in
the recent literature, see \cite{diethelm,Kilbas,Podlubny,Wang14}.

\begin{definition}[\cite{Kilbas}]\label{d2.1}\rm
The Riemann-Liouville fractional integral of order $\alpha>0$ of a
function $y:(0,+\infty)\to \mathbb{R}$ is given by
$$
I_{0+}^\alpha y(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^t(t-s)^{\alpha-1}y(s)ds
$$
provided the right side is pointwise defined on $(0,+\infty)$.
\end{definition}

\begin{definition}[\cite{Kilbas}]\label{d2.2}\rm
 The Caputo fractional derivative of order
$\alpha>0$ of a continuous function $y:(0,+\infty)\to \mathbb{R}$ is
given by
$$
D_{0+}^\alpha y(t)=\frac{1}{\Gamma(n-\alpha)}
\int_0^t\frac{y^{(n)}(s)}{(t-s)^{\alpha-n+1}}ds,
$$
where $n$ is the smallest integer greater than or equal to $\alpha$,
provided that the right side is pointwise defined on $(0,+\infty)$. 
\end{definition}

\begin{remark}[\cite{Podlubny}]\label{r2.1}\rm
By Definition \ref{d2.2}, under natural
conditions on the function $f(t)$, for $\alpha\to n$ the Caputo
derivative becomes a conventional $n$-th derivative of the function
$f(t)$.
\end{remark}

\begin{remark}[\cite{Kilbas}]\label{r2.2} \rm
As a basic example, 
$$
D_{0^+}^\alpha t^\mu=\mu(\mu-1)\dots
(\mu-n+1)\frac{\Gamma(1+\mu-n)}{\Gamma(1+\mu-\alpha)}t^{\mu-\alpha},\quad
\text {for } t\in(0,\infty).
$$ 
In particular $D_{0^+}^\alpha t^\mu=0$,
$\mu=0,1,\dots,n-1$, where $D^\alpha_{0^+}$ is the Caputo
fractional derivative, $n$ is the smallest integer greater than or
equal to $\alpha$.
\end{remark}

From the definition of the Caputo derivative and Remark \ref{r2.2}, we can
obtain the following statement.

\begin{lemma}[\cite{Kilbas}]\label{l2.1}\rm
 Let $\alpha>0$. Then the fractional differential equation
$$
D_{0+}^\alpha u(t)=0
$$
has a unique solution
$$
u(t)=c_0+c_1t+c_2t^2+\dots+c_{n-1}t^{n-1}, \quad c_i\in \mathbb{R},\,
i=0,1,2,\dots,n-1, 
$$
where $n$ is the smallest integer greater than or equal to $\alpha$.
\end{lemma}

\begin{lemma}[\cite{Kilbas}]\label{l2.2}\rm
Let $\alpha>0$. Assume that $u\in C^n[0,1]$. Then
$$
I_{0+}^\alpha D_{0+}^\alpha u(t)=u(t)+c_0+c_1t+c_2t^2+\dots+c_{n-1}t^{n-1},
$$
for some $c_i\in \mathbb{R}$, $i=0,1,2,\dots,n-1$, where $n$ is the
smallest integer greater than or equal to $\alpha$.
\end{lemma}

\begin{lemma}\label{l2.3}
 Let $y\in C[0,1]$ and $0<\alpha\leqslant1$.
Then fractional differential equation boundary-value problem
\begin{gather}\label{e2.1}
D_{0+}^\alpha u(t)=y(t),\quad 0<t<1,\\
\label{e2.2}
u(0)=\gamma u(\xi)+\lambda
\end{gather}
has a unique solution
$$
u(t)=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds
+ \frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds
+\frac{\lambda}{1-\gamma}.
$$
\end{lemma}

\begin{proof}
 We  apply Lemma \ref{l2.2} to reduce \eqref{e2.1} to an equivalent
 integral equation,
$$
u(t)=I_{0+}^\alpha y(t)+c_0, \quad c_0\in \mathbb{R}.
$$
Consequently, the general solution of \eqref{e2.1} is
$$
u(t)=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds+c_0, \quad
 c_0\in \mathbb{R}.
$$
By \eqref{e2.2}, we has
$$
c_0=\frac{\gamma}{1-\gamma}\int_0^\xi
\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds+\frac{\lambda}{1-\gamma}.
$$
Therefore, the unique solution of problem \eqref{e2.1} and \eqref{e2.2} is
$$
u(t)=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds+
\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds
+\frac{\lambda}{1-\gamma}. 
$$
\end{proof}

\begin{lemma}\label{l2.4}
Let $y\in C[0,1]$ and $0<\alpha\leqslant1$, $2<\beta\leqslant3$.
Then fractional differential equation boundary-value problem
\begin{gather}\label{e2.3}
D_{0+}^\beta(\phi_p(D_{0+}^\alpha u(t)))+y(t)=0, \quad 0<t<1,\\
\label{e2.4}
u(0)=\gamma u(\xi)+\lambda, \quad
 \phi_p(D_{0+}^\alpha u(0))=(\phi_p(D_{0+}^\alpha u(1)))'
=(\phi_p(D_{0+}^\alpha u(0)))''=0
\end{gather}
has a unique solution
\begin{equation}\label{e2.5}
\begin{aligned}
u(t)&=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)y(\tau)d\tau\Big)ds\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)y(\tau)d\tau\Big)ds+
\frac{\lambda}{1-\gamma},
\end{aligned}
\end{equation}
where
\begin{equation}\label {e2.6}
H(t,s )=\begin{cases}
\frac{(\beta-1)t(1-s)^{\beta-2}-(t-s)^{\beta-1}}{\Gamma(\beta)},
&0\leqslant s \leqslant t \leqslant 1,\\[3pt]
\frac{(\beta-1)t(1-s)^{\beta-2}}{\Gamma(\beta)},&
0\leqslant t \leqslant s \leqslant 1.
\end{cases}
\end{equation}
\end{lemma}

\begin{proof} 
From Lemma \ref{l2.2}, the boundary-value problem \eqref{e2.3} 
and \eqref{e2.4} is equivalent to the integral equation
$$
\phi_p(D_{0+}^\alpha u(t))=-I_{0+}^\beta y(t)+c_0+c_1t+c_2t^2,
$$
for some  $c_0, c_1, c_2 \in \mathbb{R}$; that is,
$$
\phi_p(D_{0+}^\alpha u(t))
=-\int_0^t\frac{(t-\tau)^{\beta-1}}{\Gamma(\beta)}y(\tau)d\tau
+c_0+c_1t+c_2t^2.
$$
By the boundary conditions 
$ \phi_p(D_{0+}^\alpha u(0))=(\phi_p(D_{0+}^\alpha u(1)))'
=(\phi_p(D_{0+}^\alpha u(0)))''=0$, we have
$$
c_0=c_2=0, \quad
 c_1=\int_0^1\frac{(\beta-1)(1-\tau)^{\beta-2}}{\Gamma(\beta)}y(\tau)d\tau.
$$
Therefore, the solution $u(t)$ of fractional differential equation
boundary-value problem \eqref{e2.3} and \eqref{e2.4} satisfies
\begin{align*}
\phi_p(D_{0+}^\alpha u(t))
&=-\int_0^t\frac{(t-\tau)^{\beta-1}}{\Gamma(\beta)}y(\tau)d\tau+
\int_0^1\frac{(\beta-1)t(1-\tau)^{\beta-2}}{\Gamma(\beta)}y(\tau)d\tau\\
&=\int_0^1H(t,\tau)y(\tau)d\tau.
\end{align*}
Consequently, $D_{0+}^\alpha u(t)
=\phi_q\Big(\int_0^1H(t,\tau)y(\tau)d\tau\Big)$. 
Thus, fractional differential equation boundary-value problem \eqref{e2.3} 
and \eqref{e2.4} is equivalent to the  problem
\begin{gather*}
D_{0+}^\alpha u(t)=\phi_q\Big(\int_0^1H(t,\tau)y(\tau)d\tau\Big),\quad 0<t<1,
\\
u(0)=\gamma u(\xi)+\lambda.
\end{gather*}
Lemma \ref{l2.3} implies that fractional differential equation boundary-value 
problem \eqref{e2.3} and \eqref{e2.4} has a unique solution,
\begin{align*}
u(t)&=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\phi_q
\Big(\int_0^1H(s,\tau)y(\tau)d\tau\Big)ds\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)y(\tau)d\tau\Big)ds+
\frac{\lambda}{1-\gamma}.
\end{align*}
The proof is complete.
\end{proof}

\begin{lemma}[\cite{Wang14}]\label{l2.5}
Let $0<\alpha\leqslant1,2<\beta\leqslant3$. The function $H(t,s)$ 
defined by \eqref{e2.6} is
continuous on $[0,1]\times[0,1]$ and satisfies
\begin{itemize}
\item[(1)] $H(t,s)\geqslant0, H(t,s)\leqslant H(1,s)$, \; for $t,s\in[0,1]$;

\item[(2)] $H(t,s)\geqslant t^{\beta-1}H(1,s)$, \; for $t,s\in(0,1)$.
\end{itemize}
\end{lemma}

\begin{lemma}[Schauder fixed point theorem \cite{diethelm}] \label{l2.6}
Let $(E,d)$ be a complete metric space, $U$ be
a closed convex subset of $E$, and $A : U\to U$ be a mapping such that the 
set $\{Au : u\in U\}$ is relatively compact in $E$. Then $A$ has at 
least one fixed point.
\end{lemma}

To prove our main results, we use the following assumptions.
\begin{itemize}
\item[(H1)] $0<\int_0^1H(1,\tau)a(\tau)d\tau<+\infty$;

\item[(H2)] there exist $0<\sigma<1$ and $c>0$ such that
\begin{equation}\label {e2.7}
f(x)\leqslant\sigma L\phi_p(x), \quad \text{for }  0\leqslant x\leqslant c,
\end{equation}
where $L$ satisfies
\begin{equation}\label {e2.8}
0<L\leqslant\Big[\phi_p
\Big(\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}\Big)
\int_0^1H(1,\tau)a(\tau)d\tau\Big]^{-1};
\end{equation}

\item[(H3)] there exist $d>0$ such that
\begin{equation} \label{e2.9}
f(x)\leqslant M\phi_p(x), \quad \text{for } d<x<+\infty,
\end{equation}
where $M$ satisfies
\begin{equation}\label {e2.10}
0<M<\Big[\phi_p
\Big(\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}2^{q-1}\Big)
\int_0^1H(1,\tau)a(\tau)d\tau\Big]^{-1};
\end{equation}

\item[(H4)] there exist $0<\delta<1$ and $e>0$ such that
\begin{equation}\label {e2.11}
f(x)\geqslant N\phi_p(x), \quad \text{for }  e<x<+\infty,
\end{equation}
where $N$ satisfies
\begin{equation}\label {e2.12}
N>\Big[\phi_p
\Big(c_\delta\int_0^1\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q(s^{\beta-1})ds\Big)
\int_\delta^1H(1,\tau)a(\tau)d\tau\Big]^{-1};
\end{equation}
with
\begin{equation}\label {e2.13}
c_\delta=\int_0^\delta\alpha(1-s)^{\alpha-1}\phi_q(s^{\beta-1})ds\in(0,1);
\end{equation}

\item[(H5)] $f(x)$ is nondecreasing in $x$;

\item[(H6)] there exist $0\leqslant\theta<1$ such that
\begin{equation}\label {e2.14}
f(kx)\geqslant(\phi_p(k))^\theta f(x), \quad
\text{for any $0<k<1$ and  $0<x<+\infty$}.
\end{equation}
\end{itemize}

\begin{remark}\label{r2.1b} \rm
Let
$$
f_0=\lim_{x\to0^+}\frac{f(x)}{\phi_p(x)}, \quad 
f_\infty=\lim_{x\to+\infty}\frac{f(x)}{\phi_p(x)}.
$$
Then, (H2) holds if $f_0=0$, (H3) holds if $f_\infty=0$,
 and (H4) holds if $f_\infty=+\infty$.
\end{remark}

\section{Existence}

\begin{theorem}\label{t3.1} Assume that {\rm (H1), (H2)} hold. Then
the fractional differential equation boundary-value problem \eqref{e1.1}
and \eqref{e1.2} has at least one positive solution for 
$0<\lambda\leqslant(1-\gamma)(1-\phi_q(\sigma))c$.
\end{theorem}

\begin{proof} 
Let $c>0$ be given in (H2). Define
$$
K_1=\{u\in C[0,1] : 0\leqslant u(t)\leqslant c \text{ on } [0,1]\}
$$
and an operator $T_\lambda : K_1\to C[0,1]$ by
\begin{equation}\label{e3.1}
\begin{aligned}
T_\lambda u(t)
&=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)a(\tau)f(u(\tau))d\tau\Big)ds\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)a(\tau)f(u(\tau))d\tau\Big)ds+
\frac{\lambda}{1-\gamma}.
\end{aligned}
\end{equation}
Then, $K_1$ is a closed convex set. From Lemma \ref{l2.4},
 $u$ is a solution of fractional differential
equation boundary-value problem \eqref{e1.1} and \eqref{e1.2} 
if and only if $u$ is a fixed point of $T_\lambda$.
Moreover, a standard argument can be used to show that $T_\lambda$ is compact.

For any $u\in K_1$, from \eqref{e2.7} and \eqref{e2.8}, we obtain
$$
f(u(t))\leqslant\sigma L\phi_p(u(t))\leqslant\sigma L\phi_p(c), \quad
 \text{on } [0,1],
$$
and
$$
\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}\phi_q(L)
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)d\tau\Big)\leqslant1.
$$
Let $0<\lambda\leqslant(1-\gamma)(1-\phi_q(\sigma))c$. 
Then, from Lemma \ref{l2.5} and \eqref{e3.1}, it follows that
\begin{align*}
0\leqslant T_\lambda u(t)
&\leqslant\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)ds\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)ds+
\frac{\lambda}{1-\gamma}\\
&\leqslant\frac{1}{\Gamma(\alpha+1)}
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)\\
&\quad +\frac{\gamma\xi^\alpha}{\Gamma(\alpha+1)(1-\gamma)}
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)+
(1-\phi_q(\sigma))c\\
&=\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)+
(1-\phi_q(\sigma))c\\
&\leqslant\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}\phi_q(L)
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)d\tau\Big)\phi_q(\sigma)c+
(1-\phi_q(\sigma))c\\
&\leqslant\phi_q(\sigma)c+(1-\phi_q(\sigma))c=c,\quad  t\in[0,1].
\end{align*}
Thus, $T_\lambda(K_1)\subseteq K_1$, By Schauder fixed point theorem, 
$T_\lambda$ has a fixed point $u\in K_1$; that is,
the fractional differential equation boundary-value 
problem \eqref{e1.1} and \eqref{e1.2} has at least one positive solution.
The proof is complete.
\end{proof}

\begin{corollary}\label{c3.1}
 Assume that {\rm (H1)} holds and $f_0=0$. Then
the fractional differential equation boundary-value problem \eqref{e1.1}
and \eqref{e1.2} has at least one positive solution for sufficiently
 small $\lambda>0$.
\end{corollary}

\begin{theorem}\label{t3.2} Assume that {\rm (H1), (H3)} hold. Then
the fractional differential equation boundary-value problem \eqref{e1.1}
and \eqref{e1.2} has at least one positive solution for all $\lambda>0$.
\end{theorem}

\begin{proof}  Let $\lambda>0$ be fixed and $d>0$ be given in (H3). Define
$D=\max_{0\leqslant x\leqslant d}f(x)$. Then
\begin{equation}\label{e3.2}
f(x)\leqslant D, \quad \text{for }  0\leqslant x\leqslant d.
\end{equation}
From \eqref{e2.10}, we have
$$
\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}2^{q-1}\phi_q(M)
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)d\tau\Big)<1.
$$
Thus, there exists $d^*>d$ large enough so that
\begin{equation}\label{e3.3}
\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}2^{q-1}
(\phi_q(D)+\phi_q(M)d^*)
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)d\tau\Big)+\frac{\lambda}{1-\gamma}
\leqslant d^*.
\end{equation}
Let
$$
K_2=\{u\in C[0,1] : 0\leqslant u(t)\leqslant d^* \text{ on } [0,1]\}.
$$
For $u\in K_2$, define
\begin{gather*}
I_1^u=\{t\in [0,1] : 0\leqslant u(t)\leqslant d\},\\
I_2^u=\{t\in [0,1] : d<u(t)\leqslant d^*\}.
\end{gather*}
Then, $I_1^u\cup I_2^u=[0,1], I_1^u\cap I_2^u=\emptyset$, and 
 in view of \eqref{e2.9}, we have
\begin{equation}\label{e3.4}
f(u(t))\leqslant M\phi_p(u(t))\leqslant M\phi_p(d^*), \quad \text{for }
 t\in I_2^u.
\end{equation}
Let the compact operator $T_\lambda$ be defined by \eqref{e3.1}. Then from
Lemma \ref{l2.5}, \eqref{e2.9} and \eqref{e3.2}, we have
\begin{align*}
0&\leqslant T_\lambda u(t)\\
&\leqslant\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)ds\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)ds+
\frac{\lambda}{1-\gamma}\\
&\leqslant\frac{1}{\Gamma(\alpha+1)}
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)\\
&\quad +\frac{\gamma\xi^\alpha}{\Gamma(\alpha+1)(1-\gamma)}
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)+
\frac{\lambda}{1-\gamma}\\
&=\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}
\phi_q\Big(\int_{I_1^u}H(1,\tau)a(\tau)f(u(\tau))d\tau+
\int_{I_2^u}H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)\\
&\quad +\frac{\lambda}{1-\gamma}\\
&\leqslant\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}
\phi_q\Big(D\int_{I_1^u}H(1,\tau)a(\tau)d\tau+
M\phi_p(d^*)\int_{I_2^u}H(1,\tau)a(\tau)d\tau\Big)\\
&\quad + \frac{\lambda}{1-\gamma}\\
&\leqslant\frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}\phi_q(D+
M\phi_p(d^*))\phi_q\Big(\int_0^1H(1,\tau)a(\tau)d\tau\Big)+
\frac{\lambda}{1-\gamma}.
\end{align*}
From \eqref{e3.3} and the inequality $(a+b)^r\leqslant2^r(a^r+b^r)$ 
for any $a, b, r>0$ (see, for example, \cite{hardy}), we obtain
\begin{align*}
0&\leqslant T_\lambda u(t)\\
&\leqslant \frac{1+\gamma(\xi^\alpha-1)}{\Gamma(\alpha+1)(1-\gamma)}2^{q-1}
(\phi_q(D)+\phi_q(M)d^*) \phi_q\Big(\int_0^1H(1,\tau)a(\tau)d\tau\Big)
+\frac{\lambda}{1-\gamma}\leqslant d^*.
\end{align*}
Thus, $T_\lambda : K_2\to K_2$. Consequently, by Schauder fixed point theorem,
$T_\lambda$ has a fixed point $u\in K_2$, that is,
the fractional differential equation boundary-value problem \eqref{e1.1} 
and \eqref{e1.2} has at least one positive solution.
The proof is complete.
\end{proof}

\begin{corollary}\label{c3.2} 
Assume that {\rm (H1)} holds and $f_\infty=0$. Then
the fractional differential equation boundary-value problem \eqref{e1.1}
and \eqref{e1.2} has at least one positive solution for all $\lambda>0$.
\end{corollary}

\begin{example} \label{examp3.1} \rm
Consider the boundary-value problem
\begin{gather} \label{e3.5}
D_{0+}^{5/2}(\phi_p(D_{0+}^{1/2} u(t)))+tu^2=0, \quad 0<t<1, \\
 \label{e3.6}
u(0)=\frac{1}{2} u(\frac{1}{2})+\lambda, \quad
\phi_p(D_{0+}^{1/2} u(0))=(\phi_p(D_{0+}^{1/2} u(1)))'
=(\phi_p(D_{0+}^{1/2} u(0)))''=0.
\end{gather}
Let $p=2$. We have $\alpha=1/2$, $\beta=\frac{5}{2}$,
$\gamma=\xi=1/2$, $a(t)=t$, $f(u)=u^2$. Clearly, (H1) holds.

By a simple computation, we obtain $0<L\leqslant2.4155$. 
Choosing $\sigma=1/2$, $c=1$, $L=2$, (H2) is satisfied. Thus, by
Theorem \ref{t3.1} the fractional differential equation boundary-value 
problem \eqref{e3.5} and \eqref{e3.6} has at least one positive 
solution for $0<\lambda\leqslant 1/4$.
\end{example}

\begin{example} \label{examp3.2}\rm
 Consider the boundary-value problem
\begin{gather} \label{e3.7}
D_{0+}^{5/2}(\phi_p(D_{0+}^{1/2} u(t)))+t\sqrt{u}=0, \quad  0<t<1,\\
 \label{e3.8}
u(0)=\frac{1}{2} u(\frac{1}{2})+\lambda, \quad
 \phi_p(D_{0+}^{1/2} u(0))=(\phi_p(D_{0+}^{1/2} u(1)))'
=(\phi_p(D_{0+}^{1/2} u(0)))''=0.
\end{gather}
Let $p=2$. We have $\alpha=1/2$, $\beta=5/2$, $\gamma=\xi=1/2$, 
$a(t)=t$, $f(u)=\sqrt{u}$. Clearly, (H1) holds.
By a simple computation, we obtain $0<M<1.2077$. 
Choosing $d=1, M=1$, then (H3) is satisfied. Thus, by
Theorem \ref{t3.2} the fractional differential equation boundary-value 
problem \eqref{e3.7}
and \eqref{e3.8} has at least one positive solution for all $\lambda>0$.
\end{example}


\section{Uniqueness}

\begin{definition}[\cite{guo27}] \label{d4.1}\rm
 A cone $P$ in a real Banach space $X$ is called solid if its interior
 $P^o$ is not empty.
\end{definition}

\begin{definition}[\cite{guo27}] \label{d4.2} \rm
Let $P$ be a solid cone in a real Banach space $X, T : P^o\to P^o$ be
an operator, and $0\leqslant\theta< 1$. Then T is called a
 $\theta$-concave operator if
$$
T(ku)\geqslant k^\theta Tu\quad \text{for any $0<k<1$ and $u\in P^o$}.
$$
\end{definition}

\begin{lemma}[{\cite[Theorem 2.2.6]{guo27}}] \label{l4.1}
  Assume that $P$ is a normal solid cone in a real Banach space $X$,
$0\leqslant\theta< 1$, and $T : P^o\to P^o$ is a $\theta$-concave 
increasing operator. Then $T$ has only one fixed point in $P^o$.
\end{lemma}

\begin{theorem}\label{t4.1} Assume that {\rm (H1), (H5), (H6)} hold. Then
the fractional differential equation boundary-value problem \eqref{e1.1}
and \eqref{e1.2} has a unique positive solution for any $\lambda>0$.
\end{theorem}

\begin{proof} Define $P=\{u\in C[0,1] : u(t)\geqslant0 \text {on } [0,1]\}$. 
Then $P$ is a normal solid cone in $C[0,1]$ with
$$
P^o=\{u\in C[0,1] : u(t)>0\ \text{ on } [0,1]\}.
$$
For any fixed $\lambda>0$, let $T_\lambda : P\to C[0,1]$ be defined 
by \eqref{e3.1}. Define $T : P\to C[0,1]$ by
\begin{align*}
Tu(t)&=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)a(\tau)f(u(\tau))d\tau\Big)ds\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)a(\tau)f(u(\tau))d\tau\Big)ds
\end{align*}
Then from (H5), we have $T$ is increasing in $u\in P^o$ and
$$
T_\lambda u(t)=Tu(t)+\frac{\lambda}{1-\gamma}.
$$
Clearly, $T_\lambda : P^o\to P^o$. Next, we prove that $T_\lambda$ 
is a $\theta$-concave increasing operator. 
In fact, for $u_1, u_2\in P$ with $u_1(t)\geqslant u_2(t)$ on $[0,1]$,
 we obtain
\begin{equation*}
T_\lambda u_1(t)\geqslant Tu_2(t)+\frac{\lambda}{1-\gamma}=T_\lambda u_2(t);
\end{equation*}
i.e., $T_\lambda$ is increasing. Moreover, (H6) implies 
\begin{align*}
T_\lambda (ku)(t)
&\geqslant k^\theta\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)a(\tau)f(u(\tau))d\tau\Big)ds\\
&\quad +k^\theta\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)a(\tau)f(u(\tau))d\tau\Big)ds+
\frac{\lambda}{1-\gamma}\\
&=k^\theta Tu(t)+\frac{\lambda}{1-\gamma}\\
&\geqslant k^\theta (Tu(t)+\frac{\lambda}{1-\gamma})=k^\theta T_\lambda u(t);
\end{align*}
i.e., $T_\lambda$ is $\theta$-concave. By Lemma \ref{l4.1},
 $T_\lambda$ has a unique fixed point $u_\lambda$ in $P^o$,
that is, the fractional differential equation boundary-value problem \eqref{e1.1}
and \eqref{e1.2} has a unique positive solution. The proof is complete.
\end{proof}

\begin{example} \label{examp4.1} \rm 
Consider the boundary-value problem
\begin{gather} \label{e4.1}
D_{0+}^{5/2}(\phi_p(D_{0+}^{1/2} u(t)))+t^2\sqrt[3]{u}=0, \quad 0<t<1,\\
 \label{e4.2}
u(0)=\frac{1}{2} u(\frac{1}{2})+\lambda, \quad
 \phi_p(D_{0+}^{1/2} u(0))=(\phi_p(D_{0+}^{1/2} u(1)))'
=(\phi_p(D_{0+}^{1/2} u(0)))''=0.
\end{gather}
Let $p=2$. We have $\alpha=1/2$, $\beta=5/2$, $\gamma=\xi=1/2$, 
$a(t)=t^2$, $f(u)=\sqrt[3]{u}$. Clearly, (H1) and $(H_5)$ hold.
Choosing $\theta=1/2$, then (H6) is satisfied. Thus, by
Theorem \ref{t4.1} the fractional differential equation boundary-value
 problem \eqref{e4.1}
and \eqref{e4.2} has a unique positive solution for any $\lambda>0$.
\end{example}

\section{Nonexistence}

 In this section, we let the Banach space $C[0,1]$ be endowed with the 
norm $\|u\|=\max_{0\leqslant t\leqslant1}|u(t)|$.

\begin{lemma}\label{l5.1} 
Assume {\rm (H1)} holds and let $0<\delta<1$ be given in {\rm (H4)}. 
Then the unique solution $u(t)$ of
fractional differential equation boundary-value problem \eqref{e2.3} 
and \eqref{e2.4} satisfies
$$
u(t)\geqslant c_\delta\|u\|\quad \text{for }  \delta\leqslant t\leqslant1,
$$
where $c_\delta$ is defined by \eqref{e2.13}.
\end{lemma}

\begin{proof} 
In view of Lemma \ref{l2.5} and \eqref{e2.5}, we have
\begin{align*}
u(t)
&\leqslant\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(1,\tau)y(\tau)d\tau\Big)ds\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)y(\tau)d\tau\big)ds+
\frac{\lambda}{1-\gamma}\\
&\leqslant\frac{1}{\Gamma(\alpha+1)}
\phi_q\Big(\int_0^1H(1,\tau)y(\tau)d\tau\Big)\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)y(\tau)d\tau\Big)ds+
\frac{\lambda}{1-\gamma}
\end{align*}
for $t\in[0,1]$, and
\begin{align*}
u(t)
&\geqslant\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1s^{\beta-1}H(1,\tau)y(\tau)d\tau\Big)ds\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)y(\tau)d\tau\Big)ds+
\frac{\lambda}{1-\gamma}\\
&=\int_0^t\alpha(t-s)^{\alpha-1}\phi_q(s^{\beta-1})ds\frac{1}{\Gamma(\alpha+1)}
\phi_q\Big(\int_0^1H(1,\tau)y(\tau)d\tau\Big)\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)y(\tau)d\tau\Big)ds+
\frac{\lambda}{1-\gamma}\\
&\geqslant c_\delta\frac{1}{\Gamma(\alpha+1)}
\phi_q\Big(\int_0^1H(1,\tau)y(\tau)d\tau\Big)\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)y(\tau)d\tau\Big)ds+
\frac{\lambda}{1-\gamma}\\
&\geqslant c_\delta\big[\frac{1}{\Gamma(\alpha+1)}
\phi_q\Big(\int_0^1H(1,\tau)y(\tau)d\tau\Big)\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)y(\tau)d\tau\Big)ds+
\frac{\lambda}{1-\gamma}\big]
\end{align*}
for $t\in[\delta,1]$. 
Therefore, $u(t)\geqslant c_\delta\|u\|$ for $\delta\leqslant t\leqslant1$.
 The proof is complete.
\end{proof}

\begin{theorem}\label{t5.1}
 Assume that {\rm (H1), (H4)} hold. Then
the fractional differential equation boundary-value problem \eqref{e1.1}
and \eqref{e1.2} has no positive solution for $\lambda>(1-\gamma)e$.
\end{theorem}

\begin{proof} 
Assume, to the contrary, the fractional differential equation boundary-value 
problem \eqref{e1.1} and \eqref{e1.2} has a positive solution $u(t)$ 
for $\lambda>(1-\gamma)e$. Then by Lemma \ref{l2.4}, we have
\begin{align*}
u(t)&=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)a(\tau)f(u(\tau))d\tau\Big)ds\\
&\quad +\frac{\gamma}{1-\gamma}\int_0^\xi\frac{(\xi-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)a(\tau)f(u(\tau))d\tau\Big)ds+
\frac{\lambda}{1-\gamma}
\end{align*}
Therefore, $u(t)>e$ on [0,1]. In view of \eqref{e2.11} and \eqref{e2.12}, 
we obtain
\begin{gather*}
f(u(t))\geqslant N\phi_p(u(t))\quad \text{on }  [0,1],\\
c_\delta \phi_q(N)\phi_q\Big(\int_\delta^1H(1,\tau)a(\tau)d\tau\Big)
\int_0^1\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}\phi_q(s^{\beta-1})ds>1.
\end{gather*}
Then by Lemmas \ref{l2.5} and \ref{l5.1}, we obtain
$$
\begin{array}{ll}
\|u\|=u(1)
&>\int_0^1\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}
\phi_q\Big(\int_0^1H(s,\tau)a(\tau)f(u(\tau))d\tau\Big)ds\\
&\geqslant\int_0^1\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}\phi_q(s^{\beta-1})ds
\phi_q\Big(\int_0^1H(1,\tau)a(\tau)f(u(\tau))d\tau\Big)\\
&\geqslant\int_0^1\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}\phi_q(s^{\beta-1})ds\phi_q(N)
\phi_q\Big(\int_\delta^1H(1,\tau)a(\tau)\phi_p(u(\tau))d\tau\big)\\
&\geqslant\|u\|c_\delta\int_0^1\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}\phi_q(s^{\beta-1})ds\phi_q(N)
\phi_q\Big(\int_\delta^1H(1,\tau)a(\tau)d\tau\Big)\\
&>\|u\|.
\end{array}
$$
This contradiction  completes the proof 
\end{proof}

\begin{corollary}\label{c5.1} 
Assume that {\rm (H1)} holds and $f_\infty=+\infty$. Then
the fractional differential equation boundary-value problem \eqref{e1.1}
and \eqref{e1.2} has no positive solution for sufficiently large $\lambda>0$.
\end{corollary}

\begin{example} \label{examp5.1} \rm
Consider the boundary-value problem
\begin{gather} \label{e5.1}
D_{0+}^{5/2}(\phi_p(D_{0+}^{1/2} u(t)))+tu^2=0, \quad 0<t<1,\\
 \label{e5.2}
u(0)=\frac{1}{2} u(\frac{1}{2})+\lambda, \quad
 \phi_p(D_{0+}^{1/2} u(0))=(\phi_p(D_{0+}^{1/2} u(1)))'
=(\phi_p(D_{0+}^{1/2} u(0)))''=0.
\end{gather}
Let $p=2$. We have $\alpha=1/2$, $\beta=5/2$, $\gamma=\xi=1/2$,
$a(t)=t$, $f(u)=u^2$. Clearly, (H1) holds.
Choosing $\delta=1/2$, by a simple computation, we obtain
 $c_\delta=0.04455, N>222.2104$.
Let $N=e=223$. Then, (H4) is satisfied. Thus, by
Theorem \ref{t5.1} the fractional differential equation boundary-value 
problem \eqref{e5.1} and \eqref{e5.2} has no positive solution 
for $\lambda>111.5$.
\end{example}

\subsection*{Acknowledgments}
 This research is supported by grants: 11071143, 60904024, 61174217
 from the Natural Science Foundation of China; JQ201119
 from the Natural Science Outstanding Youth Foundation of Shandong Province;
 ZR2012AM009, ZR2010AL002 from the Shandong Provincial Natural Science
 Foundation; J11LA01 from the Natural Science Foundation of Educational
 Department of Shandong Province.
 

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