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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 37, 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 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/37\hfil Uniqueness and asymptotic behavior]
{Uniqueness and asymptotic behavior of positive solutions
 for a fractional-order integral boundary-value problem}

\author[X. Zhou, W. Wang \hfil EJDE-2013/37\hfilneg]
{Xiangbing Zhou, Wenquan Wu}  

\address{Xiangbing Zhou \newline
 Department of Computer Science, Aba Teachers College,
 WenChuan 623002, Sichuan, China}
 \email{studydear@gmail.com}

 \address{Wenquan Wu \newline
 Department of Computer Science, Aba Teachers College,
 WenChuan 623002, Sichuan, China}
\email{wenquanwu@163.com} 	


\thanks{Submitted December 11, 2012. Published February 1, 2013.}
\subjclass[2000]{26A33, 34B10}
\keywords{Upper and lower solution; fractional differential equation;
\hfill\break\indent Schauder's fixed point theorem; positive solution}

\begin{abstract}
 In this note, we extend the results by  Jia et al \cite{Jia}
 to a more general case. By refining the conditions imposed on $f$ and
 finding  more suitable upper and lower solution, we remove some key
 conditions used in \cite{Jia}, and still establish their results.
\end{abstract}

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

\section{Introduction}

In the recent years, there has been a significant development in ordinary
and partial differential equations involving fractional derivatives, see
\cite{Yuan, SZhang, Jia, BAhmad, MFeng, Zhang2, Zhang3, Zhang4,ZBai}.
 Yuan \cite{Yuan} studied the  $(n-1,1)$-type  conjugate  boundary-value problem
\begin{equation}\label{1.1}
\begin{gathered}
\mathscr{D}_t^{\alpha}u(t)+f(t,u(t))=0,\quad 0<t<1,
n-1 <\alpha \leq n, n\ge3,\\
 u(0)=u'(0)=\dots =u^{(n-2)}(0)=0, \quad  u(1)=0,
\end{gathered}.
\end{equation}
where $f$ is continuous and semipositone, $\mathscr{D}_t^{\alpha}$
is the standard Riemann-Liouville derivative. By giving properties of
Green's function and using the Guo-Krasnosel'skii fixed point theorem on cones,
 the existence of multiple positive solutions of \eqref{1.1} were obtained.
 Zhang \cite{SZhang} considered the  existence and uniqueness of higher-order
fractional differential equation
 \begin{equation}\label{1.2}
\begin{gathered}
\mathscr{D}_t^{\alpha}x(t)+q(t)f(x,x',\dots,x^{(n-2)})=0,\quad
 0<t<1,\; n-1<\alpha\le n, \\
 x(0)=x'(0)=\dots=x^{(n-2)}(0)=x^{(n-2)}(1)=0,
\end{gathered}
\end{equation}
 where $\mathscr{D}_t^{\alpha}$ is the standard Riemann-Liouville fractional
 derivative of order $\alpha$, $q$ may be singular at $t=0$ and $f$ may be
singular at $x=0, x'=0, \dots,x^{(n-2)}=0$. By using fixed point theorem
of the mixed monotone operator, the author established the existence and
uniqueness result of positive solution for the above problem \eqref{1.2}.
Recently,  Jia et al \cite{Jia},
 considered the existence,  uniqueness and  asymptotic behavior of positive
solutions for the following higher nonlocal fractional differential equation
with Riemann-Stieltjes integral  condition
\begin{equation}\label{1.3}
\begin{gathered}
-\mathscr{D}_t^{\alpha} x(t)
= f(t,x(t),x'(t),x''(t),\dots,x^{(n-2)} (t)), \quad 0<t<1, \\
 x(0)=x'(0)=\dots=x^{(n-2)}(0)=0, \quad
x^{(n-2)}(1)=\int^1_0 x^{(n-2)}(s)dA(s),
\end{gathered}
\end{equation}
where $n-1<\alpha\le n,n\in \mathbb{N}$ and $n\ge2$,
$\mathscr{D}_t^{\alpha} $ is the standard Riemann-Liouville derivative,
$\int^1_0 x^{(n-2)}(s)dA(s)$ is linear functionals given by Riemann-Stieltjes
integrals,  $A$ is a function of bounded variation and $dA$ can be a
changing-sign measure, and  $f:(0,1)\times(0, +\infty)^{n-1}\to [0,+\infty)$ is
continuous,  $f$ may be singular at $x_i=0$  and $t = 0, 1$.
By using  upper and lower solution  method and Schauder's fixed point theorem,
the existence,  uniqueness and  asymptotic behavior of positive
 solutions of \eqref{1.3} are obtained provided that $f$  satisfies
suitable growth condition and integral conditions.

Motivated by the results mentioned above, in this paper, we study
the existence,  uniqueness and  asymptotic behavior of positive solutions
 for the   fractional differential equation
with Riemann-Stieltjes integral  condition
\begin{equation}\label{1.4}
\begin{gathered}
-\mathscr{D}_t^{\mu}x(t)=f(t,x(t),\mathscr{D}_t^{\mu_1}x(t),
 \mathscr{D}_t^{\mu_2}x(t),\dots,\mathscr{D}_t^{\mu_{n-2}}x(t)),\quad
  0<t<1,  \\
x(0)=\mathscr{D}_t^{\mu_1}x(0)=\dots=\mathscr{D}_t^{\mu_{n-2}}x(0)=0,  \quad
\mathscr{D}_t^{\mu_{n-2}}x(1)=\int^1_0 \mathscr{D}_t^{\mu_{n-2}}x(s)dA(s),
\end{gathered}
\end{equation}  where $n-1<\mu\le n,n\in \mathbb{N}$ and $n\ge2$ with
 $0<\mu_1<\mu_2<\dots<\mu_{n-2} $ and $n-2<\mu_{n-2}<\mu-1$,
$  \mathscr{D}_t^{\mu}$ is the standard Riemann-Liouville derivative,
and  $f:(0,1)\times(0, +\infty)^{n-1}\to [0,+\infty)$ is
continuous,  $f$ may be singular at $x_i=0$  and $t = 0, 1$.
By refining the conditions imposed on $f$ and finding  more suitable
upper and lower solution,  we remove some key conditions which are
required  in the works of Jia et al \cite{Jia}, but a similar
result is still established for  the more  general form \eqref{1.4}.

\section{Preliminaries}

  In this section,  we present  the necessary definitions from fractional
calculus theory.

\begin{definition}[\cite{KMiller,IPodlubny}] \label{def2.1}  \rm
 The Riemann-Liouville fractional integral of order $\alpha >0$ of a function
$x:(0,+\infty)\to \mathbb{R}$ is given by
$$
I^{\alpha}x(t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}x(s)ds
$$
provided that the right-hand
side is pointwise defined on $(0,+\infty)$.
\end{definition}


\begin{definition}[\cite{KMiller,IPodlubny}] \label{def2.2} \rm
The Riemann-Liouville fractional derivative  of order $\alpha >0$ of
a function $x:(0,+\infty)\to \mathbb{R}$ is given by
$$
\mathscr{D}_t^{\alpha}x(t)=\frac{1}{\Gamma(n-\alpha)}
(\frac{d}{dt})^{n}\int^{t}_{0}(t-s)^{n-\alpha-1}x(s)ds,
$$
where $n=[\alpha]+1$, $[\alpha]$ denotes the integer part of number
$\alpha$, provided that the right-hand side is pointwise defined
on $(0,+\infty)$.
\end{definition}

\begin{proposition}[\cite{KMiller,IPodlubny}] \label{prop2.1}
 (1) If $x\in  L^1(0, 1), \nu>\sigma> 0$,  then
$$
I^{\nu}I^{\sigma}x(t)=I^{\nu+\sigma}x(t), \quad
\mathscr{D}_t^{\sigma}I^{\nu} x(t)=I^{\nu-\sigma} x(t),\quad
\mathscr{D}_t^{\sigma}I^{\sigma} x(t)=x(t).
$$
(2)  If  $\alpha>0, \sigma>0$, then
$$
\mathscr{D}_t^{\alpha} t^{\sigma-1}
=\frac{\Gamma(\sigma)}{\Gamma(\sigma-\alpha)}t^{\sigma-\alpha-1}.
$$
\end{proposition}

\begin{proposition}[\cite{KMiller,IPodlubny}]  \label{prop2.2}
Let $\alpha > 0$, and $f(x)$ be integrable, then
$$
I^{\alpha}\mathscr{D}_t^{\alpha}f(x)
=f(x)+c_{1}x^{\alpha-1}+c_{2}x^{\alpha-2}+\cdot\cdot\cdot+c_{n}x^{\alpha-n},
$$
where $c_{i}\in \mathbb{R}$ $(i=1,2,\dots,n)$, $n$ is the smallest integer
greater than or equal to $\alpha$.
\end{proposition}

 Let
$$
x(t)=I^{\mu_{n-2}}y(t),\quad  y(t)\in C[0,1],
$$
by  Propositions \ref{prop2.1}-\ref{prop2.2} and a discussion similar to \cite{{Jia}},
we easily reduce the order  of \eqref{1.4} to the  equivalent problem
\begin{equation}
\begin{gathered}
-\mathscr{D}_t^{\mu-\mu_{n-2}} y(t)
= f(t,I^{\mu_{n-2}}y(t),I^{\mu_{n-2}-\mu_1}y(t),\dots,
I^{\mu_{n-2}-\mu_{n-3}}y(t), y(t)),   \\
y(0)=0, \quad  y(1)=\int^1_0 y(s)dA(s).
\end{gathered}\label{2.1}
\end{equation}

\begin{lemma}[\cite{ZBai}] \label{lem2.1}
 Given $h\in L^1(0, 1)$, then the problem
\begin{equation}
\begin{gathered}
\mathscr{D}_t^{\mu-\mu_{n-2}}y(t)+h(t)=0 ,\quad  0<t<1 ,\\
 y(0)=0, \quad  y(1)=0,
\end{gathered}\label{2.2}
\end{equation}
has the unique solution
$$
y(t)= \int_{0}^{1}G(t,s)h(s)ds,
$$
where $G(t,s)$ is the Green function of \eqref{2.2},  given by
\begin{equation}
G(t,s)=\begin{cases}
\frac{t^{\mu-\mu_{n-2}-1}(1-s)^{\mu-\mu_{n-2}-1}-(t-s)
 ^{\mu-\mu_{n-2}-1}}{\Gamma(\mu-\mu_{n-2})},&  0\leq s\leq t\leq 1,  \\
\frac{t^{\mu-\mu_{n-2}-1}(1-s)^{\mu-\mu_{n-2}-1}}{\Gamma(\mu-\mu_{n-2})},
& 0\leq t\leq s\leq 1.
\end{cases}
\label{2.3} 
\end{equation}
\end{lemma}

By  Proposition \ref{prop2.2}, the unique solution of the problem
\begin{equation}
\begin{gathered}
\mathscr{D}_t^{\mu-\mu_{n-2}}y(t)=0,\quad  0<t<1,\\
y(0)=0, \quad   y(1)=1,
\end{gathered}\label{2.4}
\end{equation}
is $t^{\mu-\mu_{n-2}-1}$.
Let
\begin{equation}
\mathcal{C}=\int_0^1t^{\mu-\mu_{n-2}-1}dA(t),\label{2.5}
\end{equation}
and define
$$
\mathcal{G}_A(s)=\int_0^1G(t, s)dA(t).
$$
Then the Green
function for the nonlocal BVP  \eqref{2.1} is
(for details see \cite{LWebb}  or \cite{Zhang1})
\begin{equation}
K(t,s)=\frac{t^{\mu-\mu_{n-2}-1}}{1-\mathcal{C}}\mathcal{G}_A(s)+G(t,s).
\label{2.6}
\end{equation}

In this article we use the following assumption
\begin{itemize}

\item[(H0)]    $A$ is a  function of bounded variation such that
 $\mathcal{G}_A(s)\ge0$ for $s\in [0, 1]$ and $0\le\mathcal{C} < 1$,
 where $\mathcal{C}$ is defined by \eqref{2.5}.

\end{itemize}
The following Lemma follows from \eqref{2.3} and \eqref{2.6}.


\begin{lemma} \label{lem2.2}
 Suppose {\rm (H0)}  holds. Then the Green function
defined by \eqref{2.6} satisfies:
\vskip0.1cm
(1) $ K(t,s) > 0$, for  all $t,s\in (0,1)$.

(2)  \begin{equation}
\frac{t^{\mu-\mu_{n-2}-1}}{1-\mathcal{C}}\mathcal{G}_A(s)
\le K(t,s)\le\mathcal{H}(s)t^{\mu-\mu_{n-2}-1},   \label{2.7} \end{equation}
 where
   $$
\mathcal{H}(s)=\frac{(1-s)^{\mu-\mu_{n-2}-1}}{\Gamma(\mu-\mu_{n-2})}
+ \frac{\mathcal{G}_A(s)}{1-\mathcal{C}}.
$$
\end{lemma}

\begin{definition} \label{def2.3}
 A  continuous function $\Psi(t)$ is called a lower solution of
  \eqref{2.1}, if it satisfies
\begin{gather*}
-\mathscr{D}_t^{\mu-\mu_{n-2}} \Psi(t)(t)
 \le f(t,I^{\mu_{n-2}}\Psi(t),I^{\mu_{n-2}-\mu_1}\Psi(t),
 \dots,I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t)),   \\
\Psi(0)\ge0, \quad  \Psi(1)\ge\int^1_0 \Psi(s)dA(s).
\end{gather*}
\end{definition}

\begin{definition} \label{def2.4}
 A  continuous function $\Phi(t)$ is called a upper solution of \eqref{2.1},
if it satisfies
\begin{gather*}
-\mathscr{D}_t^{\mu-\mu_{n-2}} \Phi(t)(t)
\ge f(t,I^{\mu_{n-2}}\Phi(t),I^{\mu_{n-2}-\mu_1}\Phi(t),
\dots,I^{\mu_{n-2}-\mu_{n-3}}\Phi(t), \Phi(t)),   \\
\Phi(0)\le0, \quad  \Phi(1)\le\int^1_0 \Phi(s)dA(s).
\end{gather*}
\end{definition}

\section{Main results}

 Let  $E= C[0,1]$. Define the following continuous functions on $E$:
\begin{gather*}
\kappa_0(t)=I^{\mu_{n-2}}s^{\mu-\mu_{n-2}-1}
  =\int_0^t\frac{(t-s)^{\mu_{n-2}-1}s^{\mu-\mu_{n-2}-1}}{\Gamma(\mu_{n-2})}ds
=\frac{\Gamma(\mu-\mu_{n-2})}{\Gamma(\mu)}s^{\mu-1},
  \\
\begin{aligned}
\kappa_1(t)=I^{\mu_{n-2}-\mu_1}s^{\mu-\mu_{n-2}-1}
&=\int_0^t\frac{(t-s)^{\mu_{n-2}-\mu_1-1}s^{\mu-\mu_{n-2}-1}}
  {\Gamma(\mu_{n-2}-\mu_1)}ds\\
&=\frac{\Gamma( {\mu-\mu_{n-2}})}{\Gamma(\mu-\mu_1)}t^{\mu-1-\mu_{1}},
\end{aligned}\\
\dots \\
\begin{aligned} \kappa_{n-3}(t)=I^{\mu_{n-2}-\mu_{n-3}}s^{\mu-\mu_{n-2}-1}
 &=\int_0^t\frac{(t-s)^{\mu_{n-2}-\mu_{n-3}-1}s^{\mu-\mu_{n-2}-1}}
  {\Gamma(\mu_{n-2}-\mu_{n-3})}ds\\
 &=\frac{\Gamma( {\mu-\mu_{n-2}})}{\Gamma(\mu-\mu_{n-3})}t^{\mu-1-\mu_{n-3}},
\end{aligned}\\
\kappa_{n-2}(t)=t^{\mu-1-\mu_{n-2}}.
\end{gather*}
Set
\begin{equation}
\begin{aligned}
 P&=\Big\{y\in E:\text{ there exist positive numbers } 0<l_y<1,\;
 L_y>1  \text{  such that }\\
&\quad  l_y  \kappa_{n-2}(t)\le y(t)\le L_y \kappa_{n-2}(t), \;
 t\in[0,1\Big\}.
\end{aligned}\label {3.1}
\end{equation}
Clearly,  $P$ is nonempty since $\kappa_{n-2}(t)\in P$.
 For any  $y\in P$,  define an operator $T$   by
 \begin{equation}
(T y)(t)= \int_0^1 K(t,s)f(s,I^{\mu_{n-2}}y(s),I^{\mu_{n-2}
 -\mu_1}y(s),\dots,I^{\mu_{n-2}-\mu_{n-3}}y(s), y(s))ds.
\label{3.2}
\end{equation}


In this note, we will use  the following  conditions:
\begin{itemize}
\item[(H1)]
 $f\in C((0,1)\times(0,\infty)^{n-1},[0,+\infty))$, and
$f(t,x_0,x_1,x_2,\dots,x_{n-2})$ is nonincreasing in $x_i>0$
 for $i=0,1,2,\dots,n-2$;

\item[(H2)] For any $\lambda_i>0$,
$$
0<\int_0^1\mathcal{H}(s)f(s,\lambda_0 \kappa_0(s), \lambda_1 \kappa_1(s),
 \lambda_2 \kappa_2(s),\dots, \lambda_{n-2} \kappa_{n-2}(s))ds<+\infty.
$$
\end{itemize}

\begin{lemma} \label{lem3.1}
Suppose {\rm (H0)--(H2)} hold. Then  $T$ is well defined,
$T (P)\subset P$, and  $T$ is nonincreasing relative to $y$.
\end{lemma}

\begin{proof}
For any $y\in P$, by the definition of $P$, there exist  two positive
numbers $0<l_y<1,\ L_y>1$  such that
\begin{equation}
l_y  \kappa_{n-2}(s)\le y(t)\le L_y \kappa_{n-2}(s)\label{3.3}
\end{equation}
for any $s\in [0,1]$.  It follows from \eqref{2.7} and
(H1)--(H2) that
\begin{equation}
\begin{aligned}
&(Ty)(t)\\
&= \int_0^1 K(t,s)f(s,I^{\mu_{n-2}}y(s),I^{\mu_{n-2}
  -\mu_1}y(s),\dots,I^{\mu_{n-2}-\mu_{n-3}}y(s), y(s))ds\\
&\le \kappa_{n-2}(s)\int_0^1 \mathcal{H}(s)f(s,l_y\kappa_0(s),
 l_y\kappa_1(s),\dots,l_y\kappa_{n-3}(s), l_y\kappa_{n-2}(s))ds\\
&<+\infty.
\end{aligned}\label{3.4}
\end{equation}
By \eqref{2.7}, \eqref{3.3} and \eqref{3.4}, we have
\begin{equation}
\begin{aligned}
& (Ty)(t)\\
&=\int_0^1 K(t,s)f(s,I^{\mu_{n-2}}y(s),I^{\mu_{n-2}
 -\mu_1}y(s),\dots,I^{\mu_{n-2}-\mu_{n-3}}y(s), y(s))ds\\
&\ge \frac{t^{\mu-\mu_{n-2}-1}}{1-\mathcal{C}}
\int_0^1 \mathcal{G}_A(s)f(s,L_y\kappa_0(s),L_y\kappa_1(s),\dots,
 L_y\kappa_{n-3}(s), L_y\kappa_{n-2}(s))ds.
\end{aligned}\label{3.5}
\end{equation}
Take
 \begin{equation}
\begin{gathered}
l'_y=\min\Big\{1,\, \frac1{1-\mathcal{C}}\int_0^1 \mathcal{G}_A(s)
 f(s,L_y\kappa_0(s),L_y\kappa_1(s),\dots, L_y\kappa_{n-3}(s),
  L_y\kappa_{n-2}(s))ds\Big\},\\
L'_y=\max\Big\{1,\, \int_0^1 \mathcal{H}(s)
 f(s,l_y\kappa_0(s),l_y\kappa_1(s),\dots, l_y\kappa_{n-2}(s))ds
\Big\}.
\end{gathered} \label{3.6}
\end{equation}
It follows from \eqref{3.3}-\eqref{3.6} that $T $ is well defined and
 $T (P)\subset P$. Moreover, by (H1), $T$ is nonincreasing relative to $y$.
\end{proof}


\begin{theorem}[Existence] \label{thm3.1}
Suppose {rm(H0)--(H2)} hold.
Then  \eqref{1.4} has at least one positive solution $x(t)$.
\end{theorem}

\begin{proof}
 From \eqref{3.2} and simple computation, we have
\begin{equation}
\begin{gathered}
-\mathscr{D}_t^{\mu-\mu_{n-2}} (Ty)(t)
= f(t,I^{\mu_{n-2}}y(t),I^{\mu_{n-2}-\mu_1}y(t),\dots,
I^{\mu_{n-2}-\mu_{n-3}}y(t), y(t)),   \\
(Ty)(0)=0, \quad  (Ty)(1)=\int^1_0 (Ty)(s)dA(s).
\end{gathered}\label{3.7}
\end{equation}
Let
\begin{equation}
\alpha(t) =\min\{\kappa_{n-2}(t), (T\kappa_{n-2})(t)\},\quad
\beta(t)=\max\{\kappa_{n-2}(t), (T\kappa_{n-2})(t)\},\label{3.8}
\end{equation}
then, if $\kappa_{n-2}(t)=(T\kappa_{n-2})(t)$, the conclusion of
Theorem \ref{thm3.1} holds. If $\kappa_{n-2}(t)\not=(T\kappa_{n-2})(t)$,
 clearly,  $\alpha(t), \beta(t)\in P$
and
\begin{equation}
\alpha(t)\le \kappa_{n-2}(t)\le \beta(t).\label{3.9}
\end{equation}
Set
$$
\Phi(t)=(T\beta)(t),  \Psi(t)=(T\alpha)(t),
$$
 then by  \eqref{3.8}-\eqref{3.9}  and Lemma \ref{lem3.1}, one has
\begin{equation}
\begin{gathered}
\Phi(t)=(T\beta)(t)\le (T\kappa_{n-2})(t)\le T(\alpha)(t)=\Psi(t),\\
\Phi(t)\le (T\kappa_{n-2})(t)\le \beta(t), \quad
\Psi(t)\ge (T\kappa_{n-2})(t)\ge \alpha(t),
\end{gathered}\label{3.10}
\end{equation}
and  $\Phi(t),  \Psi(t)\in P$.

 On the other hand,  by \eqref{3.7}, \eqref{3.10} and Lemma \ref{lem3.1}, we have
 \begin{equation}
\begin{aligned}
&\mathscr{D}_t^{\mu-\mu_{n-2}}\Phi(t)+ f(t, I^{\mu_{n-2}}\Phi(t),
 I^{\mu_{n-2}-\mu_1}\Phi(t),\dots,I^{\mu_{n-2}-\mu_{n-3}}\Phi(t), \Phi(t))\\
&\ge\mathscr{D}_t^{\mu-\mu_{n-2}}(T\beta)(t)+ f(t,I^{\mu_{n-2}}\beta(t),
 I^{\mu_{n-2}-\mu_1}\beta(t),\dots,I^{\mu_{n-2}-\mu_{n-3}}\beta(t), \beta(t))\\
&=-f(t,I^{\mu_{n-2}}\beta(t),I^{\mu_{n-2}-\mu_1}\beta(t),\dots,
 I^{\mu_{n-2}-\mu_{n-3}}\beta(t), \beta(t))\\
&\quad + f(t,I^{\mu_{n-2}}\beta(t),I^{\mu_{n-2}-\mu_1}\beta(t),\dots,
 I^{\mu_{n-2}-\mu_{n-3}}\beta(t), \beta(t))=0,\\
& (T\Phi)(0)=0, \quad  (T\Phi)(1)=\int^1_0 (T\Phi)(s)dA(s).
\end{aligned} \label{3.11}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\mathscr{D}_t^{\mu-\mu_{n-2}}\Psi(t)
 + f(t, I^{\mu_{n-2}}\Psi(t),I^{\mu_{n-2}-\mu_1}\Psi(t),
 \dots,I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t))\\
&\le\mathscr{D}_t^{\mu-\mu_{n-2}}(T\alpha)(t)
 + f(t,I^{\mu_{n-2}}\alpha(t),I^{\mu_{n-2}-\mu_1}\alpha(t),
 \dots,I^{\mu_{n-2}-\mu_{n-3}}\alpha(t), \alpha(t))\\
&=- f(t,I^{\mu_{n-2}}\alpha(t),I^{\mu_{n-2}-\mu_1}\alpha(t),
\dots,I^{\mu_{n-2}-\mu_{n-3}}\alpha(t)\\
&\quad + f(t,I^{\mu_{n-2}}\alpha(t),I^{\mu_{n-2}
 -\mu_1}\alpha(t),\dots,I^{\mu_{n-2}-\mu_{n-3}}\alpha(t)=0,\\
& (T\Psi)(0)=0, \quad  (T\Psi)(1)=\int^1_0 (T\Psi)(s)dA(s).
 \end{aligned} \label{3.12}
\end{equation}
Inequalities  \eqref{3.10}-\eqref{3.12} imply that $\Phi(t),\Psi(t)$
are lower and upper solution of  \eqref{2.1}, respectively.

Define the function $F$ and the operator $A$ in $E$ by
\begin{equation}
\begin{aligned}
&F(t,y)\\
&=\begin{cases} 
f(t, I^{\mu_{n-2}}\Phi(t),I^{\mu_{n-2}-\mu_1}\Phi(t),\dots,
I^{\mu_{n-2}-\mu_{n-3}}\Phi(t), \Phi(t)), & y<\Phi(t),\\
f(t,I^{\mu_{n-2}}y(t),I^{\mu_{n-2}-\mu_1}y(t),\dots,I^{\mu_{n-2}
-\mu_{n-3}}y(t), y(t)), &\Phi(t)\le y\le \Psi(t),\\
f(t, I^{\mu_{n-2}}\Psi(t),I^{\mu_{n-2}-\mu_1}\Psi(t),\dots,
 I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t)),&  y>\Psi(t),
\end{cases}
\end{aligned}\label{3.13}
\end{equation}
and 
$$
(\mathfrak{A}y)(t)=  \int_0^1 K(t,s)F(s,y(s))ds, \quad \forall y\in E.
$$
Clearly, $F:[0,1]\times[0,+\infty)\to [0,+\infty)$ is continuous  
by \eqref{3.13}. Consider the following boundary value problem
\begin{equation} 
\begin{gathered} 
 -\mathscr{D}_t^{\mu-\mu_{n-2}} y(t)=F(t,y),\quad 0<t<1 ,\\
 y(0)=0, \quad  y(1)=\int^1_0 y(s)dA(s).
\end{gathered}
\label{3.14}
\end{equation}
Obviously, a fixed point of the operator  $\mathfrak{A}$ is a solution 
of  \eqref{3.14}.
As in \cite{Jia}, $\mathfrak{A}$ has at least a fixed point $y$ such 
that $y=\mathfrak{A}y$.

In the end, we claim
 $$
\Phi(t)\le y(t)\le \Psi(t), \quad t\in [0,1].
$$
 In fact, since  $y$ is fixed point of $\mathfrak{A}$ and \eqref{3.12}, 
we obtain
\begin{equation}
 y(0)=0, \quad  y(1)=\int^1_0 y(s)dA(s),\quad \Psi(0)=0, \quad
  \Psi(1)=\int^1_0 \Psi(s)dA(s).
\label{3.15}
 \end{equation}
We firstly claim $y(t)\le \Psi(t)$.  Otherwise, suppose  $x(t)>\Psi(t)$. 
According to the definition of $F$, we have
 \begin{equation} 
\begin{aligned}
&-\mathscr{D}_t^{\mu-\mu_{n-2}} y(t)=F(t,y(t))\\
&=f(t, I^{\mu_{n-2}}\Psi(t),I^{\mu_{n-2}-\mu_1}\Psi(t),
\dots,I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t)).
\end{aligned} \label{3.16}
\end{equation}
 On the other hand,  it follows from $\psi$ is an upper solution to  
\eqref{2.1} that
 \begin{equation}
-\mathscr{D}_t^{\mu-\mu_{n-2}}\Psi(t)\ge f(t, I^{\mu_{n-2}}\Psi(t),
I^{\mu_{n-2}-\mu_1}\Psi(t),\dots,I^{\mu_{n-2}-\mu_{n-3}}\Psi(t), \Psi(t)).
\label{3.17}
\end{equation}
 Let $z(t)=\Psi(t)-y(t)$, \eqref{3.15}-\eqref{3.17} imply  that
$$
 \mathscr{D}_t^{\mu-\mu_{n-2}}z(t)=\mathscr{D}_t^{\mu-\mu_{n-2}}\Psi(t)
-\mathscr{D}_t^{\mu-\mu_{n-2}}y(t)\le 0,  
$$
and 
$$
z(0)=0, z(1)=\int^1_0 z(s)dA(s).
$$ 
It follows from \eqref{2.6} that
$$
z(t)\ge 0,
$$ 
i.e.,  $y(t) \le \Psi(t)$ on $[0, 1]$, which contradicts  $y(t)>\Psi(t)$. 
Hence, $y(t)>\Psi(t)$ is impossible.

By the same way, we also have  $y(t) \ge \Phi(t)$ on $[0, 1]$. So
\begin{equation}
\Phi(t)\le y(t)\le \Psi(t), \quad t\in [0,1]. \label{3.18}
\end{equation}
Consequently, $F(t,y(t))=f(t,I^{n-2}y(t),I^{n-3}y(t),\dots,I^{1}y(t),y(t))$,
$t\in [0,1]$. Then $y(t)$ is a positive solution of the problem  \eqref{2.1}.
It follows from  \eqref{2.1} that $x(t)=I^{\mu_{n-2}}y(t)$ is positive 
solution of \eqref{1.4}.
\end{proof}

\begin{remark} \label{rmk3.1}\rm
 In this work, we not only extend the main result of  \cite{Jia} to more 
general form with fractional derivatives in nonlinearity and boundary condition,
but also by finding more suitable upper and lower solution, we omit the 
following key conditions of \cite{Jia}:
\begin{itemize}
\item[(i)] For any $\lambda_i>0$, 
$ f(t,\lambda_0t^{n-2},\lambda_1t^{n-3},\dots,\lambda_{n-3}t,\lambda_{n-2})
\not\equiv0$, $t\in(0,1)$.

\item[(ii)] 
$$
\int_0^1\mathcal{G}_A(s)f\Big(s,\frac Ll\kappa_0(s),\frac Ll\kappa_1(s),
\dots, \frac Ll\kappa_{n-3}(s),\frac Ll\kappa_{n-2}(s)\Big)ds\ge{1-\mathcal{C}}.
$$
\end{itemize}
This implies our result   essentially improves those of \cite{Jia}.
\end{remark}

\begin{theorem}[Asymptotic Behavior] \label{thm3.2} 
Suppose  Suppose {\rm (H0)--(H2)}  hold.
 Then there exist two constants $\mathcal{B}_1, \mathcal{B}_2$ such that the 
positive solution $x(t)$  of \eqref{1.4}  satisfies
  \begin{equation}
\mathcal{B}_1\kappa_{n-2}(t)\le x(t)\le  \mathcal{B}_2\kappa_{n-2}(t).\label{3.19}
\end{equation}
\end{theorem}

\begin{proof}
By \eqref{3.18},  and  $\Phi,\Psi\in P$, we know that there exist two 
positive constants $0<l_\Phi<1,\ L_\Psi>1$ such that
$$
l_\Phi\kappa_{n-2}(t)\le \Phi(t)\le y(t)\le \Psi(t)\le L_\Psi \kappa_{n-2}(t).
$$
Notice that  $ x(t)=I^{\mu_{n-2}}y(t)$, we have 
\begin{align*}
\frac{l_\Phi\Gamma( {\mu}-\mu_{n-2})}{\Gamma(\mu)}t^{\mu-1}
&=l_\Phi I^{\mu_{n-2}}\kappa_{n-2}(s)\le x(t)\\
&\le  L_\Psi I^{\mu_{n-2}}\kappa_{n-2}(s)
=\frac{L_\Psi \Gamma( {\mu}-\mu_{n-2})}{\Gamma(\mu)}t^{\mu-1}.
\end{align*}
Let 
$$
\mathcal{B}_1=\frac{l_\Phi\Gamma( {\mu}-\mu_{n-2})}{\Gamma(\mu)},\quad
\mathcal{B}_2=\frac{L_\Psi \Gamma( {\mu}-\mu_{n-2})}{\Gamma(\mu)},
$$
then \eqref{3.19} holds.
\end{proof}

If $\mu>1$ is a integer, then we also have the following uniqueness result 
similar to \cite{Jia}.

\begin{theorem}[Uniqueness] \label{thm3.3}  
 Suppose  Suppose {\rm (H0)--(H2)} hold, and  $\mu=n>1$.
 Then  the positive solution $x(t)$  of  \eqref{1.4}  is unique.
\end{theorem}

\begin{proof} Notice that
\begin{align*}
& f(t,I^{\mu_{n-2}}w_2(t),I^{\mu_{n-2}-\mu_1}w_2(t),\dots,
 I^{\mu_{n-2}-\mu_{n-3}}w_2(t),w_2(t))\\
& \le f(t,I^{\mu_{n-2}}w_1(t),I^{\mu_{n-2}-\mu_1}w_1(t),\dots,
 I^{\mu_{n-2}-\mu_{n-3}}w_1(t),w_2(t)), 
\end{align*}
for
 $$
w_2(t)\ge w_1(t),\quad t\in [a,b].
$$ 
Thus similar to \cite{Jia}, the proof is completed.
\end{proof}

\begin{example} \label{examp3.1} \rm
  Consider the existence of positive solutions  for the nonlinear 
fractional differential equation
\begin{equation}
\begin{gathered} 
-\mathscr{D}_t^{11/3} x(t)=t^{2/3}
\big[x^{-1/2}+(\mathscr{D}_t^{2/3}x )^{-1/3}
+(\mathscr{D}_t^{7/3}x )^{-2/3}\big], \quad 0<t<1, \\
 x(0)=\mathscr{D}_t^{2/3}x (0)=\mathscr{D}_t^{7/3}x(0)=0, \quad
\mathscr{D}_t^{7/3}x(1)=\int^1_0 \mathscr{D}_t^{7/3}x(s)dA(s),
\end{gathered} \label{3.20}
\end{equation}
where
$$
A(t)=\begin{cases} 0, &t\in [0,1/2) ,\\
3/2, & t\in [1/2,3/4),\\
1,  & t\in [3/4,1]. 
\end{cases}
$$
Therefore, \eqref{3.20} has at least a positive solution.
\end{example}

\begin{proof} 
Clearly, \eqref{3.20} is equivalent to the following 4-point BVP with 
coefficients of both signs
\begin{gather*} 
-\mathscr{D}_t^{11/3} x(t)=t^{2/3}[x^{-1/2}+(\mathscr{D}_t^{2/3}x )^{-1/3}
+(\mathscr{D}_t^{7/3}x )^{-2/3}], \quad 0<t<1, \\
 x(0)=\mathscr{D}_t^{2/3}x (0)=\mathscr{D}_t^{7/3}x(0)=0, \quad
 \mathscr{D}_t^{7/3}(1)=\frac32\mathscr{D}_t^{7/3}(\frac12)
-\frac12\mathscr{D}_t^{7/3}(\frac34),
\end{gather*}
Thus $f(t,x_0,x_1,x_2)=t^{2/3}[x_0^{-1/2}+ x_1^{-1/3}+ x_2^{-2/3}]$,
$\kappa_2(t)=t^{1/3}$, and
 $$
0\le \mathcal{C}=\int_0^1t^{1/3}dA(t)=1-\Big[\int_{1/2}^{3/4}\frac{3}2dt^{1/3}
+\int_{3/4}^1dt^{1/3}\Big]\approx 0.7363<1,$$
and
$$
G(t,s)=\begin{cases} 
 G_1(t,s)=\frac{t^{1/3}(1-s)^{1/3}}{\Gamma(4/3)}, & 0\leq t\leq s\leq 1,\\
 G_2(t,s)=\frac{t^{1/3}(1-s)^{1/3}-(t-s)^{1/3}}{\Gamma(4/3)},
 & 0\leq s\leq t\leq 1.
\end{cases} 
$$
Thus,
\begin{align*}
\mathcal{G}_A(s)
&=\begin{cases} 
 \frac32G_2(1/2,s)-\frac12G_2(3/4,s), &    0\leq s<\frac12, \\
 \frac32G_1(1/2,s)-\frac12G_2(3/4,s),&   \frac12\leq s<\frac34,\\
 \frac32G_1(1/2,s)-\frac12G_1(3/4,s),&   \frac34\leq s\le1,
\end{cases}\\
& =\begin{cases} 
\frac{(\frac32\times(\frac12)^{1/3} -\frac12\times(\frac34)^{1/3} )(1-s)^{1/3}
 +\frac12(\frac34-s)^{1/3}-\frac32(\frac12-s)^{1/3}}{\Gamma(4/3)},
 & 0\leq s<\frac12, \\
 \frac{(\frac32\times(\frac12)^{1/3} -\frac12\times(\frac34)^{1/3} )(1-s)^{1/3}
 +\frac12(\frac34-s)^{1/3}}{\Gamma(4/3)},& \frac12\leq s<\frac34,\\
\frac{(\frac32\times(\frac12)^{1/3}
  -\frac12\times(\frac34)^{1/3} )(1-s)^{1/3}}{\Gamma(4/3)},
 & \frac34\leq s\le1,
\end{cases}
\end{align*}
and   
$$
\mathcal{H}(s)=\frac{(1-s)^{1/3}}{\Gamma(4/3)}
 + \frac{\mathcal{G}_A(s)}{0.2637}.
$$
Clearly, (H0) and (H1) hold.

 On the other hand,  since 
$$
\kappa_0(t)=\frac{\Gamma(4/3)}{\Gamma(11/3)}t^{\frac83},\quad
\kappa_1(t)=\frac{\Gamma(4/3)}{\Gamma(3)}t^{2},\quad
\kappa_{2}(t)=t^{1/3},
$$
  for any $\lambda_i>0,\ i=0,1,2$,  we have
\begin{align*}
 0&<\int_0^1\mathcal{H}(s)f(s,\lambda_0\kappa_0(s),\lambda_1\kappa_1(s),
\lambda_2\kappa_2(s))ds\\
&=\int_0^1 \big[\frac{(1-s)^{1/3}}{\Gamma(4/3)}
 + \frac{\mathcal{G}_A(s)}{0.2637}\big]s^{2/3}
\Big[(\frac{\Gamma(4/3)}{\Gamma(11/3)}\lambda_0)^{-1/2}
s^{-\frac4{3}}\\
&+(\frac{\Gamma(4/3)}{\Gamma(3)}\lambda_1)^{-1/3}s^{-2/3}
 +\lambda_2^{-2/3}s^{-\frac29}\Big] ds \\
&<+\infty.
\end{align*}
Thus (H2) is satisfied.
Then by Theorem \ref{thm3.1},  the BVP \eqref{3.20} has at least a positive solution.
\end{proof}

\subsection*{Acknowledgments}
 This work was supported by the Natural Sciences of Education and 
Science Office Bureau of Sichuan Province of China,
under Grant no. 2010JY0J41,11ZB152.

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\end{document}