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

\AtBeginDocument{{\noindent\small \emph{Electronic Journal of
Differential Equations}, Vol. 2008(2008), No. 52, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
 http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/52\hfil Fourth-order boundary-value problem]
{Existence of solutions for a fourth-order boundary-value problem}

\author[Y. Liu\hfil EJDE-2008/52\hfilneg]
{Yang Liu}

\address{Yang Liu\newline
  Department of Mathematics\\
  Yanbian University \\
  Yanji, Jilin 133000, China. \newline
  Department of  Mathematics \\
  Huaiyin Teachers College\\
  Huaian, Jiangsu 223300,  China}
\email{liuyang19830206@yahoo.com.cn}

\thanks{Submitted October 19, 2007. Published April 10, 2008.}
\subjclass[2000]{34B15} 
\keywords{Fourth-order boundary-value problem; upper and lower  solution;
\hfill\break\indent
 Riemann-Stieltjies integral; Nagumo-type condition}

\begin{abstract}
 In this paper, we use the lower and upper solution method to
 obtain an existence theorem for the fourth-order
 boundary-value problem
 \begin{gather*}
 u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),\quad 0<t<1,\\
 u(0)=u'(1)=u''(0)=0,\quad u'''(1)=g\big(\int^1_0u''(t)d\theta(t)\big),
 \end{gather*}
 where $f : [0,1]\times \mathbb{R}^4 \to \mathbb{R}$, $g :
 \mathbb{R}\to \mathbb{R}$ are continuous and may be nonlinear, and
 $\int^1_0u''(t)d\theta(t)$ denotes the Riemann-Stieltjes integral.
\end{abstract}

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


\section{Introduction}

It is well known that the bending of an elastic beam can be
described with  fourth-order boundary-value problems. Recently, many
authors have investigated the existence of solutions for
fourth-order boundary-value problems subject to a variety of
boundary-value conditions, see for example \cite{b1, b3, c1,l1,l2}.

Very recently,  Bai \cite{b2} used the lower and upper solution
method to obtain the existence of solutions for the problem
\begin{gather*}
u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),\quad 0<t<1,\\
u(0)=u'(1)=u''(0)=u'''(1)=0,
\end{gather*}
where $f:[0,1]\times \mathbb{R}^4 \to \mathbb{R}$ is increasing.

Motivated  by the above-mentioned papers and the main ideas in
\cite{k1}, in this paper, we use the lower and upper solution method
to establish the existence of solutions for the
 fourth-order boundary-value problem
\begin{equation}
\begin{gathered}
u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),\quad 0<t<1,\\
u(0)=u'(1)=u''(0)=0,\quad
u'''(1)=g\big(\int^1_0u''(t)d\theta(t)\big),
\end{gathered}\label{e1.1}
\end{equation}
where $f : [0,1]\times \mathbb{R}^4 \to \mathbb{R}$, $g
:\mathbb{R}\to \mathbb{R}$ are continuous and may be nonlinear.
$\theta: [0,1]\to \mathbb{R}$ is increasing nonconstant function
defined on [0,1] and $\theta(0)=0$.

If $g(x)=x$, $x\in \mathbb{R}$, then the problem \eqref{e1.1} educes
a three-point boundary-value problem by applying the following
property of the Riemann-Stieltjes integral.

\begin{lemma}\label{lem1.1}
Assume that
\begin{enumerate}
\item  $u(t)$ is a bounded function value on $C^2[a,b]$,
i.e., there exist $c$, $C\in \mathbb{R}$ such that $c\le u''(t)\le
C$, $\forall t\in [a,b]$;

\item  $\theta(t)$ is increasing on $[a,b]$;

\item the Riemann-Stieltjes integral
$\int_a^bu''(t)d\theta(t)$ exists.

\end{enumerate}
Then there is a number $v \in \mathbb{R}$ with $c\le v\le C$ such
that
$$\int_a^bu''(t)d\theta(t)=v(\theta(b)-\theta(a)).$$
\end{lemma}

For any continuous solution $u(t)$ of \eqref{e1.1}, by Lemma
\ref{lem1.1}, there exists $\eta\in(0,1)$ such that
$$
\int_a^b u''(t)d\theta(t)=u''(\eta)(\theta(1)-\theta(0))
=u''(\eta)\theta(1).
$$
Let $\sigma=\theta(1)$ and $g(x)=x$, $x\in \mathbb{R}$. Then problem
\eqref{e1.1} can be rewritten as the  three-point boundary-value
problem
\begin{equation}
\begin{gathered}
u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),\quad 0<t<1,\\
u(0)=u'(1)=u''(0)=0,\quad u'''(1)=\sigma u''(\eta),
\end{gathered}\label{e1.2}
\end{equation}

The paper is organized as follows. In the next section, we present
some preliminaries and lemmas. Section 3 is devote to our main
results.

\section{Preliminaries}

\begin{definition}\label{def2.1} \rm
Let $\alpha\in C^3[0,1]\cap C^4(0,1)$. We say $\alpha$ is a lower
solution of \eqref{e1.1} if
\begin{gather*}
\alpha^{(4)}(t)\le
f(t,\alpha(t),\alpha'(t),\alpha''(t),\alpha'''(t)),
 \quad 0<t<1,\\
\alpha(0)\le0,\quad \alpha'(1)\le 0,\\
\alpha''(0)\ge0,\quad \alpha'''(1)\ge
g(\int^1_0\alpha''(t)d\theta(t)).
\end{gather*}
Similarly, $\beta\in C^3[0,1]\cap C^4(0,1)$ is an upper solution of
\eqref{e1.1}, if $\beta$ satisfies similar inequalities in the
reverse order.
\end{definition}

If we denote by $k(t,s)$ the Green's function of
\begin{equation}
\begin{gathered}
-u''(t)=0,\quad 0<t<1,\\
u(0)=u'(1)=0,
\end{gathered}\label{e2.1}
\end{equation}
then
\begin{equation*}
k(t,s)= \begin{cases}
t,& 0\le t\le s\le1,\\
s,& 0\le s\le t\le1.
\end{cases}
\end{equation*}
Setting $-u''=v$, by standard calculation, we get
\begin{gather*}
u(t)=\int^1_0k(t,s)v(s)ds=:(Av)(t),\\
u'(t)=\int^1_tv(s)ds=:(Bv)(t).
\end{gather*}
Obviously, $A, B$ are monotone increasing operators.

We assume that $g$ is an odd function on $\mathbb{R}$. Then problem
\eqref{e1.1} is equivalent to the following integral-differential
boundary-value problem
\begin{equation}
\begin{gathered}
-v''(t)=f(t,(Av)(t),(Bv)(t),-v(t),-v'(t)),\quad 0<t<1,\\
v(0)=0,\quad  v'(1)=g(\int^1_0v(t)d\theta(t)).
\end{gathered}\label{e2.2}
\end{equation}
For $v\in C[0,1]$, we define the operator $\hat{f}$ by
$$
\hat{f}(v(t),v'(t))=f(t,(Av)(t),(Bv)(t),-v(t),-v'(t)).
$$
Then  \eqref{e2.2} is equivalent to
\begin{equation}
\begin{gathered}
-v''(t)=\hat{f}(v(t),v'(t)),\quad 0<t<1,\\
v(0)=0,\quad  v'(1)=g(\int^1_0v(t)d\theta(t)).
\end{gathered}\label{e2.3}
\end{equation}
Suppose $\alpha,\beta$ are the lower and upper solutions of BVP
\eqref{e1.1} such that $\alpha''\ge \beta''$ and let
$\psi=-\beta'',\phi=-\alpha''$. Then we have
\begin{gather*}
-\phi''(t)\le \hat{f}(\phi(t),\phi'(t)),\quad
\phi(0)\le0,\quad\phi'(1)\le g(\int^1_0\phi(t)d\theta(t)),\\
-\psi''(t)\ge \hat{f}(\psi(t),\psi'(t)),\quad
\psi(0)\ge0,\quad\psi'(1)\ge g(\int^1_0\psi(t)d\theta(t)).
\end{gather*}
Since $A, B$ are monotone continuous operators, there exists $M$
such that
$$
M=\sup\limits_{\phi\le v\le \psi}\{\|Av\|_\infty,\|Bv\|_\infty\}>0.
$$

\begin{definition}[\cite{b2}] \label{def2.2}\rm
 Let $f\in C([0,1]\times
\mathbb{R}^4,\mathbb{R})$, $\phi, \psi\in C([0,1],\mathbb{R})$ and
$\phi(t)\le \psi(t), t\in [0,1]$. We say that $f(t,x_1,x_2,x_3,x_4)$
satisfies a Nagumo-type condition with respect to $\phi,\psi$ if
there exists a positive continuous function $h(s)$ on $[0,\infty)$
satisfying
\begin{equation}
|f(t,x_1,x_2,x_3,x_4)|\le h(|x_4|),\label{e2.4}
\end{equation}
for all $(t,x_1,x_2,x_3,x_4)\in[0,1]\times[-M, M]^2
\times[\phi(t),\psi(t)]\times \mathbb{R}$, and
\begin{equation}
\int^\infty_\lambda \frac{s}{h(s)}ds>\max\limits_{0\le t\le
1}\psi(t) -\min\limits_{0\le t\le 1}\phi(t),\label{e2.5}
\end{equation}
where $\lambda=\max\{|\psi(1)-\phi(0)|,|\psi(0)-\phi(1)|\}$.
\end{definition}

\begin{lemma}\label{lem2.3}
Suppose $f$ satisfies the Nagumo-type condition with respect to
$\phi,\psi\in C^2[0,1]$ and $\phi\le \psi$. If BVP (\ref{e2.3}) has
a solution $v(t)$ such that $\phi(t)\le v(t)\le \psi(t)$, then there
exists $N>0$ such that $|v'(t)|\le N$, for $t\in[0,1]$.
\end{lemma}

The proof of the above lemma is similar to that in \cite{b2},
therefore, we omit it.


\section{Main results}

\begin{theorem}\label{thm3.1}
Suppose $\alpha,\beta$ are lower and upper solutions to BVP
\eqref{e1.1} such that $\alpha''(t)\ge\beta''(t)$ and $f$ satisfies
a Nagumo-type condition with respect to $\alpha'',\beta''$. In
addition, we assume that $g$ is odd, continuous and increasing on
$\mathbb{R}$, $\theta$ is increasing on $[0,1]$ and $\theta(0)=0$.
Then BVP \eqref{e1.1} has a solution $u(t)$ such that
$$
\alpha(t)\le u(t)\le \beta(t),\quad\alpha''(t)\ge u''(t)\ge
\beta''(t).
$$
\end{theorem}

\begin{proof} Since $f$ satisfies the Nagumo-type condition with
 respect to
$\phi=-\alpha'',\psi=-\beta''$, there exists a constant $N>0$
depending on $\phi,\psi,h$ such that
\begin{equation}
\int^N_\lambda \frac{s}{h(s)}ds>\max\limits_{0\le t\le 1}\psi(t)
-\min\limits_{0\le t\le 1}\phi(t).\label{e3.1}
\end{equation}
Take $C> \max\{N,\|\phi'\|,\|\psi'\|\}$ and
$p(v')=\max\{-C,\min\{v',C\}\}$. By modifying $\hat{f}$ and $g$ with
respect to $\phi, \psi$, we aim at obtaining a second-order
boundary-value problem and reformulating the new problem as an
integral equation. We show that solutions of the modified problem
lie in the region where $\hat{f},g$ are unmodified and hence are
solutions of problem (\ref{e2.3}). Let $\varepsilon>0$ be a fixed
small number and $F(v(t),v'(t)), G(\int_0^1v(t)d\theta(t))$ are the
modifications of $\hat{f}(v(t),v'(t))$ and
$g(\int_0^1v(t)d\theta(t))$ as follows
\begin{align*}
&F(v(t),v'(t))\\
&=
\begin{cases}
\hat{f}(\psi(t),\psi'(t))+\frac{v(t)-\psi(t)}{1+|v(t)-\psi(t)|},
&\text{if }  v(t)\ge \psi(t)+\varepsilon,\\[3pt]
\hat{f}(\psi(t),p(v'))+[\hat{f}(\psi(t),\psi'(t))-\hat{f}(\psi(t),p(v'(t)))\\
+\frac{v(t)-\psi(t)}{1+|v(t)-\psi(t)|}]\times
\frac{v(t)-\psi(t)}{\varepsilon},
&\text{if } \psi(t)\le v(t)<\psi(t)+\varepsilon,\\[3pt]
\hat{f}(v(t),p(v'(t))),& \text{if }   \phi(t)\le v(t)\le
 \psi(t),\\[3pt]
\hat{f}(\phi(t),p(v'(t)))+[\hat{f}(\phi(t),\phi'(t))
-\hat{f}(\phi(t),p(v'(t)))\\
+\frac{\phi(t)-v(t)}{1+|\phi(t)-v(t)|}]\times
\frac{\phi(t)-v(t)}{\varepsilon},
& \text{if } \phi(t)-\varepsilon< v(t)\le \phi(t),\\[3pt]
\hat{f}(\phi(t),\phi'(t))+\frac{\phi(t)-v(t)}{1+|\phi(t)-v(t)|}, &
\text{if }   v(t)\le \phi(t)-\varepsilon,
\end{cases}
\end{align*}
and
\begin{align*}
&G\big(\int_0^1v(t)d\theta(t)\big)\\
&=\begin{cases} g(\int^1_0\psi(t)d\theta(t))+
\frac{\int_0^1v(t)d\theta(t)-\int^1_0\psi(t)d\theta(t)
}{1+|\int_0^1v(t)d\theta(t)-\int^1_0\psi(t)d\theta(t)|},
& \text{if }  v(t)>\psi(t),\\
g(\int_0^1v(t)d\theta(t)),
& \text{if }  \phi(t)\le v(t)\le\psi(t),\\
g(\int^1_0\phi(t)d\theta(t))+
\frac{\int^1_0\phi(t)d\theta(t)-\int_0^1v(t)d\theta(t)}
{1+|\int^1_0\phi(t)d\theta(t)-\int_0^1v(t)d\theta(t)|}, & \text{if }
v(t)<\phi(t).
\end{cases}
\end{align*}
Obviously, $F: \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ and $G :
\mathbb{R}\to \mathbb{R}$ are continuous and bounded. Consider the
modified problem
\begin{equation}
\begin{gathered}
-v''(t)=F(v(t),v'(t)),\quad 0<t<1,\\
v(0)=0,\quad  v'(1)=G(\int^1_0v(t)d\theta(t)).
\end{gathered}\label{e3.2}
\end{equation}
Then, the BVP \eqref{e3.2} is equivalent to the integral equation
\begin{equation}
v(t)=G(\int^1_0v(t)d\theta(t))t+\int^1_0k(t,s)F(v(s),v'(s))ds.\label{e3.3}
\end{equation}
Since $F$ and $G$ are continuous and bounded, there exist $M>C$,
$m>0$ such that
\begin{gather*}
|F(v(t),v'(t))|<M\quad {\rm on} \quad\mathbb{R}\times
\mathbb{R},\\
|G(\int_0^1v(t)d\theta(t))|<m\quad {\rm on}\quad\mathbb{R}.
\end{gather*}
Choose $\tilde{M}\ge \delta+m+M$ and consider the open bounded and
convex set
$$
\Omega=\{v\in C^1[0,1]: \|v\|<\tilde{M},\|v'\|<\tilde{M}\}.
$$
Define $\tilde{F}: C^1[0,1]\times \mathbb{R}\to C[0,1]$ and
$\tilde{G}: C^1[0,1]\to C[0,1]$ by
\begin{gather*}
\tilde{F}(v)(t)=\int^1_0k(t,s)F(v(s),v'(s))ds,
\\
\tilde{G}(v)(t)=G(\int^1_0v(t)d\theta(t))t.
\end{gather*}
It is obvious that $\tilde{F},\tilde{G}$ are compact. Let
$\tilde{T}=\tilde{G}+\tilde{F}$, it is easy to see that \eqref{e3.2}
is equivalent to the fixed point equation
\begin{equation}
\tilde{T}v=v. \label{e3.4}
\end{equation}
Then, it follows from Schauder fixed point theorem that the integral
equation (\ref{e3.4}) has a fixed point $v_*$. In other words, the
BVP \eqref{e3.2} has a solution $v_*$. Also, from the definitions of
$\phi,\psi,F$ and $G$ and the choice of $C$, we have
\begin{gather*}
-\phi''(t)\le \hat{f}(\phi(t),\phi'(t))=F(\phi(t),\phi'(t)),\quad
0\le t\le1,\\
\phi(0)\le 0,\quad\phi'(1)\le g(\int^1_0\phi(t)d\theta(t))
=G(\int^1_0\phi(t)d\theta(t))
\end{gather*}
and
\begin{gather*}
-\psi''(t)\ge\hat{f}(\psi(t),\psi'(t))=F(\psi(t),\psi'(t)),\quad
0\le
 t\le1,
\\
\psi(0)\ge 0,\quad\psi'(1)\ge g(\int^1_0\psi(t)d\theta(t))
=G(\int^1_0\psi(t)d\theta(t)).
\end{gather*}
That is, $\phi$ and $\psi$ are the lower and upper solutions of
\eqref{e3.2}.

We claim that the solution $v_*$ of \eqref{e3.2} satisfies
$\phi(t)\le v_*(t)\le\psi(t)$ for $t\in [0,1]$. We only prove
$\phi(t)\le v_*(t)$, $t\in[0,1]$, the other part is proved in a
similar way. Let $w(t)=\phi(t)- v_*(t)$ for $t\in[0,1]$. Assume that
$w(t_0)=\max\limits_{0\le t\le1} w(t)>0$. We divide the proof into
three cases.
\smallskip

\noindent\textbf{Case 1.} $t_0=0$. Then we have $w(0)=\phi(0)-
v_*(0)=\phi(0)>0$. It contradict the definition of $\phi$.
\smallskip

\noindent\textbf{Case 2.} $t_0=1$. Then $w(1)>0$ and $w'(1)\ge 0$.
The boundary value conditions of  \eqref{e3.2} imply
$$
w'(1)=\phi'(1)-{v_*}'(1)\le g(\int^1_0\phi(t)d\theta(t))
-G(\int^1_0v_{*}(t)d\theta(t)).
$$
If $v_*(t)<\phi(t)$, then
\begin{align*}
G(\int^1_0v_*(t)d\theta(t)) &=g(\int^1_0\phi(t)d\theta(t))+
\frac{\int^1_0\phi(t)d\theta(t)-\int^1_0v_*(t)d\theta(t)}
{1+\int^1_0\phi(t)d\theta(t)-\int^1_0v_*(t)d\theta(t)}\\
&>g(\int^1_0\phi(t)d\theta(t)),
\end{align*}
which implies $w'(1)<0$. It is a contradiction. If $v_*(t)>\psi(t)$,
then
\begin{align*}
G(\int^1_0v_*(t)d\theta(t)) &=g(\int^1_0\psi(t)d\theta(t))+
\frac{\int^1_0v_*(t)d\theta(t)-\int^1_0\psi(t)d\theta(t)}
{1+\int^1_0v_*(t)d\theta(t)-\int^1_0\psi(t)d\theta(t)}\\
&>g(\int^1_0\psi(t)d\theta(t))\\
&\ge g(\int^1_0\phi(t)d\theta(t)),
\end{align*}
we can also get $w'(1)<0$, which is a contradiction. Hence,
$\phi(t)\le v_*(t)\le\psi(t).$ So
$$
G(\int^1_0v_*(t)d\theta(t))=g(\int^1_0v_*(t)d\theta(t)) \ge
g(\int^1_0\phi(t)d\theta(t)),
$$
which implies $w'(1)\le 0$. If $w'(1)<0$, it is a contradiction. So
we have $w'(1)=0$. Since $t_0\neq 0$, there exists $t_1\in[0,1)$
such that $w(t_1)=0$ and $w(t)>0$ on $(t_1,1]$. Then for each
$t\in[t_1,1]$, we have
$$
w''(t)=\phi''(t)-{v_*}''(t)\ge-\hat{f}(\phi(t),\phi'(t))
+\Big[\hat{f}(\phi(t),\phi'(t))+ \frac{w(t)}{1+w(t)}\Big]>0.
$$
Thus, by $w'(1)=0$, we get $w'(t)\le0\quad {\rm on} \quad[t_1,1]$,
which implies that $w$ is decreasing on $[t_1,1]$ and hence $w(1)\le
0$, it is a contradiction.
\smallskip

\noindent\textbf{Case 3.} $t_0\in(0,1)$. Then, we have $w'(t_0)=0$
and $w''(t_0)\le 0$. However, for $0<w(t_0)<\varepsilon$, we have
\begin{align*}
w''(t_0)&=\phi''(t_0)-{v_*}''(t_0)\\
&\ge-\hat{f}(\phi(t_0),\phi'(t_0))+F(v_*(t_0),{v_*}'(t_0))\\
&= \frac{w^2(t_0)}{(1+w(t_0))\varepsilon}>0,
\end{align*}
a contradiction. For $w(t_0)\ge \varepsilon$, we obtain
$$
w''(t_0)=\phi''(t_0)-{v_*}''(t_0)\ge\frac{w(t_0)}{1+w(t_0)}>0,
$$
it is also a contradiction. Thus, $\phi(t)\le v_*(t), t\in [0,1]$.
By the similar discussion, we can get $v_*(t)\le \psi(t)$.

According to the Lemma \ref{lem2.3} and the choice of $C$, for the
solution $v_*$ of \eqref{e3.2} with $\phi(t)\le v_*(t)\le \psi(t)$,
$t\in [0,1]$, we have
$$
|{v_*}'(t)|\le N<C.
$$
Thus,
\begin{gather*}
F(v_*(t),{v_*}'(t))=\hat{f}(v_*(t),{v_*}'(t)), \\
G(\int_0^1v_*(t)d\theta(t))=g(\int_0^1v_*(t)d\theta(t)).
\end{gather*}
Hence, the solution $v_*$ of \eqref{e3.2} with $\phi(t)\le v_*(t)\le
\psi(t)$, $t\in [0,1]$, is a solution of
 (\ref{e2.3}).
The proof is complete.
\end{proof}

Using the Theorem \ref{thm3.1}, we can prove the following result.

\begin{corollory}\label{coro3.2}
 Suppose $\alpha,\beta$ are lower and
upper solutions to  \eqref{e1.2} such that
$\alpha''(t)\ge\beta''(t)$ and $f$ satisfies a Nagumo-type condition
with respect to $\alpha'',\beta''$. Then \eqref{e1.2} has a solution
$u(t)$ such that
$$
\alpha(t)\le u(t)\le \beta(t),\quad\alpha''(t)\ge u''(t)\ge
\beta''(t).
$$
\end{corollory}

\begin{thebibliography}{00}

\bibitem{b1} C. Bai, D. Yang, H. Zhu; \emph{Existence of
solutions for fourth order differential equation with four-point
boundary conditions}, Appl. Math. Lett., 20 (2007) 1131-1136.

\bibitem{b2} Z. Bai; \emph{The upper and lower solution method for some
fourth-order boundary-value problem}, Nonlinear Anal., 67 (2007)
1704-1709.

\bibitem{b3} Z. Bai, H. Wang; \emph{On positive solutions of some
nonlinear fourth-order beam equations}, J. Math. Appl., 270 (2002)
357-368.

\bibitem{c1} A. Cabada, J.$\rm\acute{A}$. Cid, L. Sanchez;
\emph{Positivity and lower and upper solutions for fourth order
boundary-value problems}, Nonlinear Anal., 67 (2007) 1599-1612.

\bibitem{k1} R. A. Khan, M. Rafique; \emph{Existence and multiplicity
 results for some
three-point boundary-value problems}, Nonlinear Anal., 66 (2007)
1686-1697.

\bibitem{l1} B. Liu; \emph{Positive solutions of fourth-order two point
boundary-value problems}, Appl. Math. Comput., 148 (2004) 407-420.

\bibitem{l2} F. Li, Q. Zhang, Z. Liang; \emph{Existence and
multiplicity of solutions of a kind of fourth-order boundary-value
 problem}, Nonlinear Anal., 16 (2005) 803-816.

\end{thebibliography}
\end{document}