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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 116, 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 (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/116\hfil Upper and lower solutions]
{The Method of upper and lower solutions for
second-order non-homogeneous two-point boundary-value problem}
\author[M. Jia, X. Liu\hfil EJDE-2007/116\hfilneg]
{Mei Jia, Xiping Liu} % in alphabetical order
\address{Mei Jia \newline
College of Science, University of Shanghai for Science
and Technology, Shanghai 200093, China}
\email{jiamei-usst@163.com}
\address{Xiping Liu \newline
College of Science, University of Shanghai for Science and
Technology, Shanghai 200093, China}
\email{xipingliu@163.com}
\thanks{Submitted June 7, 2007. Published August 30, 2007.}
\thanks{Supported by grant 05EZ52 from the Foundation of
Educational Department of Shanghai}
\subjclass[2000]{34B15, 34B27}
\keywords{Upper and lower solutions; cone; monotone iterative method}
\begin{abstract}
This paper studies the existence and uniqueness of
solutions for a type of second-order two-point boundary-value
problem depending on the first-order derivative through a non-linear
term. By constructing a special cone and using the upper and lower
solutions method, we obtain the sufficient conditions of the
existence and uniqueness of solutions, and a monotone
iterative sequence solving the boundary-value problem.
An error estimate formula is also given under the condition of
a unique solution.
\end{abstract}
\maketitle
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\section{Introduction}
In this paper, we study the existence and uniqueness of
solutions to the second-order non-homogeneous two-point
boundary-value problem
\begin{equation} \label{e1.1}
\begin{gathered}
x''(t)+f(t,x(t),x'(t))=0,\quad t\in(0,1),\\
x'(0)=a,\quad x(1)=b,
\end{gathered}
\end{equation}
where $f\in C([0,1]\times \mathbb{R}^2,\mathbb{R})$, and $a$,
$b\in \mathbb{R}$.
It is well known that the upper and lower solutions method is an
important tool in studying boundary-value problem of ordinary
differential equation. Recently, there are numerous results of the
problem by means of the method (see the references in this article).
We notice that most of these papers study the existence and
uniqueness of solutions of the boundary-value problem with
nonlinear term $f(t,u)$. The nonlinear term $f$, however, usually
satisfies Nagumo condition when the $f$ depends on the first order
derivative (see for example \cite{b1,d1,j1,j2}), which weakens the role of
the first order derivative term.
In this paper, the nonlinear term $f$ depends on the first order
derivative and does not need to satisfy the Nagumo condition. By
constructing a special cone and using the upper and lower
solutions method, we obtain the sufficient conditions of the
existence and uniqueness of solutions, as well as the monotone
iterative sequence which is used to solve the boundary-value
problem. The error estimate formula is also given under the
condition of unique solution. And the method we adopt is new and
so are the conclusions we obtain.
\section{Preliminaries}
Throughout this paper, we assume that $N$ satisfies the hypothesis
\begin{itemize}
\item[(H1)] $0\frac{1-\cos N}{N}$.
\end{lemma}
\begin{theorem} \label{thm4.1}
Suppose that the hypotheses of Theorem \ref{thm3.1} hold, and
\begin{itemize}
\item[(H4)] There exists a constant $M_1$ with $0