\documentclass[reqno]{amsart} \usepackage{hyperref} \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 \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this paper, we study the existence and uniqueness of solutions to the second-order non-homogeneous two-point boundary-value problem $$\label{e1.1} \begin{gathered} x''(t)+f(t,x(t),x'(t))=0,\quad t\in(0,1),\\ x'(0)=a,\quad x(1)=b, \end{gathered}$$ 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