\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 70, pp. 1--6.\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-2007/70\hfil Existence of positive pseudo-symmetric solutions] {Existence of positive pseudo-symmetric solutions for one-dimensional $p$-Laplacian boundary-value problems} \author[Y. Yang\hfil EJDE-2007/70\hfilneg] {Yitao Yang} % in alphabetical order \address{Yitao Yang \newline Institute of Automation, Qufu Normal University, Qufu, Shandong 273165, China} \email{yitaoyangqf@163.com} \thanks{Submitted March 12, 2007. Published May 10, 2007.} \subjclass[2000]{34B15, 34B18} \keywords{Iterative; pseudo-symmetric positive solution; $p$-Laplacian} \begin{abstract} We prove the existence of positive pseudo-symmetric solutions for four-point boundary-value problems with $p$-Laplacian. Also we present an monotone iterative scheme for approximating the solution. The interesting point here is that the nonlinear term $f$ involves the first-order derivative. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this paper, we consider the four-point boundary value problem \begin{gather}\label{eq1} (\phi_p(u'))'(t)+ q(t)f(t,u(t),u'(t))=0,\quad t\in (0,1), \\ \label{eq2} u(0)-\alpha u'(\xi)=0,\quad u(\xi)-\gamma u'(\eta)=u(1)+\gamma u'(1+\xi-\eta), \end{gather} where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$, $(\phi_{p})^{-1}=\phi_{q}$, $\frac{1}{p}+\frac{1}{q}=1$, $\alpha,\gamma \geq 0$, $\xi, \eta\in (0,1)$ are prescribed and $\xi<\eta$. The study of multipoint boundary-value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseer \cite{i1,i2}. Since then, the more general nonlinear multipoint boundary-value problems have been studied by many authors by using the Leray-Schauder continuation theorem, nonlinear alternative of Leray-Schauder and coincidence degree theory, we refer the reader to \cite{a1,a2,a3,h1} for some recent results. Recently, Avery and Henderson \cite{a4} consider the existence of three positive solutions for the problem \begin{gather}\label{eq3} (\phi_p(u'))'(t)+ q(t)f(t,u(t))=0,\quad t\in (0,1), \\ \label{eq4} u(0)=0,\quad u(\eta)=u(1). \end{gather} The definition of pseudo-symmetric was introduced in their paper. Based on this definition, Ma \cite{m1} studied the existence and iteration of positive pseudo-symmetric solutions for the problem (\ref{eq3})-(\ref{eq4}). However, to the best of our knowledge, no work has been done for BVP (\ref{eq1})-(\ref{eq2}) using the monotone iterative technique. The aim of this paper is to fill the gap in the relevant literatures. We obtain not only the existence of positive solutions for (\ref{eq1})-(\ref{eq2}), but also give an iterative scheme for approximating the solutions. It is worth stating that the first term of our iterative scheme is a constant function or a simple function. Therefore, the iterative scheme is significant and feasible. At the same time, we give a way to find the solution which will be useful from an application viewpoint. We consider the Banach space $E=C^{1}[0,1]$ equipped with norm $$\|u\|:=\max\{\|u\|_{0}, \|u'\|_{0}\},$$ where $\|u\|_{0}=\max_{0\leq t\leq 1}|u(t)|$, $\|u'\|_{0}=\max_{0\leq t\leq 1}|u'(t)|$. In this paper, a positive solution $u(t)$ of BVP (\ref{eq1}), (\ref{eq2}) means a solution $u(t)$ of (\ref{eq1}), (\ref{eq2}) satisfying $u(t)>0$, for $00$ and $\upsilon^{*}$ is a nonnegative concave function on $[0,1]$, we conclude that $\upsilon^{*}(t)>0$, $t\in (0,1)$. Therefore, $\upsilon^{*}$ is a positive pseudo-symmetric solution of (\ref{eq1})-(\ref{eq2}). \end{proof} \begin{corollary}\label{corF25} Assume {\rm (H1),(H2)} hold. If $$\label{eq15} \limsup_{l\to 0}\inf_{t\in [\xi,\frac{1+\xi}{2}]}\frac{f(t,l,0)} {\phi_{p}(l)}\geq \phi_{p}(\frac{1+\xi}{2\xi}B),$$ particularly, $\limsup_{l\to 0}\inf_{t\in [\xi,\frac{1+\xi}{2}]}\frac{f(t,l,0)} {\phi_{p}(l)}=+\infty$, $$\label{eq16}\liminf_{l\to +\infty}\sup_{t\in [0,1]}\frac{f(t,l,l)} {\phi_{p}(l)}\leq \phi_{p}(A),$$ particularly, $\liminf_{l\to +\infty}\sup_{t\in [0,1]}\frac{f(t,l,l)} {\phi_{p}(l)}=0$. Where $A,B$ are defined as (\ref{eq9}), (\ref{eq10}). Then there exist two positive numbers $a,b$ with \$\frac{2b}{1+\xi}