\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil A non-resonant multi-point boundary-value problem \hfil EJDE/Conf/10} {EJDE/Conf/10 \hfil Chaitan P. Gupta \hfil} \begin{document} \setcounter{page}{143} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent Fifth Mississippi State Conference on Differential Equations and Computational Simulations, \newline Electronic Journal of Differential Equations, Conference 10, 2003, pp 143--152. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A non-resonant multi-point boundary-value problem for a $p$-Laplacian type operator % \thanks{ {\em Mathematics Subject Classifications:} 34B10, 34B15, 34G20. \hfil\break\indent {\em Key words:} Multi-point boundary-value problem, three-point boundary-value problem, \hfil\break\indent $p$-Laplacian, Leray Schauder Continuation theorem, Caratheodory's conditions. \hfil\break\indent \copyright 2003 Southwest Texas State University. \hfil\break\indent Published February 28, 2003. } } \date{} \author{Chaitan P. Gupta} \maketitle \begin{abstract} Let $\phi$ be an odd increasing homeomorphism from $\mathbb{R}$ onto $\mathbb{R}$ with $\phi (0)=0$, $f:[0$,$1]\times \mathbb{R}^{2}\to \mathbb{R}$ be a function satisfying Caratheodory's conditions and $e(t)\in L^{1}[0,1]$. Let $\xi_{i}\in (0,1)$, $a_{i}\in \mathbb{R}$, $i=1,2, \dots , m-2$, $\sum_{i=1}^{m-2}a_{i}\neq 1$, $0<\xi_{1}<\xi_{2}<\dots<\xi_{m-2}<1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary-value problem \begin{gather*} (\phi (x'(t)))'=f(t,x(t),x'(t))+e(t),\quad 00, \end{cases}\\ F(\alpha ,\eta ) =\frac{1}{2(\alpha -1)^{2}}[\alpha ^{2}(1-\eta )^{2}+(\alpha ^{2}-2\alpha )\eta ^{2}+1]. \end{gather*} \end{theorem} \paragraph{Proof} If $\alpha \leq 0$, we note from $\phi (x'(1))=\alpha \phi(x'(\eta ))$ that there exists an $\xi \in (\eta ,1)$ such that $\phi (x'(\xi ))=0$. It follows from the Wirtinger's inequality \cite[Theorem 256]{hardy} that $$\| \phi (x'(t))\|_{2}\leq \frac{2}{\pi }\| (\phi (x'(t)))'\|_{2}. \label{Eq0}$$ Next, we note, again, from $\phi (x'(1))=\alpha \phi (x'(\eta ))$ that for $00$, $\alpha \neq 1$, by (\ref{EQ6}), (\ref{EQ7}). This completes the proof of the theorem. \hfill$\square$ \paragraph{Remark} %2 It is easy to see that $C(-0.1,\eta )=2/\pi$, for all $\eta \in (0,1)$. Indeed, $\sqrt{F(-0.1,\eta )}\geq 0.648986183$ and $2/\pi\approx 0.6366197724$. Also $C(-2,1/3)=\sqrt{11/54}$ and $C(-2,15/16)=2/\pi$, since $\sqrt{F(-2,15/16)}=\sqrt{1030}/48>2/\pi$. \section{Existence Theorems} \paragraph{Definition} A function $f:[0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ satisfies Caratheodory's conditions if \begin{itemize} \item[(i)] For each $(x$,$y)\in \mathbb{R}^{2}$, the function $t\in [0,1]\to f(t,x,y)\in \mathbb{R}$ is measurable on $[0,1]$ \item[(ii)] for a.e. $t\in [0,1]$, the function $(x,y)\in \mathbb{R}^{2}\to f(t,x,y)\in \mathbb{R}$ is continuous on $\mathbb{R}^{2}$ \item[(iii)] for each $r>0$, there exists $\alpha_{r}(t)\in L^{1}[0,1]$ such that $| f(t,x,y)|\leq \alpha_{r}(t)$ for a.e. $t\in [0,1]$ and all $(x,y)\in \mathbb{R}^{2}$ with $\sqrt{x^{2}+y^{2}}\leq r$. \end{itemize} \begin{theorem} \label{one} Let $f:[0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ be a function satisfying Caratheodory's conditions. Assume that there exist functions $p(t)$, $q(t)$, $r(t)$ in $L^{1}(0,1)$ such that $$| f(t,x_{1},x_{2})| \leq p(t)\phi (| x_{1}|)+q(t)\phi (| x_{2}| )+r(t) \label{cond1}$$ for a.e. $t\in [ 0,1]$ and all $(x_{1},x_{2})\in \mathbb{R}^{2}$. Also let $a_{i}\in \mathbb{R}$, $\xi_{i}\in (0,1)$, $i=1,2,\dots, m-2$, $0<\xi_{1}<\xi_{2}<\dots<\xi_{m-2}<1$, with $\alpha=\sum_{i=1}^{m-2}a_{i}\neq 1$ be given. Then the boundary-value problem (\ref{mbp}) has at least one solution in $C^{1}[0,1]$ provided $$\| p(t)\|_{1}+\| q(t)\|_{1}+\tau<1. \label{cond2}$$ where $\tau$ is as defined in Theorem \ref{ONE}. \end{theorem} \paragraph{Proof} It is easy to see that the boundary-value problem (\ref{mbp}) is equivalent to the fixed point problem $$x(t)=\int_{0}^{t}\phi ^{-1}\Big(\int_{0}^{s}[f(\tau ,x(\tau ),x'(\tau ))+e(\tau )]d\tau +A\Big)ds, \label{EQ8}$$ where \begin{eqnarray*} A&=&\sum_{i=1}^{m-2}(\frac{a_{i}}{1-\sum_{i=1}^{m-2}a_{i}})\int_{0}^{\xi _{i}}[f(\tau ,x(\tau ),x'(\tau ))+e(\tau )]d\tau \\ &&-\frac{1}{1-\sum_{i=1}^{m-2}a_{i}}\int_{0}^{1}[f(\tau ,x(\tau ),x'(\tau ))+e(\tau )]d\tau . \end{eqnarray*} It is standard to check that the mapping $x(t)\in C^{1}[0,1]\mapsto \int_{0}^{t}\phi ^{-1}(\int_{0}^{s}[f(\tau ,x(\tau ),x'(\tau ))+e(\tau )]d\tau +A)ds\in C^{1}[0,1],$ is a compact mapping. We apply the Leray-Schauder Continuation theorem (see, e.g. \cite{mawhin}) to obtain the existence of a solution for (\ref{EQ8}) or equivalently to the boundary-value problem (\ref{mbp}). To do this, it suffices to verify that the set of all possible solutions of the family of equations \begin{gathered} (\phi (x'(t)))'=\lambda f(t,x(t),x'(t))+\lambda e(t),\quad 01$, the odd increasing homeomorphism$\phi :\mathbb{R}\to \mathbb{R}$defined by $\phi (t)=| t| ^{p-2}t \quad \text{for }t\in \mathbb{R},$ then Theorems \ref{one}, \ref{two} give existence theorems for the analogous three-point boundary-value problems for the one-dimensional analogue of the p-Laplacian. However, Theorems \ref{one}, \ref{two} apply to more general differential operators than a p-Laplacian, since Theorems \ref{one}, \ref{two} do not require the homeomorphism$\phi \$ to be homogeneous as happens to be the case for the p-Laplacian. \begin{thebibliography}{0} \frenchspacing \bibitem{gnt1} C. P. Gupta; S. K. Ntouyas and P. Ch. Tsamatos; {\it Existence Results for m-Point Boundary Value Problems,} Differential Equations Dynam. Systems, Vol. 2 (1994), 289-298. \bibitem{gt3} C. P. Gupta, S. I. Trofimchuk, {\it Solvability of a multi-point boundary-value problem of Neumann type,} Abstr. Anal. \& Appl., 4, \# 2, (1999) 71-81.. \bibitem{hardy} G. H. Hardy, J. E. Littlewood, and G. Polya, {\it Inequalities,} Cambridge University Press, London/New York, 1967. \bibitem{mawhin} J. Mawhin, {\it Topological degree methods in nonlinear boundary value problems,} in ''NSF-CBMS Regional Conference Series in Math.'' No. 40, Amer. Math. Soc., Providence, RI, 1979. \bibitem{VA} V.A. Il'in, and E. I. Moiseev, {\it Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator,} Differential Equations, Vol. 23, No. 8, (1988), pp. 979-987. \end{thebibliography} \noindent\textsc{Chaitan P. Gupta}\\ Department of Mathematics, 084 \\ University of Nevada, Reno \\ Reno, NV 89557, USA \\ email: gupta@unr.edu \end{document}