JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATlONS 168, 540-551 (1992) Solvability of a Three-Point Nonlinear Boundary Value Problem for a Second Order Ordinary Differential Equation CHAITAN P. GUPTA* Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois, 601 IS Submitted by V. Lashmikantham Received March 1, 1991 1. INTRODUCTION Let f : [0, 1] x R2 -+ R be a given function satisfying Caratheodory’s conditions, e : [0, l] + R be a function in L’(0, 1), and let q E (0, 1) be given. This paper is devoted to studying the following second-order three-point nonlinear boundary value problem: un =f(x, u(x), u’(x)) - e(x), O<x<l, (1.1) u(0) = 0, 4tl) = 41). (1.2) In case f(x, U, u’) =ps(x) +p,(x) u +pJx) U’ with pk : [0, l] 4 R locally integrable, the boundary value problem (l.lt( 1.2) was studied by Kiguradze and Lomtatidze [I]. The purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem (1.1 t( 1.2) under natural conditions on f using degree-theoretic arguments. We note that if u is a solution of (1.1 )-( 1.2) then there exists at least one 5 E (q, l), such that u’(c) = 0. Accordingly, the boundary value problem uN =f(x, u(x), u’(x)) -e(x), O<x<l, (1.3) u(0) = u'( 1) = 0 (1.4) can be considered as a limiting case of the problem (l.l)-( 1.2) when q= 1. Our results for (1.1 t(1.2) give the sharpest possible results for (1.3t( 1.4) when q = 1. * Part of the work was done while visiting Panjab University, Chandigarh, India as a Fullbright Scholar during December 1990. 540 0022-241X/92 $5.00 Copyright 0 1992 by Academic Press. Inc. All rights of reproduction in any form reserved. CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector