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International Journal of Mathematics and Mathematical Sciences
Volume 18, Issue 4, Pages 705-710
http://dx.doi.org/10.1155/S0161171295000901

Existence theorems for a second order m-point boundary value problem at resonance

Department of Mathematics, University of Nevada, Reno, NV 89557, USA

Received 15 October 1993; Revised 3 May 1994

Copyright © 1995 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let f:[0,1]×R2R be function satisfying Caratheodory's conditions and e(t)L1[0,1]. Let η(0,1), ξi(0,1), ai0, i=1,2,,m2, with i=1m2ai=1, 0<ξ1<ξ2<<ξm2<1 be given. This paper is concerned with the problem of existence of a solution for the following boundary value problems x(t)=f(t,x(t),x(t))+e(t),0<t<1,x(0)=0,x(1)=x(η),x(t)=f(t,x(t),x(t))+e(t),0<t<1,x(0)=0,x(1)=i=1m2aix(ξi).

Conditions for the existence of a solution for the above boundary value problems are given using Leray Schauder Continuation theorem.