International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 4, Pages 705-710
doi: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 Chaitan P. Gupta. 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]×R2→R be function satisfying Caratheodory's conditions and e(t)∈L1[0,1]. Let η∈(0,1), ξi∈(0,1), ai≥0, i=1,2,…,m−2, with ∑i=1m−2ai=1, 0<ξ1<ξ2<…<ξm−2<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=1m−2aix(ξi).
Conditions for the existence of a solution for the above boundary value problems are given
using Leray Schauder Continuation theorem.