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

Abstract

Let f [0, I] R R be function satisfying Caratheodory’s conditions and e(t) L[0, 1]. Let r/( (0,1), ’, (0,1), a, >_ 0, 1,2,- ,m- 2, with ,,2 ai 1, 0 < f < f2 < < ,-2 < be given. This paper is concerned with the problem of existence of a solution for the following boundary value problems x"(t) f(t,z(t),z’(t)) + (t),O < < 1, x’(O) O, x(1) x(r/), :"() =/(t, :(), :’())+ (t),o < < , x’(0) 0, x() ET a,x(,). Conditions for the existence of a solution for the above boundary -alue problems are given using Leray Schauder Continuation theorem.