Abstract

Let ? and ? be two odd increasing homeomorphism from R onto R with ?(0)=0,?(0)=0, and let f:[0,1]×R×R?R be a function satisfying Caratheodory's conditions. Let a i??,?i(0,1),i=1,2,...,m-2,0<? 1<?2<?<?m-2<1 be given. We are interested in the problem of existence of solutions for the m-point boundary value problem:(?(x?))?=f(t,x,x?),t?(0,1),x?(0)=0, ?(x?(1))=?i=1m-2ai?(x?(? i)). We note that this non-linear m-point boundary value problem is always at resonance since the associated m-point boundary value problem (?(x?))?=0,t?(0,1),x?(0)=0,?(x?(1))= ?i=1m-2ai?(x?(?i)) has non-trivial solutions x(t)=?,??R (an arbitrary constant). Our results are obtained by a suitable homotopy, Leray-Schauder degree properties, and a priori bounds. © 2003 Elsevier Ltd. All rights reserved.