International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 4, Pages 725-740
doi:10.1155/S0161171294001031

Nonresonance conditions for fourth order nonlinear boundary value problems

C. De Coster, C. Fabry, and F. Munyamarere

Département de Mathématique, Université Catholique de Louvain, Chemin du Cyclotron 2, Louvain-la-Neuve B-1348, Belgium

Received 6 February 1992

Copyright © 1994 C. De Coster et al. 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

This paper is devoted to the study of the problemu(4)=f(t,u,u′,u″,u‴),u(0)=u(2π),   u′(0)=u′(2π),   u″(0)=u″(2π),   u‴(0)=u‴(2π).We assume that f can be written under the formf(t,u,u′,u″,u‴)=f2(t,u,u′,u″,u‴)u″+f1(t,u,u′,u″,u‴)u′+f0(t,u,u′,u″,u‴)u+r(t,u,u′,u″,u‴)where r is a bounded function. We obtain existence conditions related to uniqueness conditions for the solution of the linear problemu(4)=au+bu″,u(0)=u(2π),   u′(0)=u′(2π),   u″(0)=u″(2π),   u‴(0)=u‴(2π).