Nonresonance conditions for fourth order nonlinear boundary value problems

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 thatfcan 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‴)whereris 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).