Solvability of a fourth-order boundary value problem with periodic boundary conditions II

Abstract

Let f:[0,1]×R4→R be a function satisfying Caratheodory's conditions and e(x)∈L1[0,1]. This paper is concerned with the solvability of the fourth-order fully quasilinear boundary value problem d4udx4+f(x,u(x),u′(x),u″(x),u‴(x))=e(x), 0<x<1, with u(0)−u(1)=u′(0)−u′(1)=u″(0)-u″(1)=u‴(0)-u‴(1)=0. This problem was studied earlier by the author in the special case when f was of the form f(x,u(x)), i.e., independent of u′(x), u″(x), u‴(x). It turns out that the earlier methods do not apply in this general case. The conditions need to be related to both of the linear eigenvalue problems d4udx4=λ4u and d4udx4=−λ2d2udx2 with periodic boundary conditions.