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International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 1, Pages 127-137

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

Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4801, USA

Received 1 August 1989; Revised 17 December 1989

Copyright © 1991 Hindawi Publishing Corporation. 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.


Let f:[0,1]×R4R 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.