Copyright © 2010 Fuyi Xu and Jian Liu. 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

We investigate nonlinear singular fourth-order eigenvalue problems with nonlocal boundary condition u(4)(t)-λh(t)f(t,u,u′′)=0, 0<t<1, u(0)=u(1)=∫01a(s)u(s)ds, u′′(0)=u′′(1)=∫01b(s)u′′(s)ds, where a,b∈L1[0,1], λ>0, h may be singular at t=0 and/or 1. Moreover f(t,x,y) may also have singularity at x=0 and/or y=0. By using fixed point theory in cones, an explicit interval for λ is derived such that for any λ in this interval, the existence of at least one symmetric positive solution to the boundary value problem is guaranteed. Our results extend and improve many known results including singular and nonsingular cases. The associated Green's function for the above problem is also given.