Nonlinear discrete Sturm-Liouville problems

In this paper we study nonlinear boundary value problems of the form Delta[p(t - 1)Delta y(t - 1)] + q(t)y(t) + lambda y(t) = f (y(t)); t = a + 1,..., b + 1, subject to a(11)y(a)+a(12)Delta y(a)=0 and a(21)y(b+1)+a(22)Delta y(b+1)=0. The parameter lambda is an eigenvalue of the associated linear problem; that is, there is a nontrivial function u that satisfies the boundary conditions and also Delta[p(t - 1)Delta u(t - 1)] + q(t)u(t) + lambda u(t) = 0 for t in {a + 1,a + 2,...,b + 1}. We establish sufficient conditions for the solvability of such problems. In addition, in those cases where the nonlinearity is "small," we provide a qualitative analysis of the relation between solutions of the nonlinear problem and eigenfunctions of the linear one. (c) 2005 Elsevier Inc. All rights reserved.