Nonlinear discrete Sturm–Liouville problems

In this paper we study nonlinear boundary value problems of the form Δ p(t −1)Δy(t − 1) +q(t)y(t) +λy(t ) = f y(t) ; t = a +1, . . . , b +1, subject to a11y(a)+ a12Δy(a) =0 and a21y(b +1) +a22Δy(b +1) = 0. The parameter λ is an eigenvalue of the associated linear problem; that is, there is a nontrivial function u that satisfies the boundary conditions and also Δ p(t −1)Δu(t −1) + q(t)u(t) +λu(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.  2005 Elsevier Inc. All rights reserved