Nonlinear discrete Sturm–Liouville problems

Abstract

In this paper we study nonlinear boundary value problems of the form Δ [ p ( t − 1 ) Δ y ( t − 1 ) ] + q ( t ) y ( t ) + λ y ( t ) = f ( y ( t ) ) ; t = a + 1 , … , b + 1 , subject to a 11 y ( a ) + a 12 Δ y ( a ) = 0 and a 21 y ( b + 1 ) + a 22 Δ y ( 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 ) Δ u ( t − 1 ) ] + q ( t ) u ( t ) + λ 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.