Discrete third-order three-point right-focal boundary value problems

Abstract

We are concerned with the discrete right-focal boundary value problem Δ 3 x( t) = f( t, x( t+1)), x( t 1) = Δx( t 2) = Δ 2 x( t 3) = 0, and the eigenvalue problem Δ 3 x( t) = λa( t) f( x( t+1)) with the same boundary conditions, where t 1 < t 2 < t 3. Under various assumptions on f, a, and λ, we prove the existence of positive solutions of both problems by applying a fixed-point theorem.