Existence of Solutions of a Discrete Fourth‐Order Boundary Value Problem

Abstract

Let a, b be two integers with b − a ≥ 5 and let 𝕋2 = {a + 2, a + 3, …, b − 2}. We show the existence of solutions for nonlinear fourth‐order discrete boundary value problem Δ4u(t − 2) = f(t, u(t), Δ2u(t − 1)), t ∈ 𝕋2, u(a + 1) = u(b − 1) = Δ2u(a) = Δ2u(b − 2) = 0 under a nonresonance condition involving two‐parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.