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

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