Existence of positive solution for nonlinear fourth-order difference equations

Let TT be an integer with T≥5T≥5 and let T2={2,3,…,T}T2={2,3,…,T}. We show the existence and multiplicity of positive solutions of the boundary value problem of nonlinear fourth-order difference equation Δ4u(t−2)−λf(t,u(t))=0,t∈T2,u(1)=u(T+1)=Δ2u(0)=Δ2u(T)=0,u(1)=u(T+1)=Δ2u(0)=Δ2u(T)=0, where λλ is a parameter, f:T2×[0,∞)→[0,∞)f:T2×[0,∞)→[0,∞) is continuous.