Existence of solutions for some discrete boundary value problems with a parameter

In this paper, the existence and multiplicity results of solutions are obtained for the discrete nonlinear two point boundary value problem (BVP) -Δ2u(k-1)=λf(k,u(k))k∈Z(1,T); u(0)=0=Δu(T)u(0)=0=Δu(T), where T   is a positive integer, Z(1,T)={1,2,…,T}Z(1,T)={1,2,…,T}, ΔΔ is the forward difference operator defined by Δu(k)=u(k+1)-u(k)Δu(k)=u(k+1)-u(k) and f:Z(1,T)×R→Rf:Z(1,T)×R→R is continuous, λ∈R+λ∈R+ is a parameter. By using the critical point theory and Morse theory, we obtain that the above (BVP) has solutions for λλ being in some different intervals.