  A monotone iterative method is proposed to solve nonlinear discrete boundary
value problems with the support of upper and lower solutions. We establish some
new existence results. Under some sufficient conditions, we establish maximum
principle for linear discrete boundary value problem, which relies on Green's
function and its constant sign. We then use it to establish existence of unique
solution for the nonlinear discrete boundary value problem $\Delta^2 y(t-1)=
f(t, y(t)),~t\in[1, T]$, $y(0)=0,~y(T+1)=0$.
