Existence uniqueness for a class of Nonlinear Discrete Boundary Value Problems
Provides theoretical existence and uniqueness results for discrete boundary value problems, which is incremental for mathematicians working on discrete analogues of continuous problems.
The paper proposes a monotone iterative method with upper/lower solutions to prove existence and uniqueness of solutions for a class of nonlinear discrete boundary value problems, establishing a maximum principle via Green's function.
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 $Δ^2 y(t-1)= f(t, y(t)),~t\in[1, T]$, $y(0)=0,~y(T+1)=0$.