Existence and Uniqueness of Positive Solution for Discrete Multipoint Boundary Value Problems

It is expected in this paper to investigate the existence and uniqueness of positive solution for the following difference equation: -Δ(2) u(t - 1) = f(t, u(t)) + g(t, u(t)), t ∈ ℤ 1,  T , subject to boundary conditions either u(0) - βΔu(0) = 0, u(T + 1) = αu(η) or Δu(0) = 0, u(T + 1) = αu(η), where 0 < α < 1, β > 0,  and η ∈ ℤ 2,T-1. The proof of the main result is based upon a fixed point theorem of a sum operator. It is expected in this paper not only to establish existence and uniqueness of positive solution, but also to show a way to construct a series to approximate it by iteration.