Existence and uniqueness of positive solution for discrete multipoint boundary value problems (Q1727827)

Summary: It is expected in this paper to investigate the existence and uniqueness of positive solution for the following difference equation: \(- \Delta^2 u(t - 1) = f(t, u(t)) + g(t, u(t))\), \(t \in \mathbb{Z}_{1, T}\), subject to boundary conditions either \(u(0) - \beta \Delta u(0) = 0\), \(u(T + 1) = \alpha u(\eta)\) or \(\Delta u(0) = 0\), \(u(T + 1) = \alpha u(\eta)\), where \(0 < \alpha < 1\), \(\beta > 0\), and \(\eta \in \mathbb{Z}_{2, T - 1}\). 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.