ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A FIRST AND SECOND-ORDER DISCRETE BOUNDARY VALUE PROBLEM

To purpose of the first part of this paper is to study the nonlinear first order discrete boundary value problem subjected to two-point boundary condition. We establish a new sufficient condition for the existence and uniqueness of the nonlinear first order difference evolution equation. The existence and uniqueness of solutions are proven by means of Banach contraction principle. We define an iteration procedure and apply our results to show that iterated solutions converge to the solutions of the discrete problem. We prove the iteration procedure is terminated when the residual is acceptably small. To illustrate the existence of a unique solution of the contraction mapping theorem some examples are included. In the second part of this study, we consider the following second-order difference boundary value problem Delta(2)x(i)/tau(2) = f (t(i), x(i)), i = 0, 1, 2, ..., n - 1, ux(0) + vx(n) = 0, (u) over barx(0)/tau + (v) over bar Delta x(n)/tau = 0 where f : [0, T] x R -> R and the difference operator is defined as follows Delta(2)x(i) = {x(i+1) - 2x(i) + x(i-1), for i = 1, 2, ..., n - 2, 0, for i = 0 or i = n - 1, By using Banach fixed point theorem we establish some sufficient conditions for the existence and uniqueness of the above second-order difference boundary value problem. To illustrate the existence of a unique solution of the contraction mapping theorem an example is included.