ON THE RATE OF CONVERGENCE OF VISCOSITY IMPLICIT ITERATIVE ALGORITHMS

Abstract

An essential numerical method for solving ordinary differential and differential algebraic equations is the implicit midpoint rule. Comparing the rate of convergence of the implicit midpoint rules by using numerical examples is common in the literatures. Under suitable conditions imposed on the control parameters, it is shown in this paper that certain two implicit iterative sequences converge to the same fixed point of a nonexpansive mapping in uniformly smooth Banach spaces. Moreover, analytical comparison for the rate of convergence of the implicit iterative sequences to a fixed point of a nonexpansive mapping in uniformly smooth Banach spaces is presented. The implicit iterative sequence which converges faster is determined by an analytical method which is more general than the numerical methods.