A Characterization of Local Convergence for Fixed Point Iterations in $R^1 $

Abstract

This paper develops necessary and sufficient conditions for the iteration $x_{k + 1} = f(x_k )$ to converge locally to a fixed point $x^ * $ of f, where f is a real-valued continuous function on $R^1 $ satisfying the condition that $f(x) = x^ * $ has the unique solution $x = x^ * $ in some interval about $x^ * $. Applications are made to monotone functions and functions with at least one-sided derivatives at $x^ * $. These appear to be the first such results for nonlinear iterations in the literature.