The Necessary and Sufficient Condition for the Convergence of a New Fixed Point Approximation Method for Continuous Functions on an Arbitrary Interval

Wantana Suwana-adth

Abstract

In this paper, we consider a new two step iterations for approximating a fixed point of continuous functions on an arbitrary interval. Then, necessary and sufficient condition for the convergence of the proposed iterative process of continuous functions on an arbitrary interval is given. Our results extend and generalized the results of Borwein and Borwein [D. Borwein and J. Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl., 157 (1991), 112-126]. Finally, we show the numerical examples for the proposed iterative process to compare with Mann, Ishikawa iterations.