Convergence Analysis of a Picard–CR Iteration Process for Nonexpansive Mappings

This paper proposes a novel hybrid iteration process, namely the Picard–CR iteration process. We apply the proposed iteration process for the numerical reckoning of fixed points of generalized α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-nonexpansive mappings. We establish weak and strong convergence results of generalized α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-nonexpansive mappings. This study demonstrates the superiority of the hybrid approach in terms of convergence speed. Moreover, we numerically compare the proposed iteration process with other well-known ones from the literature. In the comparison, we consider two problems: finding a fixed point of a generalized α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-nonexpansive mapping and finding roots of a complex polynomial. In the second problem, we use the so-called polynomiography in the analysis. The results showed that the proposed iteration scheme is better than other three-parameter iteration schemes from the literature. Using the proven fixed-point results, we also obtain solutions to fractional differential equations.