A note on a faster fixed point iterative method

Abstract

Very recently, Piri et al. (Numer Algorithms 81:1129–1148, 2019) introduced an iterative process to approximate a fixed point of generalized \(\alpha\)-nonexpansive mappings and discussed its convergence analysis. In this paper, we introduce an iteration process to approximate a fixed point of a contractive self-mapping that is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, and Thakur processes for contractive mappings including the recent one by Piri et al. We also obtain convergence and stability theorems of this iterative process for a contractive self-mapping. Numerical examples show that our iteration process for approximating a fixed point of a contractive self-mapping is faster than the method proposed by Piri et al. Based on this process, we finally present a new modified Newton–Raphson method for finding the roots of a function and generate some nice polynomiographs.