On the convergence of the multi-step Noor fixed point iterative scheme with errors in the class of Zamfirescu operators

Abstract

The purpose of this paper is to establish a general theorem to approximate fixed points of Zamfirescu operators on an arbitrary normed space through the multi-step Noor fixed point iterative scheme with errors in the sense of Plubtieng and Wangkeeree [S. Plubtieng and R. Wangkeeree, Strong convergence theorem for multi-step Noor iterations with errors in Banach spaces, J. Math. Anal. Appl. 321 (2006), 10-23, 2006]. Our result generalizes and improves the corresponding results of Rafiq [A. Rafiq, A Convergence Theorem for Mann Fixed Point Iteration Procedure, Appl. Math. E-Notes, 6, 289-293], Xu [Y. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998), 91-101], Liu [L. S. Liu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), 114-125], Osilike [ M. O. Osilike, Ishikawa and Mann iteration methods with errors for nonlinear equations of the accretive type, J. Math. Anal. Appl. 213 (1997), 91-105] and several authors in literature.