Abstract
Let K be a nonempty subset of a Banach space E. A mapping T : K -> K is said to satisfy (RCSC) condition if each a, b is an element of K, (1/2)parallel to a - Fa parallel to <= parallel to a - b parallel to double right arrow parallel to Fa - Fb parallel to <= (1/3) (parallel to a - b parallel to + parallel to a - Fb parallel to + parallel to b - Fa parallel to). In this paper, we study, under some appropriate conditions, weak and strong convergence for this class of maps through M iterates in uniformly convex Banach space. We also present a new example of mappings with condition (RCSC). We connect M iteration and other well-known processes with this example to show the numerical efficiency of our results. The presented results improve and extend the corresponding results of the literature.