Convergence to common fixed points of multi-step iteration process for generalized asymptotically quasi-nonexpansive mappings in convex metric spaces

Abstract

In this paper, we study strong convergence of multi-step iterations with errors for a finite family of generalized asymptotically quasinonexpansive mappings in the framework of convex metric spaces. The new iteration scheme includes modified Mann and Ishikawa iterations with errors, the three-step iteration scheme of Xu and Noor as special cases in Banach spaces. Our results extend and generalize many known results from the current literature.

Multi-Step Iteration Algorithm of Total Asymptotically Quasi-Nonexpansive Maps

“…From (5) and (6) for any , ∀ = 1, 2, … , , we obtain Next, to prove ( ) converges weakly to * . Assume there is other subsequence ( ) ( ) is weak convergence to * ∈ ( , ) and * ≠ * .…”

“…As well as, approximating methods for finding their fixed points are studied. In 2014, G. S. Saluja [5] established the strong and weak convergence for approximating common fixed point for generalized asymptotically quasi-nonexpansive maps in a Banach space.…”

Approximating Fixed Points of The General Asymptotic Set Valued Mappings

“…As well as approximating methods for finding their fixed points are studied. Saluja [11] strong and weak convergence for approximating common fixed point for generalized asymptotically quasinonexpansive mappings in a Banach space are established. One of the first results of iterative procedures was obtained by Browder [12].…”

Strong convergence theorems for two finite families of generalized asymptotically quasi-nonexpansive mappings with applications

“…Later on, many authors discussed the existence of the fixed point and the convergence of the iterative process for various mappings in convex metric spaces (see, for example, [1,3,4,8,11,12,16,18,20,23]). …”