Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings

Abstract

We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space . We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial's condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.