Unification of some iterative and proximal like methods for asymptotically nonexpansive and quasi-nonexpansive mappings

In this paper, we introduce the concept of a strongly asymptotically quasi-nonexpansive sequence of mappings in the context of a Hilbert space. Firstly, we prove the weak convergence of a Picard type iterative method to a common fixed point for the sequence as well as the strong convergence when a Halpern type regularization scheme is considered. Among other features, our results are applied to get convergence to a fixed point of iterative procedure of Ishikawa and Halpern-Ishikawa for a lipschitzian asymptotically quasi-nonexpansive mappings (resp. as the Picard type iteration and the Halpern type iteration for the sequence of strongly asymptotically quasi-nonexpansive mappings) and, the convergence of proximal like methods for asymptotically nonexpansive mappings. Finally, we show an example of an asymptotically nonexpansive mappings and compute some of the methods studied in the paper.