Iterative Methods for Some Generalizations of Nonexpansive Maps

Some generalizations of nonexpansive mappings which have been studied extensively include the (i) quasi-nonexpansive mappings; (ii) asymptotically nonexpansive mappings; (iii) asymptotically quasi-nonexpansive mappings.

For the past 30 years or so, iterative algorithms for approximating fixed points of operators belonging to subclasses of these classes of nonlinear mappings and defined in appropriate Banach spaces have been flourishing areas of research for many mathematicians. For the classes of mappings mentioned here in (i) to (iii), we show in this chapter that modifications of the Mann iteration algorithm and of the Halpern-type iteration process studied in chapter 6 can be used to approximate fixed points (when they exist).