SUCCESSIVE AVERAGES OF FIRMLY NONEXPANSIVE-MAPPINGS

The problem considered here is to find common fixed points of (possibl y infinitely) many firmly nonexpansive selfmappings in a Hilbert space . For this purpose we use averaged relaxations of the original mapping s, the averages being Bochner integrals with respect to chosen measure s. Judicious choices of such measures serve to enhance the convergence towards common fixed points. Since projection operators onto closed c onvex sets are firmly nonexpansive, the methods explored are applicabl e for solving convex feasibility problems. In particular, by varying t he measures, our analysis encompasses recent developments of so-called block-iterative algorithms. We demonstrate convergence theorems which cover and extend many known results.