Successive Averages of Firmly Nonexpansive Mappings

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