Near equality, near convexity, sums of maximally monotone operators, and averages of firmly nonexpansive mappings
We study nearly equal and nearly convex sets, ranges of maximally monotone operators, and ranges and fixed points of convex combinations of firmly nonexpansive mappings. The main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings. A striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular. Throughout, examples are provided to illustrate the theory. We also obtain detailed information on the domain and range of the resolvent average.
monotone operator firmly nonexpansive mapping nearly convex set asymptotic regularity resolvent 47h05 47h09 47h10 52a20 90c25