A weak convergence theorem for mean nonexpansive mappings

TL;DR

This paper establishes weak convergence of iterates for mean nonexpansive maps on certain convex sets, extending classical fixed point theorems to broader spaces under Opial's property.

Contribution

It generalizes classical fixed point theorems for nonexpansive maps to mean nonexpansive maps in weakly compact and uniformly convex Opial spaces.

Findings

Weak convergence of iterates in weakly compact convex sets

Extension of Browder-Petryshyn and Opial theorems

Results apply to unbounded convex subsets in uniformly convex Opial spaces

Abstract

In this paper, we prove first that the iterates of a mean nonexpansive map defined on a weakly compact, convex set converge weakly to a fixed point in the presence of Opial's property and asymptotic regularity at a point. Next, we prove the analogous result for closed, convex (not necessarily bounded) subsets of uniformly convex Opial spaces. These results generalize the classical theorems for nonexpansive maps of Browder and Petryshyn in Hilbert space and Opial in reflexive spaces satisfying Opial's condition.