Ces\`aro means of firmly nonexpansive iterates need not converge strongly

Abstract

Firmly nonexpansive operators arise naturally as resolvents of monotone operators and as generalizations of projections and proximal mappings in convex optimization and fixed point theory. While their iterates are known to converge weakly to a fixed point, strong convergence is not guaranteed (Genel and Lindenstrauss, 1975). Strong convergence of Ces\`aro means of iterates is also known to fail for general nonlinear nonexpansive mappings (Krengel and Lin, 1987). In this paper, we show that this failure persists in the much smaller class of firmly nonexpansive mappings. Using suitable meshes, we construct a new explicit family of counterexamples in infinite-dimensional Hilbert spaces with the origin as the unique fixed point. In the harmonic case, the Ces\`aro means of the iterates remain bounded away from the origin. Another variant yields Ces\`aro means that converge strongly to the origin. A third variant presents Ces\`aro means whose norms oscillate in the sense that their liminf is zero while their limsup is positive. Thus the strong convergence conclusion in von Neumann's linear mean ergodic theorem does not extend to Baillon's nonlinear mean ergodic theorem, even for firmly nonexpansive mappings.