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).
In this paper, we provide a new explicit family of counterexamples in infinite-dimensional Hilbert spaces. In the harmonic case the Cesàro means of the iterates remain bounded away from the unique fixed point. A block-construction variant yields Cesàro means whose norms oscillate in the sense that their liminf is zero while their limsup is positive. These results show that von Neumann's classical mean ergodic theorem for linear operators does not extend to Baillon's nonlinear mean ergodic theorem even in the firmly nonexpansive setting, and they illustrate inherent limitations of averaging techniques in infinite-dimensional optimization.
| Subjects: | Optimization and Control (math.OC); Functional Analysis (math.FA) |
| Cite as: | arXiv:2605.25491 [math.OC] |
| (or arXiv:2605.25491v1 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25491 arXiv-issued DOI via DataCite (pending registration) |
From: Tran Thanh Tung [view email]
[v1]
Mon, 25 May 2026 06:51:26 UTC (20 KB)
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