On the minimal displacement vector of compositions and convex combinations of nonexpansive mappings

DOI10.1007/S10208-020-09449-WarXiv1809.01196MaRDI QIDQ6306248

Publication date: 4 September 2018

Abstract: Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. Five years ago, it was shown that if finitely many firmly nonexpansive mappings have or "almost have" fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex combinations of firmly nonexpansive mappings was obtained in terms of the displacement vectors of the underlying operators. Using a new proof technique based on the Brezis-Haraux theorem and reflected resolvents, we extend these results from firmly nonexpansive to general averaged nonexpansive mappings. Various examples illustrate the tightness of our results.

Mathematics Subject Classification ID

Convex programming (90C25) Monotone operators and generalizations (47H05) Fixed-point theorems (47H10) Contraction-type mappings, nonexpansive mappings, (A)-proper mappings, etc. (47H09)

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