Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 2012; no. 1; pp. 1 - 11
• Main Authors: Bauschke, Heinz H, Martín-Márquez, Victoria, Moffat, Sarah M, Wang, Xianfu
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 26.03.2012; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Approximation; Asymptotic properties; Classification; Differential Geometry; Equivalence; Mapping; Mathematical analysis; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Operators; Optimization; Topology; firmly nonexpansive mapping; strongly nonexpansive mapping; asymptotic regularity; nonexpansive mapping; Hilbert space; resolvent; maximally monotone operator
• ISSN: 1687-1812; 1687-1820; 1687-1812; 2730-5422
• DOI: 10.1186/1687-1812-2012-53

Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or "almost have" fixed points, then the same is true for their composition. This significantly generalizes the result by Bauschke from 2003 for the case of projectors (nearest point mappings). The proof resides in a Hilbert product space and it relies upon the Brezis-Haraux range approximation result. By working in a suitably scaled Hilbert product space, we also establish the asymptotic regularity of convex combinations. 2010 Mathematics Subject Classification: Primary 47H05, 47H09; Secondary 47H10, 90C25.