Fixed points of asymptotically regular nonexpansive mappings onnonconvex sets

Kaczor, Wiesława

doi:10.1155/s1085337503205054

Cited by 2 publications

(1 citation statement)

References 31 publications

(38 reference statements)

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“…equivalently [13,14], if 2T − Id is nonexpansive (Lipschitz continuous with constant 1), i.e., Firmly nonexpansive mappings play an important role in various contexts; see, e.g., [1,2,3,7,8,9,10,15,17,21,22,25]. The Kirszbraun-Valentine theorem (see, e.g., [5,13,16,20,26]) states that any nonexpansive mapping can be extended to a nonexpansive mapping defined on the whole space.…”

Section: ∀X ∈ D)(∀y ∈ D) T X − T Y 2 + (Id −T )X − (Id −T )Ymentioning

confidence: 99%

Fenchel duality, Fitzpatrick functions and the extension of firmly nonexpansive mappings

Bauschke

2006

Proc. Amer. Math. Soc.

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Abstract. Recently, S. Reich and S. Simons provided a novel proof of the Kirszbraun-Valentine extension theorem using Fenchel duality and Fitzpatrick functions. In the same spirit, we provide a new proof of an extension result for firmly nonexpansive mappings with an optimally localized range.Throughout this paper, we assume that X is a real Hilbert space, with inner product p = · | · and induced norm · , and we denote the identity mapping on X by Id. A mapping T from a subset D of X to X is called firmly nonexpansive ifequivalently [13,14], if 2T − Id is nonexpansive (Lipschitz continuous with constant 1), i.e.,Firmly nonexpansive mappings play an important role in various contexts; see, e.g., [1,2,3,7,8,9,10,15,17,21,22,25]. The Kirszbraun-Valentine theorem (see, e.g., [5,13,16,20,26]) states that any nonexpansive mapping can be extended to a nonexpansive mapping defined on the whole space. A beautiful proof of this result, based on Fenchel duality and Fitzpatrick functions, was recently provided by Reich and Simons [23]. (For further applications of Fitzpatrick functions, see, e.g., [4,24].) In this note, we refine their technique to obtain a new proof of an extension theorem for firmly nonexpansive mappings where the range of the extension is optimally localized. This extension theorem easily implies the Kirszbraun-Valentine result.Notation not explicitly defined in the following is standard in convex analysis; see, e.g., [27].

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