Abstract

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 if (1) (∀x ∈ D)(∀y ∈ D) ‖Tx − Ty ‖ 2 + ‖(Id −T)x − (Id −T)y ‖ 2 ≤‖x − y ‖ 2; equivalently [13, 14], if 2T −Id is nonexpansive (Lipschitz continuous with constant 1), i.e., (2) (∀x ∈ D)(∀y ∈ D) ‖(2T − Id)x − (2T − Id)y ‖ ≤‖x − y‖ or if

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