Fenchel Duality, Fitzpatrick Functions and the Extension of Firmly Nonexpansive Mappings

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‖ + ‖(Id−T )x− (Id−T )y‖ ≤ ‖x− y‖; 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‖