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

Abstract

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.

On a result of Pazy concerning the asymptotic behaviour of nonexpansive mappings

“…By [1] (see also [4] for extensions to firmly nonexpansive operators), we always have 0 ∈ ran (Id −T), and this implies the result.…”

Cardiac C‐arm computed tomography using a 3D + time ROI reconstruction method with spatial and temporal regularization

“…Using the same notations as in Sec. 2.A, a few definitions from the theory of nonexpansive mappings are reminded: r T is strongly nonexpansive if T is nonexpansive and whenever (x n ) n ∈ N and (y n ) n ∈ N are sequences in R MN such that (x n − y n ) n ∈ N is bounded and x n − y n 2 − Tx n − Ty n 2 → 0, it follows that (x n − y n ) − (Tx n − Ty n ) → 0. r Positivity enforcement is a projector onto a nonempty closed convex set, therefore it is firmly nonexpansive, and therefore strongly nonexpansive (see fact 4.2 of Bauschke et al 35 ).…”

Prior image based temporally constrained reconstruction algorithm for magnetic resonance guided high intensity focused ultrasound

“…Then the following (x n − y n ) − (K x n − K y n ) → 0 also holds. 28,29 On the other hand, when K is firmly nonexpansive if ∀x, y ∈ R MNLT , ∥K x − K y ∥ 2 2 ≤ ⟨(K x − K y),(x − y)⟩, where the definition of ⟨(K x − K y),(x − y)⟩ =  MNLT i=1 ((K x) i − (K y) i )(x i − y i ). [28][29][30] The operators used for the PITCR algorithms are nonexpansive.…”

“…28,29 On the other hand, when K is firmly nonexpansive if ∀x, y ∈ R MNLT , ∥K x − K y ∥ 2 2 ≤ ⟨(K x − K y),(x − y)⟩, where the definition of ⟨(K x − K y),(x − y)⟩ =  MNLT i=1 ((K x) i − (K y) i )(x i − y i ). [28][29][30] The operators used for the PITCR algorithms are nonexpansive. The gradient descent with enough iterations is strongly nonexpansive 27,31 and the temporal derivatives are proximal mappings and hence are firmly nonexpansive.…”

“…32 All the operators used here are strongly nonexpansive and hence if K has at least one fixed point, then PITCR algorithm converges to that fixed point. 29,30…”