Firmly nonexpansive mappings in classes of geodesic spaces

Effective results on compositions of nonexpansive mappings

“…More precisely, we introduce (r, δ)-convex spaces, a class of metric spaces which also includes Busemann spaces (and, hence, CAT(0) spaces), hyperconvex spaces, CAT(κ) spaces with κ > 0, as well as the so-called W -hyperbolic spaces (see [11]). Consequently, even when N = 1 and so (1) reduces in fact to the Halpern iteration, our results generalize ones obtained previously by the authors for CAT(κ) spaces with κ > 0 [17] and by the first author for normed [15] or W -hyperbolic spaces [16].…”

“…A related example of metric spaces with a convex geodesic bicombing are the so-called W -hyperbolic spaces, defined in [11] as metric spaces together with a convexity mapping W : X × X × [0, 1] → X satisfying suitable properties. As it was remarked in [1], Busemann spaces are exactly the uniquely geodesic W -hyperbolic spaces.…”

“…The general iteration defined by (1) was first studied in Hilbert spaces by Lions [18], who assumed different hypotheses on (λ n ), with the drawback that λ n = 1 n+1 does not satisfy them. Bauschke [2] proved that the iteration (x n ) given by (1) converges strongly to the common fixed point of the mappings T 1 , .…”

A new proof of the Lie–Trotter–Kato formula in Hadamard spaces

“…For the notation and terminology not explained here, the reader is referred to Section 2. Given λ > 0, define the resolvent of f as (1) J λ (x) := arg min y∈H f (y) + 1 2λ d(x, y) 2 , x ∈ H.…”

“…The limit in (2) is uniform with respect to t on bounded subintervals of [0, ∞) and (S t ) t≥0 is a strongly continuous semigroup of nonexpansive mappings on H; see [23,Theorem 1.3.13] and [31,Theorem 1.13]. Note that formula (2) was in a similar context used already in [38,Theorem 8.2].…”