Effective asymptotic regularity for one-parameter nonexpansive semigroups

Abstract

We give explicit bounds on the computation of approximate common fixed points of one-parameter strongly continuous semigroups of nonexpansive mappings on a subset C of a general Banach space. Moreover, we provide the first explicit and highly uniform rate of convergence for an iterative procedure to compute such points for convex C. Our results are obtained by a logical analysis of the proof (proof mining) of a theorem by T. Suzuki.

New effective bounds for the approximate common fixed points and asymptotic regularity of nonexpansive semigroups

“…In this paper we obtain effective, quantitative results on the approximate common fixed points of a one-parameter nonexpansive semigroup {T(t) : t ≥ 0} on a subset C of a Banach space X by logical analysis of the proof of a result by Suzuki in [20]. It is very interesting that the bound obtained here (Section 3) is completely different to the bound obtained in another recent work [13] by Kohlenbach and the author. The latter had been derived by proof mining on a proof of a statement again by Suzuki in [21] concerning, again, the common fixed points of {T(t) : t ≥ 0} that is a generalization of the corresponding statement in [20].…”

“…The latter had been derived by proof mining on a proof of a statement again by Suzuki in [21] concerning, again, the common fixed points of {T(t) : t ≥ 0} that is a generalization of the corresponding statement in [20]. We briefly discuss a comparison between the bound in [13] and the one obtained in this work. In Section 3 we will also present an adaptation of a general logical metatheorem by Gerhardy and Kohlenbach for the specific mathematical framework and the assumptions by Suzuki, in order to illustrate how the metatheorem guarantees the extractability of the bound in the situation at hand.…”

“…In the literature one may find several examples where uniform equicontinuity is fulfilled, for instance see [13]. Moreover, any nonexpansive semigroup generated from a bounded accretive operator via the Crandall-Liggett formula fulfills the property of uniform equicontinuity, as can been seen in Crandall and Liggett [2] (see in particular (1.11) there).…”

“…Definition 2.3 (Kohlenbach and Koutsoukou-Argyraki [13]) We say that a nonexpansive semigroup {T(t) : t ≥ 0} on a subset C of a Banach space X is uniformly equicontinuous if the mapping t → T(t)z is uniformly continuous on each compact interval [0, K] for all K ∈ N and given a b ∈ N it has a common modulus of continuity for all z ∈ C b . Namely if there exists a function ω :…”

Proof-Theoretic Methods in Nonlinear Analysis

“…for all M, b ∈ N, p ∈ C, ε > 0p ≤ b ∧ S(p) − p ≤ Φ(ε, M, b) → ∀t ∈ [0, M ] ( T (t)p − p ≤ ε).The main noneffective tool used in Suzuki's proof is the binary König's lemma WKL and by Remark 2.4 it is guaranteed to have a primitive recursive (in the sense of Kleene) bound Φ. In fact, the bound actually extracted in[42] is of rather low complexity:Φ(2 −m , M, b) D2 ω D,b (3+[log 2 (1+M N )]+m)+1 , φ(k, f ) := max{2f (i) + 6 : 0 < i ≤ k}. Example: α = √ 2, β = 2, λ = 1/2.…”