New Effective Bounds for the Approximate Common Fixed Points and Asymptotic Regularity of Nonexpansive Semigroups

We give an explicit, computable and uniform bound for the computation of approximate common fixed points of one-parameter nonexpansive semigroups on a subset C of a Banach space, by proof mining on a proof by Suzuki. The bound obtained here is different to the bound obtained in a very recent work by Kohlenbach and the author which had been derived by proof mining on the -completely differentproof of a generalized version of the particular theorem by Suzuki. We give an adaptation of a logical metatheorem by Gerhardy and Kohlenbach for the given mathematical context, illustrating how the extractability of a computable bound is guaranteed. For uniformly convex C , as a corollary to our result we moreover give a computable rate of asymptotic regularity with respect to Kuhfittig’s classical iteration schema, by applying a theorem by Khan and Kohlenbach. 2010 Mathematics Subject Classification 47H20 (primary); 03F03 (secondary)


Introduction
Proof mining is a research program in applied proof theory originally initiated by Georg Kreisel in the 1950's under the name unwinding of proofs (see [16] or [17]), after he suggested a shift of focus for the application of proof interpretations: from producing relative consistency proofs for foundational purposes to a tool for extracting constructive information from actual mathematical proofs. The program involves the extraction of new quantitative constructive information by logical analysis even of proofs that appear to be nonconstructive. This information is "hidden" behind an implicit use of quantifiers in the proof, and its extraction is guaranteed by certain logical metatheorems (based on variations of Gödel's functional Dialectica interpretation [3]), see for example Kohlenbach [9], provided that the statement proved is of a certain logical form and the assumptions and general mathematical setup fit a specific formal framework. Starting in largest integer not exceeding x. Moreover, by x ∈ Z we denote the ceiling function, ie the smallest integer exceeding x or equal to x. Definition 2.1 Given a Banach space X and C ⊆ X , a mapping T on C is nonexpansive if ∀x, y ∈ C Tx − Ty ≤ x − y .
Definition 2.2 A family {T(t) : t ≥ 0} of self-mappings T(t) : C → C for a subset C of a Banach space X is called a one-parameter strongly continuous semigroup of nonexpansive mappings (or nonexpansive semigroup for short) if the following conditions hold: (1) for all t ≥ 0, T(t) is a nonexpansive mapping on C, for each x ∈ C, the mapping t → T(t)x from [0, ∞) into C is continuous.
In the following we will need to make use of the concepts of uniform equicontinuity for a nonexpansive semigroup and modulus of uniform equicontinuity introduced in [13]: 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 : N × N × N → N so that ∀b ∈ N ∀z ∈ C b ∀m ∈ N ∀K ∈ N ∀t, t ∈ [0, K] where C b := {z ∈ C : z ≤ b}. We call ω a modulus of uniform equicontinuity for the nonexpansive semigroup {T(t) : t ≥ 0}.
For our bound extractions we will assume uniform equicontinuity as defined above for the nonexpansive semigroup {T(t) : t ≥ 0}. The motivation from introducing equicontinuity, ie the property of having a common modulus of continuity for all z that are norm-bounded by a specific b ∈ N, and assuming this requirement for our semigroup, comes from the need to fit the framework of the logical metatheorems that will guarantee the extractability of the bounds as in order to achieve the majorizability of the semigroup equicontinuity is required. This slight strengthening is harmless as in praxis one may a posteriori remove equicontinuity but then the bound would be less uniform as it would depend on each point which would not be desirable. Moreover it would then not be possible to obtain the results on asymptotic regularity.
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).
Because, as we will later see, in Suzuki [20] an irrationality assumption is made, we will need a quantitative version of this assumption in our quantitative analysis of his proof. For that we will make use of the following: The Skolem normal form of the above is and f γ is the corresponding Skolem function.

Definition 2.4
The function f γ as in (i) above is called an effective irrationality measure for γ .

Proof Mining in Praxis, a Metatheorem Adaptation and Results
Our main result is a quantitative version of the following result by Suzuki: [20,Proposition 2]) Let X be a Banach space and let {T(t) : t ≥ 0} be a one-parameter nonexpansive semigroup on a subset C of X . Let α, β be positive real numbers so that α/β ∈ R \ Q. Then we have t≥0 F(T(t)) = F(T(α)) ∩ F(T(β)).

The inclusion
is proved. Our main result which we will show here, constitutes in particular a quantitative version of the latter statement. As this can be written as we have a ∀∃(∀ → ∃) ie ∀∃ statement, so it is possible to extract a computable bound on m ∈ N (see Kohlenbach [10]). This will be done by performing proof mining on Suzuki's proof of the above.
Before the main result we will firstly obtain quantitative versions of Proposition 1 and Lemma 3 in [20] by proof mining on Suzuki's proofs.
A formalized version of the above statement is: By prenexing the above we have The goal is to extract computable bounds on δ and n from the proof of the statement. This bound extraction would then amount to obtaining a quantitative version of the statement.
We can show that we may write a version of a general logical metatheorem by Gerhardy and Kohlenbach (see [10,Theorem 17.52 and Corollary 17.71]) adapted in particular explicitly for the mathematical setting and assumptions of Suzuki's proposition above.
Note that by A ω [X, · , C] −b we denote the theory involving the system A ω , defined as A ω :≡ WE-PA ω +QF-AC+WKL (denoting the weakly extensional Peano arithmetic in all finite types, the quantifier-free axiom of choice and weak König's lemma respectively), where the type system is extended over two ground types N, X for a normed space X (see [10]). Our adaptation reads: Then one can extract from the proof primitive recursive in the sense of Gödel functionals W,W so that Journal of Logic & Analysis 10:7 (2018) Effective Bounds for Approximate Common Fixed Points holds (in the sense of Kohlenbach [10,Definition 17.68]) for any nontrivial normed space X with a nonempty C ⊆ X .
The bound depends only on general bounding information (majorants, see [10,Definition 17.50]) on the input data. It is easy to show that a one-parameter nonexpansive semigroup is majorizable (see Koutsoukou-Argyraki [14]). Here the displacement assumption for some arbitrary element of the sequence is trivially covered because of the premise T(α n )z − z ≤ R 2 −k . Also note that the input number theoretic functions Φ, Ψ, ω are replaced with the nondecreasing functions Φ , Ψ , ω so that the latter are their own majorants and we have ensured that all the assumptions introduced are formulated as universal statements. Notice that the bound will not depend on representatives of the sequence of reals {α n } ⊆ R + nor its limit α ∞ ∈ R + but on L ∈ N as we can instead write ∀L ∈ N ∀α n ∈ [0, L] N ∀α ∞ ∈ [0, L] and they can therefore be seen as elements not of the Polish space R + but of the compact spaces [0, L] N and [0, L] respectively. Therefore the bound will depend only on such a parameter L ∈ N. This approach has similarly been followed for t ∈ R + , as instead of ∀t ∈ R + we write ∀M ∈ N ∀t ∈ [0, M] so that t will be considered an element of the compact space [0, M] and the bound will only depend on the parameter M ∈ N. The above metatheorem is a direct adaptation of the aforementioned metatheorem by Gerhardy and Kohlenbach, as we have written the assumptions of the above given in Suzuki [20,Proposition 1] as universal statements. In particular, the assertion of Suzuki's proposition is written as a ∀∃(∀ → ∃) ie a ∀∃ statement (considering the representation of real numbers as in [10] according to which statements involving ≤, ≥, = are seen as universal and statements involving <, > are seen as existential). Indeed, as the above metatheorem predicts we do succeed to extract a computable and highly uniform bound by performing proof mining on Suzuki's proof. In particular, we obtain the following result (note that Ψ, Φ, ω below are as in the above metatheorem-the bound will depend on their majorants Ψ , Φ , ω so for notation simplicity one may a priori choose Ψ, Φ, ω to be nondecreasing thus identifiable with their majorants Ψ , Φ , ω ): .
Proof For ( ), which is of the logical form ∀∃, the above stated metatheorem guarantees the extraction of computable bounds on δ and n. We will extract such bounds by proof mining on Proposition 1 and therefore obtain ( * ).
As is done in the proof of [20, Proposition 1], we define By this definition, clearly the rate of convergence of {β n } to 0 is the same as the rate of convergence Φ of {α n } to α ∞ and, moreover, by the assumption we have Now, in both cases α n = α ∞ + β n and α ∞ = α n + β n we claim that we have, for all n ∈ N, The above claim follows directly by the semigroup properties and the triangle inequality, ie if α n = α ∞ + β n we have and analogously if α ∞ = α n + β n we have

Now the triangle inequality gives
We moreover have and by the triangle inequality for any arbitrary m ∈ N, so overall we have calculated that By the construction of [20,Lemma 2], it is So, as for all n ∈ N, δ n − δ n+1 = k n β n ≥ 0, {δ n } is nonincreasing. Therefore, for all n ∈ N, Therefore we may write the above calculated estimate as: Now consider, together with the uniform equicontinuity assumption for the semigroup (as m ∈ N was arbitrary) the convergence assumption Now, the convergence statement (as already mentioned by [20, Lemma 2] here we have again the same rate of convergence Φ) combined with the uniform equicontinuity assumption for the semigroup (notice that Substituting in ( * * ), for a given j ∈ N which satisfies In total, as M ∈ N was arbitrary: Now let the above arbitrary j ∈ N be such that for a yet to be determined k ∈ N, . Choosing we thus have ie we have extracted the bounds:

13
• α 2 = min{α, β} • k n = [α n /α n+1 ] for all n ∈ N • α n+2 = α n − k n α n+1 for all n ∈ N Then the following hold: We show the following: Then: where Ψ(n) is defined simultaneously with f αn α n+1 : N × N → N as follows: Proof The proof of (A) is carried out by induction, it is in fact the proof of Lemma 3.5 that is given in [20] and we present it here as for the next step we will write down a quantitative version of it. By definition α 1 /α 2 = β/α ∈ R \ Q and α 1 = β > α 2 = α > 0 so A(n = 1) holds. Consider the induction hypothesis: By the definition of the sequence we have and by the definition of the floor function [·] for any it is α j+2 α j+1 ∈ R \ Q and thus α j+1 α j+2 ∈ R \ Q. So we have shown (A(j + 1)) and thus by induction we have shown (A(n)), ie 0 < α n+1 < α n and α n+1 α n+2 ∈ R \ Q for all n ∈ N. Showing (B) amounts to writing down a quantitative version of (A). Since (A) was shown by induction and the statements on the irrationality of αn α n+1 for all n ∈ N and the fact that for all n ∈ N α n > 0 were shown simultaneously, f αn α n+1 is defined recursively and simultaneously with Ψ(n) (the latter is in fact the quantitative information that is of interest here) thus the proof will be carried out again by induction. For n = 1 by the definition of {α n } we have ie the effective irrationality measure of β α with the domain restricted to exclude zero (compare with Definition 2.4); moreover, α 1 = β > 2 −Ψ(1) (respectively α 2 = α > 2 −Ψ(2) ) are clearly fulfilled when Ψ(1) := G, Ψ(2) := G as by assumption β > α > 2 −G . Consider the following induction hypothesis. Let us assume that, for some j ∈ N, we have and α j+1 > 2 −Ψ( j+1) .
Notice that for all p, q ∈ N, .
We will now show that α j+2 > 2 −Ψ( j+2) . To this end, in (B(j)) we make the choice (recall that [α j /α j+1 ] ≥ 1) and thus obtain (as by the definition of the floor function [·] for any x ∈ R we have [x] ≤ x and moreover α j α j+1 ≤ β 2 Ψ( j+1) ): (l, 1)} By recalling that by the definition of {α n } we have and by (B(j)) we obtain Therefore, as by having set We have shown by the above that so the proof of (B(j + 1)) is complete and by induction ∀n ∈ N α n > 2 −Ψ(n) .
We will now show (C). By (A) {α n } is convergent, and thus Cauchy. We will show that the limit of {α n } is zero, ie that ∀m ∈ N ∃n ∀i, j ≥ n |α i − α j | < 2 −m → ∀k ∈ N ∃l ∈ N ∀n ≥ l α n < 2 −k and we will moreover find a computable bound on l ∈ N. Because by (A) {α n } is decreasing, it is enough to show that Note that, in order to derive the quantitative (and effective) information of interest, ie the bound Φ, we will apply proof mining to the entire statement (!), and not to just its conclusion. This is because the premise will be weakened to a metastable Cauchy statement in order to apply Kohlenbach [10, Proposition 2.27].
We claim that the negation of (!) will give a contradiction, that is, we claim that will give a contradiction. To show this claim, in (I) above we make the choices 2 m := k where k ∈ N is as in (II), i := n and j := n + 1, ie: The assumption (II) for such a k, together with (III) in which k is as in (II) give Dividing the above by α n+1 > 0 we have Now, by the definition of the sequence {α n }, substituting the above we obtain and clearly α n+2 < 2 −k gives a contradiction to the assumption (II) for l := n + 2.
Thus the bound Φ on l corresponds to a bound on n shifted by 2. The latter is obtained by applying [10, Proposition 2.27] (also see Remark 2.29 there). Since We now show our main result. Then where Ψ(1) := G, Ψ(2) := G and for n > 2 where {α n } is a sequence defined by α 1 := β , α 2 := α, α n+2 := α n − αn α n+1 α n+1 , Proof Define a sequence {α n } ∈ (0, ∞) as in Lemma 3.6. For convenience set We have: We have therefore shown that (because by Lemma 3.6 0 < α n+1 < α n for all n ∈ N, thus k n = [α n /α n+1 ] ≥ 1 for all n ∈ N). Now let b ∈ N and z ∈ C b such that for some δ > 0. Let us consider the sequence for some δ > 0. We will estimate the nth term as follows. Observing the form of the first terms: . . . Now note that because, as mentioned above, for all n ∈ N, k n ≥ 1, for n ≤ m we have Moreover, note that the number of summands in the respective bound of each term of the above sequence (where each summand is a product of k i s) clearly follows the Fibonacci sequence, ie: number of summands of products of k i s as each term approximation involves the sum of the two previous term approximations.
In particular, the nth term of the sequence { T(α n )z − z } n∈N has a factor which is the n − 1th Fibonacci number. The nth Fibonacci number is given by the well-known Binet's formula: Hence, for the nth term of the sequence { T(α n )z − z } n∈N we have: Moreover, note that, for each i ∈ N: Therefore we may write: Thus, for n :=W withW as it was previously obtained in our Theorem 3.4 with L := β , as the above obtained bound on T(α n )z − z is nondecreasing on n we have that By choosing δ > 0 to be such that where W is the bound extracted in Theorem 3.4, ie by choosing the premise of Theorem 3.4 is now fulfilled, and therefore by Theorem 3.4 we obtain that where Φ, Ψ for the particular sequence {α n } are as extracted in Lemma 3.6.
Remark 1 Corollary to the proof. It would be possible to remove the equicontinuity assumption for the semigroup {T(t) : t ≥ 0}. Then the modulus of continuity of {T(t) : t ≥ 0}, and thus also the final bound, would depend on z ∈ C instead of the input b ∈ N, so that C b := {z ∈ C : z ≤ b }. That would, strictly speaking, constitute a direct quantitative version of Suzuki's result Theorem 3.1. However, omitting our supplementary equicontinuity assumption would have the following disadvantages: • Clearly the result would be less uniform.
• It would not be possible to derive the corollary on asymptotic regularity that we will derive in the end of this section.
In [13] a quantitative version of [21, Theorem 1] was given: semigroup on C ⊆ X for some Banach space X . Let α, β ∈ R + with 0 < α < β . Let γ := α/β ∈ R + \ Q + with an effective irrationality measure f γ . Let be a mapping of C into X with λ ∈ (0, 1). Moreover assume that {T(t) : t ≥ 0} is uniformly equicontinuous with a modulus of uniform equicontinuity ω . Let Λ ∈ N be such that Let us assume that we have z ∈ C b so that, for some δ > 0, In the above theorem let us make the choice λ := 1 2 ∈ (0, 1) and Λ := 2. Therefore the bound could be stated as: where k ∈ N and φ(k, f ) ∈ N are as before.
Comparing the bound Ψ that would follow from the above result shown in [13] to the bound X obtained in Theorem 3.7 here we make the interesting observation that proof mining on Suzuki's two completely different proofs of essentially the same statement gave us a completely different result. We cannot a priori determine in general which bound gives a result that is numerically better, this may differ given different examples of semigroups and/or different choice of input data. Note that both proofs by Suzuki (in [21] and [20] respectively), although completely different to each other, used an irrationality assumption on the ratio of α and β , thus both the quantitative analyses presented for the first and second approach made use of the notion of effective irrationality measure for an irrational number. (In [13] the effective irrationality measure was taken to depend only on one variable for reasons of simplicity; see the discussion in [13]).

Asymptotic Regularity
Finally, under the assumption that the Banach space X is moreover uniformly convex, we will now give a corollary to Theorem 3.7 using a result by Khan   Let U 0 := I where I denotes the identity mapping. Let λ ∈ (0, 1). Consider the mappings: We recall that: Definition 4.2 (Clarkson [1], also see Kohlenbach [8]) A Banach space X is called uniformly convex if A mapping η : (0, 2] → (0, 1] giving such a δ := η( ) is called a modulus of uniform convexity.
For example, as a modulus of uniform convexity one may consider Clarkson's modulus of convexity (see [1]) defined for any Banach space X as the function η X : (0, 2] → (0, 1] given by Moreover, we recall: [15], also see Kohlenbach [10]) Let C be a convex subset of a Banach space X and let T : C → C nonexpansive. The sequence for all x 0 ∈ C, where {x n } is a given iteration starting at x 0 , then T , or more precisely is called asymptotically regular. A rate of convergence for the above convergence is called a rate of asymptotic regularity for T .
In the following we will refer to convergence results of the above form also for different iterations {x n } as asymptotic regularity results.
By proof mining on the proof of a theorem by Kuhfittig (implicit) in [18] (also see Theorem 1.2 in [5]), Khan and Kohlenbach showed in [5] the following theorem which is a quantitative version of Kuhfittig's theorem. Note that in [5] Theorem 4.4 is actually shown in the more general context of UCW -hyperbolic spaces 3 , but here we state it adapted to the special case of Banach spaces:

Remark 2
In the case where the Banach space has a modulus of convexity η that can be written as η( ) = η( ) whereη( ) is increasing, (for instance, in the case of the Banach spaces L p , that, for p ≥ 2 have an asymptotically optimal modulus of convexity p p 2 p , see Hanner [4], also Kohlenbach [8] and [7] ) then η can be replaced withη in the bound (see [5,Remark 3.3]).
We show the following corollary to Theorem 3.7 by making use of the above Theorem 4.4. Corollary 4.5 Let C be a nonempty convex subset of a uniformly convex Banach space X with a modulus of uniform convexity η and let {T(t) : t ≥ 0} be a one-parameter uniformly equicontinuous semigroup of nonexpansive mappings on C with a modulus of uniform equicontinuity ω . Let α, β ∈ R + with 2 −G < α < β for some G ∈ N and satisfying β/α ∈ R \ Q with effective irrationality measure (with domain restricted to N × N) f β α and let F(T(α)) ∩ F(T(β)) = ∅. Let p ∈ F(T(α)) ∩ F(T(β)) and let D > 0 such that x 0 − p ≤ D for some x 0 ∈ C. Then for the sequence {x n } generated by the iteration schema of Definition 4.1, we have ∀k ∈ N ∀M ∈ N ∀b ∈ N ∀n ≥Θ ∀x n ∈ C b ∀t ∈ [0, M] T(t)x n − x n ≤ 2 −k with a rate of asymptotic regularityΘ := max By setting := X ∈ (0, 2] in the above, where X is as in Theorem 3.7, the premise of Theorem 3.7 is fulfilled, and we thus directly obtain: , G, β ,W, b, M, k, Φ, Ψ, ω) is as in Theorem 3.7.

Acknowledgements
The author is most grateful to her PhD supervisor Prof. Dr. Ulrich Kohlenbach for several suggestions and corrections that significantly improved an earlier version of this paper. This paper was written as a part of the PhD work of the author [14] at the Department of Mathematics, Technische Universität Darmstadt, Germany and the author was supported by the IRTG 1529.