Quantitative translations for viscosity approximation methods in hyperbolic spaces
aa r X i v : . [ m a t h . F A ] F e b Quantitative translations for viscosityapproximation methods in hyperbolic spaces
Ulrich Kohlenbach † and Pedro Pinto ‡ Department of Mathematics, Technische Universit¨at Darmstadt,Schlossgartenstraße 7, 64289 Darmstadt, Germany † [email protected] ‡ [email protected] February 9, 2021
Abstract
In the setting of hyperbolic spaces, we show that the convergenceof Browder-type sequences and Halpern iterations respectively entail theconvergence of their viscosity version with a Rakotch map. We also showthat the convergence of a hybrid viscosity version of the Krasnoselskii-Mann iteration follows from the convergence of the Browder type se-quence. Our results follow from proof-theoretic techniques (proof min-ing). From an analysis of theorems due to T. Suzuki, we extract a trans-formation of rates for the original Browder type and Halpern iterationsinto rates for the corresponding viscosity versions. We show that thesetransformations can be applied to earlier quantitative studies of these iter-ations. From an analysis of a theorem due to H.-K. Xu, N. Altwaijry andS. Chebbi, we obtain similar results. Finally, in uniformly convex Banachspaces we study a strong notion of accretive operator due to Brezis andSibony and extract an uniform modulus of uniqueness for the property ofbeing a zero point. In this context, we show that it is possible to obtainCauchy rates for the Browder type and the Halpern iterations (and hencealso for their viscosity versions).Keywords: Viscosity method, Rates of convergence, Rates of metasta-bility, Proof mining. MSC2020: 47H09 47J25 03F10 53C23
In this paper we study strongly convergent iteration schemes for nonexpansiveself mappings T : C → C of bounded closed convex subsets of Banach spacesand - more generally - complete ( W )-hyperbolic spaces X in the sense of [27].In the framework of Hilbert spaces, Browder [8] and Halpern [19] provedstrong convergence theorems for implicit and explicit iterations, respectively.1rowder’s implicit scheme is y n ( u ) = (1 − α n ) T ( y n ( u )) ⊕ α n u, u ∈ C, (1)while Halpern introduced the explicit iteration scheme w n +1 ( u ) = (1 − α n ) T ( w n ( u )) ⊕ α n u, w ( u ) = u ∈ C. (2)Here ‘ ⊕ ’ is the result of applying the convexity operator W from our hyperbolicspace (see the next section for details) which is the usual linear convex combi-nation in the case of normed spaces.The implicit schema ( y n ( u )) was introduced by Browder in [8] and shown to bestrongly convergent in Hilbert spaces to the metric projection of u onto the set F ix ( T ) of fixed points of T (see also [19] for a more elementary proof). In [47],Reich extended the convergence of the Browder sequence to uniformly smoothBanach spaces where the sequence converges to the unique sunny nonexpansiveretraction of C onto F ix ( T ) applied to u. For the explicit scheme ( w n ( u )), Halpern showed the strong convergence to themetric projection of the anchor point u onto the set F ix ( T ) . The conditionsconsidered by Halpern prevented the natural choice α n = n +1 , which was laterovercome by Wittmann [55]. [50] generalized Wittmann’s theorem in particularto uniformly smooth Banach spaces. In [57, 56], Xu proved the strong con-vergence of Halpern type iterations in uniformly smooth Banach spaces underconditions which are incomparable to those used by Wittmann but which alsoallow for the choice α n = n +1 . Both schemes have also been considered for families of nonexpansive mappings( S n ) instead of the single map T : the sequence implicitly defined by y n ( u ) = (1 − α n ) S n ( y n ( u )) ⊕ α n u, (3)is called the ( S n )-Browder sequence with anchor point u . The original Browdersequence is (3) for a constant sequence ( S n ).Likewise, the sequence explicitly defined by w ( u ) = u ∈ C and w n +1 ( u ) = (1 − α n ) S n ( w n ( u )) ⊕ α n u, (4)is called the ( S n )-Halpern iteration (with anchor and starting point u ). TheHalpern iteration [19] is the particular case of (4) for constant ( S n ). In [4],Bauschke considered a finite sequence of nonexpansive maps (not necessarilycommutative but under a condition on the fixed point sets for ordered compo-sitions) to define in a cyclic manner an infinite sequence of nonexpansive maps( S n ) from an initial finite list of maps. In this way, Bauschke’s result is a par-ticular instance of convergence for an iteration in the style of (4). Bauschke’sresult was further generalized in [21].A different instance of convergence for iterations (4) can be found in the methodsfor finding zeros of accretive operators, the so-called proximal point algorithms.2he Halpern-type proximal point algorithm (HPPA), introduced by Kamimuraand Takahashi [22] and independently by Xu [57], is the particular case of (4)where the sequence of nonexpansive maps is given by resolvent functions of anaccretive operator. The HPPA has been extensively studied in the literatureand several results give conditions that guarantee the strong convergence of thealgorithm both in Hilbert spaces as well as e.g. in Banach spaces which areboth uniformly smooth and uniformly convex (e.g. [22, 57, 5, 23, 1]).In [43], Moudafi introduced in the context of Hilbert spaces the so-called vis-cosity algorithms in which the fixed anchor u is replaced by the value of a strictcontraction φ applied to the current iteration. Moudafi’s viscosity algorithmswere extended to uniformly smooth Banach spaces by Xu in [58]. While Moudafionly considered a single mapping T this has also subsequently been generalizedto families of such mappings ( S n ) : the sequence implicitly defined by x n = (1 − α n ) S n ( x n ) ⊕ α n φ ( x n ) , (5)is called the ( S n )-viscosity-Browder sequence (for ( α n )).The ( S n )-viscosity-Halpern iteration is an explicit counterpart of (5), namely,given x ∈ C , the iteration is defined inductively by x n +1 = (1 − α n ) S n ( x n ) ⊕ α n φ ( x n ) . (6)Viscosity generalizations of Bauschke’s result for finite families of mappings andof the results in [21] are given in [20] and - under very general conditions - in[10] which is closely related to the prior Hybrid Steepest Descent Method, in-troduced by Yamada in [60] for finite sets of nonexpansive functions. In fact,the proof in [60] can be adapted to provide a strong convergence result for theviscosity version of Bauschke’s theorem (see [38]).In the important paper [51], Suzuki showed in the context of normed spacesthat under very general conditions the convergence of the Browder and Halpernschemes (for families of mappings) implies the convergence of the correspondingviscosity schemes. Moreover, he showed that the limit of ( x n ) is the uniquefixed point of P ◦ φ, where P ( u ) := lim y n ( u ) in the Browder case and P ( u ) :=lim w n ( u ) in the Halpern case respectively. Suzuki’s result actually appliesto a larger class of mappings φ than only strict contractions, namely to so-called Meir-Keeler contractions (MKC) introduced in [42] which in the case ofconvex sets (even of hyperbolic spaces) coincide with the uniformly contrac-tive mappings introduced earlier by Rakotch in [46]. With φ, also the map x (1 − α n ) S n ( x ) ⊕ α n φ ( x ) is an MKC mapping and so – by [42, 46] – has aunique fixed point. Thus the implicit Browder scheme (5) (even for φ a MKC)is well-defined.In section 3 of this paper, we give a complete quantitative analysis of Suzuki’sreduction technique both in terms of rates of convergence as well as in termsof rates of metastability in the sense of Tao. Since in many important cases,explicit rates of metastability have been construced in recent years for bothBrowder-type sequences as well as for Halpern iterations, we now get in all3hese situations also rates of metastability for the corresponding viscosity gen-eralizations. Usually, effective rates of convergence can be ruled out for Browderand Halpern sequences (see e.g. [44]) and so it is important that our quanti-tative analysis of Suzuki’s theorem allows us to operate on the level of rates ofmetastability, by which we mean for a sequence ( x n ) in a metric space ( X, d )any bound ϕ : (0 , ∞ ) × N N → N such that ∀ ε > ∀ f : N → N ∃ n ≤ ϕ ( ε, f ) ∀ i, j ∈ [ n, f ( n )] ( d ( x i , x j ) ≤ ε )(see [26, 54]). Note that noneffectively, the property bounded by ϕ is equiva-lent to the usual Cauchy property. So a rate of metastability provides a fini-tary quantitative statement which (noneffectively) implies back the convergencestatement. Moreover, we augment the rate of metastability with further bound-ing information which entails that the limits of ( x n ) is the unique fixed point of P ◦ φ with P as above (see Theorems 3.6 and 3.11).Our analysis uses ideas from the logic-based approach of ‘proof mining’ by whichthe extractability of such quantitative data follows from certain logical trans-formations of the given proof (see [28] for more on this). However, while ideasfrom logic played a key role in arriving at our results, the final proofs make noreferences to logic. As a common by-product of such a logical analysis of proofsone often obtains in addition to quantitative information also new qualitativegeneralizations of the original theorems. E.g. the generalization to the settingof hyperbolic spaces is such an offspring of the logic-based approach.If one is in a special situation where an actual rate of convergence for the Brow-der of Halpern sequences is available, then our quantitative transformation alsoyields a rate of convergence for the corresponding viscosity version.A different method for finding fixed points of nonexpansive mappings is theKrasnoselskii-Mann iteration x n +1 = (1 − β n ) x n ⊕ β n T ( x n ) . (7)This iteration has nice properties, notably Fej´er monotonicity w.r.t. thefixed point set of T, but in general only approximates a fixed point weakly. Thismotivated several modifications to (7) to ensure strong convergence. Here weconsider a hybrid version of the Krasnoselskii-Mann iteration and the viscosityapproximation method. Namely, for ( β n ) ⊂ (0 ,
1] and x ∈ C , the vKM iterationis defined by x n +1 = (1 − β n ) x n ⊕ β n ((1 − α n ) T ( x n ) ⊕ α n φ ( x n )) , (8)introduced in [18] and shown there to be strongly convergent in Banach spacesunder very strong conditions. In [59] this is studied further and much improved.Note that for α n ≡
0, (8) reduces to the original schema (7), and for β n ≡ T : C → C used in these sequences is of the form I − A, where A is an accretive operator which satisfies a condition of beinguniformly accretive which was introduced in [6] (see also [18] and [59]) and whichis more general than other notions of uniform Ψ or Φ-accretivity studied in theliterature. In uniformly convex Banach spaces X , this liberal notion sufficesto conclude that A has at most one zero. From this uniqueness proof due to[6] we extract a so-called modulus of uniqueness (see [28] and the referencesgiven there) which only depends on a general modulus function witnessing theuniform accretivity of A and a modulus of uniform convexity of X. From this weobtain explicit and low complexity rates of convergence for the iterations listedabove involving the mapping T = I − A. Let (
X, d ) be a metric space and C a nonempty subset of X . Definition 2.1.
Consider a mapping T : C → C . We say • T is nonexpansive if for all x, y ∈ Cd ( T ( x ) , T ( y )) ≤ d ( x, y ) ; • T is a strict contraction if there is r ∈ [0 , such that for all x, y ∈ Cd ( T ( x ) , T ( y )) ≤ rd ( x, y ) , in which case we say that T is an r -contraction; • T is a Meir-Keeler contraction (MKC) if for any ε > there is δ > suchthat d ( x, y ) < ε + δ → d ( T ( x ) , T ( y )) < ε, for all x, y ∈ C. Clearly, any contraction is a MKC mapping and any MKC mapping is anonexpansive map. Meir-Keeler contraction were introduced in [42] as a gen-eralization of metric contractions. Actually, [42] considers a condition with thestronger premise ε ≤ d ( x, y ) < ε + δ , which ends up being equivalent to the oneabove using the nonexpansivity of T .We now recall the notion of hyperbolic space. There are several distinctnotion of hyperbolic space in the literature [24, 15, 16, 48]. Our results willbe in the setting of W -hyperbolic spaces as introduced by the first author [27],here labeled simply by hyperbolic spaces . This setting is slightly more restrictivethan the notion of hyperbolic space by Goebel/Kirk [15], but more general thanthe hyperbolic spaces in the sense of Reich/Shafrir [48].The triple ( X, d, W ) is called a hyperbolic space if (
X, d ) is a metric spaceand W : X × X × [0 , → X is a function satisfying5W1) ∀ x, y ∈ X ∀ λ ∈ [0 ,
1] ( d ( z, W ( x, y, λ )) ≤ (1 − λ ) d ( z, x ) + λd ( z, y )),(W2) ∀ x, y ∈ X ∀ λ , λ ∈ [0 ,
1] ( d ( W ( x, y, λ ) , W ( x, y, λ ) = | λ − λ | d ( x, y ))),(W3) ∀ x, y ∈ X ∀ λ ∈ [0 ,
1] ( W ( x, y, λ ) = W ( y, x, − λ )),(W4) ( ∀ x, y, z, w ∈ X ∀ λ ∈ [0 , d ( W ( x, z, λ ) , W ( y, w, λ )) ≤ (1 − λ ) d ( x, y ) + λd ( z, w )) . The convexity function W was first considered by Takahashi [53], where atriple ( X, d, W ) with W satisfying (W1) is called a convex metric space. Theclasse of hyperbolic spaces includes normed spaces and their convex subsets,the Hilbert ball [16] and CAT(0)-spaces in the sense of Gromov. (see e.g. [7]for a detailed treatment). In turn, hyperbolic spaces are CAT(0)-spaces if theysatisfy the property CN − (which in the presence of other axioms is equivalent tothe Bruhat-Tits CN-inequality [9] but in contrast to the latter purely universal):CN − : ∀ x, y, z ∈ X (cid:18) d ( z, W ( x, y,
12 )) ≤ d ( z, x ) + 12 d ( z, y ) − d ( x, y ) (cid:19) . If x, y ∈ X and λ ∈ [0 , W ( x, y, λ ) by (1 − λ ) x ⊕ λy . Using(W1), it is easy to see that d ( x, (1 − λ ) x ⊕ λy ) = λd ( x, y ) and d ( y, (1 − λ ) x ⊕ λy ) = (1 − λ ) d ( x, y ) . For x, y ∈ X , the set { (1 − λ ) x ⊕ λy : λ ∈ [0 , } is called the metric segmentwith endpoints x, y and is denote by [ x, y ]. A nonempty subset X ⊆ X is called convex if [ x, y ] ⊆ C , for all x, y ∈ C .For convex C , MKC have the useful property that for each ε > ε -apart. This was shown for convex subsets ofBanach spaces in [51, Proposition 2]. We generalize this result to the geodesicsetting: Lemma 2.2.
Let C be a convex subset of X and φ a MKC on C . Then thereis a function δ : (0 , ∞ ) → (0 , such that for all ε > and x, y ∈ Cd ( x, y ) ≥ ε → d ( T ( x ) , T ( y )) ≤ (1 − δ ( ε )) · d ( x, y ) . Proof.
Let ε > φ is a MKC mapping, there is σ ∈ (0 , ε )satisfying for all x, y ∈ Cd ( x, y ) < ε σ → d ( φ ( x ) , φ ( y )) ≤ ε . We claim that for all x, y ∈ C , d ( x, y ) ≥ ε → d ( φ ( x ) , φ ( y )) ≤ (cid:16) − σ ε (cid:17) · d ( x, y ) , (9)and thus for each ε > δ ( ε ) by14 ε sup n σ ∈ (0 , ε ) : ∀ x, y ∈ C (cid:16) d ( x, y ) ≥ ε → d ( φ ( x ) , φ ( y )) ≤ (1 − σ ε ) · d ( x, y ) (cid:17)o .
6t remains to prove (9). Take x, y ∈ C such that d ( x, y ) ≥ ε and write a := d ( x, y ). For each j ∈ { , · · · , ⌊ aε ⌋ + 1 } , consider λ j ∈ [0 ,
1] defined by (cid:0) ε + σ a (cid:1) j . For j ∈ { , · · · ⌊ aε ⌋} , by (W2) d ( W ( x, y, λ j ) , W ( x, y, λ j +1 )) = ( λ j +1 − λ j ) d ( x, y ) < ε σ, and hence, by the hypothesis on σ , d ( φ ( W ( x, y, λ j )) , φ ( W ( x, y, λ j +1 ))) ≤ ε . Since φ is nonexpansive, we have d ( φ ( x ) , φ ( y )) ≤ ⌊ aε ⌋ X j =0 d ( φ ( W ( x, y, λ j )) , φ ( W ( x, y, λ j +1 )))+ d ( φ ( W ( x, y, λ ⌊ aε ⌋ +1 )) , φ ( y )) ≤ ε (cid:16)j aε k + 1 (cid:17) + (1 − λ ⌊ aε ⌋ +1 ) d ( x, y )= a − σ (cid:16)j aε k + 1 (cid:17) ≤ a (cid:16) − σ ε (cid:17) = (cid:16) − σ ε (cid:17) d ( x, y ) , showing (9) and concluding the proof.In [46], Rakotch generalized Banach’s contraction principle to maps satisfy-ing the condition x = y → d ( φ ( x ) , φ ( y )) ≤ α ( d ( x, y )) · d ( x, y ) , (10)where α : (0 , ∞ ) → (0 ,
1) is a decreasing function. However, the assumptionthat α is decreasing implies that condition (10) is equivalent to d ( x, y ) ≥ ε → d ( φ ( x ) , φ ( y )) ≤ α ( ε ) · d ( x, y ) . (11)We may always assume that α in (11) is decreasing, switching if necessary to α ′ ( ε ) := max { / , inf { α ( δ ) : δ ∈ (0 , ε ] }} . This argument shows that [51, Proposition 2] actually entails that in normedspaces MKC restricted to convex sets are Rakotch maps. The converse is alsotrue: take δ := (1 − α ( ε )) · ε to satisfy the MKC-condition.Our Lemma 2.2 generalizes this result to the setting of hyperbolic spaces. Inour results, C is always convex and so MKC are only seemingly more generalthan Rakotch maps. In what follows, we call a function δ : (0 , ∞ ) → (0 ,
1) aRakotch-modulus (see [35]) if it satisfies ∀ ε > ∀ x, y ∈ C ( d ( x, y ) ≥ ε → d ( φ ( x ) , φ ( y )) ≤ (1 − δ ( ε )) d ( x, y ))The particular case where φ is an r -contraction coincides with the case wherethere is a constant Rakotch-modulus (i.e. for all ε ∈ (0 , ∞ , δ ( ε ) = δ , where δ ∈ (0 ,
1) is such that r ≤ − δ ). Notational convention:
Throuhgout this paper ⌈ x ⌉ is defined as max { , ⌈ x ⌉} with the usual definition of ⌈·⌉ . .2 Quantitative notions Let ( x n ) be a Cauchy sequence in a metric space ( X, d ). Definition 2.3.
A function ρ : (0 , ∞ ) → N is a Cauchy rate of ( x n ) if ∀ ε > ∀ i, j ≥ ρ ( ε ) ( d ( x i , x j ) ≤ ε ) . If (
X, d ) is complete so that ( x n ) converges, then clearly ρ is a rate of con-vergence of ( x n ) towards its limit x, i.e. ∀ ε > ∀ n ≥ ρ ( ε ) ( d ( x n , x ) ≤ ε ) . Definition 2.4 ([26, 54]) . A function ϕ : (0 , ∞ ) × N N → N is a rate of metasta-bility for ( x n ) if ∀ ε > ∀ f : N → N ∃ n ≤ ϕ ( ε, f ) ∀ i, j ∈ [ n, f ( n )] ( d ( x i , x j ) ≤ ε ) . Remark 2.5.
In the official definition of ‘rate of stability’, one states that ∀ i, j ∈ [ n, n + f ( n )] ( d ( x i , x j ) ≤ ε ) which clearly implies by the formulation aboveand which in turn is implied by the latter taking ˜ f ( n ) := n + f ( n ) instead of f. Proposition 2.6. ρ : (0 , ∞ ) → (0 , ∞ ) is a Cauchy rate iff ϕ ( ε, f ) := ρ ( ε ) is arate of metastability.Proof. If ρ is a Cauchy rate, then clearly ϕ ( ε, f ) := ρ ( ε ) is a rate of metastability.Let ϕ ( ε, f ) be a rate of metastability which is a function ρ ( ε ) which does notdepend on f. Let ˜ m ≥ m ≥ ρ ( ε ) . Take f ( k ) := ˜ m for all k. Then there exists an n ≤ ρ ( ε ) such that ∀ i, j ∈ [ n, f ( n )] ⊆ [ ρ ( ε ) , ˜ m ] ( d ( x i , x j ) ≤ ε )and so, in particular, d ( x ˜ m , x m ) ≤ ε. A well-known lemma by Xu (e.g. [56, Lemma 2.1]) is used in the original proofs.
Lemma 2.7.
Consider sequences of nonnegative real numbers ( λ n ) ⊆ (0 , , ( a n ) and a sequence ( b n ) ⊂ R such that a n +1 ≤ (1 − λ n ) a n + λ n b n , for all n ∈ N . If ( i ) ∞ X n =0 λ n = ∞ (or equivalently Q ∞ n =0 (1 − λ n ) = 0) , ( ii ) lim sup b n ≤ , then lim a n = 0 . We say that A : N → N is a rate of divergence for P λ n = ∞ if ∀ k ∈ N A ( k ) X i =0 λ i ≥ k . (QΣ)8 quantitative treatment is also possible using instead the equivalent condition Q (1 − λ n ) = 0, if we have a function A ′ : N × (0 , → N such that for all m ∈ N , A ′ ( m, · ) is a rate of convergence for Q ∞ i = m (1 − λ i ) = 0, i.e. ∀ m ∈ N ∀ ε ∈ (0 , A ′ ( m,ε ) Y i = m (1 − λ i ) ≤ ε . (QΠ)For our analysis, we will need a quantitative version of this lemma, whichwe give bellow. The proof is identical to several similar previous results (seee.g. [33, Lemmas 5.2,5.3] and [45, Lemmas 12,13]). Lemma 2.8.
Consider sequences of real numbers ( λ n ) ⊂ [0 , , ( a n ) , ( b n ) ⊂ R and let B ∈ N ∗ be an upper bound on ( a n ) and assume that for all n ∈ N a n +1 ≤ (1 − λ n ) a n + λ n b n . Consider ε > , p, N ∈ N such that ∀ i ∈ [ N, p ] (cid:0) b n ≤ ε (cid:1) .(i) If A : N → N satisfies (QΣ) , then ∀ i ∈ [ σ ( ε, N ) , p ] ( a i ≤ ε ) , where σ ( ε, N ) = σ [ A, B ]( ε, N ) := A (cid:0) N + ⌈ ln (cid:0) Bε (cid:1) ⌉ (cid:1) + 1 ;(ii) If A ′ : N × (0 , → N satisfies (QΠ) , then ∀ i ∈ [ σ ( ε, N ) , p ] ( a i ≤ ε ) , where σ ( ε, N ) = σ [ A ′ , B ]( ε, N ) := max { A ′ ( N, ε B ) , N } + 1 . The conclusion of Lemma 2.8 still holds true if the main inequality is onlyvalid in the interval [
N, p ]. Lemma 2.9.
Consider sequences of real numbers ( λ n ) ⊂ [0 , , ( a n ) , ( b n ) ⊂ R . Let B ∈ N ∗ be an upper bound on ( a n ) and A : N → N a function satisfying (QΣ) . Let ε ∈ (0 , and p, N ∈ N be given. If ∀ i ∈ [ N, p ] (cid:16) a i +1 ≤ (1 − λ i ) a i + λ i b i and b i ≤ ε (cid:17) , then ∀ i ∈ [ σ ( ε, N ) , p ] ( a i ≤ ε ) , where σ ( ε, N ) = σ [ A, B ]( ε, N ) := A (cid:0) N + ⌈ ln (cid:0) Bε (cid:1) ⌉ (cid:1) + 1 . In [51], Suzuki starts by giving perspicuous proofs of Xu’s strong convergence re-sults for the viscosity method (with strict contractions) of Browder and Halperntype iterations in uniformly smooth Banach spaces.9 heorem 3.1 ([58, Thm.4.1]) . Let X be an uniformly smooth Banach spaceand C a nonempty, closed, convex subset of X . Let T and φ be a nonexpansiveand a strict contraction on C , respectively. Assume that F ix ( T ) = ∅ . Considerthe net ( x α ) satisfying x α = (1 − α ) T ( x α ) + αφ ( x α ) , for α ∈ (0 , . Then ( x α ) strongly converges, as α → , to the unique z satisfying P φ ( z ) = z , where P is the unique sunny retraction of C onto F ix ( T ) . Theorem 3.2 ([58, Thm.4.2]) . Under the same conditions as before, let x ∈ C ,and ( α n ) ⊂ (0 , satisfy ( C
1) lim α n = 0 , ( C X α n = ∞ , ( C
3) lim α n +1 α n = 1 . Consider the the iteration ( x n ) defined by x n +1 = (1 − α n ) T ( x n ) + α n φ ( x n ) . Then ( x n ) strongly converges to the unique z satisfying P φ ( z ) = z . The new proofs compare the viscosity iteration with the convergent originalone. Suzuki, then goes on to employ similar arguments to obtain extendedconvergence results to MKC-mappings φ and with T replaced by a sequence ofnonexpansive maps ( S n ) together with a stability condition that the originaliteration defined on that sequence of maps is convergent. From a logical pointof view, the only complicated aspect in Suzuki’s proofs is the use of the sunnyretraction (metric projection in Hilbert spaces) required to characterize thelimit point z . We avoid this step by considering Cauchy points of the relevantiterations instead of the actual limit. This entails that a quantitative analysisof these results will extract a function that given a metastability rate for theassumed convergence (and some other parameters that illustrate the uniformityof the bound), outputs a rate of metastability for the the viscosity iterationin question. In this section, we elaborate on the quantitative conditions of ourmain results given in the next two sections. We then apply our main theorems inparticular cases where metastability rates have been extracted and witness theassumption, to obtain several rates of metastability for viscosity-type iterations. Notation 3.3.
Consider a function ϕ on tuples of variables ¯ x , ¯ y . If we wishto consider the variables ¯ x as parameters we write ϕ [¯ x ](¯ y ) . For simplicity ofnotation we may then even omit the parameters and simply write ϕ (¯ y ) . The next lemma states that we can always transform a given rate of metasta-bility such that the interval of metastability is to the right of a given N ∈ N . Lemma 3.4. If ( x n ) is a Cauchy sequence with rate of metastability ϕ , then ∀ ε > ∀ f : N → N ∀ N ∈ N ∃ n ∈ [ N, θ ( ε, f, N )] ∀ i, j ∈ [ n, f ( n )] ( d ( x i , x j ) ≤ ε ) , where θ ( ε, f, N ) := θ [ ϕ ]( ε, f, N ) := max { N, ϕ ( ε, f N ) } , where for all m ∈ N f N ( m ) := f (max { N, m } ) . roof. Let ε, f and N be given. Since ϕ is a rate of metastability for ( x n ), wehave that for some n ′ ≤ ϕ ( ε, f N ), ∀ i, j ∈ [ n ′ , f N ( n ′ )] ( d ( x i , x j ) ≤ ε ) . Let n = max { N, n ′ } . Then n ∈ [ N, max { N, ϕ ( ε, f N ) } ] = [ N, θ ( ε, f, N )]. Since n ′ ≤ n and f N ( n ′ ) = f ( n ), the result follows.Let C ⊆ X be a bounded convex subset of a hyperbolic space X and N ∗ ∈ b ≥ diam ( C ) . For u ∈ C let ( y n ( u )) be the ( S n )-Browder sequence with anchor point u satis-fying (3). We say that a monotone function θ b satisfies the condition (B[ S n ]) iffor all u ∈ C , ∀ ε > ∀ f : N → N ∀ N ∈ N ∃ n ∈ [ N, θ b ( ε, f, N )] ∀ i, j ∈ [ n, f ( n )] ( d ( y i ( u ) , y j ( u )) ≤ ε ) . (B[ S n ]) Remark 3.5.
Note that the function θ b is uniform on u ∈ C , depending only onthe bound b ∈ N ∗ for the diameter of C . Furthermore, the monotonicty conditionhere is to be understood in the sense that the function θ b is nondecreasing in N .Such assumption is by convenience, as one can always consider θ majb ( ε, f, N ) :=max N ′ ≤ N { θ b ( ε, f, N ′ ) } . Finally, we remark that by Lemma 3.4 , such a function θ b exists whenever we have a uniform metastability rate ϕ b for ( y n ( u )) . Similarly, for u ∈ C let ( w n ( u )) be the ( S n )-Halpern iteration with anchorand starting point u defined by (4). We say that a monotone function θ ′ b satisfiesthe condition (H[ S n ]) if for all u ∈ C , ∀ ε > ∀ f : N → N ∀ N ∈ N ∃ n ∈ [ N, θ ′ b ( ε, f, N )] ∀ i, j ∈ [ n, f ( n )] ( d ( w i ( u ) , w j ( u )) ≤ ε ) . (H[ S n ])We also write the previous conditions for constant sequences ( S n ) to discussparticular instances of the main theorems. Let T be a nonexpansive map.A monotone function θ b satisfies the condition (B[ T ]) if for all u ∈ C , ∀ ε > ∀ f : N → N ∀ N ∈ N ∃ n ∈ [ N, θ b ( ε, f, N )] ∀ i, j ∈ [ n, f ( n )] ( d ( y i ( u ) , y j ( u )) ≤ ε ) , (B[ T ])where ( y n ( u )) is the ( S n )-Browder sequence for S n ≡ T , i.e. the original Brow-der type iteration.A monotone function θ ′ b satisfies the condition (H[ T ]) if for all u ∈ C , ∀ ε > ∀ f : N → N ∀ N ∈ N ∃ n ∈ [ N, θ ′ b ( ε, f, N )] ∀ i, j ∈ [ n, f ( n )] ( d ( w i ( u ) , w j ( u )) ≤ ε ) , (H[ T ])where ( w n ( u )) is the ( S n )-Halpern iteration for S n ≡ T , i.e. the original Halperntype iteration. 11 .2 Browder-viscosity In this section we prove a quantitative version of Theorem 7 in [51].
Theorem 3.6.
Let C be a nonempty bounded convex subset of a hyperbolicspace and let b ∈ N ∗ be a bound on the diameter of C . Consider ( S n ) a family ofnonexpansive maps on C , an MKC mapping φ : C → C with Rakotch-modulus δ ,and a sequence ( α n ) ⊂ (0 , . Let θ b be a monotone function satisfying (B[ S n ]) .Define Ψ( ε, f, N ) = Ψ[ b, δ, θ b ]( ε, f, N ) = θ b ( ε , f, Ψ M ) , where Ψ = N and Ψ m +1 = θ b ( ε , f M − m , Ψ m ) , with ˜ ε = εδ ( ε/ , ε = ˜ εδ (˜ ε )2 , M = (cid:24) log − δ (˜ ε ) (cid:18) ˜ ε b (cid:19)(cid:25) andfor all p ∈ N , ( f ( p ) = f ( p ) ,f m +1 ( p ) = max { f ( p ) , θ b ( ε , f m , p ) } for m < M. Then for ( x n ) satisfying (5) we have ∀ N ∈ N ∀ ε > ∀ f : N → N ∃ n ∈ [ N, Ψ( ε, f, N )] ∃ z ∈ C (cid:0) d ( z, y n ( φ ( z ))) ≤ ε ∧ ∀ i ∈ [ n, f ( n )] ( d ( x i , z ) ≤ ε ) (cid:1) . In particular, Ψ( ε, f, is a rate of metastability for ( x n ) . Proof.
Let N ∈ N , ε > f : N → N be given. Note that the definitionof the natural number M implies (1 − δ (˜ ε )) M ≤ ˜ ε b . Consider the functions f , · · · , f M as in the theorem. Define z = x and n = N . For m ≤ M ,assume that z m and n m are already defined and let ( y mn ) n be the ( S n )-Browdersequence with anchor point φ ( z m ), i.e. y mn = y n ( φ ( z m )). By (B[ S n ]), we cantake n m +1 ∈ [ n m , θ b ( ε , f M − m , n m )] such that ∀ i ∈ [ n m +1 , f M − m ( n m +1 )] (cid:16) d ( y mi , y mn m +1 ) ≤ ε (cid:17) . Define z m +1 = y mn m +1 .Notice that for m ≤ M −
1, we have n m +2 ∈ [ n m +1 , θ b ( ε , f M − m − , n m +1 )] ⊂ [ n m +1 , f M − m ( n m +1 )], and so d ( y mn m +2 , y mn m +1 ) ≤ ε . For m ∈ [1 , M ], by (W4) and the fact that S n m +1 is nonexpansive, we have d ( y mn m +1 , y m − n m +1 ) ≤ (1 − α n m +1 ) d ( y mn m +1 , y m − n m +1 ) + α n m +1 d ( φ ( z m ) , φ ( z m − )) , which gives d ( y mn m +1 , y m − n m +1 ) ≤ d ( φ ( z m ) , φ ( z m − )). Thus, for m ∈ [1 , M ], d ( z m +1 , z m ) ≤ d ( y mn m +1 , y m − n m +1 ) + d ( y m − n m +1 , y m − n m ) ≤ d ( φ ( z m ) , φ ( z m − )) + ε . m ≤ M such that d ( z m +1 , z m ) ≤ ˜ ε . Assumethat for all m ≤ M − d ( z m +1 , z m ) > ˜ ε , which gives d ( φ ( z m +1 ) , φ ( z m )) ≤ (1 − δ (˜ ε )) d ( z m +1 , z m ), since δ is a Rakotch-modulus for φ . Hence, for m ∈ [1 , M ] d ( z m +1 , z m ) ≤ (1 − δ (˜ ε )) d ( z m , z m − ) + ε . By induction d ( z M +1 , z M ) ≤ (1 − δ (˜ ε )) M d ( z , z ) + ε M − X i =0 (1 − δ (˜ ε )) i ≤ ˜ ε b b + ε δ (˜ ε ) = ˜ ε. Let m ≤ M be such that d ( z m +1 , z m ) ≤ ˜ ε . We now write y n ≡ y m n and z = z m +1 . We have [ n m +1 , f ( n m +1 )] ⊂ [ n m +1 , f M − m ( n m +1 )] and so ∀ i ∈ [ n m +1 , f ( n m +1 )] ( d ( y i , z ) ≤ ε ≤ ˜ ε ) . (12)By (W4) we have for i ∈ [ n m +1 , f ( n m +1 )] (using again that S i is nonexpan-sive), d ( x i , y i ) ≤ (1 − α i ) d ( x i , y i ) + α i d ( φ ( x i ) , φ ( z m )) , which implies, using the fact that φ is nonexpansive, d ( x i , y i ) ≤ d ( φ ( x i ) , φ ( z m )) ≤ d ( φ ( x i ) , φ ( z )) + d ( φ ( z ) , φ ( z m )) ≤ d ( φ ( x i ) , φ ( z )) + d ( z m +1 , z m ) ≤ d ( φ ( x i ) , φ ( z )) + ˜ ε. (13)An inductive argument and the monotoniticty of θ b is the last variable, entail N ≤ n m +1 ≤ n M +1 ≤ Ψ( ε, f, N ). We now argue that for i ∈ [ n m +1 , f ( n m +1 )], d ( x i , z ) ≤ ε . Assume that for some i ∈ [ n m +1 , f ( n m +1 )], d ( x i , z ) > ε . Since δ is a Rakotch-modulus for φ , d ( φ ( x i ) , φ ( z )) ≤ (1 − δ ( ε/ d ( x i , z ) , and then, using (12) and (13), d ( x i , z ) ≤ d ( x i , y i ) + d ( y i , z ) ≤ d ( φ ( x i ) , φ ( z )) + ˜ ε + d ( y i , z ) ≤ (1 − δ ( ε/ d ( x i , z ) + 2˜ ε. This implies the contradiction d ( x i , z ) ≤ εδ ( ε/ = ε .Using d ( z, z m ) ≤ ˜ ε ≤ ε, (W4) and the nonexpansivity of S n m and φ , we get d ( z, y n m ( φ ( z ))) = d ( y n m ( φ ( z m )) , y n m ( φ ( z ))) ≤ ε, which entails that the theorem is satisfied with z and n := n m +1 .13 emark 3.7. Theorem 3.6 implies that (already for N = 0 ) Ψ( ε, f, N ) is arate of metastability for ( x n ) . We wrote the statement in a more informativeform so that it implies in an elementary way that (if X is complete and C isclosed) the limit x of ( x n ) actually is the (unique) fixed point of P ◦ φ, where P ( u ) := lim y n ( u ) , which is an important further information in Suzuki’s theo-rem: let ε > and N ∈ N be so large that (1) ∀ k ≥ N ( d ( x k , x ) , d ( y k ( φ ( x )) , P ( φ ( x ))) ≤ ε ) . Then by the theorem, we get an n ≥ N and a z ∈ C such that (2) d ( z, y n ( φ ( z ))) , d ( x n , z ) ≤ ε. Using (1) , (2) d ( x, z ) ≤ ε and so by (W4) and the nonexpansivity of S n and φ (3) d ( y n ( φ ( x )) , y n ( φ ( z ))) ≤ ε. Hence using (2) , (4) d ( x, y n ( φ ( x ))) ≤ d ( z, y n ( φ ( z ))) + 4 ε ≤ ε and so by (1) (5) d ( x, P ( φ ( x ))) ≤ ε. Since ε > was arbitrary, we have that x = P ( φ ( x )) . Remark 3.8.
Theorem 3.6 can be adapted to dispense with the boundedness as-sumption on the set C , and assume the existence of a natural number b ∈ N ∗ thatonly satisfies b ≥ max { d ( x , φ ( x )) , d ( x , y n ( φ ( x ))) : n ∈ N } . In the originalproof by Suzuki, one considers that all the ( S n ) -Browder type sequences con-verge and thus, the particular fact that ( y n ( φ ( x ))) is bounded trivially follows.In this situation, one assumes that for each r ∈ N ∗ , the function θ r satisfies theformula (B[ S n ]) only for anchor points u ∈ C ∩ B r ( x ) , where B r ( x ) denotesthe closed ball centred at x with radius r . Then, Theorem 3.6 can be adaptedto hold with the bound Ψ( ε, f, N ) = θ ( M +1) b ( ε , f, Ψ M ) , where ˜ ε , ε , M are asbefore and Ψ = N and Ψ m +1 = θ ( m +1) b ( ε , f M − m , Ψ m ) , with, for all p ∈ N ( f ( p ) = f ( p ) ,f m +1 ( p ) = max (cid:8) f ( p ) , θ ( M − m +1) b ( ε , f m , p ) (cid:9) for m < M. Indeed, the proof can be carried out with similar arguments. The definition ofthe natural numbers n m and the points z m is similar to before. Assume n m and z m are already defined, with d ( x , z m ) ≤ m · b . From the hypothesis on θ ( · ) andthe fact that d ( x , φ ( z m )) = d ( x , φ ( x )) + d ( φ ( x ) , φ ( z m )) ≤ ( m + 1) b, e can define n m +1 ∈ [ n m , θ ( m +1) b ( ε , f M − m , n m )] satisfying ∀ i ∈ [ n m +1 , f M − m ( n m +1 )] (cid:16) d ( y mi , y mn m +1 ) ≤ ε (cid:17) . Moreover, with z m +1 = y mn m +1 , we have d ( x , z m +1 ) ≤ d ( x , y n m +1 ( φ ( x ))) + d ( y n m +1 ( φ ( x )) , y n m +1 ( z m )) ≤ b + d ( x , z m ) ≤ ( m + 1) b, which ensures that this construction holds for all m ≤ M . Since it still holdsthat [ n m +1 , θ ( m +2) b ( ε , f M − m − , n m +1 )] ⊂ [ n m +1 , f M − m ( n m +1 )] , we again have d ( y mn m +2 , y mn m +1 ) ≤ ε . The remainder of the proof follows the same argumentsas before, noticing that the new condition on the natural number b sufficesto show the existence of z m satisfying d ( z m +1 , z m ) ≤ ˜ ε , as it still entails (1 − δ (˜ ε )) M d ( z , z ) ≤ ˜ ε/ . An application of Theorem 3.6 gives a rate of metastability for the Browder-viscosity iteration from a function θ b satisfying (B[ T ]). Corollary 3.9.
Let C be a nonempty bounded convex subset of a hyperbolicspace and let b ∈ N ∗ be a bound on the diameter of C . Consider T a nonexpan-sive map on C , an r -contraction φ : C → C with δ ∈ (0 , such that r ≤ − δ ,and a sequence ( α n ) ⊂ (0 , . Let θ b be a monotone function satisfying (B[ T ]) .Then ( x n ) satisfying x n = (1 − α n ) T ( x n ) ⊕ α n φ ( x n ) , n ∈ N is a Cauchy sequence with metastability rate Ψ( ε, f ) = Ψ[ b, δ, θ b ]( ε, f ) = θ b ( ε , f, Ψ M ) , where Ψ = 0 and Ψ m +1 = θ b ( ε , f M − m , Ψ m ) , with ε = εδ , M = (cid:24) log − δ (cid:18) εδ b (cid:19)(cid:25) andfor all p ∈ N , ( f ( p ) = f ( p ) ,f m +1 ( p ) = max { f ( p ) , θ b ( ε , f m , p ) } for m < M. Corollary 3.10.
Let
X, C, T, φ, b, r, δ, ( α n ) , ( x n ) be as in Corollary 3.9. Let ρ be a common Cauchy rate for the T -Browder iteration ( y n ( u )) for all u ∈ C used as anchor points. Then Ψ[ b, d, ρ ]( ε ) := ρ (cid:18) εδ (cid:19) is a Cauchy rate for ( x n ) . roof. By Proposition 2.6, θ b ( ε, f, N ) := ρ ′ ( ε, N ) := max { N, ρ ( ε ) } is a mono-tone function satisfying (B[ T ]). Hence by Corollary 3.9Ψ[ b, d, ρ ]( ε ) := ρ ′ ( ε , Ψ M ) , whereΨ = 0 and Ψ m +1 = ρ ′ ( ε , Ψ m ) , with ε = εδ , M = ⌈ log − δ (cid:18) εδ b (cid:19) ⌉ , is a rate of metastability for ( x n ) which does not depend on f and hence - againby Proposition 2.6 - is a rate of convergence. An easy induction shows thatΨ m ≤ ρ ( ε ) for all m and so ρ ′ ( ε , Ψ M ) = ρ ( ε ) . The Browder type sequence has been the focus of many quantitative studiesusing techniques from proof mining. In [29], the first author analysed Browder’sfixed point theorem in Hilbert spaces showing that it was possible to avoid se-quential weak compactness used in the original proof by Browder. A simplerproof of Browder’s result due to Halpern in [19] that already avoids weak com-pactness, was also analyzed and a metastability bound was obtained. Kirk [25]showed that Halpern’s proof can be extended to CAT(0) spaces (essentially un-changed), and thus the same holds true for its quantitative version [33, Prop.9.3].By Theorem 3.6, using the rates of metastability extracted in [33], we obtainrates of metastability for the viscosity-Browder type sequences in the setting ofCAT(0) spaces. In [38] and [39, Theorem 6.7], for the special case where X is aHilbert space, similar rates of metastability have already been extracted - againusing methods of proof mining - from a direct proof of the strong convergenceof ( x n ) (i.e. without a reduction to Browder’s theorem) from [60].In [36], a quantitative treatment of Reich’s extension of Browder fixed pointtheorem to uniformly smooth Banach spaces was carried out under the addi-tional hypothesis that the space is uniformly convex. Using the rate of metasta-bility extracted in [36], once again by an application of Theorem 3.6, we obtaina rate of metastability for the viscosity-Browder type sequence in Banach spacesthat are both uniformly convex and uniformly smooth. In this section we establish a quantitative version of Theorem 8 in [51].
Theorem 3.11.
Let C be a nonempty bounded convex subset of a hyperbolicspace and let b ∈ N ∗ be a bound on the diameter of C . Consider ( S n ) a familyof nonexpansive maps on C , a MKC φ : C → C with Rakotch-modulus δ , and asequence ( α n ) ⊂ [0 , satisfying P α n = ∞ with a monotone rate of divergence A . Let θ ′ b be a monotone function satisfying (H[ S n ]) and σ be as in Lemma2.8. Define Ψ( ε, f, N ) = Ψ[ b, δ, A, θ ′ b ]( ε, f, N ) = σ [ e A ε , b ] (cid:16) ε , Ψ M +1 (cid:17) , here e A ε ( k ) := A (cid:18)(cid:24) kδ ( ε/ (cid:25)(cid:19) for all k ∈ N Ψ = max { N, A (1) + 1 } and Ψ m +1 = θ ′ b ( ε , f M − m , Ψ m ) , with ˜ ε = 2 εδ ( ε/ , ε = ˜ εδ (˜ ε )4 , M = (cid:24) log − δ (˜ ε )2 (cid:18) ˜ ε b (cid:19)(cid:25) andfor all p ∈ N , ( f ( p ) = max n f (cid:16) σ [ e A ε , b ]( ε , p ) (cid:17) , σ [ e A ε , b ]( ε , p ) o ,f m +1 ( p ) = max { f ( p ) , θ ′ b ( ε , f m , p ) } for m < M. Then for ( x n ) satisfying (6) we have ∀ N ∈ N ∀ ε > ∀ f : N → N ∃ n ∈ [ N, Ψ( ε, f, N )] ∃ k ∈ [ N, n ] , z ∈ C (cid:0) d ( z, w k ( φ ( z ))) ≤ ε ∧ ∀ i ∈ [ n, f ( n )] ( d ( x i , z ) ≤ ε ) (cid:1) . In particular, Ψ( ε, f, is a rate of metastability for ( x n ) . Proof.
Let N ∈ N , ε > f : N → N be given. From the definition ofthe natural number M , we have (cid:18) − δ (˜ ε )2 (cid:19) M ≤ ˜ ε b . Consider the functions f , · · · , f M as in the theorem. Define z = x and n = max { N, A (1) + 1 } .Notice that n − Y i =0 (1 − α i ) ≤ exp( − n − X i =0 α i ) ≤ . For all m ≤ M , assume that z m and n m are already defined and let ( w mn ) n denote the ( S n )-Halpern sequence with anchor and starting point φ ( z m ), i.e. w mn = w n ( φ ( z m )). By (H[ S n ]), we consider n m +1 ∈ [ n m , θ ′ b ( ε , f M − m , n m )] suchthat ∀ i ∈ [ n m +1 , f M − m ( n m +1 )] (cid:16) d ( w mi , w mn m +1 ) ≤ ε (cid:17) , and define z m +1 = w mn m +1 .Notice that for all m ≤ M , since n m ≥ n ≥ N , we have n m − Y i =0 (1 − α i ) ≤ . Moreover, for m ≤ M −
1, [ n m +1 , θ ′ b ( ε , f M − m − , n m +1 )] ⊆ [ n m +1 , f M − m ( n m +1 )]and so d ( w mn m +2 , z m +1 ) = d ( w mn m +2 , w mn m +1 ) ≤ ε .We start by showing that there is m ≤ M such that d ( z m +1 , z m ) ≤ ˜ ε .Assume that for all m ≤ M − d ( z m +1 , z m ) > ˜ ε . For m ∈ [1 , M ], d ( z m +1 , z m ) ≤ d ( w mn m +1 , w m − n m +1 ) + d ( w m − n m +1 , w m − n m ) ≤ d ( w mn m +1 , w m − n m +1 ) + ε , and by induction on n , we have (using ( W
4) and the nonexpansivity of S n ) d ( w mn , w m − n ) ≤ (1 − δ (˜ ε )) + δ (˜ ε ) n − Y i =0 (1 − α i ) ! d ( z m , z m − ) . m ∈ [1 , M ], d ( w mn m +1 , w m − n m +1 ) ≤ (cid:16) − δ (˜ ε )2 (cid:17) d ( z m , z m − ). Hence, d ( z m +1 , z m ) ≤ (cid:18) − δ (˜ ε )2 (cid:19) d ( z m , z m − ) + ε ≤ (cid:18) − δ (˜ ε )2 (cid:19) m d ( z , z ) + ε m − X i =0 (cid:18) − δ (˜ ε )2 (cid:19) i . We conclude, d ( z M +1 , z M ) ≤ (cid:18) − δ (˜ ε )2 (cid:19) M b + 2 ε δ (˜ ε ) ≤ ˜ ε b b + ˜ ε ε. Thus, we can take m ≤ M such that d ( z m +1 , z m ) ≤ ˜ ε . Write w n ≡ w m n , z = z m +1 = w m n m and σ = σ [ e A ε , b ]( ε , n m +1 ) ≥ n m +1 . Assume towards acontradiction that ∀ i ∈ [ n m +1 , σ ] (cid:16) d ( x i , w i ) > ε (cid:17) . For all i ∈ [ n m +1 , σ ], by (W4) d ( x i +1 , w i +1 ) ≤ (1 − α i ) d ( S i ( x i ) , S i ( w i )) + α i d ( φ ( x i ) , φ ( z m )) ≤ (1 − α i ) d ( x i , w i ) + α i ( d ( φ ( x i ) , φ ( w i )) + d ( φ ( w i ) , φ ( z )) + d ( φ ( z ) , φ ( z m ))) ≤ (1 − δ ( ε/ α i ) d ( x i , w i ) + δ ( ε/ α i d ( w i , z ) + d ( z, z m ) δ ( ε/ ≤ (1 − e α i ) d ( x i , w i ) + ˜ α i b i , where ˜ α i = δ ( ε/ α i and b i = d ( w i , z ) + d ( z, z m ) δ ( ε/
3) .Using the monotonicity of the function θ ′ b it is easy to see that n m ≤ Ψ m ,for all m ≤ M + 1. Since A is a rate of divergence for P α n = ∞ , we havethat e A ε is a rate of divergence for P δ ( ε/ α n = ∞ . Furthermore, by themonotonicity of the function A it follows that for all ε >
0, the function σ [ e A ε , b ]is also monotone. Thus from n m +1 ≤ n M +1 , we have σ ≤ Ψ( ε, f, N ). By thedefinition of the functions f m , it follows σ ≤ f M − m ( n m +1 ). Hence ∀ i ∈ [ n m +1 , σ ] (cid:18) d ( w i , z ) ≤ ε ≤ ˜ ε (cid:19) , and so, b i ≤ ˜ ε/ εδ ( ε/
3) = ε ε/ . By Lemma 2.9, we have that d ( x σ , w σ ) ≤ ε , which contradicts the assumption.Consider i such that i ∈ [ n m +1 , σ ] and d ( x i , w i ) ≤ ε . u, v ∈ C , it holds d ( φ ( u ) , φ ( v )) ≤ max { (1 − δ ( ε/ d ( u, v ) , ε/ } .For any i ∈ [ i , f M − m ( n m +1 ) −
1] such that d ( x i , w i ) ≤ ε , by (W4) and thenonexpansivity of S i , we have d ( x i +1 , w i +1 ) ≤ (1 − α i ) d ( x i , w i ) + α i d ( φ ( x i ) , φ ( z m )) ≤ (1 − α i ) d ( x i , w i ) + α i max n(cid:16) − δ (cid:16) ε (cid:17)(cid:17) d ( x i , z m ) , ε o = max n (1 − α i ) d ( x i , w i ) + α i (cid:16) − δ (cid:16) ε (cid:17)(cid:17) d ( x i , z m ) , (1 − α i ) d ( x i , w i ) + α i ε o ≤ max (cid:26)(cid:16) − δ (cid:16) ε (cid:17) α i (cid:17) d ( x i , w i ) + α i δ (cid:16) ε (cid:17) (cid:16) − δ (cid:16) ε (cid:17)(cid:17) d ( w i , z ) + d ( z, z m ) δ ( ε ) , (1 − α i ) d ( x i , w i ) + α i ε o ≤ max { (1 − ˜ α i ) ε α i ε , (1 − α i ) ε α i ε } = ε , using the inequalities d ( x i , w i ) ≤ ε − δ ( ε/ ≤ d ( w i , z ) + d ( z, z m ) δ ( ε/ ≤ ε + ˜ εδ ( ε/ ≤ ε . Hence, d ( x i , w i ) ≤ ε i ∈ [ i , f M − m ( n m +1 )] and in particular, theinequality holds for i ∈ [ σ , f ( σ )]. The theorem is now satisfied with n := σ , k := n m +1 and z as defined above. Indeed, since d ( z, z m ) = d ( z m +1 , z m ) ≤ ε and [ n, f ( n )] ⊆ [ n m +1 , f M − m ( n m +1 )], using (W4) and the nonexpansivity of S n m and φ , we have d ( z, w k ( φ ( z ))) = d ( w n m ( φ ( z m )) , w n m ( φ ( z m +1 ))) ≤ ε and for all i ∈ [ n, f ( n )] d ( x i , z ) ≤ d ( x i , w i ) + d ( w i , z ) ≤ ε ε ≤ ε . Remark 3.12.
As in
Remark 3.7 , it follows from
Theorem 3.11 in an elemen-tary way that (for X being complete and C being closed) the limit x := lim x n is the unique fixed point of P ◦ φ, where P ( u ) := lim w n ( u ) . Remark 3.13.
As in
Remark 3.8 , instead of assuming that C is bounded andworking with a bound on its diameter, we can adapt Theorem 3.11 to the situa-tion where we only have b ∈ N ∗ satisfying b ≥ max { d ( x , w n ( φ ( x ))) : n ∈ N } .We also require bounding information on the initial displacement of the relevantmappings w.r.t. x , i.e. for each n ∈ N let c n ∈ N be such that c n ≥ max { d ( x , φ ( x )) , d ( x , S n ( x )) } . In this case, the bound takes the form Ψ( ε, f, N ) = σ [ e A ε , b ′ ] (cid:0) ε , Ψ M +1 (cid:1) , where ˜ ε, ε , e A ε and M are as before, Ψ = max { N, A (1) + 1 } and Ψ m +1 = θ ′ ( m +1) b ( ε , f M − m , Ψ m ) , ′ = ( M + 1) · b + Ψ M +1 − X k =0 c k , and for all p ∈ N , f ( p ) = max { f (cid:16) σ [ e A ε , b ′ ]( ε , p ) (cid:17) , σ [ e A ε , b ′ ]( ε , p ) } ,f m +1 ( p ) = max n f ( p ) , θ ′ ( M − m +1) b ( ε , f m , p ) o for m < M. We can define n m and z m in a way similar to Remark 3.8 . Since it still holdsthat (1 − δ (˜ ε ) / M d ( z , z ) ≤ ˜ ε/ , we can conclude the existence of z m satisfy-ing d ( z m +1 , z m ) ≤ ˜ ε . The next argument requires an application of Lemma 2.9 .Let w n ≡ w m n be as before, write σ = σ [ ˜ A ε , b ′ ]( ε , n m +1 ) and assume that ∀ i ∈ [ n m +1 , σ ] (cid:16) d ( x i , w i ) > ε (cid:17) . To apply
Lemma 2.9 , it is enough to see that for all i ∈ [ n m +1 , σ + 1] , wehave d ( x i , w i ) ≤ b ′ . First, inductively it holds d ( x i , w i ) ≤ d ( x n m , w n m ) ,for i ∈ [ n m +1 , σ + 1] . The base case is trivial. In the induction step we argueas before: For i ∈ [ n m +1 , σ ] , we have d ( x i +1 , w i +1 ) ≤ (1 − e α i ) d ( x i , w i ) + e α i b i ≤ (1 − e α i ) d ( x i , w i ) + e α i d ( x n m , w n m ) ≤ d ( x n m , w n m ) , using the induction hypothesis and b i ≤ ε/ < ε/ < d ( x n m , w n m ) . Now,by induction, one shows d ( x , x n ) ≤ P n − k =0 c k , for all n ∈ N . Then, from n m +1 ≤ n M +1 ≤ Ψ M +1 and d ( x , z m +1 ) ≤ ( m + 1) b ≤ ( M + 1) b , we conclude d ( x n m , w n m ) ≤ d ( x , x n m ) + d ( x , z m +1 ) ≤ b ′ . The remainder of the proof follows unchanged.
An application of Theorem 3.11 gives a rate of metastability for the Halpern-viscosity iteration from a function θ ′ b satisfying (H[ T ]). Corollary 3.14.
Let C be a nonempty bounded convex subset of a hyperbolicspace and let b ∈ N ∗ be a bound on the diameter of C . Consider T a nonexpan-sive map on C , an r -contraction φ : C → C with δ ∈ (0 , such that r ≤ − δ ,and a sequence ( α n ) ⊂ [0 , satisfying P α n = ∞ with a monotone rate ofdivergence A . Let θ ′ b be a monotone function satisfying (H[ T ]) and σ be as inLemma 2.8. Then ( x n ) defined by x ∈ C, x n +1 = (1 − α n ) T ( x n ) ⊕ α n φ ( x n ) , n ∈ N , is a Cauchy sequence with metastability rate Ψ( ε, f ) = Ψ[ b, δ, A, θ ′ b ]( ε, f ) = σ [ e A, b ] (cid:16) ε , Ψ M +1 (cid:17) here e A ( k ) := A (cid:18)(cid:24) kδ (cid:25)(cid:19) for all k ∈ N Ψ = A (1) + 1 and Ψ m +1 = θ ′ b ( ε , f M − m , Ψ m ) , with ε = εδ , M = (cid:24) log − δ (cid:18) εδ b (cid:19)(cid:25) andfor all p ∈ N , ( f ( p ) = f (cid:16) σ [ e A, b ]( ε , p ) (cid:17) ,f m +1 ( p ) = max { f ( p ) , θ ′ b ( ε , f m , p ) } for m < M. Just as in the case of Corollary 3.10 we obtain the following consequence ofCorollary 3.14:
Corollary 3.15.
Let
X, C, b, T, φ, r, δ, ( α n ) , A, ( x n ) , σ be as in Corollary 3.14.Let ρ ( ε ) be a common (for all anchor points u ∈ C ) rate of convergence for thesequence ( w n ( u )) of Halpern iterations of T. Then Ψ( ε, f ) = Ψ[ b, δ, A, ρ ]( ε, f ) = σ [ e A, b ] (cid:16) ε , max { ρ ( ε ) , A (1) + 1 } (cid:17) where e A ( k ) := A (cid:18)(cid:24) kδ (cid:25)(cid:19) for all k ∈ N and ε = εδ . is a Cauchy rate for ( x n ) . Rates of metastability for the Halpern iterations have been extracted be-fore. In the setting of CAT(0) spaces, under appropriate conditions, rates ofmetastability were obtained in [33]. The convergence of Halpern iterations wasalso studied by proof mining methods in the setting of Banach spaces. Theresults [34] in and [37] (under different sets of conditions, both including thenatural choice α n = n +1 ) extracted a transformation of a rate of metastabilityfor a certain Browder type sequence into a rate of metastability for the Halperniteration. Thus, the recent rate obtained in [36] entails a rates of metastabilityfor Halpern iterations in the setting of Banach spaces that are simultaneouslyuniformly convex and uniformly smooth. Subsequently, by an application ofTheorem 3.11, instantiated with the corresponding rates of metastability forHalpern iterations, we obtain rates of metastability for the viscosity-Halperniterations in the setting of CAT(0) spaces and in the setting of Banach spacesthat are both uniformly convex and uniformly smooth.In Hilbert spaces, a different generalization of Wittmann’s theorem was thatof Bauschke [4] to a finite family of nonexpansive maps T , · · · T ℓ satisfying: ℓ \ i =1 F ix ( T i ) = F ix ( T ℓ · · · T ) = · · · = F ix ( T ℓ − · · · T T ℓ ) (+)Bauschke’s schema is the particular case of (4) when one considers the se-quence of nonexpansive maps ( S n ) defined cyclically by S n := T [ n +1] , for n ∈ N ,21here [ n ] := n mod ℓ . Similarly to Wittmann’s proof, Bauschke’s generaliza-tion also follows from a sequential weak compactness argument. In [13], a the-oretical approach to eliminate this principle was developed, and subsequently aquantitative version of Bauschke’s theorem was obtained. From and applicationof Theorem 3.11, we then obtain a rate of metastability for a viscosity versionof Bauschke’s schema. In [38, 39], K¨ornlein also obtained rates of metastabilityfor the viscosity version of Bauschke’s schema. The rates obtained are different,since K¨ornlein’s extraction follows from a direct proof (i.e. without a reduc-tion to Bauschke’s theorem). In the end, however, both rates look to be of asimilar complexity: K¨ornlein still needed to carry out a convoluted quantita-tive treatment for the existence of (a proxy to) z = P F φz (see [38, Problem8.4.4.]). K¨ornlein also considered Suzuki’s comment on Bauschke’s condition[52]: without any commutativity assumption, the condition (+) already followsfrom having T ℓi =1 F ix ( T i ) = F ix ( T ℓ · · · T ). In [38, Theorem 8.7.1], K¨ornleinproved a finitary version of this remark, and thus [13, Theorem 6.10] can easilybe adapted to the situation where the function τ instead satisfies k x − T ℓ · · · T ( x ) k ≤ τ ( k ) + 1 → ∀ i < ℓ k x − S i ( x ) k ≤ k + 1 . Our result entails the strong convergence of the viscosity version of Bauschke’sschema with a Rakotch map towards a common fixed point. In [20], Jung ex-tended the convergence of the viscosity approximation method of finite familiesto the setting of (suitable) Banach spaces under a slightly more general condi-tion than ( C φ .As a last application we look at proximal point algorithms, a setting inwhich sequences of nonexpansive maps are also naturally considered. Let X bea Hilbert space, and T : D ( T ) ⊂ X → X a monotone operator, i.e. h x − x ′ , y − y ′ i ≥ y ∈ T ( x ) , y ′ ∈ T ( x ′ ). Furthermore, assume that T is maximalin the sense that its graph is not properly contained in the graph of any othermonotone operator. For a real number λ >
0, the resolvent function J λ is thesingle-valued function (Id + λ T) − . It is well known that J λ is nonexpansiveand its fixed point set coincides with the set of zeros of T . First introducedby Martinet in [41], the so-called proximal point algorithm (PPA) is defined by x n +1 = J λ n ( x n ), where ( λ n ) is a sequence of positive real numbers. It is knownthat the PPA is only weakly convergent ([49] and [17]), and so modificationsto the algorithm were proposed. Introduced in [22], and independently in [57],we recall one such generalization: For u, w ∈ X , sequences of real numbers( α n ) ⊂ [0 ,
1] and ( λ n ) ⊂ R + , the iteration ( w n ) defined by w n +1 = α n u + (1 − α n ) J λ n ( w n ) , (HPPA)is the Halpern type proximal point algorithm. Note that when w = u thisiteration is the particular case of (4) with S n = J λ n for all n ∈ N .Quantitative versions of the strong convergence of this iteration were carryout in [45], [40] and [12]. Recently a generalization of HPPA to Banach spaces22nd accretive operators was studied in [1] and a rate of metastability was sub-sequently obtained in [30]. Pertaining to the Halpern type proximal point algo-rithm, this result is the most general so far, as strong convergence is establishedonly asking that ( α n ) ⊂ (0 ,
1] be a sequence slowly converging to zero and thatinf λ n >
0. (The rate extracted in [30], receives as an input a rate of metasta-bility for a Browder type sequence, which can be instantiated with that of [36].)Hence, Theorem 3.11 entails a rate of metastability for a viscosity version of theHalpern type proximal point algorithm under appropriate conditions (see also[2]).
In this subsection, we give a rate of metastability for the vKM iteration (8),provided one has a rate of metastability for Browder sequences. Our approachmakes use of a quantitative analysis of Theorem 5 in [59] (for the case with noerror terms). The following result computes a rate of metastability for the vKMiteration from a rate of metastability for viscosity-Browder sequences.
Theorem 3.16.
Let C be a nonempty bounded convex subset of a hyperbolicspace and let b ∈ N ∗ be a bound on the diameter of C . Consider T a nonexpan-sive map on C , an r -contraction φ : C → C with δ ∈ (0 , such that r ≤ − δ ,and sequences ( α n ) , ( β n ) ⊂ (0 , satisfying P ∞ n =1 α n β n = ∞ with a monotonerate of divergence µ , and lim | α n − α n − | α n β n = 0 with a monotone rate of conver-gence µ . Let (˜ x n ) be the viscosity-Browder sequence defined with T, φ and thesequence ( α n ) and ( x n ) be defined by (8) . Then d ( x n +1 , ˜ x n ) → with rate ofconvergence Ξ( ε ) := σ [˜ µ , b ] (cid:18) ε, µ (cid:18) δ ε b (cid:19)(cid:19) , where σ is as in Lemma 2.8 , and f µ ( k ) := µ (cid:18)(cid:24) kδ (cid:25)(cid:19) . In particular, if ( ˜ x n ) is a Cauchy sequence with rate of metastability Ψ then ( x n ) is a Cauchy sequence with metastability rate Ω( ε, f ) = Ω[ b, δ, Ψ , µ , µ ]( ε, f ) = max { Ξ( ε/ , Ψ( ε/ , ˆ f ) } + 1 , where ˆ f ( n ) := f (max { Ξ( ε/ , n } + 1) . Proof.
Let (˜ x n ) denote for the viscosity-Browder sequence, i.e.˜ x n = (1 − α n ) T (˜ x n ) ⊕ α n φ (˜ x n ) , n ∈ N . d (˜ x n +1 , ˜ x n ) ≤ d (˜ x n +1 , W ( T (˜ x n ) , φ (˜ x n ) , α n +1 )) + d ( W ( T (˜ x n ) , φ (˜ x n ) , α n +1 ) , ˜ x n ) W2 ≤ d (˜ x n +1 , W ( T (˜ x n ) , φ (˜ x n ) , α n +1 )) + | α n +1 − α n | d ( T (˜ x n ) , φ (˜ x n )) W4 ≤ (1 − α n +1 + α n +1 r ) d (˜ x n +1 , ˜ x n ) + | α n +1 − α n | d ( T (˜ x n ) , φ (˜ x n )) ≤ (1 − δα n +1 ) d (˜ x n +1 , ˜ x n ) + | α n +1 − α n | · b, which implies d (˜ x n +1 , ˜ x n ) ≤ b | α n +1 − α n | δα n +1 . Then, by (8) and using (W1) and (W4), we have for n ≥ d ( x n +1 , ˜ x n ) W1 ≤ (1 − β n ) d ( x n , ˜ x n ) + β n d ((1 − α n ) T ( x n ) ⊕ α n φ ( x n ) , ˜ x n ) W4 ≤ (1 − β n ) d ( x n , ˜ x n ) + β n ((1 − α n ) d ( x n , ˜ x n ) + α n rd ( x n , ˜ x n )) ≤ (1 − δα n β n ) d ( x n , ˜ x n ) ≤ (1 − λ n ) d ( x n , ˜ x n − ) + λ n γ n , where λ n = δα n β n and γ n = b | α n − α n − | δ α n β n .Since µ is a rate of divergence for ( P α n β n ), we have that f µ is a rate ofdivergence for ( P λ n ). Hence by Lemma 2.8 (for an arbitrary p ), we concludethat d ( x n +1 , ˜ x n ) converges to zero with rate of convergence Ξ.Let ε > f : N → N be given. Since Ψ is a rate of metastability for(˜ x n ), there exists n ′ ≤ Ψ( ε/ , ˆ f ) be such that d (˜ x i , ˜ x j ) ≤ ε , for i, j ∈ [ n ′ , ˆ f ( n ′ )].Take n := max { Ξ( ε/ , n ′ ). We have n ≤ Ω( ε, f ) − i, j ∈ [ n, f ( n + 1)] d ( x i +1 , x j +1 ) ≤ d ( x i +1 , ˜ x i ) + d (˜ x i , ˜ x j ) + d ( x j +1 , ˜ x j ) ≤ ε, since [ n, f ( n + 1)] ⊂ [ n ′ , ˆ f ( n ′ )]. This entails the result.Due to Corollary 3.9, we can just use a rate of metastability for the Browdersequence in the previous result (see also Lemma 3.4). Corollary 3.17.
Let
X, C, b, T, φ, r, δ, ( α n ) , ( β n ) , µ , µ , ( x n ) , σ be as in Theo-rem 3.16. Let θ b be a monotone function satisfying (B[ T ]) for ( y n ) the Browdersequence defined with T and the sequence ( α n ) . Then ( x n ) defined by (8) is aCauchy sequence with metastability rate Ω( ε, f ) = Ω[ b, δ, Ψ , µ , µ ]( ε, f ) where Ψ = Ψ[ b, δ, θ b ] is as in Corollary 3.9 , Ω is as in Theorem 3.16 . As before we also get the following corollary24 orollary 3.18.
Let
X, C, b, T, φ, r, δ, ( α n ) , ( β n ) , µ , µ , ( x n ) , σ be as in Theo-rem 3.16. Let ρ ( ε ) be a common (for all anchor points u ∈ C ) rate of conver-gence for the Browder sequence ( y n ( u )) (w.r.t. ( α n ) ) of T. Then ( x n ) definedby (8) is a Cauchy sequence with Cauchy rate Ω( ε ) = Ω[ b, δ, θ b , µ , µ ]( ε ) = max { Ξ( ε/ , Ψ( ε/ } + 1 , where Ξ( ε ) := σ [˜ µ , b ] (cid:18) ε, µ (cid:18) δ ε b (cid:19)(cid:19) ,σ is as in Lemma 2.8 , f µ ( k ) := µ (cid:18)(cid:24) kδ (cid:25)(cid:19) , Ψ = Ψ[ b, δ, θ b ] is as in Corollary 3.10 . In many applications of the algorithms studied above it is useful to allow forthe iteration to contain error terms. In this subsection we provide rates ofmetastability of such relaxed versions under appropriate assumptions on thesequence of error terms. In the following, consider ( S n ) a family of nonexpansivemaps, φ an r -contraction, ( α n ) ⊂ (0 ,
1] and ( ε n ) ⊂ R +0 a sequence of allowederrors. Assume that δ ∈ (0 ,
1) is such that r ≤ − δ , and b ∈ N ∗ is a bound onthe diameter of C . Proposition 3.19.
Let ( x n ) be a ( S n ) -viscosity Browder sequence (5) and ( x ′ n ) be a sequence in C satisfying for all n ∈ N , d ( x ′ n , (1 − α n ) S n ( x ′ n ) ⊕ α n φ ( x ′ n )) ≤ ε n . If lim ε n α n = 0 , then lim d ( x n , x ′ n ) = 0 . Furthermore, if Ψ is a rate of metastabil-ity for ( x n ) and ρ is a rate of convergence for lim ε n α n = 0 , then ρ δ ( ε ) := ρ ( εδ ) is a rate of convergence for lim d ( x n , x ′ n ) = 0 , and Ψ ′ ( ε, f ) = max { ρ δ ( ε/ , Ψ( ε/ , f ρ δ ,ε ) } , with f ρ δ ,ε ( n ) := f (max { ρ δ ( ε/ , n } ) , is a metastability rate for ( x ′ n ) .Proof. Using (W4) we have d ( x n , x ′ n ) ≤ (1 − α n ) d ( S n ( x n ) , S n ( x ′ n )) + α n d ( φ ( x n ) , φ ( x ′ n )) + ε n ≤ (1 − α n ) d ( x n , x ′ n ) + α n rd ( x n , x ′ n ) + ε n . So d ( x n , x ′ n ) ≤ ε n δα n and ρ δ is a rate of convergence for lim d ( x n , x ′ n ) = 0.For a given ε > f : N → N , consider n ′ ≤ Ψ( ε/ , f ρ δ ,ε ) such that d ( x i , x j ) ≤ ε , for all i, j ∈ [ n ′ , f ρ δ ,ε ( n ′ )]; and define n := max { ρ δ ( ε/ , n ′ } .Then n ≤ Ψ ′ ( ε, f ) and d ( x ′ i , x ′ j ) ≤ d ( x ′ i , x i ) + d ( x i , x j ) + d ( x j , x ′ j ) ≤ ε, for all i, j ∈ [ n, f ( n )]. This shows that Ψ ′ is a rate of metastability for ( x ′ n ).25imilar one can study the metastability of a relaxed Halpern type iteration. Proposition 3.20.
Let ( x n ) be a ( S n ) -viscosity Halpern iteration (6) and ( x ′ n ) be a sequence in C satisfying x ′ = x and for all n ∈ N , d ( x ′ n +1 , (1 − α n ) S n ( x ′ n ) ⊕ α n φ ( x ′ n )) ≤ ε n . If P α n = ∞ and lim ε n α n = 0 , then lim d ( x n , x ′ n ) = 0 . Furthermore, if Ψ is arate of metastability for ( x n ) , A is a rate of divergence for ( P α n ) and ρ is arate of convergence for lim ε n α n = 0 , then Γ( ε ) := σ [ ˜ A, b ]( ε, ρ ( δε )) is a rate ofconvergence for lim d ( x n , x ′ n ) = 0 , and Ψ ′ ( ε, f ) = max { Γ( ε/ , Ψ( ε/ , f Γ ,ε ) } , is a metastability rate for ( x ′ n ) , where f Γ ,ε ( n ) := f (max { Γ( ε/ , n } ) , σ is as inLemma 2.8, and ˜ A ( k ) := A (cid:0) ⌈ kδ ⌉ (cid:1) .Proof. We have for all n ∈ N , d ( x n +1 , x ′ n +1 ) ≤ (1 − α n ) d ( S n ( x n ) , S n ( x ′ n )) + α n d ( φ ( x n ) , φ ( x ′ n )) + ε n ≤ (1 − α n ) d ( x n , x ′ n ) + α n rd ( x n , x ′ n ) + ε n ≤ (1 − δα n ) d ( x n , x ′ n ) + δα n ε n δα n . Since P δα n = ∞ and lim ε n δα n = 0, by Lemma 2.7 we conclude that d ( x n , x ′ n )converges to zero. From the fact that A is a rate of divergence for ( P α n )it follows that e A is a rate of divergence for ( P δα n ). From an application ofLemma 2.8 (for an arbitrary p ) it follows that Γ is a rate of convergence towardszero for d ( x n , x ′ n ). One then argues that Ψ ′ is a rate of metastability for ( x ′ n )in the same way as in the proof of Proposition 3.19. Remark 3.21.
One can also conclude lim d ( x n , x ′ n ) = 0 under the assumption P ε n < ∞ , using a different version of Xu’s lemma – e.g. [57, Lemma 2.5] . Inthat case, a metastability rate can easily be obtained using a Cauchy modulusfor ( P ε n ) – see [40, Lemma 3.4] – instead of the convergence rate ρ . We now consider additionally a sequence ( β n ) ⊂ (0 ,
1] and discuss themetastability of the relaxed version of vKM.
Proposition 3.22.
Let ( x n ) be a vKM iteration given by (8) and ( x ′ n ) be asequence in C satisfying x ′ = x and for all n ∈ N , d ( x ′ n +1 , (1 − β n ) x ′ n ⊕ β n ((1 − α n ) T ( x ′ n ) ⊕ α n φ ( x ′ n ))) ≤ ε n . If P α n β n = ∞ and lim ε n α n β n = 0, then lim d ( x n , x ′ n ) = 0. Furthermore, if Ψ isa rate of metastability for ( x n ), A is a rate of divergence for ( P α n β n ) and ρ isa rate of convergence for lim ε n α n β n = 0, then Γ( ε ) := σ [ ˜ A, b ]( ε, ρ ( δε )) is a rateof convergence for lim d ( x n , x ′ n ) = 0, andΨ ′ ( ε, f ) = max { Γ( ε/ , Ψ( ε/ , f Γ ,ε ) } , is a metastability rate for ( x ′ n ), where f Γ ,ε ( n ) := f (max { Γ( ε/ , n } ), σ is as inLemma 2.8, and ˜ A ( k ) := A (cid:0) ⌈ kδ ⌉ (cid:1) . 26 roof. The proof of this result is similar to that of Proposition 3.20.
Remark 3.23.
Similar to
Remark 3.21 , it is also possible to conclude that lim d ( x n , x ′ n ) = 0 under the assumption P ε n < ∞ , and to obtain a rate ofmetastability rate for ( x ′ n ) using a Cauchy modulus for ( P ε n ) .Furthermore, both Propositions 3.20 and can be adapted to use a function A ′ satisfying (QΠ) instead of the rate of divergence A , in which case one makesuse of the function σ from Lemma 2.8 , instead of the function σ . Let ( X, k · k ) be a normed linear space and C ⊆ X some subset. Suppose that T : C → C has at most one fixed point, i.e.(1) ∀ p , p ∈ C ( p = T p ∧ p = T p → p = p ) . We say that T has uniformly at most one fixed point with modulus of uniqueness ω : (0 , ∞ ) → (0 , ∞ ) if(2) ∀ ε > ∀ p , p ∈ C ( k p − T p k , k p − T p k ≤ ω ( ε ) → k p − p k ≤ ε ) . If T is continuous and C is compact, (1) implies the existence of a modulus ω such that (2) but in general (2) is stronger than (1) . However, for largeclasses of uniqueness proofs one can actually extract from a proof of (1) anexplicit effective modulus ω satisfying (2) . We refer to [28] for discussions onall this. Suppose now that we have for some iterative algorithm ( x n ) ⊂ C withlim k x n − T x n k → τ : (0 , ∞ ) → N of convergence, i.e. ∀ ε > ∀ n ≥ τ ( ε ) ( k x n − T x n k ≤ ε ) . If ω is a modulus of uniqueness for T in the sense above, then ρ ( ε ) := τ ( ω ( ε )) isa Cauchy rate for ( x n ) . Hence if X is complete, C closed and T is continuous,then ( x n ) converges to a (unique) fixed point p of T with rate of convergence ρ. In the following we describe a class of nonexpansive operators T for which amodulus ω can be computed if X is uniformly convex. Consequently, we canthen compute rates of convergence for ( x n ) defined by either Krasnoselskii-Mannor by Halpern iterations. Definition 4.1.
An operator A : D ( A ) → X is accretive if for all u ∈ Ax and v ∈ Ay there exists some j ∈ J ( x − y ) such that h u − v, j i ≥ , where J is thenormalized duality mapping of X. Various strengthened forms of this notion have been considered in the liter-ature which guarantee that A has at most one zero (see e.g. [14]): Definition 4.2. (i) Let ψ : [0 , ∞ ) → [0 , ∞ ) be a continuous function with ψ (0) = 0 and ψ ( x ) > for x > . Then an operator A : D ( A ) → X is ψ -strongly accretive if ∀ ( x, u ) , ( y, v ) ∈ A ∃ j ∈ J ( x − y ) ( h u − v, j i ≥ ψ ( k x − y k ) k x − y k ) . ii) Let φ : [0 , ∞ ) → [0 , ∞ ) be a continuous function with φ (0) = 0 and φ ( x ) > for x > . Then an operator A : D ( A ) → X is uniformly φ -accretive if ∀ ( x, u ) , ( y, v ) ∈ A ∃ j ∈ J ( x − y ) ( h u − v, j i ≥ φ ( k x − y k )) . In the case of ψ -strongly accretive operators, ψ is often assumed to be strictlyincreasing in addition (see e.g. [59]).In [31, Section 2.1] it has been exhibited that all what is needed to get a modulusof uniqueness for the property of being a zero of A is the following (which isimplied by the aforementioned concepts of ψ, φ -accretivity): Definition 4.3.
An accretive operator A : D ( A ) → X is uniformly accretive if ( ∗ ) ( ∀ ε, K > ∃ δ > ∀ ( x, u ) , ( y, v ) ∈ A ( k x − y k ∈ [ ε, K ] → ∃ j ∈ J ( x − y ) ( h u − v, j i ≥ δ )) . Any function Θ ( · ) ( · ) : (0 , ∞ ) × (0 , ∞ ) → (0 , ∞ ) such that δ := Θ K ( ε ) satisfies ( ∗ ) for all ε, K > is called a modulus of uniform accretivity for A . Remark 4.4. If A is assumed to have a zero q , then the definition above canbe modified to ‘uniform accretivity at zero’ ( ∗∗ ) ( ∀ ε, K > ∃ δ > ∀ ( x, u ) ∈ A ( k x − q k ∈ [ ε, K ] → ∃ j ∈ J ( x − q ) ( h u, j i ≥ δ )) which still is sufficient to get a modulus of uniqueness for the property of beinga zero of A (see [31, Section 2.1] ). In [6], the following concept of uniform accretivity is introduced and shownto imply the uniqueness of zeroes if X is a uniformly convex normed space (seealso [18] and [59] for results involving this and related notions): Definition 4.5. A is uniformly accretive in the sense of [6] if there is a strictlyincreasing function ϕ : [0 , ∞ ) → R with lim t →∞ ϕ ( t ) = ∞ such that ∀ ( x, u ) , ( y, v ) ∈ A ∃ j ∈ J ( x − y ) ( h u − v, j i ≥ ( ϕ ( k x k ) − ϕ ( k y k )) · ( k x k − k y k )) . Consider in the following a nonexpansive selfmapping T : C → C of abounded convex subset C ⊂ X of a normed space X such that A := I − T isan (single-valued) accretive operator with D ( A ) = C. If A admits a modulusof uniqueness for the property of being a zero, then the same modulus is alsoa modulus of uniqueness for being a fixed point of T. We now extract fromthe uniqueness proof given in [6] a modulus of uniqueness (again in the case ofuniformly convex normed spaces) which only depends on a modulus functionΩ : (0 , ∞ ) → (0 , ∞ ) such that(+) ( ∀ ( x, u ) , ( y, v ) ∈ A ∀ b, ε > ∃ j ∈ J ( x − y )(max {k x k , k y k ≤ b ∧ |k x k − k y k| ≥ ε → h u − v, j i > Ω( ε, b )) . roposition 4.6. If A is uniformly accretive in the sense of [6] , then thereexists a modulus Ω satisfying (+) . Proof.
Let ϕ be as in Definition 4.5. Let b, ε > . It suffices to show that ∃ δ > ∀ x ∈ [0 , b ] ( ϕ ( x + ε ) − ϕ ( x ) > δ )since ϕ ( y ) ≥ ϕ ( x + ε ) for y ≥ x + ε and we can then take Ω( ε, b ) := ε · δ . Supposeotherwise, i.e. ∀ n ∈ N ∃ x n ∈ [0 , b ] ( ϕ ( x n + ε ) − ϕ ( x n ) ≤ n + 1 ) . Let x be a limit point of ( x n ) . Then for arbitrary large n we have x n ≤ x + ε and x + ε ≤ x n + ε and so ϕ (cid:18) x + 2 ε (cid:19) − ϕ (cid:16) x + ε (cid:17) ≤ ϕ ( x n + ε ) − ϕ ( x n ) ≤ n + 1 . Thus ϕ (cid:0) x + ε (cid:1) − ϕ (cid:0) x + ε (cid:1) ≤ ϕ being strictly increasing.In this section, we will compute a modulus of uniqueness for approximatefixed points of T = I − A in terms of a given modulus of uniform convexity for X and a modulus Ω for A. This then gives us explicit rates of convergence forBrowder-type sequences, Krasnoselskii-Mann iterations and Halpern iterationsof T and so - by the results in the previous section - also such rates for theviscosity versions of these algorithms.In the following C ⊂ X is a bounded convex subset of a uniformly convexnormed space ( X, k · k ) . Let b ≥ k x k for all x ∈ C and η : (0 , → (0 ,
1] be amodulus of uniform convexity for X, i.e. ∀ ε ∈ (0 , ∀ x, y ∈ X (cid:18) k x k , k y k ≤ ∧ k x − y k ≥ ε → (cid:13)(cid:13)(cid:13)(cid:13)
12 ( x + y ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ − η ( ε ) (cid:19) . Note that this implication holds trivially if ε > η to(0 , ∞ ) e.g. by putting it := 1 for ε > .T : C → C is a nonexpansive mapping such that A := I − T is uniformlyaccretive in the sense of (+) above with a respective modulus Ω. In the following,we write Ω( ε ) := Ω( ε, b ). Lemma 4.7.
Let x , x ∈ C be Ω( ε )4 b -approximate fixed points of T. Then |k x k − k x k| ≤ ε. Proof.
By assumption we have k A ( x ) k , k A ( x ) k ≤ Ω( ε )4 b . W.l.o.g k x k ≥ k x k . Let j ∈ J ( x − x ) be as in (+) . Assume that k x k − k x k > ε. ThenΩ( ε ) = Ω( ε )2 b · b ≥ ( k Ax k + k Ax k ) · k x − x k≥ k Ax − Ax k · k x − x k ≥ h Ax − Ax , j i > Ω( ε ) . Hence |k x k − k x k| ≤ ε. emma 4.8. If x , x ∈ C are ε · η ( ε/ b )4 -approximate fixed points of T, then x + x is an ε -approximate fixed point of T. If η ( ε ) can be written as ε · ˜ η ( ε ) with < ε ≤ ε ≤ → ˜ η ( ε ) ≤ ˜ η ( ε ) , then wecan replace ε · η ( ε/ b )4 by ε · ˜ η ( ε/ b ) = b · η ( ε/ b ) . Proof.
The proof follows the pattern of the proofs of Lemmas 2.2 and 2.3 in[29]. Claim: ∀ ε > ∀ a, x, y ∈ C (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) a − x + y (cid:13)(cid:13)(cid:13)(cid:13) > K − ε · η ( ε/ b )2 → k x − y k < ε (cid:19) , where K := max {k a − x k , k a − y k} ≤ b. Proof of claim: w.l.o.g we may assume that K ≥ ε/ , since, otherwise, k x − y k < ε + ε = ε. Consider ˜ x := a − xK , ˜ y := a − yK . Then ˜ x, ˜ y ∈ B (0) . Assume that (cid:13)(cid:13)(cid:13)(cid:13) a − x + y (cid:13)(cid:13)(cid:13)(cid:13) > K − ε · η ( ε/ b )2 ≥ K − K · η ( ε/ b )and so (cid:13)(cid:13)(cid:13)(cid:13) ˜ x + ˜ y (cid:13)(cid:13)(cid:13)(cid:13) = 1 K (cid:13)(cid:13)(cid:13)(cid:13) a − x + y (cid:13)(cid:13)(cid:13)(cid:13) > − η ( ε/ b ) . Then by the definition of η K k x − y k = k ˜ x − ˜ y k < ε b ≤ εK . Proof of the lemma: by assumption(1) ^ i =1 k x i − T x i k ≤ ε · η ( ε/ b )4 . Define x := x + x . For i = 2 , we get(2) (cid:13)(cid:13) x − x + T x (cid:13)(cid:13) ≤ k x − x k + k x − T x k≤ k x − x k + k T x − T x k + ε · η ( ε/ b )8 ≤ k x − x k + k x − x k + ε · η ( ε/ b )8 = k x − x k + ε · η ( ε/ b )8 . This implies that(3) (cid:13)(cid:13)(cid:13)(cid:13) x − x + T x (cid:13)(cid:13)(cid:13)(cid:13) ≥ k x − x k − ε · η ( ε/ b )8 , since, otherwise, we get the contradiction k x − x k ≤ (cid:13)(cid:13) x − x + T x (cid:13)(cid:13) + (cid:13)(cid:13) x − x + T x (cid:13)(cid:13) < k x − x k − ε · η ( ε/ b )8 + k x − x k + ε · η ( ε/ b )8 = k x − x k + k x − x k = k x − x k .
30y (1) applied to i = 1 we also have(4) k x − T x k ≤ k
T x − T x k + ε · η ( ε/ b )4 ≤ k x − x k + ε · η ( ε/ b )4 . With K := max {k x − x k , k x − T x k ≤ b, (3) and (4) yield that (cid:13)(cid:13)(cid:13)(cid:13) x − x + T x (cid:13)(cid:13)(cid:13)(cid:13) ≥ k x − x k− ε · η ( ε/ b )8 ≥ K − ε · η ( ε/ b )4 − ε · η ( ε/ b )8 > K − ε · η ( ε/ b )2 . The 1st part of the lemma now follows by the claim with x , x, T x as a, x, y. For the 2nd part, one first has to observe that the proof of the claim alsoestablishes the claim for K · η ( ε/K ) = ε · ˜ η ( ε/K ) ≥ ε · ˜ η ( ε/ b )instead of ε · η ( ε/ b ) . Using this,(1) ′ ^ i = k x i − T x i k ≤ ε · ˜ η ( ε/ b )now implies that k x − T x k < ε as in the proof above.The next lemma is a slight adaption of Lemma 3.2 in [32] (we include thebrief proof for completeness): Lemma 4.9. If x , x ∈ C with k x k ≥ k x k and k x − x k > ε, then (cid:13)(cid:13) x + x (cid:13)(cid:13) < k x k − β ( b, ε ) , where β ( b, ε ) := ε · η ( ε/b ) . If η ( ε ) can be written as ε · ˜ η ( ε ) with < ε ≤ ε ≤ → ˜ η ( ε ) ≤ ˜ η ( ε ) , then wecan replace β by β ′ ( b, ε ) := ε · ˜ η ( ε/b ) = b · η ( ε/b ) . Proof. k x k ≥ k x k and k x − x k > ε implies that(1) k x k > ε > . Define ˜ x := x k x k , ˜ x := x k x k . Then k ˜ x k ≤ k ˜ x k = 1 and(2) k ˜ x − ˜ x k = 1 k x k k x − x k > ε k x k ≥ εb . Since η is a modulus of uniform convexity we get(3) (cid:13)(cid:13)(cid:13)(cid:13) x + x (cid:13)(cid:13)(cid:13)(cid:13) = k x k · (cid:13)(cid:13)(cid:13)(cid:13) ˜ x + ˜ x (cid:13)(cid:13)(cid:13)(cid:13) ≤ k x k − k x k · η ( ε/b ) (1) < k x k − ε · η ( ε/b ) . The 2nd claim follows from(3) ′ (cid:26) (cid:13)(cid:13) x + x (cid:13)(cid:13) = k x k · (cid:13)(cid:13) ˜ x +˜ x (cid:13)(cid:13) < k x k − k x k · η ( ε/ k x k )= k x k − ε · ˜ η ( ε/ k x k ) ≤ k x k − ε · ˜ η ( ε/b ) . roposition 4.10. Let ω b ( ε ) := 116 b Ω (cid:16) ε · η (cid:16) εb (cid:17)(cid:17) · η (cid:18) b Ω (cid:16) ε · η (cid:16) εb (cid:17)(cid:17)(cid:19) . If x , x ∈ C are ω b ( ε ) -approximate fixed points of T, then k x − x k ≤ ε, i.e. ω b is a modulus of uniqueness for being a fixed point of T. If η ( ε ) = ε · ˜ η ( ε ) with ˜ η as in Lemma 4.9, then we can improve ω b to ω b ( ε ) := min ( b · η Ω (cid:0) b · η (cid:0) εb (cid:1)(cid:1) b ! , Ω (cid:0) b · η (cid:0) εb (cid:1)(cid:1) b ) . Proof.
W.l.o.g. k x k ≥ k x k . Assume that k x − x k > ε. By Lemma 4.8 x + x (and by assumption also x since η ( . . . ) ≤
1) is a b Ω (cid:0) ε · η (cid:0) εb (cid:1)(cid:1) -approximatefixed point of T. Hence by Lemma 4.7 (applied to x + x as x ) k x k − (cid:13)(cid:13)(cid:13)(cid:13) x + x (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε · η (cid:16) εb (cid:17) . By Lemma 4.9, on the other hand, we have that (cid:13)(cid:13)(cid:13)(cid:13) x + x (cid:13)(cid:13)(cid:13)(cid:13) < k x k − ε · η (cid:16) εb (cid:17) which gives a contradiction. The proof of the 2nd claim is analogous makinguse of the 2nd claims in lemmas 4.8 and 4.9.A first application of Proposition 4.10 gives a Cauchy rate for the path( x a ) α ∈ (0 , of points in C such that (for given c ∈ C ) x α = (1 − α ) T ( x α ) + αc. Theorem 4.11.
Let
C, X, η, b, T, Ω be as before. Then ∀ ε > ∀ α , α ∈ (0 , ^ i =1 α i ≤ ω b ( ε )2 b → k x α − x α k ≤ ε ! . Proof.
Let for i = 1 , α i ∈ (0 ,
1) be such that α i ≤ ω b ( ε )2 b . Then k T ( x α i ) − x α i k = k T ( x α i ) − T ( x α i ) + α i T ( x α i ) − α i c k = α i k T ( x α i ) − c k ≤ α i · b ≤ ω b ( ε ) . Hence by Proposition 4.10 we get that k x α − x α k ≤ ε. As another application of Proposition 4.10 we get the following quantitativeform of Theorem 1 in [59]:
Theorem 4.12.
Let
C, X, η, b, T, Ω be as before. Let ( β n ) ⊂ [0 , be such that P ∞ n =0 β n (1 − β n ) = ∞ with a rate of divergence γ. Define for x ∈ Cx n +1 := (1 − β n ) x n + β n T x n . hen ( x n ) is a Cauchy sequence with rate ∀ ε > ∀ n, m ≥ γ (cid:18)(cid:24) b π · ( ω b ( ε )) (cid:25)(cid:19) ( k x n − x m k ≤ ε ) , where ω b is the modulus of uniqueness from Proposition 4.10. For X completeand C closed, ( x n ) converges with this rate to the unique fixed point of T. Proof.
By [11] we have for all n (1) k x n − T x n k ≤ b p π · P ni =0 β i (1 − β i ) . Now let n ≥ γ (cid:16)l b π · ( ω b ( ε )) m(cid:17) , then(2) n X i =0 β i (1 − β i ) ≥ b π · ω b ( ε ) and so by (1) k x n − T x n k ≤ ω b ( ε ) . By Proposition 4.10, the claim on the Cauchy modulus follows. If X is completeand C is closed, then - by (1) - the limit p of ( x n ) is a fixed point of T (whichby Proposition 4.10 is unique). Theorem 4.13.
Let
C, X, η, b, T, Ω be as before. Assume that α n ∈ (0 , with lim n →∞ α n = 0 , ∞ X n =1 | α n +1 − α n | converges , ∞ Y n =1 (1 − α n ) = 0 . Let α be a rate of convergence of ( α n ) , β be a Cauchy modulus of s n := P ni =1 | α i +1 − α i | and θ be a rate of convergence of Q ∞ n =1 (1 − α n +1 ) = 0 to-wards . Define the Halpern iteration of T with starting point x ∈ C and anchor u ∈ C by x := x x n +1 := (1 − α n ) T ( x n ) + α n u. Then ( x n ) is a Cauchy sequence with rate ∀ ε > ∀ n, m ≥ Φ( ω b ( ε )) ( k x n − x m k ≤ ε ) , where Φ( ε, b, θ, α, β, D ) = max (cid:26) θ (cid:18) Dε b (cid:19) + 1 , α (cid:16) ε b (cid:17)(cid:27) + 1 , and < D ≤ β ( ε/ b ) Y n =1 (1 − α n +1 ) . For the choice α n := n +1 , we may take Φ( ε, b ) := (cid:24) bε + 32 b ε (cid:25) . roof. The theorem follows from Proposition 4.10 combined with [33, Proposi-tion 6.2,Corollary 6.3].
Acknowledgments:
Both authors have been supported by the GermanScience Foundation (DFG Project KO 1737/6-1).The second author was also supported by FCT - Funda¸c˜ao para a Ciˆenciae a Tecnologia, under the projects UIDB/04561/2020 and UIDP/04561/2020,and the research center CMAFcIO - Centro de Matem´atica, Aplica¸c˜oes Funda-mentais e Investiga¸c˜ao Operacional.
References [1] K. Aoyama and M. Toyoda. Approximation of zeros of accretive operatorsin a Banach space.
Israel J. Math. , 220(2):803–816, 2017.[2] K. Aoyama and M. Toyoda. Approximation of common fixed points ofstrongly nonexpansive sequences in a Banach space.
J. Fixed Point TheoryAppl. , 21(1):1–16, 2019.[3] S. Banach. Sur les op´erations dans les ensembles abstraits et leur applica-tion aux ´equations int´egrales.
Fund. Math. , 3(1):133–181, 1922.[4] H.H. Bauschke. The approximation of fixed points of compositions of non-expansive mappings in Hilbert space.
J. Math. Anal. Appl. , 202(1):150–159,1996.[5] O. Boikanyo and G. Moro¸sanu. Inexact Halpern-type proximal point algo-rithm.
J. Global Optim. , 51(1):11–26, 2011.[6] H. Brezis and M. Sibony. Methodes d’approximation et d’iteration pourles operateurs monotones.
Arch. Rational Mech. Anal. , 28:59–82, 1968.[7] M. Bridson and A. Haefliger.
Metric spaces of non-positive curvature , vol-ume 319. Springer Science & Business Media, 2013.[8] F.E. Browder. Convergence of approximants to fixed points of nonexpan-sive nonlinear mappings in Banach spaces.
Arch. Rational Mech. Anal. ,24(1):82–90, 1967.[9] F. Bruhat and J. Tits. Groupes r´eductifs sur un corps local.
PublicationsMath´ematiques de l’Institut des Hautes ´Etudes Scientifiques , 41(1):5–251,1972.[10] S.-S. Chang. Viscosity approximation methods for a finite family of nonex-pansive mappings in Banach spaces.
J. Math. Anal. Appl. , 323:1402–1416,2006.[11] R. Cominetti, J. Soto and J. Vaisman. On the rate of convergence of Kras-nosel’ski˘ı-Mann iterations and their connection with sums of Bernoullis.
Israel J. Math. , 199:757–772, 2014.[12] B. Dinis and P. Pinto. Quantitative results on the multi-parameters Prox-imal Point Algorithm.
J. Convex Anal. , 28(3), 23pp., 2021.3413] F. Ferreira, L. Leu¸stean and P. Pinto. On the removal of weak compactnessarguments in proof mining.
Adv. Math. , 354:106728, 55pp., 2019.[14] J. Garc´ıa-Falset. The asymptotic behavior of the solutions of theCauchy problem generated by φ -accretive operators. J.Math. Anal. Appl. ,310(2):594–608, 2005.[15] K. Goebel and W.A. Kirk. Iteration processes for nonexpansive mappings.In S. P. Singh, S. Thomeier, and B. Watson, editors,
Topological methodsin nonlinear functional analysis , volume 21 of
Contemporary Mathematics ,115–123. AMS, 1983.[16] K. Goebel and S. Reich. Uniform convexity, hyperbolic geometry, andnonexpansive mappings.
Monographs and Textbooks in Pure and AppliedMathematics , 83, 1984.[17] O. G¨uler. On the convergence of the proximal point algorithm for convexminimization.
SIAM J. Control Optim. , 29(2):403–419, 1991.[18] J. Gwinner. On the convergence of some iteration processes in uniformlyconvex Banach spaces.
Proc. Am. Math. Soc. , 71(1):29–35, 1978.[19] B. Halpern. Fixed points of nonexpanding maps.
Bull. Am. Math. Soc. ,73(6):957–961, 1967.[20] J.S. Jung. Viscosity approximation methods for a family of finite nonex-pansive mappings in Banach spaces.
Nonlinear Anal. , 64(11):2536–2552,2006.[21] J.S. Jung, Y.J. Cho and R.P. Agarwal. Iterative schemes with controlconditions for a family of finite nonexpansive mappings in Banach spaces.
Fixed Point Theory Appl. , 2005(2):125–135, 2005.[22] S. Kamimura and W. Takahashi. Approximating Solutions of MaximalMonotone Operators in Hilbert Spaces.
J. Approx. Theory , 106(2):226–240, 2000.[23] H. Khatibzadeh and S. Ranjbar. On the strong convergence of Halpern typeproximal point algorithm.
J.Optim.Theory Appl. , 158(2):385–396, 2013.[24] W.A. Kirk. Krasnoselskii’s iteration process in hyperbolic space.
Numer.Funct. Anal. Optim. , 4(4):371–381, 1982.[25] W.A. Kirk. Geodesic geometry and fixed point theory. In
Seminar ofmathematical analysis (Malaga/Seville, 2002/2003) , 64:195–225, 2003.[26] U. Kohlenbach. Some computational aspects of metric fixed point theory.
Nonlinear Anal. , 61:823–837, 2005.[27] U. Kohlenbach. Some logical metatheorems with applications in functionalanalysis.
Trans. Am. Math. Soc. , 357(1):89–128, 2005.[28] U. Kohlenbach.
Applied Proof Theory: Proof Interpretations and their Usein Mathematics . Springer Monographs in Mathematics. Springer-VerlagBerlin Heidelberg, 2008. 3529] U. Kohlenbach. On quantitative versions of theorems due to F.E. Browderand R. Wittmann.
Adv. Math. , 226(3):2764–2795, 2011.[30] U. Kohlenbach. Quantitative analysis of a Halpern-type Proximal PointAlgorithm for accretive operators in Banach spaces.
J. Nonlinear ConvexAnal. , 21(9):2125–2138, 2020.[31] U. Kohlenbach and A. Koutsoukou-Argyraki. Rates of convergence andmetastability for abstract Cauchy problems generated by accretive opera-tors.
J. Math. Anal. Appl. , 423:1089–1112, 2015.[32] U. Kohlenbach and L. Leu¸stean. A quantitative mean ergodic theorem foruniformly convex Banach spaces,
Ergod. Theory & Dyn. Sys. , 29:1907–1915, 2009.[33] U. Kohlenbach and L. Leu¸stean. Effective metastability of Halpern iteratesin CAT(0) spaces.
Adv. Math. , 231:2526–2556, 2012. Addendum in:
Adv.Math. , 250:650–651, 2014.[34] U. Kohlenbach and L. Leu¸stean. On the computational content of conver-gence proofs via Banach limits.
Philos. Trans. R. Soc. A , 370:3449-3463,2012.[35] U. Kohlenbach and P. Oliva. Proof mining: a systematic way of analysingproofs in mathematics.
Proc. Steklov Inst. Math. , 242:136–164, 2003.[36] U. Kohlenbach and A. Sipo¸s. The finitary content of sunny nonexpansiveretractions.
Comm. Contemp. Math. , 23(1):1950093, 63pp., 2021.[37] D. K¨ornlein. Quantitative results for Halpern iterations of nonexpansivemappings.
J. Math. Anal. Appl. , 428(2):1161-1172, 2015.[38] D. K¨ornlein. Quantitative Analysis of Iterative Algorithms in Fixed PointTheory and Convex Optimization. PhD Thesis, TU Darmstadt 2016.[39] D. K¨ornlein. Quantitative strong convergence for the hybrid steepest de-scent method. arXiv:1610.00517v1, 2016.[40] L. Leu¸stean and P. Pinto. Quantitative results on Halpern type proximalpoint algorithms. To appear in:
Comput. Optim. Appl. , 24pp., 2021.[41] B. Martinet. R´egularisation d???in´equations variationnelles par approxi-mations successives. Rev. Fran¸caise Informat.
Recherche Op´erationnelle ,4:154–158, 1970.[42] A. Meir and E. Keeler. A theorem on contraction mappings.
J. Math. Anal.Appl. , 28(2):326–329, 1969.[43] A. Moudafi. Viscosity approximation methods for fixed-points problems.
J. Math. Anal. Appl. , 241(1):46–55, 2000.[44] E. Neumann. Computational problems in metric fixed point theory andtheir Weihrauch degrees.
Log. Method. Comput. Sci.
11, 44 pp., 2015.[45] P. Pinto. A rate of metastability for the Halpern type Proximal PointAlgorithm. To appear in:
Numer. Funct. Anal. Optim. , 19pp., 2021.3646] E. Rakotch. A note on contractive mappings.
Proc. Am. Math. Soc. ,13(3):459–465, 1962.[47] S. Reich. Strong convergence theorems for resolvents of accretive operatorsin Banach spaces.
J. Math. Anal. Appl. , 75(1):287–292, 1980.[48] S. Reich and I. Shafrir. Nonexpansive iterations in hyperbolic spaces.
Non-linear Anal. Theory Methods Appl. , 15(6):537–558, 1990.[49] R. Rockafellar. Monotone operators and the proximal point algorithm.
SIAM J. Control Optim. , 14(5):877???898, 1976.[50] N. Shioji and W. Takahashi. Strong convergence of approximated se-quences for nonexpansive mappings in Banach spaces.
Proc. Am. Math.Soc. , 125:3641–3645, 1997.[51] T. Suzuki. Moudafi’s viscosity approximations with Meir-Keeler contrac-tions.
J. Math. Anal. Appl. , 325(1):342–352, 2007.[52] T. Suzuki. Some notes on Bauschke’s condition.
Nonlinear Anal. TheoryMethods Appl. , 67.7: 2224–2231, 2007.[53] W. Takahashi. A convexity in metric space and nonexpansive mappings, I.In
Kodai Mathematical Seminar Reports , 22:142–149, 1970.[54] T. Tao. Soft analysis, hard analysis, and the finite convergence principle.Essay posted May 23, 2007. In: ‘T. Tao, Structure and Randomness: Pagesfrom Year One of a Mathematical Blog’. AMS, 298pp., 2008.[55] R. Wittmann. Approximation of fixed points of nonexpansive mappings.
Arch. Math. , 58(5):486–491, 1992.[56] H.-K. Xu. Another control condition in an iterative method for nonexpan-sive mappings.
Bull. Aust. Math. Soc. , 65(1):109–113, 2002.[57] H.-K. Xu. Iterative algorithms for nonlinear operators.
J. London Math.Soc. , 66(1):240–256, 2002.[58] H.-K. Xu. Viscosity approximation methods for nonexpansive mappings.
J. Math. Anal. Appl. , 298(1):279–291, 2004.[59] H.-K.Xu, N .Altwaijry and S .Chebbi. Strong convergence of Mann’s iter-ation process in Banach spaces.
Mathematics , 8(6):954, 2020.[60] I. Yamada. The hybrid steepest descent method for the variational in-equality problem over the intersection of fixed point sets of nonexpansivemappings.