Firmly nonexpansive mappings in classes of geodesic spaces

Firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators. In this paper we do a thorough study of fixed point theory and the asymptotic behaviour of Picard iterates of these mappings in different classes of geodesic spaces, as (uniformly convex) $W$-hyperbolic spaces, Busemann spaces and CAT(0) spaces. Furthermore, we apply methods of proof mining to obtain effective rates of asymptotic egularity for the Picard iterations.


Introduction
Let C be a closed convex subset of a Hilbert space H. Firmly contractive mappings were defined by Browder [5] as mappings T : C → H satisfying the following inequality for all x, y ∈ C: T x − T y 2 ≤ x − y, T x − T y .
As Browder points out, these mappings play an important role in the study of (weak) convergence for sequences of nonlinear operators. An example of a firmly contractive mapping is the metric projection P C : H → H, defined by P C (x) = argmin y∈C { x − y }. One can easily see that any firmly contractive mapping T is nonexpansive, i.e. satisfies T x−T y ≤ x−y for all x, y ∈ C. The converse is not true, as one can see by taking T = −Id.
In his study of nonexpansive projections on subsets of Banach spaces, Bruck [6] defined a firmly nonexpansive mapping T : C → E, where C is a closed convex subset of a real Banach space E, to be a mapping with the property that for all x, y ∈ C and t ≥ 0, In Hilbert spaces these mappings coincide with the firmly contractive ones introduced by Browder. As Bruck shows, to any nonexpansive selfmapping T : C → C that has fixed points, one can associate a 'large' family of firmly nonexpansive mappings having the same fixed point set with T . Hence, from the point of view of the existence of fixed points on convex closed sets, firmly nonexpansive mappings exhibit a similar behaviour with the nonexpansive ones. However, this is not anymore true if we consider non-convex domains [43]. Firmly nonexpansive mappings in Banach spaces have also been studied in [7] and [39].
If T is firmly nonexpansive and has fixed points, it is well known [5] that the Picard iterate (T n x) converges weakly to a fixed point of T for any starting point x, while this is not true for nonexpansive mappings (take again T = −Id). This is a first reason for the importance of firmly nonexpansive mappings.
A second reason for the importance of this class of mappings is their correspondence with maximal monotone operators, due to Minty [34].
The resolvent of a monotone operator was introduced by Minty [34] in Hilbert spaces and by Brézis, Crandall and Pazy [3] in Banach spaces. Among other applications, the resolvent has proved to be very useful in the study of the asymptotic behaviour of the solutions of the Cauchy abstract problem governed by a monotone operator, see for instance [16,36,47]. Given a maximal monotone operator A : H → 2 H and µ > 0, its associated resolvent of order µ, defined by J A µ := (Id + µA) −1 , is a firmly nonexpansive mapping from H to H and the set of fixed points of J A µ coincides with the set of zeros of A. We refer to [2] for a very nice presentation of this correspondence. Rockafellar's [42] proximal point algorithm uses the resolvent to approximate the zeros of maximal monotone operators.
The subdifferential of a proper, convex and lower semicontinuous function F : H → (−∞, ∞] is a maximal monotone operator, hence the resolvent associated to the subdifferential is a firmly nonexpansive mapping, that coincides with the proximal map introduced by Moreau [35]. The proximal point algorithm for approximating the minimizers of F is based on the weak convergence towards a fixed point of the Picard iterate of the resolvent and the fact that the minimizers of F are the fixed points of the resolvent. In the last 20 years a fruitful direction of research consists of extending techniques and results obtained in normed spaces to metric spaces without linear structure. For instance minimization problems associated to convex functionals have been solved in the setting of Riemannian manifolds [15,31], while some problems have been modelled as abstract Cauchy equations in the framework of nonpositive curvature geodesic metric spaces (see [33,45] and references therein). Although apparently the framework and the conceptual approach in the previous problems are quite different, it is possible, as in the case of normed spaces, to find a bridge between them through firmly nonexpansive mappings.
The goals of our work are twofold. First we generalize known results on firmly nonexpansive mappings in Hilbert or Banach spaces to suitable classes of geodesic spaces. Second we obtain effective results on the asymptotic behaviour of Picard iterations.
In Section 2 we give basic definitions and properties of the classes of geodesic spaces we consider in this paper: W -hyperbolic spaces, UCW -hyperbolic spaces, Busemann spaces and CAT(0) spaces. We recall properties of asymptotic centers in such spaces, that are essential for our results.
Firmly nonexpansive mappings in the Hilbert ball and, more generally, in hyperbolic spaces, have already been studied in [18,40,41] and, more recently, in the paper by Kopecká and Reich [27]. In Section 3 we extend Bruck's definition of firmly nonexpansive mapping to our class of W -hyperbolic spaces. We show that, in the setting of CAT(0) spaces, the metric projection on a closed convex set and the resolvent of a proper, convex and lower semicontinuous mapping are firmly nonexpansive. Furthermore, Bruck's association of a family of firmly nonexpansive mappings to any nonexpansive mapping is adapted to Busemann spaces. Section 4 contains a fixed point theorem for firmly nonexpansive mappings defined on finite unions of closed convex subsets of a complete UCW -hyperbolic space. Our result generalizes and strengthens Smarzewski's [43] fixed point theorem for uniformly convex Banach spaces. In this section we also obtain new results about periodic points of (firmly) nonexpansive mappings.
In the next section we study the asymptotic behaviour of Picard iterates of firmly nonexpansive mappings, extending to W -hyperbolic spaces results of Reich and Shafrir [40,41]. As a consequence, we get that any firmly nonexpansive mapping with bounded orbits is asymptotically regular.
A concept of weak convergence in geodesic spaces is the so-called ∆-convergence, defined by Lim [32]. Applying our asymptotic regularity result and general properties of Fejér monotone sequences, we prove in Section 6, in the setting of complete UCW -hyperbolic spaces, the ∆-convergence of Picard iterates of a firmly nonexpansive mapping to a fixed point. As a consequence, one gets the ∆-convergence of a proximal point like algorithm to a minimizer of a proper, convex and lower semicontinuous mapping defined on a CAT(0) space.
In the final section of the paper we obtain effective rates of asymptotic regularity for Pi-card iterations, applying methods of proof mining, similar to the ones used for Krasnoselski-Mann iterations of nonexpansive mappings by Kohlenbach [23] in Banach spaces and the second author [29] in UCW -hyperbolic spaces. We point out that our results are new even for uniformly convex Banach spaces. In the case of CAT(0) spaces we obtain a quadratic rate of asymptotic regularity. Proof mining is a paradigm of research concerned with the extraction, using tools from mathematical logic, of hidden finitary and combinatorial content, such as algorithms and effective bounds, from proofs that make use of highly infinitary principles. We refer to Kohlenbach's book [25] for details.

Classes of geodesic spaces -definitions and properties
The convexity mapping W was first considered by Takahashi in [46], where a triple (X, d, W ) satisfying (W 1) is called a convex metric space. W -hyperbolic spaces were introduced by Kohlenbach [24] and we refer to [25, p.384] for a comparison between them and other notions of 'hyperbolic space' that can be found in the literature (see for example [21,17,41]). The class of W -hyperbolic spaces includes (convex subsets of) normed spaces, the Hilbert ball (see [18] for a book treatment) as well as CAT (0) spaces [4].
We shall denote a W -hyperbolic space simply by X, when the metric d and the mapping W are clear from the context. One can easily see that d(x, W (x, y, λ)) = λd(x, y) and d(y, W (x, y, λ)) = (1 − λ)d(x, y). ( Furthermore, W (x, y, 0) = x, W (x, y, 1) = y and W (x, x, λ) = x. Let us recall now some notions concerning geodesics. Let (X, d) be a metric space. A geodesic path in X (geodesic in X for short) is a map γ : A geodesic segment in X is the image of a geodesic in X. If γ : [a, b] → X is a geodesic in X, γ(a) = x and γ(b) = y, we say that the geodesic γ joins x and y or that the geodesic segment γ([a, b]) joins x and y; x and y are also called the endpoints of γ.
A metric space (X, d) is said to be a (uniquely) geodesic space if every two distinct points are joined by a (unique) geodesic segment.
If γ([a, b]) is a geodesic segment joining x and y and λ ∈ [0, 1], z : In the sequel, we shall use the notation [x, y] for the geodesic segment γ([a, b]) and we shall denote this z by (1 − λ)x ⊕ λy, provided that there is no possible ambiguity. Given three points x, y, z in a metric space (X, d), we say that y lies between x and z if these points are pairwise distinct and if we have d(x, z) = d(x, y) + d(y, z). Obviously, if y lies between x and z, then y also lies between z and x. Furthermore, the relation of betweenness satisfies also a transitivity property (see, e.g., [38, Proposition 2.2.13]): Proposition 2.1. Let X be a metric space and x, y, z, w be pairwise distinct points of X. The following statements are equivalent: (i) y lies between x and z and z lies between x and w.
(ii) y lies between x and w and z lies between y and w.
The following betweenness property expresses another form of 'transitivity', which is not true in general metric spaces: for all x, y, z, w ∈ X, if y lies between x and z and z lies between y and w, then y and z lie both between x and w.
By induction one gets Lemma 2.2. Let X be a metric space satisfying (6). For all n ≥ 2 and all x 0 , x 1 , . . . , x n ∈ X, we have that if for all k = 1, . . . , n − 1, x k lies between x k−1 and x k+1 , then for all k = 1, . . . , n − 1, x k lies between x 0 and x k+1 .
The next lemma collects some well-known properties of geodesic spaces. We refer to [38] for details.  (iii) For every geodesic segment [x, y] in X and λ,λ ∈ [0, 1], (iv) Let γ : [a, b] → X be a geodesic that joins x and y. Define Then γ − is a geodesic that joins y and x such that (vi) The following two statements are equivalent: (a) X is uniquely geodesic.
Let (X, d, W ) be a W -hyperbolic space. For all x, y ∈ X, let us define Open and closed balls are convex sets. A nice feature of our setting is that any convex subset is itself a W -hyperbolic space.
Following [46], we call a W -hyperbolic space strictly convex if for any x = y ∈ X and any λ ∈ (0, 1) there exists a unique element z ∈ X (namely z = W (x, y, λ)) such that Proposition 2.5. Let (X, d, W ) be a W -hyperbolic space. Then (i) X is a geodesic space and for all x = y ∈ X, [x, y] W is a geodesic segment joining x and y.
(ii) X is a uniquely geodesic space if and only if it is strictly convex.
(iii) If X is uniquely geodesic, then (a) W is the unique convexity mapping that makes (X, d, W ) a W -hyperbolic space.
is a geodesic satisfying W xy An important class of W -hyperbolic spaces are the so-called Busemann spaces, used by Busemann [9,10] to define a notion of 'nonpositively curved space'. We refer to [38] for an extensive study. Let us recall that a map γ : [a, b] → X is an affinely reparametrized geodesic if γ is a constant path or there exist an interval [c, d] and a geodesic γ ′ : A geodesic space (X, d) is a Busemann space if for any two affinely reparametrized geodesics γ : is convex. Examples of Busemann spaces are strictly convex normed spaces. In fact, a normed space is a Busemann space if and only if it is strictly convex.
Proposition 2.6. Let (X, d) be a metric space. The following two statements are equivalent: , any Busemann space is uniquely geodesic. For any x, y ∈ X, let [x, y] be the unique geodesic segment that joins x and y and define CAT(0) spaces are another very important class of W -hyperbolic spaces. A CAT(0) space is a geodesic space satisfying the CN inequality of Bruhat-Tits [8]: for all x, y, z ∈ X and We refer to [4, p. 163] for a proof that the above definition is equivalent with the one using geodesic triangles. In the setting of W -hyperbolic spaces, we consider the following reformulation of the CN inequality: for all x, y, z ∈ X, We refer to [25, p. 386-388] for the proof of the following result.
Proposition 2.8. Let (X, d) be a metric space. The following statements are equivalent: (14).
Proposition 2.9. Any uniformly convex W -hyperbolic space is a Busemann space.
Following [30], we shall refer to uniformly convex W -hyperbolic spaces with a monotone modulus of uniform convexity as UCW -hyperbolic spaces. Furthermore, we shall also use the notation (X, d, W, η) for a UCW -hyperbolic space having η as a monotone modulus of uniform convexity.
As it was proved in [29], CAT (0) spaces are UCW -hyperbolic spaces with a modulus of uniform convexity η(r, ε) = ε 2 8 , that does not depend on r and is quadratic in ε. In particular, any CAT (0) space is also a Busemann space.
The following lemma collects some useful properties of UCW -hyperbolic spaces. We refer to [29,30] for the proofs.
and a, x, y ∈ X are such that d(x, a) ≤ r, d(y, a) ≤ r and d(x, y) ≥ εr.

Asymptotic centers
One of the most useful tools in metric fixed point theory is the asymptotic center technique, introduced by Edelstein [11,12]. Let (X, d) be a metric space, (x n ) be a bounded sequence in X and C ⊆ X be a nonempty subset of X. We define the following functional: The asymptotic radius of (x n ) with respect to C is given by A point c ∈ C is said to be an asymptotic center of (x n ) with respect to C if We denote with A(C, (x n )) the set of asymptotic centers of (x n ) with respect to C. When C = X, we call c an asymptotic center of (x n ) and we use the notation A((x n )) for A(X, (x n )).
The following lemma will be very useful in the sequel.
Assume that y ∈ C is such that there exist p, N ∈ N satisfying Then y = c.
A classical result is the fact that in uniformly convex Banach spaces, bounded sequences have unique asymptotic centers with respect to closed convex subsets. For the Hilbert ball, this was proved in [18,Proposition 21.1]. The following result shows that the same is true for complete UCW -hyperbolic spaces.
Proposition 2.12. [30] Let (X, d, W ) be a complete UCW -hyperbolic space. Every bounded sequence (x n ) in X has a unique asymptotic center with respect to any nonempty closed convex subset C of X.

Convex functions
Let (X, d) be a geodesic space and F : X → (−∞, ∞]. The mapping F is said to be convex if, for any geodesic γ in X, the function F • γ is convex. Let us recall that the effective domain of F is the set dom F := {x ∈ X | F (x) < ∞} and that F is proper if dom F is nonempty.
For the rest of this section F is a proper convex function. If x ∈ dom F and γ : [0, c] → X is a geodesic starting at x, the directional derivative D γ F (x) of F at x in the direction γ is defined by As F is convex, the above limit (possibly infinite) always exist. Indeed, one can easily see Proof. (i) ⇒ (ii) Let ε > 0 be such that F (x) ≤ F (x) for all x ∈ B(x, ε). Let z = x be arbitrary and γ : [0, d(x, z)] → X be a geodesic in X that joins x and z. For all t < min{ε, d(x, z)} we have that d(γ(t), x) = t < ε, so that . If F (z) = ∞, the conclusion is obvious, so we can assume that z ∈ dom F . Let γ : [0, d(x, z)] → X be a geodesic that joins x and z. As F is convex, one gets that for all t ∈ [0, d(x, z)],

Firmly nonexpansive mappings
Firmly nonexpansive mappings were introduced by Bruck [6] in the context of Banach spaces and by Browder [5], under the name of firmly contractive, in the setting of Hilbert spaces. We refer to [18,Section 24] for a study of this class of mappings in the Hilbert ball. Bruck's definition can be extended to W -hyperbolic spaces. Let (X, d, W ) be a Whyperbolic space, C ⊆ X and T : C → X. Given λ ∈ (0, 1), we say that T is λ-firmly nonexpansive if for all x, y ∈ C, If (16) holds for all λ ∈ (0, 1), then T is said to be firmly nonexpansive.
Applying (W4) one gets that any λ-firmly nonexpansive mapping is nonexpansive, i.e. it satisfies d(T x, T y) ≤ d(x, y) for all x, y ∈ C.
The first example of a firmly nonexpansive mapping is the metric projection in a CAT(0) space. Let us recall that a subset C of a metric space (X, d) is called a Chebyshev set if to each point x ∈ X there corresponds a unique point z ∈ C such that d(x, z) = d(x, C), where d(x, C) = inf{d(x, y) | y ∈ C}. If C is a Chebyshev set, the metric projection P C : X → C can be defined by assigning z to x.
By [4, Proposition II.2.4], any closed convex subset C of a CAT(0) space is a Chebyshev set, the metric projection P C is nonexpansive and P C ((1 − λ)x ⊕ λP C x) = P C (x) for all x ∈ X and all λ ∈ (0, 1). It is well known that in the setting of Hilbert spaces the metric projection is firmly nonexpansive. We remark that for the Hilbert ball this was proved in [18, p. 111]. The following result shows that the same holds in general CAT(0) spaces. Proposition 3.1. Let C be a nonempty closed convex subset of a CAT(0) space (X, d). The metric projection P C onto C is a firmly nonexpansive mapping.
Proof. Let x, y ∈ X and λ ∈ (0, 1). One gets that Bruck [6] showed for Banach spaces that one can associate to any nonexpansive mapping a family of firmly nonexpansive mappings having the same fixed points. Goebel and Reich [18] obtained the same result for the Hilbert ball. We show in the sequel that Bruck's construction can be adapted also to Busemann spaces.
Let C be a nonempty closed convex subset of a complete Busemann space X and T : C → C be nonexpansive. For t ∈ (0, 1) and x ∈ C define Using (W4), one can easily see that T x t is a contraction, so it has a unique fixed point z x t ∈ C, by Banach's Contraction Mapping Principle. Let Proposition 3.2. U t is a firmly nonexpansive mapping having the same set of fixed points as T .
Proof. Let λ ∈ (0, 1) and x, y ∈ C. Denote u := (1−λ)x⊕λU t (x) and v := (1−λ)y ⊕λU t (y). We can apply Lemma 2.4.(iii) twice to get that It follows that The fact that U t and T have the same set of fixed points is immediate.
A third example of a firmly nonexpansive mapping is the resolvent of a proper, convex and lower semicontinuous mapping in a CAT(0) space.
Let (X, d) be a CAT(0) space, F : X → (−∞, ∞] and µ > 0. Following Jost [20], the Moreau-Yosida approximation F µ of F is defined by We refer to [1,45] for applications of the Moreau-Yosida approximation in CAT(0) spaces. Jost proved [20, Lemma 2] that if F : X → (−∞, ∞] is proper, convex and lower semicontinuous, then for every x ∈ X and µ > 0, there exists a unique y µ ∈ X such that We denote this y µ with J µ (x) and call J µ the resolvent of F of order µ.

A fixed point theorem
Given a subset C of a metric space (X, d), a nonexpansive mapping T : C → C and x ∈ C, the orbit O(x) of x under T is defined by O(x) = {T n x | n = 0, 1, 2, . . .}. As an immediate consequence of the nonexpansiveness of T , if O(x) is bounded for some x ∈ C, then all other orbits O(y), y ∈ C are bounded. If this is the case, we say that T has bounded orbits. Obviously, if T has fixed points, then T has bounded orbits.
In this section we prove the following fixed point theorem. C k be a union of nonempty closed convex subsets C k of X, and T : C → C be λ-firmly nonexpansive for some λ ∈ (0, 1). The following two statements are equivalent: (i) T has bounded orbits.
(ii) T has fixed points.
Let us remark that fixed points are not guaranteed if T is merely nonexpansive, as the following trivial example shows. Let x = y ∈ X, take C 1 = {x}, C 2 = {y}, C = C 1 ∪ C 2 and T : C → C, T (x) = y, T (y) = x. Then T is fixed point free and nonexpansive. If T were λ-firmly nonexpansive for some λ ∈ (0, 1), we would get that is a contradiction.
As an immediate consequence, we get a strengthening of Smarzewski's fixed point theorem for uniformly convex Banach spaces [43], obtained by weakening the hypothesis of C k being bounded for all k = 1, . . . , p to T having bounded orbits. C k be a union of nonempty closed convex subsets C k of X, and T : C → C be λ-firmly nonexpansive for some λ ∈ (0, 1).
Then T has fixed points if and only if T has bounded orbits.  Proof. Let x be a periodic point of T and m ≥ 0 be minimal with the property that T m+1 x = x. If m = 0, then x is a fixed point of T , hence we can assume that m ≥ 1. Since T is nonexpansive, we have hence we must have equality everywhere, that is since T m x = x, by the hypothesis on m. Applying now the fact that T is λ-firmly nonexpansive, we get for all k = 1, . . . , m Hence, we must have where We have the following cases: (i) m = 1, hence k = 1. Then T m−1 x = x and It follows by (W2) that hence |2λ − 1| = 1, which is impossible, since λ ∈ (0, 1).
(ii) m ≥ 2, hence m − 1 ≥ 1. Since T k x lies between β k and α k and, furthermore, α k lies between T k x and T k+1 x, we can apply Lemma 2.7 twice to get firstly that T k x lies between β k and T k+1 x and secondly, since β k lies between T k−1 x and T k x, that T k x lies between T k−1 x and T k+1 x for all k = 1, . . . , m.
Apply now Lemma 2.2 to conclude that T m−1 x lies between x and T m x, hence We have got a contradiction.
We remark that Proposition 4.3 holds for strictly convex Banach spaces too, as they are Busemann spaces. C k be a union of nonempty subsets C k of X, and T : C → C be nonexpansive. Assume that T has bounded orbits and that for some z ∈ C, the orbit (T n z) of T has a unique asymptotic center x k with respect to every C k , k = 1, . . . , p.
Then one of x k , k = 1, . . . , p is a periodic point of T .
Proof. Since T is nonexpansive, we have that If there exists k 0 ∈ {1, . . . , p} such that T x k 0 ∈ C k 0 , then applying Lemma 2.11 with y = T x k 0 , p = 1, α n = 1, β n = 0 and x n = T n z, we have that T x k 0 = x k 0 , that is, x k 0 is a fixed point of T . In particular, x k 0 is a periodic point of T . Otherwise, assume that T x k ∈ C k for all 1 ≤ k ≤ p. It is easy to see that there exist integers {n 1 , n 2 , . . . , n m } ⊆ {1, 2, . . . , p}, with m ≥ 2, such that T x n k ∈ C n k+1 for all k = 1, . . . , m − 1 and T x nm ∈ C n 1 .
Applying repeatedly (24) and the fact that x n k is the unique asymptotic center of (T n z) with respect to C n k , we get that r(x n 1 , (T n z)) ≤ r(T x nm , (T n z)) ≤ r(x nm , (T n z)}) ≤ . . . ≤ r(T x n 1 , (T n z)) ≤ r(x n 1 , (T n z)).
Thus, we must have equality everywhere. We get that r(x n 1 , (T n z)) = r(T x nm , (T n z)) and r(T x n k , (T n z)) = r(x n k+1 , (T n z)) for all k = 1, . . . , m − 1. By the uniqueness of the asymptotic centers, we get that x n 1 = T x nm and T x n k = x n k+1 for all k = 1, . . . , m − 1.
It follows that T m x n 1 = x n 1 , hence x n 1 is a periodic point of T . C k be a union of nonempty closed convex subsets C k of X, and T : C → C be a nonexpansive mapping having bounded orbits. Then T has periodic points.
Proof. By Proposition 2.12, for all z ∈ C and for all k = 1, . . . , p, the orbit (T n z) has a unique asymptotic center x k with respect to C k . Apply Lemma 4.4 to get that one of the asymptotic centers x k , k = 1, . . . , p is a periodic point of T .

Asymptotic behaviour of Picard iterations
The second main result of the paper is a theorem on the asymptotic behaviour of Picard iterations of λ-firmly nonexpansive mappings, which generalizes results obtained by Reich and Shafrir [40] for firmly nonexpansive mappings in Banach spaces and the Hilbert ball.
Theorem 5.1. Let C be a subset of a W-hyperbolic space X and T : C → C be a λ-firmly nonexpansive mapping with λ ∈ (0, 1). Then for all x ∈ X and k ∈ Z + , The mapping T is said to be asymptotically regular at x ∈ C if lim n→∞ d(T n x, T n+1 x) = 0. If this is true for all x ∈ C, we say that T is asymptotically regular.
Before proving Theorem 5.1, we give the following immediate consequences.
Corollary 5.2. The following statements are equivalent: (i) T is asymptotically regular at some x ∈ C.
(iii) T is asymptotically regular.
Corollary 5.3. If T has bounded orbits, then T is asymptotically regular.
Remark 5.4. As Adriana Nicolae pointed out to us in a private communication, one can easily see that Proposition 4.3 is an immediate consequence of the above corollary. However, our proof of this proposition holds (with small adaptations) also in more general spaces like geodesic spaces with the betweenness property (see [37]), for which it is not known whether Corollary 5.3 is true.

Proof of Theorem 5.1
In the sequel, X is a W -hyperbolic space, C ⊆ X and T : C → C.
Lemma 5.5. Assume that T is nonexpansive and x ∈ C. (iii) L ≤ r C (T ) ≤ R 1 .
Proof. (i) Since the sequence (d(T n+k x, T n x)) n is nonincreasing, obviously R k exists. Remark that and let n → ∞ to conclude that R k ≤ kR 1 .
(ii) One has that for all m, n ≥ 1, hence the sequence (d(T n x, x)) is subadditive. Apply now Fekete's subadittive lemma [14] to get that L = inf n≥1 d(T n x, x) n . The independence of x follows from the fact that for all x, y ∈ C, < r C (T ) + ε.
As ε > 0 was arbitrary, we get that L ≤ r C (T ).
Lemma 5.6. Let T be λ-firmly nonexpansive for some λ ∈ (0, 1). Then for all x, y ∈ C, Proof. Apply (W1) more times to get that Proof of Theorem 5.1 We prove that R k = kR 1 for all k ≥ 1 by induction on k. Assume that R j = jR 1 for all j = 1, . . . , k and let ε > 0. Since (d(T n+j x, T n x)) is nonincreasing, we get N ε ≥ 1 such that for any j = 1, . . . , k and for all n ≥ N ε , Let n ≥ N ε . By (26), we get that By letting n → ∞, it follows that R k+1 ≥ (k + 1)R 1 , as ε > 0 is arbitrary. Apply now Lemma 5.5.(i) to conclude that R k+1 = (k + 1)R 1 .
Since d(T n+k x, T n x) ≤ d(T k x, x) for all k, n ≥ 1, let n → ∞ to get that R 1 ≤ d(T k x, x) k for all k ≥ 1 and, as a consequence, R 1 ≤ L. Apply now Lemma 5.5.(iii) to conclude that L = R 1 = r C (T ).

∆-convergence of Picard iterates
In 1976, Lim [32] introduced a concept of convergence in the general setting of metric spaces, which is known as ∆-convergence. Kuczumow [28] introduced an identical notion of convergence in Banach spaces, which he called almost convergence. As shown in [22], ∆-convergence could be regarded, at least for CAT(0) spaces, as an analogue to the usual weak convergence in Banach spaces. Jost [19] introduced a notion of weak convergence in CAT(0) spaces, which was rediscovered by Espínola and Fernández-León [13], who also proved that it is equivalent to ∆-convergence. We refer to [44] for other notions of weak convergence in geodesic spaces.
Let (x n ) be a bounded sequence of a metric space (X, d). We say that (x n ) ∆-converges to x if x is the unique asymptotic center of (u n ) for every subsequence (u n ) of (x n ). In this case, we write x n ∆ −→ x or ∆ − lim n→∞ x n = x and we call x the ∆-limit of (x n ).
Let (X, d) be a metric space and F ⊆ X be a nonempty subset. A sequence (x n ) in X is said to be Fejér monotone with respect to F if d(p, x n+1 ) ≤ d(p, x n ) for all p ∈ F and n ≥ 0.
Thus each point in the sequence is not further from any point in F than its predecessor. Obviously, any Fejér monotone sequence (x n ) is bounded and moreover (d(x n , p)) converges for every p ∈ F . The following lemma is very easy to prove. (ii) Every subsequence (u n ) of (x n ) is Fejér monotone with respect to F and for all p ∈ F , r(p, (u n )) = r(p, (x n )). Hence, r(F, (u n )) = r(F, (x n )) and A(F, (u n )) = A(F, (x n )).
(iii) If A(F, (x n )) = {x} and A((u n )) ⊆ F for every subsequence (u n ) of (x n ), then (x n ) ∆-converges to x ∈ F .
Furthermore, one has the following result, whose proof is very similar to the one in strictly convex Banach spaces. For the sake of completeness, we give it here. Lemma 6.2. Let C be a nonempty closed convex subset of a uniquely geodesic space (X, d) and T : C → C be nonexpansive. Then the set F ix(T ) of fixed points of T is closed and convex.
Proof. The fact that F ix(T ) is closed is immediate from the continuity of T . We shall prove its convexity. Let x, y ∈ F ix(T ) be distinct and z ∈ [x, y]. Then Thus, d(x, T z) + d(T z, y) = d(x, y), so that T z ∈ [x, y]. We apply now Lemma 2.3.(ii) to get the following cases: In both cases, it follows that T z = z. Proposition 6.3. Let (X, d, W ) be a complete UCW -hyperbolic space, C ⊆ X be nonempty closed convex and T : C → C be a nonexpansive mapping with F ix(T ) = ∅. If T is asymptotically regular at x ∈ C, then the Picard iterate (T n x) ∆-converges to a fixed point of T .
Proof. By Lemma 6.2, the nonempty set F := F ix(T ) is closed and convex. Furthermore, one can see easily that (T n x) is Fejér monotone with respect to F and, by Theorem 2.12, (T n x) has a unique asymptotic center with respect to F . Let (u n ) be a subsequence of (T n x) and u be its unique asymptotic center. Then d(T u, u n ) ≤ d(T u, T u n ) + d(T u n , u n ) ≤ d(u, u n ) + d(u n , T u n ), so we can use Lemma 2.11 to obtain that T u = u, i. e. u ∈ F . Apply Lemma 6.1.(iii) to get the conclusion.
By [30,Theorem 3.5] one can replace in the above theorem the assumption that T has fixed points with the equivalent one that T has bounded orbits.
We get the following ∆-convergence result for the Picard iteration of a firmly nonexpansive mapping.
Theorem 6.4. Let (X, d, W ) be a complete UCW -hyperbolic space, C ⊆ X be nonempty closed convex and T : C → C be a λ-firmly nonexpansive mapping for some λ ∈ (0, 1). Assume that F ix(T ) = ∅. Then for all x in C, (T n x) ∆-converges to a fixed point of T .
Proof. Since F ix(T ) = ∅, we get that r C (T ) = 0, so, by Corollary 5.2, that T is asymptotically regular. Apply now Proposition 6.3.

An application to a minimization problem
Let (X, d) be a complete CAT(0) space and F : X → (−∞, ∞] be a proper, convex and lower semicontinuous mapping. We shall apply Theorem 6.4 to approximate the minimizers of F , that is the solutions of the minimization problem min x∈X F (x).
Let argmin y∈X F (y) = {x ∈ X | F (x) ≤ F (y) for all y ∈ X} be the set of minimizers of F .
The following result is a consequence of the definition of the resolvent and Proposition 2.13. Proposition 6.5. For all µ > 0, the set F ix(J µ ) of fixed points of the resolvent associated with F coincides with the set argmin y∈X F (y) of minimizers of F .
As the resolvent is a firmly nonexpansive mapping, one can apply Theorem 6.4 and the above result to obtain Corollary 6.6. Assume that F has a minimizer. Then for all µ > 0 and all x ∈ X, the Picard iterate (J n µ (x)) ∆-converges to a minimizer of F . We remark that a more general result was obtained recently by Bačák [1] using different methods. Thus, Bačák obtained in the setting of CAT(0) spaces the following proximal point algorithm: if F has minimizers, then for all x 0 ∈ X and all sequences (λ n ) divergent in sum, the sequence ∆-converges to a minimizer of F .

Effective rates of asymptotic regularity
As we have proved in Section 5, any λ-firmly nonexpansive mapping T : C → C defined on a nonempty subset C of a W-hyperbolic space X is asymptotically regular, provided T has bounded orbits. In this section we shall obtain, for UCW -hyperbolic spaces, a rate of asymptotic regularity of T , that is a rate of convergence of the sequence (d(T n x, T n+1 )) towards 0. The methods of proof are inspired by those used by Kohlenbach [23] and the second author [29] for computing rates of asymptotic regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in uniformly convex Banach spaces and UCW -hyperbolic spaces.
If F ix ε (T, x, b) = ∅ for all ε > 0, we say that T has approximate fixed points in a bneighborhood of x.
Proof. If C is bounded, then T is asymptotically regular by Corollary 5.3. Hence, for all b ≥ d C , T has approximate fixed points in a b-neighborhood of x for all x ∈ C.
Thus, for bounded C, we get that T is asymptotically regular with a rate Φ(ε, η, λ, b) that only depends on ε, on X via the monotone modulus of uniform convexity η, on C via an upper bound b on its diameter d C and on the mapping T via λ. The rate of asymptotic regularity is uniform in the starting point x ∈ C of the iteration and other data related with X, C and T .
As we have remarked in Section 2, CAT(0) spaces are UCW -hyperbolic spaces with a quadratic (in ε) modulus of uniform convexity η(ε) = ε 2 8 , which has the form required in Remark 7.2. As an immediate consequence, we get a quadratic (in 1/ε) rate of asymptotic regularity in the case of CAT(0) spaces.
Follow now the proof above to get the conclusion.