The asymptotic behavior of the composition of firmly nonexpansive mappings

The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings David Ariza-Ruiza, Genaro López-Acedoa, Adriana Nicolaeb,c aDepartamento de Análisis Matemático - IMUS, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain bDepartment of Mathematics, Babeş-Bolyai University, Kogălniceanu 1, 400084 Cluj-Napoca, Romania cSimion Stoilow Institute of Mathematics of the Romanian Academy, Research group of the project PD-3-0152, P.O. Box 1-764, RO-014700 Bucharest, Romania E-mail addresses: [email protected] (D. Ariza-Ruiz), [email protected] (G. López-Acedo), [email protected] (A. Nicolae) Abstract In this paper we provide a unified treatment of some convex minimization problems, which allows for a better understanding and, in some cases, improvement of results in this direction proved recently in spaces of curvature bounded above. For this purpose, we analyze the asymptotic behavior of compositions of finitely many firmly nonexpansive mappings in the setting of p-uniformly convex geodesic spaces focusing on asymptotic regularity and convergence results. Keywords: Firmly nonexpansive mapping, Convex optimization, Convex feasibility problem, p-uniformly convex geodesic space, CAT(κ)space 1 Introduction Various mathematical problems can be formulated as finding a common fixed point of finitely many firmly nonexpansive mappings. These include, for instance, the convex feasibility problem [1], which consists of finding a point in the intersection of a finite number of closed convex sets, or the problem of finding a common zero of a finite family of maximal monotone operators [2]. Firmly nonexpansive mappings were introduced by Bruck [3] in order to study sunny nonexpansive retractions onto closed subsets of Banach spaces. In Hilbert spaces, these mappings coincide with firmly contractive ones defined by Browder [4]. A basic example of firmly nonexpansive mappings is the metric projection onto closed convex subsets of Hilbert spaces. This mapping or finite compositions thereof appear in different iterative methods used in the study of convex minimization problems. Results on the asymptotic behavior of some such methods follow from analyzing compositions of finitely many firmly nonexpansive mappings, as it is the case of the cyclic projection algorithm employed in solving the convex feasibilty problem (see [1] for a deep treatment of this topic). Another important example of firmly nonexpansive mappings is the resolvent of a monotone operator which was introduced by Minty [5]. Since the subdifferential of a proper convex lower semi-continuous function is a maximal monotone operator, some splitting methods which involve resolvents and are used to solve convex minimization problems can be abstracted to compositions of firmly nonexpansive mappings. Using this approach, Bauschke, Combettes and Reich [6] (see also [7]) proved, in particular, that having two proper convex lower semicontinuous functions defined on a Hilbert space, one can apply alternatively the two resolvents to obtain weak convergence to a solution (provided it exists) of a minimization problem associated to the two functions, see (10). It is noteworthy that the convex feasibilty problem for two sets can be reformulated as a problem of type (10) for both the consistent and the inconsistent case. The most straightforward way to find a common fixed point or, more generally, a fixed point of the composition of finitely many firmly nonexpansive mappings is to iterate the mappings cyclically. The particular case of just one mapping was considered in Hilbert spaces by Browder [4] who proved that, for any starting point, the Picard iterates of a firmly nonexpansive mapping converge weakly to a fixed point, whenever 1 arXiv:1411.6779v1 [math.FA] 25 Nov 2014 the fixed point set is nonempty. Although the class of firmly nonexpansive mappings is not closed under composition, in Hilbert spaces firmly nonexpansive mappings are averaged and the composition of averaged mappings is averaged. Thus, results for the general case of finitely many firmly nonexpansive mappings can be deduced from the asymptotic behavior of averaged mappings (see also [8] for a more general order of composing the mappings that still preserves weak convergence). In Banach spaces not every firmly nonexpansive mappings is averaged and so extensions of such results to this setting are not immediate. Bruck and Reich [9] introduced strongly nonexpansive mappings that generalize firmly nonexpansive ones when the space is uniformly convex. This class of mappings is closed under composition and so one can analyze compositions of firmly nonexpansive mappings using results on the asymptotic behavior of strongly nonexpansive mappings. More recently, problems where the study of compositions of firmly nonexpansive mappings plays an important role (such as minimization problems [10] or abstract Cauchy problems[11]) have also been formulated in a nonlinear setting. Since our applications focus on minimizing convex functions in geodesic spaces, we mention some of the work in this direction. The proximal point method and modifications thereof for optimization problems in linear spaces have been extended to Riemanninan manifolds in [12, 13] and references therein. Spaces of nonpositive curvature in the sense of Alexandrov (also known as CAT(0) spaces) proved to be an appropriate setting for considering such problems too. Jost studied the concept of resolvents for convex functions to develop the theory of harmonic mappings [14] which motivated Bačák [15] to extend the proximal point algorithm to this context. In [16], a proximal splitting algorithm was also approached in this setting (assuming in addition local compactness) and applied to the computation of the geometric median and the Fréchet mean of a finite set of points. Regarding the convex feasibility problem, results on the asymptotic behavior of the alternating projection method have been obtained in [17, 18]. The minimization problem (10) was studied in this setting in [19]. The goal of this work is to create a suitable theoretical framework so that previous results can be better understood and in some cases improved. To this end, we present a systematic study of the asymptotic behavior of compositions of a finite number of firmly nonexpansive mappings in p-uniformly convex geodesic spaces. Asymptotic regularity is a very important concept used in the study of the asymptotic behavior of sequences and was formally introduced by Browder and Petryshyn in [20]. In Section 3 we give asymptotic regularity results for sequences obtained by applying a finite number of firmly nonexpansive mappings in a cyclic way. We assume first that they have a common fixed point, but consider also the case of two mappings which satisfy the weaker condition that their composition is not fixed point free. Moreover, we use techniques which originate from proof mining (see [21] for more details) to provide polynomial rates of asymptotic regularity, thus giving quantitative versions of such results and extending a corresponding result proved in [18]. Section 4 deals with ∆-convergence results for these problems. Assuming appropriate compactness conditions we obtain strong convergence. When the mappings have a common fixed point we apply our findings to the cyclic projection method for finitely many sets in CAT(κ)spaces improving previous results from [17, 16]. We show that the results concerning two firmly nonexpansive mappings for which the fixed point set of the composition is nonempty can be used to study the minimization problem (10) in CAT(0) spaces which partially recovers the main result of [19] and yields a convergence result for the inconsistent convex feasibility problem for two sets. 2 Preliminaries 2.1 Some Notions on Geodesics Spaces Let (X, d)be a metric space. A geodesic path that joins two points x, y ∈Xis a mapping γ: [0, l]⊆R→X such that γ(0) = x,γ(l) = yand d(γ(t), γ(t0)) = |t−t0|for all t, t0∈[0, l]. The image γ([0, l]) of γis called a geodesic segment from xto y. We denote a point z∈Xbelonging to such a geodesic segment by z= (1 −t)x⊕ty, where t=d(z, x)/d(x, y).(X, d)is a (uniquely) geodesic space iff every two points in X are joined by a (unique) geodesic path. A subset Cof Xis convex iff it contains all geodesic segments that join any two points in C. For more details on geodesic metric spaces, see [22]. Ball, Carlen and Lieb introduced in [23] the notion of p-uniform convexity which plays an essential role in Banach space theory. Recall that a normed space (X, k·k)is said to be p-uniformly convex for 2≤p < ∞ 2 iff there exists a constant C≥1such that for any x, y, ∈X,     x+y 2    p ≤1 2kxkp+1 2kykp−1 Cp    x−y 2    p . Recently, Naor and Silberman [24] extended this notion to the setting of geodesic spaces in the following way. Definition 2.1. Fix 1<p<∞. A metric space (X, d)is called p-uniformly convex with parameter c > 0 iff (X, d)is a geodesic space and for any three points x, y, z ∈Xand all t∈[0,1], d(z, (1 −t)x⊕ty)p≤(1 −t)d(z, x)p+t d(z, y)p−c 2t(1 −t)d(x, y)p.(1) Note that inequality (1) guarantees that the space Xis uniquely geodesic (see [25, Lemma 2.2] for a proof in the case p= 2). Also, any closed convex subset of a p-uniformly convex space is again a p-uniformly convex space with the same parameter. It is well-known that any Lpspace over a measurable space is p-uniformly convex. In the setting of nonlinear spaces, every CAT(0) space is 2-uniformly convex with parameter c= 2. Actually, in this case (1) characterizes CAT(0) spaces. For κ > 0, any CAT(κ)space Xwith diam(X)< π/(2√κ)is a 2-uniformly convex space with parameter c= (π−2√κ ε) tan(√κ ε)for any 0< ε ≤π/(2√κ)−diam(X), see [25]. A detailed treatment of CAT(κ)spaces can be found, for example, in [22]. We recall next a notion of weak convergence in metric spaces. Let (xn)be a bounded sequence in a metric space (X, d). Then r((xn)) := inf lim sup n→∞ d(y, xn) : y∈X is called the asymptotic radius of (xn)and the asymptotic center of (xn)is the set A((xn)) := x∈X: lim sup n→∞ d(x, xn) = r((xn)). An element of A((xn)) will also be referred to as an asymptotic center. If Xis a complete p-uniformly convex space, then any bounded sequence has a unique asymptotic center since Xadmits a modulus of uniform convexity that does not depend on the radius of balls (see [26]). We say that a sequence (xn) ∆-converges to x∈Xiff xis the unique asymptotic center of every subsequence of (xn). In this case we call xthe ∆-limit of (xn). This concept was introduced by Lim [27]. It is easy to see that in a complete p-uniformly convex space, any bounded sequence has a ∆-convergent subsequence. In the setting of CAT(0) spaces, ∆-convergence is equivalent to another concept of weak convergence that uses projections on geodesic segments (see [28]). Reasoning as in [28], one can see that this equivalence also holds for p-uniformly convex spaces. Let (X, d)be a metric space. Given Ca nonempty subset of X, the distance of a point x∈Xto Cis dist(x, C) := inf{d(x, c) : c∈C}.If Band Care nonempty subsets of X, we denote dist(B, C) := inf{d(b, c) : b∈B, c ∈C}. The metric projection PConto Cis the mapping defined by PC(x) := {c∈C:d(x, c) = dist(x, C)}, for every x∈X. Let Xbe a compete CAT(κ)space (with diam(X)< π/(2√κ)for κ > 0) and Ca closed and convex subset of X. Then PC:X→Cis well-defined and singlevalued (see [22, Proposition 2.4, page 176] and [28, Proposition 3.5]). If κ= 0,PCis also nonexpansive. For κ > 0we have that for every x∈X, y ∈C, d(PC(x), y)≤d(x, y). Moreover, in this case PCis a uniformly continuous mapping because Xis bounded and admits a modulus of uniform convexity that does not depend on the radius of balls (see the reasoning in the proof of [29, Proposition 4.5]). In fact, PCis a Lipschitz mapping (one can easily derive this from the proof of [30, Theorem 4.1]). We will also use the following inequality which originates from the work of Reshetnyak (see, for instance, [14, Theorem 2.3.1] or [31, Lemma 2.1] for a simple proof). Let Xbe a CAT(0) space. Then for every x, y, u, v ∈X, d(x, y)2+d(u, v)2≤d(x, v)2+d(y, u)2+ 2d(x, u)d(y, v).(2) We finish this subsection with two simple results that will be used later on. 3 Lemma 2.2. Let (an)and (εn)be two sequences of nonnegative numbers. If for every n∈N an+1 ≤an+εn, and ∞ X n=0 εn<∞, then the sequence (an)is convergent. Lemma 2.3. Let l≥0and M > 0. Then there exists a constant Rdepending only on land Msuch that for all 0≤a≤Mand 0≤b≤1, (a+b)l≤al+R b. Proof. When l∈[0,2], clearly (a+b)l−al≤(1+2M)b. For l > 2, the result is an immediate consequence of a Kaczmarz-Steinhaus inequality [32, page 144] which states that there exists a constant αdepending only on lsuch that for all x∈R, |1 + x|l≤1 + l x + [l] X i=2 l ixi+α|x|l. 2.2 Firmly Nonexpansive Type Mappings in Geodesic Spaces Firmly nonexpansive mappings were first introduced by Browder [4], under the name of firmly contractive, in the setting of Hilbert spaces, and later by Bruck [3] in the context of Banach spaces. Recently Bruck’s definition was extended to a nonlinear setting in [33] (see also [18]). We recall this definition using the framework of p-uniformly convex spaces. Definition 2.4. Let Cbe a nonempty subset of a p-uniformly convex space (X, d). We say that a mapping T:C→Xis firmly nonexpansive iff d(Tx, Ty)≤d((1 −λ)x⊕λT x, (1 −λ)y⊕λTy),(3) for all x, y ∈Cand λ∈[0,1[. As in the linear case, any sunny nonexpansive retraction is firmly nonexpansive. In particular, the metric projection onto closed and convex subsets of a complete CAT(0) space is firmly nonexpansive. Having Xa complete CAT(0) space and a function f:X→]−∞,+∞]that is convex, lower semi-continuous and proper (i.e., not constantly equal to +∞), the resolvent of fdefined by Jf λ(x) := argmin z∈Xf(z) + 1 2λd(x, z)2, where λ > 0, is also firmly nonexpansive. In the subsequent section we will use the following inequality (for its proof see, for instance, [16, Lemma 3.2]): for every x, y ∈X, f(Jf λ(x)) + 1 2λd(Jf λ(x), x)2≤f(y) + 1 2λd(x, y)2−1 2λd(Jf λ(x), y)2.(4) Bruck extended in [34] the notion of strongly nonexpansive mappings and introduced the so-called strongly quasi-nonexpansive ones with the aim of proving convergence of iterations built by applying finitely many such mappings. We consider in the following a related property which is useful when studying the asymptotic behavior of sequences obtained by applying a finite number of firmly nonexpansive mappings in the setting of p-uniformly convex spaces (see also the notion of strongly attracting mappings in [1]). Definition 2.5. Let Cbe a nonempty subset of a metric space (X, d). A mapping T:C→Xis said to satisfy property (P1) iff Fix(T)6=∅and there exist l > 0and β > 0such that d(Tx, u)l≤d(x, u)l−β d(Tx, x)l,(P1) for all x∈Cand u∈Fix(T). 4 If Xis p-uniformly convex with parameter c, then every firmly nonexpansive mapping T:C→Xwith Fix(T)6=∅satisfies (P1) with l=pand β=c/2. Indeed, given x∈Cand u∈Fix(T), for any λ∈[0,1[ we have that d(u, Tx)p=d(Tu, Tx)p≤d(u, (1 −λ)x⊕λTx)p ≤(1 −λ)d(u, x)p+λ d(u, Tx)p−c 2λ(1 −λ)d(x, Tx)p. Hence, d(u, Tx)p≤d(u, x)p−(c/2)λ d(x, Tx)p. Taking limit as λ→1−, we get the conclusion. A second example of mappings satisfying (P1) is the metric projection in spaces of curvature bounded above. Let Xbe a complete CAT(κ)(with diam(X)< π/(2√κ)for κ > 0) and C⊆Xclosed and convex. Recall that if κ > 0, then the metric projection PC:X→Cneed not be nonexpansive. However, it satisfies (P1) with l= 2 and β=c/2. To see this, let x∈X,u∈Cand λ∈]0,1[. Then, d(PC(x), u)2=d(PC(1 −λ)x⊕λPC(x), u)2≤d(1 −λ)x⊕λPC(x), u2 ≤(1 −λ)d(x, u)2+λd(PC(x), u)2−c 2λ(1 −λ)d(x, PC(x)). Hence, d(PC(x), u)2≤d(x, u)2−(c/2)λ d(x, PC(x)) and we only need to let λ→1−. Given finitely many mappings satisfying (P1)one can relate the intersection of their fixed point sets with the fixed point set of their composition. Proposition 2.6. Let (X, d)be a metric space and let {Ti: 1 ≤i≤r}be a family of mappings with property (P1). If Tr i=1 Fix(Ti)6=∅, then r \ i=1 Fix(Ti) = Fix(T1◦ ··· ◦Tr). Proof. Clearly Tr i=1 Fix(Ti)⊆Fix(T1◦ ··· ◦Tr). Conversely, let x∈Fix(T1◦ ··· ◦Tr). If u∈ ∩r i=1 Fix(Ti) and y= (T2◦ ··· ◦Tr)(x), then d(T1y, u)≤d(y, u)≤ ··· ≤ d(x, u).(5) Notice that T1y=x. Then, by (5), d(T1y, u) = d(y, u). Since T1has property (P1), we deduce that β d(y, T1y)l≤d(u, y)l−d(u, T1y)l= 0, that is, y=T1y. Hence, x∈Fix(T2◦ ··· ◦Tr)and T1x=x. Continuing in this way we obtain the result. The following also purely metric condition is more particular than (P1) and is still satisfied by any firmly nonexpansive mapping in CAT(0) spaces. In this setting every mapping for which the property below holds is nonexpansive. Note also that in Hilbert spaces, this notion coincides with firm nonexpansivity (see also the remark in [19, page 658]). Definition 2.7. Let Cbe a nonempty subset of a metric space (X, d). A mapping T:C→Xsatisfies property (P2) iff 2d(Tx, Ty)2≤d(x, T y)2+d(y, Tx)2−d(x, T x)2−d(y, Ty)2,(P2) for all x, y ∈C. Any mapping that satisfies (P2) and has fixed points also satisfies (P1) with l= 2 and β= 1. Indeed, if T:C→Xsatisfies (P2) and Fix(T)6=∅, then for all x∈Cand u∈Fix(T)we have that 2d(Tx, u)2= 2d(Tx, Tu)2≤d(x, u)2+d(u, Tx)2−d(x, Tx)2, from where d(Tx, u)2≤d(x, u)2−d(x, Tx)2. However, the converse implication does not hold. To see this, it suffices to take T: [−1/4,1/3] →R,T x = 2x2. Then Tsatisfies (P1) with l= 2 and β= 1/3, but (P2) fails for x=−1/4and y= 0. Note that the metric projection and the resolvent are well-defined in less regular frameworks than complete spaces of curvature bounded above, such as complete uniform Busemann nonpositively curved spaces (see 5 [14]). However, in this setting they may fail to be nonexpansive or satisfy property (P1). Since we derive our results concerning the convex feasibility problem and the minimization problem (10) from the asymptotic behavior of mappings satisfying properties (P1) or (P2), we state these results in the context of spaces of curvature bounded above. 3 Asymptotic Regularity In this section we analyze the asymptotic behavior (also from a quantitative point of view) of the sequence generated by applying iteratively a finite number of firmly nonexpansive type mappings in a cyclic way. We study the case when the considered mappings have a common fixed point, but focus also on the situation when they satisfy the weaker assumption that the fixed point set of their composition is nonempty. Let (X, d)be a metric space. A mapping T:X→Xis said to be asymptotically regular at x∈X iff lim n→∞ d(Tnx, Tn+1x)=0and it is asymptotically regular iff it is asymptotically regular at every x∈X. Likewise, a sequence (xn)⊆Xis asymptotically regular iff lim n→∞ d(xn, xn+1)=0. A rate of convergence of (d(xn, xn+1)) towards 0will be called a rate of asymptotic regularity. As mentioned in the introductory section, asymptotic regularity is an important tool used in the study of the asymptotic behavior of sequences. In the sequel, having n, r ∈N,r≥1, we use the notation n=n(mod r) + 1. Let r∈N,r≥1and T1, . . . , Tr:X→X. We consider the following cyclic method allowing errors in the evaluation of the values of the mappings Ti: given x0∈X, define the sequence (xn)in Xby d(xn+1, Tnxn)≤εn,for each n∈N,(A1) where ∞ X n=0 εn<∞. We prove below that the sequence (xn)is asymptotically regular provided the mappings used to define it satisfy property (P1) and have a common fixed point. The result applies, in particular, to firmly nonexpansive mappings in the setting of p-uniformly convex spaces. Theorem 3.1. Let (X, d)be a metric space and T1, . . . , Tr:X→Xsatisfy property (P1). If Tr i=1 Fix(Ti)6= ∅, then given any starting point x0∈X, the sequence (xn)defined by (A1) is asymptotically regular. Proof. Let u∈Tr i=1 Fix(Ti)and m∈Nsuch that εn≤1for n≥m. Because each Tisatisfies (P1) we get that for all n≥m+ 1, β d(xn, Tnxn)l≤d(xn, u)l−d(Tnxn, u)l ≤d(xn, Tn−1xn−1) + d(Tn−1xn−1, u)l−d(Tnxn, u)l ≤R εn−1+d(Tn−1xn−1, u)l−d(Tnxn, u)lby Lemma 2.3, where Ris a constant depending only on land M≥d(x0, u) + ∞ X n=0 εn.Thus, for all n≥m+ 1, β n X k=m+1 d(xk, Tkxk)l≤d(Tmx0, u)l+R ∞ X k=0 εk. Since ∞ X k=0 εk<∞, we deduce that ∞ X k=m+1 d(xk, Tkxk)l<∞, so lim n→∞ d(xn, Tnxn)=0. This finally yields that (xn)is asymptotically regular. It is easy to see that Theorem 3.1 can be slightly generalized by relaxing property (P1) imposed on the mappings T1, . . . , Trto the condition that they satisfy the inequality used to define (P1) only for one common fixed point instead of all fixed points. However, we will not focus on such extensions because the final goal of this work is to apply the above result when the mappings are metric projections, which satisfy property (P1) in spaces of curvature bounded above. 6 Corollary 3.2. Let (X, d)be a metric space and T1, . . . , Tr:X→Xsatisfy property (P1). If Tr i=1 Fix(Ti)6= ∅,then the Picard iteration of T=T1◦ ··· ◦Tris asymptotically regular. Proof. Note that, for x0∈Xand k∈N,Tkx0=xkr, where (xn)is defined by (A1) with εn= 0 for every n∈N. Hence, for all k∈N, d(Tkx0, Tk+1x0) = d(xkr, x(k+1)r)≤d(xkr, xkr+1) + . . . +d(xkr+r−1, xkr+r), which implies, by Theorem 3.1, that Tis asymptotically regular. As a consequence, we have that the sequence obtained by the cyclic projection method for a finite number of sets is asymptotically regular in spaces of curvature bounded above. Corollary 3.3. Let κ∈R. Suppose Xis a CAT(κ)space (with diam(X)< π/(2√κ)for κ > 0). Let C1, . . . , Cr⊆Xbe nonempty closed and convex sets such that Tr i=1 Ci6=∅. Given any starting point x0∈X, let (xn)be defined by (A1) with Ti=PCi, for i= 1, . . . , r. Then the sequence (xn)and the mapping PC1◦. . . ◦PCrare asymptotically regular. We give next a rate of asymptotic regularity for the sequence (xn)generated by (A1) with εn= 0 for every n∈N. For simplicity, we prove the theorem for two mappings, but the proof method works equally well in the general case. The result extends [18, Theorem 5.2] where a quadratic rate of asymptotic regularity in 1/ε was given for the sequence generated by the alternating projection method for two sets in the context of CAT(0) spaces. Since its proof relies on particular properties of the metric projection, the obtained rate of asymptotic regularity is slightly better than the one we give here when restricting to the setting of CAT(0) spaces. However, our result holds not only in a broader framework, but also for more general mappings, namely firmly nonexpansive ones. Theorem 3.4. Let (X, d)be a p-uniformly convex space with parameter cand suppose that T1, T2:X→X are firmly nonexpansive mappings with F:= Fix T1∩Fix T26=∅.Consider x0∈X,u∈Fand b > 0such that d(x0, u)≤b. Suppose that (xn)is defined by (A1) with εn= 0 for every n∈N. Then for all ε > 0and n≥θ(ε, b, c, p), we have that d(xn, xn+1)≤ε, where θ(ε, b, c, p) :=    22 c4p bp c εppif ε < 2b, 0otherwise. Moreover, for all ε > 0,n≥˜ θ(ε, b, c, p)and i∈ {1,2}, we have that d(xn, Tixn)≤ε, where ˜ θ(ε, b, c, p) := 1 + θ(ε, b, c, p). Proof. Since for all n∈N,d(xn, xn+1)≤d(xn, u) + d(u, xn+1)≤2b, we can assume that ε < 2b. Denote α:= c εp 4p bp−1. Take N:= [2bp/(cαp)]. For each i∈N, d(x2i+2, u)p≤d(x2i+1, u)p−c 2d(x2i+1, x2i+2)p ≤d(x2i, u)p−c 2(d(x2i, x2i+1)p+d(x2i+1, x2i+2)p), from where c 2(d(x2i, x2i+1)p+d(x2i+1, x2i+2)p)≤d(x2i, u)p−d(x2i+2, u)p.(6) Hence, we have that c 2 N X i=0 (d(x2i, x2i+1)p+d(x2i+1, x2i+2)p)≤d(x0, u)p−d(x2N+2, u)p≤bp.(7) 7 Assume that for all i= 0, . . . , N,d(x2i, x2i+1)p+d(x2i+1, x2i+2)p> αp. Then, using (7), we deduce that N+ 1 <2bp/(cαp),a contradiction. This means that there exists i∈ {0, . . . , N}such that d(x2i, x2i+1)p+ d(x2i+1, x2i+2)p≤αp.Hence, d(x2i, x2i+1)≤αand d(x2i+1, x2i+2)≤α, from where d(x2i, x2i+2)≤2α. This implies that for all j≥i, d(x2j, x2j+2)≤2α. By (6) and the mean value theorem, we have that for all j≥i, c 2(d(x2j, x2j+1)p+d(x2j+1, x2j+2)p)≤d(x2j, u)p−d(x2j+2, u)p ≤p bp−1(d(x2j, u)−d(x2j+2, u)) ≤p bp−1d(x2j, x2j+2)≤p bp−12α=c 2εp, that is, d(x2j, x2j+1)p+d(x2j+1, x2j+2)p≤εp. Hence, for all j≥i,d(x2j, x2j+1)≤εand d(x2j+1, x2j+2)≤ε. Thus, for all n≥2N,d(xn, xn+1)≤ε. It is easy to see that for every n≥1and i∈ {1,2}, d(xn, Tixn)≤max{d(xn, xn+1), d(xn−1, xn)}. Thus, ˜ θis a rate of convergence for (d(xn, Tixn)) towards 0. Remark 3.5.The above result also holds if we consider a finite family of firmly nonexpansive mappings T1,··· , Trwith Tr i=1 Fix Ti6=∅.In this case, a rate of asymptotic regularity for (xn), defined by (A1) with εn= 0 for every n∈N, is θ(ε, b, c, p, r) :=    r2 c2r p bp c εppif ε < 2b, 0otherwise. Moreover, for every i∈ {1, . . . , r}, ˜ θ(ε, b, c, p, r) := [r/2] + θε 2dr/2e−1, b, c, p is a rate of convergence for (d(xn, Tixn)) towards 0. We prove in the following an asymptotic regularity result for two mappings that satisfy (P2) and for which the fixed point set of the composition is nonempty. To simplify the writing, we consider the sequences (xn)and (yn)defined by d(yn, T1xn)≤εnand d(xn+1, T2yn)≤δn,for each n∈N,(A2) where ∞ X n=0 εn<∞and ∞ X n=0 δn<∞. Theorem 3.6. Let (X, d)be a CAT(0) space and let T1, T2:X→Xsatisfy property (P2). If Fix(T2◦T1)6= ∅, then given any x0∈X, the sequences (xn)and (yn)defined by (A2) are asymptotically regular. Proof. Denote S=T2◦T1. Then, for every n∈N, d(xn+1, Sxn)≤d(xn+1, T2yn) + d(T2yn, T2(T1xn)) ≤δn+εn.(8) Hence, lim n→∞ d(xn+1, Sxn)=0.(9) We also have that for every n, k ∈N, d(Sxn+1+k, Sxn+1)≤d(T1xn+1+k, T1xn+1) ≤d(T1xn+1+k, T1Sxn+k) + d(T1Sxn+k, T1Sxn) + d(T1Sxn, T1xn+1) ≤d(xn+1+k, Sxn+k) + d(Sxn+k, Sxn) + d(Sxn, xn+1) ≤δn+k+εn+k+δn+εn+d(Sxn+k, Sxn)by (8). 8 This yields, by Lemma 2.2, that (d(Sxn+k, Sxn))nconverges to some ξk≥0. Furthermore, we have that lim n→∞ d(T1Sxn+k, T1Sxn) = ξk. Suppose ξ1>0. We prove by induction that ξk=k ξ1for all k≥1. For k= 1 this is clear. Suppose the claim holds true for all i= 1, . . . , k. We show that it also holds for k+ 1. Since T1satisfies (P2), we have that 2d(T1Sxn+k, T1Sxn)2≤d(Sxn+k, T1Sxn)2+d(Sxn, T1Sxn+k)2 −d(Sxn+k, T1Sxn+k)2−d(Sxn, T1Sxn)2. Likewise, 2d(Sxn+k, Sxn)2= 2d(T2(T1xn+k), T2(T1xn))2 ≤d(T1xn+k, Sxn)2+d(T1xn, Sxn+k)2 −d(T1xn+k, Sxn+k)2−d(T1xn, Sxn)2. Adding the two above inequalities and using (2) we obtain that 2d(T1Sxn+k, T1Sxn)2+ 2d(Sxn+k, Sxn)2 ≤2d(Sxn, Sxn+k)d(T1Sxn, T1xn+k)+2d(Sxn, Sxn+k)d(T1xn, T1Sxn+k). Dividing by 2d(Sxn, Sxn+k), using the nonexpansivity of T1, the triangle inequality and the notation γk n= d(T1Sxn, T1Sxn+k)2/d(Sxn, Sxn+k), γk n+d(Sxn+k, Sxn)≤d(Sxn, xn+k) + d(xn, Sxn+k) ≤d(Sxn, Sxn+k−1) + d(Sxn+k−1, xn+k) + d(xn, Sxn−1) + d(Sxn−1, Sxn+k). Thus, γk n+d(Sxn+k, Sxn)−d(Sxn, Sxn+k−1)−d(xn, Sxn−1)−d(Sxn+k−1, xn+k) ≤d(Sxn−1, Sxn+k)≤d(Sxn−1, Sxn) + d(Sxn, Sxn+k). Letting n→ ∞ above and using the induction hypothesis and (9), we get that ξk+1 = lim n→∞ d(Sxn, Sxn+k+1)=(k+ 1)ξ1, which finishes the induction. Thus, (Sxn)is unbounded. Let u∈Fix(T2◦T1). Since, by (8), for all n∈N, d(Sxn+1, u)≤d(xn+1, u)≤d(xn+1, Sxn) + d(Sxn, u)≤δn+εn+d(Sxn, u), it follows that d(Sxn+1, u)≤ ∞ X n=0 δn+ ∞ X n=0 εn+d(x0, u), which shows that (Sxn)is bounded, a contradiction. Therefore, ξ1= 0, that is, lim n→∞ d(Sxn+1, Sxn) = 0 which yields, using (9), that lim n→∞ d(xn+1, xn) = 0 because d(xn+1, xn)≤d(xn+1, Sxn) + d(Sxn, Sxn−1) + d(Sxn−1, xn). At the same time, d(yn+1, yn)≤d(yn+1, T1xn+1) + d(T1xn+1, T1xn) + d(T1xn, yn) ≤εn+1 +εn+d(xn+1, xn), from where lim n→∞ d(yn+1, yn)=0. Remark 3.7.Note that in the proof of Theorem 3.6 one can replace the condition that Fix(T2◦T1)6=∅by the assumption that one of the sequences (xn)or (yn)is bounded (and so the other sequence will be bounded too). We show in the next section (see Theorem 4.4) that these two conditions are in fact equivalent. 9 [21] Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. 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