Averaged alternating reflections in geodesic spaces

arXiv:1211.4958v3 [math.OC] 2 Oct 2013 Averaged alternating reflections in geodesic spaces Aurora Fern´andez-Le´on1, Adriana Nicolae2,3 1Dpto. de An´alisis Matem´atico, Universidad de Sevilla, P.O. Box 1160, 41080-Sevilla, Spain 2Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania 3Department of Mathematics, Babe¸s-Bolyai University, Kog˘alniceanu 1, 400084, Cluj-Napoca, Romania E-mail: [email protected], adriana.nicol[email protected] Abstract We study the nonexpansivity of reflection mappings in geodesic spaces and apply our findings to the averaged alternating reflection algorithm employed in solving the convex feasibility problem for two sets in a nonlinear context. We show that weak convergence results from Hilbert spaces find natural counterparts in spaces of constant curvature. Moreover, in this particular setting, one obtains strong convergence. 1 Introduction The convex feasibility problem for two sets consists of finding a point in the intersection of two nonempty closed and convex sets provided such a point exists. This problem finds remarkable applications in applied mathematics and various branches of engineering (see, for example, [3, 11, 28, 10]) which have motivated many researchers to focus on methods of solving this problem. In Hilbert spaces there exists a wide range of algorithms that use metric projections on the sets in order to obtain sequences of points that converge weakly or in norm (under more restrictive conditions) to a solution of this problem. One of the most famous algorithms is the alternating projection method which was developed by von Neumann [30] and was recently adapted to the setting of CAT(0) spaces by Baˇc´ak, Searston and Sims in [2]. Another class of algorithms considered in this respect is based on reflections instead of projections. Given a nonempty closed and convex subset Aof a Hilbert space H, the reflection of a point x∈Hwith respect to Ais the image of xby the reflection mapping RA= 2PA−I, where PA stands for the metric projection onto Aand Iis the identity mapping. In this work we focus on the averaged alternating reflection (AAR) method employed in solving the convex feasibility problem for two sets. Suppose Aand Bare two nonempty closed and convex subsets of a Hilbert space with nonempty intersection. The AAR method generates the following sequence for a starting point x0∈H:xn=Tnx0, where T=I+RARB 2. This algorithmic scheme was studied by Bauschke, Combettes and Luke in [4, 5] not only in connection with the convex feasibility problem, but also for finding a best approximation pair of the sets Aand Bin case their intersection is empty and such a pair exists. The AAR method was later modified in [6] in order to solve the problem of finding the projection of a point onto the intersection of two closed and convex sets. In fact, for the convex feasibility problem, this algorithm is a special case of one described by Lions and Mercier in [25]. One obtains weak convergence of the sequence (xn) to a fixed point of the mapping Tand the projection of this point onto the set Blies in the intersection of Aand B. Here we are interested in studying the AAR method in geodesic spaces. However, several difficulties arise when considering this algorithm in a nonlinear setting. First of all one needs to 1 find an appropriate definition for the reflection mapping. In order to guarantee the existence of the reflection of a point in the space, we consider spaces with the geodesic extension property. Moreover, the reflection mapping is not always unique. A second difficulty consists in guaranteeing certain properties of this mapping. In Hilbert spaces, the proof of the convergence of the AAR method relies on the nonexpansivity of the reflection mapping which yields the firm nonexpansivity of the mapping T. In this paper we prove that the reflection mapping is nonexpansive in spaces of constant curvature and justify why it fails to be nonexpansive in the broad setting of CAT(0) spaces. We also analyze the behavior of reflection mappings in slightly more general settings, namely gluings of model spaces. Likewise, we study the convergence of the AAR method in spaces of constant curvature proving strong convergence in this case. Furthermore, we include a rate of asymptotic regularity for the AAR method. This work is partly motivated by a communication of Ian Searston given during the 10th International Conference on Fixed Point Theory and its Applications where the problem of studying the nonexpansivity of the reflection mapping in geodesic spaces was raised. 2 Preliminaries A metric space (X, d) is said to be a (uniquely) geodesic space if every two points xand yof X are joined by a (unique) geodesic, i.e., a map c: [0, l]⊆R→Xsuch that c(0) = x,c(l) = y, and d(c(t), c(t′)) = |t−t′|for all t, t′∈[0, l]. The image c([0, l]) of a geodesic forms a geodesic segment which joins xand yand is not necessarily unique. If no confusion arises, we use [x, y] to denote a geodesic segment joining xand y. A point zin Xbelongs to a geodesic segment [x, y] if and only if there exists t∈[0,1] such that d(x, z) = td(x, y) and d(y, z) = (1 −t)d(x, y) and we write z= (1 −t)x+ty for simplicity. Notice that this point may not be unique. A subset of Xis said to be convex if it contains any geodesic segment that joins every two points of it. Ageodesic triangle △(x, y, z) consists of three points x, y, z ∈X(the vertices of △) and three geodesic segments joining each pair of vertices (the edges of △). A geodesic line in Xis a subset of Xisometric to R. A geodesic space has the geodesic extension property if each geodesic segment is contained in a geodesic line. More on geodesic metric spaces can be found for instance in [7, 27]. The metric d:X×X→Ris said to be convex if for any x, y, z ∈Xone has d(x, (1 −t)y+tz)≤(1 −t)d(x, y) + td(x, z) for all t∈[0,1]. A geodesic space (X, d) is Busemann convex (introduced in [9]) if given any pair of geodesics c1: [0, l1]→Xand c2: [0, l2]→Xone has d(c1(tl1), c2(tl2)) ≤(1 −t)d(c1(0), c2(0)) + td(c1(l1), c2(l2)) for all t∈[0,1]. It is well-known that Busemann convex spaces are uniquely geodesic and with convex metric. A very important class of geodesic metric spaces are CAT(k) spaces (where k∈R), that is, metric spaces of curvature uniformly bounded above by kin the sense of Gromov. CAT(k) spaces are defined in terms of comparisons with the model spaces Mn k, which are the complete, simply connected, Riemannian n-manifolds of constant sectional curvature k. Since these model spaces are of essential importance in this work we give their definition directly as metric spaces and recall some of their properties. For a thorough treatment of such spaces and related topics the reader can check [7, 17]. The n-dimensional sphere Snis the set x∈Rn+1 : (x|x) = 1, where (·|·) is the Euclidean scalar product. Define d:Sn×Sn→Rby assigning to each (x, y)∈Sn×Snthe unique number 2 d(x, y)∈[0, π] such that cos d(x, y) = (x|y). Then (Sn, d) is a metric space called the spherical space. This is a geodesic space and if d(x, y)< π then there is a unique geodesic joining xand y. Also, balls of radius smaller than π/2 are convex. The spherical law of cosines states that in a spherical triangle with vertices x, y, z ∈Snand γthe spherical angle between the geodesic segments [x, y] and [x, z] we have cos d(y, z) = cos d(x, y) cos d(x, z) + sin d(x, y) sin d(x, z) cos γ. For u, v ∈Rn+1, consider the quadratic form given by hu|vi=−un+1vn+1 +Pn i=1 uivi. The hyperbolic n-space Hnis the set {u= (u1, u2, ..., un+1)∈Rn+1 :hu|ui=−1, un+1 >0}. Then Hn is a metric space with the hyperbolic distance d:Hn×Hn→Rassigning to each (x, y)∈Hn×Hn the unique number d(x, y)≥0 such that cosh d(x, y) = −hx|yi. The hyperbolic space is uniquely geodesic and all its balls are convex. The hyperbolic law of cosines states that in a hyperbolic triangle with vertices x, y, z ∈Hnand γthe hyperbolic angle between the geodesic segments [x, y] and [x, z] we have cosh d(y, z) = cosh d(x, y) cosh d(x, z)−sinh d(x, y) sinh d(x, z) cos γ. Let k∈Rand n∈N. The classical model spaces Mn kare defined as follows: if k > 0, Mn kis obtained from the spherical space Snby multiplying the spherical distance with 1/√k; if k= 0, Mn 0 is the n-dimensional Euclidean space En; and if k < 0, Mn kis obtained from the hyperbolic space Hnby multiplying the hyperbolic distance with 1/√−k. The model spaces inherit their geometrical properties from the three Riemaniann manifolds that define them. Thus, if k < 0, Mn kis uniquely geodesic, balls are convex and we have a counterpart of the hyperbolic law of cosines. If k > 0, there is a unique geodesic segment joining x, y ∈Mn kif and only if d(x, y)< π/√k. Moreover, closed balls of radius smaller than π/(2√k) are convex and we have a counterpart of the spherical law of cosines. We denote the diameter of Mn kby Dk. More precisely, for k > 0, Dk=π/√kand for k≤0, Dk=∞. Now we briefly introduce CAT(k) spaces. Let (X, d) be a geodesic space. A k-comparison triangle for a geodesic triangle △(x1, x2, x3) in (X, d) is a triangle △(¯x1,¯x2,¯x3) in M2 ksuch that dM2 k(¯xi,¯xj) = d(xi, xj) for i, j ∈ {1,2,3}. For kfixed, k-comparison triangles of geodesic triangles (having perimeter less than 2Dk) always exist and are unique up to isometry. A geodesic triangle △in Xis said to satisfy the CAT(k)inequality if, given ¯ △ak-comparison triangle for △, for all x, y ∈ △ d(x, y)≤dM2 k(¯x, ¯y), where ¯x, ¯y∈¯ △are the comparison points of xand y, respectively. If k≤0, a CAT(k) space is a geodesic space for which every geodesic triangle satisfies the CAT(k) inequality. If k > 0, a metric space is called a CAT(k) space if every two points at distance less than Dkcan be joined by a geodesic and every geodesic triangle having perimeter less than 2Dksatisfies the CAT(k) inequality. Let Xbe a metric space and C⊆X. We define the distance of a point z∈Xto Cby dist(z, C) = inf y∈Cd(z, y). The metric projection PConto Cis the mapping PC(z) = {y∈C:d(z, y) = dist(z, C)}for every z∈X. The following proposition gathers important properties of the metric projection in the setting of CAT(0) spaces. In [13], a counterpart of this result was given in the setting of CAT(k) spaces with k > 0. 3 Proposition 2.1 ([7], Proposition 2.4, p. 176).Let Xbe a complete CAT(0) space, x∈Xand C⊂Xnonempty closed and convex. Then the following hold: 1. The metric projection PC(x)of xonto Cis a singleton. 2. If y∈[x, PC(x)], then PC(x) = PC(y). 3. If x /∈Cand y∈Cwith y6=PC(x)then ∠PC(x)(x, y)≥π/2. 4. The mapping PCis a nonexpansive retraction from Xonto C. Further, the mapping H: X×[0,1] →Xassociating to (x, t)the point at distance td(x, PC(x)) on the geodesic [x, PC(x)] is a continuous homotopy from the identity map of Xto PC. We give next a concept of convergence in metric spaces. Let (X, d) be a metric space, (xn)⊆X a bounded sequence and x∈X. The asymptotic radius of (xn) is given by r((xn)) = inf lim sup n→∞ d(x, xn) : x∈X and the asymptotic center of (xn) is the set A((xn)) = x∈X: lim sup n→∞ d(x, xn) = r((xn)). The sequence (xn) in Xis said to ∆-converge to x∈Xif xis the unique asymptotic center of (un) for every subsequence (un) of (xn). This notion of convergence coincides with the weak convergence in Hilbert spaces. For more on this topic and other concepts of weak convergence in metric spaces the reader may see [18, 29, 20, 13, 19]. 3 Main results 3.1 Reflecting in geodesic spaces Let (X, d) be a uniquely geodesic space with the geodesic extension property. Let Cbe a nonempty closed and convex subset of Xand suppose the metric projection onto Cis well-defined and singlevalued. The reflection of a point x∈Xwith respect to Ccan be any point zin a geodesic line containing the geodesic segment [x, PCx] for which PCx=x+z 2. If Xhas no bifurcating geodesics, then geodesics can be extended in a unique way so the reflection of a point is uniquely determined. Thus, in such a setting, the reflection mapping RC which assigns to each point its reflection is well-defined and singlevalued. Recall that two geodesics bifurcate if they have a common endpoint and coincide on an interval, but one is not an extension of the other. Note however that one may also define singlevalued reflection mappings in spaces failing this property. We study next the reflection mapping in a nonlinear setting. By [16, Example 22.1], we know that in the complex Hilbert ball the reflection mapping fails to be nonexpansive which yields that in CAT(0) spaces this mapping is not necessarily nonexpansive in general. However, in the particular setting of model spaces, the reflection mapping proves to be nonexpansive. This shows that in model spaces, as in the case of Hilbert spaces, the metric projection is an averaged mapping with constant 1/2. 4 Proposition 3.1. Let k∈Rand n∈N. Suppose Cis a nonempty closed and convex subset of Mn kand x, y ∈Mn ksuch that dist(x, C),dist(y, C)< Dk/2. Then, d(RCx, RCy)≤d(x, y). Proof. First we consider the case k=−1. For simplicity, denote cx=PCx,cy=PCy,x′=RCx and y′=RCy. Let γ=∠cx(x, cy) and γ′=∠cy(y, cx). Notice that, by Proposition 2.1, γ, γ′≥π/2. Consider the geodesic triangles △(x, cx, cy) and △(x′, cx, cy). By the hyperbolic cosine law we have that cosh d(x, cy) = cosh d(x, cx) cosh d(cx, cy)−sinh d(x, cx) sinh d(cx, cy) cos γ and cosh d(x′, cy) = cosh d(x′, cx) cosh d(cx, cy)−sinh d(x′, cx) sinh d(cx, cy) cos(π−γ). Since d(x, cx) = d(x′, cx) and cos(π−γ)≥0, we get cosh d(x, cy)≥cosh d(x′, cy) and thus d(x, cy)≥d(x′, cy). Similarly we get d(y, cx)≥d(y′, cx). Consider now the geodesic triangles △(x, cx, y) and △(x′, cx, y) and denote β=∠cx(x, y). Applying again the hyperbolic cosine law we obtain that cosh d(x, y) = cosh d(x, cx) cosh d(cx, y)−sinh d(x, cx) sinh d(cx, y) cos β and cosh d(x′, y) = cosh d(x′, cx) cosh d(cx, y)−sinh d(x′, cx) sinh d(cx, y) cos(π−β). Adding these two equalities we get that cosh d(x, y) + cosh d(x′, y) = 2 cosh d(cx, x) cosh d(cx, y).(3.1) In a similar way we have that cosh d(x′, y′) + cosh d(x, y′) = 2 cosh d(cx, x) cosh d(cx, y′),(3.2) cosh d(x, y) + cosh d(x, y′) = 2 cosh d(cy, x) cosh d(cy, y),(3.3) cosh d(x′, y′) + cosh d(x′, y) = 2 cosh d(cy, y) cosh d(cy, x′).(3.4) Suppose now that d(x′, y′)> d(x, y). From (3.1) and (3.2), cosh d(x′, y′) + cosh d(x, y′) = 2 cosh d(cx, x) cosh d(cx, y′) ≤2 cosh d(cx, x) cosh d(cx, y) <cosh d(x′, y′) + cosh d(x′, y), which implies that d(x, y′)< d(x′, y). By using (3.3) and (3.4) in a similar way, we get that d(x, y′)> d(x′, y), which is a contradiction and thus the result follows. The proof in any other model space Mn kfollows similar patterns by using the corresponding law of cosines of each space. The above result also holds in the equivalent infinite dimensional spherical and hyperbolic spaces defined using points in ℓ2(see, for instance, H∞in [14] and [13, Example 3.4]). In the sequel, we analyze the reflection mapping in some particular CAT(k) spaces. More precisely, we consider the gluing of two model spaces Mn kand Mm k′(for more details on gluings see [7, Chapter I.5]). Note that two spaces of constant curvature can only be glued through geodesic lines, geodesic segments or singletons. First we consider the gluing of two model spaces by a singleton. Although the reflection mapping is not uniquely defined in this type of gluing spaces, there exists a natural way of defining it such that it is nonexpansive. 5 Proposition 3.2. Let k, k′∈R,k6=k′and m, n ∈N. Consider the gluing space (X, d)of Mn k and Mm k′through a point θ. Suppose Bis a nonempty closed and convex subset of Xand x∈X such that dist(x, B)<min{Dk, Dk′}/2. We define the reflection of a point with respect to Bin X in the following way: (1) If θ∈B,RBx=RB∩Mp jx, where Mp j∈ {Mn k, Mm k′}is the model space that contains xand RB∩Mp jis the reflection in the model space Mp jwith respect to B∩Mp j. (2) If θ /∈B, observe that Bis strictly contained in just one of the two model spaces that define the glued space X. Suppose B⊆Mn k. Then, (2.1) if x∈Mn k,RBxis the point obtained by reflecting in Mn kwith respect to B; (2.2) if x∈Mm k′, the reflection is uniquely defined since d(θ, PBθ)>0. Specifically, RBxis the point in the geodesic line in Mn kcontaining [θ, PBθ]such that d(PBθ, RBx) = d(x, PBθ) and d(RBx, θ)> d(RBx, PBθ). Then RBis nonexpansive. Proof. (1) Suppose θ∈B. We prove that RBdefined as in (1) is nonexpansive. Let x, y ∈X. The cases x, y ∈Mn kor x, y ∈Mm k′are immediate since RBreduces to the reflection in model spaces. Suppose now x∈Mn kand y∈Mm k′. Then, RBx=RB∩Mn kxand RBy=RB∩Mm k′y. Consider the geodesic triangles △1=△(RBx, PBx, θ) and △2=△(x, PBx, θ). Notice that △1,△2⊆Mn k. For simplicity, we consider k=−1 (the proof for k6=−1 follows similar patterns). Since ∠PBx(x, θ)≥ π/2, similarly as in the proof of Proposition 3.1, d(θ, x)≥d(θ, RBx). Likewise, we can see that d(θ, y)≥d(θ, RBy). Therefore, d(RBx, RBy) = d(RBx, θ) + d(θ, RBy)≤d(x, θ) + d(y, θ) = d(x, y), and the conclusion follows. (2) Next we suppose that θ /∈Band B⊆Mn k. We prove that RBdefined as in (2) is nonexpansive. Let x, y ∈X. As we mentioned before, the case x, y ∈Mn kis immediate. Let x, y ∈Mm k′. Since dist(θ, B)>0, we have that PBx=PBy=PBθand so d(RBx, RBy) = |d(x, PBx)−d(y, PBy)|=|d(x, θ)−d(y, θ)| ≤ d(x, y). Finally, suppose x∈Mn k,y∈Mm k′. As in the previous case, PBy=PBθ. Consider z=RPBθRBy∈ Mn k, where RPBθis the reflection mapping with respect to the point PBθinside Mn k. By definition we have PBθ∈[RBy, z]. Since in a model space geodesics do not bifurcate, the geodesic line containing [RBy, PBθ] coincides with the geodesic line containing [RBy, θ]. Let cbe this geodesic line. Then, by definition of the reflection, z∈c. Moreover, d(RBy, PBθ) = d(z, PBθ)> d(PBθ, θ) and d(θ, RBy)> d(RBy, PBθ). Thus, θ∈[PBθ, z]. As a consequence, d(PBθ, y) = d(PBθ, RBy) = d(PBθ, z) = d(PBθ, θ) + d(θ, z). On the other hand, PBz=PBθand therefore RBz=RBy. Finally, since d(x, z)≤d(x, θ)+d(z, θ) = d(x, y)−d(y, θ) + d(z, θ) = d(x, y) and the reflection in Mn kis nonexpansive, we get d(RBx, RBy) = d(RBx, RBz)≤d(x, z)≤d(x, y), and the result follows. 6 Next we see that when gluing two model spaces through a geodesic segment or a geodesic line, we cannot define the reflection in the gluing in such a way that it is nonexpansive with respect to every nonempty closed and convex set. To illustrate this fact we consider the particular gluing of H2and R2. Example 3.3. Let c:R→R2be the geodesic line in the plane defined by c(t) = (t, 0) and c′:R→ H2the geodesic line in the hyperbolic plane defined by c′(t) = (0,sinh t, cosh t). We consider the gluing of these two model spaces by the correspondence given by the isometry j:c(R)→c′(R) defined as j(c(t)) = c′(t)for every t∈R. Let x= (−arccosh√2, h)and y= (arccosh√2, h), where h > 0. Take p= (√z2−1,0, z), with z > 1and B={p}. We see that there exist values of hand zsuch that d(RBx, RBy)> d(x, y) = 2arccosh√2. Notice that d(y, p) = inf t∈R{d(y, c(t)) + d(c′(t), p)}= inf t∈Rq(arccosh√2−t)2+h2+ arccosh(zcosh t). Let f(t) = d(y, c(t)) + d(c′(t), p). Then, f′(t) = zsinh t pz2cosh2t−1−arccosh√2−t q(arccosh√2−t)2+h2 . Since lim t→∞ f′(t) = 2,lim t→−∞ f′(t) = −2and f′is increasing in t, then there exists only one value t0∈Rthat gives the infimum. Moreover, since f′(0) <0and f′(arccosh√2) >0, then t0∈ (0,arccosh√2). By using the symmetry of the space, we see that if t0is the value where the previous infimum is attained, then −t0is the value that gives d(x, p). Moreover, d(x, p) = d(y, p). Let d=d(RBx, RBy)and γbe the hyperbolic angle between the segments [p, c(t0)] and [p, c(−t0)]. By the hyperbolic cosine law, we have cosh 2t0= (cosh d(p, c′(t0)))2−(sinh d(p, c′(t0)))2cos γ and cosh d= (cosh d(x, p))2−(sinh d(x, p))2cos γ. If we consider for instance h= 1/100 and 1< z = (√2 + 15)/16 <√2, then t0≃0.8392 and cos γ≃ −0.7991. Consequently, cosh d≃3.7363 >cosh d(x, y) = 3. The underlying idea in this construction is the fact that geodesics in H2diverge faster than in R2. Since the previous construction can be easily adapted to the case of gluings through segments, we also obtain the same conclusion for this type of gluings. 3.2 Convergence results We begin this section by defining the averaged alternating reflection (AAR) method in the setting of geodesic spaces. Let (X, d) be a uniquely geodesic space. Given Aand Btwo nonempty closed and convex subsets of X, suppose that the reflection mappings RAand RBare well-defined and singlevalued. Consider the mapping T:X→Xdefined by T=I+RARB 2. The AAR method generates the following sequence for a starting point x0∈X:xn=Tnx0for every n≥1. 7 We prove next the convergence of the AAR method in the setting of model spaces with k≤0 which is an analogue of a weak convergence result in Hilbert spaces [4, Fact 5.9]. Note that we obtain strong convergence since the model spaces are proper metric spaces (see also [19, Proposition 4.4]; recall that a space is proper if every closed ball is compact). Theorem 3.4. Let k≤0and n∈N. Suppose Aand Bare two nonempty closed and convex subsets of Mn kwith A∩B6=∅. Let x0∈Mn kand (xn)be the sequence starting at x0generated by the AAR method. Then (1) (xn)converges to some fixed point xof the mapping Tand PBx∈A∩B. (2) The “shadow” sequence (PBxn)is convergent and its limit belongs to A∩B. Proof. (1) By Proposition 3.1, RARBis nonexpansive which yields that Tis also nonexpansive because Mn kwith k≤0 is Busemann convex. In addition, Fix(T)6=∅since A∩B6=∅. Thus, all orbits of Tare bounded. Using [15, Proposition 2] it follows that Tis asymptotically regular. Applying [1, Proposition 6.3], we get that (xn) ∆−converges to a fixed point of T, which implies that (xn) converges to a fixed point of T. Let xbe the limit of (xn). Since x=Tx it follows that x=RARBx. Moreover, because PBx=x+RBx 2, PBx=RARBx+RBx 2=PARBx∈A. Hence, PBx∈A∩B. (2) is immediate since the metric projection PBis continuous in CAT(0) spaces. We remark that a similar result holds in the gluing spaces described in Proposition 3.2 (when k, k′≤0). In H∞one obtains an analogous ∆-convergence result. In model spaces Mn kwith k > 0 the mapping Tused in defining the AAR method may fail to be nonexpansive as the following example shows. Example 3.5. Consider the points a= (1,0,0),b= (√2/2,√2/2,0) and p= (0,0,1) in the spherical space S2. Let c∈[a, p]and c′∈[b, p]such that d(a, c) = d(b, c′) = π/8. Suppose also A= [a, b]and B= [c, c′]. To see that the mapping Tis not nonexpansive it is enough to note that Tc =PAc,Tc′=PAc′, but d(Tc, Tc′) = d(PAc, PAc′) = d(a, b) = π/4> d(c, c′). Taking Bto be the whole positive octant, we obtain a similar example for the case A∩B6=∅. However, for model spaces Mn kwith k > 0 we can prove the result below. Note that the existence of a convergent subsequence of (xn) already guarantees that A∩B6=∅. Theorem 3.6. Let k > 0and n∈N. Suppose Aand Bare two nonempty closed and convex subsets of Mn k. Let C⊆Mn kbe nonempty convex with diam(C)< Dk/2such that A, B ⊆Cand RARB(C)⊆C. Let x0∈Cand (xn)be the sequence starting at x0generated by the AAR method. Then, (1) Any convergent subsequence (xnk)of (xn)converges to a fixed point xof the mapping Tand PBx∈A∩B. (2) The “shadow” sequence (PBxn)is bounded and each of its cluster points belongs to A∩B. 8 Proof. (1) Since diam(C)< Dk/2, we know that the metric is convex on C. Reasoning as in the previous theorem, we get that d(xn, xn+1)→0 as n→ ∞. Let (xnk) be a convergent subsequence of (xn) and xbe its limit. Since RARBis continuous we have RARBxnk→RARBx. Besides, (xnk+1) also tends to x, which implies that x=Tx. The fact that PBx∈A∩Bfollows as in the proof of the previous theorem. (2) Let (PBxnk) be a convergent subsequence of (PBxn) and ybe its limit. We may suppose that (xnk) is convergent (otherwise consider a convergent subsequence of it) and denote x= lim k→∞ xnk. From the continuity of the metric projection PB, we get that y=PBxand we are done. A more general way of defining the sequence (xn) would be to use the Krasnoselski-Mann [26, 23] iteration starting at x0∈X: xn+1 = (1 −λn)xn+λnRARBxn,(3.5) where (λn)⊆[0,1]. Results similar to Theorems 3.4, 3.6 hold for the sequence (xn) generated by (3.5) when assuming for instance that (λn) is divergent in sum and bounded away by 1. In this case, consider F= Fix(RARB)⊇A∩Band note that (d(xn, p)) is decreasing for each p∈F. Apply [15, Proposition 2] as above to get that d(xn, RARBxn)→0 (if k < 0, use asymptotic center techniques similar to those considered in [1, Proposition 6.3] to obtain the convergence of (xn)). Notice that now one obtains convergence to a fixed point of the mapping RARB(see also [12]). We finish this section by giving a rate of asymptotic regularity for the sequence (xn) generated by (3.5). Using proof mining methods, Kohlenbach [21] and later Kohlenbach and Leu¸stean [22] computed exponential (in 1/ε) rates of asymptotic regularity for the Krasnoselski-Mann iteration in normed and hyperbolic spaces, respectively. The next result which follows from [22, Corollary 3.18] gives an explicit bound on the rate of asymptotic regularity for the sequence (xn). Theorem 3.7. Let k∈Rand n∈N. Suppose Aand Bare two nonempty closed and convex subsets of Mn kwith A∩B6=∅. Let C⊆Mn kbe nonempty convex and of diameter bounded above by b∈(0, Dk/2) such that A, B ⊆Cand RARB(C)⊆C. Assume K∈N,K≥2and (λn)⊆1 K,1−1 K. Then, ∀x0∈X, ∀ε > 0,∀n≥Φ(ε, b, K), d(xn, xn+1)≤ε, where Φ(ε, b, K) := KM l2beK(M+1)m, with M≥(K−1)(1 + 2b) Kε . Thus, the above rate only depends on ε, on an upper bound bon the diameter of the set Cand on (λn) via K. Rates of asymptotic regularity for the Krasnoselski-Mann iteration were further studied and improved by Leu¸stean [24] in the setting of uniformly convex hyperbolic spaces that admit a modulus of uniform convexity which decreases with respect to the radius. As a consequence, one gets a quadratic in 1/ε rate of asymptotic regularity in CAT(0) spaces for constant λn=λ∈(0,1) [24, Corollary 19]. In particular, these results can be applied in the setting of model spaces Mn k with k≤0. 9