On best proximity points in metric and Banach spaces

arXiv:0911.5263v1 [math.FA] 27 Nov 2009 On best proximity points in metric and Banach spaces Rafa Esp´ınola & Aurora Fern´andez-Le´on∗ Abstract In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We do it from two different approaches, leading each one of them to different results which complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with. MS Classification: 54H25, 47H09 1 Introduction Let Aand Bbe two nonempty closed subsets of a complete metric space X. Consider a mapping T:A∪B→A∪Bsuch that T(A)⊆Band T(B)⊆A with the additional condition that there exists k∈(0,1) such that d(Tx, Ty)≤kd(x, y) for all x∈Aand y∈B, then A∩B6=∅and Thas a unique fixed point in A∩B. In [3, 4, 5, 17] a generalization of this situation was studied under the assumption of A∩B=∅. More precisely, in [3, 17] it was assumed that there exists k∈(0,1) such that d(Tx, Ty)≤kd(x, y) + (1 −k) dist(A, B) (1.1) for all x∈Aand y∈Bto obtain existence, uniqueness and convergence of iterates to the so-called best proximity points; that is, a point xeither in Aor Bsuch that d(x, Tx) = dist(A, B). This was first studied in [3] for uniformly convex Banach spaces, see also [11] for more on related topics. Then, in [17], the property UC (see Section 3 for definition) was introduced for a pair (A, B) of subsets of a metric space so a result on existence, uniqueness and convergence of iterates stands (Theorem 2.6 in Section 2) in general metric spaces. Since, as it is also proved in [17], property UC happens for a large collection of pairs of subsets of uniformly convex Banach spaces, Theorem 2.6 actually contains the main theorem of [3] (Theorem 3.10 in [3]) as a particular case. Property UC was even proved, in [17], to happen outside the setting of uniformly convex Banach spaces. In fact, this was obtained for UCED (uniformly convex in every direction) Banach spaces and strictly convex Banach spaces but in both cases under the very strong condition (see Theorem 3.7 in Section 3) of one of the sets to be of compact closure. In this work we first introduce a new property, the so-called property WUC, which is proved to happen under far less restrictive conditions than where ∗Both authors were partially supported by the Ministery of Science and Technology of Spain, Grant BFM 20000344-CO2-01 and La Junta de Antaluc´ıa project FQM-127. 1 property UC seems to reach, and then an existence, uniqueness and convergence theorem is proved for pairs of sets verifying property WUC. Second, we focus the same problem from the new approach suggested by one of the authors in [5] to obtain still new results on the same problem. As a result, a partial answer in the positive is given to a question raised in [3]. The work is organized as follows: in Section 2 we introduce most of the definitions, notations and previous results we will need. In Section 3 we look for weaker conditions than property UC. We introduce properties WUC and W-WUC and show that similar results to those in [17] hold under conditions which are easier to verify. In Section 4, we approach the same problem by the introduction of a semimetric. This is applied in a successful way by showing that the mappings verifying the contractive condition (1.1), under suitable assumptions, are contractions with respect to a certain semimetric. We finish this work with a remark on CAT(0) spaces. In [4] it was shown that when the ambient space is a Hilbert space then the kind of mappings we are dealing with actually behave as nonexpansive ones. In [5] it is shown that the semimetric there defined coincides with the metric of the ambient space when this is a Hilbert space. Our remark on CAT(0) spaces, in a certain sense the nonlinear counterparts of Hilbert spaces, states that something similar happens in these spaces. 2 Preliminaries In this section we compile the main concepts and results we will work with along this paper. We begin with some basic definitions and notations that are needed. Let (X, d) be a metric space and let Aand Bbe two subsets of X. Define dist(x, A) = inf{d(x, y) : y∈A}; PA(x) ={y∈A:d(x, y) = dist(x, A)}; dist(A, B) = inf{d(x, y) : x∈A, y ∈B}; diam(A) = sup{d(x, y) : x, y ∈D}. Recall that the set Ais said to be a Chebyshev set with respect to Bif PA(x) is a singleton for any x∈B. A metric space (X, d) is said to be a geodesic space (D-geodesic space, respectively) if every two points xand yof X(with d(x, y)≤D) are joined by a geodesic, i.e, a map c: [0, l]⊆R→X such that c(0) = x,c(l) = y, and d(c(t), c(t′)) = |t−t′|for all t, t′∈[0, l]. Moreover, (X, d) is called uniquely geodesic (D-uniquely geodesic) if there is exactly one geodesic joining xand yfor each x, y ∈X(with d(x, y)≤D). When the geodesic between two points is unique, its image (called geodesic segment) is denoted by [x, y]. The midpoint min between two points xand yin a uniquely geodesic metric space is the only point in [x, y] such that d(x, m) = d(y, m). Any Banach space is a geodesic space with usual segments as geodesic segments. Throughout this work we will just use geodesic metric space to refer to a uniquely geodesic space since all our geodesic spaces will be uniquely geodesic. A very important class of geodesic metric spaces are the CAT(k) spaces, that is, metric spaces of curvature uniformly bounded above by k. These spaces have been the object of a lot of interest by many researches and we will get back to them, especially to CAT(0) spaces, at certain moments of our exposition. For a very thorough treatment on CAT(k)-spaces the reader can check [2]. A subset Aof a geodesic metric space (X, d) is said to be convex if the geodesic joining each pair of points xand yof Ais contained in A. We will need the notion of uniformly convex geodesic metric space (see also [8, pg. 107]). 2 Definition 2.1 A geodesic metric space (X, d)is said to be uniformly convex if for any r > 0and any ε∈(0,2] there exists δ∈(0,1] such that for all a, x, y ∈Xwith d(x, a)≤r,d(y, a)≤rand d(x, y)≥εr it is the case that d(m, a)≤(1 −δ)r where mstands for a midpoint of the geodesic segment [x, y]. A mapping δ: (0,+∞)×(0,2] →(0,1] providing such a δ=δ(r, ε)for a given r > 0and ε∈(0,2] is called a modulus of uniform convexity. If moreover δdecreases with r(for a fixed ε) we say that δis a monotone modulus of uniform convexity of X. The notion of monotone modulus of uniform convexity seems to have been studied for first time in [13]. Of course, the usual modulus of convexity of a uniformly convex Banach space is monotone in this sense. For more on geometry of Banach spaces the reader can check [1, 7, 12]. Remark 2.2 If in the above definition we drop the uniformity conditions then we find the notion of strict convexity. More precisely, if Xis a Banach space and such a δexists for each a, x and yas above with d(x, y)>0, then we will say that Xis a strictly convex Banach space. If the same condition is imposed on a geodesic metric space X, then we can find the spaces of nonpositive curvature in the sense of Busemann, see [15] for a detailed study on them. Cyclic contractions and best proximity points are defined next. Definition 2.3 Let Aand Bbe two nonempty subsets of a metric space X. A map T:A∪B→ A∪Bis a cyclic contraction map if it satisfies: (1) T(A)⊆Band T(B)⊆A. (2) There is some k∈(0,1) such that d(T x, Ty)≤kd(x, y) + (1 −k) dist(A, B), for all x∈A and y∈B. Remark 2.4 Notice that condition (2) implies that Tis a relatively nonexpansive mapping, i.e., Tsatisfies that d(Tx, T y)≤d(x, y)for all x∈Aand y∈B, which were the main object of study in [4, 5]. Next we define the notion of best proximity point. Definition 2.5 Let Aand Bbe two nonempty subsets of a metric space X. Let T:A∪B→A∪B such that T(A)⊆Band T(B)⊆A. A point x∈A∪Bis said to be a best proximity point for T if d(x, Tx) = dist(A, B). Existence, uniqueness and convergence of iterates to a best proximity point for cyclic contractions have recently been studied in [3, 17]. The goal of this work is to find improvements of main results in these works. Next we state the main result from [17] (the definition of property UC is in Section 3). Theorem 2.6 Let (X, d)be a metric space and let Aand Bbe nonempty subsets of Xsuch that (A, B)satisfies the property UC. Assume that Ais complete. Let Tbe a cyclic contraction on A∪B. Then Thas a unique best proximity point zin Aand {T2nx}converges to zfor every x∈A. 3 Main result in [3] states basically the same but with Aand Bnonempty closed and convex, X a uniformly convex Banach space and no mention to property UC. In [5] a new approach to relatively nonexpansive mappings lead to the fact that such mappings, under suitable conditions, are actually nonexpansive with respect to an adequate semimetric. In Section 4 we apply this new approach to cyclic contractions. Next we introduce the main notions and results on semimetric spaces that we will need. Definition 2.7 Let Mbe a nonempty set. A function d:M×M→[0,∞)is said to be a semimetric on Mif (1) d(x, y) = 0 if, and only if, x=y. (2) d(x, y) = d(y, x)for any x, y ∈X. In this case, (M, d)is said to be a semimetric space. Contractions with respect to semimetrics are defined in a similar way to contractions with respect to metrics. Definition 2.8 Let (X, d)be a semimetric space. A mapping T:X→Xis said to be a contraction if there is a constant k∈(0,1) such that for all x, y ∈X d(Tx, Ty)≤kd(x, y). The next definition will make easier to state some of our results. Definition 2.9 Let Xbe a nonempty set. Let dand d1be a metric and a semimetric on X respectively. We say that dand d1are compatible on Xif for every ε > 0and x∈Xthere exist fx(ε)>0and gx(ε)>0such that Bd(x, fx(ε)) ⊆Bd1(x, ε)and Bd1(x, gx(ε)) ⊆Bd(x, ε), where Bd(x, r)and Bd1(x, r)stand, respectively, for the closed balls of center xand radius rwith respect to the metric and the semimetric. In [9], different counterparts of Banach’s contraction theorem are given for semimetric spaces. We state next a particular case of those results more adequate to our context (see Theorem 1 in [9]). Theorem 2.10 Let X,dand d1be as in the definition above with dand d1compatible. Let Tbe a contraction on Xfor the semimetric d1, then Thas a unique fixed point x0. Moreover, for any x∈Xthe sequence {Tnx}∞ n=1 converges to x0. We finish this section introducing two geometrical properties for Banach spaces. We begin describing property (H). Definition 2.11 Let Xbe a Banach space. Xis said to have the property (H)if for any sequence on the unit sphere of X, weak and norm convergence coincide. Remark 2.12 This property has been very extensively studied in the literature and it is closely related to the so-called Kadec-Klee property (KK-property, for short). For more on this topic, see [1, 7, 12, 14]. 4 We will also need the following uniform version of the KK property. Definition 2.13 Let Xbe a Banach space. Xis said to have the property UKK (uniform KadecKlee property) if for any ε > 0the number η(ε) = inf{1− kxk} >0, where the infimum is taken over all points xsuch that xis a weak limit for some sequence {xn}in the unit ball of Xwith kxn−xk ≥ εfor all n. Different properties of UKK Banach spaces as well as connection among all these geometrical notions can be found in the above-mentioned references. Let us just note here, as a matter of fact, that uniformly convex Banach spaces are UKK spaces and so they also have property (H). Both notions, uniform convexity and property UKK, have to do with a certain rotoundity of the balls of the space. This is obvious for uniform convexity and far less obvious for property UKK as there exist Banach spaces which are UKK and not even strictly convex. 3 The UC and WUC properties Property UC was defined in [17] in the following way. Definition 3.1 Let Aand Bbe nonempty subsets of a metric space (X, d). Then (A, B)is said to satisfy the property UC if for {xn}and {x′ n}sequences in Aand {yn}a sequence in Bsuch that limnd(xn, yn) = limnd(x′ n, yn) = dist(A, B), then limnd(xn, x′ n) = 0. The following proposition shows the uniform nature of property UC. Proposition 3.2 For Aand Bnonempty subsets of a metric space X, the following are equivalent: (i) (A, B)has property UC. (ii) For any ε > 0there exists δ > 0such that diam(A∩B(y, dist(A, B) + δ)) ≤εfor any y∈B. Proof. First we see (i)⇒(ii). Supposing the contrary implies that there is ε0>0 such that for every δ= 1/n there exist yn∈Band xn, x′ n∈Asatisfying d(yn, xn)≤dist(A, B) + 1 n, d(yn, x′ n)≤dist(A, B) + 1 nand d(xn, x′ n)> ε0, which obviously contradicts property UC. Now we prove (ii)⇒(i). Let xn, x′ n∈Aand yn∈Bsuch that d(yn, xn) and d(yn, x′ n) both converge to dist(A, B) as n→ ∞. Then given ε > 0, there is δ > 0 such that diam(A∩B(y, dist(A, B)+ δ)) ≤εfor any y∈B. Now, it is enough to take n0∈Nsuch that d(xn, yn), d(x′ n, yn)≤ dist(A, B) + δfor any n≥n0to deduce that d(xn, x′ n)≤εfor n≥n0.2 In [17] it was shown that any pair of nonempty subsets (A, B) of uniformly convex Banach spaces with Aconvex enjoy the property UC. Next we show that something similar can be said for uniformly convex geodesic spaces under adequate conditions on the modulus of convexity. Proposition 3.3 Let (X, d)be a uniformly convex geodesic metric space with a monotone modulus of convexity δ(r, ε). Let Aand Bbe two nonempty subsets of Xwith Aconvex. Then the pair (A, B)has property UC. 5 Proof. Suppose on the contrary that there exist {xn}and {x′ n}sequences in A,{yn}in B and ε0>0 such that for every k∈N, there exist nk≥kfor which d(xnk, x′ nk)≥ε0while limn→∞ d(xn, yn) = limn→∞ d(x′ n, yn) = dist(A, B). There is no loss of generality in assuming that δ(r, ε)<1 for r, ε > 0 and that dist(A, B)>0 since otherwise the result follows in a trivial way. For γ > dist(A, B) and ε1=ε0/γ, choose ε > 0 such that ε < min γ−dist(A, B),dist(A, B)δ(γ, ε1) 1−δ(γ, ε1). Then there exists N0∈Nsuch that if nk≥N0, then d(xnk, ynk)≤dist(A, B) + εand d(x′ nk, ynk)≤ dist(A, B) + ε. Let mnkbe the mid-point of the geodesic segment [xnk, x′ nk]. Using the uniform convexity of X, we have that d(ynk, mnk)≤1−δ(dist(A, B) + ε, ε1)(dist(A, B) + ε)≤ ≤1−δ(γ, ε1)(dist(A, B) + ε)<dist(A, B). Then, for nk≥N0 d(ynk, mnk)<dist(A, B), which contradicts the fact that mnk∈Aby convexity of A.2 Remark 3.4 Notice that the same result remains true if the condition on the monotonicity of the modulus of convexity is replaced by the condition of being lower semi-continuous from the right. As it was pointed in the Introduction, property UC was also shown in [17] to happen in UCED Banach spaces and strictly convex Banach spaces but requesting Ais relatively compact. Regarding the assumption on the compactness of Athe following result from [3] is relevant. Theorem 3.5 Let Aand Bbe nonempty closed subsets of a metric space (X, d)and let T:A∪B→ A∪Bbe a cyclic contraction. If either Aor Bis boundedly compact, then there exists xin A∪B with d(x, Tx) = dist(A, B). Notice that what we miss from Theorem 2.6 in this theorem is uniqueness and convergence of iterates. We see next that this is easy to obtain by adding the very mild condition (see the remark below to support this idea) of being Aa Chebyshev set for proximinal points with respect to B. Definition 3.6 Given Aand Btwo nonempty subsets of a metric space, we say that Ais a Chebyshev set for proximinal points with respect to Bif for any x∈Bsuch that dist(x, A) = dist(A, B)we have that PA(x)is a singleton. Then we can prove the following. Theorem 3.7 If in the above theorem, Ais supposed to be boundedly compact and a Chebyshev set for proximinal points with respect to B, then the best proximity point z∈Ais unique and the sequence {T2nx}converges to zfor any x∈A. Proof. We first show it is unique. Suppose zand z′are two best proximity points in A with z6=z′. Then the Chebyshev condition on Aimplies that Tz 6=Tz′. Now, the relative nonexpansivity of Timplies that d(T2z, Tz)≤d(z, T z) = dist(A, B) 6 and so, the Chebyshev condition on Aalso implies that zand z′are fixed points for T2. If we write d∗(x, y) = d(x, y)−dist(A, B) then d∗(z, Tz′) = d∗(T2z, Tz′) ≤kd∗(z′, Tz) = kd∗(T2z′, Tz)≤k2d∗(z, T z′). Hence d∗(z, T z′) = 0 and so z=z′. Finally the converges of the iterates follows directly from the facts that Ais boundedly compact, the sequences {T2nx}are bounded for any x∈Aand that lim d(T2nx, Tz) = dist(A, B) for any x∈A.2 Remark 3.8 Notice that the condition of being Chebyshev is a very natural one in this kind of problems. Think otherwise on the sets A={(x, 0) : x∈[0,1]}and B={(x, 1) : x∈[0,1]} as subsets of the plane with the maximum norm. Then any mapping T:A∪B→A∪Bwith T(A)⊆Band T(B)⊆Ais a cyclic contraction. We suggest to replace property UC with the weaker one WUC which we define next. Definition 3.9 Let Aand Bbe nonempty subsets of a metric space (X, d). Then (A, B)is said to satisfy the property WUC if for any {xm} ⊆ Asuch that for every ε > 0there exists y∈B satisfying that d(xm, y)≤dist(A, B) + εfor m≥m0, then it is the case that {xm}is convergent. Remark 3.10 Another alternative for the above definition is to ask the sequence {xm}to be Cauchy instead of convergent. It is worthwhile to note here that this is quite a detail of a formal nature since in all our main results we always assume Ato be complete. Next proposition gives the relation between the two mentioned properties. Proposition 3.11 Let Aand Bbe nonempty subsets of a metric space (X, d)such that Ais complete. Suppose the pair (A, B)has property UC. Then (A, B)has property WUC. Proof. Let {xm} ⊆ Abe such that for every δ > 0 there exists y∈Bsatisfying that d(xm, y)≤ dist(A, B)+δfor m≥m0. It suffices to show that {xm}is a Cauchy sequence. This follows directly from (ii) of Proposition 3.2. 2 Next we show that property WUC implies a nonuniform version of the equivalence given by Proposition 3.2 for property UC. We omit its proof. Proposition 3.12 Let Aand Bbe nonempty subsets of a metric space (X, d). Suppose (A, B)has property WUC then lim ε→0diam(A∩B(y, dist(A, B) + ε)) = 0 for any y∈B. The next propositions show that property WUC is likely to happen in more situations than property UC. We first weaken the notion of uniform convex geodesic space with a monotone modulus of convexity. 7 Definition 3.13 A geodesic metric space (X, d)is said to be pointwise uniformly convex if for any a∈X,r > 0and ε∈(0,2] there exists δ=δ(a, r, ε)∈(0,1] such that for all x, y ∈Xwith d(x, a)≤r,d(y, a)≤rand d(x, y)≥εr it is the case that d(m, a)≤(1 −δ)r where mstands for the midpoint of the geodesic segment [x, y]. A mapping δ:X×(0,+∞)×(0,2] → (0,1] providing such a δfor a given a∈X,r > 0and ε∈(0,2] is called a modulus of pointwise uniformly convexity (or just modulus of convexity when confusion cannot arise). If moreover δ decreases with r(for εand a) we say that δis a monotone modulus of pointwise uniformly convexity of X. Remark 3.14 Notice that both notions of uniform convexity coincide for Banach spaces. Proposition 3.15 Let (X, d)be a complete pointwise uniformly convex geodesic metric space with monotone modulus of convexity. Let Aand Bbe two nonempty subsets of Xwith Aconvex. Then the pair (A, B)has property WUC. Proof. Let {xm}be a sequence in Asuch that for every ε > 0 there exist y∈Band m0∈N satisfying that d(xm, y)≤dist(A, B) + εfor m≥m0. Suppose {xm}is not convergent, then there exists ε0>0 such that for each k∈Nthere are nk, mk≥kfor which d(xnk, xmk)≥ε0. Thus, for k=m0, we find nm0, mm0≥m0such that d(xnm0, y)≤dist(A, B) + ε, d(xmm0, y)≤dist(A, B) + ε and d(xnm0, xmm0)≥ε0. Let zm0be the mid-point in the segment [xnm0, xmm0]. Since Xis pointwise uniformly convex, we obtain that d(zm0, y)≤(dist(A, B) + ε)(1 −δ), for some δ=δ(y, dist(A, B) + ε, ε1)∈(0,1] as in the proof of Proposition 3.3. The contradiction follows from the fact that we can repeat this reasoning for any ε > 0 with yand ε0fixed. 2 Remark 3.16 This kind of modulus has been previously used for hyperbolic spaces in [16]. Proposition 3.17 Let Xbe a UKK reflexive and strictly convex Banach space. Then, for A, B ⊆ Xnonempty and convex, it is the case that (A, B)has the property WUC. Proof. Let {xn} ⊆ Abe as in the above proof. Suppose {xn}is not convergent. First we show that this sequence needs to have a separated subsequence. Consider two convergent subsequences {xnk}and {xnl}of xnwith respective limits xand x′in the closure of A. For each n∈Nchoose yn∈Bsuch that the tales of both subsequences are in B(yn,dist(A, B) + 1/n). Then it is clear that x, x′∈\ n∈N B(yn,dist(A, B) + 1/n). Since {yn}is bounded we can assume it is weakly convergent to a point yin the closure of B. Then it must be the case that x, x′∈B(y, dist(A,B)) from where, since Xis strictly convex, x=x′. Therefore we can assume that {xn}does not have any convergent subsequence and so it is a separated sequence. Let ε > 0 such that d(xn, xm)≥εfor every n6=m. Since this sequence 8 is bounded and Xis reflexive, we can also assume {xn}is weakly convergent to a point x. Now we only have to apply the UKK property in a similar way as the uniform convexity was applied in the previous proposition to deduce that xis in the closure of Abut dist(x, B)<dist(A, B), contradicting the definition of dist(A, B). 2 The UKK property for ∆-convergent sequences has been recently studied in [6, 10] for CAT(k) spaces. If we assume that Xis a geodesic space such that bounded sequences have a unique asymptotic center which belongs to the convex hull of the sequence (see any of [6, 10] for definitions), the previous proposition finds a metric counterpart that we state next and which can be proved exactly the same. Proposition 3.18 Let Xbe a geodesic metric space with the UKK property for ∆-convergent sequences and the above-mentioned property for bounded sequences. Suppose also that the function µ(r, ε)given by the UKK property decreases with respect to the radius, then, for A, B ⊆Xnonempty and convex, it is the case that (A, B)has the property WUC. Remark 3.19 Although the situation for Banach spaces is clear in the sense that uniform convexity implies property UKK, the same seems far to be the case for geodesic spaces as defined for ∆- convergent sequences. Actually the UKK property for CAT(k)spaces as shown in [6, 10] seems to be more connected with the so-called Opial condition than with the uniform convexity. It is worth to recall at this point that only Hilbert spaces and the spaces of sequences ℓpare known to enjoy the Opial property. For more on this and related topics the interested reader can consult Chapters 3, 4, 5 and 16 in [12] or [1, p. 102]. Next we show that WUC is enough to lead to a best proximity point for a cyclic contraction. Due to notation purposes, we will denote ras the contractive constant in the definition of cyclic contraction for the remainder of this section. Theorem 3.20 Let (X, d)be a metric space and Aand Btwo nonempty subsets of Xsuch that (A, B)satisfies the property WUC. Assume that Ais complete. Let Tbe a cyclic contraction on A∪B. Then Thas a unique best proximity point zin Aand the sequence {T2nx}converges to z for every x∈A. Proof. As in [17] we consider d∗(x, y) = d(x, y)−dist(A, B). Then d∗(Tx, Ty)≤rd∗(x, y) for x∈Aand y∈B. In consequence d∗(T2x, T x)≤rd∗(x, Tx) and d∗(Ty, T 2y)≤rd∗(Ty, y) for any x∈Aand y∈B. Fix x∈A,n∈Nand let m=n+kwith k∈N. Then d∗(T2mx, T2n+1x)≤r2nd∗(T2kx, Tx) ≤r2nsup{d(Tx, T 2kx) : k∈N}=r2nM(x). Proposition 3.3 in [3] guarantees that M(x) is finite for each x. Hence, given ε > 0 and taking nsuch that r2nM(x)< ε we have that T2mx∈B(T2n+1x, dist(A, B) + ε) (3.2) for m≥nand so, by the property WUC, {T2nx}is convergent. Now the proof follows the same patterns than the proof of Theorem 3 in [17]. Let z∈Abe the limit of {T2nx}, then d∗(z, Tz) = lim n→∞ d∗(T2nx, Tz)≤lim n→∞ rd∗(z, T2n−1x) 9 Proposition 4.14 The pair (A0, B0)is a nonempty, closed and convex pair in X. Furthermore, each point b∈B0can be joined through a geodesic segment of length d= dist(A, B)to its proximinal point b−hin A0and viceversa. Proof. That they are closed follows in a straightforward way from their definition and the fact that Aand Bare both closed. The fact that A0and B0are nonempty also follows in a similar way to the linear case under the assumption of reflexivity, since it is a very well-known fact (see [6, 10]) that decreasing sequences of nonempty bounded closed and convex subsets of a CAT(0) space have nonempty intersection. Finally, the convexity of the sets A0and B0follows from the convexity of the metric of CAT(0) spaces (see Proposition 2.2 in Chapter II.2 of [2]). 2 The next thing we need to do is to define the semimetric d1on B0. We will define it in such a way that a third set C0is not needed. Definition 4.15 We define the function d1:B0×B0→[0,∞)by d1(x, y) = inf{r > 0 : y∈B(x−h, d +r)and y−h∈B(x, d +r)}, where d=dist(A, B). Remark 4.16 Notice that Definition 4.5 and Definition 4.15 coincide in linear spaces. Theorem 4.17 The semimetric d1coincides with the metric dinduced by Xon B0. Proof. This result follows as an easy application of The Flat Quadrilateral Theorem ([2, p.181]). Indeed, consider the four point x, y, x −hand y−h. Then x(respectively, y) is the proximinal point of x−h(rep., y−h) in B0, and viceversa. In consequence, the angles ∠x(x−h, y), ∠y(x, y − h), ∠y−h(x−h, y) and ∠x−h(y−h, x) are all greater than or equal to π/2. Therefore the Flat Quadrilateral Theorem implies that the convex hull of the points x, y, x −hand y−his isometric to a rectangle in the 2-dimensional Euclidean space. Now, by the Pythagorean theorem, it is immediate to deduce that B1(x, pd2+r2−d) = B0∩B(x−h, pd2+r2) = B0∩B(x, r), as we wanted to proof. 2 We close this appendix by observing that it is also possible to show that the mapping T′(b) = Tb +hfor b∈B0is actually a contraction. To see this we just need to proceed as in the proof of Theorem 4.10 and recall, at the proper moment, that the convex hull of the points x, y, x −hand y−his actually a rectangle. References [1] J. M. Ayerbe, T. Dom´ınguez, and G. L´opez, Measures of Noncompactness on Metric Fixed Point Theory, Birkh¨auser, 1997. [2] M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Springer-Verlag, Berlin Heidelberg, 1999. [3] A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. 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Departamento de An´alisis Matem´atico Facultad de Matem´aticas Universidad de Sevilla P.O.Box: 1160 41080-Sevilla emails: [email protected], [email protected] 17