On selections of the metric projection and best proximity pairs in hyperconvex spaces

ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA L U B L I N – P O L O N I A VOL. LIX, 2005 SECTIO A 9–17 RAFA ESP´ INOLA On selections of the metric projection and best proximity pairs in hyperconvex spaces Dedicated to W. A. Kirk on the occasion of his receiving an Honorary Doctorate from Maria Curie-Skłodowska University Abstract. In this work we present new results on nonexpansive retractions and best proximity pairs in hyperconvex metric spaces. We sharpen the main results of R. Esp´ınola et al. in [3] (Nonexpansive retracts in hyperconvex spaces, J. Math. Anal. Appl. 251 (2000), 557–570) on existence of nonexpansive selections of the metric projection. More precisely we characterize those subsets of a hyperconvex metric space with the property that the metric projection onto them admits a nonexpansive selection as a subclass of sets introduced in [3]. This is a rather exceptional property with a lot of applications in approximation theory, in particular we apply it to answer in the positive the main question posed by Kirk et al. in [5] (Proximinal retracts and best proximity pair theorems, Num. Funct. Anal. Opt. 24 (2003), 851–862). 1. Introduction. In [3] the author et al. introduced a subclass of subsets of a metric space, the so-called weakly externally hyperconvex subsets 2000 Mathematics Subject Classification. 47H09, 41A65, 46B20. Key words and phrases. Hyperconvex spaces, metric projections, proximinal sets, best proximity pairs. The work of the author is partially supported by the Ministry of Science and Technology of Spain Grant BFM 2003-3893-C02-01 and Junta de Andaluc´ıa project FQM127. 10 R. Esp´ınola (see Section 2 for definitions), with the goal of characterizing those subsets of a metrically convex metric space for which there exists a nonexpansive selection of the metric projection. In this work we give the definitive solution to the main problems studied in that paper. More precisely it is proved that if Mis a metrically convex metric space, Ais a weakly externally hyperconvex of Mand PAis the metric projection on A(i.e. PA(x) = {y∈A:d(x, y) = inf{d(x, u) : u∈A}}) then there exists a nonexpansive selection Rof PA, this is R(x)∈PA(x)and d(R(x), R(y)) ≤d(x, y)for x, y ∈M. This is a rather exceptional property for a subset of a metric space which has a large number of nice consequences related to best approximation results (see for instance [3] and references therein). We apply this result to answer in the positive a question on best proximity pairs posed by Kirk et al. in [5]. Best proximity pairs raise in a very natural way in approximation theory when studying the proximity of two sets. For a proper motivation on best proximity pairs and their relation to fixed point theory the reader may check [5] and references therein. A subset Eof a metric space Mis said to be proximinal if given any x∈Mthere exists px∈Esuch that d(x, px) = dist(x, E) = inf{d(x, y) : y∈E}. For Aand Bnonempty subsets of a metric space let dist(A, B) = inf{d(x, y) : x∈Aand y∈B}. Definition 1.1. Let Xbe a metric space and let Aand Bbe nonempty subsets of X. Let A0={x∈A:d(x, y) = dist(A, B)for some y∈B}; B0={x∈B:d(x, y) = dist(A, B)for some y∈A}. A pair (x, y)∈A0×B0for which d(x, y) = dist(A, B)is called a best proximity pair for Aand B. In particular, it is proved in [5] that if Mis a hyperconvex metric space and Aand Bare nonempty admissible subsets of M, then A0and B0are nonempty and hyperconvex (see also Proposition 2.14 in [5]). As a consequence of our main theorem we can extend this result to Aand Bweakly externally hyperconvex subsets of M. Next this is applied to answer in the positive a question posed in [5]. The last result of this work is another application of our main theorem, in this case we obtain the nonexpansive version of the Ky Fan’s theorem given in [3] for hyperconvex spaces. 2. Definitions and preliminary results. This section contains the definitions and results that will be needed in the sequel. Hyperconvex metric spaces were introduced in 1956 by Aronszajn and Panitchpakdi in [1], for a detailed exposition on hyperconvex spaces the reader may consult the recent survey on them by the author and Khamsi [2]. On selections of the metric projection... 11 Definition 2.1. A metric space Mis said to be hyperconvex if given any family {xα}of points of Mand any family {rα}of real numbers satisfying d(xα, xβ)≤rα+rβ it is the case that TαB(xα;rα)6=∅. Next we give the definition of two subclasses of subsets of metric spaces. Definition 2.2. A subset Aof a metric space Mis said to be admissible (in M) if it is an intersection of closed balls of M. Thus Ais admissible if A=Ti∈IB(xi, ri)where xi∈Mand ri≥0for i∈I. Definition 2.3. A subset Eof a metric space Mis said to be externally hyperconvex (relative to M) if given any family {xα}of points in Mand any family {rα}of real numbers satisfying d(xα, xβ)≤rα+rβand dist(xα, E)≤rα it follows that TαB(xα;rα)∩E6=∅. Externally hyperconvex subsets were shown in [4] to enjoy nice properties as, for instance, being always proximinal. The following theorem also gives a very important property of externally hyperconvex sets. Theorem 2.4 ([4]).Let Mbe hyperconvex, Sa metric space and T?a multivalued mapping from Sinto Msuch that T?(x)is bounded nonempty externally hyperconvex for each x∈S, then there exists a selection T:S→ Mof Tsuch that: d(T(x), T(y)) ≤dH(T?(x), T?(y)) for all x, y ∈S, where dHdenotes the usual Hausdorff metric on the family of nonempty bounded closed subsets of M. The following notion plays a crucial role in this work. Definition 2.5. A subset Eof a metric space Mis a proximinal nonexpansive retract of Mif there exists a nonexpansive retraction Rof Monto Efor which d(x, R(x)) = dist(x, E) for each x∈M. Thus d(R(x), R(y)) ≤d(x, y)for each x, y ∈M. In an effort to characterize those subsets of a hyperconvex metric space which are proximinal nonexpansive retracts, the following definition was introduced in [3]. Definition 2.6. A subset Eof a metric space Mis said to be weakly externally hyperconvex (relative to M) if Eis externally hyperconvex relative to E∪ {z}for each z∈M. Precisely, given any family {xα}of points in M 12 R. Esp´ınola all but at most one of which lies in E, and any family {rα}of real numbers satisfying d(xα, xβ)≤rα+rβ,with dist(xα, E)≤rαif xα/∈E, it follows that TαB(xα;rα)∩E6=∅. It directly follows from the definition that weakly externally hyperconvex subsets are proximinal. At this point it is interesting to note that when the three classes of subsets so far presented are subsets of the same hyperconvex metric space M, then they are related in the following way: let Abe a subset of M, then Ais admissible (in M)⇒Ais externally hyperconvex (relative to M) ⇒Ais weakly externally hyperconvex (relative to M) ⇒Ais hyperconvex. The next definition was introduced in [6]. Definition 2.7. Let Abe a subset of a metric space M. A mapping R: A→Mis said to be ε-constant if d(x, R(x)) ≤εfor each x∈A. For Aas above the ε-neighborhood of Ais defined as follows: Nε(A) = [ a∈A B(a, ε). The following fact will be needed. Lemma 2.8 ([3]).Let Abe a weakly externally hyperconvex subset of a hyperconvex metric space M, then for any ε > 0the set Nε(A)is weakly externally hyperconvex and there is an ε-constant nonexpansive retraction of Nε(A)on A. 3. Proximinal nonexpansive retracts and best proximity pairs. We begin this section by recalling Theorem 3.1 in [3]. Theorem 3.1. Suppose Ais a weakly externally hyperconvex subset of a metrically convex metric space M. Then given any ε > 0there exists a nonexpansive retraction R:M→Awith the property that if u∈M\A there exists v∈M\Awith d(v, R(v)) = dist(v, A)and d(u, v)≤ε. Given Aand Mas above the ε-level set of Awith respect to Mis defined as follows: Sε={v∈M: dist(v, A) = ε}. The following corollary, although not stated in [3], is however a consequence of the proof of the previous theorem. Corollary 3.2. Let ε > 0,Aand Mas above, and S=Sn∈NSnε, then the retraction given by Theorem 3.1 can be chosen so that d(v, R(v)) = dist(v, A) for any v∈S. On selections of the metric projection... 13 Our first result is the next technical lemma on Theorem 3.1. Lemma 3.3. Let Abe a weakly externally hyperconvex subset of a metrically convex metric space M. For each n∈Nlet εn=1 2n, then there exists a nonexpansive retraction rn(associated to εn) as in Corollary 3.2 such that the sequence of retractions {rn}satisfies that d(rn(x), rm(x)) ≤ j=m X j=n+1 1 2j for x∈Mand n<m. Proof. For n= 1 we take r1as the one given by Corollary 3.2. We prove next that given rifor 1≤i≤nas in the statement of the lemma we can construct rn+1 as required. We consider Sεn+1 and proceed as in Theorem 3.1. After applying Zorn’s Lemma we may assume that Hεn+1 is the maximal subset of Sεn+1 where rn+1 can be extended as required. Then we need to prove that Sεn+1 =Hεn+1 . Suppose that there exits v∈Sεn+1 \Hεn+1 and let P(v) = \ x∈A B(x, d (x, v))!∩ \ u∈Hεn+1 B(rn+1 (u), d (u, v))! ∩Bv, 1 2n+1 ∩Brn(v),1 2n+1 ∩A. All we need to prove is that P(v)6=∅. Since Ais weakly hyperconvex and only one of the above balls is centered outside A, it is enough to check that each two of such balls have nonempty intersection. In a case-by-case check it only rests to study those cases involving the ball centered at rn(v), other cases were already studied in [3]. For these cases it is enough to recall that d(x, rn(v)) ≤d(x, v)for x∈A, now, since Ais proximinal, let pv∈Asuch that d(v, pv) = dist(v, A), so d(v, rn(v)) ≤d(v, pv) + d(pv, rn(v)) ≤2 dist(v, A) = 1 2n, and finally, for u∈Hεn+1 , d(rn+1(u), rn(v)) ≤d(rn+1(u), rn(u))+d(rn(u), rn(v)) (by induction hypothesis) ≤1 2n+1 +d(u, v). So we can consider rn+1 defined on the whole Sεn+1 as required. Next we show how to extend rn+1 to A∪Sεn+1 ∪S2εn+1 . Let v∈S2εn+1 =Sεn, then 14 R. Esp´ınola the set P(v) = \ x∈A B(x, d (x, v))!∩ \ u∈Sεn+1 B(rn+1 (u), d (u, v))! ∩Bv, 1 2n∩Brn(v),1 2n+1 ∩A is nonempty since d(rn(v), x)≤d(v, x)for x∈A,d(rn+1(u), rn(v)) ≤ d(rn+1(u), rn(u)) + d(rn(u), rn(v)) ≤(by induction) 1 2n+1 +d(u, v), and, since the metric convexity of Mimplies that there exists ˆv∈Sεn+1 such that d(v, ˆv) = 1 2n+1 , we have d(v, rn(v)) ≤d(v, ˆv) + d(ˆv, rn(ˆv))+d(rn(ˆv), rn(v)) ≤1 2n+1 +1 2n+1 +1 2n+1 =1 2n+1 2n+1 . Now, by selecting a point in P(v)it is possible to extend rn+1 as required from Sεn+1 to Sεn+1 ∪ {v}. This same argument shows how to extend rn+1 to A∪Sεn+1 ∪S2εn+1 onto Aas required. Let S=S∞ i=1 Siεn+1 . By proceeding as above but selecting ˆv∈S(i−1)εn+1 for v∈Siεn+1 , and using induction it follows that there exists a nonexpansive retraction rn+1 of A∪Sonto Aas required. Let v∈M\(A∪S), then we consider the set P(v) = \ x∈A∪S B(rn+1(x), d(x, v))!∩Brn(v),1 2n+1 . P(v)is nonempty from the hyperconvexity of Aand the fact that, by induction hypothesis, d(rn+1(x), rn(v)) ≤d(rn+1(x), rn(x)) + d(rn(x), rn(v)) ≤d(x, v) + 1 2n+1 . Again, using induction it follows that rn+1 can defined on Mas required. Hence d(rn+1(x), rn(x)) ≤1 2n+1 for x∈M. Now let {rn}be the sequence of retractions given by the above procedure, then for m > n and x∈M d(rm(x), rn(x)) ≤ j=m X j=n+1 d(rj(x), rj−1(x)) ≤ j=m X j=n+1 1 2j. On selections of the metric projection... 15 Hence the proof of the lemma is completed.  The following corollary is an immediate consequence of the previous lemma. Corollary 3.4. Let {rn}be the sequence of retractions given by Lemma 3.3, then {rn(x)}is convergent for each x∈M. Proof. To proof this corollary it is enough to recall that hyperconvex spaces are complete, hence Ais complete.  Next we present the main result of this work. Theorem 3.5. Let Abe a complete weakly externally hyperconvex subset of a metrically convex metric space M, then Ais a proximinal nonexpansive retract of M. Proof. Let {rn}be the sequence of retractions given by Lemma 3.3, then we define the mapping r:M→Aas r(x) = lim n→∞ rn(x). Corollary 3.4 implies that ris a well-defined retraction on A. Moreover, since rnis nonexpansive for each n∈N,ris nonexpansive. Additionally we claim that d(r(x), x) = dist(x, A)for x∈M. For x∈Athere is nothing to prove, so let x∈M\A. For each n∈Nthere exists vn∈M\Asuch that d(x, vn)≤1 2nand d(vn, rn(vn)) = dist(v, A). Hence we have d(x, rn(x)) ≤d(x, vn) + d(vn, rn(vn))+d(rn(vn), rn(x)) ≤1 2n+ dist(vn, A) + 1 2n ≤1 2n+ dist(x, A) + d(vn, x) + 1 2n =3 2n+ dist(x, A). Taking limit as n→ ∞ the conclusion follows.  Since Theorem 2.1 of [3] implies that proximinal nonexpansive retracts of hyperconvex spaces are weakly externally hyperconvex, the previous theorem can be re-written in the following way. Theorem 3.6. Let Mbe a hyperconvex metric space and let A⊆Mbe nonempty. Then Ais a proximinal nonexpansive retract of Mif, and only if, Ais a weakly externally hyperconvex subset of M. 16 R. Esp´ınola Next we give applications of Theorems 3.5–3.6. We begin with an application to the existence of best proximity pairs, in particular we have the following extension of Proposition 2.8 in [5]. Corollary 3.7. Let Mbe a hyperconvex metric space and let Aand Bbe nonempty weakly externally hyperconvex subsets of M. Then A0and B0are nonempty and hyperconvex. Proof. The same proof of Proposition 2.8 in [5] carries over since Aand B are proximinal nonexpansive retracts and, from Lemma 2.8, Nε(A)is weakly externally hyperconvex and there is and ε-constant nonexpansive retraction of Nε(A)on A. This corollary allows us to answer in the positive a question raised in [5], more precisely we obtain the following extension of Theorem 2.10 in [5]. Theorem 3.8. Let Aand Bbe two weakly externally hyperconvex subsets of a hyperconvex metric space Mwith Abounded, and suppose T∗:A→2B is such that: (i) for each x∈A,T∗(x)is a nonempty admissible (more generally, externally hyperconvex) subset of B; (ii) T∗: (A, d)→(2B, dH)is nonexpansive (where dHis the Hausdorff metric); (iii) T∗(A0)⊆B0. Then there exists x0∈Asuch that dist(x0, T∗(x0)) = dist(A, B) = inf{dist(x, T∗(x)) : x∈A}. Proof. The proof of this theorem follows the same steps as that of Theorem 2.10 in [5].  We finish this work with the nonexpansive version of the Fan’s approximation principle given in [3] (Theorem 5.4). We omit its proof since it follows in a similar way as the proof of Theorem 5.4 in [3]. Corollary 3.9. Let Abe a bounded weakly externally hyperconvex subset of a hyperconvex metric space Mand suppose that T:A→Mis a nonexpansive mapping. Then there exists x∈Asuch that d(x, T(x)) = inf{d(y, T (x)) : y∈A}. References [1] Aronszajn, N., P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6(1956), 405–439. [2] Esp´ınola, R., M. A. Khamsi, Introduction to hyperconvex spaces, Handbook of Metric Fixed Point Theory, (W. A. Kirk, B. Sims, eds.), Kluwer Academic Publishers, Dordrecht–Boston–London, 2001, pp. 391–435. [3] Esp´ınola, R., W. A. Kirk and G. López, Nonexpansive retracts in hyperconvex spaces, J. Math. Anal. Appl. 251 (2000), 557–570. On selections of the metric projection... 17 [4] Khamsi, M. A., W. A. Kirk and C. Mart´ınez Y´a˜nez, Fixed point and selection theorems in hyperconvex spaces, Proc. Amer. Math. Soc. 128 (2000), 3275–3283. [5] Kirk, W. A., S. Reich and P. Veeramani, Proximinal retracts and best proximity pair theorems, Num. Funct. Anal. Opt. 24 (2003), 851–862. [6] Sine, R., Hyperconvexity and approximate fixed points, Nonlinear Anal. 13 (1989), 863–869. Rafa Esp´ınola Departamento de An´alisis Matem´atico Universidad de Sevilla P.O. Box 1160 Seville, Spain e-mail: [email protected] Received January 25, 2005