Metric fixed point theory on hyperconvex spaces: recent progress

Arab J Math (2012) 1:439–463 DOI 10.1007/s40065-012-0044-z SURVEY PAPER Rafa Espínola ·Pepa Lorenzo Metric fixed point theory on hyperconvex spaces: recent progress Received: 13 June 2012 / Accepted: 14 July 2012 / Published online: 11 August 2012 © The Author(s) 2012. This article is published with open access at Springerlink.com Abstract In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on R-trees. Finally, some considerations and new results on the extension of compact mappings will be shown. Mathematics Subject Classification 47H10 ·47H09 ·47H04 ·54E40 1 Introduction In the monograph Handbook of Metric Fixed Point Theory edited by Kirk and Sims, and published in 2001 by Kluwer Academic Publishers, we can find the chapter Introduction to hyperconvex spaces [21] by Espínola and Khamsi. This survey was mainly devoted to the study of metric hyperconvexity and its impact on metric fixed point theory which began in late seventies of the twentieth century [60,64]. It is already a decade after the Handbook on Metric Fixed Point Theory appeared, and many new facts on hyperconvex metric spaces and fixed points have been stated in these past 10 years. In this survey we will address some of these facts with a special attention to those involving metric conditions on the mappings. Therefore, it is not our intention to recall in detail basic and fundamental facts on hyperconvex metric spaces, which were already very generously explained along the first five sections of [21], but rather to present an update of some of the topics covered by the other last six sections of [21]. R. Espínola (B )·P. Lorenzo Dpto. de Análisis Matemático, IMUS, Instituto Matematico de la Universidad de Sevilla, P.O.Box 1160, 41080 Sevilla, Spain E-mail: [email protected] P. Lorenzo E-mail: [email protected] 123 440 Arab J Math (2012) 1:439–463 The study of fixed point theory for nonexpansive mappings on hyperconvex spaces was initiated, independently, by Sine [60] and Soardi [60], and very fundamental properties about hyperconvex spaces as well as the structure of the fixed point set of nonexpansive mappings were shown by Baillon in [9]. After the fundamental work of Sine on hyperconvexity and fixed points, which may be compiled in the series of works [35,50,60–63], many authors continued stating new facts during the past decade of the twentieth century. The aforementioned chapter [21]inthe Handbook of Metric Fixed Point Theory was devoted to collect many of those results. Since then new interesting facts have been obtained about fixed points and best approximation theory in hyperconvex metric spaces. These studies have been especially fruitful when focusing on R-trees. The connection between R-tress and hyperconvexity became evident for most fixed point theorists after the work [40] from 1998 by Kirk; however, it was needed to wait for some time until fixed point results on R-trees began to be studied [22] in a more systematic way. This explains why one of the most fruitful topics in the field nowadays, which is fixed point results on R-trees, was not covered in [21]. Interestingly enough, there has been a whole burst in the attention that the mathematical community has paid on metric spaces and analysis on metric spaces in the past 20 years. There are many metric structures which have called the attention of a large number of mathematicians from different branches. These structures include fractal metric spaces, metric spaces of bounded curvature, hyperbolic geodesic spaces, sub-Riemann- ian spaces, doubling metric spaces, metric spaces with the Poincaré inequality,… All of them have shown to be good structures to develop analysis from different perspectives. Hyperconvex spaces may also be considered within this class of intriguing metric structures which allow us to obtain results which one would only expect to be possible under certain linear structures. For references on the study of analysis on metric spaces or nonsmooth analysis the interested reader may check [3,30,31]. In this survey we start fixing some notation and giving some basic facts on hyperconvex spaces to make the exposition more self-contained. This will be covered in Sects. 2and 3. In Sect. 4we deal with nonexpansive proximinal retracts, best approximation, and best proximity pairs. In [21] a lot of attention was given to the study of nonexpansive proximinal retracts. By then the problem of characterizing such subsets of hyperconvex spaces was still open. In Sect. 4we find the solution to this problem and its relation to the problem of best approximation and best proximity pairs. Some other related results are also addressed in this section. Section 5 will be devoted to recent progress on R-trees and metric fixed point theory. The very peculiar structure of these spaces has provided a very suitable scenario to develop beautiful and deep fixed point results for single and multivalued mappings. We will walk the history of this topic and show some of the most surprising and far-reaching results on R-trees. One of the most remarkable features of these results is that they do not require the set to be bounded to guarantee existence of fixed points. We will show how dealing with multivalued mappings is especially fruitful on R-trees. In our last section, Sect. 6, we consider some recent advances on the theory of extension of Hölder maps and relate them to extensions of uniformly continuous mappings under ℵ0-hyperconvex conditions. In this section we include some new results on compact extensions of compact mappings. 2 Notation and basic definitions Metric spaces will be denoted as (M,d),orjustMfor simplicity, where Mis the space and dstands for the distance on M. Main elements for us in a metric space will be closed balls which will be denoted by B(x,r) meaning the closed ball of center xand radius r≥0. The following notation is also typical when dealing with metric spaces and will be used along all this survey. Let Mbe a metric space, x∈Mand Aand Bsubsets of M;then rx(A)=sup{d(x,y):y∈A}, r(A)=inf{rx(A):x∈M}, R(A)=inf{rx(A):x∈A}, diam(A)=sup{d(x,y):x,y∈A}, dist(x,A)=inf{d(x,y):y∈A}, dist(A,B)=inf{d(x,y):x∈A,y∈B}, C(A)={x∈M:rx(A)=r(A)}, CA(A)={x∈A:rx(A)=R(A)}, cov(A)={B:Bis a closed ball containing A}, 123 Arab J Math (2012) 1:439–463 441 where r(A)is the radius of Arelative to M,diam(A)is the diameter of A,R(A)is the Chebyshev radius of A,andcov(A)is the admissible cover of A. Hyperconvex metric spaces were introduced by Aronszajn and Panitchpakdi to extend Hahn–Banach’s theorem from the real line to more general spaces, see [8,21,36]. As a result they determined metric conditions guaranteeing such an extension and named spaces satisfying these conditions hyperconvex metric spaces. 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 then  α B(xα,rα)=∅. A subset of a metric space with this property will be called sometimes as a hyperconvex set. The name of hyperconvexity comes from the fact that when the ball intersecting condition holds for each pair of balls then the metric space is said to be (metrically) convex. In the literature they are also known as injective metric spaces and Banach spaces which are hyperconvex are generically denotes as P1spaces [48,51]. The study of normed spaces satisfying this property and other related ones was very important during the mid twentieth century, and the interested reader may check [48] in this regard. For more on the injective nature of hyperconvex metric spaces see [8,36]or[21, Section 4]. A fundamental fact of hyperconvex spaces is given by the next proposition which proof can be found in [8,21], see also [36, Proposition 4.4]. Proposition 2.2 If M is a hyperconvex metric space then it is complete. Next we summarize some metric properties of hyperconvex spaces which have played a major role in the structure of hyperconvex spaces and have been extensively applied in metric fixed point theory; for proofs consult [21, Lemma 3.3] or [36, Lemma 4.1]. Lemma 2.3 Let A be a bounded subset of a hyperconvex metric space M. Then 1. cov(A)=∩{B(x,rx(A)):x∈M}. 2. rx(cov(A)) =rx(A), for any x ∈M. 3. r(cov(A)) =r(A). 4. r(A)=1 2diam(A). 5. diam(cov(A)) =diam(A). 6. If A =cov(A)then r(A)=R(A). A very important property in hyperconvexity with immediate impact in fixed point and approximation theory (see [21,46]) is that of being nonexpansive retracts. Definition 2.4 Let (M,d1)and (N,d2)be two metric spaces. A map T:M→Nis said to be nonexpansive if d2(Tx,Ty)≤d1(x,y) for all x,y∈M. Definition 2.5 A subset Aof a metric space Mis said to be a nonexpansive retract (of M) if there exists a nonexpansive retraction from Monto A, that is, a nonexpansive mapping R:M→Asuch that Rx =xfor each x∈A. Hyperconvex metric spaces are nonexpansive retracts of any metric space where are isometrically embedded [21, Section 4]. Even more, this property characterizes hyperconvex subsets of a hyperconvex metric space. Lemma 2.6 Let M be a hyperconvex metric space and A ⊆M. Then A is hyperconvex (as a metric space with the induced metric)if and only if A is a nonexpansive retract of M. 123 442 Arab J Math (2012) 1:439–463 A step forward in this property is to require the set to be a proximinal nonexpansive retract . Definition 2.7 Let Mbe a metric space and A⊆Mnonempty. The metric projection onto Ais then defined by PA(x)=B(x,dist(x,A)) ∩A. If PA(x)is nonempty for each x∈M,thenAis said to be a proximinal subset of M.IfAis proximinal and PAis nonexpansive or admits a nonexpansive selection, then Ais said to be a proximinal nonexpansive retract of M. Proximinal nonexpansive retracts are relevant in the theory of fixed point and best approximation points. To study the nature of such sets for a hyperconvex space is a much more complicated task than that of characterizing nonexpansive retracts. Still some classes of proximinal nonexpansive retracts have been identified. Among them we find the classes of admissible and externally hyperconvex subsets . Definition 2.8 Let Mbe a metric space. A⊆Mis said to be an admissible subset of Mif A=cov(A).The collection of all admissible subsets of Mis then denoted by A(M). Admissiblesubsetsofhyperconvexspacesenjoy of averylargenumberofproperties.Thesepropertieswere very relevant when studying approximation problems and fixed point results; see for instance [21,33,46,61,62] among others. Obviously, the class of admissible sets is closed under arbitrary intersections. Definition 2.9 A subset Eof a metric space Mis said to be externally hyperconvex (relative to M)ifgiven any family {xα}of points in Mand any family {rα}of positive real numbers satisfying d(xα,xβ)≤rα+rβand dist(xα,E)≤rα; it follows that  α B(xα;rα)∩E=∅. The class of externally hyperconvex sets relative to Mwill be denoted as E(M). Externally hyperconvex subsets were first introduced in [8] and used for first time in fixed point theory in [37] by Khamsi, Kirk and Martínez-Yáñez. It was proved in [37] that a nonexpansive multivalued mapping with externally hyperconvex values admits nonexpansiveselections. This result was then applied to obtain existence of fixed points for multivalued mappings (see also [21, Section 9]). The next result, which is immediate to prove, relates admissible and externally hyperconvex sets. Lemma 2.10 Let M be a hyperconvex space; then any admissible subset of M is also an externally hyperconvex subset of M. It is easy to find examples that show that the converse to this lemma is not true. A proof for the next theorem can be found in [21, p. 426]. Theorem 2.11 Let E be a nonempty externallyhyperconvex subset of a metric space M.Then E isaproximinal nonexpansive retract of M. 3 Further properties and basic fixed point results in hyperconvex spaces In this section we present a brief description of some of the most fundamental fixed point theorems for single and multivalued nonexpansive mappings; some other properties of hyperconvex metric spaces will also be recalled. A more detailed exposition of these results can be found in [21](seealso[36, Chapter 4]). The first versions of Kirk’s fixed point theorem for hyperconvex spaces were independently obtained by Sine [60]and Soardi [64]. These results can be summarized in the next one. 123 Arab J Math (2012) 1:439–463 443 Theorem 3.1 Let M be a nonempty bounded hyperconvex metric space and T :M→M a nonexpansive mapping. Then T has a fixed point. Moreover, the set of fixed points of T, that is, Fix(T)={x∈M:Tx =x} is a hyperconvex subset of M. A more general and very celebrated version of this result on ε-fixed points was later proved by Sine [63] or [36, Section 4.6]. Definition 3.2 Given T:M→Mwe say that x∈Mis an ε-fixed point of Tif d(x,Tx)≤ε.Thesetof ε-fixed points is denoted by Fixε(T). Theorem 3.3 Let M be a bounded hyperconvex metric space and T :M→M a nonexpansive mapping. Then, for ε≥0,Fixε(T)is nonempty and hyperconvex. The next theorem was proved by Baillon [9] and gives an extended version of Theorem 3.1. Theorem 3.4 Let M be a bounded hyperconvex metric space. Any commuting family of nonexpansive mappings {Ti}i∈I, with Ti:M→M, has a common fixed point. Moreover, the common fixed point set is hyperconvex. Baillon proved also in [9] the next very important property on hyperconvex sets. Theorem 3.5 Let {Hα}α∈Abe decreasing (with respect to the set inclusion)family of nonempty and bounded hyperconvex sets; then  α∈A Hα=∅and hyperconvex. To obtain a counterpart of the fixed point theorem for nonexpansive mappings for multivalued nonexpansive mappings, we need first to introduce the Pompeiu–Hausdorff distance (for more on this see [36,p.24], [26, p. 19] or [3, p. 72]). Definition 3.6 Let Mbe a metric space and let Mdenote the family of all nonempty bounded closed subsets of M.ForA∈Mand ε>0definetheε-neighborhood of Ato be the set Nε(A)={ε>0:dist(x,A)<ε}. Now for A,B∈Mset H(A,B)=inf{ε>0:A⊆Nε(B)and B⊆Nε(A)}. Then (M,H)is a metric space and His called the Pompeiu–Hausdorff distance on M. A nonexpansive multivalued mapping is then defined as follows: Definition 3.7 Let Mbe a metric space and T:M→2Ma multivalued mapping with closed values. Tis said to be nonexpansive if H(Tx,Ty)≤d(x,y). The question now is under which conditions a multivalued nonexpansive mapping has fixed points. The study of this problem was initiated by Nadler in [55] for Banach spaces (see also [26, Chapter 15]). It turned out that hyperconvex spaces were very adequate to deal with multivalued mappings. More general versions of the next result can be found in [37](seealso[62]or[21]). Theorem 3.8 Let M be a hyperconvex space and T :M→E(M)a nonexpansive multivalued mapping. Then there exists a nonexpansive selection f :M→M, that is, a nonexpansive mapping such that f (x)∈Tx for all x ∈M. 123 444 Arab J Math (2012) 1:439–463 The same result was proved for admissible valued mappings in [62]; it is an open question posed in [62] whether this theorem remains true if the images are supposed to be hyperconvex instead of admissible sets. As a corollary of this theorem and Theorem 3.1, we have the fixed point result for nonexpansive multivalued mappings (for more on this see [21, Section 9]). Corollary 3.9 Let M be a nonempty bounded hyperconvex space. If T :M→E(M)is a nonexpansive multivalued map then its set of fixed points is nonempty and hyperconvex. Classical fixed point topological results as those of Darbo–Sadovski or Schauder theorem have also found their counterparts in hyperconvex spaces. This survey focuses on metric fixed point theory, so it is not our goal to cover recent advances on topological fixed point theory for hyperconvex spaces. The reader can find an exposition of this kind of results in [21, Section 7]. Theorem 3.10 Let M be a compact hyperconvex metric space and T :M→M a continuous mapping. Then T has a fixed point. Different interesting papers on extensions of the above result have appeared after the publication of [21]; some of them are [11,14,19,34,46]. Also a number of works have been published about KKM mappings on hyperconvex spaces after [33]as,forinstance,[15]. 4 Proximinal nonexpansive retracts and best proximity pairs One of the subjects that received more attention in Chapter 13 [21]ofthe Handbook in Metric Fixed Point Theory was the question of characterizing proximinal nonexpansive retracts of hyperconvex spaces. This question was not solved by then, but now we know its solution. This is a kind of result with important connections to several problems in best approximation results and best proximity pairs. We begin this section describing the problems of best approximation and best proximity pairs. WeknowthatfixedpointtheoryisanimportanttoolforsolvingequationsTx =xformappings T:D→M where Disasubsetof M.However,ifTdoesnothavefixedpoints,thenoneoftentriestofindan element xwhich is in some sense closest to Tx. A classical result in this direction is the very well known Ky Fan best approximation theorem. Best approximation results for nonself mappings try to find xsuch that d(x,Tx)=dist(Tx,D). Notice that these results need not give optimal solutions to the best approximation problem, that is, those minimizing d(x,Tx). In contrast to this, best proximity pair theorems provide approximate solutions that are optimal. A few advances have been obtained for hyperconvex spaces about existence of best proximity pairs in the last ten years [6,20,38,47,52,54]. In this section we collect some of them. Definition 4.1 Let Mbe a metric space and let Aand Bbe nonempty subsets of M.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}. Apair(x,y)∈A0×B0for which d(x,y)=dist(A,B)is called a best proximity pair for Aand B. The basic questions about the problem of best proximity pairs are 1. Whenare Aand Bmutuallyproximinalinthesensethatthereexistsabestproximitypair(a,b)∈A0×B0? 2. Given that Aand Bare mutually proximinal and given a mapping T:A→2B, when does the mapping T have a best proximity pair solution, that is, when does there exist a best proximity pair (x,y)∈A0×B0 such that y∈Tx? It was shown in [47] that this problem can be solved if some of sets under consideration are proximinal nonexpansive retracts. Being proximinal nonexpansive retract is a very unusual property in general spaces but not in hyperconvex spaces. Theorem 2.11 states that externally hyperconvex sets are proximinal nonexpansive retracts. The complete characterization of such retracts was given in [20]. To show this characterization we need to introduce a new class of sets: these sets were first introduced in [23] and received the name of weakly externally hyperconvex sets. 123 Arab J Math (2012) 1:439–463 445 Definition 4.2 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. More precisely, given any family {xα}of points in Mall 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  α B(xα,rα)∩E=∅. 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 [63]. Definition 4.3 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 was proved in [23] and already explained in [21, Section 4]: Lemma 4.4 Let A be 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. The most it was known about weakly externally hyperconvex spaces and nonexpansive proximinal retractions in [23] is summarized in the following result which was already contained in [21, Section 4]. Theorem 4.5 Suppose A is a weakly externally hyperconvex subset of a metrically convex metric space M. Then given any ε>0there exists a nonexpansive retraction R :M→A with the property that if u ∈M\A there exists v∈M\A with d(v, R(v)) =dist(v, A)and d(u,v)≤ε. For a subset Aof a metric space Mand ε>0weuseSεto denote the level set Sε={u∈M:dist(u,A)=ε}. Thenexttwotechnical results were givenin[20]. The first one isa consequence of the proof of Theorem 4.5. Lemma 4.6 Let ε>0,A and M as above, and S =n∈NSnε. Then the retraction given by Theorem 4.5 can be chosen so that d(v, R(v)) =dist(v, A)for any v∈S. We include the proof of the next lemma for completeness. Lemma 4.7 Let A be 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 with εn)as in Corollary 4.6 such that the sequence of retractions {rn}satisfies that d(rn(x), rm(x)) ≤ j=m  j=n+1 1 2j for x ∈M and n <m. Moreover, {rn(x)}is convergent for each x ∈M. 123 446 Arab J Math (2012) 1:439–463 Proof For n=1wetaker1as the one given by Lemma 4.6. We prove next that given rifor 1 ≤i≤nas in the statement of the lemma we can construct rn+1as required. We consider Sεn+1and proceed as in Theorem 4.5. After applying Zorn’s Lemma we may assume that Hεn+1is the maximal subset of Sεn+1where rn+1can 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∈AB(x,d(x, v))) ∩(∩u∈Hεn+1B(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) =∅.SinceAis 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 remains to study those cases involving the ball centered at rn(v); other cases were already studied in [23]. For these cases it is enough to recall that d(x,rn(v)) ≤d(x,v) for x∈A,now,sinceAis proximinal, let pv∈Asuch that d(v, pv)=dist(v, A),so d(v, rn(v)) ≤d(v, pv)+d(pv,rn(v)) ≤2dist(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+1defined on the whole Sεn+1as required. Next we show how to extend rn+1to A∪Sεn+1∪S2εn+1.Letv∈S2εn+1=Sεn; then the set P(v) =(∩x∈AB(x,d(x, v))) ∩(∩u∈Sεn+1B(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+1such that d(v, ˆv) =1 2n+1,wehave 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+1as required from Sεn+1to Sεn+1∪{v}.Thissame argument shows how to extend rn+1to A∪Sεn+1∪S2εn+1onto Aas required. Let S=∪ ∞ i=1Siεn+1. By proceeding as above but selecting ˆv∈S(i−1)εn+1for v∈Siεn+1, and using induction it follows that there exists a nonexpansive retraction rn+1of 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. 123 Arab J Math (2012) 1:439–463 447 Again, using induction it follows that rn+1can defined on Mas required. Hence d(rn+1(x), rn(x)) ≤1 2n+1 for x∈M.Nowlet{rn}be the sequence of retractions given by the above procedure; then for m>nand x∈M d(rm(x), rn(x)) ≤ j=m  j=n+1 d(rj(x), rj−1(x)) ≤ j=m  j=n+1 1 2j. Hence the proof of the lemma is complete.  The next result answers a question from [23] in the affirmative and basically characterizes proximinal nonexpansive retracts of hyperconvex spaces. Theorem 4.8 Let A be a complete weakly externally hyperconvex subset of a metrically convex metric space M. Then A is a proximinal nonexpansive retract of M. Proof Let {rn}be the sequence of retractions given by Lemma 4.7; then we define the mapping r:M→Aas r(x)=lim n→∞ rn(x). From the convergence given in Lemma 4.7,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,v n)≤1 2nand d(vn,rn(vn)) =dist(v, A). Hencewehave d(x,rn(x)) ≤d(x,v n)+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 [23] implies that proximinal nonexpansive retracts of hyperconvex spaces are weakly externally hyperconvex, the previous theorem can be re-written in the following way: Theorem 4.9 Let M be a hyperconvex metric space and A ⊆M nonempty. Then A is a proximinal nonexpansive retract of M if, and only if, A is a weakly externally hyperconvex subset of M. Now these results were applied, following the approach from [47], to the problem of best proximity pairs. Lemma 4.10 Let M be a hyperconvex metric space and let A and B be nonempty weakly externally hyperconvex subsets of M. Then A0and B0are nonempty and hyperconvex. 123 454 Arab J Math (2012) 1:439–463 Clearly r(x)∈Aand x∈Band Aand Bare closed. Indeed, let (tn)be a sequence of elements of Bsuch that tn→t. Assume that t∈A\B.Thend(t,ξ(t)) =ε>0. Let n∈Nbe such that d(t,tn)<ε/2. For each u∈F(t)and v∈F(tn)we obtain r(t)∈[u,ξ(t)],ξ(t)∈[r(t), ξ(tn)],ξ(tn)∈[ξ(t), r(tn)],r(tn)∈[ξ(tn), v]. Then we have [ξ(t), ξ(tn)]⊂[u,v] and finally infz∈F(tn)d(z,F(t)) ≥d(ξ(t), ξ(tn)) > ε/2 for each n∈Nsufficiently large which contradicts the ε-semicontinuity of F. Since Aiscompactthereisϕ(x)∈[x,r(x)]suchthatd(x,ϕ(x)) =inft∈Ad(x,t). Moreover, ϕ(x)∈A∩B what implies that [x,r(x)]∩[x,r(ϕ(x))]=[x,ϕ(x)].(1) Now suppose that Fhas not a fixed point in X. Therefore, we have d(x,ϕ(x)) > 0,x∈X.(2) Let us choose any x0∈X. We define a transfinite sequence (xα)α< such that d(x0,xα)= β<α dβ(3) and d(x0,ϕ(xα)) =d(x0,xα)+d(xα,ϕ(xα)) (4) where is the order type of the set {α|¯α≤ℵ 0}and dβ=d(xβ,ϕ(xβ)). Let αbe a limit ordinal number. By the geodesically boundedness of Xand (3) the countable sum β<α dβ is bounded. So there is a sequence of points xαnsuch that limn→∞ β<αn=β<α and xαn→¯x∈X.Let us define xα:= ¯x. Clearly (3)and(4) are satisfied. The proof of (4) is not different from the proof of the closedness of B. If α=β+1wedefinexα:= ϕ(xβ).By(1) with x=xβ,(3)and(4) we obtain d(x0,xα)=d(x0,xβ)+ d(xβ,xα)=γ<α dγand d(x0,ϕ(xα)) =d(x0,xα)+dα. Now let us define m:= sup α<  β<α dβ.(5) If mwere equal to infinity, points xαwould lie on the geodesic ray. Hence m<∞and one can find a sequence αnfor which d(x0,xαn)→m. Clearly there is α<such that αn<αfor each n∈N. Moreover, d(x0,xα)=mwhat implies that d(x0,xα+1)=d(x0,ϕ(xα)) > m. This contradicts (5).  The non-self version of this result is also given in [57]. Theorem 5.18 Let M and K be as in the previous theorem with F :K→2Man ε-semicontinuous mapping with nonempty closed convex values. Then there exists a point x0∈Kforwhich dist(x0,Fx0)=inf x∈Kdist(x,Fx0). Also invariant approximation theorems have been given for R-trees. The next one has been taken from [54]. Theorem 5.19 Let M be a complete R-tree and K a closed bounded convex subset of M. Assume t :K→K and T :K→2Mare nonexpansive mappings with T x closed convex and T x ∩K=∅for x ∈K. If the mappings t and T commute then there is z ∈K such that z =t(z)∈Tz. 123 Arab J Math (2012) 1:439–463 455 Selections is also a very important feature of R-trees and hyperconvex spaces [37,62]. This topic was already covered in [21, Section 9]. For the particular case of R-trees check also Theorem 5.9 above or [1]. Very recently, in [25] we can find a result providing nice selections of a kind of generalized nonexpansive notion for multifunction mappings on R-trees. The nonexpansiveness condition (C), also known as Suzuki condition, was introduced in [65] to study mild nonexpansiveness conditions which still imply existence of fixed points. Definition 5.20 Let Mbe a metric space, K⊆Mand T:K→X.Then fsatisfies condition (C)if 1 2d(x,Tx)≤d(x,y)⇒ d(Tx,Ty)≤d(x,y), for all x,y∈K. This definition has been adapted for multivalued mappings in different ways. The next one was given in [59]. Definition 5.21 Let Mbe a metric space and K⊆Mand T:K→2M. The mapping Tsatisfies condition (C)if for each x,y∈Kand ux∈Txsuch that 1 2d(x,ux)≤d(x,y), there exists uy∈Tysuch that d(ux,uy)≤d(x,y). This condition was extended to the multivalued case by different authors and studied, among other places, in [25]. It was in [25] that a new condition, weakening the multivalued version of (C)and called (C),was introduced. This condition happened to work especially well for R-trees. Definition 5.22 Let Mbe a metric space, K⊆Mand T:K→2M. The mapping Tsatisfies condition (C) if for each x,y∈Kand ux∈Txwith d(x,ux)=dist(x,Tx)and 1 2d(x,ux)≤d(x,y), there exists uy∈Tysuch that d(ux,uy)≤d(x,y). We prove next a selection theorem in R-trees for multivalued mappings satisfying condition (C). Theorem 5.23 Let M be an R-tree, K ⊆M and T :K→2Ma mapping with nonempty closed and convex values which satisfies (C). Then the mapping f :K→M defined by f (x)=PTx(x)for each x ∈Kisa selection of T that satisfies condition (C). Proof Let x,y∈Ksuch that f(x)= f(y)and (1/2)d(x,f(x)) ≤d(x,y). Consider p(x)=PTy(f(x)) and p(y)=PTx(f(y)). First, suppose p(x)= f(y)and p(y)= f(x).Sincep(x)is the projection of f(x)onto Tyit follows that d(f(x), f(y)) =d(f(x), p(x)) +d(p(x), f(y)), i.e., p(x)∈[f(x), f(y)].SinceTyis convex, [p(x), f(y)]⊆Ty. This implies [f(x), f(y)]∩[f(y), y]= {f(y)}because otherwise the minimality of f(y)would be contradicted. Thus, f(y)∈[f(x), y]. Similarly, f(x)∈[f(y), x].Then f(x), f(y)∈[x,y](otherwise supposing for example that z∈[x,f(y)]∩ [f(y), y]with z= f(y)we have that f(x)∈[z,f(y)]and f(y)∈[z,f(x)]which is false). Therefore, d(f(x), f(y)) ≤d(x,y). In fact, d(f(x), f(y)) =d(x,y)−dist(x,Tx)−dist(y,Ty). Now assume p(x)=f(y).Thend(f(x), f(y)) =dist(f(x), Ty)and so, by condition (C), d(f(x), f(y)) =dist(f(x), T(y)) ≤d(x,y). 123 456 Arab J Math (2012) 1:439–463 Finally, suppose p(x)= f(y)and p(y)=f(x).Asabove,ifp(x)= f(y),wehavethat f(y)∈[f(x), y]. If (1/2)d(y,f(y)) ≤d(x,y)then (C)yields that d(f(x), f(y)) =dist(f(y), Tx)≤d(x,y). Otherwise, if (1/2)d(y,f(y)) > d(x,y),then d(f(x), f(y)) +2d(x,y)<d(f(x), f(y)) +d(f(y), y)=d(f(x), y)≤d(f(x), x)+d(x,y) ≤2d(x,y)+d(x,y). Consequently, d(f(x), f(y)) ≤d(x,y).  An immediate consequence of this result is its fixed point partner. Corollary 5.24 Let M be a bounded complete R-tree. Suppose T :M→2Mwith nonempty closed and convex values satisfies condition (C).ThenFix(T)is a nonempty complete R-tree. Moreon fixedpoints on R-trees has been done in the past year. Of particular interest are results onuniformly Lipschitzian mappings which can be found in [2,16]. Definition 5.25 Let Mbe a metric space; then a mapping T:M→Mis said to be k-uniformly Lipschitzian if d(Tnx,Tny)≤kd(x,y) for any x,y∈Mand n∈N. The study of uniformly Lipschitzian mappings in metric fixed point theory was initiated by Goebel and Kirk in [27](seealso[26, Chapter 16]). The following result can be found in [2] where, surprisingly enough, it is shown that fixed points for uniformly Lipschitzian mappings in R-trees are guaranteed under less restrictive conditions than, for instance, in Hilbert spaces. Theorem 5.26 Let M be a nonempty complete R-tree and T :M→M a k-uniformly Lipschitzian mapping with bounded orbits and k <2. Then T has a fixed point. 6 Compact extensions of mappings One of the main motivations for the seminal work of Aronszajn and Panitchpakdi [8] about extension of uniformly continuous mappings was to find a metric characterization of injectivity, that is, a metric property that characterizes those spaces which may play the role of the real line in Hahn–Banach’s theorem. The solution to this problem was given by the intersecting condition of balls defining hyperconvex metric spaces. Therefore, hyperconvex metric spaces are intrinsically related to the notion of extension of operators. The next theorem was proved in [8] and was very generously explained in [21, Section 4]. Definition 6.1 A metric space Mis said to be injective if it has the following extension property: whenever Y is a subspace of Xand T:Y→Mis nonexpansive, then Thas a nonexpansive extension of the whole X. Theorem 6.2 Let M be a metric space. The following are equivalent: 1. M is hyperconvex 2. M is injective. Actually, Theorem 6.2 is presented in a more general form in [8]; instead of dealing with nonexpansive mappings, they work with uniformly continuous mappings with subadditive modulus of continuity and find extensions preserving the same modulus of continuity. 123 Arab J Math (2012) 1:439–463 457 Definition 6.3 An extended valued nonnegative function δ:[0,+∞)→[0,+∞] is said to be a modulus of continuity if it is nondecreasing and limε→0+δ(ε) =0. If Tis a mapping from a metric space (X,d1)into a metric space (Y,d2), we say that δ(ε) is a modulus of continuity of T if δ(ε) ≥sup{d2(T(x), T(y)) :x,y∈Xand d1(x,y)≤ε}. A modulus of continuity is called subadditive if δ(ε1+ε2)≤δ(ε1)+δ(ε2) for any ε1>0andε2>0. Extension of Lipschitz mappings has been a very important topic in linear spaces which also became very important in metric spaces after the work of Ball [10]. Lindenstrauss [51] studied compact extension of compact operators between linear spaces. The main object in this work turned out to be a certain class of linear spaces which satisfies a mild hyperconvexity condition, the so-called ℵ0-hyperconvexity. Definition 6.4 A metric space Mis said to be ℵ0-hyperconvex if given any family {xα∈}of points of M, with ||<∞, and any family {rα}α∈of real numbers satisfying d(xα,xβ)≤rα+rβ it is the case that  α∈ B(xα,rα)=∅. Under the light of [51] a characterization of ℵ0-hyperconvex metric spaces in terms of compact extensions of compact uniformly continuous mappings was intended in [24]. These efforts led to the following characterization: Theorem 6.5 Let M be a metric space, then the following are equivalent: 1. Misanℵ0-hyperconvex complete metric space. 2. If Y is a metric space, T :Y→M is a uniformly continuous compact mapping with a subadditive modulus of continuity δT, and Z is a metric space containing Y metrically, then there exists a uniformly continuous compact extension ˜ T of T to the whole of Z into M such that δ˜ T(ε) =max{(1+η)εδT(1), (1+η)δT(ε)} is a subadditive modulus of continuity of ˜ T.(Recall that T :Y→X is said to be compact if it is continuous and T(A)is relatively compact in X for every bounded subset A of Y.) In 2005 Lancien and Randrianantoanina [49] presented new results on extension of Hölder mappings on spaces of continuous functions. The fact that spaces of continuous functions treated in [49]areℵ0-hypercon- vex, and that extensions found in [49] were mainly compact extensions, make it pertinent to try to revisit [49] from the approach given in [24]. What we offer next are some consequences of this approach. Henceforth, uniformly continuous mappings are supposed to have a subadditive modulus of continuity. Definition 6.6 For K>0andα∈(0,1], a mapping Tas above is said to be (K,α)-Hölder continuous if δ(ε) =Kεαis a modulus of continuity of T. For C≥1,BC(X,Y)denotes the set of all α∈(0,1]such that any (K,α)-Hölder mapping Tfrom a subset of Xinto Ycan be extended to a (CK,α)-Hölder mapping from Xto Y.In[49]thesets A(X,Y)=B1(X,Y)and ˜ A(X,Y)= C>1 BC(X,Y) are studied for Ya space of converging real sequences. In particular, it is obtained (Theorem 2.2 in [49]) that A(X,c0)=(0,1]for Xa finite dimensional normed vector space. It is interesting to point out that the extensions given by this result share the same modulus of continuity than the extended mapping. 123 458 Arab J Math (2012) 1:439–463 It is well known that spaces of continuous functions on compact Hausdorff domains are ℵ0-hypercon- vex; hence, in particular, the spaces of null real convergent sequences c0and real convergent sequences are ℵ0-hyperconvex. Notice that, under the terminology of Lancien–Randrianantoanina in [49], it follows from Theorem 6.5 that  A(Y,X)=(0,1]for compact Hölder mappings whenever Yis a metric space and Xis an ℵ0-hyperconvex metric space. For a metric space X, Isbell defined in [32] the set of extremal functions (X)of Xas the set of all functions f:X→Rsuch that it satisfies f(x)+f(y)≥d(x,y)for all xand yin Xand it is pointwise minimal (see [32]or[21, Section 8] for details). The following theorem shows that (X)can also be regarded as the hyperconvex hull of X: Theorem 6.7 Let X be a metric space and (X)the set of extremal functions on X. Then 1. (X)is a hyperconvex metric space with the metric d(X)(f,g)=supx∈X|f(x)−g(x)|. 2. X is isometrically embedded into (X)by the mapping IX:X→(X)defined by IX(x)(·)=d(x,·). 3. If X is relatively compact, then (X)is compact. Next lemma adds some information on statement 3 from Theorem 6.7 above. Lemma 6.8 If X is a boundedly compact metric space then ε(X)is boundedly compact too. Proof Let {fn}⊆ε(X)a bounded sequence of functions. Let x0∈Xfixed and fm nbe the restriction of fnto B(x0,m). Then, since {f1 n}is a bounded sequence of equicontinuous functions on a compact set, the Alaoglu theorem implies that it has a convergent subsequence {f1 nk}. If we apply the same reasoning to {f2 nk}, iterate the process and make use of the diagonalization technique at the end, the conclusion follows:  In [24, Lemma 2.1] it was proved that if Xis a complete ℵ0-hyperconvex space and A⊆Xis relatively compact, then there exists a compact hyperconvex set h(A)such that A⊆h(A)⊆X. After Lemma 6.8 and this remark it is tempting to prove whether the same remains true for boundedly compact subsets of ℵ0-hyperconvex spaces. The following example, however, shows that this is not the case: Example 6.9 Let c0be the space of null convergent real sequences with the maximum norm and consider the set A={nen}where {en}is the standard basis of c0. Trivially Ais boundedly compact; however, there is not any hyperconvex set h(A)such that A⊆h(A)⊆c0. In fact, it is enough to consider the following intersection of balls B(e1,1)⎛ ⎝ n≥2 B(nen,n−1)⎞ ⎠=[0,2]×{1}N. The next definition will help with the exposition. Definition 6.10 A metric space Xwill be said boundedly compact hyperconvex (BCH for short) if for any boundedly compact subset Aof Xthere exists a boundedly compact hyperconvex set h(A)such that A⊆ h(A)⊆X. It trivially follows that if a metric space is BCH then it is ℵ0-hyperconvex, and, by [21, Lemma 2.1], any bounded ℵ0-hyperconvex space is BCH. The following proposition shows that hyperconvex metric spaces are BCH: Proposition 6.11 If a metric space X is hyperconvex then it is BCH. Proof Given A⊆Xnonempty it is known that for any hyperconvex set h(A)such that A⊆h(A)there exists another hyperconvex set h(A)such that A⊆h(A)⊆h(A)which can be isometrically embedded into X. Now Lemma 6.8 implies the proposition.  There are, however, nonbounded ℵ0-hyperconvex spaces which are BCH. An example of such spaces is the set of null convergent sequences with bounded tiles, i.e., c0=Rn×[−1,1]N, endowed with the maximum norm. 123 Arab J Math (2012) 1:439–463 459 Proposition 6.12 The space c0is BCH and ℵ0-hyperconvex. Proof It is immediate to see thatc0is ℵ0-hyperconvex. To see thatc0is additionally BCH it is enough to recall that Rnis hyperconvex and the set of null convergent sequences defined on [−1,1]Nwith the maximum norm is bounded and ℵ0-hyperconvex. Hence, if A⊆c0boundedly compact, we can write A⊆A1×A2where A1={x∈Rn:there exists y∈Asuch that x(i)=y(i)for 1 ≤i≤n},(6) A2={x∈[−1,1]N:there exists y∈Asuch that x(i)=y(i)for i>n}(7) are also boundedly compact. So there exist boundedly compact hyperconvex sets h(A1)and h(A2)such that A1⊆h(A1)⊆Rnand A2⊆h(A2)⊆[−1,1]N; hence h(A1)×h(A2)is boundedly compact hyperconvex andsuchthatA⊆h(A1)×h(A2)⊆c0. The following theorem states an extension property for BCH spaces: Theorem 6.13 Let X be a metric space and Y a BCH metric space. Let M ⊆X be a nonempty set. If T:M→Y is a uniformly continuous compact mapping with a subadditive modulus of continuity δ, then there exists a uniformly continuous compact extension ˜ T:X→YofTforwhichδis a modulus of continuity. Proof Let x0∈Mfixed.WefirstextendTfrom Mto M∪BX(x0,1).Lety∈M,d1=sup{δ(d(x,x0)):x∈ B(x0,1)}and d2(y)=inf{d(x,y):x∈B(x0,1)}. Then we consider the nonempty set  M={y∈M:d1+d(Tx0,Ty)≥δ(d2(y))}. Since Tis a compact mapping, by construction, T( M)is boundedly compact. Now let h(T( M)) be the boundedly compact hyperconvex between T( M)and Y. Hence we may apply Theorem 4 of [8] to obtain an extension of Tto  M∪B(x0,1)into h(T( M)) ⊆Ysuch that d(Ty,Tx)≤δ(d(y,x)) for any y∈ Mand x∈B(x0,1). We check next that T:M∪B(x0,1)→Yhas the same modulus of continuity as T. It suffices to check that d(Tx,Ty)≤δ(d(x,y)) for y∈M\ Mand x∈B(x0,1); in fact, d(Tx,Ty)≤d(Tx,Tx0)+d(Tx0,Ty)≤δ(d(x,x0)) +d(Tx0,Ty) ≤d1+d(Tx0,Ty)<δ(d2(y)) ≤δ(d(x,y)). Now the result follows by induction on B(x0,n)as n∈N. To prove the following corollary it is enough to recall Proposition 6.11. Notice that the next corollary guarantees compact extensions. Corollary 6.14 If in the previous theorem Y is hyperconvex then the same conclusion follows. To state our next result we need to introduce the following definition: Definition 6.15 Let Xbe a metric space. We say that M⊆Xis a proximately centered set in Xif for any x∈Xthere exists (x)={xα}α∈A⊆Mbounded such that for any y∈Mthere exists α∈Asuch that d(y,xα)≤d(x,y). We say that Mis boundedly proximately centered in Xif  x∈B (x) is bounded for any B⊆Xbounded. The next example shows that the above definitions are not redundant. Example 6.16 Let {en}be as in Example 6.9 and, for n∈N,xn∈c0such that xn(i)=1,if 1 ≤i≤n, 0,if n<i. Take M={nen:n∈N},B={xn:n∈N}and X=M∪B⊆c0. Then, for n∈N, xn−iei=i−1,if 1 ≤i≤n, i,if n<i, and iei−jej=max{i,j}. Thus, Mis proximately centered in M, but it is not boundedly proximately centered since (xn)={iei:1≤i≤n}. 123 460 Arab J Math (2012) 1:439–463 Our next result is an extension of [49, Theorem 2.2]. Theorem 6.17 Let X be a metric space and M ⊆X boundedly proximately centered in X. If T :M→c0 is a compact uniformly continuous mapping, then there exists a compact extension  T:X→c0of T with the same modulus of continuity. Proof Let x0∈X\Mand consider B(x0,1)\M.Wefirstfind T:M∪B(x0,1)→c0as desired. Let ={(x):x∈B(x0,1)\M};wetakex∈B(x0,1)\Mand pick ε=ε(x)such that ε<1 2δ(dist(x,M)). Then, since is bounded and Tcompact there exists N∈Nsuch that |T(y)(n)|<εfor n≥Nand y∈. Now, since ∞is hyperconvex, Corollary 6.14 implies that there exists a compact extension G:M∪ B(x0,1)→∞of Twith the same modulus of continuity. For xas above we make G(x)=(ηn). Hence, for n≤Nwe fix  T(x)(n)=ηn.Forn>N,letδn∈{−1,1}be the sign of ηn. Now we set  T(x)(n)=δnmin |ηn|,sup y∈ |T(y)(n)|. Since Tis compact,  T:B(x0,1)∪M→c0. Next we prove that it admits the same modulus of continuity as T.Wefirstshowthatd( Tx,Ty)=supn∈N| T(x)(n)−T(y)(n)|≤δ(d(x,y)) when x∈B(x0,1)\Mand y∈M.Forn≤Nit follows from the properties of G.Forn>Nwe have four cases: 1. If |T(y)(n)|≤|  T(x)(n)|,then |T(y)(n)− T(x)(n)|≤2ε≤δ(d(x,y)). 2. If |T(y)(n)|>| T(x)(n)|,sgn(T(y)(n)) =δn,and| T(x)(n)|=|ηn|, then, by construction, |T(y)(n)− T(x)(n)|≤δ(d(y,x)). 3. If |T(y)(n)|>| T(x)(n)|,sgn(T(y)(n)) =δn,and| T(x)(n)|=supy∈|T(y)(n)|,thenlety0∈such that d(y,y0)≤d(y,x)and so |T(y)(n)− T(x)(n)|=|T(y)(n)|−|  T(x)(n)|≤|T(y)(n)−T(y0)(n)| +|T(y0)(n)|−|  T(x)(n)|≤δ(d(y,y0)) ≤δ(d(y,x)). 4. If |T(y)(n)|>| T(x)(n)|and sgn(T(y)(n)) = δn,then |T(y)(n)− T(x)(n)|=|T(y)(n)|+|  T(x)(n)|≤|T(y)(n)|+|ηn|=|T(y)(n)−ηn|≤δ(d(y,x)). Now we take x,y∈B(x0,1)\Mand let G(x)=(ηn)and G(y)=(νn). We want to check that d( T(x),  T(y)) ≤δ(d(x,y)).Again,forn≤Nit follows from the properties of G.Forn>Nwe have four cases: 1. If supy∈|T(y)(n)|≤min{|ηn|,|νn|} and sgn(ηn)=sgn(νn),then | T(x)(n)− T(y)(n)|=0. 2. If supy∈|T(y)(n)|≤min{|ηn|,|νn|} and sgn(ηn)= sgn(νn),then | T(x)(n)− T(y)(n)|≤|ηn−νn|≤δ(d(x,y)). 3. If |ηn|≤supy∈|T(y)(n)|≤|νn|,then | T(x)(n)− T(y)(n)|=|ηn−sgn(νn)sup y∈ |T(y)(n)|| ≤ |ηn−νn|. 4. If supy∈|T(y)(n)|≥max{|ηn|,|νn|},then | T(x)(n)− T(y)(n)|=|ηn−νn|. Hence  T:B(x0,1)→c0is uniformly continuous with modulus of continuity δ. Moreover  Tis compact as a consequence of the fact that Gis compact. 123 Arab J Math (2012) 1:439–463 461 The proof is completed by induction on Nas we extend successively from B(x0,n−1)∪Mto B(x0,n)∪M by adding B(x0,n)\(B(x0,n−1)∪M)to the domain of the new extension as above.  The big question after this theorem is if c0can be replaced by a more general ℵ0-hyperconvex space. Remark 6.18 1. The condition of boundedness in the proximately centered character of Mis required only for the compactness of the extension. The same result holds for noncompact extensions without this uniformity condition. 2. The boundedness condition is not required at all if Xis supposed to be boundedly compact since the compactness of the extension follows from its continuity. This is the case in [49, Theorem 2.2]. 3. The condition about Mbeing centered cannot be dropped from the previous theorem. In fact, consider A={nen}where {en}is the standard basis of c0. It is readable to check that Ais not proximately centered in A∪1,where1stands for the constant unit sequence, with the supremum norm. Now Example 6.9 shows that the identity map id:A→c0cannot be extended as a nonexpansive map to A∪1. Our next result requires the notion of λ-hyperconvexity (see [29,39] for more on this topic). Although the notion we present here is slightly different to that one given in [39], we still name it the same. Definition 6.19 Let Xbe a metric space and λ≥1. We say that Xis a λ-hyperconvex metric space if for every family of closed balls {B(xα,rα)}α∈A, each of radius rα, centered at xα∈Xfor α∈A, the condition d(xα,xβ)≤rα+rβfor every α, β ∈A, implies  α∈A B(xα,λrα)=∅. In [49] some results for the so-called L∞ λnormed spaces are obtained. Our next definition is an extension of this notion to the metric setting. Definition 6.20 We say that a metric space Xis compactly almost hyperconvex if for every λ>1andM⊆X boundedly compact there exists N⊆Xboundedly compact, λ-hyperconvex and such that M⊆N. It is easy to check that L∞ λ-spaces are compactly almost hyperconvex. Our next result is related to [49, Proposition 3.1]. Theorem 6.21 Let Y be a boundedly compact metric space, M ⊆Y,X a compactly almost hyperconvex space, and T :M→X a compact uniformly continuous mapping with modulus of continuity δ. Then, for ε>0, there exists  T:Y→X uniformly continuous extension of T with modulus of continuity (1+ε)δ. Proof Let x0∈Y\Mand consider B(x0,1);thenwetake  Mas in the proof of Theorem 6.13. Hence T( M)is boundedly compact and so, for ε>0, there exists a boundedly compact and (1+ε)-hyperconvex subset Nε of Xsuch that T( M)⊆Nε.Nownoticethat  x∈ M B(T(x), (1+ε)δ(d(x,x0))), is nonempty. If we define  T(x0)as any point in the above intersection  T:M∪{x0}→Xis an extension of T with modulus (1+ε)δ. We complete the extension to the whole B(x0,1)as follows: let (xn)a dense sequence in B(x0,1)\M.Foragivenε>0wetake(εn)so that n≥1(1+εn)<1+ε. Iterating the above procedure on nand extending by continuity we finally have  T:M∪B(x0,1)→X. The theorem follows by repeating the previous procedure in each B(x0,n)\B(x0,n−1)and picking an adequate sequence (εn).  We finish this work with the following remarks: Remark 6.22 1. Since Yis supposed to be boundedly compact all the mappings in the above theorem are compact. 2. Under the terminology of [49] the above theorem states that ˜ A(X,Y)=(0,1]whenever Xis boundedly compact and Ycompactly almost hyperconvex. 123 462 Arab J Math (2012) 1:439–463 3. If in Definition 6.20 we impose that for each Nand λwe can choose Nλso that ∪λ∈Nλis boundedly compact, then it is possible to replace Yboundedly compact by Yseparable in the above theorem. We do not know, however, whether in this case being compactly almost hyperconvex implies BCH. 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