New examples of subsets of c with the FPP and stability of the FPP in hyperconvex spaces

J. Fixed Point Theory Appl. (2021) 23:45 https://doi.org/10.1007/s11784-021-00881-1 c The Author(s) 2021, corrected publication 16 October 2021 Journal of Fixed Point Theory and Applications New examples of subsets of cwith the FPP and stability of the FPP in hyperconvex spaces Rafael Esp´ınola-Garc´ıa ,Mar´ıa Jap´on and Daniel Souza Abstract. The purpose of this work is two-fold. On the one side, we focus on the space of real convergent sequences cwhere we study nonweakly compact sets with the fixed point property. Our approach brings a positive answer to a recent question raised by Gallagher et al. in (J Math Anal Appl 431(1):471–481, 2015). On the other side, we introduce a new metric structure closely related to the notion of relative uniform normal structure, for which we show that it implies the fixed point property under adequate conditions. This will provide some stability fixed point results in the context of hyperconvex metric spaces. As a particular case, we will prove that the set M=[−1,1]Nhas the fixed point property for d-nonexpansive mappings where d(·,·)isametric verifying certain restrictions. Applications to some Nakano-type norms are also given. Mathematics Subject Classification. 46A45, 46A50, 46B20, 46B25, 46B45. Keywords. Nonexpansive mappings, fixed point, spaces of convergent sequences, weak compactness, hyperconvexity, stability. 1. Introduction The main topic of this work is the study of existence of fixed points for nonexpansive selfmappings. It is well-known that c0, the Banach space of all null-convergent sequences, and c, the Banach space of all convergent sequences, satisfy that their convex weakly compact sets have the fixed point property (FPP for short) for ·∞-nonexpansive mappings [5] (for definitions see Sect. 2). In regards to (c0,·∞), it was shown in [8,10,11] that convex weakly compact sets are the only convex closed subsets of c0with the FPP, that is, if a closed and convex subset of c0has the FPP then it has to be This work was completed with the support of the Ministerio de Ciencia, Innovaci´on y Univesidades del Gobierno de Espa˜na through the grant PGC2018-098474-B-C21, Plan Estatal 2017-2020 Generaci´on Conocimiento - Proyectos I+D+i. 0123456789().: V,-vol 45 Page 2 of 19 R. Esp´ınola-Garc´ıa et al. weakly compact. This raised the question of whether the FPP for · ∞- nonexpansive mappings could lead to a similar fixed point characterization of weak compactness in (c, · ∞). This question was answered negatively in [14], although weak compactness in chas actually been characterized in terms of fixed point properties for some larger family of mappings. In fact, it was shown in [16] that a closed and convex subset of cis weakly compact if, and only if, every (so-called) cascading nonexpansive selfmapping defined on it has a fixed point. In [14] the authors gave several examples of non-weakly compact, closed and convex subsets of cwith the fixed point property for ·∞-nonexpansive mappings. The fact is that those examples were all hyperconvex and so a new question was posed: is it possible to find such a non-weakly compact set lacking the property of being hyperconvex? This question motivates the first part of the present work. We will show that the actual answer is yes using the notion of uniform normal structure in metric spaces. In the second part we will explore new generalizations of the concept of relative uniform normal structure in metric spaces in order to bring new facts on fixed points for nonexpansive mappings. This work is organized as follows: In Sect. 2we recall some notions and results that will be needed afterwards. In Sect. 3we build two examples (family of examples, actually) of bounded closed and convex subsets of c which enjoy the fixed point property for nonexpansive mappings but fail to be both weakly compact and hyperconvex. This will answer the question raised in [14] in the positive and will spark new natural questions. In particular, whether it is possible to characterize bounded closed and convex subsets of (c, · ∞) with FPP for nonexpansive mappings in terms of topological or metric properties remains open. In Sect. 4, we present an extension of the notion of relative uniform normal structure for a metric space and show that, under certain assumptions, this extension, likewise uniform normal structure and relative uniform normal structure, implies the FPP. As a consequence, we will achieve a fixed point stability result for hyperconvex metric spaces. This stability property will be applied to several situations and examples, including some kinds of Nakano sequence spaces. As a byproduct we offer a new approach to the long standing problem of the fixed point property for uniformly Lipschitzian mappings defined on hyperconvex spaces (see, for instance, [26]). We finish this work with a last section where we bring forward some remarks and we set some questions related to the content of the article. 2. Definitions and basic results In this section we recall some needed definitions and results. We begin with the definition of nonexpansive mapping and fixed point property. Definition 2.1. Let (M,d) be a metric space. A mapping T:M→Mis said to be nonexpansive if d(Tx,Ty)≤d(x, y) for every x, y ∈M.Mis said to have the fixed point property for nonexpansive mappings (FPP for short) if New examples of subsets Page 3 of 19 45 every nonexpansive mapping T:M→Mhas a fixed point, that is, there exists x∈Msuch that Tx=x. The interested reader may consult [20] for a thorough exposition of fixed point theory, in this connection, see also the early papers by S. Reich [23]and [24]. In this work, the notation Bd(x, r) will always refer to the closed ball with center x∈Mand radius r≥0 with respect to the metric d.Wenow introduce a family of subsets of a metric space which will be heavily used in the next sections. Definition 2.2. Let (M,d) be a metric space, A⊂Mis said to be admissible if it is an intersection of closed balls of M. We will denote the set of admissible subsets of Mby Ad(M). When there is no danger of confusion we will omit the subscript din notations like Ad(M), Bd(x, r) and others that might come up. A very important property for a family of subsets of a given set is that of compactness. Definition 2.3. Let Mbe a set. A family {Xλ}λ∈Γof subsets of Mis said to have the finite intersection property if  λ∈Γf Xλ=∅for all finite Γf⊂Γ. Now compactness may be defined as follows. Definition 2.4. A family Lof subsets of a set Mis said to be compact if every subset of it which satisfies the finite intersection property has nonempty intersection. Some interesting properties can be deduced from the compactness of a family of sets when combined with other topological structures as, for instance, the one stated in the next proposition. Proposition 2.5. [12, Proposition 3.2] Let (M,d)be a metric space such that A(M)is compact. Then (M,d)is complete. We give next the definition of hyperconvex metric spaces. The interested reader may find more information on these spaces in [12]. Definition 2.6. A metric space (M,d) is said to be hyperconvex if, for every subset X⊂Mand for every family of radii {rx≥0:x∈X},wehavethat  x∈X B(x, rx)=∅ whenever d(x, y)≤rx+ryfor all xand y∈X. Hyperconvex metric spaces have been extensively studied in fixed point theory as they have been shown to be a very relevant class of metric spaces and provide a particular example of metric structure for which abstract results work (see [12]). In particular, bounded hyperconvex metric spaces, as it is very well-known, enjoy the FPP for nonexpansive mappings (see [2,3,12,19]). 45 Page 4 of 19 R. Esp´ınola-Garc´ıa et al. This work is also involved with uniform normal structure and relative uniform normal structure in metric spaces. We give next these notions and recall some relevant results. Definition 2.7. Given a nonempty subset Xof a metric space (M,d) we define ra(X)=sup{d(a, x):x∈X}foranya ∈M, r(X)=inf{rx(X):x∈X}and diam (X)=sup{d(x, y):x, y ∈X}, where ra(X)andr(X) stand for the Chebyshev radius of Xwith respect to aand the Chebyshev radius of Xin Mrespectively, and diam (X)isthe diameter of X. Definition 2.8. A metric space (M,d) is said to have uniform normal structure (UNS) if there exists some c∈(0,1) such that r(A)≤c·diam (A) for every A∈A(M) with 0 <diam (A)<+∞. For short, we say that Mhas UNS or c-UNS if cis needed. Spaces with UNS have been deeply studied in the literature and uniformly convex Banach spaces are known to have UNS (see, for instance, [17]). Finite dimensional normed spaces have it too as the following proposition shows: Proposition 2.9. [1, Proposition 2.12] If Xis a normed space of dimension n, then it has UNS with constant n n+1 . The following theorem shows how the concepts of UNS and FPP are related. Theorem 2.10. [18, Theorem 10] Let (M,d)be a bounded complete metric space. If (M,d)has uniform normal structure then it has the fixed point property. Initially, the previous theorem (see [21, Theorem I]) was stated with the additional requirement that the family A(M) was compact or at least countably compact (condition (R) in [18]). It turns out that this compactness condition is superfluous as stated in the following theorem. Theorem 2.11. [17, Theorem 5.4] Suppose (M,d)is a bounded complete metric space which has UNS. Then A(M)is compact. It can be shown that if (M,d) is a hyperconvex metric space then it has 1 2-UNS (see [12, Lemma 3.3]). Thus, in particular, since hyperconvex metric spaces are complete, it follows from Theorem 2.11 that the family A(M)is compact. The following definition was introduced by Soardi in [25]. Definition 2.12. A metric space (M,d) is said to have relative uniform normal structure (RUNS or c-RUNS) if there exists some c∈(0,1) such that, for every nonempty A∈A(M) with 0 <diam (A)<+∞, the following conditions are satisfied: New examples of subsets Page 5 of 19 45 (i) There exists zA∈Mwith rzA(A)≤c·diam (A). (ii) For zAas above and x∈Mwith rx(A)≤c·diam (A), we have that d(x, zA)≤c·diam (A). There is a subtle but important difference between Definition 2.8 and Definition 2.12 since the point zAgiven in Definition 2.12 does not necessarily belong to the set A. Because of that, the extra condition ii) is needed to assure the existence of a fixed point as the following theorem states. Theorem 2.13. [17, Theorem 5.6] Let (M,d)be a bounded metric space such that A(M)is compact and (M,d)has relative uniform normal structure. Then Mhas the FPP. Note that if (M,d) has uniform normal structure then it has relative uniform normal structure. Besides, it is not known whether RUNS implies compactness of A(M) as it is the case for UNS (Theorem 2.11), in this connection see [6] and [7]. 3. On subsets of (c, · ∞)with the FPP Consider the following subset of c: W={x=(xn)∈∞:1≥x1≥x2≥...≥0}. In [14] the authors introduce the subset Was a first example of a bounded closed and convex subset of cwith the FPP but failing to be weakly compact. This closed the door to a possible characterization result for weakly compactness in cas the one provided for c0in [10] (although it opened the scenario to considering fixed point theorems for a larger family of mappings, see [16]). The set Wwas not the only example of a convex closed bounded and non-weakly compact subset of cwith the FPP provided in [14]. For q=(qn) a positive convergent sequence with positive limit, subsets of c, defined out of such sequences qas follows Wq={u=(qnyn)∈∞:1≥y1≥y2≥...≥0}, were proved to enjoy the same properties as those described for W. The key fact why all these sets, Was well as the family of sets Wq, enjoy the FPP is that they all happen to be hyperconvex. Therefore, authors of [14] wondered to what extent hyperconvexity was essential to determine the FPP for non-weakly compact sets in c. The goal of this section is to provide a deeper understanding of this situation. We give a first example showing that hyperconvexity can actually be avoided. Example. Let Q=(x1,x 2,x 3)∈[0,1]3:x1+x2+x3=1 ,letWbe as previously described. If we set M=Q, ·∞,3⊕∞(W, ·∞),where·∞,3isthesupnorminR3, 45 Page 6 of 19 R. Esp´ınola-Garc´ıa et al. then Mis a closed, non-weakly compact, bounded and convex subset of c which is not hyperconvex but still has the FPP. Proof. Since W⊂c, it is obvious that M⊂c. It is also obvious that M is closed, bounded and convex. We claim that Whas UNS. Indeed, observe that Q, ·∞,3is isometric to P,·∞,3where P=(x1,x 2,x 3)∈[0,1]2×[−1,0] : x1+x2+x3=0 . Also, Pis contained in (x1,x 2,x 3)∈R3:x1+x2+x3=0 which is a 2-dimensional subspace of R3. Thus, it follows from Proposition 2.9 that Pwith the ·∞,3norm has 2 3-UNS and therefore, by isometry, Q, ·∞,3 has 2 3-UNS too. Since (W, ·∞) is hyperconvex, it follows that (W, ·∞)has1 2-UNS and therefore, given that direct maximum sums of spaces with UNS have UNS ( [18, Proposition 12]), we have that (M,·∞)has2 3-UNS. This implies that (M,·∞) has the FPP. To show that Mis not hyperconvex we consider the family of closed balls in Mgiven by: Bei,1 2:1≤i≤3, where eistands for the i-th standard basis element. Since ei−ej∞=1 whenever i=j, this collection of balls should have nonempty intersection with Mif Mwere hyperconvex. However, it follows that, for any x=(xn) in these three balls, it must be the case that |xi−1|≤1 2and |xi|≤1 2for all 1≤i≤3. Thus, x1=x2=x3=1 2and therefore, x/∈M. Hence (M,·∞) is not hyperconvex. Finally, since Mis a direct sum with a nonweakly compact subset of c, it follows that Mis not weakly compact either.  The previous example can be modified by replacing the first term of the direct sum by any non hyperconvex, bounded, closed and convex subset of any finite dimensional normed space since such sets, as stated in Proposition 2.9, have UNS. A direct sum, as the previous example, may be seen as a too straightforward construction and one can wonder if there is any such example which cannot be obtained in that way. This will be the case for the next one. Example. Let D=x=(xn)∈[0,1]N:x1+x2+x3=1,x 3≥x4≥...≥0. Then Dis a bounded, closed, convex and non-weakly compact subset of cwhich is not hyperconvex and has the FPP. New examples of subsets Page 7 of 19 45 Proof. It is obvious that D⊂cand Dis closed, bounded and convex. Similar arguments to those used in the previous example show that Dis neither weakly compact nor hyperconvex. We will apply Theorem 2.10 to prove that Dhas the FPP. It suffices to show that Dhas 2 3-UNS. Let A∈A(D) such that δ= diam (A)>0. For each n∈N, define an=inf x∈Axn,b n=sup x∈A xnandzn=an+bn 2. Observe that |zn−xn|≤1 2δfor all x∈Aand n∈N. This implies that A⊂B(z, 1 2δ) where z=(zn). However, we cannot assure that z∈D although it is still true that 1 ≥z3≥z4≥z5≥...≥0. Consider the projection mapping: π:(∞,τ)→R3,·∞ x=(xn)→ (x1,x 2,x 3) Let τbe the coordinatewise convergence topology on ∞.Wehavethat Dis τ-compact, which gives us that any A∈A(D)isτ-compact. Since A is convex too, π(A) is a compact and convex subset of R3,·∞.From the proof of [1, Proposition 2.12] together with the argument we used in the previous example, we know that there is w=(wn)∈Asuch that π(w)−π(x)∞≤2 3diam (π(A)) ≤2 3δfor all x∈A. This implies that, for x∈A,|wn−xn|≤2 3δfor 1 ≤n≤3. For n≥4 we still know that wn∈[an,b n]. Now, we are going to show that A∩ y∈A By, 2 3δ=∅. To this end, let us define ˜wn=wnif 1 ≤n≤3, min {w3,z n}if n≥4. Observe that ˜w1+˜w2+˜w3=1and|˜wn−xn|≤2 3δfor 1 ≤n≤3and x∈A. Also, since 0 ≤an≤wn≤w3and an≤zn≤bnfor all n≥4, it follows that ˜wn∈[an,b n]forn∈N. Since A∈A(D)wehavethat A= i∈I Bpi,r i∩D for a certain collection of centers pi=pi n∈Dfor i∈I. Therefore, given x=(xn)∈A, we have that x∈Bpi,r ifor all i∈I. In particular, we have that xn−pi n≤rifor all n∈Nand i∈I. Thus, given n∈N, pi n−ri≤xn≤pi n+ri 45 Page 8 of 19 R. Esp´ınola-Garc´ıa et al. for all x∈Aand i∈I. Therefore, pi n−ri≤an≤bn≤pi n+riandwecan conclude that [an,b n]⊂ i∈Ipi n−ri,p i n+ri. Hence, ˜w∈ i∈I Bpi,r i and therefore, since ˜w∈D,˜w∈A. Let x=(xn)∈A. We already know that |˜wn−xn|≤2 3δfor 1 ≤n≤3. Let n≥4. We have the following three cases to study: •Case zn≤w3.Inthiscasewehavethat ˜wn=znwhich gives that |˜wn−xn|=|zn−xn|≤1 2δ≤2 3δ. •Case w3<z nand xn≤zn. In this case, since ˜wn,x n∈[an,b n]andzn is the middle point of [an,b n], we have that |xn−˜wn|≤1 2δ≤2 3δ. •Case w3<z nand zn<x n. In this case we have that |xn−˜wn|=|xn−w3|=xn−w3≤x3−w3≤2 3δ. Since these are the only possible cases, we can conclude that |xn−˜wn|≤2 3δfor all n≥4. Hence, |˜wn−xn|≤2 3δfor all n∈Nand x∈A. Therefore, ˜w∈ y∈A By, 2 3δ and, finally, A∩ y∈A By, 2 3δ=∅. Now, since Awas an arbitrary admissible subset of Dwith positive diameter, we have that Dhas 2 3-UNS and, by Theorem 2.10,Dhas the FPP.  4. The (p, q)-relative uniform normal structure In this section we introduce a concept related to the relative uniform normal structure that will be applied to obtain some new fixed point results as well as stability of the FPP in metric spaces. Under the scope of a metric given through a norm, a stability result for the fixed point property for nonexpansive mappings is stated as follows: Assume that Cis a subset of a Banach space (X,·) such that (C, ·) satisfies the FPP. Let |·|be an equivalent New examples of subsets Page 9 of 19 45 norm on X. What can be said regarding the fulfillment of the FPP for the metric space (C, |·|)? With the backdrop of a suitable geometry for the Banach space (X,·) itself, an extensive literature has been published leading to the fulfillment of the FPP for the new metric space (C,|·|) when the Banach-Mazur distance between the spaces (X,·) and (X,|·|) is close enough (see for instance [20, Chapter 7] and references therein). It was shown in [4] that it is still possible to obtain stability results for some specific closed convex bounded subsets of dual Banach spaces, being this property local, in the sense that it is not shared by all the closed convex bounded subsets of the Banach space. This shows that the concept of the stability of the FPP depends on geometric and topological properties of the domain Crather than on the Banach space in which it lies. Furthermore, stability can be easily translated to the more general scope of metric spaces: Assume that (M,d) is a metric space with the FPP and that d(·,·) is an equivalent metric defined on M.Isitpossible to determine whether (M,d) also fulfills the FPP in terms of how “close” distances dand dare? It must be stressed that, in particular, the Banach space (∞,· ∞) may be considered the worst case scenario when it comes to obtaining general fixed point results for nonexpansive mappings. This is due to the fact that the FPP is separably determined [15, Chapter 3] and that every separable Banach space can be isometrically embedded in (∞,·∞). In spite of that, the metric condition of hyperconvexity has turned out to be an essential tool to ensure that, for some bounded subsets of (∞,· ∞), the FPP is still possible. Thus, a question is raised in a natural way: Assume that Cis a bounded hyperconvex subset of (∞,· ∞), and therefore with the FPP. Does (C, ·∞) have stability of the FPP? At this stage two facts stand out: (i) The first one is that hyperconvexity is an extremely tight property, which shows itself very unstable by renormings (it strongly depends on the shape of the closed balls). Indeed, fix some n∈N,n>1 and, for 1<p<∞, we define the following equivalent norm on ∞: xp,∞= max{Pn(x)p,Qn(x)∞} where Pn(x)=(x1,...,x n)∈Rnand Qn(x)=(xn+1,x n+2,...) for all x=(xn)∈∞. It can be easily checked that x∞≤xp,∞≤n1 px∞ for every x∈∞. It is well known that (Rn,·p) is not hyperconvex (for 1 <p<∞) so neither is (∞,·p,∞). Thus, for every >0wecan find a norm ·p,∞with x∞≤xp,∞≤(1 + )x∞for all x∈∞ and such that (∞,·p,∞) fails to be hyperconvex. (ii) The second fact worth mentioning is that the stability of the FPP is not given for free, that is, some conditions over the equivalent metric must be imposed. For instance, suppose that we consider the subset Was in Sect. 3,thatis W={x=(xn)∈∞:1≥x1≥x2≥...≥0}, 45 Page 16 of 19 R. Esp´ınola-Garc´ıa et al. Proof. Let x=(xn)∈∞. Under the previous assumptions ∞  n=1  xn bx∞ pn ≤ ∞  n=1 1 bpn ≤1, and the identity map is an isomorphism between ∞and pn, for which x∞≤xpn≤rx∞for all x∈∞. Furthermore, the closed unit balls for the norm ·pnare σ(∞, 1)-closed, since ·pnis the dual norm of the corresponding Nakano space qnwith 1 pn +1 qn = 1 endowed with the Orlicz norm. Applying Corollary 4.11 we obtain the result.  From Corollary 4.11 we can go further in the study of the FPP for bounded non-weakly compact subsets of c. For example, if we consider the original example from [14] W={x=(xn)∈∞:1≥x1≥x2≥...≥0}, we now know that, given a metric dequivalent to ·∞such that δ(·∞,d)< √2 and for which the closed d-balls are coordinatewise closed in ∞, then (W, d) has the FPP. In particular, W, ·pnhas the FPP where (pn) satisfies the assumptions of Example 4. 5. Some comments and remarks We finish the paper by bringing forward some questions raised in the course of the elaboration of this article: 1. As we already mentioned, it was proved in [14] that, unlike the c0-case, weak compactness does not determine those closed convex subsets of the Banach space csatisfying the FPP for nonexpansive mappings (see also [16]). We wonder whether there is either a topological or geometric property that could lead to a characterization of those closed convex subsets of cwith the FPP for nonexpansive mappings. 2. Related to the above question, we wonder whether there is a convex closed subset Cof cwith the FPP for nonexpansive mappings, which additionally fails to be closed for the coordinatewise topology. In the particular case that Cis a subset of c0with the FPP, we know that C is weakly compact and therefore closed coordinatewise. If Cis a convex weakly compact subset of c, it has the FPP but it is also compact for the coordinatewise topology. The examples of non-weakly compact closed convex subsets of cwith the FPP given in [14] and in Sect. 3of this article are all coordinatewise compact. 3. The problem of existence of fixed points for k-uniformly Lipschitzian mappings (that is, d(Tnx, Tny)≤kd(x, y) for any xand yin Mand n∈N) has been studied by many authors and, in particular, it remains open if such mappings have a fixed point when defined from a bounded hyperconvex space into itself. The conjecture claims that the most plausible highest value for kto obtain a fixed point is √2. The New examples of subsets Page 17 of 19 45 interested reader may check, for instance, [26]. Corollary 4.10 provides a new approach to this problem. Indeed, consider a bounded hyperconvex metric space (M,d) and a k-uniform Lipschitzian mapping T:M→M. Define d1(x, y)=sup{d(Tnx, Tny):n≥0}. Then d1is a metric on Mthat satisfies d(x, y)≤d1(x, y)≤kd(x, y)forallx, y ∈M. Therefore, if k<√2, Corollary 4.10 implies that (M,d1) has the FPP as long as the family of d1-admissible subsets of Mis compact. Now, since Tis nonexpansive for d1, it would have a fixed point. 4. It is not clear for the authors whether the hypothesis of the compactness of Ad1(M) in Corollary 4.10 can be dropped. Likewise, whether or not the thesis of Corollary 4.11 would hold without the assumption of · being an equivalent dual norm. It is worth noting that this observation is strictly related to the above remark. Indeed, if for every b>1we could find a dual norm ·on ∞with x∞≤x≤bx∞for every x∈∞and for which the set [−1,1]Nfails to have the FPP for ·-nonexpansive mappings, we would automatically have a fixed point free mapping T:[−1,1]N→[−1,1]Nwhich is b-uniformly Lipschitzian for the ·∞norm, showing a counterexample to the conjecture that every hyperconvex bounded metric space has the FPP for k-uniformly Lipschitzian mappings with k<√2[26]. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References [1] Amir, D.: On Jung’s constant and related constants in normed linear spaces. Pac. J. Math. 118(1), 1–15 (1985) 45 Page 18 of 19 R. Esp´ınola-Garc´ıa et al. [2] Aronszajn, N., Panitchpakdi, P.: Extensions of uniformly continuous transformations and hyperconvex metric spaces. Pac. J. 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