EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6433 ISSN 1307-5543 – ejpam.com Published by New York Business Global Revisiting Best Proximity Results of Relatively Meir-Keeler Condensing Operators in Hyperconvex Spaces Moosa Gabeleh1,2, Jack Markin3, Maggie Aphane2 1 Department of Mathematics, Faculty of Basic Sciences, Ayatollah Boroujerdi University, Boroujerd, Iran 2 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa 0204, South Africa 3 1440 8th St., Golden, CO 80401, USA Abstract. We first prove that if (G,H) is a nonempty, compact and hyperconvex pair of subsets of a hyperconvex metric space (M, d), then every cyclic relatively u-continuous mapping T defined on G ∪H has a best proximity point. The same result is valid for the case that T is the noncyclic relatively u-continuous map and (G,H) is a semi-sharp proximinal pair to obtain the existence of best proximity pairs. We then consider the class of relatively H-Meir-Keeler condensing operators by applying a concept of measure of noncompactness in the framework of hyperconvex spaces and in a special case in the nonreflexive Banach space ℓ∞ and revisit the previous best proximity point (pair) results of the paper by M. Gabeleh and C. Vetro [M. Gabeleh, C. Vetro, A new extension of Darbo’s fixed point theorem using relatively Meir-Keeler condensing operators, *Bull. Aust. Math. Soc.*, 98 (2018) 286–297]. Examples are given to support our main discussions. 2020 Mathematics Subject Classifications: 47H10, 51F99, 54H25 Key Words and Phrases: Hyperconvex metric space, Best proximity point, Relatively u- continuous map, Meir-Keeler condensing operator 1. Introduction Let (G,H) be a nonempty pair of subsets of a metric space (M, d). A mapping T : G ∪ H → G∪H is said to be relatively nonexpansive if d(Tx, Ty) ≤ d(x, y) for all (x, y) ∈ G×H. In particular case, if G = H, then T is well-known as a nonexpansive self-mapping. The mapping T is cyclic on G ∪ H if T (G) ⊆ H, T (H) ⊆ G, DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6433 Email addresses: gabeleh@abru.ac.ir gab.moo@gmail.com (M. Gabeleh), jmarkin@cybermesa.com (J. Markin), maggie.aphane@smu.ac.za (M. Aphane) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 2 of 17 and T is noncyclic provided that T (G) ⊆ G, T (H) ⊆ H. It is worth mentioning that if the map T is cyclic and G ∩ H = ∅, then the fixed point equation Tx = x does not have any solution. Indeed, for any x ∈ G∪H we have d(x, Tx) ≥ D(G,H) := inf{d(u, v) : (u, v) ∈ G × H}. So, in this case, instead of finding a fixed point for the cyclic mapping T we can think about the existence of a point x∗ ∈ G ∪H for which d(x∗, Tx∗) = D(G,H). Such points are called best proximity points of the cyclic mapping T . On the other hand, if T is noncyclic, then the fixed point equation may have a fixed point, but it is interesting to find a pair of fixed points (x∗, y∗) ∈ G × H (i.e. Tx∗ = x∗, T y∗ = y∗) such that d(x∗, y∗) = D(G,H). These kinds of points are called best proximity pairs for the noncyclic mapping T . Existence of best proximity points (pairs) for cyclic (noncyclic) relatively nonexpansive mappings was first established by Eldred et al. in [1] and after that R. Espinola ([2]) used a different approach to obtain the existence results of [1] (see also [3, 4] for more information about cyclic maps which satisfy contractive conditions). Here is a main result of [1]. Theorem 1. Let (G,H) be a nonempty, compact and convex pair of subsets of a Banach space X. If T : G ∪H → G ∪H is a cyclic relatively nonexpansive mapping, then T has a best proximity point. We mention that the proof of the above theorem is based on the fact that every nonempty, compact and convex pair of subsets of a Banach space X has the proximal normal structure (see Proposition 2.2 and Theorem 2.1 of [1]). Another way to prove Theorem 1 was presented in [5] by applying a concept of proximal diametral sequences. In order to state the noncyclic version of Theorem 1 we need to recall that a Banach space X is strictly convex if for any two distinct elements u, v ∈ SX := {x ∈ X : ∥x∥ = 1} we have ∥u+v 2 ∥ < 1. Hilbert and Lp(1 < p < +∞) spaces are instances of strictly convex Banach spaces. Theorem 2. (see Theorem 2.2 of [1]) Let (G,H) be a nonempty, compact and convex pair of subsets of a strictly convex Banach space X. If T : G ∪ H → G ∪ H is a noncyclic relatively nonexpansive mapping, then T has a best proximity pair. It was announced in [6] that if the Banach space X in Theorem 1 is strictly convex, then Theorem 1 is a special case of Theorem 2. In what follows we recall the extensions of Theorem 1 and Theorem 2 to a more extensive family of cyclic (noncyclic) relatively nonexpansive mappings. Definition 1. ([7]) Let (G,H) be a nonempty pair in a metric space (M, d). A mapping T : G ∪ H → G ∪ H is called relatively u-continuous if for each ε > 0 there exists δ > 0 such that d(Tx, Ty) < ε+D(G,H), whenever d(x, y) < δ+D(G,H), for all (x, y) ∈ G×H. If moreover, T is also cyclic (noncyclic), then T is said to be cyclic (noncyclic) relatively u-continuous. M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 3 of 17 Clearly, every relatively nonexpansive map is relatively u-continuous, but the inverse implication may not be hold (see Example 2.1 of [7]). Theorem 3. (Theorem 3.1 of [7]) Let (G,H) be a nonempty, compact and convex pair of subsets of a strictly convex Banach space X. If T : G ∪ H → G ∪ H is a cyclic relatively u-continuous mapping, then T has a best proximity point. The noncyclic version of Theorem 3 is as follows. Theorem 4. (Theorem 4.2 of [8]) Let (G,H) be a nonempty, compact and convex pair of subsets of a strictly convex Banach space X. If T : G∪H → G∪H is a noncyclic relatively u-continuous mapping, then T has a best proximity pair. Motivated by Schauder’s fixed point problem for compact and continuous self-mappings defined on a bounded, closed and convex subset of a Banach space, the current authors presented the extensions of Theorem 1 and Theorem 2 by shifting the compactness as- sumption on the pair (G,H) to the cyclic (noncyclic) mapping T which may not be con- tinuous (see Theorem 3.2 and Theorem 4.1 of [9]). Indeed, the cyclic (noncyclic) mapping T : G ∪H → G ∪H is said to be compact whenever ( T (K), T (L) ) is relatively compact for any bounded pair (K,L) ⊆ (G,H). Gabeleh and Vetro ([10]) generalized the aforesaid results by relaxing the compactness assumption of the cyclic (noncyclic) mappings. To state their main corollaries, we remind the reader of some requirements. From now on, B(M) denotes the family of all nonempty and bounded subsets of M. Also K(M) displays the family of all nonempty and compact subsets of M. In the case that X is a Banach space, we use BCC(X) to denote the class of all nonempty, bounded, closed and convex subsets of X. Definition 2. A measure of noncompactness (MNC for brief) is a function ℵ : B(M) → [0,+∞) satisfying the following axioms: 1. ℵ(G) = 0 if and only if G is relatively compact, 2. ℵ(G) = ℵ(G), G ∈ B(X), 3. ℵ(G ∪ H) = max{ℵ(G),ℵ(H)}, where G,H ∈ B(M). Some useful properties of the MNC ℵ on B(M) can be listed as below (see [11] for more details). (a) G ⊆ H implies ℵ(G) ≤ ℵ(H); (b) ℵ(G ∩ H) ≤ min{ℵ(G),ℵ(H)}, for all G,H ∈ B(M). (c) If lim n→+∞ ℵ(Gn) = 0 for a nonincreasing sequence {Gn} of nonempty, bounded and closed subsets of M, then G∞ := ⋂ n≥1 Gn ∈ K(M). M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 4 of 17 Here are two well-known examples of MNCs. Example 1. Define α : B(M) → [0,+∞) as α(G) = inf { ε > 0 : G can be covered by finitely many sets with diameter ≤ ε } , for all G ∈ B(M). Then α is an MNC which was first introduced by Kuratowski ([12]). Also a function χ : B(M) → [0,+∞) which is defined as χ(G) = inf { ε > 0 : G can be covered by finitely many balls with radii ≤ ε } , for all G ∈ B(M), is a generalized version of Kuratowski MNC which was presented later by Hausdorff. For a nonempty pair (G,H) in a metric space (M, d) we set G0 = { u ∈ G : there exists v′ ∈ H such that d(u, v′) = D(G,H) } , H0 = { v ∈ H : there exists u′ ∈ G such that d(u′, v) = D(G,H) } . The pair (G0,H0) is said to be a proximal pair of (G,H). There are different conditions to ensure nonemptyness of proximal pairs. For example (G0,H0) is nonempty if one of the following conditions hold: (i) (G,H) is a nonempty and compact pair in a metric space (M, d); (ii) (G,H) is a nonempty pair in a metric space (M, d) such that G is compact and H is approximatively compact w.r.t. G. We recall that the set H is approximatively compact w.r.t. the set G whenever for any point x ∈ G and any sequence {yn} in the set H for which d(x, yn) → D({x},H), then {yn} has a convergent subsequence in H; (iii) (G,H) is a nonempty and weakly compact pair in a Banach space X; (iv) (G,H) is a nonempty, closed and convex pair in a reflexive Busemann convex metric space (M, d) such that H is bounded (see [13]); (iv) (G,H) is a nonempty and admissible pair in a hyperconvex metric space (M, d) (see [14]). We will say that the nonempty pair (G,H) is proximinal whenever G0 = G, H0 = H. Let (G,H) be a nonempty pair in a Banach space X and let T : G ∪H → G∪H be a cyclic (noncyclic) relatively nonexpansive mapping. We set MG×H(T ) = { (L1,L2) ⊆ (G,H) s.t. (L1,L2) is nonempty, bounded, closed, convex, M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 5 of 17 proximinal and T − invariant with D(L1,L2) = D(G,H) } . It is worth noticing that if for example (G,H) is a nonempty, weakly compact and convex pair in a Banach space X, then (G0,H0) ∈ MG×H(T ) (see Lemmas 2.3 and 2.4 of [15]). We are now ready to recall the concept of Meir-Keeler condensing operators which was introduced in [10]. Definition 3. Let (G,H) be a nonempty and convex pair in a Banach space X and ℵ be an MNC on X. A mapping T : G ∪ H → G ∪ H is said to be a Meir-Keeler condensing operator if T is cyclic (noncyclic) and for any ε > 0 there exists δ = δ(ε) > 0 such that for any (L1,L2) ∈ MG×H(T ) we have ε ≤ ℵ(L1 ∪ L2) < ε+ δ ⇒ ℵ ( T (L1) ∪ T (L2) ) < ε. The next best proximity point (pair) theorems are the main results of [10]. Theorem 5. Let (G,H) be a nonempty, bounded, closed and convex pair in a Banach space X such that G0 is nonempty and ℵ is an MNC on X. Let T : G ∪ H → G ∪H be a cyclic relatively nonexpansive mapping which is a Meir-Keeler condensing operator. Then T has a best proximity point. Theorem 6. Let (G,H) be a nonempty, bounded, closed and convex pair in a strictly convex Banach space X such that G0 is nonempty and ℵ is an MNC on X. Let T : G ∪ H → G ∪ H be a noncyclic relatively nonexpansive mapping which is a Meir-Keeler condensing operator. Then T has a best proximity point. The main purpose of this article is to present counterpart results of Theorem 5 and 6 in the setting of hyperconvex metric spaces under different conditions. 2. Best proximity version of Schauder’s fixed point theorem in Hyperconvex spaces Let (M, d) be a metric space. Throughout this article, B(x; r) displays a closed ball centered at x ∈ M with radius r > 0, that is, B(x; r) = {y ∈ M ; d(y, x) ≤ r}. We mention that a multivalued mapping F : M → 2M − {∅} is said to be almost lower- semicontinuous at a point x ∈ M if for each ε > 0 there exist an open neighborhood U(x) of x and a point z ∈ Fx such that B(z, ε) ∩ F (y) ̸= ∅, for all y ∈ U(x). For a given ε > 0 and G ∈ BC(M), the ε-neighborhood of G is defined with Nε(G) := { u ∈ M : D(u,G) ≤ ε } . M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 6 of 17 A concept of hyperconvexity is based on a property of closed balls in metric spaces which was introduced by Aronszajn and Panitchpakdi ([16]) in 1965 as follows. Definition 4. Let (M, d) be a metric space and ∅ ≠ G ⊆ M. Then G is called hyperconvex if for any family {xα}α∈I in G such that d(xα, xβ) ≤ rα + rβ for all α, β ∈ I, we have⋂ α∈I B(xα; rα) ∩ G ̸= ∅. Particularly, if G = M, then we say that M is a hyperconvex metric space. It is worth mentioning that hyperconvex metric spaces are complete (see [17] for more details). In order to give a suitable characterization of hyperconvexity, we need to recall the following notions. Definition 5. A metric space (M, d) is called metrically convex whenever for any two distinct points x, y ∈ M there exists an element z ∈ M− {x, y} for which d(x, y) = d(x, z) + d(z, y). Definition 6. A metric space (M, d) is said to have binary ball intersection property provided that any family of closed balls, each two of which intersect must have nonempty intersection. We now have the following result regarding hyperconvex spaces. Proposition 1. (see [17]; p. 77) A complete metric space (M, d) is hyperconvex if and only if M is metrically convex and has the binary ball intersection property. Using Proposition 1 it can be shown that the non-reflexive Banach spaces L∞, l∞ are hyperconvex. It is interesting to note that hyperconvex subsets of Banach spaces may not be convex. Example 2. In the finite dimensional Banach space (R2, ∥.∥∞), suppose G = { (x, x+ 1) : −1 ≤ x ≤ 0 }⋃{ (x,−x+ 1) : 0 ≤ x ≤ 1 } . Then G is metrically convex and has binary intersection property, which means that G is a hyperconvex subset of (R2, ∥.∥∞), but clearly G is not convex. The next result give us an appropriate condition for convexity of hyperconvex subsets of a Banach space. Proposition 2. (see Corollary 4.5.18 of [18]) Let X be a Banach space. Then the following statements are equivalent: (i) X is strictly convex; (ii) Every nonempty and hyperconvex subset of X is convex. M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 7 of 17 To state another useful characterization of convexity of hyperconvex subspaces of a Banach space X, we recall the following geometric concept. Definition 7. ([19]) Let (G,H) be a nonempty pair in a metric space (M, d) with G0 ̸= ∅. The pair (G,H) is said to have the P-property if and only if{ d(x1, y1) = D(G,H) d(x2, y2) = D(G,H) ⇒ d(x1, x2) = d(y1, y2), where x1, x2 ∈ G0 and y1, y2 ∈ H0. We now conclude the following result. Proposition 3. Let X be a Banach space. Then the following statements are equivalent: (i) Ever pair of nonempty, closed and convex subsets of X has the P-property; (ii) Every nonempty and hyperconvex subset of X is convex. Proof. It is sufficient to note that by Theorem 3.1 of [20], every pair of nonempty, closed and convex subsets of X has the P-property if and only if X is strictly convex. Now the result follows from Proposition 2. Definition 8. A subset G of a hyperconvex metric space (M, d) is called admissible if G is the nonempty intersection of a family of closed balls. The set of all admissible subsets of a hyperconvex space M will be denoted by A(M). It is well-known that if (M, d) is a hyperconvex space, then every admissible subset of M is also hyperconvex. For a nonempty and bounded subset G of a hyperconvex space M the cover of G is defined with cov(G) = ⋂{ B ⊆ M : B is a closed ball containing G } . Also, we set δx(G) := sup y∈G { d(x, y) : y ∈ G } , for all x ∈ M. Definition 9. A set G in a hyperconvex metric space (M, d) is said to be sub-admissible provided that for any finite subset E of G we have cov(E) ⊆ G. The hyperconvex version of Schauder’s fixed problem was presented by R. Espinola as follows. Theorem 7. (Lemma 3 of [2]) Let (M, d) be a compact and hyperconvex metric space. If T : M → M is continuous, then T has a fixed point. M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 8 of 17 An important observation about Theorem 7 is that if M is a hyperconvex subset of a Banach space X, then M may not be convex and so the existence of a fixed point for the continuous mapping T cannot be obtained from the Schauder’s fixed point theorem. We are now in position to state the first existence result of this paper which is not only a different version of Theorem 1 in the framework of hyperconvex spaces, but also is a generalization of Theorem 7. Theorem 8. (Compare to Theorem 1 and Theorem 7) Let (M, d) be a hyperconvex metric space and (G,H) be a nonempty, compact and hyperconvex pair in M. If T : G∪H → G∪H is a cyclic relatively u-continuous mapping, then T has a best proximity point. Proof. By statement (ii) on page 4, the proximal pair (G0,H0) is nonempty. To see that both sets are compact, assume that {xn} is a sequence in G0 converging to a point p ∈ G. Then, there exists a corresponding sequence {yn} in H such that d(xn, yn) = D(G,H). By compactness of H, {yn} has a convergent subsequence, also labeled as {yn}, with limit q, such that limn→+∞ d(xn, yn) = d(p, q) = D(G,H). Therefore, G0 and H0 are closed subsets of compact sets, hence both are compact. To prove hyperconvexity of the pair (G0,H0), choose points {xi} in G0 and {yi} in H0 such that d(xi, yi) = D(G,H). Consider the collection of balls {B(xi; ri)} and {B(yi; ri)} such that d(xi, xj) ≤ ri+ rj and d(yi, yj) ≤ ri + rj . By a result of Khamsi et al. [21], dH (⋂ i B(xi; ri), ⋂ i B(yi; ri) ) ≤ sup i d(xi, yi) = D(G,H), where dH is the Hausdorff metric derived from the metric d. This shows that B(xi; ri) ∩ G0 ̸= ∅, implying that G0 is hyperconvex, and similarly that H0 is hyperconvex. Using the fact that T is relatively u-continuous, (G0,H0) is T -invariant. Define a multivalued mapping F : G0 → 2G0 by F (v) := B ( Tv; D(G,H) )⋂ G0, for all v ∈ G0. Note that the values of F are nonempty. Indeed, for any v ∈ G0, there exists y ∈ H0 such that d(v, y) = D(G,H) and again using the relatively u-continuity of T , d(Tv, Ty) = D(G,H) and so, Ty ∈ B ( Tv; D(G,H) )⋂ G0. We claim that F is almost lower-semicontinuous. Let x0 ∈ G0. Then there is an element y0 ∈ H0 such that d(x0, y0) = D(G,H). Now for given ε > 0 there exists δ > 0 such that if z ∈ G with d(z, y0) < δ + D(G,H), then d(Tz, Ty) < ε+D(G,H). Set U(x0, δ) := { u ∈ G0 : d(u, x0) < δ } . Then for any u ∈ U(x0, δ) we have d(u, y0) ≤ d(u, x0) + d(x0, y0) < δ +D(G,H), and hence, d(Tu, Ty0) < ε+D(G,H). (1) M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 9 of 17 We claim that B(Ty0; ε) ∩ Fu ̸= ∅ for u ∈ U(x0, δ). By (1) B(Ty0; ε) ∩ B ( Tu,D(G,H) ) ̸= ∅. Consider cov(p, q), where p ∈ B(Ty0, ε) ∩ G0, and q ∈ Fu. Since G0 is hyperconvex, cov(p, q) ⊆ G0. Then, as an intersection of admissible sets with pairwise nonempty inter- sections, B(Ty0; ε) ∩ cov(p, q) ∩ B ( Tu,D(G,H) ) ̸= ∅. This shows that B(Ty0; ε)∩ Fu ̸= ∅ and therefore, that F is almost lower-semicontinuous at the point x0. To see that values of F are sub-admissible sets, let {xi} be a collection of points in F (v) and consider the set cov({xi}). Hyperconvexity of G0 implies that cov({xi}) ⊆ G0, and the relation δcov({xi})(Tv) = δ{xi}(Tv), deduces that cov({xi}) ⊆ F (v), concluding that F (v) is sub-admissible. By Theorem 1 of [22], as an almost lower- semicontinuous mapping with sub-admissible values, the mapping F has a continuous selection ℏ : G0 → G0 such that ℏv ∈ Fv for v ∈ G0. By using Theorem 7, the continuous self map ℏ on the compact hyperconvex space G0 has a fixed point x∗. The definition of the mapping F implies d(x∗, Tx∗) = D(G,H) and the proof is completed. The next result is a consequence of Theorem 8. Corollary 1. (Theorem 15 of [23]) Let (G,H) be a nonempty and admissible pair in a hyperconvex metric space (M, d) such that G0 is compact. Let T : G ∪ H → G ∪ H be a cyclic relatively u-continuous mapping. Then T has a best proximity point. Proof. By Lemma 2.5 of [14], since (G,H) is an admissible pair in a hyperconvex space M the proximal pair (G0,H0) is nonempty and admissible too, which ensures that G0 is hyperconvex. Now the result follows from Theorem 8, immediately. The next example shows the useability of Theorem 8 w.r.t. Theorem 3. Example 3. Consider the hyperconvex space ℓ∞ consists of all real bounded sequences equipped with the supremum norm and let {en}n∈N be the canonical basis of ℓ∞. Set G = { te1 + e2 ; t ∈ [0, 1] } H = { se1 + e3 ; s ∈ [0, 1] } . Then (G,H) is a compact, convex, proximinal and hyperconvex pair in ℓ∞ with D(G,H) = 1. Define a mapping T : G ∪ H → G ∪H with T (te1 + e2) = te1 + e3, T (se1 + e3) = se1 + e2, for all t, s ∈ [0, 1]. Then T is cyclic and for any x = te1 + e2 ∈ G and y = se1 + e3 we have ∥Tx− Ty∥∞ = 1 = ∥x− y∥∞, M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 10 of 17 which implies that T is relatively u-continuous. It now follows from Theorem 8 that T has a best proximity point, indeed, every point of G is a best proximity point of T . Note that the Banach space ℓ∞ is not strictly convex and so the existence of a best proximity point of T cannot be concluded from Theorem 3. It is worth noticing that the noncyclic version of Theorem 8 holds by adding a geometric property which is inspired from strictly convex Banach spaces. Definition 10. ([6]) A nonempty pair (G,H) in a metric space (M, d) is said to be a semi-sharp proximinal pair provided that for any (x, y) ∈ G × H there exists at most one point (x′, y′) ∈ G ×H for which d(x, y′) = d(x′, y) = D(G,H). It is worth noticing that every nonempty and convex pair in a strictly convex Banach space X is a semi-sharp proximinal pair (see [6]). The next example shows that strict convexity assumption of a Banach space X is not an essential condition for semi-sharp proximinality of convex pairs. Example 4. In the hyperconvex space ℓ∞, put G = con ({ e2n−1 + e2n ; n ∈ N }) H = con ({ 2e2n + e2n+1 ; n ∈ N }) , where con(A) denotes the closed convex hull of the set A ⊆ ℓ∞. Clearly, D(G,H) = 1, G0 = G, H0 = H, and that (G,H) is a semi-sharp proximinal pair. Theorem 9. (Compare to Theorem 4) Let (G,H) be a nonempty, compact and hyper- convex pair in a hyperconvex metric space (M, d) which is a semi-sharp proximinal pair. If T : G ∪ H → G ∪ H is a noncyclic relatively u-continuous mapping, then T has a best proximity pair. Proof. As in the proof of Theorem 8, (G0,H0) is nonempty and hyperconvex. T - invariance of these sets follows from relatively u-continuity of T , that is, for any pair (x, y) ∈ G0×H0, if d(x, y) = D(G,H), then d(Tx, Ty) = D(G,H), implying that (Tx, Ty) ∈ (G0,H0). Choose x0 ∈ G0 and y0 ∈ H0 such that d(x0, y0) = D(G,H). Then by relatively u- continuity of T we have d(Tx0, Ty0) = D(G,H). Given ε > 0, define an open neighborhood of x0 by U(x0, δ) := {u ∈ G0 : d(u, x0) < δ}. Then u ∈ U(x0, δ) implies that d(u, y0) < d(u, x0) + d(x0, y0) < δ +D(G,H), M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 11 of 17 and therefore, by relatively u-continuity of T , d(Tu, Ty0) < ε+D(G,H). Now define a multivalued map F : G0 → 2H0 by F (u) = B ( Tu; D(G,H)) ⋂ H0, for u ∈ G0. By analogous arguments to those in the proof of Theorem 8, F is an almost lower-semicontinuous mapping with subadmissible values. Then, as above, F has a con- tinuous selection h : G0 → H0. By semi-sharp proximinality of the pair (G,H), we can define a mapping P : H0 → G0 as a projection map that associates to each point p in H0 the unique point q in G0 such that d(p, q) = D(G,H). We claim that this correspondence is a continuous mapping. Assume that a sequence {pn} in H0 converges to a point p ∈ H, and by contrary assume that {P(pn)} does not converge to P(p). By compactness of the set G0, {P(pn)} has a convergent subsequence {qn} that converges to a point q. Then, limn→+∞ d(pn, qn) = d(p, q) = D(G,H), where q ̸= P(p), contradicting uniqueness of the proximal point. Therefore, P is a continuous mapping. As a continuous self mapping on a compact hyperconvex set, the mapping Poh : G0 → G0 has a fixed point z ∈ G0. By definition of this mapping, z = Tz, and if w is the unique point in H0 such that d(z, w) = D(G,H), it follows from relatively u-continuity of T that w = Tw and the proof is completed. The next example guarantees that the semi-sharp proximinality of the pair (G,H) in Theorem 9 is essential. Example 5. Consider the hyperconvex space ℓ∞ with the canonical basis {en}n∈N and let G = { te1 + 2e2 : 0 ≤ t ≤ 1 } , H = { te3 : 0 ≤ t ≤ 1 } . Then (G,H) is a compact and hyperconvex pair with D(G,H) = 2. Clearly, (G,H) is not a semi-sharp proximinal pair. Define T : G ∪ H → G ∪H by T (te1 + 2e2) = √ te1 + 2e2, for all t ∈ [0, 1], T (te3) = { 0, if t ̸= 0, e3, if t = 0. Then d∞(Tx, Ty) = 2 = d∞(x, y), for all (x, y) ∈ G ×H, which deduces that T is a noncyclic relatively u-continuous. Note that T does not possess any best proximity pair. M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 12 of 17 3. Hyperconvex spaces equipped with a suitable MNC In this section, we extend Theorem 8 and Theorem 9 by relaxing the compactness assumption of the hyperconvex space M and replacing a suitable MNC on M. For this purpose, we need to recall the following useful tools of hyperconvex spaces which were presented by M.A. Khamsi in [24]. Definition 11. Let G be a nonempty subset of a metric space (M, d). We say that G is a nonexpansive retract of M if there exists a nonexpansive mapping R : M → G such that Rx = x for any x ∈ G. Proposition 4. (Proposition 1 of [24]) A metric space (M, d) is hyperconvex iff for any metric space (N , ρ) which contains isometrically M, there exists a nonexpansive retract R : N → M. Now for a metric space (M, d) we define ℓ∞(M) = { (xj)j∈M ⊆ R : sup j∈M |xj | < +∞ } , and let d∞ ( (xj), (yj) ) := sup j∈M |xj − yj |, for all (xj), (yj) ∈ ℓ∞(M). Then ( ℓ∞(M), d∞ ) is a hyperconvex metric space. Let x0 ∈ M be a fixed element and define the natural isometric embedding η : M → ℓ∞(M) as follows: η(x) = ( d(x, y)− d(x0, y) ) y∈M. Notice that for any y ∈ M we have supy∈M |d(x, y) − d(x0, y)| ≤ d(x, x0) and so, η is well-defined. Also, d∞ ( η(x), η(z) ) = sup y∈M |d(x, y)− d(z, y)| = d(x, z), for all x, z ∈ M, which implies that the metric space ℓ∞(M) contains an isometric copy M. If M is a hyperconvex space, then η(M) is an isometric copy of M in ℓ∞(M) and is, therefore, a hyperconvex set. Throughout this work we will use M instead of η(M). Similarly, for any hyperconvex subset G of M we use G instead of η(G). For M or any hyperconvex subset G of M, we define M∞ = cov(M) or G∞ = cov(G) in ℓ∞(M). Clearly, M∞ and G∞ are hyperconvex subsets of ℓ∞(M) and are convex. It is worth noticing that for any hyperconvex space M there is a nonexpansive retraction R of M∞ onto M, and for any hyperconvex subset G of M, R(G∞) = G (see [17]). We are now able to define a suitable MNC on the hyperconvex space M∞. Lemma 1. Let (M, d) be a hyperconvex metric space and R be a nonexpansive retract from M∞ onto M. Suppose ℵ is an MNC on M and define a function ℵR : B(M∞) → [0,∞) as ℵR(G) := ℵ ( R(G) ) , for all G ∈ B(M∞). Then ℵR is an MNC on M∞. M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 13 of 17 Proof. The proof is trivial. The next lemmas play important roles in our next results. Lemma 2. (Lemma 2.5 of [14]) Let (G,H) be a nonempty and admissible pair in a hy- perconvex metric space (M, d). Then the proximal pair (G0,H0) is also nonempty and admissible. Lemma 3. Let (G,H) be a nonempty and admissible pair in a hyperconvex metric space (M, d). Suppose (E ,F) ⊆ ( G∞,H∞ ) is a nonempty, bounded, closed, convex, and proximi- nal pair such that D(E ,F) = D(G,H) ( = D(G∞,H∞) ) . Then ( R(E), R(F) ) is a nonempty, bounded, hyperconvex, and proximinal pair in M with D ( R(E), R(F) ) = D(G,H). Proof. As we know, in ℓ∞(M) each closed, bounded and convex set is also an admissible set. Thus (E ,F) is admissible in ℓ∞(M). Since a nonexpansive retraction preserves hyperconvexity, ( R(E), R(F) ) is a hyperconvex pair. To see the proximinality of the latter pair, choose any point Rp ∈ R(E), where p ∈ E . Then there is a point q ∈ F such that d∞(p, q) = D(E ,F) ( = D(G,H) ) , implying that d(Rp,Rq) ≤ D(E ,F). Besides, the relation ( R(E), R(F) ) ⊆ (G,H) concludes that D(E ,F) = D ( R(E), R(F) ) . Therefore, d(Rp,Rq) = D(E ,F) and so, ( R(E), R(F) ) is a proximinal pair. Motivated by Definition 3 we introduce the class of Meir-Keeler condensing operators in the setting of hyperconvex metric spaces. Let (G,H) be a nonempty and admissible pair in a hyperconvex space (M, d) and let T : G ∪ H → G ∪H be a cyclic (noncyclic) relatively u-continuous mapping. Put M′ G×H(T ) = { (L1,L2) ⊆ (G,H) s.t. (L1,L2) is nonempty, bounded, hyperconvex, proximinal and T − invariant with D(L1,L2) = D(G,H) } . By Lemma 2, under the aforesaid assumptions, the proximal pair (G0,H0) is also nonempty and admissible. Also, by the fact that T is relatively u-continuous, (G0,H0) is T -invariant. So, (G0,H0) ∈ M′ G×H(T ) ̸= ∅. Definition 12. Let (G,H) be a nonempty and admissible pair in a hyperconvex space (M, d) and ℵ be an MNC on M. A mapping T : G ∪ H → G ∪ H is said to be an H- Meir-Keeler condensing operator if T is cyclic (noncyclic) and for any ε > 0 there exists δ = δ(ε) > 0 such that for any (L1,L2) ∈ M′ G×H(T ) we have ε ≤ ℵ(L1 ∪ L2) < ε+ δ ⇒ ℵ ( T (L1) ∪ T (L2) ) < ε. Let us illustrate the concept of H-Meir-Keeler condensing operator with the following example. M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 14 of 17 Example 6. Consider the hyperconvex space ℓ∞ equipped with Kuratowski MNC α defined in Example 1. Let (G,H) be a nonempty and admissible pair in ℓ∞ for which H is compact. Suppose that T : G ∪ H → G ∪H is a cyclic mapping such that T ∣∣ H is a contraction, that is, there exists r ∈ (0, 1) such that for any v, y ∈ H d∞(Tv, Ty) ≤ rd∞(v, y). Then for any (L1,L2) ∈ M′ G×H(T ) by the compactness of H, α ( T (L1) ∪ T (L2) ) = max{α ( T (L1) )︸ ︷︷ ︸ 0 , α ( T (L2) ) } = α ( T (L2) ) ≤ rα(L2) ≤ rα(L1 ∪ L2), which ensures that T is an H-Meir-Keeler condensing operator. In order to prove the main existence results of this section, we need the following auxiliary lemmas. Lemma 4. Let (G,H) be a nonempty and admissible pair in a hyperconvex metric space (M, d) and ℵ be an MNC on M. Suppose T : G∪H → G∪H is cyclic (noncyclic) relatively u-continuous mapping which is an H-Meir-Keeler condensing operator. Then TR : G∞ ∪H∞ → G∞ ∪H∞, is a cyclic (noncyclic) relatively u-continuous mapping which a Meir-Keeler condensing operator w.r.t. the MNC ℵR. Proof. We assume that T is cyclic. Then TR(G∞) = T (G) ⊆ H ⊆ H∞, TR(H∞) = T (H) ⊆ G ⊆ G∞, that is, TR is also cyclic. Now let ε > 0 be given. Since T is relatively u-continuous, there exists δ > 0 such that for any (x, y) ∈ G × H with d(x, y) < δ + D(G,H) we have d(Tx, Ty) < ε + D(G,H). Since R : M∞ → M is a nonexpansive retract, for any (u, v) ∈ G∞ ×H∞ with d∞(u, v) < δ +D(G,H) we have d∞(Ru,Rv) < δ +D(G,H) and therefore, d∞(TRu, TRv) < ε+D(G,H), that is, TR is relatively u-continuous. To show that TR is a Meir-Keeler condensing operator consider ε > 0 and (L,K) ∈ MG∞×H∞(TR). Then (L,K) ⊆ (G∞,H∞) is a nonempty, bounded, closed, convex, proximinal and TR- invariant with D(L,K) = D(G∞,H∞). It follows from Lemma 3 that ( R(L), R(K) ) is bounded, hyperconvex and proximinal with D ( R(L), R(K) ) = D(G,H). Since T maps G into H, T (R(K)) ⊆ L ∩H ⊆ R(L). We now have T ( R(L) ) = TR(L) ⊆ R(K), T ( R(K) ) = TR(K) ⊆ R(L), M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 15 of 17 which implies that ( R(L), R(K) ) is T -invariant. Thus, ( R(L), R(K) ) ∈ M′ G×H(T ). In view of the fact that T is an H-Meir-Keeler condensing operator, there exists δ = δ(ε) > 0 such that ε ≤ ℵ ( R(L) ∪R(K) )︸ ︷︷ ︸ =ℵ ( R(L∪K) ) < ε+ δ ⇒ ℵ ( T (R(L)) ∪ T ((RK)) ) < ε. By the definition of ℵR, we have that ℵR(B) = ℵ(R(B)) = ℵ(B) for any bounded subset of M, and so, ε ≤ ℵR ( L ∪ K ) < ε+ δ ⇒ ℵR ( TR(L) ∪ TR(K) ) < ε. Thereby, TR is an ℵR-Meir-Keeler condensing operator. In the case that T is noncyclic, the result follows, similarly. The next theorem is a different version of Theorem 5 in hyperconvex spaces. Theorem 10. Let (G,H) be a nonempty, admissible pair in a hyperconvex metric space (M, d) and ℵ be an MNC on X. Suppose T : G ∪ H → G ∪ H is a cyclic relatively u- continuous mapping which is an H-Meir-Keeler condensing operator. Then T has a best proximity point. Proof. It follows from Lemma 4 that the mapping TR : G∞ ∪ H∞ → G∞ ∪ H∞ is a cyclic relatively u-continuous mapping which is a Meir-Keeler condensing operator w.r.t the MNC ℵR. It now follows from Theorem 5 that TR has a best proximity point, i.e., there exists a point p ∈ G∞ ∪ H∞ such that d∞(p, TRp) = D(G∞,H∞) = D(G,H). We now have d ( Rp, TRp ) ≤ d∞(p, TRp) = D(G,H). Therefore, Rp is a best proximity point for the mapping T and the proof is completed. The noncyclic version of Theorem 10 is as follows. Theorem 11. Let (G,H) be a nonempty, admissible and semi-sharp proximinal pair in a hyperconvex metric space (M, d) and ℵ be an MNC on X. Suppose T : G∪H → G∪H is a noncyclic relatively u-continuous mapping which is an H-Meir-Keeler condensing operator. Then T has a best proximity pair. Proof. By Lemma 4, the mapping TR : G∞ ∪H∞ → G∞ ∪H∞ is a noncyclic relatively u-continuous mapping, which is a Meier-Keeler condensing operator w.r.t. the MNC ℵR. As a mapping defined in the Banach space ℓ∞(M) that satisfies the conditions of Theorem 6, the mapping TR has a best proximity pair, that is, there exists a pair (p, q) ∈ G∞×H∞ such that d∞(p, q) = D(G∞,H∞) = D(G,H), we obtain p = TRp and q = TRq. Since TRp ∈ G and TRq ∈ H, we have Rp = TRp and Rq = TRq. Similarly, we can write d∞(Rp,Rq) = D(G∞, H∞) = D(G,H). Thus, Rp and Rq are a best proximity pair for the noncyclic mapping T and we are finished. M. Gabeleh, J. Markin, M. Aphane / Eur. J. Pure Appl. Math, 18 (3) (2025), 6433 16 of 17 4. Conclusion In this paper, we revisited the main conclusions of [10] related to the existence of best proximity points (pairs) for cyclic (noncyclic) Meir-Keeler condensing operators and obtained similar results in the framework of hyperconvex metric spaces. 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