@ Appl. Gen. Topol. 23, no. 1 (2022), 91-105 doi:10.4995/agt.2022.15739 © AGT, UPV, 2022 On w-Isbell-convexity Olivier Olela Otafudu a and Katlego Sebogodi b Dedicated to the memory of Prof. Hans-Peter Künzi a School of Mathematical and Statistical Sciences North-West University, Potchefstroom Campus, Potchefstroom 2520, South Africa. (olivier.olelaotafudu@nwu.ac.za) b Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa. (7katlego3@gmail.com) Communicated by M. A. Sánchez-Granero Abstract Chistyakov introduced and developed the concept of modular metric for an arbitrary set in order to generalise the classical notion of modular on a linear space. In this article, we introduce the theory of hyper- convexity in the setting of modular pseudometric that is herein called w-Isbell-convexity. We show that on a modular set, w-Isbell-convexity is equivalent to hyperconvexity whenever the modular pseudometric is continuous from the right on the set of positive numbers. 2020 MSC: 45E35; 46A80. Keywords: modular pseudometric; Isbell-convexity; w-Isbell-convexity. 1. Introduction Modular metric spaces were introduced by Chistyakov [8] in 2010. He de- veloped the theory of modular metric on an arbitrary set and investigated the theory of metric spaces induced by a modular metric. He defined a modu- lar metric in the following way. Let X be a nonempty set, then the function w : (0,∞) × X × X −→ [0,∞] is called a modular metric if it satisfies (a) w(λ,x,y) = 0 if and only if x = y whenever λ > 0, (b) w(λ,x,y) = w(λ,y,x) whenever x,y ∈ X and λ > 0 and (c) w(λ + µ,x,y) ≤ w(λ,x,z) + w(µ,z,y) whenever x,z,y ∈ X and λ,µ > 0. Received 06 June 2021 – Accepted 10 December 2021 http://dx.doi.org/10.4995/agt.2022.15739 O. Olela Otafudu and K. Sebogodi Furthermore, the function w is said to be modular pseudometric on X if in- stead of (a), the function satisfies (d) w(λ,x,x) = 0 for all λ > 0 and x ∈ X. For a ∈ X, the modular set Xw(a) is defined by Xw = Xw(a) = {x ∈ X : limλ→∞w(λ,a,x) = 0}. Chistyakov equipped Xw with the metric qw, where qw(x,y) = inf{λ > 0 : w(λ,x,y) ≤ λ} whenever x,y ∈ Xw. In [6], Chistyakov introduced a topology τ(w) on Xw in the following sense. A subset V of Xw is τ(w)-open if for any λ > 0 and x ∈ V , there exists µ > 0 such that the entourage set Bλ,µ(x) = {y ∈ Xw : w(λ,x,y) ≤ µ} ⊂ V . He studied the Hausdorff modular pseudometric on a power set of a nonempty set equipped with a modular pseudometric. In addition, Chistyakov provided an application of modular metrics which consists of an extended kind of Helly’s theorem on the pointwise selection principle. This was obtained by building a special modular space, the set of all bounded and regulated mappings on an interval. Furthermore, Chistyakov considered the description of superposition operators acting in modular spaces, the existence of regular selections of set- valued maps, the new interpretation of Lipschitzian and absolutely continuous maps, and the existence of solutions to the Carathéodory-type ordinary dif- ferential equations in Banach spaces with the right-hand side from the Orlicz space. Since Chistyakov developed the theory of metric spaces generated by a mod- ular metric defined on an arbitrary set, the interest of this concept has grown up in the community of mathematicians, especially in operator theory where many authors are applying modular metrics to study the existence and uniquenesse of fixed points of self-maps on a modular set that satisfy some particular prop- erties (see for instance [4, 5, 20, 28]). Let mention some important and popular recent works on different type of modular metric spaces [1, 3, 10, 11, 12, 14, 15]. In addition, the well-known and important concept of hyperconvexity in a metric space has been successfully investigated and applied in many areas of mathematics and other fields. For instance the theory of hyperconvexity has been introduced in the framework of quasi-pseudometric spaces which is called Isbell-convexity (or q-hyperconvexity) (See for instance [16, 19, 21, 23, 25]). Naturally this has led us to the believe that Isbell-convexity on a set equipped with a odular pseudometric should be investigated. The aim of this article is to introduce and study the theory of Isbell-convexity in the setting of modular pseudometrics and we call it w-Isbell-convexity. For instance, we study connections between hyperconvexity in a metric space and w-Isbell-convexity on a modular set. In addition, we discuss the boundedness (w-boundedness) of a set endowed with a modular pseudometric. We eventually show that a nonexpansive self-map (w-nonexpansive map) on a w-Isbell-convex modular set has a fixed point. In addition, its fixed points set is w-Isbell convex whenever its modular pseudometric is continuous from the right on the set of positive numbers. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 92 On w-Isbell-convexity 2. Preliminaries For the comfort of the reader and in preparation of the terminology that we are going through this article, we recall the following concepts that can be found in [5, 6, 7, 8]. Definition 2.1. Consider a nonempty set X. A function w : (0,∞)×X×X → [0,∞] is said to be a modular pseudometric on X if it satisfies the following conditions: (a) w(λ,x,x) = 0 for all x ∈ X and λ ∈ (0,∞), (b) w(λ,x,y) = w(λ,y,x) for all x,y ∈ X and λ ∈ (0,∞), (c) w(λ + µ,x,y) ≤ w(λ,x,z) + w(µ,z,y) for all x,y,z ∈ X and λ,µ ∈ (0,∞). We shall say that w is a modular metric provided that w satisfies also the following condition: for all x,y ∈ X and λ ∈ (0,∞), (d) w(λ,x,y) = 0 for all λ > 0 imply x = y. Let w be a modular pseudometric on a nonempty set X. For any x ∈ X, we denote by Xw = Xw(x) := {y ∈ X : lim λ→∞ w(λ,x,y) = 0}. Let us fix an element x0 ∈ X. Then set X∗w(x0) defined by X∗w = X ∗ w(x0) = {x ∈ X : w(λ,x,x0) < ∞ for some λ > 0} is called a modular set (around x0) and x0 is called the center of X ∗ w. It has been observed in [8] that it is easy to see that Xw(x0) ⊆ X∗w(x0). For further details on modular sets and examples of these sets, we refer the reader to [8]. The function qw defined by qw(x,y) = inf{λ > 0 : w(λ,x,y) ≤ λ} whenever x,y ∈ Xw is a (pseudo) metric on Xw whenever w is modular (pseudo) metric on X. If x,y ∈ X, then qw is an extended (pseudo) metric on X. For any x ∈ Xw and λ and µ > 0, we define the sets Bwλ,µ(x) and C w λ,µ(x) by Bwλ,µ(x) := {z ∈ Xw : w(λ,x,z) < µ} and Cwλ,µ(x) := {z ∈ Xw : w(λ,x,z) ≤ µ}. Then the set Bwλ,µ(x) is called a w<-entourage about x relative to λ and µ and the set Cwλ,µ(x) is called a w≤-entourage about x relative to λ and µ. We need the following two examples in the sequel. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 93 O. Olela Otafudu and K. Sebogodi Example 2.2 ([8, Example 2.4 (a)]). Let R be equipped with the modular metric w defined by w(λ,x,y) = { ∞ if x 6= y 0 if x = y whenever x,y ∈ R and λ > 0. It is readily checked that Xw = Xw(x0) = {x0}, where x0 ∈ R and qw(x,y) = 0 for all x,y ∈ Xw. Moreover, for any r > 0, we have Bqw (x0,r) = {y ∈ Xw : 0 = qw(x0,y) < r} = {x0} and Bwr,r(x0) = {y ∈ Xw : w(r,x0,y) < r} = {x0} = C w r,r(x0). Example 2.3 ([8, Example 2.7 (b)]). Consider a pseudometric space (X,q). If we equip X with the modular pseudometric w(λ,x,y) = q(x,y) λp whenever x,y ∈ X and λ > 0, where p is a strictly positive constant. Then it follows that Xw = X and qw(x,y) = [q(x,y)] 1 p+1 . Furthermore, for any λ > 0, we have Bqw (x,λ) = { y ∈ Xw : [q(x,y)] 1 p+1 < λ } = { y ∈ Xw : q(x,y) λp < λ } = {y ∈ Xw : w(λ,x,y) < λ} = Bwλ,λ(x). If w is a modular pseudometric on a nonempty set X, then the topology induced by w (denoted by τ(w)) is defined by the following: A subset A of Xw is said to be τ(w)-open (or w-open) if for any x ∈ A and λ > 0, there exists µ := µ(x,λ) > 0 such that Bwλ,µ(x) ⊂ A. Note that B w λ,µ(x) is not τ(w)-open, in general. Lemma 2.4 ([5]). Let w be a modular pseudometric on X and x ∈ Xw. Then, whenever λ > 0 we have (a) Bqw (x,λ) ⊆ Bwλ,λ(x), (b) Cqw (x,λ) ⊆ Cwλ,λ(x), where Bqw (x,λ) and Cqw (x,λ) are respectively open and closed balls with centre x and radius λ with respect to the pseudometric qw. The following important concept of continuity of a modular pseudometric space was introduced in [6]. Definition 2.5. Let w be a modular pseudometric on a set X. Given x,y ∈ X, (a) the limit from the right w+0(λ,x,y) of w at a point λ > 0 is defined by w+0(λ,x,y) := lim µ→λ+ w(µ,x,y) = sup{w(µ,x,y) : µ > λ}. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 94 On w-Isbell-convexity (b) The limit from the left w−0(λ,x,y) of w at a point λ > 0 is defined by w−0(λ,x,y) := lim µ→λ− w(µ,x,y) = inf{w(µ,x,y) : 0 < µ < λ}. Moreover, (c) we say that w is continuous from the right on (0,∞) if for any λ > 0 we have w(λ,x,y) = w+0(λ,x,y). (d) We say that w is continuous from the left on (0,∞) if for any λ > 0 we have w(λ,x,y) = w−0(λ,x,y). (e) We say that w is continuous on (0,∞) if w is continuous from the right and continuous from the left on (0,∞). Lemma 2.6 ([6]). Let w be a modular pseudometric on a set X. If x,y ∈ Xw and 0 < inf{λ > 0 : w(λ,x,y) ≤ λ} < ∞, then qw(x,y) = inf{λ > 0 : w(λ,x,y) ≤ λ} > λ if and only if w+0(λ,x,y) > λ. The following remark is a consequence of Definition 2.5. Remark 2.7. If w is continuous from the right on (0,∞), then for any x,y ∈ Xw and λ > 0, we have that qw(x,y) ≤ λ if and only if w(λ,x,y) ≤ λ. Let us recall the well-known concept of hyperconvexity. Definition 2.8. A pseudometric space (X,q) is called hyperconvex provided that for any (xi)i∈I and family (ri)i∈I of nonnegative real numbers satisfying q(xi,xj) ≤ ri + rj whenever i,j ∈ I, the following condition holds:⋂ i∈I [ Cq(xi,ri) ] 6= ∅. Let (X,q) be a pseudometric space. Then X is said to be metrically convex if for any points x,y ∈ X and positive numbers r and s such that q(x,y) ≤ r + s, there exists z ∈ X such that q(x,z) ≤ r and q(z,y) ≤ s. Furthermore, a family of balls (Cq(xi,ri))i∈I is said to have the mixed binary intersection property if for all indices i ∈ I, Cq(xi,ri) 6= ∅. Definition 2.9. A pseudometric space (X,q) is called hypercomplete if every family (Cq(xi,ri))i∈I of balls having the mixed binary intersection property satisfies ⋂ i∈I [ Cq(xi,ri) ] 6= ∅. The following is a well-known characterisation of hyperconvexity in terms of metric convexity and hypercompleteness. Proposition 2.10 ([26]). A pseudometric space (X,q) is hyperconvex if and only if it is metrically convex and hypercomplete. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 95 O. Olela Otafudu and K. Sebogodi 3. Isbell-convexity In this section, we introduce the concept of hyperconvexity in modular metric spaces. Definition 3.1. Let w be a modular pseudometric on a nonempty set X. We say that Xw is w-Isbell-convex if for any family of points (xi)i∈I in Xw and family of points (λi)i∈I in (0,∞) such that w(λi + λj,xi,xj) ≤ λi + λj, whenever i,j ∈ I, then ⋂ i∈I [ Cwλi,λi(xi) ] 6= ∅. Example 3.2. Let (X,d) be a metric space. For any x,y ∈ X and λ > 0, it is well-known from [8, Example 2.4] that the function w defined by w(λ,x,y) = d(x,y) ϕ(λ) , where ϕ : (0,∞) −→ (0,∞) is a bounded nondecreasing function is a modular metric on X. Furthermore, whenever x0 ∈ X the set Xw(x0) = {x0} is w-Isbell-convex. Indeed, for any λ,µ > 0 such that 0 = w(λ + µ,x0,x0) < λ + µ, then we have x0 ∈ Bλ,λ(x0) ∩Bµ,µ(x0) ⊆ Cλ,λ(x0) ∩Cµ,µ(x0). Definition 3.3. Let w be a modular pseudometric on a nonempty set X. We say that Xw is w-metrically convex if for any points x,y ∈ Xw so that w(λ + µ,x,y) ≤ λ + µ whenever λ,µ > 0, there exists z ∈ Xw such that w(λ,x,z) ≤ λ and w(µ,z,y) ≤ µ. Example 3.4. Let X = R be equipped with the modular metric w(λ,x,y) = d(x,y) λ for any x,y ∈ X, where d is the discrete metric on R. Then it is obvious that Xw = R. Moreover Xw is not w-metrically convex since w ( 1, 1 2 , 1 ) = d( 1 2 , 1) 1 = 1 ≤ 1 2 + 1 2 but there is no z ∈ R such that w ( 1 2 , 1 2 ,z ) = d( 1 2 ,z) 1 2 ≤ 1 2 and w ( 1 2 ,z, 1 ) = d(z, 1) 1 2 ≤ 1 2 . If such z exits it would satisfy z = 1 and z = 1 2 . Remark 3.5. Let w be a modular metric on a set X. It is easy to see that if (Xw,qw) is metrically convex, then Xw is w-metrically convex too. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 96 On w-Isbell-convexity Proof. Let x,y ∈ Xw and λ,µ > 0 such that w(λ + µ,x,y) ≤ λ + µ. It follows that qw(x,y) ≤ λ + µ. By metric convexity of (Xw,qw), there exists z ∈ Xw such that z ∈ Cqw (x,λ) ∩Cqw (y,µ) ⊆ Cwλ,λ(x) ∩C w µ,µ(y). Hence there exists z ∈ Xw such that w(λ,x,z) ≤ λ and w(µ,z,y) ≤ µ. Therefore, Xw is w-metrically convex. � Definition 3.6. Let w be a modular pseudometric on a nonempty set X. A family (Cwλi,λi(xi))i∈I of w≤-entourages with λi > 0 and xi ∈ Xw for all i ∈ I is said to have the mixed binary intersection property provided that Cwλi,λi(xi) 6= ∅ whenever i ∈ I. Definition 3.7. Let w be a modular pseudometric on a nonempty set X. The modular set Xw is called w-Isbell-complete if every family (C w λi,λi (xi))i∈I of w≤-entourages, where λi > 0 and xi ∈ Xw for all i ∈ I, having the mixed binary intersection property satisfies⋂ i∈I [ Cwλi,λi(xi) ] 6= ∅. Proposition 3.8. If w is a modular pseudometric on X, then Xw is w-Isbell- convex if and only if Xw is w-metrically-convex and w-Isbell-complete. Proof. (=⇒) Suppose that Xw be w-Isbell-convex. Let x1,x2 ∈ Xw and λ1,µ2 > 0 such that w(λ1 + µ2,x1,x2) ≤ λ1 + µ2. Then, we set λ2 = µ1 = w(λ2 + µ1,x2,x1). By w-Isbell-convexity of Xw, there exists a ∈ Cwλ1,λ1 (x1) ∩C w µ2,µ2 (x2). It follows that w(λ1,x1,a) ≤ λ1 and w(µ2,a,x2) ≤ µ2. Hence Xw is w- metrically convex. Consider the family (Cwλi,λi(xi))i∈I of w≤-entourages having mixed binary intersection property. Then there exists z ∈ Cwλi,λi(xi) ∩C w λj,λj (xj) whenever i,j ∈ I. Furthermore, whenever i,j ∈ I we have w(λi + λj,xi,xj) ≤ w(λi,xi,z) + w(λj,z,xj) ≤ λi + λj. Thus ⋂ i∈I [ Cwλi,λi(xi) ] 6= ∅ by w-Isbell-convexity of Xw. Therefore, Xw is w-Isbell-complete. (⇐=) Let Xw be w-metrically convex and w-Isbell-complete. Let (xi)i∈I be a family of points in Xw and (λi)i∈I be a family of points in (0,∞) such that w(λi + λj,xi,xj) ≤ λi + λj whenever i,j ∈ I. By w-metrical convexity of Xw there exists z ∈ Xw such that w(λi,xi,z) ≤ λi and w(λj,z,xj) ≤ λj whenever i,j ∈ I. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 97 O. Olela Otafudu and K. Sebogodi Then the family (Cwλi,λi(xi))i∈I has the mixed binary intersection property. Since Xw is w-Isbell-complete, we have⋂ i∈I [ Cwλi,λi(xi) ] 6= ∅. � Proposition 3.9. Let w be a modular pseudometric on a nonempty set X. If (Xw,qw) be a hyperconvex pseudometric space, then Xw is w-Isbell-convex. Proof. Suppose that (Xw,qw) be a hyperconvex pseudometric space. Let (xi)i∈I be a family of points in Xw and (λi)i∈I be a family of points in (0,∞) such that w(λi + λj,xi,xj) ≤ λi + λj whenever i,j ∈ I. Then qw(xi,xj) = inf{λ > 0 : w(λ,xi,xj) ≤ λ}≤ λi + λj whenever i,j ∈ I. By hyperconvexity of (Xw,qw), we have⋂ i∈I [ Cqw (xi,λi) ] 6= ∅. Since by Lemma 2.4, Cqw (xi,λi) ⊆ Cwλi,λi(xi), it follows that ∅ 6= ⋂ i∈I [ Cqw (xi,λi) ] ⊆ ⋂ i∈I [ Cwλi,λi(xi) ] . Therefore, Xw is w-Isbell-convex. � Remark 3.10. Let w be a modular pseudometric on a set X. If w be continuous from the right on (0,∞), then qw(x,y) ≤ λ if and only if w(λ,x,y) ≤ λ for any x,y ∈ Xw and positive number λ. It is natural to wonder about the converse of Proposition 3.9. Lemma 3.11. Let w be a modular pseudometric on X. If w be continuous from the right on (0,∞), then Xw is w-metrically convex if and only if (Xw,qw) is a metrically convex pseudometric space. Proof. (=⇒) Suppose that Xw be w-metrically convex. Let x,y ∈ Xw and λ,µ > 0 such that qw(x,y) ≤ λ + µ. Since w is continuous from the right on (0,∞) we have w(λ+µ,x,y) ≤ λ+µ. It follows that there exists a ∈ Xw such that w(λ,x,a) ≤ λ and w(µ,a,z) ≤ µ. Then qw(x,a) ≤ λ and qw(a,y) ≤ µ by the right continuity of w. So (Xw,qw) is a metrically convex. (⇐=) Follows from Remark 3.5. � We leave the proof of the following lemma to the reader. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 98 On w-Isbell-convexity Lemma 3.12. Let w be a modular pseudometric on X. If w be continuous from the right on (0,∞), then Xw is w-Isbell-complete if and only if (Xw,qw) is an Isbell-complete pseudometric space. Theorem 3.13. Let w be a modular pseudometric on X. If w be continuous from the right on (0,∞), then Xw is w-Isbell-convex if and only if (Xw,qw) is a hyperconvex pseudometric space. Proof. Suppose that Xw be w-Isbell-convex. Then, by Proposition 3.8 Xw is w-Isbell-complete and w-metrically convex. Thus (Xw,qw) is an hypercomplete and metrically convex pseudometric space by Lemma 3.11 and Lemma 3.12. Therefore, (Xw,qw) is a hyperconvex pseudometric space. The converse follows by a similar argument. � 4. Nonexpansive maps We are aware of [2], where the concept of boundedness of a subset of modular set was introduced in order to study the existence of fixed point of modular contractive maps in modular metric spaces. This is a motivation for our next definitions. Let w be a modular pseudometric on X. We say that a nonempty subset A of Xw is w-bounded if there exists x ∈ Xw such that A ⊆ Cwλ,λ(x) for some λ > 0. Remark 4.1. If w be a modular pseudometric on a set X, then boundeness on (Xw,qw) implies w-boundeness. This observation follows from the fact that Cqw (x,λ) ⊆ Cwλ,λ(x) whenever λ > 0 and x ∈ Xw. Definition 4.2. Let A be a w-bounded subset of Xw. Then we denote by diamw(A) the w-diameter of A and it is defined by diamw(A) := sup{w(λ,x,y) : x,y ∈ A} for some λ > 0. Lemma 4.3. Let w be a modular pseudometric on X. If A is a w-bounded subset of Xw, then diamw(A) < ∞. Proof. Suppose that A is w-bounded. Then for some x ∈ Xw, we have A ⊆ Cwλ,λ(x) for some λ > 0. If z,y ∈ A then w(λ,x,z) ≤ λ and w(λ,x,y) ≤ λ. It follows that w(λ + λ,z,y) ≤ w(λ,z,x) + w(λ,x,y) ≤ λ + λ. Hence for λ′ = λ + λ > 0, we have sup{w(λ′,z,y) : z,y ∈ A}≤ λ′. Therefore, diamw(A) < ∞. � © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 99 O. Olela Otafudu and K. Sebogodi Lemma 4.4. Let w be a modular pseudometric on X. If w be continuous from the right on (0,∞), then boundeness on (Xw,qw) is equivalent to w-boundeness. Proof. We only prove the sufficient condition since the necessary condition follows from Remark 4.1. Suppose that A is a w-bounded subset of Xw. Then there exists x ∈ Xw such that A ⊆ Cwλ,λ(x) for some λ > 0. Let y ∈ A. Then w(λ,x,y) ≤ λ. By the right continuity of w on (0,∞), we have qw(x,y) ≤ λ for some x ∈ Xw and λ > 0. Thus A ⊆ Cqw (x,λ). Therefore, A is bounded in (Xw,qw). � Proposition 4.5. Let w be a modular pseudometric on X. Let Xw be w- Isbell-convex and (xi)i∈I is a family of points in Xw and (λi)i∈I is a family of positive real numbers such that w(λi + λj,xi,xj) ≤ λi + λj whenever i,j ∈ I. Then the set A := ⋂ i∈I[C w λi,λi (xi)] is nonempty and w-Isbell-convex. Proof. Observe that the set A is nonempty by w-Isbell-convexity of Xw. Now we show that the set A is w-Isbell-convex. Let (xα)α∈Γ be a family of points in A and (λα)α∈Γ be a family of positive real numbers such that w(λα + λβ,xα,xβ) ≤ λα + λβ whenever α,β ∈ Γ. We have to show that the family of w≤-entourages ((Cwλα,λα(xα))α∈Γ; (C w λi,λi (xi))i∈I) satisfies the hypothesis of w-Isbell-convexity. Then, in particular, for all α ∈ Γ and i ∈ I, we have w(λi,xi,xα) ≤ λi < λi + λα since xα ∈ A. By w-Isbell-convexity of Xw, it follows that ∅ 6= ⋂ α∈Γ [ Cwλα,λα(xα) ] ∩ ⋂ i∈I [ Cwλi,λi(xi) ] = A∩ ⋂ α∈Γ [ Cwλα,λα(xα) ] . Hence A is w-Isbell-convex. � For a w-bounded subset A of Xα, we set (4.1) covw(A) := ⋂{ Cwλ,λ(x) : A ⊆ C w λ,λ(x),x ∈ Xw,λ > 0 } . Furthermore, we set: rwx,λ(A) := sup y∈A {w(λ,x,y) : for some λ > 0}, where,x ∈ Xw. Proposition 4.6 (compare [13, Lemma 3.3]). Let w be a modular pseudomet- ric on X and A be a w-bounded subset of Xw. Then we have: © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 100 On w-Isbell-convexity (a) covw(A) = ⋂ x∈X [ Cwrw x,λ (A),rw x,λ (A)(x) ] . (b) rx(covw(A)) = rx(A). Definition 4.7 (compare [13, Definition 3.4]). Let w be a modular pseudomet- ric on X. A nonempty and w-bounded subset A of Xw is called w-admissible if A = covw(A). In the sequel, we will denote by Aw(Xw), the set of all w-admissible subsets of Xw. Remark 4.8. Observe that a w-admissible subset of Xw can be written as the intersection of a family of the form Cwλ,λ(x), where x ∈ Xw and λ > 0. Lemma 4.9. Let w be a modular pseudometric on X which is continuous from the right on (0,∞). Then Cqw (x,λ) = C w λ,λ(x) whenever λ > 0 and x ∈ Xw. Proof. Since we know that Cqw (x,λ) ⊆ Cwλ,λ(x), then we only prove the reverse inclusion. If a ∈ Cwλ,λ(x), then w(λ,x,a) ≤ λ which is equivalent to qw(x,a) ≤ λ by the right continuity of w. Thus a ∈ Cqw (x,λ). � Corollary 4.10. Let w be a modular pseudometric on X which is continuous from the right on (0,∞) and A ⊆ Xw. Then A is w-admissible if and only if A is qw-admissible. Definition 4.11. Let w be a modular pseudometric on X. Given a subset A of Xw, we define for λ > 0 the λ-parallel set of A as Pλ(A) = ⋃ a∈A [ Cwλ,λ(a) ] . Proposition 4.12 (compare [17, Lemma 4.2]). Let w be a modular pseudo- metric on X. If Xw is w-Isbell-convex and A is a w-admissible subset of Xw, that is A = ⋂ i∈I C w λi,λi (xi) with xi ∈ Xw and λi > 0 for each i ∈ I 6= ∅, then Pλ(A) = ⋂ i∈I [ Cwλi+λ,λi+λ(xi) ] whenever λ > 0. Proof. Let y ∈ Pλ(A). Then, for some a ∈ A we have w(λ,a,y) ≤ λ. Further- more, for each i ∈ I, w(λi + λ,xi,y) ≤ w(λi,xi,a) + w(λ,a,y) ≤ λi + λ. It follows that y ∈ Cwλi+λ,λi+λ(xi) whenever ∈ I. Hence Pλ(A) ⊆ ⋂ i∈I [ Cwλi+λ,λi+λ(xi) ] . © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 101 O. Olela Otafudu and K. Sebogodi We now suppose that y ∈ ⋂ i∈I [ Cwλi+λ,λi+λ(xi) ] . Hence, for any i ∈ I, w(λi + λ,xi,y) ≤ λi + λ. Thus the family of w≤-entourages [(C w λi,λi (xi))i∈I,C w λ,λ(y)] satisfies the hy- pothesis of w-Isbell-convexity of Xw. Then ∅ 6= [⋂ i∈I Cwλi,λi(xi) ]⋂ Cwλ,λ(y) = A ⋂ Cwλ,λ(y). It follows that w(λ,y,a) ≤ λ for some a ∈ A. Therefore, y ∈ Pλ(A). � Definition 4.13. Let w be a modular pseudometric on a set X. Then we say that a map T : Xw → Xw is w-Lipschitz if there exists a k > 0 such that w(kλ,T(x),T(y)) ≤ w(λ,x,y) for all λ > 0 and x,y ∈ Xw. If k = 1, then the map T is called a w-nonexpansive map. Remark 4.14. Let w be a modular pseudometric on X. In light of [5, Theo- rem 5.1] and [5, Theorem 5.2], one can easily prove that for any w-Lipschitz map T : Xw → Xw, we have that w(kλ,T(x),T(y)) ≤ w(λ,x,y) implies qw(T(x),T(y)) ≤ kqw(x,y) for some k > 0 and whenever λ > 0 and x,y ∈ Xw. Theorem 4.15 (compare [17, Lemma 4.3]). Let w be a modular pseudometric on X which is continuous from the right on (0,∞) and Xw be w-Isbell-convex. If A is a w-admissible subset of Xw, then there is a w-nonexpansive retraction R of Pλ(A) onto A such that w(λ,x,R(x)) ≤ λ whenever x ∈ Pλ(A) and λ > 0. Proof. Suppose that A is w-admissible. Then A = ⋂ i∈I Cwλi,λi(xi) with I 6= ∅. Since Pλ(A) is an intersection of w≤-entourages from Proposition 4.12, then Pλ(A) is w-admissible in Xw. Moreover, from Proposition 4.5 we have Pλ(A) is w-Isbell-convex. Let us consider the family Ω defined by Ω = {(D,RD) : A ⊆ D ⊆ Pλ(A) and RD : D → A is a w-nonexpansive retraction such that w(λ,RD(x),x) ≤ λ whenever x ∈ D}. If the map IA be the identity on A, then (A,IA) ∈ Ω. Hence Ω 6= ∅. Fur- thermore, one can ordered the family Ω by the partial order (D,RD) ≤ (E,RE) if and only if D ⊆ E and the w-nonexpansive map RD is an extension of the w-nonexpansive map RE. It follows that every chain in (Ω,≤) is bounded above. Thus by Zorn’s lemma, Ω has a maximal element. Suppose that (D,RD) is the maximal ele- ment of (Ω,≤). © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 102 On w-Isbell-convexity Let us prove that D = Pλ(A). Suppose that there exists an element x ∈ Pλ(A)\D such that w(λd,d,x) = λd whenever d ∈ D and λd > 0. We consider the set C := [ ⋂ d∈D Cww(λd,d,x),w(λd,d,x)(RD(d)) ]⋂[⋂ i∈I Cwλi,λi(xi) ]⋂ [Cwλ,λ(x)]. It is easy to see that C 6= ∅ since the family [(Cwλi,λi(xi))i∈I, (C w λ,λ(x)), (C w w(λd,d,x),w(λd,d,x) (RD(d)))d∈D] of ≤-entourages has the mixed binary intersection property in light of Propo- sition 3.8. Since ∅ 6= C ⊆ A. We now suppose that z ∈ C and we define the map R′ : D ∪{x} → A by R′(d) = RD(d) if d ∈ D and R′(x) = z. Then, we have for any d ∈ D w(λd,R ′(d),R′(x)) = w(λd,RD(d),z) ≤ w(λd,d,x). So R′ is w-nonexpansive. Moreover, w(λ,R′(x),x) = w(λ,z,x) ≤ λ since z ∈C. Hence (D∪{x},R′) ∈ Ω which contradicts the maximality of D. Therefore, D = Pλ(A). � Theorem 4.16. Let w be a modular pseudometric on X. If Xw be w-bounded w-Isbell-convex and w be continuous from the right on (0,∞) and T : Xw → Xw be a w-nonexpansive map, then the fixed point set Fix(T) is nonempty and w- Isbell-convex. Proof. Since T : Xw → Xw is a w-nonexpansive map, then T : (Xw,qw) → (Xw,qw) is a nonexpansive map. Moreover, Xw is qw-bounded by Lemma 4.4. Hence for any λ > 0, we have w(T(x),T(y)) ≤ w(λ,x,y) whenever x,y ∈ Xw. Then from Remark 4.14 when k = 1, we have qw(T(x),T(y)) ≤ qw(x,y) whenever x,y ∈ Xw. Observe that (Xw,qw) is a hyperconvex pseudometric space by Theorem 3.13. Furthermore, we have Fix(T) is nonempty and Isbell-convex by [13, Theorem 6.1]. Therefore, Fix(T) is w-Isbell-convex by Theorem 3.13. � Definition 4.17 (compare [19, Theorem 5.5]). Let w be a modular pseudo- metric on X and T : Xw → Xw be a map. For λ1,λ2 > 0, we define the set Fλ1,λ2 (T) by Fλ1,λ2 (T) = {x ∈ Xw : w(λ2,x,T(x)) ≤ λ2 and w(λ1,T(x),x) ≤ λ1}. Corollary 4.18 (compare [17, Theorem 4.11]). Let w be a modular pseudo- metric on X and Xw be w-Isbell-convex. If w be continuous from the right on (0,∞) and T : Xw → Xw be a w-nonexpansive map, then the set Fλ1,λ2 (T) is w-Isbell-convex whenever Fλ1,λ2 (T) is nonempty. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 103 O. Olela Otafudu and K. 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