Penot’s Compactness Property in Ultrametric Spaces with an Application

In this work, we investigate the compactness property in the sense of Penot in ultrametric spaces. Then, we show that spherical completeness is exactly the Penot’s compactness property introduced for convexity structures. The spherical completeness property misled some mathematicians to it to hyperconvexity in metric spaces. As an application, we discuss some fixed point results in spherically complete ultrametric spaces.


Introduction
Ultrametric spaces are a special type of metric spaces which enjoy a stronger triangular inequality. This stronger triangular inequality implies some amazing properties enjoyed by ultrametric spaces, like any triangle is isosceles. This is why ultrametric spaces are known as isosceles metric spaces. Because of these properties, one may tend to belittle ultrametric spaces and considers them as futile. It happens that these metric spaces are crucial in applications. For example, they appear in a strong manner in logic programming and artificial intelligence [1]. Other areas which involve heavily ultrametric spaces are bioinformatics, distributed networks, microbiology, and learning models [2,3].
In recent years, many authors followed the work of [4] where a pseudoconnection between ultrametric spaces and hyperconvexity via the 2-Helly intersection property of balls was initiated. In this work, we show that such connection is baseless and not correct. In fact, the correct connection should be made with the work of Penot on convexity structures. Indeed, Penot [5] while investigating an abstract formulation of Kirk's fixed point theorem [6] away from Banach spaces considered an abstract form of convex sets. This abstract formulation captures the weak-compactness beautifully. It is worth mentioning that convexity structures are important in applications [1,3,7,8].
As an application, we discuss the fixed point property in light of some recent papers published on this subject. In particular, we show that many of the known results assumes strong or artificial behavior.
For the interested reader into metric fixed point theory, we recommend the book of Khamsi and Kirk [9].
(1) We have Bðy, rÞ = Bðx, rÞ whenever y ∈ Bðx, rÞ, which implies for any x, y ∈ M and r ≥ 0 (2) If Bðx, r 1 Þ and Bðy, r 2 Þ are such that r 1 ≤ r 2 , then either Bðx, r 1 Þ ∩ Bðy, r 2 Þ = ∅ or Bðx, r 1 Þ ⊆ Bðy, r 2 Þ, for any x 1 , x 2 ∈ M and r 1 , r 2 ≥ 0 In other words, for closed balls in ultrametric spaces, the radius is central not the center. Moreover, the inclusion order between balls follows the order on the radiuses. This is amazing if we look at it from the point of view of classical metric and Banach spaces.
The following property is fundamental and gives an insight into the behavior of closed balls in ultrametric spaces.
Let ðM, dÞ be a metric space. An admissible subset is an intersection of closed balls [9]. The collection of admissible subsets of M will be denoted by AðMÞ. Obviously any closed ball is an admissible subset. Moreover, AðMÞ is stable by intersection; i.e., the intersection of admissible subsets is an admissible subset. In ultrametric spaces, nonempty admissible subsets are exactly closed balls according to Lemma 2. This is surprising and may lead mathematicians to a wrong interpretation. This is the case when the authors of [4] connected the ultrametric property to hyperconvexity in metric spaces. Penot [5] is credited as the one who introduced the following definition: Definition 3. Let ðM, dÞ be a metric space. We say that AðMÞ is compact if and only if for any family of closed balls fBða α , r α Þg α∈Γ , Penot introduced this property when he gave the first abstract extension of the classical Kirk's fixed point theorem [6] to metric spaces. His approach allowed him to avoid the linear convexity problem from the underlying Banach space structure. Others stumbled with this issue and specially the weak-compactness argument used heavily by Kirk. For example, Takahashi [13] used the Menger metric convexity and assumes the metric compactness which is very strong and does not capture the weak-compactness in the linear case. In fact, Penot did not work specifically with the family of admissible subsets but used the well-known concept of convexity structures. This concept is very powerful and has applications in areas as vast apart as economics and game theory [1,3,7,8]. Recall that a convexity structure is a family of subsets stable by intersection [5,8]. A property shared by convex subsets in the linear case. It is clear that AðMÞ is the smallest convexity structure which contains the closed balls.
Before we state the main result of this work, we need the following definition: Definition 4 [12,14]. A metric space ðM, dÞ is said to be spherically complete if and only if for any sequence fBðx n , r n Þg n∈ℕ of closed balls such that r n+1 ≤ r n and Bðx n+1 , r n+1 Þ ⊂ Bðx n , r n Þ, for any n ∈ ℕ, we have T n∈ℕ Bðx n , r n Þ ≠ ∅.
Theorem 5. Let ðM, dÞ be an ultrametric space. The following are equivalent: (1) AðMÞ is compact (in the sense of Penot) (2) ðM, dÞ is spherically complete Proof. Obviously we only need to prove that if ðM, dÞ is spherically complete, then AðMÞ is compact. Let fBða α , r α Þg α∈Γ be a family of closed balls such that T α∈Γ f Bða α , r α Þ ≠ ∅, for any finite subset Γ f of Γ. Let us prove that T α∈Γ Bða α , r α Þ is not empty. Set R = inf α∈Γ r α . There exists fα n g n∈ℕ such that lim n⟶∞ r α n = R. Set R n = inf fr α 0 , r α 2 ,⋯,r α n g, for n ∈ ℕ. Then, fR n g is decreasing and converges to R. Moreover, by assumption, we have T i≤n Bða α i , r α i Þ is not empty and is a closed ball with radius R n , i.e.; there exists for any n ∈ ℕ. By construction, we have Bðx n+1 , R n+1 Þ ⊂ Bð x n , R n Þ, for any n ∈ ℕ. Since ðM, dÞ is spherically complete, we conclude that T n∈ℕ Bðx n , R n Þ is not empty. We claim that Journal of Function Spaces In order to see this is true, we only need to prove Let z ∈ T n∈ℕ Bðx n , R n Þ = T n∈ℕ ð T i≤n Bða α i , r α i ÞÞ. Fix β ∈ Γ and n ∈ ℕ. Our assumption on the family fBða α , r α Þg α∈Γ implies Bðx n , R n Þ ∩ Bða β , r β Þ is not empty. Hence, dðx n , a β Þ ≤ max fR n , r β g which implies If we let n ⟶ ∞, we get dða β , zÞ ≤ max fR, r β g = r β , i.e., z ∈ Bða β , r β Þ. Since β was chosen arbitrarily in Γ, we conclude that z ∈ T α∈Γ Bða α , r α Þ, which completes the proof.
Some mathematicians confused the spherical complete property in ultrametric spaces with the hyperconvex property (see for example [4]). Theorem 5 clarifies this connection by showing that in fact spherical completeness in ultrametric spaces coincides with Penot's compactness of convexity structures in metric spaces [5]. Recall that Penot's compactness in Banach spaces was captured in a wonderful way the compactness of the weak-topology and does not connect to the hyperconvexity in any way. For the sake of being complete, let us recall the definition of hyperconvexity in metric spaces. Definition 6. The metric space ðM, dÞ is said to be hyperconvex if for any family fBðx α , r α Þg α∈Γ of closed balls in M such that dðx α , x β Þ ≤ r α + r β , for any α, β ∈ Γ, we have For more details more on hyperconvex metric spaces, we refer the reader to [15,16]. Note that hyperconvex metric spaces are convex in the sense of Menger, i.e., for any x, y ∈ M and t ∈ ½0, 1; there exists z t ∈ M such that d ðx, z t Þ = t dðx, yÞ and dðy, z t Þ = ð1 − tÞ dðx, yÞ. Taking into account this property, it is clear that ultrametric spaces can not be convex in the sense of Menger. Therefore, ultrametric spaces can not be hyperconvex.
Remark 7. Note that if an ultrametric space ðM, dÞ is spherically complete, then, it is complete.

An Application
As we discussed in the previous section, the spherical completeness should be understood within the compactness framework introduced by Penot. His work was initiated to extend Kirk's fixed point theorem to abstract metric spaces away from the linear convexity.
for any x, y ∈ M. The smallest K will be denoted by LipðTÞ. T is said to be nonexpansive if LipðTÞ ≤ 1, i.e., for any x, y ∈ M. A point x ∈ M is said to be a fixed point of T provided TðxÞ = x. The set of fixed points of T is denoted by FixðTÞ.
The following result is known and is amazing and has no equivalence in general metric spaces. We will give its proof here. Theorem 9 [9,11,17]. Let ðM, dÞ be an ultrametric space which is spherically complete. Let T : M ⟶ M be nonexpansive.
The conclusion (2) of Theorem 9 was first proved by Petalas and Vidalis in [18] and was extended by many authors (see for instance [11,19]). Therefore, given a nonexpansive mapping defined in a spherically complete metric space, it has a minimal closed ball with two options: either the radius of the minimal ball is r = 0 or it is r > 0. If r = 0, the minimal ball is reduced to one point which is a fixed point of the map. Otherwise, T fails to have a fixed point in that minimal ball.
Theorem 10 [11,18,19]. Let ðM, dÞ be a spherically complete ultrametric space. Let T : M ⟶ M be nonexpansive. Fix x ∈ M. Then, the closed ball Bðx, dðx, TðxÞÞ contains either a fixed point of T or a minimal invariant closed ball B such that for any y, z ∈ B.
From the conclusion of Theorem 10, it is clear that a nonexpanive mapping defined in a spherically complete ultrametric space has a fixed point provided; we violate the condition for any y and z in a minimal closed ball invariant by the mapping. First, we note that a fixed point may not exist in general.
Example 11 [18]. Let K be a field with a discrete non-Archimedean valuation j:j, and let c 0 ðKÞ the set of all sequences ðx j Þ j∈ℕ in K such that It is known that c 0 ðKÞ endowed with the non-Archimedean norm is a spherically complete non-Archimedean vector space. Let a ∈ K such that 0 < jaj < 1. Define T : c 0 ðKÞ ⟶ c 0 ðKÞ by T is a nonexpansive mapping in c 0 ðKÞ with no fix point.
The above example suggests that more assumptions are needed in order to secure the existence of a fixed point for nonexpansive mappings defined in spherically complete ultrametric space. The first type of assumptions which secure a fixed point is the one introduced by Priess-Crampe and Ribenboim [20].
Definition 12 [20]. Let ðM, dÞ be a metric space. A map T : M ⟶ M is strictly contracting on orbits if and only if TðxÞ ≠ x implies for each x ∈ M.
Most of the mathematicians who came afterward used the same or similar assumption and obtained the following: Theorem 13 [11,[18][19][20]. Let ðM, dÞ be a spherically complete ultrametric space. Let T : M ⟶ M be nonexpansive. Assume T is strictly contracting on orbits. Then, T has a fixed point in any closed ball Bðx, dðx, TðxÞÞ, for any x ∈ M.