Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are now experiencing their well-deserved renaissance. The aim of this paper is to present two improvements of Frink’s metrization theorem along with some fixed point results for single-valued mappings on quasimetric spaces. Moreover, Cantor’s intersection theorem for sequences of sets which are not necessarily closed is established in a quasimetric setting. This enables us to give a new proof of a quasimetric version of the Banach Contraction Principle obtained by Bakhtin. We also provide error estimates for a sequence of iterates of a mapping, which seem to be new even in a metric setting.


Introduction
The theory of metric spaces initiated in 1906 by Fréchet [16] has developed into a huge branch of mathematics. The three axioms of a metric can be considered as three pillars of this theory. However, some of these conditions might be substituted by others or even omitted completely. This practice stems either from a purely theoretic curiosity-driven tendency to generalise already known concepts or is motivated by the necessity of practical applications.
Thus, several distinct concepts have been developed. For instance, if we omit the first axiom of a metric, then we obtain the so-called pseudometric, A myriad of possible subcategories of such a broad class of spaces is obtainable by adding some extra axioms, which can be considered as triangle-like conditions. As we have already mentioned, in the scope of this paper we will be dealing mainly with K-quasimetric spaces, which are a proper subclass of semimetric spaces.

Definition 2.2.
A semimetric space (X, d) is called a quasimetric space or, more specifically, a K-quasimetric space, where K 1 is fixed, if it satisfies the following condition: (Q3) d(x, z) K · max{d(x, y), d(y, z)} for all x, y, z ∈ X. In this case, the function d is called a K-quasimetric.
A 1-quasimetric space is known broadly in the literature as an ultrametric space. Every 1-quasimetric space is therefore a metric space. On the other hand, every metric space is, in fact, a 2-quasimetric space. The reverse, however, does not hold even for 1 < K 2, i.e., there exist K-quasimetric spaces which are not metric, which is shown in the following simple Actually, every semimetric space (X, d) is a quasimetric space, if the set X is finite.
An equivalent definition of the above notion can be given by replacing (Q3) by the following axiom: for a fixed M 1 and all x, y, z ∈ X, (Q3 ) d(x, z) M · (d(x, y) + d(y, z)) .
It is easily seen that conditions (Q3) and (Q3 ) are equivalent. In fact, (Q3 ) implies (Q3) with a constant K = 2M , whereas (Q3) implies (Q3 ) with a constant M = K. In further sections (unless stated otherwise), by a K-quasimetric we understand a semimetric satisfying the condition (Q3).
A stronger concept was given by Fagin et al. [14]: for a fixed c 1, any n ∈ N and all finite sequences x, x 1 , . . . , x n , y ∈ X.
In the same paper, the authors proved the following useful and interesting Topology in semimetric spaces can be defined in various ways. The first and most common approach is to define a topology in the following way:  [2,3,11,25]. What makes this topology particularly useful is the fact that the convergence of a sequence (x n ) ∈ X N to some point x ∈ X with respect to this topology is equivalently described by d(x n , x) → 0. This topology will be considered as a default one throughout this paper, unless explicitly stated otherwise. A topology in which each open ball is an open set (i.e., open balls form a subbasis of the topology in (X, d)) is considered in [2,11].
Many authors have also posed questions concerning the metrizability of semimetric spaces (usually equipped with some additional conditions). In our previous paper [11] we showed that, in particular, semimetric spaces satisfying Wilson's [27] axiom (W 5) are uniformly metrizable. A semimetric space is said to satisfy (W 5) if for any three sequences (x n ), (y n ), (z n ) ∈ X N , Definition 2.6. Let X be a non-empty set and d 1 , d 2 be semimetrics defined on X. We say that d 1 and d 2 are uniformly equivalent if the following two conditions hold: In the next section, the topic of metrizability is revisited, focusing on metric bounds, which can be obtained for a certain class of K-quasimetric spaces.

Refinements of Frink's theorem
In [17] Frink provided an innovative method for constructing a metric equivalent to a 2-quasimetric. We recall his result in an equivalent form indicated by Schroeder [26]. In this theorem the metric ρ is obtained by the so-called chain approach, namely, ρ is defined by where the infimum is taken over all finite sequences of points x 0 , x 1 , x 2 , . . . , x n , where x 0 = x and x n = y, thus guaranteeing that the triangle inequality is satisfied. In what follows, we will denote by ρ inf the function defined by (3.1). Theorem 3.1 was extended by Schroeder [26], who obtained the following Theorem 3.2 (Schroeder). Let (X, d) be a K-quasimetric space with K 2. Then there exists a metric ρ on X such that A natural question arises, whether one can obtain a better estimation of d than that in Theorem 3.2. We will now present two possible refinements of this theorem. The first one leaves the restriction K 2 and gives an optimal bounding constant. The other relaxes the restriction and provides an optimal bound but for the pth power of the quasimetric d and not d itself.

The first refinement
In this section we present the first of the two mentioned generalisations of Theorems 3.1 and 3.2. Note that the idea to use p := log K 2 in the proof of the following theorem originates from the paper of Paluszyński and Stempak [25]. Theorem 3.3. If (X, d) is a K-quasimetric space with K 2, then there exists a metric ρ on X for which the following inequalities hold: Proof. The case where K = 1 yields an ultrametric space which is, in fact, a metric space. Fix K ∈ (1, 2] and set p := log K 2. Then p 1 and d p is a 2-quasimetric. Applying Theorem 3.1 to d p , we obtain the existence of a metric ρ such that To proceed further, we recall the following well-known Since the function [0, +∞) x → x 1 p satisfies the assumptions of Lemma 3.4, we have that ρ := (ρ ) 1 p is also a metric. Moreover, since K 2p = 4, ρ satisfies the condition which completes the proof.
A natural question arises whether the constant K 2 is optimal in Theorem 3.3. Our next result gives a positive answer to this question.
Proof. Suppose to the contrary that there exists K ∈ [1,2] Then define a K-quasimetric d on X as follows: Thus by hypothesis, we can find a metric ρ satisfying ρ d ϕ(K)ρ = αρ. Hence which gives a contradiction. A natural question arises whether the constant K = 2 plays a special role in Theorem 3.1. More precisely, does Frink's chain approach work if we apply it to a K-quasimetric with K > 2? This question was answered negatively by Schroeder [26], who gave, for any K > 2, an interesting but complicated example of a K-quasimetric space (X, d) for which the function ρ inf defined by (3.1) is not a metric. Hence, taking into account that the metric ρ in Theorem 3.1 was constructed in fact as ρ inf , we obtain the following Corollary 3.6. Let K be a real number. Then K 2 if and only if for any On the other hand, Dung and Hang [13], answering a question of Kirk and Shahzad [22], constructed a Caristi [9] mapping T on a 16-quasimetric space such that T has no fixed point. For the reader's convenience we recall the definition of such mappings.
Following [8] we say that T is asymptotically regular if for any Now we extend Corollary 3.6 by establishing a list of seven equivalent conditions including two concerning the fixed point property for Caristi mappings. (ii) for every K-quasimetric space (X, d) there exists a metric ρ such that for some c 1, (iii) for every K-quasimetric space (X, d), the c-relaxed polygonal inequality is satisfied for some c 1; (iv) for every complete K-quasimetric space (X, d), any Caristi mapping has a fixed point; (v) for every complete K-quasimetric space (X, d), any continuous and asymptotically regular Caristi mapping on X has a fixed point; Proof. Notice that if K < 1, then all the conditions (i)-(vii) are true, since X is then a singleton. So in the remaining part of the proof we assume that K 1. (ii) =⇒ (iv): Let (X, d) be a complete K-quasimetric space and let T : X → X be a Caristi mapping with an associated lower semicontinuous, bounded from below function φ. Hence for any x ∈ X, Moreover, since ρ and d are uniformly equivalent, (X, ρ) is also complete and φ is lower semicontinuous with respect to ρ, so by Caristi's theorem [9] T has a fixed point. ( Suppose to the contrary that K > 2. Set p := log 2 K. Then p > 1 and K = 2 p . Clearly, if η(t) := t p for t 0, then η(0) = η (0) = 0, so by [21,Theorem 7], there exist a complete metric space (X, ρ) and a continuous and asymptotically regular mapping T : X → X such that By [26, Remark 1.1], d is a K-quasimetric. Moreover, it is easily seen that d and ρ are uniformly equivalent, so (X, d) is complete and T is a continuous and asymptotically regular Caristi mapping with respect to d, which contradicts (v) since T has no fixed point. Again, by [26, Remark 1.1], d is a K-quasimetric. Take a pair of arbitrary distinct points x 0 , y 0 ∈ R. We will show that ρ inf (x 0 , y 0 ) = 0. Without loss of generality we may assume that x 0 < y 0 . Consider the following sequence of paths leading from x 0 to y 0 : Two refinements of Frink's metrization theorem 285 (described by the semi-metric d). From the definition of ρ inf we obtain that Since j can be arbitrarily large, ρ inf (x 0 , y 0 ) has to be equal to 0. Due to the fact that both x 0 and y 0 were chosen arbitrarily, ρ inf vanishes everywhere on X, which yields a contradiction.

The second refinement
The restrictions imposed on the constant K in both Theorem 3.3 and Frink's Theorem 3.1 can be omitted if the quasimetric d is replaced by d raised to a proper power p > 0. The advantage of this method over the previous one is that it allows us (by manipulating the power p) to obtain arbitrarily narrow metric bounds. This is established precisely in the following Proposition 3.9. Let (X, d) be a K-quasimetric space. Then for any ε > 0, there exist p ∈ (0, 1] and a metric ρ for which Proof. The case where K = 1 is obvious, so we assume further that K > 1. Put q := min{1, log K 2}. Then, q ∈ (0, 1] and d q is a K -quasimetric with Checking whether a given function fulfills the first two axioms of a metric is, in general, not problematic. The difficulty is usually to determine if the condition (Q3) is satisfied. Sometimes it is easier to verify another condition, which is equivalent to either (Q3) or (Q3 ). The next theorem provides such conditions. However, it is worth noting here that a similar result appeared in [3] (see [3,Proposition 4.1]), where the authors proved the implication (i) =⇒ (ii) formulated below. Here we expand the list with two additional conditions. We (ii) ⇒ (iii): First note that ϕ = id satisfies the conditions in (ii), so d is a quasimetric. Fix ε > 0. By Proposition 3.9, there exist p ∈ (0, 1) and a metric ρ such that ρ(x, y) d p (x, y) (1 + ε)ρ(x, y) for all x, y ∈ X.
(iv) ⇒ (i): Assume that d p fulfills the c-relaxed polygonal inequality. Clearly, in particular, d p is then a quasimetric. Hence, d is a quasimetric as well, because ϕ(t) = t 1 p fulfills the conditions from (ii), so we may refer to (i) =⇒ (ii) with d replaced by d p .

Theorems of Cantor and Banach in a quasimetric setting
In this section we generalise the Cantor intersection theorem onto quasimetric spaces, presenting also its more general version. Moreover, we derive from it the Banach Fixed-Point Theorem for quasimetric spaces, which was first proved by Bakhtin [4]. Then in [1] the authors showed that the quasimetric version can be obtained from its metric counterpart using remetrization techniques. We prove that a more general version of the Banach Contraction Principle is also true for quasimetric spaces and present a proof of it using the generalised version of Cantor's theorem. First, let us start with a technical lemma which will be of use in the proof of the Cantor intersection theorem for quasimetric spaces. (ii) for every sequence (A n ) n∈N of subsets of X, the following equivalence holds: y) is the diameter of a set A ⊂ X with respect to the semimetric d i for i = 1, 2.

Proof. (i) =⇒ (ii):
Assume that d 1 is uniformly equivalent to d 2 and let (A n ) be a sequence of subsets of X such that diam d1 A n → 0. We will show that diam d2 A n → 0. Fix ε > 0. Then there exist δ > 0 and n 0 ∈ N such that for any x, y ∈ X, if d 1 (x, y) < δ, then d 2 (x, y) < ε and diam d1 A n < δ for each n n 0 . Hence, for n n 0 and x, y ∈ A n , we have that d 1 (x, y) diam d1 A n < δ, so d 2 (x, y) < ε, which implies that diam d2 A n ε for n n 0 . This shows that diam d2 A n → 0. Now, by interchanging the roles of d 1 and d 2 , we obtain that the stated equivalence holds.
(ii) =⇒ (i): Suppose to the contrary that d 1 and d 2 are not uniformly equivalent. Without loss of generality we may assume that the first condition from Definition 2.6 does not hold, i.e., there exists ε 0 > 0 such that for all n ∈ N, there exist points x n , y n in X such that d 1 (x n , y n ) < 1 n and d 2 (x n , y n ) ε 0 .
If we set A n := {x n , y n } for n ∈ N, then diam d1 A n → 0, but diam d2 A n ε 0 which yields a contradiction.
Now with the help of Lemma 4.1 and remetrization techniques, we will prove the Cantor intersection theorem for semimetric spaces satisfying (W 5).

Theorem 4.2. Let (X, d) be a complete semimetric space in which (W5) holds.
Let (A n ) n∈N be a descending sequence of closed nonempty subsets of X such that diam A n → 0. Then n∈N A n = {x * } for some x * ∈ X.
Proof. Due to [11,Theorem 3.2], there exists a metric ρ which is uniformly equivalent to d on X. Clearly, (X, ρ) is then complete. By Lemma 4.1, since diam d A n → 0, we get that diam ρ A n → 0, so the result follows from the classic Cantor's intersection theorem.
It is well known that the classic Cantor's intersection theorem yields the Banach Contraction Principle. This observation is due to Boyd and Wong [6] (see also [18, p. 8]), who proved that if T is a Banach contraction on a metric space (X, ρ), then sets A n defined by satisfy the assumptions of Cantor's theorem. Thus, a natural question arises whether Theorem 4.2 could be used to prove the following quasimetric version of the Banach Contraction Principle established by Bakhtin [4]. We denote by L(T ) the Lipschitz constant of a mapping T .

Theorem 4.3 (Bakhtin).
Let (X, d) be a complete quasimetric space, T : X → X be Lipschitzian with L(T ) ∈ [0, 1). Then T has a unique fixed point x * ∈ X and for any x ∈ X, T n x → x * .
However, it turns out that, in general, Bakhtin's theorem cannot be proved via Cantor's intersection theorem for quasimetric spaces. This is caused by the fact that, as shown in Example 4.4 given below, there exists a complete quasimetric space (X, d) (even satisfying a c-relaxed polygonal inequality) and a Banach contraction T : X → X such that for any ε > 0, the set is not closed. Consequently, there does not exist a sequence (α n ) such that α n 0 and the sets Fix αn T satisfy the assumptions of Theorem 4.2, so the Boyd-Wong [6] trick does not work in this case. elsewhere.
Note that |x − y| d(x, y) 2|x − y|. Hence, (X, d) satisfies the 2-rpi and if x, y ∈ X, then x ∈ X and consider the following cases: Hence we easily obtain that Fix ε T = 0, 5+ √ 5 4 ε , so Fix ε T is not closed. Now we will generalise Theorem 4.2 to get a version of Cantor's theorem which yields the Banach Contraction Principle for mappings on quasimetric spaces.

Theorem 4.5. (Generalized Cantor's intersection theorem for semimetric spaces)
Let (X, d) be a complete semimetric space satisfying (W5), (A n ) n∈N be a descending sequence of nonempty subsets of X such that diam A n → 0 and there exists a subsequence (A kn ) such that for any n ∈ N, A kn ⊂ A n . Then Proof. We will apply Theorem 4.2 to the sequence (A kn ). Of course, (A kn ) is a descending sequence of closed sets with diameters tending to 0, since diam A kn diam A n . Moreover, which implies that n∈N A n = n∈N A kn = {x * } for some x * ∈ X.
We will now present a proof of the quasimetric version of the Banach Fixed Point Theorem using Theorem 4.5. For this purpose let us recall some basic notions. For a semimetric space (X, d) and a mapping T : X → X, by Fix T we understand the set of all fixed points of T . Moreover, recall that if (α n ) n∈N is a sequence of positive numbers such that α n 0, then we denote The following result explains when the sequence (A n ) satisfies the assumptions of Theorem 4.5. For metric spaces, the condition (2) of Lemma 4.6 was established in [21].  Proof. Ad 1. ' =⇒ ': Let ε > 0. Choose n ∈ N such that α n < ε and put δ := α kn . Then ' ⇐= ': Let n ∈ N and ε = α n . From the assumption, there exists δ n > 0 such that Fix δn T ⊂ Fix αn T = A n . Choose m n ∈ N such that for all j m n , α j < δ n . We define the sequence (k n ) recursively. Put k 1 := m 1 . Having defined k n , put k n+1 := max{k n + 1, m n+1 }. Then k n+1 > k n m n , so α kn < δ n and hence Ad 2. ' =⇒ ': Assume that diam A n → 0. Let (x n ) and (y n ) be such that d(x n , T x n ) → 0 and d(y n , T y n ) → 0. Fix ε > 0. Then there exists k ∈ N such that diam A k < ε. Since α k > 0, there is p ∈ N such that for each n p, d(x n , T x n ) α k and d(y n , T y n ) α k , i.e., x n , y n ∈ A k . Hence Suppose to the contrary that r > 0. Choose any s ∈ (0, r). Then s < ∞ and diam A n > s for any n ∈ N, so there exist x n and y n in A n such that d(x n , y n ) > s. On the other hand, d(x n , T x n ) α n and d(y n , T y n ) α n , so d(x n , T x n ) → 0 and d(y n , T y n ) → 0. By hypothesis, d(x n , y n ) → 0. Hence, since d(x n , y n ) > s, we obtain, letting n tend to ∞, that 0 s > 0, which yields a contradiction.
We will also need the following two lemmas. Continuing in this fashion we obtain that which yields both (3) and (4). To show (5) we again use (Q3 ) twice to get that d(x, T x) M (d(x, y) + M (d(y, T y) + αd(x, y))) , so (5) holds. Finally, fix ε > 0 and set δ := ε M 2 . Let (x n ) be such that d(x n , T x n ) δ and x n → x. Then, by (5), Letting n tend to infinity, we obtain that d(x, T x) M 2 δ = ε.
The following result is well known for selfmaps of metric spaces. We omit the proof, since it does not differ from its metric version. We can now proceed to the aforementioned alternative proof of the Banach Theorem 4.3. It will be more convenient for us to work with condition (Q3 ). In fact, we will prove the following more general result.  d(z, y)) .