Throughout this article, the letters ℕ and ω will denote the set of positive integer numbers and the set of non-negative integer numbers, respectively.

Following the terminology of 1 , by a T ₀ quasi-pseudo-metric on a set X, we mean a function d : X × X → [0, ∞) such that for all x, y, z ∈ X :

(i) d(x, y) = d(y, x) = 0 ⇔ x = y;

(ii) d(x, z) ≤ d(x, y) + d(y, z).

A T ₀ quasi-pseudo-metric d on X that satisfies the stronger condition

(i') d(x, y) = 0 ⇔ x = y,

is called a quasi-metric on X.

Our basic references for quasi-metric spaces and related structures are 2 and 3 .

We remark that in the last years several authors used the term "quasi-metric" to refer to a T ₀ quasi-pseudo-metric and the term "T ₁ quasi-metric" to refer to a quasi-metric in the above sense. It is also interesting to recall (see, for instance, 3 ) that T ₀ quasi-pseudo-metric spaces play a crucial role in some fields of theoretical computer science, asymmetric functional analysis and approximation theory.

Hereafter, we shall simply write T ₀ qpm instead of T ₀ quasi-pseudo-metric if no confusion arises.

A T ₀ qpm space is a pair (X, d) such that X is a set and d is a T ₀ qpm on X. If d is a quasi-metric on X, the pair (X, d) is then called a quasi-metric space.

Each T ₀ qpm d on a set X induces a T ₀ topology τ_d on X which has as a base the family of open balls {B_d (x, r) : x ∈ X, ε > 0}, where B_d (x, ε) = {y ∈ X : d(x, y) < ε } for all x ∈ X and ε > 0.

Note that if d is a quasi-metric, then τ_d is a T ₁ topology on X.

Given a T ₀ qpm d on X, the function d ^-1 defined by d ^-1 (x, y) = d(y, x), is also a T ₀ qpm on X, called the conjugate of d, and the function d^s defined by d^s (x, y) = max{d(x, y), d ^-1(x, y)} is a metric on X.

It is well known (see, for instance, 3 4 ) that there exist many different notions of completeness for T ₀ qpm spaces. In our context, we shall use the following very general notion:

A T ₀ qpm space (X, d) is said to be complete if every Cauchy sequence in the metric space (X, d^s ) is -convergent. In this case, we say that d is a complete T ₀ qpm on X. (Note that this notion corresponds with the notion of a d ^-1-sequentially complete quasi-pseudo-metric space as defined in 4 .)

Matthews introduced in 5 the notion of a weightable T ₀ qpm space (under the name of a "weightable quasi-metric space"), and its equivalent partial metric space, as a part of the study of denotational semantics of dataflow networks. In fact, partial metric spaces constitute an efficient tool in raising and solving problems in theoretical computer science, domain theory, and denotational semantics for complexity analysis, among others (see 6 7 8 9 10 11 12 13 14 15 16 17 , etc.).

A T ₀ qpm space (X, d) is called weightable if there exists a function w : X → [0, ∞) such that for all x, y ∈ X, d(x, y) + w(x) = d(y, x) + w(y). In this case, we say that d is a weightable T ₀ qpm on X. The function w is said to be a weighting function for (X, d).

A partial metric on a set X is a function p : X × X → [0, ∞) such that for all x, y, z ∈ X :

(i) x = y ⇔ p(x, x) = p(x, y) = p(y, y); (ii) p(x, x) ≤ p(x, y); (iii) p(x, y) = p(y, x); (iv) p(x, z) ≤ p(x, y) + p(y, z) - p(y, y).

A partial metric space is a pair (X, p) such that X is a set and p is a partial metric on X.

Each partial metric p on X induces a T ₀ topology τ_p on X which has as a base the family of open balls {Bp(x, ε) : x ∈ X, ε > 0}, where B_p (x, ε) = {y ∈ X : p(x, y) < ε + p(x, x)} for all x ∈ X and ε > 0.

The precise relationship between partial metric spaces and weightable T ₀ qpm spaces is provided in the following result.

Theorem 1.1 (Matthews 5 ). (a) Let (X, d) be a weightable T ₀ qpm space with weighting function w. Then the function p_d : X × X → [0, ∞) defined by p_d (x, y) = d(x, y) + w(x) for all x, y ∈ X, is a partial metric on X. Furthermore τ_d = τ_pd .

(b) Conversely, let (X, p) be a partial metric space. Then, the function d_p : X × X → [0, ∞) defined by d_p (x, y) = p(x, y) - p(x, x) for all x, y ∈ X is a weightable T ₀ qpm on X with weighting function w given by w(x) = p(x, x) for all x ∈ X. Furthermore .

Kada et al. introduced in 18 the notion of w-distance on a metric space and then extended the Caristi-Kirk fixed point theorem 19 , the Ekeland variational principle 20 and the nonconvex minimization theorem 21 , for w-distances. In 22 , Park extended the notion of w-distance to quasi-metric spaces and obtained, among other results, generalized forms of Ekeland's priniciple which improve and unify corresponding results in 18 23 24 . Recently, Al-Homidan et al. 25 introduced the concept of Q-function on a quasi-metric space as a generalization of w-distances, and then obtained a Caristi-Kirk-type fixed point theorem, a Takahashi minimization theorem, and versions of Ekeland's principle and of Nadler's fixed point theorem for a Q-function on a complete quasi-metric space, generalizing in this way, among others, the main results of 22 . This approach has been continued by Hussain et al. 26 , Latif and Al-Mezel 27 , and Marín et al. 1 . In particular, the authors of 27 and 1 have obtained a Rakotch-type and a Bianchini-Grandolfi-type fixed point theorems, respectively, for multivalued maps and Q-functions on complete quasi-metric spaces and complete T ₀ qpm spaces.

In this article, we prove a T ₀ qpm version of the celebrated Boyd-Wong fixed point theorem in terms of Q-functions, which generalizes and improves, in several senses, some well-known fixed point theorems. We also discuss the extension of our result to the case of multivalued maps. Although we only obtain a partial result, it is sufficient to be able to deduce a multivalued version of Boyd-Wong's theorem for partial metrics induced by complete weightable T ₀ qpm spaces. Finally, we shall show that a multivalued extension for Q-functions on complete T ₀ qpm spaces of the famous Matkowski fixed point theorem can be obtained.

We conclude this section by highlighting some pertinent concepts and facts on w-distances and Q-functions on T ₀ qpm spaces.

Definition 1.2 ( 22 ). A w-distance on a T ₀ qpm space (X, d) is a function q : X × X → [0, ∞) satisfying the following conditions:

(W1) q(x, z) ≤ q(x, y) + q(y, z) for all x, y, z ∈ X;

(W2) q(x, ·) : X → [0, ∞) is lower semicontinuous on (X, ) for all x ∈ X;

(W3) for each ε > 0 there exists δ > 0 such that q(x, y) ≤ δ and q(x, z) ≤ δ imply d(y, z) ≤ ε.

If in Definition 1.2 above condition (W2) is replaced by

(Q2) if x ∈ X, M > 0, and (y_n ) _n∈ℕ is a sequence in X that -converges to a point y ∈ X and satisfies q(x, y_n ) ≤ M for all n ∈ ℕ, then q(x, y) ≤ M,

then q is called a Q-function on (X, d) (cf. 25 ).

Clearly, every w-distance is a Q-function. Moreover, if (X, d) is a metric space, then d is a w-distance on (X, d). However, Example 3.2 of 25 shows that there exists a T ₀ qpm space (X, d) such that d does not satisfy condition (W3), and hence it is not a Q-function on (X, d).

Remark 1.3 ( 1 ). Let q be a Q-function on a T ₀ qpm space (X, d). Then, for each ε > 0 there exists δ > 0, such that q(x, y) ≤ δ andq(x, z) ≤ δ imply d^s (y, z) ≤ ε.

Remark 1.4 ( 1 ). Let (X, d) be a weightable T ₀ qpm space. Then, the induced partial metric p_d is a Q-function on (X,d). Actually, it is a w-distance on (X,d).

Let (X, d) be a T ₀ qpm space. A selfmap T on X is called BW -contractive if there exists a function φ : [0, ∞) → [0, ∞) satisfying φ(t) < t and for all t > 0, and such that for each x, y ∈ X,

If φ(t) = rt, with r ∈ [0, 1) being constant, then T is called contractive.

In their celebrated article 28 , Boyd and Wong essentially proved the following general fixed point theorem: Let (X,d) be complete metric space. Then every BW-contractive selfmap on X has a unique fixed point.

The following easy example shows that unfortunately Boyd-Wong's theorem cannot be generalized to complete quasi-metric spaces, even for T contractive.

Example 2.1. Let X = {1/n : n ∈ ℕ} and let d be the quasi-metric on X given by d(1,/n, 1/n) = 0, and d(1/n, 1/m) = 1/m for all n, m ∈ ℕ. Clearly, (X, d) is complete (in fact, it is complete in the stronger sense of 1 22 25 27 ). Define T : X → X by T1/n = 1/2n. Then, T is contractive but it has not fixed point.

Next, we show that it is, however, possible to obtain a nice quasi-metric version of Boyd-Wong's theorem using Q-functions.

Let (X, d) be a T ₀ qpm space. A selfmap T on X is called BW-weakly contractive if there exist a Q-function q on (X, d) and a function φ : [0, ∞) → [0, ∞) satisfying φ(0) = 0, φ(t) < t and for all t > 0, and such that for each x, y ∈ X,

If φ(t) = rt, with r ∈ [0, 1) being constant, then T is called weakly contractive.

Theorem 2.2. Let (X, d) be a complete T ₀ qpm space. Then, each BW-weakly contractive selfmap on X has a unique fixed point z ∈ X. Moreover, q(z, z) = 0.

Proof. Let T : X → X be BW-weakly contractive. Then, there exist a Q-function q on (X, d) and a function φ : [0, ∞) → [0, ∞) satisfying φ(0) = 0, φ(t) < t and for all t > 0, such that for each x, y ∈ X,

Fix x ₀ ∈ X and let x _n = Tⁿ x ₀ for all n ∈ ω.

We show that q(x _n , x _n+1) → 0.

Indeed, if q(x _k , x _k+1) = 0 for some k ∈ ω, then φ(q(x _k , x _k+1)) = 0 and thus q(x _n , x _n+1) = 0 for all n ≥ k. Otherwise, (q(x _n , x _n+1)) _n∈ω is a strictly decreasing sequence in (0, ∞) which converges to 0, as in the classical proof of Boyd-Wong's theorem.

Similarly, we have that q(x _n+1, x _n ) → 0.

Now, we show that for each ε ∈ (0, 1) there exists n_ε ∈ ℕ such that q(x _n , x_m ) < ε whenever m > n > n _ε .

Assume the contrary. Then, there exists ε ₀ ∈ (0, 1) such that, for each k ∈ ℕ, there exist n(k), j(k) ∈ ℕ with j(k) > n(k) > k and q(x _n(k), x _j(k)) ≥ ε ₀.

Since q(x_n , x _n+1) → 0, there exists such that q(x_n , x _n+1) < ε ₀ for

all .

For each , we denote by m(k) the least j(k) ∈ ℕ satisfying the following three conditions:

Note that there exists such a m(k) because q(x _n(k), x _n(k)+1) < ε ₀. Then, for each , we obtain

Since q(x _m(k)-1, x _m(k)) → 0, it follows from the preceding inequalities that where r_k = q(x _n(k), x _m(k)). Hence,

Choose δ > 0 with . Let such that q(x _n(k), x _n(k)+1) < (ε ₀ - δ)/2, and q(x _m(k)+1, x _m(k)) < (ε ₀ - δ)/2,

for all k > k ₀.

Then,

for some k > k ₀, which contradicts that ε ₀ ≤ q(x _n(k), x _m(k)) for all . We conclude that for each ε ∈ (0, 1), there exists n_ε ∈ ℕ such that

Next, we show that (x_n ) _n∈ω is a Cauchy sequence in the metric space (X, d^s ). Indeed, let ε > 0, and let δ = δ (ε) > 0 as given in Definition 1.2 (W3). Then, for n, m > n_δ we obtain , and , and hence from Remark 1.3, d^s (x_n , x_m ) ≤ ε. Consequently, (x_n ) _n∈ω is a Cauchy sequence in (X, d^s ).

Now, let z ∈ X such that d(x_n , z) → 0. Then q(x_n , z) → 0 by (Q2) and condition (*) above. Hence, . From Remark 1.3, we conclude that d^s (z, Tz) = 0, i.e., z = Tz.

Next, we show the uniqueness of the fixed point. Let y = Ty. If q(y, z) > 0, q(Ty, Tz) = q(y, z) ≤ φ(q(y, z)) < q(y, z), a contradiction. Hence, q(y, z) = 0. Interchanging y and z, we also have q(z, y) = 0. Therefore, y = z from Remark 1.3.

Finally, q(z, z) = 0 since otherwise we obtain q(z, z) = q(Tz, Tz) ≤ φ(q(z, z)) < q(z, z), a contradiction.    □

The following is an example of a non-BW-contractive selfmap T on a complete T ₀ qpm space (X, d) for which Theorem 2.2 applies.

Example 2.3. Let X = [0, 1) and d be the weightable T ₀ qpm on X given by d(x, y) = max{y -x, 0} for all x, y ∈ X. Clearly (X, d) is complete because d(x, 0) = 0 for all x ∈ X, and thus every sequence in X converges to 0 with respect to .

Now, define T : X → X by Tx = x ² for all x ∈ X. Then, T is not BW-contractive because d(Tx, Ty) = y ² - x ² > y - x = d(x, y), whenever 0 < x < y < 1 < x + y. However, T is BW-weakly contractive for the partial metric p_d induced by d (recall that, from Remark 1.4, pd is a Q-function on (X, d)), and the function φ : [0, ∞) → [0, ∞) defined by φ(t) = t ² for 0 ≤ t < 1 and for t ≥ 1. Indeed, for each x, y ∈ X we have,

Hence, we can apply Theorem 2.2, so that T has a unique fixed point: in fact, 0 is the only fixed point of T, and p_d (0, 0) = 0. (Note that in this example, there exists not r ∈ [0, 1) such that p_d (Tx, Ty) ≤ rp_d (x, y) for all x, y ∈ X.)

In the light of the applications of w-distances and Q-functions to the fixed point theory for multivalued maps on metric and quasi-metric spaces, it seems interesting to investigate the extension of our version of Boyd-Wong's theorem to the case of multivalued maps. In Theorem 2.6 below, we shall prove a positive result for the case of symmetry Q-functions, which are defined as follows:

Definition 2.4. A symmetric Q-function on a T ₀ qpm space (X, d) is a Q-function q on (X, d) such that

If q is a w-distance satisfying (SY), we then say that it is a symmetric w-distance on (X, d).

Example 2.5. Of course, if (X, d) is a metric space, then d is a symmetric w-distance on (X, d). Moreover, it follows from Remark 1.4, that for every weightable T ₀ qpm space (X, d) its induced partial metric p_d is a symmetric w-distance on (X, d). Note also that the w-distance constructed in Lemma 2 of 29 is also a symmetric w-distance.

Given a T ₀ qpm space (X, d), we denote by 2 ^X and by the collection of all nonempty subsets of X and the collection of all nonempty -closed subsets of X, respectively.

Generalizing the notions of a q-contractive multivalued map [ 25 , Definition 6.1] and of a generalized q-contractive multivalued map 27 , we say that a multivalued map T from a T ₀ qpm space (X, d) to 2 ^X , is BW-weakly contractive if there exists a Q-function q on (X, d) and a function φ : [0, ∞) → [0, ∞) satisfying φ(0) = 0, φ(t) < t and for all t > 0, and such that, for each x, y ∈ X and each u ∈ Tx there exists v ∈ Ty with q(u, v) ≤ φ(q(x, y)).

Theorem 2.6. Let (X, d) be a complete T ₀ qpm space and be BW-weakly contractive for a symmetric Q-function q on (X,d). Then, there is z ∈ X such that z ∈ Tz and q(z, z) = 0.

Proof. By hypothesis, there is a function φ : [0, ∞) → [0, ∞) satisfying φ(0) = 0, φ(t) < t and for all t > 0, and such that for each x, y ∈ X and u ∈ Tx there is v ∈ Ty with

Fix x ₀ ∈ X and let x ₁ ∈ Tx ₀. Then, there exists x ₂ ∈ Tx ₁ such that q(x ₁, x ₂) ≤ φ(q(x ₀, x ₁). Following this process, we obtain a sequence (x_n ) _n∈ω with x_n ∈ Tx _{n - 1} and q(x_n , x _n+1) ≤ φ(q(x _n-1, x_n ) for all n ∈ ℕ.

As in Theorem 2.2, q(x_n , x _n+1) → 0.

Now, we show that for each ε ∈ (0, 1), there exists n_ε ∈ ℕ such that q(x_n , x_m ) < ε whenever m > n > n_ε .

Assume the contrary. Then, there exists ε ₀ ∈ (0, 1) such that for each k ∈ ℕ, there exist n(k), j(k) ∈ ℕ with j(k) > n(k) > k and q(x _n(k), x _j(k)) ≥ ε ₀.

Again, by repeating the proof of Theorem 2.2, and using symmetry of q, we derive that

a contradiction.

From Remark 1.3, it follows that (x_n ) _n∈ω is a Cauchy sequence in (X, d^s ) (compare the proof of Theorem 2.2), and so there exists z ∈ X such that d(x_n , z) → 0, and thus q(x_n , z) → 0.

Therefore, for each n ∈ ω there exists v _n+1 ∈ Tz with

Since q(x_n , z) → 0 we have q(x _n+1, v _n+1) → 0, and so d^s (z, v _n+1) → 0 from Remark 1.3. Consequently, z ∈ Tz because Tz is closed in (X, d^s ).

It remains to be shown that q(z, z) = 0. Indeed, since z ∈ Tz we can construct a sequence (z_n ) _n∈ℕ in X such that z ₁ ∈ Tz, z _n+1 ∈ Tz_n , q(z, z ₁) ≤ φ(q(z, z_n )) and q(z, z _n+1) ≤ φ(q(z, z_n )) for all n ∈ ℕ. Hence (q(z, z_n )) _n∈ℕ is a nonincreasing sequence in [0, ∞) that converges to 0. From Remark 1.3, the sequence (z_n ) _n∈ℕ is Cauchy in (X, d^s ). Let u ∈ X such that d(z_n , u) → 0. It follows from condition (Q2) that q(z, u) = 0. Since q(x_n , z) → 0, we deduce by condition (Q1) that q(x_n , u) → 0. Therefore, d^s (z, u) ≤ ε for all ε > 0, from Remark 1.3. We conclude that z = u, and thus q(z, z) = 0.    □

Although we do not know if symmetric of q can be omitted in Theorem 2.6, it can be applied directly to obtain the following fixed point result for multivalued maps on partial metric spaces, which substantially improves Theorem 5.3 of 5 .

Corollary 2.7. Let (X, p) be a partial metric space such that the induced weightable T ₀ qpm d_p is complete and be BW-weakly contractive for p. Then, there is z ∈ X such that z ∈ Tz and p(z, z) = 0.

We conclude this article by showing, nevertheless, that it is possible to prove a multivalued version of the celebrated Matkowski's fixed point theorem 30 , which provides a nice generalization of Boyd-Wong's theorem when φ is nondecreasing.

Theorem 2.8. Let (X, d) be a complete T ₀ qpm space and let . If there exist a Q-function q on (X, d) and a nondecreasing function φ : (0, ∞) → (0, ∞) satisfying φⁿ (t) → 0 for all t > 0, such that for each x, y ∈ X and each u ∈ Tx, there exists v ∈ Ty with

then, there exists z ∈ X such that z ∈ Tz and q(z, z) = 0.

Proof. Let φ(0) = 0. Fix x ₀ ∈ X and let x ₁ ∈ Tx ₀. Then, there exists x ₂ ∈ Tx ₁ such that q(x ₁, x ₂) ≤ φ(q(x ₀, x ₁). Following this process, we obtain a sequence (x_n ) _n∈ω with x_n ∈ Tx _n-1 and q(x_n , x _n+1) ≤ φ(q(x _{n - 1}, x_n ) for all n ∈ ℕ. Therefore,

for all n ∈ ℕ. Since φⁿ (q(x ₀, x ₁)) → 0, it follows that q(x_n , x _n+1) → 0.

Now, choose an arbitrary ε > 0. Since φⁿ (ε) → 0, then φ(ε) < ε, so there is n_ε ∈ ℕ such that

for all n ≥ n_ε . Note that then,

for all n ≥ n_ε , and following this process

for all n ≥ n_ε and k ∈ ℕ. Applying Remark 1.3, we deduce that (x_n ) _n∈ω is a Cauchy sequence in (X, d^s ). Then, there is z ∈ X such that d(x_n , z) → 0 and thus q(x_n , z) → 0 by condition (Q2). The rest of the proof follows similarly as the proof of Theorem 2.6. We conclude that z ∈ Tz and q(z, z) = 0.    □

Remark 2.9. The above theorem improves, among others, Theorem 3.3 of 1 (compare also Theorem 1 of 31 ).

The authors declare that they have no competing interests.

The three authors have equitably contributed in obtaining the new results presented in this article.

All authors read and approved the final manuscript.