Abstract
We obtain versions of the Boyd and Wong fixed point theorem and of the Matkowski fixed point theorem for multivalued maps and wdistances on complete quasimetric spaces. Our results generalize, in several directions, some wellknown fixed point theorems.
Keywords:
Fixed point; multivalued map; wdistance; quasimetric spaceIntroduction and preliminaries
Throughout this article, the letters ℕ and ω will denote the set of positive integer numbers and the set of nonnegative integer numbers, respectively.
Following the terminology of [1], by a T_{0 }quasipseudometric 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 }quasipseudometric d on X that satisfies the stronger condition
(i') d(x, y) = 0 ⇔ x = y,
is called a quasimetric on X.
Our basic references for quasimetric spaces and related structures are [2] and [3].
We remark that in the last years several authors used the term "quasimetric" to refer to a T_{0 }quasipseudometric and the term "T_{1 }quasimetric" to refer to a quasimetric in the above sense. It is also interesting to recall (see, for instance, [3]) that T_{0 }quasipseudometric 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 }quasipseudometric 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 quasimetric on X, the pair (X, d) is then called a quasimetric 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 quasimetric, 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 quasipseudometric space as defined in [4].)
Matthews introduced in [5] the notion of a weightable T_{0 }qpm space (under the name of a "weightable quasimetric 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 [617], 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 wdistance on a metric space and then extended the CaristiKirk fixed point theorem [19], the Ekeland variational principle [20] and the nonconvex minimization theorem [21], for wdistances. In [22], Park extended the notion of wdistance to quasimetric spaces and obtained, among other results, generalized forms of Ekeland's priniciple which improve and unify corresponding results in [18,23,24]. Recently, AlHomidan et al. [25] introduced the concept of Qfunction on a quasimetric space as a generalization of wdistances, and then obtained a CaristiKirktype fixed point theorem, a Takahashi minimization theorem, and versions of Ekeland's principle and of Nadler's fixed point theorem for a Qfunction on a complete quasimetric space, generalizing in this way, among others, the main results of [22]. This approach has been continued by Hussain et al. [26], Latif and AlMezel [27], and Marín et al. [1]. In particular, the authors of [27] and [1] have obtained a Rakotchtype and a BianchiniGrandolfitype fixed point theorems, respectively, for multivalued maps and Qfunctions on complete quasimetric spaces and complete T_{0 }qpm spaces.
In this article, we prove a T_{0 }qpm version of the celebrated BoydWong fixed point theorem in terms of Qfunctions, which generalizes and improves, in several senses, some wellknown 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 BoydWong's theorem for partial metrics induced by complete weightable T_{0 }qpm spaces. Finally, we shall show that a multivalued extension for Qfunctions 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 wdistances and Qfunctions on T_{0 }qpm spaces.
Definition 1.2 ([22]). A wdistance 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 Qfunction on (X, d) (cf. [25]).
Clearly, every wdistance is a Qfunction. Moreover, if (X, d) is a metric space, then d is a wdistance 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 Qfunction on (X, d).
Remark 1.3 ([1]). Let q be a Qfunction 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 Qfunction on (X,d). Actually, it is a wdistance on (X,d).
The results
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 BWcontractive selfmap on X has a unique fixed point.
The following easy example shows that unfortunately BoydWong's theorem cannot be generalized to complete quasimetric spaces, even for T contractive.
Example 2.1. Let X = {1/n : n ∈ ℕ} and let d be the quasimetric 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 quasimetric version of BoydWong's theorem using Qfunctions.
Let (X, d) be a T_{0 }qpm space. A selfmap T on X is called BWweakly contractive if there exist a Qfunction 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 BWweakly contractive selfmap on X has a unique fixed point z ∈ X. Moreover, q(z, z) = 0.
Proof. Let T : X → X be BWweakly contractive. Then, there exist a Qfunction 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 }= T^{n }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 BoydWong'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 nonBWcontractive 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 BWcontractive because d(Tx, Ty) = y^{2 } x^{2 }> y  x = d(x, y), whenever 0 < x < y < 1 < x + y. However, T is BWweakly contractive for the partial metric p_{d }induced by d (recall that, from Remark 1.4, pd is a Qfunction 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 wdistances and Qfunctions to the fixed point theory for multivalued maps on metric and quasimetric spaces, it seems interesting to investigate the extension of our version of BoydWong's theorem to the case of multivalued maps. In Theorem 2.6 below, we shall prove a positive result for the case of symmetry Qfunctions, which are defined as follows:
Definition 2.4. A symmetric Qfunction on a T_{0 }qpm space (X, d) is a Qfunction q on (X, d) such that
If q is a wdistance satisfying (SY), we then say that it is a symmetric wdistance on (X, d).
Example 2.5. Of course, if (X, d) is a metric space, then d is a symmetric wdistance 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 wdistance on (X, d). Note also that the wdistance constructed in Lemma 2 of [29] is also a symmetric wdistance.
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 qcontractive multivalued map [[25], Definition 6.1] and of a generalized qcontractive multivalued map [27], we say that a multivalued map T from a T_{0 }qpm space (X, d) to 2^{X}, is BWweakly contractive if there exists a Qfunction 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 BWweakly contractive for a symmetric Qfunction 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_{n1}, 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 BWweakly 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 BoydWong'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 Qfunction q on (X, d) and a nondecreasing function φ : (0, ∞) → (0, ∞) satisfying φ^{n}(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_{n1 }and q(x_{n}, x_{n+1}) ≤ φ(q(x_{n  1}, x_{n}) for all n ∈ ℕ. Therefore,
for all n ∈ ℕ. Since φ^{n}(q(x_{0}, x_{1})) → 0, it follows that q(x_{n}, x_{n+1}) → 0.
Now, choose an arbitrary ε > 0. Since φ^{n}(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]).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The three authors have equitably contributed in obtaining the new results presented in this article.
All authors read and approved the final manuscript.
Acknowledgements
The authors acknowledge the support of the Spanish Ministry of Science and Innovation, under grant MTM200912872C0201.
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