This article is part of the series S. Park's Contribution to the Development of Fixed Point Theory and KKM Theory.

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Weakly contractive multivalued maps and w-distances on complete quasi-metric spaces

Josefa Marín, Salvador Romaguera* and Pedro Tirado

Author Affiliations

Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain

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Fixed Point Theory and Applications 2011, 2011:2  doi:10.1186/1687-1812-2011-2


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2011/1/2


Received:1 March 2011
Accepted:20 June 2011
Published:20 June 2011

© 2011 Marín et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain versions of the Boyd and Wong fixed point theorem and of the Matkowski fixed point theorem for multivalued maps and w-distances on complete quasi-metric spaces. Our results generalize, in several directions, some well-known fixed point theorems.

Keywords:
Fixed point; multivalued map; w-distance; quasi-metric space

Introduction and preliminaries

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 T0 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 T0 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 T0 quasi-pseudo-metric and the term "T1 quasi-metric" to refer to a quasi-metric in the above sense. It is also interesting to recall (see, for instance, [3]) that T0 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 T0 qpm instead of T0 quasi-pseudo-metric if no confusion arises.

A T0 qpm space is a pair (X, d) such that X is a set and d is a T0 qpm on X. If d is a quasi-metric on X, the pair (X, d) is then called a quasi-metric space.

Each T0 qpm d on a set X induces a T0 topology τd on X which has as a base the family of open balls {Bd(x, r) : x X, ε > 0}, where Bd(x, ε) = {y X : d(x, y) < ε } for all x X and ε > 0.

Note that if d is a quasi-metric, then τd is a T1 topology on X.

Given a T0 qpm d on X, the function d-1 defined by d-1 (x, y) = d(y, x), is also a T0 qpm on X, called the conjugate of d, and the function ds defined by ds(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 T0 qpm spaces. In our context, we shall use the following very general notion:

A T0 qpm space (X, d) is said to be complete if every Cauchy sequence in the metric space (X, ds) is -convergent. In this case, we say that d is a complete T0 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 T0 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 [6-17], etc.).

A T0 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 T0 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 T0 topology τp on X which has as a base the family of open balls {Bp(x, ε) : x X, ε > 0}, where Bp(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 T0 qpm spaces is provided in the following result.

Theorem 1.1 (Matthews [5]). (a) Let (X, d) be a weightable T0 qpm space with weighting function w. Then the function pd : X × X → [0, ∞) defined by pd(x, y) = d(x, y) + w(x) for all x, y X, is a partial metric on X. Furthermore τd = τpd.

(b) Conversely, let (X, p) be a partial metric space. Then, the function dp : X × X → [0, ∞) defined by dp(x, y) = p(x, y) - p(x, x) for all x, y X is a weightable T0 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 T0 qpm spaces.

In this article, we prove a T0 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 T0 qpm spaces. Finally, we shall show that a multivalued extension for Q-functions on complete T0 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 T0 qpm spaces.

Definition 1.2 ([22]). A w-distance on a T0 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 (yn)n∈ℕ is a sequence in X that -converges to a point y X and satisfies q(x, yn) ≤ 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 T0 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 T0 qpm space (X, d). Then, for each ε > 0 there exists δ > 0, such that q(x, y) ≤ δ andq(x, z) ≤ δ imply ds(y, z) ≤ ε.

Remark 1.4 ([1]). Let (X, d) be a weightable T0 qpm space. Then, the induced partial metric pd is a Q-function on (X,d). Actually, it is a w-distance on (X,d).

The results

Let (X, d) be a T0 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 T0 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 T0 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 x0 X and let xn = Tn x0 for all n ω.

We show that q(xn, xn+1) → 0.

Indeed, if q(xk, xk+1) = 0 for some k ω, then φ(q(xk, xk+1)) = 0 and thus q(xn, xn+1) = 0 for all n k. Otherwise, (q(xn, xn+1))nω 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(xn+1, xn) → 0.

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

Assume the contrary. Then, there exists ε0 ∈ (0, 1) such that, for each k ∈ ℕ, there exist n(k), j(k) ∈ ℕ with j(k) > n(k) > k and q(xn(k), xj(k)) ≥ ε0.

Since q(xn, xn+1) → 0, there exists such that q(xn, xn+1) < ε0 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(xn(k), xn(k)+1) < ε0. Then, for each , we obtain

Since q(xm(k)-1, xm(k)) → 0, it follows from the preceding inequalities that where rk = q(xn(k), xm(k)). Hence,

Choose δ > 0 with . Let such that q(xn(k), xn(k)+1) < (ε0 - δ)/2, and q(xm(k)+1, xm(k)) < (ε0 - δ)/2,

for all k > k0.

Then,

for some k > k0, which contradicts that ε0 q(xn(k), xm(k)) for all . We conclude that for each ε ∈ (0, 1), there exists nε ∈ ℕ such that

Next, we show that (xn)nω is a Cauchy sequence in the metric space (X, ds). Indeed, let ε > 0, and let δ = δ (ε) > 0 as given in Definition 1.2 (W3). Then, for n, m > nδ we obtain , and , and hence from Remark 1.3, ds(xn, xm) ≤ ε. Consequently, (xn)nω is a Cauchy sequence in (X, ds).

Now, let z X such that d(xn, z) → 0. Then q(xn, z) → 0 by (Q2) and condition (*) above. Hence, . From Remark 1.3, we conclude that ds(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 T0 qpm space (X, d) for which Theorem 2.2 applies.

Example 2.3. Let X = [0, 1) and d be the weightable T0 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 = x2 for all x X. Then, T is not BW-contractive because d(Tx, Ty) = y2 - x2 > y - x = d(x, y), whenever 0 < x < y < 1 < x + y. However, T is BW-weakly contractive for the partial metric pd 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) = t2 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 pd(0, 0) = 0. (Note that in this example, there exists not r ∈ [0, 1) such that pd(Tx, Ty) ≤ rpd(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 T0 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 T0 qpm space (X, d) its induced partial metric pd 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 T0 qpm space (X, d), we denote by 2X 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 T0 qpm space (X, d) to 2X, 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 T0 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 x0 X and let x1 Tx0. Then, there exists x2 Tx1 such that q(x1, x2) ≤ φ(q(x0, x1). Following this process, we obtain a sequence (xn)nω with xn Txn - 1 and q(xn, xn+1) ≤ φ(q(xn-1, xn) for all n ∈ ℕ.

As in Theorem 2.2, q(xn, xn+1) → 0.

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

Assume the contrary. Then, there exists ε0 ∈ (0, 1) such that for each k ∈ ℕ, there exist n(k), j(k) ∈ ℕ with j(k) > n(k) > k and q(xn(k), xj(k)) ≥ ε0.

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 (xn)nω is a Cauchy sequence in (X, ds) (compare the proof of Theorem 2.2), and so there exists z X such that d(xn, z) → 0, and thus q(xn, z) → 0.

Therefore, for each n ω there exists vn+1 Tz with

Since q(xn, z) → 0 we have q(xn+1, vn+1) → 0, and so ds(z, vn+1) → 0 from Remark 1.3. Consequently, z Tz because Tz is closed in (X, ds).

It remains to be shown that q(z, z) = 0. Indeed, since z Tz we can construct a sequence (zn)n∈ℕ in X such that z1 Tz, zn+1 Tzn, q(z, z1) ≤ φ(q(z, zn)) and q(z, zn+1) ≤ φ(q(z, zn)) for all n ∈ ℕ. Hence (q(z, zn))n∈ℕ is a nonincreasing sequence in [0, ∞) that converges to 0. From Remark 1.3, the sequence (zn)n∈ℕ is Cauchy in (X, ds). Let u X such that d(zn, u) → 0. It follows from condition (Q2) that q(z, u) = 0. Since q(xn, z) → 0, we deduce by condition (Q1) that q(xn, u) → 0. Therefore, ds(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 T0 qpm dp 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 T0 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 x0 X and let x1 Tx0. Then, there exists x2 Tx1 such that q(x1, x2) ≤ φ(q(x0, x1). Following this process, we obtain a sequence (xn)nω with xn Txn-1 and q(xn, xn+1) ≤ φ(q(xn - 1, xn) for all n ∈ ℕ. Therefore,

for all n ∈ ℕ. Since φn(q(x0, x1)) → 0, it follows that q(xn, xn+1) → 0.

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

for all n nε. Note that then,

for all n nε, and following this process

for all n nε and k ∈ ℕ. Applying Remark 1.3, we deduce that (xn)nω is a Cauchy sequence in (X, ds). Then, there is z X such that d(xn, z) → 0 and thus q(xn, 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 MTM2009-12872-C02-01.

References

  1. Marín, J, Romaguera, S, Tirado, P: Q-functions on quasi-metric spaces and fixed points for multivalued maps. Fixed Point Theory Appl. 2011, 10 Article ID 603861 (2011)

    Article ID 603861

    BioMed Central Full Text OpenURL

  2. Fletcher, P, Lindgren, WF: Quasi-Uniform Spaces. Marcel Dekker, New York (1982)

  3. Künzi, HPA: Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology. In: Aull CE, Lowen R (eds.) Handbook of the History of General Topology, vol. 3, pp. 853–968. Kluwer, Dordrecht (2001)

  4. Reilly, IL, Subrhamanyam, PV, Vamanamurthy, MK: Cauchy sequences in quasi-pseudo-metric spaces. Monats Math. 93, 127–140 (1982). Publisher Full Text OpenURL

  5. Matthews, SG: Partial metric topology. Procedings 8th Summer Conference on General Topology and Applications, Annals of the New York Academy of Science. 728, 183–197 (1994)

  6. Ali-Akbari, M, Honari, B, Pourmahdian, M, Rezaii, MM: The space of formal balls and models of quasi-metric spaces. Math Struct Comput Sci. 19, 337–355 (2009). Publisher Full Text OpenURL

  7. García-Raffi, LM, Romaguera, S, Schellekens, M: Applications of the complexity space to the General Probabilistic Divide and Conquer algorithm. J Math Anal Appl. 348, 346–355 (2008). Publisher Full Text OpenURL

  8. Heckmann, R: Approximation of metric spaces by partial metric spaces. Appl Categ Struct. 7, 71–83 (1999). Publisher Full Text OpenURL

  9. Romaguera, S: On Computational Models for the Hyperspace. Advances in Mathematics Research, pp. 277–294. Nova Science Publishers, New York (2009)

  10. Romaguera, S, Schellekens, M: Partial metric monoids and semivaluation spaces. Topol Appl. 153, 948–962 (2005). Publisher Full Text OpenURL

  11. Romaguera, S, Tirado, P: The complexity probabilistic quasi-metric space. J Math Anal Appl. 376, 732–740 (2011). Publisher Full Text OpenURL

  12. Romaguera, S, Valero, O: A quantitative computational model for complete partial metric spaces via formal balls. Math Struct Comput Sci. 19, 541–563 (2009). Publisher Full Text OpenURL

  13. Romaguera, S, Valero, O: Domain theoretic characterisations of quasi-metric completeness in terms of formal balls. Math Struct Comput Sci. 20, 453–472 (2010). Publisher Full Text OpenURL

  14. Schellekens, M: The Smyth completion: a common foundation for denotational semantics and complexity analysis. Electron Notes Theor Comput Sci. 1, 535–556 (1995)

  15. Schellekens, MP: A characterization of partial metrizability. Domains are quantifiable. Theor Comput Sci. 305, 409–432 (2003). Publisher Full Text OpenURL

  16. Waszkiewicz, P: Quantitative continuous domains. Appl Categ Struct. 11, 41–67 (2003). Publisher Full Text OpenURL

  17. Waszkiewicz, P: Partial metrisability of continuous posets. Math Struct Comput Sci. 16, 359–372 (2006). Publisher Full Text OpenURL

  18. Kada, O, Suzuki, T, Takahashi, W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math Jpn. 44, 381–391 (1996)

  19. Caristi, J, Kirk, WA: Geometric fixed point theory and inwardness conditions. The Geometry of Metric and Linear Spaces, Lecture Notes in Mathematics, pp. 74–83. Springer, Berlin (1975).

  20. Ekeland, I: Nonconvex minimization problems. Bull Am Math Soc. 1, 443–474 (1979). Publisher Full Text OpenURL

  21. Takahashi, W: Existence theorems generalizing fixed point theorems for multivalued mappings. In: Théra MA, Baillon JB (eds.) Fixed Point Theory and Applications, Pitman Research Notes in Mathematics Series, vol. 252, pp. 397–406. Longman Sci. Tech., Harlow (1991)

  22. Park, S: On generalizations of the Ekeland-type variational principles. Nonlin Anal. 39, 881–889 (2000). Publisher Full Text OpenURL

  23. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math Stud. 63, 123–145 (1994)

  24. Oettli, W, Théra, M: Equivalents of Ekeland's principle. Bull Aust Math Soc. 48, 385–392 (1993). Publisher Full Text OpenURL

  25. Al-Homidan, S, Ansari, QH, Yao, JC: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlin Anal Theory Methods Appl. 69, 126–139 (2008). Publisher Full Text OpenURL

  26. Hussain, N, Shah, MH, Kutbi, MA: Coupled coincidence point theorems for nonlinear contractions in partially ordered quasi-metric spaces with a Q-function. Fixed Point Theory Appl. 2011, 21 Article ID 703938 (2011)

  27. Latif, A, Al-Mezel, SA: Fixed point results in quasimetric spaces. Fixed Point Theory Appl. 2011, 8 Article ID 178306 (2011)

    Article ID 178306

    BioMed Central Full Text OpenURL

  28. Boyd, DW, Wong, JSW: On nonlinear contractions. Proc Am Math Soc. 20, 458–464 (1969). Publisher Full Text OpenURL

  29. Suzuki, T, Takahashi, W: Fixed point theorems and characterizations of metric completeness. Topol Methods Nonlin Anal J Juliusz Schauder Center. 8, 371–382 (1996)

  30. Matkowski, J: Integrable solutions of functional equations. Diss Math. 127, 1–68 (1975)

  31. Altun, I, Sola, F, Simsek, H: Generalized contractions on partial metric spaces. Topol Appl. 157, 2778–2785 (2010). Publisher Full Text OpenURL