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International Journal of Analysis
Volume 2013 (2013), Article ID 428561, 8 pages
http://dx.doi.org/10.1155/2013/428561
Research Article

Common Fixed Points in a Partially Ordered Partial Metric Space

Dipartimento di Matematica e Informatica, Università Degli Studi di Palermo, Via Archirafi, 34-90123 Palermo, Italy

Received 29 August 2012; Revised 12 September 2012; Accepted 17 September 2012

Academic Editor: Harumi Hattori

Copyright © 2013 Daniela Paesano and Pasquale Vetro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the first part of this paper, we prove some generalized versions of the result of Matthews in (Matthews, 1994) using different types of conditions in partially ordered partial metric spaces for dominated self-mappings or in partial metric spaces for self-mappings. In the second part, using our results, we deduce a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends the Subrahmanyam characterization of metric completeness.

1. Introduction

In the mathematical field of domain theory, attempts were made in order to equip semantics domain with a notion of distance. In particular, Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of data for networks, showing that the contraction mapping principle can be generalized to the partial metric context for applications in program verification. Moreover, the existence of several connections between partial metrics and topological aspects of domain theory has been lately pointed by other authors as O'Neill [2], Bukatin and Scott [3], Bukatin and Shorina [4], Romaguera and Schellekens [5], and others (see also [614] and the references therein).

After the result of Matthews [1], the interest for fixed point theory developments in partial metric spaces has been constantly growing, and many authors presented significant contributions in the directions of establishing partial metric versions of well-known fixed point theorems for the existence of fixed points, common fixed points, and coupled fixed points in classical metric spaces (see e.g., [15, 16]). Obviously, we cannot cite all these papers but we give only a partial list [1749].

Recently, Romaguera [50] proved that a partial metric space is 0-complete if and only if every -Caristi mapping on has a fixed point. In particular, the result of Romaguera extended Kirk’s [51] characterization of metric completeness to a kind of complete partial metric spaces. Successively, Karapinar in [36] extended the result of Caristi and Kirk [52] to partial metric spaces.

In the first part of this paper, following this research direction, we prove some generalized versions of the result of Matthews by using different types of conditions in ordered partial metric spaces for dominated self-mappings or in partial metric spaces for self-mappings. The notion of dominated mapping of economics, finance, trade, and industry is also applied to approximate the unique solution of nonlinear functional equations. In the second part, using the results obtained in the first part, we deduce a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends the Subrahmanyam [53] characterization of metric completeness. For other characterizations of metric completeness in terms of fixed point theory, the reader can see, for example, [54, 55] and for partial metric completeness, [41].

2. Preliminaries

First, we recall some definitions and some properties of partial metric spaces that can be found in [1, 2, 40, 48, 50]. A partial metric on a nonempty set is a function such that for all , , , .

A partial metric space is a pair such that is a nonempty set and is a partial metric on . It is clear that if , then from and , it follows that . But if , may not be . A basic example of a partial metric space is the pair , where for all . Other examples of partial metric spaces which are interesting from a computational point of view can be found in [1].

Each partial metric on generates a topology on which has as a base the family of open -balls , where for all and .

If is a partial metric on , then the function given by is a metric on .

Let be a partial metric space. A sequence in converges to a point if and only if .

A sequence in is called a Cauchy sequence if there exists (and is finite) .

A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

A sequence in is called 0-Cauchy if . We say that is 0-complete if every 0-Cauchy sequence in converges, with respect to , to a point such that .

On the other hand, the partial metric space , where denotes the set of rational numbers and the partial metric is given by , provides an example of a 0-complete partial metric space which is not complete.

It is easy to see that every closed subset of a complete partial metric space is complete.

Lemma 1 (see [1, 40]). Let be a partial metric space. Then(a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space . (b) A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if

The following lemma is obvious.

Lemma 2. Let be a partial metric space and . If and , then for all .

Define . Then , where denotes the closure of (for details see [22, Lemma 1]). From for every , we deduce that .

Let be a nonempty set and . The mappings are said to be weakly compatible if they commute at their coincidence points (i.e., whenever ). A point is called a point of coincidence of and if there exists a point such that .

Lemma 3 (see [56]). Let be a nonempty set and the mappings have a unique pointof coincidence inIf   and are weakly compatible, then and have a unique common fixed point.

Let be a nonempty set. If is a partial metric space and is a partially ordered set, then is called a partially ordered partial metric space. are called comparable if or holds. Let be a partially ordered set and two mappings. is called an -dominated mapping if for every .

3. Main Results

Let be a partial metric space and be such that . For every we consider the sequence defined by for all and we say that is a --sequence of the initial point (see [57]).

Denote with the family of non-decreasing functions such that and for each , where is the th iterate of .

Lemma 4. For every function , the following holds, if for each , then .

The following theorem is one of our main results, and it ensures the existence of a common fixed point for two self-mappings in the setting of partially ordered partial metric spaces.

Theorem 5. Let be a partially ordered partial metric space and two mappings such that . Assume that there exists such that for all with and comparable. If the following conditions hold: (i) is a -dominated mapping, (ii)either or is a 0-complete subspace of , (iii)for a non-increasing sequence converging to , we have for all and ,then and have a point of coincidence. Moreover, if and are weakly compatible, then and have a common fixed point.

Proof. Let be fixed and be a --sequence of the initial point . As for all , then the sequence is non-increasing.
If for some , then is a point of coincidence of and . Suppose that for all . Since and are comparable for all , we have
If from we obtain a contradiction and so Then, we have and hence Fix and we choose such that for all . Let and we show that
Clearly, (12) is true for . Suppose that (12) holds for some , as and are comparable for all , then
This implies that (12) holds for and by induction, it holds for all . From (12), we deduce that there exists and hence is a 0-Cauchy sequence.
Suppose that is a 0-complete subspace of , then there exists such that This holds also if is 0-complete with .
Let be such that . We show that is a point of coincidence of and . If not, we have . This implies that there exists such that for every . By condition (iii), and are comparable for every and hence, by condition (5) with and , we deduce that for every . Letting in the previous inequality and using Lemma 2, we obtain which implies , that is, . Thus, we have shown that is a point of coincidence of and . If and are weakly compatible, then . By condition (iii), , that is, and are comparable. Using the contractive condition (5), we get which implies and hence is a common fixed point of and .

We shall give a sufficient condition for the uniqueness of the common fixed point in Theorem 5.

Theorem 6. Let all the conditions of Theorem 5 be satisfied. If the following condition holds:(iv) for all there exists such that and , where is the --sequence of the initial point , then and have a unique common fixed point.

Proof. Let be two common fixed points of and with . If and are comparable, then using the contractive condition (5), we deduce that . If and are not comparable, then there exists such that , . As is a -dominated mapping, we get that To continue, we obtain and hence and are comparable.
Using the contractive condition (5) with and , we get for all . Since, the contractive condition (5) ensures that , we have Now, by condition (iv), and hence for sufficiently large, we have Without loss of generality, assuming that (24) holds for all , it follows that Now, letting in (25), we obtain
With similar arguments, we deduce that . Hence as , which is a contradiction. Thus and have a unique common fixed point.

As a consequence of Theorem 5, we state the following result.

Theorem 7. Let be a partial metric space and two mappings such that . Assume that there exists such that for all . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. Proceeding as in the proof of Theorem 5, we get that and have a unique point of coincidence and, by Lemma 3, and have a unique common fixed point.

From Theorem 7, we can deduce the following corollaries.

Corollary 8 (Banach type). Let be a partial metric space and two mappings such that . Assume that for all , where . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.

Corollary 9 (Bianchini type). Let be a partial metric space and let be two mappings such that . Assume that for all , where . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.

Corollary 10 (Reich type [58]). Let be a partial metric space and let be two mappings such that . Assume that for all , where and . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.

The following example shows that there exist mappings that satisfy the contractive condition (28), but are not quasi-contractions [59].

Example 11. Consider the set and the function given by , and . Obviously, is a partial metric on , but it is not a metric (since for and ). Clearly, is a 0-complete partial metric space. Let be defined by and for every . Take for every .
First, we will check that and satisfy the contractive condition (28). If , then and (28) trivially holds. Let, for example, , then we have the following three cases:
Thus, all the conditions of Theorem 7 are satisfied and the existence of a common fixed point of and (which is ) follows. The same conclusion cannot be obtained by the main results from [59]. Indeed, using , and then taking instead of , in (5), we obtain Since , the conclusion follows.

The following example shows that there exist mappings that satisfy the contractive condition (5), but do not satisfy the contractive condition (28).

Example 12. Let be endowed with the partial metric Clearly, is a 0-complete partial metric space. Let be defined by and for each . As and have many common fixed points (each is a common fixed point), then it is immediate to show that and do not satisfy the contractive condition (28).
If is ordered by then and satisfy the contractive condition (5) where is defined by Using Theorem 5, we deduce that and have a common fixed point.

4. Completeness in Partial Metric Spaces and Fixed Points

In this section, we characterize those partial metric spaces for which every Bianchini mapping has a fixed point in the style of Subrahmanyam characterization of metric completeness. This will be done by means of the notion of 0-completeness which was introduced by Romaguera in [50].

Let be a partial metric space and be a mapping. We recall that is a Bianchini [60] mapping if for all , where .

Theorem 13. Let be a partial metric space. If every mapping satisfying the following conditions: (i) for all , for a fixed , (ii) TX is countable has a fixed point, then is 0-complete.

Proof. Suppose that there is a 0-Cauchy sequence of distinct points in which is not convergent in . We put and we note that for every , we have .
Now, implies that . Since is a 0-Cauchy sequence in , there exists a least positive integer such that
In particular
For fixed , since and so , there is such that
Now, let be defined by From the definition of , we deduce that satisfies the condition (ii). On the other hand, satisfies also the condition (i). In fact, (i) is verified by assuming and , and noting that It is clear that has not fixed points since , . Thus, the assumption that there is a 0-Cauchy sequence which is not convergent in leads to a contradiction to Theorem 13 and thereby establishes the same.

If in Theorem 13 we choose , by Corollary 9, we obtain the following characterization of 0-completeness for partial metric spaces.

Theorem 14. A partial metric space is 0-complete if and only if every mapping satisfying the following conditions: (i) for all , for a fixed , (ii) TX is countable has a fixed point.

In Theorem 14, the class of mappings satisfying (i) and (ii) can be replaced by the class of mappings satisfying (ii) and the following condition:(i).

Acknowledgment

The authors are grateful to the editor and referees for their valuable suggestions and critical remarks for improving this paper. P. Vetro is supported by Università degli Studi di Palermo (Local University Project R.S. ex 60%).

References

  1. S. G. Matthews, “Partial metric topology,” Annals of the New York Academy of Sciences, vol. 728, pp. 183–197, 1994, Papers on General Topology and Applications. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. S. J. O'Neill, “Partial metrics, valuations and domain theory,” Annals of the New York Academy of Sciences, vol. 806, pp. 304–315, 1996, Papers on General Topology and Applications. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. M. A. Bukatin and J. S. Scott, “Towards computing distances between programs via Scott domains,” in Logical Foundations of Computer Science, S. Adian, A. Nerode et al., Eds., vol. 1234 of Lecture Notes in Computer Science, pp. 33–43, Springer, Berlin, Germany, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  4. M. A. Bukatin and S. Y. Shorina, “Partial metrics and co-continuous valuations,” in Foundations of Software Science and Computation Structures, M. Nivat, Ed., vol. 1378 of Lecture Notes in Computer Science, pp. 125–139, Springer, Berlin, Germany, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  5. S. Romaguera and M. Schellekens, “Partial metric monoids and semivaluation spaces,” Topology and Its Applications, vol. 153, no. 5-6, pp. 948–962, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. R. Heckmann, “Approximation of metric spaces by partial metric spaces,” Applied Categorical Structures, vol. 7, no. 1-2, pp. 71–83, 1999. View at Google Scholar · View at MathSciNet · View at Scopus
  7. R. D. Kopperman, S. G. Matthews, and H. Pajoohesh, “What do partial metrics represent?” in Proceedings of the 19th Summer Conference on Topology and Its Applications, University of CapeTown, 2004.
  8. H. P. A. Künzi, H. Pajoohesh, and M. P. Schellekens, “Partial quasi-metrics,” Theoretical Computer Science, vol. 365, no. 3, pp. 237–246, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. S. Romaguera and M. Schellekens, “Duality and quasi-normability for complexity spaces,” Applied General Topology, vol. 3, no. 1, pp. 91–112, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Romaguera and O. Valero, “A quantitative computational model for complete partial metric spaces via formal balls,” Mathematical Structures in Computer Science, vol. 19, no. 3, pp. 541–563, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. M. P. Schellekens, “A characterization of partial metrizability: domains are quantifiable,” Theoretical Computer Science, vol. 305, no. 1–3, pp. 409–432, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. M. P. Schellekens, “The correspondence between partial metrics and semivaluations,” Theoretical Computer Science, vol. 315, no. 1, pp. 135–149, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. P. Waszkiewicz, “Partial metrisability of continuous posets,” Mathematical Structures in Computer Science, vol. 16, no. 2, pp. 359–372, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. P. Waszkiewicz, “Quantitative continuous domains,” Applied Categorical Structures, vol. 11, no. 1, pp. 41–67, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. V. Berinde and F. Vetro, “Common fixed points of mappings satisfying implicit contractive conditions,” Fixed Point Theory and Applications, vol. 2012, article 105, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  16. L. J. Ćirić, “On contraction type mappings,” Mathematica Balkanica, vol. 1, pp. 52–57, 1971. View at Google Scholar · View at MathSciNet
  17. M. Abbas, T. Nazir, and S. Romaguera, “Fixed point results for generalized cyclic contraction mappings in partial metric spaces,” Revista de la Real Academia de Ciencias Exactas, vol. 106, pp. 287–297, 2012. View at Google Scholar · View at MathSciNet
  18. T. Abdeljawad, “Fixed points for generalized weakly contractive mappings in partial metric spaces,” Mathematical and Computer Modelling, vol. 54, pp. 2923–2927, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. T. Abdeljawad, E. Karapinar, and K. Tas, “Existence and uniqueness of common fixed point on partial metric spaces,” Applied Mathematics Letters, vol. 24, pp. 1894–1899, 2011. View at Google Scholar · View at MathSciNet
  20. T. Abdeljawad, E. Karapinar, and K. Tas, “A generalized contraction principle with control functions on partial metric spaces,” Computers & Mathematics with Applications, vol. 63, pp. 716–719, 2012. View at Google Scholar · View at MathSciNet
  21. I. Altun and A. Erduran, “Fixed point theorems for monotone mappings on partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 508730, 10 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. I. Altun, F. Sola, and H. Simsek, “Generalized contractions on partial metric spaces,” Topology and Its Applications, vol. 157, no. 18, pp. 2778–2785, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. H. Aydi, “Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces,” Journal of Nonlinear Analysis and Optimization: Theory and Applications, vol. 2, pp. 33–48, 2011. View at Google Scholar · View at MathSciNet
  24. H. Aydi, “Common fixed point results for mappings satisfying (ψ, ϕ)-weak contractions in ordered partial metric spaces,” International Journal of Mathematics and Statistics, vol. 12, pp. 53–64, 2012. View at Google Scholar
  25. H. Aydi, E. Karapınar, and W. Shatanawi, “Coupled fixed point results for (ψ,φ)-weakly contractive condition in ordered partial metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4449–4460, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  26. H. Aydi, M. Abbas, and C. Vetro, “Partial Hausdorff metric and Nadler's fixed point theorem on partial metric spaces,” Topology and Its Applications, vol. 159, no. 14, pp. 3234–3242, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  27. H. Aydi, “A common fixed point result by altering distances involving a contractive condition of integral type in partial metric spaces,” Demonstratio Mathematica. In press. View at Google Scholar
  28. K. P. Chi, E. Karapinar, and T. D. Thanh, “A generalized contraction principle in partial metric spaces,” Mathematical and Computer Modelling, vol. 55, pp. 1673–1681, 2012. View at Google Scholar · View at MathSciNet
  29. L. J. Ćirić, B. Samet, H. Aydi, and C. Vetro, “Common fixed points of generalized contractions on partial metric spaces and an application,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2398–2406, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  30. C. Di Bari and P. Vetro, “Fixed points for weak φ-contractions on partial metric spaces,” International Journal of Engineering, Contemporary Mathematics and Sciences, vol. 1, pp. 5–13, 2011. View at Google Scholar
  31. C. Di Bari, Z. Kadelburg, H. K. Nashine, and S. Radenović, “Common fixed points of g-quasicontractions and related mappings in 0-complete partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 113, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  32. D. Ilić, V. Pavlović, and V. Rakočević, “Some new extensions of Banach's contraction principle to partial metric space,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1326–1330, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. D. Ilić, V. Pavlović, and V. Rakočević, “Extensions of the Zamfirescu theorem to partial metric spaces,” Mathematical and Computer Modelling, vol. 55, pp. 801–809, 2012. View at Google Scholar · View at MathSciNet
  34. E. Karapinar, “Weak ϕ-contraction on partial metric spaces,” Journal of Computational Analysis and Applications, vol. 14, no. 2, pp. 206–210, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  35. E. Karapinar, “A note on common fixed point theorems in partial metric spaces,” Miskolc Mathematical Notes, vol. 12, no. 2, pp. 185–191, 2011. View at Google Scholar · View at MathSciNet
  36. E. Karapinar, “Generalizations of Caristi Kirk's theorem on partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 4, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  37. E. Karapınar and M. Erhan, “Fixed point theorems for operators on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1894–1899, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  38. E. Karapınar and U. Yüksel, “Some common fixed point theorems in partial metric spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 263621, 16 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. H. K. Nashine, Z. Kadelburg, and S. Radenović, “Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces,” Mathematical and Computer Modelling. In press. View at Publisher · View at Google Scholar
  40. S. Oltra and O. Valero, “Banach's fixed point theorem for partial metric spaces,” Rendiconti dell'Istituto di Matematica dell'Università di Trieste, vol. 36, no. 1-2, pp. 17–26, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. D. Paesano and P. Vetro, “Suzuki's type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces,” Topology and Its Applications, vol. 159, no. 3, pp. 911–920, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. S. Romaguera, “Fixed point theorems for generalized contractions on partial metric spaces,” Topology and Its Applications, vol. 159, no. 1, pp. 194–199, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. S. Romaguera, “Matkowski's type theorems for generalized contractions on (ordered) partial metric spaces,” Applied General Topology, vol. 12, no. 2, pp. 213–220, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. B. Samet, “Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces,” Nonlinear Analysis, vol. 72, no. 12, pp. 4508–4517, 2010. View at Publisher · View at Google Scholar · View at Scopus
  45. B. Samet, M. Rajovic, R. Lazovic, and R. Stoiljkovic, “Common fixed point results for nonlinear contractions in ordered partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 71, 2011. View at Google Scholar
  46. W. Shatanawi, B. Samet, and M. Abbas, “Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces,” Mathematical and Computer Modelling, vol. 55, pp. 680–687, 2012. View at Google Scholar
  47. N. Shobkolaei, S. M. Vaezpour, and S. Sedghi, “A common fixed point theorem on ordered partial metric spaces,” Journal of Basic and Applied Scientific Research, vol. 1, pp. 3433–3439, 2011. View at Google Scholar
  48. O. Valero, “On Banach fixed point theorems for partial metric spaces,” Applied General Topology, vol. 6, no. 2, pp. 229–240, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. F. Vetro and S. Radenović, “Nonlinear ψ-quasi-contractions of Ćirić-type in partial metric spaces,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1594–1600, 2012. View at Google Scholar
  50. S. Romaguera, “A Kirk type characterization of completeness for partial metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 493298, 6 pages, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. W. A. Kirk, “Caristi's fixed point theorem and metric convexity,” Colloquium Mathematicum, vol. 36, no. 1, pp. 81–86, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  52. J. Caristi and W. A. Kirk, “Geometric fixed point theory and inwardness conditions,” in The Geometry of Metric and Linear Spaces, vol. 490 of Lecture Notes in Mathematics, pp. 74–83, Springer, Berlin, Germany, 1975. View at Google Scholar · View at MathSciNet
  53. P. V. Subrahmanyam, “Completeness and fixed-points,” Monatshefte für Mathematik, vol. 80, no. 4, pp. 325–330, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  54. S. Park, “Characterizations of metric completeness,” Colloquium Mathematicum, vol. 49, no. 1, pp. 21–26, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  55. T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  56. M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 416–420, 2008. View at Publisher · View at Google Scholar · View at Scopus
  57. P. Vetro, “Common fixed points in cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 56, no. 3, pp. 464–468, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  58. S. Reich, “Some remarks concerning contraction mappings,” Canadian Mathematical Bulletin, vol. 14, pp. 121–124, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  59. L. J. Ćirić, “A generalization of Banach's contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974. View at Google Scholar
  60. R. M. T. Bianchini, “Su un problema di S. Reich riguardante la teoria dei punti fissi,” Bollettino dell'Unione Matematica Italiana, vol. 5, pp. 103–108, 1972. View at Google Scholar