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International Journal of Analysis
Volume 2013 (2013), Article ID 726387, 9 pages
http://dx.doi.org/10.1155/2013/726387
Research Article

Cyclic Contractions and Fixed Point Results via Control Functions on Partial Metric Spaces

Hemant Kumar Nashine1 and Zoran Kadelburg2

1Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur, Chhattisgarh 492101, India
2Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia

Received 23 August 2012; Revised 31 October 2012; Accepted 14 November 2012

Academic Editor: Seenith Sivasundaram

Copyright © 2013 Hemant Kumar Nashine and Zoran Kadelburg. 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

Cyclic weaker-type contraction conditions involving a generalized control function (with two variables) are used for mappings on 0-complete partial metric spaces to obtain fixed point results, thus generalizing several known results. Various examples are presented showing how the obtained theorems can be used and that they are proper extensions of the known ones.

1. Introduction

The celebrated Banach contraction principle has been generalized in several directions and widely used to obtain various fixed point results, with applications in many branches of mathematics.

Cyclic representations and cyclic contractions were introduced by Kirk et al. [1] and further used by several authors to obtain various fixed point results. See, for example, papers [2–9]. Note that while a classical contraction has to be continuous, cyclic contractions might not be.

On the other hand, Matthews [10] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. In partial metric spaces, self-distance of an arbitrary point need not be equal to zero. Several authors obtained many useful fixed point results in these spaces–-we just mention [11–27]. Several results in ordered partial metric spaces have been obtained as well [28–36]. Some results for cyclic contractions in partial metric spaces have been very recently obtained in [37–41].

Khan et al. [42] addressed a new category of fixed point problems for a single self-map with the help of a control function which they called an altering distance function. This idea was further used in many papers, such as Choudhury [43] where generalized control functions were used. This approach has been very recently used in [44, 45] to obtain fixed point results in partial metric spaces.

In this paper, we extend these results further, considering cyclic weaker-type contraction conditions involving a generalized control function (with two variables) for mappings on -complete partial metric spaces (Romaguera [16]). We obtain fixed point theorems for such mappings, thus generalizing several known results. Various examples are presented showing how the obtained results can be used and that they are proper extensions of the known ones.

2. Preliminaries

In 2003, Kirk et al. introduced the following notion of cyclic representation.

Definition 1 (see [1]). Let be a nonempty set, , and let be a self-mapping. Then is a cyclic representation of with respect to if(a) are nonempty subsets of ;(b), , .

They proved the following fixed point result.

Theorem 2 (see [1]). Let be a complete metric space, , and let be a cyclic representation of with respect to . Suppose that satisfies the following condition: where and is a function, upper semicontinuous from the right and for . Then, has a fixed point .

In 2010, Păcurar and Rus introduced the following notion of cyclic weaker -contraction.

Definition 3 (see [2]). Let be a metric space, , and let be closed nonempty subsets of and . An operator is called a cyclic weaker -contraction if(1) is a cyclic representation of with respect to ;(2)there exists a continuous, nondecreasing function with for and such that for any , , where .

They proved the following result.

Theorem 4 (see [2]). Suppose that is a cyclic weaker -contraction on a complete metric space . Then, has a fixed point .

This was generalized by Karapınar in [3].

Khan et al. introduced the following notion.

Definition 5 (see [42]). A function is called an altering distance function if the following properties are satisfied: (a) is continuous and nondecreasing, (b).

Choudhury introduced a generalization of Chatterjea type contraction as follows.

Definition 6 (see [43]). A self-mapping , on a metric space , is said to be a weakly -contractive (or a weak Chatterjea type contraction) if for all , where is a continuous function such that

In [43], the author proved that every weak Chatterjea type contraction on a complete metric space has a unique fixed point.

The following definitions and details can be seen, for example, in [10, 12, 13, 15, 16].

Definition 7. A partial metric on a nonempty set is a function such that for all :  (p1) ,  (p2) ,  (p3) ,  (p4) . The pair is called a partial metric space.

It is clear that, if , then from and . But if , may not be .

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

A sequence in converges to a point (in the sense of ) if . This will be denoted as () or . Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function need not be continuous in the sense that and imply .

Example 8 (see [10]). (1) A paradigmatic example of a partial metric space is the pair , where for all .
(2) Let and let . Then is a partial metric space.

Definition 9. Let be a partial metric space. Then consider the following.(1) A sequence in is called a Cauchy sequence if exists (and is finite). The space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .(2)(see [16]) A sequence in is called -Cauchy if . The space is said to be -complete if every -Cauchy sequence in converges (in ) to a point such that .

Lemma 10. Let be a partial metric space. (a) (see [46, 47]) If as , then as for each . (b) (see [16]) If is complete, then it is -complete.

The converse assertion of (b) does not hold as the following easy example shows.

Example 11 (see [16]). The space with the partial metric is -complete, but is not complete. Moreover, the sequence with for each is a Cauchy sequence in , but it is not a -Cauchy sequence.

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

3. Main Results

In this section, we will prove some fixed point theorems for self-mappings defined on a -complete partial metric space and satisfying certain cyclic weak contractive condition involving a generalized control function. To achieve our goal, we introduce the new notion of a cyclic contraction.

Definition 12. Let be a partial metric space, , and let be nonempty subsets of and . An operator is called a cyclic contraction under weak contractive condition if: (NZ1) is a cyclic representation of with respect to ; (NZ2) for any (with ), where and is a lower semicontinuous mapping such that if and only if .

Our main result is the following.

Theorem 13. Let be a -complete partial metric space, let be nonempty closed subsets of , and let . Suppose that is a cyclic contraction as defined in Definition 12. Then, has a unique fixed point such that . Moreover, . Each Picard sequence , converges to in topology .

Proof. Let be an arbitrary point of . Then there exists some such that . Now and, similarly, for , where . In the case for some , it is clear that is a fixed point of .
Without loss of the generality, we may assume that From the condition (NZ1), we observe that for all , there exists such that .
Putting and in (NZ2) condition, we have By , we have Therefore, By (8) and (10), we have If , then from (11), we have which is a contradiction (it was used that since ). Hence, , and , which is excluded. Therefore, we have and hence By (13), we have that is a nonincreasing sequence of positive real numbers. Thus, there exists such that Passing to the limit as in (13) and using (14) and lower semicontinuity of , we have thus, and hence . Therefore
Next, we claim that is a -Cauchy sequence in the space . Suppose that this is not the case. Then there exists for which we can find two sequences of positive integers and such that for all positive integers Using (17) and (), we get Thus we have Passing to the limit as in the above inequality and using (16), we obtain On the other hand, for all , there exists such that . Then (for large enough, ) and lie in different adjacently labelled sets and for certain .
Using () and (20), we get that is, Using (16), we have Again, using (), we get Passing to the limit as in the pervious inequality, and using (24) and (22), we get Similarly, we have by () Passing to the limit as , and using (16) and (22), we obtain Similarly, we have by () Using (NZ2), we obtain for all . Passing to the limit as in the last inequality (and using the lower semicontinuity of the function ), we obtain which implies that ; that is a contradiction since . We deduce that is a -Cauchy sequence.
Since is -complete and is closed, it follows that the sequence converges to some , that is: We shall prove that From condition (NZ1), and since , we have . Since is closed, from (32), we get that . Again, from the condition (NZ1), we have . Since is closed, from (32), we get that . Continuing this process, we obtain (33) and .
Now, we shall prove that is a fixed point of . Indeed, from (33), since for all , there exists such that ; applying (NZ2) with and , we obtain for all . Passing to the limit as in (34), and using (32), we get which is impossible unless , so that is, is a fixed point of .
We claim that there is a unique fixed point of . Assume on the contrary that and with . By supposition, we can replace by and by in (NZ2) to obtain a contradiction. Hence , that is, . We conclude that has only one fixed point in . The proof is complete.

If we take and in Theorem 13, then we get the following fixed point theorem.

Corollary 14. Let be a -complete partial metric space, and let satisfy the following condition: there exists a lower semicontinuous mapping such that if and only if and that for all . Then has a unique fixed point . Moreover, .

Corollary 14 extends and generalizes many existing fixed point theorems in the literature.

By taking where in Theorem 13, we have the following result.

Corollary 15. Let be a -complete partial metric space, let , be nonempty closed subsets of , , and such that(NZ1) is a cyclic representation of with respect to ;(NZ3) for any (with ), where . Then has a unique fixed point belonging to ; moreover, .

As a special case of Corollary 15, we obtain Matthews’s version of Banach contraction principle [10].

Corollary 16. Let be a -complete partial metric space, let , be nonempty closed subsets of , , and such that(NZ1) is a cyclic representation of with respect to ;(NZ4) for any (with ), where is a positive integer and is a lower semi-continuous mapping such that if and only if . Then has a unique fixed point belonging to .

4. Examples

The following example shows how Theorem 13 can be used. It is adapted from [38, Example 2.9].

Example 17. Consider the partial metric space of Example 8 (2). It is easy to see that it is -complete. Consider the following closed subsets of : , and define a mapping by Obviously, is a cyclic representation of with respect to . We will show that satisfies the contractive condition (NZ2) of Definition 12 with the control function given by .
Let (the other possibility is treated similarly) and consider the following cases:(1), and , that is, . Then , and . Hence, the condition (NZ2) reduces to and holds true.(2), and . Then , , and ; hence (NZ2) reduces to .(3), and , that is, . Then , and . Hence, (NZ2) reduces to and holds true.(4), . Then , and . (NZ2) reduces to .(5)The case , is treated symmetrically.(6)The case is trivial.
We conclude that all conditions of Theorem 13 are satisfied. The mapping has a unique fixed point .

Here is another example showing the use of Theorem 13.

Example 18. Let be equipped with the partial metric given as Then, is a -complete partial metric space. Let , , and be given as Obviously, is a cyclic representation of with respect to . We will check the contractive condition (NZ2) of Definition 12 with the control function given by . Let (the other possibility is treated symmetrically). Consider the following possible cases.(1), . Then , . . It is easy to check that holds for the given values of and .(2), . Then , and . The condition (NZ2) reduces to and can be checked directly.
Thus, all the conditions of Theorem 13 are fulfilled, and we conclude that has a unique fixed point .

We state a more involved example that is inspired with the one from [48].

Example 19. Let , if and only if for each . Define a partial metric on by (it is easy to check that axioms ()–() hold true). Let be fixed, denote , and consider the subsets and of defined by , where Denote (obviously ).
Consider the mapping given by. Obviously, and ; hence is a cyclic representation of with respect to . Take defined by .
Let us check the contractive condition (NZ2) of Theorem 13. Take and and assume, for example, that (the case is treated similarly, as well as the case when or is equal to ). Then Hence, Obviously, has a unique fixed point .

Finally, we present an example showing that in certain situations the existence of a fixed point can be concluded under partial metric conditions, while the same cannot be obtained using the standard metric.

Example 20. Let be equipped with the usual partial metric . Suppose , , , and . Consider the mapping defined by It is clear that is a cyclic representation of with respect to . Further, consider the function given by Take an arbitrary pair with, say, (the other possibility can be treated in a similar way). Then On the other hand, Hence, condition (NZ2) is satisfied, as well as other conditions of Theorem 13 (with ). We deduce that has a unique fixed point .
On the other hand, consider the same problem in the standard metric and take and . Then and hence Thus, condition (NZ2) for does not hold and the existence of a fixed point of cannot be derived using the standard metric.

Remark 21. The results of this paper are obtained under the assumption that the given partial metric space is -complete. Taking into account Lemma 10 and Example 11, it follows that they also hold if the space is complete, but that our assumption is weaker.

Acknowledgments

The authors thank the referees for valuable suggestions that helped them to improve the text. The second author is thankful to the Ministry of Science and Technological Development of Serbia.

References

  1. W. A. Kirk, P. S. Srinivasan, and P. Veeramani, “Fixed points for mappings satisfying cyclical contractive conditions,” Fixed Point Theory, vol. 4, no. 1, pp. 79–89, 2003. View at Google Scholar · View at MathSciNet
  2. M. Păcurar and I. A. Rus, “Fixed point theory for cyclic φ-contractions,” Nonlinear Analysis, vol. 72, no. 3-4, pp. 1181–1187, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  3. E. Karapınar, “Fixed point theory for cyclic weak ϕ-contraction,” Applied Mathematics Letters, vol. 24, no. 6, pp. 822–825, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  4. E. Karapinar, M. Jleli, and B. Samet, “Fixed point results for almost generalized cyclic (Ψ, ϕ)-weak contractive type mappings with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 917831, 17 pages, 2012. View at Google Scholar · View at MathSciNet
  5. E. Karapinar and K. Sadarangani, “Fixed point theory for cyclic (φ-ψ)-contractions,” Fixed Point Theory and Applications, vol. 2011, article 69, 2011. View at Google Scholar · View at MathSciNet
  6. E. Karapinar and S. Moradi, “Fixed point theory for cyclic genetalized (ϕ, ϕ)-contraction mappings,” Annali Dell'Universita' di Ferrara. In press. View at Publisher · View at Google Scholar
  7. E. Karapinar and H. K. Nashine, “Fixed point theorem for cyclic weakly Chatterjea type contractions,” Journal of Applied Mathematics, vol. 2012, Article ID 165698, 15 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  8. E. Karapinar and H. K. Nashine, “Fixed point theorems for Kanaan type cyclic weakly contractions,” Nonlinear Analysis and Optimization. In press.
  9. H. K. Nashine, “Cyclic generalized ψ-weakly contractive mappings and fixed point results with applications to integral equations,” Nonlinear Analysis, vol. 75, no. 16, pp. 6160–6169, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. S. G. Matthews, “Partial metric topology,” Annals of the New York Academy of Sciences, vol. 728, pp. 183–197, 1994, Proceedings of the 8th Summer Conference on General Topology and Applications. View at Google Scholar · View at MathSciNet
  11. R. Heckmann, “Approximation of metric spaces by partial metric spaces,” Applied Categorical Structures, vol. 7, no. 1-2, pp. 71–83, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  12. 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 MathSciNet
  13. 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 MathSciNet
  14. 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 MathSciNet
  15. M. Bukatin, R. Kopperman, S. Matthews, and H. Pajoohesh, “Partial metric spaces,” American Mathematical Monthly, vol. 116, no. 8, pp. 708–718, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  16. 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 Publisher · View at Google Scholar · View at MathSciNet
  17. 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
  18. D. Ilić, V. Pavlović, and V. Rakočević, “Extensions of the Zamfirescu theorem to partial metric spaces,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 801–809, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. E. Karapinar, “Generalizations of Caristi Kirk's theorem on partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 4, 2011. View at Google Scholar · View at MathSciNet
  20. 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. 2010, article 113, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  21. K. P. Chi, E. Karapınar, and T. D. Thanh, “A generalized contraction principle in partial metric spaces,” Mathematical and Computer Modelling, vol. 55, no. 5-6, pp. 1673–1681, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  22. L. 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
  23. 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
  24. 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 MathSciNet
  25. 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
  26. 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 MathSciNet
  27. H. K. Nashine and E. Karapinar, “Fixed point results in orbitally complete partial metric spaces,” Bulletin of the Malaysian Mathematical Sciences Society. In press.
  28. H. Aydi, “Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces,” Journal of Nonlinear Analysis and Optimization, vol. 2, no. 2, pp. 269–284, 2011. View at Google Scholar · View at MathSciNet
  29. H. Aydi, “Common fixed point results for mappings satusfying (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 · View at MathSciNet
  30. 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
  31. 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 MathSciNet
  32. 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 MathSciNet
  33. B. Samet, M. Rajović, R. Lazović, and R. Stoiljković, “Common fixed point results for nonlinear contractions in ordered partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 71, 2011. View at Publisher · View at Google Scholar
  34. 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
  35. H. K. Nashine, Z. Kadelburg, S. Radenović, and J. K. Kim, “Fixed point theorems under Hardy-Rogers weak contractive conditions on 0-complete ordered partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 180, 2012. View at Google Scholar
  36. 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 MathSciNet
  37. C. Di Bari and P. Vetro, “Fixed points for weak ϕ-contractions on partial metric spaces,” International Journal of Contemporary Mathematical Sciences, vol. 1, pp. 5–12, 2011. View at Google Scholar
  38. 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, Físicas y Naturales A, vol. 106, pp. 287–297, 2012. View at Google Scholar
  39. R. P. Agarwal, M. A. Alghamdi, and N. Shahzad, “Fixed point theory for cyclic generalized contractions in partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 40, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  40. E. Karapınar and I. S. Yuce, “Fixed point theory for cyclic generalized weak φ-contraction on partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 491542, 12 pages, 2012. View at Google Scholar · View at MathSciNet
  41. E. Karapinar, N. Shobkolaei, S. Sedghi, and S. M. Vaezpour, “A common fixed point theorem for cyclic operators in partial metric spaces,” Filomat, vol. 26, pp. 407–414, 2012. View at Google Scholar
  42. M. S. Khan, M. Swaleh, and S. Sessa, “Fixed point theorems by altering distances between the points,” Bulletin of the Australian Mathematical Society, vol. 30, no. 1, pp. 1–9, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  43. B. S. Choudhury, “Unique fixed point theorem for weak C-contractive mappings,” Kathmandu University Journal of Science, Engineering and Technology, vol. 5, pp. 6–13, 2009. View at Google Scholar
  44. A. G. B. Ahmad, Z. M. Fadail, H. K. Nashine, Z. Kadelburg, and S. Radenović, “Some new common fixed point results through generalized altering distances on partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 120, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  45. 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.
  46. T. Abdeljawad, E. Karapınar, and K. Taş, “Existence and uniqueness of a common fixed point on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1900–1904, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  47. E. Karapınar and I. 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
  48. M. A. Petric, “Best proximity point theorems for weak cyclic Kannan contractions,” Filomat, vol. 25, no. 1, pp. 145–154, 2011. View at Publisher · View at Google Scholar