Abstract
We prove some coincidence and common fixed point results for three mappings satisfying a generalized weak contractive condition in ordered partial metric spaces. As application of the presented results, we give a unique fixed point result for a mapping satisfying a weak cyclical contractive condition. We also provide some illustrative examples.
MSC:47H10, 54H25.
Similar content being viewed by others
1 Introduction and preliminaries
In the last decades, several authors have worked on domain theory in order to equip semantics domain with a notion of distance. In 1994, Matthews [29] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks and showed that the Banach contraction principle [16] can be generalized to the partial metric context for applications in program verification. Later on, many researchers studied fixed point theorems in partial metric spaces as well as ordered partial metric spaces. For more details, see [5, 6, 9–15, 19, 20, 33, 34, 36].
Recently, there have been so many exciting developments in the field of existence of fixed points in partially ordered sets. For instance, Ran and Reurings [38] extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. For more details on fixed point theory in partially ordered sets, we refer the reader to [1–4, 7, 8, 17, 18, 24, 28, 30–32, 39, 41] and the references cited therein.
In this paper, we establish some coincidence and common fixed point results for three self-mappings on an ordered partial metric space satisfying a generalized weak contractive condition. The presented theorems extend some recent results in the literature. Moreover, as application, we give a unique fixed point theorem for a mapping satisfying a weak cyclical contractive condition.
Throughout this paper, will denote the set of all non-negative real numbers. First, we start by recalling some known definitions and properties of partial metric spaces.
Definition 1.1 ([29])
A partial metric on a nonempty set X is a function such that for all :
(p1) ,
(p2) ,
(p3) ,
(p4) .
A partial metric space is a pair such that X is a nonempty set and p is a partial metric on X.
It is clear that, if , then from (p1) and (p2), ; but if , may not be 0. 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 may be found in [22, 29].
Each partial metric p on X generates a topology on X which has as a base the family of open p-balls , where for all and .
If p is a partial metric on X, then the function given by
is a metric on X.
Definition 1.2 ([29])
Let be a sequence in X. Then
-
(i)
converges to a point if and only if . We may write this as .
-
(ii)
is called a Cauchy sequence if exists and is finite.
-
(iii)
is said to be complete if every Cauchy sequence in X converges, with respect to , to a point , such that .
Lemma 1.3 ([29])
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
Definition 1.4 ([5])
Let be a partial metric space and be a given mapping. We say that T is continuous at , if for every , there exists such that .
Lemma 1.5 (Sequential characterization of continuity)
Let be a partial metric space and be a given mapping. is continuous at if it is sequentially continuous at , that is, if and only if
Let X be a nonempty set and be a given mapping. For every , we denote by the subset of X defined by
Definition 1.6 Let X be a nonempty set. Then is called an ordered partial metric space if and only if
-
(i)
is a partial metric space,
-
(ii)
is a partially ordered set.
Definition 1.7 Let be a partially ordered set. Then are called comparable if or holds.
Definition 1.8 ([30])
Let be a partially ordered set and be given mappings such that and . We say that S and T are weakly increasing with respect to R if and only if, for all , we have
and
Remark 1.9 If is the identity mapping ( for all , shortly ), then the fact that S and T are weakly increasing with respect to R implies that S and T are weakly increasing mappings, that is, and for all . Finally, a mapping is weakly increasing if and only if for all .
Example 1.10 Consider endowed with the usual ordering of real numbers and define by
Now, and , then S and T are weakly increasing with respect to R.
Definition 1.11 Let be an ordered partial metric space. We say that X is regular if and only if the following hypothesis holds: is a non-decreasing sequence in X with respect to ⪯ such that as , then for all .
Finally, we recall the following definition of partial-compatibility introduced by Samet et al. [40].
Definition 1.12 Let be a partial metric space and be given mappings. We say that the pair is partial-compatible if the following conditions hold:
(b1) implies that .
(b2) , whenever is a sequence in X such that and for some .
Note that Definition 1.12 extends and generalizes the notion of compatibility introduced by Jungck [25].
2 Main results
We start this section with some auxiliary results (see also [37]).
Lemma 2.1 Let be a metric space and let be a sequence in X such that is non-increasing and
If is not a Cauchy sequence, then there exist and two sequences and of positive integers such that and the following four sequences tend to ε when :
As a corollary, applying Lemma 2.1 to the associated metric of a partial metric p, and using Lemma 1.3, we obtain the following lemma (see also [21]).
Lemma 2.2 Let be a partial metric space and let be a sequence in X such that is non-increasing and
If is not a Cauchy sequence, then there exist and two sequences and of positive integers such that and the following four sequences tend to ε when :
In the sequel, let Ψ be the set of functions such that ψ is continuous, strictly increasing and if and only if . Also, let Φ be the set of functions such that φ is lower semi-continuous and if and only if . Such ψ and φ are called control functions.
Our first main result is the following.
Theorem 2.3 Let be a partially ordered set. Suppose that there exists a partial metric p on X such that the partial metric space is complete. Let be given mappings satisfying
-
(a)
T, S and R are continuous,
-
(b)
the pairs and are partial-compatible,
-
(c)
T and S are weakly increasing with respect to R.
Suppose that for every such that Rx and Ry are comparable, we have
where and . Then T, S and R have a coincidence point , that is, .
Proof By Definition 1.8, it follows that . Let be an arbitrary point in X. Since , there exists such that . Since , there exists such that . Continuing this process, we can construct a sequence in X defined by
By construction, we have and . Then using the fact that S and T are weakly increasing with respect to R, we obtain
We continue this process to get
We claim that is a Cauchy sequence in the partial metric space . To this aim, we distinguish the following two cases.
Case 1. We suppose that there exists such that , so that . By (2.3), applying (2.1) with and , we get
Since ψ is strictly increasing, we have
This implies that . Continuing this process, we obtain for all . This implies that , therefore is Cauchy in . The same conclusion holds if for some .
Case 2. Now, we suppose that
Here, we have for all . Thanks to (2.3), and are comparable, then using (2.2) and taking and in (2.1), we get
Since ψ is strictly increasing, the above inequality implies that
Now, taking and in (2.1), we have
which implies that
Combining (2.5) and (2.7), we get
It follows that the sequence is non-increasing and bounded below by 0. Hence, there exists such that
We claim that . Suppose that . Taking the lim sup as in (2.6) and using the properties of the functions ψ and φ, we have
This implies that , and by a property of the function φ, we have , that is a contradiction. We deduce that , i.e.,
We shall show that is a Cauchy sequence in the partial metric space . For this, it is sufficient to prove that is Cauchy in . Suppose to the contrary that is not a Cauchy sequence. Then, having in mind that is non-increasing and (2.9), it follows by Lemma 2.2 that there exist and two sequences and of positive integers such that and the following four sequences tend to ε when :
Applying (2.1) with and , we get
Taking in the above inequality and using the continuity of ψ and the lower semi-continuity of φ, we obtain
from which a contradiction follows since . Then, we deduce that is a Cauchy sequence in the partial metric space , which is complete, so converges to some , that is, from (p3) and Definition 1.2,
But from (2.9) and condition (p2), we have
therefore, it follows that
From (2.11) and the continuity of R, we get
The triangular inequality yields
By (2.2) and (2.11), we have
Having in mind that the pair is partial-compatible, then
Also, since , then we have . The continuity of T together with (2.11) give us
Combining (2.12) and (2.15) together with (2.16) and letting in (2.13), we obtain
By condition (p2) and (2.17), one can write
Similarly, by triangular inequality, we get
By (2.2) and (2.11), we have
Since the pair is partial-compatible, then
Also, since , it follows . Thus, from (2.18), and so .
The continuity of S and (2.20) give us
Combining (2.12) and (2.21) together with (2.22) and letting in (2.19), we obtain
By condition (p2) and (2.23), we get
Applying (2.1) with , we get
This implies that
and so it follows , that is . Thus, we have obtained
that is, u is a coincidence point of T, S and R. □
Remark 2.4 We point out that the order in which the mappings in condition (b) of Theorem 2.3 are considered is crucial. Trivially, Theorem 2.3 remains true if we assume that the partial-compatible pairs are and .
Example 2.5 Let be endowed with the partial metric and the order given as follows:
Consider the mappings defined by and for all . Also, define the functions by and , for all . Clearly, condition (2.1) is satisfied. In fact, for every with , we get
All the other hypotheses of Theorem 2.3 are satisfied and T, S and R have a coincidence point . (Moreover, is the unique common fixed point of T, S and R.)
Note that Theorem 2.3 is not applicable in respect of the usual order of real numbers because T is not weakly increasing. It follows that the partial order may be fundamental.
Under different hypotheses, the conclusion of Theorem 2.3 remains true without assuming the continuity of T, S and R, and the partial-compatibility of the pairs and . This is the purpose of the next theorem.
Theorem 2.6 Let be a partially ordered set. Suppose that there exists a partial metric p on X such that is complete. Let be given mappings satisfying
-
(a)
RX is a closed subspace of ,
-
(b)
T and S are weakly increasing with respect to R,
-
(c)
X is regular.
Suppose that for every such that Rx and Ry are comparable, we have
where and . Then, T, S and R have a coincidence point , that is, .
Proof Following the proof of Theorem 2.3, we have that is a Cauchy sequence in the closed subspace RX, then there exists , with , such that
Thanks to (2.3), is a non-decreasing sequence, and so, since it converges to , from the regularity of X, we get
Therefore, and Ru are comparable. Putting and in (2.25) and using (2.2), we get
Taking in the above inequality, using (2.26) and the properties of φ and ψ, we obtain
This implies that
which is true if . This means that .
Analogously, putting and in (2.25), we have
Taking in the above inequality, using (2.26) and the properties of φ and ψ, we obtain
which yields that
We conclude that u is a coincidence point of T, S and R. □
If is the identity mapping , by Theorem 2.6, we obtain the following common fixed point result involving two mappings.
Corollary 2.7 Let be a partially ordered set. Suppose that there exists a partial metric p on X such that the partial metric space is complete. Let X be regular and be given mappings such that T and S are weakly increasing. Suppose that for every such that x and y are comparable, we have
where and . Then, T and S have a common fixed point , that is, .
The following example shows that the hypothesis ‘T and S are weakly increasing (with respect to R)’ has a key role for the validity of our results.
Example 2.8 Let be endowed with the partial metric and the order ⪯ given as follows:
Consider the mappings defined by and , for all . Also, define the functions by and , for all . It is easy to show that and , for all , that is, T and S are weakly increasing. Now, take x and y comparable and, without loss of generality, assume , so that . It is easy to show that (2.27) holds and all the other hypotheses of Corollary 2.7 are satisfied. Then, T and S have a unique common fixed point .
Note that Corollary 2.7 is not applicable in respect of the usual order of real numbers because T and S are not weakly increasing.
Now, we shall prove the existence and uniqueness of a common fixed point for three mappings.
Theorem 2.9 In addition to the hypotheses of Theorem 2.3, suppose that for any , there exists such that and . Then, T, S and R have a unique common fixed point, that is, there exists a unique such that .
Proof Referring to Theorem 2.3, the set of coincidence points of T, S and R is nonempty. Now, we shall show that if and are coincidence points of T, S and R, that is, and , then
For the coincidence points and , Theorem 2.3 gives us that
By assumption, there exists such that
Now, proceeding similarly to the proof of Theorem 2.3, we can immediately define a sequence as follows:
Since T and S are weakly increasing with respect to R, we have
Putting and in (2.1) and using (2.31), we get
Since ψ is strictly increasing, we have
This gives us
Putting and in (2.1), then similarly to the above, one can find
We combine (2.32) and (2.33) to remark that
Then, the sequence is non-increasing and bounded below, so there exists such that
Adopting the strategy used in the proof of Theorem 2.3, one can show that , i.e.,
The same idea yields
Now, and from (2.35), (2.36), we obtain , and so (2.28) holds.
Thanks to (2.30) and (2.35), one can write
From partial-compatibility of the pairs and , using (2.35) and (2.37), we obtain
Denote
Since , so again by partial-compatibility of the pairs and , we get
By triangular inequality, we have
Using (2.37), (2.38), (2.39), the continuity of T and letting in the above inequality, we get
that is, and u is a coincidence point of T and R.
Analogously, the triangular inequality gives us
Using (2.37), (2.38), (2.39), the continuity of S and letting in the above inequality, we get
By condition (p2), it follows immediately
Now, applying (2.1) with , we have
This implies that
then we deduce that , and so . Until now, we have obtained
With and from (2.28), we have
This proves that u is a common fixed point of the mappings T, S and R.
Now our purpose is to check that such a point is unique. Suppose to the contrary that there is another common fixed point of T, S and R, say q. Then, applying (2.1) with , we obtain easily that . It is immediate that q is a coincidence point of T, S and R. From (2.28), this implies that
Hence, we get
which yields the uniqueness of the common fixed point of T, S and R. This completes the proof. □
Remark 2.10 We leave, as exercise for the reader, to verify that our results hold even if we replace condition (2.1) by the following
for all such that Rx and Ry are comparable.
3 Application to cyclical contractions
In this section we use the previous results to prove a fixed point theorem for a mapping satisfying a weak cyclical contractive condition. In 2003, Kirk et al. [27] studied existence and uniqueness of a fixed point for mappings satisfying cyclical contractive conditions in complete metric spaces.
Definition 3.1 Let be a metric space, m a positive integer and nonempty subsets of X. A mapping T on is called a m-cyclic mapping if , , where .
Later on, Pacurar and Rus [35] introduced the following notion, suggested by the considerations in [27].
Definition 3.2 Let Y be a nonempty set, m a positive integer and an operator. By definition, is a cyclic representation of Y with respect to T if T is a m-cyclic mapping and are nonempty sets.
Example 3.3 Let . Assume and , so that . Define such that , for all . It is clear that is a cyclic representation of Y.
Inspired by Karapinar [26] and Gopal et al. [23], we present the notion of a cyclic weak -contraction in partial metric spaces.
Definition 3.4 Let be an ordered partial metric space, be closed subsets of X and . An operator is called a cyclic weak -contraction if the following conditions hold:
-
(i)
is a cyclic representation of Y with respect to T,
-
(ii)
there exist and such that
(3.1)
for every comparable , ().
Now, we state and prove the following result.
Theorem 3.5 Let be a partially ordered set. Suppose that there exists a partial metric p on X such that the partial metric space is complete. Let be a given mapping satisfying
-
(a)
T is a cyclic weak -contraction,
-
(b)
T is weakly increasing and continuous,
-
(c)
the pair is partial-compatible,
-
(d)
for any , there exists such that and .
Then, T has a unique fixed point , that is, .
Proof Let and set
For any , there is such that and . Then, following the lines of the proof of Theorem 2.3, it is easy to show that is a Cauchy sequence in the partial metric space , which is complete, so converges to some . On the other hand, by condition (i) of Definition 3.4, it follows that the iterative sequence has an infinite number of terms in for each . Since is complete, from each , , one can extract a subsequence of that converges to y. In virtue of the fact that each , , is closed, we conclude that and thus . Obviously, is closed and complete. Now, consider the restriction of T on , that is which satisfies the assumptions of Theorem 2.3 and thus, has a unique fixed point in , say u, which is obtained by iteration from the starting point . To conclude, we have to show that, for any initial value , we get the same limit point . Due to condition (c) and using the analogous ideas of the proof of Theorem 2.9, it can be obtained that, for any initial value , as . This completes the proof. □
References
Abbas M, Nazir T, Radenović S: Common fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett. 2011, 24: 1520–1526. 10.1016/j.aml.2011.03.038
Abbas, M, Sintunavarat, W, Kumam, P: Coupled fixed point in partially ordered G-metric spaces. Fixed Point Theory Appl. (to appear)
Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151
Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory Appl. 2010., 2010: Article ID 621469
Altun I, Erduran A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 508730
Altun I, Sola F, Simsek H: Generalized contractions on partial metric spaces. Topol. Appl. 2010, 157(18):2778–2785. 10.1016/j.topol.2010.08.017
Aydi H: Coincidence and common fixed point results for contraction type maps in partially ordered metric spaces. Int. J. Math. Anal. 2011, 5(3):631–642.
Aydi H, Nashine HK, Samet B, Yazidi H: Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations. Nonlinear Anal. 2011, 74(17):6814–6825. 10.1016/j.na.2011.07.006
Aydi H: Some coupled fixed point results on partial metric spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 647091
Aydi H: Some fixed point results in ordered partial metric. J. Nonlinear Sci. Appl. 2011, 4(3):210–217.
Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011, 4(2):1–12.
Aydi H: Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces. J. Nonlinear Analysis Optim. 2011, 2(2):33–48.
Aydi H:Common fixed point results for mappings satisfying -weak contractions in ordered partial metric spaces. Int. J. Math. Stat. 2012, 12(2):53–64.
Aydi, H: A common fixed point result by altering distances involving a contractive condition of integral type in partial metric spaces. Demonstr. Math. 46(1/2) (2013) (in press)
Aydi H, Karapınar E, Shatanawi W:Coupled fixed point results for -weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl. 2011, 62: 4449–4460. 10.1016/j.camwa.2011.10.021
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intgérales. Fundam. Math. 1922, 3: 133–181.
Bhashkar TG, Lakshmikantham V: Fixed point theorems in partially ordered cone metric spaces and applications. Nonlinear Anal. 2006, 65(7):825–832.
Ćirić LB, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294
Ćirić LB, Samet B, Aydi H, Vetro C: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput. 2011, 218: 2398–2406. 10.1016/j.amc.2011.07.005
Di Bari C, Vetro P: Fixed points for φ -weak contractions on partial metric spaces. Int. J. Eng., Contemp. Math. Sci. 2011, 1: 5–13.
Dukić D, Kadelburg Z, Radenović S: Fixed points of Geraghty-type mappings in various generalized metric spaces. Abstr. Appl. Anal. 2011., 2011: Article ID 561245
Escardo MH: PCF extended with real numbers. Theor. Comput. Sci. 1996, 162: 79–115. 10.1016/0304-3975(95)00250-2
Gopal D, Imdad M, Vetro C, Hasan M: Fixed point theory for cyclic weak ϕ -contraction in fuzzy metric spaces. J. Nonlinear Analysis Appl. 2011., 2011: Article ID jnaa-00110
Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3403–3410. 10.1016/j.na.2009.01.240
Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986, 9: 771–779. 10.1155/S0161171286000935
Karapinar E: Fixed point theory for cyclic weak ϕ -contraction. Appl. Math. Lett. 2011, 24: 822–825. 10.1016/j.aml.2010.12.016
Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4: 79–89.
Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
Matthews SG: Partial metric topology. Annals of the New York Academy of Sciences 728. Proceedings of the 8th Summer Conference on General Topology and Applications 1994, 183–197.
Nashine HK, Samet B:Fixed point results for mappings satisfying -weakly contractive condition in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 2201–2209. 10.1016/j.na.2010.11.024
Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5
Nieto JJ, López RR: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23(12):2205–2212. 10.1007/s10114-005-0769-0
Oltra S, Valero O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste 2004, 36(1–2):17–26.
O’Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026
Pacurar M, Rus IA: Fixed point theory for ϕ -contractions. Nonlinear Anal. 2010, 72: 1181–1187. 10.1016/j.na.2009.08.002
Paesano D, Vetro P: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl. 2012, 159: 911–920. 10.1016/j.topol.2011.12.008
Radenović, S, Kadelburg, Z, Jandrlić, D, Jandrlić, A: Some results on weak contraction maps. Bull. Iranian Math. Soc. (to appear)
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4
Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026
Samet B, Rajović M, Lazović R, Stoiljković R: Common fixed point results for nonlinear contractions in ordered partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 71
Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040
Acknowledgements
The authors are really thankful to the anonymous referee for his/her precious suggestions useful to improve the quality of the paper. The third author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the forth author would like to thank the Commission on Higher Education and the Thailand Research Fund under Grant MRG no. 5380044 for financial support during the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Aydi, H., Vetro, C., Sintunavarat, W. et al. Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl 2012, 124 (2012). https://doi.org/10.1186/1687-1812-2012-124
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2012-124