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.
Keywords:coincidence point; common fixed point; compatible mappings; cyclic weak -contraction; partial metric space; weakly increasing mappings
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  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  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  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.
Definition 1.1 ()
is a metric on X.
Definition 1.2 ()
Lemma 1.3 ()
Definition 1.4 ()
Lemma 1.5 (Sequential characterization of continuity)
Definition 1.8 ()
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 .
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..
Note that Definition 1.12 extends and generalizes the notion of compatibility introduced by Jungck .
2 Main results
We start this section with some auxiliary results (see also ).
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 ).
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.
(a) T, SandRare continuous,
(c) TandSare weakly increasing with respect toR.
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
We continue this process to get
Since ψ is strictly increasing, we have
Case 2. Now, we suppose that
Since ψ is strictly increasing, the above inequality implies that
which implies that
Combining (2.5) and (2.7), we get
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 :
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
By condition (p2) and (2.17), one can write
Similarly, by triangular inequality, we get
By (2.2) and (2.11), we have
The continuity of S and (2.20) give us
By condition (p2) and (2.23), we get
This implies that
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 .
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.
(b) TandSare weakly increasing with respect toR,
(c) Xis regular.
This implies that
which yields that
We conclude that u is a coincidence point of T, S and R. □
Corollary 2.7Letbe a partially ordered set. Suppose that there exists a partial metricponXsuch that the partial metric spaceis complete. LetXbe regular andbe given mappings such thatTandSare weakly increasing. Suppose that for everysuch thatxandyare comparable, we have
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.
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.
Since T and S are weakly increasing with respect to R, we have
Since ψ is strictly increasing, we have
This gives us
We combine (2.32) and (2.33) to remark that
The same idea yields
Thanks to (2.30) and (2.35), one can write
By triangular inequality, we have
Analogously, the triangular inequality gives us
By condition (p2), it follows immediately
This implies that
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
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. studied existence and uniqueness of a fixed point for mappings satisfying cyclical contractive conditions in complete metric spaces.
Now, we state and prove the following result.
(b) Tis weakly increasing and continuous,
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. □
The authors declare that they have no competing interests.
All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.
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.
Abbas, M, Nazir, T, Radenović, S: Common fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett.. 24, 1520–1526 (2011). Publisher Full Text
Agarwal, RP, El-Gebeily, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal.. 87, 109–116 (2008). Publisher Full Text
Altun, I, Sola, F, Simsek, H: Generalized contractions on partial metric spaces. Topol. Appl.. 157(18), 2778–2785 (2010). Publisher Full Text
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.. 74(17), 6814–6825 (2011). Publisher Full Text
Aydi, H, Karapınar, E, Shatanawi, W: Coupled fixed point results for -weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl.. 62, 4449–4460 (2011). Publisher Full Text
Ćirić, LB, Samet, B, Aydi, H, Vetro, C: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput.. 218, 2398–2406 (2011). Publisher Full Text
Escardo, MH: PCF extended with real numbers. Theor. Comput. Sci.. 162, 79–115 (1996). Publisher Full Text
Harjani, J, Sadarangani, K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal.. 71, 3403–3410 (2009). Publisher Full Text
Jungck, G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci.. 9, 771–779 (1986). Publisher Full Text
Karapinar, E: Fixed point theory for cyclic weak ϕ-contraction. Appl. Math. Lett.. 24, 822–825 (2011). Publisher Full Text
Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.. 70, 4341–4349 (2009). Publisher Full Text
Nashine, HK, Samet, B: Fixed point results for mappings satisfying -weakly contractive condition in partially ordered metric spaces. Nonlinear Anal.. 74, 2201–2209 (2011). Publisher Full Text
Nieto, JJ, Rodríguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 22, 223–239 (2005). Publisher Full Text
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.. 23(12), 2205–2212 (2007). Publisher Full Text
O’Regan, D, Petrusel, A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl.. 341, 1241–1252 (2008). Publisher Full Text
Pacurar, M, Rus, IA: Fixed point theory for ϕ-contractions. Nonlinear Anal.. 72, 1181–1187 (2010). Publisher Full Text
Paesano, D, Vetro, P: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl.. 159, 911–920 (2012). Publisher Full Text
Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.. 132, 1435–1443 (2004). Publisher Full Text
Samet, B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal.. 72, 4508–4517 (2010). Publisher Full Text
Sintunavarat, W, Cho, YJ, Kumam, P: Common fixed point theorems for c-distance in ordered cone metric spaces. Comput. Math. Appl.. 62, 1969–1978 (2011). Publisher Full Text