 Research
 Open Access
 Published:
Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 124 (2012)
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.
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 selfmappings 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, {\mathbb{R}}_{+} will denote the set of all nonnegative 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 p:X\times X\to {\mathbb{R}}_{+} such that for all x,y,z\in X:
(p1) x=y\u27fap(x,x)=p(x,y)=p(y,y),
(p2) p(x,x)\le p(x,y),
(p3) p(x,y)=p(y,x),
(p4) p(x,y)\le p(x,z)+p(z,y)p(z,z).
A partial metric space is a pair (X,p) such that X is a nonempty set and p is a partial metric on X.
It is clear that, if p(x,y)=0, then from (p1) and (p2), x=y; but if x=y, p(x,y) may not be 0. A basic example of a partial metric space is the pair ({\mathbb{R}}_{+},p), where p(x,y)=max\{x,y\} for all x,y\in {\mathbb{R}}_{+}.
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 {T}_{0} topology {\tau}_{p} on X which has as a base the family of open pballs \{{B}_{p}(x,\epsilon ),x\in X,\epsilon >0\}, where {B}_{p}(x,\epsilon )=\{y\in X:p(x,y)<p(x,x)+\epsilon \} for all x\in X and \epsilon >0.
If p is a partial metric on X, then the function {p}^{s}:X\times X\to {\mathbb{R}}_{+} given by
is a metric on X.
Definition 1.2 ([29])
Let \{{x}_{n}\} be a sequence in X. Then

(i)
\{{x}_{n}\} converges to a point x\in X if and only if p(x,x)={lim}_{n\to +\mathrm{\infty}}p(x,{x}_{n}). We may write this as {x}_{n}\to x.

(ii)
\{{x}_{n}\} is called a Cauchy sequence if {lim}_{n,m\to +\mathrm{\infty}}p({x}_{n},{x}_{m}) exists and is finite.

(iii)
(X,p) is said to be complete if every Cauchy sequence \{{x}_{n}\} in X converges, with respect to {\tau}_{p}, to a point x\in X, such that p(x,x)={lim}_{n,m\to +\mathrm{\infty}}p({x}_{n},{x}_{m}).
Lemma 1.3 ([29])
Let (X,p) be a partial metric space. Then

(a)
\{{x}_{n}\} is a Cauchy sequence in (X,p) if and only if it is a Cauchy sequence in the metric space (X,{p}^{s}).

(b)
A partial metric space (X,p) is complete if and only if the metric space (X,{p}^{s}) is complete. Furthermore, {lim}_{n\to +\mathrm{\infty}}{p}^{s}({x}_{n},x)=0 if and only if
p(x,x)=\underset{n\to +\mathrm{\infty}}{lim}p({x}_{n},x)=\underset{n,m\to +\mathrm{\infty}}{lim}p({x}_{n},{x}_{m}).
Definition 1.4 ([5])
Let (X,p) be a partial metric space and T:X\to X be a given mapping. We say that T is continuous at {x}_{0}\in X, if for every \epsilon >0, there exists \eta >0 such that T({B}_{p}({x}_{0},\eta ))\subseteq {B}_{p}(T{x}_{0},\epsilon ).
Lemma 1.5 (Sequential characterization of continuity)
Let (X,p) be a partial metric space and T:X\to X be a given mapping. T:X\to X is continuous at {x}_{0}\in X if it is sequentially continuous at {x}_{0}, that is, if and only if
Let X be a nonempty set and R:X\to X be a given mapping. For every x\in X, we denote by {R}^{1}(x) the subset of X defined by
Definition 1.6 Let X be a nonempty set. Then (X,\u2aaf,p) is called an ordered partial metric space if and only if

(i)
(X,p) is a partial metric space,

(ii)
(X,\u2aaf) is a partially ordered set.
Definition 1.7 Let (X,\u2aaf) be a partially ordered set. Then x,y\in X are called comparable if x\u2aafy or y\u2aafx holds.
Definition 1.8 ([30])
Let (X,\u2aaf) be a partially ordered set and T,S,R:X\to X be given mappings such that TX\subseteq RX and SX\subseteq RX. We say that S and T are weakly increasing with respect to R if and only if, for all x\in X, we have
and
Remark 1.9 If R:X\to X is the identity mapping (Rx=x for all x\in X, shortly R={I}_{X}), 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, Sx\u2aafTSx and Tx\u2aafSTx for all x\in X. Finally, a mapping T:X\to X is weakly increasing if and only if Tx\u2aafTTx for all x\in X.
Example 1.10 Consider X={\mathbb{R}}_{+} endowed with the usual ordering of real numbers and define T,S,R:X\to X by
Now, {R}^{1}(Tx)=\{3\} and {R}^{1}(Sx)=Sx, then S and T are weakly increasing with respect to R.
Definition 1.11 Let (X,\u2aaf,p) be an ordered partial metric space. We say that X is regular if and only if the following hypothesis holds: \{{z}_{n}\} is a nondecreasing sequence in X with respect to ⪯ such that {z}_{n}\to z as n\to +\mathrm{\infty}, then {z}_{n}\u2aafz for all n\in \mathbb{N}.
Finally, we recall the following definition of partialcompatibility introduced by Samet et al. [40].
Definition 1.12 Let (X,p) be a partial metric space and T,R:X\to X be given mappings. We say that the pair \{T,R\} is partialcompatible if the following conditions hold:
(b1) p(x,x)=0 implies that p(Rx,Rx)=0.
(b2) {lim}_{n\to +\mathrm{\infty}}p(TR{x}_{n},RT{x}_{n})=0, whenever \{{x}_{n}\} is a sequence in X such that T{x}_{n}\to t and R{x}_{n}\to t for some t\in X.
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 (X,d) be a metric space and let \{{x}_{n}\} be a sequence in X such that \{d({x}_{n+1},{x}_{n})\} is nonincreasing and
If \{{x}_{2n}\} is not a Cauchy sequence, then there exist \epsilon >0 and two sequences \{{m}_{k}\} and \{{n}_{k}\} of positive integers such that {m}_{k}>{n}_{k}>k and the following four sequences tend to ε when k\to +\mathrm{\infty}:
As a corollary, applying Lemma 2.1 to the associated metric {p}^{s} of a partial metric p, and using Lemma 1.3, we obtain the following lemma (see also [21]).
Lemma 2.2 Let (X,p) be a partial metric space and let \{{x}_{n}\} be a sequence in X such that \{p({x}_{n+1},{x}_{n})\} is nonincreasing and
If \{{x}_{2n}\} is not a Cauchy sequence, then there exist \epsilon >0 and two sequences \{{m}_{k}\} and \{{n}_{k}\} of positive integers such that {m}_{k}>{n}_{k}>k and the following four sequences tend to ε when k\to +\mathrm{\infty}:
In the sequel, let Ψ be the set of functions \psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+} such that ψ is continuous, strictly increasing and \psi (t)=0 if and only if t=0. Also, let Φ be the set of functions \phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+} such that φ is lower semicontinuous and \phi (t)=0 if and only if t=0. Such ψ and φ are called control functions.
Our first main result is the following.
Theorem 2.3 Let (X,\u2aaf) be a partially ordered set. Suppose that there exists a partial metric p on X such that the partial metric space (X,p) is complete. Let T,S,R:X\to X be given mappings satisfying

(a)
T, S and R are continuous,

(b)
the pairs \{R,T\} and \{S,R\} are partialcompatible,

(c)
T and S are weakly increasing with respect to R.
Suppose that for every (x,y)\in X\times X such that Rx and Ry are comparable, we have
where \psi \in \mathrm{\Psi} and \phi \in \mathrm{\Phi}. Then T, S and R have a coincidence point u\in X, that is, Tu=Su=Ru.
Proof By Definition 1.8, it follows that TX\cup SX\subseteq RX. Let {x}_{0} be an arbitrary point in X. Since TX\subseteq RX, there exists {x}_{1}\in X such that R{x}_{1}=T{x}_{0}. Since SX\subseteq RX, there exists {x}_{2}\in X such that R{x}_{2}=S{x}_{1}. Continuing this process, we can construct a sequence \{{x}_{n}\} in X defined by
By construction, we have {x}_{1}\in {R}^{1}(T{x}_{0}) and {x}_{2}\in {R}^{1}(S{x}_{1}). 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 \{R{x}_{n}\} is a Cauchy sequence in the partial metric space (X,p). To this aim, we distinguish the following two cases.
Case 1. We suppose that there exists k\in \mathbb{N} such that p(R{x}_{2k},R{x}_{2k+1})=0, so that R{x}_{2k}=R{x}_{2k+1}. By (2.3), applying (2.1) with x={x}_{2k} and y={x}_{2k+1}, we get
Since ψ is strictly increasing, we have
This implies that p(R{x}_{2k+2},R{x}_{2k+1})=0. Continuing this process, we obtain p(R{x}_{n},R{x}_{2k})=0 for all n\ge 2k. This implies that R{x}_{n}=R{x}_{2k}, therefore \{R{x}_{n}\} is Cauchy in (X,p). The same conclusion holds if R{x}_{2k+1}=R{x}_{2k+2} for some k\in \mathbb{N}.
Case 2. Now, we suppose that
Here, we have p(R{x}_{n},R{x}_{n+1})\ne 0 for all n\ge 0. Thanks to (2.3), R{x}_{2n} and R{x}_{2n+1} are comparable, then using (2.2) and taking x={x}_{2n+2} and y={x}_{2n+1} in (2.1), we get
Since ψ is strictly increasing, the above inequality implies that
Now, taking x={x}_{2n} and y={x}_{2n+1} in (2.1), we have
which implies that
Combining (2.5) and (2.7), we get
It follows that the sequence \{p(R{x}_{n},R{x}_{n+1})\} is nonincreasing and bounded below by 0. Hence, there exists r\ge 0 such that
We claim that r=0. Suppose that r>0. Taking the lim sup as n\to +\mathrm{\infty} in (2.6) and using the properties of the functions ψ and φ, we have
This implies that \phi (r)=0, and by a property of the function φ, we have r=0, that is a contradiction. We deduce that r=0, i.e.,
We shall show that \{R{x}_{n}\} is a Cauchy sequence in the partial metric space (X,p). For this, it is sufficient to prove that \{R{x}_{2n}\} is Cauchy in (X,p). Suppose to the contrary that \{R{x}_{2n}\} is not a Cauchy sequence. Then, having in mind that \{p(R{x}_{n},R{x}_{n+1})\} is nonincreasing and (2.9), it follows by Lemma 2.2 that there exist \epsilon >0 and two sequences \{{m}_{k}\} and \{{n}_{k}\} of positive integers such that {m}_{k}>{n}_{k}>k and the following four sequences tend to ε when k\to +\mathrm{\infty}:
Applying (2.1) with x={x}_{2{n}_{k}} and y={x}_{2{m}_{k}1}, we get
Taking {lim\hspace{0.17em}sup}_{k\to +\mathrm{\infty}} in the above inequality and using the continuity of ψ and the lower semicontinuity of φ, we obtain
from which a contradiction follows since \epsilon >0. Then, we deduce that \{R{x}_{n}\} is a Cauchy sequence in the partial metric space (X,p), which is complete, so \{R{x}_{n}\} converges to some u\in X, 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 \{R,T\} is partialcompatible, then
Also, since p(u,u)=0, then we have p(Tu,Tu)=0. The continuity of T together with (2.11) give us
Combining (2.12) and (2.15) together with (2.16) and letting n\to +\mathrm{\infty} 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 \{S,R\} is partialcompatible, then
Also, since p(u,u)=0, it follows p(Ru,Ru)=0. Thus, from (2.18), p(Ru,Tu)=p(Ru,Ru)=0 and so Ru=Tu.
The continuity of S and (2.20) give us
Combining (2.12) and (2.21) together with (2.22) and letting n\to +\mathrm{\infty} in (2.19), we obtain
By condition (p2) and (2.23), we get
Applying (2.1) with x=y=u, we get
This implies that
and so it follows p(Tu,Su)=0, that is Tu=Su. 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 partialcompatible pairs are \{T,R\} and \{R,S\}.
Example 2.5 Let X=[0,\frac{1}{2}] be endowed with the partial metric p(x,y)=max\{x,y\} and the order given as follows:
Consider the mappings T,S,R:X\to X defined by Tx=Sx={x}^{4} and Rx=x for all x\in X. Also, define the functions \psi ,\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+} by \psi (t)=t and \phi (t)=\frac{t}{4}, for all t\ge 0. Clearly, condition (2.1) is satisfied. In fact, for every (x,y)\in X\times X with x\ge y, we get
All the other hypotheses of Theorem 2.3 are satisfied and T, S and R have a coincidence point u=0. (Moreover, u=0 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 partialcompatibility of the pairs \{T,R\} and \{R,S\}. This is the purpose of the next theorem.
Theorem 2.6 Let (X,\u2aaf) be a partially ordered set. Suppose that there exists a partial metric p on X such that (X,p) is complete. Let T,S,R:X\to X be given mappings satisfying

(a)
RX is a closed subspace of (X,p),

(b)
T and S are weakly increasing with respect to R,

(c)
X is regular.
Suppose that for every (x,y)\in X\times X such that Rx and Ry are comparable, we have
where \psi \in \mathrm{\Psi} and \phi \in \mathrm{\Phi}. Then, T, S and R have a coincidence point u\in X, that is, Tu=Su=Ru.
Proof Following the proof of Theorem 2.3, we have that \{R{x}_{n}\} is a Cauchy sequence in the closed subspace RX, then there exists v=Ru, with u\in X, such that
Thanks to (2.3), \{R{x}_{n}\} is a nondecreasing sequence, and so, since it converges to v=Ru, from the regularity of X, we get
Therefore, R{x}_{n} and Ru are comparable. Putting x={x}_{2n} and y=u in (2.25) and using (2.2), we get
Taking {lim\hspace{0.17em}sup}_{n\to +\mathrm{\infty}} in the above inequality, using (2.26) and the properties of φ and ψ, we obtain
This implies that
which is true if p(Su,Ru)=0. This means that Su=Ru.
Analogously, putting x=u and y={x}_{2n+1} in (2.25), we have
Taking {lim\hspace{0.17em}sup}_{n\to +\mathrm{\infty}} 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 R:X\to X is the identity mapping {I}_{X}, by Theorem 2.6, we obtain the following common fixed point result involving two mappings.
Corollary 2.7 Let (X,\u2aaf) be a partially ordered set. Suppose that there exists a partial metric p on X such that the partial metric space (X,p) is complete. Let X be regular and T,S:X\to X be given mappings such that T and S are weakly increasing. Suppose that for every (x,y)\in X\times X such that x and y are comparable, we have
where \psi \in \mathrm{\Psi} and \phi \in \mathrm{\Phi}. Then, T and S have a common fixed point u\in X, that is, Tu=Su=u.
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 X=[0,1] be endowed with the partial metric p(x,y)=max\{x,y\} and the order ⪯ given as follows:
Consider the mappings T,S:X\to X defined by Tx=\frac{x}{4} and Sx=\frac{x}{3}, for all x\in X. Also, define the functions \psi ,\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+} by \psi (t)=t and \phi (t)=\frac{t}{7}, for all t\ge 0. It is easy to show that Sx\u2aafTSx and Tx\u2aafSTx, for all x\in X, that is, T and S are weakly increasing. Now, take x and y comparable and, without loss of generality, assume y\u2aafx, so that x\le y. 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 u=0.
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 (x,y)\in X\times X, there exists z\in X such that Tx\u2aafTz and Ty\u2aafTz. Then, T, S and R have a unique common fixed point, that is, there exists a unique u\in X such that u=Ru=Tu=Su.
Proof Referring to Theorem 2.3, the set of coincidence points of T, S and R is nonempty. Now, we shall show that if {x}^{\ast} and {y}^{\ast} are coincidence points of T, S and R, that is, R{x}^{\ast}=T{x}^{\ast}=S{x}^{\ast} and R{y}^{\ast}=T{y}^{\ast}=S{y}^{\ast}, then
For the coincidence points {x}^{\ast} and {y}^{\ast}, Theorem 2.3 gives us that
By assumption, there exists {z}_{0}\in X such that
Now, proceeding similarly to the proof of Theorem 2.3, we can immediately define a sequence \{R{z}_{n}\} as follows:
Since T and S are weakly increasing with respect to R, we have
Putting x={z}_{2n} and y={x}^{\ast} in (2.1) and using (2.31), we get
Since ψ is strictly increasing, we have
This gives us
Putting x={x}^{\ast} and y={z}_{2n} in (2.1), then similarly to the above, one can find
We combine (2.32) and (2.33) to remark that
Then, the sequence \{p(R{z}_{n},R{x}^{\ast})\} is nonincreasing and bounded below, so there exists r\ge 0 such that
Adopting the strategy used in the proof of Theorem 2.3, one can show that r=0, i.e.,
The same idea yields
Now, p(R{x}^{\ast},R{y}^{\ast})\le p(R{x}^{\ast},R{z}_{n})+p(R{z}_{n},R{y}^{\ast}) and from (2.35), (2.36), we obtain p(R{x}^{\ast},R{y}^{\ast})=0, and so (2.28) holds.
Thanks to (2.30) and (2.35), one can write
From partialcompatibility of the pairs \{R,T\} and \{S,R\}, using (2.35) and (2.37), we obtain
Denote
Since p(u,u)=p(R{x}^{\ast},R{y}^{\ast})=0, so again by partialcompatibility of the pairs \{R,T\} and \{S,R\}, we get
By triangular inequality, we have
Using (2.37), (2.38), (2.39), the continuity of T and letting n\to +\mathrm{\infty} in the above inequality, we get
that is, Ru=Tu 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 n\to +\mathrm{\infty} in the above inequality, we get
By condition (p2), it follows immediately
Now, applying (2.1) with x=y=u, we have
This implies that
then we deduce that p(Tu,Su)=0, and so Tu=Su. Until now, we have obtained
With {y}^{\ast}=u 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 x=y=q, we obtain easily that p(q,Tq)=p(q,Sq)=p(q,Rq)=0. 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 x,y\in X 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 (X,d) be a metric space, m a positive integer and {Y}_{1},\dots ,{Y}_{m} nonempty subsets of X. A mapping T on {\bigcup}_{i=1}^{m}{Y}_{i} is called a mcyclic mapping if T({Y}_{i})\subset {Y}_{i+1}, i=1,\dots ,m, where {Y}_{m+1}={Y}_{1}.
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 T:Y\to Y an operator. By definition, Y={\bigcup}_{i=1}^{m}{Y}_{i} is a cyclic representation of Y with respect to T if T is a mcyclic mapping and {Y}_{i} are nonempty sets.
Example 3.3 Let X=\mathbb{R}. Assume {Y}_{1}={Y}_{3}=[2,0] and {Y}_{2}={Y}_{4}=[0,2], so that Y={\bigcup}_{i=1}^{4}{Y}_{i}=[2,2]. Define T:Y\to Y such that Tx=\frac{x}{2}, for all x\in Y. It is clear that Y={\bigcup}_{i=1}^{4}{Y}_{i} is a cyclic representation of Y.
Inspired by Karapinar [26] and Gopal et al. [23], we present the notion of a cyclic weak (\psi ,\phi )contraction in partial metric spaces.
Definition 3.4 Let (X,\u2aaf,p) be an ordered partial metric space, {Y}_{1},{Y}_{2},\dots ,{Y}_{m} be closed subsets of X and Y={\bigcup}_{i=1}^{m}{Y}_{i}. An operator T:Y\to Y is called a cyclic weak (\psi ,\phi )contraction if the following conditions hold:

(i)
Y={\bigcup}_{i=1}^{m}{Y}_{i} is a cyclic representation of Y with respect to T,

(ii)
there exist \psi \in \mathrm{\Psi} and \phi \in \mathrm{\Phi} such that
\psi (p(Tx,Ty))\le \psi \left(\frac{p(Tx,x)+p(Ty,y)}{2}\right)\phi (p(x,y)),(3.1)
for every comparable x\in {Y}_{i}, y\in {Y}_{i+1} (i=1,2,\dots ,m).
Now, we state and prove the following result.
Theorem 3.5 Let (X,\u2aaf) be a partially ordered set. Suppose that there exists a partial metric p on X such that the partial metric space (X,p) is complete. Let T:{\bigcup}_{i=1}^{m}{Y}_{i}\to {\bigcup}_{i=1}^{m}{Y}_{i} be a given mapping satisfying

(a)
T is a cyclic weak (\psi ,\phi )contraction,

(b)
T is weakly increasing and continuous,

(c)
the pair \{{I}_{x},T\} is partialcompatible,

(d)
for any (x,y)\in X\times X, there exists z\in X such that Tx\u2aafTz and Ty\u2aafTz.
Then, T has a unique fixed point u\in {\bigcap}_{i=1}^{m}{Y}_{i}, that is, Tu=u.
Proof Let {x}_{0}\in Y={\bigcup}_{i=1}^{m}{Y}_{i} and set
For any n\in \mathbb{N}, there is {i}_{n}\in \{1,\dots ,m\} such that {x}_{n}\in {Y}_{{i}_{n}} and {x}_{n+1}\in {Y}_{{i}_{n}+1}. Then, following the lines of the proof of Theorem 2.3, it is easy to show that \{{x}_{n}\} is a Cauchy sequence in the partial metric space (Y,p), which is complete, so \{{x}_{n}\} converges to some y\in Y. On the other hand, by condition (i) of Definition 3.4, it follows that the iterative sequence \{{x}_{n}\} has an infinite number of terms in {Y}_{i} for each i=1,2,\dots ,m. Since (Y,p) is complete, from each {Y}_{i}, i=1,2,\dots ,m, one can extract a subsequence of \{{x}_{n}\} that converges to y. In virtue of the fact that each {Y}_{i}, i=1,2,\dots ,m, is closed, we conclude that y\in {\bigcap}_{i=1}^{m}{Y}_{i} and thus {\bigcap}_{i=1}^{m}{Y}_{i}\ne \mathrm{\varnothing}. Obviously, {\bigcap}_{i=1}^{m}{Y}_{i} is closed and complete. Now, consider the restriction of T on {\bigcap}_{i=1}^{m}{Y}_{i}, that is T{\bigcap}_{i=1}^{m}{Y}_{i}:{\bigcap}_{i=1}^{m}{Y}_{i}\to {\bigcap}_{i=1}^{m}{Y}_{i} which satisfies the assumptions of Theorem 2.3 and thus, T{\bigcap}_{i=1}^{m}{Y}_{i} has a unique fixed point in {\bigcap}_{i=1}^{m}{Y}_{i}, say u, which is obtained by iteration from the starting point {x}_{0}\in Y. To conclude, we have to show that, for any initial value x\in Y, we get the same limit point u\in {\bigcap}_{i=1}^{m}{Y}_{i}. 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 x\in Y, {x}_{n}\to u as n\to +\mathrm{\infty}. 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 Gmetric spaces. Fixed Point Theory Appl. (to appear)
Agarwal RP, ElGebeily 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 (\psi ,\varphi )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 (\psi ,\phi )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 Geraghtytype 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. 10.1016/03043975(95)002502
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 jnaa00110
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 (\psi ,\phi )weakly contractive condition in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 2201–2209. 10.1016/j.na.2010.11.024
Nieto JJ, RodríguezLópez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s1108300590185
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/s1011400507690
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/S0002993903072204
Samet B: Coupled fixed point theorems for a generalized MeirKeeler 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
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/10.1186/168718122012124
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122012124
Keywords
 coincidence point
 common fixed point
 compatible mappings
 cyclic weak (\psi ,\phi )contraction
 partial metric space
 weakly increasing mappings