Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces

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.

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])

Letbe a partial metric space. Then

(a) is a Cauchy sequence inif and only if it is a Cauchy sequence in the metric space.

(b) A partial metric spaceis complete if and only if the metric spaceis 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)

Letbe a partial metric space andbe a given mapping. is continuous atif 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].

We start this section with some auxiliary results (see also [37]).

Lemma 2.1Letbe a metric space and letbe a sequence inXsuch thatis non-increasing and

Ifis not a Cauchy sequence, then there existand two sequencesandof positive integers such thatand the following four sequences tend toεwhen:

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 2.2Letbe a partial metric space and letbe a sequence inXsuch thatis non-increasing and

Ifis not a Cauchy sequence, then there existand two sequencesandof positive integers such thatand the following four sequences tend toεwhen:

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 2.3Letbe a partially ordered set. Suppose that there exists a partial metricponXsuch that the partial metric spaceis complete. Letbe given mappings satisfying

(a) T, SandRare continuous,

(b) the pairsandare partial-compatible,

(c) TandSare weakly increasing with respect toR.

Suppose that for everysuch thatRxandRyare comparable, we have

whereand. ThenT, SandRhave 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 2.6Letbe a partially ordered set. Suppose that there exists a partial metricponXsuch thatis complete. Letbe given mappings satisfying

(a) RXis a closed subspace of,

(b) TandSare weakly increasing with respect toR,

(c) Xis regular.

Suppose that for everysuch thatRxandRyare comparable, we have

whereand. Then, T, SandRhave 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 2.7Letbe 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

whereand. Then, TandShave 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 2.9In addition to the hypotheses of Theorem 2.3, suppose that for any, there existssuch thatand. Then, T, SandRhave a unique common fixed point, that is, there exists a uniquesuch 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.

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

for every comparable , ().

Now, we state and prove the following result.

Theorem 3.5Letbe a partially ordered set. Suppose that there exists a partial metricponXsuch that the partial metric spaceis complete. Letbe a given mapping satisfying

(a) Tis a cyclic weak-contraction,

(b) Tis weakly increasing and continuous,

(c) the pairis partial-compatible,

(d) for any, there existssuch thatand.

Then, Thas 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. □

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.

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