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# On coupled fixed point theorems on partially ordered G-metric spaces

Erdal Karapınar1, Billûr Kaymakçalan2* and Kenan Taş2

Author Affiliations

1 Department of Mathematics, Atılım University, İncek, Ankara, 06836, Turkey

2 Department of Mathematics and Computer Science, Çankaya University, Ankara, Turkey

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Journal of Inequalities and Applications 2012, 2012:200 doi:10.1186/1029-242X-2012-200

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/200

 Received: 2 July 2012 Accepted: 23 August 2012 Published: 7 September 2012

© 2012 Karapınar et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this manuscript, we extend, generalize and enrich some recent coupled fixed point theorems in the framework of partially ordered G-metric spaces in a way that is essentially more natural.

MSC: 46N40, 47H10, 54H25, 46T99.

##### Keywords:
coupled fixed point; coincidence point; mixed g-monotone property; ordered set; G-metric space

### 1 Introduction and preliminaries

In [1] Aydi et al. established coupled coincidence and coupled common fixed point results for a mixed g-monotone mapping satisfying nonlinear contractions in partially ordered G-metric spaces. These results generalize those of Choudhury and Maity [2].

Here we generalize, improve, enrich and extend the above mentioned coupled fixed point results of Aydi et al.

Throughout this paper, let denote the set of nonnegative integers, and be the set of positive integers.

Definition 1.1 (See [3])

Let X be a non-empty set, and be a function satisfying the following properties:

(G1) if ,

(G2) for all with ,

(G3) for all with ,

(G4) (symmetry in all three variables),

(G5) for all (rectangle inequality).

Then the function G is called a generalized metric or, more specially, a G-metric on X, and the pair is called a G-metric space.

Every G-metric on X defines a metric on X by

(1.1)

Example 1.2 Let be a metric space. The function , defined by

or

for all , is a G-metric on X.

Definition 1.3 (See [3])

Let be a G-metric space, and let be a sequence of points of X. We say that is G-convergent to if , that is, for any , there exists such that , for all . We call x the limit of the sequence and write or .

Proposition 1.4 (See [3])

Letbe aG-metric space. The following are equivalent:

(1) isG-convergent tox,

(2) as,

(3) as,

(4) as.

Definition 1.5 (See [3])

Let be a G-metric space. A sequence is called a G-Cauchy sequence if, for any , there is such that for all , that is, as .

Proposition 1.6 (See [3])

Letbe aG-metric space. Then the following are equivalent:

(1) the sequenceisG-Cauchy,

(2) for any, there existssuch that, for all.

Proposition 1.7 (See [3])

Letbe aG-metric space. A mappingisG-continuous atif and only if it isG-sequentially continuous at, that is, wheneverisG-convergent to, the sequenceisG-convergent to.

Definition 1.8 (See [3])

A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .

Definition 1.9 (See [2])

Let be a G-metric space. A mapping is said to be continuous if for any two G-convergent sequences and converging to x, y respectively, is G-convergent to .

Let be a partially ordered set and be a G-metric space, be a mapping. A partially ordered G-metric space, , is called g-ordered complete if for each convergent sequence , the following conditions hold:

() if is a non-increasing sequence in X such that implies , ,

() if is a non-decreasing sequence in X such that implies , .

Moreover, a partially ordered G-metric space, , is called ordered complete when g is equal to identity mapping in the above conditions () and ().

Definition 1.10 (See [4])

An element is said to be a coupled fixed point of the mapping if

Definition 1.11 (See [5])

An element is called a coupled coincidence point of a mapping and if

Moreover, is called a common coupled coincidence point of F and g if

Definition 1.12 Let and be mappings. The mappings F and g are said to commute if

Definition 1.13 (See [4])

Let be a partially ordered set and be a mapping. Then F is said to have mixed monotone property if is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any ,

and

Definition 1.14 (See [5])

Let be a partially ordered set and and be two mappings. Then F is said to have mixed g-monotone property if is monotone g-non-decreasing in x and is monotone g-non-increasing in y, that is, for any ,

(1.2)

and

(1.3)

Let Φ denote the set of functions satisfying

(a) ,

(b) for all ,

(c) for all .

Lemma 1.15 (See [5])

Let. For all, we have.

Aydi et al.[1] proved the following theorems.

Theorem 1.16Letbe a partially ordered set andGbe aG-metric onXsuch thatis a completeG-metric space. Suppose that there exist, andsuch that

(1.4)

for allwithand. Suppose also thatFis continuous and has the mixedg-monotone property, andgis continuous and commutes withF. If there existsuch thatand, thenFandghave a coupled coincidence point, that is, there existssuch thatand.

Theorem 1.17Letbe a partially ordered set andGbe aG-metric onXsuch thatis regular. Suppose that there existand mappingsandsuch that

(1.5)

for allwithand. Suppose also thatis complete, Fhas the mixedg-monotone property and. If there existsuch thatand, thenFandghave a coupled coincidence point.

In this manuscript, we generalize, improve, enrich and extend the above coupled fixed point results. We also state some examples to illustrate our results. This paper can be considered as a continuation of the remarkable works of Berinde [6,7].

### 2 Main results

We begin with an example to illustrate the weakness of Theorem 1.16 and Theorem 1.17 above.

Example 2.1 Let . Define by

for all . Then is a G-metric space. Define a map by and by for all . Suppose

(2.1)

and

(2.2)

It is clear that there is no that provides the statement (1.4) of Theorem 1.16.

Notice that is the unique common coincidence point of F and g. In fact, .

For some coupled fixed point and coupled coincidence point theorems, we refer the reader to [8-34].

We now state our first result which successively guarantees a coupled fixed point.

Theorem 2.2Letbe a partially ordered set andGbe aG-metric onXsuch thatis a completeG-metric space. Suppose that there exist, andsuch that

(2.3)

for allwithand. Suppose also thatFis continuous and has the mixedg-monotone property, andgis continuous and commutes withF. If there existsuch thatand, thenFandghave a coupled coincidence point, that is, there existssuch thatand.

Proof Given satisfying and , we shall construct iterative sequences and in the following way: Since , we can choose such that and . Analogously, we choose such that and due to the same reasoning. Since F has the mixed g-monotone property, we conclude that and . By repeating this process, we derive the iterative sequence

and

If for some we have , then and , that is, F and g have a coincidence point. So, we assume that for all . Thus, we have either or . We set

(2.4)

for all . Due to the property (G2), we have for all . By using inequality (2.3), we obtain

(2.5)

Taking (2.4) into account, (2.5) becomes

(2.6)

Since for all , it follows that is monotone decreasing. Therefore, there is some such that .

Now, we assert that . Suppose, on the contrary, that . Letting in (2.6) and using the properties of the map ϕ, we get

which is contradiction. Thus . Hence

(2.7)

Next, we prove that and are Cauchy sequences in the G-metric space . Suppose, on the contrary, that at least one of and is not a Cauchy sequence in . Then there exist and sequences of natural numbers and such that for every natural number k, and

(2.8)

Now, corresponding to , we choose to be the smallest for which (2.8) holds. Hence

Using the rectangle inequality (property (G5)), we get

(2.9)

Letting in the above inequality and using (2.7) yields

(2.10)

Again, by the rectangle inequality, we have

Using the fact that for any , we obtain from properties (G2)-(G4)

Next, using inequality (2.3), we have

(2.11)

Now, using (2.7),(2.10), the properties of the function ϕ, and letting in (2.11), we get

which is a contradiction. Thus, we have proven that and are Cauchy sequences in the G-metric space . Now, since is complete, there are such that and are respectively G-convergent to x and y. That is from Proposition 1.4, we have

Using the continuity of g, we get from Proposition 1.7

(2.12)

Since and , employing the commutativity of F and g yields

(2.13)

Now, we shall show that and .

The mapping F is continuous, and since the sequences and are respectively G-convergent to x and y, using Definition 1.9, the sequence is G-convergent to . Therefore, from (2.13), is G-convergent to . By uniqueness of the limit and using (2.12), we have . Similarly, we can show that . Hence, is a coupled coincidence point of F and g. This completes the proof. □

The following example illustrates that Theorem 2.2 is an extension of Theorem 1.16.

Example 2.3 Let us reconsider Example 2.1. Define a map by

and by for all . Then . We observe that

and

(2.14)

Then, the statement (2.3) of Theorem 2.2 is satisfied for and is the desired coupled coincidence point.

In the next theorem, we omit the continuity hypothesis of F.

Theorem 2.4Letbe a partially ordered set andGbe aG-metric onXsuch thatisg-ordered complete. Suppose that there existand mappingsandsuch that

(2.15)

for allwithand. Suppose also thatis complete, Fhas the mixedg-monotone property and. If there existsuch thatand, thenFandghave a coupled coincidence point.

Proof Proceeding exactly as in Theorem 2.2, we have that and are Cauchy sequences in the complete G-metric space . Then, there exist such that and . Since is non-decreasing and is non-increasing, using the regularity of , we have and for all . If and for some , then and , which implies that and , that is, is a coupled coincidence point of F and g. Then, we suppose that for all . Using the rectangle inequality, (2.15) and property for all , we get

Letting in the above inequality, we obtain

which implies that and . Thus we proved that is a coupled coincidence point of F and g. □

Corollary 2.5Letbe a partially ordered set andGbe aG-metric onXsuch thatis a completeG-metric space. Suppose that there exist, andsuch that

for allwithand. Suppose also thatFis continuous, has the mixedg-monotone property, andgis continuous and commutes withF. If there existsuch thatand, thenFandghave a coupled coincidence point.

Proof Taking with in Theorem 2.4, we obtain Corollary 2.5. □

Corollary 2.6Letbe a partially ordered set andGbe aG-metric onXsuch thatisg-ordered complete. Suppose that there exist, andsuch that

for allwithand. Suppose also thatis complete, Fhas the mixedg-monotone property, . If there existsuch thatand, thenFandghave a coupled coincidence point.

Proof Taking with in Theorem 2.4, we obtain Corollary 2.6 □

Remark 2.7 Taking (the identity mapping) in Corollary 2.5, we obtain [[2], Theorem 3.1]. Taking in Corollary 2.6, we obtain [[2], Theorem 3.2].

Now we shall prove the existence and uniqueness theorem of a coupled common fixed point. If is a partially ordered set, we endow the product set with the partial order ∇ defined by

Theorem 2.8In addition to the hypothesis of Theorem 2.2, suppose that for all, , there existssuch thatis comparable withand. Suppose also thatϕis a non-decreasing function. ThenFandghave a unique coupled common fixed point, that is, there exists a uniquesuch that

Proof From Theorem 2.2, the set of coupled coincidences is non-empty. We shall show that if and are coupled coincidence points, that is, if , , and , then

(2.16)

By assumption, there exists such that is comparable with and . Without loss of generality, we can assume that

and

Put , and choose such that and . Then, similarly as in the proof of Theorem 2.2, we can inductively define sequences and in X by and .

Further, set , , , and, in the same way, define the sequences , , and . Since

then and . Using that F is a mixed g-monotone mapping, one can show easily that and for all . Thus from (2.15), we get

Without loss of generality, we can suppose that for all . Since ϕ is non-decreasing, from the previous inequality, we get

for each . Letting in the above inequality and using Lemma 1.15, we obtain

(2.17)

Analogously, we derive that

(2.18)

Hence, from (2.17), (2.18) and the uniqueness of the limit, we get and . Hence the equalities in (2.16) are satisfied. Since and , by commutativity of F and g, we have

(2.19)

Denote and , then by (2.19), we get

(2.20)

Thus, is a coincidence point. Then, from (2.16) with and , we have and , that is,

(2.21)

From (2.20), (2.21), we get

Then, is a coupled common fixed point of F and g.

To prove the uniqueness, assume that is another coupled common fixed point. Then by (2.16), we have and . □

Theorem 2.9Letbe a partially ordered set andGbe aG-metric onXsuch thatis a completeG-metric space andis regular. Suppose that there existandhaving the mixed monotone property such that

for allwithand. If there existsuch thatand, thenFhas a coupled fixed point. Furthermore, if, then, that is, .

Proof Following the proof of Theorem 2.4 with , we have only to show that . Since , we get .

Thus, we have . Suppose that . Using inequality (2.15), we have

a contradiction. Thus, and . □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors have contributed equally. All authors read and approved the final manuscript.

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