Research

# n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces

Müzeyyen Ertürk1* and Vatan Karakaya2

Author Affiliations

1 Department of Mathematics, The Faculty of Arts and Sciences, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, 34210, Turkey

2 Current address: Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, 34220, Turkey

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Journal of Inequalities and Applications 2013, 2013:196  doi:10.1186/1029-242X-2013-196

 Received: 4 January 2013 Accepted: 5 April 2013 Published: 22 April 2013

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 paper, we study existence and uniquennes of fixed points of operator where n is an arbitrary positive integer and X is partially ordered complete metric space.

MSC: 47H10, 54H25, 54E50.

##### Keywords:
fixed point theorems; nonlinear contraction; partially ordered metric space; n-tuplet fixed point; mixed g-monotone

### 1 Introduction

As it is known, fixed point theory is one of the oldest and most famous theory in mathematics, and it has become an important tool for other areas of science such as approximation theory, statistics, engineering and economics.

Among hundreds of fixed point theorems, the Banach contraction theorem [1] is particularly well known due to its simplicity and usefulness. It states that any contraction mapping of a complete metric space has a unique fixed point.

In 2004, the Banach contraction principle were extended to metric space endowed with partial order by Ran and Reuring [2]. They pointed out that the contractivity condition on the nonlinear and monotone map is only assumed to hold on elements which are comparable in the partial order. Afterward, Nieto and Rodriguez-Lopez [3] extended results of Ran and Reuring for non-decreasing mapping and studied existence and uniqueness of first-order differential equations.

In 2006, by following the above mentioned trend, Bhaskar and Lakshmikantham [4] introduced mixed monotone property and gave their coupled fixed point theorem for mappings with mixed monotone property. Also, they produced some applications related with the existence and uniqueness of solution for a periodic boundary value problem. This work of Bhaskar and Lakshmikantham has attracted the attention of many researchers. The concept of coupled fixed point for various contractive type mappings was studied by several authors [5-10]. Lakshmikantham and Ciric [11] extended the results of [4] for monotone non-linear contractive mapping and generalized mixed monotone concept. Berinde and Borcut [12] introduced tripled fixed point theorem for non-linear mapping in partially ordered complete metric space as a generalization and extension of the coupled fixed point theorem.

Motivated by these studies, the quadruple fixed point theorem was given for different contractive type mappings [13-16].

In this paper, we generalize mentioned trend in the above for an arbitrary positive number n, that is, we introduce the concept of n-tuplet fixed point theorem and prove some results.

### 2 Main results

Let us give new definitions for our aim.

Definition 1 Let be partially ordered set and . We say that F has the mixed monotone property if is monotone non-decreasing in its odd argument and it is monotone non-increasing in its even argument. That is, for any

(2.1)

Definition 2 Let X be a nonempty set and a given mapping. An element is called a n-tuplet fixed point of F if

(2.2)

Definition 3 Let be partially ordered set and and . We say that F has the mixed g-monotone property if is monotone g-non-decreasing in its odd argument and it is monotone g-non-increasing in its even argument. That is, for any

(2.3)

Note that if g is the identity mapping, this definition reduces to Definition 1.

Definition 4 Let X be a nonempty set and a given mapping. An element is called a n-tuplet coincidence point of and  if

(2.4)

Note that if g is the identity mapping, this definition reduces to Definition 2.

Definition 5 Let be partially ordered set and and . F and g called commutative if

(2.5)

for all .

Let Φ denote the all functions , which are continuous and satisfy that

(i) ,

(ii) for each .

Since we want to shorten expressions in the following theorem, consider Condition 1 in the following for X an F.

Condition 1 Suppose either

(i) F is continuous, or

(ii) X has the following property:

(a) if non-decreasing sequence , then for all k,

(b) if non-increasing sequence , then for all k.

Theorem 1Letbe partially ordered set and suppose thatis complete metric space. Assumeandare such thatFhas the mixedg-monotone property and

(2.6)

for allfor which, , … , (ifnis odd), (ifnis even). Assume thatandgcommutes withF. Also, suppose that Condition 1 is satisfied. If there existsuch that

(2.7)

then there existsuch that

that is, Fandghave an-tuplet coincidence point.

Proof Let be such that (2.7). Since , we construct the sequence as follows:

(2.8)

for  . We claim that

(2.9)

for all . For this, we will use the mathematical induction. The inequalities in (2.9) hold because of (2.7), that is, we have

Thus, our claim is true for . Now, suppose that the inequalities in (2.9) hold . In this case,

(2.10)

Now, we must show that the inequalities in (2.9) hold . If we consider (2.8) and mixed g-monotone property of F together with (2.10), we have

Thus, (2.9) is satisfied for all . So, we have,

(2.11)

For the simplicity, we define

We will show that

(2.12)

By (2.6), (2.8) and (2.11), we get

(2.13)

(2.14)

(2.15)

Due to (2.13)-(2.15), we conclude that

(2.16)

Hence, we get (2.12).

Since for all , then for all . So, is monotone decreasing. Since it is bounded below, there is some such that

(2.17)

We want to show that . Suppose that . Then taking the limit as of both sides of (2.12) and keeping in mind that we assume that for all , we have

(2.18)

which is a contradiction. Thus, , that is

(2.19)

Now we prove that are Cauchy sequences. Suppose that at least one of is not Cauchy. So, there exists an for which we can find subsequence of integer , with such that

(2.20)

Additionally, corresponding to , we can choose such that it is the smallest integer satisfying (2.20) and . Thus,

(2.21)

By using triangle inequality and having (2.20) and (2.21) in mind

(2.22)

Letting in (2.22) and using (2.20)

(2.23)

We apply triangle inequality to (2.20) as the following.

(2.24)

Since , then

(2.25)

So, from (2.25), (2.8) and (2.6), we get

(2.26)

(2.27)

(2.28)

Combining (2.24) with (2.26)-(2.29), we get

(2.29)

Letting , we obtain a contradiction. This show that are Cauchy sequences. Since X is complete metric space, there exist such that

(2.30)

Since g is continuous, (2.30) implies that

(2.31)

From (2.10) and by regarding commutativity of F and g

(2.32)

We shall show that

Suppose now, (i) holds. Then by (2.8), (2.32) and (2.30), we have

(2.33)

Analogously,

(2.34)

(2.35)

Thus, we have

Suppose now the assumption (b) holds. If n is odd since are non-decreasing and are non-increasing, if n is even since are non-decreasing and are non-increasing and by considering , , … , we have

(2.36)

for all k. Thus, by triangle inequality and (2.32)

(2.37)

Letting implies that . Hence, . Analogously, we can get that

Thus, we proved that F and g have a n-tuplet coincidence point. □

Corollary 1The above theorem reduces to Theorem 2.1 of[2]forandif (i) is satisfied andwhere.

The following corollary is a generalization of Corollary 2.1 in [11] and Theorem 2.1 in [4].

Corollary 2Letbe a partially ordered set and suppose thatis complete metric space. Supposeand there existsuch thatFhas the mixedg-monotone property and there exist awith

(2.38)

for allfor which, , … , (ifnis odd), (ifnis even). Assume also Condition 1 holds, and assume that, gis continuous and commutes withF. If there existsuch that

(2.39)

then there existsuch that

that is, Fandghave an-tuplet coincidence point.

Proof It is sufficient to take with in previous theorem. □

### 3 Uniqueness of n-tuplet fixed point

For all ,

(3.1)

We say that is equal to if and only if , , … , .

Theorem 2In addition to hypothesis Theorem 1, assume that for allthere existsuch that

is comparable to

and

ThenFandghave a uniquen-tuplet common fixed point, that is, there existsuch that

Proof From the Theorem 1, the set of n-tuplet coincidences is non-empty. We will show that if and are n-tuplet coincidence points, that is, if

and

then

(3.2)

By assumption there is such that

(3.3)

is comparable with

(3.4)

and

(3.5)

Define sequences such that , , … , and

(3.6)

Since (3.4) and (3.5) comparable with (3.3), we may assume that

By using (2.11), we get that

for all k. From (3.1), we have

(3.7)

By (3.7) and (2.6), we have

(3.8)

(3.9)

(3.10)

(3.11)

Hence, it follows

(3.12)

for each . It is known that and imply for each . Thus, from (3.12)

(3.13)

Analogously, we can show that

(3.14)

Combining (3.13) and (3.14) and by using the triangle inequality

(3.15)

Hence, we get , , … , . Thus, we proved claim of theorem.

By commutativity of F and g,

(3.16)

Denote , , … , . Since (3.16), we get

(3.17)

Thus, is a n-tuplet coincidence point. Then from assumption in theorem with , … , it follows , , … , , that is

(3.18)

From (3.18) and (3.17),

Therefore, is n-tuplet common fixed point of F and g. To prove the uniqueness, assume that is another n-tuplet common fixed point. Then by assumption in theorem we have

□

Corollary 3Letbe partially ordered set and suppose thatis complete metric space. Supposeand there existsuch thatFhas the mixedg-monotone property and

(3.19)

for allfor which, , … , (ifnis odd), (ifnis even). Suppose there existsuch that

(3.20)

Assume also that Condition 1 holds. Then there existsuch that

(3.21)

That isFhas an-tuplet fixed point.

Proof Take , then the assumption in Theorem 1 are satisfied. Thus, we get the result. □

Corollary 4Letbe partially ordered set and suppose thatis complete metric space. Supposeand there existsuch thatFhas the mixedg-monotone property and there existwith

(3.22)

for allfor which, , … , (ifnis odd), (ifnis even). Suppose there existsuch that

Assume also that Condition 1 holds. Then there existsuch that

(3.23)

That isFandghaven-tuplet coincidence point.

Proof Taking with in above corollary we obtain this corollary. □

### Competing interests

The authors declare that they have no competing interests.

### Acknowledgements

This work is supported by Yildiz Technical University Scientific Research Projects Coordination Unit under the project number BAPK 2012-07-03-DOP03.

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