Coincidence point theorems for G-isotone mappings in partially ordered metric spaces

Shuang Wang

Author Affiliations

School of Mathematical Sciences, Yancheng Teachers University, Yancheng, Jiangsu, 224051, P.R. China

Fixed Point Theory and Applications 2013, 2013:96  doi:10.1186/1687-1812-2013-96

 Received: 31 July 2012 Accepted: 26 March 2013 Published: 12 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

We establish coincidence and common fixed-point theorems for G-isotone mappings in partially ordered metric spaces, which include the corresponding coupled, tripled and quadruple fixed-point theorems as special cases. Our proofs are simpler and essentially different from the ones devoted to coupled, tripled and quadruple fixed-point problems that appeared in the last years.

MSC: 47H10, 54H25.

Keywords:
G-isotone mapping; coincidence point; common fixed point; partially ordered metric space

1 Introduction

The Banach contraction principle is the most celebrated fixed-point theorem. There are great number of generalizations of the Banach contraction principle. A very recent trend in metrical fixed-point theory, initiated by Ran and Reurings [1], and continued by Nieto and Lopez [2,3], Bhaskar and Lakshmikantham [4] and many other authors, is to consider a partial order on the ambient metric space and to transfer a part of the contractive property of the nonlinear operators into its monotonicity properties. This approach turned out to be very productive; see, for example, [1-9], and the obtained results found important applications to the existence of solutions for matrix equations or ordinary differential equations and integral equations, see [1-4,9] and reference therein.

In 2006, Bhaskar and Lakshmikantham [4] introduced the notion of coupled fixed point and proved some fixed-point theorems under certain conditions. Later, Lakshmikantham and Ćirić [8] extended these results by defining the mixedg-monotone property, coupled coincidence point and coupled common fixed point. On the other hand, Berinde and Borcut [6] introduced the concept of tripled fixed point and proved some related theorems. Later, Borcut and Berinde [10] extended these results by defining the mixedg-monotone property, tripled coincidence point and tripled common fixed point. These results were then extended and generalized by several authors in the last five years; see [5,6,11-23] and reference therein. Recently, Karapınar [24] introduced the notion of quadruple fixed point and proved some related fixed-point theorems in partially ordered metric space (see also [24-27]). Berzig and Samet [28] extended and generalized the mentioned fixed-point results to higher dimensions. However, they used permutations of variables and distinguished between the first and the last variables. Very recently, Roldan et al.[29] extend the mentioned previous results for non-linear mappings of any number of arguments, not necessarily permuted or ordered, in the framework of partially ordered complete metric spaces. We remind the reader of the following fact: in order to guarantee the existence of coupled (tripled or quadruple) coincidence point, the authors constructed two (three or four) Cauchy sequences using the properties of mixed monotone mappings and contractive conditions. It is not easy to prove that two (three or four) sequences are simultaneous Cauchy sequences. Then we spontaneously wonder the following questions:

Question 1.1 Can we obtain more general fixed-point theorems including the corresponding coupled, tripled and quadruple fixed-point theorems as three special cases?

Question 1.2 Can we provide a new method for approximating coupled, tripled and quadruple fixed points?

In this work, motivated and inspired by the above results, we establish more general fixed-point theorems including the coupled, tripled and quadruple fixed-point theorems as three special cases. Furthermore, we provide affirmative answers to Questions 1.1 and 1.2. The main results extend and improve the recent corresponding results in the literature. Our works bring at least two new features to coupled, tripled and quadruple fixed-point theory. First, we provide a new method for approximating coupled, tripled and quadruple fixed points. Second, our proofs are simpler and essentially different from the ones devoted to coupled, tripled and quadruple fixed-point problems that appeared in the last years.

2 Preliminaries

For simplicity, we denote from now on by , where and X is a non-empty set. Let n be a positive integer, will denote the function . If elements x, y of a partially ordered set are comparable (i.e. or holds) we will write .

Let Φ denote the set of all functions , which satisfies

(iφ) for all ;

(iiφ) for all .

Definition 2.1[12]

Let be a partially ordered set and d be a metric on R. We say that is regular if the following conditions hold:

(i) if a non-decreasing sequence is such that , then for all n,

(ii) if a non-increasing sequence is such that , then for all n.

Definition 2.2[8]

Let be a partially ordered set and and . We say F has the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for any ,

and

Definition 2.3[10]

Let be a partially ordered set and two mappings , . We say that F has the mixed g-monotone property if is g-monotone non-decreasing in x, it is g-monotone non-increasing in y and it is g-monotone non-decreasing in z, that is, for any ,

and

Note that if g is the identity mapping, then Definitions 2.2 and 2.3 reduce to Definition 1.1 in [4] and Definition 4 in [6] of mixed monotone property, respectively.

Definition 2.4[29]

Let be a mapping. We say that F has the mixed monotone property if is monotone non-decreasing in x and z, and it is monotone non-increasing in y and w, that is, for any

Some authors introduced the concept of coincidence point in different ways and with different names. Let and be two mappings.

Definition 2.5 A point is:

(i) a coupled coincidence point [8] if , and ,

(ii) a tripled coincidence point [10] if , , and ,

(iii) a coupled common fixed point [8] if , and ,

(iv) a tripled common fixed point [10] if , , and ,

(v) a quadruple fixed point [24] if , , , and ,

(vi) a Φ-coincidence point [29] if , for all i.

Similarly, note that if g is the identity mapping, then coupled coincidence point, tripled coincidence point and Φ-coincidence point reduce to coupled fixed point (Gnana-Bhaskar and Lakshmikantham [4]), tripled fixed point (Berinde and Borcut [6]) and Φ-fixed point[29], respectively.

Definition 2.6 We say that the mappings and are commutative

(i) if , for all [8],

(ii) if , for all [10].

Let be a partially ordered set and d be a metric on X. We endow the product space with the following partial order: for ,

(2.1)

which will be denoted in the sequel, for convenience, by ≤, also. Obviously, is a partially ordered set. The mapping , given by

(2.2)

where , defines a metric on . It is easy to see that

(2.3)

where . Indeed, .

In order to prove our main results, we need the following lemma.

Lemma 2.7Letbe a partially ordered set anddbe a metric onX. Ifis regular, thenis regular.

Proof Without loss of generality, we assume that the sequence is non-decreasing with (), where . From (2.3), we have

(2.4)

Now suppose that  . As the sequence is non-decreasing and (2.1), we have the sequences are non-decreasing. From (2.4), the regularity of and using Definition 2.1, we have

(2.5)

Suppose that  . Since the sequence is non-decreasing and (2.1), we have the sequences are non-increasing. From (2.4), the regularity of and using Definition 2.1, we have

(2.6)

By (2.1), (2.5) and (2.6), we have for all n. By analogy, we show that if a non-increasing sequence is such that , then for all n. Therefore, is regular. □

3 Main results

We now state and prove the main results of this paper.

Definition 3.1 We say that the mappings and are commutative if for all .

Definition 3.2 Let be a partially ordered set and , . We say that T is a G-isotone mapping if, for any

Definition 3.3 An element is called a coincidence point of the mappings and if . Furthermore, if , then we say that Y is a common fixed point of T and G.

Theorem 3.4Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letandbe aG-isotone mapping for which there existssuch that for all, with,

(3.1)

whereis defined via (2.2). Supposeand also suppose either

(a) Tis continuous, Gis continuous and commutes withTor

(b) is regular andis closed.

If there existssuch that, thenTandGhave a coincidence point.

Proof Since , it follows that there exists such that . In general, there exists such that , . We denote and

(3.2)

Obviously, if for some , then there is nothing to prove. So, we may assume that for all . Since , without loss of generality, we assume that (the case is similar), that is, . Assume that , that is, . Since T is a G-isotone mapping, we get

which shows that for all . This actually means that the sequence is non-decreasing. Since , from (3.1) and (iφ) we have

(3.3)

for all . Hence, the sequence given by is monotone decreasing and bounded below. Therefore, there exists some such that . We shall prove that . Assume that . Then by letting in (3.3) and (iiφ) we have

(3.4)

We claim that is a Cauchy sequence. Indeed, if it is false, then there exist and the sequences and of such that is the minimal in the sense that and . Therefore, .

Using the triangle inequality, we obtain

Letting in the above inequality and using (3.4), we get

(3.5)

Since , we have and hence . Now, by (3.1), we have

Observe that

Letting in the above inequality and using (3.4)-(3.5), we have

where , which is a contradiction. Hence, the sequence is a Cauchy sequence in the metric space . On the other hand, since is a complete metric space, thus the metric space is complete. Therefore, there exists such that , that is, .

Now suppose that the assumption (a) holds. By the continuity of G, we have . On the other hand, by the commutativity of T and G, we have

(3.6)

By (3.6) and the continuity of T, we have

which shows that is a coincidence point of T and G.

Suppose that the assumption (b) holds. Using Lemma 2.7, we have is regular. Since is non-decreasing sequence that converges to , in view of Definition 2.1, we have for all n. Since is closed and by (3.2), we obtain that there exists for which

Then from (3.1), we have

for all . Letting in the above inequality, we have , which implies that . Therefore, is a coincidence point of T and G. □

Remark 3.5 Different kinds of contractive conditions are studied and we use a distinct methodology to prove Theorem 3.4. The authors proved that any number of sequences are simultaneous Cauchy sequence in [29]. However, we only need to proof that one sequence is a Cauchy sequence.

Taking in Theorem 3.4, we can obtain the following result immediately.

Corollary 3.6Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letandbe aG-isotone mapping for which there existssuch that for all, with,

Supposeand also suppose either

(a) Tis continuous, Gis continuous and commutes withTor

(b) is regular andis closed.

If there existssuch that, thenTandGhave a coincidence point.

Now, we will show that Theorem 3.4 allow us to derive coupled, tripled and quadruple fixed-point theorems for mixed monotone mappings in partially ordered metric space.

Taking , and for in Theorem 3.4, we can obtain the following result.

Corollary 3.7Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letandbe a mixedg-monotone mapping for which there existssuch that for allwith, ,

(3.7)

Supposeand also suppose either

(a) Fis continuous, gis continuous and commutes withFor

(b) is regular andis closed.

If there existsuch that

(3.8)

or

(3.9)

then there existsuch thatand, that is, Fandghave a couple coincidence point.

Proof For simplicity, we denote , and for all . We endow the product space with the following partial order:

(3.10)

Consider the function defined by

(3.11)

Obviously, and are two particular cases of and defined by (2.1) and (2.2), respectively. Now consider the operators and defined by

(3.12)

and

(3.13)

Since , we have .

We claim that T is a G-isotone mapping. Indeed, suppose that , . By (3.10) and (3.13), we have and . Since F is g-mixed monotone, we have

(3.14)

From (3.10), (3.12) and (3.14), we have

Similarly, we can obtain that for any , . By (3.8)-(3.10), we have there exists such that .

From (3.11) and (3.12), we have

and

for any , . It follows from (3.7) that

Now suppose that the assumption (a) holds. By the continuity of g, we have G is continuous. From (3.12), (3.13) and using the commutativity of F and g, we have, for any

which implies that G commutes with T. It is easy to see that T is continuous. Indeed, by (3.11), we obtain that () if and only if and (). Since F is continuous, we have and (), for any (). Therefore, we have

for any ().

Suppose that the assumption (b) holds. It is easy to see that is closed.

All the hypothesis of Theorem 3.4 () are satisfied, and so we deduce the existence of a coincidence point of T and G. From (3.12) and (3.13), there exists such that and , that is, is a coupled coincidence point of F and g. □

Remark 3.8 Note that in the case of the condition (b) satisfied in Corollary 3.7, we omit the control conditions: g is continuous and commutes with F, which are needed in the proof of Theorem 2.1 in [8] and Theorem 3 in [7].

Taking , and for in Theorem 3.4, we can obtain the following result by the similar argument as we did in the proof of Corollary 3.7.

Corollary 3.9Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letandsuch thatFhas the mixedg-monotone property and. Assume there is a functionsuch that

for anyfor which, and. Suppose either

(a) Fis continuous, gis continuous and commutes withFor

(b) is regular andis closed.

If there existsuch that

or

then there existsuch that

that is, Fandghave a tripled coincidence point.

Similarly, taking , and G is the identity mapping on for in Theorem 3.4, we can obtain the following result.

Corollary 3.10Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letsuch thatFhas the mixed monotone property. Assume there is a functionsuch that

for anyfor which, , and. Suppose either

(a) Fis continuous or

(b) is regular.

If there existsuch that

or

then there existsuch that

that is, Fhave a quadruple fixed point.

Theorem 3.11In addition to the hypothesis of Theorem 3.4, suppose that for everythere existssuch thatis comparable toand to. Also, assume thatφis non-decreasing. LetGcommute withTif the assumption (b) holds. ThenTandGhave a unique common fixed point, that is, there exists a unique pointsuch that.

Proof From Theorem 3.4, the set of coincidence points of T and G is non-empty. Assume that and are two coincidence points of T and G. We shall prove that . Put and choose so that . Then, similarly to the proof of Theorem 3.4, we obtain the sequence defined as follows: , . Since and are comparable, without loss of generality, we assume that . Since T is a G-isotone mapping, we have

Recursively, we get that , . Thus, by the contractive condition (3.1), one gets

Thus, by the above inequality, we get

where . Since φ is non-decreasing, it follows that

From the definition of Φ, we get , for each . Then, we have . Thus,

(3.15)

Similarly, we obtain that

(3.16)

Combining (3.15) and (3.16) yields that . Since , by the commutativity of T and G, we have

(3.17)

Denote . By (3.17), we have , that is is a coincidence point of T and G. Thus, we have . Therefore, is a common fixed point of T and G.

To prove the uniqueness, assume is another common fixed point of T and G. Then we have

□

Corollary 3.12In addition to the hypothesis of Corollary 3.7, suppose that for everythere existssuch thatis comparable toand to. Also, assume thatφis non-decreasing. Letgcommute withFif the assumption (b) holds. ThenFandghave a unique coupled common fixed point, that is, there exists a unique pointsuch that

Proof Similarly to the proof of Corollary 3.7, we can obtain all conditions of Theorem 3.4 () are satisfied. In addition, by the commutativity of g and F, we have G commutes with T. For simplicity, we denote , and . By (3.12), we have

By hypothesis, there exists such that is comparable to and to . Hence, there is no doubt that all conditions of Theorem 3.11 are satisfied (). Therefore, there exists a unique point such that . That is, and . □

By the similar argument as we did in the proof of Corollary 3.12, we deduce the following corollary from Theorem 3.11 ().

Corollary 3.13In addition to the hypothesis of Corollary 3.9, suppose that for allandin, there existsinsuch thatis comparable toand. Also, assume thatφis non-decreasing. Letgcommute withFif the assumption (b) holds. ThenFandghave a unique tripled common fixed point, that is,

Competing interests

The author declares that she has no competing interests.

Author’s contributions

SW completed the paper herself. The author read and approved the final manuscript.

Acknowledgements

The author thanks the editor and the referees for their useful comments and suggestions. This study was supported by the Natural Science Foundation of Yancheng Teachers University under Grant (12YCKL001) and UNSF of Jiangsu province, China (09KJD110005).

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