Coupled fixed point theorems on partially ordered G-metric spaces

The purpose of this paper is to extend some recent coupled fixed point theorems in the context of partially ordered G-metric spaces in a virtually different and more natural way.

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

The notion of metric space was introduced by Fréchet [1] in 1906. In almost all fields of quantitative sciences which require the use of analysis, metric spaces play a major role. Internet search engines, image classification, protein classification (see, e.g., [2]) can be listed as examples in which metric spaces have been extensively used to solve major problems. It is conceivable that metric spaces will be needed to explore new problems that will arise in quantitative sciences in the future. Therefore, it is necessary to consider various generalizations of metrics and metric spaces to broaden the scope of applied sciences. In this respect, cone metric spaces, fuzzy metric spaces, partial metric spaces, quasi-metric spaces and b-metric spaces can be given as the main examples. Applications of these different approaches to metrics and metric spaces make it evident that fixed point theorems are important not only for the branches of mainstream mathematics, but also for many divisions of applied sciences.

Inspired by this motivation Mustafa and Sims [3] introduced the notion of a G-metric space in 2004 (see also [4-7]). In their introductory paper, the authors investigated versions of the celebrated theorems of the fixed point theory such as the Banach contraction mapping principle [8] from the point of view of G-metrics. Another fundamental aspect in the theory of existence and uniqueness of fixed points was considered by Ran and Reurings [9] in partially ordered metric spaces. After Ran and Reurings’ pioneering work, several authors have focused on the fixed points in ordered metric spaces and have used the obtained results to discuss the existence and uniqueness of solutions of differential equations, more precisely, of boundary value problems (see, e.g., [10-20]). Upon the introduction of the notion of coupled fixed points by Guo and Laksmikantham [14], Gnana-Bhaskar and Lakshmikantham [15] obtained interesting results related to differential equations with periodic boundary conditions by developing the mixed monotone property in the context of partially ordered metric spaces. As a continuation of this trend, many authors conducted research on the coupled fixed point theory and many results in this direction were published (see, for example, [21-35]).

In this paper, we prove the theorem that amalgamates these three seminal approaches in the study of fixed point theory, the so called G-metrics, coupled fixed points and partially ordered spaces.

We shall start with some necessary definitions and a detailed overview of the fundamental results developed in the remarkable works mentioned above. Throughout this paper, ℕ and denote the set of non-negative integers and the set of positive integers respectively.

Definition 1 (See [3])

Let X be a non-empty set, 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.

It can be easily verified that every G-metric on X induces a metric on X given by

Trivial examples of G-metric are as follows.

Example 2 Let be a metric space. The function , defined by

or

for all , is a G-metric on X.

The concepts of convergence, continuity, completeness and Cauchy sequence have also been defined in [3].

Definition 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, if for any , there exists such that for all . We call x the limit of the sequence and write or .

Proposition 4 (See [3])

Letbe aG-metric space. The following statements are equivalent:

(1) isG-convergent tox,

(2) as,

(3) as,

(4) as.

Definition 5 (See [3])

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

Proposition 6 (See [3])

Letbe aG-metric space. The following statements are equivalent:

(1) The sequenceisG-Cauchy.

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

Definition 7 (See [3])

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

Definition 8 Let be a G-metric space. A mapping is said to be continuous if for any three G-convergent sequences , and converging to x, y and z respectively, is G-convergent to .

We define below g-ordered complete G-metric spaces.

Definition 9 Let be a partially ordered set, be a G-metric space and be a mapping. A partially ordered G-metric space, , is called g-ordered complete if for each G-convergent sequence , the following conditions hold:

() If is a non-increasing sequence in X such that , then .

() If is a non-decreasing sequence in X such that , then .

In particular, if g is the identity mapping in () and (), the partially ordered G-metric space, , is called ordered complete.

We next recall some basic notions from the coupled fixed point theory. In 1987 Guo and Lakshmikantham [14] defined the concept of a coupled fixed point. In 2006, in order to prove the existence and uniqueness of the coupled fixed point of an operator on a partially ordered metric space, Gnana-Bhaskar and Lakshmikantham [15] reconsidered the notion of a coupled fixed point via the mixed monotone property.

Definition 10 ([15])

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

and

Definition 11 ([15])

An element is called a coupled fixed point of the mapping if

The results in [15] were extended by Lakshmikantham and Ćirić in [16] by defining the mixed g-monotone property.

Definition 12 Let be a partially ordered set, and . The function 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 ,

and

It is clear that Definition 12 reduces to Definition 10 when g is the identity mapping.

Definition 13 An element is called a coupled coincidence point of the mappings and if

and a common coupled fixed point of F and g if

Definition 14 The mappings and are said to commute if

Throughout the rest of the paper, we shall use the notation gx instead of , where and , for brevity. In [35], Nashine proved the following theorems.

Theorem 15Letbe a partially orderedG-metric space. Letandbe mappings such thatFhas the mixedg-monotone property, and let there existsuch thatand. Suppose that there existssuch that for allthe following holds:

for alland, where eitheror. Assume the following hypotheses:

(i) ,

(ii) isG-complete,

(iii) gisG-continuous and commutes withF.

ThenFandghave a coupled coincidence point, that is, there existssuch thatand. Ifand, thenFandghave a common fixed point, that is, there existssuch that.

Theorem 16If in the above theorem, we replace the condition (ii) by the assumption thatXisg-ordered complete, then we have the conclusions of Theorem 15.

We next give the definition of G-compatible mappings inspired by the definition of compatible mappings in [13].

Definition 17 Let be a G-metric space. The mappings , are said to be G-compatible if

and

where and are sequences in X such that and for all are satisfied.

In this paper, we aim to extend the results on coupled fixed points mentioned above. Our results improve, enrich and extend some existing theorems in the literature. We also give examples to illustrate our results. This paper can also be considered as a continuation of the works of Berinde [11,12].

We start with an example which shows the weakness of Theorem 15.

Example 18 Let . Define by

for all . Let ⪯ be usual order. Then is a G-metric space. Define a map by and by for all . Let . Then we have

and

It is clear that there is no for which the statement (1.4) of Theorem 15 holds. Notice, however, that is the unique coupled coincidence point of F and g. In fact, it is a common fixed point of F and g, that is, .

We now state our first result which successively guarantees the existence of a coupled coincidence point.

Theorem 19Letbe a partially ordered set andbe aG-completeG-metric space. Letandbe two mappings such thatFhas the mixedg-monotone property onXand

for allwith, . Assume that, gisG-continuous and thatFandgareG-compatible mappings. Suppose further that either

(a) Fis continuous or

(b) isg-ordered complete.

Suppose also that there existsuch thatand. If, thenFandghave a coupled coincidence point, that is, there existssuch thatand.

Proof Let be such that and . Using the fact that , we can construct two sequences and in X in the following way:

We shall prove that for all ,

Since and and and , we have and , that is, (2.5) holds for . Assume that (2.5) holds for some . Since F has the mixed g-monotone property, from (2.4), we have

and

By mathematical induction, it follows that (2.5) holds for all , that is,

and

If there exists such that , then F and g have a coupled coincidence point. Indeed, in that case we would have

We suppose that for all . More precisely, we assume that either or .

For , we set

Then by using (2.3) and (2.6), for each , we have

which yields that

Now, for all with , by using rectangle inequality (G5) of G-metric and (2.10), we get

which yields that

Then by Proposition 6, we conclude that the sequences and are G-Cauchy.

Noting that is G-complete, there exist such that and are G-convergent to x and y respectively, i.e.,

Since F and g are G-compatible mappings, by (2.11), we have

Suppose that the condition (a) holds. For all , we have

Letting in the above inequality, using (2.11), (2.12) and the continuities of F and g, we have

Hence, we derive that and , that is, is a coupled coincidence point of F and g. Suppose that the condition (b) holds. By (2.8), (2.9) and (2.11), we have

Due to the fact that F and g are G-compatible mappings and g is continuous, by (2.11) and (2.12), we have

Keeping (2.15) and (2.16) in mind, we consider now

Letting in the above inequality, by using (2.15), (2.16) and the continuity of g, we conclude that

By (G1), we have and . Consequently, the element is a coupled coincidence point of the mappings F and g. □

Corollary 20Letbe a partially ordered set andbe aG-metric space such thatisG-complete. Letandbe two mappings such thatFhas the mixedg-monotone property onXand

for allwith, . Assume that, the self-mappinggisG-continuous andFandgareG-compatible mappings. Suppose that either

(a) Fis continuous or

(b) isg-ordered complete.

Suppose also that there existsuch thatand. If, thenFandghave a coupled coincidence point.

Proof It is sufficient to take and in Theorem 19. □

Corollary 21Letbe a partially ordered set andbe aG-metric space such thatisG-complete. Letandbe two mappings such thatFhas the mixedg-monotone property onXand

for allwith, . Assume thatand that the self-mappinggisG-continuous and commutes withF. Suppose that either

(a) Fis continuous or

(b) isg-ordered complete.

Suppose further that there existsuch thatand. If, thenFandghave a coupled coincidence point.

Proof Since g commutes with F, then F and g are G-compatible mappings. Thus, the result follows from Theorem 19. □

Corollary 22Letbe a partially ordered set andbe aG-metric space such thatisG-complete. Letandbe two mappings such thatFhas the mixedg-monotone property onXand

for allwith, . Assume thatand thatgisG-continuous and commutes withF. Suppose that either

(a) Fis continuous or

(b) isg-ordered complete.

Assume also that there existsuch thatand. If, thenFandghave a coupled coincidence point.

Proof Since g commutes with F, then F and g are G-compatible mappings. Thus, the result follows from Corollary 20. □

Letting in Theorem 19 and in Corollary 20, we get the following results.

Corollary 23Letbe a partially ordered set andbe aG-metric space such thatisG-complete. Letbe a mapping having the mixed monotone property onXand

for allwith, . Suppose that either

(a) Fis continuous or

(b) is ordered complete.

Suppose also that there existsuch thatand. If, thenFhas a coupled fixed point.

Corollary 24Letbe a partially ordered set andbe aG-metric space such thatisG-complete. Letbe a mapping having the mixed monotone property onXand

for allwith, . Suppose that either

(a) Fis continuous or

(b) is ordered complete.

Suppose further that there existsuch thatand. If, thenFhas a coupled fixed point.

Example 25 Let us recall Example 18. We have

and

It is clear that there any provides the statement (2.3) of Theorem 19.

Notice that is the unique coupled coincidence point of F and g which is also common coupled fixed point, that is, .

Example 26 Let . Define by

for all . Let ⪯ be usual order. Then is a G-metric space.

Define a map by

and by for all . Then . We observe that

and

then the statement (2.3) of Theorem 19 is satisfied for any and .

Notice that if we replace the condition (2.3) of Theorem 19 with the condition (1.4) of Theorem 15 [21], that is,

where , then the coupled coincidence point exists even though the contractive condition is not satisfied.

More precisely, consider . Then we have

and

It is clear that the condition (2.27) holds for .

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

The second author gratefully acknowledges the support provided by the Department of Mathematics and Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) during his stay at the Department of Mathematical and Statistical Sciences, University of Alberta as a visitor for the short term research.

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