Abstract
The purpose of this paper is to extend some recent coupled fixed point theorems in the context of partially ordered Gmetric spaces in a virtually different and more natural way.
MSC: 46N40, 47H10, 54H25, 46T99.
Keywords:
coupled fixed point; coupled coincidence point; mixed gmonotone property; ordered set; Gmetric space1 Introduction and preliminaries
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, quasimetric spaces and bmetric 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 Gmetric space in 2004 (see also [47]). 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 Gmetrics. 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., [1020]). Upon the introduction of the notion of coupled fixed points by Guo and Laksmikantham [14], GnanaBhaskar 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, [2135]).
In this paper, we prove the theorem that amalgamates these three seminal approaches in the study of fixed point theory, the so called Gmetrics, 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 nonnegative integers and the set of positive integers respectively.
Definition 1 (See [3])
Let X be a nonempty set, be a function satisfying the following properties:
(G4) (symmetry in all three variables),
(G5) for all (rectangle inequality).
Then the function G is called a generalized metric or, more specially, a Gmetric on X, and the pair is called a Gmetric space.
It can be easily verified that every Gmetric on X induces a metric on X given by
Trivial examples of Gmetric are as follows.
Example 2 Let be a metric space. The function , defined by
or
The concepts of convergence, continuity, completeness and Cauchy sequence have also been defined in [3].
Definition 3 (See [3])
Let be a Gmetric space, and let be a sequence of points of X. We say that is Gconvergent 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 aGmetric space. The following statements are equivalent:
Definition 5 (See [3])
Let be a Gmetric space. A sequence is called GCauchy sequence if for any , there is such that for all , that is, as .
Proposition 6 (See [3])
Letbe aGmetric space. The following statements are equivalent:
(2) For any, there existssuch that, for all.
Definition 7 (See [3])
A Gmetric space is called Gcomplete if every GCauchy sequence is Gconvergent in .
Definition 8 Let be a Gmetric space. A mapping is said to be continuous if for any three Gconvergent sequences , and converging to x, y and z respectively, is Gconvergent to .
We define below gordered complete Gmetric spaces.
Definition 9 Let be a partially ordered set, be a Gmetric space and be a mapping. A partially ordered Gmetric space, , is called gordered complete if for each Gconvergent sequence , the following conditions hold:
() If is a nonincreasing sequence in X such that , then .
() If is a nondecreasing sequence in X such that , then .
In particular, if g is the identity mapping in () and (), the partially ordered Gmetric 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, GnanaBhaskar 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 nondecreasing in x and is monotone nonincreasing 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 gmonotone property.
Definition 12 Let be a partially ordered set, and . The function F is said to have mixed gmonotone property if is monotone gnondecreasing in x and is monotone gnonincreasing 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 orderedGmetric space. Letandbe mappings such thatFhas the mixedgmonotone property, and let there existsuch thatand. Suppose that there existssuch that for allthe following holds:
for alland, where eitheror. Assume the following hypotheses:
(iii) gisGcontinuous 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 thatXisgordered complete, then we have the conclusions of Theorem 15.
We next give the definition of Gcompatible mappings inspired by the definition of compatible mappings in [13].
Definition 17 Let be a Gmetric space. The mappings , are said to be Gcompatible 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].
2 Main results
We start with an example which shows the weakness of Theorem 15.
for all . Let ⪯ be usual order. Then is a Gmetric 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 aGcompleteGmetric space. Letandbe two mappings such thatFhas the mixedgmonotone property onXand
for allwith, . Assume that, gisGcontinuous and thatFandgareGcompatible mappings. Suppose further that either
(a) Fis continuous or
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:
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 gmonotone 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 .
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 Gmetric and (2.10), we get
which yields that
Then by Proposition 6, we conclude that the sequences and are GCauchy.
Noting that is Gcomplete, there exist such that and are Gconvergent to x and y respectively, i.e.,
Since F and g are Gcompatible 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 Gcompatible 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 aGmetric space such thatisGcomplete. Letandbe two mappings such thatFhas the mixedgmonotone property onXand
for allwith, . Assume that, the selfmappinggisGcontinuous andFandgareGcompatible mappings. Suppose that either
(a) Fis continuous or
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 aGmetric space such thatisGcomplete. Letandbe two mappings such thatFhas the mixedgmonotone property onXand
for allwith, . Assume thatand that the selfmappinggisGcontinuous and commutes withF. Suppose that either
(a) Fis continuous or
Suppose further that there existsuch thatand. If, thenFandghave a coupled coincidence point.
Proof Since g commutes with F, then F and g are Gcompatible mappings. Thus, the result follows from Theorem 19. □
Corollary 22Letbe a partially ordered set andbe aGmetric space such thatisGcomplete. Letandbe two mappings such thatFhas the mixedgmonotone property onXand
for allwith, . Assume thatand thatgisGcontinuous and commutes withF. Suppose that either
(a) Fis continuous or
Assume also that there existsuch thatand. If, thenFandghave a coupled coincidence point.
Proof Since g commutes with F, then F and g are Gcompatible 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 aGmetric space such thatisGcomplete. Letbe a mapping having the mixed monotone property onXand
for allwith, . Suppose that either
(a) Fis continuous or
Suppose also that there existsuch thatand. If, thenFhas a coupled fixed point.
Corollary 24Letbe a partially ordered set andbe aGmetric space such thatisGcomplete. Letbe a mapping having the mixed monotone property onXand
for allwith, . Suppose that either
(a) Fis continuous or
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, .
for all . Let ⪯ be usual order. Then is a Gmetric space.
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
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
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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