Abstract
Using a combination of techniques introduced by Jleli and Samet (Fixed Point Theory Appl. 2012:210, 2012) and Samet et al. (Int. J. Anal. 2013:917158, 2013) on the one hand, and by Kadelburg et al. (Bull. Math. Anal. Appl. 4:5163, 2012) on the other hand, we show that several coupled fixed point results in (ordered) Gmetric spaces obtained recently are simple consequences of the respective standard (ordered) metric results. The technique can be applied both in symmetric and asymmetric cases. Moreover, we show by an example that the results thus obtained are usually stronger than those presented in the literature.
MSC: 47H10, 54H25.
Keywords:
Gmetric space; coupled fixed point1 Introduction
As one of fruitful generalizations of metric spaces, Gmetric spaces were introduced by Mustafa and Sims in [1]. In several subsequent papers, these and other authors obtained many fixed point and common fixed point results, thus extending the known theory from the standard metric case. It should be noted that there exist two kinds of Gmetric spaces, symmetric and asymmetric ones, and while it was immediately clear that in the symmetric case these results can be easily reduced to their metric counterparts, in the asymmetric case, new proofs usually had to be found.
The notion of a coupled fixed point for mappings with two variables was introduced in the articles [24]. After that, a great number of mathematicians worked in this field and obtained a lot of results in metric and various abstract metric spaces (see, e.g., [510]). Coupled fixed points in Gmetric spaces were investigated, e.g., in the papers [1123].
Very recently, some new methods were presented for obtaining fixed point and coupled fixed point results. On the one hand, Jleli and Samet [24] and Samet et al.[25] showed that there is a very simple technique for reducing fixed point results in Gmetric spaces, both in symmetric and in asymmetric cases, to their metric counterparts, avoiding complicated proofs from the known papers. On the other hand, in the papers [2633], the authors presented another technique which reduces coupled fixed point results in metric and various abstract metric spaces to the results for mappings with one variable. This technique was used by Kadelburg et al.[34] and afterwards by Agarwal and Karapinar [35] to obtain coupled fixed point results in symmetric Gmetric spaces.
By combining the mentioned techniques, we show in this paper that several coupled fixed point results in (ordered) Gmetric spaces obtained in recent years and presented in the papers [1123] simply follow from the wellknown standard (ordered) metric results for mappings with one variable. The technique can be applied in both symmetric and asymmetric cases. Moreover, we will show by an example that the results obtained in this way are usually stronger and can be applied in a greater number of cases.
2 Preliminaries
For more details on the following definitions and results concerning Gmetric spaces, we refer the reader to [1].
Definition 1 Let be a nonempty set, and let 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 Gmetric on and the pair is called a Gmetric space.
Definition 2 Let be a Gmetric space, and let be a sequence of points in .
1. A point is said to be the limit of a sequence if , and one says that the sequence is gconvergent to x.
2. The sequence is said to be a GCauchy sequence if, for every , there is a positive integer N such that for all ; that is, if as .
3. is said to be Gcomplete (or a complete Gmetric space) if every GCauchy sequence in is Gconvergent in .
It was shown in [1] that a Gmetric induces a Hausdorff topology and that the convergence, as described in the above definition, is relative to this topology. The topology being Hausdorff, a sequence can converge to at most one point.
Definition 3 A Gmetric space is called symmetric if
The following are some simple examples of Gmetric spaces.
Example 1 (1) Let be an ordinary metric space. Define g by
for all . Then it is clear that is a symmetric Gmetric space.
and extend g to by using the symmetry in the variables. Then it is clear that is an asymmetric Gmetric space.
Remark 1 If is a Gmetric space, then
defines a standard metric on . If the Gmetric g is symmetric, this reduces to . This simple fact implies that most of the fixed point results in symmetric Gmetric spaces can be easily reduced to their metric counterparts. In the asymmetric case, another approach is needed.
Let be a partially ordered set, and .
1. f is said to have the hmixed monotone property if the following two conditions are satisfied:
If (the identity map), we say that f has the mixed monotone property.
2. A point is said to be a coupled coincidence point of f and h if and , and their common coupled fixed point if and .
3. The mappings f and h are called wcompatible if and whenever and .
If is a nonempty set, then the triple is called an ordered Gmetric space if:
3 Main results
We will use the following simple lemma for obtaining our results.
Lemma 1Letbe an orderedGmetric space.
(a) If a relation ⊑ is defined onby
for, thenandare orderedGmetric spaces. The spacesandare complete iffis complete.
(b) Ifis a selfmap, and a mappinghas thehmixed monotone property, then the mappinggiven by
isHnondecreasing w.r.t. ⊑, i.e.,
(c) Ifhis continuous in, thenHis continuous in bothand. Iffis continuous fromto (resp. fromto), thenFis continuous in (resp. in).
Proof We will only check the second part of assertion (a); the proofs of all other parts are straightforward.
It was stated already in [1] that if g is a Gmetric on , then
define standard metrics on , and that topologies thus generated are the same as the topology of . In particular, the completeness is satisfied simultaneously.
On the other hand, it is well known that for each (standard) metric space , the mappings
for are metrics on , also preserving the completeness property. Combining these two facts, we obtain that the mappings
satisfy all the stated properties. □
Remark 2 If the given Gmetric space is symmetric, then, using the construction given in [[34], Lemma 3.1] and afterwards in [[35], Sections 4 and 5], we can obtain the previous result in a slightly different way. Namely, in this case, one can consider the Gmetric G on given by
and then associate the metrics
It is easy to see that, in fact, and .
However, this approach cannot be used in the asymmetric case, since in this case, (3.1) does not define a Gmetric on (see [[1], Section 4] and, further, Example 2).
Now we are ready to state some of our main results. We start, as a sample, with the following theorem.
Theorem 1Letbe a complete partially orderedGmetric space, and letandbe mappings such that, is closed andfhas the mixedhmonotone property. Suppose that there existsuch thatand. Suppose also that there existssuch that
for allsatisfying (and) or (and). Let us assume also that ifis a nondecreasing sequence inconverging to some, thenfor each. Then the mappingsfandhhave a coupled coincidence point. Moreover, iffandharewcompatible, thenfandhhave a common coupled fixed point.
Proof Consider the complete partially ordered metric space and the mappings and defined in Lemma 1. Then, obviously, the following conditions hold:
2. F is Hnondecreasing;
Moreover, it follows from (3.2) that there exists such that
holds for all such that or . Now, all the conditions of a special case of [[37], Theorem 2.2] are fulfilled and it follows that the mappings F and H have a coincidence point which is, obviously, a coupled coincidence point of f and h. The last assertion also follows easily. □
Corollary 1Letbe a complete partially orderedGmetric space, and letandbe mappings such that, is closed andfhas the mixedhmonotone property. Suppose that there existsuch thatand. Suppose also that there existssuch that
for allsatisfying (and) or (and). Let us assume also that ifis a nondecreasing sequence inconverging to some, thenfor each. Then the mappingsfandhhave a coupled coincidence point. Moreover, iffandharewcompatible, thenfandhhave a common coupled fixed point.
Proof We have only to prove that condition (3.4) implies condition (3.2). Indeed, putting and in (3.4), we get that
Adding up, and taking into account the definitions of mappings F and H as well as the definition of metric , we obtain condition (3.2) (i.e., condition (3.3)). □
In an even easier way, one can obtain the following versions of the previous results in the space without order. In this case, the given technique reduces the problem simply to the Banach contraction principle.
Theorem 2Letbe a completeGmetric space, and letandbe mappings such thatandis closed. Suppose also that there existssuch that condition (3.2) holds for all. Ifhis continuous, then the mappingsfandhhave a coupled coincidence point. Moreover, iffandharewcompatible, thenfandhhave a common coupled fixed point.
Corollary 2Letbe a completeGmetric space, and letandbe mappings such thatandis closed. Suppose also that there existssuch that condition (3.4) holds for all. Ifhis continuous, then the mappingsfandhhave a coupled coincidence point. Moreover, iffandharewcompatible, thenfandhhave a common coupled fixed point.
Remark 3 The last result was obtained (in the special case ) in [[34], Corollary 4.1] and afterwards (implicitly) in the course of proof of [[35], Theorem 5.3], but in the case when the given Gmetric g is symmetric. The proof from these articles cannot be applied in the asymmetric case (see Remark 2 and, further, Example 2).
Remark 4 The obtained results are strict improvements of some results obtained earlier. In the symmetric case, this was shown by Kadelburg et al. (see [[34], Example 4.1]). We present an example in an asymmetric case.
Example 2 Let be the Gmetric space considered in [[38], Example 3.4], i.e., let and be given as
and extended by symmetry in the variables. Then it is easy to check that g is a Gmetric which is asymmetric since and .
Let be given as , and let be defined by
We will show first that f and h satisfy neither the condition
for any (which is condition (3.1) from [15]) nor (weaker) condition (3.4). Indeed, take, e.g., and and condition (3.4) becomes
which is a contradiction for any . Hence, neither [[15], Theorem 3.1] nor Corollary 2 can be used to conclude that f and h have a coupled coincidence point (i.e., that f has a coupled fixed point).
In order to show that f and h satisfy the conditions of Theorem 2, we first note that , and
Take now . By a careful calculation (there are 21 nontrivial cases) it can be checked that condition (3.2) (i.e., condition (3.3)) is satisfied for all .
Hence, Theorem 2 can be applied to conclude that f has a coupled fixed point (which is ).
We note also that the approach from the papers [34,35] cannot be used in this example since
does not define a Gmetric on . Indeed, e.g.,
although , and the property (G3) of Gmetrics is not satisfied.
For the sake of simplicity, we shall consider in the rest of the paper ‘unordered’ versions of coupled fixed point results. ‘Ordered’ versions can be formulated and proved using usual variations.
The proof of our next result uses the metric .
Theorem 3Letbe a completeGmetric space, and let. Suppose that there existssuch that for allthe following inequality holds:
Thenfhas a unique coupled fixed point inand it is of the formfor some.
Proof Consider the complete metric space and the mapping defined in Lemma 1. Putting and in (3.5), one gets
interchanging the places of x and y, as well as of s and t, this gives
Putting now and in (3.5), one gets
interchanging the places of x and y, as well as of s and t, this gives
Taking into account the definition of metric in Lemma 1, it follows from the last four inequalities that
holds for all . By a wellknown result from the theory of standard metric spaces (see, e.g., [39]), it follows that there exists a unique point such that . Obviously, is a coupled fixed point of the mapping f. Since the coupled fixed point is unique, it must be of the form for some . □
When considering situations where contractive conditions are of ‘weak’ kind, or they use the socalled comparison functions, a variation of the previous approach is needed. One possibility is to use quasimetrics. Recall that a pair is called a quasimetric space if the mapping d has all the properties of a metric except, possibly, the symmetry . For some properties of quasimetric spaces, we refer to [24]. In particular, the following fixed point result was proved in that paper.
Theorem 4 [[24], Theorem 3.2]
Letbe a complete quasimetric space, and letbe a mapping satisfying
for all, whereis continuous with. ThenThas a unique fixed point.
Remark 5 An additional nondecreasing function ψ could be added in (3.6) to become
However, it was shown in [40] that it is redundant, hence we will not use it here.
Obviously, if is a Gmetric space, then
defines a quasimetric on . Moreover,
defines a quasimetric on . The space is complete iff is Gcomplete. Now we can easily prove the following theorem.
Theorem 5Letbe a completeGmetric space, and letsatisfy
for all, whereis continuous with. Thenfhas a unique coupled fixed point.
Proof Consider the (complete) quasimetric space given by (3.7) and the selfmapping F given on by for . Then contractive condition (3.8) gives
for . Hence, Theorem 4 can be applied to conclude that F has a unique fixed point, which is then a coupled fixed point of f. □
Another possibility is to impose an additional condition on the function φ, or on the comparison function Φ. This is illustrated in the next result.
Theorem 6Letbe a completeGmetric space, and letsatisfy the condition
for all, whereis rightcontinuous, forandfor. Thenfhas a unique coupled fixed point.
Proof Consider again the space and the mapping F given in Lemma 1. Putting and in (3.9), we get
interchanging places of x and y, as well as s and t, we obtain
On the other hand, putting and in (3.9), we get
interchanging places of x and y, as well as s and t, we obtain
Adding up the last four inequalities, and using the assumed properties of function Φ, we get that
for all . Hence, by a classical result of Boyd and Wong [41], it follows that F has a unique fixed point, which is then a coupled fixed point of f. □
It is clear that a lot of other known coupled fixed point results in Gmetric spaces (both symmetric and asymmetric) can be easily obtained in this way.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors are indebted to the referees whose suggestions helped them to improve the exposition. The second and third authors are thankful to the Ministry of Education, Science and Technological Development of Serbia.
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