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Multidimensional coincidence point theorems for weakly compatible mappings with the \(CLR_{g}\)property in (fuzzy) metric spaces
 J MartínezMoreno^{1},
 A Roldán^{1},
 C Roldán^{2} and
 Yeol Je Cho^{3, 4}Email author
https://doi.org/10.1186/s1366301502972
© MartínezMoreno et al.; licensee Springer. 2015
 Received: 24 December 2014
 Accepted: 18 March 2015
 Published: 11 April 2015
Abstract
In this paper, we extend the concepts of the \(E.A\)property and the \(CLR_{g}\)property in fuzzy metric spaces to the setting of multidimensional fuzzy metric spaces and show the existence of multidimensional coincidence point and fixed point theorems for weakly compatible mappings with the \(E.A\)property and the \(CLR_{g}\)property.
Keywords
 fuzzy metric space
 coincidence point
 fixed point
 contraction
 the \(E.A\) property
 the \(CLR_{g}\)property
1 Introduction
Since Banach’s fixed point theorem in 1922, many authors have improved, extended and generalized this theorem in many different ways. One of the newest branches of this theorem is devoted to the existence of coupled fixed point, which was introduced by Guo and Lakshmikantham [1] in 1987. Later, Berinde and Borcut [2] introduced the concept of tripled fixed point and proved some tripled fixed point theorems using mixed monotone mappings (see also [3, 4]). Recently, Roldán et al. [5] proposed the notion of coincidence point for nonlinear mappings with any number of variables and showed the existence and uniqueness theorems that extended the mentioned previous results for nonlinear mappings, not necessarily permuted or ordered, in the framework of partially ordered complete metric spaces by using some weaker contractive condition, which also generalized other works by Berzig and Samet [6].
Especially in [7], the existence results of coincidence points for the nonlinear mappings in any number of variables in fuzzy metric spaces were presented.
In 2002, Aamri and Moutawakil [8] defined the notion of the \(E.A\)property for nonlinear selfmappings which contained the class of noncompatible mappings in metric spaces. It was pointed out that the \(E.A\)property allows replacing the completeness requirement of the space with a more natural condition of closedness of the range as well as relaxes the completeness of the whole space, continuity of one or more mappings and containment of the range of one mapping into the range of another, which is utilized to construct the sequence of some joint iterates. Since Aamri and Moutawakil, many authors have also proved common fixed point theorems in fuzzy metric spaces for different contractive conditions.
Recently, Sintunavarat and Kumam [9] defined the notion of the \(CLR_{g}\)property in fuzzy metric spaces and improved the results of Mihet [10] without any requirement of the closedness of the space.
In this paper, we extend the notions of the \(E.A\)property and the \(CLR_{g}\)property for nonlinear mappings with any number of variables and use these notions to present the existence results of coincidence points for weakly compatible mappings in fuzzy metric spaces. Our results improve, extend and generalize many fixed point theorems in metric spaces and fuzzy metric spaces given by some authors.
2 Preliminaries
Let n be a positive integer and let \(\Lambda_{n}=\{1,2,\ldots,n\}\). Henceforth, X denotes a nonempty set and \(X^{n}\) denotes the product space \(X\times X\times\overset{n}{\cdots}\times X\). We represent the identity mapping on X as \(I_{X}\).
Throughout this manuscript, m and p denote nonnegative integers, t is a positive real number and \(i,j,s\in\{1,2,\ldots,n\}\). Unless otherwise stated, ‘for all m’ will mean ‘for all \(m\geq 0\)’, ‘for all t’ will mean ‘for all \(t>0\)’ and ‘for all i’ will mean ‘for all \(i\in\{1,2,\ldots,n\}\)’. Let us denote \(\mathbb{R}^{+}= ( 0,\infty) \) and \(\mathbb{I}=[0,1]\).
In the sequel, let \(F:X^{n}\rightarrow X\) and \(g:X\rightarrow X\) be two mappings. For brevity, \(g(x)\) is denoted by gx.
Let \(F:X^{n}\rightarrow X\) and \(g:X\rightarrow X\) be two mappings. Henceforth, let \(\sigma_{1},\sigma_{2},\ldots,\sigma_{n}:\Lambda _{n}\rightarrow\Lambda_{n}\) be n mappings from \(\Lambda_{n}\) into itself, and let Φ be the ntuple \((\sigma_{1},\sigma_{2},\ldots,\sigma_{n})\).
Definition 1
([5])
 (1)
GnanaBhaskar and Lakshmikantham’s election [11] in \(n=2\) is \(\sigma_{1}=\tau= ( 1,2 ) \) and \(\sigma_{2}= ( 2,1 ) \);
 (2)
Berinde and Borcut’s election [2] in \(n=3\) is \(\sigma _{1}=\tau= (1,2,3 ) \), \(\sigma_{2}= ( 2,1,2 ) \) and \(\sigma_{2}= ( 3,2,1 ) \);
 (3)
Karapınar’s election in \(n=4\) is \(\sigma_{1}=\tau= (1,2,3,4 ) \), \(\sigma_{2}= ( 2,3,4,1 ) \), \(\sigma _{3}= ( 3,4,1,2 ) \) and \(\sigma_{4}= ( 4,1,2,3 )\).
There exist different notions of fuzzy metric space (see [12]). For our purposes, we use the following one.
Definition 2
(George and Veeramani [13])
 (FM1)
\(M(x,y,t)>0\);
 (FM2)
\(M(x,y,t)=1\) if and only if \(x=y\);
 (FM3)
\(M(x,y,t)=M(y,x,t)\);
 (FM4)
\(M(x,y,\cdot):\mathbb{R}^{+}\rightarrow\mathbb{I}\) is continuous;
 (FM5)
\(M(x,y,t)\ast M(y,z,s)\leq M(x,z,t+s)\).
In this case, we also say that \((X,M)\) is an FMS under ∗.
 (FM6)
\(\lim_{t\rightarrow\infty}M(x,y,t)=1\) for all \(x,y\in X\).
Lemma 3
(Grabiec [14])
If \(( X,M ) \) is an \(FMS\) under some tnorm and \(x,y\in X\), then \(M(x,y,\cdot)\) is a nondecreasing function on \((0,\infty)\).
Let \(( X,M ) \) be an \(FMS\) under some tnorm. For all \(t,r>0\), the open ball with center \(x\in X\) is \(B(x,r,t)=\{y\in X:M(x,y,t)>1r\}\). A subset \(A\subseteq X\) is open if, for all \(x\in X\), there exists an open ball \(B(x,r,t)\) such that \(B(x,r,t)\subseteq A\).
George and Veeramani [13] proved that the family of all open sets of X is a Hausdorff topology \(\tau_{M}\) on X. In this topology, we may consider the following notions.
Definition 4
(3) An \(FMS\) in which every Cauchy sequence is convergent is said to be complete.
Lemma 5
(RodríguezLópez and Romaguera [15])
If \(( X,M ) \) is an \(FMS\) under some tnorm, then M is a continuous mapping on \(X^{2}\times(0,\infty)\).
For any tnorm ∗, it is easy to prove that \(\ast\leq\min\). Therefore, if \((X,M)\) is an \(FMS\) under min, then \((X,M)\) is an \(FMS\) under any (continuous or not) tnorm. This is the case in the following examples.
Example 6
Furthermore, \((X,d)\) is a complete metric space if and only if \((X,M^{d})\) (or \((X,M^{\mathrm{e}})\)) is a complete \(FMS\). For instance, this is the case for any nonempty closed subset of ℝ provided with its Euclidean metric.
The concept of the \(E.A\)property in a metric space has been recently introduced by Aamri and Moutawakil [8] and the concept of the \(CLR_{g}\)property by Sintunavarat and Kumam [9] is as follows.
Definition 7
(2) If \(t\in g(X)\), then f and g are said to satisfy the \(CLR_{g}\) property.
Similarly, we say that two selfmappings f, g of a fuzzy metric space \((X,M,\ast)\) satisfy the \(E.A\) property if there exist a sequence \(\{x_{n}\}\) in X and z in X such that \(fx_{n}\) and \(gx_{n}\) converge to z in the sense of Definition 4. Similarly, we say that if \(t\in g(X)\), then f and g are said to satisfy the \(CLR_{g}\) property.
3 Main results

for all \((x_{1},x_{2},\ldots,x_{n}),(y_{1},y_{2},\ldots,y_{n})\in X^{n}\),for all i.$$(x_{1}, x_{2},\ldots,x_{n}) \leq(y_{1}, y_{2},\ldots, y_{n}) \quad\Longleftrightarrow\quad x_{i} \preccurlyeq_{i} y_{i} $$
Recently, Roldán et al. [7] proved the following result.
Theorem 8
([7])
We are going to give a version of the above result using a pair of mappings satisfying the \(CLR_{g}\)property. The following definitions extend previous considerations from other authors.
Definition 9
Definition 10
 (1)
Sintunavarat and Kumam’s election [9] in \(n=1\);
 (2)
Jain, Tas, Kumar and Gupta’s election [16] and Khan and Sumitra’s election [17] in \(n=2\) is \(\sigma_{1}=\tau= ( 1,2 ) \) and \(\sigma_{2}= ( 2,1 )\);
 (3)
Wairojjana, Sintunavarat and Kumam’s election [18] in \(n=3\) is \(\sigma_{1}=\tau= (1,2,3 ) \), \(\sigma_{2}= ( 2,3,1 ) \) and \(\sigma_{3}= ( 3,2,1 )\) in abstract metric spaces.
Definition 11
Example 12
Note that the \(E.A\)property does not imply the \(CLR_{g}\)property and, in [17], there is an example showing that the mappings satisfying the \(CLR_{g}\)property need not be continuous.
The following result does not require the conditions on the completeness (or the closedness) of the underlying space together with the conditions on continuity and Hadžić’s condition of t.
Theorem 13
Proof
Corollary 14
Under the hypothesis of Theorem 13, assume also that F and g are weakly compatible. If \((x_{1},x_{2},\ldots,x_{n})\in X^{n}\) is a Φcoincidence point of F and g, then \((gx_{1},gx_{2},\ldots ,gx_{n})\) is also a Φcoincidence point of F and g.
Proof
Corollary 15
Let \((X,M,\ast)\) be an \(FMS\) and \(\Phi =(\sigma_{1},\sigma _{2},\ldots,\sigma_{n})\) be an ntuple of mappings from \(\{1,2,\ldots ,n\}\) into itself. Let \(F:X^{n}\rightarrow X\) and \(g:X\rightarrow X\) be two mappings satisfying the \(E.A\)property. Assume that there exists \(k\in(0,1)\) such that inequality (4) is satisfied, where \(\gamma:[0,1]\rightarrow{}[0,1]\) is a continuous mapping such that \(\ast^{n}\gamma(a)\geq a\) for each \(a\in[0,1]\). If \(g(X)\) is a closed subspace of X, then F and g have at least one Φcoincidence point.
Proof
4 The uniqueness of Φcoincidence points
Theorem 17
Proof
From Theorem 13, the set of Φcoincidence points of F and g is nonempty. The proof is divided in two steps.
Step 2. We claim that \((z_{1},z_{2},\ldots,z_{n})\) is the unique Φcoincidence point of F and g such that \(gz_{i}=z_{i}\) for all i. Indeed, by Step 1, we observe that \(gz_{i}=gx_{i}=z_{i}\) for all i. Suppose that \((z_{1}^{\prime},z_{2}^{\prime},\ldots,z_{n}^{\prime})\in X^{n}\) is another Φcoincidence point of F and g such that \(gz_{i}^{\prime}=z_{i}^{\prime} \) for all i. Since \((z_{1},z_{2},\ldots,z_{n})\) and \((z_{1}^{\prime},z_{2}^{\prime},\ldots ,z_{n}^{\prime})\) are two Φcoincidence points of F and g, by Step 1 it follows that \(gz_{i}=gz_{i}^{\prime}\) for all i and so \(z_{i}=gz_{i}=gz_{i}^{\prime}=z_{i}^{\prime}\) for all i. Therefore, \((z_{1},z_{2},\ldots,z_{n})\) is the unique Φcoincidence point of F and g such that \(gz_{i}=z_{i}\) for all i. This completes the proof. □
Corollary 18
In addition to the hypotheses of Theorem 17, suppose that \(\ast=\min\), \(\gamma(a)=a\) for all \(a\in\mathbb{I}\) and \((z_{1},z_{2},\ldots ,z_{n})\in X^{n}\) is the unique Φcoincidence point of F and g. Then \(z_{1}=z_{2}=\cdots=z_{n}\). In particular, there exists a unique \(z\in X\) such that \(F(z,z,\ldots ,z)=z\), which verifies \(gz=z\).
Proof
5 Some results in metric spaces
It seems natural to introduce the analogous definitions and results in the setting of metric spaces by using the results in fuzzy metric spaces.
Definition 19
Remark 20
Let \((X,d)\) be a metric space, \(\Phi=(\sigma_{1},\sigma _{2},\ldots,\sigma_{n})\) be an ntuple of mappings from \(\{1,2,\ldots ,n\}\) into itself. If we consider the FMS \((X,M^{\mathrm{e}},\min)\) induced by d, see Example 6, then the \(CLR_{g}\) property on \((X,d)\) coincides with the \(CLR_{g}\) property on \((X,M^{\mathrm{e}},\min)\).
From Theorem 13, we have the following in metric spaces.
Corollary 21
Proof
If \(n=1\), then we have the following.
Corollary 22
It is well known that in the setting of metric spaces multidimensional results only depend on their first argument. Therefore, multidimensional results reduce to the one dimensional case.
Proof
 (a)
F and g satisfy the \(CLR_{g}\)property if and only if T and G satisfy the \(CLR_{g}\)property;
 (b)
\((x_{1},x_{2},\ldots,x_{n})\in X^{n}\) is a Φcoincidence point of the mappings F and g if and only if \((x_{1},x_{2},\ldots,x_{n})\in X^{n}\) is a coincidence point of the mappings T and G.
Declarations
Acknowledgements
The fourth author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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