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Multidimensional coincidence point theorems for weakly compatible mappings with the \(CLR_{g}\)property in (fuzzy) metric spaces
Fixed Point Theory and Applications volume 2015, Article number: 53 (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.
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.
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])
A point \((x_{1},x_{2},\ldots,x_{n})\in X^{n}\) is called a Φcoincidence point of the mappings F and g if
for all i. If g is the identity mapping on X, then \((x_{1},x_{2},\ldots ,x_{n})\in X^{n}\) is called a Φfixed point of the mapping F.
If \(\sigma:\Lambda_{n}\rightarrow\Lambda_{n}\) is a mapping, then, from its ordered image, i.e., \(\sigma= ( \sigma(1),\sigma (2),\ldots,\sigma(n) )\), we have the following:

(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])
A triple \((X,M,\ast)\) is called a fuzzy metric space (briefly, an \(FMS\)) if X is an arbitrary nonempty set, ∗ is a continuous tnorm and \(M:X\times X\times\mathbb{R}^{+}\rightarrow\mathbb{I}\) is a fuzzy set satisfying the following conditions: for all \(x,y,z\in X\) and \(t,s>0\):

(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 ∗.
In the sequel, we only consider an \(FMS\) satisfying the following:

(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
(1) A sequence \(\{x_{m}\}_{m\geq0}\subset X\) is a Cauchy sequence if, for any \(\varepsilon>0\) and \(t>0\), there exists \(m_{0}\in\mathbb{N}\) such that
for all \(m\geq m_{0}\) and \(p\geq1\);
(2) A sequence \(\{x_{m}\}_{m\geq0}\subset X\) is convergent to a point \(x\in X\) denoted by \(\lim_{m\rightarrow\infty}x_{m}=x\) if, for any \(\varepsilon>0\) and \(t>0\), there exists \(m_{0}\in\mathbb{N}\) such that
for all \(m\geq m_{0}\);
(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
From a metric space \((X,d)\), we can consider an FMS in different ways. For all \(t>0\) and \(x,y\in X\) with \(x\neq y\), define
It is well known that \((X,M^{d})\) is an \(FMS\) under the product ∗ = ⋅ called the standard \(FMS\) on \((X,d)\) since it is the standard way of viewing the metric space \((X,d)\) as an \(FMS\). However, it is also true that \((X,M^{d})\) and \((X,M^{\mathrm{e}})\) are \(FMS\) under min.
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
(1) Two selfmappings f and g of a metric space \((X,d)\) are said to satisfy the \(E.A\) property if there exists a sequence \(\{x_{n}\}\) in X such that
for some \(t\in X\);
(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.
Main results
Henceforth, fix a partition \(\{A,B\}\) of \(\Lambda_{n}=\{1,2,\ldots,n\}\), that is, \(A\cup B=\Lambda_{n}\) and \(A\cap B=\emptyset\). We denote the following:
and
If ≼ is a partial order on X (i.e., \((X,\preccurlyeq)\) is a partially ordered set), then we use the following notation:
Consider on the product space \(X^{n}\) the following partial order:

for all \((x_{1},x_{2},\ldots,x_{n}),(y_{1},y_{2},\ldots,y_{n})\in X^{n}\),
$$(x_{1}, x_{2},\ldots,x_{n}) \leq(y_{1}, y_{2},\ldots, y_{n}) \quad\Longleftrightarrow\quad x_{i} \preccurlyeq_{i} y_{i} $$for all i.
Recently, Roldán et al. [7] proved the following result.
Theorem 8
([7])
Let \(\{A,B\}\) be a partition of \(\Lambda_{n}=\{1,2,\ldots,n\}\) and
and
Let \((X,M,\ast,\preccurlyeq)\) be a complete partially ordered \(FMS\) such that ∗ is a tnorm of Htype. Let \(\Phi=(\sigma_{1},\sigma _{2},\ldots,\sigma_{n})\) be an ntuple of mappings from \(\{1,2,\ldots ,n\}\) into itself with \(\sigma_{i}\in\Omega_{A,B}\) if \(i\in A\) and \(\sigma _{i}\in\Omega_{A,B}^{\prime}\) if \(i\in B\). Let \(F:X^{n}\rightarrow X\) and \(g:X\rightarrow X\) be two mappings such that F has the mixed \((g,\preccurlyeq)\)monotone property on X, \(F(X^{n})\subseteq g(X)\) and g is continuous and Φcompatible with F. Assume that there exists \(k\in(0,1)\) such that
for all \(t>0\) and all \(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\in X\) with \(gx_{i}\leq_{i}gy_{i}\) for all i, where \(\gamma:[0,1]\rightarrow {}[0,1]\) is a continuous mapping such that \(\ast^{n}\gamma(a)\geq a\) for each \(a\in{}[0,1]\). Suppose that
for all \(j\in\{1,2,\ldots,n\}\) and \(x_{1},x_{2},\ldots ,x_{n},y_{1},y_{2},\ldots,y_{n}\in X\) such that \(gx_{i}\preccurlyeq_{i}gy_{i}\) for all i. Suppose that either F is continuous or \((X,\tau_{M},\preccurlyeq)\) has the sequential gmonotone property. If there exist \(x_{0}^{1},x_{0}^{2},\ldots,x_{0}^{n}\in X\) such that
for all i, then F and g have at least one Φcoincidence point in X.
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
Let \(\Phi=(\sigma_{1},\sigma _{2},\ldots,\sigma_{n})\) be an ntuple of mappings from \(\{1,2,\ldots ,n\}\) into itself. The mappings \(F:X^{n}\rightarrow X\) and \(g:X\rightarrow X\) are said to be Φweakly compatible if
whenever \(gx_{i}=F(x_{\sigma_{i}(1)},x_{\sigma_{i}(2)},\ldots,x_{\sigma _{i}(n)})\) for all i and some \((x_{1},x_{2},\ldots,x_{n})\in X^{n}\).
Definition 10
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. Two mappings \(F:X^{n}\rightarrow X\) and \(g:X\rightarrow X\) are said to satisfy the Φcommon limit in the range of the gproperty (the \(CLR_{g}\) property for short) if there exist n sequences \(\{x_{m}^{1}\}_{m\geq0}, \{x_{m}^{2}\}_{m\geq0}, \ldots, \{ x_{m}^{n}\}_{m\geq0}\) such that
for all i and some \(x_{1},\ldots,x_{n}\in X\).
If \(\sigma:\Lambda_{n}\rightarrow\Lambda_{n}\) is a mapping, then, from its ordered image, i.e., \(\sigma= ( \sigma(1),\sigma (2),\ldots,\sigma(n) )\), we have the following:

(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
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. Two mappings \(F:X^{n} \rightarrow X\) and \(g:X\rightarrow X\) are said to satisfy the Φ \(E.A\) property (shortly, the \(E.A\) property) if there exist n sequences \(\{x_{m}^{1}\}_{m\geq 0}, \{x_{m}^{2}\}_{m\geq0}, \ldots, \{x_{m}^{n}\}_{m\geq0}\) such that
for all i and some \(x_{1},\ldots,x_{n}\in X\).
Example 12
Let \((\mathbb{R},M,\ast)\) be a fuzzy metric space and ∗ be a continuous tnorm. Define \(M(x,y,t)=\frac{t}{t+xy}\) for all \(x,y\in\mathbb{R}\) and \(t>0\). Define the mappings \(F:\mathbb{R}^{n} \rightarrow\mathbb{R}\) and \(g:\mathbb{R} \rightarrow \mathbb{R}\) by
for all \(x_{1},\ldots,x_{n},x\in\mathbb{R}\) and consider the sequences
for all \(i=1,\ldots,n\). Then we have
Therefore, we have
and so on. Therefore, F and g satisfy the \(CLR_{g}\)property and the \(E.A\)property with \(\Phi=(\sigma_{1},\sigma _{2},\ldots,\sigma_{n})\) given by
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
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 \(CLR_{g}\)property. Assume that there exists \(k\in(0,1)\) such that
for all \(t>0\), \(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\in X\), where \(\gamma:[0,1]\rightarrow{}[0,1]\) is a continuous mapping such that \(\ast^{n}\gamma(a)\geq a\) for each \(a\in[0,1]\). Then F and g have at least one Φcoincidence point.
Proof
Since F and g satisfy the \(CLR_{g}\)property, there exist n sequences \(\{x_{m}^{1}\}_{m\geq0}, \{x_{m}^{2}\}_{m\geq0}, \ldots, \{x_{m}^{n}\}_{m\geq0}\) such that
for all i and some \(x_{1},\ldots,x_{n}\in X\). Then we have
By letting \(m\rightarrow\infty\), we have
Since γ verifies \(1\leq\ast^{n}\gamma(1)\leq\min(\gamma(1),\ldots ,\gamma(1))=\gamma(1)\), then we have \(\gamma(1)=1\). Therefore, it follows that
and so \((x_{1},x_{2},\ldots,x_{n})\) is a Φcoincidence point of F and g. This completes the 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
Now, we let \(z_{i}=gx_{i}=F(x_{\sigma_{i}(1)},x_{\sigma_{i}(2)},\ldots,x_{\sigma_{i}(n)})\). Since F and g are weakly compatible mappings, we have
Thus, from Theorem 13, we have the conclusion. □
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
Since F and g satisfy the \(E.A\)property, there exist n sequences \(\{x_{m}^{1}\}_{m\geq0}, \{x_{m}^{2}\}_{m\geq0}, \ldots, \{x_{m}^{n}\}_{m\geq0}\) such that
for all i and some \(x_{1},\ldots,x_{n}\in X\). Since \(g(X)\) is a closed subspace of X, it follows that \(x_{i}=g(w_{i})\) for some \(w_{i}\in X\) and for all i, and so F and g satisfy the \(CLR_{g}\)property. Therefore, by Theorem 13, F and g have at least one Φcoincidence point. □
Remark 16
Corollary 15 is a version of Theorem 13 for the \(E.A\)property. Similarly, we can write a new version by replacing the \(CLR_{g}\)property for the \(E.A\)property and the closedness of \(g(X)\) for all the result in this paper.
The uniqueness of Φcoincidence points
Theorem 17
Under the hypothesis of Corollary 14, suppose additionally that
for all \(j\in\{1,2,\ldots,n\}\) and \(x_{1},x_{2},\ldots ,x_{n},y_{1},y_{2},\ldots,y_{n}\in X\). Then F and g have a unique Φcoincidence point \((z_{1},z_{2},\ldots,z_{n})\in X^{n}\) such that \(gz_{i}=z_{i}\) for all i.
Proof
From Theorem 13, the set of Φcoincidence points of F and g is nonempty. The proof is divided in two steps.
Step 1. We claim that if \((x_{1},x_{2},\ldots,x_{n}), (y_{1},y_{2},\ldots,y_{n})\in X^{n}\) are two Φcoincidence points of F and g, then we have
for all i. Let \((x_{1},x_{2},\ldots,x_{n}),(y_{1},y_{2},\ldots,y_{n})\in X^{n}\) be two Φcoincidence points of F and g. For all t and m, define
Now, we claim that \(\delta_{m+1}(kt)\geq\delta _{m}(t)\) for all m and \(t>0\). By inequalities (4) and (7), it follows that, for all m, t and j,
Since \(\ast^{n}\gamma(a)\geq a\) for each \(a\in{}[0,1]\), it follows that
i.e., \(\delta_{m+1}(kt)\geq\delta_{m}(t)\geq\cdots\geq\delta _{0}(t/k^{m})\) and, as a consequence, \(\lim_{m\rightarrow\infty}\delta_{m}(t)=1\) for all \(t>0\). It follows that, for all i and t,
which means that \(\lim_{m\rightarrow\infty}M(gy_{m}^{i},gx_{i},t)=1\) for all t, i.e.,
for all i. Let \((x_{1},x_{2},\ldots,x_{n})\in X^{n}\) be a Φcoincidence point of F and g and define \(z_{i}=gx_{i}\) for all i. Since \((z_{1},z_{2},\ldots ,z_{n})=(gx_{1},gx_{2},\ldots,gx_{n})\), from Corollary 14 it follows that \((z_{1},z_{2},\ldots ,z_{n})\) is also a Φcoincidence point of F and g.
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
For all \(t>0\), define
Clearly, λ is continuous and nondecreasing on \(\mathbb {R}^{+}\). Take any \(j,s\in\{1,2,\ldots,n\}\). Since \(\ast=\min\) and \(\gamma(a)=a\) for all \(a\in\mathbb{I}\), applying (4) we have
for all \(t>0\). Therefore, for all \(t>0\), we have
Repeating this process, \(\lambda(t)\geq\lambda(t/k)\geq\cdots\geq \lambda(t/k^{m})\) for all m. Taking the limit \(m\rightarrow\infty\), we deduce that \(\lambda(t)=1\) for all \(t>0\). For any \(j,s\in\{1,2,\ldots ,n\}\), we note that \(M(z_{j},z_{s},t)\geq\lambda(t)=1\) for all t, i.e., \(z_{j}=z_{s}\). This completes the proof. □
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
Let \((X,d)\) be a metric space and \(\Phi=(\sigma_{1},\sigma _{2},\ldots,\sigma_{n})\) be an ntuple of mappings from \(\{1,2,\ldots ,n\}\) into itself. Two mappings \(F:X^{n}\rightarrow X\) and \(g:X\rightarrow X\) are said to satisfy the Φcommon limit in the range of the gproperty (shortly, \(CLR_{g}\) property) if there exist n sequences \(\{x_{m}^{1}\}_{m\geq 0},\{x_{m}^{2}\}_{m\geq0},\ldots,\{x_{m}^{n}\}_{m\geq0}\) such that
for all i and some \(x_{1},\ldots,x_{n}\in X\).
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
Let \((X,d)\) be a metric space 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 \(CLR_{g}\)property. Assume that there exists \(k \in[0,1[\) such that
for all i. Then F and g have at least one Φcoincidence point in X.
Proof
It is easy to show that the mapping \(\delta: X^{n}\times X^{n}\to[0,\infty )\) defined by
is a metric on \(X^{n}\). The proof follows from Theorem 13 for \(M^{\mathrm{e}}\) induced by δ, γ the identity and \(\ast=\min\). □
If \(n=1\), then we have the following.
Corollary 22
Let \((X,d)\) be a metric space. Let \(f:X\rightarrow X\) and \(g:X\rightarrow X\) be two mappings satisfying the \(CLR_{g}\)property. Assume that there exists \(k \in[0,1[\) such that
for all \(x,y\in X\). Then f and g have at least one coincidence point.
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.
Theorem 23
Corollaries 21 and 22 are equivalent.
Proof
Let \(\Phi=(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})\), \(F:X^{n}\rightarrow X\) and \(g:X\rightarrow X\) be two mappings with the conditions of Corollary 21. It is easy to show that the mapping \(\delta: X^{n}\times X^{n}\to[0,\infty )\) defined by
is a metric on \(X^{n}\).
Now, define the mappings \(T:X^{n}\to X^{n}\) and \(G:X^{n}\rightarrow X^{n}\) by
respectively. It is easy to show that

(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.
Thus, from (10) it follows that
Thus the mappings T and G satisfy the conditions of Corollary 21. This completes the proof. □
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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).
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MartínezMoreno, J., Roldán, A., Roldán, C. et al. Multidimensional coincidence point theorems for weakly compatible mappings with the \(CLR_{g}\)property in (fuzzy) metric spaces. Fixed Point Theory Appl 2015, 53 (2015). https://doi.org/10.1186/s1366301502972
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DOI: https://doi.org/10.1186/s1366301502972
Keywords
 fuzzy metric space
 coincidence point
 fixed point
 contraction
 the \(E.A\) property
 the \(CLR_{g}\)property