Abstract
In this paper, we prove some recent even coincidence theorems due to Imdad et al. (Bull Math Anal Appl 5(4): 1939, 2013) using a method of reduction from the respective coincidence theorems for mappings with one variable in ordered complete metric spaces. Our technique of proof is different, slightly simpler, shorter and more effective than the ones used in Imdad et al.
Introduction
The investigation of fixed points in ordered metric spaces is a relatively new development which appears to have its origin in the paper of Ran and Reurings [30] which was well complimented by Nieto and López [25]. Ran and Reurings’ fixed point theorem extended and refined by many authors, (for details see [8, 12, 24–27, 37]).
The concept of coupled fixed point was introduced by Guo and Lakshmikantham [11]. In [5], Bhaskar and Lakshmikantham introduced the notion of mixed monotone property for a mapping \(F : X^2 \rightarrow X\) and proved some coupled fixed point theorems for weakly linear contractions enjoying mixed monotone property in ordered complete metric spaces. In this continuation, Lakshmikantham and Ćirić [22] generalized these results for nonlinear contraction mappings by introducing two ideas namely: coupled coincidence point and mixed gmonotone property. In an attempt to extend the definition from \(X^2\) to \(X^3,\) Berinde and Borcut [4] introduced the concept of tripled fixed point and utilize the same to prove some tripled fixed point theorems. After that, Karapınar [16] introduced the quadrupled fixed point to prove some quadrupled fixed point theorems for nonlinear contraction mappings satisfying mixed gmonotone property (for more details see [17, 18]). Recently, Samet and Vetro [32] extended the idea of coupled as well as quadrupled fixed point to higher dimensions by introducing the notion of fixed point of norder (or ntupled fixed point, where \(n\in \mathbb N\) and \(n\ge 3\)) and presented some ntupled fixed point results in complete metric spaces, using a new concept of finvariant set. Here it can be pointed out that the notion of tripled fixed point due to Berinde and Borcut [4] is different from the one defined by Samet and Vetro [32] for \(n = 3\) in the case of ordered metric spaces in order to keep the mixed monotone property working. Recently, Imdad et al. [13] extended the idea of mixed gmonotone property to the mapping \(F : X^n \rightarrow X\) (where n is even natural number)and proved an eventupled coincidence point theorem for nonlinear contraction mappings satisfying mixed gmonotone property. Basically their results are true for only even n but not for odd ones (for details see [15]). Further, Imdad et al. [14] proved some eventupled coincidence theorems under nonlinear weak contractions due to Choudhury et al. [9].
Very recently, Samet et al. [34] have shown that the coupled (analogously ntupled) fixed results can be more easily obtained using wellknown fixed point theorems on ordered metric spaces (see also [10, 28, 29]). This technique of proof is different, slightly simpler, shorter and more effective than classical technique. In this paper, we prove the main results of Imdad et al. [14] following the techniques of Samet et al. [34].
Preliminaries
With a view to make, our presentation selfcontained, we collect some basic definitions and needed results which will be used frequently in the text later.
Definition 2.1
Let \(X\) be a nonempty set. A relation \(`\preceq \)’ on \(X\) is said to be a partial order if the following properties are satisfied:

(i)
reflexive: \(x\preceq x\) for all \(x\in X,\)

(ii)
antisymmetric: \(x\preceq y\) and \(y\preceq x\) imply \(x=y,\)

(iii)
transitive: \(x\preceq y\) and \(y\preceq z\) imply \(x\preceq z\) for all \(x,y,z\in X.\)
A nonempty set \(X\) together with a partial order \(`\preceq \)’ is said to be an ordered set and we denote it by \((X,\preceq ).\)
Definition 2.2
Let \((X,\preceq )\) be an ordered set. Any two elements \(x\) and \(y\) are said to be comparable elements in \(X\) if either \(x\preceq y\) or \(y\preceq x.\)
Definition 2.3
([27]) A triplet \((X,d,\preceq )\) is called an ordered metric space if \((X,d)\) is a metric space and \((X,\preceq )\) is an ordered set. Moreover, if \(d\) is a complete metric on \(X,\) then we say that \((X,d,\preceq )\) is an ordered complete metric space.
Recently, Kutbi et al. [21] introduced the concept of regular map.
Definition 2.4
([21]) An ordered metric space \((X,d,\preceq )\) is said to be nondecreasing regular (resp. nonincreasing regular) if it satisfies the following property: if \(\{x_m\}\) is a nondecreasing (resp. nonincreasing) sequence and \(x_m \rightarrow x,\) then \(x_m\preceq x\) (resp. \(x \preceq x_m)\,\forall m \in {\mathbb {N}} \cup \{0\}.\)
Definition 2.5
([21]) An ordered metric space \((X,d,\preceq )\) is said to be regular if it is both nondecreasing regular and nonincreasing regular.
Definition 2.6
Let \((X,d,\preceq )\) be an ordered metric space and \(g:X\rightarrow X\) be a mapping. Then \(X\) is said to be nondecreasing gregular (resp. nonincreasing gregular) if it satisfies the following property: if \(\{x_m\}\) is a nondecreasing (resp. nonincreasing) sequence and \(x_m \rightarrow x,\) then \(gx_m\preceq gx\) (resp. \(gx \preceq gx_m)\,\forall m \in {\mathbb {N}} \cup \{0\}.\)
Definition 2.7
An ordered metric space \((X,d,\preceq )\) is said to be gregular if it is both nondecreasing gregular and nonincreasing gregular.
Notice that, on setting \(g=I\) (identity mapping on \(X\)), Definitions 2.6 and 2.7 reduce to Definitions 2.4 and 2.5 respectively.
Throughout the paper, \(n\) stands for a general even natural number. Let us denote by \(X^n\) the product space \(X \times X \times \ldots \times X\) of \(n\) identical copies of \(X.\)
Definition 2.8
([13]) Let \((X,\preceq )\) be an ordered set and \(F: X^{n} \rightarrow X\) and \(g: X\rightarrow X\) be two mappings. Then \(F\) is said to have the mixed \(g\)monotone property if \(F\) is \(g\)nondecreasing in its odd position arguments and \(g\)nonincreasing in its even position arguments, that is, for \(x^1,x^{2},x^{3},...,x^{n}\in X,\)
if
for all \(x_{1}^{1}, x_{2}^{1}\in X\), \(gx_{1}^{1}\preceq gx_{2}^{1}\Rightarrow F(x_{1}^{1}, x^{2},x^{3},...,x^{n})\preceq F(x_{2}^{1}, x^{2},x^{3},...,x^{n})\)
for all \(x_{1}^{2}, x_{2}^{2}\in X\), \(gx_{1}^{2}\preceq gx_{2}^{2}\Rightarrow F(x^{1}, x_{2}^{2},x^{3},...,x^{n})\preceq F(x^{1}, x_{1}^{2},x^{3},...,x^{n})\)
for all \(x_{1}^{3}, x_{2}^{3}\in X\), \(gx_{1}^{3}\preceq gx_{2}^{3}\Rightarrow F(x^{1}, x^{2},x^{3}_{1},...,x^{n})\preceq F(x^{1}, x^{2},x^{3}_{2},...,x^{n})\)
\(\vdots \)
for all \(x_{1}^{n}, x_{2}^{n}\in X\), \(gx_{1}^{n}\preceq gx_{2}^{n}\Rightarrow F(x^{1}, x^{2},x^{3},...,x^{n}_{2})\preceq F(x^{1}, x^{2},x^{3},...,x^{n}_{1}).\)
For \(g=I\) (identity mapping), Definition 2.8 reduces to mixed monotone property (for details see [13]).
Definition 2.9
([32]) An element \((x^1,x^2,\ldots,x^n)\in X^n\) is called an ntupled fixed point of the mapping \(F:X^n\rightarrow X\) if
Definition 2.10
([13]) An element \((x^1,x^2,\ldots,x^n)\in X^n\) is called an ntupled coincidence point of mappings \(F:X^n\rightarrow X\) and \(g:X\rightarrow X\) if
Remark 2.1
For \(n=2,\) Definitions 2.9 and 2.10 yield the definitions of coupled fixed point and coupled coincidence point respectively while on the other hand, for \(n=4\) these definitions yield the definitions of quadrupled fixed point and quadrupled coincidence point respectively.
Definition 2.11
An element \((x^1,x^2,\ldots,x^n)\in X^n\) is called an ntupled common fixed point of mappings \(F: X^n\rightarrow X\) and \(g: X\rightarrow X\) if
Definition 2.12
([14]) Let \(X\) be a nonempty set. Then the mappings \(F: X^{n}\rightarrow X\) and \(g:X\rightarrow X\) are said to be compatible if
where \(\{x^1_{m}\},\{x^2_{m}\},\ldots,\{x^n_{m}\}\) are sequences in \(X\) such that
for some \(x^1,x^2,\ldots,x^n\in X\) are satisfied.
The following families of control functions are indicated in Choudhury et al. [9].

\(\mathfrak {I}:=\{\zeta :[0,\infty )\rightarrow [0,\infty ): \zeta \text { is continuous and } \zeta (t)=0\;{\mathrm{if\; and\; only\; if\;}}t=0 \}\)

\(\Omega :=\{\varphi :[0,\infty )\rightarrow [0,\infty ): \varphi \,\text { is continuous and monotone nondecreasing and } \varphi (t) =0 \text { if and only if }t=0\}\)
Notice that members of \(\Omega \) are called altering distance functions (cf. [20]).
Now, we state the main result of Imdad et al. [14], which is indeed ntupled extension of that of Choudhury et al. [9].
Theorem 2.1
Let \((X,d,\preceq )\) be an ordered complete metric space and \(F: X^{n}\rightarrow X\) and \(g:X\rightarrow X\) be two mappings. Suppose that the following conditions are satisfied:

(i)
\(F(X^{n})\subseteq g(X),\)

(ii)
\(F\) and \(g\) are compatible,

(iii)
\(F\) has the mixed gmonotone property,

(iv)
\(g\) is continuous,

(v)
either \(F\) is continuous or \(X\) is gregular,

(vi)
there exist \(x_0^1,x_0^2,x_0^3,...,x_0^n\in X\) such that
$$\begin{aligned} {\left\{ \begin{array}{ll} gx_0^1\preceq F(x_0^1,x_0^2,x_0^3,\ldots,x_0^n)\\ F(x_0^2,x_0^3,\ldots,x_0^n,x_0^1)\preceq gx_0^2\\ gx_0^3\preceq F(x_0^3,\ldots,x_0^n,x_0^1,x_0^2)\\ \vdots \\ F(x_0^n,x_0^1,x_0^2,\ldots,x_0^{n1})\preceq gx_0^n,\\ \end{array}\right. } \end{aligned}$$ 
(vii)
there exist \(\varphi \in \Omega \) and \(\zeta \in \mathfrak {I}\) such that
$$\begin{aligned} \varphi (d(FU,FV))\le \varphi (\max \limits _{1\le i\le n}d(gx^i, gy^i))\zeta (\max \limits _{1\le i\le n}d(gx^i, gy^i)), \end{aligned}$$for all \(U=(x^1,x^2,\ldots,x^n),\,V=(y^1,y^2,\ldots,y^n)\in X^n\) with \(gy^1\preceq gx^1,gx^2\preceq gy^2,gy^3\preceq gx^3,\ldots ,gx^n\preceq gy^n.\) Then \(F\) and \(g\) have an ntupled coincidence point.
Main results
Let \((X,\preceq )\) be an ordered set. Define the following partial order \(\sqsubseteq \) on the product space \(X^n\), for \(U=(x^1,x^2,\ldots ,x^n),\,V=(y^1,y^2,\ldots ,y^n)\in X^n\)
Let \((X,d)\) be a metric space. Define the following metric \(\tilde{D}\) on the product space \(X^n,\) for \(U=(x^1,x^2,\ldots ,x^n),\,V=(y^1,y^2,\ldots ,y^n)\in X^n,\)
The proofs of the following lemmas follow immediately. We note the same idea here, but in the case of coupled and tripled fixed point theorems, we have been first used in ([3, 28, 33]).
Lemma 3.1
Let \((X,d,\preceq )\) be an ordered complete metric space. Then \((X^n,\tilde{D},\sqsubseteq )\) is an ordered complete metric space.
Lemma 3.2
Let \((X,d,\preceq )\) be an ordered metric space and \(F: X^{n}\rightarrow X\) and \(g:X\rightarrow X\) be two mappings. Define mappings \(T_F:X^{n}\rightarrow X^n\) and \(T_g:X^{n}\rightarrow X^n\) by
and \(T_g(x^1,x^2,\ldots ,x^n)=(gx^1,gx^2,\ldots ,gx^n).\) Then the following hold:

(1)
If \(F\) has the mixed gmonotone property, then \(T_F\) is monotone \(T_g\)nondecreasing with respect to \(\sqsubseteq .\)

(2)
If \(F\) and \(g\) are compatible, then \(T_F\) and \(T_g\) are compatible.

(3)
If \(g\) is continuous, then \(T_g\) is continuous.

(4)
If \(F\) is continuous, then \(T_F\) is continuous.

(5)
If \((X,d,\preceq )\) is gregular, then \((X^n,\tilde{D},\sqsubseteq )\) is nondecreasing gregular.

(6)
A point \((x^1,x^2,\ldots ,x^n)\in X^n\) is an ntupled coincidence point of \(F\) and \(g\) iff \((x^1,x^2,\ldots ,x^n)\) is a coincidence point of \(T_F\) and \(T_g.\)
The following lemma is crucial for our main result.
Lemma 3.3
Let \((X,d,\preceq )\) be an ordered complete metric space and \(f\) and \(g\) be two selfmappings on \(X.\) Suppose that the following conditions are satisfied:

(i)
\(f(X)\subseteq g(X),\)

(ii)
\(f\) is monotone gnondecreasing,

(iii)
\(f\) and \(g\) are compatible,

(iv)
\(g\) is continuous,

(v)
either \(f\) is continuous or \(X\) is nondecreasing gregular,

(vi)
there exists \(x_0\in X\) such that \(g(x_0)\preceq f(x_0),\)

(vii)
there exist \(\varphi \in \Omega \) and \(\zeta \in \mathfrak {I}\) such that for all \(x,y\in X,\)
$$\begin{aligned} \varphi (d(f(x),f(y)))\le \varphi (d(g(x),g(y)))\zeta (d(g(x),g(y))),\,with\,g(x)\preceq g(y). \end{aligned}$$(3.1)Then \(f\) and \(g\) have a coincidence point.
Proof
In view of assumption (vi), if \(g(x_0)=f(x_0),\) then \(x_0\) is a coincidence point of \(f\) and \(g\) and hence proof is finished. On the other hand if \(g(x_0)\ne f(x_0),\) then we have \(g(x_0)\prec f(x_0)\). So according to assumption (i), that is, \(f(X)\subseteq g(X)\), we can choose \(x_1\in X\) such that \(g(x_1)=f(x_0)\). Again from \(f(X)\subseteq g(X)\), we can choose \(x_2\in X\) such that \(g(x_2)=f(x_1)\). Continuing this process, we define a sequence \(\{x_m\}\subset X\) of joint iterates such that
Now, we assert that \(\{g(x_m)\}\) is a nondecreasing sequence, that is
We prove this fact by mathematical induction. On using (3.2) for \(m=0\) and assumption (vi), we have
Thus, (3.3) holds for \(m=0.\) Suppose that (3.3) holds for \(m=r>0,\) that is,
Then we have to show that (3.3) holds for \(m=r+1\). To accomplish this we use (3.2), (3.4) and assumption (ii) so that
Thus, by induction, (3.3) holds for all \(m \in \mathbb {N} \cup \{0 \}\).
If \(g(x_m)=g(x_{m+1})\) for some \(m\in \mathbb {N},\) then using (3.2), we have \(g(x_m)=f(x_m),\) that is, \(x_m\) is a coincidence point of \(f\) and \(g\) and hence proof is finished. On the other hand if \(g(x_m)\ne g(x_{m+1})\) for each \(m\in \mathbb {N}\cup \{0\},\) we can define a sequence
On using (3.2), (3.3), (3.5) and assumption (vii), we obtain
On using the property of \(\varphi ,\) we have \(\varphi (\delta _{m+1})\le \varphi (\delta _{m}),\) which implies that \(\delta _{m+1}\le \delta _{m}.\) Therefore, \(\{\delta _m\}\) is a monotone decreasing sequence of nonnegative real numbers. Hence there exists \(\delta \ge 0\) such that \(\delta _m\rightarrow \delta \) as \(m\rightarrow \infty .\) Taking limit as \(m\rightarrow \infty \) in (3.6) and using the continuities of \(\varphi \) and \(\zeta ,\) we have \(\varphi (\delta )\le \varphi (\delta )\zeta (\delta ),\) which is a contradiction under \(r=0.\) Therefore,
Now, we show that \(\{g(x_m)\}\) is a Cauchy sequence. On the contrary, suppose that \(\{g(x_m)\}\) is not a Cauchy sequence. Then, there exists an \(\epsilon >0\) and sequences of positive integers \(\{m(k)\}\) and \(\{t(k)\}\) such that for all positive integers \(k,\,t(k)>m(k)>k,\) such that
Now,
that is,
Letting \(k\rightarrow \infty \) in above inequality and using (3.7), we get
Again,
Letting \(k\rightarrow \infty \) in above inequality and using (3.7) and (3.8), we get
Since \(t(k)>m(k),\) hence by (3.3), we get \(g(x_{m(k)})\le g(x_{t(k)}).\) Therefore, owing to (3.1) and assumption (vii), we get
that is,
Letting \(k\rightarrow \infty \) in above inequality and using (3.8), (3.9) and continuities of \(\varphi \) and \(\zeta ,\) we get
which is a contradiction by virtue of property of \(\zeta .\) Therefore, the sequence \(\{g(x_m)\}\) is Cauchy. From the completeness of \(X,\) there exists \(x\in X\) such that
Since \(F\) and \(g\) are compatible, we have from (3.10),
Now, we use assumption (v). Firstly, we assume that \(f\) is continuous. Then for all \(m \in \mathbb {N} \cup \{0 \},\) we have
Taking \(k\rightarrow \infty \) in above inequality and using (3.10), (3.11) and continuities of \(f\) and \(g,\) we get \(d(g(x),f(x))=0,\) that is, \(g(x)=f(x).\) Hence, the element \(x\in X\) is a coincidence point of \(f\) and \(g.\) Next, we suppose that \(X\) is nondecreasing \(g\)regular. From (3.3) and (3.10), we get
Since \(f\) and \(g\) are compatible and \(g\) is continuous by (3.10) and (3.11), we have
Now, using triangle inequality, we have
Taking \(k\rightarrow \infty \) in above inequality and using (3.13), we have
Since \(\varphi \) is continuous and monotone nondecreasing, from the above inequality we have
By (3.12) and assumption (vii), we get
Using (3.13) and the properties of \(\varphi \) and \(\zeta ,\) we have \(\varphi (d(f(x),g(x)))=0,\) which implies that \(d(f(x),g(x))=0,\) that is, \(g(x)=f(x).\) Hence, \(x\in X\) is a coincidence point of \(f\) and \(g.\)
Lemma 3.4
In addition to the hypotheses of Lemma 3.3, suppose that for real \(x,y\in X\) there exists, \(z\in X\) such that \(f(z)\) is comparable to \(f(x)\) and \(f(y).\) Then \(f\) and \(g\) have a unique common fixed point.
Proof
The set of coincidence points of \(f\) and \(g\) is nonempty due to Lemma 3.3. Assume now, \(x\) and \(y\) are two coincidence points of \(f\) and \(g,\) that is,
Now we will show that \(g(x)=g(y).\) By assumption, there exists \(z\in X\) such that \(f(z)\) is comparable to \(f(x)\) and \(f(y).\) Put \(z_0^1=z\) and choose \(z_1\in X\) such that \(g(z_1)=f(z_0).\) Further define sequence \(\{g(z_m)\}\) such that \(g(z_{m+1})=f(z_m).\) Further set \(x_0=x\) and \(y_0=y.\) In the same way, define the sequences \(\{g(x_m)\}\) and \(\{g(y_m)\}.\) Then, it is easy to show that
Since \(f(x)=g(x_1)=g(x)\) and \(f(z)=g(z_1)\) are comparable, we have
It is easy to show that \(g(x)\) and \(g(z_m)\) are comparable, that is, for all \(m \in \mathbb {N},\)
Thus from (3.1) we have
Let \(R_m=d(g(x),g(z_{m+1})).\) Then
Using the property of \(\varphi ,\) we have \(\varphi (R_m)\le \varphi (R_{m1}),\) which implies that \(R_{m}\le R_{m1}\,(\mathrm{by\,the\,property\,of}\,\varphi ).\) Therefore \(\{R_m\}\) is a monotone decreasing sequence of nonnegative real numbers. Hence, there exists \(r\ge 0\) such that \(R_m\rightarrow r\,\mathrm{as}\,m\rightarrow \infty .\) Taking the limit as \(m\rightarrow \infty \) in (3.14) and using the continuities of \(\varphi \) and \(\zeta ,\) we have \(\varphi (r)\le \varphi (r)\zeta (r),\) which is a contradiction unless \(r=0.\) Therefore \(R_m\rightarrow 0\,{\mathrm{as}}\,m \rightarrow \infty ,\) that is,
Similarly we can prove that
Therefore by triangle inequality
Hence
Since \(g(x)=f(x)\) and \(f\) and \(g\) are compatible, we have \(gg(x)=f(gx).\) Write \(g(x)=a,\) then we have
Thus \(a\) is the coincidence point of \(f\) and \(g.\) Then owing to (3.15) with \(y=a,\) it follows that \(g(x)=g(a),\) that is,
Using (3.16) and (3.17), we have \(a=g(a)=f(a).\) Thus \(a\) is the common fixed point of \(f\) and \(g.\) To prove the uniqueness, assume that \(b\) is another common fixed point of \(f\) and \(g.\) Then by (3.15), we have
This completes the proof of Lemma.
Theorem 3.1
Theorem 2.1 is obtained using Lemmas 3.1, 3.2 and 3.3.
Proof
Consider the product space \(Y=X^n\) equipped with the metric \(\tilde{D}\) [given by (B)] and the partial order \(\sqsubseteq \) [given by (A)]. Then by Lemma 3.1, \((Y,\tilde{D},\sqsubseteq )\) is an ordered complete metric space. Also \(F\) and \(g\) induce mappings \(T_F:Y\rightarrow Y\) and \(T_g:Y\rightarrow Y\) (defined in Lemma 3.2). Clearly,

(i) implies that \(T_F(Y)\subseteq T_g(Y),\)

(ii) implies that \(T_F\) is monotone \(T_g\)nondecreasing (by item (1) of Lemma 3.2),

(iii) implies that \(T_F\) and \(T_g\) are compatible (by item (2) of Lemma 3.2),

(iv) implies that \(T_g\) is continuous (by item (3) of Lemma 3.2),

(v) implies that either \(T_F\) is continuous [by item (4) of Lemma 3.2] or \((Y,\tilde{D},\sqsubseteq )\) is nondecreasing \(g\)regular [by item (5) of Lemma 3.2],

(vi) is equivalent to the condition: there exists \(U_0=(x_0^1,x_0^2,\ldots ,x_0^n)\in Y\) such that \(T_g(U_0)\subseteq T_F(U_0).\)
Now, in view of (vii), for given \(U,V\in Y\) such that \(T_g(U)\sqsubseteq T_g(V)\) implies that
It follows that for odd \(i,\)
and for even \(i,\)
If \(i\) is odd, then using (3.18) and (vii), we get
If \(i\) is even, then using (3.19) and (vii), we get
Hence, in both the cases, for each \(i\,(1\le i\le n),\) we have
Hence using (3.20), we have
Thus all conditions of Lemma 3.3 are satisfied for ordered complete metric space \((Y,\tilde{D},\sqsubseteq )\) and mappings \(T_F:Y\rightarrow Y\) and \(T_g:Y\rightarrow Y.\) Therefore, \(T_F\) and \(T_g\) have a coincidence point in \(Y=X^n.\) According to item (6) of Lemma 3.2, the mappings \(F\) and \(g\) have an \(n\)tupled coincidence point.
Corollary 3.1
Let \((X,d,\preceq )\) be an ordered complete metric space and \(F: X^{n}\rightarrow X\) be a mapping. Suppose that the following conditions are satisfied:

(i)
\(F\) has the mixed monotone property,

(ii)
either \(F\) is continuous or \(X\) is regular,

(iii)
there exist \(x_0^1,x_0^2,x_0^3,...,x_0^n\in X\) such that
$$\begin{aligned} {\left\{ \begin{array}{ll} x_0^1\preceq F(x_0^1,x_0^2,x_0^3,\ldots,x_0^n)\\ F(x_0^2,x_0^3,\ldots,x_0^n,x_0^1)\preceq x_0^2\\ x_0^3\preceq F(x_0^3,\ldots,x_0^n,x_0^1,x_0^2)\\ \vdots \\ F(x_0^n,x_0^1,x_0^2,\ldots,x_0^{n1})\preceq x_0^n,\\ \end{array}\right. } \end{aligned}$$ 
(iv)
there exist \(\varphi \in \Omega \) and \(\zeta \in \mathfrak {I}\) such that
$$\begin{aligned} \varphi (d(FU,FV))\le \varphi (\max \limits _{1\le i\le n}d(x^i, y^i))\zeta (\max \limits _{1\le i\le n}d(x^i, y^i)), \end{aligned}$$for all \(U=(x^1,x^2,\ldots,x^n),\,V=(y^1,y^2,\ldots,y^n)\in X^n\) with \(x^1\preceq y^1,y^2\preceq x^2,x^3\preceq y^3,\ldots ,y^n\preceq x^n.\)
Then \(F\) has an ntupled fixed point.
Proof
It is sufficient to take \(g=I\) (identity mapping) in Theorem 3.1. \(\square \)
Corollary 3.2
Corollary 3.1 remains true if condition (iv) is replaced by the following: (iv)’ there exists \(\zeta \in \mathfrak {I}\) such that
for all \(U=(x^1,x^2,\ldots,x^n),\,V=(y^1,y^2,\ldots,y^n)\in X^n\) with \(x^1\preceq y^1,y^2\preceq x^2,x^3\preceq y^3,\ldots ,y^n\preceq x^n.\)
Proof
It is sufficient to take \(\varphi \) and \(g\) to be identity mappings in Theorem 3.1.
Corollary 3.3
Corollary 3.1 remains true if condition (iv) is replaced by the following:
(iv)” there exists \(k\in (0,1)\) such that
for all \(U=(x^1,x^2,\ldots,x^n),\,V=(y^1,y^2,\ldots,y^n)\in X^n\) with \(x^1\preceq y^1,y^2\preceq x^2,x^3\preceq y^3,\ldots ,y^n\preceq x^n.\)
Proof
It is sufficient to take \(\varphi \) and \(g\) to be identity mappings and \(\zeta (t)=(1k)t,\) \(k\in (0,1)\) in Theorem 3.1.
Remark 3.1

1.
On setting \(n=2\) in Theorem 3.1, we get Theorem 3.1 of Choudhury et al. [9].

2.
On setting \(n=2\) in Corollaries 3.1–3.3, we get Corollaries 3.2–3.4 of Choudhury et al. [9].

3.
On setting \(n=4\) in Theorem 3.1 and Corollaries 3.1–3.3, we get their corresponding quadrupled fixed point results.
Now we shall prove the uniqueness of ntupled fixed point.
Theorem 3.2
In addition to the hypotheses of Theorem 3.1, suppose that for real \((x^1,x^2,\ldots,x^n)\) and \((y^1,y^2,\ldots,y^n)\in X^{n}\) there exists, \((z^1,z^2,\ldots,z^n)\in X^{n}\) such that \((F(z^1,z^2,\ldots,z^n),F(z^2,\ldots,z^n,z^1),\ldots,F(z^n,\) \(z^1,\ldots,z^{n1}))\) is comparable to \((F(x^1,x^2,\ldots,x^n),F(x^2,\ldots,x^n,x^1),\ldots,F(x^n,x^1,\ldots,x^{n1}))\) and \((F(y^1,y^2,\ldots,\) \(y^n),F(y^2,\ldots,y^n,y^1),\ldots,F(y^n,y^1,\ldots,y^{n1})).\) Then \(F\) and \(g\) have a unique ntupled common fixed point.
Proof
Set \(U=(x^1,x^2,\ldots ,x^n),\,V=(y^1,y^2,\ldots ,y^n)\) and \(W=(z^1,z^2,\ldots ,z^n).\) Then we have
and
Hence using Lemma 3.4, \(T_F\) and \(T_g\) have a unique ntupled common fixed point. \(\square \)
Remark 3.2
From Theorem 3.2, for \(n=2,\) we can get unique coupled common fixed point theorem contained in Choudhury et al. [9].
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Sharma, A., Imdad, M. & Alam, A. Shorter proofs of some recent eventupled coincidence theorems for weak contractions in ordered metric spaces. Math Sci 8, 131–138 (2014). https://doi.org/10.1007/s4009601501389
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DOI: https://doi.org/10.1007/s4009601501389
Keywords
 Partially ordered set
 Compatible mapping
 Mixed gmonotone property
 ntupled coincidence point
 ntupled fixed point
Mathematics Subject Classification
 54H10
 54H25