Since Zadeh[1] introduced the concept of fuzzy sets, many authors have extensively developed the theory of fuzzy sets and applications. George and Veeramani [2, 3] gave the concept of fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space which have very important applications in quantum particle physics particularly in connection with both string and E-infinity theory.

The notion of coupled fixed points was introduced by Guo and Lakshmikantham [4] in 1987. In a recent paper, Gnana-Bhaskar and Lakshmikantham [5] introduced the concept of mixed monotone property for contractive operators of the form $F: X \times X \rightarrow X$, where $X$ is a partially ordered metric space, and the established some coupled fixed point theorems. Lakshmikantham and Ćirić [6] discussed the mixed monotone mappings and gave some coupled fixed point theorems which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem.

Shaban Sedghi et al [7] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Jin-xuan Fang [8] gave some common fixed point theorems under $\phi$-contractions for compatible and weakly compatible mappings in Menger probabilistic metric spaces. Xin-Qi Hu [9] proved a common fixed point theorem for mappings under $\varphi$-contractive conditions in fuzzy metric spaces. B.S.Choudury et. al. [10] established coupled coincidence point and coupled fixed point results for compatible mappings in partially ordered fuzzy metric spaces and gave an example to illustrate the main theorems. In 2015, Jinxuan-Fang [11] generlized a crucial fixed point theorem for probabilistic $\varphi$-contraction on complete Menger space. Other more works on this topic can be found in [12-23].

Now we propose a notion of coincidence point between mappings cases of these results that are already known under some contractive conditions.

First we give some definitions.

Definition 2.1  (see [2])  A binary operation $ * :[0, 1]\times [0, 1] \to [0, 1]$ is continuous $t$-norm if $*$ satisfies the following conditions:

(1)   $*$ is commutative and associative;

(2)   $*$ is continuous;

(3)   $a * 1 = a$ for all $a \in [0, 1]$;

(4)   $a * b \le c * d$ whenever $a \le c$ and $b \le d$ for all $ a, b, c, d \in [0, 1]$.

Definition 2.2  (see [7])   Let $\sup\limits_{0<t<1}\Delta (t, t)=1$. A $t$-norm $\Delta$ is said to be of H-type if the family of functions $\{\Delta^m(t)\}^\infty_{m=1}$ is equicontinuous at $t=1$, where

$ \begin{equation*} \Delta^1(t)=t\Delta t, \ \ \Delta^{m+1}(t)=t\Delta(\Delta^{m}(t)), m=1, 2, \cdots, t\in [0, 1]. \end{equation*} $

The $t-$norm $\Delta_M=\min$ is an example of $t$-norm of H-type, but there are some other $t$-norms $\Delta$ of H-type.

Obviously, $\Delta$ is a H-type $t$ norm if and only if for any $\lambda\in (0, 1)$, there exists $\delta(\lambda)\in (0, 1)$ such that $\Delta^m(t)> 1-\lambda$ for all $m \in \mathbb{N}$, when $t>1- \delta$.

Definition 2.3   (see [2])   A 3-tuple $(X, M, *)$ is said to be a fuzzy metric space if $X$ is an arbitrary nonempty set, $*$ is a continuous $t$-norm and $M$ is a fuzzy set on $X^2 \times (0, + \infty )$ satisfying the following conditions, for each $x, y, z\in X$ and $t, s>0, $

(FM-1) $M(x, y, t)>0$;

(FM-2) $M\left( {x, y, t} \right) = 1$ if and only if $x = y$;

(FM-3) $M\left( {x, y, t} \right) = M\left( {y, x, t} \right)$;

(FM-4) $M\left( {x, y, t} \right) * M\left( {y, z, s} \right) \le M\left( {x, z, t + s} \right)$;

(FM-5) $M\left( {x, y, \cdot } \right): (0, \infty ) \to [0, 1]$ is continuous.

Let $(X, M, *)$ be a fuzzy metric space. For $t>0$, the open ball $B(x, r, t)$ with a center $x\in X$ and a radius $0<r<1$ is defined by $ B(x, r, t)=\{y\in X: M(x, y, t)>1-r\}. $

A subset $A\subset X$ is called open if for each $x\in A$, there exist $t>0$ and $0<r<1$ such that $B(x, r, t)\subset A$. Let $\tau$ denote the family of all open subsets of $X$. Then $\tau$ is called the topology on $X$ induced by the fuzzy metric $M$. This topology is Hausdorff and first countable.

Example 2.4   Let $(X, d)$ be a metric space. Define $t$-norm $a * b = ab $ and for all $x, y \in X$ and $t > 0$, $M \left( {x, y, t} \right) = \frac{t}{t + d\left( {x, y}\right)}$. Then $(X, M, *)$ is a fuzzy metric space. We call this fuzzy metric $M$ induced by the metric $d$ the standard fuzzy metric.

Let $n$ be a positive integer. $X$ will benote a non-empty set and $X^n$ denote the product space $X^n=\underbrace{X\times X\times \cdots\times X}_n$.

Definition 2.5  (see [6])   Let $X$ be a non-empty set, $F:X\rightarrow X$ and $g:X\rightarrow X$ be two mappings. We say $F$ and $g$ are commutative (or that $F$ and $g$ commute) if $gFx= Fgx$ for all $x \in X$.

Definition 2.6  (see [6])   The mappings $F$ and $g$ where $F: X \rightarrow X $ and $ g: X \rightarrow X$, are said to be compatible if $ \lim_{n\rightarrow\infty} d(Fgx_n, gFx_n) = 0$ whenever $\{x_n\} $ is a sequence in $ X $, such that $ \lim_{n\rightarrow\infty} F( x_n) = \lim_{n\rightarrow\infty} g( x_n) = x $ for all $x\in X $ are satisfied.

Definition 2.7  (see [6])  Two mappings $F$ and $g$ on a metric space $(X, d)$ are said to be weakly compatible if they commute at their coincidence points, that is, if $Fx= gx$ for some $x\in X$, then $Fgx= gFx$.

Let $\Lambda_{n} = \{ 1, 2, \cdots, n \}$, $A$, $B$ satisfy that $ A\cup B = \Lambda_{n}$ and $A\cap B = \emptyset $. We will denote $\Omega_{A, B} = \{ \sigma: \Lambda_{n} \rightarrow \Lambda_{n}$, $\sigma(A) \subseteq A$ and $\sigma(B) \subseteq B\}$, and $\Omega^{'}_{A, B} = \{ \sigma: \Lambda_{n} \rightarrow \Lambda_{n}$, $\sigma(A) \subseteq B$ and $\sigma(B) \subseteq A\}$.

Let $(X, \leq )$ be a partially ordered space, $ x, y \in X $ and $ i \in \Lambda_{n}$. We use the following notation

$ x\leq_{i}y \Leftrightarrow \left\{\begin{array}{cl} x\leq y, i\in A, \\ x\geq y, i\in B. \end{array}\right. $

Let $ \sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}, \tau: \Lambda_{n} \rightarrow \Lambda_{n}$ be $ n + 1$ mappings and let $\Phi$ be the $ (n + 1)$-tuple $ (\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}, \tau )$.

Definition 2.8  (see [13])   Let $F: X^{n}\rightarrow X$, $g: X \rightarrow X$. A point $ (x_{1}, x_{2}, \cdots, x_{n}) \in X^{n}$ is called a $\Phi$-coincidence point of the mappings $F$ and $g$ if

$ F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})= gx_{\tau(i)}\textrm{ for all }i\in \Lambda_n. $

If $g$ is the identity mapping on $X$, then $ (x_{1}, x_{2}, \cdots, x_{n}) \in X^{n}$ is called a $\Phi$-fixed point of the mapping $F$.

Definition 2.9   Let $(X, \leq)$ be a partially ordered space. We say that $F$ has the mixed $g$-monotone property if $F$ is $g$-monotone non-decreasing in argument of $A$ and $g$-monotone non-increasing in argument of $B$, i.e., for all $ x_{1}, x_{2}, \cdots, x_{n}, y, z \in X $ and all $i$,

$ gy \leq gz \Rightarrow F(x_{1}, \cdots, x_{i-1}, y, x_{i+1}, \cdots, x_{n}) \leq_{i} F(x_{1}, \cdots, x_{i-1}, z, x_{i+1}, \cdots, x_{n}). $

It is obvious that the above formula is equivalent to the following:

$ gy\leq_{i} gz \Rightarrow F(x_{1}, \cdots, x_{i-1}, y, x_{i+1}, \cdots, x_{n}) \leq F(x_{1}, \cdots, x_{i-1}, z, x_{i+1}, \cdots, x_{n}). $

Definition 2.10   Let $F: X^{n}\rightarrow X$ and $g: X\rightarrow X$. $F$ and $g$ are called weakly compatible mappings if for $x_{1}, x_{2}, \cdots, x_{n}, $ it satisfies

$ F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)}) = gx_{\tau(i)}, \textrm{ for all } i\in \Lambda_n, $

it implies

$ F(gx_{\sigma_i(\tau(1))}, gx_{\sigma_i(\tau(2))}, \cdots, gx_{\sigma_i(\tau(n))})= gF(x_{\sigma_{\tau(i)}(1)}, x_{\sigma_{\tau(i)}(2)}, \cdots, x_{\sigma_{\tau(i)}(n)}), \ \forall i\in \Lambda_n. $

Lemma 3.1  (see [23])  For $n \in N$, let $g_{n}:(0, +\infty ) \rightarrow (0, +\infty )$ and $F_{n}:R\rightarrow [0, 1]$. Assume that $\sup\{F(t):t>0\}=1$ and for any $t>0$,

$ \lim\limits_{n \rightarrow +\infty}g_{n}(t)=0 \ and \ F_{n}(g_{n}(t)) \geq F(t). $

If each $F_{n}$ is nondecreasing, then $\lim_{n \rightarrow +\infty}F_{n}(t)=1$ for any $t>0$.

Theorem 3.2  (see [21])  Let $(X, M, \Delta)$ be a complete fuzzy metric space with $\Delta$ a triangular norm of H-type. Let $\varphi\in\Psi_{\omega}$, where $\Psi_{\omega}$ is denoted as the class of all function $\varphi :[0, +\infty ) \rightarrow [0, +\infty )$ such that for each $t>0$ there exists an $r_{t} \geq t$ satisfying $\lim_{n \rightarrow +\infty}\varphi^{n}(r_{t})=0.$ Let $T: X\rightarrow X$ be a mapping, $M(T{x}, T{y}, \varphi(t)) \geq M(x, y, t)$ for all $x, y\in X$ and all $t>0$. Then $T$ has a unique fixed point $x^{*}$. In fact, for any $x_{0}\in X$, $\lim_{n \rightarrow +\infty} T^{n}x_{0}=x^{*}$.

Theorem 3.3   Let $(Y^*, M^*, \Delta, \preccurlyeq)$ be a complete ordered fuzzy metric space with $\Delta$ a triangular norm of H-type. Let $ \varphi : [0, +\infty ) \rightarrow [0, +\infty )$, $\varphi\in\Psi_{\omega}, $ and also suppose $\widetilde{F}, \widetilde{g}: Y \rightarrow Y$ are such that $\widetilde{F}(Y)\subseteq \widetilde{g}(Y)$, $\widetilde{g}$ is continuous and $\widetilde{g}(Y)$ is complete, $\widetilde{F}$ and $\widetilde{g}$ be weakly compatible, $\widetilde{F}$ has the mixed $\widetilde{g}$-monotone property, and $M^{*}(\widetilde{F}x, \widetilde{F}y, \varphi(t)) \geq M^{*}(\widetilde{g}x, \widetilde{g}y, t)$ for each $\widetilde{g}x \preccurlyeq \widetilde{g}y$. If there exists $x_{0} \in Y $ such that $ \widetilde{g}x_{0}\preccurlyeq \widetilde{F}x_{0}$, then $\widetilde{F}$ and $\widetilde{g}$ has a fixed point.

Proof   $\Psi$ is denoted as the class of all function $\varphi :[0, +\infty ) \rightarrow [0, +\infty )$ be continuous with $ \varphi (t) < t$ for each $t >0$. Obviously, $\Psi \subseteq \Psi_{\omega}$. First we will prove Theorem 3.3 when $\varphi\in\Psi$.

From $\widetilde{F}(Y)\subseteq \widetilde{g}(Y)$, we can choose $x_{1} \in Y $ such that $ \widetilde{g}x_{1}= \widetilde{F}x_{0}$. Again we can choose $x_{2} \in Y $ such that $ \widetilde{g}x_{2}= \widetilde{F}x_{1}$. Continuing this process we can construct sequence ${x_{n}}$ in Y such that $ \widetilde{g}x_{n+1}= \widetilde{F}x_{n}$.

Using the mathematical induction and $\widetilde{F}$ has the mixed $\widetilde{g}$-monotone property, we get

$ \widetilde{g}x_{0}\preccurlyeq \widetilde{g}x_{1}\preccurlyeq \widetilde{g}x_{2}\preccurlyeq...\preccurlyeq \widetilde{g}x_{n}\preccurlyeq \widetilde{g}x_{n+1}\preccurlyeq ... $

and

$ \widetilde{F}x_{0}\preccurlyeq \widetilde{F}x_{1}\preccurlyeq \widetilde{F}x_{2}\preccurlyeq...\preccurlyeq \widetilde{F}x_{n}\preccurlyeq \widetilde{F}x_{n+1}\preccurlyeq ... $

By putting $x=x_{n-1}$, $y=x_{n}$ in $M^{*}(\widetilde{F}x, \widetilde{F}y, \varphi(t)) \geq M^{*}(\widetilde{g}x, \widetilde{g}y, t)$, we get

$ M^{*}(\widetilde{F}x_{n-1}, \widetilde{F}x_{n}, \varphi(t)) \geq M^{*}(\widetilde{g}x_{n-1}, \widetilde{g}x_{n}, t). $

That means $M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, \varphi(t)) \geq M^{*}(\widetilde{g}x_{n-1}, \widetilde{g}x_{n}, t)$, thus

$ M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, \varphi^{n}(t)) \geq M^{*}(\widetilde{g}x_{n-1}, \widetilde{g}x_{n}, \varphi^{n}(t)). $

By Lemma 3.1, we have

$ \lim\limits_{n \rightarrow +\infty}M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, t)=1, \textrm { for any }\ t>0. $

Now let $n \in N$ and $t>0$, we show by induction that, for any $k\in \mathbb{N}$,

$ M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+k}, t )\geq \Delta^{k} (M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, t-\varphi(t))). $

This is obvious for $k=0$. Assume it holds for some $k$, by the monotonicity of $\Delta$, we have

$ \begin {eqnarray*} &&M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+k+1}, t) =M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+k+1}, t-\varphi(t)+\varphi(t))\\ &\geq& M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, t-\varphi(t)) \Delta M^{*}(\widetilde{g}x_{n+1}, \widetilde{g}x_{n+k+1}, \varphi(t))\\ &\geq& M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, t-\varphi(t)) \Delta M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+k}, t )\\ &\geq& M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, t-\varphi(t)) \Delta (\Delta^{k} (M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, t-\varphi(t))))\\ &=&\Delta^{k+1} (M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, t-\varphi(t))), \end {eqnarray*} $

which completes the induction. By $\Delta^{n}(1)=1$ and $\Delta$ is a triangular norm of H-type, for any $t>0$ and $\varepsilon>0$, there is $\delta>0$ such that if $s \in (1-\delta, 1]$, then $\Delta^{n}(s)>1-\varepsilon$ for all $n\in N$.

Since, by $\lim_{n \rightarrow +\infty}M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, t-\varphi(t))=1$, there is $n_{0} \in N$ such that, for any $n>n_{0}$, $M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+1}, t-\varphi(t)) \in (1-\delta, 1]$. Hence, we get $M^{*}(\widetilde{g}x_{n}, \widetilde{g}x_{n+k}, t-\varphi(t))>1-\varepsilon$ for any $k \in \mathbb{N}$. This proves the Cauchy condition for ${\widetilde{g}x_{n}}$.

Thus ${\widetilde{g}x_{n}}$ is a Cauchy sequence. Since $\widetilde{g}(Y)$ is complete, there exists $x\in Y$ such that $\lim_{n \rightarrow +\infty} \widetilde{g}x_{n}=x$. Similarly we get ${\widetilde{F}x_{n}}$ is a Cauchy sequence, such that $\lim_{n \rightarrow +\infty} \widetilde{F}x_{n}=\lim_{n \rightarrow +\infty} \widetilde{g}x_{n+1}=x=\widetilde{g}a$ (notice that $\widetilde{g}$ is continuous).

By putting $x=x_{n}$, $y=a$ in $M^{*}(\widetilde{F}x, \widetilde{F}y, \varphi(t)) \geq M^{*}(\widetilde{g}x, \widetilde{g}y, t)$, we get

$ M^{*}(\widetilde{F}x_{n}, \widetilde{F}a, \varphi(t)) \geq M^{*}(\widetilde{g}x_{n}, \widetilde{g}a, t). $

Letting $n \rightarrow +\infty$, we get$M^{*}(\widetilde{g}a, \widetilde{F}a, \varphi(t))=1$, that means $\widetilde{g}a=\widetilde{F}a=x$.

By the condition that $\widetilde{F}$ and $\widetilde{g}$ be weakly compatible, we get $\widetilde{g}\widetilde{F}a=\widetilde{F}\widetilde{g}a$, i.e. $\widetilde{g}x=\widetilde{F}x.$ Thus we prove that $\widetilde{F}$ and $\widetilde{g}$ has a fixed point $x$.

Let $\varphi\in\Psi_{\omega}$. Put $A=\{t>0: \lim_{n \rightarrow +\infty} \varphi^{n}(t)=0\}, $ if $t\in A$, we denote by $k_{t}$ the first integer number such that $\varphi^{k_{t}-1}(t) \geq t >\varphi^{k_{t}}(t)$ ($\varphi^{0}(t)=t$).

If $t \in [0, +\infty ) \backslash A$, take an $r_{t}>t$ such that $r_{t} \in A$, and, again, denote by $k_{t}$ the first integer number such that $\varphi^{k_{t}-1}(r_{t}) \geq t >\varphi^{k_{t}}(r_{t})$.

Now define a function $\psi : [0, +\infty ) \rightarrow [0, +\infty )$ as follows:

$ \psi(0)=0 , \psi(t)=\varphi^{k_{t}}(t) \, \text{if}\, t \in A, \, \text{and}\, \psi(t)=\varphi^{k_{t}}(r_{t})\, \text{if}\, t \in [0, +\infty ) \backslash A. $

It is proved that $\psi\in\Psi$(see [21]). Hence we can apply $\psi$ and get theorem 3.3 proved by the condition that $\varphi\in\Psi_{\omega}$.

Theorem 3.4   Let $(X, M, \Delta, \leq)$ be a complete ordered fuzzy metric space with $\Delta$ a triangular norm of H-type. Let $ \Phi = (\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}, \tau)$ be $(n+1)$-tuple of mappings from $\Lambda_n$ into itself such that $\tau \in \Omega_{A, B}$ is a permutation and verifying that $\sigma_{i} \in \Omega_{A, B}$ if $ i \in A$ and $\sigma_{i} \in \Omega^{'}_{A, B}$ if $ i \in B$. Let $ \varphi : [0, +\infty ) \rightarrow [0, +\infty )$, $\varphi\in\Psi_{\omega}$, $F: X^{n} \rightarrow X$ and $g: X \rightarrow X$ be two mappings, $F(X^{n})\subseteq g(X)$, $F$ is continuous and has the mixed $g$-monotone property, $F$ and $g$ are weakly compatible mappings and

$ M(F(x_{1}, x_{2}, \cdots, x_{n}), F(y_{1}, y_{2}, \cdots, y_{n}), \varphi(t)) \geq \min\limits_{1\leq i\leq n}M(gx_{i}, gy_{i}, t) $

for which $gx_{\tau(i)}\leq_{i}gy_{\tau(i)}$ for all $i\in \Lambda_{n}$ and all $t>0$. If there exists $(x^0_1, x^0_2, \cdots, x^0_n) \in X^n$ verifying $gx^0_{\tau(i)} \leq_{i} F(x^0_{\sigma_{i}(1)}, x^0_{\sigma_{i}(2)}, \cdots, x^0_{\sigma_{i}(n)})$ for all $i$, then $F$ and $g$ have at least one $\Phi$-coincidence point.

Proof   Let $Y = X^n$. For $(x_{1}, x_{2}, \cdots, x_{n}), (y_{1}, y_{2}, \cdots, y_{n})\in X^n$, $t>0$, $M^{*}$ and binary relation $\preccurlyeq$ on $Y$ are defined as

$ M^{*}((x_{1}, x_{2}, \cdots, x_{n}), (y_{1}, y_{2}, \cdots, y_{n}), t) = \min\limits_{1 \leq i \leq n}{M(x_{i}, y_{i}, t)}, $

$ (x_{1}, x_{2}, \cdots, x_{n}) \preccurlyeq (y_{1}, y_{2}, \cdots, y_{n})\ \Leftrightarrow x_{i} \leq_{i} y_{i}, for \ all\ i \in \Lambda_{n}. $

It is easy to verify that $(Y, \preccurlyeq)$ is a partially ordered set and $(Y, M^{*}, \Delta)$ is a complete fuzzy metric space. Then $(Y, M^{*}, \Delta, \preccurlyeq)$ is a complete ordered fuzzy metric space.

For $(x_1, x_2, \cdots, x_n)\in Y$, $\widetilde{F}: Y \rightarrow Y$, $\widetilde{g}: Y \rightarrow Y$ are defined as

$ \begin {eqnarray*} &&\widetilde{F}(x_{1}, x_{2}, \cdots, x_{n})\\ &=& (F(x_{\sigma_{1}(1)}, x_{\sigma_{1}(2)}, \cdots, x_{\sigma_{1}(n)}), F(x_{\sigma_{2}(1)}, x_{\sigma_{2}(2)}, \cdots, x_{\sigma_{2}(n)}), \cdots, F(x_{\sigma_{n}(1)}, x_{\sigma_{n}(2)}, \cdots, x_{\sigma_{n}(n)})). \\ && \widetilde{g}(x_1, x_2, \cdots, x_n) = (gx_{\tau(1)}, gx_{\tau(2)}, \cdots, gx_{\tau(n)}). \end {eqnarray*} $

For $(x_{1}, x_{2}, \cdots, x_{n})\in X^n, $ if $\widetilde{F}(x_1, x_2, \cdots, x_n)=\widetilde{g}(x_1, x_2, \cdots, x_n)$, by definition of $\widetilde{F}$ and $\widetilde{g}$ we have

$ F(x_{\sigma_{1}(1)}, x_{\sigma_{1}(2)}, \cdots, x_{\sigma_{1}(n)}) = gx_{\tau(1)}, $

$ F(x_{\sigma_{2}(1)}, x_{\sigma_{2}(2)}, \cdots, x_{\sigma_{2}(n)}) = gx_{\tau(2)}, $

$ \vdots $

$ F(x_{\sigma_{n}(1)}, x_{\sigma_{n}(2)}, \cdots, x_{\sigma_{n}(n)}) = gx_{\tau(i)}, $

which implies that

$ F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)}) = gx_{\tau(i)}, \textrm{ for all } i\in \Lambda_n. $

Since $\widetilde{F}\widetilde{g}(x_1, x_2, \cdots, x_n)=\widetilde{F}(gx_{\tau(1)}, gx_{\tau(2)}, \cdots, gx_{\tau(n)}), $ the $i$th component of $\widetilde{F}\widetilde{g}(x_1, x_2, \cdots, x_n)$ is $F(gx_{\sigma_i(\tau(1))}, gx_{\sigma_i(\tau(2))}, \cdots, gx_{\sigma_i(\tau(n))})$. And

$ \widetilde{g}\widetilde{F}(x_1, x_2, \cdots, x_n)=\widetilde{g}(F(x_{\sigma_{1}(1)}, x_{\sigma_{1}(2)}, \cdots, x_{\sigma_{1}(n)}), \cdots, F(x_{\sigma_{n}(1)}, x_{\sigma_{n}(2)}, \cdots, x_{\sigma_{n}(n)})), $

the $i$th component of $\widetilde{g}\widetilde{F}(x_1, x_2, \cdots, x_n)$ is $gF(x_{\sigma_{\tau(i)}(1)}, x_{\sigma_{\tau(i)}(2)}, \cdots, x_{\sigma_{\tau(i)}(n)}).$

Since $F$ and $g$ are weakly compatible, all the component of $\widetilde{F}\widetilde{g}(x_1, x_2, \cdots, x_n)$ and the corresponding component of $\widetilde{g}\widetilde{F}(x_1, x_2, \cdots, x_n)$ are equal, which implies that

$ \widetilde{F}\widetilde{g}(x_1, x_2, \cdots, x_n)=\widetilde{g}\widetilde{F}(x_1, x_2, \cdots, x_n). $

That is, $\widetilde{F}$ and $\widetilde{g}$ are weakly compatible.

For $(x_{1}, x_{2}, \cdots, x_{n})$, $(y_{1}, y_{2}, \cdots, y_{n})\in X^{n}$, \ if \ $\widetilde{g}(x_{1}, x_{2}, \cdots, x_{n}) \preccurlyeq \widetilde{g}(y_{1}, y_{2}, \cdots, y_{n})$, by definition of $\widetilde{g}$, we have

$ gx_{\tau(i)} \leq_{i} gy_{\tau(i)} \textrm{ for all } i\in \Lambda_{n}. $

Now we need to prove $\widetilde{F}(x_{1}, x_{2}, \cdots, x_{n})\preccurlyeq \widetilde{F}(y_{1}, y_{2}, \cdots, y_{n})$. That is

$ F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})\preccurlyeq_{i} F(y_{\sigma_{i}(1)}, y_{\sigma_{i}(2)}, \cdots, y_{\sigma_{i}(n)}) \textrm{ for all } i\in \Lambda_{n}. $

We use the following notation $\tau \in \Omega_{A, B}$, $\sigma_{i} \in \Omega_{A, B}$,

$ i \in A \Rightarrow \left\{\begin{array}{cl} \sigma_{i}(t) \in A, t\in A, \\ \sigma_{i}(t) \in B, t\in B. \end{array}\right. {\rm{and}}\;\; x\leq_{i}y \Leftrightarrow \left\{\begin{array}{cl} x\leq y, i\in A\\ x\geq y, i\in B. \end{array}\right. $

For $i \in A$, if $j \in A$, then there exists $k \in A$ such that $\sigma_{i}(j)= \tau(k)$; if $j \in B$, then there exists $k \in B$ such that $\sigma_{i}(j)= \tau(k)$. So, we have

$ gx_{\tau(k)}\leq_{k}gy_{\tau(k)} \Rightarrow gx_{\tau(k)}\leq_{j}gy_{\tau(k)} \ \textrm{for all}\ j \in \Lambda_{n}. $

(i)   If $j=1 \in A$, we have $gx_{\tau(k_{1})}\leq gy_{\tau(k_{1})}$ and

$ \begin {eqnarray*} F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})= F(x_{\tau(k_{1})}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})\\ \leq F(y_{\tau(k_{1})}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})= F(y_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)}). \end {eqnarray*} $

(ii)   If $j=1 \in B$, we have $gx_{\tau(k_{1})}\geq gy_{\tau(k_{1})}$ and

$ \begin {eqnarray*} F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})= F(x_{\tau(k_{1})}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})\\ \leq F(y_{\tau(k_{1})}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})= F(y_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)}). \end {eqnarray*} $

That is

$ F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})\leq F(y_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)}) \textrm{ for all}\ i \in A. $

(i)  If $j=2 \in A$, we have $gx_{\tau(k_{2})}\leq gy_{\tau(k_{2})}$ and

$ \begin {eqnarray*} F(y_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})= F(y_{\sigma_{i}(1)}, x_{\tau(k_{2})}, \cdots, x_{\sigma_{i}(n)})\\ \leq F(y_{\sigma_{i}(1)}, x_{\tau(k_{2})}, \cdots, x_{\sigma_{i}(n)})= F(y_{\sigma_{i}(1)}, y_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)}). \end {eqnarray*} $

(ii)  If $j=2 \in B$, we have $gx_{\tau(k_{2})}\geq gy_{\tau(k_{2})}$ and

$ \begin {eqnarray*} F(y_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})= F(y_{\sigma_{i}(1)}, x_{\tau(k_{2})}, \cdots, x_{\sigma_{i}(n)})\\ \leq F(y_{\sigma_{i}(1)}, x_{\tau(k_{2})}, \cdots, x_{\sigma_{i}(n)})= F(y_{\sigma_{i}(1)}, y_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)}). \end {eqnarray*} $

That is,

$ F(y_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})\leq F(y_{\sigma_{i}(1)}, y_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})\ \textrm{ for all }\ i \in A. $

Continuing in this way, we can get

$ F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)}) \leq F(y_{\sigma_{i}(1)}, y_{\sigma_{i}(2)}, \cdots, y_{\sigma_{i}(n)}) \ \textrm{for all}\ i \in A. $

Similarly, for $i \in B $, we can have

$ gx_{\tau(k)}\leq_{k}gy_{\tau(k)} \Rightarrow gx_{\tau(k)}\geq_{j}gy_{\tau(k)} \textrm{ for all } j \in \Lambda_{n}, $

and

$ F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})\geq F(y_{\sigma_{i}(1)}, y_{\sigma_{i}(2)}, \cdots, y_{\sigma_{i}(n)}) \textrm{ for all } i \in B. $

Then

$ F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)})\leq_{i} F(y_{\sigma_{i}(1)}, y_{\sigma_{i}(2)}, \cdots, y_{\sigma_{i}(n)})\ \textrm{ for all}\ i \in \Lambda_{n}. $

That is, $\widetilde{F}$ has the mixed $\widetilde{g}$-monotone property. According to the known conditions, we have

$ \widetilde{g}(x_{1}, x_{2}, \cdots, x_{n}) \preccurlyeq \widetilde{g}(y_{1}, y_{2}, \cdots, y_{n}) \Rightarrow gx_{\tau(i)} \leq_{i} gx_{\tau(i)}, \textrm{ for all } i\in\Lambda_{n}. $

Now we will prove from $gx_{\tau(\sigma_{i}(j))} \leq_{\sigma_{i}(j)}gy_{\tau(\sigma_{i}(j))}$ to $gx_{\tau(\sigma_{i}(j))} \leq_{j}gy_{\tau(\sigma_{i}(j))}$ for all $j \in \Lambda_{n}$.

In fact, let $i \in A$, $\sigma_{i} \in \Omega_{A, B}$, and $\tau \in A$, since $gx_{\tau(k)} \leq_{\sigma_{i}(j)}gy_{\tau(k)}$, for all $k \in \Lambda_{n}$,

(1)   If $i \in A$, there exists $k \in A$, $\sigma_{i}(j)= \tau(k)$, we have $x_{\tau(\sigma_{i}(j))} = x_{\tau(k)}$, $y_{\tau(\sigma_{i}(j))} = y_{\tau(k)}$.

(2)   If $i \in B$, there exists $k \in B$, $\sigma_{i}(j)= \tau(k)$.

Following the known conditions, we have

$ M(F(x_{\sigma_{1}(1)}, x_{\sigma_{1}(2)}, \cdots, x_{\sigma_{1}(n)}), F(y_{\sigma_{1}(1)}, y_{\sigma_{1}(2)}, \cdots, y_{\sigma_{1}(n)}), \varphi(t)) \geq \min\limits_{1\leq j \leq n} M(gx_{\sigma_{1}(j)}, gy_{\sigma_{1}(j)}, t) $

$ M(F(x_{\sigma_{2}(1)}, x_{\sigma_{2}(2)}, \cdots, x_{\sigma_{2}(n)}), F(y_{\sigma_{2}(1)}, y_{\sigma_{2}(2)}, \cdots, y_{\sigma_{2}(n)}), \varphi(t)) \geq \min\limits_{1\leq j \leq n} M(gx_{\sigma_{2}(j)}, gy_{\sigma_{2}(j)}, t) $

$ \vdots $

$ M(F(x_{\sigma_{n}(1)}, x_{\sigma_{n}(2)}, \cdots, x_{\sigma_{n}(n)}), F(y_{\sigma_{n}(1)}, y_{\sigma_{n}(2)}, \cdots, y_{\sigma_{n}(n)}), \varphi(t)) \geq\min\limits_{1\leq j \leq n} M(gx_{\sigma_{n}(j)}, gy_{\sigma_{n}(j)}, t) $

which implies that

$ \begin {eqnarray*} &&\min\limits_{1\leq i \leq n}M(F(x_{\sigma_{i}(1)}, x_{\sigma_{i}(2)}, \cdots, x_{\sigma_{i}(n)}), F(y_{\sigma_{i}(1)}, y_{\sigma_{i}(2)}, \cdots, y_{\sigma_{i}(n)}), \varphi(t))\\ &\geq& \min\limits_{1\leq i \leq n}\{\min\limits_{1\leq j \leq n} d(gx_{\sigma_{i}(j)}, gy_{\sigma_{i}(j)}, t)\}. \end {eqnarray*} $

Let $i$, such that the left side of the inequality gets minimum. Then we get

$ M^{*}(\widetilde{F}(x_{1}, x_{2}, \cdots, x_{n}), \widetilde{F}(y_{1}, y_{2}, \cdots, y_{n}), \varphi(t)) \geq \ \min\limits_{1\leq j \leq n} M(gx_{\sigma_{i}(j)}, gy_{\sigma_{i}(j)}, t), $

that is,

$ \begin {eqnarray*} &&M^{*}(\widetilde{F}(x_{1}, x_{2}, \cdots, x_{n}), \widetilde{F}(y_{1}, y_{2}, \cdots, y_{n}), \varphi(t)) \\ &\geq& \min\limits_{1\leq k \leq n}M(gx_{\tau_(k)}, gy_{\tau_(k)}, t) =M^{*}(\widetilde{g}(x_{1}, x_{2}, \cdots, x_{n}), \widetilde{g}(y_{1}, y_{2}, \cdots, y_{n}), t). \end {eqnarray*} $

It is easy to verify that $\widetilde{F}(X^{n})\subseteq \widetilde{g}(X^{n})$, $\widetilde{g}$ is continuous and $\widetilde{g}(Y)$ is complete, and there exists $(x^{1}_{0}, x^{2}_{0}, \cdots, x^{n}_{0}) \in X $ verifying

$ gx^{0}_{\tau_{k}} \leq_{k} F(x^{0}_{\tau_{k}(1))}, x^{0}_{\tau_{k}(2)}, \cdots, x^{0}_{\tau_{k}(n)}), \ \textrm{ for all}\ k. $

That is, there exists $x^{1}_{0}, x^{2}_{0}, \cdots, x^{n}_{0} \in X $ verifying

$ \widetilde{g}(x^{0}_{1}, x^{0}_{2}, \cdots, x^{0}_{n}) \preccurlyeq \widetilde{F}(x^{0}_{1}, x^{0}_{2}, \cdots, x^{0}_{n}). $

Following all the conditions of Theorem 3.3 and the proof, we can have $F$ and $g$, at least, one $\Phi$ -coincidence point.

It is obvious that, if $F$ and $g$ are compatible, then they are weakly compatible. So, we have the following theorem.

Theorem 3.5   Let $(X, M, \Delta, \leq)$ be a complete ordered fuzzy metric space with $\Delta$ a triangular norm of H-type. Let $ \Phi = (\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}, \tau)$ be $(n+1)$-tuple of mappings from $\Lambda_n$ into itself such that $\tau \in \Omega_{A, B}$ is a permutation and verifying that $\sigma_{i} \in \Omega_{A, B}$ if $ i \in A$ and $\sigma_{i} \in \Omega^{'}_{A, B}$ if $ i \in B$. Let $ \varphi : [0, +\infty ) \rightarrow [0, +\infty )$, $\varphi\in\Psi_{\omega}$, $F: X^{n} \rightarrow X$ and $g: X \rightarrow X$ be two mappings, $F(X^{n})\subseteq g(X)$, $F$ is continuous and has the mixed $g$-monotone property, $F$ and $g$ are compatible mappings and

$ M(F(x_{1}, x_{2}, \cdots, x_{n}), F(y_{1}, y_{2}, \cdots, y_{n}), \varphi(t)) \geq \min\limits_{1\leq i\leq n}M(gx_{i}, gy_{i}, t) $

for which $gx_{\tau(i)}\leq_{i}gy_{\tau(i)}$ for all $i\in \Lambda_{n}$ and all $t>0$. If there exists $(x^0_1, x^0_2, \cdots, x^0_n) \in X^n$ verifying $gx^0_{\tau(i)} \leq_{i} F(x^0_{\sigma_{i}(1)}, x^0_{\sigma_{i}(2)}, \cdots, x^0_{\sigma_{i}(n)})$ for all $i$, then $F$ and $g$ have at least one $\Phi$-coincidence point.

In theorem 3.5, let $n=2$, we have $\Lambda_{2} = \{1, 2\}, A = \{1\}, B = \{2\}, \sigma_{1} \in \Omega_{A, B}$ and $ \sigma_{2}\in \Omega^{'}_{A, B}$, then $ \sigma_{1}(1)=\{1\}, \sigma_{1}(2)=\{2\}$ and $ \sigma_{2}(1)=\{2\}, \sigma_{2}(2)=\{1\}$. Then we have the following corollary.

Corollary 3.6   Let $(X, M, \Delta, \leq)$ be a complete ordered fuzzy metric space with $\Delta$ a triangular norm of H-type. $ \varphi : [0, +\infty ) \rightarrow [0, +\infty )$, $\varphi\in\Psi_{\omega}$, Let $ F: X^{2} \rightarrow X $ and $g: X \rightarrow X$ be two mappings, $ F(X^{2}) \subseteq g(X)$, $F$ is continuous and has the mixed $g$-monotone property, $F$ and $g$ be weakly compatible mapping and

$ M(F(x_{1}, x_{2}), F(y_{1}, y_{2}), \varphi(t)) \geq \min \{M(gx_{1}, gy_{1}, t), M(gx_{2}, gy_{2}, t)\} $

for which $ gx_{1} \leq gy_{1}$ and $ gx_{2} \geq gy_{2}$. If there exists $ x^{0}_{1}, x^{0}_{2} \in X$ verifying $ gx^{0}_{1} \leq F(x^{0}_{1}, x^{0}_{2})$ and $ gx^{0}_{2} \leq F(x^{0}_{2}, x^{0}_{1})$, then $F$ and $g$ have a coupled fixed point in $X$.

Similarly, in Theorem 3.5, let $n=3$, we have $ \Lambda_{3}=\{1, 2, 3\}$, $A=\{1, 3\}$, $B=\{2\}$. $\sigma_{1}, \sigma_{3}\in \Omega_{A, B}$ and $\sigma_{2} \in \Omega^{'}_{A, B}$, then $ \sigma_{1}(1)=\{1\}$, $\sigma_{1}(2)=\{2\}$, $\sigma_{1}(3)=\{3\}$, $\sigma_{2}(1)=\{2\}$, $\sigma_{2}(2)=\{1\}$, $\sigma_{2}(3)=\{2\}$ and $ \sigma_{3}(1)=\{3\}$, $\sigma_{3}(2)=\{2\}$, $\sigma_{3}(3) =\{1\}$. Then we have the following corollary.

Corollary 3.7   Let $(X, M, \Delta, \leq)$ be a complete ordered fuzzy metric space with $\Delta$ a triangular norm of H-type. $ \varphi : [0, +\infty ) \rightarrow [0, +\infty )$, $\varphi\in\Psi_{\omega}$, Let $ F: X^{3} \rightarrow X $ and $g: X \rightarrow X$ be two mappings, $ F(X^{3}) \subseteq g(X)$, $F$ is continuous and has the mixed $g$-monotone property, $F$ and $g$ be weakly compatible mapping and

$ M(F(x_{1}, x_{2}, x_{3}), F(y_{1}, y_{2}, y_{3}), \varphi(t)) \geq \min \{M(gx_{1}, gy_{1}, t), M(gx_{2}, gy_{2}, t), M(gx_{3}, gy_{3}, t)\} $

for which $ gx_{1} \leq gy_{1}, $ $ gx_{2} \geq gy_{2}$ and $ gx_{3} \leq gy_{3}$. If there exists $ x^{0}_{1}, x^{0}_{2}, x^{0}_{3} \in X$ verifying $ gx^{0}_{1} \leq F(x^{0}_{1}, x^{0}_{2}, x^{0}_{3})$, $ gx^{0}_{2} \geq F(x^{0}_{2}, x^{0}_{1}, x^{0}_{3})$ and $ gx^{0}_{3} \leq F(x^{0}_{3}, x^{0}_{2}, x^{0}_{1})$, then $F$ and $g$ have a tripled fixed point in $X$.

Remark   When $F$ and $g$ are commutative, they are weakly compatible, so we have the following theorem.

Theorem 3.8   Let $(X, M, \Delta, \leq)$ be a complete ordered fuzzy metric space with $\Delta$ a triangular norm of H-type. Let $ \Phi = (\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}, \tau)$ be $(n+1)$-tuple of mappings from $\Lambda_n$ into itself such that $\tau \in \Omega_{A, B}$ is a permutation and verifying that $\sigma_{i} \in \Omega_{A, B}$ if $ i \in A$ and $\sigma_{i} \in \Omega^{'}_{A, B}$ if $ i \in B$. Let $ \varphi : [0, +\infty ) \rightarrow [0, +\infty )$, $\varphi\in\Psi_{\omega}$, $F: X^{n} \rightarrow X$ and $g: X \rightarrow X$ be two mappings, $F(X^{n})\subseteq g(X)$, $F$ is continuous and has the mixed $g$-monotone property, $F$ and $g$ are commutative, and

$ M(F(x_{1}, x_{2}, \cdots, x_{n}), F(y_{1}, y_{2}, \cdots, y_{n}), \varphi(t)) \geq \min\limits_{1\leq i\leq n}M(gx_{i}, gy_{i}, t) $

for which $gx_{\tau(i)}\leq_{i}gy_{\tau(i)}$ for all $i\in \Lambda_{n}$ and all $t>0$. If there exists $(x^0_1, x^0_2, \cdots, x^0_n) \in X^n$ verifying $gx^0_{\tau(i)} \leq_{i} F(x^0_{\sigma_{i}(1)}, x^0_{\sigma_{i}(2)}, \cdots, x^0_{\sigma_{i}(n)})$ for all $i$, then $F$ and $g$ have at least one $\Phi$-coincidence point.

Remark   Let $k\in[0, 1)$, taking $\varphi(t) = kt $ in Theorem 3.4, 3.5, 3.8, we obtain the following corollaries.

Corollary 3.9   Let $(X, M, \Delta, \leq)$ be a complete ordered fuzzy metric space with $\Delta$ a triangular norm of H-type. Let $ \Phi = (\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}, \tau)$ be $(n+1)$-tuple of mappings from $\Lambda_n$ into itself such that $\tau \in \Omega_{A, B}$ is a permutation and verifying that $\sigma_{i} \in \Omega_{A, B}$ if $ i \in A$ and $\sigma_{i} \in \Omega^{'}_{A, B}$ if $ i \in B$. Let $ \varphi : [0, +\infty ) \rightarrow [0, +\infty )$, $\varphi\in\Psi_{\omega}$, Let $F: X^{n} \rightarrow X$ and $g: X \rightarrow X$ be two mappings, $F(X^{n})\subseteq g(X)$, $F$ is continuous and has the mixed $g$-monotone property, $F$ and $g$ are weakly compatible mappings and

$ M(F(x_{1}, x_{2}, \cdots, x_{n}), F(y_{1}, y_{2}, \cdots, y_{n}), kt) \geq \min\limits_{1\leq i\leq n}M(gx_{i}, gy_{i}, t) $

for which $gx_{\tau(i)}\leq_{i}gy_{\tau(i)}$ for all $i\in \Lambda_{n}$ and all $t>0$. If there exists $(x^0_1, x^0_2, \cdots, x^0_n) \in X^n$ verifying $gx^0_{\tau(i)} \leq_{i} F(x^0_{\sigma_{i}(1)}, x^0_{\sigma_{i}(2)}, \cdots, x^0_{\sigma_{i}(n)})$ for all $i$, then $F$ and $g$ have at least one $\Phi$-coincidence point.

Corollary 3.10   Let $(X, M, \Delta, \leq)$ be a complete ordered fuzzy metric space with $\Delta$ a triangular norm of H-type. Let $ \Phi = (\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}, \tau)$ be $(n+1)$-tuple of mappings from $\Lambda_n$ into itself such that $\tau \in \Omega_{A, B}$ is a permutation and verifying that $\sigma_{i} \in \Omega_{A, B}$ if $ i \in A$ and $\sigma_{i} \in \Omega^{'}_{A, B}$ if $ i \in B$. Let $ \varphi : [0, +\infty ) \rightarrow [0, +\infty )$, $\varphi\in\Psi_{\omega}$, Let $F: X^{n} \rightarrow X$ and $g: X \rightarrow X$ be two mappings, $F(X^{n})\subseteq g(X)$, $F$ is continuous and has the mixed $g$-monotone property, $F$ and $g$ are compatible mappings and

$ M(F(x_{1}, x_{2}, \cdots, x_{n}), F(y_{1}, y_{2}, \cdots, y_{n}), kt) \geq \min\limits_{1\leq i\leq n}M(gx_{i}, gy_{i}, t) $

for which $gx_{\tau(i)}\leq_{i}gy_{\tau(i)}$ for all $i\in \Lambda_{n}$ and all $t>0$. If there exists $(x^0_1, x^0_2, \cdots, x^0_n) \in X^n$ verifying $gx^0_{\tau(i)} \leq_{i} F(x^0_{\sigma_{i}(1)}, x^0_{\sigma_{i}(2)}, \cdots, x^0_{\sigma_{i}(n)})$ for all $i$, then $F$ and $g$ have at least one $\Phi$-coincidence point.

Corollary 3.11   Let $(X, M, \Delta, \leq)$ be a complete ordered fuzzy metric space with $\Delta$ a triangular norm of H-type. Let $ \Phi = (\sigma_{1}, \sigma_{2}, \cdots, \sigma_{n}, \tau)$ be $(n+1)$-tuple of mappings from $\Lambda_n$ into itself such that $\tau \in \Omega_{A, B}$ is a permutation and verifying that $\sigma_{i} \in \Omega_{A, B}$ if $ i \in A$ and $\sigma_{i} \in \Omega^{'}_{A, B}$ if $ i \in B$. Let $ \varphi : [0, +\infty ) \rightarrow [0, +\infty )$, $\varphi\in\Psi_{\omega}$, $F: X^{n} \rightarrow X$ and $g: X \rightarrow X$ be two mappings, $F(X^{n})\subseteq g(X)$, $F$ is continuous and has the mixed $g$-monotone property, $F$ and $g$ are commutative, and

$ M(F(x_{1}, x_{2}, \cdots, x_{n}), F(y_{1}, y_{2}, \cdots, y_{n}), kt) \geq \min\limits_{1\leq i\leq n}M(gx_{i}, gy_{i}, t) $

for which $gx_{\tau(i)}\leq_{i}gy_{\tau(i)}$ for all $i\in \Lambda_{n}$ and all $t>0$. If there exists $(x^0_1, x^0_2, \cdots, x^0_n) \in X^n$ verifying $gx^0_{\tau(i)} \leq_{i} F(x^0_{\sigma_{i}(1)}, x^0_{\sigma_{i}(2)}, \cdots, x^0_{\sigma_{i}(n)})$ for all $i$, then $F$ and $g$ have at least one $\Phi$-coincidence point.