Multidimensional common fixed point theorems under probabilistic φ-contractive conditions in multidimensional Menger probabilistic metric spaces

Abstract

In this paper, we introduce the new concepts of multidimensional Menger probabilistic metric spaces and related fixed point for a pair of mappings T: $\underbrace{X\times X\times\cdots\times X}_{n}\rightarrow X$ and A: $X\rightarrow X$. Utilizing the properties of the related triangular norm and the compatibility of A with T, some multidimensional common fixed point problems of hybrid probabilistic contractions with a gauge function φ are studied. The obtained results generalize some coupled and triple common fixed point theorems in the corresponding literature. Finally, an example is given to illustrate our main results.

1 Introduction

Coupled fixed points were studied first by Bhaskar and Lakshmikantham [1]. Since then, some new results on the existence and uniqueness of coupled fixed points have been presented in partially ordered metric spaces, cone metric spaces, and fuzzy metric spaces [2–5]. The concept of a probabilistic metric space was initiated and studied by Menger, which is a generalization of the metric space [6]. Many results for the existence of fixed points or solutions of nonlinear equations under various types of conditions in Menger probabilistic spaces (briefly, PM-spaces) have been extensively considered by many scholars [7–22]. In 2010, Jachymski established a fixed point theorem for φ-contractions and gave a characterization of a function φ having the property that there exists a probabilistic φ-contraction, which is not a probabilistic k-contraction ($k\in [0,1)$) [23]. In 2011, Xiao et al. obtained some common coupled fixed point results for hybrid probabilistic contractions with a gauge function φ in Menger probabilistic metric spaces without assuming any continuity or monotonicity conditions for φ [24]. In 2014, Luo et al. introduced the concept of generalized Menger probabilistic metric spaces and obtained some tripled common fixed point results with a gauge function φ with the same properties in generalized Menger probabilistic metric spaces [25].

The purpose of this paper is to introduce the new concepts of multidimensional Menger probabilistic metric spaces and a related fixed point for a pair of mappings T: $\underbrace{X\times X\times\cdots\times X}_{n}\rightarrow X$ and A: $X\rightarrow X$. Utilizing the properties of the related triangular norm and the compatibility of A with T, some multidimensional common fixed point problems of hybrid probabilistic contractions with a gauge function φ are studied. The obtained results generalize some coupled and triple common fixed point theorems in the corresponding literature. Finally, an example is given to illustrate our main results.

2 Preliminaries

Denote by n any given positive integer which is not smaller than 2, $\Lambda_{n}$ the set $\{1,2,\ldots,n\}$, $X^{n}$ the product $\underbrace{X\times X\times\cdots\times X}_{n}$, $\mathbb{R}$ the set of the real numbers, $\mathbb{R}^{+}$ the set of the nonnegative real numbers, and $\mathbb {Z^{+}}$ the set of all positive integers. A mapping $F:\mathbb{R}\rightarrow\mathbb{R}^{+}$ is called a distribution function if it is nondecreasing left-continuous with $\sup_{t\in\mathbb{R}}F(t)=1$ and $\inf_{t\in \mathbb{R}}F(t)=0$.

We will denote by $\mathscr{D}$ the set of all distribution functions, by $\mathscr{D}^{+}=\{F\in\mathscr{D} : F(t)=0,\forall t\leq0\}$, while H will always denote the specific distribution function defined by

$$H(t)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} 0, & t\leq0,\\ 1, & t>0. \end{array}\displaystyle \right . $$

If φ: $\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is a function such that $\varphi(0)=0$, then φ is called a gauge function. If $t\in\mathbb{R}^{+}$, then $\varphi^{n}(t)$ denotes the nth iteration of $\varphi(t)$ and $\varphi^{-1}(\{0\})=\{t\in\mathbb{R}^{+} :\varphi(t)=0\}$.

First, we give PM-spaces introduced by Menger with the related triangular norm.

Definition 2.1

[7]

A mapping $\Delta:[0,1]\times[0,1]\rightarrow [0,1]$ is called a triangular norm (for short, a t-norm) if the following conditions are satisfied for any $a, b, c, d\in[0,1]$:

• (1) $\Delta(a,1)=a$;

• (2) $\Delta(a,b)=\Delta(b,a)$;

• (3) $\Delta(a,c)\geq\Delta(b,d)$ for $a\geq b$, $c\geq d$;

• (4) $\Delta(a,\Delta(b,c))=\Delta(\Delta(a,b),c)$.

Definition 2.2

[6]

A triplet $(X, \mathscr{F}, \Delta)$ is called a Menger probabilistic metric space (for short, a Menger PM-space) if X is a nonempty set, Δ is a t-norm, and $\mathscr{F}$ is a mapping from $X\times X$ into $\mathscr{D}^{+}$ satisfying the following conditions (we denote $\mathscr{F}(x,y)$ by $F_{x,y}$):

• (MS-1) $F_{x,y}(t)=H(t)$ for all $t\in R$ if and only if $x=y$;

• (MS-2) $F_{x,y}(t)=F_{y,x}(t)$ for all $t\in R$;

• (MS-3) $F_{x,y}(t+s)\geq\Delta(F_{x,z}(t),F_{z,y}(s))$ for all $x,y,z\in X$ and $t,s\geq0$.

Then we give the generalized Menger PM-spaces introduced by Luo et al. with the related triangular norm.

Definition 2.3

[8]

A mapping $\Delta:[0,1]\times[0,1]\times [0,1]\rightarrow[0,1]$ is called a triangular norm (for short, a t-norm) if the following conditions are satisfied for any $a, b, c, d, e, f \in[0,1]$:

• (1) $\Delta(a,1,1)=a$, $\Delta(0,0,0)=0$;

• (2) $\Delta(a,b,c)=\Delta(a,c,b)=\Delta(c,b,a)$;

• (3) $\Delta(a,b,c)\geq\Delta(d,e,f)$ for $a\geq d$, $b\geq e$, $c\geq f$;

• (4) $\Delta(a,\Delta(b,c,d),e)=\Delta(\Delta(a,b,c),d,e)=\Delta (a,b,\Delta(c,d,e))$.

Definition 2.4

[25]

A triplet $(X, \mathscr{F}, \Delta)$ is called a generalized Menger probabilistic metric space (for short, a generalized Menger PM-space) if X is a nonempty set, Δ is a t-norm, and $\mathscr{F}$ is a mapping from $X\times X$ into $\mathscr{D}^{+}$ satisfying the following conditions (we denote $\mathscr{F}(x,y)$ by $F_{x,y}$):

• (GPM-1) $F_{x,y}(t)=H(t)$ for all $t\in R$ if and only if $x=y$;

• (GPM-2) $F_{x,y}(t)=F_{y,x}(t)$ for all $t\in R$;

• (GPM-3) $F_{x,w}(t_{1}+t_{2}+t_{3})\geq\Delta (F_{x,y}(t_{1}),F_{y,z}(t_{2}),F_{z,w}(t_{3}))$ for all $x,y,z,w\in X$ and $t_{1},t_{2},t_{3}\geq0$.

Now, we introduce the definition of multidimensional Menger probabilistic metric spaces with the related triangular norm.

Definition 2.5

A mapping Δ: $\underbrace{[0,1]\times [0,1]\times\cdots\times[0,1]}_{n}\rightarrow[0,1]$ is called a triangular norm (for short, a t-norm) if the following conditions are satisfied for any $a_{1}, a_{2},\ldots, a_{n}, a_{n+1}, \ldots, a_{2n}\in[0,1]$:

• (1) $\Delta(a_{1},1,\ldots,1)=a_{1}$, $\Delta(0,0,\ldots,0)=0$;

• (2) $\Delta(a_{1},a_{2},\ldots,a_{n-2},a_{n-1},a_{n})=\Delta (a_{1},a_{n},\ldots,a_{n-2},a_{n-1})=\Delta(a_{1},a_{n},a_{n-1},\ldots ,a_{n-2})=\cdots =\Delta(a_{1},a_{n},a_{n-1},a_{n-2},\ldots,a_{2})=\Delta (a_{n},a_{n-1},a_{n-2},\ldots,a_{2},a_{1})$;

• (3) $\Delta(a_{1},a_{2},\ldots,a_{n})\geq\Delta (a_{n+1},a_{n+2},\ldots,a_{2n})$ for $a_{1}\geq a_{n+1}, a_{2}\geq a_{n+2},\ldots, a_{n}\geq a_{2n}$;

• (4) $\Delta(\Delta(a_{1},a_{2},\ldots,a_{n}),a_{n+1},\ldots ,a_{2n-1})=\Delta(a_{1},\Delta(a_{2},\ldots,a_{n+1}),a_{n+2}\cdots ,a_{2n-1})=\cdots =\Delta(a_{1},\ldots,a_{n-1},\Delta(a_{n},a_{n+1},\ldots,a_{2n-1}))$.

Two typical examples of t-norm are $\Delta_{M}(a_{1},a_{2},\ldots ,a_{n})=\min\{a_{1},a_{2},\ldots,a_{n}\}$ and $\Delta _{P}(a_{1},a_{2}, \ldots,a_{n})=a_{1}a_{2}\cdots a_{n}$ for all $a_{1},a_{2},\ldots,a_{n} \in[0,1]$.

Definition 2.6

A triplet $(X, \mathscr{F}, \Delta)$ is called a multidimensional Menger probabilistic metric space (for short, a multidimensional Menger PM-space) if X is a nonempty set, Δ is a t-norm and $\mathscr{F}$ is a mapping from $X\times X$ into $\mathscr{D}^{+}$ satisfying the following conditions (we denote $\mathscr{F}(x,y)$ by $F_{x,y}$):

• (MPM-1) $F_{x,y}(t)=H(t)$ for all $t\in R$ if and only if $x=y$;

• (MPM-2) $F_{x,y}(t)=F_{y,x}(t)$ for all $t\in R$;

• (MPM-3) $F_{x_{1},x_{n+1}}(t_{1}+t_{2}+\cdots+t_{n})\geq\Delta (F_{x_{1},x_{2}}(t_{1}),F_{x_{2},x_{3}}(t_{2}),\ldots ,F_{x_{n},x_{n+1}}(t_{n}))$ for all $x_{1},x_{2},\ldots,x_{n+1}\in X$ and $t_{1},t_{2},\ldots, t_{n}\geq0$.

Remark 2.1

If $n=2$, the multidimensional Menger PM-space is a Menger PM-space. While $n=3$, the multidimensional Menger PM-space is a generalized Menger PM-space.

Remark 2.2

If $\Delta=\Delta_{M}$, the multidimensional Menger PM-space is a Menger PM-space. In fact, let $x_{1}=x,x_{2}=z,\ldots,x_{n}=z,x_{n+1}=y$ in (MPM-3), then for any $t,s,\delta\geq0$, $(n-2)\delta\leq s$, we have

$$F_{x,y}(t+s)\geq\min\bigl\{ (F_{x,z}(t),F_{z,z}( \delta),\ldots,F_{z,z}(\delta ),F_{z,y}\bigl(s-(n-2)\delta\bigr) \bigr\} . $$

Thus we have

$$F_{x,y}(t+s)\geq\min\bigl\{ (F_{x,z}(t),F_{z,y} \bigl(s-(n-2)\delta\bigr)\bigr\} . $$

Taking $\delta\rightarrow0$, we obtain

$$F_{x,y}(t+s)\geq\min\bigl\{ (F_{x,z}(t),F_{z,y}(s) \bigr\} . $$

Therefore, if $\Delta=\Delta_{M}$, the multidimensional Menger PM-space is a Menger PM-space.

Example 2.1

Suppose that $X=[-1,1]$. Define $\mathscr{F} : X\times X \rightarrow\mathscr{D}^{+}$ by

$$\mathscr{F}_{x,y}(t) = F_{x,y}(t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{t}{t+|x-y|}, & t>0,\\ 0, & t\leq0, \end{array}\displaystyle \right . $$

for all $x,y\in X$. It is easy to verify that $(X, \mathscr{F},\Delta _{M})$ satisfies (MPM-1) and (MPM-2). Now we prove it also satisfies (MPM-3). Assume that $t_{1},t_{2},\ldots, t_{n}\geq0$ and $x_{1},x_{2},\ldots,x_{n+1}\in X$. Then we have

$$\begin{aligned} F_{x_{1},x_{n+1}}(t_{1}+\cdots+t_{n}) =&\frac{t_{1}+\cdots +t_{n}}{t_{1}+\cdots+t_{n}+|x_{1}-x_{n+1}|} \\ \geq&\frac{t_{1}+\cdots+t_{n}}{t_{1}+\cdots +t_{n}+|x_{1}-x_{2}|+\cdots+|x_{n}-x_{n+1}|} \\ \geq&\min\biggl\{ \frac{t_{1}}{t_{1}+|x_{1}-x_{2}|},\ldots,\frac {t_{n}}{t_{n}+|x_{n}-x_{n+1}|}\biggr\} \\ =&\Delta_{M}\bigl(F_{x_{1},x_{2}}(t_{1}),\ldots ,F_{x_{n},x_{n+1}}(t_{n})\bigr). \end{aligned}$$

Hence $(X, \mathscr{F},\Delta_{M})$ a multidimensional Menger PM-space.

Proposition 2.1

Let $(X, \mathscr{F}, \Delta)$ be a multidimensional Menger PM-space and Δ be a continuous t-norm. Then $(X, \mathscr{F}, \Delta)$ is a Hausdorff topological space in the $(\epsilon,\lambda)$-topology $\mathscr{T}$, i.e., the family of sets

$$\bigl\{ U_{x}(\epsilon,\lambda):\epsilon>0,\lambda\in(0,1], x \in X \bigr\} $$

is a base of neighborhoods of a point x for $\mathscr{F}$, where

$$U_{x}(\epsilon,\lambda)=\bigl\{ y\in X : F_{x,y}(\epsilon)>1- \lambda\bigr\} . $$

Proof

It suffices to prove that:

• (i) for any $x\in X$, there exists an $U=U_{x}(\epsilon,\lambda)$ such that $x\in U$;

• (ii) for any given $U_{x}(\epsilon_{1},\lambda_{1})$ and $U_{x}(\epsilon_{2},\lambda_{2})$, there exist $\epsilon>0$ and $\lambda >0$, such that $U_{x}(\epsilon,\lambda)\subset U_{x}(\epsilon_{1},\lambda_{1})\cap U_{x}(\epsilon_{2},\lambda_{2})$;

• (iii) for any $y\in U_{x}(\epsilon,\lambda)$, there exist $\epsilon ^{\prime}>0$ and $\lambda^{\prime}>0$, such that $U_{y}(\epsilon^{\prime},\lambda ^{\prime})\subset U_{x}(\epsilon,\lambda)$;

• (iv) for any $x, y\in X$, $x\neq y$, there exist $U_{x}(\epsilon _{1},\lambda_{1})$ and $U_{y}(\epsilon_{2},\lambda_{2})$, such that $U_{x}(\epsilon_{1},\lambda_{1})\cap U_{y}(\epsilon_{2},\lambda _{2})=\emptyset$.

It is easy to check that (i)-(iii) are true. Now we prove that (iv) is also true. In fact, suppose that $x,y\in X$ and $x\neq y$. Then there exist $t_{0}>0$ and $0< a<1$, such that $F_{x,y}(t_{0})=a$. Let

$$U_{x}=\biggl\{ r: F_{x,r}\biggl(\frac{t_{0}}{n}\biggr)>b \biggr\} , \qquad U_{y}=\biggl\{ r: F_{y,r}\biggl( \frac{t_{0}}{n}\biggr)>b\biggr\} , $$

where $0< b<1$ and $\Delta(b,\underbrace{1,\ldots,1}_{n-2},b)>a$ (since Δ is continuous and $\Delta(1,\ldots,1)=1$, such b exists). Now suppose that there exists a point $v\in U_{x}\cap U_{y}$, which implies that $F_{x,v}(\frac{t_{0}}{n})>b$ and $F_{y,v}(\frac{t_{0}}{n})>b$. Then we have

$$a=F_{x,y}(t_{0})\geq\Delta\biggl(F_{x,v}\biggl( \frac{t_{0}}{n}\biggr),\underbrace {F_{v,v}\biggl(\frac{t_{0}}{n} \biggr),\ldots,F_{v,v}\biggl(\frac {t_{0}}{n}\biggr)}_{n-2},F_{v,y} \biggl(\frac{t_{0}}{n}\biggr)\biggr) \geq\Delta(b,\underbrace{1, \ldots,1}_{n-2},b)>a, $$

which is a contradiction. Thus the conclusion (iv) is proved. This completes the proof. □

Definition 2.7

Let $(X,\mathscr{F},\Delta)$ be a multidimensional Menger PM-space, Δ be a continuous t-norm.

• (i) A sequence $\{x_{m}\}$ in X is said to be $\mathscr {T}$-convergent to $x\in X$ if $\lim_{m\rightarrow\infty}F_{x_{m},x}=1$ for all $t>0$;

• (ii) a sequence $\{x_{m}\}$ in X is said to be a $\mathscr {T}$-Cauchy sequence, if for any given $\epsilon>0$ and $\lambda\in (0,1]$, there exists a positive integer $N=N(\epsilon,\lambda)$, such that $F_{x_{m},x_{k}}(\epsilon)>1-\lambda$, whenever $m,k\geq N$;

• (iii) $(X,\mathscr{F},\Delta)$ is said to be $\mathscr {T}$-complete, if each $\mathscr{T}$-Cauchy sequence in X is $\mathscr{T}$-convergent to some point in X.

Definition 2.8

A t-norm Δ is said to be H-type if the family of functions $\{\Delta^{m}(t)\}_{m=1}^{\infty}$ is equi-continuous at $t=1$, where

$$\Delta^{1}(t)=\Delta(t,\ldots,t), \qquad \Delta^{m+1}(t)= \Delta \bigl(\underbrace{t,\ldots,t}_{n-1},\Delta^{m}(t)\bigr), \quad m=1,2,\ldots,t\in[0,1]. $$

Definition 2.9

Let X be a nonempty set, $T: X^{n}\rightarrow X$ and $A: X\rightarrow X$ be two mappings. A is said to be commutative with T, if $AT(x_{1},\ldots,x_{n})=T(Ax_{1},\ldots,Ax_{n})$ for all $x_{1},\ldots x_{n}\in X$. A point $u\in X$ is called a multidimensional common fixed point of T and A, if $u=Au=T(u,\ldots,u)$.

Definition 2.10

Let X be a nonempty set, $T: X^{n}\rightarrow X$ and $A: X\rightarrow X$ be two mappings. Let $\{x_{m}^{1}\},\ldots,\{x_{m}^{n}\}$ be n sequences in X and $\sigma_{1},\ldots,\sigma_{n}$ be n permutations of $\Lambda_{n}$. A and T are said to be compatible in $(X,\mathscr {F},\Delta)$ if

$$\lim_{m\rightarrow\infty} F_{AT(x^{\sigma_{i}(1)}_{m},\ldots,x^{\sigma _{i}(n)}_{m}),T(Ax^{\sigma_{i}(1)}_{m},\ldots,Ax^{\sigma_{i}(n)}_{m})}(t)=1 $$

for all $i=1,\ldots,n$ and $t>0$, whenever

$$\lim_{m\rightarrow\infty} T\bigl(x^{\sigma_{i}(1)}_{m}, \ldots,x^{\sigma _{i}(n)}_{m}\bigr)=\lim_{m\rightarrow\infty}Ax^{i}_{m} \in X $$

for all $i=1,\ldots,n$;

A and T are said to be compatible in $(X,d)$ where $(X,d)$ is a usual metric space if

$$\lim_{m\rightarrow\infty} d\bigl(AT\bigl(x^{\sigma_{i}(1)}_{m}, \ldots,x^{\sigma _{i}(n)}_{m}\bigr),T\bigl(Ax^{\sigma_{i}(1)}_{m}, \ldots,Ax^{\sigma_{i}(n)}_{m}\bigr)\bigr)=0 $$

for all $i=1,\ldots,n$ and $t>0$, whenever

$$\lim_{m\rightarrow\infty} T\bigl(x^{\sigma_{i}(1)}_{m}, \ldots,x^{\sigma _{i}(n)}_{m}\bigr)=\lim_{m\rightarrow\infty}Ax^{i}_{m} \in X $$

for all $i=1,\ldots,n$.

Obviously, if T and A are commutative, then they are compatible, but the converse does not hold.

The following lemmas play an important role in proving our main results in Section 3.

Lemma 2.1

[23]

Suppose that $F\in\mathscr{D^{+}}$. For every $m\in Z^{+}$, let $F_{m}:R\rightarrow[0,1]$ be nondecreasing and $g_{m}:(0,+\infty)\rightarrow(0,+\infty)$ satisfy $\lim_{m\rightarrow \infty}g_{m}(t)=0$ for any $t>0$. If $F_{m}(g_{m}(t))\geq F(t)$ for any $t>0$, then $\lim_{m\rightarrow\infty}F_{m}(t)=1$ for any $t>0$.

Lemma 2.2

Let X be a nonempty set, and $T: X^{n}\rightarrow X$ and $A: X\rightarrow X$ be two mappings. If $T(X^{n})\subset A(X)$, then there exist n sequences $\{x^{1}_{m}\}^{\infty}_{m=0},\ldots,\{x^{n}_{m}\}^{\infty }_{m=0}$ in X, such that $Ax^{1}_{m+1}=T({x^{1}_{m},x^{2}_{m},\ldots ,x^{n}_{m}}), Ax^{2}_{m+1}=T({x^{2}_{m},x^{3}_{m},\ldots,x^{n}_{m},x^{1}_{m}}),\ldots , Ax^{n}_{m+1}=T({x^{n}_{m},x^{1}_{m},\ldots,x^{n-1}_{m}})$.

Proof

Let $x^{1}_{0},x^{2}_{0},\ldots,x^{n}_{0}$ be any given points in X. Since $T(X^{n})\subset A(X)$, we can choose $x^{1}_{1},x^{2}_{1},\ldots,x^{n}_{1}\in X$ such that $Ax^{1}_{1}=T(x^{1}_{0},x^{2}_{0},\ldots,x^{n}_{0}), Ax^{2}_{1}=T(x^{ 2}_{0},x^{3}_{0},\ldots,x^{n}_{0},x^{1}_{0}), \ldots , Ax^{n}_{1}=T(x^{n}_{0},x^{1}_{0},\ldots,x^{n-1}_{0})$. Continuing this process, we can construct n sequences $\{x^{1}_{m}\} ^{\infty}_{m=0},\ldots, \{x^{n}_{m}\}^{\infty}_{m=0}$ in X, such that

$$\begin{aligned}& Ax^{1}_{m+1}=T\bigl({x^{1}_{m},x^{2}_{m}, \ldots ,x^{n}_{m}}\bigr),\qquad Ax^{2}_{m+1}=T \bigl({x^{2}_{m},x^{3}_{m},\ldots ,x^{n}_{m},x^{1}_{m}}\bigr),\qquad \ldots, \\& Ax^{n}_{m+1}=T\bigl({x^{n}_{m},x^{1}_{m}, \ldots,x^{n-1}_{m}}\bigr). \end{aligned}$$

 □

Lemma 2.3

[13]

Let $(X,d)$ is a usual metric space. Define $\mathscr{F}:X\times X\rightarrow\mathscr{D^{+}}$ by

$$F_{x,y}=H\bigl(t-d(x,y)\bigr), \quad \textit{for }x,y\in X\textit{ and }t>0. $$

Then $(X,\mathscr{F},\Delta_{M})$ is a Menger PM-space and is called the induced Menger PM-space by $(X,d)$. It is complete if $(X,d)$ is complete.

Lemma 2.4

[14]

Let $\varphi(t): \mathbb{R^{+}}\rightarrow\mathbb {R^{+}}$ be a function. Let $a,b,t\in\mathbb{R^{+}}$. Then we have

$$H(t-a)\geq H\bigl(\varphi(t)-b\bigr)\quad \textit{if and only if} \quad \varphi(b)\leq a. $$

3 Main results

In this section, we shall give the main results of this paper.

Theorem 3.1

Let $(X, \mathscr{F}, \Delta)$ be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, φ: $\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ be a gauge function such that $\varphi^{-1}(\{0\})=\{0\}$, $\varphi (t)< t$, and $\lim_{m\rightarrow+\infty}\varphi^{m}(t)=0$ for any $t>0$. Let T: $X^{n}\rightarrow X$ and A: $X\rightarrow X$ be two mappings satisfying the following conditions:

$$\begin{aligned} F_{T(x_{1},x_{2},\ldots,x_{n}),T(y_{1},y_{2},\ldots,y_{n})}\bigl(\varphi (t)\bigr)\geq\bigl[F_{Ax_{1},Ay_{1}}(t)F_{Ax_{2},Ay_{2}}(t) \cdots F_{Ax_{n},Ay_{n}}(t)\bigr]^{\frac{1}{n}} \end{aligned}$$

(3.1)

for all $x_{1},x_{2},\ldots,x_{n}, y_{1},y_{2},\ldots,y_{n} \in X$, and $t>0$, where $T(X^{n})\subset A(X)$, A is continuous and compatible with T. Then T and A have a unique multidimensional common fixed point in X.

Proof

By Lemma 2.2, we can construct n sequences $\{x^{1}_{m}\} ^{\infty}_{m=0},\ldots,\{x^{n}_{m}\}^{\infty}_{m=0}$ in X, such that $Ax^{1}_{m+1}=T({x^{1}_{m},x^{2}_{m},\ldots ,x^{n}_{m}}),Ax^{2}_{m+1}=T({x^{2}_{m},x^{3}_{m},\ldots ,x^{n}_{m},x^{1}_{m}}),\ldots, Ax^{n}_{m+1}=T({x^{n}_{m},x^{1}_{m},\ldots,x^{n-1}_{m}})$.

From (3.1), for all $t>0$, we have

$$ \begin{aligned} F_{Ax^{1}_{m},Ax^{1}_{m+1}}\bigl(\varphi (t) \bigr)&=F_{T(x^{1}_{m-1},x^{2}_{m-1},\ldots ,x^{n}_{m-1}),T(x^{1}_{m},x^{2}_{m},\ldots,x^{n}_{m})}\bigl(\varphi (t)\bigr) \\ &\geq \bigl[F_{Ax^{1}_{m-1},Ax^{1}_{m}}(t)F_{Ax^{2}_{m-1},Ax^{2}_{m}}(t)\cdots F_{Ax^{n}_{m-1},Ax^{n}_{m}}(t) \bigr]^{\frac{1}{n}}, \\ F_{Ax^{2}_{m},Ax^{2}_{m+1}}\bigl(\varphi (t)\bigr)&=F_{T(x^{2}_{m-1},x^{3}_{m-1},\ldots ,x^{1}_{m-1}),T(x^{2}_{m},x^{3}_{m},\ldots,x^{1}_{m})}\bigl(\varphi (t)\bigr) \\ &\geq \bigl[F_{Ax^{2}_{m-1},Ax^{2}_{m}}(t)F_{Ax^{3}_{m-1},Ax^{3}_{m}}(t)\cdots F_{Ax^{1}_{m-1},Ax^{1}_{m}}(t) \bigr]^{\frac{1}{n}}, \\ &\vdots \\ F_{Ax^{n}_{m},Ax^{n}_{m+1}}\bigl(\varphi (t)\bigr)&=F_{T(x^{n}_{m-1},x^{1}_{m-1},\ldots ,x^{n-1}_{m-1}),T(x^{n}_{m},x^{1}_{m},\ldots,x^{n-1}_{m})}\bigl(\varphi (t)\bigr) \\ &\geq \bigl[F_{Ax^{n}_{m-1},Ax^{n}_{m}}(t)F_{Ax^{1}_{m-1},Ax^{1}_{m}}(t)\cdots F_{Ax^{n-1}_{m-1},Ax^{n-1}_{m}}(t) \bigr]^{\frac{1}{n}}. \end{aligned} $$

(3.2)

Denote $P_{m}(t)=[F_{Ax^{1}_{m-1},Ax^{1}_{m}}(t)F_{Ax^{2}_{m-1},Ax^{2}_{m}}(t)\cdots F_{Ax^{n}_{m-1},Ax^{n}_{m}}(t)]^{\frac{1}{n}}$. From (3.2), we have

$$\begin{aligned} P_{m+1}\bigl(\varphi(t)\bigr) =&\bigl[F_{Ax^{1}_{m},Ax^{1}_{m+1}}\bigl(\varphi (t) \bigr)F_{Ax^{2}_{m},Ax^{2}_{m+1}}\bigl(\varphi(t)\bigr)\cdots F_{Ax^{n}_{m},Ax^{n}_{m+1}}\bigl( \varphi(t)\bigr)\bigr]^{\frac{1}{n}} \\ \geq&\bigl[\underbrace{P_{m}(t)P_{m}(t)\cdots P_{m}(t)}_{n}\bigr]^{\frac {1}{n}}=P_{m}(t), \end{aligned}$$

which implies that

$$ \begin{aligned} &F_{Ax^{1}_{m},Ax^{1}_{m+1}}\bigl(\varphi^{m}(t) \bigr)\geq P_{m}\bigl(\varphi ^{m-1}(t)\bigr)\geq\cdots P_{1}(t), \\ &F_{Ax^{2}_{m},Ax^{2}_{m+1}}\bigl(\varphi^{m}(t)\bigr)\geq P_{m}\bigl( \varphi ^{m-1}(t)\bigr)\geq\cdots P_{1}(t), \\ &\vdots \\ &F_{Ax^{2}_{m},Ax^{2}_{m+1}}\bigl(\varphi^{m}(t)\bigr)\geq P_{m}\bigl( \varphi ^{m-1}(t)\bigr)\geq\cdots P_{1}(t). \end{aligned} $$

(3.3)

Since $P_{1}(t)=[F_{Ax^{1}_{0},Ax^{1}_{1}}(t)F_{Ax^{2}_{0},Ax^{2}_{1}}(t)\cdots F_{Ax^{n}_{0},Ax^{n}_{1}}(t)]^{\frac{1}{n}} \in \mathscr{D^{+}}$ and $\lim_{m\rightarrow\infty}\varphi^{m}(t)=0$ for each $t>0$, using Lemma 2.1, we have

$$\begin{aligned} \lim_{m\rightarrow\infty}F_{Ax^{1}_{m},Ax^{1}_{m+1}}(t)=1,\qquad F_{Ax^{2}_{m},Ax^{2}_{m+1}}(t)=1, \qquad \ldots,\qquad F_{Ax^{n}_{m},Ax^{n}_{m+1}}(t)=1. \end{aligned}$$

(3.4)

Thus

$$\begin{aligned} \lim_{m\rightarrow\infty}P_{m}(t)=1,\quad \forall t>0. \end{aligned}$$

(3.5)

We claim that, for any $k\in\mathbb{Z^{+}}$ and $t>0$,

$$ \begin{aligned} &F_{Ax^{1}_{m},Ax^{1}_{m+k}}(t)\geq\Delta^{k} \biggl(P_{m}\biggl(\frac{t-\varphi (t)}{n-1}\biggr)\biggr), \\ &F_{Ax^{2}_{m},Ax^{2}_{m+k}}(t)\geq\Delta^{k}\biggl(P_{m}\biggl( \frac{t-\varphi (t)}{n-1}\biggr)\biggr), \\ &\vdots \\ &F_{Ax^{n}_{m},Ax^{n}_{m+k}}(t)\geq\Delta^{k}\biggl(P_{m}\biggl( \frac{t-\varphi(t)}{n-1}\biggr)\biggr). \end{aligned} $$

(3.6)

In fact, by (3.2) and $\varphi(t)< t$, we can conclude that (3.6) holds for $k=1$ since $F_{Ax^{1}_{m},Ax^{1}_{m+1}}(t)\geq F_{Ax^{1}_{m},Ax^{1}_{m+1}}(\varphi(t))\geq P_{m}(t)\geq P_{m}(\frac {t-\varphi(t)}{n-1})\geq\Delta^{1}(P_{m}(\frac{t-\varphi(t)}{n-1}))$. Assume that (3.6) holds for some k. Since $\varphi(t)< t$, by the first inequality of (3.2), we have $F_{Ax^{1}_{m},Ax^{1}_{m+1}}(t)\geq F_{Ax^{1}_{m},Ax^{1}_{m+1}}(\varphi(t))\geq P_{m}(t)$. By (3.1) and (3.6), we have

$$\begin{aligned} F_{Ax^{1}_{m+1},Ax^{1}_{m+k+1}}\bigl(\varphi(t)\bigr) \geq &\bigl[F_{Ax^{1}_{m},Ax^{1}_{m+k}}(t)F_{Ax^{2}_{m},Ax^{2}_{m+k}}(t) \cdots F_{Ax^{n}_{m},Ax^{n}_{m+k}}(t)\bigr]^{\frac{1}{n}} \\ \geq&\Delta^{k}\biggl(P_{m}\biggl(\frac{t-\varphi(t)}{n-1}\biggr) \biggr). \end{aligned}$$

Hence, by the monotonicity of Δ, we have

$$\begin{aligned} F_{Ax^{1}_{m},Ax^{1}_{m+k+1}}(t) =&F_{Ax^{1}_{m},Ax^{1}_{m+k+1}}\bigl(t-\varphi (t)+\varphi(t)\bigr) \\ \geq&\Delta \biggl(F_{Ax^{1}_{m},Ax^{1}_{m+1}}\biggl(\frac{t-\varphi(t)}{n-1}\biggr),\ldots ,F_{Ax^{1}_{m},Ax^{1}_{m+1}}\biggl(\frac{t-\varphi (t)}{n-1}\biggr),\\ &{}F_{Ax^{1}_{m+1},Ax^{1}_{m+k+1}}\bigl(\varphi(t) \bigr)\biggr) \\ \geq&\Delta\biggl(P_{m}\biggl(\frac{t-\varphi(t)}{n-1}\biggr), \ldots,P_{m}\biggl(\frac {t-\varphi(t)}{n-1}\biggr),\Delta^{k} \biggl(P_{m}\biggl(\frac{t-\varphi (t)}{n-1}\biggr)\biggr)\biggr) \\ =&\Delta^{k+1}\biggl(P_{m}\biggl(\frac{t-\varphi(t)}{n-1}\biggr) \biggr). \end{aligned}$$

Similarly, we have $F_{Ax^{2}_{m},Ax^{2}_{m+k+1}}(t)\geq\Delta ^{k+1}(P_{m}(\frac{t-\varphi(t)}{n-1})),\ldots, F_{Ax^{n}_{m},Ax^{n}_{m+k+1}}(t)\geq\Delta^{k+1}(P_{m}(\frac{t-\varphi (t)}{n-1}))$. Therefore, by induction, (3.6) holds for all $k\in\mathbb {Z^{+}}$ and $t>0$.

Suppose that $\lambda\in(0,1]$ is given. Since Δ is a t-norm of H-type, there exists $\delta>0$ such that

$$\begin{aligned} \Delta^{k}(s)>1-\lambda, \quad s\in(1-\delta,1], k\in \mathbb{Z^{+}}. \end{aligned}$$

(3.7)

By (3.5), there exists $M\in\mathbb{Z^{+}}$, such that $P_{m}(\frac {t-\varphi(t)}{n-1})>1-\delta$ for all $m\geq M$. Hence, from (3.6) and (3.7), we get $F_{Ax^{1}_{m},Ax^{1}_{m+k}}(t)>1-\lambda , F_{Ax^{2}_{m},Ax^{2}_{m+k}}(t)>1-\lambda, \ldots , F_{Ax^{n}_{m},Ax^{n}_{m+k}}(t)>1-\lambda$ for all $m\geq M$, $k\in \mathbb{Z^{+}}$. Therefore $\{Ax^{1}_{m}\}, \{Ax^{2}_{m}\}, \ldots, \{ Ax^{n}_{m}\}$ are n Cauchy sequences.

Since $(X, \mathscr{F}, \Delta)$ is complete, there exist $u^{1}, u^{2}, \ldots, u^{n}\in X$, such that

$$\lim_{m\rightarrow\infty}Ax^{1}_{m}=u^{1}, \qquad \lim_{m\rightarrow\infty}Ax^{2}_{m}=u^{2}, \qquad\ldots, \qquad\lim_{m\rightarrow \infty}Ax^{n}_{m}=u^{n}. $$

By the continuity of A, we have

$$\lim_{m\rightarrow\infty}AAx^{1}_{m}=Au^{1}, \qquad \lim_{m\rightarrow\infty }AAx^{2}_{m}=Au^{2}, \qquad \ldots,\qquad \lim_{m\rightarrow\infty}AAx^{n}_{m}=Au^{n}. $$

The compatibility of A with T implies that

$$\begin{aligned}& \lim_{m\rightarrow\infty} F_{AT(x^{1}_{m},x^{2}_{m},\ldots ,x^{n}_{m}),T(Ax^{1}_{m},Ax^{2}_{m},\ldots,Ax^{n}_{m})}(t)=1,\qquad\ldots,\\& \lim _{m\rightarrow\infty} F_{AT(x^{n}_{m},x^{1}_{m},\ldots ,x^{n-1}_{m}),T(Ax^{n}_{m},Ax^{1}_{m},\ldots,Ax^{n-1}_{m})}(t)=1, \end{aligned}$$

where $\sigma_{1}=(1,2,\ldots,n),\sigma_{2}=(2,3,\ldots,1),\ldots,\sigma _{n}=(n,1,\ldots,n-1)$.

From (3.1) and $\varphi(t)< t$, we obtain

$$\begin{aligned} F_{AAx^{1}_{m+1},T(u^{1},u^{2},\ldots ,u^{n})}(t) =&F_{AAx^{1}_{m+1},T(u^{1},u^{2},\ldots,u^{n})}\bigl(t-\varphi (t)+ \varphi(t)\bigr) \\ \geq&\Delta\biggl(F_{AAx^{1}_{m+1},T(Ax^{1}_{m},Ax^{2}_{m},\ldots ,Ax^{n}_{m})}\biggl(\frac{t-\varphi(t)}{n-1}\biggr), \\ &{}F_{T(Ax^{1}_{m},Ax^{2}_{m},\ldots ,Ax^{n}_{m}),T(Ax^{1}_{m},Ax^{2}_{m},\ldots,Ax^{n}_{m})}\biggl(\frac{t-\varphi (t)}{n-1}\biggr),\ldots, \\ &{}F_{T(Ax^{1}_{m},Ax^{2}_{m},\ldots ,Ax^{n}_{m}),T(Ax^{1}_{m},Ax^{2}_{m},\ldots,Ax^{n}_{m})}\biggl(\frac{t-\varphi (t)}{n-1}\biggr), \\ &{}F_{T(Ax^{1}_{m},Ax^{2}_{m},\ldots,Ax^{n}_{m}),T(u^{1},u^{2},\ldots ,u^{n})}\bigl(\varphi(t)\bigr)\biggr) \\ =&\Delta\biggl(F_{AAx^{1}_{m+1},T(Ax^{1}_{m},Ax^{2}_{m},\ldots ,Ax^{n}_{m})}\biggl(\frac{t-\varphi(t)}{n-1}\biggr),1,\ldots,1, \\ &{}F_{T(Ax^{1}_{m},Ax^{2}_{m},\ldots,Ax^{n}_{m}),T(u^{1},u^{2},\ldots ,u^{n})}\bigl(\varphi(t)\bigr)\biggr). \end{aligned}$$

(3.8)

From (3.1), we have

$$\begin{aligned} F_{T(Ax^{1}_{m},Ax^{2}_{m},\ldots,Ax^{n}_{m}),T(u^{1},u^{2},\ldots ,u^{n})}\bigl(\varphi(t)\bigr) \geq& \bigl[F_{AAx^{1}_{m},Au^{1}}(t)F_{AAx^{2}_{m},Au^{2}}(t)\cdots F_{AAx^{n}_{m},Au^{n}}(t) \bigr]^{\frac{1}{n}}. \end{aligned}$$

(3.9)

Combining (3.8) with (3.9) and letting $m\rightarrow\infty$, we obtain $\lim_{m\rightarrow\infty} AAx^{1}_{m}=T(u^{1},u^{2}, \ldots,u^{n})$. Hence $T(u^{1},u^{2},\ldots,u^{n})=Au^{1}$. Similarly, we can show that $T(u^{2},u^{3},\ldots,u^{1})=Au^{2}, T(u^{3},u^{4},\ldots ,u^{2})=Au^{3}, \ldots, T(u^{n},u^{1},\ldots,u^{n-1})=Au^{n}$.

Next we show that $Au^{1}=u^{1}, Au^{2}=u^{2}, \ldots, Au^{n}=u^{n}$. In fact, from (3.1), for all $t>0$, we have

$$\begin{aligned} F_{Au^{1},Ax^{1}_{m}}\bigl(\varphi(t) \bigr)&=F_{T(u^{1},u^{2},\ldots ,u^{n}),T(x^{1}_{m-1},x^{2}_{m-1},\ldots,x^{n}_{m-1})}\bigl(\varphi (t)\bigr) \\ &\geq\bigl[F_{Au^{1},Ax^{1}_{m-1}}(t),F_{Au^{2},Ax^{2}_{m-1}}(t),\ldots ,F_{Au^{n},Ax^{n}_{m-1}}(t) \bigr]^{\frac{1}{n}}, \\ F_{Au^{2},Ax^{2}_{m}}\bigl(\varphi(t)\bigr)&=F_{T(u^{2},u^{3},\ldots ,u^{1}),T(x^{2}_{m-1},x^{3}_{m-1},\ldots,x^{1}_{m-1})}\bigl(\varphi (t) \bigr) \\ &\geq\bigl[F_{Au^{2},Ax^{2}_{m-1}}(t),F_{Au^{3},Ax^{3}_{m-1}}(t),\ldots ,F_{Au^{1},Ax^{1}_{m-1}}(t) \bigr]^{\frac{1}{n}}, \\ &\vdots\\ F_{Au^{n},Ax^{n}_{m}}\bigl(\varphi(t)\bigr)&=F_{T(u^{n},u^{1},\ldots ,u^{n-1}),T(x^{n}_{m-1},x^{1}_{m-1},\ldots,x^{n-1}_{m-1})}\bigl(\varphi (t) \bigr) \\ &\geq\bigl[F_{Au^{n},Ax^{n}_{m-1}}(t), F_{Au^{1},Ax^{1}_{m-1}}(t),\ldots ,F_{Au^{n-1},Ax^{n-1}_{m-1}}(t) \bigr]^{\frac{1}{n}}. \end{aligned}$$

(3.10)

Denote $Q_{m}(t)=[F_{Au^{1},Ax^{1}_{m}}(t),F_{Au^{2},Ax^{2}_{m}}(t),\ldots ,F_{Au^{n},Ax^{n}_{m}}(t)]^{\frac{1}{n}}$. By (3.10), we have $Q_{m}(\varphi(t))\geq Q_{m-1}(t)$, and hence for all $t>0$

$$Q_{m}\bigl(\varphi^{m}(t)\bigr)\geq Q_{m-1}\bigl( \varphi^{m-1}(t)\bigr)\geq\cdots\geq Q_{0}(t). $$

Thus, for all $t>0$, we have

$$\begin{aligned}& F_{Au^{1},A^{1}_{m}}\bigl(\varphi^{m}(t)\bigr)\geq Q_{0}(t), \qquad F_{Au^{2},A^{2}_{m}}\bigl(\varphi^{m}(t)\bigr)\geq Q_{0}(t),\qquad \ldots ,\\& F_{Au^{n},A^{n}_{m}}\bigl( \varphi^{m}(t)\bigr)\geq Q_{0}(t). \end{aligned}$$

Since $Q_{0}(t)\in\mathscr{D^{+}}$ and $\lim_{m\rightarrow\infty }(\varphi^{m}(t))=0$ for all $t>0$, by Lemma 2.1, we conclude that

$$\begin{aligned} \lim_{m\rightarrow\infty}Ax^{1}_{m}=Au^{1}, \qquad\lim_{m\rightarrow\infty }Ax^{1}_{m}=Au^{1}, \qquad \ldots,\qquad \lim_{m\rightarrow\infty}Ax^{n}_{m}=Au^{n}. \end{aligned}$$

(3.11)

This shows that $Au^{1}=u^{1}, Au^{2}=u^{2}, \ldots,Au^{n}=u^{n}$. Hence $u^{1}=T(u^{1},u^{2},\ldots,u^{n}),u^{2}=T(u^{2},u^{3},\ldots ,u^{1}), \ldots, u^{n}=T(u^{n},u^{1},\ldots,u^{n-1})$. Finally, we prove that $u^{1}=u^{2}=\cdots=u^{n}$.

$$ \begin{aligned} F_{u^{1},u^{2}}\bigl(\varphi(t) \bigr)&=F_{T(u^{1},u^{2},\ldots ,u^{n-1},u^{n}),T(u^{2},u^{3},\ldots,u^{n},u^{1})}\bigl(\varphi(t)\bigr) \\ &\geq\bigl[F_{Au^{1},Au^{2}}(t),F_{Au^{2},Au^{3}}(t),\ldots ,F_{Au^{n-1},Au^{n}}(t),F_{Au^{n},Au^{1}}(t) \bigr]^{\frac{1}{n}} \\ &=\bigl[F_{u^{1},u^{2}}(t),F_{u^{2},u^{3}}(t),\ldots ,F_{u^{n-1},u^{n}}(t),F_{u^{n},u^{1}}(t) \bigr]^{\frac{1}{n}}, \\ F_{u^{2},u^{3}}\bigl(\varphi(t)\bigr)&=F_{T(u^{2},u^{3},\ldots ,u^{n},u^{1}),T(u^{3},u^{4},\ldots,u^{1},u^{2})}\bigl(\varphi(t)\bigr) \\ &\geq\bigl[F_{Au^{2},Au^{3}}(t),F_{Au^{3},Au^{3}}(4),\ldots ,F_{Au^{n},Au^{1}}(t),F_{Au^{1},Au^{2}}(t) \bigr]^{\frac{1}{n}} \\ &=\bigl[F_{u^{1},u^{2}}(t),F_{u^{2},u^{3}}(t),\ldots ,F_{u^{n-1},u^{n}}(t),F_{u^{n},u^{1}}(t) \bigr]^{\frac{1}{n}}, \\ &\vdots \\ F_{u^{n},u^{1}}\bigl(\varphi(t)\bigr)&=F_{T(u^{n},u^{1},\ldots ,u^{n-2},u^{n-1}),T(u^{1},u^{2},\ldots,u^{n-1},u^{n})}\bigl(\varphi (t) \bigr) \\ &\geq\bigl[F_{Au^{n},Au^{1}}(t),F_{Au^{1},Au^{2}}(t),\ldots ,F_{Au^{n-2},Au^{n-1}}(t),F_{Au^{n-1},Au^{n}}(t) \bigr]^{\frac{1}{n}} \\ &=\bigl[F_{u^{1},u^{2}}(t),F_{u^{2},u^{3}}(t),\ldots ,F_{u^{n-1},u^{n}}(t),F_{u^{n},u^{1}}(t) \bigr]^{\frac{1}{n}}. \end{aligned} $$

(3.12)

Denote $R(t)=[F_{u^{1},u^{2}}(t),F_{u^{2},u^{3}}(t),\ldots ,F_{u^{n-1},u^{n}}(t),F_{u^{n},u^{1}}(t)]^{\frac{1}{n}}$. From (3.12), we have

$$R\bigl(\varphi^{m}(t)\bigr)\geq R\bigl(\varphi^{m-1}(t)\bigr) \geq\cdots\geq R(t). $$

Since $R(t)\in\mathscr{D^{+}}$, by Lemma 2.1, we get $u^{1}=u^{2}=\cdots=u^{n}$. Hence, there exists $u\in X$, such that $u=Au=T(u,\ldots,u)$.

Finally, we show the uniqueness of the multidimensional common fixed point of T and A. Suppose that v is another the multidimensional common fixed point of T and A, i.e., $v=Av=T(v,\ldots,v)$. By (3.1), for all $t>0$, we have

$$\begin{aligned} F_{u,v}\bigl(\varphi(t)\bigr) =&F_{T(u,u,\ldots,u),T(v,v,\ldots,v)}\bigl( \varphi (t)\bigr) \\ \geq&\bigl[F_{Au,Av}(t)F_{Au,Av}(t)\cdots F_{Au,Av}(t) \bigr]^{\frac {1}{n}} \\ =&F_{Au,Av}(t)=F_{u,v}(t), \end{aligned}$$

(3.13)

which implies that $F_{u,v}(\varphi^{m}(t))\geq F_{u,v}(t)$ for all $t>0$. Using Lemma 2.1, we have $F_{u,v}(t)=1$ for all $t>0$, i.e., $u=v$. This completes the proof. □

Remark 3.1

If $n=2$, Theorem 3.1 generalizes Theorem 2.2 in [24]. While $n=3$, Theorem 3.1 generalizes Theorem 3.1 in [25].

From Theorem 3.1, we can obtain the following corollaries.

Corollary 3.1

Let $(X, \mathscr{F}, \Delta)$ be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, φ: $\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ be a gauge function such that $\varphi^{-1}(\{0\})=\{0\}$, $\varphi (t)< t$, and $\lim_{m\rightarrow\infty}\varphi^{m}(t)=0$ for any $t>0$. Let T: $X^{n}\rightarrow X$ and A: $X\rightarrow X$ be two mappings satisfying the following conditions:

$$\begin{aligned} F_{T(x_{1},x_{2},\ldots,x_{n}),T(y_{1},y_{2},\ldots,y_{n})}\bigl(\varphi (t)\bigr)\geq\bigl[F_{Ax_{1},Ay_{1}}(t)F_{Ax_{2},Ay_{2}}(t) \cdots F_{Ax_{n},Ay_{n}}(t)\bigr]^{\frac{1}{n}} \end{aligned}$$

(3.14)

for all $x_{1},x_{2},\ldots,x_{n}, y_{1},y_{2},\ldots,y_{n} \in X$, and $t>0$, where $T(X^{n})\subset A(X)$, A is continuous and commutative with T. Then T and A have a unique multidimensional common fixed point in X.

If $\varphi: \mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ be a gauge function such that $\lim_{m\rightarrow\infty}\sum_{m=1}^{\infty}\varphi^{m}(t)<\infty$ for any $t>0$, we can obtain $\lim_{m\rightarrow\infty}\varphi^{m}(t)=0$. Hence we have Corollary 3.2 as follows.

Corollary 3.2

Let $(X, \mathscr{F}, \Delta)$ be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, and $\Delta\geq\Delta_{P}$, φ: $\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ be a gauge function such that $\varphi^{-1}(\{0\})=\{0\}$, $\varphi (t)< t$, and $\lim_{m\rightarrow\infty}\sum_{m=1}^{\infty}\varphi ^{m}(t)<\infty$ for any $t>0$. Let T: $X^{n}\rightarrow X$ and A: $X\rightarrow X$ be two mappings satisfying the following conditions:

$$\begin{aligned} F_{T(x_{1},x_{2},\ldots,x_{n}),T(y_{1},y_{2},\ldots,y_{n})}\bigl(\varphi (t)\bigr)\geq\bigl[\Delta \bigl(F_{Ax_{1},Ay_{1}}(t),F_{Ax_{2},Ay_{2}}(t),\ldots, F_{Ax_{n},Ay_{n}}(t)\bigr) \bigr]^{\frac{1}{n}} \end{aligned}$$

(3.15)

for all $x_{1},x_{2},\ldots,x_{n}, y_{1},y_{2},\ldots,y_{n} \in X$, and $t>0$, where $T(X^{n})\subset A(X)$, A is continuous and commutative with T. Then T and A have a unique multidimensional common fixed point in X.

Let $A=I$ (I is the identity mapping) in Corollary 3.2, we can obtain the following corollary.

Corollary 3.3

Let $(X, \mathscr{F}, \Delta)$ be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, and $\Delta\geq\Delta_{P}$, φ: $\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ be a gauge function such that $\varphi^{-1}(\{0\})=\{0\}$, $\varphi (t)< t$, and $\lim_{m\rightarrow\infty}\sum_{m=1}^{\infty}\varphi ^{m}(t)<\infty$ for any $t>0$. Let T: $X^{n}\rightarrow X$ be a mapping satisfying the following conditions:

$$\begin{aligned} F_{T(x_{1},x_{2},\ldots,x_{n}),T(y_{1},y_{2},\ldots,y_{n})}\bigl(\varphi (t)\bigr)\geq\bigl[\Delta \bigl(F_{x_{1},y_{1}}(t),F_{x_{2},y_{2}}(t),\ldots, F_{x_{n},y_{n}}(t)\bigr) \bigr]^{\frac{1}{n}} \end{aligned}$$

(3.16)

for all $x_{1},x_{2},\ldots,x_{n}, y_{1},y_{2},\ldots,y_{n} \in X$, and $t>0$. Then T has a unique multidimensional fixed point in X.

Letting $\varphi(t)=\alpha t$ ($0<\alpha<1$) in Corollary 3.2, we can obtain the following corollary.

Corollary 3.4

Let $(X, \mathscr{F}, \Delta)$ be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, and $\Delta\geq\Delta_{P}$. Let $T: X^{n}\rightarrow X$ and $A: X\rightarrow X$ be two mappings satisfying the following conditions:

$$\begin{aligned} F_{T(x_{1},x_{2},\ldots,x_{n}),T(y_{1},y_{2},\ldots,y_{n})}(\alpha t)\geq\bigl[\Delta\bigl(F_{Ax_{1},Ay_{1}}(t),F_{Ax_{2},Ay_{2}}(t), \ldots, F_{Ax_{n},Ay_{n}}(t)\bigr)\bigr]^{\frac{1}{n}} \end{aligned}$$

(3.17)

for all $x_{1},x_{2},\ldots,x_{n}, y_{1},y_{2},\ldots,y_{n} \in X$, and $t>0$, where $T(X^{n})\subset A(X)$, A is continuous and commutative with T. Then T and A have a unique multidimensional common fixed point in X.

From the proof of Theorem 3.1, we can similarly prove the following result.

Theorem 3.2

Let $(X, \mathscr{F}, \Delta)$ be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, φ: $\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ be a gauge function such that $\varphi^{-1}(\{0\})=\{0\}$, $\varphi (t)>t$, and $\lim_{m\rightarrow\infty}\varphi^{m}(t)=+\infty$ for any $t>0$. Let $T: X^{n}\rightarrow X$ and $A: X\rightarrow X$ be two mappings satisfying the following conditions:

$$\begin{aligned} F_{T(x_{1},x_{2},\ldots,x_{n}),T(y_{1},y_{2},\ldots,y_{n})}(t)\geq\min \bigl\{ F_{Ax_{1},Ay_{1}}\bigl( \varphi(t)\bigr),F_{Ax_{2},Ay_{2}}\bigl(\varphi(t)\bigr),\ldots, F_{Ax_{n},Ay_{n}} \bigl(\varphi(t)\bigr)\bigr\} \end{aligned}$$

(3.18)

for all $x_{1},x_{2},\ldots,x_{n}, y_{1},y_{2},\ldots,y_{n} \in X$, and $t>0$, where $T(X^{n})\subset A(X)$ and A is continuous and compatible with T. Then T and A have a unique multidimensional common fixed point in X.

Remark 3.2

If $n=2$, Theorem 3.2 generalizes Theorem 2.3 in [24]. While $n=3$, Theorem 3.2 generalizes Theorem 3.2 in [25].

Letting $A=I$ (I is the identity mapping) in Theorem 3.2, we can obtain the following corollary.

Corollary 3.5

Let $(X, \mathscr{F}, \Delta)$ be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, φ: $\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ be a gauge function such that $\varphi^{-1}(\{0\})=\{0\}$, $\varphi (t)>t$, and $\lim_{m\rightarrow\infty}\varphi^{m}(t)=\infty$ for any $t>0$. Let T: $X^{n}\rightarrow X$ be a mapping satisfying the following conditions:

$$\begin{aligned} F_{T(x_{1},x_{2},\ldots,x_{n}),T(y_{1},y_{2},\ldots,y_{n})}(t)\geq\min \bigl\{ F_{x_{1},y_{1}}\bigl( \varphi(t)\bigr),F_{x_{2},y_{2}}\bigl(\varphi(t)\bigr),\ldots, F_{x_{n},y_{n}} \bigl(\varphi(t)\bigr)\bigr\} \end{aligned}$$

(3.19)

for all $x_{1},x_{2},\ldots,x_{n}, y_{1},y_{2},\ldots,y_{n} \in X$, and $t>0$. Then T and A have a unique multidimensional common fixed point in X.

Theorem 3.3

Let $(X, d)$ be a complete metric space, φ: $\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ be a gauge function such that $\varphi^{-1}(\{0\})=\{0\}$, $\varphi (t)>t$, and $\lim_{m\rightarrow\infty}\varphi^{m}(t)=+\infty$ for any $t>0$. Let T: $X^{n}\rightarrow X$ and A: $X\rightarrow X$ be two mappings satisfying the following conditions:

$$\begin{aligned} &\varphi\bigl(d\bigl(T(x_{1},x_{2}, \ldots,x_{n}),T(y_{1},y_{2},\ldots,y_{n}) \bigr)\bigr) \\ &\quad\leq \max\bigl\{ d(Ax_{1},Ay_{1}),d(Ax_{2},Ay_{2}), \ldots, d(Ax_{n},Ay_{n})\bigr\} \end{aligned}$$

(3.20)

for all $x_{1},x_{2},\ldots,x_{n}, y_{1},y_{2},\ldots,y_{n} \in X$, and $t>0$, where $T(X^{n})\subset A(X)$, A is continuous and compatible with T. Then T and A have a unique multidimensional common fixed point in X.

Proof

Take $\Delta=\Delta_{M}$ and $F_{x,y}(t)=H(t-d(x,y))$. Then by Lemma 2.3 and Remark 2.2, $(X, \mathscr{F}, \Delta_{M})$ is a complete multidimensional Menger PM-space (or a Menger PM-space). From Lemma 2.4 and (3.20), we have

$$\begin{aligned} F_{T(x_{1},x_{2},\ldots,x_{n}),T(y_{1},y_{2},\ldots ,y_{n})}(t) =&H(t-d\bigl(T(x_{1},x_{2}, \ldots,x_{n}),T(y_{1},y_{2},\ldots ,y_{n})\bigr) \\ \geq&H\bigl(\varphi(t)-\max\bigl\{ d(Ax_{1},Ay_{1}),d(Ax_{2},Ay_{2}), \ldots, d(Ax_{n},Ay_{n})\bigr\} \bigr) \\ =&\min\bigl\{ H\bigl(\varphi(t)-d(Ax_{1},Ay_{1})\bigr), \ldots,H\bigl(\varphi (t)-d(Ax_{n},Ay_{n})\bigr)\bigr\} \\ =&\min\bigl\{ F_{Ax_{1},Ay_{1}}\bigl(\varphi(t)\bigr),\ldots,F_{Ax_{n},Ay_{n}} \bigl(\varphi (t)\bigr)\bigr\} . \end{aligned}$$

(3.21)

Hence the conclusion follows from Theorem 3.2. □

4 An application

In this section, we will provide an example to exemplify the validity of the main result of this paper.

Example 4.1

Suppose that $X\in[-1,1]\subset R$, $\Delta=\Delta _{M}$. Then $\Delta_{M}$ is a t-norm of H-type and $\Delta_{M}\geq \Delta_{P}$. Define $\mathscr{F}$: $X\times X\rightarrow\mathscr{D}$ by

$$\mathscr{F}_{x,y}(t)=F_{x,y}(t)= \textstyle\begin{cases} e^{-\frac{|x-y|}{t}}, & t>0,x,y\in X,\\ 0, & t\leq0,x,y\in X. \end{cases} $$

We claim that $(X,\mathscr{F},\Delta_{M})$ is a multidimensional Menger PM-space. In fact, it is easy to verify (MPM-1) and (MPM-2). Assume that for any $t_{1},t_{2},\ldots, t_{n}>0$, and $x_{1},x_{2},\ldots, x_{n+1}\in X$,

$$\Delta_{M}\bigl(F_{x_{1},x_{2}}(t_{1}),F_{x_{2},x_{3}}(t_{2}), \ldots ,F_{x_{n},x_{n+1}}(t_{n})\bigr)=\min\bigl\{ e^{-\frac {|x_{1}-x_{2}|}{t_{1}}},e^{-\frac{|x_{2}-x_{3}|}{t_{2}}}, e^{-\frac{|x_{n}-x_{n+1}|}{t_{n}}}\bigr\} =e^{-\frac{|x_{1}-x_{2}|}{t_{1}}}. $$

Then we have $t_{1}|x_{2}-x_{3}|\leq t_{2}|x_{1}-x_{2}|, t_{1}|x_{3}-x_{4}|\leq t_{3}|x_{1}-x_{2}|, \ldots , t_{1}|x_{n}-x_{n+1}|\leq t_{n}|x_{1}-x_{2}|$, and so $\frac{t_{1}+t_{2}+\cdots+t_{n}}{t_{1}}|x_{1}-x_{2}|\geq |x_{1}-x_{2}|+|x_{2}-x_{3}|+\cdots+|x_{n}-x_{n+1}|\geq|x_{1}-x_{n+1}|$. It follows that

$$\begin{aligned} F_{x_{1},x_{n+1}}(t_{1}+t_{2}+\cdots+t_{n})&=e^{-\frac {|x_{1}-x_{n+1}|}{t_{1}+t_{2}+\cdots+t_{n}}} \geq e^{-\frac {|x_{1}-x_{2}|}{t_{1}}}\\ &= \Delta_{M}\bigl(F_{x_{1},x_{2}}(t_{1}),F_{x_{2},x_{3}}(t_{2}), \ldots ,F_{x_{n},x_{n+1}}(t_{n})\bigr). \end{aligned}$$

Hence (MPM-3) holds. It is obvious that $(X,\mathscr{F},\Delta_{M})$ is complete. Suppose that $\varphi(t)=\frac{t}{n}$, then it is easy to verify that $\varphi^{-1}(\{0\})=\{0\}$, $\varphi(t)< t$, and $\lim_{m\rightarrow \infty}\sum_{m=1}^{\infty}\varphi^{m}(t)<\infty$ for any $t>0$. For $x_{1},x_{2},\ldots,x_{n}\in X$, define T: $X^{n}\rightarrow X$ as follows:

$$T(x_{1},x_{2},\ldots,x_{n})=\frac{1}{n^{4}}- \frac {x_{1}^{2}}{n^{4}}-\frac{x_{2}^{2}}{n^{4}}-\cdots-\frac {x_{n-1}^{2}}{n^{4}}- \frac{|x_{n}|}{n^{3}}. $$

Then, for each $t>0$ and $x_{1},x_{2},\ldots,x_{n},y_{1},y_{2},\ldots ,y_{n}\in X$, we have

$$\begin{aligned} &\bigl|\bigl(x_{1}^{2}-y_{1}^{2}\bigr)+ \cdots +\bigl(x_{n-1}^{2}-y_{n-1}^{2} \bigr)+n\bigl(|x_{n}|-|y_{n}|\bigr)\bigr|\\ &\quad\leq |x_{1}-y_{1}|\bigl(|x_{1}|+|y_{1}|\bigr)+ \cdots+|x_{n-1}-y_{n-1}|\bigl(|x_{n-1}|+|y_{n-1}|\bigr) +n\bigl(|x_{n}|-|y_{n}|\bigr)\\ &\quad\leq n^{2}\max\bigl\{ |x_{1}-y_{1}|, \ldots,|x_{n}-y_{n}|\bigr\} , \end{aligned}$$

and so

$$\begin{aligned} F_{T(x_{1},x_{2},\ldots,x_{n-1},x_{n}),T(y_{1},y_{2},\ldots ,y_{n-1},y_{n})}\bigl(\varphi(t)\bigr) =&F_{T(x_{1},x_{2},\ldots ,x_{n-1},x_{n}),T(y_{1},y_{2},\ldots,y_{n-1},y_{n})}\biggl( \frac {t}{n}\biggr) \\ =&e^{-\frac{|(x_{1}^{2}-y_{1}^{2})+\cdots +(x_{n-1}^{2}-y_{n-1}^{2})+n(|x_{n}|-y_{n})|}{n^{3}t}} \\ \geq&\min\bigl\{ e^{-\frac{|x_{1}-y_{1}|}{nt}},e^{-\frac {|x_{2}-y_{2}|}{nt}},\ldots,e^{-\frac{|x_{n}-y_{n}|}{nt}}\bigr\} \\ =&\bigl[\Delta_{M}\bigl(F_{x_{1},y_{1}}(t),F_{x_{2},y_{2}}(t), \ldots ,F_{x_{n},y_{n}}(t)\bigr)\bigr]^{\frac{1}{n}}. \end{aligned}$$

Thus, all conditions of Corollary 3.3 are satisfied. Therefore, T has a unique fixed point in X.