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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.

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 [25]. 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 [722]. 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.

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. (1)

    \(\Delta(a,1)=a\);

  2. (2)

    \(\Delta(a,b)=\Delta(b,a)\);

  3. (3)

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

  4. (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 \(F_{x,y}\)):

  1. (MS-1)

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

  2. (MS-2)

    \(F_{x,y}(t)=F_{y,x}(t)\) for all \(t\in R\);

  3. (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. (1)

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

  2. (2)

    \(\Delta(a,b,c)=\Delta(a,c,b)=\Delta(c,b,a)\);

  3. (3)

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

  4. (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 \(F_{x,y}\)):

  1. (GPM-1)

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

  2. (GPM-2)

    \(F_{x,y}(t)=F_{y,x}(t)\) for all \(t\in R\);

  3. (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. (1)

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

  2. (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. (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. (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 \(F_{x,y}\)):

  1. (MPM-1)

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

  2. (MPM-2)

    \(F_{x,y}(t)=F_{y,x}(t)\) for all \(t\in R\);

  3. (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:

  1. (i)

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

  2. (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 \(U_{x}(\epsilon,\lambda)\subset U_{x}(\epsilon_{1},\lambda_{1})\cap U_{x}(\epsilon_{2},\lambda_{2})\);

  3. (iii)

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

  4. (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.

  1. (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 \(t>0\);

  2. (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 \(m,k\geq N\);

  3. (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

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

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. □

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.

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Acknowledgements

The authors would like to thank the editor and the referees for their constructive comments and suggestions. The research was supported by the National Natural Science Foundation of China (11361042, 11326099, 11461045, 11071108) and the Provincial Natural Science Foundation of Jiangxi, China (20132BAB201001, 20142BAB211016, 2010GZS0147).

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Zhu, C., Wei, Z., Wu, Z. et al. Multidimensional common fixed point theorems under probabilistic φ-contractive conditions in multidimensional Menger probabilistic metric spaces. Fixed Point Theory Appl 2015, 188 (2015). https://doi.org/10.1186/s13663-015-0434-y

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Keywords

  • multidimensional Menger probabilistic metric space
  • fixed point
  • hybrid probabilistic contractions
  • compatible