The Hausdorff–Pompeiu Distance in Gn-Menger Fractal Spaces

Abstract

This paper introduces a complete $Gn$-Menger space and defines the Hausdorff–Pompeiu distance in the space. Furthermore, we show a novel fixed-point theorem for $Gn$-Menger-$\theta$-contractions in fractal spaces.

1. Introduction and Preliminaries

We begin with the concept of a $Gn$-Menger space using distributional maps (DMs) and triangular norms. Throughout the entire paper, we let $\mathbb{I}=[0,1]$, ${\mathbb{I}}^{\circ }=(0,1)$, ${\mathbb{R}}^{•}=[-\infty ,+\infty ]$, $\mathbb{J}=[0,+\infty )$ and ${\mathbb{J}}^{\circ }=(0,+\infty )$. Define the set of distributional maps ${\mho }^{+}$ as the set of all functions $ȷ:{\mathbb{R}}^{•}\to \mathbb{I}$, denoting ${ȷ}_{ı}$$=ȷ\left(ı\right)$, which are left continuous and nondecreasing on $\mathbb{R}$ with ${ȷ}_0=0$ and ${ȷ}_{+\infty }=1$. In addition, let ${\partial }^{+}\subseteq {\mho }^{+}$ consist of all (proper) mappings $ȷ\in {\mho }^{+}$ for which ${\ell }^{-}{ȷ}_{+\infty }=1$, where ${\ell }^{-}{ȷ}_{ı}$ means the left limit at the point ı. Please refer to [1,2,3] for more details. Note all proper DMs are the DMs of real random variables (namely, we have $P\left(\right|g|=\infty )=0$ for any random variable g).

In ${\mho }^{+}$, we define “≤” as follows:

$$ȷ\le \hslash ⟺{ȷ}_{\tau }\le {\hslash }_{\tau }$$

for each $\tau$ in $\mathbb{R}$ (partially ordered). For example,

$${\hslash }_{\tau }=\left\{\begin{array}{c}\begin{array}{cc}0,& \mathrm{if}\tau \in \mathbb{R}-{\mathbb{J}}^{\circ },\\ 1-e^{-\tau },& \mathrm{if}\tau \in {\mathbb{J}}^{\circ },\end{array}\hfill \end{array}\right.$$

for $\hslash \in {\partial }^{+}$. Note that the function ${\wp }_{\tau }^u$ defined by

$${\wp }_{\tau }^u=\left\{\begin{array}{c}\begin{array}{cc}0,& \mathrm{if}\tau \le u,\\ 1,& \mathrm{if}\tau >u,\end{array}\hfill \end{array}\right.$$

is an element of ${\mho }^{+}$, and ${\wp }_{\tau }^0$ is the maximal element in this space (for more information, see [1,2,3]).

Definition 1

([1,4]). A continuous triangular norm (CTN) is a continuous binary operation ∗ from ${\mathbb{I}}^2$ to $\mathbb{I}$, such that

(a)

$\mathsf{\vartheta }\ast \mathrm{ł}=\mathrm{ł}\ast \mathsf{\vartheta }$ and $\mathsf{\vartheta }\ast (\mathrm{ł}\ast ß)=(\mathsf{\vartheta }\ast \mathrm{ł})\ast ß$ for all $\mathsf{\vartheta },\mathrm{ł},ı\in \mathbb{I}$;

(b)

$\mathsf{\vartheta }\ast 1=\mathsf{\vartheta }$ for all $\mathsf{\vartheta }\in \mathbb{I}$;

(c)

$\mathsf{\vartheta }\ast \mathrm{ł}\le {\mathsf{\vartheta }}^{\prime }\ast {\mathrm{ł}}^{\prime }$ whenever $\mathsf{\vartheta }\le {\mathsf{\vartheta }}^{\prime }$ and $\mathrm{ł}\le {\mathrm{ł}}^{\prime }$ for all $\mathsf{\vartheta },\mathrm{ł},{\mathsf{\vartheta }}^{\prime },{\mathrm{ł}}^{\prime }\in \mathbb{I}$.

Some examples of t-norms are:

(1)

$\mathsf{\vartheta }{\ast }_P\mathrm{ł}=\mathsf{\vartheta }\mathrm{ł}$ (the product CTN);

(2)

$\mathsf{\vartheta }{\ast }_M\mathrm{ł}=min\{\mathsf{\vartheta },\mathrm{ł}\}$ (the minimum CTN);

(3)

$\mathsf{\vartheta }{\ast }_L\mathrm{ł}=max\{\mathsf{\vartheta }+\mathrm{ł}-1,0\}$ (the Lukasiewicz CTN).

Assume that, for every $\mathsf{\vartheta }\in {\mathbb{I}}^{\circ }$, there exists a $\mathrm{ł}\in {\mathbb{I}}^{\circ }$ (which is independent of ℓ, but depends on $\mathsf{\vartheta }$) such that the following inequality holds

$$\begin{array}{c}\hfill \stackrel{\ell }{\overbrace{(1-\mathrm{ł})\ast \cdots \ast (1-\mathrm{ł})}}>1-\mathsf{\vartheta },\mathrm{for}\mathrm{each}\ell \in \{2,3,\dots \}.\end{array}$$

(1)

In this case, we say the CTN ∗ has the (D) property (CTND for short).

Definition 2.

Let ∗ be a CTN, $U\ne \varnothing$ and ζ be a mapping from $U^n$ to ${\partial }^{+}$. The ordered tuple $(U,\zeta ,\ast )$ is called a $Gn$-Menger space if the following conditions are satisfied:

($\zeta 1$)

${\zeta }_{\tau }^{u_1,\dots ,u_n}={\wp }_{\tau }^0$ for $\tau \in {\mathbb{J}}^{\circ }$, if and only if $u_1=u_2=\cdots =u_n$ and $\tau \in {\mathbb{J}}^{\circ }$;

($\zeta 2$)

${\zeta }_{\tau }^{u_1,\dots ,u_n}$ is invariant under any permutation of $u_1,\dots ,u_n\in U$ and $\tau \in {\mathbb{J}}^{\circ }$;

($\zeta 3$)

${\zeta }_{\tau }^{u_1,u_1\dots ,u_1,u_2}\ge {\zeta }_{\tau }^{u_1,u_2,\dots ,u_n}$ for every $u_1,\dots ,u_n\in U$ and $\tau \in {\mathbb{J}}^{\circ }$;

($\zeta 4$)

${\zeta }_{\tau +\varsigma }^{u_1,u_2,\dots ,u_n}\ge {\zeta }_{\varsigma }^{u_1,u_{n+1},\dots ,u_{n+1}}\ast {\zeta }_{\tau }^{u_{n+1},u_2,\dots ,u_n}$ for every $u_1,\dots ,u_n,u_{n+1}\in U$ and $\tau ,\varsigma \in {\mathbb{J}}^{\circ }$.

Moreover, ζ is called a $Gn$-Menger distance.

For more details about $Gn$-Menger space and distance, see [5,6,7,8,9,10,11,12,13,14,15]. Our results improve and generalize recent results in [16,17,18].

Example 1.

Define $\zeta :{\mathbb{R}}^n\to {\partial }^{+}$ by

$${\zeta }_{\tau }^{u_1,\dots ,u_n}=\left\{\begin{array}{c}\begin{array}{cc}0,& \mathrm{if}\tau \in \mathbb{R}-{\mathbb{J}}^{\circ },\\ exp(-\underset{i\ne j,i,j\in \{1,2,\dots ,n\}}{max}\left\{\right|u_i-u_j\left|\right\}/\tau ),& \mathrm{if}\tau \in {\mathbb{J}}^{\circ }.\end{array}\hfill \end{array}\right.$$

Then, the ordered tuple $(\mathbb{R},\zeta ,{\ast }_P)$ is a $Gn$-Menger space.

Clearly, ($\zeta 1$) and ($\zeta 2$) are straightforward. For ($\zeta 3$), let $\tau \in {\mathbb{J}}^{\circ }$, and since

$$\frac{|u_1-u_2|}{\tau }\le \frac{\underset{i\ne j,i,j\in \{1,2,\dots ,n\}}{max}\left\{\right|u_i-u_j\left|\right\}}{\tau },$$

we get

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{u_1,u_1,\dots ,u_1,u_2}& =& exp\left(-\frac{|u_1-u_2|}{\tau }\right)\hfill \\ & \ge & exp\left(-\frac{\underset{i\ne j,i,j\in \{1,2,\dots ,n\}}{max}\left\{\right|u_i-u_j\left|\right\}}{\tau }\right)\hfill \\ & =& {\zeta }_{\tau }^{u_1,\dots ,u_n}.\hfill \end{array}$$

Regarding ($\zeta 4$), let $\tau ,\varsigma \in {\mathbb{J}}^{\circ }$, and note

$$\begin{array}{ccc}& & {\zeta }_{\varsigma }^{u_1,u_{n+1},\dots ,u_{n+1}}{\ast }_P{\zeta }_{\tau }^{u_{n+1},u_2,\dots ,u_n}\hfill \\ & =& exp\left(-\frac{|u_1-u_{n+1}|}{\varsigma }\right).exp\left(-\frac{\underset{i\ne j,i,j\in \{2,\dots ,n,n+1\}}{max}\left\{\right|u_i-u_j\left|\right\}}{\tau }\right)\hfill \\ & \le & exp\left(-\frac{|u_1-u_{n+1}|}{\varsigma +\tau }\right).exp\left(-\frac{\underset{i\ne j,i,j\in \{2,\dots ,n,n+1\}}{max}\left\{\right|u_i-u_j\left|\right\}}{\varsigma +\tau }\right)\hfill \\ & =& exp\left(-\frac{|u_1-u_{n+1}|+\underset{i\ne j,i,j\in \{2,\dots ,n,n+1\}}{max}\left\{\right|u_i-u_j\left|\right\}}{\varsigma +\tau }\right)\hfill \\ & \le & exp\left(-\frac{\underset{i\ne j,i,j\in \{1,2,\dots ,n,n+1\}}{max}\left\{\right|u_i-u_j\left|\right\}}{\varsigma +\tau }\right)\hfill \\ & \le & exp\left(-\frac{\underset{i\ne j,i,j\in \{1,2,\dots ,n\}}{max}\left\{\right|u_i-u_j\left|\right\}}{\varsigma +\tau }\right)\hfill \\ & =& {\zeta }_{\tau +\varsigma }^{u_1,u_2,\dots ,u_n}.\hfill \end{array}$$

We would like to point out that the above example also holds for CTN ${\ast }_M$. In the following, we show every $Gn$-Menger space induces a Menger metric space in the sense of Schweizer and Sklar.

Example 2.

Let $(U,\zeta ,\ast )$ be a $Gn$-Menger space. Define the distributional function η on $U^2$ as

$${\eta }_{\tau }^{u,v}={\zeta }_{\tau }^{u,v,\dots ,v}\ast {\zeta }_{\tau }^{v,u,\dots ,u},$$

for every $u,v\in U$ and $\tau \in {\mathbb{J}}^{\circ }$. Then, $(U,\eta ,\ast )$ is a Menger metric space. In fact, it is easy to check that η is a Menger metric (for more references, see [1,9,19]).

(I)

Let $\tau \in {\mathbb{J}}^{\circ }$ and

$$\begin{array}{ccc}\hfill {\wp }_{\tau }^0& =& {\eta }_{\tau }^{u,v}\hfill \\ & =& {\zeta }_{\tau }^{u,v,\dots ,v}\ast {\zeta }_{\tau }^{v,u,\dots ,u},\hfill \end{array}$$

so we have

$$\begin{array}{c}\hfill {\wp }_{\tau }^0={\zeta }_{\tau }^{u,v,\dots ,v}\end{array}$$

and

$$\begin{array}{c}\hfill {\wp }_{\tau }^0={\zeta }_{\tau }^{v,u,\dots ,u}.\end{array}$$

Using ($\zeta 1$), we get $u=v$. Obviously, the converse is also true.

(II)

From ($\zeta 2$), we have ${\eta }_{\tau }^{u,v}={\eta }_{\tau }^{v,u}$ for every $u,v\in U$ and $\tau \in {\mathbb{J}}^{\circ }$.

(III)

Let $u,v,w\in U$ and $\tau ,\varsigma \in {\mathbb{J}}^{\circ }$. From ($\zeta 4$), we have

$$\begin{array}{ccc}\hfill {\eta }_{\tau +\varsigma }^{u,v}& =& {\zeta }_{\tau +\varsigma }^{u,v,\dots ,v}\ast {\zeta }_{\tau +\varsigma }^{v,u,\dots ,u}\hfill \\ & \ge & [{\zeta }_{\tau }^{u,w,\dots ,w}\ast {\zeta }_{\varsigma }^{w,v,\dots ,v}]\ast [{\zeta }_{\varsigma }^{v,w,\dots ,w}\ast {\zeta }_{\tau }^{w,u,\dots ,u}]\hfill \\ & =& [{\zeta }_{\tau }^{u,w,\dots ,w}\ast {\zeta }_{\tau }^{w,u,\dots ,u}]\ast [{\zeta }_{\varsigma }^{w,v,\dots ,v}\ast {\zeta }_{\varsigma }^{v,w,\dots ,w}]\hfill \\ & =& {\eta }_{\tau }^{u,w}\ast {\eta }_{\varsigma }^{w,v}.\hfill \end{array}$$

It now follows that $(U,\eta ,\ast )$ is a Menger metric space from (I), (II) and (III).

Definition 3.

Let $(U,\zeta ,\ast )$ be a $Gn$-Menger space. Assume $\rho \in {\mathbb{I}}^{\circ }$, $\tau \in {\mathbb{J}}^{\circ }$ and $u_0\in U$. We define the open ball with center $u_0$ and radius ρ as

$$O_{\rho ,\tau }^{u_0}=\{u\in U:{\zeta }_{\tau }^{u_0,u,\dots ,u}>1-\rho \mathit{and}{\zeta }_{\tau }^{u,u_0,\dots ,u_0}>1-\rho \}.$$

Definition 4.

Let $(U,\zeta ,\ast )$ be a $Gn$-Menger space.

(1) A sequence $\left\{u_k\right\}$ in U is said to be convergent to u in U if, for every $\lambda \in {\mathbb{I}}^{\circ }$, there exists a positive integer N such that ${\zeta }_{\tau }^{u,u_k,\dots ,u_k}>1-\lambda$ for every $\tau \in {\mathbb{J}}^{\circ }$ whenever $k\ge N$.

(2) A sequence $\left\{u_k\right\}$ in U is called a Cauchy sequence if, for every $\lambda \in {\mathbb{I}}^{\circ }$, there exists a positive integer N such that ${\zeta }_{\tau }^{u_{k_1},u_{k_2},\dots ,u_{k_n}}>1-\lambda$ for every $\tau \in {\mathbb{J}}^{\circ }$ whenever $k_1,\dots ,k_n\ge N$.

(3) A $Gn$-Menger space $(U,\zeta ,\ast )$ is said to be complete, if and only if every Cauchy sequence in U is convergent to a point in U.

Lemma 1.

Let $(U,\zeta ,\ast )$ be a $Gn$-Menger space. Then, ζ is continuous on $U^n$.

Proof. 

For a fixed n, we let $(u_1,\dots ,u_n)\in U^n$ and $\tau \in {\mathbb{J}}^{\circ }$. Let $\left\{(u_{1,k},\dots ,u_{n,k})\right\}$ be a sequence in $U^n$ converging to $(u_1,\dots ,u_n)$. Consider a fixed number $\alpha \in {\mathbb{J}}^{\circ }$ such that $\alpha <\frac{\tau }{n+1}$. Using ($\zeta 4$) we derive

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{u_{1,k},\dots ,u_{n,k}}& \ge & {\zeta }_{\alpha }^{u_{1,k},u_1,\dots ,u_1}\ast {\zeta }_{\tau -\alpha }^{u_1,u_{2,k},\dots ,u_{n,k}}\hfill \\ & =& {\zeta }_{\alpha }^{u_{1,k},u_1,\dots ,u_1}\ast {\zeta }_{\frac{\alpha }{2}+\tau -\frac{3}{2}\alpha }^{u_1,u_{2,k},\dots ,u_{n,k}}\hfill \\ & \ge & {\zeta }_{\alpha }^{u_{1,k},u_1,\dots ,u_1}\ast {\zeta }_{\frac{\alpha }{2}}^{u_{2,k},u_2,\dots ,u_2}\ast {\zeta }_{\tau -\frac{3}{2}\alpha }^{u_1,u_2,u_{3,k},\dots ,u_{n,k}}\hfill \\ & =& {\zeta }_{\alpha }^{u_{1,k},u_1,\dots ,u_1}\ast {\zeta }_{\frac{\alpha }{2}}^{u_{2,k},u_2,\dots ,u_2}\ast {\zeta }_{\frac{\alpha }{2}+\tau -\frac{4}{2}\alpha }^{u_1,u_2,u_{3,k},\dots ,u_{n,k}}\hfill \\ & \ge & {\zeta }_{\alpha }^{u_{1,k},u_1,\dots ,u_1}\ast {\zeta }_{\frac{\alpha }{2}}^{u_{2,k},u_2,\dots ,u_2}\ast {\zeta }_{\frac{\alpha }{2}}^{u_{3,k},u_3,\dots ,u_3}\ast {\zeta }_{\tau -\frac{4}{2}\alpha }^{u_1,u_2,u_3,u_{4,k},\dots ,u_{n,k}}\hfill \\ & .& \\ & .& \\ & .& \\ & \ge & {\zeta }_{\alpha }^{u_{1,k},u_1,\dots ,u_1}\ast {\zeta }_{\frac{\alpha }{2}}^{u_{2,k},u_2,\dots ,u_2}\ast {\zeta }_{\frac{\alpha }{2}}^{u_{3,k},u_3,\dots ,u_3}\hfill \\ & & \ast \cdots \ast {\zeta }_{\frac{\alpha }{2}}^{u_{n,k},u_n,\dots ,u_n}\ast {\zeta }_{\tau -\frac{n+1}{2}\alpha }^{u_1,u_2,u_3,u_4,\dots ,u_n},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{u_1,\dots ,u_n}& \ge & {\zeta }_{\alpha }^{u_1,u_{1,k},\dots ,u_{1,k}}\ast {\zeta }_{\tau -\alpha }^{u_{1,k},u_2,\dots ,u_n}\hfill \\ & =& {\zeta }_{\alpha }^{u_1,u_{1,k},\dots ,u_{1,k}}\ast {\zeta }_{\frac{\alpha }{2}+\tau -\frac{3}{2}\alpha }^{u_{1,k},u_2,\dots ,u_n}\hfill \\ & \ge & {\zeta }_{\alpha }^{u_1,u_{1,k},\dots ,u_{1,k}}\ast {\zeta }_{\frac{\alpha }{2}}^{u_2,u_{2,k},\dots ,u_{2,k}}\ast {\zeta }_{\tau -\frac{3}{2}\alpha }^{u_{1,k},u_{2,k},u_3,\dots ,u_n}\hfill \\ & =& {\zeta }_{\alpha }^{u_1,u_{1,k},\dots ,u_{1,k}}\ast {\zeta }_{\frac{\alpha }{2}}^{u_2,u_{2,k},\dots ,u_{2,k}}\ast {\zeta }_{\frac{\alpha }{2}+\tau -\frac{4}{2}\alpha }^{u_{1,k},u_{2,k},u_3,\dots ,u_n}\hfill \\ & \ge & {\zeta }_{\alpha }^{u_1,u_{1,k},\dots ,u_{1,k}}\ast {\zeta }_{\frac{\alpha }{2}}^{u_2,u_{2,k},\dots ,u_{2,k}}\ast {\zeta }_{\frac{\alpha }{2}}^{u_3,u_{3,k},\dots ,u_{3,k}}\ast {\zeta }_{\tau -\frac{4}{2}\alpha }^{u_{1,k},u_{2,k},u_{3,k},u_4,\dots ,u_n}\hfill \\ & .& \\ & .& \\ & .& \\ & \ge & {\zeta }_{\alpha }^{u_1,u_{1,k},\dots ,u_{1,k}}\ast {\zeta }_{\frac{\alpha }{2}}^{u_2,u_{2,k},\dots ,u_{2,k}}\ast {\zeta }_{\frac{\alpha }{2}}^{u_3,u_{3,k},\dots ,u_{3,k}}\hfill \\ & & \ast \cdots \ast {\zeta }_{\frac{\alpha }{2}}^{u_n,u_{n,k},\dots ,u_{n,k}}\ast {\zeta }_{\tau -\frac{n+1}{2}\alpha }^{u_{1,k},u_{2,k},u_{3,k},u_{4,k},\dots ,u_{n,k}}.\hfill \end{array}$$

We can do this for any n. Letting $k\to \infty$ in the above, we imply by the continuity property of a CTN that

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim}{\zeta }_{\tau }^{u_{1,k},\dots ,u_{n,k}}& \ge & {\zeta }_{\tau -\frac{n+1}{2}\alpha }^{u_1,u_2,u_3,u_4,\dots ,u_n},\hfill \end{array}$$

(2)

and

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{u_1,\dots ,u_n}& \ge & \underset{k\to \infty }{lim}{\zeta }_{\tau -\frac{n+1}{2}\alpha }^{u_{1,k},u_{2,k},u_{3,k},u_{4,k},\dots ,u_{n,k}}.\hfill \end{array}$$

(3)

From (2) and (3), we get by letting $\alpha$ tend to zero that

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim}{\zeta }_{\tau }^{u_{1,k},\dots ,u_{n,k}}& =& {\zeta }_{\tau }^{u_1,\dots ,u_n},\hfill \end{array}$$

(4)

for every $\tau >0$, which shows the continuity of $\zeta$. □

2. Fixed-Point Theorem

Lemma 2.

Consider the $Gn$-Menger space $(U,\zeta ,\ast )$ in which ∗ is a CTND. Define ${\Xi }_{\mathsf{\vartheta },\zeta }:U^n⟶\mathbb{J}$ by

$${\Xi }_{\mathsf{\vartheta },\zeta }(u_1,\dots ,u_n)=inf\{\tau \in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }\},$$

for each $\mathsf{\vartheta }\in {\mathbb{I}}^{\circ }$ and $u_1,\dots ,u_n\in U$. Then, we have the following:

(I) 

Let $u_1,\dots ,u_n,w_1,\dots ,w_n\in U$. For every $\mathrm{ł}\in {\mathbb{J}}^{\circ }$, there exists $\mathsf{\vartheta }\in {\mathbb{J}}^{\circ }$ such that

$${\Xi }_{\mathrm{ł},\zeta }(u_1,\dots ,u_n)\le \sum _{j=1}^n{\Xi }_{\mathsf{\vartheta },\zeta }(u_j,w_j,w_j,\dots ,w_j)+{\Xi }_{\mathsf{\vartheta },\zeta }(w_1,\dots ,w_n);$$

(II) 

The sequence $\left\{u_k\right\}$ is convergent with respect to the $Gn$-Menger metric ζ, if and only if ${\Xi }_{\mathsf{\vartheta },\zeta }(u,u_k,\dots ,u_k)\to 0$. Moreover, the sequence $\left\{u_k\right\}$ is a Cauchy sequence with respect to the $Gn$-Menger metric ζ, if and only if it is a Cauchy sequence in ${\Xi }_{\mathsf{\vartheta },\zeta }$;

(III) 

Let $u_{k_1},u_{k_2},\dots ,u_{k_n}\in U$, where $k_1,\dots ,k_n\in N$. For every $\mathrm{ł}\in {\mathbb{J}}^{\circ }$ there exists $\mathsf{\vartheta }\in {\mathbb{J}}^{\circ }$ such that for $n\ge 3$,

$${\Xi }_{\mathrm{ł},\zeta }(u_{k_1},u_{k_2},\dots ,u_{k_n})\le \sum _{j=1}^{n-2}j{\Xi }_{\mathsf{\vartheta },\zeta }(u_{k_j},u_{k_{j+1}},\dots ,u_{k_{j+1}})+{\Xi }_{\mathsf{\vartheta },\zeta }(u_{k_{n-1}},u_{k_n},\dots ,u_{k_n});$$

(IV) 

A sequence $\left\{u_k\right\}$ in the $Gn$-Menger space U is Cauchy, if and only if, for every $ϵ\in {\mathbb{J}}^{\circ }$, there exists a positive integer N such that for every $ϵ>0$,

$$\begin{array}{c}\hfill {\Xi }_{\mathrm{ł},\zeta }(u_{k_1},u_{k_2},\dots ,u_{k_2})\le ϵ,\end{array}$$

(5)

for all $k_1,k_2\ge N$.

Proof. 

(I). For every $\mathrm{ł}\in {\mathbb{I}}^{\circ }$, we can find a $\mathsf{\vartheta }\in {\mathbb{I}}^{\circ }$ such that

$$\stackrel{n+1}{\overbrace{(1-\mathsf{\vartheta })\ast \cdots \ast (1-\mathsf{\vartheta })}}>1-\mathrm{ł},$$

due to the (D) property. Using ($\zeta 4$), we infer

$$\begin{array}{ccc}& & {\zeta }_{{\sum }_{j=1}^n{\Xi }_{\mathsf{\vartheta },\zeta }(u_j,w_j,w_j,\dots ,w_j)+{\Xi }_{\mathsf{\vartheta },\zeta }(w_1,\dots ,w_n)+(n+1)\omega }^{u_1,\dots ,u_n}\hfill \\ & \ge & {\zeta }_{{\Xi }_{\mathsf{\vartheta },\zeta }(u_1,w_1,\dots ,w_1)+\omega }^{u_1,w_1,\dots ,w_1}\ast {\zeta }_{{\Xi }_{\mathsf{\vartheta },\zeta }(u_2,w_2,\dots ,w_2)+\omega }^{u_2,w_2,\dots ,w_2}\cdots \ast {\zeta }_{{\Xi }_{\mathsf{\vartheta },\zeta }(u_n,w_n,\dots ,w_n)+\omega }^{u_n,w_n,\dots ,w_n}\ast {\zeta }_{{\Xi }_{\mathsf{\vartheta },\zeta }(w_1,w_2,\dots ,w_n)+\omega }^{w_1,w_2,\dots ,w_n}\hfill \\ & \ge & \stackrel{n+1}{\overbrace{(1-\mathsf{\vartheta })\ast \cdots \ast (1-\mathsf{\vartheta })}}\hfill \\ & >& 1-\mathrm{ł}.\hfill \end{array}$$

for each $\omega \in {\mathbb{J}}^{\circ }$. Hence,

$${\Xi }_{\mathrm{ł},\zeta }(u_1,\dots ,u_n)\le \sum _{j=1}^n{\Xi }_{\mathsf{\vartheta },\zeta }(u_j,w_j,w_j,\dots ,w_j)+{\Xi }_{\mathsf{\vartheta },\zeta }(w_1,\dots ,w_n)+(n+1)\omega .$$

Letting $\omega$ tend to 0, we get

$${\Xi }_{\mathrm{ł},\zeta }(u_1,\dots ,u_n)\le \sum _{j=1}^n{\Xi }_{\mathsf{\vartheta },\zeta }(u_j,w_j,w_j,\dots ,w_j)+{\Xi }_{\mathsf{\vartheta },\zeta }(w_1,\dots ,w_n).$$

(II). We have ${\zeta }_{\tau }^{u_1,\dots ,u_n}>1-\mathrm{ł}⟺{\Xi }_{\mathsf{\vartheta },\zeta }(u_1,\dots ,u_n)<\mathrm{ł}$ for every $\mathrm{ł}\in {\mathbb{J}}^{\circ }$.

(III). For every $\mathrm{ł}\in {\mathbb{I}}^{\circ }$, we can find a $\mathsf{\vartheta }\in {\mathbb{I}}^{\circ }$ such that for $n\ge 3$,

$$\stackrel{\frac{n(n-1)}{2}}{\overbrace{(1-\mathsf{\vartheta })\ast \cdots \ast (1-\mathsf{\vartheta })}}>1-\mathrm{ł}.$$

Then, we use a similar method in (I) to complete the proof.

(IV). It follows immediately from (II) and (III). □

We let $\Theta$ be the family of all onto and strictly increasing mappings $\theta :{\mathbb{J}}^{\circ }\to {\mathbb{J}}^{\circ }$ such that $\theta \left(\rho \right)<\rho$ for all $\rho \in {\mathbb{J}}^{\circ }$, and let all distributional maps be in ${\partial }_{+}^{+}$. Since $\zeta \in {\partial }^{+}$ and ($\zeta 1$), we get in a $Gn$-Menger space $(U,\zeta ,\ast )$ that

$${\zeta }_{\tau }^{u_1,\dots ,u_n}=\mathcal{C},\mathrm{for}\mathrm{all}\tau \in {\mathbb{J}}^{\circ }\mathrm{implies}\mathcal{C}={\wp }_{\tau }^0.$$

Lemma 3.

Consider the $Gn$-Menger space $(U,\zeta ,\ast )$ in which ∗ is a CTND. Assume that $\theta \in \Theta$. Then, for $\tau \in {\mathbb{J}}^{\circ }$

$$\begin{array}{c}\hfill inf\{{\theta }^k\left(\tau \right)\in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }\}\le {\theta }^k(inf\{\tau \in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }\}),\end{array}$$

for each $u_1,\dots ,u_n\in U$, $\mathsf{\vartheta }\in {\mathbb{I}}^{\circ }$ and $k\in N$.

Proof. 

Let $\tau \in {\mathbb{J}}^{\circ }$ be arbitrary and fixed with ${\zeta }_{\tau }^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }$. Then, ${\theta }^k\left(\tau \right)\in {\mathbb{J}}^{\circ }$, and

$$\begin{array}{c}\hfill {\theta }^k\left(\tau \right)\ge inf\{{\theta }^k\left(\mathrm{ł}\right)\in {\mathbb{J}}^{\circ }:{\zeta }_{\mathrm{ł}}^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }\}.\end{array}$$

This implies that

$$\begin{array}{c}\hfill \tau \ge {\left({\theta }^k\right)}^{-1}(inf\{{\theta }^k\left(\mathrm{ł}\right)\in {\mathbb{J}}^{\circ }:{\zeta }_{\mathrm{ł}}^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }\}),\end{array}$$

as ${\theta }^k$ is onto and strictly increasing. Thus,

$$\begin{array}{c}\hfill inf\{\tau \in {\mathbb{J}}^{\circ }:{\zeta }_{\mathrm{ł}}^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }\}\ge {\left({\theta }^k\right)}^{-1}(inf\{{\theta }^k\left(\mathrm{ł}\right)\in {\mathbb{J}}^{\circ }:{\zeta }_{\mathrm{ł}}^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }\}),\end{array}$$

which shows that

$$\begin{array}{c}\hfill inf\{{\theta }^k\left(\tau \right)\in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }\}\le {\theta }^k(inf\{\tau \in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{u_1,\dots ,u_n}>1-\mathsf{\vartheta }\}).\end{array}$$

□

Lemma 4.

Consider the $Gn$-Menger space $(U,\zeta ,\ast )$ in which ∗ is a CTND. Assume that $\theta \in \Theta$ and $\left\{u_k\right\}\subseteq U$ such that

$${\zeta }_{{\theta }^k\left(\tau \right)}^{u_k,u_{k+1},\dots ,u_{k+1}}\ge {\zeta }_{\tau }^{u_1,u_2,\dots ,u_2},$$

for all $\tau \in {\mathbb{J}}^{\circ }$. Then, $\left\{u_k\right\}$ is a Cauchy sequence.

Proof. 

From Lemma 3 and our assumption, we arrive at

$$\begin{array}{ccc}\hfill {\Xi }_{\mathrm{ł},\zeta }(u_k,u_{k+1},\dots ,u_{k+1})& =& inf\{{\theta }^k\left(\tau \right)\in {\mathbb{J}}^{\circ }:{\zeta }_{{\theta }^k\left(\tau \right)}^{u_k,u_{k+1},\dots ,u_{k+1}}>1-\mathrm{ł}\}\hfill \\ & \le & inf\{{\theta }^k\left(\tau \right)\in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{u_1,u_2,\dots ,u_2}>1-\mathrm{ł}\}\hfill \\ & \le & {\theta }^k(inf\{\tau \in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{u_1,u_2,\dots ,u_2}>1-\mathrm{ł}\})\hfill \\ & =& {\theta }^k\left({\Xi }_{\mathrm{ł},\zeta }(u_1,u_2,\dots ,u_2)\right)\to 0,\hfill \end{array}$$

for every $\mathrm{ł}\in {\mathbb{I}}^{\circ }$. Applying Lemma 2 (II), (III) and (IV), we conclude that $\left\{u_k\right\}$ is a Cauchy sequence. □

We are now ready to present a fixed-point (FP) theorem, with a controller $\theta \in \Theta$, in a complete $Gn$-Menger space $(U,\zeta ,\ast )$ in which ∗ is a CTND. We say a mapping $\Omega :U\to U$ is a $Gn$-Menger-$\theta$-contraction if

$$\begin{array}{c}\hfill {\zeta }_{\rho }^{\Omega \left({\alpha }_1\right),\dots ,\Omega \left({\alpha }_n\right)}\ge {\zeta }_{\theta \left(\rho \right)}^{{\alpha }_1,\dots ,{\alpha }_n},\end{array}$$

(6)

for every $\rho \in {\mathbb{J}}^{\circ }$.

Theorem 1.

Consider the complete $Gn$-Menger space $(U,\zeta ,\ast )$ in which ∗ is a CTND. Let the $Gn$-Menger-θ-contraction Ω satisfy (6) in which $\theta \in \Theta$. Then, Ω has a unique fixed point in U.

Proof. 

From Lemma 4 and inequality (6), we have that, for each $\alpha \in U$, the sequence ${\left\{{\Omega }^n\left(\alpha \right)\right\}}_{n=1}^{+\infty }$ is Cauchy and $\underset{k\to +\infty }{lim}{\Omega }^k\left(\alpha \right)=\delta \in U$ since U is complete. Applying the following inequality

$$\begin{array}{ccc}\hfill {\zeta }_{\rho }^{\Omega \left({\alpha }_1\right),\dots ,\Omega \left({\alpha }_n\right)}& \ge & {\zeta }_{\theta \left(\rho \right)}^{{\alpha }_1,\dots ,{\alpha }_n}\hfill \\ & \ge & {\zeta }_{\rho }^{{\alpha }_1,\dots ,{\alpha }_n}\hfill \end{array}$$

for all ${\alpha }_1,\dots ,{\alpha }_n\in U$ and $\rho \in {\mathbb{J}}^{\circ }$, we conclude the continuity of $\Omega$ and so we get

$$\delta =\underset{n\to +\infty }{lim}{\Omega }^{n+1}\left(\alpha \right)=\underset{n\to +\infty }{lim}\Omega \left({\Omega }^n\left(\alpha \right)\right)=\Omega \left(\underset{n\to +\infty }{lim}{\Omega }^n\left(\alpha \right)\right)=\Omega \left(\delta \right).$$

In addition, inequality (6) also infers the uniqueness. □

3. Application to the $\mathit{Gn}$-Menger-Fractal Space

In [20], Hutchinson considered fractal theory, which was further investigated and generalized by Barnsley [21], Bisht [22], Imdad [23], and Ri [24]. The basic concept of fractal theory is that the iterated function system (IFS) serves as the main generator of fractals. This consists of a finite set of $Gn$-Menger-$\theta$-contractions $\left\{{\Omega }_1,{\Omega }_2,\dots ,{\Omega }_m\right\}$ with $(m\ge 2)$, defined in a complete $Gn$-Menger space $(U,\zeta ,\ast )$, satisfying inequality (6). For such an IFS, there is always a unique nonempty compact subset $\Gamma$ of the complete $Gn$-Menger space $(U,\zeta ,\ast )$, such that $\Gamma ={\displaystyle \bigcup _{i=1}^m}{\Omega }_i(\Gamma )$, wherein $\Gamma$ is a fractal set called the attractor of the respective IFS.

Now, we denote $\mathcal{H}\left(U\right)$ as the set of all nonempty compact subsets of the $Gn$-Menger space $(U,\zeta ,\ast )$.

Let $V_j\ne \varnothing$ ($j=1,\dots ,n-1$) be subsets of the $Gn$-Menger space $(U,\zeta ,\ast )$, $u\in U$ and $\tau \in {\mathbb{J}}^{\circ }$. We define the $Gn$-Menger distance between u and $\{V_1,\dots ,V_{n-1}\}$ as

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{u,V_1,\dots ,V_{n-1}}=\underset{v_j\in V_j,j=1,2,\dots ,n-1}{sup}{\zeta }_{\tau }^{u,v_1,\dots ,v_{n-1}}.\end{array}$$

(7)

Lemma 5.

Consider the $Gn$-Menger space $(U,\zeta ,\ast )$. Then, for every $u\in U$, $V_j\subset \mathcal{H}\left(U\right)$ ($j=1,\dots ,n-1$) and $\tau \in {\mathbb{J}}^{\circ }$, we can find $v_{j,0}\in V_j$ such that

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{u,V_1,\dots ,V_{n-1}}={\zeta }_{\tau }^{u,v_{1,0},\dots ,v_{n-1,0}}.\end{array}$$

(8)

Proof. 

Suppose that $u\in U$, $V_j\subset \mathcal{H}\left(U\right)$ ($j=1,\dots ,n-1$) and $\tau \in {\mathbb{J}}^{\circ }$. Since $\zeta$ is continuous from Lemma 1, the compactness of $V_j$ ($j=1,\dots ,n-1$) implies that we can find $v_{j,0}\in V_j$ such that

$$\begin{array}{c}\hfill \underset{v_j\in V_j,j=1,2,\dots ,n-1}{sup}{\zeta }_{\tau }^{u,v_1,\dots ,v_{n-1}}={\zeta }_{\tau }^{u,v_{1,0},\dots ,v_{n-1,0}},\end{array}$$

(9)

so

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{u,V_1,\dots ,V_{n-1}}={\zeta }_{\tau }^{u,v_{1,0},\dots ,v_{n-1,0}}.\end{array}$$

□

Lemma 6.

Consider the $Gn$-Menger space $(U,\zeta ,\ast )$. Let $u\in U$, $V_j\subset \mathcal{H}\left(U\right)$ ($j=1,\dots ,n-1$), $\varnothing \ne W\subseteq U$ and $\tau ,\varsigma \in {\mathbb{J}}^{\circ }$. Then,

$$\begin{array}{c}\hfill {\zeta }_{\tau +\varsigma }^{u,V_1,\dots ,V_{n-1}}\ge {\zeta }_{\tau }^{u,W,W,\dots ,W}\ast {\zeta }_{\varsigma }^{w_u,V_1,\dots ,V_{n-1}},\end{array}$$

(10)

where $w_u\in W$ satisfies ${\zeta }_{\tau }^{u,W,V_2,\dots ,V_{n-1}}={\zeta }_{\tau }^{u,w_u,V_2,\dots ,V_{n-1}}$.

Proof. 

From Lemma 5, we can find a $w_u\in W$ such that

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{u,W,\dots ,W}={\zeta }_{\tau }^{u,w_u,\dots ,w_u},\end{array}$$

for every $\tau \in {\mathbb{J}}^{\circ }$. From Lemma 5 again and ($\zeta 4$), we have

$$\begin{array}{ccc}\hfill {\zeta }_{\tau +\varsigma }^{u,V_1,\dots ,V_{n-1}}& =& {\zeta }_{\tau +\varsigma }^{u,v_1,v_2,\dots ,v_{n-1}}\hfill \\ \hfill & \ge & {\zeta }_{\tau }^{u,w_u,\dots ,w_u}\ast {\zeta }_{\varsigma }^{w_u,v_1,\dots ,v_{n-1}}\hfill \\ \hfill & =& {\zeta }_{\tau }^{u,W,\dots ,W}\ast {\zeta }_{\varsigma }^{w_u,v_1,\dots ,v_{n-1}}.\hfill \end{array}$$

(11)

Then, the result follows immediately from taking the supremum over $v_j\in V_j,j=1,2,\dots ,n-1$ and inequality (11). □

We now define the $Gn$-Menger Hausdorff–Pompeiu distance among $E_j$, $j=1,\dots ,n$, in $\mathcal{H}\left(U\right)$ as:

$$\begin{array}{ccc}\hfill & & {\rm Y}{\displaystyle \underset{\rho }{\overset{E_1,\dots ,E_n}{\zeta }}}\hfill \\ \hfill & =& \underset{{\alpha }_1\in E_1}{inf}\underset{{\alpha }_j\in E_j,j=2,3,\dots ,n}{sup}{\zeta }_{\rho }^{{\alpha }_1,\dots ,{\alpha }_n}\hfill \\ & {\ast }_M& \underset{{\alpha }_2\in E_2}{inf}\underset{{\alpha }_j\in E_j,j=1,3,4,\dots ,n}{sup}{\zeta }_{\rho }^{{\alpha }_1,\dots ,{\alpha }_n}\hfill \\ & {\ast }_M& \cdots \hfill \\ \hfill & {\ast }_M& \underset{{\alpha }_n\in E_n}{inf}\underset{{\alpha }_j\in E_j,j=1,2,\dots ,n-1}{sup}{\zeta }_{\rho }^{{\alpha }_1,\dots ,{\alpha }_n},\hfill \end{array}$$

(12)

for every $\rho \in {\mathbb{J}}^{\circ }$, which is equivalent to

$$\begin{array}{ccc}\hfill & & {\rm Y}{\displaystyle \underset{\rho }{\overset{E_1,\dots ,E_n}{\zeta }}}\hfill \\ \hfill & =& \underset{{\alpha }_1\in E_1}{inf}{\zeta }_{\rho }^{{\alpha }_1,E_2,E_3,\dots ,E_n}\hfill \\ & {\ast }_M& \underset{{\alpha }_2\in E_2}{inf}{\zeta }_{\rho }^{{\alpha }_2,E_1,E_3,\dots ,E_n}\hfill \\ & {\ast }_M& \cdots \hfill \\ \hfill & {\ast }_M& \underset{{\alpha }_n\in E_n}{inf}{\zeta }_{\rho }^{E_1,E_2,\dots ,E_{n-1},{\alpha }_n},\hfill \end{array}$$

(13)

for every $\rho \in {\mathbb{J}}^{\circ }$.

Example 3.

Consider Example 1 in which $U=\mathbb{R}$. Let $\ast ={\ast }_M$, $E_1=[e_1,f_1]$, $E_2=[e_2,f_2]$ and $E_3=[e_3,f_3]$. Define the $Gn$-Menger Hausdorff distance as

$$\begin{array}{c}\hfill {\rm Y}{\displaystyle \underset{\rho }{\overset{E_1,E_2,E_3}{\zeta }}}=exp\left(-\frac{\underset{i,j\in \{1,2,3\}}{max}\left\{\right|e_i-e_j|,|f_i-f_j\left|\right\}}{\rho }\right),\end{array}$$

for all $\rho \in {\mathbb{J}}^{\circ }$. Then, $(\mathcal{H}\left(U\right),{\rm Y}{\displaystyle \underset{.}{\overset{.}{\zeta }}},\ast )$ is a $Gn$-Menger space.

Clearly, the classical Hausdorff–Pompeiu distance for compact sets $E_1=[e_1,f_1]$, $E_2=[e_2,f_2]$ and $E_3=[e_3,f_3]$ is

$$\underset{i,j\in \{1,2,3\}}{max}\left\{\right|e_i-e_j|,|f_i-f_j\left|\right\}.$$

Now, using (12), (13), Example 1 and a similar method in ([25] Proposition 3), we have that the $Gn$-Menger Hausdorff distance ${\rm Y}{\displaystyle \underset{\rho }{\overset{E_1,E_2,E_3}{\zeta }}}$ is a $Gn$-Menger distance.

Lemma 7.

Consider the $Gn$-Menger space $(U,\zeta ,\ast )$. Then, $(\mathcal{H}\left(U\right),{\rm Y}{\displaystyle \underset{.}{\overset{.}{\zeta }}},\ast )$ is a $Gn$-Menger space.

Proof. 

Clearly, $(\zeta 1)$, $(\zeta 2)$ and $(\zeta 3)$ are straightforward. It only remains to prove $(\zeta 4)$.

Suppose that $E_j\in \mathcal{H}\left(U\right)$, $j=1,\dots ,n$, $u\in E_1$, and $\varsigma ,\tau \in {\mathbb{J}}^{\circ }$. Let $\varnothing \ne W\subseteq U$. From Lemma 6, we have

$$\begin{array}{c}\hfill {\zeta }_{\tau +\varsigma }^{u,E_2,\dots ,E_n}\ge {\zeta }_{\varsigma }^{u,W,W,\dots ,W}\ast {\zeta }_{\tau }^{w_u,E_2,\dots ,E_n},\end{array}$$

(14)

where $w_u\in W$ satisfies ${\zeta }_{\tau }^{u,W,E_2,\dots ,E_n}={\zeta }_{\tau }^{u,w_u,E_2,\dots ,E_n}$. Let ${\alpha }_j\in E_j,j=1,2,\dots ,n$, and from ($\zeta 4$) we have

$$\begin{array}{ccc}\hfill & & {\rm Y}{\displaystyle \underset{\varsigma +\tau }{\overset{E_1,\dots ,E_n}{\zeta }}}\hfill \\ \hfill & =& \underset{{\alpha }_1\in E_1}{inf}{\zeta }_{\varsigma +\tau }^{{\alpha }_1,E_2,E_3,\dots ,E_n}\hfill \\ & {\ast }_M& \underset{{\alpha }_2\in E_2}{inf}{\zeta }_{\varsigma +\tau }^{{\alpha }_2,E_1,E_3,\dots ,E_n}\hfill \\ & {\ast }_M& \cdots \hfill \\ \hfill & {\ast }_M& \underset{{\alpha }_n\in E_n}{inf}{\zeta }_{\varsigma +\tau }^{E_1,E_2,\dots ,E_{n-1},{\alpha }_n}\hfill \\ \hfill & \ge & \underset{{\alpha }_1\in E_1}{inf}[{\zeta }_{\varsigma }^{{\alpha }_1,W,W,\dots ,W}\ast {\zeta }_{\tau }^{w_{{\alpha }_1},E_2,E_3,\dots ,E_n}]\hfill \\ \hfill & {\ast }_M& \underset{{\alpha }_2\in E_2}{inf}[{\zeta }_{\varsigma }^{{\alpha }_2,W,W,\dots ,W}\ast {\zeta }_{\tau }^{w_{{\alpha }_2},E_1,E_3,\dots ,E_n}]\hfill \\ & {\ast }_M& \cdots \hfill \\ \hfill & {\ast }_M& \underset{{\alpha }_n\in E_n}{inf}[{\zeta }_{\varsigma }^{W,W,\dots ,W,{\alpha }_n}\ast {\zeta }_{\tau }^{E_1,E_2,\dots ,E_{n-1},w_{{\alpha }_n}}]\hfill \\ \hfill & \ge & [\underset{{\alpha }_1\in E_1}{inf}{\zeta }_{\varsigma }^{{\alpha }_1,W,W,\dots ,W}\ast \underset{{\alpha }_2\in E_2}{inf}{\zeta }_{\varsigma }^{{\alpha }_2,W,W,\dots ,W}\ast \cdots \ast \underset{{\alpha }_n\in E_n}{inf}{\zeta }_{\varsigma }^{W,W,\dots ,W,{\alpha }_n}]\hfill \\ \hfill & {\ast }_M& [{\zeta }_{\tau }^{w_{{\alpha }_1},E_2,E_3,\dots ,E_n}\ast {\zeta }_{\tau }^{w_{{\alpha }_2},E_1,E_3,\dots ,E_n}\ast \cdots \ast {\zeta }_{\tau }^{w_{{\alpha }_2},E_1,E_3,\dots ,E_n}],\hfill \end{array}$$

(15)

which gives

$$\begin{array}{ccc}\hfill & & {\rm Y}{\displaystyle \underset{\varsigma +\tau }{\overset{E_1,\dots ,E_n}{\zeta }}}\hfill \\ \hfill & \ge & [{\rm Y}{\displaystyle \underset{\varsigma }{\overset{E_1,W,\dots ,W}{\zeta }}}]\hfill \\ \hfill & {\ast }_M& [{\zeta }_{\tau }^{w_{{\alpha }_1},E_2,E_3,\dots ,E_n}\ast {\zeta }_{\tau }^{w_{{\alpha }_2},E_1,E_3,\dots ,E_n}\ast \cdots \ast {\zeta }_{\tau }^{w_{{\alpha }_2},E_1,E_3,\dots ,E_n}].\hfill \end{array}$$

(16)

Taking the supremum over (16) for all $w\in W$, we arrive at

$$\begin{array}{ccc}\hfill & & {\rm Y}{\displaystyle \underset{\varsigma +\tau }{\overset{E_1,\dots ,E_n}{\zeta }}}\hfill \\ \hfill & \ge & {\rm Y}{\displaystyle \underset{\varsigma }{\overset{E_1,W,\dots ,W}{\zeta }}}{\ast }_M{\rm Y}\begin{array}{c}W,E_2,\dots ,E_n\\ \zeta \\ \tau \end{array}\hfill \\ \hfill & \ge & {\rm Y}{\displaystyle \underset{\varsigma }{\overset{E_1,W,\dots ,W}{\zeta }}}\ast {\rm Y}\begin{array}{c}W,E_2,\dots ,E_n\\ \zeta \\ \tau \end{array}.\hfill \end{array}$$

(17)

□

Lemma 8.

Assume that $(U,\zeta ,\ast )$ is a complete $Gn$-Menger space. Suppose that $\theta \in \Theta$ and Ω is a $Gn$-Menger-θ-contraction. Then,

$${\rm Y}{\displaystyle \underset{\rho }{\overset{{\Gamma }_{\Omega }\left(E_1\right),\dots ,{\Gamma }_{\Omega }\left(E_n\right)}{\zeta }}}\ge {\rm Y}{\displaystyle \underset{\theta \left(\rho \right)}{\overset{E_1,\dots ,E_n}{\zeta }}},$$

for every $E_1,\dots ,E_n\in \mathcal{H}\left(U\right)$ and $\rho \in {\mathbb{J}}^{\circ }$, and ${\Gamma }_{\Omega }:\mathcal{H}\left(U\right)\to \mathcal{H}\left(U\right)$ is also a $Gn$-Menger-θ-contraction, where ${\Gamma }_{\Omega }\left(G\right):=\Omega \left(G\right)$ for every $G\in \mathcal{H}\left(U\right)$.

Proof. 

Consider $E_1,\dots ,E_n$ in $\mathcal{H}\left(U\right)$. Using inequality (6) and definition (12), we get

$$\begin{array}{ccc}\hfill {\rm Y}{\displaystyle \underset{\rho }{\overset{{\Gamma }_{\Omega }\left(E_1\right),\dots ,{\Gamma }_{\Omega }\left(E_n\right)}{\zeta }}}& =& {\rm Y}{\displaystyle \underset{\rho }{\overset{\Omega \left(E_1\right),\dots \Omega \left(E_n\right)}{\zeta }}}\hfill \\ & =& \underset{\Omega \left({\alpha }_1\right)\in \Omega \left(E_1\right)}{inf}\underset{\Omega \left({\alpha }_j\right)\in \Omega \left(E_j\right),j=2,3,\dots ,n}{sup}{\zeta }_{\rho }^{\Omega \left(E_1\right),\dots ,\Omega \left(E_n\right)}\hfill \\ & {\ast }_M& \underset{\Omega \left({\alpha }_2\right)\in \Omega \left(E_2\right)}{inf}\underset{\Omega \left({\alpha }_j\right)\in \Omega \left(E_j\right),j=1,3,4,\dots ,n}{sup}{\zeta }_{\rho }^{\Omega \left(E_1\right),\dots ,\Omega \left(E_n\right)}\hfill \\ & {\ast }_M& \cdots \hfill \\ & {\ast }_M& \underset{\Omega \left({\alpha }_n\right)\in \Omega \left(E_n\right)}{inf}\underset{\Omega \left({\alpha }_j\right)\in \Omega \left(E_j\right),j=1,2,\dots ,n-1}{sup}{\zeta }_{\rho }^{\Omega \left(E_1\right),\dots ,\Omega \left(E_n\right)}\hfill \\ & =& \underset{{\alpha }_1\in E_1}{inf}\underset{{\alpha }_j\in E_j,j=2,3,\dots ,n}{sup}{\zeta }_{\rho }^{\Omega \left(E_1\right),\dots ,\Omega \left(E_n\right)}\hfill \\ & {\ast }_M& \underset{{\alpha }_2\in E_2}{inf}\underset{\Omega \left({\alpha }_j\right)\in \Omega \left(E_j\right),j=1,3,4,\dots ,n}{sup}{\zeta }_{\rho }^{\Omega \left(E_1\right),\dots ,\Omega \left(E_n\right)}\hfill \\ & {\ast }_M& \cdots \hfill \\ & {\ast }_M& \underset{{\alpha }_n\in E_n}{inf}\underset{\Omega \left({\alpha }_j\right)\in \Omega \left(E_j\right),j=1,2,\dots ,n-1}{sup}{\zeta }_{\rho }^{\Omega \left(E_1\right),\dots ,\Omega \left(E_n\right)}\hfill \\ & \ge & \underset{{\alpha }_1\in E_1}{inf}\underset{{\alpha }_j\in E_j,j=2,3,\dots ,n}{sup}{\zeta }_{\theta \left(\rho \right)}^{{\alpha }_1,\dots ,{\alpha }_n}\hfill \\ & {\ast }_M& \underset{{\alpha }_2\in E_2}{inf}\underset{{\alpha }_j\in E_j,j=1,3,4,\dots ,n}{sup}{\zeta }_{\theta \left(\rho \right)}^{{\alpha }_1,\dots ,{\alpha }_n}\hfill \\ & {\ast }_M& \cdots \hfill \\ & {\ast }_M& \underset{{\alpha }_n\in E_n}{inf}\underset{{\alpha }_j\in E_j,j=1,2,\dots ,n-1}{sup}{\zeta }_{\theta \left(\rho \right)}^{{\alpha }_1,\dots ,{\alpha }_n}\hfill \\ & =& {\rm Y}\begin{array}{c}{E}_1,\dots ,E_n\\ \zeta \\ \theta \left(\rho \right)\end{array},\hfill \end{array}$$

for every $\rho \in {\mathbb{J}}^{\circ }$. □

Theorem 2.

Assume that $(U,\zeta ,\ast )$ is a complete $Gn$-Menger space in which ∗ is a CTND. Suppose that $\theta \in \Theta$ and Ω is $Gn$-Menger-θ-contractive. Then, ${\Gamma }_{\Omega }:\mathcal{H}\left(U\right)\to \mathcal{H}\left(U\right)$ has a unique fixed point.

Proof. 

From Lemma 8, ${\Gamma }_{\Omega }$ is $Gn$-Menger-$\theta$-contractive on $\mathcal{H}\left(U\right)$ and so by Theorem 1, ${\Gamma }_{\Omega }$ has a unique fixed point. □

Example 4.

Consider the complete $Gn$-Menger space defined in Example 1. Suppose that $\theta \left(\tau \right)=\frac{\tau }{1+\tau }$, $\Omega \left(u\right)=\frac{u}{3}$ and ${\Gamma }_{\Omega }[-u,u]=[-\frac{u}{3},\frac{u}{3}]$. It is easy to show that Ω is $Gn$-Menger-θ-contractive. Furthermore, ${\Gamma }_{\Omega }$ has a unique fixed point $\left\{0\right\}$.

4. Conclusions

We defined a new version of the probabilistic Hausdorff–Pompeiu distance using the concept of $Gn$-Menger space and we presented a new fixed-point theorem for $Gn$-Menger-$\theta$-contractions in $Gn$-Menger fractal spaces. In the future, we hope to consider our results to get more common fixed-point theorems to investigate the existence and uniqueness of solutions for differential and integral equations.