Fixed Point Results of Fuzzy Multivalued Graphic Contractive Mappings in Generalized Parametric Metric Spaces

Abstract

The aim of this paper is to introduce to a pair of fuzzy graphic rational F-contraction multivalued mappings and to study the necessary condition for the existence of common fixed points of fuzzy multivalued mappings in the setup of generalized parametric metric space endowed with a directed graph. A non-trivial example is presented to support the results presented herein. Our results improve and extend some recent results in the existing literature.

1. Introduction

The crisp set theory does not provide a suitable framework to deal with the problems of uncertainties and imprecision in the given data. Zadeh [1] initiated fuzzy set theory to deal with such problems, and this theory has been given significant attention because of its applications in different disciplines, including management, engineering, environmental and medical sciences.

The fixed point theory for multivalued operators in metric structures has attracted the attention of several mathematicians. In 1973, Markin [2] initiated the theory of fixed point of multivalued mappings to satisfy contractive and nonexpansive conditions by employing the Hausdorff metric structure.

In 1981, Heilpern [3] considered fuzzy multivalued mapping to develop the fuzzy analogue to Nadler’s fixed point theorem [4]. Afterwords, several researchers obtained results regarding the fixed points of fuzzy multivalued mappings (see [5,6,7,8]).

In 2003, Rus [9] presented the idea for R-multivalued maps by employing the idea of K-multivalued mappings in [10]. In 2009, Abbas and Rhoades [11] reconsidered R-multivalued mappings and initiated the notion of generalized R-multivalued mappings to produce common fixed points of these maps.

In 2004, Ran and Reurings [12] obtained results for fixed points in the partially ordered structure of a metric space, which was extended and modified by Nieto and Lopez [13]. In 2007, Jachymski [14] studied fixed point results by using a graphic structure on metric spaces instead of an order structure.

In 2012, Wardowski [15] extended the Banach contraction results by employing F-contraction mappings. In 2013, Sgroi and Vetro [16] proved fixed points theorems based on multivalued maps that satisfied the F-contractive conditions in metric spaces (see also [17,18]).

Recently, Anton-Sancho [19] exploited the algebraic and geometric structure of underlying space to obtain an interesting fixed point result in the automorphism of the vector bundle in Moduli space over compact Riemann surfaces (see also [20]). For results on metric fixed point theory with applications, we refer to [21,22,23] and the references mentioned therein.

Hussain et al. [24] extended the concept of ordinary metrics by introducing the parametric family of distance functions. Das and Bag [25] then introduced generalized parametric metric spaces. Enriching these concepts, we have obtained common fuzzy fixed point results of fuzzy multivalued graphic F-contractive maps in generalized parametric metric spaces equipped with directed graphs. It is important to note that these results have been proven without using the Hausdorff metric. Moreover, these results unify and extend different comparable results presented in [9,10,14,26].

2. Preliminaries

In this section, the set of natural numbers, the set of positive real numbers, the set of non-negative real numbers, and the set of real numbers will be denoted by the symbols $\mathbb{N}$, ${\mathbb{R}}^{+},$ ${\mathbb{R}}_{+},$ $\mathbb{R}$, respectively.

We now recall some definitions and results of parametric metric space.

Definition 1

([24]). Let Υ be a nonempty set. A mapping $f:\mathsf{{\rm Y}}\times \mathsf{{\rm Y}}\times (0,\infty )\to [0,\infty )$ is called a parametric metric if it satisfies the following parameters

1. 

$f({\varkappa }_1,{\varkappa }_2,\theta )=0,$ for all $\theta >0$ if and only if ${\varkappa }_1$ $={\varkappa }_2;$

2. 

$f({\varkappa }_1,{\varkappa }_2,\theta )=f({\varkappa }_2,{\varkappa }_1,\theta )$ for all ${\varkappa }_1,{\varkappa }_2\in$ $\mathsf{{\rm Y}}$ and $\theta >0;$

3. 

$f({\varkappa }_1,{\varkappa }_3,\theta )\le f({\varkappa }_1,{\varkappa }_2,\theta )+f({\varkappa }_2,{\varkappa }_3,\theta )$ for all ${\varkappa }_1,{\varkappa }_2,{\varkappa }_3\in \mathsf{{\rm Y}}$ and $\theta >0.$

The pair $(\mathsf{{\rm Y}},$ $f)$ is called a parametric metric space.

Example 1.

Let $\mathsf{{\rm Y}}=\{h:(0,\infty )\to \mathbb{R}\}.$ Define $f:\mathsf{{\rm Y}}\times \mathsf{{\rm Y}}\times (0,\infty )\to [0,\infty )$ by $f(h,g,\theta )=\left|h\right(\theta )-g(\theta \left)\right|$ for all $h,g\in \mathsf{{\rm Y}}$ and $\theta >0.$ Then f is a parametric metric on $\mathsf{{\rm Y}}$ and the pair $(f,\mathsf{{\rm Y}})$ is a parametric metric space.

Definition 2

([24]). Let $\left\{{\varkappa }_n\right\}$ be a sequence in a parametric metric space $(\mathsf{{\rm Y}},f).$

(1) 

If $\underset{n\to \infty }{lim}f({\varkappa }_n,\varkappa ,\theta )=0,$ for every $\theta >0,$ then $\left\{{\varkappa }_n\right\}$ is parametrically convergent to $\varkappa \in X$ and is written as $\underset{n\to \infty }{lim}{\varkappa }_n=\varkappa .$

(2) 

If for all $\theta >0,$ such that $\underset{n,m\to \infty }{lim}f({\varkappa }_n,{\varkappa }_m,\theta )=0,$ then $\left\{{\varkappa }_n\right\}$ is a parametric Cauchy sequence in $\mathsf{{\rm Y}}$.

(3) 

A parametric space $(\mathsf{{\rm Y}},f)$ is thought to be complete if every parametric Cauchy sequence converges in it.

Definition 3.

Suppose $(\mathsf{{\rm Y}},$ $f)$ is a parametric metric space. A mapping $T:\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ is said to be a continuous mapping at a in $\mathsf{{\rm Y}}$ if, for any sequence, $\left\{a_n\right\}$ in $\mathsf{{\rm Y}}$ such that $\underset{n\to \infty }{lim}{a}_n=a,$ we have $\underset{n\to \infty }{lim}T{a}_n=Ta.$

We now consider some definitions and results for the main results.

Definition 4

([25]). Let $\circ :[0,\infty )\times [0,\infty )\to [0,\infty )$ be a binary operation such that for all $a,b,c\in [0,\infty )$, the following hold:

(i) 

$a\circ 0=a$,

(ii) 

$a\le b,$ then $a\circ c$ $\le b\circ c$ (monotonicity),

(iii) 

$a\circ c=c\circ a$ (commutativity),

(iv) 

$a\circ (b\circ c)=(a\circ b)\circ c$ (associativity),

The examples of this binary operation are as follows.

(E₁) 

$a\circ b=max\{a,b\}$.

(E₂) 

$a\circ b=a+b$.

(E₃) 

$a\circ b={(a^n+b^n)}^{{\displaystyle \frac{1}{n}}}$, for all $n\in \mathbb{N}$.

Definition 5

([25]). A binary operation ∘ is said to be continuous if sequences $\left\{a_n\right\}$ and $\left\{b_n\right\}$ in $[0,\infty )$ converge to some a and b in $[0,\infty ),$ respectively, then $\left\{a_n\circ b_n\right\}\to a\circ b$ as $n\to \infty .$

Definition 6

([25]). Let $\mathsf{{\rm Y}}$ be a nonempty set. A mapping $d_p:\mathsf{{\rm Y}}\times \mathsf{{\rm Y}}\times (0,\infty )\to [0,\infty )$ is called a generalized parametric metric $\mathcal{G}$-parametric metric if it satisfies the following:

1. 

$d_p({\varkappa }_1,{\varkappa }_2,\theta )=0,$ for all $\theta >0$ if, and only if, ${\varkappa }_1$ $={\varkappa }_2;$

2. 

$d_p({\varkappa }_1,{\varkappa }_2,\theta )=d_p({\varkappa }_2,{\varkappa }_1,\theta )$ for all ${\varkappa }_1,{\varkappa }_2\in$ $\mathsf{{\rm Y}}$ and $\theta >0;$

3. 

$d_p({\varkappa }_1,{\varkappa }_3,{\theta }_1+{\theta }_2)\le d_p({\varkappa }_1,{\varkappa }_2,{\theta }_1)\circ d_p({\varkappa }_2,{\varkappa }_3,{\theta }_2)$ for all ${\varkappa }_1,{\varkappa }_2,{\varkappa }_3\in \mathsf{{\rm Y}}$ and for ${\theta }_1,{\theta }_2>0.$

The triple $(\mathsf{{\rm Y}},$ $d_p,\circ )$ is called a $\mathcal{G}$-parametric metric space.

The parametric metric space $(\mathsf{{\rm Y}},$ $f)$ and generalized parametric metric $(\mathsf{{\rm Y}},$ $d_p,\circ )$ are independent of each other. Das and Bag [25] presented some examples to show that both metric spaces are totally different.

Example 2

([25]). Let $d_p:\mathbb{R}\times \mathbb{R}\times (0,\infty )\to [0,\infty )$ be defined as

$$d_p(t,s,\theta )={\displaystyle \frac{{|t-s|}^p}{\theta }}\mathit{for}t,s\in \mathbb{R},\mathit{for}\mathit{all}\theta >0\mathit{and}0<p<1.$$

Then, if we take binary operation ∘ as $a\circ b=a+b$ for all $a,b\in {\mathbb{R}}_{+}$, then $d_p$ is a $\mathcal{G}$-parametric metric on $\mathsf{{\rm Y}}$ and the triple $(\mathsf{{\rm Y}},d_p,\circ )$ is a $\mathcal{G}$-parametric metric space.

Definition 7

([25]). Let $\left\{{\varkappa }_n\right\}$ be a sequence in a $\mathcal{G}$-parametric metric space $(\mathsf{{\rm Y}},d_p,\circ ).$

(1) 

If $\underset{n\to \infty }{lim}{d}_p({\varkappa }_n,\varkappa ,\theta )=0,$ for all $\theta >0,$ then $\left\{{\varkappa }_n\right\}$ is called generalized parametrically convergent to $\varkappa \in X$ and is written as $\underset{n\to \infty }{lim}{\varkappa }_n=\varkappa .$

(2) 

If for all $\theta >0,$ we have $\underset{n,m\to \infty }{lim}{d}_p({\varkappa }_n,{\varkappa }_m,\theta )=0,$ then $\left\{{\varkappa }_n\right\}$ is called $\mathcal{G}$-parametric Cauchy sequence in $\mathsf{{\rm Y}}$.

(3) 

A $\mathcal{G}$-parametric space $(\mathsf{{\rm Y}},d_p,\circ )$ is called complete if every $\mathcal{G}$-parametric Cauchy sequence converges in it.

Proposition 1

([25]). If $d_p$ is a $\mathcal{G}$-parametric metric on $X,$ then for every $t,s\in$ X, $d_p(t,s,\circ )$ is a non-increasing function.

Following Jachymski [14], consider a directed graph G with vertices $V\left(G\right)$ coinciding with $\mathsf{{\rm Y}}.$ $E\left(G\right)$ represents the set which features all edges and loops. Also, $E\left(G\right)\supseteq \Delta ,$ where Δ represents a diagonal of $\mathsf{{\rm Y}}\times \mathsf{{\rm Y}}$. Furthermore $E^{*}\left(G\right)$ represent a set which features only the edges of the graph G. Also it is supposed that there is only single edge between every two vertices in the graph G and, the pair $\left(V\right(G),E(G\left)\right)$ identifies the graph $G.$

Definition 8

([14]). In a metric space $(\mathsf{{\rm Y}},d),$ an operator $\eta :\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ is known as a Banach G-contraction if

1. 

for every $t,s\in \mathsf{{\rm Y}}$ with $(t,s)\in E\left(G\right),$ we have $\left(\eta \right(t),\eta (s\left)\right)\in E\left(G\right)$, that is, η preserves edges.

2. 

There exists $\gamma \in (0,$ $1)$ such that for all $t,s\in \mathsf{{\rm Y}}$ with $(t,s)\in E\left(G\right)$, we have $d\left(\eta \right(t),\eta (s\left)\right)\le \gamma d(t,s)$.

A (directed) path in graph G of length $l\in \mathbb{N}$ between the vertices ${\varkappa }_a$ and ${\varkappa }_b$ is a collection of elements $\{{\varkappa }_1,{\varkappa }_2,\dots ,{\varkappa }_l\},$ of vertices where ${\varkappa }_a={\varkappa }_1$, ${\varkappa }_b={\varkappa }_l$ and $({\varkappa }_{j-1},{\varkappa }_j)\in E\left(G\right)$ for $j\in \{1,2,\dots ,l\}$.

In a graph G, if there is a path in between every two vertices, then we call it a connected graph. Additionally, it is known as weakly connected if $\tilde{G}$ is connected, where $\tilde{G}$ is an undirected graph obtained by G ignoring the directions of all edges in $E\left(G\right)$. If we reverse the directions of edges in a graph G, then we denote this by $G^{-1}$. Thus,

$$E\left(G^{-1}\right)=\left\{\left({\varkappa }_1,{\varkappa }_2\right)\in \mathsf{{\rm Y}}\times \mathsf{{\rm Y}}:\left({\varkappa }_2,{\varkappa }_1\right)\in E\left(G\right)\right\}.$$

It is important to note that $\tilde{G}$ is such a directed graph that the set of its edges is symmetric; hence we can write

$$E\left(\tilde{G}\right)=E\left(G\right)\cup E\left(G^{-1}\right).$$

If the set of edges of graph G is symmetric, then for $\varkappa \in V\left(G\right)$, ${\left[\varkappa \right]}_G$ represents the class of equivalence of the relation R defined on $V\left(G\right)$ by the rule:

$${\varkappa }_{2}R{\varkappa }_3\mathrm{if}\mathrm{there}\mathrm{is}\mathrm{a}\mathrm{directed}\mathrm{path}\mathrm{in}G\mathrm{from}{\varkappa }_2\mathrm{to}{\varkappa }_3.$$

If $\varphi :\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ is an operator, set

$${\mathsf{{\rm Y}}}_{\varphi }:=\{\varkappa \in \mathsf{{\rm Y}}:(\varkappa ,\varphi \left(\varkappa \right))\in E\left(G\right)\}.$$

Jachymski [14] applied the following property:

A graph G has a property (P) if, for every sequence, $\left\{{\varkappa }_n\right\}$ in $\mathsf{{\rm Y}}$, with ${\varkappa }_n\to \varkappa$ as $n\to \infty$ and $({\varkappa }_n,{\varkappa }_{n+1})$ $\in E\left(G\right),$ then $({\varkappa }_n,\varkappa )\in E\left(G\right).$

Theorem 1

([14]). Let $(\mathsf{{\rm Y}},d)$ be a complete metric space and G a directed graph such that $V\left(G\right)=\mathsf{{\rm Y}}$. If G has property (P) and $\eta :\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ is a Banach G-contraction. Then, the following conclusions hold:

(i) 

${\mathsf{{\rm Y}}}_{\eta }\ne \varnothing$ if, and only if, η has a fixed point;

(ii) 

if ${\mathsf{{\rm Y}}}_{\eta }\ne \varnothing$ and G is weakly connected, then η is a Picard operator, that is, η has a unique fixed point, say $u^{*},$ and the sequence $\left\{{\eta }^n\left(u\right)\right\}\to u^{*}$ as $n\to \infty ,$ for all $u\in \mathsf{{\rm Y}}$;

(iii) 

for any $\varkappa \in {\mathsf{{\rm Y}}}_{\eta }$, $\eta {\mid }_{{\left[\varkappa \right]}_{\tilde{G}}}$ is a Picard operator;

(iv) 

if ${\mathsf{{\rm Y}}}_{\eta }\times {\mathsf{{\rm Y}}}_{\eta }\subseteq E\left(G\right)$, then η is a weakly Picard operator, that is, η has a unique fixed point, say $u^{*}$ and, for each $u\in \mathsf{{\rm Y}},$ we have a sequence $\left\{{\eta }^n\left(u\right)\right\}\to u^{*}$ as $n\to \infty$.

For further details about Picard operators and weak Picard operators, refer to Berinde [27].

Ref. [15] Let $\Gamma$ denotes the collection of all maps $F:{\mathbb{R}}^{+}\to \mathbb{R}$ that fulfill the following:

($F_1$)

For all ${\varkappa }_1,{\varkappa }_2\in {\mathbb{R}}^{+}$ having ${\varkappa }_1<\varkappa$ implies $F\left({\varkappa }_1\right)<F\left({\varkappa }_2\right)$.,

($F_2$)

For any sequence $\left\{{\varkappa }_n\right\}$ of positive real numbers, $\underset{n\to \infty }{lim}{\varkappa }_n=0$ and $\underset{n\to \infty }{lim}F\left({\varkappa }_n\right)=-\infty$ are equivalent.

($F_3$)

There is $\alpha \in \left(0,1\right)$ that satisfies $\underset{\varkappa \to 0^{+}}{lim}{\varkappa }^{\alpha }F\left(\varkappa \right)=0$.

Recall that an ordinary subset B of $\mathsf{{\rm Y}}$ is determined by its characteristic function ${\chi }_B,$ where ${\chi }_B:B⟶\{0,1\}$ is defined as

$${\chi }_B\left(\beta \right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\beta \in B,\hfill \\ 0,\hfill & \mathrm{if}\beta \notin B.\hfill \end{array}\right.$$

The value of ${\chi }_B\left(\beta \right)$ determines whether an element belongs to B or not. This suggests that fuzzy sets can be defined by allowing the value of the mapping in set $[0,1]$ so that a fuzzy set in $\mathsf{{\rm Y}}$ features a map of domain $\mathsf{{\rm Y}}$ with values of $[0,1]$. A set of all fuzzy sets in $\mathsf{{\rm Y}}$ is denoted by $I^{\mathsf{{\rm Y}}}$. If B is a fuzzy set in $\mathsf{{\rm Y}}$, then a value $B\left(\varkappa \right)$ is known as a grade of membership of $\varkappa$ in B. The $\alpha$-level set of a fuzzy set B is denoted by ${\left[B\right]}_{\alpha }$ and defined as

$${\left[B\right]}_{\alpha }=\left\{\begin{array}{cc}\overline{\{\varkappa \in \mathsf{{\rm Y}}:B(\varkappa )>0\}},\hfill & \mathrm{if}\alpha =0,\hfill \\ \{\varkappa \in \mathsf{{\rm Y}}:B(\varkappa )\ge \alpha \},\hfill & \mathrm{if}\alpha \in (0,1],\hfill \end{array}\right.$$

where $\overline{X}$ represents the closure of set $X.$

Example 3.

Consider $\mathsf{{\rm Y}}$ as a collection of all individuals in a region and

$$\Gamma =\left\{\varkappa \in \mathsf{{\rm Y}}|\varkappa \mathrm{is}\mathrm{an}\mathrm{old}\mathrm{person}\right\}.$$

Then, it is more appropriate to identify an individual as an elderly person using the membership function Γ on set $\mathsf{{\rm Y}}$, as the term ’old’ is not well defined.

Example 4.

Let $\mathsf{{\rm Y}}=\{1,$ $2,$ $3,$ $4\}$ be equipped with the usual metric. Let $T:\mathsf{{\rm Y}}⟶I^{\mathsf{{\rm Y}}}$ be a fuzzy multivalued map; that is, for each $\varkappa \in \mathsf{{\rm Y}}$, $T\left(\varkappa \right):\mathsf{{\rm Y}}⟶[0,1]$ is a fuzzy set. As for $\lambda \in (0,$ $1]$, we can define a fuzzy set $T\left(1\right)$ by

$$T\left(1\right)\left(\kappa \right)=\left\{\begin{array}{cc}\lambda ,\hfill & \mathrm{if}\kappa =1,\hfill \\ {\displaystyle \frac{\lambda }{2}},\hfill & \mathrm{if}\kappa =2,\hfill \\ {\displaystyle \frac{\lambda }{4}},\hfill & \mathrm{if}\kappa =3,\hfill \\ {\displaystyle \frac{\lambda }{5}},\hfill & \mathrm{if}\kappa =4.\hfill \end{array}\right.$$

Let $\left(\mathsf{{\rm Y}},d_p,\circ \right)$ be a $\mathcal{G}$-parametric metric space. The set of all nonempty closed subsets of $\mathsf{{\rm Y}}$ is denoted by $P_{cl}\left(\mathsf{{\rm Y}}\right)$.

Define

$$F_{P_{cl}}\left(\mathsf{{\rm Y}}\right)=\{A\in I^{\mathsf{{\rm Y}}}:{\left[A\right]}_{\alpha }\in P_{cl}\left(\mathsf{{\rm Y}}\right)\}\mathit{for}\mathit{some}\alpha \in (0,1].$$

A point ϰ in Υ is a fuzzy fixed point of a fuzzy mapping $T:\mathsf{{\rm Y}}\to I^{\mathsf{{\rm Y}}}$ iff $\varkappa \in {\left[T\varkappa \right]}_{\alpha }.$ $Fuz\left(T\right)$ denotes the set of all fuzzy fixed points of fuzzy mapping $T.$

Suppose that $T_1,$ $T_2:\to F_{P_{cl}}\left(\mathsf{{\rm Y}}\right)$ represents fuzzy mappings. Set

$$X_{T_1,T_2}:=\{\varkappa \in \mathsf{{\rm Y}}:\left(\varkappa ,v_{\varkappa }\right)\in E\left(G\right)\mathit{where}{v}_{\varkappa }\in {\left[T_1\left(\varkappa \right)\right]}_{\alpha }\cap {\left[T_2\left(\varkappa \right)\right]}_{\beta }\},$$

for $\alpha ,\beta \in (0,1]$ where α and β may or may not be equal.

In the further discussion, we will take $\alpha ,\beta \in (0,$ $1],$ where $\alpha ,\beta$ may or may not be equal, so, in the rest of the paper, it is not significant if $\alpha ,\beta$ are not cited as part of in $(0,1]$.

Now, we give the following definition in the setup of $\mathcal{G}$-parametric metric space.

Definition 9.

Let $T_1, T_2:\mathsf{{\rm Y}}\to F_{P_{cl}}\left(\mathsf{{\rm Y}}\right)$ be two fuzzy multivalued mappings in $\mathcal{G}$-parametric metric space $\left(\mathsf{{\rm Y}},d_p,\circ \right)$. Assume that for every vertex ϰ in G and for any $v_{\varkappa }\in {\left[T_i\left(\varkappa \right)\right]}_{\alpha },$ for $i\in \{1,2\}$ we have $(\varkappa ,v_{\varkappa })\in E\left(G\right)$. A pair of mappings $(T_1,T_2)$ is said to form a fuzzy graphic rational F-contraction if, for every ${\varkappa }_1,{\varkappa }_2\in \mathsf{{\rm Y}}$ with $({\varkappa }_1,{\varkappa }_2)\in E\left(G\right)$ and $v_{{\varkappa }_1}\in {\left[T_i\left({\varkappa }_1\right)\right]}_{\alpha }$, there exists $v_{{\varkappa }_2}\in {\left[T_j\left({\varkappa }_2\right)\right]}_{\beta }$ for $i,j\in \{1,2\}$ with $i\ne j$ such that $(v_{{\varkappa }_1},v_{{\varkappa }_2})\in E^{*}\left(G\right)$ and

$$\tau +F\left(d_p(v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )\right)\le F\left(M_G({\varkappa }_1,{\varkappa }_2;v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )\right),$$

holds, where $\tau >0$, $F\in \Gamma$, and

$$\begin{array}{ccc}& & M_G({\varkappa }_1,{\varkappa }_2;v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )=max\{d_p({\varkappa }_1,{\varkappa }_2,\theta ),d_p({\varkappa }_1,v_{{\varkappa }_1},\theta ),d_p({\varkappa }_2,v_{{\varkappa }_2},\theta ),\hfill \\ & & {\displaystyle \frac{d_p\left({\varkappa }_2,v_{{\varkappa }_2},\theta \right)[1+d_p\left({\varkappa }_1,v_{{\varkappa }_1},\theta \right)]}{1+d_p\left({\varkappa }_1,{\varkappa }_2,\theta \right)}},{\displaystyle \frac{d_p\left({\varkappa }_2,v_{{\varkappa }_1},\theta \right)[1+d_p\left({\varkappa }_1,v_{{\varkappa }_1},\theta \right)]}{1+d_p\left({\varkappa }_1,{\varkappa }_2,\theta \right)}}\}.\hfill \end{array}$$

It is important to note that for distance selections of map $F,$ we may obtain different conductivity conditions.

Remember that a fuzzy map $T:\mathsf{{\rm Y}}\to F_{P_{cl}}\left(\mathsf{{\rm Y}}\right)$ is called an upper semi-continuous if for ${\varkappa }_n\in \mathsf{{\rm Y}}$ and ${\varkappa }_n^{*}\in {\left[T{\varkappa }_n\right]}_{\alpha }$ with ${\varkappa }_n\to {\varkappa }_0$ and ${\varkappa }_n^{*}\to {\varkappa }_0^{*}$, it implies ${\varkappa }_0^{*}\in {\left[T{\varkappa }_0\right]}_{\beta }$.

A subset W of V, that is $W\subset V$, is named as a clique of graph G if for every two vertices belonging to $W,$ there is an edge that connects both. This is similar to the condition that the subgraph W is complete, that is, for every $\varkappa ,$ ${\varkappa }^{*}\in W\left(G\right)$ implies $(\varkappa ,{\varkappa }^{*})\in E\left(G\right)$.

3. Main Results

We present common fuzzy fixed point results for two fuzzy multivalued mappings on a $\mathcal{G}$-parametric metric space equipped with a directed graph. First, we consider the following result.

Theorem 2.

Let ∘ be a continuous binary operation and $(\mathsf{{\rm Y}},$ $d_p,\circ )$ be a complete $\mathcal{G}$-parametric metric space endowed with a directed graph G such that $V\left(G\right)=\mathsf{{\rm Y}}$ and $E\left(G\right)\supseteq \Delta .$ If fuzzy mappings $T_1,$ $T_2:\mathsf{{\rm Y}}\to F_{P_{cl}}\left(\mathsf{{\rm Y}}\right)$ form a fuzzy graphic rational F-contraction, then the following statements hold:

(i) 

$Fuz\left(T_1\right)=Fuz\left(T_2\right)\ne \varnothing$ if, and only if, $Fuz\left(T_i\right)\ne \varnothing$ for any $i\in \left\{1,2\right\}.$

(ii) 

${\mathsf{{\rm Y}}}_{T_1,T_2}\ne \varnothing$ allowed that $Fuz\left(T_1\right)\cap Fuz\left(T_2\right)\ne \varnothing .$

(iii) 

The graph G is weakly connected and ${\mathsf{{\rm Y}}}_{T_1,T_2}\ne \varnothing$, then $Fuz\left(T_1\right)=Fuz\left(T_2\right)\ne \varnothing$ allowed that either $\left(a\right)$ either $T_1$ or $T_2$ is upper semi-continuous or $\left(b\right)$ F is continuous, either $T_1$ or $T_2$ is bounded and G has property $\left(P\right)$.

(iv) 

$Fuz\left(T_1\right)\cap Fuz\left(T_2\right)$ is a clique of graph $\tilde{G}$ if and only if $Fuz\left(T_1\right)\cap Fuz\left(T_2\right)$ is a singleton set.

Proof. 

To validate $\left(i\right)$, let ${\varkappa }^{*}$ be any point of $\mathsf{{\rm Y}},$ we suppose that ${\varkappa }^{*}\in {\left[T_1\left({\varkappa }^{*}\right)\right]}_{\alpha }$ such that ${\varkappa }^{*}$ $\notin {\left[T_2\left({\varkappa }^{*}\right)\right]}_{\beta }$. As it is given $\left(T_1,T_2\right)$ form a fuzzy graphic rational F-contraction, this implies that there exists a $\varkappa \in {\left[T_2\left({\varkappa }^{*}\right)\right]}_{\beta }$ with $\left({\varkappa }^{*},\varkappa \right)\in E^{*}\left(G\right)$ such that

$$\tau +F\left(d_p({\varkappa }^{*},\varkappa ,\theta )\right)\le F\left(M_G({\varkappa }^{*},{\varkappa }^{*};{\varkappa }^{*},\varkappa ,\theta )\right)$$

(1)

holds, where

$$\begin{array}{ccc}& & M_G({\varkappa }^{*},{\varkappa }^{*};{\varkappa }^{*},\varkappa ,\theta )=max\{d_p({\varkappa }^{*},{\varkappa }^{*},\theta ),d_p({\varkappa }^{*},{\varkappa }^{*},\theta ),d_p(\varkappa ,{\varkappa }^{*},\theta ),\hfill \\ & & {\displaystyle \frac{d_p({\varkappa }^{*},\varkappa ,\theta )[1+d_p({\varkappa }^{*},{\varkappa }^{*},\theta )]}{1+d_p({\varkappa }^{*},{\varkappa }^{*},\theta )]}},{\displaystyle \frac{d_p({\varkappa }^{*},{\varkappa }^{*},\theta )[1+d_p({\varkappa }^{*},{\varkappa }^{*},\theta )]}{1+d_p({\varkappa }^{*},{\varkappa }^{*},\theta )]}}\}\hfill \\ & =& d_p(\varkappa ,{\varkappa }^{*},\theta ).\hfill \end{array}$$

Thus, we have

$$\tau +F\left(d_p({\varkappa }^{*},\varkappa ,\theta )\right)\le F\left(d_p({\varkappa }^{*},\varkappa ,\theta )\right),$$

a contradiction as $\tau >0.$ Hence ${\varkappa }^{*}\in {\left[T_2\left({\varkappa }^{*}\right)\right]}_{\beta }$ and so $Fuz\left(T_1\right)\subseteq Fuz\left(T_2\right).$ Similarly, $Fuz\left(T_2\right)\subseteq Fuz\left(T_1\right)$ and therefore $Fuz\left(T_1\right)=Fuz\left(T_2\right)$. Also, if ${\varkappa }^{*}\in {\left[T_2\left({\varkappa }^{*}\right)\right]}_{\beta },$ then we have ${\varkappa }^{*}\in {\left[T_1\left({\varkappa }^{*}\right)\right]}_{\alpha }.$ Converse can be proved easily by simple steps.

To prove $\left(ii\right)$, let $Fuz\left(T_1\right)\cap Fuz\left(T_2\right)\ne \varnothing$. Then there exists $\varkappa \in \mathsf{{\rm Y}}$ such that $\varkappa \in {\left[T_1\left(\varkappa \right)\right]}_{\alpha }\cap {\left[T_2\left(\varkappa \right)\right]}_{\beta }$. As $\Delta \subseteq E\left(G\right)$, we deduce that $X_{T_1,T_2}\ne \varnothing$.

To validate $\left(iii\right)$, assume that ${\varkappa }_0$ is any element in $\mathsf{{\rm Y}}.$ If ${\varkappa }_0\in {\left[T_1\left({\varkappa }_0\right)\right]}_{\alpha }$ or ${\varkappa }_0\in {\left[T_2\left({\varkappa }_0\right)\right]}_{\beta }$, then the proof is completed. So, we assume that ${\varkappa }_0$ $\notin {\left[T_1\left({\varkappa }_0\right)\right]}_{\alpha }$ and ${\varkappa }_0\notin {\left[T_2\left({\varkappa }_0\right)\right]}_{\beta }$. Now for $i,j\in \{1,2\}$ with $i\ne j$, if ${\varkappa }_1\in {\left[T_i\left({\varkappa }_0\right)\right]}_{\alpha },$ then there exists ${\varkappa }_2\in {\left[T_j\left({\varkappa }_1\right)\right]}_{\beta }$ with $({\varkappa }_1,{\varkappa }_2)\in E^{*}\left(G\right)$ such that

$$\begin{array}{ccc}& & \tau +F\left(d_p(v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )\right)\le F\left(M_G({\varkappa }_1,{\varkappa }_2;v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )\right),\hfill \\ & & M_G({\varkappa }_1,{\varkappa }_2;v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )=max\{d_p({\varkappa }_1,{\varkappa }_2,\theta ),d_p({\varkappa }_1,v_{{\varkappa }_1},\theta ),d_p({\varkappa }_2,v_{{\varkappa }_2},\theta ),\hfill \\ & & {\displaystyle \frac{d_p\left({\varkappa }_2,v_{{\varkappa }_2},\theta \right)[1+d_p\left({\varkappa }_1,v_{{\varkappa }_1},\theta \right)]}{1+d_p\left({\varkappa }_1,{\varkappa }_2,\theta \right)}},{\displaystyle \frac{d_p\left({\varkappa }_2,v_{{\varkappa }_1},\theta \right)[1+d_p\left({\varkappa }_1,v_{{\varkappa }_1},\theta \right)]}{1+d_p\left({\varkappa }_1,{\varkappa }_2,\theta \right)}}\}.\hfill \end{array}$$

$$\tau +F\left(d_p({\varkappa }_1,{\varkappa }_2,\theta )\right)\le F\left(M_G({\varkappa }_0,{\varkappa }_1;{\varkappa }_1,{\varkappa }_2,\theta )\right),$$

(2)

where

$$\begin{array}{ccc}& & M_G({\varkappa }_0,{\varkappa }_1;{\varkappa }_1,{\varkappa }_2,\theta )=max\{d_p({\varkappa }_0,{\varkappa }_1,\theta ),d_p({\varkappa }_0,{\varkappa }_1,\theta ),d_p({\varkappa }_1,{\varkappa }_2,\theta ),\hfill \\ & & {\displaystyle \frac{d_p({\varkappa }_1,{\varkappa }_2,\theta )[1+d_p({\varkappa }_0,{\varkappa }_1,\theta )]}{1+d_p({\varkappa }_0,{\varkappa }_1,\theta )}},{\displaystyle \frac{d_p({\varkappa }_1,{\varkappa }_1,\theta )[1+d_p({\varkappa }_0,{\varkappa }_1,\theta )]}{1+d_p({\varkappa }_0,{\varkappa }_1,\theta )}}\}\hfill \\ & =& max\{d_p({\varkappa }_0,{\varkappa }_1,\theta ),d_p({\varkappa }_1,{\varkappa }_2,\theta )\}.\hfill \end{array}$$

If $M_G({\varkappa }_0,{\varkappa }_1;{\varkappa }_1,{\varkappa }_2,\theta )=d_p({\varkappa }_1,{\varkappa }_2,\theta ),$ then

$$\tau +F\left(d_p({\varkappa }_1,{\varkappa }_2,\theta )\right)\le F\left(d_p({\varkappa }_1,{\varkappa }_2,\theta )\right),$$

gives a contradiction as $\tau >0$. Therefore, $M_G({\varkappa }_0,{\varkappa }_1;{\varkappa }_1,{\varkappa }_2,\theta )=d_p({\varkappa }_0,{\varkappa }_1,\theta )$ and we have

$$\tau +F\left(d_p({\varkappa }_1,{\varkappa }_2,\theta )\right)\le F\left(d_p({\varkappa }_0,{\varkappa }_1,\theta )\right).$$

Correspondingly, for the point ${\varkappa }_2$ in ${\left[T_j\left({\varkappa }_1\right)\right]}_{\alpha },$ there exists ${\varkappa }_3\in {\left[T_i\left({\varkappa }_2\right)\right]}_{\beta }$ with $\left({\varkappa }_2,{\varkappa }_3\right)\in E^{*}\left(G\right)$ such that

$$\tau +F\left(d_p({\varkappa }_2,{\varkappa }_3,\theta )\right)\le F\left(M_G({\varkappa }_1,{\varkappa }_2;{\varkappa }_2,{\varkappa }_3,\theta )\right),$$

(3)

where

$$\begin{array}{ccc}& & M_G({\varkappa }_1,{\varkappa }_2;{\varkappa }_2,{\varkappa }_3,\theta )=max\{d_p({\varkappa }_1,{\varkappa }_2,\theta ),d_p({\varkappa }_1,{\varkappa }_2,\theta ),d_p({\varkappa }_2,{\varkappa }_3,\theta ),\hfill \\ & & {\displaystyle \frac{d_p({\varkappa }_2,{\varkappa }_3,\theta )[1+d_p({\varkappa }_1,{\varkappa }_2,\theta )]}{1+d_p({\varkappa }_1,{\varkappa }_2,\theta )}},{\displaystyle \frac{d_p({\varkappa }_2,{\varkappa }_2,\theta )[1+d_p({\varkappa }_1,{\varkappa }_2,\theta )]}{1+d_p({\varkappa }_1,{\varkappa }_2,\theta )}}\}\hfill \\ & =& max\{d_p({\varkappa }_1,{\varkappa }_2,\theta ),d_p({\varkappa }_2,{\varkappa }_3,\theta )\}.\hfill \end{array}$$

In the case of $M_G({\varkappa }_1,{\varkappa }_2;{\varkappa }_2,{\varkappa }_3,\theta )=d_p({\varkappa }_2,{\varkappa }_3,\theta ),$ then

$$\tau +F\left(d_p({\varkappa }_2,{\varkappa }_3,\theta )\right)\le F\left(d_p({\varkappa }_2,{\varkappa }_3,\theta )\right),$$

gives a contradiction as $\tau >0$. Therefore, $M_G({\varkappa }_1,{\varkappa }_2;{\varkappa }_2,{\varkappa }_3,\theta )=d_p({\varkappa }_1,{\varkappa }_2,\theta )$ and we have

$$\tau +F\left(d_p({\varkappa }_2,{\varkappa }_3,\theta )\right)\le F\left(d_p({\varkappa }_1,{\varkappa }_2,\theta )\right).$$

Enduring this way, for ${\varkappa }_{2n}\in {\left[T_j\left({\varkappa }_{2n-1}\right)\right]}_{\alpha }$, there exists ${\varkappa }_{2n+1}\in {\left[T_i\left({\varkappa }_{2n}\right)\right]}_{\beta }$ with $\left({\varkappa }_{2n},{\varkappa }_{2n+1}\right)\in E^{*}\left(G\right)$ we have

$$\tau +F\left(d_p({\varkappa }_{2n},{\varkappa }_{2n+1},\theta )\right)\le F\left(M_G({\varkappa }_{2n-1},{\varkappa }_{2n};{\varkappa }_{2n},{\varkappa }_{2n+1},\theta )\right),$$

that is,

$$\tau +F\left(d_p({\varkappa }_{2n},{\varkappa }_{2n+1},\theta )\right)\le F\left(d_p({\varkappa }_{2n-1},{\varkappa }_{2n})\right).$$

In the parallel manner, for ${\varkappa }_{2n+1}\in {\left[T_j\left({\varkappa }_{2n}\right)\right]}_{\alpha }$, there exists ${\varkappa }_{2n+2}\in {\left[T_i\left({\varkappa }_{2n+1}\right)\right]}_{\beta }$ such that for $\left({\varkappa }_{2n+1},{\varkappa }_{2n+2}\right)\in E^{*}\left(G\right)$ implies

$$\tau +F\left(d_p({\varkappa }_{2n+1},{\varkappa }_{2n+2},\theta )\right)\le F\left(d_p({\varkappa }_{2n},{\varkappa }_{2n+1},\theta )\right).$$

Hence, we obtained a sequence $\left\{{\varkappa }_n\right\}$ in $\mathsf{{\rm Y}}$ such that for ${\varkappa }_n\in {\left[T_j\left({\varkappa }_{n-1}\right)\right]}_{\alpha }$, there exists ${\varkappa }_{n+1}\in {\left[T_i\left({\varkappa }_n\right)\right]}_{\beta }$ with $\left({\varkappa }_n,{\varkappa }_{n+1}\right)\in E^{*}\left(G\right)$ and it satisfies

$$\tau +F\left(d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right)\le F\left(d_p({\varkappa }_{n-1},{\varkappa }_n,\theta )\right).$$

(4)

Therefore

$$\begin{array}{ccc}\hfill F\left(d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right)& \le & F\left(d_p({\varkappa }_{n-1},{\varkappa }_n,\theta )\right)-\tau \le F\left(d_p({\varkappa }_{n-2}{,}_{\varkappa n-1},\theta )\right)-2\tau \hfill \\ & \le & \dots \le F\left(d_p({\varkappa }_0,{\varkappa }_1,\theta )\right)-n\tau .\hfill \end{array}$$

(5)

From (5), we obtained $\underset{n\to \infty }{lim}F\left(d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right)=-\infty$ that together with ($F_2$) yields

$$\underset{n\to \infty }{lim}{d}_p({\varkappa }_n,{\varkappa }_{n+1},\theta )=0.$$

Now by ($F_3$), we have $h\in \left(0,1\right)$ such that

$$\underset{n\to \infty }{lim}{\left[d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right]}^{h}F\left(d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right)=0.$$

From (5), we have

$$\begin{array}{ccc}& & {\left[d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right]}^{h}F\left(d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right)-{\left[d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right]}^{h}F\left(d_p({\varkappa }_0,{\varkappa }_1,\theta )\right)\hfill \\ & \le & -n\tau {\left[d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right]}^h\le 0.\hfill \end{array}$$

Taking the limit as $n\to \infty$, we receive

$$\underset{n\to \infty }{lim}n{\left[d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\right]}^h=0,$$

so $\underset{n\to \infty }{lim}{n}^{{\displaystyle \frac{1}{h}}}{d}_p({\varkappa }_n,{\varkappa }_{n+1},\theta )=0$ and there exists $n_1\in \mathbb{N}$ such that $n^{{\displaystyle \frac{1}{h}}}{d}_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\le 1$ for all $n\ge n_1.$ So we obtain

$$d_p({\varkappa }_n,{\varkappa }_{n+1},\theta )\le {\displaystyle \frac{1}{n^{1/h}}}$$

for all $n\ge n_1$ and for all values of $\theta .$ Now, for $n,m\in \mathbb{N}$ having $m>n\ge n_1$, we obtain

$$\begin{array}{ccc}\hfill d_p\left({\varkappa }_n,{\varkappa }_m,\theta \right)& \le & d_p({\varkappa }_n,{\varkappa }_{n+1},{\displaystyle \frac{\theta }{2}})\circ d_p({\varkappa }_{n+1},{\varkappa }_m,{\displaystyle \frac{\theta }{2}})\hfill \\ & \le & d_p({\varkappa }_n,{\varkappa }_{n+1},{\displaystyle \frac{\theta }{2}})\circ d_p\left({\varkappa }_{n+1},{\varkappa }_{n+2},{\displaystyle \frac{\theta }{4}}\right)\circ \dots \hfill \\ & & \circ d_p\left({\varkappa }_{m-2},{\varkappa }_{m-1},{\displaystyle \frac{\theta }{2^{m-n}}}\right)\circ d_p\left({\varkappa }_{m-1},{\varkappa }_m,{\displaystyle \frac{\theta }{2^{m-n}}}\right)\hfill \\ & \le & {\displaystyle \frac{1}{n^{1/h}}}\circ {\displaystyle \frac{1}{{\left(n+1\right)}^{1/h}}}\circ \dots \circ {\displaystyle \frac{1}{{\left(m-1\right)}^{1/h}}}.\hfill \end{array}$$

Taking the limit as $n,m\to \infty$ on both sides, we obtain $d_p\left({\varkappa }_n,{\varkappa }_m,\theta \right)\to 0$ as $n,m\to \infty$. So $\left\{{\varkappa }_n\right\}$ is a Cauchy sequence in $X.$ Since X is complete, there exists an element ${\varkappa }^{*}\in \mathsf{{\rm Y}}$ such that ${\varkappa }_n$ $\to {\varkappa }^{*}$ as $n\to \infty$.

Now, if $T_i$ is upper semi-continuous, then as ${\varkappa }_{2n}\in X,$ ${\varkappa }_{2n+1}\in {\left[T_i\left({\varkappa }_{2n}\right)\right]}_{\alpha }$ with ${\varkappa }_{2n}\to {\varkappa }^{*}$ and ${\varkappa }_{2n+1}\to {\varkappa }^{*}$ as $n\to \infty$ implies that ${\varkappa }^{*}\in {\left[T_i\left({\varkappa }^{*}\right)\right]}_{\beta }.$ Using (i), we have ${\varkappa }^{*}\in {\left[T_i\left({\varkappa }^{*}\right)\right]}_{\alpha }={\left[T_j\left({\varkappa }^{*}\right)\right]}_{\beta }.$ In the same fashion, the result holds when $T_j$ is upper semi-continuous.

Assume that F is continuous. Since ${\varkappa }_{2n}$ converges to ${\varkappa }^{*}$ as $n\to \infty$ and $({\varkappa }_{2n},{\varkappa }_{2n+1})\in E\left(G\right),$ we have $\left({\varkappa }_{2n},{\varkappa }^{*}\right)\in E\left(G\right).$ For ${\varkappa }_{2n}\in {\left[T_j\left({\varkappa }_{2n-1}\right)\right]}_{\alpha },$ there exists $v_n\in {\left[T_i\left({\varkappa }^{*}\right)\right]}_{\beta }$ such that $\left({\varkappa }_{2n},v_n\right)\in E^{*}\left(G\right).$ As $\left\{v_n\right\}$ is bounded, $\underset{n\to \infty }{limsup}{v}_n=v^{*}$, and $\underset{n\to \infty }{liminf}{v}_n=v^{*}$ both exist. Assume that $v^{*}\ne x^{*}.$ $(T_1,T_2)$ form a fuzzy graphic rational F-contraction,

$$\tau +F\left(d_p({\varkappa }_{2n},v_n,\theta )\right)\le F\left(M_G({\varkappa }_{2n-1},{\varkappa }^{*};{\varkappa }_{2n},v_n,\theta )\right),$$

(6)

where

$$\begin{array}{ccc}& & M_G({\varkappa }_{2n-1},{\varkappa }^{*};{\varkappa }_{2n},v_n,\theta )=max\{d_p({\varkappa }_{2n-1},{\varkappa }^{*},\theta ),d_p({\varkappa }_{2n-1},{\varkappa }_{2n},\theta ),d_p({\varkappa }^{*},v_n,\theta ),\hfill \\ & & {\displaystyle \frac{d_p({\varkappa }_{2n-1},v_n,\theta )[1+d_p({\varkappa }_{2n-1},{\varkappa }_{2n},\theta )]}{1+d_p({\varkappa }_{2n-1},{\varkappa }^{*},\theta )}},{\displaystyle \frac{d_p({\varkappa }_{2n-1},{\varkappa }_{2n-1},\theta )[1+d_p({\varkappa }_{2n-1},{\varkappa }_{2n},\theta )]}{1+d_p({\varkappa }_{2n-1},{\varkappa }^{*},\theta )}}\}.\hfill \end{array}$$

On taking $limsup$ implies

$$\tau +F\left(d_p({\varkappa }^{*},v^{*},\theta )\right)\le F\left(d_p({\varkappa }^{*},v^{*},\theta )\right),$$

a contradiction. Hence, $v^{*}={\varkappa }^{*}.$ In the same way, taking the $liminf$ yields $v^{*}={\varkappa }^{*}$. Since $v_n\in {\left[T_i\left({\varkappa }^{*}\right)\right]}_{\alpha }$ for all $n\ge 1$ and ${\left[T_i\left({\varkappa }^{*}\right)\right]}_{\alpha }$ is a closed set, it follows that ${\varkappa }^{*}\in {\left[T_i\left({\varkappa }^{*}\right)\right]}_{\alpha }.$ Now, from (i), we acquire ${\varkappa }^{*}\in {\left[T_i\left({\varkappa }^{*}\right)\right]}_{\beta }$ and hence $Fuz\left(T_1\right)=Fuz\left(T_2\right).$

Lastly, to verify $\left(iv\right)$, suppose that the set $Fuz\left(T_1\right)\cap Fuz\left(T_2\right)$ is a clique set of $\tilde{G}$. We need to verify that $Fuz\left(T_1\right)\cap Fuz\left(T_2\right)$ is singleton. Assume, in contrast, that there exists v and u such that $v,u\in Fuz\left(T_1\right)\cap Fuz\left(T_2\right)$ but $u\ne v$. As $(v,u)\in E^{*}\left(G\right)$ and $T_1$ and $T_2$ form a fuzzy graphic rational F-contraction, implies that

$$\tau +F\left(d_p(v,u,\theta )\right)\le F\left(M_G(v,u;v,u,\theta )\right),$$

where

$$\begin{array}{ccc}\hfill M_G(v,u;v,u,\theta )& =& max\{d_p(v,u,\theta ),d_p(v,v,\theta ),d_p(u,u,\theta ),\hfill \\ & & {\displaystyle \frac{d_p\left(u,u,\theta \right)[1+d_p\left(v,v,\theta \right)]}{1+d_p\left(v,u,\theta \right)}},{\displaystyle \frac{d_p\left(u,v,\theta \right)[1+d_p\left(v,v,\theta \right)]}{1+d_p\left(v,u,\theta \right)}}\}\hfill \\ & =& d_p\left(v,u,\theta \right),\hfill \end{array}$$

that is,

$$\tau +F\left(d_p(v,u,\theta )\right)\le F\left(d_p\left(v,u,\theta \right)\right),$$

a contradiction as $\tau >0$. Hence $v=u$. Conversely, if $Fuz\left(T_1\right)\cap Fuz\left(T_2\right)$ is singleton, then it follows that $Fuz\left(T_1\right)\cap Fuz\left(T_2\right)$ is a clique set of $\tilde{G}$. □

Example 5.

Let $\mathsf{{\rm Y}}=\{{\eta }_n={\displaystyle \frac{n(n+1)}{2}}:n\in \mathbb{N}\}=V\left(G\right)$,

$$\begin{array}{ccc}\hfill E\left(G\right)& =& \{(\varkappa ,{\varkappa }^{*}):\varkappa \le {\varkappa }^{*}\mathit{where}\varkappa ,{\varkappa }^{*}\in V\left(G\right)\}\mathit{and}\hfill \\ \hfill E^{*}\left(G\right)& =& \{(\varkappa ,{\varkappa }^{*}):\varkappa <{\varkappa }^{*}\mathit{where}\varkappa ,{\varkappa }^{*}\in V\left(G\right)\}.\hfill \end{array}$$

We define $d_p:\mathsf{{\rm Y}}\times \mathsf{{\rm Y}}\times (0,\infty )\to [0,\infty )$ as

$$d_p\left(\varkappa ,{\varkappa }^{*},\theta \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{e^{\theta }}}max\{\varkappa ,{\varkappa }^{*}\},\hfill & \varkappa \ne {\varkappa }^{*}\hfill \\ 0,\hfill & \varkappa ={\varkappa }^{*}.\hfill \end{array}\right.$$

for $\theta >0.$ Then $(\mathsf{{\rm Y}},d_p,\circ )$ is a complete $\mathcal{G}$-parametric metric space enriched with a directed graph $G.$

Define $T_1:\mathsf{{\rm Y}}\to F_{P_{cl}}\left(\mathsf{{\rm Y}}\right)$ as follows

$$T_1\left(\varkappa \right)\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & \mathit{if}t={\varkappa }_1,\hfill \\ \\ {\displaystyle \frac{1}{3}},\hfill & \mathit{elsewhere}.\hfill \end{array}\right.$$

Now for $\alpha ={\displaystyle \frac{1}{2}},$ we have

$$\begin{array}{ccc}\hfill {\left[T_1\left(\varkappa \right)\right]}_{{\displaystyle \frac{1}{2}}}& =& \{t:{\left[T_1\left(\varkappa \right)\left(t\right)\right]}_{\alpha }\ge {\displaystyle \frac{1}{2}}\}\hfill \\ & =& \left\{{\varkappa }_1\right\}\hfill \end{array}$$

and $T_2:\mathsf{{\rm Y}}\to F_{P_{cl}}\left(\mathsf{{\rm Y}}\right)$ as in the following cases:

(1) 

For $\varkappa ={\varkappa }_1$

$$T_2\left(\varkappa \right)\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & \mathit{if}t={\varkappa }_1,\hfill \\ \\ {\displaystyle \frac{1}{3}},\hfill & \mathit{elsewhere}.\hfill \end{array}\right.$$

(2) 

For $\varkappa ={\varkappa }_n$ with $n>1$

$$T_2\left(\varkappa \right)\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & \mathit{if}t\in \{{\varkappa }_1,{\varkappa }_{n-1}\},\hfill \\ \\ {\displaystyle \frac{1}{3}},\hfill & \mathit{elsewhere}.\hfill \end{array}\right.$$

Now for $\beta ={\displaystyle \frac{1}{2}}$,

$$\begin{array}{ccc}\hfill {\left[T_2\left(\varkappa \right)\right]}_{{\displaystyle \frac{1}{2}}}& =& \{t:{\left[T\left(\varkappa \right)\left(t\right)\right]}_{\beta }\ge {\displaystyle \frac{1}{2}}\}\hfill \\ & =& \left\{\begin{array}{cc}\left\{{\varkappa }_1\right\},\hfill & \mathit{if}\varkappa ={\varkappa }_1\hfill \\ \{{\varkappa }_1,{\varkappa }_{n-1}\},\hfill & \mathit{if}\varkappa ={\varkappa }_n\mathit{with}n>1.\hfill \end{array}\right.\hfill \end{array}$$

Now, we show that a pair of mappings $(T_1,T_2)$ forms a fuzzy graphic rational F-contraction, i.e., for every ${\varkappa }_1,{\varkappa }_2\in \mathsf{{\rm Y}}$ with $({\varkappa }_1,{\varkappa }_2)\in E\left(G\right)$ and $v_{{\varkappa }_1}\in {\left[T_i\left({\varkappa }_1\right)\right]}_{\alpha },$ there exists $v_{{\varkappa }_2}\in {\left[T_j\left({\varkappa }_2\right)\right]}_{\beta }$ for $i,j\in \{1,2\}$ with $i\ne j$ such that $(v_{{\varkappa }_1},v_{{\varkappa }_2})\in E^{*}\left(G\right)$ and

$$\tau +F\left(d_p(v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )\right)\le F\left(M_G({\varkappa }_1,{\varkappa }_2;v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )\right),$$

(7)

holds, where τ denotes a positive real number, $F\in \Gamma$, and

$$\begin{array}{ccc}& & M_G({\varkappa }_1,{\varkappa }_2;v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )=max\{d_p({\varkappa }_1,{\varkappa }_2,\theta ),d_p({\varkappa }_1,v_{{\varkappa }_1},\theta ),d_p({\varkappa }_2,v_{{\varkappa }_2},\theta ),\hfill \\ & & {\displaystyle \frac{d_p\left({\varkappa }_2,v_{{\varkappa }_2},\theta \right)[1+d_p\left({\varkappa }_1,v_{{\varkappa }_1},\theta \right)]}{1+d_p\left({\varkappa }_1,{\varkappa }_2,\theta \right)}},{\displaystyle \frac{d_p\left({\varkappa }_2,v_{{\varkappa }_1},\theta \right)[1+d_p\left({\varkappa }_1,v_{{\varkappa }_1},\theta \right)]}{1+d_p\left({\varkappa }_1,{\varkappa }_2,\theta \right)}}\}.\hfill \end{array}$$

Let $F\left(\omega \right)=ln\omega +\omega$ for $\omega >0$. Then, $F\in \Gamma$ and (7) becomes

$$d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )e^{\tau +d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )-M_G\left(\varkappa ,{\varkappa }^{*};v_{\varkappa },v_{{\varkappa }^{*}},\theta \right)}\le M_G\left(\varkappa ,{\varkappa }^{*};v_{\varkappa },v_{\varkappa *},\theta \right).$$

For $\left(u_{\varkappa },u_{{\varkappa }^{*}}\right)\in E^{*}\left(G\right),$ $\alpha =\beta ={\displaystyle \frac{1}{2}},$ and $\tau =0.1$, we consider the following cases:

(I) 

If ϰ$={\varkappa }_1,$ ${\varkappa }^{*}={\varkappa }_m,$ for $m>1,$ then for $v_{\varkappa }={\varkappa }_1\in {\left[T_1\left(\varkappa \right)\right]}_{{\displaystyle \frac{1}{2}}},$ there exists $v_{{\varkappa }^{*}}={\varkappa }_{m-1}\in {\left[T_2\left({\varkappa }^{*}\right)\right]}_{{\displaystyle \frac{1}{2}}}$ such that for $({\varkappa }_1,{\varkappa }_{m-1})\in E^{*}\left(G\right),$

$$\begin{array}{ccc}& & d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )e^{\tau +d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )-M_G\left(\varkappa ,{\varkappa }^{*};v_{\varkappa },v_{{\varkappa }^{*}},\theta \right)}\hfill \\ & \le & d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )e^{\tau +d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )-d_p\left(\varkappa ,{\varkappa }^{*},\theta \right)}\hfill \\ & =& {\displaystyle \frac{m^2-m}{2e^{\theta }}}{e}^{0.01+{\displaystyle \frac{m^2-m}{2e^{\theta }}}-{\displaystyle \frac{m^2+m}{2e^{\theta }}}}\hfill \\ & =& {\displaystyle \frac{m^2-m}{2e^{\theta }}}{e}^{0.01-{\displaystyle \frac{m}{e^{\theta }}}}\hfill \\ & <& {\displaystyle \frac{m^2+m}{2e^{\theta }}}\hfill \\ & =& d_p\left(\varkappa ,{\varkappa }^{*},\theta \right)\hfill \\ & \le & M_G\left(\varkappa ,{\varkappa }^{*};v_{\varkappa },v_{\varkappa *},\theta \right)\hfill \end{array}$$

holds, where

$$\begin{array}{ccc}& & M_G(\varkappa ,{\varkappa }^{*};v_{\varkappa },v_{{\varkappa }^{*}},\theta )=max\{d_p(\varkappa ,{\varkappa }^{*},\theta ),d_p(\varkappa ,v_{\varkappa },\theta ),d_p({\varkappa }^{*},v_{{\varkappa }^{*}},\theta ),\hfill \\ & & {\displaystyle \frac{d_p\left({\varkappa }^{*},v_{{\varkappa }^{*}},\theta \right)[1+d_p\left(\varkappa ,v_{\varkappa },\theta \right)]}{1+d_p\left(\varkappa ,{\varkappa }^{*},\theta \right)}},{\displaystyle \frac{d_p\left({\varkappa }^{*},v_{\varkappa },\theta \right)[1+d_p\left(\varkappa ,v_{\varkappa },\theta \right)]}{1+d_p\left(\varkappa ,{\varkappa }^{*},\theta \right)}}\}.\hfill \end{array}$$

(II) 

In the case of $\varkappa ={\varkappa }_n,$ ${\varkappa }^{*}={\varkappa }_m$ with $m>n>1,$ then for $v_{\varkappa }={\varkappa }_1\in {\left[T_1\left(\varkappa \right)\right]}_{{\displaystyle \frac{1}{2}}},$ there exists $v_{{\varkappa }^{*}}={\varkappa }_{m-1}\in {\left[T_2\left({\varkappa }^{*}\right)\right]}_{{\displaystyle \frac{1}{2}}},$ such that for $({\varkappa }_1,{\varkappa }_{m-1})\in E^{*}\left(G\right),$

$$\begin{array}{ccc}& & d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )e^{\tau +d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )-M_G\left(\varkappa ,{\varkappa }^{*};v_{\varkappa },v_{{\varkappa }^{*}},\theta \right)}\hfill \\ & \le & d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )e^{\tau +d_p(v_{\varkappa },v_{{\varkappa }^{*}},\theta )-d_p\left(\varkappa ,{\varkappa }^{*},\theta \right)}\hfill \\ & =& {\displaystyle \frac{m^2-m}{2e^{\theta }}}{e}^{0.01+{\displaystyle \frac{m^2-m}{2e^{\theta }}}-{\displaystyle \frac{m^2+m}{2e^{\theta }}}}\hfill \\ & =& {\displaystyle \frac{m^2-m}{2e^{\theta }}}{e}^{0.01-{\displaystyle \frac{m}{e^{\theta }}}}\hfill \\ & <& {\displaystyle \frac{m^2+m}{2e^{\theta }}}\hfill \\ & =& d_p\left(\varkappa ,{\varkappa }^{*},\theta \right)\hfill \\ & \le & M_G\left(\varkappa ,{\varkappa }^{*};v_{\varkappa },v_{\varkappa *},\theta \right)\hfill \end{array}$$

holds, where

$$\begin{array}{ccc}& & M_G(\varkappa ,{\varkappa }^{*};v_{\varkappa },v_{{\varkappa }^{*}},\theta )=max\{d_p(\varkappa ,{\varkappa }^{*},\theta ),d_p(\varkappa ,v_{\varkappa },\theta ),d_p({\varkappa }^{*},v_{{\varkappa }^{*}},\theta ),\hfill \\ & & {\displaystyle \frac{d_p\left({\varkappa }^{*},v_{{\varkappa }^{*}},\theta \right)[1+d_p\left(\varkappa ,v_{\varkappa },\theta \right)]}{1+d_p\left(\varkappa ,{\varkappa }^{*},\theta \right)}},{\displaystyle \frac{d_p\left({\varkappa }^{*},v_{\varkappa },\theta \right)[1+d_p\left(\varkappa ,v_{\varkappa },\theta \right)]}{1+d_p\left(\varkappa ,{\varkappa }^{*},\theta \right)}}\}.\hfill \end{array}$$

Thus, the pair of mappings $\left(T_1,T_2\right)$ form a fuzzy graphic rational F-contraction. Therefore, all the requirements of Theorem 3.1 are satisfied. Moreover, $\left\{{\varkappa }_1\right\}$ is the common fuzzy fixed point of $T_1$ and $T_2$ with $Fuz\left(T_1\right)=Fuz\left(T_2\right).$

The following results generalizes Theorem 3.1 in [9] in two ways:

(1)

The ordinary metric space has been replaced with a $\mathcal{G}$-parametric metric space.

(2)

The mappings in Theorem 3.4 have been extended to fuzzy graphic rational F-contraction mappings.

Corollary 1.

Let $(\mathsf{{\rm Y}},d_p,\circ )$ be a complete $\mathcal{G}$-parametric metric space endowed with a directed graph G such that $V\left(G\right)=$$\mathsf{{\rm Y}}$ and $E\left(G\right)\supseteq \Delta .$ Assume that $T:\mathsf{{\rm Y}}\to F_{P_{cl}}\left(\mathsf{{\rm Y}}\right)$ is a fuzzy mapping and for every vertex ϰ in G and for any $v_{\varkappa }\in {\left[T\left(\varkappa \right)\right]}_{\alpha },$ for $i\in \{1,2\}$ we have $(\varkappa ,v_{\varkappa })\in E\left(G\right)$. If for every ${\varkappa }_1,{\varkappa }_2\in \mathsf{{\rm Y}}$ with $({\varkappa }_1,{\varkappa }_2)\in E\left(G\right)$ and $v_{{\varkappa }_1}\in {\left[T\left({\varkappa }_1\right)\right]}_{\alpha },$ there exists $v_{{\varkappa }_2}\in {\left[T\left({\varkappa }_2\right)\right]}_{\beta }$ such that $(v_{{\varkappa }_1},v_{{\varkappa }_2})\in E^{*}\left(G\right)$ and

$$\tau +F\left(d_p(v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )\right)\le F\left(M_T({\varkappa }_1,{\varkappa }_2;v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )\right),$$

holds, where τ denotes a positive real number and

$$\begin{array}{ccc}& & M_T({\varkappa }_1,{\varkappa }_2;v_{{\varkappa }_1},v_{{\varkappa }_2},\theta )=max\{d_p({\varkappa }_1,{\varkappa }_2,\theta ),d_p({\varkappa }_1,v_{{\varkappa }_1},\theta ),d_p({\varkappa }_2,v_{{\varkappa }_2},\theta ),\hfill \\ & & {\displaystyle \frac{d_p\left({\varkappa }_2,v_{{\varkappa }_2},\theta \right)[1+d_p\left({\varkappa }_1,v_{{\varkappa }_1},\theta \right)]}{1+d_p\left({\varkappa }_1,{\varkappa }_2,\theta \right)}},{\displaystyle \frac{d_p\left({\varkappa }_2,v_{{\varkappa }_1},\theta \right)[1+d_p\left({\varkappa }_1,v_{{\varkappa }_1},\theta \right)]}{1+d_p\left({\varkappa }_1,{\varkappa }_2,\theta \right)}}\}.\hfill \end{array}$$

then the following are satisfied:

(i) 

${\mathsf{{\rm Y}}}_T\ne \varnothing$ provided that $Fuz\left(T\right)\ne \varnothing .$

(ii) 

If ${\mathsf{{\rm Y}}}_T\ne \varnothing$ and G is weakly connected, then $Fuz\left(T\right)\ne \varnothing$ provided that either (a) T is upper semi-continuous or (b) F is continuous, T is bounded and G has property (P).

(iii) 

$Fuz\left(T\right)$ is a clique set of $\tilde{G}$ if and only if $Fuz\left(T\right)$ is a singleton set.

Remark 1.

Let $(\mathsf{{\rm Y}},d_p,\circ )$ be a complete $\mathcal{G}$-parametric metric space endowed with a directed graph G. If we replace the fuzzy graphic rational F-contraction defined in Definition 9 with any of the following conditions:

$$\tau +F\left(d_p(v_a,v_b,\theta )\right)\le F({\alpha }_1{d}_p(a,b,\theta )+{\beta }_1{d}_p(a,v_b,\theta )+{\gamma }_1{d}_p(b,v_b,\theta )),$$

(8)

where ${\alpha }_1,{\beta }_1,{\gamma }_1\ge 0$ and ${\alpha }_1+{\beta }_1+{\gamma }_1\le 1,$ or

$$\tau +F\left(d_p(v_a,v_b,\theta )\right)\le F\left(h[d_p(a,v_b,\theta )+d_p(b,v_b,\theta )]\right),$$

(9)

where $h\in [0,{\displaystyle \frac{1}{2}}],$ or

$$\tau +F\left(d_p(v_a,v_b,\theta )\right)\le F\left(d_p(a,b,\theta )\right).$$

(10)

then the deductions acquired in Theorem 3.1 remain true.

Remark 2.

1. 

If $E\left(G\right):=\mathsf{{\rm Y}}\times \mathsf{{\rm Y}}$, then clearly G is connected and our Theorem 2 improves and generalizes (I) Theorem 4.1 in [10] and (II) Theorems 3.1 and 3.3 of [9].

2. 

If $E\left(G\right):=\mathsf{{\rm Y}}\times \mathsf{{\rm Y}}$, then our Remark 1 extends and generalizes (I) Theorem 3.4 in [9] and (II) Theorem 4.1 of [10].

4. Conclusions

In this paper, we introduced a pair of fuzzy graphic rational F-contraction multivalued mappings and obtained common fixed point results in the structure of generalized parametric metric space endowed with a directed graph. We provided a non-trivial example to support the concepts and results presented in this paper. Our result unify several comparable results in the existing literature and extend them in different directions. The mappings discussed in this paper can be extended to intuitionistic fuzzy mappings, L-fuzzy mappings, and soft multivalued mappings, thereby broadening the scope of the fuzzy fixed-point theory.