Common fixed points of fuzzy set-valued contractive mappings on metric spaces with a directed graph

1.   Introduction

The most celebrated fixed point theorem familiar as Banach contraction principle (BCP) (see ^[13]), is largely used to obtain the existence of a solution of linear and nonlinear functional equations. Given an initial guess of the solution, BCP provides sufficient conditions to guarantee the convergence of successive approximations to actual solution of the problem. BCP ^[13] has been modified and applied in different directions for instance ^[19,26,33] and the references therein. Existence of fixed points of certain mappings established on partially ordered metric spaces has been considered by Ran and Reurings ^[34]. For further consequences in this direction ^[21,35]. Jachymski and Jozwik ^[23] used graph structure on metric fixed point theory instead of the order structure and proved fixed point results. In this fashion the consequences proven in ordered structured upgraded and generalized (see also ^[24] and the reference therein); In 2009, Gwozdz-CLukawska and Jachymski ^[36] incorporated graph theory in metric fixed point theory and flourished the results of the Hutchinson-Barnsley theory for specific families of mappings on a metric space. Bojor ^[16] amalgamated fixed point theory on metric space with graph theory and established fixed point results for Reich type contractions on metric spaces. Abbas and Nazir ^[3] used graphic structure and proved fixed points results of power graph contraction pair on a metric space. This attracted the attention of many authors and various interesting results have been obtained in this direction (see, for example, ^{[7,8,15,17,18]}). Wardowski ^[37] introduced $F$-contraction and achieved an interesting fixed point consequence as an extension of BCP. On the domain of sets equipped with directed graph, latterly, Abbas et al. ^[1] established some fixed point results of set-valued mappings fulfilling certain graphic contraction conditions (see also, ^[2]). On the other hand, one of the obstacles in mathematical modeling of real circumstances is the indefiniteness persuaded by our inabilities to classify events with ample precision. The crisp set theory cannot cope effectively with imprecisions. As an attempt to deal with the problems of inadequate data, crisp sets were replaced with fuzzy sets ^[38] which gave a birth to Fuzzy set theory. It provides appropriate mathematical tools for handling information with non statistical uncertainty. As a result, fuzzy set theory has gained much recognition because of its utilization in several domains such as management sciences, engineering, environmental sciences, medical sciences and in other emerging fields. The fundamental notions of fuzzy sets have been modified and polished up in different fashions; for example, see ^{[4,9,20,27,28]}. In 1981, Heilpern ^[22] initiated the study of fuzzy set-valued maps and obtained a fuzzy replica of Nadler's fixed point results ^[31]. Afterwards, many authors worked on the existence of fixed points of fuzzy set-valued maps, for example, Al-Mazrooei et al. ^[5,6], Azam et al. ^[10,11,12], Bose and Sahani ^[14], Mohammed ^[29], Mohammed and Azam ^[30], Qiu and Shu ^[32], and so on.

In this paper, we develop a generalized graphic fuzzy $F$- contractive mappings on metric spaces and obtain the existence of common fuzzy coincidence and fixed point results for such contractions. We present some examples to endorse the results established herein. Our results extend and unify comparable results in the present literature.

Persistent with Jacehymski ^[24], let $(\Psi, \varphi)$ be a metric space and the diagonal of $\Psi \times \Psi$ is denoted by $\Delta. \; V (G)$ denotes the set of vertices coincides with $\Psi$ of a directed graph $G$ and $E(G)$ represents the set of edges of the graph containing all loops, that is, $\Delta \subseteq E(G)$. Also it is assumed that the graph $G$ has no multiple edges and hence, one can recognize $G$ with the pair $(V(G), E(G))$. Moreover, the number $\varphi(\xi, \zeta)$ is interpreted as the weight of the edge $(\xi, \zeta)$ of $G$.

2.   Prelimnaries

Definition 2.1. In a metric space $(\Psi, \varphi),$ a mapping $h: \Psi \rightarrow \Psi$ is defined a $G$-contraction if

● for each $\xi, \zeta \in \Psi$ with $(\xi, \zeta) \in E(G)$, we possess $(h(\xi), h(\zeta)) \in E(G)$. Viz, $h$ preserves edges of the given graph $G$;

● $h$ decreases weights of edges of $G$; there exits $\eta \in (0, 1)$ such that for all $\xi, \zeta \in \Psi$ with $(\xi, \zeta) \in G$, we have $\varphi(h(\xi), h(\zeta)) \leq \eta \varphi(\xi, \eta)$.

A directed path between $\xi$ and $\zeta$ of length $\ell \in \mathbb{N}$ in graph $G$ is a finite sequence $\{\xi_n\} (n \in \{0, 1, 2, ..., \ell\})$ of vertices such that $\xi_0 = \xi$, $\xi_{\ell} = \zeta$ and $(\xi_{i-1}, \xi_{i}) \in E(G)$ for $i \in \{1, 2, ..., \ell\}$. Remember that a graph $G$ is said to be connected if there is a directed path between every pair of vertex that is from every vertex to any other vertex whereas it is said to be weakly connected if $\tilde{G}$ is connected, where $\tilde{ G }$ represents the undirected graph acquired from $G$ by neglecting the direction of edges.

The graph obtained by reversing the direction of edges is denoted by $G^{-1},$ furthermore for the sake of convenience we treat $\tilde{G}$ as a directed graph for which the set of its edges is symmetric.

In $V(G)$ we define the relation $R$ in the following way. For $\xi, \zeta \in V(G)$, $\xi R \zeta$ if and only if, there is a path in $G$ from $\xi$ to $\zeta$. Let $h:\Psi \rightarrow \Psi$ be an operator, then by $F_{h}$ we represent the set of all fixed points of $h$. Set

A metric space $(\Psi, \varphi)$ equipped with a directed graph $G$ is said to possess the property (P) ^[23]:

$(P)$ if for any sequence $\{ \xi_{n} \}\in \Psi$ satisfying $\xi_{n} \rightarrow \xi$ as $n \rightarrow \infty$ and $(\ \xi_{n}, \xi_{n+1}) \in E(G)$, we have $(\xi_{n}, \xi) \in E(G).$

Theorem 2.2. ^[23] Let $(\Psi, \varphi)$ be a complete metric space and $G$ a directed graph such that $V(G) = \Psi$ and $h:\Psi \rightarrow \Psi$ a $G$- contraction. Assume that $E(G)$ and the triplet $(\Psi, \varphi, G)$ possess property $(P)$. Then the following statements hold.

● $F_{h} \neq \emptyset$ if and only if $\Psi_{h} \neq \emptyset;$

● if $\Psi_{h} \neq \emptyset$ and $G$ is weakly connected, then $h$ is a Picard operator, that is $F_{h} = \{\xi^{\divideontimes}\}$ and sequence $\{h^{n}(\xi)\} \rightarrow \xi^{\divideontimes}$ as $n \rightarrow \infty$ for all $\xi \in \Psi$;

● for any $\xi \in \Psi_{h}$, $h|[\xi]_{\tilde{G}}$ is picard operator;

● if $\Psi_{h} \subseteq E(G)$ then $h$ is weakly picard operator, that is $F_{h} \neq \emptyset$ and for each $\xi \in \Psi$, we have sequence $\{h^{n}(\xi)\} \rightarrow \xi^{\divideontimes} \in F_{h}$ as $n \rightarrow \infty$.

Recall that a crisp set $\Lambda$ in $\Psi$ is determined by its characteristic function $\chi_\Lambda: \Lambda\longrightarrow \{0, 1\}$ interpreted as

The value of $\chi_\Lambda$ at $\xi$ indicates whether an element $\xi$ belongs to $\Lambda$ or not. A fuzzy set is illustrated by allowing a mapping $\chi_\Lambda$ to assume any possible value in the interval $[0, 1]$. Thus, a fuzzy set $\Lambda$ in $\Psi$ is characterized by the function $\Lambda$ with domain $\Psi$ and values in $[0, 1] = I$. The collection of all fuzzy sets in $\Psi$ is denoted by $I^\Psi$. If $\Lambda$ is a fuzzy set in $\Psi$, then $\Lambda(\xi)$ is called the grade or degree of membership of an element $\xi$ in $\Lambda$. The $\alpha$-level set of a fuzzy set $\Lambda$ is represented by $[\Lambda]_\alpha$ and is explained as follows:

Where $\bar{N}$ represents the closure of the crisp set $N.$

Example 2.3. Let $\Psi$ be the set of all individuals in a certain town, and

Then, it is more appropriate to identify an individual be an old person by membership function $\Lambda$ on $\Psi$ because the term "old" is not well defined.

Example 2.4. Let $\Psi = \{1, 2, 3, 4\}$ be endowed with the usual metric. Let $\gimel:\Psi \longrightarrow I^\Psi$ be a fuzzy set-valued map, that is, for each $\xi\in \Psi$, $\gimel(\xi): \Psi \longrightarrow [0, 1]$ is a fuzzy set. For instance, for some $\alpha \in (0, 1]$, we may define one of the fuzzy set $\gimel(1)$ by

In a metric space $(\Psi, \varphi)$, $CB(\Psi)$ represents a class of all non-empty closed and bounded subsets of $\Psi.$ For $\Lambda, \Upsilon \in CB(\Psi)$, the Pompeiu-Hausdorff metric induced by metric $\varphi$ is defined as

where the distance of a point $\xi$ to the set $\Upsilon$ is defined as

Consistent with Abbas et al. ^[2], let $(CB(\Psi), \varphi)$ the Pompeiu-Hausdorff metric induced by $\varphi$ and $\Delta$ represents the diagonal of $CB(\Psi) \times CB(\Psi)$. Throughout this work, we assume that for any $\Lambda, \Upsilon \in CB(\Psi)$, there is an edge between $\Lambda$ and $\Upsilon$, which aims that there is an edge between some $\kappa \in \Lambda$ and $\varrho \in \Upsilon$ which we represent by $(\Lambda, \Upsilon) \subset E(G).$ We now identify the directed graph $G$, called a directed set graph with the pair $(V(G), E(G))$ if the set $V (G)$ of its vertices coincides with $CB(\Psi)$ and $E(G)$ the set of edges of the graph containing all loops, that is, $\Delta \subseteq E(G)$. In addition, suppose that the graph $G$ has no multiple edges. Moreover, for each $\Theta, \Phi \in CB (\Psi)$, the number $H(\Theta, \Phi)$ is interpreted as the weight of the edge $(\Theta, \Phi)$ of a directed set graph $G$.

We suppose that a directed set graph $G$ has no multiple edge and $G$ is a weighted graph in the meaning that each vertex $\Theta$ is given the weight $H(\Theta, \Theta) = 0$ and each edge $(\Theta, \Phi)$ is given the weight $H(\Theta, \Phi)$. Abbas et al. ^[2] introduced the following $(P^{\ast})$ property. A directed set graph $G$ is said to possess property

$P^{\ast}$: If for any sequence of sets $\{ \Psi_{n} \}$ in $CB(\Psi)$ with $\Psi_{n}\rightarrow \Psi$ as $n \rightarrow \infty$, there exists an edge between $\Psi_{n+1}$ and $\Psi_{n}$ for $n \in \mathbb{N}$, and it further implies that, there is a subsequence $\Psi_{n_{k}}$ of $\Psi_{n}$ with an edge between $\Psi$ and $\Psi_{n_{k}}$ for $n \in \mathbb{N}.$

Definition 2.5. Let $\Lambda, \Upsilon \in I^{\Psi}$. Then by definition $[\Lambda]_{\alpha}, [\Upsilon]_{\alpha} \subseteq \Psi.$

1) There is an edge between $[\Lambda]_{\alpha}, [\Upsilon]_{\alpha} \subseteq \Psi$ for some $\alpha \in (0, 1]$, we aim that there is an edge between some $\xi \in [\Lambda]_{\alpha}$ and $\zeta \in [\Upsilon]_{\alpha}$ which we denote by $([\Lambda]_{\alpha}, [\Upsilon]_{\alpha}) \subset E(G).$

2) There is path between $[\Lambda]_{\alpha}$ and $[\Upsilon]_{\alpha}$ we aim that there is a path between some $\xi \in [\Lambda]_{\alpha}$ and $\zeta \in [\Upsilon]_{\alpha}.$

Definition 2.6. Define the set $I_{Fc}(\Psi)$ by

A relation $R$ on $I_{Fc}(\Psi)$ is interpreted as follows: For $\Lambda, \Upsilon \in I_{Fc}(\Psi), \; [ \Lambda ]_{\alpha} R [ \Upsilon ]_{\alpha}$ if there is a path between $[\Lambda]_{\alpha}$ and $[\Upsilon]_{\alpha}$ for some $\alpha \in (0, 1].$ The relation $R$ on $I_{Fc}(\Psi)$ is said to be transitive. if for some $\alpha \in (0, 1]$ there is path between $[\Lambda]_{\alpha}$ and $[\Upsilon]_{\alpha}$ and there is a path between $[\Upsilon]_{\alpha}$ and $[\Omega]_{\alpha}$ imply that there is a path between $[\Lambda]_{\alpha}$ and $[\Omega]_{\alpha}.$

Definition 2.7. Consider the fuzzy set-valued mapping $\gimel: CB(\Psi) \rightarrow I_{Fc}(\Psi),$ the set $\Psi_{\gimel}$ is explained as

Definition 2.8. Let $\daleth, \gimel : CB(\Psi) \rightarrow I_{Fc}(\Psi)$ be two fuzzy set-valued mappings. Then $\Theta \in CB(\Psi)$ is said to be a fuzzy coincidence point of $\daleth$ and $\gimel$ if $[\daleth(\Theta)]_{\alpha} = [\gimel(\Theta)]_{\alpha}$ for some $\alpha \in (0, 1]$. Also a set $A \in CB(\Psi)$ is said to be a fuzzy fixed point of $\daleth$ if there exists $\alpha \in (0, 1]$ such that $[\daleth(\Lambda)]_{\alpha} = \Lambda.$

Note that in our work the set of all fuzzy coincidence points of $\daleth$ and $\gimel$ is represented by $C_{F}(\daleth, \gimel)$ and the set of all fuzzy fixed points of $\daleth$ is represented by $Fuz(\daleth)$.

Definition 2.9. Two fuzzy set-valued maps $\daleth, \gimel :CB(\Psi) \rightarrow I_{Fc}(\Psi)$ are called weakly compatible if they commute at their coincidence point.

Definition 2.10. A subset $\Gamma$ of $CB(\Psi)$ is said to be complete if for any two fuzzy sets $\Upsilon, \Omega \in I_{Fc(X)}$ such that $[\Upsilon]_{\alpha}, [\Omega]_{\alpha} \subseteq \Gamma$ and there is an edge between $[\Upsilon]_{\alpha}$ and $[\Omega]_{\alpha}.$

Let $\digamma$ be the collection of all continuous mappings $F: \mathbb{R^{+}} \rightarrow \mathbb{R}$ that satisfies the following requirements ^[37]:

$F_{1}$) $F$ is strictly increasing, that is, for all $\kappa, \varrho \in \mathbb{R^{+}}$ with $\kappa < \varrho$ gives that $F(\kappa) < F(\varrho)$.

$F_{2}$) for every sequence $\{\kappa _{n}\}$ of positive real numbers,

is equivalent to

$F_{3}$) there exists $k \in (0, 1)$ such that

Definition 2.11. For some $\varepsilon > 0$, a metric space $(\Psi, \varphi)$ is said to be $\varepsilon$-chainable if for given $\xi, \zeta \in \Psi$, there is $n\in \mathbb{N}$ and a finite sequence $\{ \xi_{n}\}$ in $\Psi$ such that

Lemma 2.12. Let $(\Psi, \varphi)$ be a metric space if $H(\Lambda, \Upsilon) < \varepsilon$ for $\Lambda, \Phi \in CB(\Psi)$ then for every $\kappa \in \Lambda$ we have an element $\varrho \in \Upsilon$ such that $\varphi(\kappa, \varrho) < \varepsilon.$

Motivated by the work in ^[25], we introduce the following definition.

Definition 2.13. Let $\daleth, \gimel : CB(\Psi) \rightarrow I_{Fc}(\Psi)$ be fuzzy set-valued maps. The pair $(\daleth, \gimel)$ is said to be a generalized graphic fuzzy $F$-contractive mappings if the following statements are satisfied for some $\alpha \in (0, 1]$.

1) For any $\Theta$ in $CB(\Psi)$, $([ \daleth(\Theta)]_{\alpha}, \Theta) \subseteq E(G)$ and $(\Theta, \gimel[\Theta]_{\alpha}) \subseteq E(G).$

2) There is a function $\tau : \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$ with $\liminf_{\eta \to t^{+}}\tau(\eta) > 0$ for all $t \geq 0$ such that for $F \in \digamma$ there is an edge between $\Lambda$ and $\Upsilon$ with $[\daleth(\Lambda)_{\alpha}] \neq [\daleth(\Upsilon)]_{\alpha}$ such that

holds, where

It is important to note that if any pair $(\daleth, \gimel)$ of set-valued mappings from $CB(\Psi)$ to $I_{Fc}(\Psi)$ is a generalized graphic fuzzy $F$-contractive mappings for a graph $G$, then the pair $(\daleth, \gimel)$ is also generalized graphic fuzzy $F$-contractive mappings for a graph $G^{-1}$ and $\tilde{G}.$ In addition, a pair $(\daleth, \gimel)$ of generalized graphic fuzzy $F$-contractive mappings for graph $G$ is also generalized graphic fuzzy $F$-contractive mappings for the graph $G_{0}$ where $G_{0}$ is graph with $E(G_{0}) = \Psi \times \Psi.$

3.   Main results

Here, we establish some common fuzzy coincidence and fixed point results for fuzzy set-valued maps on $I_{Fc}(\Psi)$ fulfilling generalized graphic fuzzy $F$-contractive mappings conditions.

Theorem 3.1. Let $(\Psi, \varphi)$ be a metric space equipped with a directed graph $G$ with $V(G) = \Psi$, $E(G) \supseteq \Delta$ and the relation $R$ on $I_{Fc}(\Psi)$ is transitive. Suppose that $\daleth, \gimel :CB(\Psi) \rightarrow I_{Fc}(\Psi)$ is a generalized graphic fuzzy $F$-contractive mappings pair such that the range of $\gimel$ contains the range of $\daleth$, then the following statements are satisfied.

1) $C_{F}(\daleth, \gimel)\neq \emptyset$ given that $G$ is weakly connected which holds the property $(P^{\ast})$ and there exists $\alpha \in (0, 1]$ such that $[\gimel(\Psi)]_{\alpha}$ is a complete subspace of $I_{Fc}(\Psi).$

2) If $C_{F} (\daleth, \gimel)$ is complete, then the Hausdorff weight assigned to $[\daleth(\Theta)]_{\alpha}$ and $[\daleth(\Phi)]_{\alpha}$ is $0$ for some $\alpha \in (0, 1]$ and for all $\Theta, \Phi\in C_{F}(\daleth, \gimel).$

3) If $C_{F}(\daleth, \gimel)$ is complete and $\daleth$, $\gimel$ are weakly compatible, then $Fuz(\daleth) \bigcap Fuz (\gimel)$ is singleton.

4) $Fuz(\daleth) \bigcap Fuz(\gimel)$ is complete if and only if $Fuz (\daleth) \bigcap Fuz (\gimel)$ is singleton.

Proof. To verify $(1)$: Let $\Lambda_{0}$ be an arbitrary element in $CB(\Psi)$. As the range of $\gimel$ contains the range of $\daleth$, choose $\Lambda_{1} \in CB(\Psi)$ such that $[\daleth(\Lambda_{0})]_{\alpha} = [ \gimel(\Lambda_{1})]_{\alpha}.$ and for $\Lambda_{2} \in CB(\Psi)$ such that $[S(\Lambda_{1})]_{\alpha} = [ T(\Lambda_{2})]_{\alpha}.$ Carrying on this procedure, for $\Lambda_{n} \in CB(X)$ we get an $\Lambda_{n+1}$ in $CB(\Psi)$ such that $[\daleth(\Lambda_{n})]_{\alpha} = [ \gimel(\Lambda _{n+1})]_{\alpha}$ for all $n \in \mathbb{N}$.

By the hypothesis of the theorem it is given that the pair $(\daleth, \gimel)$ is generalized graphic fuzzy $F$-contractive mappings, therefore for $\Lambda_{n+1}, \Lambda_{n} \in CB(\Psi)$ we have $(\Lambda_{n+1}, [\gimel(\Lambda_{n+1}) ]_{\alpha}) \subseteq E(G)$ and $([\daleth(\Lambda_{n})]_{\alpha}, \Lambda_{n}) \subseteq E(G)$, as $[\daleth(\Lambda_{n})]_{\alpha} = [ \gimel(\Lambda _{n+1})]_{\alpha}$ so we have $([\daleth(\Lambda_{n})]_{\alpha}, \Lambda_{n}) = ([\gimel(\Lambda_{n+1})]_{\alpha}, \Lambda_{n}) \subseteq E(G).$ By using transitivity we have $(\Lambda_{n+1}, \Lambda_{n}) \subseteq E(G)$.

Let us suppose that for $\Lambda_{n}, \Lambda_{n+1} \in CB(\Psi)$ where $n \in \mathbb{N}$ we have $[\daleth(\Lambda_{n})]_{\alpha} \neq [\daleth(\Lambda_{n+1}) ]_{\alpha}$ with $\alpha \in (0, 1 ],$ otherwise, for some $k \in \mathbb{N}$ we have $[ \daleth(\Lambda_{2k}) ]_{\alpha} = [ \daleth(\Lambda_{2k+1})]_{\alpha},$ Also as $[ \daleth(\Lambda_{2k}) ]_{\alpha} = [ \gimel(\Lambda_{2k+1}) ]_{\alpha}$ therefore we can write $[ \gimel(\Lambda_{2k+1}) ]_{\alpha} = [ \daleth(\Lambda_{2k+1}) ]_{\alpha}$ and hence $\Lambda_{2k+1}\in C_{F}(\daleth, \gimel)$. Since $(\Lambda_{n+1}, \Lambda_{n}) \subseteq E(G)$ for all $n \in \mathbb{N},$ by (2.2) we get

where

consequently, we get

If

then we obtain

This implies that

Since $F$ is strictly increasing. So, we have

a contradiction.

This means that

So, we have

that is,

for all $n \in \mathbb{N}$. Thus $\{ H([\gimel(\Lambda_{n})]_{\alpha}, [\gimel(\Lambda_{n+1})]_{\alpha}) \}$ is a decreasing sequence. We now show that

By the property of $\tau$, there exists $c > 0$ with $n_{0} \in \mathbb{N}$ such that $\tau (M(\Lambda_{n}, \Lambda_{n+1})) > c$ for all $n \geq n_{0}$. Now

this means that

By $(F_{2})$, we get

Now by $(F_{3})$, there exists $h \in (0, 1)$ such that

Consider

Now applying limit as $n \rightarrow \infty$ gives that

which means that

Thus, there exists $n_{1} \in \mathbb{N}$ such that

for all $n \geq n_{1}.$ So we get

for all $n \geq n_{1}$. For $m, n \in \mathbb{N }$ with $m > n \geq n_{1}$, we have

As the series $\sum_{i = 1}^{\infty} \frac{1}{i^{\frac{1}{h}}}$ converges, so we have that $H([\gimel(A_{n})]_{\alpha}, [\gimel(\Lambda_{m}) ]_{\alpha} \rightarrow 0$ as $n, m\rightarrow \infty$. Hence $\{[\gimel(\Lambda_{n})]_{\alpha}\}$ proves to be a Cauchy sequence in $[\gimel(\Psi)]_{\alpha}$. The completeness of $([\gimel(\Psi)]_{\alpha}, \varphi)$ in $I_{Fc}(\Psi)$ implies that $[\gimel(\Lambda_{n})]_{\alpha} \rightarrow \Phi$ as $n \rightarrow \infty$ for some $\Phi \in I_{Fc}(\Psi)$. Also $\Theta$ in $CB(\Psi)$ can be found such that $[\gimel(\Theta) ]_{\alpha} = \Phi.$

Let us suppose that $[\daleth(\Theta)]_{\alpha} = [\gimel(\Theta)]_{\alpha}.$ Otherwise, as $([ \gimel(\Lambda_{n+1})]_{\alpha}, ([ \gimel(\Lambda_{n})]_{\alpha}) \subseteq E(G)$, by property $(P^{\ast})$, there exists a subsequence $[\gimel(\Lambda_{n_{k}+1})]_{\alpha}$ of $[\gimel(\Lambda_{n+1})]_{\alpha}$ such that $([\gimel(\Theta) ]_{\alpha}, \gimel(\Lambda_{n_{k}+1})]_{\alpha}) \subseteq E(G)$ for every $n \in \mathbb{N}$. As $(\Theta, [\gimel(\Theta)]_{\alpha}) \subseteq E(G)$ and $([\gimel(\Lambda_{n_{k}+1}) ]_{\alpha}, \Lambda_{n_{k}}) = ([\daleth(\Lambda_{n_{k}}) ]_{\alpha}, \Lambda_{n_{k}}) \subseteq E(G)$, we have $(\Theta, \Lambda_{n_{k}}) \subseteq E(G)$. Since the pair $(\daleth, \gimel)$ is generalized graphic fuzzy $F$-contractive mappings, so we get

where

Now we consider the following cases:

$1)$ In case $M(\Theta, \Lambda_{n_{k}})) = H([\gimel(\Theta)]_{\alpha}), [\gimel(\Lambda_{n_{k}}) ]_{\alpha})$, then we have

applying an upper limit as $k \rightarrow \infty$ gives

a contradiction.

$2)$ When $M(\Theta, \Lambda_{n_{k}})) = H([\daleth(\Theta)]_{\alpha}, [\gimel(\Theta)]_{\alpha}),$ then

a contradiction.

$3)$ If $M(\Theta, \Lambda_{n_{k}})) = H([\gimel(\Lambda_{n_{k} + 1}) ]_{\alpha}, [\gimel(\Lambda_{n_{k}}) ]_{\alpha})$, then we have that

applying an upper limit as $k \rightarrow \infty$ gives

a contradiction.

$4)$ Lastly, if we take $M(\Theta, \Lambda_{n_{k}})) = \dfrac{ H([\daleth(\Theta)]_{\alpha}, [\gimel(\Lambda_{n_{k}}]_{\alpha}) + H([\gimel(\Lambda_{n_{k} + 1}) ]_{\alpha}, [\gimel(\Theta)]_{\alpha}))}{2}$,

so, we have

applying an upper limit as $k \rightarrow \infty$ gives

a contradiction.

All cases show that $[\daleth(\Theta)]_{\alpha} = [\gimel(\Theta)]_{\alpha},$ that is, $\Theta \in C_{F}(\daleth, \gimel).$

To verify $(2)$: Let $\Theta, \Phi \in C_{F}(\daleth, \gimel)$. Assume on contrary that the Pompeiu-Hausdorff weight assign to the $[\daleth(\Theta)]_{\alpha}$ and $[\daleth(\Phi)]_{\alpha}$ is not zero. Since the pair $(\daleth, \gimel)$ is generalized graphic fuzzy $F$-contractive mappings, we have

where

Hence

a contradiction. Consequently $(2)$ is verified.

To verify $(3)$: First we are to prove that $Fuz(\gimel) \bigcap Fuz(\daleth)$ is nonempty. If $\beth = [S(\Theta) ]_{\alpha} = [T(\Theta) ]_{\alpha}$, then we get $[\gimel(\beth)]_{\alpha} = [\gimel([\daleth(\Theta) ]_{\alpha})]_{\alpha} = [\daleth[\gimel(\Theta) ]_{\alpha}]_{\alpha} = [S(\beth)]_{\alpha}$ which implies that $\beth \in C_{F}(\daleth, \gimel)$. Hence the Pompeiu-Hausdorff weight assign to $[\daleth(\Theta) ]_{\alpha}$ and $[S(\beth) ]_{\alpha}$ is zero by $(2)$. Thus $\beth = [ \daleth (\beth)]_{\alpha} = [ \gimel (\beth)]_{\alpha}$ that is $\beth \in Fuz(\gimel) \bigcap Fuz(\daleth)$. As $C_{F } (\daleth, \gimel)$ is a singleton set, so that $Fuz(\gimel) \bigcap Fuz(\daleth)$ is also singleton.

Finally, we verify $(4)$: Let us assume that the set $Fuz(\gimel) \bigcap Fuz(\daleth)$ is complete. Now need to show that $Fuz(\gimel) \bigcap Fuz(\daleth)$ is singleton. On contrary, assume that there exists $\Theta, \Phi \in CB(\Psi)$ such that $\Theta, \Phi \in Fuz(T) \bigcap Fuz(\daleth)$ and $\Theta \neq \Phi$. By completeness of $Fuz(\daleth) \bigcap Fuz(\gimel)$, there exists an edge between $\Theta$ and $\Phi$. As the pair $(\daleth, \gimel)$ is generalized graphic fuzzy $F$-contractive mappings, so we get

where

Thus

a contradiction. Hence $\Theta = \Phi$. Conversely, if $Fuz(\daleth) \bigcap Fuz(\gimel)$ is a singleton, then since $E (G) \supseteq \Delta$, implies $Fuz(\daleth) \bigcap Fuz(\gimel)$ is a complete set.

Example 3.2. Let $\Psi = \{2n : n \in \{1, 2, 3, ..., m\} \} = V(G)$, $m \geq 1$, $E(G) = \{(i, j) \in \Psi \times \Psi : i < j \}$ and $\varphi: V(G) \times V(G) \rightarrow \mathbb{R^{+}}$ be defined by

Moreover, the Pompeiu-Hausdorff metric is stated by

The Pompeiu-Hausdorff weights $(for\; n = 4)$ assigned to $\Lambda, \Upsilon \in CB(\Psi)$ are exhibited in Figure $1$.

Now we define $\daleth, \gimel: CB(\Psi) \rightarrow I_{Fc}(\Psi)$ as follows.

For $\Theta \subseteq \{2, 4 \},$

For $\Theta \varsubsetneq \{2, 4\},$

Now

And for $\Theta = \{2\},$

For $\Theta \subseteq \{4, 6\},$

and for $\Theta\subsetneq \{2, 4, 6\},$

Now

Note that for all $\Phi \in CB(\Psi)$, $(\Phi, [\daleth(\Phi)]_{\alpha}) \subseteq E(G)$ and $(\Phi, [\gimel(\Phi)]_{\alpha}) \subseteq E(G).$

Take

and

Now for $\Theta_{1}, \Theta_{2}\in CB(\Psi)$ with $[\daleth(\Theta_{1})]_{\alpha }\neq \lbrack S(\Theta_{2})]_{\alpha }$, consider the following cases:

(1) For $\alpha = 1$ and $\Theta_{1}\subseteq \{2, 4\}$ and $\Theta_{2} = \{6\}$ with $(\Theta_{1}, \Theta_{2}) \in E(G)$, we have

$H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})-M(\Theta_{1}, \Theta_{2})}$

$\leq H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})-H([\daleth(\Theta_{2})]_{1}, [\gimel(\Theta_{2})]_{1})}$

$\leq H([\{2\}, \{2, 4\})e^{H([\{2\}, \{2, 4\})-H([\{2\}, \{2, 4, 6\})} \; < \frac{1}{2n}e^{\frac{1}{2n}-\frac{2n}{2n+1}}$

$< \frac{2n}{2n+1}e^{-\frac{8}{11}} \; = e^{-\frac{8}{11}}\frac{2n}{ 2n+1}$

$= e^{-\frac{8}{11}}H(\{2, 4\}, \{2, 4, 6\})$

$= e^{ - \frac{8}{11} }H([\daleth(\Theta_{2})]_{1}, [\gimel(\Theta_{2})]_{1})$

$\leq e^{ - \tau (M(\Theta_{1}, \Theta_{2})) } M(\Theta_{1}, \Theta_{2}).$

(2) For $\alpha = 1$ and $\Theta_{1} \subseteq \{ 2, 4\}$ and $\Theta_{2} \subseteq \{ 2, 4, 6\}$ with $(\Theta_{1}, \Theta_{2}) \in E(G)$ then consider

$H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - M(\Theta_{1}, \Theta_{2}) }$

$\leq H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - H([\daleth(\Theta_{2})]_{1}, [\gimel(\Theta_{2})]_{1}) }$

$\leq H([ \{2\}, \{2, 4\})e^{H([ \{2\}, \{2, 4\}) - H([ \{2, 4\}, \{2, 4, 6..., 2n \}) }$

$< \frac{1}{2n}e^{ \frac{1}{2n} -\frac{2n}{2n +1 } } \; < \frac{2n }{2n+1}e^{ - \frac{8}{11} } \; = e^{ - \frac{8}{11} }\frac{2n}{2n+1}$

$= e^{ - \frac{8}{11} }H(\{ 2, 4\}, \{ 2, 4, 6...2n\})$

$= e^{ - \frac{8}{11} }H([\daleth(\Theta_{2})]_{1}, [\gimel(\Theta_{2})]_{1})$

$\leq e^{ - \tau (M(\Theta_{1}, \Theta_{2})) } M(\Theta_{1}, \Theta_{2}).$

(3) For $\alpha = 1$ and $\Theta_{1} = \{ 6 \}$ and $\Theta_{2} \subseteq \{ 2, 4\}$ with $(\Theta_{1}, \Theta_{2}) \in E(G)$ then consider

$H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - M(\Theta_{1}, \Theta_{2}) }$

$\leq H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - H([\daleth(\Theta_{2})]_{1}, [\gimel(\Theta_{2})]_{1}) }$

$\leq H([ \{2, 4\}, \{2\})e^{H([ \{2, 4\}, \{2\}) - H([ \{2, 4 \}, \{2, 4, 6\}) } \; < \frac{1}{2n}e^{ \frac{1}{2n} -\frac{2n}{2n +1 } }$

$< \frac{2n}{2n+1}e^{ - \frac{8}{11} } \; = e^{ - \frac{8}{11} }H(\{ 2, 4\}, \{ 2, 4, 6\})$

$= e^{ - \frac{8}{11} }H([\daleth(\Theta_{1})]_{1}, [\gimel(\Theta_{1})]_{1})$

$\leq e^{ - \tau (M(\Theta_{1}, \Theta_{2})) } M(\Theta_{1}, \Theta_{2}).$

(4) For $\alpha = 1$ and $\Theta_{1} \subsetneq \{ 2, 4, 6\}$ and $U_{2} \subseteq \{ 2, 4\}$ with $(\Theta_{1}, \Theta_{2}) \in E(G)$ then consider

$H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - M(\Theta_{1}, \Theta_{2}) }$

$\leq H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - H([\daleth(\Theta_{2})]_{1}, [\gimel(\Theta_{2})]_{1}) }$

$\leq H([ \{2, 4\}, \{2\})e^{H([ \{2, 4\}, \{2\}) - H([ \{2, 4\}, \{2, 4, 6..., 2n\}) }$

$< \frac{1}{2n}e^{ \frac{1}{2n} -\frac{2n}{2n +1 } } \; < \frac{2n }{2n+1}e^{ - \frac{8}{11} }$

$= e^{ - \frac{8}{11} }\frac{2n}{2n+1} \; = e^{ - \frac{8}{11} }H(\{ 2, 4\}, \{ 2, 4, 6, ..., 2n\})$

$= e^{ - \frac{8}{11} }H([\daleth(\Theta_{1})]_{1}, [\gimel(\Theta_{1})]_{1})$

$\leq e^{ - \tau (M(\Theta_{1}, \Theta_{2})) } M(\Theta_{1}, \Theta_{2}).$

Hence, for all $\Theta_{1}, \Theta_{2} \in CB(\Psi)$ having edge between $\Theta_{1}$ and $\Theta_{2}$, (2.2) is satisfied. Thus all the conditions of Theorem 3.1 are satisfied. Furthermore $\{2\}$ is the common fuzzy fixed point of $\daleth$ and $\gimel$, and $Fuz (\daleth) \bigcap Fuz (\gimel)$ is complete.

The following Example will show that it is not necessary the given graph $\left(V(G), E(G) \right)$ will always be complete.

Example 3.3. Let $\Psi = \{2n : n \in \{1, 2, 3, ..., m\} \} = V(G)$, $m \geq 3$, $E(G) = \{(2, 2), (4, 4), (6, 6), ..., (2n, 2n), (2, 4), (2, 6), ..., (2, 2n) \}$ and $\varphi: V(G) \times V(G) \rightarrow \mathbb{R^{+}}$ and Pompeiu-Hausdorff metric are same as explained in Example 3. The Pompieu Hausdorff weights for $(n = 4)$ assigned to $\Lambda, \Upsilon \in CB(\Psi)$ are exhibited in the Figure 2.

Now we define $\daleth, \gimel: CB(\Psi) \rightarrow I_{Fc}(\Psi)$ as follows.

For $\Theta = \{2\},$

For $\Theta \neq \{2\},$

Now

Also for $\Theta = \{2\},$

and for $\Theta \neq\{2\},$

Now

Note that for all $\Phi \in CB(\Psi)$, $(\Phi, [\daleth(\Phi)]_{\alpha}) \subseteq E(G)$ and $(\Phi, [\gimel(\Phi)]_{\alpha}) \subseteq E(G).$

Take

and

Now, we consider the following cases:

(1) For $\alpha = 1$ and $\Theta_{1} = \{ 2\}$ and $\Theta_{2} \neq \{ 2\}$ with $(\Theta_{1}, \Theta_{2}) \in E(G)$ then consider

$H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - M(\Theta_{1}, \Theta_{2}) }$

$\leq H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - H([\daleth(\Theta_{2})]_{1}, [\gimel(\Theta_{2})]_{1}) }$

$< \frac{1}{2n}e^{ \frac{1}{2n} -\frac{2n}{2n +1 } } \; < \frac{2n }{2n+1}e^{ - \frac{2}{3} } \; = e^{ - \frac{2}{3} }\frac{2n}{2n+1}$

$= e^{ - \frac{2}{3} }H([\daleth(\Theta_{2})]_{1}, [\gimel(\Theta_{2})]_{1})$

$\leq e^{ - \tau (M(\Theta_{1}, \Theta_{2})) } M(\Theta_{1}, \Theta_{2}).$

(2) For $\alpha = 1$ and $\Theta_{1}\neq \{ 2\}$ and $\Theta_{2} = \{ 2\}$ with $(\Theta_{1}, \Theta_{2}) \in E(G)$ then consider $H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - M(\Theta_{1}, \Theta_{2}) }$

$\leq H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1})e^{H([\daleth(\Theta_{1})]_{1}, [\daleth(\Theta_{2})]_{1}) - H([\daleth(\Theta_{1})]_{1}, [\gimel(\Theta_{1})]_{1}) }$

$< \frac{1}{2n}e^{ \frac{1}{2n} -\frac{2n}{2n +1 } } \; < \frac{2n }{2n+1}e^{ - \frac{2}{3} }$

$= e^{ - \frac{2}{3} }\frac{2n}{2n+1} \; = e^{ - \frac{2}{3} }H([\daleth(\Theta_{2})]_{1}, [\gimel(\Theta_{2})]_{1})$

$\leq e^{ - \tau (M(\Theta_{1}, \Theta_{2})) } M(\Theta_{1}, \Theta_{2}).$

Hence for all $\Theta_{1}, \Theta_{2} \in CB(\Psi)$ having an edge between $\Theta_{1}$ and $\Theta_{2}$, (2.2) is fulfilled. Hence all the conditions of Theorem 3.1 are fulfilled. Furthermore, $\daleth$ and $\gimel$ have a common fuzzy fixed point and $Fuz(\daleth) \bigcap Fuz(\gimel)$ is complete in $CB(\Psi)$.

Theorem 3.4. Let $(\Psi, \varphi)$ be an $\varepsilon$-chainable complete metric space for $\varepsilon > 0$ and $\daleth, \gimel: CB(\Psi) \rightarrow I_{Fc}(\Psi)$ be fuzzy set valued-mappings. Assume that for all $\Lambda, \Upsilon \in CB(\Psi)$ and $\alpha \in (0, 1]$ such that $0 < H\left([ \daleth(\Lambda)]_{ \alpha}, [\daleth(\Upsilon) ]_{\alpha} \right) < \varepsilon$, there exists a mapping $\tau: \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$ with

for all $t > 0$ such that

holds, where $F \in \digamma$ and

Then $\daleth$ and $\gimel$ possess common fuzzy fixed point provided that $\daleth$ and $\gimel$ are weakly compatible.

Proof. We are given that $\daleth, \gimel: CB(\Psi) \rightarrow I_{Fc}(\Psi)$ be fuzzy set valued-mappings and $0 < H\left([ \daleth(\Lambda)]_{ \alpha}, [\daleth(\Upsilon) ]_{\alpha} \right) < \varepsilon$ this implies that $H(\Lambda, \Upsilon) < \varepsilon,$ now by Lemma 2.12 for each $\kappa \in \Lambda$, an element $\varrho \in \Upsilon$ can be chosen such that $\varphi(\kappa, \varrho) < \varepsilon$. Consider the graph $G$ with $V(G) = \Psi$ and $E(G) = \{ (\kappa, \varrho) \in \Psi \times \Psi : 0 < \varphi(\kappa, \varrho) < \varepsilon \}.$

Then $\varepsilon$-chainablity of $(\Psi, \varphi)$ means that $G$ is connected, the connectedness implies that for $\Lambda, \Upsilon \in CB(\Psi)$ we have $(\Lambda, \Upsilon) \subset E(G)$, therefore by hypothesis of the theorem we have

holds, where $F \in \digamma$ and

This shows that the pair $(\daleth, \gimel)$ is generalized graphic fuzzy $F$-contractive mappings. Also $G$ has $(P^{\ast})$ property. Indeed if $\{\Psi_{n}\}$ is in $CB(\Psi)$ with $\Psi_{n} \rightarrow \Psi$ as $n \rightarrow \infty$ and $(\Psi_{n}, \Psi_{n+1}) \subset E(G)$ for $n \in \mathbb{N}$ means that there is subsequence $\{ \Psi_{n_{k}} \}$ of $\{ \Psi_{n} \}$ such that $(\Psi_{n_{k}}, \Psi) \subset E(G)$ for $n \in \mathbb{N}$. By employing Theorem 3.1(3), $\daleth$ and $\gimel$ possess a common fuzzy fixed point.

Corollary 3.5. Let $(\Psi, \varphi)$ be a metric space equipped with a directed graph $G$ with $V(G) = \Psi$ and $E(G) \supseteq \Delta$. Assume that $\daleth :CB(\Psi) \rightarrow I_{Fc}(\Psi)$ holds the following:

1) For every $\Phi$ in $CB(\Psi)$, $(\Phi, [\daleth(\Phi)]_{\alpha}) \subset E(G).$

2) there exists a $\tau: \mathbb{R^{+}} \rightarrow \mathbb{ R^{+}}$ with

for all $t \geq 0$ there is an edge between $\Lambda$, $\Upsilon \in CB(\Psi)$ with $[\daleth(\Lambda)]_{\alpha} \neq [\daleth(\Upsilon)]_{\alpha}$ such that

$\tau (M (\Lambda, \Upsilon)) + F (H([\daleth(\Lambda) ]_{\alpha}, [ \daleth(\Upsilon) ]_{\alpha})) \leq F (M (\Lambda, \Upsilon))$

holds, where $F \in \digamma$

Then following statements satisfy.

(i) $Fuz(\daleth)$ is complete, then the Pompeiu-Hausdorff weight assigned to the $\Theta, \Phi \in Fuz(\daleth)$ is $0$.

(ii) If the weakly connected graph $G$ satisfies the property $P^{\ast}$, then $\daleth$ has fuzzy fixed point.

(iii) $Fuz(\daleth)$ is complete if and only if $Fuz(\daleth)$ is singleton.

Proof. Take $\gimel = I$ (Identity map) in (2.2), then the Corollary 3.5 follows from Theorem 3.1.

Remark 3.6. Next, we deduce some consequences and comparative results of our main Theorem in the frame work of both single-valued and set-valued mappings. First we present set-valued analogues of Theorem 3.1.

Corollary 3.7. Let $(\Psi, \varphi)$ be a metric space equipped with a directed graph $G$ with $V(G) = \Psi$, $E(G) \supseteq \Delta$ and the relation $R$ on $I_{Fc}(\Psi)$ is transitive. Suppose that $S, T :CB(\Psi) \rightarrow CB(\Psi)$ be set-valued mappings with $S(\Psi) \supseteq T(\Psi)$ and following statements are satisfied.

1) For any $U$ in $CB(\Psi)$, $(S(U), U) \subseteq E(G)$ and $(U, T(U)) \subseteq E(G).$

2) There is a function $\tau : \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$ with $\liminf_{\eta \to t^{+}}\tau(\eta) > 0$ for all $t \geq 0$ such that for $F \in \digamma$ there is an edge between $A$ and $B$ with $S(A) \neq S(B)$ such that

holds, where

Then the following statements hold

(i) $CP(S, T)\neq \emptyset$ given that $G$ is weakly connected which holds the property $(P^{\ast})$ and $T(\Psi)$ is a complete subspace of $CB(\Psi).$

(ii) If $CP(S, T)$ is complete, then the Hausdorff weight assigned to $S(U)$ and $S(V)$ is $0$ for some and for all $\Theta, \Phi \in CP(S, T).$

(iii) If $CP(S, T)$ is complete and $S$, $T$ are weakly compatible, then $Fix(S) \bigcap Fix (T)$ is singleton.

(iv) $Fix(S) \bigcap Fix(T)$ is complete if and only if $Fix (S) \bigcap Fix (T)$ is singleton.

Proof. Consider the mappings $\omega, \iota: \Psi \rightarrow (0, 1]$ and fuzzy set-valued maps $\daleth, \gimel: CB(\Psi) \rightarrow I_{Fc}(\Psi)$ defined by

and

Now for $\alpha(x) = \omega x \in (0, 1]$ for all $x \in \Psi,$ we have

and for $\alpha(y) = \iota y \in (0, 1]$ for all $y \in \Psi,$ we have

Consequently by using Theorem 3.1, the result follows.

Remark 3.8. Let $\mathfrak{f, g}: X\rightarrow X$ be self mappings, by $F_{\mathfrak{f}}, F_{\mathfrak{g}}$ we mean the set of all fixed points of $\mathfrak{f, g}$, respectively, while by $cp(\mathfrak{f, g})$ we mean the set of coincidence points of $\mathfrak{f}$ and $\mathfrak{g}$. With this setting, we obtain the following result for single-valued maps.

Corollary 3.9. Let $(\Psi, \varphi)$ be a metric space equipped with a directed graph $G$ with $V(G) = \Psi$, $E(G) \supseteq \Delta$ and the relation $R$ on $\Psi$ is transitive. Let $\mathfrak{f, g}:\Psi \rightarrow \Psi$ be self maps satisfy the following.

1) For any $\mu$ in $\Psi$, $(\mathfrak{f}(\mu), \mu) \in E(G)$ and $(\mu, \mathfrak{g}(\mu) \in E(G).$

2) There is a function $\tau : \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$ with $\liminf_{\eta \to t^{+}}\tau(\eta) > 0$ for all $t \geq 0$ such that for $F \in \digamma$ there is an edge between $a$ and $b$ with $\mathfrak{f}(a) \neq \mathfrak{f}(b)$ such that

holds, where

Then the following statements satisfy with $\mathfrak{f}(\Psi) \supseteq \mathfrak{g}(\Psi)$.

i) $cp(\mathfrak{f}, \mathfrak{g})\neq \emptyset$ given that $G$ is weakly connected which holds the property $(P)$ and $\mathfrak{f}(\Psi)$ is a complete subspace of $\Psi.$

ii) If $cp(\mathfrak{f}, \mathfrak{g})$ is complete, then $\mathfrak{f}(\mu) = \mathfrak{f}(\nu)$ for all $\mu, \nu \in cp(\mathfrak{f}, \mathfrak{g}).$

iii) If $cp(\mathfrak{f}, \mathfrak{g})$ is complete and $\mathfrak{f}$, $\mathfrak{g}$ are weakly compatible, then $F_{\mathfrak{f}} \bigcap F_{\mathfrak{g}}$ is singleton.

iv) $F_{\mathfrak{f}} \bigcap F_{\mathfrak{g}}$ is complete if and only if $F_{\mathfrak{f}} \bigcap F_{\mathfrak{g}}$ is singleton.

Proof. Consider the mappings $\omega, \iota: \Psi \rightarrow (0, 1]$ and a fuzzy set-valued maps $\daleth, \gimel: CB(\Psi) \rightarrow I_{Fc}{\Psi}$ defined by

and

Now for $\alpha(x) = \omega x \in (0, 1]$ for all $x \in \Psi,$ we have

and for $\alpha(y) = \iota y \in (0, 1]$ for all $y \in \Psi,$ we have

Clearly $\{\mathfrak{f}x\}, \{\mathfrak{g}y\} \in CB(\Psi)$ for all $x, y \in \Psi.$ Also, note that in this case $H([\daleth(\Lambda)]_{\alpha(x)}, [\gimel(\Upsilon)]_{\alpha(y)}) = \varphi (\mathfrak{f}x, \mathfrak{g}y)$ for all $x, y \in \Psi.$ Consequently, by using Theorem 3.1, the result follows.

4.   Application to nonlinear integral equations

Consider the following nonlinear integral equation:

for all $t \in [0, \infty),$ where $f_{1}, f_{2} \in L[a, \infty)$ are known such that $f_{1}(t) \geq f_{2}(t),$ and $m(t, s), k(t, s), \; g(s, s(x)), \; h(s, y(s))$ are real or complex valued function that are measurable both in $t$ and $s$ on $[0, \infty)$ and $\lambda, \mu$ are real or complex numbers. These functions satisfy the following.

$(C_{1})$ $\int_{a}^{\infty} \sup_{a \leq s }\vert{m(t, s)}\vert dt = M_{1} < \infty$;

$(C_{2})$ $\int_{a}^{\infty} \sup_{a \leq s }\vert{k(t, s)}\vert dt = M_{2} < \infty$;

$(C_{3})\; g(s, x(s)) \in L[a, \infty)$ for all $x \in L[a, \infty)$ and there exists $K_{1} > 0$ such that for all $s \in [a, \infty),$

$(C_{4})\; h(s, x(s)) \in L[a, \infty)$ for all $x \in L[a, \infty)$ and there exists $K_{2} > 0$ such that for all $s \in [a, \infty),$

The existence theorem regarding the solution of above nonlinear integral equation can be formulated as follows:

Theorem 4.1. With the assumption $(C_{1}) - (C_{4})$ if the following conditions are also satisfied.

$(a)\; \lambda \int_{a}^{\infty} k(t, s)h(s, \mu \int_{a}^{s}m(s, \tau)g(\tau, x(\tau))d\tau + f_{1}(s) - f_{2}(s))ds = 0.$

$(b)$ For $x \in L[a, \infty),$

$(c)$ For $\Gamma(t) \in L[a, \infty)$ there exists $\Theta(t) \in L[a, \infty)$ such that

then the Eq (4.1) has a unique solution in $L[a, \infty)$ for the pair of real or complex numbers $\lambda$ and $\mu$ with $\left\vert{\lambda}\right\vert K_{2}M_{2} < 1$ and $\frac{\left\vert{\mu}\right\vert K_{1}M_{1}}{1- \left\vert{\lambda}\right\vert K_{2}M_{2}} = \alpha < 1.$

Proof. For $x(t) \in L[a, \infty)$, we define

where $f_{1}, f_{2} \in L[0, \infty)$ are known and $I$ is the identity operator on $L[0, \infty).$ Then $\mathfrak{f, g},$ and $\mathfrak{h}$ are self maps on $L[a, \infty)$. Indeed, we have

and by using ($C_{1}$) and ($C_{3}$), we obtain

and hence $\mathfrak{f}x \in L[a, \infty).$ For mapping $\mathfrak{h},$ we apply the conditions $(C_{2})$ and $(C_{4})$ to obtain

Hence $\mathfrak{h} \in L[a, \infty)$ and so, $\mathfrak{g}$ is also the self map on $L[a, \infty).$ Now, by using $(C_{2})$ and ($C_{3})$, we have for all $x, y \in L[a, \infty)$ that

Similarly, by $(C_{2})$ and $(C_{4})$, we get

Hence we have

which implies that

From Eqs (4.2) and (4.3), we obtain

Now we prove that $\mathfrak{f}(L[a, \infty)) \subset \mathfrak{g}(L[a, \infty))$ so let $x(t) \in L[a, \infty)$ be arbitrary. Then we have

by assumption $(a)$ of the Theorem.

Now we prove that the pair $(\mathfrak{f}, \mathfrak{g})$ is weakly compitable. For this, we have

Now for $\mathfrak{f}x(t) = \mathfrak{g}x(t)$, we have

Therefore from (4.5), we get

This shows that maps $\mathfrak{f}$ and $\mathfrak{g}$ are weakly compitable. Thus all the conditions of Corollary 3.9 is satisfied. Consequently, there exists a unique $x^* \in L[a, \infty)$ such that $\mathfrak{f}x^* = \mathfrak{g}x^* = x^*$ that is the unique solution of Eq (4.1).

5.   Conclusions

The results of this paper broadened the scope of fuzzy fixed point theory and fixed point theory of multi valued mappings by incorporating the generalized fuzzy graphic $F$-contraction approaches. The ideas in this work, being discussed in the setting of metric spaces, are completely fundamental. Hence, they can be made better, when presented in the framework of generalized metric spaces such as $b$-metric spaces, $G$-metric spaces, $F$-metric spaces and some other pseudo-metric or quasi metric spaces. Also, the fuzzy set-valued map's component can be extended to $L$-fuzzy mappings, intuitionist fuzzy mappings, soft set-valued maps, and so on.

Acknowledgments

The authors would like to thank the anonymous referees for their careful reading of our manuscript and their many insightful comments and suggestions.

Conflict of interest

The authors declare that they have no competing interests concerning the publication of this article.