On Graphical Fuzzy Metric Spaces with Application to Fractional Differential Equations

Abstract

In this article, the authors introduced the concept of graphical fuzzy metric spaces which is a generalization of fuzzy metric spaces with the help of a relation. The authors discussed some topological structure, convergence criteria, and proved a Banach fixed-point result in graphical fuzzy metric space. As an application of obtained results, the authors find a solution of an integral equation and nonlinear fractional differential equations in the context of graphical fuzzy metric spaces. The authors provided some examples to illustrate the obtained results herein.

1. Introduction

The subject of graph theory is gaining popularity due to its applications. It is in the mainstream of mathematics and applied sciences, where it plays an important role. A graph $\Omega =\left(V\left(\Omega \right),E\left(\Omega \right)\right)$is consisting upon an ordered pair, where $V\left(\Omega \right)$is the set of all vertices of a graph$\Omega$ and $E\left(\Omega \right)$ represents the relation among the vertices of the graph $\Omega$, named as a set of edges. Further, $\left|V\right|$ represents the cardinality of vertex set V (known as order of graph $\Omega$), and $\left|E\right|$ represents the cardinality of edge set $E$ of graph $\Omega$. Main graph theory has two classes; one class deals with the undirected graphs if there is no direction or restriction for moving one vertex to the other if both vertices are connected, and other class deals with the directed graphs in which every edge has a particular direction. Edges in a directed graph are ordered pairs, while edges in an undirected graph are unordered pairs. Leonhard Eular (1736) solved the classic unsolved problem known as the Konigsberg Bridge Problem; he was named the “Father of Graph Theory.” Due to the graphical nature of this theory, there are several applications, including data flow diagrams, decision-making abilities, the capacity to demonstrate relationships between things, and the ability to easily adapt and modify the existing systems. Graph theory has a wide range of applications in chemical sciences [1,2], data science [3], and architecture [4].

On the other hand, Jachymski [5] used the graphical structure in fixed-point theory and proved several fixed-point results via graphical structure. Shukla et al. [6] used graphical structure in metric spaces and generalized several fixed-point results in graphical metric space with an application to integral equations. Chen et al. [7] introduced a graphical convex metric space and fixed-point results for set valued G-contraction mappings with application.

In 1965, fuzzy set theory initiated by L. A. Zadeh [8] introduced a method to deal with uncertainty and vagueness in our daily life. Kramosil and Michalek [9] used the notions of fuzzy sets with metric spaces established the notion of fuzzy metric spaces (FMSs) via continuous t-norms (CTNs), in which the notion of probabilistic metric spaces is generalized. George and Veeramani ([10,11]) modified the concept of FMSs and obtained a Hausdorff topology for this kind of FMSs. Moreover, the theory of fixed points on FMS was developed in many directions, generalizing the fuzzy metric space or using different type of operators [12,13,14,15,16,17,18,19].

In this manuscript, the authors will generalize the notion of FMSs. The main objectives of this manuscript are following:

(a)

to establish a novel notion of graphical FMSs;

(b)

to make a comparison between ordered FMS and graphical FMS;

(c)

to generalize several contraction mappings and fixed point results;

(d)

to find the existence of solution of integral equation via fixed-point theory;

(e)

to find the existence of a solution of fractional differential equations via fixed-point theory.

Some Basics from Graph Theory

In this section, the authors discuss some basic definitions from graph theory that are useful for readers to understand this work. For more details, the authors refer to the papers of Jachymski [5] and Shukla et al. [6].

Consider a non-empty set $\Xi$ and assume $\Xi \times \Xi$ to be the cartesian product; the authors denote the diagonal of $\Xi \times \Xi$ by $\Lambda .$ Suppose $\Omega$ to be a direct graph in which the edges are not parallel, such that the set of its vertices $V\left(\Omega \right)$ coincides with $\Xi$, and all loops are contained in the set of its edges $E\left(\Omega \right)$; then, call that $\Xi$ is endowed with graph $\Omega =\left(V\left(\Omega \right),E\left(\Omega \right)\right) \mathrm{if} \Lambda \subseteq E\left(\Omega \right)$. Here ${\Omega }^{-1}$ denote the graph’s $\Omega$ conversion and defined by:

$$V\left({\Omega }^{-1}\right)=V\left(\Omega \right) \mathrm{and} E\left({\Omega }^{-1}\right)=\left\{\left(ʋ,ɵ\right)\in \Xi \times \Xi :\left(ɵ,ʋ\right) \in E\left(\Omega \right)\right\}.$$

The undirected graph examined from $\Omega$ is denoted by $\widehat{\Omega }$ that contains all the edges of ${\Omega }^{-1}.$ The authors thus write:

$$V\left(\widehat{\Omega }\right)=V\left(\Omega \right) \mathrm{and} E\left(\widehat{\Omega }\right)=E\left(\Omega \right){{\displaystyle \cup }}^{ }E\left({\Omega }^{-1}\right).$$

Suppose $ʋ,ɵ$ are the vertices of a graph $\Omega$; then the path having length $l\in \mathbb{N}$ from $ʋ \mathrm{to} ɵ$ in $\Omega$ is a sequence ${\left\{{ʋ}_i\right\}}_{i=0}^l$ of $l+1$ vertices such that ${ʋ}_0 = ʋ, {ʋ}_l = ɵ \mathrm{and} \left({ʋ}_{i-1},{ʋ}_i\right)\in E\left(\Omega \right) \mathrm{for} i=1,2,\cdots ,l.$ If there exists a path between any two vertices, then the authors say that $\Omega$ is connected. If there exists a path from $ʋ \mathrm{to} ɵ \mathrm{and} ɵ \mathrm{to} ʋ$ between two vertices in a directed graph, then the authors say that vertices $ʋ \mathrm{and} ɵ$ are connected. If every one of its edges are dealt as being undirected and there is a path between every two vertices, then the authors say that $\Omega$ is weakly connected. Alternatively, if $\widehat{\Omega }$ is connected, then $\Omega$ is weakly connected.

The authors define a relation $P$ on $\Xi$ by ${\left(ʋPɵ\right)}_{\Omega }$ if and only if there is a direct path from $ʋ \mathrm{to} ɵ \mathrm{in} \Omega$ The authors say $\omega \in {\left(ʋPɵ\right)}_{\Omega }$ if $\omega$ is included in a direct path from $ʋ \mathrm{to} ɵ \mathrm{in} \Omega$. For $l\in \mathbb{N},$ the authors define:

$${[ʋ]}_{\Omega }^l=\left\{ɵ\in \Xi :\mathrm{there} \mathrm{is} \mathrm{a} \mathrm{directed} \mathrm{path} \mathrm{from} ʋ \mathrm{to} ɵ \mathrm{of} \mathrm{length} l\right\}.$$

A sequence ${\{ʋ\}}_n \mathrm{in} \Xi$ is known as $\Omega \text{-}\mathrm{termwise}$ connected if it satisfies ${\left({ʋ}_{n}P{ʋ}_{n+1}\right)}_{\Omega }$ for all $n\in \mathbb{N}$. A connected component of an undirected graph is a sub-graph that has any two vertices connected to each other by pathways and is not connected to anymore vertices in the super-graph.

In this paper, the authors allot a fuzzy weight ${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)$ to a degree of nearness between $ʋ \mathrm{and} ɵ$ at $t$, where $t$ is interpreted as a time. The authors assume that the graphs under study are directed with non-empty sets of vertices and edges throughout this paper.

2. New Notions with Examples

In this section, several novel notions are defined with some non-trivial examples.

Definition 1.

Suppose$\Xi$be a non-empty set endowed with a graph$\Omega$, $\ast$is a continuous t-norm, and${Ʋ}_{\Omega }$is a fuzzy set on$\Xi \times \Xi \times \left(0,+\infty \right)$; the mapping${Ʋ}_{\Omega }$is known as a graphical fuzzy metric (FM), if

 (a) 

${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)>0;$

 (b) 

${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)=1$if and only if$ʋ = ɵ;$

 (c) 

${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)={Ʋ}_{\Omega }\left(ɵ,ʋ,t\right);$

 (d) 

$if {\left(ʋPɵ\right)}_{\Omega } such that \omega \in {\left(ʋPɵ\right)}_{\Omega } implies {Ʋ}_{\Omega }\left(ʋ,ɵ,t+s\right)\ge {Ʋ}_{\Omega }\left(ʋ,\omega ,t\right)\ast {Ʋ}_{\Omega }\left(\omega ,ɵ,s\right);$

 (e) 

${Ʋ}_{\Omega }\left(ʋ,ɵ,·\right):\left(0,+\infty \right)\to \left[0,1\right]$is continuous,

for all$ʋ,ɵ \in \Xi$and$t,s>0$. Then$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$is known as a graphical FMS.

Example 1.

Every FMS is a graphical FMS with a graph$\Omega ,$where$V\left(\Omega \right)=\Xi$and$E\left(\Omega \right)=\Xi \times \Xi$at a time$t.$

Definition 2.

Let$\left(\Xi ,⪯\right)$be a partially ordered set,$\ast$is a continuous t-norm, and${Ʋ}_{⪯}$is a fuzzy set on$\Xi \times \Xi \times \left(0,+\infty \right);$the mapping${Ʋ}_{⪯}$is known as ordered fuzzy metric, if

 (i) 

${Ʋ}_{\prec }\left(ʋ,ɵ,t\right)>0;$

 (ii) 

${Ʋ}_{⪯}\left(ʋ,ɵ,t\right)=1$if and only if$ʋ = ɵ;$

 (iii) 

${Ʋ}_{\prec }\left(ʋ,ɵ,t\right)= {Ʋ}_{\prec }\left(ɵ,ʋ,t\right);$

 (iv) 

${Ʋ}_{\prec }\left(ʋ,ɵ,t+s\right) \ge {Ʋ}_{\prec }\left(ʋ,\omega ,t\right)\ast {Ʋ}_{\prec }\left(\omega ,ɵ,s\right);$

 (v) 

${Ʋ}_{⪯}\left(ʋ,ɵ,·\right):\left(0,+\infty \right)\to \left[0,1\right]$is continuous.

For all $ʋ,ɵ,\omega \in \Xi$ and $t,s>0$ . Then $\left(\Xi ,{ʋ}_{⪯},\ast \right)$ is known as ordered FMS.

Example 2.

Let $\left(\Xi ,{ʋ}_{⪯},\ast \right)$ be an ordered FMS and ${\Omega }_I$ be a graph defined by $V\left({\Omega }_I\right)=\Xi$ and $E\left({\Omega }_I\right)=\left\{\left(ʋ,ɵ\right)\in \Xi \times \Xi :Ʋ ⪯ ɵ\right\}$ at a time $t.$ Then $\left(\Xi ,{ʋ}_{{\Omega }_I},\ast \right)$ is a graphical FMS, where ${Ʋ}_{{\Omega }_I}\left(ʋ,ɵ,t\right)= {Ʋ}_{⪯}\left(ʋ,ɵ,t\right)$ for all $ʋ,ɵ \in \Xi$ and $t>0.$ Therefore, each ordered FMS is a graphical FMS with the given graph ${\Omega }_I,$ Where ${\Omega }_I$ is an induced graph by the partial order $⪯$.

Example 3.

Let $\Xi =\mathbb{N}$ define CTN by $a\ast b=\mathit{min}\left\{a,b\right\}$ and partial order by$⪯:=\left\{\left(ʋ,ɵ\right)\in \mathbb{N}\times \mathbb{N}:ɵ is divisible by ʋ\right\}$ . Define ${Ʋ}_{⪯}$ by:

$${Ʋ}_{⪯}\left(ʋ,ɵ,t\right)=\{\begin{array}{c}1 \mathrm{if} ʋ = ɵ\hfill \\ \frac{t}{t+ʋɵ} \mathrm{if} \mathrm{otherwise}\end{array}$$

Then ${Ʋ}_{⪯}$ is an ordered FM and $\left(\Xi ,{Ʋ}_{⪯},\ast \right)$ is an ordered FMS. Then the graphical FM ${Ʋ}_{{\Omega }_I}\left(ʋ,ɵ,t\right)= {Ʋ}_{⪯}\left(ʋ,ɵ,t\right)$ is not a fuzzy metric. Indeed, for$t = s = 1,$ $ʋ = 5, ɵ = 1$ and$\omega =3$ , the authors obtain:

$${Ʋ}_{⪯}\left(ʋ,\omega ,t+s\right)=\frac{t+s}{t+s+ʋ\omega }=\frac{2}{17}.$$

On the other hand:

$${Ʋ}_{⪯}\left(ʋ,ɵ,t\right)\ast {Ʋ}_{⪯}\left(ɵ,\omega ,s\right)=\min \left\{\frac{t}{t+ʋɵ},\frac{s}{s+ɵ\omega }\right\}=\min \left\{\frac{1}{6},\frac{1}{4}\right\}=\frac{1}{6}.$$

The authors have ${Ʋ}_{⪯}\left(ʋ,\omega ,t+s\right)=\frac{2}{17}<\frac{1}{6}={Ʋ}_{⪯}\left(ʋ,ɵ,t\right)\ast {Ʋ}_{⪯}\left(ɵ,\omega ,s\right).$

Example 4.

Suppose an undirected graph$\Omega \left(V,E\right)$with$V\left(\Omega \right)=\Xi .$Let${\Omega }_j, j=1,2,\cdots ,n$be the components of$\Omega$that are connected such that$V\left(\Omega \right)={{\displaystyle \cup }}_{j=1}^{n}V\left({\Omega }_j\right),$$E\left(\Omega \right)=\Lambda {{\displaystyle \cup }}^{ }\left\{{{\displaystyle \cup }}_{j=1}^{n}E\left(V_j\right)\right\}$at a time$t.$Let$l_{ʋɵ}$be the length of the shortest from$ʋ$to$ɵ$. Now define${Ʋ}_{\Omega }:\Xi \times \Xi \times \left(0,+\infty \right)\to \left[0,1\right]$with$a\ast b=a·b$by

$${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)=\{\begin{array}{l}1 \mathrm{if} ʋ = ɵ or ʋ\in V_i, ɵ\in V_j, i,j\in \left\{1,2,\cdots ,n\right\}, i\ne j;\\ \frac{t}{t+l_{ʋɵ}} \mathrm{if} ʋ,ɵ\in V_i, i\in \left\{1,2,\cdots ,n\right\}.\end{array}$$

Then,${Ʋ}_{\Omega }$is a graphical FM on$\Xi$and$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$is a graphical FMS.

Example 5.

Suppose$\Xi =\left[0,1\right]$and the graph$\Omega \left(V,E\right)$described by$V\left(\Omega \right)=\Xi$and$E\left(\Omega \right)=\Lambda {{\displaystyle \cup }}^{ }\left\{\left(ʋ,ɵ\right)\in \Xi \times \Xi :ʋ,ɵ\in \left(0,1\right], ʋ\le ɵ\right\}$at a time$t.$Let$a\ast b=\mathit{min}\left\{a,b\right\}$and describe a mapping${Ʋ}_{\Omega }:\Xi \times \Xi \times \left(0,+\infty \right)\to \left[0,1\right]$by

$${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)=\{\begin{array}{c}1 \mathrm{if} ʋ = ɵ;\hfill \\ \frac{t}{t+ʋɵ} \mathrm{if} ʋ,ɵ\in \left(0,1\right], ʋ \ne ɵ;\hfill \\ \frac{t}{t+\left(ʋ+ɵ\right)} \mathrm{if\; otherwise}.\hfill \end{array}$$

Then,${Ʋ}_{\Omega }$is a graphical FM on$\Xi$and$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$is a graphical FMS. Indeed,${Ʋ}_{\Omega }$is not an FM on$\Xi .$Let$ʋ =\frac{1}{2}, ɵ =\frac{1}{10}, \omega =1$and$t=s=1,$therefore deducing that:

$${Ʋ}_{\Omega }\left(ʋ,\omega ,t+s\right)=\frac{4}{5}<\frac{20}{21}={Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)\ast {Ʋ}_{\Omega }\left(ɵ,\omega ,s\right).$$

As seen in the above examples, every FMS is a graphical FMS, but the converse is not true.

Definition 3.

Let$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$be a graphical FMS. An open ball$B_{\Omega }\left(ʋ,r,t\right)$with center$ʋ \in \Xi$, radius$r>0$and$t>0$is defined by

$$B_{\Omega }\left(ʋ,r,t\right)=\left\{ɵ \in \Xi :{\left(ʋPɵ\right)}_{\Omega }, {Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)>1-r\right\}.$$

Since$\Lambda \subseteq E\left(\Omega \right),$then the authors have$ʋ\in B_{\Omega }\left(ʋ,r,t\right)\ne \varnothing$for all$ʋ \in \Xi , r>0 and t>0.$The collection is known as the neighbourhood system for the topology ${\tau }_{\Omega }$and defined by$\beta =\left\{B_{\Omega }\left(ʋ,r,t\right):ʋ \in \Xi ,r>0,t>0\right\}$on$\Xi$induced by the graphical FM${Ʋ}_{\Omega }.$Openly,$U$is a subset of$\Xi$that is known as open for each$ʋ \in U$if there exists an$r>0$such that$B_{\Omega }\left(ʋ,r,t\right)\subset U$for$t>0.$Indeed, if complement$\Xi \backslash C$is open of a subset$C$of$\Xi$then it is known as closed.

Lemma 1.

Every open ball in$\Xi$is an open set.

Proof. 

Suppose an open ball $B_{\Omega }\left(ʋ,r,t\right)$ with center $ʋ$, radius $r,$ and time $t.$ Assume $ɵ = B_{\Omega }\left(ʋ,r,t\right),$ then ${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)>1-r.$ Since ${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)>1-r$, there exist $t_0\in \left(0,t\right)$ such that ${Ʋ}_{\Omega }\left(ʋ,ɵ,t_0\right)>1-r$. Put $r_0={Ʋ}_{\Omega }\left(ʋ,ɵ,t_0\right).$ As $r_0>1-r,$ there exists $u\in \left(0,1\right)$ such that $r_0>1-u>1-r.$ At this instant for specified $r_0$ and $u$ such that $r_0>1-u$, there exists $r_1,r_2\in \left(0,1\right)$ such that $r_0\ast r_1>1-u.$ Set $r_3=\max \left\{r_1,r_2\right\}$ and let the open ball $B_{\Omega }\left(ɵ,1-r_3,t-t_0\right)$. The authors claim that $B_{\Omega }\left(ɵ,1-r_3,t-t_0\right)\subset B_{\Omega }\left(ʋ,r,t\right)$. By definition, there is ${\left(ʋPɵ\right)}_{\Omega }$ and ${\left(ɵP\omega \right)}_{\Omega }$ such that ${\left(ʋP\omega \right)}_{\Omega }.$ At this instant, suppose $\omega \in B_{\Omega }\left(ɵ,1-r_3,t-t_0\right).$ Then ${Ʋ}_{\Omega }\left(ɵ,\omega ,t-t_0\right)>r_3.$ Therefore,

$${Ʋ}_{\Omega }\left(ʋ,\omega ,t\right)\ge {Ʋ}_{\Omega }\left(ʋ,ɵ,t_0\right)\ast {Ʋ}_{\Omega }\left(ɵ,\omega ,t-t_0\right)\ge r_0\ast r_3\ge r_0\ast r_1\ge 1-u>1-r.$$

Therefore, $\omega \in B_{\Omega }\left(ʋ,r,t\right)$ and $B_{\Omega }\left(ɵ,1-r_3,t-t_0\right)\subset B_{\Omega }\left(ʋ,r,t\right).$ □

Next, in graphical FMS, the authors define the terms convergence, Cauchy sequence, and completeness.

Definition 4.

Assume$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$to be a graphical FMS and a sequence$\left\{{ʋ}_n\right\}$in$\Xi .$Then$\left\{{ʋ}_n\right\}$is known as convergent and converges to$ʋ\in \Xi$if, for known$\epsilon >0,$there exists$n_0\in \mathbb{N}$such that${Ʋ}_{\Omega }\left({ʋ}_n,ʋ,t\right)>1-\epsilon$for all$n>n_0.$It is obvious that the sequence$\left\{{ʋ}_n\right\}$is convergent and converges to$ʋ,$if and only if$\underset{n\to +\infty }{\lim }{Ʋ}_{\Omega }\left({ʋ}_n,ʋ,t\right)=1.$

Remark 1.

In a graphical FMS, the limit of a sequence may not be unique. In order to examine this, suppose$\Xi =\left\{\frac{1}{2^n}{{\displaystyle \cup }}^{ }\left\{0\right\}:n\in \mathbb{N}\right\}$, and let$\Omega$be a graph with$V\left(\Omega \right)=\Xi$and$E\left(\Omega \right)=\left\{\left(ʋ,ɵ\right)\in \Xi \times \Xi :ɵ\le ʋ\right\}$at a time$t.$Let$a\ast b=a·b$and describe a mapping${Ʋ}_{\Omega }:\Xi \times \Xi \times \left(0,+\infty \right)\to \left[0,1\right]$by:

$${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)=\{\begin{array}{c}1 \mathrm{if}ʋ=ɵ;\\ \frac{t}{t+ʋɵ} \mathrm{if} ʋ,ɵ\in \Xi \backslash \left\{0\right\}, ʋ\ne ɵ;\\ \frac{1}{2} \mathrm{if\; otherwise}.\end{array}$$

Then,${Ʋ}_{\Omega }$is a graphical FM on$\Xi$and$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$is a graphical FMS. Suppose$\left\{{ʋ}_n\right\}$to be a sequence in$\Xi ,$where${ʋ}_n=\frac{1}{2^n}$for each$n\in \mathbb{N},$and then for any fixed$k\in \mathbb{N},$ we deduce:

$${Ʋ}_{\Omega }\left(\frac{1}{2^n}, \frac{1}{2^k},t\right)=\frac{t}{t+\frac{1}{2^{n+k}}}\to 1 \mathrm{as} n\to +\infty .$$

Therefore, the sequence $\left\{\frac{1}{2^n}\right\}$ converges to $\frac{1}{2^k}$ for every fixed $k\in \mathbb{N}.$

Remark 2.

Suppose$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$to be a graphical FMS in the above remark, and$\frac{1}{2}$is a limit of the sequence$\left\{\frac{1}{2^n}\right\}$, but for any$k\in \mathbb{N}, k>1$the authors have:

$$\underset{n\to +\infty }{\lim }{Ʋ}_{\Omega }\left(\frac{1}{2^n}, \frac{1}{2^k},t\right)=1\ne {Ʋ}_{\Omega }\left(\frac{1}{2}, \frac{1}{2^k},t\right).$$

That is, a graphical FM need not be continuous.

Lemma 2.

Suppose$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$to be a graphical FMS with topology${\tau }_{\Omega }$induced by graphical FM. Then${\tau }_{\Omega }$is a$T_1$but not$T_2$generally, i.e., Hausdorff.

Proof. 

The authors will demonstrate that for each $ʋ \in \Xi$, the set $\{ʋ\}$ that is singleton is a closed subset of $\Xi ,$ i.e., the set $\Xi \backslash \{ʋ\}$ is open in $\Xi .$ Assume $ɵ \in \Xi \backslash \{ʋ\}$, then $ɵ \ne ʋ,$ i.e., ${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)>0.$ Now, let $r=\frac{{Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)}{2}>0.$ Then the authors observe that $ʋ\notin B_{\Omega }\left(ɵ,r,t\right),$ otherwise $\frac{{Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)}{2}=r>{Ʋ}_{\Omega }\left(ʋ,ɵ,t\right),$ which is a contradiction. It shows that $B_{\Omega }\left(ɵ,r,t\right)\subset \Xi \backslash \{ʋ\}$, and so $\Xi \backslash \{ʋ\}$ is open. □

Definition 5.

Let$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$ be a graphical FMS and a sequence$\left\{{ʋ}_n\right\}$in$\Xi .$Then$\left\{{ʋ}_n\right\}$is known as a Cauchy sequence if, specified$\epsilon >0,$there exists$n_0\in \mathbb{N}$such that${Ʋ}_{\Omega }\left({ʋ}_n,{ʋ}_m,t\right)>1-\epsilon$for all $n,m>n_0.$It is obvious that the sequence$\left\{{ʋ}_n\right\}$is a Cauchy sequence if and only if$\underset{n,m\to ++\infty }{\lim }{Ʋ}_{\Omega }\left({ʋ}_n,{ʋ}_m,t\right)=1.$

Definition 6.

A graphical FMS$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$is known as complete if each Cauchy sequence in$\Xi$converges in$\Xi .$Suppose${\Omega }^{\prime }$is a different graph from$\Omega$such that$V\left({\Omega }^{\prime }\right)=\Xi ,$if${\Omega }^{\prime }\text{-}termwise$connected the sequence in$\Xi$converges in$\Xi$, then$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$is known as${\Omega }^{\prime }\text{-}complete$.

3. Fixed-Point Theorems

In this section, the authors discuss the existence of fixed-point theorems under some circumstances.

Definition 7.

Suppose$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$to be a graphical FMS, a mapping$ϸ:\Xi \to \Xi$and${\Omega }^{\prime }$to be a sub-graph of$\Omega$such that$\Lambda \subseteq E\left({\Omega }^{\prime }\right).$Then,$ϸ$is known as$\left(\Omega ,{\Omega }^{\prime }\right)-$fuzzy graphical contraction ($\left(\Omega ,{\Omega }^{\prime }\right)-FGC)$on$\Xi$if satisfying the following conditions:

 (FC1) 

$\left(ʋ,ɵ\right)\in E({\Omega }^{\prime })$implies$\left(ϸʋ,ϸɵ\right)\in E({\Omega }^{\prime })$at a time$t,$that is, ϸ preserves edges in$E({\Omega }^{\prime }).$

 (FC2) 

There exists$\varpi \in \left(0,1\right)$such that:

$${Ʋ}_{\Omega }\left(ϸʋ,ϸɵ,\varpi t\right)\ge {Ʋ}_{\Omega }\left(ʋ,ɵ,t\right), \mathrm{for} \mathrm{all} ʋ,ɵ\in \Xi \mathrm{with} \left(ʋ,ɵ\right)\in E\left({\Omega }^{\prime }\right) \mathrm{at} \mathrm{time} t.$$

A sequence$\left\{{ʋ}_n\right\}$ with initial value${ʋ}_0\in \Xi$is known as the$ϸ\text{-}$Picard sequence if${ʋ}_n=ϸ{ʋ}_{n-1}$for all$n\in \mathbb{N}.$In order to continue the discussion, the authors suppose that${\Omega }^{\prime }$is a sub-graph of$\Omega$such that$\Lambda \subseteq E\left({\Omega }^{\prime }\right).$

Theorem 1.

Suppose a ${\Omega }^{\prime }\text{-}$ complete graphical FMS$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$ , and let $ϸ:\Xi \to \Xi$ be a$\left(\Omega ,{\Omega }^{\prime }\right)-FGC$ . Assume that the below circumstances fulfill:

 (A) 

$ϸ{ʋ}_0\in {\left[{ʋ}_0\right]}_{{\Omega }^{\prime }}^l$if there exists${ʋ}_0\in \Xi$for some$l\in \mathbb{N}$

 (B) 

If sequence$\left\{{ʋ}_n\right\}$, that is,${\Omega }^{\prime }\text{-}$termwise connected$ϸ\text{-}$Picard converges in$\Xi ,$then for$\left\{{ʋ}_n\right\}$there exist a limit$\omega \in \Xi$and$n_0\in \mathbb{N}$such that$\left({ʋ}_n,\omega \right)\in E\left({\Omega }^{\prime }\right)$or$\left(\omega ,{ʋ}_n\right)\in E\left({\Omega }^{\prime }\right)$at a time$t$for all$n\ge n_0.$

Then, there exists ${ʋ}^{\ast }\in \Xi$ such that a sequence $\left\{{ʋ}_n\right\}$ is a $ϸ\text{-}$ Picard is ${\Omega }^{\prime }\text{-}$ termwise connected with initial value ${ʋ}_0\in \Xi$ and converges to both ${ʋ}^{\ast }$ and $ϸ{ʋ}^{\ast }.$

Proof. 

Suppose ${ʋ}_0 \in \Xi$ to be the extent that $ϸ{ʋ}_0\in {\left[{ʋ}_0\right]}_{{\Omega }^{\prime }}^l$ for some and the $ϸ-$ Picard sequence $\left\{{ʋ}_n\right\}$ starting from ${ʋ}_0$. After that, there exists a path ${\left[{ɵ}_i\right]}_{i=0}^l$ such that ${ʋ}_0 = {ɵ}_0, ϸ{ʋ}_0 = ϸ{ɵ}_l$ and $\left({ɵ}_{i-1},{ɵ}_i\right)\in E\left({\Omega }^{\prime }\right)$ at a time for $i=1,2,\cdots ,l.$ Since $ϸ$ is a $\left(\Omega ,{\Omega }^{\prime }\right)-FGC$, by (FC1), $\left(ϸ{ɵ}_{i-1},ϸ{ɵ}_i\right)\in E\left({\Omega }^{\prime }\right)$ at a time $t$ for $i=1,2,\cdots ,l.$ Therefore, ${\left[ϸ{ɵ}_i\right]}_{i=0}^l$ is a path from $ϸ{ɵ}_0=ϸ{ʋ}_0={ʋ}_1$ to $ϸ{ɵ}_l={ϸ}^2{ʋ}_0={ʋ}_2$ of length $l$ and so ${ʋ}_2\in {\left[{ʋ}_1\right]}_{{\Omega }^{\prime }}^l.$ Continuing this process, we obtain that ${\left[{ϸ}^n{ɵ}_i\right]}_{i=0}^l$ is a path from ${ϸ}^n{ɵ}_0={ϸ}^n{ʋ}_0={ʋ}_n$ to ${ϸ}^n{ɵ}_l={ϸ}^nϸ{ʋ}_0={ʋ}_{n+1}$ of length l and so, ${ʋ}_{n+1}\in {\left[{ʋ}_n\right]}_{{\Omega }^{\prime }}^l$ for all $n\in \mathbb{N}.$ Thus $\left\{{ʋ}_n\right\}$ is a ${\Omega }^{\prime }\text{-}$ termwise connected sequence. Since $\left({ϸ}^n{ɵ}_{i-1},{ϸ}^n{ɵ}_i\right)\in E\left({\Omega }^{\prime }\right)$ at a time $t$ for $i=1,2,\cdots ,l$ and $n\in \mathbb{N},$ by (FC2), the authors obtain:

$${ʋ}_{\Omega }\left({ϸ}^n{ɵ}_{i-1},{ϸ}^n{ɵ}_i,\varpi t\right)\ge {ʋ}_{\Omega }\left({ϸ}^{n-1}{ɵ}_{i-1},{ϸ}^{n-1}{ɵ}_i,t\right)\ge \cdots \ge {ʋ}_{\Omega }\left({ɵ}_{i-1},{ɵ}_i,\frac{t}{{\varpi }^n}\right).$$

(1)

Since ${\Omega }^{\prime }$ is a sub-graph of $\Omega$ and a ${\Omega }^{\prime }\text{-}$ termwise connected sequence that is $\left\{{ʋ}_n\right\}$, for any $n\in \mathbb{N},$ the authors achieve by utilizing (d) of Definition 1, and (1), whereinthe authors have:

$$\begin{gathered} {Ʋ}_{\Omega }\left({ʋ}_n,{ʋ}_{n+1},t\right)={Ʋ}_{\Omega }\left({ϸ}^n{ʋ}_0, {ϸ}^{n+1}{ʋ}_0,t\right)={Ʋ}_{\Omega }\left({ϸ}^n{ɵ}_0, {ϸ}^n{ɵ}_l,t\right) \\ \ge {\ast }_{i=1}^l\left[{Ʋ}_{\Omega }\left({ϸ}^n{ɵ}_{i-1},{ϸ}^n{ɵ}_i,\varpi t\right)\right]\ge {\ast }_{i=1}^l\left[{Ʋ}_{\Omega }\left({ɵ}_{i-1},{ɵ}_i,\frac{t}{{\varpi }^n}\right)\right]. \end{gathered}$$

Therefore, the sequence $\left\{{ʋ}_n\right\}$ is ${\Omega }^{\prime }\text{-}$termwise connected for $n,m\in \mathbb{N}$ with $m>n.$ We examined that

$${Ʋ}_{\Omega }\left({ʋ}_n,{ʋ}_m,t\right)\ge {\ast }_{i=n}^{m-1}\left[{Ʋ}_{\Omega }\left({ʋ}_i,{ʋ}_{i+1},t\right)\right]\ge {\ast }_{i=1}^l\left[{Ʋ}_{\Omega }\left({ɵ}_{i-1},{ɵ}_i,\frac{t}{{\varpi }^n}\right)\right].$$

Since $\varpi \in \left(0,1\right),$ the authors obtain $\underset{n,m\to ++\infty }{\lim }{Ʋ}_{\Omega }\left({ʋ}_n,{ʋ}_m,t\right)=1.$ Therefore, $\left\{{ʋ}_n\right\}$ is a Cauchy sequence in $\Xi .$ Utilizing the fact that $\Xi$ have ${\Omega }^{\prime }\text{-}$ completeness, the sequence $\left\{{ʋ}_n\right\}$ converges in $\Xi$ and utilizing circumstance (B), there exist ${ʋ}^{\ast }\in \Xi , n_0\in \mathbb{N}$ such that $\left({ʋ}_n,{ʋ}^{\ast }\right)\in E\left({\Omega }^{\prime }\right)$ or $\left({ʋ}^{\ast },{ʋ}_n\right)\in E\left({\Omega }^{\prime }\right)$ at a time $t,$ for all $n>n_0$ and $\underset{n\to ++\infty }{\lim }{Ʋ}_{\Omega }\left({ʋ}_n,{ʋ}^{\ast },t\right)=1.$ Therefore $\left\{{ʋ}_n\right\}$ converges to ${ʋ}^{\ast }\in \Xi ,$ At this instant, if $\left({ʋ}_n,{ʋ}^{\ast }\right)\in E\left({\Omega }^{\prime }\right)$ at a time $t,$ for all $n>n_0,$ applying (FC2), the authors deduce:

$${Ʋ}_{\Omega }\left({ʋ}_{n+1},ϸ{ʋ}^{\ast },t\right)={Ʋ}_{\Omega }\left(ϸ{ʋ}_n,ϸ{ʋ}^{\ast },t\right)\ge {Ʋ}_{\Omega }\left({ʋ}_n,{ʋ}^{\ast },\frac{t}{\varpi }\right)$$

for all $n>n_0$. Since $\underset{n\to ++\infty }{\lim }{Ʋ}_{\Omega }\left({ʋ}_n,{ʋ}^{\ast },t\right)=1,$ the authors deduce $\underset{n\to ++\infty }{\lim }{Ʋ}_{\Omega }\left({ʋ}_{n+1},ϸ{ʋ}^{\ast },t\right)=1.$ A similar result holds if $\left({ʋ}^{\ast },{ʋ}_n\right)\in E\left({\Omega }^{\prime }\right)$ at a time t for all $n>n_0$, and this gives us the sequence $\left\{{ʋ}_n\right\}$ that converges to ${ʋ}^{\ast }$ and $ϸ{ʋ}^{\ast }.$ □

Remark 3.

The above finding explains the convergence of a Picard sequence generated by a$\left(\Omega ,{\Omega }^{\prime }\right)-FGC$on${\Omega }^{\prime }\text{-}$complete graphical FMS. This theorem should not be regarded as an existence theorem, as seen in the below example in the context of${\Omega }^{\prime }\text{-}$complete graphical FMS.

Example 6.

Assume$\Xi =\left\{\frac{1}{2^n}{{\displaystyle \cup }}^{ }\left\{0\right\}:n\in \mathbb{N}\right\}$and a graph$\Omega ={\Omega }^{\prime }$with$V\left(\Omega \right)=\Xi$and$E\left(\Omega \right)=\Lambda {{\displaystyle \cup }}^{ }\left\{\left(ʋ,ɵ\right)\in \Xi \times \Xi :ɵ \le ʋ\right\}$at a time$t.$Suppose$a\ast b=a·b$and describes a mapping${Ʋ}_{\Omega }:\Xi \times \Xi \times \left(0,+\infty \right)\to \left[0,1\right]$by:

$${Ʋ}_{\Omega }\left(ʋ,ɵ,t\right)=\{\begin{array}{c}1 \mathrm{if} ʋ = ɵ;\\ \frac{t}{t+ʋɵ} \mathrm{if} ʋ,ɵ\in \Xi \backslash \left\{0\right\}, ʋ \ne ɵ;\\ \frac{1}{2} \mathrm{if\; otherwise}.\end{array}$$

Then,${Ʋ}_{\Omega }$is a graphical FM on$\Xi$and$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$is a$\Omega -$complete graphical FMS. Define a mapping$ϸ:\Xi \to \Xi$by:

$$ϸʋ=\{\begin{array}{c}\frac{ʋ}{6}, \mathrm{if} ʋ\in \left\{\frac{1}{2^n}:n\in \mathbb{N}\right\}\\ \frac{1}{6}, \mathrm{if} ʋ=0.\end{array}$$

Observe that$ϸ$is an$FGC$where$\varpi =\frac{1}{2}$and$t=1.$For each$ʋ\in \left\{\frac{1}{2^n}:n\in \mathbb{N}\right\},$we have$\left(ʋ,ϸʋ\right)\in E\left(\Omega \right)$at$t,$that is$ϸʋ\in {[ʋ]}_{{\Omega }^{\prime }}^1$. Also, observe that all conditions of Theorem 1 and (B) hold, but$ϸ$has no fixed point in$\Xi .$

Definition 8.

Let$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$be a graphical FMS and$ϸ:\Xi \to \Xi$be a map. Then, the five-tuple$\left(\Xi ,{Ʋ}_{\Omega },\ast ,\Omega ,ϸ\right)$is known to have property (2) if:

Whenever a${\Omega }^{\prime }\text{-}$termwise connected$ϸ\text{-}$Picard sequence$\left\{{ʋ}_n\right\}$two limits${ʋ}^{\ast }$and${ɵ}^{\ast }$,

$$\mathrm{if} {ʋ}^{\ast }\in \Xi \mathrm{and} {ɵ}^{\ast }\in ϸ\left(\Xi \right), \mathrm{then}$$

(2)

Theorem 2.

Suppose a${\Omega }^{\prime }\text{-}$complete graphical FMS$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$, and let$ϸ:\Xi \to \Xi$be a$\left(\Omega ,{\Omega }^{\prime }\right)-FGC$. Assume that

 (A) 

$ϸ{ʋ}_0\in {\left[{ʋ}_0\right]}_{{\Omega }^{\prime }}^l$for there exist${ʋ}_0\in \Xi$for some$l\in \mathbb{N};$

 (B) 

If sequence $\left\{{ʋ}_n\right\}$that is${\Omega }^{\prime }\text{-}$termwise connected$ϸ\text{-}$Picard converges in$\Xi ,$then for$\left\{{ʋ}_n\right\}$there exists a limit$\omega \in \Xi$and$n_0\in \mathbb{N}$such that$\left({ʋ}_n,\omega \right)\in E\left({\Omega }^{\prime }\right)$or$\left(\omega ,{ʋ}_n\right)\in E\left({\Omega }^{\prime }\right)$at a time$t$for all$n\ge n_0.$

Then, there exists${ʋ}^{\ast }\in \Xi$such that a sequence$\left\{{ʋ}_n\right\}$that is$ϸ\text{-}$Picard is${\Omega }^{\prime }\text{-}$termwise connected with initial value${ʋ}_0\in \Xi$and converges to both${ʋ}^{\ast }$and$ϸ{ʋ}^{\ast }.$Furthermore, if five-tuple$\left(\Xi ,{Ʋ}_{\Omega },\ast ,\Omega ,ϸ\right)$contains the property (2), then$ϸ$ has a fixed point in $\Xi .$

Proof. 

Applying Theorem 1, a sequence $\left\{{ʋ}_n\right\}$ converges to ${ʋ}^{\ast }$ and $ϸ{ʋ}^{\ast }.$ Since ${ʋ}^{\ast }\in \Xi$ and $ϸ{ʋ}^{\ast }\in ϸ\left(\Xi \right),$ consequently by applying (2) the authors must have ${ʋ}^{\ast }=ϸ{ʋ}^{\ast }$. □

The authors symbolize the set of all fixed points of $ϸ$ by $\mathrm{Fix}(ϸ),$ and the perception that the authors utilize is ${\Xi }_{ϸ}=\left\{ʋ\in \Xi :\left(ʋ,ϸʋ\right)\in E\left({\Omega }^{\prime }\right) \mathrm{at} \mathrm{time} t\right\}.$

Theorem 3.

Suppose$\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$to be a${\Omega }^{\prime }\text{-}$complete graphical FMS, and let$ϸ:\Xi \to \Xi$be a$\left(\Omega ,{\Omega }^{\prime }\right)-FGC$. Assume that all conditions of Theorem 2 are satisfied; then,$ϸ$has a fixed point. Furthermore, if${\Xi }_{ϸ}$is weakly connected (as a sub-graph of${\Omega }^{\prime }$), then the fixed point of$ϸ$is unique.

Proof. 

It is easy to observe that the existence of a fixed point follows from Theorem 2. Now, assume that ${\Xi }_{ϸ}$ is weakly connected. Let ${ʋ}^{\ast },{ɵ}^{\ast }\in \mathrm{Fix}(ϸ)$ be two distinct fixed points of $ϸ.$ Therefore, $E\left({\Omega }^{\prime }\right)$ includes all the loops, that is, $\mathrm{Fix}(ϸ)\subseteq {\Xi }_{ϸ}$ and so ${ʋ}^{\ast },{ɵ}^{\ast }\in {\Xi }_{ϸ}.$ The authors must get ${\left({ʋ}^{\ast }P{ɵ}^{\ast }\right)}_{{\Omega }^{\prime }}$ or ${\left({ɵ}^{\ast }P{ʋ}^{\ast }\right)}_{{\Omega }^{\prime }}$ because ${\Xi }_{ϸ}$ is a weakly connected graph. Assume that ${\left({ʋ}^{\ast }P{ɵ}^{\ast }\right)}_{{\Omega }^{\prime }}$ such that there exist a sequence ${\left\{{ʋ}_i\right\}}_{i=0}^l, {ʋ}_0={ʋ}^{\ast }, {ʋ}_l={ɵ}^{\ast }$ and $\left({ʋ}_i,{ʋ}_{i+1}\right)\in E\left({\Omega }^{\prime }\right)$ at a time $t$ and for $i=0,1,\cdots ,l-1.$ Therefore $ϸ$ is a $\left(\Omega ,{\Omega }^{\prime }\right)-$ fuzzy graphical contraction, and by consecutively utilizing (FC1), the authors deduce $\left({ϸ}^n{ʋ}_i,{ϸ}^n{ʋ}_{i+1}\right)\in E\left({\Omega }^{\prime }\right)$ at a time and for $i=0,1,\cdots ,l-1,$ and for all $n\in \mathbb{N}.$ Therefore, by (FC2) the authors have:

$${Ʋ}_{\Omega }\left({ϸ}^n{ʋ}_i,{ϸ}^n{ʋ}_{i+1},\varpi t\right)\ge {Ʋ}_{\Omega }\left({ϸ}^{n-1}{ʋ}_i,{ϸ}^{n-1}{ʋ}_{i+1},t\right)\ge \cdots \ge {Ʋ}_{\Omega }\left({ʋ}_i,{ʋ}_{i+1},\frac{t}{{\varpi }^n}\right),$$

for $i=0,1,\cdots ,l-1$ and for all $n\in \mathbb{N}.$ Now, by (d) of Definition 1, the authors get

$${Ʋ}_{\Omega }\left({ϸ}^n{ʋ}^{\ast }, {ϸ}^n{ɵ}^{\ast },\varpi t\right)\ge {\ast }_{i=0}^{l-1}\left[{Ʋ}_{\Omega }\left({ϸ}^n{ʋ}_i,{ϸ}^n{ʋ}_{i+1},\varpi t\right)\right]\ge {\ast }_{i=0}^{l-1}\left[{Ʋ}_{\Omega }\left({ʋ}_i,{ʋ}_{i+1},\frac{t}{{\varpi }^n}\right)\right].$$

Since ${ʋ}^{\ast },{ɵ}^{\ast }\in \mathrm{Fix}(ϸ)$, the authors have ${ϸ}^n{ʋ}^{\ast } = {ʋ}^{\ast }, {ϸ}^n{ɵ}^{\ast } = {ɵ}^{\ast };$ consequently, letting $n\to ++\infty ,$ as a result of the preceding inequality, the authors have ${Ʋ}_{\Omega }\left({ʋ}^{\ast }, {ɵ}^{\ast },t\right)=1,$ which implies that ${ʋ}^{\ast }= {ɵ}^{\ast }.$ Hence, the fixed point of $ϸ$ is unique. □

4. Applications

In this section, the authors utilize the fixed-point technique to find a solution of integral equations and nonlinear fractional differential equations.

First, the authors utilize a fixed-point technique to find a solution of integral equations on $\Xi$. The authors examine that the existence of a lower or upper solution to an integral equation implies the solution of the integral equation under certain conditions.

Let $\Xi =C\left(\left[0,1\right],ℝ\right)$ be the set of all real valued continuous functions on $\left[0,1\right].$ Let:

$$B=\left\{ʋ\in \Xi :0<\underset{u\in \left[0,1\right]}{\sup }ʋ\left(u\right) \mathrm{and} ʋ\left(u\right)\le 1, u\in \left[0,1\right]\right\}$$

and describe the graph $\Omega , {\Omega }^{\prime }$ by $\Omega ={\Omega }^{\prime }, V\left(\Omega \right)=\Xi$ and:

$$E\left(\Omega \right)=\Lambda {{\displaystyle \cup }}^{ }\left\{\left(ʋ,ɵ\right)\in \Xi \times \Xi :ʋ,ɵ\in B, ʋ\left(u\right)\le ɵ\left(u\right), \mathrm{for} \mathrm{all} u\in \left[0,1\right]\right\}.$$

Now consider a graphical FM defined by ${Ʋ}_{\Omega }:\Xi \times \Xi \times \left(0,+\infty \right)\to \left[0,1\right]$ as:

$${Ʋ}_{\Omega }\left(ʋ\left(u\right),ɵ\left(u\right),t\right)=\{\begin{array}{c}1 \mathrm{if} ʋ=ɵ;\hfill \\ \underset{u\in \left[0,1\right]}{\sup }\frac{t}{t+ʋ\left(u\right)ɵ\left(u\right)} \mathrm{if} ʋ,ɵ\in B, ʋ\ne ɵ;\hfill \\ \frac{1}{2} \mathrm{if\; otherwise}.\hfill \end{array}$$

Then $\left(\Xi ,{Ʋ}_{\Omega },\ast \right)$ is a ${\Omega }^{\prime }\text{-}$ complete graphical FMS with $a\ast b=a·b$. The authors suppose the integral equation as:

$$ʋ\left(u\right)=\underset{0}{\overset{1}{{\displaystyle \int }}}S\left(u,c\right)f\left(c,ʋ\left(c\right)\right)dc,$$

(3)

where $f:\left[0,1\right]\times ℝ\to ℝ \mathrm{and} S:\left[0,1\right]\times \left[0,1\right]\to \left[0,++\infty \right)$ are continuous functions. The function $\alpha \in C\left(\left[0,1\right],ℝ\right)$ is known as a lower solution of Equation (3) if:

$$\alpha \left(u\right)\le \underset{0}{\overset{1}{{\displaystyle \int }}}S\left(u,c\right)f\left(c,\alpha \left(c\right)\right)dc, u\in \left[0,1\right].$$

It is obvious that reverse inequality is met by an upper solution of $\left(3\right)$. Now, the authors examine the existence of a lower solution of $\left(3\right)$ that reveals the existence of a solution of $\left(3\right).$ The authors assume that $ϸ:\Xi \to \Xi$ is an operator and is described by

$$ϸ(ʋ)\left(u\right)=\underset{0}{\overset{1}{{\displaystyle \int }}}S\left(u,c\right)f\left(c,ʋ\left(c\right)\right)dc,$$

(4)

which provides the necessary conditions for the existence of a fixed point of $\left(4\right)$ in $\Xi .$ Obviously, such a fixed point is a solution of the integral Equation $\left(3\right).$

Theorem 4.

Assume that the requirements listed below are met:

• $f\left(c,·\right):ℝ\to ℝ$is increasing on$\left(0,1\right],$for each$c\in \left[0,1\right].$Furthermore$S\left(u,c\right)f\left(c,1\right)\le 1;$
• $there exist \varpi \in \left(0,1\right)$such that for$ʋ,ɵ\in \Xi with \left(ʋ,ɵ\right)\in E\left(\Omega \right) at t,$for each$c,l\in \left[0,1\right]$, the authors have:

$$f\left(c,ʋ\left(c\right)\right)f\left(l,ɵ\left(l\right)\right)\le ʋ\left(c\right)ɵ\left(l\right)$$

and

$$\underset{0}{\overset{1}{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}S\left(u,c\right)S\left(u,l\right)dcdl\le \varpi , \mathrm{for} \mathrm{all} u\in \left[0,1\right].$$

Then the existence of a lower solution of $\left(3\right)$ in $B$ demonstrates the existence of the solution of $\left(3\right) in \Xi .$

Proof. 

Observe that $ϸ$ is well defined. Also, by utilizing (ii), the authors get:

$$\begin{gathered} \frac{\varpi t}{\varpi t+ϸ(ʋ)\left(u\right)ϸ(ɵ)\left(u\right)}=\frac{\varpi t}{\varpi t+{{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^{1}S\left(u,c\right)S\left(u,l\right)f\left(c,ʋ\left(c\right)\right)f\left(l,ɵ\left(l\right)\right)dcdl} \\ \ge \frac{\varpi t}{\varpi t+{{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^{1}S\left(u,c\right)S\left(u,l\right)ʋ\left(c\right)ɵ\left(l\right)dcdl} \\ \ge \frac{\varpi t}{\varpi t+\underset{u\in \left[0,1\right]}{\sup }\left[ʋ\left(c\right)ɵ\left(l\right)\right]{{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^{1}S\left(u,c\right)S\left(u,l\right)dcdl} \\ \ge \frac{\varpi t}{\varpi t+\varpi \underset{u\in \left[0,1\right]}{\sup }\left[ʋ\left(u\right)ɵ\left(u\right)\right]}\ge \frac{t}{t+\underset{u\in \left[0,1\right]}{\sup }\left[ʋ\left(u\right)ɵ\left(u\right)\right]}\ge {Ʋ}_{\Omega }\left(ʋ,ɵ,t\right). \end{gathered}$$

Further, for $ʋ,ɵ\in \Xi \mathrm{with} ʋ,ɵ\in E\left(\Omega \right) \mathrm{at} \mathrm{time} t,$ the authors have $ʋ,ɵ\in B \mathrm{and} ʋ\left(u\right)\le ɵ\left(u\right)$ for all $u\in \left[0,1\right]$ and by utilizing (i), the authors deduce $\underset{u\in \left[0,1\right]}{\sup }ϸ(ʋ)\left(u\right)>0$ and:

$$ϸ(ʋ)\left(u\right)=\underset{0}{\overset{1}{{\displaystyle \int }}}S\left(u,c\right)f\left(c,ʋ\left(c\right)\right)dc\ge \underset{0}{\overset{1}{{\displaystyle \int }}}S\left(u,c\right)f\left(c,1\right)dc\ge 1$$

and

$$ϸ(ʋ)\left(u\right)=\underset{0}{\overset{1}{{\displaystyle \int }}}S\left(u,c\right)f\left(c,ʋ\left(c\right)\right)dc\ge \underset{0}{\overset{1}{{\displaystyle \int }}}S\left(u,c\right)f\left(c,f\left(c,ɵ\left(c\right)\right)\right)dc=ϸ(ɵ)\left(u\right).$$

Therefore, the existence of a lower solution of $\left(3\right),$ called $\alpha \in B,$ implies that Theorem 2′s condition (A) is fulfilled, which implies $ϸ\left(\alpha \right)\in {\left[\alpha \right]}_{{\Omega }^{\prime }}^1$. Also, observe that condition (B) of Theorem 2 satisfies the equation and a five-tuple $\left(\Xi ,{Ʋ}_{\Omega },\ast ,{\Omega }^{\prime },ϸ\right)$ has the condition (2). All the circumstances of Theorem 2 are fulfilled and hence the operator $ϸ$ has a fixed point, which is a solution of the integral Equation $\left(3\right)$ in $\Xi .$

In the case of an upper solution of an integral Equation (3), one can construct using comparable reasoning; in this scenario, one can assume the below set:

$$E\left(\Omega \right)=\left\{\left(ʋ,ɵ\right)\in \Xi \times \Xi :ʋ,ɵ\in B, ʋ\left(u\right)\ge ɵ\left(u\right) \mathrm{for} \mathrm{all} u\in \left[0,1\right]\right\}$$

continuing as above. Then, in order to avoid repetition, the authors leave out the details. □

Now the authors aim to apply Theorem 2 to obtain the existence of a solution to a nonlinear fractional differential equation:

$$D_{0+}^pÞ\left(t\right)=g\left(t,Þ\left(t\right)\right), 0<t<1$$

(5)

with the boundary conditions:

$$Þ\left(0\right)+{Þ}^{\prime }\left(0\right)=0, Þ\left(1\right)+{Þ}^{\prime }\left(1\right)=0,$$

where $1<p\le 2$ is a fractional order, $D_{0+}^p$ is the Caputo fractional derivative, and $g :\left[0,1\right]\times \left[0,+\infty \right)\to \left[0,+\infty \right)$ is a continuous function. Let $\Xi =C\left(\left[0,1\right], \left(0,1\right]\right)$ denote the space of all continuous functions defined on $\left[0, 1\right]$ equipped with the graph $\Omega \left(V,E\right)$ described by $V\left(\Omega \right)=\Xi$ and $E\left(\Omega \right)=\Lambda {{\displaystyle \cup }}^{ }\left\{\left(Þ,\delta \right)\in \Xi \times \Xi :Þ\left(\mathrm{t}\right),\delta \left(t\right)\in \Xi , Þ\left(\mathrm{t}\right)\le \delta \left(t\right), \mathrm{for} \mathrm{all} t\in \left(0,1\right]\right\}$ at a time $t.$ Let $a\ast b=\min \left\{a,b\right\}$ for all $a,b \in \left[0,1\right]$ and define a graphical FMS as follows:

$${ʋ}_{\Omega }\left(Þ\left(\mathrm{t}\right),\delta \left(t\right),\varpi \right)=\{\begin{array}{c}1 \mathrm{if} Þ\left(t\right)=\delta \left(t\right),\\ \frac{\alpha \varpi }{\alpha \varpi +\gamma \max \left\{{\sup }_{tϵ\left[0,1\right] }Þ\left(t\right),{\sup }_{tϵ\left[0,1\right] }\delta \left(t\right)\right\}}, \mathrm{otherwise}.\end{array}$$

Observe that $Þ\in \Xi$ solves (5) whenever $Þ\in \Xi$ solves the following integral equation:

$$Þ\left(t\right)=\frac{1}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-1}\left(1-t\right)g\left(s,Þ\left(s\right)\right)ds+\frac{1}{\Gamma \left(p-1\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-2}\left(1-t\right)g\left(s,Þ\left(s\right)\right)ds+\frac{1}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^t{(t-s)}^{p-1}g\left(s,Þ\left(s\right)\right)ds.$$

Theorem 5.

The integral operator$\xi :\Xi \to \Xi$is given by

$$\xi Þ\left(t\right)=\frac{1}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-1}\left(1-t\right)g\left(s,Þ\left(s\right)\right)ds+\frac{1}{\Gamma \left(p-1\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-2}\left(1-t\right)g\left(s,Þ\left(s\right)\right)ds+\frac{1}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^t{(t-s)}^{p-1}g\left(s,Þ\left(s\right)\right)ds,$$

where$g:\left[0,1\right]\times \left[0,+\infty \right)\to \left[0,+\infty \right)$fulfilling the following criteria:

$$\max \left\{{\sup }_{s\in \left[0,1\right] } g \left(s,Þ\left(s\right)\right),{ \sup }_{s\in \left[0,1\right] } g\left(s,\delta \left(s\right)\right)\right\}\le \frac{1}{4}\max \left\{{ \sup }_{s\in \left[0,1\right] } Þ\left(s\right),{ \sup }_{s\in \left[0,1\right] } \delta \left(s\right)\right\},\mathrm{for} \mathrm{all} Þ,\delta \in \Xi ;$$

Also, suppose that:

$${\sup }_{t\in \left(0,1\right) }\frac{1}{4} \left[\frac{1-t}{\Gamma \left(p+1\right)}+\frac{1-t}{\Gamma \left(p\right)}+\frac{t^p}{\Gamma \left(p+1\right)}\right]\le ā<1.$$

Then nonlinear fractional differential equation has a unique solution in$\Xi$.

Proof. 

$$\begin{array}{c}\max \left\{\xi Þ\left(t\right), \xi \delta \left(t\right)\right\} =\frac{1-t}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-1}\max \left\{{\sup }_{s\in \left[0,1\right]}g\left(s,Þ\left(s\right)\right),{\sup }_{s\in \left[0,1\right]} g\left(s,\delta \left(s\right)\right)\right\}ds+ \frac{1-t}{\Gamma \left(p-1\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-2 }\\ \max \left\{{\sup }_{s\in \left[0,1\right]}g\left(s,Þ\left(s\right)\right),{\sup }_{s\in \left[0,1\right]} g\left(s,\delta \left(s\right)\right)\right\}ds+ \frac{1}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^t{(t-s)}^{p-1}\max \left\{{\sup }_{s\in \left[0,1\right]}g\left(s,Þ\left(s\right)\right),{sup}_{s\in \left[0,1\right]} g\left(s,\delta \left(s\right)\right)\right\}\\ \le \frac{1-t}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-1}\frac{\max \left\{{\sup }_{s\in \left[0,1\right]} Þ\left(s\right),{\sup }_{s\in \left[0,1\right]} \delta \left(s\right)\right\}}{4}ds+\frac{1-t}{\Gamma \left(p-1\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-2}\frac{\max \left\{{\sup }_{s\in \left[0,1\right]}Þ\left(s\right),{\sup }_{s\in \left[0,1\right]}\delta \left(s\right)\right\}}{4}ds\\ +\frac{1}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^t{(t-s)}^{p-1}\frac{\max \left\{{\sup }_{s\in \left[0,1\right]}Þ\left(s\right),{\sup }_{s\in \left[0,1\right]}\delta \left(s\right)\right\}}{4}ds\\ \le \frac{1}{4}\max \left\{{\sup }_{s\in \left[0,1\right]} Þ\left(s\right),{\sup }_{s\in \left[0,1\right]} \delta \left(s\right)\right\} \left(\frac{1-t}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-1}ds+\frac{1-t}{\Gamma \left(p-1\right)}{{\displaystyle \int }}_0^1{(1-s)}^{p-2}ds+\frac{1}{\Gamma \left(p\right)}{{\displaystyle \int }}_0^t{(t-s)}^{p-1}ds\right)\\ \le \frac{1}{4}\max \left\{{\sup }_{s\in \left[0,1\right]} Þ\left(s\right),{\sup }_{s\in \left[0,1\right]} \delta \left(s\right)\right\}{ \sup }_{t\in \left[0,1\right]}\left[\frac{1-t}{\Gamma \left(p+1\right)}+\frac{1-t}{\Gamma \left(p\right)}+\frac{t^p}{\Gamma \left(p+1\right)}\right]\\ = ā \max \left\{{\sup }_{s\in \left[0,1\right]} Þ\left(s\right),{\sup }_{s\in \left[0,1\right]} \delta \left(s\right)\right\} ,\end{array}$$

where:

$$ā={ \sup }_{tϵ\left(0,1\right)}\frac{1}{4 }\left[\frac{1-t}{\Gamma \left(p+1\right)}+\frac{1-t}{\Gamma \left(p\right)}+\frac{t^p}{\Gamma \left(p+1\right)}\right].$$

Therefore, the above equation:

$$\begin{gathered} \max \left\{{\sup }_{t\in \left[0,1\right]} \xi Þ\left(t\right),{\sup }_{t\in \left[0,1\right]} \xi \delta \left(t\right)\right\}\le ā\max \left\{su{p}_{t\in \left[0,1\right]} Þ\left(t\right), su{p}_{t\in \left[0,1\right]} \delta \left(t\right)\right\} \\ \Rightarrow \alpha \varpi +\frac{\gamma }{ā}\max \left\{{\sup }_{t\in \left[0,1\right]} \xi Þ\left(t\right),{\sup }_{t\in \left[0,1\right]} \xi \delta \left(t\right)\right\} \le \alpha \varpi +\gamma \max \left\{{\sup }_{t\in \left[0,1\right]} Þ\left(t\right),{ \sup }_{t\in \left[0,1\right]} \delta \left(t\right)\right\} \\ \Rightarrow \frac{\alpha \left(ā\varpi \right)}{\alpha \left(ā\varpi \right)+\gamma \max \left\{{\sup }_{t\in \left[0,1\right]} \xi Þ\left(t\right),{\sup }_{t\in \left[0,1\right]} \xi \delta \left(t\right)\right\}}\ge \frac{\alpha \varpi }{\alpha \varpi +\gamma \max \left\{{\sup }_{t\in \left[0,1\right]} Þ\left(t\right),{ \sup }_{t\in \left[0,1\right]} \delta \left(t\right)\right\}} \\ \Rightarrow {Ʋ}_{\Omega }\left(\xi Þ,\xi \delta ,ā\varpi \right)\ge {Ʋ}_{\Omega }\left(Þ,\delta ,\varpi \right), \end{gathered}$$

for the variables $\alpha ,\gamma >0.$ Observe that the conditions of the Theorem 2 are fulfilled. As a result, $\xi$ has a fixed point; accordingly, the specified nonlinear fractional differential Equation (5) has a unique solution. □

5. Open Problem

(a)

In underlying space, extend the triangle inequality by using control functions and quadrilateral inequality, and discuss topological structure along with fixed-point results.

(b)

Extend underlying space via intuitionistic fuzzy sets, picture fuzzy sets, and neutrosophic sets and discuss fixed-point results along with topological properties.

6. Conclusions

Nowadays, graph theory plays an important role in science to understand and solve different problems. Herein, the authors established a novel approach of graphical FMS with illustrative examples. The authors discussed criteria for the existence of a unique solution for contraction mapping via integral equations and nonlinear fractional differential equations. This work can be easily extendable in different fuzzy metric structures.