Abstract.

Within graphical fuzzy metric spaces, the present study addresses a formulation for fuzzy graphical $\psi$-contractions as a novel family of contractions. Various fixed point results pertaining to the proposed contractions are established, supported by illustrative examples. Additionally, a discussion of the obtained findings extend and refine existing fixed point theorems is provided. To demonstrate their applicability, the study includes an application on establishing solution solvability for fractional-order nonlinear differential equations.

1. Introduction

Fixed point analysis constitutes a pivotal component within nonlinear functional analysis. It serves as a powerful method in analysing challenging problems across multiple mathematical and scientific disciplines, such as computer science, engineering, chemistry, economics, and medical sciences [55]. Among the classical results, the Banach contraction principle [4] remains a subject of active research, with expansions attained through different approaches and investigations in various abstract metric structures.

Zadeh’s introduction of fuzzy set theory [56] in 1965 is recognized as a major advancement in modern mathematics, as it assigns membership values to all elements of a set. Motivated by this concept, Kramosil and Michalek [26] proposed fuzzy metric spaces, subsequently modified by George and Veeramani [13] to ensure that the induced topological spaces are Hausdorff. Similar to their metric counterparts, researchers have extended this framework further, for instance, fuzzy cone metric spaces [33], complex-valued fuzzy metric spaces [43, 44], intuitionistic fuzzy metric spaces [34], complex-valued intuitionistic fuzzy metric spaces [52], fuzzy bipolar metric spaces [5], and neutrosophic metric spaces [25].

Grabiec [16] pioneered fixed points exploration within the fuzzy metric framework in 1988 by extending Banach contraction principle to complete fuzzy metric settings. Following this, Gregori and Sapena [20] proposed fuzzy contractive mappings in conjunction with related fixed point investigations. Following this line of research, numerous studies have explored generalized variants of contractive mappings, followed by an analysis of their fixed points. Notable studies include Mihet [29], who introduced fuzzy $\psi$-contractive mappings; Wardowski [50], who presented fuzzy $H$-contractive mappings; Khojasteh et al. [24], who formulated simulation functions; Shukla et al. [40], who studied fuzzy $Z$-contractive mappings; and Moussaoui et al. [31], who investigated simulative fuzzy $F$-contractive mappings. Additional studies on fuzzy contractions and their applications in fuzzy metric spaces can be found in [8, 15, 32].

Meanwhile, graph theory has been widely applied across disciplines such as architecture, chemistry, genetics, and computer science [58], as it provides a structural framework for illustrating objects and their associations via vertices and edges. In recent years, researchers have attempted the approach of integrating graph theory into advancement in fixed point theory. Jachymski [23] formulated Banach contraction on metric spaces possessing a graph structure in place of an order structure. Within the same abstract setting, Dinevari and Frigon [10] investigated fixed points under set-valued contractions. In 2017, Shukla et al. [41] defined graphical metric spaces, which extend classical metric spaces by weakening the triangle inequality for points lying on some paths of the defined space’s graph. They further established fixed point findings and showcased relevant applications. Such a structure was subsequently generalized to graphical $b$-metric spaces by Chuensupantharat et al. [7].

In a contemporary study, Saleem et al. [37] devised a fuzzified variant for graphical metric structures, called graphical fuzzy metric spaces, and established several fixed point outcomes. Being an innovative emerging research direction, Shukla et al. [42] addressed certain limitations in the work of [37] resulting from the complex nature of integrating fuzzy and graphical structures. In response, they presented some new fixed point results and explored some topological characteristics of graphical fuzzy metric spaces.

Inspired by [29], [37] and [42], this paper introduces a distinct family of contractions, namely fuzzy graphical $\psi$-contractive mappings, and investigates corresponding fixed point insights under the non-Archimedean graphical fuzzy metric spaces. Instances, along with application to fractional calculus are included to illustrate the findings. The subsequent sections are arranged as presented: Section 2 outlines foundational principles. The third section presents the primary findings, while Section 4 studies an application based upon findings towards proving the solvability for nonlinear fractional-order differential models. Finally, Section 5 concludes the paper by outlining prospects for continued research.

2. Preliminaries

This section offers fundamental concepts related to graphs and graphical fuzzy metric spaces to assist in comprehending the analysis that follows. Real numbers and positive integers are denoted by $ℝ$ and $\mathbb{N}$, respectively. To maintain coherence of the framework established in the literature, we follow the approach presented in [41] and [37].

Suppose $O$ stands as nonempty set, while $\mathrm{\Delta }$ represents the diagonal subset of $O\times O$. Take $\mathrm{\Gamma }$ to be a directed graph without parallel edges, for which its vertex set $\mathrm{\Xi }(\mathrm{\Gamma })=O$ and its edge set $\mathrm{\Sigma }(\mathrm{\Gamma })$ includes all loops, i.e., $\mathrm{\Delta }\subseteq \mathrm{\Sigma }(\mathrm{\Gamma })$. Then, the set $O$ is said to be associated with the graph $\mathrm{\Gamma }=(\mathrm{\Xi }(\mathrm{\Gamma }),\mathrm{\Sigma }(\mathrm{\Gamma }))$, or simply $\mathrm{\Gamma }$. The converse graph ${\mathrm{\Gamma }}^{-1}$ is defined by

$$V({\mathrm{\Gamma }}^{-1})=\mathrm{\Xi }(\mathrm{\Gamma }),\text{ and }\mathrm{\Sigma }({\mathrm{\Gamma }}^{-1})=\{(\rho ,\varrho )\in O\times O:(\varrho ,\rho )\in \mathrm{\Sigma }(\mathrm{\Gamma })\}.$$

The undirected graph induced by graph $\mathrm{\Gamma }$, which contains all symmetric edges, is denoted by $\tilde{\mathrm{\Gamma }}$ where

$$\mathrm{\Xi }(\tilde{\mathrm{\Gamma }})=\mathrm{\Xi }(\mathrm{\Gamma })=O,\text{ and }\mathrm{\Sigma }(\tilde{\mathrm{\Gamma }})=\mathrm{\Sigma }(\mathrm{\Gamma })\cup \mathrm{\Sigma }({\mathrm{\Gamma }}^{-1}).$$

Consider vertices $\rho ,\varrho$ in $\mathrm{\Gamma }$. The path connecting $\rho$ to $\varrho$ with length $l\in \mathbb{N}$ in $\mathrm{\Gamma }$ comprises a sequence ${\{x_i\}}_{i=0}^l$ containing $l+1$ vertices satisfying $x_0=\rho$, $x_l=\varrho$ and $(x_{i-1},x_i)\in \mathrm{\Sigma }(\mathrm{\Gamma })$ where $i=1,2,3,\mathrm{\dots },l$. The graph $\mathrm{\Gamma }$ is said to be connected whenever a path exists for each pair of vertices in $\mathrm{\Gamma }$. In addition, for two vertices $\rho$ and $\varrho$ in a directed graph $\mathrm{\Gamma }$, whenever a path from $\rho$ to $\varrho$ and from $\varrho$ to $\rho$ exists, then vertices $\rho$ and $\varrho$ are connected. The graph $\mathrm{\Gamma }$ is considered weakly connected whenever its associated undirected counterpart $\tilde{\mathrm{\Gamma }}$ remains connected.

A relation $P$ over $O$ is defined by ${(\rho P\varrho )}_{\mathrm{\Gamma }}$ precisely when a directed path exists from $\rho$ to $\varrho$. Given a vertex $w\in O$, we write $w\in {(\rho P\varrho )}_{\mathrm{\Gamma }}$ to express that $w$ is located on a directed path from $\rho$ to $\varrho$. Suppose $l\in \mathbb{N}$, the notation ${[\rho ]}_{\mathrm{\Gamma }}^l$ is expressed as:

$${[\rho ]}_{\mathrm{\Gamma }}^l=\{\varrho \in O:\text{a directed path connects }\rho \text{ to }\varrho \text{ with length }l\}.$$

Furthermore, sequence $\{x_n\}\subseteq O$ is called $\mathrm{\Gamma }$-termwise connected sequence whenever ${(x_{n}P{x}_{n+1})}_{\mathrm{\Gamma }}$ holds for each $n\in \mathbb{N}$. A subgraph ${\mathrm{\Gamma }}^{\prime }$ of graph $\mathrm{\Gamma }$ constitutes a graph with vertex and edge sets contained in of those of $\mathrm{\Gamma }$, that is, $\mathrm{\Xi }({\mathrm{\Gamma }}^{\prime })\subseteq \mathrm{\Xi }(\mathrm{\Gamma })$ and $\mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })\subseteq \mathrm{\Sigma }(\mathrm{\Gamma })$.

In the remainder of this paper, all the graphs are assumed to be directed graphs and possess nonempty vertex and edge sets. Graphical metric spaces are defined hereafter.

Definition 2.1 ([41]).

Suppose $O$ is a nonempty set associated with a graph $\mathrm{\Gamma }$ and $d_{\mathrm{\Gamma }}:O\times O\to [0,\mathrm{\infty })$ denoted as function in which conditions specified below hold:

1. $d_{\mathrm{\Gamma }}(x,y)\ge 0$ whenever $x,y\in O$;

2. $d_{\mathrm{\Gamma }}(x,y)=0$ exactly when $x=y$;

3. $d_{\mathrm{\Gamma }}(x,y)=d_{\mathrm{\Gamma }}(y,x)$ over all $x,y\in O$;

4. For ${(xPy)}_{\mathrm{\Gamma }}$, if $z\in {(xPy)}_{\mathrm{\Gamma }}$ then $d_{\mathrm{\Gamma }}(x,y)\le d_{\mathrm{\Gamma }}(x,z)+d_{\mathrm{\Gamma }}(z,y)$ for each $x,y,z\in O$.

Consequently, $d_{\mathrm{\Gamma }}$ is known as graphical metric on $O$, whereby $(O,d_{\mathrm{\Gamma }})$ is known as graphical metric space.

Preceding the structures concerning fuzzy metric and graphical fuzzy metric within the literature, a foundational concept essential to these definitions is recalled.

Definition 2.2 ([39]).

Binary operation $\ast$ defined $[0,1]$ acts as continuous $t$-norm whenever subsequent properties are true:

1. $1\ast a=a$ whenever $a$ belongs to $[0,1]$;

2. $a\ast b=b\ast a$ for every $a,b\in [0,1]$;

3. $(a\ast b)\ast c=a\ast (b\ast c)$ for every $a,b,c\in [0,1]$;

4. $a\le c$ and $b\le d$ imply $a\ast b\le c\ast d$ given $a,b,c,d$ belong to $[0,1]$;

5. $\ast$ is continuous.

Example 2.3.

Consider listed binary operations ${\ast }_m,{\ast }_p,{\ast }_L$ defined with $a,b\in [0,1]$:

1. $a{\ast }_{L}b=\max \{a+b-1,0\}$;

2. $a{\ast }_{p}b=ab$;

3. $a{\ast }_{m}b=\min \{a,b\}$;

Thus, each operation specified above constitute continuous $t$-norms.

Definition 2.4 ([13]).

Let $O$ be a nonempty set, $\ast$ be a continuous $t$-norm, and $M:O\times O\times (0,\mathrm{\infty })\to [0,1]$ be a fuzzy set such where the listed conditions are satisfied:

1. $M(x,y,t)>0$;

2. $M(x,y,t)=1$ if and only if $x=y$;

3. $M(x,y,t)=M(y,x,t)$;

4. $M(x,y,t+s)\ge M(x,z,t)\ast M(z,y,s)$;

5. $M(x,y,\cdot ):(0,\mathrm{\infty })\to [0,1]$ is continuous;

considering $x,y,z\in O$ along with $t,s>0$. Accordingly, mapping $M$ represents fuzzy metric and the triple $(O,M,\ast )$ forms fuzzy metric space.

For a detailed discussion of fuzzy metric space structures, see [19], [18], [38], and [53].

This section concludes with an exposition of graphical fuzzy metric spaces and several definitions as recently proposed by Saleem et al. [37].

Definition 2.5 ([37]).

Let $O$ be a nonempty set associated with graph $\mathrm{\Gamma }$, $\ast$ denotes a continuous $t$-norm, and fuzzy set $M_{\mathrm{\Gamma }}:O\times O\times (0,\mathrm{\infty })\to [0,1]$ in which conditions specified below hold:

1. $M_{\mathrm{\Gamma }}(x,y,t)>0$;

2. $M_{\mathrm{\Gamma }}(x,y,t)=1$ if and only if $x=y$;

3. $M_{\mathrm{\Gamma }}(x,y,t)=M_{\mathrm{\Gamma }}(y,x,t)$;

4. For ${(xPy)}_{\mathrm{\Gamma }}$, $w\in {(xPy)}_{\mathrm{\Gamma }}$ implies $M_{\mathrm{\Gamma }}(x,y,t+s)\ge M_{\mathrm{\Gamma }}(x,w,t)\ast M_{\mathrm{\Gamma }}(w,y,s)$;

5. $M_{\mathrm{\Gamma }}(x,y,\cdot ):(0,\mathrm{\infty })\to [0,1]$ is continuous;

where $w,x,y\in O$ and $t,s>0$. Consequently, triple $(O,M_{\mathrm{\Gamma }},\ast )$ is known as graphical fuzzy metric space, whereas $M_{\mathrm{\Gamma }}$ represents graphical fuzzy metric on $O$.

Remark 2.6.

If condition (GFMS4) is replaced by the following:

1. For ${(xPy)}_{\mathrm{\Gamma }}$, $w\in {(xPy)}_{\mathrm{\Gamma }}$ implies $M_{\mathrm{\Gamma }}(x,y,\max \{t,s\})\ge M_{\mathrm{\Gamma }}(x,w,t)\ast M_{\mathrm{\Gamma }}(w,y,s)$;

Then, $(O,M_{\mathrm{\Gamma }},\ast )$ is called a non-Archimedean graphical fuzzy metric space. Alternatively, the inequality in (GFMS$4^{\ast }$) may be written as $M_{\mathrm{\Gamma }}(x,y,t)\ge M_{\mathrm{\Gamma }}(x,w,t)\ast M_{\mathrm{\Gamma }}(w,y,t)$. Note that each non-Archimedean graphical fuzzy metric space is a graphical fuzzy metric space.

Example 2.7.

Consider $O=(1,\mathrm{\infty })$ and the graph $\mathrm{\Gamma }$ in which $\mathrm{\Xi }(\mathrm{\Gamma })=O$ and $\mathrm{\Sigma }(\mathrm{\Gamma })=\{(x,y)\in O\times O:y\ge x\}\cup \mathrm{\Delta }$. A fuzzy set $M_{\mathrm{\Gamma }}:O\times O\times (0,\mathrm{\infty })\to [0,1]$ is then given by

where $x,y\in O$ and $t>0$. Consequently, $(O,M_{\mathrm{\Gamma }},{\ast }_p)$ is readily seen as graphical fuzzy metric space.

Example 2.8 ([42]).

Consider $(O,d_{\mathrm{\Gamma }})$, a graphical metric space associated to graph $\mathrm{\Gamma }$ defined by $\mathrm{\Xi }(\mathrm{\Gamma })=O$ and $\mathrm{\Sigma }(\mathrm{\Gamma })=O\times O$. A fuzzy set $M_{\mathrm{\Gamma }}:O\times O\times (0,\mathrm{\infty })\to [0,1]$ is then expressed as

$$M_{\mathrm{\Gamma }}(x,y,t)=\frac{\alpha t^n}{\alpha t^n+\beta d_{\mathrm{\Gamma }}(x,y)}$$

where $x,y\in O$ and $t>0$, where $\alpha ,\beta >0$ and $n\in \mathbb{N}$. Consequently, $(O,M_{\mathrm{\Gamma }},{\ast }_m)$ forms a graphical fuzzy metric space. Under $\alpha =\beta =n=1$, it is known as standard graphical fuzzy metric space produced by $d_{\mathrm{\Gamma }}$, and $M_{\mathrm{\Gamma }}$ is known as graphical fuzzy metric produced by $d_{\mathrm{\Gamma }}$.

Remark 2.9.

Graphical fuzzy metric spaces are generally broader than fuzzy metric spaces. Note that a fuzzy metric space $(O,M,\ast )$ becomes a graphical fuzzy metric space once equipped with a graph $\mathrm{\Gamma }$ in which $\mathrm{\Xi }(\mathrm{\Gamma })=O$ and $\mathrm{\Sigma }(\mathrm{\Gamma })=O\times O$. However, a graphical fuzzy metric space might fail to satisfy the conditions required by the fuzzy metric space. Example below demonstrates preceding statement.

Figure 1. The graph structure corresponding to the graphical fuzzy metric space.

Example 2.10.

Take $O=\{1,2,3,4\}$ and graph $\mathrm{\Gamma }$ in which $\mathrm{\Xi }(\mathrm{\Gamma })=O$ and $\mathrm{\Sigma }(\mathrm{\Gamma })=\mathrm{\Delta }\cup \{(1,2),(1,3),(3,2)$, $(4,1),(4,2)\}$ (see Figure 1). A fuzzy set $M_{\mathrm{\Gamma }}:O\times O\times (0,\mathrm{\infty })\to [0,1]$ is expressed as

$$M_{\mathrm{\Gamma }}(x,y,t)=\frac{\min \{x,y\}}{\max \{x,y\}}$$

with $x,y\in O$ and $t>0$. Consequently, $(O,M_{\mathrm{\Gamma }},{\ast }_m)$ forms graphical fuzzy metric space. Moreover, it is a non-Archimedean graphical fuzzy metric space.

It can be observed that criteria of fuzzy metric space are not satisfied. For instance, consider the case where $x=1,y=4$, $z=2$. Hence, at any $t>0$, $(O,M_{\mathrm{\Gamma }},{\ast }_m)$ does not constitute even a strong fuzzy metric space.

Definition 2.11 ([37]).

Consider $(O,M_{\mathrm{\Gamma }},\ast )$ graphical fuzzy metric space. An open ball $B_{\mathrm{\Gamma }}(x,r,t)$ with center $x\in O$, radius $r\in (0,1)$, and $t>0$ is defined by

$$B_{\mathrm{\Gamma }}(x,r,t)=\{y\in O:{(xPy)}_{\mathrm{\Gamma }},M_{\mathrm{\Gamma }}(x,y,t)>1-r\}.$$

Significantly, topology ${\tau }_{\mathrm{\Gamma }}$ induced by graphical fuzzy metric $M_{\mathrm{\Gamma }}$ is $T_1$ but not Hausdorff in general. Additional topological aspects of graphical fuzzy metric spaces are detailed in [42].

Definition 2.12 ([37]).

Consider $(O,M_{\mathrm{\Gamma }},\ast )$ graphical fuzzy metric space. Sequence $\{x_n\}\subseteq O$ is convergent to $x\in O$ whenever each $r\in (0,1)$ and $t>0$, there exists $n_0\in \mathbb{N}$ satisfying $M_{\mathrm{\Gamma }}(x_n,x,t)>1-r$ for every $n\ge n_0$. Equivalently, sequence $\{x_n\}$ in $O$ is convergent to $x\in O$ if ${lim}_{n\to \mathrm{\infty }}{M}_{\mathrm{\Gamma }}(x_n,x,t)=1$ for any $t>0$.

Remark 2.13.

It is well established that in fuzzy metric spaces, every convergent sequence has a unique limit. However, this property does not necessarily hold in graphical fuzzy metric spaces.

Example 2.14 ([42]).

Consider $O=[0,1]$ and a graph $\mathrm{\Gamma }$ defined by $\mathrm{\Xi }(\mathrm{\Gamma })=O$ and $\mathrm{\Sigma }(\mathrm{\Gamma })=\{(x,y)\in O\times O:y\le x\}\cup \mathrm{\Delta }$. Further, consider the graphical metric $d_{\mathrm{\Gamma }}$ defined by

Let $M_{\mathrm{\Gamma }}$ be the graphical fuzzy metric defined as in Example 2.8 with $\alpha =\beta =n=1$. Then, $(O,M_{\mathrm{\Gamma }},{\ast }_m)$ is a standard graphical fuzzy metric space. Consider the sequence $\{x_n\}$ in the form of $x_n=\frac{1}{n}$ for each $n\in \mathbb{N}$. This sequence possesses infinitely many limits in $X$. In fact, every point of $O$ is a limit of $\{x_n\}$. Take any $x\in (0,1)$ and a fixed $t>0$, then the intersection of $B_{\mathrm{\Gamma }}(x,r_1,t)$ and $B_{\mathrm{\Gamma }}(1,r_2,t)$ is nonempty for all $r_1,r_2>0$. Therefore, $M_{\mathrm{\Gamma }}$ is not Hausdorff.

Definition 2.15 ([37]).

Consider $(O,M_{\mathrm{\Gamma }},\ast )$ a graphical fuzzy metric space. Sequence $\{x_n\}$ in $O$ is Cauchy sequence whenever every $r\in (0,1)$ and $t>0$, there exists $n_0\in \mathbb{N}$ satisfying $M_{\mathrm{\Gamma }}(x_n,x_m,t)>1-r$ for each $n,m\ge n_0$. Similarly, the sequence $\{x_n\}$ is Cauchy sequence whenever ${lim}_{n,m\to \mathrm{\infty }}{M}_{\mathrm{\Gamma }}(x_n,x_m,t)=1$ for any $t>0$.

Definition 2.16 ([37]).

A graphical fuzzy metric space $(O,M_{\mathrm{\Gamma }},\ast )$ is complete whenever all Cauchy sequences of $O$ are convergent. Suppose ${\mathrm{\Gamma }}^{\prime }$ is a different graph from $\mathrm{\Gamma }$ with $\mathrm{\Xi }({\mathrm{\Gamma }}^{\prime })=O$, then $(O,M_{\mathrm{\Gamma }},\ast )$ is considered ${\mathrm{\Gamma }}^{\prime }$-complete if each ${\mathrm{\Gamma }}^{\prime }$-termwise connected Cauchy sequence in $O$ is convergent.

3. Main Results

The present section discusses numerous fixed point findings under non-Archimedean graphical fuzzy metric spaces. To commence, several definitions regarding specific contractive condition are established. Let $\mathrm{\Psi }$ denoted as collection containing mappings $\psi :[0,1]\to [0,1]$ in which $\psi$ exhibits continuous, nondecreasing and holds on every $t\in (0,1)$. Lemma below follows from the properties of the mapping $\psi$.

Lemma 3.1.

Suppose $\psi$ belongs to $\mathrm{\Psi }$. Consequently, ${lim}_{n\to \mathrm{\infty }}{\psi }^n(t)=1$ holds on every $t\in (0,1)$.

Proof 3.2.

Assume $t_0\in (0,1)$ for which ${lim}_{n\to \mathrm{\infty }}{\psi }^n(t_0)\ne 1$. Since mapping $\psi$ is nondecreasing and $\psi (t_0)>t_0$, it follows that ${\psi }^{n+1}(t_0)>{\psi }^n(t_0)$ for all $n\in \mathbb{N}$. Consequently, $\{{\psi }^n(t_0)\}$ is an increasing sequence and bounded by $[0,1]$. Thus, the sequence converges, for example, ${lim}_{n\to \mathrm{\infty }}{\psi }^n(t_0)=u$ for some $u\in [0,1]$. Due to $\psi$ is continuous, one deduces that

$$u=\underset{n\to \mathrm{\infty }}{lim}{\psi }^{n+1}(t_0)=\psi \left(\underset{n\to \mathrm{\infty }}{lim}{\psi }^n(t_0)\right)=\psi (u)>u$$

which gives rise to a contradiction. Therefore, ${lim}_{n\to \mathrm{\infty }}{\psi }^n(t)=1$ throughout $t\in (0,1)$.

The following contractive mapping, due to Mihet [29], features under fuzzy metric space.

Definition 3.3 ([29]).

Consider $(O,M,\ast )$ fuzzy metric space and $\psi \in \mathrm{\Psi }$. Mapping $T:O\to O$ is known as fuzzy $\psi$-contractive mapping if condition below applies:

$$M(x,y,t)>0⟹M(Tx,Ty,t)\ge \psi (M(x,y,t)),\text{ for all }t>0.$$

Building upon this definition, we introduce its fuzzy graphical adaptation as follows.

Definition 3.4.

Let $(O,M_{\mathrm{\Gamma }},\ast )$ be a graphical fuzzy metric space, $T:O\to O$ be a mapping and ${\mathrm{\Gamma }}^{\prime }$ is a subgraph of $\mathrm{\Gamma }$ with $\mathrm{\Delta }\subseteq \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$. Then, mapping $T$ is $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical $\psi$-contraction on $O$ whenever the criteria below are met:

1. Mapping $T$ is edge preserving in $\mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$, specifically, $(x,y)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ implies $(Tx,Ty)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$;

2. Mapping $\psi \in \mathrm{\Psi }$ exists for which each distinct pair $x,y\in O$ satisfying $(x,y)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$,

   $$M_{\mathrm{\Gamma }}(Tx,Ty,t)\ge \psi (M_{\mathrm{\Gamma }}(x,y,t))\text{ for all }t>0.$$

A sequence $\{x_n\}$ in $O$ beginning with $x_0\in O$ is called $T$-Picard sequence (i.e., the Picard sequence associated with $T$) if $x_n=T{x}_{n-1}$ is satisfied for every $n\in \mathbb{N}$. In subsequent discussion, graph ${\mathrm{\Gamma }}^{\prime }$ is assumed to be a subgraph of $\mathrm{\Gamma }$ in which $\mathrm{\Delta }\subseteq \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$.

Theorem 3.5.

Suppose $(O,M_{\mathrm{\Gamma }},\ast )$ is a ${\mathrm{\Gamma }}^{\prime }$-complete non-Archimedean graphical fuzzy metric space, and $T:O\to O$ is a $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical $\psi$-contraction. Assume following conditions are satisfied:

1. there exists $x_0\in O$ satisfying $T{x}_0\in {[x_0]}_{{\mathrm{\Gamma }}^{\prime }}^l$ for some $l\in \mathbb{N}$;

2. if a ${\mathrm{\Gamma }}^{\prime }$-termwise connected $T$-Picard sequence $\{x_n\}$ converges to some $x\in O$, then limit $x\in O$ and $n_0\in \mathbb{N}$ satisfying $(x_n,x)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ or $(x,x_n)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for all $n\ge n_0$.

Consequently, there exists ${\varpi }^{\ast }\in O$ where the $T$-Picard sequence $\{x_n\}$ begins with $x_0\in O$ is ${\mathrm{\Gamma }}^{\prime }$-termwise connected and converges to both ${\varpi }^{\ast }$ and $T{\varpi }^{\ast }$ in $O$.

Proof 3.6.

Suppose $x_0\in O$ satisfies $T{x}_0\in {[x_0]}_{{\mathrm{\Gamma }}^{\prime }}^l$ for some $l\in \mathbb{N}$. Then, construct a $T$-Picard sequence $\{x_n\}$ with starting point $x_0$. It follows that a path exists ${\{y_i\}}_{i=0}^l$ where $x_0=y_0,T{x}_0=y_l$ and $(y_{i-1},y_i)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for $i=1,2,\mathrm{\dots },l$. Given $T$ as a $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical $\psi$-contraction, the condition (FG$\psi$1) ensures that $(T{y}_{i-1},T{y}_i)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for $i=1,2,\mathrm{\dots },l$. Consequently, ${\{T{y}_i\}}_{i=0}^l$ forms a path connecting $T{y}_0=T{x}_0=x_1$ to $T{y}_l=T^2{x}_0=x_2$ with length $l$, which establishes that $x_2\in {[x_1]}_{{\mathrm{\Gamma }}^{\prime }}^l$. Continue in similar manner, it can be derived that $(T^n{y}_{i-1},T^n{y}_i)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for $i=1,2,\mathrm{\dots },l$. Accordingly, ${\{T^n{y}_i\}}_{i=0}^l$ forms a path connecting $T^n{y}_0=T^n{x}_0=x_n$ to $T^n{y}_l=T^{n+1}{x}_0=x_{n+1}$ with length $l$, which establishes that $x_{n+1}\in {[x_n]}_{{\mathrm{\Gamma }}^{\prime }}^l$ for any $n\in \mathbb{N}$. Therefore, $\{x_n\}$ is considered ${\mathrm{\Gamma }}^{\prime }$-termwise connected sequence.

As $(T^n{y}_{i-1},T^n{y}_i)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for $i=1,2,\mathrm{\dots },l$ and $n\in \mathbb{N}$, applying (FG$\psi$2), it yield

$$M_{\mathrm{\Gamma }}(T^n{y}_{i-1},T^n{y}_i,t)\ge \psi \left(M_{\mathrm{\Gamma }}(T^{n-1}{y}_{i-1},T^{n-1}{y}_i,t)\right)$$

where $t>0$. Repeating argument above yield

$M_{\mathrm{\Gamma }}(T^n{y}_{i-1},T^n{y}_i,t)$ $\ge \psi \left(M_{\mathrm{\Gamma }}(T^{n-1}{y}_{i-1},T^{n-1}{y}_i,t)\right)$

$\ge {\psi }^2\left(M_{\mathrm{\Gamma }}(T^{n-2}{y}_{i-1},T^{n-2}{y}_i,t)\right)$

$\mathrm{⋮}$

$\ge {\psi }^n(M_{\mathrm{\Gamma }}(y_{i-1},y_i,t))$ (1)

over every $t>0$. Since sequence $\{x_n\}$ is a ${\mathrm{\Gamma }}^{\prime }$-termwise connected, by applying (3.6) and (GFMS${\mathit{4}}^{\mathrm{𝑒𝑛𝑢𝑚𝑖}}$), one obtains, for any $n\in \mathbb{N}$ and $t>0$,

$M_{\mathrm{\Gamma }}(x_n,x_{n+1},t)$ $=M_{\mathrm{\Gamma }}(T^n{y}_0,T^n{y}_l,t)$

$\ge M_{\mathrm{\Gamma }}(T^n{y}_0,T^n{y}_1,t)\ast M_{\mathrm{\Gamma }}(T^n{y}_1,T^n{y}_l,t)$

$\ge M_{\mathrm{\Gamma }}(T^n{y}_0,T^n{y}_1,t)\ast M_{\mathrm{\Gamma }}(T^n{y}_1,T^n{y}_2,t)\ast M_{\mathrm{\Gamma }}(T^n{y}_2,T^n{y}_l,t)$

$\mathrm{⋮}$

$\ge M_{\mathrm{\Gamma }}(T^n{y}_0,T^n{y}_1,t)\ast M_{\mathrm{\Gamma }}(T^n{y}_1,T^n{y}_2,t)\ast \mathrm{\cdots }\ast M_{\mathrm{\Gamma }}(T^n{y}_{l-1},T^n{y}_l,t)$

$\ge {\psi }^n\left(M_{\mathrm{\Gamma }}(y_0,y_1,t)\right)\ast {\psi }^n\left(M_{\mathrm{\Gamma }}(y_1,y_2,t)\right)\ast \mathrm{\cdots }\ast {\psi }^n\left(M_{\mathrm{\Gamma }}(y_{l-1},y_l,t)\right).$

Furthermore, due to the fact that sequence $\{x_n\}$ is considered ${\mathrm{\Gamma }}^{\prime }$-termwise connected, it follows that for $n,m\in \mathbb{N}$ satisfying $m>n$ and $t>0$, we obtain

$M_{\mathrm{\Gamma }}(x_n,x_m,t)$ $\ge M_{\mathrm{\Gamma }}(x_n,x_{n+1},t)\ast M_{\mathrm{\Gamma }}(x_{n+1},x_m,t)$

$\ge M_{\mathrm{\Gamma }}(x_n,x_{n+1},t)\ast M_{\mathrm{\Gamma }}(x_{n+1},x_{n+2},t)\ast M_{\mathrm{\Gamma }}(x_{n+2},x_m,t)$

$\mathrm{⋮}$

$\ge M_{\mathrm{\Gamma }}(x_n,x_{n+1},t)\ast M_{\mathrm{\Gamma }}(x_{n+1},x_{n+2},t)\ast \mathrm{\cdots }\ast M_{\mathrm{\Gamma }}(x_{m-1},x_m,t)$

$\ge \left[{\psi }^n\left(M_{\mathrm{\Gamma }}(y_0,y_1,t)\right)\ast {\psi }^n\left(M_{\mathrm{\Gamma }}(y_1,y_2,t)\right)\ast \mathrm{\cdots }\ast {\psi }^n\left(M_{\mathrm{\Gamma }}(y_{l-1},y_l,t)\right)\right]$

$\ast \left[{\psi }^{n+1}\left(M_{\mathrm{\Gamma }}(y_0,y_1,t)\right)\ast {\psi }^{n+1}\left(M_{\mathrm{\Gamma }}(y_1,y_2,t)\right)\ast \mathrm{\cdots }\ast {\psi }^{n+1}\left(M_{\mathrm{\Gamma }}(y_{l-1},y_l,t)\right)\right]\ast \mathrm{\cdots }$

$\ast \left[{\psi }^{m-1}\left(M_{\mathrm{\Gamma }}(y_0,y_1,t)\right)\ast {\psi }^{m-1}\left(M_{\mathrm{\Gamma }}(y_1,y_2,t)\right)\ast \mathrm{\cdots }\ast {\psi }^{m-1}\left(M_{\mathrm{\Gamma }}(y_{l-1},y_l,t)\right)\right].$

Let $n,m\to \mathrm{\infty }$ in the preceding inequality. Given that $\ast$ is continuous together with Lemma 3.1, it follows that

$$\underset{n,m\to \mathrm{\infty }}{lim}{M}_{\mathrm{\Gamma }}(x_n,x_m,t)=1.$$

As a result, $\{x_n\}$ is a ${\mathrm{\Gamma }}^{\prime }$-termwise connected Cauchy sequence in $O$. Given that $(O,M_{\mathrm{\Gamma }},\ast )$ is ${\mathrm{\Gamma }}^{\prime }$-complete, the sequence $\{x_n\}$ converges to some ${\varpi }^{\ast }\in O$. Employing condition (ii), there exists ${\varpi }^{\ast }\in O$ and $n_0\in \mathbb{N}$ satisfying $(x_n,{\varpi }^{\ast })\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ or $({\varpi }^{\ast },x_n)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ over $n>n_0$ and

$$\underset{n\to \mathrm{\infty }}{lim}{M}_{\mathrm{\Gamma }}(x_n,{\varpi }^{\ast },t)=1\text{ whenever }t>0.$$

Without the loss of generality, we consider $(x_n,{\varpi }^{\ast })\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ over all $n>n_0$, as $({\varpi }^{\ast },x_n)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ can be treated in a similar manner. By (FG$\psi$1), it follows that $(x_{n+1},T{\varpi }^{\ast })=(T{x}_n,T{\varpi }^{\ast })\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ whenever $n>n_0$. Now, applying (FG$\psi$2), we yield

$$M_{\mathrm{\Gamma }}(x_{n+1},T{\varpi }^{\ast },t)=M_{\mathrm{\Gamma }}(T{x}_n,T{\varpi }^{\ast },t)\ge \psi (M_{\mathrm{\Gamma }}(x_n,{\varpi }^{\ast },t)),\text{ for all }n\ge n_0.$$

As $\psi (M_{\mathrm{\Gamma }}(x_n,{\varpi }^{\ast },t))>M_{\mathrm{\Gamma }}(x_n,{\varpi }^{\ast },t)$, the preceding inequality becomes

$$M_{\mathrm{\Gamma }}(x_{n+1},T{\varpi }^{\ast },t)>M_{\mathrm{\Gamma }}(x_n,{\varpi }^{\ast },t),\text{ whenever }n\ge n_0.$$

In view of ${lim}_{n\to \mathrm{\infty }}{M}_{\mathrm{\Gamma }}(x_n,{\varpi }^{\ast },t)=1$ for every $t>0$, taking $n\to \mathrm{\infty }$ in the above inequality leads to

$$\underset{n\to \mathrm{\infty }}{lim}{M}_{\mathrm{\Gamma }}(x_{n+1},T{\varpi }^{\ast },t)=1.$$

Therefore, $\{x_n\}$ also converges to $T{\varpi }^{\ast }\in O$. Accordingly, $\{x_n\}$ converges to both ${\varpi }^{\ast }$ and $T{\varpi }^{\ast }$ in $O$.

With the aid of the property defined below, the next result establishes the manifestation of a fixed point.

Definition 3.7.

Suppose $(O,M_{\mathrm{\Gamma }},\ast )$ is a graphical fuzzy metric space, with ${\mathrm{\Gamma }}^{\prime }$ a subgraph of $\mathrm{\Gamma }$, and $T:O\to O$ is a self-mapping. Accordingly, $(O,M_{\mathrm{\Gamma }},\ast ,{\mathrm{\Gamma }}^{\prime },T)$ possess property (S), if any ${\mathrm{\Gamma }}^{\prime }$-termwise connected $T$-Picard sequence $\{x_n\}$ possess limits ${\varpi }^{\ast }\in O$ and ${\vartheta }^{\ast }\in TO$, then ${\varpi }^{\ast }={\vartheta }^{\ast }$.

Theorem 3.8.

Suppose $(O,M_{\mathrm{\Gamma }},\ast )$ is a ${\mathrm{\Gamma }}^{\prime }$-complete non-Archimedean graphical fuzzy metric space and $T:O\to O$ is $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical $\psi$-contraction. Assume following conditions are satisfied:

1. there exists $x_0\in O$ satisfying $T{x}_0\in {[x_0]}_{{\mathrm{\Gamma }}^{\prime }}^l$, for some $l\in \mathbb{N}$;

2. if ${\mathrm{\Gamma }}^{\prime }$-termwise connected $T$-Picard sequence $\{x_n\}$ converges to some $x\in O$, then limit $x\in O$ and $n_0\in \mathbb{N}$ exist satisfying $(x_n,x)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ or $(x,x_n)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ whenever $n\ge n_0$.

Consequently, there exists ${\varpi }^{\ast }\in O$ where the $T$-Picard sequence $\{x_n\}$ begins with $x_0\in O$ is ${\mathrm{\Gamma }}^{\prime }$-termwise connected and converges to both ${\varpi }^{\ast },T{\varpi }^{\ast }\in O$. In addition, $(O,M_{\mathrm{\Gamma }},\ast ,{\mathrm{\Gamma }}^{\prime },T)$ possess property (S) implies $T$ possess at least one fixed point in $O$.

Proof 3.9.

It can be concluded from Theorem 3.5 that the $T$-Picard sequence $\{x_n\}$ begins with $x_0\in O$ converges to both ${\varpi }^{\ast }$ and $T{\varpi }^{\ast }$ in $O$. Clearly, ${\varpi }^{\ast }\in O$ and $T{\varpi }^{\ast }\in TO$. Consequently, by applying property (S), it leads to ${\varpi }^{\ast }=T{\varpi }^{\ast }$ which verifies ${\varpi }^{\ast }$ is fixed under $T$.

The example below demonstrates that Theorem 3.8 establishes sufficient conditions for the existence of a fixed point. Nevertheless, the uniqueness is not guarantee.

Example 3.10.

Suppose $O=[0,\mathrm{\infty })$ and $\mathrm{\Gamma }$ is a graph given by $\mathrm{\Xi }(\mathrm{\Gamma })=O$ and $\mathrm{\Sigma }(\mathrm{\Gamma })=\mathrm{\Delta }\cup \{(x,y)\in O\times O:y\le x\}$. Fuzzy set $M_{\mathrm{\Gamma }}:O\times O\times (0,\mathrm{\infty })\to [0,1]$ is expressed as

$$M_{\mathrm{\Gamma }}(x,y,w)=\exp \left\{-\frac{d_{\mathrm{\Gamma }}(x,y)}{w}\right\},\text{ for all }x,y\in O\text{ and }w>0$$

where $d_{\mathrm{\Gamma }}$ is defined as

Accordingly, $(O,M_{\mathrm{\Gamma }},{\ast }_m)$ constitutes a non-Archimedean graphical fuzzy metric space. Now, consider mapping $T:O\times O$ defined by

Furthermore, define a mapping $\psi \in \mathrm{\Psi }$ by $\psi (\alpha )=\frac{\alpha +1}{2}$ for $\alpha \in [0,1]$, and ${\mathrm{\Gamma }}^{\prime }$ be a subgraph of $\mathrm{\Gamma }$ such that $V({\mathrm{\Gamma }}^{\prime })=O$ and $\mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })=\mathrm{\Delta }\cup \{(x,y)\in [0,1)\times [0,1):y\le x\}$. It then follows that $(O,M_{\mathrm{\Gamma }},{\ast }_m)$ is ${\mathrm{\Gamma }}^{\prime }$-complete. The mapping $T$ satisfies the conditions of a $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical $\psi$-contraction (see Figure 2 for graphical visualization).

Figure 2. Visualization of $M_{\mathrm{\Gamma }}(Tx,Ty,t)$ (orange surface) and $\psi (M_{\mathrm{\Gamma }}(x,y,t))$ (blue surface) at $t=5$, illustrating that $M_{\mathrm{\Gamma }}(Tx,Ty,t)\ge \psi (M_{\mathrm{\Gamma }}(x,y,t))$.

For any $x_0\in [0,1)$, it follows that $T{x}_0\in [0,1)$ and $T{x}_0\le x_0$. Therefore, $(x_0,T{x}_0)\in \mathrm{\Sigma }(\mathrm{\Gamma })$, which implies $T{x}_0\in {[x_0]}_{{\mathrm{\Gamma }}^{\prime }}^1$. Therefore, there exists $x_0\in O$ such that $T{x}_0\in {[x_0]}_G^l$ for some $l\in \mathbb{N}$. Moreover, any ${\mathrm{\Gamma }}^{\prime }$-termwise connected $T$-Picard sequence $\{x_n\}$ in $O$ is either constant (e.g., $x_n=1$ for all $n\in \mathbb{N}$) or of the form $x_n=\frac{x_0}{5^n}$, with $x_0\in [0,1)$, which converges to $1$ and $0$, respectively. Consequently, $(1,1),(x_n,0)$ are contained in $\mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for every $n\in \mathbb{N}$. It is easily seen that property (S) is satisfied. Therefore, all assumptions of Theorem 3.8 are satisfied. In particular, $0$ and $1$ are fixed points of mapping $T$. Moreover, there exists a path from $1$ to $0$, whereas no path exists from $0$ to $1$. Hence, the set $\{0,1\}$ is not connected.

The collection of fixed points of a mapping $T$ is expressed as $\text{Fix}(T)$. Furthermore, notation $O_T$ is defined by $O_T=\{x\in O:(x,Tx)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })\}$. The result below provides a prerequisite for the uniqueness.

Theorem 3.11.

Suppose $(O,M_{\mathrm{\Gamma }},\ast )$ is a ${\mathrm{\Gamma }}^{\prime }$-complete non-Archimedean graphical fuzzy metric space and $T:O\to O$ is $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical $\psi$-contraction. Assuming Theorem 3.8 applies and $O_T$, as a subgraph of ${\mathrm{\Gamma }}^{\prime }$, is connected, then mapping $T$ admits a unique fixed point.

Proof 3.12.

Theorem 3.8 ensures the existence of fixed point of $T$. Assume that $O_T$ is connected subgraph of ${\mathrm{\Gamma }}^{\prime }$, and that ${\varpi }^{\ast },{\vartheta }^{\ast }$ represents two separate fixed points of $T$. The fact that $\mathrm{\Delta }\subseteq \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ validates $\text{Fix}(T)\subseteq O_T$ and hence, ${\varpi }^{\ast },{\vartheta }^{\ast }\in O_T$. Since $O_T$ is connected, we deduce ${({\varpi }^{\ast }P{\vartheta }^{\ast })}_G$. Consequently, a sequence ${\{x_i\}}_{i=0}^l$ exists in which $x_0={\varpi }^{\ast },x_l={\vartheta }^{\ast }$ and $(x_{i-1},x_i)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for $i=1,2,\mathrm{\dots },l$.

Given that $T$ is a $(G,{\mathrm{\Gamma }}^{\prime })$-graphical fuzzy $\psi$-contraction, by applying (FG$\psi$1), we yield $(T^n{x}_{i-1},T^n{x}_i)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for $i=1,2,\mathrm{\dots },l$ and for each $n\in \mathbb{N}$. Henceforth, by applying (FG$\psi$2), this yields

$M_{\mathrm{\Gamma }}(T^n{x}_{i-1},T^n{x}_i,t)$ $\ge \psi \left(M_{\mathrm{\Gamma }}(T^{n-1}{x}_{i-1},T^{n-1}{x}_i,t)\right)$

$\ge {\psi }^2\left(M_{\mathrm{\Gamma }}(T^{n-2}{x}_{i-1},T^{n-2}{x}_i,t)\right)$

$\mathrm{⋮}$

$\ge {\psi }^n(M_{\mathrm{\Gamma }}(x_{i-1},x_i,t))$

for $i=1,2,\mathrm{\dots },l$ and for each $n\in \mathbb{N}$. Applying (GFMS${\mathit{4}}^{\mathrm{𝑒𝑛𝑢𝑚𝑖}}$), it leads to

$M_{\mathrm{\Gamma }}(T^n{\varpi }^{\ast },T^n{\vartheta }^{\ast },t)$ $=M_{\mathrm{\Gamma }}(T^n{x}_0,T^n{x}_l,t)$

$\ge M_{\mathrm{\Gamma }}(T^n{x}_0,T^n{x}_1,t)\ast M_{\mathrm{\Gamma }}(T^n{x}_1,T^n{x}_l,t)$

$\ge M_{\mathrm{\Gamma }}(T^n{x}_0,T^n{x}_1,t)\ast M_{\mathrm{\Gamma }}(T^n{x}_1,T^n{x}_2,t)\ast M_{\mathrm{\Gamma }}(T^n{x}_2,T^n{x}_l,t)$

$\mathrm{⋮}$

$\ge M_{\mathrm{\Gamma }}(T^n{x}_0,T^n{x}_1,t)\ast M_{\mathrm{\Gamma }}(T^n{x}_1,T^n{x}_2,t)\ast \mathrm{\cdots }\ast M_{\mathrm{\Gamma }}(T^n{x}_{l-1},T^n{x}_l,t)$

$\ge {\psi }^n\left(M_{\mathrm{\Gamma }}(x_0,x_1,t)\right)\ast {\psi }^n\left(M_{\mathrm{\Gamma }}(x_1,x_2,t)\right)\ast \mathrm{\cdots }\ast {\psi }^n\left(M_{\mathrm{\Gamma }}(x_{l-1},x_l,t)\right)$

The fact that ${\varpi }^{\ast },{\vartheta }^{\ast }\in \text{Fix}(T)$ indicates $T^n{\varpi }^{\ast }={\varpi }^{\ast }$ and $T^n{\vartheta }^{\ast }={\vartheta }^{\ast }$. From the preceding inequality, by letting $n\to \mathrm{\infty }$ and Lemma 3.1 it leads to

$$M_{\mathrm{\Gamma }}({\varpi }^{\ast },{\vartheta }^{\ast },t)=1\text{ over each }t>0.$$

Equivalently, ${\varpi }^{\ast }={\vartheta }^{\ast }$, leading to a contradiction with the initial assumption. Therefore, fixed point of $T$ is unique.

Example below is presented to clarify the preceding findings.

Example 3.13.

Suppose $O=(0,10]$ and $\mathrm{\Gamma }$ is a graph given by $\mathrm{\Xi }(\mathrm{\Gamma })=O$ and $\mathrm{\Sigma }(\mathrm{\Gamma })=\mathrm{\Delta }\cup \{(x,y)\in O\times O:x\ge y\}$. Fuzzy set $M_{\mathrm{\Gamma }}:O\times O\times (0,\mathrm{\infty })\to [0,1]$ is expressed as

$$M_{\mathrm{\Gamma }}(x,y,t)=\frac{\min \{x,y\}}{\max \{x,y\}},\text{ for all }x,y\in O\text{ and }t>0.$$

Accordingly, $(O,M_{\mathrm{\Gamma }},{\ast }_m)$ constitutes ${\mathrm{\Gamma }}^{\prime }$-complete non-Archimedean graphical fuzzy metric space. Consider mapping $T:O\times O$ represented by

$$T(x)=\sqrt{3x+4}.$$

In addition, let a mapping $\psi \in \mathrm{\Psi }$ be defined by $\psi (\alpha )=\sqrt{\alpha }$ for $\alpha \in [0,1]$ and ${\mathrm{\Gamma }}^{\prime }$ be a subgraph of $\mathrm{\Gamma }$ such that $V({\mathrm{\Gamma }}^{\prime })=O$ and $\mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })=\mathrm{\Sigma }(\mathrm{\Gamma })$.

Take any $x,y\in O$ in which $(x,y)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$. This condition results in $y\le x$. Consequently,

$$Ty=\sqrt{3y+4}\le \sqrt{3x+4}=Tx$$

which reveals that $(Tx,Ty)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$. Hence, condition (FG$\psi$1) is fulfilled. Furthermore, given $x,y\in O$ satisfying $(x,y)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$, both inequalities $\sqrt{3x+4}\ge \sqrt{x}$ and $\sqrt{3y+4}\ge \sqrt{y}$ holds. Thus, for every $t>0$, we obtain

${\displaystyle \frac{\min \{\sqrt{3x+4},\sqrt{3y+4}\}}{\max \{\sqrt{3x+4},\sqrt{3y+4}\}}}$ $\ge {\displaystyle \frac{\min \{\sqrt{x},\sqrt{y}\}}{\max \{\sqrt{x},\sqrt{y}\}}}$

$={\displaystyle \frac{\sqrt{\min \{x,y\}}}{\sqrt{\max \{x,y\}}}}$

$=\sqrt{{\displaystyle \frac{\min \{x,y\}}{\max \{x,y\}}}}.$

Hence,

$$M_{\mathrm{\Gamma }}(Tx,Ty,t)=\frac{\min \{\sqrt{3x+4},\sqrt{3y+4}\}}{\max \{\sqrt{3x+4},\sqrt{3y+4}\}}\ge \sqrt{\frac{\min \{x,y\}}{\max \{x,y\}}}=\psi (M(x,y,t)),\text{for all }t>0$$

which verifies condition (FG$\psi$2). Therefore, mapping $T$ is qualified as a $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical $\psi$-contraction. A graphical visualization of condition (FG$\psi$2) is illustrated in Figure 3.

Figure 3. Visualization of $M_{\mathrm{\Gamma }}(Tx,Ty,t)$ (orange surface) and $\psi (M_{\mathrm{\Gamma }}(x,y,t))$ (blue surface), illustrating that $M_{\mathrm{\Gamma }}(Tx,Ty,t)\ge \psi (M_{\mathrm{\Gamma }}(x,y,t))$.

Next, take arbitrary $x_0\ge 4$. Then, $(x_0,T{x}_0)\in \mathrm{\Sigma }(\mathrm{\Gamma })$, which implies $T{x}_0\in {[x_0]}_{{\mathrm{\Gamma }}^{\prime }}^1$. Therefore, $x_0\in O$ exists where $T{x}_0\in {[x_0]}_G^l$ for some $l\in \mathbb{N}$. In addition, consider a $T$-Picard sequence $\{x_n\}$ begins from such an element $x_0$. It follows that $\{x_n\}$ is ${\mathrm{\Gamma }}^{\prime }$-termwise connected and converges to some ${\varpi }^{\ast }\in O$. For this particular choice of $x_0$, it can be observed that for each ${\varpi }^{\ast }\in O$, one can find $n_0\in \mathbb{N}$ such that either $(x_n,x)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ or $(x,x_n)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ at time $t$ for all $n\ge n_0$. For instance, by choosing $x_0=7$, one can verify that condition (ii) holds (see Table 1). Moreover, observe that $(O,M_{\mathrm{\Gamma }},\ast ,{\mathrm{\Gamma }}^{\prime },T)$ possess property (S) and $O_T$ is connected. Consequently, all assumptions of Theorem 3.11 are met. In particular, $4$ fits as the unique fixed point of mapping $T$.

n	$x_n$
0	7.000000
1	5.000000
2	4.358899
3	4.132396
$\mathrm{\cdots }$	$\mathrm{\cdots }$
13	4.000007
14	4.000003
15	4.000001
$\mathrm{\cdots }$	$\mathrm{\cdots }$
998	4.000000
999	4.000000
1000	4.000000

Table 1. The sequence $x_n$ with initial value $x_0=7$. As $n\to \mathrm{\infty }$, it is observed that $x_n\to 4$.

To end this section, some existing findings in the literature that follows directly from our main findings when extended into non-Archimedean graphical fuzzy metric setting are analyzed. To begin with, the extension of the work by Gregori and Sapena [20] is considered.

Definition 3.14.

Let $(O,M_{\mathrm{\Gamma }},\ast )$ be graphical fuzzy metric space, $T:O\to O$ be mapping and ${\mathrm{\Gamma }}^{\prime }$ be subgraph of $\mathrm{\Gamma }$ with $\mathrm{\Delta }\subseteq \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$. Consequently, the mapping $T$ is said to be a $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical contractive mapping on $O$ if conditions below are satisfied:

1. Mapping $T$ is edge preserving in $\mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$, specifically, $(x,y)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ implies $(Tx,Ty)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$;

2. There is $k\in (0,1)$ in which every $x,y\in O$ satisfying $(x,y)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$,

   $$\frac{1}{M(Tx,Ty,t)}-1\le k\left(\frac{1}{M(x,y,t)}-1\right)\text{ for all }t>0.$$ (2)

Corollary 3.15.

Suppose $(O,M_{\mathrm{\Gamma }},\ast )$ is a ${\mathrm{\Gamma }}^{\prime }$-complete non-Archimedean graphical fuzzy metric space and $T:O\to O$ is a $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical contractive mapping. Assume conditions below hold:

1. there exists $x_0\in O$ satisfying $T{x}_0\in {[x_0]}_{{\mathrm{\Gamma }}^{\prime }}^l$ for some $l\in \mathbb{N}$;

2. if a ${\mathrm{\Gamma }}^{\prime }$-termwise connected $T$-Picard sequence $\{x_n\}$ converges to some $x\in O$, then limit $x\in O$ and $n_0\in \mathbb{N}$ exist satisfying $(x_n,x)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ or $(x,x_n)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for all $n\ge n_0$;

Consequently, there exists ${\varpi }^{\ast }\in O$ where the $T$-Picard sequence $\{x_n\}$ begins with $x_0\in O$ is ${\mathrm{\Gamma }}^{\prime }$-termwise connected and converges to both ${\varpi }^{\ast }$ and $T{\varpi }^{\ast }$ in $O$.

Proof 3.16.

Consider a mapping $\psi \in \mathrm{\Psi }$ expressed by

$$\psi (t)=\frac{t}{t+k(1-t)},$$

then it immediately follows from Theorem 3.5.

In view of the preceding corollary, subsequent findings on existence and uniqueness of fixed points can be established by analogous reasoning from Theorem 3.8 and Theorem 3.11.

Corollary 3.17.

Suppose $(O,M_{\mathrm{\Gamma }},\ast )$ is a ${\mathrm{\Gamma }}^{\prime }$-complete non-Archimedean graphical fuzzy metric space and $T:O\to O$ is a $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical contractive mapping. Assume conditions below hold:

1. there exists $x_0\in O$ satisfying $T{x}_0\in {[x_0]}_{{\mathrm{\Gamma }}^{\prime }}^l$, for some $l\in \mathbb{N}$;

2. if a ${\mathrm{\Gamma }}^{\prime }$-termwise connected $T$-Picard sequence $\{x_n\}$ converges to some $x\in O$, then limit $x\in O$ and $n_0\in \mathbb{N}$ exist satifying $(x_n,x)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ or $(x,x_n)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ for all $n\ge n_0$.

Consequently, there exists ${\varpi }^{\ast }\in O$ where the $T$-Picard sequence $\{x_n\}$ begins with $x_0\in O$ is ${\mathrm{\Gamma }}^{\prime }$-termwise connected and converges to both ${\varpi }^{\ast }$ and $T{\varpi }^{\ast }$ in $O$. Moreover, if $(O,M_{\mathrm{\Gamma }},\ast ,{\mathrm{\Gamma }}^{\prime },T)$ possess property (S), then $T$ possess at least one fixed point in $O$.

Corollary 3.18.

Suppose $(O,M_{\mathrm{\Gamma }},\ast )$ is a ${\mathrm{\Gamma }}^{\prime }$-complete non-Archimedean graphical fuzzy metric space and $T:O\to O$ is a $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical contractive mapping. Assuming Corollary 3.17 applies and $O_T$, as a subgraph of ${\mathrm{\Gamma }}^{\prime }$, is connected, then mapping $T$ admits a unique fixed point.

Remark 3.19.

The original work of Mihet [29] was carried out in the framework of non-Archimedean fuzzy metric spaces. It was later addressed by Wang [48], who demonstrated that the results remain valid when non-Archimedean fuzzy metric spaces are replaced by fuzzy metric spaces. Therefore, taking Remark 2.9 into consideration, the following corollary can be deduced.

Corollary 3.20 ([48]).

Let $(O,M,\ast )$ be a complete fuzzy metric space. Suppose that $T:O\to O$ is a fuzzy $\psi$-contractive mapping. Consequently, mapping $T$ admits a unique fixed point.

Proof 3.21.

Following Remark 2.9, the graphs $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$ are defined by $\mathrm{\Xi }(\mathrm{\Gamma })=V({\mathrm{\Gamma }}^{\prime })=O$ and $\mathrm{\Sigma }(\mathrm{\Gamma })=\mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })=O\times O$. This construction results in graphical fuzzy metric space. Therefore, conditions of Theorem 3.11 are met, and the assertion holds.

Remark 3.22.

Wardowski [50] proposed a new contractive condition in fuzzy metric spaces by introducing fuzzy $H$-contractive mappings. More precisely, the class $H$ contains the mappings $\eta :(0,1]\to [0,\mathrm{\infty }\}$ where $\eta$ transforms $(0,1]$ onto $[0,\mathrm{\infty })$ and strictly decreasing. However, Gregori and Miñana [17] established that fuzzy $H$-contractive mappings are contained within the class of fuzzy $\psi$-contractive mappings. On the basis of this observation, the subsequent results are derived.

Corollary 3.23 ([50]).

Let $(O,M,\ast )$ be a complete fuzzy metric space. Suppose that $T:O\to O$ is a fuzzy $H$-contractive mapping with respect to $\eta \in H$ such that

$$\eta (M(Tx,Ty,t))\le k\eta (M(x,y,t))$$

for all $x,y\in O$ and $t>0$, where $k\in (0,1)$. Consequently, the mapping $T$ admits a unique fixed point.

Proof 3.24.

Graphs $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$ are defined by $\mathrm{\Xi }(\mathrm{\Gamma })=V({\mathrm{\Gamma }}^{\prime })=O$ and $\mathrm{\Sigma }(\mathrm{\Gamma })=\mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })=O\times O$, and mapping $\psi$ is defined by

$$\psi (t)={\eta }^{-1}(k\cdot \eta (t)).$$

In this setting, it immediately follows from Theorem 3.11.

Corollary 3.25 ([20]).

Let $(O,M,\ast )$ be a complete fuzzy metric space. Assume $T:O\to O$ is a mapping such that

$$\frac{1}{M(Tx,Ty,t)}-1\le k\left(\frac{1}{M(x,y,t)}-1\right)$$

for all $x,y\in O$ and $t>0$, with $k\in (0,1)$. Consequently, the mapping $T$ admits a unique fixed point.

4. Employing Fixed Point Technique to Establish Solutions of Nonlinear Fractional Differential Equations

Fixed point analysis and fractional calculus have become imperative and intertwined with each other in mathematical modelling, especially in practical investigations in nonlinear sciences and engineering over the past years. The former has been frequently applied to examine the existence and uniqueness of solutions for multiple forms of differential and integral equations [55], while the latter serves as an enlargement of classical calculus. In classical calculus, integrals and derivatives are limited to positive integer orders. Although this methodology has proven effective in modelling certain natural phenomena and real-life applications, the increasing complexity of systems arising from the scientific and technological developments has prompted the search for new approaches to deliver more reliable results [51].

Fractional calculus extends integrals and derivatives to arbitrary real (including fractional) orders, thereby allowing the effective modelling of complex mathematical problems. In view of the advantages of fractional-order models over their integer-order counterparts in enhancing reliability and precision, considerable attention has been directed toward their numerical study. Fractional calculus has now been widely adopted across various scientific and engineering disciplines, such as physics, biology, mathematical oncology, chemistry, economics, image processing, and electronics [2, 3, 28, 30, 45, 46, 54].

Recently, fixed point techniques were utilized to explore the existence and uniqueness of solutions for fractional differential equations [1, 9, 14, 21, 22, 47, 49]. Inspired by the study in [57], this section aims to apply the obtained result to prove the existence and uniqueness of solutions for the following nonlinear fractional differential equation:

$$D_{0^{+}}^{p}y(t)=g(t,y(t)),t\in [0,1]$$ (3)

subject to the boundary conditions

$y(0)$ $=-y^{\prime }(0),$

$y(1)$ $=-y^{\prime }(1),$

where is the fractional order, $D_{0^{+}}^p$ represents the Caputo fractional derivative of order $p$, and $g:[0,1]\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a continuous function.

Suppose $O=C([0,1],ℝ)$ is the space of all continuous function defined on $[0,1]$ to $ℝ$. Consider graphs $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$ such that $\mathrm{\Gamma }={\mathrm{\Gamma }}^{\prime }$, $\mathrm{\Xi }(\mathrm{\Gamma })=O$ and

$$\mathrm{\Sigma }(\mathrm{\Gamma })=\mathrm{\Delta }\cup \{(x,y)\in O\times O:x(t)\le y(t)\text{ for all }t\in [0,1]\}.$$

Let graphical metric $d_{\mathrm{\Gamma }}:O\times O\to ℝ$ defined as $d_{\mathrm{\Gamma }}(x,y)={sup}_{t\in [0,1]}|x(t)-y(t)|$ for all $x,y\in O$. Accordingly, a graphical fuzzy metric $M_{\mathrm{\Gamma }}:O\times O\times (0,\mathrm{\infty })\to [0,1]$ is defined by

$$M_{\mathrm{\Gamma }}(x,y,w)=\exp \left\{-\frac{d_{\mathrm{\Gamma }}(x,y)}{w}\right\}$$

for all $x,y\in O$ and $w>0$. It is straightforward to establish that $(O,M_{\mathrm{\Gamma }},{\ast }_p)$ is a ${\mathrm{\Gamma }}^{\prime }$-complete graphical fuzzy metric space where ${\ast }_p$ is a product $t$-norm.

Observe that the nonlinear fractional differential equation (3) is equivalent to integral equation below:

$y(t)$ $={\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-1}(1-t)g(s,y(s))𝑑s+{\displaystyle \frac{1}{\mathrm{\Gamma }(p-1)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-2}(1-t)g(s,y(s))𝑑s$ (4)

$+{\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^t}{(t-s)}^{p-1}g(s,y(s))𝑑s$

where $\mathrm{\Gamma }(p)$ is a Gamma function. This implies for any $y\in O$ which solves the integral equation (4) also serves as a solution for the fractional differential equation (3). The presence and the distinctness of a solution for (3) is demonstrated in the next theorem.

Theorem 4.1.

Let $(O,M_{\mathrm{\Gamma }},{\ast }_p)$ be a ${\mathrm{\Gamma }}^{\prime }$-complete graphical fuzzy metric space defined as above. Assume $T:O\to O$ to be an integral operator defined by:

$Ty(t)$ $={\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-1}(1-t)g(s,y(s))𝑑s+{\displaystyle \frac{1}{\mathrm{\Gamma }(p-1)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-2}(1-t)g(s,y(s))𝑑s$

$+{\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^t}{(t-s)}^{p-1}g(s,y(s))𝑑s$

where $t\in [0,1]$ and the function $g:[0,1]\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is continuous. Suppose that conditions below hold:

1. For all $x,y\in O$ and $s\in [0,1]$,

   $$|g(s,x(s))-g(s,y(s))|\le \frac{|x(s)-y(s)|}{2};$$

2. For $t\in [0,1]$ and ,

Then, the nonlinear fractional differential equation (3) possess a unique solution in $O$.

Proof 4.2.

Consider a mapping $\psi \in \mathrm{\Psi }$ defined by $\psi (a)=a^{\kappa }$ for $a\in [0,1]$ and . For any $x,y\in O$ such that $(x,y)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$ and $t\in [0,1]$, we have

$|Tx(t)-Ty(t)|$ $=|({\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-1}(1-t)g(s,x(s))ds+{\displaystyle \frac{1}{\mathrm{\Gamma }(p-1)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-2}(1-t)g(s,x(s))ds$

$+{\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^t}{(t-s)}^{p-1}g(s,x(s))ds)-({\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-1}(1-t)g(s,y(s))ds$

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

$=|{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-1}[g(s,x(s))-g(s,y(s))]ds+{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p-1)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-2}[g(s,x(s))$

$-g(s,y(s))]ds+{\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle \int }{}^t{}_0(t-s){}^{p-1}[g(s,x(s))-g(s,y(s))]ds|$

$\le {\displaystyle \frac{1-t}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-1}\left|g(s,x(s))-g(s,y(s))\right|𝑑s+{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p-1)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-2}|g(s,x(s))$

$-g(s,y(s))\left|ds+{\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^t}{(t-s)}^{p-1}\right|g(s,x(s))-g(s,y(s))|ds$

$\le {\displaystyle \frac{1-t}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-1}{\displaystyle \frac{|x(s)-y(s)|}{2}}𝑑s+{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p-1)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-2}{\displaystyle \frac{|x(s)-y(s)|}{2}}𝑑s$

$+{\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^t}{(t-s)}^{p-1}{\displaystyle \frac{|x(s)-y(s)|}{2}}𝑑s$

$\le {\displaystyle \frac{1-t}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-1}\underset{t\in [0,1]}{sup}{\displaystyle \frac{|x(t)-y(t)|}{2}}ds+{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p-1)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-2}\underset{t\in [0,1]}{sup}{\displaystyle \frac{|x(t)-y(t)|}{2}}ds$

$+{\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^t}{(t-s)}^{p-1}\underset{t\in [0,1]}{sup}{\displaystyle \frac{|x(t)-y(t)|}{2}}ds$

$=\underset{t\in [0,1]}{sup}{\displaystyle \frac{|x(t)-y(t)|}{2}}[{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-1}ds+{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p-1)}}{\displaystyle {\int }_0^1}{(1-s)}^{p-2}ds$

$+{\displaystyle \frac{1}{\mathrm{\Gamma }(p)}}{\displaystyle {\int }_0^t}{(t-s)}^{p-1}ds]$

$=\underset{t\in [0,1]}{sup}{\displaystyle \frac{|x(t)-y(t)|}{2}}\left[{\displaystyle \frac{1-t}{p\mathrm{\Gamma }(p)}}+{\displaystyle \frac{1-t}{(p-1)\mathrm{\Gamma }(p-1)}}+{\displaystyle \frac{t^p}{p\mathrm{\Gamma }(p)}}\right]$

$={\displaystyle \frac{1}{2}}\underset{t\in [0,1]}{sup}|x(t)-y(t)|\left[{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p+1)}}+{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p)}}+{\displaystyle \frac{t^p}{\mathrm{\Gamma }(p+1)}}\right]$

$\le \underset{t\in [0,1]}{sup}|x(t)-y(t)|\underset{t\in [0,1]}{sup}{\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p+1)}}+{\displaystyle \frac{1-t}{\mathrm{\Gamma }(p)}}+{\displaystyle \frac{t^p}{\mathrm{\Gamma }(p+1)}}\right]$

$=\underset{t\in [0,1]}{sup}\kappa |x(t)-y(t)|,$

where

$$\kappa =\underset{t\in [0,1]}{sup}\frac{1}{2}\left[\frac{1-t}{\mathrm{\Gamma }(p+1)}+\frac{1-t}{\mathrm{\Gamma }(p)}+\frac{t^p}{\mathrm{\Gamma }(p+1)}\right].$$

From the above inequality, we can deduce that, for any $x,y\in O$ satisfying $(x,y)\in \mathrm{\Sigma }({\mathrm{\Gamma }}^{\prime })$, $t\in [0,1]$ and $w>0$,

$\underset{t\in [0,1]}{sup}|Tx(t)-Ty(t)|$ $\le \underset{t\in [0,1]}{sup}\kappa |x(t)-y(t)|$

$-\underset{t\in [0,1]}{sup}|Tx(t)-Ty(t)|$ $\ge -\underset{t\in [0,1]}{sup}\kappa |x(t)-y(t)|$

$-{\displaystyle \frac{{sup}_{t\in [0,1]}|Tx(t)-Ty(t)|}{w}}$ $\ge -\underset{t\in [0,1]}{sup}{\displaystyle \frac{\kappa |x(t)-y(t)|}{w}}$

$\exp \left\{-{\displaystyle \frac{{sup}_{t\in [0,1]}|Tx(t)-Ty(t)|}{w}}\right\}$ $\ge \exp \left\{-{\displaystyle \frac{{sup}_{t\in [0,1]}\kappa |x(t)-y(t)|}{w}}\right\}$

$\exp \left\{-{\displaystyle \frac{d_{\mathrm{\Gamma }}(Tx,Ty)}{w}}\right\}$ $\ge \exp \left\{-{\displaystyle \frac{\kappa d_{\mathrm{\Gamma }}(x,y)}{w}}\right\}.$

Therefore, we reach the conclusion that

$$M_{\mathrm{\Gamma }}(Tx,Ty,w)=\exp \left\{-\frac{d_{\mathrm{\Gamma }}(Tx,Ty)}{w}\right\}\ge \exp \left\{-\frac{\kappa d_{\mathrm{\Gamma }}(x,y)}{w}\right\}={\left[\exp \left\{-\frac{d_{\mathrm{\Gamma }}(x,y)}{w}\right\}\right]}^{\kappa }=\psi (M_{\mathrm{\Gamma }}(x,y,w)).$$

Consequently, the mapping $T$ is $(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })$-fuzzy graphical $\psi$-contraction. The rest of the assumptions in Theorem 3.11 can be verified accordingly. Hence, a fixed point ${\varpi }^{\ast }\in O$ of integral operator $T$ exists, which proves the existence and the uniqueness of a solution for the nonlinear fractional differential equation (3).

5. Conclusion

In this manuscript, fuzzy graphical $\psi$-contractions in graphical fuzzy metric spaces are introduced. In this structure, fixed point results for such a contractive condition are established accompanied by supporting examples. The generality of the obtained results is discussed with reference to earlier findings in the literature. Furthermore, an application of the obtained results to nonlinear fractional differential equations is presented to demonstrate their potential applicability to suitable real-world problems.

The interplay between fixed point theory and graph theory constitutes a powerful avenue for exploring mathematical problems and applications. In recent years, the investigation of best proximity point theory under various graphical metric structures is beginning to develop; for example, see [11], [12], [27], [37]. Thus, a natural continuation of this study is the expansion to best proximity point theory, which also provides broader coverage for applications in fractional calculus to address real world phenomena in science and engineering.

The class of contractive mapping proposed in this work can be viewed as a particular member of a broader class of mappings. Notably, in a recent study, Shukla et al. [43] presented a new class of mappings, namely, fuzzy $Z$-contractive mappings under fuzzy metric spaces. They demonstrated that this class is more general and includes several other classes, including fuzzy $\psi$-contractive mappings. Thus, investigating the connection between fuzzy $Z$-contractive mappings and fuzzy $\psi$-contractive mappings within graphical fuzzy metric spaces opens a new direction for continued research.

Fixed point theory is not confined to metric structures and can be developed within different frameworks, for example $L$-fuzzy topological spaces [35], non-Archimedean Lie $C^{\ast }$-algebras [6] and intuitionistic fuzzy normed spaces [36]. Examining the connections between present results across different structures contributes to a profound understanding in this area. Moreover, the application and analysis of these results in fuzzy analysis and stability theory offers a promising direction for future research.

Drawing from the above conclusions, this paper concludes by presenting several open problems:

1. In the current underlying space, is it possible to study fuzzy graphical $\psi$-proximal contractions and establish their best proximity point results?

2. Wihin the framework of graphical fuzzy metric spaces, can the property (S) be substituted with an alternative condition for establishing the existence of fixed points?

3. Can the non-Archimedean property be substituted by a novel definition of Cauchyness in graphical fuzzy metric space?

4. How can the class of fuzzy graphical $Z$ contractions be defined and what are their corresponding fixed point results in the graphical fuzzy metric structure?

5. Can analogous $\psi$-contractive arguments be applied in intuitionistic fuzzy normed spaces?