Fixed point ($\mathcal{FP}$) theory is a powerful mathematical tool with numerous applications in various domains. It has played a crucial role in creating vital concepts and approaches and is an active field of study and development. It is commonly used in pure and applied mathematics as a bridge between topology and analysis. $\mathcal{FP}$ theory is a burgeoning scientific field with significant applications in various sectors. It is concerned with research demonstrating that, under certain conditions, a self-map on a non-empty set, M, admits one or more $\mathcal{FP}$s. $\mathcal{FP}$ theorems prove the existence and uniqueness of $\mathcal{FP}$s. Because of its application in both pure and applied mathematics, the metric $\mathcal{FP}$ theorem has a considerable effect on academia. Banach made an unexpected discovery in 1922 that became known as the Banach contraction principle. One of the most adaptive conclusions in $\mathcal{FP}$ theory and approximation theory is the Banach contraction principle. It changed and generalized in a variety of ways over time. The Banach contraction concept is frequently used in the literature on $\mathcal{FP}$ theory. Numerous authors have generalized this conclusion by employing other contraction mappings in other metric spaces. The classical principle of contraction mapping states if $(\complement ,d)$ is a complete metric space and $\beth :\complement \to \complement$ such that $d(\beth (\varpi ),\beth (\varsigma \left)\right)\le \alpha d(\varpi ,\varsigma )$ for all $\varpi \varsigma \in \complement$, where, $\alpha \in$$[0,1)$, then ℶ admits a unique $\mathcal{FP}$. Banach’s [19]$\mathcal{FP}$ hypothesis helps compute mathematics for examining the existence of solutions to nonlinear integral equations, systems of linear equations, and nonlinear differential equations, as well as manifesting algorithms related to convergence. Many variations on Banach’s $\mathcal{FP}$ theorem have been proposed; see [12].

In 1971, Cirić [20], Reich [21], and Rus [16] established a $\mathcal{FP}$ theorem for mappings $\beth :\complement \to \complement$, which has been around for decades, meets the following requirement: where $a,b\ge 0$, and $a+2b<1$. If $b=0$, condition 1.1 reduces to Banach contraction. where $a\in [0,1)$, whereas for $a=0,$ condition 1.1 reduces to Kannan’s contraction. where, $b\in [0,\frac{1}{2})$. Kannan mappings are essential in mathematics and its applications because they provide a flexible framework for establishing the existence and uniqueness of $\mathcal{FP}$s in metric spaces. They are an important instrument in the study of nonlinear analysis, numerical techniques, and various scientific and technical domains. Their significance stems from their capacity to determine the convergence of iterative algorithms and solve a wide range of mathematical and practical problems. In fact, Kannan’s fixed-point theorem is a noteworthy finding in mathematics, especially regarding complete metric spaces. The theorem gives circumstances in which it is possible to derive a unique $\mathcal{FP}$ in such spaces without continuous mapping. This is important because continuity of the mapping is a necessary assumption in many traditional fixed-point theorems.

Kannan’s theorem has several applications.

1. Eliminating the Continuity Requirement: Kannan’s theorem is applicable in a wider range of scenarios, encompassing non-continuous maps, in contrast to several previous fixed-point theorems that need continuous mapping. It may, therefore, be used to solve an enormous variety of mathematical and real-world issues.

2. Metric Completeness Characterization: Kannan’s theorem is remarkable since it describes a space’s metric completeness in terms of the presence of a $\mathcal{FP}$. That is to say, a metric space is complete if and only if there is only one $\mathcal{FP}$ for a certain kind of mapping. This offers a profound relationship between the existence of $\mathcal{FP}$s and the notion of metric completeness.

3. Valuable and Interesting Mappings: Since it illuminates the characteristics of full metric spaces and their topological structure, studying mappings that meet Kannan’s theorem requirements is important. There are several practical applications for comprehending these mappings and the corresponding $\mathcal{FP}$s in mathematics and its applications, such as functional analysis and engineering mathematics, see, eg., [??]. In conclusion, Kannan’s fixed-point theorem is a key finding in mathematics that sheds light on the relationship between metric completeness and the presence of $\mathcal{FP}$s and provides a fresh viewpoint on fixed-point theory in complete metric spaces. It is extensively researched and used in many different mathematical fields, and it has had a long-lasting influence on mathematical study. Therefore, the $\mathcal{FP}$ results established in [20,21], under slightly different forms, are true generalizations of Banach’s contraction principle [19] and Kannan’s [22]$\mathcal{FP}$ theorem.

On the other side, a significant expansion of classical metric spaces, the notion of fuzzy metric spaces (developed by Kramosil and Michalek [5]) for a more adaptable and comprehensive treatment of distances and continuity. There are uses for fuzzy metric spaces in many domains, such as fixed-point theory and topology. The concepts of fuzzy metric spaces and Hausdorff topology on fuzzy metric spaces were proposed by George and Veeramani [8]. It was demonstrated that this topology is metrizable when applied to specific fuzzy metric spaces, offering a more reliable framework for researching topological characteristics. It is observed that related fuzzy metric spaces may be generated from any classical metric space. This connection demonstrates how flexible fuzzy metric spaces represent various notions of distance. $\mathcal{FP}$ theorems in various mathematical spaces are now better understood because of the introduction of graphical structures and graph theory to $\mathcal{FP}$ theory. These advancements have applications in engineering sciences, functional analysis, and optimization. Fixed-point solutions for set-valued graphical contraction mappings were recently provided by Chen et al.[14,15]. This work extends the study of convexity to the context of graphical structures by introducing the idea of a graphical convex metric space. Younis et al. (2022) [33] discussed the graphical structure of extended b-metric spaces. Studying the graphic structure of such generalized graphical metric spaces might provide insights into fixed-point theory and allow for greater flexibility in the definition of distances. Applications for fuzzy graph $\mathcal{FP}$s are found in many technical disciplines where imprecision and uncertainty are typical. They make it possible to make the best decisions in situations like network design with erratic connections, environmental management with fluctuating parameters like weather and pollution levels, and transportation route optimization in the face of changing traffic circumstances. They also impact production scheduling in unpredictable events, power grid control in the face of variable renewable energy sources, and robotics motion planning in intricate, dynamic situations. They support the development and management of cities in the face of changing demographic and economic forces in urban planning and smart cities. Engineers may use fuzzy graph $\mathcal{FP}$s as a flexible tool to address real-world issues with ambiguous data and make defensible choices in changing environments.

This article’s primary goal is to highlight and study many engineering applications of the current $\mathcal{FP}$ theory. To do this, the study offers a set of innovative fuzzy contraction mappings and uses them to construct fixed-point outcomes. Notably, in the setting of fuzzy metric spaces, these discoveries are the first of their kind. This study advances the practical use of advanced fixed-point theory in engineering applications and emphasizes its potential to handle complicated real-world challenges in various scientific domains.

This paper has two parts. Using Kannan-type contractions, we first derive $\mathcal{FP}$ and common $\mathcal{FP}$ theorems for single-valued mappings in the domain of fuzzy metric spaces. Conversely, we offer the £-Kannan-contractions as an extension of fuzzy £-contraction to show $\mathcal{FP}$s. Our results include a generalization of well-known results of Gregori and Sapna [24] and an extension of Jachymski’s notable results [13] in fuzzy metric spaces. Our primary objective is to generate several applications of the defined fuzzy contractions to different nonlinear models originating from engineering science and technology.

To facilitate the reader, we will now review key concepts and properties that are relevant to this study. Throughout the paper, we will write fuzzy metric spaces as ${\mathbb{F}}^{\mathbb{M}}-$spaces, and the sets of natural numbers be denoted by $\mathbb{N}$. The set of integers i.e., $\mathbb{Z}=\mathbb{N}\cup (-\mathbb{N})\cup \left\{0\right\}$, and ${\mathbb{Z}}^{+}:={\mathbb{N}}_{\mathrm{and}}{\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$. The symbols ${\mathbb{R}}_0^{+}$ denotes the set of non-negative real numbers. Throughout the paper, all sets are assumed to be nonempty.

For further synthesis and basic concepts like the Cauchy sequence, the convergence of the Cauchy sequence, and completeness in fuzzy setup, we refer to the readers’ [8].

It’s worth noting that the topologies generated by the standard fuzzy metric and its corresponding measure are similar to one another.

Let £ be a directed graph with vertices $\varpi$ and $\varsigma$. In this context, a path in £ is a sequence of vertices ${\left\{{\varpi }_j\right\}}_{j=0}^{\mathbb{M}}$, where there are $(\mathbb{M}+1)$ vertices, and it satisfies the conditions: the first vertex is $\varpi$, the last vertex is $\varsigma$, and for every j from 1 to $\mathbb{M}$, there is a directed edge from $\varpi j-1$ to ${\varpi }_j$ in the graph.

A graph £ is considered linked or connected when a path exists connecting any pair of its vertices. This definition of connectivity applies to directed graphs. When the graph £ is undirected, we refer to it as weakly connected, meaning there is a path connecting every pair of vertices, and the direction of the edges is not considered in determining connectivity. Shukla et al. [28] are to be credited with the following annotations [28]:

Unless otherwise stated, we consider all the graphs to be direct.

To review different fundamental ideas related to graph-$\mathcal{FP}$s, we recommend the reader to[32,33].

In a graph £, a path of length $N(N\in \mathbb{N})$ from vertex $\varpi$ directing to another vertex $\varsigma$ is represented by the sequence ${\left(\varpi i\right)}_{i=0}^N$ of $N+1$ vertices such that ${\varpi }_0=\varpi ,{\varpi }_n=\varsigma$, and $\left({\varpi }_{n-1},{\varpi }_n\right)\in \mathbb{E}\left(\pounds \right)$, $1\le i\le N$.

It is worth noting that each Picard operator is also a weak Picard operator. The $\mathcal{FP}$ of a weak Picard operator cannot be truely unique.

The next lemma will help with what comes next.

The component of £ that contains $\varpi$ is made up of edges, vertices, and a path that starts at ${\varpi }_0$. In this instance, the relation R is defined on $\mathcal{V}\left(\pounds \right)$ by the following rule:

$\varsigma R\varkappa$ if there is a path in £ from $\varsigma$ to $\varkappa$. In this scenario, $\mathcal{V}\left({\pounds }_{\varpi }\right)={\left[\varpi \right]}_{\pounds }$, where, the equivalence class of this relation is ${\left[\varpi \right]}_{\pounds }$. Thus, the relationship between ${\pounds }_{\varpi }$ is self-evident.

We will now talk about different sorts of mapping continuity. The first is well-known in metric $\mathcal{FP}$ theory and is frequently employed.

Recent findings provide the necessary conditions for mapping to a Picard Operator (PO) when the space $(\complement ,d)$ has a graph structure. This discovery, first revealed by Jachymski [13], expands on its relationship with the Kelisky-Rivlin theorem about iterates of the Bernstein operators in the space $C[0,1]$.

The notion of £-Kannan operators is proposed in this study to evaluate the $\mathcal{FP}$s for Kannan operators in ${\mathbb{F}}^{\mathbb{M}}-$spaces given a graph £. We assume that $(\complement ,d)$ is a metric space and that £ is a directed graph with the criteria that $\mathcal{V}\left(\pounds \right)=\complement ,\mathbb{E}\left(\pounds \right)\supseteq \Delta$ and that £ has no parallel edges. An instance of $Fi{x}_{\beth }$ is a collection of all $\mathcal{FP}$s for a mapping ℶ.

The ensuing notion forms the basis for the recommended results throughout the course of the work.

In the context of ${\mathbb{F}}^{\mathbb{M}}-$spaces, we generate fixed-point and common fixed solutions for single-valued fuzzy-Kannan-type contractions.

The primary outcome about common $\mathcal{FP}$for two single-valued mappings $\mathcal{S},\beth$ within the context of ${\mathbb{F}}^{\mathbb{M}}-$spaces is as follows.

The following provides the convergence result for fuzzy-Kannan-contraction of type-I for a single-valued mapping ℶ.

The fuzzy-Kannan-contraction of type-II within the fuzzy-metric setup for two mappings $\mathcal{S}\mathrm{and}\beth$ forms the basis of the common $\mathcal{FP}$ result that follows.

The convergence result of a single mapping ℶ for fuzzy-Kannan-contraction of type-II is as follows.

Specifically, we consider $(\complement ,{\mathbb{F}}^{\mathbb{M}},\ast )$ as a ${\mathbb{F}}^{\mathbb{M}}$-space with a graph indicated by £. In this case, £ represents a directed graph, with its vertex set $V\left(\pounds \right)$ being ∁ and $\mathbb{E}\left(\pounds \right)\supseteq \Delta$. We ensure that there are no parallel edges in £. Each edge corresponds to a different element from $(\complement ,{\mathbb{F}}^{\mathbb{M}},\ast )$. The following findings are essential in demonstrating the upcoming $\mathcal{FP}$ theorems.