Fixed Point Results via $\mathcal{G}$-Transitive Binary Relation and Fuzzy $\mathcal{L}$-$\mathcal{R}$-Contraction

Abstract

In this study, we initiate the concept of fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction and establish some fixed point results involving a $\mathcal{G}$-transitive binary relation and fuzzy $\mathcal{L}$-simulation functions, by employing suitable hypotheses on a fuzzy metric space endowed with a binary relation. The presented results unify, generalize, and improve various previous findings in the literature.

1. Introduction

As it covers a broad variety of mathematical tools for addressing many sorts of issues that arise from other fields of mathematics, fixed point theory is one of the most important and fundamental study domains in nonlinear functional analysis. Since its establishment, the Banach contraction principle has been researched and improved in several abstract metric spaces using various techniques. It is a key component of the metric fixed point result. A novel direction in fixed point theory research was recently pioneered by Khojasteh et al. [1] by using a class of control functions known as simulation functions. Cho [2] discussed a new type of contraction known as the L-contraction and demonstrated some fixed point results in a generalized metric space for such a contraction.

Another significant and thriving area of fixed point theory is relation-theoretic fixed point results, which was first introduced by Turinici [3] by putting out the idea of an order-theoretic fixed point result. A natural order-theoretic interpretation of the Banach contraction principle was offered in 2004 by Ran and Reurings [4], who also suggested applying their findings to matrix equations. By merging several well-known pertinent order-theoretic ideas with an arbitrary binary relation, Alam and Imdad [5] successfully demonstrated a relation-theoretic variant of the Banach contraction principle.

The fuzzy set concept was developed by L.A. Zadeh [6] in 1965 as a novel mathematical method for interacting with ambiguity and vagueness in the physical world. The idea of fuzzy sets has become a crucial and significant modeling tool.

The theory of fuzzy sets has emerged into an important and critical modeling tool. Kramosil and Michalek [7] initiated a fuzzy metric space by expanding on the idea of a probabilistic metric space to the fuzzy frame. Moreover, in order to obtain a Hausdorff topology, George and Veeramani [8] reconfigured Kramosil and Michalek’s concept of a fuzzy metric space. In this line, Gregori and Sapena [9] presented the idea of fuzzy contractive mappings, who also obtained certain fixed point outcomes. The concept of $\psi$-contractive mappings was later proposed by Mihet [10]. Recent research by Abdelhamid Moussaoui et al. [11] (see also [12]) introduced the idea of $\mathcal{FZ}$-contractions and initiated a fuzzy metric version of the simulation function technique. Further research consequences of numerous forms of contractions in fuzzy metric spaces and other structures are provided in [11,12,13,14,15,16,17,18]

In this study, we introduce the idea of a fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction and develop some fixed point results encompassing the $\mathcal{G}$-transitive binary relation and fuzzy $\mathcal{L}$-simulation functions by using appropriate hypotheses on the fuzzy metric space equipped with a binary relation. The results combine, generalize, and enhance a number of prior research results.

2. Preliminaries

Definition 1

([19]). A continuous binary operation $⋏:[0,1]\times [0,1]⟶[0,1]$ is called a continuous t-norm if it is commutative, associative, and

1.

${\wp }_1⋏1={\wp }_1$ for all ${\wp }_1\in [0,1]$;

2.

${\wp }_1⋏{\wp }_2\le {\wp }_3⋏{\wp }_4$ whenever ${\wp }_1\le {\wp }_3$ and ${\wp }_2\le {\wp }_4$, for all ${\wp }_1,{\wp }_2,{\wp }_3,{\wp }_4\in [0,1]$.

Example  1.

The following are some classic continuous t-norm examples: minimum t-norm, that is, ${\wp }_1{⋏}_m{\wp }_2=min\{{\wp }_1,{\wp }_2\}$, Lukasiewicz t-norm, ${\wp }_1{⋏}_L{\wp }_2=max\{{\wp }_1+{\wp }_2-1,0\}$, and product t-norm ${\wp }_1{⋏}_p{\wp }_2={\wp }_1·{\wp }_2$, for all ${\wp }_1,{\wp }_2\in [0,1]$.

Definition 2

([8]). Let $\mathcal{K}$ be a nonempty set, $\beta :{\mathcal{K}}^2\times (0,+\infty )\to [0,1]$ a fuzzy set, and ⋏ a continuous t-norm. The triple $(\mathcal{K},\beta ,⋏)$ is called a fuzzy metric space (in short $\mathcal{FMS}$) if

$(\mathcal{M}1)$

$\beta (\omega ,\varpi ,\varsigma )>0$;

$(\mathcal{M}2)$

$\beta (\omega ,\varpi ,\varsigma )=1$ if and only if $\omega =\varpi$;

$(\mathcal{M}3)$

$\beta (\omega ,\varpi ,\varsigma )=\beta (\varpi ,\omega ,\varsigma )$;

$(\mathcal{M}4)$

$\beta (\omega ,\varpi ,\varsigma )\ast \beta (\varpi ,\vartheta ,\kappa )\le \beta (\omega ,\vartheta ,\varsigma +\kappa )$;

$(\mathcal{M}5)$

$\beta (\omega ,\varpi ,.):(0,+\infty )\to [0,1]$ is continuous;

for all $\omega ,\varpi ,\vartheta \in \mathcal{K}$ and $\varsigma ,\kappa >0$.

Example  2

([8]). Let $(\mathcal{K},E)$ be a metric space. Define the function $\beta :\mathcal{K}\times \mathcal{K}\times (0,+\infty )\to [0,1]$ by $\beta (\omega ,\varpi ,\varsigma )=\frac{\varsigma }{\varsigma +E(\omega ,\varpi )}$, for all $\omega ,\varpi \in \mathcal{K}$, $\varsigma >0$. Then, $(\mathcal{K},\beta ,{⋏}_m)$ is an $\mathcal{FMS}$.

Example  3

([8,20]). Let $\mathcal{K}$ be a nonempty set, $\pi :[0,+\infty )\to {\mathbb{R}}^{+}$ be an increasing continuous function and $\sigma :\mathcal{K}\to {\mathbb{R}}^{+}$ be a one-to-one function. For fixed $r,s>0$, define $\beta :\mathcal{K}\times \mathcal{K}\times (0,+\infty )\to [0,1]$ by

$$\begin{array}{c}\hfill \beta (\omega ,\varpi ,\varsigma )={\left(\frac{{(min\{\sigma (\omega ),\sigma (\varpi )\})}^r+\pi (\varsigma )}{{(max\{\sigma (\omega ),\sigma (\varpi )\})}^r+\pi (\varsigma )}\right)}^s,\mathrm{for}\mathrm{all}\omega ,\varpi \in \mathcal{K}\mathrm{and}\varsigma >0.\end{array}$$

Then, $(\mathcal{K},\beta ,{⋏}_p)$ is an $\mathcal{FMS}$.

Example  4

([8,20]). Let $(\mathcal{K},E)$ be a metric space and $g:{\mathbb{R}}^{+}\to [0,+\infty )$ be an increasing continuous function. Define $\beta :\mathcal{K}\times \mathcal{K}\times (0,+\infty )\to [0,1]$ by

$$\begin{array}{c}\hfill \beta (\omega ,\varpi ,\varsigma )=exp\left(-\frac{E(\omega ,\varpi )}{g(\varsigma )}\right),\mathrm{for}\mathrm{all}\omega ,\varpi \in \mathcal{K}\mathrm{and}\varsigma >0.\end{array}$$

Then, $(\mathcal{K},\beta ,{⋏}_p)$ is an $\mathcal{FMS}$.

Lemma 1

([13]). $\beta (\omega ,\varpi ,.)$ is nondecreasing for all $\omega ,\varpi$ in $\mathcal{K}$.

Definition 3

([8]). Let $(\mathcal{K},\beta ,⋏)$ be an $\mathcal{FMS}$.

1.

A sequence $\{{\omega }_q\}\subseteq \mathcal{K}$ is called convergent to $\omega \in \mathcal{K}$ iff ${lim}_{q\to +\infty }\beta ({\omega }_q,\omega ,\varsigma )=1$ for all $\varsigma >0.$

2.

A sequence $\{{\omega }_q\}\subseteq \mathcal{K}$ is called a Cauchy sequence iff for each $\epsilon \in (0,1)$ and $\varsigma >0$, there exists $p_0\in \mathbb{N}$ such that $\beta ({\omega }_p,{\omega }_q,\varsigma )>1-\epsilon$ for all $q,p$$\ge p_0.$

3.

A complete $\mathcal{FMS}$ is an $\mathcal{FMS}$ in which every Cauchy sequence is convergent.

The idea of fuzzy contractive mapping was first proposed by Gregori and Sapena [9] as follows.

Definition 4

([9]). Let $(\mathcal{K},\beta ,⋏)$ be an $\mathcal{FMS}$. A mapping $\mathcal{G}:\mathcal{K}\to \mathcal{K}$ is called fuzzy contractive mapping if there exists $k\in (0,1)$ such that

$$\begin{array}{c}\hfill \frac{1}{\beta (\mathcal{G}\omega ,\mathcal{G}\varpi ,\varsigma )}-1\le k\left(\frac{1}{\beta (\omega ,\varpi ,\varsigma )}-1\right),\end{array}$$

for each $\omega ,\varpi \in \mathcal{K}$ and $\varsigma >0.$

Definition 5.

We say that the function $\iota :(0,1]\times (0,1]⟶\mathbb{R}$ is a fuzzy $\mathcal{L}$-simulation function if

$(\mathcal{FL}1):$

$\iota (1,1)=1$;

$(\mathcal{FL}2):$

$\iota (\omega ,\varpi )<\frac{\omega }{\varpi }$ for all $\omega ,\varpi \in (0,1)$;

$(\mathcal{FL}3):$

if $\{{\omega }_p\},\{{\varpi }_p\}$ are sequences in $(0,1]$ such that ${lim}_{p\to +\infty }{\omega }_p={lim}_{p\to +\infty }{\varpi }_p<1$ then ${lim}_{p\to +\infty }sup\iota ({\omega }_p,{\varpi }_p)<1$.

We denote by $\mathcal{FL}$ the class of all fuzzy $\mathcal{L}$-simulation functions.

Example  5.

Let $\iota :(0,1]\times (0,1]⟶\mathbb{R}$ be the function given by

$$\iota (\omega ,\varpi )=\frac{\omega }{{(\varpi )}^k}\mathrm{for}\mathrm{all}\omega ,\varpi \in (0,1],$$

where $k\in (0,1).$ Then, $\iota \in \mathcal{FL}$.

Example  6.

Let $\iota :(0,1]\times (0,1]⟶\mathbb{R}$ be the function given by

$$\iota (\omega ,\varpi )=\frac{\omega }{\alpha (\varpi )}\mathrm{for}\mathrm{all}\omega ,\varpi \in (0,1],$$

where $\alpha :(0,1]\to (0,1]$ such that α is continuous, increasing, and $\alpha (s)>s$, for all $s\in (0,1)$. Then, $\iota \in \mathcal{FL}$.

Example  7.

Let $\iota :(0,1]\times (0,1]⟶\mathbb{R}$ be defined by

$$\iota (\omega ,\varpi )=\frac{\omega \hslash (\omega ,\varpi )}{\varpi },$$

for all $\omega ,\varpi \in (0,1]$, where $\hslash :(0,+\infty ]\times (0,+\infty ]\to (0,+\infty ]$ such that $\hslash (\omega ,\varpi )<1$ for all $\omega ,\varpi <1$ and ${lim}_{p\to +\infty }sup\hslash ({\omega }_p,{\varpi }_p)<1$ if $\{{\omega }_p\},\{{\varpi }_p\}$ are sequences in $(0,1]$ with ${lim}_{p\to +\infty }{\omega }_p={lim}_{p\to +\infty }{\varpi }_p<1$. Then, $\iota \in \mathcal{FL}$.

In 2020, Saleh et al. [15] proposed the notion of fuzzy ${\theta }_f$-contractive mappings with the help of the class $\Omega$ of the functions ${\theta }_f:(0,1)\to (0,1)$ fulfilling the following conditions:

(${\Omega }_1$)

${\theta }_f$ is non-decreasing and continuous,

(${\Omega }_2$)

${lim}_{q\to +\infty }{\theta }_f({\omega }_q)=1$ if and only if ${lim}_{q\to +\infty }{\omega }_q=1$, where $\{{\omega }_q\}$ is a sequence in $(0,1)$.

The following fundamental relation-theoretic notions, concepts, and associated results are required in order to establish our results.

Definition 6

([21]). A subset $\mathcal{R}$ of $\mathcal{K}\times \mathcal{K}$ is called a binary relation on $\mathcal{K}$. If $(\omega ,\varpi )\in \mathcal{R}$, we say that ω is related to ϖ under $\mathcal{R}$ ( or $\omega \mathcal{R}\varpi$). We write $[\omega ,\varpi ]\in \mathcal{R}$ if either $(\omega ,\varpi )\in \mathcal{R}$ or $(\varpi ,\omega )\in \mathcal{R}$.

Note that $\mathcal{K}\times \mathcal{K}$ is a binary relation on $\mathcal{K}$ termed as the universal relation. Trivially, the empty relation on $\mathcal{K}$ is represented by ∅.

A binary relation $\mathcal{R}$ on a nonempty set $\mathcal{K}$ is called reflexive if $\omega \mathcal{R}\omega$ for all $\omega \in \mathcal{K}$, transitive if $\omega \mathcal{R}\varpi$ and $\varpi \mathcal{R}\vartheta$ imply $\omega \mathcal{R}\vartheta$ for all $\omega ,\varpi ,\vartheta \in \mathcal{K}$, and $\mathcal{G}$-transitive if it is transitive in $\mathcal{G}(\mathcal{K})$.

Definition 7

([5]). Let $\mathcal{K}$ be a nonempty set and $\mathcal{R}$ be a binary relation on $\mathcal{K}$. A sequence $\{{\omega }_p\}\subseteq \mathcal{K}$ is called an $\mathcal{R}$-preserving sequence if $({\omega }_p,{\omega }_{p+1})\in \mathcal{R}$ for all $p\in \mathbb{N}$.

Remark 1.

If every $\mathcal{R}$-preserving Cauchy sequence is convergent in $\mathcal{K}$, then we say that $(\mathcal{K},\beta ,⋏)$ is $\mathcal{R}$-complete. Note that, for each given binary relation $\mathcal{R}$, every complete metric space is $\mathcal{R}$-complete. In particular, $\mathcal{R}$-completeness specifically becomes the ordinary completeness under the universal relation.

Definition 8

([22]). A binary relation $\mathcal{R}$ on $\mathcal{K}$ is called a β-self-closed if for any $\mathcal{R}$-preserving sequence $\{{\omega }_p\}\subseteq \vartheta$ such that

$${\omega }_p\stackrel{\beta }{⟶}\omega \mathrm{as}p\to +\infty ,$$

there exists a subsequence $\{{\omega }_{p_k}\}$ of $\{{\omega }_p\}$ with $({\omega }_{p_k},\omega )\in \mathcal{R}$.

Definition 9.

Let $(\mathcal{K},\beta ,⋏)$ be an $\mathcal{FMS}$, $\mathcal{R}$ is a binary relation in $\mathcal{K}$. A mapping $\mathcal{G}:\mathcal{K}\to \mathcal{K}$ is called $\mathcal{R}$-continuous at $\omega \in \mathcal{K}$ if for all $\mathcal{R}$-preserving sequence $\{{\omega }_p\}$ such that ${\omega }_p\stackrel{\beta }{⟶}\omega$, we have $\mathcal{G}{\omega }_p\stackrel{\beta }{⟶}\mathcal{G}\omega$. $\mathcal{G}$ is called $\mathcal{R}$-continuous if it is $\mathcal{R}$-continuous at each point of $\mathcal{K}$.

Remark 2.

Every continuous mapping is $\mathcal{R}$-continuous, for any binary relation $\mathcal{R}$. Particularly, under the universal relation, the notion of $\mathcal{R}$-continuity coincides with usual continuity.

Definition 10

([23]). Let $\mathcal{K}$ be a nonempty set and $\mathcal{R}$ a binary relation on $\mathcal{K}$. For $\omega ,\varpi \in \mathcal{K}$, a path of length p in $\mathcal{R}$ from ω to ϖ is a finite sequence $\{{\rho }_0,{\rho }_1,{\rho }_2,\dots ,{\rho }_p\}\subset \mathcal{K}$ fulfilling the following:

($\mathcal{P}1)$

${\rho }_0=\omega$ and ${\rho }_p=\varpi$;

($\mathcal{P}2)$

$({\rho }_q,{\rho }_{q+1})\in \mathcal{R}$ for all q$(0\le q\le p-1)$.

Observe that a path of length p involves $p+1$ elements of $\mathcal{K}$.

Definition 11

([24]). If for all $\omega ,\varpi \in E\subseteq \mathcal{K}$ there exists a path from ω to ϖ in $\mathcal{R}$, then the subset E is called $\mathcal{R}$-connected.

Let $\mathcal{K}$ be a nonempty set and $\mathcal{G}:\mathcal{K}\to \mathcal{K}$ be a self-mapping. The following notations will be used.

$$\begin{array}{cc}\hfill & \mathcal{K}(\mathcal{G},\mathcal{R}):=\{\omega \in \mathcal{K}:(\omega ,\mathcal{G}\omega )\in \mathcal{R}\},\hfill \\ \hfill & \gamma (\omega ,\varpi ,\mathcal{R}):=\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{all}\mathrm{paths}\mathrm{in}\mathcal{R}\mathrm{from}\omega \mathrm{to}\varpi .\hfill \end{array}$$

3. Main Results

In this part, we begin by defining the concept of a fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction in $\mathcal{FMS}$.

Definition 12.

Let $(\mathcal{K},\beta ,⋏)$ be an $\mathcal{FMS}$, $\mathcal{R}$ is a binary relation in $\mathcal{K}$, ${\theta }_f\in \Omega$ and $\mathcal{G}:\mathcal{K}\to \mathcal{K}$. We say that $\mathcal{G}$ is a fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction with respect to $\iota \in \mathcal{FL}$ if

$$\iota ({\theta }_f(\beta (\mathcal{G}\omega ,\mathcal{G}\varpi ,\varsigma )),{\theta }_f(\beta (\omega ,\varpi ,\varsigma )))\ge 1,\mathrm{for}\mathrm{all}\omega ,\varpi \in \mathcal{K},\varsigma >0\mathrm{with}(\omega ,\varpi )\in \tilde{\mathcal{R}},$$

(1)

where $(\omega ,\varpi )\in \tilde{\mathcal{R}}:=\{(\omega ,\varpi )\in \mathcal{R}:\mathcal{G}\omega \ne \mathcal{G}\varpi \}$

Proposition 1.

Let $(\mathcal{K},\beta ,⋏)$ be an $\mathcal{FMS}$, $\mathcal{R}$ is a binary relation in $\mathcal{K}$, and $\mathcal{G}:\mathcal{K}\to \mathcal{K}$ is a fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction with respect to $\iota \in \mathcal{FL}$. Then, the following are complement to each other:

$({\mathcal{H}}_1)$

$\iota ({\theta }_f(\beta (\mathcal{G}\omega ,\mathcal{G}\varpi ,\varsigma )),{\theta }_f(\beta (\omega ,\varpi ,\varsigma )))\ge 1,$ for all $\omega ,\varpi \in \mathcal{K}$ with $(\omega ,\varpi )\in \tilde{\mathcal{R}}$,

$({\mathcal{H}}_2)$

$\iota ({\theta }_f(\Lambda (\mathcal{G}\omega ,\mathcal{G}\varpi ,\varsigma )),{\theta }_f(\beta (\omega ,\varpi ,\varsigma )))\ge 1,$ for all $\omega ,\varpi \in \mathcal{K}$ with $[\omega ,\varpi ]\in \tilde{\mathcal{R}}$.

Proof. 

The implication $({\mathcal{H}}_2)$ implies $({\mathcal{H}}_1)$ is trivial. Conversely, suppose that $({\mathcal{H}}_1)$ holds. Consider $\omega ,\varpi \in \vartheta$ with $[\omega ,\varpi ]\in \tilde{\mathcal{R}}$, then $({\mathcal{H}}_2)$ follows directly from $({\mathcal{H}}_1)$. Otherwise, if $(\varpi ,\omega )\in \tilde{\mathcal{R}}$, taking into account the symmetry of the fuzzy metric $\beta$ and $({\mathcal{H}}_1)$, we obtain

$$\iota ({\theta }_f(\beta (\mathcal{G}\varpi ,\mathcal{G}\omega ,\varsigma )),{\theta }_f(\beta (\varpi ,\omega ,\varsigma )))=\iota ({\theta }_f(\beta (\mathcal{G}\omega ,\mathcal{G}\varpi ,\varsigma )),{\theta }_f(\beta (\omega ,\varpi ,\varsigma )))\ge 1.$$

Thus, $({\mathcal{H}}_1)$ implies $({\mathcal{H}}_2)$. □

Theorem 1.

Let $(\mathcal{K},\beta ,⋏)$ be an $\mathcal{FMS}$ endowed with a binary relation $\mathcal{R}$ and $\mathcal{G}:\mathcal{K}\to \mathcal{K}$ be a self-mapping. Suppose that

(i)

$(\mathcal{K},\beta ,⋏)$ is $\mathcal{R}$-complete;

(ii)

$\mathcal{K}(\mathcal{G},\mathcal{R})\ne \varnothing ;$

(iii)

$\mathcal{R}$ is $\mathcal{G}$-closed and $\mathcal{R}$ is $\mathcal{G}$-transitive;

(iv)

$\mathcal{G}$ is a fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction with respect to some $\iota \in \mathcal{FL}$;

(v)

Either $\mathcal{R}$ is β-self-closed or $\mathcal{G}$ is $\mathcal{R}$-continuous.

Then, $\mathcal{G}$ has a fixed point.

Proof. 

As $\mathcal{K}(\mathcal{G},\mathcal{R})\ne \varnothing$, let ${\omega }_0$ be an arbitrary point such that ${\omega }_0\in \mathcal{K}(\mathcal{G},\mathcal{R})$. Now, define a Picard sequence $\{{\omega }_p\}$ by ${\omega }_{p+1}=\mathcal{G}{\omega }_p$ for all $p\in \mathbb{N}$. Using the fact that $\mathcal{R}$ is $\mathcal{G}$-closed and $({\omega }_0,\mathcal{G}{\omega }_0)\in \mathcal{R}$, we have

$$(\mathcal{G}{\omega }_0,{\mathcal{G}}^2{\omega }_0),({\mathcal{G}}^2{\omega }_0,{\mathcal{G}}^3{\omega }_0),\dots ,({\mathcal{G}}^p{\omega }_0,{\mathcal{G}}^{p+1}{\omega }_0)\in \mathcal{R}.$$

Hence,

$$\begin{array}{c}\hfill ({\omega }_p,{\omega }_{p+1})\in \mathcal{R}.\end{array}$$

(2)

If there exists $p_0\in \mathbb{N}$ such that ${\omega }_{p_0+1}={\omega }_{p_0}$, then ${\omega }_{p_0}$ is a fixed point. Assume that ${\omega }_{p+1}\ne {\omega }_p$ for all $p\in \mathbb{N}$, that is, $\beta (\mathcal{G}{\omega }_{p-1},\mathcal{G}{\omega }_p,\varsigma )<1$ for all $\varsigma >0$, which means that $({\omega }_{p-1},{\omega }_p)\in \tilde{\mathcal{R}}$. Since $\mathcal{G}$ is a fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction with respect to $\iota \in \mathcal{FL}$, we have

$$\begin{array}{cc}\hfill 1& \le \iota ({\theta }_f(\beta (\mathcal{G}{\omega }_{p-1},\mathcal{G}{\omega }_p,\varsigma )),{\theta }_f(\beta ({\omega }_{p-1},{\omega }_p,\varsigma )))\hfill \\ \hfill & <\frac{{\theta }_f(\beta (\mathcal{G}{\omega }_{p-1},\mathcal{G}{\omega }_p,\varsigma ))}{{\theta }_f(\beta ({\omega }_{p-1},{\omega }_p,\varsigma ))}.\hfill \end{array}$$

(3)

Then,

$$\begin{array}{c}\hfill {\theta }_f(\beta ({\omega }_{p-1},{\omega }_p,\varsigma ))<{\theta }_f(\beta (\mathcal{G}{\omega }_{p-1},\mathcal{G}{\omega }_p,\varsigma )).\end{array}$$

As ${\theta }_f$ is nondecreasing, we derive

$$\begin{array}{c}\hfill \beta ({\omega }_{p-1},{\omega }_p,\varsigma )<\beta (\mathcal{G}{\omega }_{p-1},\mathcal{G}{\omega }_p,\varsigma ).\end{array}$$

We derive that $\{\beta ({\omega }_{p-1},{\omega }_p,\varsigma )\}$ is a nondecreasing sequence of positive real numbers in $[0,1]$. Thus, there exists $a(\varsigma )\le 1$ such that ${lim}_{p\to +\infty }\beta ({\omega }_{p-1},{\omega }_p,\varsigma )=a(\varsigma )\ge 1$ for all $\varsigma >0$. We prove that

$$\begin{array}{c}\hfill \underset{p\to +\infty }{lim}\beta ({\omega }_{p-1},{\omega }_p,\varsigma )=1.\end{array}$$

On the contrary, assume that $a({\varsigma }_0)<1$ for some ${\varsigma }_0>0$. Now, if we take the sequences $\{{\alpha }_p=\beta ({\omega }_p,{\omega }_{p+1},{\varsigma }_0)\}$ and $\{{\eta }_p=\beta ({\omega }_{p-1},{\omega }_p,{\varsigma }_0)\}$ and apply $(\mathcal{FL}3)$, we obtain

$$\begin{array}{c}\hfill 1\le \underset{p\to +\infty }{lim}sup\iota ({\alpha }_p,{\eta }_p)<1.\end{array}$$

Which is a contradiction. Then,

$$\begin{array}{c}\hfill \underset{p\to +\infty }{lim}\beta ({\omega }_{p-1},{\omega }_p,\varsigma )=1\mathrm{for}\mathrm{all}\varsigma >0.\end{array}$$

(4)

From $({\Omega }_2)$, we have

$$\begin{array}{c}\hfill \underset{p\to +\infty }{lim}{\theta }_f(\beta ({\omega }_{p-1},{\omega }_p,\varsigma ))=1,\end{array}$$

for all $\varsigma >0$. Next, we prove that the sequence $\{{\omega }_p\}$ is Cauchy. Reasoning by contradiction, assume that $\{{\omega }_p\}$ is not Cauchy. Then, there exists $\tau \in (0,1)$, ${\varsigma }_0>0$ and two subsequences $\{{\omega }_{p_r}\}$ and $\{{\omega }_{q_r}\}$ of $\{{\omega }_p\}$ with $p_r>q_r\ge r$ for all $r\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \beta ({\omega }_{q_r},{\omega }_{p_r},{\varsigma }_0)\le 1-\tau .\end{array}$$

(5)

Taking into consideration Lemma 1, we obtain

$$\begin{array}{c}\hfill \beta ({\omega }_{q_r},{\omega }_{p_r},\frac{{\varsigma }_0}{2})\le 1-\tau .\end{array}$$

(6)

By taking $q_r$ as the lowest index fulfilling (6), we have

$$\begin{array}{c}\hfill \beta ({\omega }_{q_r},{\omega }_{p_r-1},\frac{{\varsigma }_0}{2})>1-\tau .\end{array}$$

(7)

Taking into account that the sequence $\{{\omega }_p\}$ is $\mathcal{R}$-preserving and $\mathcal{R}$ is $\mathcal{G}$-transitive, thus $({\omega }_{q_r},{\omega }_{p_r})\in \tilde{\mathcal{R}}$, and we have

$$\begin{array}{cc}\hfill 1& \le \iota ({\theta }_f(\beta ({\omega }_{q_r},{\omega }_{p_r},{\varsigma }_0)),{\theta }_f(\beta ({\omega }_{q_r-1},{\omega }_{p_r-1},\varsigma )))\hfill \\ \hfill & <\frac{{\theta }_f(\beta ({\omega }_{q_r},{\omega }_{p_r},{\varsigma }_0))}{{\theta }_f(\beta ({\omega }_{q_r-1},{\omega }_{p_r-1},\varsigma ))}.\hfill \end{array}$$

(8)

Then,

$$\begin{array}{c}\hfill {\theta }_f(\beta ({\omega }_{q_r-1},{\omega }_{p_r-1},{\varsigma }_0))<{\theta }_f(\beta ({\omega }_{q_r},{\omega }_{p_r},{\varsigma }_0)).\end{array}$$

As ${\theta }_f$ is nondecreasing, we obtain

$$\begin{array}{c}\hfill \beta ({\omega }_{q_r-1},{\omega }_{p_r-1},{\varsigma }_0)<\beta ({\omega }_{q_r},{\omega }_{p_r},{\varsigma }_0).\end{array}$$

(9)

By (5), (7), and $(\mathcal{M}4)$, we have

$$\begin{array}{cc}\hfill 1-\tau & \ge \beta ({\omega }_{q_r},{\omega }_{p_r},{\varsigma }_0)\hfill \\ \hfill & >\beta ({\omega }_{q_r-1},{\omega }_{p_r-1},{\varsigma }_0)\hfill \\ \hfill & \ge \beta ({\omega }_{q_r-1},{\omega }_{q_r},\frac{{\varsigma }_0}{2})⋏\beta ({\omega }_{q_r},{\omega }_{p_r-1},\frac{{\varsigma }_0}{2})\hfill \\ \hfill & >\beta ({\omega }_{q_r-1},{\omega }_{q_r},\frac{{\varsigma }_0}{2})\ast (1-\tau ).\hfill \end{array}$$

Taking the limit as $r\to +\infty$ in both sides and using (4), we obtain

$$\begin{array}{c}\hfill \underset{r\to +\infty }{lim}\beta ({\omega }_{q_r},{\omega }_{p_r},{\varsigma }_0)=\underset{r\to +\infty }{lim}\beta ({\omega }_{q_r-1},{\omega }_{p_r-1},{\varsigma }_0)=1-\tau .\end{array}$$

(10)

In addition,

$$\begin{array}{c}\hfill \underset{r\to +\infty }{lim}{\theta }_f(\beta ({\omega }_{q_r},{\omega }_{p_r},{\varsigma }_0))=\underset{r\to +\infty }{lim}{\theta }_f(\beta ({\omega }_{q_r-1},{\omega }_{p_r-1},{\varsigma }_0))={\theta }_f(1-\tau ).\end{array}$$

(11)

As $\mathcal{G}$ is a fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction with respect to $\iota \in \mathcal{FL}$, and making use of $(\mathcal{FL}3)$, we obtain

$$\begin{array}{cc}\hfill 1& \le \underset{r\to +\infty }{lim}sup\iota ({\theta }_f(\beta ({\omega }_{q_r},{\omega }_{p_r},{\varsigma }_0)),{\theta }_f(\beta ({\omega }_{q_r-1},{\omega }_{p_r-1},{\varsigma }_0)))<1.\hfill \end{array}$$

A contradiction. Thus, $\{{\omega }_r\}$ is an $\mathcal{R}$-preserving Cauchy sequence in $\mathcal{K}$. As $\mathcal{K}$ is $\mathcal{R}$-complete, there exists $\widehat{\omega }\in \mathcal{K}$ such that ${\omega }_p\to \widehat{\omega }$.

$(•)$ If $\mathcal{G}$ is $\mathcal{R}$-continuous, then

$$\widehat{\omega }=\underset{p\to +\infty }{lim}{\omega }_{p+1}=\underset{p\to +\infty }{lim}\mathcal{G}{\omega }_p=\mathcal{G}[\underset{p\to +\infty }{lim}{\omega }_p]=\mathcal{G}\widehat{\omega }.$$

Thus, $\widehat{\omega }$ is a fixed point of $\mathcal{G}$.

$(•)$ Now, if $\mathcal{R}$ is $\beta$-self-closed. As $\{{\omega }_p\}$ is an $\mathcal{R}$-preserving sequence and ${\omega }_p\to \widehat{\omega }$, there exists a subsequence $\{{\omega }_{p_r}\}$ of $\{{\omega }_p\}$ such that $[{\omega }_{p_r},\widehat{\omega }]\in \mathcal{R}$ for all $p\in \mathbb{N}$ and Proposition 1, we obtain

$$1\le \iota ({\theta }_f(\beta ({\omega }_{p_r+1},\mathcal{G}\widehat{\omega },\varsigma )),{\theta }_f(\beta ({\omega }_{p_r},\widehat{\omega },\varsigma ))).$$

(12)

We prove that $\widehat{\omega }$ is a fixed point of $\mathcal{G}$. By contradiction, suppose that is not the case, that is, $\beta (\widehat{\omega },\mathcal{G}\widehat{\omega },\varsigma )<1$ for all $\varsigma >0$. From (12), we have

$$\begin{array}{cc}\hfill 1& \le \iota ({\theta }_f(\beta ({\omega }_{p_r+1},\mathcal{G}\widehat{\omega },\varsigma )),{\theta }_f(\beta ({\omega }_{p_r},\widehat{\omega },\varsigma )))\hfill \\ \hfill & =\iota ({\theta }_f(\beta (\mathcal{G}{\omega }_{p_r},\mathcal{G}\widehat{\omega },\varsigma )),{\theta }_f(\beta ({\omega }_{p_r},\widehat{\omega },\varsigma )))\hfill \\ \hfill & <\frac{{\theta }_f(\beta (\mathcal{G}{\omega }_{p_r},\mathcal{G}\widehat{\omega },\varsigma ))}{{\theta }_f(\beta ({\omega }_{p_r},\widehat{\omega },\varsigma ))}\hfill \end{array}$$

Then, ${\theta }_f(\beta ({\omega }_{p_r},\omega ,\varsigma ))<{\theta }_f(\beta (\mathcal{G}{\omega }_{p_r},\mathcal{G}\widehat{\omega },\varsigma ))$. As ${\theta }_f$ is nondecreasing, we obtain

$$\beta ({\omega }_{p_r},\widehat{\omega },\varsigma )\le \beta (\mathcal{G}{\omega }_{p_r},\mathcal{G}\widehat{\omega },\varsigma )$$

Passing to the limit as $r\to +\infty$ in the last inequality, we derive that

$$1\le \underset{r\to +\infty }{lim}\beta ({\omega }_{p_r+1},\mathcal{G}\widehat{\omega },\varsigma )=\beta (\widehat{\omega },\mathcal{G}\widehat{\omega },\varsigma ).$$

Which is a contradiction. Thus, $\beta (\widehat{\omega },\mathcal{G}\widehat{\omega },\varsigma )=1$, that is, $\mathcal{G}\widehat{\omega }=\widehat{\omega }$, which means that $\widehat{\omega }$ is a fixed point of $\mathcal{G}$. □

Theorem 2.

In addition to the assumptions of Theorem 1, if $\gamma (\omega ,\varpi ,\mathcal{R}{|}_{\mathcal{G}(\mathcal{K})})\ne \varnothing$ for all $\omega ,\varpi \in \mathcal{G}(\mathcal{K})$, then $\mathcal{G}$ has a unique fixed point.

Proof. 

We argue by contradiction, suppose that $\omega$ and ${\omega }^{\ast }$ are two distinct fixed points of $\mathcal{G}$. As $\gamma (\omega ,\varpi ,\mathcal{R}{|}_{\mathcal{G}(\mathcal{K})})\ne \varnothing$ for all $\omega ,\varpi \in \mathcal{G}(\mathcal{K})$, there exists a path $\{{\delta }_0,{\delta }_1,{\delta }_2,\dots ,{\delta }_q\}$ of some finite length q in ${\mathcal{R}|}_{\mathcal{K}}$ from $\omega$ to ${\omega }^{\ast }$ such that

$${\delta }_0=\mathcal{G}{\omega }_0=\omega ,{\delta }_q=\mathcal{G}{\omega }_q={\omega }^{\ast },({\delta }_j,{\delta }_{j+1})=(\mathcal{G}{\omega }_j,\mathcal{G}{\omega }_{j+1}){\in \mathcal{R}|}_{\mathcal{G}(\mathcal{K})},j=0,1,2,\dots ,q-1.$$

As $\mathcal{R}$ is transitive, we have

$$(\omega ,\mathcal{G}{\omega }_1)\in \mathcal{R},(\mathcal{G}{\omega }_1,\mathcal{G}{\omega }_2)\in \mathcal{R},\dots ,(\mathcal{G}{\omega }_{q-1},{\omega }^{\ast })\in \mathcal{R}\mathrm{implies}({\delta }_0,{\delta }_q)=(\omega ,{\omega }^{\ast })\in \mathcal{R}.$$

Taking into account that $\mathcal{G}$ is a fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction with respect to $\iota \in \mathcal{FL}$, we obtain

$$\begin{array}{cc}\hfill 1\le & \iota ({\theta }_f(\beta (\mathcal{G}\omega ,\mathcal{G}{\omega }^{\ast },\varsigma )),{\theta }_f(\beta (\omega ,{\omega }^{\ast },\varsigma )))\hfill \\ \hfill <& \frac{{\theta }_f(\beta (\mathcal{G}\omega ,\mathcal{G}{\omega }^{\ast },\varsigma ))}{{\theta }_f(\beta (\omega ,{\omega }^{\ast },\varsigma ))}\hfill \\ \hfill =& \frac{{\theta }_f(\beta (\omega ,{\omega }^{\ast },\varsigma ))}{{\theta }_f(\beta (\omega ,{\omega }^{\ast },\varsigma ))}.\hfill \end{array}$$

A contradiction. Thus, the fixed point of $\mathcal{G}$ is unique. □

The following corollary is a sharpened version and relation-theoretic analog of the main finding (Theorem 3.1, [15]) due to Saleh et al. [15].

Corollary 1.

Let $(\mathcal{K},\beta ,⋏)$ be an $\mathcal{FMS}$ endowed with a binary relation $\mathcal{R}$ and $\mathcal{G}:\mathcal{K}\to \mathcal{K}$ be a self-mapping. Suppose the following:

(i)

$(\mathcal{K},\beta ,⋏)$ is $\mathcal{R}$-complete;

(ii)

$\mathcal{K}(\mathcal{G},\mathcal{R})\ne \varnothing ;$

(iii)

$\mathcal{R}$ is $\mathcal{G}$-closed and $\mathcal{G}$-transitive;

(iv)

There exists ${\theta }_f\in \Omega$ such that

$$\begin{array}{c}\hfill {\theta }_f(\beta (\mathcal{G}\omega ,\mathcal{G}\varpi ,\varsigma ))\ge {\left[{\theta }_f(\beta (\omega ,\varpi ,\varsigma ))\right]}^k,\end{array}$$

for all $\omega ,\varpi \in \mathcal{K}$, $\varsigma >0$ with $(\omega ,\varpi )\in \tilde{\mathcal{R}}$ and $k\in (0,1)$;

(v)

Either $\mathcal{R}$ is β-self-closed or $\mathcal{G}$ is $\mathcal{R}$-continuous.

Then, $\mathcal{G}$ has a fixed point. Moreover, if $\gamma (\omega ,\varpi ,\mathcal{R}{|}_{\mathcal{G}(\mathcal{K})})\ne \varnothing$ for all $\omega ,\varpi \in \mathcal{G}(\mathcal{K})$, then we obtain the uniqueness of the fixed point.

Proof. 

The conclusion can be drawn from Theorems 1 and 2 by defining $\iota (\omega ,\varpi )=\frac{\omega }{{\varpi }^k}$ for all $\omega ,\varpi \in (0,1]$. □

Corollary 2.

If we assume that $\mathcal{R}$ is a transitive binary relation in Theorems 1 and 2, then the results are still valid since any transitive binary relation is $\mathcal{G}$-transitive for a certain self-map $\mathcal{G}$ on $\mathcal{K}$.

Corollary 3.

If the $\mathcal{R}$-completeness of $(\mathcal{K},\beta ,⋏)$ is substituted by completeness and $\mathcal{R}$-continuity by continuity, Theorem 1 remains true.

Proof. 

It follows as a natural consequence of Remarks 1 and 2. □

4. Conclusions

In the context of $\mathcal{FMS}$s with a binary relation, we developed the idea of the fuzzy $\mathcal{L}$-$\mathcal{R}$-contraction and explored some relevant findings about the existence and uniqueness of a fixed point for such mappings via control functions, without the completeness criterion, which in turn generalizes, extends, and combines a number of findings in the literature. It is important to note that by properly integrating many examples of the control function $\iota$, we can particularize and infer a wide range of potential outcomes from our core findings. Our research could pave the way for fresh research in fuzzy fixed point theory and it is conceivable to investigate relation-theoretic fuzzy metrical coincidence and common fixed point outcomes using the obtained results and notions. Moreover, these findings are generalizable to more abstract distance spaces, including partial fuzzy metric spaces, fuzzy b-metric spaces, and others of the same nature.