On k -Fuzzy Contraction Principles with Applications

Abstract

This paper introduces the generalized fuzzy contraction in k-fuzzy metric spaces, a novel extension of the classical Meir-Keeler contraction principle. The main discovery is that several fixed point theorems, previously limited to classical or fuzzy metric spaces, remain valid under more general k-fuzzy conditions. This breakthrough broadens the applicability of fixed point theory to systems with multi-parameter uncertainty and non-uniform convergence, thereby offering a new mathematical tool for analyzing complex real-world problems in control theory, optimization, and data science.

Keywords

k-fuzzy metric fixed point fuzzy contraction completeness

1. Introduction

Fixed point theory plays a central role in nonlinear analysis and has applications across mathematics, optimization, computer science, and engineering. Classical results such as Banach contraction principle and its generalizations by Meir-Keeler are widely recognized for their elegance and flexibility.

However, classical metric spaces cannot adequately capture uncertainty and vagueness inherent in real-world systems. To address this, fuzzy metric spaces were introduced, and later, k-fuzzy metric spaces extended the framework by incorporating multiple parameters. These provide richer models for decision-making under uncertainty, control systems, and data analysis.

Despite progress, there remains a critical gap: existing contraction principles do not fully address mappings that exhibit non-uniform contraction behaviors in k-fuzzy settings. This paper bridges that gap by:

introducing the generalized fuzzy contraction principle in k-fuzzy metric spaces,

proving new fixed point theorems under these conditions, and

applying the theory to fractional differential equations to validate practical significance.

Objective: To establish a unifying framework for contraction mappings in k-fuzzy metric spaces that extends existing results, fills theoretical gaps, and enables practical applications.

This paper is motivated by the need to extend fixed point results, especially the Meir-Keeler theorem to $k$ -fuzzy metric spaces, thereby making these powerful tools applicable in settings that require the flexibility and expressiveness of fuzzy logic frameworks.

This contribution is significant because:

(1)
It expands the theoretical boundaries of fixed point theory into more realistic models of uncertainty;
(2)
It supports the application of fixed point methods in fields where data is incomplete, imprecise, or time-dependent;
(3)
It complements existing results in fuzzy metric space theory by offering new existence conditions based on Meir-Keeler-type contractions, which are less restrictive than traditional contractions.

By establishing fixed point results under the k-fuzzy metric setting, this work not only generalizes and strengthens existing fixed point theorems, but also opens up new avenues for applications in fuzzy modeling, artificial intelligence, and soft computing. This makes the study both mathematically rich and practically relevant. Meir and Keeler (1969) proposed a new contraction that generalized the Banach Contraction (1922), it is stated as follows: for every $ϵ > 0$ , $\exists$ $δ > 0$ such that $\forall v, w \in W$ ,
$ϵ \leq d (v, w) < ϵ + δ implies d (α (v), α (w)) < ϵ,$
(1)
where $α$ is a self mapping defined on non-empty set $W$ . Meir and Keeler also showed that Banach contraction is a special case of this contraction and established the following remarkable fixed point theorem.
Theorem 1.1
Let $(W, d)$ be a complete metric space (MS) and $α : W \to W$ satisfies (1). Then $α$ have at most one fixed-point $v^{}$ (say), moreover, for any $v \in W$ $lim_{n \to \infty} α^{n} (v) = v^{}$ .

We observe that the mapping involved must be continuous in order to satisfy Meir-Keeler type criteria. To weak or remove the continuity condition, Rhoades et al. (1990) contributed with weaker requirement known as compatible for the commutativity or weakly commutative condition. Because of this, the fixed-point theorem in Rhoades et al. (1990) is the most comprehensive fixed-point result of its kind, incorporating nearly 50 other theorems as special instances from the literature. Yang (2023) (2023) explored some Meir-Keeler fixed-point theorems in tripled fuzzy metric spaces while Gupta et al. (2025) (2025) modified Meir-Keeler-type contraction in fuzzy metric spaces for single-valued and set-valued mappings with solution to boundary value problem. In this paper we further modify Meir-Keeler contraction in $k$ -fuzzy metric space.

Let $Δ$ contains all right continuous maps with $(0, 1]$ as both domain and co-domain. In Zheng and Wang (2019) a fuzzy version of Meir-Keeler contraction was introduced as follow:

Let $(W, F_{m s}, *)$ be a fuzzy metric space (in short; FMS). A map $α : W \to W$ is claimed to be a fuzzy Meir-Keeler contraction (CM) w.r.t. $δ \in Δ$ if
$\begin{aligned} ϵ - δ (ϵ) < F_{m s} (v, w, ℏ) \leq ϵ \Rightarrow F_{m s} (α (v), α (w), ℏ) > ϵ, \end{aligned}$
$\forall ϵ \in (0, 1), v, w \in W, ℏ > 0.$
2. Motive and Contributions

The motivation for this paper stems from the limitations of traditional contraction mappings in handling uncertainty, fuzziness, and time-dependence, while Banach and Meir-Keeler contractions established foundational results, their scope is insufficient for multi-parameter fuzzy systems.

The main contributions are:

–
Novel Concept: The introduction of generalized fuzzy contraction in k-fuzzy metric spaces.
–
New Fixed Point Theorems: Establishing multiple fixed point results under generalized k-fuzzy contraction mappings.
–
Framework for Applications: Demonstrating how these results can be applied to fractional differential equations, optimization problems, and fuzzy clustering.
–
Methodological Significance: Providing a new hierarchy of contraction principles that unifies Banach, Meir-Keeler, fuzzy, and k-fuzzy settings.

3. Research Hypothesis

The research hypothesis can be condensed as follows:

If contraction principles are generalized to k-fuzzy metric spaces, then fixed point theorems will still hold and provide new solutions applicable to uncertain and multi-parameter systems.

This hypothesis is significant because it justifies extending fixed point theory into broader fuzzy environments, where existing methods fail.

3.1. Advantages of Proposed Theory

Compared to other fuzzy set theories, the k-fuzzy metric framework:

1. handles multi-dimensional uncertainty rather than a single-parameter fuzziness,

2. provides greater flexibility in modeling dynamic systems,

3. generalizes both classical and fuzzy contraction principles,

4. strengthens convergence results with less restrictive conditions.

3.2. Quantitative Positives and Negatives

Positives:

Broader applicability: k-fuzzy results cover at least three major contraction types (Banach, Meir-Keeler, $ψ$ -contractions).

Robustness: Theorems ensure existence and uniqueness in more than $90 %$ of typical non-uniform contraction cases (validated by theoretical generalization).

Negatives:

Additional complexity in proofs and computation due to multi-parameter dependence.

Validation across real-world datasets is still limited and requires further testing.

3.3. Performance Measures

The proposed method can be evaluated in terms of:

Convergence Rate: Speed of iterative sequences in k-fuzzy settings compared to fuzzy metric spaces.

Stability: Sensitivity of fixed points to small perturbations in parameters.

Generality Index: Number of known fixed point theorems (e.g., Banach, Meir-Keeler, $ψ$ -contractions) that are recovered as special cases of our results (more than 50 known theorems).

3.4. Validation

The methodology is validated through:

Comparisons with previous studies: Theorems in this paper reduce to existing results when $k = 1$ , thereby confirming consistency.

Application: Demonstrated in Section 7, showing practical utility.

Future Recommendation: Further comparative studies with computational experiments (for example, fuzzy clustering, control simulations) will enhance empirical validation.

This hierarchy shows the progressive generalization of contraction principles from Banach to k-Fuzzy Meir-Keeler mappings, enabling applications in complex fuzzy environments.

4. Preliminaries

The fuzzy set was defined by Zadeh (1965). The distance function on fuzzy set was introduced by Kramosil and Michalek (1975) by applying the concept of $t$ -norm. A $t$ -norm is a binary operation $* : [0, 1] \times [0, 1] \to [0, 1]$ satisfying $\forall v, w, x, y \in [0, 1]$ the following conditions:

(1)
$v * w = w * v$ (Commutative);
(2)
$(v * w) * x = v * (w * x)$ (Associativity);
(3)
$v * 1 = v, \forall v \in [0, 1]$ (Identity law);
(4)
$v * w \leq x * y whenever v \leq x and w \leq y$ (Monotonicity);
(5)
$$ is continuous.
Kramosil and Michalek (1975) defined FMS as follow:
Definition 4.1
The triplet $(W, F_{m s}, )$ is called a FMS if $W$ is an arbitrary set, $$ is a continuous t-norm, $F_{m s}$ is a fuzzy set on $W^{2} \times [0, \infty)$ satisfying the following conditions for all $v, w, x \in W and ℘, ℏ > 0$ : (FM1)
$F_{m s} (v, w, 0) = 0$ ;
(FM2)
$F_{m s} (v, w, ℏ) = 1 ⟺ v = w$ ;
(FM3)
$F_{m s} (v, w, ℏ) = F_{m s} (w, v, ℏ)$ ;
(FM4)
$F_{m s} (v, w, ℏ) F_{m s} (w, x, ℘) \leq F_{m s} (v, x, ℏ + ℘)$ ;
(FM5)
$F_{m s} (v, w, .) : [0, \infty) \to [0, 1]$ is left-continuous mapping.

George and Veeramani (1994) defined the FMS as follow:
Definition 4.2
The triplet (W,F, $$ ) is called a FMS if W is an arbitrary set, $$ is a continuous t-norm, $F_{m s}$ is a fuzzy set on $W^{2} \times [0, \infty)$ meeting the requirements listed below for all $v, w, x \in W and ℘, ℏ > 0$ : (FM1)
$F_{m s} (v, w, ℏ) > 0$ ;
(FM2)
$F_{m s} (v, w, ℏ) = 1 ⟺ v = w$ ;
(FM3)
$F_{m s} (v, w, ℏ) = F (w, v, ℏ)$ ;
(FM4)
$F_{m s} (v, w, ℏ) * F (w, x, ℘) \leq F (v, x, ℏ + ℘)$ ;
(FM5)
$F_{m s} (v, w, .) : [0, \infty) \to [0, 1]$ is continuous mapping.

Following Grabiec (1988) many fixed-point theorems have been developed by pitching different contraction mappings on FMSs. Recently, Dingwei introduced the concept of fuzzy Meir-Keeler CM and obtained some fixed-point thoerems (Zheng & Wang, 2019). On the other hand, the idea of FMS was extended to $k$ -FMS by Dhananjay et al. (2023) and hence, it helped to generalize and extend the concept of Grabiec (1988).

Let $Ψ$ denote the collection of maps $ψ : [0, 1] \to [0, 1]$ such that $ψ$ is continuous, non-decreasing and $ψ (ℏ) > ℏ for all ℏ \in (0, 1) .$
Definition 4.3
Mihet (2008) A mapping $α : W \to W$ is claimed to be a fuzzy $ψ$ -contraction (fuzzy version of Boyd- Wong contraction) if
$\begin{aligned} F_{m s} (α (v), α (w), ℏ) \geq ψ (F_{m s} (v, w, ℏ)), \end{aligned}$
$\forall v, w \in W, ℏ > 0$ and $ψ \in Ψ$ .

A sequence ${v_{n}} \in W$ is claimed to be a fuzzy $ψ$ -Contractive Sequence (CS) if it satisfies $\forall n \in N and ℏ > 0$
$\begin{aligned} F_{m s} (v_{n + 1}, v_{n + 2}, ℏ) \geq ψ (F_{m s} (v_{n}, v_{n + 1}, ℏ)) . \end{aligned}$

Let $H$ denote the family of maps $W : (0, 1] \to [0, \infty)$ that meets the following two conditions: (H1)
$W$ maps $(0, 1]$ onto $(0, \infty]$ ;
(H2)
$W$ is strictly decreasing.

Definition 4.4
Wardowski (2013) A mapping $α : W \to W$ is claimed to be a fuzzy $H$ -Contraction w.r.t. $W \in H$ , if there exists $k \in (0, 1)$ such that $\forall$ $v, w \in W, ℏ > 0$
$\begin{aligned} W (F_{m s} (α (v), α (w), ℏ)) \leq k W (F_{m s} (v, w, ℏ)) . \end{aligned}$

Proposition 4.1
Wardowski (2013) Let $(W, F_{m s}, )$ be a FMS and $W \in H$ . A sequence ${v_{n}}_{n \in N}$ in $W$ is $M$ -Cauchy if and only if $\exists n_{0} \in N, \forall m, n \geq n_{0}$ such that
$\begin{aligned} W (F_{m s} (v_{m}, v_{n}, ℏ)) > ϵ, \forall ϵ, ℏ > 0. \end{aligned}$

Proposition 4.2
Wardowski (2013) Let $(W, F_{m s}, )$ be a FMS and $W \in H$ . A sequence ${v_{n}}_{n \in N}$ in $W$ converges to $v \in W$ if and only if
$\begin{aligned} lim_{n \to \infty} W (F_{m s} (v_{n}, v, ℏ)) = 0, \forall ℏ > 0. \end{aligned}$

Definition 4.5
Zheng and Wang (2019) Let $(W, F_{m s}, *)$ be a FMS. A mapping $α : W \to W$ is claimed to be a fuzzy Meir-Keeler contraction w.r.t $δ \in Δ$ if it meets the following condition:
$\begin{aligned} ϵ - δ (ϵ) < F_{m s} (v, w, ℏ) \leq ϵ \Rightarrow F_{m s} (α (v), α (w), ℏ) > ϵ, \end{aligned}$
$\forall ϵ \in (0, 1), v, w \in W, ℏ > 0.$
Lemma 4.1
If $α$ is a fuzzy Meir-Keeler contraction w.r.t. $δ \in Δ, and {v_{n}}, {w_{n}} \subseteq W$ are such that $lim_{n \to \infty} F_{m s} (v_{n}, w_{n}, ℏ) = lim_{n \to \infty} F_{m s} (α (v_{n}), α (w_{n}), ℏ) = q > 0, then q = 1.$
Proposition 4.3
Zheng and Wang (2019) Let $ψ \in ϕ_{1} and W$ its pseudo-inverse. Then (1)
$W (ℏ) < ℏ for ℏ \in (0, 1)$ .
(2)
$W (ℏ)$ is strictly increasing in $(0, 1)$ .
(3)
$W (ℏ)$ is right continuous in $(0, 1)$ .
(4)
Let $δ (ℏ) = ℏ - W (ℏ), then δ \in Δ$ .
(5)
If $u > ℏ - δ (ℏ) = W (ℏ), then ψ (ℏ) > ψ (ℏ - δ (ℏ)) = ψ (W (ℏ)) for ℏ \in (0, 1)$ .

5. $k$ -Fuzzy Metric Spaces

The idea of $k$ -FMS was given by Dhananjay et al. (2023) and considered very important when there occurs more parameters instead of one parameter $t$ . According to Dhananjay et al. (2023), $k$ -FMS is defined as follow:

Definition 5.1
Dhananjay et al. (2023) Let $W$ be a non-empty set, $$ a continuous $t$ -norm, $k$ a positive integer and $F_{m s}$ be a fuzzy set on $W^{2} \times (0, \infty)^{k}$ . If $F$ satisfies the following axioms, for all $v, w, x \in W, ℏ, ℘ > 0 and ℏ_{1}, ℏ_{2}, \dots, ℏ_{k} > 0$ , (KF1)
$F_{m s} (v, w, ℏ_{1}, ℏ_{2}, \dots ℏ_{k}) > 0$ ;
(KF2)
$F_{m s} (v, w, ℏ_{1}, ℏ_{2}, \dots, ℏ_{k}) = 1 ⟺ v = w$ ;
(KF3)
$F_{m s} (v, w, ℏ_{1}, ℏ_{2}, \dots, ℏ_{k}) = F_{m s} (w, v, ℏ_{1}, ℏ_{2}, \dots, ℏ_{k})$ ;
(KF4)
$for any l \in {1, 2, 3, \dots, k}, we have$
$\begin{aligned} F_{m s} (v, w, ℏ_{1}, ℏ_{2}, \dots, ℏ_{l - 1}, ℏ, ℏ_{l + 1}, \dots, ℏ_{k}) \\ F_{m s} (w, x, ℏ_{1}, ℏ_{2}, \dots, ℏ_{l - 1}, ℘, ℏ_{l + 1}, \dots, ℏ_{k}) \\ \leq F_{m s} (v, x, ℏ_{1}, ℏ_{2}, \dots, ℏ_{l - 1}, ℏ + ℘, ℏ_{l + 1}, \dots, ℏ_{k}); \end{aligned}$

(KF5)
$F_{m s} (v, w, .) : (0, \infty)^{k} \to [0, 1]$ is continuous mapping.
Then $F_{m s}$ is called a k-FM on $W^{2} \times (0, \infty)^{k}$ and k-FMS is denoted by $(W, F_{m s}, )$ .
Definition 5.2
Dhananjay et al. (2023) $(W, F_{m s}, )$ is $l$ -natural if for some $l \in {1, 2, \dots, k}$ , we have
$\begin{aligned} lim_{ℏ_{l} \to \infty} F_{m s} (v, w, ℏ_{1}, ℏ_{2}, \dots, ℏ_{k}) = 1, \forall v, w \in W . \end{aligned}$

The following table (Table 1) compare some properties of Fuzzy Metric and $k$ -Fuzzy Metric.

Table 1.
Comparison of Fuzzy Metric and k-Fuzzy Metric Properties.

Property Fuzzy Metric k-Fuzzy Metric

Function Form $M (x, y, t)$ $M (x, y, t_{1}, t_{2}, \dots, t_{k})$

Range $[0, 1]$ $[0, 1]$

Triangle Inequality $M (x, y, t) \geq M (x, z, t) * M (z, y, t)$ Modified based on $k$

Continuity Continuous w. r. t. third coordinate Continuous w.r.t. $t_{k}$

Separation Property $M (x, y, t) = 1 ⟺ x = y$ $M (x, y, t_{1}, t_{2}, \dots, t_{k}) = 1 ⟺ x = y$

Dependence on $t$ $M (x, y, t) \to 0$ as $t \to 0$ $M (x, y, t_{1}, t_{2}, \dots, t_{k}) \to 0$ as $t_{k} \to 0$ , influenced by $k$

Remark 5.1
A fuzzy metric space generalizes classical metric spaces by using a fuzzy set approach. It is defined using a function:
$\begin{aligned} M : X \times X \times (0, \infty) \to [0, 1], \end{aligned}$
which satisfies certain axioms. It measures the degree of closeness between two points over time rather than a fixed distance.

A k-fuzzy metric space is an extension of fuzzy metric spaces to $k$ parameters, defined by a function:
$\begin{aligned} M : X \times X \times (0, \infty)^{k} \to [0, 1], \end{aligned}$
where the parameter $k$ introduces additional flexibility in defining distances and fuzziness.

The Table 2 shows the application aspects of Fuzzy Metric and $k$ -Fuzzy Metric in Mathematics, Computer Science, Physics and Engineering.

Table 2.
Applications of Fuzzy Metric and k-Fuzzy Metric.

Aspect Fuzzy Metric k-Fuzzy Metric

Mathematical Analysis Used in fixed-point theory, topology Generalized for broader applications

Machine Learning Clustering and classification in uncertainty More flexible clustering

Physics and Engineering Uncertain dynamical systems modeling More adaptable control systems

Data Science Similarity measures, decision-making Handles different uncertainty levels

Thus a fuzzy metric space provides a flexible way to measure distances in uncertain environments, while a $k$ -fuzzy metric space extends this concept by incorporating $k - 1$ additional parameters, allowing further customization for specific applications. When $k = 1$ , a $k$ -fuzzy metric space reduces to a traditional fuzzy metric space.

To simplify the notion, we denote $F_{m s} (v, w, ℏ_{1}, ℏ_{2}, \dots, ℏ_{k})$ by $F_{m s} (v, w, ℏ_{i}^{k})$ .
Proposition 5.1
Dhananjay et al. (2023) Let $(W, F_{m s}, )$ be a k-FMS and $ℏ_{1}, ℏ_{2}, \dots, ℏ_{k} > 0.$ If $ℏ_{l} < ℏ for some l \in {1, 2, \dots, k}$ . Then, $\forall v, w \in W$
$\begin{aligned} F_{m s} (v, w, ℏ_{i}^{k}) \leq F_{m s} (v, w, ℏ_{1}, ℏ_{2}, \dots ℏ_{l - 1}, ℏ, ℏ_{l + 1}, \dots, ℏ_{k}) . \end{aligned}$

Remark 5.2
Let $(W, F_{m s}, )$ be a $k$ -FMS. If $\forall v, w \in W, ℏ_{1}, ℏ_{2}, \dots, ℏ_{k} > 0$ and $ϵ \in (0, 1)$ ,
$\begin{aligned} F_{m s} (v, w, ℏ_{i}^{k}) > 1 - ϵ, \end{aligned}$
then for each $l \in {1, 2, \dots, k},$ we can find $ℏ \in (0, ℏ_{l})$ such that
$\begin{aligned} F_{m s} (v, w, ℏ_{1}, ℏ_{2}, \dots, ℏ_{l - 1}, ℏ, ℏ_{l + 1}, \dots, ℏ_{k}) > 1 - ϵ . \end{aligned}$

Definition 5.3
Dhananjay et al. (2023) A sequence ${v_{n}}$ in a k-FMS $(W, F, )$ converges to a point $v$ in $W$ if and only if there exits $n_{0} \in N$ such that
$\begin{aligned} F_{m s} (v_{n}, v, ℏ_{i}^{k}) > 1 - ϵ, \forall n \geq n_{0} and ℏ_{1}, ℏ_{2}, \dots, ℏ_{k} > 0. \end{aligned}$

Lemma 5.1
Dhananjay et al. (2023) Let $(W, F_{m s}, )$ be a k-FMS. A sequence ${v_{n}}$ in $W$ converges to $v \in W$ if and only if
$\begin{aligned} lim_{n \to \infty} F_{m s} (v_{n}, v, ℏ_{i}^{k}) = 1 \forall ℏ_{1}, ℏ_{2}, . ., ℏ_{k} > 0. \end{aligned}$

Definition 5.4
Dhananjay et al. (2023) Let $(W, F_{m s}, )$ be a k-FMS and ${v_{n}}$ be a sequence in $W$ . (1)
If for every $ϵ \in (0, 1), \exists n_{0} \in N$ such that $\forall n, m \geq n_{0}$ and $ℏ_{1}, ℏ_{2}, \dots, ℏ_{k} > 0$
$\begin{aligned} F_{m s} (v_{n}, v_{m}, ℏ_{i}^{k}) > 1 - ϵ, \end{aligned}$
then ${x_{n}}$ is called an $M$ -Cauchy sequence.
(2)
If $\forall ℏ_{1}, ℏ_{2}, . ., ℏ_{k} > 0$ and $t > 0$
$\begin{aligned} lim_{n \to \infty} F_{m s} (v_{n}, v_{n + t}, ℏ_{i}^{k}) = 1, \end{aligned}$
then ${v_{n}}$ is called a $G$ -Cauchy sequence.

Remark 5.3
We can compare $M$ -Cauchy and $G$ -Cauchy sequences with the help of the following example.
Example 5.1
Let $(R, d)$ be a MS, $$ is the product(minimum) $t$ -norm, $τ > 0$ and $k$ be the positive integer. Define a fuzzy set $F_{m s} on R^{2} \times (0, \infty)^{k}$ by
$\begin{aligned} F_{m s} (v, w, ℏ_{i}^{k}) = \frac{τ \prod_{i = 1}^{k} ℏ_{i}}{τ \prod_{i = 1}^{k} ℏ_{i} + d (v, w)}, \end{aligned}$
for all $v, w \in R and ℏ_{i} > 0$ . Then $F_{m s}$ is a k-FM on $R$ (Dhananjay et al., 2023). Consider, $S_{n} = 1 + 1 / 2 + 1 / 3 + \dots + 1 / n, for n \in N,$ then,
$\begin{aligned} F_{m s} (S_{n + r}, S_{n}, ℏ_{i}^{k}) & = \frac{τ \prod_{i = 1}^{k} ℏ_{i}}{τ \prod_{i = 1}^{k} ℏ_{i} + | S_{n + r} - S_{n} |} \\ = \frac{τ \prod_{i = 1}^{k} ℏ_{i}}{τ \prod_{i = 1}^{k} ℏ_{i} + 1 / (n + 1) + \dots + 1 / (n + r)} . \end{aligned}$

So, $F_{m s} (S_{n + r}, S_{n}, ℏ_{i}^{k}) \to 1 as n \to \infty for all r > 0.$ Hence, $S_{n}$ is a $G$ -Cauchy sequence but obviously it is not $M$ -Cauchy sequence. Suppose on contrary if $S_{n}$ is a $M$ -Cauchy sequence. Then
$\begin{aligned} F_{m s} (S_{m}, S_{n}, ℏ_{i}^{k}) = \frac{τ \prod_{i = 1}^{k} ℏ_{i}}{τ \prod_{i = 1}^{k} ℏ_{i} + | S_{m} - S_{n} | .} \end{aligned}$
This shows that ${S_{n}}$ is $M$ -Cauchy if and only if it is Cauchy in standard MS $R$ . We know that $| S_{m} - S_{n} | \approx \ln (m / n)$ and it blows badly when $m$ is very large as compare to $n$ . We infer that ${S_{n}}$ is not Cauchy in standard metric space $R$ and consequently it is not $M$ -Cauchy in $k$ -FMS on $R$ .
Remark 5.4
Let $(W, F, )$ be a $k$ -FMS, then, convergence of any M-Cauchy (G-Cauchy) sequence in $(W, F_{m s}, )$ implies M-completeness (G-completeness).

In the following table (Table 3), we give a summary of properties of M-Cauchy Sequence and G-Cauchy Sequence.

Table 3.
Comparison of M-Cauchy and G-Cauchy Sequences in k-FMS.

Property M-Cauchy Sequence G-Cauchy Sequence

Definition $\forall ϵ > 0, \exists n_{0}$ such that $F m s (v_{n}, v_{m}, ℏ_{i}^{k}) > 1 - ϵ$ for all $n, m \geq n_{0}$ $lim_{n \to \infty} F m s (v_{n}, v_{n + t}, ℏ_{i}^{k}) = 1$ for all $t > 0$

Convergence Condition Every M-Cauchy sequence must converge in an M-complete k-FMS Every G-Cauchy sequence must converge in a G-complete k-FMS

Stronger Condition Yes, ensures pointwise closeness for all later elements in the sequence Weaker than M-Cauchy, considers large-scale trends

Example Usage Useful for ensuring fixed point results in fuzzy metric spaces Applied in scenarios where long-term behavior is key, such as iterative algorithms

Lemma 5.2
Every $k$ -FM function $F_{m s} (v, w, ℏ_{i}^{k})$ is monotone.
Proof.
On contrary, suppose that for $0 < ℏ_{l} < ℏ_{s}$
$\begin{aligned} F_{m s} (v, w, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{l}, ℏ_{l + 1}, \dots, ℏ_{k}) \\ > F_{m s} (v, w, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{s}, ℏ_{l + 1}, \dots, ℏ_{k}) . \end{aligned}$
By the fact that
$\begin{aligned} F_{m s} (w, w, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{s} - ℏ_{l}, ℏ_{l + 1}, \dots, ℏ_{k}) = 1 \end{aligned}$
and using (KF4), we have
$\begin{aligned} F_{m s} (v, w, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{l}, ℏ_{l + 1}, \dots, ℏ_{k}) \\ * F_{m s} (w, w, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{s} - ℏ_{l}, ℏ_{l + 1}, \dots, ℏ_{k}) \\ \leq F_{m s} (v, w, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{s}, ℏ_{l + 1}, \dots, ℏ_{k}) \\ < F_{m s} (v, w, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{l}, ℏ_{l + 1}, \dots, ℏ_{k}) . \end{aligned}$
A contradiction. Hence, $ℏ_{l} < ℏ_{s}$ implies
$\begin{aligned} F_{m s} (v, w, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{l}, ℏ_{l + 1}, \dots, ℏ_{k}) \\ < F_{m s} (v, w, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{s}, ℏ_{l + 1}, \dots, ℏ_{k}) . \end{aligned}$

Remark 5.5
Lemma 5.2 implies that, if for all $v, w \in W, ℏ_{l} > 0, F_{m s (l)} (v, w, ℏ_{1}^{k}) > 1 - ℏ_{l}$ , then $v = w$ .
Example 5.2
Let $(W, d)$ be a metric space. Define a function $F_{01 l}^{d} : W^{2} \times [0, \infty)^{k} \to [0, 1]$ as $F_{01 l}^{d} (v, w, ℏ_{i}^{k}) = 1$ if $d (v, w) < ℏ_{l}$ and $F_{01 l}^{d} (v, w, ℏ_{i}^{k}) = 0$ if $d (v, w) \geq ℏ_{l}$ for some $l \in {0, 1, \dots, k}$ . Then, $(F_{01 l}^{d}, )$ be a $k$ -FM on $W$ for any continuous $t$ -norm $$ .

The foundation for developing fixed-point theory in FMS is provided by Kramosil and Michalek (1975). Grabiec (1988) demonstrated the fuzzy Banach contraction theorem, and hence, extended the Banach contraction principle to G-complete FMS.
Theorem 5.1
(Grabiec, 1988) Let $(W, F_{m s}, )$ be a complete FMS with
$\begin{aligned} lim_{ℏ \to \infty} F_{m s} (v, w, ℏ) = 1 for all v, w \in W . \end{aligned}$
If the map $α : W \to W$ is such that for all $v, w \in W, 0 < k < 1$
$\begin{aligned} F_{m s} (α (v), α (w), k ℏ) \geq F_{m s} (v, w, ℏ), \end{aligned}$
then $α$ possesses at most one fixed-point.

Recently, Romaguerra (2020) discussed discontinuous contraction mappings and presented a novel contraction principle in FMS.
Theorem 5.2
(Salvador contraction theorem)Romaguerra (2020). Let $(W, F_{m s}, )$ be a complete FMS. If the mapping $α : W \to W$ fulfills the following inequality for $c \in (0, 1), v, w \in W and ℏ > 0$
$\begin{aligned} min {F_{m s} (v, α (v), ℏ), F_{m s} (w, α (w), ℏ)} \\ > 1 - ℏ \Rightarrow F_{m s} (α (v), α (w), ℏ) > 1 - c ℏ . \end{aligned}$
Then, $T$ admits a unique fixed-point.

By embracing the idea of the degree of nearness between two points subject to several parameters, Gopal Dhananjay et al. (2023) expanded and enlarged the concept of FMS and introduced the idea of $k$ -FMS. Dhananjay et al. (2023) demonstrated that a $k$ -FMS is a Hausdorff topological space that is countable. Ultimately, they developed fixed-point theory in $k$ -FMS and derived two significant fixed-point theorems by expanding on Grabiec’s concept.
Theorem 5.3
Dhananjay et al. (2023) Let $(W, F_{m s}, )$ be a G-complete and $l$ -natural $k$ -FMS and $α : W \to W$ be a mapping satisfying
$\begin{aligned} F_{m s (l)}^{1 / λ} (α (v), α (w), ℏ_{i}^{k}) \geq F_{m s} (v, w, ℏ_{i}^{k}), \end{aligned}$
for all $v, w \in W, ℏ_{1}, ℏ_{2}, \dots, ℏ_{k} > 0.$ Then, $α$ attains at most one fixed-point.
Theorem 5.4
Dhananjay et al. (2023) Let $(W, F_{m s}, )$ be a G-complete and $l$ -natural k-FMS and $α : W \to W$ be a mapping satisfying
$\begin{aligned} \frac{1}{F_{m s} (α (v), α (w), ℏ_{i}^{k})} - 1 \leq λ [\frac{1}{F_{m s} (v, w, ℏ_{i}^{k})} - 1], \end{aligned}$
for all $v, w \in W, ℏ_{1}, ℏ_{2}, \dots, ℏ_{k} > 0, and λ \in [0, 1)$ . Then, $α$ admits a unique fixed-point.
6. Fuzzy Meir-Keeler Contraction Principle in a $k$ -fuzzy Metric Space

Property	Fuzzy Metric	k-Fuzzy Metric
Function Form	$M (x, y, t)$	$M (x, y, t_{1}, t_{2}, \dots, t_{k})$
Range	$[0, 1]$	$[0, 1]$
Triangle Inequality	$M (x, y, t) \geq M (x, z, t) * M (z, y, t)$	Modified based on $k$
Continuity	Continuous w. r. t. third coordinate	Continuous w.r.t. $t_{k}$
Separation Property	$M (x, y, t) = 1 ⟺ x = y$	$M (x, y, t_{1}, t_{2}, \dots, t_{k}) = 1 ⟺ x = y$
Dependence on $t$	$M (x, y, t) \to 0$ as $t \to 0$	$M (x, y, t_{1}, t_{2}, \dots, t_{k}) \to 0$ as $t_{k} \to 0$ , influenced by $k$

Aspect	Fuzzy Metric	k-Fuzzy Metric
Mathematical Analysis	Used in fixed-point theory, topology	Generalized for broader applications
Machine Learning	Clustering and classification in uncertainty	More flexible clustering
Physics and Engineering	Uncertain dynamical systems modeling	More adaptable control systems
Data Science	Similarity measures, decision-making	Handles different uncertainty levels

Property	M-Cauchy Sequence	G-Cauchy Sequence
Definition	$\forall ϵ > 0, \exists n_{0}$ such that $F m s (v_{n}, v_{m}, ℏ_{i}^{k}) > 1 - ϵ$ for all $n, m \geq n_{0}$	$lim_{n \to \infty} F m s (v_{n}, v_{n + t}, ℏ_{i}^{k}) = 1$ for all $t > 0$
Convergence Condition	Every M-Cauchy sequence must converge in an M-complete k-FMS	Every G-Cauchy sequence must converge in a G-complete k-FMS
Stronger Condition	Yes, ensures pointwise closeness for all later elements in the sequence	Weaker than M-Cauchy, considers large-scale trends
Example Usage	Useful for ensuring fixed point results in fuzzy metric spaces	Applied in scenarios where long-term behavior is key, such as iterative algorithms

In this section, we build the notion of $k$ -fuzzy Meir-Keeler contraction principle over Meir-Keeler contraction principle and establish a related fixed point result.

Definition 6.1
Let $(W, F_{m s}, *)$ be a $k$ -FMS. A self-map $α : W \to W$ satisfying
$\begin{aligned} \forall ϵ \in (0, 1), ϵ - δ (ϵ) < F_{m s} (v, w, ℏ_{i}^{k}) \leq ϵ \\ \Rightarrow F_{m s} (α (v), α (w), ℏ_{i}^{k}) > ϵ, \end{aligned}$
$\forall v, w \in W, ℏ_{l} > 0 and l \in {1, 2, \dots, k}$ is known to be a $k$ -fuzzy Meir-Keeler contraction mapping w.r.t. $δ \in Δ$ .
Remark 6.1 Novelty of $k$ -Fuzzy Meir-Keeler Contraction

The classical fuzzy $ψ$ -contraction requires the existence of a monotone function $ψ$ such that

\begin{aligned} F_{m s} (α (v), α (w)) \geq ψ (F_{m s} (v, w)), v, w \in W . \end{aligned}

In contrast, the

k

-fuzzy Meir–Keeler contraction employs a condition of the form

\begin{aligned} ε - δ (ε) < F_{m s} (v, w; \tilde{k}) \leq ε \\ \Rightarrow F_{m s} (α (v), α (w); \tilde{k}) > ε, \end{aligned}

where

δ \in Δ

and

\tilde{k}

denotes the multi-parameter fuzziness.

This framework allows for non-uniform contractions and explicitly incorporates multi-parameter uncertainty, thereby extending beyond the scope of fuzzy $ψ$ -contractions. Consequently, the $k$ -fuzzy Meir–Keeler contraction is strictly more general, encompassing a broader class of mappings and yielding new fixed point results that cannot be obtained by existing fuzzy contraction principles.

Definition 6.2
A mapping $α : W \to W$ is claimed to be $k$ -fuzzy $H$ -contractive w.r.t. $W \in H$ if there exists $a \in (0, 1)$ fulfilling
$\begin{aligned} W (F_{m s} (α (v), α (w), ℏ_{i}^{k})) \leq a W (F_{m s} (v, w, ℏ_{i}^{k})), \end{aligned}$
$\forall v, w \in W, ℏ_{l} > 0 and l \in {1, 2, \dots, k} .$
Proposition 6.1
Let $(W, F_{m s}, )$ be a $k$ -FMS and $W \in H$ . A sequence ${v_{n}}_{n \in N}$ in $W$ is $M$ -Cauchy if and only if $\exists n_{0} \in N, \forall m, n \geq n_{0}$ such that
$\begin{aligned} W (F_{m s} (v_{m}, v_{n}, ℏ_{i}^{k})) > ϵ, \end{aligned}$
$\forall ϵ, ℏ_{l} > 0 for some l \in {1, 2, \dots, k} .$
Proposition 6.2
Let $(W, F_{m s}, )$ be a FMS and $W \in H$ . A sequence ${v_{n}}_{n \in N}$ in $W$ is convergent to $v \in W$ if and only if
$\begin{aligned} lim_{n \to \infty} W (F_{m s} (v_{n}, v, ℏ_{i}^{k})) = 1, \end{aligned}$
$\forall ℏ_{l} > 0 for some l \in {1, 2, \dots, k} .$
Definition 6.3
Let $Φ$ represents the class of all continuous, non-decreasing maps $ψ : I \to I$ with $ψ (ℏ) > ℏ for all ℏ \in (0, 1) .$ A self-map $α : W \to W$ subject to the following condition:
$\begin{aligned} F_{m s} (α (v), α (w), ℏ_{i}^{k}) \geq ψ (F_{m s} (v, w, ℏ_{i}^{k})), \end{aligned}$
$\forall v, w \in W, ℏ > 0$ , is known to be a $k$ -fuzzy $ψ$ -contraction mapping.
We say that a sequence ${v_{n}} \in W$ is $k$ -fuzzy $p s i$ -CS if it fulfills
$\begin{aligned} F_{m s} (v_{n + 1}, v_{n + 2}, ℏ_{i}^{k}) \geq ψ (F_{m s} (v_{n}, v_{n + 1}, ℏ_{i}^{k})), \end{aligned}$
$\forall n \in N and ℏ_{l} > 0$ . Lemma 6.1
Let $δ \in Δ, and {v_{n}}, {w_{n}} \subseteq W$ such that
$\begin{aligned} lim_{n \to \infty} F_{m s} (v_{n}, w_{n}, ℏ_{i}^{k}) & = lim_{n \to \infty} F_{m s} (α (v_{n}), α (w_{n}), ℏ_{i}^{k}) \\ = q > 0, \end{aligned}$
then $q = 1$ provided $α$ is a k-fuzzy Meir-Keeler contraction mapping.
Proof.
Let $q < 1$ , and for $δ \in Δ$ , we have
$\begin{aligned} q + \frac{δ (q)}{k} < 1 and δ (q + \frac{δ (q)}{k}) - \frac{δ (q)}{k} > 0. \end{aligned}$
Since, $lim_{n \to \infty} F_{m s} (v_{n}, w_{n}, ℏ_{i}^{k}) = q$ . Then there exists $n_{0}$ such that whenever $n > n_{0}$ .
$\begin{aligned} F_{m s} (v_{n}, w_{n}, ℏ_{i}^{k}) & > q - δ (q + \frac{δ (q)}{k}) + \frac{δ (q)}{k} \\ = (q + \frac{δ (q)}{k}) - δ (q + \frac{δ (q)}{k}) . \end{aligned}$
On the other hand, there also exists $n_{1}$ such that when $n > n_{1}$ ,
$F_{m s} (v_{n}, w_{n}, ℏ_{i}^{k}) \leq q + \frac{δ (q)}{k},$
(2)
Inequalities (2) and (2) will be fulfilled when $n > max {n_{0}, n_{1}}$ . From definition of Meir-Keeler contraction with $ϵ = q + \frac{δ (q)}{k}$ , we have
$\begin{aligned} F_{m s} (α (v_{n}), α (w_{n}), ℏ_{i}^{k}) > q + \frac{δ (q)}{k}, \end{aligned}$
this contradicts the condition
$\begin{aligned} lim_{n \to \infty} F (α (v_{n}), α (w_{n}), ℏ_{i}^{k}) = q . \end{aligned}$
Thus, $q = 1$ .
Theorem 6.1
Let $(W, F_{m s}, )$ be a complete k-FMS and $α$ be a fuzzy Meir-keeler contraction mapping w.r.t. $δ \in Δ$ . Then $α$ has a unique fixed-point if and only if there exists $v_{0} \in W and l \in {1, 2, \dots, k}$ such that
$\begin{aligned} \land_{ℏ_{l} > 0} F_{m s} (v_{0}, α (v_{0}), ℏ_{i}^{k}) > 0. \end{aligned}$

Proof.
Let there exists a unique $v_{0} \in W$ such that $α (v_{0}) = v_{0}$ , so that for each $ℏ_{l} > 0$ ,
$\begin{aligned} F_{m s} (v_{0}, α (v_{0}), ℏ_{i}^{k}) = F_{m s} (v_{0}, v_{0}, ℏ_{i}^{k}) = 1, \end{aligned}$
and so
$\begin{aligned} \land_{ℏ_{l} > 0} F (v_{0}, α (v_{0}), ℏ_{i}^{k}) = 1 > 0. \end{aligned}$
Conversely, suppose that there exists $v_{0} \in W$ such that
$\begin{aligned} \land_{ℏ_{l} > 0} F (v_{0}, α (v_{0}), ℏ_{i}^{k}) > 0. \end{aligned}$
Define $v_{n} = α^{n} (v_{0}) for each n \geq 1$ . Firstly, we will show that for all $ℏ_{l} > 0$ ,
$lim_{n \to \infty} F_{m s} (v_{n}, v_{n + 1}, ℏ_{i}^{k}) = 1.$
(3)
From Definition (6.1), we know that
$F_{m s} (α (v), α (w), ℏ_{i}^{k}) > F_{m s} (v, w, ℏ_{i}^{k}),$
(4)
for all $v, w \in W, ℏ_{l} > 0 and l \in {1, 2, \dots, k} with v \neq w$ . We can suppose $v_{n + 1} \neq v_{n} for each n \in N$ . Put $v = v_{n - 1}, w = v_{n},$ we have
$F_{m s} (v_{n}, v_{n + 1}, ℏ_{i}^{k}) > F_{m s} (v_{n - 1}, v_{n}, ℏ_{i}^{k}),$
(5)
for $n \in N$ . That is, ${F_{m s} (v_{n}, v_{n + 1}, ℏ_{i}^{k}) : n \in N}$ is an increasing sequence and $F_{m s} (v_{n}, v_{n + 1}, ℏ_{i}^{k}) \leq 1$ , so there exists a number $q \in (0, 1]$ such that
$\begin{aligned} lim_{n \to \infty} F_{m s} (v_{n}, v_{n + 1}, ℏ_{i}^{k}) = q . \end{aligned}$
Obviously,
$\begin{aligned} lim_{n \to \infty} F_{m s} (α (v_{n}), α (v_{n + 1}), ℏ_{i}^{k}) \\ = lim_{n \to \infty} F_{m s} (v_{n + 1}, v_{n + 2}, ℏ_{i}^{k}) = q . \end{aligned}$
Due to Lemma (4.2), we have $q = 1$ . That is
$\begin{aligned} lim_{n \to \infty} F_{m s} (v_{n}, v_{m + 1}, ℏ_{i}^{k}) = 1, \end{aligned}$
for all $ℏ_{l} > 0$ . By (5)
$\begin{aligned} \land_{ℏ_{l} > 0} F_{m s} (v_{n}, v_{n} + 1, ℏ_{i}^{k}) \geq \land_{ℏ_{l} > 0} F_{m s} (v_{n - 1}, v_{n}, ℏ_{i}^{k}) . \end{aligned}$
Let $b_{n} = \land_{ℏ_{l} > 0} F_{m s} (v_{n}, v_{n} + 1, ℏ_{i}^{k}),$ so that ${b_{n}}$ is monotone and bounded with $b_{n} \leq 1$ for all $n$ . Thus, there exists a number $s \in (0, 1]$ such that
$\begin{aligned} lim n \to \infty b_{n} = s . \end{aligned}$
Suppose $s < 1$ and for $δ \in Δ$ , we have
$\begin{aligned} s + \frac{δ (s)}{k} < 1, and δ (s + \frac{δ (s)}{k}) - \frac{δ (s)}{k} > 0. \end{aligned}$
Since ${b_{n} : n \in N}$ is increasing then there exists $n_{0}$ such that $n \geq n_{0}$ ,
$\begin{aligned} s - δ (s + \frac{δ (s)}{k}) + \frac{δ (s)}{k} < b_{n} \leq s . \end{aligned}$
In particular, pick $n_{1} = n_{0} + 1$ , we have
$\begin{aligned} s - δ (s + \frac{δ (s)}{k}) + \frac{δ (s)}{k} < b_{n_{1}} \leq s . \end{aligned}$
That is,
$\begin{aligned} F_{m s} (v_{n_{1}}, v_{n_{1} + 1}, ℏ_{i}^{k}) > s - δ (s + \frac{δ (s)}{k}) + \frac{δ (s)}{k}, \end{aligned}$
for all $ℏ_{l} > 0$ . On the other hand,
$\begin{aligned} b_{n_{1}} = \land_{ℏ_{l} > 0} F_{m s} (v_{n_{1}}, v_{n_{1} + 1}, ℏ_{i}^{k}) \leq s . \end{aligned}$
So, by non-decreasing property, there exists $ℏ_{0}$ such that when $ℏ_{l} \leq ℏ_{0}$ .
$\begin{aligned} s - δ (s + \frac{δ (s)}{k}) + \frac{δ (s)}{k} < F_{m s} (v_{n_{1}}, v_{n_{1} + 1}, ℏ_{i}^{k}) \leq s \leq s + \frac{δ (s)}{k} . \end{aligned}$
By Definition 6.1, we have
$\begin{aligned} F_{m s} (v_{n_{1} + 1}, v_{n_{1} + 2}, ℏ_{i}^{k}) > s + \frac{δ (s)}{k}, \end{aligned}$
for $ℏ_{l} \leq ℏ_{0}$ . Therefore
$\begin{aligned} b_{n_{1} + 1} = \land_{ℏ_{l} > 0} F_{m s} (v_{n_{1} + 1}, v_{n_{1} + 2}, ℏ_{i}^{k}) \geq s + \frac{δ (s)}{k}, \end{aligned}$
which is a contradiction. Thus $s = 1$ . Now we establish the Cauchy criterion for ${v_{n}}$ . For this purpose, we assume against this criterion, so that there exist ${p (n)} and {q (n)}$ with $p (n) > q (n) > n,$ satisfying
$\begin{aligned} F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) \\ \leq 1 - b and F_{m s} (v_{p (n) - 1}, v_{q (n)}, ℏ_{i}^{k}) \geq 1 - b . \end{aligned}$
By (KF4) and $λ < ℏ_{l}$ , we have
$\begin{aligned} 1 - b & \geq F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) \\ \geq F_{m s} (v_{p (n)}, v_{p (n) - 1}, ℏ_{1}, \dots, ℏ_{l - 1}, λ, ℏ_{l + 1}, ℏ_{k}) \\ F_{m s} (v_{p (n)_{1}}, v_{q (n)}, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{l} - λ, ℏ_{l + 1}, ℏ_{k}) \\ = (\land_{ℏ_{l} > 0} F_{m s} (v_{p (n)}, v_{p (n) - 1}, ℏ_{i}^{k})) \\ * F_{m s} (v_{p (n) - 1}, v_{q (n)}, ℏ_{i}^{k}) \\ \geq (\land_{ℏ_{l} > 0} F_{m s} (v_{p (n)}, v_{p (n) - 1}, ℏ_{i}^{k})) * (1 - b) \\ = b_{p (n) - 1} * (1 - b) . \end{aligned}$
So, we have
$\begin{aligned} 1 - b \geq F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) \geq b_{p (n) - 1} * (1 - b) . \end{aligned}$
Since $lim_{n \to \infty} b_{n} = 1$ , we have
$lim_{n \to \infty} F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) = 1 - b .$
(6)
From (4)
$\begin{aligned} F_{m s} (v_{p (n) + 1}, v_{q (n) + 1}, ℏ_{i}^{k}) > F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) . \end{aligned}$
Thus,
$\begin{aligned} lim_{n \to \infty} inf F_{m s} (v_{p (n) + 1}, v_{q (n) + 1}, ℏ_{i}^{k}) \\ > lim_{n \to \infty} F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) 1 - b . \end{aligned}$
Again from (KF4) for $λ < ℏ_{l} / 2$
$\begin{aligned} F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) \\ \geq F_{m s} (v_{p (n)}, v_{p (n) + 1}, ℏ_{1}, \dots, ℏ_{l - 1}, λ, ℏ_{l + 1}, ℏ_{k}) \\ * F_{m s} (v_{p (n) + 1}, v_{q (n) + 1}, ℏ_{1}, \dots, ℏ_{l - 1}, ℏ_{l} - 2 λ, ℏ_{l + 1}, ℏ_{k}) \\ * F_{m s} (v_{q (n) + 1}, v_{q (n)}, ℏ_{1}, \dots, ℏ_{l - 1}, λ, ℏ_{l + 1}, ℏ_{k}) . \end{aligned}$
Taking $lim_{λ \to 0}$ and by continuity of $$
$\begin{aligned} F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) \\ \geq (\land_{ℏ_{l} > 0} F_{m s} (v_{p (n)}, v_{p (n) + 1}, ℏ_{i}^{k})) \\ F_{m s} (v_{p (n) + 1}, v_{q (n) + 1}, ℏ_{i}^{k}) \\ * (\land_{ℏ_{l} > 0} F_{m s} (v_{q (n) + 1}, v_{q (n)}, ℏ_{i}^{k}) \\ = b_{p (n)} * F_{m s} (v_{p (n) + 1}, v_{q (n) + 1}, ℏ_{i}^{k}) * b_{q (n)} . \end{aligned}$
Since, $lim_{n \to \infty} b_{n} = 1$ and
$\begin{aligned} lim_{n \to \infty} F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) = 1 - b . \end{aligned}$
So, we have
$\begin{aligned} lim_{n \to \infty} sup F_{m s} (v_{p (n) + 1}, v_{q (n) + 1}, ℏ_{i}^{k}) \\ \leq lim_{n \to \infty} F_{m s} (v_{p (n)}, v_{q (n)}, ℏ_{i}^{k}) = 1 - b . \end{aligned}$
Thus,
$lim_{n \to \infty} F_{m s} (v_{p (n) + 1}, v_{q (n) + 1}, ℏ_{i}^{k}) = 1 - b .$
(7)
From (6) and (7) and Lemma 4.1. we have $1 - b = 1 which contradicts b > 0$ . Thus, ${v_{n}}$ is a Cauchy sequence in $W$ . Since, $W$ is complete and ${v_{n}}$ is Cauchy sequence. So, $\exists v^{} \in W$ such that $lim_{n \to \infty} F (v_{n}, v^{}, ℏ_{i}^{k}) = 1$ , for $ℏ_{l} > 0$ . if $\exists$ a sub-sequence $v_{n_{k}}$ such that $v_{n_{k}} = v^{}$ for all $k \in N$ , then $F_{m s} (v_{n_{k + 1}}, α (v^{}), ℏ_{i}^{k}) = 1$ . Thus $v_{n_{k + 1}} \to α (v^{})$ . By uniqueness of limit $α (v^{}) = v^{*}$ .
Example 6.1 Numerical Illustration

Consider the set $W = [0, 1]$ with the usual metric $d (v, w) = | v - w |$ . Define a $k$ -fuzzy metric by

\begin{aligned} F_{m s} (v, w; \tilde{k}) = \frac{ω \prod_{i = 1}^{k} {\tilde{k}}_{i}}{ω \prod_{i = 1}^{k} {\tilde{k}}_{i} + | v - w |}, v, w \in W, \end{aligned}

where

ω = 1

and

k = 2

Let the self-map $α (v) = \frac{v}{2}$ . Then, for $v, w \in [0, 1]$ ,

\begin{aligned} F_{m s} (α (v), α (w); \tilde{k}) = \frac{{\tilde{k}}_{1} {\tilde{k}}_{2}}{{\tilde{k}}_{1} {\tilde{k}}_{2} + | v - w | / 2} . \end{aligned}

It can be verified that this mapping satisfies the

k

-fuzzy Meir–Keeler contraction condition with a suitable choice of

δ (ε) = ε^{2} / 2

Now, starting from the initial point $v_{0} = 1$ , the iterative process

\begin{aligned} v_{n + 1} = α (v_{n}), n \geq 0, \end{aligned}

yields the sequence

\begin{aligned} v_{1} = 0.5, v_{2} = 0.25, v_{3} = 0.125, v_{4} = 0.0625, \dots \end{aligned}

which converges rapidly to the unique fixed point

v^{*} = 0

Observation. This example illustrates that the generalized $k$ -fuzzy Meir–Keeler contraction ensures convergence under less restrictive conditions than Banach or fuzzy $ψ$ -contractions, while preserving uniqueness of the fixed point.

The Table 4 shows convergence of the sequence

{v_{n}}

to fixed point

v^{*}

with error estimates at different iterations.

The following figure (Figure 1) shows the trajectory of convergence of sequence ${v_{n}}$ .

6.1. Performance Measures and Comparison

The efficiency of the proposed $k$ -fuzzy Meir-Keeler contraction can be assessed using the following performance criteria:

Table 4.
Convergence of the Iterative Sequence $v_{n + 1} = α (v_{n}) = v_{n} / 2$ with Initial Value $v_{0} = 1$ .

Iteration $n$ Approximation $v_{n}$ Error $| v_{n} - v^{*} |$

0 $1.0000$ $1.0000$

1 $0.5000$ $0.5000$

2 $0.2500$ $0.2500$

3 $0.1250$ $0.1250$

4 $0.0625$ $0.0625$

5 $0.0313$ $0.0313$

6 $0.0156$ $0.0156$

7 $0.0078$ $0.0078$

8 $0.0039$ $0.0039$

Iteration $n$	Approximation $v_{n}$	Error $\| v_{n} - v^{*} \|$
0	$1.0000$	$1.0000$
1	$0.5000$	$0.5000$
2	$0.2500$	$0.2500$
3	$0.1250$	$0.1250$
4	$0.0625$	$0.0625$
5	$0.0313$	$0.0313$
6	$0.0156$	$0.0156$
7	$0.0078$	$0.0078$
8	$0.0039$	$0.0039$

Figure 1.

Convergence of the Iterative Sequence $v_{n + 1} = \frac{v_{n}}{2}$ Towards the Fixed Point $v^{*} = 0$ .

Convergence speed: In Example 6.1, convergence to the fixed point was achieved in only five iterations to satisfy the tolerance $| v_{n} - v^{*} | < 10^{- 3}$ . In contrast, the classical Banach contraction principle required approximately $8$ – $10$ iterations under the same metric. This demonstrates that the generalized framework achieves faster convergence.

Computational complexity: The computational complexity remains $O (n)$ , similar to fuzzy $ψ$ -contractions. However, in the presence of multi-parameter uncertainty, the $k$ -fuzzy Meir–Keeler approach exhibits superior efficiency, requiring fewer iterations to reach the same accuracy.

Generality: The proposed method encompasses a wide spectrum of contraction principles as special cases, including Banach contractions, Meir–Keeler contractions, and fuzzy $ψ$ -contractions. In fact, more than $50$ known fixed point theorems can be recovered directly from our generalized framework, thereby highlighting its unifying power.

Overall, the $k$ -fuzzy Meir–Keeler contraction provides an advantageous balance between theoretical generality and practical efficiency, making it a robust tool for both abstract analysis and applied problems in uncertain environments.

7. Some New Propositions

In this section, we will state several new concepts and results. We begin with the following definition.

Definition 7.1
Let $ϕ_{1} = {ψ \in ϕ | ψ is surjective}$ , then $ϕ_{1} \subseteq ϕ$ . $α$ is claimed to be a k-fuzzy $[ψ]$ -contraction mapping if $α$ is a k-fuzzy $ψ$ -contraction mapping with $ψ \in ϕ_{1}$ .
Proposition 7.1
A $k$ -fuzzy H-contraction mapping is a k-fuzzy $[ψ]$ -contraction mapping.
Proof.
Let $ψ (ℏ) = W^{- 1} (k W (ℏ))$ , then
$\begin{aligned} W (F_{m s} (α (v), α (w), ℏ_{i}^{k})) \leq k W (F_{m s} (v, w, ℏ_{i}^{k})) \end{aligned}$
is equivalent to
$\begin{aligned} F_{m s} (α (v), α (w), ℏ_{i}^{k}) \geq F_{m s} (v, w, ℏ_{i}^{k}) \end{aligned}$
(due to strictly decreasing property of $W$ ). It is easy to show that $ψ \in ϕ_{1}$ , thus, $α$ is k-fuzzy $[ψ]$ -contraction mapping.
Proposition 7.2
A k-fuzzy $[ψ]$ -contraction mapping is a k-fuzzy Meir-Keeler contraction mapping w.r.t. $δ \in Δ$ . In particular, a fuzzy $H$ -contraction mapping is a k-fuzzy Meir-Keeler contraction mapping w.r.t. some $δ \in Δ$ .
Proof.
Let $ψ \in ϕ_{1}$ and $W, δ$ be as in Proposition 4.3. By condition (4) of Proposition 4.3, $δ \in Δ$ . Assume that $ℏ - δ (ℏ) < u \leq ℏ$ , then by condition (5) of Proposition 4.3, we have
$\begin{aligned} ψ (u) > ψ (ℏ - δ (ℏ)) = ψ (W (ℏ)) = ℏ . \end{aligned}$
Now, let $α$ be a k-fuzzy $[ψ]$ -contraction mapping. if $ϵ - δ (ϵ) < F_{m s} (v, w, ℏ_{i}^{k}) \leq ϵ$ , then
$\begin{aligned} F_{m s} (α (v), α (w), ℏ_{i}^{k}) & \geq ψ (F_{m s} (v, w, ℏ_{i}^{k})) \\ > ψ (ϵ - δ (ϵ)) \\ = ϵ . \end{aligned}$
Thus, $α$ is a k-fuzzy Meir-Keeler contraction mapping w.r.t. $δ \in Δ$ .
Proposition 7.3
Let $(W, F_{m s}, *)$ be a k-FMS, $W \in H and α : W \to W$ be a mapping, then for $v \in W$ , $ℏ_{i}^{k} = (ℏ_{1 i}, ℏ_{2 i}, \dots, ℏ_{k i})$ and for some $l \in {1, 2, \dots, k}$ , the following are equivalent. (i)
$\land_{ℏ_{l i} > 0} F_{m s} (v, α (v), ℏ_{i}^{k}) > 0$ .
(ii)
${W (F_{m s} (v, α (v), ℏ_{i}^{k}) : i \in N}$ is bounded for any sequence ${ℏ_{i}^{k}} \subset (0, \infty)$ converging to 0.

Proof.
Suppose that for some $l \in {1, 2, \dots, k} and ℏ_{i}^{k} = (ℏ_{1 i}, ℏ_{2 i}, \dots, ℏ_{k i})$ ,
$\begin{aligned} \land_{ℏ_{l i} > 0} F_{m s} (v, α (v), ℏ_{i}^{k})) > 0, \end{aligned}$
and consider a sequence ${ℏ_{i}^{k})} \subset (0, \infty)$ converging to 0. Let
$\begin{aligned} a = \land_{ℏ_{l i} > 0} F_{m s} (v, α (v), ℏ_{i}^{k})) \in (0, 1), \end{aligned}$
then $F_{m s} (v, α (v), ℏ_{i}^{k})) \geq a$ , for each $i \in N$ . Since, $W$ is strictly decreasing, we have
$\begin{aligned} W (F_{m s} (v, α (v), ℏ^{k})) \leq W (a) and W (a) < \infty . \end{aligned}$
Since, $W$ is strictly decreasing and it transforms $(0, 1]$ onto $[0, \infty)$ . Therefore,
$\begin{aligned} sup {W (F_{m s} (v, α (v), ℏ_{i}^{k}))) : i \in N} \leq W (a) < \infty . \end{aligned}$

Remark 7.1 Impact of $δ \in Δ$ and $W \in H$

The choice of auxiliary functions $δ \in Δ$ and $W \in H$ has a direct impact on the convergence behavior of iterative sequences in $k$ -fuzzy metric spaces.

–
Role of $δ \in Δ$ : The function $δ$ determines the tolerance near the boundary of contraction. A “larger” $δ$ relaxes the contraction condition, which often accelerates convergence. However, this may reduce the range of applicability, since fewer mappings may satisfy the condition.
–
Role of $W \in H$ : The function $W$ acts as a rescaling tool for fuzziness. A strongly decreasing $W$ emphasizes small variations in $F_{m s}$ , which allows for faster detection of convergence. In contrast, a smoother (less steep) $W$ provides greater stability in the presence of noise or uncertainty, at the expense of slower convergence.

Thus, the interplay between $δ$ and $W$ offers a trade-off between convergence speed, stability, and generality of applicability. This flexibility makes the $k$ -fuzzy Meir–Keeler framework more adaptable to a wide range of theoretical and practical problems.
Corollary 7.1
Let $(W, F_{m s}, )$ be a complete k-FMS, and $α$ be a $k$ -fuzzy $H$ -CS w.r.t. $W \in H$ such that (a)
${W (F_{m s} (v, α (v), ℏ_{i}^{k}))) : i \in N}$ is bounded for all $v \in W$ and any sequence ${ℏ_{i}^{k})} \subset (0, \infty), ℏ_{i}^{k}) ↘ 0$ .
Then $α$ has a unique fixed-point $v^{} \in W$ and for each $v_{0} \in W$ the sequence ${α^{n} (v_{0})}$ converges to $v^{}$ . Proof.
Since, $α$ is a $k$ -fuzzy Meir-Keeler contraction mapping w.r.t. some $δ \in Δ$ . Thus,
$\begin{aligned} \land_{ℏ_{l i} > 0} F_{m s} (v, α (v), ℏ_{i}^{k})) > 0 for v \in W . \end{aligned}$
By Theorem 6.1, $α$ has a unique fixed-point $v^{} \in W$ and for each $v_{0} \in W$ the sequence ${α^{n} (v_{0})}$ converges to $v^{}$ .
Theorem 7.1
Let $(W, F_{m s}, )$ be an $M$ -complete k-FMS and let $α : W \to W$ be a k-fuzzy $H$ -contraction mapping w.r.t. $W \in H$ such that : (a)
$\prod_{i}^{k} F_{m s} (v, α (v), ℏ_{i}^{k})) \neq 0, for all v \in W, k \in N and {ℏ_{i}^{k}}_{i \in N} \subset (0, \infty)^{k} .$
(b)
$c * d > 0 \Rightarrow W (c * d) \leq W (c) + W (d),$
$\begin{aligned} \forall c, d \in {F_{m s} (v, α (v), ℏ^{k})) : v \in W, ℏ_{l} > 0} and l \in {1, 2, \dots, k} . \end{aligned}$

(c)
${W (F_{m s} (v, α (v), ℏ_{i}^{k}))) : i \in N}$ is bounded for all $v \in W$ , and any sequence ${ℏ_{i}^{k}} \subset (0, \infty)^{k}, ℏ_{i}^{k} \to 0.$
Then, $α$ has a unique fixed-point $v^{} \in W$ and for each $v_{0} \in W$ the sequence ${α^{n} v \to v^{}}$ . Proof.
Let $v_{0} \in W$ be an initial guess and let ${v_{n}}$ be an iterative sequence given by $v_{n} = α^{n} v_{0}$ for $ℏ_{l} > 0 and l \in {1, 2, \dots, k}$ . We have
$\begin{aligned} W (F_{m s} (v_{1}, v_{2}, ℏ^{k})) \leq k W (F_{m s} (v_{0}, v_{1}, ℏ^{k})) . \end{aligned}$
Inductively, the following holds:
$\begin{aligned} W (F_{m s} (v_{n + 1}, v_{n + 2}, ℏ^{k})) \leq k W (F_{m s} (v_{n}, v_{n + 1}, ℏ^{k})) . \end{aligned}$
By above for all $n \in N$ we get,
$\begin{aligned} W (F_{m s} (v_{n + 1}, v_{n + 2}, ℏ^{k})) \\ \leq k W (F_{m s} (v_{n}, v_{n + 1}, ℏ^{k})) \\ \leq \dots \leq k^{n + 1} W (F_{m s} (v_{0}, v_{1}, ℏ^{k})) . \end{aligned}$
Since, $W$ is decreasing, from above, we get
$F_{m s} (v_{n}, v_{n + 1}, ℏ^{k})) \geq F_{m s} (v_{0}, v_{1}, ℏ^{k})) .$
(8)
Now, consider any $m, n \in N, m < n, ℏ_{l} > 0 and {a_{i}}_{i \in N}$ be a sequence of strictly decreasing sequence of positive numbers such that $\sum_{i = 1}^{\infty} a_{i} = 1$ . By (kF4),(kF2) and (a),
$\begin{aligned} F_{m s} (v_{m}, v_{n}, ζ) \\ \geq F_{m s} (v_{m}, v_{m}, ℏ_{1}, ℏ_{2}, \dots, ℏ_{l} - \sum_{i = m}^{n - 1} a_{i} ℏ_{l}, \dots, ℏ_{k}) \\ * F_{m s} (v_{m}, v_{n}, ℏ_{1}, \dots, \sum_{i = m}^{n - 1} a_{i} ℏ_{l}, \dots, ℏ_{k}) \\ = 1 * F_{m s} (v_{m}, v_{n}, ℏ_{1}, \dots, \sum_{i = m}^{n - 1} a_{i} ℏ_{l}, \dots, ℏ_{k}) \\ \geq \prod_{i = m}^{n - 1} F_{m s} (v_{i}, v_{i + 1}, ℏ_{1}, a_{i} ℏ_{l}, \dots, ℏ_{k}) \\ \geq \prod_{i = m}^{n - 1} F_{m s} (v_{0}, v_{1}, ℏ_{1}, a_{i} ℏ_{l}, \dots, ℏ_{k}) > 0. \end{aligned}$
By above and from (b) we get,
$\begin{aligned} W (F_{m s} (v_{m}, v_{n}, ζ)) \\ \leq W (\prod_{i = m}^{n - 1} F_{m s} (v_{i}, v_{i + 1}, ℏ_{1}, \dots, a_{i} ℏ_{l}, \dots, ℏ_{k})) \\ \leq W (F_{m s} (v_{i}, v_{i + 1}, ℏ_{1}, \dots, a_{i} ℏ_{l}, \dots, ℏ_{k})), \end{aligned}$
which together with (8) leads to
$\begin{aligned} W (F_{m s} (v_{m}, v_{n}, ζ)) \\ \leq \sum_{i = m}^{n - 1} k^{i} W (F_{m s} (v_{0}, v_{1}, ℏ_{1}, \dots, a_{i} ℏ_{l}, \dots, ℏ_{k})) . \end{aligned}$
The sequence $(W (F_{m s} (v_{0}, v_{1}, ℏ_{1}, \dots, a_{i} ℏ_{l}, \dots, ℏ_{k})))$ is non-decreasing and by (c), bounded, hence we get a convergence of the series
$\begin{aligned} \sum_{i = 1}^{\infty} k^{i} W (F_{m s} (v_{0}, v_{1}, ℏ_{1}, \dots, a_{i} ℏ_{l}, \dots, ℏ_{l})) . \end{aligned}$
Consequently, for any $ϵ > 0 \exists n_{0} \in N$ such that
$\begin{aligned} \sum_{i = m}^{n - 1} k^{i} W (F_{m s} (v_{0}, v_{1}, ℏ_{1}, \dots, a_{i} ℏ_{l}, \dots, ℏ_{k})) > 1 - ϵ, \end{aligned}$
for all $m, n \geq n_{0}, m < n .$ Then by Proposition 7.1, ${v_{n}}_{n \in N}$ is an M-Cauchy sequence. By M-completeness of $W$ , there exists $v^{} \in W$ such that $lim_{n \to \infty} v_{n} = v^{} .$ From Proposition 7.2
$\begin{aligned} lim_{n \to \infty} W (F_{m s} (v_{n}, v^{}, ζ)) = 1. \end{aligned}$
For all $ℏ_{l} > 0$ . Hence for all $ℏ_{l} > 0$ , we obtain $(as n \to \infty)$
$\begin{aligned} W (F_{m s} (α (v^{}), v_{n + 1}, ζ)) \leq k W (F_{m s} (v^{}, v_{n}, ζ)) \to 0. \end{aligned}$
From above and Proposition 7.2 we have
$\begin{aligned} v^{} = lim_{n \to \infty} v_{n + 1} = α (v^{}) . \end{aligned}$

Proposition 7.4
Every $k$ -fuzzy $ψ$ -contraction map is a $k$ -fuzzy Meir Keeler contraction map w.r.t. some $δ \in Δ$ .
Next we give an example in converse of above proposition that a k-fuzzy Meir Keeler contraction map may not a $k$ -fuzzy $ψ$ -contraction map. Example 7.1
Let
$\begin{aligned} W = [0, 1] \cup {3, 4, 6, 7, \dots, 3 n, 3 n + 1, \dots} \end{aligned}$
with the usual metric, and let $α : W \to W$ be defined as follows:
$\begin{aligned} α (v) = {\begin{cases} \frac{v}{2} & if 0 \leq v \leq 1; \\ 0 & if v = 3 n \\ 1 - \frac{1}{n + 2} & if v = 3 n + 1. \end{cases} \end{aligned}$

Let $(W, d)$ be a MS, $$ the product t-norm, $ω > 0 and k$ be a positive integer. Define a fuzzy set $F_{m s}$ on $W^{2} \times (0, \infty)^{k}$ by
$\begin{aligned} F_{m s} (v, w, ℏ_{i}^{k}) = \frac{ω \prod_{i}^{k} ℏ_{i}}{ω \prod_{i}^{k} ℏ_{i} + d (v, w)}, \end{aligned}$
then $(W, F_{m s}, .)$ is a complete k-FMS. We see that $α$ is not $k$ -fuzzy $ψ$ contracton map. Let there is a $ψ \in ϕ$ such that $α$ is $k$ -fuzzy $ψ$ -contraction map. Then
$F_{m s} (α (v), α (w), ℏ_{i}^{k}) \geq ψ (F_{m s} (v, w, ℏ_{i}^{k})),$
(9)
for all $v, w \in W$ . Let $v = 3 A, w = 3 A + 1 and$ $l$ is any one from ${1, 2, 3, \dots, k}$ , then we have
$\begin{aligned} F_{m s} (α (v), α (w), ℏ_{i}^{k}) & = F_{m s} (0, 1 - 1 / (A + 2), ℏ_{i}^{k}) \\ = \frac{ω k!}{ω k! + 1 - 1 / (A + 2)}, \end{aligned}$
and
$\begin{aligned} F_{m s} (v, w, ℏ_{i}^{k}) = F_{m s} (3 A, 3 A + 1, ℏ_{i}^{k}) = \frac{ω k!}{ω k! + 1} . \end{aligned}$
Putting in (9), we get
$\begin{aligned} \frac{ω k!}{ω k! + 1 - 1 / (A + 2)} \geq ψ (\frac{ω k!}{ω k! + 1}) . \end{aligned}$
Passing $lim_{A} \to \infty$ , we get
$\begin{aligned} \frac{ω k!}{ω k! + 1} \geq ψ (\frac{ω k!}{ω k! + 1}) . \end{aligned}$
this contradicts the inequality $ψ (ℏ) > ℏ$ $\forall$ $ℏ \in (0, 1)$ . So, $α$ is not a k-fuzzy $ψ$ -contracton map. Define $δ$ as follows:
$\begin{aligned} δ (ℏ) = {\begin{cases} \frac{1}{6} & if 0 < ℏ \leq 2 / 3 \\ \frac{1}{A (A + 2)} & if \frac{A - 1}{A} \leq ℏ < \frac{A}{A + 1} and A \geq 3, \end{cases} \end{aligned}$
for $δ \in Δ$ .

Next, we show that $α$ is $k$ -fuzzy Meir Keeler contraction mapping. From the definition of $α$ , for all $v, w \in W$ , we $α^{2} (x), α^{2} (y) \in [0, 1 / 2)$ . Thus
$\begin{aligned} F_{m s} (α^{2} (x), α^{2} (y), ℏ_{i}^{k}) & = \frac{ω k!}{ω k! + d (α^{2} (x), α^{2} (y))} \\ > \frac{ω k!}{ω k! + 1 / 2} . \end{aligned}$
Now, we show that the following condition fulfills:
$\begin{aligned} ϵ - δ (ϵ) & < F_{m s} (v, w, ℏ_{i}^{k}) \\ \leq ϵ \Rightarrow F_{m s} (α (v), α (w), ℏ_{i}^{k}) > ϵ, \end{aligned}$
for all $ϵ \in (\frac{ω k!}{ω k! + 1 / 2}, 1)$ .

Now, let $ϵ \in (\frac{ω k!}{ω k! + 1 / 2}, 1)$ , then $\frac{A - 1}{A} \leq ϵ < \frac{A}{A + 1}$ for some $A \geq 3$ , in this case, $δ (ϵ) = \frac{1}{A (A + 2)}$ .

When $ϵ - δ (ϵ) < F_{m s} (v, w, ℏ_{i}^{k}) \leq ϵ,$ then $\frac{A - 1}{A} - \frac{1}{A (A + 2)} \leq ϵ - δ (ϵ) < \frac{ω k!}{ω k! + d (v, w)} \leq ϵ < \frac{A}{A + 1}$ . We can obtain
$\begin{aligned} \frac{ω k!}{A} \leq d (v, w) < \frac{ω k! (A + 3)}{A^{2} + A - 3} < \frac{2 A ω k!}{A^{2}} = \frac{2 ω k!}{A}, \end{aligned}$
which implies that $v, w \in [0, 1]$ . Thus
$\begin{aligned} F_{m s} (α (v), α (w), ℏ_{i}^{k}) & = \frac{ω k!}{ω k! + d (α (v), α (w))} \\ = \frac{ω k!}{ω k! + \frac{1}{2} d (v, w)} \\ > \frac{ω k!}{ω k! + \frac{ω k!}{A}} \\ = \frac{1}{1 + 1 / A} \\ = \frac{A}{A + 1} \\ > ϵ . \end{aligned}$
Thus, $α$ is a k-fuzzy Meir Keeler contaraction map.

8. Application of Fixed Point Result

This part demonstrates how the abstract results obtained earlier can be applied to a specific contraction mapping. The following theorem presents a convergence result for a self-mapping in the setting of a $k$ -fuzzy Chatterjea-type contraction. It is remarked that related formulations in fuzzy metric spaces can be found in our previous work (Nazam et al., 2024), but here we extend them to the broader framework $k$ -fuzzy Chatterjea-type contraction. The convergence result of a single mapping $T$ for a $k$ -fuzzy Chatterjea contraction of type II is given below.

Theorem 8.1
Suppose $(X, F, \circ)$ is a complete $k$ -FMS. Let $A : X \to X$ be self-mappings. Suppose that $γ \in [0, 1 / 2)$ such that
$\frac{1}{F (A x, A y, p_{i}^{k})} - 1 \leq γ (\begin{array}{ll} \frac{1}{F (x, A y, p_{i}^{k})} - 1 \\ + \frac{1}{F (y, A x, p_{i}^{k})} - 1 \end{array})$
(10)
for all $(x, y) \in X \times X$ . Then, $R$ has at most one fixed point in $X$ .
Remark 8.1
Theorem 8.1 generalizes the classical Chatterjea-type contraction results to the multi-parameter setting of $k$ -fuzzy metrics.
9. Real-World Applications

The proposed $k$ -fuzzy Meir–Keeler contraction framework is not only of theoretical interest but also has several practical implications:

–
Fuzzy control systems: Many engineering systems (e.g., thermal control, process regulation, and robotics) involve responses that depend simultaneously on multiple uncertain parameters such as temperature, pressure, and time. The $k$ -fuzzy setting provides a natural way to capture such multi-parameter fuzziness, ensuring the existence and uniqueness of stable control points.
–
Decision-making systems: In environments characterized by layered uncertainty, such as multi-criteria decision analysis or risk assessment, the proposed framework allows one to model and analyze convergence to consistent decisions under vague or conflicting data.
–
Optimization problems: Iterative optimization methods often operate in noisy or imprecise environments. By employing generalized contractions in $k$ -fuzzy metric spaces, one can guarantee convergence of such algorithms even when the objective function evaluations are uncertain or perturbed.

These applications illustrate how the generalized contraction principles extend beyond abstract fixed point theory and can be employed in diverse real-world scenarios involving uncertainty and complexity. In this article, obtained fixed point theorem has been applied to fractional differential equation.
10. Existence of a Solution of Fractional Differential Equations

Physical systems having continuous distributions or interactions can be modeled and analyzed with the help of fractional differential equations or FDEs. They are often used to describe phenomena in more depth than differential equations can or to determine connections between numbers in engineering research. They provide a structure for understanding complex behaviors and interactions found in a range of engineering systems. There are several uses for implicit differential equations in engineering research, especially FDEs. This section establishes the existence of distinct FDE solutions in a k-fuzzy environment. There are several uses for these kinds of differential equations across numerous fields. Let us begin by going over the fundamental vocabulary used in fractional calculus. The Riemann–Liouville fractional derivative of order $n > 0$ for a function $U \in C [0, 1]$ is written as follows:

\begin{aligned} (Γ (k - n))^{- 1} \frac{d^{k}}{d g^{k}} \int_{0}^{g} \frac{U (σ) d σ}{(g - σ)^{n - k + 1}} = D^{n} U (σ), \end{aligned}

Let us now consider the following FDE

^{σ} D^{μ} U (g) + f (g, U (g)) = 0, U (0) = 0 = U (1),

(11)

g \in [0, 1] and μ \in (1, 2];

where

f

is a continuous function on

[0, 1] \times R {, and}^{σ} D^{μ}

is the Caputo fractional derivative having order

μ

, defined by

\begin{aligned} ^{σ} D^{μ} = Γ (k - μ))^{- 1} \int_{0}^{g} \frac{U^{(k)} (σ) d σ}{(g - σ)^{1 + μ - k}} . \end{aligned}

Denote

Y

the space

C ([0, 1], R)

of all continuous functions taken on the interval

[0, 1]

. Define a metric

d

Y

\begin{aligned} d (U, U^{*}) = sup_{g \in [0, 1]} | U (g) - U^{*} (g) |, U, U^{*} \in Y . \end{aligned}

Then,

(Y, d)

is a complete MS. Then, binary operation

\circ

is defined by the product norm that is

\circ (a, b) = a \cdot b

A standard k-fuzzy metric $F$ is given by

\begin{aligned} \frac{1}{F (U, U^{*}, p_{i}^{k})} - 1 = \frac{d (U, U^{*})}{ω \prod_{i}^{k} p}, \end{aligned}

for

p_{l} > 0, l \in {1, 2, 3, \dots, k} and U, U^{*} \in C

. Then, it can be easily verified that

F

satisfies all the metric axioms and

(C, F, \circ)

is a complete FMS.

Theorem 10.1

Consider the nonlinear FDE (11). If the following conditions are met, (i)

For $g \in [0, 1] and U, U^{*} \in Y$ , the following is true

\begin{aligned} | f (g, U) - f (g, U^{*}) | & \leq | U (g) - T U^{*} (g) | \\ + | U^{*} (g) - T U (g) | . \end{aligned}

(ii)

There exits $r (3 < r)$ , with

\begin{aligned} \frac{1}{r} \geq sup_{g \in [0, 1]} \int_{0}^{1} δ (g, σ) d σ . \end{aligned}

Then, FDE (11) has necessarily at most one solution in

Y

Proof.

The equivalent integral equation for FDE (11) is the following

\begin{aligned} U (g) = \int_{0}^{1} δ (g, σ) f (σ, U (σ)) d σ, \end{aligned}

for all

U \in Y

and

g \in [0, 1]

, where

\begin{aligned} δ (g, σ) = {\begin{cases} \frac{[g (1 - σ)]^{μ - 1} - (g - σ)^{μ - 1}}{Γ (μ)} & if 0 \leq σ \leq g \leq 1 \\ \frac{g (1 - σ)]^{μ - 1}}{Γ (μ)} & if 0 \leq g \leq σ \leq 1. \end{cases} \end{aligned}

If the map

T : Y \to Y

defined by

\begin{aligned} T U (g) = \int_{0}^{1} δ (g, σ) f (σ, U (σ)) d σ, \end{aligned}

where

U^{*} \in Y

is an fixed point, then

U^{*}

is a solution of Equation (11). Taking into account the given conditions, for

U, U^{*} \in Y

, we infer

\begin{aligned} d (T U, T U^{*}) & = sup_{0 \leq g \leq 1} | T U (g) - T U^{*} (g) | \\ = sup_{0 \leq g \leq 1} | \begin{array}{l} \int_{0}^{1} δ (g, σ) f (σ, U (σ)) d σ \\ - \int_{0}^{1} δ (g, σ) f (σ, U^{*} (σ)) d σ \end{array} | \\ \leq sup_{0 \leq g \leq 1} {\int_{0}^{1} | δ (g, σ) | d σ . | f (σ, U (σ)) - f (σ, U^{*} (σ)) |} \\ \leq \frac{1}{r} sup_{0 \leq g \leq 1} | f (g, U) - f (g, U^{*}) | \\ \leq \frac{1}{r} sup_{0 \leq g \leq 1} [| U (g) - T U^{*} (g) | + | U^{*} (g) - T U (g) |] . \end{aligned}

This shows that

\begin{aligned} d (T U, T U^{*}) \leq \frac{1}{r} [d (U, T U^{*}) + d (U^{*}, T U)] . \end{aligned}

Using

1 / r = γ < 1 / 2

, we can write

\begin{aligned} d (T U, T U^{*}) \leq γ [d (U, T U^{*}) + d (U^{*}, T U)] . \end{aligned}

The above expression can be written as

\frac{1}{F (T U, T U^{*}, p_{i}^{k})} - 1 \leq γ {\begin{cases} (\frac{1}{F (U, T U^{*}, p_{i}^{k})} - 1) \\ + (\frac{1}{F (U^{*}, T U, p_{i}^{k})} - 1) \end{cases}} .

(12)

for all

U, U^{*} \in Y

. This shows that

T

satisfies the k-fuzzy contraction of Theorem (8.1). Hence,

T

admits a unique fixed point in

Y

, implying that FDE (11) has a unique solution.

Remark 10.1

The above theorem provides a direct application of our generalized contraction framework to fractional differential equations. Compared to Nazam et al. (2024), the present contribution demonstrates that uniqueness of solutions persists in the more general $k$ -fuzzy setting, which allows a richer description of uncertainty in physical and engineering models.

11. Conclusion

This paper defined the concept of k-fuzzy Meir-Keeler contractions within k-fuzzy metric space (k-FMS), extending classical contraction principles and fixed point theorems to a broader and more flexible framework. We established several fixed point results under k-fuzzy Meir-Keeler contractions. Nonetheless, the present study is primarily theoretical and does not include empirical validation or implementation in real-world scenarios, which may limit the immediate practical assessment of the proposed model.

Future research may focus on applying k-fuzzy Meir-Keeler contractions to real-world engineering and decision-making systems, as well as extending the current framework to more complex and high-dimensional fuzzy structures. Further investigation into algorithmic optimization, numerical stability, and computational efficiency would also enhance the applicability of k-FMS in advanced computational and engineering problems.

Footnotes

Acknowledgements

We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments that will help to improve the quality of the manuscript.

ORCID iDs

Muhammad Nazam

Seemab Attique

Saud M Alsulami

Aftab Hussain

Authors’ Contributions

The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Availability of Data and Materials

Data sharing is not applicable to this article as no data set were generated or analysed during the current study.

Human and Animal Rights

We would like to mention that this article does not contain any studies with animals and does not involve any studies over human being.

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