Another Meir-Keeler-type nonlinear contractions

1.   Introduction

A major development in fixed-point theory was achieved by Meir and Keeler ^[32], who established a pivotal theorem that extends the renowned Banach contraction mapping principle.

Let $(X, d)$ be a complete metric space and $T: X\rightarrow X$ be a mapping such that, for each $\epsilon > 0$, $\exists$ a $\delta(\epsilon) > 0$ such that

for all $x, y\in X$. Then the fixed point of $T$ in $X$ is unique.

Meir and Keeler's fixed-point method ^[32] has become a fundamental topic in fixed-point theory. It has inspired extensive research, with numerous authors contributing new ideas and methods to further develop their work.

Maiti and Pal ^[31] introduced the following contraction mapping and provided it's validity.

For every $\epsilon > 0$, $\delta > 0$ exists such that

However, other researchers ^[36,39] enhanced the above result with the following condition:

In 1976, Jungck ^[25] demonstrated a shared fixed-point result for commute mappings, assuming that one of them is continuous. Subsequently, in 1982, Sessa ^[41] introduced the concept of weakly commuting pairs of self-mappings and established a fixed-point theorem in complete metric spaces. Sharma ^[42] later presented novel results for weakly commuting mappings in such spaces, extending several related findings. Kumam et al. ^[30] explored fixed point results under generalized contractive conditions in b-metric spaces, providing an example to highlight the practical implications of their work. Moreover, He et al. ^[23] examined the existence and uniqueness of fixed-points for weakly commutative mappings within the framework of complete multiplicative metric spaces. Alsulami et al. ^[7] introduced $\alpha$-admissible and generalized $\alpha$-admissible Meir-Keeler contractions in quasi-metric spaces and extended their findings to $G$-metric spaces, proving fixed-point theorems. Abtahi ^[2] provided a criterion for sequences in metric spaces to be Cauchy, simplifying proofs of fixed-point results for Meir-Keeler-type contractions. Canzoneri and Vetroa ^[11] studied asymptotic contractions of the integral Meir-Keeler-type and established a fixed-point theorem ensuring existence and uniqueness. Gholamian and Khanehgir ^[20] introduced generalized Meir-Keeler contractions in $b$-metric-like spaces, proving fixed-point theorems with illustrative examples. Gulyaz et al. ^[22] extended the above work to $\alpha$-Meir-Keeler and generalized $\alpha$-Meir-Keeler contractions in Branciari $b$-metric spaces, establishing the existence and uniqueness of fixed-points. Karapinar et al. ^[26] studied $(\alpha\text{-}\psi)$-Meir-Keeler contractions in generalized $b$-metric spaces, while Barootkoob et al. ^[10] introduced $(\alpha\text{-}\psi\text{-}p)$-Meir-Keeler contractions, extending fixed point results via $w$-distance and applying them to nonlinear Fredholm integral equations. Panthi ^[35] proved common fixed-point theorems for compatible mappings in metric and dislocated metric spaces, and Koti et al. ^[29] established fixed-point results using Gupta-Saxena's rational expression. Additional results are documented in ^[15,16] and some other results on common fixed-point theory can be found in ^[17,18,33]. Further results in this area can also be found in ^[1,9,13]. Aksoya et al. ^[3] studied Meir-Keeler type contractions in modular metric spaces, proving fixed point theorems with examples. Further results on such contractions in partial Hausdorff and JS-metric spaces are found in ^[12,27]. Aksoy et al. ^[4] extended the fixed-point results to a broader class of contractive and non-expansive mappings in modular metric spaces. Jleli et al. ^[24] introduced proximal quasi-contractions in modular spaces with the Fatou property, establishing the best proximity point theorems. Aksoy et al. ^[5] explored $\alpha$-admissible contractions in $b$-metric spaces, proving the fixed-point results and applying them to differential equations. Arshad et al. ^[8] generalized Jleli et al.'s work using triangular $\alpha$-orbital admissible mappings ^[38], while Alharbi et al. ^[6] combined $\alpha$-orbital admissibility with simulation functions to establish fixed-point results in $b$-metric spaces. Gholidahneh et al. ^[21] introduced modular extended $b$-metric spaces and established fixed point theorems for $\alpha$ $\hat{v}$ Meir-Keeler contractions, extending their results to various settings, including graph structured and partially ordered spaces. They also linked fuzzy $b$ metric spaces with modular extended $b$ metric spaces and provided examples and applications to illustrate their findings. Modular extended $b$-metric spaces provide a powerful and flexible framework that extends classical metric spaces and enables the study of nonlinear contractions, multi-mapping systems, and function dependent fixed point problems. Their broad applicability, as seen in ^[21], makes them an essential tool in modern mathematical analysis. There are examples of extended $b$-modular metric spaces which are not classical, $b$-metric, modular, or modular $b$-metric spaces. Example 2.2 in ^[21] is an extended modular $b$-metric space which is not a classical metric or $b$-metric space.

We analyze whether the function introduced in [21, Example 2.2], namely

defines a modular extended $b$-metric space and whether it satisfies the conditions of a classical metric or $b$-metric spaces.

1) Verification as an extended modular $b$-metric space

A modular extended $b$-metric space satisfies the following conditions:

● Non-negativity and identity of indiscernibles: Since $\sinh(t) \geq 0$ for all $t \geq 0$ and $\sinh(0) = 0$, we have

● Symmetry: Since $\nu_{\lambda}(x, y) = \nu_{\lambda}(y, x)$, we obtain

● Extended modular triangle inequality: Since $\nu_{\lambda}$ satisfies the modular $b$-metric inequality

applying $\sinh$ gives

Defining $\Omega(t) = \sinh(st)$, which is strictly increasing, we obtain

Thus, $\hat{\nu}_{\lambda}$ satisfies the modular extended $b$-metric conditions.

2) Not a classical metric space

A classical metric satisfies the triangle inequality

However, the function $\hat{\nu}_{\lambda}$ satisfies

which means that the standard triangle inequality does not hold. Hence, $\hat{\nu}_{\lambda}$ is not a classical metric space.

3) Not a $b$-metric space

A $b$-metric satisfies the inequality

However, for $\hat{\nu}_{\lambda}$, we have

which does not match the linear form of the $b$-metric inequality. Since $\sinh$ is non-linear, $\hat{\nu}_{\lambda}$ does not satisfy the $b$-metric condition.

● $\hat{\nu}_{\lambda}$ is an extended modular $b$-metric space, since it satisfies the extended modular triangle inequality.

● $\hat{\nu}_{\lambda}$ is not a classical metric space, as it violates the standard triangle inequality.

● $\hat{\nu}_{\lambda}$ is not a $b$-metric space, as it does not satisfy the linear $b$-metric triangle inequality.

Okeke et al. ^[34] further generalized these concepts by defining new types of $\alpha$-$\hat{v}$-Meir-Keeler-type contractions in modular extended $b$-metric spaces, supported by examples that validated their results.

This paper builds on these works by presenting the concept of $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type nonlinear contractions, a new class of mappings within modular extended $b$-metric spaces. We establish common unique fixed-point theorems that generalize, unify, and extend several key results in modular $b$-metric and modular extended $b$-metric spaces. These theorems bridge the gap between classical and contemporary fixed-point theories, showcasing broader applicability in nonlinear analysis. An example is provided to support the findings.

The structure of this work is organized as follows: Initially, we review the fundamental definitions and results in Section 2. Subsequently, the core findings are introduced and examined in Section 3.

2.   Preliminaries

Definition 1. ^[30,41] Let $f$ and $g$ be mappings from a $b$-metric space $(X, d)$ onto itself. The mappings $f$ and $g$ are called weakly commuting if $d(fgx, gfx)\le d(fx, gx)$ for each $x\in X$.

Definition 2. ^[30] Let $f$ and $g$ be mappings from a $b$-metric space $(X, d)$ onto itself. The mappings $f$ and $g$ are termed $R$-weakly commuting if a positive real number $R$ exists such that $d(fgx, gfx)\le Rd(fx, gx)$ for each $x\in X$.

Definition 3. ^[40] Let $T$ be a self-mapping on $X$ and $\alpha:X_{\hat{v}}\times X_{\hat{v}}\rightarrow [0, +\infty)$ be a function. Then $T$ is called an $\alpha$-admissible mapping if, for all $x, y\in X$,

Definition 4. ^[28] Let $\alpha:X_{\hat{v}}\times X_{\hat{v}}\rightarrow [0, +\infty)$ be a mapping. Then the self-mapping $f:X\rightarrow X$ is said to be triangular $\alpha$-admissible if

(1) for all $x, y\in X$, $\alpha(x, y)\ge 1\implies\alpha(fx, fy)\ge 1$;

(2) for all $x, y, z\in X$, $\alpha(x, z)\ge 1\; and\; \alpha(z, y)\ge 1\implies\alpha(x, y)\ge 1$.

Lemma 1. ^[28] Let $f$ be a triangular $\alpha$-admissible mapping. Assume that $x_{0}\in X$ exists such that $\alpha(x_{0}, fx_{0})\ge 1$. Define a sequence $\{x_{n}\}_{n\ge 1}$ as $x_{n} = f^{n}x_{0}$. Then $\alpha(x_{m}, x_{n})\ge 1$ $\forall$ distinct $n, m\in\mathbb{N}$.

Lemma 2. ^[38] Let $f$ be a triangular $\alpha$-orbital admissible mapping. Assume that $x_{1}\in X$ exists such that $\alpha(x_{1}, fx_{1})\geq 1$. Define a sequence $\{x_{n}\}_{n\geq 1}$ by $x_{n+1} = fx_{n}$. Then we have $\alpha(x_{m}, x_{n})\ge 1$ $\forall$ distinct $n, m\in\mathbb{N}$.

Definition 5. ^[38] (a) Let $h: X\to X$ be a mapping and $\alpha:X\times X\to \mathbb{R}$ be a function. Then $h$ is said to be $\alpha$-orbital admissible if $\alpha(x, hx)\geq 1\implies \alpha(hx, h^{2}x)\geq1$.

(b) Let $h: X\to X$ be a map and $\alpha:x\times X\to \mathbb{R}$ be a function. Then $h$ is termed a triangular $\alpha$-orbital admissible mapping if $\alpha(x, hx)\geq 1\implies \alpha(hx, h^{2}x)\geq1$, $\alpha(x, y)\geq 1$, and $\alpha(y, hy)\geq 1\implies\alpha(x, hy)\geq 1$.

It is evident that every mapping that is $\alpha$-admissible also qualifies as an $\alpha$-orbital admissible mapping. Additionally, any triangular mapping that is $\alpha$-admissible is inherently a triangular $\alpha$-orbital admissible mapping as well. However, an instance of a triangular $\alpha$-orbital admissible mapping exists that does not conform to the criteria of being triangular $\alpha$-admissible. An example of this can be found in ^[38]. A $b$-metric space serves as a natural extension of both classical metric space by modifying the well-known triangle inequality to the form $d(x, z) \leq s(d(x, y) + d(y, z))$ for all points $x, y, z \in X$ and a fixed parameter $s \geq 1$. Recently, the concept of a $b$-metric space has been further extended to encompass $p$-metric spaces, as elaborated in ^[37]. However, the $p$ above is not a partial metric space.

Definition 6. ^[37] Let $X$ be a non-empty set. A mapping $d:X\times X\rightarrow \mathbb{R}_{+}$ is called a $p$-metric if a strictly increasing continuous function $\Omega: [0, \infty)\rightarrow [0, \infty)$ exists with $t\le \Omega(t)$ for all $t\in [0, \infty)$ such that, for all $x, y, z\in X$, the following conditions hold:

(1) $d(x, y) = 0$ if and only if $x = y$;

(2) $d(x, y) = d(y, x)$;

(3) $d(x, z)\le \Omega(d(x, y)+d(y, z))$.

The pair $(X, d)$ is called a $p$-metric space or an extended $b$-metric space.

Definition 7. ^[14] Let $X$ be a non-empty set. A mapping $\omega: (0, +\infty)\times X\times X\to \mathbb{R}_{+}\cup\{\infty\}$ is called a metric modular on $X$ if, for all $x, y, z\in X$ and $\lambda > 0$, the following conditions hold:

(1) $\omega_{\lambda}(x, y) = 0$ for all $\lambda > 0$ if and only if $x = y$;

(2) $\omega_{\lambda}(x, y) = \omega_{\lambda}(y, x)$ for all $\lambda > 0$;

(3) $\omega_{\lambda+\mu}(x, y)\le\omega_{\lambda}(x, z)+\omega_{\mu}(z, y)$ for al $\lambda$, $\mu > 0$.

The pair $(X, \omega)$ is called a modular metric space.

Definition 8. ^[14] Let $(X, \omega)$ be a modular metric space. Fix $x_{0}\in X$ and set

and

In this context, the sets $X_{\omega}$ and $X^{\ast}_{\omega}$ are referred to as modular spaces that are based on the point $x_{0}$.

Definition 9. ^[19] Let $X$ be a non-empty set and $s\ge 1$ be a real number. A mapping $\omega: (0, +\infty)\times X\times X\to \mathbb{R}_{+}\cup\{\infty\}$ is called a modular $b$-metric on $X$ if the following statements hold: for all $x, y, z\in X$:

(1) $\omega_{\lambda}(x, y) = 0$ for all $\lambda > 0$ if and only if $x = y$;

(2) $\omega_{\lambda}(x, y) = \omega_{\lambda}(y, x)$ for all $\lambda > 0$;

(3) $\omega_{\lambda+\mu}(x, y)\le s(\omega_{\lambda}(x, z) +\omega_{\mu}(z, y))$ for all $\lambda, \mu > 0$.

The pair $(X, \omega)$ is called a modular $b$-metric space.

In the paper, take $X_{\hat{v}} = X^{\ast}_{\hat{v}}(x_{0}) = \{x\in X: \hat{v}_{\lambda}(x, x_{0}) < \infty, \; for\; all\; \lambda > 0\}.$

Definition 10. ^[21] Let $X$ be a non-empty set. A mapping $\hat{v}_{\lambda}: (0, \infty)\times X\times X\rightarrow [0, \infty]$ is called a modular extended $b$-metric if a strictly increasing continuous function ${\mathfrak{A}}:[0, \infty)\rightarrow [0, \infty)$ with $\mathfrak{A}^{-1}(t)\le t\le\mathfrak{A}(t)$ for all $t\in [0, \infty)$ exists such that, for all $x, y, z\in X$, the following conditions hold:

(1) $\hat{v}_{\lambda}(x, y) = 0$ if and only if $x = y$ for all $\lambda > 0$;

(2) $\hat{v}_{\lambda}(x, y) = \hat{v}_{\lambda}(y, x)$ for all $\lambda > 0$;

(3) $\hat{v}_{\lambda+\mu}(x, y)\le {\mathfrak{A}}(\hat{v}_{\lambda}(x, z) +\hat{v}_{\mu}(z, y))$ for all $\lambda, \mu > 0$.

Then the pair $(X, \hat{v})$ is called a modular extended $b$-metric space.

The class of modular extended $b$-metric spaces is bigger than the class of $b$-metric spaces, since a $b$-metric space is modular extended $b$-metric space whenever $\mathfrak{A}(t) = st$ and a metric space if it is a modular extended $b$-metric space with $\mathfrak{A}(t) = t$.

Example 1. ^[21] Consider $X_{\omega}$ to be a modular space equipped with a $b$-metric, where the coefficient satisfies $s \geq 1$ and $\hat{v}_{\lambda}(x, y) = \sinh(\omega_{\lambda}(x, y))$. Then $\hat{v}_{\lambda}$ is a modular extended $b$-metric space with $\mathfrak{A}(t) = \sinh(st)$ for all $t\ge 1$ and $\mathfrak{A}^{-1}(r) = \frac{1}{s}\sinh^{-1}(r)$ for all $r\ge0$.

Definition 11. ^[21] Let $X_{\hat{v}}$ be a modular extended $b$-metric space. Then a sequence $\{x_{n}\}_{n\ge 1}\subseteq X_{\hat{v}}$ is called

(1) a $\hat{v}$-Cauchy sequence if, for all $\epsilon > 0$, $n(\epsilon)\in\mathbb{N}$ exists such that, for each $n, m\ge n(\epsilon)$ and $\lambda > 0$, $\hat{v}_{\lambda}(x_{n}, x_{m}) < \epsilon$;

(2) $\hat{v}$-convergent to $x^{\ast}\in X_{\hat{v}}$ if $\hat{v}_{\lambda}(x_{n}, x^{\ast})\longrightarrow 0$ as $n\rightarrow \infty$ for all $\lambda > 0$;

(3) $\hat{v}$-complete if each $\hat{v}$-Cauchy sequence in $X_{\hat{v}}$ is $\hat{v}$-convergent and its limit is in $X_{\hat{v}}$.

Definition 12. ^[34] Let $X_{\hat{v}}$ be a modular extended $b$-metric space, $T:X_{\hat{v}}\rightarrow X_{\hat{v}}$ be a mapping, and $x_{0}\in X_{\hat{v}}$. $T$ is said to be orbitally continuous at $x_{0}$ whenever $\lim\limits_{k\to\infty}\hat{v}_{\lambda}(T^{n_{k}}u, x_{0}) = 0$ implies that $\lim\limits_{k\to\infty}\hat{v}_{\lambda}(TT^{n_{k}}u, Tx_{0}) = 0$ whenever $u\in X_{\hat{v}}$ and $\{n_{k}\}\subseteq \mathbb{N}$ is a strictly increasing sequence of non-negative integer numbers.

Definition 13. Let $X_{\hat{v}}$ be a modular extended $b$-metric space, $T_{1}, T_{2}:X_{\hat{v}}\rightarrow X_{\hat{v}}$ be mappings, and $x\in X_{\hat{v}}$. Then $T_{1}, T_{2}$ are called modular extended $b$-weakly commuting mappings in $X_{\hat{v}}$ if

Example 2. Consider the set $X = (\mathbb{R}_{+}\setminus\{0\})\cup\{\infty\}$ equipped with the modular extended $b$-metric defined by

which is complete on $X_{\hat{v}}$ for all $\lambda > 0$. Define the mappings $T_{1}, T_{2}:(\mathbb{R}^{+}\setminus\{0\}) \cup\{\infty\}\rightarrow (\mathbb{R}^{+}\setminus\{0\})\cup\{\infty\}$ as $T_{1}x = \log_{32}x^{5}$, $T_{2}x = \log_{8}x^{3}$ for all $x\in(\mathbb{R}_{+}\setminus\{0\})\cup\{\infty\}$ and $\lambda > 0$, respectively. Then $T_{1}$ and $T_{2}$ are $\hat{v}$-weakly commuting mappings.

In fact, it suffices to show that $T_{1}$ and $T_{2}$ satisfy Definition 13. For all $\lambda > 0$ and $x\in(\mathbb{R}_{+}\setminus\{0\})$, we show that

Using the above definition, we get,

Again, we have

Thus we have

which shows that $T_{1}$ and $T_{2}$ are weakly commuting.

We construct an example of the main result below related to Example 2 above.

Remark 1. For all $\lambda > 0$ and $x, y\in X_{\hat{v}}$, we have

For the details of this remark, see [34, Remark 3.3].

The definition below is essential throughout this paper.

Definition 14. $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six continuous mappings. We say that $T_{i}$ for $i = 1, 2, \cdots, 6$ follows an $\alpha$-$\hat{v}$-$A$-$B$-$C$ -Meir-Keeler-type contraction, and $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$ and $\{T_{1}, T_{6}\}$ are weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}}) \subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}),$ if a function $\alpha:X_{\hat{v}}\times X_{\hat{v}}\rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$ and $x_{0}\in X_{\hat{v}}$ exists such that $\alpha(x_{0}, Tx_{0})\ge 1, \, \, \,$ and, for each $i = 1, \cdots, 6$, the mapping $T_{i}$ qualifies as triangular $\alpha$-orbital admissible function for every $\lambda, {\epsilon, \delta} > 0$ satisfying the following conditions:

where

Remark 2. $\bullet$ Following Definitions 10, $\mathfrak{A}$ is a strictly increasing continuous function $\mathfrak{A}:[0, \infty)\rightarrow [0, \infty)$ with $\mathfrak{A}^{-1}(\epsilon) \le \epsilon\le\mathfrak{A}(\epsilon)$ for all $\epsilon\in [0, \infty)$.

$\bullet$ $F^{A}_{\lambda}(T_{1}x, T_{2}y)$, $F^{B}_{\lambda}(T_{1}x, T_{3}y)$, and $F^{C}_{\lambda}(T_{2}x, T_{3}y)$ are functions containing some mixed-metric extended modular space at the $A^{th}$, $B^{th}$, and $C^{th}$ levels, respectively, for $\lambda > 0$.

$\bullet$ An example of Definition 14 will be given after the proof of Theorem 1 below.

2.1.   Relationships between Definition 14 and Definitions 1–12

Definition 14 introduces the concept of $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction mappings in a modular extended $b$-metric space, where six self- mappings interact under certain admissibility and commutativity conditions. We analyze how it builds upon Definitions 1–12.

2.1.1.   Foundations from Definitions 1–6 (basic metric and modular concepts)

Definitions 1 and 2 introduce weakly commuting mappings in $b$-metric spaces, which are crucial for Definition 14 because the contraction conditions require specific pairs of mappings $\{T_2, T_4\}, \{T_3, T_5\}, \{T_1, T_6\}$ to be weakly commuting.

● Definition 1 (weakly commuting mappings) states that two mappings $f, g$ are weakly commuting if $d(fgx, gfx) \leq d(fx, gx).$ This ensures some level of interaction between the mappings, which is a necessary condition in Definition 14.

● Definition 2 (R-weakly commuting mappings) extends this concept by introducing a control parameter $R$, leading to $d(fgx, gfx) \leq R d(fx, gx).$ This generalization helps establish the contraction conditions in Definition 14.

● Definition 3 ($\alpha$-admissibility of a mapping) defines an auxiliary function $\alpha(x, y)$ that controls the behavior of a mapping $T$ in relation to the fixed-point process. Definition 14 relies on this property, since the contraction conditions depend on triangular $\alpha$-admissibility.

● Definition 4 (triangular $\alpha$-admissible mappings) extends the concept of admissibility to triangular structures, ensuring that if $\alpha(x, z)$ and $\alpha(z, y)$ hold, then so does $\alpha(x, y)$. This property is crucial in Definition 14 for handling sequences within the modular extended $b$-metric space.

● Definitions 5 (orbital and triangular orbital admissibility) provides further generalizations that influence the structure of Definition 14. Since the mappings in Definition 14 must satisfy orbital admissibility, these preliminary definitions establish the conditions necessary for the contraction framework.

2.1.2.   Modular extended $b$-metric space (Definitions 6–10)

The transition from metric and $b$-metric spaces to a modular extended $b$-metric space is key in Definition 14.

● Definition 6 (extended $b$ metric space) introduces the concept of an extended $b$-distance function $d$, which satisfies $d(x, y) \leq \Omega(d(x, z) + d(z, y)).$ This idea is extended in Definition 14 to work with six mappings instead of one, for a strictly increasing continuous function, $\Omega:[0, \infty)\to[0, \infty)$ with $t\leq \Omega(t)$ for every $t\in [0, \infty)$.

● Definition 7 (modular metric space) introduces the concept of a modular distance function $\omega$, which satisfies $\omega_{\lambda + \mu}(x, y) \leq \omega_{\lambda}(x, z) + \omega_{\mu}(z, y).$ This idea is extended in Definition 14 to work with six mappings instead of one.

● Definition 8 (modular spaces $X_{\omega}$ and $X^*_{\omega}$) introduces two types of modular metric spaces based on growth conditions. These modular structures are embedded into Definition 14 to ensure the existence of fixed points in a well-defined modular space.

● Definition 9 (modular $b$-metric space) extends the modular metric concept to $b$-metric spaces, introducing a relaxation factor $s$ similar to the classic $b$-metric condition. This generalization is needed in Definition 14 because the contraction inequalities involve a max operation, requiring a structure that supports modular-$b$-metric behavior.

● Definition 10 (modular extended $b$-metric space) is one of the most crucial precursors to Definition 14. It defines the function $\hat{v}_{\lambda}$ that satisfies: $\hat{v}_{\lambda+ \mu}(x, y) \leq \mathfrak{A}(\hat{v}_{\lambda}(x, z) + \hat{v}_{\mu}(z, y)),$ where $\mathfrak{A}$ is a strictly increasing function. Definition 14 applies this framework to analyze contractions under modular extended $b$-metric settings.

2.1.3.   Properties of sequences in modular extended $b$-metric spaces (Definitions 11–12)

Definitions 11 and 12 introduce key sequence properties that Definition 14 relies upon to ensure convergence and the fixed points existence.

● Definition 11 ($\hat{v}$-Cauchy sequences and $\hat{v}$-convergence) formalizes when a sequence is Cauchy and convergent in modular extended $b$-metric spaces. In Definition 14, the iterative sequences

must be $\hat{v}$-Cauchy to ensure the fixed points results hold.

● Definition 12 (orbital continuity in modular extended $b$-metric spaces) establishes the continuity conditions needed for taking the limits in Definition 14. Since the proof of Theorem 3 relies on passing those limits, these conditions ensure the stability of the mappings under iteration.

2.1.4.   The role of Definition 14: Generalization and unification

Definition 14 unifies and extends all previous definitions by combining the following:

● Weakly commuting mappings (Definitions 1, 2, and 13).

● $\alpha$-admissibility conditions (Definitions 3–5).

● Extended $b$-metric space and modular extended $b$-metric properties (Definitions 6–10).

● Convergence and continuity results (Definitions 11–12).

3.   Main results

Now, we give the main results in this paper. We start with the following lemma.

Lemma 3. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$ and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ We then have a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$ and $x_{0}\in X_{\hat{v}}$ such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for every $\lambda > 0$, provided the following conditions are met:

where

Suppose that $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ are sequences in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $\hat{v}_{\lambda}(\xi_{n}, \xi_{m}) = 0\; \forall \; \lambda > 0$.

Proof. Suppose that $X_{\hat{v}}$ is empty. In that case, there is nothing to prove. We now assume that $X_{\hat{v}}\neq\emptyset$. Suppose that the mappings, $T_{i}$ for $i = 1, \cdots, 6$ satisfy the inequalities in (3.1)–(3.6). Since $x_{0}, x_{1}$, and $x_{2}$ are points in $X_{\hat{v}}$ and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$, we can find a point $x_{1}$ in $X_{\hat{v}}$ such that $\xi_{0} = T_{1}x_{0} = T_{6}x_{1}$. For $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, we can find a point $x_{2}\in X_{\hat{v}}$ such that $\xi_{1} = T_{2}x_{1} = T_{5}x_{2}$ and for $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, we can find a point $x_{3}$ in $X_{\hat{v}}$ such that $\xi_{2} = T_{3}x_{2} = T_{4}x_{3}$. Now for all $\lambda > 0$, we induce on $n$, so that there are sequences $\{x_{n}\}_{n \in \mathbb{N}}$ and $\{\xi_{n}\}_{n \in \mathbb{N}}$ within $X_{\hat{v}}$ that satisfy the following equations:

If $n_{0}\in\mathbb{N}$ exists such that $\xi_{n_{0}} = \xi_{n_{0}+1}$, $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$ holds. In fact, if $m\in\mathbb{N}$ exists such that $\xi_{m+2} = \xi_{m+3}$, then $T_{1}u = T_{6}u$, where $u = x_{m+3}$. Therefore, the pair $\{T_{1}, T_{6}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m} = \xi_{m+1}$, then $T_{2}u = T_{4}u$, where $u = x_{m+1}$. Therefore, the pair $\{T_{2}, T_{4}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m+1} = \xi_{m+3}$, then $T_{3}u = T_{5}u$, where $u = x_{m+2}$. Thus, the pair $\{T_{3}, T_{5}\}$ has a coincidence point $u\in X_{\hat{v}}$. Again, if there is an $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1} = \xi_{n_{0}+2}$, then $\xi_{n} = \xi_{n_{0}}$ for any $n\geq n_{0}$. This implies that $\{\xi_{n}\}$ is a modular $\hat{v}$ Cauchy sequence in $X_{\hat{v}}$. Actually, if $\eta\in\mathbb{N}$ exists such that (1) $\xi_{\eta} = \xi_{\eta+1} = \xi_{\eta+2}$, (2) $\xi_{\eta}\neq\xi_{\eta+1} = \xi_{\eta+2}$, (3) $\xi_{\eta}\neq\xi_{\eta+2} = \xi_{\eta+1}$, and (4) $\xi_{\eta}\neq\xi_{\eta+1}\neq\xi_{\eta+2}$ hold. In fact the Case (1) is easy, and Case (3) is similar to Case (2); then from inequality (3.1)–(3.6), we can set $x = \xi_{\eta+2}$ and $y = \xi_{\eta+3}$. Thus Case (2) stipulates that;

where

Using inequality (3.8), Eqs (3.11) and (3.7), we get

for all $\lambda > 0$. According to Lemma 2, given that $T_{i}$ functions as triangular $\alpha$-orbital admissible mapping, for each $i = 1, \cdots, 6$, we get $1\le \alpha(\eta, \eta+1)$ for all $\eta, \eta+1\in\mathbb{N}$ with $\eta < \eta+1$.

Consider $\hat{v}_{\lambda}(T_{1}\xi_{\eta+2}, T_{2}\xi_{\eta+3}) > 0$ for all $\eta\in\mathbb{N}\cup\{0\}$. Since $T_{1}$ and $T_{2}$ are triangular $\alpha$-orbital admissible mappings, it follows from $\hat{v}$-regularity that, for all $\lambda > 0$, $\hat{v}_{\lambda}(T_{1}\xi_{\eta+2}, T_{2}\xi_{\eta+3}) > 0$ and so, by Remark 1, we get

and

for all $\eta\in\mathbb{N}\cup\{0\}$. Therefore, we have

Again, from Eq (3.14) and for all $\lambda > 0$, according to Lemma 2, given that $T_{i}$ functions as triangular $\alpha$-orbital admissible mapping, for each $i = 1, \cdots, 6$, $1\le \alpha(\eta+1, \eta+2)$ for all $\eta+1, \eta+2\in\mathbb{N}$ with $\eta+1 < \eta+2$, using inequality (3.2) and Eqs (3.5) and (3.7), we get

Finally, from Eq (3.15) and for all $\lambda > 0$, according to Lemma 2, given that $T_{i}$ functions as triangular $\alpha$-orbital admissible mapping, for each $i = 1, \cdots, 6$, $1\le \alpha(\eta, \eta+1)$ for all $\eta, \eta+1\in\mathbb{N}$ with $\eta < \eta+1$, using inequality (3.3) and Eqs (3.6) and (3.7), we get

So, it follows from (3.1)–(3.3) and (3.14)–(3.16) that, for all $\lambda > 0$,

and hence, using Eq (3.7), we get

Therefore

Therefore, after some algebra and using condition (3) of Definition 10 we get

This is a modular extended $\hat{v}$-Cauchy sequence in $\hat{v}$-complete modular extended $b$-metric space.

Again, for $\xi_{\eta}\neq\xi_{\eta+1}\neq\xi_{\eta+2}$, take $x = \xi_{\eta}$ and $y = \xi_{\eta+1}$, from Case (4). We then have

where

Using inequality (3.21), Eqs (3.24) and (3.7), we get

for all $\lambda > 0$. According to Lemma 2, given that $T_{i}$ functions as triangular $\alpha$-orbital admissible mapping, for each $i = 1, \cdots, 6$, we get $1\le \alpha(\eta, \eta+1)$ for all $\eta, \eta+1\in\mathbb{N}$ with $\eta \neq\eta+1$.

Consider $\hat{v}_{\lambda}(T_{1}\xi_{\eta}, T_{2}\xi_{\eta+1}) > 0$ for all $\eta\in\mathbb{N}\cup\{0\}$. Since $T_{1}$ and $T_{2}$ are triangular $\alpha$-orbital admissible mappings, it follows from $\hat{v}$-regularity that, for all $\lambda > 0$, $\hat{v}_{\lambda}(T_{1}\xi_{\eta}, T_{2}\xi_{\eta+1}) > 0$ and so, by Remark 1, we get

and

for all $\eta\in\mathbb{N}\cup\{0\}$. Therefore, we have

Again, from Eq (3.27) and for all $\lambda > 0$, and Lemma 2, given that $T_{i}$ functions as triangular $\alpha$-orbital admissible mapping, for each $i = 1, \cdots, 6$, $1\le \alpha(\eta-1, \eta)$ for all $\eta-1, \eta\in\mathbb{N}$ with $\eta-1\neq\eta$, using inequality (3.2) and Eqs (3.5) and (3.7), we get

Finally, from Eq (3.28) and for all $\lambda > 0$, and Lemma 2, given that $T_{i}$ functions as triangular $\alpha$-orbital admissible mapping, for each $i = 1, \cdots, 6$, $1\le \alpha(\eta-1, \eta-2)$ for all $\eta-1, \eta-2\in\mathbb{N}$ with $\eta-1\neq\eta-2$, using inequality (3.3) and Eqs (3.6) and (3.7), we get

So, it follows from (3.1)–(3.3) and (3.27)–(3.29) that, for all $\lambda > 0$,

Hence, using Eq (3.7), we get

Therefore

Therefore, after some algebra and condition (3) of Definition 10, we get the result. Hence,

□

Remark 3. Suppose that $u$ is the common fixed point of $T_{i}$ for $i = 1, 2, \cdots, 6$ when either $T_{2}$ or $T_{4}$ is $\hat{v}$-continuous and the pair $\{T_{2}, T_{4}\}$ is weakly commuting. Again suppose that $u$ is true for $T_{i}$, $i = 1, 2, \cdots, 6$ when $T_{1}$ is $\hat{v}$-continuous, it is also true when $T_{2}$ or $T_{6}$ is $\hat{v}$ continuous and the pair $\{T_{1}, T_{6}\}$ is weakly commuting. Furthermore, $T_{3}$ or $T_{5}$ is $\hat{v}$-continuous and the pair $\{T_{3}, T_{5}\}$ is weakly commuting.

Theorem 1. Suppose that Lemma 3 holds. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a unique common fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for each $i = 1, \cdots, 6$.

Proof. Suppose that $X_{\hat{v}}$ is empty, in which case, there is nothing to prove. We now assume that $X_{\hat{v}}\neq\emptyset$. Then a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$ and $x_{0}\in X_{\hat{v}}$ exists such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$, and, for each $i = 1, \cdots, 6$, $T_{i}$ remain triangular $\alpha$-orbital admissible mappings for every $\lambda > 0$. Therefore the mappings, $T_{i}$ for $i = 1, \cdots, 6$ satisfy the inequalities (3.1)–(3.6). Since $x_{0}, x_{1}$ and $x_{2}$ are points in $X_{\hat{v}}$ and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$, we can find a point $x_{1}$ in $X_{\hat{v}}$ such that $\xi_{0} = T_{1}x_{0} = T_{6}x_{1}$. For $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, we can find a point $x_{2}\in X_{\hat{v}}$ such that $\xi_{1} = T_{2}x_{1} = T_{5}x_{2}$ and for $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$; we can find a point $x_{3}$ in $X_{\hat{v}}$ such that $\xi_{2} = T_{3}x_{2} = T_{4}x_{3}$. Now for all $\lambda > 0$, we induce on $n$, so that there are sequences $\{x_{n}\}_{n \in \mathbb{N}}$ and $\{\xi_{n}\}_{n \in \mathbb{N}}$ within $X_{\hat{v}}$ that satisfy the subsequent Eq (3.7). If $n_{0}\in\mathbb{N}$ exists such that $\xi_{n_{0}} = \xi_{n_{0}+1}$, $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$ hold. In fact, if $m\in\mathbb{N}$ exists such that $\xi_{m+2} = \xi_{m+3}$, then $T_{1}u = T_{6}u$, where $u = x_{m+3}$. Therefore, the pair $\{T_{1}, T_{6}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m} = \xi_{m+1}$, then $T_{2}u = T_{4}u$, where $u = x_{m+1}$. Therefore, the pair $\{T_{2}, T_{4}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m+1} = \xi_{m+3}$, then $T_{3}u = T_{5}u$, where $u = x_{m+2}$. Thus, the pair $\{T_{3}, T_{5}\}$ has a coincidence point $u\in X_{\hat{v}}$. Again, if there is an $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1} = \xi_{n_{0}+2}$, then $\xi_{n} = \xi_{n_{0}}$ for any $n\geq n_{0}$. This implies that $\{\xi_{n}\}$ is a modular $\hat{v}$ Cauchy sequence in $X_{\hat{v}}$. Actually, if $\eta\in\mathbb{N}$ exists such that (1) $\xi_{\eta} = \xi_{\eta+1} = \xi_{\eta+2}$, (2) $\xi_{\eta}\neq\xi_{\eta+1} = \xi_{\eta+2}$, (3) $\xi_{\eta}\neq\xi_{\eta+2} = \xi_{\eta+1}$, and (4) $\xi_{\eta}\neq\xi_{\eta+1}\neq\xi_{\eta+2}$ hold. In fact Case (1) is easy, and Case (3) is similar to Case (2); then from inequality (3.1)–(3.6), we get the result by setting $x = \xi_{\eta+2}$ and $y = \xi_{\eta+3}$. By Lemma 3, we conclude that

Consequently, Eq (3.34) indicates that $\{\xi_{n}\}_{n \in \mathbb{N}}$ forms a modular extended $\hat{v}$-Cauchy sequence within a $\hat{v}$-complete modular extended $b$-metric space. Therefore, a point $u \in X_{\hat{v}}$ exists such that $\xi_{n}$ converges to $u$ as $n$ approaches infinity. Furthermore, since the sequences $\{T_{1}x_{n}\} = \{T_{6}x_{n+1}\}$, $\{T_{2}x_{n+1}\} = \{T_{5}x_{n+2}\}$, and $\{T_{3}x_{n-1}\} = \{T_{4}x_{n}\}$ for all $n \in \mathbb{N}$ are all subsequences of $\{\xi_{n}\}$, it follows that all subsequences of a convergent sequence converge to the same limit. Thus, we conclude that

Since $\{T_{2}, T_{4}\}$ is weakly commuting mappings, we have, for all $\lambda > 0$,

Taking the limit of inequality (3.35) as $n\to\infty$ and noticing that $T_{2}$ or $T_{4}$ are $\hat{v}$-continuous mappings, we get

We know that $T_{4}$ is $\hat{v}$ continuous, then $T^{2}_{4}x_{n+1}\to T_{4}u$ as $n\to\infty$, $T_{4}T_{2}x_{n+1}\to T_{4}u$ as $n\to\infty$. But we can clearly see from inequality (3.35) that $T_{2}T_{4}x_{n+1}\to T_{4}u$ as $n\to\infty$. Since $T_{2}, \, \, T_{4}$ are weakly commuting mappings, for each $i = 1, \ldots, 6$, where $T_{i}$ represents a triangular $\alpha$-orbital admissible mapping. and $1\le \alpha(x_{m}, x_{n})$ for all $n, m\in\mathbb{N}$ with $n < m$, it follows that, for all $\lambda > 0$,

and hence

Since $T_{2}, \, \, T_{4}$ are weakly commuting mappings and orbitally continuous, we have

such that

Therefore, we have

which implies that $T_{2}u = T_{4}u$ for all $\lambda > 0$.

Additionally, given that $T_{3}$ and $T_{5}$ are weakly commuting mappings, it follows that for every $\lambda > 0$,

Taking the limit of inequality (3.35) as $n\to\infty$ and noticing that $T_{3}$ or $T_{4}$ is a $\hat{v}$-continuous mapping, then we get

Since $T_{5}$ is $\hat{v}$ continuous, $T^{2}_{5}x_{n+2}\to T_{5}u$ as $n\to\infty$, and $T_{5}T_{3}x_{n+2}\to T_{5}u$ as $n\to\infty$. But we can clearly see from inequality (3.37) that $T_{3}T_{5}x_{n+2}\to T_{5}u$ as $n\to\infty$.

Now we know that $T_{3}, \, \, T_{5}$ are weakly commuting mappings and orbitally continuous, $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for each $i = 1, \cdots, 6$, and $1\le \alpha(x_{m}, x_{n})$ for all $n, m\in\mathbb{N}$ with $n < m$, it follows that, for all $\lambda > 0$

thus,

Therefore, we have

which implies that $T_{3}u = T_{5}u$ for all $\lambda > 0$.

Lastly, since $T_{1}, \, \, T_{6}$ are weakly commuting mappings and orbitally continuous, $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for each $i = 1, \cdots, 6$, and $1\le \alpha(x_{m}, x_{n})$ for all $n, m\in\mathbb{N}$ with $n < m$, it follow that, for all $\lambda > 0$

and so

Therefore, we have

which implies that $T_{1}u = T_{6}u$ for all $\lambda > 0$.

Now, we claim that, for all $u\in X_{\hat{v}}$, $T_{1}u = T_{2}u = T_{3}u$. If we suppose the contrary, then $T_{1}u\neq T_{2}u\neq T_{3}u$. So, for all $\lambda > 0$, the following cases emerge:

Case 1a. $T_{1}u\neq T_{2}u\implies\alpha(u, u) \hat{v}_{\lambda}(T_{1}u, T_{2}u) > 0$.

Case 1b. $T_{1}u\neq T_{3}u\implies\alpha(u, u) \hat{v}_{\lambda}(T_{1}u, T_{3}u) > 0$.

Case 1c. $T_{2}u\neq T_{3}u\implies\alpha(u, u) \hat{v}_{\lambda}(T_{2}u, T_{3}u) > 0$.

Indeed, since $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for each $i = 1, \cdots, 6$, and $\alpha(u, u)\neq 0$, it follows that, for all $\lambda > 0$

where

Now, we consider Case 1a. Since $T_{1}u\neq T_{2}u$, for all $\lambda > 0$, we have $\alpha(u, u)\hat{v}_{\lambda}(T_{1}u, T_{2}u) > 0$ and so, from (3.28), (3.29), and (3.31), we have

which implies $\alpha(u, u)\neq 0$. Thus we have

which implies that

for all $\lambda > 0$ since $\frac{ac}{b} < 1$ and $b\neq 0$.

For Case 1c, using Case 1a above, that is, $T_{1}u = T_{2}u$, it follows from (3.39) that

So, by using the condition (1) of Definition 10,

which implies that

Hence $T_{2}u = T_{3}u$, since $\frac{b}{a} < 1$, $a^{2}\neq0$, and $\alpha(u, u)\neq 0$.

For Case 1b, using Cases 1a and 1c in (3.39), we get $0 < a\alpha(u, u)\hat{v}_{\lambda}(T_{1}u, T_{3}u) \le 0$. Since $a\neq 0$ and $\alpha(u, u)\neq 0$, we have $T_{1}u = T_{3}u$.

So, in all the cases above, we have $T_{1}u = T_{2}u = T_{3}u$. However, since $T_{1}u = T_{6}u$, $T_{2}u = T_{4}u$ and $T_{3}u = T_{5}u$, it follows that

which implies that $u\in X_{\hat{v}}$ is the coincidence point of $T_{i}$ for each $i = 1, 2, \cdots, 6$.

We demonstrate that if a point is a fixed point of $T_{1}$, it is also a fixed point for $T_{2}, T_{3}, T_{4}, T_{5},$ and $T_{6}$. Assume that a point $p \in X_{\hat{v}}$ exists such that $p$ satisfies $p = T_{1}p$. We assert that $p = T_{2}p = T_{3}p$. Indeed, suppose that this is not true. Then $p\neq T_{2}p$ and $p\neq T_{3}p$, and so we have the following cases for all $\lambda > 0$,

Case 1. $p\neq T_{2}p\implies T_{1}p\neq T_{2}p\implies\alpha(p, p)\hat{v}_{\lambda}(T_{1}p, T_{2}p) > 0$;

Case 2. $p\neq T_{3}p\implies T_{1}p\neq T_{3}p\implies\alpha(p, p)\hat{v}_{\lambda}(T_{1}p, T_{3}p) > 0$.

Indeed, note that $\alpha(p, p)\ne 0$ and $T_{i}$ remain a triangular $\alpha$-orbital admissible mapping for each $i = 1, \cdots, 6$ such that

For Case 1, it follows from (3.39) that

From $p\in Fix(T_{1})$, we have

which implies that $\alpha(T_{1}p, T_{1}p)(1-\frac{ac}{b}) \hat{v}_{\lambda}(p, T_{2}p)\le 0$, for all $\lambda > 0$ since $\alpha(T_{1}p, T_{1}p)\neq 0$ and $\frac{ac}{b} < 1$. This is a contradiction. Therefore, $p = T_{2}p$ and hence $p = T_{1}p = T_{2}p$.

For Case 2, if $p\neq T_{3}p$, then $T_{1}p\neq T_{3}p$ and therefore

for all $\lambda > 0$. Thus, it follows from $p = T_{1}p = T_{2}p$, Case 1, and (3.39) that

which implies

for all $\lambda > 0$. Since $b\neq 0$ and $\alpha(p, p)\neq 0$, we have

and so $\hat{v}_{\lambda}(T_{1}p, T_{3}p) = 0$; that is, $\hat{v}_{\lambda}(p, T_{3}p) = 0$. Therefore, $p = T_{3}p$ and so $p = T_{1}p = T_{3}p$. Hence, from Cases 1 and 2, $p = T_{1}p = T_{2}p = T_{3}p$.

Again, suppose that $p\in X_{\hat{v}}$ exists such that $p\in Fix(T_{2})$, i.e., $p = T_{2}p$. We claim that $p = T_{1}p = T_{3}p$. Indeed, suppose that this is not true. Then $p\neq T_{1}p$ and $p\neq T_{3}p$, so we have the following cases for all $\lambda > 0$:

Case 3. $p\neq T_{1}p\implies T_{2}p\neq T_{1}p \implies\alpha(p, p)\hat{v}_{\lambda}(T_{1}p, T_{2}p) > 0$;

Case 4. $p\neq T_{3}p\implies T_{2}p\neq T_{3}p \implies\alpha(p, p)\hat{v}_{\lambda}(T_{2}p, T_{3}p) > 0$.

Indeed, note that $\alpha(p, p)\ne 0$ and $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for each $i = 1, \cdots, 6$ such that

For Case 3, it follows from (3.39) that

Since $p\in Fix(T_{2})$, we have

which implies that

for all $\lambda > 0$, since $\alpha(T_{2}p, T_{2}p)\neq 0$ and $\frac{ac}{b} < 1$, which is a contradiction. Therefore, $p = T_{1}p$ and hence $p = T_{1}p = T_{2}p$.

For Case 4, if $p\neq T_{3}p$, then $T_{2}p\neq T_{3}p$ and therefore

for all $\lambda > 0$. Thus it follows from $p = T_{2}p = T_{1}p$ and (3.39) that

which implies

for all $\lambda > 0$. Since $b\neq 0$ and $\alpha(p, p)\neq 0$, we have

and so $\hat{v}_{\lambda}(p, T_{3}p) = 0$; that is, $p = T_{3}p$. Therefore, $p = T_{2}p = T_{3}p$. Hence, from Cases 3 and 4, $p = T_{1}p = T_{2}p = T_{3}p$.

Lastly, suppose that $p\in X_{\hat{v}}$ exists such that $p\in Fix(T_{3})$, i.e., $p = T_{3}p$. We claim that $p = T_{1}p = T_{2}p$. Indeed, suppose that this is not true. Then $p\neq T_{1}p$ and $p\neq T_{2}p$, and therefore we have the following cases for all $\lambda > 0$:

Case 5. $p\neq T_{1}p\implies T_{3}p\neq T_{1}p \implies\alpha(p, p)\hat{v}_{\lambda}(T_{1}p, T_{3}p) > 0$.

Case 6. $p\neq T_{2}p\implies T_{3}p\neq T_{2}p \implies\alpha(p, p)\hat{v}_{\lambda}(T_{2}p, T_{3}p) > 0$.

Indeed, note that $\alpha(p, p)\ne 0$ and $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for each $i = 1, \cdots, 6$ such that

For Case 5, if $p\neq T_{1}p$, $T_{3}p\neq T_{1}p$, and so

for all $\lambda > 0$. Thus it follows from $p = T_{3}p$, (3.39), and Case 2 that

which implies

for all $\lambda > 0$. Since $b\neq 0$ and $\alpha(p, p)\neq 0$, we have

which implies $\hat{v}_{\lambda}(T_{1}p, p) = 0$; that is, $p = T_{1}p$. Therefore, $p = T_{1}p = T_{2}p = T_{3}p$.

For Case 6, if $p\neq T_{2}p$, then $T_{2}p\neq T_{3}p$, and therefore

for all $\lambda > 0$. Thus, using $p = T_{3}p = T_{1}p$, it follows from (3.39) that

which implies

and so

for all $\lambda > 0$. Since $b\neq 0$ and $\alpha(T_{1}p, T_{1}p)\neq 0$, we have

which implies $\hat{v}_{\lambda}(p, T_{2}p) = 0$, that is, $p = T_{2}p$. Therefore, $p = T_{1}p = T_{2}p$. Hence, it follows from Cases 5 and 6 that $p = T_{1}p = T_{2}p = T_{3}p$. In all the cases above, $p = T_{1}p = T_{2}p = T_{3}p$.

Now, using Cases 1a–1c above, we have

Therefore, we have $p = T_{1}p = T_{2}p = T_{3}p = T_{4}p = T_{5}p = T_{6}p$ or $p\in Fix(T_{i})$ for each $i = 1, 2, \cdots, 6$, which shows that $p$ is a common fixed point of the mappings $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$, $T_{5}, \; and\; T_{6}$.

To prove the uniqueness of the common fixed point $p$, suppose that another common fixed point $x^{\ast}$ ($p\ne x^{\ast}$) of $T_{1}, T_{2}, T_{3}, T_{4}, T_{5}, \; and\; T_{6}$ exists, namely

Since $\alpha(x^{\ast}, p)\ge 1$, we have the following cases:

Case 1a*. $\alpha(x^{\ast}, y^{\ast})\ge 1$, $x^{\ast}\neq y^{\ast}\implies\alpha(x^{\ast}, y^{\ast}) \hat{v}_{\lambda}(T_{1}x^{\ast}, T_{2}y^{\ast}) > 0$.

Case 1b*. $\alpha(x^{\ast}, y^{\ast})\ge 1$, $x^{\ast}\neq y^{\ast}\implies\alpha(x^{\ast}, y^{\ast}) \hat{v}_{\lambda}(T_{1}x^{\ast}, T_{3}y^{\ast}) > 0$.

Case 1c*. $\alpha(x^{\ast}, y^{\ast})\ge 1$, $x^{\ast}\neq y^{\ast}\implies\alpha(x^{\ast}, y^{\ast}) \hat{v}_{\lambda}(T_{2}x^{\ast}, T_{3}y^{\ast}) > 0$.

We can see that Case 1a* follows from Case 1a above and hence, $x^{\ast} = y^{\ast}$. Again, Case 1b* follows from Case 1b above and hence, $x^{\ast} = y^{\ast}$. Finally, Case 1c* follows from Case 1c above and therefore, $x^{\ast} = y^{\ast}$. Therefore, we have $x^{\ast} = y^{\ast}$. Hence, the common fixed-point $p$ is a unique common fixed point $T_{i}$ for each $i = 1, \cdots, 6$. We are now done with the proof. □

Remark 4. Theorem 1 is a generalization of results in Karapinar et al. ^[26], Theorem 2.8 in Gholidahneh et al. ^[21], and Theorem 3.7 in Okeke et al. ^[34].

Our result have made the following progress over the classical results in the following ways:

(a) This study extends classical metric and $b$-metric spaces by introducing modular extended $b$-metric spaces, which allow function-controlled distances.

(b) It defines the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contractions, a function-dependent contraction that generalizes Banach, Kannan, and Meir-Keeler contractions.

(c) Unlike traditional single-mapping results, this work establishes common fixed-point results for six self-mappings, significantly broadening the applicability of fixed-point theorems.

(d) The proposed contraction conditions use function-dependent inequalities instead of fixed contraction constants, making them adaptable to nonlinear and dynamic systems.

(e) This work bridges classical and modern fixed-point results by incorporating modular functions, b-metric spaces, and multi-mapping interactions, leading to a more generalized framework.

Now, we establish the following example to solidify Theorem 1.

Example 3. Let $X = (\mathbb{R}\setminus\{0\})\cup\{\infty\}$ with the modular extended $b$-metric defined by

which is complete in $X_{\hat{v}}$ for all $\lambda > 0$. Define the $\hat{v}$-weakly commuting mappings $T_{1}, T_{2}, T_{3}, T_{4}, T_{5}, T_{6}:(\mathbb{R}\setminus\{0\})\cup\{\infty\}\rightarrow (\mathbb{R}\setminus\{0\})\cup\{\infty\}$ as follows:

for all $x\in(\mathbb{R}\setminus\{0\})\cup\{\infty\}$ and $\lambda > 0$, for each $i = 1, 2, \cdots, 6$ and also for $x, y\in(\mathbb{R}\setminus\{0\})\cup\{\infty\}$, $\alpha(x, T_{i}x)\geq 1\implies \alpha(T_{i}x, T_{i}^{2}x)\geq1$, $\alpha(x, y)\geq 1$ and $\alpha(y, T_{i}y)\geq 1\implies\alpha(x, T_{i}y)\geq 1$. Then the mappings $T_{1}, T_{2}, T_{3}, T_{4}, T_{5}$, and $T_{6}$ satisfy the inequalities (3.1)–(3.3) of Theorem 1.

In fact, let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction mappings for $i = 1, 2, \cdots, 6$, and let $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$ and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings. Indeed, consider $\{T_{2}, T_{4}\}$. Now for all $\lambda > 0$, and $x\in(\mathbb{R}\setminus\{0\})$, we show that $\hat{v}_{\lambda}(T_{2}T_{4}x, T_{4}T_{2}x) \le\hat{v}_{\lambda}(T_{4}x, T_{2}x)$. Then, by the definition of a $\hat{v}$-modular extended $b$-metric, we have

Again,

Thus $\hat{v}_{\lambda}(T_{2}T_{4}x, T_{4}T_{2}x) \le\hat{v}_{\lambda}(T_{4}x, T_{2}x)$, showing that $\{T_{2}, T_{4}\}$ is weakly commuting pair of self-mappings, and $\{T_{3}, T_{5}\}$ and $\{T_{1}, T_{6}\}$ are weakly commuting pairs of self-mappings according to the above mentioned procedure. It is clear that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$, so that there exists a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0, a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$ and $x_{0}\in X_{\hat{v}}$ such that $\alpha(x_{0}, T_{i}x_{0})\ge 1$. Therefore by definition, we have

Now,

Again,

Therefore, we have $F^{A}_{\lambda}(T_{1}x, T_{2}y) = a\max_{x, y\in X_{\hat{v}}}\biggl\{\ln||\frac{x}{y}||, \ln||\frac{x}{y}||, \ln||\frac{x}{y}||, \ln||\frac{x}{y}||\biggr\}$, where $a = \dfrac{1}{2}$. Given $\frac{1}{2}\ln(\epsilon)$, we choose $\ln(\delta(\epsilon)) = \frac{1}{2}\ln(\epsilon)$. If $\epsilon\le \exp\{\ln(F^{A}_{\lambda}(T_{1}x, T_{2}y))\} < \frac{1}{2}\ln(\epsilon)+\ln(\delta(\epsilon)) = \ln(\epsilon) > \epsilon$, therefore, we have, as $\epsilon > 0$,

Again, $F^{B}_{\lambda}(T_{1}x, T_{3}y) = b\max_{x, y\in X_{\hat{v}}}\biggl\{\ln||\frac{x}{y}||, \ln||\frac{x}{y}||, \ln||x||, \ln||y||\biggr\}$, where $b = \dfrac{1}{2}$. Given $\frac{1}{2}\ln(\epsilon)$, we choose $\ln(\delta(\epsilon)) = \frac{1}{2}\ln(\epsilon)$. If $\epsilon\le exp\ln(F^{B}_{\lambda}(T_{1}x, T_{3}y)) < \frac{1}{2}\ln(\epsilon)+\ln(\delta(\epsilon)) = \ln(\epsilon) < \epsilon$, therefore, we have $\epsilon > 0$ and

Lastly, $F^{C}_{\lambda}(T_{2}x, T_{3}y) = c\max_{x, y\in X_{\hat{v}}}\biggl\{\ln||\frac{x}{y}||, \ln||\frac{x}{y}||, \ln||\frac{x}{y}||\biggr\}$, where $c = \dfrac{1}{2}$. Given $\frac{1}{2}\ln(\epsilon)$, we choose $\ln(\delta(\epsilon)) = \frac{1}{2}\ln(\epsilon)$. If $\epsilon\le exp\ln(F^{C}_{\lambda}(T_{2}x, T_{3}y)) < \frac{1}{2}\ln(\epsilon)+\ln(\delta(\epsilon)) = \ln(\epsilon) < \epsilon$, Therefore, we have

Thus we have

where

Since $T_{i}$ is orbitally continuous, all the conditions of Theorem 1 are fulfilled.

Corollary 1. Let $X_{\hat{v}}$ be a $\hat{v}$-regular and $\hat{v}$-complete modular extended $b$-metric space. Consider six mappings $T_i: X_{\hat{v}} \rightarrow X_{\hat{v}}$ for $i = 1, 2, \ldots, 6$, which are orbitally continuous and satisfy a specific type of contraction known as the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler condition. The pairs $\{T_2, T_4\}$, $\{T_3, T_5\}$, and $\{T_1, T_6\}$ are weakly commuting self-mappings, with the following inclusions holding:

Additionally, a function $\alpha: X_{\hat{v}} \times X_{\hat{v}} \rightarrow [0, +\infty)$ exists with the parameters $a \neq 0$, $a < 1$, $b \neq 0$, and the condition $\frac{ac}{b} < 1$. Let $x_0 \in X_{\hat{v}}$ be given such that

For each mapping $T_i$ (where $i = 1, \ldots, 6$), we find that $T_i$ remains triangular $\alpha$-orbital admissible mapping for all $\lambda > 0$. The following conditions are satisfied for some positive integer $m \geq 1$:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has common unique fixed-point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ for some positive integer, $m\ge 1$.

Proof. According to Theorem 1, for a certain positive integer $m \geq 1$, we have the following equalities: $p = T_1^m p = T_6^m p$, $p = T_2^m p = T_4^m p$, and $p = T_3^m p = T_5^m p$. This implies that $p$ can be expressed as:

indicating that $p$ lies in the fixed-point set $\text{Fix}(T_i)$ for each $i = 1, 2, \ldots, 6$ and the positive integer $m \geq 1$. Consequently, $p$ serves as a fixed point for each mapping $T_1^m, T_2^m, T_3^m, T_4^m, T_5^m, T_6^m$. The uniqueness of this point can be derived similarly to the argument presented in Theorem 1. Thus, it follows that the mappings $T_i$ possess a unique common fixed point located in the intersection $\bigcap_{i = 1} \text{Fix}(T_i)$ for $i = 1, \ldots, 6$ and for some positive integer $m \geq 1$. □

Corollary 2. Let $X_{\hat{v}}$ be a $\hat{v}$-regular and $\hat{v}$-complete modular extended $b$-metric space. Consider six orbitally continuous mappings $T_{i}: X_{\hat{v}} \to X_{\hat{v}}$ ($i = 1, 2, \ldots, 6$) that satisfy the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction condition. Additionally, assume that the pairs $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$, and $\{T_{1}, T_{6}\}$ are weakly commuting, with the following inclusions holding:

Assume that a function $\alpha: X_{\hat{v}} \times X_{\hat{v}} \to [0, +\infty)$ exists, along with the constants $a \neq 0$, $a < 1$, $b \neq 0$, and $\frac{ac}{b} < 1$, as well as a point $x_{0} \in X_{\hat{v}}$, such that:

Furthermore, for all $\lambda > 0$, each $T_{i}$ is assumed to be a triangular $\alpha$-orbital admissible mapping and satisfies the following conditions:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ be defined in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a unique common fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for each $i = 1, \cdots, 6$.

Proof. If we take $X_{\hat{v}}$ to be empty, then there is nothing to prove. Henceforth, we assume that $X_{\hat{v}}\neq\emptyset$. Then a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exists such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$, and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mappings for every $\lambda > 0$. Suppose that the mappings, $T_{i}$ for $i = 1, \cdots, 6$ satisfy theinequalities (3.76)–(3.78). Since $x_{0}, x_{1}$, and $x_{2}$ are points in $X_{\hat{v}}$ and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$, we can find a point $x_{1}$ in $X_{\hat{v}}$ such that $\xi_{0} = T_{1}x_{0} = T_{6}x_{1}$. For $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, we can find a point $x_{2}\in X_{\hat{v}}$ such that $\xi_{1} = T_{2}x_{1} = T_{5}x_{2}$, and for $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, we can find a point $x_{3}$ in $X_{\hat{v}}$ such that $\xi_{2} = T_{3}x_{2} = T_{4}x_{3}$. Now for all $\lambda > 0$, in general, one can find sequences $\{x_n\}_{n \in \mathbb{N}}$ and $\{\xi_n\}_{n \in \mathbb{N}}$ residing in the space $X_{\hat{v}}$ that fulfill the relationships defined in Eq (3.7). If there is an integer $n_0 \in \mathbb{N}$ such that $\xi_{n_0} = \xi_{n_0 + 1}$, it follows that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$ hold. In fact, if $m\in\mathbb{N}$ exists such that $\xi_{m+2} = \xi_{m+3}$, then $T_{1}u = T_{6}u$, where $u = x_{m+3}$. Therefore, the pair $\{T_{1}, T_{6}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m} = \xi_{m+1}$, then $T_{2}u = T_{4}u$, where $u = x_{m+1}$. Therefore, the pair $\{T_{2}, T_{4}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m+1} = \xi_{m+3}$, then $T_{3}u = T_{5}u$, where $u = x_{m+2}$. Thus, the pair $\{T_{3}, T_{5}\}$ has a coincidence point $u\in X_{\hat{v}}$. Again, if there is an $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1} = \xi_{n_{0}+2}$, then $\xi_{n} = \xi_{n_{0}}$ for any $n\geq n_{0}$. This implies that $\{\xi_{n}\}$ is a modular $\hat{v}$ Cauchy sequence in $X_{\hat{v}}$. Actually, if $\eta\in\mathbb{N}$ exists such that (1) $\xi_{\eta} = \xi_{\eta+1} = \xi_{\eta+2}$, (2) $\xi_{\eta}\neq\xi_{\eta+1} = \xi_{\eta+2}$, (3) $\xi_{\eta}\neq\xi_{\eta+2} = \xi_{\eta+1}$, and (4) $\xi_{\eta}\neq\xi_{\eta+1}\neq\xi_{\eta+2}$ hold. In fact, Case (1) is easy, and Case (3) is similar to Case (2); therefore, from the inequalities (3.76)–(3.78), we get the result by setting $x = \xi_{\eta+2}$ and $y = \xi_{\eta+3}$. Following the proof of Theorem 1, the conclusion is now evident. □

Remark 5. Corollary 2 is a generalization of [34, Corollaries 3.16 and 3.18].

Corollary 3. Let $X_{\hat{v}}$ be a modular extended $b$-metric space that is both $\hat{v}$-regular and $\hat{v}$-complete. Consider six mappings $T_i: X_{\hat{v}} \rightarrow X_{\hat{v}}$ for $i = 1, 2, \ldots, 6$ that exhibit orbital continuity and adhere to the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction conditions. The pairs $\{T_2, T_4\}$, $\{T_3, T_5\}$, and $\{T_1, T_6\}$ are considered weakly commuting self-mappings, satisfying the following inclusions:

A function $\alpha: X_{\hat{v}} \times X_{\hat{v}} \rightarrow [0, +\infty)$ exists such that $a \neq 0$, $a < 1$, $b \neq 0$, and $\frac{ac}{b} < 1$. Additionally, let $x_0 \in X_{\hat{v}}$ satisfy $\alpha(x_0, x_1) \geq 1$ for some $\epsilon, \delta > 0$. For each mapping $T_i$ with $i = 1, \ldots, 6$, it is required that $T_i$ qualifies as triangular $\alpha$-orbital admissible mapping for all $\lambda > 0$, and the following conditions hold for some positive integer $m \geq 1$:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be defined in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ for some positive integer, where $m\ge 1$.

Proof. According to Corollary 2, for a certain positive integer $m \geq 1$, we have the equalities $p = T_1^m p = T_6^m p$, $p = T_2^m p = T_4^m p$, and $p = T_3^m p = T_5^m p$. Consequently, it follows that

indicating that $p$ is a fixed point for each mapping $T_i^m$ where $i = 1, 2, \ldots, 6$ for the specified positive integer $m \geq 1$. This establishes that $p$ is, in fact, a fixed point of $T_1^m$ as well as $T_2^m, T_3^m, T_4^m, T_5^m,$ and $T_6^m$ individually. Hence, the uniqueness of this fixed point can be derived as shown in Theorem 1 above. Therefore, the mappings $T_i$ possess a unique common fixed point within the intersection $\bigcap_{i = 1} \text{Fix}(T_i)$ for $i = 1, \ldots, 6$ for some positive integer $m \geq 1$. □

Remark 6. Corollary 3 is a generalization of [34, Corollary 3.19].

Corollary 4. Consider the space $X_{\hat{v}}$, which is characterized as a $\hat{v}$-regular and $\hat{v}$-complete modular extended $b$-metric space. Within this framework, let $T_i: X_{\hat{v}} \to X_{\hat{v}}$ for $i = 1, 2, \ldots, 6$ denote six mappings that are orbitally continuous and adhere to the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction conditions. The pairs $\{T_2, T_4\}$, $\{T_3, T_5\}$, and $\{T_1, T_6\}$ are weakly commuting self-mappings, and the following inclusions are satisfied:

A function $\alpha: X_{\hat{v}} \times X_{\hat{v}} \to [0, +\infty)$ exists with parameters such that $a \neq 0$, $a < 1$, $b \neq 0$, and $\frac{ac}{b} < 1$. Additionally, let $x_0 \in X_{\hat{v}}$ satisfy $\alpha(x_0, x_1) \geq 1$ for some $\epsilon, \delta > 0$. Each mapping $T_i$ for $i = 1, \ldots, 6$ is required to be triangular $\alpha$-orbital admissible mapping for all $\lambda > 0$, fulfilling the following specific conditions:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$, and for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a unique common fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for each $i = 1, \cdots, 6$.

Proof. Suppose that $X_{\hat{v}}$ is empty; then there is nothing to prove. We now assume that $X_{\hat{v}}\neq\emptyset$. Then a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge\; 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mappings for every $\lambda > 0$. Suppose that the mappings, $T_{i}$ for $i = 1, \cdots, 6$ satisfy the inequalities (3.72)–(3.90). Since $x_{0}, x_{1}$ and $x_{2}$ are points in $X_{\hat{v}}$ and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$, we can find a point $x_{1}$ in $X_{\hat{v}}$ such that $\xi_{0} = T_{1}x_{0} = T_{6}x_{1}$. For $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, we can find a point $x_{2}\in X_{\hat{v}}$ such that $\xi_{1} = T_{2}x_{1} = T_{5}x_{2}$; for $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, we can find a point $x_{3}$ in $X_{\hat{v}}$ such that $\xi_{2} = T_{3}x_{2} = T_{4}x_{3}$. Now for all $\lambda > 0$, in general, sequences $\{x_n\}_{n \in \mathbb{N}}$ and $\{\xi_n\}_{n \in \mathbb{N}}$ exist within $X_{\hat{v}}$ that satisfy the conditions outlined in Eq (3.7). If a natural number $n_0$ exists for which $\xi_{n_0} = \xi_{n_0 + 1}$, then it can be inferred that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$ hold. In fact, $m\in\mathbb{N}$ exists such that $\xi_{m+2} = \xi_{m+3}$, then $T_{1}u = T_{6}u$, where $u = x_{m+3}$. Therefore, the pair $\{T_{1}, T_{6}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m} = \xi_{m+1}$, then $T_{2}u = T_{4}u$, where $u = x_{m+1}$. Therefore, the pair $\{T_{2}, T_{4}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m+1} = \xi_{m+3}$, then $T_{3}u = T_{5}u$, where $u = x_{m+2}$. Thus, the pair $\{T_{3}, T_{5}\}$ has a coincidence point $u\in X_{\hat{v}}$. Again, if there is $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1} = \xi_{n_{0}+2}$, then $\xi_{n} = \xi_{n_{0}}$ for any $n\geq n_{0}$. This implies that $\{\xi_{n}\}$ is a modular $\hat{v}$ Cauchy sequence in $X_{\hat{v}}$. Actually, $\eta\in\mathbb{N}$ exists such that (1) $\xi_{\eta} = \xi_{\eta+1} = \xi_{\eta+2}$, (2) $\xi_{\eta}\neq\xi_{\eta+1} = \xi_{\eta+2}$, (3) $\xi_{\eta}\neq\xi_{\eta+2} = \xi_{\eta+1}$, and (4) $\xi_{\eta}\neq\xi_{\eta+1}\neq\xi_{\eta+2}$ hold. In fact, Case (1) is easy, and Case (3) is similar to Case (2). Therefore, from the inequalities (3.72)–(3.90), we get the result by setting $x = \xi_{\eta+2}$ and $y = \xi_{\eta+3}$. By Theorem 1, $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$. □

Remark 7. Corollary 4 is a generalization of [34, Corollary 3.14].

Corollary 5. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, the pairs $\{T_2, T_4\}$, $\{T_3, T_5\}$, and $\{T_1, T_6\}$ are considered weakly commuting self-mappings that satisfy the condition that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ Moreover, a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge\; 1, \, \, \, \epsilon, \delta > 0$, and, for each $i = 1, \cdots, 6$. For all $\lambda > 0$, $T_i$ is classified as a triangular $\alpha$-orbital admissible mapping, provided that the following conditions are met for some positive integer $m \geq 1$:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be defined in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ for some positive integer, $m\ge 1$.

Proof. By Corollary 4, $p = T^{m}_{1}p = T^{m}_{6}p$, $p = T^{m}_{2}p = T^{m}_{4}p$, and $p = T^{m}_{3}p = T^{m}_{5}p$. Thus, $p = T^{m}_{1}p = T^{m}_{2}p = T^{m}_{3}p = T^{m}_{4}p = T^{m}_{5}p = T^{m}_{6}p$ or $p\in Fix(T^{m}_{i})$ for $i = 1, 2, \cdots, 6$ and some positive integer $m\ge 1$, showing that $p$ is a fixed point of $T^{m}_{1}$ and also a fixed point of $T^{m}_{2}, T^{m}_{3}, T^{m}_{4}, T^{m}_{5}, \; and\; T^{m}_{6}$ respectively. Therefore, the uniqueness follows as in Theorem 1 above. Hence, $T_{i}$ has a common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ and some positive integer, $m\ge 1$. □

Corollary 6. Consider $X_{\hat{v}}$ to be a $\hat{v}$-regular and $\hat{v}$-complete modular extended $b$-metric space. Let $T_i: X_{\hat{v}} \rightarrow X_{\hat{v}}$ represent six orbitally continuous mappings that adhere to the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \ldots, 6$. The pairs $\{T_2, T_4\}$, $\{T_3, T_5\}$, and $\{T_1, T_6\}$ are identified as weakly commuting self-mappings, satisfying the inclusions:

Additionally, a function $\alpha: X_{\hat{v}} \times X_{\hat{v}} \rightarrow [0, +\infty)$ exist with the parameters $a \neq 0$, $a < 1$, $b \neq 0$, and $\frac{ac}{b} < 1$. Let $x_0 \in X_{\hat{v}}$ be such that $\alpha(x_0, x_1) \geq 1$ and $\epsilon, \delta > 0$. For each $i = 1, \ldots, 6$, it follows that $T_i$ qualifies as triangular $\alpha$-orbital admissible mapping for every $\lambda > 0$, with the following specific conditions being satisfied:

where

Let the sequences $\{x_n\}_{n \in \mathbb{N}}$ and $\{\xi_n\}_{n \in \mathbb{N}}$ be defined in $X_{\hat{v}}$ as follows. For a given $x_n$ in $X_{\hat{v}}$, select $x_{n+1}$ such that $\xi_n = T_1 x_n = T_6 x_{n+1}$. Next, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ so that $\xi_{n+1} = T_2 x_{n+1} = T_5 x_{n+2}$. Furthermore, for the point $x_{n+2}$ in $X_{\hat{v}}$, we determine $x_{n+3}$ such that $\xi_{n+2} = T_3 x_{n+2} = T_4 x_{n+3}$, for $n = 0, 1, 2, \ldots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a unique common fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for each $i = 1, \cdots, 6$.

Proof. If we take $X_{\hat{v}}$ to be empty, then there is nothing to prove. Henceforth, we assume that $X_{\hat{v}}\neq\emptyset$. Then a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for every $\lambda > 0$. Suppose that the mappings, $T_{i}$ for $i = 1, \cdots, 6$ satisfies the inequalities (3.73)–(3.102). Since $x_{0}, x_{1}$, and $x_{2}$ are points in $X_{\hat{v}}$ and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$, we can find a point $x_{1}$ in $X_{\hat{v}}$ such that $\xi_{0} = T_{1}x_{0} = T_{6}x_{1}$. For $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, we can find a point $x_{2}\in X_{\hat{v}}$ such that $\xi_{1} = T_{2}x_{1} = T_{5}x_{2}$; for $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, we can find a point $x_{3}$ in $X_{\hat{v}}$ such that $\xi_{2} = T_{3}x_{2} = T_{4}x_{3}$. Now for all $\lambda > 0$, in general, there are sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ in $X_{\hat{v}}$ such that Eq (3.7) holds. For any $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1}$, and $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$ hold, if, $m\in\mathbb{N}$ such that $\xi_{m+2} = \xi_{m+3}$, then $T_{1}u = T_{6}u$, where $u = x_{m+3}$. Therefore, the pair $\{T_{1}, T_{6}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m} = \xi_{m+1}$, then $T_{2}u = T_{4}u$, where $u = x_{m+1}$. Therefore, the pair $\{T_{2}, T_{4}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m+1} = \xi_{m+3}$, then $T_{3}u = T_{5}u$, where $u = x_{m+2}$. Thus, the pair $\{T_{3}, T_{5}\}$ has a coincidence point $u\in X_{\hat{v}}$. Again, if there is an $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1} = \xi_{n_{0}+2}$, then $\xi_{n} = \xi_{n_{0}}$ for any $n\geq n_{0}$. This implies that $\{\xi_{n}\}$ is a modular $\hat{v}$ Cauchy sequence in $X_{\hat{v}}$. Actually, if $\eta\in\mathbb{N}$ exists such that (1) $\xi_{\eta} = \xi_{\eta+1} = \xi_{\eta+2}$, (2) $\xi_{\eta}\neq\xi_{\eta+1} = \xi_{\eta+2}$, (3) $\xi_{\eta}\neq\xi_{\eta+2} = \xi_{\eta+1}$, and (4) $\xi_{\eta}\neq\xi_{\eta+1}\neq\xi_{\eta+2}$ hold. In fact, Case (1) is easy, and Case (3) is similar to Case (2); therefore, from inequalities (3.73)–(3.102), we get the result by setting $x = \xi_{\eta+2}$ and $y = \xi_{\eta+3}$. By Theorem 1, $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$. □

Remark 8. Corollary 6 is a generalization of [34, Corollary 3.12] and the results in Karapinar et al. ^[26].

Corollary 7. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, where $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$, and $\{T_{1}, T_{6}\}$ are weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ A function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, the mapping $T_{i}$ is classified as a triangular $\alpha$-orbital admissible function for every $\lambda > 0$, provided that certain conditions are satisfied for a positive integer $m \ge 1$.

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ for some positive integer, $m\ge 1$.

Proof. By Corollary 6, we get $p = T^{m}_{1}p = T^{m}_{6}p$, $p = T^{m}_{2}p = T^{m}_{4}p$, and $p = T^{m}_{3}p = T^{m}_{5}p$. Thus, $p = T^{m}_{1}p = T^{m}_{2}p = T^{m}_{3}p = T^{m}_{4}p = T^{m}_{5}p = T^{m}_{6}p$, or $p\in Fix(T^{m}_{i})$ for $i = 1, 2, \cdots, 6$ and some positive integer, $m\ge 1$, showing that $p$ is a fixed point of $T^{m}_{1}$ and also a fixed point of $T^{m}_{2}, T^{m}_{3}, T^{m}_{4}, T^{m}_{5}, T^{m}_{6}$ respectively. Therefore, the uniqueness follows as in Theorem 1 above. Hence, $T_{i}$ has a common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ and some positive integer $m\ge 1$. □

Remark 9. Corollary 7 is a generalization of [34, Corollary 3.13].

Corollary 8. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, where $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$ and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ A function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for all $\lambda > 0$ satisfying the following conditions:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a unique common fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for each $i = 1, \cdots, 6$.

Proof. If we take $X_{\hat{v}}$ to be empty, then there is nothing to prove. Now, we assume that $X_{\hat{v}}\neq\emptyset$. Then a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge\; 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mappings for every $\lambda > 0$. Suppose that the mappings, $T_{i}$ for $i = 1, \cdots, 6$ satisfy inequalities the (3.74)–(3.114). Since $x_{0}, x_{1}$, and $x_{2}$ are points in $X_{\hat{v}}$ and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$, we can find a point $x_{1}$ in $X_{\hat{v}}$ such that $\xi_{0} = T_{1}x_{0} = T_{6}x_{1}$. For $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, we can find a point $x_{2}\in X_{\hat{v}}$ such that $\xi_{1} = T_{2}x_{1} = T_{5}x_{2}$; for $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, we can find a point $x_{3}$ in $X_{\hat{v}}$ such that $\xi_{2} = T_{3}x_{2} = T_{4}x_{3}$. Now for all $\lambda > 0$, in general, the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ are in $X_{\hat{v}}$ such that Eq (3.7) hold. If $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1}$, then $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$ hold. In fact, if there is an $m\in\mathbb{N}$ such that $\xi_{m+2} = \xi_{m+3}$, then $T_{1}u = T_{6}u$, where $u = x_{m+3}$. Therefore, the pair $\{T_{1}, T_{6}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m} = \xi_{m+1}$, then $T_{2}u = T_{4}u$, where $u = x_{m+1}$. Therefore, the pair $\{T_{2}, T_{4}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m+1} = \xi_{m+3}$, then $T_{3}u = T_{5}u$, where $u = x_{m+2}$. Thus, the pair $\{T_{3}, T_{5}\}$ has a coincidence point $u\in X_{\hat{v}}$. Again, if there is an $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1} = \xi_{n_{0}+2}$, then $\xi_{n} = \xi_{n_{0}}$ for any $n\geq n_{0}$. This implies that $\{\xi_{n}\}$ is a modular $\hat{v}$ Cauchy sequence in $X_{\hat{v}}$. Actually, if $\eta\in\mathbb{N}$ exists such that (1) $\xi_{\eta} = \xi_{\eta+1} = \xi_{\eta+2}$, (2) $\xi_{\eta}\neq\xi_{\eta+1} = \xi_{\eta+2}$, (3) $\xi_{\eta}\neq\xi_{\eta+2} = \xi_{\eta+1}$, and (4) $\xi_{\eta}\neq\xi_{\eta+1}\neq\xi_{\eta+2}$ hold. In fact, Case (1) is easy, and Case (3) is similar to Case (2); therefore, from the inequalities (3.74)–(3.114), we get the result by setting $x = \xi_{\eta+2}$ and $y = \xi_{\eta+3}$. By Theorem 1, $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$. □

Remark 10. Corollary 8 is a generalization of [20, Theorem 2.8] and results in Karapinar et al. ^[26].

Corollary 9. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, and let $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$, and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ A function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$, and, for each $i = 1, \cdots, 6$, $T_{i}$ remain a triangular $\alpha$-orbital admissible mapping for all $\lambda > 0$ satisfying the following conditions for some positive integer $m\ge 1$:

where,

Let the two sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ for some positive integer, $m\ge 1$.

Proof. By Corollary 8, $p = T^{m}_{1}p = T^{m}_{6}p$, $p = T^{m}_{2}p = T^{m}_{4}p$ and $p = T^{m}_{3}p = T^{m}_{5}p$. Thus, $p = T^{m}_{1}p = T^{m}_{2}p = T^{m}_{3}p = T^{m}_{4}p = T^{m}_{5}p = T^{m}_{6}p$ or $p\in Fix(T^{m}_{i})$ for $i = 1, 2, \cdots, 6$ and some positive integer, $m\ge 1$; showing that $p$ is a fixed point of $T^{m}_{1}$ and also a fixed point of $T^{m}_{2}, T^{m}_{3}, T^{m}_{4}, T^{m}_{5}, \; and\; T^{m}_{6}$ respectively. Therefore, the uniqueness follows as in Theorem 1 above. Hence, $T_{i}$ has common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ and some positive integer, $m\ge 1$. □

Corollary 10. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, and let $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$ and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ A function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mappings for all $\lambda > 0$ for which the following hold:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again. for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a unique common fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for each $i = 1, \cdots, 6$.

Proof. Suppose that $X_{\hat{v}}$ is empty; then there is nothing to prove. We now assume that $X_{\hat{v}}\neq\emptyset$. Then a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge\; 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mappings for every $\lambda > 0$. Suppose that the mappings $T_{i}$ for $i = 1, \cdots, 6$ satisfy the inequalities (3.124)–(3.126). Since $x_{0}, x_{1}$, and $x_{2}$ are points in $X_{\hat{v}}$ and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$, we can find a point $x_{1}$ in $X_{\hat{v}}$ such that $\xi_{0} = T_{1}x_{0} = T_{6}x_{1}$. For $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, we can find a point $x_{2}\in X_{\hat{v}}$ such that $\xi_{1} = T_{2}x_{1} = T_{5}x_{2}$, and for $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, we can find a point $x_{3}$ in $X_{\hat{v}}$ such that $\xi_{2} = T_{3}x_{2} = T_{4}x_{3}$. Now for all $\lambda > 0$, induce on $n$ so that the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ are in $X_{\hat{v}}$ such that Eq (3.7) hold. For any $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1}$, then $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$ holds. In fact, if $m\in\mathbb{N}$ exists such that $\xi_{m+2} = \xi_{m+3}$, then $T_{1}u = T_{6}u$, where $u = x_{m+3}$. Therefore, the pair $\{T_{1}, T_{6}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m} = \xi_{m+1}$, then $T_{2}u = T_{4}u$, where $u = x_{m+1}$. Therefore, the pair $\{T_{2}, T_{4}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m+1} = \xi_{m+3}$, then $T_{3}u = T_{5}u$, where $u = x_{m+2}$. Thus, the pair $\{T_{3}, T_{5}\}$ has a coincidence point $u\in X_{\hat{v}}$. Again, if there is an $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1} = \xi_{n_{0}+2}$, then $\xi_{n} = \xi_{n_{0}}$ for any $n\geq n_{0}$. This implies that $\{\xi_{n}\}$ is a modular $\hat{v}$ Cauchy sequence in $X_{\hat{v}}$. Actually, if $\eta\in\mathbb{N}$ exists such that (1) $\xi_{\eta} = \xi_{\eta+1} = \xi_{\eta+2}$, (2) $\xi_{\eta}\neq\xi_{\eta+1} = \xi_{\eta+2}$, (3) $\xi_{\eta}\neq\xi_{\eta+2} = \xi_{\eta+1}$, and (4) $\xi_{\eta}\neq\xi_{\eta+1}\neq\xi_{\eta+2}$ hold. In fact, Case (1) is easy, and Case (3) is similar to Case (2); therefore, from the inequalities (3.124)–(3.126), we get the result by setting $x = \xi_{\eta+2}$ and $y = \xi_{\eta+3}$. By Theorem 1, $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$. □

Remark 11. (1) If $T = T_{1} = \cdots = T_{6}$, then the inequalities (3.124)–(3.126) and Eqs (3.127)–(3.129) of Corollary 10 coincides, which is a modification of [20, Theorem 2.8].

(2) $T_{1} = T_{2} = T_{3} = T$ and $T_{4} = T_{5} = T_{6} = I$, $c = 0$, and $a+b = 1$, then the inequalities (3.124)–(3.126) and Eqs (3.127)–(3.129) of Corollary 10 coincide with [20, Theorem 2.8].

Corollary 11. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, and let $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$, and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ A function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$, and, for each $i = 1, \cdots, 6$, $T_{i}$ is triangular $\alpha$-orbital admissible mapping for all $\lambda > 0$ and for some positive integer $m\ge 1$, for which:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ for some positive integer, $m\ge 1$.

Proof. By Corollary 10, $p = T^{m}_{1}p = T^{m}_{6}p$, $p = T^{m}_{2}p = T^{m}_{4}p$, and $p = T^{m}_{3}p = T^{m}_{5}p$. Thus, $p = T^{m}_{1}p = T^{m}_{2}p = T^{m}_{3}p = T^{m}_{4}p = T^{m}_{5}p = T^{m}_{6}p$, or $p\in Fix(T^{m}_{i})$ for $i = 1, 2, \cdots, 6$ and some positive integer $m\ge 1$, showing that $p$ is a fixed point of $T^{m}_{1}$ and also a fixed point of $T^{m}_{2}, T^{m}_{3}, T^{m}_{4}, T^{m}_{5}, \; and\; T^{m}_{6}$ respectively. Therefore, the uniqueness follows as in Theorem 1 above. Hence, $T_{i}$ has a common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$ and some positive integer, $m\ge 1$. □

Remark 12. Corollary 11 is a generalization of [20, Theorem 2.8].

Corollary 12. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, and let $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$, and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ A function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$, and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for all $\lambda > 0$ satisfying the following conditions:

where,

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a unique common fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for each $i = 1, \cdots, 6$.

Proof. Upon appeal to Theorem 1, we can see that $T_{i}$ has a common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$. □

Corollary 13. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, and let $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$, and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ A function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for all $\lambda > 0$ satisfying the following conditions:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a unique common fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for each $i = 1, \cdots, 6$.

Proof. Thanks to Corollary 2, we can see that $T_{i}$ has a common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$. □

Corollary 14. Let $X_{\hat{v}}$ be a $\hat{v}$-regular $\hat{v}$-complete modular extended $b$-metric space and let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous mappings satisfying the $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction for $i = 1, 2, \cdots, 6$, and let $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$, and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings such that $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}}), \, \, \, T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}}), \, \, \, T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}}).$ A function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mapping for all $\lambda > 0$ satisfying the following conditions:

where

Let the sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ be in $X_{\hat{v}}$ so that for $x_{n}$ in $X_{\hat{v}}$, we choose $x_{n+1}$ such that $\xi_{n} = T_{1}x_{n} = T_{6}x_{n+1}$; again, for $x_{n+1}$ in $X_{\hat{v}}$, we choose $x_{n+2}$ such that $\xi_{n+1} = T_{2}x_{n+1} = T_{5}x_{n+2}$ and, for a point $x_{n+2}$ in $X_{\hat{v}}$, we choose $x_{n+3}$ such that $\xi_{n+2} = T_{3}x_{n+2} = T_{4}x_{n+3}$ for $n = 0, 1, 2, \cdots$. Then $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has a unique common fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for each $i = 1, \cdots, 6$.

Proof. If we take $X_{\hat{v}}$ to be empty, then there is nothing to prove. Now, we assume that $X_{\hat{v}}\neq\emptyset$. Then a function $\alpha:X_{\hat{v}}\times X_{\hat{v}} \rightarrow [0, +\infty)$, $a\neq 0$, $a < 1$, $b\neq 0$, $\frac{ac}{b} < 1$, and $x_{0}\in X_{\hat{v}}$ exist such that $\alpha(x_{0}, x_{1})\ge\; 1, \, \, \, \epsilon, \delta > 0$ and, for each $i = 1, \cdots, 6$, $T_{i}$ remains a triangular $\alpha$-orbital admissible mappings for every $\lambda > 0$. Suppose that the mappings, $T_{i}$ for $i = 1, \cdots, 6$ satisfy inequalities the (3.148)–(3.150). Since $x_{0}, x_{1}$, and $x_{2}$ are points in $X_{\hat{v}}$ and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$, we can find a point $x_{1}$ in $X_{\hat{v}}$ such that $\xi_{0} = T_{1}x_{0} = T_{6}x_{1}$. For $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, we can find a point $x_{2}\in X_{\hat{v}}$ such that $\xi_{1} = T_{2}x_{1} = T_{5}x_{2}$; for $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, we can find a point $x_{3}$ in $X_{\hat{v}}$ such that $\xi_{2} = T_{3}x_{2} = T_{4}x_{3}$. Now for all $\lambda > 0$, inductively, sequences $\{x_{n}\}_{n\in\mathbb{N}}$ and $\{\xi_{n}\}_{n\in\mathbb{N}}$ in $X_{\hat{v}}$ exist such that Eq (3.7) hold. For any integer, $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1}$, then $T_{3}(X_{\hat{v}})\subseteq T_{4}(X_{\hat{v}})$, $T_{2}(X_{\hat{v}})\subseteq T_{5}(X_{\hat{v}})$, and $T_{1}(X_{\hat{v}})\subseteq T_{6}(X_{\hat{v}})$ hold. In fact, if $m\in\mathbb{N}$ exists such that $\xi_{m+2} = \xi_{m+3}$, then $T_{1}u = T_{6}u$, where $u = x_{m+3}$. Therefore, the pair $\{T_{1}, T_{6}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m} = \xi_{m+1}$, then $T_{2}u = T_{4}u$, where $u = x_{m+1}$. Therefore, the pair $\{T_{2}, T_{4}\}$ has a coincidence point $u\in X_{\hat{v}}$. If $\xi_{m+1} = \xi_{m+3}$, then $T_{3}u = T_{5}u$, where $u = x_{m+2}$. Thus, the pair $\{T_{3}, T_{5}\}$ has a coincidence point $u\in X_{\hat{v}}$. Again, if there is an $n_{0}\in\mathbb{N}$ such that $\xi_{n_{0}} = \xi_{n_{0}+1} = \xi_{n_{0}+2}$, then $\xi_{n} = \xi_{n_{0}}$ for any $n\geq n_{0}$. This implies that $\{\xi_{n}\}$ is a modular $\hat{v}$ Cauchy sequence in $X_{\hat{v}}$. Actually, if $\eta\in\mathbb{N}$ exists such that (1) $\xi_{\eta} = \xi_{\eta+1} = \xi_{\eta+2}$, (2) $\xi_{\eta}\neq\xi_{\eta+1} = \xi_{\eta+2}$, (3) $\xi_{\eta}\neq\xi_{\eta+2} = \xi_{\eta+1}$, and (4) $\xi_{\eta}\neq\xi_{\eta+1}\neq\xi_{\eta+2}$ hold. In fact, Case (1) is easy, and Case (3) is similar to Case (2); therefore, from the inequalities (3.148)–(3.150), we get the result by setting $x = \xi_{\eta+2}$ and $y = \xi_{\eta+3}$. By Theorem 1, $T_{i}$ has a fixed point in $X_{\hat{v}}$. Moreover, if $\alpha(x^{\ast}, y^{\ast})\ge 1$ for all $x^{\ast}, y^{\ast}\in \bigcap_{i = 1} Fix(T_{i})$, then $T_{i}$ has common unique fixed point in $\bigcap_{i = 1}Fix(T_{i})$ for $i = 1, \cdots, 6$. □

Example 4. Suppose that $X_{\hat{v}} = (\mathbb{R}\setminus\{0\})\cup\{\infty\}$ with the modular extended $b$-metric as defined in Example 3, which is complete in $X_{\hat{v}}$ for all $\lambda > 0$. Define the $\hat{v}$-weakly commuting mappings $T_{1}, T_{2}, T_{3}, T_{4}, T_{5}, T_{6}:(\mathbb{R}\setminus\{0\})\cup\{\infty\}\rightarrow (\mathbb{R}\setminus\{0\})\cup\{\infty\}$ as follows: $T_{1}x = \log_{64}x^{6}, \, \, \, \, T_{2}x = \log_{32}x^{5}, \, \, \, \, T_{3}x = \log_{16}x^{4},$ $T_{4}x = \log_{8}x^{3}, \, \, \, \, T_{5}x = \log_{4}x^{2}, \, \, \, \, T_{6}x = \log_{2}x$ for all $x\in(\mathbb{R}\setminus\{0\})\cup\{\infty\}$ and $\lambda > 0$, for each $i = 1, 2, \cdots, 6$, and also for $x, y\in(\mathbb{R}\setminus\{0\})\cup\{\infty\}$, $\alpha(x, T_{i}x)\geq 1\implies \alpha(T_{i}x, T_{i}^{2}x)\geq1$, and $\alpha(x, y)\geq 1$ and $\alpha(y, T_{i}y)\geq 1\implies\alpha(x, T_{i}y)\geq 1$. Then the mappings $T_{1}, T_{2}, T_{3}, T_{4}, T_{5}$, and $T_{6}$ satisfy the inequalities (3.151)–(3.153) of Corollary 14. In fact, let $T_{i}:X_{\hat{v}}\longrightarrow X_{\hat{v}}$ be six orbitally continuous $\alpha$-$\hat{v}$-$A$-$B$-$C$-Meir-Keeler-type contraction mappings for $i = 1, 2, \cdots, 6$, and let $\{T_{2}, T_{4}\}$, $\{T_{3}, T_{5}\}$, and $\{T_{1}, T_{6}\}$ be weakly commuting pairs of self-mappings. Actually, it suffices to show that the inequalities (3.1)–(3.3) coincide with inequalities (3.148)–(3.150). Now observe that from Example 3, $\hat{v}_{\lambda}(T_{3}x, T_{4}y) = \frac{1}{1+\lambda}\frac{1}{\ln(2)}\max_{x, y\in X_{\hat{v}}}\biggl\{\ln||\frac{x}{y}||\biggr\}$ and $\hat{v}_{\lambda}(T_{2}x, T_{5}y) = \frac{1}{1+\lambda}\frac{1}{\ln(2)}\max_{x, y\in X_{\hat{v}}}\biggl\{\ln||\frac{x}{y}||\biggr\}$, and from inequality (3.151), we can see that $\frac{\hat{v}_{\lambda}(T_{3}x, T_{4}y)+\hat{v}_{\lambda}(T_{2}x, T_{5}y)}{2} = \frac{1}{1+\lambda}\frac{1}{\ln(2)}\max_{x, y\in X_{\hat{v}}}\biggl\{\ln||\frac{x}{y}||\biggr\}$, which is either $\hat{v}_{\lambda}(T_{3}x, T_{4}y)$ or $\hat{v}_{\lambda}(T_{2}x, T_{5}y)$. Similarly, the inequalities (3.152) and (3.153) hold. It is now evident that the inequalities (3.1)–(3.3) and (3.148)–(3.150) coincides.

4.   Conclusions

In this paper, we introduced the concept of $\alpha$-$\hat{v}$-A-B-C-Meir-Keeler-type nonlinear contractions in modular extended $b$-metric spaces. We established common unique fixed-point theorems that unify and extend existing results in modular $b$-metric and modular extended $b$-metric spaces. These findings provide a broader perspective on contraction mappings and deepen the understanding of fixed-point theory in modular settings. The results presented here build on and generalize classical fixed-point theorems, demonstrating the richness of modular extended $b$-metric spaces.

Author Contributions

Daniel Francis: Conceptualization, methodology, validation, formal analysis, investigation, writing-original draft preparation, writing-review and editing, visualization; Godwin Amechi Okeke: Methodology, validation, resources, writing-original draft preparation, Aviv Gibali: Formal analysis, investigation; writing-original draft preparation, and editing. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We are very grateful to the editor and to the anonymous reviewers for their insightful and careful comments and suggestions, and for taking the trouble to read the long proofs in our manuscript and for their constructive report, which has been very useful for improving our paper.

Conflict of interest

The authors declare no conflict of interest.