Intuitionistic Fuzzy Contractions over Banach Algebras and Their Applications to Fractional Volterra Integral Equations with Numerical Verification

Abstract

This paper introduces a novel analytical and numerical framework for studying nonlinear fractional Volterra integral equations by employing an intuitionistic fuzzy metric structure over a Banach algebra. The principal contribution of this work is the development of fixed-point theory for a new class of intuitionistic fuzzy Z-contractions in $\mathcal{IFM}$-spaces over $\mathcal{BA}$, which extends existing fuzzy and algebra-valued metric frameworks. Within this setting, we established existence, uniqueness, and convergence results for solutions of fractional integral equations of the Caputo type by proving that the associated fractional integral operator satisfies the proposed contractive conditions. Furthermore, we demonstrated how the algebra-valued intuitionistic fuzzy structure enhances the analytical flexibility and robustness of the model. To support the theoretical findings, a numerical simulation based on a discretized iterative scheme is presented, illustrating the rapid convergence of the approximating sequence together with the monotone behavior of intuitionistic fuzzy nearness and non-nearness measures. The numerical results are consistent with the analytical theory and confirm the effectiveness of the proposed $\mathcal{IFM}$-spaces over the $\mathcal{BA}$ approach for fractional dynamical systems.

1. Introduction

In 1965, Lotfi Zadeh [1] proposed the theory of fuzzy sets. This theory has been widely generalized by multiple authors for a variety of reasons, and it continues to be crucial to current research. Kramosil et al. [2] first proposed the idea of a fuzzy metric space (commonly referred to as an $\mathcal{FM}$-space) by incorporating fuzzy set theory, adapting the probabilistic metric space framework to suit a fuzzy context. It was demonstrated that each conventional metric can be associated with a corresponding fuzzy metric. In line with the framework established by Kramosil et al. [2], Grabiec [3] extended the study by proving many fixed point results, specifically the Banach contraction theorem on complete $\mathcal{FM}$-spaces and the Edelstein contraction theorem on compact $\mathcal{FM}$-spaces. Subsequently, George and Vermani [4] introduced a revised version of the $\mathcal{FM}$-space. Later, Gregori and Sapena [5] formulated several fixed-point results ($\mathcal{FP}$ theorems) within the framework of $\mathcal{FM}$-spaces. The concept of a Hausdorff metric on $\mathcal{FM}$-spaces was later introduced by Rodriguez-Lopez and Romaguera [6], utilizing the approach of George and Veeramani [4] for compact subsets. Several fixed-point and endpoint results were derived by Kiany et al. [7] in 2011 through the application of fuzzy multi-valued contractions in a complete framework. Lately, unique $\mathcal{FP}$ theorems have been formulated by a group of researchers that includes Shamas and collaborators [8,9], involving applications in both integral and differential frameworks. Further relevant findings within this framework are available in [10,11,12,13,14,15] and the citations within.

Huang and Zhang [16] presented the concept of cone metric spaces in 2007, offering an extension to the traditional structure of metric spaces by employing Banach spaces as the range set rather than real numbers. They established certain theorems that address fixed points of nonlinear contractive operators under the condition of cone normality. Al Rawashdeh in [17] defined E-metric spaces and characterized the concept of cone metric spaces in a general way by defining an ordered normed space. Then, he defined some of the results of Huang and Zhang [16]. Also, [18] proved some new results for E-metric spaces. Subsequently, Cakalli et al. [19] demonstrated that any cone metric space valued in a topological vector space is metrizable and provided several results in this context. The concept of a cone metric space ($\mathcal{CM}$-space) over $\mathcal{BA}$s was introduced by Liu et al. [20] in 2013, wherein Banach spaces were replaced by $\mathcal{BA}$s. (Hereafter, we use ‘$\mathcal{BA}$’ as an abbreviation for Banach algebras.) They established $\mathcal{FP}$ results through the use of employing generalized Lipschitz mappings, under less restrictive and more natural assumptions related to the spectral radius of the generalized Lipschitz constant. Yan et al. [21] extended these results to partially ordered $\mathcal{CM}$-spaces over $\mathcal{BA}$s in 2016, focusing on $\mathcal{FP}$ and C$\mathcal{FP}$ theorems. Rashid et al. [22] presented E(s)-distance spaces as a broader concept that generalizes E-metric [17,18] spaces, and established several coincidence point results within this framework.

Oner et al. [23] presented the notion of a fuzzy cone metric space ($\mathcal{FCM}$-space), establishing key properties and formulating a ‘fuzzy cone Banach contraction theorem’. According to this theorem, a self-map defined on a complete $\mathcal{FCM}$-space that involves Cauchy sequences with fuzzy cone contractive criteria ensures the existence of exactly one fixed point. Following this, Oner et al. [24] proposed the notions related to closed balls and relatively compact subsets within the structure of $\mathcal{FCM}$-spaces, and formulated a version of Baire’s theorem for complete $\mathcal{FCM}$-spaces. Rehman and Li [25] in 2017 further expanded and refined the ’fuzzy cone Banach contraction theorem’ by proving generalized $\mathcal{FP}$ theorems in complete $\mathcal{FCM}$-spaces without requiring fuzzy cone contractive sequences to be Cauchy. Following this, Jabeen and collaborators [26] established several common $\mathcal{FP}$ theorems for $\mathcal{FCM}$-spaces and explored their applications. In this sequel, many researchers established $\mathcal{FP}$ and coupled $\mathcal{FP}$ results, focusing on a variety of applications (for example, see [27,28,29,30,31,32]). In 2024, Rashid et al. [33] introduced the concept of multi-valued mappings in C*-algebra-valued metric spaces. By applying the lower bound property, their study extended the Banach, Kannan, and Chatterjea theorems for such mappings in these spaces.

This study presents the idea of an intuitionistic $\mathcal{FM}$-space over $\mathcal{BA}$, deriving its fundamental topological aspects and defining a Hausdorff metric on the space. In prior studies, the fuzzy metric was defined as $\Pi :X\times X\times (0,\bowtie )\to [0,1]$, with X being a non-void set. In their work, Oner et al. [23] substituted $(0,\bowtie )$ with a cone-based setting, defining the fuzzy cone metric ($\mathcal{FCM}$) mapping as ${\Pi }_x:X\times X\times int\left(\mathcal{P}\right)\to [0,1]$, where $\mathcal{P}$ represents a cone lying in a Banach space E over the real field. In the current manuscript, this framework was extended by replacing $(0,\bowtie )$ with a $\mathcal{BA}$$\Lambda$ in an $\mathcal{FM}$-space, introducing $\mathcal{FM}$ mapping over $\mathcal{BA}$, ${\Pi }_A:X\times X\times \Lambda \to [0,1]$. By utilizing this novel concept, we explored fundamental aspects and proved an $\mathcal{FP}$ theorem on a -complete $\mathcal{IFM}$-space over $\mathcal{BA}$$\Lambda$. This approach offers significant potential for advancing fixed-point theory. This new framework allows for the exploration of additional properties and $\mathcal{FP}$ results in the aforementioned spaces, with various possible applications.

This paper is organized as follows. After presenting the necessary preliminaries, we introduce intuitionistic fuzzy metric spaces over Banach algebras (Definition 6) and investigate their fundamental topological properties (Theorems 4–8). We then established fixed-point results for intuitionistic fuzzy Z-contractions (Theorems 9–10) and applied the developed theory to a nonlinear fractional Volterra integral equation, supported by numerical validation.

2. Preliminaries

This section aims to provide readers with essential background information, to help them understand the main results. Throughout the paper, the symbols ${\Re }_{+},\Re$, and $\mathbb{N}$ are used to represent the sets of non-negative real numbers, real numbers, and natural numbers, respectively. We begin by recalling the definitions of triangular norms (†-norms) and conorms (†-conorms), commonly referred to as the axiomatic frameworks for defining fuzzy intersections and unions.

Definition 1 ([34]).

A binary operation 🟉 mapping $[0,1]\times [0,1]$ into $[0,1]$ is referred to as a continuous †-norm whenever it fulfills the axioms listed below:

1.

🟉 satisfies commutativity and associativity;

2.

🟉 is continuous;

3.

$⋉🟉1=⋉$ for all $⋉\in [0,1]$;

4.

${⋉}_1🟉{\beta }_1\le {⋉}_2🟉{\beta }_2,$ whenever ${⋉}_1\le {⋉}_2$ and ${\beta }_1\le {\beta }_2$, and ${⋉}_1,{⋉}_2,{\beta }_1,{\beta }_2\in [0,1]$.

The three fundamental types of continuous †-norms are given as follows:

• The minimum †-norm, defined as $⋉🟉\beta =min\{⋉,\beta \}$.
• The product †-norm, given by $⋉🟉\beta =⋉\beta$.
• The Lukasiewicz †-norm, expressed as $⋉🟉\beta =max\{0,⋉+\beta -1\}$.

Definition 2 ([34]).

A binary operation ♢ mapping $[0,1]\times [0,1]$ into $[0,1]$ is said to be a continuous †-conorm whenever ♢ fulfills the properties listed below:

1.

♢ is commutative, associative, and continuous;

2.

$⋉♢0=⋉$ for all $⋉\in [0,1]$;

3.

${⋉}_1♢{\beta }_1\le {⋉}_2♢{\beta }_2,$ whenever ${⋉}_1\le {⋉}_2$ and ${\beta }_1\le {\beta }_2$, and ${⋉}_1,{⋉}_2,{\beta }_1,{\beta }_2\in [0,1]$.

Three typical examples of †-conorm are $⋉♢\beta =max\{⋉,\beta \}$, $⋉♢\beta =⋉+\beta -⋉\beta$, and $⋉♢\beta =min\{⋉+\beta ,1\}.$

The following remark is taken from [35], and it might prove helpful in proving the main results.

Remark 1.

(a) 

For any ${⋎}_1,{⋎}_2\in (0,1)$ with ${⋎}_1<{⋎}_2$, there exist ${⋎}_3,{⋎}_4\in (0,1)$ such that ${⋎}_1🟉{⋎}_3\ge {⋎}_2$ and ${⋎}_1\ge {⋎}_4♢{⋎}_2$.

(b) 

For any $⋎\in (0,1)$, there exist ${⋎}_1,{⋎}_2\in (0,1)$ such that ${⋎}_1🟉{⋎}_1\ge ⋎$ and ${⋎}_2♢{⋎}_2\le ⋎$.

Definition 3 ([35]).

A 5-tuple $(\chi ,\Pi ,\aleph ,🟉,♢)$ is considered an intuitionistic fuzzy metric space (abbreviated as an $\mathcal{IFM}$-space) if χ is a non-specific set; 🟉 and ♢ represent a continuous †-norm and †-conorm, respectively; and Π and ℵ are fuzzy sets on $\chi \times \chi \times (0,\infty )$ that satisfy the conditions listed below whenever $\varsigma ,\eta ,\upsilon$ belong to χ and $\omega ,†>0$:

(a) 

$\Pi (\varsigma ,\eta ,†)+\aleph (\varsigma ,\eta ,†)\le 1;$

(b) 

$\Pi (\varsigma ,\eta ,†)>0$;

(c) 

$\Pi (\varsigma ,\eta ,†)=1$ if, and only if, $\varsigma ,\eta$ are equal;

(d) 

$\Pi (\varsigma ,\eta ,†)=\Pi (\eta ,\varsigma ,†)$;

(e) 

$\Pi (\varsigma ,\eta ,†)🟉\Pi (\eta ,\upsilon ,\omega )\le \Pi (\varsigma ,\upsilon ,†+\omega )$;

(f) 

$\Pi (\varsigma ,\eta ,.):(0,\infty )\to (0,1]$ is continuous;

(g) 

$\aleph (\varsigma ,\eta ,†)>0$;

(h) 

$\aleph (\varsigma ,\eta ,†)=0$ if, and only if, $\varsigma =\eta$;

(i) 

$\aleph (\varsigma ,\eta ,†)=\aleph (\eta ,\varsigma ,†)$;

(j) 

$\aleph (\varsigma ,\eta ,†)♢\aleph (\eta ,\upsilon ,\omega )\ge \aleph (\varsigma ,\upsilon ,†+\omega )$;

(k) 

$\aleph (\varsigma ,\eta ,.):(0,\infty )\to (0,1]$ is a continuous mapping.

Then, $(\Pi ,\aleph )$ is referred to as an intuitionistic fuzzy metric on χ. The functions $\Pi (\varsigma ,\eta ,†)$ and $\aleph (\varsigma ,\eta ,†)$ represent the degree of nearness and non-nearness, respectively, between ς and η with respect to †.

Remark 2 ([35]). 

Each $\mathcal{FM}$-space $(\chi ,\Pi ,🟉)$ can equivalently be seen as an $\mathcal{IFM}$-space represented by $(\chi ,\Pi ,1-\Pi ,🟉,♢)$, in which 🟉 and ♢ are related by the equation $\varsigma ♢\eta =1-\left(\right(1-\varsigma )🟉(1-\eta \left)\right)$ for any $\varsigma ,\eta \in \chi$.

Example 1. 

Let $(\chi ,\parallel ·\parallel )$ be a normed linear space and let $t>0$. Define the mappings $\Pi ,\aleph :\chi \times \chi \times (0,\infty )\to [0,1]$ by

$$\Pi (x,y,t)=exp\left(-{\displaystyle \frac{\parallel x-y\parallel }{t}}\right),\aleph (x,y,t)=1-exp\left(-{\displaystyle \frac{\parallel x-y\parallel }{t}}\right),$$

for all $x,y\in \chi$ and $t>0$.

Let the continuous t-norm and t-conorm be given by

$$a🟉b=ab,a♢b=min\{1,a+b\},\mathit{for}\mathit{all}a,b\in [0,1].$$

Then, $(\chi ,\Pi ,\aleph ,🟉,♢)$ is an intuitionistic fuzzy metric space.

Definition 4 ([36]).

A complex algebra is defined as a vector space Λ over the complex field $\mathbb{C}$, satisfying the following properties:

(1) 

$\varsigma \left(\eta \upsilon \right)=\left(\varsigma \eta \right)\upsilon$

(2) 

$(\varsigma +\eta )\upsilon =\varsigma \upsilon +\eta \upsilon ,\varsigma (\eta +\upsilon )=\varsigma \eta +\varsigma \upsilon$

(3) 

$⋉\left(\varsigma \eta \right)=(⋉\varsigma )\eta =\varsigma (⋉\eta )$

for all $\varsigma ,\eta$, and υ in Λ and any scalar $⋉\in \mathbb{C}$.

In the case where Λ is a Banach space with a norm that meets the multiplicative inequality condition,

(4) 

$\parallel \varsigma \eta \parallel \le \parallel \varsigma \parallel \parallel \eta \parallel ,(\varsigma ,\eta \in \Lambda )$,

then Λ is called a $\mathcal{BA}$, and if there exists a unit element I in Λ such that

(5) 

$\varsigma I=I\varsigma =\varsigma ,(\varsigma \in \Lambda )$

and

(6) 

$\parallel I\parallel =1,$

then Λ is called unital algebra. Λ is called commutative $\mathcal{BA}$ if $\varsigma \eta =\eta \varsigma$ for all ς and η in Λ.

Example 2. 

Let $X=C\left(\right[0,1],\mathbb{C})$ denote the set of all complex-valued continuous functions on the interval $[0,1]$. Define addition and scalar multiplication pointwise, and define multiplication by

$$(f·g)\left(t\right)=f\left(t\right)g\left(t\right),\mathit{for}\mathit{all}t\in [0,1].$$

Endow X with the supremum norm

$${\parallel f\parallel }_{\infty }=\underset{t\in [0,1]}{sup}\left|f\left(t\right)\right|.$$

Then, $(X,\parallel ·{\parallel }_{\infty })$ is a complex Banach space. Moreover, X is a Banach algebra, since

$${\parallel f·g\parallel }_{\infty }\le {\parallel f\parallel }_{\infty }{\parallel g\parallel }_{\infty },\mathit{for}\mathit{all}f,g\in X.$$

The constant function $e\left(t\right)=1$ for all $t\in [0,1]$ acts as the multiplicative identity. Hence, $C\left(\right[0,1],\mathbb{C})$ is a unital commutative complex Banach algebra.

Assume that Λ is a unital algebra with the identity element I. An involution on Λ is a conjugate-linear map $\kappa \mapsto {\kappa }^{🟉}$ on Λ such that ${\kappa }^{\ast \ast }=\kappa$ and ${(\kappa ♭)}^{🟉}={♭}^{🟉}{\kappa }^{🟉}$ for all $\kappa ,♭\in \Lambda$. The pair $(\Lambda ,🟉)$ is referred to as a 🟉-algebra. A Banach 🟉-algebra (in short, $B^{\ast }$-algebra) is a 🟉-algebra Λ equipped with a complete, sub-multiplicative norm that satisfies $\parallel {\kappa }^{🟉}\parallel =\parallel \kappa \parallel$ (for all $\kappa \in \Lambda$). A $B^{\ast }$-algebra satisfying the condition $|{\kappa }^{🟉}{\kappa |=|\kappa |}^2$ is called a $C^{🟉}$-algebra.

Set ${\Lambda }_h=\{\varsigma \in \Lambda :\varsigma ={\varsigma }^{\ast }\}$. An element $\varsigma \in \Lambda$ is called a positive element, denoted by $\varsigma ⪰\theta$, if $\varsigma \in {\Lambda }_h$ and its spectrum $\sigma \left(\varsigma \right)$ lies within ${\mathbb{R}}_{+}=[0,\infty )$. A partial ordering ⪯ on ${\Lambda }_h$ can be defined through positive elements in the following way: $\varsigma ⪯\eta$ if, and only if, $\eta -\varsigma ⪰\theta$, with θ representing the zero element in Λ. Hereafter, let ${\Lambda }_{+}$ represent the set $\{\varsigma \in \Lambda :\varsigma ⪰\theta \}$ with the norm given by $\parallel \varsigma \parallel ={\left({\varsigma }^{\ast }\varsigma \right)}^{{\displaystyle \frac{1}{2}}}$.

Given below are some theorems taken from [36] that are helpful in proving the main results.

Theorem 1 (Theorem 11.5(e), [36]). 

Let Λ be a commutative $\mathcal{BA}$, and let Δ denote the set of all complex homomorphisms of Λ. Then, $\lambda \in \sigma \left(\varsigma \right)$ if, and only if, $\zeta \left(\varsigma \right)=\lambda$ for some $\zeta \in \Delta$.

Theorem 2 (Theorem 11.18, [36]). 

Suppose Λ is a commutative $B^{\ast }$-algebra with a maximal idea space Δ. The Gelfand transform is then an isometric isomorphism of Λ onto $C(\Delta )$ (the algebra of all complex continuous functions on Δ), which has the property that

$$\begin{array}{c}\hfill \zeta \left({\varsigma }^{\ast }\right)=\overline{\zeta \left(\varsigma \right)}(\varsigma \in \Lambda ,\zeta \in \Delta ).\end{array}$$

Theorem 3 (Theorem 10.7, [36]). 

Suppose Λ is a $\mathcal{BA}$, $\varsigma \in \Lambda ,\parallel \varsigma \parallel <1$. Then, $\left|\zeta \right(\varsigma \left)\right|<1$ for every complex homomorphism ζ on Λ.

According to a recent analysis, Khojasteh et al. [37] introduced a family of auxiliary (referred to as simulations) functions, in an attempt to standardize different forms of contractions.

Definition 5 ([37]).

A simulation function is defined as a mapping $\rho :{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}⟶\mathbb{R}$ that meets the requirements stated below:

(S1) 

$\rho (0,0)=0$;

(S2) 

$\rho (a,♭)<♭-a$ for all $a,♭>0$;

(S3) 

if ${\left\{a_n\right\}}_{n\in \mathbb{N}}$ and ${\left\{{♭}_n\right\}}_{n\in \mathbb{N}}$ are two sequences with terms in the interval $(0,\infty )$ in such a way that ${lim}_{n⟶\infty }{a}_n={lim}_{n⟶\infty }{♭}_n>0$, then,

$$\begin{array}{c}\hfill \underset{n⟶\infty }{lim\; sup}\rho ({\kappa }_n,{♭}_n)<0.\end{array}$$

The collection of all simulation functions is denoted by Z.

Remark 3 ([37]). 

It is clear from the definition of the simulation function that $\rho (a,♭)<0$ for all $a\ge ♭>0$.

Hausdorff Property in Fuzzy and Intuitionistic Fuzzy Metric Spaces

The Hausdorff property is a fundamental topological requirement that ensures the uniqueness of limits and plays a crucial role in fixed-point theory. In classical metric spaces, Hausdorffness follows naturally from the metric structure, while in fuzzy metric spaces, it was systematically established by George and Veeramani [4] and further developed by Park [35] and others, showing that the topology induced by a fuzzy metric is Hausdorff under mild conditions. For intuitionistic fuzzy metric spaces, the presence of both nearness and non-nearness functions necessitates the careful treatment of separation properties; nevertheless, suitable assumptions on the underlying t-norm and t-conorm guarantee the Hausdorffness of the induced topology. In the present work, this property was particularly important due to the algebra-valued nature of the intuitionistic fuzzy metric, as it ensured the uniqueness of the limits and the well-posedness of fixed points in the Banach algebra setting. This justified the explicit use of Hausdorffness in the main results, including Theorem 6.

The algebras that we consider throughout this article included the commutative unital $C^{\ast }$-algebra, represented by $\Lambda$, and we will use $†\succ \theta$ to denote $†\succ {\Lambda }_{+}\backslash \left\{\theta \right\}$.

3. Main Results

This section is dedicated to introducing the concept of $\mathcal{IFM}$-spaces within the context of $\mathcal{BA}$, along with the necessary definitions and some results.

With the framework of an $\mathcal{IFM}$-space over $\Lambda$ established, we now explore the convergence properties within such spaces. The following definitions formalize the notions of convergence and Cauchy sequences, which are essential for establishing fixed-point theorems in this generalized setting.

Definition 6.

A 6-tuple $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is referred to as an $\mathcal{IFM}$-space over algebra Λ if χ is any non-specific set, 🟉 is a continuous †-norm, ♢ is a continuous †-conorm, Λ is a unital $C^{\ast }$-algebra, and ${\Pi }_{\Lambda },{\aleph }_{\Lambda }$ are fuzzy sets on $\chi \times \chi \times \Lambda$ satisfying the following conditions: for all $\varsigma ,\eta ,\upsilon \in \chi ,\omega ,†\succ \theta$,

(a) 

${\Pi }_{\Lambda }(\varsigma ,\eta ,†)+{\aleph }_{\Lambda }(\varsigma ,\eta ,†)\le 1;$

(b) 

${\Pi }_{\Lambda }(\varsigma ,\eta ,†)>0$;

(c) 

${\Pi }_{\Lambda }(\varsigma ,\eta ,†)=1$ if, and only if, $\varsigma =\eta$;

(d) 

${\Pi }_{\Lambda }(\varsigma ,\eta ,†)={\Pi }_{\Lambda }(\eta ,\varsigma ,†)$;

(e) 

${\Pi }_{\Lambda }(\varsigma ,\eta ,†)🟉{\Pi }_{\Lambda }(\eta ,\upsilon ,\omega )\le {\Pi }_{\Lambda }(\varsigma ,\upsilon ,†+\omega )$;

(f) 

${\Pi }_{\Lambda }(\varsigma ,\eta ,.):(0,\infty )\to (0,1]$ is continuous;

(g) 

${\aleph }_{\Lambda }(\varsigma ,\eta ,†)>0$;

(h) 

${\aleph }_{\Lambda }(\varsigma ,\eta ,†)=0$ if, and only if, $\varsigma =\eta$;

(i) 

${\aleph }_{\Lambda }(\varsigma ,\eta ,†)={\aleph }_{\Lambda }(\eta ,\varsigma ,†)$;

(j) 

${\aleph }_{\Lambda }(\varsigma ,\eta ,†)♢{\aleph }_{\Lambda }(\eta ,\upsilon ,\omega )\ge {\aleph }_{\Lambda }(\varsigma ,\upsilon ,†+\omega )$;

(k) 

${\aleph }_{\Lambda }(\varsigma ,\eta ,.):(0,\infty )\to (0,1]$ is continuous.

Example 3. 

Let χ be a non-empty set and let $\Lambda =C\left(\right[0,1],\mathbb{C})$ be the unital commutative $C^{\ast }$-algebra of all complex-valued continuous functions on $[0,1]$, endowed with the supremum norm

$${\parallel f\parallel }_{\infty }=\underset{t\in [0,1]}{sup}\left|f\left(t\right)\right|.$$

The unit element of Λ is the constant function $e\left(t\right)=1$.

Let $🟉:[0,1]\times [0,1]\to [0,1]$ be the product t-norm defined by

$$a🟉b=ab,$$

and let $♢:[0,1]\times [0,1]\to [0,1]$ be the t-conorm given by

$$a♢b=min\{1,a+b\}.$$

Define the mappings ${\Pi }_{\Lambda },{\aleph }_{\Lambda }:\chi \times \chi \times (0,\infty )\to \Lambda$ by

$${\Pi }_{\Lambda }(\xi ,\eta ,t)\left(s\right)=exp\left(-{\displaystyle \frac{d(\xi ,\eta )}{t}}\right),{\aleph }_{\Lambda }(\xi ,\eta ,t)\left(s\right)=1-exp\left(-{\displaystyle \frac{d(\xi ,\eta )}{t}}\right),$$

for all $\xi ,\eta \in \chi$, $t>0$, and $s\in [0,1]$, where d is a metric on χ.

Then, the 6-tuple

$$(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$$

is an intuitionistic fuzzy metric space over the algebra Λ.

Indeed, for all $\xi ,\eta ,\nu \in \chi$ and $t,\omega >0$,

• ${\Pi }_{\Lambda }(\xi ,\eta ,t)+{\aleph }_{\Lambda }(\xi ,\eta ,t)\le e$;
• ${\Pi }_{\Lambda }(\xi ,\eta ,t)>0$ and ${\aleph }_{\Lambda }(\xi ,\eta ,t)\ge 0$;
• ${\Pi }_{\Lambda }(\xi ,\eta ,t)=e$ and ${\aleph }_{\Lambda }(\xi ,\eta ,t)=0$ if, and only if, $\xi =\eta$;
• ${\Pi }_{\Lambda }(\xi ,\eta ,t)={\Pi }_{\Lambda }(\eta ,\xi ,t)$ and ${\aleph }_{\Lambda }(\xi ,\eta ,t)={\aleph }_{\Lambda }(\eta ,\xi ,t)$;
• $${\Pi }_{\Lambda }(\xi ,\eta ,t)🟉{\Pi }_{\Lambda }(\eta ,\nu ,\omega )\le {\Pi }_{\Lambda }(\xi ,\nu ,t+\omega );$$
• $${\aleph }_{\Lambda }(\xi ,\eta ,t)♢{\aleph }_{\Lambda }(\eta ,\nu ,\omega )\ge {\aleph }_{\Lambda }(\xi ,\nu ,t+\omega );$$
• The functions ${\Pi }_{\Lambda }(\xi ,\eta ,·)$ and ${\aleph }_{\Lambda }(\xi ,\eta ,·)$ are continuous on $(0,\infty )$.

Hence, all the conditions of Definition 6 are satisfied.

Definition 7.

Assume that $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is an $\mathcal{IFM}$- space defined over Λ and consider $\left\{{\varsigma }_n\right\}$ a sequence in χ. Then $\left\{{\varsigma }_n\right\}$ converges to a point $\varsigma \in \chi$ if for any $†\succ \theta$ and $⋎\in (0,1)$, there exists $n_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill {\Pi }_{\Lambda }({\varsigma }_n,\varsigma ,†)>1-⋎\mathit{and}{\aleph }_{\Lambda }({\varsigma }_n,\varsigma ,†)<⋎,\end{array}$$

for all $n\ge n_0$. We write, ${\varsigma }_n\to \varsigma$ as $n\to \infty$ or ${lim}_{n\to \infty }{\varsigma }_n=\varsigma$.

Definition 8.

Assume that $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is an $\mathcal{IFM}$-space defined over Λ, and $\left\{{\varsigma }_n\right\}$ is a sequence contained in χ. Then,

1.

A sequence $\left\{{\varsigma }_n\right\}$ within χ is called a Cauchy sequence if, for every $†\succ \theta$ and any $⋎\in (0,1)$, there exists $n_0\in \mathbb{N}$ for which ${\Pi }_{\Lambda }({\varsigma }_n,{\varsigma }_m,†)>1-⋎$ and ${\aleph }_{\Lambda }({\varsigma }_n,{\varsigma }_m,†)<⋎$ for all $m,n\ge n_0$;

2.

The space χ is called complete if every Cauchy sequence converges to some point in χ.

Having defined convergence and completeness, we turn our attention to the topological structure induced by the intuitionistic fuzzy metric. The next result shows that open balls in this space are indeed open sets, a fundamental property that supports the development of a Hausdorff topology.

Definition 9.

Assume that $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is an $\mathcal{IFM}$-space defined over Λ. An open ball $B(\varsigma ,⋎,†)$ in $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is defined as

$$\begin{array}{c}\hfill B(\varsigma ,⋎,†)=\left\{\eta \in \chi :{\Pi }_{\Lambda }(\varsigma ,\eta ,†)>1-⋎\mathrm{and}{\aleph }_{\Lambda }(\varsigma ,\eta ,†)<⋎\right\},\end{array}$$

for any $†\in {\Lambda }_{+}$ and $⋎\in (0,1)$.

Theorem 4. 

An open ball $B(\varsigma ,⋎,†)$ in $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is an open set if, for every $†\in \Lambda$, there exists ${†}_0$ in Λ such that $\theta \prec {†}_0\prec †$.

Proof. 

Let $B(\varsigma ,⋎,†)$ denote an open ball centered at $\varsigma$ with radius ⋎ relative to † in an $\mathcal{IFM}$-space $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$. Let $\eta \in B(\varsigma ,⋎,†)$, which implies that ${\Pi }_{\Lambda }(\varsigma ,\eta ,†)>1-⋎$ and ${\aleph }_{\Lambda }(\varsigma ,\eta ,†)<⋎$.

Let us take ${†}_0$ in $\Lambda$, with $\theta \prec {†}_0\prec †$ such that ${\Pi }_{\Lambda }(\varsigma ,\eta ,{†}_0)>1-⋎$ and ${\aleph }_{\Lambda }(\varsigma ,\eta ,{†}_0)<⋎$. Let ${⋎}_0={\Pi }_{\Lambda }(\varsigma ,\eta ,{†}_0)$. Then, we have ${⋎}_0>1-⋎$. Since ${⋎}_0>1-⋎$, there exists $\omega \in (0,1)$ such that ${⋎}_0>1-\omega >1-⋎$. Now, for a given ${⋎}_0$ and $\omega$ such that ${⋎}_0>1-\omega$, there exist ${⋎}_1,{⋎}_2\in (0,1)$ such that ${⋎}_0\ast {⋎}_1>1-\omega$ and $(1-{⋎}_0)♢(1-{⋎}_2)<\omega$. Put ${⋎}_3=max\{{⋎}_1,{⋎}_2\}$ and take the open ball $B(\eta ,1-{⋎}_3,†-{†}_0)$. We have to show the following:

$$B(\eta ,1-{⋎}_3,†-{†}_0)\subset B(\varsigma ,⋎,†).$$

Let $\upsilon \in B(\eta ,1-{⋎}_3,†-{†}_0)$, which implies that ${\Pi }_{\Lambda }(\eta ,\upsilon ,†-{†}_0)>{⋎}_3$ and ${\aleph }_{\Lambda }(\eta ,\upsilon ,†-{†}_0)<1-{⋎}_3$. Therefore,

$$\begin{array}{c}\hfill {\Pi }_{\Lambda }(\varsigma ,\upsilon ,†)\ge {\Pi }_{\Lambda }(\varsigma ,\eta ,{†}_0)\ast {\Pi }_{\Lambda }(\eta ,\upsilon ,†-{†}_0)\ge {⋎}_0\ast {⋎}_3\ge {⋎}_0\ast {⋎}_1>1-\omega >1-⋎\end{array}$$

and

$$\begin{array}{c}\hfill {\aleph }_{\Lambda }(\varsigma ,\upsilon ,†)\le {\aleph }_{\Lambda }(\varsigma ,\eta ,{†}_0)♢{\aleph }_{\Lambda }(\eta ,\upsilon ,†-{†}_0)\le (1-{⋎}_0)♢(1-{⋎}_3)\le (1-{⋎}_0)♢(1-{⋎}_2)<\omega <⋎.\end{array}$$

This implies that $\upsilon \in B(\varsigma ,⋎,†)$, and hence,

$$B(\eta ,1-{⋎}_3,†-{†}_0)\subset B(\varsigma ,⋎,†),$$

which completes the proof. □

It is important to note that not every open set in $\chi$ can be represented as an open ball, due to the algebraic constraints on the parameter †. This subtlety motivates a deeper investigation into the topological properties of $\mathcal{IFM}$-spaces, including their separation axioms.

Remark 4. 

It is worth noting that not every open set in χ corresponds to an open ball. This is because the condition for the existence of ${†}_0\in \Lambda$, such that $\theta \prec {†}_0\prec †$ for every $†\in \Lambda$, is not necessarily satisfied for all † within the set Λ.

From now onwards, we assume $\Lambda$ to be a unital $C^{\ast }$-algebra.

Lemma 1. 

Let $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },🟉,♢)$ be an $\mathcal{IFM}$-space over Λ and $c\succ \theta ,c\in \Lambda$. Then, for any $\varsigma \succ \theta ,{\varsigma }^{\ast }c\varsigma \succ \theta$ and $c\succ {\varsigma }^{\ast }c\varsigma ,$ whenever $\parallel \varsigma \parallel <1$.

Proof. 

Since $c\succ \theta$, we have $c=c^{\ast }$ and $\sigma \left(c\right)\subset (0,\infty )$. For $\parallel \varsigma \parallel <1$, the element ${\varsigma }^{\ast }c\varsigma$ is self-adjoint and

$$\parallel {\varsigma }^{\ast }{c\varsigma \parallel \hspace{0.17em}\le \hspace{0.17em}\parallel \varsigma \parallel }^2\parallel c\parallel \hspace{0.17em}<\hspace{0.17em}\parallel c\parallel .$$

Hence, $c-{\varsigma }^{\ast }c\varsigma$ is self-adjoint and positive.

Let $\zeta$ be any character on $\Lambda$. Then,

$$\zeta \left({\varsigma }^{\ast }c\varsigma \right)=\zeta \left(c\right){\left|\zeta \left(\varsigma \right)\right|}^2\ge 0,$$

which implies that $\sigma \left({\varsigma }^{\ast }c\varsigma \right)\subset [0,\infty )$, and therefore, ${\varsigma }^{\ast }c\varsigma \succ \theta$.

Moreover,

$$\zeta (c-{\varsigma }^{\ast }c\varsigma )=\zeta \left(c\right)\left(1-{\left|\zeta \left(\varsigma \right)\right|}^2\right)>0,$$

since $\parallel \varsigma \parallel \hspace{0.17em}<1$ implies that $\left|\zeta \right(\varsigma \left)\right|\hspace{0.17em}<1$. Thus, $\sigma (c-{\varsigma }^{\ast }c\varsigma )\subset (0,\infty )$, and hence, $c\succ {\varsigma }^{\ast }c\varsigma$. □

Lemma 2. 

Let $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ be an $\mathcal{IFM}$-space over commutative algebra Λ. Then, for each $c_1\succ \theta ,c_2\succ \theta ,c_1,c_2\succ \theta$, there exists $c\succ \theta ,c\succ \theta$ such that $c^{\ast }{c}_{2}c\prec c_1$ and $c^{\ast }{c}_{2}c\prec c_2$.

Proof. 

As $c_2\succ \theta$, find $\varsigma \succ \theta$ with $\parallel \varsigma \parallel <1$; we have from the previous lemma that ${\varsigma }^{\ast }{c}_2\varsigma \prec c_2$. Choose $n_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill n_0>max\left\{\parallel c_1\parallel ,\underset{\zeta \in \Delta }{sup}{\left|\zeta \left(c_1{c}_2\right)\right|}^{{\displaystyle \frac{1}{2}}}\right\}.\end{array}$$

Let $c={\displaystyle \frac{c_1}{n_0}}$ so $\parallel c\parallel ={\displaystyle \frac{\parallel c_1\parallel }{n_0}}<1$, which implies that $c^{\ast }{c}_{2}c\prec c_2$. Also, we claim that $c^{\ast }{c}_{2}c\prec c_1$, because ${(c_1-c^{\ast }{c}_{2}c)}^{\ast }=c_1-c^{\ast }{c}_{2}c$ and

$$\begin{array}{ccc}\hfill \zeta (c_1-c^{\ast }{c}_{2}c)& =& \zeta (c_1-\zeta \left(c^{\ast }\right)\zeta \left(c_2\right)\zeta \left(c\right)\hfill \\ & =& \zeta (c_1-\zeta \left(c_2\right)|\zeta \left(c\right){|}^2\hfill \\ & =& \zeta \left(c_1\right)-\zeta \left(c_2\right){\left|\zeta \left({\displaystyle \frac{c_1}{n_0}}\right)\right|}^2\hfill \\ & =& \zeta \left(c_1\right)-\zeta \left(c_2\right){\displaystyle \frac{|\zeta \left(c_1\right){|}^2}{n_0^2}}\hfill \\ & =& \zeta \left(c_1\right)\left(1-{\displaystyle \frac{\zeta \left(c_1\right)\zeta \left(c_2\right)}{n_0^2}}\right)\hfill \\ & =& \zeta \left(c_1\right)\left(1-{\displaystyle \frac{\zeta \left(c_1{c}_2\right)}{n_0^2}}\right)\hfill \end{array}$$

Since $n_0^2>\zeta \left(c_1{c}_2\right)$, we get $1-{\displaystyle \frac{\zeta \left(c_1{c}_2\right)}{n_0^2}}>0$. Thus, $\zeta (c_1-c^{\ast }{c}_{2}c)=⋉\in {\mathbb{R}}_{+}$. This implies that $⋉\in \sigma (c_1-c^{\ast }{c}_{2}c)\subset {\mathbb{R}}_{+}\setminus \left\{0\right\}$. Hence, this is proven. □

With these foundational lemmas established, we now construct a topology on $\chi$ based on the family of open balls. The following theorem confirms that this collection indeed forms a topology, paving the way for further topological analyses.

Theorem 5. 

Let $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ be an $\mathcal{IFM}$-space over Λ. If we define

$$\begin{array}{c}\hfill {\tau }_{\Lambda }=\{A\subset \chi :{\varsigma }_1\in \mathit{A}\mathit{if}\mathit{there}\mathit{exist}⋎\in (0,1)\mathit{and}†\succ \theta ,\mathit{such}\mathit{that}B({\varsigma }_1,⋎,†)\subset A\},\end{array}$$

then ${\tau }_{\Lambda }$ forms a topology on χ.

Proof. 

(i)

If ${\varsigma }_1\in \varphi$, it follows that $\varphi =B({\varsigma }_1,⋎,†)\subseteq \varphi$. Therefore, $\varphi \in {\tau }_{\Lambda }$. So, for each ${\varsigma }_1\in \chi$, $†\succ \theta$ and $⋎\in (0,1)$ such that $B({\varsigma }_1,⋎,†)\subset \chi$, we have $\chi \in {\tau }_{\Lambda }$.

(ii)

Let $A_k\in {\tau }_{\Lambda }$ and ${\varsigma }_1\in {\cup }_{k\in I}{A}_k$. Then, for some $k_0\in I$, we get ${\varsigma }_1\in A_{k_0}$. So, by definition, there exist $⋎\in (0,1)$ and $†\succ \theta$ such that $B({\varsigma }_1,⋎,†)\subseteq A_{k_0}$. Since $A_{k_0}\subset {\cup }_{k\in I}{A}_k$, we have $B({\varsigma }_1,⋎,†)\subset {\cup }_{k\in I}{A}_k$, which means that ${\cup }_{k\in I}{A}_k\in {\tau }_{\Lambda }$.

(iii)

Assume that $A,B\in {\tau }_{\Lambda }$ and ${\varsigma }_1\in A\cap B$. Consequently, ${\varsigma }_1\in A$ and ${\varsigma }_1\in B$, so there exist ${⋎}_1,{⋎}_2\in (0,1)$ and ${†}_1,{†}_2\succ \theta$ such that $B({\varsigma }_1,{⋎}_1,{†}_1)\subset A$ and $B({\varsigma }_1,{⋎}_2,{†}_2)\subset B$. By choosing $†\succ \theta$ such that $†\prec {†}_1,†\prec {†}_2$ (that is, ${†}_1-†\succ \theta ,{†}_2-†\succ \theta$) and $⋎=min\{{⋎}_1,{⋎}_2\}$, we have $B({\varsigma }_1,⋎,†)\subset B({\varsigma }_1,{⋎}_1,{†}_1)$; this is because, for any $\upsilon \in B({\varsigma }_1,⋎,†)$, we have ${\Pi }_{\Lambda }({\varsigma }_1,\upsilon ,†)>1-⋎$ and ${\aleph }_{\Lambda }({\varsigma }_1,\upsilon ,†)<r.$

$$\begin{array}{c}\hfill {\Pi }_{\Lambda }({\varsigma }_1,\upsilon ,{†}_1)\ge {\Pi }_{\Lambda }({\varsigma }_1,\upsilon ,†)\ast {\Pi }_{\Lambda }(\upsilon ,\upsilon ,{†}_1-†)\ge (1-⋎)\ast \left(1\right)=1-⋎>1-{⋎}_1.\end{array}$$

Similarly, one can show that ${\aleph }_{\Lambda }({\varsigma }_1,\upsilon ,{†}_1)<{⋎}_1$, which implies that $\upsilon \in B({\varsigma }_1,{⋎}_1,{†}_1)$, and hence, $B({\varsigma }_1,⋎,†)\subset B({\varsigma }_1,{⋎}_1,{†}_1)$. Also, $B({\varsigma }_1,⋎,†)\subset B({\varsigma }_1,{⋎}_2,{†}_2)$; thus,

$$\begin{array}{c}\hfill B({\varsigma }_1,⋎,†)\subset B({\varsigma }_1,{⋎}_1,{†}_1)\cap B({\varsigma }_1,{⋎}_2,{†}_2)\subset A\cap B.\end{array}$$

Hence, $A\cap B\in {\tau }_{\Lambda }$ and the proof is completed.

□

Equipped with a topology, we next examine the separation properties of $\mathcal{IFM}$-spaces. The following theorem demonstrates that such spaces are Hausdorff, an important feature for ensuring the uniqueness of limits and fixed points. In the following, we extend the Hausdorff property given in [4,35] for the class of $\mathcal{IFM}$-spaces over $\mathcal{BA}$.

Theorem 6. 

Every $\mathcal{IFM}$-space $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is Hausdorff.

Proof. 

Let ${\varsigma }_1$ and ${\varsigma }_2$ be any two distinct points in $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,\ast ,♢)$. Since ${\varsigma }_1\ne {\varsigma }_2$, by the definition of an $\mathcal{IFM}$-space, we have ${\Pi }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)<1$ and ${\aleph }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)>0$ for $†\succ \theta$. Let ${\Pi }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)={⋎}_1$ and ${\aleph }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)={⋎}_2$ for some ${⋎}_1,{⋎}_2\in (0,1)$. Let $⋎=max\{{⋎}_1,1-{⋎}_2\}$. Then, for each ${⋎}_0\in (r,1)$, there exist ${⋎}_3,{⋎}_4\in (0,1)$ such that ${⋎}_3\ast {⋎}_3>{⋎}_0$ and $(1-{⋎}_4)♢(1-{⋎}_4)<(1-{⋎}_0)$. Let ${⋎}_5=max\{{⋎}_3,{⋎}_4\}$. Now consider two open sets $B({\varsigma }_1,1-{⋎}_5,†/2)$ and $B({\varsigma }_2,1-{⋎}_5,†/2)$. We have to show that

$$B({\varsigma }_1,1-{⋎}_5,†/2)\cap B({\varsigma }_2,1-{⋎}_5,†/2)=\varphi .$$

Suppose by contradiction,

$$B({\varsigma }_1,1-{⋎}_5,†/2)\cap B({\varsigma }_2,1-{⋎}_5,†/2)\ne \varphi .$$

Then, there exists some $\eta \in B({\varsigma }_1,1-{⋎}_5,†/2)\cap B({\varsigma }_2,1-{⋎}_5,†/2)$, which implies that $\eta \in B({\varsigma }_1,1-{⋎}_5,†/2)$ and $\eta \in B({\varsigma }_2,1-{⋎}_5,†/2)$. By definition, we can write it as

$${\Pi }_{\Lambda }({\varsigma }_1,\eta ,†/2)>{⋎}_5,{\aleph }_{\Lambda }({\varsigma }_1,\eta ,†/2)<1-{⋎}_5\mathrm{and}{\Pi }_{\Lambda }({\varsigma }_2,\eta ,†/2)>{⋎}_5,{\aleph }_{\Lambda }({\varsigma }_2,\eta ,†/2)<1-{⋎}_5$$

Now,

$$\begin{array}{ccc}\hfill {⋎}_1& =& {\Pi }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)\ge {\Pi }_{\Lambda }({\varsigma }_1,\eta ,†/2)\ast {\Pi }_{\Lambda }(\eta ,{\varsigma }_2,†/2)>{⋎}_5\ast {⋎}_5\ge {⋎}_3\ast {⋎}_3>{⋎}_0>r\ge {⋎}_1\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {⋎}_2& =& {\aleph }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)\le {\aleph }_{\Lambda }({\varsigma }_1,\eta ,†/2)♢{\aleph }_{\Lambda }(\eta ,{\varsigma }_2,†/2)<(1-{⋎}_5)♢(1-{⋎}_5)\le (1-{⋎}_4)♢(1-{⋎}_4)\hfill \\ & & <1-{⋎}_0<1-⋎\le {⋎}_2\hfill \end{array}$$

form a contradiction. Thus,

$$B({\varsigma }_1,\eta ,†/2)\cap B({\varsigma }_2,\eta ,†/2)=\varphi$$

which implies that $\chi$ is a Hausdorff space. □

We now consider boundedness and compactness in $\mathcal{IFM}$-spaces. The next theorem shows that compact subsets are both closed and bounded in the intuitionistic fuzzy sense, aligning with classical topological intuition.

Definition 10.

Let $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ denotes an $\mathcal{IFM}$-space defined on Λ. A subset B of χ is referred to as IF-bounded over Λ provided there exist $†\succ \theta$ and $⋎\in (0,1)$ such that ${\Pi }_{\Lambda }(\varsigma ,\eta ,†)>1-⋎$ and ${\aleph }_{\Lambda }(\varsigma ,\eta ,†)<⋎$ for all $\varsigma ,\eta \in B$.

Theorem 7. 

Every compact subset of an $\mathcal{IFM}$-space $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is closed and IF-bounded over Λ.

Proof. 

Let $B\subset \chi$ be compact. Fix $†\succ \theta$ and $⋎\in (0,1)$. Consider the family of open sets $\left\{\mathcal{B}\right(\varsigma ,⋎,†):\varsigma \in B\}$ forming an open cover of B. Since this collection is indeed an open cover, there exist ${\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_j\in B$ such that

$$B\subset {\cup }_{i=1}^j\mathcal{B}({\varsigma }_i,⋎,†),†\succ \theta .$$

For any $\varsigma ,\eta \in B$, there exist some $1\le k,l\le j$ such that $\varsigma \in \mathcal{B}({\varsigma }_k,⋎,†)$ and $\eta \in \mathcal{B}({\varsigma }_l,⋎,†)$. Then, by definition, we have ${\Pi }_{\Lambda }({\varsigma }_k,\varsigma ,†)>1-⋎$, ${\aleph }_{\Lambda }({\varsigma }_k,\varsigma ,†)<⋎$, and ${\Pi }_{\Lambda }({\varsigma }_l,\eta ,†)>1-⋎$${\aleph }_{\Lambda }({\varsigma }_l,\eta ,†)<⋎$ for $†\succ \theta$. Now, let $\gamma =min\{{\Pi }_{\Lambda }({\varsigma }_k,{\varsigma }_l,†):1\le k,l\le j\}$ and $\lambda =max\{{\aleph }_{\Lambda }({\varsigma }_k,{\varsigma }_l,†):1\le k,l\le j\}$. So we have

$$\begin{array}{ccc}\hfill {\Pi }_{\Lambda }(\varsigma ,\eta ,3†)& \ge & {\Pi }_{\Lambda }(\varsigma ,{\varsigma }_k,†)\ast {\Pi }_{\Lambda }({\varsigma }_k,{\varsigma }_l,†)\ast {\Pi }_{\Lambda }({\varsigma }_l,\eta ,†)\ge 1-⋎\ast \gamma \ast 1-⋎,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\aleph }_{\Lambda }(\varsigma ,\eta ,3†)& \le & {\aleph }_{\Lambda }(\varsigma ,{\varsigma }_k,†)♢{\aleph }_{\Lambda }({\varsigma }_k,{\varsigma }_l,†)♢{\aleph }_{\Lambda }({\varsigma }_l,\eta ,†)\le ⋎♢\lambda ♢⋎.\hfill \end{array}$$

Let $3†={†}^{\ast }$ and choose ${⋎}_1,{⋎}_2\in (0,1)$ such that $1-⋎\ast \gamma \ast 1-⋎>1-{⋎}_1$ and $⋎♢\lambda ♢⋎<{⋎}_2$. Let ${⋎}_0=max\{{⋎}_1,{⋎}_2\}$. Then,

$$\begin{array}{ccc}\hfill {\Pi }_{\Lambda }(\varsigma ,\eta ,{†}^{\ast })& \ge & 1-⋎\ast \gamma \ast 1-⋎>1-{⋎}_1>1-{⋎}_0\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\aleph }_{\Lambda }(\varsigma ,\eta ,{†}^{\ast })& \le & ⋎♢\lambda ♢⋎<{⋎}_2<{⋎}_0\hfill \end{array}$$

Thus, B is IF-bounded. On the basis of Theorem 6, we know that each $\mathcal{IFM}$-space over $\Lambda$ possesses the Hausdorff property. Therefore, given that compact subsets in a Hausdorff space are closed, B must also be closed. □

Having established the key topological properties, we return to the study of sequences. The following lemma reveals the monotonicity properties of the fuzzy metric functions, which are useful in analyzing contractive mappings.

Theorem 8. 

Consider an $\mathcal{IFM}$-space $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ over Λ. We say that a sequence $\left\{{\varsigma }_n\right\}$ in χ converges to $\varsigma \in \chi$ if ${\Pi }_{\Lambda }({\varsigma }_n,\varsigma ,†)\to 1$ and ${\aleph }_{\Lambda }({\varsigma }_n,\varsigma ,†)\to 0$, as $n\to \infty$ for each $†\succ \theta$.

Proof. 

Assume that the sequence $\left\{{\varsigma }_n\right\}$ converges to $\varsigma$ in $\chi$. Then, for every $†\succ \theta$ and each $⋎\in (0,1)$, there exists an index $n_0\in \mathbb{N}$ such that ${\Pi }_{\Lambda }({\varsigma }_n,\varsigma ,†)>1-⋎$ and ${\aleph }_{\Lambda }({\varsigma }_n,\varsigma ,†)<⋎$ for all $n\ge n_0$. Hence, ${\Pi }_{\Lambda }({\varsigma }_n,\varsigma ,†)\to 1$ and ${\aleph }_{\Lambda }({\varsigma }_n,\varsigma ,†)\to 0$ as $n\to \infty$.

Conversely, suppose that ${\Pi }_{\Lambda }({\varsigma }_n,\varsigma ,†)\to 1$ and ${\aleph }_{\Lambda }({\varsigma }_n,\varsigma ,†)\to 0$ as $n\to \infty$. Then, for any $†\succ \theta$ and $⋎\in (0,1)$, there exists $n_0\in \mathbb{N}$ such that $⋎>1-{\Pi }_{\Lambda }({\varsigma }_n,\varsigma ,†)$ and $⋎>{\aleph }_{\Lambda }({\varsigma }_n,\varsigma ,†)$ for all $n\ge n_0$, which implies that

$$\begin{array}{c}\hfill {\Pi }_{\Lambda }({\varsigma }_n,\varsigma ,†)>1-⋎\mathrm{and}{\aleph }_{\Lambda }({\varsigma }_n,\varsigma ,†)<⋎\end{array}$$

for all $n\ge n_0$. Thus, ${\varsigma }_n\to \varsigma \in \chi$ as $n\to \infty$. □

Lemma 3. 

Let $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ be an $\mathcal{IFM}$-space over Λ. Then, ${\Pi }_{\Lambda }(\varsigma ,\eta ,.):\Lambda \to [0,1]$ behaves as a non-decreasing function, while ${\aleph }_{\Lambda }(\varsigma ,\eta ,.):\Lambda \to [0,1]$ is non-increasing.

Proof. 

On the contrary, suppose that the function ${\Pi }_{\Lambda }(\varsigma ,\eta ,.)$ is decreasing and the function ${\aleph }_{\Lambda }(\varsigma ,\eta ,.)$ is increasing. That is, ${\Pi }_{\Lambda }(\varsigma ,\eta ,\omega )<{\Pi }_{\Lambda }(\varsigma ,\eta ,†)$ for all $\omega ,†\succ \theta$ with $†\preccurlyeq \omega$ and ${\aleph }_{\Lambda }(\varsigma ,\eta ,†)<{\aleph }_{\Lambda }(\varsigma ,\eta ,\omega )$ for all $\omega ,†\succ \theta$ with $†\preccurlyeq \omega$. Then,

$$\begin{array}{c}\hfill {\Pi }_{\Lambda }(\varsigma ,\eta ,†)={\Pi }_{\Lambda }(\varsigma ,\eta ,†)\ast 1={\Pi }_{\Lambda }(\varsigma ,\eta ,†)\ast {\Pi }_{\Lambda }(\eta ,\eta ,s-†)\le {\Pi }_{\Lambda }(\varsigma ,\eta ,\omega )<{\Pi }_{\Lambda }(\varsigma ,\eta ,†),\end{array}$$

which is a contradiction. Also,

$$\begin{array}{c}\hfill {\aleph }_{\Lambda }(\varsigma ,\eta ,†)={\aleph }_{\Lambda }(\varsigma ,\eta ,†)♢0={\aleph }_{\Lambda }(\varsigma ,\eta ,†)♢{\aleph }_{\Lambda }(\eta ,\eta ,s-†)\ge {\aleph }_{\Lambda }(\varsigma ,\eta ,\omega )>{\aleph }_{\Lambda }(\varsigma ,\eta ,†),\end{array}$$

which is again a contradiction. Thus, the functions ${\Pi }_{\Lambda }(\varsigma ,\eta ,.)$ and ${\aleph }_{\Lambda }(\varsigma ,\eta ,.)$ are non-decreasing and non-increasing, respectively. □

We now introduce the concept of ${\partial }_{\Lambda }$-Cauchy sequences and ${\partial }_{\Lambda }$-completeness, which are tailored to the algebraic structure of $\Lambda$. This leads naturally to the definition of intuitionistic fuzzy Z-contractions, a generalized class of mappings suited for fixed-point analyses.

Definition 11.

A sequence $\left\{{\varsigma }_n\right\}$ in an $\mathcal{IFM}$-space $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ over Λ is called a _Λ-Cauchy sequence if ${lim}_{n\to \infty }{\Pi }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+l},†)=1$ and ${lim}_{n\to \infty }{\aleph }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+l},†)=0,$ for each $l>0$ and $†\succ \theta$. χ is called _Λ-complete if every Cauchy sequence is convergent in χ.

Definition 12.

Let $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ be an $\mathcal{IFM}$-space over Λ and $T:\chi \to \chi$ be a self mapping. Then, T is said to satisfy an intuitionistic fuzzy Z-contraction over Λ if

$$\begin{array}{c}\hfill \rho \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(T\varsigma ,T\eta ,†)}}-1,{\displaystyle \frac{1}{{\Pi }_{\Lambda }(\varsigma ,\eta ,†)}}-1\right)\ge 0\end{array}$$

and

$$\begin{array}{c}\hfill \rho \left({\aleph }_{\Lambda }(T\varsigma ,T\eta ,†),{\aleph }_{\Lambda }(\varsigma ,\eta ,†)\right)\ge 0\end{array}$$

for each $\varsigma ,\eta \in \chi$ and $†\succ \theta$ and $\rho \in Z$.

With these definitions in place, we are now prepared to present our main fixed-point theorem. The following result guarantees the existence and uniqueness of fixed points for intuitionistic fuzzy Z-contractions in complete $\mathcal{IFM}$-spaces.

Definition 13.

Let $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ be an $\mathcal{IFM}$-space over Λ, and consider a sequence $\left\{{\varsigma }_n\right\}$ in χ. The sequence $\left\{{\varsigma }_n\right\}$ is termed as an intuitionistic fuzzy Z-contractive sequence over Λ provided that the following holds:

$$\begin{array}{c}\hfill \rho \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†)}}-1,{\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+1},†)}}-1\right)\ge 0\end{array}$$

and

$$\begin{array}{c}\hfill \rho \left({\aleph }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†),{\aleph }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+1},†)\right)\ge 0\end{array}$$

for all $†\succ \theta$, $n\in \mathbb{N}$.

Now, let us prove the following main theorem related to fixed points.

Theorem 9. 

Let $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ be a _Λ-complete $\mathcal{IFM}$-space over Λ in which intuitionistic fuzzy sequences are _Λ-Cauchy. Assume that $T:\chi \to \chi$ is an intuitionistic fuzzy Z-contraction mapping over Λ. Then, there exists a unique fixed point of T in χ.

Proof. 

Let ${\varsigma }_0$ be fixed in $\chi$, and define the iteration by ${\varsigma }_{n+1}=T\left({\varsigma }_n\right)$ for $n\in \mathbb{N}$, that is, ${\varsigma }_n=T^n\left({\varsigma }_0\right)$. For $†\succ \theta$, it holds that

$$\begin{array}{c}\hfill \rho \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(T{\varsigma }_0,T^2\left({\varsigma }_0\right),†)}}-1,{\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_0,T{\varsigma }_0,†)}}-1\right)\ge 0\end{array}$$

implying

$$\begin{array}{c}\hfill \rho \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)}}-1,{\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_0,{\varsigma }_1,†)}}-1\right)\ge 0\end{array}$$

and

$$\begin{array}{c}\hfill \rho \left({\aleph }_{\Lambda }(T{\varsigma }_0,T^2\left({\varsigma }_0\right),†),{\aleph }_{\Lambda }({\varsigma }_0,T{\varsigma }_0,†)\right)\ge 0\end{array}$$

which shows that

$$\begin{array}{c}\hfill \rho \left({\aleph }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†),{\aleph }_{\Lambda }({\varsigma }_0,{\varsigma }_1,†)\right)\ge 0\end{array}$$

Similarly, we have

$$\begin{array}{c}\hfill \rho \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_2,{\varsigma }_3,†)}}-1,{\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)}}-1\right)\ge 0\end{array}$$

and

$$\begin{array}{c}\hfill \rho \left({\aleph }_{\Lambda }({\varsigma }_2,{\varsigma }_3,†),{\aleph }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)\right)\ge 0.\end{array}$$

Generalizing for $n\in \mathbb{N}$, we get

$$\begin{array}{c}\hfill \rho \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†)}}-1,{\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+1},†)}}-1\right)\ge 0\end{array}$$

and

$$\begin{array}{c}\hfill \rho \left({\aleph }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†),{\aleph }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+1},†)\right)\ge 0.\end{array}$$

Hence, $\left\{{\varsigma }_n\right\}$ is an intuitionistic fuzzy Z-contractive sequence, and by assumption, it is a _Λ-Cauchy sequence. Since $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is a _Λ-complete space, there exists $\eta \in \chi$ such that ${lim}_{n\to \infty }{\Pi }_{\Lambda }({\varsigma }_n,\eta ,†)=1$ and ${lim}_{n\to \infty }{\aleph }_{\Lambda }({\varsigma }_n,\eta ,†)=0$ for each $†\succ \theta$. Next, we show that $\eta$ serves as a fixed point of T. By the contraction property, we can write

$$\begin{array}{ccc}\hfill 0& \le & \underset{n\to \infty }{lim}\rho \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(T{\varsigma }_n,T\eta ,†)}}-1,{\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_n,\eta ,†)}}-1\right)\hfill \\ & \le & \underset{n\to \infty }{lim}\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_n,\eta ,†)}}-{\displaystyle \frac{1}{{\Pi }_{\Lambda }(T{\varsigma }_n,T\eta ,†)}}\right)\hfill \\ & =& {\displaystyle \frac{1}{{lim}_{n\to \infty }{\Pi }_{\Lambda }({\varsigma }_n,\eta ,†)}}-{\displaystyle \frac{1}{{lim}_{n\to \infty }{\Pi }_{\Lambda }(T{\varsigma }_n,T\eta ,†)}}\hfill \\ & \le & 1-{\displaystyle \frac{1}{{lim}_{n\to \infty }{\Pi }_{\Lambda }(T{\varsigma }_n,T\eta ,†)}}\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\Pi }_{\Lambda }(T{\varsigma }_n,T\eta ,†)\ge 1.\end{array}$$

The only possibility is

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\Pi }_{\Lambda }(T{\varsigma }_n,T\eta ,†)=1.\end{array}$$

(1)

Using same arguments, we find out that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\aleph }_{\Lambda }(T{\varsigma }_n,T\eta ,†)=0.\end{array}$$

(2)

By (1), (2), and (8), ${lim}_{n\to \infty }T{\varsigma }_n=T\eta$. Since $T{\varsigma }_n={\varsigma }_{n+1}$, we get that ${lim}_{n\to \infty }{\varsigma }_{n+1}=T\eta$. But ${lim}_{n\to \infty }{\varsigma }_n=\eta$, so we have $T\eta =\eta$.

Take $\upsilon$ to be a fixed point of T different from $\eta$. So, $T\eta =\eta$ and $T\upsilon =\upsilon$. Then,

$$\begin{array}{c}\hfill \rho \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(T\eta ,T\upsilon ,†)}}-1,{\displaystyle \frac{1}{{\Pi }_{\Lambda }(\eta ,\upsilon ,†)}}-1\right)\ge 0\end{array}$$

$$\begin{array}{c}\hfill \rho \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(\eta ,\upsilon ,†)}}-1,{\displaystyle \frac{1}{{\Pi }_{\Lambda }(\eta ,\upsilon ,†)}}-1\right)\ge 0\end{array}$$

which gives a contradiction from Remark 3. Thus, $\eta =\upsilon$, which completes the proof. □

As a special case of the above result, we obtained a simpler contraction theorem in which the simulation function $\rho$ is replaced by a linear factor $\kappa \in (0,1)$. This corollary bridges our work with classical Banach contraction principles in fuzzy settings.

Theorem 10. 

Consider a _Λ-complete $\mathcal{IFM}$-space $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ over Λ. Consider a mapping $T:\chi \to \chi$ such that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Pi }_{\Lambda }(T\varsigma ,T\eta ,†)}}-1\le ⋉\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(\varsigma ,\eta ,†)}}-1\right),\end{array}$$

and

$$\begin{array}{c}\hfill {\aleph }_{\Lambda }(T\varsigma ,T\eta ,†)\le ⋉{\aleph }_{\Lambda }(\varsigma ,\eta ,†),\end{array}$$

for all $\varsigma ,\eta \in \chi ,⋉\in (0,1)$ and $†\succ \theta$. In this case, T possesses a unique fixed point in χ.

Proof. 

Let ${\varsigma }_0$ be a fixed element in $\chi$ and set ${\varsigma }_{n+1}=T\left({\varsigma }_n\right)$ for all $n\in \mathbb{N}$, that is, ${\varsigma }_n=T^n\left({\varsigma }_0\right)$. For $†\succ \theta$, it holds that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Pi }_{\Lambda }(T{\varsigma }_0,T^2\left({\varsigma }_0\right),†)}}-1\le ⋉\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_0,T{\varsigma }_0,†)}}-1\right)\end{array}$$

implying

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)}}-1\le ⋉\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_0,{\varsigma }_1,†)}}-1\right)\end{array}$$

Similarly,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_2,{\varsigma }_3,†)}}-1\le ⋉\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)}}-1\right)\end{array}$$

Generalizing for $n\in \mathbb{N}$, we get

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†)}}-1\le ⋉\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+1},†)}}-1\right).\end{array}$$

(3)

In a similar pattern,

$$\begin{array}{c}\hfill {\aleph }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†)\le ⋉{\aleph }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+1},†)\end{array}$$

(4)

Hence, $\left\{{\varsigma }_n\right\}$ is an intuitionistic fuzzy Z-contractive sequence. Now, from (3), we can further proceed as follows:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†)}}-1& \le & ⋉\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+1},†)}}-1\right)\hfill \\ & \le & {⋉}^2\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_{n-1},{\varsigma }_n,†)}}-1\right)\hfill \\ & ⋮& \\ & \le & {⋉}^n\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)}}-1\right)\hfill \\ & \to & 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\displaystyle \frac{1}{{\Pi }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†)}}-1=0\end{array}$$

which implies that

$$\begin{array}{c}\hfill {\Pi }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†)\to 1\mathrm{as}n\to \infty .\end{array}$$

and

$$\begin{array}{ccc}\hfill {\aleph }_{\Lambda }({\varsigma }_{n+1},{\varsigma }_{n+2},†)& \le & ⋉{\aleph }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+1},†)\hfill \\ & \le & {⋉}^2{\aleph }_{\Lambda }({\varsigma }_{n-1},{\varsigma }_n,†)\hfill \\ & ⋮& \\ & \le & {⋉}^n{\aleph }_{\Lambda }({\varsigma }_1,{\varsigma }_2,†)\hfill \\ & \to & 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Now,

$$\begin{array}{ccc}\hfill {\Pi }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+l},†)& \ge & {\Pi }_{\Lambda }\left({\varsigma }_n,{\varsigma }_{n+1},{\displaystyle \frac{†}{l}}\right)🟉{\Pi }_{\Lambda }\left({\varsigma }_{n+1},{\varsigma }_{n+2},{\displaystyle \frac{†}{l}}\right)\ast \dots \ast {\Pi }_{\Lambda }\left({\varsigma }_{n+l-1},{\varsigma }_{n+l},{\displaystyle \frac{†}{l}}\right)\hfill \\ & \to & 1\ast 1\ast \dots \ast 1=1\mathrm{as}n\to \infty ,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\aleph }_{\Lambda }({\varsigma }_n,{\varsigma }_{n+l},†)& \le & {\aleph }_{\Lambda }\left({\varsigma }_n,{\varsigma }_{n+1},{\displaystyle \frac{†}{l}}\right)♢{\aleph }_{\Lambda }\left({\varsigma }_{n+1},{\varsigma }_{n+2},{\displaystyle \frac{†}{l}}\right)♢\dots ♢{\aleph }_{\Lambda }\left({\varsigma }_{n+l-1},{\varsigma }_{n+l},{\displaystyle \frac{†}{l}}\right)\hfill \\ & \to & 0♢0♢\dots ♢0=0\mathrm{as}n\to \infty .\hfill \end{array}$$

Thus, it can be seen that $\left\{{\varsigma }_n\right\}$ forms a _Λ-Cauchy sequence. Given that $(\chi ,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,♢)$ is _Λ-complete, there exists $\eta \in \chi$ to which $\left\{{\varsigma }_n\right\}$ converges, that is, ${lim}_{n\to \infty }{\Pi }_{\Lambda }({\varsigma }_n,\eta ,†)=1$ and ${lim}_{n\to \infty }{\aleph }_{\Lambda }({\varsigma }_n,\eta ,†)=0$ for each $†\succ \theta$. We proceed to show that $\eta$ serves as the fixed point of T. By the contraction property, one can write

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Pi }_{\Lambda }(T\eta ,T{\varsigma }_n,†)}}-1\le ⋉\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(\eta ,{\varsigma }_n,†)}}-1\right)\to 0\mathrm{as}n\to \infty .\end{array}$$

Similarly,

$$\begin{array}{c}\hfill {\aleph }_{\Lambda }(T\eta ,T{\varsigma }_n,†)\le ⋉{\aleph }_{\Lambda }(\eta ,{\varsigma }_n,†)\to 0\mathrm{as}n\to \infty .\end{array}$$

Then, ${lim}_{n\to \infty }{\Pi }_{\Lambda }(T\eta ,T{\varsigma }_n,†)=1$ and ${lim}_{n\to \infty }{\aleph }_{\Lambda }(T\eta ,T{\varsigma }_n,†)=0$ for each $†\succ \theta$, and therefore, ${lim}_{n\to \infty }T{\varsigma }_n=T\eta$, that is, ${lim}_{n\to \infty }{\varsigma }_{n+1}=T\eta$ and then $T\eta =\eta$.

To demonstrate uniqueness, let us assume $T\upsilon =\upsilon$ for some $\upsilon \in Z$. Then, for $†\succ \theta$,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{{\Pi }_{\Lambda }(\eta ,\upsilon ,†)}}-1& =& {\displaystyle \frac{1}{{\Pi }_{\Lambda }(T\eta ,T\upsilon ,†)}}-1\hfill \\ & \le & ⋉\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(\eta ,\upsilon ,†)}}-1\right)\hfill \\ & =& ⋉\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(T\eta ,T\upsilon ,†)}}-1\right)\hfill \\ & \le & {⋉}^2\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(\eta ,\upsilon ,†)}}-1\right)\hfill \\ & ⋮& \\ & \le & {⋉}^n\left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(\eta ,\upsilon ,†)}}-1\right)\hfill \\ & \to & 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Hence, ${\Pi }_{\Lambda }(\eta ,\upsilon ,†)=1$ and then $\eta =\upsilon$. □

These results unify and extend several known theorems in fuzzy fixed point theory, as illustrated by the following corollaries, which recover earlier results by appropriate choices of parameters and structures. Consequently, we deduced the following special cases.

Corollary 1. 

By choosing ${\aleph }_{\Lambda }=1-{\Pi }_{\Lambda }$ and defining $\varsigma ♢\eta =1-\left(\right(1-\varsigma )🟉(1-\eta \left)\right)$, we see that Theorems 4–8 in this paper are reduced to the results proven in [4].

Corollary 2. 

By taking $\Lambda =\mathbb{R}$, the results mentioned in Theorems 4–8 are reduced into the ones given by Park [35].

4. Application to Fractional Volterra Integral Equations and Numerical Simulations

In this section, we unify the analytical framework and numerical investigation for the nonlinear fractional Volterra integral equation. First, we establish the existence and uniqueness of solutions using the intuitionistic fuzzy metric (IFM) structure over a Banach algebra. Then, to validate the theoretical findings, we present a numerical simulation based on a discretized iterative scheme. This combined presentation highlights how the IFM-based contraction framework not only guarantees the well-posedness of the fractional integral equation, but also ensures effective numerical convergence.

4.1. Existence and Uniqueness via IFM-Space

In this subsection, we demonstrate the applicability of our fixed-point results to the study of nonlinear fractional Volterra integral equations. Consider the Caputo fractional integral equation of order $\alpha \in (0,1)$ given by

$$x\left(t\right)=g\left(t\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}F(s,x\left(s\right))ds,t\in [0,1].$$

Let $X=C\left(\right[0,1\left]\right)$ with the supremum metric

$$d(x,y)={\parallel x-y\parallel }_{\infty }.$$

Let $\Lambda =C\left(\right[0,1\left]\right)$ with pointwise positivity, which means that $a\in \Lambda$ satisfies $a\left(t\right)\ge 0$ for all t and

$$\psi \left(a\right)=\underset{t\in [0,1]}{min}a\left(t\right)>0.$$

Define the intuitionistic fuzzy metric components

$${\Pi }_{\Lambda }(x,y,a)=exp\left(-{\displaystyle \frac{{\parallel x-y\parallel }_{\infty }}{\psi \left(a\right)}}\right),{\aleph }_{\Lambda }(x,y,a)=1-{\Pi }_{\Lambda }(x,y,a).$$

Thus, $(X,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda ,🟉,\diamond )$ is an IFM-space.

The operator associated with the integral equation is

$$\left(Tx\right)\left(t\right)=g\left(t\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}F(s,x\left(s\right))ds.$$

A function x is a solution if, and only if, $Tx=x$.

Assume that F satisfies a Lipschitz condition for $L>0$ such that

$$\left|F\right(t,u)-F(t,v\left)\right|\le L|u-v|,t\in [0,1],u,v\in \mathbb{R}.$$

Then,

$${\parallel Tx-Ty\parallel }_{\infty }\le {\displaystyle \frac{L}{\Gamma (\alpha +1)}}{\parallel x-y\parallel }_{\infty }=c{\parallel x-y\parallel }_{\infty },$$

where $c={\displaystyle \frac{L}{\Gamma (\alpha +1)}}<1$.

Hence,

$${\Pi }_{\Lambda }(Tx,Ty,a)\ge exp\left(-{\displaystyle \frac{{c\parallel x-y\parallel }_{\infty }}{\psi \left(a\right)}}\right)$$

and therefore,

$${\displaystyle \frac{1}{{\Pi }_{\Lambda }(Tx,Ty,a)}}-1\le \kappa \left({\displaystyle \frac{1}{{\Pi }_{\Lambda }(x,y,a)}}-1\right),$$

and similarly,

$${\aleph }_{\Lambda }(Tx,Ty,a)\le \kappa {\aleph }_{\Lambda }(x,y,a),$$

for some $\kappa \in (0,1)$.

Thus, T is an intuitionistic fuzzy Z-contraction. Since $(X,{\Pi }_{\Lambda },{\aleph }_{\Lambda },\Lambda )$ is complete, Theorem 10 guarantees a unique fixed point of T, i.e., a unique solution to the integral equation.

4.2. Numerical Simulation

To illustrate the convergence of the iterative process, consider again the fractional Volterra integral equation

$$x\left(t\right)=g\left(t\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}F(s,x\left(s\right))ds,$$

with the test functions

$$g\left(t\right)=t^2,F(t,x)=\lambda x,$$

where $\lambda >0$ satisfies

$$c={\displaystyle \frac{\lambda }{\Gamma (\alpha +1)}}<1.$$

4.2.1. Discretization and Iterative Algorithm

Divide $[0,1]$ into N intervals: $t_k=k/N$ for $k=0,1,\dots ,N$. Denote $x_k^{\left(n\right)}$ as the n-th iterate at node $t_k$. The discrete operator is

$${\left(T{x}^{\left(n\right)}\right)}_k=t_k^2+{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}\sum _{j=0}^{k-1}{(t_k-t_j)}^{\alpha -1}{x}_j^{\left(n\right)}\Delta t,\Delta t={\displaystyle \frac{1}{N}}.$$

Starting with $x^{\left(0\right)}\left(t\right)\equiv 0$, we compute $x^{(n+1)}=T\left(x^{\left(n\right)}\right)$ until $\parallel x^{(n+1)}-x^{\left(n\right)}{\parallel }_{\infty }<\epsilon$.

4.2.2. Numerical Results and Discussion

For the parameters $\alpha =0.75$, $\lambda =0.5$, and $N=100$, the iterative scheme exhibits rapid convergence, as shown in

Table 1. Convergence behavior of the iterative scheme in the $\mathcal{IFM}-\mathcal{BA}$ framework.

Iteration n	$\parallel {\mathit{x}}^{(\mathit{n}+1)}-{\mathit{x}}^{\left(\mathit{n}\right)}{\parallel }_{\mathit{\infty }}$	${\mathbf{\Pi }}_{\mathbf{\Lambda }}\left({\mathit{x}}^{(\mathit{n}+1)},{\mathit{x}}^{\left(\mathit{n}\right)},\mathit{a}\right)$	${\mathbf{\aleph }}_{\mathbf{\Lambda }}\left({\mathit{x}}^{(\mathit{n}+1)},{\mathit{x}}^{\left(\mathit{n}\right)},\mathit{a}\right)$
0	$1.37\times {10}^{-2}$	0.8651	0.1349
1	$5.21\times {10}^{-3}$	0.9276	0.0724
2	$1.88\times {10}^{-3}$	0.9540	0.0460
3	$6.65\times {10}^{-4}$	0.9733	0.0267
4	$2.38\times {10}^{-4}$	0.9896	0.0104

and Figure 1.

The results in Table 1 clearly demonstrate the monotonic improvement in both the fuzzy nearness measure ${\Pi }_{\Lambda }$ and the fuzzy non-nearness measure ${\aleph }_{\Lambda }$. The nearness values approach 1, while the non-nearness values tend to 0, confirming the contraction behavior predicted by Theorem 10. The error $\parallel x^{(n+1)}-x^{\left(n\right)}{\parallel }_{\infty }$ decays exponentially, consistent with the theoretical bound $\parallel x^{(n+1)}-x^{\left(n\right)}{\parallel }_{\infty } \le c^n{\parallel x^{\left(1\right)}-x^{\left(0\right)}\parallel }_{\infty }$ where $c=\lambda /\Gamma (\alpha +1)\approx 0.64<1$. This rapid decay underscores the effectiveness of the $\mathcal{IFM}$-based contraction framework in ensuring numerical stability and fast convergence for fractional integral equations.

Furthermore, the behavior of the intuitionistic fuzzy nearness measure ${\Pi }_{\Lambda }$ and the non-nearness measure ${\aleph }_{\Lambda }$ provides additional insight into the convergence mechanism. During the iteration process, ${\Pi }_{\Lambda }$ exhibits a monotonic increase towards the unit element of the Banach algebra, indicating a progressively stronger nearness between successive approximations. Conversely, ${\aleph }_{\Lambda }$ decreases monotonically towards the zero element, reflecting the gradual elimination of uncertainty and non-nearness as the iteration proceeds. This complementary behavior of ${\Pi }_{\Lambda }$ and ${\aleph }_{\Lambda }$ is in full agreement with the axiomatic structure of intuitionistic fuzzy metric spaces and confirms the stability of the numerical scheme within the IFM–BA framework.

From a practical perspective, the rapid decay observed in Figure 1 highlights the efficiency of the proposed method for solving nonlinear fractional Volterra integral equations. The fast convergence implies that accurate approximations can be achieved with relatively few iterations, thereby reducing the computational cost and improving the numerical reliability. This feature is particularly valuable for fractional dynamical systems, where memory effects often lead to an increased computational complexity. Overall, the numerical findings not only validate the theoretical results, but also demonstrate the applicability and effectiveness of the $\mathcal{IFM}–\mathcal{BA}$-based approach for real-world fractional models.

5. Conclusions

In this paper, we introduced a new structure of intuitionistic fuzzy metric spaces over a Banach algebra, which extends and unifies several existing frameworks, including classical fuzzy metric spaces and fuzzy cone metric spaces. Within this algebra-valued intuitionistic fuzzy setting, we established fundamental topological properties such as openness, boundedness, completeness, and the convergence of sequences, thereby providing a solid analytical foundation for further investigations. Building on these preliminaries, we developed fixed-point results for intuitionistic fuzzy (Z)-contractions and algebra-valued fuzzy contraction mappings, and proved that such mappings admit unique fixed points in (${\mathcal{G}}_{\Lambda }$)-complete $\mathcal{IFM}$-spaces. These results not only generalize several well-known fixed-point theorems in the existing literature, but also offer a unified and flexible approach for analyzing contraction-type operators in intuitionistic fuzzy environments endowed with algebra-valued metrics.

To demonstrate the practical relevance of the proposed theory, we applied the established fixed-point framework to a nonlinear fractional Volterra integral equation of the Caputo type. The intuitionistic fuzzy metric approach ensured the existence and uniqueness of the solution under mild contractive conditions, while the accompanying numerical simulation confirmed the rapid convergence of the corresponding iterative scheme. Moreover, the observed monotonic increase in the fuzzy nearness measure and the corresponding decay in the fuzzy non-nearness measure were fully consistent with the analytical predictions, highlighting the stability and effectiveness of the $\mathcal{IFM}$-based methodology for fractional dynamical systems.

The results presented in this work open several promising directions for future research. One natural extension is the study of multi-valued mappings and set-valued contractions in $\mathcal{IFM}–\mathcal{BA}$ spaces, which would significantly broaden the applicability of the theory to optimization problems and differential inclusions. Another important direction concerns the application of the proposed framework to fractional differential inclusions and integro-differential systems, where uncertainty and memory effects coexist and algebra-valued intuitionistic fuzzy structures may offer deeper analytical insight. Furthermore, potential connections with machine learning and data science merit exploration, particularly in the modelling and analysis of fuzzy dynamical systems, adaptive algorithms, and learning processes under uncertainty. Incorporating intuitionistic fuzzy metrics over Banach algebras into such data-driven frameworks may lead to new tools for stability analyses, convergence assessments, and uncertainty quantification in complex systems.

Overall, the proposed intuitionistic fuzzy metric framework over Banach algebras provides a powerful and versatile analytical tool, with significant potential for further theoretical development and interdisciplinary applications.