Modified intuitionistic fuzzy double controlled metric space and related results with application

In this paper, we introduce the concept of fuzzy double controlled metric space that can be regarded as the generalization of fuzzy b-metric space, extended fuzzy b-metric space and controlled fuzzy metric space. We use two non-comparable functions α and β in the triangular inequality as: M q ( x , z , t α ( x , y ) + s β ( y , z ) ) ≥ M q ( x , y , t ) ∗ M q ( y , z , s ) . We prove Banach contraction principle in fuzzy double controlled metric space and generalize the Banach contraction principle in aforementioned spaces. We give some examples to support our main results. An application to existence and uniqueness of solution for an integral equation is also presented in this work.

In this article, we introduce the concept of intuitionistic fuzzy double controlled metric spaces that generalizes the concept of intuitionistic fuzzy b-metric spaces. For this purpose, two noncomparable functions are used in triangle inequalities. We generalize the concepts of the Banach contraction principle and fuzzy contractive mappings by giving authentic proof of these mappings in the sense of intuitionistic fuzzy double controlled metric spaces. To validate the superiority of these results, examples are imparted. A possible application to solving integral equations is also set forth towards the end of this work to support the proposed results.

This paper proposes a study of fuzzy \(\tilde{L}\)p (Ω) spaces, incorporating functions with triangular fuzzy coefficients. These fuzzy functional spaces provide a better adaptation to fuzzy or imprecise functions. We establish the theoretical foundations of these spaces by examining key functional properties, such as the fuzzy scalar product and the fuzzy norm. To do this, we checked the bilinearity, symmetry, positivity, homogeneity and triangular inequality in a fuzzy environment and in the presence of functions whose coefficients are triangular fuzzy numbers, by the \(\alpha\) -cut Dubois and Prade approach. The aim of this paper is to address observations identified in the existing literature, where some functional properties of \(\tilde{L}\)p (Ω) fuzzy spaces are often addressed in a too general manner, without specifying the types of fuzzy functions. This study aims to provide a more detailed and rigorous analysis, thus enriching mathematical understanding and paving the way for practical applications in diverse fields such as: fuzzy differential equations, artificial intelligence, information processing, and decision making in uncertain environments.

This paper introduces [Formula: see text]-double fuzzy G-filter spaces as a generalization of [Formula: see text]-double fuzzy filters. The notions of sum, subspace, product, and quotient of [Formula: see text]-double fuzzy G-filters are proposed. Additionally, the paper develops the concepts of stratified [Formula: see text]-double fuzzy G-filter spaces and [Formula: see text]-double fuzzy G-filterbases. A key finding of this research is the establishment of the category of stratified [Formula: see text]-double fuzzy G-filter spaces as a bicoreflective subcategory of the category of [Formula: see text]-double fuzzy G-filter spaces. Finally, the study has explored the role of [Formula: see text]-double fuzzy G-filter spaces in mathematical modeling and decision-making processes.

Introduction Metric spaces play a crucial role in mathematical analysis and topology. In recent years, fuzzy metric spaces have been widely studied due to their applications in various fields. Alexander Sostak introduced the concept of revised fuzzy metric spaces, which extends traditional fuzzy metric spaces by incorporating revised fuzzy sets. In this paper, we introduce a further generalization called revised fuzzy metric spaces, which allows for the involvement of multiple parameters (), thereby enhancing the flexibility and applicability of the framework. Objectives The primary aim of this study is to define and explore the fundamental properties of revised fuzzy metric spaces. We investigate their topological structure and establish significant properties such as first countability and the Hausdorff condition. Additionally, we extend existing results in the literature by proving a fixed-point theorem in this new setting. Method We begin by formally defining a revised fuzzy k-metric space and developing its basic properties. Using topological arguments, we demonstrate that the topology induced by a revised fuzzy metric is first countable and that the space satisfies the Hausdorff condition. Finally, we extend the fixed-point theorem established by Muraliraj and Thangathamizh into the context of revised fuzzy metric spaces, using analytical and set-theoretic techniques. Result Our findings confirm that revised fuzzy k-metric spaces preserve essential topological characteristics such as first countability and Hausdorff separation. Furthermore, the fixed-point theorem proved in this study generalizes previous results and demonstrates the broader applicability of revised fuzzy metric spaces in fixed-point theory. Conclusion This study introduces revised fuzzy metric spaces as a generalization of revised fuzzy metric spaces, providing a more comprehensive framework for analyzing metric structures with multiple parameters. The established topological properties and fixed-point theorem contribute to the further development of fuzzy metric theory, opening new avenues for future research in mathematical analysis and its applications.

This paper proposes a study of fuzzy Sobolev spaces W ̃1,p(Ω), integrating functions with triangular fuzzy coefficients. These fuzzy functional spaces aim to better model fuzzy functions, thus generalizing classical Sobolev spaces. We establish the theoretical foundations of these spaces by analyzing the fuzzy scalar product and the fuzzy norm. We focus on verifying essential properties such as bilinearity, symmetry, positivity, homogeneity, and triangle inequality, using the Dubois and Prade α-cut approach to formalize the notion of uncertainty. This paper addresses observations identified in the literature, where the lack of suitable fuzzy functional spaces for solving fuzzy differential equations, in particular fuzzy Sobolev spaces W ̃1,p(Ω) , is often not taken into account. Moreover, the analysis of fuzzy scalar product and norm properties is often presented vaguely. We thus propose a detailed approach to the functional properties of these spaces, extending classical Sobolev spaces to a fuzzy framework. This research opens up prospects for practical applications in areas such as fuzzy differential equations and decision-making in medicine, economics, artificial intelligence, information processing and various other fields.

In this paper, we introduce the concept of double fuzzy preproximity spaces as a generalization of a fuzzy preproximity spaces and investigate some of their properties. Also we study the relationships between double fuzzy preproximity spaces, double fuzzy topological spaces and double fuzzy closure spaces. In addition to this was the introduction of the concept of double fuzzy neighborhood system and has been studying the connection with double fuzzy preproximity, which resulted in the definition of the concept double fuzzy preproximal neighborhood.

We propose the so-called fuzzy semi-metric space in which the symmetric condition is not assumed to be satisfied. In this case, there are four kinds of triangle inequalities that should be considered. The purpose of this paper is to study the common coincidence points and common fixed points in the newly proposed fuzzy semi-metric spaces endowed with the so-called ⋈-triangle inequality. The other three different kinds of triangle inequalities will be the future research, since they cannot be similarly investigated as the case of ⋈-triangle inequality.

The aim of this paper is to introduce $(L,M)$-fuzzy closurestructure where $L$ and $M$ are strictly two-sided, commutativequantales. Firstly, we define $(L,M)$-fuzzy closure spaces and getsome relations between $(L,M)$-double fuzzy topological spaces and$(L,M)$-fuzzy closure spaces. Then, we introduce initial$(L,M)$-fuzzy closure structures and we prove that the category$(L,M)$-{bf FC} of $(L,M)$-fuzzy closure spaces and$(L,M)$-$mathcal{C}$-maps is a topological category over thecategory {bf SET}. From this fact, we define products of$(L,M)$-fuzzy closure spaces. Finally, we show that an initialstructure of $(L,M)$-double fuzzy topological spaces can be obtainedby the initial structure of $(L,M)$-fuzzy closure spaces induced bythem.

The category of double fuzzy preproximity spaces

A large amount of algorithms has recently been designed for the Internet under the assumption that the distance defined by the round-trip delay (RTT) is a metric. Moreover, many of these algorithms (e.g., overlay network construction, routing scheme design, sparse spanner construction) rely on the assumption that the metric has bounded ball growth or bounded doubling dimension. This paper analyzes the validity of these assumptions and proposes a tractable model matching experimental observations. On the one hand, based on Skitter data collected by CAIDA and King matrices of Meridian and P2PSim projects, we verify that the ball growth of the Internet, as well as its doubling dimension, can actually be quite large. Nevertheless, we observed that the doubling dimension is much smaller when restricting the measures to balls of large enough radius. Moreover, by computing the number of balls of radius r required to cover balls of radius R > r, we observed that this number grows with R much slower than what is predicted by a large doubling dimension. On the other hand, based on data collected on the PlanetLab platform by the All-Sites-Pings project, we confirm that the triangle inequality does not hold for a significant fraction of the nodes. Nevertheless, we demonstrate that RTT measures satisfy a weak version of the triangle inequality: there exists a small constant p such that for any triple u, v, w, we have RTT(u,v) les rho-max{RTT(u,w),RTT(w,v)}. (Smaller bounds on p can even be obtained when the triple u, v, w is skewed). We call inframetric a distance function satisfying this latter inequality. Inframetrics subsume standard metrics and ultrametrics. Based on inframetrics and on our observations concerning the doubling dimension, we propose an analytical model for Internet RTT latencies. This model is tuned by a small set of parameters concerning the violation of the triangle inequality and the geometrical dimension of the network. We demonstrate the tractability of our model by designing a simple and efficient compact routing scheme with low stretch. Precisely, the scheme has constant multiplicative stretch and logarithmic additive stretch.

Abstract. We introduce the concepts of double bitopological spacesas a generalization of intuitionistic fuzzy topological spaces in Sostak’ssense and Kandil’s fuzzy bitopological spaces. Also we introduce theconcept of (T  ;U  )-double (r;s)(u;v)-semiopen sets and doublepairwise (r;s)(u;v)-semicontinuous mappings in double bitopologi-cal spaces and investigate some of their characteristic properties. 1. IntroductionChang [2] de ned fuzzy topological spaces. These spaces and itsgeneralizations are later studied by several authors, one of which, devel-oped by Sostak [12], used the idea of degree of openness. This type ofgeneralization of fuzzy topological spaces was later rephrased by Chat-topadhyay, Hazra, and Samanta [3], and by Ramadan [11].As a generalization of fuzzy sets, the concept of intuitionistic fuzzysets was introduced by Atanassov [1]. C˘oker and his colleagues [4, 6,7] introduced intuitionistic fuzzy topological spaces using intuitionisticfuzzy sets. Using the idea of degree of openness and degree of nonopen-ness, C˘oker and M. Demirci [5] de ned intuitionistic fuzzy topologicalspaces in Sostak’s sense as a generalization of smooth fuzzy topologicalspaces and intuitionistic fuzzy topological spaces.Kandil [8] introduced and studied the notion of fuzzy bitopologicalspaces as a natural generalization of fuzzy topological spaces.In this paper, we introduce the concepts of double bitopological spacesas a generalization of intuitionistic fuzzy topological spaces in Sostak’s

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Severe congenital glottic web

This manuscript contains several new spaces as the generalizations of fuzzy triple controlled metric space, fuzzy controlled hexagonal metric space, fuzzy pentagonal controlled metric space and intuitionistic fuzzy double controlled metric space. We prove the Banach fixed point theorem in the context of intuitionistic fuzzy pentagonal controlled metric space, which generalizes the previous ones in the existing literature. Further, we provide several non-trivial examples to support the main results. The capacity of intuitionistic fuzzy pentagonal controlled metric spaces to model hesitation, capture dual information, handle imperfect information, and provide a more nuanced representation of uncertainty makes them important in dynamic market equilibrium. In the context of changing market dynamics, these aspects contribute to a more realistic and flexible modelling approach. We present an application to dynamic market equilibrium and solve a boundary value problem for a satellite web coupling.