Fuzzy Normed Research Articles

Analysis of Functional Properties of Fuzzy \(\tilde{L}\) \(^p\Omega\) Spaces

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.

Intuitionistic Type-2 Fuzzy Normed Linear Space and Some of Its Basic Properties

Intuitionistic Type-2 Fuzzy Normed Linear Space and Some of Its Basic Properties

Rough Ideal Statistical Convergence via Generalized Difference Operators in Intuitionistic Fuzzy Normed Spaces

Rough Ideal Statistical Convergence via Generalized Difference Operators in Intuitionistic Fuzzy Normed Spaces

Some Results of Fixed Point for Single Value Mapping in Fuzzy Normed Space with Applications

Fixed point theory is not only essential in mathematics but also to a vast array of applications. In this approach, we provide some fixed point theorems for fuzzy normed space. The triangle property of a fuzzy norm is introduced first. This property is used to demonstrate some fixed point solutions for self-mappings in fuzzy normed space. To demonstrate the importance of the obtained results, an application for the existence of a solution to the ordinary differential equation and the Fredholm integral equation is constructed.

Common Fixed Point Theorems in Fuzzy Normed Space

. Fixed point theory is an intriguing topic with numerous applications in several branches of mathematics. In addition, fixed point theory offers useful methods for problem-solving that may be used across many subfields of mathematical analysis. Fixed point theorems are concerned with mappings f of a set X into itself that, under particular conditions, permit a fixed point, that is, a point such that . In this paper, we use the triangle property on fuzzy normed space(Fn-space) to show a common fixed point (CF-point) without continuity. First, we review some of the fundamental terms used in the fuzzy context. After that the notion of triangle fuzzy norm is given then we show that the self mappings Γ and Λ have a CF-point in the setting of Fn-space. Additionally, certain applications of our main results to the Fredholm integral equation are investigated.

Lacunary statistical convergence on L - fuzzy normed space

The idea of lacunary statistical convergence sequences, which is a development of statistical convergence, is examined and expanded in this study on L - fuzzy normed spaces, which is a generalization of fuzzy spaces. On L - fuzzy normed spaces, the definitions of lacunary statistical Cauchy and completeness, as well as associated theorems, are provided. The link between lacunary statistical Cauchyness and lacunary statistical boundedness with regard to L - fuzzy norm is also shown.

On α ̌–φ ̆-Fuzzy Contractive Mapping in Fuzzy Normed Space

تمثل فكرة النقاط الصامدة إحدى أقوى الأدوات الرياضية. الغرض الرئيسي من هذه الورقة هو تقديم نوع جديد من دالة الانكماش الضبابي في فضاء معياري ضبابي وهي "الدالة الانكماشية الضبابية ". نثبت بعض نتائج النقطه الصامدة لهذه الداله في سياق الفضاء المعياري الضبابي باستخدام الخاصية المثلثية للمعيار الضبابي. بالإضافة إلى ذلك ، في ظل ظروف محددة ، تم إثبات بعض النتائج الأخرى لهذا النوع من الدالة. أخيرًا ، يتم تقديم مثال لإظهار فائدة النتائج.

Multidimensional fuzzy norms and cut sets in the context of medical decision making

This study introduces the concepts of complements, t-norms, t-conorms, and cut sets in the context of multi-dimensional fuzzy sets. Some fundamental results including DeMorgan-type identities, concerning these, are obtained. The advantage of using multidimensional fuzzy sets for analyzing data is that each element in the universe can be given individual attention as per the requirement. By utilizing this scope, as an application of the introduced concepts, stage identification of certain diseases based on Multidimensional fuzzy sets and intuitionistic fuzzy sets is provided together with a comparative study on typical biomedical data.

Best Proximity Point Theorem for φ ̃–ψ ̃-Proximal Contractive Mapping in Fuzzy Normed Space

The study of fixed points on the maps fulfilling certain contraction requirements has several applications and has been the focus of numerous research endeavors. On the other hand, as an extension of the idea of the best approximation, the best proximity point (ƁƤƤ) emerges. The best approximation theorem ensures the existence of an approximate solution; the best proximity point theorem is considered for addressing the problem in order to arrive at an optimum approximate solution. This paper introduces a new kind of proximal contraction mapping and establishes the best proximity point theorem for such mapping in fuzzy normed space ( space). In the beginning, the concept of the best proximity point was introduced. The concept of proximal contractive mapping in the context of fuzzy normed space is then presented. Following that, the best proximity point theory for this kind of mapping is established. In addition, we provide an example application of the results

Fuzzy Strong ϕ-b-Normed Linear Space for Fuzzy Bounded Linear Operators

In this paper, concept of fuzzy continuous operator, fuzzy bounded linear operator are introduced in fuzzy strong [Formula: see text]-b-normed linear spaces and their relations are studied. Idea of operator fuzzy norm is developed and completeness of BF(X,Y) is established.

The Fuzzy Width Theory in the Finite-Dimensional Space and Sobolev Space

This paper aims to fuzzify the width problem of classical approximation theory. New concepts of fuzzy Kolmogorov n-width and fuzzy linear n-width are introduced on the basis of α-fuzzy distance which is induced by the fuzzy norm. Furthermore, the relationship between the classical widths in linear normed space and the fuzzy widths in fuzzy linear normed space is discussed. Finally, the exact asymptotic orders of the fuzzy Kolmogorov n-width and fuzzy linear n-width corresponding to a given fuzzy norm in finite-dimensional space and Sobolev space are estimated.

Lacunary Statistical Convergence for Double Sequences on $\mathscr{L}-$ Fuzzy Normed Space

On $\mathscr{L}-$ fuzzy normed spaces, which is the generalization of fuzzy spaces, the notion of lacunary statistical convergence for double sequences which is a generalization of statistical convergence, are studied and developed in this paper. In addition, the definitions of lacunary statistical Cauchy and completeness for double sequences and related theorems are given on $\mathscr{L}-$ fuzzy normed spaces. Also, the relationship of lacunary statistical Cauchyness and lacunary statistical boundedness for double sequences with respect to $\mathscr{L}-$ fuzzy norm is shown.

Another Type of Fuzzy Inner Product Space

In this paper, we generalize the definition of fuzzy inner product space that is introduced by Lorena Popa and Lavinia Sida on a complex linear space. Certain properties of the generalized fuzzy inner product function are shown. Furthermore, we prove that this fuzzy inner product produces a Nadaban-Dzitac fuzzy norm. Finally, the concept of orthogonality is given and some of its properties are proven.

On the Ideal Convergent Sequences in Fuzzy Normed Space

This article discusses a variety of important notions, including ideal convergence and ideal Cauchyness of topological sequences produced by fuzzy normed spaces. Furthermore, the connections between the concepts of the ideal limit and ideal cluster points of a sequence in a fuzzy normed linear space are investigated. In a fuzzy normed space, we investigated additional effects, such as describing compactness in terms of ideal cluster points and other relevant but previously unresearched ideal convergence and adjoint ideal convergence aspects of sequences and nets. The countable compactness of a fuzzy normed space and its link to it were also defined. The terms ideal and its adjoint divergent sequences are then introduced, and specific aspects of them are explored in a fuzzy normed space. Our study supports the importance of condition (AP) in examining summability via ideals. It is suggested to use a fuzzy point symmetry-based genetic clustering method to automatically count the number of clusters in a data set and determine how well the data are fuzzy partitioned. As long as the clusters have the attribute of symmetry, they can be any size, form, or convexity. One of the crucial ways that symmetry is used in fuzzy systems is in the solution of the linear Fuzzy Fredholm Integral Equation (FFIE), which has symmetric triangular (Fuzzy Interval) output and any fuzzy function input.

Best Proximity Point Results in Fuzzy Normed Spaces

Fixed point (briefly FP ) theory is a potent tool for resolving several actual problems since many problems may be simplified to the FP problem. The idea of Banach contraction mapping is a foundational theorem in FP theory. This idea has wide applications in several fields; hence, it has been developed in numerous ways. Nevertheless, all of these results are reliant on the existence and uniqueness of a FP on some suitable space. Because the FP problem could not have a solution in the case of nonself-mappings, the idea of the best proximity point (briefly Bpp) is offered to approach the best solution. This paper investigates the existence and uniqueness of the Bpp of nonself-mappings in fuzzy normed space(briefly FN space) to arrive at the best solution. Following the introduction of the definition of the Bpp, the existence, and uniqueness of the Bpp are shown in a FN space for diverse fuzzy proximal contractions such as ?????? fuzzy proximal contractive mapping and ????h ????h - fuzzy proximal contractive mapping.

Some Results on Nörlund ℐ2-Lacunary Statistical Convergent Sequences in Intuitionistic Fuzzy Normed Spaces

The intent of this paper is to investigate the intuitionistic Nörlund [Formula: see text]-lacunary statistically convergent sequence space. We present some intuitionistic fuzzy normed spaces (IFNS) in Nörlund convergent spaces. Moreover, we also put forward several topological and algebraic properties of these convergent sequence spaces.

Some Edelstein’s Fixed Point Theorems in B-S Type Fuzzy Normed Linear Spaces

In this paper, some Edelstein’s fixed point theorems involving the concept of asymptotic centre in uniformly convex Banach spaces are proved in Bag and Samanta type fuzzy normed linear spaces.

Topological Properties of Jordan Intuitionistic Fuzzy Normed Spaces

AbstractThe article delves into the task of formulation of meticulous sequence spaces whose elements’ convergence is a generalized version of the Cauchy convergence in the setting of intuitionistic fuzzy norm. The task of attaining a finite limit is attained via an infinite matrix operator, namely, Jordan totient operator. We discuss some topological and algebraic properties and establish some inclusion relations between the spaces.

Best Proximity Point Theorem for α ̃–ψ ̃-Contractive Type Mapping in Fuzzy Normed Space

The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems offer an approximate solution to the fixed point equation . It is used to solve the problem in order to come up with a good approximation. This paper's main purpose is to introduce new types of proximal contraction for nonself mappings in fuzzy normed space and then proved the best proximity point theorem for these mappings. At first, the definition of fuzzy normed space is given. Then the notions of the best proximity point and - proximal admissible in the context of fuzzy normed space are presented. The notion of α ̃–ψ ̃- proximal contractive mapping is introduced. After that, the best proximity point theorem for such type of mapping in a fuzzy normed space is state and prove. In addition, the idea of α ̃–ϕ ̃-proximal contractive mapping is presented in a fuzzy normed space and under specific conditions, the best proximity point theorem for such type of mappings is proved. Furthermore, some examples are offered to show the results' usefulness.

Some Properties Of Cartesian Product Of Two Fuzzy Normed Spaces

In this paper , the concept of the Cartesian Product of two fuzzy
 normed spaces is presented. Some basic properties and theorems on this
 concept are proved. The main goal of this paper is to prove that the
 Cartesian product of two complete fuzzy normed spaces is a complete
 fuzzy normed space.
 Key words:Fuzzy normed space , Cartesian product , Cauchy sequence ,
 complete fuzzy normed space.
 1- Introduction
 
 The fuzzy set concepts was introduced in mathematics by K.Menger
 in 1942 and reintroduced in the system theory by L.A.Zadeh in 1965.
 In 1984, Katsaras [ 1 ] , first introduced the notation of fuzzy norm on
 linear space, in the same year Wu and Fang [ 4 ] also introduced a notion of
 fuzzy normed space . Later on many other mathematicians like Felbin [ 2 ]
 , Cheng and Mordeson [ 10] , Bag and Samanta [12], J.Xiao and X.Zhu
 [8,9] , Krishna and Sarma [11] , Balopoulos and Papadopoulos [ 13] etc,
 have given different definitions of fuzzy normed spaces .
 J.Kider introduced the definition of fuzzy normed space[ 7 ] , we use this
 definition to prove that the Cartesian product of two fuzzy normed spaces
 is also fuzzy normed space.
 
 The structure of the paper is as follow : In section 2 we
 present some fundamental concepts . In section 3, the definition of fuzzy
 normed space appeared [7] is used to prove that the cartesain product of
 two fuzzy normed spaces is also fuzzy normed space, then we prove that
 the cartesain product of two complete fuzzy normed spaces is complete
 fuzzy normed space.
 
 2. Preliminaries
 In this section, we briefly recall some definitions and preliminary
 results which are used in this paper.