Normed Spaces Research Articles

Generalised Ulam-Hyers Stability Analysis for System of Additive Functional Equation in Fuzzy and Random Normed Spaces: Direct and Fixed Point Approach

In this article, a new system of Functional Equations is proposed. The Ulam-Hyers stability of this class of equations is investigated using the product, sum, and mixed product-sum of powers of norms, as well as the general control function. The stability analysis is carried out in random and fuzzy normed spaces using fixed point and direct methods. One of the unique and interesting aspects of this study is that, three new and different kinds of FEs have been introduced and the stability analysis is derived for all three equations simultaneously.

A New Structure of Random Approach Normed Space via Banach Space

The goal of this research is to define the convergent sequence in A-random approach space and sequentially convergent are discussed and the cluster point, open and closed ball and linear transformation. We are going to explain a new structure of Random approach normed space via Banach space in and discussed all the relations between metric space in this research.

Existence and density theorems of Henig global proper efficient points

In this work, we provide some novel results that establish both the existence of Henig global proper efficient points and their density in the efficient set for vector optimization problems in arbitrary normed spaces. Our results do not require the assumption of convexity, and in certain cases, can be applied to unbounded sets. However, it is important to note that a weak compactness condition on the set (or on a section of it) and a separation property between the order cone and its conical neighbourhoods remains necessary. The weak compactness condition ensures that certain convergence properties hold. The separation property enables the interpolation of a family of Bishop-Phelps cones between the order cone and each of its conic neighbourhoods. This interpolation, combined with the proper handling of two distinct types of conic neighbourhoods, plays a crucial role in the proofs of our results, which include as a particular case other results that have already been established under more restrictive conditions.

Stability of a General Quadratic-Cubic Functional Equation in Non-Archimedean 2-Normed Spaces

This research aims to investigate the Hyers-Ulam stability of the mixed-type quadratic-cubic functional equation in non-Archimedean 2-normed spaces using the fixed-point method. Additionally, some counter-examples are illustrated for instability. The exciting possibilities of this cutting-edge research and unlocking new frontiers in mathematical analysis are explored.

Integral-Type Representations for the Subdifferential of Suprema

AbstractAn integral-like representation is provided for the $$\varepsilon $$ ε -subdifferential of the supremum of an arbitrary family of convex functions. Our characterizations are expressed through appropriate discrete sums performed on the data functions $$f_{t}$$ f t ’s together with specific singular measures operating on them. Moreover, as long as the underlying space is a reflexive Banach space or a separable normed space, we substitute these additional measures with related limits that involve the data functions. All the objects involved in our characterizations rely intrinsically on the data functions that are (almost) active at the reference point.

Quasi-support hyperplanes in asymmetric normed spaces

Quasi-support hyperplanes in asymmetric normed spaces

Sufficient conditions for the preservation of polygonal-connectedness in an arbitrary normed space

Sufficient conditions for the preservation of polygonal-connectedness in an arbitrary normed space

On fuzzy inner products constructed by fuzzy numbers in linear spaces

In this paper a new definition for fuzzy inner product spaces is presented. Just as the set of real numbers can be embedded in a set of fuzzy numbers, a crisp inner product space can be considered as a special case of fuzzy inner product spaces. An example is given to demonstrate that, the new definition is a nontrivial generalization for the crisp inner product spaces. Under certain restrictions, a fuzzy inner product space can become a fuzzy normed space. Based on some elementary properties for the families of semi-inner products of endpoints, the linearly topological structure of new spaces is discussed. Moreover, the orthogonality between two vectors is considered and a fuzzy version of Pythagorean theorem is given.

Умови екстремальності допустимого елемента для задачі відшу-кання узагальненого чебишовського центра кількох точок деякого полінормованого простору відносно множини цього простору

The problems related to the need to approximate complex mathematical objects in the best possible way with simpler and more convenient ones arise in various sections of mathematical science. An important class of approximation theory problems is the best simultaneous approximation of several elements. The problem of finding the Chebyshev center of several points of a linear normalized space relative to the set of this space can be attributed to the problems of best simultaneous approximation of several elements. This task consists in finding in a given set of linear normed space such a point (the relative Chebyshev center) the maximum distance to which from several fixed points of space would be the smallest, in other words not exceeding the maximum distance from the given points to any other point of this set. The problems of the best simultaneous approximation of several elements of a linear normed space by convex sets of this space from single positions were considered, in particular, in works [1, 2]. In practice, one has to deal with such problems, in which, when finding the Chebyshev's center of several given points of a linear normed space relative to the set of this space, appear weighted distances. The task of finding the weighted distances of the Chebyshev’s center was considered, in particular, in the paper [3]. In this work, the criteria of generalized Chebyshov’s center in the sense of the weighted distances of the of several points of a linear normed space relative to the convex set of this space, based on the duality ratio for the corresponding extremal problem, are established. If in the problem of the Chebyshev's center of several points of a linear normed space, in which the distances between points are determined by weighted norms, the weighted norms are replaced, generally speaking, by different norms given on the corresponding linear space, then we obtain the problem of the Chebyshev's center of several points of some polynormed space, which is considered in this work. It is clear that the problems about the Chebyshev's center of several points of a linear normed space, which were discussed above, are partial cases of the problem about the Chebyshev center of several points of some polynormed space.

On soft refined 2-normed spaces

On soft refined 2-normed spaces

Data depth functions for non-standard data by use of formal concept analysis

In this article we introduce a notion of depth functions for data types that are not given in standard statistical data formats. We focus on data that cannot be represented by one specific data structure, such as normed vector spaces. This covers a wide range of different data types, which we refer to as non-standard data. Depth functions have been studied intensively for normed vector spaces. However, a discussion of depth functions for non-standard data is lacking. In this article, we address this gap by using formal concept analysis to obtain a unified data representation. Building on this representation, we then define depth functions for non-standard data. Furthermore, we provide a systematic basis by introducing structural properties using the data representation provided by formal concept analysis. Finally, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article presents the mathematical formalisation of centrality and outlyingness for non-standard data and increases the number of spaces in which centrality can be discussed. In particular, we provide a basis for defining further depth functions and statistical inference methods for non-standard data.

Some Results about Acts over Monoid and Bounded Linear Operators

الأثر V بالنسبة إلى sinshT و خواصه قد تم دراسته في هذا البحث حيث تم دراسة علاقة الأثر المخلص والاثر المنتهى التولد والاثر المنفصل وربطها بالمؤثرات المتباينة حيث تم بهنة العلاقات التالية ان الاثر اذا وفقط اذا مقاس في حالة كون المؤثر هو عديم القوة وكذلك في حالة كون المؤثر شامل فان الاثر هو منتهي التولد اي ان الغضاء هو منتهي التولد وايضا تم برهن ان الاثر مخلص لكل مؤثر مقيد وك\لك قد تم التحقق من انه لاي مؤثر مقيد فان الاثر منفصل وفي حالة كون الاثر منفصل فان المؤثر سوف يكون متباين وايضا تم برهنة انه في حالة كون الفضاء يمتلك قاعدة فان الاثر سوف يكون كل عنصر فيه يمكن كتابته كتركيبة خطية ومن العلاقات المهمة التي قد تم برهانها انه اذا كان المؤثر مشابه لمؤثر اخر فان الاثر سوف يكون نوثيرين بوجد شرط على الفضاء .

ROUGH IDEAL CONVERGENCE OF DOUBLE SEQUENCES IN INTUITIONISTIC FUZZY NORMED SPACES

The idea of rough statistical convergence for double sequences was studied by Ozcan and Or [34] in a intuitionistic fuzzy normed space. Recently the same has been generalized in the ideal context by Hossain and Banerjee [17] for sequences. Here in this paper we have discussed the idea of rough ideal convergence of double sequences in intuitionistic fuzzy normed spaces generalizing the idea of rough statistical convergence of double sequences. Also we have defined rough I2-cluster points for a double sequence and also investigated some of the basic properties associated with rough I2-limit set of a double sequence in a intuitionistic fuzzy normed space.

Cubic Structure on Soft Gamma-m-Normed Linear Space

Objectives: The purpose of this research is to take the lead in comprehending the fuzzy idea of cubic structure on soft Gamma-m normed linear space. According to the theory of Soft m-Normed Linear Space (SMNLS), we offer the conviction of Cauchy's sequence and convergence in cubic Soft Gamma-m Normed Linear Space (CSGMNLS). We have obtained certain results like the concept of completeness in CSGMNLS. Methods: In this paper we defined the soft gamma ring, soft gamma ideals and soft gamma vector space which are used to introduce the notion of soft gamma-2-normed linear space, Soft Gamma-m-Normed Linear Space (SGMNLS) and its properties. Also, the CSGMNLS can be analyzed by using the SGMNLS. Findings: In this research from CSGMNLS construct a norm function that satisfies the properties of SGMNLS, and additionally given that example with proof in which a sequence is cauchy sequence and convergence in SGMNLS if it is cauchy and convergence sequence in CSGMNLS. Also, provided theorem and its proof for completeness of a sequence in CSGMNLS. Novelty: Already gamma ring and fuzzy n-normed linear space has been defined. We introduced the concept of SGMNLS using this also initiated the CSGMNLS and some results obtained from its properties. We suggested a necessary conditions for completeness of a sequence in CSGMNLS. Keywords: Soft Gamma ring, Soft gamma vector space, Soft gamma normed linear space, Soft m-norm, Soft m-normed linear space, 2-Normed and m-normed right soft gamma linear space

Stability and Instability of an Apollonius-Type Functional Equation

For the inner product space, we have Appolonius’ identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius-type additive functional equation. In this work, we investigate an Apollonius-type additive functional equation in 2-normed spaces. We first investigate the stability of an Apollonius-type additive functional equation in 2-Banach spaces by using Hyers’ direct method. Then, we consider the instability of an Apollonius-type additive functional equation in 2-Banach spaces.

Moduli of uniform convexity for convex sets

AbstractLet C be a proper, closed subset with nonempty interior in a normed space X. We define four variants of modulus of convexity for C and prove that they all coincide. This result, which is classical and well-known for $$C=B_X$$ C = B X (the unit ball of X), requires a less easy proof than the particular case of $$B_X.$$ B X . We also show that if the modulus of convexity of C is not identically null, then C is bounded. This extends a result by M.V. Balashov and D. Repovš.

The norming set of a bilinear form on a certain normed space $\mathbb{R}^2$

In this paper we classify the norming set of a bilinear form on the plane with a certain norm whose unit ball has only four extreme points. We obtain the results of [6, 8] as corollary.

Some results on ultrametric 2-normed spaces

In this paper, we study the ultrametric 2-normed spaces and the ultrametric 2-Banach spaces. In particular, we establish some results on Cauchy sequences in ultrametric 2-normed spaces. Also, we introduce and study the notion of bounded linear 2-functionals on ultrametric 2-Banach spaces and we give some of its properties. On the other hand, the new norm on the ultrametric 2-normed space is constructed. The concepts of closed operators between ultrametric 2-normed spaces and $b$-linear functionals in ultrametric 2-normed spaces are introduced. Finally, a necessary and sufficient condition for a linear operator to be closed in terms of its graph is proved and some results on bounded $b$-linear functionals in ultrametric 2-normed spaces are given.

Norm and relative uniform convergence

In this paper, we show that if H is a Dedekind complete normed Riesz space with the local Egoroff property, then H is lattice isomorphic to an ideal of an AM-space if and only if for any order bounded norm null sequence in H. Based on this, several characterizations of discrete Banach lattices with order continuous norms are provided. As an application, we find the special regulator for every relative uniform convergent sequence in discrete Banach lattices with order continuous norms.

Some results on $I_2$-deferred statistically convergent double sequences in fuzzy normed spaces

The primary objective of this study is to introduce the concepts of $I_2$-deferred Cesàro summability and $I_2-$ deferred statistical convergence for double sequences in fuzzy normed spaces (FNS). Furthermore, the aim is to explore the connections between these concepts and subsequently establish several theorems pertaining to the notion of $I_2$-deferred statistical convergence in FNS for double sequences. We further define $I_2$-deferred statistical limit points and $I_2$-deferred statistical cluster points of a sequence within FNS and explore the relationships among these concepts.