non-Archimedean Normed Spaces Research Articles

On the Stability of Quadratic-Quartic (Q2Q4) Functional Equation over Non-Archimedean Normed Space

In the present work the stability of Hyers-Ulam mixed type of quadratic-quartic Cauchy functional equationg(2x+y)+g(2x−y)=4g(x+y)+4g(x−y)+2g(2x)−8g(x)−6g(y)has been proved over Non-Archimedean normed space.

Isometry and phase-isometry of non-Archimedean normed spaces

Isometry and phase-isometry of non-Archimedean normed spaces

A New Approach for the Statistical Convergence over Non-Archimedean Fields in Neutrosophic Normed Spaces

The goal of the research involves elaborating on the topics of statistical convergence, including statistical Cauchy sequences within non-Archimedean Neutrosophic normed spaces, as well as achieving specific useful conclusions. The present research shows how, within a non-Archimedean field, certain sections of statistically convergent sequences that could not be true often become true. Likewise, we created statistically complete and statistically continuous spaces for such regions that demonstrated certain essential facts. κ indicates a complete field of non-Archimedean and non-trivially valued research.

Nonlinear distance measures under the framework of Pythagorean fuzzy sets with applications in problems of pattern recognition, medical diagnosis, and COVID-19 medicine selection

BackgroundThe concept of Pythagorean fuzzy sets (PFSs) is an utmost valuable mathematical framework, which handles the ambiguity generally arising in decision-making problems. Three parameters, namely membership degree, non-membership degree, and indeterminate (hesitancy) degree, characterize a PFS, where the sum of the square of each of the parameters equals one. PFSs have the unique ability to handle indeterminate or inconsistent information at ease, and which demonstrates its wider scope of applicability over intuitionistic fuzzy sets.ResultsIn the present article, we opt to define two nonlinear distances, namely generalized chordal distance and non-Archimedean chordal distance for PFSs. Most of the established measures possess linearity, and we cannot incorporate them to approximate the nonlinear nature of information as it might lead to counter-intuitive results. Moreover, the concept of non-Archimedean normed space theory plays a significant role in numerous research domains. The proficiency of our proposed measures to overcome the impediments of the existing measures is demonstrated utilizing twelve different sets of fuzzy numbers, supported by a diligent comparative analysis. Numerical examples of pattern recognition and medical diagnosis have been considered where we depict the validity and applicability of our newly constructed distances. In addition, we also demonstrate a problem of suitable medicine selection for COVID-19 so that the transmission rate of the prevailing viral pandemic could be minimized and more lives could be saved.ConclusionsAlthough the issues concerning the COVID-19 pandemic are very much challenging, yet it is the current need of the hour to save the human race. Furthermore, the justifiable structure of our proposed distances and also their feasible nature suggest that their applications are not only limited to some specific research domains, but decision-makers from other spheres as well shall hugely benefit from them and possibly come up with some further extensions of the ideas.

Stability Estimates for an Arithmetic Functional Equation with Brzdȩk Fixed Point Approaches

We introduce an arithmetic functional equation f(x2+y2)=f(x2)+f(y2) and then investigate stability estimates of the functional equation by using the Brzdȩk fixed point theorem on a non-Archimedean fuzzy metric space and a non-Archimedean fuzzy normed space. To apply the Brzdȩk fixed point theorem, the proof uses the linear relationship between two variables, x and y.

On the Generalized Quadratic-Quartic Cauchy Functional Equation and its Stability over Non-Archimedean Normed Space

Functional equation plays a very important and interesting role in the area of mathematics, which involves simple algebraic manipulations and through which one can arrive an interesting solution. The theory of functional equations is also used in the development of other areas such as analysis, algebra, Geometry etc., the new methods and techniques are applied in solving problem in Information theory, Finance, Geometry, wireless sensor networks etc., In recent decades, the study of various types of stability of a functional equation such as HUS (Hyers-Ulam stability), HURS (Hyers-Ulam-Rassias stability) and generalized HUS of different types of functional equation and also for mixed type were discussed by many authors in various space. The problem of the stability of different functional equations has been widely studied by many authors, and more interesting results have been proved in the classical case (Archimedean). In recent years, the analogues results of the stability problem of these functional equations were investigated in non-Archimedean space. The aim of this study is to investigate the HUS of a mixed type of general Quadratic-Quartic Cauchy functional equation in non-Archimedean normed space. In this current article, we prove the generalized HUS for the following Quadratic-Quartic Cauchy functional equation over non-Archimedean Normed space.<img src=image/13428783_01.gif>

Non-Archimedean approximation of the m-Apollonius type functional equation

Throughout this paper, using the fixed point and direct methods, the generalized Hyers–Ulam stability of the following [Formula: see text]-Appolonius type functional equation [Formula: see text] has been proven, where [Formula: see text] is a positive integer greater than [Formula: see text], in non-Archimedean normed spaces.

Ulam-Hyers Stability Results of λ -Quadratic Functional Equation with Three Variables in Non-Archimedean Banach Space and Non-Archimedean Random Normed Space

In this paper, we introduce the λ -quadratic functional equation with three variables and obtain its general solution. The main aim of this work is to examine the Ulam-Hyers stability of this functional equation in non-Archimedean Banach space by using direct and fixed point techniques and examine the stability results in non-Archimedean random normed space.

Equalities and inequalities for several variables mappings

In this paper, some special mappings of several variables such as the multicubic and the multimixed quadratic–cubic mappings are introduced. Then, the systems of equations defining a multicubic and a multimixed quadratic–cubic mapping are unified to a single equation. Under some mild conditions, it is shown that a multimixed quadratic–cubic mapping can be multiquadratic, multicubic and multiquadratic–cubic. Furthermore, by applying a known fixed-point theorem, the Hyers–Ulam stability of multimixed quadratic–cubic, multiquadratic, multicubic and multiquadratic–cubic are studied in non-Archimedean normed spaces.

On the stability of multicubic-quartic and multimixed cubic-quartic mappings

In this paper, we define the multicubic-quartic and the multimixed cubic-quartic mappings and characterize them. In other words, we unify the system of functional equations defining a multimixed cubic-quartic (resp., multicubic-quartic) mapping to a single equation, namely, the multimixed cubic-quartic (resp., multicubic-quartic) functional equation. We also show that under what conditions a multimixed cubic-quartic mapping can be multicubic, multiquartic and multicubic-quartic. Moreover, by using a fixed point theorem, we study the generalized Hyers-Ulam stability of multimixed cubic-quartic functional equations in non-Archimedean normed spaces. As a corollary, we show that every multimixed cubicquartic mapping under some mild conditions can be hyperstable. Lastly, we present a non-stable example for the multiquartic mappings.

Asymptotic Stability of the Pexider–Cauchy Functional Equation in Non-Archimedean Spaces

In this paper, we investigated the asymptotic stability behaviour of the Pexider–Cauchy functional equation in non-Archimedean spaces. We also showed that, under some conditions, if ∥f(x+y)−g(x)−h(y)∥⩽ε, then f,g and h can be approximated by additive mapping in non-Archimedean normed spaces. Finally, we deal with a functional inequality and its asymptotic behaviour.

Approximation for multi-quadratic mappings in non-Archimedean spaces

In this article, we unify the multi-quadratic mappings as a single functional equation. We also apply a fixed point theorem to provide the Hyers-Ulam stability for this new multi-quadratic functional equation in non-Archimedean normed spaces.

On an equation characterizing multi-Jensen-quartic mappings and its stability

In this paper, we define and investigate the mappings of several variables which are quartic in each variable. We show that such mappings can be unified as an equation, say the multi-quartic functional equation. We also establish the Hyers-Ulam stability of a such functional equation by a fixed point theorem in non-Archimedean normed spaces. Moreover, we generalize some known stability and hyperstability results.

Ulam Stability of a Functional Equation in Various Normed Spaces

In this note, we study the Ulam stability of a general functional equation in four variables. Since its particular case is a known equation characterizing the so-called bi-quadratic mappings (i.e., mappings which are quadratic in each of their both arguments), we get in consequence its stability, too. We deal with the stability of the considered functional equations not only in classical Banach spaces, but also in 2-Banach and complete non-Archimedean normed spaces. To obtain our outcomes, the direct method is applied.

Almost Multi-Cubic Mappings and a Fixed Point Application

The aim of this paper is to introduce $n$-variables mappings which are cubic in each variable and to apply a fixed point theorem for the Hyers-Ulam stability of such mapping in non-Archimedean normed spaces. Moreover, a few corollaries corresponding to some known stability and hyperstability outcomes are presented.

On Fixed Point Theorem in Non-Archimedean Fuzzy Normed Spaces

Let (X, N) be a non-archimedean fuzzy normed space and (X, k.k), a non-archimedean normed space where X is a linear space over a linearly ordered non-archimedean field K with a non-archimedean valuation. We give a proof of the fixed point theorem in non-archimedean Fuzzy normed space.

Fixed Point Theorems for Single Valued Mappings Satisfying the Ordered non-Expansive Conditions on Ultrametric and Non-Archimedean Normed Spaces

In this paper, some fixed point theorems for nonexpansive mappings in partially ordered spherically complete ultrametric spaces are proved. In addition, we investigate the existence of fixed points for nonexpansive mappings in partially ordered non-Archimedean normed spaces. Finally, we give some examples to discuss the assumptions and support our results.

The cubic ρ-functional equation in matrix non-Archimedean random normed spaces

Using the direct method and fixed point method, we investigate the Hyers-Ulam stability of the following cubic ?-functional equation f(x+2y) + f(x-2y)- 2f(x+y)-2f(x-y)-12f(x) = ?(4f(x+y/2) + 4f(x-y/2)-f(x+y)-f(x-y)-6f(x)) in matrix non-Archimedean random normed spaces, where ? is a fixed real number with ? ? 2.

GENERALIZED STABILITY OF AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION IN VARIOUS SPACES

&#x0D; ABSTRACT. In this paper, using the direct and fixed point methods, we have established the generalized Hyers-Ulam stability of the following additive-quadratic&#x0D; functional equation&#x0D; in non-Archimedean and intuitionistic random normed spaces.&#x0D; &#x0D; AMS 2010 Subject Classification: 39B82, 39B52, 46S40.&#x0D; Keywords. generalized Hyers-Ulam stability; additive mapping; quadratic mapping;&#x0D; non-Archimedean random normed spaces; intuitionistic random normed spaces; fixed point.

On stationary points of multivalued strongly contractive mappings in partially ordered ultrametric spaces and non-Archimedean

The purpose of this paper is to present a fixed point theorem for multivalued strongly contractive mappings in partially ordered ultrametric spaces. Also a consequence is obtained for these mappings in partially ordered non-Archimedean normed spaces. Finally, two supporting examples of these results are given.