Functional Equations In Several Variables Research Articles

Stability of a General Functional Equation in m-Banach Spaces

In this note, the Ulam stability of a general functional equation in several variables is investigated. It is shown that this equation is Ulam stable in m-Banach spaces. Since a particular case of the considered equation is, among others, a functional equation introduced by Ji et al. and Zhao et al. for a characterization of the so-called multi-quadratic mapping, a result on its stability is also presented. Moreover, some other applications are provided.

On perturbations of two general equations in several variables

In this paper, we deal with perturbations of two general functional equations in several variables. Namely, we prove the generalized, in the spirit of Bourgin, Ulam stability of these equations in Banach spaces. In order to do this, we use the fixed point method. Moreover, as corollaries from our main results, we get several outcomes on approximate solutions of a few important classic equations. They include, among others, the functional equations which characterize multi-additive and multi-quadratic mappings. In consequence, the perturbation of homomorphisms of Banach spaces and quadratic mappings is also treated.

Ulam stability of functional equations in 2-Banach spaces via the fixed point method

Using the fixed point method, we prove the Ulam stability of two general functional equations in several variables in 2-Banach spaces. As corollaries from our main results, some outcomes on the stability of a few known equations being special cases of the considered ones will be presented. In particular, we extend several recent results on the Ulam stability of functional equations in 2-Banach spaces.

HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

Zhang [‘On hyperstability of generalised linear functional equations in several variables’, Bull. Aust. Math. Soc.92 (2015), 259–267] proved a hyperstability result for generalised linear functional equations in several variables by using Brzdęk’s fixed point theorem. We complete and extend Zhang’s result. We illustrate our results for general linear equations in two variables and Fréchet equations.

Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes

The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and series-parallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].

Hyers–Ulam–Rassias Stability of Set Valued Additive and Cubic Functional Equations in Several Variables

In this paper, we establish Hyers–Ulam–Rassias stability results belonging to two different set valued functional equations in several variables, namely additive and cubic. The results are obtained in the contexts of Banach spaces. The work is in the domain of set valued analysis.

Functional equations with division and regular operations

Functional equations are equations in which the unknown (or unknowns) are functions. We consider equations of generalized associativity, mediality (bisymmetry, entropy), paramediality, transitivity as well as the generalized Kolmogoroff equation. Their usefulness was proved in applications both in mathematics and in other disciplines, particularly in economics and social sciences (see J. Aczél, On mean values, Bull. Amer. Math. Soc. 54 (1948) 392–400; J. Aczél, Remarques algebriques sur la solution donner par M. Frechet a l’equation de Kolmogoroff, Pupl. Math. Debrecen 4 (1955) 33–42; J. Aczél, A Short Course on Functional Equations Based Upon Recent Applications to the Social and Behavioral Sciences, Theory and decision library, Series B: Mathematical and statistical methods (D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1987); J. Aczél, Lectures on Functional Equations and Their Applications (Supplemented by the author, ed. H. Oser) (Dover Publications, Mineola, New York, 2006); J. Aczél, V. D. Belousov and M. Hosszu, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar. 11 (1960) 127–136; J. Aczél and J. Dhombres, Functional Equations in Several Variables (Cambridge University Press, New York, 1991); J. Aczél and T. L. Saaty, Procedures for synthesizing ratio judgements, J. Math. Psych. 27(1) (1983) 93–102). We use unifying approach to solve these equations for division and regular operations generalizing the classical quasigroup case.

On a fixed point theorem in 2-Banach spaces and some of its applications

The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries corresponding to some known hyperstability outcomes are presented.

Non-archimedean stability and non-stability of quadratic reciprocal functional equation in several variables

Non-archimedean stability and non-stability of quadratic reciprocal functional equation in several variables

Non-Archimedean stability of a generalized reciprocal-quadratic functional equation in several variables by direct and fixed point methods

This study is aimed to determine various stabilities of a generalized reciprocal-quadratic functional equation of the form r (?pj=1 ?juj) = ?pj=1 r (uj) [?pj=1 ?j ?pk=1,k?j ?r(uk)] 2 connected with Ulam, Hyers, T. M. Rassias, J. M. Rassias and Gavruta in non-Archimedean fields, where ?j?0; j = 1,2,..., p are arbitrary real numbers and 0 < ?1+?2+...+?p = ?pj=1 ?j=??1 in non-Archimedean fields by direct and fixed point methods.

Non-Archimedean stability of a generalized reciprocal-quadratic functional equation in several variables by direct and fixed point methods

This study is aimed to determine various stabilities of a generalized reciprocal-quadratic functional equation of the form r (Σpj=1 βjuj) = Πpj=1 r (uj) [Σpj=1 βj Πpk=1,k≠j √r(uk)] 2 connected with Ulam, Hyers, T. M. Rassias, J. M. Rassias and Gavruta in non-Archimedean fields, where βj≠0; j = 1,2,..., p are arbitrary real numbers and 0 < β1+β2+...+βp = Σpj=1 βj=β≠1 in non-Archimedean fields by direct and fixed point methods.

A Fixed Point Approach to the Stability of a Mean Value Type Functional Equation

We prove the generalized Hyers–Ulam stability of a mean value type functional equation f ( x ) − g ( y ) = ( x − y ) h ( x + y ) by applying a method originated from fixed point theory.

Stability problem for the composite type functional equations

Composite type functional equations in several variables play a significant role in various branches of mathematics and they have several interesting applications. Therefore their stability properties are of interest. The aim of this paper is to present a survey of some results and methods concerning stability of several of such equations; especially those somehow connected to the well known Gołąb–Schinzel equation.

Erratum to: Approximation on the reciprocal functional equation in several variables in matrix non-Archimedean random normed spaces

Erratum to: Approximation on the reciprocal functional equation in several variables in matrix non-Archimedean random normed spaces

Approximation on the reciprocal functional equation in several variables in matrix non-Archimedean random normed spaces

In this paper, we investigate the generalized Hyers-Ulam stability of a reciprocal type functional equation in several variables in matrix non-Archimedean random normed spaces by direct and fixed point methods.

Solution and stability of a reciprocal type functional equation in several variables

Solution and stability of a reciprocal type functional equation in several variables

Stability of a mixed additive and cubic functional equation in several variables in non-Archimedean spaces

Consider the functional equation \({\Im_1(f ) = \Im_2(f )\,\,(\Im)}\) in a certain general setting. A function g is an approximate solution of \({(\Im)}\) if \({\Im_1(g)}\) and \({\Im_2(g)}\) are close in some sense. The Ulam stability problem asks whether or not there is a true solution of \({(\Im)}\) near g. In this paper, we achieve the general solution and the stability of the following functional equation $$\begin{array}{ll}f\left(\sum\limits^{n}_{i=1}x_{i} \right)+f\left(\sum\limits^{n-1}_{i=1} x_{i}-x_{n} \right)\\\quad=2f\left(\sum\limits^{n-1}_{i=1}x_{i} \right)+\sum\limits^{n-1}_{i=1}(f(x_{i}+x_{n}) +f(x_{i}-x_{n})-2f(x_{i}))\end{array}$$ for all xi (i = 1,2, . . . , n), in non-Archimedean spaces.

Axiomatizations of Lovász extensions of pseudo-Boolean functions

Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, these functions coincide with the Lovász extensions vanishing at the origin, which subsume the discrete Choquet integrals. We also propose a simultaneous generalization of horizontal min-additivity and horizontal max-additivity, called horizontal median-additivity, and we describe the corresponding function class. Additional conditions then reduce this class to that of symmetric Lovász extensions, which includes the discrete symmetric Choquet integrals.

On a direct method for proving the Hyers–Ulam stability of functional equations

We study the stability of an equation in a single variable of the form f ( x ) = a f ( h ( x ) ) + b f ( − h ( x ) ) with some conditions imposed on constants a, b and function h. The results are later applied for proving the stability (by use of a direct method) of a couple of functional equations in several variables. They let us not only shorten various approaches obtained during the last years but also improve the approximating constants.

Stability of a mixed functional equation in several variables on Banach modules

In this paper, we establish the general solution of the functional equation ∑ i = 1 n f ( x i − 1 n ∑ j = 1 n x j ) = ∑ i = 1 n f ( x i ) − n f ( 1 n ∑ i = 1 n x i ) ( n ≥ 2 ) , and use a fixed point method to prove its generalized Hyers–Ulam stability in Banach modules over a unital Banach algebra.