Functional Equations and Inequalities in Several Variables

10.1142/4875 ◽  
2002 ◽  
Author(s):  
Stefan Czerwik
Keyword(s):  
Functional Equations ◽  
Several Variables

HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

2020 ◽  
Vol 102 (2) ◽  
pp. 293-302
Author(s):  
THEERAYOOT PHOCHAI ◽  
SATIT SAEJUNG
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Functional Equations ◽  
Linear Functional ◽  
Linear Equations ◽  
General Linear ◽  
Several Variables ◽  
Linear Functional Equations

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.

Functional equations with division and regular operations

2018 ◽  
Vol 11 (03) ◽  
pp. 1850033
Author(s):  
Sergey Davidov ◽  
Aleksandar Krapež ◽  
Yuri Movsisyan
Keyword(s):  
New York ◽  
Functional Equations ◽  
Acta Math ◽  
Reidel Publishing Company ◽  
Mean Values ◽  
Cambridge University ◽  
Several Variables ◽  
Short Course ◽  
The Social ◽  
Generalized Associativity

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 HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

2015 ◽  
Vol 92 (2) ◽  
pp. 259-267 ◽  
Author(s):  
DONG ZHANG
Keyword(s):  
Functional Equation ◽  
Exact Solutions ◽  
Normed Space ◽  
Functional Equations ◽  
Linear Functional ◽  
Approximate Solutions ◽  
Linear Functional Equation ◽  
Several Variables ◽  
Linear Functional Equations

We obtain some results on approximate solutions of the generalised linear functional equation $\sum _{i=1}^{m}L_{i}f(\sum _{j=1}^{n}a_{ij}x_{j})=0$ for functions mapping a normed space into a normed space. We show that, under suitable assumptions, the approximate solutions are in fact exact solutions. The theorems correspond to and complement recent results on the hyperstability of generalised linear functional equations.

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