Sine functional equation 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 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.

We present a survey of outcomes on Ulam stability of functional equations in non-Archimedean normed spaces. We focus mainly on functional equations in several variables (including the Cauchy equation, the Jordan–von Neumann equation, the Jensen equation, and their generalizations), but we also report a result on a general equation in a single variable, which can be applied to the very important linear functional equation. Let us note that one can observe the symmetry between the presented results and the analogous ones obtained for both classical and two-normed spaces.

Axiomatizations of Lovász extensions of pseudo-Boolean functions

Perturbations of a system of general functional equations in several variables

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.

Equations including unknown functions are called function al equations([1]). Among them, functionalequations containing only one independent variable are cal led iterative functional equations([9]). On theother hand, functional equations, like Cauchy's equation( [10]), containing two or more independent vari-ables for unknown functions of a single variable are called f unctional equations of several variables([2]).Among iterative functional equations, the problem of itera tive roots is important. If the n -th iterateof a function f is equal to g, i.e.f

Spectral synthesis and spectral analysis.- Spectral analysis and spectral synthesis on discrete Abelian groups.- Spectral synthesis and functional equations.- Mean periodic functions.- Difference equations in several variables.- Spectral analysis and synthesis on polynomial hypergroups in a single variable.- Spectral analysis and synthesis on multivariate polynomial hypergroups.

A sine functional equation

We prove the Hyers-Ulam stability of the sine and cosine functional equations in the spaces of generalized functions such as Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions.

We generalize the super-stability result for cosine functional equation mappings with values in the field of complex numbers to the case of an arbitrary Hilbert space with the Hadamard product. On the other hand, by an example we prove that this generalization is not true for sine functional equation. Mathematics subject classification (2010): Primary: 39B72, 39B82; secondary: 46E40.

The aim of this paper is to study the stability problem of the generalized sine functional equations as follows: $$ \begin{array}{*{20}c} {g(x)f(y) = f\left( {\frac{{x + y}} {2}} \right)^2 - f\left( {\frac{{x - y}} {2}} \right)^2 ,} \\ {f(x)g(y) = f\left( {\frac{{x + y}} {2}} \right)^2 - f\left( {\frac{{x - y}} {2}} \right)^2 ,} \\ {g(x)g(y) = f\left( {\frac{{x + y}} {2}} \right)^2 - f\left( {\frac{{x - y}} {2}} \right)^2 .} \\ \end{array} $$ .