Stability of generalized Cauchy equations

Roman Badora,Barbara Przebieracz,Peter Volkmann

doi:10.1007/s00010-014-0300-8

Roman Badora, Barbara Przebieracz + Show 1 more

Open Access

https://doi.org/10.1007/s00010-014-0300-8

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Journal: Aequationes mathematicae	Publication Date: Oct 10, 2014
Citations: 5	License type: CC BY 4.0

Affiliation: University of Silesia

Abstract

We investigate the stability of the functional equation f(xy )= g(x)h(y )+ k(y) on amenable semigroups. This equation is a common generalization of two Pexider equations stemming from Cauchy's additive and multiplicative functional equations, and it is a simple case of the Levi-Civita equation. Mathematics Subject Classification. Primary 39B82; Secondary 39B22.

Highlights

A common generalization of Pexider’s equations f = g(x) + k(y) and f = g(x)h(y) is f = g(x)h(y) + k(y). (1.1)For real functions this equation has already been treated in J
In the present paper we investigate the stability in the sense of Hyers-Ulam of equation (1.1) on amenable semigroups
The main result of our paper is contained

Summary

Introduction

For real functions this equation has already been treated in J. 3. f (x) = v(a(x) + b), g(x) = a(x) + b − c, h(x) = v, k(x) = v(a(x) + c), where a : S → F solves the additive Cauchy equation a(xy) = a(x) + a(y), x, y ∈ S. 4. f (x) = v(ce(x) + b), g(x) = ce(x) + u, h(x) = ve(x), k(x) = v(b − ue(x)), where e : S → F solves the multiplicative Cauchy equation e(xy) = e(x)e(y), x, y ∈ S. In the present paper we investigate the stability in the sense of Hyers-Ulam of equation (1.1) on amenable semigroups.

Results

Conclusion