Approximate homomorphisms and derivations on random Banach algebras

A Ebadian,M Madadian,M Eshaghi Gordji,H Azadi Kenary

doi:10.1186/1029-242x-2012-157

Abstract

Abstract The motivation of this paper is to investigate the stability of homomorphisms and derivations on random Banach algebras. MSC:39B82, 39B52.

Highlights

Introduction and preliminariesThe study of stability problems originated from a famous talk Under what condition does there exist a homomorphism near an approximate homomorphism? given by S
During the last three decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings
Let (X, μ, T) and (Y, μ, T) be random normed algebras: (i) An additive mapping H : X → Y is called a random homomorphism if H(xy) = H(x)H(y) for all x, y ∈ X. (ii) An additive mapping D : X → Y is called a random derivation if D(xy) = D(x)y – xD(y) for all x, y ∈ X

Summary

Introduction

Introduction and preliminariesThe study of stability problems originated from a famous talk Under what condition does there exist a homomorphism near an approximate homomorphism? given by S. Aoki [ ] and Rassias [ ] provided a generalization of the Hyers theorem for additive and linear functions respectively, by allowing the Cauchy difference to be unbounded. Rassias) Let X be a normed space, Y be a Banach space and f : X → Y be a function such that f (x + y) – f (x) – f (y) ≤ ε x p + y p for all x, y ∈ X, where ε and p are constants with ε > and p < .

Results

Conclusion