Almost Jordan homomorphisms and Jordan derivations associated to the parametric-additive functional equation on fuzzy Banach algebras

Hassan Azadi Kenary,Madjid Eshaghi Gordji,Dong Yun Shin,Hamid Rezaei,Yosouf Gheisari

doi:10.1186/1687-1847-2012-70

Hassan Azadi Kenary, Madjid Eshaghi Gordji + Show 3 more

Open Access

https://doi.org/10.1186/1687-1847-2012-70

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Abstract

Abstract In this article, we establish the generalized Hyers-Ulam (or Hyers-Ulam-Rassais) stability of Jordan homomorphisms and Jordan derivations of the following parametric additive functional equation: ∑ i = 1 m f ( x i ) = 1 2 m ∑ i = 1 m f m x i + ∑ j = 1 , j ≠ i m x j + f ∑ i = 1 m x i for a fixed positive integer m with m ≥ 2, on fuzzy Banach algebras. The concept of Ulam-Hyers-Rassias stability originated from Rassias stability theorem that appeared in his article. Mathematics Subject Classification: Primary, 46S40; Secondary, 39B52; 39B82; 26E50; 46S50; 46H25.

Highlights

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms
Given ε >0, does there exist a δ0, such that if a mapping h: G1 ® G2 satisfies the inequality d(h(x.y), h(x) * h(y)) < δ for all x, y Î G1, there exists a homomorphism H: G1 ® G2 with d(h(x), H(x)) < ε for all x Î G1? In the other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation
We investigate the generalized Hyers-Ulam stability of Jordan homomorphisms and Jordan derivations of the following parametric-additive functional equation m f (xi) =

Summary

Introduction

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let f: E ® E’’ be a mapping between Banach spaces such that ||f(x+y)-f(x)-f(y)|| ≤ δ for all x, y E, and for some δ >0. Let f: E ® E’ be a mapping from a normed vector space E into a Banach space E’ subject to the inequality f (x + y) − f (x) − f (y) ≤ x p + y p for all x, y Î E, where and p are constants with ε >0 and p

Results

Conclusion