Approximate Homomorphisms and Derivations on Normed Lie Triple Systems

Juri Lee,Hark-Mahn Kim

doi:10.1155/2014/754291

Abstract

We investigate the generalized Hyers-Ulam stability of homomorphisms and derivations on normed Lie triple systems for the following generalized Cauchy-Jensen additive equationr0f((s∑j=1pxj+t∑j=1dyj)/r0)=s∑j=1p‍f(xj)+t∑j=1d‍f(yj), wherer0,s, and tare nonzero real numbers. As a results, we generalize some stability results concerning this equation.

Highlights

The stability problem of functional equations originated from a question of Ulam [1], concerning the stability of group homomorphisms.Let (G, ∘) be a group and let (H, ⋆, d) be a metric group with the metric d(⋅, ⋅)
We investigate the generalized Hyers-Ulam stability of homomorphisms and derivations on normed Lie triple systems for the following generalized Cauchy-Jensen additive equation r0f((s ∑pj=1 xj+t ∑dj=1 yj)/r0) = s ∑pj=1 f(xj)+t ∑dj=1 f(yj), where r0, s, and t are nonzero real numbers
In 1990, Rassias [5] during 27th international symposium on functional equations asked the question whether such a theorem can be proved for p ≥ 1

Summary

Introduction

The stability problem of functional equations originated from a question of Ulam [1], concerning the stability of group homomorphisms. A C-linear mapping D : A → A is called a Lie triple derivation if D([x, y, z]) = [D(x), y, z] + [x, D(y), z] + [x, y, D(z)] for all x, y, z ∈ A. Which is a generalization of Cauchy or Jensen additive equations, where r0, s, and t are nonzero real numbers and f is a mapping between linear spaces. We refine the generalized HyersUlam stability results of [25] for Lie triple homomorphisms and Lie triple derivations on Lie triple systems associated with the general Cauchy-Jensen additive equation (1) and we apply our results to study stability theorems of Lie triple homomorphisms and Lie triple derivations associated with. Cauchy-Jensen additive equation (1) on normed Lie triple systems, which can be regarded as ternary structures. Let BA be the set of all mappings from A to B, let L(A, B) be the set of all C-linear mappings from A to B, let H(A, B) be the set of all Lie triple homomorphisms from A to B, let D(A, A) be the set of all Lie triple derivations on A, and let R+ be the set of all nonnegative reals

Stability of Homomorphisms on Normed Lie Triple Systems

Stability of Derivations on Normed Lie Triple Systems