Approximation of a generalized Euler-Lagrange type additive mapping on Lie $C^{\ast}$-algebras

Wang, Zhihua; Sahoo, Prasanna K.

doi:10.22075/ijnaa.2017.1332.1329

Approximation of a generalized Euler-Lagrange type additive mapping on Lie $C^{\ast}$-algebras

Document Type : Research Paper

Authors

Zhihua Wang ¹

Prasanna K. Sahoo ²

¹ School of Science, Hubei University of Technology, Wuhan, Hubei 430068, P.R. China

² Department of Mathematics, University of Louisville, Louisville, KY 40292, USA

10.22075/ijnaa.2017.1332.1329

Abstract

Using fixed point method, we prove some new stability results for Lie $(\alpha,\beta,\gamma)$-derivations and Lie $C^{\ast}$-algebra homomorphisms on Lie $C^{\ast}$-algebras associated with the Euler-Lagrange type additive functional equation
\begin{align*}
\sum^{n}_{j=1}f{\bigg(-r_{j}x_{j}+\sum_{1\leq i \leq n, i\neq
j}r_{i}x_{i}\bigg)}+2\sum^{n}_{i=1}r_{i}f(x_{i})=nf{\bigg(\sum^{n}_{i=1}r_{i}x_{i}\bigg)}
\end{align*}
where $r_{1},\ldots,r_{n}\in {\mathbb{R}}$ are given and $r_{i},r_{j}\neq 0$ for some $1\leq i< j\leq n$.

Keywords

Fixed point theorem

Lie $(alpha,beta,gamma)$-derivation

Lie $C^{ast}$-algebra homomorphisms

generalized Hyers-Ulam stability