Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2011; 48(4): 673-688

Printed July 1, 2011

https://doi.org/10.4134/BKMS.2011.48.4.673

Copyright © The Korean Mathematical Society.

On stability problems with shadowing property and its application

Hahng-Yun Chu, GilJun Han, and Dong Seung Kang

Chungnam National University, Dankook University, Dankook University

Abstract

Let $n \geq 2$ be an even integer. We investigate that if an odd mapping $f:X \rightarrow Y$ satisfies the following equation \begin{align*} &\ 2\, {}_{n-2}C_{\frac{n}{2}-1} r f\left(\sum^n_{j=1}\frac{x_j}{r}\right)+\sum_{\begin{smallmatrix} i_k\in \{0,1\} \\ \sum^n_{k=1} i_k=\frac{n}{2} \end{smallmatrix} } r f\left(\sum^n_{i=1}(-1)^{i_k}\frac{x_i}{r}\right) \\ =&\ 2 \, _{n-2}C_{\frac{n}{2}-1}\sum^n_{i=1}f(x_i), \end{align*} then $f:X \rightarrow Y$ is additive, where $r \in \mathbb{R}.$ We also prove the stability in normed group by using shadowing property and the Hyers-Ulam stability of the functional equation in Banach spaces and in Banach modules over unital $C^*$-algebras. As an application, we show that every almost linear bijection $h:A \rightarrow B$ of unital $C^*$-algebras $A$ and $B$ is a $C^*$-algebra isomorphism when $h(\frac{2^s}{r^s}uy)=h(\frac{2^s}{r^s}u)h(y)$ for all unitaries $u\in A,$ all $y\in A,$ and $s=0,1,2,\ldots.$

Keywords: shadowing property, Hyers-Ulam-Rassias stability, additive mapping, C*-algebra isomorphism

MSC numbers: 39B52