Bull. Korean Math. Soc. 2009; 46(3): 499-510
Printed May 1, 2009
https://doi.org/10.4134/BKMS.2009.46.3.499
Copyright © The Korean Mathematical Society.
Won-Gil Park and Jae-Hyeong Bae
National Institute for Mathematical Sciences and Kyung Hee University
For a Borel function $\psi:\mathbb R\times\mathbb R\to\mathbb R$ satisfying the functional equation $\psi(s+t,u+v)+\psi(s-t,u-v)=2\psi(s,u)+2\psi(t,v)$, we show that it satisfies the functional equation $$\psi(s,t)=s(s-t)\psi(1,0)+st\psi(1,1)+t(t-s)\psi(0,1).$$ Using this, we prove the stability of the functional equation $$f(ax+ay,bz+bw)+f(ax-ay,bz-bw)=2abf(x,z)+2abf(y,w)$$ in Banach modules over a unital $C^\star$-algebra.
Keywords: stability, functional equation, quadratic form, Borel function
MSC numbers: Primary 39B22, 39B82
2023; 60(1): 75-81
2015; 52(6): 1759-1776
2021; 58(2): 269-275
2018; 55(2): 379-403
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd