TY - JOUR
T1 - Approximation of the Cubic Functional Equations in Lipschitz Spaces
JO - Analysis in Theory and Applications
VL - 4
SP - 354
EP - 362
PY - 2014
DA - 2014/11
SN - 30
DO - http://doi.org/10.4208/ata.2014.v30.n4.2
UR - https://global-sci.org/intro/article_detail/ata/4499.html
KW - Cubic functional equation, Lipschitz space, stability.
AB - <p style="text-align: justify;">Let $\mathcal{G}$ be an Abelian group and let $\rho:\mathcal{G} \times \mathcal{G} \rightarrow [0, \infty)$ be a metric on $\mathcal{G}$. Let $\varepsilon$ be a normed space. We prove that under some conditions if $f:\mathcal{G}\to\varepsilon$ is an odd function and $C_x:\mathcal{G}\to\varepsilon$ defined by $C_x(y):=2f(x+y)+2f(x-y)+12f(x)-$ $f(2x+y)-f(2x-y)$ is a cubic function for all $x\in \mathcal{G},$ then there exists a cubic function $C:\mathcal{G}\to\varepsilon$ such that $f-C$ is Lipschitz. Moreover, we investigate the stability of cubic functional equation $2f(x+y)+2f(x-y)+12f(x)-f(2x+y)$ $-f(2x-y)=0$ on Lipschitz spaces.</p>