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Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero


Published:20161102
Printed: Mar 2017
ChangKwon Choi,
Department of Mathematics, Jeonbuk National University, Jeonju 561756, Republic of Korea
Jaeyoung Chung,
Department of Mathematics, Kunsan National University, Kunsan 573701, Republic of Korea
Yumin Ju,
Department of Mathematics, Kunsan National University, Kunsan 573701, Republic of Korea
John Rassias,
National and Capodistrian University of Athens, Pedagogical Department E. E., Athens, Greece
Abstract
Let $X$ be a real normed space, $Y$ a Bancch space and $f:X \to
Y$.
We prove the UlamHyers stability theorem
for the cubic functional equation
\begin{align*}
f(2x+y)+f(2xy)2f(x+y)2f(xy)12f(x)=0
\end{align*}
in restricted domains. As an application we consider a measure
zero stability problem
of the inequality
\begin{align*}
\f(2x+y)+f(2xy)2f(x+y)2f(xy)12f(x)\\le \epsilon
\end{align*}
for all $(x, y)$ in $\Gamma\subset\mathbb R^2$ of Lebesgue measure
0.