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Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero

Published:2016-11-02
Printed: Mar 2017
• Chang-Kwon Choi,
Department of Mathematics, Jeonbuk National University, Jeonju 561-756, Republic of Korea
• Jaeyoung Chung,
Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea
• Yumin Ju,
Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea
• John Rassias,
National and Capodistrian University of Athens, Pedagogical Department E. E., Athens, Greece
 Format: LaTeX MathJax PDF

Abstract

Let $X$ be a real normed space, $Y$ a Bancch space and $f:X \to Y$. We prove the Ulam-Hyers stability theorem for the cubic functional equation \begin{align*} f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-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(2x-y)-2f(x+y)-2f(x-y)-12f(x)\|\le \epsilon \end{align*} for all $(x, y)$ in $\Gamma\subset\mathbb R^2$ of Lebesgue measure 0.
 Keywords: Baire category theorem, cubic functional equation, first category, Lebesgue measure, Ulam-Hyers stability
 MSC Classifications: 39B82 - Stability, separation, extension, and related topics [See also 46A22]

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