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Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero

Published online by Cambridge University Press:  20 November 2018

Chang-Kwon Choi ,
Jaeyoung Chung ,
Yumin Ju  and
John Rassias
Chang-Kwon Choi
Affiliation:
Department of Mathematics, Jeonbuk National University, Jeonju þ@Ë-6þ@, Republic of Korea e-mail: ck38@jbnu.ac.kr
Jaeyoung Chung
Affiliation:
Department of Mathematics, Kunsan National University, Kunsan þ6h-6§Ë, Republic of Korea e-mail: jychung@kunsan.ac.kr ymju7532@yahoo.com
Yumin Ju
Affiliation:
National and Capodistrian University of Athens, Pedagogical Department E. E., Athens, Greece e-mail: jrassias@primedu.uoa.gr
John Rassias
Affiliation:
Department of Mathematics, Kunsan National University, Kunsan þ6h-6§Ë, Republic of Korea e-mail: jychung@kunsan.ac.kr ymju7532@yahoo.com
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Abstract

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Let $X$ be a real normed space, $Y$ a Banach space, and $f\,:\,X\,\to \,Y$. We prove theUlam–Hyers stability theorem for the cubic functional equation

$$f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right)\,=\,0$$

in restricted domains. As an application we consider a measure zero stability problem of the inequality

$$\left\| f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right) \right\|\,\le \,\varepsilon$$

for all $\left( x,\,y \right)$ in $\Gamma \,\subset \,{{\mathbb{R}}^{2}}$ of Lebesgue measure 0.

Keywords

39B82Baire category theoremcubic functional equationfirst categoryLebesgue measureUlam-Hyers stability
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017 , pp. 95 - 103
Copyright
Copyright © Canadian Mathematical Society 2017

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