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Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero

  • Chang-Kwon Choi (a1), Jaeyoung Chung (a2), Yumin Ju (a3) and John Rassias (a2)

Abstract

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.

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Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero

  • Chang-Kwon Choi (a1), Jaeyoung Chung (a2), Yumin Ju (a3) and John Rassias (a2)

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