Baire category theorem and approximation of the Pexider quadratic functional equation on a set of Lebesgue measure zero

Iz-iddine EL-Fassi; Abdellatif Chahbi; Samir Kabbaj

doi:10.1007/s41478-018-0112-7

Baire category theorem and approximation of the Pexider quadratic functional equation on a set of Lebesgue measure zero

Iz-iddine EL-Fassi, Abdellatif Chahbi, Samir Kabbaj

Source

The Journal of Analysis > 2019 > 27 > 3 > 697-707

Abstract

Let $$\mathbb {R}$$ R be the set of real numbers and Y be a Banach space. We investigate the Hyers-Ulam stability theorem when $$f,g:\mathbb {R}\rightarrow Y$$ f , g : R → Y satisfy the following Pexider quadratic inequality $$\begin{aligned} \Vert f(x+y)+f(x-y)-2g(x)-2f(y)\Vert \le \epsilon , \end{aligned}$$ ‖ f ( x + y ) + f ( x - y ) - 2 g ( x ) - 2 f ( y ) ‖ ≤ ϵ , in a set $$\Omega \subset \mathbb {R}^2$$ Ω ⊂ R 2 of Lebesgue measure $$m(\Omega )=0$$ m ( Ω ) = 0 .