On the Drygas functional equation in restricted domains

Abstract

Let ${{\mathbb{R}}}$ and Y be the set of real numbers and a Banach space respectively, and ${f, g :{\mathbb{R}} \to Y}$. We prove the Ulam-Hyers stability theorems for the Pexider-quadratic functional equation ${f(x + y) + f(x - y) = 2f(x) + 2g(y)}$ and the Drygas functional equation ${f(x + y) + f(x - y) = 2f(x) + f(y) + f(-y)}$ in the restricted domains of form ${\Gamma_d := \Gamma \cap \{(x, y) \in {\mathbb{R}}^2 : |x| + |y| \ge d\}}$, where ${\Gamma}$ is a rotation of ${B \times B \subset {\mathbb{R}}^2}$ and ${B^c}$ is of the first category. As a consequence we obtain asymptotic behaviors of the equations in a set ${\Gamma_d \subset {\mathbb{R}}^2}$ of Lebesgue measure zero.