J. Chung [Aequationes Math. 90, No. 4, 799–808 (2016; Zbl 1353.39026)] studied the Hyers-Ulam stability of functional equations in restricted domains \(\Omega \subset X\times X\) satisfying the following condition. Let \((\gamma_j, \lambda_j) \in \mathbb{R}^2\) with \(\gamma_j^2+ \lambda_j^2\neq 0\) for all \(j = 1, 2, \dots, r\), and let for each \(p_j, q_j \in X,\ j = 1, 2, \dots, r\), there exists \(t \in X\) such that \[ \{(p_j + \gamma_j t, q_j + \lambda_jt) : j = 1, 2, \dots, r\} \subset \Omega.\tag{\(*\)} \]
In this paper, the authors investigate the Hyers-Ulam stability of mixed additive-quadratic and additive-Drygas functional equations \[ \begin{aligned} &2f(x + y) + f(x - y) - 3f(x) - 3f(y) = 0, \\ &2f(x + y) + f(x- y)-3f(x) -2f(y) - f(-y) = 0, \end{aligned} \] in restricted domains \(\Omega \subset X\times X\) satisfying the above condition.
By using the Baire category theorem, they prove the stability of the above functional equations on restricted domains of the form \(H^2\cap\{(x, y) \in X^2 : \|x\|+\|y\|\geq d\}\) with \(d > 0\), where \(H\) is a subset of \(X\) such that \(H^c=X\setminus H\) is of the first category. Constructing a subset \(\Omega_d\) of \(\{(x, y) \in \mathbb{R}^2 : |x|+|y|\geq d\}\) of \(2\)-dimensional Lebesgue measure-zero satisfying condition \((*)\), they obtained measure zero stability problems of the above functional equations, when \(X = \mathbb{R}\).