Hyperstability of a Linear Functional Equation on Restricted Domains

Let X, Y be real Banach spaces, f : X → Y and ℋ $$\mathcal H$$ be a subset of X such that ℋ c $${\mathcal H}^c$$ is of the first category. Using the Baire category theorem we prove the Ulam–Hyers stability of the linear functional equation f ( a x + b y + α ) = A f ( x ) + B f ( y ) + C $$\displaystyle f(ax+by+\alpha )=Af(x)+Bf(y)+C $$ for all x , y ∈ ℋ $$x, y\in \mathcal H$$ , such that ∥x∥ + ∥y∥≥ d with d > 0, where a, b, A, B are nonzero real numbers and α ∈ X is fixed. As a consequence we solve the hyperstability problem associated to ∥ f ( a x + b y + α ) − A f ( x ) − B f ( y ) − C ∥ ≤ δ ψ ( x , y ) $$\displaystyle \|f(ax+by+\alpha )-Af(x)-Bf(y)-C\|\le \delta \psi (x,y) $$ for all x , y ∈ K $$x, y \in \mathcal K$$ , where K $$\mathcal K$$ is a subset of ℝ $$\mathbb {R}$$ with Lebesgue measure zero and ψ(x, y) = |x|p + |y|q, p, q