On the stability of Hosszú's functional equation
Let \(X\) be a normed space and \(Y\) be a Banach space. The following two theorems are proved. 1. If \(B\) is a subset of \(X^2\) such that the first (or second) coordinates of the points of \(B\) form a bounded set and \(f:X\to Y\) satisfies \(|f(x+y)-f(x)-f(y)|\leq\varepsilon\) for all \((x,y)\in X^2\setminus B\) with an \(\varepsilon\geq 0\), then there exists exactly one additive function \(A:X\to Y\) such that \(|f(x)-A(x)|\leq 5\varepsilon\) for every \(x\in X\). 2. If \(f:\mathbb{R}\to Y\) satisfies \(|f(x+y-xy)+f(xy)-f(x)-f(y)|\leq\varepsilon\) for all \(x,y\in\mathbb{R}\) with an \(\varepsilon\geq 0\), then there exists exactly one additive function \(A:\mathbb{R}\to Y\) and exactly one constant \(b\in Y\) such that \(|f(x)-A(x)-b|\leq 20\varepsilon\) for every \(x\in X\). The first theorem generalizes a result of \textit{F. Skof} [Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 117, No. 4-6, 377-389 (1983; Zbl 0794.39008)] whereas the second one contains that of \textit{C. Borelli} [Result. Math. 26, No. 3-4, 221-224 (1994; Zbl 0828.39019)].
scientific article; zbMATH DE number 920813