Asymptotic aspect of Drygas, quadratic and Jensen functional equations in metric abelian groups
Let \((X, +)\) and \((Y, +)\) be abelian groups. A function \(f \colon X \to Y \) is said to be -- additive, if \(f (x +y) = f (x) + f (y)\) holds for all \(x, y \in X\); -- Jensen, if \(f (x + y) + f (x-y) = 2f (x)\) is satisfied for all \(x, y \in X\); -- quadratic, if \(f (x +y) + f (x-y) = 2f (x) + 2f (y)\) is fulfilled for all \(x, y \in X\). Furthermore, the functional equation \[ f (x +y) + f (x-y) = 2f (x) + f (y) + f (-y) \;\;\; \left( x, y \in X\right) \] is known as Drygas equation. The main purpose of this paper is to investigate the asymptotic stability behaviour of the Drygas equation and quadratic and Jensen functional equations. In their main result the authors show that if these equations hold approximately for large arguments with an upper bound \(\varepsilon\), then they are also valid approximately everywhere with a new upper bound which is a constant multiple of \(\varepsilon\). These results are applied to the study of asymptotic properties of Drygas, quadratic and Jensen functional equations. Some results of hyperstability character for these functional equations are also obtained.
scientific article; zbMATH DE number 6931182
scientific article