Hyers-Ulam-Rassias stability of Jensen’s equation and its application

Abstract

The Hyers-Ulam-Rassias stability for the Jensen functional equation is investigated, and the result is applied to the study of an asymptotic behavior of the additive mappings; more precisely, the following asymptotic property shall be proved: Let X X and Y Y be a real normed space and a real Banach space, respectively. A mapping f : X → Y f: X \rightarrow Y satisfying f ( 0 ) = 0 f(0)=0 is additive if and only if ‖ 2 f [ ( x + y ) / 2 ] − f ( x ) − f ( y ) ‖ → 0 \left \| 2f\left [ (x+y)/2 \right ] - f(x) - f(y) \right \| \rightarrow 0 as ‖ x ‖ + ‖ y ‖ → ∞ \| x \| + \| y \| \rightarrow \infty .