On the Stability of Quadratic Functional Equations

Abstract

Let X , Y be vector spaces and k a fixed positive integer. It is shown that a mapping f ( k x + y ) + f ( k x - y ) = 2 k 2 f ( x ) + 2 f ( y ) for all x , y ∈ X if and only if the mapping f : X → Y satisfies f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y ) for all x , y ∈ X . Furthermore, the Hyers‐Ulam‐Rassias stability of the above functional equation in Banach spaces is proven.