Lipschitz stability of the K-quadratic functional equation

Abstract

Let N be the set of all positive integers, G an Abelian group with a metric d and E a normed space. For any f : G → E we define the k-quadratic difference of the function f by the formula Q_k ƒ(x; y) := 2ƒ(x) + 2k²ƒ(y) - f(x + ky) - f(x - ky) for x; y ∈ G and k ∈ N. Under some assumptions about f and Qkƒ we prove that if Qkƒ is Lipschitz, then there exists a quadratic function K : G → E such that ƒ -K is Lipschitz with the same constant. Moreover, some results concerning the stability of the k-quadratic functional equation in the Lipschitz norms are presented. Mathematics Subject Classification (2010): Primary 39B82, 39B52.

Key words: k-quadratic functional equation, stability, Lipschitz space.