On approximation of approximately generalized quadratic functional equation via Lipschitz criteria

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic difference of the function f by the formula Q k f ( x, y ) := f ( x + ky ) + f ( x - ky ) - f ( x + y ) - f ( x - y ) - 2( k 2 - 1) f ( y ) for all x , y ∈ G and for any integer k with k ≠ 1, -1. In this paper, we achieve the general solution of equation Q k f ( x, y ) = 0; after it, we show that if Q k f is Lipschitz, then there exists a quadratic function K : G  → E such that f - K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7]. Mathematics Subject Classification (2010): Primary 39B82, 39B52. Keywords: Generalized quadratic functional equation, stability, Lipschitz space