ON STABILITY OF QUADRATIC LIE HOM-DERS IN LIE BANACH ALGEBRAS

Abstract

We study the notion of a quadratic Lie hom-der in a Lie Banach algebra associated with the functional equation

$$f\left(\frac{x+y}{2}+z\right)+f\left(\frac{x+y}{2}-z\right)+f\left(\frac{x-y}{2}+z\right)+f\left(\frac{x-y}{2}-z\right)=f(x)+f(y)+4f(z),$$

which was first introduced by Park, Hong and Kim (2006). We also present a relation between the above functional equation and the quadratic functional equation on certain groups. Finally, we prove some stability results of the quadratic Lie hom-ders in Lie Banach algebras by using Hyers’ direct method.