Hyers–Ulam Stability of Solution for Generalized Lie Bracket of Derivations

Abstract

In this work, we present a new concept of additive-Jensen s-functional equations, where s is a constant complex number with s<1, and solve them as two classes of additive functions. We then indicate that they are C-linear mappings on Lie algebras. Following this, we define generalized Lie bracket derivations between Lie algebras. Then, we investigate the properties of Jordan on Lie bracket of derivations such that we show that the condition of Jordan Lie bracket derivation-derivation does not hold, but the condition of Lie bracket of Jordan derivations does hold. Ultimately, by using the fixed point theorem, we investigate the Hyers–Ulam stability of additive-Jensen s-functional equations and generalized Lie bracket derivations on Lie algebras with two control functions of Găvruta and Rassias.