Orthogonal generalized Lie bracket of triple derivations: properties and stability

In this article, by using the concept of Jensen and additive mappings, we define the Jensen ρ-functional equation which preserves orthogonality, where $\rho \neq 0, \pm 1$ , and we show it to be orthogonally additive. Also, we investigate it as an orthogonal $\mathbb{C}$ -linear mapping between orthogonal normed spaces. In the following, we introduce orthogonal generalized triple Lie bracket (triple) derivation-derivations on orthogonal triple Banach algebras. Finally, by employing the fixed point methods, we show that the orthogonal generalized triple Lie bracket (triple) derivation-derivations on orthogonal triple Banach algebras can be Hyers–Ulam stable and hyperstable.