Generalized Ternary Hom-Derivations and Jensen $\rho$-Functional Equation‎: ‎Solving and Stability

Abstract

‎In this research‎, ‎we aim to present the concept of the new generalized Jensen $\rho$-functional equation‎. ‎Next‎, ‎by utilizing ternary homomorphisms and derivations‎, ‎we define the new generalized ternary hom-derivations linked to this equation within ternary Banach algebras‎. ‎We demonstrate that the generalized Jensen $\rho$-functional equation belongs to the category of additive functions‎. ‎Furthermore‎, ‎employing the fixed point theorem‎, ‎we establish the stability of both the generalized Jensen $\rho$-functional equation and the associated generalized ternary hom-derivations‎, ‎using control functions inspired by G$\check{a}$vruta and Rassias‎. ‎Lastly‎, ‎we investigate the Jordan property as it pertains to generalized ternary hom-derivations linked to this equation within ternary Banach algebras‎, ‎alongside the generalized ternary (Jordan) hom-derivations can be stable‎.