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Hom-derivations in $$C*$$-ternary algebras. (English) Zbl 07343745
Summary: In this paper, we introduce and solve the following additive $$(\rho_1,\rho_2)$$-functional inequalities $\begin{split}\Vert f(x+y+z)-f(x)-f(y)-f(z)\Vert\\ \leq\Vert \rho_1(f(x+z)-f(x)-f(z)\Vert +\Vert\rho_2(f(y+z)-f(y)-f(z))\Vert, \end{split}$ where $$\rho_1$$ and $$\rho_2$$ are fixed nonzero complex numbers with $$|\rho_1|+|\rho_2| < 2$$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the above additive $$(\rho_1,\rho_2)$$-functional inequality in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of hom-derivations in $$C*$$-ternary algebras.
##### MSC:
 39B62 Functional inequalities, including subadditivity, convexity, etc. 16W25 Derivations, actions of Lie algebras 39B82 Stability, separation, extension, and related topics for functional equations 47H10 Fixed-point theorems 39B52 Functional equations for functions with more general domains and/or ranges 47B47 Commutators, derivations, elementary operators, etc. 46L57 Derivations, dissipations and positive semigroups in $$C^*$$-algebras
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