Aggregate special functions to approximate permuting tri-homomorphisms and permuting tri-derivations associated with a tri-additive -functional inequality in Banach algebras

In this paper, we define a new class of control functions through aggregate special functions. These class of control functions help us to stabilize and approximate a tri-additive psi-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.