Introducing of an Orthogonally Relation for Stability of Ternary Cubic Homomorphisms and Derivations on C* -Ternary Algebras

Abstract

In this article‎, ‎we introduce a kind of binary relation on a nonempty set with  name of orthogonally relation which we  develop for sequences‎, ‎continuous maps‎, ‎metric spaces‎, ‎contraction maps‎, ‎preserving maps and etc‎. ‎All of the above concepts are generalized forms of ordinary case‎, ‎so they are very important for extension and finding new results‎. ‎we expect some of the concepts in the mathematics can be changed by orthogonally relation‎, ‎such as functional equations and some of the theorem in the fixed point theorem method‎. ‎In this research we illustrate one of the applications of orthogonally relation on ternary cubic homomorphism and ternary cubic derivations‎, ‎so we prove the stability of orthogonally ternary cubic homomorphisms and orthogonally ternary cubic derivations on $C^*$-ternary algebras for the functional equation by using fixed point method‎. ‎Also to create the stability‎, ‎we choose a suitable control function and we show ability and validity of the proposed method for the functional analysis‎.