More About Functional Equations by Generalizing a Theorem

This paper stops at $\bar{g}$-calculus for modified $\bar{g}$-functions and relations through some interesting pseudo-functional equations as their solutions. The main problem we are dealing with in this research paper is generalizing the relations formulated by Ryb á rik’s theorem for pseudo-functional equations and complementing it with some other treated cases. We have proven in detail all the cases of Rybárik’s theorem. Still, in this paper, we will generalize them and some exceptional cases related to the conditions that satisfy the set of parameters or functions $(\propto, \lambda, f, h, \bar{g})$ that participate in the relevant relations of the generalized theorem. Furthermore, some exceptional cases for each relation are presented as main results that will connect this generalization theorem with Rybárik’s theorem and lead us to new results. Also, we are again in cases of classical analysis for the particular case of generator $\bar{g}$ as $\bar{g}-$ identity.