Treatment of the Nonlinear Problems Through the Generalization of a Theorem from Parameterized Nonlinear Continuous Functions

Abstract: This paper focuses on ?̅ − ???????? for modified g̅-functions as ??̅ − ???????? and relations through some interesting pseudo-functional equations as their solutions and also expression by some parameterized nonlinear continuous functions ??̅ . The main problem we are dealing with in this research paper is generalizing the relations formulated by Rybárik’s theorem for pseudo-functional equations and complementing it with some other treated cases expressed by some parameterized nonlinear continuous functions. 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 (∝, ?, ?, ℎ, ?̅) that participate in the relevant pseudo-nonlinear relations of the generalized theorem from parameterized nonlinear continuous functions. 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. We have built a system of important pseudo-nonlinear relations for representation the ??̅ − ????????, ??̅ − ???????? by the parameterized nonlinear continuous functions ??̅ . Also, we are again in cases of classical analysis for the particular case of generator ?̅ as ?̅ − ????????.