Generalizations of the Implicit Function Theorem to Fractal Sets

Abstract

Abstract This research paper explores the extension of the Implicit Function Theorem (IFT) to fractal sets. The objective is to establish the existence and differentiability of implicitly defined functions on selected fractal sets, thus generalizing the classical IFT to the realm of fractal geometry. The methodology involves a precise characterization of fractals and the formu-lation of an extended IFT. Fractals are defined as self-similar, non-smooth geometric objects with intricate structures. The extension of the IFT ne-cessitates adaptations of the concepts of differentiability and surjectivity to fit the unique properties of fractal sets. The Fr´echet derivative on the fractal set is modified to capture the local linear behavior of functions, while the notion of surjectivity is extended by requiring every open ball in the target space to contain the image of some open ball in the fractal set. Through the validation process, the extended IFT is applied to specific fractal sets, like Sierpinski Triangle and Koch Curve. By constructing functions and computing their Fr´echet derivatives, we verify the condi-tions of our theorem, ensuring the surjectivity of the Fr´echet derivative for at least one point in the fractal set. This validation serves as concrete evidence of the applicability and robustness of the extended IFT in the fractal context. The major findings of this research demonstrate that the extended IFT provides a powerful tool for analyzing implicitly defined functions on fractal sets. It opens up new avenues for studying solutions to partial differential equations, investigating optimization problems, and exploring the dynamics of fractal systems.