Some results for conjugate equations

TL;DR

This paper studies conjugate equations, providing conditions for solutions' existence, uniqueness, and regularity, with applications to fractal functions on the Sierpinski gasket and stability analysis.

Contribution

It introduces new conditions for solutions to conjugate equations, explores their regularity, and presents novel fractal functions beyond harmonic or interpolation types.

Findings

Multiple solutions can exist for certain conjugate equations.

New class of fractal functions on Sierpinski gasket identified.

Conditions for solution regularity and stability established.

Abstract

In this paper we consider a class of conjugate equations, which generalizes de Rham's functional equations. We give sufficient conditions for existence and uniqueness of solutions under two different series of assumptions. We consider regularity of solutions. In our framework, two iterated function systems are associated with a series of conjugate equations. We state local regularity by using the invariant measures of the two iterated function systems with a common probability vector. We give several examples, especially an example such that infinitely many solutions exists, and a new class of fractal functions on the two-dimensional standard Sierpinski gasket which are not harmonic functions or fractal interpolation functions. We also consider a certain kind of stability.