  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.
