Analytic Solutions of a Polynomial-Like Iterative Functional Equation near Resonance

Abstract:

      In this paper existence of local analytic solutions of a polynomial-like iterative functional equation is studied. As well as in previous work, we reduce this problem with the Schr\"oder transformation to finding analytic solutions of a functional equation without iteration of the unknown function $f$. For technical reasons, in previous work the constant $\alpha$ given in the Schr\"oder transformation, i.e., the eigenvalue of the linearized $f$ at its fixed point $O,$ is required to fulfill that $\alpha$ is off the unit circle $S^1$ or lies on the circle with the Diophantine condition. In this paper, we obtain results of analytic solutions in the case of $\alpha$ at resonance, i.e., at a root of the unity and the case of $\alpha$ near resonance under the Brjuno condition.