Analytic solutions of a nonlinear iterative equation near neutral fixed points and poles

In this paper existence of analytic solutions of a nonlinear iterative equations is studied when given functions are all analytic and when given functions have poles. As well as in many previous works, we reduce this problem to finding analytic solutions of a functional equation without iteration of the unknown function f. For technical reasons, in previous works an indeterminate constant related to the eigenvalue of the linearized f at its fixed point O is required to fulfill the Diophantine condition that O is an irrationally neutral fixed point of f. In this paper the case of rationally neutral fixed points is also discussed, where the Diophantine condition is not required.