On the convergence of a generalized power series satisfying an ODE

In the talk I will prove a theorem that gives a sufficient condition for convergence of a generalized formal power series (i.e. a series with complex exponents) that is a formal solution of an algebraic ODE. Such generalized formal power series occur often as solutions of nonlinear differential equations, for example as solutions of the Painleve 3, 5 and 6 equations.
The proof is based on Malgrange's technique, who used it in the proof of the Maillet theorem. The keystone of the proof is the implicit mapping theorem for Banach spaces. We also shall use this implicit mapping theorem. The main theorem of the talk is not original, previously it was proved by the method of majorants. But using the Malgrange's technique I expect to obtain the Maillet theorem for divergent generalized power series that satisfy an algebraic ODE