Renat Ravilevich Gontsov

On the convergence of generalized power series satisfying an algebraic ODE

We propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.

The Maillet–Malgrange type theorem for generalized power series

There is proposed the Maillet–Malgrange type theorem for a generalized power series (having complex power exponents) formally satisfying an algebraic ordinary differential equation. The theorem describes the growth of the series coefficients.

Towards the Convergence of Generalized Power Series Solutions of Algebraic ODEs

The aim of this work is to provide another proof of the sufficient condition of the convergence of a generalized power series (with complex power exponents) formally satisfying an algebraic (polynomial) ordinary differential equation. This proof is based on the implicit mapping theorem for Banach spaces rather than on the majorant method used in our previous proof. We also discuss some examples of a such type formal solutions of Painleve equations.