Multisummable solutions of nonlinear ordinary differential equations (Q1906167)

One of the main theorems is given below: A formal power series solution \[ \widehat y(x)= \sum^\infty_{k= n_0} y_k x^{- k/p},\quad p\in \mathbb{N},\quad n_0\in \mathbb{Z}, \] of a nonlinear ordinary differential equation \(x^{1- r} y'(x)= f(x, y)\), \(x\in \mathbb{C}\), \(y\in \mathbb{C}^n\), \(r\in \mathbb{Z}^+\), is multisummable in any nonsingular direction.