Convergence or Divergence of Formal Solutions of First Order Singular Nonlinear Ordinary Differential Equations in Complex Domain

MR2668515 Hidetaka USUI Hidetaka USUI Convergence or Divergence of Formal Solutions of First Order Singular Nonlinear Ordinary Differential Equations in Complex Domain Funkcialaj Ekvacioj. Serio Internacia 53 2010 89--98 http://fe.math.kobe-u.ac.jp/FE/FullPapers/53-1/53_89.pdf http://www.ams.org/mathscinet-getitem?mr=MR2668515 We study the convergence or divergence of formal solutions of first order singular nonlinear ordinary differential equation (SE) $f(x,u,u')=0$, $u(0)=0$, where the function $f$ is holomorphic in a neighbourhood of the origin of $\mathbb{C}^3$. The equation (SE) is said to be singular if $f(0,0,\xi)\equiv0$. M. Miyake and A. Shirai [2,3] already studied this problem in case of partial differential equation and they gave a criterion of convergence or divergence of formal solutions by using the Taylor coefficients of order 1 of formal solution and equation. In this paper, we give a criterion by using the Taylor coefficients of higher order of formal solution and equation. Singular nonlinear equation, Formal solution, Convergence, Divergence, Formal Gevrey index. Primary 34M25, Secondary 34A25, 34A34. 53-89 2010 Convergence or Divergence of Formal Solutions of First Order Singular Nonlinear Ordinary Differential Equations in Complex Domain Hidetaka USUI Hidetaka USUI 1 Gérard, R.; Tahara, H. Singular nonlinear partial differential equations Vieweg 1996 MR1757086

2 Miyake, M.; Shirai, A. Convergence of formal solutions of first order singular nonlinear partial differential equations in complex domain Ann. Polon. Math. 74 2000 215-228 MR1808996

3 Miyake, M.; Shirai, A. Structure of formal solutions of nonlinear first order singular partial differential equations in complex domain Funkcial. Ekvac. 48 2005 113-136 MR2154381 2154381

4 Shirai, A. Maillet type theorem for nonlinear partial differential equations and Newton polygons J. Math. Soc. Japan 53 2001 565-587 MR1828970

5 Shirai, A. A Maillet type theorem for first order singular nonlinear partial differential equations Publ. Res. Inst. Math. Sci. 39 2003 275-296 MR1987866