Formal fundamental solutions of irregular singular differential equations depending upon parameters
The author investigates linear ordinary differential equations in complex time having a singular point at 0. Each of them has a canonical formal fundamental solution of the form \(Y(z)=H(z)z^J\exp(Q(z))\), where \(H\) is a formal Puiseux series, \(Q(z)\) is a diagonal matrix with Puiseux polynomial elements and \(J\) is a constant matrix commuting with every \(Q(z)\). Both \(J\) and \(Q\) are expressed explicitly from the right-hand side of the equation. The author studies the dependence of the formal solution on the parameter of the equation. It is known that in general the canonical formal fundamental solution is not even continuous in the parameter. It was shown in [\textit{D. G. Babbitt} and \textit{V. S. Varadarajan}, Mem. Am. Math. Soc. 325, 147 p. (1985; Zbl 0583.34007)] that if \(Q\) and \(J\) are independent of the parameter of equation, then there exists an analytic family of formal fundamental solutions. In fact, Babbitt and Varadarajan proved this statement under a weaker assumption allowing \(Q\) to be not necessarily constant, but ``well-behaved''. Their proof uses deep theorems of algebraic geometry and is not constructive: they do not give an explicit way to find the formal fundamental solution in terms of the initial equation. In the present paper, the author gives a constructive proof of the previously mentioned result by using the theory of \textit{E. Wagenführer} [J. Reine Angew. Math. 272, 150-172 (1975; Zbl 0309.34006), Analysis 3, 369-384 (1983; Zbl 0541.34003) and ibid. 9, 389-405 (1989; Zbl 0679.34005)]and by using results due to the first author and \textit{H. Volkmer} [J. Reine Angew. Math. 365, 80-96 (1986; Zbl 0569.34037)] and due to \textit{D. A. Lutz} and the first author [Linear Algebra Appl. 72, 1-46 (1985; Zbl 0577.34029)]. The author also presents a necessary and sufficient condition for a family of differential equations to have a family of canonical formal fundamental solutions with a ``well-behaved'' \(Q\) as a function of the parameter. This condition is stated directly in terms of the coefficients of the family of equations.
scientific article; zbMATH DE number 1901442