Abstract
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 Painlevé equations.
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Acknowledgements
We would like to thank the referee for pointing us a minor error in the statement of Lemma 1 of [4]. Indeed, in the statement of this lemma we have written “for any integer \(\mu \geqslant \mu ^\prime\,\)” and missed an additional assumption “Re(s μ+1 − s μ ) > 0” which was used in its proof. Taking this into consideration, in Lemma 2.1 we have written a more precise “for any integer \(\mu \geqslant \mu ^\prime\) satisfying Re(s μ+1 − s μ ) > 0”. The research is supported by the Russian Foundation for Basic Research (projects no. RFBR 14-01-00346, RFBR-CNRS 16-51-150005).
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Gontsov, R.R., Goryuchkina, I.V. (2017). Towards the Convergence of Generalized Power Series Solutions of Algebraic ODEs. In: Filipuk, G., Haraoka, Y., Michalik, S. (eds) Analytic, Algebraic and Geometric Aspects of Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-52842-7_7
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