MR3364475 Sunao ŌUCHI Sunao ŌUCHI On Some Functional Equations with Borel Summable Solutions Funkcialaj Ekvacioj. Serio Internacia 58 2015 223--251 http://fe.math.kobe-u.ac.jp/FE/FullPapers/58-2/58_223.pdf http://www.ams.org/mathscinet-getitem?mr=MR3364475 A functional equation $\sum_{i=1}^{m}a_{i}u(\varphi_{i}(z))=f(z)$ is considered, where $\{\varphi_{i}(z)\}_{i=1}^{m}$ are holomorphic functions in a neighborhood of $z=0$ with $\varphi_{i}(0)=0$ and $f(z)$ is holomorphic in a sector with vertex $z=0$. It is shown under some conditions of $\{\varphi_{i}(z)\}_{i=1}^{m}$, $f(z)$ and $\{a_i\}_{i=1}^{m}$ that the equation has a formal power series solution that is Borel summable. Borel summable, Asymptotic expansion, Functional equation. 34A25; Secondary 34E05, 34K06, 44A10. 58-223 2015 On Some Functional Equations with Borel Summable Solutions Sunao ŌUCHI Sunao ŌUCHI
1 Abel, N. H. Détermination d'une fonction au moyen d'une équation qui ne contient que'une seule variable Oeuvres complètes de N. H. Abel, Christiania 1839 pp. 246-248
2 Balser, W. From Divergent Power Series to Analytic Functions Lecture Notes in Mathematics, 1582, Springer-Verlag, Berlin 1994 MR1317343 3 Balser, W. Formal power series and linear systems of meromorphic ordinary differential equations Universitext, Springer-Verlag, New York 2000 MR1722871
4 Balser, W. Multisummability of formal power series solutions of partial differential equations with constant coefficients J. Differential Equations 201 2004 63-74 MR2057538
5 Balser, W.; Braaksma, B. L. J.; Ramis, J.-P.; Sibuya, Y. Multisummability of formal power series solutions of linear ordinary differential equations Asymptotic Anal. 5 1991 27-45 MR1132079
6 Braaksma, B. L. J. Multisummability and Stokes multipliers of linear meromorphic differential equations J. Differential Equations 92 1991 45-75 MR1113588
7 Braaksma, B. L. J. Multisummability of formal power series solutions of nonlinear meromorphic differential equations Ann. Inst. Fourier (Grenoble) 42 1992 517-540 MR1182640
8 Kawai, T.; Takei, Y. Algebraic analysis of singular perturbation theory Translations of Mathematical Monographs, 227, Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI 2005 MR2182990
9 Lutz, D. A.; Miyake, M.; Schäfke, R. On the Borel summability of divergent solutions of heat equation Nagoya Math. J. 154 1999 1-29 MR1689170
10 Ōuchi, S. Characteristic Cauchy problems and solutions of formal power series Ann. Inst. Fourier (Grenoble) 33 1983 131-176 MR0698852
11 Ōuchi, S. Genuine solutions and formal solutions with Gevrey type estimates of nonlinear partial differential equations J. Math. Sci. Univ. Tokyo 2 1995 375-417 MR1366563
12 Ōuchi, S. Multisummability of formal solutions of some linear partial differential equations J. Differential Equations 185 2002 513-549 MR1935612
13 Ōuchi, S. Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields J. Math. Soc. Japan 57 2005 415-460 MR2123239
14 Ōuchi, S. Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains Asymptot. Anal. 47 2006 187-225 MR2233920
15 Wasow, W. Asymptotic Expansions of Ordinary Differential Equations Dover Publications, Inc., New York 1987 MR0919406