Geodesic
On~the~Cauchy problem for differential equations in spaces of resurgent functions
Izvestiya. Mathematics , Tome 40 (1993) no. 1, pp. 67-94

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A new method is proposed for obtaining asymptotic expansions of the solutions of a wide class of equations, taking subdominant, i.e., exponentially small, terms into account. The method is based on the study of the problem in special spaces of multiple-valued analytic functions. 
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     title = {On~the~Cauchy problem for differential equations in spaces of resurgent functions},
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B. Yu. Sternin; V. E. Shatalov. On~the~Cauchy problem for differential equations in spaces of resurgent functions. Izvestiya. Mathematics , Tome 40 (1993) no. 1, pp. 67-94. http://geodesic.mathdoc.fr/item/IM2_1993_40_1_a2/