The Gevrey Asymptotics in the Case of Singular Perturbations

In this paper, we present results in the Gevrey asymptotics which correspond to some existing results concerning asymptotic solutions in the Poincaré asymptotics of singularly perturbed ordinary differential equations. The main idea is based on a characterization of the Gevrey flat functions and a characterizaton of the Gevrey asymptotic expansions as it had been already exhibited by Y. Sibuya (1990, Lecture Notes in Pure and Applied Math., Vol. 124, pp. 393–401, Dekker, New York) and P. F. Hsieh and Y. Sibuya (1999, “Basic Theory of Ordinary Differential Equations,” Chap. XI, Sect. XI-2 and Chap. XII, Sect. XII-5, Springer-Verlag, Berlin).