Gevrey Solutions Of Singularly Perturbed Differential Equations

this article, we present an application where the existence of ireal" solutions can not be proved (at least at this time) without the Gevrey character of formal solutions. In the present form, our method does not provide optimal estimates for the type in the Gevrey estimates (i.e. the radius of convergence of the Borel transform) these estimates, though, are good enough to justify truncation at the least term. Optimal estimates have been achieved in the analog, but very special case of [29] by a judicious choice of the norms. Our method also does only prove local existence of solutions. Compare here with the results of Beno#t, Fruchard, Sch#fke and Wallet [5]: in the one dimensional case they obtain the existence of overstable solutions on ibig" domains (satisfying some geometrical condition) and good Gevrey estimates. We also would like to mention related work of Sibuya [32], Sch#fke [26] and the proof of an important conjecture on iturning points" of W. Wasow ([42], p.48) by C. Stenger [36] based on our results. 2 The equation