Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 55, pp. 1-75.

Title: Parametric Borel summability for linear singularly perturbed
       Cauchy problems with linear fractional transforms

Authors: Alberto Lastra (Univ. de Alcala, Madrid, Spain)
         Stephane Malek (Univ. of Lille 1, France)

Abstract:
 We consider a family of linear singularly perturbed Cauchy problems which
 combines partial differential operators and linear fractional transforms.
 This work is the sequel of a study initiated in [17]. We construct
 a collection of holomorphic solutions on a full covering by sectors of
 a neighborhood of the origin in $\mathbb{C}$ with respect to
 the perturbation parameter $\epsilon$. This set is built up through
 classical and special Laplace transforms along piecewise linear paths of
 functions which possess exponential or super exponential growth/decay on
 horizontal strips.
 A fine structure which entails two levels of Gevrey asymptotics of order
 1 and so-called order $1^{+}$ is presented. Furthermore, unicity properties
 regarding the $1^{+}$ asymptotic layer are observed and follow from results
 on summability with respect to a particular strongly regular sequence
 recently obtained in [13].


Submitted April 20, 2018. Published April 29, 2019.
Math Subject Classifications: 35R10, 35C10, 35C15, 35C20.
Key Words: Asymptotic expansion; Borel-Laplace transform; Cauchy problem;
           difference equation; integro-differential equation;
           linear partial differential equation; singular perturbation.