On parametric Borel summability for linear singularly perturbed Cauchy problems with linear fractional transforms

We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. 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 witnessed. Furthermore, unicity properties regarding the $1^{+}$ asymptotic layer are observed and follow from results on summability w.r.t a particular strongly regular sequence recently obtained in a previous study.