Asymptotic solution of a singularly perturbed Cauchy problem in the presence of a rational “simple” turning point

In this paper, based on S. A. Lomov's regularization method, we construct an asymptotic solution of a singularly perturbed Cauchy problem in the case of violation of the stability conditions for the spectrum of the limit operator. In particular, we consider the problem with a “simple” turning point, i.e., where one eigenvalue vanishes for $t=0$ and has the form $t^{m/n}$ (the limit operator is discretely irreversible).