The asymptotic behavior of the approximate solution of a one-dimensional singularly perturbed Goldshtik problem

The Cauchy problem is considered for an ordinary differential equation with discontinuous phase-variable nonlinearity, in the right part of which a small parameter is included. The same parameter occurs in the initial conditions, leading to the problem going from classical to singularly perturbed. It seems possible to solve the problem in such a formulation, firstly based on the concept of an exact solution, by means of the theory of equations with discontinuous nonlinearities; and secondly, being singularly perturbed, by the method of constructing asymptotics of the boundary layer type. Since the exact solution suffers a discontinuity at the starting point, which is not justified in the physical sense, the equation is approximated in order to obtain an approximate smoothed solution. It requires a convergence to the exact solution when the small parameter tends to zero. An equation with a smoothed right-hand side gives a solution in quadratures. Then the proximity of its asymptotic to the exact solution is proved. From the exponential proximity of the asymptotic to the approximate solution, the required behavior follows for the latter.