A singular approach to a class of impulsive differential equation

In this paper, a singular approach to study the solutions of an impulsive differential equation from a qualitative and quantitative point of view is proposed. In the approach, a suitable singular perturbation term is introduced and a singularly perturbed system with infinite initial values is defined, in which, the reduced problem of the singularly perturbed system is exactly the impulsive differential equation under consideration. Then the boundary layer function method is applied to construct the uniformly valid asymptotic solutions to the singularly perturbed system. Based on the continuous asymptotic solution, the discontinuous solutions of the impulsive differential equation are described and approximated. An example, namely, a classical Lotka-Volterra prey-predator model with one pulse is carried out to illustrate the main results.