Abstract
In this paper, the initial value problem for the third-order linear differential equation with small parameters at the two highest derivatives and piecewise-constant argument was considered when the roots of additional characteristic equation have negative signs. The aim of this paper is to obtain an explicit formula and the asymptotic estimations of the solution. The fundamental system of solutions, initial functions are constructed and their asymptotic estimations are obtained. It is also shown that the solution of the given problem at the points $$t=\theta_{i}$$ , $$i=\overline{0,p}$$ have the phenomenon of an initial jump of the first order of the second degree.