Existence and impulsive stability for second order delaydifferential equation

A class of the second order delayed differential equation is studied. Using Schaefer fixed point theorem, the existence of solution to the given model with impulse is proved and the proper Lyapunov functional is conducted to obtain that the nontrivial solution of the equation can be exponentially stabilized on $[t_0,+\infty)$. Two examples are given in the end.