Oscillation of a pantograph differential equation with impulsive perturbations

Some sufficient conditions are obtained for the oscillation of all solutions of a pantograph differential equation with impulsive perturbations of the formx′(t)=P(t)x(t)-Q(t)x(αt),t⩾t0,t≠tk,(∗)x(tk+)=bkx(tk),k=1,2,…(∗∗)Our results reveal the fact that the oscillatory properties of all solutions of impulsive differential equations (∗) and (∗∗) may be caused by the impulsive perturbations (∗∗), though the corresponding differential equations without impulses admit a nonoscillatory solution. Some examples are also given to illustrate the applicability of the results obtained.