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 form {x'(t) = P(t)x(t) - Q(t)x(alpha t), t >= t(0), t not equal t(k,) (*) x(t(k)(+)) = b(k)x(t(k)), 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. (C) 2012 Elsevier Inc. All rights reserved.