Oscillation of a pantograph differential equation with impulsive perturbations

Abstract 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 ( α t ) , t ⩾ t 0 , t ≠ 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.

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