Oscillation of second order linear differential equations with impulses

Abstract

We use the associated Riccati techniques and the equivalence transformation to discuss the oscillation and the nonoscillation of the second order linear ordinary differential equation with impulses of the form { ( a ( t ) x ′ ( t ) ) ′ + p ( t ) x ( t ) = 0 , t ≥ t 0 , t ≠ t k , x ( t k + ) = b k x ( t k ) , x ′ ( t k + ) = c k x ′ ( t k ) , k = 1 , 2 , … . Several good results are obtained. Some examples are also given which show that the oscillation of impulsive differential equations can be caused by impulsive perturbations, though the corresponding classical equation admits a nonoscillatory solution.