Interval oscillation criteria for second-order linear differential equations with impulsive effects

Abstract

This paper discusses how the oscillation of solutions changes by adding an impulsive term to a second-order linear differential equation having only a restoring force term but no damping term. Due to the effect of this impulsive term, the moving speed of a mass point of the equation of motion described by the second-order linear differential equation changes discontinuously. Oscillation theorems are given for this impulsive differential equation. As is well known, when the restoring force term is small, all nontrivial solutions of the equation with no impulsive effects do not oscillate. Even when the restoring force term is small, if the action of the impulsive term compensates, the oscillation criteria obtained are satisfied and all nontrivial solutions oscillate. Several examples are given to confirm this fact and some figures are shown to depict the solution curves of these examples. As shown by the obtained oscillation theorems, there are cases that the impulsive effect promotes the oscillation of solutions, but on the contrary, it may suppress the oscillation of solutions. It is shown by a simulation that there is a case where the impulsive effect suppress the oscillation of solutions. Finally, the obtained results are compared with the previous ones.