Sturmian comparison theory for impulsive differential inequalities and equations (Q1924946)

The authors generalize the Sturmian theory to second-order impulsive differential equations \((*)\) \(x''(t)+ p(t) x(t)= 0\), \(t\neq \tau_k\), \(\Delta x(\tau_k)=0\), \(\Delta x'(\tau_k)+p_k x(\tau_k)= 0\). Particularly, a comparison theorem, oscillation and non-oscillation theorems as well as a zero-separation theorem are proved. (Note that all solutions of an impulsive system, given in the special form \((*)\), are continuous.) Using comparison results and considering various simple (e.g., periodic, with constant coefficients) test systems, the authors present various sufficient conditions for oscillation and non-oscillation in \((*)\). On the other hand, this theory is also used here in the inverse order, to construct impulsive systems of the form \((*)\) with previously known oscillatory properties. The last section of the paper contains some applications of the main results to nonlinear impulsive differential equations.