Picone Type Formula for Non-selfadjoint Impulsive Differential Equations with Discontinuous Solutions

As the impulsive differential equations are useful in modelling many real processes observed in physics, chemistry, biology, engineering, etc., see [1, 11, 13, 20, 21, 22, 25, 26, 27], there has been an increasing interest in studying such equations from the point of view of stability, asymptotic behavior, existence of periodic solutions, and oscillation of solutions. The classical theory can be found in the monographs [9, 18]. Recently, the oscillation theory of impulsive differential equations has also received considerable attention, see [2, 14] for the Sturmian theory of impulsive differential equations, and [15] for a Picone type formula and its applications. Due to difficulties caused by the impulsive perturbations the solutions are usually assumed to be continuous in most works in the literature. In this paper, we consider second order non-selfadjoint linear impulsive differential equations with discontinuous solutions. Our aim is to derive a Picone type identity for such impulsive differential equations, and hence extend and generalize several results in the literature. We consider second order linear impulsive differential equations of the form