Abstract
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
Highlights
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
A solution is called oscillatory if it has arbitrarily large generalized zeros, and a differential equation is oscillatory if every solution of the equation is oscillatory
The Picone type formula is obtained by making use of the following Picone type identity, consisting of a pair of identities
Summary
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. 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 extend and generalize several results in the literature. Picone type formula; Sturm-Picone comparison; Leighton comparison; Oscillation; Second order; Non-selfadjoint; Impulse. (ii) k, r, p, m, s, q ∈ PLC(I0) := h : I0 → R is continuous on each interval (θi, θi+1), h(θi±) exist, h(θi) = h(θi−) for i ∈ N with k(t) > 0, m(t) > 0 for all t ∈ I0. By a solution of the impulsive system (1.1) on an interval I0 ⊂ [t0, ∞), we mean a nontrivial function x which is defined on I0 such that x, x′, (kx′)′ ∈ PLC(I0) and that x satisfies (1.1) for all t ∈ I0. A solution is called oscillatory if it has arbitrarily large generalized zeros, and a differential equation is oscillatory if every solution of the equation is oscillatory
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