Abstract
In this paper, we study both the oscillation and the stability of impulsive differential equations when not only the continuous argument but also the impulse condition involves delay. The results obtained in the present paper improve and generalize the main results of the key references in this subject. An illustrative example is also provided.
Highlights
The theory of impulsive differential equations is an important area of scientific activity, since every nonimpulsive differential equation can be regarded as an impulsive differential equation with no impulse effect, i.e., the corresponding impulse factor is the unit
Most of the publications are devoted to oscillation of first-order impulsive delay differential equations with instantaneous impulse conditions
Results dealing with retarded impulse conditions are relatively scarce, for instance, we can only a few papers which only deal with the stability property of the solutions
Summary
The theory of impulsive differential equations is an important area of scientific activity, since every nonimpulsive differential equation can be regarded as an impulsive differential equation with no impulse effect, i.e., the corresponding impulse factor is the unit. Our attention in this paper centers on the qualitative behaviour of solutions of the impulsive delay differential equation x′(t) + p(t)x τ (t) = 0 for t ∈ [θ0, ∞)\{θk}k∈N0 x(θk) = λkx(θk−−l) for k ∈ N0. The paper is organized as follows: In § 2, we construct the major equipments of the paper which all the results in the sequel will depend on; in § 3, we present our main results, which combine qualitative theory of delay differential equations and qualitative theory of delay differential equations in the absence of retardations in the impulse conditions; in § 4, to conclude the paper, we make our final comments and give a simple example to mention the significance and applicability of the main results. We always assume without mentioning that ∅ := 1 and ∅ := 0
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