Second-Order Linear Delay Impulsive Differential Equations

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Abstract

In Chap. 13, second-order linear delay impulsive differential equations are considered. First, the equivalence of the four properties for a second-order impulsive delay equations is established: nonoscillation of the differential equation and of the corresponding differential inequality, positivity of the fundamental function and the existence of a solution of the generalized Riccati inequality. Further, comparison theorems, as well as explicit oscillation and nonoscillation conditions are justified. In the particular case when the values of impulses for the solution and its derivative match, a special nonimpulsive delay differential equation is constructed such that oscillation of an impulsive equation is equivalent to oscillation of the constructed nonimpulsive equation. As a consequence of this theorem, sharper nonoscillation results for this case of impulsive conditions are obtained, and oscillation properties of a second-order impulsive equation and some specially constructed nonimpulsive equation can be compared.