Nonoscillation for a Second Order Linear Delay Differential Equation with Impulses

Abstract

A group of necessary and sufficient conditions for the nonoscillation of a second order linear delay equation with impulses

$$ {\left( {r{\left( t \right)}{u}\ifmmode{'}\else$'$\fi} \right)}^{\prime } = - p{\left( t \right)}u{\left( {t - r} \right)} $$

are obtained in this paper, where $p{\left( t \right)} = {\sum\limits_{n = 1}^\infty {\alpha _{n} \delta {\left( {t - t_{n} } \right)},{\kern 1pt} \delta {\left( t \right)}} }$ is a Dirac δ−function, and for each n ∈ N, α _n > 0, t _n → ∞ as n → ∞. Furthermore, the boundedness of the solutions is also investigated if the equation is nonoscillatory. An example is given to illustrate the use of the main theorems.