Necessary and sufficient conditions for oscillation of solutions to second-order neutral differential equations with impulses
\noindent This paper deals with an impulsive neutral differential equation of the form \[ \left \{ \begin{array}{c} \left( r(t)(z^{\prime }(t))^{\gamma }\right) ^{\prime }+q(t)x^{\alpha }(\sigma (t))=0,~t\geq t_{0},~t\neq \lambda _{k}, \\ \Delta (r(\lambda _{k})(z^{\prime }(\lambda _{k}))^{\gamma })+h(\lambda _{k})x^{\alpha }(\sigma (\lambda _{k}))=0,~k\in \mathbb{N}, \end{array} \right. \] where \(z(t)=x(t)+x(\tau (t)).~\)Necessary and sufficient conditions are obtained for the oscillation of the solutions for the cases \(\gamma >\alpha \) and \(\gamma <\alpha .\) Moreover, two examples are given to illustrate the main results.
Lebesgue's dominated convergence theorem
scientific article; zbMATH DE number 7439084
scientific article