Linearized oscillation theory for a nonlinear delay impulsive equation. (Q1421212)

The authors provide sufficient conditions for the oscillatory and nonoscillatory of the solutions of the following impulsive delay nonlinear differential equation \[ \dot x(t)+\sum_{k=1}^{m}r_{k}(t)f_{k}[x(h_{k}(t))]=0, \quad t\not=\tau_{j},\quad x(\tau_{j})=I_{j}x(\tau_{j}^{-})), j=1,2,\dots, \] where \(r_{k}(t)\geq 0\), \(h_{k}(t)\leq t\), \(\lim_{j\to\infty}\tau_{j}=\infty\). It is shown that oscillatory and nonoscillatory properties of the solutions of the above nonlinear equation are deduced from the following auxiliary linear impulsive differential equation \[ \dot x(t)+\sum_{k=1}^{m}r_{k}(t)a_{k}(t)x(h_{k}(t))=0, \quad t\not=\tau_{j},\quad x(\tau_{j})=b_{j}x(\tau_{j}^{-}),\quad j=1,2,\dots\;. \] Some applications for impulsive models of mathematical biology and numerical simulations for impulsive logistic equations are presented. The results of the present paper continue recent ones of the authors on the linearized theory for nonimpulsive delay equations in [J. Comput. Appl. Math. 151, No. 1, 119--127 (2003; Zbl 1025.34069)].