Oscillatory and periodic solutions of delay differential equations with piecewise constant argument.

*(English)*Zbl 0631.34078The authors give sufficient conditions for all solutions of the scalar delay differential equation with piecewise constant argument

$${y}^{\text{'}}\left(t\right)+a\left(t\right)y\left(t\right)+b\left(t\right)y\left([t-1]\right)=0$$

to be oscillatory. In the case of constant coefficients, these conditions are shown to be also necessary, the asymptotic stability of the zero solution is studied, and periodic solutions are characterized.

Reviewer: H.Engler

##### MSC:

34K99 | Functional-differential equations |

34C25 | Periodic solutions of ODE |

34C10 | Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory |

34K10 | Boundary value problems for functional-differential equations |

34K20 | Stability theory of functional-differential equations |