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**Oscillations in systems of differential equations with piecewise constant argument.**
*(English)*
Zbl 0728.34077

The authors consider linear systems of differential equations with piecewise constant argument of the form
\[
(*)\quad x'(t)=Ax(t)+Bx([t+]),\quad x(0)=c_ 0,\quad x'(t)=Ax(t)+Bx([t]),\quad x(0)=c_ 0
\]
and the associated nonhomogeneous equations, where [. ] denotes the greatest integer function. They obtain sufficient conditions under which the above equations admit unique solutions on \([0,\infty)\). Asymptotic stability of the zero solution and oscillatory behaviour and periodicity of solutions of (*) are studied. These properties depend upon the eigenvalues of a certain matrix associated with (*).

Reviewer: N.Parhi (Berhampur)

### MSC:

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

34K20 | Stability theory of functional-differential equations |

34K05 | General theory of functional-differential equations |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

### Keywords:

linear systems of differential equations with piecewise constant argument; Asymptotic stability; oscillatory behaviour; periodicity
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\textit{J. Wiener} and \textit{K. L. Cooke}, J. Math. Anal. Appl. 137, No. 1, 221--239 (1989; Zbl 0728.34077)

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### References:

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