## Abstract

Oscillation properties of impulse functional-differential equations are studied for equations of the type ẋ(t)=∑^{m}_{i=1}p_{i}(t)x(t-τ_{i}(t))=0, t∈[a,b], x(ξ)=0, ξ∉[a,b], x(t_{j})=β_{j}x(t_{j}-0), j=1,...,k, a<t_{1}<t_{2}<⋯<t_{k}<b. The proven test for oscillation generalizes the known ones and allows consideration of the solvability of boundary value problems for the corresponding nonhomogeneous impulse equations. In particular, for the scalar impulse equation ẋ(t)+p(t)x(t-τ(t))=0, t∈[0,∞), x(ξ)=0 for ξ<0, x(t_{j})=β_{j}x(t_{j}-0), β_{j}>0,j=1,2,..., denote B(t)=Π_{j∈Dt}β_{j}, where D_{t}={i:t_{i}∈[t-τ(t),t]},p_{+}(t)=max{p(t),0}. PROPOSITION. Let 1+1nB(t)/e≥∫^{t}_{r(t)}p+(s)ds where r(t)=max{t-τ(t),0},t>0. Then the nontrivial solution of this equation has no zeros on [0, ∞).

Original language | English |
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Pages (from-to) | 254-269 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 206 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 1997 |

Externally published | Yes |