Nonoscillation of first order impulse differential equations with delay

Alexander Domoshnitsky, Michael Drakhlin

Research output: Contribution to journalArticlepeer-review

52 Scopus citations


Oscillation properties of impulse functional-differential equations are studied for equations of the type ẋ(t)=∑mi=1pi(t)x(t-τi(t))=0, t∈[a,b], x(ξ)=0, ξ∉[a,b], x(tj)=βjx(tj-0), j=1,...,k, a<t1<t2<⋯<tk<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(tj)=βjx(tj-0), βj>0,j=1,2,..., denote B(t)=Πj∈Dtβj, where Dt={i:ti∈[t-τ(t),t]},p+(t)=max{p(t),0}. PROPOSITION. Let 1+1nB(t)/e≥∫tr(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 languageEnglish
Pages (from-to)254-269
Number of pages16
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - 1 Feb 1997
Externally publishedYes


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