Nonoscillation of first order impulse differential equations with delay

Oscillation properties of impulse functional-differential equations are studied for equations of the type ẋ(t)=∑^m_i=1p_i(t)x(t-τ_i(t))=0, t∈[a,b], x(ξ)=0, ξ∉[a,b], x(t_j)=β_jx(t_j-0), j=1,...,k, a<t₁<t₂<⋯<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_jx(t_j-0), β_j>0,j=1,2,..., denote B(t)=Π_j∈Dtβ_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, ∞).