临界状态下具渐近周期系数的一阶时滞微分方程的振动性(英文)

Oscillations of First Order Delay Differential Equations with Asymptotic Periodic Coefficient in a Critical State

首先证明了在临界情形 limt→∞inf[p( t) - r( t) ]=0且∫tt-τr( s) ds≡ 1e下一阶时滞微分方程x′( t) +p( t) x( t-τ) =0 ( * )所有解振动等价于 Riccati不等式 w′( t) +r( t) w2 ( t) +2 e2 ( p( t)- r( t) )≤ 0无最终正解 .然后据此给出了方程 ( * )

First we prove that every solution of the d elay differential equation x′(t)+p(t)x(t-τ)=0 (*) oscillates if and only i f a related Riccati inequality w′(t)+r(t)w2(t)+2e2(p(t)-r(t))≤0 has no eventually positive solutions in the critical case when lim t→ ∞ inf [ p(t)-r(t)]=0, where p(t),r(t)∈C([t 0,∞),[0,∞)),r(t) is a τ - pe riodic function and its integration on interval [t,t+τ] is 1/e for t≥t 0. Next, we give two oscillation and nonoscillation criteria for Eq.(*) in the critical state.