Oscillations of first-order neutral delay differential equations
Oscillations of first-order neutral delay differential equations
复制标题
DOI：
10.1016/0022-247x(86)90172-1
复制
发表时间：
1986-12
影响因子：
1.3
通讯作者：
M. Grammatikópoulos;E. Grove;G. Ladas
中科院分区：
文献类型：
--
作者：
M. Grammatikópoulos;E. Grove;G. Ladas
考虑中立型时滞微分方程(∗)(Ddt)[y(T)+py(t−τ)]+qy(t−σ)=0，t⩾t0，其中τ，q和σ是正常数，而pϵ(−∞，−1)∪(0，+∞)。(对于pϵ[−1，0的情况，参见Ladas和Sficas，中立型时滞微分方程的振荡性(待见))。然后证明了以下结果。定理1.假设p<；−1.则方程(∗)的每个非振荡解y(T)趋于±∞为t→∞。定理2.设p<；−1，τ>；σ，q(σ−τ)(1+p)>；(1e)。然后方程(∗)的每一个解都振荡。定理3.假设p>；0。则方程(∗)的每个非振荡解y(T)都趋于零，即t→∞。定理4.假设p>；0。那么方程(∗)的所有解振动的一个必要条件是σ>；τ。定理5.设p>；0，σ>；τ，q(σ−τ)(1+p)>；(1e)。然后方程(∗)的每一个解都振荡。并将这些结果推广到变系数方程。
Consider the neutral delay differential equation (∗)(d dt)[y (t)+ py (t− τ)]+ qy (t− σ)= 0, t⩾ t 0, where τ, q, and σ are positive constants, while p ϵ (−∞,− 1)∪(0,+∞).(For the case p ϵ [− 1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p<− 1. Then every nonoscillatory solution y (t) of Eq.(∗) tends to±∞ as t→∞. Theorem 2. Assume p<− 1, τ> σ, and q (σ− τ)(1+ p)>(1 e). Then every solution of Eq.(∗) oscillates. Theorems 3. Assume p> 0. Then every nonoscillatory solution y (t) of Eq.(∗) tends to zero as t→∞. Theorem 4. Assume p> 0. Then a necessary condition for all solutions of Eq.(∗) to oscillate is that σ> τ. Theorem 5. Assume p> 0, σ> τ, and q (σ− τ)(1+ p)>(1 e). Then every solution of Eq.(∗) oscillates. Extensions of these results to equations with variable coefficients are also obtained.