Search

For the differential equation where 𝑝𝑖 ∈ 𝐿 loc (𝑅+; 𝑅+), τ 𝑖 ∈ 𝐶(𝑅+; 𝑅+), τ 𝑖(𝑡) ≤ 𝑡 for 𝑡 ∈ 𝑅+, , optimal integral conditions for the oscillation of all solutions are established.

Consider the nth-order neutral differential equation [formula] where n ≧ 1, J , K are initial segments of natural numbers, and pi, τi, qk, σk ∈ R for i ∈ J and k ∈ K . A necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation [formula] has no real roots. It is to be emphasized that the coefficients and the arguments are real numbers and also that the method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutions of Eq. (E).

Consider the nth-order neutral differential equation where n ≥ 1, δ = ±1, I, K are initial segments of natural numbers, pi, τi, σk ∈ R and qk ≥ 0 for i ∈ I and k ∈ K. Then a necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation has no real roots. The method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutionso of Equation (E).

Consider the second- order neutral delay differential equation $$\frac{{d^2 }}{{dt^2 }}[y(t) + P(t - \tau )] + Q(t)y(t - \sigma ) = 0,t \geqq t_0 ,$$ (1) where P, Q e C([t0g, ∞), R) and the delays τ and σ are nonnegative real numbers. We examined the asymptotic behavior of the nonoscillatory solutions of eq. (1) and obtained sufficient conditions for the oscillation of (i) all solutions of eq. (1); (ii) all bounded solutions of eq. (1); (iii) all unbounded solutions of eq. (1).

Consider the neutral equation with several positive and negative coefficients dd[tx(t)+∑Ipix(t−τi)−∑Jrjx(t−ϱj)]+∑Kqkx(t−σk)=0, (1) where I, J, and K are initial segments of natural numbers, pi, τi, rj, ϱj, qk are positive numbers, and σk ⩾ 0 for i ϵ I, j ϵ J, and k ϵ K. Then a necessary and sufficient condition for the oscillation of all solutions of (1) is that its characteristic equation λ+λ∑Ipie−λτi−λ∑Irje−λϱj+∑Kqke−λσk=0 has no real roots.