Functional Differential Equations Linearized Global Attractivity And The Critical Vibration,

In this paper, we give the detailed analysis of the global existence and global attract ivity of solutions of three functional differential equationsx'(t) + [1 + x(t)] [1 + cx(t)]F(t, x(·) = 0, t≥ 0,x'(t) + [1 + x(t)][1 - cx(t)]F(t, x(·))=0, t≥ 0,andx'(t) + [1 + x{t)]F{t, (1 + λx(·))x(·)) = 0, t≥ 0,which contain the delay logistic equationx'(t) + r(t)[1 + x(t)]x(t - Ï„) = 0, t ≥ 0as their special forms. And the 3/2-global attractivity results are obtained by means of the conditions of attractivity of the corresponding linearize equation of the above; By establishing the equivalence for oscillation of delay and ordinary differential equations, we investigated the oscillatory behavior of the following first order delay differential equationx'(t)+P(t)x(Ï„(t))=0, t≥t₀,in the critical state, n-th neutral delay differential equation with "integrall small" coefficients[x(t) - P(t)x{t - Ï„)]⁽ⁿ⁾ + Q(t)x{t -σ) = 0, t≥ t₀,and the first order neutral delay differential equation with positive and negative coefficients[x(t) - P(t)x(t - Ï„)]' + Q(t)x(t -σ)- R(t)x(t - r) = 0, t≥ t₀.We obtain a series of new results on oscillation and nonoscillation of solutions for the above equations. Of them, many results are "best possible" in some sense.