Consider the delay differential equation y(0 + ay(t) + Pf{y(t r)) = 0, (*) where a, /?, and r are positive constants and / is a continuous function such that uf(u) > 0 for u e [-A, B], 0, and lim = 1, v ; w—»o u where A and B are positive numbers. When /(«) = sinw, (*) is the so-called "sunflower" equation, which describes the motion of the tip of the sunflower plant. We obtain necessary and sufficient conditions for the oscillation of all solutions of (*), whose graph lies eventually in the strip R+ x [—A, B], in terms of the characteristic equation of the linearized equation z(t) + az(t) + f}z(t r) = 0.

We obtained sufficient and necessary and sufficient conditions for the oscillation of all positive solutions of where r, k, τ ∊ (0, ∞) and c ∊ (0,∞). We also obtained sufficient conditions for the global attractivity of the positive equilibrium K.

Abstract Consider the neutral delay differential equation where p ∈ R, τ ≥ 0, q1 > 0, σ1 ≥ 0, for i = 1, 2, …, k. We prove the following result. Theorem. A necessary and sufficient condition for the oscillation of all solutions of Eq. (1) is that the characteristic equation has no real roots.

Consider the equation where is the generalised derivative of x defined as follows: for every t ≧ T .