Positive periodic solutions of functional differential equations

We consider the existence, multiplicity and nonexistence of positive ω-periodic solutions for the periodic equation x′(t)=a(t) g(x)x(t)-λb(t)f(x(t-τ(t))), where a,b∈C(ℝ, [0,∞)) are ω-periodic, ∫₀^ω a(t) dt>0, ∫₀^ωb(t) 0, f,g∈ C([0,∞),[0,∞)), and f(u)>0 for u>0, g(x) is bounded, τ(t) is a continuous ω-periodic function. Define f₀ =lim_u→0⁺ f(u)/u, f_∞ =lim_u→∞ f(u)/u, i₀=number of zeros in the set {f₀,f_∞} and i_∞=number of infinities in the set {f₀, f_∞}. We show that the equation has i₀ or i_∞ positive ω-periodic solution(s) for sufficiently large or small λ>0, respectively.