Positive periodic solutions of functional differential equations

AbstractWe 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, ∫0ωa(t)dt>0, ∫0ωb(t)dt>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=limu→0+f(u)u, f∞=limu→∞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