Existence of Positive Periodic Solutions for n-Dimensional Nonautonomous System

In this paper we consider the existence, multiplicity, and nonexistence of positive periodic solutions for n-dimensional nonautonomous functional differential system x'(t)=H(t,x(t))-λB(t)F(x(t-τ(t))), where hi are ω-periodic in t and there exist ω-periodic functions αi,βi∈C(R,R+) such that αi(t)≤(hi(t,x)/xi)≤βi(t),∫0ω‍αi(t)dt>0, for x∈R+n all with xi>0, and t∈R, limxi→0+(hi(t,x)/xi) exist for t∈R; bi∈C(R,R+) are ω-periodic functions and ∫0ω‍bi(t)dt>0;fi∈C(R+n,R+), fi(x)>0 for x >0; τ∈(R,R) is an ω-periodic function. We show that the system has multiple or no positive ω-periodic solutions for sufficiently large or small λ>0, respectively.