Existence of positive periodic solutions of nonlinear first-order delayed differential equations

We consider the existence of positive ω-periodic solutions for the equationu′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))),u′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))), where a,b∈C(R,[0,∞))a,b∈C(R,[0,∞)) are ω  -periodic functions with ∫0ωa(t)dt>0, ∫0ωb(t)dt>0; f,g∈C([0,∞),[0,∞))f,g∈C([0,∞),[0,∞)) and f(s)>0f(s)>0 for s>0s>0; τ is a continuous ω  -periodic function; λ>0λ>0 is a parameter. The proofs of our main results are based upon fixed point index theory.