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

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