Positive Periodic Solutions of Systems of First Order Ordinary Differential Equations

Consider the n-dimensional nonautonomous system ẋ(t) = A(t)G(x(t)) − B(t)F(x(t − τ(t))) Let u = (u1,…,un),(Formula Presented.). Under some quite general conditions, we prove that either F0 = 0 and F∞ = ∞, or F0 = ∞ and F∞ = 0, guarantee the existence of positive periodic solutions for the system for all λ > 0. Furthermore, we show that F0 = F∞ = 0, or F∞ = F∞ = ∞ guarantee the multiplicity of positive periodic solutions for the system for sufficiently large, or small λ, respectively. We also establish the nonexistence of the system when either F0 and F∞ > 0, or F0 and F∞, < for sufficiently large, or small λ, respectively. We shall use fixed point theorems in a cone. © 2005, Birkhäuser Verlag, Basel.