Positive Periodic Solutions of Systems of First Order Ordinary Differential Equations

Consider the n-dimensional nonautonomous system ?(t) = A(t)G(x(t)) ? B(t)F(x(t ? τ(t))) Let u = (u ₁,…,u _n ), $f^{i}_{0}={rm lim}_{|{rm u}|rightarrow 0}{f^{i}(rm u)over |u|}$ , $f^{i}_{infty}={rm lim}_{|{rm u}|rightarrow infty}{f^{i}(rm u)over |u|}$ , i = l,…,n, F = (f ¹…,f ⁿ ), ${rm F_{0}}={rm max}_{i=1,ldots,n}{f^{i}_{0}}$ and ${rm F_{infty}}={rm max}_{i=1,ldots,n}{f^{i}_{infty}}$ . 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.