Results in Mathematics

, Volume 48, Issue 3–4, pp 310–325

# Positive Periodic Solutions of Systems of First Order Ordinary Differential Equations

Article

## Abstract

Consider the n-dimensional nonautonomous system ẋ(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.

34K13

## Keywords

positive periodic solutions existence fixed point theorem

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