Existence of positive periodic solutions of nonlinear neutral differential systems with variable delays (Q1743827)

The authors consider the following systems of differential equations with several delays \[ \frac{dx}{dt}=\pm x(t)[G(t,x(t-\tau _1(t)), ..., x(t-\tau_n(t)))-\varphi (t)x'(t-r(t))], \] \[ \frac{du}{dt}=a(t)u(t)-\lambda b(t)f(t,u(t-\delta (t)))-c(t)x(t-\sigma (t)), \] where \(G\in C(\mathbb R^{n+1}, \mathbb R), \) \(f\in C(\mathbb R \times \mathbb R^+, \mathbb R^+), \varphi,a,b,c\in C( \mathbb R, , \mathbb R^+), r,\sigma, \delta, \tau_i \in C (\mathbb R, \mathbb R)\). All of these functions are \(\omega\)- periodic in \(t\), and \(\lambda> 0\) is a parameter. They formulate some sufficient conditions for the existence of \(\omega\)- periodic positive solutions for small \(\lambda\). J. Mawhin's coincidence degree and a fixed point theorem on cones are basic tools.