Existence of positive periodic solutions for two kinds of neutral functional differential equations (Q926769)

The following two classes of neutral functional differential equations \[ \frac{d}{dt} [ x(t) - c\,x(t-\tau) ] = -a(t) x(t) + f(t,x(t-\tau(t))) \] and \[ \frac{d}{dt} \left[ x(t) - c \int_{-\infty}^0 Q(r) x(t+r) dr \right] = -a(t) x(t) + b(t) \int_{-\infty}^0 Q(r) f(t,x(t+r)) dr \] are considered, where \(a\), \(b \in C({\mathbb R},(0,\infty))\), \(\tau \in C({\mathbb R},{\mathbb R})\), \(f \in C({\mathbb R} \times {\mathbb R},{\mathbb R})\), and \(a(t)\), \(b(t)\), \(\tau(t)\), \(f(t,\cdot)\) are \(\omega\)-periodic functions, \(\omega>0\) and \(| c| <1\). Sufficient conditions for the existence of a positive \(\omega\)-periodic solution are obtained. The proof is based on Krasnoselskii's fixed point theorem. The results obtained here are applied to various mathematical models.