Existence of nonoscillatory solutions of first and second order neutral differential equations with distributed deviating arguments (Q609795)

The authors study first and second order neutral functional differential equations with distributed deviating arguments of the form \[ \frac{d^k}{dt^k}[x(t)+P(t)x(t-\tau)]+\int_a^bq_1(t,\xi)x(t-\xi)d\xi-\int_c^dq_2(t,\mu)x(t-\mu)d\mu,\tag{1} \] where \(k=1,2\), \(P\in C[[t_0,\infty),\mathbb{R}]\), \(q_1\in C[[t_0,\infty)\times[a,b],\mathbb{R}^+]\), \(q_2\in C[[t_0,\infty)\times[c,d],\mathbb{R}^+]\),\ \(0<a<b\) and \(0<c<d\). By using Banach's contraction principle, the authors present sufficient conditions for the existence of a nonoscillatory solution of (1).