Further results on oscillation of a class of second-order neutral equations. (Q1405205)

The authors deal with the second-order neutral differential equation with distributed deviating arguments \[ [x(t)+c(t)x(t-\tau)]''+ \int_a^bp(t,\xi)x[g(t,\xi)] \,d\sigma (\xi)=0,\tag{1} \] where \(\tau>0\) is a constant; \(c(t)\in C([t_0,\infty),[0,1])\); \(p(t,\xi)\in C([t_0,\infty)\times [a,b], \mathbb{R}_+)\), and \(p(t,\xi)\) is not eventually zero on any \([t_{\mu},\infty)\times [a,b]\), \(t_{\mu}\geq t_0\), \(\mathbb{R}_+=[0,\infty)\); \(g(t,\xi)\in C([t_0,\infty)\times [a,b], \mathbb{R})\), \(g(t,\xi)\leq t\), \(\xi\in [a,b]\); \(g(t,\xi)\) is nondecreasing with respect to \(t\) and \(\xi\), respectively, and \(\lim_{t\to\infty}\inf_{\xi\in[a,b]}\{g(t,\xi)\}=\infty\); \(\sigma(\xi): [a,b]\to \mathbb{R}\) is nondecreasing, the integral in (1) is a Stieltjes one. The authors give criteria for the oscillation of all solutions to equation (1) by introducing a parameter function and using an integral averaging technique.