Oscillation and nonoscillation in neutral equations with integrable coefficients (Q1130432)

The authors obtain new sufficient conditions for the oscillation of all solutions and the existence of a positive solution to the neutral delay differential equation \[ {d\over dt} [x(t)- p(t)x(t-\tau)]+ Q(t)x(t- \delta)= 0,\quad t\geq t_0.\tag{1} \] They also consider the following special equation \[ {d\over dt} [x(t)- x(t-\tau)]+ {a\over t^\alpha\ln^\beta t} x(t-\delta)= 0,\quad t\geq 3,\tag{2} \] with \(\tau> 0\), \(a> 0\), \(\delta\geq 0\), \(\alpha> 1\) and \(\beta\in\mathbb{R}\). Necessary and sufficient conditions for equation (2) are given: \[ \begin{matrix} \beta &\text{oscillate} &\text{nonoscillate} &\text{unbounded} &\text{bounded}\\ &&&\text{positive} &\text{positive}\\ &&&\text{solution} &\text{solution}\\ + &\alpha< 2 &\alpha\geq 2 &\alpha= 2,\beta\leq 1 & \alpha\geq 2,\beta= 1\\ - &\alpha\leq 2 &\alpha>2 &- &\alpha>2\\ 0 &\alpha\leq 2 &\alpha\geq 2 &\alpha=2 &\alpha>2\\ &4a>\tau &4a\leq\tau &4a\leq\tau\end{matrix}. \]