Oscillation and nonoscillation for a class of nonlinear neutral differential equations
The neutral differential equation \[ [x(t)-P(t)x(t-\tau)]'+ Q(t)\prod_{i=1}^n|x(t-\delta_i)|^{\alpha_i} \text{ sgn } x(t-\delta_i)=0, \tag{1} \] where \(P\in C([t_0,\infty),\mathbb{R})\), \(Q\in C([t_0,\infty),\mathbb{R}^+)\), \(\tau\in(0,\infty)\), \(\delta_i\geq 0\), \(\alpha_i\geq 0\), \(i=1,\dots,n\), is considered. Many papers are concerned with the oscillation of (1) under the hypothesis \[ \int_{t_0}^\infty Q(t)dt=\infty. \tag{2} \] Some sufficient conditions for the oscillation of all solutions of (1) are established by relaxing the condition (2). In particular, if \(\alpha_1+ \alpha_2+\dots+ \alpha_n=1\), \(P(t)\equiv 1\) and \[ \int_{t_0}^\infty \Biggl\{ Q(s)\exp \int_{t_0}^S \Biggl[ \frac{2}{\tau} u(Q(u)+ \frac{1}{\tau^2} Q(u) \int_{t_0}^u v^2Q(v)dv \Biggr] du \Biggr\} ds=\infty,\tag{3} \] then the solutions of (1) oscillate. As far as a nonoscillation is concerned (1) has a bounded positive solution if and only if \(\int_{t_0}^\infty sQ(s)ds<\infty\).
scientific article; zbMATH DE number 1014350