Oscillation of first-order neutral nonlinear differential equations with variable coefficients
The author considers first-order nonlinear differential-difference equations of the form \[ {d\over dt}[x(t)-R(t)x(t-r)]+ P(t)\prod_{i=1}^{m}| x(t-\tau_i)| ^{\alpha_i}\cdot \text{sign} x(t-\tau_i)=0, \] with \(P,R\in C([t_0,\infty),{\mathbb{R}}^+)\), \(R(t)\geq 0\), \(P(t)\geq 0\), \(P(t)\not\equiv 0\), \(r>0\), \(\tau_1,\ldots,\tau_m\geq 0\), \(\alpha_1,\ldots,\alpha_m>0\), and \(\alpha_1+\cdots,\alpha_m=1\). Oscillation criteria without any sort assumption of the type \(\int_{t_0}^{\infty}P(s) ds\) \(=\infty\) or \(\int_{t_0}^{\infty}sP(s)\int_{s}^{\infty}P(u) du ds=\infty\) are established.
neutral nonlinear differential-difference equations
scientific article; zbMATH DE number 1281894