On the oscillation of neutral differential equations (Q1206919)

Using the method of the Laplace transform it is shown that all solutions of the neutral differential equation \[ {d\over dt}\left[x(t)+\delta\int^{\tau_ 2}_{\tau_ 1}x(t+s)d\mu(s)\right]+\int^{\sigma_ 2}_ {\sigma_ 1}x(t+s)d\eta(s)=0 \] are oscillatory if and only if the characteristic equation \[ \lambda\left[1+\delta\int^{\tau_ 2}_{\tau_ 1}e^{\lambda s}d\mu(s)\right]+\int^{\sigma_ 2}_{\sigma_ 1}e^{\lambda s}d\eta(s)=0 \] has no real roots, where \(\delta\in\{0,+1,- 1\}\), \(\tau_ 1\), \(\tau_ 2\) are real numbers with \(\tau_ 1<\tau_ 2\) and \(\tau_ 1\tau_ 2>0\), \(\sigma_ 1\), \(\sigma_ 2\) are real constants with \(\sigma_ 1<\sigma_ 2\), \(\mu\) and \(\eta\) are increasing real-valued functions on \([\tau_ 1,\tau_ 2]\) and\([\sigma_ 1,\sigma_ 2]\), respectively.