Oscillatory behaviour in functional differential systems of neutral type (Q1614677)

The authors study the oscillatory behaviour of the solutions to the linear neutral functional-differential equation of the form \[ \frac{d}{dx}\Big[x(t)-\int_{-1}^{0}d[\nu(\theta)]x(t-\tau(\theta))\Big]= \Big[x(t)-\int_{-1}^{0}d[\eta(\theta)]x(t-r(\theta))\Big],\tag{*} \] where \(x(t)\in \mathbb{R}^n,\) \(\nu(\theta),\eta(\theta)\) are real \(n\times n\)-matrix-valued functions of bounded variation on the interval [-1,0] and \(\tau(\theta)\) and \(r(\theta)\) are real positive continuous functions on [-1,0]. In the real scalar case by investigating the nonexistence of real solutions to the corresponding characteristic equation of \((*)\), the authors provide interesting sufficient conditions for the total oscillatory behavior of equation \((*).\) In the \(n\times n\) case, the conditions involve the usual matrix measure function \(\mu\) defined on the \(n\times n\)-matrices \(C\) by the type \(\mu(C):=\lim_{\gamma\to 0+}\frac{1}{\gamma}(\|I+\gamma C\|-1).\) A sufficient number of illustrative examples completes the paper.