Nonoscillatory solutions for system of neutral delay equation (Q1811787)

The authors consider the following system of neutral differential equations \[ {d\over dt} (x(t)+ px(t- \tau))+ Q(t) x(t-\sigma)= 0,\tag{1} \] where \(p\in\mathbb{R}\), \(x\in\mathbb{R}^n\) and \(\tau\in (0,\infty)\), \(\sigma\in [0,\infty)\), \(Q\) is a continuous \(n\times n\)-matrix on \([t_0,\infty)\). They obtain sufficient conditions for the existence of a solution with a specific asymptotic behaviour for all values of \(p\neq -1\). They consider not only the case when \(p\) is a scalar but also the case when it is a matrix. The main result in the scalar case is the following theorem: Let \(\int^\infty\|Q(s)\|ds< \infty\), where \(\|.\|\) is any norm in \(\mathbb{R}^n\). Then equation (1) has a nonoscillatory solution. In the last section, the authors extend the result to the following equation \[ {d\over dt} (x(t)+ Bx(t-\tau))+ Q(t) x(t-\sigma)= 0, \] where \(B\) is a nonsingular constant \(n\times n\)-matrix. They also claim that the result here can be extended to general nonlinear equations.