Nonoscillations in retarded systems (Q2484187)

Consider the equation \[ x(t)+ \int^0_{-1} d[\nu(\theta)]x(t- r(\theta))= 0,\tag{1} \] where \(x(t)\in \mathbb{R}^n\), \(r\in C([-1,0], \mathbb{R}_+)\), and \(\nu(\theta)\) is a real \(n\times n\) matrix valued function of bounded variation on \([-1, 0]\). Using matrix measures, the authors obtain several criteria for the equation to be oscillatory. These criteria are extended to the difference system \[ x(t)+ \sum^p_{j=1} A_j x(t- r_j)= 0,\tag{2} \] where \(A_j\in\mathbb{R}^{n\times n}\) and \(r_j> 0\), by letting \(\nu(\theta)\) be a step function. Examples are given to demonstrate the significance of the results for both (1) and (2).