Oscillatory retarded functional systems.
The authors consider a functional system \[ x(t)= \int_{-1}^0 x(t-r(\theta))\, d[\nu(\theta)], \tag{1} \] where \(x(t)\in \mathbb{R}^n\), \(r(\theta)\) is a real continuous function on \([-1,0]\) and \(\nu(\theta)\) is a real \(n\times n\)-matrix real-valued function of bounded variation on \([-1,0]\). System (1) includes the important class of delay functional systems of the form \[ x(t)= \sum_{j=1}^p A_jx(t-r_j). \tag{2} \] Also, as it is well known that the systems (1) and (2) can be considered as a particular case of differential systems of neutral type. The authors use the spectral properties and prove some oscillation theorems. One of them is: Every solution of (1) is oscillatory if and only if \(s(\lambda)< 1\) where \(s(\lambda)\in \sigma(\int_{-1}^0 \exp(-\lambda r(\theta))\, d[\nu(\theta))])\). The authors also use the matrix measure and establish some sufficient conditions for oscillations. Some examples are considered to illustrate the main results.
scientific article; zbMATH DE number 1992562
scientific article