Nonoscillation in systems of linear differential equations with delayed and advanced arguments (Q1263727)

The paper establishes a set of sufficient conditions for the existence of a nonoscillatory solution for a linear system of the form: \[ (1)\quad \dot x_ i(t)+\sum^{n}_{j=1}a_{ij}(t)x_ j(t- \tau_{ij}(t))+\sum^{n}_{j=1}b_{ij}(t)x_ j(t+\sigma_{ij}(t))=0,\quad j=1,2,...,n, \] \(\dot x{}_ i(t)=dx_ i(t)/dt\). The main result is the theorem: If the functions \(A_{ij}(t)\equiv \{\tau_{ij},\sigma_{ij},a_{ij},b_{ij}\}\) are bounded continuous ones defined in a haf-line \([t_ 0,\infty)\) such that \(0\leq A_{ij}(t)\leq A^*_{ij}\), \(t-\tau_{ij}(t)>0\), and \(\delta =\max_{1<i,j\leq n}\{\tau^*_{ij},\sigma^*_{ij}\}<\infty,\quad (\| A\| +\| B\|)\delta <e^{-2},\) where \(\|...\|\) stands for the norms of the matrices with entries \(a_{ij}\) and \(b_{ij}\), then the system (1) is nonoscillatory. The proof is based on \textit{L. K. Goebel}'s coincidence theorem [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 16, 733-735 (1968; Zbl 0165.498)].