Necessary and sufficient conditions for the oscillation of systems of difference equations with continuous arguments (Q2573434)

This paper is devoted to the study of the oscillation of all solutions of a system of delay-difference equations of the form \[ y(t) + \sum_{i=1}^n P_{i}y(t-\tau_{i}) = 0, \] where the \(P_{i}\) are \(m \times m\) matrices. Necessary and sufficient conditions are given in terms of the eigenvalues of the characteristic equation \[ \text{det}\left(I + \sum_{i=1}^nP_{i}e^{-\lambda \tau_{i}}\right) \] and sufficient conditions are given in terms of the logarithmic norm of the coefficient matrices. The special case \[ y(t) - y(t-\tau) + Py(t-\sigma) = 0 \] is considered in detail.