Asymptotic properties of the solution of a delay system (Q542150)

The asymptotic behaviour of the delay-differential system \[ z'(t) = t_0e^t(A(t)z(t)+B(t)z(t-\sigma))\text{ for }t\geq 0 \] is investigated, where \(A\) and \(B\) are smooth \(\sigma\)-periodic \(m\times m\) matrices, and \(t_0\) and \(\sigma\) are positive constants. By transforming this system into a countable system on a finite time interval and under certain assumptions, the authors prove that the solution with the initial function \(\phi\) satisfies the estimate \[ \max_t\|z'(n\sigma +t)\|\leq Lq^n\sup_{\eta}\|\phi(\eta)\| \] for some \(L\geq 1\) and \(0<q<1\). Moreover, for large enough \(n\), the solution has the asymptotic representation \[ z((n+1)\sigma+t) \approx -A^{-1}(t)B(t)z(n\sigma+t). \]