On the asymptotics of the difference equation with a proportional delay

Abstract

This paper deals with asymptotic properties of a vector difference equation with delayed argument $\Delta x_k=Ax_k+Bx_{\lfloor\lambda k\rfloor},\qquad 0\lt\lambda\lt 1,\quad k=0,1,2,\dots,$ where $A$, $B$ are constant matrices and the term $\lfloor\lambda k\rfloor$ is the integer part of $\lambda k$. Our aim is to emphasize some resemblances between the asymptotic behaviour of this delay difference equation and its continuous counterpart.