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.