Opuscula Math. 26, no. 3 (2006), 499-506

Opuscula Mathematica

# On the asymptotics of the difference equation with a proportional delay

Petr Kundrát

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.

Keywords: asymptotics of difference equations, approximation methods for dynamical systems.

Mathematics Subject Classification: 39A11, 37M99.

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