Bulletin of the
Korean Mathematical Society BKMS

In this note we study positive solutions of the $m$th order rational difference equation $x_{n}=(a_0+\sum_{i=1}^m a_ix_{n-i})/(b_0+\sum_{i=1}^m b_ix_{n-i}),$ where $n=m,m+1,m+2,\ldots$ and $x_0,\ldots,x_{m-1}>0.$ We describe a sufficient condition on nonnegative real numbers $a_0, a_1,\ldots,a_m,b_0,b_1,\ldots,b_m$ under which every solution $x_n$ of the above equation tends to the limit $(A-b_0+\sqrt{(A-b_0)^2+4a_0B})/2B$ as $n \to \infty,$ where $A=\sum_{i=1}^m a_i$ and $B=\sum_{i=1}^m b_i.$

Keywords: difference equations, equilibrium point, convergence of sequences, upper and lower limits

MSC numbers: 39A11, 40A05