On the global asymptotic stability for a rational recursive sequence

The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of the positive solutions of the nonlinear rational difference equation / , n 0,1,2,.... 0 0 1 ƒ­ „E „E „E „?„» „¹ ƒy „E „E „E „?„» „¹ ƒ­ „¸ „¸ ƒ­ ƒ{ ƒ­ ƒy ƒ{ k i i n i k i n i n i x ƒ? x B ƒ? x where the coefficients i i B ,ƒ? , ƒ? together with the initial conditions ,.... , , 1 0 x x x ƒ{k ƒ{ are arbitrary positive real numbers, while k is a positive integer number.