Global behavior of a higher order nonlinear difference equation ✩

In this paper, we consider a higher order difference equation of the form xn+1 = f (xn,xn−k ), n = 1, 2, 3, . . . , under some certain assumptions. We obtain that the length of its finite semicycle is less than or equal to k. Moreover, we concluded that the equation is permanent under some special conditions. And a sufficient condition for its global asymptotical stability is given. When these results are applied to the difference equation xn+1 = α + xn−k xn , n= 1, 2, 3, . . . , some sufficient and necessary conditions of global asymptotical stability of the equation are obtained. We should state out that the main theorem in [J. Math. Anal. Appl. 233 (1999) 790–798] is included in one of our results.  2004 Elsevier Inc. All rights reserved