Global attractivity of a higher-order nonlinear difference equation Xiu-Mei Jia a,b, * , Lin-Xia Hu c a School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China b Department of Mathematics, Hexi University, Zhangye, Gansu 734000, People’s Republic of China c Department of Mathematics, Tianshui Normal University, Tianshui, Gansu 741001, People’s Republic of China article info Keywords: Locally asymptotically stable Period-two solution Invariant interval Global attractor abstract The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlin- ear difference equation x nþ1 ¼ 1  x n A þ x nk ; n ¼ 0; 1; ... ; where A 2 ð1; 1Þ; k is a positive integer and initial conditions x k ; ... ; x 0 2 ð1; 0. It is shown that the unique negative equilibrium of the equation is a global attractor with a ba- sin that depends on certain conditions of the coefficient. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction Recently, many researchers are interested in the boundedness, invariant intervals, periodic character and global asymp- totic stability of all positive solutions of the nonlinear difference equations, see [1,2,4–7,10–12]. In particular, He et al. [3], Li and Sun [9] and Yan et al. [13] investigated the global asymptotic stability of all positive solutions of the following rational recursive sequences x nþ1 ¼ a  bx nk A  x n ; n ¼ 0; 1; ... ; x nþ1 ¼ a  bx n c þ x nk ; n ¼ 0; 1; ... ; x nþ1 ¼ a  bx n c  x nk ; n ¼ 0; 1; ... ; respectively, where the coefficients a; b; c; A; b are nonnegative real numbers and k 2f1; 2; ...g, and showed that every po- sitive equilibrium of these equations is a global attractor with a basin that depends on certain conditions imposed on the coefficients. Furthermore, they obtained sufficient conditions for the globally asymptotically stable of the positive equilibria of these equations. In 2004, Li et al. [8] studied the difference equation x nþ1 ¼ a þ bx n A þ x nk ; n ¼ 0; 1; ... ; 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.092 * Corresponding author. Address. School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China. E-mail address: jiaxium07@lzu.cn (X.-M. Jia). Applied Mathematics and Computation 216 (2010) 857–861 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc