Advances in Difference Equations 
Volume 2008 (2008), Article ID 238068, 6 pages
doi:10.1155/2008/238068
Research Article

Stability of Solutions for a Family of Nonlinear Difference Equations

Taixiang Sun,1 Hongjian Xi,1,2 and Caihong Han1

 1College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China
2Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China
Received 8 September 2007; Accepted 31 January 2008

Recommended by Mariella Cecchi

Abstract

We consider the family of nonlinear difference equations: xn+1=(∑i=13fi(xn,…,xn−k)+f4(xn,…,xn−k)f5(xn,…,xn−k))/(f1(xn,…,xn−k)f2(xn,…,xn−k)+∑i=35fi(xn,…,xn−k)), n=0,1,…, where fi∈C((0,+∞)k+1,(0,+∞)), for i∈{1,2,4,5}, f3∈C([0,+∞)k+1,(0,+∞)), k∈{1,2,…} and the initial values x−k,x−k+1,…,x0∈(0,+∞). We give sufficient conditions under which the unique equilibrium x¯=1 of these equations is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references.