Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài : On a class of second-order nonlinear difference equation | Dongsheng et al. Advances in Difference Equations 2011 2011 46 http content 2011 1 46 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access On a class of second-order nonlinear difference equation Li Dongsheng1 Zou Shuliang1 and Liao Maoxin2 Correspondence lds1010@sina. com 1School of Economics and Management University of South China Hengyang Hunan 421001 People s Republic of China Full list of author information is available at the end of the article Springer Abstract In this paper we consider the rule of trajectory structure for a kind of second-order rational difference equation. With the change of the initial values we find the successive lengths of positive and negative semicycles for oscillatory solutions of this equation and the positive equilibrium point 1 of this equation is proved to be globally asymptotically stable. Mathematics Subject Classification 2000 39A10 Keywords rational difference equation trajectory structure rule semicycle length periodicity global asymptotic stability 1 Introduction and preliminaries Motivated by those work 1-17 especially 10 we consider in this paper the following second-order rational difference equation 1 X Xn_ 1 a Xn 1 ----A------- n -1 0 1 . 1-1 x n-1 a the initial values x-1 x0 e 0 a e 0 and k I e - . Mainly by analyzing the rule for the length of semicycle to occur successively we describe clearly out the rule for the trajectory structure of its solutions and further derive the global asymptotic stability of positive equilibrium of Equation . It is easy to see that the positive equilibrium X of Equation satisfies _ _ 1 Xk l a x X xl a From this we see that Equation possesses a positive equilibrium X 1. In this paper our work is only limited to positive equilibrium X 1. Here for readers convenience we give some corresponding definitions. Definition . A positive semicycle of a solution X In 1of Equation consists of a string of terms xr xr
