Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Global Asymptotic Behavior of yn+1 = (pyn + yn−1 )/(r + qyn + yn−1 ) | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID41541 22 pages doi 2007 41541 Research Article Global Asymptotic Behavior of yn 1 pyn yn-1 r qyn yn-1 A. Brett and M. R. S. Kulenovic Received 9 July 2007 Accepted 19 November 2007 Recommended by Elena Braverman We investigate the global stability character of the equilibrium points and the period-two solutions of yn 1 pyn yn-1 r qyn yn-1 n 0 1 . with positive parameters and nonnegative initial conditions. We show that every solution of the equation in the title converges to either the zero equilibrium the positive equilibrium or the period-two solution for all values of parameters outside of a specific set defined in the paper. In the case when the equilibrium points and period-two solution coexist we give a precise description of the basins of attraction of all points. Our results give an affirmative answer to Conjecture and the complete answer to Open Problem of Kulenovic and Ladas 2002. Copyright 2007 A. Brett and M. R. S. Kulenovic. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction We investigate the global stability character of the equilibrium points and the period-two solutions of the second order rational difference equation yn 1 pyn yn-1 r qyn yn-1 n 0 1 . where the parameters p q r are positive and the initial conditions y- 1 y0 are nonnegative real numbers. We also present one conjecture which together with the established results gives a complete picture of the nature of solutions of this equation. Our results improve and extend the asymptotic results in 1 Section . Equation is an important stepping stone in understanding the global dynamics of second-order rational 2 Advances in Difference Equations difference equation of the form a fiyn Yyn-1 y A Byn Cyn-1 n 0
