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 Behavior of Solutions to Two Classes of Second-Order Rational Difference Equations Sukanya Basu and Orlando Meri | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 128602 27 pages doi 2009 128602 Research Article Global Behavior of Solutions to Two Classes of Second-Order Rational Difference Equations Sukanya Basu and Orlando Merino Department of Mathematics University of Rhode Island Kingston RI02881 USA Correspondence should be addressed to Orlando Merino merino@ Received 9 December 2008 Accepted 7 July 2009 Recommended by Ondrej Dosly For nonnegative real numbers a f Y A B and C such that B C 0 and a f Y 0 the difference equation xn 1 a fxn Yxn-1 A Bxn Cxn-1 n 0 1 2 . has a unique positive equilibrium. A proof is given here for the following statements 1 For every choice of positive parameters a f Y A B and C all solutions to the difference equation xn 1 a fxn Yxn-1 A Bxn Cxn-1 n 0 1 2 . x-1 x0 e 0 to converge to the positive equilibrium or to a prime period-two solution. 2 For every choice of positive parameters a f Y B and C all solutions to the difference equation xn 1 a fxn Yxn-1 Bxn Cxn-1 n 0 1 2 . x-1 x0 e 0 to converge to the positive equilibrium or to a prime period-two solution. Copyright 2009 S. Basu and O. Merino. 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 and Main Results In their book 1 Kulenovic and Ladas initiated a systematic study of the difference equation a fxn fxn-1 xn 1 A TJ n 0 1 2 . . . A Bxn Cxn-1 for nonnegative real numbers a f Y A B and C such that B C 0 and a f Y 0 and for nonnegative or positive initial conditions x-1 x0. Under these conditions has a unique positive equilibrium. One of their main ideas in this undertaking was to make the task more manageable by considering separate cases when one or more of the parameters in is zero. The need for this strategy is made apparent by cases such as the
