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Báo cáo hóa học: "GLOBAL BEHAVIOR OF A HIGHER-ORDER RATIONAL DIFFERENCE EQUATION"

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: GLOBAL BEHAVIOR OF A HIGHER-ORDER RATIONAL DIFFERENCE EQUATION | GLOBAL BEHAVIOR OF A HIGHER-ORDER RATIONAL DIFFERENCE EQUATION HONGJIAN XI AND TAIXIANG SUN Received 17 January 2006 Revised 6 April 2006 Accepted 12 April 2006 We investigate in this paper the global behavior of the following difference equation Xn 1 Pk Xn-i0 Xn-iv. Xn-i2k b Qk xn-i0 Xn-iv. Xn-i2k b n 0 1 . under appropriate assumptions where G 0 oo k 1 i0 i1 . i2k 0 1 .J with i0 i1 i2k the initial conditions Xú2k Xú2k 1 . X0 G 0 00 . We prove that unique equilibrium X 1 of that equation is globally asymptotically stable. Copyright 2006 H. Xi and T. Sun. 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 For some difference equations although their forms or expressions look very simple it is extremely difficult to understand thoroughly the global behaviors of their solutions. Accordingly one is often motivated to investigate the qualitative behaviors of difference equations e.g. see 2 3 6 9 10 . In 6 Ladas investigated the global asymptotic stability of the following rational difference equation E1 Xn Xn- iXn- 2 Xn 1 --- n 0 1 . 1.1 XnXn 1 Xn 2 where the initial values X_2 X_ 1 X0 G R 0 oo . In 9 Nesemann utilized the strong negative feedback property of 1 to study the following difference equation E2 Xn- 1 XnXn-2 . Xn 1 7 n 0 1 . 1.2 XnXn 1 Xn 2 where the initial values X-2 X-1 X0 E R . Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 27637 Pages 1-7 DOI 10.1155 ADE 2006 27637 2 Global behavior of a difference equation In 10 Papaschinopoulos and Schinas investigated the global asymptotic stability of the following nonlinear difference equation E3 SieZk- j- 1 j xn-i Xn-jXn-j 1 1 Xn 1 n 0 1 . ÀiG Zk xn i 1.3 where kE 1 2 3 .J j j 1 c Zk 0 1 . k and the initial values X-k X-k 1 . X0 E R . Recently Li 7 8 studied the global asymptotic stability of the .