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Báo cáo hóa học: "Research Article Global Asymptotic Stability in a Class of Difference Equations"

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 Stability in a Class of Difference Equations | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 16249 7 pages doi 10.1155 2007 16249 Research Article Global Asymptotic Stability in a Class of Difference Equations Xiaofan Yang Limin Cui Yuan Yan Tang and Jianqiu Cao Received 29 April 2007 Accepted 5 November 2007 Recommended by John R. Graef We study the difference equation xn f Xg1 g2 h g1 f Xg2 h xn-1 . xn-r n 1 2 . x1-r . x0 0 where f g1 g2 R r R and h R r 0 to are all continuous functions and min1 i r ui 1 ui f u1 . ur max1si r ui 1 ui u1 . ur e R r. We prove that this difference equation admits c 1 as the globally asymptotically stable equilibrium. This result extends and generalizes some previously known results. Copyright 2007 Xiaofan Yang et al. 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 Ladas 1 suggested investigating the nonlinear difference equation xn-1 xn-2xn-3 xn c n 1 2 . x-2 x-1 x0 0. 1.1 xn-1 xn-2 xn-3 Since then it has been proved that c 1 is the common globally asymptotically stable equilibrium of this difference equation and all of the following difference equations where a and b are nonnegative constants xn xn x-x n 1 2 . x-2 x-1 x0 0 see 1 1.2 xn-1xn-2 xn-3 xn x x -ỉ x -3 a n 1 2 . x-2 x-1 x0 0 see 6 12 1.3 xn-1 xn-2xn-3 a xn-2 xn-1xn-3 a xn ---- ---- n 1 2 . x-2 x-1 x0 0 see 6 1.4 xn-1xn-2 xn-3 a 2 Advances in Difference Equations Xn 1 Xn 2Xn 3 a Xn ----- --- n 1 2 . X 2 X 1 X0 0 see 14 1.5 Xn-1Xn-2 Xn 3 a Xn-1Xn-2 Xn 3 a Xn 7---- J 2--------- n 1 2 . X 2 X 1 X0 0 see 14 1.6 Xn 2 Xn-1Xn-3 a Xn X ỉ ỴX Ĩ. Xb-4 a n 1 2 . X 3 X 2 X 1 X0 0 see 7 1.7 X -1 Xn 3Xn 4 a . xb kXn m xi a Xn b b n 1 2 . X1-max k m l . X0 0 see 8 . 1-8 Motivated by the above work and the work by Sun and Xi 2 this article addresses the difference equation r f X g1 g2 h . Xn Xn 1 . Xn r n 1 2 . X1 r . X0