tailieunhanh - Báo cáo hóa học: " Research Article The Periodic Character of the Difference Equation xn 1 f xn−l 1, xn−2k 1"

Tham khảo luận văn - đề án 'báo cáo hóa học: " research article the periodic character of the difference equation xn 1 f xn−l 1, xn−2k 1"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hindawi Publishing Corporation Advances in Difference Equations Volume 2008 Article ID 143723 6 pages doi 2008 143723 Research Article The Periodic Character of the Difference Equation xn 1 f xn-l 1 xn-2k 1 Taixiang Sun1 and Hongjian Xi2 1 Department of Mathematics College of Mathematics and Information Science Guangxi University Nanning 530004 Guangxi China 2 Department of Mathematics Guangxi College of Finance and Economics Nanning 530003 Guangxi China Correspondence should be addressed to Taixiang Sun stx1963@ Received 3 February 2007 Revised 18 September 2007 Accepted 27 November 2007 Recommended by H. Bevan Thompson In this paper we consider the nonlinear difference equation xn 1 f xn-i 1 xn-2k 1 n 0 1 . where k l E 1 2 . with 2k l and gcd 2k l 1 and the initial values x-a x-a 1 . x0 E 0 to with a max l - 1 2k - 1 . We give sufficient conditions under which every positive solution of this equation converges to a not necessarily prime 2-periodic solution which extends and includes corresponding results obtained in the recent literature. Copyright 2008 T. Sun and H. Xi. 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 In this paper we consider a nonlinear difference equation and deal with the question of whether every positive solution of this equation converges to a periodic solution. Recently there has been a lot of interest in studying the global attractivity the boundedness character and the periodic nature of nonlinear difference equations . see 1 2 . In 3 Grove et al. considered the following difference equation _ p Xn- 2m 1 . xn 1 -1 . 0 1 . . . E 1 1 xn-2r where p E 0 ro and the initial values x-a x-a 1 . x0 E 0 ro with a max 2r 2m 1 and proved that every positive solution of E1 converges to not necessarily prime a 2s-periodic solution with s gcd m 1 2r 1 . In 4