tailieunhanh - Báo cáo hóa học: " Research Article On a k-Order System of Lyness-Type 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 On a k-Order System of Lyness-Type Difference Equations | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 31272 13 pages doi 2007 31272 Research Article On a k-Order System of Lyness-Type Difference Equations G. Papaschinopoulos C. J. Schinas and G. Stefanidou Received 17 January 2007 Revised 24 April 2007 Accepted 14 June 2007 Recommended by John R. Graef We consider the following system of Lyness-type difference equations x1 n 1 akXk n bk xk-1 n - 1 X2 n 1 a1X1 n b1 xk n - 1 Xi n 1 ai-1xi-1 n bi-1 xi-2 n - 1 i 3 4 . k where ai bi i 1 2 . k are positive constants k 3 is an integer and the initial values are positive real numbers. We study the existence of invariants the boundedness the persistence and the periodicity of the positive solutions of this system. Copyright 2007 G. Papaschinopoulos 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 Difference equations and systems of difference equations have many applications in biology economy and other sciences. So there exist many papers concerning systems of difference equations see 1-10 and the references cited therein . In 11 Kocic and Ladas investigated the existence of invariants the boundedness the persistence the periodicity and the oscillation of the positive solutions of the Lyness difference equation xn A xn 1 n 0 1 . xn-1 where A is a positive constant and the initial conditions x-1 y-1 x0 y0 are positive real numbers. 2 Advances in Difference Equations In 6-8 the authors studied the behavior of the positive solutions of the system of two Lyness difference equations byn c dxn e Xn 1 _ yn 1 _ n 0 1 . Xn-1 yn-1 where b c d e are positive constants and the initial conditions X-1 y-1 X0 y0 are positive numbers. Now in this paper we consider the system of difference equations z 1A akXk n bk X1 n 1 2 Xk-1 n - 1 z z 1X1 n b1 X2 n 1 7 z