tailieunhanh - Báo cáo hóa học: "ON THE DIFFERENCE EQUATION xn+1 = axn − bxn /(cxn − dxn−1 )"

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: ON THE DIFFERENCE EQUATION xn+1 = axn − bxn /(cxn − dxn−1 ) | ON THE DIFFERENCE EQUATION xn 1 axn bxn cxn dXn-1 E. M. ELABBASY H. EL-METWALLY AND E. M. ELSAYED Received 14 June 2006 Revised 3 September 2006 Accepted 26 September 2006 We investigate some qualitative behavior of the solutions of the difference equation xn 1 axn bxn cxn dxn 1 n 0 1 . where the initial conditions x 1 x0 are arbitrary real numbers and a b c d are positive constants. Copyright 2006 E. M. Elabbasy 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 In this paper we deal with some properties of the solutions of the difference equation . bxn xn 1 axn -----T---- cxn dxn 1 n 0 1 . where the initial conditions x 1 x0 are arbitrary real numbers and a b c d are positive constants. Recently there has been a lot of interest in studying the global attractivity boundedness character and the periodic nature of nonlinear difference equations. For some results in this area see for example 1-13 we recall some notations and results which will be useful in our investigation. Let I be some interval of real numbers and the function f has continuous partial derivatives on Ik 1 where Ik 1 I X I X X I k 1 times . Then for initial conditions x k x k 1 . x0 e I it is easy to see that the difference equation xn 1 f xn xn 1 . xn k n 0 1 has a unique solution xn n k. Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 82579 Pages 1-10 DOI ADE 2006 82579 2 On the difference equation xn 1 axn - bxn cxn - dxn-i A point x e I is called an equilibrium point of if x f x x . x . That is xn x for n 0 is a solution of or equivalently x is a fixed point of f. Definition stability . i The equilibrium point x of is locally stable if for every e 0 there exists 8 0 such that for all x-k x-k 1 . x-1 x0 e I with I x-k - x I I x-k 1 - x I I x0