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 A Part-Metric-Related Inequality Chain and Application to the Stability Analysis of Difference Equation | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 19618 9 pages doi 2007 19618 Research Article A Part-Metric-Related Inequality Chain and Application to the Stability Analysis of Difference Equation Xiaofan Yang Maobin Yang and Huaiyi Liu Received 8 October 2006 Revised 13 December 2006 Accepted 14 December 2006 Recommended by Panayiotis D. Siafarikas We find a newpart-metric-related inequality of the form min ai 1 ai 1 i 5 1 w a1a2a3 a4 a5 a1a2 a1a3 a2a3 wa4a5 max ai 1 ai 1 i 5 where 1 w 2. We then apply this result to show that c 1 is a globally asymptotically stable equilibrium of the rational difference equation xn xn-1 xn-2 1 w xn-3xn-4xn-5 wXn-1Xn-2 x -3x -4 x -3x -5 xn-4xn-5 n 1 2 . a0 a-1 a-2 a-3 a-4 0. 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 Let f x1 . xr and g x1 . xr be polynomial functions with nonnegative coefficients and nonnegative constant terms. Suppose that for all possible positive combinations of a1 through ar the following inequality chain holds mini ai 1 i ri f 1 . r. max I a 1 i r ai g a1 . aj ai In this paper we refer to such an elegant inequality chain as a part-metric-related PMR inequality chain because it is closely related to the well-known part-metric p which is defined on R r where R stands for the whole set of positive reals in this way for X x1 . xr T e R r Y y1 . yr T e R r p X Y - log2minj x 1 i r yi xi 2 Journal of Inequalities and Applications Below there are some known PMR inequality chains 1-3 min í a 1 i 4 a1 a2 a3a4 max í a 1 i 4 a a1a2 a3 a4 ai min I a 1 i 4 a1 ak 2 ak 1ak max I a 1 i 4 a a1 a2 a3 ak ai minia 1 1 i 4 A1 A2a2 AT 4 A4 maxia 1 1 i 4 ai B1a1a2 B2a3 B3a4 B4 ai where A1 A2 A3 A4 B1 B2 B3 B4 are positive numbers A1 A2 A3 A4 B1 B2 B3
