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Báo cáo sinh học: " Research Article On Lyapunov-Type Inequalities for Two-Dimensional Nonlinear Partial Systems"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article On Lyapunov-Type Inequalities for Two-Dimensional Nonlinear Partial Systems | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 504982 12 pages doi 10.1155 2010 504982 Research Article On Lyapunov-Type Inequalities for Two-Dimensional Nonlinear Partial Systems Lian-Ying Chen 1 Chang-Jian Zhao 1 and Wing-Sum Cheung2 1 Department of Mathematics China Jiliang University Hangzhou 310018 China 2 Department of Mathematics The University of Hong Kong Pokfulam Road Hong Kong Correspondence should be addressed to Chang-Jian Zhao chjzhao@163.com Received 27 February 2010 Accepted 23 June 2010 Academic Editor Alberto Cabada Copyright 2010 Lian-Ying Chen 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. We establish a new Laypunov-type inequality for two nonlinear systems of partial differential equations and the discrete analogue is also established. As application boundness of the twodimensional Emden-Fowler-type equation is proved. 1. Introduction In a celebrated paper of 1893 Liapunov 1 proved the following well-known inequality if y is a nontrivial solution of y q t y 0 1.1 on an interval containing the points a and b a b such that y a y b 0 then fb 4 b - a q s I ds. a 1.2 Since the appearance of Liapunov s fundamental paper 1 considerable attention has been given to various extensions and improvements of the Lyapunov-type inequality from 2 Journal of Inequalities and Applications different viewpoints 2-7 . In particular the Lyapunov-type inequalities for the following nonlinear system of differential equations were given in 8 x t afffxff Pi t u t Y 2u t u t -p2 t x t p-2x t - i t u t . 1.3 In this paper we obtain new Lyapunov-type inequalities for the two-dimensional nonlinear system and discrete nonlinear system respectively. 2. The Lyapunov-Type Integral Inequality for the Two-Dimensional Nonlinear System _ __ __ a s thxis th Pl s L