tailieunhanh - Báo cáo hóa học: " Research Article Variationally Asymptotically Stable Difference Systems"

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 Variationally Asymptotically Stable Difference Systems | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 35378 21 pages doi 2007 35378 Research Article Variationally Asymptotically Stable Difference Systems Sung Kyu Choi Yoon Hoe Goo and Namjip Koo Received 3 January 2007 Revised 10 May 2007 Accepted 9 August 2007 Recommended by Leonid E. Shaikhet We characterize the fi-stability in variation and asymptotic equilibrium in variation for nonlinear difference systems via nTO -summable similarity and comparison principle. Furthermore we study the asymptotic equivalence between nonlinear difference systems and their variational difference systems by means of asymptotic equilibria of two systems. Copyright 2007 Sung Kyu Choi 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 Conti 1 introduced the notion of tTO -similarity in the set of all m X m continuous matrices A t defined on R 0 to and showed that tTO -similarity is an equivalence relation preserving strict uniform and exponential stability of linear homogeneous differential systems. Choi et al. 2 studied the variational stability of nonlinear differential systems using the notion of tTO-similarity. Trench 3 introduced a definition called tTO-quasisimilarity that is not symmetric or transitive but still preserves stability properties. Their approach included most types of stability. As a discrete analog of Conti s definition of tTO -similarity Trench 4 defined the notion of summable similarity on pairs of m X m matrix functions and showed that if A and B are summably similar and the linear system Ax n A n x n n 0 1 . is uniformly exponential or strictly stable or has linear asymptotic equilibrium then the linear system Ay n B n y n has also the same properties. Also Choi and Koo 5 introduced the notion of nTO-similarity in the set of all m X m .