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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 613038, 14 pages
http://dx.doi.org/10.1155/2012/613038
Research Article

A Generalized Nonuniform Contraction and Lyapunov Function

1Nanjing College of Information Technology, Nanjing 210046, China
2Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China

Received 19 November 2012; Accepted 1 December 2012

Academic Editor: Juntao Sun

Copyright © 2012 Fang-fang Liao 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.

Abstract

For nonautonomous linear equations , we give a complete characterization of general nonuniform contractions in terms of Lyapunov functions. We consider the general case of nonuniform contractions, which corresponds to the existence of what we call nonuniform -contractions. As an application, we establish the robustness of the nonuniform contraction under sufficiently small linear perturbations. Moreover, we show that the stability of a nonuniform contraction persists under sufficiently small nonlinear perturbations.