A Generalized Nonuniform Contraction and Lyapunov Function. (27th December 2012)

• Record Type: Journal Article
• Authors: Liao, Fang-fang; Jiang, Yongxin; Xie, Zhiting
• Other Names: Sun Juntao Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2012(2012)
• Publication Date: 2012-12-27
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2012/613038
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : For nonautonomous linear equations x ′ = A ( t ) x, 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 ( D, μ ) -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.