Weak attractors from Lyapunov functions

✍ Scribed by Mike Hurley

• Book ID: 104295735
• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 97 KB
• Volume: 109
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00158-3

✦ Synopsis

References (Hurley, 1991(Hurley, , 1992(Hurley, , 1998) ) show that if a continuous map f on a metric space X has a "weak attractor", A, then there is an associated Lyapunov function, h, which is a continuous, nonnegative, real-valued map whose zero set is A, and satisfying h • fh < 0 on a certain deleted neighborhood of A. In (1996) Kim et al. show that If X is locally compact and if the zero set Z of a Lyapunov function is compact, then Z is a weak attractor. Here we obtain the same result without the compactness assumption on Z, provided that the ambient space is σ -compact.