×

Strong cellularity and global asymptotic stability. (English) Zbl 0742.57011

A closed subset \(C\) of a Banach space \(X\) is called a cell in \(X\) if the pairs (\(B(1),\partial B(1)\)) and (\(C,\partial C\)) are homeomorphic, where \(B(1)\) (resp., \(\partial B(1)\)) is the closed ball (resp., the sphere) of radius 1 centered at \(0_ X\). A subset \(A\) of \(X\) is cellular if there is a sequence \(\{C_ n\}\) of cells in \(X\) such that \(A=\cap\{C_ n: n\in\mathbb{N}\}\). If in addition for every open set \(U\) in \(X\) containing \(A\) there is an \(n\) with \(C_ n\subset U\), then \(A\) is said to be strongly cellular. A continuous mapping \(\pi: \mathbb{R}\times X\to X\) (\(\pi: \mathbb{R}^ +\times X\to X\)) is called a dynamical (semidynamical) system if \(\pi(0,x)=x\) for all \(x\in X\) and \(\pi(t+\tau,x)=\pi(t,\pi(\tau,x))\) for all \(t,\tau\in\mathbb{R}\), \(x\in X\) (\(t,\tau\in\mathbb{R}^ +\), \(x\in X\)). A subset \(Y\) of \(X\) is invariant if \(\{\pi(t,y): y\in Y\}=Y\) for every \(t\in\mathbb{R}^ +\). For a compact invariant set \(M\) let \(A(M)=\{x\in X; d(\pi(t,x),M)\to 0\hbox{ as } t\to\infty\}\). The set \(M\) is said to be asymptotically stable if \(A(M)\) contains \(\{ y\in Y: d(y,M)\leq \eta\}\) for some \(\eta>0\) and if for every \(\varepsilon > 0\) there is \(\delta > 0\) such that \(d(\pi(t,x),M)<\varepsilon\) provided \(d(x,M)<\delta\) and \(t\geq 0\). If \(A(M)=X\) then \(M\) is called globally asymptotically stable. The main result in this paper is the following theorem: Let \(X\) be an infinite-dimensional Banach space and \(\pi: \mathbb{R}^ +\times X\to X\) be a semidynamical system on \(X\). Suppose \(M\) is a nonempty compact invariant asymptotically stable subset of \(X\). Then the following are equivalent: (i) \(M\) is strongly cellular; (ii) \(A(M)\) is homeomorphic to \(X\); (iii) there exists a neighborhood \(U\) of \(M\) in \(A(M)\) which is homeomorphic to \(X\).
Reviewer: V.Valov (Sofia)

MSC:

57N17 Topology of topological vector spaces
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37C75 Stability theory for smooth dynamical systems
PDFBibTeX XMLCite
Full Text: DOI EuDML