Description

• Title:
  Attractors in hyperspace
• Creator:
  Kapitanski, Lev;
  Živanović Gonzalez, Sanja
• Description:
  Given a map $\Phi$ defined on bounded subsets of the (base) metric space $X$ and with bounded sets as its values, one can follow the orbits $A$, $\Phi(A)$, $\Phi^2(A)$, $\ldots$, of nonempty, closed, and bounded sets $A$ in $X$. This is the system $(\Phi, X)$. On the other hand, the same orbits can be viewed as trajectories of points in the hyperspace $X^\sharp$ of nonempty, closed, and bounded subsets of $X$. This is the system $(\Phi, X^\sharp)$. We study the existence and properties of global attractors for both $(\Phi, X)$ and $(\Phi, X^\sharp)$. We give very basic conditions on $\Phi$, stated at the level of the base space $X$, that are necessary and sufficient for the existence of a global attractor for $(\Phi, X)$. Continuity is not among those conditions, but if $\Phi$ is continuous in a certain sense then the attractor and the $\omega$-limit sets are $\Phi$-invariant. If $(\Phi, X)$ has a global attractor, then $(\Phi, X^\sharp)$ has a global attractor as well. Every point of the global attractor of $(\Phi, X^\sharp)$ is a compact set in $X$, and the union of all the points of that attractor is the global attractor of $(\Phi, X)$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2016.04.12
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 44, No 1 (September 2014); 199 - 227;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2014.043/8157
• Subject:
  Hyperspace;
  global attractors;
  dynamical systems;
  iterated functions systems;
  Primary 37C70;
  54C60;
  54B20;
  Secondary 37B99;
  37B35
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2014.043
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other