TY - JOUR
AU - Kapitanski, Lev
AU - Å½ivanoviÄ‡ Gonzalez, Sanja
PY - 2016/04/12
Y2 - 2023/03/30
TI - Attractors in hyperspace
JF - Topological Methods in Nonlinear Analysis
JA - TMNA
VL - 44
IS - 1
SE -
DO -
UR - https://apcz.umk.pl/TMNA/article/view/TMNA.2014.043
SP - 199 - 227
AB - 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 $(\Phi, X)$.
ER -