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Equi-attraction and continuous dependence of strong attractors of set-valued dynamical systems on parameters. (English) Zbl 1091.37003

Let \((\Lambda, \varrho)\) be a compact metric space of parameters, \(\{G_\lambda;\,\lambda \in \Lambda\}\) a family of set-valued dynamcal systems which are upper semicontinuous in initial conditions and let \(\mathcal{A}_\lambda\) be the strong global attractor of \(G_\lambda\). Under some additional assumptions, it is shown that for any fixed \(r>0\), the family \(\{\mathcal{A}_\lambda [r] = \bigcup_{\varrho(\lambda',\lambda) \leq r}\,\mathcal{A}_{\lambda'};\,\,\lambda \in \Lambda \}\) equi-attracts each bounded set. It is also demonstrated that the continuity of \(\mathcal{A}_\lambda\) in \(\lambda\) with respect to the Hausdorff metric is equivalent to the equi-attraction of the family \(\{\mathcal{A}_\lambda; \,\lambda \in \Lambda\}\). As an example, the authors consider a family of dynamical systems generated by a family of autonomous differential inclusions of a finite-dimensional space.

MSC:

37B25 Stability of topological dynamical systems
34A60 Ordinary differential inclusions
34D45 Attractors of solutions to ordinary differential equations
47H20 Semigroups of nonlinear operators
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