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Homotopy index for multivalued flows on sleek sets. (English) Zbl 1096.54016

The author studies the differential inclusion problem \(\dot x(t)\in F(x(t))\) for \(t\in \mathbb R\), \(x(t)\in K\subset \mathbb R^n\), for every \(t\in\mathbb R\), where the set \(K\) and the map \(F\) satisfy certain conditions. To each \(K\subset\mathbb R^n\) there is a set valued map \(T_K\) assigning to every \(x\in K\) the set \(T_K(x)=\{v\in\mathbb R^n:\liminf_{h\to 0^+}(\text{dist}(x+hv,K)/h)=0\}\). The set \(K\) is said to be sleek if it is closed and if the corresponding map \(T_K\) is lower semicontinuous. The author proves that if \(K=\overline{\text{Int\,}K}\subset\mathbb R^n\) is a compact sleek set and the map \(F\) satisfies a certain very restrictive condition, then the above differential inclusion problem has a solution in the sense that there exists an invariant trajectory in \(K\).

MSC:

54H20 Topological dynamics (MSC2010)
34C25 Periodic solutions to ordinary differential equations
37B30 Index theory for dynamical systems, Morse-Conley indices
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