## 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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### References:

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