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On a generalized Poincaré-Hopf formula in infinite dimensions

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  • We prove a formula relating the fixed point index of rest points of a completely continuous semiflow defined on a (not necessarily locally compact) metric space in the interior of an isolating block $B$ to the Euler characteristic of the pair $(B,B^-)$, where $B^-$ is the exit set. The proof relies on a general concept of an approximate neighborhood extension space and a full fixed point index theory for self-maps of such spaces. As a consequence, a generalized Poincaré-Hopf type formula for the differential equation determined by a perturbation of the generator of a compact $C_0$ semigroup is obtained.
    Mathematics Subject Classification: Primary: 37C25, 37B30, 55M15, 55M20; Secondary: 47D03, 47H11.


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