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Hukuhara’s topological degree for non compact valued multifunctions. (English) Zbl 1048.47045
The aim of this paper is to construct a variant of the Leray-Schauder topological degree for maps of the form $$I- F$$, where $$F: \overline D\to 2^E$$ is a set-valued map defined on a bounded open subset of a Banach space $$E$$, taking nonempty closed bounded (and possibly) nonconvex values in the hyperspace $$B(E)$$ of all such subsets of $$E$$ equipped with the Hausdorff metric and being $$h$$-compact in the sense that the set $$\{F(x)\mid x\in\overline D\}$$ is precompact in $$B(E)$$ (contrary to the usual constructions of Cellina, Hukuhara, Górniewicz and others, the map $$F$$ is not compact in the usual sense: the set $$F(D)$$ is not assumed to be precompact in $$E$$). The presented construction relies on some approximation techniques. After some preliminaries, the authors consider set-valued maps $$F$$ having so-called regular representation (roughly speaking, $$F(x)= \overline{r\circ \phi(x)}$$, where $$r: E\to E$$ is Lipschitz continuous and $$\phi:\overline D\to 2^E$$ is Hausdorff upper semicontinuous with closed bounded and convex values and admits arbitrarily close graph approximations) and provide a construction of the degree for $$I- F$$. This degree satisfies the usual properties: homotopy invariance, existence and normalization. The standard use of topological degree methods allows to obtain some appropriate variants of the well-known results of fixed-point theory, such as versions of the nonlinear and Leray-Schauder alternative and the Borsuk antipodal theorem.

MSC:
 47H11 Degree theory for nonlinear operators 55M25 Degree, winding number 54C60 Set-valued maps in general topology 47H04 Set-valued operators
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References:
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