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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.

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