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On Constrained Open Mapping Theorems

CIBULKA, R. On Constrained Open Mapping Theorems. Paris, 2011.
Jazyk publikace: eng
Anglický název: On Constrained Open Mapping Theorems
Rok vydání: 2011
Autoři: Ing. Radek Cibulka Ph.D.
Abstrakt EN: The lecture (which is based on two recent notes by the author is divided into two main parts. The first one is devoted to an analogue of the well-known Robinson-Ursescu theorem concerning the so-called relative openness with a linear rate at the reference point (which corresponds to the restrictive metric regularity introduced by B. S. Mordukhovich and B. Wang) of a set-valued mapping between Banach spaces. Conditions guaranteeing this property will be given. Moreover, simple examples illustrate that no single assumption can be dropped in general. The latter part deals with a generalization of Lyusternik-Graves theorem when the non-linear mapping under consideration is restricted to a closed convex subset of a Banach space and can be ”well” approximated at a reference point, e.g. by a bunch of mappings from the space of all continuous linear operators or by a set-valued mapping with a star-shaped graph. It is emphasized that there is no need to have a suitable approximation of both the constraint set and the non-linear mapping in a whole neighborhood of the point in question when one is interested in the openness at this point. Also, only positive homogeneity of the approximation is required. Moreover, under mild additional assumptions on the approximating mapping (such as convexity and closeness of its graph), one obtains this property in a vicinity of the reference point.
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