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Journal of Convex Analysis 19 (2012), No. 4, 955--973
Copyright Heldermann Verlag 2012

Lyusternik-Graves Theorem and Fixed Points II

Asen L. Dontchev
Mathematical Reviews, Ann Arbor, MI 48107-8604, U.S.A., Bulgaria

Hélène Frankowska
CNRS -- Institut de Mathématiques, Université P. et M. Curie, 4 place Jussieu, 75252 Paris, France

This work continues the studies in our previous paper ["Lyusternik-Graves theorem and fixed points", Proc. Amer. Math. Soc. 139 (2011) 521--534]. It is written as a separate paper which extends the previous one in the direction of closing the gap between Lyusternik-Graves theorems and fixed point theorems. Here we introduce a new definition of global metric regularity on a set and associated definitions of Aubin continuity and linear openness that are equivalent to metric regularity on the same sets and with the same constant. When the sets are neighborhoods of a point in the graph of the mapping, these definitions reduce to the well studied properties at a point. We present Lyusternik-Graves type theorems in metric spaces for single-valued and set-valued perturbations, and show that they can be derived from, and some of them are even equivalent to, corresponding set-valued fixed point theorems.

Keywords: Set-valued analysis, metric regularity, Aubin property, linear openness, contraction mapping, Lyusternik-Graves theorem, Milyutin theorem.

MSC: 49J53, 47J07, 58C15, 49K40, 90C31

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