Lyusternik-Graves theorem and fixed points

By Asen L. Dontchev, Hélène Frankowska. 2011

58 citations recorded by the source corpus.

For set-valued mappings and acting in metric spaces, we present local and global versions of the following general paradigm which has roots in the Lyusternik-Graves theorem and the contraction principle: if is metrically regular with constant and is Aubin (Lipschitz) continuous with constant such that , then the distance from to the set of fixed points of is bounded by times the infimum distance between and . From this result we derive known Lyusternik-Graves theorems, a recent theorem by Arutyunov, as well as some fixed point theorems.

Subfields: Cardiology and Cardiovascular Medicine, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f982f20094f0f7075d36ac2a79e93692a2fc3577

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In Cardiology and Cardiovascular Medicine

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Implicit Functions and Solution Mappings: A View from Variational Analysis (2009, methodology): “Graphical derivatives of set-valued mappings were introduced by Aubin [1981]; for more, see Aubin and Frankowska [1990]. The material in Section 4B is mainly from Dontchev, Quincampoix and Zlateva [2006], where results of Aubin and Frankowska [1987, 1990] were used.”
METRIC REGULARITY—A SURVEY PART 1. THEORY (2016, background): “For that reason, in some publications (for example, [31, 35]), Theorem 4.”
Openness stability and implicit multifunction theorems: Applications to variational systems (2011, result): “Note that, in turn, these papers extend many results of Dontchev and Rockafellar ([3], [4]).”