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Lyusternik-Graves theorem and fixed points

Authors: Asen L. Dontchev and Hélène Frankowska
Journal: Proc. Amer. Math. Soc. 139 (2011), 521-534
MSC (2010): Primary 49J53; Secondary 47J22, 49J40, 49K40, 90C31
Published electronically: July 16, 2010
MathSciNet review: 2736335
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Abstract: For set-valued mappings $ F$ and $ \Psi$ 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 $ F$ is metrically regular with constant $ \kappa$ and $ \Psi$ is Aubin (Lipschitz) continuous with constant $ \mu$ such that $ \kappa\mu <1$, then the distance from $ x$ to the set of fixed points of $ F^{-1}\Psi$ is bounded by $ \kappa/(1-\kappa \mu)$ times the infimum distance between $ \Psi(x)$ and $ F(x)$. From this result we derive known Lyusternik-Graves theorems, a recent theorem by Arutyunov, as well as some fixed point theorems.

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Additional Information

Asen L. Dontchev
Affiliation: Mathematical Reviews and the University of Michigan, Ann Arbor, Michigan 48109. On leave from the Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria

Hélène Frankowska
Affiliation: Combinatoire & Optimisation, CNRS, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France

Keywords: Metric regularity, openness, Lyusternik-Graves theorem, fixed point, Ekeland principle
Received by editor(s): December 24, 2009
Received by editor(s) in revised form: March 8, 2010
Published electronically: July 16, 2010
Additional Notes: The first author was supported by National Science Foundation grant DMS 1008341.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.