Lyusternik-Graves theorem and fixed points
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- by Asen L. Dontchev and Hélène Frankowska PDF
- Proc. Amer. Math. Soc. 139 (2011), 521-534 Request permission
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.References
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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
- Email: ald@ams.org
- Hélène Frankowska
- Affiliation: Combinatoire & Optimisation, CNRS, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France
- Email: frankowska@math.jussieu.fr
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 521-534
- MSC (2010): Primary 49J53; Secondary 47J22, 49J40, 49K40, 90C31
- DOI: https://doi.org/10.1090/S0002-9939-2010-10490-2
- MathSciNet review: 2736335