# Lyusternik-Graves theorem and fixed points

1 C&O - Equipe combinatoire et optimisation
UPMC - Université Pierre et Marie Curie - Paris 6, CNRS - Centre National de la Recherche Scientifique : FRE3232
Abstract : 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.
Type de document :
Article dans une revue
Proceedings of the American Mathematical Society, American Mathematical Society, 2011, 139, pp.521-534. 〈10.1090/S0002-9939-2010-10490-2〉

https://hal.inria.fr/hal-00643231
Contributeur : Estelle Bouzat <>
Soumis le : lundi 21 novembre 2011 - 14:38:06
Dernière modification le : mercredi 21 mars 2018 - 18:57:28

### Citation

Asen L. Dontchev, Hélène Frankowska. Lyusternik-Graves theorem and fixed points. Proceedings of the American Mathematical Society, American Mathematical Society, 2011, 139, pp.521-534. 〈10.1090/S0002-9939-2010-10490-2〉. 〈hal-00643231〉

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