# Lyusternik-Graves theorem and fixed points

1 C&O - Equipe combinatoire et optimisation
IMJ-PRG - Institut de Mathématiques de Jussieu - Paris Rive Gauche
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.
Document type :
Journal articles

https://hal.inria.fr/hal-00643231
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Submitted on : Monday, November 21, 2011 - 2:38:06 PM
Last modification on : Saturday, April 30, 2022 - 3:36:08 AM

### 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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