Lyusternik-Graves theorem and fixed points
Résumé
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.