Lyusternik-Graves theorem and fixed points

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.