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The Inverse Function Theorems of Lawrence Graves

Banff International Research Station for Mathematical Innovation and Discovery

The classical inverse/implicit function theorems revolves around solving an equation in terms of a parameter and tell us when the solution mapping associated with this equation is a (differentiable) function. Already in 1927 Hildebrandt and Graves observed that one can put aside differentiability obtaining that the solution mapping is just Lipschitz continuous. The idea has evolved in subsequent extensions most known of which are various reincarnations of the Lyusternik-Graves theorem. In the last several decades it has been widely accepted that in order to derive estimates for the solution mapping and put them in use for proving convergence of algorithms, it is sufficient to differentiate what you can and leave the rest as is, hoping that the resulting problem is easier to handle. More sophisticated results have been obtained by employing various forms of metric regularity of mappings acting in metric spaces, aiming at applications to numerical analysis.

Mathematics; Operations research, mathematical programming; Operator theory; Control/optimization/operation research

video/mp4

Author affiliation: AMS and the University of Michigan

2018-03-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64942

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/