Implicit Multifunction Theorems
We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson–Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson–Ursescu theorem, we also establish a refined version of the multidirectional mean value inequality which is of independent interest.
WMU ScholarWorks Citation
Ledyaev, Yuri S. and Zhu, Qiji Jim, "Implicit Multifunction Theorems" (1999). Math Faculty Publications. 22.
Ledyaev, Y.S., Zhu, Q.J. Implicit Multifunction Theorems. Set-Valued Analysis 7, 209–238 (1999). https://doi.org/10.1023/A:1008775413250