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Implicit Multifunction Theorems

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI, 49008, U.S.A.

    Yuri S. Ledyaev & Qiji J. Zhu

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  1. Yuri S. Ledyaev
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  2. Qiji J. Zhu
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Ledyaev, Y.S., Zhu, Q.J. Implicit Multifunction Theorems. Set-Valued Analysis 7, 209–238 (1999). https://doi.org/10.1023/A:1008775413250

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