Implicit Multifunction Theorems

Yu. S. Ledyaev and Q. J. Zhu

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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