## 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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