The Poincare-Miranda theorem is an extension of the intermediate value theorem to multi-dimensional functions. It is considered to be equivalent to Brouwer's fixed-point theorem. Both these theorems deal with single-valued functions.
Now, Brouwer's theorem has a generalization - Kakutani's fixed-point theorem - that deals with multi-valued (set-valued) functions. So it is reasonable to expect that the Poincare-Miranda theorem has a generalization for multi-valued functions. Is there such a generalization?