Kakutani-Fan's fixed point theorem in hyperspaces (Q1598351)

The article deals with some variants of Kakutani-Fan's fixed point theorem in hyperspaces \({\mathcal K}\) (endowed with the Pompeiu-Hausdorff metric \(h)\) of all non-empty convex bounded closed subsets of a real Banach space \(\mathbb{K}\). In particular, the authors prove that, for each \(h\)-upper semicontinuous and \({\mathcal K}\)-compact multifunction \(F:{\mathcal D}\to 2^{\mathcal K}\) of a non-empty convex bounded closed subset \({\mathcal D}\) of \({\mathcal K}\), there exists an \(X_0\in {\mathcal D}\) such that \(X_0\in F(X_0)\). As corollaries, a hyperspace version of the Brouwer and Schauder fixed point theorems for \(h\)-continuous and \({\mathcal K}\)-compact maps are formulated. As applications, the existence theorem of set-valued solutions for a Cauchy problem for differential equations under Peano type assumptions is stated.