A fixed point theorem for weakly inward multivalued contractions (Q1576988)

Using transfinite induction the author proves the following fixed point theorem: Let \(D\) be a nonempty closed subset of a Banach space \(X\) and let \(T\) be a mapping assigning to \(x\in D\) a nonempty closed set \(T(x)\subset X\). Assume that \(T\) is contractive with respect to the Hausdorff metric and that \(Tx\) is contained in the closure of \(x+\{\lambda(z-x)\mid z\in D\), \(\lambda\geq 1\}\) for each \(x\in D\). Then there is an \(x\in D\) such that \(x\in Tx\).