On some fixed point theorems for 1-set weakly contractive multi-valued mappings with weakly sequentially closed graph (Q2893950)

The authors obtain some results of Leray-Schauder type for multivalued condensing operators. To give an example, let \(E\) be a Banach space, \(\Omega\subset E\) be nonempty, closed, convex, and let \(U\subset\Omega\) be a weakly open neighborhood of \(0\). Let \(F\) be a map from the weak closure of \(U\) into the nonempty subsets of \(\Omega\) that has a weakly sequentially closed graph and a bounded image. If \(F\) is condensing with respect to a measure of weak non-compactness, then either \(F\) has a fixed point or there is a point \(x\) in the weak boundary of \(U\) in \(\Omega\) and a \(\lambda\in(0,1)\) with \(x\in\lambda F(x)\).