A Leray-schauder Alternative for Weakly-strongly Sequentially Continuous Weakly Compact Maps

This paper presents new fixed point results for weakly sequentially upper semicontinuous maps defined on locally convex Hausdorff topological spaces which are angelic when furnished with the weak topology. Moreover, we establish an applicable Leray-Schauder alternative (Theorem 2.12) for a certain subclass of these maps. Our alternative combines the advantages of the strong topology (i.e., the sets are open in the strong topology) with the advantages of the weak topology (i.e., the maps are weakly-strongly sequentially continuous and weakly compact). In Section 3, we illustrate how easily Theorem 2.12 can be applied in practice. Finally, we recall the following definition from the literature [9].