Fixed Points of Condensing Multivalued Maps in Topological Vector Spaces

The Schauder conjecture that every continuous single-valued map from a compact convex subset of a topological vector space into itself has a fixed point was stated in [12, Problem 54]. In a recent year, Cauty [2] gave a positive answer to this question by a very complicated approximation factorization. Very recently, Dobrowolski [3] established Cauty’s proof in a more accessible form by using the fact that a compact convex set in a metric linear space has the simplicial approximation property. The aim in this paper is to obtain multivalued versions of the Schauder fixed point theorem in complete metric linear spaces. For this we consider three classes of multivalued maps; that is, admissible maps introduced by Górniewicz [4], pseudocondensing maps by Hahn [5], and countably condensing maps by Väth [15], respectively. These pseudocondensing or countably condensing maps are more general than condensing maps. The main result is that every compact convex set in a complete metric linear space has the fixed point property with respect to admissible maps. The proof is based on the simplicial approximation property and its equivalent version due to Kalton et al. [9], where the latter corresponds to admissibility of the involved set in the sense of Klee [10]; see also [11]. More generally, we apply the main result to prove that every pseudocondensing admissible map from a closed convex subset of a complete metric linear space into itself has a fixed point. Finally, we present a fixed point theorem for countably condensing admissible maps in Fréchet spaces. Here, the fact that we restrict ourselves to countable sets is important in connection with differential and integral operators. The above results include the well-known theorems of Schauder [14], Kakutani [8], Bohnenblust and Karlin [1], and Sadovskii [13].