Fixed points of condensing multivalued maps in topological vector spaces
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With the aid of the simplicial approximation property, we show that every admissible multivalued map from a compact convex subset of a complete metric linear space into itself has a fixed point. From this fact we deduce the fixed point property of a closed convex set with respect to pseudocondensing admissible maps. 1. Introduction 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 Gorniewicz [4], pseudocondensing maps by Hahn [5], and countably condensing maps by V¨ath [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 Fr´echet 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]. Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:2 (2004) 107–112 2000 Mathematics Subject Classification: 47H10, 54C60, 47H09, 46A16 URL: http://dx.doi.org/10.1155/S1687182004310041
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Fixed points of condensing maps
For a subset K of a topological vector space E, the closure, the convex hull, and the closed convex hull of K in E are denoted by K, coK, and coK, respectively. By k(K) we denote the collection of all nonempty compact subsets of K. For topological spaces X and Y , a multivalued map F : X Y is said to be upper semicontinuous on X if, for any open set V in Y , the set {x ∈ X : Fx ⊂ V } is open in X. F is said to be compact if its range F(X) is contained in a compact subset of Y . Definition 1.1. Given two topological spaces X and Y , an upper semicontinuous map F : X → k(Y ) is said to be admissible if there exist a topological s
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