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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: FIXED POINTS OF CONDENSING MULTIVALUED MAPS IN TOPOLOGICAL VECTOR SPACES | FIXED POINTS OF CONDENSING MULTIVALUED MAPS IN TOPOLOGICAL VECTOR SPACES IN-SOOK KIM Received 27 October 2003 and in revised form 28 January 2004 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 Vath 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 sim-plicial 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 .
