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Báo cáo hóa học: "FIXED POINT THEOREMS IN LOCALLY CONVEX SPACES—THE SCHAUDER MAPPING METHOD"

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 POINT THEOREMS IN LOCALLY CONVEX SPACES—THE SCHAUDER MAPPING METHOD | FIXED POINT THEOREMS IN LOCALLY CONVEX SPACES THE SCHAUDER MAPPING METHOD S. COBZAS Received 22 March 2005 Revised 22 July 2005 Accepted 6 September 2005 In the appendix to the book by F. F. Bonsal Lectures on Some Fixed Point Theorems of Functional Analysis Tata Institute Bombay 1962 a proof by Singbal of the Schauder-Tychonoff fixed point theorem based on a locally convex variant of Schauder mapping method is included. The aim of this note is to show that this method can be adapted to yield a proof of Kakutani fixed point theorem in the locally convex case. For the sake of completeness we include also the proof of Schauder-Tychonoff theorem based on this method. As applications one proves a theorem of von Neumann and a minimax result in game theory. Copyright 2006 S. Cobzas. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let Bn be the unit ball of the Euclidean space Rn. Brouwer s fixed point theorem asserts that any continuous mapping f Bn Bn has a fixed point that is there exists x e Bn such that f x x. The result holds for any nonempty convex bounded closed subset K of Rn or of any finite dimensional normed space see 8 Theorems 18.9 and 18.9 . Schauder 16 extended this result to the case when K is a convex compact subset of an arbitrary normed space X. Using some special functions called Schauder mappings the proof of Schauder s theorem can be reduced to Brouwer fixed point theorem see. e.g. 8 page 197 or 12 page 180 . A further extension of this theorem was given by Tychonoff 18 who proved its validity when K is a compact convex subset of a Hausdorff locally convex space X. The proof given in the treatise of Dunford and Schwartz 4 is based on three lemmas and with some minor modifications the same proof appears in 5 and 9 . The extension of Schauder mapping method to locally convex