Báo cáo hóa học: "Research Article Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature"

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: Research Article Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 906727 8 pages doi 2009 906727 Research Article Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature Constantin P. Niculescu and Ionel Roventa Department of Mathematics University of Craiova 200585 Craiova Romania Correspondence should be addressed to lonel RovenỊa roventaionel@ Received 3 June 2009 Accepted 9 November 2009 Recommended by Anthony To Ming Lau The Schauder fixed point theorem is extended to the context of metric spaces with global nonpositive curvature. Some applications are included. Copyright 2009 C. P. Niculescu and I. Roventa. 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 The aim of our paper is to discuss the extension of the Schauder fixed point theorem to the framework of spaces with global nonpositive curvature abbreviated global NPC spaces . A formal definition of these spaces is as follows. Definition . A global NPC space is a complete metric space E E d for which the following inequality holds true for each pair of points x0 x1 e E there exists a point y e E such that for all points z e E 72 1 72 1 72 1 72 z y 2d2 z X0 2d2 z xi - 4d2 x0 xi . M These spaces are also known as the Cat 0 spaces. See 1 . In a global NPC space each pair of points x0 x1 e E can be connected by a geodesic . by a rectifiable curve Y 0 1 E such that the length of Y s t is d Y s Y t for all 0 s t 1 . Moreover this geodesic is unique. The point y that appears in Definition is the midpoint of x0 and x1 and has the property 1 d x0 y d y xfi 2 d x0 xfi. 2 Fixed Point Theory and Applications Every Hilbert space is a global NPC space. Its geodesics are the line segments. The upper half-plane H z e C Imz 0 endowed with the Poincare metric dS rfC dy2 3 y2 .

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