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 Extensions of Minimization Theorems and Fixed Point Theorems on a Quasimetric Space | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 230101 15 pages doi 2008 230101 Research Article Extensions of Minimization Theorems and Fixed Point Theorems on a Quasimetric Space Jeong Sheok Ume Department of Applied Mathematics Changwon National University Changwon 641-773 South Korea Correspondence should be addressed to Jeong Sheok Ume jsume@ Received 16 March 2008 Accepted 9 June 2008 Recommended by Thomas Bartsch We introduce the new concepts of e-distance e-type mapping with respect to some e-distance and S-complete quasimetric space and prove minimization theorems fixed point theorems and variational principles on an S-complete quasimetric space. We also give some examples of quasimetrics e-distances and e-type mapping with respect to some e-distance. Our results extend improve and unify many known results due to Caristi Ekeland Ciric Kada-Suzuki-Takahashi Ume and others. Copyright 2008 Jeong Sheok Ume. 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 Caristi 1 proved a fixed point theorem on complete metric spaces which generalized the Banach contraction principle. Ekeland 2 also obtained a minimization theorem often called the -variational principle for a proper lower semicontinuous function bounded from below on complete metric spaces. The two theorems are very useful tools in nonlinear analysis control theory economic theory and global analysis. Later Takahashi 3 proved the following minimization theorem. Let X be a complete metric space and let f X -TO to be a proper lower semicontinuous function bounded from below. Suppose that for each u e X with f u infxeXf x there exists v e X such that v fu and f v dfu v f u . Then there exists x0 e X such that f xo infxeXf x . Many authors 3-5 have generalized and extended .
