Research Article Existence Theorems for Generalized Distance on Complete Metric Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 397150 21 pages doi 2010 397150 Research Article Existence Theorems for Generalized Distance on Complete Metric Spaces Jeong Sheok Ume Department of Applied Mathematics Changwon National University Changwon 641-773 Republic of Korea Correspondence should be addressed to Jeong Sheok Ume jsume@ Received 20 September 2009 Revised 7 May 2010 Accepted 20 May 2010 Academic Editor L. Gorniewicz Copyright 2010 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. We first introduce the new concept of a distance called u-distance which generalizes -distance Tataru s distance and T-distance. Then we prove a new minimization theorem and a new fixed point theorem by using a M-distance on a complete metric space. Our results extend and unify many known results due to Caristi Ciric Ekeland Kada-Suzuki-Takahashi Kannan Ume and others. 1. Introduction The Banach contraction principle 1 Ekeland s -variational principle 2 and Caristi s fixed point theorem 3 are very useful tools in nonlinear analysis control theory economic theory and global analysis. These theorems are extended by several authors in different directions. Takahashi 4 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 u and f v d u v f u . Then there exists x0 e X such that f x0 infxeXf x . Some authors 5-7 have generalized and extended this minimization theorem in complete metric spaces. In 1996 Kada et al. 5 introduced the concept of -distance on a metric space as follows. Let X be a metric space with metric d. Then a function p X X X 0 to is called a .
