Research Article Remarks on Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 315398 7 pages doi 2010 315398 Research Article Remarks on Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings Mohamed A. Khamsi1 2 1 Department of Mathematical Science The University of Texas at El Paso El Paso TX 79968 USA 2 Department of Mathematics and Statistics King Fahd University of Petroleum Minerals . Box 411 Dhahran 31261 Saudi Arabia Correspondence should be addressed to Mohamed A. Khamsi mohamed@ Received 20 March 2010 Accepted 4 May 2010 Academic Editor W. A. Kirk Copyright 2010 Mohamed A. Khamsi. 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 discuss the newly introduced concept of cone metric spaces. We also discuss the fixed point existence results of contractive mappings defined on such metric spaces. In particular we show that most of the new results are merely copies of the classical ones. 1. Introduction Cone metric spaces were introduced in 1 . A similar notion was also considered by Rzepecki in 2 . After carefully defining convergence and completeness in cone metric spaces the authors proved some fixed point theorems of contractive mappings. Recently more fixed point results in cone metric spaces appeared in 3-8 . Topological questions in cone metric spaces were studied in 6 where it was proved that every cone metric space is first countable topological space. Hence continuity is equivalent to sequential continuity and compactness is equivalent to sequential compactness. It is worth mentioning the pioneering work of Quilliot 9 who introduced the concept of generalized metric spaces. His approach was very successful and used by many see references in 10 . It is our belief that cone metric spaces are a special case of generalized metric spaces. In this work
