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 Best Approximations Theorem for a Couple in Cone Banach Space | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 784578 9 pages doi 2010 784578 Research Article Best Approximations Theorem for a Couple in Cone Banach Space Erdal Karapinar1 and Duran Turkoglu2 1 Department of Mathematics Attltm University 06836 Incek Ankara Turkey 2 Department of Mathematics Gazi University 06500 Teknikokullar Ankara Turkey Correspondence should be addressed to Erdal Karapinar erdalkarapinar@ Received 23 March 2010 Accepted 8 June 2010 Academic Editor A. T. M. Lau Copyright 2010 E. Karapinar and D. Turkoglu. 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. The notion of coupled fixed point is introduced by Bhaskar and Lakshmikantham 2006 . In this manuscript some result of Mitrovic 2010 extended to the class of cone Banach spaces. 1. Introduction and Preliminaries Banach valued metric space was considered by Rzepecki 1 Lin 2 and lately by Huang and Zhang 3 . Basically for nonempty set X the definition of metric d XXX R 0 to is replaced by a new metric namely by an ordered Banach space E d X X X E. Such metric spaces are called cone metric spaces in short CMSs . In 1980 by using this idea Rzepecki 1 generalized the fixed point theorems of Maia type. Seven years later Lin 2 extends some results of Khan and Imdad 4 by considering this new metric space construction. In 2007 Huang and Zhang 3 discussed some properties of convergence of sequences and proved the fixed point theorems of contractive mapping for cone metric spaces any mapping T of a complete cone metric space X into itself that satisfies for some 0 k 1 the inequality d Tx Ty kd x y for all x y X has a unique fixed point. Recently many results on fixed point theorems have been extended to cone metric spaces see . 3 5-11 . In 3 the authors extends to cone metric .
