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 Fixed Point Theorems in Cone Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 609281 9 pages doi 2009 609281 Research Article Fixed Point Theorems in Cone Banach Spaces Erdal Karapinar Department of Mathematics Attltm University 06836 Ankara Turkey Correspondence should be addressed to Erdal Karapinar erdalkarapinar@ Received 23 October 2009 Accepted 15 December 2009 Recommended by Juan Jose Nieto In this manuscript a class of self-mappings on cone Banach spaces which have at least one fixed point is considered. More precisely for a closed and convex subset C of a cone Banach space with the norm x p d x 0 if there exist a b s and T C C satisfies the conditions 0 s a -2b 2 a b and 4ad Tx Ty b d x Tx dfy Ty sd x y for all x y e C then T has at least one Fixed point. Copyright 2009 Erdal Karapinar. 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 and Preliminaries In 1980 Rzepecki 1 introduced a generalized metric dE on a set X in a way that dE X X X S where E is Banach space and S is a normal cone in E with partial order In that paper the author generalized the fixed point theorems of Maia type 2 . Let X be a nonempty set endowed in two metrics d1 d2 and T a mapping of X into itself. Suppose that d1 x y d2 x y for all x y e X and X is complete space with respect to d1 and T is continuous with respect to d1 and T is contraction with respect to d2 that is d2 Tx Ty kd2 x y for all x y e X where 0 k 1. Then f has a unique fixed point in X. Seven years later Lin 3 considered the notion of K-metric spaces by replacing real numbers with cone K in the metric function that is d X X X K. In that manuscript some results of Khan and Imdad 4 on fixed point theorems were considered for K-metric spaces. Without mentioning the papers of Lin and Rzepecki in 2007 Huang and Zhang 5
