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 Proximity Point Theorems for p-Cyclic Meir-Keeler Contractions | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 197308 9 pages doi 2009 197308 Research Article Best Proximity Point Theorems for p-Cyclic Meir-Keeler Contractions S. Karpagam and Sushama Agrawal Department of Mathematics Ramanujan Institute for Advanced Study in Mathematics University of Madras Chepauk Chennai 600 005 India Correspondence should be addressed to S. Karpagam Received 31 August 2008 Revised 21 November 2008 Accepted 5 January 2009 Recommended by Tomonari Suzuki We consider a contraction map T of the Meir-Keeler type on the union of p subsets A1 . Ap p 2 of a metric space X d to itself. We give sufficient conditions for the existence and convergence of a best proximity point for such a map. Copyright 2009 S. Karpagam and S. Agrawal. 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 Meir and Keeler in 1 considered an extension of the classical Banach contraction theorem on a complete metric space. Kirk et al. in 2 extended the Banach contraction theorem for a class of mappings satisfying cyclical contractive conditions. Eldred and Veeramani in 3 introduced the following definition. Let A and B be nonempty subsets of a metric space X. A map T A u B A u B isa cyclic contraction map if it satisfies 1 T A c B and T B c A and 2 for some k e 0 1 d Tx Tỳ kd x y 1 - k dist A B for all x e A y e B. In this case a point z e A u B such that d z Tz dist A B called a best proximity point has been considered. This notion is more general in the sense that if the sets intersect then every best proximity point is a fixed point. In 3 sufficient conditions for the existence and convergence of a unique best proximity point for a cyclic contraction on a uniformly convex Banach space have been given. Further in 4 this .
