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báo cáo hóa học:" Research Article Ray’s Theorem for Firmly Nonexpansive-Like Mappings and Equilibrium Problems in Banach Spaces"

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 Ray’s Theorem for Firmly Nonexpansive-Like Mappings and Equilibrium Problems in Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 806837 7 pages doi 10.1155 2010 806837 Research Article Ray s Theorem for Firmly Nonexpansive-Like Mappings and Equilibrium Problems in Banach Spaces Satit Saejung Department of Mathematics Khon Kaen University Khon Kaen 40002 Thailand Correspondence should be addressed to Satit Saejung saejung@kku.ac.th Received 3 July 2010 Accepted 29 September 2010 Academic Editor A. T. M. Lau Copyright 2010 Satit Saejung. 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 prove that every firmly nonexpansive-like mapping from a closed convex subset C of a smooth strictly convex and reflexive Banach pace into itself has a fixed point if and only if C is bounded. We obtain a necessary and sufficient condition for the existence of solutions of an equilibrium problem and of a variational inequality problem defined in a Banach space. 1. Introduction Let C be a subset of a Banach space E. A mapping T C E is nonexpansive if Tx - Ty x - y for all x y e C. In 1965 it was proved independently by Browder 1 Gohde 2 and Kirk 3 that if C is a bounded closed convex subset of a Hilbert space and T C C is nonexpansive then T has a fixed point. Combining the results above Ray 4 obtained the following interesting result see 5 for a simpler proof . Theorem 1.1. Let C be a closed and convex subset of a Hilbert space. Then the following statements are equivalent i Fix T x e C x Tx 0 for every nonexpansive mapping T C C ii C is bounded. It is well known that for each subset C of a Hilbert space H a mapping T C H is nonexpansive if and only if S 1 2 I T is firmly nonexpansive that is the following 2 Fixed Point Theory and Applications inequality is satisfied by all x y e C Sx - Sy x - Sx - y - Sy 0. 1.1 In this case Fix T Fix S . We can restate Ray s .