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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 Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 618767 9 pages doi 10.1155 2010 618767 Research Article Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces K. Sokhuma1 and A. Kaewkhao2 1 Department of Mathematics Faculty of Science Burapha University Chonburi 20131 Thailand 2 Department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand Correspondence should be addressed to A. Kaewkhao akaewkhao@yahoo.com Received 8 August 2010 Accepted 24 September 2010 Academic Editor T. D. Benavides Copyright 2010 K. Sokhuma and A. Kaewkhao. 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. Let E be a nonempty compact convex subset of a uniformly convex Banach space X and let t E E and T E KCfE be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping respectively. Assume in addition that Fix t n Fix T 0 and Tw w for all w e Fix t n Fix T . We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary x0 e E by yn 1 - pn xn pnzn xn 1 1 - an xn antyn where zn e Txn and an pn are sequences of positive numbers satisfying 0 a an pn b 1 converges strongly to a common fixed point of t and T that is there exists x e E such that x tx e Tx. 1. Introduction Let X be a Banach space and let E be a nonempty subset of X. We will denote by FB E the family of nonempty bounded closed subsets of E and by KC E the family of nonempty compact convex subsets of E. Let H - be the Hausdorff distance on FB X that is H A B max sup dist a B sup dist b A aeA beB where dist a B inf a - b b e B is the distance from the point a to the subset B. A B e FB X 1.1 2 Fixed Point Theory and Applications A mapping t E E is said to be nonexpansive if fx - ty x - y x y e E. 1.2
