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 Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals Satit Saejung | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 745010 7 pages doi 2008 745010 Research Article Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals Satit Saejung Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen 40002 Thailand Correspondence should be addressed to Satit Saejung saejung@ Received 28 November 2007 Revised 15 January 2008 Accepted 30 January 2008 Recommended by William A. Kirk We prove a convergence theorem by the new iterative method introduced by Takahashi et al. 2007 . Our result does not use Bochner integrals so it is different from that by Takahashi et al. We also correct the strong convergence theorem recently proved by He and Chen 2007 . Copyright 2008 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. 1. Introduction Let H be a real Hilbert space with the inner product and the norm II . Let T t t 0 be a family of mappings from a subset C of H into itself. We call it a nonexpansive semigroup on C if the following conditions are satisfied 1 T 0 x x for all x G C 2 T s t T s T t for all s t 0 3 for each x G C the mapping t T f x is continuous 4 T t x - T t y II x - y II for all x y G C and t 0. Motivated by Suzuki s result 1 and Nakajo-Takahashi s results 2 He and Chen 3 recently proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. However their proof of the main result 3 Theorem is very questionable. Indeed the existence of the subsequence Sj such that of 3 are satisfied that is s _ 0 llx - T0 F II _ 0 sj needs to be proved precisely. So the aim of this short paper is to correct He-Chen s result and also to give a new result by using the method recently introduced by .
