Báo cáo sinh học: Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems

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: Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems
Nội dung trích xuất từ tài liệu:
Báo cáo sinh học: " Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems"Fixed Point Theory andApplications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon.Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems Fixed Point Theory and Applications 2011, 2011:101 doi:10.1186/1687-1812-2011-101 Yonghong Yao (yaoyonghong@yahoo.cn) Yeol Je Cho (yjcho@gsnu.ac.kr) Yeong-Cheng Liou (simplex_liou@hotmail.com) ISSN 1687-1812 Article type Research Submission date 3 November 2010 Acceptance date 20 December 2011 Publication date 20 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/101 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Fixed Point Theory and Applications go to http://www.fixedpointtheoryandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Yao et al. ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Hierarchical convergence of an implicit double-net algorithm for nonexpansivesemigroups and variational inequality problems Yonghong Yao1 , Yeol Je Cho∗2 and Yeong-Cheng Liou3 1 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, People’s Republic of China 2 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea 3 Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan ∗ Corresponding author: yjcho@gsnu.ac.kr. E-mail addresses: YY:yaoyonghong@yahoo.cn Y-CL:simplex liou@hotmail.com. Abstract In this paper, we show the hierarchical convergence of the following implicit 1 double-net algorithm: λs 1 xs,t = s[tf (xs,t ) + (1 − t)(xs,t − µAxs,t )] + (1 − s) T (ν )xs,t dν, ∀s, t ∈ (0, 1), λs 0 where f is a ρ-contraction on a real Hilbert space H , A : H → H is an α-inverse strongly monotone mapping and S = {T (s)}s≥0 : H → H is a nonexpansive semi- group with the common fixed points set F ix(S ) = ∅, where F ix(S ) denotes the set of fixed points of the mapping S , and, for each fixed t ∈ (0, 1), the net {xs,t } con- verges in norm as s → 0 to a common fixed point xt ∈ F ix(S ) of {T (s)}s≥0 and, as t → 0, the net {xt } converges in norm to the solution x∗ of the following variational inequality:   ∗ x ∈ F ix(S );      Ax∗ , x − x∗ ≥ 0, ∀x ∈ F ix(S ). Keywords: fixed point; variational inequality; double-net algorithm; hierarchical convergence; Hilbert space. MSC(2000): 49J40; 47J20; 47H09; 65J15.1 IntroductionIn nonlinear analysis, a common approach to solving a problem with multiple solutions isto replace it by a family of perturbed problems admitting a ...