Báo cáo sinh học: Fixed point and weak convergence theorems for point-dependent lambda-hybrid mappings 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: Fixed point and weak convergence theorems for point-dependent lambda-hybrid mappings in Banach spaces
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Báo cáo sinh học: " Fixed point and weak convergence theorems for point-dependent lambda-hybrid mappings in Banach spaces"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.Fixed point and weak convergence theorems for point-dependent lambda-hybrid mappings in Banach spaces Fixed Point Theory and Applications 2011, 2011:105 doi:10.1186/1687-1812-2011-105 Young-Ye Huang (yueh@mail.stut.edu.tw) Jyh-Chung Jeng (jhychung@mail.njtc.edu.tw) Tian-Yuan Kuo (sc038@mail.fy.edu.tw) Chung-Chien Hong (chtchong10@gmail.com) ISSN 1687-1812 Article type Research Submission date 25 August 2011 Acceptance date 23 December 2011 Publication date 23 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/105 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 Huang 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.Fixed point and weak convergence theorems for point-dependent λ-hybrid mappings in Banach spacesYoung-Ye Huang1 , Jyh-Chung Jeng2 , Tian-Yuan Kuo3 and Chung-Chien Hong∗41 Center for General Education, Southern Taiwan University, 1 Nantai St., YongkangDist., Tainan 71005, Taiwan2 Nanjeon Institute of Technology, 178 Chaoqin Rd., Yenshui Dist., Tainan 73746,Taiwan3 Fooyin University, 151 Jinxue Rd., Daliao Dist., Kaohsiung 83102, Taiwan4 Department of Industrial Management, National Pingtung University of Scienceand Technology, 1 Shuefu Rd., Neopu, Pingtung 91201, Taiwan∗ Corresponding author: chong@mail.npust.edu.twEmail addresses:YYH: yueh@mail.stut.edu.twJCJ: jhychung@mail.njtc.edu.twTYK: sc038@mail.fy.edu.tw 1 AbstractThe purpose of this article is to study the fixed point and weak convergenceproblem for the new defined class of point-dependent λ-hybrid mappingsrelative to a Bregman distance Df in a Banach space. We at first extendthe Aoyama–Iemoto–Kohsaka–Takahashi fixed point theorem for λ-hybridmappings in Hilbert spaces in 2010 to this much wider class of nonlinearmappings in Banach spaces. Secondly, we derive an Opial-like inequalityfor the Bregman distance and apply it to establish a weak convergencetheorem for this new class of nonlinear mappings. Some concrete examplesin a Hilbert space showing that our extension is proper are also given.Keywords: fixed point, Banach limit, Bregman distance, Gˆteaux differ- aentiable, subdifferential.2010 MSC: 47H09; 47H10. 21 Introduction Let C be a nonempty subset of a Hilbert space H . A mapping T : C → H issaid to be(1.1) nonexpansive if T x − T y ≤ x − y , ∀x, y ∈ C , cf. [1, 2]; 2 2(1.2) nonspreading if T x − T y ≤ x−y + 2 x − T x, y − T y , ∀x, y ∈ C , cf. [3–5]; 2 2(1.3) hybrid if T x − T y ≤ x−y + x − T x, y − T y , ∀x, y ∈ C , cf. [3, 5–7]. As shown in [3], (1.2) is equivalent to 2 2 2 2 Tx − Ty ≤ Tx − y + x − Tyfor all x, y ∈ C . In 1965, Browder [1] established the followingBrowder fixed point Theorem. Let C be a nonempty closed convex subset ofa ...