Báo cáo sinh học: Convergence of the modified Mann's iterative method for asymptotically kappa-strictly pseudocontractive mappings

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: Convergence of the modified Manns iterative method for asymptotically kappa-strictly pseudocontractive mappings
Nội dung trích xuất từ tài liệu:
Báo cáo sinh học: " Convergence of the modified Manns iterative method for asymptotically kappa-strictly pseudocontractive mappings"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. Convergence of the modified Manns iterative method for asymptotically kappa-strictly pseudocontractive mappings Fixed Point Theory and Applications 2011, 2011:100 doi:10.1186/1687-1812-2011-100 Ying Zhang (spzhangying@126.com) Zhiwei Xie (betterwill@gmail.com) ISSN 1687-1812 Article type Research Submission date 4 May 2011 Acceptance date 9 December 2011 Publication date 9 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/100 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 Zhang and Xie ; 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. Convergence of the modified Mann’s iterative method for asymptotically κ-strictly pseudocontractive mappings Ying Zhang∗,1,2 and Zhiwei Xie3 1 School of Mathematics and Physics, North China Electric Power University, Baoding, Hebei 071003, P.R. China 2 School of Economics, Renmin University of China, Beijing 100872, P.R. China 3 Easyway Company Limited, Beijing 100872, P.R. China *Corresponding author: spzhangying@126.com Email address: ZX: betterwill@gmail.com Abstract Let E be a real uniformly convex Banach space which has the Fr´chet differentiable e norm, and K a nonempty, closed, and convex subset of E. Let T : K → K be an asymp- totically κ-strictly pseudocontractive mapping with a nonempty fixed point set. We prove that (I − T ) is demiclosed at 0 and obtain a weak convergence theorem of the modi- fied Mann’s algorithm for T under suitable control conditions. Moreover, we also elicit a necessary and sufficient condition that guarantees strong convergence of the modified Mann’s iterative sequence to a fixed point of T in a real Banach spaces with the Fr´chet e differentiable norm. 2000 AMS Subject Classification: 47H09; 47H10. Keywords: asymptotically κ-strictly pseudocontractive mappings; demiclosedness prin- ciple; the modified Mann’s algorithm; fixed points.1 IntroductionLet E and E ∗ be a real Banach space and the dual space of E, respectively. Let K be a ∗nonempty subset of E. Let J denote the normalized duality mapping from E into 2E givenby J (x) = {f ∈ E ∗ : x, f = x 2 = f 2 }, for all x ∈ E, where ·, · denotes the duality 1pairing between E and E ∗ . In the sequel, we will denote the set of fixed points of a mappingT : K → K by F (T ) = {x ∈ K : T x = x}. A mapping T : K → K is called asymptotically κ-strictly pseudocontractive with sequence{κn }∞ ⊆ [1, ∞) such that limn→∞ κn = 1 (see, e.g., [1–3]) if for all x, y ∈ K, there exist a n=1constant κ ∈ [0, 1) and j (x − y ) ∈ J (x − y ) such that T n x − T n y, j (x − y ) ≤ κn x − y 2 − κ x − y − (T n x − T n y ) 2 ...