On an open question of V. Colao and G. Marino presented in the paper 'Krasnoselskii-Mann method for non-self mappings'
| Autor: | Meifang Guo, Xia Li, Yongfu Su |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Nonexpansive mapping
Fixed-point theorem Lambda 01 natural sciences Combinatorics symbols.namesake Strong convergence Countable set 0101 mathematics Weak convergence Physics 47H09 Multidisciplinary 47H05 Research 010102 general mathematics Regular polygon Hilbert space 010101 applied mathematics Mann iterative scheme symbols Inward condition Non-self mapping Convex function 47H10 |
| Zdroj: | SpringerPlus |
| ISSN: | 2193-1801 |
| Popis: | Let H be a Hilbert space and let C be a closed convex nonempty subset of H and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T : C\rightarrow H$$\end{document}T:C→H a non-self nonexpansive mapping. A map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h : C\rightarrow R$$\end{document}h:C→R defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x) := \inf \{\lambda \ge 0 : \lambda x+(1-\lambda )Tx \in C \}$$\end{document}h(x):=inf{λ≥0:λx+(1-λ)Tx∈C}. Then, for a fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in C$$\end{document}x0∈C and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{\alpha _0} = \max \left\{ {\frac{1}{2},h({x_0})} \right\}\end{aligned}$$\end{document}α0=max12,h(x0), Krasnoselskii–Mann algorithm is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1}=\alpha _n+(1-\alpha _n)Tx_n,$$\end{document}xn+1=αn+(1-αn)Txn, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{n+1}=\max \{\alpha _n, h(x_{x_{n+1}})\}$$\end{document}αn+1=max{αn,h(xxn+1)}. Recently, Colao and Marino (Fixed Point Theory Appl 2015:39, 2015) have proved both weak and strong convergence theorems when C is a strictly convex set and T is an inward mapping. Meanwhile, they proposed a open question for a countable family of non-self nonexpansive mappings. In this article, authors will give an answer and will prove the further generalized results with the examples to support them. |
| Databáze: | OpenAIRE |
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