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論文名稱 Title |
Hilbert 空間中非擴張映象公共固定點之迭代方法 Iterative Methods for Common Fixed Points of Nonexpansive Mappings in Hilbert spaces |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
29 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2010-06-17 |
繳交日期 Date of Submission |
2011-05-16 |
關鍵字 Keywords |
固定點、凸集最佳化、黏質性逼近、非擴張映射、約縮映象 Nonexpansive mapping, Convex optimization, Contraction, Fixed point, Viscosity approximation |
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統計 Statistics |
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中文摘要 |
此篇論文的目標是提出以黏質性方法 尋求在 Hilbert 空間 H 的閉凸子集C 上之有限多個非擴張映象T={ T_{i} }_{i=1}^{N}特定公共固定點。 我們提出兩種模式來解決這類問題: 其一為隱式,另外一個為顯式.。 隱式模式是透過固定點方程式 x_{t}= tf (x_{t} ) + (1− t)Tx_{t} , 在 f :C →C為約縮映象下, 找到集合 {x_{t} : 0 < t < 1} 。 顯式模式是將隱式離散化,且藉由遞迴式 x_{n+1}=α\_{n}f(x_{n}) +(1−α\_{n})Tx_{n},在 n ≥ 0, {α\_{n} }⊂ (0,1) 下, 定義序列 {x_{n} }。 我們所提到的隱式模式和顯式模式兩者都範數收斂於 T 的固定點, 此部分已被文獻所證明. (在顯式中強加了另外的條件於序列{α _{n} } )。 我們將在有限多個非擴張映象集合中延伸這兩種模式。我們另外也藉由所提到的模式其範數收斂到此集合的公共固定點去求解變分不等式。 |
Abstract |
The aim of this work is to propose viscosity-like methods for finding a specific common fixed point of a finite family T={ T_{i} }_{i=1}^{N} of nonexpansive self-mappings of a closed convex subset C of a Hilbert space H.We propose two schemes: one implicit and the other explicit.The implicit scheme determines a set {x_{t} : 0 < t < 1} through the fixed point equation x_{t}= tf (x_{t} ) + (1− t)Tx_{t}, where f : C→C is a contraction.The explicit scheme is the discretization of the implicit scheme and de defines a sequence {x_{n} } by the recursion x_{n+1}=α\_{n}f(x_{n}) +(1−α\_{n})Tx_{n} for n ≥ 0, where {α\_{n} }⊂ (0,1) It has been shown in the literature that both implicit and explicit schemes converge in norm to a fixed point of T (with additional conditions imposed on the sequence {α _{n} } in the explicit scheme).We will extend both schemes to the case of a finite family of nonexpansive mappings. Our proposed schemes converge in norm to a common fixed point of the family which in addition solves a variational inequality. |
目次 Table of Contents |
Abstract---------------------------------i 1 Introduction-------------------------1 2 Preliminaries-----------------------4 3 The Main Results------------------6 3.1 An Implicit Scheme-------------7 3.2 An Explicit Scheme-------------13 Reference------------------------------22 |
參考文獻 References |
[1] F.E. Browder, Convergence of approximants to xed points of non-expansive maps in Banach spaces,Arch. Rational Mech. Anal. 24 (1967)82-90. [2] K. Geobel, W.A Kirk, Topics in Metric Fixed Point Theory, in: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge Univ. Press, 1990. [3] K. Geobel, S. Reich, Uniform Convexity, Nonexpansive Mappings, and Hyperbolic Geometry, Dekker, 1984. [4] A. Mouda , Viscosity approximation methods for xed-points problems, J. Math. Anal. Appl. 241 (2000), 46-55. [5] H.K Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002) 240-256. [6] H.K Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003)659-678. [7] H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279-291. 22 |
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