一般粘滞迭代方法及其应用
General Viscosity Iterative Methods and Applications
摘要:
本文研究内容的核心之一在于，针对非扩张映射 的不动点问题，我们深刻阐明几种求解其不动点的迭代方法的相互关系。结合目前该类问题研究重点的转变，即从不动点的存在性和唯一性转变到如何构造有效迭代方法。我们系统阐述了Moudafi提出的粘滞迭代算法的发展演变过程，为以后的工作者提供参考。对于该类方法的具体应用进行深入的总结是本文的另一重心，主要围绕Mann迭代算法的修正及关于变分不等式以及均衡问题的联立求解获得相关结论。
Abstract:
The main core of this work is to clarify the profound relationships between several kinds of iterative methods for a fixed point of a given nonexpansive mapping.Concerning about the fact that the research focus has changed from the existence and uniqueness of the fixed piont to how to construct effective iterative methods.We begin with the viscosity iterative algorithm proposed by Moudafi.In order to provide a reference for future workers,we elaborate the development and evolution. The paper also deeply summarizes concrete applications of such methods.We focus not only on the modification processes of Mann iteration,but also concern about relevant conclusions about the variational inequalities and equilibrium problems.
参考文献
| [1] |
A. Moudafi. Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications, 2000, 241: 45-55. |
| [2] |
H. K. Xu. Viscosity approximation methods for nonexpansive mapping. Journal of Mathematical Analysis and Applications, 2004, 298: 279-291. |
| [3] |
G. Marino, H. K. Xu. An general iterative method for nonexpansivemappings in Hilbert spaces. Journal of Mathematical Analysis and Applications, 2006, 318: 43-52. |
| [4] |
I. Yamada. The hybrid steepest descent method for the inequalityproblem of the intersection of fixed point sets of nonexpansivemappings, in: D. Butnariu, Y. Censor, S. Reich Eds., Inherently Parallel Algorithm for Feasibility and Optimization, Elsevier, 2001: 473-504. |
| [5] |
M. Tian. A general iterative algorithm for nonexpansive mappingsin Hilbert spaces. Nonlinear Analysis, 2010, 73: 689-694. |
| [6] |
H. K. Xu. Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society, 2002, 66: 240-256. |
| [7] |
K. Geobel, W. A. Kirk. Topics in metric fixed point theory, Cambridge study advance mathematics. Cambridge University Press, 1990. |
| [8] |
T. H. Kim, H. K. Xu. Strong convergence of modified Mann iterations. Nonlinear Analysis, 2005, 61: 51-60. |
| [9] |
K. Nakajo, W. Takahashi. Strong convergence theorems for nonex pansive mappings and none xpansive semigroups. Non- linear Analysis, 2003, 279: 372-397. |
| [10] |
Y. Yao, R. Chen, J. C. Yao. Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Analysis, 2007. |
| [11] |
H. K. Xu. An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications, 2003, 116: 659- 678. |
| [12] |
H. Zhou. Convergence theorems of fixed points for k-strict pseudo-contractionins Hilbert space. Nonlinear Analysis, 2007 |
| [13] |
X. L. Qin, M. J. Shang, and S. M. Kang. Strong convergencetheorems of modified Mann I terative process for strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis, 2009, 70: 1257-1264. |
| [14] |
P. L. Combettes, S. A. Hirstoaga. Equilibrium programming in Hilbert spaces. Nonlinear Analysis, 2005, 6: 117-136. |
| [15] |
S. Takahashi, W. Takahashi. Viscosity approximation methods for eq-uilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Analysis, 2007, 33(1): 506-515. |
| [16] |
Y. Liu. A general iterative method for equilibrium problems and strict pse udo-contractions in Hilbert spaces. Nonlinear Analysis, 2009. |