Vol.1 No.2 (July 2011)
General Viscosity Iterative Methods and Applications
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.
田明 , 金鑫 (2011) 一般粘滞迭代方法及其应用。 理论数学， 1， 136-143. doi: 10.12677/pm.2011.12027
 A. Moudafi. Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications, 2000, 241: 45-55.
 H. K. Xu. Viscosity approximation methods for nonexpansive mapping. Journal of Mathematical Analysis and Applications, 2004, 298: 279-291.
 G. Marino, H. K. Xu. An general iterative method for nonexpansivemappings in Hilbert spaces. Journal of Mathematical Analysis and Applications, 2006, 318: 43-52.
 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.
 M. Tian. A general iterative algorithm for nonexpansive mappingsin Hilbert spaces. Nonlinear Analysis, 2010, 73: 689-694.
 H. K. Xu. Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society, 2002, 66: 240-256.
 K. Geobel, W. A. Kirk. Topics in metric fixed point theory, Cambridge study advance mathematics. Cambridge University Press, 1990.
 T. H. Kim, H. K. Xu. Strong convergence of modified Mann iterations. Nonlinear Analysis, 2005, 61: 51-60.
 K. Nakajo, W. Takahashi. Strong convergence theorems for nonex pansive mappings and none xpansive semigroups. Non- linear Analysis, 2003, 279: 372-397.
 Y. Yao, R. Chen, J. C. Yao. Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Analysis, 2007.
 H. K. Xu. An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications, 2003, 116: 659- 678.
 H. Zhou. Convergence theorems of fixed points for k-strict pseudo-contractionins Hilbert space. Nonlinear Analysis, 2007
 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.
 P. L. Combettes, S. A. Hirstoaga. Equilibrium programming in Hilbert spaces. Nonlinear Analysis, 2005, 6: 117-136.
 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.
 Y. Liu. A general iterative method for equilibrium problems and strict pse udo-contractions in Hilbert spaces. Nonlinear Analysis, 2009.