scholarly journals Viscosity approximation methods for equilibrium problems and fixed point problems of nonexpansive mappings and inverse-strongly monotone mappings

2007 ◽  
Vol 14 (4) ◽  
pp. 405-420 ◽  
Author(s):  
Jianmin Song ◽  
Shenghua Wang ◽  
Haiyun Zhou
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Fixed Point Problems ◽  
Monotone Mappings ◽  
Approximation Methods ◽  
Viscosity Approximation ◽  
Inverse Strongly Monotone ◽  
Viscosity Approximation Methods ◽  
Strongly Monotone Mappings

scholarly journals An Iterative Method with Norm Convergence for a Class of Generalized Equilibrium Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Haixia Zhang ◽  
Fenghui Wang
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Monotone Mappings ◽  
Norm Convergence ◽  
Inverse Strongly Monotone ◽  
Generalized Equilibrium ◽  
Strongly Monotone Mappings

Recently, Takahashi and Takahashi proposed an iterative algorithm for solving a problem for finding common solutions of generalized equilibrium problems governed by inverse strongly monotone mappings and of fixed point problems for nonexpansive mappings. In this paper, we provide a result that allows for the removal of one condition ensuring the strong convergence of the algorithm.

scholarly journals Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Luo Yi Shi ◽  
Ru Dong Chen
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convex Subset ◽  
Unique Fixed Point ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Approximation Methods ◽  
Viscosity Approximation ◽  
Viscosity Approximation Methods

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mappingTof a closed convex subsetCof a CAT(0) spaceX. Suppose that the set Fix(T)of fixed points ofTis nonempty. For a contractionfonCandt∈(0,1), letxt∈Cbe the unique fixed point of the contractionx↦tf(x)⊕(1-t)Tx. We will show that ifXis a CAT(0) space satisfying some property, then{xt}converge strongly to a fixed point ofTwhich solves some variational inequality. Consider also the iteration process{xn}, wherex0∈Cis arbitrary andxn+1=αnf(xn)⊕(1-αn)Txnforn≥1, where{αn}⊂(0,1). It is shown that under certain appropriate conditions onαn,{xn}converge strongly to a fixed point ofTwhich solves some variational inequality.

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