scholarly journals Strong Convergence Theorems for Maximal Monotone Operators, Fixed-Point Problems, and Equilibrium Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng ◽  
De-ning Qu
Keyword(s):  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Convergence Theorems ◽  
New Iterative Method

We present a new iterative method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions to an equilibrium problem, and the set of zeros of the sum of maximal monotone operators and prove the strong convergence theorems in the Hilbert spaces. We also apply our results to variational inequality and optimization problems.

scholarly journals Iterative Methods for Equilibrium Problems and Monotone Inclusion Problems in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Inclusion Problems ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
New Iterative Method

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).

scholarly journals Strong Convergence Theorems for Maximal Monotone Operators with Nonspreading Mappings in a Hilbert Space

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Hongjie Liu ◽  
Junqing Wang ◽  
Qiansheng Feng
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Monotone Mapping ◽  
Maximal Monotone ◽  
Iterative Algorithms ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Convergence Theorems ◽  
Nonspreading Mappings

We prove the strong convergence theorems for finding a common element of the set of fixed points of a nonspreading mappingTand the solution sets of zero of a maximal monotone mapping and anα-inverse strongly monotone mapping in a Hilbert space. Manaka and Takahashi (2011) proved weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space; there we introduced new iterative algorithms and got some strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space.

scholarly journals Hybrid Proximal-Point Methods for Zeros of Maximal Monotone Operators, Variational Inequalities and Mixed Equilibrium Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-31 ◽  
Author(s):  
Kriengsak Wattanawitoon ◽  
Poom Kumam
Keyword(s):  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Zero Point ◽  
Monotone Operators ◽  
Common Element ◽  
Proximal Point ◽  
Strong And Weak Convergence ◽  
Mixed Equilibrium Problems ◽  
Inverse Strongly Monotone

We prove strong and weak convergence theorems of modified hybrid proximal-point algorithms for finding a common element of the zero point of a maximal monotone operator, the set of solutions of equilibrium problems, and the set of solution of the variational inequality operators of an inverse strongly monotone in a Banach space under different conditions. Moreover, applications to complementarity problems are given. Our results modify and improve the recently announced ones by Li and Song (2008) and many authors.

scholarly journals Approximations for Equilibrium Problems and Nonexpansive Semigroups

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng
Keyword(s):  
Iterative Method ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Common Fixed Points ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Nonexpansive Semigroup ◽  
New Iterative Method

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of all common fixed points of a nonexpansive semigroup and prove the strong convergence theorem in Hilbert spaces. Our result extends the recent result of Zegeye and Shahzad (2013). In the last part of the paper, by the way, we point out that there is a slight flaw in the proof of the main result in Shehu's paper (2012) and perfect the proof.

scholarly journals Approximating Iterations for Nonexpansive and Maximal Monotone Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Zhangsong Yao ◽  
Sun Young Cho ◽  
Shin Min Kang ◽  
Li-Jun Zhu
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Minimum Norm ◽  
Nonexpansive Operator

We present two algorithms for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive operator in Hilbert spaces. We show that these two algorithms converge strongly to the minimum norm common element of the zero of the sum of two monotone operators and the fixed point of a nonexpansive operator.

scholarly journals Strong Convergence Theorems for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-13
Author(s):  
Jian-Wen Peng ◽  
Yan Wang
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Common Element ◽  
Convergence Theorems

We introduce an Ishikawa iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems which extend and generalize S. Takahashi and W. Takahashi's results (2007).

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