Weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space

Optimization ◽  
2017 ◽  
Vol 66 (10) ◽  
pp. 1689-1698 ◽  
Author(s):  
Ming Tian ◽  
Bing-Nan Jiang
Keyword(s):  
Hilbert Space ◽  
Weak Convergence ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Convergence Theorem ◽  
Monotone Mapping ◽  
Inverse Strongly Monotone Mapping ◽  
Inverse Strongly Monotone ◽  
Weak Convergence Theorem ◽  
Zero Points

scholarly journals Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Hiroko Manaka
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Duality Mapping ◽  
Nonlinear Mapping ◽  
Normalized Duality Mapping ◽  
Strongly Nonexpansive Mapping ◽  
Weak Convergence Theorem

LetEbe a smooth Banach space with a norm·. LetV(x,y)=x2+y2-2 x,Jyfor anyx,y∈E, where·,·stands for the duality pair andJis the normalized duality mapping. We define aV-strongly nonexpansive mapping byV(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists aV-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.

scholarly journals An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 99 ◽  
Author(s):  
Nopparat Wairojjana ◽  
Habib ur Rehman ◽  
Ioannis K. Argyros ◽  
Nuttapol Pakkaranang
Keyword(s):  
Hilbert Space ◽  
Weak Convergence ◽  
Numerical Solution ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Suggested Method ◽  
Experimental Results ◽  
Weak Convergence Theorem

Several methods have been put forward to solve equilibrium problems, in which the two-step extragradient method is very useful and significant. In this article, we propose a new extragradient-like method to evaluate the numerical solution of the pseudomonotone equilibrium in real Hilbert space. This method uses a non-monotonically stepsize technique based on local bifunction values and Lipschitz-type constants. Furthermore, we establish the weak convergence theorem for the suggested method and provide the applications of our results. Finally, several experimental results are reported to see the performance of the proposed method.

scholarly journals An Extension of Subgradient Method for Variational Inequality Problems in Hilbert Space

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xueyong Wang ◽  
Shengjie Li ◽  
Xipeng Kou
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Weak Convergence ◽  
Nonexpansive Mapping ◽  
Iterative Process ◽  
Subgradient Method ◽  
Variational Inequality Problems ◽  
Mild Conditions ◽  
Weak Convergence Theorem

An extension of subgradient method for solving variational inequality problems is presented. A new iterative process, which relates to the fixed point of a nonexpansive mapping and the current iterative point, is generated. A weak convergence theorem is obtained for three sequences generated by the iterative process under some mild conditions.

scholarly journals Extension and Application of the Yamada Iteration Algorithm in Hilbert Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 215
Author(s):  
Ming Tian ◽  
Meng-Ying Tong
Keyword(s):  
Weak Convergence ◽  
Monotone Mapping ◽  
Split Feasibility Problem ◽  
Common Element ◽  
Feasibility Problem ◽  
Iteration Algorithm ◽  
Strongly Monotone ◽  
Inverse Strongly Monotone Mapping ◽  
Inverse Strongly Monotone ◽  
The Split Feasibility Problem

In this paper, based on the Yamada iteration, we propose an iteration algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping. We obtain a weak convergence theorem in Hilbert space. In particular, the set of zero points of an inverse strongly-monotone mapping can be transformed into the solution set of the variational inequality problem. Further, based on this result, we also obtain some new weak convergence theorems which are used to solve the equilibrium problem and the split feasibility problem.

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