A shrinking projection method for nonexpansive mappings with nonsummable errors in a Hadamard space

2014 ◽  
Vol 243 (1-2) ◽  
pp. 89-94 ◽  
Author(s):  
Yasunori Kimura
Keyword(s):  
Projection Method ◽  
Nonexpansive Mappings ◽  
Hadamard Space ◽  
Shrinking Projection Method

scholarly journals Convergence of a Sequence of Sets in a Hadamard Space and the Shrinking Projection Method for a Real Hilbert Ball

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Yasunori Kimura
Keyword(s):  
Projection Method ◽  
Convergence Theorem ◽  
Iterative Scheme ◽  
Equivalent Condition ◽  
Set Convergence ◽  
Hadamard Space ◽  
Shrinking Projection Method ◽  
Metric Projections ◽  
Hilbert Ball

We propose a new concept of set convergence in a Hadamard space and obtain its equivalent condition by using the notion of metric projections. Applying this result, we also prove a convergence theorem for an iterative scheme by the shrinking projection method in a real Hilbert ball.

scholarly journals Strong and Weak Convergence Criteria of Composite Iterative Algorithms for Systems of Generalized Equilibria

2014 ◽  
Vol 2014 ◽  
pp. 1-25
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen ◽  
Zhao-Rong Kong
Keyword(s):  
Weak Convergence ◽  
Iterative Algorithm ◽  
Projection Method ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Generalized Mixed Equilibrium Problem ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Convergence Criteria ◽  
Shrinking Projection Method

We first introduce and analyze one iterative algorithm by using the composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove a strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.

scholarly journals Viscosity Approximations by the Shrinking Projection Method of Quasi-Nonexpansive Mappings for Generalized Equilibrium Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-30 ◽  
Author(s):  
Rabian Wangkeeree ◽  
Nimit Nimana
Keyword(s):  
Fixed Points ◽  
Equilibrium Problem ◽  
Projection Method ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Generalized Equilibrium Problem ◽  
Shrinking Projection Method ◽  
Generalized Equilibrium

We introduce viscosity approximations by using the shrinking projection method established by Takahashi, Takeuchi, and Kubota, for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.

scholarly journals A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Mei Yuan ◽  
Xi Li ◽  
Xue-song Li ◽  
John J. Liu
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Projection Method ◽  
Convergence Theorem ◽  
Projection Operator ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive

Relatively nonexpansive mappings and equilibrium problems are considered based on a shrinking projection method. Using properties of the generalizedf-projection operator, a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems is proved in Banach spaces under some suitable conditions.

Hybrid projection method for a system of unrelated generalized mixed variational-like inequality problems

2019 ◽  
Vol 26 (1) ◽  
pp. 63-78
Author(s):  
Kaleem Raza Kazmi ◽  
Rehan Ali
Keyword(s):  
Banach Space ◽  
Projection Method ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Solution Set ◽  
Common Element ◽  
Monotone Mappings ◽  
Point Problem ◽  
Existence Of A Solution ◽  
Hybrid Projection Method

Abstract The aim of this paper is to consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems. Further, we prove some properties of a solution set of generalized mixed variational-like inequality problems. Furthermore, we use a hybrid projection method to find a common element of a solution set of a system of unrelated generalized mixed variational-like inequality problems for generalized relaxed α-monotone mappings and the set of fixed points of a common fixed point problem for a family of generalized asymptotically quasi-ϕ-nonexpansive mappings in a reflexive, uniformly smooth and strictly convex Banach space.

scholarly journals An Iterative Shrinking Projection Method for Solving Fixed Point Problems of Closed andϕ-Quasi-Strict Pseudocontractions along with Generalized Mixed Equilibrium Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Kasamsuk Ungchittrakool
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Projection Method ◽  
Equilibrium Problems ◽  
Fixed Point Problems ◽  
Generalized Mixed Equilibrium Problems ◽  
Shrinking Projection Method ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Moreover, we also find the significant inequality related to the mappings mentioned in the paper and the mappings defined from generalized mixed equilibrium problems on Banach spaces. We propose an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed andϕ-quasi-strict pseudo-contractions. Our results hold in reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the corresponding results of Zhou and Gao (2010) and many others.

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