Viscosity Approximation Methods for a Family of Nonexpansive Mappings in CAT(0) Spaces

The purpose of this paper is using the viscosity approximation method to study the strong convergence problem for a family of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of the family of nonexpansive mappings are proved which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.

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Strong convergence for nonexpansive mappings by viscosity approximation methods in Hadamard manifolds

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v4i2.4239 ◽

2015 ◽

Vol 4 (2) ◽

pp. 299

Author(s):

Mandeep Kumari ◽

Renu Chugh

Keyword(s):

Nonexpansive Mapping ◽

Strong Convergence ◽

Approximation Method ◽

Nonexpansive Mappings ◽

Approximation Methods ◽

Hadamard Manifolds ◽

Viscosity Approximation Method ◽

Viscosity Approximation ◽

Viscosity Approximation Methods

<p>In 2010, Victoria Martin Marquez studied a nonexpansive mapping in Hadamard manifolds using Viscosity approximation method. Our goal in this paper is to study the strong convergence of the Viscosity approximation method in Hadamard manifolds. Our results improve and extend the recent research in the framework of Hadamard manifolds.</p>

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

Journal of Applied Mathematics ◽

10.1155/2012/421050 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11 ◽

Cited By ~ 3

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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Viscosity Approximation Methods for Nonexpansive Multi-Valued Nonself Mappings and Equilibrium Problems

Demonstratio Mathematica ◽

10.2478/dema-2014-0030 ◽

2014 ◽

Vol 47 (2) ◽

Author(s):

P. Cholamjiak ◽

W. Cholamjiak ◽

S. Suantai

Keyword(s):

Hilbert Space ◽

Strong Convergence ◽

Approximation Method ◽

Equilibrium Problems ◽

Approximation Methods ◽

Viscosity Approximation Method ◽

Viscosity Approximation ◽

Convergence Theorems ◽

Viscosity Approximation Methods

AbstractIn this paper, strong convergence theorems by the viscosity approximation method for nonexpansive multi-valued nonself mappings and equilibrium problems are established under some suitable conditions in a Hilbert space. The obtained results extend and improve the corresponding results existed in the literature.

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Relaxed Viscosity Approximation Methods with Regularization for Constrained Minimization Problems

Journal of Applied Mathematics ◽

10.1155/2013/531859 ◽

2013 ◽

Vol 2013 ◽

pp. 1-19 ◽

Cited By ~ 1

Author(s):

Lu-Chuan Ceng ◽

Hong-Kun Xu ◽

Ching-Feng Wen

Keyword(s):

Fixed Point ◽

Nonexpansive Mappings ◽

Approximation Methods ◽

Viscosity Approximation Method ◽

Convex Minimization Problem ◽

Viscosity Approximation ◽

Minimization Problems ◽

Strict Pseudocontraction ◽

Constrained Minimization Problems ◽

Viscosity Approximation Methods

We introduce a new relaxed viscosity approximation method with regularization and prove the strong convergence of the method to a common fixed point of finitely many nonexpansive mappings and a strict pseudocontraction that also solves a convex minimization problem and a suitable equilibrium problem.

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Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

Abstract and Applied Analysis ◽

10.1155/2013/328740 ◽

2013 ◽

Vol 2013 ◽

pp. 1-18

Author(s):

Lu-Chuan Ceng ◽

Abdul Latif ◽

Abdullah E. Al-Mazrooei

Keyword(s):

Banach Space ◽

Variational Inequality ◽

Strong Convergence ◽

Banach Spaces ◽

Nonexpansive Mappings ◽

Approximation Methods ◽

Uniformly Convex ◽

Viscosity Approximation ◽

Uniformly Smooth ◽

Viscosity Approximation Methods

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

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