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

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.

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.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jinfang Tang
Keyword(s):  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Approximation Methods ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Convergence Problem ◽  
Iterative Schemes ◽  
The Family ◽  
Viscosity Approximation Methods ◽  
Implicit And Explicit

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.

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

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>

scholarly journals Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

2016 ◽  
Vol 25 (1) ◽  
pp. 107-120
Author(s):  
T. M. M. SOW ◽  
◽  
C. DIOP ◽  
N. DJITTE ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Process ◽  
Real Banach Space ◽  
Monotone Mappings ◽  
Uniformly Convex ◽  
Hammerstein Equation ◽  
Uniformly Smooth ◽  
Monotone Maps

For q > 1 and p > 1, let E be a 2-uniformly convex and q-uniformly smooth or p- uniformly convex and 2-uniformly smooth real Banach space and F : E → E∗, K : E∗ → E be bounded and strongly monotone maps with D(K) = R(F) = E∗. We construct a coupled iterative process and prove its strong convergence to a solution of the Hammerstein equation u + KF u = 0. Futhermore, our technique of proof is of independent of interest.

scholarly journals Viscosity Approximation Methods for a General Variational Inequality System and Fixed Point Problems in Banach Spaces

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 36 ◽  
Author(s):  
Yuanheng Wang ◽  
Chanjuan Pan
Keyword(s):  
Variational Inequality ◽  
Banach Spaces ◽  
Approximation Scheme ◽  
Common Element ◽  
Approximation Methods ◽  
Viscosity Approximation ◽  
Expansive Mapping ◽  
Inequality System ◽  
General Variational Inequality ◽  
Viscosity Approximation Methods

In Banach spaces, we study the problem of solving a more general variational inequality system for an asymptotically non-expansive mapping. We give a new viscosity approximation scheme to find a common element. Some strong convergence theorems of the proposed iterative method are obtained. A numerical experiment is given to show the implementation and efficiency of our main theorem. Our results presented in this paper generalize and complement many recent ones.

A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems

2018 ◽  
Vol 62 (1) ◽  
pp. 241-257 ◽  
Author(s):  
C. E. Chidume ◽  
M. O. Uba ◽  
M. I. Uzochukwu ◽  
E. E. Otubo ◽  
K. O. Idu
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Real Banach Space ◽  
Maximal Monotone ◽  
Variational Inequality Problems ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Monotone Maps

AbstractLetEbe a uniformly convex and uniformly smooth real Banach space, and letE* be its dual. LetA : E→ 2E*be a bounded maximal monotone map. Assume thatA−1(0) ≠ Ø. A new iterative sequence is constructed which convergesstronglyto an element ofA−1(0). The theorem proved complements results obtained on strong convergence ofthe proximal point algorithmfor approximating an element ofA−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.

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