scholarly journals Some Results on an Infinite Family of Nonexpansive Mappings and an Inverse-Strongly Monotone Mapping in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Peng Cheng ◽  
Anshen Zhang
Keyword(s):  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Infinite Family ◽  
Common Element ◽  
Fixed Point Set ◽  
Strongly Monotone ◽  
Inverse Strongly Monotone Mapping ◽  
Inverse Strongly Monotone ◽  
Point Set

We study the problem of approximating a common element in the common fixed point set of an infinite family of nonexpansive mappings and in the solution set of a variational inequality involving an inverse-strongly monotone mapping based on a viscosity approximation iterative method. Strong convergence theorems of common elements are established in the framework of Hilbert spaces.

Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Poom Kumam ◽  
Somyot Plubtieng
Keyword(s):  
Variational Inequality ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Common Fixed Points ◽  
Countable Family ◽  
Common Element ◽  
Monotone Mappings ◽  
Inverse Strongly Monotone Mapping ◽  
Inverse Strongly Monotone ◽  
Viscosity Approximation Methods

AbstractWe use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let C be a nonempty closed convex subset of a Hilbert space H and P C is a metric projection. We consider the iteration process {x n} of C defined by x 1 = x ∈ C is arbitrary and $$ x_{n + 1} = \alpha _n f(x_n ) + (1 - \alpha _n )S_n P_C (x_n + \lambda _n Ax_n ) $$ where f is a contraction on C, {S n} is a sequence of nonexpansive self-mappings of a closed convex subset C of H, and A is an inverse-strongly-monotone mapping of C into H. It is shown that {x n} converges strongly to a common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping which solves some variational inequality. Finally, the ideas of our results are applied to find a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem, a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. The results of this paper extend and improve the results of Chen, Zhang and Fan.

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.

scholarly journals A General Composite Algorithms for Solving General Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-25
Author(s):  
Rattanaporn Wangkeeree ◽  
Uthai Kamraksa ◽  
Rabian Wangkeeree
Keyword(s):  
Fixed Point ◽  
General Equilibrium ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Fixed Point Set ◽  
Strictly Pseudocontractive Mapping ◽  
Asymptotically Nonexpansive ◽  
Point Set ◽  
The Common

We introduce a general composite algorithm for finding a common element of the set of solutions of a general equilibrium problem and the common fixed point set of a finite family of asymptotically nonexpansive mappings in the framework of Hilbert spaces. Strong convergence of such iterative scheme is obtained which solving some variational inequalities for a strongly monotone and strictly pseudocontractive mapping. Our results extend the corresponding recent results of Yao and Liou (2010).

Iterative algorithm for a split equilibrium problem and fixed problem for finite asymptotically nonexpansive mappings in Hlbert space

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1423-1434 ◽  
Author(s):  
Sheng Wang ◽  
Min Chen
Keyword(s):  
Equilibrium Problem ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Fixed Point Set ◽  
Split Equilibrium Problem ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Point Set ◽  
The Common

In this paper, we propose an iterative algorithm for finding the common element of solution set of a split equilibrium problem and common fixed point set of a finite family of asymptotically nonexpansive mappings in Hilbert space. The strong convergence of this algorithm is proved.

scholarly journals Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Tanom Chamnarnpan ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
Common Element ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

scholarly journals S-iteration process of Halpern-type for common solutions of nonexpansive mappings and monotone variational inequalities

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1727-1746 ◽  
Author(s):  
D.R. Sahu ◽  
Ajeet Kumar ◽  
Ching-Feng Wen
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Common Element ◽  
Monotone Mappings ◽  
Monotone Variational Inequalities ◽  
Numerical Examples ◽  
Inverse Strongly Monotone ◽  
Strongly Monotone Mappings

This paper is devoted to the strong convergence of the S-iteration process of Halpern-type for approximating a common element of the set of fixed points of a nonexpansive mapping and the set of common solutions of variational inequality problems formed by two inverse strongly monotone mappings in the framework of Hilbert spaces. We also give some numerical examples in support of our main result.

scholarly journals The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Rabian Wangkeeree ◽  
Panatda Boonman
Keyword(s):  
Hilbert Space ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Solution Set ◽  
Common Element ◽  
Fixed Point Set ◽  
Mixed Equilibrium Problems ◽  
Point Set ◽  
Mixed Equilibrium

We first introduce the iterative procedure to approximate a common element of the fixed-point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptoticTJmapping and the solution set of SMEP in a real Hilbert space.

scholarly journals Strong Convergence of Generalized Projection Algorithms for Nonlinear Operators

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Chakkrid Klin-eam ◽  
Suthep Suantai ◽  
Wataru Takahashi
Keyword(s):  
Strong Convergence ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Zero Point ◽  
Common Element ◽  
Fixed Point Set ◽  
Nonlinear Operators ◽  
Point Set

We establish strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method. Moreover we apply our main results to obtain strong convergence for a maximal monotone operator and two nonexpansive mappings in a Hilbert space.

scholarly journals Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family ofki-Strictly Pseudocontractive Mapping in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Haitao Che ◽  
Meixia Li ◽  
Xintian Pan
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Element ◽  
Strictly Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Definition Of ◽  
Strictly Pseudocontractive Mappings

We first extend the definition of Wnfrom an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family ofki-strictly pseudocontractive mappings in Hilbert spaces. The results obtained in this paper extend and improve the recent ones announced by many others. Furthermore, a numerical example is presented to illustrate the effectiveness of the proposed scheme.

A CONVERGENCE THEOREM FOR COMMON ELEMENTS OF EQUILIBRIUM PROBLEMS AND MAPPINGS SATISFYING CONDITION (Φ-Eµ) IN UNIFORMLY CONVEX AND UNIFORMLY SMOOTH BANACH SPACES

2018 ◽  
Vol 1 (30) ◽  
pp. 67-77
Author(s):  
Hieu Trung Nguyen ◽  
Tien Cam Truong
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Satisfying Condition ◽  
Common Element ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Uniformly Smooth ◽  
Point Set ◽  
Uniformly Smooth Banach Spaces

In this paper, we propose a new hybrid iteration for finding a common element of solution set of equilibrium problems and the fixed point set of mappings  satisfying condi- tion (Φ-Eµ), and establish the convergence of this iteration in uniformly convex and uniformly smooth Banach spaces. From this theorem, we geta corollary for the convergence for equilibrium problems and mappings satisfying condition (Eµ) in real Hilbert spaces. In addition, an example is provided to illustrate for the convergence of equilibrium problems and mappings satisfying condition (Φ-Eµ). These results are the generations and improvements of some  existing results in the literature

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