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Weak convergence theorems for an infinite family of nonexpansive mappings and equilibrium problems

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JP Journal of Fixed Point Theory and Applications
Volume 7, Number 2, 2012, Pages 113-127
Available online at 
Published by Pushpa Publishing House, Allahabad, INDIA

WEAK CONVERGENCE THEOREMS FOR AN INFINITE
FAMILY OF NONEXPANSIVE MAPPINGS AND
EQUILIBRIUM PROBLEMS
P. N. Anh*†, L. B. Long†, N. V. Quy and L. Q. Thuy
†Department

of Scientific Fundamentals
Posts and Telecommunications Institute of Technology
Hanoi, Vietnam
e-mail: anhpn@ptit.edu.vn
Department of Scientific Fundamentals
Academy of Finance
Hanoi, Vietnam
Faculty of Applied Mathematics and Informatics
Hanoi University of Technology
Vietnam
Abstract

The purpose of this paper is to investigate a new iteration scheme for
finding a common element of the set of fixed points of an infinite
family of nonexpansive mappings and the solution set of a
pseudomonotone and Lipschitz-type continuous equilibrium problem.
The scheme is based on the extragradient-type methods and fixed
point methods. We show that the iterative sequences generated by this
algorithm converge weakly to the common element in a real Hilbert
space.
© 2012 Pushpa Publishing House
2010 Mathematics Subject Classification: 65K10, 65K15, 90C25, 90C33.
Keywords and phrases: nonexpansive mapping, pseudomonotone, Lipschitz-type continuous,
equilibrium problems, fixed point.
*Corresponding author
Received December 5, 2011

P. N. Anh, L. B. Long, N. V. Quy and L. Q. Thuy

114

1. Introduction
Let H be a real Hilbert space with inner product ⋅, ⋅ and norm ⋅ . Let
C be a closed convex subset of a real Hilbert space H and PrC be the
projection of H onto C. When {x n } is a sequence in H, then x n → x

(resp. x n

x ) will denote strong (resp. weak) convergence of the sequence

{x n } to x as n → ∞. A mapping S : C → C is said to be nonexpansive if
S (x) − S ( y) ≤ x − y ,

∀x, y ∈ C.

Fix( S ) is denoted by the set of fixed points of S. Let f : C × C → R be a
bifunction such that f ( x, x ) = 0 for all x ∈ C. We consider the equilibrium
problems in the sense of Blum and Oettli [8] which are presented as follows:
Find x∗ ∈ C such that f ( x∗ , y ) ≥ 0 for all y ∈ C.

EP( f , C )

The set of solutions of EP( f , C ) is denoted by Sol ( f , C ). The bifunction f
is called strongly monotone on C with β > 0 if

f ( x, y ) + f ( y, x ) ≤ −β x − y 2 ,

∀x, y ∈ C ;

monotone on C if

f ( x, y ) + f ( y, x ) ≤ 0,

∀x, y ∈ C ;

pseudomonotone on C if
f ( x, y ) ≥ 0 ⇒ f ( y, x ) ≤ 0,

∀x, y ∈ C ;

Lipschitz-type continuo...
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