Iterative algorithms based on hybrid method and Cesàro mean of asymptotically nonexpansive mappings for equilibrium problems
Fixed Point Theory and Applications
Iterative algorithms based on hybrid method and Cesro mean of asymptotically nonexpansive mappings for equilibrium problems
Jingxin Zhang 0
Yunan Cui 1
0 Mathematics and Applied Mathematics, Harbin University of Commerce , Harbin, 150028 , P.R. China
1 Department of Mathematics, Harbin University of Science and Technology , Harbin, 150080 , P.R. China
Using Cesro means of a mapping, we modify the progress of Mann's iteration in hybrid method for asymptotically nonexpansive mappings in Hilbert spaces. Under suitable conditions, we prove that the iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. We also introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces. MSC: 47H10; 47J25; 90C33
strong convergence theorem; asymptotically nonexpansive mapping; hybrid method; equilibrium problem; Cesro means
1 Introduction
Let H be a real Hilbert space with the inner product , and the norm
a nonempty closed convex subset of H. A mapping T : C C is said to be
asymptotilimn kn = such that
T nx T ny kn x y ,
x, y C;
when kn , T is called nonexpansive.
The concept of asymptotically nonexpansive mapping was introduced by Goebel and
known that if T : H H is asymptotically nonexpansive, then F(T ) is nonempty convex.
In , Mann [] introduced the iteration as follows: a sequence {xn} defined by
In an infinite-dimensional Hilbert space, Mann iteration could conclude only weak
convergence []. Attempts to modify the Mann iteration method (.) so that strong
convergence is guaranteed have recently been made. Nakajo and Takahashi [] proposed the
following modification of Mann iteration method for a nonexpansive mapping T in a Hilbert
n = ( n) kn (diam C) as n .
where PK denotes the metric projection from H onto a closed convex subset K of H. The
above method is also called CQ method or hybrid method.
In , Kim and Xu [] adapted the iteration (.) in a Hilbert space. More precisely,
they introduced the following iteration process for asymptotically nonexpansive
mappings:
They proved that {xn} converges in norm to PF(T)x under some conditions. Several
authors (see [, ]) have studied the convergence of hybrid method.
Baillon [] first proved that the following Cesro mean iterative sequence weakly
converges to a fixed point of a nonexpansive mapping in Hilbert spaces:
Tnx =
n + i=
Shimizu and Takahashi [] proved a strong convergence theorem of the above iteration
for an asymptotically nonexpansive mapping in Hilbert spaces.
Let C be a nonempty closed convex subset of a real Hilbert space H, let f : C C R
be a functional, where R is the set of real numbers. The equilibrium problem is to find
x C such that
f (x, y) ,
y C.
The set of solutions of (.) is denoted by EP(f ). Given a mapping T : C X*, let f (x, y) =
Tx, y x , x, y C, then z EP(f ) if and only if Tz, y z , y C, i.e., z is the solution
of the variational inequality.
There are several other problems, for example, the complementarity problem, fixed
point problem and optimization problem, which can also be written in the form of an
equilibrium problem. So, equilibrium problems provide us with a systematic framework
to study a wide class of problems arising in financial economics, optimization and
operation research etc., which motivates the extensive concern. See, for example, []. In
recent years, equilibrium problems have been deeply and thoroughly researched. See, for
example, []. Some methods have been proposed to solve the equilibrium problem
in a Hilbert space; see, for instance, []. In , Jitpeera, Katchang, and Kumam []
found a common element of the set of solutions for mixed equilibrium problem, the set
of solutions of the variational inequality for a -inverse strongly monotone mapping, and
the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space
by using the viscosity and Cesro mean approximation method.
Motivated by the above-mentioned results, in this paper we introduce the following
iteration process for asymptotically nonexpansive mappings T with C a closed convex
bounded subset of a real Hilbert space:
n = ( n) Ln (diam C) as n .
We shall prove that the above iterative sequence {xn} converges strongly to a fixed point
of T under some proper conditions. In addition, we also introduce a new hybrid iterative
scheme for finding a common element of the set of common fixed points of
asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert
spaces.
We will use the notation for weak convergence and for strong convergence.
2 Preliminaries
x y = x y x y, y
x + ( )y = x + ( ) y ( ) x y
for all x, y H and [, ]. It is also known that H satisfies
() Opials condition, that is, for any sequence {xn} with xn (...truncated)