# Strong Convergence of Monotone Hybrid Method for Maximal Monotone Operators and Hemirelatively Nonexpansive Mappings

- Chakkrid Klin-eam
^{1}and - Suthep Suantai
^{1}Email author

**2009**:261932

https://doi.org/10.1155/2009/261932

© C. Klin-eam and S. Suantai. 2009

**Received: **20 May 2009

**Accepted: **21 September 2009

**Published: **11 October 2009

## Abstract

We prove strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemirelatively nonexpansive mapping in a Banach space by using monotone hybrid iteration method. By using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemirelatively nonexpansive mappings in a Banach space.

## Keywords

## 1. Introduction

Let be a real Banach space and let be the dual space of Let be a maximal monotone operator from to . It is well known that many problems in nonlinear analysis and optimization can be formulated as follows. Find a point satisfying

We denote by the set of all points such that Such a problem contains numerous problems in economics, optimization, and physics and is connected with a variational inequality problem. It is well known that the variational inequalities are equivalent to the fixed point problems. There are many authors who studied the problem of finding a common element of the fixed point of nonlinear mappings and the set of solutions of a variational inequality in the framework of Hilbert spaces see; for instance, [1–11] and the reference therein.

A well-known method to solve problem (1.1) is called the *proximal point algorithm*:
and

where and are the resovents of . Many researchers have studies this algorithm in a Hilbert space; see, for instance, [12–15] and in a Banach space; see, for instance, [16, 17].

In 2005, Matsushita and Takahashi [18] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space : chosen arbitrarily,

where is the duality mapping on , . They proved that generated by (1.3) converges strongly to a fixed point of under condition that .

In 2008, Su et al. [19] modified the CQ method (1.3) for approximation a fixed point of a closed hemi-relatively nonexpansive mapping in a Banach space. Their method is known as the monotone hybrid method defined as the following. chosen arbitrarily, then

where is the duality mapping on , They proved that generated by (1.4) converges strongly to a fixed point of under condition that .

Note that the hybrid method iteration method presented by Matsushita and Takahashi [18] can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping.

Very recently, Inoue et al. [20] proved the following strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method.

Theorem 1.1 (Inoue et al. [20]).

for all , where is the duality mapping on , and for some . If , then converges strongly to , where is the generalized projection from onto .

Employing the ideas of Inoue et al. [20] and Su et al. [19], we modify iterations (1.4) and (1.5) to obtain strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemi-relatively nonexpansive mappings in a Banach space. The results of this paper modify and improve the results of Inoue et al. [20], and some others.

## 2. Preliminaries

Throughout this paper, all linear spaces are real. Let
and
be the sets of all positive integers and real numbers, respectively. Let
be a Banach space and let
be the dual space of
. For a sequence
of
and a point
the *weak* convergence of
to
and the *strong* convergence of
to
are denoted by
and
, respectively.

Let be a Banach space. Then the duality mapping from into is defined by

Let
be the unit sphere centered at the origin of
. Then the space
is said to be *smooth* if the limit

exists for all
. It is also said to be *uniformly smooth* if the limit exists uniformly in
. A Banach space
is said to be *strictly convex* if
whenever
and
. It is said to be *uniformly convex* if for each
, there exists
such that
whenever
and
. We know the following (see, [21]):

(iv)if is strictly convex, then is strictly monotone;

(v)if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

Let be a smooth strictly convex and reflexive Banach space and let be a closed convex subset of Throughout this paper, define the function by

Observe that, in a Hilbert space , (2.3) reduces to , for all . It is obvious from the definition of the function that for all ,

Following Alber [22], the generalized projection from onto is a map that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem

Existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping In a Hilbert space, is the metric projection of onto .

Let
be a closed convex subset of a Banach space
and let
be a mapping from
into itself. We use
to denote the set of fixed points of
that is,
Recall that a self-mapping
is *hemi-relatively nonexpansive* if
and
for all
and
.

A point
is said to be an *asymptotic* fixed point of
if
contains a sequence
which converges weakly to
and
. We denote the set of all asymptotic fixed points of
by
. A hemi-relative nonexpansive mapping
is said to be *relatively nonexpansive* if
. The asymptotic behavior of a relatively nonexpansive mapping was studied in [23].

Recall that an operator
in a Banach space is call *closed*, if
and
, then
.

We need the following lemmas for the proof of our main results.

Lemma 2.1 (Kamimura and Takahashi [13]).

Let be a uniformly convex and smooth Banach space and let and be two sequences in such that either or is bounded. If , then .

Lemma 2.2 (Matsushita and Takahashi [18]).

Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space and let be a relatively hemi-nonexpansive mapping from into itself. Then is closed and convex.

Lemma 2.3 (Alber [22], Kamimura and Takahashi [13]).

Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space, and let . Then, if and only if for all .

Lemma 2.4 (Alber [22], Kamimura and Takahashi [13]).

Let
be a smooth, strictly convex, and reflexive Banach space, and let
be a set-valued mapping from
to
with graph
, domain
and range
We denote a set-valued operator
from
to
by
is said to be *monotone* of
A monotone operator
is said to be *maximal monotone* if its graph is not properly contained in the graph of any other monotone operator. It is known that a monotone mapping
is maximal if and only if for
for every
implies that
. We know that if
is a maximal monotone operator, then
is closed and convex; see [19] for more details. The following result is well known.

Lemma 2.5 (Rockafellar [24]).

Let be a smooth, strictly convex, and reflexive Banach space and let be a monotone operator. Then is maximal if and only if for all

Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed convex subset of and let be a monotone operator satisfying

Then we can define the resolvent by

We know that consists of one point. For , the Yosida approximation is defined by for all .

Lemma 2.6 (Kohsaka and Takahashi [25]).

Let and let and be the resolvent and the Yosida approximation of , respectively. Then, the following hold:

Lemma 2.7 (Kamimura and Takahashi [13]).

## 3. Main Results

In this section, we prove a strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space by using the monotone hybrid iteration method.

Theorem 3.1.

for all , where is the duality mapping on and for some . If , then converges strongly to , where is the generalized projection from onto

Proof.

We first show that and are closed and convex for each . From the definition of and it is obvious that is closed and is closed and convex for each . Next, we prove that is convex.

From (3.13) and we obtain that

By the definition of , it follows that and , whence . Therefore, it follows from the uniqueness of the that .

As direct consequences of Theorem 3.1, we can obtain the following corollaries.

Corollary 3.2.

for all where is the duality mapping on , and for some . Then converges strongly to , where is the generalized projection from onto .

Proof.

Putting , and in Theorem 3.1, we obtain Corollary 3.2.

Let be a Banach space and let be a proper lower semicontinuous convex function. Define the subdifferential of as follows:

for each . Then, we know that is a maximal monotone operator; see [21] for more details.

Corollary 3.3 (Su et al. [19, Theorem 3.1]).

for all , where is the duality mapping on and . If then converges strongly to , where is the generalized projection from onto .

Proof.

So, we obtain the desired result by using Theorem 3.1.

Since every relatively nonexpansive mapping is a hemi-relatively one, the following theorem is obtained directly from Theorem 3.1.

Theorem 3.4.

for all , where is the duality mapping on and for some . If then converges strongly to , where is the generalized projection from onto .

Corollary 3.5 (Su et al. [19, Theorem 3.2]).

for all , where is the duality mapping on and . If , then converges strongly to , where is the generalized projection from onto .

Proof.

Set in Theorem 3.4, where is the indicator function. So, from Theorem 3.4, we obtain the desired result.

## Declarations

### Acknowledgments

The authors would like to thank the referee for valuable suggestions that improve this manuscript and the Thailand Research Fund (RGJ Project) and Commission on Higher Education for their financial support during the preparation of this paper. The first author was supported by the Royal Golden Jubilee Grant PHD/0018/2550 and by the Graduate School, Chiang Mai University, Thailand.

## Authors’ Affiliations

## References

- Ceng L-C, Lee C, Yao J-C:
**Strong weak convergence theorems of implicit hybrid steepest-descent methods for variational inequalities.***Taiwanese Journal of Mathematics*2008,**12**(1):227–244.MathSciNetMATHGoogle Scholar - Ceng L-C, Ansari QH, Yao J-C:
**Viscosity approximation methods for generalized equilibrium problems and fixed point problems.***Journal of Global Optimization*2009,**43**(4):487–502. 10.1007/s10898-008-9342-6MathSciNetView ArticleMATHGoogle Scholar - Peng J-W, Yao J-C:
**A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems.***Taiwanese Journal of Mathematics*2008,**12**(6):1401–1432.MathSciNetMATHGoogle Scholar - Peng J-W, Yao J-C:
**Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems.***Mathematical and Computer Modelling*2009,**49**(9–10):1816–1828. 10.1016/j.mcm.2008.11.014MathSciNetView ArticleMATHGoogle Scholar - Peng J-W, Yao JC: Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings. to appear in Taiwanese Journal of MathematicsGoogle Scholar
- Peng J-W, Yao J-C: Some new extragradient-like methods for generalized equilibrium problems, fixed point problems and variational inequality problems. to appear in Optimization Methods and SoftwareGoogle Scholar
- Peng J-W, Yao J-C: Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping. Journal of Global Optimization. In pressGoogle Scholar
- Schaible S, Yao J-C, Zeng L-C:
**A proximal method for pseudomonotone type variational-like inequalities.***Taiwanese Journal of Mathematics*2006,**10**(2):497–513.MathSciNetMATHGoogle Scholar - Zeng L-C, Yao J-C:
**Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems.***Taiwanese Journal of Mathematics*2006,**10**(5):1293–1303.MathSciNetMATHGoogle Scholar - Zeng LC, Lin LJ, Yao JC:
**Auxiliary problem method for mixed variational-like inequalities.***Taiwanese Journal of Mathematics*2006,**10**(2):515–529.MathSciNetMATHGoogle Scholar - Zeng L-C, Wu S-Y, Yao J-C:
**Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems.***Taiwanese Journal of Mathematics*2006,**10**(6):1497–1514.MathSciNetMATHGoogle Scholar - Kamimura S, Takahashi W:
**Approximating solutions of maximal monotone operators in Hilbert spaces.***Journal of Approximation Theory*2000,**106**(2):226–240. 10.1006/jath.2000.3493MathSciNetView ArticleMATHGoogle Scholar - Kamimura S, Takahashi W:
**Strong convergence of a proximal-type algorithm in a Banach space.***SIAM Journal on Optimization*2002,**13**(3):938–945. 10.1137/S105262340139611XMathSciNetView ArticleMATHGoogle Scholar - Rockafellar RT:
**Monotone operators and the proximal point algorithm.***SIAM Journal on Control and Optimization*1976,**14**(5):877–898. 10.1137/0314056MathSciNetView ArticleMATHGoogle Scholar - Solodov MV, Svaiter BF:
**Forcing strong convergence of proximal point iterations in a Hilbert space.***Mathematical Programming*2000,**87**(1):189–202.MathSciNetMATHGoogle Scholar - Kamimura S, Kohsaka F, Takahashi W:
**Weak and strong convergence theorems for maximal monotone operators in a Banach space.***Set-Valued Analysis*2004,**12**(4):417–429. 10.1007/s11228-004-8196-4MathSciNetView ArticleMATHGoogle Scholar - Kohsaka F, Takahashi W:
**Strong convergence of an iterative sequence for maximal monotone operators in a Banach space.***Abstract and Applied Analysis*2004,**2004**(3):239–249. 10.1155/S1085337504309036MathSciNetView ArticleMATHGoogle Scholar - Matsushita S, Takahashi W:
**A strong convergence theorem for relatively nonexpansive mappings in a Banach space.***Journal of Approximation Theory*2005,**134**(2):257–266. 10.1016/j.jat.2005.02.007MathSciNetView ArticleMATHGoogle Scholar - Su Y, Wang D, Shang M:
**Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings.***Fixed Point Theory and Applications*2008, Article ID 284613,**2008:**-8.Google Scholar - Inoue G, Takahashi W, Zembayashi K: Strong convergence theorems by hybrid methods for maximal monotone operator and relatively nonexpansive mappings in Banach spaces. to appear in Journal of Convex AnalysisGoogle Scholar
- Takahashi W:
*Nonlinear Functional Analysis, Fixed Point Theory and Its Application*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Alber YI:
**Metric and generalized projection operators in Banach spaces: properties and applications.**In*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.*.*Volume 178*. Edited by: Katrosatos AG. Marcel Dekker, New York, NY, USA; 1996:15–50.Google Scholar - Butnariu D, Reich S, Zaslavski AJ:
**Asymptotic behavior of relatively nonexpansive operators in Banach spaces.***Journal of Applied Analysis*2001,**7**(2):151–174. 10.1515/JAA.2001.151MathSciNetView ArticleMATHGoogle Scholar - Rockafellar RT:
**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149:**75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView ArticleMATHGoogle Scholar - Kohsaka F, Takahashi W:
**Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces.***SIAM Journal on Optimization*2008,**19**(2):824–835. 10.1137/070688717MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.