# Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings

- Lu-Chuan Zeng
^{1, 2}, - QH Ansari
^{3}and - S Al-Homidan
^{3}Email author

**2011**:973028

https://doi.org/10.1155/2011/973028

© Lu-Chuan Zeng et al. 2011

**Received: **24 September 2010

**Accepted: **18 October 2010

**Published: **2 November 2010

## Abstract

The purpose of this paper is to introduce and consider new hybrid proximal-type algorithms for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a relatively nonexpansive mapping , and the set of zeros of a maximal monotone operator in a uniformly smooth and uniformly convex Banach space. Strong convergence theorems for these hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequences generated by these various algorithms converge strongly to the same point in . These new results represent the improvement, generalization, and development of the previously known ones in the literature.

## Keywords

## 1. Introduction

*α*-inverse-strongly monotone, if there exists an such that

It is easy to see that if
is an *α*-inverse-strongly monotone mapping, then it is
-Lipschitzian.

This algorithm was first introduced by Martinet [12] and generally studied by Rockafellar [15] in the framework of a Hilbert space. Later many authors studied its convergence in a Hilbert space or a Banach space. See, for instance, [16–21] and the references therein.

Let
be a reflexive, strictly convex, and smooth Banach space with the dual
and
be a nonempty closed convex subset of
. Let
be a maximal monotone operator with domain
and
be a relatively nonexpansive mapping. Let
be an *α*-inverse-strongly monotone mapping and
be a bifunction satisfying (A1)–(A4): (A1)
,
; (A2)
is monotone, that is,
,
; (A3)
,
; (A4) the function
is convex and lower semicontinuous. The purpose of this paper is to introduce and investigate two new hybrid proximal-type Algorithms 1.1 and 1.2 for finding an element of
.

Algorithm 1.1.

where is a sequence in and , are sequences in .

Algorithm 1.2.

where is a sequence in and is a sequence in .

In this paper, strong convergence results on these two hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequence generated by Algorithm 1.1 and the sequence generated by Algorithm 1.2, converge strongly to the same point . These new results represent the improvement, generalization and development of the previously known ones in the literature including Solodov and Svaiter [22], Kamimura and Takahashi [23], Qin and Su [24], and Ceng et al. [25].

Throughout this paper the symbol *⇀* stands for weak convergence and → stands for strong convergence.

## 2. Preliminaries

exists for each . If is smooth, then is single valued. We still denote the single valued duality mapping by .

It is also said to be uniformly smooth if the limit is attained uniformly for . Recall also that if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of . A Banach space is said to have the Kadec-Klee property if for any sequence , whenever and , we have . It is known that if is uniformly convex, then has the Kadec-Klee property; see [26, 27] for more details.

Let be a nonempty closed convex subset of a real Hilbert space and be the metric projection of onto C, then is nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. Nevertheless, Alber [28] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

It is clear that in a Hilbert space , (2.3) reduces to , for all .

Let be a nonempty closed convex subset of , and let be a mapping from into itself. A point is called an asymptotically fixed point of [31] if contains a sequence which converges weakly to such that . The set of asymptotical fixed points of will be denoted by . A mapping from into itself is called relatively nonexpansive [32–34] if and for all and .

We remark that if is a reflexive, strictly convex and smooth Banach space, then for any if and only if . It is sufficient to show that if then . From (2.5), we have . This implies that . From the definition of , we have . Therefore, we have ; see [26, 27] for more details.

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

Lemma 2.1 (see [23]).

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

Lemma 2.4 (see [35]).

Let be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space , and let be a relatively nonexpansive mapping, then is closed and convex.

The following result is according to Blum and Oettli [36].

Lemma 2.5 (see [36]).

Motivated by Combettes and Hirstoaga [37] in a Hilbert space, Takahashi and Zembayashi [38] established the following lemma.

Lemma 2.6 (see [38]).

for all , then, the following hold:

Using Lemma 2.6, one has the following result.

Lemma 2.7 (see [38]).

Utilizing Lemmas 2.5, 2.6 and 2.7 as above, Chang [39] derived the following result.

Proposition 2.8 (see [39, Lemma 2.5]).

Let
be a smooth, strictly convex and reflexive Banach space and
be a nonempty closed convex subset of
. Let
be an *α*-inverse-strongly monotone mapping, let
be a bifunction from
to
satisfying (A1)–(A4), and let
, then there hold the following:

then the mapping has the following properties:

(iv) is a closed convex subset of ,

Proof.

Then it is easy to verify that satisfies the conditions (A1)–(A4). Therefore, The conclusions (I) and (II) of Proposition 2.8 follow immediately from Lemmas 2.5, 2.6 and 2.7.

## 3. Main Results

Throughout this section, unless otherwise stated, we assume that
is a maximal monotone operator with domain
,
is a relatively nonexpansive mapping,
is an *α*-inverse-strongly monotone mapping and
is a bifunction satisfying (A1)–(A4), where
is a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space
. In this section, we study the following algorithm.

Algorithm 3.1.

where is a sequence in and , are sequences in .

First we investigate the condition under which the Algorithm 3.1 is well defined. Rockafellar [40] proved the following result.

Lemma 3.2 (Rockafellar [40]).

Let be a reflexive, strictly convex, and smooth Banach space and let be a multivalued operator, then there hold the following:

(i) is closed and convex if is maximal monotone such that ;

(ii) is maximal monotone if and only if is monotone with for all .

Utilizing this result, we can show the following lemma.

Lemma 3.3.

Let be a reflexive, strictly convex, and smooth Banach space. If , then the sequence generated by Algorithm 3.1 is well defined.

Proof.

This implies that . Therefore, is convex and hence is closed and convex.

which implies that . Consequently, and so . Therefore is well defined, then, by induction, the sequence generated by Algorithm 3.1, is well defined for each integer .

Remark 3.4.

We are now in a position to prove the main theorem.

Theorem 3.5.

Let . If is uniformly continuous, then the sequence generated by Algorithm 3.1 converges strongly to .

Proof.

Since is uniformly continuous, it follows from (3.27), (3.31) and that .

for all and . Thus from the maximality of , we infer that . Therefore, . Further, let us show that . Since and , from we obtain that and .

So, . Therefore, we obtain that by the arbitrariness of .

So we have . Utilizing the Kadec-Klee property of , we conclude that converges strongly to . Since is an arbitrary weakly convergent subsequence of , we know that converges strongly to . This completes the proof.

Theorem 3.5 covers [25, Theorem 3.1] by taking and . Also Theorem 3.5 covers [24, Theorem 2.1] by taking , and .

Theorem 3.6.

Let
be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space
. Let
be a maximal monotone operator with domain
,
be a relatively nonexpansive mapping,
be an *α*-inverse-strongly monotone mapping and
be a bifunction satisfying (A1)–(A4). Assume that
is a sequence in
satisfying
and that
is a sequences in
satisfying
.

Algorithm 3.7.

where is the single valued duality mapping on . Let . If is uniformly continuous, then converges strongly to .

Proof.

Since is uniformly continuous, it follows from (3.58) and (3.64) that .

Finally, we prove that . Indeed, for arbitrarily fixed, there exists a subsequence of such that , then . Now let us show that . Since , we have that . Moreover, since is uniformly norm-to-norm continuous on bounded subsets of , and , we obtain that . It follows from and the monotonicity of that for all and . This implies that for all and . Thus from the maximality of , we infer that . Further, let us show that . Since and , from we obtain that and .

Dividing by , we have , for all . Letting , from (A3) it follows that , for all . So, . Therefore, we obtain that by the arbitrariness of .

It follows from the definition of that and hence . So we have . Utilizing the Kadec-Klee property of , we know that . Since is an arbitrary weakly convergent subsequence of , we know that . This completes the proof.

Theorem 3.6 covers [25, Theorem 3.2] by taking and . Also Theorem 3.6 covers [24, Theorem 2.2] by taking and .

## Declarations

### Acknowledgments

This research was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (no. DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (no. 09ZZ133), National Science Foundation of China (no. 11071169), Ph.D. Program Foundation of Ministry of Education of China (no. 20070270004), Science and Technology Commission of Shanghai Municipality Grant (no. 075105118), and Shanghai Leading Academic Discipline Project (no. S30405). Al-Homidan is grateful to KFUPM for providing research facilities.

## Authors’ Affiliations

## References

- Takahashi S, Takahashi W:
**Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space.***Nonlinear Analysis. Theory, Methods & Applications*2008,**69**(3):1025–1033. 10.1016/j.na.2008.02.042MathSciNetView ArticleMATHGoogle 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 - 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 LC, Lin LJ, Yao JC:
**Auxiliary problem method for mixed variational-like inequalities.***Taiwanese Journal of Mathematics*2006,**10**(2):515–529.MathSciNetMATHGoogle 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*2010,**46**(3):331–345. 10.1007/s10898-009-9428-9MathSciNetView ArticleMATHGoogle 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 - Peng J-W, Yao J-C:
**Some new extragradient-like methods for generalized equilibrium problems, fixed point problems and variational inequality problems.***Optimization Methods and Software*2010,**25**(5):677–698. 10.1080/10556780902763295MathSciNetView ArticleMATHGoogle Scholar - 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 - 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 - 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:
**Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings.***Taiwanese Journal of Mathematics*2009,**13**(5):1537–1582.MathSciNetMATHGoogle Scholar - Martinet B:
**Régularisation d'inéquations variationnelles par approximations successives.***Revue Franç'Informatique et de Recherche Opérationnelle*1970,**4:**154–158.MathSciNetMATHGoogle 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 - Peng JW, Yao JC:
**A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems.***Taiwanese Journal of Mathematics*2008,**12:**1401–1433.MathSciNetMATHGoogle 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 - Ceng LC, Lai TC, Yao JC:
**Approximate proximal algorithms for generalized variational inequalities with paramonotonicity and pseudomonotonicity.***Computers & Mathematics with Applications*2008,**55**(6):1262–1269. 10.1016/j.camwa.2007.06.010MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Wu S-Y, Yao J-C:
**New accuracy criteria for modified approximate proximal point algorithms in Hilbert spaces.***Taiwanese Journal of Mathematics*2008,**12**(7):1691–1705.MathSciNetMATHGoogle Scholar - Güler O:
**On the convergence of the proximal point algorithm for convex minimization.***SIAM Journal on Control and Optimization*1991,**29**(2):403–419. 10.1137/0329022MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Yao J-C:
**Generalized implicit hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space.***Taiwanese Journal of Mathematics*2008,**12**(3):753–766.MathSciNetMATHGoogle Scholar - Zeng LC, Yao JC:
**An inexact proximal-type algorithm in Banach spaces.***Journal of Optimization Theory and Applications*2007,**135**(1):145–161. 10.1007/s10957-007-9261-6MathSciNetView ArticleMATHGoogle Scholar - Ceng LC, Yao JC:
**Approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions.***Journal of Computational and Applied Mathematics*2008,**213**(2):423–438. 10.1016/j.cam.2007.01.034MathSciNetView 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, Takahashi W:
**Strong convergence of a proximal-type algorithm in a Banach space.***SIAM Journal on Optimization*2003,**13**(3):938–945.MathSciNetView ArticleMATHGoogle Scholar - Qin X, Su Y:
**Strong convergence theorems for relatively nonexpansive mappings in a Banach space.***Nonlinear Analysis. Theory, Methods & Applications*2007,**67**(6):1958–1965. 10.1016/j.na.2006.08.021MathSciNetView ArticleMATHGoogle Scholar - Ceng LC, Petruşel A, Wu SY:
**On hybrid proximal-type algorithms in Banach spaces.***Taiwanese Journal of Mathematics*2008,**12**(8):2009–2029.MathSciNetMATHGoogle Scholar - Cioranescu I:
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications*.*Volume 62*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1990:xiv+260.View ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional ASnalysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. Fixed point theory and Its applicationMATHGoogle 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 Applied Mathematics*.*Volume 178*. Edited by: Kartsatos AG. Dekker, New York, NY, USA; 1996:15–50.Google Scholar - Alber YI, Reich S:
**An iterative method for solving a class of nonlinear operator equations in Banach spaces.***Panamerican Mathematical Journal*1994,**4**(2):39–54.MathSciNetMATHGoogle Scholar - Alber Y, Guerre-Delabriere S:
**On the projection methods for fixed point problems.***Analysis*2001,**21**(1):17–39.MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**A weak convergence theorem for the alternating method with Bregman distances.**In*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics*.*Volume 178*. Edited by: Reich S. Marcel Dekker, New York, NY, USA; 1996:313–318.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 - Butnariu D, Reich S, Zaslavski AJ:
**Weak convergence of orbits of nonlinear operators in reflexive Banach spaces.***Numerical Functional Analysis and Optimization*2003,**24**(5–6):489–508. 10.1081/NFA-120023869MathSciNetView ArticleMATHGoogle Scholar - Censor Y, Reich S:
**Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization.***Optimization*1996,**37**(4):323–339. 10.1080/02331939608844225MathSciNetView ArticleMATHGoogle Scholar - Matsushita S-Y, 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 - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle Scholar - Takahashi W, Zembayashi K:
**Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces.***Nonlinear Analysis. Theory, Methods & Applications*2009,**70**(1):45–57. 10.1016/j.na.2007.11.031MathSciNetView ArticleMATHGoogle Scholar - Zhang SS: Shrinking projection method for solving generalized equilibrium problem, variational inequality and common fixed point in Banach spaces with applications. to appear in Science in China A: MathematiceGoogle 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

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