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

*Fixed Point Theory and Applications*
**volume 2011**, Article number: 973028 (2011)

## 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.

## 1. Introduction

Let be a real Banach space with the dual and be a nonempty closed convex subset of . We denote by and the sets of positive integers and real numbers, respectively. Also, we denote by the normalized duality mapping from to defined by

where denotes the generalized duality pairing. Recall that if is smooth, then is single valued and if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of . We will still denote by the single valued duality mapping. Let be a bifunction and be a nonlinear mapping. We consider the following generalized equilibrium problem:

The set of such is denoted by , that is,

Whenever a Hilbert space, problem (1.2) was introduced and studied by S. Takahashi and W. Takahashi [1]. Similar problems have been studied extensively recently. See, for example, [2–11]. In the case of is denoted by . In the case of , is also denoted by . The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, for example, [12–14]. A mapping is called nonexpansive if for all . Denote by the set of fixed points of , that is, . A mapping is called *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.

Let be a real Banach space with the dual . A multivalued operator with domain is called monotone if for each and , . A monotone operator is called maximal if its graph is not properly contained in the graph of any other monotone operator. A method for solving the inclusion is the proximal point algorithm. Denote by the identity operator on a Hilbert space. The proximal point algorithm generates, for any initial point , a sequence in , by the iterative scheme

where is a sequence in the interval . Note that this iteration is equivalent to

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

Let be a real Banach space with the dual . We denote by the normalized duality mapping from to defined by

where denotes the generalized duality pairing. A Banach space is called strictly convex if for all with and . It is said to be uniformly convex if for any two sequences such that and . Let be a unit sphere of , then the Banach space is called smooth if

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.

Next, we assume that is a smooth Banach space. Consider the functional defined as in [28, 29] by

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

The generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem

The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping (see, e.g., [30]). In a Hilbert space , . From [28], in uniformly smooth and uniformly convex Banach spaces, we have

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 .

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

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , 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]).

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let and , then, there exists such that

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

Lemma 2.6 (see [38]).

Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space , and let be a bifunction from to satisfying (A1)–(A4). For and , define a mapping as follows:

for all , then, the following hold:

(i) is single valued;

(ii) is a firmly nonexpansive-type mapping, that is, for all ,

(iii);

(iv) is closed and convex.

Using Lemma 2.6, one has the following result.

Lemma 2.7 (see [38]).

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let , then, for and ,

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:

(I)for , there exists such that

(II)if is additionally uniformly smooth and is defined as

then the mapping has the following properties:

(i) is single valued,

(ii) is a firmly nonexpansive-type mapping, that is,

(iii),

(iv) is a closed convex subset of ,

(v), for all .

Proof.

Define a bifunction as follows:

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.

Let be a reflexive, strictly convex and smooth Banach space, and let be a maximal monotone operator with , then,

## 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.

For each , define two sets and as follows:

It is obvious that is closed and are closed convex sets for each . Let us show that is convex. For and , put . It is sufficient to show that . Indeed, observe that

is equivalent to

Note that there hold the following:

Thus we have

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

On the other hand, let be arbitrarily chosen, then and . From Algorithm 3.1, it follows that

So for all . Now, from Lemma 3.2 it follows that there exists such that and . Since is monotone, it follows that , which implies that and hence . Furthermore, it is clear that , then , and therefore is well defined. Suppose that and is well defined for some . Again by Lemma 3.2, we deduce that such that and , then from the monotonicity of we conclude that , which implies that and hence . It follows from Lemma 2.4 that

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.

From the above proof, we obtain that

for each integer .

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

Theorem 3.5.

Let be a uniformly smooth and uniformly convex Banach space. Let be a sequence in and be sequences in such that

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

Proof.

First of all, if follows from the definition of that . Since , we have

Thus is nondecreasing. Also from and Lemma 2.3, we have that

for each and for each . Consequently, is bounded. Moreover, according to the inequality

we conclude that is bounded. Thus, we have that exists. From Lemma 2.3, we derive the following:

for all . This implies that . So it follows from Lemma 2.1 that . Since , from the definition of , we also have

Observe that

At the same time,

Since and , it follows that

and hence that . Further, from , we have , which yields

Then it follows from that . Hence it follows from Lemma 2.1 that . Since from (3.15) we derive

we have

Thus, from , , and , we know that . Consequently from (3.16), , and it follows that

So it follows from (3.15), , and that . Utilizing Lemma 2.1 we deduce that

Furthermore, for arbitrarily fixed, it follows from Proposition 2.8 that

Since is uniformly norm-to-norm continuous on bounded subsets of , it follows from (3.23) that and , which hence yield . Utilizing Lemma 2.1, we get . Observe that

due to (3.23). Since is uniformly norm-to-norm continuous on bounded subsets of , we have that

On the other hand, we have

Noticing that

we have

From (3.26) and , we obtain

Since is also uniformly norm-to-norm continuous on bounded subsets of , we obtain

Observe that

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

Now let us show that , where

Indeed, since is bounded and is reflexive, we know that . Take arbitrarily, then there exists a subsequence of such that . Hence . Let us show that . Since , we have that . Moreover, since is uniformly norm-to-norm continuous on bounded subsets of and , we obtain

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 . Therefore, . Further, let us show that . Since and , from we obtain that and .

Since is uniformly norm-to-norm continuous on bounded subsets of , from we derive

From , it follows that

By the definition of , we have

where

Replacing by , we have from (A2) that

Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting in the last inequality, from (3.38) and (A4) we have

For , with , and , let . Since and , then and hence . So, from (A1) we have

Dividing by , we have

Letting , from (A3) it follows that

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

Next, let us show that and .

Indeed, put . From and , we have . Now from weakly lower semicontinuity of the norm, we derive for each

It follows from the definition of that and hence

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 .

Define a sequence .

Algorithm 3.7.

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

Proof.

For each , define two sets and as follows:

It is obvious that is closed and are closed convex sets for each . Let us show that is convex and so is closed and convex. Similarly to the proof of Lemma 3.3, since

is equivalent to

we know that is convex and so is . Next, let us show that for each . Indeed, utilizing Proposition 2.8, we have, for each ,

So for all and . As in the proof of Lemma 3.3, we can obtain and hence . It follows from Lemma 2.4 that

which implies that . Consequently, and so for all . Therefore, the sequence generated by Algorithm 3.7 is well defined. As in the proof of Theorem 3.5, we can obtain . Since , from the definition of we also have

As in the proof of Theorem 3.5, we can deduce not only from that but also from , and that

Since , from the definition of , we also have

It follows from (3.55) and that

Utilizing Lemma 2.1 we have

Furthermore, for arbitrarily fixed, it follows from Proposition 2.8 that

Since is uniformly norm-to-norm continuous on bounded subsets of , it follows from (3.58) that and , which together with , yield . Utilizing Lemma 2.1, we get . Observe that

due to (3.58). Since is uniformly norm-to-norm continuous on bounded subsets of , we have

Note that

Therefore, from we get

Since is also uniformly norm-to-norm continuous on bounded subsets of , we obtain

It follows that

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 .

Since is uniformly norm-to-norm continuous on bounded subsets of , from we derive . From it follows that

By the definition of , we have

where . Replacing by , we have from (A2) that

Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting in the last inequality, from (3.66) and (A4) we have , for all . For , with , and , let . Since and , then and hence . So, from (A1) we have

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

Next, let us show that and .

Indeed, put . From and , we have . Now from weakly lower semicontinuity of the norm, we derive for each

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 .

## References

- 1.
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.2008.02.042 - 2.
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. - 3.
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. - 4.
Zeng LC, Lin LJ, Yao JC:

**Auxiliary problem method for mixed variational-like inequalities.***Taiwanese Journal of Mathematics*2006,**10**(2):515–529. - 5.
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/s10898-009-9428-9 - 6.
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. - 7.
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. 10.1080/10556780902763295 - 8.
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. - 9.
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.2008.11.014 - 10.
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/s10898-008-9342-6 - 11.
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. - 12.
Martinet B:

**Régularisation d'inéquations variationnelles par approximations successives.***Revue Franç'Informatique et de Recherche Opérationnelle*1970,**4:**154–158. - 13.
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/S1085337504309036 - 14.
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. - 15.
Rockafellar RT:

**Monotone operators and the proximal point algorithm.***SIAM Journal on Control and Optimization*1976,**14**(5):877–898. 10.1137/0314056 - 16.
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.2007.06.010 - 17.
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. - 18.
Güler O:

**On the convergence of the proximal point algorithm for convex minimization.***SIAM Journal on Control and Optimization*1991,**29**(2):403–419. 10.1137/0329022 - 19.
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. - 20.
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/s10957-007-9261-6 - 21.
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.2007.01.034 - 22.
Solodov MV, Svaiter BF:

**Forcing strong convergence of proximal point iterations in a Hilbert space.***Mathematical Programming.*2000,**87**(1):189–202. - 23.
Kamimura S, Takahashi W:

**Strong convergence of a proximal-type algorithm in a Banach space.***SIAM Journal on Optimization*2003,**13**(3):938–945. - 24.
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.2006.08.021 - 25.
Ceng LC, Petruşel A, Wu SY:

**On hybrid proximal-type algorithms in Banach spaces.***Taiwanese Journal of Mathematics*2008,**12**(8):2009–2029. - 26.
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. - 27.
Takahashi W:

*Nonlinear Functional ASnalysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. Fixed point theory and Its application - 28.
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. - 29.
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. - 30.
Alber Y, Guerre-Delabriere S:

**On the projection methods for fixed point problems.***Analysis*2001,**21**(1):17–39. - 31.
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. - 32.
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.2001.151 - 33.
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-120023869 - 34.
Censor Y, Reich S:

**Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization.***Optimization*1996,**37**(4):323–339. 10.1080/02331939608844225 - 35.
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.2005.02.007 - 36.
Blum E, Oettli W:

**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145. - 37.
Combettes PL, Hirstoaga SA:

**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136. - 38.
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.2007.11.031 - 39.
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: Mathematice* - 40.
Rockafellar RT:

**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149:**75–88. 10.1090/S0002-9947-1970-0282272-5

## 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.

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Zeng, L., Ansari, Q. & Al-Homidan, S. Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings.
*Fixed Point Theory Appl* **2011, **973028 (2011). https://doi.org/10.1155/2011/973028

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### Keywords

- Hilbert Space
- Banach Space
- Nonexpansive Mapping
- Monotone Operator
- Real Banach Space