# A General Iterative Method for Variational Inequality Problems, Mixed Equilibrium Problems, and Fixed Point Problems of Strictly Pseudocontractive Mappings in Hilbert Spaces

- Rattanaporn Wangkeeree
^{1}and - Rabian Wangkeeree
^{1}Email author

**2009**:519065

https://doi.org/10.1155/2009/519065

© R.Wangkeeree and R.Wangkeeree. 2009

**Received: **23 April 2009

**Accepted: **22 June 2009

**Published: **9 August 2009

## Abstract

We introduce an iterative scheme for finding a common element of the set of fixed points of a -strictly pseudocontractive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping, and the set of solutions of the mixed equilibrium problem in a real Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we first apply our results to study the optimization problem and we next utilize our results to study the problem of finding a common element of the set of fixed points of two families of finitely -strictly pseudocontractive mapping, the set of solutions of the variational inequality, and the set of solutions of the mixed equilibrium problem. The results presented in the paper improve some recent results of Kim and Xu (2005), Yao et al. (2008), Marino et al. (2009), Liu (2009), Plubtieng and Punpaeng (2007), and many others.

## Keywords

## 1. Introduction

Throughout this paper, we always assume that is a real Hilbert space with inner product and norm , respectively, is a nonempty closed convex subset of . Let be a real-valued function and let be an equilibrium bifunction, that is, for each . Ceng and Yao [1] considered the following mixed equilibrium problem:

The set of solutions of (1.1) is denoted by . It is easy to see that is a solution of problem (1.1) implies that .

In particular, if , the mixed equilibrium problem (1.1) becomes the following equilibrium problem:

The set of solutions of (1.2) is denoted by .

If and for all , where is a mapping form into , then the mixed equilibrium problem (1.1) becomes the following variational inequality:

The set of solutions of (1.3) is denoted by . The variational inequality has been extensively studied in literature. See, for example, [2–13] and the references therein.

The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others; see for instance, [1, 2, 14, 15].

First we recall some relevant important results as follows.

In 1997, Combettes and Hirstoaga [14] introduced an iterative method of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi [16] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of and the set of fixed point points of a nonexpansive mapping. Using the idea of S. Takahashi and W. Takahashi [16], Plubtieng and Punpaeng [17] introduced an the general iterative method for finding a common element of the set of solutions of and the set of fixed points of a nonexpansive mapping which is the optimality condition for the minimization problem in a Hilbert space. Furthermore, Yao et al. [11] introduced some new iterative schemes for finding a common element of the set of solutions of and the set of common fixed points of finitely (infinitely) nonexpansive mappings. Very recently, Ceng and Yao [1] considered a new iterative scheme for finding a common element of the set of solutions of and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem which used the following condition:

(E) is -strongly convex and its derivative is sequentially continuous from the weak topology to the strong topology.

Their results extend and improve the corresponding results in [6, 11, 14]. We note that the condition (E) for the function is a very strong condition. We also note that the condition (E) does not cover the case and . Motivated by Ceng and Yao [1], Peng and Yao [18] introduced a new iterative scheme based on only the extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of the variational inequality for a monotone Lipschitz continuous mapping. They obtained a strong convergence theorem without the condition (E) for the sequences generated by these processes.

We recall that a mapping is said to be:

(ii) -Lipschitz if there exists a constant such that

(iii) -inverse-strongly monotone [19, 20] if there exists a positive real number such that

It is obvious that any -inverse-strongly monotone mapping is monotone and Lipschitz continuous. Recall that a mapping is called a -strictly pseudocontractive mapping if there exists a constant such that

Note that the class of -strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings which are mappings on such that

That is, is nonexpansive if and only if is -strictly pseudocontractive. We denote by the set of fixed points of .

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [21–24] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of nonexpansive mapping on a real Hilbert space:

where is a linear bounded operator, is the fixed point set of a nonexpansive mapping and is a given point in . Recall that a linear bounded operator is strongly positive if there is a constant with property

Recently, Marino and Xu [25] introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi [26]:

where is a strongly positive bounded linear operator on . They proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.9) converges strongly to the unique solution of the variational inequality

which is the optimality condition for the minimization problem

where is a potential function for for ).

Recall that the construction of fixed points of nonexpansive mappings via Manns algorithm [27] has extensively been investigated in literature; see, for example [27–32] and references therein. If is a nonexpansive self-mapping of , then Mann's algorithm generates, initializing with an arbitrary , a sequence according to the recursive manner

where is a real control sequence in the interval .

If is a nonexpansive mapping with a fixed point and if the control sequence is chosen so that , then the sequence generated by Manns algorithm converges weakly to a fixed point of . Reich [33] showed that the conclusion also holds good in the setting of uniformly convex Banach spaces with a Fréhet differentiable norm. It is well known that Reich's result is one of the fundamental convergence results. However, this scheme has only weak convergence even in a Hilbert space [34]. Therefore, many authors try to modify normal Mann's iteration process to have strong convergence; see, for example, [35–40] and the references therein.

Kim and Xu [36] introduced the following iteration process:

where is a nonexpansive mapping of into itself and is a given point. They proved the sequence defined by (1.13) strongly converges to a fixed point of provided the control sequences and satisfy appropriate conditions.

In [41], Yao et al. also modified iterative algorithm (1.13) to have strong convergence by using viscosity approximation method. To be more precisely, they considered the following iteration process:

where is a nonexpansive mapping of into itself and is an -contraction. They proved the sequence defined by (1.14) strongly converges to a fixed point of provided the control sequences and satisfy appropriate conditions.

Very recently, motivated by Acedo and Xu [35], Kim and Xu [36], Marino and Xu [42], and Yao et al. [41], Marino et al. [43] introduced a composite iteration scheme as follows:

where is a -strictly pseudocontractive mapping on is an -contraction, and is a linear bounded strongly positive operator. They proved that the iterative scheme defined by (1.15) converges to a fixed point of , which is a unique solution of the variational inequality (1.10) and is also the optimality condition for the minimization problem provided and are sequences in satifies the following control conditions:

Moreover, for finding a common element of the set of fixed points of a -strictly pseudocontractive nonself mapping and the set of solutions of an equilibrium problem in a real Hilbert space, Liu [44] introduced the following iterative scheme:

where is a -strictly pseudocontractive mapping on is an -contraction and, is a linear bounded strongly positive operator. They proved that the iterative scheme defined by (1.16) converges to a common element of , which solves some variation inequality problems provided and are sequences in satifies the control conditions (C1) and the following conditions:

All of the above bring us the following conjectures?

- (i)
- (ii)
- (iii)
- (iv)
- (v)

It is our purpose in this paper that we suggest and analyze an iterative scheme for finding a common element of the set of fixed points of a -strictly pseudocontractive mapping, the set of solutions of a variational inequality and the set of solutions of a mixed equilibrium problem in the framework of a real Hilbert space. Then we modify our iterative scheme to finding a common element of the set of common fixed points of two finite families of -strictly pseudocontractive mappings, the set of solutions of a variational inequality and the set of solutions of a mixed equilibrium problem. Application to optimization problems which is one of the motivation in this paper is also given. The results in this paper generalize and improve some well-known results in [17, 36, 41, 43, 44].

## 2. Preliminaries

Let be a real Hilbert space with norm and inner product and let be a closed convex subset of . We denote weak convergence and strong convergence by notations and , respectively. It is well known that for any

For every point , there exists a unique nearest point in , denoted by , such that

is called the metric projection of onto It is well known that is a nonexpansive mapping of onto and satisfies

for every Moreover, is characterized by the following properties: and

for all . It is easy to see that the following is true:

A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies . Let be a monotone map of into and let be the normal cone to at , that is, and define

Then is the maximal monotone and if and only if ; see [45].

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1 ([46]).

where is a sequence in and is a sequence in such that

Lemma 2.2 ([47]).

Let and be bounded sequences in a Banach space and let be a sequence in with Suppose for all integers and Then,

Lemma 2.3 ([42, Proposition 2.1]).

Assume that is a closed convex subset of Hilbert space , and let be a self-mapping of

(i)if is a -strictly pseudocontractive mapping, then satisfies the Lipscchitz condition

(ii)if is a -strictly pseudocontractive mapping, then the mapping is demiclosed(at ). That is, if is a sequence in such that and ,

(iii)if is a -strictly pseudocontractive mapping, then the fixed point set of is closed and convex so that the projection is well defined.

Lemma 2.4 ([25]).

Assume is a strongly positive linear bounded operator on a Hilbert space with coefficient and Then

The following lemmas can be obtained from Acedo and Xu [35, Proposition 2.6] easily.

Lemma 2.5.

Let be a Hilbert space, be a closed convex subset of . For any integer , assume that, for each is a -strictly pseudocontractive mapping for some . Assume that is a positive sequence such that . Then is a -strictly pseudocontractive mapping with .

Lemma 2.6.

Let and be as in Lemma 2.5. Suppose that has a common fixed point in . Then .

For solving the mixed equilibrium problem, let us give the following assumptions for a bifunction and the set :

for each is convex and lower semicontinuous;

By similar argument as in [48, proof of Lemma 2.3], we have the following result.

Lemma 2.7.

for all . Then, the following conditions hold:

## 3. Main Results

In this section, we derive a strong convergence of an iterative algorithm which solves the problem of finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a -strictly pseudocontractive mapping of into itself and the set of the variational inequality for an -inverse-strongly monotone mapping of into in a Hilbert space.

Theorem 3.1.

Let C be a nonempty closed convex subset of a Hilbert space H. Let be a bifunction from to satifies (A1)–(A4) and be a proper lower semicontinuous and convex function. Let be a -strictly pseudocontractive mapping of into itself. Let be a contraction of into itself with coefficient , an -inverse-strongly monotone mapping of into such that . Let be a strongly bounded linear self-adjoint operator with coefficient and . Assume that either (B1) or (B2) holds. Given the sequences and in satisfyies the following conditions

Proof.

Next, we divide the proof into six steps as follows.

Step 1.

which implies that is nonexpansive.

Step 2.

which gives that the sequence is bounded, so are and

Step 3.

As shown in [19], from the -strict pseudocontractivity of and the conditions (D4), it follows that is a nonexpansive maping for which .

Observing that

Hence (3.16) is proved.

Step 4.

Hence (3.36) is proved.

Step 5.

Since is bounded, there exists a subsequence of which converges weakly to .

(a)We first show . In fact, using Lemma 2.3(ii) and (3.36), we obtain that .

(b)Next, we prove . For this purpose, let be the maximal monotone mapping defined by (2.6):

Since is maximal monotone, we have , and hence .

(c)We show . In fact, by , and we have,

Hence . Therefore, the conclusion is proved.

Consequently

Step 6.

Using (D1), and (3.79), we get . Now applying Lemma 2.1 to (3.82), we conclude that . From and , we obtain . The proof is now complete.

By Theorem 3.1, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows.

Setting in Theorem 3.1, we have the following result.

Corollary 3.2.

Let C be a nonempty closed convex subset of a Hilbert space H. Let be a bifunction from to satifies (A1)–(A4). Let be a -strictly pseudocontractive mapping of into itself. Let be a contraction of into itself with coefficient , an -inverse-strongly monotone mapping of into such that . Let be a strongly bounded linear self-adjoint operator with coefficient and . Given the sequences and in satisfies the following conditions

Setting and in Theorem 3.1, we have , then the following result is obtained.

Corollary 3.3.

Let C be a nonempty closed convex subset of a Hilbert space H. Let be a -strictly pseudocontractive mapping of into itself. Let be a contraction of into itself with coefficient , an -inverse-strongly monotone mapping of into such that . Let be a strongly bounded linear self-adjoint operator with coefficient and . Given the sequences and in satifies the following conditions

- (i)
Since the conditions (C1) and (C2) have been weakened by the conditions (D1) and (D3) respectively. Theorem 3.1 and Corollary 3.2 generalize and improve [44, Theorem 3.2].

- (ii)
- (iii)
Since the conditions (C1) and (C2) have been weakened by the conditions (D1) and (D3) respectively. Theorem 3.1 and Corollary 3.3 generalize and improve [43, Theorem 2.1].

Setting and is nonexpansive in Theorem 3.1, we have the following result.

Corollary 3.5.

Let C be a nonempty closed convex subset of a Hilbert space H. Let be a bifunction from to satifies (A1)–(A4). Let be a nonexpansive mapping of into itself. Let be a contraction of into itself with coefficient such that . Let be a strongly bounded linear self-adjoint operator with coefficient and . Given the sequences and in satifies the following conditions

Remark 3.6.

Since the conditions and have been weakened by the conditions and , respectively. Hence Corollary 3.5 generalize, extend and improve [17, Theorem 3.3].

## 4. Applications

First, we will utilize the results presented in this paper to study the following optimization problem:

where is a nonempty bounded closed convex subset of a Hilbert space and is a proper convex and lower semicontinuous function. We denote by Argmin the set of solutions in (4.1). Let for all , and in Theorem 3.1, then . It follows from Theorem 3.1 that the iterative sequence is defined by

where , satisfy the conditions (D1)–(D5) in Theorem 3.1. Then the sequence converges strongly to a solution .

Let for all , , , and in Theorem 3.1, then . It follows from Theorem 3.1 that the iterative sequence defined by

where , and satisfy the conditions (D1), (D2) and (D5), respectively in Theorem 3.1. Then the sequence converges strongly to a solution .

We remark that the algorithms (4.2) and (4.3) are variants of the proximal method for optimization problems introduced and studied by Martinet [49], Rockafellar [45], Ferris [50] and many others.

Next, we give the strong convergence theorem for finding a common element of the set of common fixed point of a finite family of strictly pseudocontractive mappings, the set of solutions of the variational inequality problem and the set of solutions of the mixed equilibrium problem in a Hilbert space.

Theorem 4.1.

Let C be a nonempty closed convex subset of a Hilbert space H. Let be a bifunction from to satifies (A1)–(A4) and be a proper lower semicontinuous and convex function. For each let be a -strictly pseudocontractive mapping of into itself for some . Let be a contraction of into itself with coefficient , an inverse-strongly monotone mapping of into such that . Let be a strongly bounded linear self-adjoint operator with coefficient and . Assume that either (B1) or (B2) holds. Given the sequences and in satifies the following conditions

Proof.

Let such that and define . By Lemmas 2.5 and 2.6, we conclude that is a -strictly pseudocontractive mapping with and . From Theorem 3.1, we can obtain the desired conclusion easily.

Finally, we will apply the main results to the problem for finding a common element of the set of fixed points of two finite families of -strictly pseudocontractive mappings, the set of solutions of the variational inequality and the set of solutions of the mixed equilibrium problem.

Let be a -strictly pseudocontractive mapping for some . We define a mapping where is a positive sequence such that , then is a -inverse-strongly monotone mapping with . In fact, from Lemma 2.5, we have

That is

On the other hand

Hence we have

This shows that is -inverse-strongly monotone.

Theorem 4.2.

Let be a nonempty closed convex subset of a Hilbert space . Let be a bifunction from to satifies (A1)–(A4) and be a proper lower semicontinuous and convex function. Let be a finite family of -strictly pseudocontractive mapping of into itself and be a finite family of -strictly pseudocontractive mapping of into for some such that . Let be a contraction of into itself with coefficient . Let be a strongly bounded linear self-adjoint operator with coefficient and . Assume that either (B1) or (B2) holds. Given the sequences and in satifies the following conditions

Proof.

The conclusion can be obtained from Theorem 4.1.

## Declarations

### Acknowledgments

R. Wangkeeree would like to thank The National Research Council of Thailand, Grant SC-AR-012/2552 for financial support. The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

## Authors’ Affiliations

## References

- Ceng L-C, Yao J-C:
**A hybrid iterative scheme for mixed equilibrium problems and fixed point problems.***Journal of Computational and Applied Mathematics*2008,**214**(1):186–201. 10.1016/j.cam.2007.02.022MathSciNetView 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 - Iiduka H, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings.***Nonlinear Analysis: Theory, Methods & Applications*2005,**61**(3):341–350. 10.1016/j.na.2003.07.023MathSciNetView ArticleMATHGoogle Scholar - Kumam P:
**A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping.***Journal of Applied Mathematics and Computing*2009,**29**(1–2):263–280. 10.1007/s12190-008-0129-1MathSciNetView ArticleMATHGoogle Scholar - Kumam W, Kumam P:
**Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization.***Nonlinear Analysis: Hybrid Systems*2009,**3**(4):640–656. 10.1016/j.nahs.2009.05.007MathSciNetMATHGoogle Scholar - Takahashi W, Toyoda M:
**Weak convergence theorems for nonexpansive mappings and monotone mappings.***Journal of Optimization Theory and Applications*2003,**118**(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar - Kamraksa U, Wangkeeree R:
**A general iterative method for variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings in Hilbert spaces.***Thai Journal of Mathematics*2008,**6**(1):147–170.MathSciNetMATHGoogle Scholar - Wangkeeree R, Kamraksa U:
**A general iterative method for solving the variational inequality problem and fixed point problem of an infinite family of nonexpansive mappings in Hilbert spaces.***Fixed Point Theory and Applications*2009,**2009:**-23.Google Scholar - Wangkeeree R:
**An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-17.Google Scholar - Wangkeeree R, Kamraksa U: An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems. Nonlinear Analysis: Hybrid Systems. In pressGoogle Scholar
- Yao Y, Liou Y-C, Yao J-C:
**An extragradient method for fixed point problems and variational inequality problems.***Journal of Inequalities and Applications*2007,**2007:**-12.Google Scholar - Yao J-C, Chadli O:
**Pseudomonotone complementarity problems and variational inequalities.**In*Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications*.*Volume 76*. Edited by: Crouzeix JP, Haddjissas N, Schaible S. Springer, New York, NY, USA; 2005:501–558.View ArticleGoogle Scholar - Zeng LC, Schaible S, Yao JC:
**Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities.***Journal of Optimization Theory and Applications*2005,**124**(3):725–738. 10.1007/s10957-004-1182-zMathSciNetView ArticleMATHGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetGoogle Scholar - Flåm SD, Antipin AS:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetView ArticleMATHGoogle Scholar - Takahashi S, Takahashi W:
**Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**331**(1):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleMATHGoogle Scholar - Plubtieng S, Punpaeng R:
**A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**336**(1):455–469. 10.1016/j.jmaa.2007.02.044MathSciNetView ArticleMATHGoogle 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 - Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar - Liu F, Nashed MZ:
**Regularization of nonlinear Ill-posed variational inequalities and convergence rates.***Set-Valued Analysis*1998,**6**(4):313–344. 10.1023/A:1008643727926MathSciNetView ArticleMATHGoogle Scholar - Deutsch F, Yamada I:
**Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings.***Numerical Functional Analysis and Optimization*1998,**19**(1–2):33–56.MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**An iterative approach to quadratic optimization.***Journal of Optimization Theory and Applications*2003,**116**(3):659–678. 10.1023/A:1023073621589MathSciNetView ArticleMATHGoogle Scholar - Yamada I:
**The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings.**In*Inherently Parallel Algorithm for Feasibility and Optimization*. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, London, UK; 2001:473–504.View ArticleGoogle Scholar - Marino G, Xu H-K:
**A general iterative method for nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2006,**318**(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleMATHGoogle Scholar - Moudafi A:
**Viscosity approximation methods for fixed-points problems.***Journal of Mathematical Analysis and Applications*2000,**241**(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar - Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Byrne C:
**A unified treatment of some iterative algorithms in signal processing and image reconstruction.***Inverse Problems*2004,**20**(1):103–120. 10.1088/0266-5611/20/1/006MathSciNetView ArticleMATHGoogle Scholar - Tan K-K, Xu HK:
**Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1993,**178**(2):301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleMATHGoogle Scholar - Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar - Zeng L-C:
**A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1998,**226**(1):245–250. 10.1006/jmaa.1998.6053MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Weak convergence theorems for nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*1979,**67**(2):274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar - Genel A, Lindenstrauss J:
**An example concerning fixed points.***Israel Journal of Mathematics*1975,**22**(1):81–86. 10.1007/BF02757276MathSciNetView ArticleMATHGoogle Scholar - Acedo GL, Xu H-K:
**Iterative methods for strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(7):2258–2271. 10.1016/j.na.2006.08.036MathSciNetView ArticleMATHGoogle Scholar - Kim T-H, Xu H-K:
**Strong convergence of modified Mann iterations.***Nonlinear Analysis: Theory, Methods & Applications*2005,**61**(1–2):51–60. 10.1016/j.na.2004.11.011MathSciNetView ArticleMATHGoogle Scholar - Martinez-Yanes C, Xu H-K:
**Strong convergence of the CQ method for fixed point iteration processes.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(11):2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.***Journal of Mathematical Analysis and Applications*2003,**279**(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar - Qin X, Su Y:
**Approximation of a zero point of accretive operator in Banach spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):415–424. 10.1016/j.jmaa.2006.06.067MathSciNetView ArticleMATHGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(2):456–462. 10.1016/j.na.2007.05.032MathSciNetView ArticleMATHGoogle Scholar - Yao Y, Chen R, Yao J-C:
**Strong convergence and certain control conditions for modified Mann iteration.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(6):1687–1693. 10.1016/j.na.2007.01.009MathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle Scholar - Marino G, Colao V, Qin X, Kang SM:
**Strong convergence of the modified Mann iterative method for strict pseudo-contractions.***Computers & Mathematics with Applications*2009,**57**(3):455–465. 10.1016/j.camwa.2008.10.073MathSciNetView ArticleMATHGoogle Scholar - Liu Y:
**A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(10):4852–4861. 10.1016/j.na.2009.03.060MathSciNetView 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 - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals.***Journal of Mathematical Analysis and Applications*2005,**305**(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView 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 - Martinet B:
**Perturbation des méthodes d'optimisation. Applications.***RAIRO Analyse Numérique*1978,**12**(2):153–171.MathSciNetMATHGoogle Scholar - Ferris MC:
**Finite termination of the proximal point algorithm.***Mathematical Programming*1991,**50**(3):359–366. 10.1007/BF01594944MathSciNetView ArticleMATHGoogle Scholar

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