Research Article  Open  Published:
A General Iterative Method for Variational Inequality Problems, Mixed Equilibrium Problems, and Fixed Point Problems of Strictly Pseudocontractive Mappings in Hilbert Spaces
Fixed Point Theory and Applicationsvolume 2009, Article number: 519065 (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 inversestrongly 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.
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 realvalued 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:
(i)monotone if
(ii)Lipschitz if there exists a constant such that
(iii)inversestrongly monotone [19, 20] if there exists a positive real number such that
It is obvious that any inversestrongly 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 selfmapping 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:
(C1)
(C2) for all and .
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:
(2) for all , , and ;
(C3).
All of the above bring us the following conjectures?
Question 1.

(i)
Could we weaken or remove the control condition on parameter in (C1)?

(ii)
Could we weaken or remove the control condition on parameter in (C2) and (2)?

(iii)
Could we weaken or remove the control condition on the parameter in (2)?

(iv)
Could we weaken the control condition (C3) on parameters ?

(v)
Could we construct an iterative algorithm to approximate a common element of ?
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 wellknown 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 setvalued 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]).
Assume is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence in such that
(1)
(2) or
Then
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 selfmapping 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 all
is monotone, that is, for all
for each
for each is convex and lower semicontinuous;
For each and , there exists a bounded subset and such that for any ,
is a bounded set.
By similar argument as in [48, proof of Lemma 2.3], we have the following result.
Lemma 2.7.
Let be a nonempty closed convex subset of . Let be a bifunction satifies (A1)–(A4) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows:
for all . Then, the following conditions hold:
(i)for each ,;
(ii) is single valued;
(iii) is firmly nonexpansive, that is, for any
(iv)
(v) is closed and convex.
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 inversestrongly 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 inversestrongly monotone mapping of into such that . Let be a strongly bounded linear selfadjoint operator with coefficient and . Assume that either (B1) or (B2) holds. Given the sequences and in satisfyies the following conditions
(D1)
(D2)
(D3) for all and ;
(D4) for some with and ;
(D5).
Let and be sequences generated by
Then and converge strongly to a point which is the unique solution of the variational inequality
Equivalently, one has
Proof.
Since , we may assume, without loss of generality, that for all . By Lemma 2.4, we have . We will assume that . Observe that is a contraction. Indeed, for all , we have
Since is complete, there exists a unique element such that On the other hand, since is a linear bounded selfadjoint operator, one has
Observing that
we obtain is positive. It follows that
Next, we divide the proof into six steps as follows.
Step 1.
First we prove that is nonexpansive. For all and
which implies that is nonexpansive.
Step 2.
Next we prove that and are bounded. Indeed, pick any . From (2.5), we have Setting , we obtain from the nonexpansivity of that
From (2.1), we have
so, by (3.9) and the strict pseudocontractivity of , it follows that
that is,
Observe that
From (3.8), (3.11) and the last inequality, we have
It follows that
By simple induction, we have
which gives that the sequence is bounded, so are and
Step 3.
Next we claim that
Notice that
Next, we define
As shown in [19], from the strict pseudocontractivity of and the conditions (D4), it follows that is a nonexpansive maping for which .
Observing that
we have
where is an appropriate constant such that . Substituting (3.20) into (3.17), we obtain
On the other hand, from and we note that
Putting in (3.22) and in (3.23), we have
So, from (A2) we have
and hence
Without loss of generality, let us assume that there exists a real number such that for all Then, we have
and hence
where . It follows from (3.21) and the last inequality that
where .
Define a sequence such that
Then, we have
It follows from (3.29) that
Observing the conditions (D1), (D3), (D4), (D5), and taking the superior limit as , we get
We can obtain easily by Lemma 2.2. Observing that
we obtain
Hence (3.16) is proved.
Step 4.
Next we prove that

(a)
First we prove that . Observing that
we arrive at
which implies that
Therefore, it follows from (3.16), (D1), and (D2) that

(b)
Next, we will show that for any Observe that
where
This implies that
It is easy to see that and then from (3.16), we obtain

(c)
Next we prove that . From (2.3), we have
so, we obtain
It follows that
which implies that
Applying (3.16), (3.44), , and to the last inequality, we obtain that
It follows from (3.40) and (3.49) that
Then it follows from (D1), (3.49) and (3.50) that
For any , we have from Lemma 2.7,
Hence
From (3.41) we observe that
Hence
Using (D1), (D2) and (3.16), we obtain

(d)
Next we prove that . Using Lemma 2.3 (i), we have
which implies that
By (3.16), (3.51), and (3.56), we have
Observing that
Using (3.40) and the last inequality, we obtain that
From Lemma 2.3(i), (3.59), and (3.61), we have
Hence (3.36) is proved.
Step 5.
We claim that
We choose a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to .
Next, we show that .
(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):
For any given , hence . Since we have
On the other hand, from , we have
that is,
Therefore, we obtian
Noting that as and is Lipschitz continuous, hence from (3.69), we obtain
Since is maximal monotone, we have , and hence .
(c)We show . In fact, by , and we have,
From (A2), we also have
and hence
From and we get . It follows from (A4), , and the lower semicontinuous of that
For with and let Since and we have and hence So, from (A1) and (A4) and the convexity of , we have
Dividing by , we have
Letting , it follows from the weakly semicontinuity of that
Hence . Therefore, the conclusion is proved.
Consequently
as required. This together with (3.40) implies that
Step 6.
Finally, we show that . Indeed, we note that
Since , and are bounded, we can take a constant such that
for all . It then follows that
where
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 inversestrongly monotone mapping of into such that . Let be a strongly bounded linear selfadjoint operator with coefficient and . Given the sequences and in satisfies the following conditions
(D1)
(D2)
(D3) for all and
(D4) for some with and
(D5).
Let and be sequences generated by
Then and converge strongly to a point which is the unique solution of the variational inequality
Equivalently, one has
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 inversestrongly monotone mapping of into such that . Let be a strongly bounded linear selfadjoint operator with coefficient and . Given the sequences and in satifies the following conditions
(D1)
(D2)
(D3) for all and
(D4) for some with and .
Let and be sequences generated by
Then and converge strongly to a point which is the unique solution of the variational inequality
Equivalently, one has
Remark 3.4.

(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)
We can remove the control condition on the parameter in (2).

(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 selfadjoint operator with coefficient and . Given the sequences and in satifies the following conditions
(D1)
(D2)
(D3).
Let and be sequences generated by
Then and converge strongly to a point which is the unique solution of the variational inequality
Equivalently, one has
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 inversestrongly monotone mapping of into such that . Let be a strongly bounded linear selfadjoint operator with coefficient and . Assume that either (B1) or (B2) holds. Given the sequences and in satifies the following conditions
(D1)
(D2)
(D3) for all and ;
(D4) for some with and ;
(D5).
Let and be sequences generated by
where is a positive constant such that Then both and converge strongly to a point which is the unique solution of the variational inequality
Equivalently, one has
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 inversestrongly monotone mapping with . In fact, from Lemma 2.5, we have
That is
On the other hand
Hence we have
This shows that is inversestrongly 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 selfadjoint operator with coefficient and . Assume that either (B1) or (B2) holds. Given the sequences and in satifies the following conditions
(D1)
(D2)
(D3) and for all and ;
(D4) for some with and ;
(D5).
Let and be sequences generated by
where and are positive constants such that and , respectively. Then and converge strongly to a point which is the unique solution of the variational inequality
Equivalently, we have
Proof.
Taking in Theorem 4.1, we know that is inverse strongly monotone with . Hence, is a monotone Lipschitz continuous mapping with . From Lemma 2.6, we know that is a strictly pseudocontractive mapping with and then by Lemma 2.6. Observe that
The conclusion can be obtained from Theorem 4.1.
References
 1.
Ceng LC, Yao JC: 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.2007.02.022
 2.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
 3.
Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inversestrongly monotone mappings. Nonlinear Analysis: Theory, Methods & Applications 2005,61(3):341–350. 10.1016/j.na.2003.07.023
 4.
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/s1219000801291
 5.
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.2009.05.007
 6.
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:1025407607560
 7.
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.
 8.
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.
 9.
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.
 10.
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 press
 11.
Yao Y, Liou YC, Yao JC: An extragradient method for fixed point problems and variational inequality problems. Journal of Inequalities and Applications 2007, 2007:12.
 12.
Yao JC, 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.
 13.
Zeng LC, Schaible S, Yao JC: Iterative algorithm for generalized setvalued strongly nonlinear mixed variationallike inequalities. Journal of Optimization Theory and Applications 2005,124(3):725–738. 10.1007/s109570041182z
 14.
Combettes PL, Hirstoaga SA: Equilibrium programming using proximallike algorithms. Mathematical Programming 1997,78(1):29–41.
 15.
Flåm SD, Antipin AS: Equilibrium programming using proximallike algorithms. Mathematical Programming 1997,78(1):29–41.
 16.
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.2006.08.036
 17.
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.2007.02.044
 18.
Peng JW, Yao JC: 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
 19.
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967, 20: 197–228. 10.1016/0022247X(67)900856
 20.
Liu F, Nashed MZ: Regularization of nonlinear Illposed variational inequalities and convergence rates. SetValued Analysis 1998,6(4):313–344. 10.1023/A:1008643727926
 21.
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.
 22.
Xu HK: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332
 23.
Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003,116(3):659–678. 10.1023/A:1023073621589
 24.
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.
 25.
Marino G, Xu HK: 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.2005.05.028
 26.
Moudafi A: Viscosity approximation methods for fixedpoints problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615
 27.
Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S00029939195300548463
 28.
Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 2004,20(1):103–120. 10.1088/02665611/20/1/006
 29.
Tan KK, 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.1993.1309
 30.
Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119
 31.
Xu HK: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332
 32.
Zeng LC: 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.1998.6053
 33.
Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 1979,67(2):274–276. 10.1016/0022247X(79)900246
 34.
Genel A, Lindenstrauss J: An example concerning fixed points. Israel Journal of Mathematics 1975,22(1):81–86. 10.1007/BF02757276
 35.
Acedo GL, Xu HK: Iterative methods for strict pseudocontractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2007,67(7):2258–2271. 10.1016/j.na.2006.08.036
 36.
Kim TH, Xu HK: Strong convergence of modified Mann iterations. Nonlinear Analysis: Theory, Methods & Applications 2005,61(1–2):51–60. 10.1016/j.na.2004.11.011
 37.
MartinezYanes C, Xu HK: 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.2005.08.018
 38.
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/S0022247X(02)004584
 39.
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.2006.06.067
 40.
Zhou H: Convergence theorems of fixed points for strict pseudocontractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(2):456–462. 10.1016/j.na.2007.05.032
 41.
Yao Y, Chen R, Yao JC: Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Analysis: Theory, Methods & Applications 2008,68(6):1687–1693. 10.1016/j.na.2007.01.009
 42.
Marino G, Xu HK: Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,329(1):336–346. 10.1016/j.jmaa.2006.06.055
 43.
Marino G, Colao V, Qin X, Kang SM: Strong convergence of the modified Mann iterative method for strict pseudocontractions. Computers & Mathematics with Applications 2009,57(3):455–465. 10.1016/j.camwa.2008.10.073
 44.
Liu Y: A general iterative method for equilibrium problems and strict pseudocontractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(10):4852–4861. 10.1016/j.na.2009.03.060
 45.
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 1976,14(5):877–898. 10.1137/0314056
 46.
Xu HK: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
 47.
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for oneparameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017
 48.
Peng JW, Yao JC: A new hybridextragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2008,12(6):1401–1432.
 49.
Martinet B: Perturbation des méthodes d'optimisation. Applications. RAIRO Analyse Numérique 1978,12(2):153–171.
 50.
Ferris MC: Finite termination of the proximal point algorithm. Mathematical Programming 1991,50(3):359–366. 10.1007/BF01594944
Acknowledgments
R. Wangkeeree would like to thank The National Research Council of Thailand, Grant SCAR012/2552 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.
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Keywords
 Variational Inequality
 Nonexpansive Mapping
 Iterative Scheme
 Real Hilbert Space
 Variational Inequality Problem