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:

(1.1)

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:

(1.2)

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:

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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:

(1.7)

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

(1.8)

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

(1.9)

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

(1.10)

which is the optimality condition for the minimization problem

(1.11)

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

(1.12)

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:

(1.13)

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:

(1.14)

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:

(1.15)

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:

(1.16)

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.

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (iv)

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

  5. (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 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

(2.1)

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

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

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

(2.6)

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

(2.7)

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 self-mapping of

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

(2.8)

(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 ,

(2.9)

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:

(2.10)

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 -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

(D1)

(D2)

(D3) for all and ;

(D4) for some with and ;

(D5).

Let and be sequences generated by

(3.1)

Then and converge strongly to a point which is the unique solution of the variational inequality

(3.2)

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

(3.3)

Since is complete, there exists a unique element such that On the other hand, since is a linear bounded self-adjoint operator, one has

(3.4)

Observing that

(3.5)

we obtain is positive. It follows that

(3.6)

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

Step 1.

First we prove that is nonexpansive. For all and

(3.7)

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

(3.8)

From (2.1), we have

(3.9)

so, by (3.9) and the -strict pseudocontractivity of , it follows that

(3.10)

that is,

(3.11)

Observe that

(3.12)

From (3.8), (3.11) and the last inequality, we have

(3.13)

It follows that

(3.14)

By simple induction, we have

(3.15)

which gives that the sequence is bounded, so are and

Step 3.

Next we claim that

(3.16)

Notice that

(3.17)

Next, we define

(3.18)

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

Observing that

(3.19)

we have

(3.20)

where is an appropriate constant such that . Substituting (3.20) into (3.17), we obtain

(3.21)

On the other hand, from and we note that

(3.22)
(3.23)

Putting in (3.22) and in (3.23), we have

(3.24)

So, from (A2) we have

(3.25)

and hence

(3.26)

Without loss of generality, let us assume that there exists a real number such that for all Then, we have

(3.27)

and hence

(3.28)

where . It follows from (3.21) and the last inequality that

(3.29)

where .

Define a sequence such that

(3.30)

Then, we have

(3.31)

It follows from (3.29) that

(3.32)

Observing the conditions (D1), (D3), (D4), (D5), and taking the superior limit as , we get

(3.33)

We can obtain easily by Lemma 2.2. Observing that

(3.34)

we obtain

(3.35)

Hence (3.16) is proved.

Step 4.

Next we prove that

(3.36)
  1. (a)

    First we prove that . Observing that

(3.37)

we arrive at

(3.38)

which implies that

(3.39)

Therefore, it follows from (3.16), (D1), and (D2) that

(3.40)
  1. (b)

    Next, we will show that for any Observe that

(3.41)

where

(3.42)

This implies that

(3.43)

It is easy to see that and then from (3.16), we obtain

(3.44)
  1. (c)

    Next we prove that . From (2.3), we have

(3.45)

so, we obtain

(3.46)

It follows that

(3.47)

which implies that

(3.48)

Applying (3.16), (3.44), , and to the last inequality, we obtain that

(3.49)

It follows from (3.40) and (3.49) that

(3.50)

Then it follows from (D1), (3.49) and (3.50) that

(3.51)

For any , we have from Lemma 2.7,

(3.52)

Hence

(3.53)

From (3.41) we observe that

(3.54)

Hence

(3.55)

Using (D1), (D2) and (3.16), we obtain

(3.56)
  1. (d)

    Next we prove that . Using Lemma 2.3 (i), we have

(3.57)

which implies that

(3.58)

By (3.16), (3.51), and (3.56), we have

(3.59)

Observing that

(3.60)

Using (3.40) and the last inequality, we obtain that

(3.61)

From Lemma 2.3(i), (3.59), and (3.61), we have

(3.62)

Hence (3.36) is proved.

Step 5.

We claim that

(3.63)

We choose a subsequence of such that

(3.64)

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):

(3.65)

For any given , hence . Since we have

(3.66)

On the other hand, from , we have

(3.67)

that is,

(3.68)

Therefore, we obtian

(3.69)

Noting that as and is Lipschitz continuous, hence from (3.69), we obtain

(3.70)

Since is maximal monotone, we have , and hence .

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

(3.71)

From (A2), we also have

(3.72)

and hence

(3.73)

From and we get . It follows from (A4), , and the lower semicontinuous of that

(3.74)

For with and let Since and we have and hence So, from (A1) and (A4) and the convexity of , we have

(3.75)

Dividing by , we have

(3.76)

Letting , it follows from the weakly semicontinuity of that

(3.77)

Hence . Therefore, the conclusion is proved.

Consequently

(3.78)

as required. This together with (3.40) implies that

(3.79)

Step 6.

Finally, we show that . Indeed, we note that

(3.80)

Since , and are bounded, we can take a constant such that

(3.81)

for all . It then follows that

(3.82)

where

(3.83)

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

(D1)

(D2)

(D3) for all and

(D4) for some with and

(D5).

Let and be sequences generated by

(3.84)

Then and converge strongly to a point which is the unique solution of the variational inequality

(3.85)

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 -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

(D1)

(D2)

(D3) for all and

(D4) for some with and .

Let and be sequences generated by

(3.86)

Then and converge strongly to a point which is the unique solution of the variational inequality

(3.87)

Equivalently, one has

Remark 3.4.

  1. (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].

  2. (ii)

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

  3. (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

(D1)

(D2)

(D3).

Let and be sequences generated by

(3.88)

Then and converge strongly to a point which is the unique solution of the variational inequality

(3.89)

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:

(4.1)

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

(4.2)

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

(4.3)

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

(D1)

(D2)

(D3) for all and ;

(D4) for some with and ;

(D5).

Let and be sequences generated by

(4.4)

where is a positive constant such that Then both and converge strongly to a point which is the unique solution of the variational inequality

(4.5)

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 -inverse-strongly monotone mapping with . In fact, from Lemma 2.5, we have

(4.6)

That is

(4.7)

On the other hand

(4.8)

Hence we have

(4.9)

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

(D1)

(D2)

(D3) and for all and ;

(D4) for some with and ;

(D5).

Let and be sequences generated by

(4.10)

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

(4.11)

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

(4.12)

The conclusion can be obtained from Theorem 4.1.