Strong Convergence of an Iterative Method for Equilibrium Problems and Variational Inequality Problems

Let be a real Hilbert space with inner product and norm , respectively. Suppose that is nonempty, closed convex subset of and is a bifunction from to , where is the set of real number. The equilibrium problem is to find a such that

(1.1)

The set of such solutions is denoted by EP( ). Numerous problems in physics, optimization, and economics reduce to find a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problems in Hilbert space, see, for instance, Blum and Oettli [1], Combettes and Hirstoaga [2], and Moudafi [3].

A mapping is called monotone if . is called relaxed -cocoercive, if there exist constants and such that

(1.2)

when , is called -strong monotone; when , is called relaxed -cocoercive. Let be a monotone operator, the variational inequality problem is to find a point , such that

(1.3)

The set of solutions of variational inequality problem is denoted by . The variational inequality problem has been extensively studied in literatures, see, for example, [4, 5] and references therein.

Let be a strong positive bounded linear operator on with coefficient , that is, there exists a constant such that . A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space :

(1.4)

where T is a nonexpansive mapping on and is a point on .

A mapping from into itself is called nonexpansive, if . The set of fixed points of is denoted by . Let be a finite family of nonexpansive mappings and , define the mappings

(1.5)

where for all . Such a mapping is called -mapping generated by and . We know that is nonexpansive and , see [6].

Let be a nonexpansive mapping and is a contractive with coefficient . Marino and Xu [7] considered the following general iterative scheme:

(1.6)

They proved that converges strongly to , where is the metric projection from onto .

By combining equilibrium problems and (1.6), Plutbieng and Pumpaeng [8] proposed the following algorithm:

(1.7)

They proved that if the sequences and satisfy some appropriate conditions, then sequence convergence to the unique solution of the variational inequality

(1.8)

Motivated by [8], Colao et al. [9] introduced an iterative method for equilibrium problem and finite family of nonexpansive mappings

(1.9)

and proved that converges strongly to a point and also solves the variational inequality (1.8). For equilibrium problems, also see [10, 11].

On the other hand, let be a -cocoercive mapping, for finding common element of the solution of variational inequality problems and the set of fixed point of nonexpansive mappings, Takahashi and Toyoda [12] introduced iterative scheme

(1.10)

They proved that converges weakly to . Inspired by (1.10) and [13], Y. Yao and J.-C. Yao [14] given the following iterative process:

(1.11)

and proved that converges strongly to . By combining viscosity approximation method and (1.10), Chen et al. [15] introduced the process

(1.12)

and studied the strong convergence of sequence generated by (1.12). Motivated by (1.6), (1.11), and (1.12), Qin et al. [16] introduced the following general iterative process

(1.13)

and established a strong convergence theorem of to an element of .

The purpose of this paper is to introduce the iterative process: and

(1.14)

where is defined by (1.5), is -cocoercive, and is a bounded linear operator. We should show that the sequences converge strongly to an element of . Our result extends the corresponding results of Qin et al. [16] and Colao et al. [9], and many others.

Let be a real Hilbert space and a nonempty, closed convex subset of . We denote strong convergence of to by and weak convergence by . Let is a mapping such that for every point , there exists a unique satisfying , for all . is called the metric projection of onto . It is known that is a nonexpansive mapping from onto . It is also known that and

(2.1)

(2.2)

Let be a monotone mapping of into , then if and only if . The following result is useful in the rest of this paper.

Lemma 2.1 (see [17]).

Assume is a sequence of nonegative real number such that

(2.3)

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

(1) ,

(2) or .

Then, .

Lemma 2.2 (see [18]).

Let be bounded sequences in Banach space satisfying and . Let be a sequence in with . Then, .

Lemma 2.3.

For all , there holds the inequality

(2.4)

Lemma 2.4 (see [7]).

Assume that is a strong positive linear bounded operator on a Hilbert space with coefficient and . Then .

For solving the equilibrium problem for a bifunction , we assume that satisfies the following conditions:

(A1) for all ;

(A2) is monotone: for all ;

(A3)for all ;

(A4)for all is convex and lower semicontinuous.

The following result is in Blum and Oettli [1].

Lemma 2.5 (see [1]).

Let be a nonempty closed convex subset of a Hilbert space , let be a bifunction from into satisfying (A1)–(A4), let , and let . Then there exists such that

(2.5)

We also know the following lemmas.

Lemma 2.6 (see [19]).

Let be a nonempty closed convex subset of Hilbert space , let be a bifunction from to satisfying (A1)–(A4), let , and let , define a mapping as follows:

(2.6)

for all . Then, the following holds:

(1) is single-valued;

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

(2.7)

(3) ;

(4) is closed and convex.

A monotone operator is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operators. Let be a monotone mapping of into and let be the normal cone for at a point , that is

(2.8)

Define

(2.9)

It is known that in this case is maximal monotone, and if and only if .

Theorem 3.1.

Let be a real Hilbert space and be a nonempty closed convex subset of . a finite family of nonexpansive mappings from into itself and a bifunction satisfying (A1)–(A4). Let be relaxed -cocoercive and -Lipschitzian. Let be an -contraction with and a strong positive linear bounded operator with coefficient , is a constant with . Let sequences , be in and be in , is a constant in . Assume and

(i) , ;

(ii) , ;

(iii) for some with and ;

(iv) .

Then the sequence generated by (1.14) converges strongly to and solves the variational inequality , that is,

(3.1)

Proof.

Without loss of generality, we can assume . Then from Lemma 2.4 we know

(3.2)

Since is relaxed -cocoercive and -Lipschitzian and (iii) holds, we know from [14] that for all and , the following holds:

(3.3)

We divide the proof into several steps.

Step 1.

is bounded.

Take , notice that and form Lemma 2.6(2) that is nonexpansive, we have

(3.4)

Since , we have

(3.5)

Then we have

(3.6)

Thus From (3.5) we have

(3.7)

hence is bounded, so is , .

Step 2.

.

Let , where

(3.8)

Then we have

(3.9)

where

(3.10)

Next we estimate , and . At first

(3.11)

Put , we have

(3.12)

By recursion we get

(3.13)

for some . Similarly, we also get

(3.14)

Since

(3.15)

Put in the first inequality and in the second one, we have

(3.16)

Adding both inequality, by (A2) we have

(3.17)

therefore, we have

(3.18)

which implies that

(3.19)

Hence we have

(3.20)

so, by (3.20) and the property , we arrive at

(3.21)

where

(3.22)

Therefore, by (3.14) and (3.20) we get

(3.23)

Now submitting (3.11), (3.13), and (3.23) into (3.9), we have

(3.24)

Thus conditions (ii), (iii), and (iv) imply that

(3.25)

Then, Lemma 2.2 yields

(3.26)

Step 3.

for .

Note that

(3.27)

(3.28)

hence by , we know

(3.29)

By Lemma 2.3, we have

(3.30)

for some . Submitting (3.28) into (3.30), we have

(3.31)

which implies

(3.32)

hence from conditions (i), (iii), and (3.26), we have

(3.33)

Similarly, submitting (3.29) into (3.30), we also have

(3.34)

Step 4.

.

By (2.2) we have

(3.35)

hence

(3.36)

Submitting (3.36) into (3.30), we have

(3.37)

This implies that

(3.38)

Hence by Step 3 we get

(3.39)

Put , where . We have

(3.40)

Hence

(3.41)

so, we get

(3.42)

Submitting into (3.30), we have

(3.43)

where

(3.44)

which implies

(3.45)

So, we have

(3.46)

which together with (3.39) gives

(3.47)

Since and is firmly nonexpansive, we have

(3.48)

which implies

(3.49)

which together with (3.30) gives

(3.50)

So

(3.51)

Now (3.47) and condition (i) imply that

(3.52)

Since

(3.53)

Then we get

(3.54)

hence

(3.55)

Note that

(3.56)

thus from (3.47)–(3.55), we have

(3.57)

Step 5.

, where is the unique solution of variational inequality

Take a subsequence of , such that

(3.58)

Since is bounded, without loss of generality, we assume itself converges weakly to a point . We should prove .

First, let

(3.59)

with the same argument as used in [14], we can derive , since is maximal monotone, we know .

Next, from (A2), for all we have

(3.60)

in particular

(3.61)

Condition (A4) implies that is weakly semicontinuous, then from (3.52) and let we have

(3.62)

Replacing by with , using (A1) and (A4), we get

(3.63)

Divide by in both side yields , let , by (A3) we conclude . Therefore, .

Finally, from we know that . Assume , that is, . Since Hilbert space satisfies Opial's condition, we have

(3.64)

this is a contradiction, thus , therefore, . So we know

(3.65)

Step 6.

The sequence converges strongly to .

From the definition of and Lemmas 2.3, and 2.4, we have

(3.66)

which implies that

(3.67)

Since from condition (i) we have and

(3.68)

so, by Lemma 2.1, we conclude . This completes the proof.

Putting and for all in Theorem 3.1, we obtain the following corollary.

Corollary 3.2.

Let be a real Hilbert space and be a nonempty closed convex subset of . a finite family of nonexpansive mappings from into itself. Let be relaxed -cocoercive and -Lipschitzian. Let be an -contraction with and a strong positive linear bounded operator with coefficient , be a constant with . Let sequences , in and be a constant in . Assume and

(i) , ;

(ii) , for some with and ;

(iii) .

Then the sequence generated by and

(3.69)

converges strongly to and solves the variational inequality , that is,

(3.70)

Putting and , , in Theorem 3.1, we obtain the following corollary.

Corollary 3.3.

Let be a real Hilbert space and be a nonempty closed convex subset of . a finite family of nonexpansive mappings from into itself and a bifunction satisfying (A1)–(A4). Let be an -contraction with and a strong positive linear bounded operator with coefficient , is a constant with . Let sequences , in and in . Assume and

(i) , ;

(ii) , .

Then the sequence generated by and

(3.71)

converges strongly to and solves the variational inequality , that is,

(3.72)