Strong Convergence for Mixed Equilibrium Problems of Infinitely Nonexpansive Mappings

Abstract

We introduce a new iterative scheme for finding a common element of infinitely nonexpansive mappings, the set of solutions of a mixed equilibrium problems, and the set of solutions of the variational inequality for an -inverse-strongly monotone mapping in a Hilbert Space. Then, the strong converge theorem is proved under some parameter controlling conditions. The results of this paper extend and improve the results of Jing Zhao and Songnian He(2009)and many others. Using this theorem, we obtain some interesting corollaries.

1. Introduction

Let be a real Hilbert space with norm and inner product . And let be 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.

Find such that

(1.1)

The set of solutions of (1.1) is denoted by It is easy to see that is the solution of problem (1.1) and . In particular, if , the mixed equilibrium problem (1.1) reduced to the equilibrium problem.

Find such that

(1.2)

The set of solutions of (1.2) is denoted by If and for all , where is a mapping from to , then the mixed equilibrium problem (1.1) becomes the following variational inequality.

Find such that

(1.3)

The set of solutions of (1.3) is denoted by .

The variational inequality and the mixed equilibrium problems which include fixed point problems, optimization problems, variational inequality problems have been extensively studied in literature. See, for example, [2–8].

In 1997, Combettes and Hirstoaga [9] introduced an iterative method for finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, Takahashi and Takahashi [7] introduced another iterative scheme for finding a common element of and the set of fixed points of nonexpansive mappings. Furthermore,Yao et al.[8, 10] introduced an iterative scheme for finding a common element of and the set of fixed points of finitely (infinitely) nonexpansive mappings.

Very recently, Ceng and Yao [1] considered a new iterative scheme for finding a common element of and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem.

Now, 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 if there exists a positive real number such that

It is obvious that any -inverse strongly monotone mapping is monotone and Lipscitz. A mapping is called nonexpansive if We denote by the set of fixed point of .

In 2006, Yao and Yao [11] introduced the following iterative scheme.

Let be a closed convex subset of a real Hilbert space. Let be an -inverse strongly monotone mapping of into and let be a nonexpansive mapping of into itself such that . Suppose that and and are given by

(1.4)

where , , and are sequence in and is a sequence in [0,2]. They proved that the sequence defined by (1.4) converges strongly to a common element of under some parameter controlling conditions.

Moreover, Plubtieng and Punpaeng [12] introduced an iterative scheme (1.5) for finding a common element of the set of fixed point of nonexpansive mappings, the set of solutions of an equilibrium problems, and the set of solutions of the variational of inequality problem for an -inverse strongly monotone mapping in a real Hilbert space. Suppose that and , , and are given by

(1.5)

where , , and are sequence in , is a sequence in [0,2], and . Under some parameter controlling conditions, they proved that the sequence defined by (1.5) converges strongly to .

On the other hand, Yao et al. [8] introduced an iterative scheme (1.7) for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of infinitely many nonexpansive mappings in . Let be a sequence of nonexpansive mappings of into itself and let be a sequence of real number in . For each , define a mapping of into itself as follows:

(1.6)

Such a mapping is called the -mapping generated by and . In [8], given arbitrarily, the sequences and are generated by

(1.7)

They proved that under some parameter controlling conditions, generated by (1.7) converges strongly to , where .

Subsequently, Ceng and Yao [13] introduced an iterative scheme by the viscosity approximation method:

(1.8)

where , and are sequence in (0,1) such that . Under some parameter controlling conditions, they proved that the sequence defined by (1.8) converges strongly to , where .

Recently, Zhao and He [14] introduced the following iterative process.

Suppose that = ,

(1.9)

where , , , and such that . Under some parameter controlling conditions, they proved that the sequence defined by (1.9) converges strongly to , where

Motivated by the ongoing research in this field, in this paper we suggest and analyze an iterative scheme for finding a common element of the set of fixed point of infinitely nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational of inequality problem for an -inverse strongly monotone mapping in a real Hilbert space. Under some appropriate conditions imposed on the parameters, we prove another strong convergence theorem and show that the approximate solution converges to a unique solution of some variational inequality which is the optimality condition for the minimization problem. The results of this paper extend and improve the results of Zhao and He [14] and many others. For some related works, we refer the readers to [15–22] and the references therein.

2. Preliminaries

Let be a real Hilbert space and let be a closed convex subset of . Then, for any , there exists a unique nearest point in , denoted by such that

(2.1)

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

(2.2)

Moreover, is characterized by the following properties: and

(2.3)

It is clear that

A space is said to satisfy Opials condition if for each sequence in which converges weakly to a point , we have

(2.4)

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

Lemma 2.1 (see [23]).

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

Lemma 2.2 (see [24]).

Let be a real Hilbert space, let be a closed convex subset of , and let be a nonexpansive mapping with If is a sequence in weakly converging to and if converge strongly to , then .

Lemma 2.3 (see [25]).

Assume that is a sequence of nonnegative real numbers such that

(2.5)

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

and

or.

Then

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

(A1) for all ;

(A2) is monotone, that is, for any ;

(A3) is upper-hemicontinuous, that is, for each ,

(2.6)

(A4) is convex and lower semicontinuous for each ;

(B1) for each and , there exists a bounded subset and such that for any ,

(2.7)

(B2) is a bounded set.

By a similar argument as in the proof of Lemma 2.3 in [26], we have the following result.

Lemma 2.4.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from that satisfies (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:

(2.8)

for all Then, the following conditions hold:

(1)for each ;

(2) is single-valued;

(3) is firmly nonexpansive, that is, for any , ;

(4);

(5) is closed and convex.

Let be a sequence of nonexpansive mappings of into itself, where is a nonempty closed convex subset of a real Hilbert space . Given a sequence in , we define a sequence of self-mappings on by (1.6). Then We have the following result.

Lemma 2.5 (see [27]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, for every and , exists.

Remark 2.6 (see [8]).

It can be shown from Lemma 2.5 that if is a nonempty bounded subset of , then for , there exists such that for all , , where .

Remark 2.7 (see [8]).

Using Lemma 2.5, we define a mapping as follows: , for all . is called the -mapping generated by and

Since is nonexpansive, is also nonexpansive.

Indeed, for all , .

If is a bounded sequence in , then we put . Hence it is clear from Remark 2.6 that for any arbitrary , there exists such that for all ,

This implies that

Lemma 2.8 (see [27]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then

3. Main Results

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a lower semicontinuous and convex function. Let be a bifunction from satisfying (A1)–(A4), let be an -inverse-strongly monotone mapping of into , and let be a sequence of nonexpansive self-mapping on such that . Suppose that , , , and are sequences in , is a sequence in such that for some with , and is a real sequence. Suppose that the following conditions are satisfied:

(i),

(ii) and ,

(iii),

(iv) and ,

(v),

(vi) and .

Let be a contraction of into itself with coefficient . Assume that either (B1) or (B2) holds. Let the sequences , , and be generated by, and

(3.1)

for all , where is defined by (1.6) and is a sequence in , for some . Then the sequence converges strongly to a point , where .

Proof.

For any and , we note that

(3.2)

which implies that is nonexpansive.

Let be a sequence of mappping defined as in Lemma 2.4 and let . Then and . Put . From (3.2) we have

(3.3)

Hence, we obtain that

(3.4)

Therefore is bounded. Consequently, , and are also bounded.

Next, we claim that .

Indeed, setting it follows that

(3.5)

Now, we estimate and .

From the definition of , (1.6), and since , are nonexpansive, we deduce that, for each ,

(3.6)

for some constant such that sup And we note that

(3.7)

(3.8)

where = sup.

Combining (3.7) and (3.8), we obtain

(3.9)

On the other hand, from and , we note that

(3.10)

(3.11)

Putting in (3.10) and in (3.11), we have

(3.12)

So, from (A2) we get .

Hence

Without loss of generality, we may assume that there exists a real number such that , for all . Then we get

(3.13)

and hence

(3.14)

where = sup. Hence from (3.9) and (3.14), we have

(3.15)

Combining (3.5), (3.6), and (3.15), we get

(3.16)

It follows from (3.16) and conditions (i)–(vi) and that

(3.17)

By Lemma 2.1, we have Consequently,

(3.18)

From conditions (iv)–(vi), (3.7), (3.8), (3.14), and (3.18), we also get

(3.19)

Since and from the definitionof , we have . Then we have

(3.20)

For , we have

(3.21)

and hence .

From (3.3), we have

(3.22)

That is,

(3.23)

From (ii) and (3.18), we obtain

(3.24)

From (3.2)-(3.3), we get

(3.25)

Then we get,

(3.26)

Since and , we obtain

(3.27)

We note that

(3.28)

Then we derive

(3.29)

Hence

(3.30)

which imply that

(3.31)

From condition (ii), (3.18), and (3.27), we get

(3.32)

Since

(3.33)

we have

(3.34)

and then we obtain

(3.35)

So we get

(3.36)

From condition (iv) and (3.20), (3.24), and (3.32), we have Moreover, from Remark 2.7 we get

Next, we show that

(3.37)

where . Indeed, we choose a subsequence of such that

(3.38)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that

From , we obtain

Next, we show that .

First, we show that . In fact by , we have

(3.39)

From (A2), we also have

(3.40)

and hence

(3.41)

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

(3.42)

For with and , let Since and , we have and hence . So, from (A1) and (A4), we have

(3.43)

Dividing by , we have

(3.44)

Letting , it follows from the weakly semicontinuity of that

(3.45)

Hence .

Second, we show that . Assume . Since and , by Opial's condition, we have

(3.46)

which derives a contradiction. Thus we have .

Finally, by the same argument in the proof of [28, Theorem ], we can show that

Hence .

Since and , we have

(3.47)

Therefore, (3.37) holds.

Finally, we show that . From definition of , we get

(3.48)

which implies that

(3.49)

By (3.47) and Lemma 2.3, we get that converges strongly to .

This completes the proof.

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

Corollary 3.2 (see [14, Theorem ]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from satisfying (A1)–(A4), let be an -inverse-strongly monotone mapping of into , and let be a sequence of nonexpansive self-mapping on such that . Suppose that , , , , and are sequences in , is a sequence in such that for some with and is a real sequence. Suppose that the following conditions are satisfied:

(i),

(ii) and ,

(iii),

(iv) and ,

(v),

(vi) and .

Let the sequence be generated by,

(3.50)

for all , where is defined by (1.6) and is a sequence in , for some . Then the sequence converges strongly to a point , where .

Setting in Theorem 3.1, we have the following result.

Corollary 3.3.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from satisfying (A1)–(A4), let be an -inverse-strongly monotone mapping of into , and let be a sequence of nonexpansive self-mapping on such that . Suppose that , , , and are sequences in , is a sequence in such that for some with , and is a real sequence. Suppose that the following conditions are satisfied:

(i),

(ii) and ,

(iii),

(iv) and ,

(v),

(vi) and .

Let be a contraction of into itself with coefficient and let the sequence be generated by and

(3.51)

for all , where is defined by (1.6) and is a sequence in , for some . Then the sequence converges strongly to a point , where .

By Theorem 3.1, we obtain some interesting strong convergence theorems.

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

Corollary 3.4.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a lower semicontinuous and convex function. Let be a bifunction from satisfying (A1)–(A4), and let be an -inverse-strongly monotone mapping of into such that . Suppose that , , , and are sequences in , is a sequence in such that for some with and is a real sequence. Suppose that the following conditions are satisfied:

(i),

(ii) and ,

(iii),

and ,

(v),

(vi) and .

Let be a contraction of into itself with coefficient . Assume that either (B1) or (B2) holds. Then the sequences , , and generated by, and

(3.52)

converge strongly to a point , where .

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

Corollary 3.5.

Let be a nonempty closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of into and let be a sequence of nonexpansive self-mapping on such that . Suppose that , , , and are sequences in , is a sequence in such that for some with . Suppose that the following conditions are satisfied:

(i),

(ii) and ,

(iii),

(iv) and ,

(v).

Let be a contraction of into itself with coefficient . Let the sequences and be generated by and

(3.53)

for all , where defined by (1.6) and is a sequence in , for some . Then the sequences and converge strongly to a point , where .