A New Hybrid Algorithm for Variational Inclusions, Generalized Equilibrium Problems, and a Finite Family of Quasi-Nonexpansive Mappings

Abstract

We proposed in this paper a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasi-nonexpansive mappings, the set of solutions of variational inclusion, and the set of solutions of generalized equilibrium problems. Some strong convergence results were derived by using the concept of -mappings for a finite family of quasi-nonexpansive mappings. Strong convergence results are derived under suitable conditions in Hilbert spaces.

1. Introduction

Let be a real Hilbert space with inner product and inducted norm , and let be a nonempty closed and convex subset of . Then, a mapping is said to be

(1)nonexpansive if , for all ;

(2)quasi-nonexpansive if , for all and ;

(3)-Lipschitzian if there exists a constant such that , for all . We denoted by the set of fixed points of .

In 1953, Mann [1] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping in a Hilbert space :

(1.1)

where the initial point is taken in arbitrarily and is a sequence in .

However, we note that Mann's iteration process (1.1) has only weak convergence, in general; for instance, see [2, 3].

Many authors attempt to modify the process (1.1) so that strong convergence is guaranteed that has recently been made. Nakajo and Takahashi [4] proposed the following modification which is the so-called CQ method and proved the following strong convergence theorem for a nonexpansive mapping in a Hilbert space .

Theorem 1.1 (see [4]).

Let be a nonempty closed convex subset of a Hilbert space and let be a nonexpansive mapping of into itself such that . Suppose that and is given by

(1.2)

where . Then, converges strongly to .

Let be a function and let be a bifunction from to such that , where is the set of real numbers and . The generalized equilibrium problem is to find such that

(1.3)

The set of solutions of (1.3) is denoted by ; see also [5–7].

If is replaced by a real-valued function , problem (1.3) reduces to the following mixed equilibrium problem introduced by Ceng and Yao [8]: find such that

(1.4)

Let , for all . Here denotes the indicator function of the set ; that is, if and otherwise. Then problem (1.3) reduces to the following equilibrium problem: find such that

(1.5)

The set of solutions (1.5) is denoted by . Problem (1.5) includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative games, and the vector optimization problem; see [9–12] and the reference cited therein.

Recently, Tada and Takahashi [13] proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain the following theorem.

Theorem 1.2 (see [13]).

Let be a real Hilbert space, let be a closed convex subset of , let be a bifunction, and let be a nonexpansive mapping such that . For an initial point , let a sequence be generated by

(1.6)

where and . Then, converges strongly to .

Let be a single-valued nonlinear mapping and let be a set-valued mapping. The variational inclusion is to find such that

(1.7)

where is the zero vector in . The set of solutions of problem (1.7) is denoted by . Recall that a mapping is called -inverse strongly monotone if there exists a constant such that

(1.8)

A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if its graph 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 all imply . We define the resolvent operator associated with and as follows:

(1.9)

It is known that the resolvent operator is single-valued, nonexpansive, and 1-inverse strongly monotone; see [14], and that a solution of problem (1.7) is a fixed point of the operator for all ; see also [15]. If , it is easy to see that is a nonexpansive mapping; consequently, is closed and convex.

The equilibrium problems, generalized equilibrium problems, variational inequality problems, and variational inclusions have been intensively studied by many authors; for instance, see [8, 16–43].

Motivated by Tada and Takahashi [13] and Peng et al. [7], we introduce a new approximation scheme for finding a common element of the set of fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings, the set of solutions of a generalized equilibrium problem, and the set of solutions of a variational inclusion with set-valued maximal monotone and inverse strongly monotone mappings in the framework of Hilbert spaces.

2. Preliminaries and Lemmas

Let be a closed convex subset of a real Hilbert space with norm and inner product . For each , there exists a unique nearest point in , denoted by , such that . is called the metric projection of on to . It is also known that for and , is equivalent to for all . Furthermore

(2.1)

for all , ; see also [4, 44]. In a real Hilbert space, we also know that

(2.2)

for all and .

Lemma 2.1 (see [45]).

Let be a nonempty closed convex subset of a Hilbert space . Then for points and a real number , the set

(2.3)

For solving the generalized equilibrium problem, let us give the following assumptions for , and the set :

(A1) for all ;

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

(A3) for each is weakly upper semicontinuous;

(A4) for each is convex;

(A5) for each , is lower semicontinuous;

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

(2.4)

(B2) is a bounded set.

Lemma 2.2 (see [7]).

Let be a nonempty closed convex subset of a real Hilbert . Let be a bifunction from to satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function such that . For and , define a mapping as follows:

(2.5)

Assume that either (B1) or (B2) holds. Then, the following conclusions hold:

(1)for each , ;

(2) is single-valued;

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

(2.6)

(4);

(5) is closed and convex.

Lemma 2.3 (see [14]).

Let be a maximal monotone mapping and let be a Lipshitz continuous mapping. Then the mapping is a maximal monotone mapping.

Lemma 2.4.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a quasi-nonexpansive and -Lipschitz mapping of into itself. Then, is closed and convex.

Proof.

Since is -Lipschitz, it is easy to show that is closed.

Let and where . From (2.2), we have

(2.7)

which implies ; consequently, is convex. This completes the proof.

Lemma 2.5 (see [46]).

In a strictly convex Banach space , if

(2.8)

for all and , then .

In 1999, Atsushiba and Takahashi [47] introduced the concept of the -mapping as follows:

(2.9)

where is a finite mapping of into itself and for all with .

Such a mapping is called the -mapping generated by and ; see also [48–50]. Throughout this paper, we denote .

Next, we prove some useful lemmas concerning the -mapping.

Lemma 2.6.

Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a finite family of quasi-nonexpansive and -Lipschitz mappings of into itself such that and let be real numbers such that for all , , and . Let be the -mapping generated by and . Then, the followings hold:

(i) is quasi-nonexpansive and Lipschitz;

(ii).

Proof.

• (i) For each and , we observe that

  

  (2.10)

Let , then

(2.11)

Hence,

(2.12)

This shows that is a quasi-nonexpansive mapping.

Next, we claim that is a Lipschitz mapping. Note that is -Lipschitz for all . For each , we observe

(2.13)

Let , then

(2.14)

Hence,

(2.15)

Since for all , we get that is a Lipschitz mapping.

• (ii) Since is trivial, it suffices to show that . To end this, let and . Then, we have

  

  (2.16)

This shows that

(2.17)

and hence

(2.18)

Again by (2.16), we see that . Hence

(2.19)

Applying Lemma 2.5 to (2.19), we get that and hence .

Again by (2.16), we have

(2.20)

and hence

(2.21)

From (2.16), we know that . Since , we have

(2.22)

Applying Lemma 2.5 to (2.22), we get that and hence .

By proving in the same manner, we can conclude that and for all . Finally, we also have

(2.23)

which yields that since . Hence .

Lemma 2.7.

Let be a nonempty closed convex subset of a Banach space . Let be a finite family of quasi-nonexpansive and -Lipschitz mappings of into itself and sequences in such that as . Moreover, for every , let and be the -mappings generated by and and and , respectively. Then

(2.24)

Proof.

Let and and be generated by and and and , respectively. Then

(2.25)

Let and . Then

(2.26)

It follows that

(2.27)

Since as , we obtain the result.

3. Strong Convergence Theorems

In this section, we prove a strong convergence theorem which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A5), let be a proper lower semicontinuous and convex function, let be an -inverse strongly monotone mapping, let be a maximal monotone mapping, and let be a finite family of quasi-nonexpansive and -Lipschitz mappings of into itself. Assume that and either (B1) or (B2) holds. Let be the -mapping generated by and . For an initial point with and , let , , , and be sequences generated by

(3.1)

where for some , for some and for some .

Then, , , , and converge strongly to .

Proof.

Since for all , we get that is nonexpansive for all . Hence, is closed and convex. By Lemma 2.2(5), we know that is closed and convex. By Lemma 2.4, we also know that is closed and convex. Hence, is a nonempty closed convex set; consequently, is well defined for every .

Next, we divide the proof into seven steps.

Step 1.

Show that for all .

By Lemma 2.1, we see that is closed and convex for all . Hence is well defined for every , . Let . From and for all , we have

(3.2)

It follows that , and hence for all .

Step 2.

Show that exists.

Since is a nonempty closed convex subset of , there exists a unique element . From , we obtain

(3.3)

Hence is bounded; so are , , and .

Since , we also have

(3.4)

From (3.3) and (3.4), we get that exists.

Step 3.

Show that is a Cauchy sequence.

By the construction of the set , we know that for . From (2.1), it follows that

(3.5)

as . Hence is a Cauchy sequence. By the completeness of and the closeness of , we can assume that .

Step 4.

Show that .

From (3.5), we get

(3.6)

as . Since , we have

(3.7)

as . Hence, as . By the nonexpansiveness of and the inverse strongly monotonicity of , we obtain that

(3.8)

This implies that

(3.9)

It follows from (3.7) that

(3.10)

Since is -inverse strongly monotone, we have

(3.11)

This implies that

(3.12)

It follows that

(3.13)

From (3.7) and (3.10) we get

(3.14)

It follows from (3.7) and (3.14) that

(3.15)

as . Since is firmly nonexpansive and , we have

(3.16)

which implies that

(3.17)

It follows from (3.17) that

(3.18)

which yields that

(3.19)

Hence, from (3.7) and (3.14), we also have

(3.20)

It follows from (3.15) and (3.20) that

(3.21)

By Lemma 2.7, we also get that . From Lemma 2.6(i), we know that is Lipschitz. Since as , it is easy to verify that . Moreover, by Lemma 2.6(ii), we can conclude that .

Step 5.

Show that .

Since , we have

(3.22)

From (A2), we have

(3.23)

It follows from (A5) and the weakly lower semicontinuity of , , and that

(3.24)

Put for all and . Since and , we obtain , and hence . So by (A1), (A4), and the convexity of , we have

(3.25)

Hence,

(3.26)

Letting , it follows from (A3) and the weakly semicontinuity of that

(3.27)

for all . Observe that if , then holds. Hence .

Step 6.

Show that .

First observe that is an -Lipschitz monotone mapping and . From Lemma 2.3, we know that is maximal monotone. Let , that is, . Since , we get , that is,

(3.28)

By the maximal monotonicity of , we have

(3.29)

and so

(3.30)

It follows from , and that

(3.31)

By the maximal monotonicity of , we have ; consequently, .

Step 7.

Show that .

Since and , we obtain

(3.32)

By taking the limit in (3.32), we obtain

(3.33)

This shows that .

From Steps 1–7, we can conclude that , , , and converge strongly to . This completes the proof.

4. Applications

As a direct consequence of Theorem 3.1, we obtain some new and interesting results in a Hilbert space as the following theorems. Recall that is the solution set of the classical variational inequality

(4.1)

Theorem 4.1.

Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A5), let be a proper lower semicontinuous and convex function, let be an -inverse strongly monotone mapping, and let be a finite family of quasi-nonexpansive and -Lipschitz mappings of into itself. Assume that and either (B1) or (B2) holds. Let be the -mapping generated by and . For an initial point with and , let , , , and be sequences generated by

(4.2)

where for some , for some and for some .

Then, , , , and converge strongly to .

Proof.

In Theorem 3.1, take , where is the indicator function of . It is well known that the subdifferential is a maximal monotone operator. Then, problem (1.7) is equivalent to problem (4.1) and the resolvent operator for all . This completes the proof.

Next, we give a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings. In order to do this, let us assume that

(B3) for each and , there exists a bounded subset and such that for any ,

(4.3)

Theorem 4.2.

Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A5), let be an -inverse strongly monotone mapping, let be a maximal monotone mapping, and let be a finite family of quasi-nonexpansive and -Lipschitz mappings of into itself. Assume that and either (B1) or (B3) holds. Let be the -mapping generated by and . For an initial point with and , let , , , and be sequences generated by

(4.4)

where for some , for some , and for some .

Then, , , , and converge strongly to .

Proof.

In Theorem 3.1, take , for all . Then problem (1.3) reduces to the equilibrium problem (1.5).

Remark 4.3.

Theorem 3.1 improves and extends the main results in [4, 13] and the corresponding results.