Research Article | Open Access

# Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings

**Academic Editor:**Yansheng Liu

#### Abstract

We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors.

#### 1. Introduction

Throughout this paper, we assume that is a real Hilbert space with inner product and norm are denoted by and , respectively. Let denote the family of all subsets of , and let be a closed-convex subset of . Recall that a mapping is said to be a -*strict pseudocontraction* [1] if there exists such that
where denotes the identity operator on . When , is said to be *nonexpansive* [2] if
And when , is said to be *pseudocontraction* if
Clearly, the class of -strict pseudocontraction falls into the one between classes of nonexpansive mappings and pseudocontraction mapping. We denote the set of fixed points of by .

A family of mappings of into itself is called a *nonexpansive semigroup* on if it satisfies the following conditions: (i) for all ,(ii) for all ,(iii) for all and ,(iv)for all is continuous.

We denote by the set of all common fixed points of , that is, . It is known that is closed and convex.

Let be a single-valued nonlinear mapping, and let be a set-valued mapping. We consider the following *variational inclusion problem*, which is to find a point such that
where is the zero vector in *H*. The set of solutions of problem (1.4) is denoted by .

Let the set-valued mapping be a maximal monotone. We define the *resolvent operator * associate with and as follows:
where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse-strongly monotone ([3, 4]).

Let be a bifunction of into , where is the set of real numbers, let be a mapping, and let be a real-valued function. The *generalized mixed equilibrium problem* is for finding such that
The set of solutions of (1.6) is denoted by , that is,
If , then the problem (1.6) is reduced into the mixed equilibrium problem for finding such that
The set of solutions of (1.8) is denoted by . The (generalized) mixed equilibrium problems include fixed-point problems, variational inequality problems, optimization problems, Nash equilibrium problems, noncooperative games, economics, and the equilibrium problem as special cases ([5â€“15]). In the last two decades, many papers have appeared in the literature on the existence of solutions of equilibrium problems; see, for example, [9] and references therein. Some solution methods have been proposed to solve the mixed equilibrium problems; see, for example, ([7â€“10, 12â€“20]) and references therein.

In 2006, Martinez-Yanes and Xu [21] introduced the following iterative: where is a nonexpansive mapping in a Hilbert space , and is metric projection of onto a closed and convex subset of . They proved that if the sequence of parameters satisfies appropriate conditions, then the sequence converges strongly to .

In 2008, Zhang et al. [4] introduced an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with a multivalued maximal monotone mapping and an inverse-strongly monotone mapping and the set of fixed points of nonexpansive mapping in Hilbert spaces. The following iterative scheme and for all . They proved the strong convergence theorem under some mind conditions.

Recently, Shehu [19] introduced a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of -strictly pseudocontractive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in Hilbert spaces. Starting with an arbitrary , and define sequence , and as follows: where is a -strictly pseudocontractive mapping and for some , is -inverse-strongly monotone mapping of into . He proved that if the sequence , and of parameters satisfies appropriate conditions, then generated by (1.11) converges strongly to .

In this paper, motivated by the above results, we present a new general iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Martinez-Yanes and Xu [21], Shehu [19], Zhang et al. [4], and many authors.

#### 2. Preliminaries

Let be a real Hilbert space with norm and inner product , and let be a closed-convex subset of . When is a sequence in , means that converges weakly to , and means that converges strongly to . In a real Hilbert space , we have
and . 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
Moreover, is characterized by the following properties: and
Recall that a mapping of into itself is called -*inverse-strongly monotone* if there exists a positive real number such that
It is obvious that any -inverse-strongly monotone mapping is -Lipschitz monotone and continuous mapping.

In order to prove our main results, we need the following Lemmas.

Lemma 2.1 (see [22]). * Let be a -strict pseudocontraction, then*

(1)*the fixed-point set of is closed convex, so that the projection is well defined;*(2)*define a mapping by**
If , then is a nonexpansive mapping such that .*

A family of mappings is called *a family of uniformly **-strict pseudocontractions* if there exists a constant such that
Let be a countable family of uniformly -strict pseudocontractions. Let be the sequence of nonexpansive mappings defined by (2.8), that is,

Let be a sequence of nonexpansive mappings of into itself defined by (2.10), and let be a sequence of nonnegative numbers in . For each , define a mapping of into itself as follows: Such a mapping is nonexpansive from to and it is called the -mapping generated by and .

For each , let the mapping be defined by (2.11), then we can have the following crucial conclusions concerning . You can find them in [23]. Now, we only need the following similar version in Hilbert spaces.

Lemma 2.2 (see [23]). * Let be a nonempty closed-convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for every , then*

(1)* is nonexpansive and , ,*(2)*for every and , the limit exists,*(3)*a mapping defined by**
is a nonexpansive mapping satisfying , and it is called the -mapping generated by and .*

Lemma 2.3 (see [24]). * Let be a nonempty closed-convex subset of a Hilbert space , let be a countable family of nonexpansive mappings with , and let be a real sequence such that . If is any bounded subset of , then
*

Lemma 2.4 (see [25]). *Each Hilbert space satisfies Opials condition, that is, for any sequence with , the inequality
**
holds for each with .*

Lemma 2.5 (see [3]). *Let be a maximal monotone mapping, and let be a monotone mapping, then the mapping is a maximal monotone mapping.*

*Remark 2.6. * Lemma 2.5 implies that is closed and convex if is a maximal monotone mapping and is a monotone mapping.

Lemma 2.7 (see [4]). * Let be a solution of variational inclusion (1.4) if and only if , that is,
*

Lemma 2.8 (see [20]). *Let be a nonempty bounded closed-convex subset of a Hilbert space , and let be a nonexpansive semigroup on , then for any ,
*

Lemma 2.9 (see [26]). * Let C be a nonempty bounded closed-convex subset of H, let be a sequence in C, and let be a nonexpansive semigroup on C. If the following conditions are satisfied: *

(1)*,*(2)*. **For solving the generalized mixed equilibrium problem for , one gives the following assumptions for the bifunction F, and the set C:*

(A1)* for all ,*(A2)* is monotone, that is, for all ,*(A3)*for each ,*(A4)*for each is convex and lower semicontinuous, *(A5)*for each is weakly upper semicontinuous,*(B1)*for each and , there exist a bounded subset and such that for any ,
*(B2)*C is a bounded set, **then one has the following lemma.*

Lemma 2.10 (see [18]). *Let C be a nonempty closed-convex subset of H. Let be a bifunction that satisfies (A1)â€“(A5), 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 hold:*(1)*for each ,*(2)* is single valued,*(3)* is firmly nonexpansive, that is, for any ,*(4)*,*(5)* is closed and convex.*

#### 3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.

Theorem 3.1. * Let be a nonempty closed-convex subset of a real Hilbert Space . Let be bifunctions of into real numbers satisfying (A1)â€“(A5), and let be proper lower semicontinuous and convex functions with assumption (B1) or (B2). Let be -inverse-strongly monotone mappings of into , respectively, and let be maximal monotone mappings. Let be a nonexpansive semigroup on , and let be a positive real divergent sequence. Let be a countable family of uniformly -strict pseudocontractions, let be a countable family of nonexpansive mappings defined by , and let be the -mapping defined by (2.11) and a mapping defined by (2.12) with . Suppose that . Let be a sequence generated by , and
**
for every , where , , and satisfy the following conditions:*(i)*,*(ii)*,*(iii)*,
*(iv)*,*(v)*,
**then converges strongly to .*

*Proof. *First, we show that and are nonexpansive. Indeed, for all and , we obtain
which implies that the mapping is nonexpansive, so is . Let . We observe that
Since both and are nonexpansive for each , let , then and ; by conditions (i) and (ii), we have
Therefore, we get
Next, we will divide the proof into five steps.*Step 1. *We show that is well defined. Let , then is closed and convex for each . Suppose that is closed convex for some , then, from the definition of , we know that is closed convex for the same . Hence, is closed convex for and for each . This implies that is closed convex for . Furthermore, we show that . For . For , let , then
which shows that . Thus, . Hence, it follows that . This implies that is well defined.*Step 2. *We claim that and , for . Since and , we have
Also, as by (2.1), it follows that
Form (3.7) and (3.8), we have that exists. Hence, is bounded and so are , , , , , , , , , , and . For , we have that . By (2.5), we obtain
Letting and taking the limit in (3.9), we have , which shows that is Cauchy. In particular,
Since is Cauchy, we assume that . Since , then
and it follows that
Therefore,
*Step 3. *We claim that the following statements hold: (1),(2),(3),(4).

For , from (3.4), and (3.6), we obtain
Since , we have
Hence, by condition (iii) and (3.13), we have
From (3.6), we have
On the other hand,
and hence,
Putting (3.19) into (3.17), for , we have
It follows that
Therefore, from condition (iii), (3.13), and (3.16), we have
Furthermore, from (3.4), and (3.6), we get
Since , we have
Then, by condition (iii) and (3.13), we obtain that
From (3.6), we have
On the other hand, we note that
and hence,
Putting (3.28) into (3.26), we have
It follows that
Therefore, by condition (iii), (3.13), and (3.25), we have
Condition (iii) implies that
It follows that
From (3.6), we have
Since , we have
Then, by condition (iii) and (3.13), we obtain that
From (3.6), we have
On the other hand, we note that
and hence,
Putting (3.39) into (3.37),
this implies that
Therefore, by condition (iii), (3.13), and (3.36), we have
Furthermore, from (3.6), we have
Since , we have
Then, by condition (iii) and (3.13), we obtain that
From (3.6), we have