Research Article | Open Access

# A New Hybrid Projection Algorithm for System of Equilibrium Problems and Variational Inequality Problems and Two Finite Families of Quasi--Nonexpansive Mappings

**Academic Editor:**Yongfu Su

#### Abstract

We introduce a modified Mann’s iterative procedure by using the hybrid projection method for solving the common solution of the system of equilibrium problems for a finite family of bifunctions satisfying certain condition, the common solution of fixed point problems for two finite families of quasi--nonexpansive mappings, and the common solution of variational inequality problems for a finite family of continuous monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by some mild conditions. Our result presented in this paper improves and generalizes some well-known results in the literature.

#### 1. Introduction

Throughout this paper, we denote by and the set of all real numbers and the set of all positive numbers, respectively. We also assume that is a real Banach space and is the dual space of . Let be a nonempty, closed and convex subset of a real Banach space with the dual . We recall the following definitions.

A mapping is said to be *nonexpansive* if

A mapping is said to be *monotone* if for each , such that

A mapping is said to be -*strongly monotone* if there exists a constant such that

A mapping is said to be -*inverse strongly monotone* if there exists a constant such that

If is -inverse strongly monotone, then it is *Lipschitz continuous *with constant , that is, for all , .

Clearly, the class of monotone mappings include the class of -inverse strongly monotone mappings.

Let be a monotone mapping. The variational inequality problem is to find a point such that

The set of the solution of the variational inequality problem is denoted by .

Let be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space , let be a mapping, and be the set of fixed points of .

A point is said to be a* fixed point *of if . The set of the solution of the fixed point of is denoted by .

A point is said to be an *asymptotic fixed point* of if there exists a sequence such that and . We denoted the set of all asymptotic fixed points of by .

A point is said to be a *strong asymptotic fixed point *of if there exists a sequence such that and . We denoted the set of all strong asymptotic fixed points of by .

Let be a bifunction. The equilibrium problem is to find a point such that The set of the solution of equilibrium problem is denoted by . Numerous problems in sciences, mathematics, optimizations, and economics reduced to find a solution of equilibrium problems. The equilibrium problems include variational inequality problems and fixed point problem, and optimization problems as special cases (see, e.g., [1–3]). Recently, many authors have considered the problem for finding the common solution of fixed point problems, the common solution of equilibrium problems, and the common solution of variational inequality problems.

In 1953, Mann [4] introduced the iterative sequence which is defined by
where the initial element is arbitrary, is a nonexpansive mapping, and is the sequence in such that and . The sequence of (7) is generally referred to as the *Mann iteration*.

In 2009, Takahashi and Zembayashi [5] introduced the following iterative scheme by the shrinking projection method, and they proved that converges strongly to , under appropriate conditions.

**Theorem TZ.** *Let ** be a uniformly smooth and uniformly convex real Banach space, and let ** be a nonempty, closed and convex subset of **. Let ** be a bifunction from ** to ** satisfying (A1)–(A4) and let ** be a relatively nonexpansive mapping from ** into itself such that **. Let ** be a sequence generated by**for every *,* where ** is the duality mapping on **, the sequence ** satisfies **, and ** satisfies **. Then, the sequence ** converges strongly to **, where ** is the generalized projection of ** onto **. *

In 2009, Qin et al. [6] extended the iterative process (8) from a single relatively nonexpansive mapping to two relatively quasi-nonexpansive mappings. In 2011, Zegeye and Shahzad [7] introduced an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems, and common solutions of the finite family of variational inequality problems for monotone mappings in Banach spaces.

**Theorem ZS.** *Let ** be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space ** with the dual **. Let **, **, be a finite family of bifunctions. Let **, ** be a finite family relatively quasi-nonexpansive mappings and **, ** be a finite family of continuous monotone mappings. *

*For *,
* they define mappings ** by**where **(** D), ** N), **(** L)**, and ** for some **. Assume that **. Let ** be a sequence generated by** where the real numbers ** such that *. * Then, ** converges strongly to an element of **.*

In this paper, motivated and inspired by the previously mentioned above results, we introduce a new iterative procedure for solving the common solution of system of equilibrium problems for a finite family of bifunctions satisfying certain conditions and the common solution of fixed point problems for two countable families of quasi--nonexpansive mappings and the common solution of variational inequality problems for a finite family of monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by the conditions. The results obtained in this paper extend and improve several recent results in this area.

#### 2. Preliminaries

A Banach space is said to be *strictly convex* if for all with and . It is said to be *uniformly convex *if for any two sequences and in such that , and

Let = be the *unit sphere* of . Then the Banach space is said to be *smooth* if
exists for each . It is said to be *uniformly smooth* if the limit (12) is attained uniformly for all .

Let be a Banach space. Then a function is said to be the *modulus of smoothness* of if

The space is said to be *smooth *if , and is said to be *uniformly smooth *if and only if .

The *modulus of convexity* of is the function defined by

A Banach space is said to be *uniformly convex *if and only if for all .

We recall the following definitions.

*Definition 1. *Let be a nonempty set.(1)A mapping is said to be *closed* if for each and imply .(2)A mapping is said to be *quasi*--*nonexpansive *(*relatively quasi-nonexpansive*) if , and
(3)A mapping is said to be *relatively nonexpansive* [8, 9] if , and
(4)A mapping is said to be *weak relatively nonexpansive* [10] if , and

*Remark 2. *
We here the following basic properties.(1)Each relatively nonexpansive mapping is closed.(2)The class of quasi--nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse may be not true.(3)The class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.(4)The class of quasi--nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.(5)If is a real uniformly smooth Banach space, then is uniformly continuous on each bounded subset of .(6)If is a strictly convex reflexive Banach space, then is hemicontinuous, that is, is norm-to- continuous.(7)If is a smooth and strictly convex reflexive Banach space, then is single-valued, one-to-one, and onto.(8)A Banach space is uniformly smooth if and only if is uniformly convex.(9)Each uniformly convex Banach space has the *Kadec-Klee property*, that is, for any sequence , if and , then .(10)A Banach space is strictly convex if and only if is strictly monotone, that is,
(11)Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.(12) is uniformly convex, then is uniformly norm-to-norm continuous on each bounded subset of .

Let be a real Banach space and be a sequence in . We denote by and the strong convergence and weak convergence of , respectively. The *normalized duality mapping * from to is defined by
where denotes the duality pairing. It is well known that if is smooth, then is single-valued and demicontinuous, and if is uniformly smooth, then is uniformly continuous on bounded subset of . Moreover, if is reflexive and strictly convex Banach space with a strictly convex dual, then is single-valued, one-to-one, surjective, and it is the duality mapping from to and so and (see [11, 12]). We note that in a Hilbert space , the mapping is the identity operator.

Now, let be a smooth and strictly convex reflexive Banach space. As Alber (see [13]) and Kamimura and Takahashi (see [14]) did, the *Lyapunov functional * is defined by

It follows from Kohsaka and Takahashi (see [15]) that if and only if and that

Further suppose that is nonempty, closed and convex subset of . The *generalized, projection* (Alber see [13]) is defined by for each ,

*Remark 3. *If is a real Hilbert space , then and (the metric projection of onto ).

Lemma 4 (Alber [13]). *Let be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space , and let . Then
*

Lemma 5 (Kamimura and Takahashi [14]). *Let be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space , and let and . Then,
*

Lemma 6 (Qin et al. [6] and Kohsaka and Takahashi [16]). *Let be a smooth, strictly convex and reflexive Banach space, and is a continuous monotone mapping with . Then, it is proved in [16] that the resolvent , for is quasi--nonexpansive. Moreover, if is quasi--nonexpansive, then using the definition of one can show that is closed and convex (see [6]). *

Lemma 7 (Kamimura and Takahashi [14]). *Let be a uniformly convex and smooth real Banach space and let and be two sequences of . If and either or is bounded, then .*

Lemma 8. *Let be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let and be two sequences of , and . If and , then .*

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions: (A1), for all ; (A2) is monotone, that is, ;(A3); (A4) for any , is convex and lower semicontinuous.

Lemma 9 (Blum and Oettli [1]). *Let be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space and let be a bifunction satisfying the following conditions (A1)–(A4). Let be any given number and be any point. Then, there exists a point such that
*

Lemma 10 (Takahashi and Zembayashi [5]). *Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space , and be a bifunction satisfying the following conditions (A1)–(A4). Let be any given number and be any point defined a mapping as follows. Then, there exists a point such that
*

*Then, the following conclusions hold:*(1)

*is single-valued*(2)

*;**is a firmly nonexpansive type mapping, that is,*(3)

*(4)*

*;**is a closed and convex subset of*(5)

*;**.*

Lemma 11 (Zegeye and Shahzad [7]). *Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space , and be a continuous monotone mapping. Let be any given number and be any point defined a mapping as follows:
**
Then, the following conclusions hold:*(1)* is single valued ; *(2)

*is a firmly nonexpansive type mapping, that is,*(3)

*(4)*

*;**is a closed and convex subset of*(5)

*;**.*

Lemma 12 (Xu [17]). *Let be a uniformly convex Banach space, let be a closed ball of , where . Then, there exists a continuous, strictly increasing, and convex function with such that
**
for all and all with .*

#### 3. Main Results

In this section, we prove a strong convergence theorem which solves the problem for finding a common solution of the system of equilibrium problems and variational inequality problems and fixed point problems in Banach spaces.

Theorem 13. *Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space which has the Kadec-Klee property. Suppose that*(1)* is a finite family of bifunctions satisfying conditions (A1)–(A4), where ;*(2)* and are two finite families of quasi--nonexpansive mappings from into , where ;*(3)* is a finite family of continuous monotone mappings, where ;*(4)*For , one defines the mappings by
**Assume that is a nonempty and bounded in . Let be a sequence generated by
**
where , , , , for some and and are real numbers in such that .**
Then, the sequence converges strongly to . *

*Proof. *We will complete this proof by seven steps below.*Step 1. We will show that ** is closed and convex for each **.*

From the definition of , it is obvious that is closed. Moreover, since
It follows that is convex for each . Therefore, is closed and convex for each .*Step 2. We will show that ** for each **. *

From the assumption, we see that . Suppose that for some . Now, for , Since and are quasi--nonexpansive and by Lemmas 10 and 11, we compute
Therefore, . By the induction, this implies that for each .

Hence, the sequence is well defined.*Step 3. We will show that the sequence** is bounded. *

Let . From for each and is well defined. From the assumption of , we see that is closed and convex subset of .

Let , where is the unique element that satisfies . Now, we will show that as .

From the assumption of , we know that and since and . Using Lemma 4, we get
for all , where . Then, the sequence is bounded. Hence, the sequence is also bounded.*Step 4. We will show that there exists ** such that *, *as **. *

Since and , we have
Therefore, the sequence is nondecreasing. Hence exists. By the definition of , one has that and for any positive integer .

It follows that
Since exists, by taking in (37), we have .

From Lemma 7, it follows that as . Thus is a Cauchy sequence.

Without a loss of generalization, we can assume that . Since is bounded and is reflexive. Since is closed and convex, it follows that , . Moreover, by using the weak lower semicontinuous of the norm on and (35), we obtain
This implies that
By using Lemma 5, we have and hence, . By the definition of , we get
Therefore, . Since , by the Kadec-Klee property of , we obtain
From is uniformly continuous, we also have
*Step 5. We will show that **, ** and ** as **. *

Since , we have , as . Thus, from (21), we obtain
and so
This implies that is bounded. Note that reflexivity of implies reflexivity of . Thus, we assume that . Furthermore, the reflexivity of implies there exists such that . Then, it follows that
Taking the on both sides of (45) and using weak lower semicontinuous of norm to get that
Thus , and so . It follows that . Now, from (44) and the Kadec-Klee property of , we obtain
Thus the demicontinuity of implies that . Now, from (43) and the fact that has the Kadec-Klee property, we obtain
In the fact that and as , we get
Since , it follows from Lemma 11 that
From (21), we obtain
and so is bounded. Since is reflexive, we assume that . It follows that
Taking the on both sides of (52) and using the continuity of , we get that
This implies that and hence .

Now, from (51) and the Kadec-Klee property of , we obtain
In the fact that , and as , we get
Since , it follows from Lemma 10 that
From (21), we obtain
and so is bounded. Since is reflexive, we assume that . It follows that
Taking the on both sides of (58) and using the continuity of , we get that
This implies that and hence .

Now, from (57) and the Kadec-Klee property of , we obtain
*Step 6. We will show that **. **Substep 1. We will show that *.

From the definition of of algorithm (32), we have
Let be such that , for all . Then from (61), we obtain
and that is
Now, we set , for all and .

Therefore, we get . From (63), it follows that
From the continuity of and (41) and (54), we have , , as , we obtain
Since is a monotone mapping, we also have .

Thus, it follows that
and hence
If , we obtain
This implies that .

Similarly, let be such that , for all .

Then, we have again that .

Continuing in the same way, we obtain that , where .

Hence, .*Substep 2. We will show that **.*

From the definition of of algorithm (32) and (A2), we have
Let be such that , for all . Then from (69), we obtain
From the continuity of and (41) and (60), we have , , as and we obtain
Therefore, .

Now, we set , for all and .

Consequently, we get . And so .

Therefore, from (A1), we obtain
Thus, . From (A3), if , then we get . This implies that .

Similarly, let be such that , for all . Then, we have again that .

Continuing in the same way, we obtain that , where . Hence, .*Substep 3. We will show that *.

From algorithm (32) and Lemma 12, we compute
From (73), we have
From , , and , we obtain
Therefore,
It follows from the property of that
From (42), we have , as . Then,
and so
Moreover, the demicontinuity of implies that as . Thus, the Kadec-Klee property of , we obtain
Let be such that , for all .

Then, from (60), we have , as . It follows from (80) and the closedness of that
This implies that .

Similarly, let be such that , for all . Then, we have again that .

Continuing in the same way, we obtain that , where .

Hence, .*Substep 4. We will show that *.

From (73), we have
From , , and , we obtain
Therefore,
It follows from the property of that
From (42), we have , as . Then,
and so
Moreover, the demicontinuity of implies that as . Thus, the Kadec-Klee property of , we obtain
Let be such that , for all .

Then, from (54), we have , as . It follows from (88) and the closedness of that
This implies that .

Similarly, let be such that , for all . Then, we have again that .

Continuing in the same way, we obtain that , where .

Hence, .

From Substeps (6.1)–(6.4), we can conclude that
*Step 7. Finally, we will show that **.*

From , we have
Taking in (91), one has
Now, we have and by Lemma 5, we get
This completes the proof of Theorem 13.

If we set