#### Abstract

Two countable families of hemirelatively nonexpansive mappings are considered based on a hybrid projection algorithm. Strong convergence theorems of iterative sequences are obtained in an uniformly convex and uniformly smooth Banach space. As applications, convex feasibility problems, equilibrium problems, variational inequality problems, and zeros of maximal monotone operators are studied.

#### 1. Introduction

Throughout this paper, we always assume that is a real Banach space, is the dual space of , is a nonempty closed convex subset of and is the pairing between , and . We denote by and the sets of positive integers and real numbers, respectively.

Let be a bifunction and a nonlinear mapping. The “so-called” generalized mixed equilibrium problem is to find such that

The set of solutions to (1) is denoted by , that is,

##### 1.1. Analysis of Special Cases

(1) If , the problem (1) reduces to the generalized equilibrium problem, which is to find such that The set of solutions to (3) is denoted by .

(2) If , the problem (1) reduces to the mixed equilibrium problem, which is to find such that The set of solutions to (4) is denoted by .

(3) If , the problem (1) reduces to the mixed variational inequality of Browder type, which is to find such that The set of solutions to (5) is denoted by .

(4) If in (3), the problem (3) reduces to the classic variational inequality, which is to find such that which is called the Hartmann-Stampacchia variational inequality. The set of solutions to (6) is denoted by .

(5) If in (3), the problem (3) reduces to the classic equilibrium problem, which is to find such that The set of solutions to (7) is denoted by . Given a mapping , let for all . Then if and only if for all ; that is, is a solution of the variational inequality.

(6) If in (4), the problem (4) reduces to the minimize problem, which is to find such that The set of solutions to (8) is denoted by .

The problem (1) is very general in the sense that it includes, as special case, optimization problems, variational inequalities, minimax problems, monotone inclusion problems, saddle point problems, vector equilibrium problems, and the Nash equilibrium problem in noncooperative games. Numerous problems in physics, optimization, and economics reduce to finding a solution of some special case or the problem (1). Some solution methods have been proposed to solve the problems (1), (3)–(8) in Hilbert spaces and Banach spaces; see, for example, [1–7] and references therein.

A Banach space is said to be strictly convex if for all with and . Let be the unit sphere of , and define by for and . A Banach space is said to be smooth if the limit exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for .

The modulus of convexity of is the function defined by A Banach space is uniformly convex if and only if for all . Let be a fixed real number with . A Banach space is said to be -uniformly convex if there exists a constant such that for all . Observe that every -uniformly convex is uniformly convex. One should note that no Banach space is -uniformly convex for . It is well known that or is -uniformly convex if and 2-uniformly convex if ; see [8] for more details.

For each , the generalized duality mapping is defined by In particular, if , is called the normalized duality mapping. If is a Hilbert space, then , where is the identity mapping. In this paper, We denote by the normalized duality mapping. It is known that the duality mapping has the following properties:(i)if is smooth, then is single valued; (ii)if is strictly convex, then is one to one;(iii)if is reflexive, then is surjective;(iv)if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of ;(v)if is uniformly convex, then is uniformly continuous on bounded subsets of and is single valued and also one to one (see [9–12]).

Let be a smooth Banach space. Consider the function defined by It is obvious from the definition of the function that We also know that if and only if (see [13]). Moreover, if is a Hilbert space, (12) reduces to , for any .

Let be a closed convex subset of , and let be a mapping from into itself. We denote by the set of fixed points of . A point in is said to be an asymptotic fixed point of [14] if contains a sequence which converges weakly to such that the strong . The set of asymptotic fixed points of will be denoted by . A point in is said to be a strong asymptotic fixed point of [14] if contains a sequence which converges strong to such that . The set of strong asymptotic fixed points of will be denoted by .

Let be a mapping, and recall the following definition:(a) is called nonexpansive if (b) is called relatively nonexpansive if and (c)a mapping is said to be weak relatively nonexpansive if and (d)a mapping is called hemirelatively nonexpansive if and

*Remark 1. *From the definitions, it is obvious that a relatively nonexpansive mapping is a weak relatively nonexpansive mapping, and a weak relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, but the converse is not true.

Next, we give an example which is a closed hemirelatively nonexpansive mapping.

*Example 2. *Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space onto a nonempty closed convex subset . Then is a relatively nonexpansive mapping, and then it is also a closed hemi-relatively nonexpansive mapping.

In 2005, Matsushita and Takahashi [13] obtained strong convergence theorems for a single relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space . To be more precise, they proved the following theorem.

Theorem MT (see Matsushita and Takahashi [13, Theorem 3.1]). *Let be precisely a uniformly convex and uniformly smooth Banach space and a nonempty closed convex subset of , and let be a relatively nonexpansive mapping from into itself, and let be a sequence of real numbers such that and . Suppose that is given by
**
where is the duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .*

Since then, algorithms constructed for solving the same equilibrium problem, variational inequality problems, and fixed point of relatively nonexpansive mappings (or weak relatively nonexpansive mappings or hemi-relatively nonexpanisve mappings) have been further developed by many authors. For a part of works related to these problems, please see [4, 15–18], and for the hybrid algorithm projection methods for these problems, please see [19–44] and the references therein.

Motivated and inspired by the results in the literature, in this paper we focus our attention on finding a common fixed point of two countable families of hemi-relatively nonexpansive mappings (we shall give the definition of a countable family of hemi-relatively nonexpansive mappings in the next section) by using a simple hybrid algorithm. Furthermore, we will give some applications of our main result in equilibrium problems, variational inequality problems, and convex feasibility problems.

#### 2. Preliminaries

Let be a closed convex subset of , and let be a countable family of mappings from into itself. We denote by the set of common fixed points of . That is, , where denote the set of fixed points of , for all .

Recall that is said to be uniformly closed, if , whenever converges strongly to and as (see [45] for more details).

A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by .

A point is said to be a strong asymptotic fixed point of if contains a sequence which converges strongly to such that . The set of strong asymptotic fixed points of will be denoted by .

Using the definition of (strong) asymptotic fixed point of , Su et al. [46] introduced the following definitions.

*Definition 3 (see Su et al. [46]). *Countable family of mappings is said to be countable family of relatively nonexpansive mappings if and

*Definition 4 (see Su et al. [46]). *Countable family of mappings is said to be countable family of weak relatively nonexpansive mappings if and

Now, we introduce the definition of countable family of hemi-relatively nonexpansive mappings which is more general than countable family of relatively nonexpansive mappings and countable family of weak relatively nonexpansive mappings.

*Definition 5. *Countable family of mappings is said to be countable family of hemi-relatively nonexpansive mappings if and

*Remark 6. *From Definitions 3–5, one has the following facts.(1)The definitions of relatively nonexpansive mapping, weak relatively nonexpansive mapping, and hemi-relatively nonexpansive mapping are special cases of Definitions 3, 4, and 5 as for all .(2)Countable family of hemi-relatively nonexpansive mappings, which do not need the restriction (or ), is more general than countable family of relatively nonexpansive mappings (or countable family of weak relatively nonexpansive mappings).

Next we give an example which is a countable family of hemi-relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings.

*Example 7. *Let be any smooth Banach space and any element of . Define a countable family of mappings as follows: for all ,
Then is a countable family of hemi-relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings.

*Proof. *First, it is obvious that has a unique fixed point ; that is, for all . In addition, one easily sees that
This implies that
for all . It follows from the above inequality that
for all and . That is,
for all and . Hence, is a countable family of hemi-relatively nonexpansive mappings. On the other hand, letting
from the definition of , one has
which implies that and as . That is, but , which shows that is not a countable family of relatively nonexpansive mappings.

In what follows, we will need the following lemmas.

Lemma 8 (see Alber [47]). *Let be a convex subset of a smooth real Banach space . Let and . Then if and only if
*

Lemma 9 (see Alber [47]). *Let be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth real Banach space , and let . Then for each ,
*

Lemma 10 (see Kamimura and Takahashi [48]). *Let be a uniformly convex and smooth real Banach space, and let , be two sequences of . If and either or is bounded, then .*

#### 3. Main Results

Now, we give our main results in this paper.

Theorem 11. *Let be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space . Let , be two uniformly closed countable families of hemi-relatively nonexpansive mappings from into itself such that
**
For a point chosen arbitrarily, let be a sequence generated by the following iterative algorithm:
**
where the sequences . Then the sequence converges strongly to a point , where is the generalized projection from onto .*

*Proof. *We first show that is closed and convex. It is obvious that is closed. Since
is convex. Therefore, is closed and convex for all .

Let ; from the definition of and , we have
Hence, we have . This implies that for arbitrary .

Noticing , from Lemma 8, we have
Since for all , we arrive at
From Lemma 9, we have
for each and for all . So the sequence is bounded. On the other hand, noticing that and , we have
This implies that the sequence is nondecreasing. It follows that the limit of exists. By the construction of , we have that for any positive integer . It follows that
Letting in (40), by the existence of the limit of , we have . It follows from Lemma 10 that as . Hence is a Cauchy sequence. Therefore, there exists a point such that as .

Since , we have from the definition of that
From the inequality above, we have
On the other hand, taking in (40), we have
From (42) and (43), we have that

By using Lemma 10, the inequalities (43) and (44) follow that
Respectively, noticing that
It follows from (45) and (46) that
From uniform closedness of , we get . On the other hand, noticing that , we have
It follows from (46) and (47) that
From uniform closedness of , we also have . Therefore, .

Finally, we show that . From , we have
Taking the limit as in (52), we obtain
and hence from Lemma 8. This completes the proof.

*Remark 12. *Theorem 11 improves Theorem 3.15 of Zhang et al. [49] in the following senses:(1)from the class of a countable family of weak relatively nonexpansive mappings to the one of a countable family of hemi-relatively nonexpansive mappings;(2)from a single countable family of mappings to two countable families of mappings.

When in (32), we can obtain the following corollary immediately.

Corollary 13. *Let be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex Banach space . Let be a uniformly closed countable family of hemi-relatively nonexpansive mappings from into itself such that
**
For a point chosen arbitrarily, let be a sequence generated by the following iterative algorithm:
**
Then the sequence converges strongly to a point , where is the generalized projection from onto .*

*Remark 14. *We notice that if is a countable family of weak relatively nonexpansive mappings, Corollary 13 is still held. Therefore, Corollary 13 extends and improves Theorem 3.15 in [49].

#### 4. Applications to Convex Feasibility Problems

In this section, we consider the following convex feasibility problem (CFP): where , and is an intersecting closed convex subset sequence of a Banach space . This problem is a frequently appearing problem in diverse areas of mathematical and physical sciences. There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration [50–53], computer tomography [54], and radiation therapy treatment planning [55]. In computer tomography with limited data, in which an unknown image has to be reconstructed from a priori knowledge and from measured results, each piece of information gives a constraint which in turn gives rise to a convex set to which the unknown image should belong (see [56]).

Using Theorem 11, we discuss the convex feasibility problems as an application.

Theorem 15. *Let be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space . Let , be two countable families of nonempty closed convex subset of such that
**
For a point chosen arbitrarily, let be a sequence generated by the following iterative algorithm:
**
where the sequences . Then the sequence converges strongly to a point , where is the generalized projection from onto .*

*Proof. *From Lemma 9, we easily have that and are two countable families of hemi-relatively nonexpansive mappings. In view of the continuity of and , we have that and are two uniformly closed countable families of hemi-relatively nonexpansive mappings. Thus, by using Theorem 11, we have that the sequence converges strongly to a point . This completes the proof.

If we only consider a countable family of nonempty closed convex subset of , the following corollary can be obtained by using Theorem 15.

Corollary 16. *Let be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space . Let be a countable family of nonempty closed convex subset of such that
**
For a point chosen arbitrarily, let be a sequence generated by the following iterative algorithm:
**
Then the sequence converges strongly to a point , where is the generalized projection from onto .*

*Proof. *Putting for all in algorithm (58), the conclusion can be obtained from Theorem 15 immediately.

#### 5. Applications to Generalized Mixed Equilibrium Problems

In this section, we apply our main results to prove some strong convergence theorems concerning generalized mixed equilibrium problems in a Banach space .

Let be a mapping. First, we recall the following definition:

(I) is called monotone if

(II) is called -inverse strongly monotone if there exists a constant such that We remark here that an -inverse strongly monotone is )-Lipschitz continuous.

For solving the generalized mixed equilibrium problem (1), let us assume that the nonlinear mapping is monotone and continuous, the function is convex and lower semicontinuous, and the bifunction satisfies the following conditions:, for all ; is monotone, that is, , for all ;, for all ; the function is convex and lower semicontinuous for all .

The following result can be found in Blum and Oettli [1].

Lemma 17 (see Blum and Oettli [1]). *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying ()–(), and let and . Then, there exists such that
*

Lemma 18. *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a monotone and continuous mapping, let the function be convex and lower semicontinuous, and let be a bifunction from to satisfying ()–(). Then, satisfies ()–().*

*Proof. *For convenience, we set . So, we only need to prove that satisfies ()–().

(I) We show that , for all . Since satisfies , we have

(II) We show that is monotone; that is, , for all ; since is continuous and monotone, from , we have

(III) We show that , for all ; Since is continuous and is lower semicontinuous, we have

(IV) We show that the function is convex and lower semicontinuous for each .

For each , for all and for all , since satisfies () and is convex, we have
This completes the proof.

Lemma 19 (see Takahashi and Zembayashi [17]). *Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space , and let be a bifunction from to satisfying ()–(). For and , define a mapping as follows:
**
for all . Then, the following properties hold:*(1)* is single valued;*(2)* is a firmly nonexpansive-type mapping; that is, for all ,
*(3)*;*(4)* is closed and convex;*(5)*, for all .*

Lemma 20 (see Zhang et al. [57]). *Let be a -uniformly convex with and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying ()–(). Let be a positive real sequence such that . Then the sequence of mappings is uniformly closed.*

Next, we shall apply Theorem 11 to solve two generalized mixed equilibrium problems. To accomplish this purpose, let be two monotone and continuous mappings, let the function be convex and lower semicontinuous, and let and be a bifunction from to satisfying ()–(). For and , define two mappings as follows:

Theorem 21. *Let be a -uniformly convex with and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be two monotone and continuous mappings, let the function be convex and lower semicontinuous, and let and be a bifunction from to satisfying ()–() such that . For a point chosen arbitrarily, let be a sequence generated by the following iterative algorithm:
**
where , , and . Then the sequence converges strongly to a point , where is the generalized projection from onto .*

*Proof. *From Lemmas 18 and 20, we learn that and are uniformly closed. And by Lemma 19 (5), one can easily get that and are uniformly closed countable families of hemi-relatively nonexpansive mappings. Notice that if is -uniformly convex, it must be uniformly convex. Therefore, by using Theorem 11, we can obtain the conclusion of Theorem 21. This completes the proof.

Theorem 22. *Let be a -uniformly convex with and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a monotone and continuous mappings, let the function be convex and lower semicontinuous and let be a bifunction from to satisfying ()–() such that . For a point chosen arbitrarily, let be a sequence generated by the following iterative algorithm:
**
where and . Then the sequence converges strongly to a point , where is the generalized projection from onto .*

*Proof. *From Lemmas 18 and 20, we learn that is uniformly closed. And by Lemma 19(5), one can easily get that is an uniformly closed countable family of hemi-relatively nonexpansive mappings. Notice that if is -uniformly convex, it must be uniformly convex. Therefore, by using Corollary 13, we can obtain the conclusion of Theorem 21. This completes the proof.

If we let , in (70) and , in (71), the following corollary can be obtained by using Theorem 21.

Corollary 23. *Let be a -uniformly convex with and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying ()–() and a monotone and continuous mapping. Suppose that . For a point chosen arbitrarily, let be a sequence generated by the following iterative algorithm:
**
where , , and . Then the sequence converges strongly to a point , where is the generalized projection from onto .*

*Remark 24. *By analysis of special cases for generalized mixed equilibrium problem, we can obtain the corresponding results based on Theorems 21 and 22 in sequence. Here, we do not itemize these results.

#### 6. Applications to Maximal Monotone Operators

Let be a multivalued operator from to with domain and range . An operator is said to be monotone if

A monotone operator is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. It is well known that if is a maximal monotone operator, then is closed and convex.

The following result is also well known.

Lemma 25 (see Rockafellar [58]). *Let be a reflexive, strictly convex, and smooth Banach space and a monotone operator from to . Then is maximal if and only if for all .*

Let be a reflexive, strictly convex, and smooth Banach space and a maximal monotone operator from to . Using Lemma 25 and the strict convexity of , it follows that, for all and , there exists a unique such that

If , then we can define a single-valued mapping by and such a is called the resolvent of . We know that for all (see [10, 59] for more details).

First, we give an important lemma for this section and remark that the following lemma can be as example of a countable family of hemi-relatively nonexpansive mappings.

Lemma 26. *Let be a strictly convex and uniformly smooth Banach space, let be a maximal monotone operator from to such that is nonempty, and let be a sequence of positive real numbers which is bounded away from such that . Then is a uniformly closed countable family of hemi-relatively nonexpansive mappings.*

*Proof. *One has . Firstly, we show is uniformly closed. Let be a sequence such that and . Since is uniformly norm-to-norm continuous on bounded sets, we obtain
It follows from
and the monotonicity of that
for all and . Letting , one has for all and . Therefore, from the maximality of , one obtains . Hence, is uniformly closed.

In addition, for any and , from the monotonicity of , one has
for all . This implies that is a countable family of hemi-relatively nonexpansive mappings. Hence, is a uniformly closed countable family of hemi-relatively nonexpansive mappings.

We consider the problem of strong convergence concerning maximal monotone operators in a Banach space. Such a problem has been also studied in [4, 13, 49]. Using Theorem 11, we obtain the following result.

Theorem 27. *Let be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space . Let , be two maximal monotone operators from to such that , and let be a sequence of positive real numbers which is bounded away from such that and . For a point chosen arbitrarily, let be a sequence generated by the following iterative algorithm:
**
where the sequences . Then the sequence converges strongly to a point , where is the generalized projection from onto .*

*Proof. *From Lemma 26, we know that and are two uniformly closed countable families of hemi-relatively nonexpansive mappings. Furthermore, applying Theorem 11, one sees that the sequence converges strongly to a point .

#### Acknowledgments

The authors are thankful to an anonymous referee for his useful comments on this paper. This research was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission under the Computational Science and Engineering Research Cluster (CSEC-KMUTT) (Grant Project no. NRU56000508). The first author is supported by the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J13LI51) and the Foundation of Shandong Yingcai University (Grant no. 12YCZDZR03).