#### Abstract

The purpose of this paper is to present a new hybrid block iterative scheme by the generalized - projection method for finding a common element of the fixed point set for a countable family of uniformly quasi--asymptotically nonexpansive mappings and the set of solutions of the system of generalized mixed equilibrium problems in a strictly convex and uniformly smooth Banach space with the Kadec-Klee property. Furthermore, we prove that our new iterative scheme converges strongly to a common element of the aforementioned sets. The results presented in this paper improve and extend important recent results in the literature.

#### 1. Introduction

Let be a Banach space with itβs dual space , and let be a nonempty closed convex subset of . It is well known that the metric projection operator plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming problems, game theory, variational inequality, complementarity problems, and so forth (see, e.g., [1, 2] and the references therein). In 1994, Alber [3] introduced and studied the generalized projections and from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections for approximately solving the variational inequalities and von Neumannβs intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solving the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized -projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by Abler [3] and proved some properties of the generalized -projection operator. In 2009, Fan et al. [5] presented some basic results for the generalized -projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces.

Block iterative method is a method which is often used by many authors to solve the *convex feasibility problem *(CFP) (see, [6, 7], etc.). In 2008, Plubtieng and Ungchittrakool [8] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Later, Saewan and Kumam [9, 10] introduced a new modified block hybrid projection algorithm for finding a common element of the set of solutions of the generalized equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi--asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.

On the other hand, let be a bifunction, let be a real-valued function, and let be a monotone mapping, where is an arbitrary index set. The *system of generalized mixed equilibrium problems* is to find such thatβ
If is a singleton, then problem (1.1) reduces into the *generalized mixed equilibrium problem*, which is to find such that
The set of solutions (1.2) is denoted by , that is,
If , the problem (1.2) reduces into the *mixed equilibrium problem for *, denoted by , which is to find such that
If , the problem (1.2) reduces into the *mixed variational inequality* of Browder type, denoted by , which is to find such that
If and , the problem (1.2) reduces into the *equilibrium problem for *, denoted by , which is to find such that
If , the problem (1.4) reduces into the *minimize problem*, denoted by , which is to find such that
The above formulation (1.5) was shown in [11] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In other words, the is an unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. Some solution methods have been proposed to solve the ; see, for example, [11β24] and references therein.

A point is a *fixed point *of a mapping if , by denote the set of fixed points of ; that is, . Recall that is said to be *nonexpansive* if is said to be *quasi-nonexpansive* if and is said to be *asymptotically nonexpansive* if there exists a sequence with as such that is said to be *asymptotically quasi-nonexpansive* if and there exists a sequence with as such that

Recall that a point in is said to be an *asymptotic fixed point* of [25] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by .

Let be a real Banach space with norm , let be a nonempty closed convex subset of , and let denote the dual of . Let denote the duality pairing of and . If is a Hilbert space, denotes an inner product on . Consider the functional defined by
where is the *normalized duality mapping*.

A mapping from into itself is said to be *relatively nonexpansive* [26β28] if and is said to be *relatively asymptotic nonexpansive* [29] if and there exists a sequence with as such that
The asymptotic behavior of a relatively nonexpansive mapping was studied in [30β32].

is said to be *-nonexpansive* if is said to be *quasi **-nonexpansive* [17, 33, 34] if and is said to be *-asymptotically nonexpansive* if there exists a real sequence with as such that is said to be *quasi **-asymptotically nonexpansive* [34, 35] if and there exists a real sequence with as such that
A mapping is said to be *closed* if for any sequence with and , then .

*Remark 1.1. *It is easy to know that each relatively nonexpansive mapping is closed. The class of quasi--asymptotically nonexpansive mappings contains properly the class of quasi--nonexpansive mappings as a subclass, and the class of quasi--nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true (see for more detail [30β32, 36]).

As well known if is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [1] recently introduced the *generalized projection * is a map that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem
The existence and uniqueness of the operator follows from the properties of the functional and the strict monotonicity of the mapping (see, e.g., [1, 37β40]). It is obvious from the definition of function that
If is a Hilbert space, then and becomes the metric projection of onto .

Next we recall the concept of the generalized -projection operator. Let be a functional defined as follows: where is positive number, and is proper, convex, and lower semicontinuous. From definitions of and , it is easy to see the following properties: (1) is convex and continuous with respect to when is fixed; (2) is convex and lower semicontinuous with respect to when is fixed.

Let be a real Banach space with its dual . Let be a nonempty closed convex subset of . We say that is *generalized **-projection operator* if

In 2005, Matsushita and Takahashi [36] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space E:

They proved that converges strongly to , where is the generalized projection from onto .

Motivated by the results of Takahashi and Zembayashi [41], Cholamjiak and Suantai [12] proved the following strong convergence theorem by the hybrid iterative scheme for approximation of common fixed point of countable families of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space: , , Then, they proved that under certain appropriate conditions imposed on and , the sequence converges strongly to .

In 2010, Li et al. [42] introduced the following hybrid iterative scheme for the approximation of fixed point of relatively nonexpansive mapping using the properties of generalized -projection operator in a uniformly smooth real Banach space which is also uniformly convex: , where is generalized -projection operator. They proved the strong convergence theorem for finding an element in the fixed point set of . We remark here that the results of Li et al. [42] extended and improved on the results of Matsushita and Takahashi [36].

Recently, Shehu [43] introduced a new iterative scheme by hybrid methods and proved strong convergence theorem for the approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a solution to a system of generalized mixed equilibrium problems in a uniformly convex and uniformly smooth Banach space by using the properties of the generalized -projection operator. Chang et al. [44] used the modified block iterative method to propose an iterative algorithm for solving the convex feasibility problems for an infinite family of quasi--asymptotically nonexpansive mappings. Very recently, Kim [45] and Saewan and Kumam [46] considered the shrinking projection methods for asymptotically quasi--nonexpansive mappings in a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property.

In this paper, we introduce a new hybrid block iterative scheme of the generalized -projection operator for finding a common element of the fixed point set of uniformly quasi--asymptotically nonexpansive mappings and the set of solutions of the system of generalized mixed equilibrium problems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Then, we prove that our new iterative scheme converges strongly to a common element of the aforementioned sets. The results presented in this paper improve and extend the results of Shehu [43], Chang et al. [44], Li et al. [42], Takahashi and Zembayashi [41], Cholamjiak and Suantai [12], and many authors.

#### 2. Preliminaries

A Banach space is said to be *strictly convex* if for all with and . Let be the unit sphere of . Then 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 exists uniformly in . Let be a Banach space. The *modulus of smoothness* of is the function defined by . The *modulus of convexity* of is the function defined by . The *normalized duality mapping * is defined by . If is a Hilbert space, then , where is the identity mapping.

*Remark 2.1. *If is a reflexive, strictly convex and smooth Banach space, then for , if and only if . It is sufficient to show that if , then . From (1.12), we have . This implies that . From the definition of , one has . Therefore, we have ; see [38, 40] for more details.

Recall that a Banach space has the Kadec-Klee property [38, 40, 47], if for any sequence and with and , then as . It is well known that if is a uniformly convex Banach space, then has the Kadec-Klee property.

*Remark 2.2. *Let be a Banach space. Then we know that (1)if is an arbitrary Banach space, then is monotone and bounded; (2)if is strictly convex, then is strictly monotone; (3)if is smooth, then is single valued and semicontinuous; (4)if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of ; (5) is reflexive, smooth, and strictly convex, then the normalized duality mapping is single valued, one-to-one, and onto; (6)if is uniformly smooth, then is smooth and reflexive; (7) is uniformly smooth if and only if is uniformly convex; see [38] for more details.

We also need the following lemmas for the proof of our main results.

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

For example, let be a continuous and monotone operator of into and define

Then, satisfies (A1)β(A4). The following result is in Blum and Oettli [11].

Motivated by Combettes and Hirstoaga [13] in a Hilbert space and Takahashi and Zembayashi [48] in a Banach space, Zhang [49] obtained the following lemma.

Lemma 2.3 (Liu et al. [50], Zhang [49, Lemma 1.5]). *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space . Let be a bifunction from to satisfying (A1)β(A4), let be a continuous and monotone mapping, let be a semicontinuous and convex functional, for , and let . Then, there exists such that
**
where . Furthermore, define a mapping as follows:
**
Then the following holds. *(1)* is single-valued. *(2)* is firmly nonexpansive; that is, for all .*(3)*. *(4)* is closed and convex. *(5)* and .*

For the generalized -projection operator, Wu and Huang [4] proved the following basic properties.

Lemma 2.4 (Wu and Huang [4]). *Let be a reflexive Banach space with its dual and let be a nonempty closed convex subset of . The following statements hold. *(1)* is nonempty closed convex subset of for all .*(2)*If is smooth, then for all , if and only if*(3)*If is strictly convex and is positive homogeneous (i.e., for all such that where ), then is single-valued mapping.*

Recently, Fan et al. [5] have shown that the condition which is positive homogeneous and appeared in [5, Lemma 2.1(iii)] can be removed.

Lemma 2.5 (Fan et al. [5]). *Let be a reflexive Banach space with its dual , and let be a nonempty closed convex subset of . If is strictly convex, then is single valued.*

Recall that is single value mapping when is a smooth Banach space. There exists a unique element such that where . This substitution for (1.21) gives

Now we consider the second generalized -projection operator in Banach spaces (see [42]).

*Definition 2.6. *Let be a real smooth Banach space, and let be a nonempty closed convex subset of . We say that is generalized -projection operator if

Lemma 2.7 (Deimling [51]). *Let be a Banach space, and let be a lower semicontinuous convex functional. Then there exist and such that
*

Lemma 2.8 (Li et al. [42]). *Let be a reflexive smooth Banach space, and let be a nonempty closed convex subset of . The following statements hold.*(1)* is nonempty closed convex subset of for all . *(2)*For all , if and only if
*(3)*If is strictly convex, then is single-valued mapping. *

Lemma 2.9 (Li et al. [42]). *Let be a reflexive smooth Banach space and let be a nonempty closed convex subset of , and let , . Then
*

*Remark 2.10. *Let be a uniformly convex and uniformly smooth Banach space, and let for all . Then Lemma 2.9 reduces to the property of the generalized projection operator considered by Alber [1].

Lemma 2.11 (Li et al. [42]). *Let be a Banach space, and let be a proper, convex, and lower semicontinuous mapping with convex domain . If is a sequence in such that and , then .*

Lemma 2.12 (Chang et al. [44]). *Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let be a nonempty closed convex subset of . Let be a closed and quasi--asymptotically nonexpansive mapping with a sequence , . Then is a closed convex subset of .*

Lemma 2.13 (Chang et al. [44]). *Let be a uniformly convex Banach space, let be a positive number, and let be a closed ball of . Then, for any given sequence and for any given sequence of positive number with , there exists a continuous, strictly increasing, and convex function with such that, for any positive integer with ,
*

*Definition 2.14. * (Chang et al. [44]).(1)Let be a sequence of mappings. is said to be a *family of uniformly quasi*-*-asymptotically nonexpansive mappings*, if , and there exists a sequence with such that for each (2)A mapping is said to be *uniformly **-Lipschitz continuous*, if there exists a constant such that

If , it is clearly by the definition of mappings is a family of uniformly quasi--asymptotically nonexpansive is equivalent to if and there exists a sequence with such that for each ,

#### 3. Strong Convergence Theorem

Now we state and prove our main result.

Theorem 3.1. *Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Let be an infinite family of closed uniformly -Lipschitz continuous and uniformly quasi--asymptotically nonexpansive mappings with a sequence , , and let be a convex lower semicontinuous mapping with . For each , let be a bifunction from to which satisfies conditions (A1)β(A4), let be a continuous and monotone mapping, and let be a lower semicontinuous and convex function. Assume that . For an initial point with and , we define the sequence as follows:
**
where , are sequences in , and , for all , satisfy the following conditions.*(i)* for some .*(ii)* for all, and .*(iii)*for all. and . ** Then converges strongly to , where .*

*Proof. *We split the proof into six steps.*Step 1. *We first show that is closed and convex for each .

Clearly is closed and convex. Suppose that is closed and convex for each . Since for any , we know that is equivalent to
it follow that
So, is closed and convex. This implies that is well defined for all .*Step 2. *We show that for all .

We show by induction that for all . It is obvious that . Suppose that for some . Let , by the convexity of , Lemma 2.13, and the uniformly quasi--asymptotically nonexpansive of , we compute
Since , when , , , it follows from (3.4) that
This shows that which implies that , and hence, for all . Since is nonempty, is a nonempty closed convex subset of , and hence, exist for all . This implies that the sequence is well defined.*Step 3. *We show that is bounded.

Since is convex and lower semicontinuous mapping, from Lemma 2.7, we have known that there exist and such that
Since , it follows that
For each and by the definition of that , it follows from (3.7) that
This implies that is bounded and so are .*Step 4. *We show that and .

By the fact that and , followed by Lemma 2.9, we get
This implies that is nondecreasing. So, we obtain that exist, and taking , we obtain that
Since is bounded in and is reflexive, we can assume that . From the fact that when is closed and convex for each , it is easy to see that and we get
Since is convex and lower semicontinuous, we have
By (3.11) and (3.12), we get
That is , by Lemma 2.11, we have , from the Kadec-Klee property of , we obtain that
and we also have
Since and from the definition of , we have
is equivalent to
By (3.10) and in view of , we also have
From (1.20), it follow that
Since , we also have
It follows that
This implies that is bounded in . Note that is reflexive and is also reflexive; we can assume that . In view of the reflexive of , we see that . Hence, there exists such that . It follows that
Taking on both sides of the equality above and in view of the weak lower semicontinuity of norm , it yields that
That is , which implies that . It follows that . From (3.21) and the Kadec-Klee property of ; that, is as , we known that is norm-weak*-continuous, that is, . From (3.20) and the Kadec-Klee property of , we have
Since , it follows that
From that is uniformly norm-to-norm continuous on bounded subsets of , we obtain
*Step 5. *We will show that .

We show that .

For , we note that
It follows from , and as that
For any and any , it follows from (3.4) and (3.5) that
It follows that
is in equivalence to
From (3.28), and , we see that
It follows from the property of that
Since and is uniformly continuous, it yields that . Thus from (3.33), we have
Since is norm-weak*-continuous, we also have
On the other hand, for each , we observe that
In view of (3.34), we obtain for each . Since has the Kadec-Klee property, we get
By the assumption that for each , is uniformly -Lipschitz continuous, so we have
By (3.14), (3.15), and (3.37), it yields that for all. From , we get , that is, . In view of the closeness of , we have , for all . This imply that .

We show that .

Since and from (3.5), we have
is in equivalence to
From (3.10) and , as , we see that
From (1.20), it follows that
Since , we have
It follow that
This implies that is bounded in and is reflexive; we can assume that . In view of . Hence, there exists such that . It follows that
Taking on both sides of the equality above and in view of the weak lower semicontinuous of norm , it yields that
That is , which implies that . It follows that . From (3.44) and the Kadec-Klee property of , that is, as , note that is norm-weak*-continuous, that is, . From (3.43) and the Kadec-Klee property of , we have
For , by nonexpansiveness, we observe that
By Lemma (2.12)(5), we have for
From (3.28), we get , for . From (1.20), it follow that
Since , we also have
Since is bounded and is reflexive, without loss of generality, we may assume that . From the first step, we have known that is closed and convex for each , it is obvious that . Again since
taking on both sides of the equality above, we have
This implies that for all, then it follow that
from (3.51), (3.54) and the Kadec-Klee property, we have
By using the triangle inequality, we obtain
Hence, we obtain that
Since and is uniformly norm-to-norm continuous on bounded subsets, so