#### Abstract

The purpose of this paper is first to introduce the concept of *total quasi*--*asymptotically nonexpansive mapping* which contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme for
finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a countable family of total quasi--asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in an uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

#### 1. Introduction

Throughout this paper, we denote by and the set of all real numbers and all nonnegative real numbers, respectively. We also 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 . In the sequel, we denote the strong convergence and weak convergence of a sequence by and , respectively, and is the normalized duality mapping defined by Let be a proper real-valued function, a nonlinear mapping, and a bifunction. The βso calledβ generalized mixed equilibrium problem for is to find such that We denote the set of solutions of (1.2) by , that is,

*Special Examples*

(i)If , then the problem (1.2) is reduced to the mixed equilibrium problem (MEP), and the set of its solutions is denoted by
(ii) If , then the problem (1.2) is reduced to the generalized equilibrium problem (GEP), and the set of its solutions is denoted by
(iii)If , then the problem (1.2) is reduced to the equilibrium problem (EP), and the set of its solutions is denoted by
(iv) If , then the problem (1.2) is reduced to the mixed variational inequality of Browder type (VI), and the set of its solutions is denoted by

These show that the problem (1.2) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.2). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach space (see, e.g., [1β5]).

A Banach space is said to be strictly convex if for all with . is said to be uniformly convex if, for each , there exists such that for all with . is said to be smooth if the limit exists for all . is said to be uniformly smooth if the above limit exists uniformly in .

*Remark 1.1. *The following basic properties for Banach space and for the normalized duality mapping can be found in Cioranescu [6].(i)If is an arbitrary Banach space, then is monotone and bounded;(ii)If is a strictly convex Banach space, then is strictly monotone;(iii)If is a a smooth Banach space, then is single-valued, and hemicontinuous; that is, is continuous from the strong topology of to the weak star topology of ;(iv) If is a uniformly smooth Banach space, then is uniformly continuous on each bounded subset of ;(v)If is a reflexive and strictly convex Banach space with a strictly convex dual and is the normalized duality mapping in , then and ;(vi) If is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping is single valued, one to one and onto;(vii) A Banach space is uniformly smooth if and only if is uniformly convex. If is uniformly smooth, then it is smooth and reflexive.

Recall that a Banach space has the Kadec-Klee property, 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.

Next we assume that is a smooth, strictly convex and reflexive Banach space and is a nonempty closed convex subset of . In the sequel, we always use to denote the Lyapunov functional defined by

It is obvious from the definition of that

Following Alber [7], the generalized projection is defined by

Let be a mapping and be the set of fixed points of .

Recall that 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 .

*Definition 1.2. *(1) A mapping is said to be *nonexpansive* if

(2) A mapping is said to be *relatively nonexpansive* [8] if and

(3) A mapping is said to be *weak relatively nonexpansive* [9] if and

(4) A mapping is said to be *closed*, if for any sequence with and , then .

*Definition 1.3. *(1) A mapping is said to be *quasi*--*nonexpansive* [10] if and

(2) A mapping is said to be *quasi*--*asymptotically nonexpansive *[11], if and there exists a real sequence with such that

(3) A mapping is said to be *uniformly *-*Lipschitz* continuous, if there exists a constant such that

*Definition 1.4. *(1) A mapping is said to be *total quasi*--*asymptotically nonexpansive* if and there exist nonnegative real sequences with (as ) and a strictly increasing continuous function with such that for all

(2) A countable family of mappings is said to be uniformly total quasi--asymptotically nonexpansive, if and there exist nonnegative real sequences with (as ) and a strictly increasing continuous function with such that for all

*Remark 1.5. *From the definition, it is easy to know that(1)each relatively nonexpansive mapping is closed;(2) taking and , then (1.16) can be rewritten as
This implies that each quasi--asymptotically nonexpansive mapping must be a total quasi--asymptotically nonexpansive mapping, but the converse is not true;(3) the class of quasi--asymptotically nonexpansive mappings contains properly the class of quasi--nonexpansive mappings as a subclass, but the converse is not true;(4)the class of quasi--nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse is not true;(5)the class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.

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

*Remark 1.6. *If is an -inverse strongly monotone mapping, then it is -Lipschitz continuous.

Iterative approximation of fixed points for relatively nonexpansive mappings in the setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushita and Takahashi [12] obtained some weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Ofoedu and Malonza [4], Zhang [5], Su et al. [13], Zegeye and Shahzad [14], Wattanawitoon and Kumam [15], Qin et al. [16], Takahashi and Zembayashi [17], Chang et al. [18, 19], Yao et al. [20, 21], Qin et al. [22], and Cho et al. [23, 24] extend the notions from relatively nonexpansive mappings, weakly relatively nonexpansive mappings or quasi--nonexpansive mappings to quasi--asymptotically nonexpansive mappings and also prove some strongence theorems to approximate a common fixed point of quasi--nonexpansive mappings or quasi--asymptotically nonexpansive mappings.

The purpose of this paper is first to introduce the concept of total quasi--asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases, and then by using a hybrid algorithm to introduce a new iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a countable family of total quasi--asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results improve and extend the corresponding results in [8, 11β25].

#### 2. Preliminaries

First, we recall some definitions and conclusions.

Lemma 2.1 (see [7, 26]). *Let be a smooth, strictly convex and reflexive Banach space and be a nonempty closed convex subset of . Then the following conclusions hold:*(a)* for all and ;*(b)* if and , then
*(c)* for if and only if . *

*Remark 2.2. * If is a real Hilbert space , then and is the metric projection of onto .

Lemma 2.3 (see [18]). *Let be a uniformly convex Banach space, a positive number, and a closed ball of . Then, for any given sequence and for any given sequence of positive numbers with , then there exists a continuous, strictly increasing and convex function with such that for any positive integers with ,
*

Lemma 2.4. *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 total quasi--asymptotically nonexpansive mapping with nonnegative real sequences and a strictly increasing continuous functions such that (as ) and . Then is a closed convex subset of .*

*Proof. *Letting be a sequence in with (as ), we prove that . In fact, from the definition of , we have
Therefore we have
that is, .

Next we prove that is convex. For any , putting , we prove that . Indeed, in view of the definition of , we have
Since and , we have (as ). From (1.10) we have . Consequently . This implies that is a bounded sequence. Since is reflexive, is also reflexive. So we can assume that
Again since is reflexive, we have . Therefore there exists such that . By virtue of the weakly lower semicontinuity of norm , we have
that is, which implies that . Hence from (2.6) we have . Since and has the Kadec-Klee property, we have . Since is uniformly smooth, is uniformly convex, which in turn implies that is smooth. From Remark 1.1(iii) it yields that is hemi-continuous. Therefore we have . Again since , by using the Kadec-Klee property of , we have . This implies that . Since is closed, we have .

This completes the proof of Lemma 2.4.

Lemma 2.5. *Let be a smooth, strictly convex and reflexive Banach space and be a nonempty closed convex subset of . Let be a bifunction satisfying the following conditions:**β,** is monotone, that is, ,**,** The function is convex and lower semi-continuous.**Then the following conclusions hold:** (Blum and Oettli [27]) for any given and , there exists a unique such that
** (Takahashi and Zembayashi [28]) for any given and , define a mapping by **
Then, the following conclusions hold:** is single-valued;** is firmly nonexpansive-type mapping, that is, for all,
** and is quasi--nonexpansive;** is closed and convex;**.**For solving the generalized mixed equilibrium problem (1.2), let us assume that the following conditions are satisfied:** is a smooth, strictly convex, and reflexive Banach space and is a nonempty closed convex subset of ;** is -inverse strongly monotone mapping;** is bifunction satisfying the conditions (A1), (A3), (A4) in Lemma 2.5 and the following condition :** for some with ** is a lower semicontinuous and convex function. **Under the assumptions as above, we have the following results.*

Lemma 2.6. *Let satisfy the above conditions (1)β(4). Denote by
**
For any given and , define a mapping by
**
Then, the following hold:** is single-valued;** is a firmly nonexpansive-type mapping, that is, for all ,
**;** is closed and convex;**β
*

*Proof. *It follows from Lemma 2.5 that in order to prove the conclusions of Lemma 2.6 it is sufficient to prove that the function satisfies the conditions (*A1*)β(*A4*) in Lemma 2.5.

In fact, by the similar method as given in the proof of Lemma 2.4 in [1], we can prove that the function satisfies the conditions (*A1*), (*A3*), and (*A4*). Now we prove that also satisfies the conditions (*A2*).

Indeed, for any , by condition (*A2*)β², we have
This implies that the function satisfies the conditions (*A2*). Therefore the conclusions of Lemma 2.6 can be obtained from Lemma 2.3 immediately.

*Remark 2.7. *It follows from Lemma 2.5 that the mapping is a relatively nonexpansive mapping. Thus, it is quasi--nonexpansive.

#### 3. Main Results

In this section, we shall use the hybrid method to prove some strong convergence theorems for finding a common element of the set of solutions for a system of the generalized mixed equilibrium problems (1.2) and the set of common fixed points of a countable family of total quasi--asymptotically nonexpansive mappings in Banach spaces.

In the sequel, we assume that satisfy the following conditions.Let be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and a nonempty closed convex subset of . Let be a countable family of closed and uniformly total quasi--asymptotically nonexpansive mappings with nonnegative real sequences and a strictly increasing continuous functions such that (as ) and . Suppose further that for each is a uniformly -Lipschitz mapping, that is, there exists a constant such that
Let be a finite family of -inverse strongly monotone mappings. Let be a finite family of bifunction satisfying the conditions (*A1*), (*A3*), (*A4*), and the following condition (*A2*)β²: For each there exists with such that
Let be a finite family of lower semicontinuous and convex functions.

Theorem 3.1. * Let be the same as above. Suppose that
**
is a nonempty and bounded subset of . For any given , let be the sequence generated by
**
where
** is the mapping defined by (2.13) with , and
** for some is the generalized projection of onto the set and are sequences in satisfying the following conditions:** for all ;** for all ;** for some .**Then converges strongly to , where is the generalized projection from onto .*

*Proof. *We divide the proof of Theorem 3.1 into five steps.

(i) We first prove that and both are closed and convex subset of for all .

In fact, it follows from Lemmas 2.4 and 2.6 that and both are closed and convex. Therefore is a closed and convex subset in . Furthermore, it is obvious that is closed and convex. Suppose that is closed and convex for some . Since the inequality is equivalent to
therefore, we have

This implies that is closed and convex. The desired conclusions are proved. These in turn show that and are well defined.

(ii) We prove that and for all are both bounded sequences in .

By the definition of , we have for all . It follows from Lemma 2.1 (a) that
This implies that is bounded. By virtue of (1.10), is bounded. Since for all and is bounded for all , and so is bounded in . Denote by

In view of the structure of , we have and . This implies that and
Therefore is convergent. Without loss of generality, we can assume that
(iii) Next, we prove that for all .

Indeed, it is obvious that . Suppose that for some . Since , by Lemma 2.6 and Remark 2.7, is quasi--nonexpansive. Again since is uniformly smooth, is uniformly convex. Hence, For any given and for any positive integer , from Lemma 2.3 we have
Hence and so for all . By the way, from the definition of and and (3.10), it is easy to see that
(IV) Now, we prove that converges strongly to some point

First, we prove that converges strongly to some point .

In fact, since is bounded in and is reflexive, there exists a subsequence such that . Again since is closed and convex for each , it is weakly closed, and so for each . Since , from the definition of , we have
Since
we have
This implies that , that is, . In view of the Kadec-Klee property of , we obtain that .

Now we prove that . In fact, if there exists a subsequence such that , then we have
Therefore we have . This implies that

Now we prove that . In fact, by the construction of , we have that and . Therefore by Lemma 2.1(a) we have
In view of and noting the construction of we obtain that
From (1.10) it yields . Since, we have
Hence we have

This implies that is bounded in . Since is reflexive, and so is reflexive, there exists a subsequence such that . In view of the reflexive of , we see that . Hence there exists such that . Since
taking on the both sides of above equality and in view of the weak lower semicontinuity of norm , then it yields that
That is . This implies that , and so . It follows from (3.24) and the Kadec-Klee property of that (as ). Note that is hemi-continuous, it yields that . It follows from (3.23) and the Kadec-Klee property of that .

By the similar way as given in the proof of (3.20), we can also prove that

From (3.20) and (3.27) we have that
Since is uniformly continuous on any bounded subset of , we have
For any and any , it follows from (3.13), (3.20), and (3.27) that
Since
from (3.28) and (3.29), it follows that
In view of condition (b) and condition (c), we have that
It follows from the property of that
Since and is uniformly continuous, it yields . Hence from (3.34) we have
Since is hemicontinuous, it follows that
On the other hand, for each we have
This together with (3.36) shows that

Furthermore, by the assumption that for each is uniformly -Lipschitz continuous, hence we have
This together with (3.20) and (3.38), yields (as . Hence from (3.36) we have , that is, . In view of (3.38) and the closeness of , it yields that . This implies that .

Next, we prove that . Denote that
By the similar method as in the proof of (3.13), we can prove that
It follows from Lemma 2.6, (2.15), (3.32) that for any ,
From (1.10) it yields . Since , we have

Hence we have

This implies that is bounded in . Since is reflexive, and so is reflexive, there exists a subsequence such that . In view of the reflexive of , we see that . Hence there exists such that . Since
taking on the both sides of above equality and in view of the weak lower semicontinuity of norm , it yields that
This is, . This implies that , and so . It follows from (3.44) and the Kadec-Klee property of that (as ). Note that is hemicontinuous it yields that . It follows from (3.43) and the Kadec-Klee property of that .

By the similar way as given in the proof of (3.20), we can also prove that
From (3.27) and (3.47) we have that
Since is uniformly continuous on any bounded subset of , we have

Since

By the similar way as above, we can also prove that
From (3.51) and the assumption that , we have
In the proof of Lemma 2.6 we have proved that the function defined by (3.6) satisfies the condition (A1)β(A4) and
Therefore for any we have
This implies that
for some constant . Since the function is convex and lower semi-continuous, letting in (3.55), from (3.52) and (3.55), for each , we have .

For and , letting , there are and . By condition (*A1*) and (*A4*), we have
Dividing both sides of the above equation by , we have . Letting , from condition (*A3*), we have , that is, for each , we have
This implies that . Therefore, we have that

(V) Now, we prove .

Let . From and , we have . This implies that

By the definition of and (3.59), we have . Therefore, . This completes the proof of Theorem 3.1.

Theorem 3.2. *Let be the same as above. Let be an infinite family of closed and uniformly quasi--asymptotically nonexpansive mappings with a sequence and . Suppose that for each is uniformly -Lipschitz continuous and that
**
is a nonempty and bounded subset of . For any given , let be the sequence generated by
*