#### Abstract

We introduce and study a new hybrid projection algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of relatively quasi-nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly-monotone operator in a Banach space. Under suitable assumptions, we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space. The results obtained in this paper extend and improve the several recent results in this area.

#### 1. Introduction

Let be a real Banach space with norm and the dual space of . Let be a nonempty closed and convex subset of and a monotone operator of into . A mapping is called nonexpansive if for all . We denote by the set of fixed points of . The classical variational inequality problem [1, 2], denoted by is to find such that for all . One can see that the variational inequality problem (1.1) is connected with the convex minimization problem, the complementarity problem, the problem of finding a point satisfying and so on.

Recall that an operator is called monotone if for all . An operator of into is said to be -inverse-strongly-monotone [3–5] if each . We have , for a constant .

Assume that the following hold:

(C1) is -inverse-strongly-monotone,(C2),(C3) for all and .For finding a solution of the variational inequality problem for an operator that satisfies conditions (C1)–(C3) in a 2-uniformly convex and uniformly smooth Banach space , Iiduka and Takahashi [6] introduced and studied the following algorithm: , define a sequence by for every , where is the duality mapping from into , is the generalized projection from onto and is a sequence of positive real numbers. They proved that under certain appropriate conditions imposed on and is weakly sequentially continuous, the sequence generated by (1.2) converges weakly to some element in where .

In 2004, Matsushita and Takahashi [7] introduced the following iteration: chosen arbitrarily, where is a real sequence in , is a relatively nonexpansive mapping, and denotes the generalized projection from onto a closed convex subset of . They prove that the sequence generated by (1.3) converges weakly to a fixed point of .

The problem of finding a common element of the set of the variational inequalities for monotone mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; see, for instance, [3–5] and the references cited therein.

Let be a bifunction of into where is the set of real numbers. The equilibrium problem for is to find such that for all The set of solutions of (1.4) is denoted by , that is, . Given a mapping , let for all . Then if and only if for all , that is, is a solution of the variational inequality. Many problems in physics, optimization, and economics reduce to finding a solution of (1.4). Equilibrium problems have been studied extensively; see, for instance, [8, 9]. Combettes and Hirstoaga [8] introduced an iterative scheme for finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem.

For solving the equilibrium problem, let us assume that a bifunction satisfies the following conditions:

(A1) for all ;(A2) is monotone, that is, for all ;(A3)for all , (A4)for all is convex and lower semicontinuous.The problem of finding a common element of the set of fixed points and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces has been studied by many authors; see [7, 8, 10–18].

In 2008, Takahashi and Zembayashi [15] introduced the shrinking projection method which is the modification of (1.3) for a relatively nonexpansive mapping. It is given as follows: for every , where is the duality mapping on satisfying and for some . They proved that the sequence generated by (1.6) converges strongly to , where is the generalized projection of onto .

In the same year, Qin et al. [19] extend the iteration process (1.6) from a single relatively nonexpansive mapping to two relatively quasi-nonexpansive mappings: and for every . Under appropriate conditions imposed on and they obtain that the sequence generated by (1.7) converges strongly to .

In 2009, Wattanawitoon and Kumam [17] introduced the following iterative scheme which is the modification of (1.6) and (1.7) in a Banach space: and for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two relatively quasi-nonexpansive mappings in a Banach space. They proved that under certain appropriate conditions imposed on and , the sequences and generated by (1.8) converge strongly to , where .

For finding common elements of the set of the equilibrium problem, the set of the variational inequality problem for an inverse-strongly-monotone operator and the set of common fixed points for relatively quasi-nonexpansive mappings. Cholamjiak [20] introduced an iterative scheme by using the new hybrid method in a Banach space. The scheme is defined as follows: and ,, for every , where is the duality mapping on and and are sequences in . He proved that under certain appropriate conditions imposed on and , the sequences and generated by (1.9) converge strongly to .

Motivated by the recent works, we introduce an iterative scheme by a new hybrid method as follows: and , for every , where is the duality mapping on , is the generalized projection from onto a closed convex subset of , is an equilibrium bifunction satisfying (A1)–(A4), is an operator of into satisfying (C1)–(C3), are two closed relatively quasi-nonexpansive mappings, and are sequences in such that , and for some and for some with , where is the -uniformly convexity constant of . We prove that the sequences and generated by the above iterative scheme converge strongly to .

#### 2. Preliminaries

In this section, we recall some well know concepts and results.

Let be a real Banach space with dimension . The modulus of is the function defined by
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 * p-uniformly convex* if there exists a constant such that for all ; see [21–23] for more details. A Banach space is said to be * smooth* if the limit
exists for all , where denotes the unit sphere of (i.e., ). It is also said to be * uniformly smooth* if the limit (2.2) is attained uniformly for . One should note that no Banach space is -uniformly convex for ; see [23]. It is well known that a Hilbert space is -uniformly convex, uniformly smooth. For each , the * generalized duality mapping * is defined by
for all . In particular, is called the * normalized duality mapping*. If is a Hilbert space, then , where is the identity mapping. It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of . See [24, 25] for more details.

Let be a smooth, strictly convex, and reflexive Banach space and let be a nonempty closed convex subset of We denote by the function defined by for all . Following Alber [26], 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 strict monotonicity of the mapping (see, e.g., [24, 26–29]. In Hilbert space, . It is obvious from the definition of function that for all . If is a Hilbert space, then .

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 (2.6), we have . This implies that . From the definition of , one has . Therefore, we have ; see [24, 28] for more details.

Recall that a point in a closed convex subset of is said to be an asymptotic fixed point of [30] if contains a sequence which converges weakly to such that . The set of asymptotic fixed point of will be denoted by . A mapping is called relatively nonexpansive [11, 31–33] if satisfies the following conditions:

(1);(2) for all and ;(3).The asymptotic behavior of a relatively nonexpansive mapping was studied in [31–33].

A mapping is said to be relatively quasi-nonexpansive if satisfies conditions (1) and (2). It is easy to see that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [11, 31–33] which requires the strong restriction: .

We give some examples which are closed relatively quasi-nonexpansive; see [19].

*Example 2.1. *Let be a uniformly smooth and strictly convex Banach space and be a maximal monotone mapping such that its zero set . Then, is a closed relatively quasi-nonexpansive mapping from onto and .

*Example 2.2. *Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space onto a nonempty closed convex subset of . Then, is a closed relatively quasi-nonexpansive mapping with .

An operator of into is said to be * hemicontinuous* if for all , the mapping of into defined by is continuous with respect to the topology of . We define by the * normal cone* for at a point , that is,

Let be a nonempty, closed convex subset of a Banach space and a monotone, hemicontinuous operator of into . Let be an operator defined as follows: Then, is maximal monotone and ; see [34].

In the sequel, the following lemmas are needed to prove our main results.

Lemma 2.3 ([22, 35]). *Let be a given real number with and a -uniformly convex Banach space. Then, for all , and ,
**
where is the generalized duality mapping of and is the -uniformly convexity constant of .*

Lemma 2.4 ([29]). *Let be a uniformly convex and smooth Banach space and let and be two sequences of . If and either or is bounded, then .*

Lemma 2.5 ([26]). *Let be a nonempty closed convex subset of a smooth Banach space and . Then if and only if for all .*

Lemma 2.6 ([26]). *Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space and let . Then
*

Lemma 2.7 ([19]). *Let be a uniformly convex, smooth Banach space, let be a closed convex subset of , let be a closed and relatively quasi-nonexpansive mapping from into itself. Then is a closed convex subset of .*

Lemma 2.8 ([36]). *Let be a uniformly convex Banach space and be a closed ball of . Then there exists a continuous strictly increasing convex function with such that
**
for all and with .*

Lemma 2.9 ([10]). *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let and . Then, there exists such that
*

Lemma 2.10 ([19]). *Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space , and let be a bifunction from to satisfying (A1)–(A4). For all and , define a mapping as follows:
*

Then, the following hold:

(1) is single-valued;(2) is a firmly nonexpansive-type mapping [37], that is, for all , (3);(4) is closed and convex.Lemma 2.11 ([16]). *Let be a closed convex subset of a smooth, strictly, and reflexive Banach space , let be a bifucntion from to satisfying (A1)–(A4), let . Then, for all and ,
*

We make use of the following mapping studied in Alber [26]: for all and , that is, .

Lemma 2.12 ([26]). *Let be a reflexive, strictly convex, smooth Banach space and let be as in (2.16). Then
**
for all and .*

#### 3. Main Results

In this section, we prove strong convergence theorems by hybrid methods which solves the problem of finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of relatively quasi-nonexpansive mappings and the set of solutions of the variational inequality of an -inverse-strongly-monotone mapping in a 2-uniformly convex, uniformly smooth Banach space.

Theorem 3.1. *Let be a -uniformly convex, uniformly smooth Banach space, a nonempty closed convex subset of , a bifunction from to which satisfies (A1)–(A4), an operator of into satisfying (C1)–(C3), and two closed relatively quasi-nonexpansive mappings from into itself such that the set . For an initial point with and , define sequences and of as follows:
**
for every , where is the duality mapping on . Suppose that and are sequences in satisfying the restrictions:*(B1)*;*(B2)*, and ;*(B3)* for some ;*(B4)* for some with , where is the -uniformly convexity constant of .**Then, and converge strongly to .*

*Proof. *We divide the proof into eight steps.*Step 1. *Show that and are well defined.

It is obvious that is a closed convex subset of . By Lemma 2.7, we know that is closed and convex. From Lemma 2.10 (4), we also have is closed and convex. Hence, is a nonempty, closed, and convex subset of ; consequently, is well defined.

Next, we show that is closed and convex for all .

It is obvious that, is closed and convex. Suppose that is closed and convex for some . For all , we know that is equivalent to
So, is closed and convex. Then, for any , is closed and convex. This implies that is well defined.*Step 2. *We prove by induction that for all .

Putting . First, we observe that for all and . Suppose that for some . Let . From Lemmas 2.6 and 2.12, we have
Using (C1) and , we have
By using Lemma 2.3 and (C3), we have
Replacing (3.4) and (3.5) into (3.3) and using (B4), we get

By the convexity of and (3.6), for each , we have
and so
This shows that and hence . This implies that for all .*Step 3. *Show that exists.

From and , we have
Therefore, is nondecreasing. From Lemma 2.6, we have
Then the sequence is bounded. It follows that exists.*Step 4. *Show that is a Cauchy sequence in .

Since for any positive integer , by Lemma 2.6, we also have
Letting in (3.11), we have . It follows from Lemma 2.4 that as . Hence, is a Cauchy sequence. By the completeness of and the closedness of , one can assume that as . Further, we obtain
Since , we have
Applying Lemma 2.4 to (3.12) and (3.13), we have
This implies that as . Since is uniformly norm-to-norm continuous on bounded subsets of , we also obtain
*Step 5. *Show that .

Let . From (3.7) and Lemma 2.8, we obtain
This implies that
It follows from (3.14), (3.15), and (B2) that
Since is strictly increasing and continuous at with , it follows that
Since is uniformly norm-to-norm continuous on bounded sets, so is . Then
In the same manner, we can show that
In addition, , using (3.6), we have
which leads to the following:
Since and from (3.17), we observe
which yields that
From Lemmas 2.6 and 2.12, and (3.5), we have
It follows from Lemma 2.4 and (3.25) that
Hence as and
By using (3.26), we have
Applying Lemma 2.4, we get
Hence, as .

In addition,

It follows from (3.21), (3.30), and (3.31) that
From (3.20), (3.32) and by the closedness of and , we get .*Step 6. *Show that .

From (3.16), we have
Note that . From (3.33) and Lemma 2.11, we have
Using (3.24), we have . By Lemma 2.4, we obtain
Since , we have
From we have
By using (A2), we have
From (A4) and , we get for all . For with and , let . Since and , we have and hence . So from (A1), we have . That is, . It follows from (A3) that for all and hence .*Step 7. *Show that .

Define be as in (2.8), which yields that is maximal monotone and . Let . Since , we get . From , we have
On the other hand, from and Lemma 2.5, we have and hence
It follows from (3.39) and (3.40) that
where . By taking the limit as and from (3.27) and (3.28), we obtain . By the maximality of , we have and hence . That is, .*Step 8. *Show that .

From , we have
Since , we also have
By taking the limit in (3.43), we obtain that
By Lemma 2.5, we can conclude that . From (3.14), we have , and it follows that . This completes the proof.

Finally, we prove two strong convergence theorems in a 2-uniformly convex, uniformly smooth Banach space by using Theorem 3.1.

First, we consider the problem of finding a zero point of an inverse-strongly-monotone operator of into . Assume that satisfies the conditions:

(D1) is -inverse-strongly monotone,(D2).Theorem 3.2. *Let be a -uniformly convex, uniformly smooth Banach space, a bifunction from to which satisfies (A1)–(A4), an operator of into satisfying (D1)–(D2), and two closed relatively quasi-nonexpansive mappings from into itself such that . For an initial point with and , define sequences and as follows:
*