#### Abstract

We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi--nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.

#### 1. Introduction

Let be a Banach space and let be a nonempty, closed, and convex subset of . Let be an operator. The classical variational inequality problem [1] for is to find such that

where denotes the dual space of and the generalized duality pairing between and . The set of all solutions of (1.1) is denoted by . Such a problem is connected with the convex minimization problem, the complementarity, the problem of finding a point satisfying , and so on. First, we recall that a mapping is said to be

(i)*monotone*if (ii)

*-inverse-strongly monotone*if there exists a positive real number such that

In this paper, we assume that the operator satisfies the following conditions:

(C1) is -inverse-strongly monotone, (C2) for all and .Let be the normalized duality mapping from into given by

It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of . Some properties of the duality mapping are given in [2â€“4].

Recall that a mapping is said to be nonexpansive if

If is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is a nonexpansive mapping. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [5] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Consider the functional defined by

for all , where is the normalized duality mapping from to . Observe that, in a Hilbert space , (1.5) reduces to for all . The generalized projection is a mapping 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., [2, 5â€“7]). In Hilbert spaces, , where is the metric projection. It is obvious from the definition of the function that

(1) for all ,(2) for all (3) for all , (4)if is a reflexive, strictly convex, and smooth Banach space, then for all ,
For more details see [2, 3]. Let be a closed convex subset of , and let be a mapping from into itself. We denote by the set of fixed point of . A point in is said to be an *asymptotic fixed point * of [8] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is called nonexpansive if for all and relatively nonexpansive [9â€“11] if and for all and . The asymptotic behavior of relatively nonexpansive mappings which was studied in [9â€“11] is of special interest in the convergence analysis of feasibility, optimization, and equilibrium methods for solving the problems of image processing, rational resource allocation, and optimal control. The most typical examples in this regard are the Bregman projections and the Yosida type operators which are the cornerstones of the common fixed point and optimization algorithms discussed in [12] (see also the references therein).

The mapping is said to be *-nonexpansive* if for all . is said to be *nonexpansive* if and for all and .

*Remark 1.1. *The class of quasi--nonexpansive is more general than the class of relatively nonexpansive mappings [9, 10, 13â€“15] which requires the strong restriction .

Next, we give some examples which are closed quasi--nonexpansive [16].

*Example 1.2. *(1) Let be a uniformly smooth and strictly convex Banach space and let be a maximal monotone mapping from to such that its zero set is nonempty. The resolvent is a closed quasi--nonexpansive mapping from onto and .

(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 and quasi--nonexpansive mapping from onto with .

Iiduka and Takahashi [17] introduced the following algorithm for finding a solution of the variational inequality for an operator that satisfies conditions (C1)-(C2) in a 2 uniformly convex and uniformly smooth Banach space . For an initial point , define a sequence by

where is the duality mapping on , and is the generalized projection of onto . Assume that for some with where is the 2 uniformly convexity constant of . They proved that if is weakly sequentially continuous, then the sequence converges weakly to some element in where .

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, [18â€“20] and the references cited therein.

On the other hand, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping (see [21]). More precisely, let and define a contraction by for all , where is a fixed point in . Applying Banach's Contraction Principle, there exists a unique fixed point of in . It is unclear, in general, what is the behavior of as even if has a fixed point. However, in the case of having a fixed point, Browder [21] proved that the net defined by for all converges strongly to an element of which is nearest to in a real Hilbert space. Motivated by Browder [21], Halpern [22] proposed the following iteration process:

and proved the following theorem.

Theorem 1 H. *Let be a bounded closed convex subset of a Hilbert space and let be a nonexpansive mapping on . Define a real sequence in by . Define a sequence by (1.9). Then converges strongly to the element of which is the nearest to .*

Recently, Martinez-Yanes and Xu [23] have adapted Nakajo and Takahashi's [24] idea to modify the process (1.9) for a single nonexpansive mapping in a Hilbert space :

where denotes the metric projection of onto a closed convex subset of . They proved that if and , then the sequence generated by (1.10) converges strongly to .

In [15] (see also [13]), Qin and Su improved the result of Martinez-Yanes and Xu [23] from Hilbert spaces to Banach spaces. To be more precise, they proved the following theorem.

Theorem 1 QS. *Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of and let be a relatively nonexpansive mapping. Assume that is a sequence in such that . Define a sequence in by the following algorithm:
**
where is the single-valued duality mapping on . If is nonempty, then converges to .*

In [14], Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings:

where , , and are sequences in satisfying for all and are relatively nonexpansive mappings and is the single-valued duality mapping on . They proved, under appropriate conditions on the parameters, that the sequence generated by (1.12) converges strongly to a common fixed point of and .

Very recently, Qin et al. [25] introduced a new hybrid projection algorithm for two families of quasi--nonexpansive mappings which are more general than relatively nonexpansive mappings to have strong convergence theorems in the framework of Banach spaces. To be more precise, they proved the following theorem.

Theorem 1 QCKZ. *Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let and be two families of closed quasi--nonexpansive mappings of into itself with being nonempty, where is an index set. Let the sequence be generated by the following manner:
**
where is the duality mapping on , and and are sequences in satisfying *(i)* for all , *(ii)* for all , *(iii)* and for all . ** Then the sequence converges strongly to .*

On the other hand, recently, Takahashi et al. [26] introduced the following hybrid method (1.14) which is different from Nakajo and Takahashi's [24] hybrid method. It is called the shrinking projection method. They obtained the following result.

Theorem 1 NT. *Let be a nonempty closed convex subset of a Hilbert space . Let be a nonexpansive mapping of into such that and let . For and , define a sequence of as follows:
**
where for all Then converges strongly to *

Motivated and inspired by Iiduka and Takahashi [17], Martinez-Yanes and Xu [23], Matsushita and Takahashi [13], Plubtieng and Ungchittrakool [14], Qin and Su [15], Qin et al. [25], and Takahashi et al. [26], we introduce a new hybrid projection algorithm basing on the shrinking projection method for two families of quasi--nonexpansive mappings which are more general than relatively nonexpansive mappings to have strong convergence theorems for approximating the common element of the set of common fixed points of two families of quasi--nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, the problem of finding a zero point of an inverse-strongly monotone operator and the problem of finding a solution of the complementarity problem are studied. Our results improve and extend the corresponding results announced by recent results.

#### 2. Preliminaries

Let be a real Banach space with duality mapping . We denote strong convergence of to by and weak convergence by . A multivalued operator with domain and range is said to be monotone if for each and . A monotone operator is said to be maximal if its graph is not properly contained in the graph of any other monotone operators.

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 in such that and . Let be the unit sphere of . Then the Banach space is said to be smooth provided that

exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for . It is well know that if is smooth, then the duality mapping is single valued. It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subset of . Some properties of the duality mapping are given in [2, 3, 27â€“29]. We define the function which is called the modulus of convexity of as follows:

Then is said to be 2 uniformly convex if there exists a constant such that constant for all . Constant is called the 2 uniformly convexity constant of . A 2 uniformly convex Banach space is uniformly convex; see [30, 31] for more details. We know the following lemma of 2 uniformly convex Banach spaces.

Lemma 2.1 (see [32, 33]). *Let be a 2 uniformly convex Banach, then for all from any bounded set of and ,
**
where is the 2 uniformly convexity constant of .*

Now we present some definitions and lemmas which will be applied in the proof of the main result in the next section.

Lemma 2.2 (Kamimura and Takahashi [7]). *Let be a uniformly convex and smooth Banach space and let , be two sequences of such that either or is bounded. If , then .*

Lemma 2.3 (Alber [5]). *Let be a nonempty closed convex subset of a smooth Banach space and . Then, if and only if for any .*

Lemma 2.4 (Alber [5]). *Let be a reflexive, strictly convex, and smooth Banach space, let be a nonempty closed convex subset of , and let . Then
**
for all .*

Lemma 2.5 (Qin et al. [25]). *Let be a uniformly convex and smooth Banach space, let be a closed convex subset of , and let be a closed quasi--nonexpansive mapping of into itself. Then is a closed convex subset of .*

Let be a reflexive strictly convex, smooth, and uniformly Banach space and the duality mapping from to . Then is also single valued, one to one, and surjective, and it is the duality mapping from to . We need the following mapping which is studied in Alber [5]:

for all and . Obviously, . We know the following lemma.

Lemma 2.6 (Kamimura and Takahashi [7]). *Let be a reflexive, strictly convex, and smooth Banach space, and let be as in (2.5). Then
**
for all and .*

Lemma 2.7 (see [34,Lemma ]). *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 .*

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

Lemma 2.8 (see [35]). *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 .*

#### 3. Main Results

In this section, we prove strong convergence theorem which is our main result.

Theorem 3.1. *Let be a 2 uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be an operator of into satisfying (C1) and (C2), and let and be two families of closed quasi--nonexpansive mappings of into itself with being nonempty, where is an index set. Let be a sequence generated by the following manner:
**
where is the duality mapping on , and , and are sequences in satisfying the following conditions: *(i)* for all ; *(ii)*for all , for some with , where is the 2 uniformly convexity constant of ; *(iii)* for all and if one of the following conditions is satisfied:(a) for all and for all ,(b) and for all . *

*Then the sequence converges strongly to , where is the generalized projection from onto .*

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

To this end, we prove first that is closed and convex. It is obvious that is a closed convex subset of . By Lemma 2.5, we know that is closed and convex. Hence is a nonempty, closed, and convex subset of . Consequently, is well defined.

We next show that is convex for each . From the definition of , it is obvious that is closed for each . Notice that
is equivalent to
It is easy to see that is closed and convex for all and . Therefore, is closed and convex for every . This shows that is well defined.*Step 2. *Show that for all .

Put . We have to show that for all . For all , we know from Lemmas 2.4 and 2.6 that
Since and from condition (C1), we have
From Lemma 2.1, and condition (C2), we also have
Subtituting (3.6) and (3.5) into (3.4) and using the assumption (ii), we obtain
It follows from the convexity of and (3.7) that
and hence
This show that for each . That is, for all . This show that
*Step 3. *Show that exists.

We note that for all and for all . Hence
From and , we have
This shows that is nondecreasing. On the other hand, from Lemma 2.4, we have
for each . This show that is bounded. Consequently, exists.*Step 4. *Show that is a convergent sequence in .

Since for any . It follows that
Letting in (3.14), we have . It follows from Lemma 2.2 that
Hence is a Cauchy sequence in . By the completeness of and the closedness of , we can assume that
*Step 5. *We show that .

(I) We first show that . Taking in (3.14), one arrives that
From Lemma 2.2, we obtain
Noticing that , from the definition of for every , we obtain
It follows from (3.17) and and the fact that is bounded that
From Lemma 2.2, we obtain
It follows from (3.18) that
Since is uniformly norm-to-norm continuity on bounded sets, for every , one has
For every , we obtain from the properties of that
On the other hand, for all , we have
It follows form (ii) that
Notice that
Applying (3.18), (3.20), (3.21), and (3.23) to the last inequality, we obtain
Combining (3.26) with (3.28) in (3.24), we have
From Lemma 2.2, we have
Since is uniformly norm-to-norm continuity on bounded sets, for every , one has
Let for every . Therefore Lemma 2.7 implies that there exists a continuous strictly increasing convex function satisfying and (2.7).*Case I. *Assume that (a) holds. Applying (2.7), we can calculate
This implies that
On the other hand, for every , one has
It follows from (3.30) and (3.31) that
Applying and (3.35) in (3.33) we get
It follows from the property of that
Since is also uniformly norm-to-norm continuity on bounded sets, for every , one has
In a similar way, one has
On the other hand, we observe from (3.7) that
Hence
Using (3.35), we can conclude that
From (3.6), we can calculate
It follows from (3.42) and the fact that is bounded that
From Lemma 2.2, we have
Hence as for each . From (3.39) and (3.45), we have
The closedness of and implies that *Case II. *Assume that (b) holds. We observe that