1. Introduction

Let be a smooth Banach space. Throughout this paper, we denote by the function defined by which was studied by Alber [1], Kamimura and Takahashi [2], and Reich [3], where is the normalized duality mapping from to defined by where denotes the duality pairing. It is well known that if is smooth, then is single-valued, and, if has uniformly Gâteaux differentiable norm, then is uniformly continuous on bounded subsets of . Moreover, if is a reflexive and strictly convex Banach space with a strictly convex dual, then is single valued, one-to-one, surjective, and it is the duality mapping from into , and thus and (see [4]).

It is obvious from the definition of the function that and, in a Hilbert space , (1.1) reduces to , for .

Let be a reflexive, strictly convex, and smooth Banach space, and let be a nonempty closed and convex subset of . The generalized projection mapping, introduced by Alber [1], is a mapping that assigns an arbitrary point to the minimizer, , of over , that is, , where is the solution to the minimization problem Let be a real Banach space with dual . A mapping is said to be monotone if, for each , the following inequality holds: is said to be -inverse strongly monotone if there exists positive real number such that If is -inverse strongly monotone, then it is Lipschitz continuous with constant , that is, , for all , and it is called strongly monotone if there exists such that, for all , Clearly, the class of monotone mappings include the class of strongly monotone and -inverse strongly monotone mappings.

Suppose that is monotone mapping from into . The variational inequality problem is formulated as finding a point such that , for all . The set of solutions of the variational inequality problems is denoted by .

The notion of monotone mappings was introduced by Zarantonello [5], Minty [6], and Kacurovskii [7] in Hilbert spaces. Monotonicity conditions in the context of variational methods for nonlinear operator equations were also used by Vainberg and Kachurovisky [8]. Variational inequalities were initially studied by Stampacchia [9, 10] and ever since have been widely studied in general Banach spaces (see, e.g., [2, 11–13]). Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding point satisfying .

If , a Hilbert space, one method of solving a point is the projection algorithm which starts with any point and updates iteratively as according to the formula where is the metric projection from onto and is a sequence of positive real numbers. In the case that is -inverse strongly monotone, Iiduka et al. [14] proved that the sequence generated by (3.35) converges weakly to some element of .

In the case that is a 2-uniformly convex and uniformly smooth Banach space, Iiduka and Takahashi [15] introduced the following iteration scheme for finding a solution of the variational inequality problem for an inverse strongly monotone operator : where is the generalized projection from onto , is the normalized duality mapping from into , and is a sequence of positive real numbers. They proved that the sequence generated by (1.9) converges weakly to some element of provided that satisfies , for and .

It is worth to mention that the convergence is weak convergence.

To obtain strong convergence, when , a Hilbert space and is -inverse strongly monotone; Iiduka et al. [14] studied the following iterative scheme: where is a sequence in . They proved that the sequence generated by (1.10) converges strongly to , where is the metric projection from onto provided that satisfies , for and .

In the case that is 2-uniformly convex and uniformly smooth Banach space, Iiduka and Takahashi [11] studied the following iterative scheme for a variational inequality problem for -inverse strongly monotone mapping: where is the generalized projection from onto is the normalized duality mapping from into , and is a positive real sequence satisfying certain condition. Then, they proved that the sequence converges strongly to an element of provided that and satisfies for all and .

Remark 1.1. We remark that the computation of in Algorithms (1.10) and (1.11) is not simple because of the involvement of computation of from and for each .

Let be a mapping from into itself. We denote by the fixed points set of . A point in is said to be an asymptotic fixed point of (see [3]) 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 said to be nonexpansive if for each and is called relatively nonexpansive if (R1) ; (R2) for and (R3) . is called relatively quasi-nonexpansive if and for all , and .

A mapping from into itself is said to be asymptotically nonexpansive if there exists such that and for each and is called relatively asymptotically nonexpansive if there exists such that (N1) ; (N2) for and , and (N3) , where as . A-self mapping on is called uniformly -Lipschitzian if there exists such that for all . is called closed if and , then .

Clearly, we note that the class of relatively nonexpansive mappings is contained in a class of relatively asymptotically nonexpansive mappings but the converse is not true. Now, we give an example of relatively asymptotically nonexpansive mapping which is not relatively nonexpansive.

Example 1.2 (see [16]). Let , where , and . Then is closed and convex subset of . Note that is not bounded. Obviously, is uniformly convex and uniformly smooth. Let and be sequences of real numbers satisfying the following properties:(i), , , and ,(ii) and for all and (e.g., , ). Then, the map defined by for all , is uniformly Lipschitzian which is relatively asymptotically nonexpansive but not relatively nonexpansive (see [16] for the details). Note also that .

In 2005, Matsushita and Takahashi [17] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping in a Banach space : They proved that, if the sequence is bounded above from one, then the sequence generated by (1.13) converges strongly to .

Recently, many authors have considered the problem of finding a common element of the fixed-point set of relatively nonexpansive mapping and the solution set of variational inequality problem for -inverse monotone mapping (see, e.g., [12, 13, 18–20]).

In [21], Iiduka and Takahashi studied the following iterative scheme for a common point of solution of a variational inequality problem for -inverse strongly monotone mapping and fixed point of nonexpansive mapping in a Hilbert space : where is sequences satisfying certain condition. They proved that the sequence converges strongly to an element of provided that .

In the case that is a Banach space more general than Hilbert spaces, Zegeye et al. [12] studied the following iterative scheme for a common point of solution of a variational inequality problem for -inverse strongly monotone mapping and fixed point of a closed relatively quasi-nonexpansive mapping in a 2-uniformly convex and uniformly smooth Banach space : where is sequences satisfying certain condition. They proved that the sequence converges strongly to an element of provided that and satisfies for all and .

Furthermore, Zegeye and Shahzad [22] studied the following iterative scheme for common point of solution of a variational inequality problem for -inverse strongly monotone mapping and fixed point of a relatively asymptotically nonexpansive mapping on a closed convex and bounded set which is a subset of a real Hilbert space : where is the metric projection from into and and are sequences satisfying certain condition. Then, they proved that the sequence converges strongly to an element of provided that and satisfies for all and .

Remark 1.3. We again remark that the computation of in Algorithms (1.13), (1.15), and (1.16) is not simple because of the involvement of computation of from for each .
It is our purpose in this paper to introduce an iterative scheme which converges strongly to a common point of solution of variational inequality problem for a monotone operator satisfying appropriate conditions, for some nonempty closed convex subset of a Banach space and fixed points of uniformly -Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme which converges strongly to a common zero of finite family of monotone mappings. Our scheme does not involve computation of from or , for each . Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

2. Preliminaries

Let be a normed linear space with . The modulus of smoothness of is the function defined by The space is said to be smooth if , for all , and is called uniformly smooth if and only if .

The modulus of convexity of is the function defined by is called uniformly convex if and only if , for every .

In the sequel, we will need the following results.

Lemma 2.1 (see [23]). Let be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space . If is continuous monotone mapping, then is closed and convex.

Lemma 2.2. Let be a closed convex subset of a uniformly convex and smooth Banach space , and let be continuous relatively asymptotically nonexpansive mapping from into itself. Then, is closed and convex.

Lemma 2.3 (see [1]). Let be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space , and let . Then, for all ,

Lemma 2.4 (see [2]). Let be a real smooth and uniformly convex Banach space, and let and be two sequences of . If either or is bounded and as , then , as .

We make use of the function defined by studied by Alber [1]. That is, for all and . We know the following lemma.

Lemma 2.5 (see [1]). Let be reflexive strictly convex and smooth Banach space with as its dual. Then, for all and .

Lemma 2.6 (see [1]). Let be a convex subset of a real smooth Banach space . Let . Then if and only if

Lemma 2.7 (see [12]). Let be a uniformly convex Banach space and a closed ball of . Then, there exists a continuous strictly increasing convex function with such that for such that and , for .

Let be a smooth and strictly convex Banach space, a nonempty closed convex subset of , and a monotone operator satisfying for . Then, we can define the resolvent of by In other words, for . We know that is single-valued mapping from into , for all and and , where is the set of fixed points of (see, [4]).

Lemma 2.8 (see [24]). Let be a smooth and strictly convex Banach space, a nonempty closed convex subset of , and a monotone operator satisfying (2.8) and is nonempty. Let be the resolvent of . Then, for each , for all and , that is, is relatively nonexpansive.

Lemma 2.9 (see [25]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and satisfying the following conditions: , and . Then, .

Lemma 2.10 (see [26]). Let be sequences of real numbers such that there exists a subsequence of such that for all . Then, there exists a nondecreasing sequence such that , and the following properties are satisfied by all (sufficiently large) numbers : In fact, .

3. Main Result

We note that, as it is mentioned in [27], if is a subset of a real Banach space and is a mapping satisfying for all and , then

In fact, clearly, . Now, we show that . Let , then we have by hypothesis that which implies that . Hence, . Therefore, . Now we prove the main theorem of our paper.

Theorem 3.1. Let be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space . Let be a monotone mapping satisfying (2.8) and for all and . Let be a uniformly -Lipschitzian relatively asymptotically nonexpansive mapping with sequence . Assume that is nonempty. Let be the resolvent of and a sequence generated by where such that , such that . Then, converges strongly to an element of .

Proof. Let . Then, from (3.2), Lemma 2.3, and property of , we get that Now, from (3.2) and relatively asymptotically nonexpansiveness of , relatively nonexpansiveness of , property of , and (3.3), we get that where , since there exists such that for all and for some satisfying . Thus, by induction, which implies that , and hence is bounded. Now, let . Then we have that . Using Lemmas 2.3, 2.5, and property of , we obtain that Furthermore, from (3.2), Lemma 2.7, relatively asymptotically nonexpansiveness of , relatively nonexpansiveness of , and (3.6), we have that for some , where .
Similarly, from (3.7), we obtain that for some . Note that satisfies that and .
Now, the rest of the proof is divided into two parts.
Case 1. Suppose that there exists such that is nonincreasing for all . In this situation, is then convergent. Then, from (3.8) and (*), we have that which implies, by the property of , that and, hence, since is uniformly continuous on bounded sets, we obtain that Furthermore, Lemma 2.3, property of , and the fact that as imply that and hence Therefore, from (3.12) and (3.14), we obtain that But observe that from (3.2) and (3.11), we have as . Thus, as is uniformly continuous on bounded sets, we have that which implies from (3.14) that , as , and that Furthermore, since we have from (3.17), (3.15), and uniform continuity of that Since is bounded and is reflexive, we choose a subsequence of such that and . Then, from (3.14) and (3.15) we get that Thus, since satisfies condition (N3), we obtain from (3.19) that and the fact that is relatively nonexpansive and implies that , and, hence, using (3.1), we obtain that .
Therefore, from the above discussions, we obtain that . Hence, by Lemma 2.6, we immediately obtain that . It follows from Lemma 2.9 and (3.9) that , as . Consequently, .Case 2. Suppose that there exists a subsequence of such that for all . Then, by Lemma 2.10, there exists a nondecreasing sequence such that , and for all . Then, from (3.8), (*) and the fact , we have Thus, using the same proof as in Case 1, we obtain that , , as , and, hence, we obtain that Then, from (3.9), we have that Since , (3.24) implies that In particular, since , we get Then, from (3.23) and the fact that , we obtain , as . This together with (3.24) gives , as . But , for all , thus we obtain that . Therefore, from the above two cases, we can conclude that converges strongly to and the proof is complete.