1. Introduction

Let be a real Banach space and let be a nonempty closed convex subset of . Recall that a mapping is said to be nonexpansive if , for all . We denote by the set of fixed points of .

Recently, Yao et al. [1] considered the following algorithms: and for an arbitrary point , where is a sunny nonexpansive retraction, is a strongly positive bounded linear operator and are -inverse-strongly accretive and -inverse-strongly accretive operators, respectively. They proved that the defined by (1.1) and defined by (1.2) converge strongly to a unique solution of the variational inequality . Furthermore, they proved that the above algorithms converge strongly to some solutions of a system of nonlinear inequalities, which involves finding such that For related works, please see [2–5] and the references therein.

In this paper, we introduce two general algorithms (3.3) and (3.22) (defined below) and prove that the proposed algorithms strongly converge to which solves the variational inequality , where is a -Lipschitzian and -strongly accretive operator. It is worth pointing out that our proofs contain some new techniques.

2. Preliminaries

Let be a real Banach space with norm and let be its dual space. The value of and will be denoted by . For the sequence in , we write to indicate that the sequence converges weakly to . means that converges strongly to .

Let , a mapping of into is said to be -strongly accretive if there exists such that for all . A mapping from into is said to be -Lipschitzian if, for , for all . From the definition of (see [1]), we note that a strongly positive bounded linear operator is a -Lipschitzian and -strongly accretive operator.

Let . A Banach space is said to be uniformly convex if for each , there exists such that for any , It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space is said to be smooth if the limit exists for all . It is said to be uniformly smooth if the limit (2.4) is attained uniformly for . Also, we define a function called the modulus of smoothness of as follows: It is known that is uniformly smooth if and only if . Let be a fixed real number with . Then a Banach space is said to be -uniformly smooth if there exists a constant such that for all .

In order to prove our main results, we need the following lemmas.

Lemma 2.1 (see [6]). Let be a given real number with and let be a -uniformly smooth Banach space. Then for all , where is the -uniformly smooth constant of and is the generalized duality mapping from into defined by for all .

Lemma 2.2 (see [7]). Let be a closed convex subset of a smooth Banach space , let be a nonempty subset of and be a retraction from onto . Then is sunny and nonexpansive if and only if for all and .

Lemma 2.3 (see [8]). Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and let be a nonexpansive mapping of into itself. If is a sequence of such that and , then is a fixed point of .

Lemma 2.4 (see [9, 10]). Let be a sequence of nonnegative real numbers satisfying where , and satisfy the following conditions: (i) and , (ii) or , and (iii) , . Then .

Lemma 2.5 (see [11]). Let and be bounded sequences in Banach space and be a sequence in which satisfies the following condition: Suppose that ,, and . Then .

In addition, we need the following extension of Lemma 2.5 in Wang and Hu [2] in a 2-uniformly smooth Banach space.

Lemma 2.6. Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space . Let be a -Lipschitzian and -strongly accretive operator with and . Then is a contraction with contraction coefficient .

Proof. By Lemma 2.1, we have for all . From and , we have where . Hence is a contraction with contraction coefficient .

3. Main Results

Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space . Let be a nonexpansive mapping with . Let be a -Lipschitzian and -strongly accretive operator with . Let and , consider a mapping on defined by where is a sunny nonexpansive retraction from onto . It is easy to see that is a contraction. Indeed, from Lemma 2.6, we have for all . Therefore, the following implicit method is well defined:

Theorem 3.1. The net generated by the implicit method (3.3) converges in norm, as to the unique solution of the variational inequality:

Proof. We first show that the solution set of (3.4) is singleton. As a matter of fact, we assume that and both are solutions to (3.4), then Adding (3.5) to (3.6), we get The strong accretive of implies that , and the uniqueness is proved. Below we use to denote the unique solution of (3.4).
Next, we prove that is bounded. Taking , from (3.3) and using Lemma 2.6, we have that is, Observe that From , we may assume, without loss of generality, that , where is an arbitrarily small positive number. Thus, we have to be continuous, for all . Therefore, we obtain From (3.9) and (3.11), we have bounded and so is .
On the other hand, from (3.3), we obtain
Next, we show that is relatively norm-compact as . Assume that such that as . Put . It follows from (3.12) that
For a given , by (3.3) and using Lemma 2.2, we have By (3.14) and using Lemma 2.6, we have that is, In particular,
Since is bounded, without loss of generality, we may assume that converges weakly to a point . Noticing (3.13) we can use Lemma 2.3 to get . Therefore we can substitute for in (3.17) to get Consequently, the weak convergence of to actually implies that . This has proved the relative norm compactness of the net as .
We next show that solves the variational inequality (3.4). Observe that For any , we have where .
Now replacing in (3.20) with and letting , we have That is, is a solution of (3.4), hence by uniqueness. In summary, we have shown that each cluster point of (at ) equals . Therefore, as .

Theorem 3.2. Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space with a weakly sequentially continuous duality mapping . Let be a -Lipschitzian and -strongly accretive operator with . Suppose that is a nonexpansive mapping with . Let be a sunny nonexpansive retraction from onto . Let and be two real sequences in and satisfy the conditions:(A1) and ,(A2).
For given arbitrarily, let the sequence be generated by Then the sequence strongly converges to which solves the variational inequality (3.4).

Proof. We proceed with the following steps.
Step 1. We claim that is bounded. From , we may assume, without loss of generality, that for all . In fact, let , from (3.22) and using Lemma 2.6, we have where . Then from (3.22) and (3.23), we obtain By induction, we have where . Therefore, is bounded. We also obtain that and are bounded.Step 2. We claim that . Observe that Therefore, we have From (3.22), (3.27), and using Lemma 2.5, we have .Step 3. We claim that . Observe that Hence, from Step 2 and , we have Step 4. We claim that , where and is defined by (3.3). Since is bounded, there exists a subsequence of which converges weakly to . From Step 3, we obtain . From Lemma 2.3, we have . Hence, using Theorem 3.1, we have and Step 5. We claim that converges strongly to . From (3.22) and using Lemma 2.2, we have Observe that that is, By (3.22) and (3.33), we have where , . It is easy to see that , and . Hence, by Lemma 2.4, the sequence converges strongly to . From and Theorem 3.1, we have to be the unique solution of the variational inequality (3.4).

Taking and , where and , we obtain the following theorems immediately.

Corollary 3.3 (see [1, Theorem 3.5]). The net generated by the implicit method (1.1) converges in norm, as , to the unique solution of variational inequality

Corollary 3.4 (see [1, Theorem 3.7]). Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space and let be a sunny nonexpansive retraction from onto . Let the mappings be -inverse-strongly accretive and -inverse-strongly accretive operators, respectively. Let be a strongly positive linear bounded operator with coefficient . For given , let the sequence be generated iteratively by (1.2). Suppose that the sequences and satisfy the conditions (A1) and (A2), then converges strongly to which solves the variational inequality (3.35).

Acknowledgments

Supported by the Natural Science Foundation of Yancheng Teachers University under Grant (11YCKL009) and Professor and Doctor Foundation of Yancheng Teachers University under Grant (11YSYJB0202).