Further Investigation on the Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities with 𝑘-Strict Pseudocontractions

Abstract

We modify the relaxed hybrid steepest-descent methods to the case of variational inequality for finding a solution over the set of common fixed points of a finite family of strictly pseudocontractive mappings. The strongly monotone property defined on cost operator was extended to relaxed cocoercive in convergence analysis. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.

1. Introduction

Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of and let be a nonlinear mapping. We consider the following variational inequality problem: find such that which was introduced by Stampacchia [1], has emerged as an interesting and fascinating branch of mathematical and engineering sciences. The ideas and techniques of variational inequalities are being applied in structural analysis, economics, optimization, and operations research fields. It has been shown that variational inequalities provide the most natural, direct, simple, and efficient framework for a general treatment of some unrelated problems arising in various fields of pure and applied sciences. In recent years, there have been considerable activities in the development of numerical techniques including projection methods, Wiener-Hopf equations, auxiliary principle, and descent framework for solving variational inequalities; see [2–15] and the references therein.

Recall that a self-mapping is called a -strict pseudocontraction if there exists a constant such that We use to denote the fixed point set of ; that is, . As , is said to be nonexpansive; that is, is said to be pseudocontractive if and is also said to be strongly pseudocontractive if there exists a positive constant such that is pseudocontractive. Clearly, the class of -strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. We remark also that the class of strongly pseudocontractive mappings is independent of the class of -strict pseudocontractions (see, e.g., [16, 17]).

It is well known that a variational inequality in a smooth Banach space is equivalent to a fixed-point equation containing a sunny nonexpansive retraction from any point of the space onto the feasible set, assumed usually to be closed and convex. However, the sunny nonexpansive retraction is not easy to compute, due to the complexity of the feasible set [18]. In order to overcome this drawback in a Hilbert space, where the retraction is a metric projection, Xu [19] assumed that the feasible set was the set of common fixed points of a finite family of nonexpansive mappings and introduced a hybrid steepest-descent method. To be more precise, Xu proposed the following steepest-descent method cyclically in combination with each nonexpansive mapping of a finite family: where , is a self-adjoint, linear, bounded, and strongly positive mapping, is some fixed point, and denotes the identity operator of , and proved that, under the following conditions:  (C1) ; (C2) ; (C3)   or  .Assume, in addition, that (C4) holds.Then the sequence , generated by (4), converges strongly to the unique solution of variational inequality (1) with .

Recently, Zeng et al. [11] proposed a hybrid steepest-descent method with variable parameters for variational inequalities, and Yao et al. [12] analyzed the strong convergence of three-step relaxed hybrid steepest-descent methods for variational inequalities. Very recently, Liu and Cui [20] showed that the condition is sufficient for (C4) as .

In 2011, Buong and Duong [13] proposed an explicit iterative algorithm for a class of variational inequalities. To be more precise, they proved the following theorem.

Theorem BD. Let be a real Hilbert space and be a map such that, for some positive constants and , is -Lipschitz continuous and -strongly monotone. Let be nonexpansive self-maps of such that . Then the sequence defined by

where the parameters for , , , and conditions (C1), (C2), and (C5) , are satisfied, converges strongly to the unique solution of variational inequality (1).

It is worth mentioning that almost all the results regarding the existence and convergence of the steepest-descent methods for variational inequality requires that the underlying operator must be strongly monotone and Lipschitzian continuous. These strict conditions rule out many applications of these methods and their various modifications. This fact motivates to develop other methods or modify the steepest-descent methods with more weaker or general conditions.

In this paper, inspired and motivated by research going on in this area, we introduce a new parallel relaxed hybrid steepest-descent method in combination with a finite family of strict pseudocontractive mappings, which is defined in the following way: where is a finite family of -strict pseudocontractions, , , and are some positive sequences in .

Our purpose is not only to extend the relaxed hybrid steepest-descent methods to the case of variational inequality in combination with a finite family of -strictly pseudocontractive mappings, but also to remove conditions (C3), (C4), and (C5) , in convergence analysis. Moreover, the strongly monotone property defined on cost operator was extended to relaxed -cocoercive. Our results presented in this paper improve and extend the corresponding ones of [5, 11–13, 19–21].

2. Preliminaries

Let be a nonempty closed convex subset of real Hilbert . We use and to denote strong and weak convergence of sequences in , respectively. In order to prove main results, we need the following concepts and results.

Definition 1. A mapping is called -Lipschitzian continuous if there exists constant such that

Definition 2. A mapping is called (i)-strongly monotone if there exists a constant such that (ii)-inverse strongly monotonic if there exists a constant such that (iii)relaxed -cocoercive if there exist constants and such that

Remark 3. Obviously, a -strongly monotonic mapping must be a relaxed -cocoercive mapping whenever , but the converse is not true. Therefore the class of the relaxed -cocoercive mappings is a more general class than -strongly monotone.

Lemma 4 (see [22]). Let be a real Hilbert space, for any fixed ; then (i), for all;(ii), for all.

Lemma 5 (see [21]). Let be a -strict pseudocontraction. For , define by for each . Then is a nonexpansive mapping such that .

Lemma 6 (see [23]). Let and be two bounded sequences in a Banach space such that for all integers . Let be a sequence in with . Suppose that Then .

Lemma 7 (demiclosed principle [24]). Let be a nonempty closed convex subset of a real Hilbert space and let be nonexpansive mapping. Then the mapping is demiclosed; that is, and implies .

Lemma 8 (see [19]). Assume is a sequence of nonnegative real numbers such that where is a sequence in (0,1) and is a real sequence such that (i);(ii) or .Then .

Proposition 9 (see Acedo and Xu [25]). Let be a nonempty closed convex subset of a Hilbert space . Given an integer , assume that is a finite family of -strict pseudocontractions. Suppose that is a positive sequence such that . Then is a -strict pseudocontraction with .

Proposition 10 (see Acedo and Xu [25]). Let and be given as in Proposition 9. Then .

3. Main Results

Lemma 11. Let be a real Hilbert space and let be a relaxed -cocoercive and -Lipschitzian continuous mapping. For and , we have where and .

Proof. By the properties defined on , we obtain where . From , it is easy to obtain that . We immediately conclude the desired results. This completes the proof.

Theorem 12. Let be a real Hilbert space and let be a relaxed -cocoercive and -Lipschitzian continuous mapping. Let be a finite family of -strict pseudocontractions such that . Suppose are finite sequences of positive numbers such that for all and , . In addition, for a given point , (C1), (C2), and the following control conditions are satisfied: (i);(ii) and , where ;(iii).Then the sequence generated by (7) converges strongly to the unique element of the variational inequality (1).

Proof. Putting , we have that is a -strict pseudocontraction and by Propositions 9 and 10, where .
First, we show that is nonexpansive. Indeed, for each , we have It follows from that that the mapping is nonexpansive. By Lemma 5, we see that Note that and for each as . From (7) and Lemma 11, we obtain where . It follows from induction that which shows that sequence is bounded and so are and .
Next, put . Then, from (7) and , we have that where . Moreover, we note that where . Combining (21) and (22), we obtain By (C1) and conditions (ii)-(iii), we have that It follows from (i) and Lemma 6 that From (20), we have that and
On the other hand, we note that which implies that This together with (i), (C1), and (26), we obtain that is, Furthermore, we observe that It follows from condition (ii) that
By condition (iii), we may assume that as for every . It is easily seen that each and . Define ; then is a -strict pseudocontraction such that by Propositions 9 and 10. Consequently, which implies that Combining (32) and (34), we obtain Define by . By condition (ii) again, we have . Then, is nonexpansive with by Lemma 5. Notice that It follows from (29), (34), and (35) that
Now, we show that , where . Since is bounded, there exists a subsequence of . Without loss of generality, we suppose that the sequence converges weakly to such that It follows from (37) and Lemma 7 that . Consequently, by (1), it implies that Similarly, by (30), there exists a subsequence of that converges weakly to such that
Finally, we prove that as as follows. From (7), Lemmas 4 and 11 again, we have By virtue of Lemma 8 with (C1), (C2), and (40), we obtain that that is, the sequence generated by (7) converges strongly to the unique element of the variational inequality (1). This completes the proof.

Theorem 13. Let be a real Hilbert space and let be a relaxed -cocoercive and -Lipschitzian continuous mapping with . Let be a finite family of nonexpansive mappings such that . Define sequence by in the following manner: where and are finite sequences of positive numbers such that for all . If (C1), (C2), and the following control conditions are satisfied: (i),(ii),then converges strongly to the unique element of the variational inequality (1).

Proof. Clearly, a -strict pseudocontraction reduces to a nonexpansive mapping as . Putting , iterative scheme (7) reduces to (43), and the desired conclusion follows immediately from Lemma 11 and Theorem 12. This completes the proof.

Theorem 14. Let be a real Hilbert space and let be a -strongly monotone and -Lipschitzian continuous mapping with . Let be a -strict pseudocontractions such that . Define sequence by in the following manner: where , , and are some positive sequences in . If (C1), (C2), and the following control conditions are satisfied: (i),(ii) and ,then converges strongly to the unique element of the variational inequality (1).

Proof. By Remark 3, a relaxed -cocoercive mapping includes -strongly monotonic mapping as a special case. Putting , that is, , iterative scheme (7) reduces to (44), and the desired conclusion follows immediately from Lemma 11 and Theorem 12. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Supported by the National Science Foundation of China (11001287), Natural Science Foundation Project of Chongqing (CSTC, 2012jjA00039), and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130712, KJ130731).