Journal of Applied Mathematics

Volume 2012 (2012), Article ID 634927, 15 pages

doi:10.1155/2012/634927

Research Article

Hybrid Algorithms of Nonexpansive Semigroups for Variational Inequalities

Peixia Yang,¹ Yonghong Yao,¹ Yeong-Cheng Liou,² and Rudong Chen¹

¹Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

²Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

Received 19 March 2012; Accepted 30 April 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Peixia Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Two hybrid algorithms for the variational inequalities over the common fixed points set of nonexpansive semigroups are presented. Strong convergence results of these two hybrid algorithms have been obtained in Hilbert spaces. The results improve and extend some corresponding results in the literature.

1. Introduction

Let be a real Hilbert space and a nonempty closed convex subset of . Recall that a mapping is called nonexpansive if

for every . A family of mappings from into itself is called a nonexpansive semigroup on if it satisfies the following conditions:

(i) for all ,(ii) for all ,(iii) for all ,(iv)for all is continuous.

We denote by the set of all common fixed points of , that is, . It is known that is closed and convex.

Approximation of fixed points of nonexpansive mappings has been considered extensively by many authors, see, for instance, [1–18]. Nonlinear ergodic theorem for nonexpansive semigroups have been researched by some authors, see, for example, [19–23]. Our main purpose in the present paper is devoted to finding the common fixed points of nonexpansive semigroups.

Let be a nonlinear operator. The variational inequality problem is formulated as finding a point such that

Now it is well known that VI problem is an interesting problem and it covers as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance. Several numerical methods including the projection and its variant forms have been developed for solving the variational inequalities and related problems, see [24–41].

It is clear that the is equivalent to the fixed point equation

where is the projection of onto the closed convex set and is an arbitrarily fixed constant. So, fixed point methods can be implemented to find a solution of the provided satisfies some conditions and is chosen appropriately. The fixed point formulation (1.3) involves the projection , which may not be easy to compute, due to the complexity of the convex set . In order to reduce the complexity probably caused by the projection , Yamada [24] (see also [42]) recently introduced a hybrid steepest-descent method for solving the .

Assume that is an -strongly monotone and -Lipschitzian mapping with on . An equally important problem is how to find an approximate solution of the if any. A great deal of effort has been done in this problem; see [43, 44].

Take a fixed number such that . Assume that a sequence of real numbers in satisfies the following conditions:

(C1),(C2),(C3).

Starting with an arbitrary initial guess , one can generate a sequence by the following algorithm:

Yamada [24] proved that the sequence generated by (1.4) converges strongly to the unique solution of the . Xu and Kim [30] proved the strong convergence of to the unique solution of the if satisfies conditions (C1), (C2), and (C4): , or equivalently, . Recently, Yao et al. [25] presented the following hybrid algorithm:

where is a -Lipschitzian and -strongly monotone operator on and is a -mapping. It is shown that the sequences and defined by (1.5) converge strongly to , which solves the following variational inequality:

Very recently, Wang [26] proved that the sequence generated by the iterative algorithm (1.5) converges to a common fixed point of an infinite family of nonexpansive mappings under some weaker assumptions.

Motivated and inspired by the above works, in this paper, we introduce two hybrid algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces. We prove that the presented algorithms converge strongly to a common fixed point of . Such common fixed point is the unique solution of some variational inequality in Hilbert spaces.

2. Preliminaries

In this section, we will collect some basic concepts and several lemmas that will be used in the next section.

Suppose that is a real Hilbert space with inner product and norm . For the sequence in , we write to indicate that the sequence converges weakly to . means that converges strongly to . We denote by the weak -limit set of , that is

Let be a nonempty closed convex subset of a real Hilbert space . A mapping is called -Lipschitzian if there exists a positive constant such that

is said to be -strongly monotone if there exists a positive constant such that

The following equalities are well known:

for all and (see [45]).

In the sequel, we will make use of the following well-known lemmas.

Lemma 2.1 (see [46]). Let be a nonempty bounded closed convex subset of and let be a nonexpansive semigroup on . Then, for any ,

Lemma 2.2 (see [47]). Assume that is a nonexpansive mapping. If has a fixed point, then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here, is the identity operator of .

Lemma 2.3 (see [27]). Let be a real sequence satisfying . Assume that and are bounded sequences in Banach space , which satisfy the following condition: . If , then .

Lemma 2.4 (see [48]). Let be a -Lipschitzian and -strongly monotone operator on a Hilbert space with and . Then, is a contraction with contraction coefficient .

Lemma 2.5 (see [49]). Let be a sequence of nonnegative real numbers satisfying

where and satisfy the following conditions:

(i) and ,(ii), (iii).

Then, .

3. Main Results

In this section we will show our main results.

Theorem 3.1. Let be a real Hilbert space. Let be a nonexpansive semigroup such that . Let be a -Lipschitzian and -strongly monotone operator on with . Let be a continuous net of positive real numbers such that . Putting , for each , let the net be defined by the following implicit scheme:

Then, as , the net converges strongly to a fixed point of , which is the unique solution of the following variational inequality:

Proof. First, we note that the net defined by (3.1) is well defined. We define a mapping

It follows that

Obviously, is a contraction. Indeed, from Lemma 2.4, we have

for all . So it has a unique fixed point. Therefore, the net defined by (3.1) is well defined.

We prove that is bounded. Taking and using Lemma 2.4, we have

It follows that

Observe that

Thus, (3.7) and (3.8) imply that the net is bounded for small enough . Without loss of generality, we may assume that the net is bounded for all . Consequently, we deduce that is also bounded.

On the other hand, from (3.1), we have

This together with Lemma 2.1 implies that

Let be a sequence such that as . Put . Since is bounded, without loss of generality, we may assume that converges weakly to a point . Noticing (3.10), we can use Lemma 2.2 to get .

Again, from (3.1), we have

Therefore,

It follows that

Thus, implies that .

Again, from (3.12), we obtain

It is clear that , , and . We deduce immediately from (3.14) that

which is equivalent to its dual variational inequality

That is, is a solution of the variational inequality (3.2).

Suppose that and both are solutions to the variational inequality (3.2); then

Adding up (3.17) and the last inequality yields

The strong monotonicity of implies that and the uniqueness is proved. Later, we use to denote the unique solution of (3.2).

Therefore, by uniqueness. In a nutshell, we have shown that each cluster point of equals . Hence as . This completes the proof.

Next we introduce an explicit algorithm for finding a solution of the variational inequality (3.2).

Algorithm 3.2. For given arbitrarily, define a sequence iteratively by

where and are sequences in and is a sequence in .

Theorem 3.3 3.3. Let be a real Hilbert space. Let be a -Lipschitzian and -strongly monotone operator on with . Let be a nonexpansive semigroup with . Assume that

(i) and ,(ii) and ,(iii), for some .

Then, the sequences and generated by (3.19) converge strongly to if and only if , where solves the variational inequality (3.2).

Proof. The necessity is obvious. We only need to prove the sufficiency. Suppose that . First, we show that is bounded. In fact, letting , we have

From condition (i), without loss of generality, we can assume that for all . By (3.19) and Lemma 2.4, we have

where .

Then, from (3.20) and (3.21), we obtain

Since , we have by induction

where . Hence, is bounded. We also obtain that , and are all bounded.

Define for all . Observe that

Next, we estimate . As a matter of fact, we have

where . From (3.24) and (3.25), we have

Namely,

Since and , we get

Consequently, by Lemma 2.3, we deduce . Therefore,

Next, we claim that . Observe that

Note that

It follows that

By Lemma 2.1, (3.30), and (3.32), we derive

Next, we show that , where and is defined by . Since is bounded, there exists a subsequence of that converges weakly to . It is clear that . From Lemma 2.2, we have . Hence, by Theorem 3.1, we have

Finally, we prove that converges strongly to . From (3.19), we have

where and . Obviously, we can see that and . Hence, all conditions of Lemma 2.5 are satisfied. Therefore, we immediately deduce that the sequence converges strongly to .

Observe that

Consequently, it is clear that converges strongly to . From and Theorem 3.1, we get that is the unique solution of the variational inequality

This completes the proof.

Acknowledgments

Y. Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Y.-C. Liou was partially supported by NSC 100-2221-E-230-012. R. Chen was supported in part by NSFC 11071279.

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