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Hybrid SteepestDescent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
Fixed Point Theory and Applications volume 2010, Article number: 673932 (2010)
Abstract
Let be a real Hilbert space and let be a boundedly Lipschitzian and strongly monotone operator. We design three hybrid steepest descent algorithms for solving variational inequality of finding a point such that , for all , where is the set of fixed points of a strict pseudocontraction, or the set of common fixed points of finite strict pseudocontractions. Strong convergence of the algorithms is proved.
1. Introduction
Let be a real Hilbert space with the inner product and the norm , let be a nonempty closed convex subset of and let be a nonlinear operator. We consider the problem of finding a point such that
This is known as the variational inequality problem (i.e., initially introduced and studied by Stampacchia [1] in 1964. In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics, optimization, operations research, and engineering sciences; see [1–6] and the references therein.
Yamada [7] proposed hybrid methods to solve , where is composed of fixed points of a nonexpansive mapping; that is, is of the form
where is a nonexpansive mapping (i.e., for all ), is Lipschitzian and strongly monotone.
He and Xu [8] proved that has a unique solution and iterative algorithms can be devised to approximate this solution if is a boundedly Lipschitzian and strongly monotone operator and is a closed convex subset of . In the case where is the set of fixed points of a nonexpansive mapping, they invented a hybrid iterative algorithm to approximate the unique solution of and this extended the Yamada's results.
The main purpose of this paper is to continue our research in [8]. We assume that is a boundedly Lipschitzian and strongly monotone operator as in [8], but is the set of fixed points of a strict pseudocontraction , or the set of common fixed points of finite strict pseudocontractions . For the two cases of , we will design the hybrid iterative algorithms for solving and prove their strong convergence, respectively. Relative definitions are stated as below.
Let be a nonempty closed and convex subset of a real Hilbert space , and , then
(1) is called Lipschitzian on , if there there exists a positive constant such that
(2) is called boundedly Lipschitzian on , if for each nonempty bounded subset of , there exists a positive constant depending only on the set such that
(3) is said to be strongly monotone on , if there exists a positive constant such that
(4) is said to be a strict pseudocontraction if there exists a constant such that
Obviously, the nonexpansive mapping class is a proper subclass of the strict pseudocontraction class and the Lipschitzian operator class is a proper subclass of the boundedly Lipschitzian operator class, respectively.
We will use the following notations:
(i) for weak convergence and for strong convergence,
(ii) denotes the weak limit set of
(iii) denotes a closed ball with center and radius .
2. Preliminaries
We need some facts and tools which are listed as lemmas below.
Lemma 2.1.
Let be a real Hilbert space. The following expressions hold:
(i)
(ii)
Lemma 2.2 (see [9]).
Assume that is a sequence of nonnegtive real numbers satisfying the property
If and satisfy the following conditions:
(i)
(ii)
(iii)
then
Lemma 2.3 (see [10]).
Let be a nonempty closed convex subset of a real Hilbert space and is a nonexpansive mapping. If a one has sequence in such that and then
Lemma 2.4 (see [11]).
Let be a nonempty closed convex subset of a real Hilbert space , if is a strict pseudocontraction, then the mapping is demiclosed at 0. That is, if is a sequence in such that and then
Lemma 2.5 (see [8]).
Assume that is a nonempty closed convex subset of a real Hilbert space , if is boundedly Lipschitzian and strongly monotone, then variational inequality (1.1) has a unique solution.
Lemma 2.6.
Assume that is a strict pseudocontraction, and the constant satisfies Let
then is nonexpansive and
Proof.
Using Lemma 2.1(i) and the conception of strict pseudocontraction, we get
so is nonexpansive. is obvious.
Lemma 2.7.
Assume that is a real Hilbert space, is a strict pseudocontraction such that and is a boundedly Lipschitzian and strongly monotone operator. Take arbitrarily and set . Denote by the Lipschitz constant of on and let
where the constants and are such that and , respectively, and is defined as in Lemma 2.6 above. Then restricted to is a contraction.
Proof.
If , that is, , by Lemma 2.6, we have
It suggests that . Since is Lipschitzian and strongly monotone on , using Lemma 2.6, we obtain
Therefore, restricted to that is a contraction with coefficient , where
Lemma 2.8 (see [11]).
Assume is a closed convex subset of a Hilbert space .
(i)Given an integer , assume that for each , is a strict pseudocontraction for some . Assume is a positive sequence such that . Then is a strict pseudocontraction, with

(ii)
Let , and be given as in (i) above. Suppose that , then
Lemma 2.9.
Assume that is a strict pseudocontraction for some let if , then
Proof.
We prove it by induction. For , set , . Obviously
Now we prove
, if , then the conclusion holds. In fact, we can claim that . From Lemma 2.6, we know that is nonexpansive and Take , then
Since , we get
Namely, that is,
Suppose that the conclusion holds for , we prove that
It suffices to verify
, . Using Lemma 2.6 again, take ,
Since , we have
this implies that
Namely,
From (2.19) and inductive assumption, we get
therefore
Substituting it into (2.19), we obtain Thus we assert that
3. Further Extension of Hybrid Iterative Algorithm
Yamada got the following result.
Theorem 3.1 (see [7]).
Assume that is a real Hilbert space, is nonexpansive such that and is strongly monotone and Lipschitzian. Fix a constant . Assume also that the sequence satisfies the following conditions:
(i);
(ii)
(iii), or .
Take arbitrarily and define by
then converges strongly to the unique solution of .
He and Xu [8] proved that has a unique solution if is a boundedly Lipschitzian and strongly monotone operator and is a closed convex subset of . Using this result, they were able to relax the global Lipschitz condition on in Theorem 3.1 to the weaker bounded Lipschitz condition and invented a hybrid iterative algorithm to approximate the unique solution of . Their result extended the Yamada's above theorem.
In this section, we mainly focus on further extension of our hybrid algorithm in [8]. Consider , where is composed of fixed points of a strict pseudocontraction such that and is still strongly monotone and boundedly Lipschitzian. Fix a point arbitrarily, set . Denote by the Lipschitz constant of on . Fix the constant satisfying . Assume also that the sequences and satisfy for a constant and , respectively. Let and , define by the scheme:
We have the following result.
Theorem 3.2.
If the sequences and satisfy the following conditions:
(i);
(ii)
(iii), , or ,
then generated by (3.2) converges strongly to the unique solution of.
Proof.
We prove that for all by induction. It is trivial that . Suppose we have proved , that is,
Using Lemma 2.7, We then derive from (3.2) and (3.3) that
However, since and we get
This together with (3.4) implies that
It proves that . Therefore, for all . Thus is bounded. It is not difficult to verify that the sequences and are all bounded.
By (3.2) and Lemma 2.7, we have
where . By Lemma 2.2 and conditions (i)–(iii), we conclude that
Since , it is straitforward from (3.2) that
On the other hand
By the condition and (3.8)–(3.10), we obtain
By Lemma 2.4 and (3.11), we obtain
Lemma 2.5 asserts that has a unique solution . Now we prove that . By Lemma 2.1(ii), (3.2), and Lemma 2.7, we have
Let us show that
In fact, there exists a subsequence such that
Without loss of generality, we may further assume that . Since is the unique solution of , we obtain
Finally conditions (i)–(iii) and (3.14) allow us to apply Lemma 2.2 to the relation (3.13) to conclude that
4. Parallel Algorithm and Cyclic Algorithm
In this section, we discuss the parallel algorithm and the cyclic algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite strict pseudocontractions.
Let be a real Hilbert space and a strongly monotone and boundedly Lipschitzian operator. Let be a positive integer and a strict pseudocontraction for some such that We consider the problem of finding such that
Since is a nonempty closed convex subset of , (4.1) has a unique solution. Throughout this section, is an arbitrary fixed point, , is the Lipschitz constant of on , the fixed constant satisfies , and the sequence belongs to .
Firstly we consider the parallel algorithm. Take a positive sequence such that and let
By using Lemma 2.8, we assert that is a strict pseudocontraction with and holds. Thus VI(4.1) is equivalent to VI and we can use scheme (3.2) to solve VI(4.1). In fact, taking in the scheme (3.2), we get the socalled parallel algorithm
Using Lemma 2.8 and Thorem 3.2, the following conclusion can be deduced directly.
Theorem 4.1.
Suppose that and satisfy the same conditions as in Theorem 3.2. Then the sequence generated by the parallel algorithm (4.3) converges strongly to the unique solution of (4.1).
For each let
where the constant such that . Then we turn to defining the cyclic algorithm as follows:
Indeed, the algorithm above can be rewritten as
where , namely, is one of circularly. For convenience, we denote (4.6) as
We get the following result
Theorem 4.2.
If satisfies the following conditions:
(i);
(ii);
(iii), or
then the sequence generated by (4.6) converges strongly to the unique solution of .
Proof.
We break the proof process into six steps.

(1)
. We prove it by induction. Definitely . Suppose , that is,
(4.8)
We have from , (4.8), and Lemma 2.7 that
where Observing , we get
This together with (4.9) implies that
It suggests that . Therefore, for all . We can also prove that the sequences , , are all bounded.

(2)
By (4.6) and Lemma 2.7, we have
(4.12)
where Since satisfies (i)–(iii), using Lemma 2.2, we get

(3)
By (4.3) and , we have
(4.14)
Recursively,
By Lemma 2.6, is nonexpansive, we obtain
Adding all the expressions above, we get
Using this together with the conclusion of step (2), we obtain

(4)
. Assume that such that , we prove . By the conclusion of step (3), we get
(4.19)
Observe that, for each is some permutation of the mappings , since are finite, all the full permutation are , there must be some permutation that appears infinite times. Without loss of generality, suppose that this permutation is , we can take a subsequence such that
It is easy to prove that is nonexpansive. By Lemma 2.3, we get
Using Lemmas 2.6 and 2.9, we obtain

(5)
In fact, there exists a subsequence such that
(4.23)
Without loss of generality, we may further assume that Since is the solution of , we obtain

(6)
By (4.6), Lemmas 2.1(ii), and 2.7, we obtain
(4.25)
From the conclusion of step (5) and Lemma 2.2, we get
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Acknowledgment
This research is supported by the Fundamental Research Funds for the Central Universities (GRANT:ZXH2009D021).
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He, S., Liang, X. Hybrid SteepestDescent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators. Fixed Point Theory Appl 2010, 673932 (2010). https://doi.org/10.1155/2010/673932
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Keywords
 Variational Inequality
 Parallel Algorithm
 Nonexpansive Mapping
 Monotone Operator
 Hybrid Algorithm