We introduce a new iterative scheme with a countable family of nonexpansive mappings for the variational inequality problems in Hilbert spaces and prove some strong convergence theorems for the proposed schemes.
1. Introduction
Let be a Hilbert space and be a nonempty closed convex subset of . Let be a nonlinear mapping. The classical variational inequality problem (for short, ) is to find a point such that
This variational inequality was initially studied by Kinderlehrer and Stampacchia [1]. Since then, many authors have introduced and studied many kinds of the variational inequality problems (inclusions) and applied them to many fields.
It is well known that, if is a strongly monotone and Lipschitzian mapping on , then the has a unique solution (see [2]).
Let be a mapping. Recall that a mapping is nonexpansive if
The set of fixed points of is denoted by . Recently, the iterative methods for nonexpansive mappings and some kinds of nonlinear mappings have been applied to solve the convex minimization problems (see [3–7]).
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on :
where is the fixed point set of a nonexpansive mapping on H, is a given point in and is a strongly positive operator, that is, there is a constant such that
Recently, for solving the variational inequality on , Marino and Xu [8] introduced the following general iterative scheme:
where is a strongly positive linear bounded operator on , is a contraction on and .
More precisely, they gave the following result.
Theorem 1 MX (see [8, Theorem 3.4]).
Let be generated by algorithm (1.5) with the sequence satisfying the following conditions:
(C1),
(C2),
(C3) either or .
Then the scheme defined by (1.5) converges strongly to an element which is the unique solution of the variational inequality for short, :
Let be a contraction with coefficient and let be two strongly positive linear bounded operators with coefficients and , respectively.
Motivated and inspired by the iterative sheme (1.5), Ceng et al. [9] introduced the following so-called hybrid viscosity-like approximation algorithms with variable parameters for nonexpansive mappings in Hilbert spaces.
Theorem 1 CGYone (see [9, Theorem 3.1]).
Let and . Let be a sequence in and be a sequence in . Starting with an arbitrary initial guess , generate a sequence by the following iterative scheme:
Assume that
(i),
(ii),
(iii)either or ,
(iv).
Then the scheme defined by (1.7) converges strongly to an element which is the unique solution of the variational inequality for short, :
Theorem 1 CGYtwo (see [9, Theorem 3.2]).
Let and . Let be a sequence in and be a sequence in . Starting with an arbitrary initial guess , generate a sequence by the following iterative scheme:
Assume that
(i),
(ii),
(iii)either or ,
(iv).
In addition, assume that
Then the scheme defined by (1.9) converges strongly to an element which is the unique solution of the variational inequality for short, :
In this paper, motivated and inspired by the above research results, we introduce a new iterative process with a countable family of nonexpansive mappings for the variational inequality problem in Hilbert spaces.
More precisely, let be a Hilbert space and be a countable family of nonexpansive mappings from to such that . Let be a contraction with coefficient and be strongly positive linear bounded operators with coefficients and , respectively. Let and with . Take three fixed numbers , and such that , and . For any , generate the iterative scheme by
We prove that the iterative scheme defined by (1.12) strongly converges to an element which is the unique solution of the variational inequality for short, :
2. Preliminaries
Let be a Hilbert space and be a nonexpansive mapping of into itself such that . For all and , we have
and hence
Let be a sequence in a Hilbert space and let . Throughout this paper, and denote that strongly converges to and converges weakly to a point , respectively.
Lemma 2.1 (see [10]).
Let be a closed convex subset of a Hilbert space and be a nonexpansive mapping from into itself. Then is demiclosed at zero, that is,
The following lemma is an immediate consequence of the equality:
Lemma 2.2.
Let be a real Hilbert space. Then the following identity holds:
Let , be the sequences of nonnegative real numbers and let . Suppose that is a sequence of real numbers such that
Assume that . Then the following results hold.
(1)If , where , then is a bounded sequence.
(2)If one has
then .
Lemma 2.4 (see [8]).
Let be a real Hilbert space, be a contraction with coefficient and be a strongly positive linear bounded operator with coefficient . Then, for any with ,
that is, is strongly monotone with coefficient .
Lemma 2.5.
Assume is a strongly monotone linear bounded operator on a Hilbert space with coefficient . Take a fixed number such that . Then .
Proof.
The proof method is mainly from the idea of Marino and Xu [8, Lemma 2.5]. It is known that the norm of a linear bounded self-adjoint operator on is as follows:
Now, for all with , we see that (here denotes zero point in )
This completes the proof.
Remark 2.6.
Lemma 2.5 still holds if is a strongly positive linear bounded operator (see [8, Lemma 2.5]). That is, Lemma 2.5 in this section and Lemma 2.5 in [8] both hold when is a strongly monotone linear bounded operator or a strongly positive linear bounded one because an operator on a Hilbert space is strongly monotone linear if and only if it is strongly positive linear.
In fact, if is a strongly monotone linear operator with coefficient on a Hilbert space , then, for all ,
which shows that is strongly positive linear. Assume that is a strongly positive linear operator with coefficient on . Then, for all ,
which shows that is strongly monotone and linear.
3. Main Results
Let be a Hilbert space and be a nonempty closed and convex subset of . Let be a contraction with coefficient . Let be strongly positive linear bounded operator with coefficient and , respectively. Take a fixed number such that . Then, from Lemma 2.4, it follows that is strongly monotone with coefficient . For any fixed numbers and , we have , which can be seen easily from the following:
Moreover, observe that
which implies that is Lipschitzian with coefficient .
On the other hand, from Lemma 2.4, it follows that
which implies that is strongly monotone with coefficient . Hence the variational inequality (for short, )
has the unique solution.
Let be a nonexpansive mapping. Take two fixed numbers and such that and and, for all , define a mapping by
Then we have the following results.
Lemma 3.1.
If , then is a contraction with coefficient , where , that is,
Proof.
From Lemma 2.5 and Remark 2.6, it follows that, for all ,
This completes the proof.
Let be a countable family of nonexpansive mappings from into itself such that . Since each is closed and convex, then is closed and convex.
Throughout this paper, let be a contraction with coefficient . Let be strongly positive linear bounded mapping with coefficient and , respectively. Take a fixed number such that . Suppose that , (assuming that such that ( is nonempty), with and with .
Now, we can rewrite the iterative scheme (1.12) as follows:
where . Then, by Lemma 3.1, for all , we have
where .
Lemma 3.2.
If is strictly decreasing, then the scheme defined by (3.9) is bounded.
Proof.
Since , it follows from (3.10) that, for all ,
By induction, we obtain
Hence is bounded and so are and for each . This completes the proof.
Lemma 3.3.
If is strictly decreasing and the following conditions hold:
then
Proof.
By the iterative scheme (3.9), we have
and hence
where is a constant. Since , there exists a constant such that for all . Therefore, we have
Put . Since is a strictly decreasing sequence and , we have . By Lemma 2.3, it follows that as . This completes the proof.
Lemma 3.4.
If is strictly decreasing and the following conditions hold:
then
Proof.
By the iterative scheme (3.9), we have
that is,
Hence, for any , we get
Since each is nonexpansive, it follows from (2.2) that
Hence, combining (3.21) with (3.20), it follows that
which implies that
Since each and , then we have
that is,
Since and are both bounded, there exists a constant such that
By Lemma 3.3 and the assumption condition , it follows that
This completes the proof.
Finally, we give the main result in this paper.
Theorem 3.5.
If is strictly decreasing and the following conditions hold:
then the scheme defined by (3.9) converges strongly to an element which is the unique solution of the variational inequality :
Proof.
First, we prove that .
To prove this, we pick a subsequence of such that
Without loss of generality, we may, further, assume that for some . From Lemmas 2.1 and 3.3, it follows that for each and so . Since is the unique solution of the problem , we obtain
It follows from Lemma 2.2 and (3.10) that
where is a constant such that for all . Since and , by Lemma 2.3, we conclude that the scheme converges strongly to . This completes the proof.
Remark 3.6.
() For each , a simple example on control parameters is and , where is a constant in .
() We obtain the desired results without any assumptions on the family . Foe example, in Theorem CGY2, the authors gave the strong condition (1.10).
Remark 3.7.
() If in (3.9), then we have the following iterative scheme:
and the scheme defined by (3.33) converges strongly to an element which is the unique solution of the variational inequality ():
() If and in (3.33), then we have the following iterative scheme:
and the scheme defined by (3.35) converges strongly to an element which is the unique solution of the variational inequality ():
() Furthermore, if and in (3.35), then we have the following iterative scheme:
and the scheme defined by (3.37) converges strongly to an element which is the unique solution of the variational inequality (), which is Stampacchia's variational inequality:
Acknowledgment
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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