- Research Article
- Open Access
An Iterative Scheme with a Countable Family of Nonexpansive Mappings for Variational Inequality Problems in Hilbert Spaces
© Y. J. Cho and S.Wang. 2010
- Received: 13 December 2009
- Accepted: 8 February 2010
- Published: 21 February 2010
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.
- Hilbert Space
- Unique Solution
- Variational Inequality
- Nonexpansive Mapping
- Lipschitzian Mapping
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 . 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 ).
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  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:
(C3) either or .
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.  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]).
(iii)either or ,
Theorem 1 CGYtwo (see [9, Theorem 3.2]).
(iii)either or ,
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, :
Let be a Hilbert space and be a nonexpansive mapping of into itself such that . For all and , we have
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 ).
The following lemma is an immediate consequence of the equality:
Assume that . Then the following results hold.
(1)If , where , then is a bounded sequence.
Lemma 2.4 (see ).
that is, is strongly monotone with coefficient .
Assume is a strongly monotone linear bounded operator on a Hilbert space with coefficient . Take a fixed number such that . Then .
This completes the proof.
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  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.
which shows that is strongly monotone and linear.
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.
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
If is strictly decreasing, then the scheme defined by (3.9) is bounded.
Hence is bounded and so are and for each . This completes the proof.
Put . Since is a strictly decreasing sequence and , we have . By Lemma 2.3, it follows that as . This completes the proof.
This completes the proof.
Finally, we give the main result in this paper.
First, we prove 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.
( ) 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).
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
- Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics. Volume 88. Academic Press, New York, NY, USA; 1980:xiv+313.Google Scholar
- Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004, 298(1):279–291. 10.1016/j.jmaa.2004.04.059MATHMathSciNetView ArticleGoogle Scholar
- Deutsch F, Yamada I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numerical Functional Analysis and Optimization 1998, 19(1–2):33–56.MATHMathSciNetView ArticleGoogle Scholar
- Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002, 66(1):240–256. 10.1112/S0024610702003332MATHMathSciNetView ArticleGoogle Scholar
- Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003, 116(3):659–678. 10.1023/A:1023073621589MATHMathSciNetView ArticleGoogle Scholar
- Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Stud. Comput. Math.. Volume 8. Edited by: Butnarium D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.View ArticleGoogle Scholar
- Yamada I, Ogura N, Yamashita Y, Sakaniwa K: Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space. Numerical Functional Analysis and Optimization 1998, 19(1–2):165–190. 10.1080/01630569808816822MATHMathSciNetView ArticleGoogle Scholar
- Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006, 318(1):43–52. 10.1016/j.jmaa.2005.05.028MATHMathSciNetView ArticleGoogle Scholar
- Ceng L-C, Guu S-M, Yao J-C: Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces. Computers & Mathematics with Applications 2009, 58(3):605–617. 10.1016/j.camwa.2009.02.035MATHMathSciNetView ArticleGoogle Scholar
- Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleGoogle Scholar
- Maingé P-E: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007, 325(1):469–479. 10.1016/j.jmaa.2005.12.066MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.