An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces
© Nguyen Buong and Nguyen Thi Quynh Anh. 2011
Received: 17 December 2010
Accepted: 7 March 2011
Published: 15 March 2011
We introduce a new implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces.
Variational inequalities were initially studied by Kinderlehrer and Stampacchia in  and ever since have been widely investigated, since they cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance (see [1–3]).
Let be a real Hilbert space and a nonempty closed convex subset of . Let be nonexpansive self-maps of such that , where . Let and be a sequence in such that . Then, the sequence defined implicitly by (1.5) converges weakly to a common fixed point of the mappings .
They proved the following result.
Let be a real Hilbert space and a mapping such that for some constants , is -Lipschitz continuous and -strongly monotone. Let be nonexpansive self-maps of such that . Let , and let and satisfying the conditions: and , for some . Then, the sequence defined by (1.7) converges weakly to a common fixed point of the mappings . The convergence is strong if and only if .
Recently, Ceng et al.  extended the above result to a finite family of asymptotically self-maps.
2. Main Result
We formulate the following facts for the proof of our results.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (Demiclosedness Principle ).
Assume that is a nonexpansive self-mapping of a closed convex subset of a Hibert space . 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 .
Now, we are in a position to prove the following result.
It is well known  that a mapping by with a fixed for all is a nonexpansive mapping and . Using this fact, we can extend our result to the case , where is -strictly pseudocontractive as follows.
So, we have the following result.
Theorem 3.1 ..
It is known in  that where with and for -strictly pseudocontractions . Moreover, is -strictly pseudocontractive with . So, we also have the following result.
Theorem 3.2 ..
This work was supported by the Vietnamese National Foundation of Science and Technology Development.
- 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.MATHGoogle Scholar
- Glowinski R: Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer, New York, NY, USA; 1984:xv+493.View ArticleGoogle Scholar
- Zeidler E: Nonlinear Functional Analysis and Its Applications. III. Springer, New York, NY, USA; 1985:xxii+662.MATHView ArticleGoogle Scholar
- Xu H-K, Ori RG: An implicit iteration process for nonexpansive mappings. Numerical Functional Analysis and Optimization 2001,22(5–6):767–773. 10.1081/NFA-100105317MATHMathSciNetView ArticleGoogle Scholar
- Zeng L-C, Yao J-C: Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings. Nonlinear Analysis. Theory, Methods & Applications 2006,64(11):2507–2515. 10.1016/j.na.2005.08.028MATHMathSciNetView ArticleGoogle Scholar
- Ceng L-C, Wong N-C, Yao J-C: Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption. Computers & Mathematics with Applications 2008,56(9):2312–2322. 10.1016/j.camwa.2008.05.002MATHMathSciNetView ArticleGoogle Scholar
- Marino G, Xu H-K: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,329(1):336–346. 10.1016/j.jmaa.2006.06.055MATHMathSciNetView ArticleGoogle Scholar
- Yamada Y: The hybrid steepest-descent method for variational inequalities problems over the intesectionof the fixed point sets of nonexpansive mappings. In Inhently Parallel Algorithms in Feasibility and Optimization and Their Applications. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, Holland; 2001:473–504.View 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 ArticleMATHGoogle Scholar
- Zhou H: Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis. Theory, Methods & Applications 2008,69(2):456–462. 10.1016/j.na.2007.05.032MATHMathSciNetView ArticleGoogle Scholar
- Acedo GL, Xu H-K: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis. Theory, Methods & Applications 2007,67(7):2258–2271. 10.1016/j.na.2006.08.036MATHMathSciNetView 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.