- Research Article
- Open Access
Iterative Methods for Variational Inequalities over the Intersection of the Fixed Points Set of a Nonexpansive Semigroup in Banach Spaces
© Issa Mohamadi. 2011
Received: 8 November 2010
Accepted: 19 November 2010
Published: 25 November 2010
The Erratum to this article has been published in Fixed Point Theory and Applications 2011 2011:187439
This paper presents a framework of iterative methods for finding specific common fixed points of a nonexpansive self-mappings semigroup in a Banach space. We prove, with appropriate conditions, the strong convergence to the solution of some variational inequalities.
where is an arbitrarily fixed constant and is the projection of onto . This alternative equivalence has been used to study the existence theory of the solution and to develop several iterative type algorithms for solving variational inequalities. But the fixed point formulation in (1.4) involves the projection , which may not be easy to compute, due to the complexity of the convex set . So, projection methods and their variant forms can be implemented for solving variational inequalities.
Yamada  proved that the sequence converges strongly to a unique solution of . Xu and Kim  further considered and studied the hybrid steepest-descent algorithm (1.6). Their major contribution is that the strong convergence of (1.6) holds with condition being replaced by the following condition:
It is clear that condition is strictly weaker than condition , coupled with conditions and . Moreover, includes the important and natural choice for whereas does not. For more related results, see [4, 5].
for . With some appropriate assumptions, we prove the strong convergence of (1.8), (1.9), and (1.10) to the unique solution of the variational inequality in , where is the single-valued normalized duality mapping from into .
Our main purpose is to improve some of the conditions and results in the mentioned papers, especially those of Song and Xu .
exists for each , and is said to have a uniformly Gateaux differentiable norm if for each , the limit (2.1) converges uniformly for . Further, is said to be uniformly smooth if the limit (2.1) exists uniformly for .
where denotes the generalized duality pairing. It is well known if is smooth then any duality mapping on is single valued, and if has a uniformly Gateaux differentiable norm, then the duality mapping is norm to weak* uniformly continuous on bounded sets.
The nonexpansive semigroup is an example of uniformly asymptotically regular operator semigroup .
A discussion on these and related concepts can be found in .
We make use of the following well-known results throughout the paper.
Lemma 2.1 (see [12, Lemma 4.5.4]).
Lemma 2.2 (see ).
3. Implicit Iterative Method
The following lemma will be be used to show the convergence of (1.8) and (1.9).
In the following theorem, which is the main result in this section, we establish the strong convergence of the sequence defined by (1.8).
We divide the proof into several steps.
4. Explicit Iterative Method
In this section, we will present our result of the strong convergence of (1.9), but first, we need to prove, with different approach, the following lemma.
This completes the proof.
Next, we prove the strong convergence of explicit iteration scheme (1.9).
Let be a real reflexive strictly convex Banach space with a uniformly Gateaux differentiable norm. Let also be a nonexpansive semigroup from into itself such that . Assume that defined by (1.9) satisfies condition in Theorem 4.2, then condition holds.
and it completes the proof.
5. Modified Iterative Method
This completes the proof.
We can easily see that under some restrictions all the strongly monotone and Lipschitzian nonlinear operators used in this paper are replaceable by strongly accretive and strictly pseudocontractive ones (see ).
- Yamada I: The hybrid steepest-descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithm for Feasibility and Optimization. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, New York, NY, USA; 2001:473–504.View 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 HK, Kim TH: Convergence of hybrid steepest-descent methods for variational inequalities. Journal of Optimization Theory and Applications 2003,119(1):185–201.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
- Saeidi S: Modified hybrid steepest-descent methods for variational inequalities and fixed points. Mathematical and Computer Modelling 2010,52(1–2):134–142. 10.1016/j.mcm.2010.01.023MATHMathSciNetView ArticleGoogle Scholar
- Shioji N, Takahashi W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society 1997,125(12):3641–3645. 10.1090/S0002-9939-97-04033-1MATHMathSciNetView ArticleGoogle Scholar
- Shioji N, Takahashi W: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces. Nonlinear Analysis: Theory, Methods and Applications 1998,34(1):87–99. 10.1016/S0362-546X(97)00682-2MATHMathSciNetView ArticleGoogle Scholar
- Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proceedings of the American Mathematical Society 2003,131(7):2133–2136. 10.1090/S0002-9939-02-06844-2MATHMathSciNetView ArticleGoogle Scholar
- Xu H-K: A strong convergence theorem for contraction semigroups in Banach spaces. Bulletin of the Australian Mathematical Society 2005,72(3):371–379. 10.1017/S000497270003519XMATHMathSciNetView ArticleGoogle Scholar
- Chen R, Song Y: Convergence to common fixed point of nonexpansive semigroups. Journal of Computational and Applied Mathematics 2007,200(2):566–575. 10.1016/j.cam.2006.01.009MATHMathSciNetView ArticleGoogle Scholar
- Song Y, Xu S: Strong convergence theorems for nonexpansive semigroup in Banach spaces. Journal of Mathematical Analysis and Applications 2008,338(1):152–161. 10.1016/j.jmaa.2007.05.021MATHMathSciNetView ArticleGoogle Scholar
- Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2002.Google 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
- Megginson RE: An Introduction to Banach Space Theory, Graduate Texts in Mathematics. Volume 183. Springer, New York, NY, USA; 1998:xx+596.View ArticleMATHGoogle Scholar
- Ceng L-C, Ansari QH, Yao J-C: Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces. Numerical Functional Analysis and Optimization 2008,29(9–10):987–1033. 10.1080/01630560802418391MathSciNetView ArticleMATHGoogle 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.