A new explicit iteration method for variational inequalities on the set of common fixed points for a finite family of nonexpansive mappings
© Kim and Buong; licensee Springer. 2013
Received: 8 April 2013
Accepted: 18 August 2013
Published: 3 September 2013
In this paper, we introduce a new explicit iteration method based on the steepest descent method and Krasnoselskii-Mann type method for finding a solution of a variational inequality involving a Lipschitz continuous and strongly monotone mapping on the set of common fixed points for a finite family of nonexpansive mappings in a real Hilbert space.
MSC:41A65, 47H17, 47H20.
Keywordscontraction common fixed points hybrid steepest descent method nonexpansive mappings monotone mappings
1 Introduction and preliminaries
Variational inequalities were initially studied by Kinderlehrer and Stampacchia in , and since then have been widely investigated. They cover partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance (see [1–3]).
where denotes the metric projection from onto C and μ is an arbitrarily positive constant.
The fixed point formulation (1.2) involves the metric projection . To overcome the complexity caused by , Yamada  introduced a hybrid steepest descent method for solving (1.1). His idea is stated as follows. Assume that , the set of common fixed points of a finite family of nonexpansive mappings on H with an integer .
where , for integer , with the mod-function taking values in the set , and , and proved that the sequence in (1.3) converges strongly to under the following conditions:
Further, Zeng and Yao  proved the same result with (L3) replaced by
Theorem 1.1 
Then the sequence defined by (1.3) converges strongly to the unique element in (1.1).
It is not difficult to show that (L3) implies (L4) if exists. However, in general, conditions (L3) and (L4) are not comparable, i.e., neither one of them implies the other (see  for details).
and proved the following result.
Theorem 1.2 
, where ;
for some ;
then the sequence defined by (1.5) converges strongly to the unique element in (1.1).
So, (1.6) is satisfied. They did not give another sufficient condition different from (L1), (L2) and (L4).
where I is the identity mapping of H, and proved the following result.
Theorem 1.3 
Let conditions (L1), (L2) and (L3) or (L4) be satisfied. Assume in addition that (1.4) holds. Then the sequence generated by algorithm (1.7) converges strongly to the unique solution of (1.1) with .
is sufficient for (1.4) if .
In this paper, we introduce a new algorithm based on a combination of the steepest descent method for variational inequalities with the Krasnoselskii-Mann method for fixed point problems to solve (1.1) with , where is a nonexpansive mapping on H for each i.
In Section 2, we prove the strong convergence theorem for (1.9)-(1.10) without conditions (L3), (L4) and (1.8). An application to the case that is a -strictly pseudocontractive mapping is given in Section 3.
2 Main results
We need the following lemmas for the proof of our main result.
Lemma 2.1 
, , and for any fixed .
From , we have the following lemma.
Lemma 2.2 
, and for a fixed number , , where and for .
Assume that T is a nonexpansive self-map of a closed convex subset K of a real Hilbert space H. If T has a fixed point, then is demiclosed; that is, whenever is a sequence in K weakly converging to some and the sequence strongly converges to some y, it follows that .
Lemma 2.4 
Lemma 2.5 
Now, we are in a position to prove the following main result.
Then the sequence defined by (1.9)-(1.10) converges strongly to the unique element in (1.1).
Therefore, the sequence is bounded. So, the sequences , , and () are also bounded. Without loss of generality, we assume that they are bounded by a positive constant .
By Lemma 2.4, as , i.e., .
which with and (2.1) implies that as . So, as .
Similarly, we obtain that , …, as .
and , it follows that for .
Then . So, by Lemma 2.3, . Therefore, from (1.1) it implies (2.2).
and (2.2), we have that . This completes the proof. □
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.
a nonexpansive mapping, for each . So, we have the following result.
Let , . Assume that satisfy (1.10). Then the sequence defined by (1.9) with replaced by of (3.1) converges strongly to the unique element of (1.1).
4 Numerical example
Iterations of scheme ( 1.9 ), where starting point
by algorithm (1.9)
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138).
- Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York; 1980.MATHGoogle Scholar
- Glowinski R: Numerical Methods for Nonlinear Variational Problems. Springer, New York; 1984.View ArticleMATHGoogle Scholar
- Zeidler E: Nonlinear Functional Analysis and Its Applications. Springer, New York; 1985.View ArticleMATHGoogle Scholar
- Yamada Y: The hybrid steepest-descent method for variational inequalities problems over the intersection of the fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam; 2001:473–504.View ArticleGoogle Scholar
- Zeng LC, Yao JC: Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings. Nonlinear Anal. TMA 2006, 64: 2507–2515. 10.1016/j.na.2005.08.028MathSciNetView ArticleMATHGoogle Scholar
- Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589MathSciNetView ArticleMATHGoogle Scholar
- Zeng LC, Wong NC, Yao JC: Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J. Optim. Theory Appl. 2007, 132: 51–69. 10.1007/s10957-006-9068-xMathSciNetView ArticleMATHGoogle Scholar
- Liu X, Cui Y: The common minimal-norm fixed point of a finite family of nonexpansive mappings. Nonlinear Anal. TMA 2010, 73: 76–83. 10.1016/j.na.2010.02.041MathSciNetView ArticleMATHGoogle Scholar
- Marino G, Xu HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329: 336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle Scholar
- Goebel K, Kirk WA Cambridge Studies in Advanced Math. 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar
- Li G, Kim JK: Demiclosedness principle and asymptotic behavior for nonexpansive mappings in metric spaces. Appl. Math. Lett. 2001, 14(5):645–649. 10.1016/S0893-9659(00)00207-XMathSciNetView ArticleMATHGoogle Scholar
- Suzuki T: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 2007, 135: 99–106.View ArticleGoogle Scholar
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