Hybrid Viscosity Iterative Method for Fixed Point, Variational Inequality and Equilibrium Problems
© Y.-A. Chen and Y.-P. Zhang. 2010
Received: 27 December 2009
Accepted: 1 June 2010
Published: 24 June 2010
We introduce an iterative scheme by the viscosity iterative method for finding a common element of the solution set of an equilibrium problem, the solution set of the variational inequality, and the fixed points set of infinitely many nonexpansive mappings in a Hilbert space. Then we prove our main result under some suitable conditions.
The solution set of (1.1) is denoted by .
where is a positive real number.
It is easy to know that is ( )-inverse-strongly-monotone. If , then is nonexpansive. We denote by the fixed points set of .
where is a sequence in , is an -inverse-strongly monotone mapping, is a sequence in , and is the metric projection. They proved that if , then converges weakly to some
Recently, S. Takahashi and W. Takahashi  introduced an iterative scheme for finding a common element of the solution set of (1.1) and the fixed points set of a nonexpansive mapping in a Hilbert space. If is bifunction which satisfies the following conditions:
() for all
() is monotone, that is, for all
() for each
() for each is convex and lower semicontinuous,
then they proved the following strong convergence theorem.
Theorem A (see ).
Let be a closed and convex subset of a real Hilbert space . Let be a bifunction which satisfies conditions .
where and satisfy , and
Then, and converge strongly to where
Such a mapping is called the -mapping generated by and (see ).
where and are sequences in and are sequences in , is a fixed contractive mapping with contractive coefficient , is an -inverse-strongly monotone mapping of to , is a bifunction which satisfies conditions , and is generated by (1.8). Then we proved that the sequences and converge strongly to , where .
So, if , then is nonexpansive.
Lemma 2.1 (see ).
Let and be bounded sequences in a Banach space , and let be a sequence in with . Suppose for all and . Then,
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Then, the following holds:
(i) is single-valued;
(iv) is closed and convex.
Lemma 2.5 (Opial's theorem ).
holds for each with
Let be a sequence of nonexpansive self-mappings on , where is a nonempty, closed and convex subset of a real Hilbert space . Given a sequence in , one defines a sequence of self-mappings on generated by (1.8). Then one has the following results.
Lemma 2.6 (see ).
Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and is a sequence in for some . Then, for every and the limit exists.
It can be shown from Lemma 2.6 that if is a nonempty and bounded subset of , then for there exists such that for all .
This implies that
Lemma 2.9 (see ).
Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and is a sequence in for some . Then, .
3. Strong Convergence Theorem
Let be a Hilbert space. Let be a nonempty, closed, and convex subset of . Let be a bifunction which satisfies conditions , an -inverse-strongly monotone mapping of to , a contraction of into itself, and a sequence of nonexpansive self-mappings on such that . Suppose that , and are sequences in , and and are sequences in which satisfies the following conditions:
Then and generated by (1.9) converge strongly to , where .
Hence is bounded. So , and are also bounded.
Lemma 2.1 yields that . Consequently,
we obtain and hence . Thus,
for all So is a contraction by Banach contraction principle . Since is a complete space, there exists a unique element such that .
Now we show that
Using (3.23) and Lemma 2.2, we conclude that converges strongly to Consequently, converges strongly to This completes the proof.
Using Theorem 3.1, we prove the following theorem.
where , and are given as in Theorem 3.1. Then and converge strongly to , where .
Put . Then is ( )-inverse-strongly-monotone. We have and put . So by Theorem 3.1 we obtain the desired result.
The author would like to express his thanks to Professor Simeon Reich, Technion-Israel Institute of Technology, Israel, and the anonymous referees for their valuable comments and suggestions on a previous draft, which resulted in the present version of the paper. This work was supported by the Natural Science Foundation of China (10871217) and Grant KJ080725 of the Chongqing Municipal Education Commission.
- Liu F, Nashed MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Analysis 1998,6(4):313–344. 10.1023/A:1008643727926MathSciNetView ArticleMATHGoogle Scholar
- Zeng L-C, Yao J-C: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2006,10(5):1293–1303.MathSciNetMATHGoogle Scholar
- Yao Y, Liou Y-C, Yao J-C: An extragradient method for fixed point problems and variational inequality problems. Journal of Inequalities and Applications 2007, 2007:-12.Google Scholar
- Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003,118(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar
- Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleMATHGoogle Scholar
- Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics 2001,5(2):387–404.MathSciNetMATHGoogle Scholar
- Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle 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.2004.04.059MathSciNetView ArticleMATHGoogle Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.MathSciNetMATHGoogle Scholar
- Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar
- Ciric LB: A generalization of Banach's contraction principle. Proceedings of the American Mathematical Society 1974, 45: 267–273.MathSciNetView 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.