- Research
- Open access
- Published:
Generalized viscosity approximation methods for nonexpansive mappings
Fixed Point Theory and Applications volume 2014, Article number: 68 (2014)
Abstract
We combine a sequence of contractive mappings and propose a generalized viscosity approximation method. One side, we consider a nonexpansive mapping S with the nonempty fixed point set defined on a nonempty closed convex subset C of a real Hilbert space H and design a new iterative method to approximate some fixed point of S, which is also a unique solution of the variational inequality. On the other hand, using similar ideas, we consider N nonexpansive mappings with the nonempty common fixed point set defined on a nonempty closed convex subset C. Under reasonable conditions, strong convergence theorems are proven. The results presented in this paper improve and extend the corresponding results reported by some authors recently.
MSC:47H09, 47H10, 47J20, 47J25.
1 Introduction
Let H be a real Hilbert space with an inner product and norm , and C be a nonempty closed convex subset of H.
Let be a nonlinear mapping, we use to denote the set of fixed points of S (i.e., ). A mapping is called nonexpansive if the following inequality holds:
for all .
In 1967, Halpern [1] used contractions to approximate a nonexpansive mapping and considered the following explicit iterative process:
where u is a given point and is nonexpansive. He proved the strong convergence of to a fixed point of S provided that with . In 2000, Moudafi [2] introduced the viscosity approximation method for nonexpansive mappings. Until now, in many references, viscosity approximation methods still are used and studied, which formally generates the sequence by the recursive formula:
where f is a contraction and is a slowly vanishing sequence. See, for instance, [3–6]. In fact, Yamada’s hybrid steepest descent algorithm is also a kind of viscosity approximation method (see [7]).
The variational inequality problem is to find a point such that
In recent years, the theory of variational inequality has been extended to the study of a large variety of problems arising in structural analysis, economics, engineering sciences, and so on. See [8–10] and the references cited therein.
Recently, Zhou and Wang [11] proposed a simpler explicit iterative algorithm for finding a solution of variational inequality over the set of common fixed points of a finite family nonexpansive mappings. They introduced an explicit scheme as follows.
Theorem 1.1 Let H be a real Hilbert space and be an L-Lipschitz continuous and η-strongly monotone mapping. Let be N nonexpansive self-mappings of H such that . For any point , define a sequence in the following manner:
where and for . When the parameters satisfy appropriate conditions, the sequence converges strongly to the unique solution of the variational inequality (1.1).
In this paper, motivated by the above works, we introduce a more generalized iterative method like viscosity approximation. In Section 3, we combine a sequence of contractive mappings and obtain strong convergence theorem for approximating fixed point of a nonexpansive mapping. In Section 4, we propose a new iterative algorithm for finding some common fixed point of a finite family nonexpansive mappings, which is also a unique solution for the variational inequality over the set of fixed point of these mappings on Hilbert spaces.
2 Preliminaries
In order to prove our results, we collect some facts and tools in a real Hilbert space H, which are listed as below.
Lemma 2.1 Let H be a real Hilbert space. We have the following inequalities:
-
(i)
, .
-
(ii)
, , .
Lemma 2.2 [12]
Let be -averaged on C such that . Then the following conclusions hold:
-
(i)
both and are γ-averaged, where ;
-
(ii)
.
Recall that given a nonempty closed convex subset C of a real Hilbert space H, for any , there exists a unique nearest point in C, denoted by , such that
for all . Such a is called the metric (or the nearest point) projection of H onto C.
Lemma 2.3 [13]
Let C be a nonempty closed convex subset of a real Hilbert space H. Given and , then if and only if we have the relation
Lemma 2.4 [10]
Let H be a Hilbert space and C be a nonempty closed convex subset of H, and a nonexpansive mapping with . If is a sequence in C weakly converging to x and if converges strongly to y, then .
Lemma 2.5 [5]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
Then, .
Lemma 2.6 [14]
Let and be bounded sequences in a Banach space and be a sequence of real numbers such that for all . Suppose that for all and . Then .
3 Generalized viscosity approximation method combining with a nonexpansive mapping
In this section, we combine a sequence of contractive mappings and apply a more generalized iterative method like viscosity approximation to approximate some fixed point of a nonexpansive mapping defined on a closed convex subset C of a Hilbert space H, which is also the solution of the variational inequality
Suppose the contractive mapping sequence is uniformly convergent for any , where D is any bounded subset of C. The uniform convergence of on D is denoted by (), .
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a sequence of -contractive self-maps of C with . Let be a nonexpansive mapping. Assume the set and is uniformly convergent for any , where D is any bounded subset of C. Given , let {} be generated by the following algorithm:
If the sequence satisfies the following conditions:
-
(i)
;
-
(ii)
;
-
(iii)
,
then the sequence converges strongly to a point , which is also the unique solution of the variational inequality (3.1).
Proof The proof is divided into several steps.
Step 1. Show first that is bounded.
For any , we have
From the uniform convergence of on D, it is easy to get the boundedness of . Thus there exists a positive constant , such that . By induction, we obtain . Hence, is bounded, so are and .
Step 2. Show that
Indeed, observe that
By the conditions (i)-(iii) and the uniform convergence of , we have
as . By Lemma 2.5, (3.3) holds.
Step 3. Show that
Since
By the condition (i), we have . Combining with (3.3), it is easy to get (3.4).
Step 4.
where is a unique solution of the variational inequality (3.1).
Since is uniformly convergent on D, we have .
Indeed, take a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume . From (3.4), we obtain . Using Lemma 2.4, we have . Since , we get
Combining with (3.6), the inequality (3.5) holds.
Step 5. Show that
Transform the inequality into another form, we obtain
By Schwartz’s inequality, we have
By the boundedness of , , (3.3) and (3.5), we have
It follows from Lemma 2.5 that (3.7) holds. □
Remark 3.2 In [2], Moudafi proposed the viscosity iterative algorithm as follows:
where f is a contraction on H. It is a special case of (3.2) in this paper when , and . Of course, Halpern’s iteration method is also a special case of (3.2) when , .
Remark 3.3 In [7], the following iterative process was introduced:
Rewriting the equation, we get
It is easily to verify is a contractive mapping on H when . That is, Yamada’s method is a kind of viscosity approximation method. Of course it is also a special case of Theorem 3.1.
4 Generalized viscosity approximation method combining with a finite family of nonexpansive mappings
In this section, we apply a more generalized iterative method like viscosity approximation to approximate a common element of the set of fixed points of a finite family of nonexpansive mappings on Hilbert spaces.
Let be a sequence of -contractive self-maps of C with and be N nonexpansive self-mapping of C. Assume the common fixed point set and is convergent for any . Put , since every is -contractive, we have
for any . Further we obtain . Next we prove the sequence converges strongly to a point , which also solves the variational inequality
As we know, it is equivalent to the fixed point equation .
Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a sequence of -contractive self-maps of C with . Let, for each ( be an integer), be a nonexpansive mapping. Assume the set and is convergent for any . Given , let {} be generated by the following algorithm:
If the parameters and satisfy the following conditions:
-
(i)
, and ;
-
(ii)
for some and , ,
then the sequence converges strongly to a point , which is also the unique solution of the variational inequality (4.1).
Proof We will prove the theorem in the case of . The proof is divided into several steps.
Step 1. We show first that is bounded.
For any , we have
From the convergence of , it is easy to get the boundness of . Thus there exists a positive constant , such that . By induction, we obtain . Hence, is bounded, and so are and .
Step 2. We show that
Since both and are averaged nonexpansive mappings, by Lemma 2.2, is also averaged. Rewrite , where . Then we have
Further we obtain
Write , . From the condition (iii), it is easily to get and as . We have
where . Since , , we can deduce
and
Substituting (4.5) into (4.4), we have
Combining (4.6), (4.7), and condition (i), we get
By Lemma 2.6, we conclude that . Further we have
Step 3. We show that
By (4.2), we get
We have
Combining with (4.3), (4.8) holds.
Since , we can assume that as . It is easy to get for . Write , . Then we have , and
where D is an arbitrary bounded subset including . By using (4.8) and (4.9), we obtain .
Step 4. We have
where is a unique solution of the variational inequality (4.1).
Since is convergent, we have .
The proof of Step 4 is similar to that of Theorem 3.1.
Step 5. We show that
The proof of Step 5 is similar to that of Theorem 3.1. □
Remark 4.2 In [11], put , and we rewrite Zhou and Wang’s iterative algorithm as follows:
It is easily to verify is a contractive mapping on H when . Thus it is a special case of Theorem 4.1 when , and .
References
Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
Moudafi A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615
Chen J, Zhang L, Fan T: Viscosity approximation methods for nonexpansive mappings and monotone mappings. J. Math. Anal. Appl. 2007, 334: 1450–1461. 10.1016/j.jmaa.2006.12.088
Takahashi W: Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009, 70: 719–734. 10.1016/j.na.2008.01.005
Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059
Yao Y, Noor M: On viscosity iterative methods for variational inequalities. J. Math. Anal. Appl. 2007, 325: 776–787. 10.1016/j.jmaa.2006.01.091
Yamada I: The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings. Studies in Computational Mathematics 8. Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications 2001, 473–504.
Buong D, Duong LT: An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 2011, 151: 513–524. 10.1007/s10957-011-9890-7
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Tian M, Di LY: Synchronal algorithm and cyclic algorithm for fixed point problems and variational inequality problems in Hilbert spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 21
Zhou HY, Wang P: A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 2013. 10.1007/s10957-013-0470-x
López G, Martin V, Xu HK: Iterative algorithm for the multi-sets split feasibility problem. Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems 2009, 243–279.
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.2006.06.055
Suzuki T: Strong convergence theorems for an infinite family of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 1: 103–123.
Acknowledgements
The authors would like to thank the referee for valuable suggestions to improve the manuscript and the Fundamental Research Funds for the Central Universities (GRANT: 3122013k004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Duan, P., He, S. Generalized viscosity approximation methods for nonexpansive mappings. Fixed Point Theory Appl 2014, 68 (2014). https://doi.org/10.1186/1687-1812-2014-68
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-68