New viscosity method for hierarchical fixed point approach to variational inequalities
© Deng; licensee Springer. 2013
Received: 21 September 2012
Accepted: 31 July 2013
Published: 16 August 2013
A new viscosity method for hierarchically approximating some common fixed point of an infinite family of nonexpansive mappings is presented; and some strong convergence theorems for solving variational inequality problems and hierarchical fixed point problems are obtained without the aid of the convex linear combination of a countable family of nonexpansive mappings. Solutions are sought in the set of fixed points of another nonexpansive mapping. The results improve those of the authors with the related interest.
MSC:47H09, 47H10, 65J15, 47J25.
1 Introduction and preliminaries
A fairly common method in solving some nonlinear problems is to replace the original problems by a family of regularized (or perturbed) ones. Each of these regularized problems will be obtained as a limit of these unique solutions to the regularized problems. In this paper, we will introduce a new viscosity method for the hierarchical fixed point approach to variational inequality problems.
Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping is called a ρ-contraction if there exists a constant such that for all . A mapping is said to be nonexpansive if for all .
It is easy to see that (1.1) is equivalent to the following fixed point problem: finding an such that , where is the metric projection from H onto a closed convex subset .
The existence problem of hierarchical fixed points for a single nonexpansive mapping and approximation problem in the setting of Hilbert spaces has been studied by several authors (see, e.g., [2–12]).
In 2011, Zhang et al.  proved a strong convergence theorem by projection method for solving some variational inequality problems; and under suitable conditions on parameters, they also obtained a weak convergence theorem, which can solve the hierarchical fixed point problem (1.1).
However, since the involved mapping T is defined by a convex linear combination of a countable family of nonexpansive mappings , i.e., , () with , the accurate computation of at each step of the iteration process is not easily attainable. In addition, the weak convergence was obtained on condition that the iteration sequence is bounded.
Inspired and motivated by those studies mentioned above, in this paper, we introduce a new viscosity method for hierarchically approximating some common fixed point of an infinite family of nonexpansive mappings and prove the strong convergence theorems for solving some variational inequality problems and hierarchical fixed point problems.
In what follows, we shall make use of the following definitions and lemmas.
- (1)if and only if the following relation holds
- (2)There holds the relation
This implies that is nonexpansive.
Lemma 1.2 
Lemma 1.3 
If and , then exists.
- (1)Let be a sequence of mappings, and let be a mapping. is said to be graph convergent to A if (the sequence of graph of ) converges to graph A in the sense of Kuratowski-Painlevé, i.e.,
- (2)A multi-valued mapping is said to be monotone if , . A mapping is said to be maximal monotone if it is monotone, and for any when
we have .
Let be a maximal monotone operator. Then graph converges to as , which provide that .
Let be a sequence of maximal monotone operators, whose graph converges to an operator B. If A is a Lipschitz maximal monotone operator, then graph converges to , and is maximal monotone.
the mapping is strongly monotone;
the mapping is monotone, so it is maximal monotone.
2 Main results
Proof We divide the proof into several steps.
(I) exists, .
where , and so . So by Lemma 1.3, we conclude that exists, and hence , and are bounded.
(II) as .
where and , since is bounded and .
(III) for each as .
Then, since as , it follows from (2.13) and (2.16) that (2.17) holds obviously.
where , ; and .
It follows from Lemma 1.3 that exists, and hence is bounded. Then there exists an such that , .
since . The proof is complete. □
which means that , i.e., . Since there exists some , we also have .
since . This is equivalent to , which is the unique solution to the variational inequality above. The proof is complete. □
This implies that is a solution to the fixed point problem (1.1), i.e., it is a hierarchically common fixed point of a countable family of nonexpansive mappings with respect to another nonexpansive mapping S. The proof is complete. □
Remark 2.4 Since the strong convergence theorems for solving some variational inequality problems and hierarchical fixed point problems are obtained without the aid of the convex linear combination of a countable family of nonexpansive mappings, the results in this article improve those of the authors with related interest.
The author read and approved the final manuscript.
This work is supported by the National Natural Science Foundation of China (Grant No. 11061037).
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