Generalized viscosity approximation methods for nonexpansive mappings
© Duan and He; licensee Springer. 2014
Received: 30 October 2013
Accepted: 6 March 2014
Published: 20 March 2014
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.
Let H be a real Hilbert space with an inner product and norm , and C be a nonempty closed convex subset of H.
for all .
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  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.
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.
In order to prove our results, we collect some facts and tools in a real Hilbert space H, which are listed as below.
, , .
Lemma 2.2 
both and are γ-averaged, where ;
for all . Such a is called the metric (or the nearest point) projection of H onto C.
Lemma 2.3 
Lemma 2.4 
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 
Lemma 2.6 
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
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 (), .
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.
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 .
as . By Lemma 2.5, (3.3) holds.
By the condition (i), we have . Combining with (3.3), it is easy to get (3.4).
where is a unique solution of the variational inequality (3.1).
Since is uniformly convergent on D, we have .
Combining with (3.6), the inequality (3.5) holds.
It follows from Lemma 2.5 that (3.7) holds. □
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 , .
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.
As we know, it is equivalent to the fixed point equation .
, and ;
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.
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 .
Combining with (4.3), (4.8) holds.
where D is an arbitrary bounded subset including . By using (4.8) and (4.9), we obtain .
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.
The proof of Step 5 is similar to that of Theorem 3.1. □
It is easily to verify is a contractive mapping on H when . Thus it is a special case of Theorem 4.1 when , and .
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).
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