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A New Iteration Method for Nonexpansive Mappings and Monotone Mappings in Hilbert Spaces
Journal of Inequalities and Applications volume 2010, Article number: 251761 (2010)
Abstract
We introduce a new composite iterative scheme by the viscosity approximation method for nonexpansive mappings and monotone mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of set of fixed points of nonexpansive mapping and the set of solutions of variational inequality for an inverse-strongly monotone mappings, which is a solution of a certain variational inequality. Our results substantially develop and improve the corresponding results of [Chen et al. 2007 and Iiduka and Takahashi 2005]. Essentially a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings is provided.
1. Introduction
Let be a real Hilbert space and a nonempty closed convex subset of . Recall that a mapping is a contraction on if there exists a constant such that We use to denote the collection of mappings verifying the above inequality. That is . A mapping is called nonexpansive if ; see [1, 2] for the results of nonexpansive mappings. We denote by the set of fixed points of ; that is,
Let be the metric projection of onto . A mapping of into is called monotone if for , . The variational inequality problem is to find a such that
for all ; see [3–6]. The set of solutions of the variational inequality is denoted by . A mapping of into is called inverse-strongly monotone if there exists a positive real number such that
for all ; see [7–9]. For such a case, is called -inverse-strongly monotone.
In 2005, Iiduka and Takahashi [10] introduced an iterative scheme for finding a common point of the set of fixed points of a nonexapnsive mapping and the set of solutions of the variational inequality for an inverse-strong monotone mapping as follows. For an -inverse-strongly monotone mapping of to and a nonexpansive mapping of into itself such that , , , and ,
for every . They proved that the sequence generated by (1.3) converges strongly to under the conditions on and for some with ,
On the other hand, the viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping was proposed by Moudafi [11]. In 2004, in order to extend Theorem 2.2 of Moudafi [11] to a Banach space setting, Xu [12] considered the the following explicit iterative process. For nonexpansive mappings, and ,
Moreover, in [12], he also studied the strong convergence of generated by (1.5) as in either a Hilbert space or a uniformly smooth Banach space and showed that the strong is a solution of a certain variational inequality.
In 2007, Chen et al. [13] considered the following iterative scheme as the viscosity approximation method of (1.3). For an -inverse-strongly-monotone mapping of to and a nonexpansive mapping of into itself such that , , , , and ,
and showed that the sequence generated by (1.6) converges strongly to a point in under condition (1.4) on and , which is a solution of a certain variational inequality.
In this paper, motivated by above-mentioned results, we introduce a new composite iterative scheme by the viscosity approximation method. For an -inverse-strongly monotone mapping of to and a nonexpansive mapping of into itself such that , , , , and ,
If , then the iterative scheme (1.7) reduces to the iterative scheme (1.6). Under condition (1.4) on the sequences and and appropriate condition on sequence , we show that the sequence generated by (1.7) converges strongly to a point in , which is a solution of a certain variational inequality. Using this result, we also obtain a strong convergence result for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Moreover, we investigate the problem of finding a common point of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. The main results develop and improve the corresponding results of Chen et al. [13] and Iiduka and Takahashi [10]. We point out that the iterative scheme (1.7) is a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings.
2. Preliminaries and Lemmas
Let be a real Hilbert space with inner product and norm , and a closed convex subset of . We write to indicate that the sequence converges weakly to . implies that converges strongly to . For every point , there exists a unique nearest point in , denoted by , such that
for all . is called the metric projection of to . It is well known that satisfies
for every . Moreover, is characterized by the properties
for all . In the context of the variational inequality problem, this implies that
We state some examples for inverse-strongly monotone mappings. If , where is a nonexpansive mapping of into itself and is the identity mapping of , then is -inverse-strongly monotone and . A mapping of into is called strongly monotone if there exists a positive real number such that
for all . In such a case, we say that is -strongly monotone. If is -strongly monotone and -Lipschitz continuous, that is, for all , then is -inverse-strongly monotone.
If is an -inverse-strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous. We also have that for all and ,
So, if , then is a nonexpansive mapping of into . The following result for the existence of solutions of the variational inequality problem for inverse-strongly monotone mappings was given in Takahashi and Toyoda [14].
Proposition 2.1.
Let be a bounded closed convex subset of a real Hilbert space and an -inverse-strongly monotone mapping of into . Then, is nonempty.
A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if the graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies . Let be an inverse-strongly monotone mapping of into and let be the normal cone to at , that is, , and define
Then is maximal monotone and if and only if ; see [15, 16].
We need the following lemmas for the proof of our main results.
Lemma 2.2 (see [17]).
Let be a sequence of nonnegative real numbers satisfying
where and satisfy the following conditions:
(i) and or, equivalently,
(ii) or
Then .
Lemma 2.3 (see [1], demiclosedness principle).
Let be a real Hilbert space, a nonempty closed convex subset of , and a nonexpansive mapping. Then the mapping is demiclosed on , where is the identity mapping; that is, in and imply that and .
Lemma 2.4.
In a real Hilbert space , there holds the following inequality:
for all
3. Main Results
In this section, we introduce a new composite iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings and prove a strong convergence of this scheme.
Theorem 3.1.
Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of to and a nonexpansive mapping of into itself such that , and . Let be a sequence generated by
where , , and . If , and satisfy the following conditions:
(i); ;
(ii) for all and for some ;
(iii) for some with ;
(iv); ; ,
then converges strongly to , which is a solution of the following variational inequality:
Proof.
Let and for every . Let . Since is nonexpansive and from (2.5), we have
Similarly we have .
We divide the proof into several steps.
Step 1.
We show that is bounded. In fact, since
we have
By induction, we get
This implies that is bounded and so , , , , and are bounded. Moreover, since and , and are also bounded. By condition (i), we also obtain
Step 2.
We show that . From (3.1), we have
Simple calculations show that
Since
for every we have
for every , where and .
On the other hand, from (3.1) we have
Also, simple calculations show that
Since
for every it follows that
Substituting (3.11) into (3.15), we derive
where and . From conditions (i) and (iv), it is easy to see that
Applying Lemma 2.2 to (3.16), we have
By (3.11), we also have that as .
Step 3.
We show that and . Indeed, it follows that
which implies that
Obviously, by (3.7) and Step 2, we have as . This implies that
By (3.7) and (3.21), we also have
Step 4.
We show that . To this end, let . Then, by convexity of , we have
So we obtain
Since and by condition (i) and (3.21), we have . Moreover, from (2.2) we obtain
and so
And hence
Then we have
Since , and , we get . Also by (3.21), we have
Step 5.
We show that for , where is a solution of the variational inequality
To this end, choose a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to . We may assume without loss of generality that . Since by Steps 4 and 5, we have . Then we can obtain . Indeed, let us first show that . Let
Then is maximal monotone. Let . Since and , we have
On the other hand, from , we have and hence
Therefore we have
Hence we have as . Since is maximal monotone, we have and hence .
On the another hand, by Steps 3 and 4, . So, by Lemma 2.3, we obtain and hence . Then by (3.30) we have
Thus, from (3.7) we obtain
Step 6.
We show that for , where is a solution of the variational inequality
Indeed, from Lemma 2.4, we have
where
and . It is easily seen that , , and . Thus by Lemma 2.2, we obtain . This completes the proof.
Remark 3.2.
-
(1)
Theorem 3.1 improves the corresponding results in Chen et al. [13] and Iiduka and Takahashi [10]. In particular, if and is constant in (3.1), then Theorem 3.1 reduces to Theorem 3.1 of Iiduka and Takahashi [10].
-
(2)
We obtain a new composite iterative scheme for a nonexpansive mapping if in Theorem 3.1 as follows (see also Jung [18]):
(3.41)
As a direct consequence of Theorem 3.1, we have the following result.
Corollary 3.3.
Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of to such that , and . Let be a sequence generated by
where , , and . If , , and satisfy the following conditions:
(i); ,
(ii) for all and for some ,
(iii) for some with ,
(iv); ; ,
then converges strongly to , which is a solution of the following variational inequality:
4. Applications
In this section, as in [10, 13], we obtain two theorems in a Hilbert space by using Theorem 3.1.
A mapping is called strictly pseudocontractive if there exists with such that
for every . If , then is nonexpansive. Put , where is a strictly pseudocontractive mapping with . Then is -inverse-strongly monotone; see [7]. Actually, we have, for all ,
On the other hand, since is a real Hilbert space, we have
Hence we have
Using Theorem 3.1, we first get a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping.
Theorem 4.1.
Let be a closed convex subset of a real Hilbert space . Let be an -strictly pseudocontractive mapping of into itself and a nonexpansive mapping of into itself such that , and . Let be a sequence generated by
where , , and . If , , and satisfy the conditions:
(i); ,
(ii) for all and for some ,
(iii) for some with ,
(iv); ; ,
then converges strongly to , which is a solution of the following variational inequality:
Proof.
Put . Then is -inverse-strongly monotone. We have and . Thus, the desired result follows from Theorem 3.1.
Using Theorem 3.1, we also have the following result.
Theorem 4.2.
Let be a real Hilbert space . Let be an -inverse-strongly monotone mapping of into itself and a nonexpansive mapping of into itself such that , and . Let be a sequence generated by
where , , and . If , , and satisfy the conditions:
(i); ,
(ii) for all and for some ,
(iii) for some with ,
(iv); ; ,
then converges strongly to , which is a solution of the following variational inequality:
Proof.
We have . So, putting , by Theorem 3.1, we obtain the desired result.
Remark 4.3.
If in Theorems 4.1 and 4.2, then Theorems 4.1 and 4.2 reduce to Chen et al. [13, Theorems 4.1 and 4.2]. Theorems 4.1 and 4.2 also extend in Iiduka and Takahashi [10, Theorems 4.1 and 4.2] to the viscosity methods in composite iterative schemes.
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Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0064444). The author thanks the referees for their valuable comments and suggestions, which improved the presentation of this paper.
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Jung, J. A New Iteration Method for Nonexpansive Mappings and Monotone Mappings in Hilbert Spaces. J Inequal Appl 2010, 251761 (2010). https://doi.org/10.1155/2010/251761
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DOI: https://doi.org/10.1155/2010/251761