- Research Article
- Open Access

# A New Iteration Method for Nonexpansive Mappings and Monotone Mappings in Hilbert Spaces

- JongSoo Jung
^{1}Email author

**2010**:251761

https://doi.org/10.1155/2010/251761

© Jong Soo Jung. 2010

**Received:**5 October 2009**Accepted:**20 December 2009**Published:**12 January 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.

## Keywords

- Hilbert Space
- Variational Inequality
- Monotone Mapping
- Nonexpansive Mapping
- Strong Convergence

## 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 ,

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 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]).

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.

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.

where , , and . If , and satisfy the following conditions:

(i) ; ;

(ii) for all and for some ;

(iii) for some with ;

(iv) ; ; ,

Proof.

Similarly we have .

We divide the proof into several steps.

Step 1.

Step 2.

for every , where and .

By (3.11), we also have that as .

Step 3.

Step 4.

Step 5.

Hence we have as . Since is maximal monotone, we have and hence .

Step 6.

and . It is easily seen that , , and . Thus by Lemma 2.2, we obtain . This completes the proof.

- (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]):

As a direct consequence of Theorem 3.1, we have the following result.

Corollary 3.3.

where , , and . If , , and satisfy the following conditions:

(i) ; ,

(ii) for all and for some ,

(iii) for some with ,

(iv) ; ; ,

## 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.

where , , and . If , , and satisfy the conditions:

(i) ; ,

(ii) for all and for some ,

(iii) for some with ,

(iv) ; ; ,

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.

where , , and . If , , and satisfy the conditions:

(i) ; ,

(ii) for all and for some ,

(iii) for some with ,

(iv) ; ; ,

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.

## Declarations

### 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.

## Authors’ Affiliations

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