Research Article | Open Access

Hiroko Manaka, "Convergence Theorems for a Maximal Monotone Operator and a -Strongly Nonexpansive Mapping in a Banach Space", *Abstract and Applied Analysis*, vol. 2010, Article ID 189814, 20 pages, 2010. https://doi.org/10.1155/2010/189814

# Convergence Theorems for a Maximal Monotone Operator and a -Strongly Nonexpansive Mapping in a Banach Space

**Academic Editor:**S. Reich

#### Abstract

Let *E* be a smooth Banach space with a norm . Let for any , where stands for the duality
pair and *J* is the normalized duality mapping. With respect to this bifunction
, a generalized nonexpansive mapping and a -strongly nonexpansive
mapping are defined in . In this paper, using the properties of generalized
nonexpansive mappings, we prove convergence theorems for common zero
points of a maximal monotone operator and a -strongly nonexpansive mapping.

#### 1. Introduction

Let be a smooth Banach space with a norm and let be a nonempty, closed and convex subset of . We use the following bifunction studied by Alber [1], as well as Kamimura and Takahashi [2]. Let be defined by for any , where stands for the duality pair and is the normalized duality mapping. Note that the duality mapping is single valued in a smooth Banach space (see [3]). From the definition of the following properties are trivial.

Lemma 1.1. *
(a) For all ,
**
(b) If a sequence satisfies for some , then is bounded.*

Let be the fixed points set of . Ibaraki and Takahashi defined a generalized nonexpansive mapping in a Banach space (see [4]).

*Definition 1.2. *A mapping is said to be generalized nonexpansive if and for all and *. *

In this paper, we prove strong convergence theorem for finding common fixed points of a family of generalized nonexpansive mappings. In addition, we prove strong convergence theorem for finding zeroes of a generalized nonexpansive mapping and a maximal monotone operator. Now, we define a -strongly nonexpansive mapping as follows.

*Definition 1.3. *A mapping is called *-*strongly nonexpansive if there exists a constant such that
for all , where is the identity mapping on . More explicitly, if (1.2) holds, then is said to be -strongly nonexpansive with .

If is -strongly nonexpansive with , then is -strongly nonexpansive with any . It is trivial that a -strongly nonexpansive mapping is generalized nonexpansive if . In the following section, we show that in a Hilbert space a firmly nonexpansive mapping is -strongly nonexpansive with and a -strongly nonexpansive mapping is strongly nonexpansive if . Motivated by the results of Manaka and Takahashi [5], we prove weak convergence theorem for common zero points of a maximal monotone operator and a -strongly nonexpansive mapping in a Banach space.

#### 2. Preliminaries

Let be a nonempty subset of a Banach space . A mapping is said to be sunny, if for all and , A mapping is called a retraction if for all (see [6]). It is known that a generalized nonexpansive and sunny retraction of onto is uniquely determined if is a smooth and strictly convex Banach space (cf., [7]). Ibaraki and Takahashi proved the following results in [4].

Lemma 2.1 (cf., [4]). * Let be a reflexive, strictly convex, and smooth Banach space, and let be a generalized nonexpansive mapping from into itself. Then there exists a sunny and generalized nonexpansive retraction on .*

Lemma 2.2 (cf., [4]). *Let be a nonempty subset of a reflexive, strictly convex, and smooth Banach space . Let be a retraction from onto . Then is sunny and generalized nonexpansive if and only if
**
for all and .*

A generalized resolvent of a maximal monotone operator is defined by for any real number . It is well known that is single valued if is reflexive, smooth, and strictly convex (see [8]). It is also known that satisfies This implies that Therefore, from Lemma 1.1(a), we obtain the following proposition.

Proposition 2.3. *
If a sunny retraction is generalized nonexpansive, then satisfies
**
For each , a generalized resolvent satisfies
*

*Remark 2.4. *The property in Proposition 2.3(b) means that is generalized nonexpansive for any .

We recall some nonlinear mappings in Banach spaces (see, e.g., [9–12]).

*Definition 2.5. *Let be a nonempty, closed, and convex subset of *. *A mapping is said to be firmly nonexpansive if
for all and some .

In [12], Reich introduced a class of strongly nonexpansive mappings which is defined with respect to the Bregman distance corresponding to a convex continuous function in a reflexive Banach space . Let be a convex subset of , and let be a self-mapping of . A point in the closure of is said to be an asymptotically fixed point of if contains a sequence which converges weakly to and the sequence converges strongly to 0. denotes the asymptotically fixed points set of .

*Definition 2.6. *The Bregman distance corresponding to a function is defined by
where is the Gâteaux differentiable and stands for the derivative of at the point . We say that the mapping is strongly nonexpansive if and
and if it holds that for a bounded sequence such that for any .

We remark that the symbols and mean that converges strongly and weakly to , respectively. Taking the function as the convex, continuous, and Gâteaux differentiable function , we obtain the fact that the Bregman distance coincides with . Especially in a Hilbert space, . Bruck and Reich defined strongly nonexpansive mappings in a Hilbert space as follows (cf., [10]).

*Definition 2.7. *A mapping is said to be strongly nonexpansive if is nonexpansive with and if it holds that
when and are sequences in such that is bounded and .

The relation among firmly nonexpansive mappings, strongly nonexpansive mappings and -strongly nonexpansive mappings is shown in the following proposition.

Proposition 2.8. *In a Hilbert space , the following hold.*(a)*A firmly nonexpansive mapping is -strongly nonexpansive with .*(b)*A -strongly nonexpansive mapping with is strongly nonexpansive.*

*Proof. *(a) Suppose that is firmly nonexpansive. Since in a Hilbert space, it holds that
for all . Therefore, it is obvious that is firmly nonexpansive if and only if satisfies
for all . Hence we obtain (a).

(b) Suppose that is -strongly nonexpansive with . Then, it is trivial that is nonexpansive and (2.9) holds. Suppose that the sequences and satisfy the conditions in Definition 2.7. Then is also bounded. Since is -strongly nonexpansive with , we have that
Hence, for . This means that is strongly nonexpansive.

In a Banach space, -strongly nonexpansive mappings have the following properties.

Proposition 2.9. *In a smooth Banach space , the following hold.*(a)*For , is -strongly nonexpansive. For , is -strongly nonexpansive for any . For , is -strongly nonexpansive for any .*(b)*If is -strongly nonexpansive with , then, for any with , is also -strongly nonexpansive with .*(c)*If is -strongly nonexpansive with , then is -strongly nonexpansive with .*(d)*Suppose that is -strongly nonexpansive with and that satisfies . Then is -strongly nonexpansive with . Moreover, if , then
*

*Proof. *(a) Let for any , and denote and . Since , we have
For , it holds that for all . For , we obtain
Therefore, is -strongly nonexpansive for any .

(b) If is -strongly nonexpansive with , then, for with ,
This means that is -strongly nonexpansive with .

(c) Suppose that is -strongly nonexpansive with and let . Then we have that
This inequality implies that
Thus is -strongly nonexpansive with .

(d) From (b) and the assumption, is -strongly nonexpansive with , and from (c) we have that is -strongly nonexpansive with . Furthermore we obtain that
This completes the proof.

In Banach spaces, we have the following example of -strongly nonexpansive mappings.

*Example 2.10. *Let be a Banach space with a norm defined by
The normalized duality mapping is given by
Hence, we have for that
We define a mapping as follows:
We will show that this mapping is -strongly nonexpansive for any .

(a) Suppose that with and . Then, we have
Since
we have that
Hence, we obtain that
for any . This means that is -strongly nonexpansive for any

(b) Suppose that with and . Then, we have
Hence, we have that
for any . This means that is -strongly nonexpansive for any

(c) Suppose that with . Then, we have
Hence, we have that
Now, we note that
for any and for any . Therefore, we obtain that
for any . This means that is -strongly nonexpansive for any

It is clear that if then is -strongly nonexpansive; therefore, from (a), (b), and (c), we obtain the conclusion that is -strongly nonexpansive with .

Next, we present some lemmas which are used in the proofs of our theorems. Let be the set of natural numbers.

Lemma 2.11. *Let and be sequences of nonnegative real numbers and satisfy the inequality for any and a constant . If , then exists.*

Kamimura and Takahashi showed the following useful lemmas (see [2]).

Lemma 2.12. *Let be a smooth and uniformly convex Banach space. Then, there exists a continuous, strictly increasing, and convex function such that , and for each real number ,
**
for all .*

From this lemma, it is obvious that the following lemma holds.

Lemma 2.13. *Let be a smooth and uniformly convex Banach space and let and be sequences in such that either or is bounded. If , then .*

We present the following lemma which plays an important role in our theorems (cf. Butnariu and Resmerita [13] ).

Lemma 2.14. *Let be a smooth and uniformly convex Banach space and a nonempty, convex, and closed subset of . Suppose that satisfies
**
If a weakly convergent sequence satisfies that , then .*

#### 3. Main Results

In this section, we prove three strong convergence theorems. In the first result, we prove strong convergence theorem for finding common fixed points of a family of generalized nonexpansive mappings. In the next result, we prove strong convergence theorem for finding zeroes of a generalized nonexpansive mapping and a maximal monotone operator. In the last result, we prove weak convergence theorem for finding zeroes of a maximal monotone operator and a -strongly nonexpansive mapping. As consequence, we prove convergence theorem for common zeroes of a maximal monotone operator and a firmly nonexpansive mapping in a Hilbert space.

Theorem 3.1. *Let be a reflexive, smooth, and strictly convex Banach space, and let be a family of generalized nonexpansive mappings. Suppose that and that is a sunny and generalized nonexpansive retraction from to . Let a sequence be defined as follows. For any ,
**
Then, converges strongly to a point in .*

*Proof. *Since is a point in for all , from Proposition 2.3(a), we have for all that
Since and are generalized nonexpansive, we get that
Hence, we have that
and therefore, . Furthermore, Proposition 2.3(a) implies that
This is equivalent to
Setting for all , then we have that
Since for any , Lemma 1.1(b) implies that is bounded. Thus, from Lemma 2.12, we can take the continuous and strictly increasing function with such that
Since for all , we have . Therefore, is a Cauchy sequence. Since is complete and is closed, this sequence converges strongly to point .

Noting that the generalized resolvent of a maximal monotone operator for is a generalized nonexpansive mapping (see Remark 2.4), we obtain the following result.

Theorem 3.2. *Let be a reflexive, smooth, and strictly convex Banach space. Let be generalized nonexpansive and let be a maximal monotone operator. Suppose that and that is a sunny and generalized nonexpansive retraction from to . Let an iterative sequence be defined as follows: for any ,
**
where is a sequence of nonnegative real numbers. Then, the sequence converges strongly to a point in .*

*Proof. *From Propositions 2.3(a) and 2.3(b), we have for all that
Thus . Similarly, as in the proof of the previous theorem, we show that is a Cauchy sequence, and we obtain that converges strongly to point in .

The duality mapping of a Banach space with the Gâteaux differentiable norm is said to be weakly sequentially continuous if in implies that converges weak star to in (cf., [14]). This happens, for example, if is a Hilbert space, finite dimensional and smooth, or if (cf., [15]). Next, we prove the main theorem.

Theorem 3.3. *Let be a reflexive, smooth and strictly convex Banach space. Suppose that the duality mapping of is weakly sequentially continuous. Let be a nonempty, closed, and convex subset of . Let be a maximal monotone operator and let be a generalized resolvent of for a sequence . Suppose that is a -strongly nonexpansive mapping with such that and that is a sunny and generalized nonexpansive retraction. For an such that , let an iterative sequence be defined as follows: for any and ,
**
where and satisfy that
**
Then, there exists an element such that
*

*Proof. *For simplicity, we denote and by and , respectively. Let for all . Since is generalized nonexpansive, we have for any and all that
The convexity of implies that
Thus, we obtain that
and furthermore, since is generalized nonexpansive, we have that

Let . Then, from Proposition 2.9(d), is -strongly nonexpansive with and is also generalized nonexpansive. Hence, we have that
Thus, we have from (3.14), (3.16), (3.17), and (3.18) that
From Lemma 2.11, there exists . Since , we have that
Hence, , , , , and are bounded from Lemma 1.1(b). Since is uniformly convex, the boundedness of implies that there exists a subsequence such that . Moreover, we can take the index sequence satisfies . We will show that . From Proposition 2.3(a),
and furthermore, from (3.16) and Proposition 2.3(b), we obtain that
These inequalities and (3.22) imply with and (3.21) that
that is,
Lemma 2.13 implies that
Furthermore, since
we have from (3.26) and (3.27) that
Hence, for an index sequence of such that and , we obtain that
Since , there exists such that
Since , (3.26) implies that
For , the monotonicity of implies that
and we have, since is weakly sequentially continuous, that
The maximality of implies that

Now, we will show that . From Proposition 2.9(d) and , we get that
Thus, we have from (3.17) that
This implies that
Therefore we have that
From Lemma 2.13, we get that that is, as . From Lemma 1.1(a) and the boundedness of , we have that