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In this paper, to find the fixed points of the nonexpansive nonself-mappings, we introduced two new viscosity approximation methods, and then we prove the iterative sequences defined by above viscosity approximation methods which converge strongly to the fixed points of nonexpansive nonself-mappings. The results presented in this paper extend and improve the results of Song-Chen [1] and Song-Li [2].

Let C be a closed convex subset of a Hilbert space H and

Helpern [

In 2006, Yisheng Song and Rudong Chen [

where X is a real reflexive Banach space, and C is a closed subset of X which is also a sunny nonexpansive retract of X.

In 2007, Yisheng Song and Qingchun Li [

where X is a real reflexive Banach space, and C is a closed subset of X which is also a sunny nonexpansive retract of X.

In this paper, we will study two new viscosity approximation methods for nonexpansive nonself-mappings in reflexive Banach space X, which can extend the results of Song-Chen [

Let us start by making some basic definitions.

Let X be a real Banach space with the norm

Definition 2.1. Let X be a real Banach space and J denote the normalized duality mapping from X into

where

Let

Definition 2.2. Let X ba a real Banach space and T a mapping with domain

Definition 2.3. Let X be a Banach space, C and D be nonempty subsets of X,

Definition 2.4. Let X be a real reflexive Banach space, which admits a weakly sequentially continuous duality mapping from X to

where

where

We call (2) the first type viscosity approximation method for nonexpansive nonself-mapping and call (3) the second type viscosity approximation method for nonexpansive nonself-mapping.

Let us introduce some lemmas, which play important roles in our results.

Lemma 2.1. ( [

Lemma 2.2. ( [

Then

Lemma 2.3. ( [

Lemma 2.4. ( [

First of all, let us study the first type viscosity approximation for nonexpansive nonself-mappings.

Lemma 3.1. ( [

where P is a sunny nonexpansive retract of X onto C. Then as

For all

Lemma 3.2. Let X be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J from X to

Proof. Let

while,

therefore,

since

therefore

Lemma 3.3. Let X be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J from X to

then

1)

2)

Proof by lemma 3.2, we know that the sequence

by (4), we have

Set

Set

by the lemma 2.2 we have

Now we will proof

as

Remark 3.1. From the lemma 3.1 we know that p is the unique solution in

Now, we can take a subsequence

we may assume that

Theorem 3.4. Let X be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J from X to

then the sequence

Proof. Since C is closed, by lemma 3.2,

by the lemma 3.1 as

using the remark 3.1, we have

By the definition 2.4 (2), we have

While

therefore,

where

Setting

2.1, we conclude that

Let us prove p is the unique fixed point of T.

We assume that

Remark 3.2. when

So the theorem 3.4 improves the theorem 2.4 of Song-Chen [

Now let us study the second type viscosity approximation for nonexpansive nonself-mappings.

Lemma 3.5. Let X be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J from X to

Proof. Let

while,

therefore,

since

therefore

Lemma 3.6. ( [

where P is a sunny nonexpansive retract of X onto C. Then as

for all

Lemma 3.7. Let X be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J from X to

then

1)

2)

Proof by lemma 3.5, we know that the sequence

by (8), we have

Set

Set

by the lemma 2.2 we have

Now we will proof

as

Remark 3.3. From the lemma 3.6 we know that p is the unique solution in

Now, we can take a subsequence

we may assume that

Theorem 3.8. Let X be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J from X to X^{*}. Suppose C is a nonexpansive retract of X which is also a sunny nonexpansive retract of X, and

then the sequence

Proof. Since C is closed, by lemma 3.5,

by the lemma 3.6 as

using the remark 3.3, we have

By the definition 2.4 (3), we have

While

therefore,

where

Setting

Let us prove p is the unique fixed point of T.

We assume that

Remark 3.4. When

So the theorem 3.8 improves the theorem 4.3 theorem 4.4 of Song-Li [

In this paper, we studied two new viscosity approximation methods for nonexpansive nonself-mappings, which were defined by definition 2.4. And then we proved that the sequences

Chao Liu,Meimei Song, (2016) The New Viscosity Approximation Methods for Nonexpansive Nonself-Mappings. International Journal of Modern Nonlinear Theory and Application,05,104-113. doi: 10.4236/ijmnta.2016.52011