**Abstract**

This paper introduces a *T* � ð Þ *Da* mapping that is weaker than the nonexpansive mapping. It introduces several iterations for the fixed point of the *T* � ð Þ *Da* mapping. It gives fixed point theorems and convergence theorems for the *T* � ð Þ *Da* mapping in Banach space, instead of uniformly convex Banach space. This paper gives some basic properties on the *T* � ð Þ *Da* mapping and gives the example to show the existence of *T* � ð Þ *Da* mapping. The results of this paper are obtained in general Banach spaces. It considers some sufficient conditions for convergence of fixed points of mappings in general Banach spaces under higher iterations.

**Keywords:** iteration, convergence theorems, nonexpansive mapping, fixed point **2010 MSC:** 47H09, 47H10

### **1. Introduction**

In this paper, *E* is a Banach space, *C* is a nonempty closed convex subset of *E*, and *Fix T*ð Þ¼ *x*∈*C* : *Tx* ¼ *x*.

**Definition 1.** T is contraction mapping if there is *r*∈½ Þ 0*;* 1

∥*Tx* � *Ty*∥ ≤ *r*∥*x* � *y*∥ for all *x y*∈*C:*

**Definition 2.** T is nonexpansive mapping if

∥*Tx* � *Ty*∥≤∥*x* � *y*∥ for all *x y*∈*C:*

**Definition 3.** T is quasinonexpansive mapping if

∥*Tx* � *Ty*∥≤∥*x* � *y*∥ for all *x*∈*C, y*∈*F T*ð Þ*:*

**Definition 4.** *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping on a subset C, if there is *a*∈ <sup>1</sup> <sup>2</sup> *;* <sup>1</sup> , <sup>∥</sup>*Tx* � *Ty*∥≤∥*<sup>x</sup>* � *<sup>y</sup>*<sup>∥</sup> for all *<sup>α</sup>*∈½ � *<sup>a</sup>;* <sup>1</sup> *, x*∈*C, y* <sup>∈</sup>*C T*ð Þ *; <sup>x</sup>; <sup>α</sup>* , where *C T*ð Þ¼ *; x; α* f g *y*∈*C*j*y* ¼ ð Þ 1 � *α p* þ *αTp; p* ∈*C;* ∥*Tp* � *p*∥≤∥*Tx* � *x*∥ .

In 2008 Suzuki [1] defined a mapping *T* in Banach space: <sup>1</sup> <sup>2</sup> ∥*Tx* � *Ty*∥≤∥*x* � *y*∥ implies ∥*Tx* � *Ty*∥≤∥*x* � *y*∥. And *T* is said to satisfy condition (*C*). Suzuki [1] showed that the mapping satisfying condition (*C*) is weaker than nonexpansive mapping and stronger than quasinonexpansive mapping.

Suzuki [1] proved the theorem *T* is a mapping in Banach space,*T* satisfies condition (*C*), and {*xn*} is the sequence defined by the iteration process:

$$\begin{cases} \boldsymbol{\mathfrak{x}}\_{1} = \boldsymbol{\mathfrak{x}} \in \mathsf{C}, \\ \boldsymbol{\mathfrak{x}}\_{n+1} = (1 - a\_{n})\boldsymbol{\mathfrak{x}}\_{n} + a\_{n}T\boldsymbol{\mathfrak{x}}\_{n}, \end{cases} \tag{1}$$

∥*x*∥<sup>1</sup> ¼ maxf g j*x*1jþj*x*3j*;* j*x*2jþj*x*4j

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping*

*DOI: http://dx.doi.org/10.5772/intechopen.88421*

*<sup>x</sup>* <sup>¼</sup> 3 0 0 0 � �

*<sup>y</sup>* <sup>¼</sup> <sup>2</sup>*:*5 0 0 0 � �

∥*Tx* � *Ty*∥<sup>1</sup> ¼ 1*:*1>∥*x* � *y*∥1*:*

To verify that *T* is a *T* � ð Þ *Da* mapping, consider the following cases:

� �*, x* <sup>¼</sup> *<sup>x</sup>*<sup>1</sup> *<sup>x</sup>*<sup>2</sup>

*x*<sup>3</sup> *x*<sup>4</sup>

∈*C T*ð Þ *; x; α ,*

*x*<sup>3</sup> *x*<sup>4</sup>

∈*C T*ð Þ *; x; α ,*

� �∥ ≤ <sup>1</sup>*:*<sup>11</sup> ≤ ∥*<sup>y</sup>* � *<sup>x</sup>*<sup>∥</sup>

!

� �∥≤∥ *<sup>y</sup>*<sup>1</sup> � *<sup>x</sup>*<sup>1</sup> *<sup>y</sup>*<sup>2</sup> � *<sup>x</sup>*<sup>2</sup>

� �*, x* <sup>¼</sup> *<sup>x</sup>*<sup>1</sup> *<sup>x</sup>*<sup>2</sup>

1*:*1 *y*<sup>2</sup> � *x*<sup>2</sup> *y*<sup>3</sup> � *x*<sup>3</sup> *y*<sup>4</sup> � *x*<sup>4</sup>

In this section, we prove convergence theorems for fixed point of the *T* � ð Þ *Da*

*, x*<sup>1</sup> 6¼ 3*:*

*y*<sup>3</sup> � *x*<sup>3</sup> *y*<sup>4</sup> � *x*<sup>4</sup>

*, x*<sup>1</sup> ¼ 3*:*

� �<sup>∥</sup> <sup>¼</sup> <sup>∥</sup>*<sup>y</sup>* � *<sup>x</sup>*<sup>∥</sup>

!

Set

and

We see that

Case 1:

then *y*<sup>1</sup> 6¼ 3. We have

∥*Ty* � *Tx*∥ ¼ ∥

then *y*<sup>1</sup> ∈ ½ � 0*;* 1*:*9 . We have

∥*Ty* � *Tx*∥ ¼ ∥

Case 2:

**2. Fixed point**

**39**

mapping in Banach space.

Hence,*T* is not a nonexpansive mapping.

*α* ∈ 11 <sup>19</sup> *;* <sup>1</sup>

*α* ∈ 11 <sup>19</sup> *;* <sup>1</sup>

*<sup>y</sup>* <sup>¼</sup> *<sup>y</sup>*<sup>1</sup> *<sup>y</sup>*<sup>2</sup> *y*<sup>3</sup> *y*<sup>4</sup>

0 *y*<sup>2</sup> � *x*<sup>2</sup> *y*<sup>3</sup> � *x*<sup>3</sup> *y*<sup>4</sup> � *x*<sup>4</sup>

> *<sup>y</sup>* <sup>¼</sup> *<sup>y</sup>*<sup>1</sup> *<sup>y</sup>*<sup>2</sup> *y*<sup>3</sup> *y*<sup>4</sup>

!

Hence,*T* is a *T* � ð Þ *Da* mapping, and *T* is not nonexpansive.

!

then {*xn*} converges to a fixed point of *T*.

Suzuki [1] gave this convergence theorem in an ordinary Banach space, and the mapping satisfying condition (*C*) is weaker than nonexpansive mapping.

In 2016, Thakur [2] proved the theorem *T* is a mapping in uniformly convex Banach space,*T* satisfies condition (*C*), and f g *xn* is the sequence defined by iteration process:

$$\begin{cases} \mathbf{x}\_1 = \mathbf{x} \in \mathcal{C}, \\\\ \mathbf{x}\_{n+1} = T\mathbf{y}\_n, \\\\ \mathbf{y}\_n = T\mathbf{z}\_n, \\\\ \mathbf{z}\_n = (\mathbf{1} - a\_n)\mathbf{x}\_n + a\_n T\mathbf{x}\_n, \end{cases} \tag{2}$$

then {*xn*} converges to a fixed point of *T*.

Thakur [2] claimed that the rate of iteration is fastest of known iterations. However, the disadvantage is that their results must be in uniformly convex Banach space, instead of the ordinary Banach space.

The aim of this article is there exists a generalized nonexpansive mapping, which makes the sequence generated by Thakur's iteration converge to a fixed point in a general Banach space.

The following propositions are obvious:

**Proposition 1.** If T is nonexpansive, then T satisfies condition (*Da*). **Proposition 2.** If T is *T* � ð Þ *Da* mapping, then T is quasinonexpansive. **Proposition 3.** Suppose *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping. Then, for *x, y*∈*C*: (1) ∥*T*<sup>2</sup> *x* � *Tx*∥≤∥*Tx* � *x*∥ for all *x*∈*C:* (2) ∥*T*<sup>2</sup> *<sup>x</sup>* � *Ty*∥≤∥*Tx* � *<sup>y</sup>*<sup>∥</sup> or <sup>∥</sup>*T*<sup>2</sup> *y* � *Tx*∥≤∥*Ty* � *x*∥ for all *x, y* ∈*C:*

Proof:

(1) Since <sup>∥</sup>*Tx* � *<sup>x</sup>*∥≤∥*Tx* � *<sup>x</sup>*∥*, Tx*∈*C T*ð Þ *; <sup>x</sup>;* <sup>1</sup> , we have <sup>∥</sup>*T*<sup>2</sup> *x* � *Tx*∥≤∥*Tx* � *x*∥. (2) For all *x, y*∈*C*, ∥*Tx* � *x*∥≤∥*Ty* � *y*∥ or ∥*Ty* � *y*∥≤∥*Tx* � *x*∥.

Then *Tx*∈*C T*ð Þ *; y; α* or *Ty* ∈*C T*ð Þ *; x; α* . It follows that ∥*T*<sup>2</sup> *<sup>x</sup>* � *Ty*∥≤∥*Tx* � *<sup>y</sup>*<sup>∥</sup> or <sup>∥</sup>*T*<sup>2</sup> *y* � *Tx*∥≤∥*Ty* � *x*∥. **Example 1**

$$T\mathfrak{x} = \begin{cases} \begin{pmatrix} 1.1 & \varkappa\_2 \\ \varkappa\_3 & \varkappa\_4 \end{pmatrix}, & \varkappa\_1 = \mathfrak{z}, \\\ \begin{pmatrix} 0 & \varkappa\_2 \\ \varkappa\_3 & \varkappa\_4 \end{pmatrix}, & \varkappa\_1 \neq \mathfrak{z}, \end{cases}$$

where

$$\boldsymbol{\infty} = \begin{pmatrix} \boldsymbol{\infty}\_1 & \boldsymbol{\infty}\_2 \\ \boldsymbol{\infty}\_3 & \boldsymbol{\infty}\_4 \end{pmatrix}, \boldsymbol{\infty}\_1 \in [0, 3], \boldsymbol{\infty}\_2 \in [0, 0.01], \boldsymbol{\infty}\_3 \in [0, 0.01], \boldsymbol{\infty}\_4 \in [0, 0.01].$$

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping DOI: http://dx.doi.org/10.5772/intechopen.88421*

$$\|\mathbf{x}\|\_1 = \mathbf{max}\{ |\mathbf{x}\_1| + |\mathbf{x}\_3|, |\mathbf{x}\_2| + |\mathbf{x}\_4| \}$$

Set

Suzuki [1] proved the theorem *T* is a mapping in Banach space,*T* satisfies

*xn*þ<sup>1</sup> ¼ ð Þ 1 � *α<sup>n</sup> xn* þ *αnTxn,*

Suzuki [1] gave this convergence theorem in an ordinary Banach space, and the

In 2016, Thakur [2] proved the theorem *T* is a mapping in uniformly convex

*zn* ¼ ð Þ 1 � *α<sup>n</sup> xn* þ *αnTxn,*

The aim of this article is there exists a generalized nonexpansive mapping, which makes the sequence generated by Thakur's iteration converge to a fixed point in a

*y* � *Tx*∥≤∥*Ty* � *x*∥ for all *x, y* ∈*C:*

*y* � *Tx*∥≤∥*Ty* � *x*∥.

*, x*<sup>1</sup> ¼ 3*,*

*, x*<sup>1</sup> 6¼ 3*,*

*, x*<sup>1</sup> ∈½ � 0*;* 3 *, x*<sup>2</sup> ∈½ � 0*;* 0*:*01 *, x*<sup>3</sup> ∈ ½ � 0*;* 0*:*01 *, x*<sup>4</sup> ∈½ � 0*;* 0*:*01 *:*

Thakur [2] claimed that the rate of iteration is fastest of known iterations. However, the disadvantage is that their results must be in uniformly convex Banach

**Proposition 1.** If T is nonexpansive, then T satisfies condition (*Da*). **Proposition 2.** If T is *T* � ð Þ *Da* mapping, then T is quasinonexpansive. **Proposition 3.** Suppose *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping. Then, for *x, y*∈*C*:

(1)

(2)

*x* � *Tx*∥≤∥*Tx* � *x*∥.

condition (*C*), and {*xn*} is the sequence defined by the iteration process:

mapping satisfying condition (*C*) is weaker than nonexpansive mapping.

*x*<sup>1</sup> ¼ *x*∈*C, xn*þ<sup>1</sup> ¼ *Tyn,*

*yn* ¼ *Tzn,*

Banach space,*T* satisfies condition (*C*), and f g *xn* is the sequence defined by

*x*<sup>1</sup> ¼ *x*∈*C,*

�

8 >>>>><

>>>>>:

then {*xn*} converges to a fixed point of *T*.

space, instead of the ordinary Banach space.

The following propositions are obvious:

*x* � *Tx*∥≤∥*Tx* � *x*∥ for all *x*∈*C:*

*Tx* ¼

(1) Since <sup>∥</sup>*Tx* � *<sup>x</sup>*∥≤∥*Tx* � *<sup>x</sup>*∥*, Tx*∈*C T*ð Þ *; <sup>x</sup>;* <sup>1</sup> , we have <sup>∥</sup>*T*<sup>2</sup>

(2) For all *x, y*∈*C*, ∥*Tx* � *x*∥≤∥*Ty* � *y*∥ or ∥*Ty* � *y*∥≤∥*Tx* � *x*∥.

*<sup>x</sup>* � *Ty*∥≤∥*Tx* � *<sup>y</sup>*<sup>∥</sup> or <sup>∥</sup>*T*<sup>2</sup>

8 >>>>><

>>>>>:

1*:*1 *x*<sup>2</sup> *x*<sup>3</sup> *x*<sup>4</sup>

0 *x*<sup>2</sup> *x*<sup>3</sup> *x*<sup>4</sup>

!

!

*<sup>x</sup>* � *Ty*∥≤∥*Tx* � *<sup>y</sup>*<sup>∥</sup> or <sup>∥</sup>*T*<sup>2</sup>

Then *Tx*∈*C T*ð Þ *; y; α* or *Ty* ∈*C T*ð Þ *; x; α* .

then {*xn*} converges to a fixed point of *T*.

iteration process:

*Functional Calculus*

general Banach space.

(1) ∥*T*<sup>2</sup>

(2) ∥*T*<sup>2</sup>

It follows that ∥*T*<sup>2</sup>

*<sup>x</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>1</sup> *<sup>x</sup>*<sup>2</sup> *x*<sup>3</sup> *x*<sup>4</sup> � �

**Example 1**

where

**38**

Proof:

$$\mathbf{x} = \begin{pmatrix} 3 & 0\\ 0 & 0 \end{pmatrix}.$$

and

$$\mathcal{Y} = \begin{pmatrix} 2.5 & 0\\ 0 & 0 \end{pmatrix}$$

We see that

$$\|Tx - T\mathfrak{y}\|\_1 = \mathbf{1}. \mathbf{1} > \|x - y\|\_1.$$

Hence,*T* is not a nonexpansive mapping.

To verify that *T* is a *T* � ð Þ *Da* mapping, consider the following cases: Case 1:

$$a \in \left[\frac{11}{19}, 1\right], x = \begin{pmatrix} \varkappa\_1 & \varkappa\_2 \\ \varkappa\_3 & \varkappa\_4 \end{pmatrix}, \varkappa\_1 \neq \mathbf{3}.$$

$$\mathcal{Y} = \begin{pmatrix} \mathcal{Y}\_1 & \mathcal{Y}\_2 \\ \mathcal{Y}\_3 & \mathcal{Y}\_4 \end{pmatrix} \in C(T, \varkappa, a),$$

then *y*<sup>1</sup> 6¼ 3. We have

$$\|Ty - Tx\| = \|\begin{pmatrix} 0 & y\_2 - \mathbf{x}\_2 \\ y\_3 - \mathbf{x}\_3 & y\_4 - \mathbf{x}\_4 \end{pmatrix}\| \le \|\begin{pmatrix} y\_1 - \mathbf{x}\_1 & y\_2 - \mathbf{x}\_2 \\ y\_3 - \mathbf{x}\_3 & y\_4 - \mathbf{x}\_4 \end{pmatrix}\| = \|y - \mathbf{x}\|$$

Case 2:

$$a \in \left[\frac{11}{19}, 1\right], x = \begin{pmatrix} \varkappa\_1 & \varkappa\_2 \\ \varkappa\_3 & \varkappa\_4 \end{pmatrix}, x\_1 = \mathbf{3}.$$

$$\mathcal{Y} = \begin{pmatrix} \mathcal{Y}\_1 & \mathcal{Y}\_2 \\ \mathcal{Y}\_3 & \mathcal{Y}\_4 \end{pmatrix} \in \mathcal{C}(T, \varkappa, a),$$

then *y*<sup>1</sup> ∈ ½ � 0*;* 1*:*9 . We have

$$\|Ty - Tx\| = \| \begin{pmatrix} \mathbf{1.1} & \boldsymbol{\mathcal{y}}\_2 - \boldsymbol{\mathcal{x}}\_2 \\ \boldsymbol{\mathcal{y}}\_3 - \boldsymbol{\mathcal{x}}\_3 & \boldsymbol{\mathcal{y}}\_4 - \boldsymbol{\mathcal{x}}\_4 \end{pmatrix} \| \le \mathbf{1.11} \le \|y - \boldsymbol{\mathcal{x}}\| $$

Hence,*T* is a *T* � ð Þ *Da* mapping, and *T* is not nonexpansive.

#### **2. Fixed point**

In this section, we prove convergence theorems for fixed point of the *T* � ð Þ *Da* mapping in Banach space.

**Lemma 1**. Let *C* be bounded convex subset of a Banach space *B*. Assume that *T* : *C* ! *C* is *T* � ð Þ *Da* mapping and f g *xn , yn* � �*, z*f g*<sup>n</sup>* are sequences generated by iteration:

$$\begin{cases} \mathbf{x}\_1 = \mathbf{x} \in \mathbb{C}, \\ \mathbf{x}\_{n+1} = T\mathbf{y}\_n, \\ \mathbf{y}\_n = T\mathbf{z}\_n, \\ z\_n = (\mathbf{1} - a\_n)\mathbf{x}\_n + a\_n T \mathbf{x}\_n, \end{cases} \tag{3}$$

So, for *p* ¼ 1 and all *n* ≥1

≤ ∥*Txn* � *xn*∥ þ

(4) holds.

1 þ

þ

And obviously

∥*Tu*3*n*þ1þ*<sup>p</sup>* � *u*3*n*þ<sup>1</sup>∥ ¼ ∥*Txn*þ*m*þ<sup>1</sup> � *xn*þ<sup>1</sup>∥ ¼ ∥*Txn*þ*m*þ<sup>1</sup> � *Tyn*∥ ≤ ∥*xn*þ*m*þ<sup>1</sup> � *yn*∥ <sup>¼</sup> <sup>∥</sup>*Tyn*þ*<sup>m</sup>* � *Tzn*<sup>∥</sup> ≤ ∥*yn*þ*<sup>m</sup>* � *zn*<sup>∥</sup>

3 X*n*þ*p*

*n* Y þ*p*

*k*¼*n*þ1

!

*k*¼3*n*þ1

!

¼ ∥*Tzn*þ*<sup>m</sup>* � ð Þ 1 � *α<sup>n</sup> xn* � *αnTxn*∥

¼ ð Þ 1 � *α<sup>n</sup>* ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥ þ *α<sup>n</sup>*

≤ ð Þ 1 � *α<sup>n</sup>* ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥ þ *α<sup>n</sup>*

≤ ð Þ 1 � *α<sup>n</sup>* ∥*Tzn*þ*<sup>m</sup>* � *xn*∥

It follows that

**41**

¼ ∥*Tu*3*n*�<sup>1</sup> � *u*3*n*�<sup>2</sup>∥ þ

≤ ∥*Tu*3*n*�<sup>1</sup> � *u*3*n*�<sup>2</sup>∥ þ

and obtain, upon replacing *n* with *n* þ 1

*βk*

2 1 � *α<sup>k</sup>*

Case 1: *p* ¼ 3*m, m* ≥1. From (6) and (7)

≤ ð Þ 1 � *α<sup>n</sup>* ∥*Tzn*þ*<sup>m</sup>* � *xn*∥ þ *αn*∥*Tzn*þ*<sup>m</sup>* � *Txn*∥

1 þ *β*3*n*�<sup>2</sup> ð Þ∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ ¼ ð Þ 1 þ *α<sup>n</sup>* ∥*Txn* � *xn*∥

*DOI: http://dx.doi.org/10.5772/intechopen.88421*

1 1 � *α<sup>n</sup>* � �

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping*

1 1 � *α<sup>n</sup>* � �

2 1 � *α<sup>n</sup>* � �

ð Þ ∥*Txn* � *xn*∥ � ∥*Txn*þ<sup>1</sup> � *xn*þ<sup>1</sup>∥

We make the inductive assumption that (4) holds for a given *p*> 1 and all *n*> 0

∥*Tu*3*n*þ<sup>1</sup> � *u*3*n*þ<sup>1</sup>∥≤∥*Tu*3*n*þ1þ*<sup>p</sup>* � *u*3*n*þ<sup>1</sup>∥

<sup>þ</sup>*α<sup>n</sup>* <sup>∥</sup>*Tzn*þ*<sup>m</sup>* � *Txn*þ*<sup>m</sup>*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*Txn*þ*<sup>m</sup>* � *Tyn*þ*m*�<sup>1</sup><sup>∥</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>∥</sup>*Tyn* � *Tzn*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*Tzn* � *Txn*<sup>∥</sup> � �

∥*Tuk*þ<sup>1</sup> � *Tuk*∥

*βk*∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥*:*

3*n* X�2þ*<sup>p</sup>*

*k*¼3*n*�2

*βk*∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

(8)

3*n* X�2þ*<sup>p</sup>*

3*n* X�2þ*<sup>p</sup>*

∥*Tu*3*n*þ1þ*<sup>p</sup>* � *u*3*n*þ<sup>1</sup>∥ ≤ ð Þ 1 � *α<sup>n</sup>* ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥ þ *α<sup>n</sup>*

*k*¼3*n*�2

*k*¼3*n*�2

<sup>∥</sup>*Tu*3*n*þ<sup>1</sup> � *<sup>u</sup>*3*n*þ<sup>1</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

*k*≥3*n* � 2*,* ∥*Tuk*þ<sup>1</sup> � *Tuk*∥ ≤ *βk*∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥*,* (6)

*k*>*t,* ∥*Tuk* � *Tut*∥≤∥*uk* � *ut*∥*:* (7)

ð Þ ∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ � ∥*Tu*3*n*þ<sup>1</sup> � *u*3*n*þ<sup>1</sup>∥

ð Þ ∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ � ∥*Tu*3*n*þ<sup>1</sup> � *u*3*n*þ<sup>1</sup>∥ *:*

(5)

where <sup>1</sup> <sup>2</sup> <*a* ≤ *α<sup>n</sup>* ≤ *b*<1. Then

(1) ∥*Txn*þ<sup>1</sup> � *xn*þ<sup>1</sup>∥≤∥*Tyn* � *yn*∥≤∥*Tzn* � *zn*∥≤∥*Txn* � *xn*∥. (2) lim*n*!<sup>∞</sup> ∥*Txn* � *xn*∥ ¼ lim*n*!<sup>∞</sup> ∥*Tyn* � *yn*∥ ¼ lim*n*!<sup>∞</sup> ∥*Tzn* � *zn*∥ ¼ *r*≥ 0.

Proof: (1) From Proposition 3 and *zn* ¼ ð Þ 1 � *α<sup>n</sup> xn* þ *αnTxn*, we have

$$\begin{aligned} \|\|T\mathbf{x}\_{n+1} - \mathbf{x}\_{n+1}\|\| &\le \|\|T^2\mathbf{y}\_n - T\mathbf{y}\_n\|\| \\ &\le \|\|T\mathbf{y}\_n - \mathbf{y}\_n\|\| = \|\|T^2\mathbf{z}\_n - T\mathbf{z}\_n\|\| \\ &\le \|\|T\mathbf{z}\_n - \mathbf{z}\_n\|\| \\ &= \|\|T\mathbf{z}\_n - T\mathbf{x}\_n + (\mathbf{1} - a\_n)(T\mathbf{x}\_n - \mathbf{x}\_n)\|\| \\ &\le \|\|\mathbf{z}\_n - \mathbf{x}\_n\|\| + (\mathbf{1} - a\_n)\|\|T\mathbf{x}\_n - \mathbf{x}\_n\|\| \\ &= \|\|T\mathbf{x}\_n - \mathbf{x}\_n\|\|. \end{aligned}$$

(2) From (1), we have 0 ≤ ∥*Txn*þ<sup>1</sup> � *xn*þ<sup>1</sup>∥≤∥*Txn* � *xn*∥. So lim*<sup>n</sup>*!<sup>∞</sup> ∥*Txn*� *xn*∥ ¼ *r*≥0. Now, we have lim*<sup>n</sup>*!<sup>∞</sup> ∥*Txn* � *xn*∥ ¼ lim*<sup>n</sup>*!<sup>∞</sup> ∥*Tyn* � *yn*∥ ¼ lim*<sup>n</sup>*!<sup>∞</sup> ∥*Tzn* � *zn*∥ ¼ *r*≥0.

**Lemma 2.** Assume that *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping and f g *xn , yn* � �*, z*f g*<sup>n</sup>* are sequences generated by iteration (3). <sup>1</sup> <sup>2</sup> < *a* ≤ *α<sup>n</sup>* ≤ *b*<1. Let f g *um* satisfy *u*3*n*�<sup>2</sup> ¼ *xn, u*3*n*�<sup>1</sup> ¼ *zn, u*3*<sup>n</sup>* ¼ *yn*. Then, for all *n*≥ 1*, p*≥ 1

$$\begin{split} & \left( \mathbf{1} + \sum\_{k=3n-2}^{3n+p-3} \beta\_k \right) \|Tu\_{3n-2} - u\_{3n-2}\| \le \|Tu\_{3n-2+p} - u\_{3n-2}\| \\ & + \left( \prod\_{k=n}^{n+p-1} \frac{2}{\mathbf{1} - a\_k} \right) \left( \|Tu\_{3n-2} - u\_{3n-2}\| - \|Tu\_{3n-2+3p} - u\_{3n-2+3p}\| \right), \end{split} \tag{4}$$

where

$$
\beta\_k = \begin{pmatrix} a\_n & k = 3n - 2 \\ 1 & k \neq 3n - 2 \end{pmatrix}
$$

Proof: From **Lemma 1**, we have

$$\begin{aligned} &\|\|T\mathbf{x}\_{n+1} - \mathbf{x}\_{n+1}\|\| \\ &\le \|\|T\mathbf{z}\_{n} - \mathbf{z}\_{n}\|\| \\ &= \|\|T\mathbf{z}\_{n} - (\mathbf{1} - a\_{n})\mathbf{x}\_{n} - \alpha\_{n}T\mathbf{x}\_{n}\|\| \\ &\le (\mathbf{1} - a\_{n})\|\|T\mathbf{z}\_{n} - \mathbf{x}\_{n}\|\| + a\_{n}\|\|T\mathbf{z}\_{n} - T\mathbf{x}\_{n}\|\| \\ &\le (\mathbf{1} - a\_{n})\|\|T\mathbf{z}\_{n} - \mathbf{x}\_{n}\|\| + a\_{n}\|\|\mathbf{z}\_{n} - \mathbf{x}\_{n}\|\| \\ &= (\mathbf{1} - a\_{n})\|\|T\mathbf{z}\_{n} - \mathbf{x}\_{n}\|\| + \alpha\_{n}^{2}\|\|T\mathbf{x}\_{n} - \mathbf{x}\_{n}\|\|. \end{aligned}$$

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping DOI: http://dx.doi.org/10.5772/intechopen.88421*

So, for *p* ¼ 1 and all *n* ≥1

$$\begin{split} & \quad (1 + \beta\_{3n-2}) \| \| T u\_{3n-2} - u\_{3n-2} \| \\ &= (1 + a\_n) \| \| T \mathbf{x}\_n - \mathbf{x}\_n \| \\ &\le \| T \mathbf{x}\_n - \mathbf{x}\_n \| + \left( \frac{1}{1 - a\_n} \right) (\| T \mathbf{x}\_n - \mathbf{x}\_n \| - \| T \mathbf{x}\_{n+1} - \mathbf{x}\_{n+1} \|) \\ &= \| \| T u\_{3n-1} - u\_{3n-2} \| + \left( \frac{1}{1 - a\_n} \right) (\| T u\_{3n-2} - u\_{3n-2} \| - \| T u\_{3n+1} - u\_{3n+1} \|) \\ &\le \| T u\_{3n-1} - u\_{3n-2} \| + \left( \frac{2}{1 - a\_n} \right) (\| T u\_{3n-2} - u\_{3n-2} \| - \| T u\_{3n+1} - u\_{3n+1} \|). \end{split}$$

(4) holds.

**Lemma 1**. Let *C* be bounded convex subset of a Banach space *B*. Assume that

*zn* ¼ ð Þ 1 � *α<sup>n</sup> xn* þ *αnTxn,*

(2) lim*n*!<sup>∞</sup> ∥*Txn* � *xn*∥ ¼ lim*n*!<sup>∞</sup> ∥*Tyn* � *yn*∥ ¼ lim*n*!<sup>∞</sup> ∥*Tzn* � *zn*∥ ¼ *r*≥ 0.

*yn* � *Tyn*∥ ≤ ∥*Tyn* � *yn*<sup>∥</sup> <sup>¼</sup> <sup>∥</sup>*T*<sup>2</sup>

*x*<sup>1</sup> ¼ *x*∈*C, xn*þ<sup>1</sup> ¼ *Tyn, yn* ¼ *Tzn,*

(1) ∥*Txn*þ<sup>1</sup> � *xn*þ<sup>1</sup>∥≤∥*Tyn* � *yn*∥≤∥*Tzn* � *zn*∥≤∥*Txn* � *xn*∥.

Proof: (1) From Proposition 3 and *zn* ¼ ð Þ 1 � *α<sup>n</sup> xn* þ *αnTxn*, we have

≤ ∥*Tzn* � *zn*∥

¼ ∥*Txn* � *xn*∥*:*

**Lemma 2.** Assume that *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping and f g *xn , yn*

*xn*∥ ¼ *r*≥0. Now, we have lim*<sup>n</sup>*!<sup>∞</sup> ∥*Txn* � *xn*∥ ¼ lim*<sup>n</sup>*!<sup>∞</sup> ∥*Tyn* � *yn*∥ ¼

(2) From (1), we have 0 ≤ ∥*Txn*þ<sup>1</sup> � *xn*þ<sup>1</sup>∥≤∥*Txn* � *xn*∥. So lim*<sup>n</sup>*!<sup>∞</sup> ∥*Txn*�

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥≤∥*Tu*3*n*�2þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

*<sup>β</sup><sup>k</sup>* <sup>¼</sup> *<sup>α</sup><sup>n</sup> <sup>k</sup>* <sup>¼</sup> <sup>3</sup>*<sup>n</sup>* � <sup>2</sup> 1 *k* 6¼ 3*n* � 2

�

¼ ∥*Tzn* � ð Þ 1 � *α<sup>n</sup> xn* � *αnTxn*∥

<sup>¼</sup> ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup>* <sup>∥</sup>*Tzn* � *xn*<sup>∥</sup> <sup>þ</sup> *<sup>α</sup>*<sup>2</sup>

≤ ð Þ 1 � *α<sup>n</sup>* ∥*Tzn* � *xn*∥ þ *αn*∥*Tzn* � *Txn*∥ ≤ ð Þ 1 � *α<sup>n</sup>* ∥*Tzn* � *xn*∥ þ *αn*∥*zn* � *xn*∥

*<sup>n</sup>*∥*Txn* � *xn*∥*:*

∥*Txn*þ<sup>1</sup> � *xn*þ<sup>1</sup>∥ ≤ ∥*Tzn* � *zn*∥

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*�2þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*�2þ3*<sup>p</sup>*<sup>∥</sup> � �*,*

8 >>><

>>>:

<sup>∥</sup>*Txn*þ<sup>1</sup> � *xn*þ<sup>1</sup>∥≤∥*T*<sup>2</sup>

� �*, z*f g*<sup>n</sup>* are sequences generated by

*zn* � *Tzn*∥

<sup>2</sup> < *a* ≤ *α<sup>n</sup>* ≤ *b*<1. Let f g *um* satisfy

¼ ∥*Tzn* � *Txn* þ ð Þ 1 � *α<sup>n</sup>* ð Þ *Txn* � *xn* ∥ ≤ ∥*zn* � *xn*∥ þ ð Þ 1 � *α<sup>n</sup>* ∥*Txn* � *xn*∥

(3)

� �*, z*f g*<sup>n</sup>*

(4)

*T* : *C* ! *C* is *T* � ð Þ *Da* mapping and f g *xn , yn*

<sup>2</sup> <*a* ≤ *α<sup>n</sup>* ≤ *b*<1. Then

iteration:

*Functional Calculus*

where <sup>1</sup>

lim*<sup>n</sup>*!<sup>∞</sup> ∥*Tzn* � *zn*∥ ¼ *r*≥0.

1 þ 3*n* X þ*p*�3

> *n*þ Y *p*�1

> > *k*¼*n*

þ

where

**40**

*k*¼3*n*�2

!

!

are sequences generated by iteration (3). <sup>1</sup>

*βk*

2 1 � *α<sup>k</sup>*

Proof: From **Lemma 1**, we have

*u*3*n*�<sup>2</sup> ¼ *xn, u*3*n*�<sup>1</sup> ¼ *zn, u*3*<sup>n</sup>* ¼ *yn*. Then, for all *n*≥ 1*, p*≥ 1

We make the inductive assumption that (4) holds for a given *p*> 1 and all *n*> 0 and obtain, upon replacing *n* with *n* þ 1

$$\begin{split} & \left( \mathbf{1} + \sum\_{k=3n+1}^{3n+p} \beta\_k \right) \|Tu\_{3n+1} - u\_{3n+1}\| \le \|Tu\_{3n+1+p} - u\_{3n+1}\| \\ & + \left( \prod\_{k=n+1}^{n+p} \frac{2}{\mathbf{1} - a\_k} \right) \left( \|Tu\_{3n+1} - u\_{3n+1}\| - \|Tu\_{3n+1+3p} - u\_{3n+1+3p}\| \right) . \end{split} \tag{5}$$

And obviously

$$k \ge 3n - 2,\ \|Tu\_{k+1} - Tu\_k\| \le \beta\_k \|Tu\_{3n-2} - u\_{3n-2}\|.\tag{6}$$

$$k > t, \left\| \|Tu\_k - Tu\_t\| \right\| \le \left\| u\_k - u\_t \right\|. \tag{7}$$

Case 1: *p* ¼ 3*m, m* ≥1. From (6) and (7)

$$\begin{split} & \| \| T \mathbf{u}\_{3n+1} - \mathbf{u}\_{3n+1} \| \\ &= \| \| T \mathbf{x}\_{n+m+1} - \mathbf{x}\_{n+1} \| \\ &= \| \| T \mathbf{x}\_{n+m+1} - T \mathbf{y}\_n \| \\ & \le \| T \mathbf{x}\_{n+m} - T \mathbf{z}\_n \| \\ &= \| T \mathbf{y}\_{n+m} - T \mathbf{z}\_n \| \\ & \le \| T \mathbf{z}\_{n+m} - (1 - a\_n) \mathbf{x}\_n - a\_n \mathbf{T} \mathbf{x}\_n \| \\ &= \| T \mathbf{z}\_{n+m} - (1 - a\_n) \mathbf{x}\_n - a\_n \mathbf{T} \mathbf{x}\_n \| \\ & \le (1 - a\_n) \| T \mathbf{z}\_{n+m} - \mathbf{x}\_n \| + a\_n \| T \mathbf{z}\_{n+m} - T \mathbf{x}\_n \| \\ & \le (1 - a\_n) \| T \mathbf{z}\_{n+m} - T \mathbf{x}\_{n+m} \| \\ &= (1 - a\_n) \| T \mathbf{u}\_{3n+1} - \mathbf{p}\_{3n+2} \| + a\_n \sum\_{k=3m-2}^{3n-2} \| T \mathbf{u}\_{k+1} - T \mathbf{z}\_k \| \\ & \le (1 - a\_n) \| T \mathbf{u}\_{3n-1+p} - \mathbf{u}\_{3n-2} \| + a\_n \sum\_{k=3m-2}^{3n-2} \rho\_k \| T \mathbf{u}\_{3n-2} - \mathbf{u}\_{3n-2} \| . \end{split}$$

It follows that

$$\|Tu\_{3n+1+p} - u\_{3n+1}\| \le (1 - a\_n) \|Tu\_{3n-1+p} - u\_{3n-2}\| + a\_n \sum\_{k=3n-2}^{3n-2+p} \beta\_k \|Tu\_{3n-2} - u\_{3n-2}\| \tag{8}$$

$$\begin{split} & \left( 1 + \sum\_{k=3n+1}^{3n+p} \beta\_k \right) \|Tu\_{3n+1} - u\_{3n+1}\| \\ & \le (1 - a\_n) \|Tu\_{3n-1+p} - u\_{3n-2}\| + a\_n \sum\_{k=3n-2}^{3n-2+p} \beta\_k \|Tu\_{3n-2} - u\_{3n-2}\| \\ & + \left( \prod\_{k=n+1}^{n+p} \frac{2}{1 - a\_k} \right) \left( \|Tu\_{3n+1} - u\_{3n+1}\| - \|Tu\_{3n+1+3p} - u\_{3n+1+3p}\| \right). \\ & \text{From } \left( 1 + \sum\_{k=3n+1}^{3n+p} \beta\_k \right) \le \left( \prod\_{k=n+1}^{n+p} \frac{1}{1 - a\_k} \right) \text{ and } \|Tu\_{3n+1} - u\_{3n+1}\| \le \|Tu\_{3n-2} - u\_{3n-2}\|, \\ & \text{have} \end{split}$$

$$\begin{aligned} &\left(1+\sum\_{k=3n+1}^{3n+p} \beta\_k\right) \|Tu\_{3n-2}-u\_{3n-2}\| \\ &\le \left(1-a\_n\right) \|Tu\_{3n-1+p}-u\_{3n-2}\| + a\_n \sum\_{k=3n-2}^{3n-2+p} \beta\_k \|Tu\_{3n-2}-u\_{3n-2}\| \\ &+ \left(\prod\_{k=n+1}^{n+p} \frac{2}{1-a\_k}\right) \left(\|Tu\_{3n-2}-u\_{3n-2}\| - \|Tu\_{3n+1+3p}-u\_{3n+1+3p}\|\right). \end{aligned}$$

$$\begin{aligned} &\left(\frac{1+\sum\_{k=3n+1}^{3n+p} \beta\_k - a\_n \sum\_{k=3n-2}^{3n-2+p} \beta\_k}{1-a\_n} \|Tu\_{3n-2} - u\_{3n-2}\| \\ &\le \|Tu\_{3n-1+p} - u\_{3n-2}\| \\ &+ \left(\prod\_{k=n}^{n+p} \frac{2}{1-a\_k}\right) \left(\|Tu\_{3n-2} - u\_{3n-2}\| - \|Tu\_{3n+1+3p} - u\_{3n+1+3p}\|\right) \end{aligned}$$

$$\begin{aligned} &\left(1+\sum\_{k=3n-2}^{3n-2+p} \beta\_k\right) \|Tu\_{3n-2}-u\_{3n-2}\| \\ &\le \|Tu\_{3n-1+p}-u\_{3n-2}\| \\ &+\left(\prod\_{k=n}^{n+p} \frac{2}{1-a\_k}\right) \left(\|Tu\_{3n-2}-u\_{3n-2}\| - \|Tu\_{3n+1+3p}-u\_{3n+1+3p}\|\right). \end{aligned}$$

$$\begin{aligned} &\|T\mathfrak{u}\_{3n+1+p} - \mathfrak{u}\_{3n+1}\| \\ &= \|T\mathfrak{z}\_{n+m+1} - \mathfrak{x}\_{n+1}\| \\ &= \|T\mathfrak{z}\_{n+m+1} - T\mathfrak{y}\_n\| \\ &\le \|\mathfrak{z}\_{n+m+1} - \mathfrak{y}\_n\| \\ &= \|(1 - \mathfrak{a}\_{m+n+1})\mathfrak{x}\_{m+n+1} + \mathfrak{a}\_{m+n+1}T\mathfrak{x}\_{m+n+1} - T\mathfrak{z}\_n\| \end{aligned}$$

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping DOI: http://dx.doi.org/10.5772/intechopen.88421*

Using (5) and (8), we have

3 X*n*þ*p*

!

*k*¼3*n*þ1

!

*βk*

2 1 � *α<sup>k</sup>*

*βk*

*βk*

2 1 � *α<sup>k</sup>*

*<sup>k</sup>*¼3*n*þ<sup>1</sup> *<sup>β</sup><sup>k</sup>* � *<sup>α</sup><sup>n</sup>*

≤ ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

2 1 � *α<sup>k</sup>*

!

*k*¼3*n*�2

!

!

*βk*

≤ ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

2 1 � *α<sup>k</sup>*

� �

1 � *α<sup>n</sup>*

≤

≤ ð Þ 1 � *α<sup>n</sup>* ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥ þ *α<sup>n</sup>*

*n* Q þ*p k*¼*n*þ1

1 1�*α<sup>k</sup>*

!

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

P<sup>3</sup>*n*�2þ*<sup>p</sup> <sup>k</sup>*¼3*n*�<sup>2</sup> *<sup>β</sup><sup>k</sup>*

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

Case 2: *p* ¼ 3*m* þ 1*, m* ≥ 0. From (6) and (7), we have

∥*Tu*3*n*þ1þ*<sup>p</sup>* � *u*3*n*þ<sup>1</sup>∥ ¼ ∥*Tzn*þ*m*þ<sup>1</sup> � *xn*þ<sup>1</sup>∥ ¼ ∥*Tzn*þ*m*þ<sup>1</sup> � *Tyn*∥ ≤ ∥*zn*þ*m*þ<sup>1</sup> � *yn*∥

≤ ð Þ 1 � *α<sup>n</sup>* ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥ þ *α<sup>n</sup>*

∥*Tu*3*n*þ<sup>1</sup> � *u*3*n*þ<sup>1</sup>∥

3*n* X�2þ*<sup>p</sup>*

*k*¼3*n*�2

3*n* X�2þ*<sup>p</sup>*

*k*¼3*n*�2

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

¼ ∥ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> *xm*þ*n*þ<sup>1</sup> þ *α<sup>m</sup>*þ*n*þ<sup>1</sup>*Txm*þ*n*þ<sup>1</sup> � *Tzn*∥

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

<sup>∥</sup>*Tu*3*n*þ<sup>1</sup> � *<sup>u</sup>*3*n*þ<sup>1</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

*βk*∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

and ∥*Tu*3*n*þ<sup>1</sup> � *u*3*n*þ<sup>1</sup>∥≤∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥,

*βk*∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

1 þ

*n* Y þ*p*

*k*¼*n*þ1

3 P*n*þ*p k*¼3*n*þ1

1 þ

*n* Y þ*p*

*k*¼*n*þ1

<sup>1</sup> <sup>þ</sup> <sup>P</sup><sup>3</sup>*n*þ*<sup>p</sup>*

þ

þ

It follows that

*n* Y þ*p*

*k*¼*n*

1 þ 3*n* X�2þ*<sup>p</sup>*

> *n* Y þ*p*

> *k*¼*n*

Thus, for *n, p* þ 1, (4) holds.

þ

**42**

Then

!

3 X*n*þ*p*

!

*k*¼3*n*þ1

!

þ

From 1 þ

*Functional Calculus*

we have

≤ ð Þ 1 � *αm*þ*n*þ<sup>1</sup> ∥*xm*þ*n*þ<sup>1</sup> � *Tzn*∥ þ *αm*þ*n*þ<sup>1</sup>∥*Txm*þ*n*þ<sup>1</sup> � *Tzn*∥ <sup>≤</sup> ð Þ <sup>1</sup> � *<sup>α</sup>m*þ*n*þ<sup>1</sup> <sup>∥</sup>*Tym*þ*<sup>n</sup>* � *Tzn*<sup>∥</sup> <sup>þ</sup> *<sup>α</sup>m*þ*n*þ<sup>1</sup>∥*xm*þ*n*þ<sup>1</sup> � *zn*<sup>∥</sup> <sup>≤</sup> ð Þð <sup>1</sup> � *<sup>α</sup>m*þ*n*þ<sup>1</sup> <sup>∥</sup>*Tym*þ*<sup>n</sup>* � *Tzm*þ*<sup>n</sup>*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*Tzm*þ*<sup>n</sup>* � *Txn*þ*<sup>m</sup>*<sup>∥</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>∥</sup>*Txn*þ<sup>1</sup> � *Tyn*<sup>∥</sup> þ∥*Tyn* � *Tzn*∥Þ þ *αm*þ*n*þ<sup>1</sup>∥*xm*þ*n*þ<sup>1</sup> � *zn*∥ ¼ ð Þ 1 � *αm*þ*n*þ<sup>1</sup> 3*n* X�2þ*<sup>p</sup> k*¼3*n*�1 <sup>∥</sup>*Tuk*þ<sup>1</sup> � *Tuk*<sup>∥</sup> <sup>þ</sup> *<sup>α</sup>m*þ*n*þ<sup>1</sup>∥*Tym*þ*<sup>n</sup>* � ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup> xn* � *<sup>α</sup>nTxn*<sup>∥</sup> ≤ ð Þ 1 � *αm*þ*n*þ<sup>1</sup> 3*n* X�2þ*<sup>p</sup> k*¼3*n*�1 *βk*∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ <sup>þ</sup>*αm*þ*n*þ<sup>1</sup> ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup>* <sup>∥</sup>*Tym*þ*<sup>n</sup>* � *xn*<sup>∥</sup> <sup>þ</sup> *<sup>α</sup>n*∥*Tym*þ*<sup>n</sup>* � *Txn*<sup>∥</sup> � � ≤ ð Þ 1 � *αm*þ*n*þ<sup>1</sup> 3*n* X�2þ*<sup>p</sup> k*¼3*n*�1 *<sup>β</sup>k*∥*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> <sup>þ</sup> *<sup>α</sup>m*þ*n*þ<sup>1</sup>ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup>* <sup>∥</sup>*Tym*þ*<sup>n</sup>* � *xn*<sup>∥</sup> <sup>þ</sup>*α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* <sup>∥</sup>*Tym*þ*<sup>n</sup>* � *Tzm*þ*<sup>n</sup>*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*Tzm*þ*<sup>n</sup>* � *Txm*þ*<sup>n</sup>*<sup>∥</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>∥</sup>*Tyn* � *Tzn*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*Tzn* � *Txn*<sup>∥</sup> � � ¼ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> 3*n* X�2þ*<sup>p</sup> k*¼3*n*�1 *<sup>β</sup>k*∥*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> <sup>þ</sup> *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup>ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup>* <sup>∥</sup>*Tym*þ*<sup>n</sup>* � *xn*<sup>∥</sup> þ*α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* 3*n* X�2þ*<sup>p</sup> k*¼3*n*�2 ∥*Tuk*þ<sup>1</sup> � *Tuk*∥ ≤ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> 3*n* X�2þ*<sup>p</sup> k*¼3*n*�1 *<sup>β</sup>k*∥*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> <sup>þ</sup> *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup>ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup>* <sup>∥</sup>*Tym*þ*<sup>n</sup>* � *xn*<sup>∥</sup> þ*α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* 3*n* X�2þ*<sup>p</sup> k*¼3*n*�2 *βk*∥*Tuk* � *uk*∥ ¼ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> 3*n* X�2þ*<sup>p</sup> k*¼3*n*�1 *βk*∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ þ *α<sup>m</sup>*þ*n*þ<sup>1</sup>ð Þ 1 � *α<sup>n</sup>* ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥ þ*α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* 3*n* X�2þ*<sup>p</sup> k*¼3*n*�2 *βk*∥*Tuk* � *uk*∥ ¼ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> þ *α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* 3*n* X�2þ*<sup>p</sup> k*¼3*n*�2 *βk*∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ �*αn*ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ þ *α<sup>m</sup>*þ*n*þ<sup>1</sup>ð Þ 1 � *α<sup>n</sup>* ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥*:*

It follows that

$$\begin{aligned} &\|Tu\_{3n+1+p} - u\_{3n+1}\| \\ &\le (1 - a\_{m+n+1} + a\_{m+n+1}a\_n) \sum\_{k=3n-2}^{3n-2+p} \beta\_k \|Tu\_{3n-2} - u\_{3n-2}\| \\ &- a\_n(1 - a\_{m+n+1}) \|Tu\_{3n-2} - u\_{3n-2}\| + a\_{m+n+1}(1 - a\_n) \|Tu\_{3n-1+p} - u\_{3n-2}\| \end{aligned} \tag{9}$$

Using (5) and (9), we have

$$\begin{aligned} & \left( 1 + \sum\_{k=3n+1}^{3n+p} \beta\_k \right) ||Tu\_{3n+1} - u\_{3n+1}|| \\ & \le \left( 1 - a\_{m+n+1} + a\_{m+n+1} a\_n \right) \sum\_{k=3n-2}^{3n-2+p} \beta\_k ||Tu\_{3n-2} - u\_{3n-2}|| \end{aligned}$$

$$\begin{split} & -a\_n(1 - a\_{m+n+1}) \| \| T u\_{3n-2} - u\_{3n-2} \| + a\_{m+n+1}(1 - a\_n) \| \| T u\_{3n-1+p} - u\_{3n-2} \| \\ & + \left( \prod\_{k=n+1}^{n+p} \frac{2}{1 - a\_k} \right) \left( \| T u\_{3n+1} - u\_{3n+1} \| - \| T u\_{3n+1+3p} - u\_{3n+1+3p} \| \right). \\ & \text{From } \left( 1 + \sum\_{k=3n+1}^{3n+p} \beta\_k \right) \le \left( \prod\_{k=n+1}^{n+p} \frac{1}{1 - a\_k} \right) \text{ and } \| T u\_{3n+1} - u\_{3n+1} \| \le \| T u\_{3n-2} - u\_{3n-2} \|, \\ & \text{have} \end{split}$$

≤ ð Þ 1 � *α<sup>n</sup>* ð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ∥*xn*þ*m*þ<sup>1</sup> � *xn*∥ þ *α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥ þ*αn*ð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ∥*xn*þ*m*þ<sup>1</sup> � *Txn*∥ þ *α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *Txn*∥

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping*

þð Þ ð Þ <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup> ð Þþ <sup>1</sup> � *<sup>α</sup><sup>n</sup> <sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* <sup>∥</sup>*Tyn*þ*<sup>m</sup>* � *Tzn*þ*<sup>m</sup>*<sup>∥</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>∥</sup>*Tzn* � *Txn*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*Txn* � *xn*<sup>∥</sup> � �

*βk*∥*Txn* � *xn*∥

*β<sup>k</sup>* � *αn*ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup>

*β<sup>k</sup>* � *αn*ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup>

<sup>∥</sup>*Tu*3*n*þ<sup>1</sup> � *<sup>u</sup>*3*n*þ<sup>1</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

*β<sup>k</sup>* � *αn*ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup>

∥*Tuk*þ<sup>1</sup> � *Tuk*∥ þ ∥*Tu*3*n*�<sup>2</sup> � *x*3*n*�<sup>2</sup>∥ !

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥*:*

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

and ∥*Tu*3*n*þ<sup>1</sup> � *u*3*n*þ<sup>1</sup>∥≤∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥,

(10)

<sup>þ</sup>*αn*ð Þ <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup> <sup>∥</sup>*Tyn*þ*m*þ<sup>1</sup> � *Tzn*þ*m*þ<sup>1</sup><sup>∥</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>∥</sup>*Tyn* � *Tzn*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*Tzn* � *Txn*<sup>∥</sup> � �

3*n* X�3þ*<sup>p</sup> k*¼3*n*�2

3*n* X�2þ*<sup>p</sup> k*¼3*n*�2

∥*Tuk*þ<sup>1</sup> � *Tuk*∥

*βk*∥*Txn* � *xn*∥

3*n* X�2þ*<sup>p</sup> k*¼3*n*�2

> 3*n* X�2þ*<sup>p</sup>*

!

∥*Tu*3*n*þ<sup>1</sup> � *u*3*n*þ<sup>1</sup>∥

*k*¼3*n*�2

3*n* X�2þ*<sup>p</sup>*

!

*k*¼3*n*�2

2 1�*α<sup>k</sup>*

!

!

þð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ð Þþ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* ∥*xn*þ*m*þ<sup>1</sup> � *xn*∥

*DOI: http://dx.doi.org/10.5772/intechopen.88421*

≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥

<sup>þ</sup>*αn*ð Þ <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup> <sup>∥</sup>*Tyn*þ*m*þ<sup>1</sup> � *Txn*<sup>∥</sup> ≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥

≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥

þð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> *α<sup>n</sup>*

þð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> *α<sup>n</sup>*

þð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ð Þþ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>*

3*n* X�3þ*<sup>p</sup> k*¼3*n*�2

≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥

þð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ð Þþ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>*

3*n* X�3þ*<sup>p</sup> k*¼3*n*�2

≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> þ *αnα<sup>m</sup>*þ*n*þ<sup>1</sup>

Using (5) and (10), we have

*βk*

þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> þ *αnα<sup>m</sup>*þ*n*þ<sup>1</sup>

*βk*

≤

*n* Q þ*p k*¼*n*þ1

2 1 � *α<sup>k</sup>*

3 P*n*þ*p k*¼3*n*þ1

!

!

≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> þ *αnα<sup>m</sup>*þ*n*þ<sup>1</sup>

It follows that

1 þ

*n* Y þ*p*

*k*¼*n*þ1

From 1 þ

þ

we have

**45**

3 X*n*þ*p*

!

*k*¼3*n*þ1

∥*Tu*3*n*þ1þ*<sup>p</sup>* � *u*3*n*þ<sup>1</sup>∥

we have

$$\begin{split} & \left( 1 + \sum\_{k=3n+1}^{3n+p} \beta\_k \right) \|Tu\_{3n-2} - u\_{3n-2}\| \\ & \leq \left( 1 - a\_{m+n+1} + a\_{m+n+1}a\_n \right) \sum\_{k=3n-2}^{3n-2+p} \beta\_k \|Tu\_{3n-2} - u\_{3n-2}\| \\ & - a\_n(1 - a\_{m+n+1}) \|Tu\_{3n-2} - u\_{3n-2}\| + a\_{m+n+1}(1 - a\_n) \|Tu\_{3n-1+p} - u\_{3n-2}\| \\ & + \left( \prod\_{k=n+1}^{n+p} \frac{2}{1 - a\_k} \right) \left( \|Tu\_{3n-2} - u\_{3n-2}\| - \|Tu\_{3n+1+3p} - u\_{3n+1+3p}\| \right). \end{split}$$

Then

$$\begin{split} & \left( \frac{1 + \sum\_{k=3n+1}^{3n+p} \beta\_k + a\_n (1 - a\_{m+n+1}) - (1 - a\_{m+n+1} + a\_{m+n+1} a\_n) \sum\_{k=3n-2}^{3n-2+p} \beta\_k \right) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \|Tu\_{3n-2} - u\_{3n-2}\| \\ & \qquad \qquad \qquad \|Tu\_{3n-2} - u\_{3n-2}\| \le \|Tu\_{3n-1+p} - u\_{3n-2}\| \\ & \qquad \qquad \qquad + \left( \prod\_{k=n}^{n+p} \frac{2}{1 - a\_k} \right) \left( \|Tu\_{3n-2} - u\_{3n-2}\| - \|Tu\_{3n+1+3p} - u\_{3n+1+3p}\| \right). \end{split}$$

It follows that

$$\begin{aligned} &\left(1+\sum\_{k=3n-2}^{3n-2+p} \beta\_k\right) \|Tu\_{3n-2}-u\_{3n-2}\| \\ &\le \|Tu\_{3n-1+p}-u\_{3n-2}\| \\ &+\left(\prod\_{k=n}^{n+p} \frac{2}{1-a\_k}\right) \left(\|Tu\_{3n-2}-u\_{3n-2}\| - \|Tu\_{3n+1+3p}-u\_{3n+1+3p}\|\right). \end{aligned}$$

Thus, for *n, p* þ 1, (4) holds. Case 3: *p* ¼ 3*m* þ 2*, m* ≥0. From (6) and (7), we have

$$\begin{aligned} &\|\|T u\_{3n+1+p} - u\_{3n+1}\|\| \\ &= \|\|\mathcal{T}\_{n+m+1} - \mathcal{T}\mathcal{Y}\_n\|\| \\ &\le \|\|\mathcal{T}\_{n+m+1} - \mathcal{T}\mathcal{X}\_n\|\| \\ &= \|\|\mathcal{T}\_{n+m+1} - \mathcal{T}\mathcal{Z}\_n\|\| \\ &\le \|\|z\_{n+m+1} - z\_n\|\| \\ &\le \|\|z\_{n+m+1} - (1 - a\_n)\chi\_n - a\_n \mathcal{T}\chi\_n\|\| \\ &\le (1 - a\_n)\|\|z\_{n+m+1} - \chi\_n\|\| + a\_n\|\|z\_{n+m+1} - \mathcal{T}\chi\_n\|\| \\ &= (1 - a\_n)\|\|(1 - a\_{n+m+1})\chi\_{n+m+1} + a\_{n+m+1}\mathcal{T}\chi\_{n+m+1} - \chi\_n\|\| \\ &+ a\_n\|\|(1 - a\_{n+m+1})\chi\_{n+m+1} + a\_{n+m+1}\mathcal{T}\chi\_{n+m+1} - \mathcal{T}\chi\_n\|\end{aligned}$$

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping DOI: http://dx.doi.org/10.5772/intechopen.88421*

≤ ð Þ 1 � *α<sup>n</sup>* ð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ∥*xn*þ*m*þ<sup>1</sup> � *xn*∥ þ *α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥ þ*αn*ð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ∥*xn*þ*m*þ<sup>1</sup> � *Txn*∥ þ *α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *Txn*∥ ≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥ þð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ð Þþ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* ∥*xn*þ*m*þ<sup>1</sup> � *xn*∥ <sup>þ</sup>*αn*ð Þ <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup> <sup>∥</sup>*Tyn*þ*m*þ<sup>1</sup> � *Txn*<sup>∥</sup> ≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥ þð Þ ð Þ <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup> ð Þþ <sup>1</sup> � *<sup>α</sup><sup>n</sup> <sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* <sup>∥</sup>*Tyn*þ*<sup>m</sup>* � *Tzn*þ*<sup>m</sup>*<sup>∥</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>∥</sup>*Tzn* � *Txn*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*Txn* � *xn*<sup>∥</sup> � � <sup>þ</sup>*αn*ð Þ <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup> <sup>∥</sup>*Tyn*þ*m*þ<sup>1</sup> � *Tzn*þ*m*þ<sup>1</sup><sup>∥</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>∥</sup>*Tyn* � *Tzn*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*Tzn* � *Txn*<sup>∥</sup> � � ≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥ þð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ð Þþ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* 3*n* X�3þ*<sup>p</sup> k*¼3*n*�2 ∥*Tuk*þ<sup>1</sup> � *Tuk*∥ þ ∥*Tu*3*n*�<sup>2</sup> � *x*3*n*�<sup>2</sup>∥ ! þð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> *α<sup>n</sup>* 3*n* X�3þ*<sup>p</sup> k*¼3*n*�2 ∥*Tuk*þ<sup>1</sup> � *Tuk*∥ ≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Txn*þ*m*þ<sup>1</sup> � *xn*∥ þð Þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ð Þþ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>* 3*n* X�2þ*<sup>p</sup> k*¼3*n*�2 *βk*∥*Txn* � *xn*∥ þð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> *α<sup>n</sup>* 3*n* X�3þ*<sup>p</sup> k*¼3*n*�2 *βk*∥*Txn* � *xn*∥ ≤ ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥ þ ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> þ *αnα<sup>m</sup>*þ*n*þ<sup>1</sup> 3*n* X�2þ*<sup>p</sup> k*¼3*n*�2 *β<sup>k</sup>* � *αn*ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> !∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥*:*

It follows that

�*αn*ð Þ 1 � *αm*þ*n*þ<sup>1</sup> ∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ þ *αm*þ*n*þ<sup>1</sup>ð Þ 1 � *α<sup>n</sup>* ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

<sup>∥</sup>*Tu*3*n*þ<sup>1</sup> � *<sup>u</sup>*3*n*þ<sup>1</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

*βk*∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

and ∥*Tu*3*n*þ<sup>1</sup> � *u*3*n*þ<sup>1</sup>∥≤∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥,

P<sup>3</sup>*n*�2þ*<sup>p</sup> <sup>k</sup>*¼3*n*�<sup>2</sup> *<sup>β</sup><sup>k</sup>*

þ

*Functional Calculus*

From 1 þ

1 þ

*n* Y þ*p*

*k*¼*n*þ1

<sup>1</sup> <sup>þ</sup> <sup>P</sup><sup>3</sup>*n*þ*<sup>p</sup>*

2 1 � *α<sup>k</sup>*

> 1 þ 3*n* X�2þ*<sup>p</sup>*

> > *n* Y þ*p*

> > *k*¼*n*

Thus, for *n, p* þ 1, (4) holds.

þ

*k*¼3*n*�2

!

!

!

þ

Then

þ

**44**

*n* Y þ*p*

*k*¼*n*

It follows that

3 X*n*þ*p*

!

*k*¼3*n*þ1

*βk*

≤ ð Þ 1 � *αm*þ*n*þ<sup>1</sup> þ *αm*þ*n*þ1*α<sup>n</sup>*

2 1 � *α<sup>k</sup>*

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥≤∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

*βk*

≤ ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

∥*Tu*3*n*þ1þ*<sup>p</sup>* � *u*3*n*þ<sup>1</sup>∥ <sup>¼</sup> <sup>∥</sup>*Tyn*þ*m*þ<sup>1</sup> � *Tyn*<sup>∥</sup> ≤ ∥*yn*þ*m*þ<sup>1</sup> � *yn*<sup>∥</sup> ¼ ∥*Tzn*þ*m*þ<sup>1</sup> � *Tzn*∥ ≤ ∥*zn*þ*m*þ<sup>1</sup> � *zn*∥

2 1 � *α<sup>k</sup>*

!

we have

*n* Yþ*p k*¼*n*þ1

2 1 � *α<sup>k</sup>*

*βk*

≤

*n* Q þ*p k*¼*n*þ1

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

1 1�*α<sup>k</sup>*

!

3*n* X�2þ*<sup>p</sup>*

*k*¼3*n*�2

�*αn*ð Þ 1 � *α<sup>m</sup>*þ*n*þ<sup>1</sup> ∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ þ *α<sup>m</sup>*þ*n*þ<sup>1</sup>ð Þ 1 � *α<sup>n</sup>* ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

*<sup>k</sup>*¼3*n*þ<sup>1</sup> *<sup>β</sup><sup>k</sup>* <sup>þ</sup> *<sup>α</sup>n*ð Þ� <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup> ð Þ <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup> <sup>þ</sup> *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup>*α<sup>n</sup>*

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

Case 3: *p* ¼ 3*m* þ 2*, m* ≥0. From (6) and (7), we have

≤ ∥*zn*þ*m*þ<sup>1</sup> � ð Þ 1 � *α<sup>n</sup> xn* � *αnTxn*∥

≤ ð Þ 1 � *α<sup>n</sup>* ∥*zn*þ*m*þ<sup>1</sup> � *xn*∥ þ *αn*∥*zn*þ*m*þ<sup>1</sup> � *Txn*∥

¼ ð Þ 1 � *α<sup>n</sup>* ∥ð Þ 1 � *α<sup>n</sup>*þ*m*þ<sup>1</sup> *xn*þ*m*þ<sup>1</sup> þ *α<sup>n</sup>*þ*m*þ<sup>1</sup>*Txn*þ*m*þ<sup>1</sup> � *xn*∥ þ*αn*∥ð Þ 1 � *α<sup>n</sup>*þ*m*þ<sup>1</sup> *xn*þ*m*þ<sup>1</sup> þ *α<sup>n</sup>*þ*m*þ<sup>1</sup>*Txn*þ*m*þ<sup>1</sup> � *Txn*∥

� �

*α<sup>m</sup>*þ*n*þ<sup>1</sup>ð Þ 1 � *α<sup>n</sup>*

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

!

3 P*n*þ*p k*¼3*n*þ1

!

$$\begin{split} & \| T u\_{3n+1+p} - u\_{3n+1} \| \\ & \le (1 - a\_n) a\_{m+n+1} \| T u\_{3n-1+p} - u\_{3n-2} \| \\ & + \left( (1 - a\_{m+n+1} + a\_n a\_{m+n+1}) \sum\_{k=3n-2}^{3n-2+p} \beta\_k - a\_n (1 - a\_{m+n+1}) \right) \| T u\_{3n-2} - u\_{3n-2} \| \end{split} \tag{10}$$

Using (5) and (10), we have

$$\begin{split} & \left( 1 + \sum\_{k=3n+1}^{3n+p} \beta\_k \right) \|Tu\_{3n+1} - u\_{3n+1}\| \\ & \le (1 - a\_n)a\_{m+n+1} \|Tu\_{3n-1+p} - u\_{3n-2}\| \\ & + \left( (1 - a\_{m+n+1} + a\_n a\_{m+n+1}) \sum\_{k=3n-2}^{3n-2+p} \beta\_k - a\_n (1 - a\_{m+n+1}) \right) \|Tu\_{3n-2} - u\_{3n-2}\| \\ & + \left( \prod\_{k=n+1}^{n+p} \frac{2}{1 - a\_k} \right) \left( \|Tu\_{3n+1} - u\_{3n+1}\| - \|Tu\_{3n+1+3p} - u\_{3n+1+3p}\| \right). \end{split}$$
 
$$\text{From } \left( 1 + \sum\_{k=3n+1}^{3n+p} \beta\_k \right) \le \left( \prod\_{k=n+1}^{n+p} \frac{2}{1 - a\_k} \right) \text{ and } \|Tu\_{3n+1} - u\_{3n+1}\| \le \|Tu\_{3n-2} - u\_{3n-2}\|, \text{ we have}$$
 
$$\text{we have}$$

$$\begin{split} & \left( 1 + \sum\_{k=3n+1}^{3n+p} \beta\_k \right) \|Tu\_{3n-2} - u\_{3n-2}\| \\ & \le (1 - a\_n)a\_{m+n+1} \|Tu\_{3n-1+p} - u\_{3n-2}\| \\ & + \left( (1 - a\_{m+n+1} + a\_n a\_{m+n+1}) \sum\_{k=3n-2}^{3n-2+p} \beta\_k - a\_n(1 - a\_{m+n+1}) \right) \|Tu\_{3n-2} - u\_{3n-2}\| \\ & + \left( \prod\_{k=n+1}^{n+p} \frac{2}{1 - a\_k} \right) \left( \|Tu\_{3n-2} - u\_{3n-2}\| - \|Tu\_{3n+1+3p} - u\_{3n+1+3p}\| \right). \end{split}$$

*e* 6 1�*b <sup>d</sup>* ð Þ *<sup>r</sup>* <sup>þ</sup><sup>1</sup> *ε*<*r*

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥ � ∥*Tu*3*n*�2þ3*<sup>p</sup>* � *u*3*n*�2þ3*<sup>p</sup>*∥<*ε:*

≤ *d* ≤ *r*

<sup>2</sup> <*a* ≤ *α<sup>n</sup>* ≤ *b*< 1, for every *k, t*, we have 1 þ *α<sup>k</sup>* <3*αk, α<sup>t</sup>* <2*αk*. From

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

3*n*þ P *p*�3

*k*¼3*n*�2

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*�2þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*�2þ3*<sup>p</sup>*<sup>∥</sup> � �

*βk* !

.

and choose *n* so that for every *p* >0

*DOI: http://dx.doi.org/10.5772/intechopen.88421*

**Lemma 2** and *r* ≤ ∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥, we have

3*n* X þ*p*�3

3*n* X þ*p*�3

!

!

*k*¼3*n*�2

*k*¼3*n*�2

≤ ∥*Tu*3*n*�2þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

!

2 1 � *α<sup>k</sup>*

!

2 1 � *α<sup>k</sup>*

*ε*

*ε*

*ε*

Proof: Since *C* is compact, there exists a subsequence *xnk*

This is a contradiction. So lim*<sup>k</sup>*!<sup>∞</sup> ∥*Tuk* � *uk*∥ ¼ 0. That is to say,

**Theorem 2.** Assume that *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping and f g *xn* is generated

<sup>2</sup> <*a* ≤ *α<sup>n</sup>* ≤ *b*<1. Then the sequence f g *xn* converges to a fixed

� �⊂f g *xn* which con-

<sup>þ</sup><sup>1</sup> *<sup>ε</sup>*<sup>&</sup>lt; *<sup>d</sup>* <sup>þ</sup> *<sup>r</sup>:*

3*n*þ P *p*�4

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping*

*βk*

*βk*

*k*¼3*n*�2

*βk* !

Now choose *p* so that *r*

*d* þ *r* ≤ *r* 1 þ

þ

< *d* þ

¼ *d* þ *e*

≤ *d* þ *e*

≤ *d* þ *e*

≤ *d* þ *e*

≤ *d* þ *e*

< *d* þ *e*

by iteration (3), <sup>1</sup>

point of *T*.

**47**

≤ 1 þ

*n*þ Y *p*�1

*k*¼*n*

*n*þ Y *p*�1

*k*¼*n*

Σ *n*þ*p*�1 *<sup>k</sup>*¼*<sup>n</sup>* ln 1þ1þ*α<sup>k</sup>* <sup>1</sup>�*α<sup>k</sup>* � �

Σ *n*þ*p*�1 *k*¼*n* 1þ*αk* 1�*αk ε*

3 <sup>1</sup>�*<sup>b</sup>* <sup>Σ</sup> *n*þ*p*�1 *<sup>k</sup>*¼*<sup>n</sup> <sup>α</sup><sup>k</sup> ε*

6 <sup>1</sup>�*<sup>b</sup>* <sup>Σ</sup> 3*n*þ*p*�3 *k*¼3*n*�2 *βk ε*

6 <sup>1</sup>�*<sup>b</sup>* <sup>Σ</sup> <sup>3</sup>*n*þ*p*�<sup>4</sup> *k*¼3*n*�2 *β<sup>k</sup>* þ 1 � �

6 1�*b <sup>d</sup>* ð Þ *<sup>r</sup>*

verges to some *z*∈*C*. By **Lemma 3**, we have

lim*<sup>n</sup>*!<sup>∞</sup> ∥*Txn* � *xn*∥ ¼ 0. This completes the proof.

Since <sup>1</sup>

Then

$$\begin{split} & \left( \frac{(1 + \sum\_{k=3n+1}^{3n+p} \beta\_k - \left( (1 - a\_{m+n+1} + a\_n a\_{m+n+1}) \sum\_{k=3n-2}^{3n-2+p} \beta\_k - a\_n (1 - a\_{m+n+1}) \right)}{(1 - a\_n) a\_{m+n+1}} \right) \| \| T u\_{3n-2} - u\_{3n-2} \| \\ & \le \| \| T u\_{3n-1+p} - u\_{3n-2} \| \\ & + \left( \prod\_{k=n}^{n+p} \frac{2}{1 - a\_k} \right) \left( \| T u\_{3n-2} - u\_{3n-2} \| - \| T u\_{3n+1+3p} - u\_{3n+1+3p} \| \right). \end{split}$$

It follows that

$$\begin{aligned} &\left(1+\sum\_{k=3n-2}^{3n-2+p} \beta\_k\right) \|Tu\_{3n-2}-u\_{3n-2}\| \\ &\le \|Tu\_{3n-1+p}-u\_{3n-2}\| \\ &+\left(\prod\_{k=n}^{n+p} \frac{2}{1-a\_k}\right) \left(\|Tu\_{3n-2}-u\_{3n-2}\| - \|Tu\_{3n+1+3p}-u\_{3n+1+3p}\|\right). \end{aligned}$$

Thus, for *n, p* þ 1, (4) holds. This completes the induction. **Lemma 3.** *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping, ∥*Tx* � *x*∥≤∥*Ty* � *y*∥. Then

$$\|\|\mathbf{x} - T\mathbf{y}\|\| \le 3\|\|T\mathbf{x} - \mathbf{x}\|\| + \|\mathbf{x} - \mathbf{y}\|\|.$$

Proof: Since ∥*Tx* � *x*∥≤∥*Ty* � *y*∥, we have *Tx*∈*C T*ð Þ *; y; α* . Then

$$\|T^2\mathbf{x} - T\mathbf{y}\| \le \|T\mathbf{x} - \mathbf{y}\|\dots$$

It follows that

$$\|\|\mathbf{x} - T\mathbf{y}\|\| \le \|\mathbf{x} - T\mathbf{x}\|\| + \|\|T^2\mathbf{x} - T\mathbf{x}\|\| + \|\|T^2\mathbf{x} - T\mathbf{y}\|\|.$$

From **Proposition 3**, we have

$$\|\mathbf{x} - \mathbf{T}\mathbf{y}\| \le 2\|\mathbf{T}\mathbf{x} - \mathbf{x}\| + \|\mathbf{T}\mathbf{x} - \mathbf{y}\| \le 2\|\mathbf{T}\mathbf{x} - \mathbf{x}\| + \|\mathbf{T}\mathbf{x} - \mathbf{x}\| + \|\mathbf{x} - \mathbf{y}\| = \Im\|\mathbf{T}\mathbf{x} - \mathbf{x}\| + \|\mathbf{x} - \mathbf{y}\|.$$

**Theorem 1.** Assume that *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping and f g *xn , yn* � �*, z*f g*<sup>n</sup>* are sequences generated by iteration (3), <sup>1</sup> <sup>2</sup> <*a* ≤ *α<sup>n</sup>* ≤ *b*<1. Then lim*<sup>n</sup>*!<sup>∞</sup> ∥*Txn* � *xn*∥ ¼ 0.

Proof: Since C is bounded, there must exists *d*>0*,* for every *x*∈*C,* ∥*x*∥ ≤ *d*. Let f g *um* satisfy *u*3*n*�<sup>2</sup> ¼ *xn, u*3*n*�<sup>1</sup> ¼ *zn, u*3*<sup>n</sup>* ¼ *yn*. From Lemma 1, lim*<sup>k</sup>*!<sup>∞</sup> ∥*Tuk* � *uk*∥ ¼ *r*≥0. Assume *r*> 0. Let *ε* satisfy

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping DOI: http://dx.doi.org/10.5772/intechopen.88421*

$$e^{\frac{6}{1-\delta} \binom{d+1}{r}} \varepsilon < r$$

and choose *n* so that for every *p* >0

1 þ

*Functional Calculus*

*n* Y þ*p*

*k*¼*n*þ1

≤ ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

2 1 � *α<sup>k</sup>* !

It follows that

1 þ 3*n* X�2þ*<sup>p</sup>*

> *n* Y þ*p*

> *k*¼*n*

From **Proposition 3**, we have

are sequences generated by iteration (3), <sup>1</sup>

þ

It follows that

lim*<sup>n</sup>*!<sup>∞</sup> ∥*Txn* � *xn*∥ ¼ 0.

**46**

*k*¼3*n*�2

!

≤ ∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

2 1 � *α<sup>k</sup>*

!

*βk*

þ

Then

<sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>P</sup><sup>3</sup>*n*þ*<sup>p</sup>*

þ *n* Yþ*p k*¼*n* 3 X*n*þ*p*

!

*k*¼3*n*þ1

*βk*

þ ð Þ 1 � *αm*þ*n*þ<sup>1</sup> þ *αnαm*þ*n*þ<sup>1</sup>

2 1 � *α<sup>k</sup>*

!

≤ ð Þ 1 � *α<sup>n</sup> αm*þ*n*þ<sup>1</sup>∥*Tu*3*n*�1þ*<sup>p</sup>* � *u*3*n*�<sup>2</sup>∥

*<sup>k</sup>*¼3*n*þ<sup>1</sup> *<sup>β</sup><sup>k</sup>* � ð Þ <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup> <sup>þ</sup> *<sup>α</sup>nα<sup>m</sup>*þ*n*þ<sup>1</sup>

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

3*n* X�2þ*<sup>p</sup>*

!

ð Þ 1 � *α<sup>n</sup> α<sup>m</sup>*þ*n*þ<sup>1</sup>

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

Thus, for *n, p* þ 1, (4) holds. This completes the induction.

∥*T*<sup>2</sup>

<sup>∥</sup>*<sup>x</sup>* � *Ty*∥≤∥*<sup>x</sup>* � *Tx*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*T*<sup>2</sup>

f g *um* satisfy *u*3*n*�<sup>2</sup> ¼ *xn, u*3*n*�<sup>1</sup> ¼ *zn, u*3*<sup>n</sup>* ¼ *yn*. From Lemma 1, lim*<sup>k</sup>*!<sup>∞</sup> ∥*Tuk* � *uk*∥ ¼ *r*≥0. Assume *r*> 0. Let *ε* satisfy

Proof: Since ∥*Tx* � *x*∥≤∥*Ty* � *y*∥, we have *Tx*∈*C T*ð Þ *; y; α* . Then

**Lemma 3.** *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping, ∥*Tx* � *x*∥≤∥*Ty* � *y*∥. Then

∥*x* � *Ty*∥ ≤ 3∥*Tx* � *x*∥ þ ∥*x* � *y*∥*:*

∥*x* � *Ty*∥ ≤ 2∥*Tx* � *x*∥ þ ∥*Tx* � *y*∥ ≤ 2∥*Tx* � *x*∥ þ ∥*Tx* � *x*∥ þ ∥*x* � *y*∥ ¼ 3∥*Tx* � *x*∥ þ ∥*x* � *y*∥*:*

Proof: Since C is bounded, there must exists *d*>0*,* for every *x*∈*C,* ∥*x*∥ ≤ *d*. Let

**Theorem 1.** Assume that *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping and f g *xn , yn*

*x* � *Ty*∥≤∥*Tx* � *y*∥*:*

*<sup>x</sup>* � *Tx*<sup>∥</sup> <sup>þ</sup> <sup>∥</sup>*T*<sup>2</sup>

<sup>2</sup> <*a* ≤ *α<sup>n</sup>* ≤ *b*<1. Then

*x* � *Ty*∥*:*

� �*, z*f g*<sup>n</sup>*

*k*¼3*n*�2

*β<sup>k</sup>* � *αn*ð Þ 1 � *αm*þ*n*þ<sup>1</sup>

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

*<sup>k</sup>*¼3*n*�<sup>2</sup> *<sup>β</sup><sup>k</sup>* � *<sup>α</sup>n*ð Þ <sup>1</sup> � *<sup>α</sup><sup>m</sup>*þ*n*þ<sup>1</sup>

<sup>∥</sup>*Tu*3*n*�<sup>2</sup> � *<sup>u</sup>*3*n*�<sup>2</sup><sup>∥</sup> � <sup>∥</sup>*Tu*3*n*þ1þ3*<sup>p</sup>* � *<sup>u</sup>*3*n*þ1þ3*<sup>p</sup>*<sup>∥</sup> � �*:*

P<sup>3</sup>*n*�2þ*<sup>p</sup>*

� �

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥

$$\|\|Tu\_{3n-2} - u\_{3n-2}\|\| - \|\|Tu\_{3n-2+3p} - u\_{3n-2+3p}\|\| < \varepsilon.$$

Now choose *p* so that *r* 3*n*þ P *p*�4 *k*¼3*n*�2 *βk* ! ≤ *d* ≤ *r* 3*n*þ P *p*�3 *k*¼3*n*�2 *βk* !.

Since <sup>1</sup> <sup>2</sup> <*a* ≤ *α<sup>n</sup>* ≤ *b*< 1, for every *k, t*, we have 1 þ *α<sup>k</sup>* <3*αk, α<sup>t</sup>* <2*αk*. From **Lemma 2** and *r* ≤ ∥*Tu*3*n*�<sup>2</sup> � *u*3*n*�<sup>2</sup>∥, we have

$$\begin{split} d+r &\leq r\left(1+\sum\_{k=3n-2}^{3n+p-3} \delta\_k\right) \\ &\leq \left(1+\sum\_{k=3n-2}^{3n+p-3} \delta\_k\right) \|Tu\_{3n-2}-u\_{3n-2}\| \\ &\leq \|Tu\_{3n-2+p}-u\_{3n-2}\| \\ &\quad + \left(\prod\_{k=4}^{n+p-1} \frac{2}{1-a\_k}\right) \left(\|Tu\_{3n-2}-u\_{3n-2}\| - \|Tu\_{3n-2+3p}-u\_{3n-2+3p}\|\right) \\ &\quad < d+r \left(\prod\_{k=n}^{n+p-1} \frac{2}{1-a\_k}\right)e \\ &\quad = d+e^{\sum\_{k=1}^{n+p-1} \ln\left(1+\frac{2a\_k}{1-a\_k}\right)}e \\ &\quad = d+e^{\sum\_{k=1}^{n+p-1} \ln\frac{2}{1-a\_k}e} \\ &\leq d+e^{\sum\_{k=1}^{n+\frac{p-1}{2}-a\_k}a}e \\ &\leq d+e^{\sum\_{k=1}^{\frac{p-1}{2}-a\_k}a}e \\ &\leq d+e^{\sum\_{k=1}^{\frac{p-1}{2}-a\_k}a}e \\ &\leq d+e^{\sum\_{k=1}^{\frac{p-1}{2}-a\_k}a}e \\ &\leq d+e^{\sum\_{k=1}^{\frac{p-1}{2}-a\_k}a}e < d+r. \end{split}$$

This is a contradiction. So lim*<sup>k</sup>*!<sup>∞</sup> ∥*Tuk* � *uk*∥ ¼ 0. That is to say, lim*<sup>n</sup>*!<sup>∞</sup> ∥*Txn* � *xn*∥ ¼ 0. This completes the proof.

**Theorem 2.** Assume that *T* : *C* ! *C* is a *T* � ð Þ *Da* mapping and f g *xn* is generated by iteration (3), <sup>1</sup> <sup>2</sup> <*a* ≤ *α<sup>n</sup>* ≤ *b*<1. Then the sequence f g *xn* converges to a fixed point of *T*.

Proof: Since *C* is compact, there exists a subsequence *xnk* � �⊂f g *xn* which converges to some *z*∈*C*. By **Lemma 3**, we have

∥*xnk* � *Tz*∥ ≤ 3∥*Txnk* � *xnk*∥ þ ∥*xnk* � *z*∥. Since lim*nk*!<sup>∞</sup> ∥*Txnk* � *xnk*∥ ¼ 0 and lim*nk*!<sup>∞</sup> ∥*xnk* � *z*∥ ¼ 0, we have lim*nk*!<sup>∞</sup> ∥*xnk* � *Tz*∥ ¼ 0. This implies that *z* ¼ *Tz*. On the other hand, from **Proposition 3**

**References**

**49**

[1] Suzuki T. Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and Applications. 2008;**340**:1088-1095

*DOI: http://dx.doi.org/10.5772/intechopen.88421*

*Fixed Point Theorems of a New Generalized Nonexpansive Mapping*

[2] Thakur BS, Thakur D, Postolache M. A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings. Journal of Applied Mathematics and Computing. 2016;**275**:147-155

$$\begin{aligned} \|\mathbf{x}\_{n+1} - \boldsymbol{z}\| &\le \|\mathbf{y}\_n - \boldsymbol{z}\| \le \|\mathbf{z}\_n - \boldsymbol{z}\|\\ &\le a\_n \|\mathbf{T}\mathbf{x}\_n - \boldsymbol{z}\| + (\mathbf{1} - a\_n) \|\mathbf{x}\_n - \boldsymbol{z}\|,\\ &\le \|\mathbf{x}\_n - \boldsymbol{z}\|. \end{aligned}$$

So, lim*n*!<sup>∞</sup> ∥*xn* � *z*∥ exists. Therefore, lim*n*!<sup>∞</sup> ∥*xn* � *z*∥ ¼ 0. This completes the proof.
