Fixed Point Results for K-Iteration Using Non-Linear Type Mappings
Abstract: In this paper we establish convergence and stability results using general contractive condition, quasi-nonexpansive mapping and mean non expansive mapping for K-iteration process. We shall also generalize the K-iteration process for a pair of distinct mappings and with the help of example we claim that the generalized iteration process has better convergence rate than the K-iteration process for single mapping and some of the existing iteration processes. Suitable examples are given in the support of main results. 1. Introduction and Preliminary Definitions

Let $\left(X,d\right)$ be a metric space and $T:X\to X$ be a self map defined on X. Let $F\left(T\right)=\left\{z\in X:Tz=z\right\}$ denote the set of fixed point of T. For ${x}_{0}\in X$ , the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ defined by

${x}_{n+1}=T{x}_{n},n\ge 0,$ (1.1)

is called the Picard iteration.

For ${x}_{0}\in X$ , the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ defined by

*Corrosponding author.

${x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n},n\ge 0,$ (1.2)

where ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }$ is a sequence in $\left[0,1\right]$ such that ${\sum }_{n=0}^{\infty }{\alpha }_{n}=\infty$ is called the Mann iteration process  .

In 2013, Khan  produced a new type of iteration process by introducing the concept of the following Picard-Mann hybrid iterative process for a single mapping T. For the initial value ${x}_{0}\in X$ , the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ defined by

${x}_{n+1}=T{y}_{n}$ ,

${y}_{n}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n},n\ge 0,$ (1.3)

where ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }$ is a sequence in $\left[0,1\right]$ .

Khan  showed that the rate of convergence of Picard-Mann hybrid iterative process is more than the Picard iteration scheme, Mann iteration scheme  and Ishikawa iterative schemes  .

In this direction Gursoy and Karakaya  , gave new iteration process as follows:

For the initial value ${x}_{0}\in X$ , the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ defined by

$\left\{\begin{array}{l}{z}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n},\\ {y}_{n}=\left(1-{\alpha }_{n}\right)T{x}_{n}+{\alpha }_{n}T{z}_{n},\\ {x}_{n+1}=T{y}_{n}\end{array}$ (1.4)

where ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }$ , ${\left\{{\beta }_{n}\right\}}_{n=0}^{\infty }$ is a sequence in $\left[0,1\right]$ is known as Picard-S iterative process. By giving appropriate example, Gursoy and Karakaya  proved that their iterative process has better convergence rate than Picard, Mann, Ishikawa, Noor and Normal-S iterative processes.

Karakaya et al. in their paper  , introduced a new hybrid iterative process as

$\left\{\begin{array}{l}{x}_{0}\in X,\\ {y}_{n}=T\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n},\\ {x}_{n+1}=T\left(1-{\alpha }_{n}\right){y}_{n}+{\alpha }_{n}T{y}_{n}\end{array}$ (1.5)

where ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }$ , ${\left\{{\beta }_{n}\right\}}_{n=0}^{\infty }$ is a sequence in $\left[0,1\right]$ .

With the help of suitable example it was claimed by Karakaya et al.  , that their iteration process converges faster than the iteration process of Gursoy and Karakaya  .

In 2016, Thakur et al.  introduced a new iteration scheme called Thakur New Iteration Scheme as for the initial value ${x}_{0}\in X$ , the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ defined by

$\left\{\begin{array}{l}{z}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n},\\ {y}_{n}=T\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}{z}_{n},\\ {x}_{n+1}=T{y}_{n}\end{array}$ (1.6)

where ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }$ , ${\left\{{\beta }_{n}\right\}}_{n=0}^{\infty }$ is a sequence in $\left[0,1\right]$ .

In  it was claimed that the Thakur New Iteration Scheme has higher convergence rate than the iteration process of Karakaya et al.  .

In the recent work of Hussain et al.  , a new iteration scheme has been developed and it is claimed that it has better convergence rate than the iterative process Thakur et al.  . This iteration process is called K-iteration process and is given as:

For the initial value ${x}_{0}\in X$ , the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ defined by

$\left\{\begin{array}{l}{z}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n},\\ {y}_{n}=T\left(1-{\alpha }_{n}\right)T{x}_{n}+{\alpha }_{n}T{z}_{n},\\ {x}_{n+1}=T{y}_{n}\end{array}$ (1.7)

where ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }$ , ${\left\{{\beta }_{n}\right\}}_{n=0}^{\infty }$ is a sequence in $\left[0,1\right]$ .

In the present work we shall generalize some convergence and stability results for K-iteration process. We shall also prove convergence and stability results for more general form of K-iteration process and K-iteration process for a pair of two distinct mappings.

Definition 1.1  : Let X be a real Banach space. The mapping $T:X\to X$ is said to be asymptotically quasi-nonexpansive if $F\left(T\right)\ne \varnothing$ and there exists a sequence $\left\{{\mu }_{n}\right\}\subset \left[0,\infty \right)$ with ${\mu }_{n}\to 0$ as $n\to \infty$ such that

$‖{T}^{n}x-q‖\le \left(1+{\mu }_{n}\right)‖x-q‖$ (1.8)

for all $x\in X,q\in F\left(T\right)$ and $n\ge 0$ .

Definition 1.2  : Let X be a real Banach space. The mapping $T:X\to X$ is said to be mean non-expansive if there exists two non negative real numbers $a,b$ such that $a+b\le 1$ and for all $x,y\in X$ ,

$‖Tx-Ty‖=a‖x-y‖+b‖x-Ty‖$

Definition 1.3  : Let ${\left\{{z}_{n}\right\}}_{n=0}^{\infty }$ be any sequence in X. Then the iterative process ${x}_{n+1}=f\left(T,{x}_{n}\right)$ which converges to a fixed point q, is said to be stable with respect to the mapping T if for ${\phi }_{n}=‖{z}_{n+1}-f\left(T,{z}_{n}\right)‖,n=0,1,2,\cdots$ , we have ${\mathrm{lim}}_{n\to \infty }{\phi }_{n}=0$ if and only if ${\mathrm{lim}}_{n\to \infty }{z}_{n}=q$ .

Definition 1.4  : A space X is said to satisfy Opial’s condition if for each sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ in X such that ${x}_{n}$ converges weakly to x we have for all $y\in X$ , $x\ne y$ following holds:

1) $\mathrm{lim}{\mathrm{inf}}_{n\to \infty }‖{x}_{n}-x‖<\mathrm{lim}{\mathrm{inf}}_{n\to \infty }‖{x}_{n}-y‖$ ,

2) $\mathrm{lim}{\mathrm{sup}}_{n\to \infty }‖{x}_{n}-x‖<\mathrm{lim}{\mathrm{sup}}_{n\to \infty }‖{x}_{n}-y‖$ .

Lemma 1.5  : Let ${\left\{{a}_{n}\right\}}_{n=0}^{\infty }$ and ${\left\{{b}_{n}\right\}}_{n=0}^{\infty }$ be non-negative real sequences satisfying the inequality:

${a}_{n+1}\le \left(1-{b}_{n}\right){a}_{n}+{b}_{n}$ ,

where ${b}_{n}\in \left(0,1\right)$ , for all $n\in N$ , ${\sum }_{n=1}^{\infty }{b}_{n}=\infty$ and $\frac{{b}_{n}}{{a}_{n}}\to 0$ as $n\to \infty$ , then ${\mathrm{lim}}_{n\to \infty }{a}_{n}=0$ .

Lemma 1.6  : Let $\delta$ be a real number such that $0\le \delta <1$ , and ${\left\{{ϵ}_{n}\right\}}_{n=0}^{\infty }$ be a sequence of positive numbers such that ${\mathrm{lim}}_{n\to \infty }{ϵ}_{n}=0$ . Then for any sequence of positive numbers ${\left\{{a}_{n}\right\}}_{n=0}^{\infty }$ satisfying ${a}_{n+1}\le \delta {a}_{n}+{ϵ}_{n},n=0,1,2,\cdots$ , we have ${\mathrm{lim}}_{n\to \infty }{a}_{n}=0$ .

Lemma 1.7  : Let X be a real Banach space and $\left\{{g}_{n}\right\}$ be any sequence in X such that $0<{g}_{n}<1$ for all $n\in N$ . Let ${\left\{{a}_{n}\right\}}_{n=0}^{\infty }$ and ${\left\{{b}_{n}\right\}}_{n=0}^{\infty }$ be non-negative real sequences satisfying $\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖{a}_{n}‖\le c$ , $\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖{b}_{n}‖\le c$ and $\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖{g}_{n}{a}_{n}+\left(1-{g}_{n}\right){b}_{n}‖=c$ holds for some $c\ge 0$ . Then $\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖{a}_{n}-{b}_{n}‖=0$ .

2. Main Results

Theorem 2.1: Let X be a Banach space and $T:X\to X$ be a mapping satisfying the condition

$‖Tx-q‖\le \delta ‖x-q‖$ (2.1)

where $q\in F,x\in X$ and $0\le \delta <1$ . Let ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be the sequence defined by the K-iterative process given by (1.7). Then the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges strongly to $q\in F\left(T\right)$ .

Proof: From (1.7) and (2.1) we have,

$‖{x}_{n+1}-q‖=‖T{y}_{n}-q‖\le \delta ‖T{y}_{n}-q‖$ (2.2)

And

$\begin{array}{c}‖{y}_{n}-q‖=‖T\left(\left(1-{\alpha }_{n}\right)T{x}_{n}+{\alpha }_{n}T{z}_{n}\right)-q‖\\ \le \delta ‖\left(1-{\alpha }_{n}\right)T{x}_{n}+{\alpha }_{n}T{z}_{n}-q‖\\ \le \delta ‖\left(1-{\alpha }_{n}\right)\left(T{x}_{n}-q\right)+{\alpha }_{n}\left(T{z}_{n}-q\right)‖\\ \le \delta \left[\left(1-{\alpha }_{n}\right)‖T{x}_{n}-q‖+{\alpha }_{n}‖T{z}_{n}-q‖\right]\\ \le \delta \left[\left(1-{\alpha }_{n}\right)‖T{x}_{n}-q‖+{\alpha }_{n}‖T{z}_{n}-q‖\right]\\ \le {\delta }^{2}\left[\left(1-{\alpha }_{n}\right)‖{x}_{n}-q‖+{\alpha }_{n}‖{z}_{n}-q‖\right]\end{array}$ (2.3)

Again using (1.7) and (2.1) we get,

$\begin{array}{c}‖{z}_{n}-q‖=‖\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n}-q‖\\ \le \left(1-{\beta }_{n}\right)‖{x}_{n}-q‖+{\beta }_{n}‖T{x}_{n}-q‖\\ \le \left(1-{\beta }_{n}\right)‖{x}_{n}-q‖+{\beta }_{n}\delta ‖{x}_{n}-q‖\end{array}$ (2.4)

Using (2.4) in (2.3) we get,

$\begin{array}{c}‖{y}_{n}-q‖\le {\delta }^{2}\left[\left(1-{\alpha }_{n}\right)‖{x}_{n}-q‖+{\alpha }_{n}\left(1-{\beta }_{n}\right)‖{x}_{n}-q‖+{\alpha }_{n}{\beta }_{n}\delta ‖{x}_{n}-q‖\right]\\ \le {\delta }^{2}\left(1-{\alpha }_{n}+{\alpha }_{n}\left(1-{\beta }_{n}\right)+{\alpha }_{n}{\beta }_{n}\delta \right)‖{x}_{n}-q‖\\ \le {\delta }^{2}\left(1-{\alpha }_{n}{\beta }_{n}\left(1-\delta \right)\right)‖{x}_{n}-q‖\end{array}$ (2.5)

Using (2.5) in (2.2) we get,

$‖{x}_{n+1}-q‖\le {\delta }^{3}\left(1-{\alpha }_{n}{\beta }_{n}\left(1-\delta \right)\right)‖{x}_{n}-q‖$

Since $0\le \delta <1,{\alpha }_{n}\in \left[0,1\right)$ and ${\sum }_{n=0}^{\infty }{\alpha }_{n}=\infty$ . Hence by using lemma (1.6), we have

$\underset{n\to \infty }{\mathrm{lim}}‖{x}_{n+1}-q‖=0.$

Hence the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges strongly to q.

Corollary 2.2: (Akewe and Okeke  ) Let X be a Banach space and $T:X\to X$ be a mapping satisfying the condition

$‖Tx-q‖\le \delta ‖x-q‖$

where $q\in F,x\in X$ and $0\le \delta <1$ . Let ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be the sequence defined by the Picard-Mann hybrid iterative process given by (1.3). Then the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges strongly to q.

Remark 2.3: Theorem 2.1 gives generalization to many results in the literature by considering a wider class of contractive type operators and more general iterative process, including the results of Chidume  , Bosede and Rhoades  and Akewe and Okeke  .

Theorem 2.4: Let X be a Banach space and $T:X\to X$ be a mapping satisfying the condition

$‖Tx-q‖\le \delta ‖x-q‖$

where $q\in F,x\in X$ and $0\le \delta <1$ . Let ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be the sequence defined by the K-iterative process given by (1.7). Then the iteration process (1.7) is T-stable.

Proof: By theorem 2.1, the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges strongly to q. Let ${\left\{{u}_{n}\right\}}_{n=0}^{\infty }$ , ${\left\{{v}_{n}\right\}}_{n=0}^{\infty }$ and ${\left\{{w}_{n}\right\}}_{n=0}^{\infty }$ be real sequences in X.

Let ${\phi }_{n}=‖{u}_{n+1}-T{v}_{n}‖,n=0,1,2,\cdots$ , where

${w}_{n}=\left(1-{\beta }_{n}\right){u}_{n}+{\beta }_{n}T{u}_{n}$ ,

${v}_{n}=T\left(\left(1-{\alpha }_{n}\right)T{u}_{n}+{\alpha }_{n}T{w}_{n}\right)$ ,

${u}_{n+1}=T{v}_{n}$ ,

and let ${\mathrm{lim}}_{n\to \infty }{\phi }_{n}=0$ .

We shall prove that ${\mathrm{lim}}_{n\to \infty }{u}_{n}=q$ .

Now,

$\begin{array}{c}‖{u}_{n+1}-q‖=‖{u}_{n+1}-T{v}_{n}‖+‖T{v}_{n}-q‖\\ \le {\phi }_{n}+\delta ‖{v}_{n}-q‖\end{array}$ (2.6)

$\begin{array}{c}‖{v}_{n}-q‖=‖T\left(\left(1-{\alpha }_{n}\right)T{u}_{n}+{\alpha }_{n}T{w}_{n}\right)-q‖\\ \le \delta ‖\left(1-{\alpha }_{n}\right)T{u}_{n}+{\alpha }_{n}T{w}_{n}-q‖\\ \le \delta ‖\left(1-{\alpha }_{n}\right)\left(T{u}_{n}-q\right)+{\alpha }_{n}\left(T{w}_{n}-q\right)‖\\ \le \delta \left[\left(1-{\alpha }_{n}\right)‖T{u}_{n}-q‖+{\alpha }_{n}‖T{w}_{n}-q‖\right]\\ \le \delta \left[\left(1-{\alpha }_{n}\right)‖T{u}_{n}-q‖+{\alpha }_{n}‖T{w}_{n}-q‖\right]\\ \le {\delta }^{2}\left[\left(1-{\alpha }_{n}\right)‖{u}_{n}-q‖+{\alpha }_{n}‖{w}_{n}-q‖\right]\end{array}$ (2.7)

Again using (1.7) and (2.1) we get,

$\begin{array}{c}‖{w}_{n}-q‖=‖\left(1-{\beta }_{n}\right){u}_{n}+{\beta }_{n}T{u}_{n}-q‖\\ \le \left(1-{\beta }_{n}\right)‖{u}_{n}-q‖+{\beta }_{n}‖T{u}_{n}-q‖\\ \le \left(1-{\beta }_{n}\right)‖{u}_{n}-q‖+{\beta }_{n}\delta ‖{u}_{n}-q‖\\ \le \left(1-{\beta }_{n}\left(1-\delta \right)\right)‖{u}_{n}-q‖\end{array}$ (2.8)

Using (2.8) in (2.7) we get,

$\begin{array}{c}‖{v}_{n}-q‖\le {\delta }^{2}\left[\left(1-{\alpha }_{n}\right)‖{u}_{n}-q‖+{\alpha }_{n}\left(1-{\beta }_{n}\left(1-\delta \right)\right)‖{u}_{n}-q‖\right]\\ \le {\delta }^{2}\left(1-{\alpha }_{n}{\beta }_{n}\left(1-\delta \right)\right)‖{u}_{n}-q‖\end{array}$ (2.9)

Using (2.9) in (2.6) we get,

$‖{u}_{n+1}-q‖\le {\phi }_{n}+{\delta }^{3}\left(1-{\alpha }_{n}{\beta }_{n}\left(1-\delta \right)\right)‖{u}_{n}-q‖$ (2.10)

Since $0\le \delta <1$ and since $0\le {\alpha }_{n},{\beta }_{n}\le 1$ we have by lemma (1.6)

$\underset{n\to \infty }{\mathrm{lim}}{u}_{n}=q.$

Conversely let ${\mathrm{lim}}_{n\to \infty }{u}_{n}=q$ . We shall show that ${\mathrm{lim}}_{n\to \infty }{\phi }_{n}=0$ .

Now

$\begin{array}{c}{\phi }_{n}=‖{u}_{n+1}-T{v}_{n}‖\\ \le ‖{u}_{n+1}-q‖+‖Tq-T{v}_{n}‖\\ \le ‖{u}_{n+1}-q‖+\delta ‖{v}_{n}-q‖\end{array}$ (2.11)

Substituting (2.9) in (2.11),

${\phi }_{n}\le ‖{u}_{n+1}-q‖+{\delta }^{3}\left(1-{\alpha }_{n}{\beta }_{n}\left(1-\delta \right)\right)‖{u}_{n}-q‖$ (2.12)

Since ${\mathrm{lim}}_{n\to \infty }{u}_{n}=q$ , we have from (2.12) ${\mathrm{lim}}_{n\to \infty }{\phi }_{n}=0$ . Hence the K-iteration scheme is T-stable.

From theorem 2.4, we have the following corollary.

Corollary 2.5: Let X be a Banach space and $T:X\to X$ be a mapping satisfying the condition

$‖Tx-q‖\le \delta ‖x-q‖$ ,

where $q\in F,x\in X$ and $0\le \delta <1$ . Let ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be the sequence defined by the Picard-Mann hybrid iterative process given by (1.3). Then the iteration process (1.3) is T-stable.

Example 2.6: Let $X=\left[0,1\right]$ and consider the mapping $Tx=\frac{x}{2}$ . The clearly the mapping T satisfies the inequality (2.1). Now $F\left(T\right)=0$ . Now we claim that the K-iteration scheme (1.7) is T-stable. Let us take ${\alpha }_{n}={\beta }_{n}=\frac{1}{2}$ and consider the sequences ${x}_{n}={y}_{n}={z}_{n}=\frac{1}{n}$ . Then clearly ${\mathrm{lim}}_{n\to \infty }{x}_{n}=0$ .

Now

$\begin{array}{l}{\phi }_{n}=‖{x}_{n+1}-T{y}_{n}‖=‖{x}_{n+1}-\frac{{y}_{n}}{2}‖\\ =‖{x}_{n+1}-\frac{T\left(\left(1-{\alpha }_{n}\right)T{x}_{n}+{\alpha }_{n}T{z}_{n}\right)}{2}‖\\ =‖{x}_{n+1}-\frac{\left(1-{\alpha }_{n}\right)T{x}_{n}+{\alpha }_{n}T{z}_{n}}{4}‖\\ =‖{x}_{n+1}-\left(\frac{\left(1-{\alpha }_{n}\right){x}_{n}}{8}+\frac{{\alpha }_{n}{z}_{n}}{8}\right)‖\\ =‖{x}_{n+1}-\left(\frac{\left(1-{\alpha }_{n}\right){x}_{n}}{8}+\frac{{\alpha }_{n}\left(1-{\beta }_{n}\right){x}_{n}}{8}+\frac{{\alpha }_{n}{\beta }_{n}T{x}_{n}}{8}\right)‖\\ =‖{x}_{n+1}-\left(\frac{\left(1-{\alpha }_{n}\right){x}_{n}}{8}+\frac{{\alpha }_{n}\left(1-{\beta }_{n}\right){x}_{n}}{8}+\frac{{\alpha }_{n}{\beta }_{n}{x}_{n}}{16}\right)‖\\ =‖\frac{1}{n+1}-\left(\frac{1}{16n}+\frac{1}{32n}+\frac{1}{64n}\right)‖=‖\frac{1}{n+1}-\frac{1}{8n}‖\end{array}$

$=‖\frac{7n-1}{8n\left(n+1\right)}‖=‖\frac{7-\frac{1}{n}}{8\left(n+1\right)}‖$ (2.13)

Taking limit $n\to \infty$ in (2.13), we have ${\mathrm{lim}}_{n\to \infty }{\phi }_{n}=0$ . Hence the K-iteration process is T-stable.

Now we shall prove the convergence and stability results for asymptotically quasi-nonexpansive mapping by considering the more general form of K-iteration process as:

${z}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}{T}^{n}{x}_{n}$ ,

${y}_{n}={T}^{n}\left(\left(1-{\alpha }_{n}\right){T}^{n}{x}_{n}+{\alpha }_{n}{T}^{n}{z}_{n}\right)$ ,

${x}_{n+1}={T}^{n}{y}_{n}$ , where $n=0,1,2,\cdots$ , (2.14)

Theorem 2.7: Let H be a non-empty closed convex subset of a Banach space X and $T:H\to H$ be asymptotically quasi-nonexpansive mapping with real sequence ${\mu }_{n}\subseteq \left[0,\infty \right)$ . Let ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be the sequence defined by the K-iterative process given by (2.14) and satisfies the assumption that ${\sum }_{n=0}^{\infty }{\alpha }_{n}{\beta }_{n}{\mu }_{n}=\infty$ . Then the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges strongly to some fixed point q of the mapping T.

Proof: From the iterative process (2.14) we have,

$\begin{array}{c}‖{z}_{n}-q‖=‖\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}{T}^{n}{x}_{n}-q‖\\ \le \left(1-{\beta }_{n}\right)‖{x}_{n}-q‖+{\beta }_{n}‖{T}^{n}{x}_{n}-q‖\\ \le \left(1-{\beta }_{n}\right)‖{x}_{n}-q‖+{\beta }_{n}\left(1+{\mu }_{n}\right)‖{x}_{n}-q‖\\ \le \left(1+{\beta }_{n}{\mu }_{n}\right)‖{x}_{n}-q‖\end{array}$ (2.15)

and

$\begin{array}{c}‖{y}_{n}-q‖=‖{T}^{n}\left(\left(1-{\alpha }_{n}\right){T}^{n}{x}_{n}+{\alpha }_{n}{T}^{n}{z}_{n}\right)-q‖\\ \le \left(1+{\mu }_{n}\right)‖\left(1-{\alpha }_{n}\right){T}^{n}{x}_{n}+{\alpha }_{n}{T}^{n}{z}_{n}-q‖\\ \le \left(1+{\mu }_{n}\right)‖\left(1-{\alpha }_{n}\right)\left({T}^{n}{x}_{n}-q\right)+{\alpha }_{n}\left({T}^{n}{z}_{n}-q\right)‖\\ \le \left(1+{\mu }_{n}\right)\left[\left(1-{\alpha }_{n}\right)‖{T}^{n}{x}_{n}-q‖+{\alpha }_{n}‖{T}^{n}{z}_{n}-q‖\right]\end{array}$

$\begin{array}{l}\le \left(1+{\mu }_{n}\right)\left[\left(1-{\alpha }_{n}\right)\left(1+{\mu }_{n}\right)‖{x}_{n}-q‖+{\alpha }_{n}\left(1+{\mu }_{n}\right)‖{z}_{n}-q‖\right]\\ \le {\left(1+{\mu }_{n}\right)}^{2}\left[\left(1-{\alpha }_{n}\right)‖{x}_{n}-q‖+{\alpha }_{n}‖{z}_{n}-q‖\right]\\ \le {\left(1+{\mu }_{n}\right)}^{2}\left[\left(1-{\alpha }_{n}\right)‖{x}_{n}-q‖+{\alpha }_{n}\left(1+{\beta }_{n}{\mu }_{n}\right)‖{x}_{n}-q‖\right]\\ \le {\left(1+{\mu }_{n}\right)}^{2}\left(1-{\alpha }_{n}{\beta }_{n}{\mu }_{n}\right)‖{x}_{n}-q‖\end{array}$ (2.16)

Again using (2.14) we have,

$\begin{array}{c}‖{x}_{n+1}-q‖\le ‖{T}^{n}{y}_{n}-q‖\\ \le \left(1+{\mu }_{n}\right)‖{y}_{n}-q‖\\ \le {\left(1+{\mu }_{n}\right)}^{3}\left(1-{\alpha }_{n}{\beta }_{n}{\mu }_{n}\right)‖{x}_{n}-q‖\end{array}$ (2.17)

By repeating the above process, we have the following inequalities

$‖{x}_{n+1}-q‖\le {\left(1+{\mu }_{n}\right)}^{3}\left(1-{\alpha }_{n}{\beta }_{n}{\mu }_{n}\right)‖{x}_{n}-q‖$

$‖{x}_{n}-q‖\le {\left(1+{\mu }_{n-1}\right)}^{3}\left(1-{\alpha }_{n-1}{\beta }_{n-1}{\mu }_{n-1}\right)‖{x}_{n-1}-q‖$

$‖{x}_{n-1}-q‖\le {\left(1+{\mu }_{n-2}\right)}^{3}\left(1-{\alpha }_{n-2}{\beta }_{n-2}{\mu }_{n-2}\right)‖{x}_{n-2}-q‖$

$\cdots$

$‖{x}_{1}-q‖\le {\left(1+{\mu }_{0}\right)}^{3}\left(1-{\alpha }_{0}{\beta }_{0}{\mu }_{0}\right)‖{x}_{0}-q‖$

So we can write,

$‖{x}_{n+1}-q‖\le {\left(1+{\mu }_{0}\right)}^{3\left(n+1\right)}‖{x}_{0}-q‖{\prod }_{j=0}^{n}\left(1-{\alpha }_{j}{\beta }_{j}{\mu }_{j}\right)$

Since $1-x\le {\text{e}}^{-x}$ for all $x\in \left[0,1\right]$ . Now $1-{\alpha }_{j}{\beta }_{j}{\mu }_{j}<1$ , so we can write,

$\begin{array}{c}‖{x}_{n+1}-q‖\le {\left(1+{\mu }_{0}\right)}^{3\left(n+1\right)}‖{x}_{0}-q‖{\text{e}}^{-\left(1-{\alpha }_{j}{\beta }_{j}{\mu }_{j}\right)}\\ \le {\left(1+{\mu }_{0}\right)}^{3\left(n+1\right)}‖{x}_{0}-q‖{\text{e}}^{-{\sum }_{j=0}^{n}{\alpha }_{j}{\beta }_{j}{\mu }_{j}}\end{array}$ (2.18)

Taking limit $n\to \infty$ in (2.18), we have ${\mathrm{lim}}_{n\to \infty }‖{x}_{n}-q‖=0$ , that is the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges strongly to fixed point q of the mapping T.

Theorem 2.8: Let H be a non-empty closed convex subset of a Banach space X and $T:H\to H$ be asymptotically quasi-nonexpansive mapping with real sequence ${\mu }_{n}\subseteq \left[0,\infty \right)$ . Let ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be the sequence defined by the K-iterative process given by (2.14) and satisfies the assumption that ${\sum }_{n=0}^{\infty }{\alpha }_{n}{\beta }_{n}{\mu }_{n}=\infty$ . Then the iterative process (2.14) is T-stable.

Proof: Let ${\left\{{u}_{n}\right\}}_{n=0}^{\infty }\subset X$ be any arbitrary sequence. Let the sequence generated by the iterative process (2.14) is ${x}_{n+1}=f\left(T,{x}_{n}\right)$ converging to the fixed point q.

Let ${\phi }_{n}=‖{u}_{n+1}-f\left(T,{x}_{n}\right)‖.$

We shall prove that ${\mathrm{lim}}_{n\to \infty }{\phi }_{n}=0$ if and only if ${\mathrm{lim}}_{n\to \infty }{u}_{n}=q$ .

First suppose ${\mathrm{lim}}_{n\to \infty }{\phi }_{n}=0$ . Now we have

$\begin{array}{c}‖{u}_{n+1}-q‖=‖{u}_{n+1}-f\left(T,{u}_{n}\right)‖+‖f\left(T,{u}_{n}\right)-q‖\\ ={\phi }_{n}+‖{T}^{n}\left({T}^{n}\left(1-{\beta }_{n}\right){T}^{n}{u}_{n}+{\beta }_{n}{T}^{n}\left(\left(1-{\alpha }_{n}\right){u}_{n}+{\alpha }_{n}{T}^{n}{u}_{n}\right)\right)-q‖\\ \le {\phi }_{n}+{\left(1+{\mu }_{n}\right)}^{3}\left(1-{\alpha }_{n}{\beta }_{n}{\mu }_{n}\right)‖{x}_{n}-q‖\end{array}$ (2.19)

where ${\alpha }_{n},{\beta }_{n}\in \left[0,1\right]$ , ${\mathrm{lim}}_{n\to \infty }{\phi }_{n}=0$ and ${\mathrm{lim}}_{n\to \infty }{\mu }_{n}=0$ .

Now using (2.19) together with lemma (1.5), we have ${\mathrm{lim}}_{n\to \infty }‖{u}_{n}-q‖=0$ that is ${\mathrm{lim}}_{n\to \infty }{u}_{n}=q$ .

Conversely let ${\mathrm{lim}}_{n\to \infty }{u}_{n}=q$ . we have

$\begin{array}{c}{\phi }_{n}=‖{u}_{n+1}-f\left(T,{u}_{n}\right)‖\\ \le ‖{u}_{n+1}-q‖+‖f\left(T,{u}_{n}\right)-q‖\\ \le ‖{u}_{n+1}-q‖+{\left(1+{\mu }_{n}\right)}^{3}\left(1-{\alpha }_{n}{\beta }_{n}{\mu }_{n}\right)‖{u}_{n}-q‖\end{array}$

Taking limit $n\to \infty$ both sides of (6) we have ${\mathrm{lim}}_{n\to \infty }{\phi }_{n}=0$ . Hence (2.14) is T-stable.

Now we shall prove the convergence results for mean non-expansive mapping by modifying the K-iteration process for two mappings as:

${z}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}S{x}_{n}$ ,

${y}_{n}=T\left(\left(1-{\alpha }_{n}\right)S{x}_{n}+{\alpha }_{n}T{z}_{n}\right)$ ,

${x}_{n+1}=T{y}_{n}$ , where $n=0,1,2,\cdots$ , (2.20)

Lemma 2.9: Let H be a non-empty closed convex subset of a Banach space X and $S,T:H\to H$ be two mean non-expansive mapping such that $F=F\left(T\right)\cap F\left(S\right)\ne \varphi$ . Let ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be the sequence defined by the K-iterative process given by (2.20). Then ${\mathrm{lim}}_{n\to \infty }‖{x}_{n}-q‖$ exists for some $q\in F$ .

Proof: We have

$\begin{array}{l}‖{z}_{n}-q‖=‖\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}S{x}_{n}-q‖\\ \le \left(1-{\beta }_{n}\right)‖{x}_{n}-q‖+{\beta }_{n}‖S{x}_{n}-q‖\\ \le \left(1-{\beta }_{n}\right)‖{x}_{n}-q‖+{\beta }_{n}\left({a}_{1}‖{x}_{n}-q‖+{b}_{1}‖{x}_{n}-q‖\right)\\ \le \left(1-{\beta }_{n}\right)‖{x}_{n}-q‖+{\beta }_{n}\left({a}_{1}+{b}_{1}\right)‖{x}_{n}-q‖\\ \le ‖{x}_{n}-q‖\end{array}$ (2.21)

Again using (2.20) and (2.21)

$\begin{array}{c}‖{y}_{n}-q‖=‖T\left(\left(1-{\alpha }_{n}\right)S{x}_{n}+{\alpha }_{n}T{z}_{n}\right)-q‖\\ \le {a}_{2}‖\left(\left(1-{\alpha }_{n}\right)S{x}_{n}+{\alpha }_{n}T{z}_{n}\right)-q‖+{b}_{2}‖\left(\left(1-{\alpha }_{n}\right)S{x}_{n}+{\alpha }_{n}T{z}_{n}\right)-q‖\\ \le \left({a}_{2}+{b}_{2}\right)‖\left(\left(1-{\alpha }_{n}\right)S{x}_{n}+{\alpha }_{n}T{z}_{n}\right)-q‖\\ \le \left(1-{\alpha }_{n}\right)‖S{x}_{n}-q‖+{\alpha }_{n}‖T{z}_{n}-q‖\end{array}$

$\begin{array}{l}\le \left(1-{\alpha }_{n}\right)\left({a}_{1}‖{x}_{n}-q‖+{b}_{1}‖{x}_{n}-q‖\right)+{\alpha }_{n}\left({a}_{2}‖{z}_{n}-q‖+{b}_{2}‖{z}_{n}-q‖\right)\\ \le \left(1-{\alpha }_{n}\right)\left({a}_{1}+{b}_{1}\right)‖{x}_{n}-q‖+{\alpha }_{n}\left({a}_{2}+{b}_{2}\right)‖{z}_{n}-q‖\\ \le \left(1-{\alpha }_{n}\right)‖{x}_{n}-q‖+{\alpha }_{n}‖{z}_{n}-q‖\\ \le ‖{x}_{n}-q‖\end{array}$ (2.22)

Again using (2.20) and (2.22)

$\begin{array}{c}‖{x}_{n+1}-q‖\le ‖T{y}_{n}-q‖\\ \le {a}_{2}‖{y}_{n}-q‖+{b}_{2}‖{y}_{n}-q‖\\ \le \left({a}_{2}+{b}_{2}\right)‖{y}_{n}-q‖\\ \le ‖{y}_{n}-q‖\\ \le ‖{x}_{n}-q‖\end{array}$ (2.23)

This shows that $\left\{‖{x}_{n}-q‖\right\}$ is non-increasing and bounded sequence for $q\in F$ . Hence ${\mathrm{lim}}_{n\to \infty }‖{x}_{n}-q‖$ exists.

Lemma 2.10: Let be a non-empty closed convex subset of a Banach space and $S,T:H\to H$ be two mean non-expansive mapping such that $F=F\left(T\right)\cap F\left(S\right)\ne \varphi$ . Let ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be the sequence defined by the K-iterative process given by (2.20). Also consider that ${\mathrm{lim}}_{n\to \infty }‖S{x}_{n}-q‖={\mathrm{lim}}_{n\to \infty }‖T{x}_{n}-q‖=0$ for some $q\in F$ . Then ${\mathrm{lim}}_{n\to \infty }‖T{x}_{n}-{x}_{n}‖=0$ .

Proof: Let $q\in F$ . In lemma (2.9) we have proved the existence of

${\mathrm{lim}}_{n\to \infty }‖{x}_{n}-q‖$ . Let ${\mathrm{lim}}_{n\to \infty }‖{x}_{n}-q‖=c$ . (2.24)

W.L.O.G. let $c>0$ .

Now from (2.20) and (2.24) we have,

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖{z}_{n}-q‖\le \underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖{x}_{n}-q‖=c$ (2.25)

Now

$\begin{array}{c}‖S{x}_{n}-q‖\le {a}_{1}‖{x}_{n}-q‖+{b}_{1}‖{x}_{n}-q‖\\ \le \left({a}_{1}+{b}_{1}\right)‖{x}_{n}-q‖\le ‖{x}_{n}-q‖\end{array}$

Implies that $\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖S{x}_{n}-q‖\le \underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖{x}_{n}-q‖=c$ (2.26)

Now

$\begin{array}{c}‖{x}_{n+1}-q‖\le ‖T{y}_{n}-q‖\le {a}_{2}‖{y}_{n}-q‖+{b}_{2}‖{y}_{n}-q‖\\ \le \left({a}_{2}+{b}_{2}\right)‖{y}_{n}-q‖\le ‖{y}_{n}-q‖\\ \le ‖T\left(\left(1-{\alpha }_{n}\right)S{x}_{n}+{\alpha }_{n}T{z}_{n}\right)-q‖\\ \le {a}_{2}‖\left(\left(1-{\alpha }_{n}\right)S{x}_{n}+{\alpha }_{n}T{z}_{n}\right)-q‖+{b}_{2}‖\left(\left(1-{\alpha }_{n}\right)S{x}_{n}+{\alpha }_{n}T{z}_{n}\right)-q‖\\ \le \left({a}_{2}+{b}_{2}\right)‖\left(\left(1-{\alpha }_{n}\right)S{x}_{n}+{\alpha }_{n}T{z}_{n}\right)-q‖\end{array}$

$\begin{array}{l}\le \left(1-{\alpha }_{n}\right)‖S{x}_{n}-q‖+{\alpha }_{n}‖T{z}_{n}-q‖\\ \le \left(1-{\alpha }_{n}\right)\left({a}_{1}‖{x}_{n}-q‖+{b}_{1}‖{x}_{n}-q‖\right)+{\alpha }_{n}\left({a}_{2}‖{z}_{n}-q‖+{b}_{2}‖{z}_{n}-q‖\right)\\ \le \left(1-{\alpha }_{n}\right)\left({a}_{1}+{b}_{1}\right)‖{x}_{n}-q‖+{\alpha }_{n}\left({a}_{2}+{b}_{2}\right)‖{z}_{n}-q‖\\ \le \left(1-{\alpha }_{n}\right)‖{x}_{n}-q‖+{\alpha }_{n}‖{z}_{n}-q‖\\ \le ‖{x}_{n}-q‖-{\alpha }_{n}‖{x}_{n}-q‖+{\alpha }_{n}‖{z}_{n}-q‖\end{array}$

$⇒\frac{‖{x}_{n+1}-q‖-‖{x}_{n}-q‖}{{\alpha }_{n}}=‖{z}_{n}-q‖-‖{x}_{n}-q‖$

and hence

$‖{x}_{n+1}-q‖-‖{x}_{n}-q‖\le \frac{‖{x}_{n+1}-q‖-‖{x}_{n}-q‖}{{\alpha }_{n}}=‖{z}_{n}-q‖-‖{x}_{n}-q‖$

which implies that $‖{x}_{n+1}-q‖\le ‖{z}_{n}-q‖$ (2.27)

Taking limit inferior in (2.27) we obtain

$c\le \underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}‖{z}_{n}-q‖$ (2.28)

From (2.20) and (2.28) we have

$\begin{array}{c}c=\underset{n\to \infty }{\mathrm{lim}}‖{z}_{n}-q‖\\ =\underset{n\to \infty }{\mathrm{lim}}‖\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}S{x}_{n}-q‖\\ =\underset{n\to \infty }{\mathrm{lim}}‖{\beta }_{n}\left(S{x}_{n}-q\right)+\left(1-{\beta }_{n}\right)\left({x}_{n}-q\right)‖\end{array}$ (2.29)

Now from (2.24), (2.26), (2.29) and lemma (1.7), we have $\underset{n\to \infty }{\mathrm{lim}}‖S{x}_{n}-{x}_{n}‖=0$ .

Now,

$‖T{x}_{n}-q‖\le {a}_{2}‖{x}_{n}-q‖+{b}_{2}‖{x}_{n}-q‖\le ‖{x}_{n}-q‖$

$⇒\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖T{x}_{n}-q‖\le \underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}‖{x}_{n}-q‖\le c$ (2.30)

Using the conditions of the lemma in (2.30), we can write

$C=\underset{n\to \infty }{\mathrm{lim}}‖{\beta }_{n}\left(T{x}_{n}-q\right)+\left(1-{\beta }_{n}\right)\left({x}_{n}-q\right)‖$ (2.31)

Using (2.24), (2.30), (2.31) along with the lemma (1.7), we have

${\mathrm{lim}}_{n\to \infty }‖T{x}_{n}-{x}_{n}‖=0.$

Theorem 2.11: Let H be a non-empty closed convex subset of a Banach space X satisfying Opial’s condition and S, T and ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be same as defined in the lemma (2.10) .Then the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges weakly to some $q\in F$ .

Proof: From lemma (2.10) we have, ${\mathrm{lim}}_{n\to \infty }‖T{x}_{n}-{x}_{n}‖=0$ .

Since X is uniformly convex and hence it is reflexive so there exists a subsequence $\left\{{x}_{{n}_{m}}\right\}$ of $\left\{{x}_{n}\right\}$ such that $\left\{{x}_{{n}_{m}}\right\}$ converges weakly to some ${q}_{1}\in F$ . Since H is closed so ${q}_{1}\in H$ . Now we claim the weak convergence of $\left\{{x}_{n}\right\}$ to ${q}_{1}$ . Let it is not true, then there exists a subsequence of $\left\{{x}_{{n}_{i}}\right\}$ of $\left\{{x}_{n}\right\}$ which converges weakly to ${q}_{2}$ and let ${q}_{1}\ne {q}_{2}$ . Also ${q}_{2}\in F$ . Now from lemma (2.9) ${\mathrm{lim}}_{n\to \infty }‖{x}_{n}-{q}_{1}‖$ and ${\mathrm{lim}}_{n\to \infty }‖{x}_{n}-{q}_{2}‖$ both exist. Using Opial’s condition we have,

$\begin{array}{c}\underset{n\to \infty }{\mathrm{lim}}‖{x}_{n}-{q}_{1}‖\le \underset{n\to \infty }{\mathrm{lim}}‖{x}_{{n}_{m}}-{q}_{1}‖<\underset{n\to \infty }{\mathrm{lim}}‖{x}_{{n}_{m}}-{q}_{2}‖\\ ={\mathrm{lim}}_{n\to \infty }‖{x}_{n}-{q}_{2}‖={\mathrm{lim}}_{n\to \infty }‖{x}_{{n}_{i}}-{q}_{2}‖\\ <{\mathrm{lim}}_{n\to \infty }‖{x}_{{n}_{i}}-{q}_{1}‖\le {\mathrm{lim}}_{n\to \infty }‖{x}_{n}-{q}_{1}‖\end{array}$

This is a contradiction, so we must have ${q}_{1}={q}_{2}$ . Thus the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges weakly to some $q\in F$ .

Theorem 2.12: Let H be a non-empty closed compact subset of a Banach space X and S, T and ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ be same as defined in the lemma (2.10). Then the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges strongly to some $q\in F$ .

Proof: Since H is compact and hence it is sequentially compact. So there exists a subsequence $\left\{{x}_{{n}_{i}}\right\}$ of $\left\{{x}_{n}\right\}$ which converges to $q\in H$ .

Now

$\begin{array}{c}‖{x}_{{n}_{i}}-Tq‖=‖{x}_{{n}_{i}}-T{x}_{{n}_{i}}‖+‖T{x}_{{n}_{i}}-Tq‖\\ \le ‖{x}_{{n}_{i}}-T{x}_{{n}_{i}}‖+{a}_{2}‖{x}_{{n}_{i}}-q‖+{b}_{2}‖{x}_{{n}_{i}}-q‖\\ \le ‖{x}_{{n}_{i}}-T{x}_{{n}_{i}}‖+‖{x}_{{n}_{i}}-q‖\end{array}$ (2.32)

Taking limit $n\to \infty$ in (2.32) we have, $Tq=q$ that is $q\in F$ . We have earlier proved that ${\mathrm{lim}}_{n\to \infty }‖{x}_{n}-q‖$ exists for $q\in F$ . Hence the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ converges strongly to some $q\in F$ .

In  it is proves that the K-iteration process converges faster than Picard-S, Thakur-New and Vatan two-step iterative process. Now we shall compare the rate of convergence the K-iteration process defined in  and our new modified K-iteration process for two mappings.

Table 1. Iterative values of K-iteration process and Modified K-iteration process.

Example 2.13: Let $S,T:\left[0,3\right]\to \left[0,3\right]$ be two mappings defined by $T\left(x\right)=\frac{x+2}{2}$ and $s\left(x\right)={\left(x+2\right)}^{\frac{1}{2}}$ . Let ${\alpha }_{n},{\beta }_{n}$ be the sequences defined by ${\alpha }_{n}={\beta }_{n}=\frac{1}{4}$ . Let the initial approximation be ${x}_{0}=2.25$ . Clearly S, T has

unique common fixed point 2. The convergence pattern of K-iteration process and modified K-iteration process is shown in Table 1.

Clearly we can conclude from Table 1, that the modified K-iteration process has better rate of convergence than the k-iteration process.

Cite this paper: Panwar, A. and Bhokal, R. (2019) Fixed Point Results for K-Iteration Using Non-Linear Type Mappings. Open Access Library Journal, 6, 1-14. doi: 10.4236/oalib.1105245.
References

   Mann, W.R. (1953) Mean Value Methods in Iteration. Proceedings of the American Mathematical Society, 4, 506-510.
https://doi.org/10.1090/S0002-9939-1953-0054846-3

   Khan, S.H. (2013) A Picard-Mann Hybrid Iterative Process. Fixed Point Theory and Applications, 2013, 69. https://doi.org/10.1186/1687-1812-2013-69

   Ishikawa, S. (1974) Fixed Points by a New Iteration Method. Proceedings of the American Mathematical Society, 44, 147-150.
https://doi.org/10.1090/S0002-9939-1974-0336469-5

   Gursoy, F. and Karakaya, V. (2014) A Picard-S Hybrid Type Iteration Method for Solving a Differential Equation with Retared Arguments. 1-16.

   Karakaya, V., Bouzara, N.E.H., Dogan, K. and Atalan, Y. (2015) On Different Results for a New Two Step Iteration Method under Weak Contraction Mapping in Banach Spaces. 1-10. arXiv:1507.00200v1

   Thakur, B.S., Thakur, D. and Postolache, M. (2016) A New Iterative Scheme for Numerical Reckoning Fixed Points of Suzuki’s Generalized Non-Expansive Mappings. Applied Mathematics and Computation, 275, 147-155.
https://doi.org/10.1016/j.amc.2015.11.065

   Opial, Z. (1967) Weak Convergence of the Sequence of Successive Approximations for Non-Expansive Mappings. Bulletin of the American Mathematical Society, 73, 595-597. https://doi.org/10.1090/S0002-9904-1967-11761-0

   Hussain, N., Ullah, K. and Arshad, M. (2018) Fixed Point Approximation of Suzuki Generalized Non-Expansive Mapping via New Faster Iterative Process. arxiv 1802.09888v

   Zhang, S.S. (1975) About Fixed Point Theory for Mean Non Expansive Mappings in Banach Spaces. Journal of Sichuan University, 2, 67-78.

   Harder, A.M. (1987) Fixed Point Theory and Stability Results for Fixed Point Iteration Procedure. University of Missouri-Rolla, Missouri.

   Weng, X. (1991) Fixed Point Iteration for Local Strictly Pseudo-Contractive Mapping. Proceedings of the American Mathematical Society, 113, 727-731.
https://doi.org/10.1090/S0002-9939-1991-1086345-8

   Berinde, V. (2002) On the Stability of Some Fixed Point Procedures. Buletinul ?tiin?ific al Univer-sitatii Baia Mare, Seria B, Fascicola matematic?-informatic?, 18, 7-14.

   Sahu, J. (1991) Weak and Strong Convergence to Fixed Points of Asymptotically Non-Expansive Mappings. Bulletin of the Australian Mathematical Society, 43, 153-159. https://doi.org/10.1017/S0004972700028884

   Akewe, H. and Okeke, G.A. (2015) Convergence and Stability Theorems for the Picard-Mann Hybrid Iterative Scheme for a General Class of Contractive-Like Operators. Fixed Point Theory and Applications, 2015, 66.

   Chidume, C.E. (2014) Strong Convergence and Stability of Picard Iteration Sequence for General Class of Contractive-Type Mappings. Fixed Point Theory and Applications, 2014, 233. https://doi.org/10.1186/1687-1812-2014-233

   Bosede, A.O. and Rhoades, B.E. (2010) Stability of Picard and Mann Iteration for a General Class of Functions. Journal of Advanced Mathematical Studies, 3, 23.

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