On Stability of Iterative Sequences with Error

Abstract

Iterative methods were employed to obtain solutions of linear and non-linear systems of equations, solutions of differential equations, and roots of equations. In this paper, it was proved that s-iteration with error and Picard–Mann iteration with error converge strongly to the unique fixed point of Lipschitzian strongly pseudo-contractive mapping. This convergence was almost $\mathrm{F}$-stable and $\mathrm{F}$-stable. Applications of these results have been given to the operator equations $\mathrm{F}x=\mathrm{f}$ and $x+\mathrm{F}x=\mathrm{f},$ where $\mathrm{F}$ is a strongly accretive and accretive mappings of $\mathrm{X}$ into itself.

1. Introduction and Preliminaries

Consider a normed space $\mathrm{X},\mathrm{F}$:$\mathrm{X}\to \mathrm{X}$ is a mapping, $\mathrm{M}$ is an iteration procedure and ${\mathsf{\lambda }}_n,{\mathsf{\eta }}_n\in \left(0,1\right),$ we present the following iterative sequences.

${\mathrm{w}}_0\in \mathrm{X}$,

$${\mathrm{w}}_{n+1}=\mathrm{M}\left(\mathrm{F},{\mathrm{w}}_n\right),$$

is called s-iteration [1] if:

$$\begin{array}{c}{\mathrm{w}}_{n+1}={\mathsf{\lambda }}_n{\mathrm{F}z}_n+\left(1-{\mathsf{\lambda }}_n\right){\mathrm{Fw}}_n,\\ z_n={\mathsf{\eta }}_n{\mathrm{Fw}}_n+\left(1-{\mathsf{\eta }}_n\right){\mathrm{w}}_n,\forall n\ge 0.\end{array}$$

(1)

$x_0\in \mathrm{X},$

$$x_{n+1}=\mathrm{M}\left(\mathrm{F},x_n\right)$$

is called Picard–Mann iteration [2] if:

$$\begin{array}{c}{x}_{n+1}={\mathrm{F}y}_n\\ y_n={\mathsf{\lambda }}_n{\mathrm{F}x}_n+\left(1-{\mathsf{\lambda }}_n\right)x_n,\forall n\ge 0.\end{array}$$

(2)

${\mathrm{w}}_0\in \mathrm{X}$,

$${\mathrm{w}}_{n+1}=\mathrm{M}\left(\mathrm{F},{\mathrm{w}}_n\right),$$

is called s-iteration with errors if

$$\begin{array}{c}{\mathrm{w}}_{n+1}={\mathsf{\lambda }}_n{\mathrm{F}z}_n+\left(1-{\mathsf{\lambda }}_n\right){\mathrm{Fw}}_n+a_n,\\ z_n={\mathsf{\eta }}_n{\mathrm{Fw}}_n+\left(1-{\mathsf{\eta }}_n\right){\mathrm{w}}_n+c_n,\forall n\ge 0.\end{array}$$

(3)

where ${\sum }_{n=0}^{\infty }\Vert a_n\Vert <\infty , {\sum }_{n=0}^{\infty }\Vert c_n\Vert <\infty .$

$x_0\in \mathrm{X},$

$$x_{n+1}=\mathrm{M}\left(\mathrm{F},x_n\right)$$

is called Picard–Mann iteration with errors if:

$$\begin{array}{c}{x}_{n+1}={\mathrm{F}y}_n+a_n\\ y_n={\mathsf{\lambda }}_n{\mathrm{F}x}_n+\left(1-{\mathsf{\lambda }}_n\right)x_n,\forall n\ge 0.\end{array}$$

(4)

where ${\sum }_{n=0}^{\infty }\Vert a_n\Vert <\infty .$

Throughout this paper, we studied three cases: convergence, almost stability, and stability of schemes of sequences defined in Equations (3) and (4). In the following, we recall the needed definitions and lemmas.

Definition 1

([3]). Let $x_{n+1}=M\left(F,x_n\right)$ be an arbitrary iteration procedure such that $\left\{x_n\right\}$ converges to a fixed point p of $F.$ For a sequence $\left\{q_n\right\}$ suppose that

$${\mathsf{\delta }}_n=\Vert {\mathrm{q}}_{n+1}-\mathrm{M}\left(\mathrm{F},x_n\right)\Vert ,n\ge 0.$$

Then the iteration procedure is said to be$F$–stable if$\underset{n\to \infty }{\lim }{\delta }_n=0,$implies to$\underset{n\to \infty }{\lim }{q}_n= p.$

Definition 2

([4]). Let $F,\left\{x_{n + 1}\right\}, {\delta }_n,q_n,$ and $p$ be as shown in Definition 1. Then, the iteration procedure is said to be almost F-stable if ${\sum }_{n=0}^{\infty }{\delta }_n<\infty$ implies that $\underset{n\to \infty }{\lim }{q}_n=p.$

Definition 3

([5]). Let $X$ be a normed space and $F$ : $X\to X$ be a mapping then for fixed $m$, 0 $\le m<\infty$, $F$ is said to be Lipschitzian if:

$$\Vert \mathrm{F}x-\mathrm{F}y\Vert \le m \Vert x-y\Vert \forall x,y\in \mathrm{X}.$$

(5)

Let$X^{\prime }$be the dual of$X,$a set valued mapping$J:X\to 2^{X^{\prime }}$ is said to be the normalized duality mapping [5] if:

$$\mathrm{J}\left(x\right)=\left\{\mathrm{j}\in {\mathrm{X}}^{\prime }:⟨x,\mathrm{j}⟩=\Vert \mathrm{j}\Vert \Vert x\Vert ,\Vert \mathrm{j}\Vert =\Vert x\Vert \right\},\forall x\in \mathrm{X}$$

where $⟨ ,⟩$ denotes the duality pairing, i.e., $⟨ ,⟩ :X\times X^{\prime }\to K$, $⟨x, j⟩=j\left(x\right).$

It is known that a Banach space $\mathrm{X}$ is smooth if and only if the duality mapping $\mathrm{J}$ is single [5].

Definition 4

([6]). Let X be a normed space, $F$: $X\to X$ be a mapping. Then, $F$ is called strongly pseudo-contractive if for all $x$, $y\in X$, the following inequality holds:

$$\left|\left|x-y\right|\right|\le \left|\left|\left(1+\mathrm{r}\right)\left(x-y\right)–\mathrm{rt}\left(\mathrm{F}x-\mathrm{F}y\right)\right|\right|$$

(6)

$\forall r >0$and some t > 1.

Or equivalently [7], if there exist$\mathfrak{r}=\frac{1}{l},$where,$l>1$such that

$$⟨\mathrm{F}x-\mathrm{F}y,\mathrm{j}\left(x-y\right)⟩\le \mathfrak{r}\Vert x-y{\Vert }^2,\forall x,y\in \mathrm{X}.$$

If t = 1 in inequality (6), then F is called pseudo-contractive.

Definition 5

([8]). A mapping $F:X\to X$ is said to be

i-

Strongly accretive, if there is$\mathfrak{r}>0$such that for each$x,y \in X$there exists$j\left(x-y\right)\in J\left(x-y\right)$

$$⟨\mathrm{F}x-\mathrm{F}y,\mathrm{j}\left(x-y\right)⟩\ge \mathfrak{r}\Vert x-y{\Vert }^2.$$

(7)

ii-

Accretive, if$\mathfrak{r}=0$in Equation (7).

Or equivalently [9]

$$\Vert x-y\Vert \le \Vert x-y+\mathfrak{r}\left(\mathrm{F}x-\mathrm{F}y\right)\Vert , \mathrm{for}\mathrm{som}e\mathfrak{r}>0$$

(8)

Proposition 1

([10]). The relation between (strong) pseudo-contractive mapping and (strong) accretive mapping is that: $F$ is (strong) pseudo-contractive if and only if $\left(I-F\right)$ is (strong) accretive.

Lemma 1

([11]). Let $\left\{{\rho }_n\right\}$ be a non-negative sequence such that, ${\rho }_{n+1}\le \left(1-{\gamma }_n\right){\rho }_n+{\mu }_n$, where ${\gamma }_n\in \left(0,1\right)$, $\forall n\in \mathbb{N}$, $\sum {\gamma }_n=\infty$, and ${\mu }_n=o\left({\gamma }_n\right).$ Then $\underset{n\to \infty }{\lim }{\rho }_n=0$.

A general version of Lemma 1 is:

Lemma 2

([12]). Let {${\xi }_n$} be a non-negative sequence such that ${\xi }_{n+1}\le \left(1-{\gamma }_n\right){\xi }_n+b_n+{\mu }_n, n\ge 0,$where${\gamma }_n\in \left[0,1\right]$, $\forall n\in \mathbb{N}$, $\sum {\gamma }_n=\infty$, and $b_n=o\left({\gamma }_n\right)$, ${\sum }_{n=0}^{\infty }{\mu }_n<\infty$. Then $\underset{n\to \infty }{\lim }{\xi }_n=0$.

Lemma 3

([13,14]). Let $X$be a real Banach space, $F:X\to X$ be a mapping

i-

If$F$is continuous and strongly pseudo-contractive, then$F$ has a unique fixed point.

ii-

If$F$is continuous and strongly accretive, then the equation$Fx=f$has a unique solution for any$f\in X.$

iii-

If$F$is continuous and accretive, then$F$is m-accretive and the equation$x+Fx=f$has a unique solution for any$f\in X.$

For more details about previous preliminaries and to determine the important aspects of the convergence of iterative sequences, we recommend the book by C. Chidume [5] and the paper by B.E. Rhoades and L. Saliga [15].

2. Main Results

The following condition is needed:

(${\Delta }_1$): If ${\mathsf{\lambda }}_n,{\mathsf{\eta }}_n\in \left(0,1\right),\mathfrak{r}\in \left(0,1\right)$ and $m>0,$ then

$m\left(\left(m+1\right)\left(1+{\mathsf{\eta }}_n\right)+{\mathsf{\lambda }}_n{m}^2\left(2+\left(m-1\right){\mathsf{\eta }}_n\right)\right)-\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n\left(2m+m\left(m-1\right){\mathsf{\eta }}_n\right)\le \mathfrak{r}m-e,$ where $e\in \left(0,m\right).$

Theorem 1.

Let$X$be a real Banach space and$F:X\to X$be Lipschitzian strongly pseudo-contractive mapping with Lipschitz constant$m.$Suppose that$\left\{w_n\right\}$be in (3),$\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }{c}_n=0$and (${\Delta }_1$) is verified. Then:

1-

$\left\{w_n\right\}$converges strongly to the unique fixed point$p.$

2-

$\Vert q_{n+1}-p\Vert \le {\delta }_n+\Vert a_n\Vert +(1-\frac{{\lambda }_{n}e}{1+{\lambda }_n})\Vert q_n-p\Vert +(3m+m^2)\Vert c_n\Vert , \forall n\ge 0.$

Proof. 

From Lemma 3, we obtain that $\mathrm{F}$ has a unique fixed point, and from Equations (3), (6), and Proposition 1 we have:

$$\begin{array}{l}{\mathrm{Fw}}_n={\mathrm{w}}_{n+1}+{\mathsf{\lambda }}_n{\mathrm{Fw}}_n-{\mathsf{\lambda }}_n{\mathrm{F}z}_n-a_n\\ ={\mathrm{w}}_{n+1}+{\mathsf{\lambda }}_n{\mathrm{Fw}}_n-{\mathsf{\lambda }}_n{\mathrm{F}z}_n-a_n+2{\mathsf{\lambda }}_n{\mathrm{w}}_{n+1}-2{\mathsf{\lambda }}_n{\mathrm{w}}_{n+1}-\mathfrak{r}{\mathsf{\lambda }}_n{\mathrm{w}}_{n+1}\\ +\mathfrak{r}{\mathsf{\lambda }}_n{\mathrm{w}}_{n+1}-{\mathsf{\lambda }}_n{\mathrm{Fw}}_{n+1}+{\mathsf{\lambda }}_n{\mathrm{Fw}}_{n+1}\\ =\left(1+{\mathsf{\lambda }}_n\right){\mathrm{w}}_{n+1}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathrm{rI}\right){\mathrm{w}}_{n+1}-\left(1-\mathrm{r}\right){\mathsf{\lambda }}_n{\mathrm{Fw}}_n+\left(2-\mathrm{r}\right){\mathsf{\lambda }}_n^2\left({\mathrm{Fw}}_n-{\mathrm{F}z}_n\right)+{\mathsf{\lambda }}_n\left({\mathrm{Fw}}_{n+1}-{\mathrm{F}z}_n\right)-(1+\\ \left(2-\mathrm{r}\right){\mathsf{\lambda }}_n)a_n\end{array}$$

(9)

Let $\mathrm{p}$ be a fixed point of F:

$$\mathrm{p}=\left(1+{\mathsf{\lambda }}_n\right)\mathrm{p}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)\mathrm{p}-\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\mathrm{p}$$

(10)

$$\begin{array}{c}{\mathrm{Fw}}_n-\mathrm{p}=\left(1+{\mathsf{\lambda }}_n\right)\left({\mathrm{w}}_{n+1}-\mathrm{p}\right)+{\mathsf{\lambda }}_n\left[\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right){\mathrm{w}}_{n+1}-\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)\mathrm{p}\right]-\\ \left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\left({\mathrm{Fw}}_n-\mathrm{p}\right)+\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2\left({\mathrm{Fw}}_n-{\mathrm{F}z}_n\right)+{\mathsf{\lambda }}_n\left({\mathrm{Fw}}_{n+1}-{\mathrm{F}z}_n\right)-\left(1+\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)\end{array}$$

(11)

$$\begin{array}{l}\Vert {\mathrm{Fw}}_n-\mathrm{p}\Vert \ge \left(1+{\mathsf{\lambda }}_n\right)\Vert \left({\mathrm{w}}_{n+1}-\mathrm{p}\right)+\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}\left[\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right){\mathrm{w}}_{n+1}-\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)\mathrm{p}\right]\Vert \\ -\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\Vert {\mathrm{Fw}}_n-\mathrm{p}\Vert -\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2\Vert {\mathrm{Fw}}_n-{\mathrm{F}z}_n\Vert -{\mathsf{\lambda }}_n\Vert {\mathrm{Fw}}_{n+1}-{\mathrm{F}z}_n\Vert \\ -3\Vert a_n\Vert \end{array}$$

Thus:

$$\begin{array}{l}\left(1+{\mathsf{\lambda }}_n\right)\Vert {\mathrm{w}}_{n+1}-\mathrm{p}\Vert \\ \le \left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)\Vert {\mathrm{Fw}}_n-\mathrm{p}\Vert +\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2\Vert {\mathrm{Fw}}_n-{\mathrm{F}z}_n\Vert +{\mathsf{\lambda }}_n\Vert {\mathrm{Fw}}_{n+1}-{\mathrm{F}z}_n\Vert \\ +3\Vert a_n\Vert \\ \Vert {\mathrm{w}}_{n+1}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)\Vert {\mathrm{Fw}}_n-\mathrm{p}\Vert +\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2\Vert {\mathrm{Fw}}_n-{\mathrm{F}z}_n\Vert \\ + {\mathsf{\lambda }}_n\Vert {\mathrm{Fw}}_{n+1}-{\mathrm{F}z}_n\Vert +3\Vert a_n\Vert ]\\ \Vert {\mathrm{w}}_{n+1}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)m\Vert {\mathrm{w}}_n-\mathrm{p}\Vert +\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2\Vert {\mathrm{Fw}}_n-{\mathrm{F}z}_n\Vert \\ +{\mathsf{\lambda }}_n\Vert {\mathrm{Fw}}_{n+1}-{\mathrm{F}z}_n\Vert +3\Vert a_n\Vert ]\end{array}$$

(12)

Observe that

$$\begin{array}{c}\Vert {\mathrm{Fw}}_n-{\mathrm{F}z}_n\Vert \le \Vert {\mathrm{Fw}}_n-\mathrm{p}\Vert +\Vert \mathrm{p}-{\mathrm{F}z}_n\Vert \le m\Vert {\mathrm{w}}_n-\mathrm{p}\Vert +m\Vert z_n-\mathrm{p}\Vert \\ \le 2m+m\left(m-1\right){\mathsf{\eta }}_n\Vert {\mathrm{w}}_n-\mathrm{p}\Vert +m\Vert c_n\Vert \end{array}$$

(13)

$$\begin{array}{c}\Vert {\mathrm{Fw}}_{n+1}-{\mathrm{F}z}_n\Vert \le m\Vert {\mathrm{w}}_{n+1}-z_n\Vert \le [m\left(m+1\right)+{\mathsf{\lambda }}_{n}m(2m+m\left(m-1\right){\mathsf{\eta }}_n\Vert {\mathrm{w}}_n-\\ \mathrm{p}\Vert )+{\mathsf{\eta }}_{n}m\left(m+1\right)]\Vert {\mathrm{w}}_n-\mathrm{p}\Vert +m\Vert a_n\Vert +m\Vert c_n\Vert +{\mathsf{\lambda }}_n{m}^2\Vert c_n\Vert \end{array}$$

(14)

By substituting Equations (14) and (13) in (12), we get:

$$\begin{array}{l}\Vert {\mathrm{w}}_{n+1}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)m\Vert {\mathrm{w}}_n-\mathrm{p}\Vert \\ +{\mathsf{\lambda }}_n(\left[m\left(m+1\right)+{\mathsf{\lambda }}_{n}m\left(2m+m\left(m-1\right){\mathsf{\eta }}_n\right)+{\mathsf{\eta }}_{n}m\left(m+1\right)\right]\Vert {\mathrm{w}}_n-\mathrm{p}\Vert \\ +m\Vert a_n\Vert +m\Vert c_n\Vert +{\mathsf{\lambda }}_n{m}^2\Vert c_n\Vert )+\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2((2m+m(m\\ -1){\mathsf{\eta }}_n\Vert {\mathrm{w}}_n-\mathrm{p}\Vert +m\Vert c_n\Vert )+3\Vert a_n\Vert ]\\ =\frac{1}{1+{\mathsf{\lambda }}_n}[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)m+{\mathsf{\lambda }}_{n}m\left(m+1\right)\left(1+{\mathsf{\eta }}_n\right)+2{\mathsf{\lambda }}_n^2{m}^2+{\mathsf{\lambda }}_n^2{m}^2\left(m-1\right){\mathsf{\eta }}_n\\ +\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2(\left(2m+m\left(m-1\right){\mathsf{\eta }}_n\right]\Vert {\mathrm{w}}_n-\mathrm{p}\Vert +\left[\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}m+\frac{3}{1+{\mathsf{\lambda }}_n}\right]\Vert a_n\Vert \\ +\left[\frac{\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2}{1+{\mathsf{\lambda }}_n}m+\frac{{\mathsf{\lambda }}_n^2}{1+{\mathsf{\lambda }}_n}\left(m^2+m\right)\right]\Vert c_n\Vert \\ \le [1-\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}[m\mathfrak{r}-m\left(\left(m+1\right)\left(1+{\mathsf{\eta }}_n\right)+{\mathsf{\lambda }}_n{m}^2\left(2+\left(m-1\right){\mathsf{\eta }}_n\right)\right)\\ +-\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n(\left(2m+m\left(m-1\right){\mathsf{\eta }}_n\right]\Vert {\mathrm{w}}_n-\mathrm{p}\Vert +\left[m+3\right]\Vert a_n\Vert \\ +\left[3m+m^2\right]\Vert c_n\Vert \\ =[1-\frac{{\mathsf{\lambda }}_{n}e}{1+{\mathsf{\lambda }}_n}\Vert {\mathrm{w}}_n-\mathrm{p}\Vert +\left[m+3\right]\Vert a_n\Vert +\left[3m+m^2\right]\Vert c_n\Vert \end{array}$$

Lemma 1 yied to $\underset{n\to \infty }{\lim }{\mathrm{w}}_n=\mathrm{p}.$

For part(2):

Let $\left\{{\mathrm{q}}_n\right\}$ be a sequence in $\mathrm{X},$ defined $\left\{{\mathsf{\delta }}_n\right\}$ by ${\mathsf{\delta }}_n=\Vert {\mathrm{q}}_{n+1}-{\mathfrak{g}}_{\mathfrak{n}}-a_n\Vert ,$ where

$${\mathfrak{g}}_{\mathfrak{n}}={\mathsf{\lambda }}_n{\mathrm{F}z}_n+\left(1-{\mathsf{\lambda }}_n\right){\mathrm{Fq}}_n,z_n={\mathsf{\eta }}_n{\mathrm{Fq}}_n+\left(1-{\mathsf{\eta }}_n\right){\mathrm{q}}_n+c_n,n\ge 0.$$

$$\Vert {\mathrm{q}}_{n+1}-\mathrm{p}\Vert \le \Vert {\mathrm{q}}_{n+1}-{\mathfrak{g}}_{\mathfrak{n}}-a_n\Vert +\Vert a_n\Vert +\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \le {\mathsf{\delta }}_n+\Vert a_n\Vert +\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert$$

(15)

Since:

$$\begin{array}{l}{\mathrm{Fq}}_n={\mathfrak{g}}_{\mathfrak{n}}+{\mathsf{\lambda }}_n{\mathrm{Fq}}_n-{\mathsf{\lambda }}_n{\mathrm{F}z}_n\\ =\left(1+{\mathsf{\lambda }}_n\right){\mathfrak{g}}_{\mathfrak{n}}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathrm{rI}\right){\mathfrak{g}}_{\mathfrak{n}}-\left(2-\mathrm{r}\right){\mathsf{\lambda }}_n{\mathfrak{g}}_{\mathfrak{n}}+{\mathsf{\lambda }}_n{\mathrm{Fq}}_n+{\mathsf{\lambda }}_n\left(\mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}z}_n\right)\\ =\left(1+{\mathsf{\lambda }}_n\right){\mathfrak{g}}_{\mathfrak{n}}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathrm{rI}\right){\mathfrak{g}}_{\mathfrak{n}}-\left(1-\mathrm{r}\right){\mathsf{\lambda }}_n{\mathrm{Fq}}_n+\left(2-\mathrm{r}\right){\mathsf{\lambda }}_n^2\left({\mathrm{Fq}}_n-{\mathrm{F}z}_n\right)\\ +{\mathsf{\lambda }}_n\left(\mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}z}_n\right)\end{array}$$

(16)

Thus:

$$\mathrm{p}=\left(1+{\mathsf{\lambda }}_n\right)\mathrm{p}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathrm{rI}\right)\mathrm{p}-\left(1-\mathrm{r}\right){\mathsf{\lambda }}_n\mathrm{p}$$

(17)

$$\begin{array}{c}{\mathrm{Fq}}_n-\mathrm{p}=\left(1+{\mathsf{\lambda }}_n\right)\left({\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\right)+{\mathsf{\lambda }}_n\left[\left(\mathrm{I}-\mathrm{F}-\mathrm{rI}\right){\mathfrak{g}}_{\mathfrak{n}}-\left(\mathrm{I}-\mathrm{F}-\mathrm{rI}\right)\mathrm{p}\right]-\left(1-\mathrm{r}\right){\mathsf{\lambda }}_n\left(F{\mathrm{q}}_n-\mathrm{p}\right)\\ +\left(2-\mathrm{r}\right){\mathsf{\lambda }}_n^2\left(F{\mathrm{q}}_n-{\mathrm{F}z}_n\right)+{\mathsf{\lambda }}_n\left(\mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}z}_n\right).\end{array}$$

So that:

$$\begin{array}{l}\Vert {\mathrm{Fq}}_n-\mathrm{p}\Vert \ge \left(1+{\mathsf{\lambda }}_n\right)\Vert \left({\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\right)+\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}\left[\left(\mathrm{I}-\mathrm{F}-\mathrm{rI}\right){\mathfrak{g}}_{\mathfrak{n}}-\left(\mathrm{I}-\mathrm{F}-\mathrm{rI}\right)\mathrm{p}\right]\Vert \\ -\left(1-\mathrm{r}\right){\mathsf{\lambda }}_n\times \Vert {\mathrm{Fq}}_n-\mathrm{p}\Vert -\left(2-\mathrm{r}\right){\mathsf{\lambda }}_n^2\Vert F{\mathrm{q}}_n-{\mathrm{F}z}_n\Vert -{\mathsf{\lambda }}_n\Vert \mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}z}_n\Vert \\ \ge \left(1+{\mathsf{\lambda }}_n\right)\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert -\left(1-\mathrm{r}\right){\mathsf{\lambda }}_n\Vert F{\mathrm{q}}_n-\mathrm{p}\Vert -\left(2-\mathrm{r}\right){\mathsf{\lambda }}_n^2\Vert {\mathrm{Fq}}_n-{\mathrm{F}z}_n\Vert \\ -{\mathsf{\lambda }}_n\Vert \mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}z}_n\Vert \end{array}$$

Thus:

$$\begin{array}{c}\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}\left[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)\Vert {\mathrm{Fq}}_n-\mathrm{p}\Vert +\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2\Vert {\mathrm{Fq}}_n-{\mathrm{F}z}_n\Vert +{\mathsf{\lambda }}_n\Vert \mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}z}_n\Vert \right]\\ \Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}\left[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)m\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2\Vert {\mathrm{Fq}}_n-{\mathrm{F}z}_n\Vert +{\mathsf{\lambda }}_n\Vert \mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}z}_n\Vert \right]\end{array}$$

(18)

Observe that

$$\begin{array}{l}\Vert {\mathrm{Fq}}_n-{\mathrm{F}z}_n\Vert \le \Vert {\mathrm{Fq}}_n-\mathrm{p}\Vert +\Vert \mathrm{p}-{\mathrm{F}z}_n\Vert \le m\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +m\Vert z_n-\mathrm{p}\Vert \\ \le 2m+m\left(m-1\right){\mathsf{\eta }}_n\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +m\Vert c_n\Vert \end{array}$$

(19)

$$\begin{array}{l}\Vert \mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}z}_n\Vert \le m\Vert {\mathfrak{g}}_{\mathfrak{n}}-z_n\Vert \le m\left[\Vert {\mathrm{Fq}}_n-{\mathrm{q}}_n\Vert +{\mathsf{\lambda }}_n\Vert {\mathrm{Fq}}_n-{\mathrm{F}z}_n\Vert +{\mathsf{\eta }}_n\Vert {\mathrm{q}}_n-{\mathrm{Fq}}_n\Vert +\Vert c_n\Vert \right] \\ \le \left[m\left(m+1\right)+{\mathsf{\lambda }}_{n}m\left(2m+m\left(m-1\right){\mathsf{\eta }}_n\right)+{\mathsf{\eta }}_{n}m\left(m+1\right)\right]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert \\ +m\Vert c_n\Vert \end{array}$$

(20)

By substituting Equations (20) and (19) in (18), we get:

$$\begin{array}{l}\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)m\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +{\mathsf{\lambda }}_n([m\left(m+1\right)+{\mathsf{\lambda }}_{n}m\left(2m+m\left(m-1\right){\mathsf{\eta }}_n\right)+\\ {\mathsf{\eta }}_{n}m\left(m+1\right)]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +m\Vert c_n\Vert +{\mathsf{\lambda }}_n{m}^2\Vert c_n\Vert )+\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2\\ \left(2m+m\left(m-1\right){\mathsf{\eta }}_n\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +m\Vert c_n\Vert \right)]\\ =\frac{1}{1+{\mathsf{\lambda }}_n}[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)m+{\mathsf{\lambda }}_{n}m\left(m+1\right)\left(1+{\mathsf{\eta }}_n\right)+2{\mathsf{\lambda }}_n^2{m}^2+{\mathsf{\lambda }}_n^2{m}^2\left(m-1\right){\mathsf{\eta }}_n+\\ \left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2(\left(2m+m\left(m-1\right){\mathsf{\eta }}_n\right]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +\left[\frac{\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^2}{1+{\mathsf{\lambda }}_n}m+\frac{{\mathsf{\lambda }}_n^2}{1+{\mathsf{\lambda }}_n}\left(m^2+m\right)\right]\Vert c_n\Vert \\ \le [1-\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}[m\mathfrak{r}-m\left(\left(m+1\right)\left(1+{\mathsf{\eta }}_n\right)+{\mathsf{\lambda }}_n{m}^2\left(2+\left(m-1\right){\mathsf{\eta }}_n\right)\right)\\ +\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n(\left(2m+m\left(m-1\right){\mathsf{\eta }}_n\right]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert \\ +\left[\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n^{2}m+\left(m^2+m\right){\mathsf{\lambda }}_n^2\right]\Vert c_n\Vert \\ =[1-\frac{{\mathsf{\lambda }}_{n}e}{1+{\mathsf{\lambda }}_n}\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +\left[3m+m^2\right]\Vert c_n\Vert \Vert c_n\Vert .\end{array}$$

(21)

Substituting Equation (21) in (15) we obtain:

$$\begin{array}{l}\Vert {\mathrm{q}}_{n+1}-\mathrm{p}\Vert \le \Vert {\mathrm{q}}_{n+1}-{\mathfrak{g}}_{\mathfrak{n}}-a_n\Vert +\Vert a_n\Vert +\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \le {\mathsf{\delta }}_n+\Vert a_n\Vert +\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \\ \le {\mathsf{\delta }}_n+\Vert a_n\Vert +\left[1-\frac{{\mathsf{\lambda }}_{n}e}{1+{\mathsf{\lambda }}_n}\right]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +\left[3m+m^2\right]\Vert c_n\Vert .\end{array}$$

(22)

 □

Theorem 2.

Assume that$X, F, p, m, \left\{w_n\right\}, \left\{z_n\right\}, \left\{q_n\right\}, \left\{{\lambda }_n\right\}, \left\{{\eta }_n\right\},$and$\left\{{\delta }_n\right\}$be as in Theorem 1 and (${\Delta }_1$) is satisfied. Then the sequence (3) is almost F-stable.

Proof. 

Assume that ${\sum }_{n=0}^{\infty }{\mathsf{\delta }}_n<\infty$. Then, we prove that $\underset{n\to \infty }{\lim }{\mathrm{q}}_n=\mathrm{p}$.

Now, using Equation (22) such that ${\mathsf{\xi }}_n=\Vert {\mathrm{q}}_n-\mathrm{p}\Vert ,{\mathsf{\gamma }}_n=\frac{{\mathsf{\lambda }}_{n}e}{1+{\mathsf{\lambda }}_n},$ $b_n=[3m+m^2]\Vert c_n\Vert +\Vert a_n\Vert$, and ${\mathsf{\mu }}_n={\mathsf{\delta }}_n$, $\forall$ n $\ge 0.$

Note that $\underset{n\to \infty }{\lim }{\mathrm{b}}_n=0,$ thus Lemma (1.8) holds, such that $\underset{n\to \infty }{\lim }{\mathsf{\xi }}_n=0$ yields $\underset{n\to \infty }{\lim }{\mathrm{q}}_n=\mathrm{p}.$ □

Theorem 3.

Let$X, F, p, m, \left\{q_n\right\}, \left\{{\lambda }_n\right\}, \left\{{\eta }_n\right\}, \left\{a_n\right\}, \left\{c_n\right\},$and$\left\{{\delta }_n\right\}$be as in Theorem 1 and (${\Delta }_1$) is satisfied. Then$\left\{w_n\right\}$is$F$-stable.

Proof. 

Suppose that $\underset{n\to \infty }{\lim }{\mathsf{\delta }}_n=0$, then by applying Lemma 1 on (22) of Theorem 1, we obtain $\underset{n\to \infty }{\lim }{\mathrm{q}}_n=\mathrm{p}.$ □

Example 1.

Let$X=\left(0,1\right])$,$F:X\to X$by$Fx=\frac{x}{2} ,$hence, the conditions in Equations (5) and (6) are satisfied as shown below.

$$\begin{array}{l}\Vert \mathrm{F}x-\mathrm{F}y\Vert =\Vert \frac{x}{2}-\frac{y}{2}\Vert \le \frac{1}{2}\Vert x-y\Vert ⟨\mathrm{F}x-\mathrm{F}y,\mathrm{j}\left(x-y\right)⟩\le \mathfrak{r}\Vert x-y{\Vert }^2\le \left(\mathrm{F}x-\mathrm{F}y\right)\left(x-y\right)\\ \le \left|\frac{x}{2}-\frac{y}{2}\right||x-y|=\frac{1}{2}\Vert x-y{\Vert }^2\end{array}$$

Now, put${\lambda }_n=\frac{1}{2} , q_n= \frac{1}{n} , \forall n\ge 0,$since$\underset{n\to \infty }{\lim }{q}_n=0$, to show that$\underset{n\to \infty }{\lim }{\delta }_n=p=0.$

$$\begin{array}{l}{\mathsf{\delta }}_n=\Vert {\mathrm{q}}_{n+1}-x_{n+1}\Vert =\Vert {\mathrm{q}}_{n+1}-{\mathrm{Fq}}_n+a_n\Vert =\Vert \frac{1}{n+1}-\frac{{\mathrm{q}}_n}{2}\Vert \\ =\Vert \frac{1}{n+1}-\frac{(1-{\mathsf{\lambda }}_n)}{2}{\mathrm{q}}_n-\frac{{\mathsf{\lambda }}_n}{2}\frac{{\mathrm{q}}_n}{2}\Vert \\ =\Vert \frac{1}{n+1}-\frac{1}{4n}-\frac{1}{8n}\Vert ⟹\underset{n\to \infty }{\lim }{\mathsf{\delta }}_n=0.\end{array}$$

Corollary 1.

Let$X, F, p, m, \left\{q_n\right\}, \left\{{\lambda }_n\right\}, \left\{{\eta }_n\right\}, \left\{a_n\right\}, \left\{c_n\right\}, \left\{{\delta }_n\right\}$be as in Theorem 1, and$\left\{w_n\right\}$defined by Equation (1), then$\left\{w_n\right\}:$

1. 

converges strongly to the unique fixed point$p.$

2. 

is almost F-stable

3. 

is F-stable.

To prove the next results, we replace the inequality in the condition (${\Delta }_1$) by

$$\left({\Delta }_2\right): m\left(1+m^2+{\mathsf{\lambda }}_n\left(1+m\right)\right)\le \mathfrak{r}{m}^2-e$$

Theorem 4.

Suppose that$X$is a real Banach space$F:X\to X$is Lipschitzian strongly pseudo-contractive mapping with Lipschitz constant$m$. For$w_0\in X,$let$\left\{x_n\right\}$be in Equation (4),$\underset{n\to \infty }{\lim }{a}_n=0$(${\Delta }_2$) is satisfied. Then:

1-

$\left\{x_n\right\}$converges strongly to the unique fixed point$p$.

2-

$\Vert {\mathrm{q}}_{n+1}-\mathrm{p}\Vert \le {\mathsf{\delta }}_n+[1-\frac{{\mathsf{\lambda }}_{n}e}{1+{\mathsf{\lambda }}_n}]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +\Vert a_n\Vert , \forall n\ge 0.$

Proof. 

From Lemma 3, we obtained that $\mathrm{F}$ has a unique fixed point.

$$\begin{array}{l}{\mathrm{F}y}_n=x_{n+1}-a_n\\ =x_{n+1}+2{\mathsf{\lambda }}_n{x}_{n+1}-2{\mathsf{\lambda }}_n{x}_{n+1}-\mathfrak{r}{\mathsf{\lambda }}_n{x}_{n+1}+\mathfrak{r}{\mathsf{\lambda }}_n{\mathrm{F}x}_{n+1}-{\mathsf{\lambda }}_n{\mathrm{F}x}_{n+1}\\ +{\mathsf{\lambda }}_n{\mathrm{F}x}_{n+1}-a_n\\ =\left(1+{\mathsf{\lambda }}_n\right)x_{n+1}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)x_{n+1}+{\mathsf{\lambda }}_n\left({\mathrm{F}x}_{n+1}-{\mathrm{F}y}_n\right)-\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n{\mathrm{F}y}_n\\ -\left(1+\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)a_n\end{array}$$

(23)

$$=\left(1+{\mathsf{\lambda }}_n\right)\mathrm{p}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)\mathrm{p}-\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\mathrm{p}$$

(24)

So that:

$$\begin{array}{l}{\mathrm{F}y}_n-\mathrm{p}=\left(1+{\mathsf{\lambda }}_n\right)\left(x_{n+1}-\mathrm{p}\right)+{\mathsf{\lambda }}_n\left[\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)x_{n+1}-\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)\mathrm{p}\right]-\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\left({\mathrm{F}y}_n-\mathrm{p}\right)\\ +{\mathsf{\lambda }}_n\left({\mathrm{F}x}_{n+1}-{\mathrm{F}y}_n\right)-\left(1+\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)a_n\\ \Vert {\mathrm{F}y}_n-\mathrm{p}\Vert \ge \left(1+{\mathsf{\lambda }}_n\right)\Vert \left(x_{n+1}-\mathrm{p}\right)+\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}\left[\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)x_{n+1}-\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)\mathrm{p}\right]{\Vert }_{}\\ -\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\Vert {\mathrm{F}y}_n-\mathrm{p}\Vert -{\mathsf{\lambda }}_n\Vert {\mathrm{F}x}_{n+1}-{\mathrm{F}y}_n\Vert -3\Vert a_n{\Vert }_{}\end{array}$$

Thus:

$$\begin{array}{c}\left(1+{\mathsf{\lambda }}_n\right)\Vert x_{n+1}-\mathrm{p}\Vert \le \left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)\Vert {\mathrm{F}y}_n-\mathrm{p}\Vert +{\mathsf{\lambda }}_n\Vert {\mathrm{F}x}_{n+1}-{\mathrm{F}y}_n\Vert +3\Vert a_n\Vert \\ \Vert x_{n+1}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}\left[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)\Vert {\mathrm{F}y}_n-\mathrm{p}\Vert +{\mathsf{\lambda }}_n\Vert {\mathrm{F}x}_{n+1}-{\mathrm{F}y}_n\Vert +3\Vert a_n\Vert \right]\end{array}$$

(25)

Observe that:

$$\begin{array}{ll}\Vert {\mathrm{F}y}_n-\mathrm{p}\Vert & \le m\left[\left(1-{\mathsf{\lambda }}_n\right)\Vert x_n-\mathrm{p}\Vert +{\mathsf{\lambda }}_n\Vert {\mathrm{F}x}_n-p\Vert \right]=m\left(1-{\mathsf{\lambda }}_n+m{\mathsf{\lambda }}_n\right)\Vert x_n-\mathrm{p}\Vert \\ & \le m^2\Vert x_n-\mathrm{p}\Vert \end{array}$$

(26)

$\mathrm{Since}1\le m\mathrm{yields}\left(1-{\mathsf{\lambda }}_n+m{\mathsf{\lambda }}_n\right)\le m$

$$\begin{array}{c}\Vert {\mathrm{F}x}_{n+1}-{\mathrm{F}y}_n\Vert \le m\Vert x_{n+1}-y_n\Vert \le m\left[\Vert {\mathrm{F}y}_n-x_n\Vert +{\mathsf{\lambda }}_n\Vert x_n-{\mathrm{F}x}_n\Vert +\Vert a_n\Vert \right] \\ =m\left[\left(1+m^2+{\mathsf{\lambda }}_n\left(1+m\right)\right)\Vert x_n-\mathrm{p}\Vert +\Vert a_n\Vert \right]\end{array}$$

(27)

By substituting Equations (27) and (26) in (25), we yielded:

$$\begin{array}{l}\Vert x_{n+1}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}[[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)m^2+{\mathsf{\lambda }}_{n}m\left[\left(1+m^2+{\mathsf{\lambda }}_n\left(1+m\right)\right)]\Vert x_n-\mathrm{p}\Vert +\Vert a_n\Vert \right]\\ +3\Vert a_n\Vert ]\\ \Vert x_{n+1}-\mathrm{p}\Vert =\frac{1}{1+{\mathsf{\lambda }}_n}\left[(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)m^2+{\mathsf{\lambda }}_{n}m\left(\left(1+m^2+{\mathsf{\lambda }}_n\left(1+m\right)\right)\right]\Vert x_n-\mathrm{p}\Vert \\ +\left[\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}m+\frac{3}{1+{\mathsf{\lambda }}_n}\right]\Vert a_n\Vert \\ \le [1-\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}[m^2\mathfrak{r}-m\left(1+m^2+{\mathsf{\lambda }}_n\left(1+m\right)\right)]\Vert x_n-\mathrm{p}\Vert +\left[m+3\right]\Vert a_n\Vert \\ =[1-\frac{{\mathsf{\lambda }}_{n}e}{1+{\mathsf{\lambda }}_n}\Vert x_n-\mathrm{p}\Vert +\left[m+3\right]\Vert a_n\Vert \end{array}$$

By applying Lemma 1, we get $\underset{n\to \infty }{\lim }{x}_n=\mathrm{p}.$

For prove part (2):

Let $\left\{{\mathrm{q}}_n\right\}\subset \mathrm{X},$ defined $\left\{{\mathsf{\delta }}_n\right\}$ by ${\mathsf{\delta }}_n=\Vert {\mathrm{q}}_{n+1}-{\mathfrak{g}}_{\mathfrak{n}}-a_n\Vert ,$ where

$$\begin{array}{l}{\mathfrak{g}}_{\mathfrak{n}}={\mathrm{F}y}_n, y_n={\mathsf{\lambda }}_n{\mathrm{Fq}}_n+\left(1-{\mathsf{\lambda }}_n\right){\mathrm{q}}_n+c_n,n\ge 0.\\ \Vert {\mathrm{q}}_{n+1}-\mathrm{p}\Vert \le \Vert {\mathrm{q}}_{n+1}-{\mathfrak{g}}_{\mathfrak{n}}-a_n\Vert +\Vert a_n\Vert +\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \le {\mathsf{\delta }}_n+\Vert a_n\Vert +\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \end{array}$$

(28)

Since:

$$\begin{array}{l}{\mathrm{F}y}_n={\mathfrak{g}}_{\mathfrak{n}}={\mathfrak{g}}_{\mathfrak{n}}+2{\mathsf{\lambda }}_n{\mathfrak{g}}_{\mathfrak{n}}-2{\mathsf{\lambda }}_n{\mathfrak{g}}_{\mathfrak{n}}-\mathfrak{r}{\mathsf{\lambda }}_n{\mathfrak{g}}_{\mathfrak{n}}+\mathfrak{r}{\mathsf{\lambda }}_n\mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathsf{\lambda }}_n\mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}+{\mathsf{\lambda }}_n\mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}\\ =\left(1+{\mathsf{\lambda }}_n\right){\mathfrak{g}}_{\mathfrak{n}}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right){\mathfrak{g}}_{\mathfrak{n}}-\left(2-\mathfrak{r}\right){\mathsf{\lambda }}_n{\mathrm{F}y}_n+{\mathsf{\lambda }}_n\mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}\\ =\left(1+{\mathsf{\lambda }}_n\right){\mathfrak{g}}_{\mathfrak{n}}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right){\mathfrak{g}}_{\mathfrak{n}}+{\mathsf{\lambda }}_n\left(\mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}y}_n\right)-\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n{\mathrm{F}y}_n\end{array}$$

(29)

$$=\left(1+{\mathsf{\lambda }}_n\right)\mathrm{p}+{\mathsf{\lambda }}_n\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)\mathrm{p}-\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\mathrm{p}$$

(30)

So that:

$$\begin{array}{l}{\mathrm{F}y}_n-\mathrm{p}=\left(1+{\mathsf{\lambda }}_n\right)\left({\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\right)+{\mathsf{\lambda }}_n\left[\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right){\mathfrak{g}}_{\mathfrak{n}}-\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)\mathrm{p}\right]-\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\left({\mathrm{F}y}_n-\mathrm{p}\right)\\ +{\mathsf{\lambda }}_n{\left(\mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}y}_n\right)}_{}\\ \Vert {\mathrm{F}y}_n-\mathrm{p}\Vert \ge \left(1+{\mathsf{\lambda }}_n\right)\Vert \left({\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\right)+\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}\left[\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right){\mathfrak{g}}_{\mathfrak{n}}-\left(\mathrm{I}-\mathrm{F}-\mathfrak{r}\mathrm{I}\right)\mathrm{p}\right]{\Vert }_{}\\ -\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\Vert {\mathrm{F}y}_n-\mathrm{p}\Vert -{\mathsf{\lambda }}_n\Vert \mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}y}_n{\Vert }_{}\\ \ge \left(1+{\mathsf{\lambda }}_n\right)\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert -\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\Vert {\mathrm{F}y}_n-\mathrm{p}\Vert -{\mathsf{\lambda }}_n\Vert \mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}y}_n{\Vert }_{}\end{array}$$

This implies that:

$$\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}\left[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)\Vert {\mathrm{F}y}_n-\mathrm{p}\Vert +{\mathsf{\lambda }}_n\Vert \mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}y}_n\Vert \right]$$

(31)

Hence:

$$\begin{array}{ll}\Vert {\mathrm{F}y}_n-\mathrm{p}\Vert & \le m\left[\left(1-{\mathsf{\lambda }}_n\right)\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +{\mathsf{\lambda }}_n\Vert {\mathrm{Fq}}_n-\mathrm{p}\Vert \right]=m\left(1-{\mathsf{\lambda }}_n+m{\mathsf{\lambda }}_n\right)\Vert {\mathrm{q}}_n-\mathrm{p}\Vert \\ & \le m^2\Vert {\mathrm{q}}_n-\mathrm{p}\Vert \end{array}$$

(32)

$\mathrm{Sin}\mathrm{ce}1\le m\mathrm{yields}\left(1-{\mathsf{\lambda }}_n+m{\mathsf{\lambda }}_n\right)\le m$

$$\begin{array}{c}\Vert \mathrm{F}{\mathfrak{g}}_{\mathfrak{n}}-{\mathrm{F}y}_n\Vert \le m\Vert {\mathfrak{g}}_{\mathfrak{n}}-y_n\Vert \le m\left[\Vert {\mathrm{F}y}_n-{\mathrm{q}}_n\Vert +{\mathsf{\lambda }}_n\Vert {\mathrm{q}}_n-{\mathrm{Fq}}_n\Vert \right] \\ =m\left[\left(1+m^2+{\mathsf{\lambda }}_n\left(1+m\right)\right)\right]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert \end{array}$$

(33)

Substituting Equations (33) and (32) in (31) yielded that:

$$\begin{array}{l}\Vert {\mathfrak{g}}_{\mathfrak{n}}-\mathrm{p}\Vert \le \frac{1}{1+{\mathsf{\lambda }}_n}\left[\left(1+\left(1-\mathfrak{r}\right){\mathsf{\lambda }}_n\right)m^2+{\mathsf{\lambda }}_{n}m\left(1+m^2+{\mathsf{\lambda }}_n\left(1+m\right)\right)\right]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert \\ \le [1-\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}[m^2\mathfrak{r}-m\left(1+m^2+{\mathsf{\lambda }}_n\left(1+m\right)\right)]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert \\ =[1-\frac{{\mathsf{\lambda }}_{n}e}{1+{\mathsf{\lambda }}_n}]\Vert x_n-\mathrm{p}\Vert \end{array}$$

(34)

Substitute Equation (34) in (28), to obtain:

$$\Vert {\mathrm{q}}_{n+1}-\mathrm{p}\Vert \le {\mathsf{\delta }}_n+[1-\frac{{\mathsf{\lambda }}_{n}e}{1+{\mathsf{\lambda }}_n}]\Vert {\mathrm{q}}_n-\mathrm{p}\Vert +\Vert a_n\Vert .$$

(35)

 □

Theorem 5.

Assume that$X, F, p,m, \left\{x_n\right\}, \left\{q_n\right\}, \left\{{\lambda }_n\right\},$and$\left\{{\delta }_n\right\}$be as in Theorem 4 and the hypothesis that the condition (${\Delta }_2$) is satisfied. Then$\left\{x_n\right\}$in Equation (4) is almost F-stable.

Proof. 

Let ${\sum }_{n=0}^{\infty }{\mathsf{\delta }}_n<\infty ,$ to prove that $\underset{n\to \infty }{\lim }{\mathrm{q}}_n=\mathrm{p}$.

By using the conclusion of Equation (35) of Theorem 4 and an application of Lemma 1, we get $\underset{n\to \infty }{\lim }{\mathrm{q}}_n=\mathrm{p}.$ □

Theorem 6.

Let$X, F, p,m, \left\{q_n\right\}, \left\{{\lambda }_n\right\}, \left\{a_n\right\},$and$\left\{{\delta }_n\right\}$be as in Theorem 4 and (${\Delta }_2$) is satisfied. Then$\left\{x_n\right\}$in (2) is$F$-stable.

Proof. 

Suppose that $\underset{n\to \infty }{\lim }{\mathsf{\delta }}_n=0$. □

By expressing Equation (35) in the form ${\rho }_{n+1}\le \left(1-{\mathsf{\gamma }}_n\right){\rho }_n+{\mathsf{\mu }}_n$, of Lemma 1,where ${\mathsf{\gamma }}_n=\frac{{\mathsf{\lambda }}_{n}e}{1+{\mathsf{\lambda }}_n},$ ${\rho }_n=\Vert {\mathrm{q}}_n-\mathrm{p}\Vert$ and ${\mathsf{\mu }}_n={\mathsf{\delta }}_n+\Vert a_n\Vert ,$ this implies to $\underset{n\to \infty }{\lim }{\mathrm{q}}_n=\mathrm{p}$.

Corollary 2.

Let$X, F, p, m, \left\{q_n\right\}, \left\{{\lambda }_n\right\}, \left\{a_n\right\}, and \left\{{\delta }_n\right\}$be as in Theorem 4 and$\left\{x_n\right\}$be in Equation (2).

Then$\left\{x_n\right\}$:

1. 

converges strongly to the unique fixed point$p$.

2. 

is almost$F$-stable.

3. 

is$F$-stable.

3. Applications

Theorem 7.

Let$X$be a real Banach space and$F:X\to X$be Lipschitzian strongly accretive mapping with Lipschitz constant$m.$Define$\mathcal{S}:X\to X$by$\mathcal{S}x=f+x-Fx.$Let$\left\{{\lambda }_n\right\}, \left\{{\eta }_n\right\}, \left\{a_n\right\}, and \left\{c_n\right\}$as are in Theorem 1. For$w_0, f\in X,$

$$\begin{array}{c}{\mathrm{w}}_{n+1}={\mathsf{\lambda }}_n\mathcal{S}{z}_n+\left(1-{\mathsf{\lambda }}_n\right)\mathcal{S}{\mathrm{w}}_n+a_n,\\ z_n={\mathsf{\eta }}_n\mathcal{S}{\mathrm{w}}_n+\left(1-{\mathsf{\eta }}_n\right){\mathrm{w}}_n+c_n,\forall n\ge 0.\end{array}$$

Then$\left\{w_n\right\}$:

1. 

converges strongly the unique solution$p^{*}$of the equation$Fx=f.$

2. 

is almost$\mathcal{S}$-stable.

3. 

is$\mathcal{S}$-stable.

Proof. 

The mapping $\mathcal{S}$ is Lipschitzian with a constant $m_{*}=1+m$, and from Lemma 3 the equation $\mathrm{F}x=\mathrm{f}$ has a unique solution ${\mathrm{p}}^{*}$, this implies that $\mathcal{S}$ has a unique fixed point ${\mathrm{p}}^{*}.$

From Equation (7) and Proposition (6), hence

$⟨\left(I-\mathcal{S}\right)x-\left(I-\mathcal{S}\right)y,j\left(x-y\right)⟩=⟨Fx-Fy,j\left(x-y\right)⟩\ge \mathfrak{r}\Vert x-y{\Vert }^2,$ this implies $\mathcal{S}$ is strongly pseudo-contractive, therefore, the proof follows from Theorems 1–3. □

Corollary 3.

Let$X, F, \mathcal{S}, p^{*}, m, \left\{q_n\right\}, \left\{{\lambda }_n\right\}, \left\{{\eta }_n\right\}, and\left\{{\delta }_n\right\}$be as in Theorem 7 and$\left\{w_n\right\}$defined by

$$\begin{array}{c}{\mathrm{w}}_{n+1}={\mathsf{\lambda }}_n\mathcal{S}{z}_n+\left(1-{\mathsf{\lambda }}_n\right)\mathcal{S}{\mathrm{w}}_n,\\ z_n={\mathsf{\eta }}_n\mathcal{S}{\mathrm{w}}_n+\left(1-{\mathsf{\eta }}_n\right){\mathrm{w}}_n,\forall n\ge 0.\end{array}$$

Then$\left\{w_n\right\}:$

1. 

converges strongly to the unique solution$p^{*}$of the equation$Fx=f.$

2. 

is almost$\mathcal{S}$-stable.

3. 

is$\mathcal{S}$-stable.

Theorem 8.

Let$X$be a real Banach space and$F:X\to X$be Lipschitzian accretive mapping with Lipschitz constant. Define$\mathcal{S}:X\to X$by$\mathcal{S}x=f-Fx.$Let$\left\{{\lambda }_n\right\}, \left\{{\eta }_n\right\}, \left\{a_n\right\}, and \left\{c_n\right\}$as are in Theorem 1. For$w_0, f\in X,$

$$\begin{array}{c}{\mathrm{w}}_{n+1}={\mathsf{\lambda }}_n\mathcal{S}{z}_n+\left(1-{\mathsf{\lambda }}_n\right)\mathcal{S}{\mathrm{w}}_n+a_n,\\ z_n={\mathsf{\eta }}_n\mathcal{S}{\mathrm{w}}_n+\left(1-{\mathsf{\eta }}_n\right){\mathrm{w}}_n+c_n,\forall n\ge 0.\end{array}$$

Then$\left\{w_n\right\}$:

1. 

converges strongly to the unique solution$p^{*}$of the equation$x+Fx=f.$

2. 

is almost$\mathcal{S}$-stable.

3. 

is$\mathcal{S}$-stable.

Proof. 

From Lemma 3, hence, the equation $x+\mathrm{F}x=\mathrm{f}$ has a unique fixed point ${\mathrm{p}}^{*}$, (i.e., $\mathcal{S}$ has a unique fixed point ${\mathrm{p}}^{*}$). By using Equation (8), we obtained:

$$\Vert x-y\Vert \le \Vert x-y+\mathfrak{r}\left(\mathrm{F}x-\mathrm{F}y\right)\Vert =\Vert x-y+\mathfrak{r}\left(\mathcal{S}x-\mathcal{S}y\right)\Vert$$

(36)

Since:

$$\begin{array}{l}\mathcal{S}{\mathrm{w}}_n={\mathrm{w}}_{n+1}+{\mathsf{\lambda }}_n\mathcal{S}{\mathrm{w}}_n-{\mathsf{\lambda }}_n\mathcal{S}{z}_n-a_n\\ =\left(1+{\mathsf{\lambda }}_n\right){\mathrm{w}}_{n+1}-{\mathsf{\lambda }}_n\mathcal{S}{\mathrm{w}}_{n+1}+{\mathsf{\lambda }}_n\left(\mathcal{S}{\mathrm{w}}_{n+1}-\mathcal{S}{z}_n\right)\mathcal{S}{\mathrm{w}}_n+{\mathsf{\lambda }}_n{}^2\left(\mathcal{S}{\mathrm{w}}_n-\mathcal{S}{z}_n\right)\\ - \left(1+{\mathsf{\lambda }}_n\right)a_n\\ {\mathrm{p}}^{*}=\left(1+{\mathsf{\lambda }}_n\right){\mathrm{p}}^{*}-{\mathsf{\lambda }}_n\mathcal{S}{\mathrm{p}}^{*}\end{array}$$

By using Equation (36), we obtained:

$$\begin{array}{l}\Vert \mathcal{S}{\mathrm{w}}_n-{\mathrm{p}}^{*}\Vert \ge \left(1+{\mathsf{\lambda }}_n\right)\Vert \left({\mathrm{w}}_{n+1}-{\mathrm{p}}^{*}\right)+\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}\left(\mathcal{S}{\mathrm{w}}_{n+1}-\mathcal{S}{\mathrm{p}}^{*}\right)\Vert -{\mathsf{\lambda }}_n\Vert \mathcal{S}{\mathrm{w}}_{n+1}-\mathcal{S}{z}_n\Vert \\ -{\mathsf{\lambda }}_n^2\Vert \mathcal{S}{\mathrm{w}}_n-\mathcal{S}{z}_n\Vert -\left(1+{\mathsf{\lambda }}_n\right)\Vert a_n\Vert \\ \ge \left(1+{\mathsf{\lambda }}_n\right)\Vert {\mathrm{w}}_{n+1}-{\mathrm{p}}^{*}\Vert -{\mathsf{\lambda }}_n^2\Vert \mathcal{S}{\mathrm{w}}_n-\mathcal{S}{z}_n\Vert -{\mathsf{\lambda }}_n\Vert \mathcal{S}{\mathrm{w}}_{n+1}-\mathcal{S}{z}_n\Vert -(1\\ +{\mathsf{\lambda }}_n)\Vert a_n\Vert \end{array}$$

This implies:

$$\Vert {\mathrm{w}}_{n+1}-{\mathrm{p}}^{*\Vert }\le \frac{1}{1+{\mathsf{\lambda }}_n}\Vert \mathcal{S}{\mathrm{w}}_n-{\mathrm{p}}^{*}\Vert +\frac{{\mathsf{\lambda }}_n}{1+{\mathsf{\lambda }}_n}\Vert \mathcal{S}{\mathrm{w}}_{n+1}-\mathcal{S}{z}_n\Vert +\frac{{\mathsf{\lambda }}_n^2}{1+{\mathsf{\lambda }}_n}\Vert \mathcal{S}{\mathrm{w}}_n-\mathcal{S}{z}_n\Vert +\Vert a_n\Vert$$

The proof completes by the same way as Theorems 1–3. □

Corollary 4.

Let$X, F, \mathcal{S}, p^{*}, m, \left\{q_n\right\}, \left\{{\lambda }_n\right\}, \left\{{\eta }_n\right\},\left\{{\delta }_n\right\}$be as in Theorem 8 and$\left\{w_n\right\}$defined by

$$\begin{array}{c}{\mathrm{w}}_{n+1}={\mathsf{\lambda }}_n\mathcal{S}{z}_n+\left(1-{\mathsf{\lambda }}_n\right)\mathcal{S}{\mathrm{w}}_n,\\ z_n={\mathsf{\eta }}_n\mathcal{S}{\mathrm{w}}_n+\left(1-{\mathsf{\eta }}_n\right){\mathrm{w}}_n,\forall n\ge 0.\end{array}$$

Then$\left\{w_n\right\}$:

1. 

converges strongly to the unique solution$p^{*}$of the equation$x+Fx=f.$

2. 

is almost$\mathcal{S}$-stable.

3. 

is$\mathcal{S}$-stable.

Theorem 9.

Suppose that$X$is a real Banach space and$F:X\to X$is Lipschitzian strongly accretive mapping. Define$\mathcal{S}:X\to X$by$\mathcal{S}x=f+x-Fx.$Let$\left\{{\lambda }_n\right\} and \left\{a_n\right\},$as are in Theorem 4. For$x_0, f\in X,$

$$\begin{array}{c}{x}_{n+1}=\mathcal{S}{y}_n+a_n,\\ y_n={\mathsf{\lambda }}_n\mathcal{S}{x}_n+\left(1-{\mathsf{\lambda }}_n\right)x_n,\forall n\ge 0.\end{array}$$

Then$\left\{x_n\right\}$

1. 

converges strongly to the unique solution$p^{*}$of the equation$Fx=f.$

2. 

is almost$\mathcal{S}$-stable.

3. 

is$\mathcal{S}$-stable.

Proof. 

We can prove this the same way for Theorem 7. □

Corollary 5.

Let$X, F, \mathcal{S}, p^{*}, m, \left\{q_n\right\}, \left\{{\lambda }_n\right\}, and \left\{{\delta }_n\right\}$be as in Theorem 8 and$\left\{x_n\right\}$defined by

$$\begin{array}{c}{x}_{n+1}=\mathcal{S}{y}_n,\\ y_n={\mathsf{\lambda }}_n\mathcal{S}{x}_n+\left(1-{\mathsf{\lambda }}_n\right)x_n,\forall n\ge 0.\end{array}$$

Then$\left\{x_n\right\}$:

1. 

converges strongly to the fixed point$p^{*}$the unique solution of the equation$Fx=f.$

2. 

is almost$\mathcal{S}$-stable.

3. 

is$\mathcal{S}$-stable.

Theorem 10.

Let$X$be a real Banach space$,F:X\to X$is Lipschitzian accretive mapping with Lipschitz constant$m$. Define$\mathcal{S}:X\to X$by$\mathcal{S}x=f-Fx.$Let$\left\{{\lambda }_n\right\} and \left\{a_n\right\},$be as in Theorem 4. For$x_0, f\in X,$

$$\begin{array}{c}{x}_{n+1}=\mathcal{S}{y}_n+a_n,\\ y_n={\mathsf{\lambda }}_n\mathcal{S}{x}_n+\left(1-{\mathsf{\lambda }}_n\right)x_n,\forall n\ge 0.\end{array}$$

Then$\left\{x_n\right\}$:

1. 

converges strongly to the unique solution$p^{*}$of the equation$x+Fx=f.$

2. 

is almost$\mathcal{S}$-stable.

3. 

is$\mathcal{S}$-stable.

Proof. 

The proof follows the same way as Theorem 8. □

Corollary 6.

Let$X, F, \mathcal{S}, p^{*}, m, \left\{q_n\right\}, \left\{{\lambda }_n\right\}, and\left\{{\delta }_n\right\}$be as in Theorem 10 and$\left\{x_n\right\}$defined by

$$\begin{array}{c}{x}_{n+1}=\mathcal{S}{y}_n,\\ y_n={\mathsf{\lambda }}_n\mathcal{S}{x}_n+\left(1-{\mathsf{\lambda }}_n\right)x_n,\forall n\ge 0.\end{array}$$

Then$\left\{x_n\right\}:$

1. 

converge strongly to the unique solution$p^{*}$of the equation$x+Fx=f.$

2. 

is almost$\mathcal{S}$-stable.

3. 

is$\mathcal{S}$-stable.

4. Conclusions

For real Banach spaces, very interesting results were proved which say that for a Lipschitzian strongly pseudo-contractive operator, the s-iteration with error and Picard–Mann iteration with error processes converge strongly to the unique fixed point of the operator (Theorems 1 and 4). Some applications were also given (Theorem 7).

Open Problem

Let B be a non-empty closed convex subset of a Banach space X and $\{T_i,S_i, \forall i=1, 2,\dots , k\}$ be two families of total asymptotically quasi-nonexpansive self-mappings. Abed and Hasan [16] studied the convergence of the iterative sequence $\left\{w_n\right\}$, defined as:

$$\begin{array}{l}{w}_1\in B\\ w_{n+1}=\left(1-{\alpha }_{in}\right)S_i^n{w}_n+w_{in}{T}_i^n{b}_{in}\\ b_{in}=\left(1-w_{in}\right)S_i^n{a}_n+w_{in}{T}_i^n{b}_{\left(i-1\right)n}\\ b_{\left(i-1\right)n}=\left(1-{\alpha }_{\left(i-1\right)n}\right)S_{i-1}^n{w}_n+{\alpha }_{\left(i-1\right)n}{T}_{i-1}^n{b}_{\left(i-2\right)n}\\ b_{2n}=\left(1-w_{2n}\right)S_2^n{a}_n+{\alpha }_{2n}{T}_2^n{b}_{1n}\\ b_{1n}=\left(1-{\alpha }_{1n}\right)S_1^n{w}_n+{\alpha }_{1n}{T}_1^n{b}_{0n},\end{array}$$

where $b_{0n}=w_n and {\left\{{\alpha }_n\right\}}_{n=1}^{\infty }$ are sequences in (0, 1).

We suggest studying the stability of this iterative sequence.