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\title[Ball convergence for combined three--step methods]{Ball convergence for combined three--step methods under generalized conditions in Banach space}
\author[I. K. Argyros]{Ioannis K. Argyros}
\address{Cameron University,\\
Department of Mathematics Sciences,\\
Lawton, OK 73505, USA}
\email{ioannisa@cameron.edu}
%
\author[R. Behl]{Ramandeep Behl}
\address{University of KwaZulu--Natal,\\
School of Mathematics, Statistics and Computer Sciences,\\
Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa}
\email{ramanbehl87@yahoo.in}
%
\author[D. Gonz\'alez]{Daniel Gonz\'alez}
\address{Universidad de Las Am\'ericas,\\
Escuela de Ciencias F\'isicas y Matem\'aticas,\\
Quito, 170125, Ecuador}
\email{daniel.gonzalez.sanchez@udla.edu.ec}
%
\author[S. S. Motsa]{Sandile S. Motsa}
\address{University of KwaZulu--Natal,\\
School of Mathematics, Statistics and Computer Sciences,\\
Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa}
\email{sandilemotsa@gmail.com}
%
\subjclass{65G99, 65H10, 47J25, 47J05, 65D10, 65D99}
\keywords{Iterative method, Local convergence, Banach space, Lipschitz constant, Order of convergence}
\begin{abstract}
We give a local convergence analysis for an eighth--order convergent method in order to approximate a locally unique solution of nonlinear equation for Banach space valued operators. In contrast to the earlier studies using hypotheses up to the seventh Fr\'echet--derivative, we only use hypotheses on the first--order Fr\'echet--derivative and Lipschitz constants. Therefore, we not only expand the applicability of these methods but also provide the computable radius of convergence of these methods. Finally, numerical examples show that our results apply to solve those nonlinear equations but earlier results cannot be used. 
\end{abstract}
\maketitle

\section{Introduction}
One of the most basic and important problems in Numerical Analysis concerns with approximating a locally unique solution $x^*$ of the equation of the form
\begin{equation}\label{OM-1.1}
F(x)=0,
\end{equation}
where  $F: \mathbb{D} \subset \mathbb{X} \rightarrow  \mathbb{Y}$ is a Fr\'echet--differentiable operator, $\mathbb{X},\; \mathbb{Y}$ are Banach spaces and $\mathbb{D}$ is a  convex subset of $\mathbb{X}$. Let us   also denote $L(\mathbb{X},\;\mathbb{Y})$ as  the space of bounded linear operators from $\mathbb{X}$ to $\mathbb{Y}$.\medskip

Approximating $x^*$ is very important, since numerous problems can be reduced to equation \eqref{OM-1.1} using mathematical modeling \cite{4,5,10,*,15,19,21,22}. However, it is not always possible to find the solution $x^*$ in a closed form. Therefore, most of the methods are iterative to solve such type of problems.  The convergence analysis of iterative methods is usually divided into two categories: semi--local and local convergence analysis. The semi--local convergence matter is, based on the information around an initial point, to give criteria ensuring the convergence of iteration procedures. A very important problem in the study of iterative procedures is the convergence domain.  Therefore, it is very important to propose the radius of convergence of the iterative methods.\medskip

We study the local convergence of the three step eighth--order convergent method  defined for each $n=0,\;1\;2,\; \dots$ by 
\begin{equation}\label{OM-1.2}
\begin{aligned}
y_n&=x_n- F'(x_n)^{-1} F(x_n),\\
z_n&=\phi (x_n,\; F(x_n), \; F'(x_n),\; F'(y_n),\\
x_{n+1}&=  z_n - \beta A_n^{-1} F(z_n),\\
\end{aligned}
\end{equation}
where  $x_0\in\mathbb{D}$ is an initial point,  for $\alpha, \; \beta \in S$, $A_n = (\beta-\alpha) F'(x_n)+ \alpha F'(y_n),\; (S= \mathbb{R}\text{ or }S= \mathbb{C})$ and the second sub step represents any iterative method, in which the order of convergence is at least $m=1,\;2,\;3,\; \dots$. It was shown in \cite{*}  using Taylor series expansions when $\mathbb{X}=\mathbb{Y}= \mathbb{R}$ that method \eqref{OM-1.2} is of order at least $2m$, if $m<3$ and of order at least  $m+3$, if $m\geq 3$ provided that $F$ is eighth times differentiable.   The hypotheses on the derivatives of $F$ restrict the applicability of method \eqref{OM-1.2}.
 As a motivational
example, define function $F$ on $\mathbb{X}=\mathbb{Y}=\mathbb{R}$, $D=[-\frac{3}{2},
\frac{1}{2}]$ by
\begin{equation*}
F(x)=
\left\{
\begin{array}{lcl}
x^3ln x^2+x^5-x^4,& &x\not=0,\\
0, & & x=0.\\
\end{array}
\right.
\end{equation*}
Then, we have that
$$
F'(x)=3x^2lnx^2+5x^4-4x^3+2x^2,
$$
$$
F''(x)=6xlnx^2+20x^3-12x^2+10x
$$
and
$$
F'''(x)=6lnx^2+60x^2-24x+22.
$$
Then, obviously the third--order derivative of the involved function $F'''(x)$ is not bounded on $\mathbb{D}$. Notice that, in particular there is a plethora of iterative methods for approximating solutions of nonlinear equations \cite{1,2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20,21,22}.
These results show that  initial guess should be close to the required root for the convergence of the corresponding methods. But, how close initial guess should be required for the convergence of the corresponding method? These local results give no information on the radius of the ball convergence for the corresponding method. We address this question for method \eqref{OM-1.2} in the next section 2.\medskip

In the present study, we expand the applicability of method \eqref{OM-1.2} by using only hypotheses on the first--order derivative of function $F$ and generalized Lipschitz conditions. Moreover, we we will avoid to use Taylor series  expansions and use Lipschitz parameters. In this way, there is no need to use the higher--order derivatives to show the convergence of the scheme \eqref{OM-1.2}.\medskip

The rest of the paper is organized as follows: in section 2 contains the local convergence analysis of method \eqref{OM-1.2}. The numerical examples appear in the  concluding Section 3.\medskip

\section{Local Convergence}
The local convergence uses some scalars functions and parameters. Let $v,\; w_0,\; w,\; \bar{g}_2:[0,\; +\infty) \to [0,\; +\infty)$ be  continuous, increasing functions with  $w_0(0)=w(0)=0$ and $\alpha,\; \beta \in S$. Define  parameter $r_0$  by
\begin{equation}\label{OM-2.1}
r_0= \sup \{ t \geq 0: w_0(t)<1\}.
\end{equation}
Moreover, define functions $g_1,\; h_1,\; p$ and $h_p$ on the interval $ [0,\; r_0)$ by 
\begin{equation*}
g_1(t)=\frac{\int _0^1 w((1-\theta)t) d \theta }{1-w_0(t)},
\end{equation*}
\begin{equation*}
h_1=g_1(t)-1,
\end{equation*}
\begin{equation*}
p(t) = |\beta|^{-1} [|\beta-\alpha| w_0(t)+|\alpha| w_0(g_1(t)t)],\; \beta \neq 0,
\end{equation*}
and
\begin{equation*}
h_p=p(t)-1.
\end{equation*}
We have by \eqref{OM-2.2} that $h_1(0)= h_p(0)=-1<0$ and $h_1(t) \rightarrow +\infty$, $h_q(t) \rightarrow +\infty$ as $t \rightarrow r_0^{-}$. Then,  by the intermediate value theorem, we know that the functions $h_1$ and $h_p$ have zeros in the interval $\left( 0,\;r_0 \right)$. Denote by $r_1$ and $r_p$, respectively  the smallest such zeros  of the function $h_1$ and $h_p$. Furthermore, define functions $g_2$ and $h_2$  on the  interval $\left( 0,\;r_0 \right)$ by 
\begin{equation*}
g_2(t)= \bar{g}_2(t) t^{m-1},
\end{equation*}
and
\begin{equation*}
h_2(t)= g_2(t)-1.
\end{equation*}
Suppose that
\begin{equation}\label{OM-2.2}
\bar{g}_2(0)<1, \; \text{if} \; m=1
\end{equation}
and
\begin{equation}\label{OM-2.3}
g_2(t) \to a\text{ a number greater than one or }+\infty 
\end{equation}
as $t \to \bar{r}_0^{-}$ for some $\bar{r}_0 \leq r_0$. Then, we have again by the intermediate value theorem that function $h_2$ has zeros in the interval $(0,\; \bar{r}_0)$. Denote by $r_2$ the smallest such zero. Notice that, if $m>1$ condition \eqref{OM-2.2} is not needed to show $h_2(0)<0$, since in this case 
 $h_2(0)=g_2(0)-1=0-1=-1<0$.  Finally, define functions $g_3$ and $h_3$ on the interval $[0,\;\bar{r}_p)$ by  $\bar{r}_p= \min \{r_p,\; r_2\}$,
\begin{equation*}
g_3(t)= \left(1+ \frac{\int _0^1 v(\theta g_2(t)t) d \theta}{1-p(t)} \right) g_2(t),
\end{equation*}
and
\begin{equation*}
h_3(t)= g_3(t)-1.
\end{equation*}
Suppose that 
\begin{equation}\label{OM-2.4}
(1+v(0)) \bar{g}_2(0)<1,\; \text{if}\; m=1,
\end{equation}
we get by \eqref{OM-2.4} that $h_3(0)= (1+v(0))\bar{g}_2(0)-1<0$ and $h_3(t) \to +\infty$ or positive number as $t \to \bar{r}_p ^{-}$. Denote by $r_3$ the smallest zero of function $h_3$ in the interval $(0,\;r_p)$. Define the radius of convergence $r$ by 
\begin{equation}\label{OM-2.5}
r= \min \{r_1,\; r_3 \}.
\end{equation}
Then, we have that for each $t \in [0,\;r)$
\begin{equation}\label{OM-2.6}
0 \leq g_i(t)<1, \; i=1,\;2,\;3.
\end{equation}
Let  $U(z, \; \rho)$, $\bar{U}(z,\; \rho)$, stand respectively for the open and closed balls in $\mathbb{X}$ with center $z \in  \mathbb{X}$ and of radius $ \rho>0$. Next, we present the local convergence analysis of method \eqref{OM-1.2} using the preceding notations.

\begin{theorem}\label{OM-t-2.1}
Let $F: \mathbb{D} \subseteq  \mathbb{X} \rightarrow \mathbb{Y}$ be a continuously Fr\'echet--differentiable operator. Let  $v,\;w_0,\;w,\;\bar{g}_2: [0,\; \infty) \rightarrow [0,\; \infty)$ be increasing continuous functions  with $w_0(0)=w(0)=0$ and let $r_0 \in [0,\; \infty),\;  \alpha \in S,\; \beta \in S-\{0\}$, $m \geq 1$ and $r_0$  be defined by \eqref{OM-2.1} so that  \eqref{OM-2.1} and \eqref{OM-2.2}  are satisfied. Suppose that there exists  $x^* \in \mathbb{D} $ such that for each $x \in \mathbb{D}$ parameter $r_0$ be defined by \eqref{OM-2.1}.
\begin{equation}\label{OM-2.7}
F(x^*)=0,\quad F'(x^*)^{-1} \in L(\mathbb{Y},\; \mathbb{X}) 
\end{equation}
and 
\begin{equation}\label{OM-2.8}
\|F'(x^*)^{-1}(F'(x)-F'(x^*)\|\leq w_0(\|x-x^*\|).
\end{equation}
Moreover, suppose that for each $ x,\;y \in\mathbb{D} _0:= \mathbb{D} \cap U(x^*,\;r_0)$
\begin{equation}\label{OM-2.9}
\|F'(x^*)^{-1}\big(F'(x)-F'(y)\big)\|\leq  w(\|x-y\|),
\end{equation}
\begin{equation}\label{OM-2.10}
\|F'(x^*)^{-1}F'(x)\|\leq  v(\|x-x^*\|),
\end{equation}
\begin{equation}\label{OM-2.11}
\|\phi (x,\; F(x),\; F'(x),\; F'(y) ) \| \leq \bar{g}_2 (\|x-x^*\|)\|x-x^*\|^{m} 
\end{equation}
and
\begin{equation}\label{OM-2.12}
\bar{U}(x^*,r)\subseteq \mathbb{D},
\end{equation}
where the radius of convergence  $r$ is  defined by  \eqref{OM-2.3}. Then,  sequence $\{x_n\}$ generated
for $x_0 \in U(x^*,\;r)-\{x^\ast\}$ by method \eqref{OM-1.2} is well defined,  remains in $U(x^*,\;r)$ for
each $n=0,1,2,\ldots$ and converges to $x^*$. Moreover, the following estimates hold
\begin{equation}\label{OM-2.13}
\|y_n-x^*\|\leq g_1(\|x_n-x^*\|)\|x_n-x^*\| \leq \|x_n-x^*\|<r,
\end{equation}
\begin{equation}\label{OM-2.14}
\|z_n-x^*\|\leq g_2(\|x_n-x^*\|)\|x_n-x^*\| \leq \|x_n-x^*\|
\end{equation}
and
\begin{equation}\label{OM-2.15}
\|x_{n+1}-x^*\|\leq g_3(\|x_n-x^*\|)\|x_n-x^*\| \leq \|x_n-x^*\|,
\end{equation}
where the functions $g_i,\; i=1,\;2,\;3$ are defined  above the Theorem. Furthermore, if
\begin{equation}\label{OM-2.16}
\int_0 ^1 w_0(\theta R) d\theta < 1,\; \text{for}\; R \geq r,
\end{equation}
then the point $x^*$ is the only solution of equation $F(x)=0$ in $\mathbb{D}_1: = \mathbb{D} \cap \bar{U}(x^*,\;R)$.
\end{theorem}

\begin{proof}
We shall show using mathematical induction that the sequences $\{x_n\}$ is well defined in $U(x^*,\;r)$ and converges to $x^*$. By the hypothesis $x_0\in U(x^*,r)- \{x^\ast\}$, \eqref{OM-2.1},   \eqref{OM-2.3} and \eqref{OM-2.10}, we have that
\begin{equation}\label{OM-2.17}
\|F'(x^*)^{-1}(F'(x_0)-F'(x^*))\|\leq w_0(\|x_0-x^*\|)<w_0(r)<1.
\end{equation}
In view of  \eqref{OM-2.17} and the Banach Lemma on invertible operators \cite{4,5} that $F'(x_0)^{-1} \in L(\mathbb{Y},\;\mathbb{X})$, $y_0$ is  well defined 
 by the first two sub steps of method \eqref{OM-1.2} and
\begin{equation}\label{OM-2.18}
 \|F'(x_0)^{-1}F'(x^*)\|\leq\dfrac{1}{1-w_0(\|x_0-x^*\|)}\cdot
\end{equation}
We get by  \eqref{OM-2.1}, \eqref{OM-2.5}, \eqref{OM-2.6} (for $i=1$), \eqref{OM-2.7} and \eqref{OM-2.18} that
\begin{equation}\label{OM-2.19}
\begin{aligned}
\|y_0-x^* \|&= \|(x_0-x^*-F'(x_0)^{-1}F(x_0)) \|\\
& \leq \| F'(x_0)^{-1}F(x^*)\| \bigg\| \int_0^1 F'(x^*)^{-1}(F'(x^*+\theta(x_0-x^*))\\
&-F'(x_0))(x_0-x^*) d \theta \bigg\|\\
& \leq \frac{\int_0^1 w((1-\theta) \| x_0-x^*\|)d \theta \| x_0-x^*\|}{1-w_0(\| x_0-x^*\|)}\\
&  \leq g_1(\| x_0-x^*\| )\| x_0-x^*\| \leq \| x_0-x^*\|< r,
\end{aligned}
\end{equation}
which implies \eqref{OM-2.13} for $n=0$ and $y_0 \in U(x^*,\;r)$. By  \eqref{OM-2.5}, \eqref{OM-2.6}  (for $i=2$) and \eqref{OM-2.11}, we obtain in turn that
\begin{equation}\label{OM-2.20}
\begin{aligned}
\|z_0-x^*\|&= \| \phi (x_0,\; F(x_0),\; F'(x_0),\; F'(y_0)) \|\\
& \leq \bar{g}_2 (\| x_0-x^*\| ) \| x_0-x^*\| ^{m}\\
&=g_2(\| x_0-x^*\| )\| x_0-x^*\|  \leq \| x_0-x^*\|  <r,
\end{aligned}
\end{equation}
which shows \eqref{OM-2.14} for $n=0$ and $z_0 \in U(x^*,\;r)$. We  must show that $x_1$ exists. Using \eqref{OM-2.1}, \eqref{OM-2.5} and \eqref{OM-2.8}, we obtain in turn that
\begin{equation}\label{OM-2.21}
\begin{aligned}
&\left\| (\beta F'(x^*))^{-1} \left[ (\beta-\alpha) (F'(x_0)-F'(x^*))+ \alpha  (F'(y_0)-F'(x^*))  \right]  \right\|\\
& \quad  \quad \quad \quad \quad \quad \quad \quad \leq |\beta|^{-1} \left[ |\beta-\alpha| w_0(\| x_0-x^*\|)+ |\alpha|w_0(\|y_0-x^*\|) \right]\\
& \quad  \quad \quad \quad \quad \quad \quad \quad \leq |\beta|^{-1} \left[ |\beta-\alpha| w_0(\| x_0-x^*\|)+ |\alpha|w_0(g_1(\|x_0-x^*\|)\|x_0-x^*\|) \right]\\
& \quad  \quad \quad \quad \quad \quad \quad \quad=p(\| x_0-x^*\| ) \leq p(r)<1,
\end{aligned}
\end{equation}
so
\begin{equation}\label{OM-2.22}
 \| ((\beta-\alpha)F'(x_0)+ \alpha F'(y_0))^{-1}F'(x^*)\|\leq\dfrac{1}{1-p(\|x_0-x^*\|)}\cdot
\end{equation}
Then, from the last sub step of method \eqref{OM-2.1},   \eqref{OM-2.5}, \eqref{OM-2.6} (for $i=3$), \eqref{OM-2.10}, \eqref{OM-2.19}, \eqref{OM-2.20} and \eqref{OM-2.21}, we get in turn that
\begin{equation}\label{OM-2.23}
\begin{aligned}
\|x_1-x^* \|&= \| z_0-x^* \| + |\beta| \int_0^1 v(\theta \| z_0-x^*\|) d \theta \| x_0-x^*\|\\
& \leq \left( 1+  \frac{|\beta| \int _0^1 v(\theta g_2(\| x_0-x^*\|)) d \theta}{|\beta| (1-p(\| x_0-x^*\|))} \right) g_2(\| x_0-x^*\|)\| x_0-x^*\|\\
& = g_3(\| x_0-x^*\|) \| x_0-x^*\| \leq \| x_0-x^*\|<r,
\end{aligned}
\end{equation}
which shows \eqref{OM-2.15}  and $x_1 \in U(x^*,\;r)$. By simply replacing $x_0$, $y_0, \;x_1$ by $x_k$, $y_k$, $x_{k+1}$ in the preceding estimates we arrive at \eqref{OM-2.15} and \eqref{OM-2.16}.  Then, in view of the estimates 
\begin{equation}\label{OM-2.24}
\|x_{k+1}-x^*\|\leq c\|x_k-x^*\|<r, \; c=g_2(\|x_0-x^*\|) \in [0,\;1),
\end{equation}
we deduce that $\displaystyle{\lim_{k\to\infty}x_k=x^*}$ and $x_{k+1}\in U(x^*,\; r)$.
Finally, to show the uniqueness part, let $y^*\in D_1 $ with $F(y^*)=0$. Define  $Q=\int_0^1F'(x^*+\theta(x^*-y^*))d\theta$. Using \eqref{OM-2.5} and \eqref{OM-2.12}, we get that
\begin{equation}\label{OM-2.25}
\begin{aligned}
\|F'(x^*)^{-1}(Q-F'(x^*))\|&\leq\|\int_0^1w_0 (\theta \|y^*-x^*\|) d \theta\\
&\leq \int_0^1w_0  (\theta R)d\theta <1.
\end{aligned}
\end{equation}
It follows from \eqref{OM-2.25} that $Q$ is invertible.
Then, in view of the identity
\begin{equation}\label{OM-2.26}
0=F(x^*)-F(y^*)=Q(x^*-y^*),
\end{equation}
 we conclude that $x^*=y^*$. 
$\hfill{\square}$
\end{proof}

{\bf Remark 2.2}
\begin{enumerate}
\item [(a)] It follows from \eqref{OM-2.10} that condition \eqref{OM-2.12} can be dropped and be replaced by
\begin{equation}\label{OM-2.27}
v(t)= 1+w_0(t) \; \text{or} \; v(t)= 1+w_0(r_0),
\end{equation}
since, 
\begin{equation}\label{OM-2.28}
\begin{aligned}
\|F'(x^*)^{-1} \left[ \big( F'(x)  - F'(x^*) \big) +F'(x^*)\right] \|&=1+ \|F'(x^*)^{-1}(F'(x)-F'(x^*))\|\\
& \leq 1+w_0(\|x-x^*\|)\\
&=1+w_0(t) \;\;\;\text{for}\; \;\|x-x^*\|\leq r_0.
\end{aligned}
\end{equation}
\item[(b)]  If the function $w_0$ is strictly increasing, then we can choose
\begin{equation}\label{OM-2.29}
r_0=w_0^{-1}(1)
\end{equation}
instead of \eqref{OM-2.1}.
\item[(c)] If $w_0,\; w,\; v$ are constants functions (the proof of Theorem \ref{OM-t-2.1} goes through too in this case), then
\begin{equation}\label{OM-2.30}
r_1= \frac{2}{2 w_0+w}
\end{equation}
and 
\begin{equation}\label{OM-2.31}
r \leq r_1.
\end{equation}
Therefore, the radius of convergence $r$ can be larger than the radius of convergence $r_1$ for Newton's method
\begin{equation}\label{OM-2.32}
x_{n+1}=x_n - F'(x_n)^{-1} F(x_n).
\end{equation}
Notice also that the earlier radius of convergence given independently by Rheindoldt \cite{20} and Traub \cite{22} is
\begin{equation}\label{OM-2.33}
r_{TR}=\frac{2}{3w_1}
\end{equation}
and by Argyros \cite{4,5}
\begin{equation}\label{OM-2.34}
r_A=\frac{2}{2w_0+w_1},
\end{equation}
where $w_1$ is the Lipschitz constant for \eqref{OM-2.6} on $D$. But, we have 
\begin{equation}\label{OM-2.35}
w \leq w_1,\; w_0 \leq w_1,
\end{equation}
so
\begin{equation}\label{OM-2.36}
r_{TR} \leq r_A \leq r_1
\end{equation}
and
\begin{equation}\label{OM-2.37}
\frac{r_{TR}}{r_A}\to \frac{1}{3}\quad \text{as}\quad \dfrac{w_0}{w}\to 0.
\end{equation}
The radius of convergence $q$ used in \cite{*} is smaller than the radius $r_{DS}$ given by Dennis and Schabel \cite{4}
\begin{equation}\label{OM-2.38}
q < r_{SD}=\frac{1}{2w_1} < r_{TR}.
\end{equation}
However, $q$ can not be computed using the Lipschitz constants. 

\item[(d)] The results obtained here can be used for operators $F$ satisfying the autonomous differential
 equation \cite{4,5} of the form
\begin{equation}\label{OM-2.41}
F'(x)=P(F(x))
\end{equation}
where $P$ is a known continuous operator. Since $F'(x^\ast)= P(F(x^\ast))=P(0),$ we can apply the results without actually knowing the solution $x^\ast.$ Let as an example $F(x)=e^x-1.$ Then, we can choose $P(x)=x+1$.
\item[(e)] Let us show how to choose functions $\phi,\; \bar{g_2}, \; g_2$ and $m$. In addition, we assume that $\mathbb{X}=\mathbb{Y}=\mathbb{R}$. Define function $\phi$ on $\mathbb{R} ^4$ by 
\begin{equation}\label{OM-2.42}
\phi(x_n,\; F(x_n),\; F'(x_n),\; F'(y_n))= y_n-F'(y_n)^{-1} F(y_n).
\end{equation}
Then, we can choose
\begin{equation}\label{OM-2.43}
g_2(t)= \frac{\int _0^1 w((1-\theta)g_1(t)t) d \theta g_1(t)}{1-w_0(g_1(t)t)}.
\end{equation}
If $w_0,\;w,\;v$ are given in particular by $w_0(t)=L_0t,\; w(t)=Lt$ and $v(t)=<$ for some $L>0,\; L>0$ and $M \geq 1$, then we have that
\begin{equation}\label{OM-2.44}
\begin{aligned}
\bar{g}_2(t)&= \dfrac{\frac{L^2}{8(1-L_0t)^2}}{1- \frac{L_0 L t^2}{2(1-L_0t)}} ,\\
g_2(t)& = \bar{g}_2 (t) t^3 \; \text{and} \; m=4.
\end{aligned}
\end{equation}
\item[(f)] If $\beta=0$, we can obtain the results for the two-step method
\begin{equation}\label{OM-2.45}
\begin{aligned}
y_n & =  x_n -F'(x_n)^{-1} F(x_n),\\
x_{n+1}& = \phi (x_n,\; F(x_n),\; F'(x_n),\; F'(y_n))
\end{aligned}
\end{equation}
by setting $z_n= x_{n+1}$ in Theorem \ref{OM-t-2.1}.
\end{enumerate}

\section{Numerical Examples and Applications}
In this section, we shall demonstrate the theoretical results which we have proposed in the section 2. Therefore, we consider  four numerical examples in this section, which are defined as follows:

\begin{example}\label{OM-e3.1}
Let $X=Y=C[0,\;1]$ and  consider the nonlinear integral equation of the mixed Hammerstein--type \cite{12,15}, defined by 

\begin{equation}\label{OM-3.1}
x(s)=\int_{0}^{1} G(s,\;t) \left( x(t)^{\frac{3}{2}}+ \frac{x(t)^2}{2}\right) dt
\end{equation}
where the kernel $G$ is the Green's function defined on the interval  $ [0,1] \times [0,1]$ by
\begin{equation}\label{OM-3.2}
F(s,t)=
\left\{
\begin{aligned}
&(1-s)t, \; t\leq s,\\
&s(1-t), \;s\leq t.
\end{aligned}
\right.
\end{equation}
The solution $x^*(s)=0$ is the same as the solution of equation \eqref{OM-1.1}, where $F: \subseteq  C[0,\;1] \to C[0,\;1]$ defined by 
\begin{equation}\label{OM-3.3}
F(x)(s)= x(s)- \int_0^t G(s,\;t)\left( x(t)^{\frac{3}{2}}+ \frac{x(t)^2}{2}\right) dt .
\end{equation}
Notice that 
\begin{equation}\label{OM-3.4}
\left\|\int_0^t G(s,\;t) dt \right\|  \leq \frac{1}{8}\cdot
\end{equation}
Then, we have that
\begin{equation*}
F'(x) y(s)=y(s)- \int_0^t G(s,\;t)\left( \frac{3}{2}x(t)^{\frac{1}{2}}+ x(t)\right) dt,
\end{equation*}
so since $F'(x^*(s))=I$,
\begin{equation}\label{OM-3.5}
\left\| F'(x^*)^{-1} \big( F'(x)-F'(y)\big) \right\|  \leq \frac{1}{8} \left( \frac{3}{2} \| x-y\|^{\frac{1}{2}}+\| x-y\| \right).
\end{equation}
Therefore, we can choose
\begin{equation*}
w_0(t)=w(t)= \frac{1}{8}\left( \frac{3}{2}t^{\frac{1}{2}}+ t\right) 
\end{equation*}
and by Remark 2.2(a)
\begin{equation*}
v(t)=1+w_0(t).
\end{equation*}
The results in \cite{15, *} can not be used to solve this problem, since $F'$ is not Lipschitz. However, our results can apply. 
\end{example}

\begin{example}\label{OM-e3.2}
Suppose that the motion of an object in three dimensions is governed by system of differential equations
\begin{equation}\label{OM-3.6}
\begin{aligned}
&f_1'(x)-f_1(x)-1=0\\
&f_2'(y)-(e-1)y-1=0\\
&f_3'(z)-1=0\\
\end{aligned}
\end{equation}
with $x,\;y,\;z \in \Omega$ for $f_1(0)=f_2(0)=f_3(0)=0$. then, the solution of the system is given for $w=(x,\;y,\;z)^T$ by function $F:=(f_1,\;f_2,\;f_3): \Omega \to \mathbb{R}^3$ defined by
\begin{equation}\label{OM-3.7}
F(v)=\left(e^x-1,\;\frac{e-1}{2}y^2+y, \; z\right)^T.
\end{equation}
Then the Fr\'echet--derivative is given by
$$F'(v)=\begin{bmatrix}
e^x&0&0\\
0&(e-1)y+1&0\\
0&0&1
  \end{bmatrix}.$$
Then, we have that $w_0(t)=L_0t,\; w(t)=Lt,\; w_1(t)=L_1t , w_0=L_0,\; w_1=L_1$ and $v(t)=M$, where $L_0=e-1<L=e^{\frac{1}{L_0}}=1.789572397,\; L_1=e$ and  $M=e^{\frac{1}{L_0}}=1.7896$. Then, we get
\begin{equation*}
 r=0.0039782.
\end{equation*}
\end{example}

\begin{example}\label{OM-e3.3}
Let $A_1=A_2=C[0,\;1]$, be the space of continuous functions defined on the interval $[0,\;1]$ and be equipped with max norm. Let $\Omega= \bar{U}(0,\;1)$ and $B(x)=F''(x)$ for each $x \in \Omega$. Define $F$ on $\Omega$
\begin{equation}\label{OM-3.8}
F(\varphi) (x)=\phi(x)- 5 \int _0 ^1 x \theta \varphi(\theta)^3 d\theta.
\end{equation}
We have that
\begin{equation}\label{OM-3.9}
F'(\varphi(\xi)) (x)=\xi(x)- 15 \int _0 ^1 x \theta \varphi(\theta)^2 \xi (\theta) d\theta,\; \text{for each}\; \xi \in \Omega.
\end{equation}
Then, we have that $x^*=0,\; L_0=7.5,\; L_1=L=15$ and $M=2$. Using method \eqref{OM-1.2} for $w_0(t)=L_0t,\; v(t)=2=M,\; w(t)=Lt,\; w_1=L$ and $w_0=L_0$, we get 
\begin{equation*}
r=0.0013404.
\end{equation*}

\end{example}
\begin{example}\label{OM-e4.2}
Returning back to the motivation example at the introduction on this paper, we have $L=L_0=96.662907$ and $M=2$. Using method \eqref{OM-1.2}  for $w_0(t)=L_0t,\; v(t)=2=M,\; w(t)=Lt,\; w_1(t)=L$ and $w_0=L_0$, we can choose
\begin{equation*}
 r=0.00085.
\end{equation*}
\end{example}

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\end{document}