Convergence of Higher Order Jarratt-Type Schemes for Nonlinear Equations from Applied Sciences

Abstract

Symmetries are important in studying the dynamics of physical systems which in turn are converted to solve equations. Jarratt’s method and its variants have been used extensively for this purpose. That is why in the present study, a unified local convergence analysis is developed of higher order Jarratt-type schemes for equations given on Banach space. Such schemes have been studied on the multidimensional Euclidean space provided that high order derivatives (not appearing on the schemes) exist. In addition, no errors estimates or results on the uniqueness of the solution that can be computed are given. These problems restrict the applicability of the methods. We address all these problems by using the first order derivative (appearing only on the schemes). Hence, the region of applicability of existing schemes is enlarged. Our technique can be used on other methods due to its generality. Numerical experiments from chemistry and other disciplines of applied sciences complete this study.

1. Introduction

Problems from applied sciences such as mathematics, biology, chemistry, and physics (including symmetries) to mention a few are converted to nonlinear equations which are solved by iterative methods, since exact solutions are hard to find. Let ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$ denote Banach spaces and $\mathbb{D}\subset {\mathbb{T}}_1$ stand for an open and convex set. Moreover, we use the notation $\ell ({\mathbb{T}}_1,{\mathbb{T}}_2)$ for the space of continuous linear operators mapping ${\mathbb{T}}_1$ into ${\mathbb{T}}_2$. The task of determining a solution $x_{*}$ of equation:

$$F\left(x\right)=0,$$

(1)

where $F:\mathbb{D}\to {\mathbb{T}}_2$ is differentiable as defined by Fréchet is of extreme significance in computational disciplines. Finding $x_{*}$ in a closed form is desirable but rarely attainable. That is why one resorts to develop iterative schemes approximating $x_{*}$, if certain convergence criteria hold.

One of the most basic and popular iterative methods is known as the Newton method [1], which is defined as follows:

$$x_{\sigma +1}=x_{\sigma }-F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right).$$

it has second of order convergence. However, it is a one-point method and one-point methods have several issues concerning the order and computational efficiency. For instance, if we want to attain a third order one-point iterative method then we need the evaluations of function f, first order derivative $f^{\prime }$, and second order derivative $f^{″}$ (more details can be found in [1,2]). Finding second or higher order derivatives are either time consuming or do not exist. Thus, researchers are focused on the most important class of iterative methods that is known as multi-point methods. These multi-point methods are of great practical importance since they overcome theoretical limits of one-point methods (more details can be found in [1]).

Ostrowski [2] was the first mathematician who suggested an optimal [3] multi-point scheme of fourth order which requires only three functional evaluations. Later, Jarratt [4,5], in 1966 and King [6], in 1975, gave many optimal fourth order multi-point methods. King further demonstrated that Ostrowski’s method was a special case of his scheme. A plethora of such schemes can be found in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33] and the references therein.

In particular, we study the (local) convergence of the three-step scheme [33], given for each $\sigma =0,1,2,\dots ,$

$$\begin{array}{cc}\hfill y_{\sigma }& =x_{\sigma }-\alpha F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right),\hfill \\ \hfill z_{\sigma }& =A_{\lambda }(x_{\sigma },y_{\sigma }),\hfill \\ \hfill x_{\sigma +1}& =z_{\sigma }-B_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(z_{\sigma }\right),\hfill \end{array}$$

(2)

as well as the j-step scheme [33]:

$$\begin{array}{cc}\hfill z_{\sigma }^{\left(1\right)}& =x_{\sigma }-\alpha F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right),\hfill \\ \hfill z_{\sigma }^{\left(2\right)}& =x_{\sigma }-C_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right),\hfill \\ \hfill z_{\sigma }^{\left(3\right)}& =z_{\sigma }^{\left(2\right)}-B_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(z_{\sigma }^{\left(2\right)}\right),\hfill \\ \hfill z_{\sigma }^{\left(4\right)}& =z_{\sigma }^{\left(3\right)}-B_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(z_{\sigma }^{\left(3\right)}\right),\hfill \\ \hfill ⋮& \\ \hfill z_{\sigma }^{\left(j\right)}& =z_{\sigma }^{(j-1)}-B_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(z_{\sigma }^{(j-1)}\right),j=2,3,4,\dots ,m\hfill \\ \hfill x_{\sigma +1}& =z_{\sigma }^{\left(m\right)},\hfill \end{array}$$

(3)

with $\alpha \in \mathbb{R}-\left\{0\right\}$, $A_{\lambda }:{\mathbb{T}}_1\times {\mathbb{T}}_1\to {\mathbb{T}}_1$ is a continuous operator of a scheme with order $\lambda \ge 2$, $B:\mathbb{D}\to \ell ({\mathbb{T}}_1,{\mathbb{T}}_2)$ and $C:\mathbb{D}\to \ell ({\mathbb{T}}_1,{\mathbb{T}}_2)$. We have left $A_{\lambda }$ as general as possible to include numerous special cases of it. But, as an example $A_{\lambda }$ can be $A_4=A(x_k,y_k)=y_k-F^{\prime }{\left(y_k\right)}^{-1}F\left(y_k\right)$. Other choices are given by (31), (32), and in paper [33]. Schemes (2) and (3) were shown to be of order $(\lambda +2)$ and $2j$, respectively in [33], when ${\mathbb{T}}_1={\mathbb{T}}_2={\mathbb{R}}^j,j=1,2,3,\dots$. However, they used high order derivatives in order to demonstrate the convergence order.

Moreover, we refer the reader to [33] for a plethora of choices of $\alpha ,B_{\sigma },C_{\sigma }$ leading to already studied schemes or new schemes. Some choices are also given by us in the numerical section of this study. The computational efficiencies and other benefits were also presented in [33].

We have some concerns with the aforementioned studies:

(a)

The convergence order was established by utilizing the Taylor series expansion requiring higher order derivatives (not appearing on the schemes);

(b)

Lack of computable estimates on distances $\parallel x_{\sigma }-x_{*}\parallel$;

(c)

Results related to the uniqueness of the solutions are not given;

(d)

We do not know in advance how many iterates are needed to achieve a desired error tolerance given in advance;

(e)

Earlier studies have been made only on the multidimensional Euclidean space.

Concerns (a)–(e), limit the applicability of Schemes (2) and (3) and similar ones [1,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,28,29,30,31,32]. Let us consider a motivational example. We assume the following function F on ${\mathbb{T}}_1={\mathbb{T}}_2=\mathbb{R}$, $\mathbb{D}=[-\frac{1}{2},\frac{3}{2}]$ as:

$$F\left(\tau \right)=\left\{\begin{array}{ccc}{\tau }^{3}ln{\tau }^2+{\tau }^5-{\tau }^4,\hfill & & \tau \ne 0\hfill \\ 0,\hfill & & \tau =0\hfill \end{array}.\right.$$

(4)

We yield:

$$F^{\prime }\left(\tau \right)=3{\tau }^{2}ln{\tau }^2+5{\tau }^4-4{\tau }^3+2{\tau }^2,$$

$$F^{″}\left(\tau \right)=6\tau ln{\tau }^2+20{\tau }^3-12{\tau }^2+10\tau ,$$

$$F^{‴}\left(\tau \right)=6ln{\tau }^2+60{\tau }^2-24\tau +22.$$

Thus, we see that $F^{‴}\left(\tau \right)$ is not bounded in $\mathbb{D}$. Therefore, results requiring the existence of $F^{‴}\left(\tau \right)$ or higher cannot apply for studying the convergence of (2), (3), and [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].

The novelty of our study lies in the fact that we address concerns (a)–(e) using the derivative (only appearing on the schemes) as well as very general conditions. This way, we also provide computable upper bound estimations on $\parallel x_{\sigma }-x_{*}\parallel$ and results on the uniqueness of solutions. Moreover, our results are obtained in the more general setting of a Banach space. Hence, the region of applicability is extended for these schemes. It is worth noticing that computing the convergence radii shows how difficult (limited too) it is to determine initial points. Our idea is so general that it can be used on other schemes in a similar fashion [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. We suppose from now on that $x_{*}$ is a simple solution of Equation (1).

It is worth noticing that the foundations of symmetrical principles include quantum physics and micro-world. These problems once converted to equations of the form (1), their solutions are hard to obtain in closed form or analytically. That is why such schemes are important to study.

The rest of the study includes: local convergence of these schemes in Section 2, and the numerical experiments in Section 3. In particular similar work (see Example 2 and Example 4) in our Section 3 can be done in [13,14,25]. Finally, Section 4 is devoted to the concluding remarks.

2. Analysis in the Sense of Local Convergence

It is convenient for the analysis of scheme (2) to develop some parameters and functions. Suppose that scalar equation $f\left(\tau \right)=0,$ has real solutions $s_1,s_2,\dots ,s_m\in \mathbb{D}\subset \mathbb{R}$ with $s_1\le s_2\le \dots \le s_m$. Then, $s_1$ is called the smallest or minimal solution of the equation $f\left(\tau \right)=0,$ in $\mathbb{D}$. Set $\Omega =[0,\infty )$.

Suppose equation(s):

(i)

$${\mu }_0\left(\tau \right)-1=0,$$

(5)

has a smallest solution $R_0\in \Omega -\left\{0\right\}$ for some continuous and non-decreasing function ${\mu }_0:\Omega \to \Omega$. Set ${\Omega }_0=[0,2R_0)$.

(ii)

$${\psi }_1\left(\tau \right)-1=0,$$

$${\psi }_2\left(\tau \right)-1=0,$$

have the smallest solutions ${\rho }_1,{\rho }_2\in {\Omega }_0-\left\{0\right\}$, respectively, for some continuous and non-decreasing functions $\mu :{\Omega }_0\to \Omega$, ${\mu }_1:{\Omega }_0\to \Omega$ and ${\psi }_1,{\psi }_2:{\Omega }_0\to \Omega$ with:

$${\psi }_1\left(\tau \right)=\frac{{\int }_0^1\mu \left((1-\theta )\tau \right)d\theta +|1-\alpha |{\int }_0^1{\mu }_1\left(\theta \tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}.$$

(iii)

$${\psi }_3\left(\tau \right)-1=0,$$

(6)

has a smallest solution ${\rho }_3\in {\Omega }_0-\left\{0\right\}$ for some continuous and non-decreasing function $p:{\Omega }_0\to \Omega$ with:

$$\begin{array}{cc}\hfill {\psi }_3\left(\tau \right)& =[\frac{{\int }_0^1\mu \left((1-\theta ){\psi }_2\left(\tau \right)\tau \right)d\theta }{1-{\mu }_0\left({\psi }_2\left(\tau \right)\tau \right)}+\frac{p\left(\tau \right){\int }_0^1{\mu }_1\left(\theta {\psi }_2\left(\tau \right)\tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}\hfill \\ \hfill & +\frac{\left({\mu }_0\left(\tau \right)+{\mu }_0\left({\psi }_2\left(\tau \right)\tau \right)\right){\int }_0^1{\mu }_1\left(\theta {\psi }_2\left(\tau \right)\tau \right)d\theta }{\left(1-{\mu }_0\left(\tau \right)\right)\left(1-{\mu }_0\left({\psi }_2\left(\tau \right)\tau \right)\right)}]{\psi }_2\left(\tau \right).\hfill \end{array}$$

The parameter $\rho$ defined by:

$$\rho =min\left\{{\rho }_k\right\},k=1,2,3$$

(7)

shall be shown to be a convergence radius for scheme (2). Set ${\Omega }_1=[0,\rho )$. It follows from this definition that:

$$0\le {\mu }_0\left(\tau \right)<1,$$

(8)

$$0\le {\mu }_0\left({\psi }_1\left(\tau \right)\tau \right)<1,$$

(9)

$$0\le {\psi }_k\left(\tau \right)<1,\mathrm{hold}\mathrm{for}\mathrm{all}\tau \in {\Omega }_1.$$

(10)

By $\overline{S}(x_{*},\mu )$, we denote the closure of ball $S(x_{*},\mu )$ with center $x_{*}\in {\mathbb{T}}_1$ and of radius $\mu >0$.

The local convergence analysis of scheme (2) relies on the conditions $\left(A\right)$ provided that the scalar functions are as previously defined.

Suppose:

(A₁)

For each $x\in \mathbb{D}$

$$\parallel F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(x\right)-F^{\prime }\left(x_{*}\right)\right)\parallel \le {\mu }_0(\parallel x-x_{*}\parallel ).$$

Set $S_0=\mathbb{D}\cap S(x_{*},R_0)$.

(A₂)

For each $x\in S_0$

$$\begin{array}{cc}\hfill & \parallel F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(y\right)-F^{\prime }\left(x\right)\right)\parallel \le \mu (\parallel y-x\parallel ),\hfill \\ \hfill & \parallel F^{\prime }{\left(x_{*}\right)}^{-1}{F}^{\prime }\left(x\right)\parallel \le {\mu }_1(\parallel x-x_{*}\parallel ),\hfill \\ \hfill & \parallel A_{\lambda }(x,y)-x_{*}\parallel \le {\psi }_2(\parallel x-x_{*}\parallel )\parallel x-x_{*}\parallel \hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & \parallel I-B\left(x\right)\parallel \le p(\parallel x-x_{*}\parallel ),\hfill \end{array}$$

where $y=x-\alpha F^{\prime }{\left(x\right)}^{-1}F\left(x\right)$.

(A₃)

$\overline{S}(x_{*},\rho )\subset \mathbb{D}$

and

(A₄)

There exists ${\rho }_{*}\ge \rho$ satisfying:

$${\int }_0^1{\psi }_0\left(\theta {\rho }_{*}\right)d\theta <1.$$

Set $S_1=\mathbb{D}\cap \overline{S}(x_{*},{\rho }_{*})$.

Next, the local convergence analysis of scheme (2) is developed using conditions $\left(A\right)$ with functions ${\psi }_k$ as previously defined.

Theorem 1.

Suppose conditions $\left(A\right)$ hold and choose $x_0\in S(x_{*},\rho )-\left\{x_{*}\right\}$. Then, sequence $\left\{x_{\sigma }\right\}$ starting from $x_0$ and generated by scheme (2) is well defined in $S(x_{*},\rho )$. It stays in $S(x_{*},\rho )$ for all $\sigma =0,1,2,\dots$ and converges to $x_{*}$. Moreover, the only solution of Equation (1) in the set $S_1$ is $x_{*}$.

Proof. 

The following assertions shall be shown using mathematical induction:

$$\left\{x_{\sigma }\right\}\subset S(x_{*},\rho ),lim{x}_{\sigma }=x_{*},$$

(11)

$$\parallel y_{\sigma }-x_{*}\parallel \le {\psi }_1(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel <r,$$

(12)

$$\parallel z_{\sigma }-x_{*}\parallel \le {\psi }_2(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel ,$$

(13)

and

$$\parallel x_{\sigma +1}-x_{*}\parallel \le {\psi }_3(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel <r,$$

(14)

where the functions $‘‘{\psi }_k"$ as given previously and radius $\rho$ is as defined by (7).

Let $v\in S(x_{*},\rho )-\left\{x_{*}\right\}$. Using (7), (8), and $\left(A_1\right)$, we obtain:

$$\parallel F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(v\right)-F^{\prime }\left(x_{*}\right)\right)\parallel \le {\mu }_0(\parallel v-x_{*}\parallel )\le {\mu }_0\left(\rho \right)<1.$$

Hence, by a lemma due to Banach for inverses of linear operators [8], $F^{\prime }{\left(x\right)}^{-1}\in \ell ({\mathbb{T}}_1,{\mathbb{T}}_2)$ with:

$$\parallel F^{\prime }{\left(v\right)}^{-1}{F}^{\prime }\left(x_{*}\right)\parallel \le \frac{1}{1-{\mu }_0(\parallel v-x_{*}\parallel )}.$$

(15)

Then, iterates $y_0,z_0$, and $x_1$ are well defined. Thus, we can write by the three steps of Scheme (2), respectively:

$$\begin{array}{cc}\hfill y_0-x_{*}& =x_0-x_{*}-F^{\prime }{\left(x_0\right)}^{-1}F\left(x_0\right)+(1-\alpha )F^{\prime }{\left(x_0\right)}^{-1}F\left(x_0\right)\hfill \\ \hfill & =\left(F^{\prime }{\left(x_0\right)}^{-1}F\left(x_{*}\right)\right)\left({\int }_0^1{F}^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(x_{*}+\theta (x_0-x_{*})\right)-F^{\prime }\left(x_0\right)\right)d\theta (x_0-x_{*})\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill z_0-x_{*}& =A_{\lambda }(x_0,y_0)-x_{*},\hfill \end{array}$$

(17)

and

$$\begin{array}{cc}\hfill x_1-x_{*}& =z_0-x_{*}-B_0{F}^{\prime }{\left(x_0\right)}^{-1}F\left(z_0\right)\hfill \\ \hfill & =z_0-x_{*}-F^{\prime }{\left(z_0\right)}^{-1}F\left(y_0\right)+(I-B_0)F^{\prime }{\left(x_0\right)}^{-1}F\left(z_0\right)\hfill \\ \hfill & +\left(F^{\prime }{\left(z_0\right)}^{-1}-F^{\prime }{\left(x_0\right)}^{-1}\right)F\left(z_0\right).\hfill \end{array}$$

(18)

By (7), (9), (10) (for $k=1,2,3$), (15) (for $v=x_0,y_0$), $\left(A_2\right),\left(A_3\right)$, and (16)–(18), we get respectively:

$$\begin{array}{cc}\hfill \parallel y_0-x_{*}\parallel & \le \frac{\left({\int }_0^1\mu \left((1-\theta )\parallel x_0-x_{*}\parallel \right)d\theta +|1-\alpha |{\int }_o^1{\mu }_1(\theta \parallel x_0-x_{*}\parallel )d\theta \right)\parallel x_0-x_{*}\parallel }{1-{\mu }_0(\parallel x_0-x_{*}\parallel )}\hfill \\ \hfill & \le {\psi }_1(\parallel x_0-x_{*}\parallel )\parallel x_0-x_{*}\parallel \le \parallel x_0-x_{*}\parallel <\rho ,\hfill \end{array}$$

(19)

$$\parallel z_0-x_{*}\parallel =\parallel A_{\lambda }(x_0,y_0)-x_{*}\parallel \le {\psi }_2(\parallel x_0-x_{*}\parallel )\parallel x_0-x_{*}\parallel \le \parallel x_0-x_{*}\parallel ,$$

(20)

and

$$\begin{array}{cc}\hfill \parallel x_1-x_{*}\parallel & \le [\frac{{\int }_0^1\mu \left((1-\theta )\parallel z_0-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel z_0-x_{*}\parallel )}\hfill \\ \hfill & +\frac{p(\parallel x_0-x_{*}\parallel ){\int }_0^1{\mu }_1(\theta \parallel z_0-x_{*}\parallel )d\theta }{1-{\mu }_0(\parallel x_0-x_{*}\parallel )}\hfill \\ \hfill & +\frac{\left({\mu }_0(\parallel z_0-x_{*}\parallel )+{\mu }_0(\parallel x_0-x_{*}\parallel )\right){\int }_0^1{\mu }_1(\theta \parallel z_0-x_{*}\parallel )d\theta }{\left(1-{\mu }_0(\parallel x_0-x_{*}\parallel \right)\left(1-{\mu }_0(\parallel z_0-x_{*}\parallel )\right)}]\parallel z_0-x_{*}\parallel \hfill \\ \hfill & \le {\psi }_3(\parallel x_0-x_{*}\parallel )\parallel x_0-x_{*}\parallel ,\hfill \end{array}$$

(21)

so $y_0,z_0,x_1\in S(x_{*},\rho )$, and estimations (12)–(14) hold for $\sigma =0$. The induction for (11)–(14) finishes if $y_0,z_0$, and $x_1$ are replaced by $x_{\sigma },y_{\sigma }$, and $x_{\sigma +1}$, respectively in the preceding calculations. The completion of the induction gives the estimation:

$$\parallel x_{\sigma +1}-x_{*}\parallel \le \gamma \parallel x_{\sigma }-x_{*}\parallel <\rho ,$$

(22)

with $\gamma ={\psi }_3(\parallel x_0-x_{*}\parallel )\in [0,1)$, from which we have ${\displaystyle \underset{\sigma \to \infty }{lim}{x}_{\sigma }=x_{*}}$ and $x_{\sigma +1}\in S(x_{*},\rho )$.

In order to show the uniqueness part, we set $M={\int }_0^1{F}^{\prime }\left(x_{*}+\theta (u-x_{*})\right)d\theta ,\mathrm{for}\mathrm{some}u\in S_1$ with $F\left(u\right)=0$. Then, in view of $\left(A_1\right)$ and $\left(A_4\right)$, we obtain:

$$\begin{array}{cc}\hfill \parallel F^{\prime }{\left(x_{*}\right)}^{-1}\left(M-F^{\prime }\left(x_{*}\right)\right)\parallel & \le {\int }_0^1{\mu }_0(\theta \parallel u-x_{*}\parallel )d\theta \hfill \\ \hfill & \le {\int }_0^1{\mu }_0\left(\theta {\rho }_{*}\right)d\theta <1,\hfill \end{array}$$

leading to $u=x_{*}$, since $0=F\left(u\right)-F\left(x_{*}\right)=M(u-x_{*})$, $M^{-1}\in \ell ({\mathbb{T}}_2,{\mathbb{T}}_1)$ and $u-x_{*}=M^{-1}\left(0\right)=0$. □

Remark 1.

(a) 

By $\left(A_1\right)$ and the estimation:

$$\begin{array}{ccc}\parallel F^{\prime }{\left(x_{*}\right)}^{-1}{F}^{\prime }\left(x\right)\parallel \hfill & =& \parallel F^{\prime }{\left(x_{*}\right)}^{-1}(F^{\prime }\left(x\right)-F^{\prime }\left(x_{*}\right))+I\parallel \hfill \\ & \le & 1+\parallel F^{\prime }{\left(x_{*}\right)}^{-1}(F^{\prime }\left(x\right)-F^{\prime }\left(x_{*}\right))\parallel \hfill \\ & \le & 1+{\mu }_0(\parallel x_0-x_{*}\parallel )\hfill \end{array}$$

condition $\left(A_2\right)$ can be dropped and ${\mu }_1\left(\tau \right)$ be replaced by:

$${\mu }_1\left(\tau \right)=1+{\mu }_0\left(\rho \right)or{\mu }_1\left(\tau \right)=2.$$

(b) 

The results obtained here can be used for operators F satisfying the autonomous differential equation [9,10] of the form:

$$F^{\prime }\left(x\right)=P\left(F\left(x\right)\right),$$

where P is a known continuous operator. Since $F^{\prime }\left(x^{*}\right)=P\left(F\left(x^{*}\right)\right)=P\left(0\right),$ we can apply the results without actually knowing the solution $x^{*}.$ Let us consider an example $F\left(x\right)=e^x-1.$ Then, we can choose $P\left(x\right)=x+1$.

(c) 

If ${\mu }_0\left(\tau \right)=K_0\tau$ and $\mu \left(t\right)=K\tau$ then $r_1=\frac{2}{2K_0+K}$ was shown in [9,10] to be the convergence radius for Newton’s method. It follows from (7) and the definition of $r_1$ that the convergence radius ρ of the method (2) cannot be larger than the convergence radius $r_1$ of the second order Newton’s method. As already noted in [9,10], $r_1$ is at its smallest as large as the convergence ball given by Rheinboldt [28]:

$$\begin{array}{c}{r}_R={\displaystyle \frac{2}{3K_1}}.\hfill \end{array}$$

In particular, for $K_0<K_1$ (where $K_1$ is the constant on $\mathbb{D}$), we have that:

$$r_R<r_1$$

and

$${\displaystyle \frac{r_R}{r_1}}\to {\displaystyle \frac{1}{3}}as{\displaystyle \frac{K_0}{K_1}}\to 0.$$

Therefore our convergence ball $r_1$ is at most three times larger than Rheinboldt’s. The same value for $r_R$ is given by Traub [1].

(d) 

Method (2) is not changing if we use the conditions of Theorem 1 instead of the stronger conditions given in [33]. Moreover, for the error bounds in practice we can use the Computational Order of Convergence (COC) [32]:

$$\eta =\frac{ln\frac{s_{\sigma +1}}{s_{\sigma }}}{ln\frac{s_{\sigma }}{s_{\sigma -1}}},\mathrm{for}\sigma =1,2,\dots ,if{s}_{\sigma }=\parallel x_{\sigma }-x_{*}\parallel$$

(23)

or Approximate Computational Order of Convergence $\left(ACOC\right)$ [32] by:

$${\eta }^{*}=\frac{ln\frac{v_{\sigma +1}}{v_{\sigma }}}{ln\frac{v_{\sigma }}{v_{\sigma -1}}},\mathrm{for}\sigma =2,3,\dots ,if{v}_{\sigma }=\parallel x_{\sigma }-x_{\sigma -1}\parallel .$$

(24)

So, the convergence order is obtained in this way without evaluations higher than the first Fréchet derivative.

Next, we present the local convergence of scheme (3) in an analogous way. Define functions on ${\Omega }_0$ as:

$${\overline{\psi }}_2\left(\tau \right)=\frac{{\int }_o^1\mu \left((1-\theta )\tau \right)d\theta +q\left(\tau \right){\int }_0^1{\mu }_1\left(\theta \tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}{\psi }_1\left(\tau \right),$$

$$\begin{array}{cc}\hfill {\overline{\psi }}_3\left(\tau \right)=& [\frac{{\int }_0^1\mu \left((1-\theta ){\overline{\psi }}_2\left(\tau \right)\tau \right)d\theta }{1-{\mu }_0\left({\overline{\psi }}_2\left(\tau \right)\tau \right)}+\frac{p\left(\tau \right){\int }_0^1{\mu }_1\left(\theta {\overline{\psi }}_2\left(\tau \right)\tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}\hfill \\ \hfill & +\frac{{\int }_0^1{\mu }_1\left(\theta {\overline{\psi }}_2\left(\tau \right)\tau \right)d\theta }{1-{\mu }_0\left({\overline{\psi }}_2\left(\tau \right)\tau \right)}+\frac{{\int }_0^1{\mu }_1\left(\theta {\overline{\psi }}_2\left(\tau \right)\tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}]{\overline{\psi }}_2\left(\tau \right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\overline{\psi }}_4\left(\tau \right)& =h\left(\tau \right){\overline{\psi }}_3\left(\tau \right),\hfill \\ \hfill {\overline{\psi }}_5\left(\tau \right)& =h\left(\tau \right){\overline{\psi }}_4\left(\tau \right),\hfill \\ \hfill ⋮& \\ \hfill {\overline{\psi }}_m\left(\tau \right)& =h\left(\tau \right){\overline{\psi }}_{m-1}\left(\tau \right),m=4,5,6,\dots ,j\hfill \end{array}$$

where $q:{\Omega }_0\to \Omega$ is a continuous and non-decreasing function, and

$$\begin{array}{cc}\hfill h\left(\tau \right)& =\frac{{\int }_0^1\mu \left((1-\theta )\tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}+\frac{p\left(\tau \right){\int }_0^1{\mu }_1\left(\theta \tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}+\frac{2{\int }_0^1{\mu }_1\left(\theta \tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}.\hfill \end{array}$$

Suppose equations:

$$\begin{array}{cc}\hfill & {\psi }_1\left(\tau \right)-1=0,\hfill \\ \hfill & {\overline{\psi }}_2\left(\tau \right)-1=0,\hfill \\ \hfill & {\overline{\psi }}_3\left(\tau \right)-1=0,\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\overline{\psi }}_m\left(\tau \right)-1=0,\hfill \end{array}$$

have the smallest solutions in ${\Omega }_0-\left\{0\right\}$ denoted by ${\rho }_1,{\overline{\rho }}_2,\dots {\overline{\rho }}_m$, respectively.

Define parameter:

$$\overline{\rho }=min\{{\rho }_1,{\overline{\rho }}_2,\dots {\overline{\rho }}_m\}.$$

(25)

Itshall be shown that $\overline{\rho }$ is a convergence radius for scheme (3).

Consider conditions $\left(A_1\right),\left(A_3\right),\left(A_4\right)$ with $\overline{\rho }$ replacing $\rho$. Moreover replace the second condition in $\left(A_2\right)$ by:

$$\parallel I-C\left(x\right)\parallel \le q(\parallel x-x_{*}\parallel )\mathrm{for}\mathrm{each}x\in S_0.$$

Let us call the resulting conditions $\left(A^{\prime }\right)$. Then, as in Theorem 1, we get in turn the estimates that also motivate the introduction of the $“\overline{\psi }”$ functions:

$$\parallel z_{\sigma }^{\left(1\right)}-x_{*}\parallel \le {\psi }_1(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel \le \overline{\rho },$$

(26)

$$\begin{array}{cc}\hfill \parallel z_{\sigma }^{\left(2\right)}-x_{*}\parallel & =\parallel x_{\sigma }-x_{*}-F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)+(I-C_0)F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)\parallel \hfill \\ \hfill & \le \frac{\left({\int }_0^1\mu \left((1-\theta )\parallel x_{\sigma }-x_{*}\parallel \right)d\theta +q(\parallel x_{\sigma }-x_{*}\parallel ){\int }_0^1{\mu }_1\left(\theta \parallel x_{\sigma }-x_{*}\parallel \right)d\theta \right)\parallel x_{\sigma }-x_{*}\parallel }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}\hfill \\ \hfill & ={\overline{\psi }}_2(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel ,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel & \le [\frac{{\int }_0^1\mu \left((1-\theta )\parallel z_{\sigma }^{\left(2\right)}-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel z_{\sigma }^{\left(2\right)}-x_{*}\parallel )}+\frac{p(\parallel x_{\sigma }-x_{*}\parallel ){\int }_0^1{\mu }_1\left(\theta \parallel z_{\sigma }^{\left(2\right)}-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}\hfill \\ \hfill & +\frac{{\int }_0^1{\mu }_1\left(\theta \parallel z_{\sigma }^{\left(2\right)}-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel z_{\sigma }^{\left(2\right)}-x_{*}\parallel )}+\frac{{\int }_0^1{\mu }_1\left(\theta \parallel z_{\sigma }^{\left(2\right)}-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}]\parallel z_{\sigma }^{\left(2\right)}-x_{*}\parallel \hfill \\ \hfill & ={\overline{\psi }}_3(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel ,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \parallel z_{\sigma }^{\left(4\right)}-x_{*}\parallel & \le [\frac{{\int }_0^1\mu \left((1-\theta )\parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel )}+\frac{p(\parallel x_{\sigma }-x_{*}\parallel ){\int }_0^1{\mu }_1\left(\theta \parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}\hfill \\ \hfill & +\frac{{\int }_0^1{\mu }_1\left(\theta \parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel )}+\frac{{\int }_0^1{\mu }_1\left(\theta \parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}]\parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel \hfill \\ \hfill & \le [\frac{{\int }_0^1\mu \left((1-\theta )\parallel x_{\sigma }-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}+\frac{p(\parallel x_{\sigma }-x_{*}\parallel ){\int }_0^1{\mu }_1\left(\theta \parallel x_{\sigma }-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}\hfill \\ \hfill & +\frac{{\int }_0^1{\mu }_1\left(\theta \parallel x_{\sigma }-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}+\frac{{\int }_0^1{\mu }_1\left(\theta \parallel x_{\sigma }-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}]\parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel \hfill \\ \hfill & =h(\parallel x_{\sigma }-x_{*}\parallel )\parallel z_{\sigma }^{\left(3\right)}-x_{*}\parallel \le {\overline{\psi }}_4(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel ,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & ⋮\hfill \\ \hfill & \parallel x_{\sigma +1}-x_{*}\parallel =\parallel z_{\sigma }^{\left(j\right)}-x_{*}\parallel ={\overline{\psi }}_{\sigma }^{\left(j\right)}(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel .\hfill \end{array}$$

(30)

Hence, we get the local convergence result for scheme (3).

Theorem 2.

We suppose that conditions $\left(A^{\prime }\right)$ hold and choose $x_0\in S(x_{*},\rho )-\left\{x_{*}\right\}$. Then, the sequence $\left\{x_{\sigma }\right\}$ generated by scheme (3) converges to $x_{*}$, so that estimations (26)–(30) as well as the uniqueness result of Theorem 1 hold.

3. Numerical Applications

We present the computational results based on the suggested theoretical results in this paper. We choose $A_3(x_{\sigma },y_{\sigma })=x_{\sigma }-T_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)$ and $A_4(x_{\sigma },y_{\sigma })=x_{\sigma }-H_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right),$ respectively, in scheme (2) in order to obtain fifth and sixth order iterative procedures. In particular, we have:

$$\begin{array}{cc}\hfill y_{\sigma }& =x_{\sigma }-\alpha F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right),\hfill \\ \hfill z_{\sigma }& =x_{\sigma }-T_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right),\hfill \\ \hfill x_{\sigma +1}& =z_{\sigma }-B_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(z_{\sigma }\right),\hfill \end{array}$$

(31)

where

$$\begin{array}{cc}\hfill T_{\sigma }& ={\left[\left(1-\frac{\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]}^{-1}\left[\left(1+\frac{1-\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)-\frac{1-\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right],\beta \in \mathbb{R},\hfill \\ \hfill B_{\sigma }& ={\left[\left(1+\frac{\delta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)-\frac{\delta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]}^{-1}\left[\left(1+\frac{2+\delta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)-\frac{2+\delta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right],\delta \in \mathbb{R},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill y_{\sigma }& =x_{\sigma }-\alpha F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right),\hfill \\ \hfill z_{\sigma }& =x_{\sigma }-H_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)\hfill \\ \hfill x_{\sigma +1}& =z_{\sigma }-B_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(z_{\sigma }\right),\hfill \end{array}$$

(32)

where

$$\begin{array}{cc}\hfill H_{\sigma }& =I+\frac{I-h_{\sigma }}{2\alpha }+\frac{{(I-h_{\sigma })}^2}{2{\alpha }^2}+\left(1-\frac{3}{2}\alpha \right){\Delta }_{\sigma },\hfill \\ \hfill {\Delta }_{\sigma }& =-\frac{1}{6{\alpha }^2}\left[F^{\prime }{\left(x_{\sigma }\right)}^{-1}\left(F^{\prime }\left(y_{\sigma }\right)-2F^{\prime }\left(x_{\sigma }\right)+F^{\prime }\left(w_{\sigma }\right)\right)\right],\hfill \\ \hfill w_{\sigma }& =2x_{\sigma }-y_{\sigma },\hfill \\ \hfill h_{\sigma }& =F^{\prime }{\left(x_{\sigma }\right)}^{-1}{F}^{\prime }\left(y_{\sigma }\right).\hfill \end{array}$$

For more details of these values please see the article Zhanlav and Otgondroj [33]. Next, we show how to choose the $“\psi ”$ functions for Schemes (31) and (32), respectively.

Suppose equation:

$$\begin{array}{cc}\hfill q_1\left(\tau \right)-1& =0,\hfill \end{array}$$

has a smallest solution $R_{q_1}\in {\Omega }_0-\left\{0\right\}$, where:

$$\begin{array}{cc}\hfill q_1\left(\tau \right)& =\left|1-\frac{\beta }{2\alpha }\right|{\mu }_0\left(\tau \right)+\left|\frac{\beta }{2\alpha }\right|{\mu }_0\left({\psi }_1\left(\tau \right)\tau \right).\hfill \end{array}$$

Case $\beta =0$

Set ${\Omega }_1=\left[0,min\{R_0,R_{q_1}\}\right)$. Define function ${\psi }_2:{\Omega }_1\to \Omega$ by:

$$\begin{array}{cc}\hfill {\psi }_2\left(\tau \right)& =\frac{{\int }_0^1\mu \left((1-\theta )\tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}+\frac{\left({\mu }_0\left(\tau \right)+{\mu }_0\left({\psi }_1\left(\tau \right)\tau \right)\right){\int }_0^1{\mu }_1\left(\theta \tau \right)d\theta }{2\left|\alpha \right|\left(1-q_1\left(\tau \right)\right)\left(1-{\mu }_0\left(\tau \right)\right)}.\hfill \end{array}$$

The choice of functions $q_1$ and ${\psi }_2$ is justified by the estimates:

$$\begin{array}{cc}\hfill z_{\sigma }-x_{*}& =x_{\sigma }-x_{*}-T_{\sigma }{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)\hfill \\ \hfill & =x_{\sigma }-x_{*}-F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)+F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)-[\left(1-\frac{\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)\hfill \\ \hfill & +\frac{\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right){]}^{-1}\left[\left(1+\frac{1-\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{1-\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)\hfill \\ \hfill & =x_{\sigma }-x_{*}-F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)+\left[I-{\left[\left(1-\frac{\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]}^{-1}\right]\hfill \\ \hfill & \times \left[\left(1+\frac{1-\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{1-\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)\hfill \\ \hfill & =x_{\sigma }-x_{*}-F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)+{\left[\left(1-\frac{\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]}^{-1}\hfill \\ \hfill & \times [\left(1-\frac{\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)-\left(1+\frac{1-\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)\hfill \\ \hfill & -\frac{1-\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)]F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right),\hfill \end{array}$$

so

$$\begin{array}{cc}\hfill \parallel z_{\sigma }-x_{*}\parallel & =[\frac{{\int }_0^1\mu \left((1-\theta )\parallel x_{\sigma }-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}\hfill \\ \hfill & +\frac{\left({\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )\right){\int }_0^1{\mu }_1(\theta \parallel x_{\sigma }-x_{*}\parallel )d\theta }{2\left|\alpha \right|\left(1-q_1(\parallel x_{\sigma }-x_{*}\parallel )\right)\left(1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )\right)}]\parallel x_{\sigma }-x_{*}\parallel \hfill \\ \hfill & \le {\psi }_2(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel <\rho ,\hfill \end{array}$$

where we also used:

$$\begin{array}{cc}\hfill & ∥F^{\prime }{\left(x_{\sigma }\right)}^{-1}\left[\left(1-\frac{\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)-\left(1-\frac{\beta }{2\alpha }+\frac{\beta }{2\alpha }\right)F^{\prime }\left(x_{*}\right)\right]∥\hfill \\ \hfill & \le \left|1-\frac{\beta }{2\alpha }\right|∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(x_{\sigma }\right)-F^{\prime }\left(x_{*}\right)\right)∥+\left|\frac{\beta }{2\alpha }\right|∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(y_{\sigma }\right)-F^{\prime }\left(x_{*}\right)\right)∥\hfill \\ \hfill & \le \left|1-\frac{\beta }{2\alpha }\right|{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+\left|\frac{\beta }{2\alpha }\right|{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )\hfill \\ \hfill & \le q_1(\parallel x_{\sigma }-x_{*}\parallel )\le q_1\left(\rho \right)<1,\hfill \end{array}$$

so

$$∥{\left[\left(1-\frac{\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]}^{-1}{F}^{\prime }\left(x_{*}\right)∥\le \frac{1}{1-q_1(\parallel x_{\sigma }-x_{*}\parallel )},$$

and

$$\begin{array}{cc}\hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left[\left(1-\frac{\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)-\left(1+\frac{1-\beta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)-\frac{1-\beta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]∥\hfill \\ \hfill & \le \frac{1}{2\left|\alpha \right|}∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(y_{\sigma }\right)-F^{\prime }\left(x_{\sigma }\right)\right)∥\hfill \\ \hfill & \le \frac{1}{2\left|\alpha \right|}\left[∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(y_{\sigma }\right)-F^{\prime }\left(x_{*}\right)\right)∥+∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(x_{\sigma }\right)-F^{\prime }\left(x_{*}\right)\right)∥\right]\hfill \\ \hfill & \le \frac{1}{2\left|\alpha \right|}\left[{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )\right]\hfill \\ \hfill & \le \frac{1}{2\left|\alpha \right|}\left[{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0\left({\psi }_1(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \right)\right].\hfill \end{array}$$

Case $\beta \ne 0$

The proceeding calculations for ${\psi }_2$ in this case suggest that this function can be defined by:

$$\begin{array}{cc}\hfill {\psi }_2\left(\tau \right)& =\frac{{\int }_0^1\mu \left((1-\theta )\tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}+\frac{\left({\mu }_1\left(\tau \right)+|1-2\beta |{\mu }_1\left({\psi }_1\left(\tau \right)\tau \right)\right){\int }_0^1{\mu }_1\left(\theta \tau \right)d\theta }{2\left|\alpha \right|\left(1-q_1\left(\tau \right)\right)\left(1-{\mu }_0\left(\tau \right)\right)},\hfill \end{array}$$

since:

$$\begin{array}{cc}\hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left[-\frac{\beta }{2\alpha }{F}^{\prime }\left(x_{\sigma }\right)-\frac{2\beta -1}{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]∥\hfill \\ \hfill & \le \frac{1}{2\left|\alpha \right|}∥F^{\prime }{\left(x_{*}\right)}^{-1}{F}^{\prime }\left(x_{\sigma }\right)∥+\frac{|1-2\beta |}{2\left|\alpha \right|}∥F^{\prime }{\left(x_{*}\right)}^{-1}{F}^{\prime }\left(y_{\sigma }\right)∥]\hfill \\ \hfill & \le \frac{1}{2\left|\alpha \right|}\left[{\mu }_1(\parallel x_{\sigma }-x_{*}\parallel )+\frac{|1-2\beta |}{2\left|\alpha \right|}{\mu }_1(\parallel y_{\sigma }-x_{*}\parallel )\right]\hfill \\ \hfill & \le \frac{1}{2\left|\alpha \right|}\left[{\mu }_1(\parallel x_{\sigma }-x_{*}\parallel )+|1-2\beta |{\mu }_1\left({\psi }_1(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \right)\right].\hfill \end{array}$$

Moreover, suppose equation:

$$q_2\left(\tau \right)-1=0,$$

has a smallest solution $R_{q_2}\in {\Omega }_1-\left\{0\right\}$ for:

$$q_2\left(\tau \right)=\left|1+\frac{\delta }{2\alpha }\right|{\mu }_0\left(\tau \right)+\left|\frac{\delta }{2\alpha }\right|{\mu }_0\left({\psi }_1\left(\tau \right)\tau \right).$$

Define function ${\psi }_3:{\Omega }_1\to \Omega$ by:

$$\begin{array}{cc}\hfill {\psi }_3\left(\tau \right)& =[\frac{{\int }_0^1\mu \left((1-\theta ){\psi }_2\left(\tau \right)\tau \right)d\theta }{1-{\mu }_0\left({\psi }_2\left(\tau \right)\tau \right)}+\frac{\left({\mu }_0\left(\tau \right)+{\mu }_0\left({\psi }_2\left(\tau \right)\tau \right)\right){\int }_0^1{\mu }_1\left(\theta {\psi }_2\left(\tau \right)\tau \right)d\theta }{\left(1-{\mu }_0\left(\tau \right)\right)\left(1-{\mu }_0\left({\psi }_2\left(\tau \right)\tau \right)\right)}\hfill \\ \hfill & +\frac{\left({\mu }_0\left(\tau \right)+{\mu }_0\left({\psi }_1\left(\tau \right)\tau \right)\right){\int }_0^1{\mu }_1\left(\theta {\psi }_2\left(\tau \right)\tau \right)d\theta }{\left|\alpha \right|\left(1-q_2\left(\tau \right)\right)\left(1-{\mu }_0\left(\tau \right)\right)}]{\psi }_2\left(\tau \right),\hfill \end{array}$$

and

$$p\left(\tau \right)=\frac{{\mu }_0\left(\tau \right)+{\mu }_0\left({\psi }_1\left(\tau \right)\tau \right)}{\left|\alpha \right|\left(1-q_2\left(\tau \right)\right)}.$$

The choice of functions $q_2$ and ${\psi }_3$ is justified by the estimates:

$$\begin{array}{cc}\hfill x_{\sigma +1}-x_{*}& =z_{\sigma }-x_{*}-F^{\prime }{\left(z_{\sigma }\right)}^{-1}F\left(z_{\sigma }\right)+\left(F^{\prime }{\left(z_{\sigma }\right)}^{-1}-F^{\prime }{\left(x_{\sigma }\right)}^{-1}\right)F\left(z_{\sigma }\right)\hfill \\ \hfill & +(I-B_{\sigma })F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(z_{\sigma }\right)\hfill \\ \hfill & =z_{\sigma }-x_{*}-F^{\prime }{\left(z_{\sigma }\right)}^{-1}F\left(z_{\sigma }\right)+F^{\prime }{\left(z_{\sigma }\right)}^{-1}\left(F^{\prime }\left(x_{\sigma }\right)-F^{\prime }\left(z_{\sigma }\right)\right)F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(z_{\sigma }\right)\hfill \\ \hfill & +{\left[\left(1+\frac{\delta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)-\frac{\delta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]}^{-1}[\left(1+\frac{\delta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)-\frac{\delta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\hfill \\ \hfill & -\left(1+\frac{2+\delta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{2+\delta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)]F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(z_{\sigma }\right),\hfill \end{array}$$

so

$$\begin{array}{cc}\hfill \parallel x_{\sigma +1}-x_{*}\parallel & \le [\frac{{\int }_0^1\mu \left((1-\theta )\parallel z_{\sigma }-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel z_{\sigma }-x_{*}\parallel )}\hfill \\ \hfill & +\frac{\left({\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0(\parallel z_{\sigma }-x_{*}\parallel )\right){\int }_0^1{\mu }_1(\theta \parallel z_{\sigma }-x_{*}\parallel )d\theta }{\left(1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )\right)\left(1-{\mu }_0(\parallel z_{\sigma }-x_{*}\parallel )\right)}\hfill \\ \hfill & +\frac{\left({\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )\right){\int }_0^1{\mu }_1(\theta \parallel z_{\sigma }-x_{*}\parallel )d\theta }{\left|\alpha \right|\left(1-q_2(\parallel x_{\sigma }-x_{*}\parallel )\right)\left(1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )\right)}]\parallel z_{\sigma }-x_{*}\parallel \hfill \\ \hfill & \le {\psi }_3(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel <\rho ,\hfill \end{array}$$

where we also adopted the esitmates:

$$\begin{array}{cc}\hfill & ∥F^{\prime }{\left(x_{\sigma }\right)}^{-1}\left[\left(1+\frac{\delta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)-\frac{\delta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)-\left(1+\frac{\delta }{2\alpha }\right)F^{\prime }\left(x_{*}\right)+\frac{\delta }{2\alpha }{F}^{\prime }\left(x_{*}\right)\right]∥\hfill \\ \hfill & \le \left|1+\frac{\delta }{2\alpha }\right|∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(x_{\sigma }\right)-F^{\prime }\left(x_{*}\right)\right)∥+\left|\frac{\delta }{2\alpha }\right|∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(y_{\sigma }\right)-F^{\prime }\left(x_{*}\right)\right)∥\hfill \\ \hfill & \le \left|1+\frac{\delta }{2\alpha }\right|{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+\left|\frac{\delta }{2\alpha }\right|{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )\hfill \\ \hfill & \le q_2(\parallel x_{\sigma }-x_{*}\parallel )\le q_2\left(\rho \right)<1,\hfill \end{array}$$

so

$$∥{\left[\left(1+\frac{\delta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)-\frac{\delta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]}^{-1}{F}^{\prime }\left(x_{*}\right)∥\le \frac{1}{1-q_2(\parallel x_{\sigma }-x_{*}\parallel )},$$

and

$$\begin{array}{cc}\hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left[\left(1+\frac{\delta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)-\frac{\delta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)-\left(1+\frac{2+\delta }{2\alpha }\right)F^{\prime }\left(x_{\sigma }\right)+\frac{2+\delta }{2\alpha }{F}^{\prime }\left(y_{\sigma }\right)\right]∥\hfill \\ \hfill & \le \frac{1}{\left|\alpha \right|}∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(x_{\sigma }\right)-F^{\prime }\left(y_{\sigma }\right)\right)∥\hfill \\ \hfill & \le \frac{1}{\left|\alpha \right|}\left[∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(x_{\sigma }\right)-F^{\prime }\left(x_{*}\right)\right)∥+∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(y_{\sigma }\right)-F^{\prime }\left(x_{*}\right)\right)∥\right]\hfill \\ \hfill & \le \frac{1}{\left|\alpha \right|}\left[{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )\right]\hfill \\ \hfill & \le \frac{1}{\left|\alpha \right|}\left[{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0\left({\psi }_1(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \right)\right].\hfill \end{array}$$

Next, we find functions ${\tilde{\psi }}_2$ and ${\tilde{\psi }}_3$ (with ${\tilde{\psi }}_1={\psi }_1$) but for scheme (32) in an analogous way. We can write:

$$z_{\sigma }-x_{*}=x_{\sigma }-x_{*}-F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right)+(I-H_{\sigma })F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right),$$

so

$$\begin{array}{cc}\hfill \parallel z_{\sigma }-x_{*}\parallel & =\left[\frac{{\int }_0^1\mu \left((1-\theta )\parallel x_{\sigma }-x_{*}\parallel \right)d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}+\frac{q_3(\parallel x_{\sigma }-x_{*}\parallel ){\int }_0^1{\mu }_1(\theta \parallel x_{\sigma }-x_{*}\parallel )d\theta }{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}\right]\hfill \\ \hfill & \times \parallel x_{\sigma }-x_{*}\parallel \hfill \\ \hfill & \le {\tilde{\psi }}_2(\parallel x_{\sigma }-x_{*}\parallel )\parallel x_{\sigma }-x_{*}\parallel \le \parallel x_{\sigma }-x_{*}\parallel <\delta ,\hfill \end{array}$$

where we also used the following estimate:

$$\begin{array}{cc}\hfill \parallel I-H_n\parallel & =\parallel \frac{I-F^{\prime }{\left(x_{\sigma }\right)}^{-1}{F}^{\prime }\left(y_{\sigma }\right)}{2\alpha }+\frac{{\left(I-F^{\prime }{\left(x_{\sigma }\right)}^{-1}{F}^{\prime }\left(y_{\sigma }\right)\right)}^2}{2{\alpha }^2}\hfill \\ \hfill & +\left(1-\frac{3}{2}\alpha \right)\left[-\frac{1}{6{\alpha }^2}{F}^{\prime }{\left(x_{\sigma }\right)}^{-1}\left(\left(F^{\prime }\left(y_{\sigma }\right)-F^{\prime }\left(x_{\sigma }\right)\right)+\left(F^{\prime }\left(w_{\sigma }\right)-F^{\prime }\left(x_{\sigma }\right)\right)\right)\right]\parallel \hfill \\ \hfill & \le \frac{{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )}{2\left|\alpha \right|\left(1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )\right)}\hfill \\ \hfill & +\frac{1}{2{\alpha }^2}{\left(\frac{{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )}{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}\right)}^2\hfill \\ \hfill & +\frac{\left|1-\frac{3}{2}\alpha \right|}{6{\alpha }^2}[\frac{{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )}{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}\hfill \\ \hfill & +\frac{{\mu }_0\left(\parallel x_{\sigma }-x_{*}\parallel +\parallel y_{\sigma }-x_{*}\parallel \right)+{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )}]\hfill \\ \hfill & \le q_3(\parallel x_{\sigma }-x_{*}\parallel ),\hfill \end{array}$$

since

$$\begin{array}{cc}\hfill ∥I-F^{\prime }{\left(x_{\sigma }\right)}^{-1}{F}^{\prime }\left(y_{\sigma }\right)∥& \le \parallel F^{\prime }{\left(x_{\sigma }\right)}^{-1}\left(F^{\prime }\left(x_{\sigma }\right)-F^{\prime }\left(y_{\sigma }\right)\right)\parallel \hfill \\ \hfill & \le \frac{{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )+{\mu }_0(\parallel y_{\sigma }-x_{*}\parallel )}{1-{\mu }_0(\parallel x_{\sigma }-x_{*}\parallel )},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill q_3\left(\tau \right)& =\frac{{\mu }_0\left(\tau \right)+{\mu }_0\left({\tilde{\psi }}_1\left(\tau \right)\tau \right)}{2\left|\alpha \right|\left(1-{\mu }_0\left(\tau \right)\right)}+\frac{1}{2{\alpha }^2}{\left(\frac{{\mu }_0\left(\tau \right)+{\mu }_0\left({\tilde{\psi }}_1\left(\tau \right)\tau \right)}{1-{\mu }_0\left(\tau \right)}\right)}^2\hfill \\ \hfill & +\frac{|2-3\alpha |}{12{\alpha }^2}\left[\frac{{\mu }_0\left(\tau \right)+{\mu }_0\left({\tilde{\psi }}_1\left(\tau \right)\tau \right)}{1-{\mu }_0\left(\tau \right)}+\frac{{\mu }_0\left(\tau \right)+{\mu }_0\left(\tau +{\tilde{\psi }}_1\left(\tau \right)\tau \right)}{1-{\mu }_0\left(\tau \right)}\right].\hfill \end{array}$$

Hence, we define function ${\tilde{\psi }}_2$ by:

$${\tilde{\psi }}_2\left(\tau \right)=\frac{{\int }_0^1\mu \left((1-\theta )\tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}+\frac{q_3\left(\tau \right){\int }_0^1\mu \left(\theta \tau \right)d\theta }{1-{\mu }_0\left(\tau \right)}.$$

In view of the previous calculations for finding ${\psi }_3$ and the third substep of Schemes (31) and (32), we define $\tilde{\psi }:\Omega -0\to \Omega$:

$$\begin{array}{cc}\hfill {\tilde{\psi }}_3\left(\tau \right)=& [\frac{{\int }_0^1\mu \left((1-\theta ){\tilde{\psi }}_2\left(\tau \right)\tau \right)d\theta }{1-{\mu }_0\left({\tilde{\psi }}_2\left(\tau \right)\tau \right)}\hfill \\ \hfill & +\frac{\left[{\mu }_0\left(\tau \right)+{\mu }_0\left({\tilde{\psi }}_2\left(\tau \right)\tau \right)\right]{\int }_0^1{\mu }_1\left(\theta {\tilde{\psi }}_2\left(\tau \right)\tau \right)d\theta }{\left(1-{\mu }_0\left(\tau \right)\right)\left(1-{\mu }_0\left({\tilde{\psi }}_2\left(\tau \right)\tau \right)\right)}\hfill \\ \hfill & +\frac{\left[{\mu }_0\left(\tau \right)+{\mu }_0\left({\tilde{\psi }}_1\left(\tau \right)\tau \right)\right]{\int }_0^1{\mu }_1\left(\theta {\tilde{\psi }}_2\left(\tau \right)\tau \right)d\theta }{\left|\alpha \right|\left(1-q_2\left(\tau \right)\right)\left(1-{\mu }_0\left(\tau \right)\right)}]{\tilde{\psi }}_2\left(\tau \right).\hfill \end{array}$$

Moreover, to find radius $\rho$ for Schemes (31) and (32), we use equations involving ${\psi }_1,{\psi }_2,{\psi }_3$ and ${\overline{\psi }}_1,{\overline{\psi }}_2,{\overline{\psi }}_3$ as defined previously in this section. We compare these methods on the basis of the radii of convergence. In addition, we choose $ϵ={10}^{-100}$ as the error tolerance. The terminating criteria to solve nonlinear system or scalar equations are: $\left(i\right)\parallel x_{\sigma +1}-x_{\sigma }\parallel <ϵ,$ and $\left(ii\right)\parallel F\left(x_{\sigma }\right)\parallel <ϵ$.

The computations are performed with the package $Mathematica11$ with multiple precision arithmetic.

Example 1.

Following the example presented in introduction, for $x_{*}=1$, we can set:

$${\mu }_0\left(\tau \right)=\mu \left(\tau \right)=97\tau \mathrm{and}{\mu }_1\left(\tau \right)=2.$$

This way conditions $\left(A\right)$ are satisfied. Then, by solving equations ${\psi }_k\left(\tau \right)-1=0$, we find solutions ${\rho }_k$ and using (5), we determine ρ. Hence, the conclusions of Theorem 1 hold. In Table 1, we present the numerical values of radii ${\rho }_k$ and ρ for example (1).

Table 1. Radii for example (1).

Methods	$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\delta }$	${\mathit{\rho }}_1$	${\mathit{\rho }}_2$	${\mathit{\rho }}_3$	$\mathit{\rho }$
(31)	1	0	0	0.00687	0.00316	0.00224	0.00224
	$\frac{2}{3}$	0	1	0.00229	0.00201	0.00136	0.00136
	1	1	$-1$	0.00687	0.0100	0.000164	0.000164
	$\frac{2}{3}$	1	$-1$	0.00229	0.00976	0.0000732	0.0000732
(32)	1	-	$-1$	0.00687	0.00461	0.00322	0.00322
	$\frac{2}{3}$	-	$-1$	0.00229	0.00362	0.00244	0.00229

It is clear to say on the basis of above table that method (32) (for α = 1, δ = −1) has a larger radius of convergence as compared to the other mentioned methods. So, we concluded that it is better than other mentioned methods.

Example 2.

We chose a well-known nonlinear PDE problem of molecular interaction [12,33], which is given as follows:

$$v_{xx}+v_{yy}=v^2.$$

(33)

Subject to the following conditions

$$\begin{array}{cc}\hfill & v(x,0)=2x^2-x+1,\hfill \\ \hfill & v(x,1)=2,\hfill \\ \hfill & v(0,y)=2y^2-y+1,\hfill \\ \hfill & v(1,y)=2\hfill \end{array}$$

where $(x,y)\in [0,1]\times [0,1]$.

First of all, discretize the above PDE (33) by adopting the central divided difference:

$$\begin{array}{cc}\hfill v_{xx}& =\frac{v_{i+1,j}-2v_{i,j}+v_{i-1,j}}{a^2},\hfill \\ \hfill v_{yy}& =\frac{v_{i,j+1}-2v_{i,j}+v_{i,j-1}}{a^2}\hfill \end{array}$$

which further suggest the following system of nonlinear equations:

$$v_{i+1,j}-4v_{i,j}+v_{i-1,j}+v_{i,j+1}+v_{i,j-1}-a^2{v}_{i,j}^2=0,$$

where $i=1,2,3,\dots ,l-1,j=1,2,3,\dots ,l-1.$ We choose $l=13$ in order to obtain a system of nonlinear equations of order $12\times 12$ and $a=\frac{1}{l}$. The approximated solution is for $\mathbb{D}=\overline{S}\left(x_{*},\frac{1}{10}\right)$:

$$x_{*}={(0,0,0,0,\stackrel{144}{\dots },0)}^T.$$

Then, we get by the conditions $\left(A\right)$:

$${\mu }_0\left(\tau \right)=\mu \left(\tau \right)=0.10\tau and{\mu }_1\left(\tau \right)=2.$$

The above functions clearly satisfied the conditions $\left(A\right)$. Then, by solving equations ${\psi }_k\left(\tau \right)-1=0$, we find solutions ${\rho }_k$ and using (7), we determine ρ. Hence, the conclusions of Theorem 1 hold. We provide the computational values of radii of convergence for Example 2 in Table 2.

Table 2. Radii for example 2.

Methods	$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\delta }$	${\mathit{\rho }}_1$	${\mathit{\rho }}_2$	${\mathit{\rho }}_3$	$\mathit{\rho }$	${\mathit{x}}_0$	$\mathit{\sigma }$	$\mathit{\eta }$
(31)	1	0	0	6.667	3.066	2.172	2.172	${(2,2,2,\stackrel{144}{\dots },2)}^T$	4	4.9972
	$\frac{2}{3}$	0	1	2.222	1.949	1.321	1.321	${(1.2,1.2,1.2,\stackrel{144}{\dots },1.2)}^T$	3	4.9975
	1	1	$-1$	6.667	9.742	0.159	0.159	${(0.15,0.15,0.15,\stackrel{144}{\dots },0.15)}^T$	3	4.9990
	$\frac{2}{3}$	1	$-1$	2.222	9.463	0.0710	0.0710	${(0.06,0.06,0.06,\stackrel{144}{\dots },0.06)}^T$	3	5.0000
(32)	1	-	$-1$	6.667	4.473	3.120	3.120	$(2.5,2.5,2.5,\stackrel{144}{\dots },2.5)$	3	6.0000
	$\frac{2}{3}$	-	$-1$	2.223	3.517	2.363	2.223	${(2.1,2.1,2.1,\stackrel{144}{\dots },2.1)}^T$	3	6.0000

Since the method (32) (for α = 1, δ = −1) has a larger radius of convergence as compared to other special cases of (31) and (32). So, we conclude that method (32) has a higher number of convergent points as compared to other mentioned methods.

Example 3.

The kinematic synthesis problem for steering [11,31], is given as:

$$\begin{array}{cc}\hfill & {\left[E_i\left({\nu }_{2}sin\left({\eta }_i\right)-{\nu }_3\right)-A_i\left({\nu }_{2}sin\left({\phi }_i\right)-{\nu }_3\right)\right]}^2+{\left[A_i\left({\nu }_{2}cos\left({\phi }_i\right)+1\right)-A_i\left({\nu }_{2}cos\left({\eta }_i\right)-1\right)\right]}^2\hfill \\ \hfill & -{\left[{\nu }_1\left({\nu }_{2}sin\left({\eta }_i\right)-{\nu }_3\right)\left({\nu }_{2}cos\left({\phi }_i\right)+1\right)-{\nu }_1\left({\nu }_{2}cos\left({\eta }_i\right)-{\nu }_3\right)\left({\nu }_{2}sin\left({\phi }_i\right)-{\nu }_3\right)\right]}^2=0,\mathrm{for}i=1,2,3,\hfill \end{array}$$

where

$$E_i=-{\nu }_3{\nu }_2\left(sin\left({\phi }_i\right)-sin\left({\phi }_0\right)\right)-{\nu }_1\left({\nu }_{2}sin\left({\phi }_i\right)-{\nu }_3\right)+{\nu }_2\left(cos\left({\phi }_i\right)-cos\left({\phi }_0\right)\right),i=1,2,3$$

and

$$A_i=-{\nu }_3{\nu }_{2}sin\left({\eta }_i\right)+\left(-{\nu }_2\right)cos\left({\eta }_i\right)+\left({\nu }_3-{\nu }_1\right){\nu }_{2}sin\left({\eta }_0\right)+{\nu }_{2}cos\left({\eta }_0\right)+{\nu }_1{\nu }_3,i=1,2,3.$$

In Table 3, we present the values of ${\eta }_i$ and ${\phi }_i$ (in radians).

Table 3. Values of ${\eta }_i$ and ${\phi }_i$ (in radians) for Example (3).

i	${\mathit{\eta }}_{\mathit{i}}$	${\mathit{\phi }}_{\mathit{i}}$
0	$1.3954170041747090114$	$1.7461756494150842271$
1	$1.7444828545735749268$	$2.0364691127919609051$
2	$2.0656234369405315689$	$2.2390977868265978920$
3	$2.4600678478912500533$	$2.4600678409809344550$

The approximated solution is for $\mathbb{D}=\overline{S}\left(x_{*},\frac{1}{4}\right)$:

$$x_{*}={(0.9051567\dots ,0.6977417\dots ,0.6508335\dots )}^T.$$

Then, we get:

$${\mu }_0\left(\tau \right)=\mu \left(\tau \right)=8\tau ,and{\mu }_1\left(\tau \right)=2.$$

We can easily cross verify that the above functions satisfied the conditions $\left(A\right)$. Then, by solving equations ${\psi }_k\left(\tau \right)-1=0$, we find solutions ${\rho }_k$ and using (7), we determine ρ. Hence, the conclusions of Theorem 1 hold. We provide the computational radii of convergence for Example 3 in Table 4.

Table 4. Radii for Example (3).

Methods	$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\delta }$	${\mathit{\rho }}_1$	${\mathit{\rho }}_2$	${\mathit{\rho }}_3$	$\mathit{\rho }$	${\mathit{x}}_0$	$\mathit{\sigma }$	$\mathit{\eta }$
(31)	1	0	0	0.0833	0.0383	0.0271	0.0271	${(0.88,0.67,0.63)}^T$	4	5.0313
	$\frac{2}{3}$	0	1	0.0278	0.0244	0.0165	0.0165	${(0.89,0.68,0.64)}^T$	4	5.0322
	1	1	$-1$	0.0833	0.122	0.00198	0.00198	${(0.903,0.695,0.653)}^T$	4	4.9769
	$\frac{2}{3}$	1	$-1$	0.0278	0.118	0.000888	0.000888	${(0.9051,0.6977,0.6508)}^T$	3	5.0395
(32)	1	-	$-1$	0.0833	0.0559	0.0390	0.0390	${(0.87,0.66,0.62)}^T$	4	6.0001
	$\frac{2}{3}$	-	$-1$	0.0278	0.0440	0.0295	0.0278	${(0.88,0.67,0.63)}^T$	4	6.0249

We noticed from the above table that method (32) (for α = 1, δ = −1) has better choices of staring points as compared to other sub cases of (31) and (32). Since other sub cases of (31) and (32) have a smaller domain of convergence as contrast to method (32) (for α = 1, δ = −1). It also shows the consistent computational order of convergence. No doubt that a particular case of scheme (31) (for $\alpha =\frac{2}{3},\beta =1;\delta =-1$) is consuming the lowest number of iterations. This is quite obvious because the choice of initial is very close to the required solution.

Example 4.

We choose a prominent 2D Bratu problem [7,30], which is given by:

$$\begin{array}{cc}\hfill & u_{xx}+u_{tt}+C{e}^u=0,\mathrm{o}n\hfill \\ \hfill & A:(x,t)\in 0\le x\le 1,0\le t\le 1,\hfill \\ \hfill & \mathrm{a}longboundaryhypothesisu=0\mathrm{o}nA.\hfill \end{array}$$

(34)

Let us assume that ${\Theta }_{i,j}=u(x_i,t_j)$ is a numerical result over the grid points of the mesh. In addition, we consider that ${\tau }_1$ and ${\tau }_2$ are the number of steps in the direction of x and t, respectively. Moreover, we choose that h and k are the respective step sizes in the direction of x and y, respectively. In order to find the solution of PDE (34), we adopt the following approach:

$$\begin{array}{c}\hfill u_{xx}(x_i,t_j)=\frac{{\Theta }_{i+1,j}-2{\Theta }_{i,j}+{\Theta }_{i-1,j}}{h^2},C=0.1,t\in [0,1],\end{array}$$

(35)

which further yields the succeeding SNE:

$$\begin{array}{c}\hfill {\Theta }_{i,j+1}+{\Theta }_{i,j-1}-{\Theta }_{i,j}+{\Theta }_{i+1,j}+{\Theta }_{i-1,j}+h^{2}Cexp\left({\Theta }_{i,j}\right)i=1,2,3,\dots ,{\tau }_1,j=1,2,3,\dots ,{\tau }_2.\end{array}$$

(36)

By choosing ${\tau }_1={\tau }_2=11,h=\frac{1}{11},$ and $C=0.1$, we get a large SNE of order $100\times 100$ which converges to the following required root:

$$x_{*}={\left(\begin{array}{r}\hfill 0.0011\dots ,0.0018\dots ,0.0022\dots ,0.0025\dots ,0.0026\dots ,0.0026\dots ,0.0025\dots ,0.0022\dots ,\\ \hfill 0.0018\dots ,0.0011\dots ,0.0018\dots ,0.0030\dots ,0.0038\dots ,0.0043\dots ,0.0046\dots ,0.0046\dots ,\\ \hfill 0.0043\dots ,0.0038\dots ,0.0030\dots ,0.0018\dots ,0.0022\dots ,0.0038\dots ,0.0049\dots ,0.0056\dots ,\\ \hfill 0.0059\dots ,0.0059\dots ,0.0056\dots ,0.0049\dots ,0.0038\dots ,0.0022\dots ,0.0025\dots ,0.0043\dots ,\\ \hfill 0.0056\dots ,0.0064\dots ,0.0068\dots ,0.0068\dots ,0.0064\dots ,0.0056\dots ,0.0043\dots ,0.0025\dots ,\\ \hfill 0.0026\dots ,0.0046\dots ,0.0059\dots ,0.0068\dots ,0.0072\dots ,0.0072\dots ,0.0068\dots ,0.0059\dots ,\\ \hfill 0.0046\dots ,0.0026\dots ,0.0026\dots ,0.0046\dots ,0.0059\dots ,0.0068\dots ,0.0072\dots ,0.0072\dots ,\\ \hfill 0.0068\dots ,0.0059\dots ,0.0046\dots ,0.0026\dots ,0.0025\dots ,0.0043\dots ,0.0056\dots ,0.0064\dots ,\\ \hfill 0.0068\dots ,0.0068\dots ,0.0064\dots ,0.0056\dots ,0.0043\dots ,0.0025\dots ,0.0022\dots ,0.0038\dots ,\\ \hfill 0.0049\dots ,0.0056\dots ,0.0059\dots ,0.0059\dots ,0.0056\dots ,0.0049\dots ,0.0038\dots ,0.0022\dots ,\\ \hfill 0.0018\dots ,0.0030\dots ,0.0038\dots ,0.0043\dots ,0.0046\dots ,0.0046\dots ,0.0043\dots ,0.0038\dots ,\\ \hfill 0.0030\dots ,0.0018\dots ,0.0011\dots ,0.0018\dots ,0.0022\dots ,0.0025\dots ,0.0026\dots ,0.0026\dots ,\\ \hfill 0.0025\dots ,0.0022\dots ,0.0018\dots ,0.0011\dots \end{array}\right)}^T$$

the column vector. Choose ${\mathbb{T}}_1={\mathbb{T}}_2={\mathbb{R}}^{100}$ and $\mathbb{D}=\overline{S}\left(x_{*},\frac{1}{5}\right)$. Then, we have:

$${\mu }_0\left(\tau \right)=\mu \left(\tau \right)=7\tau \mathrm{and}{\mu }_1\left(\tau \right)=2.$$

This way conditions $\left(A\right)$ are satisfied. Then, by solving equations ${\psi }_k\left(\tau \right)-1=0$, we find solutions ${\rho }_k$ and using (7), we determine ρ. Hence, the conclusions of Theorem 1 hold. The computational results are depicted in Table 5.

Table 5. Radii of convergence for Example (4).

Methods	$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\delta }$	${\mathit{\rho }}_1$	${\mathit{\rho }}_2$	${\mathit{\rho }}_3$	$\mathit{\rho }$	${\mathit{x}}_0$	$\mathit{\sigma }$	$\mathit{\eta }$
(31)	1	0	0	0.0952	0.0438	0.0310	0.0310	${(0.003,0.003,0.003,\stackrel{100}{\dots },0.003)}^T$	2	4.9978
	$\frac{2}{3}$	0	1	0.0317	0.0278	0.0189	0.0189	${(0.002,0.002,0.002,\stackrel{100}{\dots },0.002)}^T$	2	4.9913
	1	1	$-1$	0.0952	0.139	0.00226	0.00226	${(0.002,0.002,0.002,\stackrel{100}{\dots },0.002)}^T$	2	4.9937
	$\frac{2}{3}$	1	$-1$	0.0317	0.135	0.0010	0.0010	${(0.0001,0.0001,0.0001,\stackrel{100}{\dots },0.0001)}^T$	2	4.9893
(32)	1	-	$-1$	0.0952	0.0639	0.0446	0.0446	${(0.004,0.004,0.004,\stackrel{100}{\dots },0.004)}^T$	2	6.0055
	$\frac{2}{3}$	-	$-1$	0.0317	0.0502	0.0338	0.0317	${(0.003,0.003,0.003,\stackrel{100}{\dots },0.003)}^T$	2	5.9853

It is straightforward to say on the basis of above table that method (32) (for α = 1, δ = −1) has a larger domain of convergence in contrast to other particulars cases of (31) and (32). It also consumes the same number of iterations as compared to other mentioned methods.

Remark 2.

We have observed from Examples (1)–(4) that method (32) (for $\alpha =1,\delta =-1$) has a larger radius of convergence as compared to other particulars cases of (31) and (32). So, we deduce that this particular case is better than other particulars cases of (31) and (32) in terms of convergent points and domain of convergence. It also shows the consistent computational order of convergence.

4. Conclusions

A unified local convergence is presented for a family of higher order Jarratt-type schemes on Banach spaces. Our analysis uses only the derivative on these schemes in contrast to other approaches using $\lambda +3$ and $(2j+1)$ order derivatives for Schemes (2) and (3), respectively (not on these schemes). Hence, the applicability of these schemes is extended. Moreover, our analysis gives computable error distances and uniqueness of the solution answers. This was not done in the earlier work [33]. Our idea provides a new way of looking at iterative schemes. So, it can extend the applicability of this and other schemes [3,4,5,6,7,8,9,10,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. Finally, numerical experiments are conducted to solve problems from chemistry and other disciplines of applied sciences. Notice in particular that many problems from the micro-world are symmetric.