Some High-Order Iterative Methods for Nonlinear Models Originating from Real Life Problems

Abstract

We develop a sixth order Steffensen-type method with one parameter in order to solve systems of equations. Our study’s novelty lies in the fact that two types of local convergence are established under weak conditions, including computable error bounds and uniqueness of the results. The performance of our methods is discussed and compared to other schemes using similar information. Finally, very large systems of equations ($100\times 100$ and $200\times 200$) are solved in order to test the theoretical results and compare them favorably to earlier works.

1. Introduction

A plenty of problems from Biology, Chemistry, Economics, Engineering, Mathematics, and Physics are converted to a mathematical expression of the following form

$$F\left(u\right)=0.$$

(1)

Here, $F:\mathsf{\Omega }\subset \mathbb{B}\to \mathbb{B}$, is differentiable, $\mathbb{B}$ is a Banach space and $\mathsf{\Omega }$ is nonempty and open. Closed form solutions are rarely found, so iterative methods [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] are used converging to the solution $u_{*}$.

In particular, we propose the following new scheme

$$\begin{array}{cc}\hfill y_p& =u_p-{\left[u_p+F\left(u_p\right),u_p;F\right]}^{-1}F\left(u_p\right)\hfill \\ \hfill z_p& =u_p-\lambda {\left[u_p+F\left(u_p\right),u_p;F\right]}^{-1}\left(F\left(u_p\right)+F\left(y_p\right)\right)-(1-\lambda ){\left[u_p,y_p;F\right]}^{-1}F\left(u_p\right)\hfill \\ \hfill u_{p+1}& =z_p-{\left[z_p+F\left(z_p\right),z_p;F\right]}^{-1}F\left(z_p\right),\hfill \end{array}$$

(2)

$u_0\in \mathsf{\Omega }$ is an initial point and $\lambda \in \mathbb{R}$ is a free parameter. In addition to this, $[·,·;F]:\mathsf{\Omega }\times \mathsf{\Omega }\to \ell (\mathbb{B},\mathbb{B})$ is a divided difference of order one.

We shall present two convergence analyses. Later, we present the advantages over other methods using similar information.

2. Local Convergence Analysis I

We assume that $\mathbb{B}=\mathbb{R}$. We use method (2) with standard Taylor expansions [9] for studying local convergence.

Theorem 1.

Suppose that mapping F is s sufficient differentiable on Ω, with $u_{*}\in \mathsf{\Omega }$, a simple zero of F. We also consider that the inverse of F, $F^{\prime }{\left(u_{*}\right)}^{-1}\in \ell (\mathbb{B},\mathbb{B})$. Then, ${\displaystyle \underset{p\to \infty }{lim}{u}_p=u_{*}}$ provided that $u_0$ is close enough to $u_{*}$. Moreover, the convergence order is six.

Proof. 

Set ${ϵ}_p=u_p-u_{*}$ and $Q_p=\frac{F^{\prime }\left(u_{*}\right)}{p!}$, where ${\left({ϵ}_p\right)}^{\gamma }={({ϵ}_1,{ϵ}_2,\cdots ,{ϵ}_k)}^{\gamma }$, ${ϵ}_p\in {\mathbb{R}}^p$. We shall use some Taylor series expansions, first for $F\left(u_p\right)$ and $F\left(u_p+F\left(u_p\right)\right)$:

$$F\left(u_p\right)=Q_1{ϵ}_p+Q_2{ϵ}_p^2+O\left({ϵ}_p^3\right)$$

(3)

and

$$F\left(u_p+F\left(u_p\right)\right)=(Q_1+Q_1^2){ϵ}_p+(3Q_1{Q}_2+Q_2+Q_1^2{Q}_2){ϵ}_p^2+O\left({ϵ}_p^3\right),$$

(4)

respectively.

By using the expressions (3) and (4) in the first substep of scheme (2), we have

$$\widehat{{ϵ}_p}=y_p-u_{*}=b_1{ϵ}_p^2+b_2{ϵ}_p^3+b_3{ϵ}_p^4+O\left(e_p^5\right),$$

(5)

where

$$\begin{array}{cc}\hfill & b_1=\frac{Q_2}{Q_1}+Q_2,\hfill \\ \hfill & b_2=\frac{2Q_3}{Q_1}-\frac{2Q_2^2}{Q_1}-\frac{2Q_2^3}{Q_1^2}+Q_1{Q}_3+3Q_3-Q_2^2\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & b_3=\frac{3Q_2^3}{Q_1}-2Q_1{Q}_2{Q}_3+Q_2^3+Q_1^2{Q}_4+4Q_1{Q}_4+6Q_4-7Q_2{Q}_3+\frac{3Q_4}{Q_1}+\frac{4Q_2^3}{Q_1^2}+\frac{5Q_3^3}{Q_1^2}-\frac{10Q_2{Q}_3}{Q_1}.\hfill \end{array}$$

Secondly, we expand $F\left(y_p\right)$

$$F\left(y_p\right)=Q_1\widehat{{ϵ}_p}+Q_2{\widehat{{ϵ}_p}}^2+O\left({\widehat{{ϵ}_p}}^3\right).$$

(6)

In view of (3)–(6), we get in the second substep of scheme (2)

$$\overline{{ϵ}_p}=u_{p+1}-u_{*}=z_p-u_{*}=b_4{ϵ}_p^3+O\left({ϵ}_p^4\right),$$

(7)

where

$$b_4=\frac{3Q_2^2}{Q_1}+\frac{2Q_2^2}{Q_1^2}+Q_2^2-\lambda \left(\frac{4Q_2^2}{Q_1}+Q_2^2+\frac{3Q_2^2}{Q_1^2}\right).$$

Thirdly, we need the expansions for $F\left(z_p\right)$ and $F(z_p+F\left(z_p\right))$

$$F\left(z_p\right)=Q_1\overline{e_p}+Q_2{\overline{e_p}}^2+O\left({\overline{e_p}}^3\right),$$

(8)

Hence, by (5) and (8), we get

$$F(z_p+F\left(z_p\right))=b_5\overline{e_p}+b_6{\overline{e_p}}^2+O\left({\overline{e_p}}^3\right),$$

(9)

leading together with the third substep of method (2) to

$$e_{p+1}=u_{p+1}-u_{*}=b_7{e}_p^6+O\left(e_p^7\right),$$

(10)

where

$$\begin{array}{cc}\hfill & b_5=Q_1+Q_1^2,\hfill \\ \hfill & b_6=3Q_1{Q}_2+Q_2+Q_1^2{Q}_2\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & b_7=\left(Q_2+\frac{Q_2}{Q_1}\right)\left[\frac{3Q_2^2}{Q_1}+\frac{2Q_2^2}{Q_1}+Q_2^2-\lambda \left(\frac{4Q_2^2}{Q_1}+\frac{3Q_2^2}{Q_1}+Q_2^2\right)\right].\hfill \end{array}$$

 ☐

According to Theorem 1, the applicability of method (2) is limited to mappings F with derivatives up to the seventh order.

Now, we choose $\mathbb{B}=\mathbb{R}$, $\mathsf{\Omega }=[-\frac{3}{2},\frac{1}{2}]$ and define a function f, as follows:

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

(11)

We have the following derivatives of function f

$$\begin{array}{cc}\hfill f^{\prime }\left(\xi \right)& =3{\xi }^{2}ln{\xi }^2+5{\xi }^4-4{\xi }^3+2{\xi }^2,\hfill \\ \hfill f^{\prime \prime }\left(\xi \right)& =12\xi ln{\xi }^2+20{\xi }^3-12{\xi }^2+10\xi ,\hfill \\ \hfill f^{\prime \prime \prime }\left(\xi \right)& =12ln{\xi }^2+60{\xi }^2-12\xi +22.\hfill \end{array}$$

However, $f^{\prime \prime \prime }\left(\xi \right)$ is not bounded on $\mathsf{\Omega }$, so Section 2, cannot be used. In this case, we have a more general alternative given in the up coming section.

3. Local Convergence Analysis II

Consider $a\ge 0$ and $b>0$. Let $w_0:[0,\infty )\times [0,\infty )\to [0,\infty )$ be a increasingly continuous map with $w_0(0,0)=0$.

Suppose equation

$$w_0(at,t)=1$$

(12)

has ${\rho }_1$ as the smallest positive zero. In addition, we assume that $w:[0,{\rho }_1)\times [0,{\rho }_1)\to [0,\infty )$ is a increasingly continuous map with $w(0,0)=0$.

Consider functions $g_1$ and $h_1$ defined on semi open interval $[0,{\rho }_1)$ as follow:

$$\begin{array}{cc}\hfill & g_1\left(t\right)=\frac{w(bt,t)}{1-w_0(at,t)},\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & h_1\left(t\right)=g_1\left(t\right)-1.\hfill \end{array}$$

By these definitions, we have $h_1\left(0\right)=-1$ and $h_1\left(t\right)\to \infty$ as $t\to {\rho }_1^{-}$. Subsequently, the intermediate value theorem assures that function $h_1$ has minimum one solution in $(0,{\rho }_1)$. Let $r_1$ be the minimal such zero.

The expression

$$w_0(t,g_1\left(t\right)t)=1$$

(13)

has the smallest positive zero ${\rho }_2$. Set ${\rho }_3=min\{{\rho }_1,{\rho }_2\}$.

We construe the functions $g_2$ and $h_2$ on interval $[0,{\rho }_3)$ in the following way

$$\begin{array}{cc}\hfill & g_2\left(t\right)=g_1\left(t\right)+\frac{b|1-\lambda |w\left(bt,\left(1+g_1\left(t\right)\right)t\right)g_1\left(t\right)}{\left(1-w_0(at,t)\right)\left(1-w_0\left(t,g_1\left(t\right)t\right)\right)},\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & h_2\left(t\right)=g_2\left(t\right)-1.\hfill \end{array}$$

We yield $h_2\left(0\right)=-1$ and $h_2\left(t\right)\to \infty$ since $t\to {\rho }_3^{-}$. The $r_2$ stand for the minimal such zero of function $h_2$ on $(0,{\rho }_3)$.

The equation

$$w_0\left(a{g}_2\left(t\right)t,g_2\left(t\right)t\right)=1$$

(14)

has ${\rho }_4$ as the smallest positive solution. Set $\rho =min\{{\rho }_3,{\rho }_4\}$. Define functions $g_3$ and $h_3$ on $[0,\rho )$ as

$$\begin{array}{cc}\hfill & g_3\left(t\right)=\frac{w\left(b{g}_2\left(t\right)t,g_2\left(t\right)t\right)g_2\left(t\right)}{1-w_0\left(a{g}_2\left(t\right)t,g_2\left(t\right)t\right)},\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & h_3\left(t\right)=g_3\left(t\right)-1.\hfill \end{array}$$

We obtain $h_3\left(0\right)=-1$ and $h_3\left(t\right)\to \infty$ as $t\to {\rho }^{-}$. The $r_3$ imply the minimal zero of $h_3$ on $(0,\rho )$. Moreover, define

$$r=min\left\{r_i\right\},\mathrm{for}i=1,2,3.$$

(15)

Accordingly, we have

$$0\le w_0(at,t)<1,$$

(16)

$$0\le w_0\left(t,g_1\left(t\right)t\right)<1,$$

(17)

$$0\le w_0\left(a{g}_2\left(t\right)t,g_2\left(t\right)t\right)<1,$$

(18)

and

$$0\le g_i\left(t\right)<1,$$

(19)

for all $t\in [0,r).$

$S(v,c)$ denotes the open ball centered at $v\in \mathbb{B}$ and of radius $c>0$. By $\overline{S}(v,c)$, we denote the closure of $S(v,c)$

We use the following conditions $\left(A\right)$ in order to study the local convergence:

(a₁) 

$F:\mathsf{\Omega }\to \mathbb{B}$ is a differentiable operator in the Fréchet sense, $[·,·;F]:\mathsf{\Omega }\times \mathsf{\Omega }\to \ell (\mathbb{B},\mathbb{B})$ is a divided difference of order one. In addition to this, we assume that $u_{*}\in \mathsf{\Omega }$ is a simple zero of F. At last, the inverse of operator F, $F^{\prime }{\left(u_{*}\right)}^{-1}\in \ell (\mathbb{B},\mathbb{B})$.

(a₂) 

Let $w_0:[0,\infty ]\times [0,\infty )\to [0,\infty )$ be a increasingly continuous function with $w_0(0,0)=0$, parameters $a\ge 0$ and $b>0$, such that for each $u,y\in \mathsf{\Omega }$

$$\begin{array}{cc}\hfill & ∥F^{\prime }{\left(u_{*}\right)}^{-1}\left([u,y;F]-F^{\prime }\left(u_{*}\right)\right)∥\le w_0(\parallel u-u_{*}\parallel ,\parallel y-u_{*}\parallel ),\hfill \\ \hfill & \parallel I+[u,u_{*};F]\parallel \le a,\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & \parallel [u,u_{*};F]\parallel \le b.\hfill \end{array}$$

Set ${\mathsf{\Omega }}_0=\mathsf{\Omega }\cap S(u_{*},{\rho }_1)$, where ${\rho }_1$ exists and is given by (12).

(a₃) 

We assume that $w:[0,{\rho }_1)\times [0,{\rho }_1)\to [0,\infty )$ is a increasingly continuous $\forall x,y,\zeta ,\eta \in {\mathsf{\Omega }}_0$

$$∥F^{\prime }{\left(u_{*}\right)}^{-1}\left([u,y;F]-[\zeta ,\eta ;F]\right)∥\le w(\parallel u-\zeta \parallel ,\parallel y-\eta \parallel ).$$

(a₄) 

$\overline{S}(u_{*},R)\subseteq \mathsf{\Omega }$, where $R=max\{r,ar,br\}$, r is defined in (15) and ${\rho }_2,{\rho }_4$ exist and are given by (13) and (13), respectively.

(a₅) 

There exists $\overline{r}\ge r$, such that

$$w_0(0,\overline{r})<1\mathrm{or}{w}_0(\overline{r},0)<1.$$

Set ${\mathsf{\Omega }}_1=\mathsf{\Omega }\cap \overline{S}(u_{*},\overline{r})$.

Theorem 2.

Under the hypotheses $\left(A\right)$ further consider that $u_0\in S(u_{*},r)-\left\{u_{*}\right\}$. Accordingly, the proceeding assertions hold

$$\left\{u_p\right\}\subseteq S(u_{*},r),$$

(20)

$$\underset{p\to \infty }{lim}\left\{u_p\right\}=u_{*},$$

(21)

$$\parallel y_p-u_{*}\parallel \le g_1(\parallel u_p-u_{*}\parallel )\parallel u_p-u_{*}\parallel \le \parallel u_p-u_{*}\parallel <r,$$

(22)

$$\parallel z_p-u_{*}\parallel \le g_2(\parallel u_p-u_{*}\parallel )\parallel u_p-u_{*}\parallel \le \parallel u_p-u_{*}\parallel ,$$

(23)

and

$$\parallel u_{p+1}-u_{*}\parallel \le g_3(\parallel u_p-u_{*}\parallel )\parallel u_p-u_{*}\parallel \le \parallel u_p-u_{*}\parallel .$$

(24)

In addition, the $u_{*}$ is the unique solution of $F\left(u\right)=0$ in the set ${\mathsf{\Omega }}_1$ mentioned in hypothesis $\left(a_5\right)$.

Proof. 

We first show items (20)–(24) by adopting mathematical induction. Because $p\in S(u_{*},r)-\left\{u_{*}\right\}$ hold and by condition $\left(a_2\right)$, we have

$$\begin{array}{cc}\hfill \parallel p+F\left(p\right)-u_{*}\parallel & =\parallel (I+[p,u_{*};F])(p-u_{*})\parallel \hfill \\ \hfill & \le \parallel I+[p,u_{*};F]\parallel \parallel p-u_{*}\parallel \hfill \\ \hfill & \le a\parallel p-u_{*}\parallel \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \parallel F\left(p\right)\parallel & =\parallel F\left(p\right)-F\left(u_{*}\right)\parallel \hfill \\ \hfill & \le \parallel [p,u_{*};F](p-u_{*})\parallel \hfill \\ \hfill & \le \parallel [p,u_{*};F]\parallel \parallel p-u_{*}\parallel \hfill \\ \hfill & \le b\parallel p-u_{*}\parallel \hfill \end{array}$$

so $p+F\left(p\right)-u_{*}$ and $F\left(p\right)$ belong in $\overline{S}(u_{*},R)$. Afterwards, for $u,y,\in S(u_{*},r)-\left\{u_{*}\right\}$, and

$$\begin{array}{cc}\hfill \parallel F^{\prime }{\left(u_{*}\right)}^{-1}\left([u,y;F]-F^{\prime }\left(u_{*}\right)\right)\parallel & \le w_0\left(\parallel u-u_{*}\parallel ,\parallel y-u_{*}\parallel \right)\hfill \\ \hfill & \le w_0(r,r)<1,\hfill \end{array}$$

so the Banach lemma on invertible operators [3,4,5,12] gives ${[u,y;F]}^{-1}\subset \ell (\mathbb{B},\mathbb{B})$, and

$$\parallel {[u,y;F]}^{-1}{F}^{\prime }\left(u_{*}\right)\parallel \le \frac{1}{1-w_0\left(\parallel u_0-u_{*}\parallel ,\parallel y-u_{*}\parallel \right)}.$$

(25)

It also follows that $y_0$ is defined.

Adopting (15), (16), (19) (for $i=1$), $\left(a_2\right),\left(a_3\right)$, (25) and $y_0$, we get

$$\begin{array}{cc}\hfill \parallel y_0-u_{*}\parallel & =\parallel u_0-u_{*}-{[u_0+F\left(u_0\right),u_0;F]}^{-1}F\left(u_0\right)\parallel \hfill \\ \hfill & =\parallel {[u_0+F\left(u_0\right),u_0;F]}^{-1}\left([u_0+F\left(u_0\right),u_0;F]-[u_0,u_{*};F](u_0-u_{*})\right)\parallel \hfill \\ \hfill & \le \parallel {[u_0+F\left(u_0\right),u_0;F]}^{-1}{F}^{\prime }\left(u_{*}\right)\parallel \parallel F^{\prime }{\left(u_{*}\right)}^{-1}\left([u_0+F\left(u_0\right),u_0;F]-[u_0,u_{*};F]\right)\parallel \parallel u_0-u_{*}\parallel \hfill \\ \hfill & \le \frac{w\left(\parallel F\left(u_{*}\right)\parallel ,\parallel u_0-u_{*}\parallel \right)\parallel u_0-u_{*}\parallel }{1-w_0(a\parallel u_0-u_{*}\parallel ,\parallel u_0-u_{*}\parallel )}\hfill \\ \hfill & \le \frac{w\left(b\parallel u_0-u_{*}\parallel ,\parallel u_0-u_{*}\parallel \right)\parallel u_0-u_{*}\parallel }{1-w_0(a\parallel u_0-u_{*}\parallel ,\parallel u_0-u_{*}\parallel )}\hfill \\ \hfill & =g_1(\parallel u_0-u_{*}\parallel )|\parallel u_0-u_{*}\parallel <\parallel u_0-u_{*}\parallel <r,\hfill \end{array}$$

(26)

so $y_0\in S(u_{*},r)$ (for $y_0\ne u_{*}$) and (22) holds for $n=0$.

$$\begin{array}{cc}\hfill \parallel F^{\prime }\left(u_{*}\right)\left([u_0,y_0;F]-F^{\prime }\left(u_{*}\right)\right)\parallel & \le w_0\left(\parallel u_0-u_{*}\parallel ,\parallel y_0-u_{*}\parallel \right)\hfill \\ \hfill & \le w_0\left(\parallel u_0-u_{*}\parallel ),g_1(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \right)\hfill \\ \hfill & \le w_0\left(r,g_1\left(r\right)r\right)<1,\hfill \end{array}$$

so ${[u_0,y_0;F]}^{-1}\in \ell (\mathbb{B},\mathbb{B})$ and

$$\parallel {[u_0,y_0;F]}^{-1}{F}^{\prime }\left(u_{*}\right)\parallel \le \frac{1}{1-w_0\left(\parallel u_0-u_{*}\parallel ,g_1(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \right)}.$$

(27)

It also follows that $z_0$ is well defined by the second substep of method (2) for $n=0$. In particular, we have

$$\begin{array}{cc}\hfill z_0& =u_0-\lambda {[u_0+F\left(u_0\right),u_0;F]}^{-1}\left(F\left(u_0\right)+F\left(y_0\right)\right)-(1-\lambda ){[u_0+F\left(u_0\right),u_0;F]}^{-1}F\left(u_0\right)\hfill \\ \hfill & =\left(u_0-{[u_0+F\left(u_0\right),u_0;F]}^{-1}F\left(u_0\right)\right)+{[u_0+F\left(u_0\right),u_0;F]}^{-1}F\left(u_0\right)-\lambda {[u_0+F\left(u_0\right),u_0;F]}^{-1}F\left(u_0\right)\hfill \\ \hfill & -(1-\lambda ){[u_0,y_0;F]}^{-1}F\left(u_0\right)-\lambda {[u_0+F\left(u_0\right),u_0;F]}^{-1}F\left(y_0\right)\hfill \\ \hfill & =y_0+(1-\lambda )\left({[u_0+F\left(u_0\right),u_0;F]}^{-1}-{[u_0,y_0,F]}^{-1}\right)F\left(u_0\right)-\lambda {[u_0+F\left(u_0\right),u_0;F]}^{-1}F\left(y_0\right)\hfill \end{array}$$

(28)

Next, by (15), (19) (for $i=2$) and (25)–(28), we get, in turn, that

$$\begin{array}{cc}\hfill \parallel z_0-u_0\parallel & \le \parallel y_0-u_{*}\parallel +|1-\lambda |\parallel {[u_0+F\left(u_0\right),u_0;F]}^{-1}{F}^{\prime }\left(u_{*}\right)\parallel \parallel F^{\prime }{\left(u_{*}\right)}^{-1}\left([u_0,y_0;F]-[u_0+F\left(u_0\right),u_0;F]\right)\parallel \hfill \\ \hfill & \times \frac{\parallel {[u_0,y_0;F]}^{-1}{F}^{\prime }\left(u_{*}\right)\parallel \parallel F^{\prime }{\left(u_{*}\right)}^{-1}[u_0,u_{*};F]\parallel \parallel u_0-u_{*}\parallel }{\left(1-w_0\left(a\parallel u_0-u_{*}\parallel ,\parallel u_0-u_{*}\parallel \right)\right)\left(1-w_0\left(\parallel u_0-u_{*}\parallel ,g_1(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \right)\right)}\hfill \\ \hfill & +\left|\lambda \right|\parallel {[u_0+F\left(u_0\right),u_0;F]}^{-1}{F}^{\prime }\left(u_{*}\right)\parallel \parallel F^{\prime }{\left(u_{*}\right)}^{-1}[y_0,u_{*};F]\parallel \parallel y_0-u_{*}\parallel \hfill \\ \hfill & \le [g_1(\parallel u_0-u_{*}\parallel )+\frac{b|1-\lambda |w\left(b\parallel u_0-u_{*}\parallel ,(1+g_1(\parallel u_0-u_{*}\parallel \left)\right)\parallel u_0-u_{*}\parallel \right)}{\left(1-w_0\left(a\parallel u_0-u_{*}\parallel ,\parallel u_0-u_{*}\parallel \right)\right)\left(1-w_0\left(\parallel u_0-u_{*}\parallel ,g_1(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \right)\right)}\hfill \\ \hfill & +\left|\lambda \right|\frac{b{g}_1(\parallel u_0-u_{*}\parallel )}{1-w_0(a\parallel u_0-u_{*}\parallel ,\parallel u_0-u_{*}\parallel )}]\parallel u_0-u_{*}\parallel \hfill \\ \hfill & =g_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \le \parallel u_0-u_{*}\parallel ,\hfill \end{array}$$

(29)

so $z_0\in S(u_{*},r)$ (for $z_0\ne u_{*}$ ) and (23) holds for $p=0$.

We have by (15), (18) and (29)

$$\begin{array}{cc}\hfill ∥F^{\prime }{\left(u_{*}\right)}^{-1}\left([z_0+F\left(u_0\right),z_0;F]-F^{\prime }\left(u_{*}\right)\right)∥& \le w_0\left(b\parallel z_0-u_{*}\parallel ,\parallel z_0-u_{*}\parallel \right)\hfill \\ \hfill & \le w_0\left(b{g}_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel ,g_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \right)\hfill \\ \hfill & \le w_0\left(b{g}_2\left(r\right)r,g_2\left(r\right)r\right)<1\hfill \end{array}$$

Accordingly, ${[z_0+F\left(z_0\right),z_0;F]}^{-1}\in \ell (\mathbb{B},\mathbb{B})$ and

$$\parallel {[z_0+F\left(z_0\right),z_0;F]}^{-1}{F}^{\prime }\left(u_{*}\right)\parallel \le \frac{1}{1-w_0\left(b{g}_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel ,g_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \right)}.$$

(30)

It also follows that $u_1$ is well defined by (30) and the last substep of method (2) for $n=0$. Then, as in (25) and (26) (for $z=3$) and (30), we obtain in turn

$$\begin{array}{cc}\hfill \parallel u_1-u_{*}\parallel & \le \frac{w(b\parallel z_0-u_{*}\parallel ,\parallel z_0-u_{*}\parallel )\parallel z_0-u_{*}\parallel }{1-w_0(a\parallel z_0-u_{*}\parallel ,\parallel z_0-u_{*}\parallel )}\hfill \\ \hfill & \le \frac{w\left(b{g}_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel ,g_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \right)g_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel }{1-w_0\left(a{g}_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel ,g_2(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \right)}\hfill \\ \hfill & =g_3(\parallel u_0-u_{*}\parallel )\parallel u_0-u_{*}\parallel \le \parallel u_0-u_{*}\parallel ,\hfill \end{array}$$

(31)

so, $u_1\in S(u_{*},r)$ (for $u_1\ne u_{*}$) and (24) holds for $n=0$. Subsequently, substituting $u_0$, $y_0,z_0$, $u_1$ by $u_m$, $y_m,z_m$, $u_{m+1}$, respectively. Hence, the induction for (30) and (22)–(24) is complete. Using the estimation

$$\parallel u_{m+1}-u_{*}\parallel <\alpha \parallel u_m-u_{*}\parallel <r,$$

(32)

where $\alpha =g_3(\parallel u_0-u_{*}\parallel )\in [0,1]$, we deduce that ${\displaystyle \underset{m\to \infty }{lim}{u}_m=u_{*}}$ and $u_{m+1}\in S(u_{*},r)$.

Finally, we want to illustrate that the required solution is unique. Therefore, let $T=[u_{*},y_{*};F]$ for $y_{*}\in {\mathsf{\Omega }}_1$, so that $F\left(y_{*}\right)=0$. Then, by $\left(a_2\right)$ and $\left(a_5\right)$, we get

$$\begin{array}{cc}\hfill \parallel F^{\prime }{\left(u_{*}\right)}^{-1}(T-F^{\prime }\left(u_{*}\right))\parallel & \le w_0(0,\parallel u_{*}-y_{*}\parallel )\hfill \\ \hfill & \le w_0(0,\overline{r})<1,\hfill \end{array}$$

so $T^{-1}\in \ell (\mathbb{B},\mathbb{B})$. Finally, $u_{*}=y_{*}$ is deduced from $0=F\left(u_{*}\right)-F\left(y_{*}\right)=T(u_{*}-y_{*})$. ☐

Remark 1.

Another way of defining functions $g_i,h_i$ and radii $r_i$ is as follows:

Let $\alpha =max\{ı,a\},i=1,2,3$. Subsequently, as in (12)–(18), we shall have instead:

Suppose that equation

$$w_0(\alpha t,t)=1$$

(33)

has a smallest positive solution ${\overline{\rho }}_1$. Let $\overline{w}:[0,{\overline{\rho }}_1]\times [0,{\overline{\rho }}_1]$ be a increasingly continuous function with $\overline{w}(0,0)=0$.

Let functions ${\overline{g}}_1$ and ${\overline{h}}_1$ be defined in the interval $[0,{\overline{\rho }}_1]$ by

$${\overline{g}}_1\left(t\right)=\frac{\overline{w}(bt,t)}{1-w_0(\alpha t,t)}and{\overline{h}}_1\left(t\right)={\overline{g}}_1\left(t\right)-1.$$

The ${\overline{r}}_1$ stands for the smallest positive root of ${\overline{h}}_1\left(t\right)=0$ in $(0,{\overline{\rho }}_1)$. Moreover, define functions ${\overline{g}}_2,{\overline{g}}_3,{\overline{h}}_2$ and ${\overline{h}}_3$ on the closed interval $[0,{\overline{\rho }}_1]$, as follows:

$$\begin{array}{cc}\hfill & {\overline{g}}_2\left(t\right)={\overline{g}}_1\left(t\right)+\frac{b|1-\lambda |w\left(bt,\left(1+{\overline{g}}_1\left(t\right)\right)t\right){\overline{g}}_1\left(t\right)}{{\left(1-w_0(\alpha t,t)\right)}^2},\hfill \\ \hfill & {\overline{g}}_3\left(t\right)=\frac{w\left(b{\overline{g}}_2\left(t\right)t,{\overline{g}}_2\left(t\right)t\right){\overline{g}}_2\left(t\right)}{1-w_0(\alpha t,t)},\hfill \\ \hfill & {\overline{h}}_2\left(t\right)={\overline{g}}_2\left(t\right)-1\hfill \\ \hfill & and\hfill \\ \hfill & {\overline{h}}_3\left(t\right)={\overline{g}}_3\left(t\right)-1.\hfill \end{array}$$

The ${\overline{r}}_2$ and ${\overline{r}}_3$ serve as the minimal positive roots of ${\overline{h}}_2\left(t\right)=0$ and ${\overline{h}}_3\left(t\right)=0$ on closed interval $[0,{\overline{\rho }}_1]$, respectively. Subsequently, Theorem 2 can be written by using the "bar" conditions and functions, with $\overline{r}=min\left\{{\overline{r}}_i\right\}$.

Remark 2.

The convergence of method (2) to $u_{*}$ is established under the conditions of Theorem 1. However, the order convergence under the conditions of Theorem 2 can be established by using (COC) and (ACOC) (for the details, please see Section 5).

4. Numerical Examples

Here, we monitor the convergence conditions on three problems (1)–(3). We choose $[u,y;F]={\int }_0^1{F}^{\prime }(y+\theta (u-y))d\theta$ in the examples. We can confirm the verification the hypotheses of Theorem 2 for the given choices of the $‘‘w"$ functions and parameters a and b.

Example 1.

Here, we investigate the application of our results on Hammerstein integral equations (see [9], pp. 19–20) for $\mathbb{B}=C[0,1]$ as follows:

$$F\left(u\left(s_1\right)\right)=u\left(s_1\right)-\frac{1}{5}{\int }_0^{1}S(s_1,s_2)u{\left(s_2\right)}^{3}d{s}_2=0,u\in C[0,1],s_1,s_2\in [0,1],$$

(34)

where

$$S(s_1,s_2)=\left\{\begin{array}{c}s(1-s_2),s\le s_2,\hfill \\ (1-s)s_2,s_2\le s.\hfill \end{array}\right.$$

We use ${\displaystyle {\int }_0^1\varphi \left(t\right)dt\simeq \sum _{k=1}^8{w}_k\varphi \left(t_k\right)}$ in (34), where $t_k$ and $w_k$ are the abscissas and weights, respectively. Using $u\left(t_j\right)$ for $u_j(j=1,2,3,\dots ,8)$, leads to

$${\displaystyle 5u_i-5-\sum _{k=1}^8{a}_{jk}{u}_k^3=0,j=1,2,3\dots ,8,}$$

$$a_{jk}=\left\{\begin{array}{c}\hfill w_k{t}_k(1-t_j),k\le j,\\ \hfill w_k{t}_j(1-t_k),j<k.\end{array}\right.$$

The values of $t_k$ and $w_k$ when $k=8$, are illustrated in Table 1. Subsequently, we have

$$\begin{array}{c}{u}_{*}=(1.002096\dots ,1.009900\dots ,1.019727\dots ,1.026436\dots ,1.026436\dots ,\\ {1.019727\dots ,1.009900\dots ,1.002096\dots )}^T.\end{array}$$

Table 1. Abscissas and weights for k = 8.

j	${\mathit{t}}_{\mathit{j}}$	${\mathit{w}}_{\mathit{j}}$
1	$0.01985507175123188415821957\dots$	$0.05061426814518812957626567\dots$
2	$0.10166676129318663020422303\dots$	$0.11119051722668723527217800\dots$
3	$0.23723379504183550709113047\dots$	$0.15685332293894364366898110\dots$
4	$0.40828267875217509753026193\dots$	$0.18134189168918099148257522\dots$
5	$0.59171732124782490246973807$	$0.18134189168918099148257522\dots$
6	$0.76276620495816449290886952\dots$	$0.15685332293894364366898110\dots$
7	$0.89833323870681336979577696\dots$	$0.11119051722668723527217800\dots$
8	$0.98014492824876811584178043\dots$	$0.05061426814518812957626567\dots$

Accordingly, we set $w_0(s_1,s_2)=w(s_1,s_2)=\frac{3}{80}(s_1+s_2),a=\frac{163}{80}$ and $b=\frac{83}{80}$. The radii for Example 1 are listed in Table 2 and Table 3:

Table 2. Convergence radii for Example 1.

$\mathit{\lambda }$	${\mathit{r}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	r
0	5.25452	3.87208	4.09301	3.87208
0.5	5.25452	4.26006	4.42602	4.26006
1	5.25452	5.25452	5.25452	5.25452

Table 3. Convergence radii for Example 1 with bar functions.

$\mathit{\lambda }$	${\mathit{r}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	r
0	5.25452	3.67748	3.87626	3.67748
0.5	5.25452	4.07351	4.17413	4.07351
1	5.25452	5.25452	4.89162	4.89162

Example 2.

Here, we choose as integral equation [17,18], for $\mathbb{B}=C[0,1]$ as

$$\left[F\left(\mu \right)\right]\left({\gamma }_1\right)=\mu \left({\gamma }_1\right)-{\int }_0^{1}G({\gamma }_1,{\gamma }_2)\left(\mu {\left({\gamma }_2\right)}^{\frac{3}{2}}+\frac{\mu {\left({\gamma }_2\right)}^2}{2}\right)d{\gamma }_2=0,$$

(35)

where

$$G({\gamma }_1,{\gamma }_2)=\left\{\begin{array}{c}(1-{\gamma }_2){\gamma }_2,{\gamma }_2\le {\gamma }_1,\hfill \\ {\gamma }_1(1-{\gamma }_2),{\gamma }_1\le {\gamma }_2.\hfill \end{array}\right.$$

(36)

Because $\mathbb{B}=C[0,1]$ so, $F:C[0,1]\to C[0,1]$ is given as

$$\left[F\left(\mu \right)\right]\left({\gamma }_1\right)=\mu \left({\gamma }_1\right)-{\int }_0^{{\gamma }_1}G({\gamma }_1,{\gamma }_2)\left(\mu {\left({\gamma }_2\right)}^{\frac{3}{2}}+\frac{\mu {\left({\gamma }_2\right)}^2}{2}\right)d{\gamma }_2.$$

(37)

We get

$$∥{\int }_0^{{\gamma }_1}G({\gamma }_1,{\gamma }_2)d{\gamma }_2∥\le \frac{1}{8}.$$

(38)

Moreover,

$$\left[F^{\prime }\left(\mu \right)\eta \right]\left({\gamma }_1\right)=\eta \left({\gamma }_1\right)-{\int }_0^{{\gamma }_1}G({\gamma }_1,{\gamma }_2)\left(\frac{3}{2}\mu {\left({\gamma }_2\right)}^{\frac{1}{2}}+\mu \left({\gamma }_2\right)\right)\eta \left({\gamma }_2\right)d{\gamma }_2,$$

so ${\mu }_{*}\left({\gamma }_1\right)=0$, since $F^{\prime }\left({\mu }_{*}\left({\gamma }_1\right)\right)=I$,

$$∥F^{\prime }{\left({\mu }_{*}\right)}^{-1}\left(F^{\prime }\left(\mu \right)-F^{\prime }\left(\eta \right)\right)∥\le \frac{1}{8}\left(\frac{3}{2}{\parallel \mu -\eta \parallel }^{\frac{1}{2}}+\parallel \mu -\eta \parallel \right).$$

(39)

Hence, we have

$$w_0(s,t)=w(s,t)=\frac{1}{16}\left[\frac{3}{2}(\sqrt{s}+\sqrt{t})+s+t\right],a=\frac{53}{16},\mathrm{and}b=\frac{37}{16}.$$

Therefore, our results can be utilized even though $F^{\prime }$ is not bounded on Ω. The radii for Example 2 are given in Table 4.

Table 4. Convergence radii for Example 2 with bar functions.

$\mathit{\lambda }$	${\mathit{r}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	r
0	1.03137	0.502403	0.61211	0.502403
0.5	1.03137	0.61199	0.70738	0.61199
1	1.03137	1.03137	1.03137	1.03137

Example 3.

We assume the following differential equations

$$\begin{array}{cc}\hfill & q_1^{\prime }\left(\mu \right)-q_1\left(\mu \right)-1=0\hfill \\ \hfill & q_2^{\prime }\left(\eta \right)-(e-1)\eta -1=0\hfill \\ \hfill & q_3^{\prime }\left(\theta \right)-1=0\hfill \end{array}$$

(40)

characterizes the progress/movement of a molecule in 3D with $(\mu ,\eta ,\theta )\in \mathsf{\Omega }$ for $q_1\left(0\right)=q_2\left(0\right)=q_3\left(0\right)=0$. The required solution $v={(\mu ,\eta ,\theta )}^T$ describes to $K:=(q_1,q_2,q_3):\mathsf{\Omega }\to {\mathbb{R}}^3$ given as

$$K\left(v\right)={\left(e^{\mu }-1,\frac{e-1}{2}{\eta }^2+\eta ,\theta \right)}^T=0.$$

(41)

It follows from (41) that

$$K^{\prime }\left(v\right)=\left[\begin{array}{ccc}{e}^{\mu }& 0& 0\\ 0& (e-1)\eta +1& 0\\ 0& 0& 1\end{array}\right],$$

which yields

$$w_0(s,t)=\frac{1}{2}(e-1)(s+t),w(s,t)=\frac{1}{2}e(s+t),a=\frac{1}{2}(e+3),\mathrm{and}b=\frac{1}{2}(e+1).$$

We depicted the radii of Example 3 in Table 5 and Table 6.

Table 5. Convergence radii for Example 3.

$\mathit{\lambda }$	${\mathit{r}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	r
0	0.1388596	0.921375	0.083356	0.083356
0.5	0.1388596	0.921375	0.086297	0.086297
1	0.1388596	0.1388596	0.1388596	0.1388596

Table 6. Convergence radii for Example 3 with bar functions.

$\mathit{\lambda }$	${\mathit{r}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	r
0	0.1388596	0.0487471	0.1229551	0.0487471
0.5	0.1388596	0.0487471	0.1377815	0.0487471
1	0.1388596	0.1388596	0.1380780	0.1380780

Example 4.

By the example of Section 2, for $\mathsf{\Omega }=\mathbb{B}=\mathbb{R},f\left(\xi \right)=0$, we get

$$w_0(s,t)=w(s,t)=\frac{96.66297}{2}(s+t),a=\frac{5}{2},\mathrm{and}b=\frac{3}{2}.$$

The radii of method (2) for Example 4 are listed in Table 7 and Table 8.

Table 7. Convergence radii for Example 4.

$\mathit{\lambda }$	${\mathit{r}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	r
0	0.00344841	0.00239612	0.00256623	0.00239612
0.5	0.00344841	0.00267769	0.00280807	0.00267769
1	0.00344841	0.00344841	0.00344841	0.00344841

Table 8. Convergence radii for Example 4 with bar functions.

$\mathit{\lambda }$	${\mathit{r}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	r
0	0.00344841	0.00225955	0.00246765	0.00225955
0.5	0.00344841	0.00225955	0.00246765	0.00225955
1	0.00344841	0.00344841	0.00334891	0.00344841

5. Applications with Large Systems

We choose $\lambda =0,$$\lambda =0.5$ and $\lambda =1$ in our scheme (2), called by $\left(PS1\right),\left(PS2\right)$ and $\left(PS3\right)$, respectively. Now, we compare our schemes with a 6th-order iterative methods suggested by Abbasbandy et al. [19] and Hueso et al. [20], among them we picked the methods (8) and (14–15) $\left(\mathrm{for}{t}_1=-\frac{9}{4}\mathrm{and}{s}_2=\frac{9}{8}\right)$, respectively, known as $\left(AS\right)$ and $\left(HS\right)$. Moreover, a comparison of them has been done with the 6th-order iterative methods given by Wang and Li [21], among their method we chose expression (6), denoted by and $\left(WS\right)$. At the last, we contrast (2) with sixth-order scheme given by Sharma and Arora [22], we pick expression (13), known as $\left(SM\right)$. The details of all the iterative expressions are given, as follows:

method $AS$:

$$\begin{array}{cc}\hfill y_j& =u_j-\frac{2}{3}{F}^{\prime }{\left(u_j\right)}^{-1}F\left(u_j\right),\hfill \\ \hfill z_j& =u_j-\left[I+\frac{21}{8}{F}^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)-\frac{9}{2}{\left(F^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)\right)}^2+\frac{15}{8}{\left(F^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)\right)}^3\right]F^{\prime }{\left(u_j\right)}^{-1}F\left(u_j\right),\hfill \\ \hfill u_{j+1}& =z_j-\left[3I-\frac{5}{2}{F}^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)+\frac{1}{2}{\left(F^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)\right)}^2\right]F^{\prime }{\left(u_j\right)}^{-1}F\left(z_j\right).\hfill \end{array}$$

(42)

scheme $HS$:

$$\begin{array}{cc}\hfill y_j& =u_j-F^{\prime }{\left(u_j\right)}^{-1}F\left(u_j\right),\hfill \\ \hfill H(u_j,y_j)& =F^{\prime }{\left(u_j\right)}^{-1}F\left(y_j\right),H(y_j,u_j)=F^{\prime }{\left(y_j\right)}^{-1}F\left(u_j\right),\hfill \\ \hfill G_s(u_j,y_j)& =s_{1}I+s_{2}H(y_j,u_j)+s_{3}H(u_j,y_j)+s_{4}H{(y_j,u_j)}^2,\hfill \\ \hfill z_j& =u_j-G_s(u_j,y_j)F^{\prime }{\left(u_j\right)}^{-1}F\left(u_j\right),\hfill \\ \hfill u_{j+1}& =z_j.\hfill \end{array}$$

(43)

where $s_1{s}_2,s_3$ and $s_4$ are real numbers.

terative method $WS$:

$$\begin{array}{cc}\hfill y_j& =u_j-F^{\prime }{\left(u_j\right)}^{-1}F\left(u_j\right),\hfill \\ \hfill z_j& =y_j-\left[2I-F^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)\right]F^{\prime }{\left(u_j\right)}^{-1}F\left(y_j\right),\hfill \\ \hfill u_{j+1}& =z_j-\left[2I-F^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)\right]F^{\prime }{\left(u_j\right)}^{-1}F\left(z_j\right).\hfill \end{array}$$

(44)

scheme $SM$:

$$\begin{array}{cc}\hfill y_j& =u_j-\frac{2}{3}{F}^{\prime }{\left(u_j\right)}^{-1}F\left(u_j\right),\hfill \\ \hfill z_j& =u_j-\left[pI+F^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)\left(qI+r{F}^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)\right)\right]F^{\prime }{\left(u_j\right)}^{-1}F\left(u_j\right),\hfill \\ \hfill u_{j+1}& =z_j-\left[\frac{5}{2}I-\frac{3}{2}{F}^{\prime }{\left(u_j\right)}^{-1}{F}^{\prime }\left(y_j\right)\right]F^{\prime }{\left(u_j\right)}^{-1}F\left(z_j\right),\hfill \end{array}$$

(45)

where $p=\frac{23}{8},q=-3$ and $r=\frac{9}{8}$.

The $\left(j\right)$, $(\parallel F\left(u_j\right)\parallel )$, $\parallel u_{j+1}-u_j\parallel$, and ${\rho }^{*}\approx \frac{log\left[\parallel u_{j+1}-u_j\parallel /\parallel u_j-u_{j-1}\parallel \right]}{log\left[\parallel u_j-u_{j-1}\parallel /\parallel u_{j-1}-u_{j-2}\parallel \right]}$ stands for index of iteration, absolute residual errors in the function F, error between two successive iterations and computational convergence order, receptively. There values are listed in Table 9, Table 10 and Table 11. Moreover, the quantity $\eta$ is the final obtained value of $\frac{\parallel u_{j+1}-u_j\parallel }{\parallel u_j-u_{j-1}{\parallel }^6}$.

Table 9. Comparisons of different methods on a Boundary value problem in Example 5.

Methods	j	$\parallel \mathit{F}\left({\mathit{u}}_{\mathit{j}}\right)\parallel$	$\parallel {\mathit{u}}_{\mathit{j}+1}-{\mathit{u}}_{\mathit{j}}\parallel$	${\mathit{\rho }}_{*}$	$\frac{\parallel {\mathit{u}}_{\mathit{j}+1}-{\mathit{u}}_{\mathit{j}}\parallel }{\parallel {\mathit{u}}_{\mathit{j}}-{\mathit{u}}_{\mathit{j}-1}{\parallel }^6}$
$AS$	1	$1.0(-4)$	$6.1(-4)$
	2	$8.7(-27)$	$2.8(-26)$		$5.133234733(-7)$
	3	$8.1(-161)$	$2.6(-160)$	$5.9985$	$5.920693970(-7)$
$HS$	1	$1.3(-4)$	$5.7(-4)$
	2	$8.1(-23)$	$2.8(-22)$		$8.252588019(-3)$
	3	$3.5(-114)$	$1.0(-113)$	$4.9954$	$2.013368332(+16)$
$WS$	1	$2.6(-4)$	$1.1(-3)$
	2	$1.1(-25)$	$3.1(-25)$		$1.977528884(-7)$
	3	$8.6(-155)$	$2.4(-154)$	$5.9957$	$2.448277731(-7)$
$SM$	1	$7.3(-5)$	$2.7(-3)$
	2	$8.2(-29)$	$2.7(-28)$		$7.804847473(-7)$
	3	$1.1(-172)$	$3.6(-172)$	$5.9973$	$9.053257416(-7)$
$PS1$	1	$4.9(-6)$	$1.6(-5)$
	2	$1.6(-38)$	$4.8(-38)$		$2.474537279(-9)$
	3	$9.3(-234)$	$2.7(-233)$	$6.0010$	$2.302596208(-9)$
$PS2$	1	$1.1(-5)$	$3.7(-5)$
	2	$3.7(-36)$	$1.1(-35)$		$4.513404180(-9)$
	3	$2.4(-219)$	$7.2(-219)$	$6.0013$	$4.108378955(-9)$
$PS3$	1	$1.9(-5)$	$6.5(-5)$
	2	$1.9(-34)$	$5.6(-34)$		$7.168046437(-9)$
	3	$6.6(-209)$	$1.9(-208)$	$6.0016$	$6.434316717(-9)$

Table 10. Comparisons of different methods on two-dimensional (2D) Bratu problem in Example 6.

Methods	j	$\parallel \mathit{F}\left({\mathit{u}}_{\mathit{j}}\right)\parallel$	$\parallel {\mathit{u}}_{\mathit{j}+1}-{\mathit{u}}_{\mathit{j}}\parallel$	${\mathit{\rho }}_{*}$	$\frac{\parallel {\mathit{u}}_{\mathit{j}+1}-{\mathit{u}}_{\mathit{j}}\parallel }{\parallel {\mathit{u}}_{\mathit{j}}-{\mathit{u}}_{\mathit{j}-1}{\parallel }^6}$
$AS$	1	$4.4(-15)$	$2.4(-14)$
	2	$6.9(-95)$	$3.5(-94)$		$1.428095547(-12)$
	3	$7.9(-574)$	$3.9(-573)$	$5.9994$	$1.973434769(-12)$
$HS$	1	$2.1(-13)$	$1.2(-12)$
	2	$2.1(-71)$	$1.2(-70)$		$7.368055345(-11)$
	3	$1.7(-361)$	$9.3(-361)$	$4.9997$	$3.495510769(+1)$
$WS$	1	$5.0(-19)$	$2.9(-18)$
	2		$1.7(-122)$	$1.0(-121)$	$1.754949400(-16)$
	3	$3.1(-743)$	$1.8(-742)$	$5.9999$	$1.666475363(-16)$
$SM$	1	$4.4(-15)$	$2.4(-14)$
	2	$7.1(-95)$	$3.6(-94)$		$1.433541371(-12)$
	3	$9.2(-574)$	$4.5(-573)$	$5.9994$	$1.433541371(-12)$
$PS1$	1	$9.1(-21)$	$5.3(-20)$
	2	$1.2(-134)$	$7.1(-134)$		$3.060974255(-18)$
	3	$6.9(-818)$	$4.0(-817)$	$6.0000$	$3.068006721(-18)$
$PS2$	1	$1.9(-20)$	$1.1(-19)$
	2	$1.7(-132)$	$1.0(-131)$		$6.095821945(-18)$
	3	$1.1(-804)$	$6.7(-804)$	$6.0000$	$6.105210728(-18)$
$PS3$	1	$3.1(-20)$	$1.8(-19)$
	2	$6.7(-131)$	$3.9(-130)$		$1.016575545(-17)$
	3	$6.3(-795)$	$3.7(-794)$	$6.0000$	$1.017779424(-17)$

Table 11. Comparisons of different methods on Example 7.

Methods	j	$\parallel \mathit{F}\left({\mathit{u}}_{\mathit{j}}\right)\parallel$	$\parallel {\mathit{u}}_{\mathit{j}+1}-{\mathit{u}}_{\mathit{j}}\parallel$	${\mathit{\rho }}_{*}$	$\frac{\parallel {\mathit{u}}_{\mathit{j}+1}-{\mathit{u}}_{\mathit{j}}\parallel }{\parallel {\mathit{u}}_{\mathit{j}}-{\mathit{u}}_{\mathit{j}-1}{\parallel }^6}$
$AS$	1	$5.2(-3)$	$1.7(-3)$
	2	$6.2(-21)$	$2.1(-21)$		$7.686036043(-5)$
	3	$1.7(-128)$	$5.8(-129)$	$6.0000$	$7.695242316(-5)$
$HS$	1	$2.3(-3)$	$7.7(-4)$
	2	$5.4(-20)$	$1.8(-20)$		$8.659247536(-2)$
	3	$3.9(-103)$	$1.3(-103)$	$5.0000$	$3.689254113(+15)$
$WS$	1	$3.5(-3)$	$1.2(-3)$
	2	$4.0(-22)$	$1.3(-22)$		$5.299207889(-5)$
	3	$8.4(-136)$	$2.8(-136)$	$6.0000$	$5.303300859(-5)$
$SM$	1	$3.0(-3)$	$1.0(-3)$
	2	$1.4(-22)$	$4.6(-23)$		$4.671758076(-5)$
	3	$1.3(-138)$	$4.3(-139)$	$6.0000$	$4.674761498(-5)$
$PS1$	1	$5.1(-3)$	$1.7(-3)$
	2	$8.4(-21)$	$2.8(-21)$		$1.130483172(-4)$
	3	$1.6(-127)$	$5.5(-128)$	$6.000$	$1.131370850(-4)$
$PS2$	1	$1.0(-1)$	$3.3(-2)$
	2	$4.0(-12)$	$1.3(-12)$		$9.906447117(-4)$
	3	$1.7(-74)$	$5.8(-75)$	$5.9989$	$1.018233765(-4)$
$PS3$	1	$3.3(-1)$	$1.1(-1)$
	2	$1.3(-8)$	$4.3(-9)$		$2.565472254(-3)$
	3	$5.6(-53)$	$1.9(-53)$	$5.9943$	$2.828427114(-3)$

The estimation of all the above parameters have been calculated by Mathematica-9. For minimizing the round-off errors, we have chosen multiple precision arithmetic with 1000 digits of mantissa. The term $b_1(\pm b_2)$ symbolizes the $b_1\times {10}^{(\pm b_2)}$ in all mentioned tables. We adopted the command "AbsoluteTiming[]" in order to calculate the CPU time. We run our programs three times and depicted the average CPU time in Table 12, also one can observe the times used for each iterative method, where we want to point out that for big size problems the method $PS1$ uses the minimum time, so it is being very competitive. The configuration of the used computer is given below:

Table 12. CPU time of different methods on Examples 5–7.

Methods	Example 5	Example 6	Example 7	Total Time	Average Time
$AS$	0.465330	210.079553	356.906591	567.451474	189.1504913
$HS$	0.583412	189.541919	366.511753	556.637084	185.5456947
$WS$	0.274193	128.377322	182.956711	311.608226	103.8694087
$SM$	1.130812	126.641140	401.627979	529.399931	176.4666437
$PS1$	0.101071	120.094370	52.204957	172.400398	57.46679933
$PS2$	0.100071	117.901198	52.146903	170.148172	56.71605733
$PS3$	0.100083	117.923227	51.972773	169.996083	56.665361

According to the CPU time, method $PS3$ is taking the lowest time for executing the results. All of the other schemes $AS,HS,SM$; and, $SM$ consuming at least double CPU timing as compare to our methods namely $PS1,PS2$ and $PS3$. So, we conclude that our methods provide results faster than the other existing methods.

Processor: Intel(R) Core(TM) i7-4790 CPU @ 3.60 GHz

Made: HP

RAM: 8:00 GB

System type: 64-bit-Operating System, x64-based processor.

Example 5.

Here, we deal with a boundary value problem from Ortega and Rheinboldt [9], given by

$$y^{\prime \prime }=\frac{y^3+6y^{\prime }+1}{2}-\frac{3}{2-x},y\left(0\right)=0,y\left(1\right)=1.$$

(46)

We assume

$$u_0=0<u_1<u_2<u_3<\cdots <u_p,where{u}_{p+1}=u_p+h,h=\frac{1}{p},$$

(47)

partition of the interval $[0,1]$ and $y_0=y\left(u_0\right)=0,y_1=y\left(u_1\right),\cdots ,y_{n-1}=y\left(u_{n-1}\right),y_p=y\left(u_p\right)=1$.

Now, we discretize expression (46) by adopting following numerical formula for derivatives

$$y_j^{\prime }=\frac{y_{j+1}-y_{j-1}}{2h},y_j^{\prime \prime }=\frac{y_{j-1}-2y_j+y_{j+1}}{h^2},j=1,2,\cdots ,p-1,$$

which leads to

$$y_{j+1}-2y_j+y_{j-1}-\frac{h^2}{2}{y}_j^3-\frac{3}{2}h(y_{k+1}-y_{k-1})-\frac{3}{2-u_j}{h}^2-\frac{1}{h^2}=0,j=1,2,\dots ,p-1,$$

$(p-1)\times (p-1)$ system of nonlinear equations.

For specific value of $p=7$, we have a $6\times 6$ system and the required solution is

$$u_{*}={\left(0.07654393\dots ,0.1658739\dots ,0.2715210\dots ,0.3984540\dots ,0.5538864\dots ,0.7486878\dots \right)}^T.$$

The computational estimations are listed in Table 9 on the basis of initial approximation $y_j^{\left(0\right)}={\left(\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2}\right)}^T$.

Example 6.

The classical 2D Bratu problem [23,24] is given by

$$\begin{array}{cc}\hfill & u_{\mu \mu }+u_{\theta \theta }+C{e}^u=0,\hfill \\ \hfill & \mathsf{\Omega }=\left\{(\mu ,\theta )\in 0\le \mu \le 1,0\le \theta \le 1\right\},\mathrm{w}ithboundaryhypothesesu=0\mathrm{o}n\mathsf{\Omega }.\hfill \end{array}$$

(48)

By adopting finite difference discretization, we can deduced the above PDE (48) to a nonlinear system. For this purpose, we denote ${\mathsf{\Delta }}_{i,j}=u({\mu }_i,{\theta }_j)$ as numerical solution at the grid points of the mesh. In addition to this, $M_1$ and $M_2$ stand for the number of steps in the directions of μ and θ, respectively. The h and k called as the respective step sizes in the directions of μ and θ. Adopt the following central difference formula to $u_{\mu \mu }$ and $u_{\theta \theta }$

$$\begin{array}{c}\hfill u_{\mu \mu }(u_i,{\theta }_j)=\frac{{\mathsf{\Delta }}_{i+1,j}-2{\mathsf{\Delta }}_{i,j}+{\mathsf{\Delta }}_{i-1,j}}{h^2},C=0.1,\theta \in [0,1],\end{array}$$

(49)

leads to us

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{i,j+1}+{\mathsf{\Delta }}_{i,j-1}-{\mathsf{\Delta }}_{i,j}+{\mathsf{\Delta }}_{i+1,j}+{\mathsf{\Delta }}_{i-1,j}+h^{2}Cexp\left({\mathsf{\Delta }}_{i,j}\right)i=1,2,3,\cdots ,M_1,j=1,2,3,\cdots ,M_2\end{array}$$

(50)

For obtaining a large system of $100\times 100$, we choose $M_1=M_2=11,C=0.1$ and $h=\frac{1}{11}$. The numerical results are listed in Table 10 based on the initial guess $u_0=0.1{\left(sin\left(\pi hi\right)sin\left(\pi hj\right)\right)}^T,i=j=10$.

Example 7.

Let us consider the following nonlinear system

$$F\left(x\right)=\left\{\begin{array}{c}{u}_j^2{u}_{j+1}-1=0,1\le j\le p-1,\hfill \\ x_p^2{u}_1-1=0.\hfill \end{array}\right.$$

(51)

For specific value $p=200$, we have $200\times 200$ system, and chose the following starting point

$$x^{\left(0\right)}={(1.25,1.25,1.25,\cdots ,1.25)}^T.$$

The $u_{*}={(1,1,1,\cdots ,1)}^T$ is the required solution of system 7. Table 11 provides the numerical results.

Remark 3.On the basis of Table 9, Table 10 and Table 11, we conclude that our methods namely $PS1,PS2$ and $PS3$ perform better in the contrast of existing schemes $AS,HS,SM$ and $SM$ on the basis of residual errors, errors between two consecutive iterations, and asymptotic error constant. In addition, our methods also demonstrate the stable computational order of convergence. Finally, we concluded that our methods not only perform better than existing methods in numerical results, but also take half of the CPU time in contrast to other existing methods (results can be easily found in Table 12).

6. Conclusions

We presented a new family of Steffensen-type methods with one parameter. The local convergence is studied in Section 2 while using Taylor expansion and derivative up to the order seven, when $\mathbb{B}={\mathbb{R}}^j$. To extend the suitability of these iterative methods, we only use hypotheses on the first derivative in Section 3 and Banach space valued operators. This way, we also find computable error bounds on $\parallel u_p-u_{*}\parallel$ as well as uniqueness results based on generalized Lipschitz-type real functions. Numerical examples of equations, favorable comparisons to other methods can be found in Section 4.