A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Abstract

Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu’s, Fisher’s, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes.

1. Introduction

Establishment of higher-order efficient iterative schemes for finding the solutions

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

(1)

(where $F:\mathbb{D}\subset {\mathbb{R}}^m\to {\mathbb{R}}^m$ is a differentiable mapping with open domain $\mathbb{D}$) is one of the foremost tasks in the area of numerical analysis and computation methods because of its wide application in real-life situations. We can easily find several real-life problems that were phrased into the nonlinear system (1) along with the same fundamental properties. For example, transport theory, combustion, reactor, kinematics syntheses, steering, chemical equilibrium, neurophysiology, and economic modeling problems were solved by being formulated to $F\left(U\right)=0$, and details can be found in the research articles [1,2,3,4,5].

Analytical methods for these problems are rare. Therefore, many authors developed iterative schemes that are based on iteration procedures. These iterative methods depend on several things, like starting the initial guess/es, the considered problem, the body structure of the proposed method, efficiency, and so forth. (For more details, please go through [6,7,8,9,10]). Some authors [11,12,13,14,15,16] gave special concern to the development of higher-order multi-point iterative methods. Faster convergence toward the required root, better efficiency, less CPU time, and fast accuracy are some of the main reasons behind the importance of multi-point methods.

The inspiration behind this work was the thought to suggest a new sixth-order iterative technique based on the weight function approach along with lower computational costs for large nonlinear systems. The beauty of using this approach is that it gives us the flexibility to produce new, as well as some special cases of the earlier methods. A good variety of applied science problems is considered in order to investigate the authenticity of our presented methods. Finally, using numerical experiments, we show the superiority of our schemes when compared to others in regard to computational cost, residual error, and CPU time. Moreover, they also show the stable computational order of convergence and minimum asymptotic error constants in contrast with exiting iterative methods.

2. Multi-Dimensional Case

Consider the following new scheme:

$$\begin{array}{cc}\hfill V_{\zeta }& =U_{\zeta }-\frac{2}{3}{F}^{\prime }{\left(U_{\zeta }\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill W_{\zeta }& =U_{\zeta }-P\left(T\left(U_{\zeta }\right)\right){\left(F^{\prime }\left(U_{\zeta }\right)+b{F}^{\prime }\left(V_{\zeta }\right)\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill U_{\zeta +1}& =W_{\zeta }+2Q{\left(U_{\zeta }\right)}^{-1}F\left(W_{\zeta }\right),\hspace{1em}\zeta =0,1,2,\dots ,\hfill \end{array}$$

(2)

where $T:\mathbb{D}\subseteq {\mathbb{R}}^m⟶{\mathbb{R}}^m$ is sufficiently differentiable in $\mathbb{D}$ with $T\left(U_{\zeta }\right)=F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)$, where $Q\left(U_{\zeta }\right)=F^{\prime }\left(U_{\zeta }\right)-3F^{\prime }\left(V_{\zeta }\right)$. We demonstrate the sixth-order convergence in Theorem 1 by adopting the same procedure suggested in [16].

Let $F:\mathbb{D}\subseteq {\mathbb{R}}^m⟶{\mathbb{R}}^m$ be sufficiently differentiable in $\mathbb{D}$. The kth derivative of F at $u\in {\mathbb{R}}^m$, $k\ge 1$, is the k-linear function $F^{\left(k\right)}\left(u\right):{\mathbb{R}}^m\times \cdots \times {\mathbb{R}}^m⟶{\mathbb{R}}^m$ with $F^{\left(k\right)}\left(u\right)(v_1,\dots ,v_k)\in {\mathbb{R}}^m$, and we have

• $F^{\left(k\right)}\left(u\right)(v_1,\dots ,v_{k-1},·)\in \mathcal{L}\left({\mathbb{R}}^m\right)$
• $F^{\left(k\right)}\left(u\right)(v_{\sigma \left(1\right)},\dots ,v_{\sigma \left(k\right)})=F^{\left(k\right)}\left(u\right)(v_1,\dots ,v_k)$, for each permutation $\sigma$ of $\{1,2,\dots ,k\}$,

that further yields

(a)

$F^{\left(k\right)}\left(u\right)(v_1,\dots ,v_k)=F^{\left(k\right)}\left(u\right)v_1\dots v_k$

(b)

$F^{\left(k\right)}\left(u\right)v^{k-1}{F}^{\left(p\right)}{v}^p=F^{\left(k\right)}\left(u\right)F^{\left(p\right)}\left(u\right)v^{k+p-1}$.

This $\xi +h\in {\mathbb{R}}^m$ being contained in the neighborhood of the required root $\xi$ of $F\left(x\right)=0$, we have

$$F(\xi +h)=F^{\prime }\left(\xi \right)\left[h+\sum _{k=2}^{p-1}{C}_k{h}^k\right]+O\left(h^p\right),$$

(3)

where $C_k=(1/k!){\left[F^{\prime }\left(\xi \right)\right]}^{-1}{F}^{\left(k\right)}\left(\xi \right)$, $k\ge 2$, provided $F^{\prime }\left(\xi \right)$ is invertible. We recognize that $C_k{h}^k\in {\mathbb{R}}^m$, since $F^{\left(k\right)}\left(\xi \right)\in \mathcal{L}({\mathbb{R}}^m\times \cdots \times {\mathbb{R}}^m,{\mathbb{R}}^m)$, and ${\left[F^{\prime }\left(\xi \right)\right]}^{-1}\in \mathcal{L}\left({\mathbb{R}}^m\right)$.

We can also write

$$F^{\prime }(\xi +h)=F^{\prime }\left(\overline{x}\right)\left[I+\sum _{k=2}^{p-2}k{C}_k{h}^{k-1}\right]+O\left(h^{p-1}\right),$$

(4)

I being the identity and $k{C}_k{h}^{k-1}\in \mathcal{L}\left({\mathbb{R}}^m\right)$.

By (4), we get

$${\left[F^{\prime }(\xi +h)\right]}^{-1}=\left[I+U_{2}h+U_3{h}^2+U_4{h}^4+\cdots \right]{\left[F^{\prime }\left(\xi \right)\right]}^{-1}+O\left(h^p\right),$$

(5)

with

$$\begin{array}{c}{U}_2=-2C_2,\hfill \\ U_3=4C_2^2-3C_3,\hfill \\ U_4=-8C_2^3+6C_2{C}_3-4C_4+6C_3{C}_2,\hfill \\ ⋮.\hfill \end{array}$$

The $e_{\zeta }=U_{\zeta }-\xi$ denotes the error in the $\zeta$th step. Then,

$$e_{\zeta +1}=M{e}_{\zeta }^p+O\left(e_{\zeta }^{p+1}\right),$$

where M is a p-linear function $M\in \mathcal{L}({\mathbb{R}}^m\times \cdots \times {\mathbb{R}}^m,{\mathbb{R}}^m)$, which is known as the error equation, and where p is the convergence order. Observe that ${e_{\zeta }}^p$ is $(e_{\zeta },e_{\zeta },\cdots ,e_{\zeta })$.

Theorem 1.

Suppose $F:\mathbb{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^m$ is a sufficient differentiable mapping with open domain $\mathbb{D}$ that consists of the required zero ξ. Further, we assume $F^{\prime }\left(x\right)$ is invertible and continuous around ξ. Moreover, we consider the starting guess $X_0$ is close enough to ξ for sure convergence. Then, scheme (2) attains maximum sixth-order convergence, provided that

$$P\left(I\right)=(1+b)I,\hspace{1em}{P}^{\prime }\left(I\right)=\frac{b-3}{4}I, P^{″}\left(I\right)=\frac{3(b+3)}{4}I,$$

(6)

where I is the identity matrix.

Proof. 

We can write $F\left(U_{\zeta }\right)$ and $F^{\prime }\left(U_{\zeta }\right)$ as follows:

$$F\left(U_{\zeta }\right)=F^{\prime }\left(\xi \right)\left[e_{\zeta }+C_2{e}_{\zeta }^2+C_3{e}_{\zeta }^3+C_4{e}_{\zeta }^4+C_5{e}_{\zeta }^5+C_6{e}_{\zeta }^6\right]+O\left(e_{\zeta }^7\right)$$

(7)

and

$$F^{\prime }\left(U_{\zeta }\right)=F^{\prime }\left(\xi \right)\left[I+2C_2{e}_{\zeta }+3C_3{e}_{\zeta }^2+4C_4{e}_{\zeta }^3+5C_5{e}_{\zeta }^4+6C_6{e}_{\zeta }^5\right]+O\left(e_{\zeta }^6\right),$$

(8)

where I is the identity matrix of size $m\times m$ and $C_m=\frac{1}{m!}{F}^{\prime }{\left(\xi \right)}^{-1}{F}^{\left(m\right)}\left(\xi \right)$, $m=2,3,4,5,6$.

By expressions (7) and (8), we get

$$F^{\prime }{\left(U_{\zeta }\right)}^{-1}=\left[I-2C_2{e}_{\zeta }+(4C_2^2-3C_3)e_{\zeta }^2\right]F^{\prime }{\left(\xi \right)}^{-1}$$

(9)

and

$$F^{\prime }{\left(U_{\zeta }\right)}^{-1}F\left(U_{\zeta }\right)=e_{\zeta }-C_2{e}_{\zeta }^2+(2C_2^2-2C_3)e_{\zeta }^3+O\left(e_{\zeta }^4\right).$$

(10)

Using expression (10) in (2), we have

$$V_{\zeta }-\xi =\frac{1}{3}{e}_{\zeta }+\frac{2}{3}{C}_2{e}_{\zeta }^2-\frac{2}{3}(2C_2^2-2C_3)e_{\zeta }^3+O\left(e_{\zeta }^4\right),$$

(11)

which further produces

$$F^{\prime }\left(V_{\zeta }\right)=F^{\prime }\left(\xi \right)\left[I+\frac{2}{3}{C}_2{e}_{\zeta }+\frac{1}{3}(4C_2^2+C_3)e_{\zeta }^2\right]+O\left(e_{\zeta }^3\right)$$

(12)

and

$$F^{\prime }\left(U_{\zeta }\right)+b{F}^{\prime }\left(V_{\zeta }\right)=F^{\prime }\left(\xi \right)\left[(1+b)I+\frac{2(b+3)C_2}{3}{e}_{\zeta }+\frac{1}{3}\left(4b{C}_2^2+(b+9)C_3\right)e_{\zeta }^2\right]+O\left(e_{\zeta }^3\right).$$

(13)

We can easily obtain the following from the expressions (10) and (13):

$$T\left(U_{\zeta }\right)=F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)=I-\frac{4C_2{e}_{\zeta }}{3}+\left(4C_2^2-\frac{8C_3}{3}\right)e_{\zeta }^2+O\left(e_{\zeta }^3\right).$$

(14)

We deduce from expression (14) that $T\left(U_{\zeta }\right)-I=O\left(e_{\zeta }\right)$. Moreover, we can write

$$P\left(T\left(U_{\zeta }\right)\right)=P\left(I\right)+P^{\prime }\left(I\right)T\left(U_{\zeta }\right)+\frac{1}{2!}{P}^{″}\left(I\right)T{\left(U_{\zeta }\right)}^2+O\left(T{\left(U_{\zeta }\right)}^3\right),$$

(15)

so

$$P\left(T\left(U_{\zeta }\right)\right)=(1+b)I-\frac{(b-3)}{3}{C}_2{e}_{\zeta }+\frac{(5b-3)C_2^2-2(b-3)C_3}{3}{e}_{\zeta }^2+O\left(e_{\zeta }^3\right).$$

(16)

By adopting the expressions (10) and (16), we have

$$\begin{array}{cc}\hfill P\left(T\left(U_{\zeta }\right)\right){\left(F^{\prime }\left(U_{\zeta }\right)+b{F}^{\prime }\left(V_{\zeta }\right)\right)}^{-1}F\left(U_{\zeta }\right)& =\left((1+b)I-\frac{(b-3)}{3}{C}_2{e}_{\zeta }+\frac{(5b-3)C_2^2-2(b-3)C_3}{3}{e}_{\zeta }^2+O\left(e_{\zeta }^3\right)\right)\hfill \\ & \times \left(\frac{e_{\zeta }}{b+1}+\frac{(b-3)C_2}{3{(b+1)}^2}{e}_{\zeta }^2-\frac{2(7b^2+6b-9)C_2^2-6(b^2-2b-3)C_3}{9{(b+1)}^3}{e}_{\zeta }^3+O\left(e_{\zeta }^4\right)\right),\hfill \\ & =e_{\zeta }+O\left(e_{\zeta }^4\right).\hfill \end{array}$$

(17)

Then, using expression (17) in (2), we yield

$$W_{\zeta }-\xi ={\alpha }_1{e}_{\zeta }^4+{\alpha }_2{e}_{\zeta }^5+{\alpha }_3{e}_{\zeta }^6+O\left(e_{\zeta }^7\right),$$

(18)

where ${\alpha }_i, i=1,2,3$ depend on some b and $C_i, 2\le j\le 6$.

Moreover, we have

$$F\left(W_{\zeta }\right)=F^{\prime }\left(\xi \right)\left[{\alpha }_1{e}_{\zeta }^4+{\alpha }_2{e}_{\zeta }^5+{\alpha }_3{e}_{\zeta }^6\right]+O\left(e_{\zeta }^7\right).$$

(19)

After some simple algebraic calculations, we have

$$\begin{array}{cc}\hfill 2Q{\left(U_{\zeta }\right)}^{-1}F\left(W_{\zeta }\right)& =2{\left(F^{\prime }\left(U_{\zeta }\right)+F\left(V_{\zeta }\right)\right)}^{-1}F\left(W_{\zeta }\right)\hfill \\ & =-{\alpha }_1{e}_{\zeta }^4-{\alpha }_2{e}_{\zeta }^5+\left(-{\alpha }_3+2{\alpha }_1{C}_2^2-{\alpha }_1{C}_3\right)e_{\zeta }^6+O\left(e_{\zeta }^7\right).\hfill \end{array}$$

(20)

Finally, we have

$$X_{\zeta +1}-\xi =W_{\zeta }-\xi +2Q{\left(U_{\zeta }\right)}^{-1}F\left(W_{\zeta }\right)={\alpha }_1\left(2C_2^2-C_3\right)e_{\zeta }^6+O\left(e_{\zeta }^7\right),$$

(21)

where ${\alpha }_1$ is a function of only $b,C_2,C_3,C_4$. □

Specializations

Some of the fruitful cases are mentioned below:

(1) We assume

$$P\left(U\right)=\frac{3}{2}{U}^2-\frac{1}{2}U+4I$$

for $b=1$, which generates the following new sixth-order, Jarratt-type scheme:

$$\left\{\begin{array}{c}\hfill \begin{array}{cc}\hfill V_{\zeta }& =U_{\zeta }-\frac{2}{3}{F}^{\prime }{\left(U_{\zeta }\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill W_{\zeta }& =U_{\zeta }-\left(\frac{3}{2}{\left(F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right)}^2-\frac{1}{2}\left(F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right)+4I\right){\left(F^{\prime }\left(U_{\zeta }\right)+F^{\prime }\left(V_{\zeta }\right)\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill U_{\zeta +1}& =W_{\zeta }+2Q{\left(U_{\zeta }\right)}^{-1}F\left(W_{\zeta }\right).\hfill \end{array}\end{array}\right.$$

(22)

(2) Consider the following weight function for $b=0$:

$$P\left(U\right)=\frac{3U}{8}+\frac{9}{8}{U}^{-1}-\frac{1}{2}I,$$

leading to

$$\left\{\begin{array}{cc}\hfill V_{\zeta }& =U_{\zeta }-\frac{2}{3}{F}^{\prime }{\left(U_{\zeta }\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill W_{\zeta }& =U_{\zeta }-\left(\frac{3}{8}\left(F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right)+\frac{9}{8}{\left(F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right)}^{-1}-\frac{1}{2}I\right)F^{\prime }{\left(U_{\zeta }\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill U_{\zeta +1}& =W_{\zeta }+2Q{\left(U_{\zeta }\right)}^{-1}F\left(W_{\zeta }\right).\hfill \end{array}\right.$$

(23)

(3) Now, we assume another weight function (for $b=0.5$),

$$P\left(U\right)={\left(44I-84U\right)}^{-1}(41I-101U),$$

that yields

$$\left\{\begin{array}{cc}\hfill V_{\zeta }& =U_{\zeta }-\frac{2}{3}{F}^{\prime }{\left(U_{\zeta }\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill W_{\zeta }& =U_{\zeta }-{\left(44I-84F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right)}^{-1}\left(41I-101F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right){\left(F^{\prime }\left(U_{\zeta }\right)+0.5F^{\prime }\left(V_{\zeta }\right)\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill U_{\zeta +1}& =W_{\zeta }+2Q{\left(U_{\zeta }\right)}^{-1}F\left(W_{\zeta }\right),\hfill \end{array}\right.$$

(24)

which is another new sixth-order scheme.

In like manner, we can obtain many familiar and advanced sixth-order, Jarratt-type schemes by adopting different weight functions.

3. Local Convergence Analysis

It is well-known that iterative methods defined on the real line or on the $m-$dimensional Euclidean space constitute the motivation for extending these methods to more abstract spaces, such as Hilbert, Banach, or other spaces. The local convergence analysis of method (2) after defining it for Banach space operators for all $\zeta =0,1,2,3,\dots$ are as

$$\begin{array}{cc}\hfill V_{\zeta }& =U_{\zeta }-\frac{2}{3}{F}^{\prime }{\left(U_{\zeta }\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill W_{\zeta }& =U_{\zeta }-P\left(T\left(U_{\zeta }\right)\right){\left(F^{\prime }\left(U_{\zeta }\right)+b{F}^{\prime }\left(V_{\zeta }\right)\right)}^{-1}F\left(U_{\zeta }\right),\hfill \\ \hfill U_{\zeta +1}& =W_{\zeta }+2Q{\left(U_{\zeta }\right)}^{-1}F\left(W_{\zeta }\right),\hfill \end{array}$$

(25)

where ${\mathbb{E}}_1$, ${\mathbb{E}}_2$ are Banach spaces, $\mathsf{\Omega }\subseteq {\mathbb{E}}_1$ is a nonempty, convex, and open subset of ${\mathbb{E}}_1$, $T\left(U\right)=F^{\prime }{\left(U\right)}^{-1}{F}^{\prime }\left(V\right)$, $Q\left(U\right)=F^{\prime }\left(U\right)-3F^{\prime }\left(V\right)$, $b\in \mathbb{R}-\{-1\}$, and $T:{\mathbb{E}}_1\to {\mathbb{E}}_2$ is such that $P\left(I\right)=(1+b)I$. Then, under certain hypotheses given later, method (25) converges to a solution $U_{*}$ of equation

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

(26)

where $F:\mathsf{\Omega }\to {\mathbb{E}}_2$ is a continuously differentiable operator in the sense of Fréchet. For acceptable convergence analysis, we first need to define some parameters and scalar functions. Let ${\psi }_0:\mathbb{T}\to \mathbb{T}$ be a continuous and increasing function with ${\psi }_0\left(0\right)=0$, where $\mathbb{T}=[0,+\infty )$. The equation

$${\psi }_0\left(t\right)=1$$

(27)

has a minimum of one positive zero. Denote by ${\rho }_0$ the smallest such solution, and set ${\mathbb{T}}_0=[0,\rho )$. Let $\psi :{\mathbb{T}}_0\to \mathbb{T}$, ${\psi }_1:{\mathbb{T}}_0\to \mathbb{T}$ be continuous and increasing maps with $\psi \left(0\right)=0$. Consider maps ${\phi }_1$ and ${\overline{\phi }}_1$ on ${\mathbb{T}}_0$ as

$${\phi }_1\left(t\right)={\displaystyle \frac{{\int }_0^1\psi \left((1-\tau )t\right)d\tau +\frac{1}{3}{\int }_0^1{\psi }_1\left(\tau t\right)d\tau }{1-{\psi }_0\left(t\right)}}$$

and

$${\overline{\phi }}_1\left(t\right)={\phi }_1\left(t\right)-1.$$

Suppose that

$$\frac{{\psi }_1\left(t\right)}{3}<1.$$

(28)

Then, by the definition of function ${\overline{\phi }}_1$ and (26), we have ${\overline{\phi }}_1\left(0\right)=\frac{{\psi }_1\left(t\right)}{3}-1<0$ and ${\overline{\phi }}_1\left(t\right)\to +\infty$, as $t\to {\rho }_0^{-}$. Then, by the mean value theorem, there exists at least one solution of ${\overline{\phi }}_1\left(t\right)=0$ in the interval $(0,{\rho }_0)$. Denoted by $R_1$ is the smallest such solution.

Suppose

$$\lambda \left(t\right)=1$$

(29)

where

$$\lambda \left(t\right)=\frac{1}{|1+b|}\left({\psi }_0\left(t\right)+\left|b\right|{\psi }_0\left({\phi }_1\left(t\right)t\right)\right)$$

has a minimum of one positive zero. Denoted by ${\rho }_{\lambda }$ is the smallest such solution, and set ${\mathbb{T}}_1=[0,{\rho }_1), {\rho }_1=min\{{\rho }_0, {\rho }_{\lambda }\}.$

Further, we consider functions ${\phi }_2$ and ${\overline{\phi }}_2$ on ${\mathbb{T}}_1=[0,\rho )$ by

$${\phi }_2\left(t\right)=\frac{{\int }_0^1\psi \left((1-\tau )t\right)d\tau }{1-{\psi }_0\left(t\right)}+\frac{\left({\psi }_0\left(t\right)+{\psi }_0\left({\phi }_1\left(t\right)t\right)\right)A\left(t\right)+{\psi }_1\left({\phi }_1\left(t\right)t\right)B\left(t\right){\int }_0^1{\psi }_1\left(\tau t\right)d\tau }{(1-\lambda \left(t\right))(1-{\psi }_0\left(t\right))}$$

and

$${\overline{\phi }}_2\left(t\right)={\phi }_2\left(t\right)-1,$$

where $A:{\mathbb{T}}_1\to \mathbb{T}$ and $B:{\mathbb{T}}_1\to \mathbb{T}$ are continuous and increasing functions. We also get ${\overline{\phi }}_2\left(0\right)=-1$ and ${\overline{\phi }}_2\left(t\right)\to +\infty$, as $t\to {\rho }_0^{-}$. Recall by $R_2$ the smallest solution of equation ${\overline{\phi }}_2\left(t\right)=0$ in $(0,{\rho }_1)$. Suppose that the equations

$$g\left(t\right)=1,\hspace{1em}{\psi }_0\left({\phi }_2\left(t\right)t\right)=1$$

(30)

have at least one positive solution, where $g\left(t\right)=\frac{1}{2}\left({\psi }_0\left(t\right)+3{\psi }_0\left({\phi }_1\left(t\right)t\right)\right)$. Denote by ${\rho }_g,{\rho }_h$ the smallest such solutions, and set ${\mathbb{T}}_2=[0,\rho )$, $\rho =min\{{\rho }_0,{\rho }_{\lambda },{\rho }_g,{\rho }_h\}.$ Define functions ${\psi }_3$ and ${\overline{\psi }}_3$ on the interval ${\mathbb{T}}_3=[0,\rho )$ by

$${\phi }_3\left(t\right)=\left(\frac{{\int }_0^1\psi \left((1-\tau ){\phi }_2\left(t\right)t\right)d\tau }{1-{\psi }_0\left({\phi }_2\left(t\right)\right)}+\frac{\left({\psi }_0\left(t\right)+3{\psi }_0\left({\phi }_1\left(t\right)t\right)+2{\psi }_0\left({\phi }_2\left(t\right)t\right)\right){\int }_0^1{\psi }_1\left(\tau {\phi }_2\left(t\right)t\right)d\tau }{1-{\psi }_0\left({\phi }_2\left(t\right)t\right)}\right){\phi }_2\left(t\right)$$

and

$${\overline{\phi }}_3\left(t\right)={\phi }_3\left(t\right)-1.$$

We also get ${\overline{\phi }}_3\left(0\right)=-1$ and ${\overline{\phi }}_3\left(t\right)\to +\infty$, as $t\to \rho -$. Denote by $R_3$ the smallest solution of equation ${\overline{\phi }}_3\left(t\right)=0$ in $(0,\rho )$. Define a radius of convergence R by

$$R=min\left\{R_i\right\}, i=1,2,3.$$

(31)

It follows from each $t\in [0,R)$

$$0\le {\psi }_0\left(t\right)<1,$$

(32)

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

(33)

$$0\le {\psi }_0\left({\phi }_2\left(t\right)t\right)<1,$$

(34)

$$0\le {\phi }_1\left(t\right)<1,$$

(35)

$$0\le \lambda \left(t\right)<1$$

(36)

$$0\le {\phi }_2\left(t\right)<1,$$

(37)

$$0\le g\left(t\right)<1$$

(38)

and

$$0\le {\phi }_3\left(t\right)<1.$$

(39)

Define $K(U,a)=\{V\in {\mathbb{E}}_1 \mathrm{such} \mathrm{that} \parallel U-V\parallel <a\}$. Let $\overline{K}(U,a)$ to stand for the closure of $K(U,a)$. By $\mathcal{L}\mathbb{B}({\mathbb{E}}_1,{\mathbb{E}}_2)$, denote the space of bounded linear operators from ${\mathbb{E}}_1$ into ${\mathbb{E}}_2$.

The following conditions $\left(H\right)$ are the base for the study of local convergence analysis:

$\left(h_1\right)$$F:\mathsf{\Omega }\to {\mathbb{E}}_2$ with $U_{*}\in \mathsf{\Omega }$ such that $F\left(U_{*}\right)=0$ and $F^{\prime }{\left(U_{*}\right)}^{-1}$ are invertible.

$\left(h_2\right)$ There exists function ${\psi }_0:\mathbb{T}\to \mathbb{T}$ continuous, increasing with ${\psi }_0\left(0\right)=0$ such that for each $x\in \mathsf{\Omega }$

$$∥F^{\prime }{\left(U_{*}\right)}^{-1}\left(F^{\prime }\left(U\right)-F^{\prime }\left(U_{*}\right)\right)∥\le {\psi }_0(\parallel U-U_{*}\parallel ).$$

Set ${\mathsf{\Omega }}_0=\mathsf{\Omega }\cap S(U_{*},{\rho }_0),$ where ${\rho }_0$ is given in (25).

$\left(h_3\right)$ There exist functions $\psi :{\mathbb{T}}_0\to \mathbb{T}$, ${\psi }_1:{\mathbb{T}}_0\to \mathbb{T}$, $A:{\mathbb{T}}_0\to \mathbb{T}$ and $B:{\mathbb{T}}_0\to \mathbb{T}$ continuous and increasing with $\psi \left(0\right)=0$ such that for each $X,V\in {\mathsf{\Omega }}_0$

$$∥F^{\prime }{\left(U_{*}\right)}^{-1}\left(F^{\prime }\left(V\right)-F^{\prime }\left(U\right)\right)∥\le \psi (\parallel V-U\parallel ),$$

$$∥F^{\prime }{\left(U_{*}\right)}^{-1}{F}^{\prime }\left(U\right)∥\le {\psi }_1(\parallel V-U\parallel ),$$

$$∥I-K\left(T\right(U\left)\right)∥\le A(\parallel V-U\parallel ),$$

$$∥K\left(I\right)-K\left(T\right(U\left)\right)∥\le B(\parallel V-U\parallel ),$$

and

$$K\left(I\right)=(1+b)I.$$

$\left(h_4\right)$$\overline{K}(U_{*},R)\subset \mathsf{\Omega }$, ${\rho }_0,{\rho }_{\lambda },{\rho }_g$ exist, given by (25), (29), (30), respectively, and R is defined in (31).

$\left(h_5\right)$ There exists $R_{*}\ge R$ such that

$${\int }_0^1{\psi }_0\left(\tau R_{*}\right)d\tau <1.$$

Set ${\mathsf{\Omega }}_1=\mathsf{\Omega }\cap \overline{K}(U_{*},R_{*})$.

Next, we provide the local convergence analysis of method (25) using the hypotheses $\left(H\right)$ and the previously developed notations.

Theorem 2.

Assume the hypotheses $\left(H\right)$ hold and $U_0\in K(U_{*},R)-\left\{U_{*}\right\}$. Then, the sequence $\left\{U_{\zeta }\right\}\subset K(U_{*},R)$, ${\displaystyle \underset{\zeta \to \infty }{lim}{U}_{\zeta }=U_{*}}$ and

$$\parallel V_{\zeta }-U_{*}\parallel \le {\phi }_1(\parallel U_{\zeta }-U_{*}\parallel )\parallel U_{\zeta }-U_{*}\parallel \le \parallel U_{\zeta }-U_{*}\parallel <R,$$

(40)

$$\parallel W_{\zeta }-U_{*}\parallel \le {\phi }_2(\parallel U_{\zeta }-U_{*}\parallel )\parallel U_{\zeta }-U_{*}\parallel \le \parallel U_{\zeta }-U_{*}\parallel$$

(41)

and

$$\parallel X_{\zeta +1}-U_{*}\parallel \le {\phi }_3(\parallel U_{\zeta }-U_{*}\parallel )\parallel U_{\zeta }-U_{*}\parallel \le \parallel U_{\zeta }-U_{*}\parallel .$$

(42)

Moreover, $U_{*}$ is the only solution of $F\left(x\right)=0$ in the set ${\mathsf{\Omega }}_1$ given in $\left(h_5\right)$.

Proof. 

Estimates (40)–(42) are shown utilizing the hypotheses $\left(H\right)$ and mathematical induction. By $\left(h_1\right)$, $\left(h_2\right)$, (31) and (32), we have that

$$\parallel F^{\prime }{\left(U_{*}\right)}^{-1}(F^{\prime }\left(U\right)-F^{\prime }\left(U_{*}\right))\parallel \le {\psi }_0(\parallel U-U_{*}\parallel )<{\psi }_0\left(R\right)<1,$$

(43)

for each $U\in K(U_{*},R)-\left\{U_{*}\right\}$, so $F^{\prime }{\left(U\right)}^{-1}\in \mathcal{L}\mathbb{B}({\mathbb{E}}_2,{\mathbb{E}}_1)$ and

$$\parallel F^{\prime }{\left(U\right)}^{-1}{F}^{\prime }\left(U_{*}\right)\parallel \le \frac{1}{1-{\psi }_0(\parallel U-U_{*}\parallel )},$$

(44)

by the Banach perturbation lemma on invertible operators [6,7]. Then, $V_0$ is well-defined by method (25) for $\zeta =0$. By convexity, we have that $U_{*}+\tau (U-U_{*})\in K(U_{*},R)$ for each $U\in K(U_{*},R)$, by adopting $\left(h_1\right)$, we get

$$F\left(U\right)=F\left(U\right)-F\left(U_{*}\right)={\int }_0^1{F}^{\prime }(U_{*}+\tau (U-U_{*}))d\tau (U-U_{*}).$$

(45)

From the hypotheses ($h_1$) and $\left(h_3\right)$, we get

$$∥F^{\prime }{\left(U_{*}\right)}^{-1}F\left(U\right)∥\le {\int }_0^1{\psi }_1(\tau \parallel U-U_{*}\parallel )d\tau \parallel U-U_{*}\parallel .$$

(46)

Using the first substep of method (25) for $\zeta =0$, (31), (35), $\left(h_3\right)$, (44) (for $U=U_0$), and (46) (for $U=U_0$), we have in turn that

$$\begin{array}{cc}\hfill \parallel V_0-U_{*}\parallel & =∥\left(U_0-U_{*}{F}^{\prime }{\left(U_0\right)}^{-1}F\left(U_0\right)\right)+\frac{1}{3}{F}^{\prime }{\left(U_0\right)}^{-1}F\left(U_0\right)∥\hfill \\ & \le \parallel F^{\prime }{\left(U_0\right)}^{-1}{F}^{\prime }\left(U_{*}\right)\parallel ∥{\int }_0^1{F}^{\prime }{\left(U_{*}\right)}^{-1}\left(F^{\prime }(U_{*}+\tau (U_0-U_{*}))-F^{\prime }\left(U_0\right)\right)d\tau ∥\hfill \\ & +\frac{1}{3}\parallel F^{\prime }{\left(U_0\right)}^{-1}{F}^{\prime }\left(U_0\right)\parallel \parallel F^{\prime }{\left(U_0\right)}^{-1}F\left(U_0\right)\parallel \hfill \\ & \le \frac{\left[{\int }_0^1\psi ((1-\tau )\parallel U_0-U_{*}\parallel )d\tau +\frac{1}{3}{\int }_0^1{\psi }_1(\tau \parallel U_0-U_{*}\parallel )d\tau \right]}{1-{\psi }_0(\parallel U_0-U_{*}\parallel )}\hfill \\ & ={\phi }_1(\parallel U_0-U_{*}\parallel )\parallel U_0-U_{*}\parallel \le \parallel U_0-U_{*}\parallel <R,\hfill \end{array}$$

(47)

so (40) holds for $\zeta =0$ and $V_0\in K(U_{*}, R)$. We must show that ${\left(F^{\prime }\left(U_0\right)+b{F}^{\prime }\left(V_0\right)\right)}^{-1}\in \mathcal{L}\mathbb{B}({\mathbb{E}}_2,{\mathbb{E}}_1)$.

By (31), (36), $\left(h_2\right)$ and (47), we have

$$\begin{array}{c}\parallel {\left((1+b)F^{\prime }\left(U_{*}\right)\right)}^{-1}\left[(F^{\prime }\left(U_0\right)-F^{\prime }\left(U_{*}\right))+b(F^{\prime }\left(V_0\right)+F^{\prime }\left(U_{*}\right))\right]\parallel \hfill \\ \le \frac{1}{|1+b|}\left[∥F^{\prime }{\left(U_{*}\right)}^{-1}\left(F^{\prime }\left(U_0\right)-F^{\prime }\left(U_{*}\right)\right)∥+\left|b\right|∥F^{\prime }{\left(U_{*}\right)}^{-1}\left(F^{\prime }\left(V_0\right)-F^{\prime }\left(U_{*}\right)\right)∥\right]\hfill \\ \le \frac{1}{|1+b|}\left[{\psi }_0(\parallel U_0-U_{*}\parallel )+\left|b\right|{\psi }_0(\parallel V_0-U_{*}\parallel )\right]\hfill \\ \le \lambda (\parallel U_0-U_{*}\parallel )<\lambda \left(R\right)<1,\hfill \end{array}$$

(48)

so

$$∥{\left(F^{\prime }\left(U_0\right)+b{F}^{\prime }\left(V_0\right)\right)}^{-1}{F}^{\prime }\left(U_{*}\right)∥\le \frac{1}{|1+b|(1-\lambda (\parallel U_0-U_{*}\parallel \left)\right)}.$$

(49)

Then, since $W_0$ is well-defined by (24) and the second substep of method (25), we can write

$$\begin{array}{cc}\hfill W_0-U_{*}& =W_0-U_{*}-F^{\prime }{\left(U_0\right)}^{-1}F\left(U_0\right)+\left[F^{\prime }{\left(U_0\right)}^{-1}-P\left(T\left(U_0\right)\right){\left(F^{\prime }\left(U_0\right)+b{F}^{\prime }\left(V_0\right)\right)}^{-1}\right]F\left(U_0\right).\hfill \end{array}$$

(50)

We need an estimate on the expression inside the bracket in (50):

$$\begin{array}{c}{F}^{\prime }{\left(U_0\right)}^{-1}-P\left(T\left(U_0\right)\right){\left(F^{\prime }\left(U_0\right)+b{F}^{\prime }\left(V_0\right)\right)}^{-1}\hfill \\ =F^{\prime }{\left(U_0\right)}^{-1}\left[I-F^{\prime }\left(U_0\right)P\left(T\left(U_0\right)\right){\left(F^{\prime }\left(U_0\right)+b{F}^{\prime }\left(V_0\right)\right)}^{-1}\right]\hfill \\ =F^{\prime }{\left(U_0\right)}^{-1}\left[F^{\prime }\left(U_0\right)+b{F}^{\prime }\left(V_0\right)-F^{\prime }\left(U_0\right)P\left(T\left(U_0\right)\right)\right]{\left(F^{\prime }\left(U_0\right)+b{F}^{\prime }\left(V_0\right)\right)}^{-1}\hfill \\ =F^{\prime }{\left(U_0\right)}^{-1}[\left(F^{\prime }\left(U_0\right)-F^{\prime }\left(U_{*}\right)\right)+\left(F^{\prime }\left(U_{*}\right)-F^{\prime }\left(V_0\right)\right)\left(\right(I-P\left(T\left(U_0\right)\right)+F^{\prime }\left(V_0\right)\left(P\left(I\right)-P\left(T\right(U\left)\right)\right]\hfill \\ \times {\left(F^{\prime }\left(U_0\right)+b{F}^{\prime }\left(V_0\right)\right)}^{-1},\hfill \end{array}$$

(51)

so by $\left(h_3\right)$, (44) and (49), we have in turn that (51), in norm, is bounded above by

$$\begin{array}{c}\parallel F^{\prime }{\left(U_0\right)}^{-1}{F}^{\prime }\left(U_{*}\right)\parallel [\left(\parallel F^{\prime }{\left(U_{*}\right)}^{-1}(F^{\prime }\left(U_0\right)-F^{\prime }\left(U_{*}\right))\parallel +\parallel F^{\prime }{\left(U_{*}\right)}^{-1}(F^{\prime }\left(V_0\right)-F^{\prime }\left(U_{*}\right))\parallel \right)\parallel I-P\left(T\left(U_0\right)\right)\parallel \hfill \\ \times \parallel F^{\prime }{\left(U_{*}\right)}^{-1}{F}^{\prime }\left(V_0\right)\parallel \parallel P\left(I\right)-P\left(T\left(U_0\right)\right)\parallel ]\frac{1}{\lambda (\parallel U_0-U_{*}\parallel )},\hfill \end{array}$$

(52)

and (31), (37), (44)(for $U=U_0$), (46)(for $U=U_0$), (47), (49) and (52), we yield

$$\begin{array}{cc}\hfill \parallel W_0-U_{*}\parallel & =\parallel U_0-U_{*}-F^{\prime }{\left(U_0\right)}^{-1}F\left(U_0\right)\parallel +\parallel \left[F^{\prime }{\left(U_0\right)}^{-1}-P\left(T\left(U_0\right)\right){\left(F^{\prime }\left(U_0\right)+b{F}^{\prime }\left(V_0\right)\right)}^{-1}\right]F^{\prime }\left(U_{*}\right)\parallel F^{\prime }{\left(U_{*}\right)}^{-1}F\left(U_0\right)\parallel \hfill \\ & \le [\frac{{\int }_0^1\psi ((1-\tau )\parallel U_0-U_{*}\parallel )d\tau }{1-{\psi }_0(\parallel U_0-U_{*}\parallel )}+\frac{1}{(1-{\psi }_0(\parallel U_0-U_{*}\parallel \left)\right)(1-\lambda (\parallel U_0-U_{*}\parallel \left)\right)}[({\psi }_0(\parallel U_0-U_{*}\parallel \hfill \\ & +{\psi }_0({\phi }_1(\parallel U_0-U_{*}\parallel )\parallel U_0-U_{*}\parallel \left)\right)\left)A\right(\parallel U_0-U_{*}\parallel )\hfill \\ & +{\psi }_1({\phi }_1(\parallel U_0-U_{*}\parallel )\parallel U_0-U_{*}\parallel \left)B\right(\parallel U_0-U_{*}\parallel ){\int }_0^1{\psi }_1(\tau \parallel U_0-U_{*}\parallel \left)d\tau \right]]\parallel U_0-U_{*}\parallel \hfill \\ & ={\phi }_2(\parallel U_0-U_{*}\parallel )\parallel U_0-U_{*}\parallel \le \parallel U_0-U_{*}\parallel <R,\hfill \end{array}$$

(53)

so (41) holds for $\zeta =0$ and $W_0\in K(U_{*},R)$. Next, we must show that $Q{\left(U_0\right)}^{-1}\in \mathcal{L}\mathbb{B}({\mathbb{E}}_2,{\mathbb{E}}_1)$. Using (31), (38), $\left(h_2\right)$ and (47), we get that

$$\begin{array}{c}∥(-2F^{\prime }{\left(U_{*}\right)}^{-1})\left(F^{\prime }\left(U_0\right)-F^{\prime }\left(U_{*}\right)-3(F^{\prime }\left(V_0\right)-F^{\prime }\left(U_{*}\right))\right)∥\hfill \\ =\frac{1}{2}\left[\parallel F^{\prime }{\left(U_{*}\right)}^{-1}(F^{\prime }\left(U_0\right)-F^{\prime }\left(U_{*}\right))\parallel +3\parallel F^{\prime }{\left(U_{*}\right)}^{-1}(F^{\prime }\left(V_0\right)-F^{\prime }\left(U_{*}\right))\parallel \right]\hfill \\ =\frac{1}{2}\left[{\psi }_0(\parallel U_0-U_{*}\parallel )+3{\psi }_0(\parallel V_0-U_{*}\parallel )\right]\hfill \\ \le g(\parallel U_0-U_{*}\parallel )<1.\hfill \end{array}$$

(54)

Thus, $Q{\left(U_0\right)}^{-1}\in \mathcal{L}\mathbb{B}({\mathbb{E}}_2,{\mathbb{E}}_1)$ and

$$\parallel Q{\left(U_0\right)}^{-1}{F}^{\prime }\left(U_{*}\right)\parallel \le \frac{1}{2(1-g(\parallel U_0-U_{*}\parallel \left)\right)}.$$

(55)

Hence, $U_1$ is well-defined by the last substep of method (25). By adopting (31), (39), (53) and (55), we have

$$\begin{array}{cc}\hfill U_1-U_{*}& =W_0-U_{*}-F^{\prime }{\left(W_0\right)}^{-1}F\left(W_0\right)+(F^{\prime }{\left(W_0\right)}^{-1}+2Q{\left(U_0\right)}^{-1})F\left(W_0\right)\hfill \\ & =W_0-U_{*}-F^{\prime }{\left(W_0\right)}^{-1}F\left(W_0\right)+F^{\prime }{\left(W_0\right)}^{-1}[(F^{\prime }\left(U_0\right)-F^{\prime }\left(U_{*}\right))\hfill \\ & -3(F^{\prime }\left(V_0\right)-F^{\prime }\left(U_{*}\right))+2(F^{\prime }\left(W_0\right)-F^{\prime }\left(U_0\right))]Q{\left(U_0\right)}^{-1}F\left(W_0\right)\hfill \end{array}$$

(56)

from which we get that

$$\begin{array}{cc}\hfill \parallel U_1-U_{*}\parallel & \le \parallel W_0-U_{*}-F^{\prime }{\left(W_0\right)}^{-1}F\left(W_0\right)\parallel +\parallel F^{\prime }{\left(W_0\right)}^{-1}{F}^{\prime }\left(U_{*}\right)\parallel [\parallel (F^{\prime }{\left(U_{*}\right)}^{-1}(F^{\prime }\left(U_0\right)-F^{\prime }\left(U_{*}\right))\parallel \hfill \\ & +3\parallel (F^{\prime }{\left(U_{*}\right)}^{-1}(F^{\prime }\left(V_0\right)-F^{\prime }\left(U_{*}\right))\parallel +2\parallel (F^{\prime }{\left(U_{*}\right)}^{-1}(F^{\prime }\left(W_0\right)-F^{\prime }\left(U_{*}\right))\parallel ]\hfill \\ & \times \parallel Q{\left(U_0\right)}^{-1}{F}^{\prime }\left(U_{*}\right)\parallel \parallel F^{\prime }{\left(U_{*}\right)}^{-1}F\left(W_0\right)\parallel \hfill \\ & \le \frac{{\int }_0^1\psi ((1-\tau )\parallel U_0-U_{*}\parallel )d\tau }{1-{\psi }_0(\parallel U_0-U_{*}\parallel )}\hfill \\ & +\frac{\left({\psi }_0(\parallel U_0-U_{*}\parallel )+3{\psi }_0(\parallel V_0-U_{*}\parallel )+2{\psi }_0(\parallel W_0-U_{*}\parallel )\right){\int }_0^1{\psi }_1(\tau \parallel U_0-U_{*}\parallel )d\tau }{(1-{\psi }_0(\parallel W_0-U_{*}\parallel \left)\right)(1-g(\parallel U_0-U_{*}\parallel \left)\right)}\hfill \\ & \le {\phi }_3(\parallel U_0-U_{*}\parallel )\parallel U_0-U_{*}\parallel \le \parallel U_0-U_{*}\parallel <R,\hfill \end{array}$$

(57)

so (42) holds for $\zeta =0$ and $U_1\in K(U_{*},R)$.

Then, we replace $U_0$, $V_0,W_0, U_1$ by $U_j$, $V_j, W_j$, $U_{j+1}$ in the preceding estimations to finish the induction for (40)–(42). In view of the estimate

$$\parallel U_{j+1}-U_{*}\parallel \le r\parallel U_j-U_{*}\parallel <R, r={\psi }_3(\parallel U_0-U_{*}\parallel )\in [0, 1),$$

(58)

we conclude that ${\displaystyle \underset{j\to \infty }{lim}{U}_j=U_{*}}$ and $U_{j+1}\in K(U_{*}, R)$. Let us consider that $V_{*}\in \mathsf{\Omega }$ be such that $F\left(V_{*}\right)=0$. Using $K1={\int }_0^1{F}^{\prime }(V_{*}+\tau (U_{*}-V_{*}))d\tau$, $\left(h_2\right)$ and $\left(h_5\right)$, we have

$$\begin{array}{c}\parallel F^{\prime }{\left(U_{*}\right)}^{-1}(K-F^{\prime }\left(U_{*}\right))\parallel \le \parallel {\int }_0^1{\psi }_0(\tau \parallel U_{*}-V_{*}\parallel )d\tau \hfill \\ \le {\int }_0^1{\psi }_0\left(\tau R\right)d\tau <1,\hfill \end{array}$$

(59)

so $K{1}^{-1}\in \mathcal{L}\mathbb{B}({\mathbb{E}}_1,{\mathbb{E}}_2)$. Therefore, by the identity

$$0=F\left(U_{*}\right)-F\left(V_{*}\right)=K(U_{*}-V_{*}),$$

(60)

we deduce that $U_{*}=V_{*}$. □

Application 1: Let us see how functions A and B can be chosen when P is given above (22). We get

$$\begin{array}{cc}I-P\left(T\left(U_{\zeta }\right)\right)\hfill & =I-\frac{3}{2}T{\left(U_{\zeta }\right)}^2+\frac{1}{2}T\left(U_{\zeta }\right)-4I\hfill \\ & =\frac{1}{2}\left(T\left(U_{\zeta }\right)-I\right)-\frac{3}{2}\left(T{\left(U_{\zeta }\right)}^2-I\right)-I\hfill \\ & =\frac{1}{2}\left(F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)-I\right)-\frac{3}{2}\left[{\left(F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right)}^2-I\right]-I\hfill \\ & =\frac{1}{2}{F}^{\prime }{\left(U_{\zeta }\right)}^{-1}\left(F^{\prime }\left(V_{\zeta }\right)-F^{\prime }\left(U_{\zeta }\right)\right)+\frac{3}{2}\left[{\left(F^{\prime }{\left(U_{\zeta }\right)}^{-1}\left(F^{\prime }\left(V_{\zeta }\right)-F^{\prime }\left(U_{\zeta }\right)\right)\right)}^2+2F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right]-I\hfill \end{array}$$

(61)

so

$$\begin{array}{cc}\hfill \parallel I-P\left(T\left(U_{\zeta }\right)\right)\parallel & =\frac{1}{2}\frac{{\psi }_0(\parallel V_{\zeta }-U_{*}\parallel )+{\psi }_0(\parallel U_{\zeta }-U_{*}\parallel )}{1-{\psi }_0(\parallel U_{\zeta }-U_{*}\parallel )}\hfill \\ & +\frac{3}{2}\left[{\left(\frac{{\psi }_0(\parallel V_{\zeta }-U_{*}\parallel )+{\psi }_0(\parallel U_{\zeta }-U_{*}\parallel )}{1-{\psi }_0(\parallel U_{\zeta }-U_{*}\parallel )}\right)}^2+\frac{2{\psi }_0(\parallel V_{\zeta }-U_{*}\parallel )}{1-{\psi }_0(\parallel U_{\zeta }-U_{*}\parallel )}\right]+1.\hfill \end{array}$$

Hence, function A can be defined by

$$\begin{array}{cc}\hfill A\left(t\right)& =\frac{1}{2}\frac{{\psi }_0\left({\phi }_1\left(t\right)t\right)+{\psi }_0\left(t\right)}{1-{\psi }_0\left(t\right)}+\left[{\left(\frac{{\psi }_0\left({\phi }_1\left(t\right)t\right)+{\psi }_0\left(t\right)}{1-{\psi }_0\left(t\right)}\right)}^2+\frac{2{\psi }_1\left({\phi }_1\left(t\right)t\right)}{1-{\psi }_0\left(t\right)}\right]+1.\hfill \end{array}$$

(62)

Similarly,

$$\begin{array}{cc}\hfill P\left(I\right)-P\left(T\left(U_{\zeta }\right)\right)& =\frac{3}{2}{I}^2-\frac{1}{2}I+4I-\frac{3}{2}T\left(U_{\zeta }\right)+\frac{1}{2}T\left(U_{\zeta }\right)-4I\hfill \\ & =\frac{3}{2}\left(I^2-{\left(F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right)}^2\right)-\frac{1}{2}\left(I-F^{\prime }{\left(U_{\zeta }\right)}^{-1}{F}^{\prime }\left(V_{\zeta }\right)\right),\hfill \end{array}$$

(63)

so we can define B by

$$\begin{array}{cc}\hfill B\left(t\right)& =\frac{3}{2}\left[{\left(\frac{{\psi }_0\left(\phi \left(t\right)t\right)+{\psi }_0\left(t\right)}{1-{\psi }_0\left(t\right)}\right)}^2+\frac{{\psi }_1\left({\phi }_1\left(t\right)t\right)}{1-{\psi }_0\left(t\right)}\right]\hfill \\ & +\frac{1}{2}\frac{{\psi }_0\left({\phi }_1\left(t\right)t\right)+{\psi }_0\left(t\right)}{1-{\psi }_0\left(t\right)},\hfill \\ & =A\left(t\right)-1.\hfill \end{array}$$

(64)

Remark 1.

The results in this section were obtained using hypotheses only on the first derivative, in contrast to the results in Theorem 1 where hypotheses up to the seventh derivative of F were used to show the convergence order six. Hence, we have extended the usage of method (25) in Banach space valued operators. Notice also that there are even simple functions defined on the real line, where the hypotheses of Theorem 1 do not hold. Hence, the method 2 may or may not converge. As a motivational and academic example, see Example 6 in the next section. Then, notice that the third derivative of F does not exist. Using the approach of Theorem (2), we bypass the computation of higher-order-than-one derivatives. However, we assume hypotheses only on the first-order derivative of operator F. For obtaining the order of convergence, we adopted

$$\rho =\frac{ln\frac{\parallel u_{\sigma +2}-\xi \parallel }{\parallel u_{\sigma +1}-\xi \parallel }}{ln\frac{\parallel u_{\sigma +1}-\xi \parallel }{\parallel u_{\sigma }-\xi \parallel }},\hspace{1em}\mathrm{for} \mathrm{each} \sigma =0,1,2,3,4,\dots$$

(65)

or

$${\rho }^{*}=\frac{ln\frac{\parallel u_{\sigma +2}-u_{\sigma +1}\parallel }{\parallel u_{\sigma +1}-u_{\sigma }\parallel }}{ln\frac{\parallel u_{\sigma +1}-u_{\sigma }\parallel }{\parallel u_{\sigma }-u_{\sigma -1}\parallel }},\hspace{1em}\mathrm{for} \mathrm{each} \sigma =1,2,3,4,\dots$$

(66)

the computational order of convergence $COC$, and the approximate computational order of convergence $ACOC$ [17,18], respectively. These definitions can also be found in [19]. They do not require derivatives higher than one. Indeed, notice that to generate iterates $u_{\sigma }$ and therefore compute ρ and ${\rho }^{*}$, we need to use the formula (2) using only the first derivatives. It is vital to note that $ACOC$ does not need the prior information of the exact root ξ.

4. Numerical Experimentation

Here, we demonstrate the suitability of our iterative methods for real-life complications. In addition, we also want to validate our theoretical results which were presented in earlier sections. Therefore, we consider four real-life issues (namely, Bratu’s one-dimension, Fisher’s, kinematic synthesis, and Hammerstein integration problems), where the fifth one is a standard academic problem and the sixth one is a motivational problem. The corresponding starting initial approximation and zeroes are depicted in examples (1)–(6).

Next, we consider our schemes, namely, (22), (23), and (24), recalled as ($NM1$), ($NM2$), and ($NM3$), respectively to investigate the computational conduct of them with existing techniques. We contrast them with sixth-order schemes given by Hueso et al. [20] and Lotfi et al. [21], where out of them we consider the expressions, namely, (14–15) $\left(\mathrm{for} t_1=-\frac{9}{4} \mathrm{and} s_2=\frac{9}{8}\right)$ and (5), known as $\left(HU\right)$ and $\left(LO\right)$, respectively. In addition, we also compare them with an Ostrowski-type method proposed by Grau-Sánchez et al. [22], where among them we choose the iterative scheme (7), denoted by $\left(GR\right)$. Further, we contrast them with sixth-order iterative schemes presented by Sharma and Arora [23] (expression-18) and Abbasbandy et al. [24] (expression-8), notated as $\left(SA\right)$ and $\left(AB\right)$, respectively. Furthermore, we contrast them with solution techniques of order six designed by Soleymani et al. [25] (method-5) and Wang and Li [26] (method-6), known as $\left(SO\right)$ and $\left(WL\right)$, respectively.

In the Table 1, Table 2, 4, 6 and 7, we report our findings’ iteration indexes $\left(n\right)$, $(\parallel F\left(x_{\zeta }\right)\parallel )$, $\parallel U_{\zeta +1}-U_{\zeta }\parallel$, $\rho =\frac{log\left[\parallel U_{\zeta +1}-U_{\zeta }\parallel /\parallel U_{\zeta }-U_{\zeta -1}\parallel \right]}{log\left[\parallel U_{\zeta }-U_{\zeta -1}\parallel /\parallel U_{\zeta -1}-U_{\zeta -2}\parallel \right]}$, $\frac{\parallel U_{\zeta +1}-U_{\zeta }\parallel }{\parallel U_{\zeta }-U_{\zeta -1}{\parallel }^6}$ and $\eta$ by using Mathematica (Version 9) with multiple precision arithmetic and minimum 300 digits of mantissa that minimize the rounding-off errors. Further, the variable $\eta$ is the last obtained value of $\frac{\parallel U_{\zeta +1}-U_{\zeta }\parallel }{\parallel U_{\zeta }-U_{\zeta -1}{\parallel }^6}$. Furthermore, the radii of convergence and the consumption of central processing unit (CPU) time by distinct schemes are depicted in the Tables 8 and 9, respectively. The $\alpha (\pm \beta )$ indicates $\alpha \times {10}^{(\pm \beta )}$.

Table 1. Conduct of different techniques in Bratu’s problem Example 1.

Methods	$\mathit{\zeta }$	$\parallel \mathit{F}\left({\mathit{U}}_{\mathit{\zeta }}\right)\parallel$	$\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel$	${\mathit{\rho }}^{*}$	$\frac{\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel }{\parallel {\mathit{U}}_{\mathit{\zeta }}-{\mathit{U}}_{\mathit{\zeta }-1}{\parallel }^6}$	$\mathit{\eta }$
$HU$	1	$2.2 (-4)$	$1.2 (-2)$
	2	$2.1 (-11)$	$1.1 (-8)$		$1.083162484 (-4)$	$3.174505914 (+1)$
	3	$1.3 (-46)$	$7.0 (-44)$	$5.3632$	$3.174505914 (+1)$
$LO$	1	$2.2 (-4)$	$1.2 (-1)$
	2	$5.7 (-13)$	$3.1 (-10)$		$1.87422921 (-4)$	$5.428184301 (+1$
	3	$9.7 (-58)$	$2.8 (-56)$	$5.3634$	$5.428184301 (+1)$
$GR$	1	$1.3 (-5)$	$7.1 (-3)$
	2	$9.0 (-21)$	$4.8 (-18)$		$3.793760743 (-5)$	$3.818203150 (-5)$
	3	$8.9 (-112)$	$4.8 (-109)$	$5.9998$	$3.818203150 (-5)$
$SA$	1	$1.1 (-3)$	$5.5 (-1)$
	2	$7.4 (-9)$	$4.0 (-6)$		$1.486642520 (-4)$	$3.377015100 (-4)$
	3	$2.6 (-39)$	$1.4 (-36)$	$5.9306$	$3.377015100 (-4)$
$AB$	1	$1.9 (-3)$	$9.1 (-1)$
	2	$1.4 (-7)$	$7.6 (-5)$		$1.306885506 (-4)$	$5.224036587 (-4)$
	3	$1.8 (-31)$	$9.9 (-29)$	$5.8525$	$5.224036587 (-4)$
$SO$	1	$3.0 (-6)$	$1.6 (-3)$
	2	$7.3 (-25)$	$3.9 (-22)$		$2.402458013 (-5)$	$2.407289926 (-5)$
	3	$1.5 (-136)$	$8.2 (-134)$	$6.0000$	$2.407289926 (-5)$
$WL$	1	$8.2 (-4)$	$4.2 (-1)$
	2	$1.5 (-9)$	$8.1 (-7)$		$1.427111276 (-4)$	$2.712173991 (-4)$
	3	$1.4 (-43)$	$7.4 (-41)$	$5.9512$	$2.712173991 (-4)$
$NM1$	1	$6.7 (-5)$	$3.5 (-2)$
	2	$3.6 (-16)$	$1.9 (-13)$		$9.787819663 (-5)$	$1.020710352 (-4)$
	3	$9.7 (-84)$	$5.2 (-81)$	$5.9984$	$1.020710352 (-4)$
$NM2$	1	$3.7 (-5)$	$2.0 (-2)$
	2	$8.2 (-18)$	$4.4 (-15)$		$6.899443783 (-5)$	$7.050512396 (-5)$
	3	$9.3 (-94)$	$5.0 (-91)$	$5.9993$	$7.050512396 (-5)$
$NM3$	1	$1.2 (-6)$	$6.5 (-4)$
	2	$1.3 (-27)$	$7.5 (-25)$		$9.459851406 (-6)$	$9.479844812 (-6)$
	3	$2.5 (-153)$	$1.3 (-150)$	$6.0000$	$9.479844812 (-6)$

Table 2. Conduct of different techniques in Fisher’s Equation (2).

Methods	$\mathit{\zeta }$	$\parallel \mathit{F}\left({\mathit{U}}_{\mathit{\zeta }}\right)\parallel$	$\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel$	${\mathit{\rho }}^{*}$	$\frac{\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel }{\parallel {\mathit{U}}_{\mathit{\zeta }}-{\mathit{U}}_{\mathit{\zeta }-1}{\parallel }^6}$	$\mathit{\eta }$
$HU$	1	$4.0$	$8.9 (-1)$
	2	$1.7 (-5)$	$1.8 (-61)$		$3.767384807 (-6)$	$1.887278364$
	3	$8.4 (-34)$	$7.6 (-35)$	$4.9968$	$1.887278364$
$LO$	1	$6.8$	$1.3$
	2	$2.6 (-5)$	$2.7 (-6)$		$6.210769679 (-7)$	$6.028784826 (-3)$
	3	$2.7 (-35)$	$2.6 (-36)$	$5.2967$	$6.028784826 (-3)$
$GR$	1	$4.8 (-1)$	$1.1 (-1)$
	2	$1.5 (-12)$	$1.6 (-13)$		$1.337154575 (-13)$	$1.444113141 (-13)$
	3	$1.5 (-83)$	$1.4 (-84)$	$5.9996$	$1.444113141 (-13)$
$SA$	1	$2.0 (+1)$	$3.5$
	2	$4.7 (-3)$	$4.9 (-4)$		$2.612933396 (-7)$	$1.099716041 (-4)$
	3	$1.5 (-25)$	$1.5 (-26)$	$5.8382$	$1.099716041 (-4)$
$AB$	1	$1.4 (+1)$	$2.9$
	2	$2.4 (-3)$	$3.1 (-4)$		$5.378390973 (-7)$	$1.597885339 (-6)$
	3	$1.5 (-26)$	$1.4 (-27)$	$5.8809$	$1.597885339 (-6)$
$SO$	1	$2.7 (-1)$	$6.8 (-2)$
	2	$4.5 (-14)$	$4.8 (-15)$		$1.923091087 (-12)$	$2.166790648 (-12)$
	3	$4.8 (-93)$	$4.7 (-94)$	$5.9994$	$2.166790648 (-12)$
$WL$	1	$2.0 (+1)$	$3.5$
	2	$4.7 (-3)$	$4.9 (-4)$		$1.666475363 (-16)$	$1.692929278 (-16)$
	3	$1.5 (-25)$	$1.5 (-26)$	$5.9999$	$1.692929278 (-16)$
$NM1$	1	$1.4$	$4.1 (-1)$
	2	$1.3 (-8)$	$1.8 (-9)$		$2.468042987 (-7)$	$2.892790896 (-7)$
	3	$1.1 (-58)$	$1.1 (-59)$	$5.9917$	$2.892790896 (-7)$
$NM2$	1	$1.0$	$2.4 (-1)$
	2	$2.1 (-10)$	$2.1 (-11)$		$6.595078065 (+5)$	$3.781460299 (+4)$
	3	$1.9 (-70)$	$1.8 (-71)$	$6.2266$	$3.781460299 (+4)$
$NM3$	1	$3.6 (-1)$	$8.7 (-2)$
	2	$3.6 (-13)$	$3.9 (-14)$		$2.391680319 (+5)$	$7.549898932 (+3)$
	3	$4.1 (-87)$	$4.0 (-88)$	$6.1921$	$7.549898932 (+3)$

Example 1. Bratu Problem:

We find the huge applicability of the Bratu Problem [27] in the areas of thermal reaction, the Chandrasekhar model of the expansion of the universe, the chemical reactor theory, radiative heat transfer, and the fuel ignition model of thermal combustion and nanotechnology [28,29,30,31]. The mathematical formulation of this problem is given below:

$$\begin{array}{c}\hfill y^{″}+C{e}^y=0, y\left(0\right)=y\left(1\right)=0.\end{array}$$

(67)

By adopting the following central difference

$$y_{\sigma }^{″}=\frac{y_{\sigma -1}-2y_{\sigma }+y_{\sigma +1}}{h^2}, \sigma =1, 2, \cdots , 51,$$

it yields the following nonlinear system of size $50\times 50$ from BVP (67) with step size $h=1/50$

$$h^{2}Cexp\left(y_{\sigma }\right)+\left(y_{\sigma -1}+y_{\sigma +1}-2y_{\sigma }\right)=0,\hspace{1em}\sigma =1, 2, \cdots , 51.$$

(68)

We consider $C=3$ and initial value ${(sin\left(\pi h\right),sin\left(2\pi h\right),\cdots ,sin\left(50\pi h\right))}^T$ for this problem, and computational out-comings are depicted in Table 1.

Example 2.

Here, we choose another well-known Fisher’s equation [32]

$$\begin{array}{c}{\theta }_t=D{\theta }_{xx}+\theta (1-\theta )=0,\hfill \\ \mathit{with} \mathit{homogeneous} \mathit{Neumann’s} \mathit{boundary} \mathit{conditions}\hfill \\ \theta (x,0)=1.5+0.5cos\left(\pi x\right),0\le x\le 1,\hfill \\ {\theta }_x(0,t)=0,\forall t\ge 0,\hfill \\ {\theta }_x(1,t)=0,\forall t\ge 0,\hfill \end{array}$$

(69)

where D is the diffusion parameter. We adopted the finite difference discretization technique in order to convert the above differential Equation (69) into a system of nonlinear equations. Thus, we chose $w_{i,j}=\theta (x_i,t_j)$ as the required solution at the grid points of the mesh. In addition, x and t are the numbers of steps in the direction of M and N, respectively. Moreover, h and k are the corresponding step sizes of M and N, respectively. By adopting central, backward, and forward differences, it resulted in:

$$\begin{array}{c}{\theta }_{xx}(x_i,t_j)=(w_{i+1,j}-2w_{i,j}+w_{i-1,j})/h^2,\hfill \\ {\theta }_t(x_i,t_j)=(w_{i,j}-w_{i,j-1})/k,\hfill \\ \mathit{and}\hfill \\ {\theta }_x(x_i,t_j)=(w_{i+1,j}-w_{i,j})/\left(h\right),\hspace{1em}t\in [0,1],\hfill \end{array}$$

that leading to

$$\begin{array}{c}\hfill \frac{w_{1,j}-w_{i,j-1}}{k}-w_{i,j}\left(1-w_{i,j}\right)-D\frac{w_{i+1,j}-2w_{i,j}+w_{i-1,j}}{h^2},\hspace{1em}i=1,2,3,\cdots ,M,j=1,2,3,\cdots ,N,\end{array}$$

(70)

where $h=\frac{1}{M},k=\frac{1}{N}$. For particular values of $M=9$, $N=9$, $h=\frac{1}{9},k=\frac{1}{9}$ and $D=1$, which led us to a nonlinear system of size $81\times 81$, with the starting point $x_0={(i/81)}^T,i=1,2,\cdots ,8$ convergence towards the following solution:

$$u(x_i,t_j)=\left(\begin{array}{c}\hfill 1.6017\dots ,1.4277\dots ,1.3328\dots ,1.2740\dots ,1.2331\dots ,1.2022\dots ,1.1772\dots ,1.1563\dots ,1.1385\dots \\ \hfill 1.5726\dots ,1.4159\dots ,1.3277\dots ,1.2717\dots ,1.2322\dots ,1.2017\dots ,1.1770\dots ,1.1563\dots ,1.1384\dots \\ \hfill 1.5203\dots ,1.3940\dots ,1.3182\dots ,1.2676\dots ,1.2303\dots ,1.2009\dots ,1.1767\dots ,1.1561\dots ,1.1384\dots \\ \hfill 1.4521\dots ,1.3648\dots ,1.3055\dots ,1.2619\dots ,1.2278\dots ,1.1998\dots ,1.1762\dots ,1.1559\dots ,1.1383\dots \\ \hfill 1.3771\dots ,1.3321\dots ,1.2911\dots ,1.2556\dots ,1.2250\dots ,1.1985\dots ,1.1756\dots ,1.1556\dots ,1.1381\dots \\ \hfill 1.3045\dots ,1.2998\dots ,1.2768\dots ,1.2492\dots ,1.2221\dots ,1.1973\dots ,1.1750\dots ,1.1554\dots ,1.1380\dots \\ \hfill 1.2429\dots ,1.2719\dots ,1.2642\dots ,1.2436\dots ,1.2196\dots ,1.1961\dots ,1.1745\dots ,1.1551\dots ,1.1379\dots \\ \hfill 1.1990\dots ,1.2514\dots ,1.2550\dots ,1.2395\dots ,1.2178\dots ,1.1953\dots ,1.1742\dots ,1.1550\dots ,1.1379\dots \\ \hfill 1.1768\dots ,1.2406\dots ,1.2501\dots ,1.2373\dots ,1.2168\dots ,1.1949\dots ,1.1740\dots ,1.1549\dots ,1.1378\dots \end{array}\right).$$

We depicted the numerical out-coming in Table 2.

Example 3.

Here, we choose a remarkable kinematic synthesis problem that is related to steering, as mentioned in [4,5], which is defined as follows:

$$\begin{array}{c}{\left[E_i\left(u_{2}sin\left({\psi }_i\right)-u_3\right)-F_i\left(u_{2}sin\left({\varphi }_i\right)-u_3\right)\right]}^2+{\left[F_i\left(u_{2}cos\left({\varphi }_i\right)+1\right)-F_i\left(u_{2}cos\left({\psi }_i\right)-1\right)\right]}^2\hfill \\ -{\left[u_1\left(u_{2}sin\left({\psi }_i\right)-u_3\right)\left(u_{2}cos\left({\varphi }_i\right)+1\right)-u_1\left(u_{2}cos\left({\psi }_i\right)-u_3\right)\left(u_{2}sin\left({\varphi }_i\right)-u_3\right)\right]}^2=0, \mathit{for} i=1,2,3,\hfill \end{array}$$

(71)

where

$$E_i=-u_3{u}_2\left(sin\left({\varphi }_i\right)-sin\left({\varphi }_0\right)\right)-u_1\left(u_{2}sin\left({\varphi }_i\right)-u_3\right)+u_2\left(cos\left({\varphi }_i\right)-cos\left({\varphi }_0\right)\right), i=1,2,3$$

and

$$F_i=-u_3{u}_{2}sin\left({\psi }_i\right)+\left(-u_2\right)cos\left({\psi }_i\right)+\left(u_3-u_1\right)u_{2}sin\left({\psi }_0\right)+u_{2}cos\left({\psi }_0\right)+u_1{u}_3, i=1,2,3.$$

The values of ${\psi }_i$ and ${\varphi }_i$ (in radians) are depicted in Table 3 and the behavior of methods in Table 4. We chose the starting approximation $u_0=(0.7, 0.7, 0.7)$ that converges to

$$\xi ={(0.9051567\cdots , 0.6977417\cdots , 0.6508335\cdots )}^T.$$

Table 3. The parameters ${\psi }_i$ and ${\varphi }_i$ (in radians) used in Example 3.

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

Table 4. Conduct of different techniques in kinematic synthesis problem 3.

Methods	$\mathit{\zeta }$	$\parallel \mathit{F}\left({\mathit{U}}_{\mathit{\zeta }}\right)\parallel$	$\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel$	${\mathit{\rho }}^{*}$	$\frac{\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel }{\parallel {\mathit{U}}_{\mathit{\zeta }}-{\mathit{U}}_{\mathit{\zeta }-1}{\parallel }^6}$	$\mathit{\eta }$
$HU$	1	$3.0 (-4)$	$7.0 (-3)$
	2	$8.7 (-9)$	$9.3 (-7)$		$7.812154542 (+6)$	$8.147793628 (+9)$
	3	$1.4 (-28)$	$5.2 (-27)$	$5.2217$	$8.147793628 (+9)$
$LO$	1	$1.2 (-4)$	$2.7 (-3)$
	2	$4.8 (-13)$	$3.0 (-11)$		$7.926557348 (+4)$	$2.247398482 (+11)$
	3	$6.9 (-53)$	$1.6 (-52)$	$5.1887$	$2.247398482 (+11)$
$GR$	1	$1.2 (-3)$	$3.7 (-2)$
	2	$6.3 (-6)$	$6.1 (-4)$		$2.198663751 (+5)$	$2.496661657 (+8)$
	3	$2.1 (-13)$	$1.3 (-11)$	$4.2910$	$2.496661657 (+8)$
$SA$	1	$7.5 (-4)$	$4.9 (-3)$
	2	$3.3 (-9)$	$6.5 (-7)$		$4.841793597 (+7)$	$1.033201098 (+6)$
	3	$1.7 (-33)$	$7.7 (-32)$	$6.4311$	$1.033201098 (+6)$
$AB$	1	$1.1 (-3)$	$5.3 (-3)$
	2	$9.1 (-9)$	$1.3 (-6)$		$5.600775513 (+7)$	$2.359206614 (+6)$
	3	$2.0 (-31)$	$1.0 (-29)$	$6.3799$	$2.359206614 (+6)$
$SO$	1	$4.3 (-5)$	$5.1 (-3)$
	2	$3.8 (-11)$	$2.1 (-9)$		$1.188878144 (+5)$	$1.015723644 (+4)$
	3	$2.0 (-50)$	$1.0 (-48)$	$6.1675$	$1.015723644 (+4)$
$WL$	1	$5.8 (-4)$	$3.9 (-3)$
	2	$1.0 (-9)$	$2.0 (-7)$		$5.475067615 (+7)$	$8.521077783 (+5)$
	3	$1.5 (-36)$	$5.8 (-35)$	$6.4215$	$8.521077783 (+5)$
$NM1$	1	$1.8 (-4)$	$5.6 (-3)$
	2	$8.7 (-10)$	$6.0 (-8)$		$1.569574095 (+6)$	$7.322404948 (+4)$
	3	$1.2 (-40)$	$4.1 (-39)$	$6.2643$	$7.322404948 (+4)$
$NM2$	1	$1.3 (-4)$	$5.6 (-3)$
	2	$3.8 (-10)$	$2.5 (-8)$		$7.167106213 (+5)$	$4.034357383 (+4)$
	3	$2.9 (-43)$	$1.0 (-41)$	$6.2301$	$4.034357383 (+4)$
$NM3$	1	$9.9 (-5)$	$4.9 (-3)$
	2	$8.2 (-11)$	$5.2 (-9)$		$2.391680319 (+5)$	$7.549898932 (+3)$
	3	$1.2 (-47)$	$4.9 (-46)$	$6.1987$	$7.549898932 (+3)$

Example 4.

We choose here a distinguished problem of applied science that is popular as the Hammerstein integral equation (see [10], pp. 19–20), which is given as follows:

$$x\left(s\right)=1+\frac{1}{5}{\int }_0^{1}F(s,t)x{\left(t\right)}^{3}dt,$$

(72)

where $x\in C[0,1], s,t\in [0,1]$ and the kernel F is

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

To convert the expression (72) into a finite-dimensional problem, we adopt the following Gauss Legendre quadrature formula:

$${\displaystyle {\int }_0^{1}f\left(t\right)dt\simeq \sum _{j=1}^8{w}_{j}f\left(t_j\right),}$$

where $t_j$ and $w_j$ are the abscissas and the weights, respectively. We recall $x\left(t_i\right)$ by $x_i(i=1,2,\dots ,8)$, where we have

$${\displaystyle 5x_i-5-\sum _{j=1}^8{a}_{ij}{x}_j^3=0, \mathit{where}i=1,2,\dots ,8}$$

and

$$a_{ij}=\left\{\begin{array}{c}\hfill w_j{t}_j(1-t_i),j\le i,\\ \hfill w_j{t}_i(1-t_j),i<j.\end{array}\right.$$

The parameters $t_j$ and $w_j$ are mentioned as depicted in Table 5.

Table 5. (Abscissas and weights for $t=8$).

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	$0.59171732124782490246973807\dots$	$0.18134189168918099148257522\dots$
6	$0.76276620495816449290886952\dots$	$0.15685332293894364366898110\dots$
7	$0.89833323870681336979577696\dots$	$0.11119051722668723527217800\dots$
8	$0.98014492824876811584178043\dots$	$0.05061426814518812957626567\dots$

The desired root is ${\xi }^{*}={(1.002096\cdots , 1.009900\cdots , 1.019727\cdots , 1.026436\cdots ,1.026436\cdots , 1.019727\cdots , 1.009900\cdots , 1.002096\cdots )}^T$. We depicted the numerical out coming in Table 6 on the ground of the starting approximation $U_0={\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)}^T$.

Table 6. Conduct of different techniques in Hammerstein integral problem 4.

Methods	$\mathit{\zeta }$	$\parallel \mathit{F}\left({\mathit{U}}_{\mathit{\zeta }}\right)\parallel$	$\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel$	${\mathit{\rho }}^{*}$	$\frac{\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel }{\parallel {\mathit{U}}_{\mathit{\zeta }}-{\mathit{U}}_{\mathit{\zeta }-1}{\parallel }^6}$	$\mathit{\eta }$
$HU$	1	$3.0 (-5)$	$6.5 (-6)$
	2	$6.9 (-31)$	$1.5 (-31)$		$1.994799598$	$9.220736175 (+25)$
	3	$4.4 (-159)$	$9.4 (-160)$	$4.9991$	$9.220736175 (+25)$
$LO$	1	$8.6 (-6)$	$1.8 (-6)$
	2	$8.4 (-37)$	$1.7 (-37)$		$4.445532921 (-3)$	$1.232404905 (+31)$
	3	$1.4 (-189)$	$3.0 (-190)$	$4.9924$	$1.232404905 (+31)$
$GR$	1	$6.7 (-6)$	$1.4 (-6)$
	2	$1.0 (-40)$	$2.1 (-41)$		$2.526393028 (-6)$	$2.741190361 (-6)$
	3	$1.2 (-249)$	$2.6 (-250)$	$5.9990$	$2.741190361 (-6)$
$SA$	1	$9.4 (-6)$	$2.0 (-6)$
	2	$1.5 (-39)$	$3.2 (-40)$		$4.964844066 (-6)$	$5.324312398 (-6)$
	3	$2.7 (-242)$	$5.7 (-243)$	$5.9991$	$5.324312398 (-6)$
$AB$	1	$5.5 (-7)$	$1.2 (-7)$
	2	$8.3 (-47)$	$1.8 (-47)$		$6.557370741 (-6)$	$7.0685748 (-6)$
	3	$1.1 (-285)$	$2.3 (-286)$	$5.9992$	$7.0685748 (-6)$
$SO$	1	$5.4 (-6)$	$1.2 (-6)$
	2	$1.9 (-41)$	$4.1 (-42)$		$1.622600124 (-6)$	$1.775041548 (-6)$
	3	$3.7 (-254)$	$8.0 (-255)$	$5.9999$	$1.775041548 (-6)$
$WL$	1	$1.6 (-6)$	$3.3 (-7)$
	2	$9.0 (-45)$	$1.9 (-45)$		$1.377597868 (-6)$	$1.431074607 (-6)$
	3	$3.3 (-274)$	$7.1 (-275)$	$5.9996$	$1.431074607 (-6)$
$NM1$	1	$6.9 (-6)$	$1.5 (-6)$
	2	$1.4 (-40)$	$2.9 (-41)$		$2.737455233 (-6)$	$2.968285627 (-6)$
	3	$9.1 (-249)$	$2.0 (-249)$	$5.9990$	$2.968285627 (-6)$
$NM2$	1	$6.4 (-6)$	$1.4 (-6)$
	2	$7.1 (-41)$	$1.5 (-41)$		$2.315331324 (-6)$	$2.514101750 (-6)$
	3	$1.5 (-250)$	$3.1 (-251)$	$5.9990$	$2.514101750 (-6)$
$NM3$	1	$5.8 (-6)$	$1.2 (-6)$
	2	$3.3 (-41)$	$7.1 (-42)$		$2.391680319 (+5)$	$7.549898932 (+3)$
	3	$1.3 (-252)$	$2.7 (-253)$	$6.1921$	$7.549898932 (+3)$

Example 5.

Finally, we choose

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

(73)

Here, we picked $\zeta =110$ in order to deduce a huge system of $110\times 110$. In addition, we selected the starting guess $U_0={(1.25, 1.25, 1.25, \cdots , \left(110times\right))}^T$ that converges to $\xi ={(1, 1, 1, \cdots , \left(110times\right))}^T$, and the results are depicted in Table 7.

Table 7. Conduct of different techniques in Example 5.

Methods	$\mathit{\zeta }$	$\parallel \mathit{F}\left({\mathit{U}}_{\mathit{\zeta }}\right)\parallel$	$\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel$	${\mathit{\rho }}^{*}$	$\frac{\parallel {\mathit{U}}_{\mathit{\zeta }+1}-{\mathit{U}}_{\mathit{\zeta }}\parallel }{\parallel {\mathit{U}}_{\mathit{\zeta }}-{\mathit{U}}_{\mathit{\zeta }-1}{\parallel }^6}$	$\mathit{\eta }$
$HU$	1	$2.2 (-1)$	$7.4 (-3)$
	2	$1.5 (-14)$	$5.0 (-15)$		$2.953163177 (-2)$	$4.440762811 (+6)$
	3	$2.0 (-75)$	$6.6 (-76)$	$4.9998$	$4.440762811 (+6)$
$LO$	1	$2.5 (-1)$	$8.4 (-3)$
	2	$1.7 (-16)$	$5.7 (-17)$		$1.658701079 (-4)$	$1.670091725 (-4)$
	3	$1.7 (-101)$	$5.7 (-102)$	$5.9998$	$1.670091725 (-4)$
$GR$	1	$9.1 (-3)$	$3.0 (-3)$
	2	$8.0 (-20)$	$2.7 (-20)$		$3.496581094 (-5)$	$3.502158271 (-5)$
	3	$3.7 (-122)$	$1.2 (-122)$	$6.0000$	$3.502158271 (-5)$
$SA$	1	$2.9 (-2)$	$9.6 (-3)$
	2	$4.7 (-16)$	$1.6 (-16)$		$2.066540292 (-4)$	$2.083784171 (-4)$
	3	$9.6 (-99)$	$3.2 (-99)$	$5.9997$	$2.083784171 (-4)$
$AB$	1	$4.0 (-2)$	$1.3 (-2)$
	2	$6.1 (-15)$	$2.0 (-15)$		$3.387336833 (-4)$	$3.430169462 (-4)$
	3	$7.0 (-92)$	$2.3 (-92)$	$5.9996$	$3.430169462 (-4)$
$SO$	1	$2.0 (-3)$	$6.6 (-4)$
	2	$8.7 (-25)$	$2.9 (-25)$		$3.501864006 (-6)$	$3.502158271 (-6)$
	3	$6.5 (-153)$	$2.2 (-153)$	$6.0000$	$3.502158271 (-6)$
$WL$	1	$3.1 (-2)$	$1.0 (-2)$
	2	$8.7 (-16)$	$2.9 (-16)$		$2.342171924 (-4)$	$2.363956833 (-4)$
	3	$4.2 (-97)$	$1.4 (-97)$	$5.9937$	$2.363956833 (-4)$
$NM1$	1	$1.1 (-2)$	$3.8 (-3)$
	2	$4.2 (-19)$	$1.4 (-19)$		$4.950410601 (-5)$	$4.961390884 (-5)$
	3	$1.1 (-117)$	$3.6 (-118)$	$5.9999$	$4.961390884 (-5)$
$NM2$	1	$6.8 (-3)$	$2.3 (-3)$
	2	$9.3 (-21)$	3.1$(-21)$		$2.235218683 (-5)$	$2.237490006 (-5)$
	3	$5.9 (-128)$	$2.0 (-128)$	$6.0000$	$2.237490006 (-5)$
$NM3$	1	$2.5 (-3)$	$8.3 (-4)$
	2	$7.2 (-24)$	$2.4 (-24)$		$7.590662313 (-6)$	$7.588009587 (-6)$
	3	$4.3 (-147)$	$1.4 (-147)$	$6.0000$	$7.588009587 (-6)$

Example 6.

As a counter-example, we picked a function F on ${\mathbb{E}}_1={\mathbb{E}}_2=\mathbb{R}$, $\mathsf{\Omega }=[-\frac{1}{\pi }, \frac{2}{\pi }]$ by

$$F\left(x\right)=\left\{\begin{array}{cc}{x}^{3}log\left({\pi }^2{x}^2\right)+x^{5}sin\left({\displaystyle \frac{1}{x}}\right),\hfill & x\ne 0\hfill \\ 0,\hfill & x=0\hfill \end{array},\right.$$

that leads to

$$F^{\prime }\left(x\right)=2x^2-x^{3}cos\left(\frac{1}{x}\right)+3x^{2}log\left({\pi }^2{x}^2\right)+5x^{4}sin\left(\frac{1}{x}\right),$$

$$F^{″}\left(x\right)=-8x^{2}cos\left(\frac{1}{x}\right)+2x(5+3log\left({\pi }^2{x}^2\right))+x(20x^2-1)sin\left(\frac{1}{x}\right)$$

and

$$F^{‴}\left(x\right)=\frac{1}{x}\left[(1-36x^2)cos\left(\frac{1}{x}\right)+x\left(22+6log\left({\pi }^2{x}^2\right)+(60x^2-9)sin\left(\frac{1}{x}\right)\right)\right].$$

Surely, we can say that $F^{‴}\left(x\right)$ is not bounded on Ω in the neighborhood of point $x=0$. This means the study prior to Section 5 is not applicable. In particular, hypotheses on the seventh-order derivative of F or even higher are considered to demonstrate the convergence of the proposed scheme in Section 3. Because of this section, we now demand the hypotheses on the first order.

Further, we have

$$H=\frac{80+16\pi +(\pi +12log2){\pi }^2}{2\pi +1}, b=1,{\psi }_0\left(t\right)=\psi \left(t\right)=Ht, {\psi }_1\left(t\right)=1+Ht$$

and functions A and B, as given in Application 3.2. The desired solution of (6) is $x^{*}=\frac{1}{\pi }$. The distinct radii, $U_0$, COC (ρ), and $CPU time$ are stated in Table 8 and Table 9.

Table 8. Different radii of convergence.

Schemes	Distinct parameters that appease the Theorem 1					$\mathit{\rho }$
	${\mathit{R}}_{\mathbf{1}}$	${\mathit{R}}_{\mathbf{2}}$	${\mathit{R}}_{\mathbf{3}}$	$\mathit{R}$	${\mathit{U}}_{\mathbf{0}}$
$NM1$	$0.011971$	$0.0016362$	$0.0032737$	$0.0016362$	$0.3198$	$6.000$
$NM2$	$0.011971$	$0.00096269$	$0.00064786$	$0.00064785$	$0.3177$	$6.000$
$NM3$	$0.011971$	$0.00041256$	$0.0011889$	$0.00041256$	$0.3179$	$6.000$

Table 9. Consumption of CPU time by distinct schemes.

$\mathit{IM}/\mathit{Ex}$	$\mathit{HU}$	$\mathit{LO}$	$\mathit{GR}$	$\mathit{SA}$	$\mathit{AB}$	$\mathit{SO}$	$\mathit{WL}$	$\mathit{NM}1$	$\mathit{NM}2$	$\mathit{NM}3$
Example 1	39.4198	21.3660	28.0788	27.8556	42.7671	27.7756	22.3137	18.5871	13.2373	13.0102
Example 2	17.40530	8.03168	5.59796	11.14688	14.15601	10.18220	7.79853	7.95962	5.76506	5.75208
Example 3	1.29392	0.54339	0.57942	2.48477	0.90264	0.94667	0.53439	0.20717	0.19013	0.18615
Example 4	28.05384	14.07596	14.08696	82.84556	27.91375	27.78366	13.98790	0.10908	0.10708	0.10507
Example 5	51.16305	26.15242	22.25168	53.57176	53.92799	33.67275	27.28121	28.15583	14.22802	8.03466
$TT$	109.28202	70.16949	70.59480	177.90459	139.66752	100.41370	72.91573	55.01879	33.52763	27.08812
$AT$	21.856405	14.033898	11.765800	35.580918	27.933505	20.082739	14.583146	11.003757	6.705525	5.417624

TT: stands for total time for all examples to the corresponding iterative method. AT: means average time taken by corresponding iterative method. CPU: stands for central processing unit.

5. Concluding Remarks

In this paper, a new family of sixth-order schemes was introduced to produce sequences heading to a solution of an equation. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. It turns out that these schemes are superior to existing ones utilizing similar information. Numerical experiments test the convergence criteria and also numerically show the superiority of the new schemes.