2 Preliminaries

Let $E_1$ and $E_2$ stand for complete normed spaces and $L(E_1,E_2)$ denote the space of continuous linear operators sending $E_1$ into $E_2$ . Moreover, define the open ball $U(z,\mu )$ and its closure $U[z,\mu ]$ , respectively, for $\mu>0$ as

$$ \begin{align*} U(z,\mu)=\{ v \in E_1:\|z-v\| <\mu\} \nonumber \end{align*} $$

and

$$ \begin{align*} U[z,\mu]=\{ v \in E_1:\|z-v\| \leq \mu \}. \nonumber \end{align*} $$

A standard auxiliary result on the inverses of operator that are linear is useful.

Lemma 2.1 (Banach’s perturbation lemma) Assume that $T\in L(E_1, E_2)$ and satisfies $\|T-I\| <1$ , where $I:E_1 \longrightarrow E_1$ denotes the identity operator. Then, the following items are valid: the linear operator T is invertible and

$$ \begin{align*} \|T^{-1}\| \leq \frac{1}{1-\|T-I\|}. \nonumber \end{align*} $$

Some basic properties of convex functions are needed. More information about such functions can be found in [Reference Adly, Van Ngai and Nguyen1–Reference Argyros3, Reference Cibulka, Dontchev and Geoffroy7, Reference Hiriart-Verryty and Lemarechal14].

We need a relationship between different types of majorant conditions. Let

Definition 2.1 A function $h_0:[0,R)\longrightarrow (-\infty , +\infty )$ which is twice continuously differentiable is said to be a center-majorant function for G on if for each

$$ \begin{align*}\|M^{-1}(G'(y)-M)\|\leq h_0'(\|y-x^*\|)-h_0'(0)\end{align*} $$

for some operator $M\in L(E_1, E_2)$ which is invertible, independent of y, and may or may not depend on $x^*$ (see also Remark 2.3(iv)).

1. (A1) The function $h_0'$ is convex and strictly increasing.

2. (A2) $h_0(0)=0$ and $h_0'(0)=-1.$

M is chosen to be any linear invertible operator satisfying this condition (see also Remark 2.3 and Section 5).

Let be such that

Definition 2.2 A function $h:[0,\rho )\longrightarrow (-\infty , +\infty )$ which is twice continuously differentiable is said to be a restricted-majorant function for G on $U(x^*,{\rho }),$ if for each $\theta \in [0,1],\,y\in U(x^*,\rho )$

$$ \begin{align*}\|M^{-1}(G'(y)-G'(x^*+\theta(y-x^*)))\|\leq h'(\|y-x^*\|)-h'(\theta\|y-x^*\|).\end{align*} $$

Notice that h depends on the function $h_0.$

1. (A3) The function $h'$ is convex and strictly increasing.

2. (A4) $h(0)=0$ and $h'(0)=-1.$

Notice that the function $h_0$ depends on $x^*$ and whereas the function h on $x^*,\rho $ , and $h_0.$

Definition 2.3 A function $h_1:[0,R)\longrightarrow (-\infty , +\infty )$ which is twice continuously differentiable is said to be a majorant function for G on $U(x^*,{\rho }),$ if for each

$$ \begin{align*}\|M^{-1}(G'(y)-G'(x^*+\theta(y-x^*)))\|\leq h_1'(\|y-x^*\|)-h_1'(\theta\|y-x^*\|).\end{align*} $$

1. (A3)’ The function $h_1'$ is convex and strictly increasing.

2. (A4)’ $h_1(0)=0$ and $h_1'(0)=-1.$

Remark 2.3 (i) It follows by these definitions that

(2.1) $$ \begin{align} h_0'(s)\leq h_1'(s)\,\mathrm{and}\,h'(s)\leq h_1'(s)\,\mathrm{for\ each}\, s\in[0,\rho). \end{align} $$

(ii) Thus, the results in the literature using only $h_1$ (for $M=G'(x^*)$ ) (see [Reference Ferreira9]) can be replaced by the pair $(h_0,h)$ resulting to finer error distances, a larger convergence radius, and a more precise and larger uniqueness radius for the solution $x^*.$ These advantages are obtained under the same computational cost, since in practice the computation of the function $h_1$ requires that of $h_0$ and h as special cases.

(iii) A popular (but not the most flexible) choice for $M=G'(x^*).$ In this case, the solution is simple. However, assumptions allow the determination of a solution that is not necessarily simple.

(iv) The choice of the initial point can be chosen without the actual knowledge of $x^*.$ Suppose that the operator G satisfies the autonomous differential equation [Reference Argyros3, Reference Ortega and Rheinboldt18]:

$$ \begin{align*}G'(x)=P(G(x)),\end{align*} $$

where P is a continuous operator. By this definition, we obtain $G'(x^*)=P(G(x^*))=P(0),$ for any solution $x^*$ of the equation $G(x)=0.$ As an example, define ${G(x)=e^x-1}$ and choose $P(x)=x+1.$ Then, for the operator P satisfies $G'(x)=P(G(x)).$ Therefore, we can choose $M=G'(x^*)$ and $M=P(G(x^*))$ or $M=P(0)$ , which is known although $x^*$ is not known.

Definition 2.4 We define the parameters

$$ \begin{align*}c_1=\sup\{s\in[0,\rho):h_0'(s) < 0\},\end{align*} $$

$$ \begin{align*}c_2=\sup\left\{s\in[0,c_1):\frac{h(s)-sh'(s)}{sh_0'(s)} < 1\right\},\end{align*} $$

and

$$ \begin{align*}c_3=\sup\{s\in (0,c_2):h_0(s) < 0\}.\end{align*} $$

Moreover, define the Newton iteration for solving the equation $h(s)=0$ given by

(2.2) $$ \begin{align} s_0=|x_0-x^*|,\,s_1=\Bigg|\frac{s_0h_0'(s_0)-h_0(s_0)}{h_0'(s_0)}\Bigg|,\,\,s_{n+1}=\Bigg|\frac{s_nh'(s_n)-h(s_n)}{h_0'(s_n)}\Bigg| \,\, \mathrm{for\, each}\ n=1,2,\ldots.\end{align} $$

Define the radius

(2.3) $$ \begin{align} \rho^*=\min\{\rho,c_2\} \end{align} $$

and the parameter

$$ \begin{align*}c_4=\min\{\rho,c_3\}.\end{align*} $$

Next, the main local analysis convergence result for Newton’s method is stated.

Theorem 2.4 Assume that the conditions (A1)–(A5) are valid with the radius $\rho ^*$ as defined in (2.4). Then, if the starter $x_0\in U(x^*,\rho ^*)-\{x^*\},$ the following assertions hold:

(i) The scalar majorant sequence $\{s_n\}$ given by the formula (2.2) converges to zero, belongs in the interval $(0, \rho ^*),$ and the sequence $\{\frac {s_{n+1}}{s_n^2}\}$ is strictly decreasing.

(ii) The sequence $\{x_n\}\subset U(x^*,\rho ^*)$ converges Q-quadratically, to $x^*$ so that

(2.4) $$ \begin{align} \|x^*-x_{n+1}\|\leq \frac{s_{n+1}}{s_n^2}\|x_n-x^*\|^2,\,\,\mathrm{and}\,\,\frac{s_{n+1}}{s_n^2}\leq \frac{h"(s_0)}{2|h_0'(s_0)|} \end{align} $$

for each $n=0,1,2,\ldots .$

(iii) Additionally, if for $\alpha \in (0,\rho ), \frac {h(\alpha )}{(\alpha h_0'(\alpha ))-1}=1,$ then $\rho ^*=\alpha $ is the largest convergence radius. Moreover, $x^* $ is the unique solution of equation (1.1) in the ball $U(x^*, c_4).$

3 The properties of the majorant functions

The parameters related to the majorant functions $h_0,h$ and the domain D are shown to exist and be positive in the auxiliary results that follow in this section. Moreover, the properties of the sequence $\{s_n\}$ which appears in Theorem 2.4 are investigated.

Lemma 3.1 The following items are valid:

and

(3.1) $$ \begin{align} \frac{sh'(s)-h(s)}{h_0'(s)}<0 \end{align} $$

for each $s\in (0,c_1).$

Proof By hypothesis $x^*\in D$ which is an open set. Thus, it follows that By the second condition in $(A_2), h_0'(0)=-1,$ there exists a parameter $\gamma>0$ so that ${h_0'(s)<0}$ for each $s\in (0,\gamma )$ leading us to deduce that $c_1>0.$ Moreover, by the condition $(A_2), h_0(0)=0$ and $h_0'(0)=-1.$ Then, there exists a parameter $\gamma>0$ so that $h_0(s)<0$ for each $s\in (0,\gamma ),$ so $c_3>0.$ By the conditions $(A_3),(A_4)$ and Lemma 2.2,

(3.2) $$ \begin{align} h(s)-sh'(s)<h(0)=0 \end{align} $$

for each But if $s\in (0,c_1),$ then $h_0'(s)<0,$ showing (3.1) by (3.2.)

It follows by the condition $(A_1)$ and the definition of the parameter $c_1$ that the real function

(3.3) $$ \begin{align} \psi_{h_0,h}:[0,c_1)\rightarrow(-\infty,0], s\rightarrow \frac{sh'(s)-h(s)}{h_0'(s)} \end{align} $$

exists in the interval $[0,c_1).$

Lemma 3.2 The following items are valid.

The function $\frac {|\psi _{h_0,h}(s)|}{s^2}$ is strictly increasing

and

(3.4) $$ \begin{align} \frac{|\psi_{h_0,h}(s)|}{s^2}\leq\frac{h"(s)}{2|h_0'(s)|} \end{align} $$

for each $s\in (0,c_1).$

Proof The definition of the function $\psi _{h_0,h}, h_0(0)=0$ and $h_0'(s)<0$ for each $s\in [0,c_1)$ imply that

(3.5) $$ \begin{align} \frac{|\psi_{h_0,h}(s)|}{s^2}&=\left| \frac{sh'(s)-h(s)}{s^2h_0'(s)}\right|\nonumber\\ &=\frac{1}{|h_0'(s)|}\left|\frac{sh'(s)-h(s)}{s^2}\right|,\nonumber\\ &=\frac{1}{h_0'(s)}\int_{0}^{1}\frac{h'(s)-h'(\theta s)}{s}d\theta\nonumber\\ &\leq\frac{1}{|h_0'(s)|}\int_{0}^{1}h"(t)(1-\theta)d\theta. \end{align} $$

Thus, (3.4) holds, where we also used Lemma 2.2 to deduce that the function

$$ \begin{align*}s\rightarrow \frac{h'(s)-h'(\theta s)}{s}\end{align*} $$

for each $s\in (0,c_1),\theta \in (0,1)$ is strictly increasing, and positive as well as the function $s\rightarrow \frac {1}{|h_0'(s)|}$ is strictly increasing. This makes the right-hand side of (3.5) positive.

Lemma 3.3 The following items are valid:

(3.6) $$ \begin{align} c_2>0 \end{align} $$

and

(3.7) $$ \begin{align} |\psi_{h_0,h}(s)|<s \end{align} $$

for each $s\in (0,c_2).$

Proof By the last two lemmas,

(3.8) $$ \begin{align} \frac{|\psi_{h_0,h}(s)|}{s}=\frac{h(s)-sh'(s)}{h_0'(s)}>0, \end{align} $$

and the function $\frac {|\psi _{h_0,h}(s)|}{s^2}$ is bounded close enough to zero. Thus, we can write

(3.9) $$ \begin{align} \lim_{s\rightarrow 0} \frac{|\psi_{h_0,h}(s)|}{s}=\lim_{s\rightarrow 0}\left( \frac{|\psi_{h_0,h}(s)|}{s^2}\right)s=0. \end{align} $$

Hence, by the estimates (3.8) and (3.9), there exists a parameter $\gamma>0$ so that

$$ \begin{align*}0<\frac{h(s)-sh'(s)}{h_0'(s)}<1.\end{align*} $$

Consequently, by the definition of the parameter $c_2,$ and the preceding double inequality, we get $c_2>0.$ Moreover, Lemma 3.2 gives that the function $s\rightarrow \frac {|\psi _{h_0,h}(s)|}{s^2}$ is strictly increasing for each $s\in (0,c_1).$ Furthermore, the definition of the parameter $c_2$ implies (3.7) for each $s\in (0,c_2).$

It follows by (3.4) that the scalar sequence $\{s_n\}$ can also be given as

(3.10) $$ \begin{align} 0<s_0=\|x^*-x_0\|, s_{n+1}=|\psi_{h_0,h}(s_n)| \end{align} $$

for each $n=0,1,2,\ldots $ .

Lemma 3.4 The following items are valid: The scalar sequence $\{s_n\}$ given in (3.10) exists in the interval $(0,c_2)$ for each $n=0,1,2,\ldots $ ;

(3.11) $$ \begin{align} s_{n+1}<s_n, \end{align} $$

i.e., the sequence $\{s_n\}$ is strictly decreasing; the sequence $\{\frac {s_{n+1}}{s^2_n}\}$ is strictly decreasing,

(3.12) $$ \begin{align} \lim_{s\rightarrow +\infty}s_n=0 \end{align} $$

and

(3.13) $$ \begin{align} \frac{s_{n+1}}{s_n^2}\leq\frac{h"(s_0)}{2|h_0'(s_0)|}. \end{align} $$

Proof Lemma 3.3 and the definition of the sequence $\{s_n\}$ given in (3.10) imply by a simple inductive argument that $s_{m+1}=|\psi _{h_0,h}(s_m)|<s_m,$ where we also used the restriction the first formula in (3.10) to deduce $s_0<\rho ^*\leq c_2.$ Thus, the sequence $\{s_m\}$ exists in the interval $(0,c_2),$ remains in $(0,c_2)$ , and satisfies (3.11). Moreover, by the definition (3.10),

(3.14) $$ \begin{align} \frac{s_{m+1}}{s_m^2}=\frac{|\psi_{h_0,h}(s_m)|}{s_m^2}\leq\frac{|\psi_{h_0,h}(s_0)|}{s_0^2}, \end{align} $$

since by Lemma 3.2, the sequence $\{\frac {s_{m+1}}{s_m^2}\}$ is strictly decreasing. Then, the estimate (3.14) and $s_m<s_0$ imply

(3.15) $$ \begin{align} s_{m+1}\leq c s_m, m=0,1,2,\ldots, \end{align} $$

where $c=\frac {|\psi _{h_0,h}(s_0)|}{s_0}\in [0,1)$ by Lemma 2.3.

Consequently, by (3.15), we deduce the validity of the item (3.12) the second item in Lemma 2.2.

Furthermore, we obtain

(3.16) $$ \begin{align} \frac{|\psi_{h_0,h}(s_0)|}{s_0^2}\leq\frac{|h"(s_0)|}{2|h_0'(s_0)|}. \end{align} $$

Finally, the item (3.13) follows by the estimates (3.14) and (3.16).

Remark 3.5 Let us show the implications of the new conditions $(A_1)-(A_2),$ when compared to the old ones $(A_3)'$ and $(A_4)',$ given in [Reference Ferreira9] for the specializations

$$ \begin{align*} h_0(s)=\frac{l_0}{2}s^2-s, h(s)=\frac{l}{2}s^2-s \end{align*} $$

and

$$ \begin{align*} h_1(s)=\frac{l_1}{2}s^2-s \end{align*} $$

for some parameters $l_0>0, l>0$ and $l_1>0.$ Notice that

(3.17) $$ \begin{align} l_0\leq l_1 \end{align} $$

and

(3.18) $$ \begin{align} l\leq l_1. \end{align} $$

Case 1: $l_0=l=l_1$ (Lipschitz). Then, the definitions of the parameters $c_2$ and $\rho ^*$ imply that

(3.19)

where in order to obtain the second element in the preceding set, we solved for the variable s the inequality

$$ \begin{align*} \frac{\frac{l_1}{2}s^2-s-s(l_1s-1)}{s(l_0s-1)}<1. \end{align*} $$

Case 2: By the condition $(A_5)$ ,

(3.20) $$ \begin{align} l_0\leq l. \end{align} $$

Then, the corresponding radius is

(3.21)

where we again solved for s the inequality

$$ \begin{align*} \frac{\frac{l}{2}s^2-s-s(ls-1)}{s(l_0s-1)}<1. \end{align*} $$

Clearly, it follows by (3.17)–(3.20) that

(3.22) $$ \begin{align} \rho_1^*\leq\rho^*. \end{align} $$

Notice that by (3.19) and (3.21) as $\frac {\ell }{\ell _1}\longrightarrow 0,$ the new radius is at least as three times larger. The value $\rho _1^*$ is due to Rheinboldt [Reference Rheinboldt, Tikhonov, Kuhnert, Kuznecov, Moszynski and Wakulicz21] and Traub [Reference Traub and Wozniakowski24].

In the numerical section, examples are developed where (3.17), (3.18), (3.20), and (3.22) are valid as strict inequalities. Moreover, under Case 1, the corresponding sequence $\{u_n\}$ to $\{s_n\}$ is

$$ \begin{align*} u_{n+1}=\left|\frac{l_1u_n^2}{2(1-l_1u_n)}\right|,\nonumber \end{align*} $$

for $n=0,1,2,\ldots $ ,

whereas

$$ \begin{align*} s_{n+1}=\left|\frac{(2l_0-l)s_n^2}{2(1-l_0s_n)}\right|,\nonumber \end{align*} $$

for $n=0,1,2,\ldots $ . Therefore, by (3.17), (3.18), and these definitions,

(3.23) $$ \begin{align} 0\leq s_{n+1}\leq u_{n+1}, \end{align} $$

for each $n=0,1,2,\ldots $ .

Hence, we conclude that the new error bounds $\{s_n\}$ are tighter than the old ones $\{u_n\}$ given in [Reference Ferreira9]. Notice also that these advantages hold even if $h_0, h$ , and $h_1$ are chosen in ways other than the Lipschitz, since $h_0$ and h are always tighter than $h_1.$ Moreover, the aforementioned advantages require no additional computational effort, since in practice the calculation to determine the function $h_1$ also requires that of the functions $h_0$ and h as special cases. Moreover, there are advantages concerning the uniqueness of the solution $x^*$ (see Remark 3.5). Furthermore, notice that the conditions on $c_1, c_2, c_3,\rho ^*$ are always weaker than the corresponding ones given in [Reference Ferreira9] for $h_0=h=h_1.$ That is, if the old convergence conditions hold, so do the new ones but not necessarily vice versa. Finally, notice that all other results involving the sequence $\{u_n\}$ and the function can be replaced by $\{s_n\}$ and the functions $h_0$ and h (see [Reference Ferreira9] and the references therein).

4 Local Analysis

In this section, we present the association of the majorant functions $h_0$ and h to the nonlinear operator $F.$ Next, we first present a Banach perturbation lemma of inverses for linear operators.

Lemma 4.1 Under the hypotheses of Lemmas 3.1–3.4, further assume that

(4.1)

Then, the following items are valid: $G'(x)^{-1}\in L(E_2,E_1)$ and

(4.2) $$ \begin{align} \|G'(x)^{-1}M\|\leq\frac{1}{|h_0'(\|x_n-x^*\|)|} \end{align} $$

for each Thus, in particular, $G'$ is invertible on

Proof It follows by the condition (4.1) that $h_0'(\|x-x^*\|)<0.$ Thus, the condition $(A_2)$ and Definition 2.1 give

$$ \begin{align*} \|M^{-1}(G'(x)-M)\|\leq h_0'(\|x-x^*\|)-h_0'(0)<-h_0'(0)=1.\nonumber \end{align*} $$

Hence, by Lemma 2.1, $M^{-1}G'(x)\in L(E_1,E_2),$ so $G'(x)^{-1}\in L(E_2,E_1).$

Moreover,

$$ \begin{align*} \|G'(x)^{-1}M\|&\leq\frac{1}{1-\|M^{-1}(G'(x)-M)\|}\nonumber\\ &\leq \frac{1}{1-(h_0'(\|x-x^*\|)-h_0'(0))}=\frac{1}{|h_0'(\|x-x^*\|)|}.\nonumber \end{align*} $$

Finally, the last item is valid due to the inequality

It is well known that the Newton method at a single point is the solution of the linearization at the point at hand. Thus, it is important to study the linearization error at that point in D:

(4.3) $$ \begin{align} \Lambda_G(u_1,u_2)=G(u_2)-G(u_1)-G'(u_1)(u_2-u_1) \end{align} $$

for each $u_2,u_1\in D.$

This error is controlled by the error in the linearization of the majorant functions $h_0, h$ , and $h_1$ defined as follows for each $v_1,v_2\in [0,R)$ :

$$ \begin{align*} \lambda_{h_0}(v_1,v_2)&=h_0(v_2)-h_0(v_1)-h_0'(v_1)(v_2-v_1),\nonumber\\ \lambda_{h}(v_1,v_2)&=h(v_2)-h(v_1)-h'(v_1)(v_2-v_1),\nonumber \end{align*} $$

and

$$ \begin{align*} \lambda_{h_1}(v_1,v_2)=h_1(v_2)-h_1(v_1)-h_1'(v_1)(v_2-v_1).\nonumber \end{align*} $$

Notice that the pair $(\Lambda _G,\lambda _{h_1})$ are used in the local analysis of convergence in [Reference Ferreira9] (see also [Reference Ferreira9–Reference Ferreira and Svaiter13]).

Lemma 4.2 Assume that Then, the following error estimate is valid:

(4.4) $$ \begin{align} \|M^{-1}\Lambda_G(x,x^*)\|\leq\lambda_{h}(\|x-x^*\|,0) \end{align} $$

for each

The invertibility of $G'$ is guaranteed by Lemma 4.1 in the ball $U(x^*,\rho ^*).$

Thus, the Newton iteration operator

(4.5) $$ \begin{align} N_G:U(x^*,\rho^*)\rightarrow E_2,\nonumber\\ x\rightarrow x-G'(x)^{-1}G(x) \end{align} $$

is well defined on $U(x^*,\rho ^*).$

The next auxiliary lemma develops conditions to guarantee that the Newtonian iterates can be repeated indefinitely.

Lemma 4.3 Let $s\in (0,\rho ^*).$ Assume $x\in U(x^*,s).$ Then, the following error estimate is valid:

(4.6) $$ \begin{align} \|N_G(x)-x^*\|\leq\frac{|\psi_{h_0,h}(s)|}{s^2}\|x-x^*\|^2 \end{align} $$

for each $x\in U(x^*,s).$

Proof Inequality (4.6) is trivially valid for $x=x^*,$ since $G'(x^*)=0.$ Thus, we can assume $\|x-x^*\|\in (0,s).$ Lemma 4.1 assures the invertibility of the linear operator $G'(x).$ Then, we have the identity

(4.7) $$ \begin{align} x^*-N_G(x)&=-G'(x)^{-1}[G(x^*)-G(x)-G'(x)(x^*-x)]\nonumber\\ &=-G'(x)^{-1}\Lambda_G(x,x^*). \end{align} $$

Consequently, by (4.7) and Lemmas 4.1 and 4.2, we get

(4.8) $$ \begin{align} \|x^*-N_G(x)\|&\leq\|-G'(x)M\|\|M^{-1}\Lambda_G(x,x^*)\|\nonumber\\ &\leq\frac{\lambda_{h}(\|x-x^*\|,0)}{|h_0'(\|x-x^*\|)|}\leq|\Lambda_{h_0,h}(\|x-x^*\|)|, \end{align} $$

since $h_0(0)=h(0)=0.$ Then, the application of Lemma 3.2 for $x\in U(x^*,s)$ implies

(4.9) $$ \begin{align} \frac{|\Lambda_{h_0,h}(\|x-x^*\|)|}{\|x-x^*\|}\leq\frac{|\lambda_{h_0,h}(s)|}{s^2}. \end{align} $$

Next, by (4.8) and (4.9), we obtain

$$ \begin{align*} \frac{\lambda_{h_0,h}(\|x-x^*\|,0)}{|h_0'(\|x-x^*\|)|}\leq\frac{|\psi_{h_0,h}(s)|}{s^2}\|x-x^*\|,\nonumber \end{align*} $$

leading to the validation of the item (4.8).

Corollary 4.4 Let $s\in (0,\rho ^*).$ Then, the following are valid:

(4.10) $$ \begin{align} N_G(U[x^*,s])\subset U[x^*,|\psi_{h_0,h}(s)|] \end{align} $$

and

(4.11) $$ \begin{align} N_G(U[x^*,\rho^*])\subset U(x^*,\rho^*). \end{align} $$

Next, a domain is determined that contains only one solution which is $x^*$ .

Remark 4.6 If $h_0=h=h_1$ , Lemma 4.1 and Proposition 4.5 reduce to the corresponding Lemmas 2.9 and 2.13, respectively. Otherwise, since

$$ \begin{align*} |h_0(s)| \leq |h_1(s)| \nonumber \end{align*} $$

and

$$ \begin{align*} |h_0(s)| \leq |h_1(s)|. \nonumber \end{align*} $$

The new results provide tighter upper error bounds on the norms $\|G'(x)^{-1}M\|$ and a larger radius of uniqueness for the solution $x^*$ (see also Remark 3.5). In particular, under the Lipchitz case, we have

$$ \begin{align*} \|G'(x)^{-1}M\| \leq \frac{1}{1-l_0\|x-x^*\|} \leq \frac{1}{1-l_1 \|x-x^*\|}. \nonumber \end{align*} $$

In order to specify $c_3$ , we must solve the inequality $h_0(s)<0$ , leading to

$$ \begin{align*} c_3 =\frac{2}{l_0}. \nonumber \end{align*} $$

The corresponding $c_3$ given in Lemma 2.13 in [Reference Ferreira9] is obtained if we solve for s the inequality $h_1(s)<0$ , leading to

Consequently, we get

Next, the proof of Theorem 2.4 is developed based on the abovementioned auxiliary results.

Proof of Theorem 2.4

It follows by (2.2) and (4.5) that the Newton iteration $\{x_n\}$ satisfies the identity

(4.12) $$ \begin{align} x_{n+1}=N_G(x_n),\ \text{for \ each}\ n=0,1,2,....\\[-36pt]\nonumber \end{align} $$

Next, notice that all the items concerning the scalar sequence $\{s_n\}$ are shown in Lemma 3.4. Moreover, the Newton iteration $\{x_n\}$ is well defined and belongs in $U(x^*,\rho ^*)$ . Indeed, for $x_0 \in U(x^*,\rho ^*)$ and, since $\rho ^* \leq c_1$ , the formula (4.12), the item $N_G (U(x^*,\rho ^*)) \subset U(x^*, \rho ^*)$ in Corollary 4.4 (see (4.11)) and Lemma 4.1 validate the claim. Next, we are going to determine that $\lim _{m \rightarrow + \infty } x_m =x^*$ . Let us show that the scalar sequence $\{s_m\}$ majorizes the Newton sequence $\{x_m\}$ , i.e.,

(4.13) $$ \begin{align} \|x^*-x_m\| \leq s_m ,\ \text{for each}\ m=0,1,2,.... \end{align} $$

By the choice $s_0=\|x^*-x_0\|$ , inequation (4.13) holds if $m=0$ . We employ mathematical induction and assume that $\|x^*-x_m\| \leq s_m$ . Then, by Lemma 3.4, $\{s_m\} \subset (0,\rho ^*)$ . Furthermore, the application of the definition of the sequence $\{s_n\}$ given by the formula (3.10) and the identity (4.12) gives

(4.14) $$ \begin{align} \|x^*-x_{m+1}\| \leq \|x^*-N_G(x_m)\| \leq |\psi_{h_0,h}(s_m)|=s_{m+1}. \end{align} $$

The proof of the induction for inequation (4.13) is complete. Furthermore, by (4.13) and (3.12), $\lim _{m \rightarrow + \infty }x_m=x^*$ , since $\lim _{m \rightarrow +\infty } s_m =0$ . The claim about the optimality of the convergence radius $\rho ^*$ is given in Lemma 2.15 in [Reference Ferreira9]. The first inequality in (2.4) follows by Lemma 3.4 and (4.12) since

$$ \begin{align*} \|x^*-x_{m+1}\|=\|x^*-N_G(x_m)\| \leq \frac{|\psi_{h_0,h}(s_m)|}{s_m^2}\|x^*-x_m\|^2, \nonumber \end{align*} $$

for each $m=0,1,2,...$ . Thus, the last inequation and the definition of the majorant sequence $\{s_m\}$ show the validity of the first inequation in (2.4). Finally, the uniqueness of the solution $x^*$ is shown in Proposition 4.5.

5 Special cases and examples

It is worth noticing that, if we simply specialize the functions $h_0$ and h, the results extend the applicability of the classical cases [Lipchitz] (see also Remark 3.5), under the Smale [Reference Smale, Ewing, Gross and Martin23] or Wang [Reference Wang and Ouyang25, Reference Wang26] conditions and under the Nesterov et al. conditions [Reference Nesterov and Nemirovskii16]. We leave the details to the motivational reader.

Next, we present two examples to test the convergence conditions and further validate the theoretical results. We have chosen $M=G'(x^*).$

Example 5.1 Consider, the choice $B_1=B_2=\mathbb {R}^3$ and $D=P[0,1].$ Define the operator on D as

(5.1) $$ \begin{align} G(t)=\Bigg(\frac{e-1}{2}t_1^2+t_1, e^{t_2}-1, t_3\Bigg)^T\,\mathrm{for }\,\,t=(t_1, t_2, t_3)^T \end{align} $$

and $t_1, t_2,t_3\in \mathbb {R}.$ Clearly, $t^*=(0,0,0)^T$ solves the equations. Then, by the definition (5.1), it follows that the derivative according to Fréchet $G'$ of the operator F is defined as

$$ \begin{align*}G'(t)=\left[\begin{array}{ccc} (e-1)t_1+1&0&\\ 0&e^{t_2}&0\\ 0&0&1\\ \end{array}\right].\end{align*} $$

We have $h_0(s)=(e-1)s,\ h(s)= e^{\frac {l}{e-1}}s$ , and $h_1(s)=es$ . Then, $\rho _1^*=0.2453$ [Reference Ferreira9] and $\rho ^*=0.3827.$ Thus, the previous radius of convergence $\rho _1^*$ is smaller than the new one $\rho ^*.$ This allows for a wider choice of initial guesses $x_0,$ and other benefits as already mentioned under novelty in the introduction of this article.

Example 5.2 Let $K[0,1]$ stand the space of continuous functions mapping the interval $[0,1]$ into the real number system. Let $B_1=B_2=K[0,1]$ and $D=P[x^*,1]$ with $x^*(\mu )=0.$ The operator G is defined on $K[0,1]$ as

$$ \begin{align*}G(z)(\mu)=z(\mu)-6\int_0^1\mu z(\tau)^3d\tau.\end{align*} $$

Then, the definition of the derivative according to Fréchet [Reference Argyros4–Reference Bonnans6, Reference Robinson22] gives for the function G

$$ \begin{align*}G'(z(w))(\mu)=w(\mu)-18\int_0^1\mu \tau z(\tau)^2w(\tau)d\tau\end{align*} $$

for each $w\in K[0,1].$ Then, the conditions (A1)–(A5) of Theorem 2.4 are validated, since $G'(x^*(\mu ))=I$ provided that $h_0(s)=h(s)=12s$ and $h_1(s)=24s.$ Then, ${\rho _1^*=0.0278}$ and $\rho ^*=0.0556.$ Notice that as in Example 5.1, $\rho _1^* < \rho ^*.$