1 Introduction

Let \({\mathcal {U}}, {\mathcal {V}},\) be Banach spaces and \(L({\mathcal {U}},{\mathcal {V}}) \) stand for the space of all continuous linear operators mapping \({\mathcal {U}}\) into \({\mathcal {V}}.\) Consider, a differentiable mapping as per Fréchet \({\mathcal {L}}:\Omega \subseteq {\mathcal {U}}\longrightarrow {\mathcal {V}},\) and its corresponding nonlinear equation

$$\begin{aligned} {\mathcal {L}}(x)=0, \end{aligned}$$
(1.1)

with \(\Omega \) denoting a non-empty open set. The task of determining a solution \(x_*\in \Omega \) is very challenging but important since applications from numerous computational disciplines are brought in the form (1.1) (Argyros and Magréñan 2018; Argyros 2004a, b; Ezquerro et al. 2010; Ortega and Rheinboldt 1970; Verma 2019). The analytic form of \(x_*\) is rarely attainable. That is why mainly iterative processes are used to generate approximations to the solution \(x_*.\)

Among these processes, the most widely used is Newton’s and its variants. In particular, Newton’s process (NP) is defined by

$$\begin{aligned} x_0\in \Omega ,\, x_{n+1}=x_n-{\mathcal {L}}'(x_n)^{-1}{\mathcal {L}}(x_n)\,\,\text {for}\,\, n=0,1,2,\ldots \end{aligned}$$
(1.2)

There exists a plethora of results related to the study of NP (Argyros and Magréñan 2018; Argyros 2004a; Argyros and Hilout 2010). These studies are based on the theory inaugurated by Kantorovich and its variants (Argyros 2021, 2022, 2004a, b; Argyros and Magréñan 2018; Argyros and Hilout 2010; Dennis 1968; Ezquerro et al. 2010; Gragg and Tapia 1974; Hernandez 2001; Ezquerro and Hernandez 2018; Kantorovich and Akilov 1982; Ortega and Rheinboldt 1970; Potra and Pták 1980, 1984; Proinov 2010; Rheinboldt 1968; Tapia 1971).

The following conditions (A) are used in non-affine or affine invariant form.

Suppose:

(A1) \(\exists \) point \(x_0\in \Omega \) and parameter \( \lambda \ge 0:\) \({\mathcal {L}}'(x_0)^{-1}\in L({\mathcal {V}}, {\mathcal {U}})\) and

$$\begin{aligned} \Vert {\mathcal {L}}'(x_0)^{-1}{\mathcal {L}}(x_0)\Vert \le \lambda . \end{aligned}$$

(A2) \(\exists \) parameter \(M_1 > 0:\) Lipschitz condition

$$\begin{aligned} \Vert {\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(w_1)-{\mathcal {L}}'(w_2))\Vert \le M_1\Vert w_1-w_2\Vert ) \end{aligned}$$

holds \(\forall w_1\in \Omega \) and \(w_2\in \Omega .\)

(A3)

$$\begin{aligned} \lambda \le \frac{1}{2M_1}. \end{aligned}$$
(1.3)

(A4) \(B[x_0, \rho ]\subset \Omega ,\) where parameter \(\rho > 0\) is given later.

Let us denote \(B[x_0, r]:=\{x\in \Omega :\Vert x-x_0\Vert \le r\}\) for \(r > 0.\) Set \(\rho =r_1=\frac{1-\sqrt{1-2M_1\lambda }}{M_1}.\)

There are many variants of Kantorovich’s convergence result for NP. One of those follows (Chen and Yamamoto 1989; Deuflhard 2004; Kantorovich and Akilov 1982).

Theorem 1.1

Under the conditions A for \(\rho =r_1;\) the NP is contained in \(B(x_0,r_1),\) convergent to a solution \(x_*\in B[x_0, r_1]\) of Eq. (1.1) and

$$\begin{aligned} \Vert x_{n+1}-x_*\Vert \le t_{n+1}-t_n=\frac{M_1(t_n-t_{n-1})^2}{2(1-M_1t_n)}, \end{aligned}$$

where the scalar sequence \(\{t_n\}\) is given by

$$\begin{aligned} t_0=0,\, t_1=\lambda ,\, t_{n+1}=t_n+\frac{M_1(t_n-t_{n-1})^2}{2(1-M_1t_n)}. \end{aligned}$$

Moreover, the convergence is linear if \(\lambda =\frac{1}{2M_1}\) and quadratic if \(\lambda < \frac{1}{2M_1}.\) Furthermore, the solution is unique \(B[x_0, r_1]\) in the first case and in \(B(x_0, r_2)\) in the second, where \(r_2=\frac{1+\sqrt{1-2M_1\lambda }}{M_1}.\)

A plethora of studies has used conditions A (Argyros and Magréñan 2018; Argyros 2004a; Argyros and Hilout 2010).

Example 1.2

Consider cubic polynomial

$$\begin{aligned} c(x)=x^3-\mu . \end{aligned}$$

Let \(\Omega =B(x_0, 1-\mu )\) for some parameter \(\mu \in (0, \frac{1}{2}).\) Choose \(x_0=1.\) Then, the conditions A are verified for \(\lambda =\frac{1-\mu }{3}\) and \(M_1=2(2-\mu ).\) It follows that the estimate

$$\begin{aligned} \frac{1-\mu }{3} > \frac{1}{4(2-\mu )} \end{aligned}$$

holds \(\forall \mu \in (0, \frac{1}{2}).\) That is condition (A3) is not satisfied. Therefore, the convergence is not assured by this theorem also used in Chen and Yamamoto (1989), Dennis (1968), Deuflhard (2004), Ezquerro et al. (2010), Gragg and Tapia (1974), Hernandez (2001), Ezquerro and Hernandez (2018), Kantorovich and Akilov (1982), Ortega and Rheinboldt (1970), Potra and Pták (1980, 1984), Proinov (2010), Rheinboldt (1968), Tapia (1971), Yamamoto (1987a, 1987b, 2000) and Zabrejko and Bnuen (1987). But the NP converges. Hence, clearly, there is a need to improve the results based on condition A, which is only sufficient but not necessary.

In this paper, several avenues are presented for achieving this goal. The idea is to replace the Lipschitz parameter \(M_1\) with smaller ones.

Consider the center Lipschitz condition

$$\begin{aligned} \Vert {\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(w_1)-{\mathcal {L}}'(x_0))\Vert \le M_0\Vert w_1-x_0\Vert \,\,\,\forall w_1\in \Omega , \end{aligned}$$
(1.4)

the set \(\Omega _1=B[x_0, \frac{1}{M_0}]\cap \Omega \) and the Lipschitz -2 condition

$$\begin{aligned} \Vert {\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(w_1)-{\mathcal {L}}'(w_2))\Vert \le M\Vert w_1-w_2\Vert \,\,\,\forall w_1, w_2\in \Omega _1. \end{aligned}$$
(1.5)

Notice that by the definition of the set \(\Omega _1\)

$$\begin{aligned} \Omega _1\subset \Omega _0. \end{aligned}$$
(1.6)

Then, the Lipschitz parameters are related by

$$\begin{aligned} M_0\le M_1, \end{aligned}$$
(1.7)

and

$$\begin{aligned} M\le M_1. \end{aligned}$$
(1.8)

Notice also since parameters \(M_0\) and M are specializations of parameter \(M_1,\, M_1=M_1(\Omega ),\, M_0=M_0(\Omega ), \) but \(M=M(\Omega _1),\) where by \(M_1(\Omega )\) we mean that the parameter \(M_1\) depends on the set \(\Omega .\) Therefore, no additional work is required to find \(M_0\) and M (see also the numerical examples).

Moreover, the ratio \(\frac{M_0}{M}\) can be very small (arbitrarily).

Example 1.3

Define scalar function

$$\begin{aligned} {\mathcal {L}}(x)=b_0x+b_1+b_2 \sin e^{b_3x}, \end{aligned}$$

for \(x_0=0,\) where \(b_j,\, j=0,1,2,3\) are real parameters. It follows by this definition that for \(b_3\) sufficiently large and \(b_2\) sufficiently small, \(\frac{M_0}{M_1}\) can be small (arbitrarily), i.e., \(\frac{M_0}{M_1}\longrightarrow 0.\)

Other extensions involve tighter majorizing sequences for NP (see Sect. 2) and improved uniqueness results for solution \(x_*\) (Sect. 3). The applications appear in Sect. 4 followed by and the conclusions in Sect. 5.

2 Real sequences

Let \(K_0, M_0, K, M\) and \(\lambda \) be positive parameters. An important role in the study of NM is played by the majorizing sequence \(\{s_n\}\) defined for \(s_0=0,\, s_1=\lambda ,\) as

$$\begin{aligned} s_2=s_1+\frac{K(s_1-s_0)^2}{2(1-K_0s_1)},\, s_{n+2}=s_{n+1}+\frac{M(s_{n+1}-s_n)^2}{2(1-M_0s_{n+1})}. \end{aligned}$$
(2.1)

That is why some convergence results are listed for it in what follows in this section.

Lemma 2.1

Suppose conditions \(K_0\lambda <1\) and \(s_{n+1} < \frac{1}{M_0}\) hold for all \(n=1,2,\ldots \) Then, the following assertions hold

$$\begin{aligned} s_n< s_{n+1} < \frac{1}{M_0},\,\, \text {for all}\,\, n=0,1,2,\ldots \end{aligned}$$

and there exists \(s_*\in [\lambda ,\frac{1}{M_0}]\) such that \(\lim _{n\longrightarrow \infty }s_n=s_*.\)

Proof

The definition of sequence \(\{s_n\}\) and the conditions of the Lemma imply the assertion and \(\lim _{n\longrightarrow \infty }s_n=s_*\in [\lambda , \frac{1}{M_0}].\) Notice that \(s_*\) is the least upper bound (unique) of the sequence \(\{s_n\}.\) \(\square \)

Next, criteria stronger than those in Lemma 2.1 are developed for the convergence of the sequence (2.1). However, these criteria are easier to verify than those of the Lemma 2.1.

Define parameter \(\gamma \) by

$$\begin{aligned} \gamma =\frac{2M}{M+\sqrt{M^2+8M_0M}}. \end{aligned}$$

This parameter plays a role in the study of NP.

Suppose from now on that \(K_0\le M_0.\) Define the real quadratic polynomials \(q, q_1, q_2\) by

$$\begin{aligned}{} & {} q(t)=M_0(K-2K_0)t^2+2M_0t-1,\\{} & {} \quad q_1(t)=(MK+2\gamma M_0(K-2K_0))t^2+4\gamma (M_0+K_0)t-4\gamma , \end{aligned}$$

and

$$\begin{aligned} q_2(t)=M_0(K-2(1-\gamma )K_0)t^2+2(1-\gamma )(M_0+K_0)t-2(1-\gamma ). \end{aligned}$$

The discriminants \(D, D_1, D_2\) of these polynomials can be given as

$$\begin{aligned}{} & {} D=4M_0(M_0+K-2K_0)> 0,\\{} & {} \quad D_1=16\gamma (\gamma (M_0-K_0)^2+(M+2\gamma M_0)K) > 0 \end{aligned}$$

and

$$\begin{aligned} D_2=4(1-\gamma )((1-\gamma )(M_0-K_0)^2+2M_0K) > 0, \end{aligned}$$

respectively. It follows by the definition of \(\gamma , q_1\) and \(q_2\) that

$$\begin{aligned} M=\frac{2M_0\gamma ^2}{1-\gamma },\, MK+2\gamma M_0(K-2K_0)=\frac{2M_0\gamma }{1-\gamma }(K-2(1-\gamma )K_0), \end{aligned}$$

since after multiplying the polynomial \(q_2\) by \(\frac{2M_0\gamma }{1-\gamma },\) we obtain the polynomial \(q_1\) i.e.,

$$\begin{aligned} q_1(t)=\frac{2M_0\gamma }{1-\gamma }q_2(t). \end{aligned}$$

That is polynomials \(q_1\) and \(q_2\) have the same roots. Denote by \(\frac{1}{2r_1}\) the unique positive root of polynomial q. This root is given explicitly by the quadratic formula and can be written as

$$\begin{aligned} \frac{1}{2r_1}=\frac{1}{M_0+\sqrt{M^2+M_0(K-2K_0)}}. \end{aligned}$$

Moreover, denote by \(\frac{1}{2r_2}\) the common positive root of polynomials \(q_1\) and \(q_2.\) This root can also be written as

$$\begin{aligned} \frac{1}{2r_2}=\frac{2}{\gamma (M_0+K_0)+\sqrt{(\gamma (M_0+K_0))^2+\gamma (MK+2\gamma M_0(K-2K_0)}}. \end{aligned}$$

Define parameter N by

$$\begin{aligned} N^{-1}=\min \left\{ \frac{1}{2r_1}, \frac{1}{2r_2}\right\} . \end{aligned}$$

Suppose

$$\begin{aligned} \lambda \le \frac{1}{2N}. \end{aligned}$$
(2.2)

By the choice of parameters \(r_1, r_2\) polynomials \(q, q_1, q_2\) and condition (2.2), it follows that \(M_0 s_2 < 1,\) since \(q(\lambda ) < 0,\) \(K_0 \lambda <1,\) \(q_1(\lambda ) \le 0\) and \(q_2(\lambda )\le 0.\) Furthermore, the following estimate holds

$$\begin{aligned} \gamma _0\le \gamma \le 1-\frac{M_0(s_2-s_1)}{1-M_0s_1}, \end{aligned}$$
(2.3)

where the parameter \(\gamma _0=\frac{M(s_2-s_1)}{2(1-M_0s_2)}.\) Indeed, the left hand side inequality reduces to \(q_1(\lambda ) \le 0\) and the right hand side to \(q_2(\lambda )\le 0.\) These assertions are true by the choice of \(\lambda .\)

Lemma 2.2

Under condition (2.2), sequence \(\{s_n\}\) satisfies

$$\begin{aligned}{} & {} s_n < s_{n+1}\le \bar{s}_{**}=\lambda +\left( 1+\frac{\gamma _0}{1-\gamma }\right) \frac{K\lambda ^2}{2(1-K_0\lambda )}, \end{aligned}$$
(2.4)
$$\begin{aligned}{} & {} 0< s_{n+2}-s_{n+1}\le \gamma _0\gamma ^{n-1}\frac{K\lambda ^2}{2(1-K_0\lambda )}\,\,\text {for all}\,\, n=1,2,\ldots \end{aligned}$$
(2.5)

and is convergent to its least upper bound \(s_*\in (\lambda , \bar{s}_{**}]\) so that

$$\begin{aligned} s_*-s_n\le \frac{\gamma _0(s_2-s_1)\gamma ^{n-2}}{1-\gamma }\,\,\text {for all}\,\, n=2,3,\ldots \end{aligned}$$
(2.6)

Proof

Induction is used to show the estimate

$$\begin{aligned} 0< \frac{M(s_{k+1}-s_k)}{2(1-M_0s_{k+1})}\le \gamma \,\, \forall \,\, n=1,2,3,\ldots \end{aligned}$$
(2.7)

It follows by the definition of roots \(\frac{1}{2r_1}, \frac{1}{2r_2}\) and polynomial \(g_1\) that estimate (2.7) holds for \(k=1.\) Using the Definition (2.1) of sequence \(\{s_n\}\) and parameter \(\gamma _0\)

$$\begin{aligned} 0< s_3-s_2\le \gamma _0(s_2-s_1)\Rightarrow & {} s_3\le s_2+\gamma _0(s_2-s_0)\\\Rightarrow & {} s_3\le s_2+(1+\gamma _0)(s_2-s_1)-(s_2-s_1)\\\Rightarrow & {} s_3\le s_1+\frac{1-\gamma _0^2}{1-\gamma _0}(s_2-s_1) < \bar{s}_{**}. \end{aligned}$$

Suppose that estimate (2.7) holds for \(k=1,2,\ldots , n-1.\) Then, similarly by (2.7) and the induction hypotheses, we obtain in turn

$$\begin{aligned} s_{k+2}\le & {} s_{k+1}+\gamma _0\gamma ^{k-1}(s_2-s_1) \\\le & {} s_k+\gamma _0\gamma ^{k-2}(s_2-s_1)+\gamma _0\gamma ^{k-1}(s_2-s_1)\\\le & {} s_1+(1+\gamma _0(1+\gamma +\cdots +\gamma ^{k-1}))(s_2-s_1)\\= & {} \lambda +\left( 1+\gamma _0\frac{1-\gamma ^k}{1-\gamma }\right) (s_2-s_1)\\< & {} \bar{s}_{**}. \end{aligned}$$

It follows by the definition (2.1) of sequence \(\{s_n\},\) estimate (2.3) and induction hypothesis (2.7) that

$$\begin{aligned} 0 < s_{k+2}-s_{k+1}\le \gamma _0\gamma ^{k-1}(s_2-s_1)\le \gamma ^k(s_2-s_1). \end{aligned}$$

Then, the estimate (2.7) for \(k+1\) replacing k holds, if

$$\begin{aligned} \frac{M}{2}(s_{k+2}-s_{k+1})\le \gamma (1-M_0s_{k+1}), \end{aligned}$$

or

$$\begin{aligned} \frac{M}{2}(s_{k+2}-s_{k+1})+\gamma M_0s_{k+1}-\gamma \le 0, \end{aligned}$$

or

$$\begin{aligned} \frac{M}{2}\gamma ^k(s_{2}-s_{1})+\gamma M_0\left( \lambda +\frac{1-\gamma ^{k+1}}{1-\gamma }(s_{2}-s_1)\right) -\gamma \le 0, \end{aligned}$$

or

$$\begin{aligned} p_k(t)\le 0\,\,\text {at}\,\, t=\gamma , \end{aligned}$$
(2.8)

where, the polynomial \(p_k:[0,1)\longrightarrow \mathbb {R}\) is defined by

$$\begin{aligned} p_k(t)=\frac{M}{2}(s_2-s_1)t^{k+1}+t M_0(1+t+\cdots +t^k)(s_2-s_1)-(1-M_0s_1)t. \end{aligned}$$
(2.9)

There is a connection between consecutive polynomials:

$$\begin{aligned} p_{k+1}(t)-p_k(t)= & {} \frac{M}{2}(s_2-s_1)^{k+2}+tM_0(1+t+\cdots +t^{k+1})(s_2-s_1)\\{} & {} -(1-M_0\lambda )t-\frac{M}{2}(s_2-s_1)t^k\\{} & {} -tM_0(1+t+\cdots +t^k)(s_2-s_1)+(1-M_0\lambda )t\\= & {} \frac{1}{2}(2M_0t^2+Mt-1)t^k(s_2-s_1). \end{aligned}$$

It follows that

$$\begin{aligned} p_{k+1}(t)=p_k(t)+\frac{1}{2}q_3(t)t^k(s_2-s-1), \end{aligned}$$

where

$$\begin{aligned} q_3(t)=2M_0t^2+Mt-M. \end{aligned}$$

Notice that \(q_3(\gamma )=0\) by the definition of \(\gamma .\) Then, in particular

$$\begin{aligned} p_{k+1}(\gamma )=p_k(\gamma ). \end{aligned}$$

Define function \(p_\infty :[0,1)\longrightarrow \mathbb {R}\) by

$$\begin{aligned} p_\infty (t)=\lim _{k\longrightarrow \infty }p_k(t). \end{aligned}$$

By this definition and polynomials \(p_k\)

$$\begin{aligned} p_\infty (t)=\gamma (\frac{M_0}{1-\gamma }(s_2-s_1)+M_0s_1-1). \end{aligned}$$
(2.10)

Consequently, assertion (2.8) holds if

$$\begin{aligned} \frac{1}{\gamma }p_\infty (t)\le 0\,\,\text {at}\,\, t=\gamma , \end{aligned}$$

which is true by the choice of parameter \(\frac{1}{2r_2}\) and polynomial \(p_2.\) The induction for assertion (2.7) is terminated leading to the conclusions. \(\square \)

Remark 2.3

The linear convergence of sequence \(\{s_n\}\) is shown under the condition (2.2). This condition provides the smallness of \(\lambda \) to force convergence. The quadratic convergence of sequence \(\{s_n\}\) can be shown if \(\lambda \) is chosen to be bounded above from by a smaller parameter than \(\frac{1}{2N}.\) Moreover, under condition (2.2) an upper bound on iterate \(s_k\) is obtained which can then be used in the proof to show quadratic convergence.

Lemma 2.4

Under condition (2.10) further suppose that for some \(\epsilon > 0,\, \beta =\frac{\epsilon }{\epsilon +1}\)

$$\begin{aligned} M_0\left( \frac{\gamma _0(s_2-s_1)}{1-\gamma }+\lambda +s_2-s_1\right) \le \beta \end{aligned}$$
(2.11)

and

$$\begin{aligned} \lambda < \frac{2}{(1+\epsilon )M}. \end{aligned}$$
(2.12)

Then, the conclusions of Lemma 2.2 hold for sequence \(\{s_n\},\)

$$\begin{aligned} s_{n+1}-s_n\le \frac{M}{2}(1+\epsilon )(s_n-s_{n-1})^2 \end{aligned}$$
(2.13)

and

$$\begin{aligned} 0 < s_{n+1}-s_n\le \frac{1}{b}(b\lambda )^{2^n}, \end{aligned}$$
(2.14)

where \(b=\frac{M}{2}(1+\epsilon )\) and \(b\lambda <1.\)

Proof

Assertion (2.14) certainly holds if the following estimate is shown

$$\begin{aligned} 0 < \frac{M}{2(1-M_0s_{k+1})}\le \frac{M}{2}(1+\epsilon ). \end{aligned}$$
(2.15)

The estimate (2.15) holds for \(k=1,\) since it is equivalent to \(M_0 \lambda \le \beta .\) But this is true by \(M_0\le 2N,\) condition (2.2) and inequality \(\frac{\epsilon M_0}{2(1+\epsilon )N}\le \beta .\)

Define polynomials \(g_n:[0, 1)\longrightarrow \mathbb {R}\) by

$$\begin{aligned} g_n(t)=(1+\epsilon ) M_0\gamma _0(1+t+\cdots +t^{n-1})(s_2-s_1)+(1+\epsilon )M_0 (\lambda +s_2-s_1)-\epsilon . \end{aligned}$$
(2.16)

It follows from this definition that

$$\begin{aligned} g_{n+1}(t)-g_n(t)= (1+\epsilon ) M_0\gamma _0(s_2-s_1)t^n > 0, \end{aligned}$$

Evidently, estimate (2.15) holds. Define function \(g_\infty :(0,1)\longrightarrow \mathbb {R}\) by

$$\begin{aligned} g_\infty (t)=\lim _{k\longrightarrow \infty }g_k(t). \end{aligned}$$

Hence, we get \(g_\infty (t)=\frac{(1+\epsilon )M_0\gamma _0(s_2-s_1)}{1-t}+(1+\epsilon )M_0(\lambda +s_2-s_1)-\epsilon .\) Evidently, the estimate

$$\begin{aligned} g_n(t)\le 0\,\,\text {at}\,\, t=\gamma \end{aligned}$$

holds if instead

$$\begin{aligned} g_\infty (t)\le 0\,\,\text {at}\,\, t=\gamma . \end{aligned}$$

But this is identical to condition (2.11). The induction for the assertion (2.15). Then, it follows by estimate (2.15) and the definition of sequence \(\{s_n\}\) that assertion (2.13) holds. Using the definition of parameter b and estimate (2.13)

$$\begin{aligned} b(s_{k+1}-s_k)\le & {} (b(s_k-s_{k-1}))^2=(b(s_k-s_{k-1})^2\\\le & {} b^2(b(s_{k-1}-s_{k-2})^2)^2=b^2b^2(s_{k-1}-s_{k-2})^{2^2}\\\le & {} b^2b^2b^2(s_{k-2}-s_{k-3})^{2^3}\le \ldots , \end{aligned}$$

thus,

$$\begin{aligned} s_{k+1}-s_k\le & {} b^{1+2+2^2+\cdots +2^{k-1}}\lambda ^{2^k}\\= & {} b^{\frac{2^k-1}{2-1}}\lambda ^{2^k}\\= & {} b^{-1}b^{2^k}\lambda ^{2^k}=\frac{(b\lambda )^{2^k}}{b}. \end{aligned}$$

Notice that \(0< b\lambda <1\) by the condition (2.12). Hence, sequence \(\{s_k\}\) converges quadratically to \(t_*.\) \(\square \)

Remark 2.5

Condition (2.11) is left uncluttered. It can be expressed as a function of \(\lambda \) by

$$\begin{aligned} \varphi (\lambda )=\frac{M_0\gamma _0(s_2-s_1)}{1-\gamma }+M_0(\lambda +s_2-s_1)-\beta . \end{aligned}$$

Suppose

$$\begin{aligned} \lambda < \frac{\epsilon }{(\epsilon +1)M_0}. \end{aligned}$$
(2.17)

It follows by the definition of polynomial \(\varphi \) and condition (2.17) that \(\varphi (0)=-\beta < 0.\) Moreover, \(\varphi (t)\longrightarrow \infty \) as \(t\longrightarrow r_3=\min \{\frac{1}{K_0}, \frac{1}{2r_2}\}.\) Hence, function \(\varphi _1\) has zeros in interval \((0, r_3)\) as a consequence of the intermediate value theorem. Let \(\lambda _0\) stand for the smallest such zero. Then, conditions (2.2), (2.11), and (2.12) are condensed as

$$\begin{aligned} \lambda \le \frac{1}{2N_0}:=\min \left\{ \frac{1}{N}, \frac{2}{(1+\epsilon )M},\frac{\epsilon }{(1+\epsilon )M_0}, \lambda _0\right\} . \end{aligned}$$
(2.18)

If \(\frac{1}{2N_0}=\frac{\epsilon }{(1+\epsilon )M_0}\) or \(\frac{1}{2N_0}=\frac{2}{(1+\epsilon )M},\) then condition (2.18) should hold as a strict inequality.

3 Convergence of NP

The Lipschitz parameters are associated with operator \({\mathcal {L}}\) and its derivatives.

Suppose there exist parameters \(K_0> 0, K > 0\) such that

$$\begin{aligned}{} & {} \Vert {\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(x_1)-{\mathcal {L}}'(x_0))\Vert \le K_0\Vert x_1-x_0\Vert , \end{aligned}$$
(3.1)
$$\begin{aligned}{} & {} \Vert {\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(x_0+\xi (x_1-x_0))-{\mathcal {L}}'(x_0))\Vert \le K\xi \Vert x_1-x_0\Vert , \end{aligned}$$
(3.2)

for \(x_1=x_0-{\mathcal {L}}'(x_0)^{-1}{\mathcal {L}}(x_0)\) and each \(\xi \in [0, 1]\) and

$$\begin{aligned} B[x_0, t_*]\subset \Omega . \end{aligned}$$
(3.3)

Conditions (A1), (1.4), (1.5), (3.1)–(3.3) and those of Lemma 2.1 or Lemma 2.2 are summarized by (H).

Next, under conditions H, we show the main convergence result for NP.

Theorem 3.1

Under conditions H sequence NP is convergent to a solution \(x_*\in B[x_0, t_*]\) of equation \({\mathcal {L}}(x)=0.\) Moreover, upper bounds

$$\begin{aligned} \Vert x_*-x_i\Vert \le s_*-s_i \end{aligned}$$
(3.4)

hold \(\forall \,\,i=0,1,2,\ldots \)

Proof

The assertions

$$\begin{aligned} \Vert x_{j+1}-x_j\Vert \le s_{j+1}-s_j \end{aligned}$$
(3.5)

and

$$\begin{aligned} B[x_{j+1}, s_*-s_{j+1}]\subset B[x_j, s_*-s_j] \end{aligned}$$
(3.6)

are proven by induction \(\forall \,\,j=0,1,2,\ldots \) Using (A1)

$$\begin{aligned} \Vert x_1-x_0\Vert +\Vert {\mathcal {L}}'(x_0)^{-1}{\mathcal {L}}(x_0)\Vert \le \lambda =s_1-s_0. \end{aligned}$$

Let \(u\in B[x_1, s_*-s_1].\) It follows by condition (A1)

$$\begin{aligned} \Vert u-x_0\Vert \le \Vert u-x_1\Vert +\Vert x_1-x_0\Vert \le s_*-s_1+s_1-s_0=s_*, \end{aligned}$$

so \(u\in B[x_1, s_*-s_1].\) That is assertions (3.5) and (3.6) hold if \(j=0.\) Assume these assertions hold if \(j=0,1,2,\ldots ,n.\) It follows for each \(\xi \in [0,1)\)

$$\begin{aligned} \Vert x_j+\xi (x_{j+1}-x_j)-x_0\Vert \le s_j+\xi (s_{j+1}-s_j)\le t_*, \end{aligned}$$

and

$$\begin{aligned} \Vert x_{j+1}-x_0\Vert \le \sum _{i=1}^{j+1}\Vert x_i-x_{i-1}\Vert \le \sum _{i=1}^{j+1}(s_i-s_{i-1})=s_{j+1}. \end{aligned}$$

It follows by induction hypotheses, the Lemmas and conditions (3.1) and (1.1)

$$\begin{aligned} \Vert {\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(x_{j+1}-{\mathcal {L}}'(x_0))\Vert \le \bar{K}\Vert x_{j+1}-x_0\Vert \le \bar{K}(s_{j+1}-s_0)\le \bar{K}s_{j+1} < 1, \end{aligned}$$

where \(\tilde{K}=\left\{ \begin{array}{cc} K_0,&{} j=0\\ M_0,&{} j=1,2,\ldots \end{array}\right. \) Hence, the inverse of linear operator \({\mathcal {L}}'(x_{j+1})\) exists. Notice that if \(j=0, K_0\) can be used, whereas if \(j=1,2,\ldots ,\) then \(M_0\) is utilized.

$$\begin{aligned} \Vert {\mathcal {L}}'(x_{j+1})^{-1}{\mathcal {L}}'(x_0)\Vert \le \frac{1}{1-\tilde{K}s_{j+1}}, \end{aligned}$$
(3.7)

as a consequence of a lemma on linear operators that are invertible due to Banach’s perturbation lemma (Argyros and Magréñan 2018; Argyros 2004a, b; Argyros and Hilout 2010).

The identity can be given by NP

$$\begin{aligned} {\mathcal {L}}(x_{n+1})=\int _0^1({\mathcal {L}}'(x_j+\xi (x_{j+1}-x_n))-{\mathcal {L}}'(x_j))(x_{j+1}-x_j)\mathrm{{d}}\xi , \end{aligned}$$
(3.8)

since

$$\begin{aligned} {\mathcal {L}}(x_{j+1})={\mathcal {L}}(x_{j+1})-{\mathcal {L}}(x_j)-{\mathcal {L}}'(x_j)(x_{j+1}-x_j). \end{aligned}$$

Then, using induction hypotheses, identity (3.8) and condition (1.5)

$$\begin{aligned} \Vert {\mathcal {L}}'(x_0)^{-1}{\mathcal {L}}(x_{j+1})\Vert \le \tilde{M}\int _0^1\xi \Vert x_{j+1}-x_j\Vert ^2\mathrm{{d}}\xi \le \frac{\tilde{M}}{2}(s_{j+1}-s_j)^2, \end{aligned}$$
(3.9)

where \(\tilde{M}=\left\{ \begin{array}{cc} K,&{}j=0\\ M,&{}j=1,2,\ldots \end{array}\right. \)

It follows by NP, estimates (3.7), (3.9) and the definition (2.1) of sequence \(\{s_n\}\)

$$\begin{aligned} \Vert x_{j+2}-x_{j+1}\Vert= & {} \Vert {\mathcal {L}}'(x_{j+1})^{-1}{\mathcal {L}}(x_{j+1})\Vert \\= & {} \Vert {\mathcal {L}}'(x_{j+1})^{-1}{\mathcal {L}}'(x_0){\mathcal {L}}'(x_0)^{-1}{\mathcal {L}}(x_{j+1})\Vert \\\le & {} \Vert {\mathcal {L}}'(x_{j+1})^{-1}{\mathcal {L}}'(x_0)\Vert \Vert {\mathcal {L}}'(x_0)^{-1}{\mathcal {L}}(x_{j+1})\Vert \\\le & {} \frac{\bar{K}}{2}\frac{(s_{j+1}-s_j)^2}{1-\bar{M}s_{j+1}}=s_{j+2}-s_{j+1}, \end{aligned}$$

where \(\bar{K}=\left\{ \begin{array}{cc} K,&{}j=0\\ M,&{}j=1,2,\ldots \end{array}\right. \) and \(\bar{M}=\left\{ \begin{array}{cc} K_0,&{}j=0\\ M_0,&{}j=1,2,\ldots \end{array}\right. \)

Moreover, if \(v\in B[x_{j+2},s_*-s_{j+2}],\) it follows

$$\begin{aligned} \Vert v-x_{j+1}\Vert\le & {} \Vert v-x_{j+2}\Vert +\Vert x_{j+2}-x_{j+1}\Vert \\\le & {} s_*-s_{j+2}+s_{j+2}-s_{j+1}=s_*-s_{j+1}. \end{aligned}$$

Thus, the element \(v\in B[x_{j+1}, s_*-s_{j+1}]\) completing the induction for assertions (3.5) and (3.6). Notice that scalar majorizing sequence \(\{s_j\}\) is fundamental as convergent. Hence, the sequence \(\{x_j\}\) is also convergent to some \(x_*\in B[x_0, s_*].\) Furthermore, let \(j\longrightarrow \infty \) in estimate (3.9) to conclude \({\mathcal {L}}(x_*)=0.\) Finally, the proof of assertion (3.4) using estimate (3.5) as standard is omitted (Yamamoto 1987b). \(\square \)

Next, the uniqueness ball for the solution \(x_*\) is presented. Notice that not all conditions S are used.

Proposition 3.2

Under center-Lipschitz condition (1.4) further assume the existence of a solution \(p\in B(x_0, R)\subset \Omega \) of equation \({\mathcal {L}}(x)=0\) such that linear operator \({\mathcal {L}}'(p)\) is invertible for some \(R > 0;\) a parameter \(R_1 > R\) given by

$$\begin{aligned} R_1=\frac{2}{M_0}-R. \end{aligned}$$
(3.10)

Then, the element p solves uniquely equation \({\mathcal {L}}(x)=0\) in the set \(T=B(x_0, R_1)\cap \Omega .\)

Proof

Define linear operator \(Q=\int _0^1 {\mathcal {L}}'(\bar{p}+\xi (p-\bar{p}))\mathrm{{d}}\xi \) for some element \(\bar{p}\in T\) satisfying \(F(\bar{p})=0.\) By using the definition of parameter \(R_1\), set T and condition (1.4)

$$\begin{aligned} \Vert {\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(x_0)-Q)\Vert\le & {} \Vert \int _0^1{\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(p+\xi (p-\bar{p}))-{\mathcal {L}}'(x_0))\mathrm{{d}}\xi \Vert \nonumber \\\le & {} M_0 \int _0^1((1-\xi )\Vert p-x_0\Vert +\xi \Vert \bar{p}-x_0\Vert )\mathrm{{d}}\xi \nonumber \\< & {} \frac{M_0}{2}(R+R_1) = 1. \end{aligned}$$
(3.11)

Then, the estimate (4.5) and the Banach lemma on linear operators (Argyros and Magréñan 2018; Argyros 2004a, b; Argyros and Hilout 2010) with inverses, imply the invertablility of linear operator Q. Moreover, by the approximation \(0= {\mathcal {L}}(p)-{\mathcal {L}}(\bar{p})=Q(p-\bar{p}),\) we deduce \(\bar{p}=p.\) \(\square \)

Remark 3.3

Notice that not all conditions of Theorem 3.1 are used in Proposition 3.2. But if they were, then, we can set \(p=x^*\) and \(R=s^*.\)

4 Special cases and Examples

It turns out that conditions of Theorem 3.1 reduce to the ones given by the earlier studies. But first we have the observations

Remark 4.1

Let us compare conditions H to conditions A:

It follows by these conditions that \(K_0\le K\le M_0.\) Consequently, replacing \(M_0\) or \(M_1\) by these tighter parameters gives previously mentioned benefits. Moreover, notice that parameters \(K_0, K, M_0, M\) are specializations of the originally used \(M_1.\) Hence, no additional cost is required in their computation.

  1. (1)

    The condition (A1) is common.

  2. (2)

    The condition (A2) always implies the conditions (3.1) and (3.2). However, the converse implication does not hold necessarily, unless \(K_0=K=M_1.\)

  3. (3)

    The new majorizing sequence \(\{s_n\}\) is tighter than the sequence \(\{t_n\}\) used by Kantorovich. In particular, under the conditions of the Theorem 1.1 and the Theorem 3.1, a simple inductive argument gives

    $$\begin{aligned}{} & {} 0\le s_n\le t_n,\\{} & {} \quad 0\le s_{n+1}-s_n\le t_{n+1}-t_n \end{aligned}$$

    and

    $$\begin{aligned} s_*\le \rho . \end{aligned}$$

    Notice also:

  4. (4)

    The conditions of Lemma 2.2 are stronger than those of the Lemma 2.1.

  5. (5)

    The conditions of the Kantorovich Theorem 1.1 imply

    $$\begin{aligned} t_n < \frac{1}{M_1}. \end{aligned}$$

    This inequality implies the one by our Lemma 2.1 but not vice versa unless if \(M_0= M_1.\)

  6. (6)

    Next, a comparison between Lemma 2.2 and the corresponding one in Theorem 1.1 follows.

Case \(K_0=M_0=K=M.\)

(i) It follows by the definition of N that \(N=M\) and condition (2.2) reduces to

$$\begin{aligned} M\lambda \le \frac{1}{2}. \end{aligned}$$
(4.1)

Furthermore, if \(M=M_1\) it reduces to the Kantorovich condition (A3) in the conditions A. But by estimate (1.8) it follows that if \(M < M_1\) then condition (A3) implies (4.1) but not vice versa. Hence, the new convergence criterion (4.1) weakens the Kantorovich criterion (A3) (see also the examples where \(M < M_1\)). (ii) The majorizing sequence reads

$$\begin{aligned} u_0=o, u_1=\lambda ,\, u_{n+2}=u_{n+1}+\frac{M(u_{n+1}-u_n)^2}{2(1-M_0u_{n+1})}. \end{aligned}$$

This sequence is more precise than \(\{t_n\}\) but not necessarily \(\{s_n\},\) unless if \(K=M\) and \(K_0=M_0.\)

(iii) The uniqueness ball is extended, since \(M_0\) is used for \(M_1\) in the formula. (see also Proposition 3.2).

Other specializations of the Lipschitz conditions give similar benefits (Table 1).

Example 4.2

The parameters using the example of the introduction are \(K_0=\frac{\mu +5}{3}, K=M_0=\frac{\mu +11}{6}.\) Moreover,\(\Omega _0=B(1, 1-\mu )\cap B(1, \frac{1}{M_0})=B(1, \frac{1}{M_0}).\) Set \(M=2(1+\frac{1}{3-\mu })\) \(M_0 < M_1\) and \( M < M_1\) for all \(\mu \in (0,0.5).\) Criterion (2.2) is then satisfied if \(\mu \in S_0:=[0.42, 0.5).\) Hence, the range of values for \(\mu \) for which NP converges is extended. Interval \(S_0\) can be enlarged if the condition of Lemma 2.1 is verified. Then, for \(\mu =0.4,\) we have the following \(\frac{1}{M_0}=0.5263,\)

Table 1 Sequence (2.1)
Full size table

Hence, the conditions of Lemma 2.1 hold. For \(\epsilon =0.63, \) we have

$$\begin{aligned} M_0\left( \frac{\gamma _0(s_2-s_1)}{1-\gamma }+\lambda +s_2-s_1\right) =0.1143\le \beta =0.3865 \end{aligned}$$

and

$$\begin{aligned} \lambda =0.2000 < \frac{2}{(1+\epsilon )M}=0.4431. \end{aligned}$$

Thus, the conditions (2.11) and (2.12) hold, and the interval \(S_0\) is further enlarged.

Example 4.3

Let \({\mathcal {U}}={\mathcal {V}}=C[0,1]\) be the space of continuous real functions on the interval [0, 1]. The max-norm is used. Set \(\Omega =B[x_0,3].\) Define Hammerstein nonlinear integral operator \({\mathcal {L}}\) on \(\Omega \) as

$$\begin{aligned} {\mathcal {L}}(v)(w)=v(w)-y(w)-\int _0^1G(w,t)v^3(t)\mathrm{{d}}t,\,\, v\in C[0,1], w\in [0,1]. \end{aligned}$$
(4.2)

where function \(y\in C[0,1],\) and G is a kernel related by Green’s function

$$\begin{aligned} G(w,t)=\left\{ \begin{array}{cc} (1-w)t,&{}t\le w\\ w(1-t),&{}w\le t. \end{array}\right. \end{aligned}$$
(4.3)

It follows by this definition that \({\mathcal {L}}'\) is defined by

$$\begin{aligned}{}[{\mathcal {L}}'(v)(z)](w)=z(w)-3\int _0^1G(w,t)v^2(t)z(t)\mathrm{{d}}t \end{aligned}$$
(4.4)

\(z\in C[0,1], w\in [0,1].\) Pick \(x_0(w)=y(w)=1.\) It then follows from (4.2)–(4.4) that \({\mathcal {L}}'(x_0)^{-1}\in L({\mathcal {V}}, {\mathcal {U}}),\)

$$\begin{aligned}{} & {} \Vert I-{\mathcal {L}}'(x_0)\Vert < 0.375,\,\, \Vert {\mathcal {L}}'(x_0)^{-1}\Vert \le 1.6,\\{} & {} \quad \lambda =0.2, \, M_0=2.4,\, M_1=3.6, \end{aligned}$$

and \(\Omega _0=B(x_0, 3)\cap B(x_0, 0.4167)=B(x_0, 0.4167),\) so \(M=1.5.\) Notice that \(M_0 < M_1\) and \(M < M_1.\) Set \(K_0=K=M_0.\) The Kantorovich convergence criterion (A3) is not satisfied, since \(2M_1\lambda =1.44 >1.\) Therefore convergence of NP is not assured. But our condition is satisfied, since \(2M\lambda =0.6 < 1.\)

Remark 4.4

  1. (i)

    Under conditions H, set \(p=x_*\) and \(R=s_*.\)

  2. (ii)

    Lipschitz condition (1.5) can be replaced by

    $$\begin{aligned} \Vert {\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(w_1)-{\mathcal {L}}'(w_2))\Vert \le d\Vert w_1-w_2\Vert , \end{aligned}$$
    (4.5)

    \(\forall \,\,w_1\in \Omega _0\) and \(w_2=w_1-{\mathcal {L}}'(w_1)^{-1}{\mathcal {L}}(w_1)\in \Omega _0.\) This, even smaller parameter d can replace M in the aforementioned results. The existence of iterate \(w_2\) is assured by (1.4).

  3. (iii)

    Another way to reduce Lipschitz constant M is given as follows: Suppose Lipschitz condition (1.5) is replaced by

    $$\begin{aligned} \Vert {\mathcal {L}}'(x_0)^{-1}({\mathcal {L}}'(w_1)-{\mathcal {L}}'(w_2))\Vert \le d_0\Vert w_1-w_2\Vert , \end{aligned}$$
    (4.6)

    \(\forall w_1\in T_1\) and \(w_2=w_1-{\mathcal {L}}'(w_1)^{-1}{\mathcal {L}}(w_1)\in T_1,\) where \(T_1=B(x_1, \frac{1}{M_0}-\lambda )\) provided that \(\lambda \mu <1.\) Notice that \(d_0\le d\le M,\) since \(T_1\subset \Omega _0\subset \Omega _1.\) In the case of example 4.1, the parameters are \(d_0=\frac{5(4-\mu )^3+\mu (3-\mu )^3}{3(3-\mu )(4-\mu )^2}< d=\frac{6+2(3-\mu )(1+2\mu )}{3(3-\mu )} < M\,\, \forall \mu \in (0,0.5).\)

5 Conclusion

A new methodology extends the applicability of NP. The new results are finer than the earlier ones. Therefore, they can replace them. No additional conditions are used. The methodology is very general. Consequently, it can be applied to extend other procedures (Argyros 2004b; Chen and Yamamoto 1989; Deuflhard 2004; Yamamoto 1987b).