MSC 2000 Subject Classification: Primary 47H10; Secondary 65 H 10 .

Keywords: fixed points, attraction points, attraction balls.

Let $G:D\subseteq {\mathbb{R}}^n\to D$$G:D\subseteq {\mathbb{R}}^n\to D$G:D subeR^(n)rarr DG: D \subseteq \mathbb{R}^{n} \rightarrow D$G:D\subseteq {\mathbb{R}}^n\to D$ be a nonlinear mapping which has a fixed point $x^{\ast }\in \mathrm{int}D$$x^{\ast }\in \mathrm{int}D$x^(**)in int Dx^{*} \in \operatorname{int} D$x^{\ast }\in \mathrm{int}D$ :

This point is an attraction point [1, Def. 10.1.1] if, given a norm on ${\mathbb{R}}^n$${\mathbb{R}}^n$R^(n)\mathbb{R}^{n}${\mathbb{R}}^n$, there exists an open ball $B_r:=B_r\left(x^{\ast }\right)=\{x\in {\mathbb{R}}^n:\Vert x-x^{\ast }\Vert <r\}\subseteq D$$B_r:=B_r\left(x^{\ast }\right)=\left\{x\in {\mathbb{R}}^n:‖x-x^{\ast }‖<r\right\}\subseteq D$B_(r):=B_(r)(x^(**))={x inR^(n):||x-x^(**)|| < r}sube DB_{r}:=B_{r}\left(x^{*}\right)=\left\{x \in \mathbb{R}^{n}:\left\|x-x^{*}\right\|<r\right\} \subseteq D$B_r:=B_r\left(x^{\ast }\right)=\{x\in {\mathbb{R}}^n:\Vert x-x^{\ast }\Vert <r\}\subseteq D$, such that for any initial approximation $x_0\in B_r$$x_0\in B_r$x_(0)inB_(r)x_{0} \in B_{r}$x_0\in B_r$, the successive approximations

remain in $D$$D$DD$D$ and converge to $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$. Note that a finite number of iterates are allowed to lie outside $B_r$$B_r$B_(r)B_{r}$B_r$.

The following result is well known.

Theorem 1 (Ostrowski) (see, e.g., [2, Th.22.1], [1, Th.10.1.3] and [3, Th.3.5]). If $G$$G$GG$G$ is differentiable at the fixed point $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$ and the spectral radius satisfies

then $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$ is an attraction point.
Remark 1 According to [1, N.R.10.1-2], this result holds also when instead of ${\mathbb{R}}^n$${\mathbb{R}}^n$R^(n)\mathbb{R}^{n}${\mathbb{R}}^n$ one considers an arbitrary Banach space $X$$X$XX$X$, with the observation that in defining the spectral radius of a linear continuous operator from $X$$X$XX$X$ to $X$$X$XX$X$ one takes its resolvent and spectrum (see, e.g., [4, p. 795]).

Condition (2) is sharp:
Example 1 [1, E.10.1.2]. The function $G:\mathbb{R}\to \mathbb{R},G(x)=x+x^3$$G:\mathbb{R}\to \mathbb{R},G(x)=x+x^3$G:RrarrR,G(x)=x+x^(3)G: \mathbb{R} \rightarrow \mathbb{R}, G(x)=x+x^{3}$G:\mathbb{R}\to \mathbb{R},G(x)=x+x^3$, is differentiable on $\mathbb{R}$$\mathbb{R}$R\mathbb{R}$\mathbb{R}$, and $\sigma =1$$\sigma =1$sigma=1\sigma=1$\sigma =1$ at the fixed point $x^{\ast }=0$$x^{\ast }=0$x^(**)=0x^{*}=0$x^{\ast }=0$, which is not an attraction point.

The spectral radius also offers some global information regarding the convergence rate of all the sequences of the successive approximations converging toward $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$, while the spectral elements of $G^{\prime }\left(x^{\ast }\right)$$G^{\prime }\left(x^{\ast }\right)$G^(')(x^(**))G^{\prime}\left(x^{*}\right)$G^{\prime }\left(x^{\ast }\right)$ characterize the convergence rate of each individual such sequence. Indeed, $\sigma$$\sigma$sigma\sigma$\sigma$ yields in fact the worst convergence rate among all the sequences converging to $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$ (see [1, Th.10.1.4] and [3, Th.3.5]), while the zero eigenvalue and its corresponding eigenvectors characterize the high convergence orders (more precisely, the $q$$q$qq$q$-superlinear convergence and with $q$$q$qq$q$-orders $1+p,p\in (0,1])$$1+p,p\in (0,1])$1+p,p in(0,1])1+p, p \in(0,1])$1+p,p\in (0,1])$ of a single such sequence [5]. ${}^1$${}^1$^(1){ }^{1}${}^1$

Theorem 1 says nothing about the size $r$$r$rr$r$ of the attraction ball. It is interesting to note that such an estimate can be deduced by applying some existing results. Indeed, under some additional differentiability assumptions on $G$$G$GG$G$ and denoting $F(x)=x-G(x)$$F(x)=x-G(x)$F(x)=x-G(x)F(x)=x-G(x)$F(x)=x-G(x)$, the successive approximations may be regarded as inexact Newton iterates for solving the nonlinear system $F(x)=0[5]$$F(x)=0[5]$F(x)=0[5]F(x)=0[5]$F(x)=0[5]$ :

Ypma [8] has obtained an estimation for the radius of attraction of this method, and we can deduce the corresponding one for the successive approximations.

However, instead of conditions in terms of Hölder continuity constant of $I-G^{\prime }$$I-G^{\prime }$I-G^(')I-G^{\prime}$I-G^{\prime }$ required by this approach, we obtain below (sharp) estimates in a direct manner, in terms of the Hölder continuity constant of $G^{\prime }$$G^{\prime }$G^(')G^{\prime}$G^{\prime }$. Before doing that, we notice that the successive approximations may also be regarded as quasi-Newton iterates (or, more general, as inexactly solved perturbed Newton iterates [9]) [10], but not as (exact) Newton iterates; in the first case there do not exist estimates for the radius of the convergence, while in the second there exist (see, e.g., [11], [12], [13], [14]).

Theorem 2 Suppose there exist $r_1>0,p\in (0,1],K_p>0$$r_1>0,p\in (0,1],K_p>0$r_(1) > 0,p in(0,1],K_(p) > 0r_{1}>0, p \in(0,1], K_{p}>0$r_1>0,p\in (0,1],K_p>0$ and a norm $\Vert \cdot \Vert$$\Vert \cdot \Vert$||*||\|\cdot\|$\Vert \cdot \Vert$ in ${\mathbb{R}}^n$${\mathbb{R}}^n$R^(n)\mathbb{R}^{n}${\mathbb{R}}^n$ such that $G$$G$GG$G$ is differentiable on $B_{r_1}$$B_{r_1}$B_(r_(1))B_{r_{1}}$B_{r_1}$, with $G^{\prime }$$G^{\prime }$G^(')G^{\prime}$G^{\prime }$ Hölder continuous at exponent $p$$p$pp$p$ :

Moreover, assume that

and denote

Then, for any initial approximation $x_0\in B_r$$x_0\in B_r$x_(0)inB_(r)x_{0} \in B_{r}$x_0\in B_r$, the successive approximations remain in $B_r$$B_r$B_(r)B_{r}$B_r$ and converge ( $q$$q$qq$q$-)linearly:

where $t=\frac{K_p}{1+p}{\Vert x_0-x^{\ast }\Vert }^p+q<1$$t=\frac{K_p}{1+p}{‖x_0-x^{\ast }‖}^p+q<1$t=(K_(p))/(1+p)||x_(0)-x^(**)||^(p)+q < 1t=\frac{K_{p}}{1+p}\left\|x_{0}-x^{*}\right\|^{p}+q<1$t=\frac{K_p}{1+p}{\Vert x_0-x^{\ast }\Vert }^p+q<1$.
Therefore,

Proof. The continuity hypothesis on $G^{\prime }$$G^{\prime }$G^(')G^{\prime}$G^{\prime }$ implies (see, e.g., [1, 3.2.12]):

Next,

The key condition is $t<1$$t<1$t < 1t<1$t<1$, which yields $\Vert x_0-x^{\ast }\Vert <r_2$$‖x_0-x^{\ast }‖<r_2$||x_(0)-x^(**)|| < r_(2)\left\|x_{0}-x^{*}\right\|<r_{2}$\Vert x_0-x^{\ast }\Vert <r_2$. The rest of the proof follows by induction.

Remark 2 a) The relationship between condition (2) ( $\sigma <1$$\sigma <1$sigma < 1\sigma<1$\sigma <1$ ) and the existence of a norm such that ( 4$)(\Vert G^{\prime }\left(x^{\ast }\right)\Vert <1)$$)\left(‖G^{\prime }\left(x^{\ast }\right)‖<1\right)$)(||G^(')(x^(**))|| < 1))\left(\left\|G^{\prime}\left(x^{*}\right)\right\|<1\right)$)(\Vert G^{\prime }\left(x^{\ast }\right)\Vert <1)$ holds is the following. Condition (4) implies (2), since the spectral radius satisfies

Condition (2) does not imply (4) in any norm, but for any $\epsilon >0$$\epsilon >0$epsi > 0\varepsilon>0$\epsilon >0$ there exists a norm $\Vert \cdot {\Vert }_{(\epsilon )}$$\Vert \cdot {\Vert }_{(\epsilon )}$||*||_((epsi))\|\cdot\|_{(\varepsilon)}$\Vert \cdot {\Vert }_{(\epsilon )}$ on ${\mathbb{R}}^n$${\mathbb{R}}^n$R^(n)\mathbb{R}^{n}${\mathbb{R}}^n$ such that (see, e.g., [1, 2.2.8], [2, Th.19.3]):

In the case of a Banach space instead of ${\mathbb{R}}^n$${\mathbb{R}}^n$R^(n)\mathbb{R}^{n}${\mathbb{R}}^n$, the statements regarding relations (7) and (8) remain true, with the remark that the involved norms must be equivalent to the initial one (see, e.g., [4, p. 795]). Therefore, the relationship between (2) and (4) remains the same.

Corollary 1 Under the hypotheses of Theorem 2, if $G^{\prime }$$G^{\prime }$G^(')G^{\prime}$G^{\prime }$ is Lipschitz continuous, i.e., Hölder continuous with $p=1,L:=K_1$$p=1,L:=K_1$p=1,L:=K_(1)p=1, L:=K_{1}$p=1,L:=K_1$, then instead of (5) one may take

and, correspondingly, $t=\frac{L}{2}\Vert x_0-x^{\ast }\Vert +q$$t=\frac{L}{2}‖x_0-x^{\ast }‖+q$t=(L)/(2)||x_(0)-x^(**)||+qt=\frac{L}{2}\left\|x_{0}-x^{*}\right\|+q$t=\frac{L}{2}\Vert x_0-x^{\ast }\Vert +q$.
Corollary 2 The assumptions of Theorem 2 imply the necessary condition

regardless of the value of the Hölder or Lipschitz continuity constant.
If $G^{\prime }$$G^{\prime }$G^(')G^{\prime}$G^{\prime }$ is Lipschitz continuous, then (10) becomes

Proof. The statement is obtained by taking into account the triangle inequality

The assumptions we have considered do not necessarily require contractivetype nonlinear mappings on the whole estimated ball, as the following example shows.

Example 2 Let $G:\mathbb{R}\to \mathbb{R},G(x)=x^2$$G:\mathbb{R}\to \mathbb{R},G(x)=x^2$G:RrarrR,G(x)=x^(2)G: \mathbb{R} \rightarrow \mathbb{R}, G(x)=x^{2}$G:\mathbb{R}\to \mathbb{R},G(x)=x^2$, having $x^{\ast }=0$$x^{\ast }=0$x^(**)=0x^{*}=0$x^{\ast }=0$ as attraction point; $r_1>0$$r_1>0$r_(1) > 0r_{1}>0$r_1>0$ may be chosen arbitrarily large, the derivative $G^{\prime }$$G^{\prime }$G^(')G^{\prime}$G^{\prime }$ is Lipschitz continuous on $\mathbb{R}$$\mathbb{R}$R\mathbb{R}$\mathbb{R}$, with $L=2$$L=2$L=2L=2$L=2$. Since $G^{\prime }(0)=0$$G^{\prime }(0)=0$G^(')(0)=0G^{\prime}(0)=0$G^{\prime }(0)=0$, one obtains from (9) that $r=r_2=1$$r=r_2=1$r=r_(2)=1r=r_{2}=1$r=r_2=1$. Moreover,

The above example also shows that estimates (9) and (11) are sharp.
Remark 3 We notice that the predicted radius $r_2$$r_2$r_(2)r_{2}$r_2$ may vary inverse proportionally with $r_1$$r_1$r_(1)r_{1}$r_1$, since the Hölder continuity constant may increase with $r_1$$r_1$r_(1)r_{1}$r_1$. In the previous example, if we take $r_1=\frac{1}{2}$$r_1=\frac{1}{2}$r_(1)=(1)/(2)r_{1}=\frac{1}{2}$r_1=\frac{1}{2}$ we get $L=1$$L=1$L=1L=1$L=1$ and then $r_2=2$$r_2=2$r_(2)=2r_{2}=2$r_2=2$, which is too large. Analogously, we can obtain too small values for $r_2$$r_2$r_(2)r_{2}$r_2$ if we take for instance $G(x)=x^3$$G(x)=x^3$G(x)=x^(3)G(x)=x^{3}$G(x)=x^3$ and large values for $r_1$$r_1$r_(1)r_{1}$r_1$.

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