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<h1>Newton’s method and regularly smooth operators</h1>
<p class="authors">
<span class="author">Ioannis K. Argyros\(^\ast \)</span>
</p>
<p class="date">December 20, 2007.</p>
</div>
<p>\(^\ast \)Cameron University, Department of Mathematical Sciences, Lawton, OK 73505, USA, e-mail: <span class="ttfamily">iargyros@cameron.edu</span>. </p>

<div class="abstract"><p> A semilocal convergence analysis for Newton’s method in a Banach space setting is provided in this study. Using a combination of regularly smooth and center regularly smooth conditions on the operator involved, we obtain more precise majorizing sequences than in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>. It then follows that under the same computational cost and the same or weaker hypotheses than in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span> the following benefits are obtained: larger convergence domain; finer estimates on the distances involved, and an at least as precise information on the location of the solution of the corresponding equation. </p>
<p>Numerical examples are given to further validate the results obtained in this study. </p>
<p><b class="bf">MSC.</b> 65B05, 65G99, 65J15, 47H17, 49M15. </p>
<p><b class="bf">Keywords.</b> Newton’s method, Banach space, majorizing sequence, regularly smooth operators Fréchet-derivative, semilocal convergence, integral equation, radiative transfer, Newton-Kantorovich hypothesis. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>In this study we are concerned with the problem of approximating a locally unique solution \(x^{\ast }\) of equation</p>
<div class="equation" id="f1.1/578">
<p>
  <div class="equation_content">
    \begin{equation}  F(x)=0, \label{f1.1/578}\end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where \(F\) is a Fréchet-differentiable operator defined on an open convex subset \(D\) of a Banach space \(X\) with values in a Banach space \(Y.\) </p>
<p>A large number of problems in applied mathematics and also in engineering are solved by finding the solutions of certain equations. For example, dynamic systems are mathematically modeled by difference or differential equations, and their solutions usually represent the states of the systems. For the sake of simplicity, assume that a time-invariant system is driven by the equation \(\dot{x}=Q(x),\) for some suitable operator \(Q,\) where \(x\) is the state. Then the equilibrium states are determined by solving equation <a href="#f1.1/578">1</a>. Similar equations are used in the case of discrete systems. The unknowns of engineering equations can be functions (difference, differential, and integral equations), vectors (systems of linear or nonlinear algebraic equations), or real or complex numbers (single algebraic equations with single unknowns). Except in special cases, the most commonly used solution methods are iterative - when starting from one or several initial approximations a sequence is constructed that converges to a solution of the equation. Iteration methods are also applied for solving optimization problems. In such cases, the iteration sequences converge to an optimal solution of the problem at hand. Since all of theses methods have the same recursive structure, they can be introduced and discussed in a general framework. </p>
<p>Newton’s method</p>
<div class="equation" id="f1.2/578">
<p>
  <div class="equation_content">
    \begin{equation}  x_{n+1}=x_{n}-F^{\prime }(x_{n})^{-1}F(x_{n})\  (n\geq 0),\  (x_{0}\in D) \label{f1.2/578}\end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> is undoubtedly the most popular iterative procedure for generating a sequence \(\{ x_{n}\} \) \((n\geq 0)\) approximating \(x^{\ast }.\) Here, \(F^{\prime }(x)\in L(X,Y)\) the space of bounded linear operators from \(X\) into \(Y.\) </p>
<p>There is an extensive literature on local as well as semilocal convergence results for Newton’s method under various assumptions <span class="cite">
	[
	<a href="#appel1-17/1" >1</a>
	]
</span>–<span class="cite">
	[
	<a href="#argyros2007/5" >5</a>
	]
</span>, <span class="cite">
	[
	<a href="#chandrasekhar1982/6" >6</a>
	]
</span>–<span class="cite">
	[
	<a href="#potra1983/10" >10</a>
	]
</span>, <span class="cite">
	[
	<a href="#zabrejko671-684/12" >12</a>
	]
</span>. </p>
<p>The hypothesis of \(w\)-smoothness:</p>
<div class="equation" id="f1.3/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert F^{\prime }(x)-F^{\prime }(y)\right\Vert \leq w(\left\Vert x-y\right\Vert )\  \text{for all }x,y\in D, \label{f1.3/578}\end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> has been used to provide a semilocal convergence analysis for Newton’s <br />method, where \(w:[0,\infty )\rightarrow \lbrack 0,\infty )\) is a continuous non-decreasing function which vanishes at zero, and it is positive elsewhere <span class="cite">
	[
	<a href="#appel1-17/1" >1</a>
	]
</span>, <span class="cite">
	[
	<a href="#argyros374-397/4" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#argyros2007/5" >5</a>
	]
</span>, <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>, <span class="cite">
	[
	<a href="#zabrejko671-684/12" >12</a>
	]
</span>. In the case: \(w(r)=cr,\) condition (<a href="#f1.3/578">3</a>) reduces to the common Lipschitz hypothesis whereas, when \(w(r)=cr^{p}\) \(p\in \lbrack 0,1)\) we obtain the Hölder assumption. </p>
<p>Recently in <span class="cite">
	[
	<a href="#argyros103-110/3" >3</a>
	]
</span>–<span class="cite">
	[
	<a href="#argyros2007/5" >5</a>
	]
</span> we introduced the center \(w\)-smoothness condition:</p>
<div class="equation" id="f1.4/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert F^{\prime }(x)-F^{\prime }(x_{0})\right\Vert \leq w_{0}(\left\Vert x-x_{0}\right\Vert ) \label{f1.4/578}\end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> where \(w_{0}\) is a function with the same for all \(x\in D,\) properties as \(w.\) </p>
<p>Note that condition (<a href="#f1.3/578">3</a>) implies (<a href="#f1.4/578">4</a>). Using weaker (<a href="#f1.4/578">4</a>) (which is what is really needed for finding bounds on \(\left\Vert F^{\prime }(x_{n})^{-1}F^{\prime }(x_{0})\right\Vert )\) instead of condition (<a href="#f1.3/578">3</a>), leads to more precise majorizing sequences, which in turn are used to provide under the same hypotheses a finer semilocal convergence analysis with the following advantages over the earlier mentioned works: larger convergence domain; finer error bounds on the distances involved, and an at least as precise information on the location of the solution \(x^{\ast }.\) Note that the above advantages are obtained under the same computational cost, since in practice finding function \(w\) requires that of \(w_{0}.\) In this study we show that the above advantages hold true if operator \(F\) is \(w\)-regularly smooth on \(D\) <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span> (to be precised in Definition <a href="#Definition 2.1/578">1</a>). </p>
<p>Numerical examples are also provided to further validate the results obtained in this study. </p>
<h1 id="a0000000003">2 Semilocal Convergence Analysis of Newton’s method (<a href="#f1.2/578">2</a>)</h1>
<p>Let \(N\) denote the class of non-decreasing continuous functions \(w:[0,\infty )\rightarrow \lbrack 0,\infty )\) that are concave. That is they have convex subgraphs \(\{ (s,t):s\geq 0,\  \)and \(t\leq w(s)\} ,\) and vanish at zero <span class="cite">
	[
	<a href="#rockafellar1967/11" >11</a>
	]
</span>. </p>
<p>We need the definition: <div class="definition_thmwrapper " id="Definition 2.1/578">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
  </div>
  <div class="definition_thmcontent">
  <p> <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span> Let \(F:D\subseteq X\rightarrow Y\) be a Fréchet-differentiable operator. Denote by \(h\)\((F)\) the quantity \(\inf \left\Vert F^{\prime }(x)\right\Vert ,\) \(x\in D.\) Given \(w\in N,\) we say that \(F\) is \(w\)-regularly smooth on \(D,\) if there exists an \(h\)\(\in \lbrack 0,\)\(h\)\((F)]\) such that for all \(x,y\in D:\)</p>
<div class="equation" id="f2.1/578">
<p>
  <div class="equation_content">
    \begin{equation}  w^{-1}(h_{F}(x,y)+\left\Vert F^{\prime }(y)-F^{\prime }(x)\right\Vert )-w^{-1}(h_{F}(x,y))\leq \left\Vert y-x\right\Vert , \label{f2.1/578}\end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> where,</p>
<div class="equation" id="f2.2/578">
<p>
  <div class="equation_content">
    \begin{equation}  h_{F}(x,y)=\min \{ \left\Vert F^{\prime }(x)\right\Vert ,\left\Vert F^{\prime }(y)\right\Vert \} -\underline{h}. \label{f2.2/578}\end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> The operator is regularly smooth on \(D,\) if it is \(w\)-regularly smooth there for some \(w\in N.\) </p>

  </div>
</div> </p>
<p>Throughout this study \(w^{-1}\) denotes the function whose closed epigraph \(cl\{ (s,t):s\geq 0,\) and \(t\geq w^{-1}(s)\} \) is symmetrical to closed of the subgraph of \(w\) with respect to the axis \(t=s.\) Due to the convexity of \(w^{-1},\) each \(w\)-regularly smooth operator is also \(w\)-smooth but not necessarily vice versa <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>. Several properties for operators \(F\) that are \(w\)-regularly smooth can be found in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>. </p>
<p>It follows from condition (<a href="#f2.1/578">5</a>) that for \(x=\overline{x}\) fixed there exists a function \(w_{0}\) with the same properties as \(w\) such that for all \(y\in D:\)</p>
<div class="equation" id="f2.3/578">
<p>
  <div class="equation_content">
    \begin{equation}  w_{0}^{-1}(h_{F}(\overline{x},y)+\left\Vert F^{\prime }(y)-F^{\prime }(\overline{x})\right\Vert )-w^{-1}(h_{F}(\overline{x},y))\leq \left\Vert y-\overline{x}\right\Vert . \label{f2.3/578}\end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> Clearly,</p>
<div class="equation" id="f2.4/578">
<p>
  <div class="equation_content">
    \begin{equation}  w_{0}(s)\leq w(s)\  \text{for all\  }s\in \lbrack 0,\infty ) \label{f2.4/578}\end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> holds in general, and \(\tfrac {w(s)}{w_{0}(s)}\) can be arbitrarily large <span class="cite">
	[
	<a href="#argyros103-110/3" >3</a>
	]
</span>–<span class="cite">
	[
	<a href="#argyros2007/5" >5</a>
	]
</span>. </p>
<p>It is convenient for us to introduce suitable notations. The superscript \(t\) means the non-negative part of a real number: \(a^{+}:=\max \{ a,0\} .\) </p>
<p>Denote:</p>
<div class="displaymath" id="a0000000004">
  \begin{align*}  Z(p,q) &  :=\{ q,p-q\} ,\\ m(p,q,r) &  :=\min \{ p,(c-z(p-q,w))^{+}\} , \end{align*}
</div>
<p> and</p>
<div class="displaymath" id="a0000000005">
  \begin{align*}  g(p,q,r) &  :=\int \nolimits _{0}^{r}[w(m(p,q,\theta )+\theta )-w(m(p,q,\theta ))]{\rm d}\theta \\ \text{for all }p &  \geq 0,\  \text{and\  }r\geq 0. \end{align*}
</div>
<p> In order for us to apply Newton’s method (<a href="#f1.2/578">2</a>) to equation (<a href="#f1.1/578">1</a>), choose \(x_{0}\in D\) such that \(F^{\prime }(x_{0})^{-1}\in L(Y,X),\) and set \(F_{0}:=F^{\prime }(x_{0})^{-1}F.\) Clearly, equation (<a href="#f1.1/578">1</a>) is equivalent to the equation \(F_{0}(x)=0,\) and the Newton iterations for \(F\) and \(F_{0}\) starting at \(x_{0}\) are identical. </p>
<p>Let \(h\) be a lower bound for \(h\) \((F_{0}):\)</p>
<div class="equation" id="f2.5/578">
<p>
  <div class="equation_content">
    \begin{equation}  0\leq \underline{h}\leq \underline{h}(F_{0}), \label{f2.5/578}\end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> and let \(w\in N,\) \(w_{0}\in N\) satisfy (<a href="#f2.1/578">5</a>) and (<a href="#f2.2/578">6</a>), respectively with \(F_{0}\) instead of \(F\) and \(\overline{x}=x_{0}.\) Then \(F_{0}\) is \(w\)-regularly smooth, and center-\(w_{0}\)-smooth on \(D.\) </p>
<p>Let us define constants</p>
<div class="equation" id="f2.6/578">
<p>
  <div class="equation_content">
    \begin{equation}  k_{0}=W_{0}^{-1}(1-\underline{h}), \label{f2.6/578}\end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<div class="equation" id="f2.7/578">
<p>
  <div class="equation_content">
    \begin{equation}  k=w^{-1}(1-\underline{h}), \label{f2.7/578}\end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p> and denote by a an upper bound on \(\left\Vert F_{0}(x_{0}\right\Vert :\)</p>
<div class="equation" id="f2.8/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert F_{0}(x_{0})\right\Vert \leq a. \label{f2.8/578}\end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p> Define scalar sequences \(\{ \alpha _{n}\} ,\  \{ \gamma _{n}\} ,\  \{ \gamma _{n}\} \  (n\geq 0)\  \)as follows:</p>
<div class="displaymath" id="f2.9/578">
  \begin{align}  \alpha _{0} &  :=k,\  \gamma _{0}:=1,\  \delta _{0}:=a,\label{f2.9/578}\\ \alpha _{n} &  :=(\alpha _{n-1}-\delta _{n-1})^{+},\  \gamma _{n}:=1-w_{0}(\alpha _{n}+t_{n})+w_{0}(\alpha _{n}),\nonumber \\ \delta _{n} &  :=\gamma _{n}^{-1}g(\alpha _{n-1},\alpha _{n-1}-\delta _{n-1},\delta _{n-1}),\nonumber \end{align}
</div>
<p> where,</p>
<div class="displaymath" id="a0000000006">
  \[  t_{n}:=\sum _{i=0}^{n-1}\delta _{i},  \]
</div>
<p> function \(g_{\underline{h}}(t)\  \)on \([0,\infty )\) by:</p>
<div class="equation" id="f2.10/578">
<p>
  <div class="equation_content">
    \begin{equation}  g_{\underline{h}}(t):=a-t+g(k,(k-t)^{+}-t,t)\  \text{for all }t\geq 0, \label{f2.10/578}\end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p> and set</p>
<div class="displaymath" id="a0000000007">
  \[  t^{\ast }=\lim _{n\rightarrow \infty }t_{n}.  \]
</div>
<p> The triple \((\alpha _{n},\gamma _{n},\) \(\delta _{n})\) is well defined for all \(n\geq 0\) provided that</p>
<div class="equation" id="f2.11/578">
<p>
  <div class="equation_content">
    \begin{equation}  \gamma _{n}>0\  \text{for all\  }n\geq 0. \label{f2.11/578}\end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> Condition (<a href="#f2.11/578">15</a>) can be replaced by </p>
<div class="equation" id="f2.12/578">
<p>
  <div class="equation_content">
    \begin{equation}  t_{n} < w_{0}^{-1}(1)\  \text{for all\  }n\geq 0. \label{f2.12/578}\end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p> We can now state the main semilocal convergence theorem for Newton’s <br />method (<a href="#f1.2/578">2</a>) for operator \(F_{0}\) that are \(w\)-regularly smooth on \(D.\) </p>
<p><div class="theorem_thmwrapper " id="Theorem 2.2/578">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let the operator \(F_{0}\) be \(w\)-regularly smooth, and center- \(w_{0}\)-regularly smooth on \(D.\) Assume </p>
<p>\(x_{0}\in D\) satisfies condition</p>
<div class="equation" id="f2.13/578">
<p>
  <div class="equation_content">
    \begin{equation}  t_{n}\leq w^{-1}(1),\label{f2.13/578}\end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p> and</p>
<div class="equation" id="f2.14/578">
<p>
  <div class="equation_content">
    \begin{equation}  \overline{U}(x_{0},t^{\ast })=\{ x\in X:\left\Vert x-x_{0}\right\Vert \leq t^{\ast }\} \subseteq D,\label{f2.14/578}\end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> then, sequence \(\{ x_{n}\} \) generated by Newton’s method <span class="rm">(<a href="#f1.2/578">2</a>)</span> (with \(F_{0}\) replacing \(F\)) is well defined, remains in \(U(x_{0}\) ,\(t^{\ast })\) for all \(n\geq 0,\) and converges to a solution \(x^{\ast }\) of equation \(F(x)=0\) in \(\overline{U}(x_{0},t^{\ast }).\) </p>
<p>Moreover the following estimates hold true for all \(n\geq 0:\)</p>
<div class="equation" id="f2.15/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert x_{n+1}-x_{n}\right\Vert \leq t_{n+1}-t_{n}, \label{f2.15/578}\end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<div class="equation" id="f2.16/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert x_{n}-x^{\ast }\right\Vert \leq t^{\ast }-t_{n}, \label{f2.16/578}\end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<div class="equation" id="f2.17/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert F_{0}^{\prime }(x_{n})^{-1}\right\Vert \leq \gamma _{n}^{-1}, \label{f2.17/578}\end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p> and</p>
<div class="equation" id="Proof 1/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert F_{0}(x_{n})\right\Vert \leq g(\alpha _{n-1},\alpha _{n}-\delta _{n-1},\delta _{n-1}). \label{f2.18/578}\end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p> Furthermore, if a is such that \(t^{\ast }\leq k_{0},\) then the solution \(x^{\ast }\) is unique in \(\overline{U}(x_{0},g_{\underline{h},2}^{-1}(0),\) where </p>
<p>\(g_{\underline{h},2}^{-1}\) stands for the inverse of the restriction of function \(g_{\underline{h}}\) on the interval \([w^{-1}(1),\infty ).\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000008">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>  The proof is similar to the corresponding one given in Theorem 4.3 in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>, p. 83)]. Simply use (<a href="#f2.2/578">6</a>) instead of (<a href="#f2.1/578">5</a>) (used in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>), in the derivation of the estimate (<a href="#f2.17/578">21</a>). To avoid duplications we will only sketch the above mentioned differences in the proofs. </p>
<p>It is convenient to set:</p>
<div class="displaymath" id="f2.19/578">
  \begin{align}  \overline{\alpha }_{n} &  :=w^{-1}(\left\Vert F_{0}^{\prime }(x_{n})\right\Vert -\underline{h}),\  \overline{\gamma }_{n}:=\left\Vert F_{0}^{\prime }(x_{n})^{-1}\right\Vert ^{-1},\label{f2.19/578}\\ \overline{\delta }_{n} &  :=\left\Vert x_{n+1}-x_{n}\right\Vert \  (n\geq 0).\nonumber \end{align}
</div>
<p> For \(n=0,\) we have</p>
<div class="displaymath" id="a0000000009">
  \[  \overline{\alpha }_{0}=k=\alpha _{0},\  \overline{\gamma }_{0}=1=\gamma _{0},\  \text{and\  }\overline{\delta }_{0}\leq a=\delta _{0},  \]
</div>
<p> hold true. </p>
<p>Assume, that for all \(i=0,1,...,n-1\) \((n\geq 1):\)</p>
<div class="displaymath" id="a0000000010">
  \[  F_{0}^{\prime }(x_{i})\  \text{exists so that }F_{0}^{\prime }(x_{i})^{-1}\in L(Y,X),  \]
</div>
<p> and</p>
<div class="equation" id="f2.20/578">
<p>
  <div class="equation_content">
    \begin{equation}  \overline{\alpha }_{i}\geq \alpha _{i},\  \overline{\gamma }_{i}\geq \gamma _{i},\  \overline{\delta }_{i}\leq \delta _{i}.\label{f2.20/578}\end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p> It follows by the induction hypothesis that</p>
<div class="equation" id="f2.21/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert x_{i}-x_{0}\right\Vert \leq \sum _{j=0}^{i-1}\overline{\delta }_{j}\leq \sum _{j=0}^{i-1}\delta _{j}=t_{i}.\label{f2.21/578}\end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p> Hence, \(F_{0}^{\prime }(x_{i})\) exists. Using (<a href="#f2.2/578">6</a>) on \(U(x_{0},t^{\ast })\subseteq D\) we obtain in turn (for \(\overline{x}=x_{0}):\)</p>
<div class="displaymath" id="f2.22/578">
  \begin{align} &  \left\Vert F_{0}^{\prime }(x_{i})-F_{0}^{\prime }(x_{0})\right\Vert \label{f2.22/578}\\ &  \leq w_{0}^{-1}(w_{0}^{-1}(\min \{ 1,\left\Vert F_{0}^{\prime }(x_{i})\right\Vert \} -\underline{h})+\left\Vert x_{i}-x_{0}\right\Vert - \nonumber \\ &  \quad -\min \{ 1,\left\Vert F_{0}^{\prime }(x_{i})\right\Vert \} +\underline{h}\nonumber \\ &  =w_{0}(\min \{ k,\overline{\alpha }_{i}\} +\left\Vert x_{i}-x_{0}\right\Vert )-w_{0}(\min \{ k,\overline{\alpha }_{i}\} )\nonumber \end{align}
</div>
<p> By Lemma 2.1 in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>, we have:</p>
<div class="displaymath" id="a0000000011">
  \[  \overline{\alpha }_{i}\geq (k-\left\Vert x_{i}-x_{0}\right\Vert )^{+}\geq (k-t_{i})^{+}=\alpha _{i},  \]
</div>
<p> which together with (<a href="#f2.22/578">26</a>) gives </p>
<div class="displaymath" id="f2.23/578">
  \begin{align}  \left\Vert F_{0}^{\prime }(x_{i})-F_{0}^{\prime }(x_{0})\right\Vert &  \leq w_{0}(\min \{ k,\alpha _{i}\} +\left\Vert x_{i}-x_{0}\right\Vert )-w_{0}(\min \{ k,\alpha _{i}\} \label{f2.23/578}\\ &  \leq w_{0}(\alpha _{i}+t_{i})-w_{0}(\alpha _{i}).\nonumber \end{align}
</div>
<p> In view of (<a href="#f2.4/578">8</a>), (<a href="#f2.13/578">17</a>), (<a href="#f2.19/578">23</a>)-(<a href="#f2.23/578">27</a>) we obtain</p>
<div class="equation" id="f2.24/578">
<p>
  <div class="equation_content">
    \begin{equation}  \overline{\gamma }_{i}\geq 1-w_{0}(\alpha _{i}+t_{i})+w_{0}(\alpha _{i})\geq 1-w_{0}(t_{i})\geq 1-w(t_{i}).\label{f2.24/578}\end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<p> It follows from (<a href="#f2.24/578">28</a>), and the Banach Lemma on invertible operators <span class="cite">
	[
	<a href="#argyros2007/5" >5</a>
	]
</span>, <span class="cite">
	[
	<a href="#kantorovich1982/9" >9</a>
	]
</span>, that \(F_{0}^{\prime }(x_{i})^{-1}\) exists, so that (<a href="#f2.17/578">21</a>) is satisfied and</p>
<div class="displaymath" id="a0000000012">
  \[  \gamma _{i}\leq \overline{\gamma }_{i}.  \]
</div>
<p> Using (<a href="#f1.2/578">2</a>) (for \(F_{0}\) replacing \(F\)) we obtain the approximation</p>
<div class="displaymath" id="f2.25/578">
  \begin{align}  x_{2+1}-x_{i} &  =-F_{0}^{\prime }(x_{i})^{-1}[F_{0}(x_{i})-F_{0}(x_{i-1})-F_{o}^{\prime }(x_{i-1})(x_{i}-x_{i-1})]\label{f2.25/578}\\ &  =-F_{0}^{\prime }(x_{i})^{-1}\int \nolimits _{0}^{1}[F_{0}^{\prime }(x_{i-1}+\theta (x_{i}-x_{i-1}))-F_{0}^{\prime }(x_{i-1})](x_{i}-x_{i-1}d\theta .\nonumber \end{align}
</div>
<p> It follows by Lemma 2.2 in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span> that:</p>
<div class="displaymath" id="f2.26/578">
  \begin{align}  r(x_{i-1},x_{i}) &  :=\left\Vert \int _{0}^{1}[F_{0}^{\prime }(x_{i-1}+\theta (x_{i}-x_{i-1}))-F_{0}^{\prime }(x_{i-1})](x_{i}-x_{i-1})d\theta \right\Vert \label{f2.26/578}\\ &  \leq g(\overline{\alpha }_{i-1},\overline{\alpha }_{i}-\overline{\delta }_{i-1},\overline{\delta }_{i-1})\leq g(\alpha _{i-1},\alpha _{i}-\delta _{i-1},\delta _{i-1}).\nonumber \end{align}
</div>
<p> Hence, we obtain by (<a href="#f2.17/578">21</a>), (<a href="#f2.25/578">29</a>) and (<a href="#f2.26/578">30</a>) that:</p>
<div class="equation" id="f2.27/578">
<p>
  <div class="equation_content">
    \begin{equation}  \overline{\delta }_{i}\leq \gamma _{i}^{-1}r(x_{i-1},x_{i})g(\alpha _{i-1},\alpha _{i}-\delta _{i-1},\delta _{i-1})=:\delta _{i},\label{f2.27/578}\end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<p> which completes the induction for (<a href="#f2.20/578">24</a>). It now follows:</p>
<div class="equation" id="f2.28/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert x_{n+i}-x_{n}\right\Vert \leq \sum _{j=n}^{n+i-1}\overline{\delta }_{j}\leq \sum _{j=n}^{n+i-1}\delta _{j}<\sum _{j=n}^{\infty }\delta _{j}=t^{\ast }-t_{n}\label{f2.28/578}\end{equation}
  </div>
  <span class="equation_label">32</span>
</p>
</div>
<p> It also follows from (<a href="#f2.28/578">32</a>) that sequence \(\{ x_{n}\} \) is Cauchy (since \(\{ t_{n}\} \  \)is a convergent sequence) in a Banach space \(X,\) and as such it converges to some \(x^{\ast }\in \overline{U}(x_{0},t^{\ast })\) (since \(\overline{U}(x_{0},t^{\ast })\) is a closed set). By letting \(i\rightarrow \infty \) in (<a href="#f2.26/578">30</a>) we obtain</p>
<div class="displaymath" id="a0000000013">
  \[  F_{0}(x^{\ast })=0\Rightarrow F(x^{\ast })=0.  \]
</div>
<p> Estimate (<a href="#f2.16/578">20</a>) follows from (<a href="#f2.28/578">32</a>) (i.e. (<a href="#f2.15/578">19</a>)) by using standard majorization techniques <span class="cite">
	[
	<a href="#argyros374-397/4" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#kantorovich1982/9" >9</a>
	]
</span>. Moreover, estimate (<a href="#Proof 1/578">22</a>) is simply (<a href="#f2.26/578">30</a>) for</p>
<div class="displaymath" id="a0000000014">
  \[  r(x_{n-1},x_{n})=\left\Vert F_{0}(x_{n})\right\Vert .  \]
</div>
<p> The uniqueness part of the proof as identical to the corresponding one in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span> is omitted. </p>
<p>That completes the proof of the Theorem. <div class="proof_wrapper" id="a0000000015">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="Remark 2.3/578">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3</span>
  </div>
  <div class="remark_thmcontent">
  <p> In order for us to compare our Theorem <a href="#Theorem 2.2/578">2</a> with the related Theorem 4.3 in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>, let us define sequences \(\{ \alpha _{n}^{1}\} ,\{ \gamma _{n}^{1}\} ,\{ \delta _{n}^{1}\} ,\{ t_{n}^{1}\} \) as <br />\(\{ \alpha _{n}\} ,\{ \gamma _{n}\} ,\{ \delta _{n}^{1}\} ,\{ t_{n}^{1}\} \) respectively by simply setting \(w_{0}=w\) in (<a href="#f2.9/578">13</a>). Clearly, if \(w_{0}=w,\) then our Theorem <a href="#Theorem 2.2/578">2</a> reduces to Theorem 4.3 in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>. Otherwise (i.e. if (<a href="#f2.4/578">8</a>) holds as a strict inequality), then with the exception of the convergence domain, the rest of the advantages of our approach over the corresponding ones in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span> (as already stated in the Introduction of this study) hold true. </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="Remark 2.4/578.">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4</span>
  </div>
  <div class="remark_thmcontent">
  <p>The convergence domain can also be extended as follows: Define sequences \(\{ \alpha _{n}^{2}\} ,\  \{ \gamma _{n}^{2}\} ,\  \{ \delta _{n}^{2}\} ,\  \{ t_{n}^{2}\} \) as \(\{ \alpha _{n}\} ,\  \{ \gamma _{n}\} ,\  \{ \delta _{n}\} ,\) \(\{ t_{n}\} \) respectively by with \(k_{0}\) (given by (<a href="#f2.6/578">10</a>)) replacing \(k\) (given by (<a href="#f2.7/578">11</a>)) in (<a href="#f2.9/578">13</a>). Moreover, replace condition (<a href="#f2.13/578">17</a>) by weaker (<a href="#f2.12/578">16</a>) (with \(t_{n}^{2}\) replacing \(t_{n}\) in (<a href="#f2.12/578">16</a>)). It then follows from the proof of Theorem <a href="#Theorem 2.2/578">2</a> that with the above changes the conclusions of this theorem hold true with the exception of the uniqueness part which holds true on \(\overline{U}(x_{0},t_{2}^{\ast }),\) \(t_{2}^{\ast }=\lim \limits _{n\rightarrow \infty }t_{n}^{2}.\) That is we arrived at: </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Theorem 2.5/578">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let the operator \(F_{0}\) be \(w\)-regularly smooth, and center- \(w_{0}\)-regularly smooth on \(D.\) Assume: \(x_{0}\in D\) satisfies:</p>
<div class="displaymath" id="f2.29/578">
  \begin{align}  t_{n}^{2} &  \leq w_{0}^{-1}(1),\label{f2.29/578}\\ \alpha _{0} &  :=k_{0},\nonumber \end{align}
</div>
<p> and</p>
<div class="displaymath" id="a0000000016">
  \[  \overline{U}(x_{0},t_{2}^{\ast })\subseteq D,  \]
</div>
<p> then sequence \(\{ x_{n}\} \) generated by Newton’s method <span class="rm">(<a href="#f1.2/578">2</a>)</span> (with \(F_{0}\) replacing \(F\)) is well defined, remains in \(U(x_{0},t_{2}^{\ast }),\) and converges to a solution \(x\) of equation \(F(x)=0\) in \(\overline{U}(x_{0},t_{n}^{\ast }).\) </p>
<p>Moreover the following estimates hold true for all \(n\geq 0:\)</p>
<div class="equation" id="f2.30/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert x_{n+1}-x_{n}\right\Vert \leq t_{n+1}^{2}-t_{n}^{2}, \label{f2.30/578}\end{equation}
  </div>
  <span class="equation_label">34</span>
</p>
</div>
<div class="equation" id="f2.31/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert x_{n}-x^{\ast }\right\Vert \leq t_{2}^{\ast }-t_{n}^{2}, \label{f2.31/578}\end{equation}
  </div>
  <span class="equation_label">35</span>
</p>
</div>
<div class="equation" id="f2.32/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert F_{0}^{\prime }(x_{n})^{-1}\right\Vert \leq \tfrac {1}{\gamma _{n}^{2}}, \label{f2.32/578}\end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p> and</p>
<div class="equation" id="Proof.2/578">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert F_{0}(x_{n})\right\Vert \leq g(\alpha _{n-1}^{2},\alpha _{n}^{2}-\delta _{n-1}^{2},\delta _{n-1}^{2}). \label{f2.33/578}\end{equation}
  </div>
  <span class="equation_label">37</span>
</p>
</div>
<p> Furthermore the solution \(x^{\ast }\) is unique in \(U(x_{0},w_{0}^{-1}(1)).\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000017">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>  In view of the proof of Theorem <a href="#Theorem 2.2/578">2</a>, we only need to show the uniqueness part. Let \(y^{\ast }\) be a solution of equation</p>
<div class="displaymath" id="a0000000018">
  \[  F_{0}(x)=0\  \text{in \  }U(x_{0},w_{0}^{-1}(1)).  \]
</div>
<p> Let us, denote</p>
<div class="displaymath" id="a0000000019">
  \[  L:=\int \nolimits _{0}^{1}F_{0}^{\prime }(y^{\ast }+\theta (x^{\ast }-y^{\ast })){\rm d}\theta  \]
</div>
<p> Using (<a href="#f2.2/578">6</a>) for \(\overline{x}=x_{0},\) \(y=y^{\ast }+\theta (x^{\ast }-y^{\ast }),\theta \in \lbrack 0,1],\) and \(F\) replaced by \(F_{0},\) as in (<a href="#f2.23/578">27</a>) we obtain</p>
<div class="displaymath" id="a0000000020">
  \[  \left\Vert F_{0}^{\prime }(x_{0})-L\right\Vert {\lt}1-w_{0}(t_{2}^{\ast }){\lt}1.  \]
</div>
<p> Hence, the linear operator \(L\) is invertible on \(U(x_{0},w_{0}^{-1}(1)\) . Moreover using the identity</p>
<div class="displaymath" id="a0000000021">
  \[  0=F(x^{\ast })-F(y^{\ast })=L\  (x^{\ast }-y^{\ast }),  \]
</div>
<p> we obtain \(x^{\ast }=y^{\ast }.\) </p>
<p>That completes the proof of the theorem. <div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>In the next result we compare majorizing sequences \(\{ t_{n}\} ,\  \{ t_{n}^{1}\} ,\  \{ t_{n}^{2}\} :\) </p>
<p><div class="proposition_thmwrapper " id="Proposition 2.6/578">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">6</span>
  </div>
  <div class="proposition_thmcontent">
  <p> Under the hypotheses of Theorem <span class="rm"><a href="#Theorem 2.2/578">2</a></span>, the conclusions of our Theorem <span class="rm">2.4</span>, and Theorem <span class="rm">4.3</span> in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span> hold true. </p>
<p>Moreover the following estimates hold true for all \(n\geq 0:\)</p>
<div class="equation" id="f2.34/578">
<p>
  <div class="equation_content">
    \begin{equation}  0\leq t_{n+1}^{2}-t_{n}^{2}\leq t_{n+1}-t_{n}\leq t_{n+1}^{1}-t_{n}^{1}, \label{f2.34/578}\end{equation}
  </div>
  <span class="equation_label">38</span>
</p>
</div>
<div class="equation" id="f2.35/578">
<p>
  <div class="equation_content">
    \begin{equation}  0\leq t_{2}^{\ast }\leq t^{\ast }\leq t_{1}^{\ast }-\lim _{n\Rightarrow \infty }t_{n}^{1}, \label{f2.35/578}\end{equation}
  </div>
  <span class="equation_label">39</span>
</p>
</div>
<p> and</p>
<div class="equation" id="Proof.3/578">
<p>
  <div class="equation_content">
    \begin{equation}  t_{2}^{\ast }-t_{n}^{2}\leq t^{\ast }-t_{n}\leq t_{1}^{\ast }-t_{n}. \label{f2.36/578}\end{equation}
  </div>
  <span class="equation_label">40</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000023">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The proof follows immediately by induction on \(n\geq 0,\) the definition of the \(``t"\) sequences and (<a href="#f2.4/578">8</a>). </p>
<p>That completes the proof of the Proposition. </p>
<p><div class="remark_thmwrapper " id="Remark 2.7/578">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">7</span>
  </div>
  <div class="remark_thmcontent">
  <p> (a) If strict inequality holds in<b class="bfseries"> </b>(<a href="#f2.4/578">8</a>), so does in (<a href="#f2.34/578">38</a>). </p>
<p>(b) If equality holds in (<a href="#f2.4/578">8</a>), then Theorem 2.4 reduces to Theorem 4.3 in <span class="cite">
	[
	<a href="#galperin813-858/7" >7</a>
	]
</span>. Otherwise it is an improvement with advantages as stated in the Introduction of this study (see also (<a href="#f2.34/578">38</a>)-(<a href="#Proof.3/578">40</a>), and compare (<a href="#f2.12/578">16</a>), (<a href="#f2.13/578">17</a>), (<a href="#f2.29/578">33</a>) in this case). Note also that sufficient convergence conditions other than the ones given here for the satisfaction of conditions (<a href="#f2.12/578">16</a>), (<a href="#f2.13/578">17</a>), (<a href="#f2.29/578">33</a>) (or (<a href="#f2.11/578">15</a>)), weaker (in general), have been given by us in <span class="cite">
	[
	<a href="#argyros374-397/4" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#argyros2007/5" >5</a>
	]
</span> for operators \(F\) that are \(w\)-smooth. Clearly, those conditions can replace (<a href="#f2.12/578">16</a>), (<a href="#f2.13/578">17</a>), (<a href="#f2.29/578">33</a>) in the above results provided that operator \(F\) is \(w\)-regularly smooth. </p>
<p>As an application, we compare the \(``t"\) iterations in the interesting case, when</p>
<div class="equation" id="f2.37/578">
<p>
  <div class="equation_content">
    \begin{equation}  w_{0}(s)=c_{0}s\  \text{and }w(s)=cs.\label{f2.37/578}\end{equation}
  </div>
  <span class="equation_label">41</span>
</p>
</div>
<p> Using (<a href="#f2.37/578">41</a>), and the definitions of the \("t"\) iterations, we obtain:</p>
<div class="displaymath" id="a0000000024">
  \[  t_{0}=0,\  t_{1}=a,t_{n+1}+\tfrac {f(t_{n})}{2(1-c_{0}t_{n})},  \]
</div>
<div class="equation" id="f2.39/578">
<p>
  <div class="equation_content">
    \begin{equation}  t_{0}^{1}=0,\  t_{1}^{1}=a,\  t_{n+1}^{1}=t_{n}+\tfrac {f(t_{n}^{1})}{2(1-ct_{n}^{1})},\label{f2.39/578}\end{equation}
  </div>
  <span class="equation_label">42</span>
</p>
</div>
<p> and</p>
<div class="displaymath" id="a0000000025">
  \[  t_{0}^{2}=0,\  t_{2}^{1}=a,\  t_{n+1}^{2}=t_{n}^{2}+\tfrac {c(t_{n}^{2}-t_{n-1}^{2})^{2}}{2(1-c_{0}t_{n}^{2})},  \]
</div>
<p> where,</p>
<div class="equation" id="f2.41/578">
<p>
  <div class="equation_content">
    \begin{equation}  f(s)=\tfrac {c}{2}s^{2}-s+a.\label{f2.41/578}\end{equation}
  </div>
  <span class="equation_label">43</span>
</p>
</div>
<p> Condition (<a href="#f2.13/578">17</a>) is satisfied provided that the famous Newton-<br />Kantorovich hypothesis:</p>
<div class="equation" id="f2.42/578">
<p>
  <div class="equation_content">
    \begin{equation}  K=ca\leq \tfrac {1}{2}\label{f2.42/578}\end{equation}
  </div>
  <span class="equation_label">44</span>
</p>
</div>
<p> holds true <span class="cite">
	[
	<a href="#argyros103-110/3" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#argyros2007/5" >5</a>
	]
</span>, <span class="cite">
	[
	<a href="#kantorovich1982/9" >9</a>
	]
</span>. It then also follows that sequences \(\{ t_{n}\} ,\) \(\{ t_{n}^{1}\} \) given by (<a href="#f2.39/578">42</a>), and (<a href="#f2.41/578">43</a>) converge to \(t^{\ast },\) \(t_{1}^{\ast }\) respectively with</p>
<div class="displaymath" id="a0000000026">
  \[  t^{\ast }\leq t_{1}^{\ast }=\tfrac {1-\sqrt{1-2ca}}{c_{0}},  \]
</div>
<p> so that the other conclusions of Theorem <a href="#Theorem 2.2/578">2</a> also hold true. </p>

  </div>
</div> </p>
<p>It was shown by us in <span class="cite">
	[
	<a href="#argyros103-110/3" >3</a>
	]
</span> (see also <span class="cite">
	[
	<a href="#argyros374-397/4" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#argyros2007/5" >5</a>
	]
</span>) that finer sequence \(\{ t_{n}^{2}\} \) converges to \(t_{2}^{\ast }\) provided that</p>
<div class="equation" id="f2.44/578">
<p>
  <div class="equation_content">
    \begin{equation}  K_{\beta }=(c+\beta c_{0})a\leq \beta \  \text{for some }\beta \in \lbrack 0,1]\label{f2.44/578}\end{equation}
  </div>
  <span class="equation_label">45</span>
</p>
</div>
<p> or (<a href="#f2.44/578">45</a>),</p>
<div class="equation" id="f2.45/578">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {2c_{0}a}{2-\beta }\leq 1,\label{f2.45/578}\end{equation}
  </div>
  <span class="equation_label">46</span>
</p>
</div>
<p> and</p>
<div class="equation" id="f2.46/578">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {c_{0}\beta ^{2}}{2-\beta }\leq c\  \text{for some }\beta \in \lbrack 0,2),\label{f2.46/578}\end{equation}
  </div>
  <span class="equation_label">47</span>
</p>
</div>
<p> or</p>
<div class="equation" id="f2.47/578">
<p>
  <div class="equation_content">
    \begin{equation}  c_{0}a\leq 1-\tfrac {1}{2}\beta \  \text{for }\beta \in \lbrack \beta _{0},2),\label{f2.47/578}\end{equation}
  </div>
  <span class="equation_label">48</span>
</p>
</div>
<p> where</p>
<div class="displaymath" id="a0000000027">
  \[  \beta _{0}=\tfrac {-\beta _{1}+\sqrt{\beta _{1}^{2}+8\beta _{1}}}{2}, \beta _{1}=\tfrac {c}{c_{0}}.  \]
</div>
<p> or</p>
<div class="equation" id="f2.48/578">
<p>
  <div class="equation_content">
    \begin{equation}  K_{2}=c_{2}a\leq \tfrac {1}{2},\label{f2.48/578}\end{equation}
  </div>
  <span class="equation_label">49</span>
</p>
</div>
<p> where,</p>
<div class="displaymath" id="a0000000028">
  \[  c_{2}=\tfrac {1}{2}\left( c+4c_{0}+\sqrt{c^{3}+8c_{0}c}\right) .  \]
</div>
<p> The simplest from conditions (<a href="#f2.44/578">45</a>)–(<a href="#f2.48/578">49</a>) is \(K_{1}\):</p>
<div class="equation" id="f2.49/578">
<p>
  <div class="equation_content">
    \begin{equation}  K_{1}=c_{1}a\leq \tfrac {1}{2},\  c_{1}=\tfrac {c+c_{0}}{2},\label{f2.49/578}\end{equation}
  </div>
  <span class="equation_label">50</span>
</p>
</div>
<p> Note also that</p>
<div class="equation" id="f2.50/578">
<p>
  <div class="equation_content">
    \begin{equation}  K\leq \tfrac {1}{2}\Rightarrow K_{1}\leq \frac{1}{2}\Rightarrow K_{2}\leq \tfrac {1}{2}.\label{f2.50/578}\end{equation}
  </div>
  <span class="equation_label">51</span>
</p>
</div>
<p> but not necessarily vice versa unless if \(c_{0}=c.\) </p>
<p>We complete the study with three numerical examples: </p>
<p><div class="example_thmwrapper " id="Example 2.7/578">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">8</span>
  </div>
  <div class="example_thmcontent">
  <p> Let \(X=Y=\mathbb {R}\), \(x_{0}=0,\) and for given parameters \(d_{i},\) \(i=0,1,2,3,\) define function \(F\) by</p>
<div class="equation" id="f2.51/578">
<p>
  <div class="equation_content">
    \begin{equation}  F(x)=d_{0}+d_{1}x+d_{2}\sin {\rm e}^{d_{3}x}. \label{f2.51/578}\end{equation}
  </div>
  <span class="equation_label">52</span>
</p>
</div>
<p> It can easily be seen by (<a href="#f2.50/578">51</a>) that for \(d_{3}\) large, and d\(_{2}\) sufficiently small \(\tfrac {c}{c_{0}}\) can be arbitrarily large. </p>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="Example 2.8/578">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">9</span>
  </div>
  <div class="example_thmcontent">
  <p> Let \(X=Y=\mathbb {R}\), \(x_{0}=1,\) \(b\in \left[ 0,\tfrac {1}{2}\right) ,\) and define function \(F\) by</p>
<div class="equation" id="f2.52/578">
<p>
  <div class="equation_content">
    \begin{equation}  F(x)=x^{3}-b. \label{f2.52/578}\end{equation}
  </div>
  <span class="equation_label">53</span>
</p>
</div>
<p> Using (<a href="#f2.51/578">52</a>) we have: \(a=\tfrac {1}{3}(1-b),\) \(c_{0}=3-b,\) and \(c=2(2-b).\) Note that</p>
<div class="displaymath" id="a0000000029">
  \[  c_{0}{\lt}c\  \text{for all }b\in \left[ 0,\tfrac {1}{2}\right) .  \]
</div>
<p> Condition (<a href="#f2.42/578">44</a>) does not hold, since</p>
<div class="displaymath" id="a0000000030">
  \[  K=\tfrac {2}{3}(1-b)(2-b){\gt}\tfrac {1}{2}\  \text{for all }b\in \left[ 0,\tfrac {1}{2}\right) .  \]
</div>
<p> That is there is no guarantee that Newton’s method (<a href="#f1.2/578">2</a>) starting from \(x_{0}=1\) converges to the solution \(x^{\ast }=\sqrt[3]{b}.\) </p>
<p>However, condition (<a href="#f2.48/578">49</a>) holds for all \(b\in \left[ \tfrac {5-\sqrt{13}}{3},\tfrac {1}{2}\right) ,\) since</p>
<div class="displaymath" id="a0000000031">
  \[  K_{1}=\tfrac {1}{6}(1-b)[3-b+2(2-b)]\leq \tfrac {1}{2}.  \]
</div>
<p>Note that \(\tfrac {5-\sqrt{13}}{3}=.46481624...\). Finally condition (<a href="#f2.48/578">49</a>) for holds for \(b\in \left[.450339002,\tfrac {1}{2}\right) .\) </p>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="Example 2.9/578">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">10</span>
  </div>
  <div class="example_thmcontent">
  <p> Let \(X=Y=C[0,1],\) be the space of real-valued continuous functions defined on the interval \([0,1]\) with norm</p>
<div class="displaymath" id="a0000000032">
  \[  \left\Vert x\right\Vert =\max _{0\leq s\leq 1}\left\vert x(s)\right\vert .  \]
</div>
<p> Let \(d\in \lbrack 0,1]\) be a given parameter. Consider the cubic integral equation</p>
<div class="equation" id="f2.53/578">
<p>
  <div class="equation_content">
    \begin{equation}  u(s)=u^{3}(s)+\lambda u(s)\int \nolimits _{0}^{1}q(s,t)u(t){\rm d}t+y(s)-d. \label{f2.53/578}\end{equation}
  </div>
  <span class="equation_label">54</span>
</p>
</div>
<p> Here the kernel \(q(s,t)\) is a continuous function of two variables defined on \([0,1]\times \lbrack 0,1];\) the parameter \(\lambda \) is a real number called the “albedo" for scattering; \(y(s)\) is a given continuous function defined on \([0,1]\) and \(x(s)\) is the unknown function sought in \(C[0,1].\) Equations of the form (<a href="#f2.52/578">53</a>) arise in the theory of radiative transfer, neutron transport, and the kinetic theory of gasses <span class="cite">
	[
	<a href="#argyros275-292/2" >2</a>
	]
</span>, <span class="cite">
	[
	<a href="#chandrasekhar1982/6" >6</a>
	]
</span>. </p>
<p>For simplicity, we choose \(u_{0}(s)=y(s)=1\) and \(q(s,t)=s/(s+t),\) for all \(s\in \lbrack 0,1]\) and \(t\in \lbrack 0,t]\) with \(s+t\neq 0.\) If we let \(D=U(u_{0},1-d),\) and define the operator \(f\) on \(D\) by</p>
<div class="equation" id="f2.54/578">
<p>
  <div class="equation_content">
    \begin{equation}  f(x)(s)=x^{3}(s)+\lambda x(s)\int _{0}^{1}q(s,t)x(t){\rm d}t+y(s)-d\label{f2.54/578}\end{equation}
  </div>
  <span class="equation_label">55</span>
</p>
</div>
<p> for all \(s\in \lbrack 0,1],\) then every zero of \(f\) satisfies equation (<a href="#f2.52/578">53</a>). We have the estimate</p>
<div class="equation" id="f2.55/578">
<p>
  <div class="equation_content">
    \begin{equation}  \max _{0\leq s\leq 1}\left\vert \int _{0}^{1}s/(s+t){\rm d}t\right\vert =\ln 2.\label{f2.55/578}\end{equation}
  </div>
  <span class="equation_label">56</span>
</p>
</div>
<p> Therefore if we set \(b_{0}=\left\Vert f^{\prime }(u_{0})^{-1}\right\Vert ,\) then it follows from (<a href="#f2.53/578">54</a>) and (<a href="#f2.54/578">55</a>) that conditions \(a=b_{0}(\left\vert \lambda \right\vert \ln \  2+1-d),\) \(c=2b_{0}[\left\vert \lambda \right\vert \ln \  2+3(2-d)]\  \)and \(c_{0}=b_{0}[2\left\vert \lambda \right\vert \ln \  2+3(3-d)].\) Moreover, since \(c_{0}{\lt}c\) we get a wider choice of values \(\lambda \) for which our conditions (<a href="#f2.44/578">45</a>)–(<a href="#f2.48/578">49</a>) or (<a href="#f2.49/578">50</a>) hold than the ones provided by (<a href="#f2.42/578">44</a>)\(\  \)(used in <span class="cite">
	[
	<a href="#argyros275-292/2" >2</a>
	]
</span>, <span class="cite">
	[
	<a href="#chandrasekhar1982/6" >6</a>
	]
</span>, <span class="cite">
	[
	<a href="#kantorovich1982/9" >9</a>
	]
</span>.</p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="appel1-17/1">1</a></dt>
  <dd><p><i class="sc">Appel, J., DePascale, E., Lysenko, J.V.</i> and <i class="sc">Zabrejko, P.P.</i>, <i class="itshape">New results on Newton-Kantorovich approximations with applications to nonlinear integral equations</i>, Numer. Funct. Anal. and Optim., <b class="bf">18</b>, nos. 1–2, pp.&#160;1–17, 1997. </p>
</dd>
  <dt><a name="argyros275-292/2">2</a></dt>
  <dd><p><i class="sc">Argyros, I.K.</i>, <i class="itshape">Quadratic equations and applications to Chandrasekhar’s and related equations</i>, Bull. Austral. Math. Soc., <b class="bf">32</b>, pp.&#160;275–292, 1985. </p>
</dd>
  <dt><a name="argyros103-110/3">3</a></dt>
  <dd><p><i class="sc">Argyros, I.K.</i>, <i class="itshape">On the Newton Kantorovich hypothesis for solving equations</i>, J. Comput. Appl. Math., <b class="bf">11</b>, no. 1, pp.&#160;103–110, 2004. </p>
</dd>
  <dt><a name="argyros374-397/4">4</a></dt>
  <dd><p><i class="sc">Argyros, I.K.</i>, <i class="itshape">A unifying local-semilocal convergence analysis, and applications for two-point Newton-like methods in Banach space</i>, J. Math. Anal. Applic., <b class="bf">298</b>, pp.&#160;374–397, 2004. </p>
</dd>
  <dt><a name="argyros2007/5">5</a></dt>
  <dd><p><i class="sc">Argyros, I.K.</i>, <i class="itshape">Computational theory of iterative methods</i>, Series: Computational Mathematics, <b class="bf">15</b>, Editors, C.K.Chui and L. Wuytack, Elsevier Publ., New York, U.S.A., 2007. </p>
</dd>
  <dt><a name="chandrasekhar1982/6">6</a></dt>
  <dd><p><i class="sc">Chandrasekhar, S.</i>, <i class="itshape">Radiative transfer</i>, Dover Publ., 1982. </p>
</dd>
  <dt><a name="galperin813-858/7">7</a></dt>
  <dd><p><i class="sc">Galperin, A.</i> and <i class="sc">Waksman, Z.</i>, <i class="itshape">Regular smoothness and Newton’s method</i>, Numer. Funct. Anal. and Optimiz., <b class="bf">15</b>, nos.7–8, pp.&#160;813–858, 1994. </p>
</dd>
  <dt><a name="gutierrez239-247/8">8</a></dt>
  <dd><p><i class="sc">Gutierrez, T.M., Hernandez, M.A.</i> and <i class="sc">Salanova, M.A.</i>, <i class="itshape">Accesibility of solutions by Newton’s method</i>, Intern. J. Comput. Math., <b class="bf">57</b>, pp.&#160;239–247, 1995. </p>
</dd>
  <dt><a name="kantorovich1982/9">9</a></dt>
  <dd><p><i class="sc">Kantorovich, L.V.</i> and <i class="sc">Akilov, G.P.</i>, <i class="itshape">Functional analysis in normed spaces</i>, Pergamon Press, Oxford, 1982. </p>
</dd>
  <dt><a name="potra1983/10">10</a></dt>
  <dd><p><i class="sc">Potra, F.A.</i> and <i class="sc">Ptak, V.</i>, <i class="itshape">Nondiscrete induction and iterative processes</i>, Pitman. Publ., London, 1983. </p>
</dd>
  <dt><a name="rockafellar1967/11">11</a></dt>
  <dd><p><i class="sc">Rockafellar, R.T.</i>, <i class="itshape">Convex analysis</i>, Princeton University Press, Princeton, 1967. </p>
</dd>
  <dt><a name="zabrejko671-684/12">12</a></dt>
  <dd><p><i class="sc">Zabrejko, P.P.</i> and <i class="sc">Nguyen, D.F.</i>, <i class="it">The majorant method in the theory of Newton-Kantorovich approximations and the Ptak error estimates</i>, Numer. Funct. Anal. and Optim., <b class="bf">9</b>, pp.&#160;671–684, 1987. </p>
</dd>
</dl>


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