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<title>Convergence analysisthe two-step Newton method of order four: Convergence analysisthe two-step Newton method of order four</title>
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<h1>Convergence analysisthe two-step Newton method of order four</h1>
<p class="authors">
<span class="author">Ioannis K. Argyros\(^\ast \) Sanjay K. Khattri\(^\S \)</span>
</p>
<p class="date">September 12, 2012.</p>
</div>
<p>\(^\ast \)Department of Mathematical Sciences, Cameron University, Lawton, Oklahoma 73505-6377, USA, e-mail: <span class="tt">iargyros@cameron.edu</span>. </p>
<p>\(^\dag \)Department of Engineering, Stord Haugesund University College, Norway, e-mail: <span class="tt">sanjay.khattri@hsh.no</span>. </p>

<div class="abstract"><p> We provide a tighter than before convergence analysis for the two-step Newton method of order four using recurrent functions. Numerical examples are also provided in this study. </p>
<p><b class="bf">MSC.</b> 65H10; 65G99; 65J15; 47H17; 49M15 </p>
<p><b class="bf">Keywords.</b> Two-step Newton method, Newton’s method, Banach space, Kantorovich hypothesis, majorizing sequence, Lipschitz/center-Lipschitz condition. </p>
</div>
<h1 id="intro">1 Introduction</h1>
<p>  In this study, we are concerned with the problem of approximating a locally unique solution \(x^\star \) of equation </p>
<div class="equation" id="eq:11">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:11} \mathcal{F}(x)=0, \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where, \(\mathcal{F}\) is Fréchet-differentiable operator defined on a convex subset \(\mathcal{D}\) of a Banach space \(\mathcal{X}\) with values in a Banach space \(\mathcal{Y}\). </p>
<p>Many problems in computational mathematics can be brought in the form <a href="#eq:11" class="eqref">1</a>. The solutions of these equations are rarely found in closed form. Therefore most solution methods for these equations are iterative. Newton’s method </p>
<div class="equation" id="eq:12">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:12} x_{n+1} = x_n - \mathcal{F}^\prime (x_n)^{-1}\mathcal{F}(x_n)\quad (n\ge {0}), \quad (x_0\in \mathcal{D}) \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> is undoubtedly the most popular method for generating a sequence \(\{ x_n\} \) converging quadratically to \(x^\star \) <span class="cite">
	[
	<a href="#five" >5</a>
	, 
	<a href="#thirteen" >13</a>
	, 
	<a href="#fifteen" >15</a>
	]
</span>. Two-step Newton method (TSNM) </p>
<div class="equation" id="eq:13">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:13} \begin{split}  y_n & = x_n - \mathcal{F}^\prime (x_n)^{-1}\mathcal{F}(x_n)\quad (n\ge {0}), \quad (x_0\in \mathcal{D}),\\ x_{n+1} & = y_n - \mathcal{F}^\prime (y_n)^{-1}\mathcal{F}(y_n), \end{split} \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> generates a converging sequence \(\{ x_n\} \) to \(x^\star \) with order four <span class="cite">
	[
	<a href="#five" >5</a>
	, 
	<a href="#nine" >9</a>
	]
</span>. The following conditions have been used to show the semilocal convergence for the Newton’s method <a href="#eq:12" class="eqref">2</a> and consequently the semilocal convergence of (TSNM) <span class="cite">
	[
	<a href="#five" >5</a>
	, 
	<a href="#thirteen" >13</a>
	, 
	<a href="#fifteen" >15</a>
	, 
	<a href="#seventeen" >17</a>
	]
</span> (\(\textbf{C}_{\text{K}}\)): </p>
<div class="displaymath" id="a0000000002">
  \begin{align}  \mathcal{F}^\prime (x_0)^{-1} & \in L(\mathcal{Y},\mathcal{X}) \quad \textrm{for some } x_0\in {\mathcal{D}}; \nonumber \\ \big\Vert {\mathcal{F}^\prime (x_0)^{-1}\mathcal{F}(x_0)}\big\Vert &  \le {\nu } \nonumber \\ \big\Vert {\mathcal{F}^\prime (x_0)^{-1}\big[\mathcal{F}^\prime (x)-\mathcal{F}^\prime (y)\big]} \big\Vert &  \le L \left\Vert {x-y}\right\Vert \quad \textrm{for all } x,y\in {\mathcal{D}}; \nonumber \\ h_{K} &  = L{\eta } \le \tfrac {1}{2} \label{eq:14} \intertext {and} \overline{U}(x_0,\lambda )= \big\{ x \in {{{\mathcal{X}}}}\, \big\vert \,  &  \Vert {x-x_0}\Vert \le \lambda \big\}  \subseteq \mathcal{D}, \nonumber \end{align}
</div>
<p> for specified \(\lambda \ge {0}\). </p>
<p>Note that ?? is the, famous for its simplicity and clarity, Kantorovich sufficient convergence hypothesis for the Newton’s method <a href="#eq:12" class="eqref">2</a>. A current survey on Newton-type methods can be found in <span class="cite">
	[
	<a href="#five" >5</a>
	]
</span> and the references therein (see also [1–4] and [6–17]). We have shown <span class="cite">
	[
	<a href="#five" >5</a>
	]
</span> the quadratic convergence of the Newton’s method <a href="#eq:12" class="eqref">2</a> using the set of conditions (\(\textbf{C}_{\text{AH}}\)) </p>
<div class="displaymath" id="a0000000003">
  \begin{alignat}{5} \mathcal{F}^\prime (x_0)^{-1}& \in L(\mathcal{Y},\mathcal{X}) \quad & & \textrm{for some } x_0\in {\mathcal{D}}; \nonumber \\ \big\Vert {\mathcal{F}^\prime (x_0)^{-1}\mathcal{F}(x_0)}\big\Vert & \le {\eta } & {}\nonumber \\ \big\Vert {\mathcal{F}^\prime (x_0)^{-1}\big[\mathcal{F}^\prime (x)-\mathcal{F}^\prime (x_0)\big]} \big\Vert & \le L_0 \left\Vert {x-x_0}\right\Vert \quad & & \textrm{for all } x\in {\mathcal{D}}; \nonumber \\ \big\Vert {\mathcal{F}^\prime (x_0)^{-1}\big[\mathcal{F}^\prime (x)-\mathcal{F}^\prime (y)\big]} \big\Vert & \le L \left\Vert {x-y}\right\Vert \quad & & \textrm{for all } x,y\in {\mathcal{D}}; \nonumber \\ h_{AH} = \overline{L}{\eta } & \le \tfrac {1}{2} & {}\label{eq:15} \intertext {and} \overline{U}(x_0,\lambda _0) & \subseteq \mathcal{D}, & {}\nonumber \end{alignat}
</div>
<p> for some specified \(\lambda _0\ge {0}\), where </p>
<div class="displaymath" id="eq:16">
  \begin{align}  \overline{L} & = \tfrac {1}{8}\left(L+4L_0+\sqrt{L^2+8L_0L}\right).\label{eq:16} \end{align}
</div>
<p> Note that </p>
<div class="equation" id="eq:17">
<p>
  <div class="equation_content">
    \begin{equation}  L_0 \le L \label{eq:17} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> holds in general, and \(L/L_0\) can be arbitrarily large <span class="cite">
	[
	<a href="#four" >4</a>
	, 
	<a href="#five" >5</a>
	]
</span>. Moreover, the \(L_0\) Center-Lipschitz is not an additional condition, since \(L_0\) is a special case of \(L\). Furthermore, we have by ??-<a href="#eq:17" class="eqref">7</a> </p>
<div class="equation" id="eq:18">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:18} h_{K}\le \tfrac {1}{2}\quad \Longrightarrow \quad h_{AH} \le \tfrac {1}{2} \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> but not necessarily vice versa unless if \(L_0\) = \(L\). The error analysis under ?? is also tighter than ??. Hence, the applicability of Newton’s method <a href="#eq:12" class="eqref">2</a> has been extended. </p>
<p>In this study, we provide the sufficient convergence conditions for (TSNM) corresponding to ??. The paper is organized as follows: §2 contains the semilocal convergence analysis for (TSNM), whereas the numerical examples are given in §3. </p>
<h1 id="a0000000004">2 Semilocal Convergence Analysis for (TSNM)</h1>
<p>We need the following result on majorizing sequence for (TSNM). <div class="lemma_thmwrapper theorem-style-plain" id="a0000000005">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">1</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let \(L_0\), \(L\), \(\eta \) be constants. Assume: there exist parameters \(\alpha \) and \(\phi \) such that </p>
<div class="displaymath" id="eq:21">
  \begin{align}  \tfrac {L\eta }{2(1-L_0\eta )} & \le \alpha ,\label{eq:21}\\ \tfrac {L_1\eta }{2(1-L_2\eta )} & \le \phi \le \phi _0\label{eq:22} \intertext {and} \eta & \le \eta _0\label{eq:23} \end{align}
</div>
<p> where, </p>
<div class="displaymath" id="eq:24">
  \begin{gather}  L_1 = \alpha ^2\, L,\qquad L_2 = (1+\alpha )L_0,\label{eq:24}\\ \phi _1 = \tfrac {4L_0\alpha }{2(L_0+L_2)\alpha - L + \sqrt{\left[2(L_0+L)\alpha -L\right]^2+8L_0L\alpha }},\label{eq:25}\\ \phi _2 = \tfrac {2L_1}{L_1+\sqrt{L_1^2+8L_1L_2}}, \qquad \phi _3 = \tfrac {2\alpha \left[1-(L_0+L_2)\eta \right]}{L\eta },\label{eq:26}\\ \phi _0=\min \left\{ \phi _1,\phi _2,\phi _3\right\} ,\label{eq:27}\\ \eta _1 = \tfrac {2}{L_1+2L_2(1+\phi )}, \qquad \eta _2 = \tfrac {1}{L_0+L_2},\label{eq:28}\\ \eta _0 = \min \left\{ \eta _1,\eta _2\right\} . \end{gather}
</div>
<p> Then, sequences \(\{ s_n\} \), \(\{ t_n\} \) generated by </p>
<div class="equation" id="eq:210">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:210} \begin{array}{llll} t_0=0,\quad s_0 = \eta , \quad t_{n+1} = s_n + \tfrac {L(s_n-t_n)^2}{2(1-L_0s_n)},\\ s_{n+1} = t_{n+1} + \tfrac {L(t_{n+1}-{{s_n}})^2}{2(1-L_0t_{ n+1})}, \end{array} \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p> are non-decreasing, bounded from above by </p>
<div class="equation" id="eq:211">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:211} t^{\star \star } = \left(\tfrac {1+\alpha }{1-\phi }\right)\eta , \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> and converge to their common least upper bound \(t^\star \in [0,t^{\star \star }]\). Moreover, the following estimates holds </p>
<div class="displaymath" id="eq:213">
  \begin{alignat}{3} 0& \le {t_{n+1}-s_n} & \le \alpha (s_n-t_n),\label{eq:212} \intertext {and} 0& \le {s_{n+1}-t_{n+1}} & \le \phi (s_n-t_n).\label{eq:213} \end{alignat}
</div>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000006">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>We shall show using induction on \(k\): </p>
<div class="displaymath" id="eq:215">
  \begin{alignat}{3} 0& \le \tfrac {L(s_k-t_k)}{2(1-L_0s_k)} & \le \alpha ,\label{eq:214} \intertext {and} 0& \le \tfrac {L_1(s_k-t_k)}{2(1-L_0t_{k+1})} & \le \phi .\label{eq:215} \end{alignat}
</div>
<p> Note that estimates <a href="#eq:213" class="eqref">19</a> and <a href="#eq:213" class="eqref">19</a> will then follow from <a href="#eq:215" class="eqref">20</a> and <a href="#eq:215" class="eqref">20</a>, respectively. Estimates <a href="#eq:215" class="eqref">20</a> and <a href="#eq:215" class="eqref">20</a> hold by the left hand side hypotheses in <a href="#eq:21" class="eqref">9</a> and <a href="#eq:23" class="eqref">10</a>, respectively. It follows from <a href="#eq:210" class="eqref">17</a>, <a href="#eq:215" class="eqref">20</a> and <a href="#eq:215" class="eqref">20</a> that estimates <a href="#eq:213" class="eqref">19</a> and <a href="#eq:213" class="eqref">19</a> hold for \({{n=0}}\). Let us assume estimates <a href="#eq:215" class="eqref">20</a> and <a href="#eq:215" class="eqref">20</a> hold for all \(k\le {{n}}\). It then follows that estimates <a href="#eq:213" class="eqref">19</a> and <a href="#eq:213" class="eqref">19</a> hold for \(n={{k}}\). We then have: </p>
<div class="displaymath" id="eq:216">
  \begin{align}  0 &  \le s_k-t_k \le \phi (s_{k-1}-t_{k-1}) \le \phi \cdot \phi (s_{k-2}-t_{k-2})\le \cdots \le \phi ^k\eta ,\label{eq:216} \\ 0 & \le t_{k+1}-s_k \le \alpha (s_k-t_k) \le \alpha \phi ^k\eta ,\label{eq:217} \end{align}
</div>
<p> and </p>
<div class="displaymath" id="a0000000007">
  \begin{align}  t_{k+1} \le s_k + \alpha \phi ^k\eta & \le {t_k+\alpha \phi ^k\eta +\phi ^k\eta }\nonumber \\ & \le s_{k-1} + \alpha \phi ^{k-1}\eta +\alpha \phi ^k \eta +\phi ^k\eta \nonumber \\ & \le t_{k-1} + \phi ^{k-1}\eta +\alpha \phi ^{k-1} \eta +\alpha \phi ^k \eta +\phi ^k\eta \nonumber \\ & = t_{k-1} + (\phi ^{k-1}+\phi ^k)\eta +\alpha (\phi ^{k-1}+\phi ^k) \eta \nonumber \le \cdots \\ & \le s_0 + \alpha (\eta +\phi \eta +\cdots +\phi ^k\eta ) + \alpha (\phi \eta +\cdots +\phi ^k\eta )\nonumber \\ & =(1+\alpha )(1+\phi +\cdots +\phi ^k\eta ) \le t^{\star \star }.\label{eq:218} \end{align}
</div>
<p> In view of <a href="#eq:216" class="eqref">21</a> and <a href="#eq:218" class="eqref">23</a>, estimate <a href="#eq:215" class="eqref">20</a> certainly holds if </p>
<div class="displaymath" id="eq:220">
  \begin{gather}  0 \le \tfrac {L\phi ^k\eta }{2\left[1-L_2(1+\phi +\cdots +\phi ^{k-1})\eta -L_0t^{k-1}\eta \right]} \le \alpha ,\label{eq:219} \intertext {or} L\phi ^k\eta +2\alpha L_2(1+\phi +\cdots +\phi ^{k-1})\eta -2\alpha +2L_0\alpha t^{k-1}\eta \le {0}.\label{eq:220} \end{gather}
</div>
<p> Estimate <a href="#eq:220" class="eqref">24</a> motivates us to introduce functions \(f_k\) on \([0,1)\) by </p>
<div class="equation" id="eq:221">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:221} f_k(t) = L\eta t^k + 2\alpha L_2(1+t+\cdots +t^{k-1})\eta +2L_0\alpha t^{k-1}\eta -2 \alpha . \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p> We need a relationship between two consecutive functions \(f_k\): </p>
<div class="displaymath" id="a0000000008">
  \begin{align}  f_{k+1}(t) & = Lt^{k+1}\eta + +2\alpha L_0 t^{k}\eta +2\alpha L_2(1+t+\cdots +t^k)\eta -2\alpha -Lt^k\eta \nonumber \\ & {} \qquad -2\alpha L_2(1+t+\cdots +t^{k-1})\eta - 2L_0\alpha t^{k-1}\eta +2\alpha +f_k(t)\nonumber \\ & =f_k(t) + Lt^{k+1}\eta -Lt^k\eta +2\alpha L_2 t^k \eta + 2L_0\alpha t^{k}\eta - 2L_0\alpha t^{k-1}\eta \nonumber \\ & = f_k(t) +g(t)t^{k-1}\eta ,\label{eq:222} \intertext {where} g(t) & = Lt^2 + \left[2\alpha (L_2+L_0)-L\right]t-2L_0\alpha .\label{eq:223} \end{align}
</div>
<p> Using <a href="#eq:221" class="eqref">25</a>, we see that <a href="#eq:220" class="eqref">24</a> holds </p>
<div class="displaymath" id="eq:224">
  \begin{align}  \textrm{if } \quad f_k(\phi ) & \le {0} \label{eq:224} \\ \textrm{ or} \quad {{f_1}}(\phi ) & \le {0},\label{eq:225} \end{align}
</div>
<div class="displaymath" id="eq:226">
  \begin{align}  \textrm{since,} \qquad g(\phi ) & \le {0} \qquad \textrm{and}\qquad f_{k+1}(\phi ) = f_k(\phi ) + g(\phi )\phi ^k\eta \le f_k(\phi ) \label{eq:226} \end{align}
</div>
<p> where \(\phi \) is chosen as in the right hand side inequality of <a href="#eq:21" class="eqref">9</a>. But <a href="#eq:223" class="eqref">26</a> also holds by <a href="#eq:21" class="eqref">9</a>. Moreover, define function \(f_\infty \) on \([0,1)\) by </p>
<div class="displaymath" id="eq:227">
  \begin{align}  f_\infty (t) & = \lim _{k\to \infty }f_k(t). \label{eq:227} \intertext {Then, we have by \eqref{eq:224}} f_\infty (\phi ) & \le {0}.\nonumber \end{align}
</div>
<p> Hence, <a href="#eq:213" class="eqref">19</a> and <a href="#eq:215" class="eqref">20</a> hold for all \(k\). Similarly, <a href="#eq:215" class="eqref">20</a> holds if </p>
<div class="displaymath" id="eq:229">
  \begin{gather}  L_1\phi ^k\eta \le 2\phi \left[1-L_2(1+\phi +\cdots +\phi ^k)\eta \right]\label{eq:228} \intertext {or} L_1\phi ^k\eta +2\phi L_2(1+\phi +\cdots +\phi ^k)\eta -2\phi \le {0}.\label{eq:229} \end{gather}
</div>
<p> As in <a href="#eq:221" class="eqref">25</a> we define functions \(h_k\) on \([0,1)\) by </p>
<div class="equation" id="eq:230">
<p>
  <div class="equation_content">
    \begin{equation}  h_k(t) = L_1t^k\eta +2tL_2 (1+t+\cdots +t^k)\eta - 2\phi . \label{eq:230} \end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<p> We need a relationship between two consecutive functions \(h_k\): </p>
<div class="displaymath" id="a0000000009">
  \begin{align}  h_{k+1}(t) &  = L_1t^{k+1}\eta + 2tL_2(1+t+\cdots +t^{k+1})\eta -2\phi - L_1t^k\eta -\nonumber \\ & \quad -2tL_2(1+t+\cdots +t^k)\eta +2\phi + h_k(t)\nonumber \\ & = h_k(t) +L_1t^{k+1}\eta -L_1t^k\eta + 2L_2t^{k+2}\eta \nonumber \\ & = h_k(t) + g_1(t)t^k\eta \label{eq:231} \intertext {where} g_1(t) & = 2L_2t^2+L_1 t- L_1.\label{eq:232} \end{align}
</div>
<p> In view of <a href="#eq:230" class="eqref">31</a>, estimate <a href="#eq:229" class="eqref">30</a> holds if </p>
<div class="equation" id="eq:233">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:233} \textrm{if}\quad h_k(\phi ) \le {0} \quad \textrm{or}\quad h_1(\phi ) \le {0} \end{equation}
  </div>
  <span class="equation_label">33</span>
</p>
</div>
<div class="displaymath" id="234">
  \begin{align}  \textrm{since,}\quad g_1(\phi ) \le {0} \quad \textrm{and} \quad h_{k+1}(\phi ) = h_k(\phi ) + g_1(\phi ) \phi ^k \eta \le h_k(\phi )\label{234} \end{align}
</div>
<p> where \(\phi \) is chosen as in the right hand side of <a href="#eq:23" class="eqref">10</a>. Note now that <a href="#eq:233" class="eqref">33</a> holds by <a href="#eq:23" class="eqref">10</a>. Furthermore, define functions \(h_\infty \) on \([0,1)\) by </p>
<div class="equation" id="eq:235">
<p>
  <div class="equation_content">
    \begin{equation}  h_\infty (t) =\lim _{k\to \infty }h_k(t).\label{eq:235} \end{equation}
  </div>
  <span class="equation_label">35</span>
</p>
</div>
<p> We then have </p>
<div class="equation" id="eq:236">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:236} h_\infty (\phi )\le {0}. \end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p> That completes the induction for <a href="#eq:213" class="eqref">19</a> and <a href="#eq:215" class="eqref">20</a>. Finally, in view of <a href="#eq:213" class="eqref">19</a>, <a href="#eq:213" class="eqref">19</a> and <a href="#eq:218" class="eqref">23</a>, sequences \(\{ t_n\} \), \(\{ s_n\} \) converge to \(t^\star \). That completes the proof of the Lemma. </p>

  </div>
</div>
<p> We need an Ostrowski-type relationship between iterates \(\{ x_n\} \) and \(\{ y_n\} \) <span class="cite">
	[
	<a href="#five" >5</a>
	, 
	<a href="#fourteen" >14</a>
	]
</span>. <div class="lemma_thmwrapper theorem-style-plain" id="a0000000010">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let us assume iterates \(\{ x_n\} \) and \(\{ y_n\} \) in (TSNM) are well defined for all \(n\ge {0}\). Then, the following identities hold: </p>
<div class="displaymath" id="eq:238">
  \begin{align}  \mathcal{F}(x_{n+1}) & = \int _{0}^1 \big[ \mathcal{F}^\prime (y_n + \theta (x_{n+1} - y_n) )- \mathcal{F}^\prime (y_n) \big](x_{n+1}-y_n)\mathrm{d}\theta ,\label{eq:237} \intertext {and} \mathcal{F}(y_n) & = \int _{0}^1 \big[ \mathcal{F}^\prime (x_n + \theta (y_{n} - x_n) )- \mathcal{F}^\prime (x_n) \big] (y_n-x_n)\mathrm{d}\theta .\label{eq:238} \end{align}
</div>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000011">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>Identity <a href="#eq:238" class="eqref">37</a> follows from the Taylor’s theorem and the first iteration in (TSNM), whereas <a href="#eq:238" class="eqref">37</a> follows from Taylor’s theorem and the second iteration in (TSNM). That completes the proof of the Lemma. </p>

  </div>
</div>
<p> We can show the following semilocal convergence result for (TSNM). <div class="lemma_thmwrapper theorem-style-plain" id="a0000000012">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let \(\mathcal{F}:\mathcal{D} \subset \mathcal{X}\rightarrow \mathcal{Y}\) be Fréchet-differentiable operator. Assume: there exist \(x_0\in {\mathcal{D}}\), \(L_0{\gt}{0}\), \(L{\gt}0\) and \(\eta \ge 0\) such that for all \(x,y\in {\mathcal{D}}\): </p>
<div class="displaymath" id="eq:239">
  \begin{align}  \mathcal{F}^\prime (x_0)^{-1} & \in L(\mathcal{Y},\mathcal{X}), \label{eq:239} \\ \big\Vert {\mathcal{F}^\prime \left(x_0\right)^{-1} \mathcal{F}(x_0)}\big\Vert & \le \eta , \label{eq:240}\\ \big\Vert {\mathcal{F}^\prime \left(x_0\right)^{-1} \left(\mathcal{F}^\prime (x)-\mathcal{F}^\prime (x_0)\right)}\big\Vert & \le L_0\left\Vert {x-x_0}\right\Vert ,\label{eq:241} \\ \big\Vert {\mathcal{F}^\prime \left(x_0\right)^{-1} \left(\mathcal{F}^\prime (x)-\mathcal{F}^\prime (y)\right)}\big\Vert & \le L\left\Vert {x-y}\right\Vert ,\label{eq:242}\\ \overline{U}(x_0,t^\star ) & \subseteq {\mathcal{D}};\label{eq:243} \end{align}
</div>
<p> hypotheses of Lemma <span class="rm">2.1</span> hold, where \(t^\star \) is given in Lemma <span class="rm">2.1</span>. Then, sequences \(\{ x_n\} \) and \(\{ y_n\} \) generated by (TSNM) are well defined, remain in \(\overline{U}(x_0,t^\star )\) for all \(n\ge {0}\) and converge to a solution \(x^\star \in \overline{U}(x_0,t^\star )\) of equation \(\mathcal{F}(x)=0.\) Moreover, the following estimates hold </p>
<div class="displaymath" id="eq:244">
  \begin{align}  \left\Vert {y_{n} - x_n }\right\Vert & \le s_{n}-t_n, \label{eq:244} \\ \left\Vert {x_{n+1} - y_n }\right\Vert & \le t_{n+1}-s_n, \label{eq:245} \\ \left\Vert {x_{n+1} - x_n }\right\Vert & \le t_{n+1}-t_n, \label{eq:246} \\ \left\Vert {y_{n+1} - y_n }\right\Vert & \le s_{n+1}-s_n, \label{eq:247} \\ \left\Vert {x_{n} - x^\star }\right\Vert & \le t^\star -t_n, \label{eq:248} \\ \left\Vert {y_{n} - x^\star }\right\Vert & \le t^\star -s_n. \label{eq:249} \end{align}
</div>
<p> Furthermore, if there exists \(R\ge {t^\star }\) such that </p>
<div class="displaymath" id="eq:251">
  \begin{align}  \overline{U}(x_0,R) & \subseteq {\mathcal{D}} \label{eq:250} \intertext {and} L_0(t^\star +R) & {\lt} 2,\label{eq:251} \end{align}
</div>
<p> then, \(x^\star \) is the only solution of \(\mathcal{F}(x)=0\) in \(\overline{U}(x_0,R)\) </p>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000013">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>We shall show using induction on \(k\) that (TSNM) is well defined, the iterates remain in \(\overline{U}(x_0,t^\star )\) for all \(n\ge {0}\) and estimates <a href="#eq:244" class="eqref">43</a> and <a href="#eq:245" class="eqref">44</a> hold for all \(n\ge {0}\). Iterate \(y_0\) is well defined by the first equation in (TSNM) for \(n=0\) and <a href="#eq:239" class="eqref">38</a>. We also have by <a href="#eq:26" class="eqref">13</a> and <a href="#eq:240" class="eqref">39</a> </p>
<div class="equation" id="a0000000014">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber \left\Vert {y_0-x_0}\right\Vert = \left\Vert {\mathcal{F}^\prime (x_0)^{-1}\mathcal{F}(x_0)}\right\Vert \le \eta = {{s_0 = s_0-t_0}} \le t^{\star }. \end{equation}
  </div>
  <span class="equation_label">50</span>
</p>
</div>
<p> That is <a href="#eq:244" class="eqref">43</a> holds for \(n=0\) and \(y_0\in \overline{U}(x_0,t^\star )\). Using (TSNM) for \(n=0\), we see that \(x_1\) is well defined. Let \(w\in \overline{U}(x_0,t^\star )\). Then, we have by Lemma 2.1 and <a href="#eq:241" class="eqref">40</a>: </p>
<div class="equation" id="eq:252">
<p>
  <div class="equation_content">
    \begin{equation}  \big\Vert {\mathcal{F}^\prime \left(x_0\right)^{-1} \big[\mathcal{F}^\prime (w)-\mathcal{F}^\prime (x_0)\big]}\big\Vert \le L_0\left\Vert {w-x_0}\right\Vert \le L_0t^\star < 1. \label{eq:252} \end{equation}
  </div>
  <span class="equation_label">50</span>
</p>
</div>
<p> It follows from <a href="#eq:252" class="eqref">50</a> and the Banach lemma on invertible operators <span class="cite">
	[
	<a href="#five" >5</a>
	, 
	<a href="#thirteen" >13</a>
	, 
	<a href="#fifteen" >15</a>
	]
</span> that \(\mathcal{F}^\prime \left(w\right)^{-1}\) exists and </p>
<div class="displaymath" id="eq:253">
  \begin{align}  \big\Vert {\mathcal{F}^\prime \left(w\right)^{-1} \mathcal{F}^\prime (x_0)}\big\Vert & \le \tfrac {1}{1-L_0\Vert {w-x_0}\Vert }.\label{eq:253} \end{align}
</div>
<p> In particular, for \(x_1\in \overline{U}(x_0,t^\star )\), we have </p>
<div class="displaymath" id="eq:254">
  \begin{align}  \big\Vert {\mathcal{F}^\prime \left(x_1\right)^{-1} \mathcal{F}^\prime (x_0)}\big\Vert & \le \tfrac {1}{1-L_0\Vert {x_1-x_0}\Vert } \le \tfrac {1}{1-L_0(t_1-t_0)} = \tfrac {1}{1-L_0t_1}.\label{eq:254} \end{align}
</div>
<p> Moreover, in view of <a href="#eq:238" class="eqref">37</a> for \(n=0\), (TSNM), <a href="#eq:26" class="eqref">13</a> and <a href="#eq:240" class="eqref">39</a>-<a href="#eq:242" class="eqref">41</a>, we get </p>
<div class="displaymath" id="a0000000015">
  \begin{align} & \left\Vert {x_{1} - y_0 }\right\Vert =\\ & =\left\Vert {\int _0^1{ {\left[\mathcal{F}^\prime (y_0)^{-1} \mathcal{F}^\prime (x_0)\right]}\mathcal{F}^\prime (x_0)^{-1}\left[ \mathcal{F}^\prime (x_0\! +\! \theta (y_0-x_0))\! -\! \mathcal{F}^\prime (x_0)\right] }}\textrm{d}\theta (y_0-x_0)\! \! \  \right\Vert \nonumber \\ & \le \tfrac {L_0}{1-L_0\Vert {y_0-x_0}\Vert } \int _0^1 {\theta \Vert {y_0-x_0}\Vert ^2} \textrm{d}{\theta }\nonumber \\ & =\tfrac {L_0}{2({1-L_0\Vert {y_0-x_0}\Vert })} \Vert {y_0-x_0}\Vert ^2\nonumber \\ & \le \tfrac {L_0}{2(1-L_0s_0)} (s_0-t_0)^2 = t_1-s_0,\nonumber \end{align}
</div>
<p> which shows <a href="#eq:245" class="eqref">44</a> for \(n=0\). We also have </p>
<div class="equation" id="a0000000016">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber \left\Vert {x_{1} - x_0 }\right\Vert \le \left\Vert {x_{1} - y_0 }\right\Vert + \left\Vert {y_{0} - x_0 }\right\Vert \le t_1 - s_0 + s_0 - t_0 = t_1-t_0 \le t^\star , \end{equation}
  </div>
  <span class="equation_label">54</span>
</p>
</div>
<p> which implies <a href="#eq:246" class="eqref">45</a> holds for \(n=0\) and \(x_1\in \overline{U}(x_0,t^\star )\). </p>
<p>Using (TSNM), <a href="#eq:26" class="eqref">13</a>, <a href="#eq:238" class="eqref">37</a> (for \(n=0\)) and <a href="#eq:254" class="eqref">52</a>, we get </p>
<div class="displaymath" id="a0000000017">
  \begin{align}  \left\Vert {y_1-x_1}\right\Vert & = \left\Vert {\left[\mathcal{F}^\prime (x_1)^{-1}\mathcal{F}^\prime (x_0)\right] \left[\mathcal{F}^\prime (x_0)^{-1}\mathcal{F}(x_1)\right] }\right\Vert \nonumber \\ & \le \left\Vert {\mathcal{F}^\prime (x_1)^{-1}\mathcal{F}^\prime (x_0)}\right\Vert \left\Vert {\mathcal{F}^\prime (x_0)^{-1}\mathcal{F}(x_1) }\right\Vert \nonumber \\ & \le \tfrac {1}{1-L_0t_1}\left\Vert \int _0^1{\mathcal{F}^\prime (x_0)^{-1} \left[\mathcal{F}^\prime (y_0+\theta (x_1-y_0)) - \mathcal{F}^\prime (y_0)\right]\textrm{d}\theta (x_1-y_0) }\right\Vert \nonumber \\ & \le \tfrac {L_0}{1-L_0t_1}\int _0^1{ \theta \Vert {x_1-y_0}\Vert \textrm{d}\theta \Vert {x_1-y_0}\Vert }\nonumber \\ & \le \tfrac {{L}}{1-L_0t_1}{ \tfrac {1}{2}(t_1-s_0)(t_1-s_0) } = s_1-t_1,\nonumber \end{align}
</div>
<p> which implies <a href="#eq:244" class="eqref">43</a> for \(n=1\). We then have: </p>
<div class="displaymath" id="a0000000018">
  \begin{gather}  \Vert {y_1-y_0}\Vert \le \Vert {y_1-x_1}\Vert + \Vert {x_1-y_0}\Vert \le s_1-t_1 + t_1-s_0 = s_1-s_0,\nonumber \\ \Vert {y_1-x_0}\Vert \le \Vert {y_1-y_0}\Vert + \Vert {y_0-x_0}\Vert \le s_1-s_0 + s_0-t_0 = s_1 \le t^\star ,\nonumber \end{gather}
</div>
<p> which imply <a href="#eq:247" class="eqref">46</a> for \(n=0\) and \(y_1\in \overline{U}(x_0,t^\star )\). Let us now assume <a href="#eq:244" class="eqref">43</a>-<a href="#eq:247" class="eqref">46</a>, \(y_n,x_k\in \overline{U}(x_0,t^\star )\) for all \(n\le {k}\). Using (TSNM), <a href="#eq:26" class="eqref">13</a>, <a href="#eq:238" class="eqref">37</a>, <a href="#eq:238" class="eqref">37</a>, <a href="#eq:242" class="eqref">41</a>, <a href="#eq:253" class="eqref">51</a> and the induction hypotheses, we have in turn: </p>
<div class="displaymath" id="a0000000019">
  \begin{align}  \Vert {x_{k+1}-x_0}\Vert & \le \Vert {x_{k+1}-x_k}\Vert + \Vert {x_{k}-x_{k-1}}\Vert +\cdots + \Vert {x_{1}-x_0}\Vert \nonumber \\ & \le t_{k+1}-t_k + t_k-t_{k-1}+\cdots + t_1-t_0 = t_{k+1} \le t^\star , \label{eq:255}\\ \Vert {y_{k}-x_0}\Vert & \le \Vert {y_{k}-x_k}\Vert + \Vert {x_{k}-x_{0}}\Vert \label{eq:256}\\ &  \le s_k-t_k + t_k - t_0 \nonumber \\ & = s_k \le t^\star \nonumber \end{align}
</div>
<div class="displaymath" id="eq:257">
  \begin{align} & \left\Vert {y_{k+1}-x_{k+1}}\right\Vert = \label{eq:257}\\ & =\left\Vert { \big[ \mathcal{F}^\prime (x_{k+1})^{-1} \mathcal{F}^\prime (x_0)\big] \big[ \mathcal{F}^\prime (x_{0})^{-1} \mathcal{F}(x_{k+1})\big]}\right\Vert \nonumber \\ &  \le \left\Vert { \mathcal{F}^\prime (x_{k+1})^{-1} \mathcal{F}^\prime (x_0)} \right\Vert \left\Vert { \mathcal{F}^\prime (x_{0})^{-1} \mathcal{F}(x_{k+1})}\right\Vert \nonumber \\ & \le \tfrac {1}{1-L_0\Vert {x_{k+1}\! -\! x_0}\Vert }\! \int _0^1\! \big\Vert {\mathcal{F}^\prime (x_{0})^{-1}} \left[\mathcal{F}^\prime (y_k\! +\! \theta (x_{k+1}-y_k)) - \mathcal{F}^\prime (y_k)\right]\textrm{d}\theta (x_{k+1}\! -\! y_k)\big\Vert \nonumber \\ & \le \tfrac {L}{1-L_0t_{k+1}} \int _0^1\theta \Vert {x_{k+1}-y_k}\Vert ^2\textrm{d}\theta \nonumber \\ & \le \tfrac {L}{1-L_0t_{k+1}} \tfrac {1}{2}(t_{k+1}-s_k)^2\nonumber \\ & = s_{k+1}-t_{k+1}, \nonumber \end{align}
</div>
<div class="displaymath" id="eq:258">
  \begin{align} & \left\Vert {x_{k+2}-y_{k+1}}\right\Vert = \label{eq:258} \left\Vert { \big[ \mathcal{F}^\prime (y_{k+1})^{-1} \mathcal{F}^\prime (x_0)\big] \big[ \mathcal{F}^\prime (x_{0})^{-1} \mathcal{F}(y_{k+1})\big]}\right\Vert \leq \end{align}
</div>
<div class="displaymath" id="a0000000020">
  \begin{align} & \le \tfrac {1}{1\! -\! L_0s_{k+1}}\! \!  \int _0^1\! \! \big\Vert {\mathcal{F}^\prime (x_{0})^{-1}} \! \! \! \left[\mathcal{F}^\prime (x_{k+1}\! +\! \theta (y_{k+1}\! -\! x_{k+1}))\!  -\!  \mathcal{F}^\prime (x_{k+1})\right]\! \! \textrm{d}\theta (y_{k+1}\! -\! x_{k+1})\big\Vert \nonumber \\ & \le \tfrac {L}{1-L_0s_{k+1}} \int _0^1 \theta {\Vert {y_{k+1}-x_{k+1}}\Vert ^2}\textrm{d}\theta \nonumber \\ & \le \tfrac {L}{2(1-L_0s_{k+1})}(s_{k+1}-t_{k+1})^2 = t_{k+2}-s_{k+1}, \nonumber \end{align}
</div>
<div class="displaymath" id="a0000000021">
  \begin{align}  \Vert {y_{k+2}-y_{k+1}}\Vert & \le \Vert {y_{k+2}-x_{k+2}}\Vert + \Vert {x_{k+2}-y_{k+1}}\Vert \nonumber \\ & \le s_{k+2}-t_{k+2}+t_{k+2}-s_{k+1} = s_{k+2}-s_{k+1}, \label{eq:259} \\ \Vert {x_{k+2}-x_{k+1}}\Vert & \le \Vert {x_{k+2}-y_{k+1}}\Vert + \Vert {y_{k+1}-x_{k+1}}\Vert \nonumber \\ & \le t_{k+2}-s_{k+1}+s_{k+1}-t_{k+1} = {{t_{k+2}-t_{k+1}}}\label{eq:260} \end{align}
</div>
<p> which show <a href="#eq:244" class="eqref">43</a>-<a href="#eq:247" class="eqref">46</a> hold for all \(n\ge {0}\). Estimates <a href="#eq:248" class="eqref">47</a> and <a href="#eq:249" class="eqref">48</a> follow from <a href="#eq:246" class="eqref">45</a> and <a href="#eq:247" class="eqref">46</a>, respectively by using standard majorization technique <span class="cite">
	[
	<a href="#five" >5</a>
	, 
	<a href="#thirteen" >13</a>
	, 
	<a href="#fifteen" >15</a>
	]
</span>. It follows from Lemma 2.1 and <a href="#eq:244" class="eqref">43</a>-<a href="#eq:248" class="eqref">47</a> that (TSNM) is Cauchy in a Banach space \(\mathcal{X}\) and as such it converges to some \(x^\star \in \overline{U}(x_0,t^\star )\) (since \(\overline{U}(x_0,t^\star )\) is a closed set). Moreover, we have by <a href="#eq:257" class="eqref">56</a> </p>
<div class="displaymath" id="eq:261">
  \begin{align}  \big\Vert {\mathcal{F}^\prime (x_{0})^{-1} \mathcal{F}(x_{k+1}) }\big\Vert & \le \tfrac {L}{2}\Vert {x_{k+1}-y_k}\Vert \Vert {x_{k+1}-y_k}\Vert \rightarrow {0}, \quad \textrm{as} \quad k\to \infty .\label{eq:261} \end{align}
</div>
<p> That is \(\mathcal{F}(x^\star ) = 0.\) Finally to show uniqueness, let \(y^\star \in \overline{U}(x_0,R)\) be a solution of equation \(\mathcal{F}(x)=0.\) Let us define linear operator \(M\) by </p>
<div class="equation" id="eq:262">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:262} M = \int _{0}^1\mathcal{F}^\prime (y^\star +\theta (x^\star -y^\star )){\rm d\theta }. \end{equation}
  </div>
  <span class="equation_label">61</span>
</p>
</div>
<p> Then using <a href="#eq:241" class="eqref">40</a>, <a href="#eq:251" class="eqref">49</a> and <a href="#eq:251" class="eqref">49</a>, we get in turn </p>
<div class="displaymath" id="a0000000022">
  \begin{align}  \left\Vert {\mathcal{F}^\prime (x_0)\left[M-\mathcal{F}^\prime (x_0)\right]}\right\Vert & \le L_0\int _{0}^1\left\Vert {y^\star +\theta (x^\star -y^\star ) -x_0}\right\Vert {\rm d\theta }\nonumber \\ & \le L_0\int _{0}^1\left[(1-\theta )\Vert {y^\star -x_0}\Vert +\theta \Vert {x^\star -x_0}\Vert \right] {\rm d\theta }\nonumber \\ & \le \tfrac {L_0}{2}(R+t^\star ) {\lt} 1.\label{eq:263} \end{align}
</div>
<p> It follows from <a href="#eq:259" class="eqref">58</a> and the Banach Lemma on invertible operators that \(M^{-1}\) exists. Then, in view of the identity </p>
<div class="equation" id="eq:264">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:264} 0 = \mathcal{F}(x^\star ) -\mathcal{F}(y^\star ) = M(x^\star -y^\star ), \end{equation}
  </div>
  <span class="equation_label">63</span>
</p>
</div>
<p> we conclude that \(x^\star =y^\star \). That completes the proof of the Theorem. </p>

  </div>
</div>
<p><div class="remark_thmwrapper theorem-style-remark" id="a0000000023">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4</span>
  </div>
  <div class="remark_thmcontent">
  <p>1) Limit point \(t^\star \) can be replaced by \(t^{\star \star }\), given in closed form by <a href="#eq:27" class="eqref">14</a>, in hypotheses <a href="#eq:240" class="eqref">39</a> and <a href="#eq:248" class="eqref">47</a>. </p>
<p>2) The verification of conditions <a href="#eq:21" class="eqref">9</a>-<a href="#eq:23" class="eqref">10</a> require simple algebra (see also Example 3.1). </p>
<p>3) If \(L_0 = L\), then scalar sequences \(\{ s_n\} \), \(\{ t_n\} \) given by <a href="#eq:26" class="eqref">13</a> reduce essentially to the ones used in <span class="cite">
	[
	<a href="#nine" >9</a>
	]
</span>. In particular, we have in this case </p>
<div class="equation" id="eq:265">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:265} \begin{array}{llll} \overline{t}_0 =0,\quad \overline{s}_0 = \eta , \quad \overline{t}_{n+1} = \overline{s}_n + \tfrac {L(\overline{s}_n-\overline{t}_n)^2}{2(1-L\overline{s}_n)},\\ \overline{s}_{n+1} = \overline{t}_{n+1} + \tfrac {L( \overline{t}_{n+1}-\overline{s}_n)^2 }{ 2 (1-L\overline{t}_{n+1})} \end{array} \end{equation}
  </div>
  <span class="equation_label">64</span>
</p>
</div>
<p> If \(L_0 {\lt} L\) iteration <a href="#eq:26" class="eqref">13</a> is tighter than <a href="#eq:261" class="eqref">60</a>. Moreover, in view of the proof of the Theorem 2.3, we note that sequence </p>
<div class="equation" id="eq:266">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:266} \begin{array}{llll} \overline{\overline{t}}_0 =0,\quad \overline{\overline{s}}_0 = \eta , \quad \overline{\overline{t}}_{n+1} = \overline{\overline{s}}_n + \tfrac {L^\star (\overline{\overline{s}}_n-\overline{\overline{t}}_n)^2}{2(1-L_0\overline{ \overline{s}} _n) } , \\ \overline{\overline{s}}_{n+1} = \overline{\overline{t}}_{n+1} + \tfrac {L^\star (\overline{\overline{t}}_{n+1}-\overline{\overline{s}}_n)^2 }{ 2 (1-L_0\overline{\overline{t}}_{n+1})}, \end{array} \end{equation}
  </div>
  <span class="equation_label">65</span>
</p>
</div>
<p> is also majorizing for (TSNM), where </p>
<div class="equation" id="a0000000024">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber L^\star = \left\{ \begin{array}{lll} {L_0}, \quad & \textrm{if}\quad n=0 \\ {L}, \quad & \textrm{if}\quad n>0. \end{array}\right. \end{equation}
  </div>
  <span class="equation_label">66</span>
</p>
</div>
<p> In case \(L_0{\lt}L,\) <a href="#eq:226" class="eqref">29</a> is even a tighter majorizing sequence than <a href="#eq:261" class="eqref">60</a>. Furthermore, \(L, L_1\) can be replaced by \(L_0,L_1^\star = \alpha ^2L_0\) at the left hand sides of <a href="#eq:21" class="eqref">9</a> and <a href="#eq:23" class="eqref">10</a>, respectively. </p>
<p>4) If \(\alpha =0\), define \(L_1=L\), then it is simple algebra to show that conditions of Lemma 2.1 reduce to ??. Moreover, if \(L_0=L\), these conditions reduce to ??. That is we have Newton’s method <a href="#eq:12" class="eqref">2</a> and iteration <a href="#eq:26" class="eqref">13</a> reduces to </p>
<div class="equation" id="eq:267">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:267} t_0 = 0, \quad t_1=\eta , \quad t_{n+2} = t_{n+1} + \tfrac {L(t_{n+1}-t_n)^2}{2(1-L_0t_{n+1})}. \end{equation}
  </div>
  <span class="equation_label">66</span>
</p>
</div>
<p> In the case of Newton’s method for \(L_0=L\), we have the well-known Kantorovich majorizing sequence. </p>
<div class="equation" id="eq:268">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:268} \nu _0 = 0, \quad \nu _1=\eta , \quad \nu _{n+2} = \nu _{n+1} + \tfrac {L(\nu _{n+1}-\nu _n)^2}{2(1-L_0\nu _{n+1})}. \end{equation}
  </div>
  <span class="equation_label">67</span>
</p>
</div>
<p> Note that if \(L_0{\lt}L\), \(\{ t_n\} \) is a tighter majorizing sequence than \(\{ \nu _n\} \) for the Newton’s method <span class="cite">
	[
	<a href="#five" >5</a>
	, 
	<a href="#thirteen" >13</a>
	, 
	<a href="#fifteen" >15</a>
	]
</span>.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000025">3 Numerical Examples</h1>
<p> Let \(\mathcal{X}=\mathcal{Y}=\mathbb {R}^2\) be equipped with the max-norm, \(x_0=(1,1)^T\), \(\mathcal{D}=\overline{U}(x_0,1-p)\), \(p\in [0,1)\) and define \(\mathcal{F}\) on \(\mathcal{D}\) by </p>
<div class="equation" id="eq:31">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq:31} \mathcal{F}(x) = \left(\xi _1^3-p,\xi _2^3-p\right)^T, \quad x=\left(\xi _1,\xi _2\right)^T. \end{equation}
  </div>
  <span class="equation_label">68</span>
</p>
</div>
<p> Using <a href="#eq:235" class="eqref">35</a>-<a href="#eq:238" class="eqref">37</a>, we get </p>
<div class="displaymath" id="a0000000026">
  \begin{alignat}{5} \eta & = \tfrac {1-p}{3},\quad & L_0 & = 3-p \quad & & \textrm{and}\quad &  L & = 2(2-p){\gt}L_0. \nonumber \intertext {Let $p = 0.7$. Then, we get} \eta & = 0.1, & L_0 & = 2.3 & & \textrm{and} & L & = 2.6. \nonumber \end{alignat}
</div>
<p> The Newton-Kantorovich hypothesis ?? is satisfied, since </p>
<div class="equation" id="a0000000027">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber \tfrac {2}{3}(1-p)(2-p) = 0.26 < 1\quad \textrm{for all}\quad {p \in [0,{1}/{2})}. \end{equation}
  </div>
  <span class="equation_label">68</span>
</p>
</div>
<p> Using Lemma 2.1, for \(\alpha = 0.17 \quad \textrm{and}\quad \phi = 0.0052\), we get </p>
<div class="displaymath" id="a0000000028">
  \begin{gather*}  \begin{alignedat} \end{alignedat}{6} L_1 & = 0.07514, & \qquad L_2 & = 2.691 & \qquad & \phi & =& & 0.756703694,\\ \phi _2 & = 0.111383518, &  \phi _3 & = 0.666923077, & \phi _0 & =& & \phi _2 \\ \eta _1 & = 0.364622409, & \eta _2 & = 0.200360649, & \eta _0 & =& & \eta _2, \end{gather*}
</div>
<p> <br />Lη/[2(1-L_0η)] = 0.168831169 &#8195;<span class="rmfamily">and</span> &#8195;L_1η/[2(1-L_2η)] = 0.005140238. </p>
<div class="displaymath" id="a0000000029">
  \end{gather*}
</div>
<p> Hence, the hypotheses of Lemma 2.1 are satisfied. Moreover, we have by <a href="#eq:211" class="eqref">18</a> that </p>
<div class="equation" id="a0000000030">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber t^{\star \star } =0.11761158 < 1-p = 0.3. \end{equation}
  </div>
  <span class="equation_label">68</span>
</p>
</div>
<p> Furthermore, using <a href="#eq:248" class="eqref">47</a> (for \(t^\star \) replaced by \(t^{\star \star }\)), we get </p>
<div class="equation" id="a0000000031">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber t^{\star \star } < R < \tfrac {2}{L_0}-t^{\star \star } = 0.751953637. \end{equation}
  </div>
  <span class="equation_label">68</span>
</p>
</div>
<p> So, we can choose \(R=0.3\). Hence, hypotheses of Theorem 2.3 hold, and (TSNM) converges to </p>
<div class="displaymath" id="a0000000032">
  \begin{equation*}  x^\star = \left(\sqrt[3]{0.7},\sqrt[3]{0.7}\right)^T = \left(0.887904002, 0.887904002\right)^T. \end{equation*}
</div>
<p> We compare <a href="#eq:26" class="eqref">13</a> to <a href="#eq:261" class="eqref">60</a>. </p>
<p> <div class="table"  id="table:1">
   <div class="centered"><small class="footnotesize"><div class="centered"><table class="tabular">
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>  \(n\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>  \(s_n\! -\! t_n\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>  \(t_{n+1}\! -\! s_n\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>  \(\overline{s}_n\! -\! \overline{t}_n\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>  \(\overline{t}_{n+1}\! -\! \overline{s}_n\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \(\overline{\overline{s}}_n\! -\! \overline{\overline{t}}_n\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \(\overline{\overline{t}}_{n+1}\! -\! \overline{\overline{s}}_n\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(0\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\({1.00e-01}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1.69e-02}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\({1.00e-01}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\({1.76e-02}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1.00e-01}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1.49e-02}\)</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(1\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({5.07e-04}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({4.57e-07}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({5.78e-04}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({6.27e-07}\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({3.49e-04}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2.15e-07}\)</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(2\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({3.73e-13}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2.47e-25}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({5.37e-13}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1.02e-24}\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({8.19e-14}\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({1.19e-26}\)</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(3\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({1.09e-49}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2.11e-98}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({1.94e-48}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({1.09e-96}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2.49e-52}\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1.09e-103}\)</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(4\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({7.91e-196}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1.11e-390}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({9.44e-191}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({1.67e-380}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2.11e-206}\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({7.88e-412}\)</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(5\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({2.21e-780}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({8.70e-1560}\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({5.24e-760}\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\({5.15e-1519}\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1.09e-822}\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2.13e-1644}\)</p>

    </td>
  </tr>
</table> </div></small> <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">Comparison among <a href="#eq:26" class="eqref">13</a>, <a href="#eq:265" class="eqref">64</a> and <a href="#eq:266" class="eqref">65</a></span> 
</figcaption> </div>
</div>  As expected from the theoretical results iteration <a href="#eq:26" class="eqref">13</a> is faster than <a href="#eq:265" class="eqref">64</a>. </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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</dd>
  <dt><a name="two">2</a></dt>
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  <dd><p><i class="sc">I. K. Argyros</i>, <i class="itshape"> On a multistep Newton method in Banach spaces and the Ptak error estimates</i>, Adv. Nonlinear Var. Inequal., <b class="bf">6</b> (2003), no. 2, pp.&#160;121–135. </p>
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  <dt><a name="four">4</a></dt>
  <dd><p><i class="sc">I. K. Argyros</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. Appl., <b class="bf">298</b> (2004), no. 2, pp.&#160;374–397. </p>
</dd>
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  <dd><p><i class="sc">I. K. Argyros</i>, <i class="sc">J. Y. Cho</i> and <i class="sc">S. Hilout</i>, <i class="itshape"> Numerical Methods for Equations and its Applications</i>, CRC Press Taylor &amp; Francis Group 2012, New York. </p>
</dd>
  <dt><a name="six">6</a></dt>
  <dd><p><i class="sc">R. P. Brent</i>, <i class="itshape"> Algorithms for Minimization without Derivatives</i>, Prentice Hall, Englewood Cliffs, New Jersey, 1973. </p>
</dd>
  <dt><a name="seven">7</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art4"> <i class="sc">E. Cătinaş</i>, <i class="itshape"> On some iterative methods for solving nonlinear equations</i>, Rev. Anal. Numér. Théor. Approx., <b class="bf">23</b> (1994), no. 1, pp.&#160;47–53. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
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  <dt><a name="eight">8</a></dt>
  <dd><p><i class="sc">J. A. Ezquerro</i> and <i class="sc">M. A. Hernández</i>, <i class="itshape"> Multipoint super-Halley type approximation algorithms in Banach spaces</i>, Numer. Funct. Anal. Optim., <b class="bf">21</b> (2000), no. 7-8,<br />pp.&#160;845–858. </p>
</dd>
  <dt><a name="nine">9</a></dt>
  <dd><p><i class="sc">J. A. Ezquerro</i>, <i class="sc">M. A. Hernández</i> and <i class="sc">M. A. Salanova</i>, <i class="itshape"> A Newton-like method for solving some boundary value problems</i>, Numer. Funct. Anal. Optim., <b class="bf">23</b> (2002), no. 7-8, pp.&#160;791–805. </p>
</dd>
  <dt><a name="ten">10</a></dt>
  <dd><p><i class="sc">J. A. Ezquerro</i>, <i class="sc">M. A. Hernández</i> and <i class="sc">M. A. Salanova</i>, <i class="itshape"> A discretization scheme for some conservative problems</i>, J. Comput. Appl. Math., <b class="bf">115</b> (2000), no. 1-2,<br />pp.&#160;181–192. </p>
</dd>
  <dt><a name="eleven">11</a></dt>
  <dd><p><i class="sc">M. A. Hernández</i>, <i class="sc">M. J. Rubio</i> and <i class="sc">J.A. Ezquerro</i>, <i class="itshape"> Secant-like methods for solving nonlinear integral equations of the Hammerstein type</i>, J. Comput. Appl. Math., <b class="bf">115</b> (2000), no. 1-2, pp.&#160;245–254. </p>
</dd>
  <dt><a name="twelve">12</a></dt>
  <dd><p><i class="sc">M. A. Hernández</i> and <i class="sc">M. J. Rubio</i>, <i class="itshape"> Semilocal convergence of the secant method under mild convergence conditions of differentiability</i>, Comput. Math. Appl., <b class="bf">44</b> (2002), no. (3-4), pp.&#160;277-285. </p>
</dd>
  <dt><a name="thirteen">13</a></dt>
  <dd><p><i class="sc">L. V. Kantorovich</i> and <i class="sc">G. P. Akilov</i>, <i class="itshape"> Functional Analysis</i>, Pergamon Press, Oxford, 1982. </p>
</dd>
  <dt><a name="fourteen">14</a></dt>
  <dd><p><i class="sc">A. M. Ostrowski</i>, <i class="itshape"> Solutions of equations in euclidean and Banach spaces</i>, A Series of Monographs and Textbooks, Academic Press, New York, 1973. </p>
</dd>
  <dt><a name="fifteen">15</a></dt>
  <dd><p><i class="sc">J. M. Ortega</i> and <i class="sc">W. C. Rheinboldt</i>, <i class="itshape"> Iterative solution of nonlinear equations in several variables</i>, Academic Press, New York 1970. </p>
</dd>
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  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/1993-vol22-no1-art8"> <i class="sc">I. Păvăloiu</i>, <i class="itshape"> A convergence theorem concerning the method of Chord</i>, Rev. Anal. Numér. Théor. Approx., <b class="bf">21</b> (1972), no. 1, pp.&#160;59–65. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
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  <dd><p><i class="sc">F. A. Potra</i> and <i class="sc">V. Pták</i>, <i class="itshape">Nondiscrete induction and iterative processes</i>, Research Notes in Mathematics, <b class="bf">103</b>, Pitman Avanced Publ. Program, Boston, 1984. </p>
</dd>
</dl>


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