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<title>Extending the solvability of equations using secant-type methods in Banach space: Extending the solvability of equations using secant-type methods in Banach space</title>
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<h1>Extending the solvability of equations using secant-type methods in Banach space</h1>
<p class="authors">
<span class="author">Ioannis K. Argyros\(^\ast \) Santhosh George\(^{\ast \ast }\)</span>
</p>
<p class="date">September 15, 2017. Accepted: September 22, 2021; Published online: February 17, 2022.</p>
</div>
<div class="abstract"><p> We extend the solvability of equations defined on a Banach space using numerically efficient secant-type methods.The convergence domain of these methods is enlarged using our new idea of restricted convergence region. By using this approach, we obtain a more precise location where the iterates lie than in earlier studies leading to tighter Lipschitz constants. This way the semi-local convergence produces weaker sufficient convergence criteria and tighter error bounds than in earlier works. These improvements are also obtained under the same computational effort, since the new Lipschitz constants are special cases of the old ones. </p>
<p><b class="bf">MSC.</b> 65G99, 45G10, 47H99 </p>
<p><b class="bf">Keywords.</b> Banach space, Secant-type method, Semi-local convergence, restricted convergence region, Lipschitz conditions. </p>
</div>
<p>\(^\ast \)Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA, e-mail: <span class="tt">iargyros@cameron.edu</span>. </p>
<p>\(^{\ast \ast }\)Department of Mathematical and Computational Sciences, NIT Karnataka, India-575 025, e-mail: <span class="tt">sgeorge@nitk.ac.in</span>. </p>
<h1 id="a0000000002">1 Introduction</h1>
<p> Let \(F: {\Omega }\subset \mathcal{B}_1\longrightarrow \mathcal{B}_2\) be a nonlinear operator, \(\mathcal{B}_1,\mathcal{B}_2\) be Banach spaces and \(\Omega \) be a convex set. Numerous iterative methods for solving equation \(F(x)=0\) can be written like </p>
<div class="equation" id="1.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.1} x_{n+1}=x_n-L_nF(x_n)\, \, \textnormal{for each}\, \,  n=0,1,2,\ldots \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>where \(L_n\in \mathcal{L}(\mathcal{B}_2, \mathcal{B}_1)\) the space of bounded linear operators from \(\mathcal{B}_2\) into \(\mathcal{B}_1\) for each \(n\in \mathbb {N}\cup \{ 0\} .\) The most widely used methods like (<a href="#1.1">1</a>) are Newton’s method, where \(L_n= F'(x_n)^{-1},\) and the secant method, where \(L_n=\delta F(x_n, x_{n-1})^{-1}\) and \(\delta F\) stands for a consistent approximation to the Fréchet-derivative of \(F\) <span class="cite">
	[
	<a href="#6" >6</a>
	, 
	<a href="#26" >26</a>
	]
</span>. A lot of problems in control theory, optimization, inverse problems theory, Mathematical Physics, Chemistry, Economics, Biology and also in engineering can be brought in the form of equation \(F(x)=0\) using Mathematical modeling <span class="cite">
	[
	<a href="#1" >1</a>
	, 
	<a href="#2" >2</a>
	, 
	<a href="#3" >3</a>
	, 
	<a href="#6" >6</a>
	, 
	<a href="#11" >11</a>
	, 
	<a href="#12" >12</a>
	, 
	<a href="#13" >13</a>
	, 
	<a href="#14" >14</a>
	, 
	<a href="#22" >22</a>
	, 
	<a href="#23" >23</a>
	, 
	<a href="#26" >26</a>
	, 
	<a href="#29" >29</a>
	]
</span>. Closed form solutions are preferred but this is rarely possible. Consequently, mostly iterative methods are utilized say like method (<a href="#1.1">1</a>) to generate a sequence approximating a locally unique solution \(x^*\) of equation \(F(x)=0\) under some conditions. It is well known from the numerical efficiency that it is not advantageous to change the operator \(L_n\) at each step of the iterative method. If one keeps the operator piecewise constant more efficient iterative methods can be obtained. Optimal recepts can be obtained based on the dimension of the space <span class="cite">
	[
	<a href="#29" >29</a>
	]
</span>. Iterative methods of this type have been studied by Traub <span class="cite">
	[
	<a href="#29" >29</a>
	]
</span>, Potra and Pták <span class="cite">
	[
	<a href="#26" >26</a>
	]
</span>, Bosarg and Falb <span class="cite">
	[
	<a href="#7" >7</a>
	, 
	<a href="#8" >8</a>
	]
</span>, Dennis <span class="cite">
	[
	<a href="#9" >9</a>
	]
</span>, Potra <span class="cite">
	[
	<a href="#26" >26</a>
	, 
	<a href="#27" >27</a>
	]
</span>, Amat <span class="cite">
	[
	<a href="#1" >1</a>
	, 
	<a href="#2" >2</a>
	]
</span>, Ezquerro <i class="it">et al.</i> <span class="cite">
	[
	<a href="#12" >12</a>
	]
</span>, Hernandez <i class="it">et al.</i> <span class="cite">
	[
	<a href="#14" >14</a>
	, 
	<a href="#15" >15</a>
	, 
	<a href="#16" >16</a>
	]
</span>, Argyros <span class="cite">
	[
	<a href="#3" >3</a>
	, 
	<a href="#4" >4</a>
	, 
	<a href="#5" >5</a>
	, 
	<a href="#6" >6</a>
	]
</span> (see also the references in the preceding papers). </p>
<p>In this paper we are motivated by the work by Potra <span class="cite">
	[
	<a href="#25" >25</a>
	, 
	<a href="#26" >26</a>
	]
</span> who improved the work by <span class="cite">
	[
	<a href="#7" >7</a>
	, 
	<a href="#8" >8</a>
	, 
	<a href="#9" >9</a>
	, 
	<a href="#29" >29</a>
	]
</span>. Let us choose </p>
<div class="equation" id="1.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.2} L_n\in \{ \delta F(x_{p_n}, x_{q_n})^{-1}, \delta F(x_{q_n}, x_{p_n})^{-1}\} \, \,  \textnormal{for each}\, \,  n=0,1,2,\ldots , \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>where \(\{ p_n\} \) and \(\{ q_n\} \) are non-decreasing sequences of integers such that </p>
<div class="equation" id="1.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.3} q_0=-1, p_0=0, q_n\leq p_n\leq n\, \,  \textnormal{for each}\, \,  n=1,2,3,\ldots . \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>The convergence region of method (<a href="#1.1">1</a>)–(<a href="#1.2">2</a>) is small in general. That is why, we find in this paper a more accurate location containing the iterates \(\{ x_n\} \) than \(\Omega \) leading to tighter Lipschitz constants. This way we obtain: weaker sufficient convergence criteria, tighter error bounds on the distances \(\| x_{n+1}-x_n\| ,\) \(\| x_n-x^*\| \) and at least as precise information on the location of the solution \(x^*.\) It is worth noticing that the preceding improvements are obtained under the same computational effort, since in practice the computation of the old Lipschitz constants requires the computation of the new constants as a special cases. </p>
<p>The rest of the paper is structured as follows. Section <a href="#S2">2</a> and Section <a href="#S3">3</a> contain the semi-local convergence of secant-type and Newton-type methods, respectively. </p>
<h1 id="S2">2 Semi-local convergence \(p_n\neq q_n\) for each \(n\)</h1>

<p>We shall study the iterative procedure (<a href="#1.1">1</a>) and (<a href="#1.2">2</a>) in this section for the triplets \((F, x_0, x_{-1})\) belonging to the class \(A(\alpha _0, \alpha , \beta , \gamma )\) defined as follows: <div class="definition_thmwrapper " id="D2.1">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
  </div>
  <div class="definition_thmcontent">
  <p> Let \(\alpha _0 {\gt} 0,\,  \alpha {\gt} 0, \beta \geq 0, \gamma \geq 0\) satisfy </p>
<div class="equation" id="2.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.1} \alpha \beta + 2\sqrt{\alpha \beta }\leq 1. \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p>We say that the triplet \((F, x_0, x_{-1})\) belongs to the class \(A(\alpha _0, \alpha , \beta , \gamma )\) if : </p>
<ol class="enumerate">
  <li><p>\(F\) is a nonlinear operator defined on a convex subset \(\Omega \) of a Banach space \(\mathcal{B}_1\) and with values in a Banach space \(\mathcal{B}_2.\) </p>
</li>
  <li><p>\(x_0\) and \(x_{-1}\) are two points belonging to the interior \(\Omega ^\circ \) of \(\Omega \) and satisfies the inequality </p>
<div class="equation" id="2.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.2} \| x_0-x_{-1}\| \leq \beta . \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
</li>
  <li><p>\(F\) is Fréchet differentiable on \(\Omega ^\circ \) and there exists a mapping \(\delta F:\Omega ^\circ \times \Omega ^\circ \longrightarrow L(\mathcal{B}_1, \mathcal{B}_2)\) such that: </p>
<div class="displaymath" id="a0000000003">
  \[ \textnormal{ linear operator}\, \, L_0,\, \,  \textnormal{where}\, \, L_0\, \, \textnormal{is either}\, \, \delta F(x_0, x_{-1}) \]
</div>
<div class="equation" id="2.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.3} \textnormal{or}\, \, \delta F(x_{-1}, x_0), \textnormal{ is invertible, its inverse}\, \, L_0=P_0^{-1}\, \, \textnormal{is bounded} \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<div class="equation" id="2.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.4} \| L_0F(x_0)\| \leq \alpha ; \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<div class="equation" id="2.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.5} \| L_0(\delta F(x,y)-F'(x_0))\| \leq \alpha _0(\| x-x_0\| +\| y-x_0\| )\, \, \textnormal{for each}\, \, x,y\in \Omega , a_0 > 0. \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>Set \(\Omega _0=\Omega ^\circ \cap U(x_0, r_0),\,  r_0=\frac{1-a_0c}{2a_0}.\) </p>
<div class="equation" id="2.6">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.6} \| L_0(\delta F(x,y)-F'(z))\| \leq \alpha (\| x-z\| +\| y-z\| )\, \, \textnormal{for each}\, \, x,y, z\in \Omega _0 \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p>and </p>
<div class="equation" id="2.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.7} \alpha _0\leq \alpha . \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
</li>
  <li><p>The set \(\Omega _\gamma =\{ x\in \Omega : F \textnormal{is continuous at}\, \,  x\} \) contains the ball \(\bar{U}(x_1, r_1)\subset V\) with center \(x_1=x_0-L_0F(x_0)\) and radius \(r_1=\frac{1}{2\alpha }[1-\alpha (2\beta +\gamma )-\sqrt{(1-\alpha \gamma )^2-4\alpha \beta }].\) </p>
</li>
</ol>

  </div>
</div>We associate the class \(A(\alpha _0, \alpha , \beta , \gamma )\) with the constant \(\delta \) and sequence \(\{ s_n\} _{n\geq -1}\) given by the formulae: </p>
<div class="equation" id="2.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.8} s_{-1}=\frac{1+\alpha \gamma }{2\alpha },\,  s_0=\frac{1-\alpha \gamma }{2\alpha },\,  s_{n+1}=s_n-\frac{s_n^2-\delta ^2}{s_{p_n}-s_{q_n}}, \, \, n=0,1,2,\ldots \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p>and \(\delta =\frac{1}{2\alpha }\sqrt{(1-\alpha \gamma )^2-4\alpha \beta }.\) </p>
<p>Using the above notation we present the following semi-local convergence result. <div class="theorem_thmwrapper " id="T2.1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If \((F, x_0, x_{-1})\in A(\alpha _0, \alpha , \beta , \gamma )\) then the iterative algorithm <a href="#1.1" class="eqref">1</a>–<a href="#1.2" class="eqref">2</a> is well defined, the sequence \(\{ x_n\} _{n\geq 1}\) generated by it converges to a solution \(x^*\in U(x_0, r_1)\) of the equation \(F(x)=0.\) Moreover, the following items hold: </p>
<div class="displaymath" id="a0000000004">
  \begin{eqnarray} \nonumber \| x_n-x^*\| & \leq &  s_0-\| x_n-x_0\| -[(s_n-\| x_n-x_0\| ^2\\ \nonumber & & -(\| x_n-x_{p_{n-1}}\| +\| x_{n-1}-x_{p_{n-1}}\|  \\ \label{2.9} & & + \| x_{p_{n-1}}-x_{q_{n-1}}\| )\| x_{n}-x_{{n-1}}\| ]^{1/2}\leq s_n-\delta \end{eqnarray}
</div>
<p> and </p>
<div class="displaymath" id="a0000000005">
  \begin{eqnarray} \nonumber \| x_n-x^*\|  & \geq &  [(s_0-\tfrac {1}{2}(\| x_{p_n}-x_{q_n}\| +\| x_{p_n}-x_{0}\| +\| x_{q_n}-x_{0}\| )- \| x_{n}-x_{p_{n}}\| )^2 \\ \label{2.10} & & +(2s_0-\| x_{p_n}-x_0\| -\| x_{q_n}-x_0\| )\| x_n-x_{n+1}\| ]^{1/2}\\ \nonumber & & -s_0+\tfrac {1}{2}(\| x_{p_n}-x_{q_n}\| +\| x_{p_n}-x_0\| +\| x_{q_n}-x_0\| )+\| x_n-x_{p_n}\| . \end{eqnarray}
</div>

  </div>
</div><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">
  
  </div>
</div> The linear operator \(M=\delta F(u,v)\) is invertible for each \(u, v\in \Omega ^\circ \) with </p>
<div class="equation" id="2.11">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.11} \| u-x_0\| +\| v-x_0\|  < 2s_0. \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p>It follows from (<a href="#2.4">7</a>) that </p>
<div class="displaymath" id="a0000000007">
  \begin{eqnarray*}  \| I-L_0M\| & =& \| L_0(M_0-M)\| \leq \| L_0(M-F’(x_0))\| +\| L_0(F’(x_0)-M_0)\| \\ & \leq & \alpha _0(\| u-x_0\| +\| v-x_0\| )+\| x_0-x_{-1}\|  {\lt} 1. \end{eqnarray*}
</div>
<p> Hence, by the Banach Lemma <span class="cite">
	[
	<a href="#6" >6</a>
	, 
	<a href="#28" >28</a>
	, 
	<a href="#30" >30</a>
	, 
	<a href="#31" >31</a>
	, 
	<a href="#32" >32</a>
	]
</span> \(M\) is invertible and </p>
<div class="equation" id="2.12">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.12} \| (L_0M)^{-1}\| \leq [1-\alpha _0(\| u-x_0\| +\| v-x_0\| +\gamma )]^{-1}. \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p>Note that the condition (<a href="#2.4">7</a>) implies the following Lipschitz condition for \(F'\) </p>
<div class="equation" id="2.13">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.13} \| L_0(F'(u)-F'(v))\| \leq 2\alpha \| u-v\| ,\, \,  u,v\in \Omega _0^\circ . \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p>Using the integral representation </p>
<div class="equation" id="2.14">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.14} F(x)-F(y)=\int _0^1F'(y+\theta (x-y))d\theta (x-y) \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p>we obtain that </p>
<div class="equation" id="2.15">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.15} \| L_0(F(x)-F(y)-F'(u)(x-y))\| \leq \alpha (\| x-u\| +\| y-u\| )\| x-y\|  \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p>for all \(x, y \in \Omega _0^\circ .\) </p>
<p>Finally form (<a href="#2.6">9</a>) and (<a href="#2.14">17</a>) we have </p>
<div class="equation" id="2.16">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.16} \| L_0(F(x)-F(y)-\delta F(u,v)(x-y)\| \leq \alpha (\| x-u\| +\| y-u\| +\| u-v\| )\| x-y\|  \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>for all \(x, y, u,v\in \Omega _0^\circ .\) Estimates (<a href="#2.13">16</a>), (<a href="#2.14">17</a>),(<a href="#2.15">18</a>), by a continuity argument, remain valid if \(x\) and/or \(y\) belong to \(\Omega _\gamma .\) Using the above inequalities we shall prove that </p>
<div class="equation" id="2.17">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.17} \| x_n-x_{n+1}\| \leq s_n-s_{n+1} \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p>for \(n=-1,0,1,\ldots .\) </p>
<p>Clearly, the sequence \(\{ s_n\} _{n\geq 1}\) given by (<a href="#2.8">11</a>) is decreasing and converges to \(\delta .\) Hence, if (<a href="#1.1">1</a>)-(<a href="#1.2">2</a>) is well defined for \(n=0,1,2,\ldots , k, \) and if (<a href="#2.17">20</a>) holds for \(n\leq k\) than </p>
<div class="displaymath" id="a0000000008">
  \[ \| x_0-x_n\| \leq s_0-s_n {\lt} s_0-\delta  \]
</div>
<p> for \(n\leq k.\) That is (<a href="#2.11">14</a>) is satisfied for \(u=x_i\) and \(v=x_j\) with \(i, j \leq k.\) Therefore, (<a href="#1.1">1</a>)-(<a href="#1.2">2</a>) will be well defined for \(n=k+1\) as well. </p>
<p>For \(n=-1\) and \(n=0\) (<a href="#2.17">20</a>) reduces to \(\| x_{-1}-x_0\|  \leq \gamma \) and \(\| x_0-x_{-1}\| \leq \beta \) (see (<a href="#2.2">5</a>) and (<a href="#2.4">7</a>)). Suppose (<a href="#2.17">20</a>) holds for \(n=-1,0,1,\ldots , k,\) where \( k\geq 0.\) Denote \(M_n=L_n^{-1},\) where \(L_n\) is given by (<a href="#1.2">2</a>). Notice that </p>
<div class="equation" id="2.18">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.18} F(x_{k+1})=F(x_{k+1})-F(x_k)-M_k(x_{k+1}-x_k). \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p>Then, using (<a href="#2.11">14</a>) and (<a href="#2.15">18</a>) we can write </p>
<div class="displaymath" id="a0000000009">
  \begin{eqnarray*}  \| x_{k+1}-x_{k+2}\| & =& \| L_{k+1}F(x_{k+1})\| =\| (L_0M_{k+1})^{-1}L_0F(x_{k+1})\| \\ & \leq & \frac{\alpha (\| x_{k+1}-x_{p_k}\| +\| x_k-x_{p_k}\| +\| x_{p_k}-x_{q_k}\| )}{1-\alpha _0(\| x_{p_{k+1}}-x_0\| +\| x_{q_{k+1}}-x_0\| +\gamma )}\| x_k-x_{k+1}\| \\ & \leq & \frac{\alpha ((s_{p_k}-s_{k+1}+s_{p_k}-s_k+s_{q_k}-s_{p_k})}{1-\alpha _0(s_0-s_{p_{k+1}}-s_0-s_{q_{k+1}}+s_{-1}-s_0)}(s_k-s_{k+1})\\ & \leq & \frac{s_{p_k}+s_{q_k}-s_{k+1}-s_k}{s_{p_{k+1}}+s_{q_{k+1}}}(s_k-s_{k+1})=s_{k+1}-s_{k+2}. \end{eqnarray*}
</div>
<p> So, (<a href="#2.16">19</a>) holds for each \(n.\) \(\mathcal{B}_1\) is a complete space. Hence, the sequence \(\{ x_n\} _{n\geq 0}\) converges to \(x^*\) and </p>
<div class="equation" id="2.19">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.19} \| x_n-x^*\| \leq s_n-\delta . \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p>Next, by (<a href="#2.15">18</a>) and (<a href="#2.17">20</a>), we obtain </p>
<div class="equation" id="2.20">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.20} \| L_0F(x_{k+1})\| \leq \alpha (\| x_{k+1}-x_{p_k}\| +\| x_k-x_{p_k}\| +\| x_{p_k}-x_{q_k}\| )\| x_k-x_{k+1}\| , \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<p>so it follows that \(F(x^*)=0.\) </p>
<p>Let \(x=x_n\) and \(y=x^*\) in (<a href="#2.14">17</a>) and denote \(A=\int _0^1F'(x^*+\theta (x_n-x^*))d\theta .\) Then, by (<a href="#2.16">19</a>) and (<a href="#2.18">21</a>) we get in turn that </p>
<div class="displaymath" id="a0000000010">
  \begin{eqnarray*}  \| x_n-x_0\| +\| x^*-x_0\| +\| x_0-x_{-1}\| & \leq &  2\| x_n-x_0\| +\| x_n-x^*\| +\gamma \\ &  {\lt}&  2(\| x_n-x_0\| +\| x_n-x^*\| )\\ & \leq &  2(s_0-s_n+s_n-\delta )+\gamma \\ & \leq & 2s_0+\gamma =\tfrac {1}{\alpha }. \end{eqnarray*}
</div>
<p> In view of (<a href="#2.5">8</a>) and the Banach’s lemma we show as in (<a href="#2.12">15</a>) that, \(A\) is invertible and </p>
<div class="equation" id="2.20">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.20} \| (L_0A)^{-1}\| \leq [1-\alpha _0(2\| x_n-x_0\| +\| x_n-x^*\| +\gamma )^{-1}. \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p>Using (<a href="#2.19">22</a>) and (<a href="#2.20">24</a>), we have </p>
<div class="displaymath" id="a0000000011">
  \begin{eqnarray*}  \| x_n-x^*\| & =& \| A^{-1}F(x_n)\| \leq \| (L_0A)^{-1}\| \| L_0F(x_n)\| \\ & \leq & \tfrac {\alpha (\| x_n-x_{p_{n-1}}\| +\| x_{n-1}-x_{p_{n-1}}\| +\| x_{p_{n-1}}-x_{q_{n-1}}\| )}{1-\alpha _0(2\| x_n-x_0\| +\| x_n-x^*\| +\gamma )}\| x_n-x_{n-1}\| . \end{eqnarray*}
</div>
<p> It is easy to see that the above inequality together with (<a href="#2.7">10</a>) and \(\| x_n-x^*\|  {\lt} s_0\) imply the estimate (<a href="#2.9">12</a>). </p>
<p>By the identity </p>
<div class="displaymath" id="a0000000012">
  \[ x_{n+1}-x_n=x^*-x_n+(L_0M_n)^{-1}L_0(F(x^*)-F(x_n)-M_n(x^*-x_n)), \]
</div>
<p> (<a href="#2.17">20</a>) and (<a href="#2.16">19</a>), we obtain </p>
<div class="displaymath" id="a0000000013">
  \[ \| x_{n+1}-x_n\| \leq \tfrac {\alpha (2\| x_n-x_{p_n}\| +\| x^*-x_n\| +\| x_{p_n}-x_{q_n}\| )}{1-\alpha _0(\| x_{p_n}-x_0\| +\| x_{q_n}-x_0\| +\gamma }\| x_n-x^*\| +\| x_n-x^*\| , \]
</div>
<p> so (<a href="#2.9">12</a>) is shown. <div class="proof_wrapper" id="a0000000014">
  <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="corollary_thmwrapper " id="C2.3">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">3</span>
  </div>
  <div class="corollary_thmcontent">
  <p> Suppose \((F, x_0, x_{-1})\in A(\alpha _0, \alpha , \beta , \gamma ).\) Then the equation \(F(x)=0\) has a solution \(x^*\in U\) and this is the only solution of the equation in the set \(Q_1=\{ x\in \Omega _\gamma :\| x-x_0\| \leq r\} \) if \(\delta {\gt} 0, r=\delta -s_0-\gamma +\frac{1}{\alpha _0},\) or in the set \(Q_2=\{ x\in \Omega _\gamma :\| x-x_0\| \leq s_0\} \) if \(\delta =0.\) </p>

  </div>
</div><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> The existence has been shown in the Theorem<a href="#T2.1">2</a>. Suppose \(\delta {\gt} 0\) and let \(x^*\in U\) and \(y^*\in Q_1\) be solutions of the equation \(F(x)=0.\) Let us denote \(S_*=\int _0^1 F'(y^*+\theta (x^*-y^*))d\theta .\) Using (<a href="#2.5">8</a>), we have </p>
<div class="displaymath" id="a0000000016">
  \begin{eqnarray*}  \| I-L_0S_*\| & =& \| L_0(M_0-S_*)\| \leq \alpha _0(\| y^*-x_0\| +\| x^*-x_0\| +\| x_0-x_{-1}\| )\\ &  {\lt} & \alpha _0(r+s_0-\delta +\gamma )=1. \end{eqnarray*}
</div>
<p> That is \(S_*\) is invertible. Then, by (<a href="#2.14">17</a>) we deduce that \(x^*=y^*.\) </p>
<p>Case \(\delta =0.\) Let \(M_n=0\) and \(q_n=-1\) for \(n=0,1,2,\ldots \) then the iterative procedure (<a href="#1.1">1</a>)-(<a href="#1.2">2</a>) becomes </p>
<div class="equation" id="2.21">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.21} x_{n+1}=x_n-L_0F(x_n),\, \, n=0,1,2,\ldots . \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p>By Theorem <a href="#T2.1">2</a> it follows that the sequence \(\{ x_n\} _{n\geq 0}\) given by (<a href="#2.21">25</a>) converges to a solution \(x^*\) of the equation \(F(x)=0.\) It also follows that </p>
<div class="equation" id="2.22">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.22} \| x_n-x_{n+1}\| \leq s_n-s_{n+1}, \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<p>where </p>
<div class="equation" id="2.23">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.23} s_0=(\tfrac {\beta }{\alpha _0})^{1/2},\,  s_{n+1}=s_n-\alpha _0 s_n^2,\,  n=0,1,2,\ldots . \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p>Using induction we get that </p>
<div class="equation" id="2.24">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.24} s_n\geq \frac{(\frac{\beta }{\alpha _0})^{1/2}}{n+1},\, \, \,  n=0,1,2,\ldots . \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<p>Let \(y^*\in Q_2\) be a solution of the equation \(F(x)=0\) and denote \(A_n=\int _0^1F'(y^*+\theta (x_n-y^*))d \theta .\) According to (<a href="#2.5">8</a>), (<a href="#2.11">14</a>), (<a href="#2.20">24</a>) and (<a href="#2.21">25</a>) we have </p>
<div class="displaymath" id="a0000000017">
  \begin{eqnarray*}  \| x_{n+1}-y^*\| & =& \| L_0(M_0-A_n)(x_n-y^*)\| \\ & \leq & \alpha _0\| x_n-y^*\| (\| y^*-x_0\| +\| x_n-x_0\| +\| x_0-x_{-1}\| )\\ & \leq & \| x_n-y^*\| (1-\alpha _0 s_n)\leq \ldots \\ & \leq & \| x_1-y^*\| \prod _{j=1}^n(1-\alpha _0 s_j). \end{eqnarray*}
</div>
<p> By (<a href="#2.24">28</a>), we deduce that \(\lim _{n\longrightarrow \infty }\prod _{j=1}^n(1-\alpha _0 s_j) =0.\) Hence, we conclude that \(y^*=\lim _{n\longrightarrow \infty }x_n=x^*.\) <div class="proof_wrapper" id="a0000000018">
  <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>Next, we show that the results obtained in this section are sharp within the class \(A(\alpha _0, \alpha , \beta , \gamma ).\) <div class="proposition_thmwrapper " id="P2.4">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">4</span>
  </div>
  <div class="proposition_thmcontent">
  <p><span class="cite">
	[
	<a href="#27" >27</a>
	]
</span> Suppose \(\alpha _0 {\gt} 0, \alpha {\gt} 0, \beta \geq 0, \gamma \geq 0\) satisfy inequality <a href="#2.1" class="eqref">4</a>. Then, the following items hold: </p>
<ol class="enumerate">
  <li><p>There exist a function \(F: \mathbb {R}\longrightarrow \mathbb {R}\) and two points \(x_0, x_{-1} \in \mathbb {R}\) such that the triplet \((F, x_0, x_{-1})\in A(\alpha _0, \alpha , \beta , \gamma )\) and for this triplet the estimates <a href="#2.8" class="eqref">11</a> are attained at each \(n=0,1,2,3,\ldots .\) </p>
</li>
  <li><p>For each \(n=0,1,2,\ldots \) there exist a function \(f_n:\mathbb {R}\longrightarrow \mathbb {R}\) and two points \(x_0, x_{-1}\in \mathbb {R}\) such that the triplet \((F,x_0, x_{-1})\in A(\alpha _0, \alpha , \beta ,\gamma )\) and for this triplet <a href="#2.9" class="eqref">12</a> holds with equality. </p>
</li>
</ol>

  </div>
</div></p>
<p>Concerning the domain of uniqueness of the solution \(x^*\) established in the Corollary <a href="#C2.3">3</a> we have <div class="proposition_thmwrapper " id="P2.4">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">5</span>
  </div>
  <div class="proposition_thmcontent">
  <p> <span class="cite">
	[
	<a href="#27" >27</a>
	]
</span> Suppose \(\alpha _0 {\gt} 0, \alpha {\gt} 0, \beta \geq 0, \gamma \geq 0\) s satisfy(<a href="#2.1">4</a>). Let \(\delta \) be the constant defined by (<a href="#2.8">11</a>). </p>
<ol class="enumerate">
  <li><p>If \(\delta {\gt} 0\) then there exist a function \(F:\mathbb {R}\longrightarrow \mathbb {R}\) and four points \(x_0, x_{-1}, x^*, y^* \in \mathbb {R}\) such that \((F, x_0, x_{-1})\in A(\alpha _0, \alpha , \beta , \gamma ), \) \(F(x^*)=F(y^*)=0, |x_0-x^*|=s_0-\delta , |x_0-y^*|=s_0+\delta .\) </p>
</li>
  <li><p>If \(\delta =0,\) then for each \(\epsilon {\gt} 0\) there exist a function \(f_\epsilon :\mathbb {R}\longrightarrow \mathbb {R}\) and four points \(x_0, x_{-1}, x^*, y_\epsilon ^*\in \mathbb {R}\) such that \((F, x_0, x_{-1})\in A(\alpha _0, \alpha , \beta , \gamma ), \) \(F(x^*)=F(y_\epsilon ^*)=0, |x_0-x^*|=s_0, |x_0-y_\epsilon ^*|=s_0+\epsilon .\) </p>
</li>
</ol>

  </div>
</div></p>
<p>Next, we shall consider some particular cases of the iterative procedure (<a href="#1.1">1</a>)-(<a href="#1.2">2</a>) and shall compare the results obtained in the preceding section with some known results. <div class="remark_thmwrapper " id="R2.6">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">6</span>
  </div>
  <div class="remark_thmcontent">
  <p> (a1) In earlier studies on Secant-type methods condition </p>
<div class="equation" id="8''">
<p>
  <div class="equation_content">
    \begin{equation} \label{8''} \| L_0(\delta F(x,y)-F'(z))\| \leq \alpha _1(\| x-z\| +\| y-z\| ) \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<p>for each \(x,y,z\in \Omega \) together with the condition </p>
<div class="equation" id="4'">
<p>
  <div class="equation_content">
    \begin{equation} \label{4'} \alpha _1\gamma +2\sqrt{\alpha _1\beta }\leq 1 \end{equation}
  </div>
  <span class="equation_label">30</span>
</p>
</div>
<p>are used <span class="cite">
	[
	<a href="#21" >21</a>
	, 
	<a href="#22" >22</a>
	, 
	<a href="#23" >23</a>
	, 
	<a href="#24" >24</a>
	, 
	<a href="#25" >25</a>
	, 
	<a href="#26" >26</a>
	, 
	<a href="#27" >27</a>
	, 
	<a href="#28" >28</a>
	, 
	<a href="#29" >29</a>
	, 
	<a href="#30" >30</a>
	, 
	<a href="#31" >31</a>
	, 
	<a href="#32" >32</a>
	]
</span>. Notice that (<a href="#8''">29</a>) implies (<a href="#2.5">8</a>), (<a href="#2.6">9</a>), </p>
<div class="displaymath" id="a0000000019">
  \[ \alpha _0\leq \alpha _1 \]
</div>
<div class="displaymath" id="a0000000020">
  \[ \alpha \leq \alpha _1, \]
</div>
<p> and \(\frac{\alpha _1}{\alpha }\) can be arbitrarily large <span class="cite">
	[
	<a href="#3" >3</a>
	, 
	<a href="#4" >4</a>
	, 
	<a href="#5" >5</a>
	, 
	<a href="#6" >6</a>
	]
</span>. Moreover, we have by (<a href="#2.1">4</a>) and (<a href="#4'">30</a>) that </p>
<div class="displaymath" id="a0000000021">
  \[  (\ref{4'})\Longrightarrow (\ref{2.1})  \]
</div>
<p> but not necessarily vice versa, unless, if \(\alpha =\alpha _1.\) Furthermore, in <span class="cite">
	[
	<a href="#27" >27</a>
	]
</span> \(\bar{r}=s_0+\delta \) and </p>
<div class="displaymath" id="a0000000022">
  \[  \bar{r}\leq r,  \]
</div>
<p> so the uniqueness of the solution is extended under our approach. Finally, it follows from the proof of Theorem <a href="#T2.1">2</a> that sequence \(\{ s_n\} \) defined by </p>
<div class="displaymath" id="a0000000023">
  \begin{eqnarray} \nonumber s_{-1}& =& \tfrac {1+\alpha _0 \gamma }{2\alpha _0} {\gt} t_{-1},\\ \nonumber s_0& =& \tfrac {1-\alpha _0 \gamma }{2\alpha _0} {\gt} t_0\\ \label{2.28} s_{n+2}& =& s_{n+1}+\tfrac {\alpha (s_{p_n}-s_{n+1}+s_{q_n}-s_n)}{1-\alpha _0(s_0-s_{p_{n+1}}+s_{-1}-s_{q_{n+1}}} \end{eqnarray}
</div>
<p> is also a majorizing sequence for \(\{ x_n\} \) and we can have instead of the corresponding estimates given in Theorem <a href="#T2.1">2</a>, the more precise estimates </p>
<div class="displaymath" id="2.29">
  \begin{align} \| x_n-x^*\| & \leq \tfrac {\alpha (\| x_n-x_{p_{n-1}}\| +\| x_{n-1}-x_{p_{n-1}}\| +\| x_{p_{n-1}}-x_{q_{n-1}}\| )}{1-\alpha _0(2\| x_n-x_0\| +\| x_n-x^*\| +\gamma )}\| x_n-x_{n-1}\|  \label{2.29} \end{align}
</div>
<p> and </p>
<div class="displaymath" id="2.30">
  \begin{align} \| x_{n+1}-x_n\| & \leq \tfrac {\alpha (2\| x_n-x_{p_{n}}\| +\| x^*-x_n\| +\| x_{p_{n}}-x_{q_{n}}\| )\| x_n-x^*\| }{1-\alpha _0(\| x_{p_n}-x_0\| +\| x_{q_n}-x_0\| +\gamma )} \label{2.30} +\| x_n-x^*\| , \end{align}
</div>
<p> respectively. Estimates (<a href="#2.29">32</a>) and (<a href="#2.30">33</a>) are clearly more precise than the corresponding ones given in <span class="cite">
	[
	<a href="#25" >25</a>
	, 
	<a href="#26" >26</a>
	]
</span> by </p>
<div class="displaymath" id="2.31">
  \begin{align} \| x_n-x^*\| & \leq \tfrac {\alpha _1(\| x_n-x_{p_{n-1}}\| +\| x_{n-1}-x_{p_{n-1}}\| +\| x_{p_{n-1}}-x_{q_{n-1}}\| )}{1-\alpha _1(2\| x_n-x_0\| +\| x_n-x^*\| +\gamma )} \label{2.31} \| x_n-x_{n-1}\|  \end{align}
</div>
<p> and </p>
<div class="displaymath" id="2.32">
  \begin{align} \| x_{n+1}-x_n\| & \leq \tfrac {\alpha _1(2\| x_n-x_{p_{n}}\| +\| x^*-x_n\| +\| x_{p_{n}}-x_{q_{n}}\| )\| x_n-x^*\| }{1-\alpha _1(\| x_{p_n}-x_0\| +\| x_{q_n}-x_0\| +\gamma )} \label{2.32} +\| x_n-x^*\| , \end{align}
</div>
<p> respectively. The preceding results are obtained assuming that \(\alpha _0\leq \alpha .\) However, if \(\alpha \leq \alpha _0,\) then the preceding results hold with \(\alpha _0\) replacing \(\alpha .\) The advantages in this study were obtained under the same computational cost as in <span class="cite">
	[
	<a href="#7" >7</a>
	, 
	<a href="#8" >8</a>
	, 
	<a href="#9" >9</a>
	, 
	<a href="#10" >10</a>
	, 
	<a href="#11" >11</a>
	, 
	<a href="#12" >12</a>
	, 
	<a href="#13" >13</a>
	, 
	<a href="#14" >14</a>
	, 
	<a href="#15" >15</a>
	, 
	<a href="#16" >16</a>
	, 
	<a href="#17" >17</a>
	, 
	<a href="#18" >18</a>
	, 
	<a href="#21" >21</a>
	, 
	<a href="#22" >22</a>
	, 
	<a href="#23" >23</a>
	, 
	<a href="#24" >24</a>
	, 
	<a href="#25" >25</a>
	, 
	<a href="#26" >26</a>
	, 
	<a href="#27" >27</a>
	, 
	<a href="#28" >28</a>
	, 
	<a href="#29" >29</a>
	, 
	<a href="#30" >30</a>
	, 
	<a href="#31" >31</a>
	, 
	<a href="#32" >32</a>
	]
</span>, since in practice the computation of \(\alpha _1\) requires the computation if \(\alpha _0\) and \(\alpha \) as special cases. </p>
<p>The preceding results can be improved even further, if we replace \(\Omega _0\) by \(\Omega _1\) defined by </p>
<div class="displaymath" id="a0000000024">
  \[ \Omega _1=\Omega _0\cap U(x_1, r_0-\| L_0F(x_0)\| ). \]
</div>
<p> Notice that \(\Omega _1\subseteq \Omega _0,\) so the corresponding condition to (<a href="#2.6">9</a>) hold on \(\Omega _1\) and the corresponding constant \(\bar{\alpha }\) will be such that \(\bar{\alpha } \leq \alpha .\) Examples where strict inequalities \(\alpha _0 {\lt} \alpha _1, \alpha _0 {\lt} \alpha , \alpha {\lt} \alpha _1\) and \(\bar{\alpha } {\lt} \alpha \) can be found in <span class="cite">
	[
	<a href="#3" >3</a>
	, 
	<a href="#4" >4</a>
	, 
	<a href="#5" >5</a>
	, 
	<a href="#6" >6</a>
	]
</span>. In the rest of the remarks our results compare favorable to earlier ones. </p>
<p>(a2) If \(p_n=n\) and \(q_n=n-1\) for each \(n=1,2,\ldots \) then (<a href="#1.1">1</a>)–(<a href="#1.2">2</a>) reduces to the secant method. The error estimates (<a href="#2.9">12</a>) and (<a href="#2.10">13</a>) improves (<a href="#2.29">32</a>) and (<a href="#2.30">33</a>) which in turn improved the ones in <span class="cite">
	[
	<a href="#7" >7</a>
	, 
	<a href="#8" >8</a>
	, 
	<a href="#9" >9</a>
	, 
	<a href="#10" >10</a>
	, 
	<a href="#11" >11</a>
	, 
	<a href="#12" >12</a>
	, 
	<a href="#13" >13</a>
	, 
	<a href="#14" >14</a>
	, 
	<a href="#15" >15</a>
	, 
	<a href="#16" >16</a>
	, 
	<a href="#17" >17</a>
	, 
	<a href="#18" >18</a>
	, 
	<a href="#21" >21</a>
	, 
	<a href="#22" >22</a>
	, 
	<a href="#23" >23</a>
	, 
	<a href="#24" >24</a>
	, 
	<a href="#29" >29</a>
	, 
	<a href="#30" >30</a>
	, 
	<a href="#31" >31</a>
	, 
	<a href="#32" >32</a>
	]
</span>. </p>
<p>(a3) If \(p_n=0\) and \(q_n=-1\) for \(n=0,1,2,\ldots \) then (<a href="#1.1">1</a>)–(<a href="#1.2">2</a>) reduces to the simplified secant method. The result contained in Theorem <a href="#T2.1">2</a> improves in this case the result from <span class="cite">
	[
	<a href="#27" >27</a>
	]
</span>. </p>
<p>(a4) If \(p_{km+j}=km, q_{km+j}=km-1, (q_{-1}=q_0=-1), j=0,1,\ldots , m-1, k=0,1,\ldots ,\) then (<a href="#1.1">1</a>)-(<a href="#1.2">2</a>) reduces to a procedure considered by Traub <span class="cite">
	[
	<a href="#29" >29</a>
	]
</span> for scalar equations. A local analysis for this procedure has been done by Potra and Pták, <span class="cite">
	[
	<a href="#26" >26</a>
	]
</span>, Laasonen <span class="cite">
	[
	<a href="#18" >18</a>
	]
</span> made a semi-local analysis for the case \(m=2.\)When \(p_n=q_n+1\) the iterative processes (<a href="#1.1">1</a>)–(<a href="#1.2">2</a>) was studied by Dennis <span class="cite">
	[
	<a href="#9" >9</a>
	]
</span>. The results obtained in Theorem <a href="#T2.1">2</a> improves all the above mentioned results. Note that by taking \(y_n=x_{nm}\) one contains a sequence \(\{ y_n\} _{n\geq 0}\) which converges to \(x^*\) with \(R-\)order \((m+\sqrt{m^2+4})/2.\) The parameter \(m\) can be chosen according to the dimension of the space in order to maximize the numerical efficiency of the procedure (see <span class="cite">
	[
	<a href="#25" >25</a>
	, 
	<a href="#26" >26</a>
	]
</span>). <span class="qed">â–¡</span></p>

  </div>
</div></p>
<h1 id="S3">3 Semi-local convergence analysis \(\lowercase {p}_{\lowercase {n}}=\lowercase {q}_{\lowercase {n}}\) for each \(\lowercase {n}\)</h1>
<p>  If \(x_{-1}=x_0\) and \(p_n=q_n\) for each \(n=0,1,2,\ldots ,\) then the iterative procedure (<a href="#1.1">1</a>)-(<a href="#1.2">2</a>) becomes </p>
<div class="equation" id="3.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.1} x_{n+1}=x_n-F'(x_{p_n})^{-1}F(x_n),\,  n=0,1,2,\ldots . \end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p>In <span class="cite">
	[
	<a href="#9" >9</a>
	, 
	<a href="#10" >10</a>
	]
</span>, Dennis proved that this iterative procedure converges under the hypotheses of the Kantorovich theorem. This fact follows by taking \(\gamma =0\) in Theorem <a href="#T2.1">2</a>. To be more precise let us consider the class \(A'(\alpha _0,\alpha ,\beta )\) defined below. <div class="definition_thmwrapper " id="D3.1">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">7</span>
  </div>
  <div class="definition_thmcontent">
  <p> Let \(\alpha _0 {\gt} 0, \alpha {\gt}0\) and \(\beta \geq 0\) satisfy </p>
<div class="equation" id="3.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.2} 4\alpha \beta \leq 1. \end{equation}
  </div>
  <span class="equation_label">37</span>
</p>
</div>
<p>We say that a pair \((F, x_0)\) belongs to the class \(A'(\alpha _0,\alpha ,\beta ),\) if </p>
<ol class="enumerate">
  <li><p>\(F\) is a nonlinear operator defined on a convex subset \(\Omega \) of a Banach space \(\mathcal{B}_1 \) and with values in a Banach space \(\mathcal{B}_2.\) </p>
</li>
  <li><p>\(x_0\) is a point belonging to the interior \(\Omega ^\circ \) of \(\Omega .\) </p>
</li>
  <li><p>\(F\) is Fréchet differentiable on \(\Omega ^\circ , F'(x_0)\) is boundedly invertible </p>
<div class="equation" id="3.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.3} \| F'(x_0)^{-1}(F'(x)-F'(x_0))\| \leq 2\alpha _0\| x-x_0\| \, \, \textnormal{for each}\, \, x,y\in \Omega . \end{equation}
  </div>
  <span class="equation_label">38</span>
</p>
</div>
<p>Set \(\Omega _0=\Omega \cap U(x_0,r_0),\, r_0=\frac{1}{2\alpha _0}\) and </p>
<div class="equation" id="3.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.4} \| F'(x_0)^{-1}(F'(x)-F'(y))\| \leq 2 \alpha \| x-y\| \, \, \textnormal{for each}\, \, x,y\in \Omega _0. \end{equation}
  </div>
  <span class="equation_label">39</span>
</p>
</div>
</li>
  <li><p>The set \(\Omega _c=\{ x\in \Omega , F\, \, \textnormal{is continuous at}\, \,  x\} \) contains the closed ball \(U\) with center \(x_1=x_0-F'(x_0)^{-1}F(x_0)\) and radius \(r_1=\frac{1}{2\alpha }(1-2\alpha \beta -\sqrt{1-4\alpha \beta }).\) </p>
</li>
</ol>

  </div>
</div>It is easy to see that \((F, x_0)\in A'(\alpha _0,\alpha ,\beta )\) if and only if \((F, x_0,x_0)\in A(\alpha _0,\alpha ,\beta ).\) In this case from Theorem <a href="#T2.1">2</a> it follows that the iterative procedure (<a href="#3.1">36</a>) converges and the following estimates hold: </p>
<div class="displaymath" id="a0000000025">
  \begin{eqnarray} \nonumber \| x_n-x^*\|  & \leq & s_0-\| x_n-x_0\| \\ \nonumber & & -[(s_0-\| x_n-x_0\| )^2-(\| x_n-x_{p_{n-1}}\| +\| x_{n-1}-x_{p_{n-1}}\| )\\ \label{3.5} & & \times \| x_n-x_{n-1}\| ]^{1/2} \end{eqnarray}
</div>
<p> and </p>
<div class="displaymath" id="a0000000026">
  \begin{eqnarray} \nonumber \| x_n-x^*\|  & \geq & [s_0-\| x_{p_n}-x_0\| -\| x_n-x_{p_n}\| )^2\\ \nonumber & & +2(s_0-\| x_{p_n}-x_0\| )\| x_n-x_{n+1}\| ]^{1/2}-s_0+\| x_{p_n}-x_0\| )\\ \label{3.6} & & \times \| x_n-x_{p_n}\| . \end{eqnarray}
</div>
<p> <div class="remark_thmwrapper " id="R3.2">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">8</span>
  </div>
  <div class="remark_thmcontent">
  <p> Set \(\gamma =0\) in the cases of Remark <a href="#R2.6">6</a> to obtain the corresponding improvements for Newton-type methods over the ones in <span class="cite">
	[
	<a href="#25" >25</a>
	, 
	<a href="#26" >26</a>
	, 
	<a href="#27" >27</a>
	]
</span> and the works earlier than the preceding. <span class="qed">â–¡</span></p>

  </div>
</div></p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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</a> </p>
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  <dd><p><a href ="https://www.jstor.org/stable/2949510"> <i class="sc">J.E. Dennis</i>, <i class="it">On the Kantorovich hypothesis for Newton’s method</i>, SIAM. J. Numer. Anal., <b class="bf">6</b> (1969), 493–507, <a href="https://www.jstor.org/stable/2949510">https://www.jstor.org/stable/2949510</a>. <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">J.E. Dennis</i>, <i class="it">Toward a unified convergence theory for Newton-like methods</i>, In Nonlinear functional analysis and applications, L.B. Rall, Ed., Academic Press, 1971, New York. </p>
</dd>
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</a> </p>
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</a> </p>
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</a> </p>
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</dd>
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</dd>
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</a> </p>
</dd>
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</dd>
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</a> </p>
</dd>
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  <dd><p><a href ="https://doi.org/10.1002/cpa.3160270305"> <i class="sc">E.J. Zehnder</i>, <i class="it">A remark on Newton’s method</i>, Commun. Pure Appl. Math., <b class="bf">27</b> (1974), 361–366, <a href="https://doi.org/10.1002/cpa.3160270305">https://doi.org/10.1002/cpa.3160270305</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="32">32</a></dt>
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</dd>
</dl>


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