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<h1>Analytic and empirical study of the rate of convergence of some iterative methods</h1>
<p class="authors">
<span class="author">Vasile Berinde\(^{1,2}\), Abdul Rahim Khan\(^{2}\) Mădălina Păcurar\(^{3}\)</span>
</p>
<p class="date">April 15, 2015.</p>
</div>
<p>\(^{1}\)Department of Mathematics and Computer Science North University of Baia Mare Victorie1 76, 430072 Baia Mare, Romania, e-mail: <span class="tt">vberinde@ubm.ro</span>. </p>
<p>\(^{2}\)Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia, e-mail: <span class="tt">arahim@kfupm.edu.sa</span>. </p>
<p>\(^{3}\)Department of Statistics, Analysis, Forecast and Mathematics Faculty of Economics and Bussiness Administration Babeş-Bolyai University of Cluj-Napoca 56-60 T. Mihali St., 400591 Cluj-Napoca, Romania, e-mail: <span class="tt">madalina.pacurar@econ.ubbcluj.ro</span>. </p>
<p>Dedicated to prof. I. Păvăloiu on the occasion of his 75th anniversary  </p>
<div class="abstract"><p> We study analytically and empirically the rate of convergence of two \(k\)-step fixed point iterative methods in the family of methods </p>
<div class="equation" id="eq-1">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq-1} x_{n+1}=T(x_{i_0+n-k+1},x_{i_1+n-k+1},\dots , x_{{i_{k-1}+n-k+1}}),\, n\geq k-1, \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where \(T: X^k \to X\) is a mapping satisfying some Presic type contraction conditions and \((i_0,i_1,\dots ,i_{k-1})\) is a permutation of \((0,1,\dots ,k-1)\). </p>
<p>We also consider the Picard iteration associated with the fixed-point problem \(x=T(x,\dots ,x)\) and compare analytically and empirically the rate and speed of convergence of three iterative methods. Our approach opens a new perspective on the study of the rate of convergence/speed of convergence of fixed-point iterative methods and also illustrates the essential difference between them by means of some concrete numerical experiments. </p>
<p><b class="bf">MSC.</b> 47H09, 47H10, 54H25. <b class="bf">Keywords.</b> metric space, contractive mapping, fixed point, \(k\)-step fixed point iterative method, rate of convergence. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>In the book <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span> (see also <span class="cite">
	[
	<a href="#Pav05" >28</a>
	]
</span>), I. Păvăloiu studied some multistep iterative methods for solving the scalar equation </p>
<div class="equation" id="eq-1">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq-1} x=\varphi (x) \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> where \(\varphi \colon I\rightarrow I\) is a function and \(I\subset \mathbb {R}\) is an interval. In order to solve <a href="#eq-1" class="eqref">2</a>, he considers a function \(g\colon I^s\rightarrow I\), where \(s\geq 1\) is an integer, and the restriction of \(g\) to the diagonal of \(I^s\) coincides with \(\varphi \), that is, </p>
<div class="equation" id="eq-2">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq-2} g(x,x,\dots ,x)=\varphi (x), \qquad \forall x\in I. \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> Then, by choosing \(x_0,x_1,\dots ,x_{s-1}\in I\), one constructs the \(s\)-point iterative sequence \(\{ x_n\} \) defined by </p>
<div class="equation" id="eq-3">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq-3} x_{s+p}=g(x_p,x_{p+1},\dots ,x_{p+s-1}),\qquad p=0,1,\dots \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> The convergence of the iterative method <a href="#eq-3" class="eqref">4</a> is established in Theorem 4.2.1 in <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span> (Theorem 5.3.1 in <span class="cite">
	[
	<a href="#Pav05" >28</a>
	]
</span>), which essentially states that, if \(\varphi \) and \(g\) are defined as above and there exist constants \(\alpha _i\in (0,1),\, i=1,2,\dots ,s\) satisfying </p>
<div class="equation" id="eq-4">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq-4} \alpha _1+\alpha _2+\dots +\alpha _s<1 \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> such that </p>
<div class="equation" id="eq-5">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq-5} |g(u_0,\dots ,u_{s-1})-g(u_1,\dots ,u_{s})|\leq \alpha _1 |u_0-u_1|+\dots +\alpha _{s} |u_{s-1}-u_s|,\end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> for all \(u_0,u_1,\dots ,u_{k\! -\! 1}\! \in \!  I\), then the sequence \(\{ x_n\} \) given by <a href="#eq-3" class="eqref">4</a> converges to \(\overline{x}\in I\), the unique solution of equation <a href="#eq-1" class="eqref">2</a>, for any initial values \(x_0,x_1,\dots ,x_{s\! -\! 1}\in I\). </p>
<p>Subsequently, by considering the family of \(s!\) iterative methods </p>
<div class="equation" id="eq-5a">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq-5a} x_{n+1}=g(x_{i_0+n-s+1},x_{i_1+n-s+1},\dots , x_{{i_{s-1}+n-s+1}}),\qquad n\geq s-1, \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> where \((i_0,i_1,\dots ,i_{s-1})\) is a permutation of \((0,1,\dots ,s-1)\), the authors in <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span> and <span class="cite">
	[
	<a href="#Pav05" >28</a>
	]
</span> search for a certain iterative method in that family for which the best error estimate is obtained (by means of Theorem 4.2.1 <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span>). </p>
<p>The conclusion (see Theorem 5.3.3 in <span class="cite">
	[
	<a href="#Pav05" >28</a>
	]
</span>) is that the optimal method in this respect corresponds to the particular method obtained from <a href="#eq-5a" class="eqref">7</a> in case of the permutation \((i_0,i_1,\dots ,i_{s-1})\) of \((0,1,\dots ,s-1)\) for which one has </p>
<div class="displaymath" id="a0000000003">
  \[  \alpha _{i_0}\geq \alpha _{i_1}\geq \dots \geq \alpha _{i_{s-1}}.  \]
</div>
<p> Starting from the fact that, in <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span> and <span class="cite">
	[
	<a href="#Pav05" >28</a>
	]
</span>, no direct proof is given of the fact that the methods in <a href="#eq-5a" class="eqref">7</a> are also convergent, our aim in this paper is quadruple: </p>
<ul class="itemize">
  <li><p>First, to give a different proof of Theorem 4.2.1 in <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span> (Theorem 5.3.1 in <span class="cite">
	[
	<a href="#Pav05" >28</a>
	]
</span>) in the more general case of mappings defined on a complete metric space \(X\); </p>
</li>
  <li><p>Second, to consider the one-point iterative method </p>
<div class="equation" id="eq-5b">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq-5b} y_{n+1}=g(y_{n},y_{n},\ldots ,y_{n}),\qquad n\geq 0, \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> and prove that it converges to \(\overline{x}\), for any initial value \(y_0\); </p>
</li>
  <li><p>Third, to show analytically that all the three iterative methods mentioned above have linear rate of convergence; </p>
</li>
  <li><p>Fourth, to define a suitable concept of speed of convergence and to show empirically that the rate of convergence and the speed of convergence are distinct concepts and, additionally, to present some examples that show that two methods having the same rate of convergence may exhibit a different speed of convergence. </p>
</li>
</ul>
<h1 id="a0000000004">2 Preliminaries</h1>
<p>We first note that Theorem 4.2.1 in <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span> (Theorem 5.3.1 in <span class="cite">
	[
	<a href="#Pav05" >28</a>
	]
</span>) is a particular case of Prešić fixed point theorem, established in the general setting of a metric space <span class="cite">
	[
	<a href="#Presic" >29</a>
	]
</span>. </p>
<p>Indeed, let \((X,d)\) be a metric space and \(T\colon X\rightarrow X\) a self mapping. Denote by \(Fix\, (T):=\{ x\in X:\, Tx=x\} \) the set of fixed points of \(T\). </p>
<p>If \((X,d)\) is complete and \(T\) is a contraction, <i class="it">i.e.</i>, there exists a constant \(\alpha \in [0,1)\) such that </p>
<div class="equation" id="2.1">
<p>
  <div class="equation_content">
    \begin{equation}  \label{2.1} d(Tx, Ty)\leq a\, d(x,y),\qquad \text{ for all}\; x,y\in X, \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> then, by the well known Banach contraction mapping principle (see <span class="cite">
	[
	<a href="#Ber07" >4</a>
	]
</span>, for example), we know that \(Fix\, (T)=\{ p\} \) and that, for any \(x_0\in X\), the Picard iteration, that is, the sequence defined by \(x_{n+1}=Tx_{n},\, n=0,1,\dots \), converges to \(p\), as \(n\rightarrow \infty \). </p>
<p>The Banach contraction mapping principle has been extended by Prešić <span class="cite">
	[
	<a href="#Presic" >29</a>
	]
</span> (see also <span class="cite">
	[
	<a href="#Wein" >41</a>
	]
</span>), to mappings \(f\colon X^k\rightarrow X\) satisfying a contractive condition that includes <a href="#2.1" class="eqref">9</a> in the particular case \(k=1\). </p>
<p><div class="theorem_thmwrapper " id="TPresic">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.1</span>
    <span class="theorem_thmtitle">S. Prešić <span class="cite">
	[
	<a href="#Presic" >29</a>
	]
</span>, 1965</span>
  </div>
  <div class="theorem_thmcontent">
  <p>  Let \((X,d)\) be a complete metric space, \(k\) a positive integer, \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{k}\in \mathbb {R}_{+}, \) \(\overset {k}{\underset {i=1}{\sum }}\alpha _{i}=\alpha {\lt}1\) and \(f\colon X^k\rightarrow X\) a mapping satisfying </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B1537D0>">
<p>
  <div class="equation_content">
    \begin{equation}  \label{Cond_Presic} d\left(f(x_0,\dots ,x_{k-1}), f(x_1,\dots ,x_{k})\right)\leq \alpha _1 d(x_0,x_1)+\dots +\alpha _{k} d(x_{k-1},x_k),\end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p> for all \(x_0,\dots ,x_k\in X\). </p>
<p>Then: </p>
<ul class="itemize">
  <li><p>\(f\) has a unique fixed point \(\overline{x}\), that is, there exists a unique \(x^* \in X\) such that \(f(x^*,\dots ,x^*)=x^*\); </p>
</li>
  <li><p>the sequence \(\{ x_n\} _{n\geq 0}\) defined by </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B153B90>">
<p>
  <div class="equation_content">
    \begin{equation}  \label{Rel_Sirkpasi} x_{n+1}=f(x_{n},\dots , x_{n-k+1}),\qquad n=k-1,k,k+1,\dots \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p> converges to \(\overline{x}\), for any \(x_0,\dots ,x_{k-1}\in X\). </p>
</li>
</ul>

  </div>
</div> </p>
<p>It is easy to see that, subject to a change of notation, Theorem 4.2.1 in <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span> is obtained from Theorem <a href="#TPresic">2.1</a> for \(X=I\subset \mathbb {R}\) and that, in the particular case \(k=1\), from Theorem <a href="#TPresic">2.1</a>, we get exactly the well-known Banach contraction mapping principle. In this case, the \(k\)-point iterative method (??) reduces to Picard iterations: </p>
<div class="equation" id="Picard">
<p>
  <div class="equation_content">
    \begin{equation}  \label{Picard} x_{n+1}=f(x_n),\qquad n=0,1,2,3,\dots , \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p>Apart from applications in numerical analysis, Prešić fixed point theorem has other important applications in the study of global asymptotic stability of the equilibrium for nonlinear difference equations; see the paper <span class="cite">
	[
	<a href="#Chen" >11</a>
	]
</span> and the monograph <span class="cite">
	[
	<a href="#Pac-teza" >18</a>
	]
</span>. </p>
<p>On the other hand, some other Prešić type fixed point theorems have been obtained in <span class="cite">
	[
	<a href="#Ciric-Presic" >12</a>
	]
</span>, <span class="cite">
	[
	<a href="#Pac-teza" >18</a>
	]
</span>, <span class="cite">
	[
	<a href="#Pac-AMUC" >19</a>
	]
</span>, <span class="cite">
	[
	<a href="#Pac-Ovidius" >17</a>
	]
</span>, <span class="cite">
	[
	<a href="#Pac-UAIC" >21</a>
	]
</span>, <span class="cite">
	[
	<a href="#Rus-Presic" >30</a>
	]
</span>, and for more general contractive type conditions in <span class="cite">
	[
	<a href="#BerP" >8</a>
	]
</span>, <span class="cite">
	[
	<a href="#BerPac" >9</a>
	]
</span> and <span class="cite">
	[
	<a href="#Chen" >11</a>
	]
</span>, with some applications to nonlinear cyclic systems of equations and difference equations. </p>
<p>Theorem <a href="#TPresic">2.1</a> and other similar results, like the ones in <span class="cite">
	[
	<a href="#Ciric-Presic" >12</a>
	]
</span>, <span class="cite">
	[
	<a href="#Pac-AMUC" >19</a>
	]
</span>, <span class="cite">
	[
	<a href="#Pac-Ovidius" >17</a>
	]
</span>, <span class="cite">
	[
	<a href="#Rus-Presic" >30</a>
	]
</span>, have important applications in the iterative solution of nonlinear equations; see <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span> and <span class="cite">
	[
	<a href="#Pav05" >28</a>
	]
</span>, <span class="cite">
	[
	<a href="#BerPac" >9</a>
	]
</span>, <span class="cite">
	[
	<a href="#Ber11" >10</a>
	]
</span>, as well as <span class="cite">
	[
	<a href="#Sukla14" >31</a>
	]
</span>-<span class="cite">
	[
	<a href="#Geo11a" >40</a>
	]
</span>. </p>
<p>Another important generalization of Theorem <a href="#TPresic">2.1</a> was obtained by I.A. Rus <span class="cite">
	[
	<a href="#Rus-Presic" >30</a>
	]
</span>, for operators \(T\) fulfilling the more general condition </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B162F30>">
<p>
  <div class="equation_content">
    \begin{equation} \label{Cond_PresicRus} d(T(x_0,\dots ,x_{k-1}), T(x_1,\dots ,x_{k}))\leq \varphi (d(x_0,x_1),\dots ,d(x_{k-1},x_k)),\end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p> for any \(x_0,\dots ,x_k\in X\), where \(\varphi :\mathbb R^k_+\rightarrow \mathbb R_+\) satisfies certain appropriate conditions. </p>
<p>Another important generalization of Prešić’s result was recently obtained by L. Cirić and S. Prešić in <span class="cite">
	[
	<a href="#Ciric-Presic" >12</a>
	]
</span>, where, instead of (??) and its generalization (??), the following contraction condition is considered: </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B1AC230>">
<p>
  <div class="equation_content">
    \begin{equation} \label{Cond_PresicCiric} d(T(x_0,\dots ,x_{k-1}), T(x_1,\dots ,x_{k}))\leq \lambda \max \{ d(x_0,x_1),\dots ,d(x_{k-1},x_k)\} ,\end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p> for any \(x_0,\dots ,x_k\in X\), where \(\lambda \in (0,1)\). </p>
<p>Other general Prešić type fixed point results have been very recently obtained by the third author in <span class="cite">
	[
	<a href="#Pac-teza" >18</a>
	]
</span>-<span class="cite">
	[
	<a href="#Pac-UAIC" >21</a>
	]
</span> based on alternative contractive conditions which are more general than ??, ?? and ??. For other related results, we refer to <span class="cite">
	[
	<a href="#Sukla14" >31</a>
	]
</span>-<span class="cite">
	[
	<a href="#Geo11a" >40</a>
	]
</span>.</p>
<p>The following lemmas will be useful in proving our main results in this paper. </p>
<p><div class="lem_thmwrapper " id="Lema_Presic">
  <div class="lem_thmheading">
    <span class="lem_thmcaption">
    Lemma
    </span>
    <span class="lem_thmlabel">1</span>
  </div>
  <div class="lem_thmcontent">
  <p><span class="rm">(<span class="cite">
	[
	<a href="#Presic" >29</a>
	]
</span>)</span>  Let \(k\in \mathbb {N},k\neq 0\) and \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{k}\in \mathbb {R}_{+}\) such that \(\overset {k}{\underset {i=1}{\sum }}\alpha _{i}=\alpha {\lt}1\). If \(\{ \Delta _{n}\} _{n\geq 1}\) is a sequence of positive numbers satisfying </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B1AE390>">
<p>
  <div class="equation_content">
    \begin{equation} \label{LemaPresic_conditie} \Delta _{n+k}\leq \alpha _{1}\Delta _{n}+\alpha _{2}\Delta _{n+1}+\ldots +\alpha _{k}\Delta _{n+k-1},\quad n\geq 1, \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> then there exist \(L{\gt}0\) and \(\theta \in (0,1)\) such that </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B1AE750>">
<p>
  <div class="equation_content">
    \begin{equation} \label{LemaPresic_concluzie} \Delta _{n}\leq L\cdot \theta ^{n},\qquad \text{ for all } n\geq 1. \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>

  </div>
</div> </p>
<p>The next Lemma is due to Ostrowski (<span class="cite">
	[
	<a href="#Ost" >16</a>
	]
</span>) and can also be found in an extended form in <span class="cite">
	[
	<a href="#Ber07" >4</a>
	]
</span>. </p>
<p><div class="lem_thmwrapper " id="Lema_Tati">
  <div class="lem_thmheading">
    <span class="lem_thmcaption">
    Lemma
    </span>
    <span class="lem_thmlabel">2</span>
  </div>
  <div class="lem_thmcontent">
  <p> Let \(\{ a_n\} _{n\geq 0}\), \(\{ b_n\} _{n\geq 0}\) be two sequences of positive real numbers and \(q\in (0,1)\) such that: </p>
<ul class="itemize">
  <li><p>\(a_{n+1}\leq q \, a_{n}+b_{n},n\geq 0\); </p>
</li>
  <li><p>\(b_n\rightarrow 0\) as \(n\rightarrow \infty \). </p>
</li>
</ul>
<p> Then: </p>
<div class="displaymath" id="a0000000005">
  \[ \underset {n\rightarrow \infty }{\lim }a_{n}=0. \]
</div>

  </div>
</div> A more general form of the previous lemma has been obtained in <span class="cite">
	[
	<a href="#Ber00" >2</a>
	]
</span>. <div class="lem_thmwrapper " id="Lema_Tati_2">
  <div class="lem_thmheading">
    <span class="lem_thmcaption">
    Lemma
    </span>
    <span class="lem_thmlabel">3</span>
  </div>
  <div class="lem_thmcontent">
  <p> Let \(\{ a_n\} _{n\geq 0}\) be a sequence of positive real numbers and let \(\{ b_n\} _{n\geq 0}\) be a sequence of non-negative real numbers for which there exist \(q\in (0,1)\) and an integer \(k\geq 0\) such that: </p>
<ul class="itemize">
  <li><p>\(a_{n+1}\leq q \,  a_{n-k}+b_{n},\quad n\geq k\); </p>
</li>
  <li><p>\(\sum _{n=1}^{\infty } b_n {\lt}+\infty \). </p>
</li>
</ul>
<p> Then: </p>
<div class="displaymath" id="a0000000006">
  \[ \sum _{n=1}^{\infty } a_n {\lt}+\infty . \]
</div>

  </div>
</div> Note that for \(k=0\), by Lemma <a href="#Lema_Tati_2">3</a>, we actually get the conclusion of Lemma <a href="#Lema_Tati">2</a>, <i class="it">i.e.</i>, </p>
<div class="displaymath" id="a0000000007">
  \[ \underset {n\rightarrow \infty }{\lim }a_{n}=0. \]
</div>
<h1 id="a0000000008">3 Main results</h1>
<p>Our first main result is an improved version of Prešić fixed point theorem in <span class="cite">
	[
	<a href="#Presic" >29</a>
	]
</span> (Theorem <a href="#TPresic">2.1</a>); see also <span class="cite">
	[
	<a href="#Pac-teza" >18</a>
	]
</span>. </p>
<p><div class="theorem_thmwrapper " id="Th_Presic-1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \((X,d)\) be a complete metric space, \(k\) a positive integer and \(f:X^k\rightarrow X\) a mapping for which there exist \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{k}\in \mathbb {R}_{+}, \) \(\overset {k}{\underset {i=1}{\sum }}\alpha _{i}=\alpha {\lt}1\) such that </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B226630>">
<p>
  <div class="equation_content">
    \begin{equation}  \label{Cond_Presic-1} d(f(x_0,\dots ,x_{k-1}), f(x_1,\dots ,x_{k}))\leq \overset {k}{\underset {i=1}{\sum }}\alpha _i d(x_{i-1},x_i), \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p> for all \(x_0,\dots ,x_k\in X\). </p>
<p>Then: </p>
<ul class="itemize">
  <li><p>\(f\) has a unique fixed point \(\overline{x}\), <i class="it">i.e.</i>, \(f(\overline{x},\dots , \overline{x})=\overline{x}\); </p>
</li>
  <li><p>the sequence \(\{ y_{n}\} _{n\geq 0}\), defined by </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B226A50>">
<p>
  <div class="equation_content">
    \begin{equation} \label{Rel_Sir.yn.P} y_{n+1}=f(y_{n},y_{n},\ldots ,y_{n}), \quad n\geq 0,\end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> converges to \(\overline{x}\), for any \(y_0\in X\); </p>
</li>
  <li><p>the sequence \(\{ x_{n}\} _{n\geq 0}\) defined by \(x_{0},\ldots ,x_{k-1}\in X\) and </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B19CE90>">
<p>
  <div class="equation_content">
    \begin{equation} \label{Rel_xn.P} x_{n+1}=f(x_{n-k+1},x_{n-k},\ldots ,x_{n}), \quad n\geq k-1, \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p> also converges to \(\overline{x}\), for all \(x_{0},\ldots ,x_{k-1}\in X\). </p>
</li>
  <li><p>the sequence \(\{ z_{n}\} _{n\geq 0}\) defined by \(z_{0},\ldots ,z_{k-1}\in X\) and </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B19E090>">
<p>
  <div class="equation_content">
    \begin{equation} \label{Rel_zn.P} z_{n+1}=f(z_{n},z_{n-1},\ldots ,x_{n-k+1}),\quad n\geq k-1, \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p> converges to \(\overline{x}\), for all \(z_{0},\ldots ,z_{k-1}\in X\). </p>
</li>
  <li><p>The following estimates hold: </p>
<div class="equation" id="estimate-1">
<p>
  <div class="equation_content">
    \begin{equation} \label{estimate-1} d(y_{n},\overline{x})\leq \tfrac {\alpha ^{n}}{1-\alpha }\,  d(y_{1},y_{0})\, ,\quad n=1,2,\dots ; \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<div class="equation" id="estimate-2">
<p>
  <div class="equation_content">
    \begin{equation} \label{estimate-2} d(x_{n},\overline{x})\leq L\,  \tfrac {\theta ^{n}}{1-\theta },\quad n=1,2,\dots ; \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p> where \(L{\gt}0\) and \(\theta \in (0,1)\) are some constants. </p>
<div class="equation" id="estimate-3">
<p>
  <div class="equation_content">
    \begin{equation} \label{estimate-3} d(z_{n},\overline{x})\leq L_1\,  \tfrac {\theta _1^{i}}{1-\theta _1},\quad n=1,2,\dots ; \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<p> where \(L_1{\gt}0\) and \(\theta _1 \in (0,1)\) are some constants. </p>
</li>
</ul>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000009">
  <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>\(1), 2)\) By considering the associate operator \(F:X\rightarrow X, F(x)=f(x,\ldots ,x)\), for any \(x\in X\) we have: </p>
<div class="displaymath" id="a0000000010">
  \begin{eqnarray*} & & d(F(x),F(y))= d(f(x,x,\ldots ,x),f(y,y,\ldots ,y))\leq \\ & & \  \  \leq d(f(x,\ldots ,x),f(x,\ldots ,x,y))+d(f(x,\ldots ,x,y),f(x,\ldots ,x,y,y))+\\ & & \quad \  +\ldots +\\ & & \quad \  +d(f(x,x,y,\ldots ,y),f(x,y,\ldots ,y))+d(f(x,y,\ldots ,y),f(y,\ldots ,y)). \end{eqnarray*}
</div>
<p>By ?? we obtain: </p>
<div class="displaymath" id="a0000000011">
  \begin{eqnarray*}  d(F(x),F(y))& \leq & \left[\alpha _{1}d(x,x)+\alpha _{2}d(x,x)+\ldots +\alpha _{k-1}d(x,x)+\alpha _{k}d(x,y)\right]+\\ & \quad +& \left[\alpha _{1}d(x,x)+\alpha _{2}d(x,x)+\ldots +\alpha _{k-1}d(x,y)+\alpha _{k}d(y,y)\right]+\\ & \quad +& \ldots +\\ & \quad +& \left[\alpha _{1}d(x,y)+\alpha _{2}d(y,y)+\ldots +\alpha _{k-1}d(y,y)+\alpha _{k}d(y,y)\right], \end{eqnarray*}
</div>
<p> so </p>
<div class="displaymath" id="a0000000012">
  \[  d(F(x),F(y))\leq \overset {k}{\underset {i=1}{\sum }}\alpha _{i}d(x,y)=\alpha d(x,y),  \]
</div>
<p> for any \(x,y\in X\), which shows that \(F\) is a Banach contraction with constant \(\alpha \in [0,1)\). </p>
<p>Consequently, by Banach contraction mapping principle, \(F\) has a unique fixed point \(\overline{x}\in X\) that can be obtained by means of the Picard iterations corresponding to \(F\) starting from any \(x_0\in X\), which thus proves \(1)\) and \(2)\). </p>
<p>\(3)\) We prove now that the \(k\)-step iteration method \(\{ x_n\} _{n\geq 0}\), defined by (??) converges to the unique fixed point \(\overline{x}\) of \(f\). For \(n\geq k\) we have: </p>
<div class="displaymath" id="a0000000013">
  \begin{eqnarray} \nonumber & & d(x_{n},x_{n+1})=d(f(x_{n-k},x_{n-k+1},\ldots ,x_{n-1}),f(x_{n-k+1},x_{n-k+2},\ldots ,x_{n}))\leq \\ \label{Rel_delta.n.P} & & \  \leq \alpha _{1}d(x_{n-k},x_{n-k+1})+\alpha _{2}d(x_{n-k+1},x_{n-k+2})+\ldots +\alpha _{k}d(x_{n-1},x_{n}). \end{eqnarray}
</div>
<p> If </p>
<div class="displaymath" id="a0000000014">
  \[  \Delta _{n}=d(x_{n-1},x_{n}),\quad n\geq 1,  \]
</div>
<p> then, by (??), we obtain that the sequence \(\{ \Delta _n\} _{n\geq 1}\) satisfies: </p>
<div class="displaymath" id="a0000000015">
  \[  \Delta _{n+1}\leq \alpha _{1}\Delta _{n-k+1}+\alpha _{2}\Delta _{n-k+2}+\ldots +\alpha _{k}\Delta _{n},\quad n\geq 1,  \]
</div>
<p> where \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{k}{\gt}0\) and \(\overset {k}{\underset {i=1}{\sum }}\alpha _{i}=\alpha {\lt}1\). </p>
<p>By Lemma <a href="#Lema_Presic">1</a>, there exist \(L{\gt}0\) and \(\theta \in (0,1)\) such that \(\Delta _{n}\leq L\theta ^{n},n\geq 1\), that is, </p>
<div class="equation" id="bla">
<p>
  <div class="equation_content">
    \begin{equation} \label{bla} d(x_{n-1},x_{n})\leq L\theta ^{n},\quad n\geq 1. \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p>For \(n\geq 1\) and \(p\geq 1\), by (<a href="#bla">25</a>) we obtain: </p>
<div class="displaymath" id="a0000000016">
  \begin{eqnarray*}  d(x_{n},x_{n+p})& \leq & d(x_{n},x_{n+1})+d(x_{n+1},x_{n+2})+\ldots +d(x_{n+p-1},x_{n+p})\leq \\ & \leq & L\theta ^{n+1}+L\theta ^{n+2}+\ldots +L\theta ^{n+p}=\\ & =& L\theta ^{n+1}\left(1+\theta +\theta ^{2}+\ldots +\theta ^{p-1}\right), \end{eqnarray*}
</div>
<p> so </p>
<div class="equation" id="Cauchy">
<p>
  <div class="equation_content">
    \begin{equation} \label{Cauchy} d(x_{n},x_{n+p})\leq L\theta ^{n+1}\dfrac {1-\theta ^{p}}{1-\theta },\quad n\geq 1,p\geq 1. \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<p> Since \(\theta \in (0,1)\), it follows that \(\{ x_{n}\} _{n\geq 0}\) is a Cauchy sequence, which in the complete metric space \((X,d)\) is convergent. </p>
<p>We prove that \(\{ x_{n}\} _{n\geq 0}\) in fact converges to \(\overline{x}\), the unique fixed point of \(f\). Indeed, for \(n\geq 0\) we have: </p>
<div class="displaymath" id="a0000000017">
  \begin{align} & d(x_{n+1},\overline{x})\leq d(f(x_{n-k+1},x_{n-k+2},\ldots ,x_{n}),f(\overline{x},\overline{x},\ldots ,\overline{x}))\nonumber \\ & \leq d(f(x_{n-k+1},x_{n-k+2},\ldots ,x_{n}),f(x_{n-k+2},x_{n-k+3},\ldots ,x_{n},\overline{x}))+\nonumber \\ & \quad +d(f(x_{n-k+2},x_{n-k+3},\ldots ,x_{n},\overline{x}),f(x_{n-k+3},x_{n-k+4},\ldots ,x_{n},\overline{x},\overline{x}))+\nonumber \\ & \quad +\ldots +d(f(x_{n},\overline{x},\ldots ,\overline{x}),f(\overline{x},\overline{x},\ldots ,\overline{x}))\label{eq-mare}, \end{align}
</div>
<p> so by (??) we obtain: </p>
<div class="displaymath" id="a0000000018">
  \begin{eqnarray*} & & d(x_{n+1},\overline{x})\leq [\alpha _{1}d(x_{n-k+1},x_{n-k+2})+\ldots +\alpha _{k-1}d(x_{n-1},x_{n})+\alpha _{k}d(x_{n},\overline{x})]+\\ & & \  \  \  \  +[\alpha _{1}d(x_{n-k+2},x_{n-k+3})+\ldots +\alpha _{k-1}d(x_{n},\overline{x})+\alpha _{k}d(\overline{x},\overline{x})]+\\ & & \  \  \  \  +\ldots +\\ & & \  \  \  \  +[\alpha _{1}d(x_{n},\overline{x})+\alpha _{2}d(\overline{x},\overline{x})+\ldots +\alpha _{k}d(\overline{x},\overline{x})]. \end{eqnarray*}
</div>
<p> Now using (<a href="#bla">25</a>) it follows that: </p>
<div class="displaymath" id="a0000000019">
  \begin{eqnarray*} & & d(x_{n+1},\overline{x})\leq [\alpha _{1}L\theta ^{n-k+2}+\alpha _{2}L\theta ^{n-k+3}+\ldots +\alpha _{k-1}L\theta ^{n}+\alpha _{k}d(x_{n},\overline{x})]+\\ & & \  \  \  \  +[\alpha _{1}L\theta ^{n-k+3}+\alpha _{2}L\theta ^{n-k+4}+\ldots +\alpha _{k-2}L\theta ^{n}+\alpha _{k-1}d(x_{n},\overline{x})+\alpha _{k}\cdot 0]+\\ & & \  \  \  \  +\ldots \\ & & \  \  \  \  +[\alpha _{1}d(x_{n},\overline{x})+0]=\\ & & \  \  =\alpha _{1}L\theta ^{n-k+2}+(\alpha _{1}+\alpha _{2})L\theta ^{n-k+3}+\ldots +(\alpha _{1}+\alpha _{2}+\ldots +\alpha _{k-1})L\theta ^{n}+\\ & & \  \  \  \  +(\alpha _{1}+\alpha _{2}+\ldots +\alpha _{k})d(x_{n},\overline{x}). \end{eqnarray*}
</div>
<p> Finally we obtain that: </p>
<div class="displaymath" id="a0000000020">
  \begin{align*}  d(x_{n+1},\overline{x})& \leq \alpha d(x_{n},\overline{x})+L\theta ^{n}[\alpha _{1}\theta ^{2-k}+(\alpha _{1}+\alpha _{2})\theta ^{3-k}+\ldots +\notag \\ & \quad +(\alpha _{1}+\alpha _{2}+\ldots +\alpha _{k-2})\theta +(\alpha _{1}+\alpha _{2}+\ldots +\alpha _{k-1})],\quad n\geq 1, \end{align*}
</div>
<p> where \(\theta \in (0,1)\). This inequality shall lead to estimate </p>
<div class="equation" id="<plasTeX.TeXFragment object at 0x000001A89B2A2090>">
<p>
  <div class="equation_content">
    \begin{equation} \label{Rel_rate.P} d(x_{n+1},\overline{x})\leq \alpha d(x_{n},\overline{x})+M\cdot \theta ^{n},n\geq 0, \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000021">
  \[  M\! =\! L[\alpha _{1}\theta ^{2-k}+(\alpha _{1}+\alpha _{2})\theta ^{3-k}+\ldots +(\alpha _{1}+\alpha _{2}+\ldots +\alpha _{k-2})\theta +(\alpha _{1}+\alpha _{2}+\ldots +\alpha _{k-1})]  \]
</div>
<p> is a fixed positive number (since \(k\) is fixed). Considering </p>
<div class="displaymath" id="a0000000022">
  \begin{eqnarray*} & & a_{n}=d(x_{n},\overline{x}),\\ & & \  \  q=\alpha \in [0,1)\\ & & b_{n}=M\theta ^{n},\quad n\geq 1, \end{eqnarray*}
</div>
<p> the conditions of Lemma <a href="#Lema_Tati">2</a> are fulfilled, so by its conclusion </p>
<div class="displaymath" id="a0000000023">
  \[  d(x_{n},\overline{x})\rightarrow 0,\quad n\rightarrow \infty .  \]
</div>
<p>Consequently, the sequence \(\{ x_{n}\} _{n\geq 0}\) converges to \(\overline{x}\), the unique fixed point of \(f\). </p>
<p>\(4)\) First, we observe that for </p>
<div class="displaymath" id="a0000000024">
  \[  \Delta _n=d(z_n,z_{n-1}), \quad n\geq 1,  \]
</div>
<p> proceeding as in the previous case, we obtain by Lemma <a href="#Lema_Presic">1</a>, that there exist \(L_1{\gt}0\) and \(\theta _1\in (0,1)\) such that \(\Delta _{n}\leq L_1\theta _1^{n},\  n\geq 1\), that is, </p>
<div class="equation" id="bla-1">
<p>
  <div class="equation_content">
    \begin{equation} \label{bla-1} d(z_{n-1},z_{n})\leq L_1\theta _1^{n},\quad n\geq 1. \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<p> Next, in a similar way to the case of \(\{ x_{n}\} _{n\geq 0}\) but by following slightly different computations, we find that </p>
<div class="displaymath" id="a0000000025">
  \[  d(z_{n+1},\overline{x})\leq \alpha d(z_{n-k+1},\overline{x})+M_1\,  \theta _1^n,  \]
</div>
<p> for a certain constant \(M_1{\gt}0\). </p>
<p>Now, simply use Lemma <a href="#Lema_Tati_2">3</a> to get the conclusion that the sequence \(\{ z_{n}\} _{n\geq 0}\) converges to \(\overline{x}\), too. </p>
<p>\(5)\) The error estimate <a href="#estimate-1" class="eqref">21</a> follows by the Banach contraction mapping principle in the form given in <span class="cite">
	[
	<a href="#Ber07" >4</a>
	]
</span>, while the estimates <a href="#estimate-2" class="eqref">22</a> and <a href="#estimate-3" class="eqref">23</a> are obtained by <a href="#Cauchy" class="eqref">26</a> and its version for \(\{ z_{n}\} _{n\geq 0}\), respectively, by letting \(p\rightarrow \infty \). <div class="proof_wrapper" id="a0000000026">
  <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>
<h1 id="a0000000027">4 Rate of convergence versus speed of convergence</h1>
<p>As before, let \(\{ x_{n}\} _{n\geq 0}\) be a convergent sequence with limit \(\overline{x}\). If, for some \(r\), we have </p>
<div class="displaymath" id="a0000000028">
  \[  \lim _{n\rightarrow \infty }\tfrac {d(\bar{x}_{n+1},\bar{x}^*)}{[d(\bar{x}_{n},\bar{x}^*)]^r}=\lambda {\lt}+\infty ,  \]
</div>
<p> then \(r\) is called the <i class="itshape">rate of convergence</i> of \(\{ x_{n}\} _{n\geq 0}\), while \(\lambda \) is termed as its <i class="itshape">asymptotic error</i>; see <span class="cite">
	[
	<a href="#ORh70" >15</a>
	]
</span> for more details. </p>
<p>If \(r=1\), we say that the convergence of \(\{ x_{n}\} _{n\geq 0}\) is <em>linear</em>, if \(r=2\), we say that the convergence is <em>quadratic</em>, while, for \(1{\lt}r{\lt}2\), we say that the convergence is <em>superlinear</em>. </p>
<p>Now, let \(\{ x_{n}\} _{n\geq 0}\) and \(\{ y_{n}\} _{n\geq 0}\) be two convergent sequences with the same limit \(\overline{x}\). If </p>
<div class="displaymath" id="a0000000029">
  \[  \lim _{n\rightarrow \infty }\tfrac {d(\bar{x}_{n},\bar{x}^*)}{d(y_{n},\bar{x}^*)}=\beta {\lt}+\infty ,  \]
</div>
<p> exists and \(\beta =0\), then we say that \(\{ x_{n}\} _{n\geq 0}\) converges <i class="itshape">faster</i> than \(\{ y_{n}\} _{n\geq 0}\) to \(\overline{x}\), and if \(\beta \neq 0\), we say that \(\{ x_{n}\} _{n\geq 0}\) and \(\{ y_{n}\} _{n\geq 0}\) have the same speed of convergence. </p>
<p>Clearly, if \(\beta =\infty \), then \(\{ y_{n}\} _{n\geq 0}\) converges faster than \(\{ x_{n}\} _{n\geq 0}\) to \(\overline{x}\) (for more details see <span class="cite">
	[
	<a href="#Ber98" >1</a>
	]
</span>, <span class="cite">
	[
	<a href="#Ber04" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#Ber07" >4</a>
	]
</span>). </p>
<p>This concept of convergence can be defined in a more general context, when \(\{ x_{n}\} _{n\geq 0}\) and \(\{ y_{n}\} _{n\geq 0}\) are convergent sequences with different limits, \(\overline{x}\) and \(\overline{y}\), respectively; see <span class="cite">
	[
	<a href="#Ber98" >1</a>
	]
</span> and <span class="cite">
	[
	<a href="#Ber07" >4</a>
	]
</span>. </p>
<p><div class="example_thmwrapper " id="ex3">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">4.1</span>
  </div>
  <div class="example_thmcontent">
  <p> If we consider the sequences \(\{ a_{n}\} _{n\geq 0}\), \(\{ b_{n}\} _{n\geq 0}\), \(\{ c_{n}\} _{n\geq 0}\) given by </p>
<div class="displaymath" id="a0000000030">
  \[  a_n=\tfrac {1}{n+1},\, \quad b_n=\tfrac {1}{2^n},\, \quad c_n=2^{-2^n},  \]
</div>
<p> then, obviously, \(a_n\rightarrow 0\), \(b_n\rightarrow 0\) and \(c_n\rightarrow 0\), as \(n\rightarrow \infty \), and since </p>
<div class="displaymath" id="a0000000031">
  \[  a)\,  \lim _{n\rightarrow \infty }\tfrac {a_{n+1}}{a_n}=1,\quad b)\, \lim _{n\rightarrow \infty }\tfrac {b_{n+1}}{b_n}=\tfrac {1}{2},\quad c)\, \lim _{n\rightarrow \infty }\tfrac {b_{n}}{a_n}=0,  \]
</div>
<p> it follows that \(\{ a_{n}\} _{n\geq 0}\) and \(\{ b_{n}\} _{n\geq 0}\) have the same rate of convergence (linear). However, \(\{ b_{n}\} _{n\geq 0}\) converges faster than \(\{ a_{n}\} _{n\geq 0}\) to \(0\). </p>
<p>Moreover, since </p>
<div class="displaymath" id="a0000000032">
  \[  \lim _{n\rightarrow \infty }\tfrac {c_{n+1}}{(c_n)^2}=1,  \]
</div>
<p> it follows that the sequence \(\{ c_{n}\} _{n\geq 0}\) has quadratic rate of convergence and, as an immediate consequence, converges faster than both \(\{ a_{n}\} _{n\geq 0}\) and \(\{ b_{n}\} _{n\geq 0}\). </p>

  </div>
</div> </p>
<p>If we now use the proof of Theorem <a href="#Th_Presic-1">3.1</a> and the complete form of the Banach contraction principle – see for example <span class="cite">
	[
	<a href="#Ber07" >4</a>
	]
</span> – then we obtain for the sequences \(\{ x_{n}\} _{n\geq 0}\) and \(\{ y_{n}\} _{n\geq 0}\) the following error estimates: </p>
<div class="equation" id="eq-33">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq-33} d(y_{n+1},\overline{x})\leq \alpha d(y_{n},\overline{x}), \quad n\geq 0; \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<div class="equation" id="eq-34">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq-34} d(x_{n+1},\overline{x})\leq \alpha d(x_{n},\overline{x})+M\cdot \theta ^{n},\quad n\geq 0, \end{equation}
  </div>
  <span class="equation_label">30</span>
</p>
</div>
<p> where \(M{\gt}0\) and \(\theta \in (0,1)\) are constant, and also </p>
<div class="equation" id="eq-35">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq-35} d(z_{n+1},\overline{x})\leq \alpha d(z_{n},\overline{x})+M_1\cdot \theta _1^{n},\quad n\geq 0. \end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<p> Thus, the estimates <a href="#eq-33" class="eqref">29</a>-<a href="#eq-35" class="eqref">31</a> show that the sequences \(\{ x_{n}\} _{n\geq 0}\), \(\{ y_{n}\} _{n\geq 0}\) and \(\{ z_{n}\} _{n\geq 0}\) have all linear rate of convergence, while the estimates <a href="#estimate-1" class="eqref">21</a>-<a href="#estimate-3" class="eqref">23</a> offer information on the speed of convergence of these sequences. </p>
<p>In the proof of Lemma <a href="#Lema_Presic">1</a> in <span class="cite">
	[
	<a href="#Presic" >29</a>
	]
</span>, we note that \(\theta \) in <a href="#estimate-2" class="eqref">22</a> is the unique positive root of the polynomial equation </p>
<div class="displaymath" id="a0000000033">
  \[  t^k-\alpha _1 \, t^{k-1}-\dots -\alpha _{k-1}\, t-\alpha _k=0,  \]
</div>
<p> while \(\theta _1\) in <a href="#estimate-3" class="eqref">23</a> is the unique positive root of the polynomial equation </p>
<div class="displaymath" id="a0000000034">
  \[  t^k-\alpha _k \, t^{k-1}-\dots -\alpha _{2}\, t-\alpha _1=0.  \]
</div>
<p> Therefore, in view of the estimates <a href="#estimate-2" class="eqref">22</a>-<a href="#estimate-3" class="eqref">23</a>, to compare the iterative methods \(\{ x_{n}\} _{n\geq 0}\), \(\{ y_{n}\} _{n\geq 0}\) and \(\{ z_{n}\} _{n\geq 0}\), it suffices to establish the order relation between the constants </p>
<div class="displaymath" id="a0000000035">
  \[  \alpha , \theta , \theta _1\in (0,1).  \]
</div>
<h1 id="a0000000036">5 Examples and concluding remarks</h1>
<p> The following example illustrates Theorem <a href="#Th_Presic-1">3.1</a>. <div class="example_thmwrapper " id="ex2">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">5.1</span>
  </div>
  <div class="example_thmcontent">
  <p> Let \(X=\mathbb {R}\) with the usual metric and \(f\colon X^2\rightarrow X\) be defined by </p>
<div class="displaymath" id="a0000000037">
  \[  f(x,y)=\tfrac {x+2y}{4},\quad \forall (x,y)\in X^2.  \]
</div>
<p> It is easy to check that \(f\) satisfies condition ?? (with \(\alpha _1=\tfrac {1}{4}\), \(\alpha _2=\tfrac {1}{2}\)), condition ?? (with \(\varphi (t_1,t_2)=\tfrac {1}{4}t_1+\tfrac {1}{2}t_2\)), as well as condition ?? (with \(\lambda =\tfrac {3}{4}\)).</p>
<p>Consider the sequence \(\{ x_n\} \) </p>
<div class="displaymath" id="a0000000038">
  \[  x_{n+1}=\tfrac {x_n+2x_{n-1}}{4},\quad n\geq 1,  \]
</div>
<p> corresponding to the identity permutation \((0,1)\). Then we have </p>
<div class="displaymath" id="a0000000039">
  \[  x_n=c_1\Big(\tfrac {1-\sqrt{33}}{8}\Big)^n+c_2\Big(\tfrac {1+\sqrt{33}}{8}\Big)^n, \quad n\geq 1,  \]
</div>
<p> where \(c_1\) and \(c_2\) are some constants. Consider now the sequence \(\{ z_n\} \) </p>
<div class="displaymath" id="a0000000040">
  \[  z_{n+1}=\tfrac {2 z_n+z_{n-1}}{4},\quad n\geq 1,  \]
</div>
<p> corresponding to the permutation \((1,0)\) of \((0,1)\). Similarly, we obtain </p>
<div class="displaymath" id="a0000000041">
  \[  z_n=a_1\Big(\tfrac {1-\sqrt{5}}{4}\Big)^n+a_2\Big(\tfrac {1+\sqrt{5}}{4}\Big)^n, \quad n\geq 1,  \]
</div>
<p> where \(a_1\) and \(a_2\) are some constants. Now, considering the sequence \(\{ y_n\} \) given by \(y_0\in X\) and </p>
<div class="displaymath" id="a0000000042">
  \[  y_{n+1}=\tfrac {3}{4} y_n\geq 0,  \]
</div>
<p> we get </p>
<div class="displaymath" id="a0000000043">
  \[  y_n=\Big(\tfrac {3}{4}\Big)^n y_0,\quad n\geq 0.  \]
</div>
<p> Since </p>
<div class="displaymath" id="a0000000044">
  \[  \Big|\tfrac {1-\sqrt{5}}{4}\Big| {\lt}\Big|\tfrac {1-\sqrt{33}}{8} \Big|{\lt}\tfrac {3}{4} {\lt}\tfrac {1+\sqrt{5}}{4} {\lt}\tfrac {1+\sqrt{33}}{8},  \]
</div>
<p> we have </p>
<div class="displaymath" id="a0000000045">
  \[  \lim _{n\rightarrow \infty }\tfrac {x_n}{y_n}=c_1\cdot \lim _{n\rightarrow \infty }\Big(\tfrac {1-\sqrt{33}}{6}\Big)^n+c_2\cdot \lim _{n\rightarrow \infty }\Big(\tfrac {1+\sqrt{33}}{6}\Big)^n=+\infty ;  \]
</div>
<div class="displaymath" id="a0000000046">
  \[  \lim _{n\rightarrow \infty }\tfrac {z_n}{y_n}=a_1\cdot \lim _{n\rightarrow \infty }\Big(\tfrac {1-\sqrt{5}}{3}\Big)^n+a_2\cdot \lim _{n\rightarrow \infty }\Big(\tfrac {1+\sqrt{5}}{3}\Big)^n=+\infty ;  \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000047">
  \[  \lim _{n\rightarrow \infty }\tfrac {x_n}{z_n}=\lim _{n\rightarrow \infty }\tfrac {c_1\Big(\tfrac {1-\sqrt{33}}{8}\Big)^n+c_2\Big(\tfrac {1+\sqrt{33}}{8}\Big)^n}{a_1\Big(\tfrac {1-\sqrt{5}}{4}\Big)^n+a_2\Big(\tfrac {1+\sqrt{5}}{4}\Big)^n}  \]
</div>
<div class="displaymath" id="a0000000048">
  \[  =\tfrac {c_2}{a_2}\cdot \lim _{n\rightarrow \infty }\Big(\tfrac {1+\sqrt{33}}{2+2\sqrt{5}}\Big)^n =+\infty ,  \]
</div>
<p> as \(c_1,c_2,a_1,a_2\neq 0\). </p>
<p>These calculations prove that the sequence \(\{ y_n\} \) converges faster than \(\{ z_n\} \) to \(0\), \(\{ z_n\} \) converges faster than \(\{ x_n\} \), and \(\{ y_n\} \) converges faster than \(\{ z_n\} \), although all the three sequences have the same (linear) rate of convergence. </p>

  </div>
</div> </p>
<p>We thus can conclude that the above numerical tests confirm the theoretical results obtained in Păvăloiu <span class="cite">
	[
	<a href="#Pav81" >23</a>
	]
</span>. Indeed, amongst the \(k!\) iterative methods of the form </p>
<div class="equation" id="eq-5u">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq-5u} x_{n+1}=f(x_{i_0+n-k+1},x_{i_1+n-k+1},\dots , x_{{i_{k-1}+n-k+1}}),\quad n\geq k-1, \end{equation}
  </div>
  <span class="equation_label">32</span>
</p>
</div>
<p> where \((i_0,i_1,\dots ,i_{k-1})\) is a permutation of \((0,1,\dots ,k-1)\), the optimal method is \(\{ z_n\} \), which corresponds to the permutation \((i_0,i_1,\dots ,i_{k\! -\! 1})\) of \((0,1,\dots ,k\! -\! 1)\) for which one has </p>
<div class="displaymath" id="a0000000049">
  \[  \alpha _{i_0}\geq \alpha _{i_1}\geq \dots \geq \alpha _{i_{k-1}}.  \]
</div>
<p>This also shows that, especially in the case of fixed point iteration procedures, which have generally linear rate a convergence, in order to decide about the fastest iterative method, we have to take into consideration the speed of convergence, usually deduced from the error estimates of the form <a href="#estimate-1" class="eqref">21</a>-<a href="#estimate-3" class="eqref">23</a>. </p>
<p>An interesting conclusion that follows from the above example, is that, in this particular case, the one-point algorithm \(\{ y_n\} \) converges faster than the two two-step algorithms \(\{ x_n\} \) and \(\{ z_n\} \). </p>
<p>The problem is to study if this claim is valid in general. In view of <a href="#estimate-1" class="eqref">21</a>-<a href="#estimate-3" class="eqref">23</a>, it would be sufficiently to show that </p>
<div class="displaymath" id="a0000000050">
  \[  0{\lt}\alpha {\lt}\theta _1{\lt}\theta {\lt}1,  \]
</div>
<p> which is a result similar to that given by Theorem 5.3.2 in <span class="cite">
	[
	<a href="#Pav05" >28</a>
	]
</span>. </p>
<p>We invite the reader to carry out all the calculations for the function \(f\) in the next example. </p>
<p><div class="example_thmwrapper " id="ex1">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">5.2</span>
  </div>
  <div class="example_thmcontent">
  <p> Let \(X=\mathbb {R}\) with the usual metric and \(f\colon X^3\rightarrow X\) be defined by </p>
<div class="displaymath" id="a0000000051">
  \[  f(x,y,z)=\tfrac {x-2y+3z}{7},\quad \forall (x,y,z)\in X^3,  \]
</div>
<p> which obviously satisfies condition ?? with \(\alpha _1=\tfrac {1}{7}\), \(\alpha _2=\tfrac {2}{7}\) and \(\alpha _3=\tfrac {3}{7}\). </p>

  </div>
</div> </p>
<p><div class="acknowledgements_thmwrapper " id="a0000000052">
  <div class="acknowledgements_thmheading">
    <span class="acknowledgements_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgements_thmcontent">
  <p>The first author’s research was done during his visit of Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia (December 2014-January 2015). He gratefully thanks Professor Abdul Rahim Khan and Dr. Al-Attas, the Chairman of the Department of Mathematics and Statistics, for the excellent conditions they offered during the visit. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
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</a> </p>
</dd>
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</dd>
  <dt><a name="Luong">38</a></dt>
  <dd><p><i class="sc">N.V. Luong</i> and <i class="sc">N.X. Thuan</i>, <i class="it">Some fixed point theorems of Prešić-Ćirić type</i>, Acta Univ. Apulensis Math. Inform., <b class="bfseries">30</b> (2012), pp.&#160;237–249. </p>
</dd>
  <dt><a name="Khan">14</a></dt>
  <dd><p><a href ="http://www.dx.doi.org/10.1186/1687-1847-2012-38"> <i class="sc">M.S. Khan, M. Berzig</i> and <i class="sc">B. Samet</i>, <i class="it">Some convergence results for iterative sequences of Prešić type and applications</i>, Adv. Difference Equ., <b class="bfseries">2012</b>, 2012:38, 12 pp. <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="Geo11">39</a></dt>
  <dd><p><a href ="http://www.dx.doi.org/10.1186/1687-1812-2011-85"> <i class="sc">R. George, K.P. Reshma</i> and <i class="sc">R.A. Rajagopalan</i>, <i class="it">Generalised fixed point theorem of Prešić type in cone metric spaces and application to Markov process</i>, Fixed Point Theory Appl., <b class="bfseries">2011</b>, 2011:85, 8 pp. <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="Geo11a">40</a></dt>
  <dd><p><i class="sc">R. George</i> and <i class="sc">M.S. Khan</i>, <i class="it">On Prešić type extension of Banach contraction principle</i>, Int. J. Math. Anal. (Ruse), <b class="bfseries">5</b> (2011) nos. 21–24, pp.&#160;1019–1024. </p>
</dd>
  <dt><a name="Wein">41</a></dt>
  <dd><p><a href ="http://www.dx.doi.org/10.1007/BF01386089"> <i class="sc">H.J. Weinitschke</i>, <i class="it">Über eine Klasse von Iterationsverfahren</i>, Numer. Math., <b class="bfseries">6</b> (1964), pp.&#160;395–404 (in German). <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
</dl>


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