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<h1>Iterates of a modified Bernstein type operator</h1>
<p class="authors">
<span class="author">Teodora Cătinaş\(^{\ast }\)</span>
</p>
<p class="date">September 4, 2019; accepted: November 5, 2019; published online: January 21, 2020.</p>
</div>
<div class="abstract"><p> Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of some modified Bernstein type operators. </p>
<p><b class="bf">MSC.</b> 41A36, 41A25, 39B12, 47H10. </p>
<p><b class="bf">Keywords.</b> Bernstein type operators, contraction principle, weakly Picard operators. </p>
</div>

<p>\(^{\ast }\)Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Str. M. Kogălniceanu Nr. 1, RO-400084 Cluj-Napoca, Romania, E-mail: tcatinas@math.ubbcluj.ro </p>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \((X,d)\) be a metric space and \(A:X\rightarrow X\) an operator. We denote by (see, <i class="it">e.g.</i>, <span class="cite">
	[
	<a href="#Rus2001" >21</a>
	]
</span>) </p>
<div class="displaymath" id="a0000000003">
  \begin{align*}  F_{A} &  :=\{ x\in X~ |~ A(x)=x\}  \text{-the fixed points set of }A\text{;}\\ I(A) &  :=\{ Y\subset X~ |~ A(Y)\subset Y,\  Y\neq \emptyset \}  \text{-the family of the nonempty invariant }\\ &  \text{subsets of }A;\\ A^{0} &  :=1_{X},\  A^{1}:=A,\  ...,\  A^{n+1}:=A\circ A^{n},\   \  n\in \mathbb {N}\text{.}\end{align*}
</div>
<p><div class="definition_thmwrapper " id="def2.1">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
  </div>
  <div class="definition_thmcontent">
  <p>The operator \(A:X\rightarrow X\) is a Picard operator if there exists \(x^{\ast }\in X\) such that: </p>
<p>(i) \(F_{A}=\{ x^{\ast }\} ;\) </p>
<p>(ii) the sequence \((A^{n}(x_{0}))_{n\in \mathbb {N}}\) converges to \(x^{\ast }\) for all \(x_{0}\in X\). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="def2.2">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2</span>
  </div>
  <div class="definition_thmcontent">
  <p>The operator \(A\) is a weakly Picard operator if the sequence \((A^{n}(x))_{n\in \mathbb {N}}\) converges, for all \(x\in X\), and the limit (which may depend on \(x\)) is a fixed point of \(A\). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="def2.3">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">3</span>
  </div>
  <div class="definition_thmcontent">
  <p>If \(A\) is a weakly Picard operator then we consider the operator \(A^{\infty },\; A^{\infty }:X\rightarrow X\), defined<span class="rmfamily"> </span>by </p>
<div class="displaymath" id="a0000000004">
  \[  A^{\infty }(x):=\underset {n\rightarrow \infty }{{\lim }}A^{n}(x).  \]
</div>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="thRus">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4</span>
  </div>
  <div class="theorem_thmcontent">
  <p> <span class="cite">
	[
	<a href="#Rus2001" >21</a>
	]
</span> An operator \(A\) is a weakly Picard operator if and only if there exists a partition of \(X,\) \(X={\textstyle \bigcup \limits _{\lambda \in \Lambda }} X_{\lambda },\) such that </p>
<ol class="enumerate">
  <li><p>\(X_{\lambda }\in I(A),\) \(\forall \lambda \in \Lambda ;\) </p>
</li>
  <li><p>\(\left. A\right\vert _{X_{\lambda }}:X_{\lambda }\rightarrow X_{\lambda }\) is a Picard operator, \(\forall \lambda \in \Lambda .\) </p>
</li>
</ol>

  </div>
</div> </p>
<h1 id="a0000000005">2 Iterates of a modified Bernstein operator</h1>
<p>We study the convergence of the iterates of the modified Bernstein type operators, given in (<a href="#Bmx">1</a>), using the weakly Picard operators technique and the contraction principle. This approach for some other linear and positive operators lead to similar results, for example, in <span class="cite">
	[
	<a href="#AgrRus2003" >1</a>
	]
</span>, <span class="cite">
	[
	<a href="#AgrRusNAF" >2</a>
	]
</span>, <span class="cite">
	[
	<a href="#CatOtFPT" >6</a>
	]
</span>-<span class="cite">
	[
	<a href="#CatOtrRus16" >9</a>
	]
</span>, <span class="cite">
	[
	<a href="#Rus01a" >22</a>
	]
</span>-<span class="cite">
	[
	<a href="#Rus2010" >25</a>
	]
</span>. </p>
<p>The limit behavior for the iterates of some classes of positive linear operators were also studied, for example, in <span class="cite">
	[
	<a href="#GavIva10" >10</a>
	]
</span>-<span class="cite">
	[
	<a href="#Rasa10" >20</a>
	]
</span>. In the papers <span class="cite">
	[
	<a href="#GavIva10" >10</a>
	]
</span>-<span class="cite">
	[
	<a href="#GavIva11b" >12</a>
	]
</span> there were introduced new methods (<i class="it">e.g.</i>, Korovkin type technique) for the study of the asymptotic behavior of the iterates of positive linear operators, positive linear operators preserving the affine functions and defined on the space of bounded real-valued functions on \([0,1]\). These techniques enlarge the class of operators for which the limit of the iterates can be computed. In <span class="cite">
	[
	<a href="#GonKacPit06" >13</a>
	]
</span>, <span class="cite">
	[
	<a href="#GonPitRas07" >14</a>
	]
</span>, <span class="cite">
	[
	<a href="#GonRas06" >15</a>
	]
</span> there were proposed methods to determine the degree of convergence for the iterates of certain positive linear operators towards the first Bernstein operator. In <span class="cite">
	[
	<a href="#GonRas13" >16</a>
	]
</span>, <span class="cite">
	[
	<a href="#Rasa10" >20</a>
	]
</span> there were introduced new techniques (infinite products, rates of convergence), based on the results from <span class="cite">
	[
	<a href="#Gwo-LukJach" >17</a>
	]
</span>, in order to prove that infinite products of certain positive linear operators weakly converge to the first Bernstein operator. </p>
<p><b class="bfseries">A modified Bernstein operator. </b>Let \(f\) be an integrable function on \([0,1].\) The modified Bernstein operator, introduced by J. L. Durrmeyer in <span class="cite">
	[
	<a href="#Dur67" >5</a>
	]
</span>, is defined by (see <span class="cite">
	[
	<a href="#Der81" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#Der85" >4</a>
	]
</span>)</p>
<div class="equation" id="Bmx">
<p>
  <div class="equation_content">
    \begin{equation}  \left( M_{n}^{{}}f\right) \left( x\right) =(n+1){ \sum \limits _{k=0}^{n}} p_{n,k}(x){ \int _{0}^{1}} p_{n,k}(t)f(t)dt,\   \   \  n\geq 1\label{Bmx}\end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> with </p>
<div class="equation" id="pmi">
<p>
  <div class="equation_content">
    \begin{equation}  p_{n,k}\left( x\right) =\tbinom {n}{k}x^{k}(1-x)^{n-k},\   \   \text{for }0\leq k\leq n.\label{pmi}\end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> For \(f(x)=x^{m},\) with \(m\leq n\), based on the properties of beta function, <span class="mbox" style="width: ">we get <span class="cite">
	[
	<a href="#Der81" >3</a>
	]
</span></span> </p>
<div class="displaymath" id="a0000000006">
  \[  \left( M_{n}f\right) \left( x\right) =(n+1){ \sum \limits _{k=0}^{n}} p_{n,k}(x)\tfrac {(k+m)!}{k!}\tfrac {n!}{(n+m+1)!},  \]
</div>
<p> or</p>
<div class="displaymath" id="a0000000007">
  \[  \left( M_{n}f\right) \left( x\right) =\tfrac {(n+1)!}{(n+m+1)!}{ \sum \limits _{k=0}^{n}} \tbinom {m}{k}\tfrac {m!}{k!}\tfrac {n!}{(n-k)!}x^{k}.  \]
</div>
<p><div class="remark_thmwrapper " id="a0000000008">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">5</span>
  </div>
  <div class="remark_thmcontent">
  <p>The modified Bernstein polynomials (<a href="#Bmx">1</a>) are obtained by the classical Bernstein polynomials</p>
<div class="equation" id="clasB">
<p>
  <div class="equation_content">
    \begin{equation}  \left( B_{n}^{{}}f\right) \left( x\right) ={ \sum \limits _{k=0}^{n}} p_{n,k}(x)f(\tfrac {k}{n}),\label{clasB}\end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> by replacing \(f(\frac{k}{n})\) with \((n+1){\textstyle \int _{0}^{1}} p_{n,k}(t)f(t)dt.\) <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="THPropM">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6</span>
  </div>
  <div class="theorem_thmcontent">
  <p><span class="cite">
	[
	<a href="#Der85" >4</a>
	]
</span> The operator \(M_{n}\) has the following properties: </p>
<ul class="itemize">
  <li><p>it is linear and positive; </p>
</li>
  <li><p>it preserves the constants; </p>
</li>
  <li><p>it preserves the degree of polynomials when their degrees are \(\leq n;\) </p>
</li>
  <li><p>it is a contraction on \(L^{p}[0,1],\) \(p\geq 1.\) </p>
</li>
</ul>

  </div>
</div> </p>
<p>Next we give the main result of this note. </p>
<p><div class="theorem_thmwrapper " id="thbmBn">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
  </div>
  <div class="theorem_thmcontent">
  <p> <i class="itshape">The operator </i>\(M_{n}^{{}}\) is a weakly Picard operator and</p>
<div class="equation" id="ec.2">
<p>
  <div class="equation_content">
    \begin{equation}  M_{n}^{\infty }(f)=\int _{0}^{1}f(t)dt.\label{ec.2}\end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>

  </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> We define </p>
<div class="displaymath" id="a0000000010">
  \[  X_{\alpha } =\Big\{  f\in C[0,1] : \int _{0}^{1}f(t)dt=\alpha \Big\} ,\   \   \text{with }\alpha \in \mathbb {R},  \]
</div>
<p> and denote by </p>
<div class="displaymath" id="a0000000011">
  \[  F_{\alpha }^{{}}(x):=\alpha .  \]
</div>
<p> We have that \(X_{\alpha }^{{}}\) is a closed subsets of \(C[0,1]\) and \(C[0,1]=\underset {\alpha \in \mathbb {R}}{\cup }X_{\alpha }\) is a partition of \(C[0,1]\). </p>
<p>We have that \(X_{\alpha }^{{}}\) is an invariant subset of \(M_{n},\) which follows by linearity of the modified Bernstein operators and Theorem <a href="#THPropM">6</a>. </p>
<p>By Theorem <a href="#THPropM">6</a> we know that \(M_{n}\) is a contraction. </p>
<p>On the other hand, as \(M_{n}\) reproduces constant functions it follows \(F_{\alpha }^{{}}\in X_{\alpha }^{{}}\) is a fixed point of \(M_{n}\). </p>
<p>From the contraction principle, it follows that \(\alpha \) is the unique fixed point of \(M_{n}\) and \(\left. M_{n}\right\vert _{X_{\alpha }^{{}}}\) is a Picard operator, with </p>
<div class="displaymath" id="a0000000012">
  \[  M_{n}^{\infty }(f)=\int _{0}^{1}f(t)dt.  \]
</div>
<p> By Theorem <a href="#thRus">4</a> it follows that the operators \(M_{n},\) \(n\geq 1\) are weakly Picard operators. <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">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000014">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">8</span>
  </div>
  <div class="remark_thmcontent">
  <p>In <span class="cite">
	[
	<a href="#Rus2004" >24</a>
	]
</span> was shown that the classical Bernstein operator \(B_{n}\) given in (<a href="#clasB">3</a>) is a weakly Picard operator with </p>
<div class="displaymath" id="a0000000015">
  \[  B_{n}^{\infty }(f)(x)=f(0)+[f(1)-f(0)]x.  \]
</div>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
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</dd>
  <dt><a name="AgrRusNAF">2</a></dt>
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</dd>
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</dd>
  <dt><a name="Der85">4</a></dt>
  <dd><p><a href ="https://doi.org/https://doi.org/10.1016/0021-9045(85)90043-7"> <span class="scshape">M.M. Derriennic</span>, <em>On multivariate approximation by Bernstein-type polynomials</em>, J. Approx. Theory, <b class="bfseries">45</b> (1985), 155–166. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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  <dd><p><span class="scshape">J.L. Durmeyer</span>, <i class="itshape">Une formule d’inversion de la transformee de Laplace: Applications a la theorie des moments</i>, These de 3e cycle, Faculte des Sciences de l’Universite de Paris, 1967. </p>
</dd>
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</dd>
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</dd>
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</a> </p>
</dd>
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</a> </p>
</dd>
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</a> </p>
</dd>
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</dd>
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  <dd><p><a href ="https://doi.org/https://doi.org/10.1007/s10092-007-0131-2"> <span class="scshape">H. Gonska, P. Piţul, I. Raşa</span>, <em>Over-iterates of Bernstein-Stancu operators</em>, Calcolo, <b class="bfseries">44</b> (2007), 117–125. <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="GonRas06">15</a></dt>
  <dd><p><a href ="https://doi.org/https://doi.org/10.1007/s10474-006-0038-4"> <span class="scshape">H. Gonska, I. Raşa</span>, <em>The limiting semigroup of the Bernstein iterates: degree of convergence</em>, Acta Math. Hungar., <b class="bfseries">111</b> (2006) nos. 1–2, 119–130. <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="GonRas13">16</a></dt>
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</a> </p>
</dd>
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</a> </p>
</dd>
  <dt><a name="KalZie70">18</a></dt>
  <dd><p><a href ="https://doi.org/https://doi.org/10.1016/0021-9045(70)90055-9"> <span class="scshape">S. Karlin, Z. Ziegler</span>, <em>Iteration of positive approximation operators</em>, J. Approx. Theory, <b class="bfseries">3</b> (1970), 310–339. <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="KelRiv67">19</a></dt>
  <dd><p><a href ="https://doi.org/https://doi.org/10.2140/pjm.1967.21.511"> <span class="scshape">R.P. Kelisky, T.J. Rivlin</span>, <em>Iterates of Bernstein polynomials</em>, Pacific J. Math., <b class="bfseries">21</b> (1967), 511–520. <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="Rasa10">20</a></dt>
  <dd><p><span class="scshape">I. Raşa</span>, <em>\(C_{0}\)-Semigroups and iterates of positive linear operators: asymptotic behaviour</em>, Rend. Circ. Mat. Palermo, Ser. II, Suppl., <b class="bfseries">82</b> (2010), 123–142. </p>
</dd>
  <dt><a name="Rus2001">21</a></dt>
  <dd><p><span class="scshape">I.A. Rus</span>, <i class="itshape">Generalized contractions and applications,</i> Cluj Univ. Press, 2001. </p>
</dd>
  <dt><a name="Rus01a">22</a></dt>
  <dd><p><span class="scshape">I.A. Rus</span>, <i class="itshape">Fixed points and interpolation point set of a positive linear operator on </i>\(C(\overline{D}),\) Studia Univ. Babeş-Bolyai Math., <b class="bfseries"> 55</b> (2010) no. 4, 243–248. </p>
</dd>
  <dt><a name="Rus2002">23</a></dt>
  <dd><p><span class="scshape">I.A. Rus</span>, <em>Iterates of Stancu operators, via contraction principle</em>, Studia Univ. Babeş–Bolyai Math., <b class="bfseries">47</b> (2002) no. 4, 101–104. </p>
</dd>
  <dt><a name="Rus2004">24</a></dt>
  <dd><p><a href ="https://doi.org/https://doi.org/10.1016/j.jmaa.2003.11.056"> <span class="scshape">I.A. Rus</span>, <em>Iterates of Bernstein operators, via contraction principle</em>, J. Math. Anal. Appl., <b class="bfseries">292</b> (2004), 259–261. <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="Rus2010">25</a></dt>
  <dd><p><span class="scshape">I.A. Rus</span>, <em>Fixed point and interpolation point set of a positive linear operator on </em>\(C(\overline{D})\), Studia Univ. Babeş–Bolyai Math., <b class="bfseries">55</b> (2010) no. 4, 243–248. </p>
</dd>
</dl>


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