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<h1>Inverse problems via generalized contractive type operators</h1>
<p class="authors">
<span class="author">Ştefan M. Şoltuz\(^\ast \)</span>
</p>
<p class="date">February 11, 2010.</p>
</div>
<p>\(^\ast \)“T. Popoviciu” Institute of Numerical Analysis, Cluj-Napoca, Romania, e-mail:<br /><span class="ttfamily">smsoltuz@gmail.com</span>. </p>

<div class="abstract"><p> We prove a “collage” theorem for a generalized contractive type operators. </p>
<p><b class="bf">MSC.</b> 65J22, 47H10. </p>
<p><b class="bf">Keywords.</b> Generalized contractive type operators. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(X\) be a real Banach space, \(T:X\rightarrow X\) be an operator. The following result of Barnsley, see <span class="cite">
	[
	<a href="#barnsley" >1</a>
	]
</span>, became “a classic”. </p>
<p><div class="theorem_thmwrapper " id="a0000000003">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p><b class="bfseries">(Collage Theorem)</b> Let \(x\in X\) be given and \(T:X\rightarrow X\) a contraction with contraction factor \(L\in \left( 0,1\right) ,\) (i.e. \(\left\Vert Tx-Ty\right\Vert \leq L\left\Vert x-y\right\Vert ,\linebreak \forall \  x,\  y\in X),\) and fixed point \(x^{\ast }.\) Then </p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  \left\Vert x-x^{\ast }\right\Vert \leq \tfrac {1}{1-L}\left\Vert x-Tx\right\Vert . \end{equation*}
</div>

  </div>
</div> </p>
<p>Kunze <i class="it">et. al.</i>, see <span class="cite">
	[
	<a href="#kunze" >2</a>
	]
</span>, <span class="cite">
	[
	<a href="#kunze03" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#kunze04" >4</a>
	]
</span>, were able to apply the Collage Theorem to inverse problems in ODE , that is to reconstruct the field of an ODE, from a given “target" (trajectory). Our aim is to generalize the above Collage result to a larger operatorial class than contractions. Recently, similar results were introduced for other operatorial classes, see <span class="cite">
	[
	<a href="#stefan" >5</a>
	]
</span> and <span class="cite">
	[
	<a href="#stefanrevue" >6</a>
	]
</span>. </p>
<p>We shall consider the following class of operators: let \(T\) be such that there exist \(\alpha ,\beta ,\gamma \in \lbrack 0,1)\), not simultaneous zero, satisfying, </p>
<div class="equation" id="cond">
<p>
  <div class="equation_content">
    \begin{equation}  0<\alpha +\beta +\gamma <1 \label{cond} \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>and for each \(x,y\in X,\)</p>
<div class="equation" id="ineqweak1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert Tx-Ty\right\Vert \leq \alpha \left\Vert x-y\right\Vert +\beta \left\Vert y-Tx\right\Vert +\gamma \left\Vert x-Ty\right\Vert . \label{ineqweak1} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>For simplicity, let us denote this class by \(WR.\) </p>
<p><div class="remark_thmwrapper " id="a0000000005">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2</span>
  </div>
  <div class="remark_thmcontent">
  <p>Clearly, the contractions are included in this class Let \(F(T)\) denote the fixed point set with respect to \(X\) for the operator \(T.\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>If an operator belongs to the above operatorial class and it has a fixed points then the successive approximation converges to the unique fixed point. </p>
<p><div class="theorem_thmwrapper " id="a0000000006">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(X\) be a real Banach space and \(\alpha ,\beta ,\gamma \in \lbrack 0,1)\), not simultaneous zero, such that conditions \(\left( \ref{cond}\right) \) and \(\left( \ref{ineqweak1}\right) \) are satisfied, then the successive approximation iteration converges strongly to the unique fixed point of \(T\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000007">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(x^{\ast }\  \)be the fixed point. </p>
<div class="displaymath" id="a0000000008">
  \begin{eqnarray*}  \left\Vert x_{n+1}-x^{\ast }\right\Vert & \leq & \left\Vert Tx_{n}-Tx^{\ast }\right\Vert \\ & \leq & \alpha \left\Vert x_{n}-x^{\ast }\right\Vert +\beta \left\Vert x^{\ast }-Tx_{n}\right\Vert +\gamma \left\Vert Tx^{\ast }-x_{n}\right\Vert \\ & =& \left( \alpha +\gamma \right) \left\Vert x_{n}-x^{\ast }\right\Vert +\beta \left\Vert Tx^{\ast }-Tx_{n}\right\Vert \\ & =& \left( \alpha +\gamma \right) \left\Vert x_{n}-x^{\ast }\right\Vert +\beta \left\Vert x^{\ast }-x_{n+1}\right\Vert , \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000009">
  \begin{align*}  \left\Vert x_{n+1}-x^{\ast }\right\Vert & \leq \tfrac {\alpha +\gamma }{1-\beta }\left\Vert x_{n}-x^{\ast }\right\Vert .\\ \tfrac {\alpha +\gamma }{1-\beta }& =1-\tfrac {1-\alpha -\beta -\gamma }{1-\beta }. \end{align*}
</div>
<p>Therefore, </p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  \left\Vert x_{n+1}-x^{\ast }\right\Vert \leq \left( 1-\tfrac {1-\alpha -\beta -\gamma }{1-\beta }\right) ^{n}\left\Vert x_{0}-x^{\ast }\right\Vert \rightarrow 0\  \text{as\  }n\rightarrow \infty , \end{equation*}
</div>
<p>since \(\tfrac {1-\alpha -\beta -\gamma }{1-\beta }\in \left( 0,1\right) .\) Uniqueness easily results from condition \(\left( \ref{cond}\right) \) and \(\left( \ref{ineqweak1}\right) .\) </p>
<p><div class="remark_thmwrapper " id="a0000000011">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4</span>
  </div>
  <div class="remark_thmcontent">
  <p>It is straightforward to see that the above result holds in a complete metric space. <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000012">2 The inverse problem</h1>
<p>Suppose that \(x^{\ast }\in F(T).\) Kunze <i class="it">et. al.</i>, see <span class="cite">
	[
	<a href="#kunze" >2</a>
	]
</span>, <span class="cite">
	[
	<a href="#kunze03" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#kunze04" >4</a>
	]
</span>, have considered a class of inverse problems for ordinary differential equations and provided a mathematical basis for solving them within the framework of Banach spaces and contractions. We shall consider the same framework of Banach spaces and the larger class of operators which satisfy \(\left( \ref{cond}\right) \). </p>
<p>\(\  \)A typical inverse problem is the following, as formulated in <span class="cite">
	[
	<a href="#kunze" >2</a>
	]
</span>: </p>
<p><div class="problem_thmwrapper " id="prbl1">
  <div class="problem_thmheading">
    <span class="problem_thmcaption">
    Problem
    </span>
    <span class="problem_thmlabel">5</span>
  </div>
  <div class="problem_thmcontent">
  <p> For given \(\varepsilon {\gt}0\) and a “target” \(\bar{x},\; \)find \(T_{\varepsilon }\in WR\) such that \(\left\Vert \bar{x}-x_{T_{\varepsilon }}^{\ast }\right\Vert {\lt}\varepsilon ,\; \)where \(x_{T_{\varepsilon }}^{\ast }=T_{\varepsilon }\left( x_{T_{\varepsilon }}^{\ast }\right) \) is the unique fixed point of the operator \(T_{\varepsilon }\). </p>

  </div>
</div> </p>
<p>Consider now the following problem which we shall fit in our framework and which is very useful for practitioners, see <span class="cite">
	[
	<a href="#kunze" >2</a>
	]
</span>. </p>
<p><div class="problem_thmwrapper " id="prbl2">
  <div class="problem_thmheading">
    <span class="problem_thmcaption">
    Problem
    </span>
    <span class="problem_thmlabel">6</span>
  </div>
  <div class="problem_thmcontent">
  <p> Let \(\bar{x}\in X\) be a target and let \(\delta {\gt}0\) be given. Find \(T_{\delta }\in WR,\) such that \(\left\|  \bar{x}-T_{\delta }\bar{x}\right\|  {\lt}\delta .\) </p>

  </div>
</div> </p>
<p>In other words, instead of searching for \(WR\) maps whose fixed points lie close to target \(\bar{x},\) we search for \(WR\) maps that send \(\bar{x}\) close to itself. </p>
<p><div class="theorem_thmwrapper " id="coll">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>(<b class="bfseries">Collage theorem for WR Operators</b>) Let \(X\) be a real Banach space and \(T\) an operator satisfying \(\left( \ref{cond}\right) \) with contraction factors \(\alpha ,\beta ,\gamma \in \lbrack 0,1),\) and fixed point \(x^{\ast }\in X\). Then for any \(x\in X,\)</p>
<div class="displaymath" id="a0000000013">
  \begin{equation*}  \left\Vert x^{\ast }-x\right\Vert \leq \tfrac {\left( \gamma +1\right) }{1-\left( \alpha +\beta +\gamma \right) }\left\Vert x-Tx\right\Vert . \end{equation*}
</div>

  </div>
</div> </p>
<p><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>The condition \(\left( \ref{cond}\right) \) assures that the fixed point \(x^{\ast }\) is unique. If \(x=x^{\ast },\) the above inequality holds. If \(x\neq x^{\ast },\forall x\in X,\) then one obtains </p>
<div class="displaymath" id="a0000000015">
  \begin{align*}  \left\Vert x^{\ast }-x\right\Vert &  \leq \left\Vert Tx^{\ast }-Tx\right\Vert +\left\Vert Tx-x\right\Vert \\ &  \leq \alpha \left\Vert x^{\ast }-x\right\Vert +\beta \left\Vert x-x^{\ast }\right\Vert +\gamma \left\Vert x^{\ast }-Tx\right\Vert +\left\Vert Tx-x\right\Vert \\ &  =\alpha \left\Vert x^{\ast }-x\right\Vert +\beta \left\Vert x-x^{\ast }\right\Vert +\gamma \left\Vert x^{\ast }-x\right\Vert +\gamma \left\Vert Tx-x\right\Vert +\left\Vert Tx-x\right\Vert \\ &  =\alpha \left\Vert x^{\ast }-x\right\Vert +\beta \left\Vert x-x^{\ast }\right\Vert +\gamma \left\Vert x^{\ast }-x\right\Vert +\left( \gamma +1\right) \left\Vert Tx-x\right\Vert \\ &  =\left( \alpha +\beta +\gamma \right) \left\Vert x^{\ast }-x\right\Vert +\left( \gamma +1\right) \left\Vert Tx-x\right\Vert . \end{align*}
</div>
<p>From which one gets the conclusion by using \(\left( \ref{cond}\right) .\) </p>
<p>The above “Collage Theorem” allows us to reformulate the inverse Problem <a href="#prbl1">5</a> in the particular and more convenient Problem <a href="#prbl2">6</a>. </p>
<p><div class="theorem_thmwrapper " id="a0000000016">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">8</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If Problem <span class="rm"><a href="#prbl2">6</a></span> has a solution, then Problem <span class="rm"><a href="#prbl1">5</a></span> has a solution too. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000017">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(\varepsilon {\gt}0\) and \(\bar{x}\in X\) be given. For \(\delta :=\left( \left( 1\! -\! \left( \alpha \! +\! \beta \! +\! \gamma \right) \right) /\left( \gamma +1\right) \right) \varepsilon ,\) let \(T_{\delta }\in WR\) be such that \(\left\|  \bar{x}-T_{\delta }\bar{x}\right\|  {\lt}\delta .\) If \(x_{T_{\delta }}^{\ast }\) is the unique fixed point of the \(CL\) mapping \(T_{\delta },\) then, by Theorem <a href="#coll">7</a>, </p>
<div class="displaymath" id="a0000000018">
  \begin{equation*}  \left\|  \bar{x}-x_{T_{\delta }}^{\ast }\right\|  \leq \tfrac {\left( \gamma +1\right) }{1-\left( \alpha +\beta +\gamma \right) }\left\|  \bar{x}-T_{\delta }\bar{x}\right\|  \leq \tfrac {\left( \gamma +1\right) }{1-\left( \alpha +\beta +\gamma \right) }\delta =\varepsilon . \end{equation*}
</div>

<p>Note that shrinking the distance between two operators, one of them from WR\(, \) reduces the distance between their fixed points. </p>
<p><div class="proposition_thmwrapper " id="useful">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">9</span>
  </div>
  <div class="proposition_thmcontent">
  <p> Let \(X\) be a real Banach space and \(T_{1}\in WR\) with contraction factor \(\alpha _{1},\beta _{1},\gamma _{1}\in \left( 0,1\right) \) and \(T_{2}:X\rightarrow X\) a map such that \(x_{1}^{\ast },x_{2}^{\ast }\in X\) are distinct fixed points for \(T_{1}\) and \(T_{2}\). Then, </p>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  \left\Vert x_{1}^{\ast }-x_{2}^{\ast }\right\Vert \leq \tfrac {\left( \gamma _{1}+1\right) }{1-\left( \alpha _{1}+\beta _{1}+\gamma _{1}\right) }\sup _{x\in X}\left\Vert T_{1}x-T_{2}x\right\Vert . \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000020">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>One obtains </p>
<div class="displaymath" id="a0000000021">
  \begin{align*}  \left\Vert x_{1}^{\ast }-x_{2}^{\ast }\right\Vert &  =\left\Vert T_{1}x_{1}^{\ast }-T_{2}x_{2}^{\ast }\right\Vert \\ &  \leq \left\Vert T_{1}x_{1}^{\ast }-T_{1}x_{2}^{\ast }\right\Vert +\left\Vert T_{1}x_{2}^{\ast }-T_{2}x_{2}^{\ast }\right\Vert \\ &  \leq \alpha _{1}\left\Vert x_{1}^{\ast }-x_{2}^{\ast }\right\Vert +\beta _{1}\left\Vert x_{2}^{\ast }-T_{1}x_{1}^{\ast }\right\Vert +\gamma _{1}\left\Vert x_{1}^{\ast }-T_{1}x_{2}^{\ast }\right\Vert +\left\Vert T_{1}x_{2}^{\ast }-T_{2}x_{2}^{\ast }\right\Vert \\ &  \leq \alpha _{1}\left\Vert x_{1}^{\ast }-x_{2}^{\ast }\right\Vert +\beta _{1}\left\Vert x_{2}^{\ast }-x_{1}^{\ast }\right\Vert +\gamma _{1}\left\Vert x_{1}^{\ast }-x_{2}^{\ast }\right\Vert +\\ & \quad +\gamma _{1}\left\Vert x_{2}^{\ast }-T_{1}x_{2}^{\ast }\right\Vert +\left\Vert T_{1}x_{2}^{\ast }-T_{2}x_{2}^{\ast }\right\Vert \\ &  =\alpha _{1}\left\Vert x_{1}^{\ast }-x_{2}^{\ast }\right\Vert +\beta _{1}\left\Vert x_{2}^{\ast }-x_{1}^{\ast }\right\Vert +\gamma _{1}\left\Vert x_{1}^{\ast }-x_{2}^{\ast }\right\Vert +\\ & \quad +\left( \gamma _{1}+1\right) \left\Vert T_{1}x_{2}^{\ast }-T_{2}x_{2}^{\ast }\right\Vert \\ &  \leq \alpha _{1}\left\Vert x_{1}^{\ast }-x_{2}^{\ast }\right\Vert +\beta _{1}\left\Vert x_{2}^{\ast }-x_{1}^{\ast }\right\Vert +\gamma _{1}\left\Vert x_{1}^{\ast }-x_{2}^{\ast }\right\Vert +\\ & \quad +\left( \gamma _{1}+1\right) \sup _{x\in X}\left\Vert T_{1}x\- T_{2}x\right\Vert , \end{align*}
</div>
<p> from which we get the conclusion. </p>
<p><div class="theorem_thmwrapper " id="m">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">10</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(X\) be a real Banach space, \(T:X\rightarrow X,\) \(\bar{x}=T\bar{x}\) and suppose there exists \(T_{1}\in WR\) such that \(\sup _{x\in X}\left\|  T_{1}x-Tx\right\|  \leq \varepsilon .\) Then </p>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  \left\|  \bar{x}-T_{1}\bar{x}\right\|  \leq \left( 1+\tfrac {\alpha +\beta }{1-\gamma }\right) \tfrac {\left( \gamma +1\right) }{1-\left( \alpha +\beta +\gamma \right) }\varepsilon . \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000023">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(x^{\ast }=T_{1}x^{\ast }\), and by use of Proposition <a href="#useful">9</a> we obtain </p>
<div class="displaymath" id="a0000000024">
  \begin{equation*}  \left\Vert \bar{x}-x^{\ast }\right\Vert \leq \tfrac {\left( \gamma +1\right) }{1-\left( \alpha +\beta +\gamma \right) }\left( \sup _{x\in X}\left\Vert T_{1}x-Tx\right\Vert \right) . \end{equation*}
</div>
<p> We have </p>
<div class="displaymath" id="a0000000025">
  \begin{align*}  \left\Vert \bar{x}-T_{1}\bar{x}\right\Vert &  \leq \left\Vert \bar{x}-x^{\ast }\right\Vert +\left\Vert x^{\ast }-T_{1}\bar{x}\right\Vert \\ &  \leq \left\Vert \bar{x}-x^{\ast }\right\Vert +\left\Vert T_{1}x^{\ast }-T_{1}\bar{x}\right\Vert , \end{align*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000026">
  \begin{eqnarray*}  \left\Vert T_{1}x^{\ast }-T_{1}\bar{x}\right\Vert & \leq & \alpha \left\Vert \bar{x}-x^{\ast }\right\Vert +\beta \left\Vert \bar{x}-T_{1}x^{\ast }\right\Vert +\gamma \left\Vert x^{\ast }-T_{1}\bar{x}\right\Vert \\ & =& \alpha \left\Vert \bar{x}-x^{\ast }\right\Vert +\beta \left\Vert \bar{x}-T_{1}x^{\ast }\right\Vert +\gamma \left\Vert T_{1}x^{\ast }-T_{1}\bar{x}\right\Vert , \\ & & i.e. \\ \left\Vert T_{1}x^{\ast }-T_{1}\bar{x}\right\Vert & \leq & \tfrac {\alpha +\beta }{1-\gamma }\left\Vert \bar{x}-x^{\ast }\right\Vert . \end{eqnarray*}
</div>
<p> Thus </p>
<div class="displaymath" id="a0000000027">
  \begin{eqnarray*}  \left\Vert \bar{x}-T_{1}\bar{x}\right\Vert & \leq & \left\Vert \bar{x}-x^{\ast }\right\Vert +\tfrac {\alpha +\beta }{1-\gamma }\left\Vert \bar{x}-x^{\ast }\right\Vert \\ & \leq & \left( 1+\tfrac {\alpha +\beta }{1-\gamma }\right) \left\Vert \bar{x}-x^{\ast }\right\Vert \leq \left( 1+\tfrac {\alpha +\beta }{1-\gamma }\right) \tfrac {\left( \gamma +1\right) }{1-\left( \alpha +\beta +\gamma \right) }\varepsilon . \end{eqnarray*}
</div>

<p><div class="remark_thmwrapper " id="a0000000028">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">11</span>
  </div>
  <div class="remark_thmcontent">
  <p>The quantity \(\left( 1+\tfrac {\alpha +\beta }{1-\gamma }\right) \tfrac {\left( \gamma +1\right) }{1-\left( \alpha +\beta +\gamma \right) }\) can be written as <br />\(\tfrac {1+\gamma }{1-\gamma }\tfrac {\left( 1+\gamma \right) +\left( \alpha +\beta \right) }{\left( 1-\gamma \right) -\left( \alpha +\beta \right) }.\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="barnsley">1</a></dt>
  <dd><p><i class="sc">M.F. Barnsley</i>, <i class="it">Fractals everywhere</i>, New York, Academic Press, 1988. </p>
</dd>
  <dt><a name="kunze">2</a></dt>
  <dd><p><i class="sc">H.E. Kunze</i> and <i class="sc">E.R. Vrscay</i>, <i class="itshape">Solving inverse problems for ordinary differential equations using the Picard contraction mapping</i>, Inverse Problems, <b class="bfseries">15</b>, pp.&#160;745–770, 1999. </p>
</dd>
  <dt><a name="kunze03">3</a></dt>
  <dd><p><i class="sc">H.E. Kunze</i> and <i class="sc">S. Gomes</i>, <i class="itshape">Solving an inverse problem for Urison-type integral equations using Banach’s fixed point theorem</i>, Inverse Problems, <b class="bfseries">19</b>, pp.&#160;411–418, 2003. </p>
</dd>
  <dt><a name="kunze04">4</a></dt>
  <dd><p><i class="sc">H.E. Kunze, J.E. Hicken</i> and <i class="sc">E.R. Vrscay</i>, <i class="itshape">Inverse problems for ODEs using contraction maps and suboptimality for the ’collage method’</i>, Inverse Problems, <b class="bfseries">20</b>, pp.&#160;977-991, 2004. </p>
</dd>
  <dt><a name="stefan">5</a></dt>
  <dd><p><i class="sc">Ş.M. Şoltuz</i>, <i class="itshape">Solving inverse problems via hemicontractive maps</i>, Nonlinear Analysis, textbf71, pp.&#160;2387–2390, 2009. </p>
</dd>
  <dt><a name="stefanrevue">6</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2008-vol37-no2-art14"> <i class="sc">Ş.M. Şoltuz</i>, <i class="itshape">Solving inverse problems via weak-contractive maps</i>, Rev. Anal. Numer. Theor. Approx., <b class="bf">37</b>, no. 2, pp.&#160;217–220, 2008. <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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