Document Zbl 1242.54027

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A, Theory Methods</a> <a href="/?q=in%3A295743" title="Articles in this Issue">75, No. 4, 2154-2165 (2012)</a>. </div> <div class="abstract">Let $\Psi$ be the family of nondecreasing functions $\psi:[0,+\infty)\to[0,+\infty)$ such that $\sum_{n=1}^{\infty}\psi^n(t)<+\infty$ for each $t\geq0$, where $\psi^n$ is the $n$-th iterate of $\psi$. Let $(X,d)$ be a metric space and $T:X\to X$. The authors call the mapping $T$ $\alpha$-$\psi$-contractive if there exist functions $\alpha:X\times X\to[0,+\infty)$ and $\psi\in\Psi$ such that $\alpha(x,y)\,d(Tx,Ty)\leq\psi(d(x,y))$ for all $x,y\in X$. Furthermore, $T$ is called $\alpha$-admissible if $\alpha(x,y)\geq1$ implies that $\alpha(Tx,Ty)\geq1$.<br class="zbmathjax-paragraph">The authors prove that, if the space $(X,d)$ is complete and the mapping $T$ is continuous, $\alpha$-admissible and $\alpha$-$\psi$-contractive, then $T$ has a fixed point in $X$, provided that $\alpha(x_0,Tx_0)\geq1$ for some $x_0\in X$. Continuity of $T$ can be omitted if, for each sequence $\{x_n\}$ in $X$, $\alpha(x_n,x_{n+1})\geq1$ for all $n$ and $x_n\to x\in X$ as $n\to\infty$ imply that $\alpha(x_n,x)\geq1$ for all $n$. The uniqueness of the fixed point holds if, for all $x,y\in X$, there is some $z\in X$ such that $\alpha(x,z)\geq1$ and $\alpha(y,z)\geq1$.<br class="zbmathjax-paragraph">Several known (and some new) fixed and coupled fixed point theorems in metric, as well as in ordered metric spaces can be obtained from these results. Some examples are presented, showing how the obtained theorems can be used. Also, some applications to ordinary differential equations are given.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10637">Zoran Kadelburg (Beograd)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6030602">Cited in <strong>45</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A6030602">Cited in <strong>687</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A54H25" title="MSC2020">54H25</a> </td> <td class="space"> Fixed-point and coincidence theorems (topological aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A54E40" title="MSC2020">54E40</a> </td> <td class="space"> Special maps on metric spaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A54E50" title="MSC2020">54E50</a> </td> <td class="space"> Complete metric spaces </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afixed+point">fixed point</a>; <a href="/?q=ut%3Acoupled+fixed+point">coupled fixed point</a>; <a href="/?q=ut%3Acontractive+mapping">contractive mapping</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1242.54027" data-ciurl="/ci/06030602" data-biburl="/bibtex/06030602.bib" data-amsurl="/amsrefs/06030602.bib" data-xmlurl="/xml/06030602.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06030602.pdf" title="Zbl 1242.54027 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.na.2011.10.014" aria-label="DOI for “Fixed point theorems for $\alpha$-$\psi$-contractive type mappings”" title="10.1016/j.na.2011.10.014">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://polipapers.upv.es/index.php/AGT/article/view/6694" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. 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Document Zbl 1242.54027 - zbMATH Open