Document Zbl 07971735

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Nurul</a>; <a href="/authors/celik.emine" title="Author Profile">Çelik, Emine</a>; <a href="/authors/telyakovskiy.aleksey-s" title="Author Profile">Telyakovskiy, Aleksey S.</a></div> <h2 class="title"> <strong>An example of fractional ODE loss of maximum principle and Hopf’s lemma.</strong> <i>(English)</i> <a class="label nowrap" href="/7971735">Zbl 07971735</a> </h2> <div class="source"> <a href="/serials/659" title="Journal Profile">Rend. Circ. Mat. Palermo (2)</a> <a href="/?q=in%3A522220" title="Articles in this Issue">74, No. 1, Paper No. 27, 11 p. (2025)</a>. </div> <div class="abstract">Summary: It has been known that the fractional ODEs involving Riemann-Liouville fractional derivatives maintain many “analogies” of the classic elliptic-type properties, including maximum principles, Hopf’s Lemma, the existence of real eigenvalues, etc. However, in this short work, we construct a counterexample demonstrating that Caputo-type fractional operators (and their compositions) fail to uphold the maximum principle and Hopf’s lemma. This finding is somewhat unexpected and serves as a benchmark problem or counterexample, reminding us to remain cautious when developing parallel theories for fractional differential equations and applying them in modeling. Since the maximum principle and Hopf’s lemma are often used to construct convex cones in Banach spaces and guarantee the existence of a principal eigenvalue in classical elliptic theory, this result leads us to conjecture that fractional Sturm-Liouville problems involving the Caputo fractional derivative may lack a conventional principal eigenvalue. This conjecture is partially proved in a following-up work.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A08" title="MSC2020">34A08</a> </td> <td class="space"> Fractional ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34B05" title="MSC2020">34B05</a> </td> <td class="space"> Linear boundary value problems for ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34B09" title="MSC2020">34B09</a> </td> <td class="space"> Boundary eigenvalue problems for ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amaximum+principle">maximum principle</a>; <a href="/?q=ut%3AHopf+lemma">Hopf lemma</a>; <a href="/?q=ut%3Acounterexample">counterexample</a>; <a href="/?q=ut%3Amixed+fractional+derivative">mixed fractional derivative</a>; <a href="/?q=ut%3Afractional+ODE">fractional ODE</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 07971735" data-ciurl="/ci/07971735" data-biburl="/bibtex/07971735.bib" data-amsurl="/amsrefs/07971735.bib" data-xmlurl="/xml/07971735.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07971735.pdf" title="Zbl 07971735 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/10.1007/s12215-024-01184-3" aria-label="DOI for “An example of fractional ODE loss of maximum principle and Hopf's lemma”" title="10.1007/s12215-024-01184-3">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Abdeljawad, T.; Thabet, ST; Kedim, I.; Vivas-Cortez, M., On a new structure of multi-term Hilfer fractional impulsive neutral Levin-Nohel integrodifferential system with variable time delay, AIMS Math., 9, 3, 7372-7395, 2024; Abdeljawad, T.; Thabet, ST; Kedim, I.; Vivas-Cortez, M., On a new structure of multi-term Hilfer fractional impulsive neutral Levin-Nohel integrodifferential system with variable time delay, AIMS Math., 9, 3, 7372-7395, 2024 · <a href="https://doi.org/10.3934/math.2024357" class="nowrap">doi:10.3934/math.2024357</a></td> </tr><tr> <td>[2]</td> <td class="space">Aleroev, T.S., Li, Y.: Fractional Sturm-Liouville problem with Caputo derivative loses the principal eigenvalue. 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