Document Zbl 1390.34002

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Existence and stability.</strong> <i>(English)</i> <a class="label nowrap" href="/1390.34002">Zbl 1390.34002</a> </h2> <div class="source"> <a href="/serials/3847" title="Series Profile">De Gruyter Series in Nonlinear Analysis and Applications</a> 26. Berlin: De Gruyter (ISBN 978-3-11-055313-0/hbk; 978-3-11-055381-9/ebook). xxiii, 336 p. (2018). </div> <div class="abstract">The book focuses on the existence and stability of solutions to initial and boundary value problems (BVP) for functional differential and integral equations and inclusions that involve Caputo fractional derivative and Hadamard fractional integral. The existence results discussed in the chapters are proved by using appropriate fixed point techniques. The stability results are of global, Ulam-Hyers and Ulam-Hyers-Rassias type. The book is a collection of results published by the authors in different journals. The contents are detailed below.<br class="zbmathjax-paragraph">The first chapter introduces the basic concepts like fractional derivative and integral operators, multivalued analysis, measure of noncompactness and phase spaces. Further, different kinds of fixed point techniques are also discussed. The second chapter deals with the existence and stability of solutions for some classes of nonlinear implicit fractional differential equations (NIFDE) including nonlocal condition. Further the existence and stability results for perturbed and neutral NIFDE with finite delay are studied.<br class="zbmathjax-paragraph">The third chapter is devoted to the existence and stability of solutions for some classes of impulsive NIFDE with finite delay. The fourth chapter establishes sufficient conditions for the existence and stability of solutions for some classes of BVP of NIFDE. Here the existence results for BVP for NIFDE with $0<\alpha\leq 1$ and $1<\alpha\leq 2$ are discussed. Also the authors established the existence of $L^1$-solutions of BVPs for NIFDEs with local and nonlocal conditions.<br class="zbmathjax-paragraph">The chapter five deals with the existence and stability results for some classes of BVP for impulsive NIFDE. Chapter six presents the results on integrable solutions of implicit fractional differential equations and the $L^1$-solutions with nonlocal condition. The infinite delay case is also discussed and the existence results are established by using measure of noncompactness. Chapter seven deals with the existence results of partial Hadamard fractional integral equations and inclusions. The results are obtained by using the upper and lower solutions method.<br class="zbmathjax-paragraph">Chapter eight mainly focusses on the Ulam stability results for partial Hadamard fractional integral equations and inclusions. Chapter nine discusses the existence and stability of Hadamard-Stieltjes fractional integral equations of Volterra type. The existence and attractivity results to Volterra-type nonlinear multidelay Hadamard-Stieltjes fractional integral equations are presented in the later part of the chapter. The main focus of chapter ten is on the stability results for random partial Hadamard fractional integral equations. Further the existence and stability results of multidelay Hadamard fractional integral equations in Frechét spaces with random effects are also discussed.<br class="zbmathjax-paragraph">The book is concise and includes many examples to illustrate different approaches of existence and stability of solutions of NIFDE. It may be recommended for graduate students or for researchers who work on analysis of solutions of fractional dynamical systems.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A13328">Krishnan Balachandran (Coimbatore)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6827301">Cited in <strong>130</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34-02" title="MSC2020">34-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K37" title="MSC2020">34K37</a> </td> <td class="space"> Functional-differential equations with fractional derivatives </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R11" title="MSC2020">35R11</a> </td> <td class="space"> Fractional partial differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A45G10" title="MSC2020">45G10</a> </td> <td class="space"> Other nonlinear integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26A33" title="MSC2020">26A33</a> </td> <td class="space"> Fractional derivatives and integrals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47N20" title="MSC2020">47N20</a> </td> <td class="space"> Applications of operator theory to differential and integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K32" title="MSC2020">34K32</a> </td> <td class="space"> Implicit functional-differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K27" title="MSC2020">34K27</a> </td> <td class="space"> Perturbations of functional-differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K20" title="MSC2020">34K20</a> </td> <td class="space"> Stability theory of functional-differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K40" title="MSC2020">34K40</a> </td> <td class="space"> Neutral functional-differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K45" title="MSC2020">34K45</a> </td> <td class="space"> Functional-differential equations with impulses </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K10" title="MSC2020">34K10</a> </td> <td class="space"> Boundary value problems for functional-differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aimplicit+fractional+differential+equations">implicit fractional differential equations</a>; <a href="/?q=ut%3Aintegral+equations">integral equations</a>; <a href="/?q=ut%3ACaputo+fractional+derivative">Caputo fractional derivative</a>; <a href="/?q=ut%3AHadamard+fractional+integral">Hadamard fractional integral</a>; <a href="/?q=ut%3Aexistence+of+solutions">existence of solutions</a>; <a href="/?q=ut%3AUlam+type+stability">Ulam type stability</a>; <a href="/?q=ut%3Afixed+point+techniques.">fixed point techniques.</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 1390.34002" data-ciurl="/ci/06827301" data-biburl="/bibtex/06827301.bib" data-amsurl="/amsrefs/06827301.bib" data-xmlurl="/xml/06827301.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06827301.pdf" title="Zbl 1390.34002 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/10.1515/9783110553819" aria-label="DOI for “Implicit fractional differential and integral equations. 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