Document Zbl 0874.34054

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(1993). </div> <div class="abstract">From the preface: This book addresses the need for the study of generalized solutions to broad classes of FDE. In the first three chapters we concentrate on differential equations with piecewise continuous arguments (EPCA). These equations arise in an attempt to extend the theory of FDE with continuous arguments to differential equations with discontinuous arguments. This task is also of considerable applied interest since EPCA include, as particular cases, impulsive and loaded equations of control theory and are similar to those found in some biomedical models. Underlying each EPCA is a dynamical system governed by a difference equation of a discrete argument which describes its stability, oscillation and periodic properties. Boundary and initial-value problems for some EPCA with partial derivatives are considered and the behavior of their solutions investigated. The results are also extended to equations with positive definite operators in Hilbert spaces. It is well known that profound and close links exist between functional and functional differential equations. Thus the study of the first often enables one to predict properties of differential equations of neutral type. Functional equations are directly related to difference equations of a discrete argument, and bordering on difference equations are impulsive FDE with impacts and switching and loaded equations. In the last two chapters we turn from mildly weakened solutions of EPCA to generalized-function solutions of ordinary differential and functional differential equations.<br class="zbmathjax-paragraph">The unifying theme of the book is the development of theoretically meaningful and potentially applicable generalized concepts of solutions for important classes of FDE. Contrary to general functional differential equations, EPCA of all types (retarded, advanced, mixed, neutral) have two-sided solutions, and FDE with linearly transformed arguments possess, under certain conditions, analytic or entire solutions. Some methods in the theory of entire solutions are applied to prove stability theorems for linear EPCA with variable coefficients. Integral transformations establish close connections between entire and generalized functions (distributions). Therefore, a unified approach may be used in the study of both distributional and entire solutions to some classes of linear ordinary and functional differential equations. A special role is played by the chapter on differential equations with periodic transformations of the argument. These FDE can be reduced to ordinary differential equations and are very important in a number of biological models. Reducible FDE naturally appear in the construction of Lyapunov functionals for retarded differential equations. They represent a rich source of analytic solutions and provide an insight into the structure of solutions for more general FDE, especially equations with linearly transformed arguments. Finally, a list of new research topics and open problems is included.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A825873">Cited in <strong>191</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K05" title="MSC2020">34K05</a> </td> <td class="space"> General theory of functional-differential equations </td> </tr><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%3A92C50" title="MSC2020">92C50</a> </td> <td class="space"> Medical applications (general) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aboundary+and+initial-value+problems">boundary and initial-value problems</a>; <a href="/?q=ut%3Aretarded">retarded</a>; <a href="/?q=ut%3Aadvanced">advanced</a>; <a href="/?q=ut%3Amixed">mixed</a>; <a href="/?q=ut%3Aneutral">neutral</a>; <a href="/?q=ut%3Astability+properties">stability properties</a>; <a href="/?q=ut%3Aoscillation+properties">oscillation properties</a>; <a href="/?q=ut%3Aimpulsive+and+loaded+equations+of+control+theory">impulsive and loaded equations of control theory</a>; <a href="/?q=ut%3Abiomedical+models">biomedical models</a>; <a href="/?q=ut%3Adynamical+system">dynamical system</a>; <a href="/?q=ut%3Aperiodic+properties">periodic properties</a>; <a href="/?q=ut%3Ageneralized-function+solutions">generalized-function solutions</a>; <a href="/?q=ut%3Aperiodic+transformations+of+the+argument">periodic transformations of the argument</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 0874.34054" data-ciurl="/ci/00825873" data-biburl="/bibtex/00825873.bib" data-amsurl="/amsrefs/00825873.bib" data-xmlurl="/xml/00825873.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00825873.pdf" title="Zbl 0874.34054 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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