Document Zbl 1281.46001

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New York, NY: Springer (ISBN 978-1-4614-8476-9/hbk; 978-1-4614-8477-6/ebook). xix, 246 p. (2013). </div> <div class="abstract">The essence of the stability of functional equations is “When is it true that a function satisfying a functional equation ${\mathcal E}$ approximately must be close to an exact solution of ${\mathcal E}$?”. In 1940, <span class="zbmathjax-textit">S. M. Ulam</span> [Problems in modern mathematics. First published under the title ‘A collection of mathematical problems.’ (Science Editions.) New York: John Wiley and Sons (1964; <a href="/0137.24201">Zbl 0137.24201</a>)] raised the first stability problem, which was partially solved in the next year for the Cauchy functional equation $f(x+y)=f(x)+f(y)$ by <span class="zbmathjax-textit">D. H. Hyers</span> [Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; <a href="/0061.26403">Zbl 0061.26403</a>)]. <span class="zbmathjax-textit">T. Aoki</span> [J. Math. Soc. Japan 2, 64–66 (1950; <a href="/0040.35501">Zbl 0040.35501</a>)] and <span class="zbmathjax-textit">Th. M. Rassias</span> [Proc. Am. Math. Soc. 72, 297–300 (1978; <a href="/0398.47040">Zbl 0398.47040</a>)] extended Hyers’ theorem for additive and linear mappings, respectively, when the Cauchy difference is allowed to be unbounded. The result of Rassias has influenced the development of what is now called the Hyers-Ulam-Rassias stability theory. Since then, many mathematicians have investigated the stability of various functional equations and mappings in different settings. The first attempt for establishing the stability of functional equations in “Fuzzy Mathematics” in the spirit of Hyers-Ulam-Rassias was done by <span class="zbmathjax-textit">A. K. Mirmostafaee</span> and the reviewer [Fuzzy Sets Syst. 159, No. 6, 720–729 (2008; <a href="/1178.46075">Zbl 1178.46075</a>)] in which they established three versions of the stability of the Cauchy functional equation in a linear space $X$ together with a fuzzy norm $N: X\times \mathbb{R} \to [0,1]$, where one may regard $N(x, t)$ as the truth value of the statement ‘the norm of $x$ is less than or equal to the real number $t$’. This fact was not included in the short literature review of the main theme of the book. Inspired by this paper, many mathematicians have investigated the stability of the Cauchy, Jensen, quadratic, cubic functional equations and their mixed types in other similar settings such as random normed spaces and non-Archimedean random spaces equipped with a triangular norm. In particular, <span class="zbmathjax-textit">D. Miheţ</span> and <span class="zbmathjax-textit">V. Radu</span> [J. Math. Anal. Appl. 343, No. 1, 567–572 (2008; <a href="/1139.39040">Zbl 1139.39040</a>)] initiated the study of the stability of the Cauchy functional equation in random normed spaces by using the fixed point alternative theorem first developed by <span class="zbmathjax-textit">V. Radu</span> [Fixed Point Theory 4, No. 1, 91–96 (2003; <a href="/1051.39031">Zbl 1051.39031</a>)].<br class="zbmathjax-paragraph">The book under review is essentially a collection of several recent papers related to the stability of functional equations in the framework of fuzzy and random normed spaces. The first two chapters nicely provide background of the theory of random normed spaces. The literature on the Cauchy, Jensen, quadratic and cubic functional equations is significant. The Cauchy and cubic equations are almost completely investigated in the book. The Jensen equation is not considered. The quadratic equation is only investigated in Chapter 3 for random normed spaces under special triangular norms. In Chapter 8, some notions such as ‘non-Archimedean Lie $C^*$-algebras’ are introduced which appear to be artificial. The book could prove to be useful for graduate students who are interested in the Hyers-Ulam-Rassias stability of functional equations.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10123">Mohammad Sal Moslehian (Mashhad)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6195709">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A6195709">Cited in <strong>42</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A46-02" title="MSC2020">46-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to functional analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A39-02" title="MSC2020">39-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to difference and functional equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A39B52" title="MSC2020">39B52</a> </td> <td class="space"> Functional equations for functions with more general domains and/or ranges </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46S10" title="MSC2020">46S10</a> </td> <td class="space"> Functional analysis over fields other than $\mathbb{R}$ or $\mathbb{C}$ or the quaternions; non-Archimedean functional analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46S40" title="MSC2020">46S40</a> </td> <td class="space"> Fuzzy functional analysis </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AHyers-Ulam-stability">Hyers-Ulam-stability</a>; <a href="/?q=ut%3Afunctional+equation">functional equation</a>; <a href="/?q=ut%3Arandom+normed+space">random normed space</a>; <a href="/?q=ut%3Atriangular+norm">triangular norm</a>; <a href="/?q=ut%3Anon-Archimedean+space">non-Archimedean space</a>; <a href="/?q=ut%3Afuzzy+normed+space">fuzzy normed space</a>; <a href="/?q=ut%3ACauchy+functional+equation">Cauchy functional equation</a>; <a href="/?q=ut%3Aquadratic+functional+equation">quadratic functional equation</a>; <a href="/?q=ut%3Acubic+functional+equation">cubic functional equation</a>; <a href="/?q=ut%3Aquartic+functional+equation">quartic functional equation</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0137.24201">Zbl 0137.24201</a>; <a href="/0061.26403">Zbl 0061.26403</a>; <a href="/0040.35501">Zbl 0040.35501</a>; <a href="/0398.47040">Zbl 0398.47040</a>; <a href="/1178.46075">Zbl 1178.46075</a>; <a href="/1139.39040">Zbl 1139.39040</a>; <a href="/1051.39031">Zbl 1051.39031</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 1281.46001" data-ciurl="/ci/06195709" data-biburl="/bibtex/06195709.bib" data-amsurl="/amsrefs/06195709.bib" data-xmlurl="/xml/06195709.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06195709.pdf" title="Zbl 1281.46001 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/10.1007/978-1-4614-8477-6" aria-label="DOI for “Stability of functional equations in random normed spaces”" title="10.1007/978-1-4614-8477-6">DOI</a> </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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