CINXE.COM

Document Zbl 1459.39053 - zbMATH Open

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1459.39053 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">&times;</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools &amp; Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/1459.39053?"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open &mdash; the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span>&nbsp;<a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="an:1459.39053" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span>&nbsp;</span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>&nbsp;&nbsp;</span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>&nbsp;&nbsp;</span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>&nbsp;&nbsp;</span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>&nbsp;&nbsp;</span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>&nbsp;&nbsp;</span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>&nbsp;&nbsp;</span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>&nbsp;&nbsp;</span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>&nbsp;&nbsp;</span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>&nbsp;&nbsp;</span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>&nbsp;&nbsp;</span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>&nbsp;&nbsp;</span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>&nbsp;&nbsp;</span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>&nbsp;&nbsp;</span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>&nbsp;&nbsp;</span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>&nbsp;&nbsp;</span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>&nbsp;&nbsp;</span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>&nbsp;&nbsp;</span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a&nbsp;<span class="item">&</span>&nbsp;b&nbsp;</span><span>&nbsp;</span>logical and (default)</a></li> <li><a href="#"><span class="token">a&nbsp;<span class="item">|</span>&nbsp;b&nbsp;</span><span>&nbsp;</span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab&nbsp;&nbsp;&nbsp;</span><span>&nbsp;</span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>&nbsp;&nbsp;</span><span>&nbsp;</span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab&nbsp;c<span class="item">"</span></span><span>&nbsp;</span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab&nbsp;c<span class="item">)</span></span><span>&nbsp;</span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki &amp; ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &amp;</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b &amp; au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&amp;</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/luo.danfeng" title="Author Profile">Luo, Danfeng</a>; <a href="/authors/luo.zhiguo" title="Author Profile">Luo, Zhiguo</a>; <a href="/authors/qiu.hongjun" title="Author Profile">Qiu, Hongjun</a></div> <h2 class="title"> <strong>Existence and Hyers-Ulam stability of solutions for a mixed fractional-order nonlinear delay difference equation with parameters.</strong> <i>(English)</i> <a class="label nowrap" href="/1459.39053">Zbl 1459.39053</a> </h2> <div class="source"> <a href="/serials/2235" title="Journal Profile">Math. Probl. Eng.</a> <a href="/?q=in%3A450504" title="Articles in this Issue">2020, Article ID 9372406, 12 p. (2020)</a>. </div> <div class="abstract">Summary: This paper focuses on a kind of mixed fractional-order nonlinear delay difference equations with parameters. Under some new criteria and by applying the Brouwer theorem and the contraction mapping principle, the new existence and uniqueness results of the solutions have been established. In addition, we deduce that the solution of the addressed equation is Hyers-Ulam stable. Some results in the literature can be generalized and improved. As an application, three typical examples are delineated to demonstrate the effectiveness of our theoretical results.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7283737">Cited in <strong>14</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A39B82" title="MSC2020">39B82</a> </td> <td class="space"> Stability, separation, extension, and related topics for functional equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A08" title="MSC2020">34A08</a> </td> <td class="space"> Fractional ordinary differential equations </td> </tr></table> </div> <!-- Modal used to show zbmath metadata in different output formats--> <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">&times;</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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1459.39053" data-ciurl="/ci/07283737" data-biburl="/bibtex/07283737.bib" data-amsurl="/amsrefs/07283737.bib" data-xmlurl="/xml/07283737.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07283737.pdf" title="Zbl 1459.39053 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/10.1155/2020/9372406" aria-label="DOI for “Existence and Hyers-Ulam stability of solutions for a mixed fractional-order nonlinear delay difference equation with parameters”" title="10.1155/2020/9372406">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">Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Hoboken, NJ, USA: A Wiley-Interscience Publication, Hoboken, NJ, USA &middot; <a href="/0789.26002" class="nowrap">Zbl 0789.26002</a></td> </tr><tr> <td>[2]</td> <td class="space">Podlubny, I.; Thimann, K. V., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (1999), Cambridge, UK: Academic Press, Cambridge, UK &middot; <a href="/0924.34008" class="nowrap">Zbl 0924.34008</a></td> </tr><tr> <td>[3]</td> <td class="space">Jalilian, Y., Fractional integral inequalities and their applications to fractional differential equations, Acta Mathematica Scientia, 36, 5, 1317-1330 (2016) &middot; <a href="/1374.35015" class="nowrap">Zbl 1374.35015</a>&nbsp;&middot; <a href="https://doi.org/10.1016/s0252-9602(16)30071-6" class="nowrap">doi:10.1016/s0252-9602(16)30071-6</a></td> </tr><tr> <td>[4]</td> <td class="space">Grace, S. R., On the oscillatory behavior of solutions of nonlinear fractional differential equations, Applied Mathematics and Computation, 266, 259-266 (2015) &middot; <a href="/1410.34103" class="nowrap">Zbl 1410.34103</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.amc.2015.05.062" class="nowrap">doi:10.1016/j.amc.2015.05.062</a></td> </tr><tr> <td>[5]</td> <td class="space">He, Y., Existence results and the monotone iterative technique for nonlinear fractional differential systems involving fractional integral boundary conditions, Advances in Difference Equations, 2017, 1 (2017) &middot; <a href="/1422.34034" class="nowrap">Zbl 1422.34034</a>&nbsp;&middot; <a href="https://doi.org/10.1186/s13662-017-1304-1" class="nowrap">doi:10.1186/s13662-017-1304-1</a></td> </tr><tr> <td>[6]</td> <td class="space">Zada, A.; Ali, S.; Li, Y., Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Advances in Difference Equations, 2017, 1 (2017) &middot; <a href="/1444.34083" class="nowrap">Zbl 1444.34083</a>&nbsp;&middot; <a href="https://doi.org/10.1186/s13662-017-1376-y" class="nowrap">doi:10.1186/s13662-017-1376-y</a></td> </tr><tr> <td>[7]</td> <td class="space">Xie, W.; Xiao, J.; Luo, Z., Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions, Applied Mathematics Letters, 41, 46-51 (2015) &middot; <a href="/1320.34013" class="nowrap">Zbl 1320.34013</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.aml.2014.10.014" class="nowrap">doi:10.1016/j.aml.2014.10.014</a></td> </tr><tr> <td>[8]</td> <td class="space">Xie, W.; Xiao, J.; Luo, Z., Existence of solutions for fractional boundary value problem with nonlinear derivative dependence, Abstract and Applied Analysis, 2014 (2014) &middot; <a href="/1474.34059" class="nowrap">Zbl 1474.34059</a>&nbsp;&middot; <a href="https://doi.org/10.1155/2014/812910" class="nowrap">doi:10.1155/2014/812910</a></td> </tr><tr> <td>[9]</td> <td class="space">Chaudhary, N. I.; Latif, R.; Zahoor Raja, M. A.; Tenreiro Machado, J. A., An innovative fractional order LMS algorithm for power signal parameter estimation, Applied Mathematical Modelling, 83, 703-718 (2020) &middot; <a href="/1481.94050" class="nowrap">Zbl 1481.94050</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.apm.2020.03.014" class="nowrap">doi:10.1016/j.apm.2020.03.014</a></td> </tr><tr> <td>[10]</td> <td class="space">Khan, A.; Khan, H.; Gómez-Aguilar, J. F.; Abdeljawad, T., Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos, Solitons &amp; Fractals, 127, 422-427 (2019) &middot; <a href="/1448.34046" class="nowrap">Zbl 1448.34046</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.chaos.2019.07.026" class="nowrap">doi:10.1016/j.chaos.2019.07.026</a></td> </tr><tr> <td>[11]</td> <td class="space">Khan, A.; Gómez-Aguilar, J. F.; Saeed Khan, T.; Khan, H., Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chaos, Solitons &amp; Fractals, 122, 119-128 (2019) &middot; <a href="/1448.92307" class="nowrap">Zbl 1448.92307</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.chaos.2019.03.022" class="nowrap">doi:10.1016/j.chaos.2019.03.022</a></td> </tr><tr> <td>[12]</td> <td class="space">Khan, H.; Gómez-Aguilar, J. F.; Khan, A.; Khan, T. S., Stability analysis for fractional order advection-reaction diffusion system, Physica A: Statistical Mechanics and Its Applications, 521, 737-751 (2019) &middot; <a href="/1514.35461" class="nowrap">Zbl 1514.35461</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.physa.2019.01.102" class="nowrap">doi:10.1016/j.physa.2019.01.102</a></td> </tr><tr> <td>[13]</td> <td class="space">Morales-Delgado, V. F.; Gómez-Aguilar, J. F.; Saad, K. M.; Khan, M. A.; Agarwal, P., Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: a fractional calculus approach, Physica A: Statistical Mechanics and Its Applications, 523, 48-65 (2019) &middot; <a href="/07563353" class="nowrap">Zbl 07563353</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.physa.2019.02.018" class="nowrap">doi:10.1016/j.physa.2019.02.018</a></td> </tr><tr> <td>[14]</td> <td class="space">Luo, D. F.; Luo, Z. G., Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses, Advances in Difference Equations, 2019 (2019) &middot; <a href="/1459.34032" class="nowrap">Zbl 1459.34032</a>&nbsp;&middot; <a href="https://doi.org/10.1186/s13662-019-2101-9" class="nowrap">doi:10.1186/s13662-019-2101-9</a></td> </tr><tr> <td>[15]</td> <td class="space">Goodrich, C.; Peterson, A. C., Discrete Fractional Calculus (2015), New York City, NY, USA: Springer International Publishing, New York City, NY, USA &middot; <a href="/1350.39001" class="nowrap">Zbl 1350.39001</a></td> </tr><tr> <td>[16]</td> <td class="space">Goodrich, C. S., Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Computers &amp; Mathematics with Applications, 61, 2, 191-202 (2011) &middot; <a href="/1211.39002" class="nowrap">Zbl 1211.39002</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.camwa.2010.10.041" class="nowrap">doi:10.1016/j.camwa.2010.10.041</a></td> </tr><tr> <td>[17]</td> <td class="space">Goodrich, C. S., A convexity result for fractional differences, Applied Mathematics Letters, 35, 58-62 (2014) &middot; <a href="/1314.26010" class="nowrap">Zbl 1314.26010</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.aml.2014.04.013" class="nowrap">doi:10.1016/j.aml.2014.04.013</a></td> </tr><tr> <td>[18]</td> <td class="space">Goodrich, C. S., Continuity of solutions to discrete fractional initial value problems, Computers &amp; Mathematics with Applications, 59, 11, 3489-3499 (2010) &middot; <a href="/1197.39002" class="nowrap">Zbl 1197.39002</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.camwa.2010.03.040" class="nowrap">doi:10.1016/j.camwa.2010.03.040</a></td> </tr><tr> <td>[19]</td> <td class="space">Abdeljawad, T.; Torres, D. F. M., Symmetric duality for left and right Riemann-Liouville and Caputo fractional differences, Arab Journal of Mathematical Sciences, 23, 2, 157-172 (2017) &middot; <a href="/1366.26012" class="nowrap">Zbl 1366.26012</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.ajmsc.2016.07.001" class="nowrap">doi:10.1016/j.ajmsc.2016.07.001</a></td> </tr><tr> <td>[20]</td> <td class="space">Abdeljawad, T.; Al-Mdallal, Q. M., Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall’s inequality, Journal of Computational and Applied Mathematics, 339, 218-230 (2018) &middot; <a href="/1472.39006" class="nowrap">Zbl 1472.39006</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.cam.2017.10.021" class="nowrap">doi:10.1016/j.cam.2017.10.021</a></td> </tr><tr> <td>[21]</td> <td class="space">Abdeljawad, T.; Alzabut, J.; Baleanu, D., A generalized q-fractional Gronwall inequality and its applications to nonlinear delay q-fractional difference systems, Journal of Inequalities and Applications, 2016, 1 (2016) &middot; <a href="/1353.39009" class="nowrap">Zbl 1353.39009</a>&nbsp;&middot; <a href="https://doi.org/10.1186/s13660-016-1181-2" class="nowrap">doi:10.1186/s13660-016-1181-2</a></td> </tr><tr> <td>[22]</td> <td class="space">Dassios, I. K.; Baleanu, D. I., Duality of singular linear systems of fractional nabla difference equations, Applied Mathematical Modelling, 39, 14, 4180-4195 (2015) &middot; <a href="/1443.65436" class="nowrap">Zbl 1443.65436</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.apm.2014.12.039" class="nowrap">doi:10.1016/j.apm.2014.12.039</a></td> </tr><tr> <td>[23]</td> <td class="space">Atici, F.; Eloe, P., Two-point boundary value problems for finite fractional difference equations, Journal of Difference Equations and Applications, 17, 4, 445-456 (2011) &middot; <a href="/1215.39002" class="nowrap">Zbl 1215.39002</a></td> </tr><tr> <td>[24]</td> <td class="space">Goodrich, C., On positive solutions to nonlocal fractional and integer-order difference equations, Applicable Analysis and Discrete Mathematics, 5, 1, 122-132 (2011) &middot; <a href="/1289.39008" class="nowrap">Zbl 1289.39008</a>&nbsp;&middot; <a href="https://doi.org/10.2298/aadm110111001g" class="nowrap">doi:10.2298/aadm110111001g</a></td> </tr><tr> <td>[25]</td> <td class="space">Chen, F.; Zhou, Y., Existence and Ulam stability of solutions for discrete fractional boundary value problem, Discrete Dynamics in Nature and Society, 2013 (2013) &middot; <a href="/1417.39056" class="nowrap">Zbl 1417.39056</a>&nbsp;&middot; <a href="https://doi.org/10.1155/2013/459161" class="nowrap">doi:10.1155/2013/459161</a></td> </tr><tr> <td>[26]</td> <td class="space">Kaewwisetkul, B.; Sitthiwirattham, T., On nonlocal fractional sum-difference boundary value problems for Caputo fractional functional difference equations with delay, Advances in Difference Equations, 2017 (2017) &middot; <a href="/1422.34220" class="nowrap">Zbl 1422.34220</a>&nbsp;&middot; <a href="https://doi.org/10.1186/s13662-017-1283-2" class="nowrap">doi:10.1186/s13662-017-1283-2</a></td> </tr><tr> <td>[27]</td> <td class="space">Ouncharoen, R.; Chasreechai, S.; Sitthiwirattham, T., On nonlinear fractional difference equation with delay and impulses, Symmetry, 12, 6 (2020)</td> </tr><tr> <td>[28]</td> <td class="space">Ulam, S., A Collection of Mathematical Problem (1960), New York, NY, USA: Interscience, New York, NY, USA &middot; <a href="/0086.24101" class="nowrap">Zbl 0086.24101</a></td> </tr><tr> <td>[29]</td> <td class="space">Hyers, D. H., On the stability of the linear functional equation, Proceedings of the National Academy of Sciences, 27, 4, 222-224 (1941) &middot; <a href="/67.0424.01" class="nowrap">JFM 67.0424.01</a>&nbsp;&middot; <a href="https://doi.org/10.1073/pnas.27.4.222" class="nowrap">doi:10.1073/pnas.27.4.222</a></td> </tr><tr> <td>[30]</td> <td class="space">Wang, J.; Li, X., Ulam-Hyers stability of fractional Langevin equations, Applied Mathematics and Computation, 258, 72-83 (2015) &middot; <a href="/1338.39047" class="nowrap">Zbl 1338.39047</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.amc.2015.01.111" class="nowrap">doi:10.1016/j.amc.2015.01.111</a></td> </tr><tr> <td>[31]</td> <td class="space">Sousa, J.; Oliveira, E., Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Applied Mathematics Letters, 81, 50-56 (2018) &middot; <a href="/1475.45020" class="nowrap">Zbl 1475.45020</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.aml.2018.01.016" class="nowrap">doi:10.1016/j.aml.2018.01.016</a></td> </tr><tr> <td>[32]</td> <td class="space">Guo, Y.; Shu, X. B.; Li, Y.; Xu, F., The existence and Hyers-Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order \(1&lt;\beta&lt;2\), Boundary Value Problems, 2019 (2019) &middot; <a href="/1524.34205" class="nowrap">Zbl 1524.34205</a>&nbsp;&middot; <a href="https://doi.org/10.1186/s13661-019-1172-6" class="nowrap">doi:10.1186/s13661-019-1172-6</a></td> </tr><tr> <td>[33]</td> <td class="space">Niu, P.; Shu, X. B.; Li, Y., The existence and Hyers-Ulam stability for second order random impulsive differential equations, Dynamic Systems &amp; Applications, 28, 3, 673-690 (2019)</td> </tr><tr> <td>[34]</td> <td class="space">Li, S.; Shu, L.; Shu, X.-B.; Xu, F., Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastics, 91, 6, 857-872 (2019) &middot; <a href="/1506.37074" class="nowrap">Zbl 1506.37074</a>&nbsp;&middot; <a href="https://doi.org/10.1080/17442508.2018.1551400" class="nowrap">doi:10.1080/17442508.2018.1551400</a></td> </tr><tr> <td>[35]</td> <td class="space">Ouncharoen, R.; Chasreechai, S.; Sitthiwirattham, T., Existence and stability analysis for fractional impulsive Caputo difference-sum equations with periodic boundary condition, Mathematics, 8, 5 (2020)</td> </tr><tr> <td>[36]</td> <td class="space">Liu, X.; Jia, M.; Ge, W., The method of lower and upper solutions for mixed fractional four-point boundary value problem with \(p\)-Laplacian operator, Applied Mathematics Letters, 65, 56-62 (2017) &middot; <a href="/1357.34018" class="nowrap">Zbl 1357.34018</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.aml.2016.10.001" class="nowrap">doi:10.1016/j.aml.2016.10.001</a></td> </tr><tr> <td>[37]</td> <td class="space">Bai, Z.; Sun, W., Existence and multiplicity of positive solutions for singular fractional boundary value problems, Computers &amp; Mathematics with Applications, 63, 9, 1369-1381 (2012) &middot; <a href="/1247.34006" class="nowrap">Zbl 1247.34006</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.camwa.2011.12.078" class="nowrap">doi:10.1016/j.camwa.2011.12.078</a></td> </tr><tr> <td>[38]</td> <td class="space">Yang, L.; Shen, C.; Xie, D., Multiple positive solutions for nonlinear boundary value problem of fractional order differential equation with the Riemann-Liouville derivative, Advances in Difference Equations, 2014 (2014) &middot; <a href="/1353.34011" class="nowrap">Zbl 1353.34011</a>&nbsp;&middot; <a href="https://doi.org/10.1186/1687-1847-2014-284" class="nowrap">doi:10.1186/1687-1847-2014-284</a></td> </tr><tr> <td>[39]</td> <td class="space">Dassios, I. K., A practical formula of solutions for a family of linear non-autonomous fractional nabla difference equations, Journal of Computational and Applied Mathematics, 339, 317-328 (2018) &middot; <a href="/1464.65284" class="nowrap">Zbl 1464.65284</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.cam.2017.09.030" class="nowrap">doi:10.1016/j.cam.2017.09.030</a></td> </tr><tr> <td>[40]</td> <td class="space">Dassios, I. K., Stability and robustness of singular systems of fractional nabla difference equations, Circuits, Systems, and Signal Processing, 36, 1, 49-64 (2017) &middot; <a href="/1368.93592" class="nowrap">Zbl 1368.93592</a>&nbsp;&middot; <a href="https://doi.org/10.1007/s00034-016-0291-x" class="nowrap">doi:10.1007/s00034-016-0291-x</a></td> </tr><tr> <td>[41]</td> <td class="space">Dassios, I.; Baleanu, D., On a singular system of fractional nabla difference equations with boundary conditions, Boundary Value Problems, 2013 (2013) &middot; <a href="/1296.39002" class="nowrap">Zbl 1296.39002</a>&nbsp;&middot; <a href="https://doi.org/10.1186/1687-2770-2013-148" class="nowrap">doi:10.1186/1687-2770-2013-148</a></td> </tr><tr> <td>[42]</td> <td class="space">Lizama, C., The Poisson distribution, abstract fractional difference equations, and stability, Proceedings of the American Mathematical Society, 145, 9, 3809-3827 (2017) &middot; <a href="/1368.39001" class="nowrap">Zbl 1368.39001</a>&nbsp;&middot; <a href="https://doi.org/10.1090/proc/12895" class="nowrap">doi:10.1090/proc/12895</a></td> </tr><tr> <td>[43]</td> <td class="space">Lizama, C.; Murillo-Arcila, M.; Leal, C., Lebesgue regularity for differential difference equations with fractional damping, Mathematical Methods in the Applied Sciences, 41, 7, 2535-2545 (2018) &middot; <a href="/1391.35405" class="nowrap">Zbl 1391.35405</a>&nbsp;&middot; <a href="https://doi.org/10.1002/mma.4757" class="nowrap">doi:10.1002/mma.4757</a></td> </tr><tr> <td>[44]</td> <td class="space">Lv, W., Existence and uniqueness of solutions for a discrete fractional mixed type sum-difference equation boundary value problem, Discrete Dynamics in Nature and Society, 2015 (2015) &middot; <a href="/1418.39016" class="nowrap">Zbl 1418.39016</a>&nbsp;&middot; <a href="https://doi.org/10.1155/2015/376261" class="nowrap">doi:10.1155/2015/376261</a></td> </tr><tr> <td>[45]</td> <td class="space">Ma, W.; Luo, X.; Zhu, Q., Practical exponential stability of stochastic age-dependent capital system with Lévy noise, Systems &amp; Control Letters, 144 (2020) &middot; <a href="/1454.91371" class="nowrap">Zbl 1454.91371</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.sysconle.2020.104759" class="nowrap">doi:10.1016/j.sysconle.2020.104759</a></td> </tr><tr> <td>[46]</td> <td class="space">Gao, L.; Cao, Z.; Zhang, M.; Zhu, Q., Input-to-state stability for hybrid delayed systems with admissible edge-dependent switching signals, Journal of the Franklin Institute, 357, 13, 8823-8850 (2020) &middot; <a href="/1448.93278" class="nowrap">Zbl 1448.93278</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.jfranklin.2020.06.008" class="nowrap">doi:10.1016/j.jfranklin.2020.06.008</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH&nbsp;Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> &copy; 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 &amp; 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 = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none">&nbsp;<span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

Pages: 1 2 3 4 5 6 7 8 9 10