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The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. <a href="https://mathscinet.ams.org/mathscinet/search/journal/profile?groupId=3285" target="new">Subscribers to MathSciNet</a> may click through for more detailed information. </div> </p> </section> </div> </header> <main> <section class="row"> <nav class="col-md-3">  <nav class="journals-list-group"> <script type="text/javascript" src="/journals/javascript/proc_nav.js"></script> <a href="/journals/journals"><i class="fa-sharp fa-solid fa-house"></i> Journals Home</a> <a href="/epubsearch">eContent Search</a> <a href="/aboutproc" class="">About Proceedings of the American Mathematical Society</a> <a href="/procedit" class="">Editorial Board</a> <a href="/proc/procsubmit.html" class="">Author and Submission Information</a> <a href="/policies/policies" class="" target="_blank">Journal Policies</a> <a href="/subscriberinfo" class="" target="_blank">Subscription Information</a> <a href="/jemail" class="" target="_blank"> Journal Email Notifications</a> <hr> <a href="/pairing/pair_my_device.html"><i class="fa fa-globe" aria-hidden="true"></i> Remote Access</a> </nav> </nav> <article class="col-md-9"> <section>   <div class="row">  <ul class="nav nav-tabs"> <li role="presentation" class=""> <script type="text/javascript" src="/journals/javascript/proc_nav.js"></script> <a href="javascript: goto_most_recent_proc()" aria-controls="currissue" role="tab"> Current volume </a> </li> <li role="presentation" class="">  <a href="/cgi-bin/mstrack/accepted_papers/proc?active=press" aria-controls="press" role="tab">Articles in press </a> </li> <li role="presentation" class=""> <a href="/journals/proc/0000-000-00?active=recent"> Recently published</a> </li> <li role="presentation" class=""> <a href="/journals/proc/all_issues.html?active=allissues"> All issues : 1950 – Present </a> </li> </ul> <script src="/js/abstracts/ajaxcall.js"></script> </div> <p> </p> <style> #navCell { display: none;} </style> <script src="/journals/javascript/references.js" type="text/javascript"></script> <script async src="/js/mj.js"></script>  <script src="https://platform-api.sharethis.com/js/sharethis.js#property=598c93d1e2587a001253a011&product=unknown" async="async"></script>  <div class="sharethis-inline-share-buttons"></div>   <article class="col-md-12"> <header> <h2>The Poisson distribution, abstract fractional difference equations, and stability<br> <small>HTML articles powered by AMS MathViewer <a href="/AMSMathViewer"><i class="fa fa-info-circle" aria-label="about MathViewer"></i></a></small></h2> <dl> <dd>by <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?mrauthid=114975">Carlos Lizama</a>  </dd> <dd>Proc. Amer. Math. Soc. <strong>145</strong> (2017), 3809-3827 </dd> <dd> DOI: https://doi.org/10.1090/proc/12895 </dd> <dd> Published electronically: May 24, 2017 </dd> <dd class="mathviewerPDFLink"> <a href="S0002-9939-2017-12895-0.pdf"><small>PDF</small></a> | <a href="https://www.copyright.com/openurl.do?issn=1088-6826&WT.mc.it=American%20Mathematical%20Society" target="_blank">Request permission</a> </dd> </dl> </header> <section id="Abstract"> <h2>Abstract:</h2> We study the initial value problem \begin{equation*} \tag {$*$} \left \{\begin {array}{rll} _C\Delta ^{\alpha } u(n) &= Au(n+1), \quad n \in \mathbb {N}_0; \\ u(0) &= u_0 \in X, \end{array}\right . \end{equation*} when $A$ is a closed linear operator with domain $D(A)$ defined on a Banach space $X$. We introduce a method based on the Poisson distribution to show existence and qualitative properties of solutions for the problem $(*)$, using operator-theoretical conditions on $A$. We show how several properties for fractional differences, including their own definition, are connected with the continuous case by means of sampling using the Poisson distribution. We prove necessary conditions for stability of solutions, that are only based on the spectral properties of the operator $A$ in the case of Hilbert spaces. </section> <section id="references"> <details> <summary>References</summary> <ul><li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Abdeljawad%2C%20Thabet">Thabet Abdeljawad</a>, <em>On Riemann and Caputo fractional differences</em>, Comput. Math. 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I, II</em>, Cambridge University Press, New York, 1959. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0107776">MR <strong>0107776</strong></a></li> </ul> </details> </section> <section id="similar"> <details> <summary>Similar Articles</summary> <ul> <li> Retrieve articles in <em>Proceedings of the American Mathematical Society</em> with MSC (2010): <a href="/joursearch/servlet/DoSearch?f1=msc&v1=39A13&jrnl=one&onejrnl=proc">39A13</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=39A14&jrnl=one&onejrnl=proc">39A14</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=39A06&jrnl=one&onejrnl=proc">39A06</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=39A60&jrnl=one&onejrnl=proc">39A60</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=47D06&jrnl=one&onejrnl=proc">47D06</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=47D99&jrnl=one&onejrnl=proc">47D99</a> </li> <li> Retrieve articles in all journals with MSC (2010): <a href="/joursearch/servlet/DoSearch?f1=msc&v1=39A13">39A13</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=39A14">39A14</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=39A06">39A06</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=39A60">39A60</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=47D06">47D06</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=47D99">47D99</a> </ul> </details> </section> <section id="additionalinformation"> <details open> <summary>Bibliographic Information</summary> <ul> <li><strong>Carlos Lizama</strong></li><li>Affiliation: Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile</li> <li>MR Author ID: <a href='https://mathscinet.ams.org/mathscinet/search/authors.html?mrauthid=114975'>114975</a></li><li>Email: carlos.lizama@usach.cl</li> <li>Received by editor(s): June 30, 2014</li> <li>Received by editor(s) in revised form: June 27, 2015, July 8, 2015, and July 13, 2015</li> <li>Published electronically: May 24, 2017</li> <li>Additional Notes: The author was partially supported by FONDECYT grant number 1140258</li> <li>Communicated by: Mark M. Meerschaert</li> <li>© Copyright 2017 American Mathematical Society</li> <li>Journal: Proc. Amer. Math. Soc. <strong>145</strong> (2017), 3809-3827 </li> <li>MSC (2010): Primary 39A13, 39A14, 39A06, 39A60, 47D06, 47D99 <li>DOI: https://doi.org/10.1090/proc/12895</li> <li>MathSciNet review: <a href = "https://mathscinet.ams.org/mathscinet-getitem?mr=3665035">3665035</a></li> </ul> </details> </section> </article> </section> </article> </section> </main> </div>   <hr class="mx-auto"> <footer id="pagefooter" class="text-center"> <div class="container text-center"> <div class="row">  <div class="col-12 mb-1"> <p> American Mathematical Society · <a href="https://goo.gl/maps/ZvJdkQEC79p" title="Google map 201 Charles Street Providence, Rhode Island" aria-label="Google map to AMS headquarters"> 201 Charles Street Providence, Rhode Island 02904-2213 </a> · <a href="/reach" aria-label="Contact Us">Contact Us</a> </p> </div>  <script src="https://kit.fontawesome.com/2ad6d7fbd6.js" crossorigin="anonymous" defer></script>  <nav class="col-12 mb-1" aria-label="Social media links"> <ul class="list-inline"> <li class="list-inline-item"> <a href="https://www.facebook.com/amermathsoc" class="link-dark" title="Follow us on Facebook" aria-label="View the AMS on Facebook"> <i class="fa-brands fa-facebook-f" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://bsky.app/profile/amermathsoc.bsky.social" class="link-dark" title="Follow us on Bluesky" aria-label="View the AMS on Bluesky"> <i class="fa-brands fa-bluesky"></i></a></li> <li class="list-inline-item"> <a href="https://twitter.com/amermathsoc" class="link-dark" title="Follow us on X" aria-label="View the AMS on X"> <i class="fa-brands fa-x-twitter" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.linkedin.com/company/51001" class="link-dark" title="Follow us on LinkedIn" aria-label="View the AMS on LinkedIn"> <i class="fa-brands fa-linkedin-in" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.instagram.com/amermathsoc" class="link-dark" title="Follow us on Instagram" aria-label="View the AMS on Instagram"> <i class="fa-brands fa-instagram" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.youtube.com/amermathsoc" class="link-dark" title="Follow us on YouTube" aria-label="View the AMS on YouTube"> <i class="fa-brands fa-youtube" aria-hidden="true"></i></a></li> </ul> </nav>  <div class="col-12"> <p><small> © <a href="/copyright" aria-label="Copyright policy">Copyright</a> <span id="copyYear">2024</span> American Mathematical Society · <a href="/privacy" aria-label="Privacy statement">View Our Privacy Statement</a> · <a href="/tou" aria-label="Terms of use">Terms of Use</a> · <a href="/accessibility" aria-label="Accessibility policy">Accessibility and AMS Online Content</a> </small></p> <p><small> AMS, American Mathematical Society, the tri-colored AMS logo, and Advancing research, Creating connections, are trademarks and service marks of the American Mathematical Society and registered in the U.S. Patent and Trademark Office. </small></p> </div> </div> </div> <script> // Dynamically update the copyright year document.getElementById('copyYear').textContent = new Date().getFullYear(); </script> </footer>