{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T23:43:21Z","timestamp":1772495001787,"version":"3.50.1"},"reference-count":16,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,11,27]],"date-time":"2021-11-27T00:00:00Z","timestamp":1637971200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag\u2013Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.<\/jats:p>","DOI":"10.3390\/axioms10040322","type":"journal-article","created":{"date-parts":[[2021,11,29]],"date-time":"2021-11-29T01:49:58Z","timestamp":1638150598000},"page":"322","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1305-2411","authenticated-orcid":false,"given":"Ricardo","family":"Almeida","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0634-2370","authenticated-orcid":false,"given":"Ravi P.","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4922-641X","authenticated-orcid":false,"given":"Snezhana","family":"Hristova","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Informatics, University of Plovdiv \u201cPaisii Hilendarski\u201d, 4000 Plovdiv, Bulgaria"}]},{"given":"Donal","family":"O\u2019Regan","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, National University of Ireland, H91 TK33 Galway, Ireland"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3457","DOI":"10.1140\/epjst\/e2018-00021-7","article-title":"Generalized fractional derivatives generated by a class of local proportional derivatives","volume":"226","author":"Jarad","year":"2017","journal-title":"Eur. 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Hybrid Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2140029","DOI":"10.1142\/S0218348X21400296","article-title":"Mittag\u2013Leffler stability analysis of tempered fractional neural networks with short memory and variable-order","volume":"8","author":"Gu","year":"2021","journal-title":"Fractals"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/322\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:36:40Z","timestamp":1760168200000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/322"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,27]]},"references-count":16,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["axioms10040322"],"URL":"https:\/\/doi.org\/10.3390\/axioms10040322","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,27]]}}}