{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T11:24:17Z","timestamp":1782127457474,"version":"3.54.5"},"reference-count":22,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,12,15]],"date-time":"2023-12-15T00:00:00Z","timestamp":1702598400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201148"],"award-info":[{"award-number":["12201148"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["QKHJC-ZK[2022]YB069"],"award-info":[{"award-number":["QKHJC-ZK[2022]YB069"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Guizhou Provincial Science and Technology Projects","award":["12201148"],"award-info":[{"award-number":["12201148"]}]},{"name":"Guizhou Provincial Science and Technology Projects","award":["QKHJC-ZK[2022]YB069"],"award-info":[{"award-number":["QKHJC-ZK[2022]YB069"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper introduces a novel concept of the impulsive delayed Mittag\u2013Leffler-type vector function, an extension of the Mittag\u2013Leffler matrix function. It is essential to seek explicit formulas for the solutions to linear impulsive fractional differential delay equations. Based on explicit formulas of the solutions, the finite-time stability results of impulsive fractional differential delay equations are presented. Finally, we present four examples to illustrate the validity of our theoretical results.<\/jats:p>","DOI":"10.3390\/axioms12121129","type":"journal-article","created":{"date-parts":[[2023,12,15]],"date-time":"2023-12-15T11:26:52Z","timestamp":1702639612000},"page":"1129","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Finite-Time Stability of Impulsive Fractional Differential Equations with Pure Delays"],"prefix":"10.3390","volume":"12","author":[{"given":"Tingting","family":"Xie","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6926-8716","authenticated-orcid":false,"given":"Mengmeng","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2001). Application of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/3779"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kilbas, A., Srivastava, H., and Trujillo, J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier.","DOI":"10.3182\/20060719-3-PT-4902.00008"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sabatier, J., Agrawal, O., and Machado, J. (2007). Advances in Fractional Calculus, Springer.","DOI":"10.1007\/978-1-4020-6042-7"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/S0895-7177(00)00040-6","article-title":"Theoretical examination of the pulse vaccination policy in the SIR epidemic model","volume":"31","author":"Stone","year":"2000","journal-title":"Math. Comput. Model."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1080\/02331930500530401","article-title":"Nonlinear impulsive integro-differential equations of mixed type and optimal controls","volume":"55","author":"Wei","year":"2006","journal-title":"Optimization"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1016\/j.matcom.2008.02.007","article-title":"Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate","volume":"79","author":"Zhao","year":"2008","journal-title":"Math. Comput. Simul."},{"key":"ref_7","first-page":"1396","article-title":"Dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate","volume":"10","author":"Cao","year":"2020","journal-title":"J. Appl. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.neucom.2020.09.010","article-title":"Global exponential synchronization of interval neural networks with mixed delays via delayed impulsive control","volume":"420","author":"Wang","year":"2021","journal-title":"Neurocomputing"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.aml.2018.02.004","article-title":"Novel Mittag\u2013Leffler stability of linear fractional delay difference equations with impulse","volume":"82","author":"Wu","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_10","first-page":"254","article-title":"Exploring delayed Mittag\u2013Leffler type matrix functions to study finite time stability of fractional delay differential equations","volume":"324","author":"Li","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s12190-016-1072-1","article-title":"Stability of impulsive delay differential equations","volume":"56","author":"You","year":"2018","journal-title":"J. Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Chen, C., and Li, M. (2022). Existence and ulam type stability for impulsive fractional differential systems with pure delay. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6120742"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2334","DOI":"10.1002\/mma.8648","article-title":"Exact solutions and finite time stability for higher fractional-order differential equations with pure delay","volume":"46","author":"Liu","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1515\/ms-2023-0030","article-title":"Existence and Finite-Time Stability Results for Impulsive Caputo-Type Fractional Stochastic Differential Equations with Time Delays","volume":"73","author":"Tian","year":"2023","journal-title":"Math. Slovaca"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/j.mechrescom.2005.08.010","article-title":"Finite time stability analysis of PD\u03b1 fractional control of robotic time-delay systems","volume":"33","year":"2006","journal-title":"Mech. Res. Commun."},{"key":"ref_16","first-page":"475","article-title":"Finite-time stability analysis of fractional order time-delay system: Gronwall\u2019s approach","volume":"49","year":"2009","journal-title":"Mech. Res. Commun."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.neucom.2014.07.060","article-title":"Finite-time stability of fractional delayed neural networks","volume":"149","author":"Wu","year":"2015","journal-title":"Neurocomputing"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"6607","DOI":"10.1002\/mma.5765","article-title":"Finite time stability and relative controllability of Riemann Liouville fractional delay differential equations","volume":"42","author":"Li","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2823","DOI":"10.1007\/s11071-014-1628-2","article-title":"Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays","volume":"78","author":"Rakkiyappan","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4285","DOI":"10.1016\/j.apm.2015.11.012","article-title":"Finite-time stability of impulsive fractional-order systems with time-delay","volume":"40","author":"Hei","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.cnsns.2017.09.001","article-title":"Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion","volume":"57","author":"Wu","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.aml.2018.06.003","article-title":"Representation of solution of a Riemann\u2013Liouville fractional differential equation with pure delay","volume":"85","author":"Li","year":"2018","journal-title":"Appl. Math. Lett."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/12\/1129\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:39:46Z","timestamp":1760132386000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/12\/1129"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,15]]},"references-count":22,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["axioms12121129"],"URL":"https:\/\/doi.org\/10.3390\/axioms12121129","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,15]]}}}