{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:56:21Z","timestamp":1760237781188,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T00:00:00Z","timestamp":1658275200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Institutional Funding Program","award":["IFP-A-2022-2-1-09"],"award-info":[{"award-number":["IFP-A-2022-2-1-09"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>An implicit time\u2013fractal\u2013fractional differential equation involving the Atangana\u2019s fractal\u2013fractional derivative in the sense of Caputo with the Mittag\u2013Leffler law type kernel is studied. Using the Banach fixed point theorem, the well-posedness of the solution is proved. We show that the solution exhibits an exponential growth bound, and, consequently, the long-time (asymptotic) property of the solution. We also give examples to illustrate our problem.<\/jats:p>","DOI":"10.3390\/axioms11070348","type":"journal-article","created":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T11:22:24Z","timestamp":1658316144000},"page":"348","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Implicit Time\u2013Fractal\u2013Fractional Differential Equation"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5163-229X","authenticated-orcid":false,"given":"McSylvester Ejighikeme","family":"Omaba","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Hafr Al Batin, P.O. Box 1803, Hafr Al Batin 31991, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6668-1107","authenticated-orcid":false,"given":"Soh Edwin","family":"Mukiawa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Hafr Al Batin, P.O. Box 1803, Hafr Al Batin 31991, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1375-1474","authenticated-orcid":false,"given":"Eze R.","family":"Nwaeze","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36101, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.chaos.2017.04.027","article-title":"Fractal\u2013fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system","volume":"102","author":"Atangana","year":"2017","journal-title":"Chaos Solit. Fract."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1016\/j.aej.2020.01.005","article-title":"Analysis of Frac-tal\u2013fractional differential equations","volume":"59","author":"Atangana","year":"2020","journal-title":"Alex. Eng. 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