{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,22]],"date-time":"2026-10-22T14:59:29Z","timestamp":1792681169708,"version":"4.5.0"},"reference-count":23,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,26]],"date-time":"2021-01-26T00:00:00Z","timestamp":1611619200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Some generalizations of the Gronwall\u2013Bellman (G\u2013B) inequality are presented in this paper in continuous form and on time scales. After S. Hilger introduced the time scales theory in 1988, over the years many mathematicians have studied new versions of this inequality according to new results; the purpose of this paper is to present some of them. Therefore, in the Introduction, some generalizations of G\u2013B inequality in continuous forms, linear and nonlinear are presented. In the second section, some important and interesting results on time scales theory are given. In the third and main part of our paper, G\u2013B inequalities on time scales and their possible connection with G\u2013B inequalities presented in the introduction are investigated. In particular, in the third section of this work, more attention is given to G\u2013B type inequalities on time scales discussed in the last four years.<\/jats:p>","DOI":"10.3390\/sym13020198","type":"journal-article","created":{"date-parts":[[2021,1,26]],"date-time":"2021-01-26T12:03:57Z","timestamp":1611662637000},"page":"198","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Some Gronwall\u2013Bellman Inequalities on Time Scales and Their Continuous Forms: A Survey"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1361-2175","authenticated-orcid":false,"given":"Francesca","family":"Barich","sequence":"first","affiliation":[{"name":"Alma Mater Studiorum, Department of Mathematics, University of Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"292","DOI":"10.2307\/1967124","article-title":"Note on the Derivatives with Respect to a Parameter of the Solutions of a System of Differential Equations","volume":"20","author":"Gronwall","year":"1919","journal-title":"Ann. Math."},{"key":"ref_2","unstructured":"Bellman, R.E. (2008). Stability, Boundedness, and Asymptotic Behavior of Solutions of Linear Systems. Stability Theory of Differential Equations, Dover Publications."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bellman, R.E., and Kenneth, L.C. (1959). Differential-difference equations with constant coefficients. Asymptotic Behavior of Solutions of Differential-Difference Equations, American Mathematical Society. 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Inequal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/198\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:15:43Z","timestamp":1760159743000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/198"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,26]]},"references-count":23,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["sym13020198"],"URL":"https:\/\/doi.org\/10.3390\/sym13020198","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,26]]}}}