{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:06:23Z","timestamp":1760231183762,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T00:00:00Z","timestamp":1662076800000},"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>As a new usage of Leibniz integral rule on time scales, we proved some new extensions of dynamic Gronwall\u2013Pachpatte-type inequalities on time scales. Our results extend some existing results in the literature. Some integral and discrete inequalities are obtained as special cases of the main results. The inequalities proved here can be used in the analysis as handy tools to study the stability, boundedness, existence, uniqueness and oscillation behavior for some kinds of partial dynamic equations on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.<\/jats:p>","DOI":"10.3390\/sym14091823","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T09:51:09Z","timestamp":1662630669000},"page":"1823","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some Generalizations of (\u2207\u2207)\u0394\u2013Gronwall\u2013Bellman\u2013Pachpatte Dynamic Inequalities on Time Scales with Application"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2822-4092","authenticated-orcid":false,"given":"Ahmed A.","family":"El-Deeb","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"}]},{"given":"Alaa A.","family":"El-Bary","sequence":"additional","affiliation":[{"name":"Basic and Applied Science Institute, Arab Academy for Science, Technology and Maritime Transport, P.O. Box 1029, Alexandria 21532, Egypt"},{"name":"National Committee for Mathematics, Academy of Scientific Research and Technology, Cairo 11516, Egypt"},{"name":"Council of Future Studies and Risk Management, Academy of Scientific Research and Technology, Cairo 11516, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0387-921X","authenticated-orcid":false,"given":"Jan","family":"Awrejcewicz","sequence":"additional","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1\/15 Stefanowski St., 90-924 Lodz, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1215\/S0012-7094-43-01059-2","article-title":"The stability of solutions of linear differential equations","volume":"10","author":"Bellman","year":"1943","journal-title":"Duke Math. 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