{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:08:30Z","timestamp":1760231310121,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,11]],"date-time":"2022-09-11T00:00:00Z","timestamp":1662854400000},"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>In the present paper, some new generalizations of dynamic inequalities of Gronwall\u2013Bellman\u2013Pachpatte-type on time scales are established. Some integral and discrete Gronwall\u2013Bellman\u2013Pachpatte-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Leibniz integral rule on time scales. To highlight our research advantages, several implementations of these findings are presented. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.<\/jats:p>","DOI":"10.3390\/sym14091902","type":"journal-article","created":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T22:37:28Z","timestamp":1663108648000},"page":"1902","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Some Dynamic (\u0394\u0394)\u2207- Gronwall\u2013Bellman\u2013Pachpatte-Type Inequalities on Time Scales and Its Applications"],"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, Cairo 11884, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8846-0487","authenticated-orcid":false,"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":[[2022,9,11]]},"reference":[{"key":"ref_1","unstructured":"Hilger, S. (1988). Ein ma\u00dfkettenkalk\u00fcl mit Anwendung auf Zentrumsmannigfaltigkeiten. [Ph.D. Thesis, Universitat Wurzburg]."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2001). 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