{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:07:18Z","timestamp":1760231238074,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T00:00:00Z","timestamp":1661904000000},"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 this article, with the help of Leibniz integral rule on time scales, we prove some new dynamic inequalities of Gronwall\u2013Bellman\u2013Pachpatte-type on time scales. These inequalities can be used as handy tools to study the qualitative and quantitative properties of solutions of the initial boundary value problem for partial delay dynamic equation.<\/jats:p>","DOI":"10.3390\/sym14091804","type":"journal-article","created":{"date-parts":[[2022,9,1]],"date-time":"2022-09-01T03:55:38Z","timestamp":1662004538000},"page":"1804","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["\u0394\u2013Gronwall\u2013Bellman\u2013Pachpatte Dynamic Inequalities and Their Applications on Time Scales"],"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\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Institute of Space Science, 077125 Magurele, Romania"},{"name":"Department of Mathematics, Cankaya University, Ankara 06530, Turkey"}]},{"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":[[2022,8,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1186\/s13660-015-0837-7","article-title":"On some delay nonlinear integral inequalities in two independent variables","volume":"2015","author":"Boudeliou","year":"2015","journal-title":"J. Inequalities Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4737","DOI":"10.1002\/mma.4927","article-title":"On some dynamic inequalities of Steffensen type on time scales","volume":"41","author":"Abdeldaim","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Agarwal, R., O\u2019Regan, D., and Saker, S. (2014). Dynamic Inequalities on Time Scales, Springer.","DOI":"10.1007\/978-3-319-11002-8"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Akdemir, A.O., Butt, S.I., Nadeem, M., and Ragusa, M.A. (2021). New general variants of chebyshev type inequalities via generalized fractional integral operators. Mathematics, 9.","DOI":"10.3390\/math9020122"},{"key":"ref_5","first-page":"23","article-title":"Pachpatte inequalities on time scales","volume":"6","author":"Bohner","year":"2005","journal-title":"JIPAM J. Inequal. Pure Appl. Math."},{"key":"ref_6","first-page":"1","article-title":"The Gr\u00fcss inequality on time scales","volume":"3","author":"Bohner","year":"2007","journal-title":"Commun. Math. Anal."},{"key":"ref_7","first-page":"8","article-title":"Ostrowski inequalities on time scales","volume":"9","author":"Bohner","year":"2008","journal-title":"JIPAM J. Inequal. Pure Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"287947","DOI":"10.1155\/2008\/287947","article-title":"Hermite-Hadamard inequality on time scales","volume":"24","author":"Dinu","year":"2008","journal-title":"J. Inequal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.21608\/JOMES.2018.9457","article-title":"Some Gronwall-bellman type inequalities on time scales for Volterra-Fredholm dynamic integral equations","volume":"26","year":"2018","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_10","first-page":"130","article-title":"Some dynamic inequalities on time scales and their applications","volume":"19","author":"Xu","year":"2019","journal-title":"Adv. Difference Equ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1186\/s13662-021-03282-3","article-title":"On some new double dynamic inequalities associated with leibniz integral rule on time scales","volume":"2021","author":"Rashid","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1186\/s13662-019-2268-0","article-title":"On some generalizations of dynamic Opial-type inequalities on time scales","volume":"2019","author":"Kh","year":"2019","journal-title":"Adv. Differ. Eq."},{"key":"ref_13","first-page":"5841985","article-title":"Some nonlinear delay Volterra-Fredholm type dynamic integral inequalities on time scales","volume":"8","author":"Tian","year":"2018","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"ref_14","unstructured":"Hilger, S. (1988). Ein Ma\u00dfkettenkalk\u00fcl mit Anwendung auf Zentrumsmannigfaltigkeiten. [Ph.D. Thesis, Universitat Wurzburg]."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2001). Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser Boston, Inc.","DOI":"10.1007\/978-1-4612-0201-1"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2003). Advances in Dynamic Equations on Time Scales, Birkhauser.","DOI":"10.1007\/978-0-8176-8230-9"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1016\/j.aml.2008.08.022","article-title":"Generalized retarded integral inequalities","volume":"22","author":"Ferreira","year":"2009","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.na.2007.05.027","article-title":"Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities","volume":"69","author":"Ma","year":"2008","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_19","first-page":"239","article-title":"A generalization of retarded integral inequalities in two independent variables and their applications","volume":"221","author":"Tian","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_20","first-page":"1260","article-title":"On retarded integral inequalities in two independent variables and their applications","volume":"182","author":"Xu","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.jmaa.2004.07.020","article-title":"On retarded integral inequalities and their applications","volume":"301","author":"Sun","year":"2005","journal-title":"J. Math. Anal. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1804\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:20:43Z","timestamp":1760142043000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1804"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,31]]},"references-count":21,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091804"],"URL":"https:\/\/doi.org\/10.3390\/sym14091804","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,31]]}}}