{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:07:23Z","timestamp":1760231243277,"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>We established several novel inequalities of Gronwall\u2013Pachpatte type on time scales. Our results can be used as handy tools to study the qualitative and quantitative properties of the solutions of the initial boundary value problem for a partial delay dynamic equation. The Leibniz integral rule on time scales has been used in the technique of our proof. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.<\/jats:p>","DOI":"10.3390\/sym14091806","type":"journal-article","created":{"date-parts":[[2022,9,1]],"date-time":"2022-09-01T03:55:38Z","timestamp":1662004538000},"page":"1806","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some Generalizations of Novel (\u0394\u2207)\u0394\u2013Gronwall\u2013Pachpatte Dynamic Inequalities on Time Scales with 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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Institute of Space Science, Magurele, 077125 Bucharest, Romania"},{"name":"Department of Mathematics, Cankaya University, Ankara 06530, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,31]]},"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). Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser Boston, Inc.","DOI":"10.1007\/978-1-4612-0201-1"},{"key":"ref_3","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_4","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_5","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_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":"6","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":"2018","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. Differ. 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","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":"Discret. Dyn. Nat. Soc."},{"key":"ref_13","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_14","first-page":"6","article-title":"Pachpatte inequalities on time scales","volume":"6","author":"Bohner","year":"2005","journal-title":"JIPAM J. Inequal. Pure Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"163","DOI":"10.7153\/jmi-02-16","article-title":"Dynamic double integral inequalities in two independent variables on time scales","volume":"2","author":"Anderson","year":"2008","journal-title":"J. Math. Ineq."},{"key":"ref_16","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. Inequal. Appl."},{"key":"ref_17","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_18","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_19","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_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\/1806\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:20:59Z","timestamp":1760142059000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1806"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,31]]},"references-count":21,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091806"],"URL":"https:\/\/doi.org\/10.3390\/sym14091806","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,31]]}}}