{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:07:13Z","timestamp":1760231233497,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,7]],"date-time":"2022-09-07T00:00:00Z","timestamp":1662508800000},"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 prove some new dynamic inequalities of the Gronwall\u2013Bellman\u2013Pachpatte type on time scales. Our results can be used in analyses as useful tools for some types of partial dynamic equations on time scales and in their applications in environmental phenomena and physical and engineering sciences that are described by partial differential equations.<\/jats:p>","DOI":"10.3390\/sym14091867","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T09:51:09Z","timestamp":1662630669000},"page":"1867","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["(\u0394\u2207)\u2207-Pachpatte Dynamic Inequalities Associated with Leibniz Integral Rule 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 11884, Cairo, 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":[[2022,9,7]]},"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":"8","article-title":"Ostrowski inequalities on time scales","volume":"9","author":"Bohner","year":"2008","journal-title":"J. Inequalities 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":"2008","author":"Dinu","year":"2008","journal-title":"J. Inequalities 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","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. Equ."},{"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":"1","article-title":"Pachpatte inequalities on time scales","volume":"6","author":"Bohner","year":"2005","journal-title":"J. Inequal. Pure Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zakarya, M., Altanji, M., AlNemer, G., Abd El-Hamid, H.A., Cesarano, C., and Rezk, H.M. (2021). Fractional reverse coposn\u2019s inequalities via conformable calculus on time scales. Symmetry, 13.","DOI":"10.3390\/sym13040542"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Rezk, H.M., AlNemer, G., Saied, A.I., Bazighifan, O., and Zakarya, M. (2022). Some New Generalizations of Reverse Hilbert-Type Inequalities on Time Scales. Symmetry, 14.","DOI":"10.3390\/sym14040750"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"AlNemer, G., Zakarya, M., Abd El-Hamid, H.A., Agarwal, P., and Rezk, H.M. (2020). Some dynamic Hilbert-type inequalities on time scales. Symmetry, 12.","DOI":"10.3390\/sym12091410"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"El-Deeb, A.A., Makharesh, S.D., Askar, S.S., and Awrejcewicz, J. (2022). A variety of Nabla Hardy\u2019s type inequality on time scales. Mathematics, 10.","DOI":"10.3390\/math10050722"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"El-Deeb, A.A., and Baleanu, D. (2022). Some new dynamic Gronwall-Bellman-Pachpatte type inequalities with delay on time scales and certain applications. J. Inequalities Appl., 45.","DOI":"10.1186\/s13660-022-02778-0"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"11382","DOI":"10.3934\/math.2022635","article-title":"A variety of dynamic \u03b1-conformable Steffensen-type inequality on a time scale measure space","volume":"7","author":"Moaaz","year":"2022","journal-title":"AIMS Math."},{"key":"ref_21","first-page":"1909","article-title":"Generalization of Mitrinovi\u0107-Pe\u010dari\u0107 inequalities on time scales","volume":"51","author":"Akin","year":"2021","journal-title":"Rocky Mt. J. Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"El-Deeb, A.A., Makharesh, S.D., Nwaeze, E.R., Iyiola, O.S., and Baleanu, D. (2021). On nabla conformable fractional Hardy-type inequalities on arbitrary time scales. J. Inequalities Appl., 192.","DOI":"10.1186\/s13660-021-02723-7"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"El-Deeb, A.A., and Awrejcewicz, J. (2021). Novel Fractional Dynamic Hardy\u2013Hilbert-Type Inequalities on Time Scales with Applications. Mathematics, 9.","DOI":"10.3390\/math9222964"},{"key":"ref_24","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. J."},{"key":"ref_25","first-page":"1","article-title":"On some fundamental integral inequalities and their discrete analogues","volume":"2","author":"Pachpatte","year":"2001","journal-title":"J. Inequalities Pure Appl. Math."},{"key":"ref_26","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_27","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. Inequalities"},{"key":"ref_28","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_29","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_30","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_31","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_32","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. 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