{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T00:54:59Z","timestamp":1767833699705,"version":"3.49.0"},"reference-count":22,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,27]],"date-time":"2022-10-27T00:00:00Z","timestamp":1666828800000},"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>Throughout this article, generalizations of some Gr\u00f3nwall\u2013Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper can be utilized as a useful way to investigate the uniqueness, boundedness, continuousness, dependence and stability of nonlinear hyperbolic partial integro-differential equations. To highlight our research advantages, several implementations of these findings will be presented. Young\u2019s method, which depends on a Riemann method, will follow to prove the key results. Symmetry plays an essential role in determining the correct methods for solving dynamic inequalities.<\/jats:p>","DOI":"10.3390\/sym14112257","type":"journal-article","created":{"date-parts":[[2022,10,27]],"date-time":"2022-10-27T04:35:17Z","timestamp":1666845317000},"page":"2257","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Some Generalizations of Integral Inequalities in n Independent Variables and Their Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5435-1736","authenticated-orcid":false,"given":"Waleed","family":"Abuelela","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt"},{"name":"Department of Mathematics and Natural Sciences, Faculty of Engineering Technology and Science, Higher Colleges of Technology, Abu Dhabi P.O. Box 25035, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2822-4092","authenticated-orcid":false,"given":"Ahmed A.","family":"El-Deeb","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, 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, 077125 Bucharest, Romania"},{"name":"Department of Mathematics, Cankaya University, Ankara 06530, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,27]]},"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. J."},{"key":"ref_2","first-page":"1","article-title":"On some fundamental integral inequalities and their discrete analogues","volume":"2","author":"Pachpatte","year":"2001","journal-title":"J. Ineq. Pure Appl. Math."},{"key":"ref_3","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. Ineq. Appl."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","unstructured":"Beckenbacha, E.F., and Bellman, R. (1961). Inequalities, Springer.","DOI":"10.1007\/978-3-642-64971-4"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1016\/0022-247X(80)90293-0","article-title":"On some new integro-differential inequalities of the Wendroff type","volume":"73","author":"Pachpatte","year":"1980","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","first-page":"37","article-title":"On Wendroff type integral inequalities in n independent variables","volume":"7","author":"Bondge","year":"1979","journal-title":"Chin. J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"292","DOI":"10.2307\/1967124","article-title":"Note on the Derivatives with Respect to a Parameter of the Solutions of a System of Differential Equations","volume":"20","author":"Gronwall","year":"1919","journal-title":"Ann. Math."},{"key":"ref_9","unstructured":"Dragomir, S.S., Agarwal, P., Jleli, M., and Samet, B. (2018). 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