{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T01:23:58Z","timestamp":1767835438208,"version":"3.49.0"},"reference-count":20,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T00:00:00Z","timestamp":1716768000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhejiang Normal University Postdoctoral Research Fund","award":["ZC304022938"],"award-info":[{"award-number":["ZC304022938"]}]},{"name":"Zhejiang Normal University Postdoctoral Research Fund","award":["61976196"],"award-info":[{"award-number":["61976196"]}]},{"name":"Zhejiang Normal University Postdoctoral Research Fund","award":["LZ22F030003"],"award-info":[{"award-number":["LZ22F030003"]}]},{"name":"Natural Science Foundation of China","award":["ZC304022938"],"award-info":[{"award-number":["ZC304022938"]}]},{"name":"Natural Science Foundation of China","award":["61976196"],"award-info":[{"award-number":["61976196"]}]},{"name":"Natural Science Foundation of China","award":["LZ22F030003"],"award-info":[{"award-number":["LZ22F030003"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["ZC304022938"],"award-info":[{"award-number":["ZC304022938"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["61976196"],"award-info":[{"award-number":["61976196"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LZ22F030003"],"award-info":[{"award-number":["LZ22F030003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-differential equations, and these inequalities can be used in the symmetrical properties of functions. These inequalities also generalize some former famous inequalities in the literature. Two examples as applications will be provided to demonstrate the strength of our inequalities in estimating the boundedness and global existence of the solution to initial value problems for nonlinear integro-differential equations and differential equations which can be seen in graphs. This research work will ensure opening new opportunities for studying nonlinear dynamic inequalities on a time-scale structure of a varying nature.<\/jats:p>","DOI":"10.3390\/axioms13060356","type":"journal-article","created":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T09:33:31Z","timestamp":1716802411000},"page":"356","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["New Nonlinear Retarded Integral Inequalities and Their Applications to Nonlinear Retarded Integro-Differential Equations"],"prefix":"10.3390","volume":"13","author":[{"given":"Mahvish","family":"Samar","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinzhong","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdul","family":"Shakoor","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4773-1452","authenticated-orcid":false,"given":"Mawia","family":"Osman","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bainov, D., and Simeonov, P. (1992). Integral Inequalities and Applications, Kluwer Academic.","DOI":"10.1007\/978-94-015-8034-2"},{"key":"ref_2","unstructured":"Hardy, G.H., Littlewood, J.E., and Polya, G. (1952). Inequalities, Cambridge University Press. [2nd ed.]."},{"key":"ref_3","unstructured":"Pachpatte, B.G. (1998). Inequalities for Differential and Integral Equations, Academic Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Boichuk, A.A., and Samoilenko, A.M. (2016). Generalized Inverse Operators and Fredholm Boundary Value Problems, Walter de Gruyter. [2nd ed.].","DOI":"10.1515\/9783110378443"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1007\/s11253-023-02211-9","article-title":"Results on retarded nonlinear integral inequalities with mix powers and their applications in delay integro-differential","volume":"75","author":"Shakoor","year":"2023","journal-title":"Ukr. Math. 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