{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:29:00Z","timestamp":1760236140995,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,23]],"date-time":"2021-10-23T00:00:00Z","timestamp":1634947200000},"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>The aim of this paper is to deduce the asymptotic and Hille-type criteria of the dynamic equations of third order on time scales. Some of the presented results concern the sufficient condition for the oscillation of all solutions of third-order dynamical equations. Additionally, compared with the related contributions reported in the literature, the Hille-type oscillation criterion which is derived is superior for dynamic equations of third order. The symmetry plays a positive and influential role in determining the appropriate type of study for the qualitative behavior of solutions to dynamic equations. Some examples of Euler-type equations are included to demonstrate the finding.<\/jats:p>","DOI":"10.3390\/sym13112007","type":"journal-article","created":{"date-parts":[[2021,10,25]],"date-time":"2021-10-25T21:42:05Z","timestamp":1635198125000},"page":"2007","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Asymptotics and Hille-Type Results for Dynamic Equations of Third Order with Deviating Arguments"],"prefix":"10.3390","volume":"13","author":[{"given":"Taher S.","family":"Hassan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A. Othman","family":"Almatroud","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammed M.","family":"Al-Sawalha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2434-1929","authenticated-orcid":false,"given":"Ismoil","family":"Odinaev","sequence":"additional","affiliation":[{"name":"Department of Automated Electrical Systems, Ural Power Engineering Institute, Ural Federal University, 620002 Yekaterinburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0377-0427(01)00432-0","article-title":"Dynamic equations on time scales: A survey","volume":"141","author":"Agarwal","year":"2002","journal-title":"J. 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Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1090\/S0002-9947-1948-0027925-7","article-title":"Non-oscillation theorems","volume":"64","author":"Hille","year":"1948","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Wong, J.S. (1971). Second order oscillation with retarded arguments. Ordinary Differential Equations, Academic Press.","DOI":"10.1016\/B978-0-12-743650-0.50054-X"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"49","DOI":"10.4153\/CMB-1973-011-1","article-title":"Oscillation criteria for second order nonlinear delay equations","volume":"16","author":"Erbe","year":"1973","journal-title":"Canad. Math. Bull."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.jmaa.2006.06.033","article-title":"Hille and Nehari type criteria for third-order dynamic equations","volume":"329","author":"Erbe","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1563","DOI":"10.1080\/10236198.2013.766729","article-title":"Hille and Nehari type criteria for third order delay dynamic equations","volume":"19","author":"Agarwal","year":"2013","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1016\/j.mcm.2008.12.011","article-title":"Oscillation of third-order nonlinear delay dynamic equations on time scales","volume":"49","author":"Hassan","year":"2009","journal-title":"Math. Comput. Model."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.aml.2018.11.021","article-title":"Oscillation of second-order nonlinear noncanonical differential equations with deviating argument","volume":"91","author":"Baculikova","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"499","DOI":"10.4134\/CKMS.2011.26.3.499","article-title":"Oscillation behavior of solutions of third-order nonlinear delay dynamic equations on time scales","volume":"26","author":"Han","year":"2011","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_14","first-page":"639","article-title":"Oscillation results for third-order nonlinear delay dynamic equations on time scales","volume":"34","author":"Li","year":"2011","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/s12190-010-0406-7","article-title":"Asymptotic behavior of solutions for third-order half-linear delay dynamic equations on time scales","volume":"36","author":"Li","year":"2011","journal-title":"J. Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"715981","DOI":"10.1155\/2012\/715981","article-title":"On oscillations of solutions of third-order dynamic equation","volume":"2012","author":"Hovhannisy","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_17","first-page":"1","article-title":"Oscillation theorems for certain third-order nonlinear delay dynamic equations on time scales","volume":"75","author":"Sun","year":"2011","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1186\/1029-242X-2013-47","article-title":"Behavior of solutions of a third-order dynamic equation on time scales","volume":"2013","author":"Senel","year":"2013","journal-title":"Senel J. Inequal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2354","DOI":"10.1016\/j.cam.2011.11.021","article-title":"Asymptotic properties of solutions of certain third-order dynamic equations","volume":"236","author":"Wang","year":"2012","journal-title":"J. Comput. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1016\/j.cam.2008.08.017","article-title":"Asymptotic behavior of solutions of third-order dynamic equations on time scales","volume":"255","author":"Yu","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1186\/s13660-019-1967-0","article-title":"Oscillatory and asymptotic properties of third-order quasilinear delay differential equations","volume":"2019","author":"Chatzarakis","year":"2019","journal-title":"J. Inequalit. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/s12591-010-0005-y","article-title":"Oscillation of third-order nonlinear functional dynamic equations on time scales","volume":"18","author":"Erbe","year":"2010","journal-title":"Differ. Eq. Dynam. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Moaaz, O., El-Nabulsi, R.A., Muhsin, W., and Bazighifan, O. (2020). Improved Oscillation Criteria for 2nd-Order Neutral Differential Equations with Distributed Deviating Arguments. Mathematics, 8.","DOI":"10.3390\/math8020197"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Hassan, T.S., Sun, Y., and Abdel Menaem, A. (2020). Improved oscillation results for functional nonlinear dynamic equations of second order. Mathematics, 8.","DOI":"10.3390\/math8111897"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2597","DOI":"10.1007\/s11425-011-4304-8","article-title":"Oscillation of third-order functional dynamic equations on time scales","volume":"54","author":"Saker","year":"2011","journal-title":"Sci. China Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1016\/j.indag.2017.10.006","article-title":"Fite-Hille-Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments","volume":"29","author":"Bohner","year":"2018","journal-title":"Indag. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Trean\u0163\u0103, S. (2020). Gradient Structures Associated with a Polynomial Differential Equation. Mathematics, 8.","DOI":"10.3390\/math8040535"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s00245-014-9244-6","article-title":"Filtering for Non-Markovian SDEs Involving Nonlinear SPDEs and Backward Parabolic Equations","volume":"70","author":"Ijacu","year":"2014","journal-title":"Appl. Math. Optim."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1186\/s13662-017-1164-8","article-title":"Oscillation criteria for third-order functional half-linear dynamic equations","volume":"2017","author":"Agarwal","year":"2017","journal-title":"Adv. Difference Equ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"369","DOI":"10.2140\/pjm.1976.64.369","article-title":"Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations","volume":"64","author":"Erbe","year":"1976","journal-title":"Pac. J. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2007\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:21:49Z","timestamp":1760167309000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2007"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,23]]},"references-count":30,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112007"],"URL":"https:\/\/doi.org\/10.3390\/sym13112007","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,10,23]]}}}