{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:16:34Z","timestamp":1760235394990,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,13]],"date-time":"2021-08-13T00:00:00Z","timestamp":1628812800000},"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>Symmetry plays an essential role in determining the correct methods for the oscillatory properties of solutions to differential equations. This paper examines some new oscillation criteria for unbounded solutions of third-order neutral differential equations of the form (r2(\u03c2)((r1(\u03c2)(z\u2032(\u03c2))\u03b21)\u2032)\u03b22)\u2032 + \u2211i=1nqi(\u03c2)x\u03b23(\u03d5i(\u03c2))=0. New oscillation results are established by using generalized Riccati substitution, an integral average technique in the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems.<\/jats:p>","DOI":"10.3390\/sym13081485","type":"journal-article","created":{"date-parts":[[2021,8,13]],"date-time":"2021-08-13T05:34:46Z","timestamp":1628832886000},"page":"1485","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Symmetry and Its Role in Oscillation of Solutions of Third-Order Differential Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2756-0089","authenticated-orcid":false,"given":"M. Sathish","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Paavai Engineering College (Autonomous), Namakkal 637 018, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-9608","authenticated-orcid":false,"given":"Omar","family":"Bazighifan","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy"},{"name":"Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Khalifa","family":"Al-Shaqsi","sequence":"additional","affiliation":[{"name":"Department of Information Technology, Nizwa College of Technology, University of Technology and Applied Science, P.O. Box 75, Nizwa 612, Oman"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9570-3010","authenticated-orcid":false,"given":"Fongchan","family":"Wannalookkhee","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hale, J.K. (1977). Theory of Functional Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-9892-2"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.mcm.2010.02.011","article-title":"Oscillation of third-order neutral differential equations","volume":"52","author":"Dzurina","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_3","first-page":"1","article-title":"Oscillation of solutions to third-order half-linear neutral differential equations","volume":"29","author":"Dzurina","year":"2012","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"569201","DOI":"10.1155\/2012\/569201","article-title":"Oscillation of Third-Order Neutral Delay Differential Equations","volume":"2012","author":"Li","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5691758","DOI":"10.1155\/2019\/5691758","article-title":"Oscillation criteria for third-order Emden\u2013Fowler differential equations with unbounded neutral coefficients","volume":"2019","author":"Chatzarakis","year":"2019","journal-title":"Complexity"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-017-1187-1","article-title":"Oscillatory and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments","volume":"2017","author":"Tunc","year":"2017","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Bazighifan, O., and Ragusa, M.A. (2021). Nonlinear Neutral Delay Differential Equations of Fourth-Order: Oscillation of Solutions. Entropy, 23.","DOI":"10.3390\/e23020129"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1186\/s13662-015-0604-6","article-title":"Oscillation and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments","volume":"2015","author":"Tian","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1007\/BF01324723","article-title":"Oscillation theorems for linear differential equations of second order","volume":"53","author":"Philos","year":"1989","journal-title":"Arch. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"839","DOI":"10.7494\/OpMath.2017.37.6.839","article-title":"Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation","volume":"37","author":"Graef","year":"2017","journal-title":"Opusc. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1186\/s13662-016-0902-7","article-title":"Oscillatory behavior of third-order nonlinear neutral delay differential equations","volume":"2016","author":"Jiang","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bazighifan, O., Mofarreh, F., and Nonlaopon, K. (2021). On the Qualitative Behavior of Third-Order Differential Equations with a Neutral Term. Symmetry, 13.","DOI":"10.3390\/sym13071287"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"106293","DOI":"10.1016\/j.aml.2020.106293","article-title":"On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations","volume":"105","author":"Li","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Chatzarakis, G.E., Dzurina, J., and Jadlovski, I. (2019). Oscillatory properties of third-order neutral delay differential equations with noncanonical operators. Mathematics, 7.","DOI":"10.3390\/math7121177"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3851","DOI":"10.3934\/math.2020250","article-title":"Asymptotic behavior of solutions of third-order neutral differential equations with discrete and distributed delay","volume":"5","author":"Kumar","year":"2020","journal-title":"AIMS Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kumar, M.S., Bazighifan, O., Almutairi, A., and Chalishajar, D.N. (2021). Philos-type oscillation results for third-order differential equation with mixed neutral terms. Mathematics, 9.","DOI":"10.3390\/math9091021"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"AlGhamdi, A., Bazighifan, O., and El-Nabulsi, R.A. (2021). Important Criteria for Asymptotic Properties of Nonlinear Differential Equations. Mathematics, 9.","DOI":"10.3390\/math9141659"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1007\/s40819-018-0508-8","article-title":"Some new oscillatory behavior of certain third-order nonlinear neutral differential equations of mixed type","volume":"4","author":"Kumar","year":"2018","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_19","first-page":"1","article-title":"Oscillatory behavior of solutions to third-order nonlinear differential equations with a superlinear neutral term","volume":"2020","author":"Tunc","year":"2020","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_20","first-page":"268","article-title":"Oscillation criteria of third order nonlinear neutral differential equations","volume":"1","author":"Soliman","year":"2012","journal-title":"Int. J. Appl. Math. Res."},{"key":"ref_21","first-page":"1796","article-title":"Oscillation and asymptotic behavior of third-order nonlinear neutral delay differential equations with distributed deviating arguments","volume":"8","author":"Sun","year":"2018","journal-title":"J. Appl. Anal. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1485\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:45:11Z","timestamp":1760165111000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1485"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,13]]},"references-count":21,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081485"],"URL":"https:\/\/doi.org\/10.3390\/sym13081485","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,8,13]]}}}