{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:11:21Z","timestamp":1760235081213,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,17]],"date-time":"2021-07-17T00:00:00Z","timestamp":1626480000000},"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>In this paper, we analyze the asymptotic behavior of solutions to a class of third-order neutral differential equations. Using different methods, we obtain some new results concerning the oscillation of this type of equation. Our new results complement related contributions to the subject. The symmetry plays a important and fundamental role in the study of oscillation of solutions to these equations. An example is presented in order to clarify the main results.<\/jats:p>","DOI":"10.3390\/sym13071287","type":"journal-article","created":{"date-parts":[[2021,7,18]],"date-time":"2021-07-18T21:18:52Z","timestamp":1626643132000},"page":"1287","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On the Qualitative Behavior of Third-Order Differential Equations with a Neutral Term"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-9608","authenticated-orcid":false,"given":"Omar","family":"Bazighifan","sequence":"first","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2116-7382","authenticated-orcid":false,"given":"Fatemah","family":"Mofarreh","sequence":"additional","affiliation":[{"name":"Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia"}],"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":[[2021,7,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Grace, S.R., and O\u2019Regan, D. (2002). Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic.","DOI":"10.1007\/978-94-017-2515-6"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Grace, S.R., and O\u2019Regan, D. (2003). Oscillation Theory for Second Order Dynamic Equations, Taylor & Francis.","DOI":"10.4324\/9780203222898"},{"key":"ref_3","unstructured":"Bainov, D.D., and Mishev, D.P. (1991). Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bazighifan, O., and Postolache, M. (2020). Improved conditions for oscillation of functional nonlinear differential equations. Mathematics, 8.","DOI":"10.3390\/math8040552"},{"key":"ref_5","first-page":"125475","article-title":"On the oscillation of certain fourth-order differential equations with p-Laplacian like operator","volume":"386","author":"Bazighifan","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1186\/s13660-021-02595-x","article-title":"On oscillation of second-order noncanonical neutral differential equations","volume":"2021","author":"Muhib","year":"2021","journal-title":"J. Inequal. Appl."},{"key":"ref_7","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_8","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":"6","author":"Graef","year":"2017","journal-title":"Opuscula Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1186\/1029-242X-2014-512","article-title":"Asymptotic behavior of a third-order nonlinear neutral delay differential equation","volume":"2014","author":"Jiang","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Moaaz, O., Awrejcewicz, J., and Muhib, A. (2020). Establishing new criteria for oscillation of odd-order nonlinear differential equations. Mathematics, 8.","DOI":"10.3390\/math8060937"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Elabbasy, E.M., Qaraad, B., Abdeljawad, T., and Moaaz, O. (2020). Oscillation criteria for a class of third-order damped neutral differential equations. Symmetry, 12.","DOI":"10.3390\/sym12121988"},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1007\/BF01223686","article-title":"On the existence of nonoscillatory solutions tending to zero at \u221e for differential equations with positive delays","volume":"36","author":"Philos","year":"1981","journal-title":"Arch. Math."},{"key":"ref_14","first-page":"545","article-title":"Oscillation of third-order nonlinear delay differential equations","volume":"17","author":"Agarwal","year":"2013","journal-title":"Taiwan Math. Soc."},{"key":"ref_15","first-page":"181","article-title":"On the oscillation of third-order quasi-linear neutral functional differential equations","volume":"47","author":"Thandapani","year":"2011","journal-title":"Arch. Math. (BRNO) Tomus."},{"key":"ref_16","first-page":"1","article-title":"Oscillatory and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments","volume":"16","author":"Tunc","year":"2017","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.jmaa.2006.01.001","article-title":"Oscillation criteria of a certain class of third-order nonlinear delay differential equations with damping","volume":"325","author":"Tiryaki","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_18","first-page":"2837","article-title":"Oscillation of second-order nonlinear neutral dynamic equations on time scales","volume":"216","author":"Zhang","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"225","DOI":"10.21136\/MB.2010.140700","article-title":"On the oscillation of certain class of third-order nonlinear delay differential equations","volume":"135","author":"Saker","year":"2010","journal-title":"Math. Bohemica"},{"key":"ref_20","first-page":"102","article-title":"On the oscillation of certain third order nonlinear functional differential bequations","volume":"202","author":"Grace","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Moaaz, O., Qaraad, B., El-Nabulsi, R., and Bazighifan, O. (2020). New results for Kneser solutions of third-order nonlinear neutral differential equations. Mathematics, 8.","DOI":"10.3390\/math8050686"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.aml.2018.08.016","article-title":"On nonexistence ofKneser solutions of third-order neutral delay differential equations","volume":"88","author":"Dzurina","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Vidhyaa, K.S., Graef, J.R., and Thandapani, E. (2020). New oscillation results for third-order half-linear neutral differential equations. Mathematics, 8.","DOI":"10.3390\/math8030325"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1287\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:31:16Z","timestamp":1760164276000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1287"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,17]]},"references-count":23,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071287"],"URL":"https:\/\/doi.org\/10.3390\/sym13071287","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,7,17]]}}}