{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:08:06Z","timestamp":1760234886999,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,21]],"date-time":"2021-06-21T00:00:00Z","timestamp":1624233600000},"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>Differential equations with delay arguments are one of the branches of functional differential equations which take into account the system\u2019s past, allowing for more accurate and efficient future prediction. The symmetry of the equations in terms of positive and negative solutions plays a fundamental and important role in the study of oscillation. In this paper, we study the oscillatory behavior of a class of odd-order neutral delay differential equations. We establish new sufficient conditions for all solutions of such equations to be oscillatory. The obtained results improve, simplify and complement many existing results.<\/jats:p>","DOI":"10.3390\/sym13061095","type":"journal-article","created":{"date-parts":[[2021,6,21]],"date-time":"2021-06-21T13:29:58Z","timestamp":1624282198000},"page":"1095","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["New Results for Oscillation of Solutions of Odd-Order Neutral Differential Equations"],"prefix":"10.3390","volume":"13","author":[{"given":"Clemente","family":"Cesarano","sequence":"first","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Roma, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3850-1022","authenticated-orcid":false,"given":"Osama","family":"Moaaz","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Roma, Italy"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"given":"Belgees","family":"Qaraad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"},{"name":"Department of Mathematics, Faculty of Science, Amran University, Amran 999101, Yemen"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8373-1428","authenticated-orcid":false,"given":"Nawal A.","family":"Alshehri","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}]},{"given":"Sayed K.","family":"Elagan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}]},{"given":"Mohammed","family":"Zakarya","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,21]]},"reference":[{"key":"ref_1","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 delay","volume":"36","author":"Philos","year":"1981","journal-title":"Arch. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hale, J.K. (1977). Theory of Functional Differential Equations, Spring.","DOI":"10.1007\/978-1-4612-9892-2"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1016\/j.aml.2010.10.043","article-title":"Oscillation of third-order nonlinear differential equations","volume":"24","author":"Baculikova","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Grace, S.R., and Regan, D. (2000). Oscillation Theory for Difference and Functional Differential Equations, Marcel Dekker, Kluwer Academic.","DOI":"10.1007\/978-94-015-9401-1"},{"key":"ref_5","unstructured":"Erbe, L., Kong, Q., and Zhang, B.G. (1995). Oscillation Theory for Functional Differential Eqautions, Marcel Dekker."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/0022-247X(84)90066-0","article-title":"Oscillation theorems for nth-order differential equations with deviating arguments","volume":"101","author":"Grace","year":"1984","journal-title":"J. Math. Appl. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.jmaa.2003.11.054","article-title":"Integral averaging technique and oscillation of certain even order delay differential equations","volume":"292","author":"Xu","year":"2004","journal-title":"J. Math. Appl. Anal."},{"key":"ref_8","first-page":"1","article-title":"Oscillatory behavior of solutions of odd-order nonlinear delay differential equations","volume":"2020","author":"Moaaz","year":"2020","journal-title":"Adv. Differ. Eqs."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1006\/jmaa.2001.7571","article-title":"Oscillation criteria for certain nth order differential equations with deviating arguments","volume":"262","author":"Agarwal","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2010\/727486","article-title":"Oscillation of even-order neutral delay differential equations","volume":"2010","author":"Li","year":"2010","journal-title":"Adv. Differ. Eq."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Moaaz, O., Furuichi, S., and Muhib, A. (2020). New comparison theorems for the nth order neutral differential equations with delay inequalities. Mathematics, 8.","DOI":"10.3390\/math8030454"},{"key":"ref_12","first-page":"359","article-title":"Oscillation of solutions of non-linear neutral delay differential equations of higher order for p(t)=\u00b11","volume":"40","author":"Rath","year":"2004","journal-title":"Arch. Math."},{"key":"ref_13","first-page":"1","article-title":"Oscillation of higher-order quasi-linear neutral differential equations","volume":"2011","author":"Xing","year":"2011","journal-title":"Adv. Differ. Eqs."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.ajmsc.2017.07.001","article-title":"Oscillation criteria for a class of third order damped differential equations","volume":"24","author":"Moaaz","year":"2018","journal-title":"Arab J. Math. Sci."},{"key":"ref_16","first-page":"226","article-title":"Functions of \u03b1-slow increase","volume":"4","author":"Shang","year":"2012","journal-title":"Bull. Math. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1017\/S0017089509990188","article-title":"Comparison theorems on the oscillation and asymptotic behavior of higher-order neutral differential equations","volume":"52","author":"Karpuz","year":"2010","journal-title":"Glasgow Math J."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1095\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:20:02Z","timestamp":1760163602000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1095"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,21]]},"references-count":17,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["sym13061095"],"URL":"https:\/\/doi.org\/10.3390\/sym13061095","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,6,21]]}}}