{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T07:04:48Z","timestamp":1780556688019,"version":"3.54.1"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,8]],"date-time":"2021-01-08T00:00:00Z","timestamp":1610064000000},"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 purpose of this article is to explore the asymptotic properties for a class of fourth-order neutral differential equations. Based on a comparison with the differential inequality of the first-order, we have provided new oscillation conditions for the solutions of fourth-order neutral differential equations. The obtained results can be used to develop and provide theoretical support for and to further develop the study of oscillation for a class of fourth-order neutral differential equations. Finally, we provide an illustrated example to demonstrate the effectiveness of our new criteria.<\/jats:p>","DOI":"10.3390\/sym13010101","type":"journal-article","created":{"date-parts":[[2021,1,10]],"date-time":"2021-01-10T23:03:42Z","timestamp":1610319822000},"page":"101","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":38,"title":["Neutral Delay Differential Equations: Oscillation Conditions for the Solutions"],"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":"Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512, Yemen"},{"name":"Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hammad","family":"Alotaibi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6337-0760","authenticated-orcid":false,"given":"Abd Allaah A.","family":"Mousa","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,8]]},"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","unstructured":"Bazighifan, O., and Kumam, P. (2020). Oscillation Theorems for Advanced Differential Equations with p-Laplacian Like Operators. Mathematics, 8.","DOI":"10.3390\/math8050821"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bazighifan, O., and Abdeljawad, T. (2020). Improved Approach for Studying Oscillatory Properties of Fourth-Order Advanced Differential Equations with p-Laplacian Like Operator. Mathematics, 8.","DOI":"10.3390\/math8050656"},{"key":"ref_4","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. Appl. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"10276","DOI":"10.1002\/mma.6694","article-title":"Oscillatory applications of some fourth-order differential equations","volume":"43","author":"Bazighifan","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_6","first-page":"1","article-title":"Kamenev and Philos-types oscillation criteria for fourth-order neutral differential equations","volume":"201","author":"Bazighifan","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1186\/s13662-020-02875-8","article-title":"Oscillatory and asymptotic behavior of advanced differential equations","volume":"2020","author":"Bazighifan","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_8","first-page":"1","article-title":"An oscillation criterion in 4th-order neutral differential equations with a continuously distributed delay","volume":"336","author":"Chatzarakis","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1186\/s13662-019-2240-z","article-title":"Oscillation criteria for even-order neutral differential equations with distributed deviating arguments","volume":"2019","author":"Moaaz","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1007\/BF01223686","article-title":"On the existence of non-oscillatory solutions tending to zero at \u221e for differential equations with positive delays","volume":"36","author":"Philos","year":"1981","journal-title":"Arch. Math."},{"key":"ref_11","unstructured":"Ladde, G.S., Lakshmikantham, V., and Zhang, B.G. (1987). Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker."},{"key":"ref_12","first-page":"125475","article-title":"On the qualitative behavior of the solutions to second-order neutral delay differential equations","volume":"256","author":"Santra","year":"2020","journal-title":"J. Inequal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"125475","DOI":"10.1016\/j.amc.2020.125475","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_14","doi-asserted-by":"crossref","first-page":"2191","DOI":"10.1016\/j.camwa.2010.09.011","article-title":"Oscillation of even-order half-linear functional differential equations with damping","volume":"61","author":"Liu","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Chatzarakis, G.E., Grace, S.R., Jadlovska, I., Li, T., and Tunc, E. (2019). Oscillation criteria for third-order Emden-Fowler differential equations with unbounded neutral coefficients. Complexity, 2019.","DOI":"10.1155\/2019\/5691758"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0893-9659(98)00028-7","article-title":"Oscillation criteria for even order neutral differential equations","volume":"11","author":"Zafer","year":"1998","journal-title":"Appl. Math. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1202","DOI":"10.1016\/j.aml.2006.01.003","article-title":"Oscillation behavior of even order neutral differential equations with variable coefficients","volume":"19","author":"Zhang","year":"2006","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1186\/1687-1847-2011-45","article-title":"Oscillation of higher-order quasi linear neutral differential equations","volume":"2011","author":"Xing","year":"2011","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Moaaz, O., Awrejcewicz, J., and Bazighifan, O. (2020). A New Approach in the Study of Oscillation Criteria of Even-Order Neutral Differential Equations. Mathematics, 8.","DOI":"10.3390\/math8020197"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4472","DOI":"10.1016\/j.camwa.2011.10.024","article-title":"Oscillation theorems for second-order nonlinear neutral differential equations","volume":"62","author":"Baculikova","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Agarwal, R., Grace, S., and O\u2019Regan, D. (2000). Oscillation Theory for Difference and Functional Differential Equations, Kluwer Acad. Publ.","DOI":"10.1007\/978-94-015-9401-1"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.aml.2012.08.004","article-title":"New results for oscillatory behavior of even-order half-linear delay differential equations","volume":"26","author":"Zhang","year":"2013","journal-title":"Appl. Math. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1618","DOI":"10.1016\/j.aml.2011.04.015","article-title":"On the oscillation of higher-order half-linear delay differential equations","volume":"24","author":"Zhang","year":"2011","journal-title":"Appl. Math. Lett."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/101\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:08:31Z","timestamp":1760159311000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/101"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,8]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010101"],"URL":"https:\/\/doi.org\/10.3390\/sym13010101","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,8]]}}}