{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T21:10:33Z","timestamp":1775509833985,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,6]],"date-time":"2020-05-06T00:00:00Z","timestamp":1588723200000},"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 study was to examine the asymptotic properties and oscillation of the even-order neutral differential equations. The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order delay equations. Our results improve and complement some well-known results. We obtain Hille and Nehari type oscillation criteria to ensure the oscillation of the solutions of the equation. One example is provided to illustrate these results.<\/jats:p>","DOI":"10.3390\/sym12050764","type":"journal-article","created":{"date-parts":[[2020,5,7]],"date-time":"2020-05-07T04:46:07Z","timestamp":1588826767000},"page":"764","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["New Comparison Theorems for the Even-Order Neutral Delay Differential Equation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3850-1022","authenticated-orcid":false,"given":"Osama","family":"Moaaz","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"given":"Rami Ahmad","family":"El-Nabulsi","sequence":"additional","affiliation":[{"name":"Athens Institute for Education and Research, Mathematics and Physics Divisions, 10671 Athens, Greece"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-9608","authenticated-orcid":false,"given":"Omar","family":"Bazighifan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen"},{"name":"Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512, Yemen"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3844-7820","authenticated-orcid":false,"given":"Ali","family":"Muhib","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"},{"name":"Department of Mathematics, Faculty of Education\u2013Al-Nadirah, Ibb University, Ibb, Yemen"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,6]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0895-7177(97)00141-6","article-title":"Oscillation criteria for second order retarde ddifferential equations","volume":"26","author":"Agarwal","year":"1997","journal-title":"Math. Comput. Model."},{"key":"ref_3","first-page":"387","article-title":"On the oscillation of higher-order delay differential equations","volume":"187","author":"Baculikova","year":"2012","journal-title":"Math. Slovaca"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bazighifan, O., and Cesarano, C. (2020). A Philos-Type Oscillation Criteria for Fourth-Order Neutral Differential Equations. Symmetry, 12.","DOI":"10.3390\/sym12030379"},{"key":"ref_5","first-page":"1","article-title":"Oscillation of higher-order differential equations with distributed delay","volume":"55","author":"Bazighifan","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"106171","DOI":"10.1016\/j.aml.2019.106171","article-title":"Pairs of nontrivial smooth solutions for nonlinear Neumann problems","volume":"103","author":"Vetro","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Vetro, C. (2020). An elliptic equation on n-dimensional manifolds. Complex Var. Elliptic Equ.","DOI":"10.1080\/17476933.2020.1711745"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Cesarano, C., Pinelas, S., Al-Showaikh, F., and Bazighifan, O. (2019). Asymptotic Properties of Solutions of Fourth-Order Delay Differential Equations. Symmetry, 11.","DOI":"10.3390\/sym11050628"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Cesarano, C., and Bazighifan, O. (2019). Oscillation of fourth-order functional differential equations with distributed delay. Axioms, 7.","DOI":"10.3390\/math7070619"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Cesarano, C., and Bazighifan, O. (2019). Qualitative behavior of solutions of second order differential equations. Symmetry, 11.","DOI":"10.3390\/sym11060777"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"191","DOI":"10.30538\/oms2019.0062","article-title":"Oscillation of solutions to fourth-order delay differential equations with middle term","volume":"3","author":"Elabbasy","year":"2019","journal-title":"Open J. Math. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Elabbasy, E.M., Cesarano, C., Bazighifan, O., and Moaaz, O. (2019). Asymptotic and oscillatory behavior of solutions of a class of higher-order differential equations. Symmetry, 11.","DOI":"10.3390\/sym11121434"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"El-Nabulsi, R.A., Moaaz, O., and Bazighifan, O. (2020). New Results for Oscillatory Behavior of Fourth-Order Differential Equations. Symmetry, 12.","DOI":"10.3390\/sym12010136"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kiguradze, I.T., and Chanturiya, T.A. (1993). Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Acad. Publ.","DOI":"10.1007\/978-94-011-1808-8"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Gyori, I., and Ladas, G. (1991). Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press.","DOI":"10.1093\/oso\/9780198535829.001.0001"},{"key":"ref_17","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_18","first-page":"41","article-title":"Oscillation of fourth order neutral differential equations with p-Laplacian like operators","volume":"56","author":"Li","year":"2014","journal-title":"Bound. Value Probl."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","unstructured":"Moaaz, O., Dassios, I., Bazighifan, O., and Muhib, A. (2020). Oscillation Theorems for Nonlinear Differential Equations of Fourth-Order. Mathematics, 8.","DOI":"10.3390\/math8040520"},{"key":"ref_21","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_22","first-page":"1","article-title":"Oscillation criteria for even-order neutral differential equations with distributed deviating arguments","volume":"297","author":"Moaaz","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1090\/S0002-9947-1957-0087816-8","article-title":"Oscillation criteria for second order linear differential equations","volume":"85","author":"Nehari","year":"1957","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_24","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_25","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_26","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_27","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_28","doi-asserted-by":"crossref","first-page":"1709","DOI":"10.1016\/j.aml.2011.04.025","article-title":"Oscillation criteria for even-order half-linear functional differential equations with damping","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\/12\/5\/764\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:25:26Z","timestamp":1760361926000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/764"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,6]]},"references-count":28,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["sym12050764"],"URL":"https:\/\/doi.org\/10.3390\/sym12050764","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,6]]}}}