{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:30:04Z","timestamp":1760149804134,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T00:00:00Z","timestamp":1694390400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We investigate the oscillation of the fourth-order differential equation for a class of functional differential equations of the neutral type. We obtain a new single-oscillation criterion for the oscillation of all the solutions of our equation. We establish new monotonic properties for some cases of positive solutions of the studied equation. Moreover, we improve these properties by using an iterative method. This development of monotonic properties contributes to obtaining new and more efficient criteria for verifying the oscillation of the equation. The results obtained extend and improve previous findings in the literature by using an Euler-type equation as an example. The importance of the results was clarified by applying them to some special cases of the studied equation. The fourth-order delay differential equations have great practical importance due to their wide applications in civil, mechanical, and aeronautical engineering. Research on this type of equation is still ongoing due to its remarkable importance in many fields.<\/jats:p>","DOI":"10.3390\/sym15091744","type":"journal-article","created":{"date-parts":[[2023,9,12]],"date-time":"2023-09-12T04:38:48Z","timestamp":1694493528000},"page":"1744","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimizing the Monotonic Properties of Fourth-Order Neutral Differential Equations and Their Applications"],"prefix":"10.3390","volume":"15","author":[{"given":"Hend","family":"Salah","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3850-1022","authenticated-orcid":false,"given":"Osama","family":"Moaaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"},{"name":"Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1167-2430","authenticated-orcid":false,"given":"Sameh S.","family":"Askar","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7103-8872","authenticated-orcid":false,"given":"Ahmad M.","family":"Alshamrani","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"given":"Elmetwally M.","family":"Elabbasy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,11]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Rihan, F.A. (2021). Delay Differential Equations and Applications to Biology, Springer Nature Singapore Pte Ltd.","key":"ref_1","DOI":"10.1007\/978-981-16-0626-7"},{"unstructured":"Hale, J.K. (1971). Oxford Applied Mathematical Sciences, Springer.","key":"ref_2"},{"doi-asserted-by":"crossref","unstructured":"Gyori, I., and Ladas, G. (1991). Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press.","key":"ref_3","DOI":"10.1093\/oso\/9780198535829.001.0001"},{"unstructured":"Erbe, L.H., Kong, Q., and Zhong, B.G. (1995). Oscillation Theory for Functional Differential Equations, Marcel Dekker.","key":"ref_4"},{"unstructured":"Ladde, G.S., Lakshmikantham, V., and Zhang, B.G. (1987). Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker.","key":"ref_5"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1007\/s00009-020-01538-y","article-title":"On oscillation of second order delay differential equations with a sublinear neutral term","volume":"17","author":"Grace","year":"2020","journal-title":"Mediterr. J. Math."},{"doi-asserted-by":"crossref","unstructured":"Jadlovsk\u00e1, I. (2021). New criteria for sharp oscillation of second-order neutral delay differential equations. 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An improved criterion for the oscillation of fourth-order differential equations. Mathematics, 8.","key":"ref_18","DOI":"10.3390\/math8040610"},{"doi-asserted-by":"crossref","unstructured":"Kiguradze, I., and Chanturia, T.A. (1993). Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic, Publishers.","key":"ref_19","DOI":"10.1007\/978-94-011-1808-8"},{"key":"ref_20","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_21","first-page":"787","article-title":"A new approach in the study of oscillatory behavior of even-order neutral delay differential equations","volume":"225","author":"Agarwal","year":"2013","journal-title":"Appl. Math. 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